Weisstein. Concise encyclopedia of mathematics (CRC)(3236s)

How to Use This Edition The second edition of the CRC Concise Encyclopedia of Mathematics has been designed with the user in mind and for ease of accessibility. Listed below are various changes in the new edition that will make the book easier for the reader to use while navigating to different areas of interest Alphabetization All entries are listed in alphabetical order. There is a separate section appearing before the A’s to cover the entriesthat are numerals. The alphabetizing of letters is not affected by dashes, apostrophes, or any other punctuation fallingwithin a word. For example, you will find A-Integrable listed in the Ai section of the book. Following the samelogic, all entries for Abel will precede entries for Abel’s. Cross-References In many cases, a particular entry of interest can be located from a cross-reference. Cross-references are indicatedin SMALL CAPS typeface in the text. In addition, for some main listings, you will be re-directed to a different entry (or multiple entries) as indicated in small caps underneath the main listing. For example, A BEL ’ S T EST A BEL ’ S U NIFORM C ONVERGENCE T EST Finally, most articles are followed by a “ See also” list of related entries. References All Reference listings follow the text of the corresponding entry. Note that in this reference style, page ranges may be abbreviated. Accordingly, a page range of 132-136 will be indicated as 132-36. Another example of this is apage range of 96-100 that is indicated by 96-00. Entries Many new entries have been added for the user to the new edition. However, because this is a work in progress,some of the new entries have not been completed with appropriate definitions or textual description. Followingmany of these kinds of entries, the reader is referred to other items of interest that are closely related or similar tothe article in question. The MathWorld website, produced by Wolfram Research, Inc. and Dr. Eric Weisstein, can be found at http://mathworld.wolfram.com . Wolfram Research, Inc. retains the copyright in certain entries therein; CRC Press LLC has certain exclusive rights to publish all of said entries in all media and formats other than free distribution over the internet. Numerals ( /C281, 0, 1)-Matrix The number of distinct (/C281 ; 0; 1)/-/n /C29n matrices (counting row and column permutations, the trans- pose, and multiplication by /C281 as equivalent) having 2n different row and column sums for n /C302, 4, 6, ... are 1, 4, 39, 2260, 1338614, ... (Kleber). For example, the 2 /C292 matrix is given by /C281 /C281 01/C20/C21 ; To get the total number from these counts (assuming that 0 is not the missing sum, which is true for n 5 10) ; multiply by (2n!)2 : In general, if an -matrix which has different column and row sums (collectively called line sums), then 1. n is even, 2. The number in f/C28n; 1 /C28n ; 2 /C28n; ... ; ng that does not appear as a line sum is either /C28n or , and 3. Of the largest line sums, half are column sums and half are row sums (Bodendiek and Burosch 1995, F. Galvin). See also ALTERNATING SIGN MATRIX , C-MATRIX , INTEGER MATRIX References Bodendiek, R. and Burosch, G. "Solution to the Antimagic 0; 1;/C281 Matrix Problem." Aufgabe 5.30 in Streifzu ¨ge durch die Kombinatorik: Aufgaben und Lo¨sungen aus dem Schatz der Mathematik-Olympiaden. Heidelberg, Germany: Spektrum Akademischer Verlag, pp. 250 /C1/253, 1995. ( /C281, 1)-Matrix See also HADAMARD MATRIX ,INTEGER MATRIX References Kahn, J.; Komlo ´s, J.; and Szemeredi, E. "On the Probability that a Random 91 Matrix is Singular." J. Amer. Math. Soc. 8, 223 /C1/240, 1995. 0-Free ZEROFREE 0 DIVISION BY ZERO,FALLACY ,N AUGHT ,ZERO,ZERO DIVISOR ,ZERO-FORM,ZERO MATRIX ,ZERO-SUM GAME, ZEROFREE 0 /C301 FALLACY(0, 1)-Matrix A(0 ; 1)/-INTEGER MATRIX , i.e., a matrix each of whose elements is 0 or 1, also called a binary matrix. The numbers of binary matrices with no adjacent 1s (in either columns or rows) for n /C301, 2, ..., are given by 2, 7, 63, 1234, ... (Sloane’s A006506). For example, the binary matrices with no adjacent 1s are 00 00/C20/C21 ;00 01/C20/C21 ;0010/C20/C21 ;0100/C20/C21 0110/C20/C21 ;1000/C20/C21 ;1001/C20/C21 ; These numbers are closely related to the HARD SQUARE ENTROPY CONSTANT . The numbers of binary matrices with no three adjacent 1s for , 2, ..., are given by 2, 16, 265, 16561, ... (Sloane’s A050974). Wilf (1997) considers the complexity of transforming anm/C29nbinary matrix Ainto a TRIANGULAR MATRIX by permutations of the rows and columns of , and concludes that the problem falls in difficulty between a known easy case and a known hard case of thegeneral NP -COMPLETE PROBLEM . See also ADJACENCY MATRIX ,FROBENIUS- KO¨ NIG THE- OREM ,GALE-RYSER THEOREM ,HADAMARD’S MAXIMUM DETERMINANT PROBLEM ,H ARD SQUARE ENTROPY CONSTANT ,IDENTITY MATRIX ,INCIDENCE MATRIX , INTEGER MATRIX ,LAM’S PROBLEM , S-CLUSTER , S-RUN References Brualdi, R. A. "Discrepancy of Matrices of Zeros and Ones." Electronic J. Combinatorics 6, No. 1, R15, 1 /C1/12, 1999. http://www.combinatorics.org/Volume_6/v6i1toc.html. Ehrlich, H. "Determinantenabscha ¨tzungen fu ¨r bina ¨re Ma- trizen." Math. Z. 83, 123/C1/132, 1964. Ehrlich, H. and Zeller, K. "Bina ¨re Matrizen." Z. angew. Math. Mechanik 42, T20/C1/21, 1962. Komlo ´s, J. "On the Determinant of -Matrices." Studia Math. Hungarica 2,7/C1/21 1967. Metropolis, N. and Stein, P. R. "On a Class of Matrices with Vanishing Determinants." J. Combin Th. 3, 191/C1/198, 1967. Ryser, H. J. "Combinatorial Properties of Matrices of Zeros and Ones." Canad. J. Math. 9, 371/C1/377, 1957. Sloane, N. J. A. Sequences A006506/M1816 and A050974 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference inHonor of Erdos’ 80th Birthday Held at Trinity College,Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press,pp. 557 /C1 /562, 1997. Williamson, J. "Determinants Whose Elements Are 0 and 1." Amer. Math. Monthly 53, 427/C1/434, 1946. 1 The number one (1), also called "unity" is the first POSITIVE INTEGER .I ti sa n ODD NUMBER . Although the number 1 used to be considered a PRIME NUMBER ,i t requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own (Wells 1986, p. 31). The number 1 is sometimes also called "unity," so the th roots of 1 are often called the th ROOTS OF UNITY . FRACTIONS having 1 as a NUMERATOR are called UNIT FRACTIONS . If only one root, solution, etc., exists to a given problem, the solution is called UNIQUE . The GENERATING FUNCTION having all COEFFICIENTS 1 is given by 1 1 /C28 x /C301 /C27x /C27x2 /C27x3 /C27x4 /C27... : See also FALLACY ,ONE-FORM,ONE-MOUTH THEOREM , ONE-NINTH CONSTANT ,ONE-SHEETED HYPERBOLOID , ONE-TO- ONE,ONE-WAY FUNCTION , 2, 3,COMPLEXITY (NUMBER ), EXACTLY ONE,ROOT OF UNITY,UNIQUE , UNIT FRACTION ,ZERO References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 30 /C1/ 32, 1986. 2 The number two (2) is the second POSITIVE INTEGER and the first PRIME NUMBER .Itis EVEN , and is the only EVEN PRIME (the PRIMES other than 2 are called the ODD PRIMES ). The number 2 is also equal to its FACTORIAL since 2! /C302 : A quantity taken to the POWER 2 is said to be SQUARED . The number of times k a given BINARY number bn /C1/C1/C1b2b1b0 is divisible by 2 is given by the position of the first bk /C301 ; counting from the right. For example, 12 /C301100 is divisible by 2 twice, and 13 /C301101 is divisible by 2 zero times. The only known solutions to the CONGRUENCE 2n /C133 (mod n) are n /C304700063497 (Sloane’s A050259; Guy 1994) and 63130707451134435989380140059866138830623361447484274774099906755 (P.-L. Montgomery 1999). In general, the least satis- fying 2n /C13k (mod n) for k /C302, 3, ... are n /C303, 4700063497, 6, 19147, 10669, 25, 9, 2228071, ... (Sloane’s A036236). See also 1,BINARY , 3,RULER FUNCTION ,SQUARED , TWO-EARS THEOREM ,TWO-FORM,TWO-GRAPH ,TWO- SCALE EXPANSION ,T WO-SHEETED HYPERBOLOID , ZEROReferences Daiev, V. "Problem 636: Greatest Divisors of Even Integers." Math. Mag. 40, 164 /C1/165, 1967. Guy, R. K. "Residues of Powers of Two." §F10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 250, 1994. Montgomery, P.-L. "New solution to 2^n /C30/C30 3 (mod n)." [email protected] posting, 24 Jun 1999. Sloane, N. J. A. Sequences A036236 and A050259 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 41 /C1/ 44, 1986. 2x mod 1 Map Let x0 be a RATIONAL NUMBER in the CLOSED INTERVAL [0; 1]; and generate a SEQUENCE using the MAP xn/C271 /C132xn (mod 1): (1) Then the number of periodic ORBITS of period p (for PRIME ) is given by Np /C302p /C28 2 p (2) (i.e, the number of period- repeating bit strings, modulo shifts). Since a typical ORBIT visits each point with equal probability, the NATURAL INVARIANT is given by r(x) /C301: (3) See also TENT MAP References Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, pp. 26 /C1/31, 1993. 3 3 is the only INTEGER which is the sum of the preceding POSITIVE INTEGERS (1/C272/C303) and the only number which is the sum of the FACTORIALS of the preceding POSITIVE INTEGERS (/1!/C272!/C303):It is also the first ODD PRIME . A quantity taken to the POWER 3 is said to be CUBED . The sequence 1, 31, 331, 3331, 33331, ... (Sloane’s A033175) consisting of n/C300, 1, ... 3s followed by a 1. The th tern is given by a(n)/C3010n/C271/C287 3: The result is prime for , 2, 3, 4, 5, 6, 7, 17, 39, ... (Sloane’s A055520); i.e., for 3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, ... (Sloane’s A051200), afact which Gardner (1997) calls "a remarkable pat- tern that is entirely accidental and leads nowhere." See also 1, 2, 3X /C271 MAPPING ,CUBED ,PERIOD THREE THEOREM ,T ERNARY ,T HREE- CHOICE POLYGON , THREE- CHOICE WALK,T HREE- COLORABLE ,T HREE CONICS THEOREM ,T HREE JUG PROBLEM ,T HREE- VALUED LOGIC ,TREFOIL KNOT,W IGNER 3J-SYMBOL , ZERO References Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, p. 194, 1997. Sloane, N. J. A. Sequences A033175, A051200, and A055520 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Smarandache, F. Properties of Numbers. University of Craiova, 1973. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 46 /C1/ 48, 1986. 3x /C271 Mapping COLLATZ PROBLEM 4 See also FOUR COINS PROBLEM ,FOUR- COLOR THEO- REM,FOUR CONICS THEOREM ,FOUR EXPONENTIALS CONJECTURE ,FOUR TRAVELERS PROBLEM ,FOUR- VEC- TOR,F OUR- VERTEX THEOREM ,L AGRANGE’S FOUR- SQUARE THEOREM References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 55 /C1/ 58, 1986. 4-D Geometry 4-DIMENSIONAL GEOMETRY 4-Dimensional Geometry 4-dimensional geometry is Euclidean geometry ex- tended into one additional DIMENSION . The prefix "hyper-" is usually used to refer to the 4- (and higher-) dimensional analogs of 3-dimensional objects, e.g. HYPERCUBE , HYPERPLANE , HYPERSPHERE . -dimen- sional POLYHEDRA are called POLYTOPES . the 4-dimen- sional cases of general -dimensional objects are often given special names, such as those summarized in the following table. 2-D 3-D 4-D General CIRCLE SPHERE GLOME HYPERSPHERE SQUARE CUBE TESSERACT HYPERCUBEEQUILATERAL TRIANGLETETRAHEDRON PENTATOPE SIMPLEX POLYGON POLYHEDRON POLYCHORON POLYTOPE LINE SEG- MENTPLANE HYPERPLANE HYPERPLANE SQUARE OCTAHEDRON 16-CELL CROSS POLY- TOPE EDGE FACE FACET FACET AREA VOLUME CONTENT CONTENT The SURFACE AREA of a HYPERSPHERE in -D is given by Sn /C302pn=2 G1 2 n/C16/C17 ; and the VOLUME by Vn /C30pn=2Rn G 1 /C271 2 n/C16/C17 ; where G(n) is the GAMMA FUNCTION . See also DIMENSION ,HYPERCUBE ,HYPERSPHERE References Hinton, C. H. The Fourth Dimension. Pomeroy, WA: Health Research, 1993. Manning, H. The Fourth Dimension Simply Explained. Magnolia, MA: Peter Smith, 1990. Manning, H. Geometry of Four Dimensions. New York: Dover, 1956. Neville, E. H. The Fourth Dimension. Cambridge, England: Cambridge University Press, 1921. Rucker, R. von Bitter. The Fourth Dimension: A Guided Tour of the Higher Universes. Boston, MA: Houghton Mifflin, 1984. Sommerville, D. M. Y. An Introduction to the Geometry of Dimensions. New York: Dover, 1958. 5 See also FIVE DISKS PROBLEM ,MIQUEL FIVE CIRCLES THEOREM ,P ENTAGON ,P ENTAGRAM ,P ENTAHEDRON , TETRAHEDRON 5-COMPOUND References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 58 /C1/ 67, 1986. 5-Cell PENTATOPE 6 See also 6-SPHERE COORDINATES ,HEXAGON ,HEXAHE- DRON ,SIX CIRCLES THEOREM ,SIX-COLOR THEOREM , SIX EXPONENTIALS THEOREM ,W IGNER 6J-SYMBOL References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 67 /C1/ 69, 1986. 6-Sphere Coordinates The coordinate system obtained by INVERSION of CARTESIAN COORDINATES , with u; v ; w /C23 (/C28/C12;/C12): The transformation equations are x /C30u u2 /C27 v2 /C27 w2 (1) yv u2 /C27 v2 /C27 w2 (2) zw u2 /C27 v2 /C27 w2 : (3) The equations of the surfaces of constant coordinates are given by x /C281 2u !2 /C27y2 /C27z2 /C301 4u2 ; (4) which gives spheres tangent to the yz-plane at the origin for u constant, x2 /C27 y /C281 2v !2 /C27z2 /C301 4v2 ; (5) which gives spheres tangent to xz-plane at the origin for v constant, and x2 /C27y2 /C27 z /C281 2w !2 /C301 4w2 : (6)which gives spheres tangent to the xy-plane at the origin for w constant. The metric coefficients are guu /C30gvv /C30gww /C301 u2 /C27 v2 /C27 w2 ðÞ2 : (7) See also CARTESIAN COORDINATES ,INVERSION References Moon, P. and Spencer, D. E. "6-Sphere Coordinates (u; v; w):/" Fig. 4.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 122 /C1/ 123, 1988. 7 See also SEVEN CIRCLES THEOREM References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 70 /C1/ 71, 1986. 8 See also EIGHT CURVE ,EIGHT- POINT CIRCLE THEO- REM,EIGHT SURFACE References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 71 /C1/ 73, 1986. 8-Cell TESSERACT 9 See also NINE-POINT CENTER ,N INE-POINT CIRCLE , NINE-POINT CONIC ,W IGNER 9J-SYMBOL References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 73 /C1/ 76, 1986. 10 The number 10 (ten) is the basis for the DECIMAL system of notation. In this system, each "decimal place" consists of a DIGIT 0/C1/9 arranged such that each DIGIT is multiplied by a POWER of 10, decreasing from left to right, and with a decimal place indicating the10 0/C301/s place. For example, the number 1234.56 specifies 1 /C29103 /C272 /C29102 /C273 /C29101 /C274 /C29100 /C275 /C2910 /C281 /C276 /C2910/C282 : The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, ... (Sloane’s A011557), called one, ten, HUNDRED , THOUSAND , ten thousand, hundred thou- sand, MILLION , 10 million, 100 million, and so on. The names of subsequent decimal places for LARGE NUM- BERS differ depending on country. Any POWER of 10 which can be written as the PRODUCT of two numbers not containing 0s must be OF THE FORM 2n /C215 5n /C3010n for an INTEGER such that neither 2n nor 5n contains any ZEROS . The largest known such number is 1023 /C30233 /C215 533 /C308 ; 589 ; 934 ; 592 /C215116 ; 415 ; 321 ; 826 ; 934 ; 814 ; 453 ; 125 : A complete list of known such numbers is 101 /C3021 /C215 51 102 /C3022 /C215 52 103 /C3023 /C215 53 104 /C3024 /C215 54 105 /C3025 /C215 55 106 /C3026 /C215 56 107 /C3027 /C215 57 109 /C3029 /C215 59 1018 /C30218 /C215 518 1033 /C30233 /C215 533 (Madachy 1979). Since all POWERS of 2 with expo- nents 86 Bn 54:6 /C29107contain at least one ZERO (M. Cook), no other POWER of ten less than 46 million can be written as the PRODUCT of two numbers not containing 0s. See also BILLION ,DECIMAL ,HUNDRED ,LARGE NUM- BER,MILLIARD ,MILLION ,THOUSAND ,TRILLION ,ZERO References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 127 /C1/128, 1979. Pickover, C. A. Keys to Infinity. New York: Wiley, p. 135, 1995. Sloane, N. J. A. Sequences A011557 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 76 /C1/ 82, 1986. 11 References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.12 One DOZEN , or a twelfth of a GROSS . See also DOZEN ,GROSS References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. 13 A NUMBER traditionally associated with bad luck. A so-called BAKER’S DOZEN is equal to 13. Fear of the number 13 is called TRISKAIDEKAPHOBIA . There are 13 ARCHIMEDEAN SOLIDS . Mazur and Tate (1973/74) proved that there is no ELLIPTIC CURVE over the rationals Q having a RATIONAL POINT of order 13. See also BAKER’S DOZEN ,TRISKAIDEKAPHOBIA References Mazur, B. and Tate, J. "Points of Order 13 on Elliptic Curves." Invent. Math. 22,41/C1/49, 1973/74. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. 14 References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. 15 See also 15 PUZZLE ,FIFTEEN THEOREM 15 Puzzle A puzzle introduced by Sam Loyd in 1878. It consists of 15 squares numbered from 1 to 15 which are placedin a 4/C294 box leaving one position out of the 16 empty. The goal is to reposition the squares from a givenarbitrary starting arrangement by sliding them oneat a time into the configuration shown above. For some initial arrangements, this rearrangement is possible, but for others, it is not.To address the solubility of a given initial arrange-ment, proceed as follows. If the SQUARE containing the number iappears "before" (reading the squares in the box from left to right and top to bottom) numberswhich are less than , then call it an inversion of order , and denote it n i:Then define N /C13X15 i/C301ni /C30X15 i/C302ni ; where the sum need run only from 2 to 15 rather than 1 to 15 since there are no numbers less than 1 (so n1 must equal 0). If N is EVEN , the position is possible, otherwise it is not. This can be formally proved using ALTERNATING GROUPS . For example, in the following arrangement /n2 /C301 (2 precedes 1) and all other ni /C300 ; so N /C301 and the puzzle cannot be solved. Johnson (1879) proved that odd permutations of the puzzle are impossible, which Story (1879) proved that all even permutations are possible. While Herstein and Kaplansky (1978) wrote that "no really easy proof seems to be known," Archer (1999) presented a simple proof. A more general result due to Wilson (1974) showed that for any CONNECTED GRAPH on nodes, with the exception of CYCLE GRAPHS Cnand the THETA-0 GRAPH , either exactly half or all of the n! possible labelings are obtainable by sliding labels, depending on whether the graph is BIPARTITE (Archer 1999). u0has six inequivalent labelings, which has (n /C282)! inequivalent labelings. Reversing the order of the "8 Puzzle" made on a 3 /C293 board can be proved to require at least 26 moves, although the best solution requires 30 moves (Gard- ner 1984, pp. 200 and 206 /C1/207). The number of distinct solutions in 28, 30, 32, ... moves are 0, 10, 112, 512, ... (Sloane’s A046164), giving 634 solutions better than the 36-move solution given by Dudeney (1949). References Archer, A. F. "A Modern Treatment of the 15 Puzzle." Amer. Math. Monthly 106, 793 /C1/799, 1999. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 312 /C1/316, 1987. Beasley, J. D. The Mathematics of Games. Oxford, England: Oxford University Press, pp. 80 /C1/81, 1990. Bogomolny, A. "Sam Loyd’s Fifteen." http://www.cut-the- knot.com/pythagoras/fifteen.html. Bogomolny, A. "Sam Loyd’s Fifteen [History]." http:// www.cut-the-knot.com/pythagoras/history15.html. Davies, A. L. "Rotating the 15 Puzzle." Math. Gaz. 54, 237 /C1/ 240, 1970. Dudeney, H. E. Problem 253 in The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 64 /C1/65, 200 /C1/201, and 206 /C1/207, 1984. Herstein, I. N. and Kaplansky, I. Matters Mathematical, 2nd ed. New York: Chelsea, pp. 114 /C1/115, 1978.Hurd, S. and Trautman, D. "The Knight’s Tour on the 15- Puzzle." Math. Mag. 66, 159 /C1/166, 1993. Johnson, W. W. "Notes on the ‘15 Puzzle. I."’ Amer. J. Math. 2, 397 /C1/399, 1879. Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Tempus Books, pp. 177 /C1/180, 1989. Kraitchik, M. "The 15 Puzzle." §12.2.1 in Mathematical Recreations. New York: W. W. Norton, pp. 302 /C1/308, 1942. Liebeck, H. "Some Generalizations of the 14 /C1/15 Puzzle." Math. Mag. 44, 185 /C1/189, 1971. Loyd, S. Mathematical Puzzles of Sam Loyd, Vol. 1. New York: Dover, pp. 19 /C1/20, 1959. Loyd, S. Jr. Sam Loyd’s Cyclopedia of 5,000 Puzzles, Tricks, and Conundrums. Lamb Pub., 1993. Mallison, H. V. "An Array of Squares." Math. Gaz. 24, 119 /C1/ 121, 1940. Sloane, N. J. A. Sequences A046164 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Spitznagel, E. L. Jr. Selected Topics in Mathematics. New York: Holt, Rinehart and Winston, pp. 143 /C1/148, 1971. Spitznagel, E. L. Jr. "A New Look at the Fifteen Puzzle." Math. Mag. 40, 171 /C1/174, 1967. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 14 /C1/16, 1999. Story, W. E. "Notes on the ‘15 Puzzle. II."’ Amer. J. Math. 2, 399/C1/404, 1879. Whipple, F. J. W. "The Sign of a Term in the Expansion of a Determinant." Math. Gaz. 13, 126, 1926. Wilson, R. M. "Graph Puzzles, Homotopy, and the Alternat- ing Group." J. Combin. Th. Ser. B 16,8 6/C1/96, 1974. 15 Schoolgirl Problem KIRKMAN’S SCHOOLGIRL PROBLEM 16-Cell The finite regular 4-D CROSS POLYTOPE with S CHLA ¨- FLI SYMBOL f3;3;4gand VERTICES which are the PERMUTATIONS of (, 0, 0, 0). The 16-cell is the dual of the TESSERACT . Its graph is isomorphic to the CIRCU- LANT GRAPH Ci1;2;3(8):/ See also 24-CELL, 120-CELL, 600-CELL,CELL,CROSS POLYTOPE ,H YPERCUBE ,P ENTATOPE ,P OLYCHORON , POLYTOPE ,TESSERACT References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 17 is a FERMAT PRIME which means that the 17-sided REGULAR POLYGON (the HEPTADECAGON )is CONSTRUC- TIBLE using COMPASS and STRAIGHTEDGE (as proved by Gauss). See also CONSTRUCTIBLE POLYGON ,FERMAT PRIME , HEPTADECAGON References Lefevre, V. "Properties of 17." http://www.ens-lyon.fr/~vle- fevre/d17_eng.html. 17-gon HEPTADECAGON 18-Point Problem Place a point somewhere on a LINE SEGMENT . Now place a second point and number it 2 so that each of the points is in a different half of the LINE SEGMENT . Continue, placing every th point so that all points are on different (1=N)/th of the LINE SEGMENT . Formally, for a given , does there exist a sequence of real numbers x1 ; x2 ; ..., xNsuch that for every n /C23 f1; ... ; N g and every k /C23f1 ; ... ; ng; the inequality k /C28 1 n5xi Bk n holds for some i /C23f1 ; ... ; n g/? Surprisingly, it is only possible to place 17 points in this manner (Berlekamp and Graham 1970, Warmus 1976). Steinhaus (1979) gives a 14-point solution (0.06, 0.55, 0.77, 0.39, 0.96, 0.28, 0.64, 0.13, 0.88, 0.48, 0.19, 0.71, 0.35, 0.82), and Warmus (1976) gives the 17-point solution 4 7 5x1 B7 12;27 5x2 B5 17;1617 5x3 B1 ;1 14 5x4 B1 13; 8 11 5x5 B11 15;5 11 5x6 B6 13 ;17 5x7 B2 13 ;1417 5x8 B56; 38 5x9 B5 13;1117 5x10 B23 ;3 14 5x11 B3 13; 15 17 5x12 B1112;12 5x12 B9 17 ; 0 5x14 B1 17; 1317 5x15 B45;5 16 5x16 B6 17 ;1017 5x17 B1117; Warmus (1976) states that there are 768 patterns of 17-point solutions (counting reversals as equivalent). See also DISCREPANCY THEOREM ,POINT PICKING References Berlekamp, E. R. and Graham, R. L. "Irregularities in the Distributions of Finite Sequences." J. Number Th. 2, 152 /C1/ 161, 1970. Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, pp. 34 /C1/36, 1997. Steinhaus, H. "Distribution on Numbers" and "General- ization." Problems 6 and 7 in One Hundred Problems in 1979. Warmus, M. "A Supplementary Note on the Irregularities of Distributions." J. Number Th. 8, 260 /C1/263, 1976. 24-Cell A finite regular 4-D POLYTOPE with SCHLA ¨ FLI SYMBOL f3; 4; 3g: Coxeter (1969) gives a list of the VERTEX positions. The EVEN coefficients of the /D4/ lattice are 1, 24, 24, 96, ... (Sloane’s A004011), and the 24 shortest vectors in this lattice form the 24-cell (Coxeter 1973, Conway and Sloane 1993, Sloane and Plouffe 1995). The 24-cell is self-dual, and is the unique regular convex POLYCHORON which has no direct 3-D analog. One construction for the 24-cell evokes comparison with the RHOMBIC DODECAHEDRON . Given two equal cubes, we construct this dodecahedron by cutting one cube into six congruent square pyramids, and attach- ing these to the six squares bounding the other cube. Similarly, given two equal tesseracts, we can con- struct the 24-cell by cutting one tesseract into eight congruent cubic pyramids, and attaching these to the eight cubes bounding the other tesseract (Towle). See also 16-CELL, 120-CELL, 600-CELL,CELL,H YPER- CUBE ,PENTATOPE ,POLYCHORON ,POLYTOPE References Conway, J. H. and Sloane, N. J. A. Sphere-Packings, Lat- tices and Groups, 2nd ed. New York: Springer-Verlag, 1993. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Sloane, N. J. A. Sequences A004011/M5140 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M5150 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 36 Officer Problem How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, and major, a captain, a lieutenant, and a sub-lieutenant be ar- ranged in a regular 6 /C296 array such that no row or column duplicates a rank or a regiment? The answer is that no such arrangement is possible. See also EULER’S GRAECO- ROMAN SQUARES CONJEC- TURE ,LATIN SQUARE References Bose, R. C.; Shrikhande, S. S.; and Parker, E. T. "Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler’s Conjecture." Canad. J. Math. 12, 189, 1960. Bruck, R. H. and Ryser, H. J. "The Nonexistence of Certain Finite Projective Planes." Canad. J. Math. 1,88/C1/93, 1949. Parker, E. T. "Orthogonal Latin Squares." Not. Amer. Math. Soc. 6, 276, 1959. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 31, 1999. Tarry, G. "Le proble `me de 36 officiers." Compte Rendu de l’Assoc. Franc ¸ais Avanc. Sci. Naturel 1, 122 /C1/123, 1900. Tarry, G. "Le proble `me de 36 officiers." Compte Rendu de l’Assoc. Franc ¸ais Avanc. Sci. Naturel 2, 170 /C1/203, 1901. 42 According to Adams (1997), 42 is the ultimate answer to life, the universe, and everything, although it is left as an exercise to the reader to determine the actual question leading to this result. References Adams, D. The Hitchhiker’s Guide to the Galaxy. New York: Ballantine Books, 1997. 72 Rule RULE OF 72 120-Cell A finite regular 4-D POLYTOPE with SCHLA ¨ FLI SYMBOL f5; 3; 3g: The 120-cell has 600 vertices (Coxeter 1969), and consists of 120 DODECAHEDRA and 720 PENTAGONS (Coxeter 1973, p. 264). In the plate following p. 176, Coxeter (1973) illustrates the poly- tope. The dual of the 120-cell is the 600-CELL . See also 16-CELL, 24-CELL, 600-CELL,CELL,H YPER- CUBE ,PENTATOPE ,POLYCHORON ,POLYTOPE ,SIMPLEX References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, H. S. M. "Stellating ." §14.2 in Regular Polytopes, 3rd ed. New York: Dover, pp. 136 /C1/137, 157, 264 /C1/267, and 292, 1973.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 144 A DOZEN DOZEN , also called a GROSS . 144 is a SQUARE NUMBER and a SUM-PRODUCT NUMBER . See also DOZEN 163 The number 163 is very important in number theory, since d /C30163 is the largest number such that the IMAGINARY QUADRATIC FIELD Q /C28ffiffiffi dp/C16/C17 has CLASS NUMBER h(/C28d) /C301 : It also satisfies the curious iden- tities 163X4 i/C3008 i/C18/C19 (1) 1 244 /C278 4/C18/C19/C20/C21 (2) 1 244 /C27X4 i /C3004 i/C18/C192"# ; (3) wheren k/C0/C1 is a BINOMIAL COEFFICIENT (Stoschek). An approximation due to Stoschek is given by p :29 163 /C30512163 :3:1411043 ; (4) which is good to 3 digits. See also R AMANUJAN CONSTANT References Stoschek, E. "Modul 33: Algames with Numbers." http:// marvin.sn.schule.de/~inftreff/modul33/task33.htm. 196-Algorithm Take any POSITIVE INTEGER of two DIGITS or more, reverse the DIGITS , and add to the original number. Now repeat the procedure with the SUM so obtained. This procedure quickly produces PALINDROMIC NUM- BERS for most INTEGERS . For example, starting with the number 5280 produces (5280, 6105, 11121, 23232). The end results of applying the algorithm to 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, ... (Sloane’s A033865). The valuefor 89 is especially large, being 8813200023188. The first few numbers not known to produce PALIN- DROMES are 196, 887, 1675, 7436, 13783, ... (Sloane’s A006960), which are simply the numbers obtained by iteratively applying the algorithm to the number 196.This number therefore lends itself to the name of the ALGORITHM . In 1990, John Walker computed 2,415,836 iterations of the algorithm on 196 andobtained a number having 1,000,000 digits. This was extended in 1995 by Tim Irvin, who obtained a number having 2,000,000 digits. The rec.puzzles archive states that a 3,924,257-digit nonpalindromic number is obtained after 9,480,000 iterations. The number of terms a(n) in the iteration sequence required to produce a PALINDROMIC NUMBER from (i.e., a(n) /C301 for a PALINDROMIC NUMBER , a(n) /C302ifa PALINDROMIC NUMBER is produced after a single iteration of the 196-algorithm, etc.) for , 2, ... are 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, ... (Sloane’s A030547). The smallest numbers which require, 1, 2, ... iterations to reach a palindrome are 0, 10, 19, 59, 69, 166, 79, 188, ... (Sloane’s A023109). The 196-algorithm can be implemented in Mathema- tica as PalindromicQ[n_Integer?Positive]: /C30 Module[ {sn /C30ToString[n]}, sn /C30/C30StringReverse[sn] ] Algorithm196[n_Integer?PalindromicQ,it_:0]:- /C30{n} Algorithm196[n_Integer?Positive, it_:Infinity]: /C30 FixedPointList[# /C27 ToExpression[StringRe- verse[ToString[#]]]&, n, it, SameTest- /C21(PalindromicQ[#2]&) ] M. Sofroniou gives an efficient Mathematica imple- mentation which has complexity O k2ðÞ for steps, requiring approximately 10.6 hours on a 450 MHz Pentium II to compute 250,000 iterations. Extrapo- lating the timing data suggests that approximately 42 days would be needed on this same machine to match Walker’s 2,415,836 iterations. See also ADDITIVE PERSISTENCE ,D IGITADDITION , MULTIPLICATIVE PERSISTENCE ,P ALINDROMIC NUM- BER,P ALINDROMIC NUMBER CONJECTUR E, RATS SEQUENCE ,RECURRING DIGITAL INVARIANT References Brown, K. S. "Digit Reversal Sums Leading to Palindromes." http://www.seanet.com/~ksbrown/kmath004.htm. De Geest, P. "Websources about ‘196’ Becoming Palindromic by Using Reversal Sums." http://www.ping.be/~ping6758/ weblinks.htm. Eddins, S. "The Palindromic Order of a Number." IMSA Math. J. 4, Spring 1996. http://www.imsa.edu/edu/math/ journal/volume4/webver/palinord.html. Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 242 /C1/245, 1979. Gruenberger, F. "How to Handle Numbers with Thousands of Digits, and Why One Might Want to." Sci. Amer. 250, 19 /C1/26, Apr. 1984. Irving, T. "About Two Months of Computing, or, An Addendum to Mr. Walker’s Three Years of Computing" http://www.fourmilab.ch/documents/threeyears/two_- months_more.html. Math Forum. "Ask Dr. Math: Making Numbers into Palin- dromic Numbers." http://forum.swarthmore.edu/dr.math/ problems/barnes10.11.html. Peters, I. J. "Search for the Biggest Numeric Palindrome." http://www.floot.demon.co.uk/palindromes.html.rec.puzzles archive. 1996. ftp://rtfm.mit.edu/pub/usenet/ news.answers/puzzles/archive/arithmetic/part1. Safroniou, M. "Palindromic Numbers: The 196-Algorithm." MATHEMATICA NOTEBOOK ALGORITHM196.NB . Sloane, N. J. A. Sequences A006960/M5410, A023109, A030547, and A033865 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Walker, J. "Three Years of Computing: Final Report on the Palindrome Quest." http://www.fourmilab.ch/documents/ threeyears/threeyears.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . 239 Some interesting properties (as well as a few arcane ones not reiterated here) of the number 239 are discussed in Beeler et al. (1972, Item 63). 239 appears in MACHIN’S FORMULA 1 4 p /C304 tan/C28115/C16/C17 /C28tan /C2811 239/C16/C17 ; which is related to the fact that 2 /C215 134 /C281 /C302392 ; which is why 239/169 is the 7th CONVERGENT offfiffiffi 2p : Another pair of INVERSE TANGENT FORMULAS invol- ving 239 is tan/C2811 239/C16/C17 tan/C2811 70/C16/C17 /C28tan/C2811 99/C16/C17 tan/C281 1 408/C16/C17 /C27tan/C281 1 577/C16/C17 : 239 needs 4 SQUARES (the maximum) to express it, 9 CUBES (the maximum, shared only with 23) to express it, and 19 fourth POWERS (the maximum) to express it (see WARING’S PROBLEM ). However, 239 doesn’t need the maximum number of fifth POWERS (Beeler et al. 1972, Item 63). References Schroeppel, R. Item 63 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 24, Feb. 1972. 243 Feynman (1997) noticed the curious fact that the decimal expansion 1 243/C300:004115226337448559 . . . repeats pairs of the digits 0, 1, 2, 3, ... separated by the digits 4, 5, 6, 7, .... Just after this point, the pattern breaks, since the fraction is given exactly bythe repeating decimal 1 243/C300:004115226337448559670781893 : This pattern is related to the fact that 1 9/C300:¯1 and 1 81 /C300:0123456789 : References Feynman, R. P. and Leighton, R. ‘Surely You’re Joking, Mr. Feynman!’: Adventures of a Curious Character. New York: W. W. Norton, p. 99, 1997. 257-gon 257 is a FERMAT PRIME , and the 257-gon is therefore a CONSTRUCTIBLE POLYGON using COMPASS and STRAIGHTEDGE , as proved by Gauss. An illustration of the 257-gon is not included here, since its 257 segments so closely resemble a CIRCLE . Richelot and Schwendenwein found constructions for the 257-gon in 1832 (Coxeter 1969). De Temple (1991) gives a construction using 150 CIRCLES (24 of which are CARLYLE CIRCLES ) which has GEOMETROGRAPHY sym- bol 94S1 /C2747S2 /C27275C1 /C270C2 /C27150C3and SIMPLI- CITY 566. See also 65537-GON ,CONSTRUCTIBLE POLYGON ,FER- MAT PRIME ,HEPTADECAGON ,PENTAGON References Bachmann, P. Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie. Leipzig, Germany: Teub- ner, 1872. Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97 /C1/108, 1991. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352 /C1/386, 1955. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Klein, F. "The Construction of the Regular Polygon of 17 Sides." Part I, Ch. 4 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 24 /C1/41, 1980. Pascal, E. "Sulla costruzione del poligono regolare di 257 lati." Rendiconto dell Accad. della scienze fisiche e mate- mat. sezione della Soc. a reale di Napoli, Ser. 2 1,33/C1/39, 1887. Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964. Richelot, F. J. "De resolutione algebraica aequationis X257 /C30 1; sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata." J. reine angew. Math. 9,1/C1/26, 146 /C1/161, 209 /C1/ 230, and 337 /C1/358, 1832. Trott, M. " cos(2 p=257) a` la Gauss." Mathematica Educ. Res. 4,31/C1/36, 1995.600-Cell A finite regular 4-D POLYTOPE with SCHLA ¨ FLI SYMBOL f3; 3; 5g: The 600-cell has 120 VERTICES (Coxeter 1969). In the plate following p. 160, Coxeter (1973) gives two illustrations of the polytope. The dual of the 600-cell is the 120-CELL . See also 16-CELL, 24-CELL, 120-CELL,CELL,H YPER- CUBE ,PENTATOPE ,POLYCHORON ,POLYTOPE ,SIMPLEX References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, H. S. M. "Gosset’s Construction for . §8.5 in Regular Polytopes, 3rd ed. New York: Dover, pp. 136 /C1/137, 153 /C1/ 154, and 157, 1973. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 666 A number known as the BEAST NUMBER appearing in the Bible and ascribed various numerological proper- ties. See also APOCALYPTIC NUMBER ,B EAST NUMBER , LEVIATHAN NUMBER References De Geest, P. "The Number of the Best 666." http:// www.ping.be/~ping6758/weblinks.htm. Hardy, G. H. A Mathematician’s Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 96, 1993. 1729 1729 is sometimes called the HARDY- RAMANUJAN NUMBER . It is the smallest TAXICAB NUMBER , i.e., the smallest number which can be expressed as the sum of two cubes in two different ways: 1729/C3013/C27123/C3093/C27103: See also HARDY- RAMANUJAN NUMBER ,TAXICAB NUM- BER 2187 The digits in the number 2187 form the two VAMPIRE NUMBERS :2 1/C2987/C301827 and 2187 /C3027/C2981:2187 is also given by 37. See also VAMPIRE NUMBER References Gardner, M. "Lucky Numbers and 2187." Math. Intell. 19, 26 /C1/29, Spring 1997. 65537-gon 65537 is the largest known FERMAT PRIME , and the 65537-gon is therefore a CONSTRUCTIBLE POLYGON using COMPASS and STRAIGHTEDGE , as proved by Gauss. The 65537-gon has so many sides that it is, for all intents and purposes, indistinguishable from a CIRCLE using any reasonable printing or display methods. Hermes spent 10 years on the construction of the 65537-gon at Ko¨nigsberg around (1900). After the Second World War, his manuscripts were moved to the Mathematical Institute in Go¨ttingen, where they can now be viewed (Coxeter 1969).De Temple (1991) notes that a GEOMETRIC CONSTRUC- TION can be done using 1332 or fewer C ARLYLE CIRCLES . See also 257-GON ,CONSTRUCTIBLE POLYGON ,HEPTA- DECAGON ,PENTAGON References Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97/C1/108, 1991. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352 /C1/386, 1955. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Hermes, J. "Ueber die Teilung des Kreises in 65537 gleiche Teile." Nachr. Ko ¨nigl. Gesellsch. Wissensch. Go ¨ttingen, Math.-Phys. Klasse , pp. 170 /C1/186, 1894. A AAA Theorem Specifying three ANGLES A, B, and C does not uniquely define a TRIANGLE , but any two TRIANGLES with the same ANGLES are SIMILAR . Specifying two ANGLES of a TRIANGLE automatically gives the third since the sum of ANGLES in a TRIANGLE sums to 180 8 (/ p RADIANS ), i.e., C /C30 p /C28A /C28B : See also AAS THEOREM , ASA THEOREM , ASS THEO- REM, SAS THEOREM , SSS THEOREM ,TRIANGLE AAS Theorem Specifying two angles A and B and a side a uniquely determines a TRIANGLE with AREA K /C30a2 sin B sin C 2 sin A/C30a2 sin B sin( p /C28 A /C28 B) 2 sin A: (1) The third angle is given by C /C30 p /C28A /C28B ; (2) since the sum of angles of a TRIANGLE is 1808 (/p RADIANS ). Solving the LAW OF SINES a sin A /C30b sin B (3) for b gives b /C30asin B sin A : (4) Finally, c /C30b cos A /C27a cos B /C30a(sin B cot A /C27cos B) (5) /C30a sin B(cot A /C27cot B): (6) See also AAA THEOREM , ASA THEOREM , ASS THEO- REM, SAS THEOREM , SSS THEOREM ,TRIANGLEAbacus A mechanical counting device consisting of a frame holding a series of parallel rods on each of which beads are strung. Each bead represents a counting unit, and each rod a place value. The primary purpose of the abacus is not to perform actual computations, but to provide a quick means of storing numbers during a calculation. Abaci were used by the Japa- nese and Chinese, as well as the Romans. See also ROMAN NUMERAL ,SLIDE RULE References Boyer, C. B. and Merzbach, U. C. "The Abacus and Decimal Fractions." A History of Mathematics, 2nd ed. New York: Wiley, pp. 199 /C1/01, 1991. Fernandes, L. "The Abacus: The Art of Calculating with Beads." http://www.ee.ryerson.ca/~elf/abacus/. Gardner, M. "The Abacus." Ch. 18 in Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 232 /C1/41, 1979. Pappas, T. "The Abacus." In The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 209, 1989. Pullan, J. M. The History of the Abacus. New York: Prager, 1968. Smith, D. E. "Mechanical Aids to Calculation: The Abacus." Ch. 3 §1i n History of Mathematics, Vol. 2. New York: Dover, pp. 156 /C1/96, 1958. Yoshino, Y. The Japanese Abacus Explained. New York: Dover, 1963. abc Conjecture ACONJECTURE due to J. Oesterle ´and D. W. Masser. It states that, for any INFINITESIMAL e>0;there exists a CONSTANT Cesuch that for any three RELATIVELY PRIME INTEGERS a,b,csatisfying a/C27b/C30c; (1) the INEQUALITY max(½a½;½b½;½c½)5CeY p½abcp1/C27e(2) holds, where p½abcindicates that the PRODUCT is over PRIMES pwhich DIVIDE the PRODUCT abc. If this CONJECTURE were true, it would imply F ERMAT’S LAST THEOREM for sufficiently large POWERS (Goldfeld 1996). This is related to the fact that the abc conjecture implies that there are at least Clnx WIEFERICH PRIMES 5xfor some constant C(Silver- man 1988, Vardi 1991). The conjecture can also be stated by defining the height and radical of the sum P : a /C27 b /C30 c as h(P) /C30 max fln½a½; ln½b½; ln ½c ½g (3) r(P) /C30X p ½abcln p ; (4) where p runs over all prime divisors of a, b, and c. Then the abc conjecture states that for all e > 0; there exists a constant K such that for all P : a /C27b /C27c ; h(P) 5r(P) /C27 eh(P) /C27K (5) (van Frankenhuysen 2000). van Frankenhuysen (2000) has shown that there exists an infinite se- quence of sums P : a /C27b /C30c or RATIONAL INTEGERS with large height compared to the radical, h(p) ]r(P) /C274Klffiffiffiffiffiffiffiffiffiffi h(P)p ln[h(P)]; (6) with Kl/C302l=22p e !1=4 >1:517 (7) forl/C300:5990 ;improving a result of Stewart and Tijdeman (1986). See also FERMAT’S LAST THEOREM ,M ASON’S THEO- REM,M ORDELL CONJECTURE ,ROTH’S THEOREM ,W IE- FERICH PRIME References Cox, D. A. "Introduction to Fermat’s Last Theorem." Amer. Math. Monthly 101,3/C1/4, 1994. Elkies, N. D. "ABC Implies Mordell." Internat. Math. Res. Not. 7,9 9/C1/09, 1991. Goldfeld, D. "Beyond the Last Theorem." The Sciences 36, 34/C1/0, March/April 1996. Goldfeld, D. "Beyond the Last Theorem." Math. Horizons , 26/C1/1 and 24, Sept. 1996. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 75 /C1/6, 1994. Lang, S. "Old and New Conjectures in Diophantine Inequal- ities." Bull. Amer. Math. Soc. 23,3 7/C1/5, 1990. Lang, S. Number Theory III: Diophantine Geometry. New York: Springer-Verlag, pp. 63 /C1/7, 1991. Mason, R. C. Diophantine Equations over Functions Fields. Cambridge, England: Cambridge University Press, 1984. Mauldin, R. D. "A Generalization of Fermat’s Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer. Math. Soc. 44, 1436 /C1/437, 1997. Nitaq, A. "The abc Conjecture Home Page." http:// www.math.unicaen.fr/~nitaj/abc.html. Silverman, J. "Wieferich’s Criterion and the abc Conjecture." J. Number Th. 30, 226/C1/37, 1988. Stewart, C. L. and Tijdeman, R. "On the Oesterle ´-Masser Conjecture." Mh. Math. 102, 251/C1/57, 1986. Stewart, C. L. and Yu, K. "On the ABC Conjecture." Math. Ann. 291, 225/C1/30, 1991. van Frankenhuysen, M. "The ABC Conjecture Implies Roth’s Theorem and Mordell’s Conjecture." Mat. Contemp. 16,4 5/C1/2, 1999. van Frankenhuysen, M. "A Lower Bound in the abc Conjecture." J. Number Th. 82,9 1/C1/5, 2000.Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 66, 1991. Vojta, P. Diophantine Approximations and Value Distribu- tion Theory. Berlin: Springer-Verlag, p. 84, 1987. Abel Polynomial A polynomial An(x;a) given by the associated S HEF- FER SEQUENCE with f(t)/C30teat; (1) given by An(x;a)/C30x(x/C28an)n/C281: (2) The GENERATING FUNCTION is X/C12 k/C300Ak(x;a) k!tk/C30exW(at)=a; (3) where W(x)i sL AMBERT’S W-FUNCTION . The asso- ciated BINOMIAL IDENTITY is (x/C27y)(x/C27y/C28an)n/C281 /C30Xn k/C300n krC1+rC1D xy(x/C28ak)k/C281[y/C28a(n/C28k)]n/C28k/C281; (4) wheren krC0rC1 is a BINOMIAL COEFFICIENT , a formula originally due to Abel (Riordan 1979, p. 18; Roman 1984, pp. 30 and 73). The first few Abel polynomials are A0(x;a)/C301 A1(x;a)/C30x A2(x;a)/C30x(x/C282a) A3(x;a)/C30x(x/C283a)2 A4(x;a)/C30x(x/C284a)3: References Riordan, J. Combinatorial Identities. New York: Wiley, p. 18, 1979. Roman, S. "The Abel Polynomials." §4.1.5 in The Umbral Calculus. New York: Academic Press, pp. 29 /C1/0 and 72 /C1/5, 1984. Abel Transform The following INTEGRAL TRANSFORM relationship, known as the Abel transform, exists between two functions f(x) and g(t) for 0BaB1; f(x)/C30gx 0g(t)dt (x/C28t)a(1) g(t)/C30/C28sin(pa) pd dtgt 0f(x)dx (x/C28t)1/C28a(2) /C30/C28sin(pa) pgt 0df dxdx (t/C28x)1/C28a/C27f(0) t1/C28a"# : (3) The Abel transform is used in calculating the radial mass distribution of galaxies (Binney and Tremaine 1987) and inverting planetary radio occultation data to obtain atmospheric information as a function of height. Bracewell (1999, p. 262) defines a slightly different form of the Abel transform given by g(x) /C30A[f(r)] /C302g/C12 xf(r)rdrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28 x2p : (4) The following table gives a number of common Abel transform pairs (Bracewell 1999, p. 264). Here, Pa(x) /C13Px 2a /C281 2 ! /C301 for 0 Bx B0 0 otherwiserC06 (5) where P(x) is the RECTANGLE FUNCTION , and M(x) /C302 p x /C283gx 0J0(x) dx /C28x /C282J0(x)rC00rC01 (6) /C30p2 x2 [J1(x)H0(x) /C28J0(x)H1(x)]; (7) where Jn(x)isaB ESSEL FUNCTION OF THE FIRST KIND and Hn(x)isaS TRUVE FUNCTION . /f(r)// g(x)/ conditions / Pa(r)// 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28x2p // a2 > x2/ /(a2 /C28r2)/C281=2 Pa(r)//p// a2 > x2/ /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28r2p Pa(r)//1 2p(a2 /C28x2)// a2 > x2/ /(a2 /C28r2)Pa(r)//4 3(a2 /C28x2)3=2 // a2 > x2/ /(a2 /C28r2)3=2 Pa(r)//3 8p(a2 /C28x2)2 // a2 > x2/ /(a /C28 r)Pa(r)// affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28x2p /C28x2 cosh /C281a xrC1+rC1D / /1 pcosh /C281a rrC1+rC1D // a /C28x/ / d(r /C28a)//2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28 x2p Pa(x)/ /e /C28r2 =s2 // sffiffiffippe /C28x2 =s2 // s > 0/ /r2e /C28r2 =s2 // s(x2 /C271 2s2)ffiffiffippe /C28x2 =s2 // s > 0/ /e /C28r2 =s2 sffiffiffipp (r2 /C281 2s2)//x2e /C28x2 = s2 // s > 0/ /1 b2 /C27 r2//pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C27 r2p // b2 /C27x2 > 0/ /J0(vr)//2 cos( vx) v// v > 0/ /M(r)//8p4 v2x2sin2xv 2 prC1+rC1D // v > 0/ See also FOURIER TRANSFORM ,HILBERT TRANSFORM , INTEGRAL EQUATIONReferences Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 11 and 97, 1988. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 875 /C1/76, 1985. Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 651, 1987. Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 262 /C1/66, 1999. Hilfer, R. (Ed.). Applications of Fractional Calculus in Physics. Singapore: World Scientific, pp. 3 /C1/, 2000. Liouville, J. "Memoire sur quelques que´stions de ge´ome´trie et de me´canique, et sur un nouveau genre pour re´spondre ces que´stions." J. E´ cole Polytech. 13,1/C1/9, 1832. Lu¨tzen, J. Joseph Liouville, 1809 /C1/882. Master of Pure and Applied Mathematics. New York: Springer-Verlag, p. 314, 1990. Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 376 /C1/77, 1967. Abel’s Binomial Theorem The identity Xm y/C300m yrC1+rC1D (w /C28y)m/C28y /C281(z /C27y)y /C30w /C281(z /C27w /C27m)m (Bhatnagar 1995, p. 51). There are a host of other such BINOMIAL IDENTITIES . See also BINOMIAL IDENTITY , Q-ABEL’S THEOREM References Abel, N. H. "Beweis eines Ausdrucks, von welchem die Binomial-Formel ein einzelner Fall ist." J. reine angew. Math. 1, 159/C160, 1826. Reprinted in Euvres Comple `tes, 2nd ed., Vol. 1. pp. 102 /C103, 1881. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U (n) Extensions. Ph.D. thesis. Ohio State University, p. 51, 1995. Riordan, J. Combinatorial Identities. New York: Wiley, p. 18, 1979. Abel’s Convergence Theorem Given a T AYLOR SERIES f(z)/C30X/C12 n/C300Cnzn/C30X/C12 n/C300Cnrneinu; (1) where the COMPLEX NUMBER zhas been written in the polar form z/C30reiu;examine the REAL and IMAGINARY PARTS u(r;u)/C30X/C12 n/C300Cnrncos(nu) (2) v(r;u)/C30X/C12 n/C300Cnrnsin(nu): (3) Abel’s theorem states that, if u(1;u) and v(1;u) are CONVERGENT , then u(1;u)/C27iv(1;u)/C30lim r01f(reiu): (4) Stated in words, Abel’s theorem guarantees that, if a REAL POWER SERIES CONVERGES for some POSITIVE value of the argument, the DOMAIN of UNIFORM CONVERGENCE extends at least up to and including this point. Furthermore, the continuity of the sum function extends at least up to and including this point. References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 773, 1985. Abel’s Curve Theorem The sum of the values of an INTEGRAL of the "first" or "second" sort gx1 ; y1 x0 ; y0Pdx Q/C27.../C27gxN ; yN x0 ; y0Pdx Q/C30F(z) and P(x1 ; y1) Q(x1 ; y1)dx1 dz/C27.../C27P(xN ; yN) Q(xN ; yN)dxN dz/C30dF dz; from a FIXED POINT to the points of intersection with a curve depending rationally upon any number of parameters is a RATIONAL FUNCTION of those para- meters. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 277, 1959. Abel’s Differential Equation The Abel equation of the first kind is given by y ?/C30f0(x) /C27f1(x)y /C27f2(x)y2 /C27f3(x)y3 /C27... (Murphy 1960, p. 23; Zwillinger 1997, p. 120), and the Abel equation of the second kind by [g0(x) /C27g1(x)y]y?/C30f0(x) /C27f1(x)y /C27f2(x)y2 /C27f3(x)y3 (Murphy 1960, p. 25; Zwillinger 1997, p. 120). References Murphy, G. M. Ordinary Differential Equations and Their Solution. Princeton, NJ: Van Nostrand, 1960. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Abel’s Differential Equation Identity Given a homogeneous linear SECOND-ORDER ORDIN- ARY DIFFERENTIAL EQUATION , yƒ/C27 P(x)y?/C27Q(x)y /C30 0 ; (1) call the two linearly independent solutions y1(x) and y2(x) : Then yƒ1 /C27P(x)y?1 /C27Q(x)y1 /C300 (2) yƒ2 /C27P(x)y?2 /C27Q(x)y2 /C300: (3)Now, take y1/C29 (3) minus y2/C29 (2), y1[yƒ2 /C27P(x)y?2 /C27Q(x)y2] /C28y2[yƒ1 /C27P(x)y?1 /C27Q(x)y1] /C300 (4) (y1yƒ2 /C28y2yƒ1) /C27P(y1y?2 /C28y?1y2) /C27Q(y1y2 /C28y1y2) /C300 (5) (y1yƒ2 /C28y2yƒ1) /C27P(y1y?2 /C28y?1y2) /C300 : (6) Now, use the definition of the WRONSKIAN and take its DERIVATIVE , W /C13y1y?2 /C27y?1y2 (7) W ?/C30(y?y ?2 /C27y1yƒ2) /C28(y?1y?2 /C27y ƒ1y2) y1yƒ2 /C28yƒ1y2 : (8) Plugging W and W ? into (6) gives W ?/C27PW /C300 : (9) This can be rearranged to yield dW W/C30/C28P(x) dx (10) which can then be directly integrated to lnW(x) W0"# /C30/C28gP(x) dx; (11) where lnx is the NATURAL LOGARITHM . Exponentiat- ing then yields Abel’s identity W(x) /C30W0e /C28gP(x) dx ; (12) where W0 is a constant of integration. See also ORDINARY DIFFERENTIAL EQUATION– SECOND- ORDER References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 118, 262, 277, and 355, 1986. Abel’s Duplication Formula The duplication formula for ROGERS L-FUNCTION follows from A BEL’S FUNCTIONAL EQUATION and is given by 1 2L(x2)/C30L(x)/C28Lx 1/C27x ! : See also ABEL’S FUNCTIONAL EQUATION ,D ILOGA- RITHM References Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431/C1/48, 1997. Abel’s Functional Equation Let L(x) denote the ROGERS L-FUNCTION defined in terms of the usual DILOGARITHM by L(x) /C306 p2Li2(x) /C271 2 ln x ln(1 /C28x)hi /C306 p2X/C12 n /C301xn n2 /C271 2 ln x ln(1 /C28x)"# ; then L(x) satisfies the functional equation L(x) /C27L(y) /C30L(xy) /C27Lx(1 /C28 y) 1 /C28 xy ! /C27Ly(1 /C28 x) 1 /C28 xy ! : ABEL’S DUPLICATION FORMULA follows from this iden- tity. See also ABEL’S DUPLICATION FORMULA ,D ILOGA- RITHM ,F UNCTIONAL EQUATION ,P OLYLOGARITHM , RIEMANN ZETA FUNCTION ,ROGERS L-FUNCTION References Abel, N. H. Oeuvres Completes, Vol. 2 (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 189 /C1/92, 1988. Bytsko, A. G. Two-Term Dilogarithm Identities Related to Conformal Field Theory. 9 Nov 1999. http://xxx.lanl.gov/ abs/math-ph/9911012/. Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431 /C1/48, 1997. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 14 and 21, 1999. Rogers, L. J. "On Function Sum Theorems Connected with the Series a/C12 1 xn =n2:/" Proc. London Math. Soc. 4, 169 /C189, 1907. Abel’s Impossibility Theorem In general, POLYNOMIAL equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of ADDITIONS , SUBTRACTIONS , MULTI- PLICATIONS , DIVISIONS , and ROOT EXTRACTIONS . This was also shown by Ruffini in 1813 (Wells 1986, p. 59). See also CUBIC EQUATION ,GALOIS’S THEOREM ,POLY- NOMIAL ,QUADRATIC EQUATION ,QUARTIC EQUATION , QUINTIC EQUATION References Abel, N. H. "Beweis der Unmo ¨glichkeit, algebraische Glei- chungen von ho¨heren Graden als dem vierten allgemein aufzulo ¨sen." J. reine angew. Math. 1, 65, 1826. Reprinted in Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 66 /C17, 1988. Artin, E. Galois Theory, 2nd ed. Notre Dame, IN: Edwards Brothers, 1944. Faucette, W. M. "A Geometric Interpretation of the Solution of the General Quartic Polynomial." Amer. Math. Monthly 103,51/C17, 1996. Fraleigh, J. B. A First Course in Abstract Algebra. Reading, MA: Addison-Wesley, 1982. Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975.Hungerford, T. W. Algebra. New York: Springer-Verlag, 1980. van der Waerden, B. L. A History of Algebra: From al- Khwarizmi to Emmy Noether. New York: Springer-Verlag, pp. 85 /C18, 1985. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 59, 1986. Abel’s Inequality Let ffn g and fan g be SEQUENCES with fn ]fn/C271 > 0 for n /C30 1, 2, ..., then jXm n/C301anfnj5Af1 ; where A /C30max f½a1 ½;½a1 /C27a2 ½;...;½a1 /C27a2 /C27.../C27am ½g: Abel’s Irreducibility Theorem If one ROOT of the equation f(x) /C300; which is irredu- cible over a FIELD K, is also a ROOT of the equation F(x) /C300in K, then all the ROOTS of the irreducible equation f(x) /C300 are ROOTS of F(x) /C300: Equivalently, F(x) can be divided by f(x) without a REMAINDER , F(x) /C30f(x)F1(x); where F1(x) is also a POLYNOMIAL over K. See also ABEL’S LEMMA ,KRONECKER’S POLYNOMIAL THEOREM ,SCHO¨ NEMANN’S THEOREM References Abel, N. H. "Me´moire sur une classe particulie `re d’e´quations re´solubles alge´briquement." J. reine angew. Math. 4, 1829. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 120, 1965. Abel’s Lemma The pure equation xp /C30C of PRIME degree p is irreducible over a FIELD when C is a number of the FIELD but not the pth POWER of an element of the FIELD . Jeffreys and Jeffreys (1988) use the term "Abel’s lemma" for another LEMMA related to A BEL’S UNIFORM CONVERGENCE TEST . See also ABEL’S IRREDUCIBILITY THEOREM ,G AUSS’S POLYNOMIAL THEOREM ,K RONECKER’S POLYNOMIAL THEOREM ,SCHO¨ NEMANN’S THEOREM References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 118, 1965. Jeffreys, H. and Jeffreys, B. S. "Abel’s Lemma." §1.1153 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 41 /C1/2, 1988. Abel’s Test ABEL’S UNIFORM CONVERGENCE TEST Abel’s Theorem ABEL’S BINOMIAL THEOREM ,A BEL’S CONVERGENCE THEOREM ,ABEL’S CURVE THEOREM ,ABEL’S IMPOSSI- BILITY THEOREM ,ABEL’S IRREDUCIBILITY THEOREM , ABELIAN THEOREM , Q-ABEL’S THEOREM Abel’s Uniform Convergence Test Let fun(x) g be a SEQUENCE of functions. If 1. un(x) can be written un(x) /C30anfn(x);/ 2. aan is CONVERGENT , 3. fn(x)isa MONOTONIC DECREASING SEQUENCE (i.e., fn/C271(x) 5fn(x)) for all n, and 4. fn(x)is BOUNDED in some region (i.e., 0 5fn(x) 5 M for all x e [a ; b])/ then, for all x /C23 [a ; b]; the SERIES aun(x) CONVERGES UNIFORMLY . See also CONVERGENCE TESTS ,CONVERGENT SERIES , UNIFORM CONVERGENCE References Bromwich, T. J. I’a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 59, 1991. Jeffreys, H. and Jeffreys, B. S. "Abel’s Lemma" and "Abel’s Test." §1.1153 /C1/.1154 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 41 /C1/2, 1988. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 17, 1990. Abelian A group or other algebraic object is said to be Abelian is the law of commutativity always holds. If an algebraic object is not Abelian, it is said to be NON- ABELIAN . See also ABELIAN CATEGORY ,ABELIAN DIFFERENTIAL , ABELIAN FUNCTION ,ABELIAN GROUP ,ABELIAN INTE- GRAL ,A BELIAN VARIETY ,C OMMUTATIVE ,N ON-ABE- LIAN Abelian Category An Abelian category is an abstract mathematical CATEGORY which displays some of the characteristic properties of the CATEGORY of all ABELIAN GROUPS . See also ABELIAN GROUP ,CATEGORYReferences Freyd, P. Abelian Categories: An Introduction to the Theory of Functors. New York: Harper & Row, 1964. Grothendieck, A. "Sur quelques points d’alge `bre homologi- que." Toˆhoku Math. J. 9, 119 /C1/21, 1957. Mac Lane, S. and Gehring, F. W. Categories for the Working Mathematician, 2nd ed. New York: Springer-Verlag, 1998. Abelian Differential An Abelian differential is an ANALYTIC or MERO- MORPHIC DIFFERENTIAL on a COMPACT or closed RIEMANN SURFACE . Abelian Extension This entry contributed by NICOLAS BRAY If F is an ALGEBRAIC GALOIS EXTENSION of K such that the GALOIS GROUP of the extension is ABELIAN , then F is said to be an Abelian extension of K. See also ALGEBRAIC EXTENSION ,GALOIS EXTENSION , GALOIS GROUP Abelian Function An INVERSE FUNCTION of an ABELIAN INTEGRAL . Abelian functions have two variables and four peri- ods, and can be defined by U y ; t;q ? qrC1+rC1D /C30X/C12 l /C30/C28/C1222piy(l/C27q ?)/C27 pit(l /C27q ?)2/C272piq(l/C27q?) Baker (1907, p. 21). Abelian functions are a general- ization of ELLIPTIC FUNCTIONS , and are also called hyperelliptic functions. See also ABELIAN INTEGRAL ,E LLIPTIC FUNCTION , THETA FUNCTIONS References Baker, H. F. Abelian Functions: Abel’s Theorem and the Allied Theory, Including the Theory of the Theta Func- tions. New York: Cambridge University Press, 1995. Baker, H. F. An Introduction to the Theory of Multiply Periodic Functions. London: Cambridge University Press, 1907. Weisstein, E. W. "Books about Abelian Functions." http:// www.treasure-troves.com/books/AbelianFunctions.html. Abelian Group N.B. A detailed online essay by S. Finch was the starting point for this entry. AGROUP for which the elements COMMUTE (i.e., AB/C30 BAfor all elements Aand B) is called an Abelian group. All CYCLIC GROUPS are Abelian, but an Abelian group is not necessarily CYCLIC . All SUBGROUPS of an Abelian group are NORMAL . In an Abelian group, each element is in a CONJUGACY CLASS by itself, and the CHARACTER TABLE involves POWERS of a single ele- ment known as a GENERATOR . No general formula is known for giving the number of nonisomorphic FINITE GROUPS of a given ORDER . However, the number of nonisomorphic Abelian FINITE GROUPS a(n) of any given ORDER n is given by writing n as n /C30Y ip ai i; (1) where the pi are distinct PRIME FACTORS , then a(n) /C30Y iP( ai) ; (2) where P(k) is the PARTITION FUNCTION . This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (Sloane’s A000688). The smallest orders for which n /C30 1, 2, 3, ... noniso- morphic Abelian groups exist are 1, 4, 8, 36, 16, 72, 32, 900, 216, 144, 64, 1800, 0, 288, 128, ... (Sloane’s A046056), where 0 denotes an impossible number (i.e., not a product of partition numbers) of noniso- morphic Abelian, groups. The "missing" values are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, ... (Sloane’s A046064). The incrementally largest num- bers of Abelian groups as a function of order are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (Sloane’s A046054), which occur for orders 1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ... (Sloane’s A046055). The KRONECKER DECOMPOSITION THEOREM states that every FINITE Abelian group can be written as a GROUP DIRECT PRODUCT of CYCLIC GROUPS of PRIME POWER ORDER . If the ORDER of a FINITE GROUP is a PRIME p, then there exists a single Abelian group of order p (denoted Zp) and no non-Abelian groups. If the ORDER is a prime squared p2 then there are two Abelian groups (denoted Zp2 and Zp /C29Zp : If the ORDER is a prime cubed p3 ; then there are three Abelian groups (denoted Zp /C29Zp /C29Zp ; Zp /C29Zp2 ; and Zp3 ) ; and five groups total. If the order is a PRODUCT of two primes p and q, then there exists exactly one Abelian group of ORDER pq (denoted Zp /C29Zq) :/ Another interesting result is that if a(n) denotes the number of nonisomorphic Abelian groups of ORDER n, then X/C12 n/C301a(n)n /C28s /C30 z(s) z(2s) z(3s) /C1/C1/C1; (3) where z(s) is the RIEMANN ZETA FUNCTION . Srinivasan (1973) has also shown that XN n/C301a(n) /C30A1N /C27A2N1=2 /C27A3N1 =3 /C27O[x105=407(ln x)2] ; (4) whereAk /C13Y j/C301 j"kzj k ! /C302:294856591... for k /C301 /C2814:6475663 ... for k /C302 118:6924619 . . . for k/C303;8 < :(5) and z(s) is again the R IEMANN ZETA FUNCTION . [Richert (1952) incorrectly gave A3/C30114:/] DeKoninck and Ivic (1980) showed that XN n/C3011 a(n)/C30BN/C27O[ffiffiffiffiffi Np (lnN)/C281=2]; (6) where B/C13Y 1/C28X/C12 k/C3021 P(k/C282)/C281 P(k)"# 1 pk() /C300:752 . . . (7) is a product over PRIMES . Bounds for the number of nonisomorphic non-Abelian groups are given by Neumann (1969) and Pyber (1993). See also FINITE GROUP ,GROUP THEORY ,KRONECKER DECOMPOSITION THEOREM ,PARTITION FUNCTION P, RING References Arnold, D. M. and Rangaswamy, K. M. (Eds.). Abelian Groups and Modules. New York: Dekker, 1996. DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Func- tions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields. Amsterdam, Netherlands: North-Holland, 1980. Erdos, P. and Szekeres, G. "U ¨ber die Anzahl abelscher Gruppen gegebener Ordnung und u ¨ber ein verwandtes zahlentheoretisches Problem." Acta Sci. Math. (Szeged) 7, 95/C1/02, 1935. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/abel/abel.html. Fuchs, L. and Go ¨bel, R. (Eds.). Abelian Groups. New York: Dekker, 1993. Kendall, D. G. and Rankin, R. A. "On the Number of Abelian Groups of a Given Order." Quart. J. Oxford 18, 197/C1/08, 1947. Kolesnik, G. "On the Number of Abelian Groups of a Given Order." J. reine angew. Math. 329, 164/C1/75, 1981. Neumann, P. M. "An Enumeration Theorem for Finite Groups." Quart. J. Math. Ser. 2 20, 395/C1/01, 1969. Pyber, L. "Enumerating Finite Groups of Given Order." Ann. Math. 137, 203/C1/20, 1993. Richert, H.-E. "U ¨ber die Anzahl abelscher Gruppen gegeb- ener Ordnung I." Math. Zeitschr. 56,2 1/C1/2, 1952. Sloane, N. J. A. Sequences A000688/M0064 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Srinivasan, B. R. "On the Number of Abelian Groups of a Given Order." Acta Arith. 23, 195/C1 /05, 1973. Abelian Integral An INTEGRAL OF THE FORM gx 0dtffiffiffiffiffiffiffiffiffi R(t)p ; where R(t)i sa POLYNOMIAL of degree >4:They are also called HYPERELLIPTIC INTEGRALS . See also ABELIAN FUNCTION ,ELLIPTIC INTEGRAL References Siegel, C. L. Topics in Complex Function Theory, Vol. 2: Automorphic Functions and Abelian Integrals. New York: Wiley, 1988. Abelian Theorem A theorem which asserts that if a sequence or function behaves regularly, then some average of it behaves regularly. For example, A(x) /C2x implies A1(x) /C30gx 0A(t) dt /C21 2x2 for any A(x) : The converse is false, but can be made into a correct TAUBERIAN THEOREM if A(x) is subjected to an appropriate additional condition (Hardy 1999, p. 46). See also TAUBERIAN THEOREM References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 46, 1999. Abelian Variety An Abelian variety is an algebraic GROUP which is a complete ALGEBRAIC VARIETY . An Abelian variety of DIMENSION 1isan ELLIPTIC CURVE . See also ALBANESE VARIETY References Murty, V. K. Introduction to Abelian Varieties. Providence, RI: Amer. Math. Soc., 1993. Shimura, G. Abelian Varieties With Complex Multiplication and Modular Functions. Princeton, NJ: Princeton Uni- versity Press, 1999. Shimura, G. and Taniyama, Y. Complex Multiplication of Abelian Varieties and Its Applications to Number Theory. Tokyo: Mathematical Society of Japan, 1961. Abelianization In general, groups are not ABELIAN . However, there is always a GROUP HOMOMORPHISM h : G 0 G? to an ABELIAN GROUP , and this homomorphism is called Abelianization. The homomorphism is abstractly described by its kernel, the COMMUTATOR SUBGROUP [G, G]. So G ?/C30G=[G; G] : Roughly speaking, in any expression, every product becomes commutative after Abelianization. As a consequence, some previously unequal expressions may become equal, or even represent the IDENTITY ELEMENT . For example, in the eight-element QUATERNION GROUP /G /C30f91;9i ;9j;9kg/, the COMMUTATOR SUB- GROUP is f91g: The Abelianization of G is a copy ofZ2 /C29Z2 ; and for instance, i ?j?/C30j?i ? in the Abelianiza- tion. See also ABELIAN ,GROUP ,HOMOMORPHISM Abel-Plana Formula This entry contributed by DAVID ANDERSON The Abel-Plana formula gives an expression for the difference between a discrete sum and the corre- sponding integral. The formula can be derived from the ARGUMENT PRINCIPLE Ggf(z)g ?(z) g(z)dz /C30X nf( mn) /C28X mf( nm); (1) where mnare the zeros of g(z) and nmare the poles contained within the CONTOUR g : An appropriate choice of g and g then yields X/C12 n/C300f(n) /C28g/C12 0f(x) dx /C301 2 f(0) /C2812g/C12 0[f(it) /C28f(/C28it)][cot( pit) /C27i] dt; (2) or equivalently X/C12 n/C300f(n) /C28g/C12 0f(x) dx /C301 2 f(0) /C27ig/C12 0f(it) /C28 f( /C28it) e2 pt /C28 1dt : (3) The formula is particularly useful in Casimir effect calculations involving differences between quantized modes and free modes. See also ARGUMENT PRINCIPLE References Mostepanenko, V. M. and Trunov, N. N. §2.2 in The Casimir Effect and Its Applications. Oxford, England: Clarendon Press, 1997. Saharian, A. A. "The Generalized Abel-Plana Formula. Applications to Bessel Functions and Casimir Effect." http://www.ictp.trieste.it/~pub_off/preprints-sources/2000/IC2000014P.pdf. Abhyankar’s Conjecture For a FINITE GROUP G, let p(G) be the SUBGROUP generated by all the S YLOW P-SUBGROUPS ofG.I fXis a projective curve in characteristic p/C210, and if x0;..., xtare points of X(fort/C210), then a NECESSARY and SUFFICIENT condition that Goccur as the G ALOIS GROUP of a finite covering YofX, branched only at the points x0;...,xt;is that the QUOTIENT GROUP G=p(G) has 2 g/C27tgenerators. Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the projective line with a point deleted), and Harbater (1994) proved thefull Abhyankar conjecture by building upon this special solution. See also FINITE GROUP ,G ALOIS GROUP ,Q UOTIENT GROUP ,SYLOW P-SUBGROUP References Abhyankar, S. "Coverings of Algebraic Curves." Amer. J. Math. 79, 825 /C1/56, 1957. American Mathematical Society. "Notices of the AMS, April 1995, 1995 Frank Nelson Cole Prize in Algebra." http:// www.ams.org/notices/199504/prize-cole.pdf. Harbater, D. "Abhyankar’s Conjecture on Galois Groups Over Curves." Invent. Math. 117,1/C1/5, 1994. Raynaud, M. "Reve ˆtements de la droite affine en caracte ´r- istique p /C210 et conjecture d’Abhyankar." Invent. Math. 116, 425 /C1/62, 1994. Ablowitz-Ramani-Segur Conjecture The Ablowitz-Ramani-Segur conjecture states that a nonlinear PARTIAL DIFFERENTIAL EQUATION is solva- ble by the INVERSE SCATTERING METHOD only if every nonlinear ORDINARY DIFFERENTIAL EQUATION ob- tained by exact reduction has the PAINLEVE ´ PROP- ERTY . See also INVERSE SCATTERING METHOD References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 351, 1989. Abnormal Number A hypothetical number which can be factored into primes in more than one way. Hardy and Wright (1979) prove the FUNDAMENTAL THEOREM OF ARITH- METIC by showing that no abnormal numbers exist. See also FUNDAMENTAL THEOREM OF ARITHMETIC References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 21, 1979. Abs ABSOLUTE VALUE Abscissa The x- (horizontal) coordinate of a point in a two dimensional coordinate system. Physicists and as- tronomers sometimes use the term to refer to the axis itself instead of the distance along it. See also AXIS,ORDINATE ,REAL LINE, X-AXIS, Y-AXIS, Z-AXIS Absolute Convergence A SERIES anunis said to CONVERGE absolutely if the SERIES an unjj CONVERGES , where unjj denotes the ABSOLUTE VALUE .Ifa SERIES is absolutely convergent, then the sum is independent of the order in which terms are summed. Furthermore, if the SERIES ismultiplied by another absolutely convergent series, the product series will also converge absolutely. See also CONDITIONAL CONVERGENCE ,CONVERGENT SERIES ,RIEMANN SERIES THEOREM References Bromwich, T. J. I’a. and MacRobert, T. M. "Absolute Con- vergence." Ch. 4 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 69 /C1/7, 1991. Jeffreys, H. and Jeffreys, B. S. "Absolute Convergence." §1.051 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 16, 1988. Absolute Deviation Let ¯u denote the MEAN of a SET of quantities ui ; then the absolute deviation is defined by Dui /C13 ui /C28 ¯u jj : See also DEVIATION ,M EAN DEVIATION ,SIGNED DE- VIATION ,STANDARD DEVIATION Absolute Error The DIFFERENCE between the measured or inferred value of a quantity x0 and its actual value x, given by Dx /C13x0 /C28x (sometimes with the ABSOLUTE VALUE taken) is called the absolute error. The absolute error of the SUM or DIFFERENCE of a number of quantities is less than or equal to the SUM of their absolute errors. See also ERROR PROPAGATION ,PERCENTAGE ERROR , RELATIVE ERROR References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Absolute Frequency The number of data points which fall within a given CLASS in a FREQUENCY DISTRIBUTION . See also CUMULATIVE FREQUENCY ,FREQUENCY DIS- TRIBUTION ,RELATIVE FREQUENCY ,RELATIVE CUMU- LATIVE FREQUENCY References Kenney, J. F. and Keeping, E. S. "Frequency Distributions." §1.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 12 /C1/9, 1962. Absolute Geometry GEOMETRY which depends only on the first four of EUCLID’S POSTULATES and not on the PARALLEL POSTULATE . Euclid himself used only the first four postulates for the first 28 propositions of the ELE- MENTS , but was forced to invoke the PARALLEL POSTULATE on the 29th. See also AFFINE GEOMETRY , ELEMENTS ,E UCLID’S POSTULATES ,GEOMETRY ,ORDERED GEOMETRY ,PAR- ALLEL POSTULATE References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 90 /C1/1, 1989. Absolute Moment The absolute moment of Mnof a probability function P(x) taken about a point a is defined by Mn /C30g x /C28a jjnP(x) dx: See also CENTRAL MOMENT ,MOMENT ,RAW MOMENT References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 146, 1984. Absolute Monotonic Sequence See also ABSOLUTELY MONOTONIC SEQUENCE References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 224, 1971. Absolute Pseudoprime CARMICHAEL NUMBER Absolute Square Also known as the squared norm. The absolute square of a COMPLEX NUMBER z is written zjj2 ; where zjjis the MODULUS and is defined as zjj2/C13z¯z; (1) where ¯z denotes the COMPLEX CONJUGATE of z. For a REAL NUMBER , (1) simplifies to zjj2/C30z2 : (2) If the COMPLEX NUMBER is written z /C30x /C27iy; then the absolute square can be written x /C27iy jj2/C30x2 /C27y2 : (3) An absolute square can be computed in terms of x and y using the Mathematica command ComplexExpan- d[Abs[z]2,TargetFunctions- /C21{Conjugate} ]. An important identity involving the absolute square is given bya 9be /C28i drC10rC10rC10rC102/C30(a 9be /C28id)(a 9beid) /C30a2 /C27b2 9ab(ei d /C27e /C28id) /C30a2 /C27b2 92ab cos d : (4) If a /C301, then (4) becomes 1 9be /C28idrC10rC10rC10rC10 2/C301 /C27b2 92b cos d /C30(1 9b)2 /C144b sin2(1 2 d) : (5) If a /C301, and b /C301, then 1 /C28e /C28idrC10rC10rC10rC102/C304 sin2(1 2 d): (6) Finally, ½eif1 /C27eif2 ½2 /C30(eif1 /C27eif2 )(e /C28if1 /C27e /C28i f2 ) /C302[1 /C27cos(f2/C28f1)] /C304 cos2[1 2(f2/C28f1)]: (7) See also ARGUMENT (COMPLEX NUMBER ), COMPLEX NUMBER ,MODULUS (COMPLEX NUMBER ) Absolute Value The absolute value of a REAL NUMBER xis denoted xjj and given by the "unsigned" portion of x, xjj/C30xsgn(x)/C30/C28xforx50 xforx]0;rC06 where sgn xis the sign function SGN. The absolute value is therefore always greater than or equal to 0. The same notation is used to denote the MODULUS of a COMPLEX NUMBER z/C30x/C27iy;zjj/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p ;aP-ADIC NORM , or a general VALUATION . The NORM of a VECTOR xis also denoted xjj;although xjjis more commonly used. Other NOTATIONS similar to the absolute value are the FLOOR FUNCTION /C28x/C29bc ;NINT function [ x];and CEILING FUNCTION /C26x/C27de :/ The integral of the absolute value of the different of two variables is given by g1 0 g1 0x /C28y jjndx dy /C302 (n /C27 1)(n /C27 2) ; which has values 1/3, 1/6, 1/10, 1/15, 1/21, ... for n /C301, 2, ..., i.e., the inverses of the TRIANGULAR NUMBERS (Sloane’s A000217). See also ABSOLUTE SQUARE ,C EILING FUNCTION , FLOOR FUNCTION ,M ODULUS (COMPLEX NUMBER ), NINT,RECTANGLE FUNCTION ,SGN,TRIANGLE FUNC- TION ,VALUATION References Sloane, N. J. A. Sequences A000217/M2535 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Absolutely Continuous A MEASURE l is absolutely continuous with respect to another measure m if l(E) /C300 for every set with m(E) /C300: This makes sense as long as m is a POSITIVE MEASURE , such as LEBESGUE MEASURE , but l can be any measure, possibly a COMPLEX MEASURE . By the RADON- NIKODYM THEOREM , this is equivalent to saying that l(E) /C30gEfdm where the integral is the LEBESGUE INTEGRAL , for some INTEGRABLE function f. The function f is like a derivative, and is called the RADON- NIKODYM DERI- VATIVE dl=d m:/ The measure supported at 0 (/m(E) /C301 iff 0 /C23 E) is not absolutely continuous with respect to LEBESGUE MEASURE , and is a SINGULAR MEASURE . See also COMPLEX MEASURE ,CONCENTRATED ,HAAR MEASURE ,L EBESGUE DECOMPOSITION (MEASURE ), LEBESGUE MEASURE ,M UTUALLY SINGULAR ,POLAR REPRESENTATION (MEASURE ), SINGULAR MEASURE References Rudin, W. Functional Analysis, 2nd ed. New York: McGraw- Hill, pp. 121 /C1/25, 1991. Absolutely Fair A sequence of random variates X0 ; X1 ; ... is called absolutely fair if for n /C301, 2, ..., (X1) /C300 and (Xn/C271 ½X1 ; ...; Xn) /C300 (Feller 1971, p. 210). See also MARTINGALE References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971.Absolutely Monotonic Function This entry contributed by RONALD M. AARTS A function f(x) is absolutely monotonic in the interval a Bx Bb if it has nonnegative derivatives of all orders in the region, i.e., f (k)(x) ]0 (1) for a Bx Bb and k /C300, 1, 2, .... For example, the functions f(x) /C30/C28ln(/C28x)(/C281 5x B0) (2) and f(x) /C30sin /C281 x (0 5x 51) (3) are absolutely monotonic functions (Widder 1941). See also ABSOLUTELY MONOTONIC SEQUENCE References Widder, D. V. Ch. 4 in The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Absolutely Monotonic Sequence See also ABSOLUTE MONOTONIC SEQUENCE ,A BSO- LUTELY MONOTONIC FUNCTION References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 224, 1971. Absorption Law The law appearing in the definition of a BOOLEAN ALGEBRA which states a ffl(a /C150b) /C30a /C150(a fflb) /C30a for binary operators /C150 and ffl (which most commonly are logical OR and logical AND). See also BOOLEAN ALGEBRA ,LATTICE References Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, p. 317, 1996. Abstract Algebra That portion of ALGEBRA dealing with theoretical as opposed to applied topics. Ash (1998) includes the following areas in his definite of abstract algebra: logic and foundations, counting, elementary NUMBER THEORY , informal SET THEORY , LINEAR ALGEBRA , and the theory of linear operators. See also ALGEBRA References Ash, R. B. A Primer of Abstract Mathematics. Washington, DC: Math. Assoc. Amer., 1998. Abstract Manifold An abstract manifold is a MANIFOLD in the context of an abstract space with no particular embedding, or representation in mind. It is a TOPOLOGICAL SPACE with an ATLAS of COORDINATE CHARTS . For example, the SPHERE S2 can be considered a SUBMANIFOLD of R3 or a QUOTIENT SPACE O(3) =O(2): But as an abstract manifold, it is just a MANIFOLD , which can be covered by two coordinate charts / f1: R2 0 S2 / and /f2: R2 0 S2 /, with the single TRANSITION FUNCTION , f/C281 2( f1 : R2 /C28(0; 0) 0 R2 /C28(0; 0) defined by f/C281 2(f1(x; y) /C30(x=r2 ; y=r2) where /r2 /C30x2 /C27y2/. It can also be thought of as two disks glued together at their boundary. See also ALGEBRAIC MANIFOLD ,H OMOGENEOUS SPACE ,M ANIFOLD ,S UBMANIFOLD ,T OPOLOGICAL SPACE Abstract Mathematics ABSTRACT ALGEBRA Abstract Simplicial Complex An abstract simplicial complex is a collection S of finite nonempty sets such that if A is an element of S, then so is every nonempty subset of A (Munkres 1993, p. 15). See also SIMPLICIAL COMPLEX References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Abstract Vector Space See also QUOTIENT VECTOR SPACE ,VECTOR SPACE Abstraction Operator LAMBDA CALCULUS Abundance The abundance of a number n is the quantity A(n) /C13 s(n) /C282n; where s(n) is the DIVISOR FUNCTION . Kravitz has conjectured that no numbers exist whose abundance is an ODD SQUARE (Guy 1994). The following table lists special classifications given to a number n based on the value of A(n) ://A(n)/ Number /B0/ DEFICIENT NUMBER -1 ALMOST PERFECT NUMBER 0 PERFECT NUMBER 1 QUASIPERFECT NUMBER / > 0/ ABUNDANT NUMBER See also ABUNDANCY ,DEFICIENCY References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45 /C1/6, 1994. Abundancy The ratio s(n)=n ; where s(n) is the DIVISOR FUNCTION . See also ABUNDANCE ,ABUNDANT NUMBER References Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63,3/C1/0, 1990. Abundant Number An abundant number is an INTEGER n which is not a PERFECT NUMBER and for which s(n) /C13 s(n) /C28n > n; (1) where s(n) is the DIVISOR FUNCTION . The quantity s(n) /C282n is sometimes called the ABUNDANCE . The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (Sloane’s A005101). Abundant numbers are some- times called EXCESSIVE NUMBERS . There are only 21 abundant numbers less than 100, and they are all EVEN . The first ODD abundant number is 945/C3033/C2157/C2155: (2) That 945 is abundant can be seen by computing s(945)/C30975 >945: (3) Any multiple of a PERFECT NUMBER or an abundant number is also abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers. Define the density function A(x)/C13lim n0/C12½fn:s(n)]xng½ n(4) for a POSITIVE REAL NUMBER x, then Davenport (1933) proved that A(x) exists and is continuous for all x, and Erdos (1934) gave a simplified proof (Finch). Wall (1971) and Wall et al. (1977) showed that 0:2441BA(2)B0:2909 ; (5) and Dele´glise (1998) showed that 0:2474 BA(2) B0 :2480 : (6) A number which is abundant but for which all its PROPER DIVISORS are DEFICIENT is called a PRIMITIVE ABUNDANT NUMBER (Guy 1994, p. 46). See also ALIQUOT SEQUENCE ,D EFICIENT NUMBER , HIGHLY ABUNDANT NUMBER ,M ULTIAMICABLE NUM- BERS ,PERFECT NUMBER ,PRACTICAL NUMBER ,PRIMI- TIVE ABUNDANT NUMBER ,W EIRD NUMBER References Dele´glise, M. "Bounds for the Density of Abundant Integers." Exp. Math. 7, 137/C1/43, 1998. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3 /C1/3, 1952. Erdos, P. "On the Density of the Abundant Numbers." J. London Math. Soc. 9, 278/C1/82, 1934. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/abund/abund.html. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45 /C1/6, 1994. Singh, S. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker, pp. 11 and 13, 1997. Sloane, N. J. A. Sequences A005101/M4825 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Souissi, M. Un Texte Manuscrit d’Ibn Al-Banna’ Al-Marra- kusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975. Wall, C. R. "Density Bounds for the Sum of Divisors Function." In The Theory of Arithmetic Functions: Pro- ceedings of the Conference at Western Michigan Univer- sity, April 29-May 1, 1971. (Ed. A. A. Gioia and D. L. Goldsmith). New York: Springer-Verlag, pp. 283 /C1/87, 1971. Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds for the Sum of Divisors Function." Math. Comput. 26, 773/C1/77, 1972. Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds for the Sum of Divisors Function." Math. Comput. 31, 616, 1977. Acceleration Let a particle travel a distance s(t) as a function of time t(here, scan be thought of as the ARC LENGTH of the curve traced out by the particle). The SPEED (the SCALAR NORM of the VECTOR VELOCITY ) is then given by ds dt/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx dt !2 /C27dy dt !2 /C27dz dt !2vuut: (1) The acceleration is defined as the time DERIVATIVE of the VELOCITY , so the SCALAR acceleration is given by a/C13dv dt(2)/C30d2s dt2(3) /C30dx dtd2x dt2/C27dy dtd2y dt2/C27dz dtd2z dt2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx dt !2 /C27dy dt !2 /C27dz dt !2vuut(4) /C30dx dsd2x dt2/C27dy dsd2y dt2/C27dz dsd2z dt2(5) /C30dr ds/C215d2r dt2: (6) The VECTOR acceleration is given by a/C13dv dt/C30d2r dt2/C30d2s dt2ˆT/C27kds dt !2 ˆN; (7) where ˆTis the UNIT TANGENT VECTOR ,kthe CURVA- TURE ,sthe ARC LENGTH , and ˆNthe UNIT NORMAL VECTOR . Let a particle move along a straight LINE so that the positions at times t1;t2;and t3ares1;s2;and s3; respectively. Then the particle is uniformly acceler- ated with acceleration aIFF a/C132(s2/C28s3)t1/C27(s3/C28s1)t2/C27(s1/C28s2)t3 (t1/C28t2)(t2/C28t3)(t3/C28t1)"# (8) is a constant (Klamkin 1995, 1996). Consider the measurement of acceleration in a rotat- ing reference frame. Apply the ROTATION OPERATOR ˜R/C13d dt ! body/C27v/C29 (9) twice to the RADIUS VECTOR rand suppress the body notation, aspace/C30˜R2r/C30d dt/C27v/C29 !2 r/C30d dt/C27v/C29 ! dr dt/C27v/C29r ! /C30d2r dt2/C27d dt(v/C29r)/C27v/C29dr dt/C27v/C29(v/C29r) /C30d2r dt2/C27v/C29dr dt/C27r/C29dv dt/C27v/C29dr dt /C27v/C29(v/C29r): (10) Grouping terms and using the definitions of the VELOCITY v/C13dr=dtand ANGULAR VELOCITY a/C13 dv=dtgive the expression aspace /C30d2r dt2 /C272v /C29v /C27 v /C29(v /C29r) /C27r /C29 a: (11) Now, we can identify the expression as consisting of three terms abody /C13d2r dt2 ; (12) aCoriolis /C132v /C29v; (13) acentrifugal /C13 v /C29( v /C29r) ; (14) a "body" acceleration, centrifugal acceleration, and Coriolis acceleration. Using these definitions finally gives aspace /C30abody /C27aCoriolis /C27acentrifugal /C27r /C29 a; (15) where the fourth term will vanish in a uniformly rotating frame of reference (i.e., a /C300): The centrifu- gal acceleration is familiar to riders of merry-go- rounds, and the Coriolis acceleration is responsible for the motions of hurricanes on Earth and necessi- tates large trajectory corrections for intercontinental ballistic missiles. See also ANGULAR ACCELERATION ,A RC LENGTH , JERK,VELOCITY References Klamkin, M. S. "Problem 1481." Math. Mag. 68, 307, 1995. Klamkin, M. S. "A Characteristic of Constant Acceleration." Solution to Problem 1481. Math. Mag. 69, 308, 1996. Accidental Cancellation ANOMALOUS CANCELLATION Accretion CUMULATION Accumulation Point An accumulation point is a POINT which is the limit of a SEQUENCE , also called a LIMIT POINT . For some MAPS , periodic orbits give way to CHAOTIC ones beyond a point known as the accumulation point. See also BOLZANO- WEIERSTRASS THEOREM Bolzano- Weierstrass Theorem, CANTOR’S INTERSECTION THE- OREM ,CHAOS ,FRACTIONAL PART,HEINE- BOREL THE- OREM ,LIMIT POINT ,LOGISTIC MAP,M ODE LOCKING , PERIOD DOUBLING ,P ISOT- VIJAYARAGHAVAN CON- STANT Achilles and the Tortoise Paradox ZENO’S PARADOXES Achiral AMPHICHIRALAckermann Function The Ackermann function is the simplest example of a WELL DEFINED TOTAL FUNCTION which is COMPUTABLE but not PRIMITIVE RECURSIVE , providing a counter- example to the belief in the early 1900s that every COMPUTABLE FUNCTION was also PRIMITIVE RECUR- SIVE (Do¨tzel 1991). It grows faster than an exponen- tial function, or even a multiple exponential function. The Ackermann function A(x;y) is defined by A(x;y)/C13y/C271i f x/C300 A(x/C281;1) if y/C300 A(x/C281;A(x;y/C281)) otherwise :8 < :(1) Special values for INTEGER xinclude A(0;y)/C30y/C271 (2) A(1;y)/C30y/C272 (3) A(2;y)/C302y/C273 (4) A(3;y)/C302y/C273/C283 (5) A(4;y)/C3022U2 |{z} y/C273/C283: (6) Expressions of the latter form are sometimes called POWER TOWERS .A(0;y) follows trivially from the definition. A(1;y) can be derived as follows, A(1;y)/C30A(0;A(1;y/C281))/C30A(1;y/C281)/C271 /C30A(0;A(1;y/C282))/C271/C30A(1;y/C282)/C272 /C30.../C30A(1;0)/C27y/C30A(0;1)/C27y/C30y/C272: (7) /A(2;y) has a similar derivation, A(2;y)/C30A(1;A(2;y/C281))/C30A(2;y/C281)/C272 /C30A(1;A(2;y/C282))/C272/C30A(2;y/C282)/C274/C30... /C30A(2;0)/C272y/C30A(1;1)/C272y/C302y/C273: (8) Buck (1963) defines a related function using the same fundamental RECURRENCE RELATION (with arguments flipped from Buck’s convention) F(x;y)/C30F(x/C281;F(x;y/C281)); (9) but with the slightly different boundary values F(0;y)/C30y/C271 (10) F(1;0)/C302 (11) F(2;0)/C302 (12) F(x;0)/C301 for x/C303;4;:. . . (13) Buck’s recurrence gives F(1;y)/C302/C27y (14) F(2;y)/C302y (15) F(3;y)/C302y(16) F(4; y) /C30 22U2 |ffl{zffl} y: (17) Taking F(4; n) gives the sequence 1, 2, 4, 16, 65536, 265536, ... (Sloane’s A006263). Defining ah(x) /C30F(x; x) for x /C300, 1, ... then gives 1, 3, 4, 8, 65536, 22U2 |ffl{zffl} m; ... (Sloane’s A001695), where m /C30 2U2 |{z} 65536; a truly huge number! See also ACKERMANN NUMBER ,COMPUTABLE FUNC- TION ,GOODSTEIN SEQUENCE ,POWER TOWER ,PRIMI- TIVE RECURSIVE FUNCTION , TAK FUNCTION ,TOTAL FUNCTION References Buck, R. C. "Mathematical Induction and Recursive Defini- tions." Amer. Math. Monthly 70, 128 /C1/35, 1963. Do¨tzel, G. "A Function to End All Functions." Algorithm: Recreational Programming 2.4,16/C1/7, 1991. Kleene, S. C. Introduction to Metamathematics. New York: Elsevier, 1971. Pe´ter, R. Rekursive Funktionen. Budapest: Akad. Kiado, 1951. Reingold, E. H. and Shen, X. "More Nearly Optimal Algo- rithms for Unbounded Searching, Part I: The Finite Case." SIAM J. Comput. 20, 156 /C1/83, 1991. Rose, H. E. Subrecursion, Functions, and Hierarchies. New York: Clarendon Press, 1988. Sloane, N. J. A. Sequences A001695/M2352 and A006263/ M1310 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Smith, H. J. "Ackermann’s Function." http://pweb.netcom.- com/~hjsmith/Ackerman.html. Spencer, J. "Large Numbers and Unprovable Theorems." Amer. Math. Monthly 90, 669 /C1/75, 1983. Tarjan, R. E. Data Structures and Network Algorithms. Philadelphia PA: SIAM, 1983. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 11, 227, and 232, 1991. Ackermann Number A number OF THE FORM n /C160/C1/C1/C1/C160n|fflfflfflfflffl{zfflfflfflfflffl} n; where ARROW NOTATION has been used. The first few Ackermann numbers are 1 /C1601 /C301; 2 /C160/C160 2 /C304; and 3 /C160/C160/C160 3 /C30 33U3 |ffl{zffl} 7 ;625;507;484;987:/ See also ACKERMANN FUNCTION ,ARROW NOTATION , POWER TOWER References Ackermann, W. "Zum hilbertschen Aufbau der reellen Zahlen." Math. Ann. 99, 118 /C1/33, 1928. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 60 /C1/1, 1996. Crandall, R. E. "The Challenge of Large Numbers." Sci. Amer. 276,74/C1/9, Feb. 1997. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 11, 227, and 232, 1991.Acnode Another name for an ISOLATED POINT . See also CRUNODE ,SPINODE ,TACNODE Acoptic Polyhedron A term invented by B. Gru¨nbaum in an attempt to promote concrete and precise POLYHEDRON terminol- ogy. The word "coptic" derives from the Greek for "to cut," and acoptic polyhedra are defined as POLYHEDRA for which the FACES do not intersect (cut) themselves, making them 2-MANIFOLDS . See also HONEYCOMB ,NOLID ,POLYHEDRON ,SPONGE Action Let M(X) denote the GROUP of all invertible MAPS X 0 X and let G be any GROUP .AHOMOMORPHISM u : G 0 M(X) is called an action of G on X. Therefore, u satisfies 1. For each g /C23 G ; u(g)isa MAP X 0 X : x /C2 u(g)x ;/ 2. u(gh)x /C30 u(g)( u(h)x);/ 3. u(e)x /C30x; where e is the group identity in G, 4. u(g /C281)x /C30 u(g) /C281x:/ See also CASCADE ,F LOW,S EMIDIRECT PRODUCT , SEMIFLOW Actuarial Polynomial The polynomials a(b) n(x) given by the S HEFFER SE- QUENCE with g(t)/C30(1/C28t)/C28b(1) f(t)/C30ln(1/C28t); (2) giving GENERATING FUNCTION X/C12 k/C300a(b) n k!tk/C30ex(1/C28et)/C27bt: (3) The Sheffer identity is a(b) n(x/C27y)/C30Xn k/C300n krC1+rC1D a(b) k(y)fn/C28k(/C28x); (4) where fn(x)i sa n EXPONENTIAL POLYNOMIAL . The actuarial polynomials are given in terms of the EXPONENTIAL POLYNOMIALS fn(x)b y a(b) n(x)/C30(1/C28t)bfn(/C28x) (5) /C30Xn k/C300b krC1+rC1D f(k) n(/C28x): (6) They are related to the S TIRLING NUMBERS OF THE SECOND KIND S(n;m)b y a(b) n(x)/C30Xn k/C300b krC1+rC1DXn j/C30kS(n;j)(j)k(/C28x)j/C28k; (7) wheren krC0rC1 is a BINOMIAL COEFFICIENT and (x)nis a FALLING FACTORIAL . The actuarial polynomials also satisfy the identity a(b) n(/C28x) /C30e /C28xX/C12 k/C300(k /C27 b)n k!xk (8) (Roman 1984, p. 125; Whittaker and Watson 1990, p. 336). The first few polynomials are a( b) 0(x) /C301 a( b) 1(x) /C30/C28x /C27 b a( b) 2(x) /C30x2 /C28x(1 /C272b) /C27 b2 a( b) 3(x) /C30/C28x3 /C273x2( b /C271) /C28x(3b2 /C273b /C271) /C27 b3 : See also SHEFFER SEQUENCE References Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 42, 1964. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 254, 1981. Roman, S. "The Actuarial Polynomial." §4.3.4 in The Umbral Calculus. New York: Academic Press, pp. 123 /C1/25, 1984. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Acute Angle An ANGLE of less than p=2 RADIANS (90 8) is called an acute angle. See also ACUTE TRIANGLE ,A NGLE ,F ULL ANGLE , OBTUSE ANGLE ,R EFLEX ANGLE ,R IGHT ANGLE , STRAIGHT ANGLE Acute Triangle A TRIANGLE in which all three ANGLES are ACUTE ANGLES .A TRIANGLE which is neither acute nor a RIGHT TRIANGLE (i.e., it has an OBTUSE ANGLE )is called an OBTUSE TRIANGLE . From the LAW OF CO-SINES , for a triangle with side lengths a, b, and c, cos C /C30a2 /C27 b2 /C28 c2 2ab; with C the angle opposite side C. For an angle to be acute, cos C > 0 : Therefore, an acute triangle satisfies a2 /C27b2 > c2 ; b2 /C27c2 > a2 ; and c2 /C27a2 > b2 :/ The smallest number of acute triangles into which an arbitrary OBTUSE TRIANGLE can be dissected is seven if B > 90 /C14; B /C28A; B /C28C B90 /C14; and otherwise eight (Manheimer 1960, Gardner 1981, Wells 1991). A SQUARE can be dissected into as few as 9 acute triangles (Gardner 1981, Wells 1991). See also OBTUSE TRIANGLE ,ONO INEQUALITY ,RIGHT TRIANGLE References Gardner, M. "Mathematical Games: A Fifth Collection of ‘Brain-Teasers."’ Sci. Amer. 202, 150 /C1/54, Feb. 1960. Gardner, M. "Mathematical Games: The Games and Puzzles of Lewis Carroll and the Answers to February’s Problems." Sci. Amer. 202, 172 /C1/82, Mar. 1960. Gardner, M. "Mathematical Games: The Inspired Geome- trical Symmetries of Scott Kim." Sci. Amer. 244,22/C1/1, Jun. 1981. Goldberg, G. "Problem E1406." Amer. Math. Monthly 67, 923, 1960. Hoggatt, V. E. Jr. "Acute Isosceles Dissection of an Obtuse Triangle." Amer. Math. Monthly 68, 912 /C1/13, 1961. Johnson, R. S. "Problem 256 [1977: 155]." Crux Math. 4,53/C1/ 4, 1978. Nelson, H. L. "Solution to Problem 256." Crux Math. 4, 102 /C1/ 04, 1978. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 1 /C1/, 1991. Acyclic Digraph An acyclic digraph is a DIRECTED GRAPH containing no directed cycles, also known as a directed acyclic graph or a "DAG." Every acyclic digraph has at least one node of OUTDEGREE 0. The numbers of acyclic digraphs on n/C301, 2, ... vertices are 1, 2, 6, 31, 302, 5984, ... (Sloane’s A003087). See also DIRECTED GRAPH ,FOREST References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 200, 1994. Robinson, R. W. "Counting Unlabeled Acyclic Digraphs." In Combinatorial Mathematics V (Melbourne 1976) . Provi- dence, RI: Amer. Math. Soc., pp. 28 /C1/3, 1976. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 190, 1990. Sloane, N. J. A. Sequences A003087/M1696 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Acyclic Graph FOREST Ad ADJOINT REPRESENTATION ,A DJOINT REPRESENTA- TION (LIE GROUP ) Adams’ Circle Given a TRIANGLE DABC ; construct the CONTACT TRIANGLE DTATBTC : Now extend lines parallel to the sides of the CONTACT TRIANGLE from the GERGONNE POINT . These intersect the triangle DABC in the six points P, Q, R, S, T, and U. As C. Adams proved in 1843, these points are CONCYCLIC in a CIRCLE now known as Adams’ circle. Moreover, Adams’ circle is concentric with the INCIRCLE of DABC (Honsberger 1995, pp. 62 /C1/4). Extend the segments UP, TS, and RQ to form a TRIANGLE DXYZ : Then the GERGONNE POINT of DABC is the SYMMEDIAN POINT ofDXYZ ;and Adams’ circle ofDABC is the L EMOINE CIRCLE ofDXYZ (Honsberger 1995, p. 98). See also CONTACT TRIANGLE ,GERGONNE POINT References Honsberger, R. "A Real Gem." §7.4 (v) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 62 /C1/4 and 98, 1995. Adams’ Method Adams’ method is a numerical METHOD for solving linear FIRST-ORDER ORDINARY DIFFERENTIAL EQUA- TIONS OF THE FORM dy dx/C30f(x;y): (1) Let h/C30xn/C271/C28xn (2) be the step interval, and consider the M ACLAURIN SERIES ofyabout xn; yn/C271/C30yn/C27dydx ! n(x/C28xn)/C2712d2y dx2 ! n(x/C28xn)2/C27... (3) dydx ! n/C271/C30dydx ! n/C27d2y dx2 ! n(x/C28xn)2/C27...: (4) Here, the DERIVATIVES ofyare given by the BACK- WARD DIFFERENCES qn/C13dydx ! n/C30Dyn xn/C271/C28xn/C30yn/C271/C28yn h(5) 9qn/C13d2y dx2 ! n/C30qn/C28qn/C281 (6) 92qn/C13d3y dx3 ! n/C309qn/C289qn/C281; (7) etc. Note that by (1), qnis just the value of f(xn;yn):/ For first-order interpolation, the method proceeds by iterating the expression yn/C271/C30yn/C27qnh (8) where qn/C13f(xn;yn):The method can then be ex- tended to arbitrary order using the finite differenceintegration formula from Beyer (1987) g1 0fpdp/C30 1/C271 29/C275 1292/C273893/C2725172094/C2795 28895/C27190876048096/C27...rC16rC1* fp (9) to obtain yn/C271 /C28yn /C30h(qn /C271 2 9qn/C281 /C275 12 92qn /C282 /C2738 93qn/C283 /C27251 720 94qn/C284 /C2795 288 95qn/C285 /C27...Þ: (10) Note that von Ka´rma´n and Biot (1940) confusingly use the symbol normally used for FORWARD DIFFER- ENCES d to denote BACKWARD DIFFERENCES 9:/ See also GILL’S METHOD ,M ILNE’S METHOD ,PREDIC- TOR-CORRECTOR METHODS ,RUNGE- KUTTA METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 896, 1972. Bashforth, F. and Adams, J. C. Theories of Capillary Action. London: Cambridge University Press, 1883. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 455, 1987. Jeffreys, H. and Jeffreys, B. S. "The Adams-Bashforth Method." §9.11 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 292 /C1/93, 1988. Ka´rma´n, T. von and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treat- ment of Engineering Problems . New York: McGraw-Hill, pp. 14 /C1/0, 1940. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 741, 1992. Whittaker, E. T. and Robinson, G. "The Numerical Solution of Differential Equations." Ch. 14 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 363 /C1/67, 1967. Adams-Bashforth-Moulton Method ADAMS’ METHOD Addend A quantity to be ADDED to another, also called a SUMMAND . For example, in the expression a /C27b /C27c; a, b, and c are all addends. The first of several addends, or "the one to which the others are added" (a in the previous example), is sometimes called the AUGEND . See also ADDITION ,AUGEND ,PLUS,RADICAND Addition The combining of two or more quantities using the PLUS operator. The individual numbers being com- bined are called ADDENDS , and the total is called the SUM. The first of several ADDENDS , or "the one to which the others are added," is sometimes called the AUGEND . The opposite of addition is SUBTRACTION . While the usual form of adding two n-digit INTEGERS (which consists of summing over the columns right to left and "CARRYING " a 1 to the next column if the sum exceeds 9) requires n operations (plus carries), two n-digit INTEGERS can be added in about 2 lg n steps by n processors using carry-lookahead addition (McGeoch 1993). Here, lg x is the LG function, the LOGARITHM to the base 2. See also ADDEND ,A MENABLE NUMBER ,A UGEND , CARRY ,D IFFERENCE ,D IVISION ,M ULTIPLICATION , PLUS,SUBTRACTION ,SUM References McGeoch, C. C. "Parallel Addition." Amer. Math. Monthly 100, 867 /C1/71, 1993. Addition Chain An addition chain for a number n is a SEQUENCE 1 /C30 a0 Ba1 B...Bar /C30n; such that each member after a0 is the SUM of two earlier (not necessarily distinct) ones. The number r is called the length of the addition chain. For example, 1; 1 /C271 /C302; 2 /C272 /C304; 4 /C272 /C306 ; 6 /C272 /C308; 8 /C276 /C3014 is an addition chain for 14 of length r /C305 (Guy 1994). See also BRAUER CHAIN ,H ANSEN CHAIN ,SCHOLZ CONJECTURE References Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111 /C1/13, 1994. Addition-Multiplication Magic Square A square which is simultaneously a MAGIC SQUARE and MULTIPLICATION MAGIC SQUARE . The top square shown above has order eight, with addition MAGIC CONSTANT 840 and multiplicative magic constant 2,058,068,231,856,000 (Horner 1955, Hunter and Madachy 1975). The bottom two squares have order nine with addition MAGIC CONSTANTS 848 and 1200 and multiplicative magic constants 5,804,807,833,440,000 and 1,619,541,385,529,760, 000, respectively (Hunter and Madachy 1975, Mada- chy 1979). L. Sallows has constructed an interesting 3 /C293 magic square in which the products of corresponding pairs of 2 /C292 diagonals are 12, 24, 36, and 72, while the products of the numbers in the pair of 3 /C293 diagonals also give 72. See also MAGIC SQUARE References Horner, W. W. "Addition-Multiplication Magic Square of Order 8." Scripta Math. 21,23/C1/7, 1955. Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, pp. 30 /C1/1, 1975. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 89 /C1/1, 1979. Additive Number Theory The portion of NUMBER THEORY concerned with expressing an integer as a sum of integers from some given set. See also CIRCLE METHOD ,M ULTIPLICATIVE NUMBER THEORY ,NUMBER THEORY Additive Persistence Consider the process of taking a number, adding its DIGITS , then adding the DIGITS of the number derived from it, etc., until the remaining number has only one DIGIT . The number of additions required to obtain a single DIGIT from a number n is called the additive persistence of n, and the DIGIT obtained is called the DIGITAL ROOT of n. For example, the sequence obtained from the starting number 9876 is (9876, 30, 3), so 9876 has an additive persistence of 2 and a DIGITAL ROOT of 3. The additive persistences of the first few positive integers are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, ... (Sloane’s A031286). The smallest numbers of additive persistence n for n /C300, 1, ... are 0, 10, 19, 199, 19999999999999999999999, ... (Sloane’s A006050). See also ADDITIVE PERSISTENCE ,D IGITADDITION , DIGITAL ROOT,M ULTIPLICATIVE PERSISTENCE ,N AR- CISSISTIC NUMBER ,RECURRING DIGITAL INVARIANTReferences Hinden, H. J. "The Additive Persistence of a Number." J. Recr. Math. 7, 134 /C1/35, 1974. Sloane, N. J. A. Sequences A006050/M4683 and A031286 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sloane, N. J. A. "The Persistence of a Number." J. Recr. Math. 6,97/C1/8, 1973. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Ade´le An element of an ADE´ LE GROUP , sometimes called a REPARTITION in older literature (e.g., Chevalley 1951, p. 25). Ade´les arise in both NUMBER FIELDS and FUNCTION FIELDS . The ade´les of a NUMBER FIELD are the additive SUBGROUPS of all elements inQ kv ; where v is the PLACE , whose ABSOLUTE VALUE is B1 at all but finitely many v/s. Let F be a FUNCTION FIELD of algebraic functions of one variable. Then a MAP r which assigns to every PLACE P of F an element r(P)ofF such that there are only a finite number of PLACES P for which vp(r(P)) B 0 is called an ade´le (Chevalley 1951, p. 1951). See also FUNCTION FIELD,IDELE References Chevalley, C. C. Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc., p. 25, 1951. Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Ade´le Group The restricted topological GROUP DIRECT PRODUCT of the GROUP Gkvwith distinct invariant open subgroups G0v:/ References Weil, A. Ade´les and Algebraic Groups. Princeton, NJ: Princeton University Press, 1961. Adem Relations Relations in the definition of a STEENROD ALGEBRA which state that, for i B2j; Sqi ( Sqj(x) /C30X/C28i /C29 k /C300j /C28k /C281 i /C282krC1+rC1D Sqi/C27j/C28k ( Sqk(x) ; where f ( g denotes function COMPOSITION and /C28i /C29 is the FLOOR FUNCTION . See also STEENROD ALGEBRA Adequate Knot A class of KNOTS containing the class of ALTERNATING KNOTS . Let c(K) be the CROSSING NUMBER . Then for KNOT SUM K1#K2which is an adequate knot, c(K1#K2) /C30c(K1) /C27c(K2) : This relationship is postulated to hold true for all KNOTS . See also ALTERNATING KNOT,C ROSSING NUMBER (LINK) Adiabatic Invariant A property of motion which is conserved to exponen- tial accuracy in the small parameter representing the typical rate of change of the gross properties of the body. See also ALGEBRAIC INVARIANT ,LYAPUNOV CHARAC- TERISTIC NUMBER Adjacency List The adjacency list representation of a GRAPH consists of n lists one for each vertex vi ; 1 5i 5n ; which gives the vertices to which viis adjacent. The adjacency lists of a graph g may be computed using ToAdja- cencyLists [g] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). A graph may be constructed from adjacency lists using FromAdjacencyLists [e]. See also ADJACENCY MATRIX References Skiena, S. "Adjacency Lists." §3.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 86 /C1/7, 1990. Adjacency Matrix The adjacency matrix of a simple GRAPH is a MATRIX with rows and columns labeled by VERTICES , with a 1 or 0 in position (vi ; vj) according to whether vi and vj are ADJACENT or not. For a simple graph with no self- loops, the adjacency matrix must have 0s on the diagonal. For an undirected graph, the adjacency matrix is symmetrical. The adjacency matrix of a graph can be computed using Edges [g] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). See also ADJACENCY LIST,INCIDENCE MATRIX ,IN- TEGER MATRIXReferences Chartrand, G. Introductory Graph Theory. New York: Dover, p. 218, 1985. Skiena, S. "Adjacency Matrices." §3.1.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 81 /C1/5, 1990. Adjacency Relation The SET E of EDGES of a GRAPH (V, E), being a set of unordered pairs of elements of V, constitutes a RELATION on V. Formally, an adjacency relation is any RELATION which is IRREFLEXIVE and SYMMETRIC . See also IRREFLEXIVE ,RELATION ,SYMMETRIC Adjacent Fraction Two FRACTIONS are said to be adjacent if their difference has a unit NUMERATOR . For example, 1/3 and 1/4 are adjacent since 1 =3 /C281 =4 /C301=12 ; but 1=2 and 1=5 are not since 1 =2 /C281 =5 /C303=10 : Adjacent fractions can be adjacent in a FAREY SEQUENCE . See also FAREY SEQUENCE ,FORD CIRCLE ,FRACTION , NUMERATOR References Pickover, C. A. Keys to Infinity. New York: Wiley, p. 119, 1995. Adjacent Value The value nearest to but still inside an inner FENCE . References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 667, 1977. Adjacent Vertices In a GRAPH G, two VERTICES are adjacent if they are joined by an EDGE . See also EDGE (GRAPH ), GRAPH ,VERTEX (GRAPH ) Adjoint Given a SECOND-ORDER ORDINARY DIFFERENTIAL EQUATION ˜Lu(x)/C13p0d2u dx2/C27p1du dx/C27p2u; (1) where pi/C13pi(x) and u/C13u(x);the adjoint operator ˜L/C31 is defined by ˜L/C31u/C13d dx2(p0u)/C28d dx(p1u)/C27p2u /C30p0d2u dx2/C27(2p?0/C28p1)du dx/C27(pƒ0/C28p?1/C27p2)u:(2) Write the two LINEARLY INDEPENDENT solutions as y1(x) and y2(x):Then the adjoint operator can also be written ˜L/C31u /C30g(y2 ˜Ly1 /C28y1 ˜Ly2)dx /C30p1 p0(y ?2y2 /C28y1y ?2)"# : (3) In general, given two adjoint operators ˜A and ˜B ; ( ˜A ˜B) /C31/C30 ˜B/C31 ˜A/C31; (4) which can be generalized to ( ˜A ˜B /C1/C1/C1 ˜Z) /C31/C30 ˜Z /C31/C1/C1/C1 ˜B/C31 ˜A/C31: (5) Note that many older physics text use the a DAGGER notation A $ to denote the adjoint (Arfken 1985). For example, (Dirac 1982, p. 26) denotes the adjoint of the BRA vector /C142P ½a as a $½P /C143; or ¯a½P/C143: The term Hermitian conjugate is sometimes also used instead of adjoint (Griffiths 1987, p. 22) See also ADJOINT CURVE ,ADJOINT MATRIX ,DAGGER , HERMITIAN OPERATOR ,SELF-ADJOINT ,STURM- LIOU- VILLE THEORY References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985. Dirac, P. A. M. "Conjugate Relations." §8in Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, pp. 26 /C1/9, 1982. Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, p. 220, 1987. Adjoint Curve A curve which has at least multiplicity ri /C281 at each point where a given curve (having only ordinary singular points and cusps) has a multiplicity riis called the adjoint to the given curve. When the adjoint curve is of order n /C283 ; it is called a special adjoint curve. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 30, 1959. Adjoint Matrix The adjoint matrix, sometimes also called the adju- gate matrix or conjugate transpose (Golub and van Loan 1996, p. 14), of an m /C29n MATRIX A is the n /C29m matrix defined by A /C31/C13 ¯AT ; (1) where the ADJOINT operator is denoted with a star, T denotes the TRANSPOSE , and ¯A denotes the CONJU- GATE MATRIX . Unfortunately, several different nota- tions are in use. Older physics text commonly use A $ (Arfken 1985, p. 210), mathematicians commonly use A/C31 (Courant and Hilbert 1989, p. 9), and computer scientists sometimes use AH (Golub and van Loan 1996, p. 14). In this work, a star is used to denote the adjoint operator, so care must be taken not to confusethis with the star used in older physics and engineer- ing texts to denote the COMPLEX CONJUGATE . If a MATRIX is SELF-ADJOINT , it is said to be HERMI- TIAN. The adjoint matrix of a MATRIX product is given by (ab)/C31 ij/C13[(ab)T]ij: (2) Using the identity for the product of TRANSPOSE gives [(ab)T]ij/C30[bTaT]ij/C30bT ikaTkj/C30[bT]ik[aT]kj/C30b/C31 ika/C31kj /C30[b/C31a/C31]ij; (3) where E INSTEIN SUMMATION has been used here to sum over repeated indices, it follows that (AB)/C31/C30B/C31A/C31: (4) See also ADJOINT ,C OMPLEX CONJUGATE ,D AGGER , HERMITIAN MATRIX ,SCHUR DECOMPOSITION ,TRANS- POSE References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 210, 1985. Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 49, 1962. Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 14, 1996. Adjoint Operator Given a SECOND-ORDER ORDINARY DIFFERENTIAL EQUATION pi/C13pi(x) (1) where u/C13u(x) and ˜L/C31;the adjoint operator ˜L/C31u (denoted by a DAGGER ), is defined by d dx2(p0u)/C28d dx(p1u)/C27p2u(y1yƒ2/C28y2yƒ1)/C27P(y1y?2/C28y?1y2) /C27Q(y1y2/C28y1y2)/C300p0 p0d2u dx2/C27(2p?0/C28p1)du dx/C27(pƒ0/C28p?1/C27p2)u/C30˜L/C31u /C30g(y2˜Ly1/C28y1˜Ly2)dx/C30p1 p0(y?1y2/C28y1y?2)"# :(2) Write the two LINEARLY INDEPENDENT solutions as y?/C30f0(x)/C27f1(x)y/C27f0(x)y2/C27f3(x)y3/C27... a n d / [g0(x)/C27g1(x)y]y?/C30f0(x)/C27f1(x)y/C27f2(x)y2/C27f3(x)y3/. Then the adjoint operator can also be written ˜A: (3) In general, given two adjoint operators ˜Band ( ˜A˜B)/C31/C30 ˜B/C31˜A/C31; ( ˜A ˜B /C1/C1/C1 ˜Z) /C31/C30 ˜Z/C31/C1/C1/C1 ˜B /C31 ˜A/C31: (4) which can be generalized to A $: (5) The adjoint of the BRA vector /C142P½ a is denoted a $½P/C143; or ¯a½P/C143 (Dirac 1982, p. 26). The term Hermitian con- jugate is sometimes also used (Griffiths 1987, p. 22) See also ADJOINT MATRIX ,D AGGER ,H ERMITIAN OPERATOR ,S ELF-ADJOINT OPERATOR ,S TURM- LIOU- VILLE THEORY References Dirac, P. A. M. "Conjugate Relations." §8in Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, pp. 26 /C1/9, 1982. Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, p. 220, 1987. Adjoint Representation AL IE ALGEBRA is a VECTOR SPACE g with a LIE BRACKET [X, Y], satisfying the JACOBI IDENTITY . Hence any element X gives a linear transformation given by ad(X)(Y) /C30[X ; Y] ; (1) which is called the adjoint representation of g : It is a LIE ALGEBRA REPRESENTATION because of the JACOBI IDENTITY , [ad(X1) ; ad(X2)](Y) /C30[X1 ; [X2 ; Y]] /C28[X2 ; [X1 ; Y]] /C30[[X1 ; X2] ; Y] /C30ad([X1 ; X2])(Y): (2) A REPRESENTATION is given by matrices. The simplest LIE ALGEBRA is glnthe set of matrices. Consider the adjoint representation of gl2 ; which has four dimen- sions and so will be a four dimensional representa- tion. The matrices e1 /C30 10 00rC00rC01 (3) e2 /C30 01 00rC00rC01 (4) e3 /C30 0010rC00rC01 (5) e 4 /C30 0001rC00rC01 (6) give a basis for gl 2 : Using this basis, the adjoint representation is described by the following matrices, ad e1 /C3000 00 01 00 00 /C2810 00 002 6643 775 (7)ad e2 /C3000 10 /C2810 01 00 00 00 /C28102 6643 775 (8) ad e 3 /C300 /C2810 0 0000 100 /C281 01002 6643 775 (9) ad e 4 /C300000 0 /C28100 0010 00002 6643 775: (10) The following Mathematica function gives the adjoint representation of the matrix min the Lie algebra, given by a basis, the list of matrices g. ad[g_List, m_List?MatrixQ]: /C30Transpose[Li- nearSolve[Transpose[Flatten/@g], Flatten[m.#1-#1.m]]&/@g] See also COMMUTATOR ,LIE ALGEBRA ,LIE GROUP ,LIE BRACKET ,N ILPOTENT LIE ALGEBRA ,R EPRESENTA- TION ,SEMISIMPLE LIE ALGEBRA References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Adjugate Matrix ADJOINT MATRIX Adjunction Ifais an element of a FIELD Fover the PRIME FIELD P, then the set of all RATIONAL FUNCTIONS ofawith COEFFICIENTS inPis a FIELD derived from Pby adjunction of a. Adleman-Pomerance-Rumely Primality Test A modified M ILLER’S PRIMALITY TEST which gives a guarantee of PRIMALITY orCOMPOSITENESS . The ALGO- RITHM ’s running time for a number nhas been proved to be as O((lnn)cln ln ln n) for some c/C210. It was simplified by Cohen and Lenstra (1984), implemented by Cohen and Lenstra (1987), and subsequently optimized by Bosma and van der Hulst (1990). References Adleman, L. M.; Pomerance, C.; and Rumely, R. S. "On Distinguishing Prime Numbers from Composite Number." Ann. Math. 117, 173/C1/06, 1983. Bosma, W. and van der Hulst, M.-P. "Faster Primality Testing." In Advances in Cryptology, Proc. Eurocrypt ’89, Houthalen, April 10 /C1/3, 1989 (Ed. J.-J. Quisquater). New York: Springer-Verlag, 652 /C1/56, 1990. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn 91; b /C302, 3; 5; 6; 7; 10; 11; 12 Up to High Powers, rev. ed. Provi- dence, RI: Amer. Math. Soc., pp. lxxxiv-lxxxv, 1988. Cohen, H. and Lenstra, A. K. "Primality Testing and Jacobi Sums." Math. Comput. 42, 297 /C1/30, 1984. Cohen, H. and Lenstra, A. K. "Implementation of a New Primality Test." Math. Comput. 48, 103 /C1/21, 1987. Mihailescu, P. "A Primality Test Using Cyclotomic Exten- sions." In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Proc. AAECC-6, Rome, July 1988). New York: Springer-Verlag, pp. 310 /C1/23, 1989. Adleman-Rumely Primality Test ADLEMAN- POMERANCE- RUMELY PRIMALITY TEST Admissible A string or word is said to be admissible if that word appears in a given SEQUENCE . For example, in the SEQUENCE aabaabaabaabaab ... ; a, aa, baab are all admissible, but bb is inadmissible. See also BLOCK GROWTH Ado’s Theorem Every finite-dimensional LIE ALGEBRA of character- istic p /C300 has a FAITHFUL finite-dimensional repre- sentation. See also IWASAWA’S THEOREM ,LIE ALGEBRA References Jacobson, N. Lie Algebras. New York: Dover, pp. 202 /C1/03, 1979. Affine Complex Plane The set A2 of all ORDERED PAIRS of COMPLEX NUM- BERS . See also AFFINE CONNECTION ,A FFINE EQUATION , AFFINE GEOMETRY ,A FFINE GROUP ,A FFINE HULL, AFFINE PLANE ,AFFINE SPACE ,AFFINE TRANSFORMA- TION ,AFFINITY ,COMPLEX PLANE ,COMPLEX PROJEC- TIVE PLANE Affine Connection CONNECTION COEFFICIENT Affine Equation A nonhomogeneous LINEAR EQUATION or system of nonhomogeneous LINEAR EQUATIONS is said to be affine. See also AFFINE COMPLEX PLANE ,AFFINE CONNEC- TION ,A FFINE GEOMETRY ,A FFINE GROUP ,A FFINE HULL,AFFINE PLANE ,AFFINE SPACE ,AFFINE TRANS- FORMATION ,AFFINITYAffine Geometry A GEOMETRY in which properties are preserved by PARALLEL PROJECTION from one PLANE to another. In an affine geometry, the third and fourth of EUCLID’S POSTULATES become meaningless. This type of GEO- METRY was first studied by Euler. See also ABSOLUTE GEOMETRY ,A FFINE COMPLEX PLANE ,A FFINE CONNECTION ,A FFINE EQUATION , AFFINE GROUP ,AFFINE HULL,AFFINE PLANE ,AFFINE SPACE ,A FFINE TRANSFORMATION ,A FFINITY ,O R- DERED GEOMETRY References Birkhoff, G. and Mac Lane, S. "Affine Geometry." §9.13 in A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 268 /C175, 1996. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 179 /C182, 1930. Leichtweiß, K. Affine Geometry of Convex Bodies. Heidel- berg, Germany: Barth Verlag, 1998. Affine Group The set of all nonsingular AFFINE TRANSFORMATIONS of a TRANSLATION in SPACE constitutes a GROUP known as the affine group. The affine group contains the full linear group and the group of TRANSLATIONS as SUBGROUPS . See also AFFINE COMPLEX PLANE ,AFFINE CONNEC- TION ,AFFINE EQUATION ,AFFINE GEOMETRY ,AFFINE HULL,AFFINE PLANE ,AFFINE SPACE ,AFFINE TRANS- FORMATION ,AFFINITY References Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, p. 237, 1996. Affine Hull The IDEAL generated by a SET in a VECTOR SPACE . See also AFFINE COMPLEX PLANE ,AFFINE CONNEC- TION ,AFFINE EQUATION ,AFFINE GEOMETRY ,AFFINE GROUP ,AFFINE PLANE ,AFFINE SPACE ,AFFINE TRANS- FORMATION ,AFFINITY ,CONVEX HULL,HULL Affine Plane A 2-D AFFINE GEOMETRY constructed over a FINITE FIELD . For a FIELD F of size n, the affine plane consists of the set of points which are ordered pairs of elements in F and a set of lines which are themselves a set of points. Adding a POINT AT INFINITY and LINE AT INFINITY allows a PROJECTIVE PLANE to be con- structed from an affine plane. An affine plane of order n is a BLOCK DESIGN OF THE FORM (/n2 ; n, 1). An affine plane of order nexists IFFaPROJECTIVE PLANE of order nexists. See also AFFINE COMPLEX PLANE ,AFFINE CONNEC- TION ,AFFINE EQUATION ,AFFINE GEOMETRY ,AFFINE GROUP ,AFFINE HULL,AFFINE SPACE ,AFFINE TRANS- FORMATION ,AFFINITY ,PROJECTIVE PLANE References Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. Affine Scheme Let P be the set of PRIME IDEALS of a COMMUTATIVE RING A. Then an affine scheme is a technical mathematical object defined as the SPECTRUM s(A) of P, regarded as a local-ringed space with a structure sheaf. A local-ringed space that is locally isomorphic to an affine scheme is called a SCHEME (Itoˆ 1986, p. 69). See also PRIME IDEAL ,SCHEME ,SPECTRUM (RING) References Itoˆ, K. (Ed.). "Schemes." §16D in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, p. 69, 1986. Affine Space Let V be a VECTOR SPACE over a FIELD K, and let A be a nonempty SET. Now define addition p /C27a /C23 A for any VECTOR a /C23 V and element p /C23 A subject to the condi- tions 1. p /C270 /C30p ;/ 2. (p /C27a) /C27b /C30p /C27(a /C27b) ;/ 3. For any q /C23 A; there EXISTS a unique VECTOR a /C23 V such that q /C30p /C27a:/ Here, a, b /C23 V : Note that (1) is implied by (2) and (3). Then A is an affine space and K is called the COEFFICIENT FIELD . In an affine space, it is possible to fix a point and coordinate axis such that every point in the SPACE can be REPRESENTED AS an n-tuple of its coordinates. Every ordered pair of points A and B in an affine space is then associated with a VECTOR AB. See also AFFINE COMPLEX PLANE ,AFFINE CONNEC- TION ,AFFINE EQUATION ,AFFINE GEOMETRY ,AFFINE GROUP ,AFFINE HULL,AFFINE PLANE ,AFFINE SPACE , AFFINE TRANSFORMATION ,AFFINITY Affine Transformation Any TRANSFORMATION preserving COLLINEARITY (i.e., all points lying on a LINE initially still lie on a LINE after TRANSFORMATION ) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). An affine transformation may also be thought of as a shearing transformation (Croft et al. 1991). An affine transformation is also called an AFFINITY . An affine transformation of Rn is a MAP F : Rn 0 Rn OF THE FORMF(p) /C30 Ap /C27 q (1) for all p /C23 Rn; where A is a linear transformation of Rn : If det(A) /C301; the transformation is ORIENTATION- PRESERVING ; if det(A) /C30/C281 ; it is ORIENTATION-REVER- SING. CONTRACTION , EXPANSION , DILATION , REFLECTION , SIMILARITY TRANSFORMATIONS , SPIRAL SIMILARITIES , ROTATION , and TRANSLATION are all affine transfor- mations, as are their combinations. A particular example combining ROTATION and EXPANSION is the rotation-enlargement transformation x? y?rC00rC01 /C30scos a sin a /C28sin a cos arC00rC01 x /C28x0 y /C28y0rC00rC01 /C30scos a(x /C28x0) /C27sin a(y /C28y0) /C28sin a(x /C28x0) /C27cos a(y /C28y0)rC00rC01 : (2) Separating the equations, x?/C30(s cos a)x /C27(s sin a)y /C28s(x0 cos a /C27y0 sin a) (3) y?/C30(/C28s sin a)x /C27(s cos a)y /C27s(x0 sin a /C28y0 cos a) : (4) This can be also written as x?/C30ax/C27by/C27c (5) y?/C30bx/C27ay/C27d; (6) where a/C30scosa (7) b/C30/C28ssina: (8) The scale factor sis then defined by s/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2p ; (9) and the rotation ANGLE by a/C30tan/C281/C28b a ! : (10) See also AFFINE COMPLEX PLANE ,AFFINE CONNEC- TION ,AFFINE EQUATION ,AFFINE GEOMETRY ,AFFINE GROUP ,AFFINE HULL,AFFINE PLANE ,AFFINE SPACE , AFFINE TRANSFORMATION ,A FFINITY ,E QUIAFFINITY , EUCLIDEAN MOTION References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 130, 1997. Zwillinger, D. (Ed.). "Affine Transformations." §4.3.2 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 265 /C1/66, 1995. Affine Variety An affine variety V is a VARIETY contained in AFFINE SPACE . For example, f(x; y; z):x2 /C27y2 /C28z2 /C300 g (1) is the CONE , and f(x ; y; z):x2 /C27y2 /C28z2 /C300; ax /C27by /C27cz /C300 g (2) is a CONIC SECTION , which is a SUBVARIETY of the cone. The cone can be written V(x2 /C27y2 /C28z2)to indicate that it is the variety corresponding to x2 /C27 y2 /C28z2 /C300: Naturally, many other polynomials van- ish on V(x2 /C27y2 /C28z2); in fact all polynomials in I(C) /C30 fx2 /C27y2 /C28z2 g: The set I(C)isan IDEAL in the POLY- NOMIAL RING C[x; y; z] : Note also, that the ideal of polynomials vanishing on the conic section is the IDEAL generated by x2 /C27y2 /C28z2 and ax /C27 by /C27 cz:/ A MORPHISM between two affine varieties is given by polynomial coordinate functions. For example, the map f(x; y; z) /C30 (x2 ; y2 ; z2)isa MORPHISM from X /C30 V(x2 /C27y2 /C27z2)toY /C30V(x /C27y /C27z) : Two affine varieties are ISOMORPHIC if there is a MORPHISM which has an inverse morphism. For example, the affine variety V(x2 /C27y2 /C27z2) is isomorphic to the cone V(x2 /C27y2 /C28 z2) via the coordinate change f(x; y; z) /C30(x; y; iz):/ Many polynomials f may be factored, for instance f /C30 x2 /C27y2 /C30(x /C27iy)(x /C28iy) ; and then V(f) /C30V(x /C27iy) @ V(x /C28iy) : Consequently, only IRREDUCIBLE POLYNO- MIALS , and more generally only PRIME IDEALS p are used in the definition of a variety. An affine variety V is the set of common zeros of a collection of poly- nomials p1 ; ..., pk ; i.e., V /C30fx /C30(x1 ; ... ; xn):p1(x) /C30.../C30pk(x) /C300 g (3) as long as the IDEAL I /C30(p1 ; ...; pk)isa PRIME IDEAL . More classically, an affine variety is defined by any set of polynomials, i.e., what is now called an ALGEBRAIC SET. Most points in V will have dimension n /C28k ; but V may have singular points like the origin in the cone. When V is one-dimensional generically (at almost all points), which typically occurs when k /C30n /C281; then V is called a curve. When V is two-dimensional, it is called a surface. In the case of COMPLEX affine space, a curve is a RIEMANN SURFACE , possibly with some singularities. Mathematica has a built-in functionImplicitPlot in the Mathematica add-on package Graphics‘Im-plicitPlot‘ (which can be loaded with the com- mand BBGraphics‘ ) that will graph affine varieties in the real affine plane. For example, the following graphs a hyperbola and a circle. BBGraphics‘; Show[GraphicsArray[{ ImplicitPlot[x^2 - y^2 /C30/C30 1, {x, -2, 2}, DisplayFunction - /C21 Identity], ImplicitPlot[x^2 /C27 y^2 /C30/C30 1, {x, -2, 2}, DisplayFunction - /C21 Identity] }]] An extension to this function called Implicit- Plot3D can be downloaded from MathSource and used to plot affine varieties in three-dimensional space. See also ALGEBRAIC SET,CATEGORY THEORY ,COM- MUTATIVE ALGEBRA ,C ONIC SECTION ,G ROEBNER BASIS,PROJECTIVE VARIETY ,SCHEME ,STACK (MOD- ULI SPACE ), INTRINSIC VARIETY ,ZARISKI TOPOLOGY References Bump, D. Algebraic Geometry. Singapore: World Scientific, pp. 1 /C1/, 1998. Cox, D.; Little, J.; and O’Shea, D. Ideals, Varieties, and Algorithms. New York: Springer-Verlag, pp. 5 /C1/9, 1997. Hartshorne, R. Algebraic Geometry. New York: Springer- Verlag, 1977. Affinity AFFINE TRANSFORMATION Affix In the archaic terminology of Whittaker and Watson (1990), the COMPLEX NUMBER z representing x /C27iy :/ References Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Aggregate An archaic word for infinite SETS such as those considered by Georg Cantor.See also C LASS (SET), SET AGM ARITHMETIC- GEOMETRIC MEAN Agnesi’s Witch WITCH OF AGNESI Agne ´sienne WITCH OF AGNESI Agonic Lines SKEW LINES Ahlfors Five Island Theorem Let f(z)bea TRANSCENDENTAL MEROMORPHIC FUNC- TION , and let D1 ; D2 ; ..., D5 be five SIMPLY CONNECTED domains in C with disjoint closures (Ahlfors 1932). Then there exists j /C23f1; 2 ; ...; 5 g and, for any R /C210, a SIMPLY CONNECTED domain G ƒfz /C23C : ½z½> Rg such that f(z)isa CONFORMAL MAP of G onto Dj : If f(z) has only finitely many POLES , then "five" may be replaced by "three" (Ahlfors 1933). See also MEROMORPHIC FUNCTION ,TRANSCENDENTAL FUNCTION References Ahlfors, L. "Sur les fonctions inverses des fonctions me ´r- omorphes." C. R. Acad. Sci. 194, 1145 /C1/147, 1932. Rep- rinted in Lars Valerian Ahlfors: Collected Papers Volume 1, 1929 /C1/955(Ed. R. M. Shortt). Boston, MA: Birkha ¨user, 149/C1/51, 1982. Ahlfors, L. "U ¨ber die Kreise die von einer Riemannschen Fla¨che schlicht u ¨berdeckt werden." Comm. Math. Helv. 5, 28/C1/8, 1933. Reprinted in Lars Valerian Ahlfors: Collected Papers Volume 1, 1929 /C1/955 (Ed. R. M. Shortt). Boston, MA: Birkha ¨user, 163 /C1/73, 1982. Bergweiler, W. "Iteration of Meromorphic Functions." Bull. Amer. Math. Soc. (N. S.) 29, 151/C1/88, 1993. Hayman, W. K. Meromorphic Functions. Oxford, England: Oxford University Press, 1964. Nevanlinna, R. Analytic Functions. New York: Springer- Verlag, 1970. Ahlfors-Bers Theorem The R IEMANN’S MODULI SPACE gives the solution to RIEMANN’S MODULI PROBLEM , which requires an ANALYTIC parameterization of the compact R IEMANN SURFACES in a fixed HOMEOMORPHISM . A-Integrable A generalization of the L EBESGUE INTEGRAL .A MEA- SURABLE FUNCTION f(x) is called A-integrable over the CLOSED INTERVAL [a, b]i f mfx:½f(x)½>ng/C30O(n/C281); (1) where mis the L EBESGUE MEASURE , and I/C30lim n0/C12gb a[f(x)]ndx (2) exists, where [f(x)]n/C30f(x)i f½f(x)½5n 0i f ½f(x)½>n:rC06 (3) References Titchmarsh, E. C. "On Conjugate Functions." Proc. London Math. Soc. 29,4 9/C1/0, 1928. Airy Differential Equation Some authors define a general Airy differential equation asyƒ9k2xy/C300: (1) This equation can be solved by series solution using the expansions y/C30X/C12 n/C300anxn(2) y?/C30X/C12 n/C300nanxn/C281/C30X/C12 n/C301nanxn/C281 /C30X/C12 n/C300(n/C271)an/C271xn(3) yn/C30X/C12 n/C300(n/C271)nan/C271xn/C281/C30X/C12 n/C301(n/C271)nan/C271xn/C281 /C30X/C12 n/C300(n/C272)(n/C271)an/C272xn: (4) Specializing to the "conventional" Airy differentialequation occurs by taking the MINUS SIGN and setting k2/C301:Then plug (4) into yƒ/C28xy/C300 (5) to obtain X/C12 n/C300(n/C272)(n/C271)an/C272xn/C28xX/C12 n/C300anxn/C300 (6) X/C12 n/C300(n/C272)(n/C271)an/C272xn/C28X/C12 n/C300anxn/C271/C300 (7) 2a2/C27X/C12 n/C301(n/C272)(n/C271)an/C272xn/C28X/C12 n/C301an/C281xn/C300 (8) 2a2/C27X/C12 n/C301[(n/C272)(n/C271)an/C272/C28an/C281]xn/C300: (9) In order for this equality to hold for all x, each term must separately be 0. Therefore, a2/C300 (10) (n/C272)(n/C271)an/C272/C30an/C281: (11) Starting with the n/C303 term and using the above RECURRENCE RELATION , we obtain 5/C2154a5/C3020a5/C30a2/C300: (12) Continuing, it follows by INDUCTION that a2/C30a5/C30a8/C30a11/C30...a3n/C281/C300 (13) forn/C301, 2, .... Now examine terms OF THE FORM a3n: a3/C30a0 3 /C2152(14) a6 /C30a3 6 /C215 5 /C30a0 (6 /C215 5)(3 /C215 2) (15) a9 /C30a6 9 /C215 8 /C30a0 (9 /C215 8)(6 /C215 5)(3 /C215 2) : (16) Again by INDUCTION , a3n /C30a0 [(3n)(3n /C28 1)][(3 n /C28 3)(3n /C28 4)] /C1/C1/C1[6 /C215 5][3 /C215 2] (17) for n /C301, 2, .... Finally, look at terms OF THE FORM a3n /C271 ; a4 /C30a1 4 /C215 3 (18) a7 /C30a4 7 /C215 6 /C30a1 (7 /C215 6)(4 /C215 3) (19) a10 /C30a7 10 /C215 9 /C30a1 (10 /C215 9)(7 /C215 6)(4 /C215 3) : (20) By INDUCTION , a3n /C271 /C30a1 [(3n /C27 1)(3n)][(3n /C28 2)(3n /C28 3)] /C1/C1/C1[7 /C215 6][4 /C215 3] (21) for n /C30 1, 2, .... The general solution is therefore y /C30a01 /C27X/C12 n/C301x3n (3n)(3n /C28 1)(3n /C28 3)(3n /C28 4) /C1/C1/C13 /C215 2"# /C27a1x /C27X/C12 n/C301x3n/C271 (3n /C27 1)(3n)(3n /C28 2)(3n /C28 3) /C1/C1/C14 /C215 3"# : (22) For a general k2 with a MINUS SIGN, equation (1) is yƒ/C28 k2xy /C300 ; (23) and the solution is y(x) /C301 3ffiffiffixp[AI/C281 =3(2 3 kx3=2 Þ/C28BI1 =3(23 kx3 =2 Þ/C138; (24) where I is a MODIFIED BESSEL FUNCTION OF THE FIRST KIND . This is usually expressed in terms of the AIRY FUNCTIONS Ai(x) and Bi(x) y(x) /C30A? Ai(k2=3x) /C27B ?Bi(k2=3x) : (25) If the PLUS SIGN is present instead, then yƒ/C27k2xy /C300 (26) and the solutions are y(x) /C301 3ffiffiffixp[AJ/C281=3(2 3kx3 =2 Þ/C27BJ1=3(23kx3 =2 Þ/C138; (27) where J(z)isaB ESSEL FUNCTION OF THE FIRST KIND .A generalization of the Airy differential equation is given by y§/C284xy?/C282y/C300; (28) which has solutions y/C30C1[Ai(x)]2/C27C2Ai(x) Bi(x)/C27C3[Bi(x)]2(29) (Abramowitz and Stegun 1972, p. 448; Zwillinger1997, p. 128). See also A IRY-FOCK FUNCTIONS ,A IRY FUNCTIONS , BESSEL FUNCTION OF THE FIRST KIND,M ODIFIED BESSEL FUNCTION OF THE FIRST KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Airy Functions." §10.4.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 446 /C152, 1972. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Airy Functions The Ai( x) and Bi( x) functions are defined as the two LINEARLY INDEPENDENT solutions to yƒ/C28yz/C300: (1) (Abramowitz and Stegun 1972, pp. 446 /C147; illu- strated above), written in the form y(z)/C30AAi(z)/C27BBi(z); (2) where Ai(z)/C301 3ffiffiffixpI/C281=32 3z3=2rC16rC1* /C28I1=323z3=2rC16rC1* hi /C30ffiffiffiffiffiffi z 3ps K1=323z3=2rC16rC1* (3) Bi(z)/C30ffiffiffi z 3s I/C281=323z3=2rC16rC1* /C27I1=323z3=2rC16rC1* hi ; (4) where I(z)i sa MODIFIED BESSEL FUNCTION OF THE FIRST KIND andK(z)i sa MODIFIED BESSEL FUNCTION OF THE SECOND KIND . The functions are implemented inMathematica asAiryAi [z] andAiryBi [z]. Their derivatives are implemented as AiryAiPrime [z] and AiryBiPrime [z]. Plots of Ai( z) in the COMPLEX PLANE are illustrated above, and Bi( z) is illustrated below. The Airy Ai( x) function is given by the integral Ai(z)/C301 2pg/C12 /C28/C12ei(zt/C27t3=3)dt (5) and the INFINITE SERIES Ai(x)/C301 32=3pX/C12 n/C300G1 3(n/C271)rC16rC1* n! /C2(31=3x)nsin2(n/C271)p 3"# (6) (Banderier et al. ). A generalization of the Airy function has been constructed by Hardy. Forz/C300, Ai(0)/C301 32=3G(2 3)(7) Bi(0)/C301 31=6G(23); (8) where G(z) is the GAMMA FUNCTION . The ASYMPTOTIC SERIES of Ai( z) has a different form in different QUADRANTS of the COMPLEX PLANE , a fact known as the STOKES PHENOMENON . Functions related to the Airy functions have been defined as Gi(z)/C131 pg/C12 0sin(1 3t3/C27ztÞdt (9) Hi(z)/C131 pg/C12 0exp/C2813t3/C27ztrC16rC1* dt; (10) where Gi( z) is defined for I[z]"0 and Hi( z) forR[z]] 0:The can be expressed in terms of the Airy functions by Gi(z)/C30/C28z2 2p1F41:2 3;56;76;43;1 1296z6rC16rC1* /C27[sgn( z)]6 360pz61F41:7 6;43;53;11 6:1 1296z6rC16rC1* /C27z6 6½z½6 /C2[Bi(/C28½z½)/C27Bi(½z½)]/C28iffiffiffi 3p ½z½3 6z4[Ai(/C28½z½)/C28Ai(½z½)] /C271 6z4½z½6I[z]/C27R[z][Bi(½z½)/C28Bi(/C28½z½)] fg (11) Hi(z)/C302 3ffiffiffiffiffiffi /C2823q J/C281=323/C28zðÞ3=2rC16rC1* /C28J1=323/C28zðÞ3=2rC16rC1* hi /C27z2 2p1F21:43;53;19z3rC16rC1* ; (12) where pFqis a GENERALIZED HYPERGEOMETRIC FUNC- TION ,SGNis the sign function, zjjis the MODULUS ofz, R[z] is the REAL PART ,I[z] is the IMAGINARY PART , and Jn(z)i saB ESSEL FUNCTION OF THE FIRST KIND . Watson (1966, pp. 188 /C1/90) gives a slightly more general definition of the Airy function as the solution to the AIRY DIFFERENTIAL EQUATION Fƒ9k2 Fx /C300 (13) which is FINITE at the ORIGIN , where F? denotes the DERIVATIVE dF=dx; k2 /C301=3; and either SIGN is permitted. Call these solutions (1=p)F(9k2 ; x) ; then 1 p F91 3; xrC16rC1* /C13g/C12 0cos t3 9xtrC0rC1 dt (14) F13; xrC16rC1* /C3013 pffiffiffi x 3s J/C281 =32x3 =2 33 =2 ! /C27J1 =32x3 =2 33=2 ! "# (15) F/C281 3; xrC16rC1* /C3013 pffiffiffi x 3s I/C281 =32x3 =2 33=2 ! /C28I1 =32x3 =2 33=2 ! "# ; (16) where J(z)isaB ESSEL FUNCTION OF THE FIRST KIND . Using the identity Kn(x) /C30p 2I/C28n(x) /C28 In(x) sin(np); (17) where K(z)isa MODIFIED BESSEL FUNCTION OF THE SECOND KIND , the second case can be re-expressed F(/C281 3; x) /C3013 pffiffiffi x 3s 2 psin13prC16rC1* K1=32x3=2 33=2 ! (18) /C30p 3ffiffiffi x 3s 2 pffiffiffi 3p 2K1=32x3=2 33=2 ! (19) /C301 3ffiffiffixpK1=32x3=2 33=2 ! : (20) See also AIRY-FOCK FUNCTIONS ,BESSEL FUNCTION OF THE FIRST KIND,M AP-AIRY DISTRIBUTION ,M ODIFIED BESSEL FUNCTION OF THE FIRST KIND,M ODIFIED BESSEL FUNCTION OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Airy Functions." §10.4 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 446 /C1/52, 1972. Banderier, C.; Flajolet, P.; Schaeffer, G.; and Soria, M. "Planar Maps and Airy Phenomena." Preprint. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Bessel Functions of Fractional Order, AiryFunctions, Spherical Bessel Functions." §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 234 /C1 /45, 1992.Spanier, J. and Oldham, K. B. "The Airy Functions Ai( x) and Bi( x)." Ch. 56 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 555 /C1/62, 1987. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Airy Projection AMAP PROJECTION . The inverse equations for fare computed by iteration. Let the ANGLE of the projection plane be ub:Define a/C300 for ub/C301 2p ln[12cos (12p/C28ub)] tan [1 2(12p/C28ub)]otherwise :8 >< >:(1) For proper convergence, let xi/C30p=6 and compute the initial point by checking xi/C30½exp[/C28(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p /C27atanxi) tan xi]½: (2) As long as xi>1;take xi/C271/C30xi=2 and iterate again. The first value for which xiB1 is then the starting point. Then compute xi/C30cos/C281fexp[/C28(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2/C27y2p /C27atanxi) tan xi]g (3) until the change in xibetween evaluations is smaller than the acceptable tolerance. The (inverse) equa- tions are then given by f/C301 2p/C282xi (4) l/C30tan/C281/C28x y ! : (5) AiryAi AIRYFUNCTIONS AiryAiPrime AIRYFUNCTIONS AiryBi AIRYFUNCTIONS AiryBiPrime AIRYFUNCTIONS Airy-Fock Functions The three Airy-Fock functions are v(z)/C301 2ffiffiffippAi(z) (1) w1(z)/C302eip=6v(vz) (2) w2(z)/C302e/C28ip=6v(v/C281z); (3) where Ai( z)i sa nA IRY FUNCTION . These functions satisfy v(z) /C30v1(z) /C28 v2(z) 2i (4) w1(z) /C30w2(¯z); (5) where ¯z is the COMPLEX CONJUGATE of z. See also AIRY FUNCTIONS References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 65, 1988. Aitken Interpolation An algorithm similar to NEVILLE’S ALGORITHM for constructing the LAGRANGE INTERPOLATING POLYNO- MIAL . Let f(x½x0 ; x1 ; ...; xk) be the unique POLYNO- MIAL of kth ORDER coinciding with f(x)atx0 ; ..., xk : Then f(x½x0 ; x1) /C301 x1 /C28 x0jf0 x0 /C28x f1 x1 /C28x j f(x½x0 ; x2) /C301 x2 /C28 x0jf0 x0 /C28x f2 x2 /C28x j f(x½x0 ; x1 ; x2) /C301 x2 /C28 x1jf(x½x0 ; x1)x1 /C28x f(x½x0 ; x2)x2 /C28xj f(x½x0 ; x1 ; x2 ; x3) /C301 x3 /C28 x2jf(x½x0 ; x1)x2 /C28x f(x½x0 ; x1)x3 /C28xj: See also LAGRANGE INTERPOLATING POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 879, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 93 /C1/4, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 102, 1992. Aitken’s Delta Squared Process An ALGORITHM which extrapolates the partial sums sn of a SERIES Sn anwhose CONVERGENCE is approxi- mately geometric and accelerates its rate of CONVER- GENCE . The extrapolated partial sum is given by s?n/C13sn/C271/C28(sn/C271/C28sn)2 sn/C271/C282sn/C27sn/C281: See also EULER’S SERIES TRANSFORMATIONReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 18, 1972. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 160, 1992. Ajima-Malfatti Points The lines connecting the vertices and corresponding circle-circle intersections in M ALFATTI’S TANGENT TRIANGLE PROBLEM coincide in a point Ycalled the first Ajima-Malfatti point (Kimberling and MacDo-nald 1990, Kimberling 1994). Similarly, letting Aƒ;Bƒ; andCƒbe the excenters of ABC , then the lines A?Aƒ; B?Bƒ;andC?Cƒare coincident in another point called the second Ajima-Malfatti point. The points aresometimes simply called the malfatti points (Kimber- ling 1994). References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Kimberling, C. "1st and 2nd Ajima-Malfatti Points." http:// cedar.evansville.edu/~ck6/tcenters/recent/ajmalf.html. Kimberling, C. and MacDonald, I. G. "Problem E 3251 and Solution. " Amer. Math. Monthly 97, 612/C1/13, 1990. Akinetor Moon, P. and Spencer, D. E. Theory of Holors: A Generalization of Tensors. Cambridge, England: Cambridge University Press, 1986. Akisation CUMULATION Albanese Variety An A BELIAN VARIETY which is canonically attached to an ALGEBRAIC VARIETY which is the solution to a certain universal problem. The Albanese variety isdual to the P ICARD VARIETY . References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 67 /C1/8, 1988. Albers Conic Projection ALBERS EQUAL- AREA CONIC PROJECTION Albers Equal-Area Conic Projection An EQUAL-AREA PROJECTION . Let f0be the LATITUDE for the origin of the CARTESIAN COORDINATES and l0 its LONGITUDE . Let f1and f2be the standard parallels. Then x /C30 r sin u (1) y /C30 r0 /C28 r cos u; (2) where r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C28 2n sin fp n (3) u /C30n( l/C28l0) (4) r0 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C28 2n sin f0p n (5) C /C30cos2 f1 /C272n sin f1 (6) n /C301 2(sin f1 /C27sin f2): (7) The inverse FORMULAS are f /C30sin /C281C /C28 r2n2 2n ! (8) l /C30 l0 /C27u n ; (9) where r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27( r0 /C28y)2q (10) u /C30tan /C281 x r0 /C28 y ! : (11) See also EQUAL- AREA PROJECTION References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 98 /C1/03, 1987.Alcuin’s Sequence The INTEGER SEQUENCE 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, ... (Sloane’s A005044) given by the COEFFICIENTS of the MACLAURIN SERIES for 1=(1 /C28x2)(1 /C28x3)(1 /C28x4): The number of different TRIANGLES which have INTEGRAL sides and PERIMETER n is given by T(n) /C30P3(n) /C30X 1 5j5/C28n=2 /C29P2(j) (1) /C30n2 12"# /C28n 4$% n /C27 2 4$% (2) /C30[n2 48] for n even [(n /C27 3)2 48] for n odd:8 >>>< >>>:(3) where P 2(n) and P3(n) are PARTITION FUNCTIONS , with Pk(n) giving the number of ways of writing n as a sum of k terms, [x] is the NINT function, and xbcis the FLOOR FUNCTION (Jordan et al. 1979, Andrews 1979, Honsberger 1985). Strangely enough, T(n) for n /C303, 4, ... is precisely Alcuin’s sequence. See also PARTITION FUNCTION P,TRIANGLE References Andrews, G. "A Note on Partitions and Triangles with Integer Sides." Amer. Math. Monthly 86, 477, 1979. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 39 /C1/7, 1985. Jordan, J. H.; Walch, R.; and Wisner, R. J. "Triangles with Integer Sides." Amer. Math. Monthly 86, 686 /C1/89, 1979. Sloane, N. J. A. Sequences A005044/M0146 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Aleksandrov’s Uniqueness Theorem A convex body in EUCLIDEAN n-space that is centrally symmetric with center at the ORIGIN is determined among all such bodies by its brightness function (the VOLUME of each projection). See also TOMOGRAPHY References Gardner, R. J. "Geometric Tomography." Not. Amer. Math. Soc. 42, 422/C1/29, 1995. Aleksandrov-Cech Cohomology A theory which satisfies all the E ILENBERG- STEENROD AXIOMS with the possible exception of the LONG EXACT SEQUENCE OF A PAIR AXIOM , as well as a certain additional continuity CONDITION . References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 68, 1988. Aleph The SET THEORY symbol (/ /C210) for the CARDINALITY of an INFINITE SET. See also ALEPH-0 ,ALEPH-1 ,COUNTABLE SET,COUN- TABLY INFINITE ,FINITE ,INFINITE ,TRANSFINITE NUM- BER,UNCOUNTABLY INFINITE Aleph-0 The SET THEORY symbol /C2100 for a SET having the same CARDINAL NUMBER as the "small" INFINITE SET of INTEGERS . The ALGEBRAIC NUMBERS also belong to /C2100 : Rather surprising properties satisfied by /C2100 include /C210r 0 /C30/C2100 (1) r /C2100 /C30/C2100 (2) /C2100 /C27f /C30/C2100 ; (3) where f is any FINITE SET. However, /C210/C2100 0/C30C ; (4) where C is the CONTINUUM . See also ALEPH-1 ,CARDINAL NUMBER ,CONTINUUM , CONTINUUM HYPOTHESIS ,COUNTABLY INFINITE ,FI- NITE,INFINITE ,TRANSFINITE NUMBER ,UNCOUNTABLY INFINITE Aleph-1 The SET THEORY symbol /C2101 for the smallest INFINITE SET larger than ALEPH-0 , and equal to the CARDIN- ALITY of the set of countable ORDINAL NUMBERS . The CONTINUUM HYPOTHESIS asserts that /C2101 /C30c ; where c is the CARDINALITY of the "large" INFINITE SET of REAL NUMBERS (called the CONTINUUM in SET THEORY ). However, the truth of the CONTINUUM HYPOTHESIS depends on the version of SET THEORY you are using and so is UNDECIDABLE . Curiously enough, n-D SPACE has the same number of points (c) as 1-D SPACE , or any FINITE INTERVAL of 1-D SPACE (a LINE SEGMENT ), as was first recognized by Georg Cantor. See also ALEPH-0 ,CARDINALITY ,CONTINUUM ,CON- TINUUM HYPOTHESIS ,COUNTABLY INFINITE ,FINITE , INFINITE ,ORDINAL NUMBER ,TRANSFINITE NUMBER , UNCOUNTABLY INFINITE Alethic A term in LOGIC meaning pertaining to TRUTH and FALSEHOOD . See also FALSE ,PREDICATE ,TRUEAlexander Ideal The order IDEAL in L; the RING of integral LAURENT POLYNOMIALS , associated with an ALEXANDER MATRIX for a KNOT K. Any generator of a principal Alexander ideal is called an ALEXANDER POLYNOMIAL . Because the ALEXANDER INVARIANT of a TAME KNOT in S3 has a SQUARE presentation MATRIX , its Alexander ideal is PRINCIPAL and it has an ALEXANDER POLYNOMIAL D(t):/ See also ALEXANDER INVARIANT ,ALEXANDER MATRIX , ALEXANDER POLYNOMIAL References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206 /C107, 1976. Alexander Invariant The Alexander invariant H/C31( ˆX)ofa KNOT K is the HOMOLOGY of the INFINITE cyclic cover of the comple- ment of K, considered as a MODULE over L; the RING of integral LAURENT POLYNOMIALS . The Alexander in- variant for a classical TAME KNOT is finitely presen- table, and only H1 is significant. For any KNOT Kn in Sn/C272 whose complement has the homotopy type of a FINITE COMPLEX , the Alexander invariant is finitely generated and therefore finitely presentable. Because the Alexander invariant of a TAME KNOT in S3 has a SQUARE presentation MATRIX , its ALEXANDER IDEAL is PRINCIPAL and it has an ALEXANDER POLYNOMIAL denoted D(t) :/ See also ALEXANDER IDEAL ,A LEXANDER MATRIX , ALEXANDER POLYNOMIAL References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206 /C107, 1976. Alexander Matrix A presentation matrix for the ALEXANDER INVARIANT H1( ˜X)ofa KNOT K.IfV is a SEIFERT MATRIX for a TAME KNOT K in S3 ; then VT /C28tV and VT /C28tVT are Alexander matrices for K, where VTdenotes the MATRIX TRANSPOSE . See also ALEXANDER IDEAL ,ALEXANDER INVARIANT , ALEXANDER POLYNOMIAL ,SEIFERT MATRIX References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206 /C107, 1976. Alexander Polynomial APOLYNOMIAL invariant of a KNOT discovered in 1923 by J. W. Alexander (Alexander 1928). In technical language, the Alexander polynomial arises from the HOMOLOGY of the infinitely cyclic cover of a KNOT ’s complement. Any generator of a PRINCIPAL ALEXAN- DER IDEAL is called an Alexander polynomial (Rolfsen 1976). Because the ALEXANDER INVARIANT of a TAME KNOT in S3 has a SQUARE presentation MATRIX , its ALEXANDER IDEAL is PRINCIPAL and it has an Alex- ander polynomial denoted D(t) :/ Let C be the MATRIX PRODUCT of BRAID WORDS of a KNOT , then det(1 /C28C) 1 /C27 t /C27 ... /C27 tn /C281 /C30DL ; (1) where DL is the Alexander polynomial and det is the DETERMINANT . The Alexander polynomial of a TAME KNOT in S3 satisfies D(t) /C30 det(VT /C28tV) : (2) where V is a SEIFERT MATRIX , det is the DETERMI- NANT , and VT denotes the MATRIX TRANSPOSE . The Alexander polynomial also satisfies D(1) /C3091: (3) The Alexander polynomial of a splittable link is always 0. Surprisingly, there are known examples of nontrivial KNOTS with Alexander polynomial 1. An example is the ( /C283; 5; 7) PRETZEL KNOT . The Alexander polynomial remained the only known KNOT POLYNOMIAL until the JONES POLYNOMIAL was discovered in 1984. Unlike the Alexander polynomial, the more powerful JONES POLYNOMIAL does, in most cases, distinguish HANDEDNESS . A normalized form of the Alexander polynomial symmetric in t and t /C281 and satisfying D(unknot) /C30 1 (4) was formulated by J. H. Conway and is sometimes denoted 9L : The NOTATION [a /C27b /C27c /C27... is an ab- breviation for the Conway-normalized Alexander polynomial of a KNOT a /C27b(x /C27x /C281) /C27c(x2 /C27x /C282) /C27... (5) For a description of the NOTATION for LINKS , see Rolfsen (1976, p. 389). Examples of the Conway- Alexander polynomials for common KNOTS include 9TK /C30[1 /C281 /C30/C28x /C281 /C271 /C28x (6) 9FEK /C30[3 /C281 /C30/C28x /C281 /C273 /C28x (7) 9SSK /C30[1 /C281 /C271 /C30x /C282 /C28x /C281 /C271 /C28x /C27x2 (8) for the TREFOIL KNOT , FIGURE-OF-EIGHT KNOT , and SOLOMON’S SEAL KNOT , respectively. Multiplying through to clear the NEGATIVE POWERS gives the usual Alexander polynomial, where the final SIGN is determined by convention. Let an Alexander polynomial be denoted D; then there exists a SKEIN RELATIONSHIP (discovered by J. H. Conway) DL/C27(t) /C28DL/C28(t) /C27(t /C281 =2 /C28t1=2) DL0(t) /C300 (9) corresponding to the above LINK DIAGRAMS (Adams 1994). A slightly different SKEIN RELATIONSHIP con- vention used by Doll and Hoste (1991) is 9L/C27/C289L/C28/C30z9L0: (10) These relations allow Alexander polynomials to be constructed for arbitrary knots by building them up as a sequence of over- and undercrossings. For a KNOT , DK(/C281) /C131(mod 8) if Arf(K) /C300 ; 5(mod 8) if Arf(K) /C301 ;rC06 (11) where Arf is the ARF INVARIANT (Jones 1985). If K is a KNOT and jDK(i) j/C213: (12) then Kcannot be REPRESENTED AS a closed 3- BRAID . Also, if DK(e2pi=5)>13 2; (13) then Kcannot be REPRESENTED AS a closed 4-braid (Jones 1985). The HOMFLY POLYNOMIAL P(a;z) generalizes the Alexander polynomial (as well at the J ONES POLY- NOMIAL ) with 9(z)/C30P(1;z) (14) (Doll and Hoste 1991).Rolfsen (1976) gives a tabulation of Alexander poly- nomials for KNOTS up to 10 CROSSINGS and LINKS up to 9 CROSSINGS . See also BRAID GROUP ,JONES POLYNOMIAL ,KNOT, KNOT DETERMINANT ,LINK,SKEIN RELATIONSHIP References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 165 /C1/69, 1994. Alexander, J. W. "Topological Invariants of Knots and Links." Trans. Amer. Math. Soc. 30, 275/C1/06, 1928. Alexander, J. W. "A Lemma on a System of Knotted Curves." Proc. Nat. Acad. Sci. USA 9,9 3/C1/5, 1923. Casti, J. L. "The Alexander Polynomial." Ch. 1 in Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics. New York: Wiley, pp. 1/C1/4, 2000. Doll, H. and Hoste, J. "A Tabulation of Oriented Links." Math. Comput. 57, 747/C1/61, 1991. Jones, V. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103/C1/11, 1985. Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Rolfsen, D. "Table of Knots and Links." Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 280 /C1/87, 1976. Stoimenow, A. "Alexander Polynomials." http://guests.mpim- bonn.mpg.de/alex/ptab/a10.html. Stoimenow, A. "Conway Polynomials." http://guests.mpim- bonn.mpg.de/alex/ptab/c10.html. Alexander’s Horned Sphere The above solid, composed of a countable UNION of COMPACT SETS , is called Alexander’s horned sphere. It is HOMEOMORPHIC with the BALL B3 ; and its boundary is therefore a SPHERE . It is therefore an example of a wild embedding in E3 : The outer complement of the solid is not SIMPLY CONNECTED , and its fundamental GROUP is not finitely generated. Furthermore, the set of nonlocally flat ("bad") points of Alexander’s horned sphere is a CANTOR SET. The complement in R3 of the bad points for Alex- ander’s horned sphere is SIMPLY CONNECTED , making it inequivalent to ANTOINE’S HORNED SPHERE . Alex- ander’s horned sphere has an uncountable infinity of WILD POINTS , which are the limits of the sequences of the horned sphere’s branch points (roughly, the "ends" of the horns), since any NEIGHBORHOOD of a limit contains a horned complex. A humorous drawing by Simon Frazer (Guy 1983, Schroeder 1991, Albers 1994) depicts mathematician John H. Conway with Alexander’s horned sphere growing from his head. See also ANTOINE’S HORNED SPHEREReferences Albers, D. J. Illustration accompanying "The Game of ‘Life’." Math Horizons, p. 9, Spring 1994. Guy, R. "Conway’s Prime Producing Machine." Math. Mag. 56,26/C13, 1983. Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 80 /C11, 1976. Schroeder, M. Fractals, Chaos, Power Law: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 58, 1991. Alexander’s Theorem Any LINK can be represented by a closed BRAID . Alexander-Conway Polynomial CONWAY POLYNOMIAL Alexander-Spanier Cohomology A fundamental result of DE RHAM COHOMOLOGY is that the kth DE RHAM COHOMOLOGY VECTOR SPACE of a MANIFOLD M is canonically isomorphic to the Alexander-Spanier cohomology VECTOR SPACE Hk(M;R) (also called cohomology with compact sup- port). In the case that M is COMPACT , Alexander- Spanier cohomology is exactly "singular" COHOMOL- OGY. Algebra The branch of mathematics dealing with such topics as GROUP THEORY , invariant theory, and COHOMOL- OGY which studies number systems and operations within them. The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwarizmi about algebraic methods. Note that mathematicians refer to the "school algebra" generally taught in middle and high school as "ARITHMETIC ," reserving the word "algebra" for the more advanced aspects of the subject. Formally, an algebra is a VECTOR SPACE V, over a FIELD F with a MULTIPLICATION which turns it into a RING defined such that, if f /C23 F and x; y /C23 V ; then f(xy) /C30(fx)y /C30x(fy) : In addition to the usual algebra of REAL NUMBERS , there are :1151 additional CONSISTENT algebras which can be formulated by weakening the FIELD AXIOMS , at least 200 of which have been rigorously proven to be self- CONSISTENT (Bell 1945). Algebras which have been investigated and found to be of interest are usually named after one or more oftheir investigators. This practice leads to exotic- sounding (but unenlightening) names which algebra- ists frequently use with minimal or nonexistentexplanation. See also A BSTRACT ALGEBRA ,ALTERNATIVE ALGEBRA , ASSOCIATIVE ALGEBRA ,B*-ALGEBRA ,BANACH ALGE- BRA,BOOLEAN ALGEBRA ,BOREL SIGMA ALGEBRA ,C*- ALGEBRA ,C AYLEY ALGEBRA ,C LIFFORD ALGEBRA , COMMUTATIVE ALGEBRA ,DERIVATION ALGEBRA ,EX- TERIOR ALGEBRA ,FUNDAMENTAL THEOREM OF ALGE- BRA,GRADED ALGEBRA ,GRASSMANN ALGEBRA ,HECKE ALGEBRA ,H EYTING ALGEBRA ,H OMOLOGICAL ALGE- BRA,HOPF ALGEBRA ,JORDAN ALGEBRA ,LIE ALGEBRA , LINEAR ALGEBRA ,M EASURE ALGEBRA ,N ONASSOCIA- TIVE ALGEBRA ,POWER ASSOCIATIVE ALGEBRA ,QUA- TERNION ,R OBBINS ALGEBRA ,S CHUR ALGEBRA , SEMISIMPLE ALGEBRA ,SIGMA ALGEBRA ,SIMPLE AL- GEBRA ,STEENROD ALGEBRA ,UMBRAL ALGEBRA , VON NEUMANN ALGEBRA References Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991. Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, pp. 35 /C1/6, 1945. Bhattacharya, P. B.; Jain, S. K.; and Nagpu, S. R. (Eds.). Basic Algebra, 2nd ed. New York: Cambridge University Press, 1994. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, 1996. Brown, K. S. "Algebra." http://www.seanet.com/~ksbrown/ ialgebra.htm. Cardano, G. Ars Magna or The Rules of Algebra. New York: Dover, 1993. Chevalley, C. C. Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc., 1951. Chrystal, G. Textbook of Algebra, 2 vols. New York: Dover, 1961. Connell, E. H. Elements of Abstract and Linear Algebra. http://www.cs.miami.edu/~ec/book/. Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL: University of Chicago Press, 1923. Dickson, L. E. Modern Algebraic Theories. Chicago, IL: H. Sanborn, 1926. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. Edwards, H. M. Galois Theory, corrected 2nd printing. New York: Springer-Verlag, 1993. Euler, L. Elements of Algebra. New York: Springer-Verlag, 1984. Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lexington, MA: D. C. Heath, 1994. Grove, L. Algebra. New York: Academic Press, 1983. Hall, H. S. and Knight, S. R. Higher Algebra, A Sequel to Elementary Algebra for Schools. London: Macmillan, 1960. Harrison, M. A. "The Number of Isomorphism Types of Finite Algebras." Proc. Amer. Math. Soc. 17, 735 /C1/37, 1966. Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., 1996. Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975. Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, 1989. Kaplansky, I. Fields and Rings, 2nd ed. Chicago, IL: University of Chicago Press, 1995. Lang, S. Undergraduate Algebra, 2nd ed. New York: Springer-Verlag, 1990. Spiegel, M. R. Schaum’s Outline of Theory and Problems of College Algebra, 2nd ed. New York: McGraw-Hill, 1997. Uspensky, J. V. Theory of Equations. New York: McGraw- Hill, 1948.van der Waerden, B. L. Algebra, Vol. 2. New York: Springer-Verlag, 1991. van der Waerden, B. L. Geometry and Algebra in Ancient Civilizations. New York: Springer-Verlag, 1983. van der Waerden, B. L. A History of Algebra: From al- Khwarizmi to Emmy Noether. New York: Springer-Verlag, 1985. Varadarajan, V. S. Algebra in Ancient and Modern Times. Providence, RI: Amer. Math. Soc., 1998. Weisstein, E. W. "Books about Algebra." http://www.trea- sure-troves.com/books/Algebra.html. Algebraic Closure The FIELD ¯F is called an algebraic closure of F if ¯F is algebraic over F and if every polynomial f(x) /C23 F[x] SPLITS completely over ¯F ; so that ¯F can be said to contain all the elements that are algebraic over F. For example, the FIELD of COMPLEX NUMBERS C is the algebraic closure of the FIELD of REALS R :/ See also ALGEBRAICALLY CLOSED ,SPLITTING FIELD References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 455, 1998. Algebraic Coding Theory CODING THEORY Algebraic Combinatorics The use of techniques from algebra, topology, and geometry in the solution of combinatorial problems, or the use of combinatorial methods to attack pro- blems in these areas (Billera et al. 1999, p. ix). See also COMBINATORICS References Billera, L. J.; Bjo¨rner, A.; Greene, C.; Simion, R. E.; and Stanley, R. P. (Eds.). New Perspectives in Algebraic Combinatorics. Cambridge, England: Cambridge Univer- sity Press, 1999. Algebraic Congruence A CONGRUENCE OF THE FORM f(x) /C130 (mod n) where f(x)isan INTEGER POLYNOMIAL (Nagell 1951, p. 73). See also CONGRUENCE ,FUNCTIONAL CONGRUENCE References Nagell, T. "Algebraic Congruences and Functional Con- gruences," "Algebraic Congruences to a Prime Modulus," "Algebraic Congruences to a Composite Modulus," "Alge-braic Congruences to a Prime-Power Modulus," and"Numerical Examples of Solution of Algebraic Con-gruences." §22, 24, and 26 /C1 /8i n Introduction to Number Theory. New York: Wiley, pp. 73 /C1/6, 79/C1/1, and 83 /C1/3, 1951. Algebraic Connectivity The second smallest EIGENVALUE of the LAPLACIAN MATRIX of a graph G. This eigenvalue is greater than 0 IFF G is a CONNECTED GRAPH . See also CONNECTED GRAPH ,FIEDLER VECTOR ,LA- PLACIAN MATRIX References Chung, F. R. K. Spectral Graph Theory. Providence, RI: Amer. Math. Soc., 1997. Demmel, J. "CS 267: Notes for Lecture 23, April 9, 1999. Graph Partitioning, Part 2." http://www.cs.berkeley.edu/ ~demmel/cs267/lecture20/lecture20.html. Algebraic Curve An algebraic curve over a FIELD K is an equation f(X ; Y) /C300; where f(X ; Y)isa POLYNOMIAL in X and Y with COEFFICIENTS in K. A nonsingular algebraic curve is an algebraic curve over K which has no SINGULAR POINTS over K. A point on an algebraic curve is simply a solution of the equation of the curve. A K-RATIONAL POINT is a point (X, Y) on the curve, where X and Y are in the FIELD K. See also ALGEBRAIC GEOMETRY ,ALGEBRAIC VARIETY , CURVE References Griffiths, P. A. Introduction to Algebraic Curves. Provi- dence, RI: Amer. Math. Soc., 1989. Algebraic Expression An algebraic expression in variables fx1 ; ... ; xn g is an expression constructed with the variables and ALGEBRAIC NUMBERS using addition, multiplication, and rational powers. References Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525 /C1/41, 2000. Algebraic Extension This entry contributed by NICOLAS BRAY An extension F of a FIELD K is said to be algebraic if every element of F is algebraic over K (i.e., is the root of a nonzero polynomial with coefficients in K). See also GALOIS EXTENSION Algebraic Function A function which can be constructed using only a finite number of ELEMENTARY OPERATIONS together with the INVERSES of functions capable of being so constructed. Nonalgebraic functions are called TRANS- CENDENTAL FUNCTIONS . See also ELEMENTARY FUNCTION ,ELEMENTARY OP- ERATION ,TRANSCENDENTAL FUNCTIONReferences Knopp, K. "Algebraic Functions." Ch. 5 in Theory of Func- tions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 119 /C1/34, 1996. Koch, H. "Algebraic Functions of One Variable." Ch. 6 in Number Theory: Algebraic Numbers and Functions. Pro- vidence, RI: Amer. Math. Soc., pp. 141 /C1/70, 2000. Algebraic Function Field FUNCTION FIELD Algebraic Geometry Algebraic geometry is the study of geometries that come from algebra, in particular, from RINGS .In CLASSICAL ALGEBRAIC GEOMETRY , the algebra is the RING of POLYNOMIALS , and the geometry is the set of zeros of polynomials, called an ALGEBRAIC VARIETY . For instance, the UNIT CIRCLE is the set of zeros of x2 /C27y2 /C301 and is an ALGEBRAIC VARIETY , as are all of the CONIC SECTIONS . In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any COMMUTATIVE RING with a unit, such as the INTEGERS . The geometry of such a ring is determined by its algebraic structure, in particular its PRIME IDEALS . Grothendieck defined SCHEMES as the basic geometric objects, which have the same relationship to the geometry of a ring as a MANIFOLD to a COORDINATE CHART . The language of CATEGORY THEORY evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry. As a consequence, algebraic geometry became very useful in other areas of mathematics, most notably in ALGEBRAIC NUMBER THEORY . For instance, Deligne used it to prove a variant of the RIEMANN HYPOTH- ESIS. Also, Andrew Wiles’ proof of FERMAT’S LAST THEOREM used the tools developed in algebraic geometry. In the latter part of the twentieth century, research- ers have tried to extend the relationship between algebra and geometry to arbitrary NONCOMMUTATIVE RINGS . The study of geometries associated to non- commutative rings is called NONCOMMUTATIVE GEO- METRY . See also ALGEBRAIC CURVE ,A LGEBRAIC NUMBER THEORY ,A LGEBRAIC VARIETY ,C ATEGORY THEORY , COMMUTATIVE ALGEBRA ,CONIC SECTION ,DIFFEREN- TIAL GEOMETRY ,GEOMETRY ,NONCOMMUTATIVE GEO- METRY ,P LANE CURVE ,S CHEME ,S PACE CURVE , ZARISKI TOPOLOGY References Abhyankar, S. S. Algebraic Geometry for Scientists and Engineers. Providence, RI: Amer. Math. Soc., 1990. Bump, D. Algebraic Geometry. Singapore: World Scientific, 1998. Cox, D.; Little, J.; and O’Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer- Verlag, 1996. Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry. New York: Springer-Verlag, 1995. Eisenbud, D. (Ed.). Commutative Algebra, Algebraic Geome- try, and Computational Methods. Singapore: Springer- Verlag, 1999. Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, 1978. Greuel, G.-M. Computer Algebra and Algebraic Geometry-- Achievements and Perspectives. 29 Feb 2000. http:// xxx.lanl.gov/abs/math.AG/0002247/. Harris, J. Algebraic Geometry: A First Course. New York: Springer-Verlag, 1992. Hartshorne, R. Algebraic Geometry, rev. ed. New York: Springer-Verlag, 1997. Hulek, K.; Catanese, F.; Peters, C.; and Reid, M. (Eds.). New Trends in Algebraic Geometry: EuroConference on Alge- braic Geometry, Warwick, July 1996. Cambridge, Eng- land: Cambridge University Press, 1999. Lang, S. Introduction to Algebraic Geometry. New York: Interscience, 1958. Newstead, P. E. (Ed.). Algebraic Geometry. New York: Dekker, 1999. Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, Vol. 1. Cambridge, England: Cambridge University Press, 1994. Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, Vol. 2. Cambridge, England: Cambridge University Press, 1994. Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, Vol. 3. Cambridge, England: Cambridge University Press, 1994. Pragacz, P.; Szurek, M.; and Wisniewski, J. Algebraic Geometry: Hirzenbruch 70. Providence, RI: Amer. Math. Soc., 1999. Seidenberg, A. (Ed.). Studies in Algebraic Geometry. Wa- shington, DC: Math. Assoc. Amer., 1980. Serto¨z, S. (Ed.). Algebraic Geometry. New York: Dekker, 1998. van Oystaeyen, F. Algebraic Geometry for Associative Alge- bras. New York: Dekker, 2000. Weil, A. Foundations of Algebraic Geometry, enl. ed. Providence, RI: Amer. Math. Soc., 1962. Weisstein, E. W. "Books about Algebraic Geometry." http:// www.treasure-troves.com/books/AlgebraicGeometry.html. Yang, K. Complex Algebraic Geometry: An Introduction to Curves and Surfaces, 2nd ed. New York: Dekker, 1999. Algebraic Integer If r is a ROOT of the POLYNOMIAL equation xn /C27an/C281xn/C281 /C27/C1/C1/C1/C27a1x /C27a0 /C300; where the ais/ are INTEGERS and r satisfies no similar equation of degree Bn; then r is called an algebraic integer of degree n. An algebraic integer is a special case of an ALGEBRAIC NUMBER (for which the leading COEFFICIENT an need not equal 1). RADICAL INTEGERS are a SUBRING of the algebraic integers. A SUM or PRODUCT of algebraic integers is again an algebraic integer. However, ABEL’S IMPOSSIBILITY THEOREM shows that there are algebraic integers of degree ]5 which are not expressible in terms of ADDITION , SUBTRACTION , MULTIPLICATION , DIVISION , and ROOT EXTRACTION (the ELEMENTARY OPERATIONS )on COMPLEX NUMBERS . In fact, if ELEMENTARY OPERA- TIONS are allowed on real numbers only, then there are real numbers which are algebraic integers of degree 3 which cannot be so expressed. The GAUSSIAN INTEGERS are algebraic integers of Q(ffiffiffiffiffiffi /C281p ) ; since a /C27bi are roots of z2 /C282az /C27a2 /C27b2 /C300: See also ALGEBRAIC NUMBER ,CASUS IRREDUCIBILUS , ELEMENTARY OPERATION ,EUCLIDEAN NUMBER ,RADI- CAL INTEGER References Ferreiro ´s, J. "Algebraic Integers." §3.3.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 97 /C1/9, 1999. Hancock, H. Foundations of the Theory of Algebraic Num- bers, Vol. 1: Introduction to the General Theory. New York: Macmillan, 1931. Hancock, H. Foundations of the Theory of Algebraic Num- bers, Vol. 2: The General Theory. New York: Macmillan, 1932. Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num- ber Theory. Cambridge, England: Cambridge University Press, 1989. Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in Action. New York: W. H. Freeman, pp. 347 /C1/53, 1991. Algebraic Invariant A quantity such as a DISCRIMINANT which remains unchanged under a given class of algebraic transfor- mations. Such invariants were originally called HY- PERDETERMINANTS by Cayley. See also DISCRIMINANT (POLYNOMIAL ), INVARIANT , QUADRATIC INVARIANT References Grace, J. H. and Young, A. The Algebra of Invariants. New York: Chelsea, 1965. Gurevich, G. B. Foundations of the Theory of Algebraic Invariants. Groningen, Netherlands: P. Noordhoff, 1964. Hermann, R. and Ackerman, M. Hilbert’s Invariant Theory Papers. Brookline, MA: Math Sci Press, 1978. Hilbert, D. Theory of Algebraic Invariants. Cambridge, England: Cambridge University Press, 1993. Mumford, D.; Fogarty, J.; and Kirwan, F. Geometric Invar- iant Theory, 3rd enl. ed. New York: Springer-Verlag, 1994. Weisstein, E. W. "Books about Invariants." http://www.trea- sure-troves.com/books/Invariants.html. Algebraic Knot A single component ALGEBRAIC LINK . Most knots up to 11 crossings are algebraic, but they quickly become outnumbered by nonalgebraic knots for more cross- ings (Hoste et al. 1998). See also ALGEBRAIC LINK,KNOT,LINK References Bonahon, F. and Siebermann, L. "The Classification of Algebraic Links." Unpublished manuscript. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Algebraic K-Theory K-THEORY Algebraic Language Let X be an alphabet (i.e., a finite and nonempty set), and call its member letters. A word on X is a finite sequence of letters a1 ...an ; where a1 ; ...; an /C23 X : Denote the empty word by e, and the set of all words in X by X /C31: Define the concatenation (also called product) of a word u /C30a1 ...anwith a word v /C30 b1 ...bm as uv /C30a1 ...anb1 ...bm : In general, concate- nation is not commutative. Use the notation ½u½ato mean the number of letters a in the word u.A language L is then a subset of X /C31; and L is said to be algebraic when a set of rewriting rules, applied recursively, forms all the words of L and no others. See also DYCK LANGUAGE References Bousquet-Me ´lou, M. "Convex Polyominoes and Algebraic Languages." J. Phys. A: Math. Gen. 25, 1935 /C1/944, 1992. Delest, M.-P. and Viennot, G. "Algebraic Languages and Polyominoes [sic] Enumeration." Theoret. Comput. Sci. 34, 169 /C1/06, 1984. Algebraic Link A class of fibered knots and links which arises in ALGEBRAIC GEOMETRY . An algebraic link is formed by connecting the NW and NE strings and the SW and SE strings of an ALGEBRAIC TANGLE (Adams 1994). See also ALGEBRAIC KNOT,A LGEBRAIC TANGLE , FIBRATION ,TANGLE References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 48 /C1/9, 1994. Bonahon, F. and Siebermann, L. "The Classification of Algebraic Links." Unpublished manuscript. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 335, 1976. Algebraic Manifold An algebraic manifold is another name for a smooth ALGEBRAIC VARIETY . It can be covered by COORDINATE CHARTS so that the TRANSITION FUNCTIONS are given by RATIONAL FUNCTIONS . Technically speaking, the coordinate charts should be to all of affine space Cn :/ For example, the SPHERE is an algebraic manifold, with a chart given by STEREOGRAPHIC PROJECTION to C ; and another chart at /C12; with the TRANSITION FUNCTION given by 1=z: In this setting, it is called theRIEMANN SPHERE . The TORUS is also an algebraic manifold, in this setting called an ELLIPTIC CURVE , with charts given by ELLIPTIC FUNCTIONS such as the WEIERSTRASS ELLIPTIC FUNCTION . See also ABSTRACT MANIFOLD ,ALGEBRAIC GEOMETRY , ALGEBRAIC VARIETY ,ELLIPTIC CURVE ,MANIFOLD Algebraic Number If r is a ROOT of the POLYNOMIAL equation a0xn /C27a1xn/C281 /C27/C1/C1/C1/C27an/C281x /C27an /C300 ; (1) where the ais/ are INTEGERS and r satisfies no similar equation of degree Bn; then r is an algebraic number of degree n.Ifr is an algebraic number and a0 /C301; then it is called an ALGEBRAIC INTEGER . It is also true that if the cis/ in a0xn /C27c1xn/C281 /C27/C1/C1/C1/C27cn/C281x /C27cn /C300 (2) are algebraic numbers, then any ROOT of this equa- tion is also an algebraic number. If a is an algebraic number of degree n satisfying the POLYNOMIAL a(x /C28 a)(x /C28 b)(x /C28 g)... ; (3) then there are n /C281 other algebraic numbers b; g ; ... called the conjugates of a: Furthermore, if a satisfies any other algebraic equation, then its conjugates also satisfy the same equation (Conway and Guy 1996). Any number which is not algebraic is said to be TRANSCENDENTAL . The set of algebraic numbers is denoted A (Mathematica ), or sometimes ¯Q (Nester- enko 1999), and is implemented in Mathematica as Algebraics . A number x can then be tested to see if it is algebraic using the command Element[ x, Algebraics]. See also ALGEBRAIC INTEGER ,EUCLIDEAN NUMBER , HERMITE- LINDEMANN THEOREM ,R ADICAL INTEGER , Q-BAR,TRANSCENDENTAL NUMBER References Conway, J. H. and Guy, R. K. "Algebraic Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 189 /C1/ 90, 1996. Courant, R. and Robbins, H. "Algebraic and Transcendental Numbers." §2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 103 /C1/07, 1996. Ferreiro ´s, J. "The Emergence of Algebraic Number Theory." §3.3 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 94 /C1/9, 1999. Hancock, H. Foundations of the Theory of Algebraic Num- bers. Vol. 1: Introduction to the General Theory. New York: Macmillan, 1931. Hancock, H. Foundations of the Theory of Algebraic Num- bers. Vol. 2: The General Theory. New York: Macmillan, 1932. Koch, H. Number Theory: Algebraic Numbers and Func- tions. Providence, RI: Amer. Math. Soc., 2000. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 35, 1951. Narkiewicz, W. Elementary and Analytic Number Theory of Algebraic Numbers. Warsaw: Polish Scientific Publishers, 1974. Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. http://www.math.jussieu.fr/~neste- ren/. Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in Action. New York: W. H. Freeman, pp. 347 /C1/53, 1991. Algebraic Number Field NUMBER FIELD Algebraic Number Theory NUMBER THEORY Algebraic Projective Geometry PROJECTIVE GEOMETRY Algebraic Set An algebraic set is the locus of zeros of a collection of POLYNOMIALS . For example, the circle is the set of zeros of x2 /C27y2 /C281 and the point at (a, b) is the set of zeros of x and y. The algebraic set f(x; 0)g@f(0; y)g is the set of solutions to xy /C300. It decomposes into two irreducible algebraic sets, called ALGEBRAIC VARI- ETIES . In general, an algebraic set can be written uniquely as the finite union of ALGEBRAIC VARIETIES . The intersection of two algebraic sets is an algebraic set corresponding to the union of the polynomials. For example, x /C300 and y /C300 intersect at (0; 0); i.e., where x /C300 and y /C300. In fact, the intersection of an arbitrary number of algebraic sets is itself an alge- braic set. However, only a finite union of algebraic sets is algebraic. If X is the set of solutions to fi /C300 and Y is the set of solutions to gj /C300; then X @ Y is the set of solutions to figj /C300: Consequently, the algebraic sets are the closed sets in a TOPOLOGY , called the ZARISKI TOPOLOGY . The set of polynomials vanishing on an algebraic set X is an IDEAL in the POLYNOMIAL RING . Conversely, any IDEAL defines an algebraic set since it is a collection of polynomials. HILBERT’S NULLSTELLEN- SATZ describes the precise relationship between IDEALS and algebraic sets. See also ALGEBRAIC VARIETY ,C ATEGORY THEORY , COMMUTATIVE ALGEBRA ,CONIC SECTION ,H ILBERT’S NULLSTELLENSATZ ,IDEAL ,PRIME IDEAL ,PROJECTIVE VARIETY ,SCHEME ,ZARISKI TOPOLOGY References Bump, D. Algebraic Geometry. Singapore: World Scientific, pp. 1 /C1/, 1998. Hartshorne, R. Algebraic Geometry. New York: Springer- Verlag, 1977.Algebraic Surface The set of ROOTS of a POLYNOMIAL f(x; y ; z) /C300: An algebraic surface is said to be of degree n /C30max( i /C27 j /C27k) ; where n is the maximum sum of powers of all terms amxim yjm zkm : The following table lists the names of algebraic surfaces of a given degree. Order Surface 3 CUBIC SURFACE 4 QUARTIC SURFACE 5 QUINTIC SURFACE 6 SEXTIC SURFACE 7 HEPTIC SURFACE 8 OCTIC SURFACE 9 NONIC SURFACE 10 DECIC SURFACE 12 DODECIC SURFACE See also BARTH DECIC,BARTH SEXTIC ,BOY SURFACE , CAYLEY CUBIC ,C HAIR ,CLEBSCH DIAGONAL CUBIC , CUSHION ,DERVISH ,ENDRA ss OCTIC,HEART SURFACE , HENNEBERG’S MINIMAL SURFACE ,KUMMER SURFACE , ORDER (ALGEBRAIC SURFACE ), ROMAN SURFACE ,SAR- TI DODECIC SURFACE ,TOGLIATTI SURFACE References Banchoff, T. F. "Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order." In Geo- metric Analysis and Computer Graphics: Proceedings of a Workshop Held May 23 /C1/5, 1988 (Eds. P. Concus, R. Finn, D. A. Hoffman). New York: Springer-Verlag, pp. 31 /C1/7, 1991. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 7, 1986. Algebraic Tangle Any TANGLE obtained by additions and multiplica- tions of rational TANGLES (Adams 1994). See also ALGEBRAIC LINK,TANGLE References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 41 /C1/1, 1994. Algebraic Topology The study of intrinsic qualitative aspects of spatial objects (e.g., SURFACES ,SPHERES ,TORI,CIRCLES , KNOTS ,LINKS , configuration spaces, etc.) that remain invariant under both-directions continuous ONE-TO- ONE (HOMEOMORPHIC ) transformations. The disci- pline of algebraic topology is popularly known as "RUBBER-SHEET GEOMETRY " and can also be viewed as the study of DISCONNECTIVITIES . Algebraic topology has a great deal of mathematical machinery for studying different kinds of HOLE structures, and it gets the prefix "algebraic" since many HOLE struc- tures are represented best by algebraic objects like GROUPS and RINGS . A technical way of saying this is that algebraic topology is concerned with FUNCTORS from the topo- logical CATEGORY of GROUPS and HOMOMORPHISMS . Here, the FUNCTORS are a kind of filter, and given an "input" SPACE , they spit out something else in return. The returned object (usually a GROUP or RING ) is then a representation of the HOLE structure of the SPACE , in the sense that this algebraic object is a vestige of what the original SPACE was like (i.e., much informa- tion is lost, but some sort of "shadow" of the SPACE is retained–just enough of a shadow to understand some aspect of its HOLE -structure, but no more). The idea is that FUNCTORS give much simpler objects to deal with. Because SPACES by themselves are very compli- cated, they are unmanageable without looking at particular aspects. COMBINATORIAL TOPOLOGY is a special type of alge- braic topology that uses COMBINATORIAL methods. See also CATEGORY ,COMBINATORIAL TOPOLOGY ,DIF- FERENTIAL TOPOLOGY ,FUNCTOR ,HOMOTOPY THEORY , TOPOLOGY References Dieudonne ´,J. A History of Algebraic and Differential Topology: 1900 /C1/960. Boston, MA: Birkha ¨user, 1989. Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, 1997. Massey, W. S. A Basic Course in Algebraic Topology. New York: Springer-Verlag, 1991. Maunder, C. R.F. Algebraic Topology. New York: Dover, 1997. May, J. P. A Concise Course on Algebraic Topology. Chicago, IL: University of Chicago Press, 1999. May, J. P. Simplicial Objects in Algebraic Topology. Chi- cago, IL: University of Chicago Press, 1982. Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Sato, H. Algebraic Topology: An Intuitive Approach. Provi- dence, RI: Amer. Math. Soc., 1999. Weisstein, E. W. "Books about Topology." http://www.trea- sure-troves.com/books/Topology.html. Algebraic Unknotting Number The algebraic unknotting number of a knot K in S3 is defined as the algebraic unknotting number of the S- equivalence class of a SEIFERT MATRIX of K. The algebraic unknotting number of an element in an S- equivalent class is defined as the minimum number of algebraic unknotting operations necessary to trans- form the element to the S-equivalence class of the zero matrix (Saeki 1999). See also SEIFERT MATRIX ,UNKNOTTING NUMBERReferences Fogel, M. "Knots with Algebraic Unknotting Number One." Pacific J. Math. 163, 277 /C195, 1994. Murakami, H. "Algebraic Unknotting Operation, Q&A." Gen. Topology 8, 283 /C192, 1990. Saeki, O. "On Algebraic Unknotting Numbers of Knots." Tokyo J. Math. 22, 425 /C143, 1999. Algebraic Variety A generalization to n-D of ALGEBRAIC CURVES . More technically, an algebraic variety is a reduced SCHEME of FINITE type over a FIELD K. An algebraic variety V is defined as the SET of points in the REALS Rn (or the COMPLEX NUMBERS Cn /) satisfying a system of POLY- NOMIAL equations fi(x1 ; ...; xn) /C300 for i /C301, 2, .... According to the HILBERT BASIS THEOREM ,a FINITE number of equations suffices. A variety is the set of common zeros to a collection of POLYNOMIALS . In classical algebraic geometry, the polynomials have COMPLEX NUMBERS for coefficients. Because of the FUNDAMENTAL THEOREM OF ALGEBRA , such polynomials always have zeros. For example, f(x ; y; z):x2 /C27y2 /C28z2 g is the CONE , and f(x; y; z):x2 /C27y2 /C28z2 ; ax /C27by /C27cz /C300 g is a CONIC SECTION , which is a SUBVARIETY of the cone. Actually, the cone and the conic section are examples of AFFINE VARIETIES because they are in AFFINE SPACE . A general variety is comprised of affine varieties glued together, like the COORDINATE CHARTS of a MANIFOLD . The FIELD of coefficients can be any ALGEBRAICALLY CLOSED field. When a variety is embedded in projective space, it is a PROJECTIVE ALGEBRAIC VARIETY . Also, an INTRINSIC VARIETY can be thought of as an abstract object, like a MANIFOLD , independent of any particular embedding. A SCHEME is a generalization of a variety, which includes the possibility of replacing C[x; y; z] by any COMMUTA- TIVE RING with a unit. A further generalization is a STACK . See also ABELIAN VARIETY ,AFFINE VARIETY ,ALBA- NESE VARIETY ,ALGEBRAIC NUMBER THEORY ,BRAUER- SEVERI VARIETY ,CATEGORY THEORY ,CHOW VARIETY , COMMUTATIVE ALGEBRA ,CONIC SECTION ,INTRINSIC VARIETY ,PICARD VARIETY ,PROJECTIVE ALGEBRAIC VARIETY ,SCHEME ,STACK (MODULI SPACE ), ZARISKI TOPOLOGY References Bump, D. Algebraic Geometry. Singapore: World Scientific, pp. 79 /C1/6, 1998. Ciliberto, C.; Laura, E.; and Somese, A. J. (Eds.). Classifica- tion of Algebraic Varieties. Providence, RI: Amer. Math. Soc., 1994. Hartshorne, R. Algebraic Geometry. New York: Springer- Verlag, 1977. Algebraically Closed A FIELD K is said to be algebraically closed if every POLYNOMIAL with coefficients in K has a ROOT in K. See also ALGEBRAIC CLOSURE ,FIELD References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 455, 1998. Algebraically Independent This entry contributed by JOHNNY CHEN Let K be a FIELD , and A a K-algebra. Elements y1 ; ..., yn are algebraically independent over K if the natural surjection K[Y1 ; ... ; Yn] 0 K[y1 ; ...yn] is an iso- morphism. In other words, there are no polynomial relations F(y1 ; ...; yn) /C300 with coefficients in K. References Reid, M. Undergraduate Commutative Algebra. Cambridge, England: Cambridge University Press, 1995. See also IRRATIONAL NUMBER ,L INDEMANN- WEIER- STRASS THEOREM ,S CHANUEL’S CONJECTURE ,S HI- DLOVSKII THEOREM ,TRANSCENDENTAL NUMBER Algebraics ALGEBRAIC NUMBER Algebroidal Function An ANALYTIC FUNCTION f(z) satisfying the irreducible algebraic equation A0(z)f k /C27A1(z)f k /C281 /C27/C1/C1/C1/C27Ak(z) /C300 with single-valued MEROMORPHIC FUNCTIONS Aj(z)in a COMPLEX DOMAIN G is called a k-algebroidal function in G. See also MEROMORPHIC FUNCTION References Iyanaga, S. and Kawada, Y. (Eds.). "Algebroidal Functions." §19 in Encyclopedic Dictionary of Mathematics. Cam- bridge, MA: MIT Press, pp. 86 /C1/8, 1980. Algorithm A specific set of instructions for carrying out a procedure or solving a problem, usually with the requirement that the procedure terminate at some point. Specific algorithms sometimes also go by the name METHOD , PROCEDURE ,or TECHNIQUE . The word "algorithm" is a distortion of al-Khwarizmi, an Arab mathematician who wrote an influential treatise about algebraic methods. See also 196-ALGORITHM ,ALGORITHMIC COMPLEXITY , ARCHIMEDES ALGORITHM ,BHASKARA- BROUCKNER AL- GORITHM ,B ORCHARDT- PFAFF ALGORITHM ,B RELAZ’S HEURISTIC ALGORITHM ,BUCHBERGER’S ALGORITHM ,BULIRSCH- STOER ALGORITHM ,BUMPING ALGORITHM , COMPUTABLE FUNCTION ,CONTINUED FRACTION FAC- TORIZATION ALGORITHM ,D ECISION PROBLEM ,D IJK- STRA’S ALGORITHM ,E UCLIDEAN ALGORITHM , FERGUSON- FORCADE ALGORITHM ,F ERMAT’S ALGO- RITHM ,FLOYD’S ALGORITHM ,G AUSSIAN APPROXIMA- TION ALGORITHM ,G ENETIC ALGORITHM ,G OSPER’S ALGORITHM ,G REEDY ALGORITHM ,H ASSE’S ALGO- RITHM , HJLS ALGORITHM ,JACOBI ALGORITHM ,KRUS- KAL’S ALGORITHM ,L EVINE- O’SULLIVAN GREEDY ALGORITHM , LLL ALGORITHM ,M ARKOV ALGORITHM , MILLER’S ALGORITHM ,N EVILLE’S ALGORITHM ,N EW- TON’S METHOD ,PRIME FACTORIZATION ALGORITHMS , PRIMITIVE RECURSIVE FUNCTION ,P ROGRAM , PSLQ ALGORITHM ,PSOSA LGORITHM ,Q UOTIENT- DIFFER- ENCE ALGORITHM ,R ISCH ALGORITHM ,S CHRAGE’S ALGORITHM ,S HANKS’ ALGORITHM ,S PIGOT ALGO- RITHM ,S YRACUSE ALGORITHM ,T OTAL FUNCTIO N, TURING MACHIN E,Z ASSENHAUS- BERLEKAMP ALGO- RITHM ,ZEILBERGER’S ALGORITHM References Aho, A. V.; Hopcroft, J. E.; and Ullman, J. D. The Design and Analysis of Computer Algorithms. Reading, MA: Addison-Wesley, 1974. Atallah, M. J. Algorithms and Theory of Computation Handbook. Boca Raton, FL: CRC Press, 1998. Baase, S. Computer Algorithms. Reading, MA: Addison- Wesley, 1988. Bellman, R. E.; Cooke, K. L.; and Lockett, J. A. Algorithms, Graphs, and Computers. New York: Academic Press, 1970. Brassard, G. and Bratley, P. Fundamentals of Algorithmics. Englewood Cliffs, NJ: Prentice-Hall, 1995. Chabert, J.-L. (Ed.). A History of Algorithms: From the Pebble to the Microchip. New York: Springer-Verlag, 1999. Collberg, C. "A /l/goVista." http://www.algovista.com/. Cormen, T. H.; Leiserson, C. E.; and Rivest, R. L. Introduc- tion to Algorithms. Cambridge, MA: MIT Press, 1990. Greene, D. H. and Knuth, D. E. Mathematics for the Analysis of Algorithms, 3rd ed. Boston, MA: Birkha ¨user, 1990. Harel, D. Algorithmics: The Spirit of Computing, 2nd ed. Reading, MA: Addison-Wesley, 1992. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. Kozen, D. C. Design and Analysis and Algorithms. New York: Springer-Verlag, 1991. Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Sedgewick, R. Algorithms in C, 3rd ed. Reading, MA: Addison-Wesley, 1998. Sedgewick, R. and Flajolet, P. An Introduction to the Analysis of Algorithms. Reading, MA: Addison-Wesley, 1996. Skiena, S. S. The Algorithm Design Manual. New York: Springer-Verlag, 1997. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Skiena, S. S. "The Stony Brook Algorithm Repository." http://www.cs.sunysb.edu/~algorith/. Wilf, H. Algorithms and Complexity. Englewood Cliffs, NJ: Prentice Hall, 1986. http://www.cis.upenn.edu/~wilf/Alg- Comp2.html. Algorithmic Complexity BIT COMPLEXITY ,KOLMOGOROV COMPLEXITY Alhazen’s Billiard Problem In a given CIRCLE , find an ISOSCELES TRIANGLE whose LEGS pass through two given POINTS inside the CIRCLE . This can be restated as: from two POINTS in the PLANE of a CIRCLE , draw LINES meeting at the POINT of the CIRCUMFERENCE and making equal ANGLES with the NORMAL at that POINT . The problem is called the billiard problem because it corresponds to finding the POINT on the edge of a circular "BILLIARD " table at which a cue ball at a given POINT must be aimed in order to carom once off the edge of the table and strike another ball at a second given POINT . The solution leads to a BIQUADRATIC EQUATION OF THE FORM H(x2 /C28y2) /C282Kxy /C27(x2 /C27y2)(hy /C28kx) /C300 : The problem is equivalent to the determination of the point on a spherical mirror where a ray of light will reflect in order to pass from a given source to an observer. It is also equivalent to the problem of finding, given two points and a CIRCLE such that the points are both inside or outside the CIRCLE , the ELLIPSE whose FOCI are the two points and which is tangent to the given CIRCLE . The problem was first formulated by Ptolemy in 150 AD, and was named after the Arab scholar Alhazen, who discussed it in his work on optics. It was not until 1997 that Neumann proved the problem to be insoluble using a COMPASS and RULER construction because the solution requires extraction of a CUBE ROOT (Neumann 1998). This is the same reason that the CUBE DUPLICATION problem is insoluble. See also BILLIARDS ,BILLIARD TABLE PROBLEM ,CUBE DUPLICATION References Do¨rrie, H. "Alhazen’s Billiard Problem." §41 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 197 /C1/00, 1965. Hogendijk, J. P. "Al-Mutaman’s Simplified Lemmas for Solving ‘Alhazen’s Problem’." From Baghdad to Barce- lona/De Bagdad a` Barcelona, Vol. I, II (Zaragoza, 1993), pp. 59 /C1/01, Anu. Filol. Univ. Barc., XIX B-2, Univ. Barce- lona, Barcelona, 1996. Lohne, J. A. "Alhazens Spiegelproblem." Nordisk Mat. Tidskr. 18,5/C1/5, 1970. Neumann, P. M. " Reflections on Reflection in a Spherical Mirror." Amer. Math. Monthly 105, 523 /C1/28, 1998.Riede, H. "Reflexion am Kugelspiegel. Oder: das Problem des Alhazen." Praxis Math. 31,65/C1/0, 1989. Sabra, A. I. "ibn al-Haytham’s Lemmas for Solving ‘Alha- zen’s Problem’." Arch. Hist. Exact Sci. 26, 299 /C1/24, 1982. Alhazen’s Problem ALHAZEN’S BILLIARD PROBLEM Alias Transformation A transformation in which the coordinate system is changed, leaving vectors in the original coordinate system "fixed" while changing their representation in the new coordinate system. In contrast, a transforma- tion in which vectors are transformed in a fixed coordinate system is called an ALIBI TRANSFORMA- TION . See also ALIBI TRANSFORMATION ,ROTATION FORMULA Aliasing Given a power spectrum (a plot of power vs. fre- quency), aliasing is a false translation of power falling in some frequency range (/C28fc ; fc) outside the range. Aliasing can be caused by discrete sampling below the NYQUIST FREQUENCY . The sidelobes of any INSTRU- MENT FUNCTION (including the simple SINC SQUARED function obtained simply from FINITE sampling) are also a form of aliasing. Although sidelobe contribution at large offsets can be minimized with the use of an APODIZATION FUNCTION , the tradeoff is a widening of the response (i.e., a lowering of the resolution). See also APODIZATION FUNCTION ,N YQUIST FRE- QUENCY Alibi Transformation A transformation in which vectors are transformed in a fixed coordinate system. In contrast, a transforma- tion in which the coordinate system is changed, leaving vectors in the original coordinate system "fixed" while changing their representation in the new coordinate system, is called an ALIAS TRANSFOR- MATION . See also ALIAS TRANSFORMATION ,ROTATION FORMULA Aliquant Divisor A number which does not DIVIDE another exactly. For instance, 4 and 5 are aliquant divisors of 6. A number which is not an aliquant divisor (i.e., one that does DIVIDE another exactly) is said to be an ALIQUOT DIVISOR . See also ALIQUOT DIVISOR ,DIVISOR ,PROPER DIVISOR Aliquot Cycle ALIQUOT SEQUENCE ,SOCIABLE NUMBERS Aliquot Divisor A number which DIVIDES another exactly. For in- stance, 1, 2, 3, and 6 are aliquot divisors of 6. A number which is not an aliquot divisor is said to be an ALIQUANT DIVISOR . The term "aliquot" is frequently used to specifically mean a PROPER DIVISOR , i.e., a DIVISOR of a number other than the number itself. See also ALIQUANT DIVISOR ,DIVISOR ,PROPER DIVISOR Aliquot Sequence Let s(n) /C13 s(n) /C28n where s(n) is the DIVISOR FUNCTION and s(n) is the RESTRICTED DIVISOR FUNCTION . Then the SEQUENCE of numbers s0(n) /C13n ; s1(n) /C30s(n) ; s2(n) /C30s(s(n)) ;/C1/C1/C1 is called an aliquot sequence. If the SEQUENCE for a given n is bounded, it either ends at s(1) /C300or becomes periodic. 1. If the SEQUENCE reaches a constant, the con- stant is known as a PERFECT NUMBER . 2. If the SEQUENCE reaches an alternating pair, it is called an AMICABLE PAIR. 3. If, after k iterations, the SEQUENCE yields a cycle of minimum length t OF THE FORM sk/C271(n); sk /C272(n); ..., sk/C271(n); then these numbers form a group of SOCIABLE NUMBERS of order t. It has not been proven that all aliquot sequences eventually terminate and become period. The smal- lest number whose fate is not known is 276, which has been computed up to s628(276) (Guy 1994). There are five such sequences less than 1000, namely 276, 552, 564, 660, and 966, sometimes called the "Lehmer five." Furthermore, there are 934 open sequences 5105;and 9710 open sequences 5106(Creyaufmu ¨l- ler). See also 196-ALGORITHM ,A DDITIVE PERSISTENCE , AMICABLE NUMBERS ,CATALAN’S ALIQUOT SEQUENCE CONJECTURE ,M ULTIAMICABLE NUMBERS ,M ULTIPER- FECT NUMBER ,M ULTIPLICATIVE PERSISTENCE ,PER- FECT NUMBER ,S OCIABLE NUMBERS ,U NITARY ALIQUOT SEQUENCE References Creyaufmu ¨ller, W. "Aliquot Sequences." http://home.t-onli- ne.de/home/Wolfgang.Creyaufmueller/aliquote.htm. Guy, R. K. "Aliquot Sequences." §B6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 60 /C12, 1994. Guy, R. K. and Selfridge, J. L. "What Drives Aliquot Sequences." Math. Comput. 29, 101/C107, 1975. Sloane, N. J. A. Sequences A003023/M0062 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Sloane, N. J. A. and Plouffe, S. Figure M0062 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Alladi-Grinstead Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let N(n) be the number of ways in which the FACTORIAL n! can be decomposed into nFACTORS of the form Pbk karranged in nondecreasing order. Also define m(n)/C13max( pb1 1); (1) i.e., m(n) is the LEAST PRIME FACTOR raised to its appropriate POWER in the factorization. Then define a(n)/C13lnm(n) lnn(2) where ln( x) is the NATURAL LOGARITHM . For instance, 9!/C302/C2152/C2152/C2152/C2152/C21522/C2155/C2157/C21534 /C302/C2152/C2152/C2152/C2153/C2155/C2157/C21523/C21533 /C302/C2152/C2152/C2152/C2155/C2157/C21523/C21532/C21532 /C302/C2152/C2152/C2153/C21522/C21522/C2155/C2157/C21533 /C302/C2152/C2152/C21522/C21522/C2155/C2157/C21532/C21532 /C302/C2152/C2152/C2153/C2153/C2155/C2157/C21532/C21524 /C302/C2152/C2153/C2153/C21522/C2155/C2157/C21523/C21532 /C302/C2152/C2153/C2153/C2153/C2153/C2155/C2157/C21525 /C302/C2153/C2153/C21522/C21522/C21522/C2155/C2157/C21532 /C302/C2153/C2153/C2153/C2153/C21522/C2155/C2157/C21524 /C302/C2153/C2153/C2153/C2153/C2155/C2157/C21523/C21523 /C303/C2153/C2153/C2153/C21522/C21522/C2155/C2157/C21523; (3) so a(9)/C30ln 3 ln 9/C30ln 3 2ln 3/C3012: (4) For large n, lim n0/C12a(n)¼ec/C281¼0:809394020534 :::; (5) where c/C13X/C12 k/C3021klnk k/C281 ! : (6) References Alladi, K. and Grinstead, C. "On the Decomposition of n! into Prime Powers." J. Number Th. 9, 452/C1/58, 1977. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/aldgrns/aldgrns.html. Guy, R. K. "Factorial nas the Product of nLarge Factors." §B22 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 79, 1994. Allais Paradox Choose between the following two alternatives: 1. 90% chance of an unknown amount x and a 10% chance of $1 million, or 2. 89% chance of the same unknown amount x, 10% chance of $2.5 million, and 1% chance of nothing. The PARADOX is to determine which choice has the larger EXPECTATION VALUE ,0:9x/C27/$/100;000 or 0:89x/C27/ //$/250;000: However, the best choice depends on the unknown amount, even though it is the same in both cases! This appears to violate the INDEPENDENCE AXIOM . See also INDEPENDENCE AXIOM ,M ONTY HALL PRO- BLEM ,NEWCOMB’S PARADOX References Allais, M. "Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’e´cole ame´ricaine." Econometrica 21, 503 /C1/46, 1953. Kreps, D. M. Notes on the Theory of Choice. Boulder, CO: Westview Press, p. 192, 1988. Fishburn, P. C. Utility Theory for Decision Making. New York: Wiley, 1970. Savage, L. J. The Foundations of Statistics, 2nd ed. New York: Dover, 1972. Allegory A technical mathematical object which bears the same resemblance to binary relations as CATEGORIES do to FUNCTIONS and SETS . See also CATEGORY References Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster- dam, Netherlands: North-Holland, 1990. Allometric Mathematical growth in which one population grows at a rate PROPORTIONAL to the POWER of another population. References Coffey, W. J. Geography Towards a General Spatial Systems Approach. London: Routledge, Chapman & Hall, 1981. All-Pairs Shortest Path The shortest distance between any pair of vertices in the shortest-path spanning tree, as long as the path giving the shortest path does not pass through the root of the spanning tree (Skiena 1990, p. 228). The problem can be solved using n applications of DIJK- STRA’S ALGORITHM or FLOYD’S ALGORITHM . The latter also works in the case of a weighted graph where the edges have negative weights.See also FLOYD’S ALGORITHM ,DIJKSTRA’S ALGORITHM , GRAPH GEODESIC References Skiena, S. "All Pairs Shortest Paths." §6.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 228 /C1/29, 1990. All-Poles Model MAXIMUM ENTROPY METHOD All-to-All Communication GOSSIPING Almost All Given a property P,if P(x) /C2x as x 0/C12 (so the number of numbers less than x not satisfying the property P is s(x)) ; then P is said to hold true for almost all numbers. For example, almost all positive integers are COMPOSITE NUMBERS (which is not in conflict with the second of EUCLID’S THEOREMS that there are an infinite number of PRIMES ). See also FOR ALL,NORMAL ORDER References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 50, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 8, 1979. Almost Alternating Knot An ALMOST ALTERNATING LINK with a single compo- nent. See also ALMOST ALTERNATING LINK Almost Alternating Link Call a projection of a LINK an almost alternating projection if one crossing change in the projection makes it an alternating projection. Then an almost alternating link is a LINK with an almost alternating projection, but no alternating projection. Every AL- TERNATING KNOT has an almost alternating projec- tion. A PRIME KNOT which is almost alternating is either a TORUS KNOT or a HYPERBOLIC KNOT . There- fore, no SATELLITE KNOT is an almost alternating knot. All nonalternating 9-crossing PRIME KNOTS are almost alternating. Of the 393 nonalternating knots and links with 11 or fewer crossings, all but five are known to be almost alternating (and 3 of these have 11 crossings). The fate of the remaining five is not known. The ( q;2);(4;3);and (5 ;3)/-TORUS KNOTS are almost alternating (Adams 1994, p. 142). See also ALTERNATING KNOT,LINK References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 139 /C1/46, 1994. Almost Everywhere A property of X is said to hold almost everywhere if the SET of points in X where this property fails has MEASURE ZERO . See also ALMOST EVERYWHERE CONVERGENCE ,M EA- SURE ZERO References Jeffreys, H. and Jeffreys, B. S. "‘Measure Zero’: ‘Almost Everywhere’." §1.1013 in Methods of Mathematical Phy- sics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 29 /C1/0, 1988. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 1, 1991. Almost Everywhere Convergence A weakened version of POINTWISE CONVERGENCE hypothesis which states that, for X a MEASURE SPACE , fn(x) 0 f(x) for all x /C23 Y ; where Y is a measurable subset of Xsuch that m(X_Y)/C300:/ See also POINTWISE CONVERGENCE References Browder, A. Mathematical Analysis: An Introduction. New York: Springer-Verlag, 1996. Almost Integer A number which is very close to an INTEGER . One surprising example involving both Eand PIis ep/C28p/C3019:999099979 . . . (1) which can also be written as (p/C2720)i/C30/C280:9999999992 /C280:0000388927 i:/C281 (2) cos(ln( p/C2720)):/C280:9999999992 : (3) Applying COSINE a few more times gives cos(pcos(pcos(ln( p/C2720)))) :/C281/C273:9321609261 /C2910/C2835: (4) This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. - Sloane, J. H. Conway, and S. Plouffe, but no satisfy- ing explanation as to "why" it has been true has yetbeen discovered. An interesting near-identity is given by 1 4cos1 10rC16rC1* /C27cosh1 10rC16rC1* /C272cos1 20ffiffiffi 2prC16rC1* cosh1 20ffiffiffi2prC16rC1* hi /C301/C272:480 . . . /C2910/C2813(5) (W. Dubuque). Other remarkable near-identities are given by5(1/C27ffiffiffi 5p )[G3 4rC16rC1* ]2 e5x=6ffiffiffipp /C301/C274:5422 . . . /C2910/C2814(6) where G(z) is the GAMMA FUNCTION (S. Plouffe), e6/C28p4/C28p5/C300:000017673 . . . (7) (D. Wilson), r160 p !1=13 :0:9999996766 ; (8) where r:0:739085 is the root of x/C30cosx(L. A. Broukhis), ln 2/C27log102/C300:994177 . . . (9) (D. Davis), 163 ln 163/C3031:9999983738 . . . (10) (posted to sci.math ; origin unknown), eK5=7/C28gp/C28(2=7/C27g):1:00014678 (11) Kg/C2819=7p2=7/C27g 2f:1:00105 (12) egf(Kp)/C28(2=7/C27g):1:01979 ; (13) where Kis C ATALAN’S CONSTANT ,gis the E ULER- MASCHERONI CONSTANT , and fis the GOLDEN RATIO (D. Barron), and 163(p/C28e)/C3068:999664 . . . (14) 53453 ln 53453/C304910 :00000122 . . . (15) (2/C281)2/C27(52/C281)2 62/C271"# e/C28(2/C271)2/C27(52/C271)2 62/C281"#/C281 /C30613 37e/C2835 991/C3044:99999999993962 . . . (16) (Stoschek). Stoschek also gives an interesting near- identity involving the fine structure constant aand FEIGENBAUM CONSTANT d; (28/C28d/C281)(a/C281/C28137):0:999998 : (17) The near identity 3ffiffiffi 2p (ffiffiffi 5p /C282)/C301:0015516 . . . (18) arises by noting that the stellation ratio 3(ffiffiffi 5p /C282) in the CUMULATION of the DODECAHEDRON to form the GREAT DODECAHEDRON is approximately equal toffiffiffi2p : / A set of almost integers due to D. Hickerson are those OF THE FORM hn/C30n! 2(ln 2)n/C271: (19) for 15n515;as summarized in the following table. n /hn/ 0 0.72135 1 1.04068 2 3.00278 3 12.996294 74.99874 5 541.00152 6 4683.001257 47292.998738 545834.99791 9 7087261.00162 10 102247563.00527 11 1622632572.9975512 28091567594.98157 13 526858348381.00125 14 10641342970443.0845315 230283190977853.0374416 5315654681981354.51308 17 130370767029135900.45799 These numbers are close to integers due to the fact that the quotient is the dominant term in an infiniteseries for the number of possible outcomes of a race between npeople (with ties are allowed). Calling this number f(n);it follows that f(n)/C30X n k/C301n krC1+rC1D f(n/C28k) (20) forn]1;wheren krC0rC1 is a BINOMIAL COEFFICIENT . From this, we obtain the exponential generating function forf X/C12 n/C300f(n) n!zn/C301 2/C28ez; (21) and then by CONTOUR INTEGRATION it can be shown that f(n)/C301 2n!X/C12 k/C30/C28/C121 (ln 2/C272pik)n/C271(22) forn]1;where iis the square root of -1 and the sum is over all integers k(here, the imaginary parts of the terms for kand/C28kcancel each other, so this sum is real.) The k/C300 term dominates, so f(n) is asympto- tic to n!=(2(ln 2)n/C271):In fact, the other terms are quitesmall for nfrom 1 to 15, so f(n) is the nearest integer ton!=(2(ln 2)n/C271) for these values (Hickerson), given by the sequence 1, 3, 13 75, 541, 4683, ... (Sloane’s A034172). A large class of IRRATIONAL "almost integers" can be found using the theory of MODULAR FUNCTIONS , and a few rather spectacular examples are given by Rama- nujan (1913 /C1/4). Such approximations were also studied by Hermite (1859), Kronecker (1863), and Smith (1965). They can be generated using some amazing (and very deep) properties of the J-FUNC- TION . Some of the numbers which are closest approx- imations to INTEGERS areepffiffiffiffiffiffi 163p (sometimes known as the RAMANUJAN CONSTANT and which corresponds to the field Q(ffiffiffiffiffiffiffiffiffiffiffiffiffi /C28163p ) which has CLASS NUMBER 1 and is the IMAGINARY QUADRATIC FIELD of maximal discri- minant), epffiffiffiffi 22p ;epffiffiffiffi 37p ;andepffiffiffiffi 58p ;the last three of which have CLASS NUMBER 2 and are due to Ramanujan (Berndt 1994, Waldschmidt 1988). The properties of the J-FUNCTION also give rise to the spectacular identity ln(6403203/C27744) p"#2 /C30163/C272:32167 . . . /C2910/C2829(23) (Le Lionnais 1983, p. 152).The list below gives numbers OF THE FORM x/C13epffiffinp forn51000 for which [ x]/C28x50:01:/ epffiffi 6p /C302;197:990869543 . . . epffiffiffiffi 17p /C30422;150:997675680 . . . epffiffiffiffi 18p /C30614;551:992885619 . . . epffiffiffiffi 22p /C302;508;951:998257424 . . . epffiffiffiffi 25p /C306;635;623:999341134 . . . epffiffiffiffi 37p /C30199;148;647:999978046551 . . . epffiffiffiffi 43p /C30884;736;743:999777466 . . . epffiffiffiffi 58p /C3024;591;257;751:999999822213 . . . epffiffiffiffi 59p /C3030;197;683;486:993182260 . . . epffiffiffiffi 67p /C30147;197;952;743:999998662454 . . . epffiffiffiffi 74p /C30545;518;122;089:999174678853 . . . epffiffiffiffiffiffi 149p /C3045;116;546;012;289;599:991830287 . . . epffiffiffiffiffiffi 163p /C30262;537;412;640;768;743:999999999999250072 . . . epffiffiffiffiffiffi 177p /C301;418;556;986;635;586;485:996179355 . . . epffiffiffiffiffiffi 232p /C30604;729;957;825;300;084;759:999992171526 . . . epffiffiffiffiffiffi 267p /C3019;683;091;854;079;461;001;445:992737040 . . . epffiffiffiffiffiffi 326p /C304;309;793;301;730;386;363;005;719:996011651 . . . epffiffiffiffiffiffi 386p /C30639;355;180;631;208;421;212;174;016:997669832 . . . epffiffiffiffiffiffi 522p /C3014;871;070;263;238;043;663;567;... . . . 627 ;879;007:999848726 . . . epffiffiffiffiffiffi 566p /C30288;099;755;064;053;264;917;867;... . . . 975 ;825;573:993898311 . . . epffiffiffiffiffiffi 638p /C3028;994;858;898;043;231;996;779;... ...7 7 1 ;804;797;161:992372939 . . . epffiffiffiffiffiffi 719p /C303;842;614;373;539;548;891;490;... ...2 9 4 ;277;805;829;192:999987249 . . . e pffiffiffiffiffiffi 790p /C30223 ; 070 ; 667 ; 213 ; 077 ; 889 ; 794; 379 ; ... ...623 ; 183 ; 838 ; 336 ; 437 :992055117 ... e pffiffiffiffiffiffi 792p /C30249 ; 433 ; 117 ; 287 ; 892 ; 229 ; 255; 125 ; ... ...388 ; 685 ; 911 ; 710 ; 805 :996097323 ... e pffiffiffiffiffiffi 928p /C30365 ; 698 ; 321 ; 891 ; 389 ; 219 ; 219; 142 ; ... ...531 ; 076 ; 638 ; 716 ; 362 ; 775 :998259747 . .. e pffiffiffiffiffiffi 986p /C306; 954 ; 830 ; 200 ; 814 ; 801 ; 770 ; 418 ; 837 ; ... ...940 ; 281 ; 460 ; 320 ; 666 ; 108 :994649611 . .. Gosper noted that the expression 1 /C28262537412640768744 e /C28pffiffiffiffiffiffi 163p /C28196884 e /C282 pffiffiffiffiffiffi 163p /C27103378831900730205293632 e /C283 pffiffiffiffiffiffi 163p : (24) differs from an INTEGER by a mere 10 /C2859: / See also CLASS NUMBER , J-FUNCTION ,P I,P ISOT- VIJAYARAGHAVAN CONSTANT References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 90 /C11, 1994. Cohen, H. In From Number Theory to Physics (Ed. M. Waldschmidt, P. Moussa, J.-M. Luck, and C. Itzyk- son). New York: Springer-Verlag, 1992. Hermite, C. "Sur la the´orie des e´quations modulaires." C. R. Acad. Sci. (Paris) 48, 1079 /C1084 and 1095 /C1102, 1859. Hermite, C. "Sur la the´orie des e´quations modulaires." C. R. Acad. Sci. (Paris) 49,16/C14, 110 /C118, and 141 /C144, 1859. Kronecker, L. "U¨ ber die Klassenzahl der aus Werzeln der Einheit gebildeten komplexen Zahlen." Monatsber. K. Preuss. Akad. Wiss. Berlin , 340 /C145. 1863. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Ramanujan, S. "Modular Equations and Approximations to p:/" Quart. J. Pure Appl. Math. 45, 350 /C172, 1913 /C1914. Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., 1992. Sloane, N. J. A. Sequences A034172 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Smith, H. J. S. Report on the Theory of Numbers. New York: Chelsea, 1965. Stoschek, E. "Modul 33: Algames with Numbers." http:// marvin.sn.schule.de/~inftreff/modul33/task33.htm. Waldschmidt, M. "Some Transcendental Aspects of Rama- nujan’s Work." In Ramanujan Revisited: Proceedings of the Centenary Conference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). New York: Academic Press, pp. 57 /C16, 1988. Waldschmidt, M. In Ramanujan Centennial International Conference (Ed. R. Balakrishnan, K. S. Padmanabhan, and V. Thangaraj). Ramanujan Math. Soc., 1988. Almost Perfect Number A number n for which the DIVISOR FUNCTION satisfies s(n) /C30 2n /C28 1 is called almost perfect. The only known almost perfect numbers are the POWERS of 2, namely 1, 2, 4, 8, 16, 32, ... (Sloane’s A000079). Singh (1997) calls almost perfect numbers SLIGHTLY DEFEC- TIVE. See also QUASIPERFECT NUMBERReferences Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 16 and 45 /C13, 1994. Singh, S. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker, p. 13, 1997. Sloane, N. J. A. Sequences A000079/M1129 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Almost Periodic Function This entry contributed by RONALD M. AARTS A function representable as a generalized Fourier series. Let R be a METRIC SPACE with metric r(x; y): Following Bohr (1947), a CONTINUOUS FUNCTION x(t) for ( /C28/C12B t B/C12 ) with values in R is called an almost periodic function if, for every e > 0 ; there exists l /C30 l( o) > 0 such that every interval [t0 ; t0 /C27 l( o)] contains at least one number t for which r[x(t) ; x(t /C27 t)] B o (/C28/C12B t B/C12): (1) Another formal description can be found in Krasno- sel’skii et al. (1973). Every almost periodic function is bounded and uni- formly continuous on the entire REAL LINE. In addi- tion, the range of an almost period function is compact in R:/ See also FOURIER SERIES ,PERIODIC FUNCTION References Bohr, H. Almost Periodic Functions. New York: Chelsea, 1947. Besicovitch, A. S. Almost Periodic Functions. New York: Dover, 1954. Corduneanu, C. Almost Periodic Functions. New York: Wiley Interscience, 1961. Krasnosel’skii, M. A.; Burd, V. Sh.; and Kolesov, Yu. S. Nonlinear Almost Periodic Oscillations. New York: Wiley, 1973. Levitan, B. M. Almost-Periodic Functions. Moscow, 1953. Almost Prime A number nwith prime factorization n/C30Yr i/C301pai i is called k-almost prime when the sum of the POWERS ar i/C301ai/C30k:The set of k-almost primes is denoted Pk:/ The PRIMES correspond to the "1-almost prime" numbers 2, 3, 5, 7, 11, ... (Sloane’s A000040). The 2- almost prime numbers correspond to SEMIPRIMES 4, 6, 9, 10, 14, 15, 21, 22, ... (Sloane’s A001358). The first few 3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42,44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... (Sloane’s A014612). The first few 4-almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ...(Sloane’s A014613). The first few 5-almost primes are 32, 48, 72, 80, ... (Sloane’s A014614). See also CHEN’S THEOREM ,PRIME NUMBER ,SEMI- PRIME References Sloane, N. J. A. Sequences A000040/M0652, A001358/ M3274, A014612, A014613, and A014614 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Almost Unit An almost unit is a nonunit in the INTEGRAL DOMAIN of FORMAL POWER SERIES with a nonzero first coeffi- cient, P /C30a1x /C27z2x2 /C27...; where a1 "0: Under the operation of composition, the almost units in the INTEGRAL DOMAIN of FORMAL POWER SERIES over a FIELD F form a GROUP (Henrici 1988, p. 45). See also SCHUR- JABOTINSKY THEOREM References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, p. 45, 1988. Alon-Tarsi Conjecture See also LATIN SQUARE References Drisko, A. A. "Proof of the Alon-Tarsi Conjecture for n /C30/ /2rp/." Electronic J. Combinatorics 5, No. 1, R28, 1 /C1/, 1998. http://www.combinatorics.org/Volume_5/ v5i1toc.html. Alpha Alpha is the name for the first letter in the Greek alphabet: a:/ In finance, alpha is a financial measure giving the difference between a fund’s actual return and its expected level of performance, given its level of risk (as measured by BETA ). A POSITIVE alpha indicates that a fund has performed better than expected based on its BETA , whereas a NEGATIVE alpha indicates poorer performance. See also ALPHA FUNCTION ,A LPHA- TEST,A LPHA VALUE ,BETA,SHARPE RATIOAlpha Function an(z) /C13g/C12 1tne /C28zt dt /C30n!z /C28(n /C271)e /C28zXn k /C300zk k!: It is equivalent to an(z) /C30E/C28n(z) ; where En(z) is the EN-FUNCTION . See also BETA EXPONENTIAL FUNCTION , EN-FUNCTION Alpha Value An alpha value is a number 0 5 a 51 such that P(z ] zobserved ) 5 a is considered "SIGNIFICANT ," where P is a P-VALUE . See also CONFIDENCE INTERVAL , P-VALUE ,SIGNIFI- CANCE Alphabet A SET (usually of letters) from which a SUBSET is drawn. A sequence of letters is called a WORD , and a set of WORDS is called a CODE . See also CODE,STRING ,W ORD Alpha-Beta Conjecture MANN’S THEOREM Alphamagic Square AMAGIC SQUARE for which the number of letters in the word for each number generates another MAGIC SQUARE . This definition depends, of course, on the language being used. In English, for example, 52 2 1 8 28 15 2 12 8 25498 11 7 3 65 1 0; where the MAGIC SQUARE on the right corresponds to the number of letters in five twenty -two eighteen twenty -eight fifteen two twelve eight twenty -five References Sallows, L. C. F. "Alphamagic Squares." Abacus 4,28/C1/5, 1986. Sallows, L. C. F. "Alphamagic Squares. 2." Abacus 4,20/C1/9 and 43, 1987. Sallows, L. C. F. "Alpha Magic Squares." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994. Alphametic A CRYPTARITHM in which the letters used to represent distinct DIGITS are derived from related words or meaningful phrases. The term was coined by Hunter in 1955 (Madachy 1979, p. 178). References Brooke, M. One Hundred & Fifty Puzzles in Crypt-Arith- metic. New York: Dover, 1963. Hunter, J. A. H. and Madachy, J. S. "Alphametics and the Like." Ch. 9 in Mathematical Diversions. New York: Dover, pp. 90 /C1/5, 1975. Madachy, J. S. "Alphametics." Ch. 7 in Madachy’s Mathe- matical Recreations. New York: Dover, pp. 178 /C1/00, 1979. Alpha-Test For some constant a0 ; a(f ; z) B a0 implies that z is an APPROXIMATE ZERO of f, where a(f ; z) /C30½f(z) ½ ½f ?(z)½sup k>1f (k)(z) k!f ?(z)rC10rC10rC10rC10rC10rC10rC10rC10rC10rC101 =(k /C281) Smale (1986) found a constant a : 0 :130707 for the test, and this value was subsequently improved to a0 /C303 /C282ffiffiffi 2p :0 :171573 by Wang and Han (1989), and further improved by Wang and Zhao (1995; Petkovic et al. 1997, p. 2). See also APPROXIMATE ZERO,N EWTON’S METHOD , POINT ESTIMATION THEORY References Kim, M. Ph.D. thesis. New York: City University of New York, 1985. Petkovic, M. S.; Herceg, D. D.; and Ilic, S. M. Point Estima- tion Theory and Its Applications. Novi Sad, Yugoslavia: Institute of Mathematics, 1997. Smale, S. "Newton’s Method Estimates from Data at One Point." In The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Ed. R. E. Ewing, K. I. Gross, and C. F. Martin). New York: Springer-Verlag, pp. 185 /C1/96, 1986. Wang, X. and Han, D. "On Dominating Sequence Method in the Point Estimate and Smale’s Theorem." Scientia Sinica Ser. A, 905 /C1/13, 1989. Wang, D. and Zhao, F. "The Theory of Smale’s Point Estimation and Its Application." J. Comput. Appl. Math. 60, 253 /C1/69, 1995.Alternating Algebra EXTERIOR ALGEBRA Alternating Group A PERMUTATION GROUP of an even number of permu- tations on a set of length n, denoted Anor Alt(n) (Scott 1987, p. 267). An alternating group is a NORMAL SUBGROUP of the PERMUTATION GROUP , and has ORDER n!=2;the first few values of which for n/C302, 3, ... are 1, 3, 12, 60, 360, 2520, ... (Sloane’s A001710). Alternating groups are FINITE analogs of the families of simple L IE GROUPS . Alternating groups with n]5 are non-A BELIAN SIM- PLE GROUPS (Scott 1987, p. 295). The number of conjugacy classes in the alternating groups Anfor n/C302, 3, ... are 1, 3, 4, 5, 7, 9, ... (Sloane’s A000702). See also 15 PUZZLE ,FINITE GROUP ,GROUP ,JORDAN’S SYMMETRIC GROUP THEOREM ,LIE GROUP ,PERMUTA- TION GROUP ,SIMPLE GROUP ,SYMMETRIC GROUP References Scott, W. R. Group Theory. New York: Dover, pp. 267 and 295, 1987. Sloane, N. J. A. Sequences A000702/M2307 and A001710/ M2933 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#alt. Alternating Knot An alternating knot is a KNOT which possesses a knot diagram in which crossings alternate between under- and overpasses. Not all knot diagrams of alternating knots need be alternating diagrams. The TREFOIL KNOT and FIGURE-OF-EIGHT KNOT are alternating knots. The number of PRIME alternating and nonalternating knots of ncrossings are summar- ized in the following table. type Sloane counts alternating A002864 0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, 1288, 4878, 19536, 85263, 379799, ... nonalternating A051763 0, 0, 0, 0, 0, 0, 0, 3, 8, 42, 185, 888, 5110, 27436, 168030,1008906, ... The 3 nonalternating knots of eight crossings are 08/C1/ 19,08/C1/20, and 08/C1/21, illustrated below (Wells 1991). One of TAIT’S KNOT CONJECTURES states that the number of crossings is the same for any diagram of a reduced alternating knot. Furthermore, a reduced alternating projection of a knot has the least number of crossings for any projection of that knot. Both of these facts were proved true by Kauffman (1988), Thistlethwaite (1987), and Murasugi (1987). FLYPE moves are sufficient to pass between all minimal diagrams of a given alternating knot (Hoste et al. 1998). If K has a reduced alternating projection of n cross- ings, then the SPAN of K is An: Let c(K) be the CROSSING NUMBER . Then an alternating knot K1#K2 (a KNOT SUM) satisfies c(K1#K2) /C30c(K1) /C27c(K2) : In fact, this is true as well for the larger class of ADEQUATE KNOTS and postulated for all KNOTS . It is conjectured that the proportion of knots which are alternating tends exponentially to zero with increasing crossing number (Hoste et al. 1998), a statement which has been proved true for alternating links. See also ADEQUATE KNOT,A LMOST ALTERNATING LINK,A LTERNATING LINK,F LYPING CONJECTURE , TAIT’S KNOT CONJECTURES References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 159 /C1/64, 1994. Arnold, B.; Au, M.; Candy, C.; Erdener, K.; Fan, J.; Flynn, R.; Muir, J.; Wu, D.; and Hoste, J. "Tabulating Alternating Knots through 14 Crossings." ftp://chs.cusd.claremon- t.edu/pub/knot/paper.TeX.txt. Arnold, B.; Au, M.; Candy, C.; Erdener, K.; Fan, J.; Flynn, R.; Muir, J.; Wu, D.; and Hoste, J. ftp://chs.cusd.clare- mont.edu/pub/knot/AltKnots/. Erdener, K. and Flynn, R. "Rolfsen’s Table of all Alternating Diagrams through 9 Crossings." ftp://chs.cusd.claremon- t.edu/pub/knot/Rolfsen_table.final. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Kauffman, L. "New Invariants in the Theory of Knots." Amer. Math. Monthly 95, 195 /C1/42, 1988. Little, C. N. "Non Alternate 9 Knots of Orders Eight and Nine." Trans. Roy. Soc. Edinburgh 35, 663 /C1/64, 1889. Little, C. N. "Alternate 9 Knots of Order 11." Trans. Roy. Soc. Edinburgh 36, 253 /C1/55, 1890. Little, C. N. "Non-Alternate 9 Knots." Trans. Roy. Soc. Edinburgh 39, 771 /C1/78, 1900. Murasugi, K. "Jones Polynomials and Classical Conjectures in Knot Theory." Topology 26, 297 /C1/07, 1987. Sloane, N. J. A. Sequences A002864/M0847 and A051763 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html.Thistlethwaite, M. "A Spanning Tree Expansion for the Jones Polynomial." Topology 26, 297 /C1/09, 1987. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 160, 1991. Alternating Knot Diagram A KNOT DIAGRAM which has alternating under- and overcrossings as the KNOT projection is traversed. The first KNOT which does not have an alternating diagram has 8 crossings. Alternating Link A LINK which has a LINK DIAGRAM with alternating underpasses and overpasses. The proportion of links which are alternating tends exponentially to zero with increasing crossing num- ber (Sundberg and Thistlethwaite 1998, Thistle- thwaite 1998). See also ALMOST ALTERNATING LINK,ALTERNATING KNOT References Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Menasco, W. and Thistlethwaite, M. "The Classification of Alternating Links." Ann. Math. 138, 113 /C1/71, 1993. Sundberg, C. and Thistlethwaite, M. "The Rate of Growth of the Number of Prime Alternating Links and Tangles." Pacific J. Math. 182, 329 /C1/58, 1998. Thistlethwaite, M. "On the Structure and Scarcity of Alter- nating Links and Tangles." J. Knot Th. Ramifications 7, 981 /C1/004, 1998. Alternating Multilinear Form An alternating multilinear form on a REAL VECTOR SPACE V is a MULTILINEAR FORM F : V /C156/C1/C1/C1/C156V 0 R (1) such that F(x1 ; ...; xi ; xi /C271 ; ...; xn) /C30/C28F(x1 ; ...; xi/C271 ; xi ; ...; xn) (2) for any index i. For example, F((a1 ; a2 ; a3) ; (b1 ; b2 ; b3); (c1 ; c2 ; c3)) /C30a1b2c3 /C28a1b3c2 /C27a2b3c1 /C28a2b1c3 /C27a3b1c2 /C28a3b2c1 (3) is an alternating form on R3:/ An alternating multilinear form is defined on a MODULE in a similar way, by replacing Rwith the RING . See also DUAL SPACE ,EXTERIOR ALGEBRA ,M ODULE , MULTILINEAR FORM,VECTOR SPACE Alternating Permutation An arrangement of the elements c1 ; ..., cn such that no element cihas a magnitude between ci/C281and ci/C271is called an alternating (or ZIGZAG ) permutation. The determination of the number of alternating permuta- tions for the set of the first n INTEGERS f1; 2; ...; ng is known as ANDRE ´ ’S PROBLEM . An example of an alternating permutation is (1, 3, 2, 5, 4). As many alternating permutations among n elements begin by rising as by falling. The magnitude of the cn/s does not matter; only the number of them. Let the number of alternating permutations be given by Zn /C30 2An : This quantity can then be computed from 2nan /C30X aras ; (1) where r and s pass through all INTEGRAL numbers such that r /C27s /C30n /C281 ; (2) /a0 /C30a1 /C301; and An /C30n!an : (3) The numbers Anare sometimes called the EULER ZIGZAG NUMBERS , and the first few are given by 1, 1, 1, 2, 5, 16, 61, 272, ... (Sloane’s A000111). The EVEN - numbered An/s are called EULER NUMBERS , SECANT NUMBERS ,or ZIG NUMBERS , and the ODD-numbered ones are sometimes called TANGENT NUMBERS or ZAG NUMBERS . Curiously enough, the SECANT and TANGENT MA- CLAURIN SERIES can be written in terms of the An/sas sec x /C30A0 /C27A2x2 2! /C27A4x4 4! /C27... (4) tan x /C30A1x /C27A3x3 3! /C27A5x5 5! /C27...; (5) or combining them, sec x /C27tan x /C30A0 /C27A1x /C27A2x2 2! /C27A3x3 3! /C27A4x4 4! /C27A5x5 5! /C27...: (6) See also ENTRINGER NUMBER ,EULER NUMBER ,EULER ZIGZAG NUMBER ,SECANT NUMBER ,SEIDEL- ENTRIN- GER-ARNOLD TRIANGLE ,TANGENT NUMBER References Andre ´, D. "Developments de sec x et tan x:/" C. R. Acad. Sci. Paris 88, 965 /C1/67, 1879. Andre ´, D. "Memoire sur les permutations alterne ´es." J. Math. 7, 167 /C1/84, 1881. Arnold, V. I. "Bernoulli-Euler Updown Numbers Associated with Function Singularities, Their Combinatorics and Arithmetics." Duke Math. J. 63, 537 /C1/55, 1991.Arnold, V. I. "Snake Calculus and Combinatorics of Ber- noulli, Euler, and Springer Numbers for Coxeter Groups." Russian Math. Surveys 47,3/C1/5, 1992. Bauslaugh, B. and Ruskey, F. "Generating Alternating Permutations Lexicographically." BIT 30,17/C1/6, 1990. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 110 /C1/11, 1996. Do¨rrie, H. "Andre ´’s Deviation of the Secant and Tangent Series." §16 in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 64 /C1/9, 1965. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 69 /C1/5, 1985. Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663 /C1/ 88, 1967. Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Operation on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76,44/C1/4, 1996. Ruskey, F. "Information of Alternating Permutations." http://www.theory.csc.uvic.ca/~cos/inf/perm/Alterna- ting.html. Sloane, N. J. A. Sequences A000111/M1492 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Alternating Representation See also REPRESENTATION Alternating Series A SERIES OF THE FORM X/C12 k /C301(/C281)k /C271ak (1) or X/C12 k /C301(/C281)kak : (2) Rather surprisingly, the alternating series X/C12 k/C301( /C281)k /C281 k/C30ln 2 (3) converges to the natural logarithm of 2. See also SERIES References Arfken, G. "Alternating Series." §5.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 293 /C1/94, 1985. Bromwich, T. J. I’a. and MacRobert, T. M. "Alternating Series." §19 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 55 /C1/7, 1991. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 170, 1984. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.New York: Hyperion, p. 218, 1998. Pinsky, M. A. "Averaging an Alternating Series." Math. Mag. 51, 235/C1 /37, 1978. Alternating Series Test Also known as the LEIBNIZ CRITERION .An ALTERNAT- ING SERIES CONVERGES if a1 ]a2 ]...and lim k 0/C12ak /C300: See also CONVERGENCE TESTS Alternating Sign Matrix A MATRIX of 0s, 1s, and -1s in which the entries in each row or column sum to 1 and the nonzero entries in each row and column alternate in sign. The number of n /C29n alternating sign matrices for n /C301, 2, ... are 1, 2, 21, 1344, 628080, ...(Sloane’s A050204), illustrated below: A?1 /C30[1] (1) A ?2 /C30 10 01rC00rC01 ;0110rC00rC01 (2) A? 3 /C30/C28111 1 /C2811 11 /C2812 435;/C28111 100 1002 435;/C28111 11 /C281 1 /C28112 435 00 1 00 1 11 /C2812435;001 010 1002 435;001 100 0102 435; ...: (3) If the additional restriction is added that any -1s in a row or column must have a /C271 "outside" it (i.e., all -1s are "bordered" by /C271 /s), then the number of these "Robins and Rumsey" n /C29n alternating sign matrices Anare given by 1, 2, 7, 42, 429, 7436, 218348, ... (Sloane’s A005130). The single A1and two A2/s are identical to A?1 and A?2 ; but only seven of the 21 A?3/s are A3/s: A3 /C30001 0101002 435;001 1000102 435;010 0011002 435;010 1 /C2811 0102 435; (4) 010 1000012 435;100 0010102 435;100 0100012 435 (5) The conjecture that the number A n of An is explicitly given by the formula AnYn/C281 j /C300(3j /C27 1)! (n /C27 j)!; (6) now proven to be true, was known as the ALTERNAT- ING SIGN MATRIX CONJECTURE . Let A(n; k) be the number of n /C29n alternating sign matrices with one in the top row occurring in the kth position. ThenAn /C30Xn k /C301A(n ; k): (7) The result A(n; k /C27 1) A(n ; k)/C30(n /C28 k)(n /C27 k /C28 1) k(2n /C28 k /C28 1) (8) for 0 Bk Bn implies (7) (Mills et al. 1983). Making a triangular array of the number of A?n with a 1 at the top of column k gives 1 11 232 71 41 47 42 105 135 105 42 (Sloane’s A048601), and taking the ratios of adjacent terms gives the array 2=2 2=33 =2 2=45 =54 =2 2=57 =99 =75 =2 (Sloane’s A029656 and A029638). The fact that these numerators and denominators are respectively thenumbers in the (2, 1)- and (1, 2)-Pascal triangleswhich are different from 1 is known as the REFINED ALTERNATING SIGN MATRIX CONJECTURE . See also ALTERNATING SIGN MATRIX CONJECTURE , CONDENSATION ,DESCENDING PLANE PARTITION ,IN- TEGER MATRIX ,PERMUTATION MATRIX References Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math. 53, 193/C1/25, 1979. Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999. Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637/C1/46. Kuperberg, G. "Another Proof of the Alternating-Sign Matrix Conjecture." Internat. Math. Res. Notes , No. 3, 139/C1/50, 1996. Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr. "Proof of the Macdonald Conjecture." Invent. Math. 66,7 3/C1/7, 1982. Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr. "Alternat- ing Sign Matrices and Descending Plane Partitions." J. Combin. Th. Ser. A 34, 340/C1/59, 1983. Robbins, D. P. "The Story of 1, 2, 7, 42, 429, 7436, ...." Math. Intell. 13,1 2/C1/9, 1991. Robbins, D. P. and Rumsey, H. Jr. "Determinants and Alternating Sign Matrices." Adv. Math. 62, 169/C1/84, 1986. Sloane, N. J. A. Sequences A005130/M1808, A029638, A029656, A048601, and A050204 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Stanley, R. P. "A Baker’s Dozen of Conjectures Concerning Plane Partitions." In Combinatoire E´ nume ´rative. Proceed- ings of the colloquium held at the Universite ´ du Que´bec, Montreal, May 28-June 1, 1985 (Ed. G. Labelle and P. Leroux). New York: Springer-Verlag, pp. 285 /C1/93, 1986. Zeilberger, D. "Proof of the Alternating Sign Matrix Con- jecture." Electronic J. Combinatorics 3, No. 2, R13, 1 /C1/4, 1996. http://www.combinatorics.org/Volume_3/volu- me3_2.html. Zeilberger, D. "Proof of the Refined Alternating Sign Matrix Conjecture." New York J. Math. 2,59/C1/8, 1996. Zeilberger, D. "A Constant Term Identity Featuring the Ubiquitous (and Mysterious) Andrews-Mills-Robbins- Rumsey numbers 1, 2, 7, 42, 429, ...." J. Combin. Theory A 66,17/C1/7, 1994. Alternating Sign Matrix Conjecture The conjecture that the number of ALTERNATING SIGN MATRICES "bordered" by /C271/s Anis explicitly given by the formula An /C30Yn/C281 j/C300(3j /C27 1)! (n /C27 j)!: This conjecture was proved by Doron Zeilberger in 1995 (Zeilberger 1996a). This proof enlisted the aid of an army of 88 referees together with extensive computer calculations. A beautiful, shorter proof was given later that year by Kuperberg (Kuperberg 1996), and the REFINED ALTERNATING SIGN MATRIX CONJECTURE was subsequently proved by Zeilberger (Zeilberger 1996b) using Kuperberg’s method to- gether with techniques from q-calculus and orthogo- nal polynomials. See also ALTERNATING SIGN MATRIX ,REFINED ALTER- NATING SIGN MATRIX CONJECTURE References Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999. Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/46. Kuperberg, G. "Another Proof of the Alternating-Sign Matrix Conjecture." Internat. Math. Res. Notes , No. 3, 139 /C1/50, 1996. Zeilberger, D. "A Constant Term Identity Featuring the Ubiquitous (and Mysterious) Andrews- Mills-Robbins-Rumsey numbers 1, 2, 7, 42, 429, ...." J. Combin. Theory A 66,17/C1/7, 1994. Zeilberger, D. "Proof of the Alternating Sign Matrix Con- jecture." Electronic J. Combinatorics 3, No. 2, R13, 1 /C1/4, 1996a. http://www.combinatorics.org/Volume_3/volu- me3_2.html. Zeilberger, D. "Proof of the Refined Alternating Sign Matrix Conjecture." New York J. Math. 2,59/C1/8, 1996b. Alternating Tensor ANTISYMMETRIC TENSORAlternative Algebra Let A denote an R/-ALGEBRA , so that A is a VECTOR SPACE over R and A /C29A 0 A (1) (x; y) /C2x /C215 y : (2) Then A is said to be alternative if, for all x; y /C23 A (x /C215 y) /C215 y /C30x /C215 (y /C215 y) (3) (x /C215 x) /C215 y /C30x /C215 (x /C215 y): (4) Here, VECTOR MULTIPLICATION x /C215 y is assumed to be BILINEAR . The ASSOCIATOR (x; y; z) is an alternating function, and the SUBALGEBRA generated by two elements is associative. See also ASSOCIATOR References Finch, S. "Zero Structures in Real Algebras." http:// www.mathsoft.com/asolve/zerodiv/zerodiv.html. Schafer, R. D. An Introduction to Non-Associative Algebras. New York: Dover, p. 5, 1995. Alternative Denial The term used in PROPOSITIONAL CALCULUS for the NAND CONNECTIVE . The notation A½B is used for this connective, a most unfortunate choice in light of modern usage of A½B or A½½B to denote OR. See also JOINT DENIAL , NAND References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 26, 1997. Alternative Link A category of LINK encompassing both ALTERNATING KNOTS and TORUS KNOTS . See also ALTERNATING KNOT,LINK,TORUS KNOT References Kauffman, L. "Combinatorics and Knot Theory." Contemp. Math. 20, 181/C1/00, 1983. Altitude The altitudes of a TRIANGLE are the CEVIANS AiHi which are PERPENDICULAR to the LEGS AjAkopposite Ai : The three altitudes of any TRIANGLE are CONCUR- RENT at the ORTHOCENTER H (Durell 1928). This fundamental fact did not appear anywhere in Euclid’s ELEMENTS . The altitudes have lengths hi /C13AiHi given by hi /C30ai/C271 sin ai /C272 /C30ai/C272 sin ai/C271 (1) h1 /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s /C28 a1)(s /C28 a2)(s /C28 a3)p a1; (2) where s is the SEMIPERIMETER and ai /C13AjAk : Another pair of interesting FORMULAS are sh /C30D R (3) where D is the AREA of the TRIANGLE DA1A2A3 and sh is the SEMIPERIMETER of the ALTITUDE TRIANGLE DH1H2H3 ; and h1h2h3 /C302sh D/C302D2 R; (4) where R is the CIRCUMRADIUS of DA1A2A3(Johnson 1929, p. 191). Other formulas satisfied by the altitude include 1 h1/C271 h2/C271 h3/C301 r (5) 1 r1/C301 h2/C271 h3/C271 h1(6) 1 r2/C271 r3/C301 r /C281 r1/C302 h1; (7) where r is the INRADIUS and riare the EXRADII (Johnson 1929, p. 189). In addition, HA1/C215 HH1 /C30HA2/C215 HH2 /C30HA3/C215 HH3 (8) HA1 /C215 HH1 /C301 2a2 1 /C27a22 /C27a23rC0rC1 /C284R2 ; (9)where R is the CIRCUMRADIUS . The points A1 ; A3 ; H1 ; and H3 (and their permutations with respect to indices) all lie on a CIRCLE , as do the points A3 ; H3 ; H, and H1(and their permutations with respect to indices). TRIANGLES DA1A2A3and DA1H2H3 are inversely similar. The triangle H1H2H3 has the minimum PERIMETER of any TRIANGLE inscribed in a given ACUTE TRIANGLE (Johnson 1929, pp. 161 /C1/65). Additional properties involving the FEET of the altitudes are given by Johnson (1929, pp. 261 /C1/62). The line joining the feet to two altitudes of a triangle is ANTIPARALLEL to the third side (Johnson 1929, p. 172). See also CEVIAN ,FOOT,M ALTITUDE ,ORTHOCENTER , PERPENDICULAR ,PERPENDICULAR FOOT,TAYLOR CIR- CLE References Coxeter, H. S. M. and Greitzer, S. L. "More on the Altitude and Orthocentric Triangle." §2.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 36 /C1/0, 1967. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 20, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Altitude Plane The plane through an edge of a TRIHEDRAL ANGLE drawn perpendicularly to the opposite face. The term was first used by J. Neuberg (Altshiller-Court 1979, p. 298). References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 27, 1979. Altitude Triangle The TRIANGLE DH1H2H3formed by connecting the three feet H1 ; H2 ; and H3of the altitudes of a given triangle DA1A2A3 :/ See also ALTITUDE Alysoid CATENARY Ambient Isotopy An ambient isotopy from an embedding of a MANI- FOLD M in N to another is a HOMOTOPY of self DIFFEOMORPHISMS (or ISOMORPHISMS , or piecewise- linear transformations, etc.) of N, starting at the IDENTITY MAP, such that the "last" DIFFEOMORPHISM compounded with the first embedding of M is the second embedding of M. In other words, an ambient isotopy is like an ISOTOPY except that instead of distorting the embedding, the whole ambient SPACE is being stretched and distorted and the embedding is just "coming along for the ride." For SMOOTH MANI- FOLDS ,aMAP is ISOTOPIC IFF it is ambiently isotopic. For KNOTS , the equivalence of MANIFOLDS under continuous deformation is independent of the embed- ding SPACE .KNOTS of opposite CHIRALITY have ambi- ent isotopy, but not REGULAR ISOTOPY . See also ISOTOPY ,REGULAR ISOTOPY References Hirsch, M. W. Differential Topology. New York: Springer- Verlag, 1988. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Ambiguous An expression is said to be ambiguous (or poorly defined) if its definition does not assign it a unique interpretation or value. An expression which is not ambiguous is said to be WELL DEFINED . See also ILL DEFINED ,W ELL DEFINEDAmbiguous Rectangle FAULT- FREE RECTANGLE Ambrose-Kakutani Theorem For every ergodic FLOW on a nonatomic PROBABILITY SPACE , there is a MEASURABLE SET intersecting almost every orbit in a discrete set. Amenable Number A number n which can be built up from INTEGERS a1 ; a2 ; ..., ak by either ADDITION or MULTIPLICATION such that Xk i/C301ai /C30Yk i/C301ai /C30n: The numbers fa1 ; ...; an g in the SUM are simply a PARTITION of n. The first few amenable numbers are 2 /C272 /C302 /C292 /C304 1 /C272 /C273 /C301 /C292 /C293 /C306 1 /C271 /C272 /C274 /C301 /C291 /C292 /C294 /C308 1 /C271 /C272 /C272 /C272 /C301 /C291 /C292 /C292 /C292 /C308: In fact, all COMPOSITE NUMBERS are amenable. See also COMPOSITE NUMBER ,PARTITION ,SUM References Tamvakis, H. "Problem 10454." Amer. Math. Monthly 102, 463, 1995. Amicable Numbers AMICABLE PAIR,A MICABLE QUADRUPLE ,A MICABLE TRIPLE ,M ULTIAMICABLE NUMBERS ,RATIONAL AMIC- ABLE PAIR Amicable Pair An amicable pair ( m, n ) consists of two INTEGERS m, n for which the sum of PROPER DIVISORS (the DIVISORS excluding the number itself) of one number equals the other. Amicable pairs are occasionally called FRIENDLY PAIRS (Hoffman 1998, p. 45), although this nomenclature is to be discouraged since the numbers more commonly known as FRIENDLY PAIRS are defined by a different, albeit related, criterion. Symbolically,amicable pairs satisfy s(m)/C30n (1) s(n)/C30m; (2) where s(n)/C13s(n)/C28n (3) is the RESTRICTED DIVISOR FUNCTION . Equivalently, an amicable pair ( m, n ) satisfies s(m)/C30s(n)/C30s(m)/C27s(n)/C30m/C27n: (4) where s(n) is the DIVISOR FUNCTION . The smallest amicable pair is (220, 284) which has factorizations 220/C3011 /C2155/C21522(5) 284/C3071 /C21522(6) giving RESTRICTED DIVISOR FUNCTIONS s(220)/C30X f1;2;4;5;10;11;20;22;44;55;110g /C30284 (7) s(284)/C30X f1;2;4;71;142g/C30220: (8) The quantity s(m)/C30s(n)/C30s(m)/C27s(n); (9) in this case, 220 /C27284/C30504, is called the PAIR SUM . The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744,10856), (12285, 14595), (17296, 18416), (63020,76084), ... (Sloane’s A002025 and A002046). An exhaustive tabulation is maintained by D. Moews. In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056), althoughthese results were actually rediscoveries of numbers known to Arab mathematicians. By 1747, Euler had found 30 pairs, a number which he later extended to60. In 1866, 16-year old B. Nicolo `I. Paganini found the small amicable pair (1184, 1210) which hadeluded his more illustrious predecessors (Paganini1866/C1867; Dickson 1952, p. 47). There were 390 known amicable pairs as of 1946 (Escott 1946). Thereare a total of 236 amicable pairs below 10 8(Cohen 1970), 1427 below 1010(te Riele 1986), 3340 less than 1011(Moews and Moews 1993), 4316 less than 2 :01/C29 1011(Moews and Moews), and 5001 less than /:3:06/C291011(Moews and Moews). Rules for producing amicable pairs include theT HAˆBIT IBN KURRAH RULE rediscovered by Fermat and Descartes and extended by Euler to E ULER’S RULE . A further extension not previously noticed was discovered by Borho (1972). Pomerance (1981) has proved that [amicable numbers 5n]Bne/C28[ln(n)]1=2(10) for large enough n(Guy 1994). No nonfinite lower bound has been proven.Let an amicable pair be denoted ( m, n ), and take mB n.(m, n ) is called a regular amicable pair of type ( i, j) if (m;n)/C30(gM;gN); (11) where /g/C13GCD( m;n)/is the GREATEST COMMON DIVISOR , GCD( g;M)/C30GCD( g;N)/C301; (12) MandNare SQUAREFREE , then the number of PRIME FACTORS ofMandNareiandj. Pairs which are not regular are called irregular or exotic (te Riele 1986). There are no regular pairs of type (1 ;j) for j]1:If m/C130 (mod 6) andn/C30s(m)/C28m (13) isEVEN , then ( m, n ) cannot be an amicable pair (Lee 1969). The minimal and maximal values of m=nfound by te Riele (1986) were 938304290 =1344480478 /C300:697893577 . . . (14) and 4000783984 =4001351168 /C300:9998582518 . . . (15) te Riele (1986) also found 37 pairs of amicable pairshaving the same PAIR SUM . The first such pair is (609928, 686072) and (643336, 652664), which hasthe PAIR SUM s(m)/C30s(n)/C30m/C27n/C301;296;000: (16) te Riele (1986) found no amicable n-tuples having the same PAIR SUM forn/C212. However, Moews and Moews found a triple in 1993, and te Riele found a quadruplein 1995. In November 1997, a quintuple and sextuple were discovered. The sextuple is (1953433861918, 2216492794082), (1968039941816, 2201886714184),(1981957651366, 2187969004634), (1993501042130,2176425613870), (2046897812505, 2123028843495), (2068113162038, 2101813493962), all having PAIR SUM 4169926656000. Amazingly, the sextuple is smaller than any known quadruple or quintuple, and is likely smaller than any quintuple. The earliest known odd amicable numbers all were divisible by 3. This led Bratley and McKay(1968) to conjecture that there are no amicable pairs coprime to 6 (Guy 1994, p. 56). However, Battiato and Borho (1988) found a counter-example, and now many amicable pairs are knownwhich are not divisible by 6 (Pedersen). The smallest known example of this kind is the amic- able pair (42262694537514864075544955198125,42405817271188606697466971841875), each number of which has 32 digits. A search was then begun for amicable pairs coprime to 30. The first example was found by Y. Kohmoto in1997, consisting of a pair of numbers each having 193 digits (Pedersen). Kohmoto subsequently found two other examples, and te Riele and Pedersen used twoof Kohmoto’s examples to calculated 243 type- /(3;2) pairs coprime to 30 by means of a method whichgenerates type- /(3;2) pairs from a type- /(2;1) pairs. No amicable pairs which are coprime to 2 /C2153 /C2155 /C215 7/C30210 are currently known. On October 4, 1997, Mariano Garcia found the largest known amicable pair, each of whose members has 4829 DIGITS . The new pair is N1/C30CM[(P/C27Q)P89/C281] (17) N2/C30CQ[(P/C28M)P89/C281]; (18) where C/C30211P89(19) M /C30 287155430510003638403359267 (20) P /C30 574451143340278962374313859 (21) Q /C30 136272576607912041393307632916794623 : (22) P, Q,(P /C27 Q)P89 /C281; and (P /C28M)P89 /C281 are PRIME . See also AMICABLE QUADRUPLE ,AMICABLE TRIPLE , AUGMENTED AMICABLE PAIR,BREEDER ,CROWD ,EU- LER’S RULE,FRIENDLY PAIR,M ULTIAMICABLE NUM- BERS ,P AIR SUM,Q UASIAMICABLE PAIR,R ATIONAL AMICABLE PAIR,SOCIABLE NUMBERS ,SUPER UNITARY AMICABLE PAIR,THAˆ BIT IBN KURRAH RULE,UNITARY AMICABLE PAIR References Alanen, J.; Ore, Ø.; and Stemple, J. "Systematic Computa- tions on Amicable Numbers." Math. Comput. 21, 242/C1/45, 1967. Battiato, S. and Borho, W. "Are there Odd Amicable Numbers not Divisible by Three?" Math. Comput. 50, 633/C1/37, 1988. Borho, W. "On Thabit ibn Kurrah’s Formula for Amicable Numbers." Math. Comput. 26, 571/C1/78, 1972. Borho, W. "Some Large Primes and Amicable Numbers." Math. Comput. 36, 303/C1/04, 1981. Borho, W. "Befreundete Zahlen: Ein zweitausend Jahre altes Thema der elementaren Zahlentheorie." In Mathema- tische Miniaturen 1: Lebendige Zahlen: Fu ¨nf Exkursionen. Basel, Switzerland, Birkha ¨user, pp. 5 /C1/8, 1981. Borho, W. and Hoffmann, H. "Breeding Amicable Numbers in Abundance." Math. Comput. 46, 281/C1/93, 1986. Bratley, P.; Lunnon, F.; and McKay, J. "Amicable Numbers and Their Distribution." Math. Comput. 24, 431/C1/32, 1970. Bratley, P. and McKay, J. "More Amicable Numbers." Math. Comput. 22, 677/C1/78, 1968. Cohen, H. "On Amicable and Sociable Numbers." Math. Comput. 24, 423/C1/29, 1970. Costello, P. "Amicable Pairs of Euler’s First Form." J. Rec. Math. 10, 183/C1/89, 1977 /C1/978. Costello, P. "Amicable Pairs of the Form ( i;1):/"Math. Comput. 56, 859/C1/65, 1991. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 38 /C1/0, 1952. Erdos, P. "On Amicable Numbers." Publ. Math. Debrecen 4, 108/C1/11, 1955 /C1/956. Erdos, P. "On Asymptotic Properties of Aliquot Sequences." Math. Comput. 30, 641/C1/45, 1976. Escott, E. B. E. "Amicable Numbers." Scripta Math. 12,6 1/C1/ 2, 1946. Garcı ´a, M. "New Amicable Pairs." Scripta Math. 23, 167/C1/71, 1957. Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math- ematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind fromScientific American. New York: Vintage, pp. 160 /C1 /71, 1978. Guy, R. K. "Amicable Numbers." §B4 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 55 /C1/9, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.New York: Hyperion, 1998. Lee, E. J. "Amicable Numbers and the Bilinear Diophantine Equation." Math. Comput. 22, 181/C1 /97, 1968.Lee, E. J. "On Divisibility of the Sums of Even Amicable Pairs." Math. Comput. 23, 545/C1/48, 1969. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, I." J. Rec. Math. 5,7 7/C1/3, 1972. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, II." J. Rec. Math. 5, 153/C1/73, 1972. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, III." J. Rec. Math. 5, 231/C1/49, 1972. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 145 and 155 /C1/56, 1979. Moews, D. and Moews, P. C. "A Search for Aliquot Cycles and Amicable Pairs." Math. Comput. 61, 935/C1/38, 1993. Moews, D. and Moews, P. C. "A List of Amicable Pairs Below 2:01/C291011:/" Rev. Jan. 8, 1993. http://xraysgi.ims.ucon- n.edu:8080/amicable.txt. Moews, D. and Moews, P. C. "A List of the First 5001 Amicable Pairs." Rev. Jan. 7, 1996. http://xraysgi.ims.u-conn.edu:8080/amicable2.txt. Ore, Ø.Number Theory and Its History. New York: Dover, pp. 96 /C100, 1988. Paganini, B. N. I. Atti della R. Accad. Sc. Torino 2, 362, 1866/C1867. Pedersen, J. M. "Known Amicable Pairs." http://www.vej- lehs.dk/staff/jmp/aliquot/knwnap.htm. Pedersen, J. M. "Various Amicable Pair Lists and Statis- tics." http://www.vejlehs.dk/staff/jmp/aliquot/apstat.htm. Pomerance, C. "On the Distribution of Amicable Numbers." J. reine angew. Math. 293/294 , 217/C122, 1977. Pomerance, C. "On the Distribution of Amicable Numbers, II."J. reine angew. Math. 325, 182/C188, 1981. Root, S. Item 61 in Beeler, M.; Gosper, R. W.; and Schroep- pel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelli- gence Laboratory, Memo AIM-239, p. 23, Feb. 1972. Sloane, N. J. A. Sequences A002025/M5414 and A002046/ M5435 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Souissi, M. Un Texte Manuscrit d’Ibn Al-Banna’ Al-Marra- kusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975. Speciner, M. Item 62 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 24, Feb. 1972. te Riele, H. J. J. "Four Large Amicable Pairs." Math. Comput. 28, 309/C112, 1974. te Riele, H. J. J. "On Generating New Amicable Pairs from Given Amicable Pairs." Math. Comput. 42, 219/C123, 1984. te Riele, H. J. J. "Computation of All the Amicable Pairs Below 10 10."Math. Comput. 47, 361/C1/68 and S9-S35, 1986. te Riele, H. J. J.; Borho, W.; Battiato, S.; Hoffmann, H.; and Lee, E. J. "Table of Amicable Pairs Between 1010and 1052." Centrum voor Wiskunde en Informatica, Note NM- N8603. Amsterdam: Stichting Math. Centrum, 1986. te Riele, H. J. J. "A New Method for Finding Amicable Pairs." In Mathematics of Computation 1943 /C1/993: A Half-Century of Computational Mathematics (Vancouver,BC, August 9 /C1 /3, 1993) (Ed. W. Gautschi). Providence, RI: Amer. Math. Soc., pp. 577 /C1/81, 1994. Weisstein, E. W. "Sociable and Amicable Numbers." M ATH- EMATICA NOTEBOOK SOCIABLE.M . Amicable Quadruple An amicable quadruple as a QUADRUPLE (a;b;c;d) such that s(a)/C30s(b)/C30s(c)/C30s(d)/C30a/C27b/C27c/C27d (1) where s(n) is the DIVISOR FUNCTION . If (a, b) and (x, y) are amicable pairs and GCD (a; x) /C30GCD (a; y) /C30GCD (b; x) /C30GCD (a; y) /C301 ; (2) then (ax; ay ; bx ; by) is an amicable quadruple. This follows from the identity s(ax) /C30 s(a)s(x) /C30(a /C27b)(x /C27y) /C30ax /C27ay /C27bx /C27by : (3) The smallest known amicable quadruple is (842448600, 936343800, 999426600, 1110817800). Large amicable quadruples can be generated using the formula a b c d2 6643 775/C30C n173 /C215 1933058921 /C215 149 /C215 103540742849 173 /C215 1933058921 /C215 15531111427499 336352252427 /C215 149 /C215 103540742849 336352252427 /C215 155311114274992 6643 775; (4) where C n /C302n/C281Mn/C215 59 /C215 72 /C215 114 /C215 172 /C215 19 /C215 292 /C215 67 /C215 712 /C215 109 /C215 131 /C215 139 /C215 179 /C215 307 /C215 431 /C215 521 /C215 653 /C215 1019 /C215 1279 /C215 2557 /C215 3221 /C215 5113 /C215 5171 /C215 6949 (5) and Mnis a MERSENNE PRIME with n a prime > 3 (Y. Kohmoto; Guy 1994, p. 59). See also AMICABLE PAIR,AMICABLE TRIPLE References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 59, 1994. Amicable Triple Dickson (1913, 1952) defined an amicable triple to be a TRIPLE of three numbers (l ; m; n) such that s(l) /C30m /C27n s(m) /C30l /C27n s(n) /C30l /C27m; where s(n) is the RESTRICTED DIVISOR FUNCTION (Madachy 1979). Dickson (1913, 1952) found eight sets of amicable triples with two equal numbers, and two sets with distinct numbers. The latter are (123228768, 103340640, 124015008), for which s(123228768) /C30103340640 /C27124015008 /C30227355648 s(103340640) /C30123228768 /C27124015008 /C30247243776 s(124015008) /C30123228768 /C27103340640 /C30226569408 ; and (1945330728960, 2324196638720, 2615631953920), for which s(1945330728960) /C302324196638720 /C272615631953920 /C304939828592640s(2324196638720) /C301945330728960 /C272615631953920 /C304560962682880 s(2615631953920) /C301945330728960 /C272324196638720 /C304269527367680 : A second definition (Guy 1994) defines an amicable triple as a TRIPLE (a ; b; c) such that s(a) /C30 s(b) /C30 s(c) /C30a /C27b /C27c ; where s(n) is the DIVISOR FUNCTION . An example is ( 22325/C21511;25327;223271):/ See also AMICABLE PAIR,AMICABLE QUADRUPLE References Borho, W. "U ¨ber die Fixpunkte der k-fach iterierten Teiler- summenfunktionen." Mitt. Math. Gesellsch. Hamburg 9, 34/C1/8, 1969. Dickson, L. E. "Amicable Number Triples." Amer. Math. Monthly 20,8 4/C1/2, 1913. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 50, 1952. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 59, 1994. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 156, 1979. Mason, T. E. "On Amicable Numbers and Their General- izations." Amer. Math. Monthly 28, 195/C1/00, 1921. Weisstein, E. W. "Sociable and Amicable Numbers." M ATH- EMATICA NOTEBOOK SOCIABLE.M . Amortization The payment of a debt plus accrued INTEREST by regular payments. Ampersand Curve The PLANE CURVE with Cartesian equation (y2/C28x2)(x/C281)(2x/C283)/C304(x2/C27y2/C282x)2: References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Amphicheiral AMPHICHIRAL Amphichiral An object is amphichiral (also called REFLEXIBLE )ifit is superposable with its MIRROR IMAGE (i.e., its image in a plane mirror). See also AMPHICHIRAL KNOT,CHIRAL ,DISSYMMETRIC , HANDEDNESS ,MIRROR IMAGE Amphichiral Knot An amphichiral knot is a KNOT which is capable of being continuously deformed into its own MIRROR IMAGE . More formally, a knot K is amphichiral (also called achiral or amphicheiral) if there exists an orientation-reversing homeomorphism of R3 mapping K to itself (Hoste et al. 1998). (If the words "orienta- tion-reversing" are omitted, all knots are equivalent to their mirror images.) There are 20 amphichiral knots having ten or fewer crossings, illustrated above, which correspond to 04 /C101 (the FIGURE-OF-EIGHT KNOT ), 06 /C103, 08 /C103, 08 /C109, 08 /C112, 08 /C117, 08 /C118, 10 /C117,10 /C133, 10 /C137, 10 /C143, 10 /C145, 10 /C179, 10 /C181, 10 /C188, 10 /C199, 10 /C109, 10 /C115, 10 /C118, and 10 /C123 (Jones 1985). The following table gives the total number of amphichiral knots, number of /C27 amphichiral noninvertible knots, /C28 amphichiral non- invertible knots, and fully amphichiral invertible knots a with n crossings, starting with n /C303. type Sloane counts amph. A052401 0, 1, 0, 1, 0, 5, 0, 13, 0, 58, 0, 274, 1, ... //C27/ A051767 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 65, ... //C28/ A051768 0, 0, 0, 0, 0, 1, 0, 6, 0, 40, 0, 227, 1, ... a A052400 0, 1, 0, 1, 0, 4, 0, 7, 0, 17, 0, 41, 0, 113, ... Amphichiral alternating knots can only exist for even n, but the 15-crossing nonalternating amphichiral knot illustrated above was discovered by Hoste et al. (1998). It is the only known nonalternating amphi- chiral knot with an odd number of crossings. The HOMFLY POLYNOMIAL is good at identifying amphichiral knots, but sometimes fails to identify knots which are not. No KNOT INVARIANT which always definitively determines if a KNOT isAMPHI- CHIRAL is known. Letb/C27be the SUM ofPOSITIVE exponents, and b/C28the SUM ofNEGATIVE exponents in the BRAID GROUP Bn:If b/C27/C283b/C28/C28n/C271>0; then the KNOT corresponding to the closed BRAID bis not amphichiral (Jones 1985). See also AMPHICHIRAL ,BRAID GROUP ,CHIRAL KNOT, INVERTIBLE KNOT,KNOT SYMMETRY ,MIRROR IMAGE References Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, pp. 311 /C1/19, 1985. Haseman, M. G. "On Knots, with a Census of the Amphi- cheirals with Twelve Crossings." Trans. Roy. Soc. Edin- burgh 52, 235/C1/55, 1917. Haseman, M. G. "Amphicheiral Knots." Trans. Roy. Soc. Edinburgh 52, 597/C1/02, 1918. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,3 3/C1/8, Fall 1998. Jones, V. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103/C1/11, 1985. Jones, V. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335/C1/88, 1987. Sloane, N. J. A. Sequences A051767, A051768, A052400, and A052401 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Amplitude The variable f(also denoted am u) used in ELLIPTIC FUNCTIONS and ELLIPTIC INTEGRALS , which can be defined by f/C30amu/C30am(u;k)/C30gu 0dn(u;k)du; (1) where dn( u;k)/C30dn(u)i saJ ACOBI ELLIPTIC FUNCTION with MODULUS . As is common with J ACOBI ELLIPTIC FUNCTIONS , the modulus kis often suppressed for conciseness. The amplitude is the inverse function of the ELLIPTIC INTEGRAL OF THE FIRST KIND . The amplitude function is implemented in Mathematica as JacobiAmplitude [u, m], where m /C30k2 is the PARAMETER . The DERIVATIVE of the amplitude is given by d duam(u; k) /C30d duam(u) /C30dn(u; k) /C30dn(u) ; (2) or using the notation f; df du /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2 sin2 fp /C30dn(u; k) /C30dn(u) : (3) The amplitude function has the special values am(0 ; k) /C30am(0) /C300 (4) am(K(k) ; k) /C301 2 p; (5) where K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND . In addition, it obeys the identities sin f /C30sin(am( u; k)) /C30sin(am u) /C30sn(u ; k) /C30sn(u) (6) cos f /C30cos(am( u; k)) /C30cos(am u) /C30cn(u; k) /C30cn(u) (7) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2 sin2 fp /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2 sin2(am(u; k))p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2 sn2 up /C30dn(u; k) /C30dn(u) ; (8) which serve as definitions for the JACOBI ELLIPTIC FUNCTIONS . The term "amplitude" is also used to refer to the magnitude of an oscillation, so the amplitude of the sinusoidal curve y /C30A cos(vt) (9) is A. See also ARGUMENT (ELLIPTIC INTEGRAL ), CHARAC- TERISTIC (ELLIPTIC INTEGRAL ), DELTA AMPLITUDE , ELLIPTIC FUNCTION ,ELLIPTIC INTEGRAL OF THE FIRST KIND,JACOBI ELLIPTIC FUNCTIONS ,MODULAR ANGLE , MODULUS (ELLIPTIC INTEGRAL ), NOME,PARAMETER References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. New York: Dover, p. 590, 1972. Fischer, G. (Ed.). Plate 132 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 129, 1986. Anaglyph A STEREOGRAM made of two pictures, one red and one blue, taken from offset positions. When the pictures are viewed through glasses with one lens of each color, the picture appears to be three-dimensional. See also STEREOGRAM References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 166, 1999. Anallagmatic Curve A curve which is invariant under INVERSION . Exam- ples include the CARDIOID , CARTESIAN OVALS ,CASSINI OVALS , LIMAC ¸ ON, STROPHOID , and MACLAURIN TRISEC- TRIX. Anallagmatic Pavement HADAMARD MATRIX Analogy Inference of the TRUTH of an unknown result obtained by noting its similarity to a result already known to be TRUE . In the hands of a skilled mathematician, analogy can be a very powerful tool for suggesting new and extending old results. However, subtleties can render results obtained by analogy incorrect, so rigorous PROOF is still needed. See also GAUSS’S FORMULAS ,INDUCTION ,N APIER’S ANALOGIES Analysis The study of how continuous mathematical struc- tures (FUNCTIONS ) vary around the NEIGHBORHOOD of a point on a SURFACE . Analysis includes CALCULUS , DIFFERENTIAL EQUATIONS , etc. See also ANALYSIS (LOGIC ), ANALYSIS SITUS,CALCU- LUS,C OMPLEX ANALYSIS ,F UNCTIONAL ANALYSIS , NONSTANDARD ANALYSIS ,REAL ANALYSIS References Bottazzini, U. The "Higher Calculus": A History of Real and Complex Analysis from Euler to Weierstrass. New York: Springer-Verlag, 1986. Bressoud, D. M. A Radical Approach to Real Analysis. Washington, DC: Math. Assoc. Amer., 1994. Ehrlich, P. Real Numbers, Generalization of the Reals, & Theories of Continua. Norwell, MA: Kluwer, 1994. Hairer, E. and Wanner, G. Analysis by Its History. New York: Springer-Verlag, 1996. Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, 1988. Weisstein, E. W. "Books about Analysis." http://www.trea- sure-troves.com/books/Analysis.html. Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: Dekker, 1977. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Analysis (Logic) Logicians often call second-order arithmetic "analy- sis." Unfortunately, this term conflicts with the more usual definition of ANALYSIS as the study of functions. This terminology problem is discussed briefly by Enderton (1977, p. 287). See also SET THEORY References Enderton, H. B. Elements of Set Theory. New York: Aca- demic Press, 1977. Analysis of Variance ANOVA Analysis Situs An archaic name for TOPOLOGY . Analytic A solution to a problem that can be written in "closed form" in terms of known functions, constants, etc., is often called an analytic solution. Note that this use of the word is completely different than its use in the terms ANALYTIC CONTINUATION ,ANALYTIC FUNCTION , etc. See also ANALYTIC CONTINUATION ,ANALYTIC FUNC- TION Analytic Continuation An ANALYTIC FUNCTION is determined near a point z0 by a POWER SERIES f(z)/C30X/C12 k/C300ak(z/C28z0)k: (1) Such a power series expansion is in general valid only within its RADIUS OF CONVERGENCE . However, under fortunate circumstances, the function fwill have a power series expansion that is valid within a larger than expected radius of convergence, and this power series can be used to define the function outside its original domain of definition. Letf1andf2beANALYTIC FUNCTIONS on domains V1 andV2;respectively, and suppose that the intersec- tionV1SV2is not empty and that f1/C30f2onV1SV2: Then f2is called an analytic continuation of f1toV2; and vice versa (Flanigan 1983, p. 234). If it exists, the analytic continuation of f1toV2is unique.By means of analytic continuation, starting from a representation of a function by any one POWER SERIES , any number of other POWER SERIES can be found which together define the value of the functionat all points of the domain. Furthermore, any point can be reached from a point without passing through a singularity of the function, and the aggregate of allthe power series thus obtained constitutes the analy- tic expression of the function (Whittaker and Watson 1990, p. 97). Analytic continuation can lead to some interesting phenomenon such as MULTIVALUED FUNCTIONS . For example, consider analytic continuation of the SQUARE ROOT function f(z)/C30ffiffiffizp:Although this func- tion is not globally well-defined (since every nonzero number has two square roots), fhas a well-defined TAYLOR SERIES around z0/C301; f(z)/C30f(z0)/C27(z/C28z0)f?(z0)/C27(z/C28z0)2 2!f??(z0)/C27... /C301/C271 2(z/C281)/C2818(z/C281)3/C271 16(z/C281)3/C285 128(z/C281)4 /C27... which can be used to extend the domain over which f is defined. Note that when ½z½/C301;the POWER SERIES forfhas a RADIUS OF CONVERGENCE of 1. The animation above shows the analytic continuation off(z)/C30ffiffiffizpalong the path eit:Note that when the function goes all the way around, fis the negative of the original function, so going around twice returns the function to its original value. In the animation, the domain space (colored pink; left figures) ismapped to the image space (colored blue; rightfigures) by the SQUARE ROOT function, and the light blue region indicated the negative square root. How-ever, by continuing the function around the circle, thesquare root function takes values in what used to be the light blue region, so the roles of the blue and light blue region are reversed. This can be interpreted asgoing from one branch of the multivalued SQUARE ROOT function to the other. This illustrates that analytic continuation extends a function using thenearby values that provide the information on the power series. It is possible for the function to never return to the same value. For example, f(z)/C30lnzincreased by 2 pi every time it is continued around zero. The naturaldomain of a function is the maximal chain of domains on which a function can be analytically continued to a single-valued function. For ln z;it is the connected infinite COVER of the punctured plane, and for z/C281=2it is the connected double COVER . If there is a boundary across which the function cannot be extended, then is called the natural boundary. For instance, there exists a MEROMORPHIC FUNCTION f in the unit disk where every point on the unit circle is a limit point of the set of poles. Then the circle is a natural boundary for f. See also ANALYTIC FUNCTION ,D IRECT ANALYTIC CONTINUATION ,G LOBAL ANALYTIC CONTINUATION , MONODROMY THEOREM ,PERMANENCE OF ALGEBRAIC FORM,PERMANENCE OF MATHEMATICAL RELATIONS PRINCIPLE ,SCHWARZ REFLECTION PRINCIPLE References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 378 /C180, 1985. Davis, P. J. and Pollak, H. "On the Analytic Continuation of Mapping Functions." Trans. Amer. Math. Soc. 87, 198 /C125, 1958. Flanigan, F. J. Complex Variables: Harmonic and Analytic Functions. New York: Dover, 1983. Knopp, K. "Analytic Continuation and Complete Definition of Analytic Functions." Ch. 8 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 83 /C111, 1996. Krantz, S. G. "Uniqueness of Analytic Continuation" and "Analytic Continuation." §3.2.3 and Ch. 10 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 38 /C19 and 123 /C141, 1999. Levinson, N. and Raymond, R. Complex Variables. New York: McGraw-Hill, pp. 398 /C102, 1970. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 389 /C190 and 392 /C198, 1953. Needham, T. "Analytic Continuation." §5.XI in Visual Com- plex Analysis. New York: Clarendon Press, pp. 247 /C157, 2000. Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 319 /C127, 1987. Whittaker, E. T. and Watson, G. N. "The Process of Con- tinuation." §5.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 96 /C18, 1990. Analytic Function A COMPLEX FUNCTION is said to be analytic on a region R if it is COMPLEX DIFFERENTIABLE at every point in R. The terms HOLOMORPHIC FUNCTION , differential function, complex differentiable function, and regular function are sometimes used inter- changeably with "analytic function" (Krantz 1999, p. 16). Many mathematicians prefer the term "holo- morphic function" (or "holomorphic map") to "analytic function" (Krantz 1999, p. 16), while "analytic" ap- pears to be in widespread use among physicists, engineers, and in some older texts (Morse and Feshbach 1953, pp. 356 /C174; Knopp 1996, pp. 83 /C111; Whittaker and Watson 1990, p. 83). If a FUNCTION is analytic, it is infinitely DIFFERENTI- ABLE .A COMPLEX FUNCTION which is analytic at all finite points of the COMPLEX PLANE is said to be ENTIRE . See also BERGMAN SPACE ,COMPLEX DIFFERENTIABLE ,DIFFERENTIABLE ,ENTIRE FUNCTION ,H OLOMORPHIC FUNCTION ,M EROMORPHIC FUNCTION ,PSEUDOANALY- TIC FUNCTION ,REAL ANALYTIC FUNCTION ,SEMIANA- LYTIC ,SUBANALYTIC References Knopp, K. "Analytic Continuation and Complete Definition of Analytic Functions." Ch. 8 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 83 /C111, 1996. Krantz, S. G. "Alternative Terminology for Holomorphic Functions." §1.3.6 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 16, 1999. Morse, P. M. and Feshbach, H. "Analytic Functions." §4.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 356 /C174, 1953. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Analytic Geometry The study of the GEOMETRY of figures by algebraic representation and manipulation of equations de- scribing their positions, configurations, and separa- tions. Analytic geometry is also called COORDINATE GEOMETRY since the objects are described as n-tuples of points (where n /C302 in the PLANE and 3 in SPACE )in some COORDINATE SYSTEM . See also ARGAND DIAGRAM ,CARTESIAN COORDINATES , CARTESIAN GEOMETRY ,COMPLEX PLANE ,GEOMETRY , PLANE ,QUADRANT ,SPACE , X-AXIS, Y-AXIS, Z-AXIS References Courant, R. and Robbins, H. "Remarks on Analytic Geome- try." §2.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 72 /C17, 1996. Analytic Set A DEFINABLE SET, also called a SOUSLIN SET. See also COANALYTIC SET,SOUSLIN SET Analytic Solution ANALYTIC Anarboricity Given a GRAPH G, the anarboricity is the maximum number of line-disjoint nonacyclic SUBGRAPHS whose UNION is G. See also ARBORICITY Anchor An anchor is the BUNDLE MAP rfrom a VECTOR BUNDLE Ato the TANGENT BUNDLE TBsatisfying 1. [ r(X); r(Y)] /C30 r([X ; Y]) and 2. [X ; fY] /C30 f[X ; Y] /C27 ( r(X) /C215 f)Y ;/ where X and Y are smooth sections of A, f is a smooth function of B, and the bracket is the "Jacobi- Lie bracket" of a VECTOR FIELD . See also BUNDLE ,LIE ALGEBROID References Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744 /C152, 1996. Anchor Ring An archaic name for the TORUS . References Eisenhart, L. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, p. 314, 1960. Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley, p. 239, 1977. Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles & Rigid Bodies, 4th ed. Cambridge, England: Cambridge University Press, p. 21, 1959. And A term (PREDICATE )in LOGIC which yields TRUE if one or more conditions are TRUE , and FALSE if any condition is FALSE . A AND B is denoted N1 ; CM[(P /C27 Q)]P80 /C281]; or simply A/C31: The BINARY AND operator has the following TRUTH TABLE : /A//B//CM[(P /C27Q)]P80 /C281]/ FF F FT F TF F TT T A PRODUCT of ANDs (the AND of J0( vr) conditions) is called a CONJUNCTION , and is denoted N2 Two binary numbers can have the operation AND performed bitwise with 1 representing TRUE and 0 FALSE . Some computer languages denote this opera- tion on A;B;andCasA&&B&&C orlogand(A,B,C) . See also BINARY OPERATOR ,INTERSECTION ,NOT,OR, PREDICATE ,TRUTH TABLE , XORAND ACONNECTIVE inLOGIC which yields TRUE if all conditions are TRUE , and FALSE if any condition is FALSE .AAND Bis denoted AfflB(Mendelson 1997, p. 12), A&B;ASB(Simpson 1987, p. 538), A /C215B; A:B(Carnap 1958, p. 7), or simply AB(Simpson 1987, p. 538). The way to distinguish the similar symbols ffl(AND) and /C150(OR) is to note that the symbol for AND is oriented in the same direction as the capital letter ‘A." The AND operation is imple-mented in Mathematica asAnd[A,B, ...]. The circuit diagram symbol for an AND gate is illustrated above.The AND operation can be written in terms of NOTand AND as AfflB/C30!(!A/C150!B): The BINARY AND operator has the following TRUTH TABLE (Carnap 1958, p. 10; Simpson 1987, p. 545; Mendelson 1997, p. 12). AB /AfflB/ TTT TFFFTFFFF A PRODUCT of ANDs (the AND of nconditions) is called a CONJUNCTION , and is denoted Ln k/C301Ak: For example, the TRUTH TABLE forAAND BAND C is given below (Simpson 1987, p. 545). ABC /AfflBfflC/ TTTTTTFFTFTFTFFF FTTF FTFF FFTFFFFF Two binary numbers can have the operation AND performed bitwise with 1 representing TRUE and 0 FALSE . Some computer languages denote this opera- tion on A, B, and C asA&&B&&C orlogand(A,B,C) . See also BINARY OPERATOR ,CONJUNCTION ,CONNEC- TIVE,INTERSECTION , NAND, NOR, NOT, OR, TRUTH TABLE ,W EDGE , XNOR, XOR References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, pp. 7 and 10, 1958. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 12, 1997. Simpson, R. E. "The AND Gate." §12.5.2 in Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 538 and 544 /C1/46, 1987. Anderson-Darling Statistic A statistic defined to improve the KOLMOGOROV- SMIRNOV TEST in the TAIL of a distribution. See also KOLMOGOROV- SMIRNOV TEST,KUIPER STA- TISTIC References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 621, 1992. Andre ´’s Problem The determination of the number of ALTERNATING PERMUTATIONS having elements f1; 2; ...; ng:/ See also ALTERNATING PERMUTATION Andre ´’s Reflection Method A technique used by Andre ´ (1887) to provide an elegant solution to the BALLOT PROBLEM (Hilton and Pederson 1991) and in study of WIENER PROCESSES (Doob 1953; Papoulis 1984, p. 505). See also BALLOT PROBLEM ,W IENER PROCESS References Andre ´, D. "Solution directe du proble `me re´solu par M. Ber- trand." Comptes Rendus Acad. Sci. Paris 105, 436 /C1/37, 1887. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 22, 1974. Doob, J. L. Stochastic Processes. New York: Wiley, 1953.Hilton, P. and Pederson, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13,64/C1/5, 1991. Papoulis, A. "The Reflection Principle and Its Applications." Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 505 /C1/10, 1984. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 185, 1991. Andrew’s Sine The function c(z) /C30sinrC1+z crC1D ½z ½Bc p 0;½z½> c p8 < : which occurs in estimation theory. See also SINE References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 697, 1992. Andrews Cube SEMIPERFECT MAGIC CUBE Andrews-Curtis Link The LINK of 2-spheres in R4 obtained by SPINNING intertwined arcs. The link consists of a knotted 2- sphere and a SPUN TREFOIL KNOT . See also SPUN KNOT,TREFOIL KNOT References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 94, 1976. Andrews-Schur Identity Xn k/C300qk2/C27ak2n/C28k/C27a krC00rC01 /C30X/C12 k/C30/C28/C12q10k2/C27(4a/C281)k2n/C272a/C272 n/C285krC00rC01 /C2[10k/C272a/C272] [2n/C272a/C272]; (1) where [ x]i saG AUSSIAN POLYNOMIAL .I ti sa POLY- NOMIAL identity for a/C300, 1 which implies the ROGERS- RAMANUJAN IDENTITIES by taking n0/C12 and applying the J ACOBI TRIPLE PRODUCT identity. A variant of this equation is Xn k /C30/C28/C28a=2 /C29qk2/C272ak n /C27k /C27a n /C28krC00rC01 /C30X[n=5] /C28[(n/C272a /C272)=5]q15k2/C27(6a/C271)k 2n /C272a /C272 5 /C285krC00rC01 /C2[10k /C27 2a /C27 2] [2n /C27 2a /C27 2]; (2) where the symbol xbcin the SUM limits is the FLOOR FUNCTION (Paule 1994). The RECIPROCAL of the identity is X/C12 k /C300qk2 /C272ak (q; q)2k /C27a /C30Y/C12 j/C3001 (1 /C28 q2j/C271)(1 /C28 q20j/C274a /C274)(1 /C28 q20j/C284a/C2716)(3) for a /C300, 1 (Paule 1994). For q /C301, (1) and (2) become Xn /C28/C28a =2 /C29n /C27k /C27a n /C28krC1+rC1D /C30X/C28n=5 /C29 /C28/C28(n/C272a /C272)=5 /C292n /C272a /C272 n /C285krC1+rC1D5k /C27 q /C27 1 n /C27 a /C27 1: (4) References Andrews, G. E. "A Polynomial Identity which Implies the Rogers-Ramanujan Identities." Scripta Math. 28, 297 /C1/05, 1970. Paule, P. "Short and Easy Computer Proofs of the Rogers- Ramanujan Identities and of Identities of Similar Type." Electronic J. Combinatorics 1, R10 1 /C1/, 1994. http:// www.combinatorics.org/Volume_1/volume1.html#R10. Andrica’s Conjecture Andrica’s conjecture states that, for pn the nth PRIME NUMBER , the INEQUALITY An /C13ffiffiffiffiffiffiffiffiffiffipn/C271p/C28ffiffiffiffiffipnpB1 holds, where the discrete function An is plotted above. The largest value among the first 1000 PRIMES is forn /C304, givingffiffiffiffiffiffi 11p /C28ffiffiffi 7p :0 :670873 : Since the Andrica function falls asymptotically as n increases so a PRIME GAP of increasing size is needed at large n,it seems likely the CONJECTURE is true. However, it has not yet been proven. /Anbears a strong resemblance to the PRIME DIFFER- ENCE FUNCTION , plotted above, the first few values of which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, ... (Sloane’s A001223). A generalization of Andrica’s conjecture considers the equation px n/C271 /C28pxn /C301 and solves for x. The smallest such x is x :0 :567148 (Sloane’s A038458), known as the SMARANDACHE CONSTANT , which occurs for pn /C30113 and pn/C271 /C30127 (Perez). See also BROCARD’S CONJECTURE ,G OOD PRIME , FORTUNATE PRIME ,PO´ LYA CONJECTURE ,PRIME DIF- FERENCE FUNCTIO N,S MARANDACHE CONSTANTS , TWIN PEAKS References Golomb, S. W. "Problem E2506: Limits of Differences of Square Roots." Amer. Math. Monthly 83,60/C1/1, 1976. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 21, 1994. Perez, M. L. (Ed.). "Five Smarandache Conjectures on Primes." http://www.gallup.unm.edu/~smarandache/con- jprim.txt. Rivera, C. "Problems & Puzzles: Conjecture Andrica’s Con- jecture.-008." http://www.primepuzzles.net/conjectures/ conj_008.htm. Sloane, N. J. A. Sequences A001223/M0296 and A038458 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Anger Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27y? x/C271/C28v2 x2 ! y/C30x/C28v px2sin(vx) whose solutions are A NGER FUNCTIONS . See also ANGER FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Anger and Weber Functions." §12.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 498 /C1/99, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 989, 2000. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Anger Function A generalization of the BESSEL FUNCTION OF THE FIRST KIND defined by Jv(z) /C131 p g p 0cos (vu /C28z sin u) du: If v is an INTEGER n, then Jn(z) /C30Jn(z) ; where Jn(z)is aB ESSEL FUNCTION OF THE FIRST KIND . Anger’s original function had an upper limit of 2 p; but the current NOTATION was standardized by Watson (1966). See also ANGER DIFFERENTIAL EQUATION ,B ESSEL FUNCTION ,M ODIFIED STRUVE FUNCTION ,PARABOLIC CYLIN DER FUNCTION ,S TRUVE FUNCTION ,W EBER FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Anger and Weber Functions." §12.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 498 /C1/99, 1972. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Anger Function Jv(x) and Weber Function Ev(x):/" §1.5 in Integrals and Series, Vol. 3: More Special Func- tions. Newark, NJ: Gordon and Breach, p. 28, 1990. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Angle Given two intersecting LINES or LINE SEGMENTS , the amount of ROTATION about the point of intersection (the VERTEX ) required to bring one into correspon- dence with the other is called the angle u between them. Angles are usually measured in DEGREES (denoted /C14) ; RADIANS (denoted rad, or without a unit), or sometimes GRADIANS (denoted grad). One full rotation in these three measures corresponds to 3608,2p rad, or 400 grad. Half a full ROTATION is called a STRAIGHT ANGLE , and a QUARTER of a full rotation is called a RIGHT ANGLE . An angle less than a RIGHT ANGLE is called an ACUTE ANGLE , and an anglegreater than a RIGHT ANGLE is called an OBTUSE ANGLE . The use of DEGREES to measure angles harks back to the Babylonians, whose SEXAGESIMAL number system was based on the number 60. 360 8 likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The DEGREE is further divided into 60 ARC MINUTES , and an ARC MINUTE into 60 ARC SECONDS . A more natural measure of an angle is the RADIAN . It has the property that the ARC LENGTH around a CIRCLE is simply given by the radian angle measure times the CIRCLE RADIUS . The RADIAN is also the most useful angle measure in CALCULUS because the DERIVATIVE of TRIGONOMETRIC functions such as d dxsin x /C30cos x does not require the insertion of multiplicative con- stants like p=180: GRADIANS are sometimes used in surveying (they have the nice property that a RIGHT ANGLE is exactly 100 GRADIANS ), but are encountered infrequently, if at all, in mathematics. The concept of an angle can be generalized from the CIRCLE to the SPHERE . The fraction of a SPHERE subtended by an object is measured in STERADIANS , with the entire SPHERE corresponding to 4 pSTERA- DIANS . A ruled SEMICIRCLE used for measuring and drawing angles is called a PROTRACTOR .ACOMPASS can also be used to draw circular ARCS of some angular extent. See also ACUTE ANGLE ,ARC MINUTE ,ARC SECOND , CENTRAL ANGLE ,COMPLEMENTARY ANGLE ,DEGREE , DIHEDRAL ANGLE ,DIRECTED ANGLE ,EULER ANGLES , EXTERIOR ANGLE ,F ULL ANGLE ,G RADIAN ,H ORN ANGLE ,INSCRIBED ANGLE ,OBLIQUE ANGLE ,OBTUSE ANGLE ,P ERIGON ,P ROTRACTOR ,R ADIAN ,R EFLEX ANGLE ,R IGHT ANGLE ,S OLID ANGLE ,S TERADIAN , STRAIGHT ANGLE ,SUBTEND ,SUPPLEMENTARY ANGLE , VERTEX ANGLE References Dixon, R. Mathographics. New York: Dover, pp. 99 /C1/00, 1991. Harris, J. W. and Stocker, H. "Angle." §3.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 62 /C1/4, 1998. Angle Bisector The (interior) bisector of an ANGLE is the LINE orLINE SEGMENT which cuts it into two equal ANGLES on the same "side" as the ANGLE . The length of the bisector of ANGLE A1in the above TRIANGLE DA1A2A3 is given by t2 1 /C30a2a31 /C28a2 1 (a2 /C27 a3)2"# ; where ti /C13AiTiand ai /C13AjAk : The angle bisectors meet at the INCENTER I, which has TRILINEAR CO- ORDINATES 1:1:1. See also ANGLE BISECTOR THEOREM ,CYCLIC QUAD- RANGLE ,E XTERIOR ANGLE BISECTOR ,ISODYNAMIC POINTS ,O RTHOCENTRIC SYSTEM ,S TEINER- LEHMUS THEOREM ,TRISECTION References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 /C1/0, 1967. Dixon, R. Mathographics. New York: Dover, p. 19, 1991. Mackay, J. S. "Properties Concerned with the Angular Bisectors of a Triangle." Proc. Edinburgh Math. Soc. 13, 37 /C1/02, 1895. Angle Bisector Theorem The ANGLE BISECTOR of an ANGLE in a TRIANGLE divides the opposite side in the same RATIO as the sides adjacent to the ANGLE . Angle Bracket The combination of a BRA and KET (bra/C27ket /C30bracket) which represents the INNER PRO- DUCT of two functions or vectors, f ½ghi/C30gf(x)g(x) dx v½whi/C30v /C215w: By itself, the BRA is a COVARIANT 1-VECTOR , and the KET is a CONTRAVARIANT ONE-FORM . These terms are commonly used in quantum mechanics. See also BRA,BRACE ,D IFFERENTIAL K-FORM,K ET, ONE-FORM,PARENTHESIS ,SQUARE BRACKET References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 271, 1997.Angle of Parallelism Given a point P and a LINE AB, draw the PERPENDI- CULAR through P and call it PC. Let PD be any other line from P which meets CB in D.Ina HYPERBOLIC GEOMETRY ,asD moves off to infinity along CB, then the line PD approaches the limiting line PE, which is said to be parallel to CB at P. The angle /C218CPE which PE makes with PC is then called the angle of parallelism for perpendicular distance x, and is given by Y (x) /C302 tan /C281(e /C28x) : This is known as LOBACHEVSKY’S FORMULA . See also HYPERBOLIC GEOMETRY ,L OBACHEVSKY’S FORMULA References Coxeter, H. S. M. "The Angle of Parallelism." §16.3 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 291 /C1/95, 1969. Manning, H. P. Introductory Non-Euclidean Geometry. New York: Dover, pp. 31 /C1/2 and 58, 1963. Angle Trisection TRISECTION Angle-Preserving Transformation CONFORMAL MAPPING Angular Acceleration The angular acceleration a is defined as the time DERIVATIVE of the ANGULAR VELOCITY v; a /C13dv dt/C30d2 u dt2 ˆz /C30a r: See also ACCELERATION ,ANGULAR DISTANCE ,ANGU- LAR VELOCITY Angular Defect The DIFFERENCE between the SUM of face ANGLES Ai at a VERTEX of a POLYHEDRON and 2p; d/C302p/C28X iAi: See also DESCARTES TOTAL ANGULAR DEFECT ,JUMP ANGLE ,SPHERICAL DEFECT Angular Distance The angular distance traveled around a CIRCLE is the number of RADIANS the path subtends, u /C13l 2 pr2p /C30l r : See also ANGULAR ACCELERATION ,ANGULAR VELO- CITY Angular Velocity The angular velocity v is the time DERIVATIVE of the ANGULAR DISTANCE u with direction ˆz PERPENDICULAR to the plane of angular motion, v /C13du dtˆz /C30v r: See also ANGULAR ACCELERATION ,A NGULAR DIS- TANCE Anharmonic Ratio CROSS- RATIO Animal 1. A FIXED POLYOMINO . 2. The set of points obtained by taking the centers of a FIXED POLYOMINO . See also POLYOMINO References Delest, M.-P. and Viennot, G. "Algebraic Languages and Polyominoes [sic] Enumeration." Theoret. Comput. Sci. 34, 169 /C1/06, 1984. Read, R. C. "Contributions to the Cell Growth Problem." Canad. J. Math. 14,1/C1/0, 1962. Anisohedral Tiling A k-anisohedral tiling is a tiling which permits no n- ISOHEDRAL TILING with n Bk. References Berglund, J. "Is There a k-Anisohedral Tile for k ]5/?" Amer. Math. Monthly 100, 585 /C1/88, 1993. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., 1991. Annealing SIMULATED ANNEALING Annihilator The term annihilator is used in several different ways in various aspects of mathematics. It is most com-monly used to mean the SET of all functions satisfying a given set of conditions which is zero on every member of a given SET. Annuity PRESENT VALUE Annulus The region in common to two concentric CIRCLES of RADII a and b. The AREA of an annulus is Aannulus /C30 p(b2 /C28a2): In the above figure, the area of the circle whose diameter is tangent to the inner circle and has endpoints at the outer circle is equal to the area of the annulus. See also ANNULUS THEOREM ,B ULLSEYE ILLUSION , CHORD ,CIRCLE ,CONCENTRIC CIRCLES ,LUNE,SPHE- RICAL SHELL References Harris, J. W. and Stocker, H. "Annulus, Circular Ring." §3.8.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 91, 1998. Pappas, T. "The Amazing Trick." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 69, 1989. Annulus Conjecture ANNULUS THEOREM Annulus Theorem LetKn 1andKn2be disjoint bicollared KNOTS inRn/C271or Sn/C271and let Udenote the open region between them. Then the closure of Uis a closed annulus Sn/C29[0;1]: Except for the case n/C303, the theorem was proved by Kirby (1969). References Kirby, R. C. "Stable Homeomorphisms and the Annulus Conjecture." Ann. Math. 89, 575/C1/82, 1969. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 38, 1976. Anomalous Cancellation The simplification of a FRACTION a=bwhich gives a correct answer by "canceling" DIGITS ofaand b. There are only four such cases for NUMERATOR and DENOMINATORS of two DIGITS in base 10: 64 =16/C30 4=1 /C304 ; 98 =49 /C308 =4 /C302; 95=19 /C305=1 /C305; and 65 =26 /C305=2 (Boas 1979). The concept of anomalous cancellation can be ex- tended to arbitrary bases. PRIME bases have no solutions, but there is a solution corresponding to each PROPER DIVISOR of a COMPOSITE b. When b /C281is PRIME , this type of solution is the only one. For base 4, for example, the only solution is 324 =134 /C3024 : Boas gives a table of solutions for b 539 : The number of solutions is EVEN unless b is an EVEN SQUARE . bN bN 4 1 26 4 6 2 27 6 8 2 28 10 9 2 30 6 10 4 32 4 12 4 34 6 14 2 35 6 15 6 36 21 16 7 38 2 18 4 39 6 20 4 21 10 22 6 24 6 See also FRACTION ,P RINTER’S ERRORS ,R EDUCED FRACTION References Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathemati- cal Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113 /C1/29, 1979. Moessner, A. Scripta Math. 19. Moessner, A. Scripta Math. 20. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 86 /C1/7, 1988. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 26 /C1/7, 1986. Anomalous Number BENFORD’S LAW Anonymous A term in SOCIAL CHOICE THEORY meaning invariance of a result under permutation of voters. See also DUAL VOTING ,MONOTONIC VOTINGAnosov Automorphism A HYPERBOLIC linear map Rn 0 Rnwith INTEGER entries in the transformation MATRIX and DETERMI- NANT 9 1isanA NOSOV DIFFEOMORPHISM of the n- TORUS , called an Anosov automorphism (or HYPER- BOLIC AUTOMORPHISM ). Here, the term automorphism is used in the GROUP THEORY sense. Anosov Diffeomorphism An Anosov diffeomorphism is a C1 DIFFEOMORPHISM f such that the MANIFOLD M is HYPERBOLIC with respect to f: Very few classes of Anosov diffeomorph- isms are known. The best known is ARNOLD’S CAT MAP. A HYPERBOLIC linear map Rn 0 Rnwith INTEGER entries in the transformation MATRIX and DETERMI- NANT 9 1 is an Anosov diffeomorphism of the n- TORUS . Not every MANIFOLD admits an Anosov diffeo- morphism. Anosov diffeomorphisms are EXPANSIVE , and there are no Anosov diffeomorphisms on the CIRCLE . It is conjectured that if f : M 0 M is an Anosov diffeomorphism on a COMPACT RIEMANNIAN MANI- FOLD and the NONWANDERING SET V(f)off is M, then f is TOPOLOGICALLY CONJUGATE to a FINITE-TO- ONE FACTOR of an ANOSOV AUTOMORPHISM of a NILMANIFOLD . It has been proved that any Anosov diffeomorphism on the n-TORUS is TOPOLOGICALLY CONJUGATE to an ANOSOV AUTOMORPHISM , and also that Anosov diffeomorphisms are C1 STRUCTURALLY STABLE . See also ANOSOV AUTOMORPHISM ,AXIOM AD IFFEO- MORPHISM ,DYNAMICAL SYSTEM References Anosov, D. V. "Geodesic Flow on Closed Riemannian Mani- folds of Negative Curvature." Trudy Mat. Inst. Steklov 90, 1 /C109, 1970. Smale, S. "Differentiable Dynamical Systems." Bull. Amer. Math. Soc. 73, 747 /C117, 1967. Anosov Flow A FLOW defined analogously to the ANOSOV DIFFEO- MORPHISM , except that instead of splitting the TAN- GENT BUNDLE into two invariant sub- BUNDLES , they are split into three (one exponentially contracting, one expanding, and one which is 1-dimensional and tangential to the flow direction). See also DYNAMICAL SYSTEM Anosov Map An important example of a A NOSOV DIFFEOMORPHISM . xn/C271 yn/C271rC00rC01 /C3021 11rC00rC01 xn ynrC00rC01 ; where xn/C271;yn/C271are computed mod 1. See also ARNOLD’S CAT MAP ANOVA "Analysis of Variance." A STATISTICAL TEST for hetero- geneity of MEANS by analysis of group VARIANCES .To apply the test, assume random sampling of a variate y with equal VARIANCES , independent errors, and a NORMAL DISTRIBUTION . Let n be the number of REPLICATES (sets of identical observations) within each of K FACTOR LEVELS (treatment groups), and yij be the jth observation within FACTOR LEVEL i. Also assume that the ANOVA is "balanced" by restricting n to be the same for each FACTOR LEVEL . Now define the sum of square terms SST /C13Xk i /C301Xn j/C301(yij /C28 ˜y)2 (1) /C30Xk i/C301Xn j/C301y2 ij /C28Pk i /C301Pnj/C301yijrC16rC1*2 Kn (2) SSA /C131 nXk i/C301Xn j/C301yij ! 2 /C281 KnXk i/C301Xn j/C301yij ! 2 (3) SSE /C13Xk i /C301Xn j/C301(yij /C28 ¨yi)2 (4) /C30 SST /C28 SSA ; (5) which are the total, treatment, and error sums of squares. Here, ¨yiis the mean of observations within FACTOR LEVEL i, and ˜y is the "group" mean (i.e., mean of means). Compute the entries in the following table, obtaining the P-VALUE corresponding to the calcu- lated F-RATIO of the mean squared values F /C30MSA MSE : (6) Category SS / /C14/Freedom Mean Squared F-RATIO Treatment SSA /K /C281// MSA /C13SSA K /C28 1//MSA MSE/ Error SSE /K(n /C281)// MSE /C13SSE K(n /C28 1)/ Total SST /Kn /C281// MST /C13SST Kn /C28 1/ If the P-VALUE is small, reject the NULL HYPOTHESIS that all MEANS are the same for the different groups. See also FACTOR LEVEL ,MANOVA,R EPLICATE , VARIANCE References Miller, R. G. Beyond ANOVA: Basics of Applied Statistics. Boca Raton, FL: Chapman & Hall, 1997.Anthropomorphic Polygon A SIMPLE POLYGON with precisely two EARS and one MOUTH . References Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122,31/C1/5, 1991. Anthyphairetic Ratio An archaic term for a CONTINUED FRACTION . References Fowler, D. H. The Mathematics of Plato’s Academy: A New Reconstruction, 2nd ed. New York: Oxford University Press, 1987. Antiautomorphism If a MAP f : G 0 G ? from a GROUP G to a GROUP G? satisfies f(ab) /C30f(a)f(b) for all a; b /C23 G; then f is said to be an antiautomorphism. See also AUTOMORPHISM Anticenter The point of concurrence of the three MALTITUDES of a CYCLIC QUADRILATERAL . Let MACand MBDbe the MIDPOINTS of the diagonals of a CYCLIC QUADRILAT- ERAL ABCD , and let P be the intersection of the diagonals. Then the ORTHOCENTER of TRIANGLE DPMACMBD is the anticenter T of ABCD (Honsberger 1995, p. 39). See also CYCLIC QUADRILATERAL ,MALTITUDE References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 36 /C1/7, 1995. Anticevian Triangle Given a center a : b : g ; the anticevian triangle is defined as the TRIANGLE with VERTICES /C28a : b : g ; a : /C28b : g; and a : b : /C28g : If A?B ?C? is the CEVIAN TRIANGLE of X and AƒB ƒCƒ is an anticevian triangle, then X and Aƒare HARMONIC CONJUGATE POINTS with respect to AandA?:/ See also CEVIAN TRIANGLE References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Antichain Let P be a finite PARTIALLY ORDERED SET. An antic- hain in P is a set of pairwise incomparable elements (e.g., a family of SUBSETS such that, for any two of them, neither is a SUBSET of the other). Antichains are also called Sperner systems in older literature (Comtet 1974). The following table gives the antichains on n-set f1; 2; ...; ng for small n. n antichains 1 / ¥;f(1)g/ 2 / ¥;ff1gg;ff2gg;ff1 g;f2gg;ff1; 2gg/ 3 / ¥;ff1gg;ff2gg;ff3 gg;ff1; 2gg;/ / ff1; 3gg;ff2; 3gg;ff1g;f2gg;ff1g;f3gg;/ / ff2g;f3gg;ff1; 2; 3gg;ff1g;f2; 3gg;ff1 ; 2g;f2 ; 3gg;/ / ff1; 2g;f1; 3gg;ff1; 2g;f3gg;ff2g;f1; 3gg;ff2 ; 3g;f1;3 gg;/ / ff1g;f2g;f3gg;ff1; 2g;f2; 3g;f1; 3gg/ The number of antichains on the n-set f1; 2; ...; ng for n /C30 1, 2, ..., are 1, 2, 5, 19, 167, ... (Sloane’s A014466). If the EMPTY SET is not considered a valid antichain, then these reduce to 0, 1, 4, 18, 166, ... (Sloane’s A007153; Comtet 1974, p. 273). The num- bers obtained by adding one to Sloane’s A014466, 2, 3, 6, 20, 168, 7581, 7828354, ... (Sloane’s A000372), are also frequently encountered (Speciner 1972). The number of antichains on the n-set are equal to the number of monotonic increasing Boolean func- tions of n variables, and also the number of free distributive lattices with n generators (Comtet 1974, p. 273). Determining these numbers is known as DEDEKIND’S PROBLEM , and the numbers in each of these sequences are sometimes called Dedekind numbers (Sloane). The WIDTH of P is the maximum CARDINALITY of an ANTICHAIN inP. For a PARTIAL ORDER , the size of the longest ANTICHAIN is called the WIDTH w(P):Sperner (1928) proved that the maximum width of an antic- hain containing nelements is wmax( n)/C30n n=2bcrC1+rC1D ; wheren krC0rC1 is a BINOMIAL COEFFICIENT and nbcis the FLOOR FUNCTION . See also BOOLEAN FUNCTION ,C HAIN ,D ILWORTH’S LEMMA ,PARTIALLY ORDERED SET,W IDTH (PARTIAL ORDER ) References Agnew, R. P. "Minimax Functions, Configuration Functions, and Partitions." J. Indian Math. Soc. 24,1/C1/1, 1961.Anderson, I. Combinatorics of Finite Sets. Oxford, England: Oxford University Press, p. 38, 1987. Arocha, J. L. "Antichains in Ordered Sets" [Spanish]. Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico 27,1/C1/1, 1987. Berman, J. "Free Spectra of 3-Element Algebras." In Uni- versal Algebra and Lattice Theory (Puebla, 1982) (Ed. R. S. Freese and O. C. Garcia). New York: Springer-Ver- lag, 1983. Berman, J. and Koehler, P. "Cardinalities of Finite Dis- tributive Lattices." Mitteilungen aus dem Mathematischen Seminar Giessen 121, 103/C1/24, 1976. Birkhoff, G. Lattice Theory, 3rd ed. Providence, RI: Amer. Math. Soc., p. 63, 1967. Church, R. "Numerical Analysis of Certain Free Distributive Structures." Duke Math. J. 6, 732/C1/33, 1940. Church. "Enumeration by Rank of the Elements of the Free Distributive Lattice with Seven Generators." Not. Amer. Math. Soc. 12, 724, 1965. Comtet, L. "Sperner Systems." §7.2 in Advanced Combina- torics: The Art of Finite and Infinite Expansions, rev. enl. ed.Dordrecht, Netherlands: Reidel, pp. 271 /C1/73, 1974. Dedekind, R. "U ¨ber Zerlegungen von Zahlen durch ihre gro¨ssten gemeinsammen Teiler." In Gesammelte Werke, Bd. 1. pp. 103 /C1/48, 1897. Erdos, P.; Ko, Chao; and Rado, R. "Intersection Theorems for Systems of Finite Sets." Quart. J. Math. Oxford 12, 313/C1/ 20, 1961. Gilbert, E. N. "Lattice Theoretic Properties of Frontal Switching Networks." J. Math. Phys. 33,5 7/C1/7, 1954. Hansel, G. "Proble `mes de de ´nombrement et d’e ´valuation de bornes concernant les e ´le´ments du trellis distributif libre." Publ. Inst. Statist. Univ. Paris 16, 163/C1/94, 1967. Harrison, M. A. Introduction to Switching and Automata Theory. New York: McGraw-Hill, p. 188, 1965. Hilton, A. J. W. and Milner, E. C. "Some Intersection The- orems of Systems of Finite Sets." Quart. J. Math. Oxford 18, 369/C1/84, 1967. Katona, G. "On a Conjecture of Erdos and a Stronger Form of Sperner’s Theorem." Studia Sci. Math. Hung. 1,5 9/C1/3, 1966. Katona, G. "A Theorem of Finite Sets." In Theory of Graphs, Proceedings of the Colloquium Held at Tihany, Hungary (Ed. P. Erdos and G. Katona). New York: Academic Press, pp. 187 /C1/07, 1968. Kleitman, D. "A Conjecture of Erdos-Katona on Commen- surable Pairs Among Subsets of a n-Set." In Theory of Graphs, Proceedings of the Colloquium Held at Tihany,Hungary (Ed. P. Erdos and G. Katona). New York: Aca- demic Press, pp. 215 /C1 /18, 1968. Kleitman, D. "On Dedekind’s Problem: The Number of Monotone Boolean Functions." Proc. Amer. Math. Soc. 21, 677/C1/82, 1969. Kleitman, D. and Markowsky, G. "On Dedekind’s Problem: The Number of Isotone Boolean Functions. II." Trans. Amer. Math. Soc. 213, 373/C1/90, 1975. Lunnon, W. F. "The IU Function: The Size of a Free Distributive Lattice." In Combinatorial Mathematics and Its Applications (Ed. D. J. A. Welsh). New York: Aca- demic Press, pp. 173 /C1/81, 1971. Mesalkin, L. D. "A Generalization of Sperner’s Theorem on the Number of Subsets of a Finite Set." Theory Prob. 8, 203/C1/04, 1963. Milner, E. C. "A Combinatorial Theorem on Systems of Sets." J. London Math. Soc. 43, 204/C1/06, 1968. Muroga, S. Threshold Logic and Its Applications. New York: Wiley, p. 38 and 214, 1971. Rivie`re, N. M. "Recursive Formulas on Free Distributive Lattices." J. Combin. Th. 5, 229/C1/34, 1968. Shapiro. "On the Counting Problem for Monotone Boolean Functions." Comm. Pure Appl. Math. 23, 299/C1/12, 1970. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 241, 1990. Sloane, N. J. A. Sequences A006826/M2469, A007153/ M3551, and A014466 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Speciner, M. Item 18 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 10, Feb. 1972. Sperner, E. "Ein Satz u¨ber Untermengen einer endlichen Menge." Math. Z. 27, 544 /C1/48, 1928. Ward, M. "Note on the Order of the Free Distributive Lattice." Bull. Amer. Math. Soc. 52, 423, 1946. Yamamoto, K. "Logarithmic Order of Free Distributive Lattice." J. Math. Soc. Japan 6, 343 /C1/53, 1954. Anticlastic When the GAUSSIAN CURVATURE K is everywhere NEGATIVE ,a SURFACE is called anticlastic and is saddle-shaped. A SURFACE on which K is everywhere POSITIVE is called SYNCLASTIC . A point at which the GAUSSIAN CURVATURE is NEGATIVE is called a HYPER- BOLIC POINT . See also ELLIPTIC POINT ,G AUSSIAN QUADRATURE , HYPERBOLIC POINT ,P ARABOLIC POINT ,P LANAR POINT ,SYNCLASTIC Anticommutative An OPERATOR +forwhich a + b /C30/C28b + a issaidtobe anticommutative. See also COMMUTATIVE Anticommutator For OPERATORS ˜A and ˜B ; the anticommutator is defined by f ˜A; ˜B g/C13 ˜A ˜B /C27 ˜B ˜A: See also COMMUTATOR ,JORDAN ALGEBRA ,JORDAN PRODUCT Anticomplementary Triangle A TRIANGLE DA?B ?C? which has a given TRIANGLE DABC as its MEDIAL TRIANGLE . The TRILINEAR CO-ORDINATES of the anticomplementary triangle are A?/C30/C28 a /C281 : b /C281 : c /C281 B ?/C30a /C281 : /C28b /C281 : c /C281 C?/C30a /C281 : b /C281 : /C28c /C281 : See also MEDIAL TRIANGLE Anticross-Stitch Curve BOX FRACTAL Antiderivative INTEGRAL Antidifferentiation INTEGRATION Antigonal Points Given /C218AXB/C27/C218AYB /C30 p RADIANS in the above fig- ure, then X and Y are said to be antigonal points with respect to A and B. Antihomography A CIRCLE -preserving TRANSFORMATION composed of anODD number of INVERSIONS . See also HOMOGRAPHY Antihomologous Points Two points which are COLLINEAR with respect to a SIMILITUDE CENTER but are not HOMOLOGOUS POINTS . Four interesting theorems from Johnson (1929) fol- low. 1. Two pairs of antihomologous points form in- versely similar triangles with the HOMOTHETIC CENTER . 2. The PRODUCT of distances from a HOMOTHETIC CENTER to two antihomologous points is a con- stant. 3. Any two pairs of points which are antihomolo- gous with respect to a SIMILITUDE CENTER lie on a CIRCLE . 4. The tangents to two CIRCLES at antihomologous points make equal ANGLES with the LINE through the points. See also HOMOLOGOUS POINTS ,HOMOTHETIC CENTER , SIMILITUDE CENTER References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 19 /C1/1, 1929. Antilaplacian The antilaplacian of u with respect to x is a function whose LAPLACIAN with respect to x equals u. The antilaplacian is never unique. See also LAPLACIAN Antilinear An antilinear OPERATOR ˜A satisfies the following two properties: ˜A[f1(x) /C27f2(x)] /C30 ˜Af1(x) /C27 ˜Af2(x) ˜Acf(x) /C30 ˜c ˜Af(x) ; where ˜c is the COMPLEX CONJUGATE of c. See also ANTIUNITARY ,LINEAR OPERATOR References Sakurai, J. J. Modern Quantum Mechanics. Menlo Park, CA: Benjamin/Cummings, 1985. Antilinear Operator An antilinear OPERATOR ˜L/C31u /C30g(y2 ˜Ly1 /C28y1 ˜Ly2) dx /C30p1 p0(y?1y2 /C28y1y?2)"# satisfies the following two properties: PD /C30CB D /C30PE where /C218CPE is the COMPLEX CONJUGATE of Ce :/ See also ANTIUNITARY OPERATOR ,LINEAR OPERATOR References Sakurai, J. J. Modern Quantum Mechanics. Menlo Park, CA: Benjamin/Cummings, 1985.Antilogarithm The INVERSE FUNCTION of the LOGARITHM , defined such that logb(antilogb z) /C30z /C30antilogb(logb z) : The antilogarithm in base b of z is therefore bz: / See also COLOGARITHM ,LOGARITHM ,POWER Antimagic Graph A GRAPH with e EDGES labeled with distinct elements f1; 2 ...; c g so that the SUM of the EDGE labels at each VERTEX differ. See also LABELED GRAPH ,MAGIC GRAPH References Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction. San Diego, CA: Academic Press, 1990. Antimagic Square An antimagic square is an n/C29nARRAY of integers from 1 to n2such that each row, column, and main diagonal produces a different sum such that thesesums form a SEQUENCE of consecutive integers. It is therefore a special case of a HETEROSQUARE . Anti- magic squares of orders 4 /C1are illustrated above (Madachy 1979). For the 4 /C294 square, the sums are 30, 31, 32, ..., 39; for the 5 /C295 square they are 59, 60, 61, ..., 70; and so on.Let an antimagic square of order nhave entries 0, 1, ...,n 2/C282;n2/C281;and let M(n)/C131 2n(n2/C271) be the magic constant. Then if and antimagic square of order nexists, it is either positive with sums [M(n)/C28n;M(n)/C27n/C271];or negative with sums [M(n)/C28n/C281;M(n)/C27n] (Madachy 1979). Antimagic squares of orders one, two, and three are impossible. In the case of the 3 /C293 square, there is no known method of proof of this fact except by case analysis or enumeration by computer. There are 18 families of antimagic squares of order four. The total number of antimagic squares of orders 1, 2, ... modulo the full group of symmetries (reflection, rotation, complementation, and exchanges) are 0, 0, 0, 299710, ... (Sloane’s A050257; Cormie). Abe (1994) and Madachy (1979) ask for methods of constructing antimagic squares of every order. Re- cently, J. Cormie and V. Linek have developed gen- eral constructions for squares of order n for all n /C21 3, as well as for bordering antimagic squares. See also HETEROSQUARE ,M AGIC SQUARE ,TALISMAN SQUARE References Abe, G. "Unsolved Problems on Magic Squares." Disc. Math. 127,3/C1/3, 1994. Cormie, J. "The Anti-Magic Square Project." http://www.u- winnipeg.ca/~jcormie/. Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in Madachy’s Mathematical Recreations. New York: Dover, pp. 103 /C1/13, 1979. Sloane, N. J. A. Sequences A050257 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Magic Squares." MATHEMATICA NOTEBOOK MAGICSQUARES.M . Antimorph A number which can be represented both in the form x2 0 /C28Dy20and in the form Dx21 /C28y2 : 1This is only possible when the PELL EQUATION x2 /C28Dy2 /C30/C281 is solvable. Then x2/C28Dy2/C30/C28(x0/C28Dy2 0)(x2n/C28Dy2n) /C30D(x0yn/C28y0xn)2/C28(x0xn/C28Dy0yn)2: See also IDONEAL NUMBER ,POLYMORPH References Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. New York: Dover, 1964. Antimorphic Number ANTIMORPH Antinomy APARADOX or contradiction.Antiparallel Two lines PQandRSare said to be antiparallel with respect to the sides of an ANGLE Aif they make the same angle in the opposite senses with the BISECTOR of that angle. If PQand RSare antiparallel with respect to PR and QS, then the latter are also antiparallel with respect to the former. Furthermore, ifPQandRSare antiparallel, then the points P,Q, R, and Sare CONCYCLIC (Johnson 1929, p. 172; Honsberger 1995, pp. 87 /C1/8). There are a number of fundamental relationshipsinvolving a triangle and antiparallel lines (Johnson 1929, pp. 172 /C1 /73). 1. The line joining the feet to two ALTITUDES of a triangle is antiparallel to the third side.2. The tangent to a triangle’s CIRCUMCIRCLE at a vertex is antiparallel to the opposite side. 3. The radius of the CIRCUMCIRCLE at a vertex is perpendicular to all lines antiparallel to the opposite sides. In a TRIANGLE DABC ;aSYMMEDIAN BKbisects all segments antiparallel to a given side AC(Honsberger 1995, p. 88). Furthermore, every antiparallel to BCin DABC isPARALLEL to the tangent to the CIRCUMCIR- CLEofDABC atA(Honsberger 1995, p. 98). See also ANGLE ,CONCYCLIC ,COSINE CIRCLE ,COSINE HEXAGON ,H YPERPARALLEL ,L EMOINE CIRCLE ,L E- MOINE HEXAGON ,PARALLEL ,TUCKER CIRCLES ,TUCK- ER HEXAGON References Casey, J. "Theory of Isogonal and Isotomic Points, and of Antiparallel and Symmedian Lines." Supp. Ch. §1in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 165 /C1/73, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 65, 1971. Honsberger, R. "Parallels and Antiparallels." §9.1 in Epi- sodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 87 /C1/ 8, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 172, 1929. Lachlan, R. §113 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 63, 1893. Phillips, A. W. and Fisher, I. Elements of Geometry. New York: American Book Co., 1896. Antipedal Triangle The antipedal triangle A of a given TRIANGLE T is the TRIANGLE of which T is the PEDAL TRIANGLE . For a TRIANGLE with TRILINEAR COORDINATES a : b : g and ANGLES A, B, and C, the antipedal triangle has VERTICES with TRILINEAR COORDINATES /C28( b /C27 a cos C)(g /C27 a cos B):(g /C27 a cos B)(a /C27 b cos C): ( b /C27 a cos C)(a /C27 g cos B) ( g /C27 b cos A)(b /C27 a cos C):/C28( g /C27 b cos A)(a /C27 b cos C): (a /C27 b cos C)( b /C27 g cos A) ( b /C27 g cos A)( g /C27 a cos B):(a /C27 g cos B)( g /C27 b cos A): /C28( a /C27 g cos B)(b /C27 g cos A): The ISOGONAL CONJUGATE of the ANTIPEDAL TRIANGLE of a given TRIANGLE is HOMOTHETIC with the original TRIANGLE . Furthermore, the PRODUCT of their AREASequals the SQUARE of the AREA of the original TRIANGLE (Gallatly 1913). See also PEDAL TRIANGLE References Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 56 /C1/8, 1913. Antipersistent Process A FRACTAL PROCESS for which H B1 =2; so r B0. See also PERSISTENT PROCESS Antipodal Map The MAP which takes points on the surface of a SPHERE S2 to their ANTIPODAL POINTS . Antipodal Points Two points are antipodal (i.e., each is the ANTIPODE of the other) if they are diametrically opposite. Exam- ples include endpoints of a LINE SEGMENT , or poles of a SPHERE . Given a point on a SPHERE with LATITUDE d and LONGITUDE l; the antipodal point has LATITUDE /C28d and LONGITUDE l 9180/C14 (where the sign is taken so that the result is between /C281808 and /C27180/C14):/ See also ANTIPODE ,B ORSUK- ULAM THEOREM ,D IA- METER ,G REAT CIRCLE ,LYUSTERNIK- SCHNIRELMANN THEOREM ,METEOROLOGY THEOREM ,SPHERE Antipode Given a point A, the point B which is the ANTIPODAL POINT of A is said to be the antipode of A. See also ANTIPODAL POINTS References Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, p. 25, 1965. Antiprism ASEMIREGULAR POLYHEDRON constructed with 2 n- gons and 2 nTRIANGLES . The nets are particularly simple, consisting of two n-gons on top and bottom, separated by a ribbon of 2 ntriangles, with the two n- gons being offset by one ribbon segment. The SAGITTA of a regular n-gon of side length ahas length s/C301 2atanp 2n ! (1) Letdbe the length of a lateral edge when the top and bottom bases separated by a distance h, then s2/C27(1 2a)2/C27h2/C30d2; (2) so d/C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4h2/C27a2sec2p 2n !vuut: (3) For an antiprism of side lengths 1, a/C30d/C301;and solving for hgives h/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281 4sec2p 2n !vuut: (4) TheCIRCUMRADIUS Rcircof an antiprism is given by Rcirc/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2hrC16rC1*2 /C27R2r /C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 csc2p 2n !vuut; (5) where R/C301 2cscp n ! (6) is the CIRCUMRADIUS of one of the bases. The TETRAHEDRON can be considered a degenerate 2- antiprism and the 3-antiprism of heightffiffiffi 6p a=3 (for side length a) is simply the OCTAHEDRON . The first few heights hnproducing unit antiprisms for a/C301 are h3/C301 2ffiffiffi 6p (7) h4/C3021=4(8) h5/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10(5/C27ffiffiffi5p )q (9)h 6/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3p /C281q (10) h 8/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C277 2ffiffiffi 2pq /C281/C28ffiffiffi2pr : (11) The DUALS are the TRAPEZOHEDRA . The SURFACE AREA of an-gonal antiprism is S/C302An/C28gon/C272nAD /C3021 4na2cotp n !"# /C272n12arC16rC1*ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2/C27h2p /C301 2na a cotp n ! /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C271 4a2tan2p 2n !vuut2 435:(12) Ifh/C30a, this simplifies to S/C30 1 2na2cotp n ! /C27ffiffiffi 3p"# : (13) The first few are S3/C302ffiffiffi3p (14) S 4/C302(1/C27ffiffiffi 3p ) (15) S5/C301 25ffiffiffi 3p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pqrC1+rC1D (16) S6/C306ffiffiffi3p (17) S 8/C304(1/C27ffiffiffi 2p /C27ffiffiffi 3p ): (18) To find the volume, label vertices as in the above figure. Then the vectors v1andv2are given by v1/C30(/C28s;1 2a;h) (19) v2/C30(/C28s;/C281 2a;h); (20) so the normal to one of the lateral facial planes is n/C30v1/C29v2/C30(ah;0;as); (21) and the unit normal is ˆn/C30v1/C29v2 ½v1/C29v2½ /C30ahffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2(h2/C27s2)p ;0;asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2(h2/C27s2)p ! : (22) The height of a pyramid with apex at the center and having the triangle determined by x1and x2as the base is then given by the projection of a vector from the origin to a point on the plane onto the normal, hpyr /C30 ˆu /C215 (R /C28s ;/C281 2 a ;12 h) /C30 ˆu /C215 (R /C28s ;/C2812 a;12 h) /C30 ˆu /C215 (R; 0 ;12 h) (23) /C30a2h cotrC1+p 2nrC1D 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2rC00 h2 /C271 4 a2 tan2rC1+p 2nrC1DsrC01: (24) The total volume of the 2n pyramids having the lateral faces as bases is therefore Vpyr /C30(2n)13 hpyr(12 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 /C27h2)phi /C301 12 a2h cotrC1+p 2nrC1D (25) Plugging in h and setting a /C301 gives Vpyr /C301 12 n cotp 2n !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C281 4sec2p 2n !vuut: (26) The two pyramids having the upper and lower surfaces as bases contribute a volume V hase /C3021 2rC16rC1* 12 hrC16rC1* 14 na2 cotp n !"# /C301 12 na2 h cotp n ! : (27) Combining the two, setting a /C301, and plugging in the height h to get unit lateral edges gives the total volume as the somewhat complicated expression V /C301 12 n cotp 2n ! /C27cotp n ! "#ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C281 4sec2p 2n !vuut: (28) The volumes of the first few unit antiprisms are therefore given by V 3 /C301 3ffiffiffi 2p (29) V4 /C301 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C273ffiffiffi 2pq (30) V5 /C301 6(5 /C272ffiffiffi 5p ) (31)v6 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 21/C27ffiffiffi 3prC16rC1*r (32) See also GYROELONGATED PYRAMID ,O CTAHEDRON , PRISM ,PRISMOID ,TRAPEZOHEDRON References Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 130, 1987. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 149, 1969. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 85 /C1/6, 1997. Pedagoguery Software. Poly . http://www.peda.com/poly/. Weisstein, E. W. "SolidGeometry." MATHEMATICA NOTEBOOK SOLIDGEOMETRY.M . Antiquity GEOMETRIC PROBLEMS OF ANTIQUITY Antiset A SET which transforms via converse functions. Antisets usually arise in the context of CHU SPACES . See also CHU SPACE ,SET References Stanford Concurrency Group. "Guide to Papers on Chu Spaces." http://boole.stanford.edu/chuguide.html. Antisnowflake KOCH ANTISNOWFLAKE Antisphere PSEUDOSPHERE Antisquare Number A number OF THE FORM pa /C215 A is said to be an antisquare if it fails to be a SQUARE NUMBER for the two reasons that a is ODD and A is a nonsquare modulo p. See also SQUARE NUMBER ,SQUAREFREE ,SQUAREFUL Antisymmetric A quantity which changes SIGN when indices are reversed. For example, Aij /C13ai /C28ajis antisymmetric since Aij/C30/C28Aji:/ See also ANTISYMMETRIC MATRIX ,A NTISYMMETRIC TENSOR ,SYMMETRIC Antisymmetric Matrix An antisymmetric matrix is a MATRIX which satisfies the identity A/C30/C28AT(1) where ATis the matrix TRANSPOSE . A matrix mmay be tested to see if it is antisymmetric using the Mathematica function AntisymmetricQ[m_List?MatrixQ] : /C30 (m /C30/C30/C30 - Transpose[m]) In component notation, this becomes aij /C30/C28aji : (2) Letting k /C30i /C30j; the requirement becomes akk /C30/C28akk ; (3) so an antisymmetric matrix must have zeros on its diagonal. The general 3 /C293 antisymmetric matrix is OF THE FORM 0 a12 a13 /C28a12 0 a23 /C28a13/C28a23 02 435: (4) Applying A /C281to both sides of the antisymmetry condition gives /C28A /C281AT /C301 : (5) Any SQUARE MATRIX can be expressed as the sum of symmetric and antisymmetric parts. Write A /C301 2(A /C27AT) /C2712(A /C28AT): (6) But A /C30a11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n nn::: n an1an2/C1/C1/C1 ann2 6643 775 (7) AT/C30a11a21 /C1/C1/C1 an1 a12a22 /C1/C1/C1 an2 nn::: n a1na2n/C1/C1/C1 ann2 6643 775; (8) so A /C27AT /C302a11 a12 /C27a21 /C1/C1/C1 a1n /C27an1 a12 /C27a21 2a22 /C1/C1/C1 a2n /C27an2 nn::: n a1n /C27an1a2n /C27an2/C1/C1/C1 2ann2 6643 775; (9) which is symmetric, and A /C28A T /C300 a12 /C28a21 /C1/C1/C1 a1n /C28an1 /C28(a12 /C28a21)0 /C1/C1/C1 a2n /C28an2 nn::: n /C28(a1n /C28an1) /C28(a2n /C28an2) /C1/C1/C1 02 6643 775; (10) which is antisymmetric. See also SKEW SYMMETR IC MATRIX ,S YMMETR IC MATRIX Antisymmetric Relation A RELATION R on a SET S is antisymmetric provided that distinct elements are never both related to oneanother. In other words xRy and yRx together imply that x /C30y. Antisymmetric Tensor An antisymmetric (also called alternating) tensor is a TENSOR which changes sign when two indices are switched. For example, a tensor Ax1 ;/C1/C1/C1;xn such that Ax1 ;/C1/C1/C1; xi ;/C1/C1/C1; xj ;/C1/C1/C1; xn /C30/C28Ax1 ;/C1/C1/C1; xj ;/C1/C1/C1; xi ;/C1/C1/C1; xn (1) is antisymmetric. The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies Amn /C30/C28Anm : (2) Furthermore, any rank-2 TENSOR can be written as a sum of SYMMETRIC and antisymmetric parts as Amn /C301 2(Amn /C27Anm) /C2712(Amn /C28Anm) : (3) The antisymmetric part of a tensor Aab is sometimes denoted using the special notation A ab½/C138/C301 2(Aab /C28Aba) : (4) For a general rank- n TENSOR , A a1 /C1/C1/C1an ½/C138/C131 n!ea1 /C1/C1/C1anX permutationsAa1 /C1/C1/C1an ; (5) where ea1 /C1/C1/C1anis the PERMUTATION SYMBOL . Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example T(ab)c d½/C138/C3014(Tabc de/C27Tbacde/C28Tabced/C28Tbaced) : (6) (Wald 1984, p. 26). See also ALTERNATING MULTILINEAR FORM,EXTERIOR ALGEBRA ,SYMMETRIC TENSOR ,W EDGE PRODUCT References Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984. Antiunitary An operator ˜A which satisfies: ˜Af1 ½ ˜Af2rC10rC11 /C30f1 ½f2 hi ˜A[f1(x) /C27f2(x)] /C30 ˜Af1(x) /C27 ˜Af2(x) ˜Acf(x) /C30 ˜c ˜Af(x) ; where f ½ghi is the INNER PRODUCT and ˜c is the COMPLEX CONJUGATE ofc. See also ANTILINEAR ,UNITARY References Sakurai, J. J. Modern Quantum Mechanics. Menlo Park, CA: Benjamin/Cummings, 1985. Antiunitary Operator An operator ˜B which satisfies: 2ffiffiffi 3p /C30S4 91 /C30C1 f : M 0 M /C30V( f) where 2(1 /C27ffiffiffi 3p ) is the INNER PRODUCT andxn/C271 yn/C271rC00rC01 /C30 21 11rC00rC01 xn ynrC00rC01 is the COMPLEX CONJUGATE of Ce :/ See also ANTILINEAR OPERATOR ,UNITARY OPERATOR References Sakurai, J. J. Modern Quantum Mechanics. Menlo Park, CA: Benjamin/Cummings, 1985. Antoine’s Horned Sphere A topological 2-sphere in 3-space whose exterior is not SIMPLY CONNECTED . The outer complement of Antoine’s horned sphere is not SIMPLY CONNECTED . Furthermore, the group of the outer complement is not even finitely generated. Antoine’s horned sphere is inequivalent to ALEXANDER’S HORNED SPHERE sin- ce the complement in R3 of the bad points for ALEX- ANDER’S HORNED SPHERE is SIMPLY CONNECTED . See also ALEXANDER’S HORNED SPHERE References Alexander, J. W. "An Example of a Simply-Connected Sur- face Bounding a Region which is not Simply-Connected." Proc. Nat. Acad. Sci. 10,8/C10, 1924. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 76 /C19, 1976. Antoine’s Necklace Construct a chain C of 2n components in a solid TOR- US V. Now form a chain C1of 2n solid tori in V, where p1(V /C28C1) $ p1(V /C28C) via inclusion. In each component of C1 ; construct a smaller chain of solid tori embedded in that compo- nent. Denote the union of these smaller solid tori C2 : Continue this process a countable number of times,then the intersection A /C30S/C12 i/C301Ci which is a nonempty compact SUBSET of R3 is called Antoine’s necklace. Antoine’s necklace is HOMEO- MORPHIC with the CANTOR SET. See also ALEXANDER’S HORNED SPHERE ,NECKLACE References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 73 /C14, 1976. Apeirogon The REGULAR POLYGON essentially equivalent to the CIRCLE having an infinite number of sides and denoted with SCHLA ¨ FLI SYMBOL f/C12g:/ See also CIRCLE ,REGULAR POLYGON References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Schwartzman, S. The Words of Mathematics: An Etymologi- cal Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994. Ape´ry Number The numbers defined by An/C30Xn k/C300n krC1+rC1D2n/C27k krC1+rC1D2 /C30Xn k/C300[(n/C27k!]2 (k!)4[(n/C28k)!]2; (1) wheren krC0rC1 is a BINOMIAL COEFFICIENT . The first few for n/C300, 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (Sloane’s A005259). They are also given by the RECURRENCE RELATION an/C30(34n3/C2851n2/C2727n/C285)an/C281/C27(n/C281)3an/C282 n3 (2) (Beukers 1987). There is also an associated set of numbers Bn/C30Xn k/C300n krC1+rC1D2n/C27k krC1+rC1D (3) (Beukers 1987). The values for n/C300, 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (Sloane’s A005258). Both AnandBnarose in Ape ´ry’s irrationality proof of z(2) and z(3) (van der Poorten 1979, Beukers 1987). They satisfy some surprising congruence properties, Ampr/C281/C13Ampr/C281/C281(mod p3r) (4) Bmpr/C281/C13Bmpr/C281/C281(mod p3r) (5) forpaPRIME]5 and m;reN(Beukers 1985, 1987), as well as B(p/C281)=2/C134a2/C282p(mod p)i f p/C30a2/C27b2;aodd 0 (mod p)i f p/C133 (mod 4)rC06 (Stienstra and Beukers 1985, Beukers 1987). Defin- inggnfrom the GENERATING FUNCTION X/C12 n/C301gnqn/C30qY/C12 n/C301(1/C28q2n)4(1/C28q4n)4(6) gives gnof 1, -4, -2, 24, -11, -44, ... (Sloane’s A030211; Koike 1984) for n/C301, 3, 5, ..., and A(p/C281)=2/C13gp(mod p) (7) forpanODD PRIME (Beukers 1987). Furthermore, for panODD PRIME andm;reN; A(mpr/C281)=2/C28gpA(mpr/C281/C281)=2/C27p3Ampr/C282/C281)=2/C130 (mod pr) (8) (Beukers 1987). The Ape ´ry numbers are given by the diagonal elements An/C30Annin the identity Amn/C30X/C12 k/C30/C28/C12X/C12 j/C30/C28/C12m krC1+rC1D2m krC1+rC1D22m/C27n/C28j/C28k 2mrC1+rC1D /C30X/C12 k/C30/C28/C12m/C27n/C28k krC1+rC1D2m/C27n/C282k m/C28krC1+rC1D2 /C30X/C12 k/C30/C28/C12m krC1+rC1D n krC1+rC1D m/C27k krC1+rC1D n/C27k krC1+rC1D (9) (Koepf 1998, p. 119). References Ape´ry, R. "Irrationalite ´dez(2) et z(3):/"Aste´risque 61,1 1/C1/3, 1979. Ape´ry, R. "Interpolation de fractions continues et irrationa- lite´de certaines constantes." Mathe ´matiques, Ministe `re universite ´s (France), Comite ´travaux historiques et scien- tifiques. Bull. Section Sciences 3, 243/C1/46, 1981. Beukers, F. "Some Congruences for the Ape ´ry Numbers." J. Number Th. 21, 141/C1/55, 1985. Beukers, F. "Another Congruence for the Ape ´ry Numbers." J. Number Th. 25, 201/C1/10, 1987. Chowla, S.; Cowles, J.; and Cowles, M. "Congruence Proper- ties of Ape ´ry Numbers." J. Number Th. 12, 188/C1/90, 1980. Gessel, I. "Some Congruences for the Ape ´ry Numbers." J. Number Th. 14, 362/C1/68, 1982. Koepf, W. "Hypergeometric Identities." Ch. 2 in Hypergeo- metric Summation: An Algorithmic Approach to Summa- tion and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 29 and 119, 1998. Koike, M. "On McKay’s Conjecture." Nagoya Math. J. 95, 85/C1/9, 1984. Sloane, N. J. A. Sequences A005258/M3057, A005259/ M4020, and A030211 in "An On-Line Version of theEncyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Stienstra, J. and Beukers, F. "On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic K3 Surfaces." Math. Ann. 271, 269/C1 /04, 1985. van der Poorten, A. "A Proof that Euler Missed... Ape ´ry’s Proof of the Irrationality of z(3):/"Math. Intel. 1, 196/C1/03, 1979.Ape´ry’s Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Ape´ry’s constant is defined by z(3)/C301:2020569 . . . ; (1) (Sloane’s A002117) where z(z) is the R IEMANN ZETA FUNCTION . Ape ´ry (1979) proved that z(3) is IRRA- TIONAL , although it is not known if it is TRANSCEN- DENTAL . Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for theirrationality of z(3) (Hata 2000). z(3) arises naturally in a number of physical problems, including in thesecond- and third-order terms of the electron’s gyro- magnetic ratio, computed using quantum electrody- namics. The CONTINUED FRACTION forz(3) is [1, 4, 1, 18, 1, 1, 1, 4, 1, ...] (Sloane’s A013631). The positions at which the numbers 1, 2, ... occur in the continued fraction are 1, 12, 25, 2, 64, 27, 17, 140, 10, ... (Sloane’sA033165). The incrementally maximal terms are 1, 4, 18, 30, 428, 458, 527, ... (Sloane’s A033166), which occur at positions 1, 2, 4, 29, 63, 572, ... (Sloane’sA033167). The following table summarized progress in comput- ing upper bounds on the IRRATIONALITY MEASURE for z(3):Here, the exact values for two of the numerical bounds are given by m1/C301/C276l nc0/C27d0 6l nc0/C28d0:7:377956 (2) m4/C301/C274 ln(ffiffiffi 2p /C271)/C273 4 ln(ffiffiffi 2p /C271)/C283:13:4178202 ; (3) where c0/C301 9(362/C27133ffiffiffi 7p ) (4) d0/C3026/C27pffiffiffi 3p /C28cot(1 9p)/C28cot(29p)hi (5) (Hata 2000). index upper boundreference 1 7.377956 Hata (2000) 2 8.830284 Hata (1990) 3 12.74359 Dvornicich and Viola (1987)4 13.41782 Sorokin (1994), Nesterenko (1996), Pre ´vost (1996) Beukers (1979) reproduced Ape ´ry’s rational approx- imation to z(3) using the triple integral of the form g1 0g1 0g1 0Ln(x)Ln(y) 1/C28(1/C28xy)udx dy du ; (6) where Ln(x)i saL EGENDRE POLYNOMIAL . This inte- gral is closely related to z(3) using the curious identity g1 0g1 0g1 0xrys 1/C28(1/C28xy)udx dy du /C302z(3)/C28Pr l/C3012 l3forr/C30s Pmax( r;s) 1/C30min( r;s)/C2711 r/C28sl2forr"srC10rC10rC10rC108 >>>< >>>: /C302z(3)/C28H (3) r forr/C30s c1(1/C27min( r;s))/C28c1(1/C27max( r;s)) r/C28s jjforr"s;8 < : where H(n) ris a generalized HARMONIC NUMBER and ck(x)i sa POLYGAMMA FUNCTION (Hata 2000). Sums related to z(3) are z(3)/C305 2X/C12 n/C301(/C281)n/C281 n32n nrC1+rC1D/C3052X /C12 k/C301(/C281)k/C271(k!)2 (2k)!k3(7) (used by Ape ´ry), and l(3)/C30X/C12 k/C3001 (2k/C271)3/C307 8z(3) (8) X/C12 k/C3001 (3k/C271)3/C302p3 81ffiffiffi 3p/C2713 27z(3) (9) X/C12 k/C3001 (4k/C271)3/C30p3 64/C277 16z(3) (10) X/C12 k/C3001 (6k/C271)3/C30p3 36ffiffiffi 3p/C2791 216z(3); (11) where l(z) is the D IRICHLET LAMBDA FUNCTION . The above equations are special cases of a general result due to Ramanujan (Berndt 1985). Ape ´ry’s proof relied on showing that the sum a(n)/C13Xn k/C300n krC1+rC1D2n/C27k krC1+rC1D2 ; (12) wheren krC0rC1 is a BINOMIAL COEFFICIENT , satisfies the RECURRENCE RELATION (n/C271)3a(n/C271)/C28(34n3/C2751n2/C2727n/C275)a(n) /C27n3a(n/C281)/C300 (13) (van der Poorten 1979, Zeilberger 1991). The char- acteristic polynomial x2/C2834x/C271 has roots (1 /C27 9ffiffiffi 2p )4;so lim n0/C12an/C271 an/C30(1/C27ffiffiffi 2p )4(14)is irrational and ancannot satisfy a two-term recur- rence (Jin and Dickinson 2000). Ape´ry’s constant is also given by z(3)/C30X/C12 n/C301Sn;2 n!n; (15) where Sn;mis a S TIRLING NUMBER OF THE FIRST KIND . This can be rewritten as z(3)/C301 2X/C12 n/C3011 n21/C2712/C27.../C271 n ! /C3012X /C12 n/C301Hn n2;(16) where Hnis the nthHARMONIC NUMBER (Castellanos 1988). INTEGRALS forz(3) include z(3)/C301 2g/C12 0t2 et/C281dt (17) /C30871 4p2ln 2/C272gx=4 0xln(sin x)dx"# : (18) Gosper (1990) gave z(3)/C301 4X/C12 k/C30130k/C2811 (2k/C281)k32k krC1+rC1D2: (19) ACONTINUED FRACTION involving Ape ´ry’s constant is 6 z(3)/C305/C2816 117/C2826 535/C28/C1/C1/C1n6 34n3/C2751n2/C2727n/C275/C28/C1/C1/C1 (20) (Ape´ry 1979, Le Lionnais 1983). Amdeberhan (1996) used W ILF-ZEILBERGER PAIRS (F, G ) with F(n;k)/C30(/C281)kk!2(sn/C28k/C281)! (sn/C27k/C271)!(k/C271); (21) s/C301 to obtain z(3)/C3052X /C12 n/C301(/C281)n/C281 1 2n nrC1+rC1D n3; (22) Fors/C302, z(3)/C301 4X/C12 n/C301(/C281)n/C28156n2/C2832n/C275 (2n/C281)21 3n nrC1+rC1D 2n nrC1+rC1D n3 (23) and for s/C303, z(3)/C30X/C12 n/C300(/C281)n 724n nrC1+rC1D 3n nrC1+rC1D /C26120 n/C275265 n4/C2713761 n2/C2713878 n3/C271040 (4n/C271)(4n/C273)(n/C271)(3n/C271)2(3n/C272)2(24) (Amdeberhan 1996). The corresponding G(n;k) for s /C301 and 2 are G(n; k) /C302(/C281)kk!2(n /C28 k)! (n /C27 k /C27 1)!(n /C27 1)2 (25) and G(n; k) /C30( /C281)kk!2(2n /C28 k)!(3 /C27 4n)(4n2 /C27 6n /C27 k /C27 3) 2(2n /C27 k /C27 2)!(n /C27 1)2(2n /C27 1)2 : (26) Gosper (1996) expressed z(3) as the MATRIX PRODUCT lim N 0/C12YN n/C301Mn /C30 0 z(3) 01rC00rC01 ; (27) where Mn /C13 (n /C27 1)4 4006( n /C275 4)2(n /C2774)224570 n4 /C27 64101 n3 /C27 62152 n2 /C27 26427 n /C27 4154 31104( n /C2713)(n /C2712)(n /C2723) 012 643 75 (28) which gives 12 bits per term. The first few terms are M1 /C301 106002077 1728 012 435 (29) M 2 /C301 98017501 4320 012 435 (30) M 3 /C309 6760050501 20160 012 435; (31) which gives z(3) : 423203577229 352066176000 /C301 :20205690315732... (32) Given three INTEGERS chosen at random, the prob- ability that no common factor will divide them all is z(3)½/C138/C281:1 :20206 /C281 :0:831907 : (33) B. Haible and T. Papanikolaou computed z(3) to 1,000,000 DIGITS using a WILF-ZEILBERGER PAIR identity with F(n; k) /C30(/C281)kn!6(2n /C28 k /C28 1)!k!3 2(n /C27 k /C27 1)!2(2n)!3 ; (34) s /C301, and t /C301, giving the rapidly converging z(3) /C30X/C12 n/C300(/C281)nn!10(205n2/C27250n/C2777) 64(2n/C271)!5(35) (Amdeberhan and Zeilberger 1997). The record as of Dec. 1998 was 128 million digits, computed by S. Wedeniwski. See also RIEMANN ZETA FUNCTION ,TRILOGARITHM , WILF-ZEILBERGER PAIRReferences Amdeberhan, T. "Faster and Faster Convergent Series for z(3):/"Electronic J. Combinatorics 3, R13 1 /C1/, 1996. http:// www.combinatorics.org/Volume_3/volume3.html#R13. Amdeberhan, T. and Zeilberger, D. "Hypergeometric Series Acceleration via the WZ Method." Electronic J. Combina- torics 4, No. 2, R3, 1 /C1/, 1997. http://www.combinatoric- s.org/Volume_4/wilftoc.html#R03. Also available at http:// www.math.temple.edu/~zeilberg/mamarim/mamar-imhtml/accel.html. Ape´ry, R. "Irrationalite ´dez(2) et z(3): /"Aste´risque 61,1 1/C1/3, 1979. Berndt, B. C. Ramanujan’s Notebooks: Part I. New York: Springer-Verlag, 1985. Beukers, F. "A Note on the Irrationality of z(3):/"Bull. London Math. Soc. 11, 268/C1/72, 1979. Beukers, F. "Another Congruence for the Ape ´ry Numbers." J. Number Th. 25, 201/C1/10, 1987. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.New York: Wiley, 1987. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1 /8, 1988. Conway, J. H. and Guy, R. K. "The Great Enigma." In The Book of Numbers. New York: Springer-Verlag, pp. 261 /C1/ 62, 1996. Dvornicich, R. and Viola, C. "Some Remarks on Beukers’ Integrals." In Number Theory, Colloq. Math. Soc. Ja ´nos Bolyai, Vol. 51 . Amsterdam, Netherlands: North-Holland, pp. 637 /C1/57, 1987. Ewell, J. A. "A New Series Representation for z(3):/"Amer. Math. Monthly 97, 219/C1/20, 1990. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/apery/apery.html. Gosper, R. W. "Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics." In Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Dekker, 1990. Gutnik, L. A. "On the Irrationality of Some Quantities Containing z(3):/"Acta Arith. 42, 255/C1/64, 1983. English translation in Amer. Math. Soc. Transl. 140,4 5/C1/5, 1988. Haible, B. and Papanikolaou, T. "Fast Multiprecision Eva- luation of Series of Rational Numbers." Technical ReportTI-97/C1 /. Darmstadt, Germany: Darmstadt University of Technology, Apr. 1997. Hata, M. "A New Irrationality Measure for z(3):/"Acta Arith. 92,4 7/C1/7, 2000. Jin, Y. and Dickinson, H. "Ape ´ry Sequences and Legendre Transforms." J. Austral. Math. Soc. Ser. A 68, 349/C1/56, 2000. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 36, 1983. Nesterenko, Yu. V. "A Few Remarks on z(3):/"Mat. Zametki 59, 865/C1/80, 1996. English translation in Math. Notes 59, 625/C1/36, 1996. Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u-qam.ca/pi/records.html. Pre´vost, M. "A New Proof of the Irrationality of z(2) and z(3) using Pade ´Approximants." J. Comput. Appl. Math. 67, 219/C1 /35, 1996. Sloane, N. J. A. Sequences A002117/M0020, A013631, A033165, A033166, and A033167 in "An On-Line Versionof the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Sorokin, V. N. "Hermite-Pade ´Approximations for Nikishin Systems and the Irrationality of z(3): /"Uspekhi Mat. Nauk 49, 167/C1/68, 1994. English translation in Russian Math. Surveys 49, 176/C1/77, 1994. van der Poorten, A. "A Proof that Euler Missed... Ape´ry’s Proof of the Irrationality of z(3):/" Math. Intel. 1, 196 /C1/03, 1979. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 33, 1986. Zeilberger, D. "The Method of Creative Telescoping." J. Symb. Comput. 11, 195 /C1/04, 1991. Aphylactic Projection A term sometimes used to describe a MAP PROJECTION which is neither EQUAL-AREA nor CONFORMAL (Lee 1944; Snyder 1987, p. 4). See also CONFORMAL MAPPING ,EQUAL- AREA PROJEC- TION ,MAP PROJECTION References Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190 /C1/00, 1944. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987. Apoapsis The greatest radial distance of an ELLIPSE as mea- sured from a FOCUS . Taking v /C30 p in the equation of an ELLIPSE r /C30a(1 /C28 e2) 1 /C27 e cos v gives the apoapsis distance r/C27/C30a(1 /C27e): Apoapsis for an orbit around the Earth is called apogee, and apoapsis for an orbit around the Sun is called aphelion. See also ECCENTRICITY ,ELLIPSE ,FOCUS ,PERIAPSIS Apocalypse Number A number having 666 DIGITS (where 666 is the BEAST NUMBER ) is called an apocalypse number. The FIBO- NACCI NUMBER F3184 is an apocalypse number. See also APOCALYPTIC NUMBER ,B EAST NUMBER , LEVIATHAN NUMBER References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97 /C1/ 02, 1995. Apocalyptic Number A number OF THE FORM 2n which contains the digits 666 (the BEAST NUMBER ) is called an APOCALYPTICNUMBER .2157 is an apocalyptic number. The first few such powers are 157, 192, 218, 220, ... (Sloane’s A007356). See also APOCALYPSE NUMBER ,B EAST NUMBER , LEVIATHAN NUMBER References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97 /C1/ 02, 1995. Sloane, N. J. A. Sequences A007356/M5405 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M5405 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Apodization The application of an APODIZATION FUNCTION . Apodization Function A function (also called a TAPERING FUNCTION ) used to bring an interferogram smoothly down to zero at the edges of the sampled region. This suppresses side-lobes which would otherwise be produced, but at the expense of widening the lines and therefore decreas- ing the resolution. The following are apodization functions for symme- trical (2-sided) interferograms, together with the INSTRUMENT FUNCTIONS (or APPARATUS FUNCTIONS ) they produce and a blowup of the INSTRUMENT FUNCTION sidelobes. The INSTRUMENT FUNCTION I(k) corresponding to a given apodization function A(x) can be computed by taking the finite F OURIER COSINE TRANSFORM , I(k)/C30ga /C28acos(2 pkx)A(x)dx: (1) Type Apodization FunctionINSTRUMENT FUNCTION BARTLETT /1/C28xjj a// asinc2(pka)/ BLACKMAN /BA(x)// B1(k)/ CONNES /1/C28x2 a2rC1+rC1D 2 // 8affiffiffiffiffiffi 2pp J5=2(2pka) (2pka)5=2/ COSINE /cospx 2arC1+rC1D //4acos(2 pak) p(1/C2816a2k2)/ GAUSSIAN /e/C28x2=(2a2)//2fa 0cos(2 pkx)e/C28x2=(2s2)dx/ HAMMING /HmA(x)// HmI(k)/ HANNING /HnA(x)// HnI(k)/ UNIFORM 1 /2asinc(2 pka)/ WELCH /1/C28x2 a2// WI(k)/ where BA(x)/C300:42/C270:5cospx a ! /C270:08cos2px a ! (2) BI(k) /C30a(0:84/C280:36a2k2/C282:17/C2910/C2819a4k4)sinc(2 pak) (1/C28a2k2)(1/C284a2k3) (3) HmA(x)/C300:54/C270:46cospx a ! (4) HmI(k)/C30a(1:08/C280:64a2k2)sinc(2 pak) 1/C284a2k2(5) HnA(x)/C30cos2px 2a ! (6) /C301 21/C27cospx a !"# (7) HnI(k)/C30asinc(2 pak) 1/C284a2k2(8) /C30a[sinc(2 pka)/C2712sinc(2 pka/C28p)/C2712sinc(2 pka}p)] (9)W I(k)/C30a2ffiffiffiffiffiffi 2pp J3=2(2pka) (2pka)3=2(10) /C30asin(2pka)/C282pakcos(2 pak) 2a3k3p3: (11) Type Instrument Function FWHMIF Peak/Peak(/C28)Sidelobe Peak//Peak(/C27)Sidelobe Peak/ Bartlett 1.77179 1 0.00000000 /0:0471904 / Blackman 2.29880 0.84 //C280:00106724 / 0.00124325 Connes 1.90416 /16 15///C280:0411049 // 0:0128926 / Cosine 1.63941 /4 p///C280:0708048 // 0:0292720 / Gaussian – 1 – – Hamming 1.81522 1.08 //C280:00689132 / 0.00734934 Hanning 2.00000 1 //C280:0267076 / 0.00843441 Uniform 1.20671 2 //C280:217234 // 0:128375 / Welch 1.59044 /4 3///C280:0861713 // 0:356044 / A general symmetric apodization function A(x) can be written as a F OURIER SERIES A(x)/C30a0/C272X/C12 n/C301ancosnpx b ! : (12) where the COEFFICIENTS satisfy a0/C272X/C12 n/C301an/C301: (13) The corresponding apparatus function is I(t)/C13gb /C28bA(x)e/C282pikxdx/C302bfa0sinc(2 pkb) /C27X/C12 n/C301[sinc(2 pkb/C27np)/C27sinc(2 pkb/C28np)]g: (14) To obtain an APODIZATION FUNCTION with zero at ka/C303=4;use a0sinc(3 2pÞ/C27a1[sinc(52p)/C27sinc(12p)/C300: (15) Plugging in (14), /C28(1/C282a1)2 3p/C27a12 5p/C272 p ! /C30/C281 3(1/C282a1)/C27a1(15/C271)/C300 (16) a1(6 5/C2723)/C3013 (17) a1 /C301 3 6 5 /C2723/C305 6 /C215 3 /C27 2 /C215 5 /C305 28 (18) a0 /C301 /C282a1 /C3028 /C28 2 /C215 5 28/C301828 /C309 14 : (19) The HAMMING FUNCTION is close to the requirement that the APPARATUS FUNCTION goes to 0 at ka /C305=4; giving a0 /C302546 :0:5435 (20) a1 /C302192 :0:2283 : (21) The BLACKMAN FUNCTION is chosen so that the APPARATUS FUNCTION goes to 0 at ka /C305=4 and ka /C30 9=4 ; giving a0 /C303969 9304 :0:42659 (22) a1 /C3011554652 :0:24828 (23) a 2 /C30715 18608 :0:38424 ; (24) See also BARTLETT FUNCTION ,BLACKMAN FUNCTION , CONNES FUNCTION ,COSINE APODIZATION FUNCTION , FULL WIDTH AT HALF MAXIMUM ,G AUSSIAN FUNC- TION ,H AMMING FUNCTION ,H ANN FUNCTION ,H AN- NING FUNCTION ,M ERTZ APODIZATION FUNCTION , PARZEN APODIZATION FUNCTION ,UNIFORM APODIZA- TION FUNCTION ,W ELCH APODIZATION FUNCTION References Ball, J. A. "The Spectral Resolution in a Correlator System" §4.3.5 in Methods of Experimental Physics, Vol. 12C (Ed. M. L. Meeks). New York: Academic Press, pp. 55 /C1/7, 1976. Blackman, R. B. and Tukey, J. W. "Particular Pairs of Windows." In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 95 /C1/01, 1959. Brault, J. W. "Fourier Transform Spectrometry." In High Resolution in Astronomy: 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics (Ed. A. Benz, M. Huber, and M. Mayor). Geneva Observatory, Sauverny, Switzerland, pp. 31 /C1/2, 1985. Harris, F. J. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform." Proc. IEEE 66,51/C1/ 3, 1978. Norton, R. H. and Beer, R. "New Apodizing Functions for Fourier Spectroscopy." J. Opt. Soc. Amer. 66, 259 /C1/64, 1976. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 547 /C1/48, 1992. Schnopper, H. W. and Thompson, R. I. "Fourier Spectro- meters." In Methods of Experimental Physics 12A (Ed. M. L. Meeks). New York: Academic Press, pp. 491 /C1/29, 1974.Apollonian Gasket Consider three mutually tangent circles, and draw their inner SODDY CIRCLES . Then draw the inner SODDY CIRCLES of this circle with each pair of the original three, and continue iteratively. The points which are never inside a circle form a set of measure 0 having fractal dimension approximately 1.3058 (Man- delbrot 1983, p. 172). See also BOWL OF INTEGERS ,FORD CIRCLE ,SODDY CIRCLES References Boyd, D. W. "Improved Bounds for the Disk Packing Con- stants." Aeq. Math. 9,9 9/C1/06, 1973. Boyd, D. W. "The Residual Set Dimension of the Apollonian Packing." Mathematika 20, 170/C1/74, 1973. Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, pp. 169 /C1/72, 1983. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 3 /C1/, 1991. Apollonius Circles There are two completely different definitions of the so-called Apollonius circles: 1. The set of all points whose distances from two fixed points are in a constant ratio 1 : m(Durell 1928, Ogilvy 1990).2. The eight CIRCLES (two of which are nondegene- rate) which solve A POLLONIUS’ PROBLEM for three CIRCLES . Given one side of a TRIANGLE and the ratio of the lengths of the other two sides, the LOCUS of the third VERTEX is the Apollonius circle (of the first type) whose CENTER is on the extension of the given side. For a given TRIANGLE , there are three circles of Apollonius. Denote the three Apollonius circles (of the first type) of a TRIANGLE byk1;k2;andk3;and their centers L1; L2;and L3:The center L1is the intersection of the side A2A3with the tangent to the CIRCUMCIRCLE at A1:L1is also the pole of the SYMMEDIAN POINT Kwith respect to CIRCUMCIRCLE . The centers L1;L2;andL3 are COLLINEAR on the POLAR ofKwith regard to its CIRCUMCIRCLE , called the L EMOINE LINE . The circle of Apollonius k1is also the locus of a point whose PEDAL TRIANGLE isISOSCELES such that P1P2/C30P1P3:/ Let U and V be points on the side line BC of a TRIANGLE DABC met by the interior and exterior ANGLE BISECTORS of ANGLES A. The CIRCLE with DIAMETER UV is called the A-Apollonian circle. Similarly, construct the B- and C-Apollonian circles. The Apollonian circles pass through the VERTICES A, B, and C, and through the two ISODYNAMIC POINTS S and S?: The VERTICES of the D-TRIANGLE lie on the respective Apollonius circles. See also APOLLONIUS’ PROBLEM ,APOLLONIUS PURSUIT PROBLEM ,CASEY’S THEOREM ,HART’S THEOREM ,HEX- LET,ISODYNAMIC POINTS ,SODDY CIRCLES ,TANGENT CIRCLES ,TANGENT SPHERES References Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 16, 1928. Herrmann, M. "Eine Verallgemeinerung des Apollonischen Problems." Math. Ann. 145, 256 /C1/64, 1962. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 40 and 294 /C1/99, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 14 /C1/3, 1990. Apollonius Point Consider the EXCIRCLES GA ;GB ; and GC of a TRIANGLE , and the CIRCLE G internally TANGENT to all three. Denote the contact point of G and GA by A?; etc. Then the LINES AA?; BB?; and CC? CONCUR in this point. It has TRIANGLE CENTER FUNCTION a /C30sin2 A cos2[1 2(B /C28C)]: References Kimberling, C. "Apollonius Point." http://cedar.evansvil- le.edu/~ck6/tcenters/recent/apollon.html. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Kimberling, C.; Iwata, S.; and Hidetosi, F. "Problem 1091 and Solution." Crux Math. 13, 128 /C1/29 and 217 /C1/18, 1987.Apollonius Pursuit Problem Given a ship with a known constant direction and speed v, what course should be taken by a chase ship in pursuit (traveling at speed V) in order to intercept the other ship in as short a time as possible? The problem can be solved by finding all points which can be simultaneously reached by both ships, which is an APOLLONIUS CIRCLE with m /C30v=V : If the CIRCLE cuts the path of the pursued ship, the intersection is the point towards which the pursuit ship should steer. If the CIRCLE does not cut the path, then it cannot be caught. See also APOLLONIUS CIRCLES ,A POLLONIUS’ PRO- BLEM ,PURSUIT CURVE References Ogilvy, C. S. Solved by M. S. Klamkin. "A Slow Ship Intercepting a Fast Ship." Problem E991. Amer. Math. Monthly 59, 408, 1952. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 17, 1990. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 126 /C1/35, 1999. Warmus, M. "Un the ´ore`me sur la poursuite." Ann. de la Soc. Polonaise de Math. 19, 233/C1/34, 1946. Apollonius Spheres TANGENT SPHERES Apollonius’ Problem Given three objects, each of which may be a POINT , LINE,o r CIRCLE , draw a CIRCLE that is TANGENT to each. There are a total of ten cases. The two easiest involve three points or three LINES , and the hardest involves three CIRCLES . Euclid solved the two easiest cases in his Elements , and the others (with the exception of the three CIRCLE problem), appeared in the Tangencies of Apollonius which was, however, lost. The general problem is, in principle, solvable by STRAIGHTEDGE and COMPASS alone. The three- CIRCLE problem was solved by Vie`te (Boyer 1968), and the solutions are called APOLLONIUS CIRCLES . There are eight total solutions. The simplest solution is obtained by solving the three simultaneous quadratic equations (x /C28x1)2 /C27(y /C28y1)2 /C28(r 9r1)2 /C300 (1) (x /C28x2)2 /C27(y /C28y2)2 /C28(r 9r2)2 /C300 (2) (x /C28x3)2 /C27(y /C28y3)2 /C28(r 9r3)2 /C300 (3) in the three unknowns x, y, r for the eight triplets of signs (Courant and Robbins 1996). Expanding the equations gives (x2 /C27y2 /C28r2) /C282xxi /C282yyi /C142rri /C27(x2 i /C27y2i /C28r2i ) /C300 (4) for i /C301, 2, 3. Since the first term is the same for each equation, taking (2) /C28(1) and (3) /C28(1) gives ax /C27by /C27cr /C30d (5) a ?x /C27b?y /C27c?r /C30d?; (6)where a /C302(x1 /C28x2) (7) b /C302(y1 /C28y2) (8) c /C3092(r1 /C28r2) (9) d /C30(x21 /C27y21 /C28r21) /C28(x22 /C27y22 /C28r22) (10) and similarly for a ?; b?; c ? and d? (where the 2 subscripts are replaced by 3s). Solving these two simultaneous linear equations gives x /C30b?d /C28 bd?/C28b ?cr /C27 bc?r ab ?/C28ba ? (11) y /C30/C28a ?d /C27 ad ?/C27a ?cr /C28 ac ?r ab ?/C28a 0b; (12) which can then be plugged back into the QUADRATIC EQUATION (1) and solved using the QUADRATIC FOR- MULA . Perhaps the most elegant solution is due to Gergonne. It proceeds by locating the six HOMOTHETIC CENTERS (three internal and three external) of the three given CIRCLES . These lie three by three on four lines (illustrated above). Determine the POLES of one of these with respect to each of the three CIRCLES and connect the POLES with the RADICAL CENTER of the CIRCLES . If the connectors meet, then the three pairs of intersections are the points of tangency of two of the eight circles (Petersen 1879, Johnson 1929, Do¨rrie 1965). To determine which two of the eight Apollo- nius circles are produced by the three pairs, simply take the two which intersect the original three CIRCLES only in a single point of tangency. The procedure, when repeated, gives the other three pairs ofCIRCLES . If the three CIRCLES are mutually tangent, then the eight solutions collapse to two, known as the SODDY CIRCLES . Larmor (1891) and Lachlan (1893, pp. 244 /C1/51) con- sider the problem of four circles having a commontangent circle. See also A POLLONIUS PURSUIT PROBLEM ,B END (CURVATURE ), CASEY’S THEOREM ,C IRCULAR TRIAN- GLE,D ESCARTES CIRCLE THEOREM ,F OUR COINS PROBLEM ,H ART CIRCLE ,H ART’S THEOREM ,SODDY CIRCLES References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, p. 226, 1952. Boyer, C. B. A History of Mathematics. New York: Wiley, p. 159, 1968. Courant, R. and Robbins, H. "Apollonius’ Problem." §3.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 117 and 125 /C1/27, 1996. Do¨rrie, H. "The Tangency Problem of Apollonius." §32 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 154 /C1/60, 1965. F. Gabriel-Marie. Exercices de ge´ome´trie. Tours, France: Maison Mame, pp. 18 /C1/0 and 663, 1912. Gauss, C. F. Werke, Band 4. New York: George Olms, p. 399, 1981. Gergonne, M. "Recherche du cercle qui en touche trois autres sur une sphe`re." Ann. math. pures appl. 4, 1813 /C1/ 814. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 118 /C1/21, 1929. Lachlan, R. "Circles with Touch Three Given Circles" and "Systems of Four Circles Having a Common Tangent Circle." §383 /C1/96 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 241 /C1/51, 1893. Larmor, A. "Contacts of Systems of Circles." Proc. London Math. Soc. 23, 136 /C1/57, 1891. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 48 /C1/1, 1990. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 151, 1989. Petersen, J. Example 403 in Methods and Theories for the Solution of Problems of Geometrical Constructions, Ap- plied to 410 Problems. London: Sampson Low, Marston, Searle & Rivington, pp. 94 /C1/5, 1879. Rouche ´, E. and de Comberousse, C. Traite ´ de ge´ome´trie plane. Paris: Gauthier-Villars, pp. 297 /C1/03, 1900. Salmon, G. Conic Sections, 6th ed. New York: Chelsea, pp. 88 /C1/35, 1960. Simon, M. U¨ ber die Entwicklung der Elementargeometrie im XIX Jahrhundert. Berlin, pp. 97 /C1/05, 1906. Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 4 /C1/, 1991. Apollonius’ Theorem STEWART’S THEOREM Apothem Given a CIRCLE , the PERPENDICULAR distance a from the MIDPOINT of a CHORD to the CIRCLE ’s center is called the apothem. It is also equal to the RADIUS r minus the SAGITTA s, a /C30r /C28s : See also CHORD ,RADIUS ,SAGITTA ,SECTOR ,SEGMENT Apparatus Function INSTRUMENT FUNCTIONAppell Cross Sequence A sequence s(l) n(x) /C30[h(t)]lsn(x) ; where sn(x)isaS HEFFER SEQUENCE , h(t) is invertible, and l ranges over the real numbers is called a STEFFENSEN SEQUENCE .If sn(x) is an associated SHEFFER SEQUENCE , then s(l) nis called a CROSS SEQUENCE .Ifsn(x) /C30xn ; then s l n(x) /C30[h(t)]lxn is called an Appell cross sequence. Examples include the BERNOULLI POLYNOMIAL ,EU- LER POLYNOMIAL , and HERMITE POLYNOMIAL . See also APPELL SEQUENCE ,CROSS SEQUENCE ,SHEF- FER SEQUENCE ,STEFFENSEN SEQUENCE References Roman, S. "Cross Sequences and Steffensen Sequences." §5.3 inThe Umbral Calculus. New York: Academic Press, pp. 140 /C143, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684/C160, 1973. Appell Hypergeometric Function A formal extension of the HYPERGEOMETRIC FUNCTION to two variables, resulting in four kinds of functions (Appell 1925; Whittaker and Watson 1990, Ex. 22, p. 300), F1(a;b;b?;g;x;y)/C30X/C12 m/C300X/C12 n/C300(a)m/C27n(b)m(b?)n m!n!(g)m/C27nxmyn (1) F2(a;b;b?;g;g?;x;y) /C30X/C12 m/C300X/C12 n/C300(a)m/C27n(b)m(b?)n m!n!(g)m(g?)nxmyn (2) F3(a;a?;b;b?;g;x;y) /C30X/C12 m/C300X/C12 n/C300(a)m(a?)n(b)m(b?)n m!n!(g)m/C27nxmyn (3) F4(a;b;g;g?;x;y)/C30X/C12 m/C300X/C12 n/C300(a)m/C27n(b)m/C27n m!n!(g)m(g?)nxmyn: (4) Appell defined the functions in 1880, and Picardshowed in 1881 that they may all be expressed by INTEGRALS OF THE FORM g1 0u a(1 /C28u)b(1 /C28xu)g(1 /C28yu) ddu (5) (Bailey 1934, pp. 76 /C1/9). The Appell functions are special cases of the KAMPE ´ DE FE´ RIET FUNCTION , and are the first four in the set of HORN FUNCTIONS . In particular, the general integral g(a /C27b sin x /C27c cos x)v dx /C30CF1n /C271;1 2 ;12; n /C272;a /C27 c cos x /C27 b sin x a /C28 bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27c2 b2s ;0 BBBB@ a /C27 c cos x /C27 b sin x a /C27 bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27c2 b2srC1D ; (6) where C /C30sec[x /C27tan /C281(c b)](a /C27c cos x /C27b sin x)n/C271 /C2 b(n /C271)ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27c2 b2s"# /C281 /C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27c2 b2s /C28 sin x) /C28 c cosx bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 c2 b2s /C27 avuuuuuuut /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27c2 b2s /C27 sin x) /C27 c cos x bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 c2 b2s /C28 avuuuuuuut; (7) has a closed form in terms of F 1 :/ /F1( a; b; b?; g; x; y) reduces to the HYPERGEOMETRIC FUNCTION in the cases F1( a; b; b?; g;0; y) /C302F1( a; b?; g; y) (8) F1(a; b; b?; g; x; 0) /C302F1(a; b; g; x) (9) The F1function is built into Mathematica 4.0 as AppellF1 [a, b1, b2, c, x, y]. See also ELLIPTIC INTEGRAL ,H ORN FUNCTION ,H Y- PERGEOMETRIC FUNCTION ,KAMPE ´ DE FE´ RIET FUNC- TION ,LAURICELLA FUNCTIONS References Appell, P. "Sur les fonctions hyperge ´ome´triques de plusieurs variables." In Me´moir. Sci. Math. Paris: Gauthier-Villars, 1925. Appell, P. and Kampe ´ de Fe´riet, J. Fonctions hyperge ´o- me´triques et hypersphe ´riques: polynomes d’Hermite. Paris: Gauthier-Villars, 1926.Bailey, W. N. "A Reducible Case of the Fourth Type of Appell’s Hypergeometric Functions of Two Variables." Quart. J. Math. (Oxford) 4, 305 /C1/08, 1933. Bailey, W. N. "On the Reducibility of Appell’s Function F4 :/" Quart. J. Math. (Oxford) 5, 291 /C1/92, 1934. Bailey, W. N. "Appell’s Hypergeometric Functions of Two Variables." Ch. 9 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 73 /C1/3 and 99 /C1/01, 1935. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 222 and 224, 1981. Exton, H. Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, p. 27, 1978. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1461, 1980. Watson, G. N. "The Product of Two Hypergeometric Func- tions." Proc. London Math. Soc. 20, 189 /C1/95, 1922. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Wolfram, S. The Mathematica Book, 4th ed. Cambridge, England: Cambridge University Press, pp. 771 /C1/72, 1999. Appell Polynomial References Suetin, P. K. "Classical Appell’s Orthogonal Polynomials." Ch. 3 in Orthogonal Polynomials in Two Variables. Amsterdam, Netherlands: Gordon and Breach, pp. 63 /C1/6, 1999. Appell Sequence An Appell sequence is a SHEFFER SEQUENCE for (g(t) ; t) : Roman (1984, pp. 86 /C1/06) summarizes prop- erties of Appell sequences and gives a number of specific examples. The sequence sn(x) is Appell for g(t) IFF 1 g(t) ey(t) /C30X/C12 k/C300sk(y) k!tk (1) for all y in the field C of characteristic 0, and IFF sn(x) /C30xn g(t)(2) (Roman 1984, p. 27). The Appell identity states that the sequence sn(x) is an Appell sequence IFF sn(x/C27y)/C30Xn k/C300n krC1+rC1D sk(y)xn/C28k(3) (Roman 1984, p. 27). The B ERNOULLI POLYNOMIALS ,EULER POLYNOMIALS , and H ERMITE POLYNOMIALS are Appell sequences (in fact, more specifically, they are A PPELL CROSS SE- QUENCES ). See also APPELL CROSS SEQUENCE ,S HEFFER SE- QUENCE ,UMBRAL CALCULUS References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 209 /C110, 1988. Roman, S. "Appell Sequences." §2.5 and §2in The Umbral Calculus. New York: Academic Press, pp. 17 and 26 /C18 and 86 /C106, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C160, 1973. Appell Transformation A HOMOGRAPHIC transformation x1 /C30ax /C27 by /C27 c a ƒx /C27 b ƒy /C27 c ƒ y1 /C30a ?x /C27 b ?y /C27 c ? a ƒx /C27 b ƒy /C27 c ƒ with t1 substituted for t according to kdt1 /C30dt (a ƒx /C27 bƒy /C27 c ƒ)2 : References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 210 /C1/11, 1988. AppellF1 APPELL HYPERGEOMETRIC FUNCTION Apple A SURFACE OF REVOLUTION defined by Kepler. It consists of more than half of a circular ARC rotated about an axis passing through the endpoints of the ARC. The equations of the upper and lower boundaries in the x-z PLANE are z9/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28(x /C28r)2q for R /C21r and /x /C23 [/C28(r /C27R) ; r }R]/. It is the outside surface of a SPINDLE TORUS . See also BUBBLE ,LEMON ,OBLATE SPHEROID ,SPHERE-SPHERE INTERSECTION ,SPINDLE TORUS Approximate Zero An initial point that provides safe convergence of NEWTON’S METHOD (Smale 1981; Petkovic et al. 1997, p. 1). See also ALPHA- TEST,N EWTON’S METHOD ,P OINT ESTIMATION THEORY References Petkovic, M. S.; Herceg, D. D.; and Ilic, S. M. Point Estima- tion Theory and Its Applications. Novi Sad, Yugoslavia: Institute of Mathematics, 1997. Smale, S. "The Fundamental Theorem of Algebra and Complexity Theory." Bull. Amer. Math. Soc. 4,1/C1/5, 1981. Approximately Equal If two quantities A and B are approximately equal, this is written A :B:/ See also DEFINED ,EQUAL Approximately Equal To APPROXIMATELY EQUAL Approximation Theory The mathematical study of how given quantities can be approximated by other (usually simpler) ones under appropriate conditions. Approximation theory also studies the size and properties of the ERROR introduced by approximation. Approximations are often obtained by POWER SERIES expansions in which the higher order terms are dropped. See also LAGRANGE REMAINDER References Achieser, N. I. Theory of Approximation. New York: Dover, 1992. Cheney, E. W. Introduction to Approximation Theory, 2nd ed.New York: Chelsea, 1982. Golomb, M. Lectures on Theory of Approximation. Argonne, IL: Argonne National Laboratory, 1962. Jackson, D. The Theory of Approximation. New York: Amer. Math. Soc., 1930. Natanson, I. P. Constructive Function Theory, Vol. 1: Uni- form Approximation. New York: Ungar, 1964. Petrushev, P. P. and Popov, V. A. Rational Approximation of Real Functions. New York: Cambridge University Press, 1987. Rivlin, T. J. An Introduction to the Approximation of Func- tions. New York: Dover, 1981. Timan, A. F. Theory of Approximation of Functions of a Real Variable. New York: Dover, 1994. Weisstein, E. W. "Books about Approximation Theory." http://www.treasure-troves.com/books/Approxima- tionTheory.html. Arakelov Theory A formal mathematical theory which introduces "components at infinity" by defining a new type of divisor class group of INTEGERS of a NUMBER FIELD . The divisor class group is called an "arithmetic surface." See also ARITHMETIC GEOMETRY Arbelos The term "arbelos" means SHOEMAKER’S KNIFE in Greek, and this term is applied to the shaded AREA in the above figure which resembles the blade of a knife used by ancient cobblers (Gardner 1979). Archimedes himself is believed to have been the firstmathematician to study the mathematical propertiesof this figure. The position of the central notch is arbitrary and can be located anywhere along the DIAMETER . The arbelos satisfies a number of unexpected iden-tities (Gardner 1979, Schoch). 1. Call the diameters of the left and right SEMI- CIRCLES rB1 and 1 /C28r;respectively, so the dia- meter of the enclosing SEMICIRCLE is 1. Then the arc length along the bottom of the arbelos is L/C30pr/C27p(1/C28r)/C30p1 so the arc length along the enclosing semicircle isthe same as the arc length along the two smaller semicircles.2. Draw the PERPENDICULAR BDfrom the tangent of the two SEMICIRCLES to the edge of the large CIRCLE . Then the AREA of the arbelos is the same as the AREA of the CIRCLE with DIAMETER BD. Let AC/C301 and r/C30AB, then simultaneously solve the equations r2/C27h2/C30x2(1) (1/C28r)2/C27h2/C30y2(2) x2/C27y2/C3012(3) for the sides x/C30AD/C30ffiffiffirp(4) y/C30CD/C30ffiffiffiffiffiffiffiffiffiffiffiffi 1/C28rp (5) h/C30BD/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(1/C28r)p : (6) 3. The CIRCLES C1andC?1inscribed on each half of BDon the arbelos (called A RCHIMEDES’ CIRCLES ) each have DIAMETER (AB)(BC)=(AC):/ IfAC/C301 and AB/C30r, then the radius of the Archimedes’ circles is R/C301 2r(1/C28r): (7) The positions of the circles can be found using the triangles shown above. The lengths of the horizo- nal legs and hypotenuses are known as indicated, so the vertical legs can be found using theP YTHAGOREAN THEOREM . This then gives the centers of the circles as x1/C30r/C28R/C301 2r(1/C27r) (8) y1/C30ffiffiffiffiffiffiffiffiffi 2rRp /C30rffiffiffiffiffiffiffiffiffiffiffi 1/C28rp (9) and x?1/C30r/C27R/C301 2r(3/C28r) (10) y?1/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R(1/C28r)p /C30(1/C28r)ffiffiffirp: (11) 4. Let A?be the point at which the CIRCLE centered atAand of RADIUS r/C30ABintersects the enclosing SEMICIRCLE , and let C?be the point at which the CIRCLE centered at CofRADIUS 1/C28r/C30BCinter- sects the enclosing SEMICIRCLE . Then the smallest CIRCLE C2passing through A?and tangent to BDis equal to the smallest CIRCLE C?2passing through C? and tangent to BD(Schoch). Moreover, the radii R of these circles are the same as A RCHIMEDES’ CIRCLES . Solving (x/C281 2)2/C27y2/C30(12)2(12) x2/C27y2/C30r2(13) gives ( x;y)/C30(r2;rffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p );so the center of C2is x2/C30r2/C271 2r(1/C28r)/C3012r(r/C271) (14) y2/C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2:p (15) Similarly, solving (x/C281 2)2/C27y2/C30(12)2(16) (x/C281)2/C27y2/C30(1/C28r)2(17) gives ( x;y)/C30(r(2/C28r);(1/C28r)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(2/C28r)p );so the center of C?2is x?2/C30r(2/C28r)/C281 2r(1/C28r)/C3012r(r/C283) (18) y?2/C30(1/C28r)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(2/C28r)p : (19) 5. The A POLLONIUS CIRCLE C3of the circles with arcs BA?;BC?;andAA?DC?Cis located at a position x/C301 2r(1/C273r/C282r2) (20) y/C30r(1/C28r)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2/C28r)(1/C27r)p (21) and has radius Requal to that of A RCHIMEDES’ CIRCLES (Schoch), as does the smallest circle C?3 passing through Band tangent to C3:/ Furthermore, letting B?D?be the line parallel to BDthrough the center of CIRCLE C3;the CIRCLE Cƒ3 with center on B?D?and tangent to the small semicircles of the arbelos also has radius R (Schoch). The position of the center of Cƒ3is given by xƒ3/C30x/C301 2r(1/C273r/C282r2) (22) yƒ3/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1 2r/C27R)/C28(x/C2812r)2q /C30r(1/C28r)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27r/C28r2p : (23) The vertical h?position of D?is h?/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4/C2814(2r3/C283r2/C28r/C271)2q /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(1/C28r)(2r2/C283r/C281)(2r2/C28r/C282)p : (24) 6. Let Pbe the MIDPOINT ofAB, and let Qbe the MIDPOINT ofBC. Then draw the SEMICIRCLE hav- ingPQas a DIAMETER with center M. This CIRCLE has RADIUS RPQ/C301 2f1/C2812[r/C27(1/C28r)]g/C3014: (25) The smallest circle C4through D?touching arc PQ then has radius R(Schoch). Using similar trian- gles, the center of this circle is at x4/C30r(2r4/C285r3/C273r/C271) 1/C274r/C284r2(26) y4/C302r2/C282r/C281 2(4r2/C284r/C281) /C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(1/C28r)(2r2/C283r/C281)(2r2/C28r/C282)p : (27) Similarly, let Ube the point of intersection of B?D? and the SEMICIRCLE PQ, then the CIRCLE through B,B?;and Ualso has RADIUS R(Schoch). The center of this CIRCLE is at x?4/C301 4r(3/C273r/C282r2) (28) y?4/C3014r(1/C28r) /C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2r/C271)(3/C282r)p : (29) Consider the circle XofRADIUS rXwhich is tangent to the two interior semicircles. Its position and radius are obtained by solving the simultaneous equations h2/C27z2/C30(1 2r/C27rX)2(30) h2/C27(1 2/C28z)2/C30[12(1/C28r)/C27rX]2(31) (1 2r/C27rX)2/C27[12(1/C28r)/C27rX]2/C30(14)2: (32) giving z/C3014/C2714(2r/C281)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274r/C284r2p (33) h/C30r(1/C28r) (34) rX/C301 4(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274r/C284r2p /C281): (35) Letting Cƒ4be the smallest CIRCLE through Xand tangent to ABC , the radius of Cƒ4is therefore h=2/C30 r(1/C28r)=2/C30R(Schoch), and its center is located at xƒ4/C301 4/C2712r/C2714(2r/C281)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274r/C284r2p (36) yƒ4/C301 2r(1/C28r): (37) 7. Within each small semicircle of an arbelos, construct arbeloses similar to the original. Thenthe circles C 5and C?5are congruent and have radius R(Schoch). Moreover, connect the mid- points of the arcs and their cusp points to form the RECTANGLES uEFGH anduE?F?G?H?:Then these rectangles are similar with respect to the point Cƒ5 (Schoch). This point lies on the line B?D?;and the circle with center Cƒ5and radius Cƒ5B?also has radius R,s o Cƒ5has coordinates (1 2r(1/C273r/C28 2r2);1 2r(1/C28r)):The following tables summarized the positions of the rectangle vertices. XCoordinates /X?/Coordinates E /(1 2r;12r)// E?//(r(2/C28r);0)/ F /(12r(1/C27r);12r(1/C28r))//F?//(12r(3/C28r);12r(1/C28r))/ G /(r2;0)// G?//(12(1/C27r);12(1/C28r))/ H /(12r2;12r2)// H?//(12(1/C272r/C28r2);12(1/C28r)2)/ 8. Let MM?be the PERPENDICULAR BISECTOR ofAC, letBbe the cusp of the arbelos and Dlie above it, letEand G?be the tops of the large and small semicircles, respectively. Let EG?intersect the lines MM?andBDin points IandJ, respectively. Then the smallest circle C6passing through Iand tangent to arc ACatM?;the smallest circle C?6 through Jand tangent to the outside semicircle at PC;and the circle Cƒ6with diameter JBare all equal to the Archimedean circles (Schoch). The circle Cƒ6is called the B ANKOFF CIRCLE , and is also the CIRCUMCIRCLE of the point Band tangent points PAand PCof the first Pappus circle. The centers of the circles C6;C?6;andCƒ6are given by x6/C301 2 y6/C301 2(1/C28r/C27r2) (38) x?6 /C30r(1 /C28 r /C27 2r2) 2(1 /C28 2r /C27 2r2) (39) y?6 /C30r(1 /C28 r)(1 /C28 r /C27 r2) 1 /C28 2r /C27 2r2 (40) x ƒ6 /C30r (41) yƒ6 /C301 2r(1 /C28r) : (42) Rather amazingly, the points E, M, B, G?; PC ; D, and M ? are CONCYCLIC (Schoch) in a circle with center ((1 /C272r)=4 ; 1 =4) and radius REMBG ?PCDM ?/C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(1 /C282r /C272r2)p : (43) 9. The smallest CIRCUMCIRCLE of the Archimedean circles has an area equal to that of the arbelos. 10. The line tangent to the semicircles AB and BC contains the point E and F which lie on the lines AD and CD, respectively. Furthermore, BD and EF bisect each other, and the points B, D, E, and F are CONCYCLIC . 11. Construct a chain of TANGENT CIRCLES starting with the CIRCLE TANGENT to the two small ones and large one (a so-called PAPPUS CHAIN ). The centers of the CIRCLES lie on an ELLIPSE , and the DIAMETER of the nth CIRCLE Cnis (/(1=n))/th PER- PENDICULAR distance to the base of the SEMICIR- CLE. This result is most easily proven using INVERSION , but was known to Pappus, who re- ferred to it as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981). 12. If Bdivides ACin the GOLDEN RATIO f;then the circles in the chain satisfy a number of other special properties (Bankoff 1955). See also ARCHIMEDES’ CIRCLES ,B ANKOFF CIRCLE , COXETER’S LOXODROMIC SEQUENCE OF TANGENT CIRCLES ,GOLDEN RATIO,INVERSION ,PAPPUS CHAIN , STEINER CHAIN References Allanson, B. "Pappus’s Arbelos" java applet. http://www.a- delaide.net.au/~allanson/arbelos.html. Bankoff, L. "The Fibonacci Arbelos." Scripta Math. 20, 218, 1954. Bankoff, L. "The Golden Arbelos." Scripta Math. 21,7 0/C1/6, 1955. Bankoff, L. "Are the Twin Circles of Archimedes Really Twins?" Math. Mag. 47, 214/C1/18, 1974. Bankoff, L. "How Did Pappus Do It?" In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 112 /C1/18, 1981. Bankoff, L. "The Marvelous Arbelos." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Wa- shington, DC: Math. Assoc. Amer., 1994. Cadwell, J. H. Topics in Recreational Mathematics. Cam- bridge, England: Cambridge University Press, 1966. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 35 /C1/6, 1971. Dodge, C. W.; Schoch, T.; Woo, P. Y.; and Yiu, P. "Those Ubiquitous Archimedean Circles." Math. Mag. 72, 202/C1/ 13, 1999. Gaba, M. G. "On a Generalization of the Arbelos." Amer. Math. Monthly 47,1 9/C1/4, 1940. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,1 8/C1/8, Jan. 1979. Heath, T. L. The Works of Archimedes with the Method of Archimedes. New York: Dover, p. 307, 1953. Hood, R. T. "A Chain of Circles." Math. Teacher 54, 134/C1/37, 1961. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 116 /C1/17, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 54 /C1/5, 1990. Schoch, T. "A Dozen More Arbelos Twins." http://www.bio- la.edu/academics/undergrad/math/woopy/arbel2.htm. Soddy, F. "The Bowl of Integers and the Hexlet." Nature 139,7 7/C1/9, 1937. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 5 /C1/, 1991. Woo, P. "The Arbelos." http://www.biola.edu/academics/un- dergrad/math/woopy/arbelos.htm. Yiu, P. "The Archimedean Circles in the Shoemaker’s Knife." Lecture at the 31st Annual Meeting of the Florida Section of the Math. Assoc. Amer., Boca Raton, FL, March 6 /C1/, 1998. Arborescence ADIRECTED GRAPH is called an arborescence if, from a given node xknown as the ROOT NODE , there is exactly one elementary path from x to every other node y. See also ARBORICITY ,DIRECTED GRAPH ,ROOT NODE Arboricity Given a GRAPH G, the arboricity is the MINIMUM number of line-disjoint acyclic SUBGRAPHS whose UNION is G. See also ANARBORICITY Arc In general, any smooth curve joining two points. In particular, any portion (other than the entire curve) of a CIRCLE or ELLIPSE . As Archimedes proved, for CHORDS AC and BD which are PERPENDICULAR to each other, arc AB /C27arc CD /C30arc BC /C27arc DA (Wells 1991). The prefix "arc" is also used to denote the INVERSE FUNCTIONS of TRIGONOMETRIC FUNCTIONS and HYPER- BOLIC FUNCTIONS . Finally, any path through a graph which passes through no vertex twice is called an arc (Gardner 1984, p. 96). See also APPLE ,ARC LENGTH ,CHORD ,CIRCLE- CIRCLE INTERSECTION ,C IRCULAR TRIANGLE ,F IVE DISKS PROBLEM ,FLOWER OF LIFE,LEMON ,LENS,PIECEWISE CIRCULAR CURVE ,R EULEAUX POLYGON ,R EULEAUX TRIANGLE ,SALINON ,SEED OF LIFE,TRIANGLE ARCS, VENN DIAGRAM ,YIN-YANG References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 118, 1991. Arc Length Arc length is defined as the length along a curve, s /C13gb adljj: (1) Defining the line element ds2 /C13 dljj2; parameterizing the curve in terms of a parameter t, and noting thatds =dt is simply the magnitude of the VELOCITY with which the end of the RADIUS VECTOR r moves gives s /C30gb ads /C30gb ads dtdt /C30gb ar?(t) jj dt: (2) In POLAR COORDINATES , dl /C30ˆr dr /C27r ˆu du /C30dr duˆr /C27r ˆu ! d u; (3) so ds /C30 dljj/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27dr d u !2 duvuut(4) s /C30g dljj/C30g02 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27dr du !2 d uvuut: (5) In CARTESIAN COORDINATES , dl /C30dyˆx /C27dyˆy (6) ds /C30 dl:dl jj /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 /C27dy2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy dx !2 /C271 dxvuut: (7) Therefore, if the curve is written r(x) /C30xˆx /C27f(x)ˆy; (8) then s /C30gb affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27f ?2(x)q dx: (9) If the curve is instead written r(t) /C30x(t)ˆx /C27y(t)ˆy; (10) then s /C30gb affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2(t) /C27y?2(t)q dt: (11) Or, in three dimensions, r(t)/C30x(t)ˆx/C27y(t)ˆy/C27z(t)ˆz; (12) so s/C30gb affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix? 2(t)/C27y?2(t)/C27z?2(t)q dt: (13) See also CURVATURE ,G EODESIC ,N ORMAL VECTOR , RADIUS OF CURVATURE ,RADIUS OF TORSION ,SPEED , SURFACE AREA,TANGENTIAL ANGLE ,TANGENT VEC- TOR,TORSION (DIFFERENTIAL GEOMETRY ), VELOCITY Arc Minute A unit of ANGULAR measure equal to 60 ARC SECONDS , or 1/60 of a DEGREE . The arc minute is denoted0(not to be confused with the symbol for feet ). See also ARC SECOND ,DEGREE Arc Second A unit of ANGULAR measure equal to 1/60 of an ARC MINUTE , or 1/3600 of a DEGREE . The arc second is denoted (not to be confused with the symbol for inches ). See also ARC MINUTE ,DEGREE Arccos INVERSE COSINE ArcCos INVERSE COSINE Arccosecant INVERSE COSECANT ArcCosh INVERSE HYPERBOLIC COSINE Arccosine INVERSE COSINE ArcCot INVERSE COTANGENT Arccot INVERSE COTANGENT Arccotangent INVERSE COTANGENT Arccoth INVERSE HYPERBOLIC COTANGENT ArcCoth INVERSE HYPERBOLIC COTANGENT ArcCsc INVERSE COSECANT Arccsc INVERSE COSECANT Arccsch INVERSE HYPERBOLIC COSECANT ArcCsch INVERSE HYPERBOLIC COSECANTArch A4- POLYHEX (Gardner 1978, p. 147). The term is also used by Gradshteyn and Ryzhik (2000, p. xxx) to denote Arch z /C30i cos /C281 z; where cos/C281zis the INVERSE COSINE . See also ARCTH ,ARSH,ARTH,INVERSE COSINE References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, 1978. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Archimedean Dual The DUALS of the A RCHIMEDEAN SOLIDS , sometimes called the C ATALAN SOLIDS , are given in the following table. Hume (1986) gives exact solutions for the side lengths, angles, and DIHEDRAL ANGLES of the Archi- medean duals. nARCHIMEDEAN SOLID DUAL 1CUBOCTAHEDRON RHOMBIC DODECAHEDRON 2GREAT RHOMBICOSIDODECA- HEDRONDISDYAKIS TRIACONTAHE-DRON 3GREAT RHOMBICUBOCTAHE- DRONDISDYAKIS DODECAHEDRON 4ICOSIDODECAHEDRON RHOMBIC TRIACONTAHEDRON 5SMALL RHOMBICOSIDODECA- HEDRONDELTOIDAL HEXECONTAHE-DRON 6SMALL RHOMBICUBOCTAHE- DRONDELTOIDAL ICOSITETRAHE-DRON 7SNUB CUBE (laevo) PENTAGONAL ICOSITETRAHE- DRON (dextro) 8SNUB DODECAHEDRON (lae- vo)PENTAGONAL HEXECONTAHE-DRON (dextro) 9TRUNCATED CUBE SMALL TRIAKIS OCTAHEDRON 10 TRUNCATED DODECAHEDRON TRIAKIS ICOSAHEDRON 11 TRUNCATED ICOSAHEDRON PENTAKIS DODECAHEDRON 12 TRUNCATED OCTAHEDRON TETRAKIS HEXAHEDRON 13 TRUNCATED TETRAHEDRON TRIAKIS TETRAHEDRON Here are the Archimedean DUALS (Pearce 1978, Holden 1991) displayed in the order listed above (left to right, then continuing to the next row). Here are the Archimedean solids paired with their DUALS . See also ARCHIMEDEAN SOLID,CATALAN SOLID References Holden, A. Shapes, Space, and Symmetry. New York: Dover, p. 54, 1991. Hume, A. "Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals." Computing Science Tech. Rep. , No. 130. Murray Hill, NJ: AT&T Bell Laboratories, 1986. Pearce, P. Structure in Nature Is a Strategy for Design. Cambridge, MA: MIT Press, pp. 34 /C1/5, 1978. Archimedean Solid The Archimedean solids are convex POLYHEDRA which have a similar arrangement of nonintersecting reg-ular plane CONVEX POLYGONS of two or more different types arranged in the same way about each VERTEX with all sides the same length (Cromwell 1997, pp. 91 /C1/2). The Archimedean solids are distinguished from the regular PRISMS and ANTIPRISMS by having very high symmetry, thus excluding solids belonging to a DIHEDRAL GROUP of symmetries (e.g., prisms and antiprisms with unit side lengths) and the ELON- GATED SQUARE GYROBICUPOLA (because that surface’s symmetry-breaking twist allows vertices "near theequator" and those "in the polar regions" to bedistinguished; Cromwell 1997, p. 92). The Archime-dean solids are sometimes also referred to as the SEMIREGULAR POLYHEDRA . Nine of the Archimedean solids can be obtained by TRUNCATION of a P LATONIC SOLID , and two further can be obtained by a second truncation. The remain-ing two solids, the SNUB CUBE and SNUB DODECAHE- DRON , are obtained by moving the faces of a CUBE and DODECAHEDRON outward while giving each face a twist. The resulting spaces are then filled withribbons of EQUILATERAL TRIANGLES (Wells 1991). Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular TETRAHEDRON so that four of their faces lie on the faces of that TETRAHEDRON . A method of constructing the Archimedean solids using a method known as "expansion" has been enumerated by Stott (Stott1910; Ball and Coxeter 1987, pp. 139 /C1 /40). Let the cyclic sequence S/C30(p1;p2;...pq) represent the degrees of the faces surrounding a vertex (i.e., S is a list of the number of sides of all polygonssurrounding any vertex). Then the definition of anArchimedean solid requires that the sequence mustbe the same for each vertex to within ROTATION and REFLECTION . Walsh (1972) demonstrates that Sre- presents the degrees of the faces surrounding each vertex of a semiregular convex polyhedron or TESSEL- LATION of the plane IFF 1.q]3 and every member of Sis at least 3, 2.aq i/C3011 pi]1 2q/C281;with equality in the case of a plane TESSELLATION , and 3. for every ODD NUMBER p/C23S;Scontains a subsequence ( b,p,b). Condition (1) simply says that the figure consists of two or more polygons, each having at least threesides. Condition (2) requires that the sum of interiorangles at a vertex must be equal to a full rotation for the figure to lie in the plane, and less than a full rotation for a solid figure to be convex. The usual way of enumerating the semiregular polyhedra is to eliminate solutions of conditions (1)and (2) using several classes of arguments and thenprove that the solutions left are, in fact, semiregular(Kepler 1864, pp. 116 /C1 /26; Catalan 1865, pp. 25 /C1/2; Coxeter 1940, p. 394; Coxeter et al. 1954; Lines 1965, pp. 202 /C1/03; Walsh 1972). The following table gives all possible regular and semiregular polyhedra and tessellations. In the table, ‘P’ denotes P LATONIC SOLID , ‘M’ denotes a PRISM orANTIPRISM , ‘A’ denotes an Archimedean solid, and ‘T’ a plane tessellation. S Figure Solid S CHLA ¨FLI SYMBOL (3, 3, 3) P TETRAHEDRON /f3;3g/ (3, 4, 4) M Triangular PRISM /tf2;3g/ (3, 6, 6) A TRUNCATED TETRAHEDRON t/f3;3g/ (3, 8, 8) A TRUNCATED CUBE /tf4;3g/ (3, 10, 10) A TRUNCATED DODECAHE- DRON/tf5;3g/ (3, 12, 12) T (Plane TESSELLATION ) /tf6;3g/ (4, 4, n)M n-gonal PRISM /tf2;ng/ (4, 4, 4) P CUBE /f4;3g/ (4, 6, 6) A TRUNCATED OCTAHEDRON /tf3;4g/ (4, 6, 8) A GREAT RHOMBICUBOCTA-HEDRON t3 4fg/ (4, 6, 10) A GREAT RHOMBICOSIDODE- CAHEDRONt3 5fg/ (4, 6, 12) T (Plane TESSELLATION )t3 6fg/ (4, 8, 8) T (Plane TESSELLATION ) /tf4;4g/ (5, 5, 5) P DODECAHEDRON /f5;3g/ (5, 6, 6) A TRUNCATED ICOSAHEDRON /tf3;5g/ (6, 6, 6) T (Plane TESSELLATION ) /f6;3g/ (3, 3, 3, n)M n-gonal ANTIPRISM s2 nfg/ (3, 3, 3, 3) P OCTAHEDRON /f3;4g/ (3, 4, 3, 4) A CUBOCTAHEDRON /34fg/ (3, 5, 3, 5) A ICOSIDODECAHEDRON /35fg/ (3, 6, 3, 6) T (Plane TESSELLATION ) /36fg/ (3, 4, 4, 4) A SMALL RHOMBICUBOCTA- HEDRONr3 4fg/ (3, 4, 5, 4) A SMALL RHOMBICOSIDODE- CAHEDRONr3 5fg/ (3, 4, 6, 4) T (Plane TESSELLATION )r36fg/ (4, 4, 4, 4) T (Plane TESSELLATION ) /f4;4g/ (3, 3, 3, 3, 3) P ICOSAHEDRON /f3;5g/ (3, 3, 3, 3, 4) A SNUB CUBE s34fg/ (3, 3, 3, 3, 5) A SNUB DODECAHEDRON s35fg/ (3, 3, 3, 3, 6) T (Plane TESSELLATION )s36fg/ (3, 3, 3, 4, 4) T (Plane TESSELLATION )– (3, 3, 4, 3, 4) T (Plane TESSELLATION )s44fg/ (3, 3, 3, 3, 3) T (Plane TESSELLATION ) /f3;6g/ As shown in the above table, there are exactly 13 Archimedean solids (Walsh 1972, Ball and Coxeter 1987). They are called the CUBOCTAHEDRON ,GREATRHOMBICOSIDODECAHEDRON ,GREAT RHOMBICUBOCTA- HEDRON ,ICOSIDODECAHEDRON ,SMALL RHOMBICOSIDO- DECAHEDRON ,SMALL RHOMBICUBOCTAHEDRON ,SNUB CUBE ,SNUB DODECAHEDRON ,TRUNCATED CUBE ,TRUN- CATED DODECAHEDRON ,TRUNCATED ICOSAHEDRON (soccer ball), TRUNCATED OCTAHEDRON , and TRUN- CATED TETRAHEDRON . The Archimedean solids satisfy (2p/C28s)V/C304p; where sis the sum of face-angles at a vertex and Vis the number of vertices (Steinitz and Rademacher 1934, Ball and Coxeter 1987). Here are the Archimedean solids shown in alphabe- tical order (left to right, then continuing to the next row). The following table lists the symbols for the Archi-medean solids (Wenninger 1989, p. 9). nSolid S CHLA ¨FLI SYMBOLWYTHOFF SYMBOLC&R Symbol 1CUBOCTAHEDRON /3 4fg/ 22½34 3 4 (3.4)2 2GREAT RHOMBICOSIDODECA- HEDRONt3 5fg/ 2352 ½34/ 3GREAT RHOMBICUBOCTAHE- DRONt3 4fg/ 2342 ½34/ 4ICOSIDODECAHEDRON /35fg/ 22½34 3 5 (3.5)2 5SMALL RHOMBICOSIDODECA- HEDRONt3 5fg/ 352½34 2 3.4.5.4 6SMALL RHOMBICUBOCTAHE- DRONr3 4fg/ 342½34 2 3.43 7SNUB CUBE s3 4fg//2½34 2 3 4 34.4 8SNUB DODECAHEDRON s35fg//2½34 2 3 5 34.5 9TRUNCATED CUBE /tf4;3g/232½34 4 3.82 10 TRUNCATED DODECAHEDRON t/f5;3g/232½34 5 3.102 11 TRUNCATED ICOSAHEDRON /tf3;5g/252½34 3 5.62 12 TRUNCATED OCTAHEDRON t/f3;4g/242½34 3 4.62 13 TRUNCATED TETRAHEDRON t/f3;3g/232½34 3 3.62 The following table gives the number of vertices v, edges e, and faces f, together with the number of n- gonal faces fnfor the Archimedean solids. nSolid ve f /f3//f4//f5//f6//f8//f10/ 1CUBOCTAHEDRON 12 24 14 8 6 2GREAT RHOMBICOSIDODECAHEDRON120 180 62 30 20 12 3GREATRHOMBICUBOCTAHEDRON 48 72 26 12 8 6 4ICOSIDODECAHEDRON 30 60 32 20 12 5SMALLRHOMBICOSIDODECAHEDRON 60 120 62 20 30 12 6SMALL RHOMBICUBOCTAHEDRON24 48 26 8 18 7SNUB CUBE 24 60 38 32 6 8SNUB DODECAHEDRON 60 150 92 80 12 9TRUNCATED CUBE 24 36 14 8 6 10 TRUNCATED DODECAHEDRON 60 90 32 20 12 11 TRUNCATED ICOSAHEDRON 60 90 32 12 20 12 TRUNCATED OCTAHEDRON 24 36 14 6 8 13 TRUNCATED TETRAHEDRON 12 18 8 4 4 Let rbe the INRADIUS of the dual polyhedron (corresponding to the INSPHERE , which touches the faces of the dual solid), rbe the MIDRADIUS of both the polyhedron and its dual (corresponding to the MID- SPHERE , which touches the edges of both the poly- hedron and its duals), and Rthe CIRCUMRADIUS(corresponding to the CIRCUMSPHERE of the solid which touches the vertices of the solid). Since the CIRCUMSPHERE and INSPHERE are dual to each other, they obey the relationship Rr/C30r2(1) (Cundy and Rollett 1989, Table II following p. 144). The following tables give the analytic and numericalvalues of r,r;andRfor the Archimedean solids with EDGES of unit length (Coxeter et al. 1954; Cundy and Rollett 1989, Table II following p. 144). Hume (1986)gives approximate expressions for the DIHEDRAL ANGLES of the Archimedean solid (and exact expres- sions for their duals). nSolid r /r/ R 1 CUBOCTAHEDRON /3 4//12ffiffiffi 3p / 1 2 GREAT RHOMBICOSIDODECAHEDRON/1 241105/C276ffiffiffi 5prC0rC1 / //C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31/C2712ffiffiffi 5pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C2712ffiffiffi 5pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31/C2712ffiffiffi 5pp / 3 GREAT RHOMBICUBOCTAHEDRON/3 9714/C27ffiffiffi 2prC0rC1 / //C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13/C276ffiffiffi 2pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C276ffiffiffi 2pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13/C276ffiffiffi 2pp / 4 ICOSIDODECAHEDRON /1 85/C273ffiffiffi 5prC0rC1 //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pp //1 2(1/C27ffiffiffi 5p )/ 5 SMALL RHOMBICOSIDODECAHE- DRON/1 4115/C272ffiffiffi 5prC0rC1 / //C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11/C274ffiffiffi 5pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C274ffiffiffi 5pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11/C274ffiffiffi 5pp / 6 SMALL RHOMBICUBOCTAHEDRON /1 176/C27ffiffiffi 2prC0rC1 / //C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 2pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C272ffiffiffi 2pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 2pp / 7 SNUB CUBE ** * 8 SNUB DODECAHEDRON ** * 9 TRUNCATED CUBE /1 175/C272ffiffiffi2prC0rC1 / //C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C274ffiffiffi 2pp //1 22/C27ffiffiffi 2prC0rC1 //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C274ffiffiffi 2pp / 10 TRUNCATED DODECAHEDRON /5 48817ffiffiffi2p /C273ffiffiffiffiffiffi10prC0rC1 / //C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 37/C2715ffiffiffi 5pp //1 45/C273ffiffiffi 5prC0rC1 //1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 74/C2730ffiffiffi 5pp / 11 TRUNCATED ICOSAHEDRON /9 87221/C27ffiffiffi 5prC0rC1 / //C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 58/C2718ffiffiffi 5pp //3 41/C27ffiffiffi 5prC0rC1 //1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 58/C2718ffiffiffi 5pp / 12 TRUNCATED OCTAHEDRON /9 20ffiffiffiffiffiffi 10p //3 2//12ffiffiffiffiffiffi 10p / 13 TRUNCATED TETRAHEDRON /9 44ffiffiffiffiffiffi22p //3 4ffiffiffi 2p //1 2ffiffiffiffiffiffi 22p / *The complicated analytic expressions for the CIR- CUMRADII of these solids are given in the entries for the SNUB CUBE and SNUB DODECAHEDRON . nSolid r /r/ R 1CUBOCTAHEDRON 0.75 0.86603 1 2GREAT RHOMBICOSIDODECAHEDRON3.73665 3.76938 3.80239 3GREAT RHOMBICUBOCTAHEDRON2.20974 2.26303 2.31761 4ICOSIDODECAHEDRON 1.46353 1.53884 1.61803 5SMALL RHOMBICOSIDODECAHEDRON2.12099 2.17625 2.23295 6SMALL RHOMBICUBOCTAHEDRON1.22026 1.30656 1.39897 7SNUB CUBE 1.15763 1.24719 1.34371 8SNUB DODECAHEDRON 2.03969 2.09688 2.15583 9 TRUNCATED CUBE 1.63828 1.70711 1.77882 10 TRUNCATED DODECAHEDRON 2.88526 2.92705 2.96945 11 TRUNCATED ICOSAHEDRON 2.37713 2.42705 2.47802 12 TRUNCATED OCTAHEDRON 1.42302 1.5 1.58114 13 TRUNCATED TETRAHEDRON 0.95940 1.06066 1.17260 The Archimedean solids and their DUALS are all CANONICAL POLYHEDRA . Since the Archimedean solids of convex, the CONVEX HULL of each Archimedean solid is the solid itself. See also ARCHIMEDEAN SOLID STELLATION ,CATALAN SOLID,DELTAHEDRON ,ISOHEDRON ,JOHNSON SOLID , KEPLER- POINSOT SOLID,PLATONIC SOLID,Q UASIRE- GULAR POLYHEDRO N,S EMIREGULAR POLYHEDRON , UNIFORM POLYHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 136, 1987. Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2: Geometry. Cam- bridge, MA: MIT Press, pp. 269 /C186, 1974. Catalan, E. "Me´moire sur la The´orie des Polye`dres." J. l’E´ cole Polytechnique (Paris) 41,1/C11, 1865. Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc. 24, 1 /C1, 1928. Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z. 46, 380 /C107, 1940. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401 /C150, 1954. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 79 /C16, 1997. Cundy, H. and Rollett, A. "Stellated Archimedean Polyhe- dra." §3.9 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 123 /C128 and Table II following p. 144, 1989. Fejes To´th, L. Ch. 4 in Regular Figures. Oxford, England: Pergamon Press, 1964. Holden, A. Shapes, Space, and Symmetry. New York: Dover, p. 54, 1991. Hume, A. "Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals." Computing Science Tech. Rep. , No. 130. Murray Hill, NJ: AT&T Bell Laboratories, 1986. Kepler, J. "Harmonice Mundi." Opera Omnia, Vol. 5. Frankfurt, pp. 75 /C134, 1864. Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 199 /C107, 1942. Le, Ha. "Archimedean Solids." http://daisy.uwaterloo.ca/ ~hqle/Polyhedra/archimedean.html. Lines, L. Solid Geometry. New York: Dover, 1965. Maehara, H. "On the Sphericity of the Graphs of Semi- Regular Polyhedra." Discr. Math. 58, 311 /C115, 1986. Nooshin, H.; Disney, P. L.; and Champion, O. C. "Properties of Platonic and Archimedean Polyhedra." Table 12.1 in "Computer-Aided Processing of Polyhedric Configura- tions." Ch. 12 in Beyond the Cube: The Architecture ofSpace Frames and Polyhedra (Ed. J. F. Gabriel). New York: Wiley, pp. 360 /C161, 1997. Pearce, P. Structure in Nature Is a Strategy for Design. Cambridge, MA: MIT Press, pp. 34 /C15, 1978. Pedagoguery Software. Poly . http://www.peda.com/poly/. Pugh, A. Polyhedra: A Visual Approach. Berkeley: Univer- sity of California Press, p. 25, 1976. Rawles, B. A. "Platonic and Archimedean Solids--Faces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios." http://www.intent.com/sg/polyhedra.html. Robertson, S. A. and Carter, S. "On the Platonic and Archimedean Solids." J. London Math. Soc. 2, 125 /C132, 1970. Rorres, C. "Archimedean Solids: Pappus." http:// www.mcs.drexel.edu/~crorres/Archimedes/Solids/Pap- pus.html. Steinitz, E. and Rademacher, H. Vorlesungen u¨ber die Theorie der Polyheder. Berlin, p. 11, 1934. Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3 /C14, 1910. Vichera, M. "Archimedean Polyhedra." http://alpha.ujep.cz/ ~vicher/puzzle/telesa/telesa.htm. Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods of Semi-Regular Polyhedra." Geometriae Dedicata 1, 117 /C123, 1972. Weisstein, E. W. "Archimedean Solids with Analytic Ver- tices." MATHEMATICA NOTEBOOK ARCHIMEDEAN.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 6 /C1, 1991. Wenninger, M. J. "The Thirteen Semiregular Convex Poly- hedra and Their Duals." Ch. 2 in Dual Models. Cam- bridge, England: Cambridge University Press, pp. 14 /C15, 1983. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, 1989. Archimedean Solid Stellation A large class of POLYHEDRA which includes the DODECADODECAHEDRON and GREAT ICOSIDODECAHE- DRON . No complete enumeration (even with restric- tive uniqueness conditions) has been worked out. There are at least four stellations of the CUBOCTAHE- DRON (Wenninger 1989), although the exact number depends on what type of cells formed by planeintersections are allowed. There are also many stellations of the Archimedean solid duals. The RHOMBIC DODECAHEDRON has three stellations (Wells 1991, pp. 216 /C117). See also ARCHIMEDEAN SOLID ,CATALAN SOLID References Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401/C150, 1954. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Wenninger, M. J. "Commentary on the Stellation of the Archimedean Solids." In Polyhedron Models. New York: Cambridge University Press, pp. 66 /C12, 1989. Archimedean Spiral A SPIRAL with POLAR equation r /C30 a u1 =n ; (1) where r is the radial distance, u is the polar angle, and n is a constant which determines how tightly the spiral is "wrapped." The CURVATURE of an Archime- dean spiral is given by k /C30nu1 /C281=n(1 /C27 n /C27 n2 u2) a(1 /C27 n2 u2)3 =2 ; (2) and the ARC LENGTH by s /C30au1 =n 2F1((2n)/C281 ;/C281 2;1/C27(2n) /C281; /C28n2 u2) ; (3) where2F1(a ; b; c; x)isa HYPERGEOMETRIC FUNC- TION . Various special cases are given in the following table. Name n LITUUS -2 HYPERBOLIC SPIRAL -1 ARCHIMEDES’ SPIRAL 1 FERMAT’S SPIRAL 2 If a fly crawls radially outward along a uniformly spinning disk, the curve it traces with respect to a reference frame in which the disk is at rest is an Archimedean spiral (Steinhaus 1999, p. 137). Furthermore, a heart-shaped frame composed of two arcs of an Archimedean spiral which is fixed to a rotating disk converts uniform rotational motion to uniform back-and-forth motion (Steinhaus 1999, pp. 136 /C1/37). See also ARCHIMEDES’ SPIRAL ,D AISY,F ERMAT’S SPIRAL ,HYPERBOLIC SPIRAL ,LITUUS ,SPIRAL References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 90 /C1/2, 1997. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 59 /C1/0, 1991. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 and 189, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967. MacTutor History of Mathematics Archive. "Spiral of Archi- medes." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Spiral.html. Pappas, T. "The Spiral of Archimedes." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 149, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 136 /C1/37, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 8 /C1/, 1991. Archimedean Spiral Inverse Curve The INVERSE CURVE of the A RCHIMEDEAN SPIRAL r/C30au1=n with INVERSION CENTER at the origin and inversion RADIUS kis the A RCHIMEDEAN SPIRAL r/C30kau1=n: Archimedean Tessellation TESSELLATION Archimedean Valuation AVALUATION for which xjj51IMPLIES 1/C27x jj5Cfor the constant C/C301 (independent of x). Such a VALUA- TION does not satisfy the strong TRIANGLE INEQUALITY x/C27y jj5max( xjj;yjj): Archimedes Algorithm Successive application of A RCHIMEDES’ RECURRENCE FORMULA gives the Archimedes algorithm, which can be used to provide successive approximations to p(PI). The algorithm is also called the B ORCHARDT- PFAFF ALGORITHM . Archimedes obtained the first rigorous approximation of pbyCIRCUMSCRIBING and INSCRIB- INGn/C30G /C2152k /-gons on a CIRCLE . From A RCHIMEDES’ RECURRENCE FORMULA , the CIRCUMFERENCES aandb of the circumscribed and inscribed POLYGONS are a(n)/C302ntanp n ! (1) b(n)/C302nsinp n ! ; (2) where b(n)BC/C302pr/C302p/C2151/C302pBa(n): (3) For a HEXAGON ,n/C306 and a0/C13a(6)/C304ffiffiffi 3p (4) b0/C13b(6)/C306; (5) where ak/C13a(6 /C2152k):The first iteration of A RCHI- MEDES’ RECURRENCE FORMULA then gives a1/C302 /C2156 /C2154ffiffiffi 3p 6/C274ffiffiffi3p/C3024ffiffiffi3p 3/C272ffiffiffi3p/C3024 2/C28ffiffiffi 3prC16rC1* (6) b 1/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24 2/C28ffiffiffi 3prC16rC1* /C2156r /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 3pq /C306ffiffiffi 6p /C28ffiffiffi 2prC16rC1* : (7) Additional iterations do not have simple closed forms, but the numerical approximations for k /C300, 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are 3:00000 B p B3:46410 (8) 3:10583 B p B3:21539 (9) 3 :13263 B p B3 :15966 (10) 3 :13935 B p B3 :14609 (11) 3:14103 B p B3:14271 : (12) By taking k /C304 (a 96-gon) and using strict inequal- ities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result 223 71 /C303:14084... B p B22 7 /C303 :14285... : (13) See also PI References Miel, G. "Of Calculations Past and Present: The Archime- dean Algorithm." Amer. Math. Monthly 90,17/C1/5, 1983. Phillips, G. M. "Archimedes in the Complex Plane." Amer. Math. Monthly 91, 108 /C1/14, 1984. Archimedes’ Axiom An AXIOM actually attributed to Eudoxus (Boyer and Merzbach 1991, pp. 89 /C1/0) which states that a b /C30c d IFF the appropriate one of following conditions is satisfied for INTEGERS m and n: 1. If ma Bnb, then mc Bnd. 2. If ma /C30nb, then mc /C30nd. 3. If ma /C21nb, then mc /C21nd. Also known as the continuity axiom or Archimedes’ lemma, this axiom survives in the writings of Eu- doxus (Boyer and Merzbach 1991). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the EXHAUSTION METHOD which Archimedes invented to solve problems of AREA and VOLUME . Formally, Archimedes’ axiom states that if AB and CD are two line segments, then there exist a finite number of points A1 ; A2 ; ..., An on A @ B such that CD /C13AA1 /C13AA2 /C13.../C13An/C281An ; and B is between A and An(Itoˆ 1986, p. 611). A geometry in which Archimedes’ lemma does not hold is called a NON- ARCHIMEDEAN GEOMETRY .See also CONTINUITY AXIOMS ,FRACTION ,INEQUALITY , NON-ARCHIMEDEAN GEOMETRY References Boyer, C. B. and Merzbach, U. C. "The Abacus and Decimal Fractions." A History of Mathematics, 2nd ed. New York: Wiley, p. 100, 1991. Itoˆ, K. (Ed.). §155B and 155D in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 611, 1986. Archimedes’ Cattle Problem Also called the BOVINUM PROBLEMA . It is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus onethird the number of the black greater than the brown;the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh thenumber of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number ofthe black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number ofthe brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?" Solution consists of solving the simultaneous D IO- PHANTINE EQUATIONS inINTEGERS W,X,Y,Z(the number of white, black, spotted, and brown bulls) and w,x,y,z(the number of white, black, spotted, and brown cows), W/C305 6X/C27Z (1) X/C309 20Y/C27Z (2) Y/C301342W/C27Z (3) w/C307 12(X/C27x) (4) x/C309 20(Y/C27y) (5) y/C301130(Z/C27z) (6) z/C301342(W/C27w): (7) The smallest solution in INTEGERS is W/C3010;366;482 (8) X/C307;460;514 (9) Y/C307;358;060 (10) Z/C304;149;387 (11) w /C307 ;206;360 (12) x /C304 ;893;246 (13) y /C303 ;515;820 (14) z /C305;439;213: (15) A more complicated version of the problem requires that W /C27X be a SQUARE NUMBER and Y /C27Z a TRIANGULAR NUMBER . The solution to this PROBLEM are numbers with 206544 or 206545 digits. References Amthor, A. and Krumbiegel B. "Das Problema bovinum des Archimedes." Z. Math. Phys. 25, 121 /C1/71, 1880. Archibald, R. C. "Cattle Problem of Archimedes." Amer. Math. Monthly 25, 411 /C1/14, 1918. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, pp. 249 /C1/52, 1966. Bell, A. H. "Solution to the Celebrated Indeterminate Equa- tion x2 /C28ng2 /C301 :/" Amer. Math. Monthly 1, 240, 1894. Bell, A. H. "‘Cattle Problem.’ By Archimedes 251 BC." Amer. Math. Monthly 2, 140, 1895. Bell, A. H. "Cattle Problem of Archimedes." Math. Mag. 1, 163, 1882 /C1/884. Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, p. 391, 1989. Calkins, K. G. "Archimedes’ Problema Bovinum. " http:// www2.andrews.edu/~calkins/profess/cattle.htm. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 342 /C1/45, 1952. Do¨rrie, H. "Archimedes’ Problema Bovinum ." §1in100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 3 /C1/, 1965. Grosjean, C. C. and de Meyer, H. E. "A New Contribution to the Mathematical Study of the Cattle-Problem of Archi- medes." In Constantin Carathe ´odory: An International Tribute, Vols. 1 and 2 (Ed. T. M. Rassias). Teaneck, NJ: World Scientific, pp. 404 /C1/53, 1991. Merriman, M. "Cattle Problem of Archimedes." Pop. Sci. Monthly 67, 660 /C1/65, 1905. Rorres, C. "The Cattle Problem." http://www.mcs.drexel.edu/ ~crorres/Archimedes/Cattle/Statement.html. Stewart, I. "Mathematical Recreations: Counting the Cattle of the Sun." Sci. Amer. 282, 112 /C1/13, Apr. 2000. Vardi, I. "Archimedes’ Cattle Problem." Amer. Math. Monthly 105, 305 /C1/19, 1998. Archimedes’ Circles Draw the PERPENDICULAR LINE from the intersection of the two small SEMICIRCLES in the ARBELOS . The two CIRCLES C1and C2TANGENT to this line, the largeSEMICIRCLE , and each of the two SEMICIRCLES are then congruent and known as Archimedes’ circles. See also ARBELOS ,BANKOFF CIRCLE ,SEMICIRCLE Archimedes’ Constant PI Archimedes’ Hat-Box Theorem Enclose a SPHERE in a CYLINDER and cut out a SPHERICAL SEGMENT by slicing twice PERPENDICU- LARLY to the CYLINDER ’s axis. Then the lateral SUR- FACE AREA of the SPHERICAL SEGMENT S1is equal to the lateral SURFACE AREA cut out of the CYLINDER S2 by the same slicing planes, i.e., S /C13S1 /C30S2 /C302pRh; where R is the RADIUS of the CYLINDER (and tangent SPHERE ) and his the height of the cylindrical (and spherical) segment. See also ARCHIMEDES’ PROBLEM ,CYLINDER ,SPHERE , SPHERICAL SEGMENT References Cundy, H. and Rollett, A. "Sphere and Cylinder--Archi- medes’ Theorem." §4.3.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 172 /C1/73, 1989. Archimedes’ Lemma ARCHIMEDES’ AXIOM Archimedes’ Midpoint Theorem LetMbe the MIDPOINT of the ARC AMB . Pick Cat random and pick Dsuch that MD/C222AC(where /C222 denotes PERPENDICULAR ). Then AD /C30DC /C27BC: See also MIDPOINT References Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 31 /C1/2, 1991. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 1 /C1/, 1995. Archimedes’ Postulate ARCHIMEDES’ LEMMA Archimedes’ Problem Cut a SPHERE by a PLANE in such a way that the VOLUMES of the SPHERICAL SEGMENTS have a given RATIO . See also ARCHIMEDES’ HAT-BOX THEOREM ,SPHERICAL SEGMENT Archimedes’ Recurrence Formula Let anand bnbe the PERIMETERS of the CIRCUM- SCRIBED and INSCRIBED n-gon and a2nand b2nthe PERIMETERS of the CIRCUMSCRIBED and INSCRIBED 2n/- gon. Then a2n /C302anbn an /C27 bn(1) b2n /C30ffiffiffiffiffiffiffiffiffiffiffiffi a2nbnp : (2) The first follows from the fact that side lengths of the POLYGONS on a CIRCLE of RADIUS r /C301 are sR /C302 tanp n ! (3) sr /C302 sinp n ! ; (4)so an /C302n tanp n ! (5) bn /C302n sinp n ! : (6) But 2anbn an /C27 bn/C302 /C215 2n tanp n ! /C215 2n sinp n ! 2n tanp n ! /C27 2n sinp n ! /C304ntanp n ! sinp n ! tanp n ! /C27 sinp n ! : (7) Using the identity tan1 2xrC16rC1* /C30tan x sin x tan x /C27 sin x (8) then gives 2anbn an /C27 bn/C304n tanp 2nrC16rC1* /C30a2n : (9) The second follows from ffiffiffiffiffiffiffiffiffiffiffiffi a2nbnp /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4n tanp 2n ! /C215 2n sinp n !vuut(10) Using the identity sin x /C302 sin 1 2 xrC16rC1* cos12 xrC16rC1* (11) gives ffiffiffiffiffiffiffiffiffiffiffiffi a2nbnp /C302nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tanx 2n ! /C2152 sinp 2n ! cosp 2n !vuut /C304nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2p 2n !vuut/C304n sin p 2n ! /C30b2n : (12) Successive application gives the ARCHIMEDES ALGO- RITHM , which can be used to provide successive approximations to PI(/p):/ See also ARCHIMEDES ALGORITHM ,PI References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 186, 1965. Archimedes’ Spiral An ARCHIMEDEAN SPIRAL with POLAR equation r /C30a u: This spiral was studied by Conon, and later by Archimedes in On Spirals about 225 BC. Archimedes was able to work out the lengths of various tangents to the spiral. Archimedes’ spiral can be used for COMPASS and STRAIGHTEDGE division of an ANGLE into n parts (including ANGLE TRISECTION ) and can also be used for CIRCLE SQUARING . In addition, the curve can be used as a cam to convert uniform circular motion into uniform linear motion (Steinhaus 1983, p. 137; Brown). The cam consists of one arch of the spiral above the X-AXIS together with its reflection in the X- AXIS. Rotating this with uniform angular velocity about its center will result in uniform linear motion of the point where it crosses the Y-AXIS . See also ARCHIMEDEAN SPIRAL References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987. Brown, H. T. 507 Mouvements me´caniques. Lie`ge, Belgium: Desoer, p. 28, 1923. Gardner, M. The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: Chicago University Press, pp. 106 /C1/07, 1991. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 90 /C1/2, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 /C1/87, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 173 /C1/64, 1967. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 137, 1999. Archimedes’ Spiral Inverse Taking the ORIGIN as the INVERSION CENTER ,ARCHI- MEDES’ SPIRAL r /C30a u inverts to the HYPERBOLIC SPIRAL r /C30a=u :/ ArcSec INVERSE SECANTArcsec INVERSE SECANT Arcsecant INVERSE SECANT ArcSech INVERSE HYPERBOLIC SECANT Arcsech INVERSE HYPERBOLIC SECANT ArcSin INVERSE SINE Arcsin INVERSE SINE Arcsine INVERSE SINE Arcsinh INVERSE HYPERBOLIC SINE ArcSinh INVERSE HYPERBOLIC SINE Arctan INVERSE TANGENT ArcTan INVERSE TANGENT Arctangent INVERSE TANGENT Arctangent Integral INVERSE TANGENT INTEGRAL Arctanh INVERSE HYPERBOLIC TANGENT ArcTanh INVERSE HYPERBOLIC TANGENT Arcth Arcth z/C301 icot/C281(/C28iz); where cot/C281zis the INVERSE COTANGENT . See also ARCH,ARSH,ARTH,INVERSE COTANGENT References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. xxx, 2000. Arcwise-Connected See also CONNECTED SET,LOCALLY PATHWISE- CON- NECTED ,PATH-CONNECTED ,PATHWISE- CONNECTED Arcwise-Connected Set See also CONNECTED SET,PATH-CONNECTED SET Area The AREA of a SURFACE is the amount of material needed to "cover" it completely. The AREA of a TRIANGLE is given by AD/C301 2 lh ; (1) where l is the base length and h is the height, or by HERON’S FORMULA AD/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s /C28a)(s /C28b)(s /C28c)p ; (2) where the side lengths are a, b, and c and s the SEMIPERIMETER . The AREA of a RECTANGLE is given by Arectangle /C30ab ; (3) where the sides are length a and b. This gives the special case of Asquare /C30a2 (4) for the SQUARE . The AREA of a REGULAR POLYGON with n sides and side length s is given by An/C28gon /C301 4 ns2 cotp n ! : (5) CALCULUS and, in particular, the INTEGRAL , are powerful tools for computing the AREA between a curve f(x) and the X-AXIS over an INTERVAL [a, b], giving A /C30gb af(x) dx : (6) The AREA of a POLAR curve with equation r /C30r( u)is A /C301 2gr2 d u: (7) Written in CARTESIAN COORDINATES , this becomes A /C301 2 g xdy dt /C28ydx dt ! dt (8) /C3012 g(xdy/C28ydx) : (9) For the AREA of special surfaces or regions, see the entry for that region. The generalization of AREA to 3-D is called VOLUME , and to higher DIMENSIONS is called CONTENT . See also ARC LENGTH ,A REA ELEMENT ,C ONTENT , SURFACE AREA,VOLUME References Gray, A. "The Intuitive Idea of Area on a Surface." §15.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 351 /C1/53, 1997. Area Element The area element for a SURFACE with RIEMANNIAN METRIC ds2 /C30Edu2 /C272Fdudv /C27Gdv2 is dA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG /C28 F2p du ffldv ; where du ffldv is the WEDGE PRODUCT . See also AREA,LINE ELEMENT ,RIEMANNIAN METRIC , VOLUME ELEMENT References Gray, A. "The Intuitive Idea of Area on a Surface." §15.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 351 /C1/53, 1997. Area Integral A double integral over three coordinates giving the AREA within some region R, A /C30ggRdx dy: If a plane curve is given by /y /C30f(x)/, then the area between the curve and the X-AXIS from x /C30 a to x /C30 bis given by A/C30gb af(x)dx: See also INTEGRAL ,L INE INTEGRAL ,L USIN AREA INTEGRAL ,M ULTIPLE INTEGRAL ,SURFACE INTEGRAL , VOLUME INTEGRAL Area Principle There are at least two results known as "the area principle." The geometric area principle states that A1Pjj A2Pjj/C30A1BCjj A2BCjj: (1) This can also be written in the form A1Pjj A2Pjj"# /C30A1BCjj A2BCjj"# ; (2) where AB CD"# (3) is the ratio of the lengths [A, B] and [C, D] for AB ½½CD with a PLUS or MINUS SIGN depending on if these segments have the same or opposite directions, and ABC DEF"# (4) is the RATIO of signed AREAS of the TRIANGLES . Gru¨nbaum and Shepard (1995) show that CEVA’S THEOREM ,HOEHN’S THEOREM , and MENELAUS’ THEO- REM are the consequences of this result. The area principle of complex analysis states that if f is a SCHLICHT FUNCTION and if h(z) /C301 f(z) /C301 z/C27X/C12 j/C300bjzj ; (5) then X/C12 j/C301jbjrC10rC10rC10rC10251 (6) (Krantz 1999, p. 150). See also CEVA’S THEOREM ,H OEHN’S THEOREM ,M E- NELAUS’ THEOREM ,SCHLICHT FUNCTION ,SELF-TRANS- VERSALITY THEOREM References Gru¨nbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254 /C1/68, 1995. Krantz, S. G. "Schlicht Functions." §12.1.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 149, 1999.Areal Coordinates BARYCENTRIC COORDINATES (t1 ; t2 ; t3) normalized so that they become the AREAS of the TRIANGLES PA1A2 ; PA1A3 ; and PA2A3 ; where P is the point whose coordinates have been specified, normalized by the area of the original triangle DA1A2A3 : This is equiva- lent to application of the normalization relation t1 /C27t2 /C27t3 /C301 (Coxeter 1969, p. 218). See also BARYCENTRIC COORDINATES ,TRILINEAR CO- ORDINATES References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 218, 1969. Area-Preserving Map A MAP F from Rn to Rn is AREA -preserving if m(F(A)) /C30 m(A) for every subregion A of Rn ; where m(A) is the n-D MEASURE of A. A linear transformation is AREA - preserving if its corresponding DETERMINANT is equal to 1. See also CONFORMAL MAP,SYMPLECTIC MAP Arf Invariant ALINK invariant which always has the value 0 or 1. A KNOT has A RF INVARIANT 0 if the KNOT is "pass equivalent" to the UNKNOT and 1 if it is pass equivalent to the TREFOIL KNOT .I fK/C27;K/C28;and L are projections which are identical outside the region of the crossing diagram, and K/C27and K/C28are KNOTS while lis a 2-component LINK with a nonintersecting crossing diagram where the two left and right strandsbelong to the different LINKS , then a(K/C27)/C30a(K/C28)/C27l(L1;L2); (1) where lis the LINKING NUMBER ofL1andL2:The Arf invariant can be determined from the ALEXANDER POLYNOMIAL or J ONES POLYNOMIAL for a KNOT . ForDK the A LEXANDER POLYNOMIAL ofK, the Arf invariant is given by DK(/C281) /C131(mod 8) if Arf(K) /C300 5(mod 8) if Arf(K) /C301rC06 (2) (Jones 1985). For the JONES POLYNOMIAL WKof a KNOT K, Arf(K) /C30WK(i) (3) (Jones 1985), where I is the IMAGINARY NUMBER . References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 223 /C1/31, 1994. Jones, V. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103 /C1/11, 1985. Weisstein, E. W. "Knots." MATHEMATICA NOTEBOOK KNOTS.M . Arg ARGUMENT (COMPLEX NUMBER ) Argand Diagram A plot of COMPLEX NUMBERS as points z /C30x /C27iy using the X-AXIS as the REAL AXIS and Y-AXIS as the IMAGINARY AXIS. An Argand diagram is also called the COMPLEX PLANE or ARGAND PLANE . The Argand plane was described by C. Wessel prior to Argand. See also COMPLEX PLANE ,IMAGINARY NUMBER ,REAL NUMBER References Argand, R. Essai sur une manie `re de repre´senter les quantite ´s imaginaires dans les constructions ge´o- me´triques. Paris: Albert Blanchard, 1971. Reprint of the 2nd ed., published by G. J. Hoel in 1874. First edition published Paris, 1806. Argand Plane ARGAND DIAGRAM Argoh’s Conjecture Let Bk be the kth BERNOULLI NUMBER . Then does nBn/C281 /C13/C281 (mod n) IFF n is PRIME ? For example, for n /C30 1, 2, ..., nBn/C281 (mod n) is 0, -1, -1, 0, -1, 0, -1, 0, -3, 0, -1, ... (Sloane’s A046094). There are no counterexamples less than n /C305; 600: Any counterexample to Argoh’s conjecture would be a contradiction to GIUGA’S CONJECTURE , and vice versa. See also BERNOULLI NUMBER ,GIUGA’S CONJECTURE References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,40/C1/0, 1996.Sloane, N. J. A. Sequences A046094 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Argument (Complex Number) A COMPLEX NUMBER z may be REPRESENTED AS z /C13x /C27iy /C30 zjjei u ; (1) where zjjis called the MODULUS of z, and u is called the argument (or PHASE ) and is given by arg(x /C27iy) /C13tan /C281y x ! : (2) Here, u; sometimes also denoted f; corresponds to the counterclockwise ANGLE from the POSITIVE REAL AXIS, i.e., the value of u such that x /C30cos u and y /C30sin u: The special kind of INVERSE TANGENT used here takes into account the quadrant in which z lies and is returned by theFORTRAN command ATAN2(X,Y) and the Mathematica command ArcTan [x, y], and is often restricted to the range /C28p B u 5 p: In the degenerate case when x /C30 0, f /C30/C281 2 p if y B0 undefined if y /C300 1 2 p if y > 0:8 >< >:(3) From the definition of the argument, arg(zw) /C30arg( zjjeiuz wjjeiuw ) /C30arg(ei uz eiuw ) /C30arg ei(uz/C27uw)rC0rCB /C30arg(z) /C27arg(w) : (4) Extending this procedure gives arg(zn) /C30n arg(z) : (5) The argument of a COMPLEX NUMBER is sometimes called the PHASE . See also AFFIX,C OMPLEX NUMBER , DE MOIVRE’S IDENTITY ,E ULER FORMULA ,IMAGINARY PART,IN- VERSE TANGENT ,M ODULUS (COMPLEX NUMBER ), PHASE ,PHASOR ,REAL PART References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Krantz, S. G. "The Argument of a Complex Number." §1.2.6 nHandbook of Complex Analysis. Boston, MA: Birkha ¨u- ser, p. 11, 1999. Silverman, R. A. Introductory Complex Analysis. New York: Dover, 1984. Argument (Elliptic Integral) Given an AMPLITUDE fin an ELLIPTIC INTEGRAL , the argument uis defined by the relation f/C13am u : See also AMPLITUDE ,ELLIPTIC INTEGRAL Argument (Function) An argument of a FUNCTION f(x1 ; ... ; xn) is one of the n parameters on which the function’s value depends. For example, the SINE sin x is a one-argument function, the BINOMIAL COEFFICIENTn mrC0rC1 is a two- argument function, and the HYPERGEOMETRIC FUNC- TION 2F1(a ; b; c; z) is a four-argument function. Argument Addition Relation A mathematical relationship relating f(x /C27y)tof(x) and f(y) :/ See also ARGUMENT MULTIPLICATION RELATION ,RE- CURRENCE RELATION ,REFLECTION RELATION ,TRANS- LATION RELATION Argument Multiplication Relation A mathematical relationship relating f(nx)tof(x) for INTEGER n. See also ARGUMENT ADDITION RELATION ,R ECUR- RENCE RELATION ,REFLECTION RELATION ,TRANSLA- TION RELATION Argument Principle If f(z)is MEROMORPHIC in a region R enclosed by a CONTOUR g ; let N be the number of COMPLEX ROOTS of f(z)in g; and P be the number of POLES in g ; then N /C28P /C301 2pi g gf ?(z) dz f(z) Defining w /C13f(z) and s /C13f( g) gives N /C28P /C301 2 pi g sdw w: See also CAUCHY INTEGRAL FORMULA ,CAUCHY INTE- GRAL THEOREM ,H URWITZ’S ROOT THEOREM ,M ERO- MORPHIC FUNCTION ,P OLE,R OOT,R OUCHE ´ ’S THEOREM ,VARIATION OF ARGUMENT References Duren, P.; Hengartner, W.; and Laugessen, R. S. "The Argument Principle for Harmonic Functions." Math. Mag. 103, 411 /C1/15, 1996. Knopp, K. Theory of Functions, Parts I and II. New York: Dover, pp. 132 /C1/34, 1996. Krantz, S. G. "The Argument Principle." Ch. 5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 69 /C1/8, 1999. Argument Variation VARIATION OF ARGUMENTAristotle’s Wheel Paradox A PARADOX mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two con- centric CIRCLES of different DIAMETERS (a wheel within a wheel). there is a 1:1 correspondence of points on the large CIRCLE with points on the small CIRCLE , so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. this seems to imply that the two CIRCUMFERENCES of different sized CIRCLES are equal, which is impossible. The fallacy lies in the assumption that a 1:1 corre- spondence of points means that two curves must have the same length. In fact, the CARDINALITIES of points in a LINE SEGMENT of any length (or even an INFINITE LINE,aPLANE , a 3-D SPACE , or an infinite dimensional EUCLIDEAN SPACE ) are all the same: /C2101(ALEPH-1 ), so the points of any of these can be put in a ONE-TO-ONE correspondence with those of any other. See also ZENO’S PARADOXES References Ballew, D. "The Wheel of Aristotle." Math. Teacher 65, 507/C1/ 09, 1972. Costabel, P. "The Wheel of Aristotle and French Considera- tion of Galileo’s Arguments." Math. Teacher 61, 527/C1/34, 1968. Drabkin, I. "Aristotle’s Wheel: Notes on the History of the Paradox." Osiris 9, 162/C1/98, 1950. Gardner, M. Wheels, Life, and other Mathematical Amuse- ments. New York: W. H. Freeman, pp. 2 /C1/, 1983. Pappas, T. "The Wheel of Paradox Aristotle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 202, 1989. vos Savant, M. The World’s Most Famous Math Problem. New York: St. Martin’s Press, pp. 48 /C1/0, 1993. Arithmetic The branch of mathematics dealing with INTEGERS or, more generally, numerical computation. Arithmetical operations include ADDITION ,CONGRUENCE calcula- tion, DIVISION ,FACTORIZATION ,MULTIPLICATION , POWER computation, ROOT EXTRACTION , and SUBTRAC- TION . Arithmetic was part of the QUADRIVIUM taught in medieval universities. The FUNDAMENTAL THEOREM OF ARITHMETIC , also called the UNIQUE FACTORIZATION THEOREM , states that any POSITIVE INTEGER can be represented in exactly one way as a PRODUCT ofPRIMES . The L O¨WENHEIM- SKOLEM THEOREM , which is a funda- mental result in MODEL THEORY , establishes the existence of "nonstandard" models of arithmetic. See also ALGEBRA ,CALCULUS ,FLOATING- POINT AR- ITHMETIC ,FUNDAMENTAL THEOREM OF ARITHMETIC , GROUP THEORY ,HIGHER ARITHMETIC ,LINEAR ALGE- BRA,LO¨ WENHEIM- SKOLEM THEOREM ,MODEL THEORY , NUMBER THEORY ,TRIGONOMETRY References Karpinski, L. C. The History of Arithmetic. Chicago, IL: Rand, McNally, & Co., 1925. Maxfield, J. E. and Maxfield, M. W. Abstract Algebra and Solution by Radicals. Philadelphia, PA: Saunders, 1992. Thompson, J. E. Arithmetic for the Practical Man. New York: Van Nostrand Reinhold, 1973. Weisstein, E. W. "Books about Arithmetic." http://www.trea- sure-troves.com/books/Arithmetic.html. Arithmetic Function A function c(n) such that c(n/C27m)/C30c(c(n)/C27c(m)) and c(n;m)/C30c(c(n)c(m)): See also ARITHMETICAL FUNCTION References Atanassov, K. Bull. Number Th. 9, 18, 1985. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Arithmetic Geometry A vaguely defined branch of mathematics dealing with VARIETIES , the M ORDELL CONJECTURE ,ARAKE- LOV THEORY , and ELLIPTIC CURVES . References Cornell, G. and Silverman, J. H. (Eds.). Arithmetic Geome- try.New York: Springer-Verlag, 1986. Lorenzini, D. An Invitation to Arithmetic Geometry. Provi- dence, RI: Amer. Math. Soc., 1996. Arithmetic Mean For a CONTINUOUS DISTRIBUTION FUNCTION , the ar- ithmetic mean of the population, denoted m;˜x;xhi;or A(x);is given by m/C30f(x) hi/C13g/C12 /C28/C12P(x)f(x)dx; (1) where xhiis the EXPECTATION VALUE . For a DISCRETE DISTRIBUTION , m/C30f(x) hi/C13PN n/C300P(xn)f(xn)PN n/C300P(xn)/C30XN n/C300P(xn)f(xn): (2) The population mean satisfies f(x)/C27g(x) hi /C30f(x) hi/C27g(x) hi (3)cf(x) hi /C30cf(x) hi ; (4) and f(x)g(y) hi /C30f(x) hi g(y) hi (5) ifxandyare INDEPENDENT STATISTICS . The "sample mean," which is the mean estimated from a statistical sample, is an UNBIASED ESTIMATOR for the population mean. For small samples, the mean is more efficient than the MEDIAN and approximately p=2 less (Kenney and Keeping 1962, p. 211). A general expression which often holds approximately is mean/C28mode:3(mean /C28median) : (6) Given a set of samples fxig;the arithmetic mean is A(x)/C13˜x/C13m/C13xhi/C301 NXN i/C301xi: (7) Hoehn and Niven (1985) show that A(a1/C27c;a2/C27c;...;an/C27c) /C30c/C27A(a1;a2;...;an) (8) for any POSITIVE constant c. For positive arguments, the arithmetic mean satisfies A]G]H; (9) where Gis the GEOMETRIC MEAN and His the HARMONIC MEAN (Hardy et al. 1952; Mitrinovic 1970; Beckenbach and Bellman 1983; Bullen et al. 1988; Mitrinovic et al. 1993; Alzer 1996). This can be shown as follows. For a;b>0; 1ffiffiffiap/C281ffiffiffi bp !2 ]0 (10) 1 a/C282ffiffiffiffiffiffi abp/C271 b]0 (11) 1 a/C271 b]2ffiffiffiffiffiffi abp (12) ffiffiffiffiffiffi abp ]2 1 a/C271 b(13) G]H; (14) with equality IFFb/C30a. To show the second part of the inequality, (ffiffiffiap/C28ffiffiffi bp )2/C30a/C282ffiffiffiffiffiffiabp /C27b]0 (15) a/C27b 2]ffiffiffiffiffiffiabp (16) A]G; (17) with equality IFFa/C30b. Combining (14) and (17) then gives (9). Given n independent random GAUSSIAN DISTRIBUTED variates xi ; each with population mean mi /C30 m and VARIANCE s2 i /C30 s2 ; ˜x /C131 NXN i/C301xi (18) xhi/C301 NXN i/C301xi*+ /C301 NXN i/C301xihi /C301 NXN i/C301m /C301 N(N m) /C30 m; (19) so the sample mean is an UNBIASED ESTIMATOR of population mean. However, the distribution of ˜x depends on the sample size. For large samples, ˜x is approximately NORMAL . For small samples, STU- DENT’S T-DISTRIBUTION should be used. The VARIANCE of the sample mean is independent of the distribution. var( ˜x) /C30var1 nXN i/C301xi ! /C301 N2varXN i/C301xi ! /C301 N2Xn i/C301var(xi) /C301 N2 !XN i/C301s2 /C30s2 N: (20) From K-STATISTIC for a GAUSSIAN DISTRIBUTION , the UNBIASED ESTIMATOR for the VARIANCE is given by s2 /C30N N /C28 1s2 ; (21) where s /C131 NXN i/C301(xi /C28 ¯x)2 ; (22) so var( ˜x) /C30s2 N /C28 1 : (23) The SQUARE ROOT of this, sx /C30sffiffiffiffiffiffiffiffiffiffiffiffiffiffi N /C28 1p ; (24) is called the STANDARD ERROR . var( ˜x) /C13 ˜x2rC10rC11 /C28 ˜xhi2 ; (25) so ˜x2rC10rC11 /C30var( ˜x) /C27(˜x)2 /C30s2 N/C27 m2 : (26) See also ARITHMETIC- GEOMETRIC MEAN,ARITHMETIC- HARMONIC MEAN,C ARLEMAN’S INEQUALITY ,C UMU-LANT ,GENERALIZED MEAN,GEOMETRIC MEAN,HAR- MONIC MEAN,H ARMONIC- GEOMETRIC MEAN, KURTOSIS ,MEAN,MEAN DEVIATION ,MEDIAN (STATIS- TICS), MODE,M OMENT ,Q UADRATIC MEAN,R OOT- MEAN-SQUARE ,SAMPLE VARIANCE ,SKEWNESS ,STAN- DARD DEVIATION ,TRIMEAN ,VARIANCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Alzer, H. "A Proof of the Arithmetic Mean-Geometric Mean Inequality." Amer. Math. Monthly 103, 585, 1996. Beckenbach, E. F. and Bellman, R. Inequalities. New York: Springer-Verlag, 1983. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 471, 1987. Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means & Their Inequalities. Dordrecht, Netherlands: Reidel, 1988. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. Inequalities. Cambridge, England: Cambridge University Press, 1952. Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151 /C1/56, 1985. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Mitrinovic, D. S. Analytic Inequalities. New York: Springer- Verlag, 1970. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical and New Inequalities in Analysis. Dordrecht, Nether- lands: Kluwer, 1993. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 601, 1995. Arithmetic Progression ARITHMETIC SEQUENCE Arithmetic Sequence A SEQUENCE of n numbers fd0 /C27kdgn /C281 k /C300 such that the differences between successive terms is a constant d. See also ARITHMETIC SERIES ,BAUDET’S CONJECTURE , NONARITHMETIC PROGRESSION SEQUENCE ,S E- QUENCE ,SZEMERE ´ DI’S THEOREM Arithmetic Series An arithmetic series is the SUM of a SEQUENCE fakg;k /C301, 2, ..., in which each term is computed from the previous one by adding (or subtracting) a constant d. Therefore, for k/C211, ak/C30ak/C281/C27d/C30ak/C282/C272d/C30.../C30a1/C27d(k/C281):(1) The sum of the sequence of the first nterms is then given by Sn/C13Xn k/C301ak/C30Xn k/C301[a1/C27(k/C281)d]/C30na1/C27dXn k/C301(k/C281) /C30na1/C27dXn k/C302(k/C281) /C30na1 /C27dXn/C281 k /C301k (2) Using the SUM identity Xn k /C301k /C301 2n(n /C271) (3) then gives Sn /C30na1 /C2712dn(n /C281) /C3012n[2ai /C27d(n /C281)] : (4) Note, however, that a1 /C27an /C30a1 /C27[a1 /C27d(n /C281)] /C302a1 /C27d(n /C281); (5) so Sn /C301 2 n(a1 /C27an) ; (6) or n times the AVERAGE of the first and last terms! This is the trick Gauss used as a schoolboy to solve the problem of summing the INTEGERS from 1 to 100 given as busy-work by his teacher. While his class- mates toiled away doing the ADDITION longhand, Gauss wrote a single number, the correct answer 12(100)(1 /C27100) /C3050 /C215 101 /C305050 (7) on his slate (Burton 1989, pp. 80 /C1/1; Hoffman 1998, p. 207). When the answers were examined, Gauss’s proved to be the only correct one. See also ARITHMETIC SEQUENCE ,GEOMETRIC SERIES , HARMONIC SERIES ,PRIME ARITHMETIC PROGRESSION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987. Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989. Courant, R. and Robbins, H. "The Arithmetical Progression." §1.2.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 12 /C1/3, 1996. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 164, 1989. Arithmetical Function INTEGER FUNCTION Arithmetic-Geometric Mean The arithmetic-geometric mean (often abbreviated AGM) M(a;b) of two numbers aandbis defined by starting with a0/C13aandb0/C13b;then iterating an/C271/C301 2(an/C27bn) (1)bn/C271/C30ffiffiffiffiffiffiffiffiffiffi anbnp (2) until an/C30bn:anandbnconverge towards each other since an/C271/C28bn/C271/C301 2(an/C27bn)/C28ffiffiffiffiffiffiffiffiffiffi anbnp /C30an/C282ffiffiffiffiffiffiffiffiffiffi anbnp /C27bn 2: (3) Butffiffiffiffiffib np Bffiffiffiffiffianp;so 2bnB2ffiffiffiffiffiffiffiffiffiffi anbnp : (4) Now, add an/C28bn/C282ffiffiffiffiffiffiffiffiffiffi anbnp to each side an/C27bn/C282ffiffiffiffiffiffiffiffiffiffi anbnp Ban/C28bn; (5) so an/C271/C28bn/C271B1 2(an/C28bn): (6) The AGM is very useful in computing the values of complete ELLIPTIC INTEGRALS and can also be used for finding the INVERSE TANGENT . In terms of the com- plete ELLIPTIC INTEGRAL OF THE FIRST KIND K(k); M(a;b)/C30(a/C27b)p 4Ka/C28b a/C27b ! : (7) The special value 1 =M(1;ffiffiffiffiffi 2)p is called G AUSS’S CON- STANT . The AGM has the properties lM(a;b)/C30M(la;lb) (8) M(a;b)/C30M1 2(a/C27b);ffiffiffiffiffiffi abprC16rC1* (9) M(1;ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p )/C30M(1/C27x;1/C28x) (10) M(1;b)/C301/C27b 2M1;2ffiffiffi bp 1/C27b ! : (11) The Legendre form is given by M(1;x)/C30Y/C12 n/C3001 2(1/C27kn); (12) where k0/C13xand kn/C271/C132ffiffiffiffiffi knp 1/C27kn: (13) Solutions to the differential equation (x3/C28x)d2y dx2/C27(3x2/C281)dy dx/C27xy/C300 (14) are given by [ M(1/C27x;1/C28x)]/C281and [ M(1;x)]/C281: / A generalization of the ARITHMETIC-GEOMETRIC MEAN is Ip(a ; b) /C30g/C12 0xp /C282 dx (xp /C27 ap)1 =p(xp /C27 bp)(p /C281)=p (15) which is related to solutions of the differential equation x(1 /C28xp)Y ƒ/C27[1 /C28(p /C271)xp]Y ?/C28(p /C281)xp /C281Y /C300: (16) When p /C302or p /C303, there is a modular transforma- tion for the solutions of (16) that are bounded as x 0 0: Letting Jp(x) be one of these solutions, the transformation takes the form Jp(l) /C30 mJp(x) ; (17) where l /C301 /C28 u 1 /C27 (p /C28 1)u (18) m /C301 /C27 (p /C28 1)u p (19) and xp /C27up /C301: (20) The case p /C302 gives the ARITHMETIC-GEOMETRIC MEAN , and p /C303 gives a cubic relative discussed by Borwein and Borwein (1990, 1991) and Borwein (1996) in which, for a ; b > 0 and I(a ; b) defined by I(a ; b) /C30g/C12 0tdt [(a3 /C27 t3)(b3 /C27 t3)2]1 =3 ; (21) I(a; b) /C30Ia /C27 2b 3;b 3 (a2 /C27ab /C27b2)"# ! (22) For iteration with a0 /C30a and b0 /C30b and an /C271 /C30an /C27 2bn 3 (23) bn/C271 /C30bn 3(a2 n /C27anbn /C27b2n) ; (24) lim n0/C12an /C30 lim n0/C12bn /C30I(1; 1) I(a; b) : (25) Modular transformations are known when p /C304 and p /C306, but they do not give identities for p /C306 (Borwein 1996). See also ARITHMETIC- HARMONIC MEAN References Abramowitz, M. and Stegun, C. A. (Eds.). "The Process of the Arithmetic-Geometric Mean." §17.6 in Handbook of Math- ematical Functions with Formulas, Graphs, and Mathe- matical Tables, 9th printing. New York: Dover, pp. 571 ad 598 /C1/99, 1972. Borwein, J. M. Problem 10281. "A Cubic Relative of the AGM." Amer. Math. Monthly 103, 181 /C1/83, 1996. Borwein, J. M. and Borwein, P. B. "A Remarkable Cubic Iteration." In Computational Method & Function Theory:Proc. Conference Held in Valparaiso, Chile, March 13 /C1/8, 1989 (Ed. A. Dold, B. Eckmann, F. Takens, E. B Saff, S. Ruscheweyh, L. C. Salinas, L. C., and R. S. Varga). New York: Springer-Verlag, 1990. Borwein, J. M. and Borwein, P. B. "A Cubic Counterpart of Jacobi’s Identity and the AGM." Trans. Amer. Math. Soc. 323, 691 /C1/01, 1991. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 906 /C1/07, 1992. Arithmetic-Harmonic Mean Let an/C271 /C301 2(an /C27bn) (1) bn/C271 /C302anbn an /C27 bn: (2) Then A(a0 ; b0) /C30 lim n0/C12an /C30 lim n 0/C12bnffiffiffiffiffiffiffiffiffiffi a0b0p ; (3) which is just the GEOMETRIC MEAN . Arithmetic-Logarithmic-Geometric Mean Inequality a /C27 b 2>b /C28 a ln b /C28 ln a>ffiffiffiffiffiffi abp : See also NAPIER’S INEQUALITY References Nelson, R. B. "Proof without Words: The Arithmetic-Loga- rithmic-Geometric Mean Inequality." Math. Mag. 68, 305, 1995. Armstrong Number The n-digit numbers equal to sum of nth powers of their digits (a finite sequence), also called plus perfect numbers. They first few are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (Sloane’s A005188). See also HARSHAD NUMBER ,NARCISSISTIC NUMBER References Sloane, N. J. A. Sequences A005188/M0488 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Arnold Diffusion The nonconservation of ADIABATIC INVARIANTS which arises in systems with three or more DEGREES OF FREEDOM . References Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion, 2nd ed. New York: Springer-Verlag, 1994. Rasband, S. N. "Arnold Diffusion." §8.6 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 179 /C1/81, 1990. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 74, 1989. Arnold Tongue Consider the CIRCLE MAP.IfK is NONZERO , then the motion is periodic in some FINITE region surrounding each rational V: This execution of periodic motion in response to an irrational forcing is known as MODE LOCKING . If a plot is made of K versus V with the regions of periodic MODE-LOCKED parameter space plotted around rational V values (the WINDING NUM- BERS ), then the regions are seen to widen upward from 0 at K /C30 0 to some FINITE width at K /C30 1. The region surrounding each RATIONAL NUMBER is known as an ARNOLD TONGUE . At K /C30 0, the Arnold tongues are an isolated set of MEASURE zero. At K /C30 1, they form a general CANTOR SET of dimension d /C300:8700 93:7 /C2910 /C284(Rasband 1990, p. 131). In general, an Arnold tongue is defined as a resonance zone emanating out from RATIONAL NUMBERS in a two-dimensional parameter space of variables. See also CIRCLE MAP,DEVIL’S STAIRCASE References Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 130 /C131, 1990. Arnold’s Cat Map The best known example of an ANOSOV DIFFEOMORPH- ISM. It is given by the TRANSFORMATION xn /C271 yn /C271rC00rC01 /C3011 12rC00rC01 xn ynrC00rC01 ; (1) where xn/C271 and yn/C271 are computed mod 1. The Arnold cat mapping is non-Hamiltonian, nonanalytic, and mixing. However, it is AREA-PRESERVING since the DETERMINANT is 1. The LYAPUNOV CHARACTERISTIC EXPONENTS are given by j1 /C28 s 1 12 /C28 s j/C30s2 /C283s /C271 /C300; (2) so s9/C301 2(3 9ffiffiffi 5p ) : (3) The EIGENVECTORS are found by plugging s9 into the MATRIX EQUATION 1 /C28 s9 1 12 /C28 s9rC00rC01 x yrC00rC01 /C3000rC00rC01 : (4) For s /C27; the solution is y /C301 2(1 /C27ffiffiffi 5p )x /C13 fx; (5) where f is the GOLDEN RATIO , so the unstable(normalized) EIGENVECTOR is j/C27/C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 /C2810ffiffiffi 5pq 1 2(1 /C27ffiffiffi 5p )1"# : (6) Similarly, for s/C28; the solution is y /C30/C281 2(ffiffiffi 5p /C281)x /C13 f /C281x; (7) so the stable (normalized) EIGENVECTOR is j/C28/C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 /C2710ffiffiffi 5pq 1 2(1 /C28ffiffiffi 5p )1"# : (8) See also ANOSOV MAP Aronhold Process The process used to generate an expression for a covariant in the first degree of any one of the equivalent sets of COEFFICIENTS for a curve. See also CLEBSCH- ARONHOLD NOTATION ,J OA- CHIMSTHAL’S EQUATION References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 74, 1959. Aronson’s Sequence The sequence whose definition is: "t is the first, fourth, eleventh, ... letter of this sentence." The first few values are 1, 4, 11, 16, 24, 29, 33, 35, 39, ... (Sloane’s A005224). References Hofstadter, D. R. Metamagical Themas: Questing of Mind and Pattern. New York: BasicBooks, p. 44, 1985. Sloane, N. J. A. Sequences A005224/M3406 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Arrangement In general, an arrangement of objects is simply a grouping of them. The number of "arrangements" of n items is given either by a COMBINATION (order is ignored) or PERMUTATION (order is significant). The division of SPACE into cells by a collection of HYPERPLANES (Agarwal and Sharir 2000) is also called an arrangement. See also COMBINATION ,C ONFIGURATION ,C UTTING , HYPERPLANE ,ORDERING ,PERMUTATION References Agarwal, P. K. and Sharir, M. "Arrangements and Their Applications." Ch. 2 in Handbook of Computational Geo- metry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Neth- erlands: North-Holland, pp. 49 /C1/19, 2000. Arrangement Number PERMUTATION Array An array is a "list of lists" with the length of each level of list the same. The size (sometimes called the "shape") of a d-dimensional array is then indicated as m /C29n /C29x /C1/C1/C1/C29p|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} d: The most common type of array encountered is the 2-D m /C29n rectangular array having m columns and n rows. If m /C30n, a square array results. Sometimes, the order of the elements in an array is significant (as in a MATRIX ), whereas at other times, arrays which are equivalent modulo reflections (and rotations, in the case of a square array) are considered identical (as in a MAGIC SQUARE or PRIME ARRAY ). In order to exhaustively list the number of distinct arrays of a given shape with each element being one of k possible choices, the naive algorithm of running through each case and checking to see whether it’s equivalent to an earlier one is already just about as efficient as can be. The running time must be at least the number of answers, and this is so close to kmn/C1/C1/C1p that the difference isn’t significant. However, finding the number of possible arrays of a given shape is much easier, and an exact formula can be obtained using the POLYA ENUMERATION THEOREM . For the simple case of an m /C29 n array, even this proves unnecessary since there are only a few possible symmetry types, allowing the possibilities to be counted explicitly. For example, consider the case of m and n EVEN and distinct, so only reflections need be included. To take a specific case, let m /C306 and n /C304 so the array looks like abc n def ghi n jkl /C1/C1/C1/C1/C1/C1/C1/C1/C1 /C27 /C1/C1/C1/C1/C1/C1/C1/C1/C1 mn on pqr stu n vwx where each a, b, ..., x can take a value from 1 to k. The total number of possible arrangements is k24 (/kmn in general). The number of arrangements which are equivalent to their left-right mirror images is k12 (in general, kmn=2) ; as is the number equal to their up- down mirror images, or their rotations through 180 8. There are also k6 arrangements (in general, kmn=4) with full symmetry. In general, it is therefore true that kmn=4 with full symmetry kmn=2 /C28kmn =4with only left-right reflection kmn=2 /C28kmn =4with only up-down reflection kmn=2/C28kmn=4with only 180/C14rotation ;8 >>< >>: so there arekmn/C283kmn=2/C272kmn=4 arrangements with no symmetry. Now dividing by the number of images of each type, the result, for m"nwith m, n EVEN ,i s N(m;n;k) /C301 4kmn/C27(12)(3)(kmn=2/C28kmn=4) /C2714(kmn/C283kmn=2/C272kmn=4) /C3014kmn/C2734kmn=2/C2712kmn=4: The number is therefore of order O(kmn=4);with "correction" terms of much smaller order. See also ANTIMAGIC SQUARE ,EULER SQUARE ,KIRK- MAN’S SCHOOLGIRL PROBLEM ,L ATIN RECTANGLE , LATIN SQUARE ,M AGIC SQUARE ,M ATRIX ,M RS. PER- KINS’ QUILT,M ULTIPLICATION TABLE ,O RTHOGONAL ARRAY ,PERFECT SQUARE ,PRIME ARRAY ,QUOTIENT- DIFFERENCE TABLE ,ROOM SQUARE ,STOLARSKY AR- RAY,TRUTH TABLE ,W YTHOFF ARRAY Arrow Notation ANOTATION invented by Knuth (1976) to represent LARGE NUMBERS in which evaluation proceeds from the right (Conway and Guy 1996, p. 60). For example, m/C160n/C30mn(1) m/C160/C160n/C30m/C160/C1/C1/C1/C160m|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} n/C30mmUm |fflffl{zfflffl} n m/C160/C1602/C30m/C160m|fflffl{zfflffl} 2/C30m/C160m/C30mm(2) m/C160/C1603/C30m/C160m/C160m|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 3/C30m/C160(m/C160m) /C30m/C160mm/C30mmm(3) m/C160/C160/C1602/C30m/C160/C160m|fflffl{zfflffl} 2/C30m/C160/C160m/C30mmUm |fflffl{zfflffl} m(4) m/C160/C160/C1603/C30m/C160/C160mm/C160/C160m|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 3/C30m/C160/C160m/C30mmUm |fflffl{zfflffl} m /C30m /C160/C1/C1/C1/C160m|fflfflfflfflfflffl{zfflfflfflfflfflffl}/C30 mmUm |fflffl{zfflffl} mmUm |fflffl{zfflffl} mmmUm |fflffl{zfflffl} m(5) /m /C160/C160 m/ is sometimes called a POWER TOWER . The values n /C160/C1/C1/C1/C160n|fflfflfflfflffl{zfflfflfflfflffl} nare called ACKERMANN NUMBERS . See also ACKERMANN NUMBER ,C HAINED ARROW NOTATION ,DOWN ARROW NOTATION ,LARGE NUMBER , POWER TOWER ,STEINHAUS- MOSER NOTATION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 59 /C1/2, 1996. Guy, R. K. and Selfridge, J. L. "The Nesting and Roosting Habits of the Laddered Parenthesis." Amer. Math. Monthly 80, 868 /C1/76, 1973. Knuth, D. E. "Mathematics and Computer Science: Coping with Finiteness. Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limita- tions." Science 194, 1235 /C1/242, 1976. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 11 and 226 /C1/29, 1991. Arrow’s Paradox Perfect democratic VOTING is, not just in practice but in principle, impossible. See also SOCIAL CHOICE THEORY ,VOTING References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 13 /C1/5, 1998. Gardner, M. Time Travel and Other Mathematical Bewil- derments. New York: W. H. Freeman, p. 56, 1988. Arrowhead Curve SIERPINSKI ARROWHEAD CURVE Arsh Arsh z /C301 isin /C281(iz) ; where sin /C281 z the INVERSE SINE. See also ARCH,ARCTH ,ARTH,INVERSE SINE References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. xxx, 2000. Art Gallery Theorem Also called Chva´tal’s art gallery theorem. If the walls of an art gallery are made up of n straight LINE SEGMENTS , then the entire gallery can always be supervised by n=3bc watchmen placed in corners, where xbcis the FLOOR FUNCTION . This theorem was proved by Chva´tal (1975). It was conjectured that an art gallery with n walls and h HOLES requires(n /C27h) =3 bc watchmen, which has now been proven by Bjorling-Sachs and Souvaine (1991, 1995) and Hoffman et al. (1991). See also ILLUMINATION PROBLEM ,TRIANGULATION , VORONOI DIAGRAM References Bjorling-Sachs, I. and Souvaine, D. L. "A Tight Bound for Guarding Polygons with Holes." Report LCSR-TR-165. New Brunswick, NJ: Lab. Comput. Sci. Res., Rutgers Univ., 1991. Bjorling-Sachs, I. and Souvaine, D. L. "An Efficient Algo- rithm for Guard Placement in Polygons with Holes." Disc. Comput. Geom. 13,77/C1/09, 1995. Chva´tal, V. "A Combinatorial Theorem in Plane Geometry." J. Combin. Th. 18,39/C1/1, 1975. de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. Computational Geometry: Algorithms and Appli- cations, 2nd rev. ed. Berlin: Springer-Verlag, pp. 48 and 59, 2000. Fisk, S. "A Short Proof of Chva´tal’s Watchman Theorem." J. Combin. Th. Ser. B 24, 374, 1978. Fournier, A. and Montuno, D. Y. "Triangulating Simple Polygons and Equivalent Problems." ACM Trans. Gra- phics 3, 153 /C1/74, 1984. Garey, M. R.; Johnson, D. S.; Preparata, F. P.; and Tarjan, R. E. "Triangulating a Simple Polygon." Inform. Process. Lett. 7, 175 /C1/79, 1978. Hoffmann, F.; Kaufmann, M.; and Kriegel, K. "The Art Gallery Theorem for Polygons with Holes." Proc. 32nd Annual IEEE Sympos. Found. Comput. Sci.,39/C1/8, 1991. Honsberger, R. "Chva ´tal’s Art Gallery Theorem." Ch. 11 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 104 /C1/10, 1976. Kahn, J.; Klawe, M.; and Kleitman, D. "Traditional Galleries Require Fewer Watchmen." SIAM J. Alg. Disc. Math. 4, 194 /C1/06, 1993. Klee, V. "On the Complexity of d-Dimensional Voronoi Diagrams." Archiv. Math. 34,75/C1/0, 1980. O’Rourke, J. Art Gallery Theorems and Algorithms. New York: Oxford University Press, 1987. O’Rourke, J. §2.3 in Computational Geometry in C, 2nd ed. Cambridge, England: Cambridge University Press, 1998. Stewart, I. "How Many Guards in the Gallery?" Sci. Amer. 270, 118 /C1/20, May 1994. Tucker, A. "The Art Gallery Problem." Math Horizons, pp. 24 /C1/6, Spring 1994. Urrutia, J. "Art Gallery and Illumination Problems." Ch. 22 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Hol- land, pp. 973 /C1/027, 2000. Wagon, S. "The Art Gallery Theorem." §10.3 in Mathematica in Action. New York: W. H. Freeman, pp. 333 /C1/45, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 9, 1991. Arth Arth z/C301 itan/C281(iz): where tan/C281zis the INVERSE TANGENT . See also ARCH,ARSH,ARCTH ,INVERSE TANGENT References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. xxx, 2000. Articulation Vertex An articulation of a CONNECTED GRAPH is a node whose removal will disconnect the graph (Chartrand 1985). In general, an articulation vertex is node of a GRAPH whose removal increases the number of com- ponents (Harary 1994, p. 26). Articulation vertices are also called cut-vertices or "cutpoints" (Harary 1994, p. 26). A GRAPH with no articulation vertices is called a BICONNECTED GRAPH . See also BICONNECTED GRAPH ,BLOCK ,BRIDGE ,CUT SET,NONSEPARABLE GRAPH ,VERTEX (GRAPH ) References Chartrand, G. "Cut-Vertices and Bridges." §2.4 in Introduc- tory Graph Theory. New York: Dover, pp. 45 /C1/9, 1985. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 175, 1990. Artin Braid Group BRAID GROUP Artin L-Function An Artin L-function over the RATIONALS Q encodes in a GENERATING FUNCTION information about how an irreducible MONIC POLYNOMIAL over factors when reduced modulo each PRIME . For the POLYNOMIAL x2 /C271; the Artin L-function is L(s ; Q(i)=Q ; sgn) /C30Y p odd prime1 1 /C28/C281 p ! p /C28s; where (/C281 =p)isaL EGENDRE SYMBOL , which is equivalent to the EULER L-FUNCTION . The definition over arbitrary POLYNOMIALS generalizes the above expression. See also LANGLANDS RECIPROCITY References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Artin Reciprocity ARTIN’S RECIPROCITY THEOREM Artin’s Conjecture There are at least two statements which go by the name of Artin’s conjecture. The first is the RIEMANN HYPOTHESIS .The second states that every INTEGER not equal to /C281 or a SQUARE NUMBER is a primitive root modulo p for infinitely many p and proposes a density for the set of such p which are always rational multiples of a constant known as ARTIN’S CONSTANT . There is an analogous theorem for functions instead of numbers which has been proved by Billharz (Shanks 1993, p. 147).See also A RTIN’S CONSTANT ,RIEMANN HYPOTHESIS References Matthews, K. R. "A Generalization of Artin’s Conjecture for Primitive Roots." Acta Arith. 29, 113/C1/46, 1976. Moree, P. "A Note on Artin’s Conjecture." Simon Stevin 67, 255/C1/57, 1993. Ram Murty, M. "Artin’s Conjecture for Primitive Roots." Math. Intell. 10,5 9/C1/7, 1988. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 31, 80 /C1/3, and 147, 1993. Artin’s Constant Ifn"/C281 and nis not a PERFECT SQUARE , then Artin conjectured that the SETS(n) of all PRIMES for which nis a PRIMITIVE ROOT is infinite. Under the assump- tion of the EXTENDED RIEMANN HYPOTHESIS , Artin’s conjecture was solved by Hooley (1967). If, in addition, nis not an rthPOWER for any r/C211 then let n?be the SQUAREFREE PART ofnand suppose that n?/C13=1 (mod 4). Then Artin conjectured that the density of S(n) relative to the PRIMES is given by CArtin;where CArtin/C30Y/C12 k/C3011/C281 pk(pk/C281)"# /C300:3739558136 . . . ;(1) andpkis the kthPRIME , independently of the choice of n. /CArtinis connected with the PRIME ZETA FUNCTION P(n) by lnCArtin/C30/C28X/C12 n/C302(un/C281)P(n) n; (2) where un/C30un/C281/C27un/C282 (3) with u1/C301;u2/C303 (Ribenboim 1998, Gourdon and Sebah). Wrench (1961) gave 45 digits of CArtin ;and Gourdon and Sebah give 60.Ifn?/C131 (mod 4) and nis still restricted not to be an rth power, then the density is not C Artinitself, but a rational multiple thereof. The explicit formula for computing the density in this case is conjectured to be C?Artin/C301/C28m(n?)Y prime q qjn?1 q2/C28q/C2812 643 75CArtin (4) (Finch, Matthews 1976), where m(n) is the M O¨BIUS FUNCTION . Special cases can be written down expli- citly for n?/C30p a PRIME , C?Artin /C30 1 /C271 p2 /C28 p /C28 1 ! CArtin (5) or n?/C30pq ; where p, q are both PRIMES with u; v /C13 1 (mod 4); C?Artin /C30 1 /C271 p2 /C28 p /C28 11 q2 /C28 q /C28 1 ! CArtin ; (6) If n is a perfect cube (which is not a perfect square), a perfect fifth power (which is not a perfect square or perfect cube), etc., other formulas apply (Hooley 1967, Western and Miller 1968). The significance of Artin’s constant is more easily seen by describing it as the fraction of PRIMES p for which 1=p has a maximal DECIMAL EXPANSION , i.e., p is a FULL REPTEND PRIME , (Conway and Guy 1996). See also ARTIN’S CONJECTURE ,DECIMAL EXPANSION , FULL REPTEND PRIME ,PRIMITIVE ROOT,STEPHENS’ CONSTANT References Artin, E. Collected Papers (Ed. S. Lang and J. T. Tate). New York: Springer-Verlag, pp. viii-ix, 1965. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 169, 1996. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/artin/artin.html. Finch, S. "Correction Factors for Artin’s Constant." http:// www.mathsoft.com/asolve/constant/artin/factor.html. Gourdon, X. and Sebah, P. "Some Constants from Number Theory." http://xavier.gourdon.free.fr/Constants/Miscella- neous/constantsNumTheory.html. Hooley, C. "On Artin’s Conjecture." J. reine angew. Math. 225, 209 /C1/20, 1967. Hooley, C. Applications of Sieve Methods to the Theory of Numbers. Cambridge, England: Cambridge University Press, 1976. Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, 1990. Lehmer, D. H. and Lehmer, E. "Heuristics Anyone?" In Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Po´lya (Ed. G. Szego, C. Loew- ner, S. Bergman, M. M. Schiffer, J. Neyman, D. Gilbarg, and H. Solomon). Stanford, CA: Stanford University Press, 1962. Lenstra, H. W. Jr. "On Artin’s Conjecture and Euclid’s Algorithm in Global Fields." Invent. Math. 42, 201 /C1/24, 1977. Matthews, K. R. "A Generalization of Artin’s Conjecture for Primitive Roots." Acta Arith. 29, 113 /C1/46, 1976. Plouffe, S. "Artin’s Constant." http://www.lacim.uqam.ca/ piDATA/artin.txt. Ram Murty, M. "Artin’s Conjecture for Primitive Roots." Math. Intell. 10,59/C1/7, 1988. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 80 /C1/3, 1993. Western, A. E. and Miller, J. C. P. Tables of Indices and Primitive Roots. Cambridge, England: Cambridge Uni- versity Press, pp. xxxvii-xlii, 1968.Wrench, J. W. "Evaluation of Artin’s Constant and the Twin Prime Constant." Math. Comput. 15, 396 /C1/98, 1961. Artin’s Reciprocity Theorem A general RECIPROCITY THEOREM for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. If R is a NUMBER FIELD and R? a finite integral extension, then there is a SURJECTION from the group of fractional IDEALS prime to the discriminant, given by the Artin symbol. For some cycle c, the kernel of this SURJECTION contains each PRINCIPAL fractional IDEAL generated by an element congruent to 1 mod c. See also LANGLANDS PROGRAM Artinian Group A GROUP in which any decreasing CHAIN of distinct SUBGROUPS terminates after a FINITE number. Artinian Ring A noncommutative SEMISIMPLE RING satisfying the "descending chain condition." See also GORENSTEIN RING,SEMISIMPLE RING References Artin, E. "Zur Theorie der hyperkomplexer Zahlen." Hamb. Abh. 5, 251 /C1/60, 1928. Artin, E. "Zur Arithmetik hyperkomplexer Zahlen." Hamb. Abh. 5, 261 /C1/89, 1928. Artistic Sequence A SERIES is called artistic if every three consecutive terms have a common three-way ratio P[ai ; ai/C271 ; ai/C272] /C30(ai /C27 ai/C271 /C27 ai/C272)ai /C271 aiai/C272: A SERIES is also artistic IFF its BIAS is a constant. A GEOMETRIC SERIES with RATIO r /C21 0 is an artistic series with P/C301 r/C271/C27r]3: See also BIAS (SERIES ), GEOMETRIC SERIES ,MELODIC SEQUENCE References Duffin, R. J. "On Seeing Progressions of Constant Cross Ratio." Amer. Math. Monthly 100,3 8/C1/7, 1993. ASA Theorem Specifying two adjacent ANGLES A and B and the side between them c uniquely determines a TRIANGLE with AREA K /C30c2 2 (cot A /C27 cot B) (1) The angle C is given in terms of A and B by C /C30 p /C28A /C28B ; (2) and the sides a and b can be determined by using the LAW OF SINES a sin A /C30b sin B /C30c sin C (3) to obtain a /C30sin A sin( p /C28 A /C28 B)c (4) b /C30sin B sin( p /C28 A /C28 B)c : (5) See also AAA THEOREM , AAS THEOREM , ASS THEO- REM, SAS THEOREM , SSS THEOREM ,TRIANGLE Aschbacher’s Component Theorem Suppose that E(G) (the commuting product of all components of G)is SIMPLE and G contains a semisimple INVOLUTION . Then there is some semi- simple INVOLUTION x such that CG(x) has a NORMAL SUBGROUP K which is either QUASISIMPLE or ISO- MORPHIC to O /C27(4; q) ? and such that Q /C30CG(K)is TIGHTLY EMBEDDED . See also INVOLUTION (GROUP ), ISOMORPHIC GROUPS , NORMAL SUBGROUP ,Q UASISIMPLE GROUP ,S IMPLE GROUP ,TIGHTLY EMBEDDED A-Sequence N.B. A detailed online essay by S. Finch was the starting point for this entry. An INFINITE SEQUENCE of POSITIVE INTEGERS aiS satisfying 1 5a1 Ba2 Ba3 B... (1) is an A-sequence if no akis the SUM of two or more distinct earlier terms (Guy 1994). Such sequences are sometimes also known as sum-free sets.Erdos (1962) proved S(A) /C13 sup all A sequencesX/C12 k /C3011 akB103: (2) Any A-sequence satisfies the CHI INEQUALITY (Levine and O’Sullivan 1977), which gives S(A) B3:9998 : Abbott (1987) and Zhang (1992) have given a bound from below, so the best result to date is 2:0649 BS(A) B3 :9998 : (3) Levine and O’Sullivan (1977) conjectured that the sum of RECIPROCALS of an A-sequence satisfies S(A) 5X/C12 k /C3011 xk/C303 :01... ; (4) where xi are given by the LEVINE- O’SULLIVAN GREEDY ALGORITHM . See also B2-SEQUENCE ,M IAN-CHOWLA SEQUENCE , SUM-FREE SET References Abbott, H. L. "On Sum-Free Sequences." Acta Arith. 48,93/C1/ 6, 1987. Erdos, P. "Remarks on Number Theory III. Some Problems in Additive Number Theory." Mat. Lapok 13,28/C1/8, 1962. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/erdos/erdos.html. Guy, R. K. "/B2/-Sequences." §E28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 228 /C1/29, 1994. Levine, E. and O’Sullivan, J. "An Upper Estimate for the Reciprocal Sum of a Sum-Free Sequence." Acta Arith. 34, 9 /C1/4, 1977. Zhang, Z. X. "A Sum-Free Sequence with Larger Reciprocal Sum." Unpublished manuscript, 1992. ASS Theorem Specifying two adjacent side lengths a and c of a TRIANGLE (with a Bc) and one ACUTE ANGLE A opposite a does not, in general, uniquely determine a triangle. If sin A Ba=c ; there are two possible TRIANGLES satisfying the given conditions. If sin A /C30 a =c; there is one possible TRIANGLE . If sin A > a=c; there are no possible TRIANGLES . Remember: don’t try to prove congruence with the ASS theorem or you will make an ASS out of yourself. See also AAA THEOREM , AAS THEOREM , SAS THEO- REM, SSS THEOREM ,TRIANGLE Associate Letpbe an ODD PRIME ,aa positive number such that p ½a(i.e., pdoes not DIVIDE a), and let xbe one of the numbers 1, 2, 3, ..., p/C281:Then there is a unique x?; called the associate of x, such that xx ?/C13a (mod p) with 0 Bx?Bp (Hardy and Wright 1979, p. 67). If x?/C30 x; then a is called a QUADRATIC RESIDUE of p. See also QUADRATIC RESIDUE References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 67, 1979. Associated Fiber Bundle Given a GROUP ACTION G /C29F 0 F and a PRINCIPAL BUNDLE p : A 0 M ; the associated fiber bundle on M is ˜p : A /C29F =G 0 M : (1) In particular, it is the QUOTIENT SPACE A /C29F =G where (a ; x) /C2(ga; g /C281x) ::/ For example, the torus T /C30f(eis ; eit) has a S1 action given by f(eiu)(eis ; eit) /C30(ei(s/C27 u) ; ei(t/C27 u)) (2) and the frame bundle on the sphere, p : SO(3) 0 S2 ; (3) is a principal S1 bundle. The associated fiber bundle is a fiber bundle on the sphere, with fiber the torus. It is an example of a four-dimensional MANIFOLD . See also BUNDLE ,FIBER BUNDLE ,G ROUP ACTION , PRINCIPAL BUNDLE ,QUOTIENT SPACE Associated Laguerre Polynomial LAGUERRE POLYNOMIAL Associated Legendre Polynomial LEGENDRE POLYNOMIAL Associated Principal Bundle See also BUNDLE Associated Sequence AS HEFFER SEQUENCE for (1; f(t)) is called the associated sequence for f(t) ; and a sequence sn(x)of polynomials satisfying the orthogonality conditions [f(t)]k ½sn(x)DE /C30n!dnk ; where dnkis the DELTA FUNCTION , is said to be associated to f(t) :/ See also SHEFFER SEQUENCE References Roman, S. The Umbral Calculus. New York: Academic Press, 1984.Associated Stirling Number of the First Kind STIRLING NUMBER OF THE FIRST KIND Associated Triangles The three CIRCULAR TRIANGLES A?B?C ?; AB ?C ?; A?BC ?; and A?B ?C obtained by extending the arcs of a CIRCULAR TRIANGLE ABC into complete circles. See also CIRCULAR TRIANGLE References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 251 /C1/52, 1893. Associated Vector Bundle Given a PRINCIPAL BUNDLE p:A0M;with fiber a LIE GROUP Gand BASE MANIFOLD M, and a REPRE- SENTATION ofG, say f:G/C29V0V;then the asso- ciated vector bundle is ˜p:A/C29V=G0M: (1) In particular, it is the QUOTIENT SPACE A/C29V=G where ( a;v)/C2(ga;g/C281v):/ This construction has many uses. For instance, any REPRESENTATION of the ORTHOGONAL GROUP gives rise to a BUNDLE ofTENSORS on a R IEMANNIAN MANIFOLD as the vector bundle associated to the FRAME BUNDLE . For example, p:SO(3)0S2is the frame bundle on S2;where pw1 w2 w32 6643 7750 BB@1 CCA/C30w1; (2) writing the special orthogonal matrix with rows wi:It is aSO(2) bundle with the action defined by cos u /C28sin u sin u cos urC00rC01 /C215 A /C3010 0 0 cos u /C28sin u 0 sin u cos u2 435A; (3) which preserves the map p: / The TANGENT BUNDLE is the associated vector bundle with the standard REPRESENTATION of SO(2) on V /C30 R2; given by pairs (v, A), with v /C30 (a; b) /C23 R2 and A /C23 SO(3) : Two pairs (v1 ; A1) and (v2 ; A2) represent the same tangent vector IFF there is a g /C23 SO(2) such that v2 /C30gv1 and A1 /C30g /C215 A2 :/ See also ASSOCIATED FIBER BUNDLE ,FRAME BUNDLE , GROUP ACTION ,L IE GROUP ,P RINCIPAL BUNDLE , REPRESENTATION ,QUOTIENT SPACE Associative Three elements x, y and z of a set S are said to be associative under a binary operation /C31 if they satisfy x/C31(y/C31z) /C30(x/C31y) /C31z : Real numbers are associative under addition x /C27(y /C27z) /C30(x /C27y) /C27z and multiplication x /C215(y /C215 z) /C30(x /C215 y) /C215 z: See also ASSOCIATIVE ALGEBRA ,COMMUTATIVE ,DIS- TRIBUTIVE ,TRANSITIVE Associative Algebra In simple terms, let x, y, and z be members of an ALGEBRA . Then the ALGEBRA is said to be associative if x /C215 (y /C215 z) /C30(x /C215 y) /C215 z ; (1) where /C215 denotes MULTIPLICATION . More formally, let A denote an R/-algebra, so that A is a VECTOR SPACE over R and A /C29A 0 A (2) (x; y) 0 x /C215 y: (3) Then A is said to be m-associative if there exists an m-dimensional SUBSPACE S of A such that (y /C215 x) /C215 z /C30y /C215(x /C215 z) (4) for all y; z /C23 A and x /C23 S : Here, VECTOR MULTIPLICA- TION x /C215 y is assumed to be BILINEAR .An n-dimen- sional n-associative ALGEBRA is simply said to be "associative." See also ASSOCIATIVE References Finch, S. "Zero Structures in Real Algebras." http:// www.mathsoft.com/asolve/zerodiv/zerodiv.html.Associative Magic Square An n /C29n MAGIC SQUARE for which every pair of numbers symmetrically opposite the center sum to n2 /C271: The LO SHU is associative but not PANMAGIC . Order four squares can be PANMAGIC or associative, but not both. Order five squares are the smallest which can be both associative and PANMAGIC , and 16 distinct associative PANMAGIC SQUARES exist, one of which is illustrated above (Gardner 1988). See also MAGIC SQUARE ,PANMAGIC SQUARE References Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213 /C1/25, 1988. Associator For an ALGEBRA A, the associator is the trilinear map A /C29A /C29A 0 A given by (x; y; z) /C30(xy)z /C28x(yz) : The associator is identically zero IFF A is associative. See also ALTERNATIVE ALGEBRA ,C OMMUTATOR , POWER ASSOCIATIVE ALGEBRA References Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 13, 1996. Asterisk STAR Astroid A 4-cusped HYPOCYCLOID which is sometimes also called a TETRACUSPID ,CUBOCYCLOID ,o r PARACYCLE . The PARAMETRIC EQUATIONS of the astroid can be obtained by plugging in n/C13a=b/C304o r4 =3 into the equations for a general HYPOCYCLOID , giving x/C303bcosf/C27bcos(3 f)/C304bcos3f/C30acos3f (1) y/C303bsinf/C28bsin(3f)/C304bsin3f/C30asin3f:(2) In C ARTESIAN COORDINATES , x2=3/C27y2=3/C30a2=3: (3) InPEDAL COORDINATES with the PEDAL POINT at the center, the equation is r2/C273p2/C30a2(4) The ARC LENGTH ,CURVATURE , and TANGENTIAL ANGLE are s(t)/C303 2gt 0½sin(2 t?)jdt?/C3032sin2t (5) k(t)/C30/C282 3csc(2 t) (6) f(t)/C30/C28t: (7) As usual, care must be taken in the evaluation of s(t) fort>p=2:Since (5) comes from an integral involving the ABSOLUTE VALUE of a function, it must be monotonic increasing. Each QUADRANT can be treated correctly by defining n/C302t p"# /C271; (8) where xbcis the FLOOR FUNCTION , giving the formula s(t)/C30(/C281)/C271[n(mod 2)] 3 2sin2t/C273[12n]: (9) The overall ARC LENGTH of the astroid can becomputed from the general HYPOCYCLOID formula sn/C30Sa(n/C281) n(10) with n/C304, s4/C306a: (11) The AREA is given by An/C30(n/C281)(n/C282) n2pa2(12) with n/C304, A4/C3038pa2: (13) The EVOLUTE of an ELLIPSE is a stretched HYPOCY- CLOID . The gradient of the TANGENT Tfrom the point with parameter pis/C28tanp:The equation of this TANGENT Tis xsinp/C27ycosp/C3012asin(2 p) (14) (MacTutor Archive). Let Tcut the X-AXIS and the Y- AXIS atXandY, respectively. Then the length XYis a constant and is equal to a. The astroid can also be formed as the ENVELOPE produced when a LINE SEGMENT is moved with each end on one of a pair of PERPENDICULAR axes (e.g., it is the curve enveloped by a ladder sliding against a wall or a garage door with the top corner moving along a vertical track; left figure above). The astroid is therefore a GLISSETTE . To see this, note that for a ladder of length L, the points of contact with the wall and floor are ( x0;0) and (0 ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2/C28x2 0p );respectively. The equation of the LINE made by the ladder with its foot at ( x0;0) is therefore y/C280/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2/C28x2 0p /C28x0(x/C28x0) (15) which can be written U(x;y;x0)/C30y/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2/C28x2 0p x0(x/C28x0): (16) The equation of the ENVELOPE is given by the simultaneous solution of U(x; y; x0) /C30y /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 /C28 x2 0p x0(x /C28x0) /C300 @U @x0/C30x2 0 /C28 L2x x2 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 /C28 x2 0p /C300;8 >>>< >>>:(17) which is x /C30x3 0 L2 (18) y /C30(L2 /C28 x20)3 =2 L2 (19) Noting that x2=3 /C30x20 L4 =3 (20) y2 =3 /C30L2 /C28 x20 L4 =3 (21) allows this to be written implicitly as x2 =3 /C27y2=3 /C30L2 =3 ; (22) the equation of the astroid, as promised. The related problem obtained by having the "garage door" of length L with an "extension" of length DL move up and down a slotted track also gives a surprising answer. In this case, the position of the "extended" end for the foot of the door at horizontal position x0 and ANGLE u is given by x /C30/C28DL cos u (23) y /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 /C28x2 0q /C27DL sin u: (24) Using x0 /C30L cos u (25) then gives x /C30/C28DL Lx0 (26) y /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 /C28x2 0q 1 /C27DL L ! (27) Solving (26) for x0 ; plugging into (27) and squaring then givesy2 /C30L2 /C28L2x2 ( DL)21 /C27DL L !2 : (28) Rearranging produces the equation x2 ( DL)2 /C27y2 (L /C27DL)2 /C301 ; (29) the equation of a (QUADRANT of an) ELLIPSE with SEMIMAJOR and SEMIMINOR AXES of lengths dland l/C27dl:/ the astroid is also the ENVELOPE of the family of ELLIPSES x2 c2/C27y2 (1/C28c)2/C281/C300; (30) illustrated above (Wells 1991). See also DELTOID ,ELLIPSE ENVELOPE ,LAME´ CURVE , NEPHROID ,RANUNCULOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 219, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 172 /C1/75, 1972. Lockwood, E. H. "The Astroid." Ch. 6 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 52 /C1/1, 1967. MacTutor History of Mathematics Archive. "Astroid." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Astro-id.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 146 /C1 /47, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 10 /C1/1, 1991. Yates, R. C. "Astroid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 1 /C1/, 1952. Astroid Evolute A HYPOCYCLOID EVOLUTE for n /C304 is another ASTROID scaled by a factor n=(n /C282) /C304=2 /C302 and rotated 1=(2 /C215 4) /C301=8 of a turn. Astroid Involute A HYPOCYCLOID INVOLUTE for n /C304 is another ASTRO- ID scaled by a factor (n /C282)=n /C302 =4 /C301=2 and rotated 1=(2 /C215 4) /C301=8 of a turn. Astroid Pedal Curve The PEDAL CURVE of an ASTROID with PEDAL POINT at the center is a QUADRIFOLIUM . Astroid Radial Curve The QUADRIFOLIUM x /C30x0 /C273a cos t /C283a cos(3 t) y /C30y0 /C273a sin t /C273 sin(3 t): Astroidal Ellipsoid The surface which is the inverse of the ELLIPSOID in the sense that it "goes in" where the ELLIPSOID "goes out." It is given by the PARAMETRIC EQUATIONS x /C30(a cos u cos v)3 y /C30(b sin u cos v)3 z /C30 (c sin v)3 for u /C23 [ /C28p=2 ; p=2] and v /C23 [ /C28p; p] : The special case a /C30 b /C30 c /C30 1 corresponds to the HYPERBOLIC OCTA- HEDRON . See also ELLIPSOID ,HYPERBOLIC OCTAHEDRON References Nordstrand, T. "Astroidal Ellipsoid." http://www.uib.no/peo- ple/nfytn/asttxt.htm. Asymptosy ASYMPTOTIC behavior. A useful yet endangered word, found rarely outside the captivity of the Oxford English Dictionary. See also ASYMPTOTE ,ASYMPTOTIC Asymptote A curve approaching a given curve arbitrarily closely, as illustrated in the above diagram. See also ASYMPTOSY ,A SYMPTOTIC ,A SYMPTOTIC CURVE References Giblin, P. J. "What is an Asymptote?" Math. Gaz. 56, 274/C184, 1972. Asymptotic Approaching a value or curve arbitrarily closely (i.e., as some sort of LIMIT is taken). A CURVE Awhich is asymptotic to given CURVE Cis called the ASYMPTOTE of C. Hardy and Wright (1979, p. 7) use the symbol 7 to denote that one quantity is asymptotic to another. If f7 f; then Hardy and Wright say that f and f are of the same ORDER OF MAGNITUDE . See also ASYMPTOSY ,A SYMPTOTE ,A SYMPTOTIC CURVE ,ASYMPTOTIC DIRECTION ,ASYMPTOTIC NOTA- TION ,ASYMPTOTIC SERIES ,LANDAU SYMBOL ,LIMIT, ORDER OF MAGNITUDE References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Asymptotic Curve Given a REGULAR SURFACE M, an asymptotic curve is formally defined as a curve x(t)on M such that the NORMAL CURVATURE is 0 in the direction x?(t) for all t in the domain of x. The differential equation for the parametric representation of an asymptotic curve is eu ?2 /C272fu ?v?/C27gv ?2 /C300; (1) where e, f, and g are coefficients of the SECOND FUNDAMENTAL FORM . The differential equation for asymptotic curves on a MONGE PATCH (u; v; h(u; v)) is huuu?2 /C272huuu ?v ?/C27hvvv?2 /C300 ; (2) and on a polar patch (r cos u; r sin u; h(r)) is hƒ(r)r ?2 /C27h?(r)ru ?2 /C300: (3) The images below show asymptotic curves for the ELLIPTIC HELICOID , FUNNEL , HYPERBOLIC PARABO- LOID , and MONKEY SADDLE . See also RULED SURFACE References Gray, A. "Asymptotic Curves," "Examples of Asymptotic Curves," and "Using Mathematica to Find Asymptotic Curves." §18.1, 18.2, and 18.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 417 /C1/29, 1997. Asymptotic Direction An asymptotic direction at a point p of a REGULAR SURFACE M /C23R3 is a direction in which the NORMAL CURVATURE of M vanishes.1. There are no asymptotic directions at an ELLIPTIC POINT . 2. There are exactly two asymptotic directions at a HYPERBOLIC POINT . 3. There is exactly one asymptotic direction at a PARABOLIC POINT . 4. Every direction is asymptotic at a PLANAR POINT . See also ASYMPTOTIC CURVE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 364 and 418, 1997. Asymptotic Equipartition Property This entry contributed by ERIK G. MILLER A theorem from INFORMATION THEORY that is a simple consequence of the WEAK LAW OF LARGE NUMBERS .It states that if a set of values X1 ; X2/, ..., Xnis drawn independently from a random variable X distributed according to P(x) then the joint probability P(X1 ; ...; Xn) satisfies /C281 nln P(X1 ; X2 ; ...; Xn) 0 H(X) ; where H(X) is the ENTROPY of the random variable X. See also ENTROPY References Cover, T. M. and Thomas, J. A. Elements of Information Theory. New York: Wiley, 1991. Asymptotic Expansion ASYMPTOTIC SERIES Asymptotic Notation Let n be a integer variable which tends to infinity and let x be a continuous variable tending to some limit. Also, let f(n)orf(x) be a positive function and f(n)or f(x) any function. Then Hardy and Wright (1979) define 1. f /C30O( f) to mean that ½f ½BAf for some constant A and all values of n and x, 2. f /C30o(f) to mean that f =f 0 0 ;/ 3. f /C2 f to mean that f =f 0 1;/ 4. f ) f to mean the same as f /C30o( f) ;/ 5. f ) f to mean f = f 0/C12; and 6. f7f to mean A1 f Bf BA2 f for some positive constants A1 and A2 :/ /f /C30o( f) implies and is stronger than f /C30O( f) :/ The term L ANDAU SYMBOL is sometimes used to indicate the notation o(f);and in general, O(x) and o(x) are read as "is of order x." See also LANDAU SYMBOL References Hardy, G. H. and Wright, E. M. "Some Notations." §1.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 7 /C1/, 1979. Jeffreys, H. and Jeffreys, B. S. "Increasing and Decreasing Functions." §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988. Asymptotic Series An asymptotic series is a SERIES EXPANSION of a FUNCTION in a variable x which may converge or diverge (Erde ´lyi 1987, p. 1), but whose partial sums can be made an arbitrarily good approximation to a given function for large enough x. To form an asymptotic series R(x)of f(x) /C2R(x); (1) take xnRnxðÞ/C30xn[f(x) /C28Sn(x)] ; (2) where SnxðÞ/C13a0 /C27a1 x/C27a2 x2 /C27/C1/C1/C1/C27an xn : (3) The asymptotic series is defined to have the proper- ties lim x0/C12xnRn(x) /C300 for fixed n (4) lim x0/C12xnRn(x) /C30/C12 for fixed x (5) Therefore, f(x) :X/C12 n/C300anx /C28n (6) in the limit x 0/C12: If a function has an asymptotic expansion, the expansion is unique. The symbol /C2 is also used to mean directly SIMILAR . See also HYPERASYMPTOTIC SERIES ,SUPERASYMPTO- TIC SERIES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 15, 1972. Arfken, G. "Asymptotic of Semiconvergent Series." §5.10 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 339 /C146, 1985. Bleistein, N. and Handelsman, R. A. Asymptotic Expansions of Integrals. New York: Dover, 1986. Boyd, J. P. "The Devil’s Invention: Asymptotic, Superasymp- totic and Hyperasymptotic Series." Acta Appl. Math. 56, 1 /C18, 1999. Copson, E. T. Asymptotic Expansions. Cambridge, England: Cambridge University Press, 1965. de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, 1982. Dingle, R. B. Asymptotic Expansions: Their Derivation and Interpretation. London: Academic Press, 1973. Erde´lyi, A. Asymptotic Expansions. New York: Dover, 1987.Morse, P. M. and Feshbach, H. "Asymptotic Series; Method of Steepest Descent." §4.6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 434 /C143, 1953. Olver, F. W. J. Asymptotics and Special Functions. New York: Academic Press, 1974. Wasow, W. R. Asymptotic Expansions for Ordinary Differ- ential Equations. New York: Dover, 1987. Weisstein, E. W. "Books about Asymptotic Series." http:// www.treasure-troves.com/books/AsymptoticSeries.html. Atiyah-Singer Index Theorem A theorem which states that the analytic and topolo- gical "indices" are equal for any elliptic differential operator on an n-D COMPACT DIFFERENTIABLE C /C12 boundaryless MANIFOLD . See also COMPACT MANIFOLD ,DIFFERENTIABLE MANI- FOLD References Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Operators on Compact Manifolds." Bull. Amer. Math. Soc. 69, 322 /C133, 1963. Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Operators I, II, III." Ann. Math. 87, 484 /C104, 1968. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 4, 1996. Atkin-Goldwasser-Kilian-Morain Certificate A recursive PRIMALITY CERTIFICATE for a PRIME p. The certificate consists of a list of 1. A point on an ELLIPTIC CURVE C y2 /C30x3 /C27g2x /C27g3 (mod p) for some numbers g2 and g3 :/ 2. A PRIME q with q > (p1 =4 /C271)2; such that for some other number k and m /C30 kq with k " 1; mC(x; y; g2 ; g3 ; p) is the identity on the curve, but kC(x; y; g2 ; g3 ; p) is not the identity. This guar- antees PRIMALITY of p by a theorem of Goldwasser and Kilian (1986). 3. Each q has its recursive certificate following it. So if the smallest q is known to be PRIME , all the numbers are certified PRIME up the chain. AP RATT CERTIFICATE is quicker to generate for small numbers. The Mathematica task ProvablePri- meQ[n] in the Mathematica add-on package Num- berTheory‘PrimeQ‘ (which can be loaded with the command BBNumberTheory‘ ) therefore generates an Atkin-Goldwasser-Kilian-Morain certificate only for numbers above a certain limit (1010by default), and a P RATT CERTIFICATE for smaller numbers. See also ELLIPTIC CURVE PRIMALITY PROVING ,ELLIP- TIC PSEUDOPRIME ,P RATT CERTIFICATE ,P RIMALITY CERTIFICATE ,W ITNESS References Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Math. Comput. 61,29/C1/8, 1993. Bressoud, D. M. Factorization and Prime Testing. New York: Springer-Verlag, 1989. Goldwasser, S. and Kilian, J. "Almost All Primes Can Be Quickly Certified." Proc. 18th STOC. pp. 316 /C1/29, 1986. Morain, F. "Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm." Rapport de Recherche 911, INRIA, Octobre 1988. Schoof, R. "Elliptic Curves over Finite Fields and the Computation of Square Roots mod p." Math. Comput. 44, 483 /C1/94, 1985. Wunderlich, M. C. "A Performance Analysis of a Simple Prime-Testing Algorithm." Math. Comput. 40, 709 /C1/14, 1983. Atlas An atlas is a collection of consistent COORDINATE CHARTS on a MANIFOLD , where "consistent" most commonly means that the TRANSITION FUNCTIONS of the charts are SMOOTH . As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a MANIFOLD and looks like flat two-dimensional Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas. The overlapping maps from one chart to another are called transition functions. They represent the tran- sition from one chart’s point of view to that of another. Let the open unit ball in Rn be denoted B1 : Then if f : U 0 B1 and c : V 0 B1 are two coordinate charts, the composition f(c /C281 is a function defined on c(U S V) : That is, it is a function from an open subset of B1 to B1 ; and given such a function from Rn to Rn ; there are conditions for it to be smooth or have k smooth derivatives (i.e., it is a C-K FUNCTION ). Furthermore, when R2n is isomorphic to Cn (in the even DIMENSIONAL case), a function can be HOLO- MORPHIC . A smooth atlas has transition functions that are C- INFINITY smooth (i.e., infinitely differentiable). The consequence is that a smooth function on one chart is smooth in any other chart (by the CHAIN RULE for higher derivatives). Similarly, one could have an atlas in class Ck; where the transition functions are in class C-K. In the even-dimensional case, one may ask whether the transition functions are HOLOMORPHIC . In this case, one has a holomorphic atlas, and by the chain rule, it makes sense to ask if a function on the manifold is holomorphic. It is possible for two atlases to be compatible, mean- ing the union is also an atlas. By ZORN’S LEMMA , there always exists a maximal atlas, where a maximal atlas is an atlas not contained in any other atlas. However, in typical applications, it is not necessary to use amaximal atlas and any sufficiently refined atlas will do. See also COORDINATE CHART ,H OLOMORPHIC FUNC- TION ,M ANIFOLD ,S MOOTH FUNCTION ,T RANSITION FUNCTION ,ZORN’S LEMMA Atom ATOMIC STATEMENT ,URELEMENT Atomic Statement In LOGIC , a statement which cannot be broken down into smaller statements. Attraction Basin BASIN OF ATTRACTION Attractor An attractor is a SET of states (points in the PHASE SPACE ), invariant under the dynamics, towards which neighboring states in a given BASIN OF ATTRACTION asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct BASINS OF ATTRACTION . This restriction is necessary since a DYNAMICAL SYSTEM may have multiple attractors, each with its own BASIN OF ATTRACTION . Conservative systems do not have attractors, since the motion is periodic. For dissipative DYNAMICAL SYSTEMS , however, volumes shrink exponentially so attractors have 0 volume in n-D phase space. A stable FIXED POINT surrounded by a dissipative region is an attractor known as a SINK. Regular attractors (corresponding to 0 LYAPUNOV CHARACTER- ISTIC EXPONENTS ) act as LIMIT CYCLES , in which trajectories circle around a limiting trajectory which they asymptotically approach, but never reach. STRANGE ATTRACTORS are bounded regions of PHASE SPACE (corresponding to POSITIVE LYAPUNOV CHARAC- TERISTIC EXPONENTS ) having zero MEASURE in the embedding PHASE SPACE and a FRACTAL DIMENSION . Trajectories within a STRANGE ATTRACTOR appear to skip around randomly. See also BARNSLEY’S FERN,BASIN OF ATTRACTION , CHAOS GAME,F RACTAL DIMENSION ,L IMIT CYCLE , LYAPUNOV CHARACTERISTIC EXPONENT ,M EASURE , SINK (MAP), STRANGE ATTRACTOR Aubel’s Theorem VON AUBEL’S THEOREM Auction A type of sale in which members of a group of buyers offer ever increasing amounts. The bidder making the last bid (for which no higher bid is subsequently made within a specified time limit: "going once, going twice, sold") must then purchase the item in question at this price. Variants of simple bidding are also possible, as in a VICKREY AUCTION . See also VICKREY AUCTION Augend The first of several ADDENDS , or "the one to which the others are added," is sometimes called the augend. Therefore, while a, b, and c are ADDENDS in a /C27 b /C27 c ; a is the augend. See also ADDEND ,ADDITION Augmented Amicable Pair A PAIR of numbers m and n such that s(m) /C30 s(n) /C30 m /C27 n /C28 1; where s(m) is the DIVISOR FUNCTION . Beck and Najar (1977) found 11 augmented amicable pairs. See also AMICABLE PAIR,DIVISOR FUNCTION ,QUASIA- MICABLE PAIR References Beck, W. E. and Najar, R. M. "More Reduced Amicable Pairs." Fib. Quart. 15, 331/C132, 1977. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 59, 1994. Augmented Dodecahedron JOHNSON SOLID J58:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Augmented Hexagonal Prism JOHNSON SOLID J54:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Augmented Pentagonal Prism JOHNSON SOLID J52:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Augmented Polyhedron AUNIFORM POLYHEDRON with one or more other solids adjoined. Augmented Sphenocorona JOHNSON SOLID J87:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Augmented Triangular Prism JOHNSON SOLID J49:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Augmented Tridiminished Icosahedron JOHNSON SOLID J64:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Augmented Truncated Cube JOHNSON SOLID J66:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Augmented Truncated Dodecahedron JOHNSON SOLID J68:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Augmented Truncated Tetrahedron JOHNSON SOLID J65:/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Augmenting Path A path constructed by repeatedly finding a path of positive capacity from a source to a sink and then adding it to the flow (Skiena 1990, p. 237). See also BERGE’S THEOREM References Ford, L. R. and Fulkerson, D. R. Flows in Networks. Princeton, NJ: Princeton University Press, 1962. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Aureum Theorema Gauss’s name for the QUADRATIC RECIPROCITY THEO- REM. Aurifeuillean Factorization A factorization OF THE FORM 24n /C272 /C271 /C30(22n/C271 /C282n /C271 /C271)(22n/C271 /C272n /C271 /C271): (1) The factorization for n /C3014 was discovered by Aur- ifeuille, and the general form was subsequently discovered by Lucas. The large factors are sometimes written as L and M as follows 24k /C282 /C271 /C30(22k /C281 /C282k /C271)(22k /C281 /C272k /C271) (2) 36k/C283 /C271 /C30(32k /C281 /C271)(32k /C281 /C283k /C271) /C2(32k /C281 /C273k /C271); (3) which can be written 22h /C271 /C30L2hM2h (4) 33h /C271 /C30(3h /C271)L3hM3h (5) 55k /C281 /C30(5h /C271)L5hM5h ; (6) where h /C302k /C281 and L2h ; M2h /C302h /C271 /C142k (7) L3h ; M3h /C303h /C271 /C143k (8) L5h ; M5h /C3052h /C273 /C215 5h /C271 /C145k(5k /C271): (9) See also GAUSS’S CYCLOTOMIC FORMULA References Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn 91; b /C302, 3; 5; 6; 7; 10; 11; 12 Up to High Powers, rev. ed. Provi- dence, RI: Amer. Math. Soc., pp. lxviii-lxxii, 1988.Riesel, H. "Aurifeullian Factorization" in Appendix 6. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 309 /C1/15, 1994. Wagstaff, S. S. Jr. "Aurifeullian Factorizations and the Period of the Bell Numbers Modulo a Prime." Math. Comput. 65, 383 /C1/91, 1996. Ausdehnungslehre EXTERIOR ALGEBRA Aut "Aut" is the term applied in PROPOSITIONAL CALCULUS to the XOR connective. "Aut" is Latin form for "either/ or (but not both)," e.g., "Aut Caesar aut nihil" (Cesare Borgia; 1476 /C1/507). The symbol Aut is also commonly used for the completely different purpose of denoting an AUTO- MORPHISM . See also AUTOMORPHISM , XOR References Oxford University Press. The Oxford Dictionary of Quota- tions, 3rd ed. Oxford, England: Oxford University Press, p. 89, 1980. Authalic Latitude An AUXILIARY LATITUDE which gives a SPHERE equal SURFACE AREA relative to an ELLIPSOID . The authalic latitude is defined by b/C30sin/C281q qp ! ; (1) where q/C30(1/C28e2)sinf 1/C28e2sin2f/C281 2eln1/C28esinf 1/C27esinf ! "# (2) andqpisqevaluated at the north pole ( /f/C3090/C14):Let Rqbe the RADIUS of the SPHERE having the same SURFACE AREA as the ELLIPSOID , then Rq/C30affiffiffiffiffi qp 2s : (3) The series for bis b/C30f/C28(1 3e2/C2731 180e4/C2759 560e6/C27. . .) sin( f) /C27(17 360e4/C2761 1260e6/C27. . .) sin(4 f) /C28(383 45360e6/C27. . .) sin(6 f)/C27...: (4) The inverse FORMULA is found from Df /C30(1 /C28 e2 sin2 f)2 2 cos f /C2q 1 /C28 e2 /C28sin f 1 /C28 e2 sin2 f /C271 2eln1 /C28 e sin f 1 /C27 e sin f ! "# ; (5) where q /C30qp sin b (6) and f0 /C30sin /C281(q=2): This can be written in series form as f /C30 b /C27(1 3 e2 /C2731 180 e4 /C27517 5040 e6 /C27...) sin(2b) : /C27(23 360 e4 /C27251 3780 e6 /C27...) sin(4b) /C27(761 45360 e6 /C27...) sin(6b) /C27...: (7) See also LATITUDE References Adams, O. S. "Latitude Developments Connected with Geo- desy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, 1921. Snyder, J. P. Map Projections */A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 16, 1987. Authalic Projection Lee (1944) defines an authalic MAP PROJECTION to be one in which at any point the scales in two orthogonal directions are inversely proportional. See also EQUAL- AREA PROJECTION References Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Review 7, 190 /C1/00, 1944. Autocorrelation The autocorrelation function Rf (t) of a real function f(t) is defined by Rf (t) /C13 lim T 0/C121 2T gT /C28Tf( t)f(T /C27 t) dt (1) (Papoulis 1962, p. 241). For a complex function, the autocorrelation rf (t) is defined by rf (t) /C13f w f /C30 ¯f(/C28t) + f(t) /C30g/C12 /C28/C12f(t /C27 t) ¯f( t) dt : (2) where + denotes CONVOLUTION , w denotes CROSS- CORRELATION , and ¯f is the COMPLEX CONJUGATE (Papoulis 1962, pp. 241 /C1/42). The autocorrelation discards phase information, returning only the power, and is therefore an irreversible operation.There is also a somewhat surprising and extremely important relationship between the autocorrelation and the FOURIER TRANSFORM known as the WIENER- KHINTCHINE THEOREM . Let F[f(x)] /C30F(k) ; and ¯F denote the COMPLEX CONJUGATE of F, then the FOURIER TRANSFORM of the ABSOLUTE SQUARE of F(k) is given by F[ ½F(k) ½2] /C30g/C12 /C28/C12¯f(t)f( t /C27x) d t: (3) The autocorrelation is a HERMITIAN OPERATOR since rf (/C28t) /C30 ¯rf (t):/ /f w f is MAXIMUM at the ORIGIN ; in other words, g/C12 /C28/C12f(u)f(u /C27x) du 5g/C12 /C28/C12f 2(u) du : (4) To see this, let e be a REAL NUMBER . Then g/C12 /C28/C12[f(u) /C27 ef(u /C27x)]2 du > 0 (5) g/C12 /C28/C12f 2(u) du /C272eg/C12 /C28/C12f(u)f(u /C27x) du /C27e2g/C12 /C28/C12f 2(u /C27x) du > 0 (6) g/C12 /C28/C12f 2(u) du /C272eg/C12 /C28/C12f(u)f(u /C27x) du /C27e2g/C12 /C28/C12f 2(u /C27x) du > 0: (7) Define a /C13g/C12 /C28/C12f 2(u) du (8) b /C132g/C12 /C28/C12f(u)f(u /C27x) du: (9) Then plugging into above, we have ae2 /C27be /C27c > 0: This QUADRATIC EQUATION does not have any REAL ROOT ,sob2 /C284ac 50; i.e., b =2 5a: It follows that g/C12 /C28/C12f(u)f(u/C27x)du5g/C12 /C28/C12f2(u)du; (10) with the equality at x/C300. This proves that fwfis MAXIMUM at the ORIGIN . See also AVERAGE POWER ,C ONVOLUTION ,C ROSS- CORRELATION ,Q UANTIZATION EFFICIENCY ,W IENER- KHINTCHINE THEOREM References Bracewell, R. "The Autocorrelation Function." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 40 /C1/5, 1999. Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Correlation and Autocorrelation Using the FFT." §13.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 538 /C1/39, 1992. Autogonal Projection CONFORMAL PROJECTION Automata Theory The mathematical study of abstract computing ma- chines (especially TURING MACHINES ) and the analy- sis of algorithms used by such machines. See also CELLULAR AUTOMATON ,TURING MACHINE References Harrison, M. A. Introduction to Switching and Automata Theory. New York: McGraw-Hill, p. 188, 1965. Simon, M. Automata Theory. Singapore: World Scientific, 1999. Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2001. Automatic Set A k-automatic set is a set of integers whose base- k representations form a regular language, i.e., a language accepted by a finite automaton or state machine. If bases a and b are incompatible (do not have a common power) and if an a-automatic set Sa and b-automatic set Sb are both of density 0 over the integers, then it is believed that Sa S Sbis finite. However, this problem has not been settled. Some automatic sets, such as the 2-automatic con- sisting of numbers whose BINARY representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (Sloane’s A048645) have a simple arithmetic expression. However, this is not the case for general k-automatic sets. See also TURING MACHINE References Cobham, A. "On the Base-Dependence of Sets of Numbers Recognizable by Finite Automata." Math. Systems Th. 3, 186 /C1/92, 1969. Cobham, A. "Uniform Tag Sequences." Math. Systems Th. 6, 164 /C1/92, 1972. Sloane, N. J. A. Sequences A048645 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Automaton AUTOMATIC SET,C ELLULAR AUTOMATON ,T URING MACHINE Automorphic Form See also AUTOMORPHIC FUNCTION ,LANGLANDS PRO- GRAMAutomorphic Function An automorphic function f(z)ofa COMPLEX variable z is one which is analytic (except for POLES ) in a domain D and which is invariant under a DENUMERABLY INFINITE group of LINEAR FRACTIONAL TRANSFORMA- TIONS (also known as MO¨ BIUS TRANSFORMATIONS ) z?/C30az /C27 b cz /C27 d : Automorphic functions are generalizations of TRIGO- NOMETRIC FUNCTIONS and ELLIPTIC FUNCTIONS . See also AUTOMORPHIC FORM,M ODULAR FUNCTION , MO¨ BIUS TRANSFORMATION ,ZETA FUCHSIAN References Hadamard, J.; Gray, J. J.; and Shenitzer, A. Non-Euclidean Geometry in the Theory of Automorphic Forms. Provi- dence, RI: Amer. Math. Soc., 1999. Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, NJ: Princeton Uni- versity Press, 1971. Siegel, C. L. Topics in Complex Function Theory, Vol. 2: Automorphic Functions and Abelian Integrals. New York: Wiley, 1988. Automorphic Number A number ksuch that nk2has its last digits equal to k is called n-automorphic. For example, 1 /C215¯52/C132¯5 (Wells 1986, pp. 58 /C1/9) and 1 /C215¯62/C133¯6 (Wells 1986, p. 68) are 1-automorphic and 2 /C215¯82/C1312¯8 and 2 /C215 882/C3015488 are 2-automorphic. de Guerre and Fair- bairn (1968) give a history of automorphic numbers. The first few 1-automorphic numbers are 1, 5, 6, 25, 76, 376, 625, 9376, 90625, ... (Sloane’s A003226, Wells1986, p. 130). There are two 1-automorphic numberswith a given number of digits, one ending in 5 and one in 6 (except that the 1-digit automorphic numbers include 1), and each of these contains the previousnumber with a digit prepended. Using this fact, it is possible to construct automorphic numbers having more than 25,000 digits (Madachy 1979). The firstfew 1-automorphic numbers ending with 5 are 5, 25, 625, 0625, 90625, ... (Sloane’s A007185), and the first few ending with 6 are 6, 76, 376, 9376, 09376, ...(Sloane’s A016090). The 1-automorphic numbers a(n) ending in 5 are IDEMPOTENT (mod 10n) since [a(n)]2/C13a(n)(mod 10n) (Sloane and Plouffe 1995). The following table gives the 10-digit n-automorphic numbers. nn -Automorphic NumbersSloane 1 0000000001, 8212890625, 1787109376–, A007185, A016090 2 0893554688 A030984 3 6666666667, 7262369792, 9404296875–, A030985, A030986 4 0446777344 A030987 5 3642578125 A030988 6 3631184896 A030989 7 7142857143, 4548984375,1683872768A030990, A030991, A030992 8 0223388672 A030993 9 5754123264, 3134765625,8888888889A030994, A030995, – The infinite 1-automorphic number ending in 5 is given by ...56259918212890625 (Sloane’s A018247), while the infinite 1-automorphic number ending in 6 is given by ...740081787109376 (Sloane’s A018248). See also IDEMPOTENT ,NARCISSISTIC NUMBER ,NUM- BER PYRAMID ,TRIMORPHIC NUMBER References Fairbairn, R. A. "More on Automorphic Numbers." J. Recr. Math. 2, 170 /C1/74, 1969. Fairbairn, R. A. Erratum to "More on Automorphic Num- bers." J. Recr. Math. 2, 245, 1969. de Guerre, V. and Fairbairn, R. A. "Automorphic Numbers." J. Recr. Math. 1, 173 /C1/79, 1968. Hunter, J. A. H. "Two Very Special Numbers." Fib. Quart. 2, 230, 1964. Hunter, J. A. H. "Some Polyautomorphic Numbers." J. Recr. Math. 5, 27, 1972. Kraitchik, M. "Automorphic Numbers." §3.8 in Mathematical Recreations. New York: W. W. Norton, pp. 77 /C1/8, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 34 /C1/4 and 175 /C1/76, 1979. Schroeppel, R. Item 59 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 23, Feb. 1972. Sloane, N. J. A. Sequences A003226/M3752, A007185/ M3940, A016090, A018247, and A018248 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 59 and 171, 178, 191 /C1/92, 1986. Automorphism An ISOMORPHISM of a system of objects onto itself. The term derives from the Greek prefix a yto (auto ) "self" and mor 8 vsi& (morphosis ) "to form" or "to shape." The automorphisms of a GRAPH always describe a GROUP (Skiena 1990, p. 19). An automorphism of a region of the COMPLEX PLANE is a conformal SELF-MAP (Krantz 1999, p. 81).See also ANOSOV AUTOMORPHISM ,G RAPH AUTO- MORPHISM References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 81, 1999. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Automorphism Group The GROUP of functions from an object Gto itself which preserve the structure of the object, denoted Aut(G):The automorphism group of a GROUP pre- serves the MULTIPLICATION table, the automorphism group of a GRAPH the INCIDENCE MATRICES , and that of a FIELD the ADDITION and MULTIPLICATION tables. Autonomous A differential equation or system of ORDINARY DIFFER- ENTIAL EQUATIONS is said to be autonomous if it does not explicitly contain the independent variable(usually denoted t). A second-order autonomous differential equation is OF THE FORM F(y;y?;yƒ)/C300; where y?/C13dy=dt/C13v:By the CHAIN RULE ,yƒcan be expressed as yƒ/C30v?/C30dv dt/C30dv dydy dt/C30dv dyv: For an autonomous ODE, the solution is independent of the time at which the initial conditions are applied. This means that all particles pass through a givenpoint in phase space. A nonautonomous system of n first-order ODEs can be written as an autonomoussystem of n/C271 ODEs by letting t/C13x n/C271and increas- ing the dimension of the system by 1 by adding theequation dxn/C271 dt/C301: Autoregressive Model MAXIMUM ENTROPY METHOD Auxiliary Circle The CIRCUMCIRCLE of an ELLIPSE , i.e., the CIRCLE whose CENTER concurs with that of the ELLIPSE and whose RADIUS is equal to the ELLIPSE ’s SEMIMAJOR AXIS. See also CIRCLE ,ECCENTRIC ANGLE ,ELLIPSE References Montenbruck, O. and Pfleger, T. Astronomy on the Personal Computer, 4th ed. Berlin: Springer-Verlag, p. 62, 2000. Auxiliary Latitude AUTHALIC LATITUDE ,C ONFORMAL LATITUDE ,G EO- CENTRIC LATITUDE ,ISOMETRIC LATITUDE ,LATITUDE , PARAMETRIC LATITUDE ,R ECTIFYING LATITUDE ,R E- DUCED LATITUDE Auxiliary Triangle MEDIAL TRIANGLE Average MEAN Average Absolute Deviation a /C301 NXN i /C301xi /C28 m jj /C30 xi /C28 m jjhi : See also ABSOLUTE DEVIATION ,DEVIATION ,STANDARD DEVIATION ,VARIANCE Average Function If f is CONTINUOUS on a CLOSED INTERVAL [a, b], then there is at least one number x /C31 in [a, b] such that gb a f(x)dx /C30f(xƒ)(b /C28a): The average value of the FUNCTION (f /C28) on this interval is then given by f(x/C31) :/ See also MEAN-VALUE THEOREMAverage Power The average power of a complex signal f(t)asa function of time t is defined as f 2(t)rC10rC11 /C30 lim T 0/C121 2T gT /C28Tf(t)2dtrC10rC10rC10rC10; where zjjis the MODULUS (Papoulis 1962, p. 240). See also AUTOCORRELATION References Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. Average Seek Time POINT- POINT DISTANCE–1- D Avoided Pattern A pattern t /C30( t1 ; ...; tn) is said to avoid a /C30 ( a1 ; ...; ak)ifa is not CONTAINED in t : In other words, t avoids a IFF no K-SUBSET of t is ORDER ISOMORPHIC to a:/ See also CONTAINED PATTERN ,ORDER ISOMORPHIC , PERMUTATION PATTERN ,W ILF CLASS,W ILF EQUIVA- LENT References Mansour, T. Permutations Avoiding a Pattern from Skand at Least Two Patterns from S3 : 31 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007194/. Axial Vector PSEUDOVECTOR Axiom A PROPOSITION regarded as self-evidently TRUE with- out PROOF . The word "axiom" is a slightly archaic synonym for POSTULATE . Compare CONJECTURE or HYPOTHESIS , both of which connote apparently TRUE but not self-evident statements. See also ARCHIMEDES’ AXIOM ,A XIOM OF CHOICE , AXIOMATIC SYSTEM ,CANTOR- DEDEKIND AXIOM ,CON- GRUENCE AXIOMS ,CONJECTURE ,CONTINUITY AXIOMS , COUNTABLE ADDITIVITY PROBABILITY AXIOM ,D EDE- KIND’S AXIOM ,DIMENSION AXIOM ,EILENBERG- STEEN- ROD AXIOMS ,E UCLID’S AXIOMS ,E XCISION AXIOM , FANO’S AXIOM ,FIELD AXIOMS ,H AUSDORFF AXIOMS , HILBERT’S AXIOMS ,HOMOTOPY AXIOM ,INACCESSIBLE CARDINALS AXIOM ,INCIDENCE AXIOMS ,INDEPEN- DENCE AXIOM ,INDUCTION AXIOM ,L AW,L EMMA , LONG EXACT SEQUENCE OF A PAIR AXIOM ,ORDERING AXIOMS ,PARALLEL AXIOM ,PASCH’S AXIOM ,PEANO’S AXIOMS ,P LAYFAIR’S AXIOM ,P ORISM ,P OSTULATE , PROBABILITY AXIOMS ,P ROCLUS’ AXIOM ,R ULE,T 2- SEPARATION AXIOM ,THEOREM ,ZERMELO’S AXIOM OF CHOICE ,ZERMELO- FRAENKEL AXIOMS Axiom A Diffeomorphism Let f : M 0 M be a C1 DIFFEOMORPHISM on a com- pact RIEMANNIAN MANIFOLD M. Then f satisfies Axiom A if the NONWANDERING set V( f)of f is hyperbolic and the PERIODIC POINTS of f are DENSE in v( f) : although it was conjectured that the first of these conditions implies the second, they were shown to be independent in or around 1977. examples include the ANOSOV DIFFEOMORPHISMS and SMALE HORSESHOE MAP. In some cases, Axiom A can be replaced by the condition that the DIFFEOMORPHISM is a hyperbolic diffeomorphism on a hyperbolic set (Bowen 1975, Parry and Pollicott 1990). See also ANOSOV DIFFEOMORPHISM ,AXIOM AF LOW, DIFFEOMORPHISM ,DYNAMICAL SYSTEM ,RIEMANNIAN MANIFOLD ,SMALE HORSESHOE MAP References Bowen, R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. New York: Springer-Verlag, 1975. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 143, 1993. Parry, W. and Pollicott, M. "Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics." Aste´risque No. 187 /C188, 1990. Smale, S. "Differentiable Dynamical Systems." Bull. Amer. Math. Soc. 73, 747 /C117, 1967. Axiom A Flow A FLOW defined analogously to the AXIOM A DIFFEO- MORPHISM , except that instead of splitting the TAN- GENT BUNDLE into two invariant sub- BUNDLES , they are split into three (one exponentially contracting, one expanding, and one which is 1-dimensional and tangential to the flow direction). See also DYNAMICAL SYSTEM Axiom of Choice An important and fundamental axiom in SET THEORY sometimes called ZERMELO’S AXIOM OF CHOICE . It was formulated by Zermelo in 1904 and states that, given any SET of mutually exclusive nonempty SETS, there exists at least one SET that contains exactly one element in common with each of the nonempty SETS . The axiom of choice is related to the first of HILBERT’S PROBLEMS . In ZERMELO- FRAENKEL SET THEORY (in the form omitting the axiom of choice), the ZORN’S LEMMA , TRICHOTOMY LAW, and the WELL ORDERING PRINCIPLE are equivalent to the axiom of choice (Mendelson 1997, p. 275). In contexts sensitive to the axiom of choice, the notation "ZF" is often used to denote Zermelo-Fraenkel without the axiom of choice, while "ZFC" is used if the axiom of choice is included.In 1940, Go¨del proved that the axiom of choice is CONSISTENT with the axioms of VON NEUMANN- BER- NAYS- GO¨ DEL SET THEORY (a conservative extension of ZERMELO- FRAENKEL SET THEORY ). However, in 1963, Cohen (1963) unexpectedly demonstrated that the axiom of choice is also independent of ZERMELO- FRAENKEL SET THEORY (Mendelson 1997; Boyer and Merzbacher 1991, pp. 610 /C111). See also HILBERT’S PROBLEMS ,SET THEORY , VON NEUMANN- BERNAYS- GO¨ DEL SET THEORY ,W ELL OR- DERED SET,W ELL ORDERING PRINCIPLE ,ZERMELO- FRAENKEL AXIOMS ,ZERMELO- FRAENKEL SET THEORY , ZORN’S LEMMA References Boyer, C. B. and Merzbacher, U. C. A History of Mathe- matics, 2nd ed. New York: Wiley, 1991. Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, pp. 178 /C179, 1958. Cohen, P. J. "The Independence of the Continuum Hypoth- esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143 /C1148, 1963. Cohen, P. J. "The Independence of the Continuum Hypoth- esis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105 /C110, 1964. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 274 /C176, 1996. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Axiom of Comprehension AXIOM OF SEPARATION Axiom of Extensionality The axiom of ZERMELO- FRAENKEL SET THEORY which asserts that sets formed by the same elements are equal, /C214x(x/C23a/C13x/C23b)[a/C30b: Using the notation aƒb(ais a SUBSET ofb) for x/C23 a(x/C23b);the axiom can be rewritten aƒbfflbƒa[a/C30b: See also ZERMELO- FRAENKEL SET THEORY References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Axiom of Foundation One of the Z ERMELO- FRAENKEL AXIOMS , also known the axiom of regularity (Rubin 1967, Suppes 1972). In the formal language of SET THEORY , it states that x"0[/C215y(y/C23xfflySx/C30f); where [means IMPLIES ,/C215means EXISTS ,fflmeans AND,Sdenotes INTERSECTION , and fis the EMPTY SET (Mendelson 1997, p. 288). More descriptively, "every nonempty set is disjoint from one of its elements." The axiom of foundation can also be stated as "A set contains no infinitely descending (membership) se- quence," or "A set contains a (membership) minimal element," i.e., there is an element of the set that shares no member with the set (Ciesielski 1997, p. 37; Moore 1982, p. 269; Rubin 1967, p. 81; Suppes 1972, p. 53). Mendelson (1958) proved that the equivalence of these two statements necessarily relies on the AXIOM OF CHOICE . The dual expression is called e/-induction, and is equivalent to the axiom itself (Itoˆ 1986, p. 147). See also AXIOM OF CHOICE ,Z ERMELO- FRAENKEL AXIOMS References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Mendelson, E. "The Axiom of Fundierung and the Axiom of Choice." Archiv fu¨r math. Logik und Grundlagenfors. 4, 67 /C10, 1958. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997. Mirimanoff, D. "Les antinomies de Russell et de Burali-Forti et le proble `me fondamental de la the´orie des ensembles." Enseign. math. 19,37/C12, 1917. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Neumann, J. von. "U¨ ber eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre." J. reine angew. Math. 160, 227 /C141, 1929. Neumann, J. von. "Eine Axiomatisierung der Mengenlehre." J. reine angew. Math. 154, 219 /C140, 1925. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Zermelo, E. "U¨ ber Grenzzahlen und Mengenbereiche." Fund. Math. 16,29/C17, 1930. Axiom of Infinity The axiom of ZERMELO- FRAENKEL SET THEORY which asserts the existence of a set containing all the natural numbers, /C215 r( ¥/C23 x /C150/C214 y /C23 x(y?/C23 x)): Here, following von Neumann, 0 /C30 f; 1 /C30 0?/C30 f0g; 2 /C30 1 ?/C30 f0 ; 1 g; 3 /C30 2?/C30 f0; 1; 2g; .... See also ZERMELO- FRAENKEL SET THEORY References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986.Axiom of Regularity AXIOM OF FOUNDATION Axiom of Replacement One of the ZERMELO- FRAENKEL AXIOMS which asserts the existence for any set a of a set x such that, for any y of a, if there exists a z satisfying A(y; z) ; then such z exists in x. This axiom was introduced by Fraenkel. See also ZERMELO- FRAENKEL AXIOMS References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Axiom of Separation The axiom of ZERMELO- FRAENKEL SET THEORY which asserts the existence for any set a and a formula A(y) of a set x consisting of all elements of a satisfying A(y); /C215 x /C214 y(y /C23 x /C13 y /C23 a ffl A(y)) : This axiom is also called the axiom of comprehension or axiom of subsets, and was introduced by Zermelo. See also ZERMELO- FRAENKEL SET THEORY References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Axiom of the Empty Set One of the ZERMELO- FRAENKEL AXIOMS which asserts the existence of the EMPTY SET f: The axiom may be stated symbolically as /C215 x /C214 y(!y /C23 x): See also ZERMELO- FRAENKEL AXIOMS References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Axiom of the Power Set One of the ZERMELO- FRAENKEL AXIOMS which asserts the existence for any set a of the POWER SET x consisting of all the SUBSETS of a. The axiom may be stated symbolically as /C214x/C215y(y/C23x/C13/C214z/C23y(z/C23a)): See also POWER SET,ZERMELO- FRAENKEL AXIOMS References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Axiom of the Sum Set The axiom of ZERMELO- FRAENKEL SET THEORY which asserts the existence for any set a of the sum (union) x of all sets that are elements of a. The axiom may be stated symbolically as /C215 x /C214 y(y /C23 x /C13/C215 z /C23 a(y /C23 z)): See also ZERMELO- FRAENKEL SET THEORY References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Axiom of the Unordered Pair The axiom of ZERMELO- FRAENKEL SET THEORY which asserts the existence for any sets a and b of a set x having a and b as its only elements. x is called the unordered pair of a and b, denoted fa ; b g: The axiom may be stated symbolically as /C215 x /C214 y(y /C23 x /C13 y /C30 a /C150 y /C30 b): See also ZERMELO- FRAENKEL SET THEORY References Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C148, 1986. Axiomatic Set Theory A version of SET THEORY in which axioms are taken as uninterpreted rather than as formalizations of pre- existing truths. See also AXIOMATIC SYSTEM ,COMPLETE AXIOMATIC THEORY ,NAIVE SET THEORY ,SET THEORY References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 22 /C13, 1977. Axiomatic System A logical system which possesses an explicitly stated SET of AXIOMS from which THEOREMS can be derived. See also AXIOMATIC SET THEORY ,COMPLETE AXIO- MATIC THEORY ,CONSISTENCY ,M ODEL THEORY ,THE- OREM Axioms of Subsets This entry contributed by NICOLAS BRAYFor any set theoretic formula f(x; t1 ; t2 ; ...; tn); (/C214t1)(/C214t2) /C1/C1/C1(/C214tn)(/C214A)(/C215B)(/C214x) : (x /C23 B Ux /C23 A fflf(x; t1 ; ...; tn)) In other words, for any formula and set A there is a SUBSET of A consisting exactly of those elements which satisfy the formula. Axis A LINE with respect to which a curve or figure is drawn, measured, rotated, etc. The term is also used to refer to a LINE through a SHEAF OF PLANES (Woods 1961; Altshiller-Court 1979, p. 12). See also ABSCISSA ,BROCARD AXIS,HOMOLOGY AXIS, LEMOINE AXIS,L INE,M AJOR AXIS,M EDIAL AXIS, MINOR AXIS,ORDINATE ,ORTHIC AXIS,PERSPECTIVE AXIS,R ADICAL AXIS,R EAL AXIS,SEMIMAJOR AXIS, SEMIMINOR AXIS,SHEAF OF PLANES ,SIMILARITY AXIS, X-AXIS, Y-AXIS, Z-AXIS References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 8, 1961. Ax-Kochen Isomorphism Theorem LetPbe the SETofPRIMES , and let QpandZp(t) be the FIELDS ofP-ADIC NUMBERS and formal POWER SERIES overZp/C30(0;1;...;p/C281):Further, suppose that D is a "nonprincipal maximal filter" on P. ThenQ p/C23pQp=DandQ p/C23qZp(t)=Dare ISOMORPHIC . See also HYPERREAL NUMBER ,NONSTANDARD ANALY- SIS Axonometry A METHOD for mapping 3-D figures onto the PLANE . See also CROSS SECTION ,MAP PROJECTION ,POHLKE’S THEOREM ,PROJECTION ,STEREOLOGY References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 313, 1973. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 322 /C1/23, 1988. Azimuthal Equidistant Projection An AZIMUTHAL PROJECTION which is neither EQUAL- AREA nor CONFORMAL . Let f1 and l0 be the LATITUDE and LONGITUDE of the center of the projection, then the transformation equations are given by x /C30k? cos f sin( l /C28 l0) (1) y /C30k?[cos f1 sin f /C28sin f1 cos f cos(l /C28 l0)]: (2) Here, k ?/C30c sin c (3) and cos c /C30sin f1 sin f /C27cos f1 cos f cos(l /C28 l0) ; (4) where c is the angular distance from the center. Theinverse FORMULAS are f /C30sin /C281cos c sin f1 /C27y sin c cos f1 c ! (5) and l /C30l0 /C27tan /C281x sin c c cos f1 cos c /C28 y sin f1 sin c ! for f1 "990( l0 /C27tan /C281 /C28x y ! for f1 /C3090( l0 /C27tan /C281xy ! for f 1 /C30/C2890( :8 >>>>>>>>>< >>>>>>>>>: (6) with the angular distance from the center given by c /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2 :p (7) See also AZIMUTHAL PROJECTION ,EQUIDISTANT PRO- JECTION References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 191 /C1/02, 1987. Azimuthal Projection A MAP PROJECTION on which the azimuths of all points are shown correctly with respect to the center (Snyder 1987, p. 4). A plane tangent to one of the Earth’s poles is the basis for polar azimuthal projec- tion. The term "zenithal" is an older one for azimuthal projections (Hinks 1921, Lee 1944). See also AZIMUTHAL EQUIDISTANT PROJECTION ,LAMBERT AZI- MUTHAL EQUAL- AREA PROJECTION ,O RTHOGRAPHIC PROJECTION ,STEREOGRAPHIC PROJECTION References Hinks, A. R. Map Projections, 2nd rev. ed. Cambridge, England: Cambridge University Press, 1921. Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190/C1/00, 1944. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987. B B2-Sequence N.B. A detailed online essay by S. Finch was the starting point for this entry. Also called a SIDON SEQUENCE .An INFINITE SEQUENCE of POSITIVE INTEGERS 1 5b1 Bb2 Bb3 B... (1) such that all pairwise sums bi /C27bj (2) for i 5j are distinct (Guy 1994). An example is 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, ... (Sloane’s A005282). Zhang (1993, 1994) showed that S(B2) /C13 SUP all B2 sequencesX/C12 k /C3011 bk> 2:1597 ; (3) which has been increased to S(B2) > 2:16086 by R. Lewis using the sequence 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 291, 324, ... (Sloane’s A046185). The definition can be extended to Bn/-sequences (Guy 1994). See also A-SEQUENCE ,MIAN-CHOWLA SEQUENCE References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/erdos/erdos.html. Guy, R. K. "Packing Sums of Pairs," "Three-Subsets with Distinct Sums," and "/B2/-Sequences," and B2/-Sequences Formed by the Greedy Algorithm." §C9, C11, E28, and E32 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 115 /C1/118, 121 /C1/123, 228 /C1/229, and 232 /C1/233, 1994. Mian, A. M. and Chowla, S. D. "On the B2/-Sequences of Sidon." Proc. Nat. Acad. Sci. India A14,3/C1/4, 1944. Sloane, N. J. A. Sequences A005282/M1094 and A046185 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Zhang, Z. X. "A B2-Sequence with Larger Reciprocal Sum." Math. Comput. 60, 835 /C1/839, 1993. Zhang, Z. X. "Finding Finite B2-Sequences with Larger m /C28a1=2 m:/" Math. Comput. 63, 403 /C1/414, 1994. Baby Monster Group Also known as FISCHER’S BABY MONSTER GROUP . The SPORADIC FINITE GROUP B. It has ORDER 241 /C215 313 /C215 56 /C215 72 /C215 11 /C215 13 /C215 17 /C215 19 /C215 23 /C215 31 /C215 47: See also FINITE GROUP ,MONSTER GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/BM.html.BAC-CAB Identity The VECTOR TRIPLE PRODUCT identity A /C29(B /C29C) /C30B(A /C215 C) /C28C(A /C215 B) : This identity can be generalized to n-D a2 /C29/C1/C1/C1/C29an/C281 /C29(b1 /C29/C1/C1/C1/C29bn/C281) /C30(/C281)n/C271b1 /C1/C1/C1 bn/C281 a2/C215 b1 /C1/C1/C1 a2/C215 bn/C281 n::: n an/C281/C215 b1/C1/C1/C1 an/C281/C215bn/C281/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2: See also L AGRANGE’S IDENTITY BAC-CAB Rule BAC -CAB I DENTITY Bachelier Function BROWN FUNCTION Bachet Equation The D IOPHANTINE EQUATION x2/C27k/C30y3: which is also an ELLIPTIC CURVE . The general equa- tion is still the focus of ongoing study. Bachet’s Conjecture LAGRANGE’S FOUR- SQUARE THEOREM Bachet’s Theorem LAGRANGE’S FOUR- SQUARE THEOREM Backhouse’s Constant LetP(x) be defined as the POWER SERIES whose nth term has a COEFFICIENT equal to the nthPRIME , P(x)/C13X/C12 k/C300pkxk/C301/C272x/C273x2/C275x3/C277x4/C2711x5/C27...; and let Q(x) be defined by Q(x)/C301 P(x)/C30X/C12 k/C300qkxk: Then N. Backhouse conjectured that lim n0/C12jqn/C271 qnj/C301:4560749485826896713995953511116 . . . : This list was subsequently shown to exist by P. Flajo- let. References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/backhous/back- hous.html. Ba¨cklund Transformation A method for solving classes of nonlinear PARTIAL DIFFERENTIAL EQUATIONS . See also INVERSE SCATTERING METHOD ,SOLITON References Anderson, R. L. and Ibragimov, N. H. Lie-Ba ¨cklund Trans- formation in Applications. Philadelphia, PA: SIAM, 1979. Dodd, R. K.; Eilbeck, J. C.; and Morris, H. C. Solitons and Nonlinear Equations. London: Academic Press, 1984. Infeld, E. and Rowlands, G. "Ba¨cklund Transformations." §7.5 in Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge University Press, pp. 175 /C1/77, 2000. Lamb, G. L. Jr. Elements of Soliton Theory. New York: Wiley, 1980. Miura, R. M. (Ed.). Ba¨cklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications . New York: Springer-Verlag, 1974. Olver, P. J. Applications of Lie Groups to Differential Equations. New York: Springer-Verlag, 1986. Rogers, C. and Shadwick, W. F. Ba¨cklund Transformations and Their Applications. New York: Academic Press, 1982. Whitham, G. B. Linear and Nonlinear Waves. New York: Wiley, pp. 609 /C1/11, 1974. Zwillinger, D. "Ba¨cklund Transformations." §87 in Hand- book of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 321 /C1/24, 1997. Backtracking A method of solving combinatorial problems by means of an algorithm which is allowed to run forward until a dead end is reached, at which point previous steps are retraced and the algorithm is allowed to run forward again. Backtracking can greatly reduce the amount of work in an exhaustive search. Backtrack- ing is implemented asBacktrack [s, partialQ , solu- tionQ ] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). Backtracking also refers to a method of drawing FRACTALS by appropriate numbering of the corre- sponding tree diagram which does not require storage of intermediate results (Lauwerier 1991). References Baumert, L. D. and Golomb, S. W. "Backtrack Program- ming." J. Ass. Comp. Machinery 12, 516 /C1/24, 1965. Lauwerier, H. A. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press, 1991. Skiena, S. "Backtracking and Distinct Permutations." §1.1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 12 /C1/4, 1990. Wilf, H. "Backtrack: An i(1) Expected Time Algorithm for the Graph Coloring Problem." Info. Proc. Let. 18, 119 /C1/21, 1984.Backus-Gilbert Method A method which can be used to solve some classes of INTEGRAL EQUATIONS and is especially useful in implementing certain types of data inversion. It has been applied to invert seismic data to obtain density profiles in the Earth. References Backus, G. and Gilbert, F. "The Resolving Power of Growth Earth Data." Geophys. J. Roy. Astron. Soc. 16, 169 /C1/05, 1968. Backus, G. E. and Gilbert, F. "Uniqueness in the Inversion of Inaccurate Gross Earth Data." Phil. Trans. Roy. Soc. London Ser. A 266, 123 /C1/92, 1970. Loredo, T. J. and Epstein, R. I. "Analyzing Gamma-Ray Burst Spectral Data." Astrophys. J. 336, 896 /C1/19, 1989. Parker, R. L. "Understanding Inverse Theory." Ann. Rev. Earth Planet. Sci. 5,35/C1/4, 1977. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Backus-Gilbert Method." §18.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 806 /C1/09, 1992. Backward Difference The backward difference is a FINITE DIFFERENCE defined by 9p /C139fp /C13fp /C28fp /C281 : (1) Higher order differences are obtained by repeated operations of the backward difference operator, so 92 p/C309(9p)/C309(fp/C28fp/C281)/C309fp/C289fp/C281 (2) /C30(fp/C28fp/C281)/C28(fp/C281/C28fp/C282) /C30fp/C282fp/C281/C27fp/C282 (3) In general, 9kp/C139kfp/C13Xk m/C300(/C281)mk m/CP8/CP9 fp/C28m; (4) wherek m/CP8/CP9 is a BINOMIAL COEFFICIENT . NEWTON’S BACKWARD DIFFERENCE FORMULA ex- presses fpas the sum of the nth backward differences fp/C30f0/C27p90/C271 2!p(p/C271)920/C271 3!p(p/C271)(p/C272)930 /C27...:; (5) where 9n0is the first nth difference computed from the difference table. See also ADAMS’ METHOD ,D IFFERENCE EQUATION , DIVIDED DIFFERENCE ,FINITE DIFFERENCE ,FORWARD DIFFERENCE ,NEWTON’S BACKWARD DIFFERENCE FOR- MULA ,RECIPROCAL DIFFERENCE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 429 and 433, 1987. Backward Stability The property of certain algorithms that accurate answers are returned for well-conditioned problems, and the inaccuracy of the answers returned for ill- conditioned problems is proportional to the sensitiv- ity. Bader-Deuflhard Method A generalization of the BULIRSCH- STOER ALGORITHM for solving ORDINARY DIFFERENTIAL EQUATIONS . References Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations." Numer. Math. 41, 373 /C198, 1983. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 730, 1992. Baer Differential Equation The Baer differential equation is given by (x /C28 a1)(x /C28a2)yƒ/C271 2 2x /C28(a1 /C27a2) ½/C138 y?/C28(p2x /C27q2)y /C300; while the Baer "wave equation" is (x /C28a1)(x /C28a2)yƒ/C271 2 2x /C28(a1 /C27a2) ½/C138 y?/C28(k2x2 /C28p2x /C27q2)y /C300 (Moon and Spencer 1961, pp. 156 /C1/57; Zwillinger 1997, p. 121). References Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Bagging See also RESAMPLING STATISTICS Baguenaudier A PUZZLE involving disentangling a set of rings from a looped double rod, originally used by French peasants to lock chests (Steinhaus 1983). The word "bague- naudier" means "time-waster" in French, and the puzzle is also called the Chinese rings or Devil’s needle puzzle. ("Bague" also means "ring," but thisappears to be an etymological coincidence. Interest- ingly, the bladder-senna tree is also known as "baguenaudier" in French.) Culin (1965) attributes the puzzle to Chinese general Hung Ming (A.D. 181 /C1/ 34), who gave it to his wife as a present to occupy her while he was away at the wars. The solution of the baguenaudier is intimately related to the theory of GRAY CODES . The minimum number of moves a(n) needed for n rings is a(n) /C30[2 3 (2n /C281)] /C301 3 (2n/C271 /C282) n even 13 (2n/C271 /C281) n odd;( (1) where xdeis the CEILING FUNCTION , giving 1, 2, 5, 10, 21, 42, 85, 170, 341, 682, ... (Sloane’s A000975). The GENERATING FUNCTION for these numbers is 1 (1 /C28 2x)(1 /C28 x2) /C301 /C272x /C275x2 /C2710x3 /C2721x4 /C27...: (2) They are also given by the RECURRENCE RELATION a(n) /C30a(n /C281) /C272a(n /C282) /C271 (3) with a(1) /C301 and a(2) /C302:/ By simultaneously moving the two end rings, the number of moves for n rings can be reduced to b(n) /C302n/C281 /C281 n even 2n/C281n odd;/C26 (4) giving 1, 1, 4, 7, 16, 31, 64, 127, 256, 511, ... (Sloane’s A051049). Defining the complexity of a solution as the minimal number of times the ring passes through the arc from the last ring to the base of the puzzle, the minimal complexity of a solution if 2n/C281 ; as conjectured by Kauffman (1996) and proved by Przytycki and Sikora (2000). See also GRAY CODE,HABIRO MOVE References Culin, S. "Ryou-Kaik-Tjyo--Delay Guest Instrument (Ring Puzzle)." §20 in Games of the Orient: Korea, China, Japan. Rutland, VT: Charles E. Tuttle, pp. 31 /C1/2, 1965. Dubrovsky, V. "Nesting Puzzles, Part II: Chinese Rings Produce a Chinese Monster." Quantum 6,6 1/C1/5 (Mar.) and 58/C1/9 (Apr.), 1996. Gardner, M. "The Binary Gray Code." In Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 15 /C1/7, 1986. Kauffman, L. H. "Tangle Complexity and the Topology of the Chinese Rings." In Mathematical Approaches to Biomole- cular Structure and Dynamics. New York: Springer- Verlag, pp. 1 /C1/0, 1996. Kraitchik, M. "Chinese Rings." §3.12.3 in Mathematical Recreations. New York: W. W. Norton, pp. 89 /C1/1, 1942. Przytycki, J. H. and Sikora, A. S. Topological Insights from the Chinese Rings. 21 Jul 2000. http://xxx.lanl.gov/abs/ math.GT/0007134/. Sloane, N. J. A. Sequences A000975 and A051049 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Slocum, J. and Botermans, J. Puzzles Old and New: How to Make and Solve Them. Seattle, WA: University of Wa- shington Press, p. 105, 1988. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 268 /C1/69, 1999. University of Waterloo. "Wire and RIng Puzzles." http:// www.ahs.uwaterloo.ca/~museum/vexhibit/puzzles/wire/ wire.html. Bailey’s Lemma If, for n ]0 ; bn /C30Xn r/C300ar (q; q)n/C28r(aq; q)n/C27r; (1) then b?n /C30Xn r/C300a?r (q; q)n/C28r(aq; q)n/C27r; (2) where a?r /C30( r1; q)r( r2; q)r(aq =r1 r2)r ar (aq =r1; q)r(aq =r2; q)r(3) b?n /C30X j]0( r1; q)j( r2; q)j(aq =r11 r2; q)n/C28j(aq =r1 r2)j bj (q; q)n /C28j(aq =r1; q)n(aq=r2; q)n: (4) References Andrews, G. E. "Multiple Series Rogers-Ramanujan Type Identities." Pacific J. Math. 114, 267 /C1/83, 1984. Andrews, G. E. "Bailey’s Lemma" and "Bailey’s Lemma in Computer Algebra." §3.4 and 10.4 in q-Series: Their Development and Application in Analysis, Number The- ory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 25 /C1/7 and 99 /C1/00, 1986. Bailey, W. N. "Identities of the Rogers-Ramanujan Type." Proc. London Math. Soc. 50,1/C1/0, 1949. Bailey’s Method LAMBERT’S METHOD Bailey’s Theorem Let G(z) be the GAMMA FUNCTION , thenG(m /C271 2) G(m)"#2 /C21 m /C271 2 !21 m /C27 1 /C271 /C215 3 2 /C215 4 !21 m /C27 2 /C27...2 435 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n /C30G(n /C271 2) G(n)"#2 /C21 n /C271 2 !21 n /C27 1 /C271 /C215 3 2 /C215 4 !21 n /C27 2 /C27...2 435 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} m: Writing the sums explicitly, Bailey’s theorem states G(m/C271 2) G(m)"#2Xn/C281 k/C3001 m/C27k(2k/C281)!! (2k)!!"#2 G(n/C271 2) G(n)"#2Xm/C281 k/C3001 n/C27k(2k/C281)!! (2k)!!"#2 : See also GAMMA FUNCTION References Bailey, W. N. "The Partial Sum of the Coefficients of the Hypergeometric Series." J. London Math. Soc. 6,4 0/C1/1, 1931. Bailey, W. N. "On One of Ramanujan’s Theorems." J. London Math. Soc. 7,3 4/C1/6, 1932. Darling, H. B. C. "On a Proof of One of Ramanujan’s Theorems." J. London Math. Soc. 5,8/C1/, 1930. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 106 /C1/07 and 112, 1999. Hodgkinson, J. "Note on One of Ramanujan’s Theorems." J. London Math. Soc. 6,4 2/C1/3, 1931. Watson, G. N. "Theorems Stated by Ramanujan (VIII): Theorems on Divergent Series." J. London Math. Soc. 4, 82/C1/6, 1929. Watson, G. N. Quart. J. Math. (Oxford) 1, 310/C1/18, 1930. Whipple, F. J. W. "The Sum of the Coefficients of a Hyper- geometric Series." J. London Math. Soc. 5, 192, 1930. Bailey’s Transformation The very general transformation 9F8/C20a;1/C271 2a; b; c; d 12a 1/C27a/C28b;1/C27a/C28c;1/C27a/C28d: e; f; g; /C28m; 1/C27a/C28e;1/C27a/C28f;1/C27a/C28g;1/C27a/C27m/C2P /C30(1/C27a)m(1/C27k/C28e)m(1/C27k/C28f)m(1/C27k/C28g)m (1/C27k)m(1/C27a/C28e)m(1/C27a/C28f)m(1/C27a/C28g)m /C299F8k; 1 /C271 2k; k /C27b /C28a ; k /C27c /C28a ; k /C27d /C28a ; 12k; 1 /C27a /C28b; a /C27a /C28c; 1 /C27a /C28d;" e ; f ; g ; /C28m; 1 /C27k /C28e ; 1 /C27k /C28f ; 1 /C27k /C28g ; 1 /C27k /C27m/C2P ; where k /C301 /C272a /C28b /C28c /C28d; and the parameters are subject to the restriction b /C27c /C27d /C27e /C27f /C27g /C28m /C302 /C273a (Bailey 1935, p. 27). Bhatnagar (1995, pp. 17 /C1/8) defines the Bailey trans- form as follows. Let (a; q)nbe the Q-POCHHAMMER SYMBOL , and let a be an indeterminate, and let the LOWER TRIANGULAR MATRICES F /C30(F(n; k)) and F /C30 (G(n; k)) be defined as F(n; k) /C301 (q; q)n/C28k(aq; q)n/C27k and G(n ; k) /C30(1 /C28 aq2n)(a; q)n /C27k (1 /C28 a)(q; q)n/C28kÞ(/C281)n/C28kqn /C28k 2ðÞ Then F and G are MATRIX INVERSES . See also DOUGALL- RAMANUJAN IDENTITY ,GENERAL- IZED HYPERGEOMETRIC FUNCTION References Bailey, W. N. "Some Identities Involving Generalized Hy- pergeometric Series." Proc. London Math. Soc. 29, 503 /C1/ 16, 1929. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: University Press, 1935. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, 1995. Milne, S. C. and Lilly, G. M. "The Aland ClBailey Trans- form and Lemma." Bull. Amer. Math. Soc. 26, 258 /C1/63, 1992. Bailey-Borwein-Plouffe Algorithm The DIGIT-EXTRACTION ALGORITHM for calculating the digits of PI given by the formula p /C30X/C12 n/C3004 8n /C27 1 /C282 8n /C27 4 /C281 8n /C27 5 /C281 8n /C27 6 ! 1 16 !n : See also PI,PI FORMULAS References Adamchik, V. and Wagon, S. "A Simple Formula for p:/" Amer. Math. Monthly 104, 852 /C1/55, 1997. Adamchik, V. and Wagon, S. "Pi: A 2000-Year Search Changes Direction." http://members.wri.com/victor/arti- cles/pi.html. Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps.Finch, S. "Unsolved Mathematics Problems: The Miraculous Bailey-Borwein-Plouffe Pi Algorithm." http://www.math- soft.com/asolve/plouffe/plouffe.html. Baire Category Theorem A nonempty complete METRIC SPACE cannot be RE- PRESENTED AS the UNION of a COUNTABLE family of NOWHERE DENSE SUBSETS . See also COUNTABLE SET,M ETRIC SPACE ,NOWHERE DENSE Baire Function References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 104 /C1/ 06, 1971. Baire Space ATOPOLOGICAL SPACE Xin which each SUBSET ofXof the "first category" has an empty interior. A TOPOLO- GICAL SPACE which is HOMEOMORPHIC to a complete METRIC SPACE is a Baire space. Bairstow’s Method A procedure for finding the quadratic factors for the COMPLEX CONJUGATE ROOTS of a POLYNOMIAL P(x) with REAL COEFFICIENTS . x/C28(a/C28ib) ½/C138 x/C28(a/C28ib) ½/C138 /C30x2/C272ax/C27(a2/C27b2) /C13x2/C27Bx/C27C: (1) Now write the original POLYNOMIAL as P(x)/C30(x2/C27Bx/C27C)Q(x)/C27Rx/C27S (2) R(B/C27dB;C/C27dC):R(B;C)/C27@R @BdB/C27@R @CdC (3) S(B/C27dB;C/C27dC):S(B;C)/C27@S @BdB/C27@S @CdC (4) @P @C/C300/C30(x2/C27Bx/C27C)@Q @C/C27Q(x)/C27@R @C/C27@S @C(5) /C28Q(x)/C30(x2/C27Bx/C27C)@Q @C/C27@R @C/C27@S @C(6) @P @B/C300/C30(x2/C27Bx/C27C)@Q @B/C27xQ(x)/C27@R @B/C27@S @B(7) /C28xQ(x)/C30(x2/C27Bx/C27C)@Q @B/C27@R @B/C27@S @B: (8) Now use the 2-D N EWTON’S METHOD to find the simultaneous solutions. References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, pp. 277 and 283 /C1/84, 1989. Baker’s Dozen The number 13. See also 13,DOZEN Baker’s Map The MAP xn/C271 /C302 mxn ; (1) where x is computed modulo 1. A generalized Baker’s map can be defined as xn/C271 /C30laxn yn B a (1 /C28 lb) /C27 lbxnyn > a/C26 (2) yn/C271 /C30yn ayn B a yn /C28 a byn > a;8 >>>< >>>:(3) where b /C131 /C28 a; l a /C27 lb 51 ; and x and y are computed mod 1. The q /C301 Q-DIMENSION is D1 /C301 /C27a ln1 a ! /C27 b ln1 b ! a ln1 ga ! /C27 b ln1 gb ! : (4) If la /C30 lb ; then the general Q-DIMENSION is Dq /C301 /C271 q /C28 1ln(aq /C27 bq) ln la: (5) References Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, p. 60, 1983. Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, pp. 81 /C1/2, 1993. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 32, 1990. Bakos’ Compound CUBE 4-COMPOUND Balanced ANOVA An ANOVA in which the number of REPLICATES (sets of identical observations) is restricted to be the same for each FACTOR LEVEL (treatment group). See also ANOVABalanced Binomial Coefficient An integer n is p-balanced for p a prime if, among all nonzero binomial coefficientsn k/C0/CP ; for k /C300, ..., n (mod p), there are equal numbers of quadratic resi- dues and nonresidues (mod p). Let Tpbe the set of integers n,05n 5p /C281; that are p-balanced. Among all the primes B1;000;000; only those with p /C302, 3, and 11 have Tp /C30¥:/ p /Tp/ 2 /¥/ 3 /¥/ 5 /f3g/ 7 /f3g/ 11 /¥/ 13 /f7; 11g/ 17 /f3; 15g/ See also BINOMIAL COEFFICIENT References Garfield, R. and Wilf, H. S. "The Distribution of the Binomial Coefficients Modulo p." J. Number Th. 41,1, 1992. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos’ 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press, pp. 557 /C1/62, 1997. Balanced Incomplete Block Design BLOCK DESIGN Ball The n-ball, denoted Bn ; is the interior of a SPHERE Sn/C281 ; and sometimes also called the n-DISK. (Although physicists often use the term "SPHERE "to mean the solid ball, mathematicians definitely do not!) Let Vol(Bn) denote the volume of an n-D ball of RADIUS r. Then X/C12 n/C300Vol(Bn) /C30e pr2 [1 /C27erf(rffiffiffipp)]; where erf(x) is the ERF function. See also ALEXANDER’S HORNED SPHERE ,BALL LINE PICKING ,B ALL TRIANGLE PICKING ,B ANACH- TARSKI PARADOX ,B ING’S THEOREM ,B ISHOP’S INEQUALITY , BOUNDED SET,DISK,H YPERSPHERE ,SPHERE ,W ILD POINT References Freden, E. Problem 10207. "Summing a Series of Volumes." Amer. Math. Monthly 100, 882, 1993. Ball Line Picking Given an n-ball Bn of radius R, find the distribution of the lengths s of the lines determined by two points chosen at random within the ball. The probability distribution of lengths is given by Pn(s) /C30nsn/C281 RnIx(1 2(n /C271);12) ; (1) where x /C131 /C28s2 4R2 (2) and Ix(p; q) /C30B(x; p ; q) B(p; q) (3) is a REGULARIZED BETA FUNCTION , with B(x; p ; q)is an INCOMPLETE BETA FUNCTION and B(p; q)isa BETA FUNCTION (Tu and Fischbach 2000). The first few are P1(s) /C301 R /C28s 2R (4) P2(s) /C304s pR2cos/C281s 2R ! /C282s2 pR3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28s2 4R2s (5) P3(s) /C303s2 R3 /C289s3 4R4 /C273s5 16R6 (6) P4(s) /C308s3 pR4cos/C281s 2R ! /C288s4 3pR5 /C2 1 /C28s2 4R2 !3=2 /C284s4 pR5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C281s2 4R2s : (7) The average lengths are given by ¯s1 /C302R 3 (8) ¯s2 /C30128R 45 p (9) ¯s3 /C3036R 35 (10) ¯s4 /C3016384 R 4725p: (11) See also BALL POINT PICKING ,SPHERE LINE PICKINGReferences Kendall, M. G. and Moran, P. A. P. Geometrical Probability. New York: Hafner, 1963. Santalo ´,L.A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Tu, S.-J. and Fischbach, E. A New Geometric Probability Technique for an N.-Dimensional Sphere and Its Applica- tions 17 Apr 2000. http://xxx.lanl.gov/abs/math-ph/ 0004021/. Ball Point Picking See also BALL LINE PICKING ,DISK POINT PICKING , NOISE SPHERE ,SPHERE POINT PICKING Ball Tetrahedron Picking The mean volume of a TETRAHEDRON formed by four random points in a UNIT SPHERE is¯V /C3012p=715 (Hostinsky 1925; Solomon 1978, p. 124). See also SPHERE TETRAHEDRON PICKING References Hostinsky, B. "Sur les probabilite ´sg e´ome´triques." Publ. Fac. Sci. Univ. Masaryk , No. 50. Brno, Czechoslovakia, 1925. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978. Ball Triangle Picking The determination of the probability for obtaining an OBTUSE TRIANGLE by picking three points at random in the unit DISK was generalized by Hall (1982) to the n-dimensional BALL . Buchta (1986) subsequently gave closed form evaluations for Hall’s integrals. Let Pnbe the probability that that three points chosen independently and uniformly from the n-BALL form an ACUTE TRIANGLE , then P2m/C271 /C30/C281 2 /C2822m/C2812m m/CP8/CP9 4m 2m/CP8/CP9 4m m/CP8/CP9 6m /C27 1 2m/CP8/CP9 /C27m2m m/CP8/CP92 22m /C29Xm k /C3002k k/CP8/CP9 2m /C27 k m/CP8/CP9 4m /C27 2k 2m /C27 k/CP8/CP9 /C293m /C27 k /C27 1 (m /C27 k)(3m /C27 2k /C27 1) (1) P2m/C272 /C3014 /C283 22m/C2744m /C27 4 m /C27 1/CP8/CP9 2m /C27 2 m /C27 1/CP8/CP9 /C2724m 2m m/CP8/CP9 p2 /C21 (2m /C27 1)2 2m m/CP8/CP92 664 /C27Xm k /C30022k(3m /C27 k /C28 3) (2k /C27 1)2k k/CP8/CP9 2m /C27 k m/CP8/CP9 2m /C27 k /C27 2 m/CP8/CP9/C2P ; (2) the first few being P2 /C304 p2 /C281 8 :0 :280285 (3) P3 /C3033 70 :0:471429 (4) P4 /C30256 45p2 /C271 32 :0 :607655 (5) P5 /C301415 2002 :0:706793 (6) P6 /C302048 315p2 /C2731 256 :0:779842 (7) P7 /C30231161 277134 :0:834113 (8) P8 /C304194304 606375 p2 /C2789 512 :0:874668 (9) P9 /C309615369 10623470 :0:905106 : (10) The case P2corresponds to DISK TRIANGLE PICKING case. See also CUBE TRIANGLE PICKING ,OBTUSE TRIANGLE , SPHERE POINT PICKINGReferences Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653 /C1/59, 1986. Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob. 19, 712 /C1/15, 1982. Ballantine BORROMEAN RINGS Ballieu’s Theorem Let the CHARACTERISTIC POLYNOMIAL of an /n /C29n/ COMPLEX MATRIX A bewrittenintheform P(l) /C30½l1 /C28A ½/C30 l n/C27b1 l n/C281/C27b2 l n/C282/C27.../C27bn/C281 l /C27bn: Then for any set m /C30( m1 ; m2 ; ...; mn)of POSITIVE numbers with m0 /C300 and M /C28/C30 max 05k 5n/C281mk /C27 mn ½bn/C28k ½ mk /C271; all the EIGENVALUES li(for i /C301, ..., n) lie on the CLOSED DISK ½z½5M /C28 in the COMPLEX PLANE . References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1153, 2000. Ballot Problem Suppose A and B are candidates for office and there are 2n voters, n voting for A and n for B. In how many ways can the ballots be counted so that A is always ahead of or tied with B? The solution is a CATALAN NUMBER Cn :/ A related problem also called "the" ballot problem is to let A receive a votes and Bbvotes with a /C21b. This version of the ballot problem then asks for the probability that A stays ahead of B as the votes are counted (Vardi 1991). The solution is (a /C28b) =(a /C27b); as first shown by M. Bertrand (Hilton and Pedersen 1991). Another elegant solution was provided by Andre ´ (1887) using the so-called ANDRE ´ ’S REFLECTION METHOD . The problem can also be generalized (Hilton and Pedersen 1991). Furthermore, the TAK FUNCTION is connected with the ballot problem (Vardi 1991). See also ANDRE ´ ’S REFLECTION METHOD ,C ATALAN NUMBER ,STAIRCASE WALK, TAK FUNCTION References Andre ´, D. "Solution directe du proble `me re ´solu par M. Ber- trand." Comptes Rendus Acad. Sci. Paris 105, 436/C1/37, 1887. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 49, 1987. Carlitz, L. "Solution of Certain Recurrences." SIAM J. Appl. Math. 17, 251/C1/59, 1969. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 22, 1974. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 67 /C1/7, 1968. Hilton, P. and Pedersen, J. "The Ballot Problem and Catalan Numbers." Nieuw Archief voor Wiskunde 8, 209 /C1/16, 1990. Hilton, P. and Pedersen, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13,64/C1/5, 1991. Kraitchik, M. "The Ballot-Box Problem." §6.13 in Mathema- tical Recreations. New York: W. W. Norton, p. 132, 1942. Motzkin, T. "Relations Between Hypersurface Cross Ratios, and a Combinatorial Formula for Partitions of a Polygon, for Permanent Preponderance, and for Non-Associative Products." Bull. Amer. Math. Soc. 54, 352 /C1/60, 1948. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 185 /C1/87, 1991. Balthasart Projection A CYLINDRICAL EQUAL-AREA PROJECTION which uses a standard parallel of fs /C3050 /C14:/ See also CYLINDRICAL EQUAL- AREA PROJECTION , BEHRMANN CYLINDRICAL EQUAL- AREA PROJECTION , GALL ORTHOGRAPHIC PROJECTION ,L AMBERT AZI- MUTHAL EQUAL- AREA PROJECTION ,PETERS PROJEC- TION ,TRISTAN EDWARDS PROJECTION Banach Algebra A Banach algebra is an ALGEBRA B over a FIELD F endowed with a NORM /C215kk such that B is a BANACH SPACE under the norm /C215kk and multiplication is continuous in the sense that if x; y /C23 B then xykk5 xkk ykk: Continuity of multiplication is the most important property. F is frequently taken to be the COMPLEX NUMBERS in order to assure that the SPECTRUM fully characterizes an OPERATOR (i.e., the spectral theorems for normal or compact normal operators do not, in general, hold in the SPECTRUM over the REAL NUMBERS ).If B has a unit, then x /C23 B is invertible if and only if ˆx( f) "0 for all f; where x /C2 ˆx is the GELFAND TRANS- FORM . See also B*-ALGEBRA ,B ANACH SPACE ,G ELFAND TRANSFORM References Helemskii, A. Ya. Banach and Locally Convex Algebras. Oxford, England: Oxford University Press, 1993. Katznelson, Y. An Introduction to Harmonic Analysis. New York: Dover, 1976. Rudin, W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, 1987. Banach Fixed Point Theorem Let f be a contraction mapping from a closed SUBSET F of a BANACH SPACE E into F. Then there exists a unique z /C23 F such that f(z) /C30z :/ See also FIXED POINT THEOREM References Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990. Banach Measure An " AREA " which can be defined for every set–even those without a true geometric AREA –which is rigid and finitely additive. Banach Space A Banach space is a COMPLETE VECTOR SPACE Bwith a norm vkk:Its topology is determined by its norm, and the vector space operations of addition and scalar multiplication are required to be continuous. Two norms /C142v/C1431and/C142v/C1432are called equivalent if they give the same TOPOLOGY , which is equivalent to the existence of constants candCsuch that c/C142v/C14315/C142v/C14325C/C142v/C1431 (1) holds for all v. In the finite dimensional case, all norms are equivalent. An infinite dimensional spacecan have many different norms. A basic example is ndimensional E UCLIDEAN SPACE with the Euclidean norm. Usually, the notion of Banach space is only used in the infinite dimensionalsetting, typically as a VECTOR SPACE of functions. For example, the set of continuous functions on the realline with the norm of a function fgiven by fkk/C30sup x/C23Rf(x)jj (2) is a Banach space, where sup denotes the SUPREMUM . On the other hand, the set of continuous functions onthe unit interval [0 ;1] with the norm of a function f given by fkk/C30g1 0f(x)jj dx (3) is not a Banach space because it is not complete. For instance, the CAUCHY SEQUENCE of functions fn1 for x 51=2 1 2n /C271 /C28nx for x 51=2 /C271=n 0 for x > 1=2 /C271=n8 < : (4) does not converge to a continuous function. HILBERT SPACES with their norm given by the inner product are examples of Banach spaces. While a HILBERT SPACE is always a Banach space, the con- verse need not hold. Therefore, it is possible for a Banach space not to have a norm given by an inner product. For instance, the supremum norm cannot be given by an INNER PRODUCT . See also BESOV SPACE ,COMPLETE SPACE ,H ILBERT SPACE ,SCHAUDER FIXED POINT THEOREM ,VECTOR SPACE Banach-Hausdorff-Tarski Paradox BANACH- TARSKI PARADOX Banach-Steinhaus Theorem UNIFORM BOUNDEDNESS PRINCIPLE Banach-Tarski Paradox First stated in 1924, the Banach-Tarski paradox states that it is possible to dissect a BALL into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by R. M. Robinson in 1944, although the pieces are extremely complicated. (Actually, four pieces are sufficient as long as the single point at the center is neglected.) A generalization of this theorem is that any two bodies in R3 which do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (i.e., they are EQUIDECOM- POSABLE ). See also BALL,CIRCLE SQUARING ,DISSECTION ,EQUI- DECOMPOSABLE References Banach, S. and Tarski, A. "Sur la de´composition des ensembles de points en parties respectivement con- gruentes." Fund. Math. 6, 244 /C177, 1924. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 16 /C17, 1998. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 48, 1984. Hertel, E. "On the Set-Theoretical Circle-Squaring Pro- blem." http://www.minet.uni-jena.de/Math-Net/reports/ sources/2000/00 /C16report.ps. Stromberg, K. "The Banach-Tarski Paradox." Amer. Math. Monthly 86, 3, 1979.Wagon, S. "A Hyperbolic Interpretation of the Banach- Tarski Paradox." Mathematica J. 3,58/C10, 1993. Wagon, S. The Banach-Tarski Paradox. New York: Cam- bridge University Press, 1993. Bandwidth The bandwidth of a MATRIX M//C30 (mij) isthemaximum valueof i /C28j jj suchthat mij isnonzero. The bandwidth of a GRAPH G is the minimum bandwidth among ADJACENCY MATRICES of GRAPHS isomorphic to G. Bounds for the bandwidth of a graph have been considered by (Harper 1964), and the bandwidth of the k-cube was determined by Harper (1966). References Chva´talova ´, J. "Optimal Labelling of a Product of Two Paths." Disc. Math. 11, 249 /C1/53, 1975. Harper, L. H. "Optimal Assignments of Numbers to Ver- tices." J. Soc. Indust. Appl. Math. 12, 131 /C1/35, 1964. Harper, L. H. "Optimal Numberings and Isoperimetric Problems on Graphs." J. Combin. Th. 1, 385 /C1/93, 1966. Bang’s Theorem The lines drawn to the VERTICES of a face of a TETRAHEDRON from the point of contact of the FACE with the INSPHERE form three ANGLES at the point of contact which are the same three ANGLES in each FACE . See also TETRAHEDRON References Altshiller-Court, N. §245 in Modern Pure Solid Geometry. New York: Chelsea, p. 74, 1979. Bang, A. S. Tidskrift f. Math. , p. 48, 1897. Brown, B. H. "Theorem of Bang. Isosceles Tetrahedra." Amer. Math. Monthly 33, 224/C1/26, 1926. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 93, 1976. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 13, 1991. White, H. S. "Two Tetrahedron Theorems." Nouvelles Ann. de Math 14, 220/C1/22, 1907 /C1/908. Bankoff Circle The circle through the cusp of the ARBELOS and the tangent points of the first Pappus circle, which is congruent to the two A RCHIMEDES’ CIRCLES .I fAB/C30r and AC /C301, then the radius of the Bankoff circle is R /C301 2r(1 /C28r) : See also ARCHIMEDES’ CIRCLES ,A RBELOS ,P APPUS CHAIN References Bankoff, L. "Are the Twin Circles of Archimedes Really Twins?" Math. Mag. 47, 214 /C1/18, 1974. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,18/C1/8, Jan. 1979. Banzhaf Power Index The number of ways in which a group of n with weights an i/C301 wi /C301 can change a losing coalition (one with a wi B1=2)) to a winning one, or vice versa. It was proposed by the lawyer J. F. Banzhaf in 1965. References Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, pp. 9 /C1/0, 1995. Bar A bar (also called an overbar) is a horizontal line written above a mathematical symbol to give it some special meaning. If the bar is placed over a single symbol, as in ¯x (voiced "x-bar"), it is sometimes called a MACRON . If placed over multiple symbols (especially in the context of a RADICAL ), it is known as a VINCULUM . Common uses of the bar symbol include the following. 1. The MEAN ¯x /C131 nXn i/C301xi of a set xifgn i/C301 :/ 2. The COMPLEX CONJUGATE ¯z /C13x /C28iy for z /C30x /C27iy:/ 3. The COMPLEMENT ¯F of a set F. 4. A SET stripped of any structure besides order, hence the ORDER TYPE of the set. In conventional typography, "bar" refers to a vertical (instead a horizontal) bar, such as those used to denote ABSOLUTE VALUE / xjjðÞ (Bringhurst 1997, p. 271). See also DOUBLE BAR,HAT,MACRON ,VINCULUM References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 271, 1997.Bar (Edge) The term in rigidity theory for the EDGES of a GRAPH . See also CONFIGURATION ,FRAMEWORK Bar Chart A bar graph is any plot of a set of data such that the number of data elements falling within one or more categories is indicated using a rectangle whose height or width is a function of the number of elements. See also HISTOGRAM ,PIE CHART References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 23, 1962. Bar Graph BARCHART Bar Graph Polygon A column-convex SELF-AVOIDING POLYGON which con- tains the bottom edge of its minimal bounding rectangle. The anisotropic perimeter and area gen- erating function G(x;y;q)/C30X m]1X n]1X a]aC(m;n;a)xmynqa; where C(m;n;a) is the number of polygons with 2 m horizonal bonds, 2 nvertical bonds, and area a, has been computed exactly for the bar graph polygons (Bousquet-Me ´lou 1996, Bousquet-Me ´louet al. 1999). The anisotropic area and perimeter generating func- tion G(x;y;q) and partial generating functions Hm(y; q) ; connected by G(x; y; q) /C30X m]1Hm(y; q)xm ; satisfy the self-reciprocity and inversion relations Hm(1=y; 1=q) /C30( /C281)m yqmHm(y; q) and G(x; y; q) /C28yG(/C28xq ; 1 =y; 1 =q) /C300 (Bousquet-Me ´lou et al. 1999). See also LATTICE POLYGON ,SELF-AVOIDING POLYGON References Bousquet-Me ´lou, M. "A Method for Enumeration of Various Classes of Column-Convex Polygons." Disc. Math. 154,1/C1/ 5, 1996. Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/ 9908123/. Bar Polyhex A POLYHEX consisting of HEXAGONS arranged along a line. See also BAR POLYIAMOND References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, p. 147, 1978. Bar Polyiamond A POLYIAMOND consisting of EQUILATERAL TRIANGLES arranged along a line. See also BAR POLYHEX References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Barber Paradox A man of Seville is shaved by the Barber of Seville IFF the man does not shave himself. Does the barbershave himself? This PSEUDOPARADOX was proposed by Bertrand Russell. See also PSEUDOPARADOX ,RUSSELL’S PARADOX References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 4 /C1/, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 17 /C1/8, 1998. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 116, 1998. Barbier’s Theorem All CURVES OF CONSTANT WIDTH of width w have the same PERIMETER pw :/ Bare Angle Center The TRIANGLE CENTER with TRIANGLE CENTER FUNC- TION a /C30A: References Kimberling, C. "Major Centers of Triangles." Amer. Math. Monthly 104, 431 /C1/38, 1997. Barlow Packing A face-centered cubic SPHERE PACKING obtained by placing layers of spheres one on top of another. Because there are two distinct ways to place each layer on top of the previous one, there are an infinite number of such packings as the number of layers is increased. See also KEPLER CONJECTURE ,SPHERE PACKING References Barlow, W. "Probable Nature of the Internal Symmetry of Crystals." Nature 29, 186/C1/88, 1883. Sloane, N. J. A. "Kepler’s Conjecture Confirmed." Nature 395, 435/C1/36, 1998. Barnes’ G-Function Barnes’ G-function is defined by G(z/C271) /C13(2p)z=2e/C28z(z/C271)/C27gz2½/C138 =2Y/C12 n/C3011/C27z n !n e/C28z/C27z2=(2n)"# (1) where gis the E ULER- MASCHERONI CONSTANT (Whit- taker and Watson 1990, p. 264; Voros 1987). It is an ENTIRE FUNCTION analogous to 1 =G(z);where G(z)i s the GAMMA FUNCTION , except that it has order 2 instead of 1. This is an ANALYTIC CONTINUATION of the G-function defined in the construction of the G LAISHER- KINKELIN CONSTANT G(n)/C13G(n) ½/C138n/C281 Kn; (2) where Kn/C1300112233/C1/C1/C1(n/C281)n/C281; (3) which has the special values G(n)/C300i f n/C300;/C281;/C282;... 1i f n/C301 0!1!2! /C1/C1/C1(n/C282)! if n/C302;3;4...8 < :(4) for INTEGER n. This function is what Sloane and Plouffe (1995) call the SUPERFACTORIAL , and the first few values for n/C301, 2, ... are 1, 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... (Sloane’s A000178).Barnes’ G-function satisfies the functional equation G(z/C271)/C30G(z)G(z); (5) and has the T AYLOR SERIES lnG(1/C27z)/C301 2ln(2p)/C281 ½/C138 z/C28(1/C27g)z2 2 /C27X/C12 n/C303(/C281)n/C281z(n/C281)zn n(6) inzjjB1:It also gives an analytic solution to the finite product Yn i/C301G(k/C27i)/C30G(n/C27k/C271) G(k/C271); (7) has the identities G(n) ½/C138n G(n)/C30K(n); (8) where K(n) is the K-FUNCTION , and the equivalent reflection formulas G0(z/C271) G(z/C271)/C301 2ln(2p)/C2812/C28z/C27zG0(z) G(z)(9) lnG(1/C28z) G(1/C27z)"# /C30pgz 0zcot(pz)dz/C28zln(2p) (10) G(12/C27z) (1 2/C28z)/C30(2p)2 G(12/C27z)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p cos(pz)s exppgz 0tan(pz)dz/C20/C2P (11) (Voros 1987; Whittaker and Watson 1990, p. 264). A Stirling-like ASYMPTOTIC SERIES asz0/C12is given by lnG(1/C27z)/C2z21 2lnz/C2834/CP6/CP7 /C2712ln(2p)z/C281 12lnz/C28lnA /C27O1 z ! (12) (Voros 1987). /G(n) has the special values G(12)/C30p/C281=4exp1 24ln 2/C2732z?(/C281)hi (13) /C30A/C283=2p/C281=4e1=821=24(14) G(32)/C30A/C283=2p1=4e1=821=24; (15) and so on, where z?(/C281) is the derivative of the RIEMANN ZETA FUNCTION evaluated at -1 and the GLAISHER- KINKELIN CONSTANT Ais defined by A/C30exp[1 12/C28z?(/C281)]/C301:28242712 . . . (16) (Voros 1987). Mathematica 4.0 implements the con- stant AasGlaisher . In general, for odd n/C302k/C271; G(1 2(2k /C271)) /C30ckA/C283 =2 p/C28(2k/C283)=4e1 =821 =24 2(k/C281)(k /C282)=2; (17) where ck /C30Yk /C282 i/C3012i G(1 2 /C27 i) ffiffiffipp (18) for k /C211, of which the first few terms are 1, 1, 1, 3, 45, 4725 4465125, ... (Sloane’s A057863). Barnes’ G-function can arise in spectral functions in mathematical physics (Voros 1987). Another G-FUNCTION is defined by Erde´lyi et al. (1981, p. 20) as G(z) /C13 c01 2 /C27hz/CP6/CP7 /C28 c0(12z) ; (19) where c0(z) is the DIGAMMA FUNCTION . An unrelated pair of functions are denoted gnand Gnand are known as RAMANUJAN G- AND G-FUNCTIONS . See also EULER- MASCHERONI CONSTANT , G-FUNC- TION ,G LAISHER- KINKELIN CONSTANT , K-FUNCTION , MEIJER’S G-FUNCTION ,RAMANUJAN G- AND G-FUNC- TIONS ,SUPERFACTORIAL References Barnes, E. W. "The Theory of the G-Function." Quart. J. Pure Appl. Math. 31, 264 /C1/14, 1900. Dyson, F. J. "Fredholm Determinants and Inverse Scatter- ing Problems." Commun. Math. Phys. 47, 171 /C1/83, 1976. Glaisher, J. W. L. "On a Numerical Continued Product." Messenger Math. 6,71/C1/6, 1877. Glaisher, J. W. L. "On the Product 112233 /C1/C1/C1nn :/" Messenger Math. 7,43/C1/7, 1878. Glaisher, J. W. L. "On Certain Numerical Products." Mes- senger Math. 23, 145 /C1/75, 1893. Glaisher, J. W. L. "On the Constant Which Occurs in the Formula for 112233 /C1/C1/C1nn :/" Messenger Math. 24,1/C1/6, 1894. Kinkelin. "U¨ ber eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integral- rechnung." J. reine angew. Math. 57, 122 /C1/58, 1860. Lenard, A. "Some Remarks on Large Toeplitz Matrices." Pacific J. Math. 42, 137 /C1/45, 1972. McCoy, B. and Wu, T. T. The Two-Dimensional Ising Model. Cambridge, MA: Harvard University Press, p. 264 and Appendix B, 1973. Sloane, N. J. A. Sequences A000178/M2049 and A057863 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439 /C1/ 65, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 264, 1990. Widom, H. "The Strong Szego Limit Theorem for Circular Arcs." Indiana Univ. Math. J. 21, 277 /C1/83, 1971. Widom, H. "Toeplitz Determinants with Singular Generat- ing Functions." Amer. J. Math. 95, 333 /C1/83, 1973.Barnes’ Lemma If a CONTOUR in the COMPLEX PLANE is curved such that it separates the increasing and decreasing sequences of POLES , then 1 2pi gi/C12 /C28i/C12G( a /C27s)G(b /C27s) G( g /C28s) G /C2( d /C28s) dsG( a /C27 g) G( a /C27 d) G( b /C27 g) G( b /C27 d) G( a /C27 b /C27 g /C27 d); where G(z) is the GAMMA FUNCTION (Bailey 1935, p. 7). Barnes’ second lemma states that g2 2piG(a1 /C27 s) G(a2 /C27 s) G( a3 /C27 s) G(1 /C28 b1 /C28 s) G( /C28s) ds G( b2 /C27 s) /C30G( a1)G( a2) G( a3)G(1 /C28 b1 /C27 a1) G(1 /C28 b1 /C27 a2) G(1 /C28 b1 /C27 a3) G( b2 /C28 a1) G( b2 /C28 a2)G( b2 /C28 a3) provided that b1 /C27 b2 /C30 a1 /C27 a2 /C27 a3 /C271 (Bailey 1935, pp. 42 /C1/3). References Bailey, W. N. "Barnes’ Lemma" and "Barnes’ Second Lemma." §1.7 and 6.2 in Generalised Hypergeometric Series. Cambridge, England: University Press, pp. 7 and 42 /C1/3, 1935. Barnes, E. W. "A New Development in the Theory of the Hypergeometric Functions." Proc. London Math. Soc. 6, 141 /C1/77, 1908. Barnes-Wall Lattice A lattice which can be constructed from the LEECH LATTICE A24 :/ See also COXETER- TODD LATTICE ,L ATTICE POINT , LEECH LATTICE References Barnes, E. S. and Wall, G. E. "Some Extreme Forms Defined in Terms of Abelian Groups." J. Austral. Math. Soc. 1,47/C1/ 3, 1959. Conway, J. H. and Sloane, N. J. A. "The 16-Dimensional Barnes-Wall Lattice A16 :/" §4.10 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 127 /C1/29, 1993. Barnette’s Conjecture The conjecture that every 3-connected BIPARTITE CUBIC PLANAR GRAPH is H AMILTONIAN . See also BIPARTITE GRAPH ,CUBIC GRAPH ,HAMILTO- NIAN GRAPH References Barnette, D. Conjecture 5 in Recent Progress in Combina- torics (Ed. W. T. Tutte). New York: Academic Press, 1969. Owens, P. J. "Bipartite Cubic Graphs and a Shortness Exponent." Disc. Math. 44, 327/C1/30, 1983. Barnsley’s Fern The ATTRACTOR of the ITERATED FUNCTION SYSTEM given by the set of "fern functions" f1(x; y) /C300:85 0:04 /C280:04 0:85/C20/C2P x y/C20/C2P /C270:00 1:60/C20/C2P (1) f2(x; y) /C30/C280:15 0:28 0:26 0:24/C20/C2P xy/C20/C2P /C270:00 0:44/C20/C2P (2) f 3(x; y) /C30 0:20 /C280:26 0:23 0:22/C20/C2P x y/C20/C2P /C270:00 1:60/C20/C2P (3) f4(x; y) /C30 0:00 0 :00 0:00 0 :16/C20/C2P xy/C20/C2P (4) (Barnsley 1993, p. 86; Wagon 1991). These AFFINE TRANSFORMATIONS are contractions. The tip of the fern (which resembles the black spleenwort variety of fern) is the fixed point of f1 ; and the tips of the lowest two branches are the images of the main tip under f2 andf3(Wagon 1991). See also DYNAMICAL SYSTEM ,F RACTAL ,ITERATED FUNCTION SYSTEM References Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, pp. 86, 90, 102 and Plate 2, 1993. Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. 238, 1988. Wagon, S. "Biasing the Chaos Game: Barnsley’s Fern." §5.3 inMathematica in Action. New York: W. H. Freeman, pp. 156 /C1/63, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Barrel ASOLID OF REVOLUTION composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane. For sides consisting of an arc of an ELLIPSE , the equation of the side is given by x(z)/C30r2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28(z/C281 2h)2 a2vuut; (1) with x(0)/C30r 1:Solving for agives a/C30hr2 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 2/C28r21p ; (2) so the sides have equation x(z)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 2/C27(r1/C28r2)(r1/C27r2)(h/C282z)2 h2s : (3) Using the equation for a SOLID OF REVOLUTION then gives V/C30pgh 0[x(z)]2dx/C301 3ph(2r2 2/C27r21): (4) For sides consisting of a PARABOLIC SEGMENT , the equation of the side is given by x(z)/C30r2/C27a(z/C281 2h)2(5) with x(0)/C30r1:Solving for agives a /C304(r1 /C28 r2) h2; (6) so the sides have equation x(z) /C30r2 /C27(r1 /C28 r2)(h /C28 2z)2 h2 : (7) Using the equation for a SOLID OF REVOLUTION then gives V /C30 pgh 0[x(z)]2dx /C301 15 ph(3r2 1 /C274r1r2 /C278r22) : (8) See also CYLINDER References Harris, J. W. and Stocker, H. "Barrel." §4.10.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998. Barrier A number n is called a barrier of a number-theoretic function f(m) if, for all m Bn, m /C27f(m) 5n: Neither the TOTIENT FUNCTION f(n) nor the DIVISOR FUNCTION s(n) has a barrier. Let U ⁄C be an OPEN SET and x0 /C23@U ; then a function b : ¯U 0 R is called a barrier for U at a point x0 if 1. b is continuous, 2. b is SUBHARMONIC on U, 3. b ½@U 50 ;/ 4. fz /C23@U : b(z) /C300g/C30fz0 g/ (Krantz 1999, pp. 100 /C1/01). See also SUBHARMONIC FUNCTION References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 64 /C1/5, 1994. Krantz, S. G. "The Concept of a Barrier." §7.7.9 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 100 /C1/ 01, 1999.Barth Decic The Barth decic is a DECIC SURFACE in complex three- dimensional projective space having the maximum possible number of ORDINARY DOUBLE POINTS (345). It is given by the implicit equation 8(x2 /C28 f4y2)(y2 /C28 f4z2)(z2 /C28 f4x2) /C29(x4 /C27y4 /C27z4 /C282x2y2 /C282x2z2 /C282y2z2) /C27(3 /C275f) /C2(x2 /C27y2 /C27z2 /C28w2)2[x2 /C27y2 /C27z2 /C28(2 /C28 f)w2]2w2 /C300; where f is the GOLDEN MEAN and w is a parameter (Endraß, Nordstrand), taken as w /C301 in the above plot. The Barth decic is invariant under the ICOSAHE- DRAL GROUP . See also ALGEBRAIC SURFACE ,BARTH SEXTIC ,DECIC SURFACE ,ORDINARY DOUBLE POINT References Barth, W. "Two Projective Surfaces with Many Nodes Admitting the Symmetries of the Icosahedron." J. Alg. Geom. 5, 173/C1/86, 1996. Endraß, S. "Fla ¨chen mit vielen Doppelpunkten." DMV- Mitteilungen 4,1 7/C1/0, 4/1995. Endraß, S. "Barth’s Decic." http://enriques.mathematik.uni- mainz.de/kon/docs/Ebarthdecic.shtml. Nordstrand, T. "Batch Decic." http://www.uib.no/people/ nfytn/bdectxt.htm. Barth Sextic The Barth-sextic is a SEXTIC SURFACE in complex three-dimensional projective space having the max- imum possible number of ORDINARY DOUBLE POINTS (65). Of these, 20 nodes are at the vertices of a regular DODECAHEDRON of side length 2 =f; and 30 are at the midpoints of the edges of a concentric DODECAHEDRON of side length 2=f2 ; where f is the GOLDEN RATIO . The surface was discovered by W. Barth in 1994, and is given by the implicit equation 4(f2x2 /C28y2)( f2y2 /C28z2)( f2z2 /C28x2) /C28(1 /C272f) /C2(x2 /C27y2 /C27z2 /C28w2)2w2 /C300; where f is the GOLDEN MEAN , and w is a parameter (Endraß, Nordstrand), taken as w /C301 in the above plot. The Barth sextic is invariant under the ICOSAHEDRAL GROUP . Under the map (x;y;z;w)0(x2;y2;z2;w2); the surface is the eightfold cover of the C AYLEY CUBIC (Endraß). See also ALGEBRAIC SURFACE ,BARTH DECIC,CAYLEY CUBIC ,ORDINARY DOUBLE POINT ,SEXTIC SURFACE References Barth, W. "Two Projective Surfaces with Many Nodes Admitting the Symmetries of the Icosahedron." J. Alg. Geom. 5, 173/C1/86, 1996. Dominici, P. "Flight Through Barth’s Sextic." http:// www.mi.uni-erlangen.de/~bauerth/flight/. Endraß, S. "Fla ¨chen mit vielen Doppelpunkten." DMV- Mitteilungen 4,1 7/C1/0, 4/1995. Endraß, S. "Barth’s Sextic." http://enriques.mathematik.uni- mainz.de/kon/docs/Ebarthsextic.shtml. Knapp, A. W. (Ed.). Notices Amer. Math. Soc. 46, cover and p. 318, 1999. Nordstrand, T. "Barth Sextic." http://www.uib.no/people/ nfytn/sexttxt.htm. Bartlett Function The APODIZATION FUNCTION f(x)/C301/C28xjj a(1) which is a generalization of the one-argument TRIAN- GLE FUNCTION . Its FULL WIDTH AT HALF MAXIMUM isa.It has INSTRUMENT FUNCTION I(x)/C30ga /C28ae/C282pikx1/C28xjj a ! dx /C30g0 /C28ae/C282pikx1/C27x a ! dx /C27ga 0e/C282pikx1/C28x a ! dx: (2) Letting x?/C13/C28xin the first part therefore gives g0 /C28ae/C282pikx1/C27x a ! dx/C30g0 ae/C282pikx?1/C28x? a ! (/C28dx?) /C30ga 0e/C282pikx1/C28x a ! dx: (3) Rewriting (2) using (3) gives I(x)/C30(e2pikx/C27e/C282pikx)1/C28x a ! dx /C302ga 0cos(2 pkx)1/C28x a ! dx: (4) Integrating the first part and using the integral gxcos(bx)dx/C301 b2cos(bx)/C27x bsin(bx) (5) for the second part gives I(x)/C302sin(2pkx) 2pk/C281 a1 4p2k2cos(2 pkx)/C27x 2pksin(2pkx)() "#a 0 /C302sin(2pka) 2pk/C280"# /C281 acos(2 pka)/C281 4p2k2/C27asin(2pka) 2pk"# () /C301 2p2ak2[cos(2 pka)/C281]/C30asin2(pka) p2k2a2/C30asinc2(pka) (6) where sinc xis the SINC FUNCTION . The peak (in units ofa) is 1. The function I(x) is always positive, so there are no NEGATIVE sidelobes. The extrema are given by letting b/C13pkaand solving d dbsinb b !2 /C302sinb bsinb/C28bcosb b2/C300 (7) sinb(sinb/C28bcosb)/C300 (8) sinb/C28bcosb/C300 (9) tanb/C30b: (10) Solving this numerically gives b/C304:49341 for the first maximum, and the peak POSITIVE sidelobe is 0.047190. The full width at half maximum is given by setting x /C13 pka and solving sinc2 x /C301 2 (11) for x1 =2 ; yielding x1 =2 /C30 pk1=2a /C301:39156 : (12) Therefore, with L /C132a; FWHM /C302k1 =2 /C300:885895 a/C301:77179 L: (13) See also APODIZATION FUNCTION ,PARZEN APODIZA- TION FUNCTION ,TRIANGLE FUNCTION References Bartlett, M. S. "Periodogram Analysis and Continuous Spectra." Biometrika 37,1/C1/6, 1950. Blackman, R. B. and Tukey, J. W. The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 98 /C1/9, 1959. Barycentric Coordinates Barycentric coordinates are triples of numbers (t1;t2;t3) corresponding to masses placed at the vertices of a reference triangle DA1A2A3:These masses then determine a point P, which is the centroid of the three masses, and is identified with coordinates ( t1;t2;t3):The vertices of the triangle are given by (1 ;0;0);(0;1;0);and (0 ;0;1):Barycentric coordinates were discovered by Mo ¨bius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993). To find the barycentric coordinates for an arbitrary point P, find t2and t3from the point Qat the intersection of the line A1Pwith the side A2A3;and then determine t1as the mass at Anthat will balance a mass t2/C27t3atQ, thus making Pthe centroid (left figure). Furthermore, the areas of the triangles DA1A2P,DA1A3P, and DA2A3Pare proportional to the barycentric coordinates t3;t2;and t1ofP(right figure; Coxeter 1969, p. 217). Barycentric coordinates are homogeneous, so (t1;t2;t3)/C30(mt1;mt2;mt3) (1) form"0. Barycentric coordinates normalized so that they become the actual areas of the subtriangles are called homogeneous barycentric coordinates, andbarycentric coordinates normalized so that t1/C27t2/C27t3/C301; (2) so that the coordinates give the areas of the sub- triangles normalized by the area of the original triangle are called AREAL COORDINATES (Coxeter 1969, p. 218). Barycentric and areal coordinates can provide particular elegant proofs of geometric theo- rems such as R OUTH’S THEOREM ,CEVA’S THEOREM , and M ENELAUS’ THEOREM (Coxeter 1969, pp. 219 /C1/21). The homogeneous barycentric coordinates corre- sponding to TRILINEAR COORDINATES /a:b:g/are / (aa;bb;cg)/, and the TRILINEAR COORDINATES corre- sponding to homogeneous barycentric coordinates(t 1;t2;t3) are /t1=a:t2=b:t3=c/. The homogeneous barycentric coordinates for some common trianglecenters are summarized in the following table, where / s/C30(a/C27b/C27c)=2/is the SEMIPERIMETER . triangle center homogeneous barycentric coordinates CENTROID (TRIANGLE )(1, 1, 1) CIRCUMCENTER (a2(b2/C27c2/C28a2),b2(c2/C27a2/C28b2),c3(a2/C27b2/C28c2)) EXCENTERS (/C28a,b,c) (a,/C28b,c) (a,b,/C28c) GERGONNE POINT ((s/C28b)(s/C28c), (s/C28c)(s/C28a), (s/C28a)(s/C28b)) INCENTER (a,b,c) NAGEL POINT (s/C28a,s/C28b,s/C28c) ORTHOCENTER /((a2/C27b2/C28c2)(c2/C27a2/C28b2);(b2/C27c2/C28a2)(a2/C27b2/C28c2));/ /(c2/C27a2/C28b2)(b2/C27c2/C28a2))/ SYMMEDIAN POINT /(a2;b2;c2)/ In barycentric coordinates, a line has a linear homo-geneous equation. In particular, the line joiningpoints ( r 1;r2;r3) and ( s1;s2;s3) has equation jr1r2r3 s1s2s3 t1t2t3j(3) (Loney 1962, pp. 39 and 57; Coxeter 1969, p. 219;Bottema 1982). If the vertices P iof a triangle DP1P2P3 have barycentric coordinates ( xi;yi;zi);then the area of the triangle is DP1P2P3/C30jx1y1z1 x2y2z2 x3y3z3jDABC (4) (Bottema 1982, Yiu 2000). See also AREAL COORDINATES ,TRILINEAR COORDI- NATES References Bottema, O. "On the Area of a Triangle in Barycentric Coordinates." Crux. Math. 8, 228 /C1/31, 1982. Coxeter, H. S. M. "Barycentric Coordinates." §13.7 in Intro- duction to Geometry, 2nd ed. New York: Wiley, pp. 216 /C1/ 21, 1969. Fauvel, J.; Flood, R.; and Wilson, R. J. (Eds.) Mo¨bius and his Band: Mathematics and Astronomy in Nineteenth-Century Germany. Oxford, England: Oxford University Press, 1993. Loney, S. L. The Elements of Coordinate Geometry, 2 vols. in 1. Part II: Trilinear Coordinates. London: Macmillan, 1962. Yiu, P. "The Uses of Homogeneous Barycentric Coordinates in Plane Euclidean Geometry." Int. J. Educ. Math. Sci. Tech. 2000. Base (Logarithm) The number used to define the number system in which a LOGARITHM is computed. In general, the logarithm of a number x in base b is written logb x: The symbol log x is an abbreviation regrettably used both for the COMMON LOGARITHM log10 x (by engineers and physicists and indicated on pocket calculators) and for the NATURAL LOGARITHM loge x (by mathema- ticians). ln x denotes the NATURAL LOGARITHM loge x (as used by engineers and physicists and indicated on pocket calculators), and lg x denotes log2 x: In this work, the notations log x /C30log10 x and ln x /C30loge x are used. To convert between logarithms in different bases, the formula logb x /C30ln x ln b can be used. See also COMMON LOGARITHM , E,LG,LN,LOGARITHM , NAPIERIAN LOGARITHM ,NATURAL LOGARITHM ,BASE (NUMBER ) Base (Neighborhood System) A base for a neighborhood system of a point x is a collection N of OPEN SETS such that x belongs to every member of N, and any OPEN SET containing x also contains a member of N as a SUBSET . Base (Number) A REAL NUMBER x can be represented using any INTEGER number b as a base (sometimes also called a RADIX or SCALE ). The choice of a base yields to a representation of numbers known as a NUMBER SYSTEM . In base b, the DIGITS 0, 1, ..., b /C281 are used (where, by convention, for bases larger than 10, the symbols A, B, C, ...are generally used as symbols representing the DECIMAL numbers 10, 11, 12, ...).Base Name 2 BINARY 3 TERNARY 4 QUATERNARY 5 Quinary 6 Senary 7 Septenary 8 OCTAL 9 Nonary 10 DECIMAL 11 Undenary 12 DUODECIMAL 16 HEXADECIMAL 20 VIGESIMAL 60 SEXAGESIMAL Let the base b representation of a number x be written (anan/C281 ...a0 :a/C281 ...)b ; (1) (e.g., 123:45610) ; then the index of the leading DIGIT needed to represent the number is n /C13 logb x bc ; (2) where xbcis the FLOOR FUNCTION . Now, recursively compute the successive DIGITS ai /C30ri bi$% ; (3) where rn /C13x and ri/C281 /C30ri /C28aibi (4) for i /C30n, n /C28 1; ..., 1, 0, .... This gives the base b representation of x. Note that if x is an INTEGER , then i need only run through 0, and that if x has a fractional part, then the expansion may or may not terminate. For example, the HEXADECIMAL represen- tation of 0.1 (which terminates in DECIMAL notation) is the infinite expression 0:19999...h/. Some number systems use a mixture of bases for counting. Examples include the Mayan calendar and the old British monetary system (in which ha’pen- nies, pennies, threepence, sixpence, shillings, half crowns, pounds, and guineas corresponded to units of1/2, 1, 3, 6, 12, 30, 240, and 252, respectively). Knuth (1998) has considered using TRANSCENDENTAL bases. This leads to some rather unfamiliar results, such as equating pto 1 in "base p;/"p/C3010p/. See also BINARY ,D ECIMAL ,D UODECIMAL ,H EREDI- TARY REPRESENTATION ,HEXADECIMAL ,OCTAL ,QUA- TERNARY ,SEXAGESIMAL ,TERNARY ,VIGESIMAL References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 28, 1972. Bogomolny, A. "Base Converter." http://www.cut-the-knot.- com/binary.html. Knuth, D. E. "Positional Number Systems." §4.1 in The Art of Computer Programming, Vol. 2: Seminumerical Algo- rithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 195 /C1/ 13, 1998. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 6 /C1/1, 1991. Weisstein, E. W. "Bases." MATHEMATICA NOTEBOOK BASES.M . Base Curve DIRECTRIX (RULED SURFACE ) Base Manifold The base manifold in a BUNDLE is analogous to the domain for a set of functions. In fact, a bundle, by definition, comes with a map to the base manifold, often called p or projection. For example, the base manifold to the TANGENT BUNDLE of a MANIFOLD M is the MANIFOLD M.A VECTOR FIELD is a function from the manifold to the TANGENT BUNDLE , with the restriction that every point gets mapped to a vector at that point. In general, a BUNDLE has SECTIONS , at least locally, which are maps from the base manifold to the BUNDLE . See also BUNDLE ,M ANIFOLD ,S ECTION (BUNDLE ), TANGENT BUNDLE ,VECTOR BUNDLE Base Space The SPACE B of a FIBER BUNDLE given by the MAP f : E 0 B; where E is the TOTAL SPACE of the FIBER BUNDLE . See also FIBER BUNDLE ,TOTAL SPACE Baseball The numbers three and four appear prominently in the game of baseball. There are three strikes for an out, and three outs per inning, 3 /C215 3 /C309 innings in a game, giving 33 /C3027 outs per game (assuming no extra innings). In addition, there are 3 /C293 players per team. Four balls are needed for a walk. The number of bases can either be regarded as three (excluding HOME PLATE ) or four (including it). See also BASEBALL COVER ,HOME PLATEBaseball Cover A pair of identical plane regions (mirror symmetric about two perpendicular lines through the center) which can be stitched together to form a baseball (or tennis ball). A baseball has a CIRCUMFERENCE of 9 1/8 inches. The practical consideration of separating the regions far enough to allow the pitcher a good grip requires that the "neck" distance be about 1 3/16 inches. The baseball cover was invented by Elias Drake as a boy in the 1840s. (Thompson’s attribution of the current design to trial and error development by C. H. Jackson in the 1860s is apparently unsub- stantiated, as discovered by George Bart.) One way to produce a baseball cover is to draw the regions on a SPHERE , then cut them out. However, it is difficult to produce two identical regions in this manner. Thompson (1996) gives mathematical ex- pressions giving baseball cover curves both in the plane and in 3-D. J. H. Conway has humorously proposed the following "baseball curve conjecture:" no two definitions of "the" baseball curve will give the same answer unless their equivalence was obvious from the start. See also BASEBALL ,H OME PLATE ,T ENNIS BALL THEOREM ,YIN-YANG References Thompson, R. B. "Designing a Baseball Cover. 1860’s: Patience, Trial, and Error. 1990’s: Geometry, Calculus, and Computation." http://www.mathsoft.com/asolve/base- ball/baseball.html. Rev. March 5, 1996. Basepoint See also LOOP Basic Polynomial Sequence APOLYNOMIAL SEQUENCE pn(x) is called the basic polynomial sequence for a DELTA OPERATOR Qif 1. p0(x) /C301;/ 2. pn(0) /C300 for all n /C210, 3. Qpn(x) /C30npn /C281(x):/ If pn(x) is a basic polynomial sequence for some DELTA OPERATOR Q, then it is a BINOMIAL-TYPE SEQUENCE of polynomials. Furthermore, if pn(x)isa BINOMIAL-TYPE SEQUENCE of polynomials, then it is a basic polyno- mial sequence for some DELTA OPERATOR . See also BINOMIAL- TYPE SEQUENCE ,D ELTA OPERA- TOR,POLYNOMIAL SEQUENCE ,UMBRAL OPERATOR References Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Basin of Attraction The set of points in the space of system variables such that initial conditions chosen in this set dynamically evolve to a particular ATTRACTOR . See also WADA BASIN Basis The word basis can arise in several different contexts. Speaking in general terms, an object is "generated" by a basis in whatever manner is appropriate. For example, a VECTOR SPACE can have a BASIS which SPANS the vector space by finite LINEAR COMBINA- TIONS . See also BASIS POINT ,B ASIS (TOPOLOGY ), BASIS (VECTOR SPACE ), HAMEL BASIS,H ILBERT BASIS, ORTHONORMAL BASIS,VECTOR BASIS Basis (Topology) If X is a SET, a basis for a TOPOLOGY on X is a collection B of SUBSETS of X (called basis elements) satisfying the following properties. 1. For each x /C23 X ; there is at least one basis element B containing X. 2. If x belongs to the intersection of two basis elements B1 and B2 ; then there is a basis element B3 containing x such that B3 ƒB1 S B2/. References Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975. Basis (Vector Space) A basis of a VECTOR SPACE V is defined as a subset v1 ; ... ; vnof vectors in V that are LINEARLY INDE- PENDENT and SPAN V. Consequently, if (v1 ; v2 ; ...; vn) is a list of vectors in V, then these vectors form a basis if and only if every v /C23 V can beuniquely written as v /C30a1b1 /C27a2b2 /C27.../C27anbn ; where a1 ; ...; apare elements of R or C: A VECTOR SPACE V will have many different bases, but there are always the same number of basis vectors in each of them. The number of basis vectors in V is called the DIMENSION of V. Every spanning list in a vector space can be reduced to a basis of the vector space. The simplest example of a basis is the standard basis in Rn consisting of the coordinate axes. For example, in R2 ; the standard basis consists of two VECTORS e1 /C30 (1; 0) and e2 /C30(0; 1): Any VECTOR w /C30(a; b) can be written uniquely as the LINEAR COMBINATION / w /C30ae1 /C27be2/. Indeed, a vector is defined by its coordinates. The VECTORS v1 /C30(3; 2) and v2 /C30(2; 1) are also a basis for R2 because any VECTOR w /C30 (a ; b) can be uniquely written as w /C30 (/C28a /C27 2b)v1 /C27(2a /C28 3b)v2 : The above figure shows (0:6 ;/C280:5)n /C27 (0:9;:02)m; which are linear combinations of the basis f(0:6;/C280 :5); (0:9; 0:2)g:/ Here is a Mathematica function which will return the coefficients ai given a basis vi : LinearCombination[v_List?MatrixQ, w_] : /C30 LinearSolve[Transpose[v], w] For example, LinearCombo [{{1, 2}, {0, 1}}, {-3, 4}] yields f3;/C282g;since 3( /C281;2)/C282(0;1)/C30(/C283;4)/. When a VECTOR SPACE is infinite dimensional, then a basis exists, as long as one assumes the AXIOM OF CHOICE . A subset of the basis which is linearly independent and whose span is DENSE is called a complete set, and is similar to a basis. When Vis a HILBERT SPACE , a complete set is called a H ILBERT BASIS . See also BASIS,DIMENSION ,HILBERT BASIS,LINEAR COMBINATION ,ORTHONORMAL BASIS,SPAN (VECTOR SPACE ), VECTOR SPACE Basis Element A collection B of subsets of a set X forming a topological BASIS . See also BASIS (TOPOLOGY ) Basis Point One basis point is defined to be 0.01 PERCENTAGE POINTS . Therefore, a change of 0.21% could also be expressed as a change by 21 "basis points." See also PERCENTAGE POINT Basis Theorem HILBERT BASIS THEOREM Basler Problem The problem of analytically finding the value of z(2); where z(n) is the RIEMANN ZETA FUNCTION . See also APE´ RY’S CONSTANT ,RIEMANN ZETA FUNC- TION References Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67 /C1/8, 1988. Basset Function MODIFIED BESSEL FUNCTION OF THE SECOND KIND Bat CHEVRON Batch A set of values of similar meaning obtained in any manner. References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 667, 1977. Bateman Equation References Fairlie, D. B. and Leznov, A. N. The Complex Bateman Equation in a Space of Arbitrary Dimension. 16 Sep 1999. http://xxx.lanl.gov/abs/solv-int/9909013/. Bateman Function kn(x) /C13e/C28x G(1 /C271 2n) U(/C281 2n ; 0 ; 2x) for x /C210, where U is a CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND . See also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATION ,HYPERGEOMETRIC FUNCTIONReferences Bateman, H. "The k-Function, a Particular Case of the Confluent Hypergeometric Function." Trans. Amer. Math. Soc. 33, 817 /C1/31, 1931. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 179, 1998. Koepf, W. and Schmersau, D. "Bounded Nonvanishing Functions are Bateman Functions." Complex Variables 25, 237 /C1/59, 1994. Batrachion A class of CURVE defined at INTEGER values which hops from one value to another. Their name derives from the Greek word batraxi on batrachion , which means "small frog." Many batrachions are FRACTAL . Examples include the BLANCMANGE FUNCTION ,HOF- STADTER- CONWAY $10,000 SEQUENCE ,HOFSTADTER’S Q- SEQUENCE , and MALLOWS’ SEQUENCE . References Pickover, C. A. "The Crying of Fractal Batrachion 1,489." Ch. 25 in Keys to Infinity. New York: W. H. Freeman, pp. 183 /C1/91, 1995. Baudet’s Conjecture If C1 ; C2 ; ...; f > Cr are sets of positive integers and @r i /C301Ci /C30N ; where N is the set of positive integers, then some Ci contains arbitrarily long ARITHMETIC SEQUENCES . The conjecture was proved in 1928 by B. L. van der Waerden. See also ARITHMETIC SEQUENCE , VAN DER WAERDEN’S THEOREM References van der Waerden, B. L."How the Proof of Baudet’s Con- jecture Was Found." Studies in Pure Mathematics (Pre- sented to Richard Rado). London: Academic Press, pp. 251 /C1/60, 1971. Bauer’s Identical Congruence LetT(m) denote the set of the f(m) numbers less than and RELATIVELY PRIME tom, where f(n) is the TOTIENT FUNCTION . Define fm(x)/C30Y t/C23T(m)(x/C28t): (1) Then a theorem of Lagrange states that fp(x)/C13xf(p)/C281 (mod p) (2) forpanODD PRIME (Hardy and Wright 1979, p. 98). This can be generalized as follows. Let pbe an ODD PRIME DIVISOR ofmandpathe highest POWER which divides m, then fm(x)/C13(xp/C281/C281)f(m)=(p/C281)(mod pa) (3) and, in particular, fpa (x) /C13(xp/C281 /C281)pa /C281 (mod pa) : (4) Now, if m /C212is EVEN and 2a is the highest POWER of 2 that divides m, then fm(x) /C13(x2 /C281)f(m) =2 (mod 2a) (5) and, in particular, f2a (x) /C13(x2 /C281)2a/C282 (mod 2a): (6) See also CONGRUENCE ,LEUDESDORF THEOREM References Bauer. Nouvelles annales 2, 256 /C1/64, 1902. Hardy, G. H. and Wright, E. M. J. London Math. Soc. 9,38/C1/ 1 and 240, 1934. Hardy, G. H. and Wright, E. M. "Bauer’s Identical Congru- ence." §8.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 98 /C1/00, 1979. Bauer’s Theorem Let m ]3 be an integer and let f(x) /C30Xn k/C300akxn/C28k be an INTEGER POLYNOMIAL that has at least one real zero. Then f(x) has infinitely many PRIME DIVISORS that are not congruent to 1 (mod m) (Nagell 1951, p. 168). See also BAUER’S IDENTICAL CONGRUENCE ,P RIME DIVISOR References Nagell, T. "A Theorem of Bauer on the Prime Divisors of Certain Polynomials." §49 in Introduction to Number Theory. New York: Wiley, pp. 168 /C169, 1951. Bauer-Muir Transformation A transformation formula for CONTINUED FRACTIONS (Lorentzen and Waadeland 1992) which can, for example, be used to prove identities such as 1 1/C272/C27q 1 /C272/C27q2 1/C272 /C27 q3 1 /C27/C1/C1/C1/C301 2/C27q 2 /C27 q /C27q2 2 /C27 q2 /C27q3 2 /C27 q3 /C27/C1/C1/C1 (Berndt et al.). See also CONTINUED FRACTION References Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fractionin Ramanujan’s Lost Notebook." To appears in Trans. Amer. Math. Soc. Lorentzen, L. and Waadeland, H. Continued Fractions with Applications. Amsterdam, Netherlands: North-Holland, p. 76, 1992. Bauspiel A construction for the RHOMBIC DODECAHEDRON . References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 26 and 50, 1973. Baxter-Hickerson Function In April 1999, Ed Pegg conjectured onsci.math that there were only finitely many ZEROFREE cubes, to which D. Hickerson responded with a counterexam- ple. A few days later, Lew Baxter posted the slightly simpler example f(n) /C301 3(2 /C215 105n /C28104n /C272 /C215 103n /C27102n /C2710n /C271); which produces numbers whose cubes lack zeros. The first few terms for n /C300, 1, ... are 2, 64037, 6634003367, 666334000333667, ... (Sloane’s A052- 427). Primes occur for n /C300, 1, 7, 133, ... (Sloane’s A051832) with no others 5470 (Weisstein, Dec. 15, 1999), corresponding to 2, 64037, ... (Sloane’s A051833). See also NUMBER PATTERN ,ZEROFREE References Pegg, E. Jr. "Fun with Numbers." http://www.mathpuzzle.- com/numbers.html. Sloane, N. J. A. Sequences A051832, A051833, and A052427 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Bayes’ Formula BAYES’ THEOREM Bayes’ Theorem Let Aand Bjbe SETS.C ONDITIONAL PROBABILITY requires that PASBj/C0/CP /C30P(A)P(Bj½A); (1) whereSdenotes INTERSECTION ("and"), and also that PASBj/C0/CP /C30PBjSA/C0/CP /C30P(Bj)P(A½Bj): (2) Therefore, P(Bj½A)/C30P(Bj)P(A½Bj) P(A): (3) Now, let S/C13@N i/C301Ai; (4) so Ai is an event in S and Ai S Aj /C30¥ for i "j; then A /C30A S S /C30A S@N i/C301Ai/CP8/CP9 /C30@N i/C301A S Ai ðÞ (5) P(A) /C30P @N i /C301A S Ai ðÞ/CP8/CP9 /C30XN i/C301PAS Ai ðÞ : (6) But this can be written P(A) /C30XN i/C301P(Ai)P(A½Ai); (7) so P(Ai ½A) /C30P(Ai)P(A½Ai) XN j/C301P(Aj)P(A½Aj)(8) (Papoulis 1984, pp. 38 /C1/9). See also CONDITIONAL PROBABILITY ,INCLUSION- EX- CLUSION PRINCIPLE ,INDEPENDENT STATISTICS ,TOTAL PROBABILITY THEOREM References Papoulis, A. "Bayes’ Theorem in Statistics" and "Bayes’ Theorem in Statistics (Reexamined)." §3 /C1/ and 4 /C1/ in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 38 /C1/9, 78 /C1/1, and 112 /C1/14, 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 810, 1992. Bayesian Analysis A statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution. Begin with a "PRIOR DIS- TRIBUTION " which may be based on anything, includ- ing an assessment of the relative likelihoods of parameters or the results of non-Bayesian observa- tions. In practice, it is common to assume a UNIFORM DISTRIBUTION over the appropriate range of values for the PRIOR DISTRIBUTION . Given the PRIOR DISTRIBUTION , collect data to obtain the observed distribution. Then calculate the LIKE- LIHOOD of the observed distribution as a function of parameter values, multiply this likelihood function by the PRIOR DISTRIBUTION , and normalize to obtain a unit probability over all possible values. This is called the POSTERIOR DISTRIBUTION . The MODE of the dis- tribution is then the parameter estimate, and "prob- ability intervals" (the Bayesian analog of CONFIDENCE INTERVALS ) can be calculated using the standard procedure. Bayesian analysis is somewhat controver- sial because the validity of the result depends on how valid the PRIOR DISTRIBUTION is, and this cannot be assessed statistically.See also MAXIMUM LIKELIHOOD ,PRIOR DISTRIBUTION , UNIFORM DISTRIBUTION References Gelman, A.; Carlin, J.; Stern, H.; and Rubin, D. Bayesian Data Analysis. Boca Raton, FL: Chapman & Hall, 1995. Hoel, P. G.; Port, S. C.; and Stone, C. J. Introduction to Statistical Theory. New York: Houghton Mifflin, pp. 36 /C1/2, 1971. Iversen, G. R. Bayesian Statistical Inference. Thousand Oaks, CA: Sage Pub., 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 799 /C1/06, 1992. Sivia, D. S. Data Analysis: A Bayesian Tutorial. New York: Oxford University Press, 1996. Bays’ Shuffle A shuffling algorithm used in a class of RANDOM NUMBER generators. References Knuth, D. E. §3.2 and 3.3 in The Art of Computer Program- ming, Vol. 2: Seminumerical Algorithms, 2nd ed. Read- ing, MA: Addison-Wesley, 1981. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 270 /C1/71, 1992. Beal’s Conjecture A generalization of FERMAT’S LAST THEOREM which states that if ax /C27by /C30cz ; where a, b, c, x, y, and z are POSITIVE INTEGERS and x;y;z>2;then a,b, and c have a common factor. The conjecture was announced in Mauldin (1997), and a cash prize of $75,000 hasbeen offered for its proof or a counterexample. See also ABC CONJECTURE ,FERMAT’S LAST THEOREM References Brun, V. "U ¨ber hypothesesenbildungen." Arc. Math. Nat- urvidenskab 34,1/C1/4, 1914. Darmon, H. and Granville, A. "On the Equations zm/C30F(x;y) andAxp/C27Byq/C30cZr:/"Bull. London Math. Soc. 27, 513/C1/43, 1995. Mauldin, R. D. "A Generalization of Fermat’s Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer. Math. Soc. 44, 1436 /C1/437, 1997. Mauldin, R. D. "The Beal Conjecture and Prize." http:// www.math.unt.edu/~mauldin/beal.html. Beam Detector N.B. A detailed online essay by S. Finch was the starting point for this entry. A "beam detector" for a given curve Cis defined as a curve (or set of curves) through which every LINE tangent to or intersecting Cpasses. The shortest 1- arc beam detector, illustrated in the upper left figure, has length L1/C30p/C272:The shortest known 2-arc beam detector, illustrated in the right figure, has angles u1:1:286 rad (1) u2:1:191 rad ; (2) given by solving the simultaneous equations 2 cos u1/C28sin(1 2u2)/C300 (3) tan(12u1)cos(12u2)/C27sin(12u2)[sec2(12u2)/C271]/C302: (4) The corresponding length is L2/C302p/C282u1/C28u2/C272 tan1 2u1/CP6/CP7 /C27sec12u2/CP6/CP7 /C28cos12u2/CP6/CP7 /C27tan12u1/CP6/CP7 sin12u2/CP6/CP7 /C304:8189264563 . . . : (5) A more complicated expression gives the shortest known 3-arc length L3/C304:799891547 . . . /. Finch de- fines L/C30inf n]1Ln (6) as the beam detection constant, or the TRENCH DIGGERS’ CONSTANT . It is known that L]p:/ References Croft, H. T.; Falconer, K. J.; and Guy, R. K. §A30 in Unsolved Problems in Geometry. New York: Springer- Verlag, 1991. Faber, V.; Mycielski, J.; and Pedersen, P. "On the Shortest Curve which Meets All Lines which Meet a Circle." Ann. Polon. Math. 44, 249/C1/66, 1984. Faber, V. and Mycielski, J. "The Shortest Curve that Meets All Lines that Meet a Convex Body." Amer. Math. Monthly 93, 796/C1/01, 1986. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/beam/beam.html. Makai, E. "On a Dual of Tarski’s Plank Problem." In Diskrete Geometrie. 2 Kolloq., Inst. Math. Univ. Salzburg, 127 /C1/32, 1980. Stewart, I. "The Great Drain Robbery." Sci. Amer. 273, 206/C1/ 07, Sep. 1995. Stewart, I. Sci. Amer. 273, 106, Dec. 1995. Stewart, I. Sci. Amer. 274, 125, Feb. 1996.Bean Curve The PLANE CURVE given by the Cartesian equation x4/C27x2y2/C27y4/C30x(x2/C27y2): References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Beast Number The occult "number of the beast" associated in the Bible with the Antichrist. It has figured in many numerological studies. It is mentioned in Revelation 13:18: "Here is wisdom. Let him that hath under-standing count the number of the beast: for it is thenumber of a man; and his number is 666." The origin of this number is not entirely clear, although it may be as simple as the number containing the concatena-tion of one symbol of each type (exclude M/C301000) in R OMAN NUMERALS :DCLXVI /C30666 (Wells 1986). The first few numbers containing the beast number intheir digits are 666, 1666, 2666, 3666, 4666, 5666,6660, ...(Sloane’s A051003). The beast number has several interesting properties which numerologists may find particularly interest- ing (Keith 1982 /C1 /3). In particular, the beast number is equal to the sum of the squares of the first 7 PRIMES 22/C2732/C2752/C2772/C27112/C27132/C27172/C30666; (1) satisfies the identity f(666)/C306/C2156/C2156; (2) where fis the TOTIENT FUNCTION , as well as the sum X6 /C215 6 i /C301i /C30666 (3) which is the sum of numbers on a roulette wheel (Emanouilidis 1998). Emanouilidis (1998) also gives additional more obscure connections between 666 and the numbers on a roulette wheel. The number 666 is a sum and difference of the first three 6th POWERS , 666 /C3016 /C2826 /C2736 (4) (Keith). Another curious identity is that there are exactly two ways to insert " /C27" signs into the sequence 123456789 to make the sum 666, and exactly one way for the sequence 987654321, 666 /C30 1 /C27 2 /C27 3 /C27 4 /C27 567 /C27 89 /C30 123 /C27 456 /C27 78 /C27 9 (5) 666 /C30 9 /C27 87 /C27 6 /C27 543 /C27 21 (6) (Keith). 666 is a REPDIGIT , and is also a TRIANGULAR NUMBER T6 /C215 6 /C30T36 /C30666: (7) In fact, it is the largest REPDIGIT TRIANGULAR NUMBER (Bellew and Weger 1975 /C1/6). 666 is also a SMITH NUMBER . The first 144 DIGITS of p /C283; where p is PI, add to 666. In addition 144 /C30(6 /C276) /C29(6 /C276) (Blatner 1997). Finally, X5 i /C3002048i /C13691 (mod 666) : (8) A number OF THE FORM 2i which contains the digits of the beast number "666" is called an APOCALYPTIC NUMBER , and a number having 666 digits is called an APOCALYPSE NUMBER . See also APOCALYPSE NUMBER ,APOCALYPTIC NUM- BER,BIMONSTER ,MONSTER GROUP ,ROMAN NUMERAL References Bellew, D. W. and Weger, R. C. "Repdigit Triangular Num- bers." J. Recr. Math. 8,96/C1/7, 1975 /C1/6. Blatner, D. The Joy of Pi. New York: Walker, back jacket, 1997. Castellanos, D. "The Ubiquitous p:/" Math. Mag. 61, 153 /C1/54, 1988. Eco, U. Foucault’s Pendulum. San Diego: Harcourt Brace Jovanovich, p. 31, 1989. Emanouilidis, E. "Roulette and the Beastly Number." J. Recr. Math. 29, 246 /C1/47, 1998. Gardner, M. "Mathematical Games: A Fanciful Dialogue About the Wonders of Numerology." Sci. Amer. 202, 150 /C1/ 56, Feb. 1960. Hardy, G. H. A Mathematician’s Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 96, 1993. Keith, M. "The Number of the Beast." http://member.aol.- com/s6sj7gt/mike666.htm. Keith, M. "The Number 666." J. Recr. Math. 15,85/C1/7, 1982 /C1/983.Sloane, N. J. A. Sequences A051003 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. Beatty Sequence The Beatty sequence is a SPECTRUM SEQUENCE with an IRRATIONAL base. In other words, the Beatty sequence corresponding to an IRRATIONAL NUMBER u is given by ubc; 2 ubc ; 3 ubc ; ..., where xbcis the FLOOR FUNCTION .If a and b are POSITIVE IRRATIONAL NUMBERS such that 1 a /C271 b /C301; then the Beatty sequences abc; 2abc ; ... and bbc; 2bbc ; ... together contain all the POSITIVE INTEGERS without repetition. The sequences for particular values of a and b are given in the following table (Sprague 1963; Wells 1986, pp. 35 and 40), where f is the GOLDEN RATIO . parameter Sloane sequence /a /C30ffiffiffi 2p / A001951 1, 2, 4, 5, 7, 8, 9, 11, 12, ... /b /C302 /C27ffiffiffi 2p / A001952 3, 6, 10, 13, 17, 20, 23, 27, 30, ... /a /C30ffiffiffi3p / A022838 1, 3, 5, 6, 8, 10, 12, 13, 15, 17, ... /b /C301 2(3 /C27ffiffiffi 3p )/ A054406 2, 4, 7, 9, 11, 14, 16, 18, 21, 23, 26, ... /a/C30e/ A022843 2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, ... /b/C30e=(e/C281)/A054385 1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, ... /a/C30p/ A022844 3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, ... /b/C30p=(p/C281)/A054386 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, ... /a/C30f/ A000201 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, ... /b/C30f2/ A001950 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, ... See also FRACTIONAL PART,W YTHOFF ARRAY , WYTHOFF’S GAME References Gardner, M. Penrose Tiles and Trapdoor Ciphers...and the Return of Dr. Matrix, reissue ed. New York: W. H. Free- man, p. 21, 1989. Graham, R. L.; Lin, S.; and Lin, C.-S. "Spectra of Numbers." Math. Mag. 51, 174/C176, 1978. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 227, 1994. Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, pp. 29 /C10, 1973. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 18, 1995. Sprague, R. Recreations in Mathematics: Some Novel Puz- zles. London: Blackie and Sons, 1963. Sloane, N. J. A. Sequences A000201/M2322, A001950/ M1332, A001951/M0955, A001952/M2534, A022838, A022843, A022844, A054406, A054385, and A054386 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences /eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 35, 1986. Beauzamy and De´got’s Identity For P, Q, R, and S POLYNOMIALS in n variables [P /C215 Q ; R /C215 S] /C30X i1 ; ... ; in ]0A i1! /C1/C1/C1in! ; where A /C13[R(i1 ; ... ; in)(D1 ; ...; Dn)Q(x1 ; ...; xn) /C2P(i1 ; ... ; in)(D1 ; ...; Dn)S(x1 ; ...; xn)]; /Di /C30@=@xiis the DIFFERENTIAL OPERATOR ,[X, Y]is the BOMBIERI INNER PRODUCT , and P(i1 ; ... ; in) /C30Di1 1/C1/C1/C1Din n P : See also REZNIK’S IDENTITY Bed-of-Nails Function SHAH FUNCTION Bee A4- POLYHEX . References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, p. 147, 1978. Behrens-Fisher Test FISHER- BEHRENS PROBLEMBehrmann Cylindrical Equal-Area Projection A CYLINDRICAL EQUAL-AREA PROJECTION which uses a standard parallel of fs/C3030/C14:/ See also BALTHASART PROJECTION ,C YLINDRICAL EQUAL- AREA PROJECTION ,EQUAL- AREA PROJECTION , GALL ORTHOGRAPHIC PROJECTION ,L AMBERT AZI- MUTHAL EQUAL- AREA PROJECTION ,PETERS PROJEC- TION ,TRISTAN EDWARDS PROJECTION References Dana, P. H. "Map Projections." http://www.colorado.edu/ geography/gcraft/notes/mapproj/mapproj_f.html. Bei The IMAGINARY PART of Jn(xe3pi=4)/C30bern(x)/C27ibein(x): (1) The function bein(x) has the series expansion bein(x)/C30(1 2x)nX/C12 k/C300sin[(34n/C2712k)p] k!G(n/C27k/C271)(14x2)k; (2) where G(x) is the GAMMA FUNCTION (Abramowitz and Stegun 1972, p. 379). The special case n /C300 gives J0iffiffi ip x/CP6/CP7 /C13ber(x) /C27i bei(x) ; (3) where J0(x) is the zeroth order BESSEL FUNCTION OF THE FIRST KIND . The function bei0(x) /C13bei(x) has the series expansion bei(x) /C13X/C12 n/C300( /C281)n(1 2x)2/C274n [(2n/C271)!]2: (4) See also BER,BESSEL FUNCTION ,KEI,KELVIN FUNC- TIONS ,KER References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 379 /C1/81, 1972. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions bern(x);bein(x);kern(x) and kein(x):/"§1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29 /C1/0, 1990. Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 543 /C1/54, 1987. Bell Curve GAUSSIAN DISTRIBUTION ,NORMAL DISTRIBUTION Bell Number The number of ways a SET ofnelements can be PARTITIONED into nonempty SUBSETS is called a B ELL NUMBER and is denoted Bn:For example, there are five ways the numbers f1;2;3gcan be partitioned:{{1},{2},{3}}, {{1, 2},{3}}, {{1, 3},{2}}, {{1}, {2, 3}}, and {{1, 2, 3}}, so B3/C305:B0/C301 and the first few Bell numbers for n/C301, 2, . . . are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, . . . (Sloane’s A000110). Bell numbers are closely related to C ATALAN NUM- BERS . The diagram above shows the constructions giving B3/C305 and B4/C3015;with line segments repre- senting elements in the same SUBSET and dots representing subsets containing a single element(Dickau). The INTEGERS Bncan be defined by the sum Bn/C30Xn k/C301S(n;k); (1) where S(n;k)i saS TIRLING NUMBER OF THE SECOND KIND , i.e., as the S TIRLING TRANSFORM of the sequence 1, 1, 1, . . . The Bell number are given by the EXPONENTIAL GENERATING FUNCTION een/C281/C30X/C12 n/C300Bn n!xn: (2) The Bell numbers can also be generated using the BELL TRIANGLE , using the RECURRENCE RELATION Bn/C271/C30Xn k/C300Bkn k/CP8/CP9 ; (3) wherea b/C0/CP is a BINOMIAL COEFFICIENT , or using the formula of Comtet (1974) Bn/C30e/C281X2n m/C301mn m!&’ ; (4) where xdedenotes the CEILING FUNCTION . The Bell number Bnis also equal to fn(1);where fn(x) is an EXPONENTIAL POLYNOMIAL .DOBINSKI’S FORMULA gives the nth Bell number Bn/C301 eX/C12 k/C300kn k!: (5) Lova´sz (1993) showed that this formula gives the asymptotic limit Bn/C2n/C281=2[l(n)]n/C271=2el(n)/C28n/C281; (6) where l(n) is defined implicitly by the equation l(n) log[ l(n)] /C30n: (7) A variation of DOBINSKI’S FORMULA gives Bn /C30Xn k /C301kn k!Xn/C28k j/C300( /C281)j j! (8) (Pitman 1997). de Bruijn (1958) gave the asymptotic formula ln Bn n/C30ln n /C28ln ln n /C281 /C27ln ln n ln n/C271 ln n /C271 2ln ln n ln n !2 /C27Oln ln n (ln n)2"# (9) TOUCHARD’S CONGRUENCE states Bp/C27k /C13Bk /C27Bk /C271 (mod p) ; (10) when p is PRIME . The only PRIME Bell numbers for n 51000 are B2 ; B3 ; B7 ; B13 ; B42 ; and B55 : The Bell numbers also have the curious property that B0 B1 B2 /C1/C1/C1 Bn B1 B2 B3 /C1/C1/C1 Bn/C271 nn n::: n BnBn/C271Bn/C272/C1/C1/C1 B2n/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/C30Y n i/C301i! (11) (Lenard 1986), where the product is simply a SUPER- FACTORIAL , the first few of which for n /C300, 1, 2, ... are 1, 1, 2, 12, 288, 34560, 24883200, ... (Sloane’s A000178). See also BELL TRIANGLE ,DOBINSKI’S FORMULA ,EX- PONENTIAL POLYNOMIAL ,STIRLING NUMBER OF THE SECOND KIND,TOUCHARD’S CONGRUENCE References Bell, E. T. "Exponential Numbers." Amer. Math. Monthly 41, 411 /C1/19, 1934. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 91 /C1/4, 1996. de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, pp. 102 /C1/09, 1958. Dickau, R. M. "Bell Number Diagrams." http://forum.s- warthmore.edu/advanced/robertd/bell.html. Dickau, R. "Visualizing Combinatorial Enumeration." Math- ematica in Educ. Res. 8,11/C1/8, 1999. Gardner, M. "The Tinkly Temple Bells." Ch. 2 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 24 /C1/8, 1992. Gould, H. W. Bell & Catalan Numbers: Research Bibliogra- phy of Two Special Number Sequences, 6th ed. Morgan- town, WV: Math Monongliae, 1985. Lenard, A. In Fractal Music, Hypercards, and More Math- ematical Recreations from Scientific American Magazine. (M. Gardner). New York: W. H. Freeman, pp. 35 /C1/6, 1992. Levine, J. and Dalton, R. E. "Minimum Periods, Modulo p,of First Order Bell Exponential Integrals." Math. Comput. 16, 416 /C1/23, 1962.Lova´sz, L. Combinatorial Problems and Exercises, 2nd ed. Amsterdam, Netherlands: North-Holland, 1993. Pitman, J. "Some Probabilistic Aspects of Set Partitions." Amer. Math. Monthly 104, 201 /C1/09, 1997. Rota, G.-C. "The Number of Partitions of a Set." Amer. Math. Monthly 71, 498 /C1/04, 1964. Sloane, N. J. A. Sequences A000110/M1484 and A000178/ M2049 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Bell Polynomial The Bell polynomial are defined by Bn; k(x1 ; x2 ; ...)/C30X j1 /C27j2 /C27/C1/C1/C1/C30k j1 /C272j2 /C27/C1/C1/C1/C30nn! j1!j2! /C1/C1/C1x1 1! !j1x2 2! !j2 /C1/C1/C1: They have GENERATING FUNCTION X/C12 k /C300bk(x; x1 ; x2 ; ...) k!tk /C30exX/C12 k /C301xk k!tk ! : See also EXPONEN TIAL POLYNOMIAL ,IDEMPOTENT NUMBER ,LAH NUMBER References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 133, 1974. Roman, S. "The Bell Polynomials." §4.1.8 in The Umbral Calculus. New York: Academic Press, pp. 82 /C1/6, 1984. Bell Triangle A triangle of numbers which allow the BELL NUMBERS to be computed using the RECURRENCE RELATION Bn/C271/C30Xn k/C300Bkn k/CP8/CP9 : See also BELL NUMBER ,CLARK’S TRIANGLE ,LEIBNIZ HARMONIC TRIANGLE ,LOSSNITSCH’S TRIANGLE ,NUM- BER TRIANGLE ,PASCAL’S TRIANGLE ,SEIDEL- ENTRIN- GER-ARNOLD TRIANGLE Bellows Conjecture The conjecture proposed by Dennis Sullivan that all FLEXIBLE POLYHEDRA keep a constant VOLUME as they are flexed (Cromwell 1997). This conjecture was proven by Connelly et al. (1997). See also FLEXIBLE POLYHEDRON References Connelly, R.; Sabitov, I.; and Walz, A. "The Bellows Conjecture." Contrib. Algebra Geom. 38,1/C1/0, 1997. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 245 and 247, 1997. Mackenzie, D. "Polyhedra Can Bend But Not Breathe." Science 279, 1637, 1998. Beltrami Differential Equation For a MEASURABLE FUNCTION m; the Beltrami differ- ential equation is given by f˜z /C30 mfz ; where fzis a PARTIAL DERIVATIVE and ˜z denotes the COMPLEX CONJUGATE of z. See also QUASICONFORMAL MAP References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1087, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Beltrami Field A VECTOR FIELD u satisfying the vector identity u /C29(9 /C29u) /C300 where A /C29B is the CROSS PRODUCT and 9/C29A is the CURL is said to be a Beltrami field. See also DIVERGENCELESS FIELD ,IRROTATIONAL FIELD,SOLENOIDAL FIELD Beltrami Identity An identity in CALCULUS OF VARIATIONS discovered in 1868 by Beltrami. The EULER- LAGRANGE DIFFEREN- TIAL EQUATION is @f @y /C28d dx@f @yx ! /C300: (1) Now, examine the DERIVATIVE of f with respect to x df dx /C30@f @yyx /C27@f @yxyxx /C27@f @x : (2) Solving for the @f/@y term gives @f @yyx /C30df dx /C28@f @yxyxx /C28@f @x : (3) Now, multiplying (1) by yx givesyx@f @y /C28yxd dx@f @yx ! /C300: (4) Substituting (3) into (4) then gives df dx /C28@f @yxyxx /C28@f @x /C28yxd dx@f @yx ! /C300 (5) /C28@f @x /C27d dxf /C28yx@f @yx ! /C300 : (6) This form is especially useful if fx /C300, since in that case d dxf /C28yx@f @yx ! /C300 ; (7) which immediately gives f /C28yx@f @yx/C30C ; (8) where C is a constant of integration (Weinstock 1974, pp. 24 /C1/5; Arfken 1985, pp. 928 /C1/29; Fox 1988, pp. 8 /C1/). The Beltrami identity greatly simplifies the solution for the minimal AREA SURFACE OF REVOLUTION about a given axis between two specified points. It also allows straightforward solution of the BRACHISTO- CHRONE PROBLEM . See also BRACHISTOCHRONE PROBLEM ,CALCULUS OF VARIATIONS ,EULER- LAGRANGE DIFFERENTIAL EQUA- TION ,SURFACE OF REVOLUTION References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985. Fox, C. An Introduction to the Calculus of Variations. New York: Dover, 1988. Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974. Beltrami’s Theorem Let f : M 0 N be a GEODESIC MAPPING . If either M or N has constant curvature, then both surfaces have constant curvature (Ambartzumian 1982, p. 26; Kreyszig 1991). See also GEODESIC MAPPING References Ambartzumian, R. V. Combinatorial Integral Geometry. Chichester, England: Wiley, 1982. Kreyszig, E. §91 in Differential Geometry. New York: Dover, 1991. Bend (Curvature) The bend of a circle Cmutually tangent to three other circles is defined as the signed CURVATURE ofC. If the contacts are all external, the signs of the bends of all four circles are taken as POSITIVE , whereas if one circle surrounds the other three, the sign of this circle is taken as NEGATIVE (Coxeter 1969). Bends can also be defined for spheres. See also CURVATURE ,DESCARTES CIRCLE THEOREM , SODDY CIRCLES References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 13 /C1/4, 1969. Bend (Knot) AKNOT used to join the ends of two ropes together to form a longer length. References Owen, P. Knots. Philadelphia, PA: Courage, p. 49, 1993. Benford’s Law A phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford’s law states that in listings, tables of statis-tics, etc., the DIGIT 1 tends to occur with PROBABILITY ~30%, much greater than the expected 10% (i.e., one digit out of 10). Benford’s law can be observed, for instance, by examining tables of LOGARITHMS and noting that the first pages are much more worn and smudged than later pages (Newcomb 1881). While Benford’s law unquestionably applies to many situa-tions in the real world, a satisfactory explanation has been given only recently through the work of Hill (1996). Benford’s law applies to data that are notdimension- less, so the numerical values of the data depend on the units. If there exists a universal probability distribution P(x) over such numbers, then it must be invariant under a change of scale, so P(kx)/C30f(k)P(x): (1) IffP(x)dx/C301, then fP(kx)dx/C301/k, and normal- ization implies /f(k)/C301=k/. Differentiating with re- spect to kand setting k/C301 gives xP?(x)/C30/C28P(x); (2) having solution /P(x)/C301=x/. Although this is not a proper probability distribution (since it diverges),both the laws of physics and human convention impose cutoffs. For example, if street addresses are distributed uniformly over the range of 1 to somemaximum cutoff value, then they’ll obey something close to Benford’s law. If many powers of 10 lie between the cutoffs, then the probability that the first (decimal) digit is Dis given by the LOGARITHMIC DISTRIBUTION PD/C30gD/C271 DP(x)dx g10 1P(x)dx/C30lnD/C271 D ! ln 10/C30ln(D/C271)/C28ln(D) ln 10(3) forD/C301, . . ., 9, illustrated above and tabulated below. DP D DP D 1 0.30103 6 0.0669468 2 0.176091 7 0.0579919 3 0.124939 8 0.05115254 0.09691 9 0.04575755 0.0791812 However, Benford’s law applies not only to scale- invariant data, but also to numbers chosen from a variety of different sources. Explaining this fact requires a more rigorous investigation of CENTRAL LIMIT -like theorems for the MANTISSAS of random variables under MULTIPLICATION . As the number of variables increases, the density function approachesthat of a LOGARITHMIC DISTRIBUTION . Hill (1996) rigorously demonstrated that the "distribution ofdistributions" given by random samples taken froma variety of different distributions is, in fact, Ben- ford’s law (Matthews 1999). One striking example of Benford’s law is given by the 54 million real constants in Plouffe’s "Inverse Sym-bolic Calculator" database, 30% of which begin with the DIGIT 1. Taking data from several disparate sources, the table below, shows the distribution of first digits as compiles by Benford (1938) in his original paper. First Digit Col. Title 1 2 3 4 5 6 7 8 9 Samples A Rivers, Area 31.0 16.4 10.7 11.3 7.2 8.6 5.5 4.2 5.1 335 B Population 33.9 20.4 14.2 8.1 7.2 6.2 4.1 3.7 2.2 3259C Constants 41.3 14.4 4.8 8.6 10.6 5.8 1.0 2.9 10.6 104D Newspapers 30.0 18.0 12.0 10.0 8.0 6.0 6.0 5.0 5.0 100 E Specific Heat 24.0 18.4 16.2 14.6 10.6 4.1 3.2 4.8 4.1 1389 F Pressure 29.6 18.3 12.8 9.8 8.3 6.4 5.7 4.4 4.7 703G H.P. Lost 30.0 18.4 11.9 10.8 8.1 7.0 5.1 5.1 3.6 690 H Mol. Wgt. 26.7 25.2 15.4 10.8 6.7 5.1 4.1 2.8 3.2 1800 I Drainage 27.1 23.9 13.8 12.6 8.2 5.0 5.0 2.5 1.9 159J Atomic Wgt. 47.2 18.7 5.5 4.4 6.6 4.4 3.3 4.4 5.5 91 K /n/C281;ffiffiffinp / 25.7 20.3 9.7 6.8 6.6 6.8 7.2 8.0 8.9 5000 L Design 26.8 14.8 14.3 7.5 8.3 8.4 7.0 7.3 5.6 560 M Reader’s Digest33.4 18.5 12.4 7.5 7.1 6.5 5.5 4.9 4.2 308 N Cost Data 32.4 18.8 10.1 10.1 9.8 5.5 4.7 5.5 3.1 741 O X-Ray Volts 27.9 17.5 14.4 9.0 8.1 7.4 5.1 5.8 4.8 707 P Am. League 32.7 17.6 12.6 9.8 7.4 6.4 4.9 5.6 3.0 1458Q Blackbody 31.0 17.3 14.1 8.7 6.6 7.0 5.2 4.7 5.4 1165 R Addresses 28.9 19.2 12.6 8.8 8.5 6.4 5.6 5.0 5.0 342 S /n1;n2/C1/C1/C1n!/25.3 16.0 12.0 10.0 8.5 8.8 6.8 7.1 5.5 900 T Death Rate 27.0 18.6 15.7 9.4 6.7 6.5 7.2 4.8 4.1 418 Average 30.6 18.5 12.4 9.4 8.0 6.4 5.1 4.9 4.7 1011 Probable Error9 0.89 0.49 0.49 0.39 0.29 0.29 0.29 0.3 The following table gives the distribution of the first digit of the mantissa following Benford’s Law using a number of different methods. method Sloane sequence Sainte-Lague A055439 1, 2, 3, 1, 4, 5, 6, 1, 2, 7, 8, 9, ... d’Hondt A055440 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 3, 1, ... largest remainder, Hare quotasA055441 1, 2, 3, 4, 1, 5, 6, 7, 1, 2, 8, 1, ... largest remainder,Droop quotasA055442 1, 2, 3, 1, 4, 5, 6, 1, 2, 7, 8, 1, ... References Barlow, J. L. and Bareiss, E. H. "On Roundoff Error Dis- tributions in Floating Point and Logarithmic Arithmetic." Computing 34, 325/C1/47, 1985. Benford, F. "The Law of Anomalous Numbers." Proc. Amer. Phil. Soc. 78, 551/C1/72, 1938. Bogomolny, A. "Benford’s Law and Zipf’s Law." http:// www.cut-the-knot.com/do_you_know/zipfLaw.html. Boyle, J. "An Application of Fourier Series to the Most Significant Digit Problem." Amer. Math. Monthly 101, 879/C1/86, 1994.Flehinger, B. J. "On the Probability that a Random Integer Has Initial Digit A."Amer. Math. Monthly 73, 1056 /C1/061, 1966. Franel, J. Naturforschende Gesellschaft, Vierteljahrsschrift (Zu¨rich) 62, 286/C1/95, 1917. Hill, T. P. "Base-Invariance Implies Benford’s Law." Proc. Amer. Math. Soc. 12, 887/C1/95, 1995. Hill, T. P. "The Significant-Digit Phenomenon." Amer. Math. Monthly 102, 322/C1/27, 1995. Hill, T. P. "A Statistical Derivation of the Significant-Digit Law." Stat. Sci. 10, 354/C1/63, 1996. Hill, T. P. "The First Digit Phenomenon." Amer. Sci. 86, 358/C1/63, 1998. Knuth, D. E. "The Fraction Parts." §4.2.4B in The Art of Computer Programming, Vol. 2: Seminumerical Algo- rithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 254 /C1/ 62, 1998. Ley, E. "On the Peculiar Distribution of the U.S. Stock Indices Digits." Amer. Stat. 50, 311/C1/13, 1996. Matthews, R. "The Power of One." http://www.newscientist.- com/ns/19990710/thepowerof.html. Newcomb, S. "Note on the Frequency of the Use of Digits in Natural Numbers." Amer. J. Math. 4,3 9/C1/0, 1881. Nigrini, M. "A Taxpayer Compliance Application of Ben- ford’s Law." J. Amer. Tax. Assoc. 18,7 2/C1/1, 1996. Nigrini, M. "I’ve Got Your Number." J. Accountancy , pp. 79 /C1/3, May 1999. Plouffe, S. "Graph of the Number of Entries in Plouffe’s Inverter." http://www.lacim.uqam.ca/plouffe/statis- tics.html. Raimi, R. A. "The Peculiar Distribution of First Digits." Sci. Amer. 221, 109/C1/19, Dec. 1969. Raimi, R. A. "On the Distribution of First Significant Digits." Amer. Math. Monthly 76, 342/C1/48, 1969. Raimi, R. A. "The First Digit Phenomenon." Amer. Math. Monthly 83, 521/C1/38, 1976. Schatte, P. "Zur Verteilung der Mantisse in der Gleitkom- madarstellung einer Zufallsgro ¨ße." Z. Angew. Math. Mech. 53, 553/C1/65, 1973. Schatte, P. "On Mantissa Distributions in Computing and Benford’s Law." J. Inform. Process. Cybernet. 24, 443/C1/55, 1988. Sloane, N. J. A. Sequences A055439, A055440, A055441, and A055442 in "An On-Line Version of the Encyclopediaof Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Benham’s Wheel An optical ILLUSION consisting of a spinnable top marked in black with the pattern shown above. When the wheel is spun (especially slowly), the black broken lines appear as green, blue, and red colored bands! References Cohen, J. and Gordon, D. A. "The Prevost-Fechner-Benham Subjective Colors." Psycholog. Bull. 46,97/C1/36, 1949. Festinger, L.; Allyn, M. R.; and White, C. W. "The Percep- tion of Color with Achromatic Stimulation." Vision Res. 11, 591 /C1/12, 1971. Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 148 /C1/51, 1996. Trolland, T. L. "The Enigma of Color Vision." Amer. J. Physiology 2,23/C1/8, 1921. Benjamin-Bona-Mahony Equation The PARTIAL DIFFERENTIAL EQUATION ut /C28uxxx /C27uux /C300 (Arvin and Goldstein 1985; Zwillinger 1997, p. 130). A generalized version is given by ut /C2892ut /C27}( f(u)) /C300 (Goldstein and Wichnoski 1980; Zwillinger 1997, p. 132). References Arvin, J. and Goldstein, J. A. "Global Existence for the Benjamin-Bona-Mahony Equation in Arbitrary Dimen- sions." Nonlinear Anal. 9, 861 /C1/65, 1985. Goldstein, J. A. and Wichnoski, B. J. "On the Benjamin- Bona-Mahony Equation in Higher Dimensions." Non- linear Anal. 4, 665 /C1/75, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 130 and 132, 1997. Bennequin’s Conjecture A BRAID with M strands and R components with P positive crossings and N negative crossings satisfies ½P /C28N ½52U /C27M /C28R 5P /C27N ; where U is the UNKNOTTING NUMBER . While the second part of the INEQUALITY was already known to be true (Boileau and Weber, 1983, 1984) at the time the conjecture was proposed, the proof of the entire conjecture was completed using results of Kronhei- mer and Mrowka on MILNOR’S CONJECTURE (and, independently, using MENASCO’S THEOREM ). See also BRAID ,M ENASCO’S THEOREM ,M ILNOR’S CONJECTURE ,UNKNOTTING NUMBER References Bennequin, D. "L’instanton gordien (d’apre `s P. B. Kronhei- mer et T. S. Mrowka)." Aste´risque 216, 233 /C1/77, 1993. Birman, J. S. and Menasco, W. W. "Studying Links via Closed Braids. II. On a Theorem of Bennequin." Topology Appl. 40,71/C1/2, 1991. Boileau, M. and Weber, C. "Le proble `me de J. Milnor sur le nombre gordien des n//uds alge´briques." Enseign. Math. 30, 173 /C1/22, 1984.Boileau, M. and Weber, C. "Le proble `me de J. Milnor sur le nombre gordien des n//uds alge´briques." In Knots, Braids and Singularities (Plans-sur-Bex, 1982). Geneva, Switzer- land: Monograph. Enseign. Math. Vol. 31, pp. 49 /C1/8, 1983. Cipra, B. What’s Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Soc., pp. 8 /C1/3, 1994. Kronheimer, P. B. "The Genus-Minimizing Property of Algebraic Curves." Bull. Amer. Math. Soc. 29,63/C1/9, 1993. Kronheimer, P. B. and Mrowka, T. S. "Gauge Theory for Embedded Surfaces. I." Topology 32, 773 /C1/26, 1993. Kronheimer, P. B. and Mrowka, T. S. "Recurrence Relations and Asymptotics for Four-Manifold Invariants." Bull. Amer. Math. Soc. 30, 215 /C1/21, 1994. Menasco, W. W. "The Bennequin-Milnor Unknotting Con- jectures." C. R. Acad. Sci. Paris Se´r. I Math. 318, 831 /C1/36, 1994. Benson’s Formula An equation for a LATTICE SUM with n /C303 /C28b3(1) /C30X ?/C12 i; j; k /C30/C28/C12( /C281)i/C27j/C27k/C271 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 /C27 j2 /C27 k2p /C3012pX/C12 m; n/C301 ; 3 ; ...sech2(1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 /C27n2p ) : Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is numerically equal to /C281:74756 . . . ;a value known as "the" M ADELUNG CONSTANT . See also MADELUNG CONSTANTS References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 301, 1987. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/mdlung/mdlung.html. Ber The REAL PART of Jn(xe3pi=4)/C30bern(x)/C27ibein(x): (1) The function bern(x) has the series expansion bern(x)/C30(1 2x)nX/C12 k/C300cos[(34n/C2712k)p] k!G(n/C27k/C271)(14x2)k; (2) where G(x) is the GAMMA FUNCTION (Abramowitz and Stegun 1972, p. 379). The special case n /C300 gives J0iffiffi ip x/CP6/CP7 /C13ber(x) /C27i bei(x); (3) where J0(x) is the zeroth order BESSEL FUNCTION OF THE FIRST KIND . The function ber0(x) /C13ber(x) has the series expansion ber(x) /C13X/C12 n/C300( /C281)n(1 2x)4n [(2n)!]2: (4) See also BEI,BESSEL FUNCTION ,KEI,KELVIN FUNC- TIONS ,KER References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 379 /C1/81, 1972. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions bern(x) ; bei n(x); kern(x) and kein(x):/" §1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29 /C1/0, 1990. Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 543 /C1/54, 1987. Beraha Constants The nth Beraha constant (or number) is given by B(n) /C132 /C272 cos2 p n ! : They appear to be ROOTS of the CHROMATIC POLY-NOMIALS of planar triangular GRAPHS . B(5) is f /C271; where f is the GOLDEN RATIO , B(7) is the SILVER CONSTANT , and B(10) /C30 f /C272: The following table summarizes the first few Beraha numbers. n /B(n)/ Approx. 14 203142 5 /1 2(3 /C27ffiffiffi 5p )/ 2.618 63 7 /2 /C272 cos(2 7 p)/ 3.247 8 /2 /C27ffiffiffi 2p / 3.414 9 /2 /C272 cos(2 9 p)/ 3.532 10 /1 2(5/C27ffiffiffi 5p )/3.618 See also CHROMATIC POLYNOMIAL ,G OLDEN RATIO, SILVER CONSTANT References Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 160 /C1/63, 1986. Tutte, W. T. "Chromials." University of Waterloo, 1971. Tutte, W. T. "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications. New York: Gordon and Breach, p. 439, 1969. Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case l/C301:/" Research Report COPR 72 /C1/, University of Waterloo, 1972a. Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case l/C30/C12:/" Research Report COPR 72 /C1/, University of Waterloo, 1972b. Berezin Transform The operator ˜Bdefined by ˜Bf(x)/C30gD(1/C28½z½2)2 ½1/C28z¯w½4f(w)dA(w) forz/C23D;where Dis the unit open disk and ¯wis the COMPLEX CONJUGATE (Hedenmalm et al. 2000, p. 29). References Hedenmalm, H.; Korenblum, B.; and Zhu, K. "The Berezin Transform." Ch. 2 in Theory of Bergman Spaces. New York: Springer-Verlag, pp. 28 /C1/1, 2000. Berge’s Theorem A MATCHING is maximal IFF it contains no AUGMENT- ING PATH . See also MATCHING References Berge, C. "Two Theorems in Graph Theory." Proc. Nat. Acad. Sci. USA 43, 842 /C1/44, 1957. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Berger-Kazdan Comparison Theorem Let M be a compact n-D MANIFOLD with INJECTIVITY radius inj(M): Then Vol(M) ]cn inj(M) p; with equality IFF M is ISOMETRIC to the standard round SPHERE Sn with RADIUS inj(M) ; where cn(r)is the VOLUME of the standard n-HYPERSPHERE of RADIUS r. See also BLASCHKE CONJECTURE ,H YPERSPHERE , INJECTIVE ,ISOMETRY References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Bergman Kernel A Bergman kernel is a function of a COMPLEX VARIABLE with the "reproducing kernel" property defined for any DOMAIN in which there exist NONZERO ANALYTIC FUNCTIONS of class l2(d) with respect to the LEBESGUE MEASURE dv. References HazewinKel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 356 /C1/57, 1988. Bergman Space Let G be an open subset of the COMPLEX PLANE C ; and let L2 a(G) denote the collection of all ANALYTIC FUNC- TIONS f : G 0 C whose MODULUS is square integrable with respect to AREA measure. Then L2a(G); sometimes also denoted A2(G) ; is called the Bergman space for G. Thus, the Bergman space consists of all the ANALYTIC FUNCTIONS in L2(G): The Bergman space can also be generalized to LP a (G); where 0 Bp B/C12 :/ See also HARDY SPACE References Hedenmalm, H.; Korenblum, B.; and Zhu, K. Theory of Bergman Spaces. New York: Springer-Verlag, 2000.Shields, A. L. "Weighted Shift Operators and Analytic Function Theory." In Topics in Operator Theory. Provi- dence, RI: Amer. Math. Soc., pp. 49 /C1/28, 1974. Zhu, K. Operator Theory in Function Spaces. New York: Dekker, 1990. Berlekamp-Massey Algorithm If a sequence takes only a small number of different values, then by regarding the values as the elements of a FINITE FIELD , the Berlekamp-Massey algorithm is an efficient procedure for finding the shortest linear recurrence from the field that will generate the sequence. See also REED-SLOANE ALGORITHM References Berlekamp, E. R. Ch. 7 in Algorithmic Coding Theory. New York: McGraw-Hill, 1968. Berlekamp, E. R.; Fredricksen, H. M.; and Proto, R. C. "Minimum Conditions for Uniquely Determining the Generator of a Linear Sequence." Util. Math. 5, 305/C1/15, 1974. Brent, R. P.; Gustavson, F. G.; and Yun, D. Y. Y. "Fast Solution of Toeplitz Systems of Equations and Computa-tion of Pade ´Approximants." J. Algorithms 1, 259/C1 /95, 1980. Dickinson, B. W.; Morf, M.; and Kailath, T. "A Minimal Realization Algorithm for Matrix Sequences." IEEE Trans. Automatic Control 18,3 1/C1/8, 1974. Gustavson, F. G. "Analysis of the Berlekamp-Massey Linear Feedback Shift-Register Synthesis Algorithm." IBM J. Res. Dev. 20, 204/C1/12, 1976. MacWilliams, F. J. and Sloane, N. J. A. Ch. 9 in The Theory of Error-Correcting Codes. New York: Elsevier, 1978. Massey, J. L. "Shift-Register Synthesis and BCH Decodin- g."IEEE Trans. Information Th. 15, 122/C1/27, 1969. McEliece, R. J. The Theory of Information Coding. Reading, MA: Addison-Wesley, 1977. Mills, W. H. "Continued Fractions and Linear Recurrences." Math. Comput. 29, 173/C1/80, 1975. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 25 /C1/6, 1995. Berlekamp-Zassenhaus Algorithm An algorithm that can be used to find subsets Sof a set for which the product of elements of Sof a set of monic irreducible polynomials in ZPfor which the product of the elements of Shas integer coefficients (van Hoeij 2000). References van Hoeij, M. "Factoring Polynomials and the Knapsack Problem." Preprint. http://www.math.fsu.edu/~aluffi/ar-chive/paper124.ps.gz. Zassenhaus, H. "On Hensel Factorization, I." J. Number Th. 1, 291/C1 /11, 1969. Bernays-Go ¨del Set Theory VON NEUMANN- BERNAYS- GO¨DELSETTHEORY Bernoulli Differential Equation dy dx/C27p(x)y/C30q(x)yn: (1) Letv/C13y1/C28nforn"1;then dv dx/C30(1/C28n)y/C28ndy dx: (2) Rewriting (1) gives y/C28ndydx/C30q(x)/C28p(x)y 1/C28n/C30q(x)/C28vp(x): (3) Plugging (3) into (2), dv dx/C30(1/C28n)[q(x)/C28vp(x)]: (4) Now, this is a linear FIRST-ORDER ORDINARY DIFFER- ENTIAL EQUATION OF THE FORM dv dx/C27vP(x)/C30Q(x); (5) where P(x)/C13(1/C28n)p(x) and Q(x)/C13(1/C28n)q(x):It can therefore be solved analytically using an INTEGRAT- ING FACTOR v/C30gegP(x)dxQ(x)dx/C27C egP(x)dx /C30(1/C28n)ge(1/C28n)gp(x)dxq(x)dx/C27C e(1/C28n)gp(x)dx; (6) where Cis a constant of integration. If n/C301, then equation (1) becomes dy dx/C30y(q/C28p) (7) dy y/C30(q/C28p)dx (8) y/C30C2eg[q(x)/C28p(x)]dx: (9) The general solution is then, with C1and C2con-stants, y/C30(1/C28n)ge(1/C28n)gp(x)dxq(x)dx/C27C1 e(1/C28n)gp(x)dx2 66643 77751=(1/C28n) forn"1 C2eg[(q(x)/C28p(x)]dxforn/C301:8 >>>>>>< >>>>>>:(10) References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, p. 28, 1992. Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 22, 1956. Rainville, E. D. and Bedient, P. E. Elementary Differential Equations. New York: Macmillian, pp. 69 /C1/1, 1964. Simmons, G. F. Differential Equations, With Applications and Historical Notes. New York: McGraw-Hill, p. 49, 1972. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995. Zwillinger, D. "Bernoulli Equation." §II.A.37 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 120 and 157 /C1/58, 1997. Bernoulli Distribution ASTATISTICAL DISTRIBUTION given by P(n)/C30q/C131/C28pforn/C300 p forn/C301/C26 (1) /C30pn(1/C28p)1/C28nforn/C300;1: (2) The distribution of heads and tails in COIN TOSSING is a Bernoulli distribution with p/C30q/C301=2:The MO- MENT-GENERATING FUNCTION of the Bernoulli distri- bution is M(t)/C30etnhi/C30X1 n/C300etnpn(1/C28p)1/C28n/C30e0(1/C28p)/C27etp;(3) so M(t)/C30(1/C28p)/C27pet(4) M?(t)/C30pet(5) Mƒ(t)/C30pet(6) M(n)(t)/C30pet; (7) and the MOMENTS about 0 are m?1/C30m/C30M?(0)/C30p (8) m?2/C30Mƒ(0)/C30p (9) m?n/C30M(n)(0)/C30p: (10) The MOMENTS about the MEAN are m2/C30m?2/C28(m?1)2/C30p/C28p2/C30p(1/C28p) (11) m3 /C30 m?3 /C283m?2 m?1 /C272(m?1)3 /C30p /C283p2 /C272p3 /C30p(1 /C28p)(1 /C282p) (12) m4 /C30 m?4 /C284m?3 m?1 /C276m?2(m ?1)2 /C283(m?1)4 /C30p /C284p2 /C276p3 /C283p4 /C30p(1 /C28p)(3p2 /C283p /C271): (13) The MEAN , VARIANCE , SKEWNESS , and KURTOSIS are then m /C30p (14) s2 /C30 m2 /C30p(1 /C28p) (15) g1 /C30m3 s3/C30p(1 /C28 p)(1 /C28 2p) [p(1 /C28 p)]3 =2/C301 /C28 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p(1 /C28 p)p (16) g2 /C30m4 s4/C283 /C30p(1 /C28 2p)(2p2 /C28 2p /C27 1) p2(1 /C28 p)2 /C283 /C306p2 /C28 6p /C27 1 p(1 /C28 p): (17) To find an estimator ˆp for the mean of a Bernoulli population with actual mean p, let N trials be made and suppose n successes are obtained. Assume an estimator given by §n N; (18) so that the probability of obtaining the observed n successes in N trials is then N n/CP8/CP9 pn(1 /C28p)N /C28n : (19) The expectation value of the estimator ˆp is therefore given by ˆphi$XN n/C300pN n/CP8/CP9 pn(1/C28p)N/C28n /C30(1/C28p)N 1 1/C28p !N p/C30p; (20) sophiis indeed an UNBIASED ESTIMATOR for the population mean p. See also BERNOULLI TRIAL,BINOMIAL DISTRIBUTION , COIN TOSSING ,RUN References Evans, M.; Hastings, N.; and Peacock, B. "Bernoulli Dis- tribution." Ch. 4 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 31 /C1/3, 2000. Bernoulli Function BERNOULLI POLYNOMIALBernoulli Inequality (1/C27x)n>1/C27nx; (1) where x>/C281"0i sa REAL NUMBER and n/C211a n INTEGER . This inequality can be proven by taking a MACLAURIN SERIES of (1/C27x)n; (1/C27x)n/C301/C27nx/C271 2n(n/C281)x2/C2716n(n/C281)(n/C282)x3/C27/C1/C1/C1:(2) Since the series terminates after a finite number of terms for INTEGRAL n, the Bernoulli inequality for x/C210 is obtained by truncating after the first-order term.When /C281BxB0;slightly more finesse is needed. In this case, let y/C30½x½/C30/C28x>0 so that 0 ByB1;and take (1/C28y)n/C301/C28ny/C271 2n(n/C281)y2/C2816n(n/C281)(n/C282)y3/C27/C1/C1/C1:(3) Since each POWER ofymultiplies by a number B1 and since the ABSOLUTE VALUE of the COEFFICIENT of each subsequent term is smaller than the last, it follows that the sum of the third order and subse- quent terms is a POSITIVE number. Therefore, (1/C28y)n>1/C28ny; (4) or (1/C27x)n>1/C27nx;for/C281BxB0; (5) completing the proof of the INEQUALITY over all ranges of parameters. Forx>/C281"0;the following generalizations of Ber- noulli inequality are valid for real exponents: (1/C27x)a>1/C27axifa/C211o r aB0; (6) and (1/C27x)aB1/C27axif 0BaB1 (7) (Mitrinovic 1970). References Mitrinovic, D. S. Analytic Inequalities. New York: Springer- Verlag, 1970. Bernoulli Lemniscate LEMNISCATE Bernoulli Number There are two definitions for the Bernoulli numbers. In modern usage, the Bernoulli numbers are writtenB n;while the Bernoulli numbers encountered in older literature (where they are confusingly also denoted Bn) are distinguished by writing them as B/C31: nIn each case, the Bernoulli numbers are a special case of the BERNOULLI POLYNOMIALS Bn(x)o r B/C31 n(x) with Bn/C30 Bn(0) and B/C31n/C30B/C31n(0):/ The older definition of the Bernoulli numbers, no longer in widespread use, defines B/C31 nusing the equations x ex/C281/C27x 2/C281/C13X/C12 n/C301(/C281)n/C281B/C31nx2n (2n)! /C30B/C311x2 2!/C28B/C312x4 4!/C27B/C313x6 6!/C27/C1/C1/C1 (1) for½x½B2p,o r 1/C28x 2cotx 2 ! /C13X/C12 n/C301B/C31nx2n (2n)! /C30B/C311x2 2!/C28B/C312x4 4!/C27B/C313x6 6!/C27/C1/C1/C1 (2) for½x½Bp(Whittaker and Watson 1990, p. 125). Gradshteyn and Ryzhik (2000) denote these numbers B/C31 n;while Bernoulli numbers defined by the newer (National Bureau of Standards) definition are de- noted Bn:The B/C31 nBernoulli numbers may be calcu- lated from the integral B/C31 n/C304ng/C12 0t2n/C281dt e2pt/C281; (3) and analytically from B/C31n/C302(2n)! (2p)2nX/C12 p/C301p/C282n/C302(2n)! (2p)2nz(2n) (4) forn/C301, 2, . . ., where z(z) is the R IEMANN ZETA FUNCTION . The first few Bernoulli numbers b/C31 nare B/C31 1/C301 6 B/C31 2/C301 30 B/C313/C301 42 B/C314/C301 30 B/C315/C305 66 B/C316/C30691 2;730 B/C317/C307 6 B/C31 8/C303;617 510 B/C319/C3043;867 798 B/C3110/C30174;611 330 B/C3111/C30854;513 138: Bernoulli numbers defined by the modern definitionare denoted Bnand sometimes called "even-index" Bernoulli numbers. These are the Bernoulli numbers returned, by example, by the Mathematica function BernoulliB [n]. The first few are B0/C301 B1/C30/C281 2 B2/C301 6 B4/C30/C281 30 B6/C301 42 B8/C30/C281 30 B10/C305 66 B12/C30/C28691 2;730 B14/C3076 B16/C30/C283;617 510 B18/C3043;867 798 B20/C30/C28174;611 330 B22/C30854;513 138 (Sloane’s A000367 and A002445), with B2n/C271/C300 (5) forn/C301, 2, . . . The Bernoulli numbers Bnare a superset of the archaic ones B/C31 nsince Bn/C131 for n/C300 /C281 2forn/C301 (/C281)(n=2)/C281B/C31 n=2forneven 0 for nodd:8 >>< >>:(6) The Bncan be defined by the identity x ex/C281/C13X/C12 n/C300Bnxn n!: (7) These relationships can be derived using the gener- ating function F(x;t)/C30X/C12 n/C300Bn(x)tn n!; (8) which converges uniformly for ½t½B2pand all x (Castellanos 1988). Taking the partial derivativegives @F(x;t) @x/C30X/C12 n/C300Bn/C281(x)tn (n/C281)!/C30tX/C12 n/C300Bn(x)tn n!/C30tF(x;t):(9) The solution to this differential equation can be found using SEPARATION OF VARIABLES as F(x;t)/C30T(t)ext; (10) so integrating gives g1 0F(x;t)dx/C30T(t)g1 0extdx/C30T(t)et/C281 t: (11) But integrating (11) explicitly gives g1 0F(x;t)dx/C30X/C12 n/C300tn n!g1 0Bn(x)dx /C301/C27X/C12 n/C300tn n!g1 0Bn(x)dx/C301; (12) so T(t)et/C281 t/C301: (13) Solving for T(t) and plugging back into (10) then gives text et/C281/C30X/C12 n/C300Bn(x)tn n!(14) (Castellanos 1988). Setting x/C300 and adding t=2t o both sides then gives 1 2tcoth(12t)/C30X/C12 n/C300B2nt2n (2n)!: (15) Letting t/C302ixthen gives xcotx/C30X/C12 n/C300(/C281)nB2n(2x)2n (2n)!(16) forx/C23[/C28p;p]:The Bernoulli numbers may also be calculated from the integral Bn/C30n! 2pigz ez/C281dz zn/C271; (17) or from Bn/C30lim x00dn dxnx ex/C281: (18) The Bernoulli numbers satisfy the identity k/C271 1/CP8/CP9 Bk/C27k/C271 2/CP8/CP9 Bk/C281/C27/C1/C1/C1/C27k/C271 k/CP8/CP9 B1/C27B0/C300;(19) where (n k)i s a BINOMIAL COEFFICIENT . They also sa- tisfy the nice sum identity Xn i/C300(1/C2821/C28i)(1/C282i/C28n/C271)Bn/C28iBi (n/C28i)!i!/C30(1/C28n)Bn n!(20) (Gosper).An ASYMPTOTIC SERIES for the even Bernoulli num- bers is B2n/C2(/C281)n/C2814ffiffiffiffiffiffipnp n pe !2n : (21) Bernoulli numbers appear in expressions OF THE FORM an k/C301kp;where p/C301, 2, . . . Bernoulli numbers also appear in the series expansions of functions involving tan x;cotx;cscx;ln½sinx½;ln½cosx½; ln½tanx½;tanh x;coth x;and csch x:An analytic solution exists for EVEN orders, B2n/C30(/C281)n/C2812(2n)! (2p)2nX/C12 p/C301p/C282n/C30(/C281)n/C2812(2n)! (2p)2nz(2n) (22) forn/C301, 2, . . ., where z(2n) is the R IEMANN ZETA FUNCTION . Another intimate connection with the RIEMANN ZETA FUNCTION is provided by the identity Bn/C30(/C281)n/C271nz(1/C28n): (23) The DENOMINATOR ofB2kis given by the VON STAUDT- CLAUSEN THEOREM denom( B2k)/C30Y2k/C271 pprime (p/C281)½2kp; (24) which also implies that the DENOMINATOR ofB2kis SQUAREFREE (Hardy and Wright 1979). Another curious property is that the fraction part of Bnin DECIMAL has a DECIMAL PERIOD which divides n, and there is a single digit before that period (Conway 1996). Bernoulli first used the Bernoulli numbers while computing an k/C301kp/. He used the property of the FIGURATE NUMBER TRIANGLE that Xn i/C300aij/C30(n/C271)anj j/C271; (25) along with a form for anjwhich he derived inductively to compute the sums up to n/C3010 (Boyer 1968, p. 85). Forp/C23Z>0;the sum is given by Xn k/C301kp/C30(B/C27n/C271)[p/C271]/C28Bp/C271 p/C271; (26) where the NOTATION B[k]means the quantity in question is raised to the appropriate POWER k, and all terms OF THE FORM Bmare replaced with the corresponding Bernoulli numbers Bm:Written expli- citly in terms of a sum of POWERS , Xn k/C301kp/C30np/C27Xp k/C300Bkp! k!(p/C28k/C271)!np/C28k/C271: (27) It is also true that the COEFFICIENTS of the terms in such an expansion sum to 1 (which Bernoulli stated without proof). Ramanujan gave a number of curious infinite sum identities involving Bernoulli numbers (Berndt 1994). G. J. Fee and S. Plouffe have computed B200;000 ; which has /C2800; 000 DIGITS (Plouffe). Plouffe and collaborators have also calculated Bnfor n up to 72,000. See also ARGOH’S CONJECTURE ,B ERNOULLI FUNC- TION ,B ERNOULLI NUMBER OF THE SECOND KIND, BERNOULLI POLYNOMIAL ,DEBYE FUNCTIONS ,EULER- MACLAURIN INTEGRATION FORMULAS ,E ULER NUM- BER,FIGURATE NUMBER TRIANGLE ,GENOCCHI NUM- BER,M ODIFIED BERNOULLI NUMBER ,P ASCAL’S TRIANGLE ,RIEMANN ZETA FUNCTION , VON STAUDT- CLAUSEN THEOREM References Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804 /C1/06, 1972. Arfken, G. "Bernoulli Numbers, Euler-Maclaurin Formula." §5.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 327 /C1/38, 1985. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 71, 1987. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 81 /C1/5, 1994. Boyer, C. B. A History of Mathematics. New York: Wiley, 1968. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67 /C1/8, 1988. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 107 /C1/10, 1996. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Bernoulli Numbers." §6.5 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 283 /C1/90, 1994. Hardy, G. H. and Wright, W. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 91 /C1/3, 1979. Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen fu¨r Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995. Ireland, K. and Rosen, M. "Bernoulli Numbers." Ch. 15 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 228 /C1/48, 1990. Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663 /C1/ 88, 1967. Nielsen, N. Traite ´ e´le´mentaire des nombres de Bernoulli. Paris: Gauthier-Villars, 1923. Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u- qam.ca/pi/records.html. Ramanujan, S. "Some Properties of Bernoulli’s Numbers." J. Indian Math. Soc. 3, 219 /C1/34, 1911. Roman, S. The Umbral Calculus. New York: Academic Press, p. 31, 1984.Sloane, N. J. A. Sequences A000367/M4039 and A002445/ M4189 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Bernoulli Numbers, Bn :/" Ch. 4 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 35 /C1/8, 1987. Wagstaff, S. S. Jr. "Ramanujan’s Paper on Bernoulli Num- bers." J. Indian Math. Soc. 45,49/C1/5, 1981. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Woon, S C. Generalization of a Relation Between the Riemann Zeta Function and Bernoulli Numbers. 24 Dec 1998. http://xxx.lanl.gov/abs/math.NT/9812143/. Young, P. T. "Congruences for Bernoulli, Euler, and Stirling Numbers." J. Number Th. 78, 204 /C1/27, 1999. Bernoulli Number of the Second Kind A number defined by bn/C30bn(0);where bn(x)i sa BERNOULLI POLYNOMIAL OF THE SECOND KIND (Roman 1974, p. 294), also called Cauchy numbers of the first kind. The first few for n/C300, 1, 2, . . . are 1, 1/2, /C281=6; 1/4,/C2819=30;9/4, . . . (Sloane’s A006232 and A006233). They are given by bn/C30g1 0(x)ndx; where ( x)nis a FALLING FACTORIAL , and have EXPO- NENTIAL GENERATING FUNCTION E(x)/C30x ln(1/C27x)/C301/C271! 2x/C282! 6x2/C273! 4x3/C27/C1/C1/C1: See also BERNOULLI NUMBER ,BERNOULLI POLYNO- MIAL OF THE SECOND KIND References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 294, 1974. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, p. 259, 1988. Roman, S. The Umbral Calculus. New York: Academic Press, p. 114, 1984. Sloane, N. J. A. Sequences A006232/M5067 and A006233/ M1558 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Bernoulli Polynomial There are two definitions of Bernoulli polynomials in use. The nth Bernoulli polynomial is denoted here by Bn(x) (Abramowitz and Stegun 1972), and the archaic form of the Bernoulli polynomial by B/C31 n(x) (or some- times fn(x)):When evaluated at zero, these defini- tions correspond to the B ERNOULLI NUMBERS , Bn/C13Bn(0) (1) B/C31 n/C13B/C31n(0): (2) The Bernoulli polynomials are an A PPELL SEQUENCE with g(t)/C30et/C281 t(3) (Roman 1984, p. 31), giving the GENERATING FUNC- TION tetx et/C281/C13X/C12 n/C300Bn(x)tn n!(4) (Abramowitz and Stegun 1972, p. 804), first obtained by Euler (1738). The first few Bernoulli polynomialsare B 0(x)/C301 B1(x)/C30x/C281 2 B2(x)/C30x2/C28x/C271 6 B3(x)/C30x3/C283 2x2/C2712x B4(x)/C30x4/C282x3/C27x2/C281 30 B5(x)/C30x5/C2852x4/C2753x3/C2816x B6(x)/C30x6/C283x5/C275 2x4/C2812x2/C271 42: Whittaker and Watson (1990, p. 126) define an older type of "Bernoulli polynomial" by writing tezt/C281 et/C281/C30X/C12 n/C301fn(z)tn n!(5)instead of (5). This gives the polynomials fn(x)/C30Bn(x)/C28Bn; (6) where Bnis a B ERNOULLI NUMBER , the first few of which are f1(x)/C30x f2(x)/C30x2/C28x f3(x)/C30x3/C283 2x2/C2712x f4(x)/C30x4/C282x3/C27x2 f5(x)/C30x5/C2852x4/C2753x3/C2816x: The Bernoulli polynomials also satisfy Bn(1)/C30(/C281)nBn(0) (7) and Bn(1/C28x)/C30(/C281)nBn(x) (8) (Lehmer 1988), as well as the relation Bn(x/C271)/C28Bn(x)/C30nxn/C281(9) (Whittaker and Watson 1990, p. 127). Bernoulli (1713) defined the polynomials in terms of sums of the POWERS of consecutive integers, Xm/C281 k/C300kn/C281/C301 n[Bn(m)/C28Bn(0)]: (10) The Bernoulli polynomials satisfy the RECURRENCE RELATION dBn dx/C30nBn/C281(x) (11) (Appell 1882), and obey the identity Bn(x)/C30(B/C27x)n; (12) where Bkis interpreted here as Bk(x):Hurwitz gave the F OURIER SERIES Bn(x)/C30/C28n! (2pi)nX ?/C12 k/C30/C28/C12k/C28ne2pikx; (13) for 0BxB1;where the prime in the summation indicates that the term k/C300 is omitted. Performing the sum gives Bn(x)/C30/C28n! (2pi)n[(/C281)nLin(e/C282pix)/C27Lin(e2pix)];(14) where Lin(x) is the POLYLOGARITHM function. Raabe (1851) found 1 mXm/C281 k/C300Bnx/C27k m ! /C30m/C28nBn(mx): (15) A sum identity involving the Bernoulli polynomials is Xm k /C300m k/CP8/CP9 Bk( a)Bm/C28k(b) /C30/C28(m /C281)Bm( a /C27 b) /C27m(a /C27 b /C281)Bm/C281(a /C27 b) (16) for m an INTEGER . A sum identity due to S. M. Ruiz is Xn k /C300(/C281)k/C27n n k/CP8/CP9 Bn(k) /C30n!; (17) where (n k)isa BINOMIAL COEFFICIENT . The Bernoulli polynomials are also given by the formula Bn(x) /C30Bn(0) /C27Xn k/C301n kS(n /C281; k /C281)(x)k ; (18) where S(n; m)isaS TIRLING NUMBER OF THE SECOND KIND and (x)kis a FALLING FACTORIAL (Roman 1984, p. 94). A general identity is given by (n)mxn/C28m /C30Xn k /C30m(n)k (k /C28 m /C27 1)!Bn/C28k(x) ; (19) which simplifies to nxn/C281 /C30Xn k/C301n k/CP8/CP9 Bn/C28k(x) (20) (Roman 1984, p. 97). Gosper gave the identity Xi j/C300[2(i /C28 j) /C28 1]32f (2(2f /C271)/C271)B2(i /C28j)B2j/C271(1 3) [2(i /C28 j)]!(2j /C27 1)! /C302 /C215 32(i/C281)(22i/C281 /C27 1)B2i /C281(13) /C28 (i /C2812)B2i (2i)! : (21) Roman (1984, p. 93) defines a generalization B( a) n(x)of the Bernoulli numbers with an additional free para- meter such that Bn(x) /C30B(1) n(x) :/ See also BERNOULLI NUMBER ,BERNOULLI POLYNO- MIAL OF THE SECOND KIND,E ULER- MACLAURIN INTEGRATION FORMULAS ,EULER POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804 /C1/06, 1972. Appell, P. E. "Sur une classe de polynomes." Annales d’E´ cole Normal Superieur, Ser. 2 9, 119 /C1/44, 1882. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 330, 1985. Bernoulli, J. Ars conjectandi. Basel, Switzerland, p. 97, 1713. Published posthumously. Euler, L. "Methodus generalis summandi progressiones." Comment. Acad. Sci. Petropol. 6,68/C1/7, 1738. Lehmer, D. H. "A New Approach to Bernoulli Polynomials." Amer. Math. Monthly. 95, 905 /C1/11, 1988. Lucas, E. Ch. 14 in The´orie des Nombres. Paris, 1891.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Zeta Function z(s; x) ; Bernoulli Poly- nomials Bn(x); Euler Polynomials En(x) ; and Polyloga- rithms Liv(x) :/" §1.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 23 /C1/4, 1990. Raabe, J. L. "Zuru ¨ckfu¨hrung einiger Summen und bestimm- ten Integrale auf die Jakob Bernoullische Function." J. reine angew. Math. 42, 348 /C1/76, 1851. Roman, S. "The Bernoulli Polynomials." §4.2.2 in The Umbral Calculus. New York: Academic Press, pp. 93 /C1/ 00, 1984. Spanier, J. and Oldham, K. B. "The Bernoulli Polynomial Bn(x) :/" Ch. 19 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 167 /C1/73, 1987. Bernoulli Polynomial of the Second Kind Polynomials bn(x) which form a SHEFFER SEQUENCE with g(t) /C30t et /C28 1 (1) f(t) /C30et /C281; (2) giving GENERATING FUNCTION X/C12 k /C300bk(x) k!tk /C30t(t /C27 1)x ln(1 /C27 t) : (3) Roman (1984) defines BERNOULLI NUMBERS OF THE SECOND KIND as bn /C30bn(0): They are related to the STIRLING NUMBERS OF THE FIRST KIND s(n ; m)by bn(x)/C30bn(0)/C27Xn k/C301n ks(n/C281;k/C281)xk(4) (Roman 1984, p. 115), and obey the reflection formula bn(1 2n/C281/C28x)/C30(/C281)nbn(12n/C281/C27x) (5) (Roman 1984, p. 119). The first few Bernoulli polynomials of the second kind are b0(x)/C301 b1(x)/C301 2(2x/C271) b2(x)/C3016(6x2/C281) b3(x)/C3014(4x3/C286x2/C271) b4(x)/C301 30(30x4/C28120x3/C27120x2/C2819): See also BERNOULLI NUMBER OF THE SECOND KIND, BERNOULLI POLYNOMIAL ,SHEFFER SEQUENCE ,STIR- LING NUMBER OF THE FIRST KIND References Roman, S. "The Bernoulli Polynomials of the Second Kind." §5.3.2 in The Umbral Calculus. New York: Academic Press, pp. 113 /C1/19, 1984. Bernoulli Scheme References Petersen, K. Ergodic Theory. Cambridge, England: Cam- bridge University Press, 1983. Bernoulli Trial An experiment in which s TRIALS are made of an event, with probability p of success in any given TRIAL . See also BERNOULLI DISTRIBUTION ,COIN TOSSING , RUN References Papoulis, A. "Bernoulli Trials." §3 /C1/ in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 57 /C1/3, 1984. Bernoulli’s Method In order to find a root of a polynomial equation a0xn /C27a1xn/C281 /C27/C1/C1/C1/C27an /C300 ; (1) consider the difference equation a0y(t /C27n) /C27a1y(t /C27n /C281) /C27/C1/C1/C1/C27any(t); which is known to have solution y(t) /C30w1xt 1 /C27w2xt2 /C27/C1/C1/C1/C27wnxtn /C27/C1/C1/C1; (2) where w1 ; w2 ;... /, are arbitrary functions of t with period 1, and x1 ;...; xnare roots of (1). In order to find the absolutely greatest root (1), take any arbi- trary values for y(0) ; y(1) ;...; y(n /C281): By repeated application of (2), calculate in succession the values y(n) ; y(n /C271); y(n /C272); ... Then the ratio of two successive members of this sequence tends in general to a limit, which is the absolutely greatest root of (1). See also ROOT References Whittaker, E. T. and Robinson, G. "A Method of Daniel Bernoulli." §52 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 98 /C1/9, 1967. Bernoulli’s Paradox Suppose the HARMONIC SERIES converges to h: X/C12 k/C3011 k /C30h: Then rearranging the terms in the sum gives h /C281 /C30h; which is a contradiction. See also HARMONIC SERIESReferences Boas, R. P. "Some Remarkable Sequences of Integers." Ch. 3 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 39 /C1/0, 1979. Bernoulli’s Theorem WEAK LAW OF LARGE NUMBERS BernoulliB BERNOULLI NUMBER ,BERNOULLI POLYNOMIAL Bernstein Minimal Surface Theorem If a MINIMAL SURFACE is given by the equation z /C30 f(x; y) and f has CONTINUOUS first and second PARTIAL DERIVATIVES for all REAL x and y, then f is a PLANE . See also MINIMAL SURFACE References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 369, 1988. Osserman, R. "Bernstein’s Theorem." §5in A Survey of Minimal Surfaces. New York: Dover, pp. 34 /C1/2, 1986. Bernstein Polynomial The POLYNOMIALS defined by Bi ; n(t) /C30n i/CP8/CP9 ti(1 /C28t)n/C28i ; where (n k)isa BINOMIAL COEFFICIENT . The Bernstein polynomials of degree n form a basis for the POWER - POLYNOMIALS of degree n. Another form of Bernstein polynomials is given by Bn(f ; x) /C30Xn j/C300n j/CP8/CP9 xj(1 /C28x)n /C28jfj n ! (Gzyl and Palacios 1997, Mathe ´ 1999). See also BE´ ZIER CURVE References Bernstein, S. "De ´monstration du the ´ore`me de Weierstrass fonde´e sur le calcul des probabilities." Comm. Soc. Math. Kharkov 13,1/C1/, 1912. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 222, 1971. Gzyl, H. and Palacios, J. L. "The Weierstrass Approximation Theorem and Large Deviations." Amer. Math. Monthly 104, 650/C1/53, 1997. Kac, M. "Une remarque sur les polynomes de M. S. Bern- stein." Studia Math. 7,4 9/C1/1, 1938. Kac, M. "Reconnaissance de priorite ´relative a `ma note, ‘Une remarque sur les polynomes de M. S. Bernstein."’ Studia Math. 8, 170, 1939. Lorentz, G. G. Bernstein Polynomials. Toronto: University of Toronto Press, 1953. Mathe ´, P. "Approximation of Ho ¨lder Continuous Functions by Bernstein Polynomials." Amer. Math. Monthly 106, 568/C1/74, 1999. Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, p. 101, 1941. Bernstein’s Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. LetEn(f) be the error of the best uniform approxima- tion to a REAL function f(x) on the INTERVAL [/C281;1] by REAL POLYNOMIALS of degree at most n.I f a(x)/C30xjj; (1) then Bernstein showed that 0:267 . . .Blim n0/C122nE2n(a)B0:286: (2) He conjectured that the lower limit ( /b) was b/C301=(2/C2ffiffiffipp):However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed b/C300:2801694990 . . . : (3) For rational approximations p(x)=q(x) for pand qof degree mandn, D. J. Newman (1964) proved 1 2e/C289ffiffinp 5En;n(a)53e/C28ffiffinp (4) forn]4:Gonchar (1967) and Bulanov (1975) im- proved the lower bound to e/C28pffiffiffiffiffiffiffi n/C271p 5En;n(a)53e/C28ffiffinp : (5) Vjacheslavo (1975) proved the existence of POSITIVE constants mandMsuch that m5epffiffinp En;n(a)BM (6) (Petrushev 1987, pp. 105 /C1/06). Varga et al. (1993) conjectured and Stahl (1993) proved that lim n0/C12epffiffiffiffi 2np E2n;2n(a)/C308: (7) References Bulanov, A. P. "Asymptotics for the Best Rational Approx- imation of the Function Sign x."Mat. Sbornik 96, 171/C1/78, 1975. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/brnstn/brnstn.html. Gonchar, A. A. "Estimates for the Growth of Rational Functions and their Applications." Mat. Sbornik 72, 489/C1/03, 1967. Newman, D. J. "Rational Approximation to xjj:/"Michigan Math. J. 11,1 1/C1/4, 1964. Petrushev, P. P. and Popov, V. A. Rational Approximation of Real Functions. New York: Cambridge University Press, 1987. Stahl, H. "Best Uniform Rational Approximation of xjjon [/C281;1]:/"Russian Acad. Sci. Sb. Math. 76, 461/C1/87, 1993. Varga, R. S. Scientific Computations on Mathematical Pro- blems and Conjectures. Philadelphia, PA: SIAM, 1990.Varga, R. S. and Carpenter, A. J. "On a Conjecture of S. Bernstein in Approximation Theory." Math. USSR Sbor- nik57, 547/C1/60, 1987. Varga, R. S.; Ruttan, A.; and Carpenter, A. J. "Numerical Results on Best Uniform Rational Approximations to xjjon [/C281;/C271]:Math. USSR Sbornik 74, 271/C1/90, 1993. Vjacheslavo, N. S. "On the Uniform Approximation of xjjby Rational Functions." Dokl. Akad. Nauk SSSR 220, 512/C1/ 15, 1975. Bernstein’s Inequality LetPbe a POLYNOMIAL of degree nwith derivative P?: Then P?kk/C125nPkk/C12; where Pkk/C12/C13max zjj/C301P(z) jj : Bernstein’s Polynomial Theorem Ifg(u) is a trigonometric POLYNOMIAL of degree m satisfying the condition g(u) jj51 where uis arbitrary and real, then g?(u)5m:/ References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 5, 1975. Bernstein-Be ´zier Curve BE´ZIER CURVE Bernstein-Szego Polynomials The POLYNOMIALS on the interval [ /C281;1] associated with the WEIGHT FUNCTIONS w(x)/C30(1/C28x2)/C281=2 w(x)/C30(1/C28x2)1=2 w(x)/C30ffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x 1/C27xs ; also called B ERNSTEIN POLYNOMIALS . References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 31 /C1/3, 1975. Berry Conjecture The longstanding conjecture that the nonimaginary solutions Enof z(1 2/C27iEn)/C300; where z(z) is the R IEMANN ZETA FUNCTION , are the EIGENVALUES of an "appropriate" H ERMITIAN OPERA- TOR H. Berry and Keating (1999) further conjecture that this operator is H /C30xp /C30/C28ixd dx /C271 2 ! ; where x and p are the position and conjugate momentum operators, respectively. See also RIEMANN HYPOTHESIS ,RIEMANN ZETA FUNC- TION References Berry, M. V. and Keating, J. P. "H /C30xp and the Riemann Zeros." In Supersymmetry and Trace Formulae: Chaos and Disorder (Ed. I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii). New York: Kluwer, pp. 355 /C1/67, 1999. Berry Paradox There are several versions of the Berry paradox, the original version of which was published by Bertrand Russell and attributed to Oxford University librarian Mr. G. Berry. In one form, the paradox notes that the number "one million, one hundred thousand, one hundred and twenty one" can be named by the description: "the first number not nameable in under ten words." However, this latter expression has only nine words, so the number can be named in under ten words, so there is an inconsistency in naming it in this manner! References Chaitin, G. J. "The Berry Paradox." Complexity 1,26/C1/0, 1995. Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 6, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 20 /C1/1, 1998. Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, p. 60, 1927. Berry-Esse ´en Theorem If F(x) is a probability distribution with zero mean and r /C30g/C12 /C28/C12½x½3 dF(x) B/C12 ; (1) where the above integral is a STIELTJES INTEGRAL , then for all x and n, ½Fn(x) /C28F(x) /C281 2½B33 4r s3ffiffiffinp; (2) where F(x) is the NORMAL DISTRIBUTION FUNCTION , F(x) /C271 =2 /C30N(x) in Feller’s notation, and Fn(x) /C30Fn/C31(xsffiffiffinp) (3) is the normalized n-fold CONVOLUTION of F(x) (Wal- lace 1958, Feller 1971). See also CENTRAL LIMIT THEOREMReferences Bergstro ¨m, H. "On the Central Limit Theorem." Skand. Aktuarietidskr. 27, 139 /C1/53, 1944. Bergstro ¨m, H. "On the Central Limit Theorem in the Space Rk ; k /C211." Skand. Aktuarietidskr. 28, 106 /C1/27, 1945. Bergstro ¨m, H. "On the Central Limit Theorem in the Case of not Equally Distributed Random Variables." Skand. Ak- tuarietidskr. 32,37/C1/2, 1949. Berry, A. C. "The Accuracy of the Gaussian Approximation to the Sum of Independent Variates." Trans. Amer. Math. Soc. 49, 122 /C1/36 1941. Esseen, C. G. "On the Liapounoff Limit of Error in the Theory of Probability." Ark. Mat. Astr. och Fys. 28A, No. 9, 1 /C1/9, 1942. Esseen, C. G. "Fourier Analysis of Distribution Functions." Acta Math. 77,1/C1/25, 1945. Esseen, C. G. "A Moment Inequality with an Application to the Central Limit Theorem." Skand. Aktuarietidskr. 39, 160 /C1/70, 1956. Feller, W. "The Berry-Esse ´en Theorem." §16.5 in An Intro- duction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 542 /C1/46, 1971. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 369, 1988. Hsu, P. L. "The Approximate Distribution of the Mean and Variance of a Sample of Independent Variables." Ann. Math. Stat. 16,1/C1/9, 1945. Wallace, D. L. "Asymptotic Approximations to Distribu- tions." Ann. Math. Stat. 29, 635 /C1/54, 1958. Bertelsen’s Number An erroneous value of p(109) ; where p(x) is the PRIME COUNTING FUNCTION . Bertelsen’s value of 50,847,478 is 56 lower than the correct value of 50,847,534. See also PRIME COUNTING FUNCTION References Brown, K. S. "Bertelsen’s Number." http://www.seanet.com/ ~ksbrown/kmath049.htm. Bertini’s Theorem The general curve of a system which is LINEARLY INDEPENDENT on a certain number of given irreduci- ble curves will not have a singular point which is not fixed for all the curves of the system. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 115, 1959. Bertrand Curves Two curves which, at any point, have a common principal NORMAL VECTOR are called Bertrand curves. The product of the TORSIONS of Bertrand curves is a constant. Bertrand’s Paradox BERTRAND’S PROBLEM Bertrand’s Postulate If n /C213, there is always at least one PRIME between n and 2n /C282: Equivalently, if n /C211, then there is always at least one PRIME between n and 2n: The conjecture was first made by Bertrand in 1845 (Nagell 1951, p. 67). It was proved in 1850 /C1/1by Chebyshev, and is therefore sometimes known as CHEBYSHEV’S THEOREM . An extension of this result is that if n /C21k, then there is a number containing a PRIME divisor /C21k in the sequence n, n /C271 ; ...; n /C27 k /C281: (The case n /C30k /C271 then corresponds to Ber- trand’s postulate.) This was first proved by Sylvester, independently by Schur, and a simple proof was given by Erdos (Hoffman 1998, p. 37) A related problem is to find the least value of u so that there exists at least one PRIME between n and n /C27 O(nu) for sufficiently large n (Berndt 1994). The smallest known value is u /C306 =11 /C27 e (Lou and Yao 1992). See also CHOQUET THEORY , DE POLIGNAC’S CONJEC- TURE ,PRIME NUMBER References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, p. 135, 1994. Erdos, P. "Ramanujan and I." In Proceedings of the Inter- national Ramanujan Centenary Conference held at Anna University, Madras, Dec. 21, 1987. (Ed. K. Alladi). New York: Springer-Verlag, pp. 1 /C1/0, 1989. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Lou, S. and Yau, Q. "A Chebyshev’s Type of Prime Number Theorem in a Short Interval (II)." Hardy-Ramanujan J. 15,1/C1/3, 1992. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 70, 1951. Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. 7 /C1/, 2000. Bertrand’s Problem What is the PROBABILITY that a CHORD drawn at random on a CIRCLE of RADIUS r (i.e., CIRCLE LINE PICKING ) has length ]r (or sometimes greater than or equal to the side length of an inscribed equilateral triangle; Solomon 1978, p. 2)? The answer depends on the interpretation of "two points drawn at random," or more specifically on the "natural" measure for the problem. In the most commonly considered measure, the ANGLES u1and u2are picked at random on the CIRCUMFERENCE of the circle. Without loss of general- ity, this can be formulated as the probability that the chord length of a single point at random angle u measured from the X-AXIS on the unit circle. Since the length as a function of u (CIRCLE LINE PICKING )is given by s( u) /C302 sin(1 2 u)/CP2/CP2/CP2/CP2/CP2/CP2; (1)solving for s( u) /C301 gives p=3; so the fraction of the top unit semicircle having chord length greater than 1 is P /C30p /C28p 3 p/C302 3 : (2) However, if a point is instead placed at random on a RADIUS of the CIRCLE and a CHORD drawn PERPENDI- CULAR to it, then P /C30ffiffi 3p 2r r/C30ffiffiffi 3p 2: (3) The latter interpretation is more satisfactory in the sense that the result remains the same for a rotated CIRCLE , a slightly smaller CIRCLE INSCRIBED in the first, or for a CIRCLE of the same size but with its center slightly offset. Jaynes (1983) shows that the interpretation of "random" as a continuous UNIFORM DISTRIBUTION over the RADIUS is the only one posses- sing all these three invariances. See also CHORD ,CIRCLE LINE PICKING ,GEOMETRIC PROBABILITY References Bogomolny, A. "Bertrand’s Paradox." http://www.cut-the- knot.com/bertrand.html. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 21 /C1/3, 1998. Isaac, R. The Pleasures of Probability. New York: Springer- Verlag, 1995. Jaynes, E. T. Papers on Probability, Statistics, and Statis- tical Physics. Dordrecht, Netherlands: Reidel, 1983. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 42 /C1/5, 1995. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 11 /C1/2, 1984. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, p. 2, 1978. Bertrand’s Test A CONVERGENCE TEST also called DE MORGAN’S AND BERTRAND’S TEST . If the ratio of terms of a SERIES fan g/C12 n/C301can be written in the form an an/C271/C301/C271 n/C27rn nlnn; then the series converges if limn0/C12rn/C211 and di- verges if limn0/C12rnB1;where limn0/C12is the LOWER LIMIT andlimn0/C12is the UPPER LIMIT . See also KUMMER’S TEST References Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 40, 1991. Bertrand’s Theorem BERTRAND’S POSTULATE Besov Space A type of abstract SPACE which occurs in SPLINE and RATIONAL FUNCTION approximations. The Besov space B a p ;qis a complete quasinormed space which is a BANACH SPACE when 1 5 p ; q 5/C12 (Petrushev and Popov 1987). See also BANACH SPACE References Bergh, J. and Lo¨fstro ¨m, J. Interpolation Spaces. New York: Springer-Verlag, 1976. Peetre, J. New Thoughts on Besov Spaces. Durham, NC: Duke University Press, 1976. Petrushev, P. P. and Popov, V. A. "Besov Spaces." §7.2 in Rational Approximation of Real Functions. New York: Cambridge University Press, pp. 201 /C103, 1987. Triebel, H. Interpolation Theory, Function Spaces, Differen- tial Operators. New York: Wiley, 1998. Bessel Differential Equation x2d2y dx2 /C27xdy dx /C27(x2 /C28m2)y /C300 : (1) Equivalently, dividing through by x2 ; d2y dx2 /C271 xdy dx /C27 1 /C28m2 x2 ! y /C300; (2) The solutions to this equation define the BESSEL FUNCTIONS . The equation has a regular SINGULARITY at 0 and an irregular SINGULARITY at /C12:/ A transformed version of the Bessel differential equation given by Bowman (1958) is x2d2y dx2 /C27(2p /C271)xdy dx /C27(a2x2r /C27 b2)y /C300: (3) The solution is y /C30x/C28p C1Jq=ra rxr ! /C27C2Yq=ra rxr ! "# ; (4) where q /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 /C28 b2q ; (5) /Jn(x) and Yn(x) are the BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS , and C1and C2are con- stants. Another form is given by letting y /C30xaJn( bxg); h /C30yx /C28a ; and /j /C30 bxg / (Bowman 1958, p. 117), then d2y dx2 /C282a /C28 1 xdy dx /C27 b2 g2x2g/C282 /C27a2 /C28 n2 g2 x2 ! y /C300 : (6) The solution isy /C30xa[AJn(bxg) /C27BYn( bxg)] for integer n AJn( bxg) /C27BJ /C28n( bxg) for noninteger n :/C26 (7) See also AIRY FUNCTIONS ,A NGER FUNCTION ,B EI, BER,B ESSEL FUNCTION ,B OURGET’S HYPOTHESI S, CATALAN INTEGRALS ,CYLINDRICAL FUNCTION ,D INI EXPANSION ,HANKEL FUNCTION ,HANKEL’S INTEGRAL , HEMISPHERICAL FUNCTION ,K APTEYN SERIES , LIPSCHITZ’S INTEGRAL ,LOMMEL DIFFERENTIAL EQUA- TION ,L OMMEL FUNCTION ,L OMMEL’S INTEGRALS , NEUMANN SERIES (BESSEL FUNCTION ), PARSEVAL’S INTEGRAL ,P OISSON INTEGRAL ,R AMANUJAN’S INTE- GRAL ,R ICCATI DIFFERENTIAL EQUATION ,S ONINE’S INTEGRAL ,STRUVE FUNCTION ,W EBER FUNCTIONS , WEBER’S DISCONTINUOUS INTEGRALS References Abramowitz, M. and Stegun, C. A. (Eds.). §9.1.1 in Hand- book of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 550, 1953. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Bessel Function A function Zn(x) defined by the RECURRENCE RELA- TIONS Zn/C271 /C27Zn/C2812n xZn and Zn/C271 /C28Zn /C281 /C30/C282dZn dx: The Bessel functions are more frequently defined as solutions to the DIFFERENTIAL EQUATION x2d2y dx2 /C27xdydx /C27(x 2 /C28n2)y /C300 : There are two classes of solution, called the BESSEL FUNCTION OF THE FIRST KIND Jn(x) and BESSEL FUNCTION OF THE SECOND KIND Yn(x) : (A BESSEL FUNCTION OF THE THIRD KIND is a special combination of the first and second kinds.) Several related func- tions are also defined by slightly modifying the defining equations. See also BESSEL FUNCTION OF THE FIRST KIND, BESSEL FUNCTION OF THE SECOND KIND,B ESSEL FUNCTION OF THE THIRD KIND,CYLINDER FUNCTION , HEMICYLINDRICAL FUNCTION ,M ODIFIED BESSEL FUNCTION OF THE FIRST KIND,M ODIFIED BESSEL FUNCTION OF THE SECOND KIND,SPHERICAL BESSEL FUNCTION OF THE FIRST KIND,SPHERICAL BESSEL FUNCTION OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Functions of Integer Order," "Bessel Functions of Fractional Order," and "Integrals of Bessel Functions." Chs. 9 /C1/1i n Hand- book of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 355 /C1/89, 435 /C1/56, and 480 /C1/91, 1972. Adamchik, V. "The Evaluation of Integrals of Bessel Func- tions via G-Function Identities." J. Comput. Appl. Math. 64, 283/C1/90, 1995. Arfken, G. "Bessel Functions." Ch. 11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573 /C1/36, 1985. Bickley, W. G. Bessel Functions and Formulae. Cambridge, England: Cambridge University Press, 1957. Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Byerly, W. E. "Cylindrical Harmonics (Bessel’s Functions)." Ch. 7 in An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 219 /C1/37, 1959. Gray, A. and Mathews, G. B. A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed. New York: Dover, 1966. Luke, Y. L. Integrals of Bessel Functions. New York: McGraw-Hill, 1962. McLachlan, N. W. Bessel Functions for Engineers, 2nd ed. with corrections. Oxford, England: Clarendon Press, 1961. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Bessel Functions of Integral Order" and "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions." §6.5 and 6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing,2nd ed. Cambridge, England: Cambridge University Press, pp. 223 /C1 /29 and 234 /C1/45, 1992. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Weisstein, E. W. "Books about Bessel Functions." http:// www.treasure-troves.com/books/BesselFunctions.html. Bessel Function Fourier Expansion Letn]1=2 and a1;a2;. . . be the POSITIVE ROOTS of Jn(x)/C300:An expansion of a function in the interval (0, 1) in terms of B ESSEL FUNCTIONS OF THE FIRST KIND f(x)/C30X/C12 l/C301ArJn(xar); (1) has COEFFICIENTS found as follows: g1 0xf(x)Jn(xal)dx/C30X/C12 r/C301Arg1 0xJn(xar)Jn(xal)dx:(2) But ORTHOGONALITY of B ESSEL FUNCTION ROOTS gives g1 0xJn(xal)Jn(xar)dx/C301 2dl;rJ2 n/C271(ar) (3) (Bowman 1958, p. 108), sog1 0xf(x)Jn(xal)dx/C3012X/C12 r/C301Ardl;rJ2 n/C271(xar) /C301 2AlJ2 n/C271(al); (4) and the COEFFICIENTS are given by Al/C302 J2 n/C271(al)g1 0xf(x)Jn(xal)dx: (5) References Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Bessel Function of the First Kind The Bessel functions of the first kind Jn(x) are defined as the solutions to the B ESSEL DIFFERENTIAL EQUA- TION x2d2y dx2/C27xdy dx/C27(x2/C28m2)y/C300 (1) which are nonsingular at the origin. They are some- times also called CYLINDER FUNCTIONS orCYLINDRI- CAL HARMONICS . The above plot shows Jn(x) for n/C301, 2 , ... , 5 .To solve the differential equation, apply F ROBENIUS METHOD using a series solution OF THE FORM y/C30xkX/C12 n/C300anxn/C30X/C12 n/C300anxn/C27k: (2) Plugging into (1) yields x2X/C12 n/C300(k/C27n)(k/C27n/C281)anxk/C27n/C282 /C27xX/C12 n/C300(k/C27n)anxk/C27n/C281 /C27x2X/C12 n/C300anxk/C27n/C28m2X/C12 n/C300anxn/C27k/C300 (3) X/C12 n/C300(k/C27n)(k/C27n/C281)anxk/C27n/C27X/C12 n/C300(k/C27n)anxk/C27n /C27X/C12 n/C302an/C282xk/C27n/C28m2X/C12 n/C300anxn/C27k/C300: (4) The INDICIAL EQUATION , obtained by setting n/C300, is a0[k(k/C281)/C27k/C28m2]/C30a0(k2/C28m2)/C300: (5) Since a0is defined as the first NONZERO term, k2/C28 m2/C300;sok/C309m:Now, if k/C30m, X/C12 n/C300[(m/C27n)(m/C27n/C281)/C27(m/C27n)/C28m2] /C2anxm/C27n/C27X/C12 n/C302an/C282xm/C27n/C300 (6) X/C12 n/C300[(m/C27n)2/C28m2]anxm/C27n/C27X/C12 n/C302an/C282xm/C27n/C300 (7) X/C12 n/C300n(2m/C27n)anxm/C27n/C27X/C12 n/C302an/C282xm/C27n/C300 (8) a1(2m/C271)/C27X/C12 n/C302[ann(2m/C27n)/C27an/C282]xm/C27n/C300:(9) First, look at the special case m/C30/C281=2;then (9) becomes X/C12 n/C302[ann(n/C281)/C27an/C282]xm/C27n/C300; (10) so an/C30/C281 n(n/C281)an/C282: (11) Now let n/C132l;where l/C301, 2, . . . a2l/C30/C281 2l(2l/C281)a2l/C282 /C30(/C281)l [2l(2l/C281)[2(l/C281)(2l/C283)]/C1/C1/C1[2 /C2151 /C2151]a0 /C30(/C281)l 2ll!(2l/C281)!!a0; (12) which, using the identity 2ll!(2l/C281)!!/C30(2l)!;gives a2l/C30(/C281)l (2l)!a0: (13) Similarly, letting n/C132l/C271; a2l/C271/C30/C281 (2l/C271)(2l)a2l/C281 /C30(/C281)l [2l(2l/C271)][2( l/C281)(2l/C281)]/C1/C1/C1[2 /C2151 /C2153][1]a1; (14) which, using the identity 2ll!(2l/C271)!!/C30(2l/C271)!;givesa2l/C271/C30(/C281)l 2ll!(2l/C271)!!a1/C30(/C281)l (2l/C271)!a1: (15) Plugging back into (2) with k/C30m/C30/C281=2 gives y/C30x/C281=2X/C12 n/C300anxn /C30x/C281=2X/C12 n/C301;3;5;...anxn/C27X/C12 n/C300;2;4;...anxn"# /C30x/C281=2X/C12 l/C300a2lx2l/C27X/C12 l/C300a2l/C271x2l/C271"# /C30x/C281=2a0X/C12 I/C300(/C281)l (2l)!x2l/C27a1X/C12 I/C300(/C281)l (2l/C271)!x2l/C271"# /C30x/C281=2(a0cosx/C27a1sinx): (16) The B ESSEL FUNCTIONS of order 91=2 are therefore defined as J/C281=2(x)/C13ffiffiffiffiffiffi 2 pxs cosx (17) J1=2(x)/C13ffiffiffiffiffiffi 2 pxs sinx; (18) so the general solution for m/C3091=2i s y/C30a?0J/C281=2(x)/C27a?1J1=2(x): (19) Now, consider a general m"/C281=2:Equation (9) requires a1(2m/C271)/C300 (20) [ann(2m/C27n)/C27an/C282]xm/C27n/C300 (21) forn/C302, 3, . . ., so a1/C300 (22) an/C30/C281 n(2m/C27n)an/C282 (23) forn/C302, 3, . . . Let n/C132l/C271;where l/C301, 2, . . ., then a2l/C271/C30/C281 (2l/C271)[2(m/C271)/C271]a2l/C281/C30/C1/C1/C1 /C30.../C30f(n;m)a1/C300; (24) where f(n;m) is the function of landmobtained by iterating the recursion relationship down to a1:Now letn/C132l;where l/C301, 2, . . ., so a2l/C30/C281 2l(2m/C272l)a2l/C282/C30/C281 4l(m/C27l)a2l/C282 /C30(/C281)l [4l(m/C27l)][4(l/C281)(m/C27l/C281)]/C1/C1/C1[4 /C215(m/C271)]a0: (25) Plugging back into (9), y/C30X/C12 n/C300anxn/C27m/C30X/C12 n/C301;3;5;...anxn/C27m/C27X/C12 n/C300;2;4;...anxn/C27m /C30X/C12 l/C300a2l/C271x2l/C27m/C271/C27X/C12 l/C300a2lx2l/C27m /C30a0X/C12 l/C300(/C281)l [4l(m/C27l)][4(l/C281)(m/C27l/C281)]/C1/C1/C1[4(m/C271)]x2l/C27m /C30a0X/C12 l/C300[(/C281)lm(m/C281)/C1/C1/C11]x2l/C27m [4l(m/C27l)][4(l/C281)(m/C27l/C281)]/C1/C1/C1[4(m/C271)m(m/C281)/C1/C1/C11] /C30a0X/C12 l/C300(/C281)lm! 22ll!(m/C27l)!x2l/C27m; (26) Now define Jm(x)/C13X/C12 l/C300(/C281)l 22l/C27ml!(m/C27l)!x2l/C27m; (27) where the factorials can be generalized to GAMMA FUNCTIONS for nonintegral m. The above equation then becomes y/C30a02mm!Jm(x)/C30a?0Jm(x): (28) Returning to equation (5) and examining the case k/C30 /C28m; a1(1/C282m)/C27X/C12 n/C302[ann(n/C282m)/C27an/C282]xn/C28m/C300:(29) However, the sign of mis arbitrary, so the solutions must be the same for /C27mand/C28m:We are therefore free to replace /C28mwith/C28mjj;so a1(1/C272mjj)/C27X/C12 n/C302[ann(n/C272mjj)/C27an/C282]xmjj/C27n/C300; (30) and we obtain the same solutions as before, but with mreplaced by mjj: Jm(x)/C30X/C12 l/C300(/C281)l 22l/C27mjjl!(mjj/C27l)lx2l/C27mjjformjj"/C281 2 ffiffiffiffiffiffi 2 pxs cosx form/C30/C2812 ffiffiffiffiffiffi 2 pxs sinx form/C3012:8 >>>>>>>>>>< >>>>>>>>>>: (31) We can relate J mandJ/C28m(when mis an INTEGER )b y writing J/C28m(x)/C30X/C12 l/C300(/C281)l 22l/C28ml!(l/C28m)!x2l/C28m: (32) Now let l/C13l?/C27m:ThenJ/C28m(x)/C30X/C12 l?/C27m/C300(/C281)l?/C27m 22l?/C27m(l?/C27m)!l!x2l?/C27m /C30X/C281 l?/C30/C28 m(/C281)l?/C27m 22l?/C27ml?!(l?/C27m)!x2l?/C27m /C27X/C12 l?/C300(/C281)l?/C27m 22l?/C27ml?!(l?/C27m)!x2?l/C27m: (33) Butl?!/C30/C12forl?/C30/C28 m;...;/C281;so the DENOMINATOR is infinite and the terms on the right are zero. We therefore have J/C28m(x)/C30X/C12 l/C300(/C281)l/C27m 22l/C27ml!(l/C27m)!x2l/C27m/C30(/C281)mJm(x):(34) Note that the B ESSEL DIFFERENTIAL EQUATION is second-order, so there must be two linearly indepen-dent solutions. We have found both only for mjj/C301=2: For a general nonintegral order, the independentsolutions are J mandJ/C28m:When mis an INTEGER , the general (real) solution is OF THE FORM Zm/C13C1Jm(x)/C27C2Ym(x); (35) where Jmis a Bessel function of the first kind, Ym (a.k.a. Nm) is the B ESSEL FUNCTION OF THE SECOND KIND (a.k.a. N EUMANN FUNCTION or W EBER FUNC- TION ), and C1and C2are constants. Complex solu- tions are given by the H ANKEL FUNCTIONS (a.k.a. BESSEL FUNCTIONS OF THE THIRD KIND ). The Bessel functions are ORTHOGONAL in [0 ;1] with respect to the weight factor x. Except when 2 nis a NEGATIVE INTEGER , Jm(z)/C30z/C281=2 22m/C271=2im/C271=2G(m/C271)M0;m(2iz); (36) where G(x) is the GAMMA FUNCTION and M0;mis a WHITTAKER FUNCTION . In terms of a CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND , the Bessel function is written Jn(z)/C30(1 2z)n G(n/C271)0F1(n/C271;/C2814z2): (37) A derivative identity for expressing higher order Bessel functions in terms of J0(x)i s Jn(x)/C30inTnid dx ! J0(x); (38) where Tn(x)i saC HEBYSHEV POLYNOMIAL OF THE FIRST KIND . Asymptotic forms for the Bessel functions are Jm(x):1 G(m/C271)x 2 !m (39) forx/C261 and Jm(x):ffiffiffiffiffiffi 2 pxs cos x/C28mp 2/C28p 4 ! (40) forx/C271:/ A derivative identity is d dx[xmJm(x)]/C30xmJm/C281(x): (41) An integral identity is gu 0u?J0(u?)du?/C30uJ1(u): (42) Some sum identities are 1/C30[J0(x)]2/C272X/C12 k/C301[Jk(x)]2(43) (Abramowitz and Stegun 1972, p. 363), 1/C30J0(x)/C272X/C12 k/C301J2k(x) (44) (Abramowitz and Stegun 1972, p. 361), 0/C30X2n k/C300(/C281)kJk(z)J2n/C28k(z)/C272X/C12 k/C301Jk(z)J2n/C27k(z) (45) forn]1 (Abramowitz and Stegun 1972, p. 361), Jn(2z)/C30Xn k/C300Jk(z)Jn/C28k(z) /C272X/C12 k/C301(/C281)kJk(z)Jn/C27k(z) (46) (Abramowitz and Stegun 1972, p. 361), and the JACOBI- ANGER EXPANSION eizcosu/C30X/C12 n/C30/C28/C12inJn(z)einu; (47) which can also be written eizcosu/C30J0(z)/C272X/C12 n/C301inJn(z) cos( nu): (48) The Bessel function addition theorem states Jn(y/C27z)/C30X/C12 m/C30/C28/C12Jm(y)Jn/C28m(z): (49) The first kroots x1;... ,xkof the Bessel function Jn(x) can be found in Mathematica (Wolfram Research, Urbana, IL) using the command BesselJZeros [n, k] in the Mathematica add-on package Numerical- Math‘BesselZeros‘ (which can be loaded with the command BBNumericalMath‘ ). R OOTS of the FUNCTION Jn(x) are given in the following table.zero /J0(x)// J1(x)// J2(x)// J3(x)// J4(x)// J5(x)/ 1 2.4048 3.8317 5.1336 6.3802 7.5883 8.7715 2 5.5201 7.0156 8.4172 9.7610 11.0647 12.3386 3 8.6537 10.1735 11.6198 13.0152 14.3725 15.7002 4 11.7915 13.3237 14.7960 16.2235 17.6160 18.98015 14.9309 16.4706 17.9598 19.4094 20.8269 22.2178 The first kroots x1;... ,xkof the derivative of the Bessel function J?n(x) can be found in Mathematica using the command BesselJPrimeZeros [n,k]i n theMathematica add-on package NumericalMath‘- BesselZeros‘ (which can be loaded with the com- mandBBNumericalMath‘ ). The first few such ROOTS are given in the following table. zero /J?0(x)// J?1(x)// J?2(x)// J?3(x)// J?4(x)// J?5(x)/ 1 3.8317 1.8412 3.0542 4.2012 5.3175 6.41562 7.0156 5.3314 6.7061 8.0152 9.2824 10.5199 3 10.1735 8.5363 9.9695 11.3459 12.6819 13.9872 4 13.3237 11.7060 13.1704 14.5858 15.9641 17.3128 5 16.4706 14.8636 16.3475 17.7887 19.1960 20.5755 Various integrals can be expressed in terms of Bessel functions Jn(z)/C301 pgp 0cos(zsinu/C28nu)du; (50) which is B ESSEL’S FIRST INTEGRAL , Jn(z)/C30i/C28n pgp 0eizcosucos(nu)du (51) Jn(z)/C301 2ping2p 0eizcosfeinfdf (52) forn/C301, 2, . . ., Jn(z)/C302 pxn (2rn/C281)!!gp=2 0sin2nucos(xcosu)du(53) forn/C301, 2, . . ., Jn(x)/C301 2pigge(x=2)(z/C281=z)z/C28n/C281dz (54) forn/C21/C281=2:The Bessel functions are normalized so that g/C12 0Jn(x)dx/C301 (55) for positive integral (and real) n. Integrals involving J1(x) include g/C12 0J1(x) x"#2 dx /C304 3 p (56) g/C12 0J1(x) x"#2 xdx/C301 2 : (57) The special case of n /C300 gives J0(z) as the series J0(z) /C30X/C12 k /C300(/C281)k(1 4z2)k (k!)2 (58) (Abramowitz and Stegun 1972, p. 360), or the integral J0(z)/C301 pgp 0eizcosudu: (59) See also BESSEL FUNCTION OF THE SECOND KIND, DEBYE’S ASYMPTOTIC REPRESENTATION ,D IXON- FER- RAR FORMULA ,H ANSEN- BESSEL FORMULA ,KAPTEYN SERIES ,K NESER- SOMMERFELD FORMULA ,M EHLER’S BESSEL FUNCTION FORMULA ,NICHOLSON’S FORMULA , POISSON’S BESSEL FUNCTION FORMULA ,R AYLEIGH FUNCTION ,SCHLA ¨ FLI’S FORMULA ,SCHLO ¨ MILCH’S SER- IES,SOMMERFELD’S FORMULA ,SONINE- SCHAFHEITLIN FORMULA ,W ATSON’S FORMULA ,W ATSON- NICHOLSON FORMULA ,W EBER’S DISCONTINUOUS INTEGRALS ,W E- BER’S FORMULA ,WEBER- SONINE FORMULA ,WEYRICH’S FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Functions Jand Y."§9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358 /C1/64, 1972. Arfken, G. "Bessel Functions of the First Kind, Jn(x)/" and "Orthogonality." §11.1 and 11.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573 /C1/91 and 591 /C1/96, 1985. Lehmer, D. H. "Arithmetical Periodicities of Bessel Func- tions." Ann. Math. 33, 143/C1/50, 1932. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 619 /C1/22, 1953. Spanier, J. and Oldham, K. B. "The Bessel Coefficients J0(x) and J1(x)/" and "The Bessel Function Jn(x):/" Chs. 52 /C1/3i n An Atlas of Functions. Washington, DC: Hemisphere, pp. 509 /C1/20 and 521 /C1/32, 1987. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Bessel Function of the Second Kind A Bessel function of the second kind Yn(x)i sa solution to the B ESSEL DIFFERENTIAL EQUATION which is singular at the origin. Bessel functions of the second kind are also called N EUMANN FUNCTIONS or WEBER FUNCTIONS . The above plot shows Yn(x) for n/C301, 2, . . ., 5. Letv/C13Jm(x) be the first solution and ube the other one (since the B ESSEL DIFFERENTIAL EQUATION is second-order, there are two LINEARLY INDEPENDENT solutions). Then xuƒ/C27u?/C27xu/C300 (1) xvƒ/C27v?/C27xv/C300: (2) Take v/C29(1) minus u/C29(2), x(uƒv/C28uvƒ)/C27u?v/C28uv?/C300 (3) d dx[x(u?v/C28uv?)]/C300; (4) sox(u?v/C28uv?)/C30B;where Bis a constant. Divide by xv2; u?v/C28uv? v2/C30d dxu v ! /C30B xv2(5) u v/C30A/C27Bgdx xv2: (6) Rearranging and using v/C13Jm(x) gives u/C30AJm(x)/C27BJm(x)gdx xJ2 m(x) /C13A?Jm(x)/C27B?Ym(x); (7) where Ymis the so-called Bessel function of the second kind. /Yn(z) can be defined by Yn(z) /C30Jv(z) cos(np) /C28 J/C28 n(z) sin( np) (8) (Abramowitz and Stegun 1972, p. 358), where Jn(z)is aB ESSEL FUNCTION OF THE FIRST KIND and, for n an integer n by the SERIES Yn(z) /C30/C28(1 2z) /C28n pXn/C281 k /C300(n /C28 k /C28 1)! k!(14z2)k /C272 pln(12z)Jn(z) /C28(12z)n pX/C12 k /C300[c0(k /C271) /C27 c0(n /C27k /C271)](/C2814z2)k k!(n /C27 k)! ; (9) where c0(x) is the DIGAMMA FUNCTION (Abramowitz and Stegun 1972, p. 360). The function has the integral representations Yn(z) /C301 p1ntp 0 sin(z sin u /C28 nu) du /C281 p1 nt/C120 [e nt /C27e /C28 nt(/C281)n]e /C28z sin ht dt: (10) /C30/C282(1 2 x) /C28v ffiffiffippG(1 2 /C28 n) g/C12 1cos(xt) dt (t2 /C28 1)n/C271 =2 (11) (Abramowitz and Stegun 1972, p. 360). ASYMPTOTIC SERIES are Ym(x) /C22 p[ln(1 2 x) /C27 g] m /C300; x /C261 /C28G(m) p2 x !m m "0; x /C2618 >>>>< >>>>:(12) Y m(x) /C2ffiffiffiffiffiffi 2 pxs sin x /C28mp 2/C28p 4 ! x /C271 ; (13) where G(z)isa GAMMA FUNCTION . For the special case n /C300, Y0(x) is given by the series Y0(z) /C302 p[ln(1 2z) /C27 g]J0(z) /C27X/C12 k/C301(/C281)k/C271Hk(1 4z2)k (k!)2() ;(14)(Abramowitz and Stegun 1972, p. 360), where gis the EULER- MASCHERONI CONSTANT andHnis a HARMONIC NUMBER . See also BESSEL FUNCTION OF THE FIRST KIND, BOURGET’S HYPOTHESIS ,HANKEL FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Functions Jand Y."§9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358 /C1/64, 1972. Arfken, G. "Neumann Functions, Bessel Functions of the Second Kind, Nn(x):/"§11.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596 /C1/ 04, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 625 /C1/27, 1953. Spanier, J. and Oldham, K. B. "The Neumann Function Yn(x):/" Ch. 54 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 533 /C1/42, 1987. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Bessel Function of the Third Kind HANKEL FUNCTION Bessel Polynomial Krall and Find (1948) defined the Bessel polynomials as the function yn(x)/C30Xn k/C300(n/C27k!) (n/C28k)!k!x 2 !k (1) which satisfies the differential equation x2yƒ/C27(2x/C272)y?/C27n(n/C271)y/C300: (2) Carlitz (1957) subsequently considered the related polynomials pn(x)/C30xnyn/C2811 x ! : This polynomial forms an associated S HEFFER SE- QUENCE with f(t)/C30t/C281 2t2: (3) This gives the GENERATING FUNCTION X/C12 k/C300pk(x) k!tk/C30ex(1/C28ffiffiffiffiffiffiffiffi 1/C282tp ): (4) The explicit formula is pn(x)/C30X/C12 k/C301(2n/C28k/C281)! 2n/C28k(k/C281)!(n/C28k)!xk: (5) The polynomials satisfy the recurrence formula pƒn(x) /C282p ?n(x) /C272npn/C281(x) /C300 : (6) The first few polynomials are p0(x) /C301 p1(x) /C30x p2(x) /C30x2 /C27x p3(x) /C30x3 /C273x2 /C273x p4(x) /C30x4 /C276x3 /C2715x2 /C2715x: See also BESSEL FUNCTION ,SHEFFER SEQUENCE References Carlitz, L. "A Note on the Bessel Polynomials." Duke Math. J. 24, 151 /C1/62, 1957. Grosswald, E. Bessel Polynomials. New York: Springer- Verlag, 1978. Krall, H. L. and Fink, O. "A New Class of Orthogonal Polynomials: The Bessel Polynomials." Trans. Amer. Math. Soc. 65, 100 /C1/15, 1948. Roman, S. "The Bessel Polynomials." §4.1.7 in The Umbral Calculus. New York: Academic Press, pp. 78 /C1/2, 1984. Bessel Transform HANKEL TRANSFORM Bessel’s Correction The factor (N /C281)=N in the relationship between the VARIANCE s and the EXPECTATION VALUES of the SAMPLE VARIANCE , s2/CP0/CPP /C30N /C28 1 Ns2 ; (1) where s2 /C13/C142x2 /C143/C28/C142x/C1432 : (2) For two samples, ˆs2 /C30N1s2 1 /C27 N2s22 N1 /C27 N2 /C28 2 : (3) See also SAMPLE VARIANCE ,VARIANCE References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 161, 1951. Bessel’s Finite Difference Formula An INTERPOLATION formula also sometimes known as fp /C30f0 /C27p d1 =2 /C27B2( d2 0 /C27 d21) /C27B3 d31 =2 /C27B4( d40 /C27 d41) /C27B5 d51=2 /C27/C1/C1/C1; (1) for p /C23 [0; 1]; where d is the CENTRAL DIFFERENCE and B2n /C131 2 G2n /C1312 (E2n /C27F2n) (2)B2n/C271 /C13G2n/C271 /C281 2 G2n /C1312(F2n /C28E2n) (3) E2n /C13G2n /C28G2n/C271 /C13B2n /C28B2n /C271 (4) F2n /C13G2n/C271 /C13B2n /C28B2n/C271 ; (5) where Gkare the COEFFICIENTS from GAUSS’S BACK- WARD FORMULA and GAUSS’S FORWARD FORMULA and Ekand Fkare the COEFFICIENTS from EVERETT’S FORMULA . The Bk/s also satisfy B2n(p) /C30B2n(q) (6) B2n /C271(p) /C30/C28B2n/C271(q) ; (7) for q/C131/C28p: (8) See also EVERETT’S FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 90 /C1/1, 1990. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Whittaker, E. T. and Robinson, G. "The Newton-Bessel Formula." §24 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 39 /C1/0, 1967. Bessel’s First Integral Jn(x)/C301 pgp 0cos(nu/C28xsinu)du; where Jn(x)i saB ESSEL FUNCTION OF THE FIRST KIND . Bessel’s Formula BESSEL’S FINITE DIFFERENCE FORMULA ,B ESSEL’S INTERPOLATION FORMULA ,BESSEL’S STATISTICAL FOR- MULA Bessel’s Inequality Iff(x)i s PIECEWISE CONTINUOUS and has a general FOURIER SERIES X iaifi(x) (1) with WEIGHTING FUNCTION w(x);it must be true that gf(x)/C28X iaifi(x)"#2 w(x)dx]0 (2) gf2(x)w(x)dx/C282X iaigf(x)fi(x)w(x)dx /C27X ia2 i f f2 i (x)w(x) dx ]0: (3) But the COEFFICIENT of the generalized FOURIER SERIES is given by am /C13g f(x) fm(x)w(x) dx; (4) so g f2(x)w(x) dx /C282X ia2 i /C27X ia2i ]0 (5) g f2(x)w(x) dx ]X ia2i : (6) Equation (6) is an inequality if the functions fiare not COMPLETE . If they are COMPLETE , then the inequality (2) becomes an equality, so (6) becomes an equality and is known as PARSEVAL’S THEOREM .If f(x) has a simple FOURIER SERIES expansion with COEFFICIENTS a0 ; a1 ; an , a p and b1 ; ...,bn ; then 1 2 a2 0 /C27X/C12 k ¼1(a2k /C27b2k) 51 p g p /C28 p[f(x)]2 dx: (7) The inequality can also be derived from SCHWARZ’S INEQUALITY ½/C142f ½g /C143½2 5/C142f ½f /C143/C142g ½g /C143 (8) by expanding g in a superposition of EIGENFUNCTIONS of f, g /C30ai aifi : Then /C142f ½g/C143/C30X iai /C142f ½fi /C1435X iai (9) ½/C142f ½g /C143½2 5X iai/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP22 /C30X iai !X i¯ai ! /C30X iai ¯ai 5/C142f ½f /C143/C142g ½g /C143; (10) where ¯f is the COMPLEX CONJUGATE .Ifg is normal- ized, then /C142g ½g/C143/C301 and /C142f ½f /C143]X iai ¯ai (11) See also SCHWARZ’S INEQUALITY ,TRIANGLE INEQUAL- ITY References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 526 /C1/27, 1985. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1102, 2000. Bessel’s Interpolation Formula BESSEL’S FINITE DIFFERENCE FORMULABessel’s Second Integral POISSON INTEGRAL Bessel’s Statistical Formula Let ¯x1 and s2 1 be the observed mean and variance of a sample of N1drawn from a normal universe with unknown mean m(1) and let ¯x2 and s22be the observed mean and variance of a sample of N2drawn from a normal universe with unknown mean m(2) : Assume the two universes have a common variance s2 ; and define ¯w /C13 ˆx1 /C28 ¯x2 (1) v /C13 m(1) /C28 m(2) (2) N /C13N1 /C27N2 (3) Then t /C30¯w /C28 v sw =ffiffiffiffiffi Np/C30¯w /C28 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i/C301(wi /C28 ¯w)2 N(N /C28 1)s (4) is distributed as STUDENT’S T-DISTRIBUTION fn(t) with n /C30N /C282 :/ See also STUDENT’S T-DISTRIBUTION References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 186, 1951. BesselI MODIFIED BESSEL FUNCTION OF THE FIRST KIND BesselJ BESSEL FUNCTION OF THE FIRST KIND BesselK MODIFIED BESSEL FUNCTION OF THE SECOND KIND BesselY BESSEL FUNCTION OF THE SECOND KIND Beta A financial measure of a fund’s sensitivity to market movements which measures the relationship between a fund’s excess return over Treasury Bills and the excess return of a benchmark index (which, by definition, has b /C301): A fund with a beta of b has performed r /C13 b /C281 ðÞ /C29100% better (or rjjworse if r B0) than its benchmark index (after deducting the T-bill rate) in up markets and rjjworse (or rjjbetter if rB0) in down markets. See also ALPHA ,BETA DISTRIBUTION ,BETA FUNCTION , BETA INTEGRAL ,SHARPE RATIO Beta Distribution A general type of STATISTICAL DISTRIBUTION which is related to the GAMMA DISTRIBUTION . Beta distribu- tions have two free parameters, which are labeled according to one of two notational conventions. The usual definition calls these a and b; and the other uses b?/C13 b /C281 and a?/C13 a /C281 (Beyer 1987, p. 534). The above plots are for various values of ( a; b): The domain is [0; 1]; and the probability function P(x) and DISTRIBUTION FUNCTION D(x) are given by P(x) /C30(1 /C28 x)b/C281xa/C281 B( a; b)/C30G( a /C27 b) G(a) G(b)(1 /C28x) b/C281x a/C281(1) D(x) /C30 I(x; a ; b); (2) where B(a ; b) is the BETA FUNCTION , I(x; a; b) is the REGULARIZED BETA FUNCTION , and a; b > 0 : The distribution is normalized since g1 0P(x) dx /C30G( a /C27 b) G( a) G( b) g1 0xa /C281(1 /C28x) b/C281 dx (3) /C30G( a /C27 b) G(a) G(b)B( a; b) /C301: (4) The CHARACTERISTIC FUNCTION is f(t) /C30Fxa /C281(1 /C28 x)b /C281 b(a ; b)[1 2sgn(1 /C28x) /C27sgn x]() /C301F1(a; a /C27b; it) ; (5) where F[f]isaF OURIER TRANSFORM with parameters a /C30b /C301 and1F1(a; b; z)isa CONFLUENT HYPERGEO- METRIC FUNCTION . The MEAN is m /C30G(a /C27 b) G( a) G( b) g1 0xa /C281(1 /C28x) b/C281xdx /C30G(a /C27 b) G( a) G( b)B(a /C271; b) /C30G( a /C27 b) G( a) G( b)G( a /C27 1)G(b) G( a /C27 b /C27 1) /C30a a /C27 b : (6) The RAW MOMENTS are given by m?r /C30g1 0P(x)(x /C28 m)r dx /C30G( a /C27 b) G( a /C27 r) G( a /C27 b /C27 r) G(a)(7) (Papoulis 1984, p. 147), and the CENTRAL MOMENTS bymr /C30/C28a a /C27 b !r 2F1/C28r ; a; a /C27 b;a /C27 b a ! ; (8) where 2F1(a; b; c; x)isa HYPERGEOMETRIC FUNC- TION . The VARIANCE ,SKEWNESS , and KURTOSIS are therefore given by s2/C30ab (a/C27b)2(a/C27b/C271)(9) g1/C302(b/C28a)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27a/C27bp ffiffiffiffiffiffiabp(2/C27a/C27b)(10) g 2/C306[a3/C27a2(1/C282b)/C27b2(1/C27b)/C282ab(2/C27b)] ab(a/C27b/C272)(a/C27b/C273): (11) The MODE of a variate distributed as b(a;b)i s ˆx/C30a/C281 a/C27b/C282: (12) See also GAMMA DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 944 /C1/45, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 534 /C1/35, 1987. Jambunathan, M. V. "Some Properties of Beta and Gamma Distributions." Ann. Math. Stat. 25, 401/C1/05, 1954. Kolarski, I. "On Groups of nIndependent Random Variables whose Product Follows the Beta Distribution." Colloq. Math. IX Fasc. 2, 325/C1/32, 1962. Krysicki, W. "On Some New Properties of the Beta Distribu- tion." Stat. Prob. Let. 42, 131/C1/37, 1999. Beta Exponential Function Another " BETA FUNCTION " defined in terms of an integral is the "exponential" beta function, given by bn(z)/C13g1 /C281tne/C28ztdt (1) /C30n!z/C28(n/C271)ezXn k/C300(/C281)kzk k!/C28e/C28zXn k/C300zk k!"# : (2) If n is an integer, then bn(z) /C30(/C281)n/C271E /C28n(/C28z) /C28E /C28n(z) ; (3) where En(z) is the EN-FUNCTION . The exponential beta function satisfies the RECURRENCE RELATION zbn(z) /C30(/C281)nez /C28e /C28z /C27nbn/C281(z) : (4) The values for n /C300, 1, and 2 are b0(z) /C302 sinh z z (5) b1(z) /C302(sinh z /C28 z cosh z) z2: (6) b2(z)/C302(2/C27z2) sinh z/C284zcosh z z3: (7) See also ALPHA FUNCTION , EN-FUNCTION Beta Function The beta function is the name used by Legendre and Whittaker and Watson (1990) for the BETA INTEGRAL (also called the Eulerian integral of the first kind). Toderive the integral representation of the beta func-tion, write the product of two FACTORIALS as m!n!/C30g/C12 0e/C28uumdug/C12 0e/C28vvndv: (1) Now, let u/C13x2;v/C13y2;so m!n!/C304g/C12 0e/C28x2x2m/C271dxg/C12 0e/C28y2y2n/C271dy /C304g/C12 /C28/C12g/C12 /C28/C12e/C28(x2/C27y2)x2m/C271y2n/C271dx dy : (2) Transforming to POLAR COORDINATES with x/C30rcosu; y/C30rsinu m!n!/C304gp=2 0g/C12 0e/C28r2(rcosu)2m/C271(rsinu)2n/C271rd rd u /C304g/C12 0e/C28r2r2m/C272n/C273drgp=2 0cos2m/C271usin2n/C271udu /C302(m/C27n/C271)!gp=2 0cos2m/C271usin2n/C271udu:(3) The beta function is then defined by B(m/C271;n/C271)/C30B(n/C271;m/C271) /C132gp=2 0cos2m/C271usin2n/C271udu/C30m!n! (m/C27n/C271)!:(4) Rewriting the arguments,B(p;q)/C30G(p)G(q) G(p/C27q)/C30(p/C281)!(q/C281)! (p/C27q/C281)!: (5) The general trigonometric form is gp=2 0sinnxcosmxd x/C301 2B(12(n/C271);12(m/C271)): (6) Equation (6) can be transformed to an integral over POLYNOMIALS by letting u/C13cos2u; B(m/C271;n/C271)/C13m!n! (m/C27n/C271)!/C30g1 0um(1/C28u)ndu(7) B(m;n)/C13G(m)G(n) G(m/C27n)/C30g1 0um/C281(1/C28u)n/C281du: (8) The beta function is implemented in Mathematica as Beta [a,b]. For any z1;z2with /R[z1];R[z2]>0; B(z1;z2)/C30B(z2;z1) (9) (Krantz 1999, p. 158). The INCOMPLETE BETA FUNCTION B(z;a;b);imple- mented in Mathematica asBeta [z,a,b], is defined by the integral in (8) with an upper limit of zinstead of 1. The REGULARIZED BETA FUNCTION I(z;a;b); implemented in Mathematica asBetaRegulari- zed[z,a,b] is defined by I(z;a;b)/C30B(z;a;b) B(a;b): (10) To put it in a form which can be used to derive the LEGENDRE DUPLICATION FORMULA , let x/C13ffiffiffiup;sou/C30 x2anddu/C302xd x ;and B(m;n)/C30g1 0x2(m/C281)(1/C28x2)n/C281(2xd x) /C302g1 0x2m/C281(1/C28x2)n/C281dx: (11) To put it in a form which can be used to develop integral representations of the B ESSEL FUNCTIONS and HYPERGEOMETRIC FUNCTION , let u/C13x=(1/C27x);so B(m/C271;n/C271)/C30g/C12 0umdu (1/C27u)m/C27n/C272: (12) Derivatives of the beta function are given by d daB(a;b)/C30B(a;b)[c0(a)/C28c0(a/C27b)] (13) d dbB(a;b)/C30B(a;b)[c0(b)/C28c0(a/C27b)] (14) d2 da2B(a;b)/C30B(a;b) /C2 [ c0(a) /C28 c0(a /C27b)]2 /C27 c1(a) /C28 c1(a /C27b)/C8/C9 ; (15) d2 db2B(a ; b) /C30B(a ; b) /C2 [c0(b) /C28 c0(a /C27b)]2 /C27 c1(b) /C28 c1(a /C27b)/C8/C9 ; (16) d2 da dbB(a; b) /C30B(a ; b)[c0(a) /C28 c0(a /C27b)][ c0(b) /C28 c0(a /C27b)] f /C28c1(a /C27b) (17) where cn(x) is the POLYGAMMA FUNCTION . Various identities can be derived using the GAUSS MULTIPLICATION FORMULA B(np ; nq) /C30G(np) G(nq) G[n(p /C27 q)] /C30n/C28nqB(p; q)Bp/C271 n ; q ! /C1/C1/C1Bp/C27n /C28 1 n; q ! B(q; q)B(2q ; q) /C1/C1/C1B([n /C28 1]q; q): (18) Additional identities include B(p; q /C271) /C30G(p)G(q /C27 1) G(p /C27 q /C27 1) /C30q pG(p /C27 1)G(q) G([p /C27 1]q) /C30q pB(p /C271; q) (19) B(p; q) /C30B(p /C271; q) /C27B(p ; q /C271) (20) B(p; q /C271) /C30q p /C27 qB(p; q) : (21) If n is a POSITIVE INTEGER , then B(p ; n /C271) /C301 /C215 2 /C1/C1/C1n p(p /C27 1) /C1/C1/C1(p /C27 n)(22) B(p ; p)B(p /C271 2 ; p /C2712) /C30p 24p /C281p (23) B(p /C27q)B(p /C27q; r) /C30B(q; r)B(q /C27r ; p) : (24) Gosper gives the general formulas Y2n i/C300Bi 2n /C27 1 /C27a ;i 2n /C27 1 /C27b ! /C30(2n /C27 1)(2n/C271) =2 pnB(n;1 2[(b /C27 a)(2n /C27 1) /C27 1])B(a(2n /C27 1); b(2n /C27 1)) (n /C28 1)! (25) for ODD n, andY2n/C281 i/C300Bi 2n /C27a ;i 2n /C27b ! /C30nn pnB(n; 2(a /C27 b)n)B(2an; 2bn) 22(a /C27b)n/C30n /C301(n /C28 1)!B((a /C27 b)n ; (a /C27 b /C27 1)n) ; (26) which are an immediate consequence of the analo- gous identities for GAMMA FUNCTIONS . Plugging n /C301 andn/C302 into the above give the special cases B(a;b)B(a/C271 3;b/C2713)B(a/C2723;b/C2723) /C306pffiffiffi 3p B(3a;3b) 1/C273(a/C27b)(27) B(a;b)B(a/C271 4;b/C2714)B(a/C2712;b/C2712)B(a/C2734;b/C2734) /C3023/C284(a/C27b)p2B(4a;4b) (a/C27b)[1/C274(a/C27b)]B(2(a/C27b);2(a/C27b/C271): (28) See also BETA INTEGRAL ,CENTRAL BETA FUNCTION , DIRICHLET INTEGRALS ,G AMMA FUNCTION ,INCOM- PLETE BETA FUNCTION ,REGULARIZED BETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Beta Function" and "Incomplete Beta Function." §6.2 and 6.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 258 and 263, 1972. Arfken, G. "The Beta Function." §10.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 560 /C1/65, 1985. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Beta Function." §1.5 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 9 /C1/3, 1981. Jeffreys, H. and Jeffreys, B. S. "The Beta Function." §15.02 inMethods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 463 /C1/64, 1988. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities.Braunschweig, Germany: Vieweg, pp. 6 /C1 /, 1998. Krantz, S. G. "The Beta Function." §13.1.11 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 157 /C1/58, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 425, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gamma Function, Beta Function, Factorials,Binomial Coefficients" and "Incomplete Beta Function,Student’s Distribution, F-Distribution, Cumulative Bino-mial Distribution." §6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206 /C1 /09 and 219 /C1/23, 1992. Spanier, J. and Oldham, K. B. "The Incomplete Beta Func- tion B(v;m;x):/" Ch. 58 in An Atlas of Functions. Wa- shington, DC: Hemisphere, pp. 573 /C1/80, 1987. Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Beta Function (Exponential) mr /C30/C28a a /C27 b !r 2F1/C28r ; a; a /C27 b;a /C27 b a ! ; Another "BETA FUNCTION " defined in terms of an integral is the "exponential" beta function, given by 2F1(a; b; c; x)u2ab ( a /C27 b)2( a /C27 b /C27 1)(1) /C302(b /C28 a)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 a /C27 bp ffiffiffiffiffiffiabp(2 /C27 a /C27 b) (2) The exponential beta function satisfies the RECUR- RENCE RELATION 6[a3 /C27 a2(1 /C28 2b) /C27 b2(1 /C27 b) /C28 2ab(2 /C27 b)] ab( a /C27 b /C27 2)(a /C27 b /C27 3) : (3) The first few integral values are b( a; b) /C30 ˆx /C30a /C28 1 a /C27 b /C28 2 : (4) (5) g1 /C281tne /C28zt dt /C30n!z/C28(n/C271) ezXn k /C300(/C281)kzk k!/C28e /C28zXn k /C300zk k!"# : (6) See also ALPHA FUNCTION Beta Integral The integral g1 0xp(1 /C28x)q dx called the EULERIAN INTEGRAL OF THE FIRST KIND by Legendre and Whittaker and Watson (1990). The solution is the BETA FUNCTION B(p /C271; q /C271):/ See also BETA FUNCTION ,EULERIAN INTEGRAL OF THE FIRST KIND,E ULERIAN INTEGRAL OF THE SECOND KINDReferences Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Beta Prime Distribution A distribution with probability function P(x) /C30xa/C281(1 /C27 x) /C28 a/C28 b B(a; b); where B is a BETA FUNCTION . The MODE of a variate distributed as b?(a; b)is ˆx /C30a /C28 1 b /C27 1 : If x is a b?( a; b) variate, then 1 =x is a b?( b; a) variate. If x is a b( a; b) variate, then (1 /C28x)=x and x=(1 /C28x) are b?( b; a) and b?( a; b) variates. If x and y are g( a1) and g( a2) variates, then x=y is a b?(a1 ; a2) variate. If x2 =2 and y2 =2 are g(1=2) variates, then z2 /C13 x=yðÞ2is a b?(1=2; 1=2) variate. BetaRegularized REGULARIZED BETA FUNCTION Bethe Lattice CAYLEY TREE Betrothed Numbers QUASIAMICABLE PAIR Betti Group The free part of the HOMOLOGY GROUP with a domain of COEFFICIENTS in the GROUP of INTEGERS (if this HOMOLOGY GROUP is finitely generated). See also HOMOLOGY GROUP References Alexandrov, P. S. Combinatorial Topology. New York: Do- ver, 1998. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 380, 1988. Betti Number Betti numbers are topological objects which were proved to be invariants by Poincare ´, and used by him to extend the POLYHEDRAL FORMULA to higher dimensional spaces. Informally, the Betti number isthe maximum number of cuts that can be madewithout dividing a surface into two separate pieces (Gardner 1984, pp. 9 /C10). Formally, the nth Betti number is the rank of the nth HOMOLOGY GROUP of aTOPOLOGICAL SPACE . The following table gives the Betti number of some common surfaces. SURFACE Betti number CROSS-CAP 1 CYLINDER 1 KLEIN BOTTLE 2 MO¨ BIUS STRIP 1 plane lamina 0 PROJECTIVE PLANE 1 SPHERE 0 TORUS 2 Let prbe the RANK of the HOMOLOGY GROUP Hrof a TOPOLOGICAL SPACE K. For a closed, orientable sur- face of GENUS g, the Betti numbers are p0 /C301; p1 /C302g; and p2 /C301: For a NONORIENTABLE SURFACE with k CROSS-CAPS , the Betti numbers are p0 /C301; p1 /C30k /C281/, and p2 /C300:/ See also CHROMATIC NUMBER ,EULER CHARACTERIS- TIC,GENUS (SURFACE ), HOMOLOGY GROUP ,POINCARE ´ DUALITY ,TOPOLOGICAL SPACE References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 9 /C1/1 and 15 /C1/6, 1984. Be´zier Curve Given a set of n /C271 control points P0 ; P1 ; ..., Pn ; the corresponding Be´zier curve (or Bernstein-Be ´zier curve) is given by C(t) /C30Xn i /C300PiBi ; n(t); where Bi; n(t)isaB ERNSTEIN POLYNOMIAL and t /C23 [0; 1]:/ A "rational" Be´zier curve is defined by C(t) /C30Pn i/C300Bi; p(t)wiPiPni /C300Bi ; p(t)wi;where p is the order, Bi; pare the BERNSTEIN POLY- NOMIALS , Piare control points, and the weight wiof Piis the last ordinate of the homogeneous point P v: i These curves are CLOSED under perspective transfor- mations, and can represent CONIC SECTIONS exactly. The Be´zier curve always passes through the first and last control points and lies within the CONVEX HULL of the control points. The curve is tangent to P1 /C28P0 and Pn /C28Pn/C281at the endpoints. The "variation diminishing property" of these curves is that no line can have more intersections with a Be´zier curve than with the curve obtained by joining consecutive points with straight line segments. A desirable property of these curves is that the curve can be translated and rotated by performing these operations on the control points. Undesirable properties of Be´zier curves are their numerical instability for large numbers of control points, and the fact that moving a single control point changes the global shape of the curve. The former is sometimes avoided by smoothly patching together low-order Be´zier curves. A generalization of the Be´zier curve is the B-SPLINE . See also B-SPLINE , NURBS CURVE Be´zier Spline BE´ ZIER CURVE ,SPLINE Be´zout Numbers Integers ( l; m) for a and b such that la /C27 mb /C30GCD( a ; b) : For INTEGERS a1 ; ...,ap ; the Be´zout numbers are a set of numbers k1 ; ..., kn such that k1a1 /C27k2a2 /C27/C1/C1/C1/C27knan /C30d; where dis the GREATEST COMMON DIVISOR ofa1;... , ap:/ See also GREATEST COMMON DIVISOR Be´zout’s Theorem In general, two algebraic curves of degrees mandn intersect in m /C215npoints and cannot meet in more than m /C215npoints unless they have a component in common (i.e., the equations defining them have a common factor). This can also be stated: if Pand Q are two POLYNOMIALS with no roots in common, then there exist two other POLYNOMIALS AandBsuch that AP/C27BQ/C301:Similarly, given NPOLYNOMIAL equa- tions of degrees n1;n2;... , /nNinNvariables, there are in general n1n2/C1/C1/C1nNcommon solutions. Se´roul (2000, p. 10) uses the term Be ´zout’s theorem for the following two theorems. 1. Let a;b/C23Zbe any two integers, then there exist u;v/C23Zsuch that au /C27bv /C30GCD( a ; b) : 2. Two integers a and b are RELATIVELY PRIME if there exist u; v /C23Z such that au /C27bv /C301: See also BLANKINSHIP ALGORITHM ,GREATEST COM- MON DIVISOR ,POLYNOMIAL References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 10, 1959. Se´roul, R. "The Be´zout Theorem." §2.4.1 in Programming for Mathematicians. Berlin: Springer-Verlag, p. 10, 2000. Shub, M. and Smale, S. "Complexity of Be´zout’s Theorem. I. Geometric Aspects." J. Amer. Math. Soc. 6, 459 /C1/01, 1993. Shub, M. and Smale, S. "Complexity of Be´zout’s Theorem. II. Volumes and Probabilities." In Computational Algebraic Geometry (Nice, 1992) . Boston, MA: Birkha ¨user, pp. 267 /C1/ 85, 1993. Shub, M. and Smale, S. "Complexity of Be´zout’s Theorem. III. Condition Number and Packing." J. Complexity 9,4/C1/ 4, 1993. Shub, M. and Smale, S. "Complexity of Be´zout’s Theorem. IV. Probability of Success; Extensions." SIAM J. Numer. Anal. 33, 128 /C1/48, 1996. Shub, M. and Smale, S. "Complexity of Be´zout’s Theorem. V. Polynomial Time." Theoret. Comput. Sci. 134, 141 /C1/64, 1994. Bhargava’s Theorem Let the nth composition of a function f(x) be denoted f(n)(x) ; such that f(0)(x) /C30f(x) and f(1)(x) /C30f(x) : Denote the COMPOSITION of f and g by f(g(x) /C30f(g(x)); and define X F(a ; b; c) /C30F(a ; b ; c) /C27F(b; c ; a) /C27F(c ; b ; a) : (1) Let u /C13(a ; b; c) (2) ½½u ½½/C13a /C27b /C27c (3) ½½u½½/C13a4 /C27b4 /C27c4 ; (4) and f(u) /C30(a(b /C28c) ; b(c /C28a) ; c(a /C28b)) (5) g(u) /C30X a2b;X ab2 ; 3abc/CP6/CP7 : (6) Then if ½u ½/C300 (i.e., c /C30/C28a /C28b) ; ½½f(m)(g(n)(u) ½½/C30½½g(n)(f(m)(u)½½ /C302(ab /C27bc /C27ca)2m/C2713n ; (7) where m; n /C23f0; 1; ...g and COMPOSITION is done in terms of components. See also DIOPHANTINE EQUATION–4TH POWERS , FORD’S THEOREMReferences Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 97 /C1/00, 1994. Bhargava, S. "On a Family of Ramanujan’s Formulas for Sums of Fourth Powers." Ganita 43,63/C1/7, 1992. Bhaskara-Brouckner Algorithm SQUARE ROOT Bialtitude The common perpendicular to two opposite edges of a TETRAHEDRON . See also ALTITUDE ,BIMEDIAN ,TETRAHEDRON References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 50, 1979. Bianchi Identities The RIEMANN TENSOR is defined by Rlmv k; h /C301 2@ @xh /C2@2glv @xk @xm /C28@2gmv @xk @xl /C28@2glk @xm @xv /C27@2gmk @xv @xl ! : (1) Permuting n ; k ; and h (Weinberg 1972, pp. 146 /C1/47) gives the Bianchi identities Rlmv k; h /C27Rlmhv; k /C27Rlmkh ; v /C300; (2) which can be written concisely as Ra b[lm; v] /C300 (3) (Misner et al. 1973, p. 221), where T[a1...an] denoted the ANTISYMMETRIC TENSOR part. Wald (1984, p. 39) calls 9[aRo bc]d /C300 (4) the Bianchi identity, where 9 is the COVARIANT DERIVATIVE , and Rd? abcis the RIEMANN TENSOR . See also BIANCHI IDENTITIES (CONTRACTED ), RIEMANN TENSOR References Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravita- tion. San Francisco: W. H. Freeman, 1973. Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972. Bianchi Identities (Contracted) CONTRACTING lwith nin the B IANCHI IDENTITIES Rlmnk ;h/C27Rlmhn;k/C27Rlmkh ;n/C300 (1) gives Rmk; h /C28Rmh; k /C27Rn mkh; n /C300: (2) CONTRACTING again, R; h /C28Rm h; m /C28Rn h; n /C300; (3) or (Rm h /C281 2 dm hR); m /C300; (4) or (Rmn /C2812 gmnR); m /C300: (5) Bias (Estimator) The bias of an ESTIMATOR ˜u is defined as B( ˜u) /C13 ˜u/CP0/CPP /C28 u : It is therefore true that ˜u /C28 u /C30( ˜u /C28/C142 ˜u /C143) /C27( /C142 ˜u/C143/C28 u) /C30( ˜u /C28/C142 ˜u/C143) /C27B( ˜u) : An ESTIMATOR for which B /C300 is said to be UNBIASED ESTIMATOR . See also BIASED ESTIMATOR ,ESTIMATOR ,U NBIASED ESTIMATOR Bias (Series) The bias of a SERIES is defined as Q[ai ; ai/C271 ; ai /C272] /C13aiai/C272 /C28 a2 i/C271 a1ai/C271ai/C272: A SERIES is GEOMETRIC IFF Q /C300. A SERIES is ARTISTIC IFF the bias is constant. See also ARTISTIC SEQUENCE ,GEOMETRIC SEQUENCE References Duffin, R. J. "On Seeing Progressions of Constant Cross Ratio." Amer. Math. Monthly 100,38/C1/7, 1993. Biased Estimator An ESTIMATOR which exhibits BIAS. See also BIAS (ESTIMATOR ), ESTIMATOR ,U NBIASED ESTIMATORBiaugmented Pentagonal Prism JOHNSON SOLID J53:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Biaugmented Triangular Prism JOHNSON SOLID J50:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Biaugmented Truncated Cube JOHNSON SOLID J67:/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . BIBD BLOCK DESIGN Bicentered Tree A TREE (also called a bicentral tree) having two nodes that are GRAPH CENTERS . The numbers of bicentered trees on n /C301, 2, ... nodes are 0, 1, 0, 1, 1, 3, 4, 11, 20, 51, 108 ... (Sloane’s A000677). See also CENTERED TREE,GRAPH CENTER ,TREE References Biggs, N. L.; Lloyd, E. K.; and Wilson, R. J. Graph Theory 1736 /C1/936. Oxford, England: Oxford University Press, p. 49, 1976. Cayley, A. "On the Analytical Forms Called Trees, with Application to the Theory of Chemical Combinations." Reports Brit. Assoc. Advance. Sci. 45, 237 /C1/05, 1875. Reprinted in Math Papers, Vol. 9, pp. 427 /C1/60. Sloane, N. J. A. Sequences A000677/M2366 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Bicentral Tree BICENTERED TREE Bicentric Perspective Bicentric perspective is the study of the projection of 3D space from a pair of fiducial points instead of a single one, the latter of which may be called "centric" or "natural" PERSPECTIVE by way of distinction. See also PERSPECTIVE ,PROJECTION References Koenderink, J. J. "Fundamentals of Bicentric Perspective." In Future Tendencies in Computer Science, Control and Applied Mathematics. Proceedings of the International Conference on Research in Computer Science and Control held on the occasion of the 25th Anniversary of INRIA inParis, December 8 /C1/1, 1992 (Ed. A. Bensoussan and J.- P. Verjus). New York: Springer-Verlag, 233 /C1/51, 1992. Bicentric Polygon A POLYGON which has both a CIRCUMCIRCLE (which touches each vertex) and an INCIRCLE (which is tangent to each side). All TRIANGLES are bicentric with R2 /C28x2 /C302Rr ; (1) where R is the CIRCUMRADIUS , r is the INRADIUS , and x is the separation of centers. For BICENTRIC QUAD- RILATERALS (Fuss’s problem), the CIRCLES satisfy 2r2(R2 /C27x2) /C30(R2 /C28x2)2 (2) (Do¨rrie 1965) or, in another form, 1 (R /C28 x)2 /C271 (R /C27 x)2 /C301 r2 (3) (Davis; Dure´ge; Casey 1888, pp. 109 /C1/10; Johnson 1929; Do¨rrie 1965). If the circles permit successive tangents around the INCIRCLE which close the POLYGON for one starting point on the CIRCUMCIRCLE , then they do so for all points on the CIRCUMCIRCLE , a result known as PONCELET’S PORISM . See also BICENTRIC QUADRILATERAL ,BICENTRIC TRI- ANGLE ,CIRCUMCIRCLE ,INCIRCLE ,POLYGON ,PONCE- LET’S PORISM ,PONCELET TRANSVERSE ,TANGENTIAL QUADRILATERAL ,TRIANGLE ,W EILL’S THEOREM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124, 1987. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Do¨rrie, H. "Fuss’ Problem of the Chord-Tangent Quadrilat- eral." §39 in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 188 /C1/93, 1965. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 91 /C1/6, 1929. Bicentric Quadrilateral A 4-sided BICENTRIC POLYGON , also called a CYCLIC- INSCRIPTABLE QUADRILATERAL . The INRADIUS r, CIR- CUMRADIUS R, and offset s are connected by the equation 1 (R /C28 s)2 /C271 (R /C27 s)2 /C301 r2 (1) (Davis; Dure´ge; Casey 1888, pp. 109 /C1/10; Johnson 1929; Do¨rie 1965; Coolidge 1971, p. 46). In addition r /C30ffiffiffiffiffiffiffiffiffiffiffi abcdp s (2) R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ac /C27 bd)(ad /C27 bc)(ad /C27 cd) abcds (3) (Beyer 1987), and a /C27c /C30b /C27d: (4) The AREA of a bicentric quadrilateral is A /C30ffiffiffiffiffiffiffiffiffiffiffiffi abcd :p (5) See also BICENTRIC POLYGON ,BICENTRIC TRIANGLE , CYCLIC QUADRILATERAL ,PONCELET’S PORISM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124, 1987. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971. Davis, M. A. Educ. Times 32. Do¨rrie, H. "Fuss’ Problem of the Chord-Tangent Quadrilat- eral." §39 in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 188 /C1/93, 1965.Dure´ge, H. Theorie der elliptischen Functionen: Versuch einer elementaren Darstellung. Leipzig, Germany: Teub- ner, p. 185, 1861. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 91 /C1/6, 1929. Bicentric Triangle All triangles are bicentric, i.e., possess both an INCIRCLE and a CIRCUMCIRCLE . This is not necessarily the case for polygons with four or more sides. The INRADIUS r and CIRCUMRADIUS R are connected by 1 r /C27 d /C271 r /C28 d /C301 R ; where d is the distance between the INCENTER and CIRCUMCENTER (Coolidge 1971, p. 45). See also BICENTRIC POLYGON ,BICENTRIC QUADRILAT- ERAL References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971. Bichromatic Graph A GRAPH with EDGES of two possible "colors," usually identified as red and blue. For a bichromatic graph with R red EDGES and B blue EDGES , R /C27B ]2: See also BLUE-EMPTY GRAPH ,EXTREMAL COLORING , EXTREMAL GRAPH ,M ONOCHROMATIC FORCED TRIAN- GLE,RAMSEY NUMBER Bicollared A SUBSET X ƒY is said to be bicollared in Y if there exists an embedding b : X /C29[/C281; 1] 0 Y such that b(x; 0) /C30x when x /C23 X : The MAP b or its image is then said to be the bicollar. References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 34 /C1/5, 1976. Biconditional The CONNECTIVE in A UB (also denoted A /C13B) that returns a true result IFF A and B are either both true or both false. The biconditional is also called an EQUIVALENCE . See also CONDITIONAL ,EQUIVALENT References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 8, 1958. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 14, 1997. Bicone Two cones placed base-to-base. See also DIPYRAMID ,CONE,D OUBLE CONE,N APPE , SPHERICON Bi-Connected Component A maximal SUBGRAPH of an undirected graph such that any two edges in the SUBGRAPH lie on a common simple cycle. See also STRONGLY CONNECTED COMPONENT Biconnected Component BLOCK Biconnected Graph A GRAPH with no ARTICULATION VERTICES is called biconnected (Skiena 1990, p. 175), block, or "nonse- parable graph" (Harary 1994, p. 26). The numbers of biconnected simple graphs on n /C301, 2, ... nodes are 0, 1, 1, 3, 10, 56, 468, ... (Sloane’s A002218). A graph can be tested for biconnectivity using BiconnectedQ [g] in the Mathematica add-on package Discrete-Math‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). Any graph containing a node of degree 1 cannot be biconnected. All HAMILTONIAN GRAPHS are bicon- nected (Skiena 1990, p. 177). See also ARTICULATION VERTEX ,BLOCK ,CONNECTED GRAPH , K-CONNECTED GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A002218/M2873 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Bicorn The bicorn is the name of a collection of QUARTIC CURVES studied by Sylvester in 1864 and Cayley in 1867 (MacTutor Archive). The bicorn is given by the PARAMETRIC EQUATIONS x/C30asint (1) y/C30acos2t(2/C27cost) 3/C27sin2t(2) and Cartesian equation y2(a2/C28x2)/C30(x2/C272ay/C28a2)2(3) (Mactutor, with the final asquared instead of to the first power). The graph of the bicorn is similar to that of the COCKED HAT CURVE . The CURVATURE is given by k/C306ffiffiffi 2p (cost/C282)3(3 cos t/C282) sec t a[73/C2880 cos t/C279 cos(2 t)]3=2: (4) References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 147 /C1/49, 1972. MacTutor History of Mathematics Archive. "Bicorn." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Bi- corn.html. Bicubic Graph A BIPARTITE CUBIC GRAPH . Tutte (1971) conjectured that all 3-connected bicubic graphs are Hamiltonian (the TUTTE CONJECTURE ). The Horton graph on 96 nodes provided the first counterexample (Bondy and Murty 1976, p. 240; illustrated above). Horton subsequently found a counterexample on 92 nodes (Horton 1982). Two smaller (nonisomorphic) counterexamples on 78 nodes have since been found (Ellingham 1981, 1982b; Owens 1983). Ellingham and Horton (1983) subsequently found a nonhamilto- nian 3-connected bicubic graph on 54 vertices, illu- strated above. See also BIPARTITE GRAPH ,C UBIC GRAPH ,T UTTE CONJECTURE References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 61 and 240, 1976. Ellingham, M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs." Research Report No. 28, Dept. of Math., Univ. Melbourne, Melbourne, 1981.Ellingham, M. N. Cycles in 3-Connected Cubics Graphs. M.Sc. thesis. Melbourne, Australia: University of Mel- bourne, June 1982a. Ellingham, M. N. "Constructing Certain Cubic Graphs." In Combinatorial Mathematics, IX: Proceedings of the Ninth Australian Conference held at the University of Queens- land, Brisbane, August 24 /C1/8, 1981) (Ed. E. J. Billington, S. Oates-Williams, and A. P. Street). Berlin: Springer- Verlag, pp. 252 /C1/74, 1982b. Ellingham, M. N. and Horton, J. D. "Non-Hamiltonian 3- Connected Cubic Bipartite Graphs." J. Combin. Th. Ser. B 34, 350 /C1/53, 1983. Gropp, H. "Configurations and the Tutte Conjecture." Ars. Combin. A 29, 171 /C1/77, 1990. Horton, J. D. "On Two-Factors of Bipartite Regular Graphs." Discr. Math. 41,35/C1/1, 1982. Owens, P. J. "Bipartite Cubic Graphs and a Shortness Exponent." Disc. Math. 44, 327 /C1/30, 1983. Tutte, W. T. "On the 2-Factors of Bicubic Graphs." Discr. Math. 1, 203 /C1/08, 1971. Bicubic Spline A bicubic spline is a special case of bicubic interpola- tion which uses an interpolation function OF THE FORM y(x1 ; x2) /C30X4 i /C301X4 j/C301cijti /C281uj/C281 yx1(x1 ; x2) /C30X4 i /C301X4 j/C301(i /C281)cijti/C282uj/C281 yx2(x1 ; x2) /C30X4 i/C301X4 j/C301(j /C281)cijti/C281uj/C282 yx1x2/C30X4 i/C301X4 j/C301(i /C281)(j /C281)cijti/C282uj /C282 ; where cijare constants and u and t are parameters ranging from 0 to 1. For a bicubic spline, however, the partial derivatives at the grid points are determined globally by 1-D SPLINES . See also B-SPLINE ,SPLINE References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 118 /C1/22, 1992. Bicupola Two adjoined CUPOLAS . See also CUPOLA ,ELONGATED GYROBICUPOLA ,ELON- GATED ORTHOBICUPOLA ,GYROBICUPOLA ,ORTHOBICU- POLA Bicuspid Curve The PLANE CURVE given by the Cartesian equation (x2 /C28a2)(x /C28a)2 /C27(y2 /C28a2)2 /C300: Bi-Cyclide Coordinates BICYCLIDE COORDINATES Bicyclide Coordinates A coordinate system which is similar to BISPHERICAL COORDINATES but having fourth-degree surfaces in- stead of second-degree surfaces for constant m : The coordinates are given by the transformation equa- tions x /C30a Lcn m dn m sn n cn n cos c (1) y /C30a Lcn m dn m sn n cn n sin c (2) z /C30a Lsin m dn n ; (3) where L/C131 /C28dn2 m sn2 n ; (4) /m /C23 [0; K] ; n /C23 [0; K ?] ; c /C23 [0; 2p) ; and cn x; dn x; andsn x are JACOBI ELLIPTIC FUNCTIONS . Surfaces of constant m are given by the bicyclides (x2 /C27y2 /C27z2)2 /C27a2 k4(1 /C28 k2)2 /C28 2(1 /C28 k2)dn2 m /C27 (1 /C27 k2)dn4 m dn2 m cn2 m /C2(x2 /C27y2) /C28a2sn2 m /C271 k2 sn2 m ! z2 /C27a4 k2 /C300 ; (5) surfaces of constant n by the cyclides of rotation cn2 n a2 sn2 n(x2 /C27y2) /C27dn2 n a2z2"#2 /C282cn2 n a2 sn2 n(x2 /C27y2) /C282dn2 n a2z2 /C271 /C300 ; (6) and surfaces of constant c by the half-planes tan c /C30y x : (7) See also BISPHERICAL COORDINATES ,C AP-CYCLIDE COORDINATES ,CYCLIDIC COORDINATES References Moon, P. and Spencer, D. E. "Bicyclide Coordinates ( m; n ; c):/ " Fig. 4.08 in Field Theory Handbook, Including Coordi- nate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 124 /C1/26, 1988. Bicylinder STEINMETZ SOLID Bidiakis Cube The 12-VERTEX graph consisting of a CUBE in which two opposite faces (say, top and bottom) have edges drawn across them which connect the centers of opposite sides of the faces in such a way that the orientation of the edges added on top and bottom are PERPENDICULAR to each other. See also BISLIT CUBE,CUBE,CUBICAL GRAPH Bieberbach Conjecture The nth COEFFICIENT in the POWER SERIES of a UNIVALENT FUNCTION should be no greater than n. In other words, if f(z) /C30a0 /C27a1z /C27a2z2 /C27.../C27anzn /C27... is a CONFORMAL MAP of a UNIT DISK on any domain, then ½an ½5n½a1 ½: In more technical terms, "geometric extremality implies metric extremality." An alternate formulation is that ½aj ½leqj for any SCHLICHT FUNCTION f (Krantz 1999, p. 150). The conjecture had been proven for the first six terms (the cases n /C302, 3, and 4 were done by Bieberbach, Lowner, and Garabedian and Schiffer, respectively), was known to be false for only a finite number of indices (Hayman 1954), and true for a convex or symmetric domain (Le Lionnais 1983). The general case was proved by Louis de Branges (1985). de Branges proved the MILIN CONJECTURE , which estab- lished the ROBERTSON CONJECTURE , which in turn established the Bieberbach conjecture (Stewart 1996). author result Bieberbach (1916) / ½a2 ½52/ Lo¨wner (1923) / ½a3 ½53/ Garabedian and Schiffer (1955) / ½a4 ½54/ Pederson (1968), Ozawa (1969) / ½a6 ½56/ Pederson and Schiffer (1972) / ½a5½55/ de Branges (1985) /½aj½leqjfor all j The sum Xn j/C30k(/C281)k/C27j2j j/C28k/CP8/CP9 n/C27j/C271 n/C28j/CP8/CP9 e/C28jt was an essential tool in de Branges’ proof (Koepf 1998, p. 29). See also MILIN CONJECTURE ,R OBERTSON CONJEC- TURE ,SCHLICHT FUNCTION ,UNIVALENT FUNCTION References Bieberbach, L. "U ¨ber die Koeffizienten derjenigen Potenz- reihen, welche eine schlichte Abbildung des Einheit- skreises vermitteln." Sitzungsber. Preuss. Akad. Wiss. , pp. 940 /C155, 1916. Charzynski, Z. and Schiffer, M. "A New Proof of the Bieberbach Conjecture for the Fourth Coefficient." Arch. Rational Mech. Anal. 5, 187/C193, 1960. de Branges, L. "A Proof of the Bieberbach Conjecture." Acta Math. 154, 137/C152, 1985. Duren, P.; Drasin, D.; Bernstein, A.; and Marden, A. The Bieberbach Conjecture: Proceedings of the Symposium onthe Occasion of the Proof. Providence, RI: Amer. Math. Soc., 1986. Garabedian, P. R. "Inequalities for the Fifth Coefficient." Comm. Pure Appl. Math. 19, 199/C114, 1966. Garabedian, P. R.; Ross, G. G.; and Schiffer, M. "On the Bieberbach Conjecture for Even n."J. Math. Mech. 14, 975/C189, 1965.Garabedian, R. and Schiffer, M. "A Proof of the Bieberbach Conjecture for the Fourth Coefficient." J. Rational Mech. Anal. 4, 427/C165, 1955. Gong, S. The Bieberbach Conjecture. Providence, RI: Amer. Math. Soc., 1999. Hayman, W. K. Multivalent Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1994. Hayman, W. K. and Stewart, F. M. "Real Inequalities with Applications to Function Theory." Proc. Cambridge Phil. Soc. 50, 250/C160, 1954. Kazarinoff, N. D. "Special Functions and the Bieberbach Conjecture." Amer. Math. Monthly 95, 689/C196, 1988. Koepf, W. "Hypergeometric Identities." Ch. 2 in Hypergeo- metric Summation: An Algorithmic Approach to Summa-tion and Special Function Identities. Braunschweig, Germany: Vieweg, p. 29, 1998. Korevaar, J. "Ludwig Bieberbach’s Conjecture and its Proof." Amer. Math. Monthly 93, 505/C113, 1986. Krantz, S. G. "The Bieberbach Conjecture." §12.1.2 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, pp. 149 /C150, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 53, 1983. Lo¨wner, K. "Untersuchungen u ¨ber schlichte konforme Ab- bildungen des Einheitskreises. I." Math. Ann. 89, 103/C121, 1923. Ozawa, M. "On the Bieberbach Conjecture for the Sixth Coefficient." Kodai Math. Sem. Rep. 21,9 7/C128, 1969. Pederson, R. N. "On Unitary Properties of Grunsky’s Ma- trix." Arch. Rational Mech. Anal. 29, 370/C177, 1968. Pederson, R. N. "A Proof of the Bieberbach Conjecture for the Sixth Coefficient." Arch. Rational Mech. Anal. 31, 331/C151, 1968/1969. Pederson, R. and Schiffer, M. "A Proof of the Bieberbach Conjecture for the Fifth Coefficient." Arch. Rational Mech. Anal. 45, 161/C193, 1972. Stewart, I. "The Bieberbach Conjecture." In From Here to Infinity: A Guide to Today’s Mathematics. Oxford, Eng- land: Oxford University Press, pp. 164 /C166, 1996. Weinstein, L. "The Bieberbach Conjecture." Internat. Math. Res. Not. 5,6 1/C14, 1991. Bienayme ´-Chebyshev Inequality CHEBYSHEV INEQUALITY Bifoliate The PLANE CURVE given by the Cartesian equation x4/C27y4/C302axy2: References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Bifolium A FOLIUM with b /C300. The bifolium is the PEDAL CURVE of the DELTOID , where the PEDAL POINT is the MIDPOINT of one of the three curved sides. The Cartesian equation is (x2 /C27y2)2 /C304axy2 and the POLAR equation is r /C30 4a sin2 u cos u: See also FOLIUM ,QUADRIFOLIUM ,TRIFOLIUM References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 214, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152 /C1/53, 1972. MacTutor History of Mathematics Archive. "Double Folium." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Dou- ble.html. Bifurcation A period doubling, quadrupling, etc., that accompa- nies the onset of CHAOS . It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. Bifurcations come in four basic varieties: FLIP BIFUR- CATION , FOLD BIFURCATION , PITCHFORK BIFURCATION , and TRANSCRITICAL BIFURCATION (Rasband 1990). See also CODIMENSION ,F EIGENBAUM CONSTANT , FEIGENBAUM FUNCTION ,F LIP BIFURCATION ,H OPF BIFURCATION ,L OGISTIC MAP,P ERIOD DOUBLING , PITCHFORK BIFURCATION ,T ANGENT BIFURCATION , TRANSCRITICAL BIFURCATION References Guckenheimer, J. and Holmes, P. "Local Bifurcations." Ch. 3 in Nonlinear Oscillations, Dynamical Systems, and Bifur- cations of Vector Fields, 2nd pr., rev. corr. New York: Springer-Verlag, pp. 117 /C1/65, 1983. Lichtenberg, A. J. and Lieberman, M. A. "Bifurcation Phe- nomena and Transition to Chaos in Dissipative Systems." Ch. 7 in Regular and Chaotic Dynamics, 2nd ed. New York: Springer-Verlag, pp. 457 /C1/69, 1992. Rasband, S. N. "Asymptotic Sets and Bifurcations." §2.4 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 25 /C1/1, 1990.Weisstein, E. W. "Books about Chaos." http://www.treasure- troves.com/books/Chaos.html. Wiggins, S. "Local Bifurcations." Ch. 3 in Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, pp. 253 /C1/19, 1990. Bifurcation Theory The study of the nature and properties of BIFURCA- TIONS . See also CHAOS ,DYNAMICAL SYSTEM References Chen, Z.; Chow, S.-N.; and Li, K. (Eds.) Bifurcation Theory and Its Numerical Analysis: Proceedings of the 2ndInternational Conference, Xi’an China, June 29-July 3, 1998. Singapore: Springer-Verlag, 1999. Bigraph BIPARTITE GRAPH Bigyrate Diminished Rhombicosidodecahedron JOHNSON SOLID J79:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Biharmonic Equation The differential equation obtained by applying the BIHARMONIC OPERATOR and setting to zero. 94f/C300: (1) In C ARTESIAN COORDINATES , the biharmonic equation is 94f/C3092(92)f /C30@2 @x2/C27@2 @y2/C27@2 @z2 ! @2 @x2/C27@2 @y2/C27@2 @z2 ! f /C30@4f @x4/C27@4f @y4/C27@4f @z4/C272@4f @x2@y2/C272@4f @y2@z2/C272@4f @x2@z2 /C300: (2) In POLAR COORDINATES (Kaplan 1984, p. 148) 94 f /C30 frrrr /C272 r2frruu /C271 r4fuuuu /C272 rfrrr /C282 r3fruu /C281 r2frr /C274 r4fuu /C271 r3fr /C300: (3) For a radial function f(r) ; the biharmonic equation becomes 94 f /C301 rd drrd dr1 rd drrdf dr !"#() /C30 frrrr /C272 rfrrr /C281 r2frr /C271 r3fr /C300: (4) Writing the inhomogeneous equation as 94 f /C3064 b; (5) we have 64brdr/C30drd dr1 rd drrdf dr !"#() (6) 32br2 /C27C1 /C30rd dr1 rd drrdf dr !"# (7) 32br /C27C1 r ! dr /C30d1 rd drrdf dr !"# (8) 16br2 /C27C1 ln r /C27C2 /C301 rd drrdf dr ! (9) (16 br3 /C27C1r ln r /C27C2r) dr /C30drdf dr ! : (10) Now use gr ln rdr/C301 2 r2 ln r /C2814 r2 (11) to obtain 4 br4 /C27C1(12 r2 ln r /C2814 r2) /C2712 C2r2 /C27C3 /C30rdf dr(12) 4 br3 /C27C ?1r ln r /C27C?2r /C27C3 r ! dr /C30df (13) f(r) /C30 br4 /C27C?1(12 r2 ln r /C2814 r2) /C2712 C ?2r2 /C27C3 ln r /C27C4 /C30 br4 /C27ar2 /C27b /C27(cr2 /C27d)lnr R ! : (14) The homogeneous biharmonic equation can be sepa- rated and solved in 2-D BIPOLAR COORDINATES . See also BIHARMONIC OPERATOR , VON KA´ RMA´ N EQUA- TIONSReferences Kantorovich, L. V. and Krylov, V. I. Approximate Methods of Higher Analysis. New York: Interscience, 1958. Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1991. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997. Biharmonic Operator Also known as the BILAPLACIAN . 94 /C30( 92)2 : In n-D space, 941 r ! /C303(15 /C28 8n /C27 n2) r5 : See also BIHARMONIC EQUATION , D’ALEMBERTIAN , LAPLACIAN , VON KA´ RMA´ N EQUATIONS Biholomorphic Function CONFORMAL MAPPING Biholomorphic Map CONFORMAL MAPPING Biholomorphic Transformation CONFORMAL MAPPING Bijection A transformation which is ONE-TO-ONE and ONTO . See also DOMAIN ,ONE-TO- ONE,ONTO,PERMUTATION , RANGE (IMAGE ) Bilaplacian BIHARMONIC OPERATOR Bilinear Basis A bilinear basis is a BASIS , which satisfies the conditions (ax/C27by)/C215z/C30a(x /C215z)/C27b(y /C215z) z /C215 (ax /C27by) /C30a(z /C215 x) /C27b(z /C215 y) ; See also BASIS,B ILINEAR FUNCTION ,M ULTILINEAR BASIS Bilinear Form A bilinear form on a REAL VECTOR SPACE is a function b : V /C29V 0 R that satisfies the following axioms for any scalar a and any choice of vectors v; w; v1 ; v2 ; w1 and w2 : 1. b(av ; w) /C30b(v ; aw) /C30 ab(v ; w)/ 2. b(v1 /C27v2 ; w) /C30b(v1 ; w) /C27b(v2 ; w)/ 3. b(v ; w1 /C27w2) /C30b(v; w1) /C30/C27b(v; w2):/ For example, the function b((x1 ; x2) ; (y1 ; y2)) /C30x1y2 /C27 x2y1 is a bilinear form on R2 :/ On a COMPLEX VECTOR SPACE , a bilinear form takes values in the COMPLEX NUMBERS . In fact, a bilinear form can take values in any VECTOR SPACE , since the axioms make sense as long as VECTOR ADDITION and SCALAR MULTIPLICATION are defined. See also BILINEAR FUNCTION ,M ULTILINEAR FORM, SYMMETRIC BILINEAR FORM,VECTOR SPACE Bilinear Function A function of two variables is bilinear if it is linear with respect to each of its variables. The simplest example is f(x;y)/C30xy:/ See also BILINEAR BASIS,LINEAR FUNCTION ,SYM- METRIC BILINEAR FORM Billiard Table Problem BILLIARDS Billiards The game of billiards is played on a RECTANGULAR table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck withthe end of a "cue" stick, causing it to bounce into other balls and REFLECT off the sides of the table. Realbilliards can involve spinning the ball so that it does not travel in a straight LINE, but the mathematical study of billiards generally consists of REFLECTIONS in which the reflection and incidence angles are thesame. However, strange table shapes such as CIRCLES and ELLIPSES are often considered. Many interesting problems can arise in the detailedstudy of billiards trajectories. For example, any smooth plane convex set has at least two DOUBLE NORMALS , so there are always two distinct "to and fro" paths for any smoothly curved table. More amazingly, there are always f(k) distinct k-gonal periodic orbits on smooth billiard table, where f(k) is the TOTIENT FUNCTION (Croft et al. 1991, p. 16). This gives Steinhaus’s result that there are always two distinct periodic triangular orbits (Croft and Swinnerton- Dyer 1963) as a special case. Analysis of billiardspath can involve sophisticated use of ERGODIC THEORY and DYNAMICAL SYSTEMS . Given a rectangular billiard table with only corner pockets and sides of INTEGER lengths mandn(with mandnRELATIVELY PRIME ), a ball sent at a 45 8angle from a corner will be pocketed in another corner after m/C27n/C282 bounces (Steinhaus 1983, p. 63; Gardner 1984, pp. 211 /C1/14). Steinhaus (1983, p. 64) also gives a method for determining how to hit a billiard ballsuch that it caroms off all four sides before hitting a second ball (Knaster and Steinhaus 1946, Steinhaus 1948). A LHAZEN’S BILLIARD PROBLEM seeks to find the point at the edge of a circular "billiards" table at which a cue ball at a given point must be aimed in order tocarom once off the edge of the table and strike another ball at a second given point. It was not until 1997 that Neumann proved that the problem isinsoluble using a COMPASS and RULER construction. On an ELLIPTICAL billiard table, the ENVELOPE of a trajectory is a smaller ELLIPSE ,aHYPERBOLA ,aLINE through the FOCI of the ELLIPSE , or a closed polygon (Steinhaus 1983, pp. 239 and 241; Wagon 1991). The closed polygon case is related to PONCELET’S PORISM . The only closed billiard path of a single circuit in an ACUTE TRIANGLE is the PEDAL TRIANGLE . There are an infinite number of multiple-circuit paths, but all segments are parallel to the sides of the PEDAL TRIANGLE . There exists a closed billiard path inside a CYCLIC QUADRILATERAL if its CIRCUMCENTER lies inside the quadrilateral (Wells 1991). There are four identical closed billiard paths inside and touching each face of a CUBE such that each leg on the path has the same length (Hayward 1962; Steinhaus 1979; Steinhaus 1983; Gardner 1984, pp. 33 /C1/5; Wells 1991). This path is in the form of a chair-shaped hexagon, and each leg has lengthffiffiffi 3p =3: For a unit cube, one such path has vertices (0, 2/3, 2/ 3), (1/3, 1, 1/3), (2/3, 2/3, 0), (1, 1/3, 1/3), (2/3, 0, 2/3), (1/3, 1/3, 1). Lewis Carroll (Charles Dodgson ) also considered this problem (Weaver 1954). There are three identical closed billiard paths inside and touching each face of a TETRAHEDRON such that each leg of the path has the same length (Gardner 1984, pp. 35 /C1/6; Wells 1991). These were discovered by J. H. Conway and independently by Hayward (1962). The vertices of the path are appropriately chosen vertices of equilateral triangles in each facialplane which are scaled by a factor of 1/10. For a tetrahedron with unit side lengths, each leg has lengthffiffiffiffiffiffi10p =10 : For a tetrahedron with vertices (0, 0, 0), (0,ffiffiffi 2p =2 ;ffiffiffi2p =2); ( /ffiffiffi2p =2 ; 0,ffiffiffi2p =2); ( /ffiffiffi2p =2;ffiffiffi2p =2; 0), the vertices of one such path are ( /3ffiffiffi2p =20;7ffiffiffi2p =20;ffiffiffi2p =5);( /3ffiffiffi2p =20;3ffiffiffi2p =20;3ffiffiffi2p =10);( /7ffiffiffi2p =20;3ffiffiffi2p =20;ffiffiffi2p =5);( /7ffiffiffi2p =20;7ffiffiffi2p =20;3ffiffiffi2p =10): / Conway has shown that period orbits exist in all TETRAHEDRA , but it is not known if there are periodic orbits in every POLYHEDRON (Croft et al. 1991, p. 16). See also ALHAZEN’S BILLIARD PROBLEM ,B ILLIARD TABLE PROBLEM ,PONCELET’S PORISM ,R EFLECTION PROPERTY ,SALMON’S THEOREM References Altshiller Court, N. "Pouring Problems: The Robot Method." Mathematics in Fun and Earnest. New York: Dial Press, pp. 223 /C1/31, 1958. Bakst, A. Mathematical Puzzles and Pastimes. New York: Van Nostrand, pp. 10 /C1/1, 1954. Bellman, R. E.; Cooke, K. L.; and Lockett, J. A. Ch. 5 in Algorithms, Graphs, and Computers. New York: Aca- demic Press, 1970. Boldrighini, C.; Keane, M.; and Marchetti, F. "Billiards in Polygons." Ann. Probab. 6, 532/C1/40, 1978. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 89 /C1/3, 1967. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Billiard Ball Trajectories in Convex Regions." §A4 in Unsolved Pro- blems in Geometry. New York: Springer-Verlag, pp. 15 /C1/8, 1991. Croft, H. T. and Swinnerton, H. P. F. "On the Steinhaus Billiard Table Problem." Proc. Cambridge Philos. Soc. 59, 37/C1/1, 1963. Davis, D.; Ewing, C.; He, Z.; and Shen, T. "The Billiards Simulation." http://serendip.brynmawr.edu/chaos/ home.html. De Temple, D. W. and Robertson, J. M. "A Billiard Path Characterization of Regular Polygons." Math. Mag. 54, 73/C1/5, 1981. De Temple, D. E. and Robertson, J. M. "Convex Curves with Periodic Billiard Polygons." Math. Mag. 58,4 0/C1/2, 1985. Dullin, H. R.; Richter, P. H.; and Wittek, A. "A Two-Para- meter Study of the Extent of Chaos in a Billiard System." Chaos 6,4 3/C1/8, 1996. Gardner, M. "Bouncing Balls in Polygons and Polyhedrons." Ch. 4 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 29 /C1/8 and 211 /C1/14, 1984. Gutkin, E. "Billiards in Polygons." Physica D 19, 311/C1/33, 1986. Halpern, B. "Strange Billiard Tables." Trans. Amer. Math. Soc. 232, 297/C1/05, 1977. Hayward, R. "The Bouncing Billiard Ball." Recr. Math. Mag. , No. 9, 16 /C1/8, June 1962. Klamkin, M. S. "Problem 116." Pi Mu Epsilon J. 3, 410/C1/11, Spring 1963. Knaster, B. and Steinhaus, H. Ann. de la Soc. Polonaise de Math. 19, 228/C1/31, 1946. Knuth, D. E. "Billiard Balls in an Equilateral Triangle." Recr. Math. Mag. 14,2 0/C1/3, Jan. 1964. Madachy, J. S. "Bouncing Billiard Balls." In Madachy’s Mathematical Recreations. New York: Dover, pp. 231 /C1/ 41, 1979. Marlow, W. C. The Physics of Pocket Billiards. Philadelphia, PA: AIP, 1995. Mauldin, R. D. (Ed.). Problem 147 in The Scottish Book: Math at the Scottish Cafe. Boston, MA: Birkha ¨user, 1982. Neumann, P. Submitted to Amer. Math. Monthly. O’Beirne, T. H. Ch. 4 in Puzzles and Paradoxes: Fascinating Excursions in Recreational Mathematics. New York: Dover, 1984. Pappas, T. "Mathematics of the Billiard Table." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 43, 1989. Peterson, I. "Billiards in the Round." http://www.science- news.org/sn_arc97/3_1_97/mathland.htm. Sine, R. and Kre / `i?/novic, V. "Remarks on Billiards." Amer. Math. Monthly 86, 204 /C1/06, 1979. Steinhaus, H. Econometrica 16, 101 /C1/04, 1948. Steinhaus, H. "Problems P.175, P.176, and P.181." Colloq. Math. 4, 243 and 262, 1957. Steinhaus, H. Problem 33 in One Hundred Problems in Elementary Mathematics. New York: Dover, 1979. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Tabachnikov, S. Billiards. Providence, RI: Amer. Math. Soc., 1995. Turner, P. H. "Convex Caustics for Billiards in R2 and R3 :/" In Conference on Convexity and Related Combinatorial Geometry, Oklahoma, 1980 (Ed. D. C. Kay and M. Breen). New York: Dekker, 1982. Tweedie, M. C. K. "A Graphical Method of Solving Tarta- glian Measuring Problems." Math. Gaz. 23, 278 /C1/82, 1939. Wagon, S. "Billiard Paths on Elliptical Tables." §10.2 in Mathematica in Action. New York: W. H. Freeman, pp. 330 /C1/33, 1991. Weaver, W. "The Mathematical Manuscripts of Lewis Car- roll." Proc. Amer. Philosoph. Soc. 98, 377 /C1/81, 1954. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 13 /C1/5, 1991. Billion The word billion denotes different numbers in Amer- ican and British usage. In the American system, one billion equals 109. In the British, French, and Ger- man systems, one billion equals 1012. Fortunately, in recent years, the "American" system has become common in both the United States and Britain. See also LARGE NUMBER ,M ILLIARD ,M ILLION ,TRIL- LION Bilunabirotunda JOHNSON SOLID J91 :/References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Bimagic Cube A bimagic cube of order 25 is known. See also MAGIC CUBE References Hendricks, J. R. A Bimagic Cube: Order 25. Published by the author, 2000. Bimagic Square If replacing each number by its square in a MAGIC SQUARE produces another MAGIC SQUARE , the square is said to be a bimagic square. Bimagic squares are also called DOUBLY MAGIC SQUARES , and are 2-MULTI- MAGIC SQUARES . The first known bimagic square (shown above) has order 8 with magic constant 260 for addition and 11,180 after squaring. It is believed that no bimagic squares of order less than 8 exists (Benson and Jacoby 1976), and Hendricks (1998) shows that a bimagic square of order 3 is impossible for any set of numbers except the trivial case of using the same number 9 times. See also MAGIC SQUARE ,M ULTIMAGIC SQUARE ,TRI- MAGIC SQUARE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 212, 1987. Benson, W. H. and Jacoby, O. New Recreations with Magic Squares. New York: Dover, 1976. Hendricks, J. R. "Note on the Bimagic Square of Order 3." J. Recr. Math. 29, 265/C1/67, 1998. Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 inMathematical Diversions. New York: Dover, p. 31, 1975. Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical Recreations. New York: W. W. Norton, pp. 143 and 176 /C1/ 78, 1942. Bimedian A LINE SEGMENT joining the MIDPOINTS of opposite sides of a QUADRILATERAL or TETRAHEDRON . VARIGNON’S THEOREM states that the bimedians of a QUADRILATERAL bisect each other (left figure). In addition, the three bimedians of a tetrahedron are CONCURRENT and bisect each other (right figure; Altshiller-Court 1979, p. 48). See also COMMANDINO’S THEOREM ,M EDIAN (TRIAN- GLE), VARIGNON’S THEOREM References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979. Neuberg, J. "Notes Mathe ´matiques: 49. Proble ´me sur les te´trae`dres." Mathesis 38, 446 /C1/48, 1924. Bimodal Distribution A STATISTICAL DISTRIBUTION having two separated peaks. See also UNIMODAL DISTRIBUTION Bimonster The wreathed product of the MONSTER GROUP by Z2 :The bimonster is a quotient of the COXETER GROUP with the above COXETER- DYNKIN DIAGRAM . This had been conjectured by Conway, but was proven around 1990 by Ivanov and Norton. If the parameters p ; q ; r in Coxeter’s NOTATION [3p ; q ; r] are written side by side, the bimonster can be denoted by the BEAST NUMBER 666. Bin An interval into which a given data point does or does not fall. See also BIN-PACKING PROBLEM ,HISTOGRAM Binary The BASE 2 method of counting in which only the digits 0 and 1 are used. In this BASE , the number 1011 equals 1 /C21520/C271/C21521/C270/C21522/C271/C21523/C3011:This BASE is used in computers, since all numbers can be simply REPRESENTED AS a string of electrically pulsed ons and offs. The following table gives the binary equiva- lents of the first few decimal numbers. 1 1 11 1011 21 10101 2 10 12 1100 22 101103 11 13 1101 23 101114 100 14 1110 24 11000 5 101 15 1111 25 11001 6 110 16 10000 26 110107 111 17 10001 27 110118 1000 18 10010 28 11100 9 1001 19 10011 29 11101 10 1010 20 10100 30 11110 A NEGATIVE /C28nis most commonly REPRESENTED AS the complement of the POSITIVE number n/C281;so /C2811/C30000010112would be written as the complement of 10/C30000010102;or 11110101. This allows addition to be carried out with the usual carrying and the left- most digit discarded, so 17 /C1/1/C306 gives 00010001 17 11110101 /C2811 00000110 6 The number of times ka given binary number bn...b2b1b0is divisible by 2 is given by the position of the first bk/C301 counting from the right. For example, 12 /C301100 is divisible by 2 twice, and 13/C301101 is divisible by 2 0 times. The number of 1s N(1; n) in the binary representa- tion of a number is given by N(1; n) /C30n /C28gde(n!; 2) /C30n /C28X/C28log2n/C29 k /C301n 2k$% ; (1) where gde(n!; 2) is the GREATEST DIVIDING EXPONENT of 2 with respect to n!: This is a special application of the general result that the POWER of a PRIME p dividing a FACTORIAL (Graham et al. 1990, Vardi 1991). Writing a(n) for N(1; n); the number of 1s is also given by the RECURRENCE RELATION a(2n) /C30a(n) (2) a(2n /C271) /C30a(n) /C271; (3) with a(0) /C300; and by N(1; n) /C302n /C28log2(d) ; (4) where d is the DENOMINATOR of 1 n!dn dxn (1 /C28x) /C281=2"# x/C300: (5) For n /C30 1, 2, ..., the first few values are 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, ... (Sloane’s A000120; Smith 1966, Graham 1970, McIlroy 1974). Unfortunately, the storage of binary numbers in computers is not entirely standardized. Because computers store information in 8-bit bytes (where a bit is a single binary digit), depending on the "word size" of the machine, numbers requiring more than 8 bits must be stored in multiple bytes. The usual FORTRAN77 integer size is 4 bytes long. However, a number REPRESENTED AS (byte1 byte2 byte3 byte4) in a VAX would be read and interpreted as (byte4 byte3 byte2 byte1) on a Sun. The situation is even worse for floating point (real) numbers, which are represented in binary as a MANTISSA and CHARACTERISTIC , and worse still for long (8-byte) reals! Binary multiplication of single bit numbers (0 or 1) is equivalent to the AND operation, as can be seen in the following MULTIPLICATION TABLE . //C29/ 01 000 101 See also BASE (NUMBER ), BINARY CARRY SEQUENCE , DECIMAL ,F ACTORIAL ,H EXADECIMAL ,M OSER-DE BRUIJN SEQUENCE ,N EGABINARY ,O CTAL ,Q UATERN- ARY,R UDIN- SHAPIRO SEQUENCE ,S TOLARSKY- HAR- BORTH CONSTANT ,TERNARYReferences Graham, R. L. "On Primitive Graphs and Optimal Vertex Assignments." Ann. New York Acad. Sci. 175, 170 /C1/86, 1970. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Factorial Factors." §4.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 111--115, 1994. Heath, F. G. "Origin of the Binary Code." Sci. Amer. , Aug. 1972. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 6 /C1/, 1991. McIlroy, M. D. "The Number of 1’s in Binary Integers: Bounds and Extremal Properties." SIAM J. Comput. 3, 255 /C1/61, 1974. Pappas, T. "Computers, Counting, & Electricity." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 24 /C1/5, 1989. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Error, Accuracy, and Stability" and "Diagnos- ing Machine Parameters." §1.2 and §20.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 18 /C1/1, 276, and 881 /C1/86, 1992. Sloane, N. J. A. Sequences A000120/M0105 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Smith, N. "Problem B-82." Fib. Quart. 4, 374 /C1/65, 1966. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 67, 1991. Weisstein, E. W. "Bases." MATHEMATICA NOTEBOOK BASES.M . Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 42 /C1/4, 1986. Binary Bracketing A binary bracketing is a BRACKETING built up entirely of binary operations. The number of binary bracket- ings of n letters (CATALAN’S PROBLEM ) are given by the CATALAN NUMBERS Cn/C281 ; where Cn /C131 n /C27 12n n/CP8/CP9 /C301 n /C27 1(2n)! n!2/C30(2n)! (n /C27 1)!n! ; where (2n n ) denotes a BINOMIAL COEFFICIENT and n!is the usual FACTORIAL , as first shown by Catalan in 1838. For example, for the four letters a, b, c, and d there are five possibilities: ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d; and a(b(cd)) ; written in shorthand as ((xx)x)x; (x(xx))x;(xx)(xx);x((xx)x;andx(x(xx)):/ See also BRACKETING ,CATALAN NUMBER ,CATALAN’S PROBLEM References Schro ¨der, E. "Vier combinatorische Probleme." Z. Math. Physik 15, 361/C1/76, 1870. Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M1459 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Stanley, R. P. "Hipparchus, Plutarch, Schro ¨der, and Hough." Amer. Math. Monthly 104, 344/C1/50, 1997. Binary Carry Sequence The sequence a(n) given by the exponents of the highest power of 2 dividing n, i.e., the number of trailing 0s in the BINARY representation of n. For n /C301, 2, ..., the first few are 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, ... (Sloane’s A007814). Amazingly, this corresponds to one less than the number of disk to be moved at nth step of optimal solution to TOWERS OF HANOI problem, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, ... (Sloane’s A001511). The anti- PARITY of this sequence is given by 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, ... (Sloane’s A035263) which, amazingly, also corresponds to the ACCUMULATION POINT of 2n cycles through successive bifurcations. See also DOUBLE- FREE SET,TOWERS OF HANOI References Atanassov, K. "On the 37th and the 38th Smarandache Problems. Notes on Number Theory and Discrete Mathe- matics, Sophia, Bulgaria 5,83/C15, 1999. Atanassov, K. On Some of the Smarandache’s Problems. Lupton, AZ: American Research Press, pp. 16 /C11, 1999. Derrida, B.; Gervois, A.; and Pomeau, Y. "Iteration of Endomorphisms on the Real Axis and Representation of Number." Ann. Inst. Henri Poincare ´, Section A: Physique The´orique 29, 305 /C156, 1978. Karamanos, K. and Nicolis, G. "Symbolic Dynamics and Entropy Analysis of Feigenbaum Limit Sets." Chaos, Solitons, Fractals 10, 1135 /C1150, 1999. Metropolis, M.; Stein, M. L.; and Stein, P R. "On Finite Limit Sets for Transformations on the Unit Interval." J. Combin. Th. A 15,25/C14, 1973. Sloane, N. J. A. Sequences A001511/M0127, A007814, and A035263 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993. Vitanyi, P. M. B. " An Optimal Simulation of Counter Machines." SIAM J. Comput. 14,1/C13, 1985. Binary Goldbach Conjecture GOLDBACH CONJECTURE Binary Heap HEAP Binary Matrix (0,1)-MATRIX Binary Operation This entry contributed by J. BRAD WEATHERLY A binary operation on a nonempty set A is a map f : A /C29 A 0 A; such that f is defined for every element in A and the image of f is unique. Examples of binary operations on A from A /C29 A to A include /C27 and -. See also BINARY OPERATORBinary Operator An OPERATOR defined on a set S which takes two elements from S as inputs and returns a single element of S. Binary operators are called composi- tions by Rosenfeld (1968). Sets possessing a binary multiplication operation include the GROUP , GROUP- OID, MONOID , QUASIGROUP , and SEMIGROUP . Sets possessing both a binary multiplication and a binary addition operation include the DIVISION ALGEBRA , FIELD , RING , RINGOID , SEMIRING , and UNIT RING . See also AND, BINARY OPERATION ,BOOLEAN ALGE- BRA,CLOSURE (SET), CONNECTIVE ,D IVISION ALGE- BRA,FIELD,GROUP ,GROUPOID ,M ONOID ,OPERATOR , OR, MONOID , NOT, QUASIGROUP ,R ING,R INGOID , SEMIGROUP ,SEMIRING , XNOR, XOR, UNIT RING References Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968. Binary Quadratic Form A QUADRATIC FORM in two variables having the form Q(x; y) /C30 a11x2 /C272a12xy /C27a22y2 : (1) Consider a binary quadratic form with real coeffi- cients a11 ; a12 ; and a22 ; determinant D /C13a11a22 /C28a2 12 /C301; (2) and a11 > 0: Then Q(x; y)is POSITIVE DEFINITE .An important result states that exist two integers x and y not both 0 such that Q(x; y) 52ffiffiffi 3p (3) for all values of aijsatisfying the above constraint (Hilbert and Cohn-Vossen 1999, p. 39). See also PELL EQUATION ,POSITIVE DEFINITE QUAD- RATIC FORM,Q UADRATIC FORM,Q UADRATIC INVAR- IANT References Hilbert, D. and Cohn-Vossen, S. "The Minimum Value of Quadratic Forms." §6.2 in Geometry and the Imagination. New York: Chelsea, pp. 39 /C1/1, 1999. Binary Relation Given a set of objects S, a binary relation is a subset of the CARTESIAN PRODUCT S /C156S:/ See also RELATION References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 161, 1990. Binary Remainder Method An ALGORITHM for computing a UNIT FRACTION (Stew- art 1992). References Eppstein, D. Egypt.ma Mathematica notebook. http:// www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma. Stewart, I. "The Riddle of the Vanishing Camel." Sci. Amer. 266, 122 /C1/24, June 1992. Binary Search A SEARCHING algorithm which works on a sorted table by testing the middle of an interval, eliminating the half of the table in which the key cannot lie, and then repeating the procedure iteratively. See also SEARCHING References Lewis, G. N.; Boynton, N. J.; and Burton, F. W. "Expected Complexity of Fast Search with Uniformly Distributed Data." Inform. Proc. Let. 13,4/C1/, 1981. Skiena, S. "Backtracking and Distinct Permutations." §1.1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 12 /C1/4, 1990. Binary Splitting References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Brent, R. P. "The Complexity of Multiple-Precision Arith- metic." Complexity of Computational Problem Solving (Ed. R. S. Andressen and R. P. Brent). Brisbane, Austra- lia: University of Queensland Press, 1976. Gourdon, X. and Sebah, P. "Binary Splitting Method." http:// xavier.gourdon.free.fr/Constants/Algorithms/split- ting.html. Haible, B. and Papanikolaou, T. "Fast Multiprecision Eva- luation of Series of Rational Numbers." Report TI-97 /C1/.TH Darmstadt. Binary Tree A TREE with two BRANCHES at each FORK and with one or two LEAVES at the end of each BRANCH . (This definition corresponds to what is sometimes known as an "extended" binary tree.) The height of a binary tree is the number of levels within the TREE . For a binary tree of height H with n nodes, H 5n 52H /C281: These extremes correspond to a balanced tree (each node except the LEAVES has a left and right CHILD , and all LEAVES are at the same level) and a degen- erate tree (each node has only one outgoing BRANCH ), respectively. For a search of data organized into a binary tree, the number of search steps S(n) neededto find an item is bounded by lg n 5S(n) 5n: Partial balancing of an arbitrary tree into a so-called AVL binary search tree can improve search speed. The number of binary trees with n internal nodes is the CATALAN NUMBER Cn (Sloane’s A000108), and the number of binary trees of height b is given by Sloane’s A001699. The numbers of binary trees on n /C301, 2, ... nodes (i.e., n-node trees having VERTEX DEGREE either 1 or 3; also called 3-Cayley trees, 3- valent trees, or boron trees) are 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0 ,4, 0, 6, 0, 11, ... (Sloane’s A052120). See also B-TREE,CAYLEY TREE,COMPLETE BINARY TREE,E XTENDED BINARY TREE,H EAP,Q UADTREE , QUATERNARY TREE,RAMUS TREE,RED-BLACK TREE, SPLAY TREE,STERN- BROCOT TREE,W EAKLY BINARY TREE References Lucas, J.; Roelants van Baronaigien, D.; and Ruskey, F. "Generating Binary Trees by Rotations." J. Algorithms 15, 343/C1/66, 1993. Ranum, D. L. "On Some Applications of Fibonacci Num- bers." Amer. Math. Monthly 102, 640/C1/45, 1995. Ruskey, F. "Information on Binary Trees." http://www.theor- y.csc.uvic.ca/~cos/inf/tree/BinaryTrees.html. Ruskey, F. and Proskurowski, A. "Generating Binary Trees by Transpositions." J. Algorithms 11,6 8/C1/4, 1990. Skiena, S. S. The Algorithm Design Manual. New York: Springer-Verlag, pp. 177 /C1/78, 1997. Sloane, N. J. A. Sequences A000108/M1459, A001699/ M3087, and A052120 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Binet Forms The two RECURRENCE SEQUENCES Un/C30mUn/C281/C27Un/C282 (1) Vn/C30mVn/C281/C27Vn/C282 (2) with U0/C300;U1/C301 and V0/C302;V1/C30m;can be solved for the individual UnandVn:They are given by Un/C30an/C28bn D(3) Vn/C30an/C27bn; (4) where D/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2/C274p (5) a/C13m/C27D 2(6) b/C13m/C28D 2: (7) A useful related identity is Un/C281 /C27Un/C271 /C30Vn : (8) BINET’S FIBONACCI NUMBER FORMULA is a special case of the Binet form for Un corresponding to m /C301. See also BINET’S FIBONACCI NUMBER FORMULA , FIBONACCI Q-MATRIX Binet’s Fibonacci Number Formula A special case of the UnBINET FORM with m /C301, corresponding to the nth FIBONACCI NUMBER , Fn /C30(1 /C27ffiffiffi 5p )n /C28 (1 /C28ffiffiffi5p )n 2nffiffiffi5p : It was derived by Binet in 1843, although the result was known to Euler and to Daniel Bernoulli more than a century earlier. See also B INET FORMS ,FIBONACCI NUMBER References Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 21, 2000. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 62, 1986. Binet’s Log Gamma Formulas Binet’s first formula for ln G(z); where G(z)isa GAMMA FUNCTION , is given by ln G(z) /C30(z /C281 2)lnz /C28z /C2712ln(2p) /C27g/C12 0[(et /C281)/C281 /C28t/C281 /C2712]t/C281e/C28tz dt for R[z] > 0 (Erde ´lyi et al. 1981, p. 21). Binet’s second formula is ln G(z) /C30 z /C2812/CP6/CP7 ln z /C28z /C2712 ln (2p) /C272g/C12 0tant 2 ! e2 pt /C28 1dt for R[z] > 0 (Erde ´lyi et al. 1981, p. 22; Whittaker and Watson 1990, p. 251). See also GAMMA FUNCTION ,MALMSTE ´ N’S FORMULA References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, 1981. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Binet-Cauchy Identity The algebraic identity Xn i /C301aici !Xn i/C301bidi ! /C28Xn i /C301aidi !Xn i/C301bici !/C30X 1 5i 5j 5n(aibj /C28ajbi)(cidj /C28cjdi): (1) Letting ci /C30ai and di /C30bi gives LAGRANGE’S IDENTITY . The identity can be coded in Mathematica as follows. BBDiscreteMath‘Combinatorica‘; BinetCauchyId[n_] : /C30 Module[{ aa /C30 Array[a, n], bb /C30 Array[b, n], cc /C30 Array[c, n], dd /C30 Array[d, n] }, aa.cc bb.dd - aa.dd bb.cc /C30/C30 Plus @@ ((a[#1]b[#2] - a[#2]b[#1])(c[#1]d[#2] - c[#2]d[#1]) & @@@ KSubsets[Range[n], 2]) ] The n /C302 case then gives (a1c1 /C27a2c2)(b1d1 /C27b2d2) /C28(b1c1 /C27b2c2)(a1d1 /C27a2d2) /C30(a1b2 /C28a2b1)(c1d2 /C28c2d1): (2) The n /C303 case is equivalent to the vector identity (A /C29B) /C215(C /C29D) /C30(A /C215C)(B /C215D) /C28(A /C215D)(B /C215C); (3) where A /C215B is the DOT PRODUCT and A /C29B is the CROSS PRODUCT . Note that this identity itself is sometimes known as LAGRANGE’S IDENTITY . See also LAGRANGE’S IDENTITY References Mitrinovic, D. S. Analytic Inequalities. New York: Springer- Verlag, p. 42, 1970. Bing’s Theorem If M3 is a closed oriented connected 3-MANIFOLD such that every simple closed curve in M lies interior to a BALL in M, then M is HOMEOMORPHIC with the HYPERSPHERE , S3 :/ See also BALL,HYPERSPHERE References Bing, R. H. "Necessary and Sufficient Conditions that a 3- Manifold be S3 :/" Ann. Math. 68,17/C1/7, 1958. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 251 /C1/57, 1976. Binomial APOLYNOMIAL with 2 terms. See also BINOMIAL COEFFICIENT ,M ONOMIAL ,POLY- NOMIAL ,TRINOMIAL Binomial Coefficient The number of ways of picking nunordered outcomes from Npossibilities, also known as a COMBINATION or combinatorial number. The symbolsNCnandN n/C0/CP are used to denote a binomial coefficient, and are some- times read as " N CHOOSE n." The value of the binomial coefficient is given by NCn/C13N n/CP8/CP9 /C13N! (N/C28n)!n!; (1) where n! denotes a FACTORIAL . Writing the FACTORIAL as a GAMMA FUNCTION n!/C30G(n/C271) allows the bino- mial coefficient to be generalized to non-integral arguments. The binomial coefficients form the rows of P ASCAL’S TRIANGLE , and the number of LATTICE PATHS from the ORIGIN (0;0) to a point ( a, b ) is the binomial coefficienta/C27b a/C0/CP (Hilton and Pedersen 1991). For a POSITIVE INTEGER n, the BINOMIAL THEOREM gives (x/C27a)n/C30Xn k/C300n k/CP8/CP9 xkan/C28k: (2) The FINITE DIFFERENCE analog of this identity is known as the C HU-VANDERMONDE IDENTITY . A simi- lar formula holds for NEGATIVE INTEGERS , (x/C27a)/C28n/C30X/C12 k/C300/C28n k/CP8/CP9 xka/C28n/C28k: (3) There are a number of elegant BINOMIAL SUMS . The binomial coefficients satisfy the identities n 0/CP8/CP9 /C30n n/CP8/CP9 /C301 (4) n k/CP8/CP9 /C30n n/C28k/CP8/CP9 /C30(/C281)kk/C28n/C281 k/CP8/CP9 (5) n/C271 k/CP8/CP9 /C30n k/CP8/CP9 /C27n k/C281/CP8/CP9 : (6) As shown by Kummer in 1852, if pkis the largest power of a PRIME pthat dividesn/C27k k/C0/CP ;where nandk are nonnegative integers, then kis the number of carries that occur when kis added to nin base p (Graham et al. 1989, Exercise 5.36, p. 245; Ribenboim 1989; Vardi 1991, p. 68). Kummer’s result can also be stated in the form that the exponent of a PRIME p dividingn m/C0/CP is given by the number of integers j]0 for which frac( m=pj)>frac( n=pj); (7) where frac( x) denotes the FRACTIONAL PART ofx. This inequality may be reduced to the study of the exponential sums anL(n)e(x=n);where L(n) is the MANGOLDT FUNCTION . Estimates of these sums are given by Jutila (1974, 1975), but recent improvementshave been made by Granville and Ramare (1996). R. W. Gosper showed that f(n)/C30n/C281 1 2(n/C281)/CP8/CP9 /C13(/C281)(n/C281)=2(mod n) (8) for all PRIMES , and conjectured that it holds only forPRIMES . This was disproved when Skiena (1990) found it also holds for the COMPOSITE NUMBER n/C30 3/C2911/C29179:Vardi (1991, p. 63) subsequently showed that n/C30p2is a solution whenever pis a W IEFERICH PRIME and that if n/C30pkwith k/C213 is a solution, then so is n/C30pk/C281:This allowed him to show that the only solutions for COMPOSITE nB1:3/C29107are 5907, 10932, and 35112, where 1093 and 3511 are W IEFER- ICH PRIMES . Consider the binomial coefficients f(n)/C302n/C281 n/C0/CP ;the first few of which are 1, 3, 10, 35, 126, ... (Sloane’s A001700). The GENERATING FUNCTION is 1 21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C284xp /C281"# /C30x/C273x2/C2710x3/C2735x4/C27...:(9) These numbers are SQUAREFREE only for n/C302, 3, 4, 6, 9, 10, 12, 36, ... (Sloane’s A046097), with no others known. It turns out that f(n) is divisible by 4 unless n belongs to a 2- AUTOMATIC SET S2;which happens to be the set of numbers whose BINARY representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (Sloane’s A048645). Similarly, f(n)i s divisible by 9 unless nbelongs to a 3- AUTOMATIC SET S3;consisting of numbers nfor which the representation of 2 ninTERNARY consists entirely of 0s and 2s (except possibly for a pair of adjacent 1s; D. Wilson, A. Karttunen). The initial elements of S3are 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 18, 19, 21, 22, 27, ... (Sloane’s A051382). If f(n) is squarefree, then nmust belong to S/C30S 2SS3:It is very probable that Sis finite, but no proof is known. Now, squares larger than 4 and 9 might also divide f(n);but by eliminat- ing these two alone, the only possible nforn5264 are 1, 2, 3, 4, 6, 9, 10, 12, 18, 33, 34, 36, 40, 64, 66, 192,256, 264, 272, 513, 514, 516, 576 768, 1026, 1056, 2304, 16392, 65664, 81920, 532480, and 545259520.All of these but the last have been checked (D. Wil-son), establishing that there are no other nsuch that f(n) is squarefree for n5545;259;520: / Erdos showed that the binomial coefficientn k/C0/CP ;with 35k5n=2i sa POWER of an INTEGER for the single case50 3/C0/CP /C301402(Le Lionnais 1983, p. 48). Binomial coefficients Tn/C281/C30n 2/C0/CP are squares a2when a2is a TRIANGULAR NUMBER , which occur for a/C301, 6, 35, 204, 1189, 6930, ... (Sloane’s A001109). These values ofahave the corresponding values n/C302, 9, 50, 289, 1682, 9801, ... (Sloane’s A052436). The binomial coefficientsn n=2bc/CP6/CP7 are called CENTRAL BINOMIAL COEFFICIENTS , where xbc is the FLOOR FUNCTION , although the subset of coefficients2n n/C0/CP is sometimes also given this name. Erdos and Graham (1980, p. 71) conjectured that the CENTRAL BINOMIAL COEFFICIENT2n n/C0/CP isnever SQUAREFREE forn/C214, and this is sometimes known as the E RDOS SQUAREFREE CONJECTURE .S A´RKOZY’S THEOREM (Sa´rkozy 1985) provides a partial solution which states that the BINOMIAL COEFFICIENT2n n/C0/CP is never SQUAREFREE for all sufficiently large n ] n0(Vardi 1991). Granville and Ramare (1996) proved that the only SQUAREFREE values are n /C302 and 4. Sander (1992) subsequently showed that2n9d n/C0/CP are also never SQUAREFREE for sufficiently large n as long as d is not "too big." For p, q, and r distinct PRIMES , then the function (8) satisfies f(pqr)f(p)f(q)f(r) /C13f(pq)f(pr)f(qr) (mod pqr) (10) (Vardi 1991, p. 66). Most binomial coefficients (n k) with n ]2k have a prime factor p 5n=k; and Lacampagne et al. (1993) conjecture that this inequality is true for all n /C21 17 :125k ; or more strongly that any such binomial coefficient has LEAST PRIME FACTOR p 5n =k or p 5 17 with the exceptions62 6/C0/CP ;959 56/C0/CP ;474 66/C0/CP ;284 28/C0/CP for which p /C3019, 19, 23, 29 (Guy 1994, p. 84). The binomial coefficientm n/C0/CP (mod 2) can be computed using the XOR operation n XOR m, making PASCAL’S TRIANGLE mod 2 very easy to construct. The binomial coefficient "function" can be defined as C(x; y) /C13x! y!(x/C28y)(11) (Fowler 1996), shown above. It has a very complicated GRAPH for NEGATIVE xand ywhich is difficult to render using standard plotting programs. See also APE´ RY NUMBER ,BALANCED BINOMIAL COEF- FICIENT ,BALLOT PROBLEM ,BINOMIAL DISTRIBUTION , BINOMIAL IDENTITY ,BINOMIAL SUMS,BINOMIAL THE- OREM ,C ENTRAL BINOMIAL COEFFICIENT ,C HOOSE , CHU-VANDERMONDE IDENTITY ,C OMBINATION ,D EFI- CIENCY ,E RDOS SQUAREFREE CONJECTURE ,E XCEP- TIONAL BINOMIAL COEFFICIENT ,FACTORIAL ,GAMMA FUNCTION ,GAUSSIAN COEFFICIENT ,GAUSSIAN POLY- NOMIAL ,GOOD BINOMIAL COEFFICIENT ,KINGS PRO- BLEM ,KLEE’S IDENTITY ,LAH NUMBER ,MULTICHOOSE , MULTINOMIAL COEFFICIENT ,PERMUTATION ,R OMAN COEFFICIENT ,SA´ RKOZY’S THEOREM ,STANLEY’S IDEN- TITY,S TAR OF DAVID THEOREM ,S TOLARSKY- HAR- BORTH CONSTANT ,S TREHL IDENTITIES ,S ZE´ KELY IDENTITY ,W OLSTENHOLME’S THEOREMReferences Abramowitz, M. and Stegun, C. A. (Eds.). "Binomial Coeffi- cients." §24.1.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 10 and 822 /C1/23, 1972. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 66 /C1/4, 1996. Erdos, P.; Graham, R. L.; Nathanson, M. B.; and Jia, X. Old and New Problems and Results in Combinatorial Number Theory. New York: Springer-Verlag, 1998. Erdos, P.; Lacampagne, C. B.; and Selfridge, J. L. "Esti- mates of the Least Prime Factor of a Binomial Coefficient."Math. Comput. 61, 215/C1 /24, 1993. Feller, W. "Binomial Coefficients" and "Problems and Iden- tities Involving Binomial Coefficients." §2.8 and 2.12 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 48 /C1/0 and 61 /C1/4, 1968. Fowler, D. "The Binomial Coefficient Function." Amer. Math. Monthly 103,1/C1/7, 1996. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial Coefficients." Ch. 5 in Concrete Mathematics: A Founda- tion for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 153 /C1/42, 1994. Granville, A. and Ramare ´, O. "Explicit Bounds on Exponen- tial Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43,7 3/C1/07, 1996. Guy, R. K. "Binomial Coefficients," "Largest Divisor of a Binomial Coefficient," and "Series Associated with the &/- Function." §B31, B33, and F17 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 84 /C1/5, 87/C1/9, and 257 /C1/58, 1994. Harborth, H. "Number of Odd Binomial Coefficients." Not. Amer. Math. Soc. 23, 4, 1976. Hilton, P. and Pedersen, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13,6 4/C1/5, 1991. Jutila, M. "On Numbers with a Large Prime Factor." J. Indian Math. Soc. 37,4 3/C1/3, 1973. Jutila, M. "On Numbers with a Large Prime Factor. II." J. Indian Math. Soc. 38, 125/C1/30, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Ogilvy, C. S. "The Binomial Coefficients." Amer. Math. Monthly 57, 551/C1/52, 1950. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gamma Function, Beta Function, Factorials, Binomial Coefficients." §6.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.Cambridge, England: Cambridge University Press,pp. 206 /C1 /09, 1992. Prudnikov, A. P.; Marichev, O. I.; and Brychkow, Yu. A. Formula 41 in Integrals and Series, Vol. 1: Elementary Functions. Newark, NJ: Gordon & Breach, p. 611, 1986. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 23 /C1/4, 1989. Riordan, J. "Inverse Relations and Combinatorial Identi- ties." Amer. Math. Monthly 71, 485/C1/98, 1964. Sander, J. W. "On Prime Divisors of Binomial Coefficients." Bull. London Math. Soc. 24, 140/C1/42, 1992. Sa´rkozy, A. "On the Divisors of Binomial Coefficients, I." J. Number Th. 20,7 0/C1/0, 1985. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 262, 1990. Sloane, N. J. A. Sequences A001109/M4217, A001700/ M2848, A046097, A048645, A051382, and A052436, in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Spanier, J. and Oldham, K. B. "The Binomial Coefficients n m/C0/CP :/" Ch. 6 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 43 /C1/2, 1987. Sved, M. "Counting and Recounting." Math. Intel. 5,2 1/C1/6, 1983. Vardi, I. "Application to Binomial Coefficients," "Binomial Coefficients," "A Class of Solutions," "Computing BinomialCoefficients," and "Binomials Modulo an Integer." §2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 25 /C1 /8 and 63 /C1/1, 1991. Wolfram, S. "Geometry of Binomial Coefficients." Amer. Math. Monthly 91, 566/C1/71, 1984. Binomial Differential Equation The ORDINARY DIFFERENTIAL EQUATION (y?)m/C30f(x;y) (Hille 1969, p. 675; Zwillinger 1997, p. 120). References Hille, E. Lectures on Ordinary Differential Equations. Reading, MA: Addison-Wesley, 1969. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Binomial Distribution The binomial distribution gives the probability dis- tribution Pp(n½N) of obtaining exactly nsuccesses out ofNBERNOULLI TRIALS (where the result of each BERNOULLI TRIAL is true with probability pand false with probability q/C301/C28p):The binomial distribution is therefore given by Pp(n½N)/C30N n/CP8/CP9 pn(1/C28p)N/C28n/C30N! n!(N/C28n)!pnqN/C28n:(1) The above plot shows the distribution of nsuccesses out of N/C3020 trials with p/C30q/C301=2:Steinhaus (1983, pp. 25 /C1/8) considers the expected number of squaresS(n;N;s) containing a given number of grains non board of size safter random distribution of Nof grains, S(n;N;s)/C30sP1=s(n½N): (2) Taking N/C30s/C3064 gives the results summarized in the following table. Sn 0 23.3591 1 23.7299 2 11.8650 3 3.892214 0.9421625 0.179459 6 0.0280109 7 0.00368408 4.16639 /C2910 /C284 9 4.11495 /C2910/C285 10 3.59242 /C2910/C286 The probability of obtaining more successes than the nobserved in a binomial distribution is P/C30XN k/C30n/C271N k/CP8/CP9 pk(1/C28p)N/C28k/C30Ip(n/C271;N/C28n);(3) where Ix(a;b)/C13B(x;a;b) B(a;b); (4) /B(a;b) is the BETA FUNCTION , and B(x;a;b) is the incomplete BETA FUNCTION . The CHARACTERISTIC FUNCTION for the binomial dis- tribution is f(t)/C30(q/C27peit)n(5) (Papoulis 1984, p. 154). The MOMENT-GENERATING FUNCTION Mfor the distribution is M(t)/C30/C142etn/C143/C30XN n/C300etnN n/CP8/CP9 pnqN/C28n /C30XN n/C300N n/CP8/CP9 (pet)(1/C28p)N/C28n/C30[pet/C27(1/C28p)]N(6) M?(t)/C30N[pet/C27(1/C28p)]N/C281(pet) (7) M??(t)/C30N(N/C281)[pet/C27(1/C28p)]N/C282(pet)2 /C27N[pet/C27(1/C28p)]N/C281(pet): (8) The MEAN is m/C30M?(0)/C30N(p/C271/C28p)p/C30Np: (9) The MOMENTS about 0 are m?1/C30m/C30Np (10) m?2/C30Np(1/C28p/C27Np) (11) m?3/C30Np(1/C283p/C273Np/C272p2/C283NP2/C27N2p2) (12) m?4/C30Np(1/C287p/C277Np/C2712p2/C2818Np2/C276N2p2/C286p3 /C2711Np3/C286N2p3/C27N3p3); (13) so the MOMENTS about the MEAN are m2/C30s2/C30[N(N/C281)p2/C27Np]/C28(Np)2 /C30N2p2/C28Np2/C27Np/C28N2p2/C30Np(1/C28p)/C30Npq (14) m3/C30m?3/C283m?2m?1/C272(m1)3/C30Np(1/C28p)(1/C282p) (15) m4/C30m?4/C284m?3m?1/C276m?2(m?1)2/C283(m1)4 /C30Np(1/C28p)[3p2(2/C28N)/C273p(N/C282)/C271]: (16) The SKEWNESS and KURTOSIS are g1/C30m3 s3/C30Np(1/C28p)(1/C282p) [Np(1/C28p)]3=2/C301/C282pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Np(1/C28p)p /C30q/C28pffiffiffiffiffiffiffiffiffiffi Npqp (17) g2/C30m4 s4/C283/C306p2/C286p/C271 Np(1/C28p)/C301/C286pq Npq: (18) An approximation to the Bernoulli distribution for large Ncan be obtained by expanding about the value ˜nwhere P(n) is a maximum, i.e., where dP=dn/C300: Since the LOGARITHM function is MONOTONIC , we can instead choose to expand the LOGARITHM . Let n/C13 ˜n/C27h;then ln[P(n)]/C30ln[P(˜n)]/C27B1h/C271 2B2h2/C271 3!B3h3/C27...;(19) where Bk/C30dkln[P(n)] dnk"# n/C30˜n: (20) But we are expanding about the maximum, so, by definition, B1/C30dln[P(n)] dn"# n/C30˜n/C300: (21) This also means that B2is negative, so we can write B2/C30/C28½B2½:Now, taking the LOGARITHM of (1) gives ln[P(n)]/C30lnN!/C28lnn!/C28ln(N/C28n)!/C27nlnp /C27(N/C28n)l nq: (22)For large nand N/C28nwe can use S TIRLING’S APPROXIMATION ln(n!):nlnn/C28n; (23) so d[ln(n!)] dn:(lnn/C271)/C281/C30lnn (24) d[ln(N/C28n)!] dn:d dn[(N/C28n) ln(N/C28n)/C28(N/C28n)] /C30/C28 ln(N/C28n)/C27(N/C28n)/C281 N/C28n/C271"# /C30/C28ln(N/C28n); (25) and dln[P(n)] dn:/C28lnn/C27ln(N/C28n)lnp/C28lnq: (26) To find ˜n;set this expression to 0 and solve for n, lnN/C28˜n ˜np q ! /C300 (27) N/C28˜n ˜np q/C301 (28) (N/C28˜n)p/C30˜nq (29) ˜n(q/C27p)/C30˜n/C30Np; (30) since p/C27q/C301:We can now find the terms in the expansion B2/C30d2ln[P(n)] dn2"# n/C30˜n/C30/C281 ˜n/C281 N/C28˜n /C30/C281 Np/C281 N(1/C28p)/C30/C281 N1 p/C271 q ! /C30/C281 Np/C27q pq ! /C30/C281 Npq/C30/C281 N(1/C28p)(31) B3/C13d3ln[P(n)] dn3"# n/C30˜n/C30/C281 ˜n2/C281 (N/C28˜n)2/C301 N2p2/C281 N2q2 /C30q2/C28p2 N2p2q2/C30(1/C282p/C27p2)/C28p2 N2p2(1/C28p)2 /C301/C282p N2p2(1/C28p)2(32) B4 /C13d4 ln[P(n)] dn4"# n/C30˜n/C30/C282 ˜n3 /C282 (n /C28 ˜n)3 /C30/C2821 N3p3 /C271 N3q3 ! /C302(p3 /C27 q3) N3p3q3 /C302(p2 /C28 pq /C27 q2) N3p3q3 /C302[p2 /C28 p(1 /C28 p) /C27 (1 /C28 2p /C27 p2)] N3p3(1 /C28 p3) /C302(3p2 /C28 3p /C27 1) N3p3(1 /C28 p3): (33) Now, treating the distribution as continuous, lim N 0/C12XN n/C300P(n) :g P(n) dn /C30g/C12 /C28/C12P(˜n /C27 h) d h /C301: (34) Since each term is of order 1=N /C21 =s2 smaller than the previous, we can ignore terms higher than B2 ; so P(n) /C30P(˜n)e /C28½B2 ½ h2 =2 : (35) The probability must be normalized, so g/C12 /C28/C12P(˜n) e /C28½B2 ½ h2 =2 dh /C30P(˜n)ffiffiffiffiffiffiffiffi 2p ½B2 ½s /C301 ; (36) and P(n) /C30ffiffiffiffiffiffiffiffi ½B2 ½ 2 ps e /C28½B2 ½(n/C28˜n)2 =2 /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pNpqp exp /C28(n /C28 Np)2 2Npq"# : (37) Defining s2 /C13Npq ; P(n) /C301 sffiffiffiffiffiffi 2pp exp /C28(n /C28 ˜n)2 2s2"# ; (38) which is a GAUSSIAN DISTRIBUTION . For p /C101; a different approximation procedure shows that the binomial distribution approaches the POISSON DIS- TRIBUTION . The first CUMULANT is k1 /C30np; (39) and subsequent CUMULANTS are given by the RECUR- RENCE RELATION kr/C271 /C30pqdkr dp: (40) Let x and y be independent binomial RANDOM VARI- ABLES characterized by parameters n, p and m, p. The CONDITIONAL PROBABILITY of x given that x /C27y /C30 k isP(x /C30i ½x /C27y /C30k) /C30P(x /C30 i ; x /C27 y /C30 k) P(x/C27y/C30k) /C30P(x/C30i;y/C30k/C28i) P(x/C27y/C30k)/C30P(x/C30i)P(y/C30k/C28i) P(x/C27y/C30k) /C30n i/CP8/CP9 pi(1/C28p)n/C28im k/C28i/CP8/CP9 pk/C28i(1/C28p)m/C28(k/C28i) n/C27m k/CP8/CP9 pk(1/C28p)n/C27m/C28k /C30n i/CP8/CP9 m k/C28i/CP8/CP9 n/C27m k/CP8/CP9 : (41) Note that this is a HYPERGEOMETRIC DISTRIBUTION . See also DE MOIVRE- LAPLACE THEOREM ,HYPERGEO- METRIC DISTRIBUTION ,NEGATIVE BINOMIAL DISTRIBU- TION References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 531, 1987. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 102 /C1/03, 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Incomplete Beta Function, Student’s Distribu- tion, F-Distribution, Cumulative Binomial Distribution."§6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 219 /C1 /23, 1992. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 108 /C1/09, 1992. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Binomial Expansion BINOMIAL SERIES Binomial Formula BINOMIAL SERIES ,BINOMIAL THEOREM Binomial Identity Roman (1984, p. 26) defines "the" binomial identity as the equation pn(x/C27y)/C30Xn k/C300n k/CP8/CP9 pk(y)pn/C28k(x): (1) IFFthe sequence pn(x) satisfies this identity for all y in a FIELD Cof characteristic 0, then pn(x)i sa n ASSOCIATED SEQUENCE known as a BINOMIAL-TYPE SEQUENCE . In general, a binomial identity is a formula expres- sing products of factors as a sum over terms, eachincluding a BINOMIAL COEFFICIENT (n k):The prototypi- cal example is the BINOMIAL THEOREM (x /C27a)n /C30Xn k/C300n k/CP8/CP9 xkan/C28k (2) for n /C210. Abel (1826) gave a host of such identities (Riordan 1979, Roman 1984), some of which include (x /C27 y)(x /C27 y /C28 an)n/C281 /C30Xn k /C300n k/CP8/CP9 xy(x /C28ak)k /C281[y /C28a(n /C28k)]n /C28k /C281 ; (3) x/C281(x /C27y /C28na)n /C30Xn k/C300Xn k /C300n k/CP8/CP9 (x /C28ak)k /C281[y /C28a(n /C28k)]n/C28k(4) (Abel 1826, Riordan 1979, p. 18; Roman 1984, pp. 30 and 73), and x/C281(x /C27y)n /C30Xn k /C300n k/CP8/CP9 (x /C28ak)k /C281(y /C27ak)n/C28k(5) (Saslaw 1989). See also ABEL’S BINOMIAL THEOREM ,ABEL POLYNO- MIAL ,BINOMIAL COEFFICIENT ,D ILCHER’S FORMULA , Q-ABEL’S THEOREM References Abel, N. H. "Beweis eines Ausdrucks, von welchem die Binomial-Formel ein einzelner Fall ist." J. reine angew. Math. 1, 159 /C1/60, 1826. Reprinted in /(E/uvres Comple `tes, 2nd ed., Vol. 1. pp. 102 /C1/03, 1881. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 61, 1995. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 128, 1974. Ekhad, S. B. and Majewicz, J. E. "A Short WZ-Style Proof of Abel’s Identity." Electronic J. Combinatorics 3, No. 2, R16, 1, 1996. http://www.combinatorics.org/Volume_3/vo- lume3_2.html. Foata, D. "Enumerating k-Trees." Discr. Math. 1, 181 /C1/86, 1971. Riordan, J. Combinatorial Identities. New York: Wiley, p. 18, 1979. Roman, S. "The Abel Polynomials." §4.1.5 in The Umbral Calculus. New York: Academic Press, pp. 29 /C1/0 and 72 /C1/5, 1984. Saslaw, W. C. "Some Properties of a Statistical Distribution Function for Galaxy Clustering." Astrophys. J. 341, 588 /C1/ 98, 1989. Strehl, V. "Binomial Sums and Identities." Maple Technical Newsletter 10,37/C1/9, 1993. Strehl, V. "Binomial Identities--Combinatorial and Algorith- mic Aspects." Discrete Math. 136, 309 /C1/46, 1994. Binomial Number A number OF THE FORM an 9bn ; where a, b, and n are INTEGERS . They can be factored algebraically an /C28bn /C30(a /C28b)(an/C281 /C27an /C282b /C27.../C27abn/C282 /C27bn/C281) (1) for all n,an /C27bn /C30(a /C27b)(an/C281 /C28an /C282b /C27.../C28abn/C282 /C27bn/C281) (2) for n not a power of 2, and anm /C28bnm /C30(am /C28bm) /C2[am(n/C281) /C27am(n/C282)bm /C27.../C27bm(n/C281)] : (3) for all positive integers m, n. For example, a2 /C28b2 /C30(a /C28b)(a /C27b) (4) a3 /C28b3 /C30(a /C28b)(a2 /C27ab /C27b2) (5) a4 /C28b4 /C30(a /C28b)(a /C27b)(a2 /C27b2) (6) a5 /C28b5 /C30(a /C28b)(a4 /C27a3b /C27a2b2 /C27ab3 /C27b4) (7) a6 /C28b6 /C30(a /C28b)(a /C27b)(a2 /C28ab /C27b2)(a2 /C27ab /C27b2) (8) a7 /C28b7 /C30(a /C28b)(a6 /C27a5b /C27a4b2 /C27a3b3 /C27a2b4 /C27ab5 /C27b6) (9) a8 /C28b8 /C30(a /C28b)(a /C27b)(a2 /C27b2)(a4 /C27b4) (10) a9 /C28b9 /C30(a /C28b)(a2 /C27ab /C27b2)(a6 /C27a3b3 /C27b6) (11) a10 /C28b10 /C30(a /C28b)(a /C27b)(a4 /C28a3b /C27a2b2 /C28ab3 /C27b4) /C2 (a4 /C27a3b /C27a2b2 /C27ab3 /C27b4) (12) and a2 /C27b2 /C30a2 /C27b2 (13) a3 /C27b3 /C30(a /C27b)(a2 /C28ab /C27b2) (14) a4 /C27b4 /C30a4 /C27b4 (15) a5 /C27b5 /C30(a /C27b)(a4 /C28a3b /C27a2b2 /C28ab3 /C27b4) (16) a6 /C27b6 /C30(a2 /C27b2)(a4 /C28a2b2 /C27b4) (17) a7 /C27b7 /C30(a /C27b)(a6 /C28a5b /C27a4b2 /C28a3b3 /C27a2b4 /C28ab5 /C27b6) (18) a8 /C27b8 /C30a8 /C27b8 (19) a9/C27b9/C30(a/C27b)(a2/C28ab/C27b2)(a6/C28a3b3/C27b6) (20) a10/C27b10/C30(a2/C27b2)(a8/C28a6b2/C27a4b4/C28a2b6/C27b8): (21) In 1770, Euler proved that if ( a;b)/C301;then every FACTOR of a2n/C27b2n(22) is either 2 or OF THE FORM 2n/C271K/C271:(A number OF THE FORM 22n/C271 is called a F ERMAT NUMBER .) Ifpandqare PRIMES , then (apq/C281)(a/C281) (ap/C281)(aq/C281)/C281 (23) isDIVISIBLE by every PRIME FACTOR ofap/C281not dividing aq/C281:/ See also CUNNINGHAM NUMBER ,FERMAT NUMBER , MERSENNE NUMBER ,R IESEL NUMBER ,S IERPINSKI NUMBER OF THE SECOND KIND References Guy, R. K. "When Does 2a /C282bDivide na /C28nb :/" §B47 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 102, 1994. Qi, S and Ming-Zhi, Z. "Pairs where 2a /C282b Divides na /C28nb for All n." Proc. Amer. Math. Soc. 93, 218 /C120, 1985. Schinzel, A. "On Primitive Prime Factors of an /C28bn :/" Proc. Cambridge Phil. Soc. 58, 555 /C1/62, 1962. Binomial Polynomial FALLING FACTORIAL Binomial Series For ½x½B1; (1 /C27x)n /C30Xn k /C300n k/CP8/CP9 xk (1) /C30n 0/CP8/CP9 x0 /C27n 1/CP8/CP9 x1 /C27n 2/CP8/CP9 x2 /C27/C1/C1/C1 (2) /C301 /C27n! 1!(n /C28 1)!x /C27n! (n /C28 2)!2!x2 /C27... (3) /C301 /C27nx /C27n(n /C28 1) 2x2 /C27...: (4) The binomial series also has the CONTINUED FRAC- TION representation (1/C27x)n/C301 1/C28nx 1/C271 /C215(1/C27n) 1 /C2152x 1/C271 /C215(1/C28n) 2 /C2153x 1/C272(2/C27n) 3 /C2154x 1/C272(2/C28n) 4 /C2155x 1/C273(3/C27n) 5 /C2156x 1/C27...:(5) See also BINOMIAL IDENTITY ,B INOMIAL THEOREM , MULTINOMIAL SERIES ,NEGATIVE BINOMIAL SERIES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 14 /C1/5, 1972. Pappas, T. "Pascal’s Triangle, the Fibonacci Sequence & Binomial Formula." The Joy of Mathematics. San Carlos,CA: Wide World Publ./Tetra, pp. 40 /C1/1, 1989. Binomial Sums The important BINOMIAL THEOREM states that Xn k/C300n k/CP8/CP9 rk/C30(1/C27r)n: (1) Sums of powers of BINOMIAL COEFFICIENTS ar(n)/C30Xn k/C300n k/CP8/CP9r (2) are given by a1(n)/C302n(3) a2(n)/C302n n/CP8/CP9 (4) /a1(n) and a2(n) obey the RECURRENCE RELATION a1(n/C271)/C282a1(n)/C300 (5) (n/C271)a2(n/C271)/C28(4n/C272)a2(n)/C300: (6) Franel (1894, 1895) was the first to obtain recur- rences for a3n(Riordan 1948, p. 193) and a4(n); (n/C271)2a3(n/C271)/C28(7n2/C277n/C272)a3(n)/C288n2a3(n/C281) /C300 (7) (Barrucand 1975, Cusick 1989, Jin and Dickinson 2000) (n/C271)3a4(n/C271)/C282(2n/C271)(3n2/C273n/C271)a4(n) /C284n(4n/C271)(4n/C281)a4(n/C281)/C300: (8) (Jin and Dickinson 2000). Therefore, a3nare some- times called F RANEL NUMBERS . The sequence for a3n cannot be expressed as a fixed number of hypergeo- metric terms (Petkovsek et al. 1996, p. 160), and therefore has no closed-form hypergeometric expres- sion. Perlstadt (1987) found recurrences of length 4 forr/C305 and 6, while Schmidt and Yuan (1995) showed that the give recurrences for r/C303, 4, 5, and 6 are minimal, are the minimal lengths for r/C216 are at least 3. The following table summarizes the firstfew values of a r(n) for small r. kSloane /ak(n)/ 1 A000079 1, 2, 4, 8, 16, 32, 54, ... 2 A000984 1, 2, 6, 20, 70, 252, 924, ... 3 A000172 1, 2, 10, 56, 346, 2252, ... 4 A005260 1, 2, 18, 164, 1810, 21252, ... 5 A005260 1, 2, 34, 488, 9826, 206252, ... The corresponding alternating series is br/C13Xn k/C300(/C281)kn k/CP8/CP9k /C300: (9) The first few values are b1(n)/C300 (10) b2(n)/C302nffiffiffipp G(1 2/C2812n)G(1/C2712n); (11) /C300 for n/C302k (/C281)k(n k) for n/C302k/C281/C26 (12) b3(n)/C302nffiffiffippG(1/C273 2n) n!G(1 2(1/C28n))G(1/C2712n)2(13) /C300 for n/C302k/C281 (/C281)k(3k)! (k!)3forn/C302k;8 < :(14) where G(z) is the GAMMA FUNCTION , and the odd terms of b3(n) are given by de Bruijn’s s(3;n) with alternating signs. de Bruijn (1982) has considered the sum s(m;n)/C30X2n k/C300(/C281)k/C27n2n k !m (15) form;n]1:This sum has closed form for m/C301, 2, and 3, s(1;n)/C300 (16) s(2;n)/C30(2n)! (n!)2; (17) the CENTRAL BINOMIAL COEFFICIENT , giving 1, 2, 6, 20, 70, 252, 924, . . . (Sloane’s A000984), and s(3;n)/C30(3n)! (n!)3; (18) giving 1, 6, 90, 1680, 36450, 756756, . . . (Sloane’s A006480; Aizenberg and Yuzhakov 1984). However,there is no similar formula for m]4 (Finch). The first few terms of s(4;n) are 1, 14, 786, 61340, 5562130, . . . (Sloane’s A050983), and for s(5;n) are 1, 30, 5730, 1696800, 613591650, . . . (Sloane’s A050984). An interesting generalization of b 1(n) was found by Ruiz (1996), X/C12 k/C300(/C281)kn k/CP8/CP9 (x/C28k)n/C30n! (19) andXn k/C300(/C281)kn k/CP8/CP9 (x/C28k)n/C30n! (20) for positive integer nand all x. The infinite sum of inverse binomial coefficients has the analytic form X/C12 k/C3001 n k/CP8/CP9/C302F1(1;1;/C28n;/C281) (21) /C30/C28(n/C271)g1 0dx (1/C28x)n/C272(x/C271); (22) where2F1(a;b;c;x)i sa HYPERGEOMETRIC FUNC- TION . In fact, in general, X/C12 k/C3001 n k/CP8/CP9p/C30p/C271Fp(1;...;1|fflfflfflfflfflffl{zfflfflfflfflfflffl} p/C271;/C28n;...;/C28n|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} p;(/C281)k) (23) and X/C12 k/C300(/C281)k n k/CP8/CP9p/C30p/C271Fp(1;...;1|fflfflfflfflfflffl{zfflfflfflfflfflffl} p/C271);/C28n;...;/C28n|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} p;(/C281)k/C271): (24) A fascinating series of identities involving inverse central binomial coefficients times small powers are given by X/C12 n/C3011 2n n/CP8/CP9/C301 27(2pffiffiffi 3p /C279)/C300:7363998587 . . . (25) X/C12 n/C3011 n2n n/CP8/CP9/C301 9pffiffiffi 3p /C300:6045997881 . . . (26) X/C12 n/C3011 n22n n/CP8/CP9 /C301 3z(2)/C3018p2(27) X/C12 n/C3011 n42n n/CP8/CP9/C3017 36z(4)/C3017 3240p4(28) (Comtet 1974, p. 89; Le Lionnais 1983, pp. 29, 30, 41, 36), which follow from the beautiful formula X/C12 n/C3011 nk2n n/CP8/CP9/C301 2k/C271Fk(1;...;1|fflfflfflfflfflffl{zfflfflfflfflfflffl} k/C271;32;2;...;2|fflfflfflfflfflffl{zfflfflfflfflfflffl} k/C281;14) (29) fork]1;where mFn(a1;...;am;b1;...;bn;x)i sa GENERALIZED HYPERGEOMETRIC FUNCTION . Additional sums of this type include X/C12 n /C3011 n32n n/CP8/CP9/C301 18pffiffiffi 3p [ c1(1 3) /C28 c1(23)] /C2843 z(3) (30) X/C12 n /C3011 n52n n/CP8/CP9 /C301 432pffiffiffi 3p [ c3(1 3) /C28 c3(23)] /C2819 3 z(5) /C2719 z(3)p2(31) X/C12 n/C3011 n72n n/CP8/CP9/C3011 311040pffiffiffi 3p [c5(1 3) /C28 c5(23)] /C28493 24 z(7) /C2713 z(5)p2 /C2717 1620z(3)p4 ; (32) where cn(x) is the POLYGAMMA FUNCTION and z(x)is the RIEMANN ZETA FUNCTION (Plouffe 1998). Sums OF THE FORM X/C12 n/C301( /C281)n/C271 nk2n n/CP8/CP9 /C301 2 k /C271Fk(1; ...; 1|fflfflfflfflfflffl{zfflfflfflfflfflffl} k /C271;32 ; 2; ...; 2|fflfflfflfflfflffl{zfflfflfflfflfflffl} k /C281; /C2814) (33) can also be simplified (Plouffe) to give the special cases X/C12 n /C301( /C281)n/C281 n2n n/CP8/CP9 /C302 5ffiffiffi 5p sinh/C281(1 2) (34) X/C12 n/C301( /C281)n/C281 n22n n/CP8/CP9 /C302[sinh/C281(12)]2 (35) X/C12 n/C301( /C281)n/C281 n32n n/CP8/CP9 /C302 5 z(3) : (36) Other general identities include (a /C27 b)n a/C30Xn k /C300n k/CP8/CP9 (a /C28kc)k /C281(b /C27kc)n/C28k(37) (Prudnikov et al. 1986), which gives the BINOMIAL THEOREM as a special case with c /C300, and X/C12 n /C3002n /C27s n/CP8/CP9 xn /C302 F1(1 2(s /C271);12(s /C272); s /C271; 4x) /C302? (ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 4xp /C27 1)?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28 4xp ; (38) where 2F1(a ; b; c; z)isa HYPERGEOMETRIC FUNCTION (Abramowitz and Stegun 1972, p. 555; Graham et al. 1994, p. 203). For NONNEGATIVE INTEGERS n and r with r 5n /C271 ; Xn k /C300( /C281)k k /C27 1n k/CP8/CP9Xr/C281 j/C300(/C281)j n j/CP8/CP9 (r /C28j)n/C28k"/C27Xn/C28r j /C300(/C281)j n j/CP8/CP9 (n /C271 /C28r /C28j)n/C28k/C2P /C30n!: (39) Taking n /C302r /C281 gives Xn k/C300(/C281)k K /C28 1n k/CP8/CP9Xr/C281 j/C300n j/CP8/CP9 (r /C28j)n/C28k /C301 2n!: (40) Other identities are Xn k /C300n /C27k k/CP8/CP9 [xn/C271(1 /C28x)k /C27(1 /C28x)n/C271xk] /C301 (41) (Gosper 1972) and X ini 2/CP8/CP9 /C27X i>jninj /C30n 2/CP8/CP9 ; (42) where n /C13X ini : (43) The latter is the umbral analog of the multinomial theorem for n2 (a /C27 b /C27 c)2 2/C30a2 2/C27b2 2/C27c2 2/C27ab /C27ac /C27bc (44) using the lower-factorial polynomial (n)2/C30n(n/C281)=2; giving a/C27b/C27c 2/CP8/CP9 /C30a 2/CP8/CP9 /C27b 2/CP8/CP9 /C27c 2/CP8/CP9 /C27ab/C27ac/C27bc:(45) The identity holds true not only for ( n)2andn2=2;but also for any quadratic polynomial OF THE FORM n(n/C27 a)=2 (Dubuque). See also APE´ RY NUMBER ,B INOMIAL COEFFICIENT , CENTRAL BINOMIAL COEFFICIENT ,H YPERGEOMETRIC IDENTITY ,H YPER GEOMETRIC SERIES ,IDEMPOTENT NUMBER ,JONAH FORMULA KLEE’S IDENTITY ,LUCAS CORRESPONDENCE THEOREM ,MARRIED COUPLES PRO- BLEM ,M ORLEY’S FORMULA ,N EXUS NUMBER ,STAN- LEY’S IDENTITY ,S TREHL IDENTITIES ,S ZE´ KELY IDENTITY ,W ARING FORMULA ,W ORPITZKY’S IDENTITY References Aizenberg, I. A. and Yuzhakov, A. P. Integral Representa- tions and Residues in Multidimensional Complex Analy- sis.Providence, RI: Amer. Math. Soc., p. 194, 1984. Barrucand, P. "Problem 75 /C1/: A Combinatorial Identity." SIAM Rev. 17, 168, 1975. Beukers, F. "Another Congruence for the Ape ´ry Numbers." J. Number Th. 25, 201/C1/10, 1987. Cusick, T. W. "Recurrences for Sums of Powers of Binomial Coefficients." J. Combin. Th. Ser. A 52,7 7/C1/3, 1989. de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, 1982. Egorychev, G. P. Integral Representation and the Computa- tion of Combinatorial Sums. Providence, RI: Amer. Math. Soc., 1984. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/nielram/nielram.html. Franel, J. "On a Question of Laisant." L’interme ´diaire des mathe ´maticiens 1,45/C1/7, 1894. Franel, J. "On a Question of J. Franel." L’interme ´diaire des mathe ´maticiens 2,33/C1/5, 1895. Gosper, R. W. Item 42 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 16, Feb. 1972. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial Coefficients." Ch. 5 in Concrete Mathematics: A Founda- tion for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 153 /C1/42, 1994. Jin, Y. and Dickinson, H. "Ape´ry Sequences and Legendre Transforms." J. Austral. Math. Soc. Ser. A 68, 349 /C1/56, 2000. MacMahon P. A. "The Sums of the Powers of the Binomial Coefficients." Quart. J. Math. 33, 274 /C1/88, 1902. McIntosh, R. J. "Recurrences for Alternating Sums of Powers of Binomial Coefficients." J. Combin. Th. A 63, 223 /C1/33, 1993. Perlstadt, M. A. "Some Recurrences for Sums of Powers of Binomial Coefficients." J. Number Th. 27, 304 /C1/09, 1987. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Plouffe, S. "The Art of Inspired Guessing." Aug. 7, 1998. http://www.lacim.uqam.ca/plouffe/inspired.html. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980. Ruiz, S. Math. Gaz. 80, 579 /C1/82, Nov. 1996. Schmidt, A. L. and Yuan, J. "On Recurrences for Sums of Powers of Binomial Coefficients." Tech. Rep., 1995. Shanks, E. B. "Iterated Sums of Powers of the Binomial Coefficients." Amer. Math. Monthly 58, 404 /C1/07, 1951. Sloane, N. J. A. Sequences A000079/M1129, A000172/ M1971, A000984/M1645, A005260/M2110, A005261/ M2156, A006480/M4284, A050983, and A050984 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Strehl, V. "Binomial Identities--Combinatorial and Algorith- mic Aspects. Trends in Discrete Mathematics." Disc. Math. 136, 309 /C1/46, 1994. Binomial Theorem The theorem that, for POSITIVE INTEGERS n, (x /C27a)n /C30Xn k/C300n! k!(n /C28 k)!xkan /C28k /C30Xn k /C300n k/CP8/CP9 xkan/C28k ; the so-called BINOMIAL SERIES , where (n k) are BINO- MIAL COEFFICIENTS . The theorem was known for the case n /C302 by Euclid around 300 BC, and stated in its modern form by Pascal in a posthumous pamphlet published in 1665. Newton (1676) showed that a similar formula (with INFINITE upper limit) holds for NEGATIVE INTEGERS n, (x /C27a) /C28n /C30X/C12 k /C300/C28n k/CP8/CP9 xka/C28n/C28k ; the so-called NEGATIVE BINOMIAL SERIES , which con- verges for xjj> ajj:/ See also BINOMIAL COEFFICIENT ,BINOMIAL IDENTITY , BINOMIAL SERIES ,CAUCHY BINOMIAL THEOREM ,CHU- VANDERMON DE IDENTITY ,L OGARITHMIC BINOMIALFORMULA ,NEGATIVE BINOMIAL SERIES , Q-BINOMIAL THEOREM ,RANDOM WALK References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 307 /C1/08, 1985. Boyer, C. B. and Merzbach, U. C. "The Binomial Theorem." A History of Mathematics, 2nd ed. New York: Wiley, pp. 393 /C1/94, 1991. Conway, J. H. and Guy, R. K. "Choice Numbers Are Bino- mial Coefficients." In The Book of Numbers. New York: Springer-Verlag, pp. 72 /C1/4, 1996. Coolidge, J. L. "The Story of the Binomial Theorem." Amer. Math. Monthly 56, 147 /C1/57, 1949. Courant, R. and Robbins, H. "The Binomial Theorem." §1.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 16 /C1/8, 1996. Pascal, B. Traite du Triangle Arithmetic. 1665. Whittaker, E. T. and Robinson, G. "The Binomial Theorem." §10 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 15 /C1/9, 1967. Binomial Transform The binomial transform takes the sequence a0 ; a1 ; a2 ; ... to the sequence b0 ; b1 ; b2 ; ... via the transforma- tion bn /C30Xn k /C300(/C281)n/C28k n k/CP8/CP9 ak : The inverse transform is an /C30Xn k/C300n k/CP8/CP9 bk : (Sloane and Plouffe 1995, pp. 13 and 22). The inverse binomial transform of bn /C301 for prime n and bn /C300 for composite n is 0, 1, 3, 6, 11, 20, 37, 70, ... (Sloane’s A052467). The inverse binomial transform of bn /C301 for even n and bn /C300 for odd n is 0, 1, 2, 4, 8, 16, 32, 64, ... (Sloane’s A000079). Similarly, the inverse binomial transform of bn /C301 for odd n and bn /C300 for even n is 1, 2, 4, 8, 16, 32, 64, ... (Sloane’s A000079). The inverse binomial transform of the B ELL NUMBERS 1, 1, 2, 5, 15, 52, 203, . . . (Sloane’s A000110) is a shifted version of the same numbers: 1, 2, 5, 15, 52, 203, . . . (Bernstein and Sloane 1995, Sloane and Plouffe 1995, p. 22). The CENTRAL and RAW MOMENTS of statistical dis- tributions are also related by the binomial transform. See also CENTRAL MOMEN T,E ULER TRANSFORM , EXPONENTIAL TRANSFORM ,M O¨ BIUS TRANSFORM , RAW MOMENT References Bernstein, M. and Sloane, N. J. A. "Some Canonical Se- quences of Integers." Linear Algebra Appl. 226//228 ,57/C1/ 2, 1995. Sloane, N. J. A. Sequences A000079/M1129, A000110/ M1484, and A052467 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Binomial Triangle PASCAL’S TRIANGLE Binomial-Type Sequence A sequence of POLYNOMIALS pnsatisfying the identi- ties pn(x /C27y) /C30X k ]0n k/CP8/CP9 pk(x)pn/C28k(y) : See also BINOMIAL IDENTITY ,SHEFFER SEQUENCE , UMBRAL CALCULUS References Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Binormal Developable A RULED SURFACE M is said to be a binormal developable of a curve y if M can be parameterized by x(u ; v) /C30y(u) /C27v ˆB(u) ; where B is the BINORMAL VECTOR . See also NORMAL DEVELOPABLE ,TANGENT DEVELOP- ABLE References Gray, A. "Developables." §17.6 in Modern Differential Geo- metry of Curves and Surfaces with Mathematica. Boca Raton, FL: CRC Press, pp. 352 /C1/54, 1993. Binormal Vector ˜B /C13 ˆT /C29 ˆN (1) /C30r?/C29r ƒ r?/C29r ƒ jj; (2) where the unit TANGENT VECTOR T and unit "princi- pal" NORMAL VECTOR N are defined by ˆT /C13r?(s) ˆr(s)jj (3) ˆN /C13rƒ(s) rƒ(s) jj (4) Here, r is the RADIUS VECTOR , s is the ARC LENGTH , tis the TORSION , and k is the CURVATURE . The binormal vector satisfies the remarkable identity [ ˙B; ¨B; /C5B] /C30 t 5d dsk t ! : (5) See also FRENET FORMULAS ,NORMAL VECTOR ,TAN- GENT VECTOR References Kreyszig, E. "Binormal. Moving Trihedron of a Curve." §13 in Differential Geometry. New York: Dover, pp. 36 /C1/7, 1991. Bin-Packing Problem The problem of packing a set of items into a number of bins such that the total weight, volume, etc. does not exceed some maximum value. A simple algorithm (the first-fit algorithm) takes items in the order they come an places them in the first bin in which they fit. In 1973, J. Ullman proved that this algorithm can differ from an optimal packing by as much at 70% (Hoffman 1998, p. 171). An alternative strategy first orders the items from largest to smallest, then places them sequentially in the first bin in which they fit. In 1973, D. Johnson showed that this strategy is never suboptimal by more than 22%, and furthermore that no efficient bin-packing algorithm can be guaranteed to do better than 22% (Hoffman 1998, p. 172). There exist arrangements of items such that applying the packing algorithm after removing an item results inone more bin being required than the number obtained if the item is included (Hoffman 1998, pp. 172 /C1/73). See also COOKIE- CUTTER PROBLEM ,TILING PROBLEM References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Bioche’s Theorem If two complementary P LU¨CKER CHARACTERISTICS are equal, then each characteristic is equal to its comple- ment except in four cases where the sum of order andclass is 9. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 101, 1959. Biotic Potential LOGISTIC EQUATION Bipartite Graph A set of VERTICES decomposed into two disjoint sets such that no two VERTICES within the same set are adjacent. A bigraph is a special case of a K-PARTITE GRAPH with k /C302. Bipartite graphs are equivalent to two-colorable graphs, and a graph is bipartite IFF all its cycles are of even length (Skiena 1990, p. 213). The numbers of bipartite graphs on n /C301, 2, ...nodes are 1, 2, 3, 7, 13, 35, 88, 303, ... (Sloane’s A033995). A graph can be tested for bipartiteness using Bipar- titeQ [g] in the Mathematica add-on package Dis- creteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The numbers of CONNECTED bipartite graphs on n/C301, 2 ...nodes are 1, 1, 1, 3, 5, 17, 44, 182, ... (Sloane’s A005142). All TREES are bipartite (Skiena 1990, p. 213). See also BICUBIC GRAPH ,C OMPLETE BIPARTITE GRAPH , K-PARTITE GRAPH ,KO¨ NIG-EGEVA ´ RY THEOREMReferences Chartrand, G. Introductory Graph Theory. New York: Dover, p. 116, 1985. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 12, 1986. Skiena, S. "Coloring Bipartite Graphs." §5.5.2 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 213, 1990. Sloane, N. J. A. Sequences A033995 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Steinbach, P. Field Guide to Simple Graphs. Albuquerque, NM: Design Lab, 1990. Biplanar Double Point ISOLATED SINGULARITY Bipolar Coordinates Bipolar coordinates are a 2-D system of coordinates. There are two commonly defined types of bipolarcoordinates, the first of which is defined by x/C30asinh v cosh v/C28cosu(1) y/C30asinu cosh v/C28cosu; (2) where u/C23[0;2p);v/C23(/C28/C12;/C12):The following identi- ties show that curves of constant uandvare CIRCLES inxy-space. x2/C27(y/C28acotu)2/C30a2csc2u (3) (x/C28acoth v)2/C27y2/C30a2csch2v: (4) The SCALE FACTORS are hu/C30a cosh v/C28cosu(5) hv/C30a cosh v/C28cosu(6) The L APLACIAN is 92/C30(cosh v/C28cosu)2 a2@2 @u2/C27@2 @v2 ! : (7) LAPLACE’S EQUATION is separable. Two-center bipolar coordinates are two coordinatesgiving the distances from two fixed centers r 1andr2; sometimes denoted rand r?:For two-center bipolar coordinates with centers at ( 9c;0); r2 1/C30(x/C27c)2/C27y2(8) r22/C30(x/C28c)2/C27y2: (9) Combining (8) and (9) gives r2 1 /C28r22 /C304cx : (10) Solving for CARTESIAN COORDINATES x and y gives x /C30r2 1 /C28 r22 4c (11) y /C3091 4cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16c2r2 1 /C28(r21 /C28r22 /C274c2)2q : (12) Solving for POLAR COORDINATES gives r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 /C27 r22 /C28 2c2 2s (13) u /C30tan/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r4 2 /C28 2(4c2 /C27 r21)r22 /C28 (4c2 /C28 r21)2q r21 /C28 r222 435: (14) See also B IPOLAR CYLINDRICAL COORDINATES ,POLAR COORDINATES References Lockwood, E. H. "Bipolar Coordinates." Ch. 25 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 186 /C1/90, 1967. Bipolar Cylindrical Coordinates A set of CURVILINEAR COORDINATES defined by x /C30a sinh v cosh v /C28 cos u (1) y /C30a sin u cosh v /C28 cos u (2) z /C30z ; (3) where u /C23 [0; 2 p); v /C23 (/C28/C12;/C12); and z /C23 (/C28/C12;/C12): There are several notational conventions, and whereas (u; v; z) is used in this work, Arfken (1970) prefers ( h ; j; z) : The following identities show thatcurves of constant u and v are CIRCLES in xy-space. x2 /C27(y /C28a cot u)2 /C30a2 csc2 u (4) (x /C28a coth v)2 /C27y2 /C30a2 csch2 v: (5) The SCALE FACTORS are hu /C30a cosh v /C28 cos u (6) hv /C30a cosh v /C28 cos u (7) hz /C301: (8) The LAPLACIAN is 92 /C30(cosh v /C28 cos u)2 a2@2 @u2 /C27@2 @v2 ! /C27@2 @z2 : (9) LAPLACE’S EQUATION is not separable in BIPOLAR CYLINDRICAL COORDINATES , but it is in 2-D BIPOLAR COORDINATES . See also BIPOLAR COORDINATES ,POLAR COORDINATES References Arfken, G. "Bipolar Coordinates (/j; h; z)." §2.9 in Mathema- tical Methods for Physicists, 2nd ed. Orlando, FL: Aca- demic Press, pp. 97 /C1/02, 1970. Bipolyhedral Group The image of A5 /C29A5in the SPECIAL ORTHOGONAL GROUP SO(4); where A5 is the ICOSAHEDRAL GROUP . See also ICOSAHEDRAL GROUP ,SPECIAL ORTHOGONAL GROUP References Endraß, S. "The Sarti Surface." http://enriques.mathemati- k.uni-mainz.de/kon/docs/Esarti.shtml. Biprism Two slant triangular PRISMS fused together. See also PRISM ,SCHMITT- CONWAY BIPRISM Bipyramid DIPYRAMID Biquadratefree A number is said to be biquadratefree (or quarticfree) if its PRIME FACTORIZATION contains no quadrupled factors. All PRIMES and PRIME POWERS pn with n 5 3 are therefore trivially biquadratefree. The biquadra- tefree numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, ... (Sloane’s A046100). The biquadrate- ful numbers (i.e., those that contain at least one biquadrate) are 16, 32, 48, 64, 80, 81, 96, ... (Sloane’s A046101). The number of biquadratefree numbers less than 10, 100, 1000, ... are 10, 93, 925, 9240, 92395, 923939, ..., and their asymptotic density is 1=z(4)/C3090=p4:0:923938 ;where z(n) is the R IE- MANN ZETA FUNCTION . See also CUBEFREE ,PRIME NUMBER ,RIEMANN ZETA FUNCTION ,SQUAREFREE References Sloane, N. J. A. Sequences A046100 and A046101 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Biquadratic Equation QUARTIC EQUATION Biquadratic Number A biquadratic number is a fourth POWER ,n4:The first few biquadratic numbers are 1, 16, 81, 256, 625, . . . (Sloane’s A000583). The minimum number of biqua-dratic numbers needed to represent the numbers 1, 2, 3, . . . are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, . . . (Sloane’s A002377), and the number ofdistinct ways to represent the numbers 1, 2, 3, . . . in terms of biquadratic numbers are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, . . . A brute-forcealgorithm for enumerating the biquadratic permuta- tions of nis repeated application of the GREEDY ALGORITHM . Every POSITIVE integer is expressible as a SUM of (at most) g(4)/C3019 biquadratic numbers (W ARING’S PRO- BLEM ). Davenport (1939) showed that G(4)/C3016; meaning that all sufficiently large integers require only 16 biquadratic numbers. It is also known thatevery integer is a sum of at most 10 signed biqua- drates ( eg(4)510; although it is not known if 10 canbe reduced to 9). The following table gives the first few numbers which require 1, 2, 3, . . ., 19 biquadratic numbers to represent them as a sum, with the sequences for 17, 18, and 19 being finite. # Sloane Numbers 1 Sloane’s A0002901, 16, 81, 256, 625, 1296, 2401, 4096, ... 2 Sloane’s A0033362, 17, 32, 82, 97, 162, 257,272, ... 3 Sloane’s A0033373, 18, 33, 48, 83, 98, 113, 163, ... 4 Sloane’s A0033384, 19, 34, 49, 64, 84, 99, 114,129, ... 5 Sloane’s A0033395, 20, 35, 50, 65, 80, 85, 100,115, ... 6 Sloane’s A0033406, 21, 36, 51, 66, 86, 96, 101, 116, ... 7 Sloane’s A0033417, 22, 37, 52, 67, 87, 102, 112,117, ... 8 Sloane’s A0033428, 23, 38, 53, 68, 88, 103, 118,128, ... 9 Sloane’s A0033439, 24, 39, 54, 69, 89, 104, 119,134, ... 10 Sloane’s A00334410, 25, 40, 55, 70, 90, 105,120, 135, ... 11 Sloane’s A00334511, 26, 41, 56, 71, 91, 106,121, 136, ... 12 Sloane’s A00334612, 27, 42, 57, 72, 92, 107,122, 137, ... 13 Sloane’s A04604413, 28, 43, 58, 73, 93, 108, 123, 138, ... 14 Sloane’s A04604514, 29, 44, 59, 74, 94, 109,124, 139, ... 15 Sloane’s A04604615, 30, 45, 60, 75, 95, 110,125, 140, ... 16 Sloane’s A04604731, 46, 61, 76, 111, 126, 141, 156, ... 17 Sloane’s A04604847, 62, 77, 127, 142, 157, 207,222, ... 18 Sloane’s A04604963, 78, 143, 158, 223, 238,303, 318, ... 19 Sloane’s A04605079, 159, 239, 319, 399 The following table gives the numbers which can be represented in ndifferent ways as a sum of k biquadrates. kn Sloane Numbers 1 1 Sloane’s A0002901, 16, 81, 256, 625, 1296, 2401, 4096, ... 2 2 Sloane’s A018786635318657, 3262811042, 8657437697, ... The numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, ... (Sloane’s A046039) cannot be represented using distinct biquadrates. See also CUBIC NUMBER ,PARTITION ,SQUARE NUM- BER,W ARING’S PROBLEM References Davenport, H. "On Waring’s Problem for Fourth Powers." Ann. Math. 40, 731 /C1/47, 1939. Hardy, G. H. and Wright, E. M. "The Representation of a Number by Two or Four Squares." Ch. 20 in An Introduc- tion to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 297 /C1/16, 1979. Sloane, N. J. A. Sequences A000290, A000583/M5004, A002377, A003336, A003337, A003338, A003339, A003340, A003341, A003342, A003343, A003344, A003345, A003346, A018786, and A046039 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Biquadratic Reciprocity Theorem Gauss stated the reciprocity theorem for the case n /C304 x4 /C13q (mod p) (1) can be solved using the GAUSSIAN INTEGERS as p s ! 4s p ! 4/C30(/C281)[(N( p) /C281)=4][(N(s) /C281)=4] : (2) Here, p and s are distinct GAUSSIAN INTEGER PRIMES , and N(a /C27bi) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2p (3) is the norm. The symbola p/CP6/CP7 means a p ! 4 /C301i f x4 /C13 a (mod p) is solvable /C281; i ; or /C28i otherwise/C26 (4) where "solvable" means solvable in terms of GAUS- SIAN INTEGERS . 2 is a quartic residue (mod p) IFF there are integers x, y such thatx2 /C2764y2 /C30p: (5) This is a generalization of the GENUS THEOREM . See also BIQUADRATIC RESIDUE ,G ENUS THEOREM , RECIPROCITY THEOREM References Ireland, K. and Rosen, M. "Cubic and Biquadratic Recipro- city." Ch. 9 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108 /C1/37, 1990. Biquadratic Residue If there is an INTEGER x such that x4 /C13q (mod p) ; (1) then q is said to be a biquadratic residue (mod p). If not, q is said to be a biquadratic nonresidue (mod p). See also BIQUADRATIC RECIPROCITY THEOREM ,CUBIC RESIDUE ,QUADRATIC RESIDUE References Nagell, T. Introduction to Number Theory. New York: Wiley, p. 115, 1951. Biquaternion A QUATERNION with COMPLEX coefficients. The ALGE- BRA of biquaternions is isomorphic to a full matrix ring over the complex number field (van der Waerden 1985). See also QUATERNION References Clifford, W. K. "Preliminary Sketch of Biquaternions." Proc. London Math. Soc. 4, 381 /C1/95, 1873. Hamilton, W. R. Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method. Dublin: Hodges and Smith, 1853. Study, E. "Von den Bewegung und Umlegungen." Math. Ann. 39, 441 /C1/66, 1891. van der Waerden, B. L. A History of Algebra from al- Khwarizmi to Emmy Noether. New York: Springer-Verlag, pp. 188 /C1/89, 1985. Birational Transformation A transformation in which coordinates in two SPACES are expressed rationally in terms of those in another. See also RIEMANN CURVE THEOREM ,W EBER’S THEO- REM Birch Conjecture SWINNERTON- DYERCONJECTURE Birch-Swinnerton-Dyer Conjecture SWINNERTON- DYERCONJECTURE Birkhoff’s Ergodic Theorem Let T be an ergodic ENDOMORPHISM of the PROBABIL- ITY SPACE X and let f : X 0 R be a real-valued MEASURABLE FUNCTION . Then for ALMOST EVERY x /C23 X ; we have 1 nXn j/C301f(Tj(x) 0g fdm (1) as n 0/C12: To illustrate this, take f to be the characteristic function of some SUBSET A of X so that f(x) /C301i f x /C23 A 0i f x QA:/C26 (2) The left-hand side of (1) just says how often the orbit of x (that is, the points x, Tx, T2x; ...) lies in A, and the right-hand side is just the MEASURE of A. Thus, for an ergodic ENDOMORPHISM , "space-avera- ges /C30time-averages almost everywhere." Moreover, if T is continuous and uniquely ergodic with BOREL PROBABILITY MEASURE m and f is continuous, then we can replace the ALMOST EVERYWHERE convergence in (1) with "everywhere." See also BIRKHOFF’S THEOREM ,ERGODIC THEORY References Cornfeld, I.; Fomin, S.; and Sinai, Ya. G. Appendix 3 in Ergodic Theory. New York: Springer-Verlag, 1982. Birkhoff-Khinchin Ergodic Theorem BIRKHOFF’S ERGODIC THEOREM Birkhoff-Witt Theorem POINCARE ´ -BIRKHOFF- WITT THEOREM Birotunda Two adjoined ROTUNDAS . See also BILUNABIROTUNDA ,CUPOLAROTUNDA ,ELON- GATED GYROCUPOLAROTUNDA ,ELONGATED ORTHOCU- POLAROTUNDA ,E LONGATED ORTHOBIROTUNDA , GYROCUPOLAROTUNDA ,G YROELONGATED ROTUNDA , ORTHOBIROTUNDA ,TRIANGULAR HEBESPHENOROTUN- DA Birthday Attack Birthday attacks are a class of brute-force techniques used in an attempt to solve a class of CRYPTOGRAPHIC HASH FUNCTION problems. These methods take ad- vantage of functions which, when supplied with a random input, return one of k equally likely values. By repeatedly evaluating the function for different inputs, the same output is expected to be obtained after about 1:2ffiffiffi kp evaluations.See also BIRTHDAY PROBLEM ,CRYPTOGRAPHIC HASH FUNCTION References RSA Laboratories. "Question 95. What is a Birthday Attack" and "Question 96. How Does the Length of a Hash Value Affect Security?" http://www.rsasecurity.com/rsalabs/faq/. van Oorschot, P. and Wiener, M. "A Known Plaintext Attack on Two-Key Triple Encryption." In Advances in Cryptol- ogy--Eurocrypt ’90. New York: Springer-Verlag, pp. 366 /C1/ 77, 1991. Yuval, G. "How to Swindle Rabin." Cryptologia 3, 187/C1/89, Jul. 1979. Birthday Problem Consider the probability Q1(n;d) that no two people out of a group of nwill have matching birthdays out ofdequally possible birthdays. Start with an arbi- trary person’s birthday, then note that the probability that the second person’s birthday is different is ( d/C28 1)=d;that the third person’s birthday is different from the first two is [( d/C281)=d][(d/C282)=d];and so on, up through the nth person. Explicitly, Q1(n;d)/C30d/C281 dd/C282 d/C1/C1/C1d/C28(n/C281) d /C30(d/C281)(d/C282)/C1/C1/C1[d/C28(n/C281)] dn/C281: (1) But this can be written in terms of FACTORIALS as Q1(n;d)/C30d! (d/C28n)!dn; (2) so the probability P2(n;365) that two people out of a group of n do have the same birthday is therefore P2(n;d)/C301/C28Q1(n;d)/C301/C28d! (d/C28n)!dn: (3) If 365-day years have been assumed, i.e., the exis-tence of leap days is ignored, then the number ofpeople needed for there to be at least a 50% chance that two share birthdays is the smallest nsuch that P 2(n;365)]1=2:This is given by n/C3023, since P2(23;365) /C303809390470229739078524370829105639051888645406094 7509188326851535012542620742522314756326980590820 :0:507297 : (4) The number nof people needed to obtain P2(n;d)] 1=2 for d/C301, 2, . . ., are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, . . . (Sloane’s A033810). The probability P2(n;d) can be estimated as P2(n;d):1/C28e/C28n(n/C281)=2d(5) :1/C281/C28n 2d !n/C281 ; (6) where the latter has error eBn3 6(d/C28n/C271)2(7) (Sayrafiezadeh 1994). In general, let Qi(n;d) denote the probability that a birthday is shared by exactly i(and no more) people out of a group of npeople. Then the probability that a birthday is shared by k or more people is given by Pk(n;d)/C301/C28Xk/C281 i/C301Qi(n;d): (8) /Q2can be computed explicitly as Q2(n;d)/C30n! dnXn=2bc i/C3021 2id i/CP8/CP9 d/C28i n/C282i/CP8/CP9 /C30n! dnXn=2bc i/C301d! 2ii!(n/C282i)!(d/C28n/C27i)! /C30(/C281)n dn/C20 2/C28n=2G(1/C27n)P(/C28d) n(1 2ffiffiffi 2p ) /C28G(1/C27d) G(1/C27d/C28n)/C2P ; (9) wheren m/C0/CP is a BINOMIAL COEFFICIENT ,G(n)i sa GAMMA FUNCTION , and P(l) n(x)i sa n ULTRASPHERICAL POLYNOMIAL . This gives the explicit formula for P3(n;d)a s P3(n;d)/C301/C28Q1(n;d)/C28Q2(n;d) /C301/C27(/C281)n/C271G(n/C271)P(/C28d) n(2/C281=2) 2n=2dn:(10) /Q3(n;d) cannot be computed in entirely closed form, but a partially reduced form is Q3(n;d)/C30G(d/C271) dn(/C281)nF(9 8)/C28F(/C2898) G(d/C28n/C271)/C27(/C281)nG" /C2(1/C27n)Xn=3bc i/C301(/C283)/C28i2(i/C28n)=2P(i/C28d) n/C283i(1 2ffiffiffi 2p ) G(d/C28i/C271)G(i/C271)/C2P ; (11)where F/C30F(n;d;a)/C131/C283F21 3(1/C28n);13(2/C28n);/C2813 12(d/C28n/C271);12(d/C28n/C272);a"# (12) and3F2(a;b;c;d;e;z)i sa GENERALIZED HYPER- GEOMETRIC FUNCTION . In general, Qk(n;d) can be computed using the RECURRENCE RELATION Qk(n;d)/C30Xn=kbc i/C301/C20n!d! diki!(k!)i(n/C28ik)!(d/C28i)! /C29Xk/C281 j/C301Qj(n/C28k;d/C28i)(d/C28i)n/C28ik dn/C28ik/C2P (13) (Finch). However, the time to compute this recursive function grows exponentially with kand so rapidly becomes unwieldy. The minimal number of people to give a 50% probability of having at least ncoincident birthdays is 1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 1385, 1596, 1813, ... (Sloane’s A014088; Diaco- nis and Mosteller 1989). A good approximation to the number of people nsuch that p/C30Pk(n;d) is some given value can be given by solving the equation ne/C28n=(dk)/C30dk/C281k!l n1 1/C28p ! 1/C28n d(k/C271) ! "#1=k (14) fornand taking nde;where nde is the CEILING FUNCTION (Diaconis and Mosteller 1989). For p/C30 0:5 and k/C301, 2, 3, ..., this formula gives n/C301, 23, 88, 187, 313, 459, 622, 797, 983, 1179, 1382, 1592, 1809, ... (Sloane’s A050255), which differ from the true values by from 0 to 4. A much simpler but also poorer approximation for nsuch that /p/C300:5/forkB20 is given by n/C3047(k/C281:5)3=2(15) (Diaconis and Mosteller 1989), which gives 86, 185,307, 448, 606, 778, 965, 1164, 1376, 1599, 1832, ... fork/C303, 4, ... (Sloane’s A050256). The "almost" birthday problem, which asks the number of people needed such that two have a birthday within a day of each other, was consideredby Abramson and Moser (1970), who showed that 14people suffice. An approximation for the minimum number of people needed to get a 50 /C1 /0 chance that two have a match within kdays out of dpossible is given by n(k;d)/C301:2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2k/C271s (16) (Sevast’yanov 1972, Diaconis and Mosteller 1989). See also BIRTHDAY ATTACK ,C OINCIDENCE ,SMALL WORLD PROBLEM ,SULTAN’S DOWRY PROBLEM References Abramson, M. and Moser, W. O. J. "More Birthday Sur- prises." Amer. Math. Monthly 77, 856 /C1/58, 1970. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 45 /C1/6, 1987. Bloom, D. M. "A Birthday Problem." Amer. Math. Monthly 80, 1141 /C1/142, 1973. Bogomolny, A. "Coincidence." http://www.cut-the-knot.com/ do_you_know/coincidence.html. Clevenson, M. L. and Watkins, W. "Majorization and the Birthday Inequality." Math. Mag. 64, 183 /C1/88, 1991. Diaconis, P. and Mosteller, F. "Methods of Studying Coin- cidences." J. Amer. Statist. Assoc. 84, 853 /C1/61, 1989. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 31 /C1/2, 1968. Finch, S. "Puzzle #28 [June 1997]: Coincident Birthdays." http://www.mathsoft.com/mathcad/library/puzzle/soln28/ soln28.html. Gehan, E. A. "Note on the ‘Birthday Problem."’ Amer. Stat. 22, 28, Apr. 1968. Heuer, G. A. "Estimation in a Certain Probability Problem." Amer. Math. Monthly 66, 704 /C1/06, 1959. Hocking, R. L. and Schwertman, N. C. "An Extension of the Birthday Problem to Exactly k Matches." College Math. J. 17, 315 /C1/21, 1986. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 102 /C1/03, 1975. Klamkin, M. S. and Newman, D. J. "Extensions of the Birthday Surprise." J. Combin. Th. 3, 279 /C1/82, 1967. Levin, B. "A Representation for Multinomial Cumulative Distribution Functions." Ann. Statistics 9, 1123 /C1/126, 1981. McKinney, E. H. "Generalized Birthday Problem." Amer. Math. Monthly 73, 385 /C1/87, 1966. Mises, R. von. "U¨ ber Aufteilungs--und Besetzungs- Wahrscheinlichkeiten." Revue de la Faculte ´ des Sciences de l’Universite ´ d’Istanbul, N. S. 4, 145 /C1/63, 1939. Rep- rinted in Selected Papers of Richard von Mises, Vol. 2 (Ed. P. Frank, S. Goldstein, M. Kac, W. Prager, G. Szego, and G. Birkhoff). Providence, RI: Amer. Math. Soc., pp. 313 /C1/ 34, 1964. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 179 /C1/ 80, 1994. Sayrafiezadeh, M. "The Birthday Problem Revisited." Math. Mag. 67, 220 /C1/23, 1994. Sevast’yanov, B. A. "Poisson Limit Law for a Scheme of Sums of Dependent Random Variables." Th. Prob. Appl. 17, 695 /C1/99, 1972. Sloane, N. J. A. Sequences A014088, A033810, A050255, and A050256 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Stewart, I. "What a Coincidence!" Sci. Amer. 278,95/C1/6, June 1998. Tesler, L. "Not a Coincidence!" http://www.nomodes.com/ coincidence.html. Bisected Perimeter Point NAGEL POINTBisection Procedure A simple procedure for iteratively converging on a solution which is known to lie inside some interval [a, b]. Let ap and bn be the endpoints at the nth iteration and rnbe the nth approximate solution. Then, the number of iterations required to obtain an error smaller than e is found as follows. bn /C28an /C301 2n/C281 (b /C28a) (1) rn /C131 2(an /C27bn) (2) ½rn /C28r ½51 2(bn /C28an) /C302 /C28n(b /C28a) B e (3) /C28n ln 2 Bln e /C28ln(b /C28a) ; (4) so n >ln(b /C28 a) /C28 ln e ln 2: (5) See also ROOT References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 964 /C1/65, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Bracketing and Bisection." §9.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 343 /C1/47, 1992. Bisector Bisection is the division of a given curve or figure into two equal parts (halves). See also ANGLE BISECTOR ,BISECTION PROCEDURE , EXTERIOR ANGLE BISECTOR ,HALF,HEMISPHERE ,LINE BISECTOR ,PERPENDICULAR BISECTOR ,TRISECTION Bishop’s Inequality Let V(r) be the volume of a BALL of radius r in a complete n-D RIEMANNIAN MANIFOLD with RICCI CURVATURE ](n /C281)k : Then V(r) ]Vk(r) ; where Vk is the volume of a BALL in a space having constant SECTIONAL CURVATURE . In addition, if equality holds for some BALL , then this BALL is ISOMETRIC to the BALL of radius rin the space of constant SECTIONAL CURVATURE k:/ See also BALL,ISOMETRY References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Bishops Problem Find the maximum number of bishops B(n) which can be placed on an n /C29n CHESSBOARD such that no two attack each other. The answer is 2n /C282 (Dudeney 1970, Madachy 1979), giving the sequence 2, 4, 6, 8, ... (the EVEN NUMBERS ) for n /C302, 3, .... One maximal solution for n /C30 8 is illustrated above. The number of distinct maximal arrangements of bishops for n /C301, 2, ... are 1, 4, 26, 260, 3368, ... (Sloane’s A002465). The number of rotationally and reflectively distinct solu- tions on an n /C29n board for n ]2is B(n) /C302(n/C284)=2[2(n/C282)=2 /C271] for n even 2(n/C283)=2[2(n/C283)=2 /C271] for n odd/C26 (Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent formula is B(n) /C302n/C283 /C272[(n/C281)=2]/C281 ; where nbcis the FLOOR FUNCTION , giving the se- quence for n /C301, 2, ... as 1, 1, 2, 3, 6, 10, 20, 36, ... (Sloane’s A005418). The minimum number of bishops needed to occupy or attack all squares on an n /C29n CHESSBOARD is n, arranged as illustrated above. See also CHESS ,KINGS PROBLEM ,KNIGHTS PROBLEM , QUEENS PROBLEM ,ROOKS PROBLEM References Ahrens, W. Mathematische Unterhaltungen und Spiele, Vol. 1, 3rd ed. Leipzig, Germany: Teubner, p. 271, 1921. Dudeney, H. E. "Bishops--Unguarded" and "Bishops-- Guarded." §297 and 298 in Amusements in Mathematics. New York: Dover, pp. 88 /C1/9, 1970.Guy, R. K. "The n Queens Problem." §C18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 133 /C1/35, 1994. Madachy, J. Madachy’s Mathematical Recreations. New York: Dover, pp. 36 /C1/6, 1979. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 74 /C1/5, 1995. Sloane, N. J. A. Sequences A002465/M3616 and A005418/ M0771 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Bislit Cube The 8-VERTEX graph consisting of a CUBE in which two opposite faces have DIAGONALS oriented PERPENDICU- LARto each other. See also BIDIAKIS CUBE,CUBE,CUBICAL GRAPH Bispherical Coordinates A system of CURVILINEAR COORDINATES variously denoted ( j;h;f) (Arfken 1970) or ( u;h;c) (Moon and Spencer 1988). Using the notation of Arfken, the bispherical coordinates are defined by x/C30asinjcosf cosh h/C28cosj(1) y/C30asinjsinf cosh h/C28cosj(2) z/C30asinh h cosh h/C28cosj: (3) Surfaces of constant h are given by the spheres x2 /C27y2 /C27(z /C28a coth h)2 /C30a2 sinh2 h ; (4) surfaces of constant j by the APPLES /( j B p=2) or LEMONS /( j > p=2) x2 /C27y2 /C27z2 /C282affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p cot j /C30a2 ; (5) and surface of constant c by the half-planes tan f /C30y=x: (6) The SCALE FACTORS are hj /C30a cos h /C28 cos j (7) hh /C30a cosh h /C28 cos j (8) hf /C30a sin j cosh h /C28 cos j : (9) The LAPLACIAN is given by 92f /C30(cosh h /C28 cos j)2 a2 sin j /C2 sin j@ @ h1 cosh h /C28 cos j@f @ h !( /C27@ @ jsin j cosh h /C28 cos j@f @ j !/C27 /C27(cosh h /C28 cos j)2 a2 sin2 j@2f @ f2 : In bispherical coordinates, LAPLACE’S EQUATION is separable (Moon and Spencer 1988), but the HELM- HOLTZ DIFFERENTIAL EQUATION is not. See also BICYCLIDE COORDINATES ,LAPLACE’S EQUA- TION– BISPHERICAL COORDINATES ,SPHERICAL COORDI- NATES ,TOROIDAL COORDINATES References Arfken, G. "Bispherical Coordinates ( j; h; f) :/" §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115 /C1/17, 1970. Moon, P. and Spencer, D. E. "Bispherical Coordinates ( h; u ; c) :/" Fig. 4.03 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 110 /C1/ 12, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665 /C1/66, 1953.Bisymmetric Matrix A SQUARE MATRIX is called bisymmetric if it is both CENTROSYMMETRIC and either SYMMETRIC or SKEW SYMMETRIC (Muir 1960, p. 19). See also CENTROSYMMETRIC MATRIX ,S KEW SYM- METRIC MATRIX ,SYMMETRIC MATRIX References Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Bit Complexity The number of single operations (of ADDITION , SUB- TRACTION , and MULTIPLICATION ) required to complete an algorithm. See also STRASSEN FORMULAS References Borodin, A. and Munro, I. The Computational Complexity of Algebraic and Numeric Problems. New York: American Elsevier, 1975. Bit Length The number of binary bits necessary to represent a number, given explicitly by BL(n)/C30lgnde ; where xdeis the CEILING FUNCTION and lg nisLG, the LOGARITHM to base 2. For n/C300, 1, 2, ..., the first few values are 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, ... (Sloane’sA036377). The function is given by the Mathematica 4.0 function BitLength[ n] in theDeveloper con- text. References Sloane, N. J. A. Sequences A036377 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Bitangent A LINE which is TANGENT to a curve at two distinct points. There exist plane QUARTIC CURVES X i /C27j 54aijxiyj /C300 that have 28 real bitangents (Shioda 1995, Trott 1997), for example 122(x4 /C27y4) /C28152(x2 /C27y2) /C27350x2y2 /C2781 /C300 (Trott 1997), illustrated above. See also KLEIN’S EQUATION ,PLU¨ CKER CHARACTERIS- TICS,SECANT LINE,SOLOMON’S SEAL LINES,TANGENT LINE References Shioda, F. Comm. Math. Univ. Sancti Pauli 44, 109, 1995. Trott, M. "Applying GroebnerBasis to Three Problems in Geometry." Mathematica Educ. Res. 6,15/C1/8, 1997. Bitwin Chain A bitwin chain of length one consists of two pairs of TWIN PRIMES with the property that they are related by being of the form: (n /C281; n /C271) and (2n /C281; 2n /C271): In general a chain of length i consists of i /C271 pairs of TWIN PRIMES , (n /C281; n /C271); (2n /C281 ; 2n /C271); ...; (2i /C215 n /C281 ; 2i /C215 n /C271): Bitwin chains can also be viewed as consisting of two related CUNNINGHAM CHAINS of the first and second kinds, (n /C281; 2n /C281; 4n /C281 ; ...) and(n /C271; 2n /C271; 4n /C271 ; ...): P. Jobling (1999) found the largest known chain of length six, 337190719854678690 /C215 2n 91; where n /C300to6. See also CUNNINGHAM CHAIN ,TWIN PRIMES References Jobling, P. "A BiTwin chain of length 6 discovered." [email protected] posting, 4 Oct 1999. Biunitary Divisor A divisor d of a positive integer n is biunitary if the greatest common unitary divisor of d and n=d is 1. For a prime power py ; the biunitary divisors are the powers 1, p, p2 ; ..., py ; except for py=2 when y is EVEN (Cohen 1990). See also DIVISOR , K-ARY DIVISOR ,UNITARY DIVISOR References Cohen, G. L. "On an Integer’s Infinary Divisors." Math. Comput. 54, 395 /C111, 1990. Suryanarayana, D. "The Number of Bi-Unitary Divisors of an Integer." The Theory of Arithmetic Functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1971. New York: Springer-Verlag, pp. 273 /C182, 1972. Suryanarayana, D. and Rao, R. S. R. C. "The Number of Bi- Unitary Divisors of an Integer. II." J. Indian Math. Soc. 39, 261 /C180, 1975. Bivalent Capable of taking on one out of two possible values. See also EXCLUDED MIDDLE LAW,UNIVALENT Bivalent Range If the CROSS-RATIO k of fAB ; CDg satisfy k2 /C28 k /C271 /C300 ; (1) then the points are said to form a bivalent range, and fAB ; CDg/C30fAC; DBg/C30fAD; BCg/C30 k (2) fAC; BDg/C30fAD ; BCg/C30fAB ; DCg/C30/C28k2 : (3) See also HARMONIC RANGE References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 268, 1893. Bivariate Distribution See also GAUSSIAN BIVARIATE DISTRIBUTION Bivariate Normal Distribution GAUSSIAN BIVARIATE DISTRIBUTION Bivector An antisymmetric TENSOR of second RANK (a.k.a. 2- form). /C0X /C30Xab va ffl vb ; where fflis the WEDGE PRODUCT (or OUTER PRODUCT ). See also TENSOR ,VECTOR Biweight TUKEY’S BIWEIGHT Bjo¨rling Curve Let a(z) ; g(z):(a ; b) 0 R3 be curves such that ½½g ½½/C30 1 and a /C215 g /C30 0; and suppose that a and g have holo- morphic extensions a; g :(a ; b) /C29 (c ; d) 0 C3such that ½½g ½½/C30 1 and a /C215 g /C30 0 also for z /C23 (a; b) /C29 (c ; d): Fix z0 /C23 (a ; b) /C29(c; d) : Then the Bjo¨rling curve, de- fined by B(z) /C30 a(z) /C28igz z0g(z) /C29 a?(z) dz ; is a minimal curve (Gray 1997, p. 762). References Bjo¨rling, E. G. "In integrationem aequationis derivatarum partialum superficiei, cujus in puncto, unoquoque princi- pales ambo radii curvedinis aequales sunt signoque con- trario." Arch. Math. Phys. 4, 290 /C1/15, 1844. Dierkes, U.; Hildebrand, S.; Ku¨ster, A.; and Wohlrab, O. Minimal Surfaces, 2 vols. New York: Springer-Verlag, pp. 120 /C1/35, 1992. Gray, A. "Minimal Surfaces via Bjo¨rling’s Formula." Ch. 33 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 761 /C1/72, 1997. Nitsche, J. C. C. Lectures on Minimal Surfaces, Vol. 1: Introduction, Fundamentals, Geometry and Basic Bound- ary Value Problems. Cambridge, England: Cambridge University Press, pp. 139 /C1/45, 1989. Schwarz, H. A. Gesammelte Mathematische Abhandlungen, Vols. 1 /C1/. New York: Chelsea, pp. 179 /C1/89, 1972.Black Dot Illusion In the above illustration, black dots appear to form and vanish at the intersections of the gray horizontal and vertical lines. When focusing attention on a single white dot, some gray dots nearby and some black dots a little further away also seem to appear. More black dots seem to appear as the eye is scanned across the image (as opposed to focusing on a single point). Strangely, the effect seems to be reduced, but not eliminated, when the head is cocked at a 45 8 angle. The effect seems to exist only at intermediate distances; if the eye is moved very close to or very far away from the figure, the phantom black dots do notappear. See also I LLUSION References Gephart, J. "Find the Black Dot." http://udel.edu/~jgephart/ fun2.htm. Black Spleenwort Fern BARNSLEY’S FERN Blackboard Bold DOUBLESTRUCK Blackman Function An APODIZATION FUNCTION given by A(x)/C300:42/C270:5 cospx a ! /C270:08 cos2px a ! : (1) Its FULL WIDTH AT HALF MAXIMUM is 0:810957 a:The APPARATUS FUNCTION is I(k)/C30a(0:84/C280:36a2k2/C282:17/C2910/C2819a4k4)sin(2 pak) (1/C28a2k2)(1/C284a2k2):(2) The COEFFICIENTS are approximations in the general expansion A(x) /C30a0 /C272X/C12 n /C301an cosnpx b ! ; (3) to a0 /C303969 9304 :0:42659 (4) a1 /C301155 4652 :0:24828 (5) a2 /C30715 18608 :0:38424 ; (6) which produce zeros of I(k)atka /C307=4 and ka /C309=4 :/ See also APODIZATION FUNCTION References Blackman, R. B. and Tukey, J. W. "Particular Pairs of Windows." In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 98 /C1/9, 1959. Black-Scholes Theory The theory underlying financial derivatives which involves "stochastic calculus" and assumes an uncor- related LOG NORMAL DISTRIBUTION of continuously varying prices. A simplified "binomial" version of the theory was subsequently developed by Sharpe et al. (1995) and Cox et al. (1979). It reproduces many results of the full-blown theory, and allows approx- imation of options for which analytic solutions are not known (Price 1996). See also GARMAN- KOHLHAGEN FORMULA References Black, F. and Scholes, M. S. "The Pricing of Options and Corporate Liabilities." J. Political Econ. 81, 637 /C1/59, 1973. Cox, J. C.; Ross, A.; and Rubenstein, M. "Option Pricing: A Simplified Approach." J. Financial Economics 7, 229 /C1/63, 1979. Price, J. F. "Optional Mathematics is Not Optional." Not. Amer. Math. Soc. 43, 964 /C1/71, 1996. Sharpe, W. F.; Alexander, G. J.; Bailey, J. V.; and Sharpe, W. C. Investments, 6th ed. Englewood Cliffs, NJ: Prentice- Hall, 1998.Blanche’s Dissection The simplest dissection of a SQUARE into rectangles of the same AREAS but different shapes, composed of the seven pieces illustrated above. The square is 210 units on a side, and each RECTANGLE has AREA 2102=7/C306300 :/ See also PERFECT SQUARE DISSECTION ,RECTANGLE References Descartes, B. "Division of a Square into Rectangles." Eur- eka, No. 34, 31 /C1/5, 1971. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 14 /C1/5, 1991. Blancmange Function ACONTINUOUS FUNCTION which is nowhere DIFFER- ENTIABLE . The iterations towards the continuous function are BATRACHIONS resembling the H OFSTAD- TER-CONWAY $10,000 SEQUENCE . The first six iterations are illustrated below. The dth iteration contains N/C27 1 points, where N/C302d;and can be obtained by setting b(0)/C30b(N)/C300;letting b(m/C272n/C281)/C302n/C271 2[b(m)/C27b(m/C272n)]; and looping over n/C30dto 1 by steps of /C281 and m/C300 to N /C281 by steps of 2n :/ Peitgen and Saupe (1988) refer to this curve as the TAKAGI FRACTAL CURVE . See also HOFSTADTER- CONWAY $10,000 SEQUENCE , WEIERSTRASS FUNCTION References Dixon, R. Mathographics. New York: Dover, pp. 175 /C1/76 and 210, 1991. Peitgen, H.-O. and Saupe, D. (Eds.). "Midpoint Displacement and Systematic Fractals: The Takagi Fractal Curve, Its Kin, and the Related Systems." §A.1.2 in The Science of Fractal Images. New York: Springer-Verlag, pp. 246 /C1/48, 1988. Takagi, T. "A Simple Example of the Continuous Function without Derivative." Proc. Phys. Math. Japan 1, 176 /C1/77, 1903. Tall, D. O. "The Blancmange Function, Continuous Every- where but Differentiable Nowhere." Math. Gaz. 66,11/C1/2, 1982. Tall, D. "The Gradient of a Graph." Math. Teaching 111, 48 /C1/2, 1985. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 16 /C1/7, 1991. Blankinship Algorithm A method for finding solutions u and v to a linear congruence au /C27bv /C30d by constructing a matrix formed by adjoining a vector containing a and b with a UNIT MATRIX , M /C30a 10 b 01/C20/C2P ; and applying the EUCLIDEAN ALGORITHM to the first column, while extending the operations to all rows. The algorithm terminates when the first column contains the GREATEST COMMON DIVISOR GCD( a; b) :/ See also EUCLIDEAN ALGORITHM ,GREATEST COMMON DIVISOR References Blankinship, W. A. "A New Version of the Euclidean Algo- rithm." Amer. Math. Monthly 70, 742 /C1/45, 1963. Se´roul, R. "The Blankinship Algorithm." §8.2 in Program- ming for Mathematicians. Berlin: Springer-Verlag, pp. 161 /C1/63, 2000.Blaschke Condition If faj g⁄D(0; 1) (with possible repetitions) satisfies X/C12 j/C301(1 /C28½aj ½) 5/C12 ; where D(0; 1) is the unit open disk, and no aj /C300; then there is a bounded ANALYTIC FUNCTION on D(0; 1) which has ZERO SET consisting precisely of the aj/s, counted according to their MULTIPLICITIES . More specifically, the INFINITE PRODUCT Y/C12 j/C301/C28¯aj ½aj ½Baj(z) ; where Baj(z)isaB LASCHKE FACTOR and ˜z is the COMPLEX CONJUGATE , converges uniformly on com- pact subsets of D(0; 1) to a bounded analytic function B(z) :/ See also BLASCHKE FACTOR ,BLASCHKE FACTORIZA- TION ,BLASCHKE PRODUCT References Krantz, S. G. "The Blaschke Condition." §9.1.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 118 /C1/ 19, 1999. Blaschke Conjecture The only WIEDERSEHEN MANIFOLDS are the standard round spheres. The conjecture has been proven by combining the BERGER- KAZDAN COMPARISON THEO- REM with A. Weinstein’s results for n EVEN and C. T. Yang’s for n ODD. See also WIEDERSEHEN MANIFOLD References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Blaschke Factor If a is a point in the open UNIT DISK, then the Blaschke factor is defined by Ba(z) /C30z /C28 a 1 /C28 ¯az ; where ¯a is the COMPLEX CONJUGATE of a. Blaschke factors allow the manipulation of the zeros of a HOLOMORPHIC FUNCTION analogously to factors of /(z/C28a) for complex polynomials (Krantz 1999, p. 117). See also BLASCHKE CONDITION ,BLASCHKE FACTOR- IZATION References Krantz, S. G. "Blaschke Factors." §9.1.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 117, 1999. Blaschke Factorization Let f be a bounded ANALYTIC FUNCTION on D(0; 1) vanishing to order m ]0 at 0 and let faj g be its other zeros, listed with multiplicities. Then f(z) /C30zmF(z)Y/C12 j/C301/C28¯aj ½aj ½Baj(z) ; where F is a bounded ANALYTIC FUNCTION on D(0; 1); F is zerofree, ˜z is the COMPLEX CONJUGATE , and sup z /C23D(0; 1)½f(z) ½/C30 sup z /C23D(0; 1)½F(z) ½: See also BLASCHKE FACTOR References Krantz, S. G. "Blaschke Factorization." §9.1.7 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 119, 1999. Blaschke Product A Blaschke product is an expression of the form B(z) /C30zmY/C12 j/C301/C28¯aj ½aj ½Baj(z) ; where m is a nonnegative integer and ˜z is the COMPLEX CONJUGATE . See also BLASCHKE FACTOR References Krantz, S. G. "Blaschke Products." §9.1.6 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 119, 1999. Blaschke’s Theorem A convex planar domain in which the minimal GENERALIZED DIAMETER is /C211 always contains a CIRCLE of RADIUS 1/3. See also GENERALIZED DIAMETER References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 17 /C1/8, 1991. Blasius Differential Equation The third-order ORDINARY DIFFERENTIAL EQUATION 2y§/C27yyƒ/C300: This equation arises in the theory of fluid boundary layers, and must be solved numerically (Rosenhead 1963; Schlichting 1979; Tritton 1989, p. 129). The velocity profile produced by this differential equation is known as the Blasius profile.References Meyer, G. H. Initial Value Methods for Boundary Value Problems: Theory and Application of Invariant Imbed- ding. New York: Academic Press, 1973. Rosenhead, L. (Ed.). Laminar Boundary Layers. Oxford, England: Oxford University Press, 1963. Schlichting, H. Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979. Tritton, D. J. Physical Fluid Dynamics, 2nd ed. Oxford, England: Clarendon Press, p. 129, 1989. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997. Blecksmith-Brillhart-Gerst Theorem A generalization of SCHRO ¨ TER’S FORMULA . References Berndt, B. C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, p. 73, 1985. Blichfeldt’s Lemma BLICHFELDT’S THEOREM Blichfeldt’s Theorem Any bounded planar region with POSITIVE AREA > A placed in any position of the UNIT SQUARE LATTICE can be TRANSLATED so that the number of LATTICE POINTS inside the region will be at least A /C271 (Blichfeldt 1914, Steinhaus 1983) The theorem can be general- ized to n-D. See also LATTICE POINT ,M INKOWSKI CONVEX BODY THEOREM ,PICK’S THEOREM References Blichfeldt, H. F. "A New Principle in the Geometry of Numbers, with Some Applications." Trans. Amer. Math. Soc. 15, 227 /C1/35, 1914. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 97 /C1/9, 1999. B-Line A line which simultaneously bisects a triangle’s perimeter and area. See also CLEAVER ,SPLITTER References Todd, A. "Bisecting a Triangle." Pi Mu Epsilon J. 11,3 1/C1/7, Fall 1999. Todd, A. "Bisecting a Triangle." http://www.math.colosta- te.edu/~todd/triangle.html. BLM/Ho Polynomial A 1-variable unoriented KNOT POLYNOMIAL Q(x):It satisfies Qunknot/C301 (1) and the SKEIN RELATIONSHIP QL/C27/C27QL/C28/C30x(QL0/C27QL/C12): (2) It also satisfies QL1#L2/C30QL1QL2; (3) where is the KNOT SUM and QL/C31/C30QL ; (4) where L /C31 is the MIRROR IMAGE of L. The BLM/Ho polynomials of MUTANT KNOTS are also identical. Brandt et al. (1986) give a number of interesting properties. For any LINK L with ]2 components, QL /C281 is divisible by 2(x /C281): If L has c components, then the lowest POWER of x in QL(x)is1/C28c ; and lim x00xc/C281QL(x) /C30 lim (l; m)0(1; 0)(/C28m)c/C281PL(l; m) ; (5) where PLis the HOMFLY POLYNOMIAL . Also, the degree of QL is less than the CROSSING NUMBER of L. If L is a 2-BRIDGE KNOT , then QL(z) /C302z /C281VL(t)VL(t/C281 /C271 /C282z /C281) ; (6) where z /C13/C28t /C28t/C281 (Kanenobu and Sumi 1993). The POLYNOMIAL was subsequently extended to the 2- variable KAUFFMAN POLYNOMIAL F, which satisfies Q(x) /C30F(1; x) : (7) Brandt et al. (1986) give a listing of Q POLYNOMIALS for KNOTS up to 8 crossings and links up to 6 crossings. References Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C. "A Polynomial Invariant for Unoriented Knots and Links." Invent. Math. 84, 563 /C1/73, 1986. Ho, C. F. "A New Polynomial for Knots and Links--Pre- liminary Report." Abstracts Amer. Math. Soc. 6, 300, 1985. Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- Bridge Knots through 22-Crossings." Math. Comput. 60, 771 /C1/78 and S17-S28, 1993. Stoimenow, A. "Brandt-Lickorish-Millett-Ho Polynomials." http://guests.mpim-bonn.mpg.de/alex/ptab/blmh10.html. Weisstein, E. W. "Knots." MATHEMATICA NOTEBOOK KNOTS.M . Bloch Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let F be the set of COMPLEX ANALYTIC FUNCTIONS f defined on an open region containing the CLOSURE of the UNIT DISK D /C30 fz : ½z ½B1 g satisfying f(0) /C300 and df =dz(0) /C301 : For each f in F, let b(f) be the SUPREMUM of all numbers r such that there is a disk S in D on which f is ONE-TO- ONE and such that f(S) contains a disk of radius r.In 1925, Bloch (Conway 1978) showed that b(f) ]1 =72: Define Bloch’s constant by B /C13inf fb(f): f /C23 F g: Ahlfors and Grunsky (1937) derived0 :433012701 ... /C301 4ffiffiffi 3p 5B B1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27ffiffiffi 3ppG(1 3)G(1112) G(14)B0:4718617 : They also conjectured that the upper limit is actually the value of B, B /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27ffiffiffi 3ppG(1 3) G(1112) G(1 4) /C30ffiffiffipp21 =4G(1 3) G(14)ffiffiffiffiffiffiffiffiffiffi G(1112) G(1 12)vuut /C300:4718617 ... (Le Lionnais 1983). See also LANDAU CONSTANT References Conway, J. B. Functions of One Complex Variable I, 2nd ed. New York: Springer-Verlag, 1989. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/bloch/bloch.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983. Minda, C. D. "Bloch Constants." J. d’Analyse Math. 41,54/C1/ 4, 1982. Bloch-Landau Constant LANDAU CONSTANT Block A maximal BICONNECTED SUBGRAPH of a given GRAPH G. In the illustration above, the blocks are f2; 5; 6g; f3; 4; 6; 7g; and f1; 7g:/ If a graph G is biconnected, then G itself is called a block (Harary 1994, p. 26) or a BICONNECTED GRAPH (Skiena 1990, p. 175). See also BICONNECTED GRAPH ,BLOCK DESIGN ,DIGIT BLOCK ,SQUARE POLYOMINO References Aho, A. V.; Hopcroft, J. E.; and Ullman, J. D. The Design and Analysis of Computer Algorithms. Reading, MA: Addison-Wesley, 1974. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. "Biconnected Components." §5.1.4 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 175 /C1/77, 1990. Block (Group Action) A GROUP ACTION G /C29V0V might preserve a special kind of PARTITION of V called a system of blocks. A block is a SUBSET D of V such that for any group element g either 1. g preserves D; i.e., gD/C30D; or 2. g translates everything in D out of D; i.e., g DSD/C30 f:/ For example, the GENERAL LINEAR GROUP GL(2; R) acts on the plane minus the origin, R2 /C28(0; 0): The lines A /C30f(at; bt) g are blocks because either a line is mapped to itself, or to another line. Of course, the points on the line may be rescaled, so the lines in A are minimal blocks. In fact, if two blocks intersect then their intersection is also a block. Hence, the minimal blocks form a PARTITION of V: It is important to avoid confusion with the notion of a block in a BLOCK DESIGN , which is different. See also GROUP ,PRIMITIVE GROUP ACTION , STEINER SYSTEM References Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Block (Set) One of the disjoint SUBSETS making up a SET PARTI- TION . A block containing n elements is called an n- block. The partitioning of sets into blocks can be denoted using a RESTRICTED GROWTH STRING . See also BLOCK DESIGN ,R ESTRICTED GROWTH STRING ,SET PARTITION Block Design An incidence system (v, k, l ; r, b) in which a set X of v points is partitioned into a family A of b subsets (blocks) in such a way that any two points determine l blocks with k points in each block, and each point is contained in r different blocks. It is also generally required that k Bv, which is where the "incomplete" comes from in the formal term most often encoun- tered for block designs, BALANCED INCOMPLETE BLOCK DESIGNS (BIBD). The five parameters are not inde- pendent, but satisfy the two relations vr /C30bk (1) l(v /C281) /C30r(k /C281): (2) A BIBD is therefore commonly written as simply (v, k, l); since b and r are given in terms of v, k, and l by b /C30v(v /C28 1)l k(k /C28 1) (3)r /C30l(v /C28 1) k /C28 1: (4) A BIBD is called SYMMETRIC if b /C30v (or, equivalently, r /C30k). Writing X /C30fxi gv i/C301and A /C30fAj gb; j/C301then the INCI- DENCE MATRIX of the BIBD is given by the v /C29 b MATRIX M defined by mij /C301if xi /C23 A 0 otherwise :/C26 (5) This matrix satisfies the equation MM T/C30(r/C28l)I /C27lJ; (6) where I is a v /C29v IDENTITY MATRIX and J is the v /C29v UNIT MATRIX (Dinitz and Stinson 1992). Examples of BIBDs are given in the following table. Block Design ( v,k,l)/ AFFINE PLANE(/n2;n,1 ) FANO PLANE (7, 3, 1) HADAMARD DESIGNSYMMETRIC (/4n/C273;2n/C271;n) PROJECTIVEPLANESYMMETRIC (/n2/C27n/C271;n/C271;1) STEINER TRI- PLE SYSTEM(v,3 ,1 ) UNITAL (/q3/C271;q/C271;1) See also AFFINE PLANE ,D ESIGN ,F ANO PLANE , HADAMARD DESIGN ,P ARALLEL CLASS,P ROJECTIVE PLANE ,RESOLUTION ,RESOLVABLE ,STEINER TRIPLE SYSTEM ,SYMMETRIC BLOCK DESIGN ,UNITAL References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1 /C1/2, 1992. Ryser, H. J. "The ( b;v;r;k;l)/-Configuration." §8.1 in Com- binatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 96 /C1/02, 1963. Block Diagonal Matrix A block diagonal matrix, also called a diagonal block matrix, is a SQUARE DIAGONAL MATRIX in which the diagonal elements are SQUARE MATRICES of any size (possibly even 1 /C291);and the off-diagonal elements are 0. A block diagonal matrix is therefore a BLOCK MATRIX in which the blocks off the diagonal are the ZERO MATRICES , and the diagonal matrices are SQUARE . Block diagonal matrices can be constructed in Math- ematica using the following code snippet. BBLinearAlgebra‘MatrixManipulation‘ BlockDiagonal[a_List]: /C30 Module[{n /C30Length[a],lens /C30Length/@a,i,k,tmp}, k /C30Outer[List,lens,lens]; tmp /C30Map[ZeroMatrix[#1[[1]],#1[[2]]]&,k,{2}]; BlockMatrix@ ReplacePart[tmp,a,Table[{i,i},{i,Length[a]}], Table[{i},{i,Length[a]}]]] See also BLOCK MATRIX ,C AYLEY- HAMILTON THEO- REM,DIAGONAL MATRIX ,DIRECT SUM,JORDAN CANO- NICAL FORM,L INEAR TRANSFORMATION ,M ATRIX , MATRIX DIRECT SUM Block Growth Let ( x0x1x2. . .) be a sequence over a finite ALPHABET A (all the entries are elements of A). Define the block growth function B(n) of a sequence to be the number ofADMISSIBLE words of length n. For example, in the sequence aabaabaabaabaab ...;the following words are ADMISSIBLE Length Admissible Words 1 a, b 2 /aa;ab;ba / 3 /aab;aba;baa / 4 /aaba abaa ;baab / soB(1)/C302;B(2)/C303;B(3)/C303;B(4)/C303;and so on. Notice that B(n)5B(n/C271);so the block growth function is always nondecreasing. This is because any ADMISSIBLE word of length ncan be extended rightwards to produce an ADMISSIBLE word of length n/C271:Moreover, suppose B(n)/C30B(n/C271) for some n. Then each admissible word of length nextends to a unique ADMISSIBLE word of length n/C271:/ For a SEQUENCE in which each substring of length n uniquely determines the next symbol in the SE- QUENCE , there are only finitely many strings of length n, so the process must eventually cycle and the SEQUENCE must be eventually periodic. This gives us the following theorems: 1. If the SEQUENCE is eventually periodic, with least period p, then B(n) is strictly increasing until it reaches p, and B(n) is constant thereafter. 2. If the SEQUENCE is not eventually periodic, then B(n) is strictly increasing and so B(n)]n/C271 foralln.I fa SEQUENCE has the property that B(n)/C30 n/C271 for all n, then it is said to have minimal block growth, and the SEQUENCE is called a S TURMIAN SEQUENCE . The block growth is also called the GROWTH FUNCTION or the COMPLEXITY of a SEQUENCE . Block Matrix A block matrix is a MATRIX that is defined using smaller matrices, called blocks. For example, AB CD/C20/C2P ; (1) where A, B, C , and Dare themselves matrices, is a block matrix. In the specific example A/C300220/C20/C2P (2) B/C30333 333/C20/C2P (3) C/C304444 442 435 (4) D/C30505 050 5052 435; (5) it is the matrix 02333 2033344505 44050 445052 666643 77775: (6) Block matrices can be created using BlockMa- trix [blocks ] in the Mathematica add-on package LinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). When two block matrices have the same shape and their diagonal blocks are square matrices, then they multiply similarly to MATRIX MULTIPLICATION . For example, A1B1 C1D1/C20/C2P A2B2 C2D2/C20/C2P /C30A1A2/C27B1C2 A1B2 C1A2/C27D1C2C1B2/C27D1D2/C20/C2P : (7) When the blocks are SQUARE MATRICES , the set of invertible block matrices form a group, which is a special case of the GENERAL LINEAR GROUP . In this case, it is GL2(R/C31);the invertible two by two matrices with entries in the UNITS of a RING R, where here Ris the ring of square matrices. See also BLOCK DIAGONAL MATRIX ,CAYLEY- HAMIL- TON THEOREM ,MATRIX ,RING Blow-Up A common mechanism which generates SINGULARI- TIES from smooth initial conditions. See also BLOW- UP LEMMA Blow-Up Lemma The blow-up lemma essentially says that regular pairs in SZEMERE ´ DI’S REGULARITY LEMMA behave like COMPLETE BIPARTITE GRAPHS from the point of view of embedding bounded degree subgraphs. In particular, given a graph R of order r, minimal VERTEX DEGREE d and maximal VERTEX DEGREE D; then there exists an e > 0 such that the following holds. Let N be an arbitrary positive integer, and replace the vertices of R with pairwise disjoint N-sets V1 ; V2 ; ..., Vr (blowing up). Now construct two graphs on the same vertex set V /C30@ Vi : The graph R(N)is obtained by replacing all edges of R with copies of the complete bipartite graph KN ; N ; and construct a sparser graph by replacing the edges of R with some ( e; d)/-superregular pair. If a graph H with D(H) 5D is embeddable into R(N) ; then it is already embeddable into G (Komlo ´s et al. 1998). See also SZEMERE ´ DI’S REGULARITY LEMMA References Komlo ´s, J.; Sa´rkozy, G. N.; and Szemere ´di, E. "Blow-Up Lemma." Combinatorica 17, 109 /C1/23, 1997. Komlo ´s, J.; Sa´rkozy, G. N.; and Szemere ´di, E. "Proof of the Seymour Conjecture for Large Graphs." Ann. Comb. 2, 43 /C1/0, 1998. Blue-Empty Coloring BLUE-EMPTY GRAPH Blue-Empty Graph An EXTREMAL GRAPH in which the forced TRIANGLES are all the same color. Call R the number of red MONOCHROMATIC FORCED TRIANGLES and B the num- ber of blue MONOCHROMATIC FORCED TRIANGLES , then a blue-empty graph is an EXTREMAL GRAPH with B /C300. For EVEN n, a blue-empty graph can be achieved by coloring red two COMPLETE SUBGRAPHS of n=2 points (the RED NET method). There is no blue- empty coloring for ODD n except for n /C307 (Lorden 1962). See also COMPLETE GRAPH ,EXTREMAL GRAPH ,MONO- CHROMATIC FORCED TRIANGLE ,RED NET References Lorden, G. "Blue-Empty Chromatic Graphs." Amer. Math. Monthly 69, 114 /C1/20, 1962. Sauve ´, L. "On Chromatic Graphs." Amer. Math. Monthly 68, 107 /C1/11, 1961.Board A board is a subset of the polygons determined by a number of (usually regularly spaced and oriented) lines. These polygons form the spaces on which "pieces" can be placed and move in many games (called board games). The simplest division the plane is into equal squares. The 3 /C293 square board is used in TIC-TAC-TOE . The 8 /C298 square board is used in CHECKERS and CHESS . Hexagonal boards are used in some games. Chinese checkers uses a board in the space of a pentagram with spaces at the vertices of a regular triangular tiling. See also CHECKERS ,CHESS ,CHESSBOARD ,GRID,ROOK NUMBER ,TIC-TAC-TOE References Bell, R. C. Board and Table Games from Many Civilizations. New York: Dover, 1980. Gardner, M. "Four Unusual Board Games." Ch. 5 in The Sixth Book of Mathematical Games from Scientific Amer- ican. Chicago, IL: University of Chicago Press, pp. 39 /C1/7, 1984. Murray, H. J. R. A History of Board-Games Other than Chess. New York: Oxford University Press, 1952. Parlett, D. The Oxford History of Board Games. Oxford, England: Oxford University Press, 1999. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 10, 1999. Boatman’s Knot CLOVE HITCH Boˆcher Equation A second-order ORDINARY DIFFERENTIAL EQUATION OF THE FORM yƒ/C271 2m1 x/C28a1/C27.../C27mn/C281 x/C28an/C281"# y? /C2714A0/C27A1x/C27.../C27A1x1 (x/C28a1)m1(x/C28a2)m2...(x/C28an/C281)mn/C281"# y/C300: References Moon, P. and Spencer, D. E. "Differential Equations." §6i n Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 144 /C1/62, 1988. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995. Bochner Identity For a smooth HARMONIC MAP u : M 0 N ; D( ½9u½2) /C30½9(du) ½2 /C27 RicM 9u;9u hi /C28 RiemN(u)( 9u;9u) 9u;9u hi ; where 9 is the GRADIENT , Ric is the RICCI TENSOR , and Riem is the RIEMANN TENSOR . References Eels, J. and Lemaire, L. "A Report on Harmonic Maps." Bull. London Math. Soc. 10,1/C1/8, 1978. Bochner’s Theorem Among the continuous functions on Rn ; the POSITIVE DEFINITE FUNCTIONS are those functions which are the FOURIER TRANSFORMS of finite measures. Bode’s Rule Let the values of a function f(x) be tabulated at points xiequally spaced by h /C30xi/C271 /C28xi ; so f1 /C30f(x1) ; f2 /C30 f(x2) ; ..., f5 /C30f(x5) : Then Bode’s rule approximating the integral of f(x) is given by the NEWTON- COTES -like formula gx5 x1f(x) dx /C302 45h(7f1 /C2732f2 /C2712f3 /C2732f4 /C277f5) /C288 945h7f(6)(j) : See also HARDY’S RULE,NEWTON- COTES FORMULAS , SIMPSON’S 3/8 RULE,SIMPSON’S RULE,TRAPEZOIDAL RULE,W EDDLE’S RULE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972. Bogdanov Map A 2-D MAP which is conjugate to the HE´ NON MAP in its nondissipative limit. It is given by x?/C30x /C27y? y?/C30y /C27 ey /C27kx(x /C281) /C27 mxy: See also HE´ NON MAP References Arrowsmith, D. K.; Cartwright, J. H. E.; Lansbury, A. N.; and Place, C. M. "The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System." Int. J. Bifurcation Chaos 3, 803 /C1/42, 1993. Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373 /C1/ 88, 1981.Bogomolov-Miyaoka-Yau Inequality Relates invariants of a curve defined over the IN- TEGERS . If this inequality were proven true, then FERMAT’S LAST THEOREM would follow for sufficiently large exponents. Miyaoka claimed to have proven this inequality in 1988, but the proof contained an error. See also FERMAT’S LAST THEOREM References Cox, D. A. "Introduction to Fermat’s Last Theorem." Amer. Math. Monthly 101,3/C1/4, 1994. Bohemian Dome A QUARTIC SURFACE which can be constructed as follows. Given a CIRCLE C and PLANE E PERPENDICU- LAR to the PLANE of C, move a second CIRCLE K of the same RADIUS as C through space so that its CENTER always lies on C and it remains PARALLEL to E. Then K sweeps out the Bohemian dome. It can be given by the PARAMETRIC EQUATIONS x /C30a cos u y /C30b cos v /C27a sin u z /C30c sin v where u; v /C23 [0; 2p) : In the above plot, a /C300:5; b /C301:5; andc/C301. See also QUARTIC SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 19 /C1/0, 1986. Fischer, G. (Ed.). Plate 50 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 50, 1986. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 389, 1997. Nordstrand, T. "Bohemian Dome." http://www.uib.no/people/ nfytn/bodtxt.htm. Bohr Matrix A finite or infinite SQUARE MATRIX with RATIONAL entries. (If the matrix is infinite, all but a finite number of entries in each row must be 0.) The sum or product of two Bohr matrices is another Bohr matrix. References Apostol, T. M. "Bohr Matrices." §8.4 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 167 /C1/68, 1997. Bohr-Favard Inequalities If f has no spectrum in [/C28l;l] ; then fkk/C125p 2lf ?kk/C12 (Bohr 1935). A related inequality states that if Akis the class of functions such that f(x) /C30f(x /C272p) ; f(x) ; f ?(x); ...; f(k /C281)(x) are absolutely continuous and f2 p 0f(x) dx /C300; then fkk/C1254 pX/C12 n/C300( /C281)n(k /C271) (2n /C27 1)k/C271 f(k)(x)/CP3/CP3/CP3/CP3 /C12 (Northcott 1939). Further, for each value of k, there is always a function f(x) belonging to Akand not identically zero, for which the above inequality becomes an equality (Favard 1936). These inequal- ities are discussed in Mitrinovic et al. (1991). References Bohr, H. "Ein allgemeiner Satz u¨ber die Integration eines trigonometrischen Polynoms." Prace Matem.-Fiz. 43, 1935. Favard, J. "Application de la formule sommatoire d’Euler a` la de´monstration de quelques proprie ´te´s extre´males des inte´grale des fonctions pe´riodiques ou presquepe ´riodi- ques." Mat. Tidsskr. B,81/C1/4, 1936. Reviewed in Zentral- blatt f. Math. 16,58/C1/9, 1939. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, pp. 71 /C1/2, 1991. Northcott, D. G. "Some Inequalities Between Periodic Func- tions and Their Derivatives." J. London Math. Soc. 14, 198 /C1/02, 1939. Tikhomirov, V. M. "Approximation Theory." In Analysis II. Convex Analysis and Approximation Theory (Ed. R. V. Gamkrelidze). New York: Springer-Verlag, pp. 93 /C1/ 55, 1990. Bohr-Mollerup Theorem If a function 8 :(0;/C12) 0 (0;/C12) satisfies 1. ln[ 8(x)] is convex, 2.8(x/C271)/C30x8(x) for all x/C210, and 3.8(1)/C301;/ then8(x) is the GAMMA FUNCTION G(x):Therefore, by ANALYTIC CONTINUATION ,G(z) is the only MERO- MORPHIC FUNCTION onCsatisfying the functionalequation zG(z)/C30G(z/C271) withG(1)/C301 and which is logarithmically convex on the positive REAL AXIS . See also GAMMA FUNCTION References Krantz, S. G. "The Bohr-Mollerup Theorem." §13.1.10 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 157, 1999. Bolyai-Gerwein Theorem WALLACE- BOLYAI- GERWEIN THEOREM Bolza Problem Given the functional U/C30gt1 t0f(y1;...;yn;y?1;...;y?n)dt /C27G(y10;...;ynr;y11;...;yn1); (1) find in a class of arcs satisfying pdifferential and q finite equations fa(y1;...;yn;y?1;...;y?n)/C300 for a/C301;...;p(2) cb(y1;...;yn)/C300 for b/C301;...;q (3) as well as the requations on the endpoints xg(y10;...;ynr;y11;...;yn1)/C300 forg/C301;...;r;(4) one which renders Ua minimum. References Goldstine, H. H. A History of the Calculus of Variations from the 17th through the 19th Century. New York: Springer- Verlag, p. 374, 1980. Bolzano Theorem BOLZANO- WEIERSTRASS THEOREM Bolzano-Weierstrass Theorem Every BOUNDED infinite set in Rnhas an ACCUMULA- TION POINT . Forn/C301, an infinite subset of a closed bounded set S has an ACCUMULATION POINT inS. For instance, given a bounded SEQUENCE ap;with/C28C5an5Cfor all n,i t must have a MONOTONIC subsequence ank:The SUB- SEQUENCE ankmust converge because it is monotonic and bounded. Because Sis closed, it contains the limit of ank:/ The Bolzano-Weierstrass theorem is closely related to the H EINE- BOREL THEOREM and C ANTOR’S INTERSEC- TION THEOREM , each of which can be easily derived from either of the other two. See also ACCUMULATION POINT ,CANTOR’S INTERSEC- TION THEOREM ,HEINE- BOREL THEOREM ,INTERMEDI- ATE VALUE THEOREM References Jeffreys, H. and Jeffreys, B. S. §1.034 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cam- bridge University Press, pp. 9 /C1/0, 1988. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 7, 1996. Bombieri Inner Product For HOMOGENEOUS POLYNOMIALS P and Q of degree n, [P; Q] /C13X i1 ; ... ; in ]0(i1!...in!)(ai; ...; inbi1 ; ... ; in) : Bombieri Norm This entry contributed by KEVIN O’B RYANT The Bombieri p-norm of a polynomial Q(x) /C30Xn i/C300aixi (1) is defined by [Q]p /C13Xn i /C300n i/CP8/CP91 /C28p ½ai ½p"# 1=p ; (2) where (n k)isa BINOMIAL COEFFICIENT . The most re- markable feature of Bombieri’sn norm is that given polynomials R and S such that RS /C30Q ; then BOM- BIERI’S INEQUALITY [R]2[S]2 5n m/CP8/CP91 =2 [Q]2 (3) holds, where n is the degree of Q, and m is the degree of either R or S. This theorem captures the heuristic that if R and S have big coefficients, then so does RS; i.e., there can’t be too much cancellation. See also NORM,BOMBIERI’S INEQUALITY ,POLYNOMIAL NORM References Beauzamy, B.; Bombieri, E.; Enflo, P.; and Montgomery, H. L. "Products of Polynomials in Many Variables." J. Number Th. 36, 219 /C1/45, 1990. Borwein, P. and Erde´lyi, T. "Bombieri’s Norm." §5.3.E.7 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 274, 1995. Reznick, B. "An Inequality for Products of Polynomials." Proc. Amer. Math. Soc. 117, 1063 /C1/073, 1993. Bombieri’s Inequality For HOMOGENEOUS POLYNOMIALS P and Q of degree m and n, then[P /C215 Q]2 ]ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m!n! (m /C27 n)! [P]2[Q]2 ;s where [P /C215 Q]2is the BOMBIERI NORM .Ifm /C30n, this becomes [P/C215Q]2][P]2[Q]2; See also BOMBIERI NORM,BEAUZAMY AND DE´ GOT’S IDENTITY ,REZNIK’S IDENTITY References Borwein, P. and Erde ´lyi, T. "Bombieri’s Norm." §5.3.E.7 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 274, 1995. Bombieri’s Theorem Define E(x;q;a)/C13c(x;q;a)/C28x f(q); (1) where c(x;q;a)/C30X n5x n/C13a(mod q)L(n) (2) (Davenport 1980, p. 121), L(n) is the M ANGOLDT FUNCTION , and f(q) is the TOTIENT FUNCTION . Now define E(x;q)/C30max a (a;q)/C301½E(x;q;a)½ (3) where the sum is over aRELATIVELY PRIME toq, (a;q)/C301;and E/C31(x;q)/C30max y5xE(y;q): (4) Bombieri’s theorem then says that for fixed A/C210, X q5QE/C31(x;q)/C10ffiffiffixpQ(lnx)5; (5) provided that /ffiffiffixp(lnx)/C284BQBffiffiffixp /. References Bombieri, E. "On the Large Sieve." Mathematika 12, 201/C1/ 25, 1965. Davenport, H. "Bombieri’s Theorem." Ch. 28 in Multiplica- tive Number Theory, 2nd ed. New York: Springer-Verlag, pp. 161 /C1/68, 1980. Bond Percolation A PERCOLATION which considers the lattice edges as the relevant entities (left figure). See also PERCOLATION THEORY ,SITE PERCOLATION Bonferroni Correction The Bonferroni correction is a multiple-comparison correction used when several independent STATISTI- CAL TESTS are being performed simultaneously (since while a given ALPHA VALUE a may be appropriate for each individual comparison, it is not for the set of all comparisons). In order to avoid a lot of spurious positives, the ALPHA VALUE needs to be lowered to account for the number of comparisons being per- formed. The simplest and most conservative approach is the Bonferroni correction, which sets the ALPHA VALUE for the entire set of n comparisons equal to a by taking the ALPHA VALUE for each comparison equal to a=n: Explicitly, given n tests Ti for hypotheses Hi (/1 5i 5 n) under the assumption H0that all hypotheses Hi are false, and if the individual test critical values are 5 a=n; then the experiment-wide critical value is 5 a: In equation form, if P(Ti passes ½H0) 5a n for 1 5i 5n; then P(some Ti passes ½H0) 5 a; which follows from BONFERRONI’S INEQUALITIES . Another correction instead uses 1 /C28(1 /C28 a)1 =n : While this choice is applicable for two-sided hypotheses, multivariate normal statistics, and positive orthant dependent statistics, it is not, in general, correct(Shaffer 1995). See also A LPHA VALUE ,HYPOTHESIS TESTING ,STATIS- TICAL TEST References Bonferroni, C. E. "Il calcolo delle assicurazioni su gruppi di teste." In Studi in Onore del Professore Salvatore Ortu Carboni. Rome: Italy, pp. 13 /C1/0, 1935. Bonferroni, C. E. "Teoria statistica delle classi e calcolo delle probabilita `."Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 8,3/C1/2, 1936.Dewey, M. "Carlo Emilio Bonferroni: Life and Works." http:// www.nottingham.ac.uk/~mhzmd/life.html. Miller, R. G. Jr. Simultaneous Statistical Inference. New York: Springer-Verlag, 1991. Perneger, T. V. "What’s Wrong with Bonferroni Adjust- ments." Brit. Med. J. 316, 1236 /C1/238, 1998. Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev. Psych. 46, 561/C1/84, 1995. Bonferroni Test BONFERRONI CORRECTION Bonferroni’s Inequalities Let P(Ei) be the probability that Eiis true, and P@n i/C301Ei ðÞ be the probability that at least one of E1; E2;...,Enis true. Then P@n i/C301Ei/CP8/CP9 5Xn i/C301P(Ei): A slightly wider class of inequalities are also known as "Bonferroni inequalities." References Comtet, L. "Bonferroni Inequalities." §4.7 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 193 /C1/94, 1974. Galambos, J.; and Simonelli, I. Bonferroni-Type Inequalities with Applications. New York: Springer-Verlag, 1996. Bonne Projection AMAP PROJECTION which resembles the shape of a heart. Let f1be the standard parallel, l0the central meridian, fbe the LATITUDE , and lthe LONGITUDE on aUNIT SPHERE . Then x/C30rsinE (1) y/C30cotf1/C28rcosE; (2) where r/C30cotf1/C27f1/C28f (3) E/C30(l/C28l0) cos f r: (4) The inverse FORMULAS are f /C30cot f1 /C27 f1 /C28 r (5) l/C30l0 /C27r cos ftan/C281 x cot f1 /C28 y ! ; (6) where r /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27(cot f1 /C28y)2q : (7) The WERNER PROJECTION is a special case of the Bonne projection. See also MAP PROJECTION ,W ERNER PROJECTION References MathWorks. "Mapping Toolbox: Bonne Projection." http:// www.mathworks.com/access/helpdesk/help/toolbox/map/ bonneprojection.shtml. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington,DC: U. S. Government Printing Office, pp. 138 /C1 /40, 1987. Book Stacking Problem How far can a stack of nbooks protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible dnfornbooks (in terms of book lengths) is half the nth partial sum of the HARMONIC SERIES , given explicitly by dn/C301 2Xn k/C3011k/C30 1 2[g/C27C(1/C27n)] where C(z) is the DIGAMMA FUNCTION andgis the EULER- MASCHERONI CONSTANT . The first few valuesare d1/C3012/C300:5 d2/C303 4/C300:75 d3/C301112:0:91667 d4/C3025 24:1:04167 ; (Sloane’s A001008 and A002805). In order to find the number of stacked books required to obtain dbook-lengths of overhang, solve the dn equation for d, and take the CEILING FUNCTION . For n/C301, 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (Sloane’s A014537) books are needed. References Dickau, R. M. "The Book-Stacking Problem." http:// www.prairienet.org/~pops/BookStacking.html. Eisner, L. "Leaning Tower of the Physical Review." Amer. J. Phys. 27, 121, 1959. Gamow, G. and Stern, M. Puzzle Math. New York: Viking, 1958. Gardner, M. Martin Gardner’s Sixth Book of Mathematical Games from Scientific American. New York: Scribner’s, pp. 167 /C1/69, 1971. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Read- ing, MA: Addison-Wesley, pp. 272 /C1/74, 1990. Johnson, P. B. "Leaning Tower of Lire." Amer. J. Phys. 23, 240, 1955. Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 1, 322, 1953. Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 2, 411, 1954. Sloane, N. J. A. Sequences A001008/M2885, A002805/ M1589, and A014537 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Boole IVERSON BRACKET Boole Polynomial Polynomials sk(x;l) which form a S HEFFER SE- QUENCE with g(t)/C301/C27elt(1) f(t)/C30et/C281 (2) and have GENERATING FUNCTION X/C12 k/C300sk(x;l) k!tk/C30(1/C27t)x 1/C27(1/C27t)l: (3) The first few are s0(x;l)/C301 2 s1(x;l)/C3014(2x/C28l)t x2(x;l)/C3014[2x(x/C28l/C281)/C27l]: Jordan (1950) considers the related polynomials rn(x) which form a SHEFFER SEQUENCE with g(t) /C301 2(1 /C27et) (4) f(t) /C30et /C281: (5) These polynomials have GENERATING FUNCTION X/C12 k /C300rn(x) k!tk /C302(1 /C27 t)x 2 /C27 t: (6) The first few are r0(x) /C301 r1(x) /C301 2(2x /C281) r2(x) /C3012(2x2 /C284x /C271) r3(x) /C3014(4x3 /C2818x2 /C2720x /C283): The PETERS POLYNOMIALS are a generalization of the Boole polynomials. See also PETERS POLYNOMIAL References Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 37, 1964. Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Boole’s Inequality Let P(Ei) be the probability of an event Ei occurring. Then P @N i /C301Ei/CP8/CP9 5XN i/C301P(Ei) ; where @ denotes the UNION .IfEi and Ej are DISJOINT SETS for all iandj, then the INEQUALITY becomes an equality. See also DISJOINT SETS,UNION Boolean Algebra A mathematical structure which is similar to a BOOLEAN RING , but which is defined using the meet and join operators instead of the usual addition and multiplication operators. Explicitly, a Boolean alge- bra is the PARTIAL ORDER on subsets defined by inclusion (Skiena 1990, p. 207), i.e., the Boolean algebra b(A) of a set Ais the set of subsets of A that can be obtained by means of a finite number ofthe set operations UNION (OR), INTERSECTION (AND), and COMPLEMENTATION (NOT) (Comtet 1974, p. 185). A Boolean algebra also forms a LATTICE (Skiena 1990, p. 170), and each of the elements of b(A) is called a BOOLEAN FUNCTION . There are 22nBOOLEAN FUNC- TIONS in a Boolean algebra of order n(Comtet 1974, p. 186).In 1938, Shannon proved that a two-valued Booleanalgebra (whose members are most commonly denoted 0 and 1, or false and true) can describe the operation of two-valued electrical switching circuits. In moderntimes, Boolean algebra and B OOLEAN FUNCTIONS are therefore indispensable in the design of computerchips and integrated circuits. Boolean algebras have a recursive structure apparentin the H ASSE DIAGRAMS illustrated above for Boolean algebras of orders n/C302, 3, 4, and 5. These figures illustrate the partition between left and right halves of the lattice, each of which is the Boolean algebra on n/C281 elements (Skiena 1990, pp. 169 /C1/70). A Boolean algebra can be formally defined as a SETB of elements a,b, ... with the following properties: 1.Bhas two binary operations, ffl(logical AND, or "WEDGE ") and /C150(logical OR, or " VEE"), which satisfy the IDEMPOTENT laws affla/C30a/C150a/C30a; (1) the COMMUTATIVE laws afflb/C30bffla (2) a/C150b/C30b/C150a; (3) and the ASSOCIATIVE laws affl(bfflc)/C30(afflb)fflc (4) a/C150(b/C150c)/C30(a/C150b)/C150c: (5) 2. The operations satisfy the ABSORPTION LAW affl(a/C150b)/C30a/C150(afflb)/C30a: (6) 3. The operations are mutually distributive affl(b/C150c)/C30(afflb)ffl(afflc) (7) a/C150(bfflc)/C30(a/C150b)ffl(afflc): (8) 4.Bcontains universal bounds ¥and Iwhich satisfy ¥ffla/C30¥ (9) ¥/C150a/C30a (10) Iffla/C30a (11) I/C150a/C30I: (12) 5.Bhas a unary operation a0a?of complementa- tion which obeys the laws affla?/C30¥ (13) a /C150a?/C30I (14) (Birkhoff and Mac Lane 1965). In the slightly archaic terminology of (Bell 1937, p. 444), a Boolean algebra can be defined as a set B of elements a, b, ... with BINARY OPERATORS /C150 (or /C27; logical OR) and ffl(or : ; logical AND) such that 1a. If a and b are in the set B, then a /C150b is in the set B. 1b. If a and b are in the set B, then a fflb is in the set B. 2a. There is an element Z (zero) such that a /C150Z /C30 a for every element a. 2b. There is an element U (unity) such that a ffl U /C30a for every element a. 3a. a /C150b /C30b /C150a :/ 3b. a fflb /C30b ffla :/ 4a. a /C150b fflc /C30(a /C150b) ffl(a /C150c) :/ 4b. a ffl(b /C150c) /C30(a fflb) /C150(a fflc):/ 5. For every element a there is an element a ? such that a /C150a ?/C30U and a ffla ?/C30Z:/ 6. There are at least two distinct elements in the set B. Huntington (1933ab) presented the following basis for Boolean algebra: 1. Commutativity. x /C150y /C30y /C150x:/ 2. Associativity. (x /C150y) /C150z /C30x /C150(y /C150z):/ 3. HUNTINGTON AXIOM . !(!x /C150y) /C150!(!x /C150!y) /C30x:/ H. Robbins then conjectured that the HUNTINGTON AXIOM could be replaced with the simpler ROBBINS AXIOM , !(!(x /C150y) /C150!(x /C150!y)) /C30x (15) The ALGEBRA defined by commutativity, associativity, and the ROBBINS AXIOM is called ROBBINS ALGEBRA . Computer theorem proving demonstrated that every ROBBINS ALGEBRA satisfies the second WINKLER CON- DITION , from which it follows immediately that all ROBBINS ALGEBRAS are Boolean (McCune, Kolata 1996). See also BOOLEAN FUNCTION ,BOOLEANS ,H UNTING- TON AXIOM ,M AXIMAL IDEAL THEOREM ,R OBBINS ALGEBRA ,R OBBINS AXIOM ,W INKLER CONDITIONS , WOLFRAM AXIOM References Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1986. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, p. 317, 1996. Comtet, L. "Boolean Algebra Generated by a System of Subsets." §4.4 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 185 /C1/89, 1974.Halmos, P. Lectures on Boolean Algebras. Princeton, NJ: Van Nostrand, 1963. Huntington, E. V. "New Sets of Independent Postulates for the Algebra of Logic." Trans. Amer. Math. Soc. 35, 274 /C1/ 04, 1933a. Huntington, E. V. "Boolean Algebras: A Correction." Trans. Amer. Math. Soc. 35, 557 /C1/58, 1933. Kolata, G. "Computer Math Proof Shows Reasoning Power." New York Times , Dec. 10, 1996. McCune, W. "Robbins Algebras are Boolean." http://www- unix.mcs.anl.gov/~mccune/papers/robbins/. Mendelson, E. Introduction to Boolean Algebra and Switch- ing Circuits. New York: McGraw-Hill, 1973. Sikorski, R. Boolean Algebra, 3rd ed. New York: Springer- Verlag, 1969. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Wells, C. F. "Boolean Expression Manipulation." http:// www.mathsource.com/cgi-bin/msitem?0204 /C1/69. Boolean Connective One of the LOGIC operators AND ffl; OR /C150; and NOT /C15:/ See also QUANTIFIER Boolean Function Consider a Boolean algebra of subsets b(A) generated by a set A, which is the set of subsets of Athat can be obtained by means of a finite number of the set operations union, intersection, and complementation. Then each of the elements of b(A) is called a Boolean function generated by A(Comtet 1974, p. 185). Each Boolean function has a unique representation (up to order) as a union of COMPLETE PRODUCTS . It follows that there are 22pinequivalent Boolean functions for a setAwith cardinality p(Comtet 1974, p. 187). In 1938, Shannon proved that a two-valued Booleanalgebra (whose members are most commonly denoted 0 and 1, or false and true) can describe the operationof two-valued electrical switching circuits. The follow- ing table gives the TRUTH TABLE for the 222/C3016 possible Boolean functions of two binary variables. AB /F0//F1//F2//F3//F4//F5//F6//F7/ 0 000000000 0 100001111 1 0001100111 101010101 AB /F8//F9//F10//F11//F12//F13//F14//F15/ 0 0 1 1111111 0 1 0 0001111 1000110011 1101010101 The names and symbols for these functions are given in the following table (Simpson 1987, p. 539). operation symbol name /F0/ 0 FALSE /F1// A fflB/ AND /F2// A ffl!B/ A AND NOT B /F3/ AA /F4// !A fflB/ NOT A AND B /F5/ BB /F6// A/C150B/ XOR /F7// A /C150B/ OR /F8// A/C150B/ NOR /F9/ A XNOR B XNOR /F10// !B/ NOT B /F11// A /C150!B/ A OR NOT B /F12// !A/ NOT A /F13// !A /C150B/ NOT A OR B /F14// AfflB/ NAND /F15/ 1 TRUE Determining the number of monotone Boolean func- tions of n variables is known as DEDEKIND’S PROBLEM and is equivalent to the number of ANTICHAINS on the n-set f1; 2; ... ; ng: Boolean functions can also be thought of as colorings of a Boolean n-cube. The numbers of inequivalent monotone Boolean functions in n /C301, 2, ... variables are given by 2, 3, 5, 10, 30, ...(Sloane’s A003182). Let M(n; k) denote the number of distinct monotone Boolean functions of n variables with k MINCUTS . Then M(n; 0) /C301 M(n; 1) /C302n M(n ; 2) /C30 2n/C281(2n /C281) /C283n /C272n M(n; 3) /C301 6(2n)(2n /C281)(2n /C282) /C286n /C275n /C274n /C283n : See also ANTICHAIN ,BOOLEAN ALGEBRA ,BOOLEANS ,COMPLETE PRODUCT ,CONJUNCTION ,DEDEKIND’S PRO- BLEM ,MINCUT ,MONOTONE FUNCTION References Comtet, L. "Boolean Algebra Generated by a System of Subsets." §4.4 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 185 /C1/89, 1974. Shapiro. "On the Counting Problem for Monotone Boolean Functions." Comm. Pure Appl. Math. 23, 299 /C1/12, 1970. Simpson, R. E. Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, 1987. Sloane, N. J. A. Sequences A003182/M0729 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Boolean Representation Theorem Every BOOLEAN ALGEBRA is isomorphic to the BOO- LEAN ALGEBRA of sets. It is equivalent to the MAXIMAL IDEAL THEOREM , which can be proved without using the AXIOM OF CHOICE (Mendelson 1997, p. 121). See also BOOLEAN ALGEBRA ,M AXIMAL IDEAL THEO- REM References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 121, 1997. Stone, M. "The Representation Theorem for Boolean Alge- bras." Trans. Amer. Math. Soc. 40,37/C1/11, 1936. Boolean Ring A RING with a unit element in which every element is IDEMPOTENT . See also BOOLEAN ALGEBRA Booleans The domain of Booleans, sometimes denoted B; consisting of the elements TRUE and FALSE , imple- mented in Mathematica asBooleans .InMathema- tica, a quantity can be tested to determine if it is in the domain of Booleans usingElement[ e, Booleans]. See also BOOLEAN ALGEBRA ,B OOLEAN FUNCTION , FALSE ,TRUE Boomeron Equation The system of PARTIAL DIFFERENTIAL EQUATIONS ut/C30b /C215vx bxt/C30uxxb/C27a/C29vx/C282v/C29(v/C29b): References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 57, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Boosting See also RESAMPLING STATISTICS Bootstrap Methods A set of methods that are generally superior to ANOVA for small data sets or where sample distribu- tions are non-normal. See also ANOVA, JACKKNIFE ,PERMUTATION TESTS , RESAMPLING STATISTICS References Chernick, M. R. Bootstrap Methods: A Practitioner’s Guide. New York: Wiley, 1999. Davison, A. C. and Hinkley, D. V. Bootstrap Methods and Their Application. Cambridge, England: Cambridge Uni- versity Press, 1997. Efron, B. and Tibshirani, R. J. An Introduction to the Bootstrap. Boca Raton, FL: CRC Press, 1994. Mooney, C. Z. and Duval, R. D. Bootstrapping: A Nonpara- metric Approach to Statistical Inference. Sage, 1993. Borchardt-Pfaff Algorithm ARCHIMEDES ALGORITHM Border Square A MAGIC SQUARE that remains magic when its border is removed. A nested magic square remains magic after the border is successively removed one ring at a time. An example of a nested magic square is the order 7 square illustrated above (i.e., the order 7, 5, and 3 squares obtained from it are all magic). See also MAGIC SQUARE References Chabert, J.-L. (Ed.). "Squares with Borders" and "Arnauld’s Borders Method." §2.1 and 2.4 in A History of Algorithms: From the Pebble to the Microchip. New York: Springer- Verlag, pp. 53 /C1/8 and 70 /C1/0, 1999. Kraitchik, M. "Border Squares." §7.7 in Mathematical Recreations. New York: W. W. Norton, pp. 167 /C1/70, 1942. Bordism A relation between COMPACT boundaryless MANI- FOLDS (also called closed MANIFOLDS ). Two closed MANIFOLDS are bordant IFF their disjoint union is the boundary of a compact (n /C271)/-MANIFOLD . Roughly, two MANIFOLDS are bordant if together they form the boundary of a MANIFOLD . The wordbordism is now used in place of the original term COBORDISM . References Budney, R. "The Bordism Project." http://www.math.cornel- l.edu/~rybu/bordism/bordism.html. Bordism Group There are bordism groups, also called COBORDISM GROUPS or COBORDISM RINGS , and there are singular bordism groups. The bordism groups give a frame- work for getting a grip on the question, "When is a compact boundaryless MANIFOLD the boundary of another MANIFOLD ?" The answer is, precisely when all of its STIEFEL- WHITNEY CLASSES are zero. Singular bordism groups give insight into STEENROD’S REALI- ZATION PROBLEM : "When can homology classes be realized as the image of fundamental classes of manifolds?" That answer is known, too. The machinery of the bordism group winds up being important for HOMOTOPY THEORY as well. References Budney, R. "The Bordism Project." http://www.math.cornel- l.edu/~rybu/bordism/bordism.html. Borel Algebra See also BOREL SIGMA ALGEBRA ,BOREL SUBALGEBRA Borel Determinacy Theorem Let T be a TREE defined on a metric over a set of paths such that the distance between paths p and q is 1=n; where n is the number of nodes shared by p and q. Let A be a BOREL SET of paths in the topology induced by this metric. Suppose two players play a game by choosing a path down the tree, so that they alternate and each time choose an immediate successor of the previously chosen point. The first player wins if the chosen path is in A. Then one of the players has a winning STRATEGY in this GAME . See also GAME THEORY ,TREE Borel Field If a FIELD has the property that, if the sets An ; ..., An ; ... belong to it, then so do the sets A1 /C27.../C27An /C27... and A1 ...An ... ; then the field is called a Borel field (Papoulis 1984, p. 29). See also FIELD References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Borel Measure If F is the BOREL SIGMA ALGEBRA on some TOPOLOGI- CAL SPACE , then a MEASURE m : F 0 R is said to be a Borel measure (or BOREL PROBABILITY MEASURE ). For a Borel measure, all continuous functions are MEA- SURABLE . Borel Probability Measure BOREL MEASURE Borel Set A Borel set is an element of a BOREL SIGMA ALGEBRA . Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable unions and intersections. Formally, the class B of Borel sets in Euclidean Rnis the smallest collection of sets that includes the open and closed sets such that if E, E1 ; E2 ; ... are in B, then so are @/C12 i/C301Ei ;S/C12i/C301Ei ; and Rn_E ; where F_E is a SET DIFFERENCE (Croft et al. 19991). The set of rational numbers is a Borel set, as is the CANTOR SET. See also CLOSED SET,OPEN SET,STANDARD SPACE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991. Borel Sigma Algebra A SIGMA ALGEBRA which is related to the TOPOLOGY of a SET. The Borel s/-algebra is defined to be the SIGMA ALGEBRA generated by the OPEN SETS (or equiva- lently, by the CLOSED SETS ). See also BOREL ALGEBRA ,BOREL MEASURE ,BOREL SUBALGEBRA Borel Space A SET equipped with a SIGMA ALGEBRA of SUBSETS . Borel Subalgebra See also BOREL ALGEBRA ,BOREL SIGMA ALGEBRA Borel’s Expansion Letf(t)/C30a/C12 n/C300Antnbe any function for which the integral I(x)/C13g/C12 0e/C28txtpf(t)dt converges. Then the expansion I(x)G(p/C271) xp/C271A0/C27(p/C271)A1 x/C27(p/C271)(p/C272)A2 x2/C27..."# ;where G(z) is the GAMMA FUNCTION , is usually an ASYMPTOTIC SERIES forI(x):/ Borel-Cantelli Lemma Let fAng/C12 n/C300be a SEQUENCE of events occurring with a certain probability distribution, and let Abe the event consisting of the occurrence of a finite number of events An;n/C301, .... Then if X/C12 n/C301P(An)B/C12 ; then P(A)/C301: References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 435 /C1/36, 1988. Borel-Weyl Theorem LetG/C30SL(n;C):Ifl/C23Znis the highest weight of an irreducible holomorphic representation VofG, (i.e., l is a dominant integral weight), then the G-map f: V/C310G(l) defined by a/C2Fa;where Fa(g)/C30a;gvhi ;is anISOMORPHISM . Thus, V$G(l)/C31:/ References Huang, J.-S. "The Borel-Weyl Theorem." §8.7 in Lectures on Representation Theory. Singapore: World Scientific, pp. 105 /C1/07, 1999. Born-Infeld Equation The PARTIAL DIFFERENTIAL EQUATION (1/C28u2 t)uxx/C272uxutuxt/C28(1/C27u2x)utt/C300: References Whitham, G. B. Linear and Nonlinear Waves. New York: Wiley, p. 617, 1974. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 132, 1997. Boron Tree BINARY TREE Borromean Rings Three mutually interlocked rings, named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (illustrated above) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1983, Wells 1991). The Borromean rings have LINK symbol 06 /C1/3 /C1/2, BRAID WORD s/C281 1s2 s/C281 1s2 s/C281 1s2 ; and are also the simplest BRUNNIAN LINK . See also BRUNNIAN LINK,CIRCLE- CIRCLE INTERSEC- TION ,TRIQUETRA ,VENN DIAGRAM References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 58 /C1/9, 1989. Gardner, M. The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: University of Chicago Press, 1991. Jablan, S. "Borromean Triangles." http://members.tripod.- com/~modularity/links.htm. Pappas, T. "Trinity of Rings--A Topological Model." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 31, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 266 /C1/67, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 18, 1991.Borrow The procedure used in SUBTRACTION to "borrow" 10 from the next higher DIGIT column in order to obtain a POSITIVE DIFFERENCE in the column in question. See also CARRY Borsuk’s Conjecture Borsuk conjectured that it is possible to cut an n-D shape of GENERALIZED DIAMETER 1 into n /C271 pieces each with diameter smaller than the original. It is true for n /C302, 3 and when the boundary is "smooth." However, the minimum number of pieces required has been shown to increase as /C21 :1ffiffinp : Since 1 :1ffiffinp > n /C271at n /C309162, the conjecture becomes false at high dimensions. In fact, the conjecture is false for every n /C21561. See also GENERALIZED DIAMETER ,KELLER’S CONJEC- TURE ,LEBESGUE MINIMAL PROBLEM References Borsuk, K. "U¨ ber die Zerlegung einer Euklidischen n- dimensionalen Vollkugel in n Mengen." Verh. Internat. Math.-Kongr. Zu¨rich 2, 192, 1932. Borsuk, K. "Drei Sa¨tze u¨ber die n-dimensionale euklidische Spha¨re." Fund. Math. 20, 177 /C1/90, 1933. Cipra, B. "If You Can’t See It, Don’t Believe It...." Science 259,2 6/C1/7, 1993. Cipra, B. What’s Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 21 /C1/5, 1993. Gru¨nbaum, B. "Borsuk’s Problem and Related Questions." In Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, Heldat the University of Washington, Seattle, June 13 /C1 /5, 1961. Providence, RI: Amer. Math. Soc., pp. 271 /C1/84, 1963. Kalai, J. K. G. "A Counterexample to Borsuk’s Conjecture." Bull. Amer. Math. Soc. 329,6 0/C1/2, 1993. Lyusternik, L. and Schnirel’mann, L. Topological Methods in Variational Problems. Moscow, 1930. Lyusternik, L. and Schnirel’mann, L. "Topological Methods in Variational Problems and Their Application to the Differential Geometry of Surfaces." Uspehi Matem. Nauk (N.S.) 2, 166/C1/17, 1947. Borsuk-Ulam Theorem Every continuous map /f:Sn0Rn /must identify a pair of ANTIPODAL POINTS . References Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 121 and 284, 1997. Borwein Conjectures Use the definition of the Q-SERIES (a; q)n /C13Yn/C281 j/C300(1 /C28aqj) (1) and define N M/C20/C2P /C13(qN /C28M /C271; q)M (q; q)m: (2) Then P. Borwein has conjectured that (1) the POLY- NOMIALS An(q) ; Bn(q) ; and Cn(q) defined by (q; q3)n(q2; q3)n /C30An(q3) /C28qBn(q3) /C28q2Cn(q3) (3) have NONNEGATIVE COEFFICIENTS , (2) the POLYNO- MIALS A/C31 n(q); B /C31n(q) ; and C/C31n(q) defined by (q; q3)2 n(q2; q3)2n /C30A/C31 n(q3) /C28qB/C31 n(q3) /C28q2C/C31n(q3) (4) have NONNEGATIVE COEFFICIENTS , (3) the POLYNO- MIALS A/C31 n(q) ; B/C31n(q) ; C /C31n(q); D/C31n(q) ; and E /C31n(q) defined by (q; q5)n(q2; q5)n(q3; q5)n(q4; q5)n /C30 A/C31 n(q5) /C28qB/C31n(q5) /C28q2C /C31n(q5) /C28q3D /C31n(q5) /C28q4E /C31n(q5) (5) have NONNEGATIVE COEFFICIENTS , (4) the POLYNO- MIALS A$ n(m; n; t; q); B $n(m; n; t; q) ; and C $n(m; n; t; q) defined by (q; q3)m(q2; q3)m(zq; q3)n(zq2; q3)n /C30X2m t/C300zt[A$(m; n; t; q3) /C28qB$(m; n; t; q3) /C28q2C$(m; n; t; q3)] (6) have NONNEGATIVE COEFFICIENTS , (5) for k ODD and 1 5a 5k =2; consider the expansion (qa; qk)m(qk /C28a; qk)n /C30X(k/C281)=2 n/C30(1/C28k) =2(/C281)nqk( n2/C27n)=2/C28a nFn(qk) (7) with Fn(q) /C30X/C12 j/C30/C28/C12(/C281)jqj(k2j/C272k n/C27k/C282a)=2 m /C27n m /C27 n /C27kj/C20/C2P ; (8) then if a is RELATIVELY PRIME to k and m /C30n, the COEFFICIENTS of Fn(q) are NONNEGATIVE , and (6) given a /C27 b B2K and /C28K /C27 b 5n /C28m 5K /C28 a; consider G( a; b; K; q) /C30X q(/C281)jqj[K(a /C27b)j/C27K(a /C27 b)] =2 m /C27n m /C27Kj/C20/C2P ; (9) the GENERATING FUNCTION for partitions inside an m /C29n rectangle with hook difference conditions spe- cified by a; b; and K. Let a and b be POSITIVE RATIONAL NUMBERS and k /C211an INTEGER such thatak and bk are integers. then if 1 5 a /C27 b 52k /C281 (with strict inequalities for k /C30 2) and /C28k /C27 b 5n /C28m 5 k /C28 a; then g( a; b; k; q) has NONNEGATIVE COEFFI- CIENTS . See also Q-SERIES References Andrews, G. E. et al. "Partitions with Prescribed Hook Differences." Europ. J. Combin. 8, 341 /C1/50, 1987. Bressoud, D. M. "The Borwein Conjecture and Partitions with Prescribed Hook Differences." Electronic J. Combi- natorics 3, No. 2, R4, 1 /C1/4, 1996. http://www.combinator- ics.org/Volume_3/volume3_2.html#R4. Bott Periodicity Theorem Define O /C30lim 0O(n) ; F /C30R (1) U /C30lim 0U(n); F /C30C (2) Sp /C30lim 0Sp(n) ; F /C30H: (3) Then V2BU $BU /C29Z (4) V4BO $BSp /C29Z (5) V4BSp $BO /C29Z: (6) References Atiyah, M. F. K-Theory. New York: Benjamin, 1967. Bott, R. "The Stable Homotopy of the Classical Groups." Ann. Math. 70, 313 /C1/37, 1959. Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, p. 229, 1997. Milnor, J. W. Morse Theory. Princeton, NJ: Princeton University Press, 1963. Bottle Imp Paradox In Robert Louis Stevenson’s "bottle imp paradox," you are offered the opportunity to buy, for whatever price you wish, a bottle containing a genie who will fulfill your every desire. The only catch is that the bottle must thereafter be resold for a price smaller than what you paid for it, or you will be condemned to live out the rest of your days in excruciating torment. Obviously, no one would buy the bottle for 1c since he would have to give the bottle away, but no one would accept the bottle knowing he would be unable to get rid of it. Similarly, no one would buy it for 2c, and soon. However, for some reasonably large amount, it will always be possible to find a next buyer, so the bottle will be bought (Paulos 1995). See also U NEXPECTED HANGING PARADOX References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 25 /C1/7, 1998. Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, p. 97, 1995. Bouligand Dimension MINKOWSKI- BOULIGAND DIMENSION Bound GREATEST LOWER BOUND ,INFIMUM ,LEAST UPPER BOUND ,SUPREMUM Bound Variable An occurrence of a variable in a LOGIC which is not FREE . Bound variables are also called DUMMY VARI- ABLES . See also DUMMY VARIABLE ,SENTENCE References Comtet, L. "Bound Variables." §1.11 in Advanced Combina- torics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 30 /C1/4, 1974. Boundary The set of points, known as BOUNDARY POINTS , which are members of the CLOSURE of a given set S and the CLOSURE of its complement set. The boundary is sometimes called the FRONTIER . See also BOUNDARY CONDITIONS ,B OUNDARY MAP, BOUNDARY POINT ,BOUNDARY SET,NATURAL BOUND- ARY,SURGERY Boundary Conditions There are several types of boundary conditions commonly encountered in the solution of PARTIAL DIFFERENTIAL EQUATIONS . 1. DIRICHLET BOUNDARY CONDITIONS specify the value of the function on a surface T /C30f(r; t) :/ 2. NEUMANN BOUNDARY CONDITIONS specify the normal derivative of the function on a surface, @T @n /C30ˆn /C2159T /C30f(r ; y) : 3. CAUCHY BOUNDARY CONDITIONS specify a weighted average of first and second kinds. 4. ROBIN BOUNDARY CONDITIONS . For an elliptic partial differential equation in a region V; Robin boundary conditions specify the sum of and the normal derivative of u /C30 f at all points of the boundary of V; with a and f being prescribed. See also BOUNDARY VALUE PROBLEM ,D IRICHLET BOUNDARY CONDITIONS ,GOURSAT PROBLEM ,INITIALVALUE PROBLEM ,NEUMANN BOUNDARY CONDITIONS , PARTIAL DIFFERENTIAL EQUATION ,ROBIN BOUNDARY CONDITIONS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 502 /C1/04, 1985. Morse, P. M. and Feshbach, H. "Boundary Conditions and Eigenfunctions." Ch. 6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 495 /C1/98 and 676 /C1/90, 1953. Boundary Map The MAP Hn(X ; A) 0 Hn/C281(A) appearing in the LONG EXACT SEQUENCE OF A PAIR AXIOM . See also LONG EXACT SEQUENCE OF A PAIR AXIOM Boundary Point A point which is a member of the CLOSURE of a given set S and the CLOSURE of its complement set. If A is a subset of Rn ; then a point x /C23Rn is a boundary point of A if every NEIGHBORHOOD of x contains at least one point in A and at least one point not in A. See also BOUNDARY Boundary Set A (symmetrical) boundary set of RADIUS r and center x0 is the set of all points x such that x /C28x0 jj /C30r : Let x0 be the ORIGIN .InR1 ; the boundary set is then the pair of points x /C30r and x /C30/C28r : In R2 ; the boundary set is a CIRCLE .InR3 ; the boundary set is a SPHERE . See also CIRCLE ,C OMPACT SET,D ISK,O PEN SET, SPHERE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Boundary Value Problem A boundary value problem is a problem, typically an ORDINARY DIFFERENTIAL EQUATION or a PARTIAL DIFFERENTIAL EQUATION , which has values assigned on the physical boundary of the DOMAIN in which the problem is specified. For example, @2u @t2/C2892u/C30finV u(0;t)/C30u1 on@V @u @t(0;t)/C30u2on@V;8 >>>>>< >>>>>: where @Vdenotes the boundary of V;is a boundary problem. See also BOUNDARY CONDITIONS ,INITIAL VALUE PROBLEM References Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Computational Differential Equations. Lund: Studentlit- teratur, 1996. Powers, D. L. Boundary Value Problems, 4th ed. San Diego, CA: Academic Press, 1999. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Two Point Boundary Value Problems." Ch. 17 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 745 /C1/78, 1992. Bounded A mathematical object (such as a set or function) is said to bounded if it possesses a BOUND , i.e., a value which all members of the set, functions, etc., are less than. See also BOUNDED SET Bounded Set A SET in a METRIC SPACE (X, d) is bounded if it has a FINITE GENERALIZED DIAMETER , i.e., there is an R B/C12 such that d(x; y) 5R for all x; y /C23 X : A SET in Rn is bounded if it is contained inside some BALL x2 1 /C27.../C27 x2n 5R2 of FINITE RADIUS R (Adams 1994). See also BOUND ,FINITE References Adams, R. A. Calculus: A Complete Course. Reading, MA: Addison-Wesley, p. 707, 1994. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Jeffreys, H. and Jeffreys, B. S. "Bounded, Unbounded, Convergent, Oscillatory." §1.041 in Methods of Mathema- tical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 11 /C12, 1988. Bounded Variation A FUNCTION f(x) is said to have bounded variation if, over the CLOSED INTERVAL x /C23 [a ; b]; there exists an M such that f(xi) /C28f(a) jj /C27 f(x2) /C28f(x1) jj /C27.../C27 f(b) /C28f(xn /C281) jj 5M (1) for all a Bx1 Bx2 B...Bxn/C281 Bb:/ The space of functions of bounded variation is denoted "BV," and has the SEMINORM F(f) /C30supg fdf dx; (2) where f ranges over all COMPACTLY SUPPORTED functions bounded by -1 and 1. The seminorm is equal to the SUPREMUM over all sums above, and is also equal to f df =dx jj dx (when this expressionmakes sense). On the interval [0; 1]; the function x2 sin(1 =x) (pur- ple) is of bounded variation, but x sin 1=x (red) is not. More generally, a function f is locally of bounded variation in a domain U if f is LOCALLY INTEGRABLE , f /C23 L1 loc ; and for all open subsets W, with COMPACT CLOSURE in U, and all SMOOTH VECTOR FIELDS g COMPACTLY SUPPORTED in W, gWf div gdx 5c(W) sup½g½; (3) div denotes DIVERGENCE and c is a constant which only depends on the choice of W and f. Such functions form the space BVloc(U) : They may not be DIFFERENTIABLE , but by the RIESZ REPRESENTA- TION THEOREM , the derivative of a BVloc/-function f is a REGULAR BOREL MEASURE Df. Functions of bounded variation also satisfy a compactness theorem. Given a sequence fn of functions in BVloc(U); such that sup nfnkkL1(W)/C27gW½Dfn½dx/CP8/CP9 B/C12 ; that is the TOTAL VARIATION of the functions is bounded, in any COMPACTLY SUPPORTED open subset W, there is a SUBSEQUENCE fnkwhich converges to a function f/C23BVlocin the topology of L1 loc:Moreover, the limit satisfies gW½Df½dx5lim infgW½Dfnk½dx: (4) They also satisfy a version of P OINCARE ´’S LEMMA . See also DIFFERENTIABLE ,W EAKLY DIFFERENTIABLE References Jeffreys, H. and Jeffreys, B. S. "Functions of Bounded Variation." §1.09 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 24 /C1/6, 1988. Simon, L. §2.6 in Lectures on Geometric Measure Theory Canberra: Centre for Mathematical Analysis, Australian National University, 1984. Bour’s Minimal Surface Gray (1997) defines Bour’s minimal curve over com- plex z by x?/C30zm/C281 m /C28 1 /C28zm/C271 m /C27 1 (1) y?/C30izm/C281 m /C28 1 /C27zm/C271 m /C27 1 ! (2) z ?/C302zm m; (3) and then derives a family of MINIMAL SURFACES . The order three Bour surface resembles a CROSS-CAP and is given using ENNEPER- WEIERSTRASS PARAME- TERIZATION by (4) g /C30ffiffiffizp(5) or explicitly by the PARAMETRIC EQUATIONS x /C30r cos u /C281 2 r2 cos(2 u) (6) y /C30/C28r sin u /C2812 r2 sin(2u) ; (7) z /C304 3 r3 =2 cos(32u) (8) (Maeder 1997). The coefficients of the FIRST FUNDA- MENTAL FORM are given by E /C301 /C27r2 (9) F /C30 0 (10) G /C30 r2(r2 /C271) (11) and the coefficients of the SECOND FUNDAMENTAL FORM by e /C30/C28r /C281 =2 cos(32 f) (12) f /C30ffiffiffirpsin(3 2 f) (13)g /C30r3 =2 cos(3 2 f): (14) The AREA ELEMENT is dA /C30r(r /C271)2 dr ffld f: (15) The GAUSSIAN and MEAN CURVATURES are given by K /C30/C281 r(r /C27 1)4 (16) H /C300 : (17) See also CROSS- CAP,ENNEPER- WEIERSTRASS PARAME- TERIZATION ,MINIMAL SURFACE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 732 /C1/33, 1997. Maeder, R. Programming in Mathematica, 3rd ed. Reading, MA: Addison-Wesley, pp. 29 /C1/0, 1997. Bourget Function The function defined by the CONTOUR INTEGRAL Jn; k(z) /C301 2 pi g(0/C27) t/C28n/C281 t /C271 t !k exp1 2zt/C281 t !"# dt; where f(0/C27)denotes the CONTOUR encircling the point z /C30 0 once in a counterclockwise direction. It is equal to Jn;k(z)/C301 pgp 0(2 cos u)kcos(nu/C28zsinu)du (Watson 1966, p. 326). See also BESSEL FUNCTION OF THE FIRST KIND References Bourget, J. "Me ´moire sue les nombres de Cauchy et leur application a `divers proble `mes de me ´canique ce ´leste." J. de Math. 6,3 3/C1/4, 1861. Giuliani, G. "Alcune osservazioni sopra le funzioni spheriche di ordine superiore al secondo e sopra altre funzioni che se ne possono dedurre (April, 1888)." Giornale di Mat. 26, 155/C1/71, 1888. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of theSoviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 465, 1988. Watson, G. N. "The Functions of Bourget and Giuliani." §10.31 in A Treatise on the Theory of Bessel Functions, 2nd ed.Cambridge, England: Cambridge University Press, pp. 326 /C1 /27, 1966. Bourget’s Hypothesis When nis an INTEGER ]0;then Jn(z) and Jn/C27m(z) have no common zeros other than at z/C300 for man INTEGER ]1 ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . The theorem has been proved true for m /C301 2, 3, and 4. References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Bourque-Ligh Conjecture Bourque and Ligh (1992) conjectured that the LEAST COMMON MULTIPLE MATRIX on a GCD -CLOSED SET S is nonsingular. This conjecture was shown to be false by Hong (1999). See also GCD -CLOSED SET,LEAST COMMON MULTIPLE MATRIX References Bourque, K. and Ligh, S. "On GCD and LCM Matrices." Linear Algebra Appl. 174,65/C1/4, 1992. Hong, S. "On the Bourque-Ligh Conjecture of Least Common Multiple Matrices." J. Algebra 218, 216 /C1/28, 1999. Boussinesq Equation The linear Boussinesq equation is the PARTIAL DIF- FERENTIAL EQUATION utt /C28 a2uxx /C30 b2uxxtt (1) (Whitham 1974, p. 9; Zwillinger 1997, p. 129). The nonlinear Boussinesq equation is utt /C28uxx /C28uxxxx /C273(u2)xx /C300 (2) (Calogero and Degasperis 1982; Zwillinger 1997, p. 130). The modified Boussinesq equation is 1 3 utt /C28utuxx /C2832 u2 xuxx /C27uxxxx /C300 (3) (Clarkson 1986; Zwillinger 1997, p. 132). References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, 1982. Clarkson, P. A. "The Painleve ´ Property, a Modified Boussi- nesq Equation and a Modified Kadomtsev-Petviashvili Equation." Physica D 19, 447 /C1/50, 1986. Whitham, G. B. Linear and Nonlinear Waves. New York: Wiley, 1974. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 129 /C1/30, 1997. Boustrophedon Transform The boustrophedon ("ox-plowing") transform b of a sequence a is given by bn /C30Xn k /C300n k/CP8/CP9 akEn/C28k (1)an /C30Xn k/C300(/C281)n/C28k n k/CP8/CP9 bkEn/C28k (2) for n ]0; where Enis a SECANT NUMBER or TANGENT NUMBER defined by X/C12 n/C300Enxn n!/C30secx/C27tanx: (3) The exponential generating functions of aandbare related by B(x)/C30(secx/C27tanx)A(x); (4) where the exponential generating function is defined by A(x)/C30X/C12 n/C300Anxn n!: (5) See also ALTERNATING PERMUTATION ,E NTRINGER NUMBER ,S ECANT NUMBER ,S EIDEL- ENTRINGER- AR- NOLD TRIANGLE ,TANGENT NUMBER References Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Operation on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76,4 4/C1/4, 1996. Bovinum Problema ARCHIMEDES’ CATTLE PROBLEM Bow x4/C30x2y/C28y3: References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Bowditch Curve LISSAJOUS CURVE Bowl of Integers Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller circles touch each other at the center of the large circle and are tangent to the large circle on the extremities ofone of its diameters. This arrangement is called the "bowl of integers" (Soddy 1937) since the BEND of each of the infinite chain of spheres that can be packed into it such that each successive sphere is tangent to its neighbors is an integer. The first few bends are then /C281, 2, 5, 6, 9, 11, 14, 15, 18, 21, 23, ... (Sloane’s A046160). The sizes and positions of the first few rings of spheres are given in the table below. n /kn//zn//Rn// fn/ 1- 10 0 – 22 /1 2/ 0– 35 /2 5//25ffiffiffi 3p //1 6p/ 46 /12//23/ 0 59 /23//29ffiffiffi 7p //9tan/C281(1 2ffiffiffi 3p )/ 61 1 /8 11//6 11/ 0 71 4 /11 14//27ffiffiffi 3p //1 6p/ 81 5 /4 5//2 15ffiffiffiffiffiffi 13p //9tan/C281(2ffiffiffi3p ) / 91 8 /5 6//49/ 0 10 21 /67//1 21ffiffiffiffiffiffi 19p //9tan/C281(3 7ffiffiffi 3p )/ 11 23 /20 23//2 23ffiffiffiffiffiffi 21p //9tan/C281(1 9ffiffiffi 3p )/ 12 27 /8 9//1027/0,9tan/C281(13ffiffiffi 3p )/ 13 30 /9 10//2 15ffiffiffi7p //9tan/C281(1 5ffiffiffi 3p )/ 14 33 /10 11//2 33ffiffiffiffiffiffi 31p //9tan/C281(1 11ffiffiffi3p ) / 15 38 /35 38//6 19/ 0 Spheres can also be packed along the plane tangent to the two spheres of radius 2 (Soddy 1937). The sequence of integers for can be found using theequation of five TANGENT SPHERES . Letting k3/C30k4/C30 2 gives k(k1;k2) /C301 2(4/C27k1/C27k2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3[k2(8/C28k2)/C272k1(k2/C274)/C283k2 1q ): For example, k(3;3)/C3011;k(3;11)/C3015;k(11;15)/C30 27;k(15;27)/C3035;k(27;27)47 ;and so on, giving the sequence -1, 2, 3, 11, 15, 27, 35, 47, 51, 63, 75, 83, ... (Sloane’s A046159). The sizes and positions of the first few rings of spheres are given in the table below. n /kn//Rn// fn/ 1- 1 0 – 220 –33 /2 3/ 0 41 1 /16p/ 51 5 /4 15/ 0 62 7 /2 27ffiffiffi 7p //9tan/C281(3ffiffiffi3p ) / 73 5 /6 35/ 0 84 7 /4 47ffiffiffi3p //1 6p/ 95 1 /2 51ffiffiffiffiffiffi 13p //9tan/C281(3 5ffiffiffi 3p )/ 10 63 /8 63/ 0 11 75 /2 75ffiffiffiffiffiffi19p //9tan/C281(5ffiffiffi3p ) / 12 83 /2 83ffiffiffiffiffiffi 21p //9tan/C281(5 3ffiffiffi 3p )/ 13 99 /10 99/ 0 14 107 /6 107ffiffiffi 3p //1 6p/ 15 111 /4 111ffiffiffi 7p //9tan/C281(1 2ffiffiffi 3p )/ 16 123 /2 123ffiffiffiffiffiffi31p //9tan/C281(5 7ffiffiffi 3p )/ 17 143 /12 143/ 0 18 147 /2 147ffiffiffiffiffiffi37p //9tan/C281(7ffiffiffi3p ) / 19 155 /2 155ffiffiffiffiffiffi39p //9tan/C281(1 6ffiffiffi 3p )/ 20 171 /2 171ffiffiffiffiffiffi43p //9tan/C281(7 5ffiffiffi 3p )/ The analogous problem of placing two circles of bend 2 inside a circle of bend -1 and then constructing chains of mutually tangent circles was considered by B. L. Galebach and A. R. Wilks. The circle have integral bends given by -1, 2, 3, 6, 11, 14, 15, 18, 23, 26, 27, 30, 35, 38, ... (Sloane’s A042944). Of these, the only known numbers congruent to 2, 3, 6, 11 (mod 12) missing from this sequence are 78, 159, 207, 243, 246, 342, ... (Sloane’s A042945), a sequence which is conjectured to be finite. See also APOLLONIAN GASKET ,BEND (CURVATURE ), COXETER’S LOXODROMIC SEQUENCE OF TANGENT CIRCLES ,HEXLET ,SPHERE ,TANGENT SPHERES References Borkovec, M.; de Paris, W.; and Peikert, R. "The Fractal Dimension of the Apollonian Sphere Packing." Fractals 2, 521 /C1/26, 1994. Sloane, N. J. A. Sequences A042944, A042945, A046159, and A046160 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Soddy, F. "The Bowl of Integers and the Hexlet." Nature 139,77/C1/9, 1937. Bowley Index The statistical INDEX PB /C131 2(PL /C27PP) ; where PLis LASPEYRES’ INDEX and PPis PAASCHE’S INDEX . See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962. Bowley Skewness Also known as QUARTILE SKEWNESS COEFFICIENT , (Q3 /C28 Q2) /C28 (Q2 /C28 Q1) Q3 /C28 Q1/C30Q1 /C28 2Q2 /C27 Q3 Q3 /C28 Q1; where the Qs denote the INTERQUARTILE RANGES .See also INTERQUARTILE RANGE ,SKEWNESS References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 102, 1962. Bowling Bowling is a game played by rolling a heavy ball down a long narrow track and attempting to knock down ten pins arranged in the form of a TRIANGLE with its vertex oriented towards the bowler. The number 10 is, in fact, the TRIANGULAR NUMBER T4/C304(4/C271)=2/C3010:/ Two "bowls" are allowed per "frame." If all the pins are knocked down in the two bowls, the score for that frame is the number of pins knocked down. If some ornone of the pins are knocked down on the first bowl, then all the pins knocked down on the second, it is called a "spare," and the number of points tallied is 10plus the number of pins knocked down on the bowl of the next frame. If all of the pins are knocked down on the first bowl, the number of points tallied is 10 plusthe number of pins knocked down on the next two bowls. Ten frames are bowled, unless the last frame is a strike or spare, in which case an additional bowl isawarded. The maximum number of points possible, correspond- ing to knocking down all 10 pins on every bowl, is 300. References Cooper, C. N. and Kennedy, R. E. "A Generating Function for the Distribution of the Scores of All Possible Bowling Games." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994. Cooper, C. N. and Kennedy, R. E. "Is the Mean Bowling Score Awful?" In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994. Box CUBOID Box Counting Dimension CAPACITY DIMENSION Box Fractal AFRACTAL also called the anticross-stitch curve which can be constructed using STRING REWRITING by creating a matrix with 3 times as many entries as the current matrix using the rules line 1 : ‘‘ +000 ‘‘ ++00; ‘‘ 000 ‘‘00 line 2 : ‘‘ +000 ‘‘ +00; ‘‘ 000 ‘‘00 line 3 : ‘‘ +000 ‘‘ ++00; ‘‘ 000 ‘‘00 Let Nn be the number of black boxes, Ln the length of a side of a white box, and Anthe fractional AREA of black boxes after the nth iteration. Nn /C305n (1) Ln /C30(1 3)n /C303/C28n (2) An /C30L2 n Nn /C30(5 9)n : (3) The CAPACITY DIMENSION is therefore dcap /C30/C28 lim n0/C12ln Nn ln Ln/C30/C28 lim n0/C12ln(5n) ln(3 /C28n) /C30ln 5 ln 3 /C301:464973521... : (4) See also CANTOR DUST,CROSS- STITCH CURVE ,SIER- PINSKI CARPET ,SIERPINSKI SIEVE References Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Box-and-Whisker Plot A HISTOGRAM -like method of displaying data invented by J. Tukey (1977). Draw a box with ends at the QUARTILES Q1and Q3 : Draw the MEDIAN as a horizontal line in the box. Extend the "whiskers" to the farthest points. For every point that is more than 3/2 times the INTERQUARTILE RANGE from the end of a box, draw a dot on the corresponding top or bottom of the whisker. If two dots have the same value, draw them side by side. References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, pp. 39 /C1/1, 1977.Boxcar Function The function Be(a; b) /C30c[H(x /C28a) /C28H(x /C28b)] which is equal to c for a 5x 5b and 0 otherwise. Here H(x) is the HEAVISIDE STEP FUNCTION . The special case B1(/C281=2 ; 1 =2) gives the unit RECTANGLE FUNC- TION . See also HEAVISIDE STEP FUNCTION ,R ECTANGLE FUNCTION References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 324, 1993. Boxcars A roll of two 6s (the highest roll possible) on a pair of 6-sided DICE. The probability of rolling boxcars in a single roll of two dice is 1/36, or 2.777...%. In order to have a 50% chance of obtaining at least one boxcars in n rolls of two dice, it must be true that 135 36 !n /C3012 ; (1) so solving for n gives n /C30ln 2 ln 36 /C28 ln 35 /C3024 :605... : (2) In fact, rolling two dice 25 times gives a probability of 1/C283536 ! 25 :0:505532 (3) that at least once boxcars will occur. See also DICE, DE ME´ RE´ ’S PROBLEM ,SNAKE EYES Box-Counting Dimension CAPACITY DIMENSION Box-Muller Transformation A transformation which transforms from a 2-D con- tinuous UNIFORM DISTRIBUTION to a 2-D GAUSSIAN BIVARIATE DISTRIBUTION (or COMPLEX GAUSSIAN DIS- TRIBUTION ). If x1and x2are uniformly and indepen- dently distributed between 0 and 1, then z1 and z2 as defined below have a GAUSSIAN DISTRIBUTION with MEAN m /C300 and VARIANCE s2 /C301: z1 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C282ln x1p cos(2 px2) (1) z2 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C282ln x1p sin(2px2) : (2) This can be verified by solving for x1 and x2 ; x1 /C30e/C28(z2 1/C27z22) =2 (3) x2 /C301 2ptan /C281z2 z1 ! : (4) Taking the JACOBIAN yields @(x1 ; x2) @(z1 ; z2) /C30@x1 @z1@x1 @z2 @x2 @z1@x2 @z2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2 /C30/C28 1ffiffiffiffiffiffi 2 pp e /C28z2 1 =2"# 1ffiffiffiffiffiffi 2pp e/C28z2 2 =2"# : (5) See also GAUSSIAN BIVARIATE DISTRIBUTION ,GAUS- SIAN DISTRIBUTION ,NORMAL DEVIATES References Box, G. E. P. and Muller, M. E. "A Note on the Generation of Random Normal Deviates." Ann. Math. Stat. 28, 610 /C1/611, 1958. Box-Packing Theorem The number of "prime" boxes is always finite, where a set of boxes is prime if it cannot be built up from one or more given configurations of boxes. See also CONWAY PUZZLE ,C UBOID , DE BRUIJN’S THEOREM ,K LARNER’S THEOREM ,S LOTHOUBER- GRAATSMA PUZZLE References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 74, 1976. Boy Surface ANONORIENTABLE SURFACE which is one of the three possible SURFACES obtained by sewing a M O¨BIUS STRIP to the edge of a DISK. The other two are the CROSS-CAP and R OMAN SURFACE . The Boy surface is a model of the PROJECTIVE PLANE without singularities and is a SEXTIC SURFACE . The Boy surface can be described using the general method for NONORIENTA-BLE SURFACES , but this was not known until the analytic equations were found by Ape ´ry (1986). Based on the fact that it had been proven impossible to describe the surface using quadratic polynomials,Hopf had conjectured that quartic polynomials were also insufficient (Pinkall 1986). Ape ´ry’s IMMERSION proved this conjecture wrong, giving the equations explicitly in terms of the standard form for a NONORIENTABLE SURFACE , f1(x;y;z)/C301 2[(2x2/C28y2/C28z2)(x2/C27y2/C27z2)/C272yz(y2/C28z2) /C27zx(x1/C28z2)/C27xy(y2/C28x2)] (1) f2(x;y;z)/C301 2ffiffiffi 3p [(y2/C28z2)(x2/C27y2/C27z2) /C27zx(z2/C28x2)/C27xy(y2/C28x2)] (2) f3(x;y;z)/C301 8(x/C27y/C27z) /C2[(x/C27y/C27z)3/C274(y/C28x)(z/C28y)(x/C28z)]:(3) Plugging in x/C30cosusinv (4) y/C30sinusinv (5) z/C30cosv (6) and letting u/C23[0;p] and v/C23[0;p] then gives the Boy surface, three views of which are shown above. TheR3parameterization can also be written as x/C30ffiffiffi 2p cos2vcos(2 u)/C27cosusin(2 v) 2/C28ffiffiffi 2p sin(3 u) sin(2 v)(7) y/C30ffiffiffi 2p cos2vcos(2 u)/C27cosusin(2 v) 2/C28ffiffiffi 2p sin(3 u) sin(2 v)(8) z/C302 cos2v 2/C28ffiffiffi2p sin(3 u) sin(2 v)(9) (Nordstrand) for u/C23[/C28p=2;p=2] and v/C23[0;p]: / Three views of the surface obtained using this parameterization are shown above. In fact, a HOMOTOPY (smooth deformation) between the R OMAN SURFACE and Boy surface is given by the equations x(u ; v) /C30ffiffiffi 2p cos(2 u) cos2 v /C27 cos u sin(2 v) 2 /C28 affiffiffi2p sin(3 u) sin(2 v)(10) y(u; v) /C30ffiffiffi2p sin(2 u) cos2 v /C28 sin u sin(2 v) 2 /C28 affiffiffi2p sin(3 u) sin(2 v)(11) z(u; v) /C30 3 cos2v 2 /C28 affiffiffi2p sin(3 u) sin(2 v)(12) as a varies from 0 to 1, where a /C300 corresponds to the R OMAN SURFACE and a /C301 to the Boy surface (Wang), shown below. In R4 ; the parametric representation is x0 /C303[(u2 /C27v2 /C27w2)(u2 /C27v2) /C28ffiffiffi 2p vw(3u2 /C28v2)] (13) x1 /C30ffiffiffi 2p (u2 /C27v2)(u2 /C28v2 /C27ffiffiffi2p uw) (14) x 2 /C30ffiffiffi 2p (u2 /C27v2)(2uv /C28ffiffiffi2p vw) (15) x 3 /C303(u2 /C27v2)2 ; (16) and the algebraic equation is 64(x0 /C28x3)3x3 3 /C2848(x0 /C28x3)2x23(3x21 /C273x22 /C272x23) /C2712(x0 /C28x3)x3[27(x21 /C27x22)2 /C2824x23(x21 /C27x22) /C2736ffiffiffi 2p x2x3(x2 2 /C283x21) /C27x43] /C27(9x21 /C279x22 /C282x23) /C2[/C2881(x21 /C27x22)2 /C2872x23(x21 /C27x22) /C27108ffiffiffi 2p x1x3(x2 1 /C283x22) /C274x43] /C300 (17) (Ape´ry 1986). Letting x0 /C301 (18) x1 /C30x (19) x2 /C30y (20) x3 /C30z (21) gives another version of the surface in R3 : / See also CROSS- CAP,IMMERSION ,M O¨ BIUS STRIP, NONORIENTABLE SURFACE ,REAL PROJECTIVE PLANE , ROMAN SURFACE ,SEXTIC SURFACEReferences Ape´ry, F. "The Boy Surface." Adv. Math. 61, 185 /C1266, 1986. Ape´ry, F. Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces. Braunschweig, Germany: Vieweg, 1987. Boy, W. "U¨ ber die Curvatura integra und die Topologie geschlossener Fla¨chen." Math. Ann 57, 151 /C1184, 1903. Brehm, U. "How to Build Minimal Polyhedral Models of the Boy Surface." Math. Intell. 12,51/C156, 1990. Carter, J. S. "On Generalizing Boy Surface--Constructing a Generator of the 3rd Stable Stem." Trans. Amer. Math. Soc. 298, 103 /C1122, 1986. Fischer, G. (Ed.). Plates 115 /C1120 in Mathematische Mod- elle/Mathematical Models, Bildband/Photograph Vo- lume. Braunschweig, Germany: Vieweg, pp. 110 /C1115, 1986. Hilbert, D. and Cohn-Vossen, S. §46 /C147 in Geometry and the Imagination. New York: Chelsea, 1999. Nordstrand, T. "Boy’s Surface." http://www.uib.no/people/ nfytn/boytxt.htm. Petit, J.-P. and Souriau, J. "Une repre´sentation analytique de la surface de Boy." C. R. Acad. Sci. Paris Se´r. 1 Math 293, 269 /C1272, 1981. Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, pp. 64 /C165, 1986. Stewart, I. Game, Set and Math. New York: Viking Penguin, 1991. Bp-Theorem If Op ?(G) /C301 and if x is a p-element of G, then Lp ?(CG(x) 5E(CG(x)); where Lp ? is the P-LAYER . Bra A(COVARIANT )1-VECTOR denoted c ½:h The bra is DUAL to the CONTRAVARIANT KET, denoted ½ ci: Taken together, the bra and KET form an ANGLE BRACKET (bra/C27ket /C30bracket). The bra is commonly encoun- tered in quantum mechanics. See also ANGLE BRACKET ,BRACKET PRODUCT ,COVAR- IANT VECTOR ,DIFFERENTIAL K-FORM,KET,ONE-FORM References Dirac, P. A. M. "Bra and Ket Vectors." §6i n Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, pp. 18 /C1/22, 1982. Brace One of the symbols fand gused in many different contexts in mathematics. Braces are used 1. To denote grouping of mathematical terms, usually as the outermost delimiter in a complex expression such as fa/C27b[c/C27d(e/C27f)]g;/ 2. To delineate a SET,a si n fa1;...;ang;/ 3. Using a left bracket only, to denote differentcases for an expression, such as p(n) /C301 for n even 0 for n odd;/C26 4. Using a single horizontal underbrace, to indicate the number of items in a list with not all elements shown explicitly, as in 1 ; 1 ; ...; 1|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} n:/ 5. As an alternate notation to the FRACTIONAL PART function, fxg/C30frac x:/ See also ANGLE BRACKET ,P ARENTHESIS ,S QUARE BRACKET References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 273, 1997. Braced Square The braced square problem asks: given a hinged SQUARE composed of four equal rods (indicated by the thick lines above), how many more hinged rods must be added in the same plane (with no two rods crossing) so that the original square is rigid in the plane. The best solution known, illustrated in the left figure above, uses a total of 27 rods, where A, B, and C are COLLINEAR . If rods are allowed to cross, the best known solution, discovered by E. Friedman in Jan. 2000, requires 21 rods, as illustrated in the right figure above. Friedman has also considered the minimum number of rods needed to construct RIGID regular n-gons (with overlapping permitted). The best known solutions for n/C303, 4, ... are 3, 21, 69, 11, 45, 99, 51, .... See also HINGED TESSELLATION ,R IGID GRAPH , SQUARE References Friedman, E. "Problem of the Month (January 2000)." http:// www.stetson.edu/~efriedma/mathmagic/0100.html. Gardner, M. "The Rigid Square." §6.1 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 48 /C1/49 and 54 /C1/55, 1984. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 19, 1991. Brachistochrone Problem Find the shape of the CURVE down which a bead sliding from rest and ACCELERATED by gravity will slip (without friction ) from one point to another in the least time. The term derives from the Greekbraxist o&(brachistos ) "the shortest" and xrono& (chronos ) "time, delay." The brachistochrone problem was one of the earliest problems posed in the CALCULUS OF VARIATIONS . The solution, a segment of a CYCLOID , was found by Leibniz, L’Hospital, Newton, and the two Bernoullis.Johann Bernoulli solved the problem using the analogous one of considering the path of light re-fracted by transparent layers of varying density (Mach 1893, Gardner 1984, Courant and Robbins 1996). Note that bead may actually travel uphillalong the cycloid for a distance, but the path is nonetheless faster than a straight line or any other line. The time to travel from a point P 1to another point P2 is given by the INTEGRAL t12/C30g2 1ds v; (1) The VELOCITY at any point is given by a simple application of energy conservation equating kinetic energy to gravitational potential energy, 1 2mv2/C30mgy ; (2) so v/C30ffiffiffiffiffiffiffiffi 2gyp : (3) Plugging this into (1) then gives t12/C30g2 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2p ffiffiffiffiffiffiffiffi2gyp dx/C30g2 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2 2gys dx: (4) The function to be varied is thus f/C30(1/C27y?2)1=2(2gy)/C281=2; (5) To proceed, one would normally have to apply the full-blown E ULER- LAGRANGE DIFFERENTIAL EQUATION @f @y/C28d dx@f @y? ! /C300: (6) However, the function f(y;y?;x) is particularly nice since xdoes not appear explicitly. Therefore, @f=@x/C30 0;and we can immediately use the B ELTRAMI IDEN- TITY f/C28y?@f @y?/C30C: (7) Computing @f @y?/C30y?(1/C27y?2)/C281=2(2gy)/C281=2; (8) subtracting y?(@f=@y?) from f, and simplifying then gives 1ffiffiffiffiffiffiffiffiffiffi2 gypffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 y?2p /C30C : (9) Squaring both sides and rearranging slightly results in 1 /C27dy dx !22 435y /C30 1 2gC2 /C30k2 ; (10) where the square of the old constant C has been expressed in terms of a new (POSITIVE ) constant k2 : This equation is solved by the PARAMETRIC EQUATIONS x /C301 2k2( u /C28sin u) (11) y /C301 2k2(1 /C28cos u) ; (12) which are–lo and behold–the equations of a CYCLOID . If kinetic friction is included, the problem can also be solved analytically, although the solution is signifi- cantly messier. In that case, terms corresponding to the normal component of weight and the normal component of the ACCELERATION (present because of path CURVATURE ) must be included. Including both terms requires a constrained variational technique (Ashby et al. 1975), but including the normal compo- nent of weight only gives an elementary solution. The TANGENT and NORMAL VECTORS are T /C30dx dsˆx /C27dy dsˆy (13) N /C30/C28dy dsˆx /C27dx dsˆy; (14) gravity and friction are then Fgravity /C30mg ˆy (15) Ffriction /C30/C28m(Fgravity˙N)T /C30/C28mmgdx dsT ; (16) and the components along the curve are Fgravity˙T /C30mgdy ds (17) Ffriction˙T /C30/C28mmgdx ds; (18) so Newton’s Second Law gives mdv dt /C30mgdy ds /C28 mmgdx ds : (19) But dv dt /C30vdv ds /C301 2d ds(v2) (20)1 2v2 /C30g(y /C28 mx) (21) v /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2g(y /C28 mx) ;p (22) so t /C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 (y?)2 2g(y /C28 mx)s dx : (23) Using the EULER- LAGRANGE DIFFERENTIAL EQUATION gives [1 /C27y?2](1 /C27 my?) /C272(y /C28 mx)yƒ/C300: (24) This can be reduced to 1 /C27 (y?)2 (1 /C27 my?)2 /C30C y /C28 mx : (25) Now letting y?/C30cot(1 2 u) ; (26) the solution is x /C301 2k2[(u /C28sin u) /C27 m(1 /C28cos u)] (27) y /C301 2k2[(1 /C28cos u) /C27 m(u /C28sin u)]: (28) See also CALCULUS OF VARIATIONS ,CYCLOID ,TAUTO- CHRONE PROBLEM References Ashby, N.; Brittin, W. E.; Love, W. F.; and Wyss, W. "Brachistochrone with Coulomb Friction." Amer. J. Phys. 43, 902 /C1/905, 1975. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, 1996. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 130 /C1/131, 1984. Haws, L. and Kiser, T. "Exploring the Brachistochrone Problem." Amer. Math. Monthly 102, 328 /C1/336, 1995. Mach, E. The Science of Mechanics. Chicago, IL: Open Court, 1893. Phillips, J. P. "Brachistochrone, Tautochrone, Cycloid--Ap- ple of Discord." Math. Teacher 60, 506 /C1/508, 1967. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 148 /C1/149, 1999. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 60 /C1/66 and 385 /C1/389, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 46, 1991. Bracket Mathematicians often use the term "bracket" to mean "COMMUTATOR ," which is denoted using SQUARE BRACKETS . See also ANGLE BRACKET ,B RA,B RACE ,B RACKET POLYNOMIAL ,BRACKET PRODUCT ,IVERSON BRACKET , KET,LAGRANGE BRACKET ,POISSON BRACKET ,SQUARE BRACKET Bracket Polynomial A one-variable KNOT POLYNOMIAL related to the JONES POLYNOMIAL . The bracket polynomial, how- ever, is not a topological invariant, since it is changed by type I REIDEMEISTER MOVES . However, the SPAN of the bracket polynomial is a knot invariant. The bracket polynomial is occasionally given the grand- iose name REGULAR ISOTOPY INVARIANT . It is defined by Lhi(A; B; d) /C13X sL½ shi d½½ s½½; (1) where A and B are the "splitting variables," s runs through all "states" of L obtained by SPLITTING the LINK , L½ shi is the product of "splitting labels" corre- sponding to s; and ½½ s½½/C13NL /C281; (2) where NL is the number of loops in s: Letting B /C30A/C281 (3) d /C30/C28 A2 /C28A/C282 (4) gives a KNOT POLYNOMIAL which is invariant under REGULAR ISOTOPY , and normalizing gives the KAUFF- MAN POLYNOMIAL X which is invariant under AMBI- ENT ISOTOPY . The bracket POLYNOMIAL of the UNKNOT is 1. The bracket POLYNOMIAL of the MIRROR IMAGE K /C31 is the same as for K but with A replaced by A/C281 : In terms of the one-variable KAUFFMAN POLYNOMIAL X, the two-variable KAUFFMAN POLYNOMIAL F and the JONES POLYNOMIAL V, X(A) /C30 (/C28A3) /C28w(L) Lhi; (5) Lhi(A) /C30F(/C28A3 ; A /C27A/C281) (6) Lhi(A) /C30V(A/C284) ; (7) where w(L) is the WRITHE of L. See also JONES POLYNOMIAL ,S QUARE BRACKET POLYNOMIAL References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 148 /C1/155, 1994. Kauffman, L. "New Invariants in the Theory of Knots." Amer. Math. Monthly 95, 195 /C1/242, 1988. Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, pp. 26 /C1/29, 1991. Weisstein, E. W. "Knots and Links." MATHEMATICA NOTE- BOOK KNOTS.M . Bracket Product L2-INNER PRODUCTBracketing Take x itself to be a bracketing, then recursively define a bracketing as a sequence B /C30(B1 ; ...; Bk) where k ]2 and each Bi is a bracketing. A bracketing can be REPRESENTED AS a parenthesized string of xs, with parentheses removed from any single letter x for clarity of notation (Stanley 1997). Bracketings built up of binary operations only are called BINARY BRACKETINGS . For example, four letters have 11 possible bracketings: xxxx (xx)xx x(xx)xx x (xx) (xxx)xx (xxx)( ( xx)x)x (x(xx))x (xx)(xx) x((xx)x) x(x(xx)); the last five of which are binary. The number of bracketings on n letters is given by the GENERATING FUNCTION 1 4(1 /C27x /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C286x /C27x2p ) /C30x /C27x2 /C273x3 /C2711x4 /C2745x5 (Schro ¨der 1870, Stanley 1997) and the RECURRENCE RELATION sn /C303(2n /C28 3)sn/C281 /C28 (n /C28 3)sn/C282 n (Sloane), giving the sequence for sn as 1, 1, 3, 11, 45, 197, 903, ... (Sloane’s A001003). The numbers are also given by sn /C30X i1 /C27.../C27ik /C30ns(i1) /C1/C1/C1s(ik) for n ]2 (Stanley 1997). The first PLUTARCH NUMBER 103,049 is equal to s10 (Stanley 1997), suggesting that Plutarch’s problem of ten compound propositions is equivalent to the number of bracketings. In addition, Plutarch’s secondnumber 310,954 is given by ( s 10/C27s11)=2/C30310;954 (Habsieger et al. 1998). See also BINARY BRACKETING ,PLUTARCH NUMBERS References Comtet, L. "Bracketing Problems." §1.15 in Advanced Com- binatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 52 /C1/57, 1974. Habsieger, L.; Kazarian, M.; and Lando, S. "On the Second Number of Plutarch." Amer. Math. Monthly 105, 446, 1998. Schro ¨der, E. "Vier combinatorische Probleme." Z. Math. Physik 15, 361/C1/376, 1870. Sloane, N. J. A. Sequences A001003/M2898 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Stanley, R. P. "Hipparchus, Plutarch, Schro ¨der, and Hough." Amer. Math. Monthly 104, 344/C1 /350, 1997. Bradley’s Theorem Let S( a; b; m; z) /C13mX/C12 j/C300G(m /C27 j(z /C27 1)) G( b /C27 1 /C27 jz) G(m /C27 jz /C27 1)G(a /C27 b /C27 1 /C27 j(z /C27 1))( a)j j!; where ( a)jis a POCHHAMMER SYMBOL , and let a be a NEGATIVE INTEGER . Then S( a; b; m; z) /C30G(b /C27 1 /C28 m) G(a /C27 b /C27 1 /C28 m) ; where G(z) is the GAMMA FUNCTION . References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 346 /C1/348, 1994. Bradley, D. "On a Claim by Ramanujan about Certain Hypergeometric Series." Proc. Amer. Math. Soc. 121, 1145 /C1/1149, 1994. Brahmagupta Identity Let b /C13det B /C30x2 /C28ty2 ; where B is the BRAHMAGUPTA MATRIX , then det[B(x1 ; y1)B(x2 ; y2)] /C30det[B(x1 ; y1)] det[B(x2 ; y2)] /C30 b1 b2 : References Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. 34,30/C1/39, 1996. Brahmagupta Matrix B(x; y) /C30xy 9ty 9x/C20/C2P : It satisfies B(x1 ; y1)B(x2 ; y2) /C30B(x1x2 9ty1y2 ; x1y2 9y1x2) : Powers of the matrix are defined by Bn /C30xy ty x/C20/C2Pn /C30xnyn tynxn/C20/C2P /C13Bn: Thexnandynare called B RAHMAGUPTA POLYNOMIALS . The Brahmagupta matrices can be extended to NEGATIVE INTEGERS B/C28n/C30xy ty x/C20/C2P/C28n /C30x/C28ny/C28n ty/C28nx/C28n/C20/C2P /C13B/C28n: See also BRAHMAGUPTA IDENTITYReferences Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. 34,3 0/C1/39, 1996. Brahmagupta Polynomial One of the POLYNOMIALS obtained by taking POWERS of the B RAHMAGUPTA MATRIX . They satisfy the RE- CURRENCE RELATION xn/C271/C30xxn/C27tyyn (1) yn/C271/C30xyn/C27yxn: (2) A list of many others is given by Suryanarayan (1996). Explicitly, xn/C30xn/C27tn 2/CP8/CP9 xn/C282y2/C27t2n 4/CP8/CP9 xn/C284y4/C27... ( 3 ) yn/C30nxn/C281y/C27tn 3/CP8/CP9 xn/C283y3/C27t2n 5/CP8/CP9 xn/C285y5/C27... (4) The Brahmagupta POLYNOMIALS satisfy @xn @x/C30@yn @y/C30nxn/C281 (5) @xn @y/C30t@yn @y/C30ntyn/C281: (6) The first few POLYNOMIALS are x0/C300 x1/C30x x2/C30x2/C27ty2 x3/C30x3/C273txy2 x4/C30x4/C276tx2y2/C27t2y4 and y0/C300 y1/C30y y2/C302xy y3/C303x2y/C27ty3 y4/C304x3y/C274txy3: Taking x/C30y/C301 and t/C302 gives ynequal to the P ELL NUMBERS and xnequal to half the Pell-Lucas num- bers. The Brahmagupta POLYNOMIALS are related to the M ORGAN- VOYCE POLYNOMIALS , but the relation- ship given by Suryanarayan (1996) is incorrect. References Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. 34,3 0/C1/39, 1996. Brahmagupta’s Formula For a QUADRILATERAL with sides of length a, b, c, and d, the AREA K is given by K /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s /C28a)(s /C28b)(s /C28c)(s /C28d) /C28abcd cos2[1 2(A /C27B)];q (1) where s /C131 2(a /C27b /C27c /C27d) (2) is the SEMIPERIMETER , A is the ANGLE between a and d, and B is the ANGLE between b and c. For a CYCLIC QUADRILATERAL (i.e., a QUADRILATERAL inscribed in a CIRCLE ), A /C27B /C30 p; so K /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s /C28a)(s /C28b)(s /C28c)(s /C28d)p (3) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (bc /C27 ad)(ac /C27 bd)(ab /C27 cd)p 4R ; (4) where R is the RADIUS of the CIRCUMCIRCLE . If the QUADRILATERAL is INSCRIBED in one CIRCLE and CIRCUMSCRIBED on another, then the AREA FORMULA simplifies to K /C30ffiffiffiffiffiffiffiffiffiffiffi abcdp : (5) See also BRETSCHNEIDER’S FORMULA ,H ERON’S FOR- MULA ,QUADRILATERAL References Brown, K. S. "Heron’s FOrmula and Brahmagupta’s Gen- eralization." http://www.seanet.com/~ksbrown/kmath19 6.htm. Coxeter, H. S. M. and Greitzer, S. L. "Cyclic Quadrangles; Brahmagupta’s Formula." §3.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56 /C1/60, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 81 /C1/82, 1929. Brahmagupta’s Problem Solve the PELL EQUATION x2 /C2892y2 /C301 in INTEGERS . The smallest solution is x /C301151, y /C30120. See also DIOPHANTINE EQUATION ,PELL EQUATIONBrahmagupta’s Theorem In a CYCLIC QUADRILATERAL ABCD having perpendi- cular diagonals AC /C222BD; the perpendiculars to the sides through point T of intersection of the diagonals (the ANTICENTER ) always bisects the opposite side (so MAB ; MBC ; MCD ; and MDAare the MIDPOINTS of the corresponding sides of the QUADRILATERAL ). See also ANTICENTER ,CYCLIC QUADRILATERAL ,M ID- POINT References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 37, 1995. Braid An intertwining of strings attached to top and bottom "bars" such that each string never "turns back up." In other words, the path of each string in a braid could be traced out by a falling object if acted upon only by gravity and horizontal forces. See also BRAID GROUP References Christy, J. "Braids." http://www.mathsource.com/cgi-bin/ msitem?0202 /C1/228. Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Braid Group Also called A RTIN BRAID GROUPS . Consider nstrings, each oriented vertically from a lower to an upper "bar." If this is the least number of strings needed tomake a closed braid representation of a LINK ,nis called the BRAID INDEX . Now enumerate the possible braids in a group, denoted Bn:A general n-braid is constructed by iteratively applying the si(/i/C30 1;...;n/C281) operator, which switches the lower endpoints of the ith and ( i/C271)/th strings–keeping the upper endpoints fixed–with the ( i/C271)/th string brought above theith string. If the ( i/C271)/th string passes below theith string, it is denoted s/C281 i:/ Topological equivalence for different representations of a BRAID WORD Pi siand Pi s?iis guaranteed by the conditions si sj /C30 sj si for ½i /C28j½]2 si si/C271 si /C30 si/C271 si si/C271for all i/C26 as first proved by E. Artin. Any n-braid is expressed as a BRAID WORD , e.g., s1 s2 s3 s /C281 2s1is a BRAID WORD for the braid group B3 : When the opposite ends of the braids are connected by nonintersecting lines, KNOTS are formed which are identified by their braid group and BRAID WORD . The BURAU REPRESENTATION gives a matrix representation of the braid groups. References Birman, J. S. "Braids, Links, and the Mapping Class Groups." Ann. Math. Studies , No. 82. Princeton, NJ: Princeton University Press, 1976. Birman, J. S. "Recent Developments in Braid and Link Theory." Math. Intell. 13,52/C1/60, 1991. Christy, J. "Braids." http://www.mathsource.com/cgi-bin/ msitem?0202 /C1/228. Jones, V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335 /C1/388, 1987. Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Weisstein, E. W. "Knots and Links." MATHEMATICA NOTE- BOOK KNOTS.M . Braid Index The least number of strings needed to make a closed braid representation of a LINK . The braid index is equal to the least number of SEIFERT CIRCLES in any projection of a KNOT (Yamada 1987). Also, for a nonsplittable LINK with CROSSING NUMBER c(L) and braid index i(L); c(L) ]2[i(L) /C281] (Ohyama 1993). Let E be the largest and e the smallest POWER of l in the HOMFLY POLYNOMIAL of an oriented LINK , and i be the braid index. Then the MORTON-FRANKS-WILLIAMS INEQUALITY holds, i ]1 2(E /C28e) /C271 (Franks and Williams 1987). The inequality is sharp for all PRIME KNOTS up to 10 crossings with the exceptions of 09 /C1/042, 09 /C1/049, 10 /C1/132, 10 /C1/150, and 10 /C1/156. References Franks, J. and Williams, R. F. "Braids and the Jones Polynomial." Trans. Amer. Math. Soc. 303,97/C1/108, 1987.Jones, V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335 /C1/388, 1987. Ohyama, Y. "On the Minimal Crossing Number and the Brad Index of Links." Canad. J. Math. 45, 117 /C1/131, 1993. Yamada, S. "The Minimal Number of Seifert Circles Equals the Braid Index of a Link." Invent. Math. 89, 347 /C1/356, 1987. Braid Word Any n-braid is expressed as a braid word, e.g., s1 s2 s3 s /C281 2s1 is a braid word for the BRAID GROUP B3 : By ALEXANDER’S THEOREM , any LINK is representable by a closed braid, but there is no general procedure for reducing a braid word to its simplest form. However, MARKOV’S THEOREM gives a procedure for identifying different braid words which represent the same LINK . Let b/C27 be the sum of POSITIVE exponents, and b/C28 the sum of NEGATIVE exponents in the BRAID GROUP Bn : If b/C27/C283b /C28]n; then the closed braid b is not AMPHICHIRAL (Jones 1985). See also BRAID GROUP References Jones, V. F. R. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103 /C1/111, 1985. Jones, V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335 /C1/388, 1987. Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Braikenridge-Maclaurin Construction Let An ; B2 ; C1 ; A2 ; and B1 be five points determining a CONIC . Then the CONIC is the LOCUS of the point C2/C30A1(L /C215C1A2)/C215B1(L /C215C1B2); where Lis a line through the point A1B2/C215B1A2:/ See also BRAIKENRIDGE- MACLAURIN THEOREM ,CONIC SECTION Braikenridge-Maclaurin Theorem The converse of PASCAL’S THEOREM , which states that if the three pairs of opposite sides of (an irregular) HEXAGON meet at three COLLINEAR points, then the six vertices lie on a conic, which may degenerate into a pair of lines (Coxeter and Greitzer 1967, p. 76). See also BRAIKENRIDGE- MACLAURIN CONSTRUCTION , CONIC SECTION ,PASCAL’S THEOREM References Coxeter, H. S. M. Projective Geometry, 2nd ed. New York: Springer-Verlag, p. 85, 1987. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 76, 1967. Branch A branch at a point u in a TREE is a maximal SUBTREE containing u as an ENDPOINT (Harary 1994, p. 35). See also FORK,LEAF (TREE), LIMB,TREE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Lu, T. "The Enumeration of Trees with and without Given Limbs." Disc. Math. 154, 153 /C1/165, 1996. Schwenk, A. "Almost All Trees are Cospectral." In New Directions in the Theory of Graphs (Ed. F. Harary). New York: Academic Press, pp. 275 /C1/307, 1973. Branch Cut A line in the COMPLEX PLANE across which a MULTI-VALUED FUNCTION is discontinuous. Some functions have a relatively simple branch cut structure, but branch cuts for some functions are extremely compli- cated. The illustrations above show the single branch cut present in the definition of the square root function in the complex plane. In general, branch cuts are not unique, but are chosen by convention to give simple analytic properties. An alternative to branch cuts is the use of RIEMANN SURFACES . function branch cut(s) /cos/C281 z// (/C28/C12;/C281) and (1;/C12)/ /cosh/C281 // ( /C28/C12; 1)/ /cot/C281 z// (/C28i ; i)/ /coth/C281 // [ /C281 ; 1]/ /csc/C281 z// (/C281; 1)/ /csch/C281 // ( /C28i ; i)/ /ln z// ( /C28/C12; 0]/ /sec/C281 z// (/C281; 1)/ /sech/C281 // ( /C12; 0] and (1;/C12)/ /sin/C281 z// (/C28/C12;/C281) and (1;/C12)/ /sinh/C281 // (/C28i/C12;/C28i) and ( i;i/C12)/ /ffiffiffizp // (/C28/C12;0)/ /tan/C281z// (/C28i/C12;/C28i) and ( i;i/C12)/ /tanh/C281 // (/C28/C12;/C281] and [1 ;/C12)/ /zn;nQZ//(/C28/C12;0) forR[n]50; (/C28/C12;0] forR[n]>0/ See also BRANCH POINT ,CUT,M ULTIVALUED FUNC- TION ,RIEMANN SURFACE References Kahan, W. "Branch Cuts for Complex Elementary Func- tions, or Much Ado About Nothing’s Sign Bit." In The State of the Art in Numerical Analysis: Proceedings of the Joint IMA/SIAM Conference on the State of the Art in Numer-ical Analysis Held at the UN (Ed. A. Iserles and M. J. D. Powell). New York: Clarendon Press, pp. 165 /C1 / 211, 1987. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 399 /C1/401, 1953. Branch Line BRANCH CUT Branch Point An argument at which identical points in the COM- PLEX PLANE are mapped to different points. For example, consider f(z)/C30za: Then f(e0i) /C30f(1) /C301 ; but f(e2 pi) /C30e2pia ; despite the fact that ei0 /C30e2 pi : PINCH POINTS are also called branch points. See also BRANCH CUT,PINCH POINT References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 397 /C1/399, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 391 /C1/392 and 399 /C1/401, 1953. Brauer Chain A Brauer chain is an ADDITION CHAIN in which each member uses the previous member as a summand. A number n for which a shortest chain exists which is a Brauer chain is called a BRAUER NUMBER . See also ADDITION CHAIN ,BRAUER NUMBER ,HANSEN CHAIN References Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111 /C1/113, 1994. Brauer Group The GROUP of classes of finite dimensional central simple ALGEBRAS over k with respect to a certain equivalence. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 479, 1988. Brauer Number A number n for which a shortest chain exists which is aBRAUER CHAIN is called a Brauer number. There are infinitely many non-Brauer numbers. See also BRAUER CHAIN ,HANSEN NUMBER References Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111 /C1/113, 1994. Brauer’s Theorem If, in the GERSGORIN CIRCLE THEOREM for a given m, ½ajj /C28amm ½>Lj /C27Lm for all j "m; then exactly one EIGENVALUE of A lies in the DISK Gm :/ References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000.Brauer-Severi Variety An ALGEBRAIC VARIETY over a FIELD K that becomes ISOMORPHIC to a PROJECTIVE SPACE . References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 480 /C1/481, 1988. Braun’s Conjecture Let B /C30fb1 ; b2 ; ...g be an INFINITE ABELIAN SEMI- GROUP with linear order b1 Bb2 B... such that b1is the unit element and a Bb IMPLIES ac Bbc for a ; b; c /C23 B : Define a MO¨ BIUS FUNCTION m on B by m(b1) /C301 and X bd ½bnm(bd) /C300 for n /C302, 3, .... Further suppose that m(bn) /C30 m(n) (the true MO¨ BIUS FUNCTION ) for all n ]1: Then Braun’s conjecture states that bmn /C30bm bn for all m; n ]1:/ See also MO¨ BIUS PROBLEM References Flath, A. and Zulauf, A. "Does the Mo¨bius Function Determine Multiplicative Arithmetic?" Amer. Math. Monthly 102, 354 /C1/256, 1995. Breadth-First Traversal A search algorithm of a GRAPH which explores all nodes adjacent to the current node before moving on. For cyclic graphs, care must be taken to make sure that no nodes are repeated. When properly imple- mented, all nodes in a given connected component are explored. See also DEPTH- FIRST TRAVERSAL References Skiena, S. "Breadth-First and Depth-First Search." §3.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 95 /C1/97, 1990. Breeder A pair of POSITIVE INTEGERS (a1;a2) such that the equations a1/C27a2x/C30s(a1)/C30s(a2)(x/C271) have a POSITIVE INTEGER solution x, where s(n) is the DIVISOR FUNCTION .I fxisPRIME , then ( a1;a2x)i sa n AMICABLE PAIR (te Riele 1986). ( a1;a2) is a "special" breeder if a1 /C30au a2 /C30a; where a and u are RELATIVELY PRIME ,(a ; u) /C301: If regular amicable pairs of type (i ; 1) with i ]2 are OF THE FORM (au, ap) with p PRIME , then (au, a) are special breeders (te Riele 1986). See also AMICABLE PAIR References te Riele, H. J. J. "Computation of All the Amicable Pairs Below 1010." Math. Comput. 47, 361 /C1/368 and S9-S35, 1986. Brelaz’s Heuristic Algorithm An ALGORITHM which can be used to find a good, but not necessarily minimal, EDGE or VERTEX COLORING for a GRAPH . However, the algorithm does minimally color COMPLETE K-PARTITE GRAPH . See also CHROMATIC NUMBER ,E DGE COLORING , VERTEX COLORING References Brelaz, D. "New Methods to Color the Vertices of a Graph." Comm. ACM 22, 251/C1/256, 1979. Skiena, S. "Finding a Vertex Coloring." §5.5.3 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 214 /C1/215, 1990. Brent’s Factorization Method A modification of the P OLLARD RHO FACTORIZATION METHOD which uses xi/C271/C30x2 i/C28c(mod n): References Brent, R. "An Improved Monte Carlo Factorization Algo- rithm." Nordisk Tidskrift for Informationsbehandlung (BIT) 20, 176/C1/184, 1980. Brent’s Method AROOT -finding ALGORITHM which combines root bracketing, bisection, and INVERSE QUADRATIC INTER- POLATION . It is sometimes known as the VAN WIJN- GAARDEN-DEKER-BRENT METHOD . Brent’s method uses a L AGRANGE INTERPOLATING POLYNOMIAL of degree 2. Brent (1973) claims that this method will always converge as long as the values of the function are computable within a given region containing a ROOT . Given three points x1;x2; andx3;Brent’s method fits xas a quadratic function ofy, then uses the interpolation formulax/C30[y/C28f(x1)][y/C28f(x2)]x3 [f(x3)/C28f(x1)][f(x3)/C28f(x2)] /C27[y/C28f(x2)][y/C28f(x3)]x1 [f(x1)/C28f(x2)][f(x1)/C28f(x3)] /C27[y/C28f(x3)][y/C28f(x1)]x2 [f(x2)/C28f(x3)][f(x2)/C28f(x1)]: (1) Subsequent root estimates are obtained by setting y/C300, giving x/C30x2/C27P Q; (2) where P/C30S[R(R/C28T)(x3/C28x2)/C28(1/C28R)(x2/C28x1)] (3) Q/C30(T/C281)(R/C281)(S/C281) (4) with R/C13f(x2) f(x3)(5) S/C13f(x2) f(x1)(6) T/C13f(x1) f(x3)(7) (Press et al. 1992). References Brent, R. P. Ch. 3 /C1/4i n Algorithms for Minimization With- out Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973. Forsythe, G. E.; Malcolm, M. A.; and Moler, C. B. §7.2 in Computer Methods for Mathematical Computations. Eng- lewood Cliffs, NJ: Prentice-Hall, 1977. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Van Wijngaarden-Dekker-Brent Method." §9.3 inNumerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 352 /C1/355, 1992. Brent-Salamin Formula A formula which uses the ARITHMETIC-GEOMETRIC MEAN to compute PI. It has quadratic convergence and is also called the G AUSS-SALAMIN FORMULA and SALAMIN FORMULA . Let an/C271/C301 2(an/C27bn) (1) bn/C271/C30ffiffiffiffiffiffiffiffiffiffi anbnp (2) cn/C271/C301 2(an/C28bn) (3) dn/C13a2 n/C28b2n; (4) and define the initial conditions to be a0/C301;b0/C30 1=ffiffiffi 2p :Then iterating apandbngives the ARITHMETIC- GEOMETRIC MEAN , and p is given by p /C304[M(1; 2/C281=2)]2 1 /C28P/C12 j/C3012j/C271dj(5) /C304[M(1; 2/C281 =2)]2 1 /C28P/C12 j/C3012j/C271c2 j: (6) King (1924) showed that this formula and the LEGENDRE RELATION are equivalent and that either may be derived from the other. See also ARITHMETIC- GEOMETRIC MEAN,PI References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 48 /C1/51, 1987. Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148 /C1/163, 1988. King, L. V. On the Direct Numerical Calculation of Elliptic Functions and Integrals. Cambridge, England: Cambridge University Press, 1924. Lord, N. J. "Recent Calculations of p : The Gauss-Salamin Algorithm." Math. Gaz. 76, 231 /C1/242, 1992. Salamin, E. "Computation of p Using Arithmetic-Geometric Mean." Math. Comput. 30, 565 /C1/570, 1976. Bretschneider’s Formula Given a general QUADRILATERAL with sides of lengths a, b, c, and d (Beyer 1987), the AREA is given by Aquadrilateral /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p2q2 /C28(b2 /C27d2 /C28a2 /C28c2)2q ; where p and q are the diagonal lengths. See also BRAHMAGUPTA’S FORMULA ,HERON’S FORMU- LA References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Brianchon Point The point of CONCURRENCE of the joins of the VERTICES of a TRIANGLE and the points of contact of a CONIC SECTION INSCRIBED in the TRIANGLE .ACONIC INSCRIBED in a TRIANGLE has an equation OF THE FORM f u /C27g v /C27h w /C300 ; so its Brianchon point has TRILINEAR COORDINATES (1=f ; 1 =g; 1=h): For KIEPERT’S PARABOLA , the Bran- chion point has TRIANGLE CENTER FUNCTION a /C301 a(b2 /C28 c2) ; which is the STEINER POINT .See also HEPTAGON THEOREM ,KIEPERT’S PARABOLA , STEINER POINTS References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Heptagon Theorem." §2.1 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8 /C1/11, 1974. Brianchon’s Theorem The DUAL of PASCAL’S THEOREM (Casey 1888, p. 146). It states that, given a HEXAGON CIRCUMSCRIBED on a CONIC SECTION , the lines joining opposite VERTICES (DIAGONALS ) meet in a single point. See also DUALITY PRINCIPLE ,PASCAL’S THEOREM References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 146 /C1/147, 1888. Coxeter, H. S. M. and Greitzer, S. L. "Brianchon’s Theo- rem." §3.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 77 /C1/79, 1967. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "Extensions of Pascal’s and Brianchon’s Theorems." Ch. 2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8 /C1/30, 1974. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 261, 1930. Johnson, R. A. §387 in Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 237, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 110, 1990. Smogorzhevskii, A. S. The Ruler in Geometrical Construc- tions. New York: Blaisdell, pp. 33 /C1/34, 1961. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 20 /C1/21, 1991. Brick A RECTANGULAR PARALLELEPIPED . See also CANONICAL BRICK,EULER BRICK,HARMONIC BRICK,RECTANGULAR PARALLELEPIPED Bride’s Chair One name for the figure used by Euclid to prove the PYTHAGOREAN THEOREM . See also PEACOCK’S TAIL,W INDMILL References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 203, 1991. Bridge The bridges of a CONNECTED GRAPH are the EDGES whose removal disconnects the GRAPH (Chartrand 1985, p. 45; Skiena 1990, p. 177). More generally, a bridge is an edge of a GRAPH G whose removal increases the number of components of G (Harary 1994, p. 26). An edge of a CONNECTED GRAPH is a bridge IFF is does not lie on any cycle. The bridges of a graph can be found using Bridges [g] in the Math- ematica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ ). Every edge of a TREE is a bridge. A CUBIC GRAPH contains a bridge IFF it contains an ARTICULATION VERTEX (Skiena 1990, p. 177). See also ARTICULATION VERTEX ,BLOCK References Chartrand, G. "Cut-Vertices and Bridges." §2.4 in Introduc- tory Graph Theory. New York: Dover, pp. 45 /C1/49, 1985. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 171 and 177, 1990. Bridge Card Game Bridge is a CARD game played with a normal deck of 52 cards. The number of possible distinct 13-card hands is N /C3052 13/CP8/CP9 /C30635;013;559;600: where (n k)isa BINOMIAL COEFFICIENT . While the chances of being dealt a hand of 13 CARDS (out of 52) of the same suit are4 52 13/CP8/CP9/C301 158;753; 389; 900 ; the chance that one of four players will receive a hand of a single suit is 1 39 ;688;347;497 : There are special names for specific types of hands. A ten, jack, queen, king, or ace is called an "honor." Getting the three top cards (ace, king, and queen) of three suits and the ace, king, and queen, and jack of the remaining suit is called 13 top honors. Getting all cards of the same suit is called a 13-card suit. Getting 12 cards of same suit with ace high and the 13th card not an ace is called 2-card suit, ace high. Getting no honors is called a Yarborough. The probabilities of being dealt 13-card bridge hands of a given type are given below. As usual, for a hand with probability P, the ODDS against being dealt it are (1=P) /C281:1 :/ Hand Exact ProbabilityProbability ODDS 13 top honors/4 N /C30/ /1 158 ; 753 ; 389 ; 900//6:30 /C2910 /C2812/ 158,753,389,899:1 13-cardsuit/4 N /C30/ /1 158 ; 753 ; 389 ; 900//6:30 /C2910 /C2812/ 158,753,389,899:1 12-cardsuit, ace high/4 /C21512 /C21536 N/C30/ /4 1;469;938;705//2:72/C2910/C289/ 367,484,697.8:1 Yarborough /32 13/C0/CP N/C305;394 9;860;459//5:47/C2910/C284/ 1,827.0:1 four aces /48 9/C0/CP N/C3011 4;165// 2:64/C2910/C283/ 377.6:1 nine honors /20 9/C0/CP32 4/C0/CP N/C30/ // /888;212 93;384;347//9:51/C2910/C283/ 104.1:1 See also CARDS ,POKER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 48 /C149, 1987. Kraitchik, M. "Bridge Hands." §6.3 in Mathematical Recrea- tions. New York: W. W. Norton, pp. 119 /C1121, 1942. Reese, T. Bridge for Bright Beginners. New York: Dover, 1973. Rubens, J. The Secrets of Winning Bridge. New York: Dover, 1981. Bridge Index A numerical KNOT invariant. For a TAME KNOT K, the bridge index is the least BRIDGE NUMBER of all planar representations of the KNOT . The bridge index of the UNKNOT is defined as 1. See also BRIDGE NUMBER ,CROOKEDNESS References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 114, 1976. Schubert, H. "U¨ ber eine numerische Knotteninvariante." Math. Z. 61, 245 /C1288, 1954. Bridge Knot An n-bridge knot is a knot with BRIDGE NUMBER n. The set of 2-bridge knots is identical to the set of rational knots. If L is a 2-BRIDGE KNOT , then the BLM /HO POLYNOMIAL Q and JONES POLYNOMIAL V satisfy QL(z) /C302z /C281VL(t)VL(t/C281 /C271 /C282z /C281) ; where z /C13/C28t /C28t/C281 (Kanenobu and Sumi 1993). Kane- nobu and Sumi also give a table containing the number of distinct 2-bridge knots of n crossings for n /C3010 to 22, both not counting and counting MIRROR IMAGES as distinct. n /Kn//Kn /C27K + n/ 30 0 40 0 5 6789 10 45 85 11 91 182 12 176 341 13 352 704 14 693 1365 15 1387 2774 16 2752 5461 17 5504 11008 18 10965 21845 19 21931 43862 20 43776 87381 21 87552 175104 22 174933 349525References Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- Bridge Links through 20 Crossings." Adv. Studies Pure Math. 20, 125 /C1145, 1992. Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- Bridge Knots through 22-Crossings." Math. Comput. 60, 771 /C1778 and S17-S28, 1993. Schubert, H. "Knotten mit zwei Bru¨cken." Math. Z. 65, 133 /C1170, 1956. Bridge Number The least number of unknotted arcs lying above the plane in any projection. The knot 05 /C1002 has bridge number 2. Such knots are called 2-BRIDGE KNOTS . There is a one-to-one correspondence between 2- BRIDGE KNOTS and rational knots. The knot 08 /C1010 is a 3-bridge knot. A knot with bridge number b is an n- EMBEDDABLE KNOT where n 5 b:/ See also BRIDGE INDEX References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 64 /C167, 1994. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 115, 1976. Bridge of Ko¨nigsberg KO¨ NIGSBERG BRIDGE PROBLEM Brightness The area of the SHADOW of a body on a plane, also called the "outer quermass." See also INNER QUERMASS ,SHADOW References Blaschke, W. Kreis und Kugel. New York: Chelsea, p. 140, 1949. Bonnesen, T. "Om Minkowski’s uligheder fur konvexer legemer." Mat. Tidsskr. B, 80, 1926. Bonnesen, R. and Fenchel, W. Theorie der Konvexer Ko ¨rper. New York: Chelsea, p. 140, 1971. Chakerian, G. D. "Is a Body Spherical If All Its Projections Have the Same I.Q.?" Amer. Math. Monthly 77, 989/C1992, 1970. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 23, 1991. Firey, W. J. "Blaschke Sum of Convex Bodies and Mixed Bodies." In Proceedings of the Colloquium on Convexity (Ed. W. Fenchel). Copenhagen, Denmark: Københavns Univ. Math. Inst., pp. 94 /C1101, 1967. Brill-Noether Theorem If the total group of the canonical series is divided into two parts, the difference between the number ofpoints in each part and the double of the dimension of the complete series to which it belongs is the same. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 263, 1959. Bring Quintic Form AT SCHIRNHAUSEN TRANSFORMATION can be used to take a general QUINTIC EQUATION to the form x5 /C28x /C28a /C300 ; where a may be COMPLEX . See also BRING- JERRARD QUINTIC FORM,Q UINTIC EQUATION References Bring, E. S. Quart. J. Math. 6, 1864. Grunert, J. A. "VIII. Miscellen von dem Herausgeber." Archiv der Math. Phys. 41, 105 /C1/112, 1864. Harley, R. "A Contribution to the History of the Problem of the Reduction of the General Equation of the Fifth Degree to a Trinomial Form." Quart. J. Math. 6,38/C1/47, 1864. Ruppert, W. M. "On the Bring Normal Form of a Quintic in Characteristic 5." Arch. Math. 58,44/C1/46, 1992. Tortolini, B. "Rivista bibliografica sopra a transformazione del Sig. Jerrard per l’equazioni di quinto grado." Annali di Mat. pura appl. 6,33/C1/42, 1864. Bring-Jerrard Quintic Form AT SCHIRNHAUSEN TRANSFORMATION can be used to algebraically transform a general QUINTIC EQUATION to the form z5 /C27c1z /C27c0 /C300: (1) In practice, the general quintic is first reduced to the PRINCIPAL QUINTIC FORM y5 /C27b2y2 /C27b1y /C27b0 /C300 (2) before the transformation is done. Then, we require that the sum of the third POWERS of the ROOTS vanishes, so s3(yj) /C300 : We assume that the ROOTS zi of the Bring-Jerrard quintic are related to the ROOTS yi of the PRINCIPAL QUINTIC FORM by zi /C30 ay4 i /C27 by3i /C27gy2i /C27 dyi /C27 e: (3) In a similar manner to the PRINCIPAL QUINTIC FORM transformation, we can express the COEFFICIENTS cj in terms of the bj :/ See also BRING QUINTIC FORM,PRINCIPAL QUINTIC FORM,QUINTIC EQUATION References Grunert, J. A. "VIII. Miscellen von dem Herausgeber." Archiv der Math. Phys. 41, 105 /C1/112, 1864. Klein, F. "U¨ ber die Transformation der elliptischen Funk- tionen und die Auflo¨sung der Gleichungen fu¨nften Grades." Math. Ann. 14, 1878/79. Tortolini, B. "Rivista bibliografica sopra a transformazione del Sig. Jerrard per l’equazioni di quinto grado." Annali di Mat. pura appl. 6,33/C1/42, 1864.Brioschi Formula For a curve with METRIC ds2 /C30Edu2 /C27Fdudv /C27Gdv2 ; (1) where E, F, and G is the first FUNDAMENTAL FORM , the GAUSSIAN CURVATURE is K /C30M1 /C27 M2 (EG /C28 F2)2 ; (2) where M1 /C13/C281 2Euv /C27Fuv /C2812Guu12EuFu /C2812Ev Fv /C2812Gu EF 12Gv FG/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2(3) M 2 /C1301 2Ev12Gu 12Ev EF 12Gu FG/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2/CP2; (4) which can also be written K /C30/C28 1ffiffiffiffiffiffiffiffi EGp@ @u1ffiffiffiffiEp @ffiffiffiffiGp @u ! /C27@ @v1ffiffiffiffiGp @ffiffiffiffiEp @v ! "# (5) /C30/C28 1ffiffiffiffiffiffiffiffiEGp @ @uGuffiffiffiffiffiffiffiffiEGp ! /C27@ @vEvffiffiffiffiffiffiffiffiEGp ! "# : (6) See also F UNDAMENTAL FORMS ,G AUSSIAN CURVA- TURE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 504 /C1/507, 1997. Briot-Bouquet Equation An ORDINARY DIFFERENTIAL EQUATION OF THE FORM xmy?/C30f(x;y); where mis a POSITIVE INTEGER ,fisANALYTIC atx/C30 y/C300;f(0;0)/C300;andf?y(0;0)"0:/ Zwillinger (1997, p. 120), citing Ince (1956, p. 295), define the Briot-Bouquet equation as xy?/C28ly/C30a10x/C27a20x2/C27a11yx/C27a02y2/C27/C1/C1/C1 References Briot and Bouquet. "Proprie ´te´s des fonctions de ´finie par des e´quations diffe ´rentielles." J. l’Ecole Polytechnique , Cah. 36. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 481 /C1/482, 1988. Ince, E. L. Ordinary Differential Equations. New York: Dover, 1956. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Brjuno Number Let pn =qnbe the sequence of CONVERGENTS of the CONTINUED FRACTION of a number a: Then a Brjuno number is an IRRATIONAL NUMBER such that X/C12 n/C300ln qn/C271 qnB/C12 (Marmi et al. 1999). Brjuno numbers arise in the study of one-dimensional analytic small divisors problems, and Brjuno (1971, 1972) proved that all "germs" with linear part l/C30e2 pia are linearizable if a is a Brjuno number. Yoccoz (1995) proved that this condition is also NECESSARY . References Brjuno, A. D. "Analytical Form of Differential Equations." Trans. Moscow Math. Soc. 25, 131 /C1288, 1971. Brjuno, A. D. "Analytical Form of Differential Equations. II." Trans. Moscow Math. Soc. 26, 199 /C1239, 1972. Marmi, S.; Moussa, P.; and Yoccoz, J.-C. "The Brjuno Functions and Their Regularity Properties." Comm. Math. Phys. 186, 265 /C1293, 1997. Marmi, S.; Moussa, P.; and Yoccoz, J.-C. "Complex Brjuno Functions." Preprint. 5 Dec 1999. http://rene.ma.utexa- s.edu/mp_arc/index-99.html. Moussa, P. and Marmi, S. "Diophantine Conditions and Real of Complex Brjuno Functions." Preprint. 5 Dec 1999. http://rene.ma.utexas.edu/mp_arc/index-99.html. Siegel, C. L. "Iteration of Analytic Functions." Ann. Math. 43, 807 /C1812, 1942. Yoccoz, J.-C. "The´ore`me de Siegel, nombres de Bruno et polyno ˆmes quadratiques." Aste´rique 231,3/C188, 1995. Broadcasting GOSSIPING Brocard Angle Define the first BROCARD POINT as the interior point V of a TRIANGLE for which the ANGLES /C218VAB ;/C218VBC; and /C218VCA are equal to an angle v: Similarly, define the second BROCARD POINT as the interior point V? for which the ANGLES /C218V?AC ;/C218V?CB; and /C218V?BA areequal to an angle v?: Then v /C30 v?; and this angle is called the Brocard angle. The Brocard angle v of a TRIANGLE DA1A2A3 is given by the formulas cot v /C30cot A1 /C27cot A2 /C27cot A3 (1) /C30a2 1 /C27 a22 /C27 a23 4D ! (2) /C301 /C27 cos a1 cos a2 cos a3 sin a1 sin a2 sin a3(3) /C30sin2 a1 /C27 sin2 a2 /C27 sin2 a3 2 sin a1 sin a2 sin a3(4) /C30a1 sin a1 /C27 a2 sin a2 /C27 a3 sin a3 a1 cos a1 /C27 a2 cos a2 /C27 a3 cos a3(5) csc2 v /C30csc2 a1 /C27csc2 a2 /C27csc2 a3 (6) sin v /C302 Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 1a22 /C27 a22a23 /C27 a23a21p (7) where D is the TRIANGLE AREA , A, B, and C are ANGLES , and a, b, and c are side lengths (Johnson 1929), where (6) is due to Neuberg (Tucker 1883). If an ANGLE a of a TRIANGLE is given, the maximum possible Brocard angle is given by cot v /C303 2tan(12 a) /C2712cos(12 a) (8) (Johnson 1929, p. 289). If v is specified, that the largest possible value amaxand minimum possible value aminof any possible triangle having Brocard angle v are given by cot(1 2 amax) /C30cot v /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cot2/C283p (9) cot(1 2amin)/C30cotv/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cot2/C283;q (10) where the square rooted quantity is the radius of the corresponding N EUBERG CIRCLE (Johnson 1929, p. 288). The maximum possible Brocard angle for any triangle is 30 8(Honsberger 1995, pp. 102 /C1/103). Let a TRIANGLE have ANGLES A,B, and C. Then sinAsinBsinC5kABC ; (11) where k/C303ffiffiffi 3p 2p !3 (12) (Le Lionnais 1983). This can be used to prove that 8v3BABC (13) (Abi-Khuzam 1974). See also BROCARD CIRCLE ,B ROCARD LINE,E QUI- BROCARD CENTER ,FERMAT POINTS ,NEUBERG CIRCLE References Abi-Khuzam, F. "Proof of Yff’s Conjecture on the Brocard Angle of a Triangle." Elem. Math. 29, 141 /C1/142, 1974. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 172, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 61, 1971. Emmerich, A. Die Brocardschen Gebilde und ihre Beziehun- gen zu den verwandten merkwu ¨rdigen Punkten und Kreisen des Dreiecks. Berlin: Georg Reimer, 1891. Honsberger, R. "The Brocard Angle." §10.2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 101 /C1/106, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263 /C1/286 and 289 /C1/294, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 65 /C1/66, 1893. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983. Tucker, R. "The ‘Triplicate Ratio’ Circle." Quart. J. Pure Appl. Math. 19, 342 /C1/348, 1883. Brocard Axis The LINE KO passing through the SYMMEDIAN POINT K and CIRCUMCENTER O of a TRIANGLE . The distance OK is called the BROCARD DIAMETER . The Brocard axis is PERPENDICULAR to the LEMOINE AXIS and is the ISOGONAL CONJUGATE of KIEPERT’S HYPERBOLA . It has equations sin(B /C28C) a /C27sin(C /C28A) b /C27sin(A /C28B) g/C300 bc(b2 /C28c2) a /C27ca(c2 /C28a2) b /C27ab(a2 /C28b2)g/C300: The SYMMEDIAN POINT K, CIRCUMCENTER O, ISODY- NAMIC POINTS S and S ?; and BROCARD MIDPOINT MB all lie along the Brocard axis. Note that the Brocard axis is not equivalent to the BROCARD LINE. See also BROCARD CIRCLE ,B ROCARD DIAMETER , BROCARD LINEBrocard Circle The CIRCLE passing through the first and second BROCARD POINTS V and V?; the LEMOINE POINT K, and the CIRCUMCENTER O of a given TRIANGLE . The BROCARD POINTS V and V? are symmetrical about the LINE KO ; which is called the BROCARD LINE. The LINE SEGMENT KO is called the BROCARD DIAMETER , and it has length OK /C30OV cos v /C30Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 4 sin2 vp cos v; where R is the CIRCUMRADIUS and v is the BROCARD ANGLE . The distance between either of the BROCARD POINTS and the SYMMEDIAN POINT is VK /C30V?K /C30VO tan v: The Brocard circle and LEMOINE CIRCLE are con- centric. See also BROCARD ANGLE ,B ROCARD DIAMETER , BROCARD POINTS References Brocard, M. H. "Etude d’un nouveau cercle du plan du triangle." Assoc. Franc ¸ais pour l’Academie des Sciences- Congre ´s d’Alger , 1881. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 75, 1971. Emmerich, A. Die Brocardschen Gebilde und ihre Beziehun- gen zu den verwandten merkwu ¨rdigen Punkten und Kreisen des Dreiecks. Berlin: Georg Reimer, 1891. Honsberger, R. "The Brocard Circle." §10.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 106 /C1/110, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 272, 1929. Lachlan, R. "The Brocard Circle." §134/C1/135 in An Elemen- tary Treatise on Modern Pure Geometry. London: Macmil- lian, pp. 78 /C1/81, 1893. Brocard Diameter The LINE SEGMENT KO joining the SYMMEDIAN POINT K and CIRCUMCENTER O of a given TRIANGLE . It is the DIAMETER of the TRIANGLE’S BROCARD CIRCLE , and lies along the BROCARD AXIS. The Brocard diameter has length OK /C30OV cos v /C30Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 4 sin2 vp cos v; where V is the first BROCARD POINT , R is the CIRCUMRADIUS , and v is the BROCARD ANGLE . See also BROCARD AXIS,BROCARD CIRCLE ,BROCARD LINE,BROCARD POINTS Brocard Line A LINE from any of the VERTICES Ai of a TRIANGLE to the first V or second V? BROCARD POINT . Let the ANGLE at a VERTEX Ai also be denoted Ai ; and denote the intersections of A1 V and A1 V? with A2A3as W1 and W2 : Then the ANGLES involving these points are /C218A1 VW3 /C30A1 (1) /C218W3 VA2 /C30A3 (2) /C218A2 VW1 /C30A2 (3)Distances involving the points Wi and W ?i are given by A2 V/C30a3 sin A2sin v (4) A2 V A3 V/C30a2 3 a1a2/C30sin(A3 /C28 v) sin v (5) W3A1 W3A2/C30a2 sin v a1 sin(A3 /C28 v) /C30a2 a3 !2 ; (6) where v is the BROCARD ANGLE (Johnson 1929, pp. 267 /C1/268). The Brocard line, MEDIAN M, and SYMMEDIAN POINT K are concurrent, with A1 V1 ; A2K ; and A3M meeting at a point P. Similarly, A1 V?; A2M ; and A3K meet at a point which is the ISOGONAL CONJUGATE point of P (Johnson 1929, pp. 268 /C1/269). See also BROCARD AXIS,BROCARD DIAMETER ,BRO- CARD POINTS ,ISOGONAL CONJUGATE ,S YMMEDIAN POINT ,MEDIAN (TRIANGLE ) References Emmerich, A. Die Brocardschen Gebilde und ihre Beziehun- gen zu den verwandten merkwu ¨rdigen Punkten und Kreisen des Dreiecks. Berlin: Georg Reimer, 1891. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263 /C1/286, 1929. Brocard Midpoint The MIDPOINT of the B ROCARD POINTS . It has TRIAN- GLE CENTER FUNCTION a/C30a(b2/C27c2)/C30sin(A/C27v); where vis the B ROCARD ANGLE . It lies on the BROCARD AXIS . References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/187, 1994. Brocard Points The first Brocard point is the interior point V(ort1or Z1)ofa TRIANGLE for which the ANGLES /C218VAB ;/C218 VBC; and /C218VCA are equal to an angle v: The second Brocard point is the interior point V? (or t2 or Z2) for which the ANGLES /C218V?AC ;/C218V?CB; and /C218V?BA are equal to an angle v?: The two angles v /C30 v? are equal, and this angle is called the BROCARD ANGLE , v /C30/C218VAB /C30/C218VBC /C30/C218VCA /C30/C218V?AC /C30/C218V?CB /C30/C218V?BA : The first two Brocard points are ISOGONAL CONJU- GATES (Johnson 1929, p. 266). They were described by French army officer Henri Brocard in 1875, although they had previously been investigated by Jacobi and, in 1816, Crelle (Wells 1991; Honsberger 1995, p. 98). The satisfy VO /C30V?O and /C218VO V?/C302v; where O is the CIRCUMCENTER and v is the BROCARD ANGLE (Honsberger 1995, p. 106). If three dogs start at the vertices of a triangle and chase either their left or right neighbor at a constant speed, that the three will meet at either V or V? (Wells 1991). One BROCARD LINE, MEDIAN , and SYMMEDIAN (out of the three of each) are CONCURRENT , with AV; CK, and BG meeting at a point, where G is the CENTROID and K is the SYMMEDIAN POINT . Similarly, AV?; BG, and CK meet at a point which is the ISOGONAL CONJUGATE of the first (Johnson 1929, pp. 268 /C1/269; Honsberger 1995, pp. 121 /C1/124). Let CBCbe the CIRCLE which passes through the vertices B and C and is TANGENT to the line AC at C, and similarly for CAB and CBC : Then the CIRCLES CAB ; CBC ; and CACintersect in the first Brocard point V: Similarly, let C ?BC be the CIRCLE which passes through the vertices B and C and is TANGENT to the line AB at B, and similarly for C?ABand C ?AC : Then the CIRCLES C ?AB ; C ?BC ; and C ?ACintersect in the second Brocard points V? (Johnson 1929, pp. 264 /C1/265; Honsberger1995, pp. 99 /C1/100). The PEDAL TRIANGLES of V and V? are congruent, and SIMILAR to the TRIANGLE DABC (Johnson 1929, p. 269). Lengths involving the Brocard points include OV/C30OV?/C30Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C284 sin2 vp (1) VV?/C302R sin vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C284 sin2 vp : (2) Extend the segments AV; B V; and CV to the CIRCUM- CIRCLE of DABC to form DC?A?B ?; and the segments AV?;BV?;andCV?to form DBƒCƒAƒ:Then DA?B?C?and DAƒBƒCƒare congruent to DABC (Honsberger 1995, pp. 104 /C1/106). Brocard’s third point is related to a given TRIANGLE by the TRIANGLE CENTER FUNCTION a/C30a/C283(3) (Casey 1893, Kimberling 1994). The third Brocard point Vƒ(ort3orZ3)i s COLLINEAR with the S PIEKER CENTER and the ISOTOMIC CONJUGATE POINT of its TRIANGLE’S INCENTER . See also BROCARD ANGLE ,BROCARD MIDPOINT ,EQUI- BROCARD CENTER ,YFF POINTS References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples,2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 66, 1893. Coolidge, J. L. "The Brocard Figures." §1.5 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 60 /C1 /84, 1971. Emmerich, A. Die Brocardschen Gebilde und ihre Beziehun- gen zu den verwandten merkwu ¨rdigen Punkten und Kreisen des Dreiecks. Berlin: Georg Reimer, 1891. Honsberger, R. "The Brocard Points." Ch. 10 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 99 /C1/124, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263 /C1/286, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 65 /C1/66 and 79 /C1/80, 1893. Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes’ Folium." Math. Mag. 61, 172 /C1/187, 1988. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 21 /C1/22, 1991. Brocard Triangles Given TRIANGLE DA1A2A3 ; let the point of intersection of A2 V and A3 V? be B1 ; where V and V? are the BROCARD POINTS , and similarly define B2and B3 : Then B1B2B3 is called the first Brocard triangle, and is INVERSELY SIMILAR to A1A2A3(Honsberger 1995, p. 112). It is inscribed in the BROCARD CIRCLE drawn with OK as the DIAMETER . The triangles B1A2A3 ; B2A3A1 ; and B3A1A2are ISOSCELES TRIANGLES with base angles v; where v is the BROCARD ANGLE . The sum of the areas of the ISOSCELES TRIANGLES is D; the AREA of TRIANGLE A1A2A3 : The first Brocard triangle is in perspective with the given TRIANGLE , with A1B1 ; A2B2 ; and A3B3 CONCURRENT . The CENTROID of the first brocard triangle is the CENTROID G of the original triangle (Honsberger 1995, pp. 112 /C1/116). Let perpendiculars be drawn from the midpoints MA ; MB ; and MC of each side of the first Brocard triangleto the opposite sides of the triangle DABC : Then the extensions of these lines CONCUR in the NINE-POINT CENTER (Honsberger 1995, pp. 116 /C1/118). Let c1 ; c2 ; and c3 be the CIRCLES through the vertices A2and A3 ; Anand A3 ; and Anand A2 ; respectively, which intersect in the first BROCARD POINT V: Similarly, define c ?1 ; c ?2 ; and c ?3with respect to the second BROCARD POINT V?: Let the two circles c1and c ?1tangent at Anto A1A2and A1A3 ; and passing respectively through A3 and A2 ; meet again at C1 ; and similarly for C2 and C3 : Then the triangle DC1C2C3 is called the second Brocard triangle. The second Brocard triangle is also the triangle obtained as the intersections of the lines A1K;A2K; and A3Kwith the B ROCARD CIRCLE , where Kis the SYMMEDIAN POINT . Let P1;P2;andP3be the intersec- tions of the lines A1K;A2K;and A3Kwith the CIRCUMCIRCLE ofDA1A2A3:Then C1;C2;and C3are the midpoints of A1P1;A2P2;and A3P3;respectively (Lachlan 1893). The two Brocard triangles are in PERSPECTIVE atM. See also BROCARD CIRCLE ,CIRCLE- CIRCLE INTERSEC- TION ,M CCAY CIRCLE ,NINE-POINT CENTER ,STEINER POINTS ,TARRY POINT References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 75, 1971. Emmerich, A. Die Brocardschen Gebilde und ihre Beziehun- gen zu den verwandten merkwu ¨rdigen Punkten und Kreisen des Dreiecks. Berlin: Georg Reimer, 1891. Honsberger, R. "The Brocard Triangles." §10.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 110 /C1/118, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 277 /C1/281, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 78 /C1/81, 1893. Brocard’s Conjecture p(p2 n/C271) /C28 p(p2n) ]4 for n ]2 where p(n) is the PRIME COUNTING FUNCTION and pn is the nth PRIME . For n /C301, 2, ..., the first few values are 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, ... (Sloane’s A050216). See also ANDRICA’S CONJECTURE References Sloane, N. J. A. Sequences A050216 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Brocard’s Problem Find the values of n for which n! /C271isa SQUARE NUMBER m2 ; where n! is the FACTORIAL (Brocard 1876, 1885). Pairs of numbers (m, n) are called BROWN NUMBERS . The only known solutions are n /C304, 5, and 7, and there are no other solutions with n 5107 (Wells 1986, p. 70; D. Wilson). It is virtually certain that there are no more solutions (Guy 1994). In fact, Dabrowski (1996) has shown that n! /C27 A /C30 k2 has only finitely many solutions for general A, although this result requires assumption of a weak form of the ABC CONJECTURE if A is SQUARE ). Wilson has also computed the least k such that n! /C27 k2 is square starting at n /C304, giving 1, 1, 3, 1, 9, 27, 15, 18, 288, 288, 420, 464, 1856, ... (Sloane’s A038202).See also BROWN NUMBERS ,F ACTORIAL ,S QUARE NUMBER References Brocard, H. Question 166. Nouv. Corres. Math. 2, 287, 1876. Brocard, H. Question 1532. Nouv. Ann. Math. 4, 391, 1885. Dabrowski, A. "On the Diophantine Equation x! /C27 A /C30 y2 :/" Nieuw Arch. Wisk. 14, 321 /C1324, 1996. Erdos, P. and Obla´th, R. "U¨ ber diophantische Gleichungen der Form n! /C30 xp 9yp und n! 9 m! /C30 xp/" Acta Szeged 8, 241 /C1255, 1937. Gupta. Math. Student 3, 71, 1935. Guy, R. K. "Equations Involving Factorial n." §D25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 193 /C1194, 1994. Hardy, G. H.; Aiyar, S.; Venkatesvara, P.; and Wilson, B. M. (Eds.). Collected Papers of Srinivasa Ramanujan. Cam- bridge, England: The University Press, p. 327, 1927. Overholt, M. "The Diophantine Equation n! /C27 1 /C30 m2 :/" Bull. London Math. Soc. 25, 104, 1993. Sloane, N. J. A. Sequences A038202 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 70, 1986. Bromwich Integral The inverse of the LAPLACE TRANSFORM , given by F(t) /C301 2pi g g /C27i /C12 g/C28i/C12epif(s) ds ; where g is a vertical CONTOUR in the COMPLEX PLANE chosen so that all singularities of f(s) are to the left of it. See also LAPLACE TRANSFORM References Arfken, G. "Inverse Laplace Transformation." §15.12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 853 /C1/861, 1985. Brooks’ Theorem The CHROMATIC NUMBER of a graph is at most the maximum VERTEX DEGREE D; unless the graph is COMPLETE or an odd cycle. See also CHROMATIC NUMBER References Brooks, R. L. "On Coloring the Nodes of a Network." Proc. Cambridge Philos. Soc. 37, 194 /C1/197, 1941. Lova´sz, L. "Three Short Proofs in Graph Theory." J. Combin. Th. Ser. B 19, 111 /C1/113, 1975. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 215, 1990. Brothers APAIR of consecutive numbers. See also PAIR,SMITH BROTHERS ,TWINS Brouwer Fixed Point Theorem Any continuous FUNCTION G : B 0 Bn has a FIXED POINT , where Bn /C30fx /C23Rn : x2 1 /C27/C1/C1/C1/C27x2n 51 g is the unit n-BALL . See also BALL,FIXED POINT THEOREM References Kannai, Y. "An Elementary Proof of the No Retraction Theorem." Amer. Math. Monthly 88, 264 /C1/268, 1981. Milnor, J. W. Topology from the Differentiable Viewpoint. Princeton, NJ: Princeton University Press, p. 14, 1965. Munkres, J. R. Elements of Algebraic Topology. Perseus Press, p. 117, 1993. Samelson, H. "On the Brouwer Fixed Point Theorem." Portugal. Math. 22, 189 /C1/191, 1963. Browkin’s Theorem For every POSITIVE INTEGER n, there exists a SQUARE in the plane with exactly n LATTICE POINTS in its interior. This was extended by Schinzel and Kuli- kowski to all plane figures of a given shape. The generalization of the SQUARE in 2-D to the CUBE in 3- D was also proved by Browkin. See also CUBE,SCHINZEL’S THEOREM ,SQUARE References Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 121 /C1/125, 1973. Brown Function For a FRACTAL PROCESS with values y(t /C28Dt) and y(t /C27 Dt) ; the correlation between these two values is given by the Brown function r /C3022H /C281 /C281; also known as the BACHELIER FUNCTION ,LE´ VY FUNC- TION ,orW IENER FUNCTION . Brown Numbers Brown numbers are PAIRS (m, n)of INTEGERS satisfy- ing the condition of BROCARD’S PROBLEM , i.e., such that n! /C271 /C30m2 where n! is the FACTORIAL and m2is a SQUARE NUMBER . Only three such PAIRS of numbers are known: (5, 4), (11, 5), (71, 7), and Erdos conjectured that these are the only three such PAIRS . See also BROCARD’S PROBLEM ,FACTORIAL ,SQUARE NUMBER ,W ILSON PRIME References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 193, 1994.Pickover, C. A. Keys to Infinity. New York: Wiley, p. 170, 1995. Brown’s Criterion A SEQUENCE fni g of nondecreasing POSITIVE INTEGERS is COMPLETE IFF 1. n1 /C301 :/ 2. For all k /C302, 3, ..., sk/C281 /C30 n1 /C27 n2 /C27/C1/C1/C1/C27 nk/C281 ] nk /C281: A corollary states that a SEQUENCE for which n1 /C301 and nk/C271 52 nk is COMPLETE (Honsberger 1985). See also COMPLETE SEQUENCE References Brown, J. L. Jr. "Notes on Complete Sequences of Integers." Amer. Math. Monthly 68, 557 /C1/560, 1961. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 123 /C1/130, 1985. Broyden’s Method An extension of the SECANT METHOD of root finding to higher dimensions. See also SECANT METHOD References Broyden, C. G. "A Class of Methods for Solving Nonlinear Simultaneous Equations." Math. Comput. 19, 577 /C1/593, 1965. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 382 /C1/385, 1992. Bruck-Ryser Theorem BRUCK- RYSER- CHOWLA THEOREM Bruck-Ryser-Chowla Theorem If n /C131; 2 (mod 4); and the SQUAREFREE part of n is divisible by a PRIME p /C133 (mod 4); then no DIFFER- ENCE SET of ORDER n exists. Equivalently, if a PROJECTIVE PLANE of order n exists, and n /C301or2 (mod 4), then n is the sum of two SQUARES . Dinitz and Stinson (1992) give the theorem in the following form. If a symmetric (v; k; l)/-BLOCK DESIGN exists, then 1. If v is EVEN , then k /C28l is a SQUARE NUMBER , 2. If visODD, then the D IOPHANTINE EQUATION x2/C30(k/C28l)y2/C27(/C281)(v/C281)=2lz2 has a solution in integers, not all of which are 0. See also BLOCK DESIGN ,DIFFERENCE SET,FISHER’S BLOCK DESIGN INEQUALITY References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1 /C112, 1992. Gordon, D. M. "The Prime Power Conjecture is True for n B 2 ;000; 000:/" Electronic J. Combinatorics 1,R61 /C17, 1994. http://www.combinatorics.org/Volume_1/volume 1.html#R6. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963. Bruhat Order References Bjo¨rner, A. and Wachs, M. "Bruhat Order of Coxeter Groups and Shellability." Adv. Math. 43,87/C1100, 1982. Stanley, R. P. Exercise 3.75(a) in Enumerative Combinato- rics, Vol. 1. Cambridge, England: Cambridge University Press, 1999. Stanley, R. P. Exercises 6.47 and 7.103d in Enumerative Combinatorics, Vol. 2. Cambridge, England: Cambridge University Press, pp. 243 and 485, 1999. Brun’s Constant The number obtained by adding the reciprocals of the odd TWIN PRIMES , B /C13 (1 3 /C2715) /C27 (15 /C2717) /C27 (1 11 /C271 13) /C27 (1 17 /C271 19) /C27/C1/C1/C1; (1) By BRUN’S THEOREM , the constant converges to a definite number as p 0/C12: Any finite sum under- estimates B. Shanks and Wrench (1974) used all the TWIN PRIMES among the first 2 million numbers. Brent (1976) calculated all TWIN PRIMES up to 100 billion and obtained (Ribenboim 1989, p. 146) B : 1:90216054 ; (2) assuming the truth of the first HARDY- LITTLEWOOD CONJECTURE . Using TWIN PRIMES up to 1014, Nicely (1996) obtained B :1:9021605778 92:1 /C2910/C289 (3) (Cipra 1995, 1996), in the process discovering a bug in Intel’s†PentiumTMmicroprocessor. Using TWIN PRIMES up to 2:5515 ; Nicely subsequently obtained the result B :1:9021605820 92:4 /C2910/C289 : (4) (Note that the value given by Le Lionnais 1983 is incorrect) Segal (1930) proved that Brun-type sums Bdof 1 =p over consecutive primes separated by d are finite (Halberstam and Richert 1983, p. 92). Wolf suggests that Bdis roughly equal to 4=d which, in the d /C302 case of twin primes, gives B2 :2 instead of 1:902:/... Wolf also considers the "COUSIN PRIMES " Brun’s constant B4 :/ See also COUSIN PRIMES ,TWIN PRIMES ,TWIN PRIMECONJECTURE ,TWIN PRIMES CONSTANT References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 64, 1987. Brent, R. P. "Tables Concerning Irregularities in the Dis- tribution of Primes and Twin Primes Up to 1011." Math. Comput. 30, 379, 1976. Brun, V. "La serie 1 =5 /C271=7 /C27/C1/C1/C1 est convergente ou finie." Bull. Sci. Math. 43, 124 /C1/128, 1919. Cipra, B. "How Number Theory Got the Best of the Pentium Chip." Science 267, 175, 1995. Cipra, B. "Divide and Conquer." What’s Happening in the Mathematical Sciences, 1995 /C1/1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 38 /C1/47, 1996. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/brun/brun.html. Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 41, 1983. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 67, 1951. Nicely, T. "Enumeration to 1014 of the Twin Primes and Brun’s Constant." Virginia J. Sci. 46, 195 /C1/204, 1996. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, 1989. Segal, B. "Ge´ne´ralisation du the´ore`me de Brun." Dokl. Akad. Nauk SSSR , 501 /C1/507, 1930. Shanks, D. and Wrench, J. W. "Brun’s Constant." Math. Comput. 28, 293 /C1/299, 1974. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 40 /C1/ 41, 1986. Brun’s Sieve See also SIEVE References Blecksmith, R.; Erdos, P.; and Selfridge, J. L. "Cluster Primes." Amer. Math. Monthly 106,43/C1/48, 1999. Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974. Brun’s Sum BRUN’S CONSTANT Brun’s Theorem The series producing BRUN’S CONSTANT CONVERGES even if there are an infinite number of TWIN PRIMES . Proved in 1919 by V. Brun. Brunnian Link A Brunnian link is a set of n linked loops such that each proper sublink is trivial, so that the removal of any component leaves a set of trivial unlinked UNKNOTS . The B ORROMEAN RINGS are the simplest example and have n/C303. See also BORROMEAN RINGS References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976. Brunn-Minkowski Inequality The nth root of the CONTENT of the set sum of two sets in Euclidean n-space is greater than or equal to the sum of the nth roots of the CONTENTS of the individual sets. See also TOMOGRAPHY References Cover, T. M. "The Entropy Power Inequality and the Brunn- Minkowski Inequality" §5.10 in Open Problems in Com- munications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, p. 172, 1987. Schneider, R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge, England: Cambridge University Press, 1993. Brusselator Equations The system of ordinary differential equations u?/C30A /C27u2v /C28(B /C271)u (1) v ?/C30Bu /C28u2v (2) (Hairer et al. 1987, p. 112; Zwillinger 1997, p. 136). The so-called full Brusselator equations are given by u?/C301 /C27u2v /C28(w /C271)u (3) v ?/C30uw /C28u2v (4) w?/C30/C28 uw /C27 a (5) (Hairer et al. 1987, p. 114; Zwillinger 1997, p. 136). References Hairer, E.; Nørsett, S. P.; and Wanner, G. Solving Ordinary Differential Equations I. New York: Springer-Verlag, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 136, 1997. Brute Force Factorization DIRECT SEARCH FACTORIZATION B-Spline A generalization of the BE´ ZIER CURVE . Let a vector known as the KNOT VECTOR be defined T /C30ft0 ; t1 ; ...; tm g; (1) where T is a nondecreasing SEQUENCE with ti /C23 [0; 1];and define control points P0 ; ..., Pn : Define the degree as p /C13m /C28n /C281: (2) The "knots" tp /C271 ; ..., tm/C28p/C281are called INTERNAL KNOTS . Define the basis functions as Ni;0(t) /C301i f ti 5t Bti/C271and ti Bti/C271 0 otherwise/C26 (3) Ni ; p(t) /C30t /C28 ti ti/C27p /C28 tiNi; p /C281(t) /C27ti /C27p /C271 /C28 t ti/C27p /C271 /C28 ti/C271Ni/C271; p /C281(t): (4) Then the curve defined by C(t) /C30Xn i /C300PiNi;p(t) (5) is a B-spline. Specific types include the nonperiodic B- spline (first p /C271 knots equal 0 and last p /C271 equal to 1) and uniform B-spline (INTERNAL KNOTS are equally spaced). A B-spline with no INTERNAL KNOTS is a BE´ ZIER CURVE . A curve is p /C28k times differentiable at a point where k duplicate knot values occur. The knot values determine the extent of the control of the control points. See also BE´ ZIER CURVE , NURBS CURVE B-Tree B-trees were introduced by Bayer (1972) and McCreight. They are a special m-ary balanced tree used in databases because their structure allows records to be inserted, deleted, and retrieved withguaranteed worst-case performance. An n-node B- tree has height O(1g2);where LGis the LOGARITHM to base 2. The Apple†Macintosh†(Apple Computer, Cupertino, CA) HFS filing system uses B-trees to store disk directories (Benedict 1995). A B-tree satisfies the following properties: 1. The ROOT is either a LEAF (TREE ) or has at least two CHILDREN . 2. Each node (except the ROOT and LEAVES ) has between m=2 de andmCHILDREN , where xdeis the CEILING FUNCTION . 3. Each path from the ROOT to a LEAF (TREE ) has the same length. Every 2 /C1/TREE is aB-tree of order 3. The number of B-trees of order-3 with n/C301, 2, ... leaves are 1, 1, 1, 1, 2, 2, 3, 4, 5, 8, 14, 23, 32, 43, 63, ... (Ruskey, Sloane’s A014535). The number of order-4 B-trees with n/C301, 2, ... leaves are 1, 1, 1, 2, 2, 4, 5, 9, 15, 28, 45, ... (Sloane’s A037026). See also RED-BLACK TREE,TREE References Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Struc- tures and Algorithms. Reading, MA: Addison-Wesley, pp. 369 /C1/374, 1987. Bayer, R. and McCreight, E. "Organization and Mainte- nance of Large Ordered Indexes." Acta Informatica 1, 173 /C1/189, 1972. Benedict, B. Using Norton Utilities for the Macintosh. Indianapolis, IN: Que, pp. B-17-B-33, 1995. Beyer, R. "Symmetric Binary B-Trees: Data Structures and Maintenance Algorithms." Acta Informat. 1, 290 /C1/306, 1972. Knuth, D. E. "B-Trees." The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison-Wesley, pp. 482 /C1/485 and 490 /C1/491, 1998. Ruskey, F. "Information on B-Trees." http://www.theory.cs- c.uvic.ca/~cos/inf/tree/BTrees.html. Skiena, S. S. The Algorithm Design Manual. New York: Springer-Verlag, p. 178, 1997. Sloane, N. J. A. Sequences A014535 and A037026 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Bubble A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single SPHERE , illustrated above (courtesy of J. M. Sullivan). More complicated forms occur when multiple bubbles are joined together. The simplest example is the DOUBLE BUBBLE , and beautiful configurations can form when three or more bubbles are conjoined (Sullivan). An outstanding problem involving bubbles is the determination of the arrangements of bubbles with the smallest SURFACE AREA which enclose and sepa- rate n given volumes in space. See also DOUBLE BUBBLE ,P LATEAU’S LAWS,P LA- TEAU’S PROBLEM ,SPHERE References Morgan, F. "Mathematicians, Including Undergraduates, Look at Soap Bubbles." Amer. Math. Monthly 101, 343 /C1/ 351, 1994. Pappas, T. "Mathematics & Soap Bubbles." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 219, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 214 /C1/216, 1999. Sullivan, J. M. "Generating and Rendering Four-Dimen- sional Polytopes." Mathematica J. 1,76/C1/85, Winter 1991. Sullivan, J. M. "Polytope Bubble Images." http:// www.math.uiuc.edu/~jms/Images/polyt.html. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, pp. 44 /C1/ 45, 1979.Buchberger’s Algorithm The algorithm for the construction of a GRO¨ BNER BASIS from an arbitrary ideal basis. See also GRO¨ BNER BASIS References Becker, T. and Weispfenning, V. Gro¨bner Bases: A Computa- tional Approach to Commutative Algebra. New York: Springer-Verlag, pp. 213 /C1/214, 1993. Buchberger, B. "Theoretical Basis for the Reduction of Polynomials to Canonical Forms." SIGSAM Bull. 39, 19/C1/24, Aug. 1976. Cox, D.; Little, J.; and O’Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer- Verlag, 1996. Buchowski Paradox A paradox arising in the use of comparative adjec- tives. Suppose you have exactly two brothers, both of whom are older than you are. Then the followingapparently false statement is actually true: "My younger brother is older than I am." Buckminster Fuller Dome GEODESIC DOME Buffon’s Needle Problem Find the probability P(l;d) that a needle of length l will land on a line, given a floor with equally spaced PARALLEL LINES a distance dapart. The problem was first posed by the French naturalist Buffon in 1733, and reproduced with the solution by Buffon in 1777. Forl5d; P(l;d)/C30g2p 0lcosu jj ddu 2p/C30l 2pd4gp=2 0cosudu /C302l pd[sinu]p=2 0/C302l pd: (1) For l ]d; the solution is slightly more complicated, P(l; d) /C301 pdd p /C282 sin/C281d l !"# /C272l 1 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28d2 l2s ! () (2) (Uspensky 1937, p. 252; Kunkel). Several attempts have been made to experimentally determine p by needle-tossing. For a discussion of the relevant statistics and a critical analysis of one of the more accurate (and least believable) needle-tossings, see Badger (1994). Uspensky (1937, pp. 112 /C1/113) discusses experiments conducted with 2520, 3204, and 5000 trials. An asymptotically unbiased estima- tor for p from the needle-tossing experiment is ˆp /C302rn N; (3) where r /C30l=d ; n is the number of throws, and N is the number of line crossings, which has asymptotic variance var( ˆp) /C30p2 n(1 2p /C281) :5:63 n (4) (Mantel 1953; Solomon 1978, p. 7). If the needle is longer than the distance between two lines, then the probability that it intersects at least one line is P(l) /C302l pd(1 /C28sin f0) /C272f0 p; (5) where cos f0 /C30d =l (Uspensky 1937, p. 258). The problem can be extended to a "needle" in the shape of a CONVEX POLYGON with GENERALIZED DIAMETER less than d. The probability that the boundary of the polygon will intersect one of the lines is given by P /C30p pd ; (6) where p is the PERIMETER of the polygon (Uspensky 1937, p. 253; Solomon 1978, p. 18). A further general- ization obtained by throwing a needle on a board ruled with two sets of perpendicular lines is called the BUFFON- LAPLACE NEEDLE PROBLEM . See also BUFFON- LAPLACE NEEDLE PROBLEM References Badger, L. "Lazzarini’s Lucky Approximation of p:/"Math. Mag. 67,8 3/C1/91, 1994. Buffon, G. Proc. Paris Acad. Sci. 1733. Buffon, G. Essai d’arithme ´tique morale. Supple ´ment a l’Histoire Naturelle, Vol. 4, 1777. Diaconis, P. "Buffon’s Needle Problem with a Long Needle." J. Appl. Prob. 13, 614/C1/618, 1976.Do¨rrie, H. "Buffon’s Needle Problem." §18 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 73 /C1/77, 1965. Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32,1/C1/37, 1995. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.New York: Hyperion, p. 209, 1998. Isaac, R. The Pleasures of Probability. New York: Springer- Verlag, 1995. Klain, Daniel A. and Rota, G.-C. Introduction to Geometric Probability. New York: Cambridge University Press, 1997. Kraitchik, M. "The Needle Problem." §6.14 in Mathematical Recreations. New York: W. W. Norton, p. 132, 1942. Kunkel, P. "Buffon’s Needle." http://www.nas.com/~kunkel/ buffon/buffon.htm. Mantel, L. "An Extension of the Buffon Needle Problem." Ann. Math. Stat. 24, 674/C1 /677, 1953. Perlman, M. and Wichura, M. "On Sharpening Buffon’s Needle." Amer. Stat. 20, 157/C1/163, 1975. Santalo ´,L .A . Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Schuster, E. F. "Buffon’s Needle Experiment." Amer. Math. Monthly 81,2 6/C1/29, 1974. Solomon, H. "Buffon Needle Problem, Extensions, and Estimation of p:/" Ch. 1 in Geometric Probability. Phila- delphia, PA: SIAM, pp. 1 /C1/24, 1978. Stoka, M. "Problems of Buffon Type for Convex Test Bodies." Conf. Semin. Mat. Univ. Bari, No. 268, 1 /C1/17, 1998. Uspensky, J. V. "Buffon’s Needle Problem," "Extension of Buffon’s Problem," and "Second Solution of Buffon’sProblem." §12.14/C1 /12.16 in Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 112 /C1/115, 251 /C1/ 255, and 258, 1937. Wegert, E. and Trefethen, L. N. "From the Buffon Needle Problem to the Kreiss Matrix Theorem." Amer. Math. Monthly 101, 132/C1/139, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 53, 1986. Buffon-Laplace Needle Problem Find the probability P(l;a;b) that a needle of length lwill land on a line, given a floor with a grid of equally spaced PARALLEL LINES distances aand b apart, with lBa;b:The position of the needle can be specified with points ( x, y) and its orientation with coordinate f:By symmetry, we can consider a single rectangle of the grid, so 0 BxBaand 0ByBb:In addition, since opposite orientations are equivalent, we can take /C28p=2 B f B p=2 :/ The probability is given by P(l; a; b) /C301 /C28g p =2 /C28 p=2F(f) df pab; (1) where F( f) /C30ab /C28bl cos f /C28la sin f jj /C271 2l2 sin(2f) jj (2) (Uspensky 1937, p. 256; Solomon 1978, p. 4), giving P(l; a ;b) /C302l(a /C27 b) /C28 l2 pab: (3) If the plane is instead tiled with congruent triangles with sides a, b, c, and a needle with length l less than the shortest altitude is thrown, the probability that the needle is contained entirely within one of the triangles is given by P /C301 /C27(Aa2 /C27 Bb2 /C27 Cc2)l2 2pK2 /C28(4a /C27 4b /C27 4c /C28 3l)l 2pK; (4) where A, B, and C are the angles opposite a, b, and c, respectively, and K is the AREA of the triangle. For equilateral triangles, this simplifies to P /C301 /C272 3l a !2 /C28lffiffiffi 3p pa4 /C28l a ! (5) (Uspensky 1937, p. 258). See also BUFFON’S NEEDLE PROBLEM References Schuster, E. F. "Buffon’s Needle Experiment." Amer. Math. Monthly 81,26/C1/29, 1974. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, pp. 3 /C1/6, 1978. Uspensky, J. V. "Laplace’s Problem." §12.17 in Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 255 /C1/257, 1937. Bug Problem MICE PROBLEM Building A highly structured geometric object used to study GROUPS which act upon them. See also COXETER GROUP ,GROUP References Garrett, P. Buildings and Classical Groups. Boca Raton, FL: Chapman and Hall, 1997.Bulirsch-Stoer Algorithm An algorithm which finds RATIONAL FUNCTION extra- polations OF THE FORM Ri(i/C271)...(i/C27m)/C30Pm(x) Pn(x)/C30p0/C27p1x/C27.../C27pmxm q0/C27q1x/C27.../C27qnxn and can be used in the solution of ORDINARY DIFFER- ENTIAL EQUATIONS . References Bulirsch, R. and Stoer, J. §2.2 in Introduction to Numerical Analysis. New York: Springer-Verlag, 1991. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Richardson Extrapolation and the Bulirsch- Stoer Method." §16.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 718 /C1/725, 1992. Bullet Nose A plane curve with implicit equation a2 x2/C28b2 y2/C301: (1) In parametric form, x/C30acost (2) y/C30bcott: (3) The CURVATURE is k/C303abcottcsct (b2csc4t/C27a2sin2t)3=2(4) and the TANGENTIAL ANGLE is f/C30tan/C281bcsc3t a ! : (5) References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 127 /C1/129, 1972. Bullseye Illusion Although the inner shaded region has the same area as the outer shaded ANNULUS , it appears to be larger. Since the rings are equally spaced, Ainner /C30 p /C215 32 /C309p Aouter /C30 p /C215 52 /C28 p /C215 42 /C309p: See also ILLUSION References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 87, 1991. Bump Function Given any OPEN SET U in Rn with COMPACT CLOSURE K /C30 ¯U ; there exists SMOOTH FUNCTIONS which are identically one on U and vanish arbitrarily close to U. One way to express this more precisely is that for any OPEN SET V containing K, there is a SMOOTH FUNC- TION f such that 1. f(x) /C30 1 for all x /C23 U and 2. f(x) /C30 0 for all x Q V :/ A function f that satisfies (1) and (2) is called a bump function. If f f /C30 1 then by rescaling f, namely fk(x) /C30 knf(kx) ; one gets a sequence of smooth functions which converges to the DELTA FUNCTION . See also COMPACT SUPPORT ,C ONVOLUTION ,D IRAC DISTRIBUTION ,SMOOTH FUNCTIONBumping Algorithm Given a PERMUTATION fp1 ; p2 ; ...; pn g of f1; ...; ng; the bumping algorithm constructs a standard YOUNG TABLEAU by inserting the pione by one into an already constructed YOUNG TABLEAU . To apply the bumping algorithm, start with ffp1 gg; which is a YOUNG TABLEAU .Ifp1through pkhave already been inserted, then in order to insert pk/C271 ; start with the first line of the already constructed YOUNG TABLEAU and search for the first element of this line which is greater than pk /C271 : If there is no such element, append pk/C271to the first line and stop. If there is such an element (say, pp); exchange ppfor pk /C271 ; search the second line using pp ; and so on. See also TABLEAU CLASS,YOUNG TABLEAU References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Bundle The term "bundle" is an abbreviated form of the full term FIBER BUNDLE . Depending on context, it may mean one of the special cases of FIBER BUNDLES , such as a VECTOR BUNDLE or a PRINCIPAL BUNDLE . Bundles are so named because they contain a collection of objects which, like a bundle of hay, are held together in a special way. All of the fibers line up–or at least they line up to nearby fibers. LOCALLY , a bundle looks like a PRODUCT MANIFOLD in a TRIVIALIZATION . The graph of a function f sits inside the product as (x ; f(x)): The SECTIONS of a bundle generalize functions in this way. It is necessary to use bundles when the range of a function only makes sense locally, as in the case of a VECTOR FIELD on the SPHERE . Bundles are a special kind of SHEAF . See also FIBER BUNDLE ,JET BUNDLE ,LINE BUNDLE , PRINCIPAL BUNDLE ,SHEAF ,TANGENT BUNDLE ,VEC- TOR BUNDLE Bundle Map A bundle map is a map between bundles along with a compatible map between the BASE MANIFOLDS . Sup- pose p:X0Mandq:Y0Nare two BUNDLES , then F : X 0 Y is a bundle map if there is a map f : M 0 N such that q(F(x)) /C30f(p(x)) for all x /C23 X : In particular, the FIBER of X over a point m /C23 M ; gets mapped to the fiber of Y over f(m) /C23 N :/ In the language of CATEGORY THEORY , the above diagram COMMUTES . To be more precise, the induced map between fibers has to be a map in the category of the fiber. For instance, in a bundle map between VECTOR BUNDLES the fiber over m /C23 M is mapped to the fiber over f(m) /C23 M by a LINEAR TRANSFORMATION . For example, when f : M 0 N is a SMOOTH MAP between SMOOTH MANIFOLDS then df : TM 0 TN is the differential, which is a bundle map between the tangent bundles. Over any point in m /C23 M ; the tangent vectors at m get mapped to tangent vectors at f(m) /C23 N by the JACOBIAN . See also BUNDLE ,C OMMUTATIVE DIAGRAM ,F IBER (BUNDLE ), JACOBIAN ,P RINCIPAL BUNDLE ,V ECTOR BUNDLE Buniakowsky Inequality SCHWARZ’S INEQUALITY Burali-Forti Paradox In the theory of transfinite ORDINAL NUMBERS , 1. Every WELL ORDERED SET has a unique ORDINAL NUMBER , 2. Every segment of ordinals (i.e., any set of ordinals arranged in natural order which contains all the predecessors of each of its elements) has an ORDINAL NUMBER which is greater than any ordinal in the segment, and 3. The set B of all ordinals in natural order is well ordered. Then by statements (3) and (1), B has an ordinal b: Since bis in B, it follows that bBbby (2), which is a contradiction. See also ORDINAL NUMBER References Copi, I. M. "The Burali-Forti Paradox." Philos. Sci. 25, 281/C1/ 286, 1958.Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 5, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 29 /C1/30, 1998. Mirimanoff, D. "Les antinomies de Russell et de Burali-Forti et le proble `me fondamental de la the ´orie des ensembles." Enseign. math. 19,3 7/C1/52, 1917. Burau Representation Gives a MATRIX representation biof a BRAID GROUP in terms of ( n/C281)/C29(n/C281) MATRICES .A/C28talways appears in the ( i, i) position. b1/C30/C28t0 0 ... 0 /C28110. . .0 001. . .0 nnn:::n 001. . .12 666643 77775(1) b i/C301 ... 0 0 ... 0 n:::nn:::n 0 ... /C28t0 ... 0 0 ... /C28t0 ... 0 0 ... /C281 1 ... 0 0:::00:::n 0 ... 0 0 ... 12 6666666643 777777775(2) b n/C281/C301 0 ... 0 0 0 1 ... 0 0 nn:::nn 0 0 ... 0 /C28t 0 0 ... 0 /C28t2 666643 77775(3) LetCbe the MATRIX PRODUCT ofBRAID WORDS , then det(1/C28C) 1/C27t/C27.../C27tn/C281/C30DL; (4) where DLis the A LEXANDER POLYNOMIAL and det is the DETERMINANT . References Burau, W. "U ¨ber Zopfgruppen und gleichsinnig verdrilte Verkettungen." Abh. Math. Sem. Hanischen Univ. 11, 171/C1/178, 1936. Jones, V. "Hecke Algebra Representation of Braid Groups and Link Polynomials." Ann. Math. 126, 335/C1/388, 1987. Burgers’ Equation The PARTIAL DIFFERENTIAL EQUATION ut/C27uux/C30nuxx (Benton and Platzman 1972; Zwillinger 1995, p. 417; Zwillinger 1997, p. 130). The so-called nonplanar Burgers equation is given by ut/C27uux/C27Ju 2t/C301 2duxx (Sachdev and Nair 1987; Zwillinger 1997, p. 131). References Benton, E. R. and Platzman, G. W. "A Table of Solutions of the of the One-Dimensional Burgers Equation." Quart. Appl. Math. , 195 /C1/212, Jul. 1972. Sachdev, P. L. and Nair, K. R. C. "Generalized Burgers Equations and Euler-Painleve ´ Transcendents. II." J. Math. Phys. 28, 997 /C1/1004, 1987. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, 1997. Burkhardt Quartic The VARIETY which is an invariant of degree four and is given by the equation y4 0 /C28y0(y31 /C27y32 /C27y33 /C27y34) /C273y1y2y3y4 /C300 : See also QUARTIC EQUATION References Burkhardt, H. "Untersuchungen aus dem Gebiet der hyper- elliptischen Modulfunctionen. II." Math. Ann. 38, 161 /C1/ 224, 1890. Burkhardt, H. "Untersuchungen aus dem Gebiet der hyper- elliptischen Modulfunctionen. III." Math. Ann. 40, 313 /C1/ 343, 1892. Hunt, B. "The Burkhardt Quartic." Ch. 5 in The Geometry of Some Special Arithmetic Quotients. New York: Springer- Verlag, pp. 168 /C1/221, 1996. Bu¨ rmann’s Theorem Bu¨rmann’s theorem deals with the expansion of functions in powers of another function. Let f(z)be a function of z which is analytic in a closed region S, of which a is an interior point, and let f(a) /C30b: Suppose also that f?(a) "0: Then TAYLOR’S THEOREM furnishes the expansion f(z) /C28b /C30 f?(a)(z /C28a) /C27fƒ(a) 2!(z /C28a)2 /C27...; (1) and if it is legitimate to revert this series, we obtain z /C28a /C30f(z) /C28 b f?(a)/C281 2f ƒ(a) [ f?(a)]3 [ f(z) /C28b]2 /C27...; (2) which expresses z as an ANALYTIC FUNCTION of the variable f(z) /C28b for sufficiently small values of z /C28a jj : If then f(z) is analytic near z /C30a, it follows that f(z)isan ANALYTIC FUNCTION of f(z) /C28b when z /C28a jj is sufficiently small, and so there will be an expansion in the form f(z) /C30f(a) /C27a1[ f(z) /C28b] /C27a2 2![ f(z) /C28b]2 /C27a3 3![ f(z) /C28b]3 /C27... (3) The actual coefficients in the expansion are given by the following theorem, which is generally known as Bu¨rmann’s theorem. Let c(z) be a function of zdefined by the equation c(z) /C30z /C28 a f(z) /C28 b : (4) Then an ANALYTIC FUNCTION f(z) can, in a certain domain of values of z, be expanded in the form f(z) /C30f(a) /C27Xn/C281 m/C301[ f(z) /C28 b]m m!dm/C281 dam/C281 ff ?(a)[ c(a)]m g /C27Rn ; (5) where the remainder term is Rn /C301 2pi gx aggf(z) /C28 b f(t) /C28 b"#n/C281f ?(t) f?(z) dt dz f(t)/C28f(z); (6) andgis a CONTOUR in the t-plane enclosing the points aand zsuch that if zis any point inside g;the equation f(t)/C30f(z) has no roots on or inside the CONTOUR except a simple root t/C30z:/ TEIXEIRA’S THEOREM is extended form of Bu ¨rmann’s theorem. The L AGRANGE EXPANSION gives another such extension. See also DARBOUX’S FORMULA ,LAGRANGE EXPANSION , LAGRANGE INVERSION THEOREM ,T AYLOR SERIES , TEIXEIRA’S THEOREM References Bu¨rmann. "Rapport sur deux me ´moirs d’analyse." Me´moires de l’Institut National des Sci. et Arts: Sci. Math. Phys. 2, 13/C1/17, 1799. Dixon, A. C. "On Burmann’s Theorem." Proc. London Math. Soc. 34, 151/C1/153, 1902. Whittaker, E. T. and Watson, G. N. "Bu ¨rmann’s Theorem" and "Teixeira’s Extended Form of Bu ¨rmann’s Theorem." §7.3 and 7.3.1 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 128 /C1/132, 1990. Burnside Problem A problem originating with W. Burnside (1902), who wrote, "A still undecided point in the theory ofdiscontinuous groups is whether the ORDER of a GROUP may be not finite, while the order of every operation it contains is finite." This question wouldnow be phrased as "Can a finitely generated group be infinite while every element in the group has finite order?" (Vaughan-Lee 1990). This question was an-swered by Golod (1964) when he constructed finitely generated infinite P-GROUP . These GROUPS , however, do not have a finite exponent. LetFrbe the FREE GROUP ofRANK rand let Nbe the NORMAL SUBGROUP generated by the set of nth POWERS fgng/C23Frg: j Then Nis a normal subgroup of Fr:We define B(r;n)/C30Fr=Nto be the QUOTIENT GROUP . We call B(r;n) the r-generator Burnside group of exponent n. It is the largest r-generator group of exponent n, in the sense that every other such group is a HOMOMORPHIC image of B(r ; n) : The Burnside problem is usually stated as: "For which values of r and n is B(r; n)aFINITE GROUP ?" An answer is known for the following values. For r /C301, B(1; n)isa CYCLIC GROUP of ORDER n. For n /C302, B(r ; 2) is an elementary ABELIAN 2-group of ORDER 2r : For n /C303, B(r ; 3) was proved to be finite by Burnside. The ORDER of the B(r ; 3) groups was established by Levi and van der Waerden (1933), namely 3a where a /C13 r /C27r 2/CP8/CP9 /C27r 3/CP8/CP9 ; (1) where (n k)isa BINOMIAL COEFFICIENT . For n /C304, B(r ; 4) was proved to be finite by Sanov (1940). Groups of exponent four turn out to be the most complicated for which a POSITIVE solution is known. The precise nilpotency class and derived length are known, as are bounds for the ORDER . For example, B(2; 4) jj /C30212 (2) B(3; 4) jj /C30269 (3) B(4; 4) jj /C302422 (4) B(5; 4) jj /C3022728 ; (5) while for larger values of r the exact value is not yet known. For n /C306, B(r ; 6) was proved to be finite by Hall (1958) with ORDER 2a3b ; where a /C131 /C27(r /C281)3c (6) b /C13 1 /C27 (r /C28 1)2r (7) c /C13 r /C27r 2/CP8/CP9 /C27r 3/CP8/CP9 : (8) No other Burnside groups are known to be finite. On the other hand, for r /C212 and n ] 665; with n ODD, B(r ; n) is infinite (Novikov and Adjan 1968). There is a similar fact for r /C212 and n a large POWER of 2. E. Zelmanov was awarded a FIELDS MEDAL in 1994 for his solution of the "restricted" Burnside problem. See also FREE GROUP References Burnside, W. "On an Unsettled Question in the Theory of Discontinuous Groups." Quart. J. Pure Appl. Math. 33, 230 /C1238, 1902. Golod, E. S. "On Nil-Algebras and Residually Finite p- Groups." Isv. Akad. Nauk SSSR Ser. Mat. 28, 273 /C1276, 1964. Hall, M. "Solution of the Burnside Problem for Exponent Six." Ill. J. Math. 2, 764 /C1786, 1958. Levi, F. and van der Waerden, B. L. "U¨ ber eine besondere Klasse von Gruppen." Abh. Math. Sem. Univ. Hamburg 9, 154 /C1158, 1933. Novikov, P. S. and Adjan, S. I. "Infinite Periodic Groups I, II, III." Izv. Akad. Nauk SSSR Ser. Mat. 32, 212 /C1244, 251 /C1524, and 709 /C1731, 1968.Sanov, I. N. "Solution of Burnside’s problem for exponent four." Leningrad State Univ. Ann. Math. Ser. 10, 166 /C1170, 1940. Vaughan-Lee, M. The Restricted Burnside Problem, 2nd ed. New York: Clarendon Press, 1993. Burnside’s Conjecture This entry contributed by NICOLAS BRAY In Note M, Burnside (1955) states, "The contrast that these results shew between groups of odd and of even order suggests inevitably that simple groups of odd order do not exist." Of course, SIMPLE GROUPS of prime order do exist, namely the groups Zp for any prime p. Therefore, Burnside conjectured that every FINITE SIMPLE GROUP of non-prime order must have even order. The conjecture was proven true by Feit and Thompson (1963). See also ABELIAN GROUP ,FEIT-THOMPSON CONJEC- TURE ,FEIT-THOMPSON THEOREM ,SIMPLE GROUP References Burnside, W. Theory of Groups of Finite Order, 2nd ed. New York: Dover, 1955. Feit, W. and Thompson, J. G. "Solvability of Groups of Odd Order." Pacific J. Math. 13, 775/C1/1029, 1963. Burnside’s Lemma CAUCHY- FROBENIUS LEMMA Buschman Transform The INTEGRAL TRANSFORM defined by (Kf)(x)/C30g/C12 /C28/C12(x2/C28t2)l=2 /C27Pl nt x ! f(t)dt; where ya /C27is the TRUNCATED POWER FUNCTION and Pln(x) is an associated L EGENDRE POLYNOMIAL . References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993. Busemann-Petty Problem If the section function of a centered convex body in Euclidean n-space /(n]3) is smaller than that of another such body, is its volume also smaller? References Gardner, R. J. "Geometric Tomography." Not. Amer. Math. Soc. 42, 422/C1/429, 1995. Busy Beaver A busy beaver is an n-state, 2-symbol, 5-tuple T URING MACHINE which writes the maximum possible number BB(n) of 1s on an initially blank tape before halting. Forn/C300, 1, 2, ..., BB(n) is given by 0, 1, 4, 6, 13, ]4098 ;]136612 ; .... The busy beaver sequence is also known as RADO’S SIGMA FUNCTION . See also HALTING PROBLEM ,TURING MACHINE References Chaitin, G. J. "Computing the Busy Beaver Function." §4.4 in Open Problems in Communication and Computation (Ed. T. M. Cover and B. Gopinath). New York: Springer- Verlag, pp. 108 /C1/112, 1987. Dewdney, A. K. "A Computer Trap for the Busy Beaver, the Hardest-Working Turing Machine." Sci. Amer. 251,19/C1/ 23, Aug. 1984. Marxen, H. and Buntrock, J. "Attacking the Busy Beaver 5." Bull. EATCS 40, 247 /C1/251, Feb. 1990. Sloane, N. J. A. Sequences A028444 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Butterfly Catastrophe A CATASTROPHE which can occur for four control factors and one behavior axis. The butterfly cata- strophe is the universal unfolding of the singularity f(x) /C30x6 of codimension 4, i.e., with four unfolding parameters. It has the form F(x; u; v; w ; t) /C30 x6 /C27ux4 /C27vx3 /C27wx2 /C27tx:/ The equations x /C30c(8at3 /C2724t5) y /C30c(/C286at2 /C2815t4) display such a catastrophe (von Seggern 1993). References Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 94, 1993. Butterfly Curve A PLANE CURVE given by the implicit equation y6 /C30(x2 /C28x6) : See also DUMBBELL CURVE ,EIGHT CURVE ,PIRIFORM References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Butterfly Effect Due to nonlinearities in weather processes, a butter- fly flapping its wings in Tahiti can, in theory, produce a tornado in Kansas. This strong dependence of outcomes on very slightly differing initial conditions is a hallmark of the mathematical behavior known as CHAOS . See also CHAOS ,LORENZ SYSTEM Butterfly Fractal The FRACTAL -like curve generated by the 2-D function f(x;y)/C30(x2/C28y2) sinx/C27y a ! x2/C27y2: Butterfly Polyiamond A6- POLYIAMOND . References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Butterfly Theorem Given a CHORD PQ of a CIRCLE , draw any other two CHORDS AB and CD passing through its MIDPOINT . Call the points where AD and BC meet PQ X and Y. Then M is also the MIDPOINT of XY. There are a number of proofs of this theorem, including those by W. G. Horner, Johnson (1929, p. 78), and Coxeter (1987, pp. 78 and 144). The latter concise proof employs PROJECTIVE GEOMETRY . The following proof is given by Coxeter and Greitzer (1967, p. 46). In the figure at right, drop perpendi- culars x1and y1from X and Y to AB, and x2and y2from X and Y to CD. Write a /C30PM /C30MQ ; x /C30XM, and y /C30MY, and then note that by SIMILAR TRIAN- GLES x y /C30x1 y1/C30x2 y2(1) x1 y2/C30AXCY(2) x2 y1/C30XD YB; (3) so x2 y2/C30x1 y1x2 y2/C30x1 y2x2 y1/C30AX /C215XD CY /C215YB/C30PX /C215XQ PY /C215YQ /C30(a/C28x)(a/C27x) (a/C27y)(a/C28y)/C30a2/C28x2 a2/C28y2/C30a2 a2/C301; (4) sox/C30y.Q.E.D. See also CHORD ,C IRCLE ,C YCLIC QUADRILATERAL , MIDPOINT ,QUADRILATERAL References Coxeter, H. S. M. Projective Geometry, 2nd ed. New York: Springer-Verlag, pp. 78 and 144, 1987. Coxeter, H. S. M. and Greitzer, S. L. "The Butterfly." §2.8 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 45 /C1/46, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 78, 1929. C Cable TENSEGRITY Cable Knot Let K1be a TORUS KNOT . Then the SATELLITE KNOT with COMPANION KNOT K2 is a cable knot on K2 :/ See also SATELLITE KNOT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 118, 1994. Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, 1985. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 112 and 283, 1976. Cactus Fractal AM ANDELBROT SET-like FRACTAL obtained by iterat- ing the map zn/C271/C30z3 n/C27(z0/C281)zn/C28z0: See also FRACTAL ,JULIA SET,MANDELBROT SET Cage Graph A 3-regular g-cage for g]3i sa CUBIC GRAPH ofGIRTH gwith the minimum possible number of points. More generally, an ( v, g)-cage graph is a smallest v-regular graph with GIRTH g. Cubic cages were first discussed by Tutte (1947), but the intensive study of cage graphs did not begin until publication of an article by Erdos and Sachs (1963). There exists a (3 ;g)/-cagefor all g]3;and the (3 ;g)/-cages are unique for g/C303 to 8. The number of nonisomorphic (3 ;g) cages for g/C301, 2, ... are given by 0, 0, 1, 1, 1, 1, 1, 1, 18, 3, ... (Sloane’s A052453; Gould 1988, Royle). The number of vertices in the (3 ;g) cages for g/C303, 4, ... are 4, 6, 10, 14, 22, 30, 46, 62, 94, ... (Sloane’s A052454). Aselection of known (3 ;g) /-cages are illustrated above. There are a number of special cases (Wong 1982). The (2;g)/-cage is the CYCLE GRAPH Cg;the ( v;2)/-cage is the MULTIGRAPH ofvedges on two vertices, the ( v;3)/- cage is the COMPLETE GRAPH Kv/C271;and the ( v;4)/-cage is the BIPARTITE GRAPH Kv;v:/ Computing the number of vertices in a ( v, g)-cage is very difficult for g]5 and n]3 (Wong 1982). The following table summarizes known cages. A lower bound for the number of vertices f(v;g)i na( v, g)- cage is given by fl(v;g)/C30v(v/C281)r/C282 v/C282forg/C302r/C271 2(v/C281)r/C282 v/C282forg/C302r8 >>>< >>>: (Tutte 1967, p. 70; Bolloba ´s 1978, p. 105; Wong 1982). Sauer (1967ab) has obtained the best known upper bounds f u(3;g)/C304 3/C2729122g/C282forgodd 23/C2729122g/C282forgeven( (1) fu(n;g)/C302(n/C281)g/C282forgodd 2(n/C281)g/C283forgeven ;l12) (2) with v]4 (Wong 1982). In the table, Kndenotes a COMPLETE GRAPH , and Km;n a complete bipartite graph. g /(3;g)// (4;g)// (5;g)// (6;g)//(7;g)/-cage 3/K4// K5// K6// K7// K8/ 4/K3;3// K4;4// K5;5// K6;6//K7;7/ 5P ETERSEN GRAPHROBERTSON GRAPHROBERTSON- WEGNER GRAPHHOFFMAN- SINGLETON GRAPH 6H EAWOOD GRAPH 7M CGEE GRAPH 8L EVI GRAPH g /f(3; g)//f(4; g)//f(5; g)//f(6; g)//f(7; g)/ 3 45678 4 6 81 01 21 4 5 1 01 93 04 05 0 6 1 42 64 26 29 0 72 4 83 0 9 /[54; 58] / 10 70 11 /B112 / The first (3; 9)/-cage was found by Biggs and Hoare (1980), and Brinkmann et al. (1995) completed an exhaustive search yielding all 18 (3; 9)/-cages (Royle). The three (3; 10) /-cages were found by O’Keefe and Wong (1980). Computations by McKay and W. Myr- vold have demonstrated that a (3; 11) /-cage must have 112 vertices (Royle). The single known example was found by Balaban (1973). The known (4 ;g)/- and (5 ;g)/-cages are shown above (Wong 1982). See also CAYLEY GRAPH ,C UBIC GRAPH ,E XCESS , HOFFMAN- SINGLETON GRAPH ,M OORE GRAPH ,REGU- LAR GRAPH ,ROBERTSON GRAPH ,ROBERTSON- WEGNER GRAPH ,UNITRANSITIVE GRAPH References Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships among the Cages." Rev. Roumaine Math. Pures Appl. 18, 1033 /C1/1043, 1973. Biggs, N. L. Ch. 23 in Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993. Biggs, N. L. "Constructions for Cubic Graphs of Large Girth." LSE Tech Report 97 /C1/11. Biggs, N. L. and Hoare, M. J. "A Trivalent Graph with 58 Vertices and Girth 9." Disc. Math. 30, 299/C1/301, 1980.Bolloba ´s, B. Extremal Graph Theory. New York: Academic Press, 1978. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 236 /C1/239, 1976. Brinkmann, G.; McKay, B. D.; and Saager, C. "The Smallest Cubic Graphs of Girth Nine." Combin., Probability, and Computing 5,1/C1/13, 1995. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. §6.9 in Distance Regular Graphs. New York: Springer-Verlag, 1989. Erdos, P. and Sachs, H. "Regula ¨re graphen gegebener Taillenweite mit minimaler Knotenzahl." Wiss. Z. Uni. Halle (Math. Nat.) 12, 251/C1/257, 1963. Friedman, E. "Cages." http://www.stetson.edu/~efriedma/ girth/. Gould, R. (Ed.). Graph Theory. Menlo Park, CA: Benjamin- Cummings, 1988. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 174 /C1/175, 1994. Holton, D A. and Sheehan, J. (Eds.). Ch. 6 in The Petersen Graph. Cambridge, England: Cambridge University Press, 1993. O’Keefe, M. and Wong, P. K. "A Smallest Graph of Girth 10 and Valency 3." J. Combin. Th. B 29,9 1/C1/105, 1980. Royle, G. "Cubic Cages." http://www.cs.uwa.edu.au/~gordon/ cages/. Sauer, N. ‘Extremaleigneschaften regula ¨rer Graphen gegeb- ener Taillenweite, I." O¨sterreich. Akad. Wiss. Math. Natur. Kl. S.-B. II 176,9/C1/25, 1967. Sauer, N. ‘Extremaleigneschaften regula ¨rer Graphen gegeb- ener Taillenweite, II." O¨sterreich. Akad. Wiss. Math. Natur. Kl. S.-B. II 176,2 7/C1/43, 1967. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 191 and 221, 1990. Sloane, N. J. A. Sequences A052453 and A052454 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/eisonline.html. Tutte, W. T. "A Family of Cubical Graphs." Proc. Cambridge Philos. Soc. , 459/C1 /474, 1947. Tutte, W. T. The Connectivity of Graphs. Toronto, Canada: Toronto University Press, pp. 71 /C1/83, 1967. Weisstein, E. W. "Graphs." M ATHEMATICA NOTEBOOK GRAPHS.M . Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/22, 1982. Cahn-Hilliard Equation The PARTIAL DIFFERENTIAL EQUATION ut/C309 /C215M(u)9@f @u/C28K92u !"# : References Novick-Cohen, A. and Segal, L. A. "Nonlinear Aspects of the Cahn-Hilliard Equation." Physica D 10, 277/C1/298, 1984. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 132, 1997. Cairo Tessellation A TESSELLATION appearing in the streets of Cairo and in many Islamic decorations. Its tiles are obtained by projection of a DODECAHEDRON , and it is the DUAL TESSELLATION of the semiregular tessellation of squares and equilateral triangles. See also DODECAHEDRON ,TESSELLATION References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 23, 1991. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, p. 38, 1979. Cake Cutting It is always possible to "fairly" divide a cake among n people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes that everyone has received 1=n of the cake according to his own measure (Steinhaus 1983, pp. 65 /C1/71). Finally, if there is some piece on which two people disagree, then there is a way of partition- ing and dividing a cake such that each participant believes that he has obtained more than 1 =n of the cake according to his own measure. There are also similar methods of dividing collections of individually indivisible objects among two or more people when cash payments are used to even up the final division (Steinhaus 1983, pp. 67 /C1/68). Ignoring the height of the cake, the cake-cutting problem is really a question of fairly dividing a CIRCLE into n equal AREA pieces using cuts in its plane. One method of proving fair cake cutting to always be possible relies on the FROBENIUS- KO¨ NIG THEOREM . See also CIRCLE DIVISION BY CHORDS ,CIRCLE DIVI- SION BY LINES ,C YLINDER CUTTING ,E NVYFREE , FROBENIUS- KO¨ NIG THEOREM ,HAM SANDWICH THEO- REM,PANCAKE THEOREM ,PIZZA THEOREM ,SQUARE DIVISION BY LINES,TORUS CUTTING ,VOTING References Beck, A. "Constructing a Fair Share." Amer. Math. Monthly 94, 157 /C1/162, 1987. Brams, S. J. and Taylor, A. D. "An Envy-Free Cake Division Protocol." Amer. Math. Monthly 102,9/C1/19, 1995.Brams, S. J. and Taylor, A. D. Fair Division: From Cake- Cutting to Dispute Resolution. New York: Cambridge University Press, 1996. Dubbins, L. "Group Decision Devices." Amer. Math. Monthly 84, 350 /C1/356, 1997. Dubbins, L. and Spanier, E. "How to Cut a Cake Fairly." Amer. Math. Monthly 68,1/C1/17, 1961. Gale, D. "Dividing a Cake." Math. Intel. 15, 50, 1993. Hill, T. "Determining a Fair Border." Amer. Math. Monthly 90, 438 /C1/442, 1983. Hill, T. P. "Mathematical Devices for Getting a Fair Share." Amer. Sci. 88, 325 /C1/331, Jul.-Aug. 2000. Jones, M. L. "A Note on a Cake Cutting Algorithm of Banach and Knaster." Amer. Math. Monthly 104, 353 /C1/355, 1997. Knaster, B. "Sur le proble `me du partage pragmatique de H. Steinhaus." Ann. de la Soc. Polonaise de Math. 19, 228 /C1/230, 1946. Rebman, K. "How to Get (At Least) a Fair Share of the Cake." In Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 22 /C1/37, 1979. Robertson, J. and Webb, W. Cake Cutting Algorithms: Be Fair If You Can. Natick, MA: Peters, 1998. Steinhaus, H. "Remarques sur le partage pragmatique." Ann. de la Soc. Polonaise de Math. 19, 230 /C1/231, 1946. Steinhaus, H. "The Problem of Fair Division." Econometrica 16, 101 /C1/104, 1948. Steinhaus, H. "Sur la division pragmatique." Ekonometrika (Supp.) 17, 315 /C1/319, 1949. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 64 /C1/67, 1999. Stromquist, W. "How to Cut a Cake Fairly." Amer. Math. Monthly 87, 640 /C1/644, 1980. Cal WALSH FUNCTION Calabi’s Triangle The one TRIANGLE , in addition to the EQUILATERAL TRIANGLE , for which the largest inscribed SQUARE can be inscribed in three different ways. The ratio of the sides to that of the base is given by x/C30 1:55138752454 . . . (Sloane’s A046095), where x/C301 3/C27(/C2823/C273iffiffiffiffiffiffiffiffi 237p )1=3 3 /C21522=3/C2711 3[2(/C2823/C273iffiffiffiffiffiffiffiffi237p )]1=3 is the largest POSITIVE ROOT of 2x3/C282x2/C283x/C272/C300; which has CONTINUED FRACTION [1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, ...] (Sloane’s A046096). See also GRAHAM’S BIGGEST LITTLE HEXAGON ,TRIAN- GLE References Conway, J. H. and Guy, R. K. "Calabi’s Triangle." In The Book of Numbers. New York: Springer-Verlag, p. 206, 1996. Sloane, N. J. A. Sequences A046095 and A046096 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Calabi-Yau Manifold CALABI- YAU SPACE Calabi-Yau Space Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space OF THE FORM M /C29V ; where M is a four dimensional manifold (space-time) and V is a six dimensional COMPACT Calabi-Yau space. They are related to KUMMER SURFACES . Although the main application of Calabi- Yau spaces is in theoretical physics, they are also interesting from a purely mathematical standpoint. Consequently, they go by slightly different names, depending mostly on context, such as Calabi-Yau manifolds or Calabi-Yau varieties. Although the definition can be generalized to any dimension, they are usually considered to have three complex dimensions. Since their COMPLEX STRUCTURE may vary, it is convenient to think of them as having six real dimensions and a fixed SMOOTH STRUCTURE . A Calabi-Yau space is characterized by the existence of a NONVANISHING HARMONIC SPINOR f: This condi- tion implies that its CANONICAL BUNDLE is TRIVIAL . Consider the local situation using coordinates. In R6 ; pick coordinates x1 ; x2 ; x3 and y1 ; y2 ; y3 so that zj /C30xj /C27iyj (1) gives it the structure of C3 : Then fz /C30dz1 ffldz2 ffldz3 (2) is a local section of the canonical bundle. A unitary change of coordinates w /C30Az, where A is a UNITARY MATRIX , transforms f by det A; i.e. fw /C30det Afz : (3) If the linear transformation A has DETERMINANT 1, that is, it is a special unitary transformation, then f is consistently defined as fz or as fw :/ On a Calabi-Yau manifold V, such a f can be defined globally, and the LIE GROUP SU(3) is very important in the theory. In fact, one of the many equivalent definitions, coming from RIEMANNIAN GEOMETRY , says that a Calabi-Yau manifold is a 2n/-dimensional manifold whose HOLONOMY GROUP reduces to SU(n): Another is that it is a CALIBRATED MANIFOLD with a CALIBRATION FORM c; which is algebraically the sameas the REAL PART of dz1 ffl...ffldzn : (4) Often, the extra assumptions that V is SIMPLY CONNECTED and/or COMPACT are made. Whatever definition is used, Calabi-Yau manifolds, as well as their MODULI SPACES , have interesting proper- ties. One is the symmetries in the numbers forming the HODGE DIAMOND of a compact Calabi-Yau mani- fold. It is surprising that these symmetries, called MIRROR SYMMETRY , can be realized by another Calabi- Yau manifold, the so-called mirror of the original Calabi-Yau manifold. The two manifolds together form a MIRROR PAIR. Some of the symmetries of the geometry of mirror pairs have been the object of recent research. See also CALIBRATED MANIFOLD ,CANONICAL BUNDLE , COMPLEX MANIFOLD ,DOLBEAULT COHOMOLOGY ,HAR- MONIC ,HODGE DIAMOND ,KA¨ HLER FORM,LIE GROUP , MIRROR PAIR,MODULI SPACE ,SPINOR ,VARIETY Calabi-Yau Variety CALABI- YAU SPACE Calculus In general, "a" calculus is an abstract theory devel- oped in a purely formal way. "The" calculus, more properly called ANALYSIS (or REAL ANALYSIS or, in older literature, INFINITESIMAL ANALYSIS ) is the branch of mathematics studying the rate of change of quantities (which can be interpreted as SLOPES of curves) and the length, AREA , and VOLUME of objects. The calculus is sometimes divided into DIFFERENTIAL and INTEGRAL CALCULUS , con- cerned with DERIVATIVES d dxf(x) and INTEGRALS g f(x) dx; respectively. While ideas related to calculus had been known for some time (Archimedes’ EXHAUSTION METHOD was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed. Even so, many years elapsed until the subject was put on a mathematically rigorous footing by mathematicians such as Weierstrass. See also ARC LENGTH ,AREA,CALCULUS OF VARIA- TIONS ,CHANGE OF VARIABLES THEOREM ,DERIVATIVE , DIFFERENTIAL CALCULUS ,E LLIPSOIDAL CALCULUS , EXTENSIONS CALCULUS ,F LUENT ,F LUXION ,F RAC- TIONAL CALCULUS ,FUNCTIONAL CALCULUS ,FUNDA- MENTAL THEOREMS OF CALCULUS ,HEAVISIDE CALCU- LUS,INTEGRAL ,INTEGRAL CALCULUS ,JACOBIAN , LAMBDA CALCULUS ,K IRBY CALCULUS ,M ALLIAVIN CALCU LUS,P REDICATE CALCULUS ,P ROPOSITIONAL CALCULUS ,SLOPE ,STOCHASTIC CALCULUS ,TENSOR CALCULUS ,UMBRAL CALCULUS ,VOLUME References Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999. Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Wal- tham, MA: Blaisdell, 1967. Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications to Differ- ential Equations and Probability. Waltham, MA: Blais- dell, 1969. Apostol, T. M.; Chrestenson, H. E.; Ogilvy, C. S.; Richmond, D. E.; and Schoonmaker, N. J. A Century of Calculus, Part I: 1894 /C1/1968. Washington, DC: Math. Assoc. Amer., 1992. Apostol, T. M.; Mugler, D. H.; Scott, D. R.; Sterrett, A. Jr.; and Watkins, A. E. A Century of Calculus, Part II: 1969 /C1/ 1991. Washington, DC: Math. Assoc. Amer., 1992. Ayres, F. Jr. and Mendelson, E. Schaum’s Outline of Theory and Problems of Differential and Integral Calculus, 3rd ed. New York: McGraw-Hill, 1990. Borden, R. S. A Course in Advanced Calculus. New York: Dover, 1998. Boyer, C. B. A History of the Calculus and Its Conceptual Development. New York: Dover, 1989. Brown, K. S. "Calculus and Differential Equations." http:// www.seanet.com/~ksbrown/icalculu.htm. Courant, R. and John, F. Introduction to Calculus and Analysis, Vol. 1. New York: Springer-Verlag, 1999. Courant, R. and John, F. Introduction to Calculus and Analysis, Vol. 2. New York: Springer-Verlag, 1990. Hahn, A. Basic Calculus: From Archimedes to Newton to Its Role in Science. New York: Springer-Verlag, 1998. Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1992. Marsden, J. E. and Tromba, A. J. Vector Calculus, 4th ed. New York: W. H. Freeman, 1996. Mendelson, E. 3000 Solved Problems in Calculus. New York: McGraw-Hill, 1988. Strang, G. Calculus. Wellesley, MA: Wellesley-Cambridge Press, 1991. Weisstein, E. W. "Books about Calculus." http://www.trea- sure-troves.com/books/Calculus.html. Calculus of Variations A branch of mathematics which is a sort of general- ization of CALCULUS . Calculus of variations seeks to find the path, curve, surface, etc., for which a given FUNCTION has a STATIONARY VALUE (which, in physi- cal problems, is usually a MINIMUM or MAXIMUM ). Mathematically, this involves finding STATIONARY VALUES of integrals OF THE FORM i /C30ga bf(y; ˙y; x) dx : (1) i has an extremum only if the EULER- LAGRANGEDIFFERENTIAL EQUATION is satisfied, i.e., if @f @y /C28d dx@f @ ˙y ! /C300: (2) the FUNDAMENTAL LEMMA OF CALCULUS OF VARIA- TIONS states that, if gb aM(x)h(x) dx /C300 (3) for all h(x) with CONTINUOUS second PARTIAL DERIVA- TIVES , then M(x) /C300 (4) on (a, b). A generalization of calculus of variations known as MORSE THEORY (and sometimes called "calculus of variations in the large" uses nonlinear techniques to address variational problems. See also BELTRAMI IDENTITY ,BOLZA PROBLEM ,BRA- CHISTOCHRONE PROBLEM ,CATENARY ,ENVELOPE THE- OREM ,E ULER- LAGRANGE DIFFERENTIAL EQUATION , ISOPERIMETRIC PROBLEM ,ISOVOLUME PROBLEM ,LIN- DELOF’S THEOREM ,M ORSE THEORY ,PLATEAU’S PRO- BLEM ,P OINT- POINT DISTANCE–2- D, POINT- POINT DISTANCE–3- D, ROULETTE ,S KEW QUADRILATERAL , SPHERE WITH TUNNEL ,S URFACE OF REVOLUTION , UNDULOID ,W EIERSTRASS- ERDMAN CORNER CONDI- TION References Arfken, G. "Calculus of Variations." Ch. 17 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 925 /C1/962, 1985. Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, 1925. Forsyth, A. R. Calculus of Variations. New York: Dover, 1960. Fox, C. An Introduction to the Calculus of Variations. New York: Dover, 1988. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, 1992. Jeffreys, H. and Jeffreys, B. S. "Calculus of Variations." Ch. 10 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 314 /C1/332, 1988. Menger, K. "What is the Calculus of Variations and What are Its Applications?" In The World of Mathematics (Ed. K. Newman). Redmond, WA: Microsoft Press, pp. 886 /C1/ 890, 1988. Sagan, H. Introduction to the Calculus of Variations. New York: Dover, 1992. Smith, D. R. Variational Methods in Optimization. New York: Dover, 1998. Todhunter, I. History of the Calculus of Variations During the Nineteenth Century. New York: Chelsea, 1962. Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974. Weisstein, E. W. "Books about Calculus of Variations." http://www.treasure-troves.com/books/CalculusofVariatio ns.html. Calcus 1 calcus /C131 2304: See also HALF,Q UARTER ,SCRUPLE ,U NCIA ,U NIT FRACTION Caldero ´n’s Formula f(x) /C30Ccg/C12 /C28/C12g/C12 /C28/C12f ; ca; bl11ml111 ca; b(x)a /C282 da db ; where ca ; b(x) /C30 ajj/C281 =2cx /C28 b a ! : This result was originally derived using HARMONIC ANALYSIS , but also follows from a WAVELETS view- point. C*-Algebra A special type of B*-ALGEBRA in which the INVOLU- TION is the ADJOINT operator in a HILBERT SPACE . See also B*-ALGEBRA , K-THEORY References Davidson, K. R. -Algebras by Example. Providence, RI: Amer. Math. Soc., 1996. Wegge-Olsen, N. E. K-Theory and -Algebras: A Friendly Approach. Oxford, England: Oxford University Press, 1993. Caliban Puzzle A puzzle in LOGIC in which one or more facts must be inferred from a set of given facts. Calibration Form A calibration form on a RIEMANNIAN MANIFOLD M is a DIFFERENTIAL K-FORM f such that 1. f is a CLOSED FORM . 2. The COMASS of f; sup v /C23fflpTM ; vjj/C301f(v) jj (1) defined as the largest value of f on a p vector of p- volume one, equals 1. A p-dimensional submanifold is calibrated when f restricts to give the VOLUME FORM . It is not hard to see that a calibrated submanifold N minimizes its volume among objects in its HOMOLOGY CLASS .ByS TOKES’ THEOREM ,ifN ? represents thesame homology class, then gNf /C30gN ?f: (2) Since vol(N) /C30gNf (3) and vol(N ?) ]gN ?f; (4) it follows that the volume of N is less than or equal to the volume of N ?:/ A simple example is dx on the plane, for which the lines y /C30c are calibrated submanifolds. In fact, in this example, the calibrated submanifolds give a FOLIA- TION .OnaK A¨ HLER MANIFOLD , the KA¨ HLER FORM v is a calibration form, which is INDECOMPOSABLE . For example, on C2 /C30 (x1 /C27y1i ; x2 /C27y2i) fg ; (5) the Ka¨hler form is dx1 ffldy1 /C27dx2 ffldy2 : (6) On a KA¨ HLER MANIFOLD , the calibrated submanifolds are precisely the complex submanifolds. Conse- quently, the complex submanifolds are locally volume minimizing. See also KA¨ HLER FORM,KA¨ HLER MANIFOLD ,VOLUME FORM Calogero-Degasperis-Fokas Equation The PARTIAL DIFFERENTIAL EQUATION uxxx /C281 8 u3 x /C27ux(Aeu /C27Be/C28u) /C300: References Gerdt, V. P.; Shvachka, A. B.; and Zharkov, A. Y. "Computer Algebra Applications for Classification of Integrable Non- Linear Evolution Equations." J. Symb. Comput. 1, 101 /C1/ 107, 1985. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 132, 1997. Calugareanu Theorem Letting Lk be the LINKING NUMBER of the two components of a ribbon, Tw be the TWIST , and Wr be the WRITHE , then Lk(K)/C30Tw(K)/C27Wr(K): (Adams 1994, p. 187). See also GAUSS INTEGRAL ,LINKING NUMBER ,TWIST , WRITHE References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. Calugareanu, G. "L’inte ´grale de Gauss et l’Analyse des n/œ/ uds tridimensionnels." Rev. Math. Pures Appl. 4,5/C1/20, 1959. Calugareanu, G. "Sur les classes d’isotopie des noeuds tridimensionnels et leurs invariants." Czech. Math. J. 11, 588 /C1/625, 1961. Calugareanu, G. "Sur les enlacements tridimensionnels des courbes ferme ´es." Comm. Acad. R. P. Romı ˆne 11, 829 /C1/ 832, 1961. Kaul, R. K. Topological Quantum Field Theories--A Meeting Ground for Physicists and Mathematicians. 15 Jul 1999. http://xxx.lanl.gov/abs/hep-th/9907119/. Pohl, W. F. "The Self-Linking Number of a Closed Space Curve." J. Math. Mech. 17, 975 /C1/985, 1968. Calvary Cross See also CROSS Cameron’s Sum-Free Set Constant A set of POSITIVE INTEGERS S is sum-free if the equation x /C27y /C30z has no solutions x, y, z /C23 S: The probability that a random sum-free set S consists entirely of ODD INTEGERS satisfies 0 :21759 5c 50 :21862 : References Cameron, P. J. "Cyclic Automorphisms of a Countable Graph and Random Sum-Free Sets." Graphs and Combi- natorics 1, 129 /C1/135, 1985. Cameron, P. J. "Portrait of a Typical Sum-Free Set." In Surveys in Combinatorics 1987 (Ed. C. Whitehead). New York: Cambridge University Press, 13 /C1/42, 1987. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cameron/camer- on.html. Campbell’s Theorem Any n-dimensional RIEMANNIAN MANIFOLD can be locally EMBEDDED into an (n /C271)/-dimensional mani- fold with RICCI CURVATURE Rab /C300: A similar version of the theorem for a PSEUDO- RIEMANNIAN MANIFOLD states that any n-dimensional PSEUDO- RIEMANNIAN MANIFOLD can be locally and isometrically embedded in an n(n /C271)=2/-dimensional PSEUDO- EUCLIDEAN SPACE .See also EMBEDDING ,P SEUDO- EUCLIDEAN SPACE , PSEUDO- RIEMANNIAN MANIFOLD ,RICCI CURVATURE , RIEMANNIAN MANIFOLD References Eisenhart, L. P. Riemannian Geometry. Princeton, NJ: Princeton University Press, 1964. Cancellation ANOMALOUS CANCELLATION Cancellation Law If bc /C13bd (mod a) and (b; a) /C301 (i.e., a and b are RELATIVELY PRIME ), then c /C13d (mod a) :/ See also CONGRUENCE References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 36, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 56, 1993. Cannonball Problem Find a way to stack a SQUARE of cannonballs laid out on the ground into a SQUARE PYRAMID (i.e., find a SQUARE NUMBER which is also SQUARE PYRAMIDAL ). This corresponds to solving the DIOPHANTINE EQUA- TION Xk i /C301i2 /C301 6 k(1 /C27k)(1 /C272k) /C30N2 for some pyramid height k. The only solution is k /C3024, N /C3070, corresponding to 4900 cannonballs (Ball and Coxeter 1987, Dickson 1952), as conjectured by Lucas (1875, 1876) and proved by Watson (1918). See also SPHERE PACKING ,SQUARE NUMBER ,SQUARE PYRAMID ,SQUARE PYRAMIDAL NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 59, 1987. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, p. 25, 1952. Lucas, E ´. Question 1180. Nouvelles Ann. Math. Ser. 2 14, 336, 1875. Lucas, E ´. Solution de Question 1180. Nouvelles Ann. Math. Ser. 2 15, 429/C1/432, 1876. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 77 and 152, 1988. Pappas, T. "Cannon Balls & Pyramids." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 93, 1989. Watson, G. N. "The Problem of the Square Pyramid." Messenger. Math. 48,1/C1/22, 1918. Canonical The word canonical is used to indicate a particular choice from of a number of possible conventions. This convention allows a mathematical object or class of objects to be uniquely identified or standardized. For example, the RIGHT-HAND RULE for the CROSS PRO- DUCT is a convention, which corresponds to the canonical ORIENTATION in R3 :/ See also BASIS (VECTOR SPACE ), CANONICAL BRICK, CANONICAL BUNDLE ,CANONICAL TRANSFORMATION , RATIONAL CANONICAL FORM Canonical Box Matrix JORDAN BLOCK Canonical Brick A1/C292 /C294 RECTANGULAR PARALLELEPIPED . See also BRICK References Gardner, M. "Mathematical Games: In Which a Mathema- tical Aesthetic is Applied to Modern Minimal Art." Sci. Amer. 239,22/C1/32, Nov. 1978. Canonical Bundle The canonical bundle is a HOLOMORPHIC LINE BUNDLE on a COMPLEX MANIFOLD which is determined by its COMPLEX STRUCTURE . On a coordinate chart (z1 ; ...zn) ; it is spanned by the nonvanishing section dz1 ffl...ffldzn : The TRANSITION FUNCTION between COORDINATE CHARTS is given by the determinant of the JACOBIAN of the coordinate change. The canonical bundle is defined in a similar way to the HOLOMORPHIC TANGENT BUNDLE . In fact, it is the nth EXTERIOR POWER of the DUAL BUNDLE to the HOLOMORPHIC TANGENT BUNDLE . Canonical Form A clear-cut way of describing every object in a class in a ONE-TO-ONE manner. See also NORMAL FORM,ONE-TO- ONE References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 7, 1996.Canonical Polygon A closed polygon whose vertices lie on a POINT LATTICE and whose edges consist of vertical and horizontal steps of unit length or diagonal steps (at angles which are multiples of 45 8 with respect to the lattice axes) of lengthffiffiffi 2p : In addition, no two steps may be taken in the same direction, no edge inter- sections are allowed, and no point may be a vertex of two edges. The numbers of distinct canonical poly- gons of n /C301, 2, ... sides are 0, 0, 1, 3, 3, 9, 13, 48, 125, ... (Sloane’s A052436). There are exactly eight distinct convex canonical polygons, illustrated above. The concept can also be generalized to diagonals rotated with respect to the lattice axes. See also GOLYGON ,LATTICE POLYGON References Kyrmse, R. E. "Canonical Polygons." http://users.sti.com.br/ rkyrmse/canonic-e.htm. Sloane, N. J. A. Sequences A052436 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Canonical Polyhedron APOLYHEDRON is said to be canonical if all its EDGES touch a SPHERE and the center of gravity of their contact points is the center of that SPHERE . Each combinatorial type of ( GENUS zero) polyhedron con- tains just one canonical version. The A RCHIMEDEAN SOLIDS and their DUALS are all canonical. References Hart, G. W. "Calculating Canonical Polyhedra." Mathema- tica Educ. Res. 6,5/C1/10, Summer 1997. Hart, G. "Calculating Canonical Polyhedra." http:// www.georgehart.com/canonical/canonical-supple- ment.html. Hart, G. "Canonical Polyhedra." http://www.georgehart.com/ virtual-polyhedra/canonical.html. Canonical Transformation SYMPLECTIC DIFFEOMORPHISM Cantor Comb CANTOR SET Cantor Diagonal Argument CANTOR DIAGONAL METHOD Cantor Diagonal Method A clever technique used by Georg Cantor to show that the INTEGERS and REALS cannot be put into a ONE-TO- ONE correspondence (i.e., the UNCOUNTABLY INFINITE set of REAL NUMBERS is "larger" than the COUNTABLY INFINITE set of INTEGERS ). It proceeds by first considering a countably infinite list of elements from a set S, each of which is an infinite set (in the case of the REALS , the decimal expansion of each REAL ). A new member S? of S is then created by arranging its nth term to differ from the nth term of the nth member of S. This shows that S is not COUNTABLE , since any attempt to put it in one-to-one correspondence with the integers will fail to include some elements of S. The argument is rather subtle, and requires some care to describe clearly. See also CARDINALITY ,C ONTINUUM HYPOTHESIS , COUNTABLE SET,COUNTABLY INFINITE References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 81 /C1/83, 1996. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 220 /C1/223, 1998. Penrose, R. The Emperor’s New Mind: Concerning Compu- ters, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 84 /C1/85, 1989. Cantor Diagonal Slash CANTOR DIAGONAL METHOD Cantor Dust A FRACTAL which can be constructed using STRING REWRITING by creating a matrix three times the sizeof the current matrix using the rules line 1 : "+" 0 "++"; ""0 "" line 2 : "+" 0 "" ; ""0 "" line 3 : "+" 0 "++"; ""0 "" Let Nn be the number of black boxes, Ln the length of a side of a box, and Anthe fractional AREA of black boxes after the nth iteration. Nn /C304n (1) Ln /C30(1 3)n /C303/C28n (2) An /C30L2 nNn /C30(4 9)n : (3) The CAPACITY DIMENSION is therefore dcap /C30/C28 lim n0/C12ln Nn ln Ln/C30/C28 lim n0/C12ln (4n) ln (3/C28n)/C302l n2 ln 3 :1:26186 : (4) See also BOX FRACTAL ,SIERPINSKI CARPET ,SIERPINS- KI SIEVE References Dickau, R. M. "Cantor Dust." http://forum.swarthmore.edu/ advanced/robertd/cantor.html. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 103 /C1/104, 1993. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Cantor Function The function whose values are 1 2c1 2/C27.../C27cm/C281 2m/C281/C272 2m ! for any number between a/C13c1 3/C27.../C27cm/C281 3m/C281/C271 3m and b/C13c1 3/C27.../C27cm/C281 3m/C281/C272 3m: Chalice (1991) shows that any real-valued function F(x)o n[ 0 ;1] which is MONOTONE INCREASING and satisfies 1.F(0)/C300;/ 2.F(x=3)/C30F(x)=2;/ 3.F(1/C28x)/C301/C28F(x)/ is the Cantor function. The DEVIL’S STAIRCASE is sometimes also called the Cantor function (Devaney 1987, p. 110). See also CANTOR SET,DEVIL’S STAIRCASE References Chalice, D. R. "A Characterization of the Cantor Function." Amer. Math. Monthly 98, 255 /C1/258, 1991. Devaney, R. L. An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, 1987. Wagon, S. "The Cantor Function" and "Complex Cantor Sets." §4.2 and 5.1 in Mathematica in Action. New York: W. H. Freeman, pp. 102 /C1/108 and 143 /C1/149, 1991. Cantor Set The Cantor set /(T/C12) is given by taking the interval [0; 1] (set T0) ; removing the middle third (/T1); removing the middle third of each of the two remain- ing pieces (/T2) ; and continuing this procedure ad infinitum. It is therefore the set of points in the INTERVAL [0; 1] whose ternary expansions do not contain 1, illustrated above. This produces the SET of REAL NUMBERS fxg such that x /C30c1 3/C27.../C27cn 3n /C27...; (1) where cnmay equal 0 or 2 for each n. This is an infinite, PERFECT SET. The total length of the LINE SEGMENTS in the nth iteration is ln /C302 3 !n ; (2) and the number of LINE SEGMENTS is Nn /C302n ; so the length of each element is en /C13l N /C3013 ! n (3) and the CAPACITY DIMENSION is dcap /C13/C28 lim e00/C27ln N ln e/C30/C28 lim n0/C12n ln 2 /C28n ln 3 /C30ln 2 ln 3 /C300 :630929... : (4) The Cantor set is nowhere DENSE , so it has LEBESGUE MEASURE 0. A general Cantor set is a CLOSED SET consisting entirely of BOUNDARY POINTS . Such sets are UNCOUN- TABLE and may have 0 or POSITIVE LEBESGUE MEA- SURE . The Cantor set is the only totally disconnected, perfect, COMPACT METRIC SPACE up to a HOMEO- MORPHISM (Willard 1970). See also ALEXANDER’S HORNED SPHERE ,ANTOINE’S NECKLACE ,C ANTOR FUNCTIO N,C LOSED SET, SCRAWNY CANTOR SETReferences Boas, R. P. Jr. A Primer of Real Functions. Washington, DC: Amer. Math. Soc., 1996. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 15 /C1/ 20, 1991. Harris, J. W. and Stocker, H. "Cantor Set." §4.11.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 114, 1998. Willard, S. §30.4 in General Topology. Reading, MA: Addi- son-Wesley, 1970. Cantor Square Fractal A FRACTAL which can be constructed using STRING REWRITING by creating a matrix three times the size of the current matrix using the rules line 1 : "+" 0 "+++"; ""0 "" line 2 : "+" 0 "++"; ""0 "" line 3 : "+" 0 "+++"; ""0 "" The first three steps are illustrated above. The size of the unit element after the nth iteration is Ln /C3013 ! n and the number of elements is given by the RECUR- RENCE RELATION Nn /C304Nn/C281 /C275(9n) where N1 /C135; and the first few numbers of elements are 5, 65, 665, 6305, .... Expanding out gives Nn /C305Xn k /C3004n/C28k9k /C281 /C309n /C284n : The CAPACITY DIMENSION is therefore D /C30/C28 lim n0/C12lnNn lnLn/C30/C28lim n0/C12ln(9n/C284n) ln(3/C28n)/C30/C28lim n0/C12ln(9n) ln(3/C28n) /C30ln 9 ln 3/C302l n3 ln 3/C302: Since the DIMENSION of the filled part is 2 (i.e., the SQUARE is completely filled), Cantor’s square fractal is not a true FRACTAL . See also BOX FRACTAL ,CANTOR DUST References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 82 /C1/ 83, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Cantor-Dedekind Axiom The points on a line can be put into a ONE-TO-ONE correspondence with the REAL NUMBERS . See also CARDINAL NUMBER ,CONTINUUM HYPOTH- ESIS,DEDEKIND CUT Cantor’s Equation ve /C30 e ; where v is an ORDINAL NUMBER and e is an INACCES- SIBLE CARDINAL . See also CARDINAL NUMBER ,INACCESSIBLE CARDINAL , ORDINAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 274, 1996. Cantor’s Intersection Theorem A theorem about (or providing an equivalent defini- tion of)) COMPACT SETS , originally due to Georg Cantor. Given a decreasing sequence of bounded nonempty CLOSED SETS C1 ‡C2 ‡C3 ‡... in the real numbers, then Cantor’s intersection theorem states that there must exist a point p in their intersection, p /C23 Cn for all n. For example, 0 /C23S [0; 1=n] : It is also true in higher DIMENSIONS of EUCLIDEAN SPACE . Note that the hypotheses stated above are crucial. The infinite intersection of open intervals may be empty, for instance S (0; 1=n) : Also, the infinite intersection of unbounded closed sets may be EMPTY , e.g., S [n;/C12] :/ Cantor’s intersection theorem is closely related to the HEINE- BOREL THEOREM and BOLZANO- WEIERSTRASS THEOREM , each of which can be easily derived from either of the other two. It can be used to show that the CANTOR SET is nonempty. See also BOLZANO- WEIERSTRASS THEOREM ,BOUNDED SET,CANTOR SET,CLOSED SET,COMPACT SET,HEINE- BOREL THEOREM ,INTERSECTION ,R EAL NUMBER , TOPOLOGICAL SPACECantor’s Paradox The SET of all SETS is its own POWER SET. Therefore, the CARDINALITY of the SET of all SETS must be bigger than itself. See also CANTOR’S THEOREM ,POWER SET References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 5, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 32 /C1/33, 1998. Cantor’s Theorem The CARDINAL NUMBER of any set is lower than the CARDINAL NUMBER of the set of all its subsets. A COROLLARY is that there is no highest /C210 (ALEPH ). See also CANTOR’S PARADOX Cap A topological object produced by puncturing a surface a single time, attaching two ZIPS around the puncture in opposite directions, distorting the hole so that the zips line up, and then zipping up. The cap is topologically trivial in the sense that a surface with a cap is topologically equivalent to a surface without one. See also CROSS- CAP,CROSS- HANDLE ,CUP,H ANDLE , SPHERICAL CAP References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 104, 1971. Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393/C1/399, 1999. Capacity TRANSFINITE DIAMETER Capacity Dimension ADIMENSION also called the FRACTAL DIMENSION , HAUSDORFF DIMENSION , and H AUSDORFF- BESICOV- ITCH DIMENSION in which nonintegral values are permitted. Objects whose capacity dimension is dif- ferent from their TOPOLOGICAL DIMENSION are called FRACTALS . The capacity dimension of a compact METRIC SPACE Xis a REAL NUMBER dcapicity such that if n( e) denotes the minimum number of open sets of diameter less than or equal to e; then n( e) is propor- tional to e/C28D as e 0 0: Explicitly, dcapacity /C13/C28lim e00/C27ln N ln e (if the limit exists), where N is the number of elements forming a finite COVER of the relevant METRIC SPACE and e is a bound on the diameter of the sets involved (informally, e is the size of each element used to cover the set, which is taken to approach 0). If each element of a FRACTAL is equally likely to be visited, then dcapacity /C30dinformation ; where dinformation is the INFORMATION DIMENSION . The capa- city dimension satisfies dcorrelation 5dinformation 5dcapacity where dcorrelation is the CORRELATION DIMENSION , and is conjectured to be equal to the LYAPUNOV DIMEN- SION. See also CORRELATION EXPONENT ,DIMENSION ,HAUS- DORFF DIMENSION ,KAPLAN- YORKE DIMENSION References Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 538 /C1/541, 1995. Peitgen, H.-O. and Richter, D. H. The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag, 1986. Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: Dekker, 1977. Cap-Cyclide Coordinates A coordinate system obtained by INVERSION of the BICYCLIDE COORDINATES . They are given by thetransformation equations x /C30L a Ysn m dn n cos c (1) y /C30L aYsn m dn n sin c (2) z /C30ffiffiffi kp Pi 2a Y; (3) where L/C301 /C28dn2 m sn2 n (4) Y/C30sn2 m dn2 n /C27Lffiffiffi kp/C27cn m dn m sn n cn n"#2 (5) P/C30L2 k/C28(sn2 m dn2 n /C27cn2 m dn2 m sn2 n cn2 n) ; (6) and cn x; dn x; and sn x are JACOBI ELLIPTIC FUNC- TIONS . Surfaces of constant m are ring cyclides with complicated equations (Moon and Spencer 1988, p. 133), surfaces of constant n are cap-cyclides with complicated equations (Moon and Spencer 1988, p. 133), and surfaces of constant c are half-planes tan c /C30y x : (7) See also BICYCLIDE COORDINATES ,CYCLIDIC COORDI- NATES ,D ISK-CYCLIDE COORDINATES ,FLAT-RING CY- CLIDE COORDINATES References Moon, P. and Spencer, D. E. "Cap-Cyclide Coordinates ( m; n ; c) :/" Fig. 4.11 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 132 /C1/ 135, 1988. Capping CUMULATION Carathe ´odory Derivative A function f is Carathe ´odory differentiable at a if there exists a function f which is CONTINUOUS at a such that f(x) /C28f(a) /C30 f(x)(x /C28a) : Every function which is Carathe ´odory differentiable is also F RE´CHET DIFFERENTIABLE . See also DERIVATIVE ,FRE´ CHET DERIVATIVE Carathe ´odory’s Fundamental Theorem Each point in the CONVEX HULL of a set S in Rn is in the convex combination of n /C271 or fewer points of S. See also CONVEX HULL,HELLY’S THEOREM References Eckhoff, J. "Helly, Radon, and Carathe ´odory Type Theo- rems." Ch. 2.1 in Handbook of Convex Geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 389 /C1/448, 1993. Carathe ´odory’s Theorem If V1and V2are bounded domains, @V1 ;@V2are JORDAN CURVES , and 8 : V1 0V2is a CONFORMAL MAPPING , then 8 (respectively, 8/C281) extends one-to- one and continuously to @V1 (respectively, @V2):/ References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 152, 1999. Cardano’s Formula CUBIC EQUATION Cardinal Addition Let A and B be any sets with empty INTERSECTION , and let ½X ½ denote the CARDINAL NUMBER of a SET X. Then ½A½/C27½B ½/C30½A @ B ½ (Ciesielski 1997, p. 68; Dauben 1990, p. 173; Rubin 1967, p. 274; Suppes 1972, pp. 112 /C1/113). It is an interesting exercise to show that cardinal addition is WELL DEFINED . The main steps are to show that for any CARDINAL NUMBERS a and b, there exist disjoint sets A and B with CARDINAL NUMBERS a and b, and to show that if A and B are disjoint and C and D disjoint with ½A½/C30½C½ and ½B ½/C30½D ½ then ½A @ B½/C30 ½C @ D½: The second of these is easy. The first is a little tricky and requires an appeal to the axioms of SET THEORY . Also, one needs to restrict the definition of cardinal to guarantee if a is a cardinal, then there is a set A satisfying ½A½/C30a :/ See also CARDINAL MULTIPLICATION ,CARDINAL EX- PONENTIATION References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.Cardinal Comparison For any sets A and B, their CARDINAL NUMBERS satisfy ½A½5½B½ IFF there is a one-to-one function f from A into B (Rubin 1967, p. 266; Suppes 1972, pp. 94 and 116). It is easy to show this satisfies the reflexive and transitive axioms of a PARTIAL ORDER . However, it is difficult to show the antisymmetry property, whose proof is known as the SCHRO ¨ DER- BERNSTEIN THEOREM . To show the trichotomy prop- erty, one must use the AXIOM OF CHOICE . Although an order type can be defined similarly, it does not seem usual to do so. See also SCHRO ¨ DER-BERNSTEIN THEOREM References Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Cardinal Exponentiation Let A and B be any sets, and let ½X ½ be the CARDINAL NUMBER of a set X. Then cardinal exponentiation is defined by ½A½½B ½/C30½set of all function from B into A½ (Ciesielski 1997, p. 68; Dauben 1990, p. 174; Moore 1982, p. 37; Rubin 1967, p. 275, Suppes 1972, p. 116). It is easy to show that the CARDINAL NUMBER of the POWER SET of A is 2 ½A ½; sine ½f0; 1 g½/C302 and there is a natural BIJECTION between the SUBSETS of A and the functions from A into f0; 1g:/ See also CARDINAL ADDITION ,CARDINAL MULTIPLICA- TION ,CARDINAL NUMBER ,POWER SET References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Cardinal Multiplication LetAandBbe any sets. Then the product of ½A½and ½B½is defined as the C ARTESIAN PRODUCT ½A½+½B½/C30½A/C29B½ (Ciesielski 1997, p. 68; Dauben 1990, p. 173; Moore 1982, p. 37; Rubin 1967, p. 274; Suppes 1972, pp. 114 /C1/115). See also CARDINAL ADDITION ,CARDINAL EXPONENTIA- TION References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Cardinal Number In common usage, a cardinal number is a number used in counting (a COUNTING NUMBER ), such as 1, 2, 3, .... In formal SET THEORY , a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting SETS using it gives the same result. (This is not true for the ORDINAL NUMBERS .) In fact, the cardinal numbers are obtained by collecting all ORDINAL NUMBERS which are obtain- able by counting a given set. A set has /C2100(ALEPH-0 ) members if it can be put into a ONE-TO-ONE corre- spondence with the finite ORDINAL NUMBERS . The cardinality of a set is also frequently referred to as the "power" of a set (Moore 1982, Dauben 1990, Suppes 1972). In Cantor’s original notation, the symbol for a SET A annotated with a single overbar ¯A indicated A stripped of any structure besides order, hence it represented the ORDER TYPE of the set. A double overbar ¯¯A then indicated stripping the order from the set and thus indicated the cardinal number of the set. However, in modern notation, the symbol ½A½ is used to denote the cardinal number of set. Cantor, the father of modern SET THEORY , noticed that while the ORDINAL NUMBERS v /C271 ; v /C272; ... were bigger than omega in the sense of order, they were not bigger in the sense of EQUIPOLLENCE . This led him to study what would come to be called cardinal numbers. He called the ordinals v; v /C271; ... that are equipollent to the integers "the second number class" (as opposed to the finite ordinals, which he called the "first number class"). Cantor showed 1. The second number class is bigger than the first. 2. There is no class bigger than the first number class and smaller than the second. 3. The class of real numbers is bigger than the first number class. One of the first serious mathematical definitions of cardinal was the one devised by Gottlob Frege and Bertrand Russell, who defined a cardinal number ½A½ as the set of all sets EQUIPOLLENT to A. (Moore 1982, p. 153; Suppes 1972, p. 109). Unfortunately, the objects produced by this definition are not sets in the sense of ZERMELO- FRAENKEL SET THEORY , butrather "PROPER CLASSES " in the terminology of von Neumann. Tarski (1924) proposed to instead define a cardinal number by stating that every set A is associated with a cardinal number ½A½; and two sets A and B have the same cardinal number IFF they are EQUIPOLLENT (Moore 1982, pp. 52 and 214; Rubin 1967, p. 266; Suppes 1972, p. 111). The problem is that this definition requires a special axiom to guarantee that cardinals exist. A. P. Morse and Dana Scott defined cardinal number by letting A be any set, then calling ½A½ the set of all sets EQUIPOLLENT to A and of least possible RANK (Rubin 1967, p. 270). It is possible to associate cardinality with a specific set, but the process required either the AXIOM OF FOUNDATION or the AXIOM OF CHOICE . However, these are two of the more controversial ZERMELO- FRAENKEL AXIOMS . With the AXIOM OF CHOICE , the cardinals can be enumerated through the ordinals. In fact, the two can be put into one-to-one correspondence. The AXIOM OF CHOICE implies that every set can be WELL ORDERED and can therefore be associated with an ORDINAL NUMBER . This leads to the definition of cardinal number for a SETAas the least ORDINAL NUMBER bsuch that A and bare EQUIPOLLENT . In this model, the cardinal numbers are just the INITIAL ORDINALS . This defini- tion obviously depends on the AXIOM OF CHOICE , because if the AXIOM OF CHOICE is not true, then there are sets that cannot be well ordered. Cantor believed that every set could be well ordered and used this correspondence to define the /C210/s ("alephs"). For any ORDINAL NUMBER a;/C210a/C30va:/ An INACCESSIBLE CARDINAL cannot be expressed in terms of a smaller number of smaller cardinals. See also ALEPH ,ALEPH-0 ,ALEPH-1 ,CANTOR- DEDEKIND AXIOM ,CANTOR DIAGONAL SLASH ,CARDINAL ADDI- TION ,CARDINAL EXPONENTIATION ,CARDINAL MULTI- PLICATION ,C ONTINUUM ,C ONTINUUM HYPOTHESIS , EQUIPOLLENT ,INACCESSIBLE CARDINALS AXIOM ,IN- FINITY ,O RDINAL NUMBER ,P OWER SET,S URREAL NUMBER ,UNCOUNTABLE SET References Cantor, G. U¨ber unendliche, lineare Punktmannigfaltigkei- ten, Arbeiten zur Mengenlehre aus dem Jahren 1872 /C1/ 1884. Leipzig, Germany: Teubner, 1884. Conway, J. H. and Guy, R. K. "Cardinal Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 277 /C1/ 282, 1996. Courant, R. and Robbins, H. "Cantor’s ‘Cardinal Numbers."’ §2.4.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 83 /C1/86, 1996. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Ferreiro ´s, J. "The Notion of Cardinality and the Continuum Hypothesis." Ch. 6 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 171 /C1/214, 1999. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Tarski, A. "Sur quelques the ´ore`mes qui e ´quivalent a `l’ax- iome du choix." Fund. Math. 5, 147/C1/154, 1924. Cardinality CARDINAL NUMBER Cardioid The curve given by the POLAR equation r/C30a(1/C27cosu); (1) sometimes also written r/C302b(1/C27cosu); (2) where b/C13a=2;the C ARTESIAN equation (x2/C27y2/C28ax)2/C30a2(x2/C27y2); (3) and the PARAMETRIC EQUATIONS x/C30acost(1/C27cost) (4) y/C30asint(1/C27cost): (5) The cardioid is a degenerate case of the LIMAC ¸ON.I ti s also a 1-CUSPED EPICYCLOID (with r/C30r) and is the CAUSTIC formed by rays originating at a point on the circumference of a CIRCLE and reflected by the CIRCLE . the name cardioid was first used by de castillon in philosophical transactions of the royal society in 1741. its ARC LENGTH was found by la hire in 1708. there are exactly three PARALLEL TANGENTS to the cardioid with any given gradient. also, the TANGENTS at the ends of any CHORD through the CUSP point are atRIGHT ANGLES . The length of any CHORD through the CUSP point is 2 a:/ The cardioid may also be generated as follows. Drawa CIRCLE Cand fix a point Aon it. Now draw a set of CIRCLES centered on the CIRCUMFERENCE ofCand passing through A. The ENVELOPE of these CIRCLES is then a cardioid (Pedoe 1995). Let the CIRCLE Cbe centered at the origin and have RADIUS 1, and let the fixed point be A/C30(1;0):Then the RADIUS of a CIRCLE centered at an ANGLE ufrom (1, 0) is r2/C30(0/C28cosu)2/C27(1/C28sinu)2 /C30cos2u/C271/C282 sin u/C27sin2u/C302(1/C28sinu):(6) If the fixed point Ais not on the circle, then the resulting ENVELOPE is a LIMAC ¸ONinstead of a cardi- oid. The ARC LENGTH ,CURVATURE , and TANGENTIAL ANGLE are s/C30gt 02½cos(1 2t)½dt/C304asin(12u) (7) k/C303½sec(12u)½ 4a(8) f/C303 2u: (9) As usual, care must be taken in the evaluation of s(t) fort>p:Since (7) comes from an integral involving the ABSOLUTE VALUE of a function, it must be monotonic increasing. Each QUADRANT can be treated correctly by defining n/C30t p$% /C271; (10) where xbcis the FLOOR FUNCTION , giving the formula s(t)/C30(/C281)1/C27[n(mod 2)]4 sin(1 2t)/C27812njk : (11) The PERIMETER of the curve is L /C30g2 p 0½2a cos(1 2 u) ½ du /C304ag p 0cos(12 u) d u /C304ag p =2 0cos f(2 df) /C308ag p=2 0cos f d f /C308a[sin f]p =2 0/C308a : (12) The AREA is A /C301 2g2 p 0r2 du /C3012 a2g2p 0(1 /C272 cos u /C27cos2 u) du /C3012 a2g2 p 0f1 /C272 cos u /C2712[1 /C27cos(2 u)]g du /C3012 a2g2 p 0[32 /C272 cos u /C2712cos(2 u)] du /C3012 a2[32 u /C272 sin u /C2714sin(2u)]2p 0/C3032 pa2 : (13) See also CARDIOID COORDINATES ,CIRCLE ,CISSOID , COIN PARADOX ,C ONCHOID ,E QUIANGULAR SPIRAL , LEMNISCATE ,LIMAC ¸ ON,MANDELBROT SET References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 214, 1987. Gray, A. "Cardioids." §3.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 54 /C1/55, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 118 /C1/121, 1972. Lockwood, E. H. "The Cardioid." Ch. 4 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 34 /C1/43, 1967. MacTutor History of Mathematics Archive. "Cardioid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Car- dioid.html. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., pp. xxvi-xxvii, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 24 /C1/25, 1991. Yates, R. C. "The Cardioid." Math. Teacher 52,10/C1/14, 1959. Yates, R. C. "Cardioid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 4 /C1/7, 1952. Cardioid Caustic The CATACAUSTIC of a CARDIOID for a RADIANT POINT at the CUSP is a NEPHROID . The CATACAUSTIC for PARALLEL rays crossing a CIRCLE is a CARDIOID .Cardioid Coordinates A coordinate system (m ; n ; c) defined by the coordi- nate transformation x /C30mn ( m2 /C27 n2)2 cos c (1) y /C30mn ( m2 /C27 n2)2 sin c (2) z /C301 2n2 /C28 n2 ( m2 /C27 n2)2 (3) with m; n 50 and c /C23 0; 2p ½Þ : Surfaces of constant m are given by the cardioids of revolution intersecting the positive half of the z-axis x2 /C27y2 /C27z2 /C301 4m2 [ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2 /C27z2p /C271]; (4) surfaces of constant n by the cardioids of revolution intersecting the negative half of the z-axis x2/C27y2/C27z2/C301 4n2[ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2/C27z2p /C28z]; (5) and surfaces of constant cby the half-planes tanc/C30y x: (6) The metric coefficients are gmm/C301 (m2/C27n2)3(7) gnn/C301 (m2/C27n2)3(8) gcc/C30m2n2 (m2/C27n2)4(9) See also CARDIOID References Moon, P. and Spencer, D. E. "Cardioid Coordinate ( m; n ; c):/" Fig. 4.02 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 107 /C1/109, 1988. Cardioid Evolute x /C302 3 a /C2713 a cos u(1 /C28cos u) y /C301 3 a sin u(1 /C28cos u) : This is a mirror-image CARDIOID with a?/C30a=3 :/ Cardioid Inverse Curve If the CUSP of the cardioid is taken as the INVERSION CENTER , the cardioid inverts to a PARABOLA . Cardioid Involute x /C302a /C273a cos u(1 /C28cos u) y /C303a sin u(1 /C28cos u) : This is a mirror-image CARDIOID with a?/C303a :/ Cardioid Pedal Curve The PEDAL CURVE of the CARDIOID where the PEDAL POINT is the CUSP is CAYLEY’S SEXTIC .Cards Cards are a set of n rectangular pieces of cardboard with markings on one side and a uniform pattern on the other. The collection of all cards is called a "deck," and a normal deck of cards consists of 52 cards having 14 distinct values for each of four different "suits." The suits are called clubs (/$); diamonds (/2) ; hearts / ( +) ; and spades (/&): Spades and clubs are colored black, while hearts and diamonds are colored red. The cards of each suit are numbered 1 through 13, where the special terms ace (1), jack (11), queen (12), and king (13) are used instead of numbers 1 and 11 /C1/13. However, in BRIDGE and a number of other games, the ace is considered the highest card, and so would be assigned a value of 14 instead of 1. The randomization of the order of cards in a deck is called SHUFFLING . Cards are used in many gambling games (such as POKER ), and the investigation of the probabilities of various outcomes in card games was one of the original motivations for the development of modern PROBABILITY theory. See also BRIDGE CARD GAME,CLOCK SOLITAIRE ,COIN, COIN TOSSING ,CRIBBAGE ,DICE,POKER ,SHUFFLE References Chatto, W. A. Facts and Speculations on the Origin and History of Playing Cards. Saint Clair Shores, MI: Scho- larly Press, 1977. Hargrave, C. P. History of Playing Cards and a Bibliogra- phy of Cards and Gaming. New York: Dover, 1986. Horr, N. T. Bibliography of Card Games and of the History of Playing Cards. Montclair, NJ: Patterson Smith, 1972. Jessel, F. and Horr, N. T. Bibliographies of Works on Playing Cards and Gaming. Montclair, NJ: Patterson Smith, 1972. Leeming, J. Games and Fun with Playing Cards. New York: Dover, 1980. Parlett, D. S. A Dictionary of Card Games. Oxford, England: Oxford University Press, 1992. Parlett, D. S. The Oxford Guide to Card Games: A History of Card Games. Oxford, England: Oxford University Press, 1991. Parlett, D. S. Solitaire: Aces Up and 399 Other Card Games. New York: Pantheon, 1991. Sackson, S. Card Games Around the World. New York: Dover, 1994. University of Waterloo. "Playing Cards." http://www.ahs.u- waterloo.ca/~museum/vexhibit/plcards/plcards.html. Caret The symbol ffl which is used to denote partial conjunction in symbolic logic. It also appears in several other contexts in mathematics and is some- times called a " WEDGE ". The shape of the caret is similar to that of the HAT. See also HAT,W EDGE References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 274, 1997. Carleman Equation The system of PARTIAL DIFFERENTIAL EQUATIONS ut /C27ux /C30v2 /C28u2 vt /C28vx /C30u2 /C28v2 : References Kaper, H. G. and Leaf, G. K. "Initial Value Problems for the Carleman Equation." Nonlinear Anal. 4, 343 /C1/362, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Carleman’s Inequality Let fai gn i/C301be a SET of POSITIVE numbers. Then Xn i/C301(a1a2 ...ai)1=i 5eXn i/C301ai (which is given incorrectly in Gradshteyn and Ryzhik 1994). Here, the constant E is the best possible, in the sense that counterexamples can be constructed for any stricter INEQUALITY which uses a smaller con- stant. The theorem is suggested by writing a ?i/C30ap iin HARDY’S INEQUALITY Xn i/C301a1/C27.../C27ai i !p Bp p/C281 !pXn i/C301api(1) and letting p0/C12:/ See also ARITHMETIC MEAN, E,G EOMETRIC MEAN, HARDY’S INEQUALITY References Carleman, T. "Sur les fonctions quasi-analytiques." Confe ´r- ences faites au cinqui‘eme congre `s des mathe ´maticiens scandinaves. Helsingfors, pp. 181 /C1/196, 1923. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1126, 2000. Hardy, G. H.; Littlewood, J. E.; and Po ´lya, G. "Carleman’s Inequality." §9.12 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 249 /C1/250, 1988. Kaluza, T. and Szego, G. "U ¨ber Reihen mit lauter positiven Gliedern." J. London Math. Soc. 2, 266/C1/272, 1927. Knopp, K. "U ¨ber Reihen mit positiven Gliedern." J. London Math. Soc. 3, 205/C1/211, 1928. Mitrinovic, D. S. Analytic Inequalities. New York: Springer- Verlag, p. 131, 1970. Ostrowski, A. "U ¨ber quasi-analytischen Funktionen und Bestimmtheit asymptotischer Entwicklungen." Acta Math. 53, 181/C1/266, 1929. Po´lya, G. "Proof of an Inequality." Proc. London Math. Soc. 24, lvii, 1926. Valiron, G. §3, Appendix B in Lectures on the General Theory of Integral Functions. New York: Chelsea, pp. 186 /C1/187, 1949. Carlson-Levin Constant N.B. A detailed online essay by S. Finch was the starting point for this entry.Assume that fis a NONNEGATIVE REAL function on 0;/C12½Þ and that the two integrals g/C12 0xp/C281/C28l[f(x)]pdx (1) g/C12 0xq/C281/C27m[f(x)]qdx (2) exist and are FINITE .I f p/C30q/C302 and l/C30m/C301; Carlson (1934) determined g/C12 0f(x)dx 5ffiffiffippg/C12 0[f(x)]2dxl11sl11n 1=4g/C12 0x2[f(x)]2dxl11sl11n 1=4 (3) and showed thatffiffiffippis the best constant (in the sense that counterexamples can be constructed for any stricter INEQUALITY which uses a smaller constant). For the general case g/C12 0f(x)dx 5Cg/C12 0xp/C281/C28l[f(x)]pdxl11sl11n sg/C12 0xq/C281/C27m[f(x)]qdxl11sl11n t ; (4) and Levin (1948) showed that the best constant C/C301 (ps)s(qt)tGs a ! Gt a ! (l/C27m)Gs/C27t a !2 666643 77775a ; (5) where s/C13m pm/C27ql(6) t/C13l pm/C27ql(7) a/C131/C28s/C28t (8) andG(z) is the GAMMA FUNCTION . References Beckenbach, E. F.; and Bellman, R. Inequalities. New York: Springer-Verlag, 1983. Boas, R. P. Jr. Review of Levin, V. I. "Exact Constants in Inequalities of the Carlson Type." Math. Rev. 9, 415, 1948. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/crlslvn/crlslvn.html. Levin, V. I. "Exact Constants in Inequalities of the Carlson Type." Doklady Akad. Nauk. SSSR (N. S.) 59, 635/C1/638, 1948. English review in Boas (1948). Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Amsterdam, Netherlands: Kluwer, 1991. Carlson’s Theorem If f(z) is regular and OF THE FORM O(ekzjj) where k B p; for R[z] ]0; and if f(z) /C300 for z /C300, 1, ..., then f(z)is identically zero. See also GENERALIZED HYPERGEOMETRIC FUNCTION References Bailey, W. N. "Carlson’s Theorem." §5.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 36 /C1/40, 1935. Carlson, F. "Sur une classe de se´ries de Taylor." Disserta- tion. Uppsala, Sweden, 1914. Hardy, G. H. "On Two Theorems of F. Carlson and S. Wi- gert." Acta Math. 42, 327 /C1/339, 1920. Riesz, M. "Sur le principe de Phragme ´n-Lindelo ¨f." Proc. Cambridge Philos. Soc. 20, 205 /C1/207, 1920. Riesz, M. Erratum to "Sur le principe de Phragme ´n- Lindelo ¨f." Proc. Cambridge Philos. Soc. 21, 6, 1921. Titchmarsh, E. C. Ch. 5 in The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, 1960. Wigert, S. "Sur un the´ore`me concernant les fonctions entie`res." Archiv fo¨r Mat. Astr. o Fys. 11, No. 22, 1916. Carlyle Circle Consider a QUADRATIC EQUATION x2 /C28sx /C27p /C300 where s and p denote signed lengths. The CIRCLE which has the points A /C30(0; 1) and B /C30(s ; p)asa DIAMETER is then called the Carlyle circle Cs;pof the equation. The CENTER of Cs;p is then at the MIDPOINT of AB, M /C30(s =2 ; (1 /C27p) =2); which is also the MID- POINT of S /C30(s ; 0) and Y /C30(0; 1 /C27p) : Call the points at which Cs;p crosses the X-AXIS H1 /C30(x1 ; 0) and H2 /C30 (x2 ; 0) (with x1 ]x2) : Then s /C30x1 /C27x2 p /C30x1x2 (x /C28x1)(x /C28x2) /C30x2 /C28sx /C27p ; so x1 and x2 are the ROOTS of the quadratic equation. See also 257-GON , 65537-GON ,HEPTADECAGON ,PENTA- GON References Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 4 /C1/5, 1982. De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97 /C1/108, 1991.Eves, H. An Introduction to the History of Mathematics, 6th ed. Philadelphia, PA: Saunders, 1990. Leslie, J. Elements of Geometry and Plane Trigonometry with an Appendix and Very Copious Notes and Illustra- tions, 4th ed., improved and exp. Edinburgh: W. & G. Tait, 1820. Carmichael Condition A number n satisfies the Carmichael condition IFF (p /C281) (n=p /C281) j for all PRIME DIVISORS p of n. This is equivalent to the condition (p /C281) (n /C281) j for all PRIME DIVISORS p of n. See also CARMICHAEL NUMBER References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,40/C1/50, 1996. Carmichael Function There are two definitions of the Carmichael function. One is the reduced totient function (also called the least universal exponent function), defined as the smallest integer m such that kn /C131 (mod n) for all k RELATIVELY PRIME to n. The ORDER of a (mod n)isat most l(n) (Ribenboim 1989). The first few values of this function, implemented in Mathematica 4.0 as CarmichaelLambda [n], are 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, ... (Sloane’s A002322). It can be defined recur- sively as l(n) /C30f(n) for n /C30p a ; p /C302 and a 52; or p ]3 1 2 f(n) for n /C302 a and a ]3 LCM[ l(pai i)]ifor n /C30Q ipai i:8 < : Some special values are l(1) /C301 l(2) /C301 l(4) /C302 l(2r) /C302r/C282 for r ]3; and l ?(pr) /C30 f(pr) for p an ODD PRIME and r ]1:/ The second Carmichael’s function l ?(n) is given by the LEAST COMMON MULTIPLE (LCM) of all the FACTORS of the TOTIENT FUNCTION f(n);except that if 8 n;jthen 2a/C282is a FACTOR instead of 2a/C281:The values of l?(n) for the first few nare 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, ... (Sloane’s A011773). See also MODULO MULTIPLICATION GROUP ,TOTIENT FUNCTION References Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 27, 1989. Riesel, H. "Carmichael’s Function." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 273 /C1/275, 1994. Sloane, N. J. A. Sequences A002322/M0298 and A011773 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences /eisonline.html. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, p. 226, 1991. Carmichael Lambda CARMICHAEL FUNCTION Carmichael Number A Carmichael number is an ODD COMPOSITE NUMBER n which satisfies FERMAT’S LITTLE THEOREM an /C281 /C281 /C130 (mod n) (1) for every choice of a satisfying (a ; n) /C301 (i.e., a and n are RELATIVELY PRIME ) with 1 Ba Bn : A Carmichael number is therefore a PSEUDOPRIME to any base. Carmichael numbers therefore cannot be found to be COMPOSITE using FERMAT’S LITTLE THEOREM . How- ever, if (a; n) "1; the congruence of FERMAT’S LITTLE THEOREM is sometimes NONZERO , thus identifying a Carmichael number n as COMPOSITE . Carmichael numbers are sometimes called "absolute pseudoprimes" and also satisfy KORSELT’S CRITERION . R. D. Carmichael first noted the existence of such numbers in 1910, computed 15 examples, and con- jectured that there were infinitely many. In 1956, Erdos sketched a technique for constructing large Carmichael numbers (Hoffman 1998, p. 183), and a proof was given by Alford et al. (1994). The first few Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ... (Sloane’s A002997). The number of Carmichael num- bers less than 102,103, ... are 0, 1, 7, 16, 43, 105, ... (Sloane’s A055553; Pinch 1993). The smallest Carmi- chael numbers having 3, 4, ... factors are 561 /C303 /C29 11 /C2917; 41041 /C307 /C2911 /C2913 /C2941; 825265, 321197185, ... (Sloane’s A006931). Carmichael numbers have at least three PRIME FACTORS . For Carmichael numbers with exactly three PRIME FACTORS , once one of the PRIMES has been specified, there are only a finite number of Carmi- chael numbers which can be constructed. Indeed, for Carmichael numbers with k prime factors, there are only a finite number with the least k /C282 specified. Numbers OF THE FORM (6k /C271)(12 k /C271)(18 k /C271) are Carmichael numbers if each of the factors is PRIME (Korselt 1899, Ore 1988, Guy 1994). This can be seen since for N /C13(6k /C271)(12 k /C271)(18 k /C271) /C301296 k3 /C27396k2 /C2736k /C271; (2) /N /C281 is a multiple of 36k and the LEAST COMMONMULTIPLE of 6k; 12k; and 18k is 36k; so aN /C281 /C131 modulo each of the PRIMES 6k /C271; 12k /C271 ; and 18k /C27 1; hence aN /C281 /C131 modulo their product. The first few such Carmichael numbers correspond to k /C301, 6, 35, 45, 51, 55, 56, ... (Sloane’s A046025) and are 1729, 294409, 56052361, 118901521, ... (Sloane’s A033502). In Jan. 1999, Dubner found the largest known Carmichael of this form, having 4848 digits and index k /C30133752260 /C215 3003 /C215 101604 (3) The prime factors of N have 1616, 1616, and 1617 digits. Let C(n) denote the number of Carmichael numbers less than n. Then, for all sufficiently large n, C(n) > n2 =7 (4) (Alford et al. 1994), which proves that there are infin- itely many Carmichael numbers. The upper bound C(n) Bn expln n ln ln ln n ln ln n ! (5) has also been proved (R. G. E. Pinch). The Carmichael numbers have the following proper- ties: 1. If a PRIME pdivides the Carmichael num- ber n, then /n/C131 (mod p/C281)/implies that n/C13p(mod p(p/C281)). 2. Every Carmichael number is SQUAREFREE . 3. An ODD COMPOSITE SQUAREFREE number nis a Carmichael number IFFndivides the DENOMINA- TORof the B ERNOULLI NUMBER Bn/C281:/ The largest known Carmichael numbers having a given number of factors are summarized in thefollowing table (Dubner 1989, Dubner 1998). Factors Digits Discoverer 3 10200 Dubner 4 2467 Caldwell and Dubner 5 1015 Caldwell and Dubner 6 827 Caldwell and Dubner See also C ARMICHAEL CONDITION ,PSEUDOPRIME References Alford, W. R.; Granville, A.; and Pomerance, C. "There are Infinitely Many Carmichael Numbers." Ann. Math. 139, 703/C1/722, 1994. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 87, 1987. Carlini, A. and Hosoya, A. Carmichael Numbers on a Quantum Computer. 5 Aug 1999. http://xxx.lanl.gov/abs/ quant-ph/9908022/. Dubner, H. "A New Method for Producing Large Carmichael Numbers." Math. Comput. 53, 411 /C1/414, 1989. Dubner, H. "Carmichael Number Record." Posting to [email protected] . Sep. 11, 1998. Dubner, H. "3-Component Carmichael Number." Posting to [email protected] . Jan. 15, 1999. Guy, R. K. "Carmichael Numbers." §A13 in Unsolved Pro- blems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 30 /C1/32, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 182 /C1/183, 1998. Korselt, A. "Proble `me chinois." L’interme ´diaire math. 6, 143 /C1/143, 1899. Ore, Ø. Number Theory and Its History. New York: Dover, 1988. Pinch, R. G. E. "The Carmichael Numbers up to 1015." Math. Comput. 55, 381 /C1/391, 1993. Pinch, R. G. E. ftp://ftp.dpmms.cam.ac.uk/pub/Carmichael/. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimes to 25 /C215 109 :/" Math. Comput. 35, 1003 /C1/1026, 1980. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 118 /C1/125, 1996. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkha ¨user, pp. 89 /C1/90 and 94 /C1/95, 1994. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 116, 1993. Sloane, N. J. A. Sequences A002997/M5462, A006931/ M5463, A033502, A046025, and A055553 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Carmichael Sequence A FINITE , INCREASING SEQUENCE of INTEGERS fa1 ; ...; am g such that (ai /C281) (a1 ...ai /C281) j for i /C301, ..., m, where mnj indicates that m DIVIDES n. A Carmichael sequence has exclusive EVEN or ODD elements. There are infinitely many Carmichael sequences for every order. See also GIUGA SEQUENCE References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,40/C1/50, 1996. Carmichael’s Conjecture Carmichael’s conjecture asserts that there are an INFINITE number of CARMICHAEL NUMBERS . This was proven by Alford et al. (1994). See also CARMICHAEL NUMBER ,CARMICHAEL’S TOTI- ENT FUNCTION CONJECTURE References Alford, W. R.; Granville, A.; and Pomerance, C. "There Are Infinitely Many Carmichael Numbers." Ann. Math. 139, 703 /C1/722, 1994. Cipra, B. What’s Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., 1993.Guy, R. K. "Carmichael’s Conjecture." §B39 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 94, 1994. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimes to 25 /C215 109 :/" Math. Comput. 35, 1003 /C1/1026, 1980. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 29 /C1/31, 1989. Schlafly, A. and Wagon, S. "Carmichael’s Conjecture on the Euler Function is Valid Below 1010 ;000;000 :/" Math. Comput. 63, 415 /C1/419, 1994. Carmichael’s Theorem If a and n are RELATIVELY PRIME so that the GREAT- EST COMMON DIVISOR GCD( a ; n) /C301 ; then a l(n) /C131 (mod n) where l is the CARMICHAEL FUNCTION . Carmichael’s Totient Function Conjecture It is thought that the TOTIENT VALENCE FUNCTION Nf(n) ]2 ; i.e., if there is an n such that f(x) /C30n; then there are at least two solutions x. This assertion is called Carmichael’s totient function conjecture and is equivalent to the statement that there exists an m "n such that f(n) /C30 f(m) (Ribenboim 1996, pp. 39 /C1/40). Dickson 1952 (p. 137) states that the conjecture was proved by Carmichael (1907), who also developed a method of finding the solution (Carmichael 1909). The result also appears as in exercise in Carmichael (1914). However, Carmichael (1922) subsequently discovered an error in the proof, and the conjecture currently remain open. Any counterexample to the conjecture must have more than 10,000,000 DIGITS (Schlafly and Wagon 1994; conservatively given as 10,000 in Conway and Guy 1996, p. 155). Ford (1998ab) showed that if there is a counter- example to Carmichael’s conjecture, then a positive proportion of totients are counterexamples. SIERPINSKI’S CONJECTURE states that all integers > 1 appear as multiplicities of the TOTIENT VALENCE FUNCTION . See also TOTIENT FUNCTION ,SIERPINSKI’S CONJEC- TURE ,TOTIENT VALENCE FUNCTION References Carmichael, R. D. "On Euler’s f/-Function." Bull. Amer. Math. Soc. 13, 241/C1/243, 1907. Carmichael, R. D. "Notes on the Simplex Theory of Num- bers." Bull. Amer. Math. Soc. 15, 217/C1/223, 1909. Carmichael, R. D. The Theory of Numbers. New York: Wiley, 1914. Carmichael, R. D. "Note on Euler’s f/-Function." Bull. Amer. Math. Soc. 28, 109/C1/110, 1922. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, 1952. Ford, K. "The Distribution of Totients." Ramanujan J. 2, 67/C1/151, 1998a. Ford, K. "The Distribution of Totients, Electron. Res. Announc. Amer. Math. Soc. 4,27/C1/34, 1998b. Guy, R. K. "Carmichael’s Conjecture." §B39 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 94 /C1/95, 1994. Klee, V. "On a Conjecture of Carmichael." Bull. Amer. Math. Soc. 53, 1183 /C1/1186, 1947. Masai, P. and Valette, A. "A Lower Bound for a Counter- example to Carmichael’s Conjecture." Boll. Un. Mat. Ital. 1, 313 /C1/316, 1982. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Schlafly, A. and Wagon, S. "Carmichael’s Conjecture on the Euler Function is Valid Below 1010 ;000;000 :/" Math. Comput. 63, 415 /C1/419, 1994. Carnot’s Polygon Theorem If a PLANE cuts the sides AB, BC, CD, and DA of a SKEW QUADRILATERAL ABCD in points P, Q, R, and S, then AP PB/C215BQQC/C215CR RD/C215DS SA /C301 both in magnitude and sign (Altshiller-Court 1979, p. 111). More generally, if P1 ; P2 ; ..., are the VERTICES of a finite POLYGON with no "minimal sides" and the side PiPj meets a curve in the POINTS Pij1 and Pij2 ; then Q iP1P12iQ iP2P23i/C1/C1/C1Q iPNPN1iQ iPNPN1i/C1/C1/C1Q iP2P2i1/C301 ; where AB denotes the DISTANCE from POINT A to B. References Altshiller-Court, N. "Carnot’s Theorem." §329 in Modern Pure Solid Geometry. New York: Chelsea, p. 111, 1979. Carnot, L. N. M. Ge´ome´trie de position. Paris: Duprat, p. 287, 1803. Carnot, L. N. M. Me´moir sur la relation qui existe entre les distances respectives de cinq points quelconques pris dans l’espace; suivi d’un Essai sur la the´orie des transversales. Paris: Courcier, p. 71, 1806. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 160, 1888. Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 190, 1959. Carnot’s Theorem Given any TRIANGLE A1A2A3 ; the signed sum of PERPENDICULAR distances from the CIRCUMCENTER O to the sides is OO1 /C27OO2 /C27OO3 /C30R /C27r ; where r is the INRADIUS and R is the CIRCUMRADIUS . The sign of the distance is chosen to be POSITIVE IFF the entire segment OOi lies outside the TRIANGLE . See also JAPANESE TRIANGULATION THEOREMReferences Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, pp. 256 and 262, 1972. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 25, 1985. Carotid-Kundalini Fractal A fractal-like structure is produced for x B0by superposing plots of CAROTID- KUNDALINI FUNCTIONS cknof different orders n. the region /C281 Bx B0is called FRACTAL LAND by pickover (1995), the central region the GAUSSIAN MOUNTAIN RANGE , and the region 0 Bx B1 OSCILLATION LAND . The plot above shows n /C301 to 25. Gaps in FRACTAL LAND occur whenever x cos/C281 x /C302 pp q for p and q RELATIVELY PRIME INTEGERS . At such points x, the functions assume the (q /C271)=2 de values cos(2 pr =q) for r /C300, 1, ..., q=2bc ; where zdeis the CEILING FUNCTION and zbcis the FLOOR FUNCTION . References Pickover, C. A. "Are Infinite Carotid-Kundalini Functions Fractal?" Ch. 24 in Keys to Infinity. New York: Wiley, pp. 179 /C1/181, 1995. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Carotid-Kundalini Function The FUNCTION given by CKn(x)/C13cos(nxcos/C281x); where nis an INTEGER and/C281BxB1:/ See also CAROTID- KUNDALINI FRACTAL Carry The operating of shifting the leading DIGITS of an ADDITION into the next column to the left when the SUM of that column exceeds a single DIGIT (i.e., 9 in base 10). See also ADDEND ,ADDITION ,BORROW Carrying Capacity LOGISTIC GROWTH CURVE Cartan Decomposition References Huang, J.-S. "Linear Reductive Groups and Cartan Decom- position." §10.1 in Lectures on Representation Theory. Singapore: World Scientific, pp. 129 /C1/130, 1999. Cartan Matrix A Cartan matrix is a SQUARE INTEGER MATRIX who elements (Aij) satisfy the following conditions. 1. Aij is an integer, one of f/C283;/C282;/C281; 0 ; 2 g:/ 2. Aii /C302 the diagonal entries are all 2. 3. Aij 50 off of the diagonal. 4. Aij /C300 iff Aji /C300:/ 5. There exists a DIAGONAL MATRIX D such that DAD/C281 gives a SYMMETRIC and POSITIVE DEFINITE QUADRATIC FORM . A Cartan matrix can be associated to a SEMISIMPLE LIE ALGEBRA g: It is a k /C29k SQUARE MATRIX , where k is the RANK of g: The SIMPLE ROOTS are the basis vectors, and Aij is determined by their inner product, using the KILLING FORM . Aij /C302 /C142ai ; aj /C143=/C142aj ; aj /C143 (1) In fact, it is more a table of values than a matrix. By reordering the basis vectors, one gets another Cartan matrix, but it is considered equivalent to the original Cartan matrix. The Lie algebra g can be reconstructed, up to ISOMORPHISM , by the 3k generators fej ; fi ; hi g which satisfy the SERRE RELATIONS . In fact, g/C30h/C154e/C154f (2) where h;e;f are the LIE SUBALGEBRAS generated by the generators of the same letter. For example, A /C302 /C281 /C2812l12ml121 (3)is a Cartan matrix. The LIE ALGEBRA g has six generators fh1 ; h2 ; e1 ; e2 ; f1 ; f2 g: They satisfy the following relations. 1. [h1 ; h2] /C300:/ 2. [e1 ; f1] /C30h1/ and /[e2 ; f2] /C30h2/ while [e1, f2] /C30 [e2, f1] /C300. 3. [hi ; ej] /C30/C28Aijej :/ 4. [hi ; fj] /C30/C28Aijfj :/ 5. e12 /C30[e1 ; e2] "0 and f12 /C30[f1 ; f2] "0 :/ 6. [ei ; e12] /C300 and [fi ; f12] /C300 :/ From these relations, it is not hard to see that g/C30sl3 with the standard REPRESENTATION h1 /C30101 0 /C2810 0002 435 (4) h 2 /C3000 0 01 0 00 /C2812 435 (5) e 1 /C30010 0000002 435 (6) e 2 /C30000 0010002 435 (7) e 12 /C30001 000 0002 435 (8) f 1 /C30000 1000002 435 (9) f 2 /C30000 000 0102 435 (10) f 12 /C30000 000 /C281002 435: (11) In addition, the W EYL GROUP can be constructed directly from the Cartan matrix. Its rows determine the reflections against the simple roots. The following Mathematica command converts a Cartan matrix to alist of generators for the Weyl group, in its represen- tation on the ROOT LATTICE . In particular, its output represents the matrices of the Weyl group as INTEGER MATRICES . See also DYNKIN DIAGRAM ,LIE ALGEBRA ,ROOT (LIE ALGEBRA ), ROOT SYSTEM ,SEMISIMPLE LIE ALGEBRA , SPECIAL LINEAR LIE ALGEBRA ,W EYL GROUP References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Jacobson, N. "The Determination of the Cartan Matrices." §4.5 in Lie Algebras. New York: Dover, pp. 121 and 128 /C1/ 135, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Cartan Relation The relationship Sqi(x%y) /C30aj/C27k/C30i Sqj(x)%Sqk(y) en- countered in the definition of the STEENROD ALGEBRA . Cartan Subgroup A type of maximal ABELIAN SUBGROUP . References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/549, 1996. Cartan Torsion Coefficient The ANTISYMMETRIC parts of the CHRISTOFFEL SYM- BOL OF THE SECOND KIND Gl mn :/ Cartesian Coordinates Cartesian coordinates are rectilinear 2-D or 3-D coordinates (and therefore a special case of CURVI- LINEAR COORDINATES ) which are also called rectan- gular coordinates. The three axes of 3-D Cartesian coordinates, conventionally denoted the X-, Y-, and Z- AXES (a NOTATION due to Descartes ) are chosen to be linear and mutually PERPENDICULAR . In 3-D, the coordinates x, y, and z may lie anywhere in the INTERVAL (/C28/C12;/C12) :/ The INVERSION of 3-D Cartesian is called 6-SPHERE COORDINATES coordinates. The SCALE FACTORS of Cartesian coordinates are all unity, hi /C301: The LINE ELEMENT is given by ds /C30dx ˆx /C27dy ˆy /C27dz ˆz ; (1) and the VOLUME ELEMENT by dV /C30dx dy dz: (2) The GRADIENT has a particularly simple form, 9/C13ˆx@ @x /C27ˆy@ @y /C27ˆz@ @z ; (3) as does the LAPLACIAN92 /C13@2 @x2 /C27@2 @y2 /C27@2 @z2 : (4) The LAPLACIAN is 92F /C139 /C215 ( 9F) /C30@2F @x2 /C27@2F @y2 /C27@2F @z2 /C30ˆx@2Fx @x2 /C27@2Fx @y2 /C27@2Fx @z2 ! /C27ˆy@2Fy @x2 /C27@2Fy @y2 /C27@2Fy @z2 ! /C27ˆz@2Fz @x2 /C27@2Fz @y2 /C27@2Fz @z2 ! : (5) The DIVERGENCE is 9 /C215 F /C30@Fx @x/C27@Fy @y/C27@Fz @z; (6) and the CURL is 9/C29F /C13ˆx ˆy ˆz @ @x@ @y@ @z FxFyFzl112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112/C30 @Fz @y/C28@Fy @z ! ˆx /C27@Fx @z/C28@Fz @x ! ˆy /C27@Fy @x/C28@Fx @y ! ˆz: (7) The GRADIENT of the DIVERGENCE is 9(9 /C215u)/C30@ @x@uz @x/C27@uy @y/C27@uz @z ! @ @y@uz @x/C27@uy @y/C27@uz @z ! @ @z@uz @x/C27@uy @y/C27@uz @z !2 66666666643 7777777775 /C30@ @x @ @y @ @z2 6666666643 777777775@ux @x/C27@uy @y/C27@uz @z ! : (8) LAPLACE’S EQUATION is separable in Cartesian coor- dinates. See also CARTESIAN GEOMETRY ,COORDINATES ,HELM- HOLTZ DIFFERENTIAL EQUATION– CARTESIAN COORDI- NATES , 6-SPHERE COORDINATES References Arfken, G. "Special Coordinate Systems--Rectangular Car- tesian Coordinates." §2.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 94 /C1/ 95, 1985. Moon, P. and Spencer, D. E. "Rectangular Coordinates (x; y; z) :/" Table 1.01 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 9 /C1/11, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 656, 1953. Cartesian Geometry The use of coordinates (such as CARTESIAN COORDI- NATES ) in the study of GEOMETRY . Cartesian geometry is named after Rene´ Descartes (Bell 1986, p. 48), although Descartes may have been anticipated by Fermat (Coxeter and Greitzer 1967, p. 31). See also ANALYTIC GEOMETRY ,CARTESIAN COORDI- NATES References Bell, E. T. Men of Mathematics. New York: Simon and Schuster, p. 48, 1986. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 31, 1967. Cartesian Ovals A curve consisting of two ovals which was first studied by Descartes in 1637. It is the locus of a point P whose distances from two FOCI F1and F2in two- center BIPOLAR COORDINATES satisfy mr 9nr ?/C30k ; (1) where m, n are POSITIVE INTEGERS , k is a POSITIVE real, and r and r ? are the distances from F1 and F2 : If m /C30n, the oval becomes an ELLIPSE .InC ARTESIAN COORDINATES , the Cartesian ovals can be written mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x /C28a)2 /C27y2q /C27nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x /C27a) 2 /C27y2q /C30k2 (2) (x2 /C27y2 /C27a2)(m2 /C28n2) /C282ax(m2 /C27n2) /C28k2 /C30/C282nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x /C27a) 2 /C27y2q ; (3) [(m2 /C28n2)(x2 /C27y2 /C27a2) /C282ax(m2 /C27n2)]2 /C302(m2 /C27n2)(n2 /C27y2 /C27a2) /C284ax(m2 /C28n2) /C28k2 : (4) Now define b /C13m2 /C28n2 (5) c /C13m2 /C27n2 ; (6)and set a /C301. Then [b(x2 /C27y2) /C282cx /C27b]2 /C274bx /C27k2 /C282c /C302c(x2 /C27y2) : (7) If c ? is the distance between F1and F2 ; and the equation r /C27mr ?/C30a (8) is used instead, an alternate form is [(1 /C28m2)(x2 /C27y2) /C272m2c ?x /C27a?2 /C28m2c ?2]2 /C304a?2(x2 /C27y2) : (9) The curves possess three FOCI.Ifm /C301, one Carte- sian oval is a central CONIC , while if m /C30a/c, then the curve is a LIMAC ¸ ON and the inside oval touches the outside one. Cartesian ovals are ANALLAGMATIC CURVES . References Baudoin, P. Les ovales de Descartes et le limac ¸on de Pascal. Paris: Vuibert, 1938. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 35, 1989. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 155 /C1/157, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 188, 1967. MacTutor History of Mathematics Archive. "Cartesian Oval." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Cartesian.html. Cartesian Product The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all points (a, b) where a /C23 A and b /C23 B : It is denoted A /C29B ; and is called the Cartesian product since it originated in Descartes’ formulation of analytic geometry. In the Cartesian view, points in the plane are specified by their vertical and horizontal coordinates, with points on a line being specified by just one coordinate. The main examples of direct products are E UCLIDEAN 3- space ( /R/C29R/C29R;where Rare the REAL NUMBERS ), and the plane ( /R/C29R):/ The GRAPH PRODUCT is sometimes called the Carte- sian product (Vizing 1963, Cark and Suen 2000). See also DIRECT PRODUCT ,DISJOINT UNION ,EXTER- NAL DIRECT PRODUCT ,EXTERNAL DIRECT SUM,GRAPH PRODUCT ,GROUP DIRECT PRODUCT ,PRODUCT SPACE References Clark, W. E. and Suen, S. "An Inequality Related to Vizing’s Conjecture." Electronic J. Combinatorics 7, No. 1, N4, 1 /C1/ 3, 2000. http://www.combinatorics.org/Volume_7/ v7i1toc.html#N4. Comtet, L. "Product Sets." §1.2 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed.Dordrecht, Netherlands: Reidel, pp. 3 /C1 /4, 1974. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 49 /C1/50, 1984. Vizing, V. G. "The Cartesian Product of Graphs." Vycisl. Sistemy 9,30/C1/43, 1963. Cartesian Space EUCLIDEAN SPACE Cartesian Trident TRIDENT OF DESCARTES Cartography The study of MAP PROJECTIONS and the making of geographical maps. See also MAP PROJECTION Cascade A Z/-ACTION or N/-ACTION . A cascade and a single MAP X 0 X are essentially the same, but the term "cas- cade" is preferred by many Russian authors. See also ACTION ,FLOW Casey’s Theorem Four CIRCLES c1;c2;c3;andc4are TANGENT to a fifth CIRCLE or a straight LINE IFF T12T349T13T429T14T23/C300: (1) where Tijis the length of a common TANGENT to CIRCLES iand j(Johnson 1929, pp. 121 /C1/122). The following cases are possible: 1. If all the Ts are direct common tangents, then c5 has like contact with all the circles, 2. If the Ts from one circle are transverse while the other three are direct, then this one circle hascontact with c 5unlike that of the other three, 3. If the given circles can be so paired that the common tangents to the circles of each pair are direct, while the other four are transverse, thenthe members of each pair have like contact with c 5/ (Johnson 1929, p. 125). The special case of Casey’s theorem shown above wasgiven in a S ANGAKU PROBLEM from 1874 in the Gumma Prefecture. In this form, a single circle is drawn inside a square, and four circles are thendrawn around it, each of which is tangent to the square on two of its sides. For a square of side length awith lower left corner at (0 ;0) containing a central circle of radius rwith center ( x, y), the radii and positions of the four circles can be found by solving (1/C28r4/C28x)2/C27(y/C28r4)2/C30(r/C27r4)2(2) (1/C28r1/C28x)2/C27(1/C28r1/C28y)2/C30(r/C27r1)2(3) (x/C28r3)2/C27(y/C28r3)2/C27(r/C27r3)2(4) (x/C28r2)2/C27(1/C28r2/C28y)2/C30(r/C27r2)2: (5) Four of the Tijfor the theorem are given immediately for the figure as T12/C30a/C28r1/C28r2 (6) T34/C30a/C28rr/C28r4 (7) T14/C30a/C28r1/C28r4 (8) T23/C30a/C28r2/C28r3: (9) The remaining T13andT24can be found as shown in the above right figure. Let cijbe the distance from Oi toOj;then c2 13/C30(a/C28r1/C28r3)2/C27(a/C28r1/C28r3)2/C302(a/C28r1/C28r3)2(10) c224/C30(a/C28r2/C28r4)2/C27(a/C28r2/C28r4)2 /C302(a/C28r2/C28r4)2; (11) so T13/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 13/C28(r3/C28r1)2q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(a/C28r1/C28r3)2/C28(r3/C28r1)2q (12) T24/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 24/C28(r2/C28r4)2q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(a/C28r2/C28r4)2/C28(r2/C28r4)2q : (13) Since the four circles are all externally tangent to c5; the relevant form of Casey’s theorem to use has signs (/C27;/C28);so we have the equation (a/C28r1/C28r2)(a/C28r3/C28r4)/C27(a/C28r1/C28r4)(a/C28r2/C28r3) /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [2(a/C28r1/C28r3)2/C28(r3/C28r1)2][2(a/C28r2/C28r4)2/C28(r2/C28r4)2]q /C300(14) (Rothman 1998). Solving for athen gives the relation- ship a/C302(r1r3/C28r2r4)/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(r1/C28r2)(r1/C28r4)(r3/C28r2)(r3/C28r4)p r1/C28r2/C27r3/C28r4 (15) Durell (1928) calls the following Casey’s theorem: if t is the length of a common tangent of two circles of radii aand b,t?is the length of the corresponding common tangent of their inverses with respect to any point, and a?and b?are the radii of their inverses, then t2 ab /C30t?2 a ?b ?: (16) See also PURSER’S THEOREM ,TANGENT CIRCLES References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 103, 1888. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 125, 1893. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 37, 1971. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 117, 1928. Fukagawa, H. and Pedoe, D. "Many Circles and Squares (Casey’s Theorem)." §3.3 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 41 /C1/42 and 120 /C1/1989. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 121 /C1/127, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 244 /C1/251, 1893. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/91, May 1998. Casimir Operator An OPERATOR G/C30Xm i/C301eR i uiR on a representation R of a LIE ALGEBRA . References Jacobson, N. Lie Algebras. New York: Dover, p. 78, 1979. Casoratian The Casoratian of sequences x(1) n; x(2)n; ..., x(k) nis defined by the k /C29k DETERMINANT C(x(1) n; x(2)n; x(k) n) /C30x(1) n x(2)n ... x(k) n x(1) n/C271 x(2)n/C271 ... x(k) n/C271 nn:::::: x(1) n/C27k /C281x(2)n/C27k /C281... x(k) n/C27k/C281l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112: The solutions x (1) n;x(2)n;...,x(k) nof the linear difference equation xn/C27k/C27b(k/C281) nxn/C27(k/C281)/C27.../C27b(1) nxn/C271/C27b(0)nxn/C300 forn/C300, 1, ..., are linearly independent sequences IFF their Casoratian is nonzero for n/C300 (Zwillinger 1995). See also LINEARLY DEPENDENT SEQUENCESReferences Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 229, 1995. Casorati-Weierstrass Theorem WEIERSTRASS- CASORATI THEOREM Cassini Ellipses CASSINI OVALS Cassini Ovals The curves, also called Cassini ellipses, described by a point such that the product of its distances from two fixed points a distance 2 aapart is a constant b2:The shape of the curve depends on b=a:IfaBb, the curve is a single loop with an OVAL (left figure above) or dog bone (second figure) shape. The case a/C30bproduces a LEMNISCATE (third figure). If a/C21b, then the curve consists of two loops (right figure). Cassini ovals are ANALLAGMATIC CURVES . The curve was first investigated by Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun traveled around the Earth on one of these ovals, withthe Earth at one FOCUS of the oval. The Cassini ovals are defined in two-center BIPOLAR COORDINATES by the equation r1r2/C30b2; (1) with the origin at a FOCUS . Even more incredible curves are produced by the locus of a point the product of whose distances from 3 or more fixed points is a constant. The Cassini ovals have the C ARTESIAN equation [(x/C28a)2/C27y2][(x/C27a)2/C27y2]/C30b4(2) or the equivalent form (x2/C27y2/C27a2)2/C284a2x2/C30b4(3) and the polar equation r4/C27a4/C282a2r2cos(2 u)/C30b4: (4) Solving for r2using the QUADRATIC EQUATION gives r2 /C302a2 cos(2 u) 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4a4 cos2(2u) /C28 4(a4 /C28 b4)p 2 /C30a2 cos(2 u) 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 cos2(2u) /C27b4 /C28a4p /C30a2 cos(2 u) 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4[cos2(2u) /C281] /C27b4p /C30a2 cos(2 u) 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b4 /C28a4 sin2(2u)q /C30a2cos(2 u) 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b a !4 /C28sin2(2u)vuut2 643 75: (5) Let a TORUS of tube radius a be cut by a plane perpendicular to the plane of the torus’s centroid. Call the distance of this plane from the center of the torus hole r, let a /C30r, and consider the intersection of this plane with the torus as r is varied. The resulting curves are Cassini ovals, with a LEMNISCATE occur- ring at r /C301=2 (Gosper). Cassini ovals are therefore TORIC SECTIONS . If a Bb, the curve has AREA A /C301 2 r2 d u /C302(12)g p =4 /C28p =4r2 d u /C30a2 /C27b2Ea4 b4 ! ; (6) where the integral has been done over half the curve and then multiplied by two and E(x) is the complete ELLIPTIC INTEGRAL OF THE SECOND KIND .Ifa /C30b, the curve becomes r2 /C30a2 cos(2 u) /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28sin2 uphi /C302a2 cos(2 u) ; (7) which is a LEMNISCATE having AREA A /C302a2 (8) (two loops of a curveffiffiffi 2p the linear scale of the usual lemniscate r2 /C30a2 cos(2 u) ; which has area A /C30a2 =2 for each loop). If a /C21b, the curve becomes two disjoint ovals with equations r/C309affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos(2 u)9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b a !2 /C28sin2(2u)vuutvuuut ; (9) where u/C23[/C28u 0;u0] and u0/C131 2sin/C281b a !22 435: (10) See also C ASSINI SURFACE ,LEMNISCATE ,M ANDEL- BROT SET,OVAL,TORUS References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 221, 1987. Gray, A. "Cassinian Ovals." §4.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 82 /C1/86, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 153 /C1/155, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 187 /C1/188, 1967. MacTutor History of Mathematics Archive. "Cassinian Ovals." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cassinian.html. Piziak, R. and Turner, D. "Exploring Gerschgorin Circles and Cassini Ovals." Mathematica Educ. 3,1 3/C1 /21, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 25 /C1/26, 1991. Yates, R. C. "Cassinian Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 8 /C1/ 11, 1952. Cassini Projection AMAP PROJECTION defined by x/C30sin/C281B (1) y/C30tan/C281 tanf cos(l/C28l0)"# ; (2) where B/C30cosfsin(l/C28l0): (3) The inverse FORMULAS are f/C30sin/C281(sinDcosx) (4) l/C30l0/C27tan/C281tanx cosD ! ; (5) where D/C30y/C27f0: (6) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 92 /C1/95, 1987. Cassini Surface The QUARTIC SURFACE obtained by replacing the constant b in the equation of the CASSINI OVALS with b /C30z, obtaining [(x /C28a)2 /C27y2][(x /C27a)2 /C27y2] /C30z4 : (1) As can be seen by letting y /C300 to obtain (x2 /C28a2)2 /C30z4 (2) x2 /C27z2 /C30a2 ; (3) the intersection of the surface with the y /C300 PLANE is a CIRCLE of RADIUS a. Let a TORUS of tube radius a be cut by a plane perpendicular to the plane of the torus’s centroid. Call the distance of this plane from the center of the torus hole r, let a /C30r, and consider the intersection of this plane with the torus as r is varied. The resulting curves are CASSINI OVALS , and the surface having these curves as CROSS SECTIONS is the Cassini surface (x /C272 /C27z2 /C27c2) /C284c2x2 /C304c2r2 ; which has a scaled r2 on the right side instead of z4 (Gosper). See also CASSINI OVALS ,TORUS References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 20, 1986. Fischer, G. (Ed.). Plate 51 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 51, 1986.Cassini’s Identity For Fn the nth FIBONACCI NUMBER , Fn/C281Fn/C271 /C28F2 n /C30(/C281)n : This identity was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165 /C1/168). It is a special case of CATALAN’S IDENTITY with r /C30 1. See also D’OCAGNE’S IDENTITY ,CATALAN’S IDENTITY , FIBONACCI NUMBER References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 41, 1967. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, p. 12, 1996. Casson Invariant References Akbulut, S. and McCarthy, J. Casson’s Invariant for Or- iented Homology 3-Spheres--An Exposition. Princeton, NJ: Princeton University Press, 1990. Saveliev, N. Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant. Berlin: de Gruyter, 1999. Castillon’s Problem Inscribe a TRIANGLE in a CIRCLE such that the sides of the TRIANGLE pass through three given POINTS A,B, andC. References Do¨rrie, H. "Castillon’s Problem." §29 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 144 /C1/147, 1965. F. Gabriel-Marie. Exercices de ge ´ome´trie. Tours, France: Maison Mame, pp. 20 /C1/22, 1912. Rouche ´, E. and de Comberousse, C. Traite ´de ge ´ome´trie plane. Paris: Gauthier-Villars, pp. 310 /C1/311, 1900. Casting Out Nines An elementary check of a MULTIPLICATION which makes use of the CONGRUENCE 10n/C131 (mod 9) for n] 2:From this CONGRUENCE ,a MULTIPLICATION ab/C30c must give a /C13X ai /C30a /C31 b /C13X bi /C30b /C31 c /C13X ci /C30c /C31; so ab /C13a/C31b /C31 must be /C13c /C31 (mod 9). Casting out nines was transmitted to Europe by the Arabs, but was probably an Indian invention and is therefore some- times also called "the Hindu check." The procedure was described by Fibonacci in his Liber Abaci (Wells 1986, p. 74). References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 28 /C1/29, 1996. Hilton, P.; Holton, D.; and Pedersen, J. "Casting Out 9’s and 11’s: Tricks of the Trade." Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 53 /C1/57, 1997. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 74, 1986. Casus Irreducibilus If P(x) is an irreducible CUBIC EQUATION all of whose roots are real, then to obtain them by radicals, you must take roots of nonreal numbers at some point. See also ALGEBRAIC INTEGER References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 547 and 551, 1998. Cat Map ARNOLD’S CAT MAP Catacaustic The curve which is the ENVELOPE of reflected rays. CARDIOID CUSP of CARDIOIDNEPHROID CIRCLE not on CIRCUM- FERENCELIMAC ¸ ON CIRCLE on CIRCUMFER- ENCECARDIOID CIRCLE point at /C12/ NEPHROID CISSOID OF DIOCLESFOCUS CARDIOID one arch of a CYCLOIDrays PERPENDI- CULAR axistwo arches of a CYCLOID DELTOID point at infinity ASTROID /ln x/ rays PARALLEL axisCATENARYLOGARITHMIC SPIRALORIGIN equal LOGARITH- MIC SPIRAL PARABOLA rays PERPENDI- CULAR axisTSCHIRNHAUSEN CUBIC QUADRIFOLIUM center ASTROID TSCHIRNHAUSEN CUBICFOCUS SEMICUBICAL PARABOLA See also CAUSTIC ,CIRCLE CAUSTIC ,DIACAUSTIC References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 60 and 207, 1972. Catafusene POLYHEX Catalan CATALAN’S CONSTANT Catalan Integrals Special cases of general FORMULAS due to Bessel. J0(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 /C28y2p ) /C301 p g p 0ey cos u cos(z sin u) du; where J0(z)isaB ESSEL FUNCTION OF THE FIRST KIND . Now, let z /C131 /C28z ? and y /C131 /C27z ?: Then J0(2iffiffiffizp)/C301 pgp 0e(1/C27z) cos ucos[(1/C28z) sin u]du: See also BESSEL FUNCTION OF THE FIRST KIND Catalan Number The Catalan numbers are an INTEGER SEQUENCE fCng which appears in TREE enumeration problems of the type, "In how many ways can a regular n-gon be divided into n/C282TRIANGLES if different orientations are counted separately?" (E ULER’S POLYGON DIVISION PROBLEM ). The solution is the Catalan number Cn/C282 (Do¨rrie 1965, Honsberger 1973), as graphically illu- strated above (Dickau). The first few Catalan num- bers for n/C301, 2, ... are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ... (Sloane’s A000108). The only ODD Catalan numbers are those OF THE FORM C2k/C281;and the last DIGIT is five for k/C309 to 15. The only PRIME Catalan numbers for n5215/C281 are C2/C302 and C3/C305:/ The Catalan numbers turn up in many other related types of problems. Cn/C281can also be defined as the number of ( /C281;1)/-sequences fs1;s2;...;sngsuch that a2n i/C301sj/C300 and ai j/C301sj]0 for i52n/C281 (Mays and Wojciechowski 2000). The following table gives the first few such sequences. nlists 1 /f1;/C281g/ 2 /f1;1;/C281;/C281g/ 3 /f1;1;/C281;1;/C281;/C281g;f1;1;1;/C281;/C281;/C281g/ 4 /f1;1;/C281;1;/C281;1;/C281;/C281g;/ /f1;1;/C281;1;1;/C281;/C281;/C281g;/ /f1;1;1;/C281;/C281;1;/C281;/C281g;/ /f1;1;1;/C281;1;/C281;/C281;/C281g;/ /f1;1;1;1;/C281;/C281;/C281;/C281g/ The Catalan number Cn/C281also gives the number of BINARY BRACKETINGS ofnletters (C ATALAN’S PRO- BLEM ), the solution to the BALLOT PROBLEM , thenumber of trivalent PLANTED PLANAR TREES (Dickau; illustrated above), the number of states possible in an n-FLEXAGON , the number of different diagonals pos- sible in a FRIEZE PATTERN with n/C271 rows, the number of ways of forming an n-fold exponential, the number of rooted planar binary trees with n internal nodes, the number of rooted plane bushes with nEDGES , the number of extended BINARY TREES with ninternal nodes, the number of mountains which can be drawn with nupstrokes and ndown- strokes, the number of noncrossing handshakes possible across a round table between npairs of people (Conway and Guy 1996), and the number of SEQUENCES with NONNEGATIVE PARTIAL SUMS which can be formed from n1s and n/C281s (Bailey 1996, Brualdi 1992)! An explicit formula for Cnis given by Cn/C131 n/C2712n nl11sl11n /C301 n/C271(2n)! n!2/C30(2n)! (n/C271)!n!; (1) where2n nl1ml11 denotes a BINOMIAL COEFFICIENT andn!i s the usual FACTORIAL .ARECURRENCE RELATION forCn is obtained from Cn/C271 Cn/C30(2n/C272)! (n/C272)[(n/C271)!]2(n/C271)(n!)2 (2n)! /C30(2n/C272)(2n/C271)(n/C271) (n/C272)(n/C271)2/C302(2n/C271)(n/C271)2 (n/C271)2(n/C272) /C302(2n/C271) n/C272; (2) so Cn/C271/C302(2n/C271) n/C272Cn: (3) Other forms include Cn/C302 /C2156 /C21510/C1/C1/C1(4n/C282) (n/C271)!(4) /C302n(2n/C281)!! (n/C271)!(5) /C30(2n)! n!(n/C271)!: (6) SEGNER’S RECURRENCE FORMULA , given by Segner in 1758, gives the solution to E ULER’S POLYGON DIVISION PROBLEM En/C30E2En/C281/C27E3En/C282/C27.../C27En/C281E2: (7) With E1/C30E2/C301;the above RECURRENCE RELATION gives the Catalan number Cn/C282/C30En:/ The GENERATING FUNCTION for the Catalan numbers is given by 1 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 4xp 2x/C30X/C12 n/C300Cnxn /C301 /C27x /C272x2 /C275x3 /C27...: (8) The asymptotic form for the Catalan numbers is Ck /C24k ffiffiffippk3 =2 (9) (Vardi 1991, Graham et al. 1994). A generalization of the Catalan numbers is defined by pdk /C301 kpk k /C281l11sl11n /C301 (p /C28 1)k /C27 1pk kl11sl11n (10) for k ]1 (Klarner 1970, Hilton and Pederson 1991). The usual Catalan numbers Ck /C30 2 dkare a special case with p /C302.pdk gives the number of p-ary TREES with k source-nodes, the number of ways of associat- ing k applications of a given p-ary OPERATOR , the number of ways of dividing a convex POLYGON into k disjoint (p /C271)/-gons with nonintersecting DIAGONALS , and the number of P-GOOD PATHS from (0, /C281) to (k;(p /C281)k /C281) (Hilton and Pederson 1991). A further generalization is obtained as follows. Let p be an INTEGER > 1; let Pk /C30(k ;(p /C281)k /C281) with k ]0; and q 5p /C281: Then definepdq0 /C301 and let pdqk be the number ofP-GOOD PATHS from (1, q /C281) to Pk(Hilton and Pederson 1991). Formulas forpdqiinclude the generalized JONAH FORMULA n /C28q k /C281l11sl11n /C30Xk i /C301p dqin /C28pi k /C28il11sl11n (11) and the explicit formula pdqk/C30p/C28q pk/C28qpk/C28q k/C281l11sl11n : (12) ARECURRENCE RELATION is given by pdqk/C30X i;jpdp/C28r;ipdq/C27r;j (13) where i;j;r]1;k]1;qBp/C28r;and i/C27j/C30k/C271 (Hilton and Pederson 1991). See also BALLOT PROBLEM ,B INARY BRACKETING , BINARY TREE,CATALAN’S PROBLEM ,CATALAN’S TRI- ANGLE ,DELANNOY NUMBER ,EULER’S POLYGON DIVI- SION PROBLEM ,F LEXAGON ,F RIEZE PATTERN , MOTZKIN NUMBER , P-GOOD PATH,PLANTED PLANAR TREE,S CHRO ¨ DER NUMBER ,S TAIRCASE POLYGON , SUPER CATALAN NUMBER References Alter, R. "Some Remarks and Results on Catalan Numbers." Proc. 2nd Louisiana Conf. Comb., Graph Th., and Com- put., 109/C1/132, 1971. Alter, R. and Kubota, K. K. "Prime and Prime Power Divisibility of Catalan Numbers." J. Combin. Th. A 15, 243/C1/256, 1973.Bailey, D. F. "Counting Arrangements of 1’s and -1’s." Math. Mag. 69, 128/C1/131, 1996. Brualdi, R. A. Introductory Combinatorics, 3rd ed. New York: Elsevier, 1997. Campbell, D. "The Computation of Catalan Numbers." Math. Mag. 57, 195/C1/208, 1984. Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration of Certain Sets of Planted Trees." J. Combin. Th. Ser. B 18, 209/C1/221, 1975. Chu, W. "A New Combinatorial Interpretation for General- ized Catalan Numbers." Disc. Math. 65,9 1/C1/94, 1987. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 96 /C1/106, 1996. Dershowitz, N. and Zaks, S. "Enumeration of Ordered Trees." Disc. Math. 31,9/C1/28, 1980. Dickau, R. M. "Catalan Numbers." http://forum.swarthmor- e.edu/advanced/robertd/catalan.html. Do¨rrie, H. "Euler’s Problem of Polygon Division." §7i n 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 21 /C1/27, 1965. Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How Likely is a Function to be Convex?" Math. Mag. 61, 211/C1/ 219, 1988. Gardner, M. "Catalan Numbers." Ch. 20 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 253 /C1/266, 1988. Gardner, M. "Catalan Numbers: An Integer Sequence that Materializes in Unexpected Places." Sci. Amer. 234, 120/C1/ 125, June 1976. Gould, H. W. Bell & Catalan Numbers: Research Bibliogra- phy of Two Special Number Sequences, 6th ed. Morgan- town, WV: Math Monongliae, 1985. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Exercise 9.8 inConcrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Guy, R. K. "Dissecting a Polygon Into Triangles." Bull. Malayan Math. Soc. 5,5 7/C1/60, 1958. Hilton, P. and Pederson, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Int. 13,6 4/C1/75, 1991. Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 130 /C1/134, 1973. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 146 /C1/150, 1985. Klarner, D. A. "Correspondences Between Plane Trees and Binary Sequences." J. Comb. Th. 9, 401/C1/411, 1970. Mays, M. E. and Wojciechowski, J. "A Determinant Property of Catalan Numbers." Disc. Math. 211, 125/C1/133, 2000. Rogers, D. G. "Pascal Triangles, Catalan Numbers and Renewal Arrays." Disc. Math. 22, 301/C1/310, 1978. Sands, A. D. "On Generalized Catalan Numbers." Disc. Math. 21, 218/C1/221, 1978. Singmaster, D. "An Elementary Evaluation of the Catalan Numbers." Amer. Math. Monthly 85, 366/C1/368, 1978. Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, pp. 18 /C1/20, 1973. Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M1459 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 187 /C1 /188 and 198 /C1/ 199, 1991. Wells, D. G. The Penguin Dictionary of Curious and Inter- esting Numbers. London: Penguin, pp. 121 /C1/122, 1986. Catalan Solid The DUAL POLYHEDRA of the ARCHIMEDEAN SOLIDS , given in the following table. They are known as Catalan solids in honor of the French mathematician who first published them in 1862 (Wenninger 1983, p. 1). n ARCHIMEDEAN SOLID DUAL 1 CUBOCTAHEDRON RHOMBIC DODECAHEDRON 2 GREAT RHOMBICOSIDODECA- HEDRONDISDYAKIS TRIACONTAHEDRON 3 GREAT RHOMBICUBOCTAHE- DRONDISDYAKISDODECAHEDRON 4 ICOSIDODECAHEDRON RHOMBIC TRIACONTAHEDRON 5 RHOMBICOSIDODECAHEDRON DELTOIDAL HEXE- CONTAHEDRON 6 SMALL RHOMBICUBOCTAHE- DRONDELTOIDAL ICOSITE- TRAHEDRON 7 SNUB CUBE (laevo) PENTAGONAL ICOSI- TETRAHEDRON (dextro) 8 SNUB DODECAHEDRON (laevo)PENTAGONAL HEXE- CONTAHEDRON (dextro) 9 TRUNCATED CUBE SMALL TRIAKIS OCTAHEDRON 10 TRUNCATED DODECAHEDRON TRIAKIS ICOSAHEDRON 11 TRUNCATED ICOSAHEDRON PENTAKIS DODECAHEDRON 12 TRUNCATED OCTAHEDRON TETRAKIS HEXAHEDRON 13 TRUNCATED TETRAHEDRON TRIAKIS TETRAHEDRON Here are the ARCHIMEDEAN DUALS (Pearce 1978, Holden 1991) displayed in the order listed above (left to right, then continuing to the next row). Here are the Archimedean solids paired with the corresponding Catalan solids. See also ARCHIMEDEAN SOLID,D UAL POLYHEDRON , SEMIREGULAR POLYHEDRON References Catalan, E. "Me´moire sur la The´orie des Polye`dres." J. l’E´ cole Polytechnique (Paris) 41,1/C1/71, 1865. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Pedagoguery Software. Poly . http://www.peda.com/poly/. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Catalan’s Aliquot Sequence Conjecture The conjecture proposed by Catalan in 1888 and extended by E. Dickson that each ALIQUOT SEQUENCE ends in a PRIME ,a PERFECT NUMBER , or a set of SOCIABLE NUMBERS . The conjecture remains open to this day. See also ALIQUOT SEQUENCE ,SOCIABLE NUMBERS References Creyaufmu ¨ller, W. "Aliquot Sequences." http://home.t-onli- ne.de/home/Wolfgang.Creyaufmueller/aliquote.htm. Catalan’s Conjecture 8 and 9 (23 and 32) are the only consecutive POWERS (excluding 0 and 1), i.e., the only solution to CATA- LAN’S DIOPHANTINE PROBLEM . Solutions to this pro- blem (CATALAN’S DIOPHANTINE PROBLEM ) are equivalent to solving the simultaneous DIOPHANTINE EQUATIONS X2 /C28Y3 /C301 X3 /C28Y2 /C301: This CONJECTURE has not yet been proved or refuted, although it has been shown to be decidable in a FINITE (but more than astronomical) number of steps. In particular, if n and n /C271 are POWERS , then n B exp exp exp exp 730 (Guy 1994, p. 155), which follows from R. Tijdeman’s proof that there can be only a FINITE number of exceptions should the CONJECTURE not hold. Hyyro and Makowski proved that there do not exist three consecutive POWERS (Ribenboim 1996), and it is also known that 8 and 9 are the only consecutive CUBIC and S QUARE NUMBERS (in either order). See also CATALAN’S DIOPHANTINE PROBLEM References Guy, R. K. "Difference of Two Power." §D9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 155 /C1/157, 1994. Ribenboim, P. Catalan’s Conjecture: Are 8 and 9 the only Consecutive Powers? Boston, MA: Academic Press, 1994. Ribenboim, P. "Catalan’s Conjecture." Amer. Math. Monthly 103, 529/C1/538, 1996. Ribenboim, P. "Consecutive Powers." Expositiones Mathe- maticae 2, 193/C1/221, 1984. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 71 and 73, 1986. Catalan’s Constant A constant which appears in estimates of combina- torial functions. It is usually denoted K,b(2);orG.I t is not known if KisIRRATIONAL . Numerically, K/C300:915965594177 . . . (1) (Sloane’s A006752). The CONTINUED FRACTION forK is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (Sloane’s A014538). K can be given analytically by the following expres-sions, K/C13b(2) (2) /C30/C28ix 2(i) (3) /C30X/C12 k/C300(/C281)k (2k/C271)2/C301 12/C281 32/C271 52/C27... ( 4 ) /C301/C27X/C12 n/C3011 (4n/C271)2/C281 9/C28X/C12 n/C3011 (4n/C273)2(5)/C30g1 0tan/C281xd x x(6) /C30/C28g1 0lnxd x 1/C27x2; (7) where b(z) is the D IRICHLET BETA FUNCTION andxn(z) is L EGENDRE’S CHI-FUNCTION . In terms of the POLY- GAMMA FUNCTION C1(x); K/C301 16C1(1 4)/C281 16C1(34) (8) /C301 80C1(5 12)/C271 80C1(1 12)/C281 10p2(9) /C301 32C1(1 8)/C281 32C1(38)/C281 16ffiffiffi 2p : (10) Applying CONVERGENCE IMPROVEMENT to (4) gives K/C301 16X/C12 m/C301(m/C271)3m/C281 4mz(m/C272); (11) where z(z) is the R IEMANN ZETA FUNCTION and the identity 1 (1/C283z)2/C281 (1/C28z)2/C30X/C12 m/C301(m/C271)3m/C281 4mzm(12) has been used (Flajolet and Vardi 1996). The Flajolet and Vardi algorithm also gives K/C301ffiffiffi 2pY/C12 k/C3011/C281 22k ! z(2k) b(2k)"#1=(2k/C271) ; (13) where b(z) is the D IRICHLET BETA FUNCTION . Glaisher (1913) gave K/C301/C28X/C12 n/C301nz(2n/C271) 16n(14) (Vardi 1991, p. 159). W. Gosper used the related FORMULA K/C301ffiffiffi 2p1 C(2)/C281"#21=2Y/C12 k/C3021 /C28C(2k)/C281"#1=(2k/C271) ;(15) where C(m)/C30mcm/C281(1 4) pm(2m/C281)4m/C281Bm; (16) where Bnis a B ERNOULLI NUMBER and c(x)i sa POLYGAMMA FUNCTION (Finch). The Catalan constant may also be defined by K/C131 2g1 0K(k)dk; (17) where K(k) (not to be confused with Catalan’s constant itself, denoted K) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND . K /C30p ln 2 8/C27X/C12 i/C301ai 2 (i/C271)=2 bc i2 ; (18) where fai g/C30f1; 1 ; 1 ; 0 ;/C281;/C281;/C281; 0g (19) is given by the periodic sequence obtained by append- ing copies of f1 ; 1 ; 1 ; 0;/C281;/C281;/C281; 0g (in other words, ai /C13a[i/C281 (mod8)] /C271for i /C218) and xbcis the FLOOR FUNCTION (Nielsen 1909). See also DIRICHLET BETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807 /C1/808, 1972. Adamchik, V. "Integral and Series Representations for Catalan’s Constant." http://members.wri.com/victor/arti- cles/catalan.html. Adamchik, V. "Thirty-Three Representations of Catalan’s Constant." http://library.wolfram.com/demos/v4/Catalan- Formulas.nb. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 551 /C1/552, 1985. Fee, G. J. "Computation of Catalan’s Constant using Rama- nujan’s Formula." ISAAC ’90. Proc. Internat. Symp. Symbolic Algebraic Comp., Aug. 1990. Reading, MA: Addison-Wesley, 1990. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/catalan/catalan.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Glaisher, J. W. L. "Numerical Values of the Series 1 /C28 1=3n /C271 =5n /C281 =7n /C271 =9n /C28&c for n /C302, 4, 6." Messenger Math. 42,35/C1/58, 1913. Gosper, R. W. "A Calculus of Series Rearrangements." In Algorithms and Complexity: New Directions and Recent Results (Ed. J. F. Traub). New York: Academic Press, 1976. Nielsen, N. Der Eulersche Dilogarithms. Leipzig, Germany: Halle, pp. 105 and 151, 1909. Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u- qam.ca/pi/records.html. Sloane, N. J. A. Sequences A006752/M4593 and A014538 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Srivastava, H. M. and Miller, E. A. "A Simple Reducible Case of Double Hypergeometric Series involving Catalan’s Constant and Riemann’s Zeta Function." Int. J. Math. Educ. Sci. Technol. 21, 375 /C1/377, 1990. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 159, 1991. Yang, S. "Some Properties of Catalan’s Constant G." Int. J. Math. Educ. Sci. Technol. 23, 549 /C1/556, 1992. Catalan’s Diophantine Problem Find consecutive POWERS , i.e., solutions to ab /C28cd /C301; excluding 0 and 1. CATALAN’S CONJECTURE is that the only solution is 32 /C2823 /C301; so 8 and 9 (23 and 32) arethe only consecutive POWERS (again excluding 0 and 1). See also CATALAN’S CONJECTURE References Cassels, J. W. S. "On the Equation ax /C28by /C301: II." Proc. Cambridge Phil. Soc. 56,97/C1/103, 1960. Inkeri, K. "On Catalan’s Problem." Acta Arith. 9, 285 /C1/290, 1964. Catalan’s Identity F2 n /C28Fn/C27rFn /C28r /C30(/C281)n/C28rF2 r ; where Fn is a FIBONACCI NUMBER . Letting r /C301 gives CASSINI’S IDENTITY . See also CASSINI’S IDENTITY , D’OCAGNE’S IDENTITY , FIBONACCI NUMBER Catalan’s Problem The problem of finding the number of different ways in which a PRODUCT of n different ordered FACTORS can be calculated by pairs (i.e., the number of BINARY BRACKETINGS of n letters). For example, for the four FACTORS a, b, c, and d, there are five possibilities: ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d) ; and a(b(cd)) : The solution was given by Catalan in 1838 as C ?n /C30(4n /C28 6)!!!! n!/C302 /C215 6 /C215 10 /C1/C1/C1(4n /C28 6) n! ; where n!!!! is a MULTIFACTORIAL and n! is the usual FACTORIAL , which is equal to the CATALAN NUMBER Cn/C281/C30C?n:/ See also BINARY BRACKETING ,CATALAN’S DIOPHAN- TINE PROBLEM ,CATALAN NUMBER ,EULER’S POLYGON DIVISION PROBLEM References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 23, 1965. Catalan’s Surface AMINIMAL SURFACE given by the PARAMETRIC EQUA- TIONS x(u ; v) /C30u /C28sin u cosh v (1) y(u; v) /C301 /C28cos u cosh v (2) z(u; v) /C304 sin(1 2u) sinh(12v) (3) (Gray 1997), or x(r; f) /C30a sin(2f) /C282a f /C2712av2 cos(2 f) (4) y(r ; f) /C30/C28a cos(2 f) /C2812av2 cos(2 f) (5) z(r ; f) /C302av sin f ; (6) where v /C30/C28r /C271 r (7) (do Carmo 1986). References Catalan, E. "Me´moire sur les surfaces dont les rayons de courbures en chaque point, sont e´gaux et les signes contraires." C. R. Acad. Sci. Paris 41, 1019 /C1/1023, 1855. do Carmo, M. P. "Catalan’s Surface" §3.5D in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 45 /C1/46, 1986. Fischer, G. (Ed.). Plates 94 /C1/95 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 90 /C1/91, 1986. Gray, A. "Catalan’s Minimal Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 692 /C1/693, 1997. JavaView. "Classic Surfaces from Differential Geometry: Catalan Surface." http://www-sfb288.math.tu-berlin.de/ vgp/javaview/demo/surface/common/PaSurface_Cata- lan.html. Catalan’s Triangle A triangle of numbers with entries given by cnm /C30(n /C27 m)!(n /C28 m /C27 1) m!(n /C27 1)! for 0 5m 5n; where each element is equal to the one above plus the one to the left. Furthermore, the sum of each row is equal to the last element of the next row and also equal to the CATALAN NUMBER Cn : 1 11 12 2 13 5 5 14 91 41 4 1 5 14 28 42 42 1 6 20 48 90 132 132 (Sloane’s A009766). See also BELL TRIANGLE ,CLARK’S TRIANGLE ,EULER’S TRIANGLE ,L EIBNIZ HARMONIC TRIANGLE ,N UMBER TRIANGLE ,P ASCAL’S TRIANGLE ,P RIME TRIANGLE ,SEIDEL- ENTRINGER- ARNOLD TRIANGLE References Sloane, N. J. A. Sequences A009766 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Catalan’s Trisectrix TSCHIRNHAUSEN CUBIC Catalogue Paradox Consider a library which compiles a bibliographic catalog of all (and only those) catalogs which do not list themselves. Then does the library’s catalog list itself? See also PSEUDOPARADOX ,RUSSELL’S PARADOX References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 5, 1977. Gonseth, F. "La structure du paradoxe des catalogues." §106 in Les mathe ´matiques et la re´alite´: Essai sur la me´thode axiomatique. Paris: Fe´lix Alcan, pp. 255 /C1/257, 1936. Catastrophe For any system that seeks to minimize a function, only seven different local forms of CATASTROPHE "typically" occur for four or fewer variables: 1. F OLD CATASTROPHE , 2. C USP CATASTROPHE , 3. S WALLOWTAIL CATASTROPHE , 4. B UTTERFLY CATASTROPHE , 5. E LLIPTIC UMBILIC CATASTROPHE , 6. H YPERBOLIC UMBILIC CATASTROPHE , and 7. P ARABOLIC UMBILIC CATASTROPHE . More specifically, for any system with fewer than fivecontrol factors and fewer than three behavior axes, these are the only seven catastrophes possible. The following tables gives the possible catastrophes as afunction of control factors and behavior axes (Goetz). ControlFactors1 BehaviorAxis2 Behavior Axes 1 FOLD 2 CUSP 3 SWALLOWTAIL HYPERBOLIC UMBILIC , ELLIPTIC UMBILIC 4 BUTTERFLY PARABOLIC UMBILIC The following table gives prototypical examples for equations showing each type of catastrophe. equation catastrophe /x3 /C27ux/ FOLD CATASTROPHE /x4 /C27ux2 /C27vx/ CUSP CATA- STROPHE , Rie- mann-Hugoniot catastrophe /x5 /C27ux3 /C27vx2 /C27wx / SWALLOWTAIL CATASTROPHE /x3 /C27y3 /C27uxy /C27vx /C27wy / HYPERBOLIC UMBI- LIC CATASTROPHE /x3 /C28xy2 /C27u(x2 /C27y2) /C27vx /C27wy / ELLIPTIC UMBILIC CATASTROPHE /x6 /C27ux4 /C27vx3 /C27wx2 /C27tx/ BUTTERFLY CATASTROPHE /x2y /C27y4 /C27ux2 /C27vy2 /C27wx /C27ty/ PARABOLIC UMBI- LIC CATASTROPHE See also BUTTERFLY CATASTROPHE ,C ATASTROPHE THEORY ,C USP CATASTROPHE ,E LLIPTIC UMBILIC CATASTROPHE ,FOLD CATASTROPHE ,HYPERBOLIC UM- BILIC CATASTROPHE ,P ARABOLIC UMBILIC CATA- STROPHE ,SWALLOWTAIL CATASTROPHE References Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. Catastrophe Theory Catastrophe theory studies how the qualitative nat- ure of equation solutions depends on the parameters that appear in the equations. Subspecializations include bifurcation theory, nonequilibrium thermo- dynamics, singularity theory, synergetics, and topo- logical dynamics. For any system that seeks to minimize a function, only seven different local forms of CATASTROPHE "typically" occur for four or fewer variables. See also CATASTROPHE References Arnold, V. I. Catastrophe Theory, 3rd ed. Berlin: Springer- Verlag, 1992. Dujardin, L. "Catastrophe Teacher: An Introduction for Experimentalists." http://perso.wanadoo.fr/l.d.v.dujardin/ ct/eng_index.html. Gilmore, R. Catastrophe Theory for Scientists and Engi- neers. New York: Dover, 1993. Goetz, P. "Phil’s Good Enough Complexity Dictionary." http://www.cs.buffalo.edu/~goetz/dict.html. Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. Saunders, P. T. An Introduction to Catastrophe Theory. Cambridge, England: Cambridge University Press, 1980.Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, p. 211, 1987. Thom, R. Structural Stability and Morphogenesis: An Out- line of a General Theory of Models. Reading, MA: Addison- Wesley, 1993. Thompson, J. M. T. Instabilities and Catastrophes in Science and Engineering. New York: Wiley, 1982. Weisstein, E. W. "Books about Catastrophe Theory." http:// www.treasure-troves.com/books/CatastropheTheory.html. Woodcock, A. E. R. and Davis, M. Catastrophe Theory. New York: E. P. Dutton, 1978. Zeeman, E. C. Catastrophe Theory--Selected Papers 1972 /C1/ 1977. Reading, MA: Addison-Wesley, 1977. Categorical Game A GAME in which no DRAW is possible. All CATEGO- RICAL GAMES are unfair (Steinhaus 1983, p. 16). See also DRAW,GAME References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 16 1999. Categorical Variable A variable which belongs to exactly one of a finite number of CATEGORIES . See also CATEGORY Category A category consists of two things: a collection of OBJECTS and, for each pair of OBJECTS , a collection of MORPHISMS (sometimes called "arrows") from one to another. In most concrete categories over sets, an OBJECT is some mathematical structure (e.g., a GROUP , VECTOR SPACE ,or DIFFERENTIABLE MANIFOLD ) and a MORPH- ISM is a MAP between two OBJECTS . The MORPHISMS are then required to satisfy some fairly natural conditions; for instance, the IDENTITY MAP between any object and itself is always a MORPHISM , and the composition of two MORPHISMS (if defined) is always a MORPHISM . One usually requires the MORPHISMS to preserve the mathematical structure of the objects. So if the objects are all groups, a good choice for a MORPHISM would be a group HOMOMORPHISM . Similarly, for vector spaces, one would choose linear maps, and for differentiable manifolds, one would choose differ- entiable maps. In the category of TOPOLOGICAL SPACES , homomorph- isms are usually continuous maps between topologi- cal spaces. However, there are also other category structures having TOPOLOGICAL SPACES as objects, but they are not nearly as important as the "stan- dard" category of TOPOLOGICAL SPACES and continu- ous maps. See also ABELIAN CATEGORY ,ALLEGORY ,EILENBERG- STEENROD AXIOMS ,G ROUPOID ,H OLONOMY ,LOGOS , MONODROMY ,TOPOS References Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster- dam, Netherlands: North-Holland, 1990. Getzler, E. and Kapranov, M. (Eds.). Higher Category Theory. Providence, RI: Amer. Math. Soc., 1998. Lawvere, F. W. and Schanuel, S. H. Conceptual Mathe- matics: A First Introduction to Categories. Cambridge, England: Cambridge University Press, 1997. Mac Lane, S. and Gehring, F. W. Categories for the Working Mathematician, 2nd ed. New York: Springer-Verlag, 1998. Munkres, J. R. "Categories and Functors." §28 in Elements of Algebraic Topology. Perseus Press, pp. 154 /C1/160, 1993. Category Theory The branch of mathematics which formalizes a number of algebraic properties of collections of transformations between mathematical objects (such as binary relations, groups, sets, topological spaces, etc.) of the same type, subject to the constraint that the collections contain the identity mapping and are closed with respect to compositions of mappings. The objects studied in category theory are called CATE- GORIES . See also CATEGORY Catenary The curve a hanging flexible wire or chain assumeswhen supported at its ends and acted upon by auniform gravitational force. The word catenary isderived from the Latin word for "chain." In 1669, Jungius disproved Galileo’s claim that the curve of a chain hanging under gravity would be a PARABOLA (MacTutor Archive). The curve is also called the alysoid and chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 inresponse to a challenge by Jakob Bernoulli. Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote atreatise on the catenary in 1690 (MacTutor Archive).If you roll a PARABOLA along a straight line, its FOCUS traces out a catenary. As proved by Euler in 1744, thecatenary is also the curve which, when rotated, givesthe surface of minimum SURFACE AREA (the CATE- NOID ) for the given bounding CIRCLE . The PARAMETRIC EQUATIONS for the catenary are given by x(t)/C30t (1)y(t)/C301 2a(et=a/C27e/C28t=a)/C30acosht a ! ; (2) where t/C300 corresponds to the vertex, and the CESA`RO EQUATION is (s2/C27a2)k/C30/C28a: (3) The ARC LENGTH ,CURVATURE , and TANGENTIAL ANGLE are s(t)/C30asinht a ! ; (4) k(t)/C30/C281 asech2t a ! ; (5) f(t)/C30/C282 tan/C281tanht 2a !"# : (6) The slope is proportional to the ARC LENGTH as measured from the center of symmetry. The St. Louis Arch closely approximates an inverted catenary, but it has a finite thickness and varyingcross sectional area (thicker at the base; thinner at the apex). The centroid has half-length of L/C30299.2239 feet at the base, height of 625.0925 feet, top cross sectional area 125.1406 square feet, and bottom cross sectional area 1262.6651 square feet. The catenary also gives the shape of the road ( ROULETTE ) over which a regular polygonal "wheel" can travel smoothly. For a regular n-gon, the Carte- sian equation of the corresponding catenary is y/C30/C28Acoshx A ! ; (7) where A /C13R cosp n ! : (8) See also CALCULUS OF VARIATIONS ,CATENOID ,LINDE- LOF’S THEOREM ,ROULETTE ,SURFACE OF REVOLUTION References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 214, 1987. Gray, A. "The Evolute of a Tractrix is a Catenary." §5.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 102 /C1/103, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 199 /C1/200, 1972. Lockwood, E. H. "The Tractrix and Catenary." Ch. 13 in A Book of Curves. Cambridge, England: Cambridge Univer- sity Press, pp. 118 /C1/124, 1967. MacTutor History of Mathematics Archive. "Catenary." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cate-nary.html. National Park Service. "Arch History and Architecture: Catenary Curve Equation." http://www.nps.gov/jeff/equa-tion.htm. Pappas, T. "The Catenary & the Parabolic Curves." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 34, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 247 /C1 /249, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 26 /C1/27, 1991. Yates, R. C. "Catenary." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 12 /C1/14, 1952. Catenary Evolute x/C30a[x/C281 2sinh(2 t)] y/C302acosh t:Catenary Involute The parametric equation for a CATENARY is r(t)/C30at cosh tl12ml121 ; (1) so dr dt/C30a1 sinh tl12ml121 (2) dr dtl112l112l112l112l112l112l112l112l112l112/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27sinh 2tp /C30acosh t (3) and ˆT/C30dr dt dr dtl112l112l112l112l112l112l112l112l112l112/C30sech t tanh tl12ml121 (4) ds 2/C30½dr2½/C30a2(1/C27sinh2t)dt2/C30a2cosh2dt2(5) ds dt/C30acosh t: (6) Therefore, s/C30agcosh td t/C30asinh t (7) and the equation of the INVOLUTE is x/C30a(t/C28tanh t) (8) y/C30asech t: (9) This curve is called a TRACTRIX . Catenary Radial Curve The KAMPYLE OF EUDOXUS . Catenoid A CATENARY of REVOLUTION . The catenoid and PLANE are the only SURFACES OF REVOLUTION which are also MINIMAL SURFACES . The catenoid can be given by the PARAMETRIC EQUATIONS x /C30c coshv c ! cos u (1) y /C30c coshv c ! sin u (2) z /C30v; (3) where u /C23 [0; 2p): The differentials are dx /C30sinhv c ! cos udv/C28coshv c ! sin udu (4) dy /C30sinhv c ! sin udv/C27coshv c ! cos udu (5) dz /C30du; (6) so the LINE ELEMENT is ds2 /C30dx2 /C27dy2 /C27dz2 /C30 sinh2v c ! /C271"# dv2 /C27cosh2v c ! du2 /C30cosh2v c ! dv2 /C27cosh2v c ! du2 : (7) The PRINCIPAL CURVATURES are k1 /C30/C281 csech2v c ! (8)k2 /C301 csech2v c ! : (9) The MEAN CURVATURE of the catenoid is H /C300 (10) and the GAUSSIAN CURVATURE is K /C30/C281 c2sech4v c ! : (11) The HELICOID can be continuously deformed into a catenoid with c /C301 by the transformation x(u ; v) /C30cos a sinh v sin u /C27sin a cosh v cos u (12) y(u;v)/C30/C28cosasinh vcosu/C27sinacosh vsinu(13) z(u;v)/C30ucosa/C27vsina; (14) where a/C300 corresponds to a HELICOID anda/C30p=2t o a catenoid. See also CATENARY ,COSTA MINIMAL SURFACE ,HELI- COID ,MINIMAL SURFACE ,SURFACE OF REVOLUTION References do Carmo, M. P. "The Catenoid." §3.5A in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43,1986. Fischer, G. (Ed.). Plate 90 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume.Braunschweig, Germany: Vieweg, p. 86, 1986. Gray, A. "The Catenoid." §20.4 Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 467 /C1 /469, 1997. JavaView. "Classic Surfaces from Differential Geometry: Catenoid/Helicoid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Catenoid- Helicoid.html. Meusnier, J. B. "Me ´moire sur la courbure des surfaces." Me´m. des savans e ´trangers 10(lu 1776), 477 /C1 /510, 1785. Ogawa, A. "Helicatenoid." Mathematica J. 2, 21, 1992. Osserman, R. A Survey of Minimal Surfaces. New York: Dover, p. 18 1986. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 247 /C1/249, 1999. Caterpillar Graph A TREE with every NODE on a central stalk or only one EDGE away from the stalk. A tree is a caterpillar graph IFF all nodes of degree ]3 are surrounded by at most two nodes of degree two or greater. The number of caterpillar graphs on n /C301, 2, ... nodes are 1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 136, ... (Sloane’s A005418), giving the number of noncaterpillar graphs on n /C307, 8, ... as 1, 3, 11, 34, 99, ... (Sloane’s A052471). The non- caterpillar graphs on n 59 nodes are illustrated above. See also TREE References Gardner, M. Wheels, Life, and other Mathematical Amuse- ments. New York: W. H. Freeman, p. 160, 1983. Hoffman, N. "Binary Grids and a Related Counting Pro- blem." Two Year Coll. Math. J. 9, 267 /C1/272, 1978. Sloane, N. J. A. Sequences A005418/M0771 and A052471 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sulanke, R. A.. "Moments of Generalized Motzkin Paths." J. Integer Sequences 3, No. 00.1.1, 2000. http://www.re- search.att.com/~njas/sequences/JIS/SULANKE/sulan- ke.html. Cattle Problem of Archimedes ARCHIMEDES’ CATTLE PROBLEM Cauchy Binomial Theorem Yn k/C301(1 /C27yqk) /C30Xn m /C300ymqm(m /C271)=2 n ml12ml121 q /C30Xn m /C300ymqm(m /C271)=2 (q)n (q)m(q)n/C28m; where [nrm]q is a Q-BINOMIAL COEFFICIENT . See also Q-BINOMIAL COEFFICIENT , Q-BINOMIAL THE- OREM Cauchy Boundary Conditions BOUNDARY CONDITIONS of a PARTIAL DIFFERENTIAL EQUATION which are a weighted AVERAGE of DIRICH- LET BOUNDARY CONDITIONS (which specify the value of the function on a surface) and NEUMANN BOUNDARYCONDITIONS (which specify the normal derivative of the function on a surface). See also BOUNDARY CONDITIONS ,CAUCHY PROBLEM , DIRICHLET BOUNDARY CONDITIONS ,N EUMANN BOUNDARY CONDITIONS References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 678 /C1/679, 1953. Cauchy Condition UNIFORMLY CAUCHY Cauchy Criterion ANECESSARY and SUFFICIENT condition for a SE- QUENCE SitoCONVERGE . The Cauchy criterion is satisfied when, for all e>0;there is a fixed number N such that Sj/C28Sil112l112l112l112Befor all i;j>N: / Cauchy Distribution The Cauchy distribution, also called the L ORENTZIAN DISTRIBUTION , is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a LINE SEGMENT tilted at a random ANGLE cuts the X-AXIS . Let u represent the ANGLE that a line, with fixed point of rotation, makes with the vertical axis, as shownabove. Then tanu/C30x b(1) u/C30tan/C281x b ! (2) du/C30/C281 1/C27x2 b2dx b/C30/C28bd x b2/C27x2; (3) so the distribution of ANGLE uis given by du p/C30/C281 pbd x b2/C27x2: (4) This is normalized over all angles, since gp=2 /C28p=2du p/C301 (5) and /C28g/C12 /C28/C121 pbdx b2 /C27 x2 /C301 ptan/C281b x !"#/C12 /C28/C12 /C301 p[1 2 p /C28(/C2812 p)] /C301 : (6) The general Cauchy distribution and its cumulative distribution can be written as P(x) /C301 p1 2 G (x /C28 m)2 /C27 (1 2 G)2 (7) D(x) /C301 2 /C271 ptan/C281x /C28 m b ! ; (8) where G is the FULL WIDTH AT HALF MAXIMUM (/ G/C302b in the above example) and m is the MEDIAN (m /C300in the above example). The CHARACTERISTIC FUNCTION is f(t) /C301 p g/C12 /C28/C12eitx1 2 G (1 2 G)2 /C27 (x /C28 m)2 dx /C30eimt /C28G tjj=2 : (9) The MOMENTS mnof the distribution are undefined since the integrals mn /C30g/C12 /C28/C12G 2pxn (x /C28 m)2 /C27 (12 G)2 (10) diverge for n ]1:/ If X and Y are variates with a NORMAL DISTRIBUTION , then Z /C13X =Y has a Cauchy distribution with MEDIAN m /C300 and full width G/C302sy sx: (11) The sum of n variates each from a Cauchy distribu- tion has itself a Cauchy distribution, as can be seen from Pn(x) /C30F/C281 f[ f(t)]n g/C30(12 nG) p[(1 2 nG)2 /C27 (x /C28 nm)2] ; (12) where f(t) is the CHARACTERISTIC FUNCTION and F/C281 fjjis the inverse FOURIER TRANSFORM , taken with parameters a /C30b /C301 :/ See also GAUSSIAN DISTRIBUTION ,NORMAL DISTRIBU- TIONReferences Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 104, 1984. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 114 /C1/115, 1992. Cauchy Equation EULER EQUATION Cauchy Functional Equation The fifth of HILBERT’S PROBLEMS is a generalization of this equation. See also HILBERT’S PROBLEMS Cauchy Integral Formula Given a CONTOUR INTEGRAL OF THE FORM Ggf(z)dz z/C28z0; (1) define a path gras an infinitesimal clockwise CIRCLE around the point z0(the dot in the above illustration), and define the path g0as an arbitrary loop with a cut line (on which the forward and reverse contributions cancel each other out) so as to go around z0:/ The total path is then g/C30g0/C27gr; (2) so Ggf(z)dz z/C28z0/C30Gg0f(z)dz z/C28z0/C27Ggrf(z)dz z/C28z0: (3) From the C AUCHY INTEGRAL THEOREM , the CONTOUR INTEGRAL along any path not enclosing a POLE is 0. Therefore, the first term in the above equation is 0 since g0does not enclose the POLE , and we are left with Ggf(z)dz z/C28z0/C30Ggrf(z)dz z/C28z0: (4) Now, let z/C13z0/C27reiu;sodz/C30ireiudu:Then Ggf(z)dz z/C28z0/C30Ggrf(z0/C27reiu) reiuireiudu /C30Ggrf(z0/C27reiu)idu: (5) But we are free to allow the radius rto shrink to 0, so Ggf(z) dz z /C28 z0/C30lim r00 Ggrf(z0 /C27reiu)idu /C30Ggrf(z0)idu /C30if(z0)Ggrdu /C302pif(z0) ; (6) and f(z0) /C301 2pi G gf(z) dz z /C28 z0: (7) If multiple loops are made around the POLE , then equation (7) becomes n(g ; z0)f(z0) /C301 2 pi G gf(z) dz z /C28 z0; (8) where n(g ; z0) is the WINDING NUMBER . A similar formula holds for the derivatives of f(z) ; f ?(z0) /C30lim h00f(z0 /C27 h) /C28 f(z0) h /C30lim h001 2 pih G gf(z) dz z /C28 z0 /C28 h /C28G gf(z) dz z /C28 z0"# /C30lim h001 2 pih G gf(z)[(z /C28 z0) /C28 (z /C28 z0 /C28 h] dz (z /C28 z0 /C28 h)(z /C28 z0) /C30lim h001 2 pih G ghf(z) dz (z /C28 z0 /C28 h)(z /C28 z0) /C301 2pi G gf(z) dz (z /C28 z0)2 : (9) Iterating again, f ƒ(z0) /C302 2pi G gf(z)dz (z/C28z0)3: (10) Continuing the process and adding the WINDING NUMBER n, n(g;z0)f(r)(z0)/C30r! 2piGgf(z)dz (z/C28z0)r/C271: (11) See also ARGUMENT PRINCIPLE ,CONTOUR INTEGRAL , MORERA’S THEOREM References Arfken, G. "Cauchy’s Integral Formula." §6.4 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 371 /C1/376, 1985. Kaplan, W. "Cauchy’s Integral Formula." §9.9 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 598 /C1/ 599, 1991. Knopp, K. "Cauchy’s Integral Formulas." Ch. 5 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 61 /C1/66, 1996. Krantz, S. G. "The Cauchy Integral Theorem and Formula." §2.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 26 /C1/29, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 367 /C1/372, 1953. Woods, F. S. "Cauchy’s Theorem." §146 in Advanced Calcu- lus: A Course Arranged with Special Reference to theNeeds of Students of Applied Mathematics. Boston, MA: Ginn, pp. 352 /C1 /353, 1926. Cauchy Integral Test INTEGRAL TEST Cauchy Integral Theorem Iff(z) is analytic in some simply connected region R, then Ggf(z)dz/C300 (1) for any closed CONTOUR gcompletely contained in R. Writing zas z/C13x/C27iy (2) andf(z)a s f(z)/C13u/C27iv (3) then gives Ggf(z)dz/C30gg(u/C27iv)(dx/C27id y) /C30ggud x/C28vd y/C27iggvd x/C27ud y : (4) From G REEN’S THEOREM , ggf(x;y)dx/C28g(x;y)dy/C30/C28gg@g @x/C27@f @y ! dx dy ;(5) ggf(x;y)dx/C27g(x;y)dy/C30gg@g @x/C28@f @y ! dx dy (6) so (4) becomes Ggf(z)dz/C30/C28gg@v @x/C27@u @y ! dx dy /C27igg@u @x/C27@v @y ! dx dy : (7) But the C AUCHY- RIEMANN EQUATIONS require that @u @x/C30@v @y(8) @u @y/C30/C28@v @x; (9) so Ggf(z) dz /C300; (10) Q.E.D. For a MULTIPLY CONNECTED region, GC1f(z) dz /C30GC2f(z) dz : (11) See also ARGUMENT PRINCIPLE ,C AUCHY INTEGRAL THEOREM ,CONTOUR INTEGRAL ,M ORERA’S THEOREM , RESIDUE THEOREM References Arfken, G. "Cauchy’s Integral Theorem." §6.3 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 365 /C1/371, 1985. Kaplan, W. "Integrals of Analytic Functions. Cauchy Inte- gral Theorem." §9.8 in Advanced Calculus, 4th ed. Read- ing, MA: Addison-Wesley, pp. 594 /C1/598, 1991. Knopp, K. "Cauchy’s Integral Theorem." Ch. 4 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 47 /C1/60, 1996. Krantz, S. G. "The Cauchy Integral Theorem and Formula." §2.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 26 /C1/29, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 363 /C1/367, 1953. Woods, F. S. "Integral of a Complex Function." §145 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathe- matics. Boston, MA: Ginn, pp. 351 /C1/352, 1926. Cauchy Mean Theorem CAUCHY’S FORMULA Cauchy Number of the First Kind BERNOULLI NUMBER OF THE SECOND KIND Cauchy Principal Value PVg/C12 /C28/C12f(x) dx /C13limR0/C12gR /C28Rf(x) dx PVgb af(x) dx /C13lime00gc/C28 e af(x) dx /C27gb c/C27 ef(x) dx"# ; where e > 0 and a 5c 5b : Russian authors use the notation P(x) instead of PVx for the principal value of x. References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 401 /C1/403, 1985. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 158, 1991. Cauchy Problem If f(x; y)isan ANALYTIC FUNCTION in a NEIGHBOR- HOOD of the point (x0 ; y0) (i.e., it can be expanded in aseries of NONNEGATIVE INTEGER POWERS of (x /C28x0) and (y /C28y0)) ; find a solution y(x) of the DIFFERENTIAL EQUATION dy dx /C30f(x) ; with initial conditions y /C30y0 and x /C30x0 : The existence and uniqueness of the solution were proven by Cauchy and Kovalevskaya in the CAUCHY- KOVALEVS- KAYA THEOREM . The Cauchy problem amounts to determining the shape of the boundary and type of equation which yield unique and reasonable solutions for the CAUCHY BOUNDARY CONDITIONS . See also CAUCHY BOUNDARY CONDITIONS ,CAUCHY- KOVALEVSKAYA THEOREM Cauchy Product The Cauchy product of two sequences f(n) and g(n) defined for nonnegative integers n is defined by (f(g)(n) /C30Xn k /C300f(k)g(n /C28k) : See also CONVOLUTION References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 24, 1997. Cauchy Ratio Test RATIO TEST Cauchy Remainder The remainder after n terms of a TAYLOR SERIES is given by Rn /C30(x /C28 x/C31)n(x /C28 x0)n/C271 n!f(n/C271)(x/C31); where x/C31/C23 (x0 ; x):/ Note that the Cauchy remainder Rn is also sometimes taken to refer to the remainder when terms up to the (n /C281)/st power are taken in the TAYLOR SERIES , and that a notation in which h 0 x /C28x0 ; x/C310 a /C27 uh; and x/C28x/C3101/C28uis sometimes used (Blumenthal 1926; Whittaker and Watson 1990, pp. 95 /C1/96). See also LAGRANGE REMAINDER ,SCHLO ¨ MILCH RE- MAINDER ,TAYLOR SERIES References Beesack, P. R. "A General Form of the Remainder in Taylor’s Theorem." Amer. Math. Monthly 73,6 4/C1/67, 1966. Blumenthal, L. M. "Concerning the Remainder Term in Taylor’s Formula." Amer. Math. Monthly 33, 424 /C1/426, 1926. Hamilton, H. J. "Cauchy’s Form of Rnfrom the Iterated Integral Form." Amer. Math. Monthly 59, 320, 1952. Whittaker, E. T. and Watson, G. N. "Forms of the Remain- der in Taylor’s Series." §5.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 95 /C1/96, 1990. Cauchy Root Test ROOT TEST Cauchy Sequence A SEQUENCE a1 ; a2 ; ... such that the METRIC d(am ; an) satisfies lim min( m; n) 0/C12d(am ; an) /C300: Cauchy sequences in the rationals do not necessarily CONVERGE , but they do CONVERGE in the REALS . REAL NUMBERS can be defined using either DEDEKIND CUTS or Cauchy sequences. See also DEDEKIND CUT Cauchy Test RATIO TEST Cauchy-Davenport Theorem Let t be a NONNEGATIVE INTEGER and let x1 ; ..., xt be nonzero elements of Zpwhich are not necessarily distinct. Then the number of elements of Zp that can be written as the sum of some SUBSET (possibly empty) of the xiis at least min fp ; t /C271g: In particu- lar, if t ]p /C281; then every element of Zpcan be so written. References Martin, G. "Dense Egyptian Fractions." Trans. Amer. Math. Soc. 351, 3641 /C1/3657, 1999. Vaughan, R. C. Lemma 2.14 in The Hardy-Littlewood Method, 2nd ed. Cambridge, England: Cambridge Uni- versity Press, 1997. Cauchy-Frobenius Lemma Let J be a FINITE GROUP and the image R(J)bea representation which is a HOMEOMORPHISM of J into a PERMUTATION GROUP S(X) ; where S(X) is the GROUP of all permutations of a SET X. Define the orbits of R(J) as the equivalence classes under x /C2y; which is true if there is some permutation p in R(J) such that p(x) /C30y: Define the fixed points of p as the elements x of X for which p(x) /C30x: Then the AVERAGE number of FIXED POINTS of permutations in R(J) is equal to the number of orbits of R(J) :/ The LEMMA was apparently known by Cauchy (1845) in obscure form and Frobenius (1887) prior to Burn- side’s (1900) rediscovery. It is sometimes also called BURNSIDE’S LEMMA , the PO´ LYA-BURNSIDE LEMMA ,oreven "the LEMMA THAT IS NOT BURNSIDE’S !" Whatever its name, the lemma was subsequently extended and refined by Po´lya (1937) for applications in COMBINA- TORIAL counting problems. In this form, it is known as PO´ LYA ENUMERATION THEOREM . See also PO´ LYA ENUMERATION THEOREM References Cauchy, A. "Me´moire sur diverses proprie ´te´s remarquables des substitutions re´gulie`res ou irre´gulie`res, et des sys- te´mes de substitutiones conjuge ´es." C. R. Acad. Sci. Paris 21, 835, 1845. Reprinted in/Œ/uvres Comple `tes d’Augustin Cauchy, Tome IX. Paris: Gauthier-Villars, 342 /C1/360, 1896. Frobenius, F. G. "U¨ ber die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul." J. reine angew. Math. 101, 273 /C1/299, 1887. Reprinted in Ferdi- nand Georg Frobenius Gesammelte Abhandlungen, Band II. Berlin: Springer-Verlag, pp. 304 /C1/330, 1968. Neumann, P. M. "A Lemma that is not Burnside’s." Math. Scientist 4, 133 /C1/141, 1979. Khan, M. R. "A Counting Formula for Primitive Tetrahedra in Z3 :/" Amer. Math. Monthly 106, 525 /C1/533, 1999. Po´lya, G. "Kombinatorische Anzahlbestimmungen fu¨r Grup- pen, Graphen, und chemische Verbindungen." Acta Math. 68, 145 /C1/254, 1937. Rotman, J. A First Course in Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000. Cauchy-Hadamard Theorem The RADIUS OF CONVERGENCE of the TAYLOR SERIES a0 /C27a1z /C27a2z2 /C27... is r /C301 lim n0/C12( anjj)1 =n : See also RADIUS OF CONVERGENCE ,TAYLOR SERIES Cauchy-Kovalevskaya Theorem The theorem which proves the existence and unique- ness of solutions to the CAUCHY PROBLEM . See also CAUCHY PROBLEM Cauchy-Lagrange Identity LAGRANGE’S IDENTITY Cauchy-Maclaurin Theorem MACLAURIN- CAUCHY THEOREM Cauchy-Riemann Equations Let f(x;y)/C13u(x;y)/C27iv(x;y); (1) where z/C13x/C27iy; (2) so dz /C30dx /C27idy: (3) The total derivative of f with respect to z may then be computed as follows. y /C30z /C28 x i (4) x /C30z /C28iy ; (5) so @y @z /C301 i /C30/C28i (6) @x @z /C301; (7) and df dz /C30@f @x@x @z /C27@f @y@y @z /C30@f @x /C28i@f @y : (8) In terms of u and v, (8) becomes df dz /C30@u @x /C27i@v @x ! /C28i@u @y /C27i@v @y ! /C30@u @x /C27i@v @x ! /C27/C28 i@u @y /C27@v @y ! : (9) Along the real, or X-AXIS , @f =@y /C300 ; so df dz /C30@u @x /C27i@v @x : (10) Along the imaginary, or Y-AXIS , @f =@x /C300; so df dz /C30/C28i@u @y /C27@v @y : (11) If f is COMPLEX DIFFERENTIABLE , then the value of the derivative must be the same for a given dz, regardless of its orientation. Therefore, (10) must equal (11), which requires that @u @x /C30@v @y (12) and @v @x /C30/C28@u @y: (13) These are known as the Cauchy-Riemann equations. They lead to the condition @2u @x @y /C30/C28@2v @x @y : (14) The Cauchy-Riemann equations may be conciselywritten as df d¯z /C30@f @x /C27i@f @y /C30@u @x /C27i@v @x ! /C27i@u @y /C27i@v @y ! /C30@u @x /C28@v @y ! /C27i@u @y /C27@v @x ! /C300 ; (15) where ¯z is the COMPLEX CONJUGATE . If z /C30reiu ; then the Cauchy-Riemann equations be- come @u @r /C301 r@v @ u (16) 1 r@u @ u /C30/C28@v @r (17) (Abramowitz and Stegun 1972, p. 17). If u and v satisfy the Cauchy-Riemann equations, they also satisfy LAPLACE’S EQUATION in 2-D, since @2u @x2 /C27@2u @y2 /C30@ @x@v @y ! /C27@ @y/C28@v @x ! /C300 (18) @2v @x2 /C27@2v @y2 /C30@ @x/C28@u @y ! /C27@ @y@u @x ! /C300 : (19) By picking an arbitrary f(z) ; solutions can be found which automatically satisfy the Cauchy-Riemann equations and L APLACE’S EQUATION . This fact is used to use CONFORMAL MAPPINGS to find solutions to physical problems involving scalar potentials such as fluid flow and electrostatics. See also ANALYTIC FUNCTION ,C AUCHY INTEGRAL THEOREM ,COMPLEX DERIVATIVE ,CONFORMAL TRANS- FORMATION ,E NTIRE FUNCTION ,M ONOGENIC FUNC- TION ,POLYGENIC FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972. Arfken, G. "Cauchy-Riemann Conditions." §6.2 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 3560 /C1/365, 1985. Knopp, K. "The Cauchy-Riemann Differential Equations." §7 inTheory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 28 /C1/31, 1996. Krantz, S. G. "The Cauchy-Riemann Equations." §1.3.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 13, 1999. Levinson, N. and Redheffer, R. M. Complex Variables. San Francisco, CA: Holden-Day, 1970. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Cauchy’s Cosine Integral Formula g p=2 /C28 p=2cos m/C27 n/C282 ueiu(m/C28 n/C272j) du /C30p G(m /C27 n /C28 1) 2 m/C27 n/C282 G(m /C27 j) G( n /C28 j) ; where G(z) is the GAMMA FUNCTION . Cauchy’s Determinant Theorem Any row r and column s of a DETERMINANT being selected, if the element common to them be multiplied by its COFACTOR in the DETERMINANT , and every product of another element of the row by another element of the columns be multiplied by its COFAC- TOR, the sum of the results is equal to the given DETERMINANT . Symbolically, D/C30ars@D @ars/C27X ariaks@2 D @ari @aks(1) /C30(/C281)r/C27sarsArs /C27X 9ariaksArk ; is ; (2) where i ; k /C301 ; 2, ..., n; i "s; k "r; and the sign before ariaksArk ; is is determined by the formula (/C281)n1/C27n2 ; with n1the total number of PERMUTATION INVERSIONS in the suffix and n2 /C30r /C27i /C27k /C27s :/ See also DETERMINANT References Muir, T. "Cauchy’s Theorem." §110 in A Treatise on the Theory of Determinants. New York: Dover, pp. 95 /C1/96, 1960. Cauchy’s Formula The GEOMETRIC MEAN is smaller than the ARITHMETIC MEAN , YN i/C301ni ! 1 =N 5PN i /C301ni N; with equality in the cases (1) N /C301 or (2) ni /C30nj for all i, j. See also ARITHMETIC MEAN,GEOMETRIC MEAN Cauchy’s Inequality A special case of HO¨ LDER’S SUM INEQUALITY with p /C30 q /C302; Xn k /C301akbk ! 2 5Xn k /C301a2 k !Xn k /C301b2k ! ; (1) where equality holds for ak /C30cbk : The inequality is sometimes also called Lagrange’s inequality (Mitri- novic 1970, p. 42), and can be written in vector form asa /C215 b jj5 ajjbjj: (2) In 2-D, it becomes (a2 /C27b2)(c2 /C27d2) ](ac /C27bd)2 : (3) It can be proven by writing Xn i /C301(aix /C27bi)2 /C30Xn i /C301a2 ix /C27bi ai !2 /C300 : (4) If bi =aiis a constant c, then x /C30/C28c : If it is not a constant, then all terms cannot simultaneously van- ish for REAL x, so the solution is COMPLEX and can be found using the QUADRATIC EQUATION x /C30/C282P aibi 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4(P aibi)2 /C28 4P a2 iPb2iq 2P a2i: (5) In order for this to be COMPLEX , it must be true that X iaibi !2 5X ia2 i !X ib2i ! ; (6) with equality when bi =aiis a constant. The VECTOR derivation is much simpler, (a /C215 b)2 /C30a2b2 cos2 u 5a2b2 ; (7) where a2 /C13a /C215 a /C30X ia2i ; (8) and similarly for b. See also CHEBYSHEV INEQUALITY ,HO¨ LDER’S INEQUAL- ITIES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Wal- tham, MA: Blaisdell, pp. 42 /C1/43, 1967. Cauchy, A. L. Cours d’analyse de l’E ´cole Royale Polytechni- que, 1e `re partie: Analyse alge ´brique. Paris: p. 373, 1821. Reprinted in /Œ/uvres comple `tes, 2e se ´rie, Vol. 3. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1092, 2000. Hardy, G. H.; Littlewood, J. E.; and Po ´lya, G. "Cauchy’s Inequality." §2.4 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 16 /C1/18, 1952. Jeffreys, H. and Jeffreys, B. S. "Cauchy’s Inequality." §1.16 inMethods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 54, 1988. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 12, 1999. Mitrinovic, D. S. "Cauchy’s and Related Inequalities." §2.6 in Analytic Inequalities. New York: Springer-Verlag, pp. 41 /C1/48, 1970. Cauchy’s Rigidity Theorem RIGIDITY THEOREM Cauchy’s Theorem CAUCHY BINOMIAL THEOREM ,C AUCHY- DAVENPORT THEOREM ,CAUCHY’S DETERMINANT THEOREM ,CAU- CHY’S FORMULA ,CAUCHY- HADAMARD THEOREM ,CAU- CHY INTEGRAL THEOREM ,C AUCHY- KOVALEVSKAYA THEOREM ,M ACLAURIN- CAUCHY THEOREM ,R IGIDITY THEOREM Cauchy-Schwarz Inequality SCHWARZ’S INEQUALITY Cauchy-Schwarz Integral Inequality Let a1 and a2 by any two REAL integrable functions in [a, b], then lim min( m; n) 0/C12d(am ; an) /C300: with equality IFF F with k real. References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000. Cauchy-Schwarz Sum Inequality p "2 u1 Equality holds IFF the sequences u2 ; u8 ; ... and m1 ; m2 ; ... are proportional. See also FIBONACCI IDENTITY References Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Wal- tham, MA: Blaisdell, pp. 42 /C1/43, 1967. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1092, 2000. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 12, 1999. Caudrey-Dodd-Gibbon-Sawada-Kotera Equation The PARTIAL DIFFERENTIAL EQUATION ut /C27uxxxxx /C2730uuxxx /C2730uxuxx /C27180u2ux /C300 : See also SAWADA- KOTERA EQUATION References Aiyer, R. N.; Fuchssteiner, B.; and Oevel, W. "Solitons and Discrete Eigenfunctions of the Recursion Operator of Non- Linear Evolution Equations: I. The Caudrey-Dodd-Gib-bon-Sawada-Kotera Equations." J. Phys. A: Math. Gen. 19, 3755 /C1/3770, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 132, 1997. Caustic The curve which is the ENVELOPE of reflected (CATA- CAUSTIC ) or refracted (DIACAUSTIC ) rays of a given curve for a light source at a given point (known as the RADIANT POINT ). The caustic is the EVOLUTE of the ORTHOTOMIC . See also CATACAUSTIC ,CIRCLE CAUSTIC ,DIACAUSTIC , ENVELOPE ,EVOLUTE ,ORTHOTOMIC ,RADIANT POINT References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 60, 1972. Lockwood, E. H. "Caustic Curves." Ch. 24 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 182 /C1/185, 1967. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 28, 1991. Yates, R. C. "Caustics." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 15 /C1/20, 1952. Cavalieri’s Principle 1. If the lengths of every one-dimensional slice are equal for two regions, then the regions have equal AREAS . 2. If the AREAS of every two-dimensional SECTION are equal for two SOLIDS , then the SOLIDS have equal VOLUMES . See also CROSS SECTION ,PAPPUS’S CENTROID THEO- REM,SECTION ,VOLUME THEOREM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 126 and 132, 1987. Harris, J. W. and Stocker, H. "Cavalieri’s Theorem." §4.1.1 inHandbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 95, 1998. Kern, W. F. and Bland, J. R. "Cavalieri’s Theorem" and "Proof of Cavalieri’s Theorem." §11 and 49 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 25 /C1/27 and 145 /C1/146, 1948. Cavalieri’s Theorem CAVALIERI’S PRINCIPLE Cayley Algebra The only NONASSOCIATIVE DIVISION ALGEBRA with REAL SCALARS . There is an 8-square identity corre- sponding to this algebra. The elements of a Cayley algebra are called C AYLEY NUMBERS or OCTONIONS , and the MULTIPLICATION TABLE for any Cayley algebra over a FIELD Fwith characteristic p"2 may be taken as shown in the following table, where u1 ; u2 ; ..., u8are a bases over F and m1 ; m2 ; and m3are nonzero elements of F (Schafer 1996, pp. 5 /C1/). /u1//u2// u3// u4// u5// u6// u7// u8/ /u1//u1//u2// u3// u4// u5// u6// u7// u8/ /u2//u2//m1u1///C28u4///C28m1u3///C28u6///C28m1u5// u8// m1u7/ /u3//u3//u4//m2u1//m2u2///C28u7///C28u8///C28m2u5///C28m2u6/ /u4//u4//m1u3///C28m2u2///C28m1 m2u1///C28u8///C28m1u7// m2u6/ m1 m2 m5 /u5//u5//u6// u7// u8// m3u1//m3u2// m3u3// m3u4/ /u6//u6//m1u5// u8// m1u7///C28m3u2///C28m1 m3u1///C28m3u4/ /C28m1m2m3 /u7//u7///C28u8//m2u5///C28m2u6///C28m3u3//m3u4///C28m2 m3u1/ m2 m3 m2 /u8//u8///C28m1u7//m2u6///C28m1 m2u5///C28m3u4//m1 m3u3///C28m2 m3u2/ m1m2m3m1 See also CAYLEY NUMBER ,DIVISION ALGEBRA ,OCTO- NION ,NONASSOCIATIVE ALGEBRA References Kurosh, A. G. General Algebra. New York: Chelsea, pp. 226 /C1/28, 1963. Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, pp. 5 /C1/6, 1996. Cayley Cubic A CUBIC RULED SURFACE (Fischer 1986) in which the director line meets the director CONIC SECTION . Cayley’s surface is the unique cubic surface having four ORDINARY DOUBLE POINTS (Hunt), the maximum possible for CUBIC SURFACE (Endraß). The Cayley cubic is invariant under the TETRAHEDRAL GROUP and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (Endraß). If the ORDINARY DOUBLE POINTS in projective 3-space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), then the equation of the surface in projective coordinates is 1 x0/C271 x1/C271 x2/C271 x3/C300 (1) (Hunt). Defining "affine" coordinates with plane at infinity v /C30x0 /C27x1 /C27x2 /C272x3 andx /C30x0 v (2) y /C30x1 v (3) z /C30x2 v (4) then gives the equation /C285(x2y /C27x2z /C27y2x /C27y2z /C27z2y /C27z2x) /C272(xy /C27xz /C27yz) /C300 (5) plotted in the left figure above (Hunt). The slightly different form 4(x3 /C27y3 /C27z3 /C27w3) /C28(x /C27y /C27z /C27w)3 /C300 (6) is given by Endraß which, when rewritten in TETRA- HEDRAL COORDINATES , becomes x2 /C27y2 /C28x2z /C27y2z /C27z2 /C281 /C300; (7) plotted in the right figure above. The Hessian of the Cayley cubic is given by 0 /C30x2 0(x1x2 /C27x1x3 /C27x2x3) /C27x21(x0x2 /C27x0x3 /C27x2x3) /C27x22(x0x1 /C27x0x3 /C27x1x3) /C27x23(x0x1 /C27x0x2 /C27x1x2) (8) in homogeneous coordinates x0 ; x1 ; x2 ; and x3 : Taking the plane at infinity as v /C305(x0 /C27x1 /C27x2 /C272x3)=2 and setting x, y, and z as above gives the equation 25[x3(y/C27z)/C27y3(x/C27z)/C27z3(x/C27y)]/C2750(x2y2/C27x2z2/C27y2z2) /C28125(x2yz/C27y2xz/C27z2xy)/C2760xyz/C284(xy/C27xz/C27yz)/C300;(9) plotted above (Hunt). The Hessian of the Cayley cubic has 14 ORDINARY DOUBLE POINTS , four more than a the general Hessian of a smooth CUBIC SURFACE (Hunt). See also CAYLEY SURFACE References Endraß, S. "Fla¨chen mit vielen Doppelpunkten." DMV- Mitteilungen 4,17/C1/20, Apr. 1995. Endraß, S. "The Cayley Cubic." http://enriques.mathemati- k.uni-mainz.de/kon/docs/Ecayley.shtml. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 14, 1986. Fischer, G. (Ed.). Plate 33 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 33, 1986. Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Hunt, B. The Geometry of Some Special Arithmetic Quoti- ents. New York: Springer-Verlag, pp. 115 /C1/122, 1996. Nordstrand, T. "The Cayley Cubic." http://www.uib.no/peo- ple/nfytn/cleytxt.htm. Cayley Graph The Cayley graph of a GROUP G is a DIRECTED GRAPH determined by a set of generators g1 ; ..., gk : The vertices correspond to the elements of the group, and whenever gia /C30b; an edge is drawn between a and b. For example, the DIHEDRAL GROUP D7(left figure) is generated by the two elements, flips (red) and rota- tions (blue). The Cayley graph depends on the choice of a generating set. The right figure above illustrates the Cayley graph for the ALTERNATING GROUP A4 :/ Royle has constructed all cubic Cayley graphs up to 1000 vertices, excluding those on 512 and 768 vertices. The Cayley graphs of infinite groups provide inter- esting geometries. For example, the Cayley graphs of the FREE GROUP on two generators are illustrated above (drawn out to successive levels), representing horizontal and vertical displacement respectively. Each new edge is drawn at half the size to give FRACTAL images.See also CAGE GRAPH ,C AYLEY TREE,D ISCRETE GROUP ,FREE GROUP ,GRAPH ,GROUP ,TREE References Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Grossman, I. and Magnus, W. Groups and Their Graphs. New York: Random House, p. 45, 1964. Royle, G. "Cubic Cages." http://www.cs.uwa.edu.au/~gordon/ cages/. Cayley Lines The 60 PASCAL LINES of a hexagon inscribed in a conic intersect three at a time through 20 STEINER POINTS , and also three at a time in 60 KIRKMAN POINTS . Each STEINER POINT lies together with three KIRKMAN POINTS on a total of 20 lines known as Cayley lines. The 20 Cayley lines pass four at a time though 15 points known as SALMON POINTS (Wells 1991). There is a dual relationship between the 20 Cayley lines and the 20 STEINER POINTS . See also KIRKMAN POINTS ,PASCAL LINES,PASCAL’S THEOREM ,PLU¨ CKER LINES,SALMON POINTS ,STEINER POINTS References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 236 /C1/237, 1929. Salmon, G. "Notes: Pascal’s Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 379 /C1/ 382, 1960. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 172, 1991. Cayley Number There are two completely different definitions of Cayley numbers. The first and most commonly encountered type of Cayley number is the eight elements in a CAYLEY ALGEBRA , also known as octonions. The set of octonions is sometimes denoted O: A typical Cayley number is OF THE FORM a /C27bi0 /C27ci1 /C27di2 /C27ei3 /C27fi4 /C27gi5 /C27hi6 ; where each of the triples (i0 ; i1 ; i3) ; (i1 ; i2 ; i4); (i2 ; i3 ; i5); (i3 ; i4 ; i6); (i4 ; i5 ; i0); (i5 ; i6 ; i1); (i6 ; i0 ; i2) behaves like the QUATERNIONS (i ; j ; k): Cayley num- bers are not ASSOCIATIVE . They have been used in the study of 7- and 8-D space, and a general rotation in 8- D space can be written x?0((((((xc1)c2)c3)c4)c5)c6)c7: A quantity which describes a D ELPEZZO SURFACE is sometimes also called a Cayley number (Coxeter 1973, p. 211). See also COMPLEX NUMBER ,D EL PEZZO SURFACE , QUATERNION ,REAL NUMBER References Conway, J. H. and Guy, R. K. "Cayley Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 234 /C1/ 235, 1996. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Okubo, S. Introduction to Octonion and Other Non-Associa- tive Algebras in Physics. New York: Cambridge University Press, 1995. Cayley Surface In affine 3-space the Cayley surface is given by x3 /C30x1x2 /C281 3x3 1 (Nomizu and Sasaki 1994). The surface has been generalized by Eastwood and Ezhov (2000) to FN(x1 ; x2 ; ...; xN) /C13XN d/C301(/C281)dX i/C27j /C27.../C27m /C30Nxixj ...xm|fflfflfflfflfflffl{zfflfflfflfflfflffl} d/C300: This gives the first few hypersurfaces as x4 /C30x1x3 /C271 2 x2 2 /C28x21x2 /C271 4 x4 1 x5 /C30x1x4 /C27x2x3 /C28x21x3 /C28x1x22 /C27x31x2 /C281 5 x5 1 : See also CAYLEY CUBIC References Eastwood, M. and Ezhov, V. Cayley Hypersurfaces. 25 Jan 2000. http://xxx.lanl.gov/abs/math.DG/0001134/. Nomizu, K. and Sasaki, T. Affine Differential Geometry: Geometry of Affine Immersions. Cambridge, England: Cambridge University Press, 1994. Nomizu, K. and Pinkall, U. "Cayley Surfaces in Affine Differential Geometry." Toˆhoku Math. J. 41, 589 /C1/596, 1989. Cayley Transform The LINEAR FRACTIONAL TRANSFORMATION z /C2i /C28 z i /C27 zthat maps the UPPER HALF-PLANE fz : I[z] > 0 g CON- FORMALLY onto the UNIT DISK fz : ½z ½B1g:/ See also CONFORMAL MAPPING ,LINEAR FRACTIONAL TRANSFORMATION References Krantz, S. G. "The Cayley Transform." §6.3.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 85, 1999. Cayley Tree A TREE in which each non-leaf NODE has a constant number of branches n is called an n-Cayley tree. 2- Cayley trees are PATH GRAPHS . The unique n-Cayley tree on n /C271 nodes is the STAR GRAPH . The illustra- tion above shows the first few 3-Cayley trees (also called trivalent trees, binary trees, or boron trees). The numbers of binary trees on n /C301, 2, ... nodes (i.e., n-node trees having VERTEX DEGREE either 1 or 3; also called 3-Cayley trees, 3-valent trees, or boron trees) are 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0 ,4, 0, 6, 0, 11, ... (Sloane’s A052120). The illustrations above show the first few 4-Cayley and 5-Cayley trees. The PERCOLATION THRESHOLD for a Cayley tree having z branches is pc /C301 z /C28 1 : See also CAYLEY GRAPH ,PATH GRAPH ,STAR GRAPH , TREE References Sloane, N. J. A. Sequences A052120 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Cayley-Bacharach Theorem Let X1 ; X2 ƒP2 be CUBIC plane curves meeting in nine points p1 ; ..., p9 : If X ƒP2 is any CUBIC contain- ing p1 ; ..., p8 ; then X contains p9 as well. It is related to GORENSTEIN RINGS , and is a generalization of PAPPUS’S HEXAGON THEOREM and P ASCAL’S THEOREM . See also PASCAL’S THEOREM ,P APPUS’S HEXAGON THEOREM References Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach Theorems and Conjectures." Bull. Amer. Math. Soc. 33, 295/C1/324, 1996. Cayley-Dickson Algebra CAYLEY ALGEBRA Cayley-Hamilton Theorem Given a11/C28xa12 /C1/C1/C1 a1m a21 a22/C28x/C1/C1/C1 a2m nn:::n am1 am2 /C1/C1/C1 amm/C28xl112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112 /C30x m/C27cm/C281xm/C281/C27.../C27c0; (1) then Am/C27cm/C281Am/C281/C27.../C27c0I/C300; (2) where Iis the IDENTITY MATRIX . Cayley verified this identity for m/C302 and 3 and postulated that it was true for all m. For m/C302, direct verification gives a/C28xb cd /C28xl112l112l112l112l112l112l112l112/C30(a/C28x)(d/C28x)/C28bc /C30x 2/C28(a/C27d)x/C27(ad/C28bc)/C13x2/C27c1x/C27c2 (3) A/C30ab cdl12ml121 (4) A2/C30ab cdl12ml121 ab cdl12ml121 /C30a2/C27bc ab /C27bd ac/C27cd bc /C27d2l12ml121 (5) /C28(a/C27d)A/C30/C28a2/C28ad/C28ab/C28bd /C28ac/C28dc/C28ad/C28d2l12ml121 (6) (ad/C28bc)I/C30ad/C28bc 0 0 ad/C28bcl12ml121 ; (7) so A2/C28(a/C27d)A/C27(ad/C28bc)I/C3000 00l12ml121 : (8) The Cayley-Hamilton theorem states that a n/C29n MATRIX Ais annihilated by its CHARACTERISTIC POLY- NOMIAL det(xI/C28A);which is monic of degree n. References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 181, 1962. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1117, 2000. Segercrantz, J. "Improving the Cayley-Hamilton Equation for Low-Rank Transformations." Amer. Math. Monthly 99, 42/C1/44, 1992.Cayleyian Curve The ENVELOPE of the lines connecting corresponding points on the J ACOBIAN CURVE and S TEINERIAN CURVE . The Cayleyian curve of a net of curves of order nhas the same GENUS (CURVE ) as the J ACOBIAN CURVE and S TEINERIAN CURVE and, in general, the class 3 n(n/C281):/ References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 150, 1959. Cayley-Klein Parameters The parameters a;b;g;anddwhich, like the three EULER ANGLES , provide a way to uniquely character- ize the orientation of a solid body. These parameters satisfy the identities a¯a/C27g¯g/C301 (1) a¯a/C27b¯b/C301 (2) b¯b/C27d¯d/C301 (3) ¯ab/C27¯gd/C300 (4) ad/C28bg/C301 (5) and b/C30/C28 ¯g (6) d/C30¯a; (7) where ¯zdenotes the COMPLEX CONJUGATE . In terms of the E ULER ANGLES u;f;and c;the Cayley-Klein parameters are given by a/C30ei(c/C27f)=2cos(1 2u) (8) b/C30iei(c/C27f)=2sin(1 2u) (9) g/C30iei(c/C27f)=2sin(12u) (10) d/C30ei(c/C27f)=2cos(1 2u) (11) (Goldstein 1960, p. 155). The transformation matrix is given in terms of the Cayley-Klein parameters by A/C301 2(a2/C28g2/C27d2/C28b2)12i(g2/C28a2/C27d2/C28b2)gd/C28ab 1 2i(a2/C27g2/C28b2/C28d2)12(a2/C27g2/C27b2/C27d2)/C28i(ab/C27gd) bd/C28ag i(ag/C27bd) ad/C27bg2 643 75 (12) (Goldstein 1960, p. 153). The Cayley-Klein parameters may be viewed as parameters of a matrix (denoted Qfor its close relationship with QUATERNIONS ) Q/C30ab gdl12ml121 (13) which characterizes the transformations u?/C30 au /C27 bv (14) v ?/C30gu /C27 dv: (15) of a linear space having complex axes. This matrix satisfies Q /C31Q /C30QQ/C31/C30I ; (16) where I is the IDENTITY MATRIX and A/C31 the ADJOINT MATRIX , as well as ½Q ½/C31½Q ½/C301: (17) In terms of the EULER PARAMETERS ei and the PAULI MATRICES si ; the Q/-matrix can be written as Q /C30e0I /C27i( e1 s1 /C27e2 s2 /C27e3 s3) (18) (Goldstein 1980, p. 156). See also EULER ANGLES ,EULER PARAMETERS ,PAULI MATRICES ,QUATERNION ,ROTATION References Goldstein, H. "The Cayley-Klein Parameters and Related Quantities." §4 /C1/5in Classical Mechanics, 2nd ed. Read- ing, MA: Addison-Wesley, pp. 148 /C1/158, 1980. Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Description of Rotations in Terms of Unitary 2 /C292 Matrices. Cayley-Klein Parameters." §1.4.3 in Quantum Theory of Angular Momentum. Singapore: World Scien- tific, pp. 24 /C1/27, 1988. Cayley-Klein-Hilbert Metric The METRIC of Felix Klein’s model for HYPERBOLIC GEOMETRY , g11 /C30a2(1 /C28 x2 2) (1 /C28 x2 1 /C28 x22)2 g12 /C30a2x1x2 (1 /C28 x2 1 /C28 x22)2 g22 /C30a2(1 /C28 x2 1) (1 /C28 x2 1 /C28 x22)2 : See also HYPERBOLIC GEOMETRY Cayley-Menger Determinant This entry contributed by KAREN D. COLLINS A DETERMINANT that gives the volume of a SIMPLEX in j dimensions. If S is a j-simplex in Rn with vertices v1 ; ... ; vj /C271 and B//C30( bik) denotesthe (j /C271) /C29(j /C271) matrixgivenby bik /C30 vi /C28vk kk2 2 ; (1) then the CONTENT Vj is given byV2 j (S) /C30(/C281)j /C271 2j(j!)2det( ˆB) ; (2) where ˆB is the (j /C272) /C29(j /C272) matrix obtained from B by bordering B with a top row (0; 1; ... ; 1) and a left column (0; 1; ... ; 1)T : Here, the vector L2-NORMS ½½vi /C28vk ½½2 are the edge lengths and the DETERMINANT in (2) is the Cayley-Menger determinant (Sommer- ville 1958, Gritzmann and Klee 1994). The first few coefficients for j /C300, 1, ... are /C281, 2, /C2816, 288, /C289216, 460800, ... (Sloane’s A055546). For j /C302, (2) becomes /C2816 D2 /C3001 1 1 10 c2b2 1 c20 a2 1 b2a20l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112; (3) which gives the AREA for a plane triangle with side lengths a, b, and c, and is a form of HERON’S FORMULA . For j /C303, the content of the 3-simplex (i.e., volume of the general TETRAHEDRON ) is given by the determi- nant 288V2 /C3001 1 1 1 10 d2 12d213d214 1 d2 21 0 d223d224 1 d2 31d232 0 d234 1 d241d242d243 0l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112; (4) where the edge between vertices i and j has length d ij : Setting the left side equal to 0 (corresponding to a TETRAHEDRON of volume 0) gives a relationship between the DISTANCES between vertices of a planar QUADRILATERAL (Uspensky 1948, p. 256). See also HERON’S FORMULA ,QUADRILATERAL ,TETRA- HEDRON References Gritzmann, P. and Klee, V. §3.6.1 in "On the Complexity of Some Basic Problems in Computational Convexity II. Volume and Mixed Volumes." In Polytopes: Abstract, Convex and Computational (Ed. T. Bisztriczky, P. McMullen, R. Schneider, R.; and A. W. Weiss). Dor- drecht, Netherlands: Kluwer, 1994. Sloane, N. J. A. Sequences A055546 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sommerville, D. M. Y. An Introduction to the Geometry of N Dimensions. New York: Dover, p. 124, 1958. Uspensky, J. V. Theory of Equations. New York: McGraw- Hill, p. 256, 1948. Cayley’s Group Theorem Every FINITE GROUP of order n can be REPRESENTED ASaPERMUTATION GROUP onnletters, as first proved by Cayley in 1878 (Rotman 1995). See also FINITE GROUP ,PERMUTATION GROUP References Rotman, J. J. An Introduction to the Theory of Groups, 4th ed. New York: Springer-Verlag, p. 52, 1995. Cayley’s Hypergeometric Function Theorem If (1 /C28z)a /C27b /C28c 2F1(2a; 2b;2c; z) /C30X/C12 n /C300anzn ; then 2F1(a; b; c /C271 2; z)2F1(c /C28a ; c /C28b; c12; z) /C30X/C12 n /C300(c)n (c /C271 2) anzn ; where2F1(a ; b; c; z)isa HYPERGEOMETRIC FUNC- TION . See also HYPERGEOMETRIC FUNCTION Cayley’s Ruled Surface CAYLEY CUBIC Cayley’s Sextic A plane curve discovered by Maclaurin but first studied in detail by Cayley. The name Cayley’s sextic is due to R. C. Archibald, who attempted to classify curves in a paper published in Strasbourg in 1900(MacTutor Archive). Cayley’s sextic is given in POLAR COORDINATES by r/C304acos3(1 3u): (1) Parametric equations can be given by x(t)/C304acos4(12t)(2 cos t/C281) (2) y(t)/C304acos3(12t) sin(32t) (3) (Gray 1997, p. 119). Calculating rgives r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p /C304 cos3(1 2t); (4)andtis related to uby u/C30tan/C281y x ! /C303 2t; (5) thus recovering (1). The C ARTESIAN equation is 4(x2/C27y2/C28ax)3/C3027a2(x2/C27y2)2: (6) The ARC LENGTH ,CURVATURE , and TANGENTIAL ANGLE for the curve with a/C301 are s(t)/C303(t/C27sint); (7) k(t)/C3013sec2(12t); (8) f(t)/C302t: (9) References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 119 /C1/120, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178 and 180, 1972. MacTutor History of Mathematics Archive. "Cayley’s Sex- tic." http://www-groups.dcs.st-and.ac.uk/~history/Curves/ Cayleys.html. Cayley’s Sextic Evolute The EVOLUTE of Cayley’s sextic is x/C301 8a/C271 16a[3 cos(23t)/C28cos(2 t)] y/C301 16a[3 sin(2 3t)/C28sin(2 t)]; which is a NEPHROID . C-Curve LE´VYFRACTAL C-Determinant A DETERMINANT appearing in PADE´APPROXIMANT identities: Cr=s /C30ar/C28s /C271ar/C28s/C272/C1/C1/C1 ar nn::: n ar ar/C271 /C1/C1/C1 ar/C27s/C281l112l112l112l112l112l112l112l112l112l112l112l112: See also P ADE´ APPROXIMANT Cech Cohomology The direct limit of the COHOMOLOGY groups with COEFFICIENTS in an ABELIAN GROUP of certain cover- ings of a TOPOLOGICAL SPACE . Ceiling CEILING FUNCTION Ceiling Function The function xdewhich gives the smallest INTEGER ]x; shown as the thick curve in the above plot. Schroeder (1991) calls the ceiling function symbols the "GALLOWS " because of the similarity in appear- ance to the structure used for hangings. The name and symbol for the ceiling function were coined by K. E. Iverson (Graham et al. 1990). Although some authors used the symbol ]x[ to denote the ceiling function (by analogy with the older notation [x] for the FLOOR FUNCTION ), this practice is strongly dis- couraged (Graham et al. 1990, p. 67). Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used (D. W. Cantrell). Here, S&O indicates Spanier and Oldham (1987). notation name S&O Graham et al.Mathematica / xbc/ integer- value/Int(x)/ floor or inte- ger partFloor [ x]/sgn(x) xjjbc / integer-part /Ip(x)/ no name IntegerPart [ x ] /x/C28 xbc/ fractional-value/frac(x)/ fractionalpart or fxg /no name /sgn(x)(½x ½/C28½x ½bc)/ fractional- part/Fp(x)/ no name Fractional- Part [ x] Odlyzko and Wilf (1991) have shown that the se- quence fxn g defined by x0 /C301 and xn/C271 /C303 2xnlm satisfies xn /C30 K(32)njk for all n, where K /C301:6222705028... is analogous to MILLS’ CONSTANT in the sense that the formula is useless unless K is known exactly ahead of time (Finch). See also FLOOR FUNCTION ,INTEGER PART,M ILLS’ CONSTANT ,NEAREST INTEGER FUNCTION ,STAIRCASE FUNCTION References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Finch, S. "Powers of 3/2 Modulo One." http://www.mathsoft.- com/asolve/pwrs32/pwrs32.html. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer Functions." Ch. 3 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 67 /C1/101, 1994. Iverson, K. E. A Programming Language. New York: Wiley, p. 12, 1962. Odlyzko, A. M. and Wilf, H. S. "Functional Iteration and the Josephus Problem." Glasgow Math. J. 33, 235 /C1/240, 1991. Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 57, 1991. Cell A finite regular POLYTOPE . See also 16-CELL, 24-CELL, 120-CELL, 600-CELL Cellular Automaton A cellular automaton is a grid (possibly 1-D) of cells which evolves according to a set of rules based on thestates of surrounding cells. von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his "universalconstructor." von Neumann proved that an automa-ton consisting of cells with four orthogonal neighbors and 29 possible states would be capable of simulating aT URING MACHINE for some configuration of about 200,000 cells (Gardner 1983, p. 227). 1-D automata called " ELEMENTARY CELLULAR AUTO- MATA " are represented by a row of pixels with states either 0 or 1. These can be indexed with an 8-bit binary number, as shown by Stephen Wolfram. Wolfram further restricted the number from 28 /C30 256 to 32 by requiring certain symmetry conditions. The most well-known cellular automaton is Conway’s game of LIFE, popularized in Martin Gardner’s Scien- tific American columns. Although the computation of successive LIFE generations was originally done by hand, the computer revolution soon arrived and allowed more extensive patterns to be studied and propagated. See also AUTOMATA THEORY ,ELEMENTARY CELLULAR AUTOMATON ,LIFE,LANGTON’S ANT,TOTALISTIC CEL- LULAR AUTOMATON ,TURING MACHINE References Adami, C. Artificial Life. Cambridge, MA: MIT Press, 1998. Buchi, J. R. and Siefkes, D. (Eds.). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. New York: Springer-Verlag, 1989. Burks, A. W. (Ed.). Essays on Cellular Automata. Urbana- Champaign, IL: University of Illinois Press, 1970. Cipra, B. "Cellular Automata Offer New Outlook on Life, the Universe, and Everything." In What’s Happening in the Mathematical Sciences, 1995 /C1/1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 70 /C1/81, 1996. Dewdney, A. K. The Armchair Universe: An Exploration of Computer Worlds. New York: W. H. Freeman, 1988. Gardner, M. "The Game of Life, Parts I-III." Chs. 20 /C1/22 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 219 and 222, 1983. Goles, E. and Martı ´nez, S. (Eds.). Cellular Automata and Complex Systems. Amsterdam, Netherlands: Kluwer, 1999. Gutowitz, H. (Ed.). Cellular Automata: Theory and Experi- ment. Cambridge, MA: MIT Press, 1991. Hopcroft, J. E. and Ullman, J. D. Introduction to Automata Theory, Languages, and Computation. Reading, MA: Addison Wesley, 1979. Hopcroft J. E. "An n log n Algorithm for Minimizing the States in a Finite Automaton." In The Theory of Machines and Computations (Ed. Z. Kohavi.) New York: Academic Press, pp. 189 /C1/196, 1971. Levy, S. Artificial Life: A Report from the Frontier Where Computers Meet Biology. New York: Vintage, 1993. Martin, O.; Odlyzko, A.; and Wolfram, S. "Algebraic Aspects of Cellular Automata." Communications in Mathematical Physics 93, 219 /C1/258, 1984. Preston, K. Jr. and Duff, M. J. B. Modern Cellular Auto- mata: Theory and Applications. New York: Plenum, 1985. Sigmund, K. Games of Life: Explorations in Ecology, Evolu- tion and Behaviour. New York: Penguin, 1995. Sloane, N. J. A. Sequences A006977/M2497 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M2497 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Toffoli, T. and Margolus, N. Cellular Automata Machines: A New Environment for Modeling. Cambridge, MA: MIT Press, 1987. Weisstein, E. W. "Books about Cellular Automata." http:// www.treasure-troves.com/books/CellularAutomata.html. Wolfram, S. "Statistical Mechanics of Cellular Automata." Rev. Mod. Phys. 55, 601 /C1/644, 1983. Wolfram, S. "Twenty Problems in the Theory of Cellular Automata." Physica Scripta T9, 170 /C1/183, 1985.Wolfram, S. (Ed.). Theory and Application of Cellular Automata. Reading, MA: Addison-Wesley, 1986. Wolfram, S. Cellular Automata and Complexity: Collected Papers. Reading, MA: Addison-Wesley, 1994. Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2001. Wuensche, A. and Lesser, M. The Global Dynamics of Cellular Automata: An Atlas of Basin of Attraction Fields of One-Dimensional Cellular Automata. Reading, MA: Addison-Wesley, 1992. Cellular Space AH AUSDORFF SPACE which has the structure of a so- called CW -COMPLEX . Center A special POINT which usually has some symmetric placement with respect to points on a curve or in a SOLID . The center of a CIRCLE is equidistant from all points on the CIRCLE and is the intersection of any two distinct DIAMETERS . The same holds true for the center of a SPHERE . See also CENTER (GROUP ), CENTER OF MASS,CIRCLE , CIRCUMCENTER ,C LEAVANCE CENTER ,C URVATURE CENTER ,ELLIPSE ,EQUI-BROCARD CENTER ,EXCENTER , FUHRMANN CENTER ,H OMOTHETIC CENTER ,INCEN- TER,INVERSION CENTER ,M AJOR TRIANGLE CENTER , NINE-POINT CENTER ,O RTHOCENTER ,P ERSPECTIVE CENTER ,POINT ,RADICAL CENTER ,SIMILITUDE CEN- TER,S PHERE ,S PIEKER CENTER ,T AYLOR CENTER , TRIANGLE CENTER ,T RIANGLE CENTER FUNCTION , YFF CENTER OF CONGRUENCE Center (Group) The center of a GROUP is the set of elements which commute with every element of the GROUP . It is equal to the intersection of the CENTRALIZERS of the GROUP elements. See also CENTRALIZER ,ISOCLINIC GROUPS ,NILPOTENT GROUP Center Function TRIANGLE CENTER FUNCTION Center of Gravity CENTROID (GEOMETRIC ) Center of Mass CENTROID (GEOMETRIC ) Center of Similitude SIMILITUDE CENTER Centered Cube Number A FIGURATE NUMBER OF THE FORM , CCubn /C30n3 /C27(n /C281)3 /C30(2n /C281)(n2 /C28n /C271): The first few are 1, 9, 35, 91, 189, 341, ... (Sloane’s A005898). The GENERATING FUNCTION for the cen- tered cube numbers is x(x3 /C27 5x2 /C27 5x /C27 1) (x /C28 1)4 /C30x /C279x2 /C2735x3 /C2791x4 /C27...: See also CUBIC NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 51, 1996. Sloane, N. J. A. Sequences A005898/M4616 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Centered Hexagonal Number HEX NUMBER Centered Pentagonal Number A CENTERED POLYGONAL NUMBER consisting of a central dot with five dots around it, and then additional dots in the gaps between adjacent dots. The general term is (5n2 /C285n /C272)=2 ; and the first few such numbers are 1, 6, 16, 31, 51, 76, ... (Sloane’s A005891). The GENERATING FUNCTION of the centeredpentagonal numbers is x(x2 /C27 3x /C27 1) (1 /C28 x)3/C30x /C276x2 /C2716x3 /C2731x4 /C27...: See also CENTERED POLYGONAL NUMBER ,CENTERED SQUARE NUMBER ,CENTERED TRIANGULAR NUMBER , HEX NUMBER References Sloane, N. J. A. Sequences A005891/M4112 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Centered Polygonal Number A FIGURATE NUMBER in which layers of POLYGONS are drawn centered about a point instead of with the point at a VERTEX . See also CENTERED PENTAGONAL NUMBER ,CENTERED SQUARE NUMBER ,CENTERED TRIANGULAR NUMBER References Sloane, N. J. A. Sequences A001844/M3826 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M3826 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Centered Square Number ACENTERED POLYGONAL NUMBER consisting of a central dot with four dots around it, and then additional dots in the gaps between adjacent dots. The general term is n2/C27(n/C281)2;and the first few such numbers are 1, 5, 13, 25, 41, ... (Sloane’s A001844). Centered square numbers are the sum of two consecutive SQUARE NUMBERS and are congruent to 1 (mod 4). The GENERATING FUNCTION giving the centered square numbers is x(x /C27 1)2 (1 /C28 x)3 /C30x /C275x2 /C2713x3 /C2725x4 /C27...: See also CENTERED PENTAGONAL NUMBER ,CENTERED POLYGONAL NUMBER ,CENTERED TRIANGULAR NUM- BER,SQUARE NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 41, 1996. Sloane, N. J. A. Sequences A001844/M3826 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Centered Tree A TREE (also called a central tree) having a single node that is a GRAPH CENTER . The numbers of centered trees on n /C301, 2, ... nodes are 1, 1, 0, 1, 1, 2, 3, 7, 12, 27, 55, ... (Sloane’s A000676). See also BICENTERED TREE,GRAPH CENTER ,TREE References Biggs, N. L.; Lloyd, E. K.; and Wilson, R. J. Graph Theory 1736 /C1/1936. Oxford, England: Oxford University Press, p. 49, 1976. Cayley, A. "On the Analytical Forms Called Trees, with Application to the Theory of Chemical Combinations." Reports Brit. Assoc. Advance. Sci. 45, 237 /C1/305, 1875. Reprinted in Math Papers, Vol. 9, pp. 427 /C1/460. Sloane, N. J. A. Sequences A000676/M0831 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Centered Triangular Number A CENTERED POLYGONAL NUMBER consisting of a central dot with three dots around it, and then additional dots in the gaps between adjacent dots. The general term is (3n2 /C283n /C272)=2 ; and the first few such numbers are 1, 4, 10, 19, 31, 46, 64, ... (Sloane’s A005448). The GENERATING FUNCTION giving the centered triangular numbers is x(x2 /C27 x /C27 1) (1 /C28 x)3/C30x /C274x2 /C2710x3 /C2719x4 /C27...: See also CENTERED PENTAGONAL NUMBER ,CENTERED SQUARE NUMBER References Sloane, N. J. A. Sequences A005448/M3378 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Centillion In the American system, 10303. See also LARGE NUMBER Central Angle An ANGLE having its VERTEX at a CIRCLE ’s center which is formed by two points on the CIRCLE’S CIRCUMFERENCE . For angles with the same endpoints, uc/C302ui; where uiis the INSCRIBED ANGLE . References Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., pp. xxi-xxii, 1995. Central Beta Function The central beta function is defined by b(p) /C13B(p; p); (1) where B(p; q) is the BETA FUNCTION . It satisfies the identities b(p) /C3021/C282pB(p ;1 2) (2) /C3021 /C282p cos(pp)(1 2 /C28p ; p) (3) /C30g1 0tpdt (1 /C27 t)2p (4) 2 pY/C12 n/C301n(n /C27 2p) (n /C27 p)(n /C27 p) : (5) With p /C301 =2; the latter gives the WALLIS FORMULA . When p /C30a =b; bb(a=b) /C3021/C282a =bJ(a ; b); (6) where J(a ; b) /C13g1 0ta/C281 dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 tbp : (7) The central beta function satisfies (2 /C274x)b(1 /C27x) /C30xb(x) (8) (1 /C282x) b(1 /C28x) b(x) /C302p cot(px) (9) b(1 2 /C28x) /C3024x /C281 tan(px)b(x) (10) b(x)b(x /C271 2) /C3024x /C271 pb(2x) b(2x /C2712): (11) For p an ODD POSITIVE INTEGER , the central betafunction satisfies the identity b(px)/C301 ffiffiffippY(p/C281)=2 k/C3012x/C272k/C281 p 2pYp/C281 k/C300bx/C27k p ! :(12) See also BETA FUNCTION ,REGULARIZED BETA FUNC- TION References Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua- tion of the Gamma Function at Rational Values of Small Denominators." IMA J. Numerical Analysis 12, 519/C1/526, 1992. Central Binomial Coefficient The nth central binomial coefficient is defined as n n=2bcl11)l117 ;wheren kl1ml11 is a BINOMIAL COEFFICIENT and nbc is the FLOOR FUNCTION . The first few values are 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, ... (Sloane’s A001405). The central binomial coefficients have GENERATING FUNCTION 1/C284x2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C284x2p 2(2x3/C28x2)/C301/C272x/C273x2/C276x3/C2710x4/C27...: The central binomial coefficients are SQUAREFREE only for n/C301, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, ... (Sloane’s A046098), with no others less than 7320. The above coefficients are a superset of the alter- native "central" binomial coefficients 2n nl11sl11n /C30(2n)! (n!)2; which have GENERATING FUNCTION 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C284xp /C301/C272x/C276x2/C2720x3/C2770x4/C27...: The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (Sloane’s A000984). A fascinating series of identities involving inverse central binomial coefficients times small powers aregiven by X /C12 n¼11 2n nl11sl11n/C301 27(2pffiffiffi 3p /C279)/C300:7363998587 . . . (1) X/C12 n¼11 n2n nl11sl11n/C301 9pffiffiffi 3p /C300:6045997881 . . . (2) X/C12 n¼11 n22n nl11sl11n/C301 3z(2)/C3018p2(3) X/C12 n/C3011 n42n nl11sl11n/C3017 36 z(4) /C3017 3240p4 (4) (Comtet 1974, p. 89; Le Lionnais 1983, pp. 29, 30, 41, 36), which follow from the beautiful formula X/C12 n /C3011 nk2n nl11sl11n/C301 2k /C271Fk (1; ...; 1|fflfflfflfflfflffl{zfflfflfflfflfflffl} k /C271;32 ; 2; ...; 2|fflfflfflfflfflffl{zfflfflfflfflfflffl} k /C281;14): (5) for k ]1 ; wheremFn(a1 ; ...; am; b1 ; ...; bn; x)isa GENERALIZED HYPERGEOMETRIC FUNCTION . Additional sums of this type include X/C12 n/C3011 n32n nl11sl11n/C301 18pffiffiffi 3p [ c1(1 3) /C28 c1(23)] /C2843 z(3) (6) X/C12 n/C3011 n52n nl11sl11n /C301 432pffiffiffi 3p [ c3(1 3) /C28 c3(23)] /C2819 3 z(5) /C2719 z(3) p3 ; (7) X/C12 n/C3011 n72n nl11sl11n/C3011 311040 pffiffiffi 3p [ c5(1 3) /C28 c5(23)] /C28493 24 z(7) /C2713 z(5)p2 /C2717 1620z(3) p4 ; (8) where cn(x) is the POLYGAMMA FUNCTION and z(x)is the RIEMANN ZETA FUNCTION (Plouffe 1998). Similarly, we have X/C12 n/C301(/C281)n /C281 2n nl11sl11n /C301 25[5 /C274ffiffiffi 5p csch/C281(2)] (9) X/C12 n/C301( /C281)n/C281 n2n nl11sl11n /C302 5ffiffiffi 5p csch/C281(2) (10) X/C12 n/C301(/C281)n/C281 n22n nl11sl11n/C302[csch/C281(2)]2 (11) X/C12 n/C301( /C281)n/C281 n32n nl11sl11n/C302 5 z(3) (12) (Le Lionnais 1983, p. 35; Guy 1994, p. 257), where z(z) is the RIEMANN ZETA FUNCTION . These follow from the analogous identity X/C12 n/C301( /C281)n/C281 nk2n nl11sl11n/C301 2k/C271Fk (1 ; ...; 1|fflfflfflfflfflffl{zfflfflfflfflfflffl} k/C271;32; 2; ...; 2|fflfflfflfflfflffl{zfflfflfflfflfflffl} k /C281; /C2814) : (13) Erdos and Graham (1980, p. 71) conjectured that thecentral binomial coefficient2n nl1ml11 is never SQUAREFREE for n /C214, and this is sometimes known as the ERDOS SQUAREFREE CONJECTURE .SA´ RKOZY’S THEOREM (Sa´r- kozy 1985) provides a partial solution which states that the BINOMIAL COEFFICIENT2n nl1ml11 is never SQUARE- FREE for all sufficiently large n ]n0(Vardi 1991). Granville and Ramare (1996) proved that the only SQUAREFREE values are n /C302 and 4. Sander (1992) subsequently showed that /2n9d nl1ml11 / are also never SQUAREFREE for sufficiently large n as long as d is not "too big." See also BINOMIAL COEFFICIENT ,B INOMIAL SUMS, CENTRAL TRINOMIAL COEFFICIENT ,ERDOS SQUARE- FREE CONJECTURE ,S TAIRCASE WALK,S A´ RKO¨ ZY’S THEOREM ,QUOTA SYSTEM References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Granville, A. and Ramare, O. "Explicit Bounds on Exponen- tial Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43,73/C1/107, 1996. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Plouffe, S. "The Art of Inspired Guessing." Aug. 7, 1998. http://www.lacim.uqam.ca/plouffe/inspired.html. Sander, J. W. "On Prime Divisors of Binomial Coefficients." Bull. London Math. Soc. 24, 140 /C1/142, 1992. Sa´rkozy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20,70/C1/80, 1985. Sloane, N. J. A. Sequences A000984/M1645, A001405/ M0769, and A046098 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Vardi, I. "Application to Binomial Coefficients," "Binomial Coefficients," "A Class of Solutions," "Computing Binomial Coefficients," and "Binomials Modulo and Integer." §2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 25 /C1/28 and 63 /C1/71, 1991. Central Conic An ELLIPSE orHYPERBOLA . See also CONIC SECTION References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 146 /C1/150, 1967. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 77, 1990. Central Difference The central difference for a function tabulated at equal intervals fnis defined by d(fn)/C30dn/C30d1 n/C30fn/C271=2/C28fn/C281=2: (1) First and higher order central differences arranged so as to involve integer indices are then given by dn/C271=2/C30d1 n/C271=2/C30fn/C271/C28fn (2) d2 n /C30 d1n/C271=2 /C28 d1n/C281 =2 /C30fn/C271 /C282fn /C27fn /C281 (3) d3n/C271 =2 /C30 d2n/C271 /C28 d2n /C30fn/C272 /C283fn/C271 /C273fn /C28fn/C281 : (4) Higher order differences may be computed for EVEN and ODD powers, d2k n/C271 =2 /C30X2k j/C300(/C281)j 2k jl11sl11n fn/C27k /C28j (5) d2k/C271 n/C271 =2 /C30X2k /C271 j/C300(/C281)j 2k /C271 jl11sl11n fn/C27k /C271/C28j : (6) See also BACKWARD DIFFERENCE ,D IVIDED DIFFER- ENCE ,FORWARD DIFFERENCE References Abramowitz, M. and Stegun, C. A. (Eds.). "Differences." §25.1 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877 /C1/878, 1972. Jeffreys, H. and Jeffreys, B. S. "Central Differences For- mula." §9.084 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 284 /C1/286, 1988. Sheppard, W. F. Proc. London Math. Soc. 31, 459, 1899. Whittaker, E. T. and Robinson, G. "Central-Difference For- mulae." Ch. 3 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 35 /C1/52, 1967. Central Dilation A DILATION that is not merely a TRANSLATION . Two triangles related by a central dilation are said to be PERSPECTIVE TRIANGLES because the lines joining corresponding vertices CONCUR . See also DILATION ,PERSPECTIVE TRIANGLES ,SPIRAL SIMILARITY ,TRANSLATION References Coxeter, H. S. M. and Greitzer, S. L. "Dilation." §4.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 94 /C1/95, 1967. Central Factorial The central factorials x[k] form an associated SHEFFER SEQUENCE with f(t) /C30et =2 /C28e/C28t =2 /C302 sinh(1 2 t) ;giving the GENERATING FUNCTION X/C12 k /C300x[k] k!tk /C30e2x sinh /C281(t=2) : The first central factorials are x[0] /C301 x[1] /C30x x[2] /C30x2 x[3] /C301 4(4x3 /C28x) /C30/C2814(1 /C282x)x(1 /C272x) x[4] /C30x4 /C28x2 /C30/C28(1 /C28x)x2(1 /C27x) x[5] /C301 16(16x5 /C2840x3 /C279x) /C301 16(1 /C282x)(3 /C282x)x(1 /C272x)(3 /C272x) : See also FACTORIAL ,F ALLING FACTORIAL ,G OULD POLYNOMIAL ,RISING FACTORIAL References Roman, S. The Umbral Calculus. New York: Academic Press, pp. 133 /C1/134, 1984. Central Limit Theorem Letx1;x2;...;xNbe a set of NINDEPENDENT random variates and each xihave an arbitrary probability distribution P(x1;...;xN) with MEAN miand a finite VARIANCE s2 i:Then the normal form variate Xnorm/C13PN i/C301xi/C28PNi/C301miffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN i/C301s2 iq (1) has a limiting cumulative distribution function which approaches a NORMAL (GAUSSIAN ). Under additional conditions on the distribution of the summand, the probability density itself is also G AUS- SIAN (Feller 1971) with MEAN m/C300 and VARIANCE s2/C301:If conversion to normal form is not performed, then the variate X/C131 NXN i/C301xi (2) is NORMALLY DISTRIBUTED with mX/C30mxand sX/C30sx=ffiffiffiffiffi Np :/ Kallenberg (1997) gives a six-line proof of the central limit theorem. An elementary, but slightly morecumbersome proof of the central limit theorem, consider the INVERSE FOURIER TRANSFORM ofPX(f): F/C281[PX(f)]/C13g/C12 /C28/C12e2pifXP(X)dX /C30g/C12 /C28/C12X/C12 n/C300(2pifX)n n!P(X)dX /C30X/C12 n/C300(2pif)n n!g/C12 /C28/C12XnP(X)dX /C30X/C12 n/C300(2pif)n n!/C142Xn/C143: (3) Now write /C142Xn/C143/C30/C142N/C28n(x1/C27x2/C27.../C27xN)n/C143 /C30g/C12 /C28/C12N/C28n(x1/C27... /C27xN)nP(x1)/C1/C1/C1P(xN)dx1/C1/C1/C1dxN; ð4Þ so we have F/C281[PX(f)]/C30X/C12 n/C300(2pif)n n!/C142Xn/C143 /C30X/C12 n/C300(2pif)n n!g/C12 /C28/C12N/C28n(x1/C27.../C27xN)n /C29P(x1)/C1/C1/C1P(xN)dx1/C1/C1/C1dxN /C30g/C12 /C28/C12X/C12 n/C3002pif(x1/C27.../C27xN) N"#n1 n! /C29P(x1)/C1/C1/C1P(xN)dx1/C1/C1/C1dxN /C30g/C12 /C28/C12e2pif(x1/C27.../C27xN)=NP(x1)/C1/C1/C1P(xN)dx1/C1/C1/C1dxN /C30g/C12 /C28/C12e2pifx1=NP(x1)dx1l12ml121 /C29/C1/C1/C1/C29g/C12 /C28/C12e2pifxN=NP(xN)dxNl12ml121 /C30g/C12 /C28/C12e2pifx=NP(x)dxl12ml121 N /C30g/C12 /C28/C121/C272pif N ! x/C271 22pif N !2 x2/C27...2 435P(x)dx8 < :9 = ;N /C30g/C12 /C28/C12P(x)dx/C272pif Ng/C12 /C28/C12xP(x)dx"#/C28(2pf)2 2N2g/C12 /C28/C12x2P(x)dx/C27O(N/C283)/C138N /C301/C272pif N/C142x/C143/C28(2pf)2 2N2/C142x2/C143/C27O(N/C283)"#N /C30exp Nln 1/C272pif N/C142x/C143/C28(2pf)2 2N2/C142x2/C143/C27O(N/C283)"#() (5) Now expand ln(1/C27x)/C30x/C281 2x2/C2713x3/C27...; (6) so F/C281[PX(f)]:exp N2pif N/C142x/C143/C28(2pf)2 2N2/C142x2/C143"( /C271 2(2pif)2 N2/C142x/C1432/C27O(N/C283)l121l127 /C30exp 2 pif/C142x/C143/C28(2pf)2(/C142x2/C143/C28/C142x/C1432) 2N/C27O(N/C282)"# :exp 2 pifmx/C28(2pf)2s2 x 2N"# ; (7) since mx/C13/C142x/C143 (8) s2 x/C13/C142x2/C143/C28/C142x/C1432: (9) Taking the F OURIER TRANSFORM , PX/C13g/C12 /C28/C12e/C282pifxF/C281[PX(f)]df /C30g/C12 /C28/C12e2pif(mz/C28x)/C28(2pf)2s2 z=2Ndf: (10) This is OF THE FORM g/C12 /C28/C12eiaf/C28bf2df; (11) where a/C132p(mx/C28x) and b/C13(2psx)2=2N:But, from Abramowitz and Stegun (1972, p. 302, equation 7.4.6), g/C12 /C28/C12eiaf/C28bf2df/C30e/C28a2=4bffiffiffi p bs : (12) Therefore, PX /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p (2psz)2 2Nvuuuutexp/C28[2p( mx /C28 x)]2 4(2psz)2 2N8 >>>< >>>:9 >>>= >>>; /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pN 4p2 s2 xs exp /C284p2(mx /C28 x)22N 4 /C215 4p2 s2x"# /C30ffiffiffiffiffi Np sxffiffiffiffiffiffi2pp e /C28(mz/C28x)2N =2 s2 z : (13) But mX /C30 mx and mX /C30 mx ; so PX /C301 sXffiffiffiffiffiffi 2pp e/C28(mX/C28x)2 =2 s2 X : (14) The "fuzzy" central limit theorem says that data which are influenced by many small and unrelated random effects are approximately NORMALLY DISTRIB- UTED . See also BERRY- ESSE´ EN THEOREM ,LINDEBERG CON- DITION ,L INDEBERG- FELLER CENTRAL LIMIT THEO- REM,LYAPUNOV CONDITION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Feller, W. "The Fundamental Limit Theorems in Probabil- ity." Bull. Amer. Math. Soc. 51, 800 /C1/832, 1945. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, p. 229, 1968. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971. Kallenberg, O. Foundations of Modern Probability. New York: Springer-Verlag, 1997. Lindeberg, J. W. "Eine neue Herleitung des Exponentialge- setzes in der Wahrscheinlichkeitsrechnung." Math. Z. 15, 211 /C1/225, 1922. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 112 /C1/113, 1992. Trotter, H. F. "An Elementary Proof of the Central Limit Theorem." Arch. Math. 10, 226 /C1/234, 1959. Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483 /C1/494, 1995. Central Moment A MOMENT mnof a probability function P(x) taken about the mean m; mn /C30g(x /C28 m)nP(x) dx: (1) The central moments mn can be expressed as terms of the RAW MOMENTS m?n(i.e., those taken about zero) using the BINOMIAL TRANSFORM mn /C30Xn k /C300n kl11sl11n (/C281)n/C28k m?k m?1n/C28k; (2) with m?0 /C301 (Papoulis 1986, p. 146). The first fewvalues are therefore m1 /C300 (3) m2 /C30/C28m?12/C27 m?2 (4) m3 /C302m ?13/C283m?1 m?2 /C27 m?3 (5) m4 /C30/C283m?14/C276m?12m?2 /C284m ?1 m?3 /C27 m?4 (6) m5 /C304m?15/C2810m?13m?2 /C2710m?12m ?3 /C285 m?1 m ?4 /C27 m ?5 : (6) See also ABSOLUTE MOMENT ,CUMULANT ,KURTOSIS , MOMENT ,PEARSON KURTOSIS ,RAW MOMENT ,SKEW- NESS References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 146, 1984. Kenney, J. F. and Keeping, E. S. "Moments About the Mean." §7.3 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 92 /C1/93, 1962. Central Point A point v is a central point of a graph if the eccentricity of the point equals the GRAPH RADIUS . The set of all central points is called the GRAPH CENTER . See also CENTROID POINT ,G RAPH CENTER ,G RAPH ECCENTRICITY ,GRAPH RADIUS References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 35, 1994. Central Tree CENTERED TREE Central Trinomial Coefficient The nth central trinomial coefficient is defined as the coefficient of xn in the expansion of (1 /C27x /C27x2)n : It is also the number of permutations of n symbols, each /C281, 0, or 1, which sum to 0. For example, there are seven such permutations of three symbols: f/C281; 0; 1g;f/C281; 1; 0g;f0;/C281 ; 1 g;f0 ; 0 ; 0g; and f0; 1;/C281g;f1;/C281 ; 0 g;f1; 0;/C281g: The first few central binomial coefficients are 1, 3, 7, 19, 51, 141, 393, ... (Sloane’s A002426). This sequence cannot be expressed as a fixed number of hypergeometric terms (Petkovsek et al. 1996, p. 160). The GENERATING FUNCTION is given by f(x)/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27x)(1/C283x)p /C301/C27x/C273x2/C277x3/C27...: See also CENTRAL BINOMIAL COEFFICIENT ,TRINOMIAL COEFFICIENT References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Sloane, N. J. A. Sequences A002426/M2673 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Central Value CLASS MARK Centralizer The centralizer of an element z of a GROUP G is the set of elements of G which commute with z, CG(z) /C30fx /C23 G ; xz /C30zx g: Likewise, the centralizer of a SUBGROUP H of a GROUP G is the set of elements of G which commute with every element of H, CG(H) /C30fx /C23 G;/C214h /C23 H ; xh /C30hxg: The centralizer always contains the CENTER of the group and is contained in the corresponding NORMAL- IZER.InanA BELIAN GROUP , the centralizer is the whole group. See also ABELIAN GROUP ,CENTER (GROUP ), GROUP , NORMALIZER ,SUBGROUP Centrally Symmetric Set CENTROSYMMETRIC SET Centric Perspective PERSPECTIVE Centrode C/C13tT/C27kB; where tis the TORSION ,kis the CURVATURE ,Tis the TANGENT VECTOR , and Bis the BINORMAL VECTOR . Centroid (Function) By analogy with the GEOMETRIC CENTROID , the centroid of an arbitrary function f(x) is defined as /C142x/C143/C30g/C12 /C28/C12xf(x)dx g/C12 /C28/C12f(x)dx: References Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 139 /C1/140 and 156, 1999.Centroid (Geometric) The CENTER OF MASS of a 2-D planar LAMINA or a 3-D solid. The mass of a LAMINA with surface density function s(x;y)i s M/C30ggs(x;y)dA; (1) and the coordinates of the centroid (also called the CENTER OF GRAVITY ) are ¯x/C30ggxs(x;y)dA M(2) ¯y/C30ggys(x;y)dA M: (3) The centroid of a lamina is the point on which it would balance when placed on a needle. The centroidof a solid is the point on which the solid would "balance." The centroid of a set of npoint masses m ilocated at positions xiis ¯x/C30Pn i/C301mixiPni/C301mi; (4) which, if all masses are equal, simplifies to ¯x/C30Pni/C301xi n: (5) The centroid of npoint masses also gives the location at which a school should be built in order to minimize the distance travelled by children from ncities, located at the positions of the masses, and with mi equal to the number of students from city i(Stein- haus 1983, pp. 113 /C1/116). The centroid of the vertices of a quadrilateral occursat the point of intersection of the BIMEDIANS (i.e., the lines MABMCDand MADMBCjoining pairs of opposite MIDPOINTS ) (Honsberger 1995, pp. 36 /C1/37). In addi- tion, it is the MIDPOINT of the line MACMBDconnecting the midpoints of the diagonals ACandBD(Honsber- ger 1995, pp. 39 /C1/40). Given an arbitrary HEXAGON , connecting the cen- troids of each consecutive three sides gives the so- called CENTROID HEXAGON , a hexagon with equal and parallel sides (Wells 1991). The centroids of several common laminas along the nonsymmetrical axis are summarized in the following table. Figure / ¯y/ PARABOLIC SEGMENT /2 5h/ SEMICIRCLE /4r 3p/ In 3-D, the mass of a solid with density function r(x; y; z)is M /C30gggr(x; y; z) dV ; (6) and the coordinates of the center of mass are ¯x /C30gggxr(x; y; z) dV M (7) ¯y /C30gggyr(x; y; z) dV M (8) ¯z /C30gggz r(x; y; z) dV M: (9) Figure / ¯z/ CONE /1 4h/ CONICAL FRUSTUM /h(R2 1 /C27 2R1R2 /C27 3R22) 4(R2 1/C27R1R2/C27R22)/ HEMISPHERE /3 8R/ PARABOLOID /2 3h/ PYRAMID /14h/ See also CENTROID HEXAGON ,PAPPUS’S CENTROID THEOREM References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., 1995. Kern, W. F. and Bland, J. R. "Center of Gravity." §39 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 110, 1948. McLean, W. G. and Nelson, E. W. "First Moments and Centroids." Ch. 9 in Schaum’s Outline of Theory andProblems of Engineering Mechanics: Statics and Dy- namics, 4th ed. New York: McGraw-Hill, pp. 134 /C1/162, 1988. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 53 /C1/54, 1991. Centroid (Orthocentric System) The centroid of the four points constituting an ORTHOCENTRIC SYSTEM is the center of the common NINE-POINT CIRCLE (Johnson 1929, p. 249). This fact automatically guarantees that the centroid of the INCENTER and EXCENTERS of a TRIANGLE is located at the CIRCUMCENTER . References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Centroid (Triangle) The CENTROID (CENTER OF MASS ) of the VERTICES of a TRIANGLE is the point G(sometimes also denoted M) which is also the intersection of the TRIANGLE’S three MEDIANS (Johnson 1929, p. 249; Wells 1991, p. 150). The point is therefore sometimes called the median point. The centroid is always in the interior of the TRIANGLE . It has TRILINEAR COORDINATES 1 a:1 b:1 c; (1) or cscA: csc B: csc C; (2) and homogeneous BARYCENTRIC COORDINATES (1; 1; 1):/ If the sides of a TRIANGLE DA1A2A3are divided by points P1 ; P2 ; and P3 so that A2P1 P1A3/C30A3P2 P2A1/C30A1P3 P3A2/C30p q ; (3) then the centroid of the TRIANGLE DP1P2P3 is M, the centroid of the original triangle DA1A2A3(Johnson 1929, p. 250). One BROCARD LINE, MEDIAN , and SYMMEDIAN (out of the three of each) are CONCURRENT , with AV; CK, and BG meeting at a point, where V is the first BROCARD POINT and K is the SYMMEDIAN POINT . Similarly, AV?; BG, and CK, where V? is the second BROCARD POINT , meet at a point which is the ISOGONAL CONJUGATE of the first (Johnson 1929, pp. 268 /C1/269). Pick an interior point X. The TRIANGLES BXC , CXA , and AXB have equal areas IFF X corresponds to the centroid. The centroid is located one third of the way from each VERTEX to the MIDPOINT of the opposite side. Each median divides the triangle into two equal areas; all the medians together divide it into six equal parts, and the lines from the MEDIAN POINT to the VERTICES divide the whole into three equivalent TRIANGLES . In general, for any line in the plane of a TRIANGLE ABC , d /C301 3(dA /C27dB /C27dC) ; (4) where d, dA ; dB ; and dCare the distances from the centroid and VERTICES to the line. A TRIANGLE will balance at the centroid, and along any line passing through the centroid. The TRILINEAR POLAR of the centroid is called the LEMOINE AXIS. The PERPENDICULARS from the centroid are proportionalto s /C281 i; a1p2 /C30a2p2 /C30a3p3 /C302 3 D; (5) where D is the AREA of the TRIANGLE . Let P be an arbitrary point, the VERTICES be A1 ; A2 ; and A3 ; and the centroid G. Then PA12/C27PA22/C27PA32 /C30GA12/C27GA22/C27GA32/C273PG2 : (6) If O is the CIRCUMCENTER of the triangle’s centroid, then OG2 /C30R2 /C2819(a2 /C27b2 /C27c2) : (7) The centroid lies on the EULER LINE and NAGEL LINE. The centroid of the PERIMETER of a TRIANGLE is the triangle’s SPIEKER CENTER (Johnson 1929, p. 249). The SYMMEDIAN POINT of a triangle is the centroid of its PEDAL TRIANGLE (Honsberger 1995, pp. 72 /C1/74). Given a triangle DABC ; construct circles through each pair of vertices which also pass through the CENTROID G. The TRIANGLE DA?B?C ? determined by the center of these circles then satisfies a number of interesting properties. The first is that the CIRCUM- CIRCLE Oand CENTROID GofDABC are, respectively, the CENTROID G?and SYMMEDIAN POINT K?of the triangle DA?B?C?(Honsberger 1995, p. 77). In addi- tion, the MEDIANS ofDABC andDA?B?Cintersect in the midpoints of the sides of DABC :/ See also CIRCUMCENTER ,E ULER LINE,E XMEDIAN POINT ,INCENTER ,NAGEL LINE,ORTHOCENTER References Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 622, 1970. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 7, 1967. Dixon, R. Mathographics. New York: Dover, pp. 55 /C1/57, 1991. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 72 /C1/74 and 77, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 173 /C1/176 and 249, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Kimberling, C. "Centroid." http://cedar.evansville.edu/~ck6/ tcenters/class/centroid.html. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 62 /C1/63, 1893. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 150, 1991. Centroid Hexagon The hexagon obtained from an arbitrary HEXAGON by connecting the centroids of each consecutive three sides. This hexagon has equal and parallel sides (Wells 1991). References Cadwell, J. H. Topics in Recreational Mathematics. Cam- bridge, England: Cambridge University Press, 1966. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 53 /C1/54, 1991. Centroid Point A point in a WEIGHTED TREE that has minimum weight for the tree. The set of all centroid points is called a TREE CENTROID (Harary 1994, p. 36). The largest possible values for a centroid point (i.e., the maximum minimum weight) for a tree on n /C302, 3, ... nodes are 1, 1, 2, 2, 3, 3, .... See also TREE CENTROID ,W EIGHTED TREE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Centroidal Line The three planes determined by the edges of a TRIHEDRON and the internal bisectors of the respec-tively opposite faces are coaxal, and the common line of these planes is called the centroidal line. See also TRIHEDRON References Altshiller-Court, N. "Centroidal Lines." §2.5 in Modern Pure Solid Geometry. New York: Chelsea, pp. 40 /C1/41, 1979. Centrosymmetric Matrix A SQUARE MATRIX is called centrosymmetric if it is symmetric with respect to the center (Muir 1960, p. 19). See also BISYMMETRIC MATRIX ,SYMMETRIC MATRIX References Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Centrosymmetric Set A CONVEX SET K is centro-symmetric, sometimes also called centrally symmetric, if it has a center p that bisects every CHORD of K through p. References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 7, 1991. Certificate of Compositeness COMPOSITENESS CERTIFICATE Certificate of Primality PRIMALITY CERTIFICATE Cesa`ro Equation An INTRINSIC EQUATION which expresses a curve in terms of its ARC LENGTH s and RADIUS OF CURVATURE R(or equivalently, the CURVATURE k):/ See also ARC LENGTH ,INTRINSIC EQUATION ,NATURAL EQUATION ,RADIUS OF CURVATURE ,W HEWELL EQUA- TION References Yates, R. C. "Intrinsic Equations." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 123 /C1/126, 1952. Cesa`ro Fractal A FRACTAL also known as the TORN SQUARE FRACTAL . The base curves and motifs for the two fractals illustrated above are shown below. See also FRACTAL ,KOCH SNOWFLAKE References Cesa`ro, E. "Remarques sur la courbe de von Koch." Atti della R. Accad. della Scienze fisiche e matem. Napoli 12, No. 15, 1905. Reprinted as §228 in Opere scelte, a cura dell’Unione matematica italiana e col contributo del Consiglio nazio- nale delle ricerche, Vol. 2: Geometria, analisi, fisica matematica. Rome: Edizioni Cremonese, pp. 464 /C1/479, 1964. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, p. 43, 1991. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 79, 1989. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Cesa`ro Mean FEJES TO´ TH’S INTEGRAL Cesa`ro’s Theorem The three points determined on three coplanar edges of a TETRAHEDRON by the external bisecting planes of the opposite DIHEDRAL ANGLES are COLLINEAR . Furthermore, this line belongs to the plane deter- mined by the three points in which the remaining three (concurrent) edges of the TETRAHEDRON are met by the internal bisecting planes of the respectively opposite DIHEDRAL ANGLE . References Altshiller-Court, N. "Gergonne’s Theorem." §235 in Modern Pure Solid Geometry. New York: Chelsea, p. 71, 1979.Ceva’s Theorem Given a TRIANGLE with VERTICES A, B, and C and points along the sides D, E, and F,aNECESSARY and SUFFICIENT condition for the CEVIANS AD, BE, and CF to be CONCURRENT (intersect in a single point) is that BD /C215 CE /C215 AF /C30DC /C215 EA /C215 FB: (1) This theorem was first published by Giovanni Cevian 1678. Let P /C30[V1 ; ...; Vn] be an arbitrary n-gon, C a given point, and k a POSITIVE INTEGER such that 1 5k 5 n=2 : For i /C301, ..., n, let Wi be the intersection of the lines CVi and Vi /C28kVi/C27k ; then Yn i /C301Vi /C28kWi WiVi/C27k"# /C301: (2) Here, AB ½½CD and AB CD"# (3) is the RATIO of the lengths [A, B] and [C, D] with a plus or minus sign depending on whether these segments have the same or opposite directions (Gru¨nbaum and Shepard 1995). Another form of the theorem is that three CONCUR- RENT lines from the VERTICES of a TRIANGLE divide the opposite sides in such fashion that the product ofthree nonadjacent segments equals the product of theother three (Johnson 1929, p. 147). See also H OEHN’S THEOREM ,MENELAUS’ THEOREM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 122, 1987. Coxeter, H. S. M. and Greitzer, S. L. "Ceva’s Theorem." §1.2 inGeometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 4 /C1/5, 1967. Durell, C. V. A Course of Plane Geometry for Advanced Students, Part I. London: Macmillan, p. 54, 1909. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 40 /C1/41, 1928. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 81, 1930. Gru¨nbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254/C1/268, 1995. Honsberger, R. "Ceva’s Theorem." §12.1 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 136 /C1/138, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 145 /C1/151, 1929. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., p. xx, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 28 /C1/29, 1991. Cevian A line segment which joins a VERTEX of a TRIANGLE with a point on the opposite side (or its extension). In the above figure, s /C30b sin a? sin(g /C27 a?) : The condition for Cevians from the three sides of a TRIANGLE to CONCUR is known as CEVA’S THEOREM . If AD, BE, and CF are cevians of a TRIANGLE DABC through an arbitrary point P inside DABC ; then the ratios AP PD ;BP PE ;CP PF into which P divides the Cevians have a sum ]6 and a product ]8 (Ramler 1958; Honsberger 1995, pp. 138 /C1/141). See also ANGLE BISECTOR ,CEVA’S THEOREM ,CEVIAN CIRCLE ,CEVIAN TRIANGLE ,MEDIAN (TRIANGLE ), PED- AL-CEVIAN POINT ,ROUTH’S THEOREM ,SPLITTER References Honsberger, R. "On Cevians." Ch. 12 in Episodes in Nine- teenth and Twentieth Century Euclidean Geometry. Wa- shington, DC: Math. Assoc. Amer., pp. 13 and 137 /C1/146, 1995. Ramler, O. J. Solved by C. W. Trigg. "Problem E1043." Amer. Math. Monthly 65, 421, 1958. The´bault, V. "On the Cevians of a Triangle." Amer. Math. Monthly 60, 167 /C1/173, 1953.Cevian Circle The CIRCUMCIRCLE of the CEVIAN TRIANGLE DA?B ?C ? of a given TRIANGLE DABC with respect to a point P. See also CEVIAN TRIANGLE ,CIRCUMCIRCLE Cevian Conjugate Point ISOTOMIC CONJUGATE POINT Cevian Transform Vandeghen’s (1965) name for the transformation taking points to their ISOTOMIC CONJUGATE POINTS . See also ISOTOMIC CONJUGATE POINT References Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091 /C1/1094, 1965. Cevian Triangle Given a point Pand a TRIANGLE DABC ;the Cevian triangle DA?B?C?is defined as the triangle composed of the endpoints of the CEVIANS though P. If the point Phas TRILINEAR COORDINATES a:b:g, then the Cevian triangle has VERTICES 0:b:g,a:0:g, and a:b:0. If A?B?C?is the C EVIAN TRIANGLE ofXand AƒBƒCƒis the ANTICEVIAN TRIANGLE , then XandAƒare HARMO- NIC CONJUGATE POINTS with respect to A and A?:/ If DA?B ?C ? is the Cevian triangle of DABC ; then the triangle DAƒB ƒCƒ obtained by reflecting A?; B?; and C? across the midpoints of their sides is also a Cevian triangle of DABC (Honsberger 1995, p. 141; left figure). Furthermore, if the CEVIAN CIRCLE crosses the sides of DABC in three points Aƒ; B ƒ; and Cƒ; then DAƒB ƒC ƒ is also a Cevian triangle of DABC (Honsber- ger 1995, pp. 141 /C1/142; right figure). See also ANTICEVIAN TRIANGLE ,C EVIAN ,C EVIAN CIRCLE References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 141 /C1/143, 1995. CG Given a GROUP G, the algebra CG is a VECTOR SPACE CG /C30X aigi ½ai /C23C ; gi /C23 Gno of finite sums of elements of G, with multiplication defined by g /C215 h /C30gh ; the group operation. It is an example of a GROUP RING . For example, when the group is the SYMMETRIC GROUP on three letters, S3 ; the GROUP RING CS3is a six-dimensional algebra. An example of the product of elements is (3f1; 3; 2g/C27if1 ; 2 ; 3g)(/C282f2 ; 1; 3g/C27f3; 2; 1g) /C30/C286f2; 3; 1g/C282i f2; 1; 3 g/C27i f3; 2; 1g/C273 f3; 1; 2 g: MODULES over CG correspond to complex REPRESEN- TATIONS of G. When G is a FINITE GROUP then CG is a finite-dimensional algebra. See also ALGEBRA ,GROUP ,GROUP RING,PERMUTA- TION ,REPRESENTATION ,RING Ch HYPERBOLIC COSINE Chain Let P be a finite PARTIALLY ORDERED SET. A chain in P is a set of pairwise comparable elements (i.e., a TOTALLY ORDERED subset). The LENGTH of P is the maximum CARDINALITY of a chain in P. For a PARTIALORDER , the size of the longest chain is called the LENGTH . See also ADDITION CHAIN ,ANTICHAIN ,BRAUER CHAIN , CHAIN (GRAPH ), CHAIN OF CIRCLES ,D ILWORTH’S LEMMA ,H ANSEN CHAIN ,LENGTH (PARTIAL ORDER ), PAPPUS CHAIN ,PARTIAL ORDER References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 272, 1974. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 241, 1990. Chain (Graph) A chain of a GRAPH is a SEQUENCE fx1 ; x2 ; ...; xn g such that (x1 ; x2) ; (x2 ; x3) ; ..., (xn/C281 ; xn) are EDGES of the GRAPH . See also GRAPH Chain Complex A chain complex is a sequence of maps /C1/C1/C10@i/C271Ci0@iCi/C2810@i/C281/C1/C1/C1; (1) where the spaces Cimay be GROUPS orMODULES . The maps must satisfy @i/C281(@i/C300:Making the domain implicitly understood, the maps are denoted by @; called the BOUNDARY OPERATOR or the differential. Chain complexes are an algebraic tool for computing or defining HOMOLOGY and have a variety of applica- tions. A COCHAIN COMPLEX is used in the case of COHOMOLOGY . Elements of Cpare called CHAINS . For each p, the kernel of @p:Cp0Cp/C281is called the group of cycles, Zp/C30fc/C23Cp:@(c)/C300g: (2) The letter Zis short for the German word for cycle, "Zyklus." The image @(Cp/C271) is contained in the group of cycles because @(@/C300:It is called the group of boundaries. Bp/C30fc/C23Cp: there exists b/C23Cp/C271such that @(b)/C30cg:(3) The quotients Hp/C30Zp=Bpare the HOMOLOGY GROUPS of the chain. For example, the sequence /C1/C1/C10/C294Z=8Z0/C294Z=8Z0/C294/C1/C1/C1; (4) where every space is Z=8Zand each map is given by multiplication by 4 is a chain complex. The cycles at each stage are Zp/C30f0;2;4;6gand the boundaries areBp/C30f0;4g:So the homology at each stage is the group of two elements Z=2Z:A simpler example is given by a LINEAR TRANSFORMATION a:V0W;which can be extended to a chain complex by the zero vector space and the ZERO MAP. Then the nontrivial homol- ogy groups are ker a and W =im(a) :/ The terminology of chain complexes comes from the calculation for HOMOLOGY of geometric objects in a TOPOLOGICAL SPACE , like a MANIFOLD . For example, the figure above is the circle as a SIMPLICIAL COM- PLEX . Let A and B denote the points, and C and D denote the oriented segments, which are the chains. The boundary of C is B /C28A; and the boundary of D is A /C28B :/ The group C1 is the FREE ABELIAN GROUP C; D hi and the group C0 is the FREE ABELIAN GROUP A; B hi : The BOUNDARY OPERATOR is @(nC /C27mD) /C30n(B /C28A) /C27m(A /C28B) /C30(m /C28n)A /C27(n /C28m)B : (5) The other groups Cpare the TRIVIAL GROUP , and the other maps are the ZERO MAP. Then Z1 is generated by C /C27D and B1 is the trivial subgroup. So H1 is the rank one FREE ABELIAN GROUP isomorphic to Z: The zero- dimensional case is slightly more interesting. Every element of C0has no boundary and so is in Z0while the boundaries B0are generated by A /C28B: Hence, H0 /C30Z0 =B0is also isomorphic to Z: Note that the result is not affected by how the circle is cut into pieces, or by how many cuts are used. See also CHAIN EQUIVALENCE ,CHAIN HOMOMORPH- ISM,CHAIN HOMOTOPY ,COCHAIN COMPLEX ,COHO- MOLOGY ,F REE ABELIAN GROUP ,H OMOLOGY , HOMOLOGY (CHAIN ), SIMPLICIAL HOMOLOGY References Hilton, P. and Stammbach, U. A Course in Homological Algebra. New York: Springer-Verlag, pp. 117 /C1/118, 1997. Munkres, J. Elements of Algebraic Topology. Reading, MA: Addison-Wesley, pp. 58 and 71 /C1/76, 1984. Chain Equivalence Chain equivalences give an EQUIVALENCE RELATION on the space of CHAIN HOMOMORPHISMS . Two CHAIN COMPLEXES are chain equivalent if there are chain maps f : C/C310 D/C31 and g : D/C310 C /C31 such that f( g is CHAIN HOMOTOPIC to the identity on D/C31 and g( f is CHAIN HOMOTOPIC to the identity on C/C31:/ See also CHAIN COMPLEX .C HAIN HOMOMORPHISM , CHAIN HOMOTOPY ,HOMOTOPY EQUIVALENCE ,SNAKE LEMMAReferences Hilton, P. and Stammbach, U. A Course in Homological Algebra. New York: Springer-Verlag, pp. 117 /C1/118, 1997. Munkres, J. Elements of Algebraic Topology. Reading, MA: Addison-Wesley, pp. 58 and 71 /C1/76, 1984. Chain Fraction CONTINUED FRACTION Chain Homomorphism Also called a chain map. Given two CHAIN COMPLEXES C/C31and D/C31; a chain homomorphism is given by homomorphisms ai : Ci 0 Di such that a( @C /C30@D(a; where @C and @D are the BOUNDARY OPERATORS . See also CHAIN COMPLEX ,C HAIN EQUIVALENCE , CHAIN HOMOTOPY ,HOMOMORPHISM (MODULE ) References Hilton, P. and Stammbach, U. A Course in Homological Algebra. New York: Springer-Verlag, pp. 117 /C1/118, 1997. Munkres, J. Elements of Algebraic Topology. Addison- Wesley, pp. 58 and 71 /C1/76, 1984. Chain Homotopy Suppose a : C/C310 D /C31 and b : C/C310 D /C31 are two CHAIN HOMOMORPHISMS . Then a chain homotopy is given by a sequence of maps dp : Cp 0 Dp /C271 such that @D(d /C27 d(@C /C30 a /C28 b; where @ denotes the BOUNDARY OPERATOR . See also CHAIN COMPLEX ,C HAIN EQUIVALENCE , CHAIN HOMOMORPHISM ,HOMOTOPY ,SNAKE LEMMA References Hilton, P. and Stammbach, U. A Course in Homological Algebra. New York: Springer-Verlag, p. 124, 1997. Munkres, J. Elements of Algebraic Topology. Reading, MA: Addison-Wesley, pp. 58 and 71 /C1/76, 1984. Chain Map CHAIN HOMOMORPHISM Chain of Circles A sequence of circles which closes (such as a STEINER CHAIN or the circles inscribed in the ARBELOS )is called a chain. See also ARBELOS ,COXETER’S LOXODROMIC SEQUENCE OF TANGENT CIRCLES ,N INE CIRCLES THEOREM , PAPPUS CHAIN ,SEVEN CIRCLES THEOREM ,SIX CIR- CLES THEOREM ,STEINER CHAIN ,STEINER’S PORISM References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "Chains of Circles." Ch. 3 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 31 /C1/68, 1974. Chain Rule If g(x)is DIFFERENTIABLE at the point x and f(x)is DIFFERENTIABLE at the point g(x) ; then f(g is DIFFER- ENTIABLE at x. Furthermore, let y /C30f(g(x)) and u /C30 g(x) ; then dy dx /C30dy du/C215du dx: (1) There are a number of related results which also go under the name of "chain rules." For example, if z /C30 f(x; y) ; x /C30g(t); and y /C30h(t) ; then dz dt /C30@z @xdx dt /C27@z @ydy dt: (2) The "general" chain rule applies to two sets of functions y1 /C30f1(u1 ; ... ; up) n (3) ym /C30fm(u1 ; ...; up) and u1 /C30g1(x1 ; ...; xn) n (4) up /C30gp(x1 ; ... ; xn): Defining the m /C29n JACOBI MATRIX by @yi @xj ! /C30@y1 @x1@y1 @x2/C1/C1/C1@y1 @xn nn::: n @ym @x1@ym @x2/C1/C1/C1@ym @xn2 6666643 777775; (5) and similarly for (@y i =@uj) and ( @ui =@xj) then gives @yi @xj ! /C30@yi @uj ! @ui @xj ! : (6) In differential form, this becomes dy1 /C30@y1 @u1@u1 @x1/C27.../C27@y1 @up@up @x1 ! dx1 /C27@y1 @u1@u1 @x2/C27.../C27@y1 @up@up @x2 ! dx2 /C27... (7) (Kaplan 1984).See also DERIVATIVE ,JACOBIAN ,POWER RULE,PRO- DUCT RULE References Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, p. 165, 1999. Kaplan, W. "Derivatives and Differentials of Composite Functions" and "The General Chain Rule." §2.8 and 2.9 in Advanced Calculus, 3rd ed. Reading, MA: Addison- Wesley, pp. 101 /C1/105 and 106 /C1/110, 1984. Chained Arrow Notation A NOTATION which generalizes ARROW NOTATION and is defined as a /C160/C1/C1/C1/C160b|fflfflfflfflffl{zfflfflfflfflffl} c/C13a 0 b 0 c: See also ARROW NOTATION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 61, 1996. Chainette CATENARY Chair A SURFACE with tetrahedral symmetry which, accord- ing to Nordstrand, looks like an inflatable chair from the 1970s. It is given by the implicit equation (x2 /C27y2 /C27z2 /C28ak2)2 /C28b[(z /C28k)2 /C282x2][(z /C27k)2 /C282y2] /C300: The surface illustrated above has k /C305, a /C300:95 ; and b/C300:8:/ See also BRIDE’S CHAIR References Nordstrand, T. "Chair." http://www.uib.no/people/nfytn/ chairtxt.htm. Chaitin’s Constant An IRRATIONAL NUMBER V which gives the probability that for any set of instructions, a UNIVERSAL TURING MACHINE will halt. The digits in V are random and cannot be computed ahead of time. See also HALTING PROBLEM ,TURING MACHINE ,UNI- VERSAL TURING MACHINE References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/chaitin/chaitin.html. Gardner, M. "The Random Number V Bids Fair to Hold the Mysteries of the Universe." Sci. Amer. 241,20/C1/34, Nov. 1979. Gardner, M. "Chaitin’s Omega." Ch. 21 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 307 /C1/319, 1992. Kobayashi, K. "Sigma(N)O-Complete Properties of Programs and Lartin-Lof Randomness." Information Proc. Let. 46, 37 /C1/42, 1993. Chaitin’s Number CHAITIN’S CONSTANT Chaitin’s Omega CHAITIN’S CONSTANT Champernowne Constant Champernowne’s constant 0.1234567891011... (Sloa- ne’s A033307) is the number obtained by concatenat- ing the POSITIVE INTEGERS and interpreting them as decimal digits to the right of a decimal point. It is NORMAL in base 10. In 1961, Mahler showed it to also be TRANSCENDENTAL . The first few terms in the CONTINUED FRACTION of the Champernowne constant are 0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 457540111391031076483646628242956118599603939... 710457555000662004393090262659256314937953207...747128656313864120937550355209460718308998457... 5801469863148833592141783010987 ; 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ... (Sloane’s A030167). The next term of the CONTINUED FRACTION is huge, having 2504 digits. In fact, the coefficients eventually become un- bounded, making the continued fraction difficult to calculate for too many more terms. Large terms greater than 105 occur at positions 5, 19, 41, 102, 163, 247, 358, 460, ... and have 6, 166, 2504, 140, 33102, 109, 2468, 136, ... digits, respectively (Plouffe). The 527th partial quotient of the continued fraction expansion has 411,100 decimal digits and the 1709th partial quotient has 4,911,098 decimal digits, as computed using Mathematica 4.0. This result was obtained by Mark Sofroniou and Giulia Spaletta andpresented at the conference on Foundations of Com- putational Mathematics in Oxford, UK, July 1999. Interestingly, the C OPELAND- ERDOS CONSTANT , which is the decimal number obtained by concatenating the PRIMES (instead of all the positive integers), has a well-behaved CONTINUED FRACTION that does not show the "large term" phenomenon. See also COPELAND- ERDOS CONSTANT ,SMARANDACHE SEQUENCES References Champernowne, D. G. "The Construction of Decimals Nor- mal in the Scale of Ten." J. London Math. Soc. 8, 1933. Copeland, A. H. and Erdos, P. "Note on Normal Numbers." Bull. Amer. Math. Soc. 52, 857/C1/860, 1946. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html. Sloane, N. J. A. Sequences A030167 and A033307 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 26, 1986. Change of Variables Theorem A theorem which effectively describes how lengths, areas, volumes, and generalized n-dimensional vo- lumes ( CONTENTS ) are distorted by DIFFERENTIABLE FUNCTIONS . In particular, the change of variables theorem reduces the whole problem of figuring outthe distortion of the content to understanding theinfinitesimal distortion, i.e., the distortion of the DERIVATIVE (a linear MAP), which is given by the linear MAP’sDETERMINANT .S of:Rn0Rnis an AREA- PRESERVING linear MAP IFF det(f) jj /C301;and in more generality, if Sis any subset of Rn;the CONTENT of its image is given by det( f) jj times the CONTENT of the original. The change of variables theorem takes thisinfinitesimal knowledge, and applies CALCULUS by breaking up the DOMAIN into small pieces and adds up the change in AREA , bit by bit. The change of variable formula persists to thegenerality of DIFFERENTIAL FORMS on MANIFOLDS , giving the formula gM(f/C31v)/C30gW(v) (1) under the conditions that Mand Ware compact connected oriented MANIFOLDS with nonempty bound- aries, f:M0Wis a smooth map which is an orientation-preserving DIFFEOMORPHISM of the boundaries. In 1-D, the explicit statement of the theorem for fa continuous function of yis gsf(f(x))df dxdx/C30gTf(y)dy; (2) where y /C30 f(x) is a differential mapping on the interval [c, d] and T is the interval [a, b] with f(c) /C30 a and f(d) /C30b (Lax 1999). In 2-D, the explicit statement of the theorem is gRf(x ; y) dx dy /C30gR/C31f[x(u; v) ; y(u ; v)]@(x; y) @(u ; v)l112l112l112l112l112l112l112l112l112l112 du dv and in 3-D, it is gRf(x; y; z) dx dy dz /C30gR/C31f[x(u; v; w) ; y(u; v; w);z(u; v ; w)] /C2@(x; y; z) @(u; v ; w)l112l112l112l112l112l112l112l112l112l112 du dv dw ; (3) where R /C30f(R/C31) is the image of the original region R/C31; @(x; y; z) @(u; v; w)l112l112l112l112l112l112l112l112l112l112 (4) is the J ACOBIAN , and f is a global orientation-preser- ving DIFFEOMORPHISM of R and R/C31 (which are open subsets of Rn):/ The change of variables theorem is a simple conse- quence of the CURL THEOREM and a little DE RHAM COHOMOLOGY . The generalization to n-D requires no additional assumptions other than the regularity conditions on the boundary. See also IMPLICIT FUNCTION THEOREM ,JACOBIAN References Jeffreys, H. and Jeffreys, B. S. "Change of Variable in an Integral." §1.1032 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 32 /C1/33, 1988. Kaplan, W. "Change of Variables in Integrals." §4.6 in Advanced Calculus, 3rd ed. Reading, MA: Addison-Wes- ley, pp. 238 /C1/245, 1984. Lax, P. D. "Change of Variables in Multiple Integrals." Amer. Math. Monthly 106, 497 /C1/501, 1999. Chaos A DYNAMICAL SYSTEM is chaotic if it1. Has a DENSE collection of points with periodic orbits, 2. Is sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), and 3. Is TOPOLOGICALLY TRANSITIVE . Chaotic systems exhibit irregular, unpredictable be- havior (the BUTTERFLY EFFECT ). The boundary be- tween linear and chaotic behavior is often characterized by PERIOD DOUBLING , followed by quad- rupling, etc., although other routes to chaos are also possible (Abarbanel et al. 1993; Hilborn 1994; Stro- gatz 1994, pp. 363 /C1/365). An example of a simple physical system which dis- plays chaotic behavior is the motion of a magnetic pendulum over a plane containing two or more attractive magnets. The magnet over which the pendulum ultimately comes to rest (due to frictional damping) is highly dependent on the starting position and velocity of the pendulum (Dickau). Another such system is a double pendulum (a pendulum with another pendulum attached to its end). See also ACCUMULATION POINT ,ATTRACTOR ,BASIN OF ATTRACTION ,BUTTERFLY EFFECT ,CHAOS GAME,DY- NAMICAL SYSTEM ,FEIGENBAUM CONSTANT ,FRACTAL DIMENSION ,GINGERBREADMAN MAP,H E´ NON- HEILES EQUATION ,H E´ NON MAP,L IMIT CYCLE ,L OGISTIC EQUATION ,L YAPUNOV CHARACTERISTIC EXPONENT , PERIOD THREE THEOREM ,PHASE SPACE ,Q UANTUM CHAOS ,RESONANCE OVERLAP METHOD ,SARKOVSKII’S THEOREM ,S HADOWING THEOREM ,S INK (MAP), STRANGE ATTRACTOR References Abarbanel, H. D. I.; Rabinovich, M. I.; and Sushchik, M. M. Introduction to Nonlinear Dynamics for Physicists. Singa- pore: World Scientific, 1993. Bai-Lin, H. Chaos. Singapore: World Scientific, 1984. Baker, G. L. and Gollub, J. B. Chaotic Dynamics: An Introduction, 2nd ed. Cambridge, England: Cambridge University Press, 1996. Smith, P. Explaining Chaos. Cambridge, England: Cam- bridge University Press, 1998. Cvitanovic, P. Universality in Chaos: A Reprint Selection, 2nd ed. Bristol: Adam Hilger, 1989. Devaney, R. L. An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, 1987. Dickau, R. M. "Magnetic Pendulum." http://forum.swarth- more.edu/advanced/robertd/magneticpendulum.html. Drazin, P. G. Nonlinear Systems. Cambridge, England: Cambridge University Press, 1992. Field, M. and Golubitsky, M. Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature. Oxford, England: Oxford University Press, 1992. Gleick, J. Chaos: Making a New Science. New York: Penguin, 1988. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed.New York: Springer-Verlag, 1997. Hall, N. (Ed.). Exploring Chaos: A Guide to the New Science of Disorder. New York: W. W. Norton, 1994. Hilborn, R. C. Chaos and Nonlinear Dynamics. New York: Oxford University Press, 1994. Kapitaniak, T. and Bishop, S. R. The Illustrated Dictionary of Nonlinear Dynamics and Chaos. New York: Wiley, 1998. Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion, 2nd ed. New York: Springer-Verlag, 1994. Lorenz, E. N. The Essence of Chaos. Seattle, WA: University of Washington Press, 1996. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993. Ott, E.; Sauer, T.; and Yorke, J. A. Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems. New York: Wiley, 1994. Peitgen, H.-O.; Ju¨rgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer- Verlag, 1992. Poon, L. "Chaos at Maryland." http://www-chaos.umd.edu. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, 1990. Strogatz, S. H. Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineer- ing. Reading, MA: Addison-Wesley, 1994. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Tufillaro, N.; Abbott, T. R.; and Reilly, J. An Experimental Approach to Nonlinear Dynamics and Chaos. Redwood City, CA: Addison-Wesley, 1992. Wiggins, S. Global Bifurcations and Chaos: Analytical Methods. New York: Springer-Verlag, 1988. Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, 1990. Chaos Game Pick a point at random inside a regular n-gon. Then draw the next point a fraction r of the distance between it and a VERTEX picked at random. Continue the process (after throwing out the first few points). The result of this "chaos game" is sometimes, but not always, a FRACTAL . The case (n; r) /C30(4; 1=2) gives the interior of a SQUARE with all points visited with equal probability. The above plots show the chaos game for 10,000 points in the regular 3-, 4-, 5-, and 6-gons with r/C301=2:/ The above plots show the chaos game for 10,000 points in the square with r/C300:25;0.4, 0.5, 0.6, 0.75, and 0.9. See also BARNSLEY’S FERN References Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993. Dickau, R. M. "The Chaos Game." http://forum.swarthmor- e.edu/advanced/robertd/chaos_game.html. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 149 /C1/163, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Chaplygin’s Equation The PARTIAL DIFFERENTIAL EQUATION uxx /C27y2 1 /C28y2 c2uyy /C27yuy /C300: References Landau, L. D. and Lifschitz, E. M. Fluid Mechanics, 2nd ed. Oxford, England: Pergamon Press, p. 432, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997. Chapman-Kolmogorov Equation The equation f(xn ½xs) /C30g/C12 /C28/C12f(xn ½xr)f(xr ½xs) dxr which gives the transitional densities of a MARKOV SEQUENCE . Here, n > r > s are any integers (Papoulis 1984, p. 531). See also MARKOV PROCESS References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Character (Group) The GROUP THEORETICAL term for what is known to physicists, by way of its connection with matrix TRACES , as the trace. The powerful GROUP ORTHOGON- ALITY THEOREM gives a number of important proper- ties about the structures of GROUPS , many of which are most easily expressed in terms of characters. In essence, group characters can be thought of as the TRACES of a special set of matrices (a so-called IRREDUCIBLE REPRESENTATION ) used to represent group elements and whose multiplication corresponds to the multiplication table of the group. The explicit construction of a set of characters (CHARACTER TABLE ) is illustrated for the FINITE GROUP D3. All members of the same CONJUGACY CLASS in the same representation have the same character. Mem- bers of other CONJUGACY CLASSES may also have the same character, however. An (abstract) GROUP can be uniquely identified by a listing of the characters of its various representations, known as a CHARACTER TABLE . Some of the SCHO¨ NFLIES SYMBOLS denote different sets of symmetry operations but correspondto the same abstract GROUP and so have the same CHARACTER TABLES . See also CHARACTER TABLE ,C ONJUGACY CLASS , GROUP ORTHOGONALITY THEOREM ,TRACE (MATRIX ) Character (Number Theory) A number theoretic function xk(n) for POSITIVE inte- gral n is a character modulo k if xk(1) /C301 xk(n) /C30 xk(n /C27k) xk(m)xk(n) /C30 xk(mn) for all m, n, and xk(n) /C300 if (k ;n) "1 : xk can only assume values which are f(k) ROOTS OF UNITY , where f is the TOTIENT FUNCTION . See also DIRICHLET L-SERIES ,MULTIPLICATIVE CHAR- ACTER ,PRIMITIVE CHARACTER Character Table AFINITE GROUP Ghas a finite number of CONJUGACY CLASSES and a finite number of distinct IRREDUCIBLE REPRESENTATIONS . The CHARACTER of a REPRESENTA- TION is constant on a CONJUGACY CLASS . Hence, the values of the characters can be written as an array, known as a character table. Typically, the rows are given by the IRREDUCIBLE REPRESENTATIONS and the columns are given the CONJUGACY CLASSES . A char- acter table contains enough information to uniquelyidentify a given abstract group and distinguish it from others. For example, the SYMMETRIC GROUP on three letters S3has three CONJUGACY CLASSES , represented by the PERMUTATIONS f1;2;3g;f2;1;3g;and f2;3;1g:It also has three IRREDUCIBLE REPRESENTATIONS ; two are one-dimensional and the third is two-dimen- sional: 1. The TRIVIAL REPRESENTATION f1(g)(a)/C30a:/ 2. The ALTERNATING REPRESENTATION , given by the signature of the PERMUTATION , f2(g)(a)/C30sgn(g)a:/ 3. The STANDARD REPRESENTATION on V/C30 z1;z2;z3 ðÞ :azi/C300 fg with f3({a, b, c })(z1,z2,z3) /C30(za,zb,zc). The STANDARD REPRESENTATION can be described on C2via the matrices ˜f3(f2;1;3g)/C3001 10l12ml121 ˜f3(f2;3;1g)/C300/C281 1/C281l12ml121 ; and hence the CHARACTER of the first matrix is 0 and that of the second is /C281. The CHARACTER of the identity is always the dimension of the VECTOR SPACE . The trace of the alternating representation is just the SIGNATURE of the PERMUTATION . Consequently, the character table for S3is shown below. 12 3 /S3/ e(12) (123) trivial 1 1 1 alternating 1 /C2811 standard 2 0 /C281 Chemists and physicists use a special convention for representing character tables which is applied espe- cially to the so-called POINT GROUPS , which are the 32 finite symmetry groups possible in a lattice. In the example above, the numbered regions contain the following contents (Cotton 1990 pp. 90 /C1/92). 1. The symbol used to represent the group in question (in this case C3v):/ 2. The CONJUGACY CLASSES , indicated by number and symbol, where the sum of the coefficients givesthe ORDER of the group. 3. M ULLIKEN SYMBOLS , one for each IRREDUCIBLE REPRESENTATION . 4. An array of the CHARACTERS of the IRREDUCIBLE REPRESENTATION of the group, with one column for each CONJUGACY CLASS , and one row for each IRREDUCIBLE REPRESENTATION . 5. Combinations of the symbols x,y,z,Rx;Ry;and Rz;the first three of which represent the coordi- nates x,y, and z, and the last three of which stand for rotations about these axes. These are related to transformation properties and basis representa- tions of the group.6. All square and binary products of coordinates according to their transformation properties. The character tables for many of the POINT GROUPS are reproduced below using this notation. /C1/E A1/Cs/E /sh/ A11 /x;y;Rz//x2;y2;z2;xy/ B1/C281 /z;Rx;Ry/yz, xz /Ci/Ei /Ag/11 /Rx;Ry;Rz//x2;y2;z2;xy;xz;yz/ /Au/1/C281 /x;y;z/ /C2/E /C2/ A11 /z;Rz// x2;y2;z2;xy/ B1/C281 /x;y;Rx;Rz/yz, xz /C3/E /C3//C2 3// o/C30exp(2 pi=3)/ A111 /z;Rz// x2;y2;z2;xy/ E /1 1l1s /o* /og//(x;y)(Rx;Ry)//(x2/C28y2;xy)(yz;xz)/ /C4/E /C3//C2/C43 A11 1 1 /z;Rz// x2/C27y2;z2/ B1/C2811 /C281 /x2/C28y2;xy/ E /11l1s //C28i1 i}/(x;y)(Rx;Ry)/(yz, xz ) /C5/E /C5//C2 5/C53C54 /o/C30exp(2 pi=5)/ A11 1 1 1 /z;Rz// x2/C27y2;z2/ /E1//1 1l1s /o*o2*o2o} /(x;y)(Rx;Ry)/(yz, xz ) /E2//11l1s /o2*oo *o2} /(x2/C28y2;xy)/ /C6/E /C6//C3//C2//C2 3/C65/o/C30exp(2 pi=6)/ A11 1 1 1 1 /z;Rz// x2/C27y2;z2/ B1/C2811 /C2811 /C281 /E1//1 1l1s /o*/C28o/C281/C28o* o}/(Rx;Ry)/(yz, xz ) /E2//11l1s //C28o*/C28o*1 /C28oo *} /(x2/C28y2;xy)/ /D2/E /C2(z)//C2(y)//C2(x)/ /A1/1111 /x2/C27y2;z2 / /B1/11 /C281/C281 /z;Rz/xy /B2/1/C2811 /C281y, R yxz /B3/1/C281/C2811 /z;Rz/yz /D3/E /2C3//3C2/ /A1/111 /x2/C27y2;z2/ /A2/11 /C281 /z;Rz/ xy E2/C2810 /(x;y)(Rx;Ry)//(x2/C28y2;xy)(xz;yz)/ /D4/E /2C4//C2//2C?2//2Cƒ2/ /A1/11111 /x2/C27y2;z2/ /A2/111 /C281/C281 /z;Rz/ /B1/1/C28111 /C281 /x2/C28y2/ /B2/1/C2811 /C2811 xy E20 /C28200 /(x;y)(Rx;Ry)/(xz, yz ) /D5/E /2C5// 2C2 5//5C2/ /A1/111 1 /x2/C27y2;z2/ /B1/111 /C281/z;Rz/ /B2/2 /2 cos 72/C14//2 cos 144/C14/ 0/(x;y)(Rx;Ry)/(xz, yz ) /B3/2 /2 cos 144/C14//2 cos 72/C14/ 0 /(x2/C28y2;xy)/ /D6/E /2C6//2C3//C2//3C?2//3Cƒ2/ /A1/111111 /x2/C27y2;z2/ /A2/1111 /C281/C281/z;Rz//B1/1/C2811 /C2811 /C281 /B2/1/C2811 /C281/C2811 /(x;y)(Rx;Ry)/ /E1/21 /C281/C28200 ( xz, yz ) /E2/2/C281/C281200 /(x2/C28y2;xy)/ /C2v/E /C2//sv(xz)//s?v(yz)/ /A1/1 111 z /x2;y2;z2 / /A2/11 /C281/C281 /Rz/xy /B1/1/C2811 /C281 /x;Ry/xz /B2/1/C281/C2811 /y;Rx/yz /C3v/E /2C3//3sv/ /A1/111 z /x2/C27y2;z2/ /A2/11 /C281 /Rz/ E2/C2810 /(x;y)(Rx;Ry)//(x2/C28y2;xy)(xz;yz)/ /C4v/E /2C4//C2//2sv//2sd/ /A1/11111 z /x2/C27y2;z2 / /A2/111 /C281/C281 /Rz/ /B1/1/C28111 /C281 /x2/C28y2/ /B2/1/C2811 /C2811 xy E20 /C28200 /(x;y)(Rx;Ry)/(xz, yz ) /C5v/E /2C5// 2C2 5//5sv/ /A1/111 1 z /x2/C27y2;z2/ /B1/111 /C281/Rz/ /B2/2 /2 cos 72/C14//2 cos 144/C14/0/(x;y)(Rx;Ry)/(xz, yz ) /B3/2 /2 cos 144/C14//2 cos 72/C14/0 /(x2/C28y2;xy)/ /C6v/E /2C6//2C3//C2//3sv//3sd/ /A1/111111 z /x2/C27y2;z2/ /A2/1111 /C281/C281/Rz/ /B1/ 1 /C2811 /C2811 /C281 /B2/ 1 /C2811 /C281 /C2811 /E1/ 21 /C281 /C28200 /(x ; y)(Rx ; Ry)/ (xz, yz) /E2/ 2 /C281 /C281200 /(x2 /C28y2 ; xy)/ /C/C12v/ E /C F /C12 / ... / /C12sv/ /A1 /C13S/C27/ 1 1 ... 1 z /x2 /C27y2 ; z2/ /A2 /C13S/C28/ 1 1 ... /C281 /Rz/ /E1 /C13P/ 2 /2 cos F/ ... 0 /(x; y); (Rx ; Ry)/ (xz, yz) /E2 /C13D/ 2 /2 cos 2F/ ... 0 /(x2 /C28y2 ; xy)/ /E3 /C13F/ 2 /2 cos 3F/ ... 0 /n//n//n//::://n/ See also CHARACTER (GROUP ), CONJUGACY CLASS , GROUP ,IRREDUCIBLE REPRESENTATION ,P OINT GROUPS ,REPRESENTATION References Bishop, D. M. "Character Tables." Appendix 1 in Group Theory and Chemistry. New York: Dover, pp. 279 /C1/288, 1993. Cotton, F. A. "Character Tables." §4.4 in Chemical Applica- tions of Group Theory, 3rd ed. New York: Wiley, pp. 90 /C1/ 95, 1990. Huang, J.-S. "Characters of Representations." §2.2 in Lec- tures on Representation Theory. Singapore: World Scien- tific, pp. 9 /C1/11, 1999. Iyanaga, S. and Kawada, Y. (Eds.). "Characters of Finite Groups." Appendix B, Table 5 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1496 /C1/ 1503, 1980. Sosnovsky, A. and Demarco, G. L. "Character Tables of Finite Groups." Mathematica Educ. Res. 6,5/C1/8, 1997. Characteristic (Elliptic Integral) A parameter n used to specify an ELLIPTIC INTEGRAL OF THE THIRD KIND P(n; f, k). See also AMPLITUDE ,ELLIPTIC INTEGRAL ,M ODULAR ANGLE ,MODULUS (ELLIPTIC INTEGRAL ), NOME,PARA- METER References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 590, 1972. Characteristic (Euler) EULER CHARACTERISTICCharacteristic (Field) For a FIELD K with multiplicative identity 1, consider the numbers 2 /C301 /C271; 3 /C301 /C271 /C271 ; 4 /C301 /C271 /C271 /C271; etc. Either these numbers are all different, in which case we say that K has characteristic 0, or two of them will be equal. In the latter case, it is straightfor- ward to show that, for some number p, we have 1 /C271 /C27.../C271|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} p times/C300: If p is chosen to be as small as possible, then p will be a PRIME , and we say that K has characteristic p. The characteristic of a field K is sometimes denoted ch(K). The FIELDS Q (rationals), R (reals), C (complex numbers), and the P-ADIC NUMBERS Qphave char- acteristic 0. For p a PRIME , the FINITE FIELD GF( /pn) has characteristic p. If H is a SUBFIELD of K, then H and K have the same characteristic. See also FIELD,FINITE FIELD,SUBFIELD References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 422, 1998. Characteristic (Partial Differential Equation) Paths in a 2-D plane used to transform PARTIAL DIFFERENTIAL EQUATIONS into systems of ORDINARY DIFFERENTIAL EQUATIONS . They were invented by Riemann. For an example of the use of characteris- tics, consider the equation u1/C286uux/C300: Now let u(s)/C30u(x(s);t(s)):Since du ds/C30dx dsux/C27dt dsut; it follows that dt=ds/C301;dx=ds/C30/C286u;anddu=ds/C300: Integrating gives t(s)/C30s;x(s)/C30/C286su0(x);and u(s)/C30 u0(x);where the constants of integration are 0 and u0(x)/C30u(x;0):/ References Farlow, S. J. Partial Differential Equations for Scientists and Engineers. New York: Dover, pp. 205 /C1/212, 1993. Landau, L. D. and Lifschitz, E. M. Fluid Mechanics, 2nd ed. Oxford, England: Pergamon Press, pp. 310 /C1/346, 1982. Moon, P. and Spencer, D. E. Partial Differential Equations. Lexington, MA: Heath, pp. 27 /C1/29, 1969. Whitham, G. B. Linear and Nonlinear Waves. New York: Wiley, pp. 113 /C1/142, 1974. Zauderer, E. Partial Differential Equations of Applied Mathematics, 2nd ed. New York: Wiley, pp. 78 /C1/121, 1989. Zwillinger, D. "Method of Characteristics." §88 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 325 /C1/330, 1997. Characteristic (Real Number) For a REAL NUMBER x, xbc/C30int(x) is called the characteristic, where xbcis the FLOOR FUNCTION . See also MANTISSA ,SCIENTIFIC NOTATION Characteristic Class Characteristic classes are COHOMOLOGY classes in the BASE SPACE of a VECTOR BUNDLE , defined through OBSTRUCTION theory, which are (perhaps partial) obstructions to the existence of k everywhere linearly independent vector FIELDS on the VECTOR BUNDLE . The most common examples of characteristic classes are the CHERN ,PONTRYAGIN , and STIEFEL- WHITNEY CLASSES . Characteristic Equation The equation which is solved to find a matrix’s EIGENVALUES , also called the characteristic polyno- mial. For a general k /C29k MATRIX M ; the characteristic equation in variable t is defined by det(M /C28tI) /C300; (1) where I is the IDENTITY MATRIX and det(A) is the DETERMINANT of the MATRIX A: Writing M out ex- plicitly gives M /C13a11a12 /C1/C1/C1 a1k a21a22 /C1/C1/C1 a2k nn::: n ak1ak2/C1/C1/C1 akk2 6643 775; (2) so the characteristic equation is given by a11 /C28ta12 /C1/C1/C1 a1k a21 a22 /C28t /C1/C1/C1 a2k nn::: n ak1 ak2 /C1/C1/C1 akk /C28tl112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112/C300 (3) The solutions t of the characteristic equation are called EIGENVALUES , and are extremely important in the analysis of many problems in mathematics and physics. See also BALLIEU’S THEOREM ,C AYLEY- HAMILTON THEOREM ,DIAGONAL MATRIX ,EIGENVALUE ,PARODI’S THEOREM ,ROUTH- HURWITZ THEOREM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1117 /C1/1119, 2000. Characteristic Factor A characteristic factor is a factor in a particular factorization of the TOTIENT FUNCTION f(n) such that the product of characteristic factors gives the repre- sentation of a corresponding abstract GROUP as a GROUP DIRECT PRODUCT . By computing the character- istic factors, any ABELIAN GROUP can be expressed asa GROUP DIRECT PRODUCT of CYCLIC SUBGROUPS , for example, the FINITE GROUP Z2/N /Z4or Z2/N /Z2/N /Z2. There is a simple algorithm for determining the characteristic factors of MODULO MULTIPLICATION GROUPS . See also CYCLIC GROUP ,G ROUP DIRECT PRODUCT , MODULO MULTIPLICATION GROUP ,TOTIENT FUNCTION References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 94, 1993. Characteristic Function (Probability) The characteristic function f(t) is defined as the FOURIER TRANSFORM of the PROBABILITY DENSITY FUNCTION using FOURIER TRANSFORM parameters (a; b) /C30(1; 1); f(t) /C30F[P(x)] /C30g/C12 /C28/C12eitxP(x) dx (1) /C30g/C12 /C28/C12P(x) dx /C27itg/C12 /C28/C12xP(x) dx /C271 2(it)2g/C12 /C28/C12x2P(x) dx /C27... (2) /C30X/C12 k /C300(it)k k!m?k (3) /C301 /C27it m?1 /C281 2 t2 m?2 /C281 3!it3 m?3 /C271 4!t4 m?4 /C27...; (4) where m?n(sometimes also denoted nn) is the nth MOMENT about 0 and m?0 /C131 (Abramowitz and Stegun 1972, p. 928). A DISTRIBUTION is not uniquely speci- fied by its MOMENTS , but is uniquely specified by its characteristic function, P(x) /C30F/C281[f(t)] /C301 2 p g/C12 /C28/C12e /C28itx f(t) dt (5) (Papoulis 1984, p. 155). The characteristic function can therefore be used to generate RAW MOMENTS , f(n)(0)/C13dnf dtn"# t/C300/C30inm?n (6) or the CUMULANTS kn; lnf(t)/C13X/C12 n/C30okn(it)n n!: (7) See also CUMULANT ,MOMENT ,MOMENT- GENERATING FUNCTION ,PROBABILITY DENSITY FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972. Kenney, J. F. and Keeping, E. S. "Moment-Generating and Characteristic Functions," "Some Examples of Moment- Generating Functions," and "Uniqueness Theorem for Characteristic Functions." §4.6 /C1/4.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 72 /C1/77, 1951. Papoulis, A. "Characteristic Functions." §5 /C1/5inProbability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 153 /C1/162, 1984. Characteristic Function (Set) Given a SUBSET A of a larger set, the characteristic function xAis identically one on A, and is zero elsewhere. These kinds of functions get their own name because they are useful tools. It is easier to say "the char- acteristic function of the rationals" or "the character- istic function of PRIMES " than to keep repeating the definition.A characteristic function is a special case of a SIMPLE FUNCTION . See also SET,SIMPLE FUNCTION References Lukacs, E. Characteristic Functions. London: Griffin, 1970. Characteristic Polynomial The expanded form of the CHARACTERISTIC EQUATION , det(xI /C28A); where A is an n /C29n MATRIX and I is the IDENTITY MATRIX . The characteristic polynomial of a GRAPH G takes Aas the ADJACENCY MATRIX ofA:/ See also CAYLEY- HAMILTON THEOREM ,EIGENVALUE , SPECTRUM (MATRIX ) References Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 310, 1996. Hagos, E. M. "The Characteristic Polynomial of a Graph is Reconstructible from the Characteristic Polynomials of its Vertex-Deleted Subgraphs and Their Complements." Elec- tronic J. Combinatorics 7, No. 1, R12, 1 /C1/9, 2000. http:// www.combinatorics.org/Volume_7/v7i1toc.html. Characteristic Root EIGENVALUE Characteristic Vector EIGENVECTORCharlier A-Series CHARLIER SERIES Charlier Differential Series CHARLIER SERIES Charlier Polynomial The orthogonal polynomials defined by c(m) n(x)/C302F0(/C28n;/C28x;;/C28m/C281) (1) /C30(/C281)n mn(x/C28n/C271)n1F1(/C28n;x/C28n/C271;m) (2) /C302F0(/C28n;/C28x;;/C281=m) (3) where ( x)nis the P OCHHAMMER SYMBOL (Koekoek and Swarttouw 1998). The first few are given by c(m) 0(x)/C301 c(m) 1(x)/C301/C28x m c(m) 2(x)/C30x2/C27m2/C28x(1/C272m) m2: References Koekoek, R. and Swarttouw, R. F. "Charlier." §1.12 in The Askey-Scheme of Hypergeometric Orthogonal Polynomialsand its q -Analogue. Delft, Netherlands: Technische Uni- versiteit Delft, Faculty of Technical Mathematics andInformatics Report 98 /C1 /17, pp. 49 /C1/50, 1998. ftp:// www.twi.tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1 /17.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities.Braunschweig, Germany: Vieweg, p. 115, 1998. Charlier Series A class of formal series expansions in derivatives of a distribution C(t) which may (but need not) be the NORMAL DISTRIBUTION FUNCTION F(t)/C131ffiffiffiffiffiffi 2pp e/C28t2=2 and moments or other measured parameters. Edge- worth series are known as the Charlier series or Gram-Charlier series. Let c(t) be the CHARACTERISTIC FUNCTION of the function C(t);andgritsCUMULANTS . Similarly, let F(t) be the distribution to be approxi- mated, f(t) its CHARACTERISTIC FUNCTION , and krits CUMULANTS . By definition, these quantities are con- nected by the formal series f(t)/C30expX/C12 r/C301(kr/C28gr)(it)r r!"# c(t) (Wallace 1958). Integrating by parts gives (it)r c(t)as the CHARACTERISTIC FUNCTION of (/C281)r C(r)(x) ; so the formal identity corresponds pairwise to the identity F(x) /C30expX/C12 r/C301( kr /C28 gr)( /C28D)r r!"# C(x); where D is the DIFFERENTIAL OPERATOR . The most important case C(t) /C30F(t) was considered by Cheby- shev (1890), Charlier (1905), and Edgeworth (1905). Expanding and collecting terms according to the order of the derivatives gives the so-called Gram- Charlier A-Series, which is identical to the formal expansion of F /C28C in Hermite polynomials. The A- series converges for functions F whose tails approach zero faster than C?1=2 (Crame ´r 1925, Wallace 1958, Szego 1975). See also CORNISH- FISHER ASYMPTOTIC EXPANSION , EDGEWORTH SERIES References Charlier, C. V. L. "U¨ ber das Fehlergesetz." Ark. Math. Astr. och Phys. 2, No. 8, 1 /C1/9, 1905 /C1/06. Chebyshev, P. L. "Sur deux the´ore`mes relatifs aux probabil- ite´s." Acta Math. 14, 305 /C1/315, 1890. Crame ´r, H. "On Some Classes of Series Used in Mathema- tical Statistics." Proceedings of the Sixth Scandinavian Congress of Mathematicians, Copenhagen. pp. 399 /C1/425, 1925. Edgeworth, F. Y. "The Law of Error." Cambridge Philos. Soc. 20,36/C1/66 and 113 /C1/141, 1905. Gram, J. P. "U¨ ber die Entwicklung reeler Funktionen in Reihen mittelst der Methode der kleinsten Quadrate." J. reine angew. Math. 94,41/C1/73, 1883. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. Wallace, D. L. "Asymptotic Approximations to Distribu- tions." Ann. Math. Stat. 29, 635 /C1/654, 1958. Charlier’s Check A check which can be used to verify correct computa- tions in a table of grouped classes. For example, consider the following table with specified class limits and frequencies f. The class marks xiare then computed as well as the rescaled frequencies ui ; which are given by ui /C30fi /C28 x0 c; (1) where the class mark is taken as x0 /C3074 :5 and the class interval is c /C3010. The remaining quantities are then computed as follows. class limits /xi// fi//(m)n//fiui//fiu2 i//fi(ui /C271)2 / 30 /C1/39 34.5 2 /C284 /C2883 2 1 8 40 /C1/49 44.5 3 /C283 /C2892 7 1 250 /C1/59 54.5 11 /C282 /C2822 44 11 60 /C1/69 64.5 20 /C281 /C2820 20 0 70 /C1/79 74.5 32 0 0 0 32 80 /C1/89 84.5 25 1 25 25 100 90 /C1/99 94.5 7 2 14 28 63 total 100 /C2820 176 236 In order to compute the VARIANCE , note that s2 u /C30P i fiu2 iP i fi/C28P i fiuiP i fi !2 (2) /C30176 100 /C28/C2820 100 !2 /C301:72; (3) so the VARIANCE of the original data is s2 x /C30c2s2u /C30172: (4) Charlier’s check makes use of the additional column fi(ui /C271)2 added to the right side of the table. By noting that the identity X ifi(ui/C271)2/C30X ifi(u2i/C272ui/C271) /C30X ifiu2i/C272X ifiui/C27X ifi; (5) connects columns five through seven, it can be checked that the computations have been done correctly. In the example above, 236/C30176/C272(/C2820)/C27100; (6) so the computations pass Charlier’s check. See also VARIANCE References Kenney, J. F. and Keeping, E. S. "Charlier Check." §6.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 47 /C1/48, 81, 94 /C1/95, and 104, 1962. Chart COORDINATE CHART Chasles-Cayley-Brill Formula The number of coincidences of a ( n;n?) correspon- dence of value gon a curve of GENUS pis given by n/C27n?/C272pg: See also ZEUTHEN’S THEOREM References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 129, 1959. Chasles’s Contact Theorem If a one-parameter family of curves has index N and class M, the number tangent to a curve of order n1 and class m1 in general position is m1N /C27n1M : References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 436, 1959. Chasles’s Polars Theorem If the TRILINEAR POLARS of the VERTICES of a TRIAN- GLE are distinct from the respectively opposite sides, they meet the sides in three COLLINEAR points. See also COLLINEAR ,TRIANGLE ,TRILINEAR POLAR Chasles’s Theorem If two projective PENCILS of curves of orders n and n 0 have no common curve, the LOCUS of the intersections of corresponding curves of the two is a curve of order n /C27n 0through all the centers of either PENCIL . Conversely, if a curve of order n /C27n0 contains all centers of a PENCIL of order n to the multiplicity demanded by NOETHER’S FUNDAMENTAL THEOREM , then it is the LOCUS of the intersections of correspond- ing curves of this PENCIL and one of order n0 projective therewith. See also NOETHER’S FUNDAMENTAL THEOREM ,PENCIL References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 33, 1959. Chebyshev This entry contributed by RONALD M. AARTS A number of spellings of "Chebyshev" (which is the spelling used exclusively in this work) are commonly found in the literature. These include Tchebicheff, Cebysev, Tschebyscheff, Chebishev, and Tsche- byscheff (Clenshaw). References Clenshaw, C. W. Mathematical Tables, Vol. 5: Chebyshev Series for Mathematical Functions. Department of Scien- tific and Industrial Research.Chebyshev Approximation Formula Using a CHEBYSHEV POLYNOMIAL OF THE FIRST KIND T(x) ; define cj /C132 NXN k /C301f(xk)Tj(xk) /C302 NXN k /C301f cosp(k /C281 2) N()"# cospj(k /C2812) N() : Then f(x) :XN /C281 k /C300ckTk(x) /C2812 c0 : It is exact for the N zeros of TN(x) : This type of approximation is important because, when truncated, the error is spread smoothly over [/C281; 1]: The Cheby- shev approximation formula is very close to the MINIMAX POLYNOMIAL . References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Chebyshev Approximation," "Derivatives or Integrals of a Chebyshev-Approximated Function," and "Polynomial Approximation from Chebyshev Coefficients." §5.8, 5.9, and 5.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 184 /C1/188, 189 /C1/190, and 191 /C1/192, 1992. Chebyshev Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. The constants lm;n/C30inf r/C23Rm;nsup x]0½e/C28x/C28r(x)½; where r(x)/C30p(x) q(x); pand qaremth and nth order POLYNOMIALS , and Rm;nis the set all RATIONAL FUNCTIONS with REAL coefficients. See also ONE-NINTH CONSTANT ,RATIONAL FUNCTION References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/onenin/onenin.html. Petrushev, P. P. and Popov, V. A. Rational Approximation of Real Functions. New York: Cambridge University Press, 1987. Varga, R. S. Scientific Computations on Mathematical Pro- blems and Conjectures. Philadelphia, PA: SIAM, 1990. Philadelphia, PA: SIAM, 1990. Chebyshev Deviation max a5x5b½f(x)/C28r(x)½w(x) fg : References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 41, 1975. Chebyshev Differential Equation (1/C28x2)d2y dx2/C28xdy dx/C27a2y/C300 (1) for½x½B1:The Chebyshev differential equation has regular SINGULARITIES at/C281, 1, and /C12:It can be solved by series solution using the expansions y/C30X/C12 n/C300anxn(2) y?/C30X/C12 n/C300nanxn/C281/C30X/C12 n/C301nanxn/C281/C30X/C12 n/C300(n/C271)an/C271xn(3) yƒ/C30X/C12 n/C300(n/C271)nan/C271xn/C281/C30X/C12 n/C301(n/C271)nan/C271xn/C281 /C30X/C12 n/C300(n/C272)(n/C271)an/C272xn: (4) Now, plug (2 /C1/4) into the original equation (1) to obtain (1/C28x2)X/C12 n/C300(n/C272)(n/C271)an/C272xn /C28xX/C12 n/C300(n/C271)nn/C271xn/C27a2X/C12 n/C300anxn/C300 (5)X/C12 n/C300(n/C272)(n/C271)an/C272xn/C28X/C12 n/C300(n/C272)(n/C271)an/C272xn/C272 /C28X/C12 n/C300(n/C271)an/C272xn/C271/C27a2X/C12 n/C300anxn/C300 (6) X/C12 n/C300(n/C272)(n/C271)an/C272xn/C28X/C12 n/C302n(n/C281)anxn/C272 /C28X/C12 n/C301nanxn/C27a2X/C12 n/C300anxn/C300 (7) 2/C2151a2/C273/C2152a3x/C281/C215ax/C27a2a0/C27a2a1x /C27X/C12 n/C302[(n/C272)(n/C271)an/C272/C28n(n/C281)an/C28nan/C27a2an]xn /C300 (8) (2a2/C27a2a0)/C27[(a2/C281)a1/C276a3]x /C27X/C12 n/C302[(n/C272)(n/C271)an/C272/C27(a2/C28n2)an]xn/C300; (9) so 2a2/C27a2a0/C300 (10) (a2/C281)a1/C276a3/C300; (11) and by induction, an/C272/C30n2/C28a2 (n/C271)(n/C272)an (12) forn/C302, 3, .... Since (10) and (11) are special cases of (12), the general RECURRENCE RELATION can be written an/C272/C30n2/C28a2 (n/C271)(n/C272)an (13) forn/C300, 1, .... From this, we obtain for the EVEN COEFFICIENTS a2/C30/C28a2 2a0 (14) a4/C3022/C28a2 3 /C2154a2/C30(22/C28a2)(/C28a2) 1 /C2152 /C2153 /C2154a0 (15) a2n/C30[(2n)2/C28a2][(2n/C282)2/C28a2]/C1/C1/C1(/C28a2) (2n)!a0: (16) and for the ODD COEFFICIENTS a3/C301/C28a2 6a0 (17) a5 /C3032 /C28 a2 4 /C215 5a3 /C30(32 /C28 a2)(12 /C28 a2) 5!a1 (18) a2n /C281 /C30 [(2n /C28 1)2 /C28 a2][(2n /C28 3)2 /C28 a2] /C1/C1/C1[12 /C28 a2] (2n /C27 1)! a1 : (19) The even coefficients k /C302n can be given in closed form by as ak even /C30a0Yk =2 j /C301(k /C282j)2 /C28 a2 /C302k /C281 pa csc(1 2 pa) G(1 /C281 2 k /C2812 a) G(1 /C2812 k /C2712 a)a0 ; (20) and the odd coefficients k /C302n /C281as ak odd /C30a1Y(k /C281)=2 j/C301(k /C282j)2 /C28 a2 /C302k /C281 pa sec(1 2 pa) G(1 /C281 2 k /C2812 a)G(1 /C2812 k /C2712 a) a1 : (21) The general solution is then given by summing over all indices, y /C30a01 /C27X/C12 k/C302 ;4...ak even k!xk"# /C27 x /C27X/C12 k/C303;5...ak odd k!xk"# ; (22) which can be done in closed form as y /C30a0 cos(a sin/C281 x) /C27a1 asin(a sin/C281 x): (23) Performing a change of variables gives the equivalent form of the solution y /C30b1 cos(a cos/C281 x) /C27b2 sin( a cos/C281 x) (24) /C30b1T a(x) /C27b2ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p Ua /C281(x); (25) where Tn(x)isaC HEBYSHEV POLYNOMIAL OF THE FIRST KIND and Un(x)isaC HEBYSHEV POLYNOMIAL OF THE SECOND KIND . Another equivalent form of the solution is given by y /C30c1 cosh[ a ln(x /C27ffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C281p )] /C27ic2 sinh[ a ln(x /C27ffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C281p )]: (26) See also CHEBYSHEV POLYNOMIAL OF THE FIRST KIND, CHEBYSHEV POLYNOMIAL OF THE SECOND KIND References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 735, 1985. Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 232 and 252, 1986.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Chebyshev Functions The function defined by u(n) /C13Xn i/C301ln pi /C30lnY p 5np ! ; (1) where pi is the ith PRIME (left figure), so lim x 0/C12x u(x) /C301 (2) (right figure). The function has asymptotic behavior u(n) /C2n (3) (Bach and Shallit 1996; Hardy 1999, p. 28). The notation q(n) is also commonly used for this function (Hardy 1999, p. 27). Chebyshev also defined the related function c(n) /C13X p; n p n 5nln p ; (4) which is equal to the summatory MANGOLDT FUNC- TION and is given by the logarithm of the LEAST COMMON MULTIPLE of the numbers from 1 to n. The values of LCM(1 ; 2 ;/C1/C1/C1; n) for n /C301, 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, ... (Sloane’s A003418). For example, c(10) /C30ln 2520 /C303ln2/C272ln3/C27ln 5 /C27ln 7: (5) The function has asymptotic behavior c(n) /C2n (6) (Hardy 1999, p. 27). According to Hardy (1999, p. 27), the functions u(n) andc(n) are in some ways more natural than the PRIME COUNTING FUNCTION p(x) since they deal with multiplication of primes instead of the counting of them. See also MANGOLDT FUNCTION ,P RIME COUNTING FUNCTION ,PRIME NUMBER THEOREM References Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, pp. 206 and 233, 1996. Costa Pereira, N. "Estimates for the Chebyshev Function c(x)/C28u(x):/"Math. Comp. 44, 211/C1/221, 1985. Costa Pereira, N. "Corrigendum: Estimates for the Cheby- shev Function c(x) /C28 u(x) :/" Math. Comp. 48, 447, 1987. Costa Pereira, N. "Elementary Estimates for the Chebyshev Function c(x) and for the Mo¨bius Function M(x):/" Acta Arith. 52, 307 /C1/337, 1989. Dusart, P. "Ine´galite ´s explicites pour c(X) ; u(X); p(X) et les nombres premiers." C. R. Math. Rep. Acad. Sci. Canad 21, 53 /C1/59, 1999. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 27, 1999. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 60, 1951. Panaitopol, L. "Several Approximations of p(x):/" Math. Ineq. Appl. 2, 317 /C1/324, 1999. Robin, G. "Estimation de la foction de Tchebychef u sur le kie`me nombre premier er grandes valeurs de la fonctions v(n); nombre de diviseurs premiers de n." Acta Arith. 42, 367 /C1/389, 1983. Rosser, J. B. and Schoenfeld, L. "Sharper Bounds for Cheby- shev Functions u(x) and c(x) :/" Math. Comput. 29, 243 /C1/ 269, 1975. Schoenfeld, L. "Sharper Bounds for Chebyshev Functions u(x) and c(x) ; II." Math. Comput. 30, 337 /C1/360, 1976. Selmer, E. S. "On the Number of Prime Divisors of a Binomial Coefficient." Math. Scand. 39, 271 /C1/281, 1976. Sloane, N. J. A. Sequences A003418/M1590 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Chebyshev Inequality Apply MARKOV’S INEQUALITY with a /C13k2 to obtain P[(x /C28 m)2 ]k2] 5/C142(x /C28 m)2 /C143 k2/C30s2 k2 : (1) Therefore, if a RANDOM VARIABLE x has a finite MEAN m and finite VARIANCE s2 ; then /C214k ]0 ; P( ½x /C28 m½]k) 5s2 k2 (2) P( ½x /C28 m½]k s) 51 k2 : (3) See also CHEBYSHEV SUM INEQUALITY References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "Tchebychef’s Inequality." §2.17 and §5.8 in Inequalities, 2nd ed. Cam- bridge, England: Cambridge University Press, pp. 43 /C1/45 and 123, 1988. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 149 /C1/151, 1984.Chebyshev Integral g xp(1 /C28x)q dx /C30x1/C27p 2F1(p/C271;/C28q;p/C272;x) p/C271: See also CHEBYSHEV INTEGRAL INEQUALITY Chebyshev Integral Inequality gb af1(x)dxgb af2(x)dx/C1/C1/C1gb afn(x)dx 5(b/C28a)n/C281gb af1(x)f2(x)/C1/C1/C1fn(x)dx where f1;f2;...,fnare NONNEGATIVE integrable functions on [ a, b] which are alleither monotonic increasing or monotonic decreasing. References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1092, 2000. Chebyshev Phenomenon PRIME QUADRATIC EFFECT Chebyshev Polynomial of the First Kind A set of ORTHOGONAL POLYNOMIALS defined as the solutions to the C HEBYSHEV DIFFERENTIAL EQUATION and denoted Tn(x):They are used as an approxima- tion to a LEAST SQUARES FIT , and are a special case of the ULTRASPHERICAL POLYNOMIAL with a/C300:They are also intimately connected with trigonometric MULTIPLE-ANGLE FORMULAS . The Chebyshev polyno- mials of the first kind are denoted Tn(x);and are implemented in Mathematica asChebyshevT [n,x]. They are normalized such that Tn(1)/C301:The first few polynomials are illustrated above for x/C23[/C281;1] and n/C301, 2, ..., 5. The Chebyshev polynomials of the first kind can be obtained from the GENERATING FUNCTIONS g1(t;x)/C131/C28t2 1/C282xt/C27t2/C30T0(x)/C272X/C12 n/C301Tn(x)tn(1) and g2(t;x)/C131/C28xt 1/C282xt/C27t2/C30X/C12 n/C300Tn(x)tn(2) for½x½51 and ½t½B1 (Beeler et al. 1972, Item 15). (A closely related GENERATING FUNCTION is the basis for the definition of C HEBYSHEV POLYNOMIAL OF THE SECOND KIND .) The polynomials can also be defined in terms of the sums Tn(x)/C30n 2Xn=2bc r/C300(/C281)r n/C28rn/C28r rl11sl11n (2x)n/C282r(3) Tn(x)/C30cos(cos/C281x)/C30Xn=2bc m/C300n 2ml11sl11n xn/C282m(x2/C281)m;(4) wheren kl1ml11 is a BINOMIAL COEFFICIENT and xbcis the FLOOR FUNCTION , or the product Tn(x)/C302n/C281Yn k/C301x/C28cos(2k/C281)p 2n"#() (5) (Zwillinger 1995, p. 696). /Tnalso satisfy the curious DETERMINANT equation Tn/C30x100 /C1/C1/C1 00 12x10:::00 012 x1:::00 00 12 x:::00 00 0 1:::10 n:::::::::::::::1 00 0 0 /C1/C1/C1 12 xl112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112: (6) The Chebyshev polynomials of the first kind are a special case of the J ACOBI POLYNOMIALS P(a;b) nwith a/C30 b/C30/C281=2; Tn(x)/C30P(/C281=2;/C281=2) n (x) P(/C281=2;/C281=2) n (1)/C302F1(/C28n;/C28n;1 2;12(1/C28x));(7) where2F1(a;b;c;x)i sa HYPERGEOMETRIC FUNCTION (Koekoek and Swarttouw 1998). Zeros occur when x/C30cospk/C281 2l11)l117 n2 435 (8) fork/C301, 2, ..., n. Extrema occur for x/C30cos pk n ! ; (9) where k/C300;1;...;n:At maximum, Tn(x)/C301;and at minimum, Tn(x)/C30/C281:The Chebyshev POLYNOMIALS are ORTHONORMAL with respect to the WEIGHTING FUNCTION (1/C28x2)/C281=2g1 /C281Tm(x)Tn(x)dxffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C301 2pdnmform"0;n"0 p form/C30n/C300;l12) (10) where /dmn/is the K RONECKER DELTA . Chebyshev polynomials of the first kind satisfy the additional discrete identity Xm k/C301Ti(xk)Tj(xk)/C301 2mdijfori"0;j"0 m fori/C30j/C300;l12) (11) where xkfork/C301, ..., mare the mzeros of Tm(x): They also satisfy the RECURRENCE RELATIONS Tn/C271(x)/C302xTn(x)/C28Tn/C281(x) (12) Tn/C271(x)/C30xTn(x)/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C28x2)f1/C28[Tn(x)]2gq (13) forn]1:They have a COMPLEX integral representa- tion Tn(x)/C301 4pigg(1/C28z2)z/C28n/C281dz 1/C282xz/C27z2(14) and a Rodrigues representation Tn(x)/C30(/C281)nffiffiffipp(1/C28x2)1=2 2n(n/C281 2)!dn dxn[(1/C28x2)n/C281=2]:(15) Using a FAST FIBONACCI TRANSFORM with multiplica- tion law (A;B)(C;D)/C30(AD/C27BC/C272xAC ;BD/C28AC) (16) gives Tn/C271(x);/C28Tn(x)/C30(T1(x);/C28T0(x))(1;0)n: (17) Using G RAM- SCHMIDT ORTHONORMALIZATION in the range ( /C281,1) with WEIGHTING FUNCTION (1/C28x2)(/C281=2) gives p0(x)/C301 (18) p1(x)/C30x/C28g1 /C281x(1/C28x2)/C281=2dx g1 /C281(1/C28x2)/C281=2dx2 66643 7775 /C30x/C28 [/C281(1/C28x2)1=2]1 /C281 [sin/C281x]1/C281/C30x (19) p2(x) /C30 x /C28g1 /C281x3(1 /C28 x2)/C281 =2 dx g1 /C281x2(1 /C28 x2)/C281 =2 dx2 66643 7775x /C28g1 /C281x2(1 /C28 x2) /C281=2 dx g1 /C281(1 /C28 x2) /C281=2 dx2 66643 7775/C215 1 /C30 x /C280 ½/C138 x /C28p 2 p /C30x2 /C281 2; (20) etc. Normalizing such that Tn(1) /C301 gives T0(x) /C301 T1(x) /C30x T2(x) /C302x2 /C281 T3(x) /C304x3 /C283x T4(x) /C308x4 /C288x2 /C271 T5(x) /C3016x5 /C2820x3 /C275x T6(x) /C3032x6 /C2848x4 /C2718x2 /C281 The Chebyshev polynomial of the first kind is related to the BESSEL FUNCTION OF THE FIRST KIND Jn(x) and MODIFIED BESSEL FUNCTION OF THE FIRST KIND In(x) by the relations Jn(x) /C30inTnid dx ! J0(x) (21) In(x) /C30Tnd dx ! I0(x): (22) Letting x /C13cos u allows the Chebyshev polynomials of the first kind to be written as Tn(x) /C30cos(n u) /C30cos(n cos/C281 x) : (23) The second linearly dependent solution to the trans- formed differential equation d2Tn du2 /C27n2Tn /C300 (24) is then given by Vn(x) /C30sin(nu) /C30sin(n cos /C281 x) ; (25) which can also be written Vn(x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p Un/C281(x) ; (26) where Unis a CHEBYSHEV POLYNOMIAL OF THE SECOND KIND . Note that Vn(x) is therefore not a POLYNOMIAL .The triangle of RESULTANTS r(Tn(x); Tk(x)) is given by f0g;//f/C281; 0g;//f0 ;/C284; 0g;//f1; 16 ; 64 ; 0g;/ {0, /C2816, 0, 4096, 0}, ... (Sloane’s A054375). The POLYNOMIALS pn(x) /C30xn /C2821 /C28nTn(x) (27) of degree n /C282; the first few of which are p1(x) /C300 p2(x) /C301 2 p3(x) /C3034 x p4(x) /C30x2 /C281 8 p5(x)/C305 16(4x3/C28x) are the POLYNOMIALS of degree Bnwhich stay closest toxnin the interval ( /C281;1):The maximum deviation is 21/C28nat the n/C271 points where x/C30coskp n ! ; (28) fork/C300, 1, ..., n(Beeler et al. 1972). See also CHEBYSHEV APPROXIMATION FORMULA ,CHE- BYSHEV POLYNOMIAL OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771 /C1/802, 1972. Arfken, G. "Chebyshev (Tschebyscheff) Polynomials" and "Chebyshev Polynomials--Numerical Applications." §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731 /C1/748, 1985. Beeler et al. . Item 15 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 9, Feb. 1972. Iyanaga, S. and Kawada, Y. (Eds.). "Cebysev (Tschebyscheff) Polynomials." Appendix A, Table 20.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1478 /C1/1479, 1980. Koekoek, R. and Swarttouw, R. F. "Chebyshev." §1.8.2 in The Askey-Scheme of Hypergeometric Orthogonal Polyno-mials and its q -Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1 /17, pp. 41 /C1/43, 1998. ftp:// www.twi.tudelft.nl/publications/tech-reports/1998/DUT- TWI-98 /C1/17.ps.gz. Koepf, W. "Efficient Computation of Chebyshev Polyno- mials." In Computer Algebra Systems: A Practical Guide (Ed. M. J. Wester). New York: Wiley, pp. 79 /C1/99, 1999. Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990. Shohat, J. The´orie ge ´ne´rale des polynomes orthogonaux de Tchebichef. Paris: Gauthier-Villars, 1934. Sloane, N. J. A. Sequences A054375 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Chebyshev Polynomials Tn(x) and Un(x):/" Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193 /C1/207, 1987. Vasilyev, N. and Zelevinsky, A. "A Chebyshev Polyplay- ground: Recurrence Relations Applied to a Famous Set of Formulas." Quantum 10,2 0/C1/26, Sept./Oct. 1999. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995. Chebyshev Polynomial of the Second Kind A modified set of Chebyshev POLYNOMIALS defined by a slightly different GENERATING FUNCTION . They arise in the development of four-dimensional SPHERICAL HARMONICS in angular momentum theory. They are a special case of the ULTRASPHERICAL POLYNOMIAL with a/C301:They are also intimately connected with trigo- nometric MULTIPLE-ANGLE FORMULAS . The Chebyshev polynomials of the second kind are denoted Un(x);and implemented in Mathematica asChebyshevU [n,x]. The polynomials Un(x) are illustrated above for x/C23 [/C281;1] and n/C301, 2, ..., 5. The defining GENERATING FUNCTION of the Chebyshev polynomials of the second kind is g2(t;x)/C301 1/C282xt/C27t2/C30X/C12 n/C300Un(x)tn(1) for½x½B1 and ½t½B1:To see the relationship to a CHEBYSHEV POLYNOMIAL OF THE FIRST KIND T(x);take @g=@t; @g @t/C30/C28(1/C282xt/C27t2)/C282(/C282x/C272t) /C302(t/C28x)(1/C282xt/C27t2)/C282/C30X/C12 n/C300nUn(x)tn/C281:(2) Multiply (2) by t, (2t2/C282xt)(1/C282xt/C27t2)/C282/C30X/C12 n/C300nUn(x)tn(3) and take (3) minus (2),(2t2/C282tx)/C28(1/C282xt/C27t2) (1/C282xt/C27t2)2/C30t2/C281 (1/C282xt/C27t2)2 /C30X/C12 n/C300(n/C281)Un(x)tn: (4) The Rodrigues representation is Un(x)/C30(/C281)n(n/C271)ffiffiffipp 2n/C271(n/C271 2)!(1/C28x2)1=2dn dxn[(1/C28x2)n/C271=2]:(5) The polynomials can also be defined in terms of the sums Un(x)/C30Xn=2bc r/C300(/C281)rn/C28r rl11sl11n (2x)n/C282r /C30Xn=2de m/C300n/C271 2m/C271l11sl11n xn/C282m(x2/C281)m; (6) where xbcis the FLOOR FUNCTION and xdeis the CEILING FUNCTION , or in terms of the product Un(x)/C302nYn k/C301x/C28coskp n/C271 !"# (7) (Zwillinger 1995, p. 696). /Un(x) also obey the interesting DETERMINANT identity Un/C302x100 /C1/C1/C1 00 12 x10:::00 01 2 x1:::00 001 2 x:::00 0001:::10 n:::::::::::::::1 0000 /C1/C1/C1 12 xl112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112: (8) The Chebyshev polynomials of the second kind are a special case of the J ACOBI POLYNOMIALS P(a;b) nwith / a/C30b/C301=2/, Un(x)/C30(n/C271)P(1=2;1=2) n (x) P(1=2;1=2) n (1) /C302F1(/C28n;n/C272;3 2;12(1/C28x)); (9) where2F1(a;b;c;x)i sa HYPERGEOMETRIC FUNCTION (Koekoek and Swarttouw 1998). The first few POLYNOMIALS are U0(x)/C301 U1(x)/C302x U2(x)/C304x2/C281 U3(x)/C308x3/C284x U4(x)/C3016x4/C2812x2/C271 U5(x)/C3032x5/C2832x3/C276x U6(x) /C3064x6 /C2880x4 /C2724x2 /C281 : Letting x /C13cos u allows the Chebyshev polynomials of the second kind to be written as Un(x) /C30sin[(n /C27 1)] u] sin u: (10) The second linearly dependent solution to the trans- formed differential equation is then given by Wn(x) /C30cos[(n /C27 1)u] sin u; (11) which can also be written Wn(x) /C30(1 /C28x2) /C281 =2Tn/C271(x) ; (12) where Tn(x)isaC HEBYSHEV POLYNOMIAL OF THE FIRST KIND . Note that Wn(x) is therefore not a POLYNOMIAL . The triangle of RESULTANTS r(Un(x) ; Uk(x)) is given by f0g;f/C284; 0 g;f0 ;/C2864 ; 0g;f16 ; 256; 4096 ; 0g; f0;0;0;1048576 ;0g;... (Sloane’s A054376). See also CHEBYSHEV APPROXIMATION FORMULA ,CHE- BYSHEV POLYNOMIAL OF THE FIRST KIND,U LTRA- SPHERICAL POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771 /C1/802, 1972. Arfken, G. "Chebyshev (Tschebyscheff) Polynomials" and "Chebyshev Polynomials--Numerical Applications." §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731 /C1/748, 1985. Koekoek, R. and Swarttouw, R. F. "Chebyshev." §1.8.2 in The Askey-Scheme of Hypergeometric Orthogonal Polyno-mials and its q -Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1 /17, pp. 41 /C1/43, 1998. ftp:// www.twi.tudelft.nl/publications/tech-reports/1998/DUT- TWI-98 /C1/17.ps.gz. Koepf, W. "Efficient Computation of Chebyshev Polyno- mials." In Computer Algebra Systems: A Practical Guide (Ed. M. J. Wester). New York: Wiley, pp. 79 /C1/99, 1999. Pegg, E. Jr. "ChebyshevU." http://www.mathpuzzle.com/ ChebyshevU.html. Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990. Sloane, N. J. A. Sequences A054376 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Chebyshev Polynomials T n(x) and Un(x):/" Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193 /C1/207, 1987. Vasilyev, N. and Zelevinsky, A. "A Chebyshev Polyplay- ground: Recurrence Relations Applied to a Famous Set ofFormulas." Quantum 10,2 0/C1 /26, Sept./Oct. 1999. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.Chebyshev Quadrature AG AUSSIAN QUADRATURE -like FORMULA for numer- ical estimation of integrals. It uses WEIGHTING FUNC- TION W(x)/C301 in the interval [ /C281;1] and forces all the weights to be equal. The general FORMULA is g1 /C281f(x)dx/C302 nXn i/C301f(xi): The ABSCISSAS are found by taking terms up to ynin the M ACLAURIN SERIES of sn(y)/C30exp1 2n/C282/C27ln(1/C28y)1/C281 y ! /C27ln(1/C27y)1/C271 y ! "#() ; and then defining Gn(x)/C13xnsn1 x ! : The ROOTS ofGn(x) then give the ABSCISSAS . The first few values are G0(x)/C301 G1(x)/C30x G2(x)/C301 3(3x2/C281) G3(x)/C3012(2x3/C28x) G4(x)/C301 45(45x4/C2830x2/C271) G5(x)/C301 72(72x5/C2860x3/C277x) G6(x)/C301 105(105x6/C28105x4/C2721x2/C281) G7(x)/C301 6480(6480 x7/C287560 x5/C272142 x3/C28149x) G8(x)/C301 42525(42525 x8/C2856700 x6/C2720790 x4/C282220 x2/C2843) G9(x)/C301 22400(22400 x9/C2833600 x7/C2715120 x5/C282280 x3/C2753x): Because the ROOTS are all REAL forn57 and n/C309 only (Hildebrand 1956), these are the only permissi- ble orders for Chebyshev quadrature. The error term is En/C30cnf(n/C271)(j) (n/C271)!nodd cnf(n/C272)(j) (n/C272)!neven ;8 >>>< >>>: where c n/C30g1 /C281xGn(x)dx n odd g1 /C281x2Gn(x)dx n even :8 >>>< >>>: The first few values of c nare 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) gives abscissas up to n/C307 and Hildebrand (1956) up ton/C309. n /xi/ 2 9 0.57735 30 9 0.707107 4 9 0.187592 9 0.794654 50 9 0.374541 9 0.832497 6 9 0.266635 9 0.422519 9 0.866247 70 9 0.323912 9 0.529657 9 0.883862 90 9 0.167906 9 0.528762 9 0.601019 9 0.911589 The ABSCISSAS and weights can be computed analy- tically for small n. n /xi/ 2 /91 3ffiffiffi 3p / 30 /91 2ffiffiffi 2p / 49ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5p /C28 2 3ffiffiffi5ps 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi5p /C27 2 3ffiffiffi5ps 50 9 1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffiffiffiffi 11p 3s 91 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffiffiffiffi 11p 3s See also GAUSSIAN QUADRATURE ,LOBATTO QUADRA- TUREReferences Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 345 /C1/351, 1956. Chebyshev Sum Inequality If a1 ]a2 ]...]an b1 ]b2 ]...]bn ; then nXn k /C301akbk ]Xn k /C301ak !Xn k /C301bk ! : This is true for any distribution. See also CAUCHY’S INEQUALITY ,H O¨ LDER’S INEQUAL- ITIES References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1092, 2000. Hardy, G. H.; Littlewood, J. E.; and Po ´lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 43 /C1/44, 1988. Chebyshev-Gauss Quadrature Also called C HEBYSHEV QUADRATURE .AG AUSSIAN QUADRATURE over the interval [ /C281;1] with WEIGHT- ING FUNCTION W(x)/C30(1/C28x2)/C281=2(Abramowitz and Stegun 1972, p. 889). The ABSCISSAS for quadrature order nare given by the roots of the C HEBYSHEV POLYNOMIAL OF THE FIRST KIND Tn(x);which occur symmetrically about 0. The WEIGHTS are wi/C30/C28An/C271gn AnT?n(xi)Tn/C271(xi)/C30An An/C281gn/C281 Tn/C281(xi)T?n(xi);(1) where Anis the COEFFICIENT ofxninTn(x):For HERMITE POLYNOMIALS , An/C302n/C281; (2) so An/C271 An/C302: (3) Additionally, gn/C301 2p; (4) so wi/C30/C28p Tn/C271(xi)T?n(xi): (5) Since Tn(x)/C30cos(ncos/C281x); (6) the ABSCISSAS are given explicitly by xi/C30cos(2i/C281)p 2n"# : (7) Since T?n(xi)/C30(/C281)i/C271n ai(8) Tn/C271(xi)/C30(/C281)isinai; (9) where ai/C30(2i/C281)p 2n; (10) all the WEIGHTS are wi/C30p n: (11) The explicit FORMULA is then g1 /C281f(x)dxffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C30p nXn k/C301fcos2k/C281 2np !"# /C272p 22n(2n)!f(2n)(j):(12) The following two tables give the numerical and analytic values for the first few points and weights. n /xi// wi/ 290.707107 1.5708 3 0 1.0472 90.866025 1.0472 490.382683 0.785398 90.92388 0.785398 5 0 0.628319 90.587785 0.628319 90.951057 0.628319 2 /91 2ffiffiffi 2p //1 2p/ 30 /13p/ 3 /91 2ffiffiffi 3p //1 3p/ 4 /912ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pp //1 4p/ 4 /91 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pp //1 4p/50 /15p/ 5 /912ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(5/C28ffiffiffi 5p )q //1 5p/ 5 /91 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(5/C27ffiffiffi 5p )q //1 5p/ References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 889, 1972. Bronwin, B. "On the Determination of the Coefficients in Any Series of Sines and Cosines of Multiples of a Variable Angle from Particular Values of that Series." Phil. Mag. 34, 260/C1/268, 1849. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 330 /C1/331, 1956. Tchebicheff, P. "Sur les quadratures." J. de math. pures appliq. 19,1 9/C1/34, 1874. Whittaker, E. T. and Robinson, G. "Chebyshef’s Formulae." §79 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 158 /C1/159, 1967. Chebyshev-Radau Quadrature AG AUSSIAN QUADRATURE -like FORMULA over the interval [ /C281;1] which has WEIGHTING FUNCTION W(x)/C30x:The general FORMULA is g1 /C281xf(x)dx/C30Xn i/C301wi[f(xi)/C28f(/C28xi)]: n /xi// wi/ 1 0.7745967 0.4303315 2 0.5002990 0.2393715 0.8922365 0.2393715 3 0.4429861 0.1599145 0.7121545 0.1599145 0.9293066 0.1599145 4 0.3549416 0.1223363 0.6433097 0.1223363 0.7783202 0.12233630.9481574 0.1223363 References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987. Chebyshev’s Formula CHEBYSHEV- GAUSS QUADRATURE Chebyshev’s Theorem There are at least two theorems known as Cheby- shev’s theorem. The first is BERTRAND’S POSTULATE , and the second is a weak form of the PRIME NUMBER THEOREM stating that the ORDER OF MAGNITUDE of the PRIME COUNTING FUNCTION p(x)is p(x)7x ln x ; where 7 denotes "is ASYMPTOTIC to" (Hardy and Wright 1979, p. 9). See also BERTRAND’S POSTULATE ,PRIME COUNTING FUNCTION ,PRIME NUMBER THEOREM References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Chebyshev-Sylvester Constant In 1891, Chebyshev and Sylvester showed that for sufficiently large x, there exists at least one PRIME NUMBER p satisfying x Bp B(1 /C27 a)x; where a /C300 :092... : Since the PRIME NUMBER THEO- REM shows the above inequality is true for all a > 0 for sufficiently large x, this constant is only of historical interest. References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 22, 1983. ChebyshevT CHEBYSHEV POLYNOMIAL OF THE FIRST KIND ChebyshevU CHEBYSHEV POLYNOMIAL OF THE SECOND KIND Checkerboard CHESSBOARD Checker-Jumping Problem Seeks the minimum number of checkers placed on a board required to allow pieces to move by a sequence of horizontal or vertical jumps (removing the piece jumped over) n rows beyond the forward-most initial checker. The first few cases are 2, 4, 8, 20. It is, however, impossible to reach level five. See also CHECKERSReferences Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 23 /C1/28, 1976. Checkers Schroeppel (1972) estimated that there are about 1012 possible positions. However, this disagrees with the estimate of Jon Schaeffer of 5 /C291020 plausible posi- tions, with 1018 reachable under the rules of the game. Because "solving" checkers may require only the SQUARE ROOT of the number of positions in the search space (i.e., 109), there is hope that some day checkers may be solved (i.e., it may be possible to guarantee a win for the first player to move before the game is even started; Dubuque 1996). Depending on how they are counted, the number of EULERIAN CIRCUITS on an n /C29n checkerboard are either 1, 40, 793, 12800, 193721, ... (Sloane’s A006240) or 1, 13, 108, 793, 5611, 39312, ... (Sloane’s A006239). See also BOARD ,CHECKER- JUMPING PROBLEM ,CHESS- BOARD References Dubuque, W. "Re: number of legal chess positions." math- [email protected] posting, Aug 15, 1996. Hopper, M. Win at Checkers. New York: Dover, 1956. Kraitchik, M. "Chess and Checkers" and "Checkers (Draughts)." §12.1.1 and 12.1.10 in Mathematical Recrea- tions. New York: W. W. Norton, pp. 267 /C1/276 and 284 /C1/ 287, 1942. Parlett, D. S. Oxford History of Board Games. Oxford, England: Oxford University Press, 1999. Schaeffer, J. One Jump Ahead: Challenging Human Supre- macy in Checkers. New York: Springer-Verlag, 1997. Schroeppel, R. Item 93 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 35, Feb. 1972. Sloane, N. J. A. Sequences A006239/M4909 and A006240/ M5271 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Checksum A sum of the digits in a given transmission modulo some number. The simplest form of checksum is a parity bit appended on to 7-bit numbers (e.g., ASCII characters) such that the total number of 1s is always EVEN ("even parity") or ODD ("odd parity"). A signifi- cantly more sophisticated checksum is the CYCLIC REDUNDANCY CHECK (or CRC), which is based on the algebra of polynomials over the integers (mod 2). It is substantially more reliable in detecting transmission errors, and is one common error-checking protocol used in modems. See also CYCLIC REDUNDANCY CHECK ,ERROR- COR- RECTING CODE References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Cyclic Redundancy and Other Checksums." Ch. 20.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 888 /C1/895, 1992. Cheeger’s Finiteness Theorem Consider the set of compact n-RIEMANNIAN MANI- FOLDS M with diameter /(M) 5d; Volume /(M) ]V ; and ½K½5 k where k is the SECTIONAL CURVATURE . Then there is a bound on the number of DIFFEOMORPHISMS classes of this set in terms of the constants n, d, V, and k :/ References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Chefalo Knot A fake KNOT created by tying a SQUARE KNOT , then looping one end twice through the KNOT such that when both ends are pulled, the KNOT vanishes. Chen’s Theorem Every "large" EVEN NUMBER may be written as 2n /C30 p /C27m where p is a PRIME and m /C23 P2is the SET of SEMIPRIMES (i.e., 2-ALMOST PRIMES ). See also ALMOST PRIME ,G OLDBACH CONJECTURE , PRIME NUMBER ,SCHNIRELMANN’S THEOREM ,SEMI- PRIME References Chen, J. R. "On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes." Kexue Tongbao 17, 385 /C1/386, 1966. Chen, J. R. "On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes. I." Sci. Sinica 16, 157 /C1/176, 1973. Chen, J. R. "On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes. II." Sci. Sinica 16, 421 /C1/430, 1978. Hardy, G. H. and Wright, W. M. "Unsolved Problems Con- cerning Primes." Appendix §3in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 415 /C1/416, 1979. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 297, 1996. Rivera, C. "Problems & Puzzles: Conjecture Chen’s Con- jecture.-002." http://www.primepuzzles.net/conjectures/ conj_002.htm. Ross, P. M. "On Chen’s Theorem that Each Large Even Number has the Form /p1 /C27p2/ or /p1 /C27p2p3/." J. London Math. Soc. 10, 500 /C1/506, 1975. Chern Class A GADGET defined for COMPLEX VECTOR BUNDLES . The Chern classes of a COMPLEX MANIFOLD are the Chern classes of its TANGENT BUNDLE . The ith Chern class is an OBSTRUCTION to the existence of (n /C28i /C271) every- where COMPLEX linearly independent VECTOR FIELDSon that VECTOR BUNDLE . The ith Chern class is in the (2i)/th cohomology group of the base SPACE . See also CHERN NUMBER ,OBSTRUCTION ,PONTRYAGIN CLASS,STIEFEL- WHITNEY CLASS Chern Number The Chern number is defined in terms of the CHERN CLASS of a MANIFOLD as follows. For any collection CHERN CLASSES such that their cup product has the same DIMENSION as the MANIFOLD , this cup product can be evaluated on the MANIFOLD ’s FUNDAMENTAL CLASS . The resulting number is called the Chern number for that combination of Chern classes. The most important aspect of Chern numbers is that they are COBORDISM invariant. See also CHERN CLASS ,PONTRYAGIN NUMBER ,STIE- FEL-WHITNEY NUMBER Chernoff Face A way to display nvariables on a 2-D surface. For instance, let xbe eyebrow slant, ybe eye size, zbe nose length, etc. The above figures show faces produced using 10 characteristics–head eccentricity, eye size, eye spacing, eye eccentricity, pupil size,eyebrow slant, nose size, mouth shape, mouth size, and mouth opening)–each assigned one of 10 possible values, generated using Mathematica (S. Dickson). References Dickson, S. "Faces" Mathematica notebook. http:// mathworld.wolfram.com/notebooks/ChernoffFaces.nb. Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, p. 212, 1993. Chess Chess is a game played on an 8 /C298BOARD , called a CHESSBOARD , of alternating black and white squares. Pieces with different types of allowed moves are placed on the board, a set of black pieces in the first two rows and a set of white pieces in the last two rows. The pieces are called the bishop (2), king (1), knight (2), pawn (8), queen (1), and rook (2). The object of the game is to capture the opponent’s king. It is believed that chess was played in India as early as the sixth century AD. Hardy (1999, p. 17) estimated the number of possible games of chess as 101050 : In a game of 40 moves, the number of possible board positions is at least 10120according to Peterson (1996). However, this value does not agree with the 1040 possible positions given by Beeler et al. (1972). This value was obtained by estimating the number of pawn positions (in the no-captures situation, this is 158), times all pieces in all positions, dividing by 2 for each of the (rook, knight) which are interchangeable, dividing by 2 for each pair of bishops (since half the positions will have the bishops on the same color squares). There are more positions with one or two captures, since the pawns can then switch columns (Schroeppel 1996). Shannon (1950) gave the value P(40) :64! 32!(8!)2(2!)6 :1043 : The number of chess games which end in exactly n plies (including games that mate in fewer than n plies) for n /C301, 2, 3, ... are 20, 400, 8902, 197742, 4897256, 120921506, 3284294545, ... (K. Thompson, Sloane’s A006494). Rex Stout’s fictional detective Nero Wolfe quotes the number of possible games after ten moves as follows: "Wolfe grunted. One hundred and sixty-nine million, five hundred and eighteen thousand, eight hundred and twenty-nine followed by twenty-one ciphers. The number of ways the first ten moves, both sides, may be played" (Stout 1983). The number of chess positions after n moves for n /C301, 2, ... are 20, 400, 5362, 71852, 809896?, 9132484?, ... (Schwarzkopf 1994, Sloane’s A019319). Cunningham (1889) incorrectly found 197,299 games and 71,782 positions after the fourth move. C. Flye St. Marie was the first to find the correct number of positions after four moves: 71,852. Dawson (1946) gives the source as Intermediare des Mathematiques (1895), but K. Fabel writes that Flye St. Marie corrected the number 71,870 (which he found in 1895) to 71,852 in 1903. The history of the determina- tion of the chess sequences is discussed in Schwarz- kopf (1994). The analysis of chess is extremely complicated due to the many possible options at each move. Steinhaus (1983, pp. 11 /C1/14), as well as many entire books, consider clever end-game positions which may be analyzed completely. Two problems in recreational mathematics ask1. How many pieces of a given type can be placed on a CHESSBOARD without any two attacking. 2. What is the smallest number of pieces needed tooccupy or attack every square. The answers are given in the following table (Mada-chy 1979). Piece Max. Min. BISHOPS 14 8 KINGS 16 9 KNIGHTS 32 12 QUEENS 85 ROOKS 88 See also BISHOPS PROBLEM ,BOARD ,CHECKERBOARD , CHECKERS ,FAIRY CHESS ,G O,G OMORY’S THEOREM , HARD HEXAGON ENTROPY CONSTANT ,K INGS PRO- BLEM ,K NIGHT’S TOUR,M AGIC TOUR,Q UEENS PRO- BLEM ,ROOKS PROBLEM ,TOUR References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 124 /C1/127, 1987. Beeler, M. et al. Item 95 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 35, Feb. 1972. Culin, S. "Tjyang-keui--Chess." §82 in Games of the Orient: Korea, China, Japan. Rutland, VT: Charles E. Tuttle, pp. 82 /C1/91, 1965. Dawson, T. R. "A Surprise Correction." The Fairy Chess Review 6, 44, 1946. Dickins, A. "A Guide to Fairy Chess." p. 28, 1967/1969/1971. Dudeney, H. E. "Chessboard Problems." Amusements in Mathematics. New York: Dover, pp. 84 /C1/109, 1970. Fabel, K. "Nu ¨sse." Die Schwalbe 84, 196, 1934. Fabel, K. "Weihnachtsnu ¨sse." Die Schwalbe 190, 97, 1947. Fabel, K. "Weihnachtsnu ¨sse." Die Schwalbe 195, 14, 1948. Fabel, K. "Ero ¨ffnungen." Am Rande des Schachbretts ,3 4/C1/ 35, 1947. Fabel, K. "Die ersten Schritte." Rund um das Schachbrett , 107/C1/109, 1955. Fabel, K. "Ero ¨ffnungen." Schach und Zahl 8, 1966/1971. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 86 /C1/89, 1975. Kraitchik, M. "Chess and Checkers." §12.1.1 in Mathema- tical Recreations. New York: W. W. Norton, pp. 267 /C1/276, 1942. Lasker, E. Lasker’s Manual of Chess. New York: Dover, 1960. Madachy, J. S. "Chessboard Placement Problems." Ch. 2 in Madachy’s Mathematical Recreations. New York: Dover, pp. 34 /C1/54, 1979. Parlett, D. S. Oxford History of Board Games. Oxford, England: Oxford University Press, 1999. Peterson, I. "The Soul of a Chess Machine: Lessons Learned from a Contest Pitting Man Against Computer." Sci. News 149, 200 /C1/201, Mar. 30, 1996. Petkovic, M. Mathematics and Chess. New York: Dover, 1997. Schroeppel, R. "Reprise: Number of legal chess positions." [email protected] posting, Aug. 18, 1996. Schwarzkopf, B. "Die ersten Zu¨ge." Problemkiste , 142 /C1/143, No. 92, Apr. 1994. Shannon, C. "Programming a Computer for Playing Chess." Phil. Mag. 41, 256 /C1/275, 1950. Sloane, N. J. A. Sequences A006494, A007545/M5100, and A019319 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 11 /C1/14, 1999. Stout, R. "Gambit." In Seven Complete Nero Wolfe Novels. New York: Avenic Books, p. 475, 1983. Velucchi, M. "Some On-Line PostScript MathChess Papers." http://anduin.eldar.org/~problemi/papers.html. Chessboard A board containing 8 /C298 squares alternating in color between black and white on which the game of CHESS is played. The checkerboard is identical to the chess- board except that chess’s black and white squares are colored red and white in CHECKERS . It is impossible to cover a chessboard from which two opposite corners have been removed with DOMINOES . Sprague (1963) considered the problem of "rolling" five cubes, each which an upright letter "A" on its top, on a chessboard. Here "rolling" means the cubes are moved from square to adjacent square by being tipped over along an edge (as one might move a heavy box) in a series of quarter turns. If five such cubes are initially arranged in the shape of a plus sign with the edges of the of plus sign aligned with the upper and left corners of a chessboard (top left in above figure), then it is impossible to obtain a straight row or column with all "A"s on top and oriented identi- cally. The best that can be done is to place four out of the five "A"s in the same orientation and facing upward, with the remaining "A" also facing upward and rotated a quarter turn, illustrated above in the bottom row (Gardner 1984, pp. 75 /C1/78). The above plot shows a chessboard centered at (0, 0) and its INVERSE about a small circle also centered at (0, 0) (Gardner 1984, pp. 244 /C1/245; Dixon 1991). See also CHECKERS ,CHESS ,CIRCULAR CHESSBOARD , DOMINO ,G OMORY’S THEOREM ,INVERSION ,K INGS PROBLEM ,K NIGHTS PROBLEM ,K NIGHT’S TOUR, QUEENS PROBLEM ,R OOKS PROBLEM ,W HEAT AND CHESSBOARD PROBLEM References Dixon, R. "Inverse Points and Mid-Circles." §1.6 in Matho- graphics. New York: Dover, pp. 62 /C1/73, 1991. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984. Pappas, T. "The Checkerboard." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 136 and 232, 1989. Sprague, R. Recreations in Mathematics: Some Novel Puz- zles. London: Blackie and Sons, 1963. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 29 /C1/30, 1999. Chevalley Groups Finite SIMPLE GROUPS of L IE-TYPE . They include four families of linear SIMPLE GROUPS :PSL(n;q); PSU (n;q);PSp(2n;q);orPVe(n;q):/ See also TWISTED CHEVALLEY GROUPS References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#exc. Chevalley’s Theorem Let f(x) be a member of a FINITE FIELD F[x1 ; x2 ...; xn] and suppose f(0; 0; ...; 0) /C300 and n is greater than the degree of f, then f has at least two zeros in An(F) :/ References Chevalley, C. "De´monstration d’une hypothe `se de M. Artin." Abhand. Math. Sem. Hamburg 11,73/C1/75, 1936. Ireland, K. and Rosen, M. "Chevalley’s Theorem." §10.2 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 143 /C1/144, 1990. Chevron A6- POLYIAMOND . References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Chi The Chi function is defined by Chi(z) /C30 g /C27ln z /C27gz 0cosh t /C28 1 tdt; where g is the EULER- MASCHERONI CONSTANT . Thefunction is given by the Mathematica command CoshIntegral [z]. See also COSINE INTEGRAL ,SHI,SINE INTEGRAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Cosine Integrals." §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231 /C1/233, 1972. Chi Distribution The probability density function and cumulative distribution function are Pn(x) /C3021 /C28n=2xn/C281e/C28x2 =2 G(1 2 n) (1) Dn(x) /C30Q(12 n;12 x2); (2) where Q is the REGULARIZED GAMMA FUNCTION . m /C30ffiffiffi 2p G(1 2(n /C27 1)) G(1 2 n) (3) s2 /C302[G(1 2 n) G(1 /C2712 n) /C28G2(12(n /C27 1))] G2(12 n) (4) g1 /C302G3(1 2(n /C27 1)) /C28 3 G(12 n) G(12(n /C27 1))G(1 /C2712 n) [ G(1 2 n) G(1 /C2712 n) /C28G2(12(n /C27 1))]3 =2 /C27G2(1 2 n)G3 /C27 n 2 ! [G(1 2 n)G(1 /C2712 n) /C28G2(12(n /C27 1))]3 =2 (5) g2 /C30/C283G4(1 2(n /C27 1)) /C27 6 G(12 n) G2(12(n /C27 1))G(1 /C2712 n) G(12 n) G2 /C27 n 2 ! /C28G2(12(n /C27 1))"#2 /C27/C284 G2(1 2n)G(12(n /C27 1)) G3 /C27 n 2 ! /C27G3(12n) G4 /C27 n 2 ! G(12n) G2 /C27 n 2 ! /C28G2(12(n /C27 1))"#2 ; (6) where m is the MEAN , s2the VARIANCE ,g1the SKEWNESS , and g2the KURTOSIS . For n/C301, the x distribution is a HALF-NORMAL DISTRIBUTION with u/C30 1:Forn/C302, it is a R AYLEIGH DISTRIBUTION with s/C301:/ See also CHI-SQUARED DISTRIBUTION ,HALF-NORMAL DISTRIBUTION ,RAYLEIGH DISTRIBUTION Chi Inequality The inequality (j /C271)aj /C27ai ](j /C271)i; which is satisfied by all A-SEQUENCE . References Levine, E. and O’Sullivan, J. "An Upper Estimate for the Reciprocal Sum of a Sum-Free Sequence." Acta Arith. 34, 9 /C1/24, 1977. Child A node which is one EDGE further away from a given node in a ROOTED TREE . See also ROOT NODE,ROOTED TREE,SIBLING Chinese Hypothesis A PRIME p always satisfies the condition that 2p /C282is divisible by p. However, this condition is not true exclusively for PRIMES (e.g., 2341 /C282 is divisible by 341 /C3011 /C215 31): COMPOSITE NUMBERS n (such as 341) for which 2n /C282 is divisible by n are called POULET NUMBERS , and are a special class of FERMAT PSEUDO- PRIMES . The Chinese hypothesis is a special case of FERMAT’S LITTLE THEOREM . See also CARMICHAEL NUMBER ,EULER’S THEOREM , FERMAT’S LITTLE THEOREM ,FERMAT PSEUDOPRIME , POULET NUMBER ,PSEUDOPRIME References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 19 /C1/20, 1993. Chinese Postman Problem A problem asking for the shortest tour of a graph which visits each edge at least once (Kwan 1962; Skiena 1990, p. 194). For an EULERIAN GRAPH ,an EULERIAN CIRCUIT is the optimal solution. In a TREE , however, the path crosses each twice. See also EULERIAN CIRCUIT ,TRAVELING SALESMAN PROBLEM References Edmonds, J. and Johnson, E. L. "Matching, Euler Tours, and the Chinese Postman." Math. Programm. 5,88/C1/124, 1973. Kwan, M. K. "Graphic Programming Using Odd or Even Points." Chinese Math. 1, 273 /C1/277, 1962. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Chinese Remainder Theorem Let r and s be POSITIVE INTEGERS which are RELA- TIVELY PRIME and let a and b be any two INTEGERS . Then there is an INTEGER N such thatN /C13a (mod r) (1) and N /C13b (mod s) : (2) Moreover, N is uniquely determined modulo rs.An equivalent statement is that if (r ; s) /C301; then every pair of RESIDUE CLASSES modulo r and s corresponds to a simple RESIDUE CLASS modulo rs. The theorem can also be generalized as follows. Given a set of simultaneous CONGRUENCES x /C13ai (mod mi) (3) for i /C301, ..., r and for which the miare pairwise RELATIVELY PRIME , the solution of the set of CON- GRUENCES is x /C30a1b1M m1/C27.../C27arbrM mr(mod M) ; (4) where M /C13m1m2 /C1/C1/C1mr (5) and the bi are determined from biM mi/C131 (mod mi): (6) References Ireland, K. and Rosen, M. "The Chinese Remainder Theo- rem." §3.4 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 34 /C1/38, 1990. Se´roul, R. "The Chinese Remainder Theorem." §2.6 in Programming for Mathematicians. Berlin: Springer-Ver- lag, pp. 12 /C1/14, 2000. Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, pp. 189 /C1/191, 1939. Wagon, S. "The Chinese Remainder Theorem." §8.4 in Mathematica in Action. New York: W. H. Freeman, pp. 260 /C1/263, 1991. Chinese Rings BAGUENAUDIER Chiral Having forms of different HANDEDNESS which are not mirror-symmetric. See also DISSYMMETRIC ,ENANTIOMER ,HANDEDNESS , MIRROR IMAGE ,REFLEXIBLE Chiral Knot A chiral knot is a KNOT which is not capable of being continuously deformed into its own MIRROR IMAGE . See also AMPHICHIRAL KNOT,KNOT SYMMETRY Chi-Squared Distribution Ax2distribution is a GAMMA DISTRIBUTION with u/C132 anda/C13r=2;where ris the number of DEGREES OF FREEDOM .I fYihave NORMAL INDEPENDENT distribu- tions with MEAN 0 and VARIANCE 1, then x2/C13Xr i/C301Y2 i (1) is distributed as x2with rDEGREES OF FREEDOM .I fx2 i are independently distributed according to a x2 distribution with r1;r2;...,rkDEGREES OF FREEDOM , then Xk j/C301x2 j (2) is distributed according to x2with r/C13ak j/C301rjDEGREES OF FREEDOM . The probability density function is Pr(x)/C30xr=2/C281e/C28x=2 G(1 2r)2r=2(3) for /x/C23[0;/C12)/. The cumulative distribution function is then Dr(x2)/C30gx2 0tr=2/C281e/C28t=2dt G(1 2r)2r=2/C30g(1 2r;12x2) G(1 2r) /C30P(1 2r;12x2); (4) where P(a;z)i sa REGULARIZED GAMMA FUNCTION . The CONFIDENCE INTERVALS can be found by finding the value of xfor which Dr(x) equals a given value. The MOMENT-GENERATING FUNCTION of the x2distri- bution is M(t)/C30(1/C282t)/C28r=2(5) R(t)/C13lnM(t)/C30/C281 2rln(1/C282t) (6) R?(t)/C30r 1/C282t(7) Rƒ(t)/C302r (1/C282t)2; (8) so m/C30R?(0)/C30r (9) s2/C30Rƒ(0)/C302r (10) g1/C302ffiffiffi 2 rs (11) g2/C3012 r: (12) ThenthMOMENT about zero for a distribution with r DEGREES OF FREEDOM ism?n/C302nG(n/C2712r) G(1 2r)/C30r(r/C272)/C1/C1/C1(r/C272n/C282); (13) and the moments about the MEAN are m2/C302r (14) m3/C308r (15) m4/C3012r(r/C274): (16) The nthCUMULANT is kn/C302nG(n)(12r)/C302n/C281(n/C281)!r: (17) The MOMENT-GENERATING FUNCTION is M(t)/C30ert=ffiffiffiffi 2rp 1/C282tffiffiffiffiffi 2rp !/C28r=2 /C30etffiffiffiffiffi 2=rp 1/C28ffiffiffi 2 rs t !"# /C28r=2 /C301/C28t2 r/C281 32 r !3=2 t3/C28...2 435/C28r=2 : (18) Asr0/C12; lim r0/C12M(t)/C30et2=2; (19) so for large r, ffiffiffiffiffiffiffiffi 2x2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X i(xi/C28mi)2 s2 ivuut(20) is approximately a GAUSSIAN DISTRIBUTION with MEANffiffiffiffiffi 2rp and VARIANCE s2/C301:Fisher showed that x2/C28rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2r/C281p (21) is an improved estimate for moderate r. Wilson and Hilferty showed that x2 r !1=3 (22) is a nearly G AUSSIAN DISTRIBUTION with MEAN m/C30 1/C282=(9r) and VARIANCE s2/C302=(9r):/ In a G AUSSIAN DISTRIBUTION , P(x)dx/C301 sffiffiffiffiffiffi 2pp e/C28(x/C28m)2=2s2dx; (23) let z/C13(x/C28m)2=s2: (24) Then dz /C302(x /C28 m)2 s2dx /C302ffiffiffizp sdx (25) so dx /C30s 2ffiffiffizp dz : (26) But P(z) dz /C302P(x) dx; (27) so P(x) dx /C3021 sffiffiffiffiffiffi 2pp e /C28z=2 dz /C301 sffiffiffipp e /C28z=2 dz: (28) This is a x2 distribution with r /C301, since P(z) dz /C30z1 =2 /C281e /C28z=2 G(1 2)21=2dz /C30x/C281=2e /C281 =2 ffiffiffiffiffiffi 2pp dz : (29) If Xiare independent variates with a NORMAL DISTRIBUTION having MEANS mi and VARIANCES s2 ifor i /C301, ..., n, then 1 2 x2 /C13Xn i/C301(xi /C28 mi)2 2s2 i(30) is a GAMMA DISTRIBUTION variate with a /C30n =2; P(1 2 x2)d(12 x2) /C301 G(1 2 n) e /C28 x2 =2(1 2 x2)(n=2)/C281d(12 x2) : (31) The noncentral chi-squared distribution is given by P(x) /C302/C28n=2e /C28(l/C27x)=2xn=2 /C281F(1 2 n;14 lx) ; (32) where F(a; z) /C130F1(; a; z) G(a); (33) /0F1is the CONFLUENT HYPERGEOMETRIC LIMIT FUNC- TION and G is the GAMMA FUNCTION . The MEAN , VARIANCE ,SKEWNESS , and KURTOSIS are m/C30l/C27n (34) s2/C302(2l/C27n) (35) g1/C302ffiffiffi 2p (3l/C27n) (2l/C27n)3=2(36) g2/C3012(4l/C27n) (2l/C27n)2: (37) See also CHI DISTRIBUTION ,SNEDECOR’S F-DISTRIBU- TION ,STATISTICAL DISTRIBUTIONReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 940 /C1/943, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 535, 1987. Kenney, J. F. and Keeping, E. S. "The Chi-Square Distribu- tion." §5.3 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 98 /C1/100, 1951. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Incomplete Gamma Function, Error Function,Chi-Square Probability Function, Cumulative PoissonFunction." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209 /C1 /214, 1992. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 115 /C1/116, 1992. Chi-Squared Test Let the probabilities of various classes in a distribu- tion be p1;p2;...,pk;with means m1;m2;.... The expected frequency x2 s/C30Xk i/C301(mi/C28Npi)2 Npi is a measure of the deviation of a sample from expectation. Karl Pearson proved that the limiting distribution of x2 sisx2(Kenney and Keeping 1951, pp. 114 /C1/116). Pr(x2]x2 s)/C30g/C12 x2 sf(x2)d(x2) /C301 2g/C12 x2 sx2 2 !(k/C283)=2 Gk/C281 2 ! e/C28x2=2d(x2) /C301/C28G1 2x2 s;k/C281 2 ! Gk/C281 2 ! /C301/C28Ix2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(k/C281)p ;k/C283 2 ! ; where I(x;n)i sP EARSON’S FUNCTION . There are some subtleties involved in using the x2test to fit curves (Kenney and Keeping 1951, pp. 118 /C1/119). When fitting a one-parameter solution using x2;the best-fit parameter value can be found by calculating x2at three points, plotting against the parameter values of these points, then finding the minimum of a PARABOLA fit through the points (Cuzzi 1972, pp. 162 /C1/168). See also CHI-SQUARED DISTRIBUTION References Cuzzi, J. The Subsurface Nature of Mercury and Mars from Thermal Microwave Emission. Ph.D. Thesis. Pasadena, CA: California Institute of Technology, 1972. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Chmutov Surface An ALGEBRAIC SURFACE with affine equation Pd(x1 ; x2) /C27Td(x3) /C300; (1) where Td(x)isaC HEBYSHEV POLYNOMIAL OF THE FIRST KIND and Pd(x1 ; x2) is a polynomial defined by Pd(x1 ; x2) /C30x1 10 /C1/C1/C1 000 2x2x11::: 000 3 x2x1::::::::: n 01 x2::: 100 001::: x110 n::::::::: x2x11 000 /C1/C1/C1 1 x2x1l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112 /C27x 2 10 /C1/C1/C1 000 2x1x21::: 000 3 x1x2::::::::: n 01 x1::: 100 001::: x210 n::::::::: x1x21 000 /C1/C1/C1 1 x1x2l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112; (2) where the matrices have dimensions d /C29d: These represent surfaces in CP 3 with only ORDINARY DOU- BLE POINTS as singularities. The first few surfaces are given by x /C27y /C27z /C300 (3) x2 /C27y2 /C272z2 /C301 /C272x /C272y (4) 6 /C27x3 /C27y3 /C274z3 /C303(2xy /C27z): (5) The dth order such surface has N(d) /C301 12(5d3 /C2813d2 /C2712d)i f d /C130 (mod 6) 1 12(5d3 /C2813d2 /C2716d /C288) if d /C132; 4 (mod 6) 1 12(5d3 /C2813d2 /C2713d /C284) if d /C131; 5 (mod 6) 1 12(5d3 /C2814d2 /C279d)i f d /C133 (mod 6)8 >>>>< >>>>: singular points (Chmutov 1992), giving the sequence 0, 1, 3, 14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... for d /C301, 2, .... For a number of orders d, Chmutov surfaces have more ordinary double points than any other known equations of the same degree. Based on Chmutov’s equations, Banchoff (1991) defined the simpler set of surfaces Tn(x) /C27Tn(y) /C27Tn(z) /C300 ; (6) where n is EVEN and Tn(x) is again a CHEBYSHEV POLYNOMIAL OF THE FIRST KIND . For example, the surfaces illustrated above have orders 2, 4, and 6 are given by the equations 2(x2 /C27y2 /C27z2) /C303 (7) 3 /C278(x4 /C27y4 /C27z4) /C308(x2 /C27y2 /C27z2) (8) 2[x2(3 /C284x2)2 /C27y2(3 /C284y2)2 /C27z2(3 /C284z2)2] /C303: (9) See also GOURSAT’S SURFACE ,O RDINARY DOUBLE POINT ,SUPERELLIPSE References Banchoff, T. F. "Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order." In Geo- metric Analysis and Computer Graphics: Proceedings of a Workshop Held May 23 /C1/25, 1988 (Eds. P. Concus, R. Finn, D. A. Hoffman). New York: Springer-Verlag, pp. 31 /C1/37, 1991. Chmutov, S. V. "Examples of Projective Surfaces with Many Singularities." J. Algebraic Geom. 1, 191 /C1/196, 1992. Hirzebruch, F. "Singularities of Algebraic Surfaces and Characteristic Numbers." In The Lefschetz Centennial Conference, Part I: Proceedings of the Conference on Algebraic Geometry, Algebraic Topology, and Differential Equations, Held in Mexico City, December 10 /C1/14, 1984 (Ed. S. Sundararaman). Providence, RI: Amer. Math. Soc., pp. 141 /C1/155, 1986. Trott, M. The Mathematica Guidebook, Vol. 2: Graphics. New York: Springer-Verlag, 2000. Choice Axiom AXIOM OF CHOICE Choice Number COMBINATION Cholesky Decomposition Given a symmetric POSITIVE DEFINITE MATRIX A ; the Cholesky decomposition is an UPPER TRIANGULAR MATRIX U such that A /C30UTU : Cholesky decomposition is implemented as Choles- kyDecomposition [m] in the Mathematica add-on packageLinearAlgebra‘Cholesky‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also LU DECOMPOSITION ,M ATRIX DECOMPOSI- TION ,QRD ECOMPOSITION References Gentle, J. E. "Cholesky Factorization." §3.2.2 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer-Verlag, pp. 93 /C1/95, 1998. Nash, J. C. "The Choleski Decomposition." Ch. 7 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 84 /C1/93, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Cholesky Decomposition." §2.9 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 89 /C1/91, 1992. Choose An alternative term for a BINOMIAL COEFFICIENT ,in whichn kl1ml11 is read as "n choose k." R. K. Guy sug- gested this pronunciation around 1950, when the notationsnCrandnCrwere commonly used. Leo Moser liked the pronunciation and he and others spread it around. It got the final seal of approval from Donald Knuth when he incorporated it into the TEX mathematical typesetting language as fn_choose k g:/ See also BINOMIAL COEFFICIENT ,MULTICHOOSE Choquet Theory Erdos proved that there exist at least one PRIME OF THE FORM 4k /C271 and at least one PRIME OF THE FORM 4k /C273 between n and 2n for all n /C216. See also EQUINUMEROUS ,PRIME NUMBER Chord The LINE SEGMENT joining two points on a curve. The term is often used to describe a LINE SEGMENT whose ends lie on a CIRCLE . In the above figure, r is the RADIUS of the CIRCLE , a is called the APOTHEM , and s the SAGITTA . The shaded region in the left figure is called a SECTOR , and the shaded region in the right figure is called a SEGMENT . All ANGLES inscribed in a CIRCLE and subtended by the same chord are equal. The converse is also true: The LOCUS of all points from which a given segment subtends equal ANGLES is a CIRCLE .Given any closed convex curve, it is possible to find a point P through which three chords, inclined to one another at angles of 60 8, pass such that P is the MIDPOINT of all three (Wells 1991). Let a CIRCLE of RADIUS R have a CHORD at distance r. The AREA enclosed by the CHORD , shown as the shaded region in the above figure, is then A /C302gffiffiffiffiffiffiffiffiffiffi R2 /C28r2p 0x(y) dy: (1) But y2 /C27(r /C27x)2 /C30R2 ; (2) so x(y) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28y2p /C28r (3) and A /C302gffiffiffiffiffiffiffiffiffiffi R2 /C28r2p 0(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2 /C28y2p /C28r) dy (4) /C30R2 tan/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R r !2 /C281vuut2 643 75/C28rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28r2p : (5) Checking the limits, when r /C30R, A /C300 and when r 0 0; A /C301 2 pR2 ; (6) the expected area of the SEMICIRCLE . See also ANNULUS ,APOTHEM ,BERTRAND’S PROBLEM , CONCENTRIC CIRCLES ,HOLDITCH’S THEOREM ,RADIUS , SAGITTA ,SECTOR ,SEGMENT ,SEMICIRCLE References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 29, 1991. Chord Diagram See also ALGEBRA OF CHORD DIAGRAMS ,KONTSEVICH INTEGRAL Chordal RADICAL AXIS Chordal Theorem The LOCUS of the point at which two given CIRCLES possess the same POWER is a straight line PERPENDI- CULAR to the line joining the MIDPOINTS of the CIRCLE and is known as the chordal (or, more commonly, the RADICAL AXIS) of the two CIRCLES . See also POWER (CIRCLE ), RADICAL LINE References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 153, 1965. Chow Coordinates A generalization of GRASSMANN COORDINATES to m-D ALGEBRAIC VARIETIES of degree d in Pn ; where Pn is an n-D projective space. To define the Chow coordi- nates, take the intersection of an m-D ALGEBRAIC VARIETY Z of degree d by an (n /C28m)/-D SUBSPACE U of Pn : Then the coordinates of the d points of intersec- tion are algebraic functions of the GRASSMANN CO- ORDINATES of U, and by taking a symmetric function of the algebraic functions, a HOMOGENEOUS POLYNO- MIAL known as the Chow form of Z is obtained. The Chow coordinates are then the COEFFICIENTS of the Chow form. Chow coordinates can generate the smallest field of definition of a divisor. See also CHOW RING,CHOW VARIETY References Chow, W.-L. and van der Waerden., B. L. "Zur algebraische Geometrie IX." Math. Ann. 113, 692 /C1/704, 1937. Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 43, 1117 /C1/1124, 1996. Chow Ring The intersection product for classes of rational equivalence between cycles on an ALGEBRAIC VARIETY . See also CHOW COORDINATES ,CHOW VARIETY References Chow, W.-L. "On Equivalence Classes of Cycles in an Algebraic Variety." Ann. Math. 64, 450 /C1/479, 1956. Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 43, 1117 /C1/1124, 1996.Chow Variety The set Cn; m; d of all m-D varieties of degree d in an n-D projective space Pn into an M-D projective space PM:/ See also CHOW COORDINATES ,CHOW RING References Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 43, 1117/C1/1124, 1996. Christoffel Formula Let fpn(x)gbe orthogonal POLYNOMIALS associated with the distribution da(x) on the interval [ a, b]. Also let r/C13c(x/C28x1)(x/C28x2)/C1/C1/C1(x/C28xl) (for c"0) be a POLYNOMIAL of order lwhich is NONNEGATIVE in this interval. Then the orthogonal polynomials fq(x)gassociated with the distribution r(x)da(x) can be represented in terms of the poly- nomials pn(x)a s r(x)qn(x)/C30pn(x)pn/C271(x)/C1/C1/C1 pn/C27l(x) pn(x1)pn/C271(xl)/C1/C1/C1 pn/C27l(x1) nn:::n pn(xl)pn/C271(xl)/C1/C1/C1 pn/C27l(xl)l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112: In the case of a zero x kof multiplicity m/C211, we replace the corresponding rows by the derivatives of order 0, 1, 2, ..., m/C281 of the POLYNOMIALS pn(xl);..., pn/C27l(xl)a tx/C30xk:/ References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 29 /C1/0, 1975. Christoffel Number One of the quantities liappearing in the G AUSS- JACOBI MECHANICAL QUADRATURE . They satisfy l1/C27l2/C27.../C27ln/C30gb ada(x)/C30a(b)/C28a(a) (1) and are given by ln/C30gb apn(x) p?n(xn)(x/C28xn)"#2 da(x) (2) ln/C30/C28kn/C271 kn1 pn/C271(xn)p?n(xn)(3) /C30kn kn/C2811 pn/C281(xn)P?n(xn)(4) (ln)/C281/C30[p0(xn)]2/C27.../C27[pn(xn)]2; (5) where knis the higher COEFFICIENT ofpn(x):/ See also COTES NUMBER ,HERMITE’S INTERPOLATING POLYNOMIAL References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 47 /C1/8, 1975. Christoffel Symbol The Christoffel symbols are TENSOR -like objects de- rived from a RIEMANNIAN METRIC g. They are used to study the geometry of the metric and appear, for example, in the GEODESIC EQUATION . There are two closely related kinds of Christoffel symbols, the FIRST KIND Gi; j; k ; and the SECOND KIND Gk i; j :/ It is always possible to pick a coordinate system on a RIEMANNIAN MANIFOLD such that the Christoffel symbol vanishes at a chosen point. In general relativity, Christoffel symbols are "gravitational forces," and the preferred coordinate system referred to above would be one attached to a body in free fall. See also CHRISTOFFEL SYMBOL OF THE FIRST KIND, CHRISTOFFEL SYMBOL OF THE SECOND KIND,GEODE- SIC,LEVI-CIVITA CONNECTION ,RIEMANNIAN GEOME- TRY References Carmo, M. Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall, pp. 441 /C1/42, 1976. Sternberg, S. Differential Geometry. New York: Chelsea, pp. 353 /C1/54, 1983. Christoffel Symbol of the First Kind The first type of TENSOR derived from a RIEMANNIAN METRIC g which is used to study the geometry of the metric. Christoffel symbols of the first kind are variously denoted [ij, k],ij k/C138;l12 Gabc ; or fab ; c g: [ij; k] /C30gmk Gmij (1) /C30gmk /C0em /C215@ /C0ei @qi (2) /C30 /C0ek/C215@ /C0ei @qj ; (3) where gmk is the METRIC TENSOR , Gmijis a CHRISTOFFEL SYMBOL OF THE SECOND KIND , and /C0ei /C13@ /C0r @qi /C30hi ˆei : (4) But @gij @qk /C30@ @qk ( /C0ei /C215 /C0ej) /C30@ /C0ei @qk /C215 /C0ej /C27 /C0ei /C215@ /C0ej @qk /C30[ik; j] /C27[jk ; i] ; (5) so[ab ; c] /C301 2(gac;b/C27gbc;a/C28gab;c): (6) See also CHRISTOFFEL SYMBOL ,CHRISTOFFEL SYMBOL OF THE SECOND KIND References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 160 /C1/67, 1985. Christoffel Symbol of the Second Kind The second type of TENSOR -like object derived from a RIEMANNIAN METRIC gwhich is used to study the geometry of the metric. Christoffel symbols of the second kind are variously denoted asm ijno orGm ij:In the latter case, they are sometimes known as connec- tion coefficients. Gm ij/C13 /C0em/C215@ /C0ei @qj(1) /C30gkm[ij;k] (2) /C301 2gkm @gik @qj/C27@gjk @qi/C28@gij @qk ! ; (3) where gkmis the METRIC TENSOR . The Christoffel symbol of the second kind is related to the C HRIS- TOFFEL SYMBOL OF THE FIRST KIND [bc, d ]b y Ga bc/C30gadfbc;dg: (4) Christoffel symbols of the second kind can also be defined by G/C0ea /C0eb /C0eg/C13 /C0ea/C215(9/C0eg /C0eb) (5) (long form) or Ga bg/C13 /C0ea/C215(9g /C0eb); (6) (abbreviated form), and satisfy 9/C0eg /C0eb/C30G/C0ea /C0eb /C0eg /C0ea (7) (long form) and 9g /C0eb/C30Ga bg /C0ea (8) (abbreviated form). Christoffel symbols of the second kind are not TENSORS , but have TENSOR -like CONTRAVARIANT and COVARIANT indices. Christoffel symbols of the second kind also do not transform as tensors. In fact, changing coordinates from x1;...;xntoy1;...;yn gives Gk? ij/C30X @2xl @yi@yj@yk @xl/C27X GT rs@xr @yi@xs @yj@yk @xt: (9) However, a fully COVARIANT Christoffel symbol of the second kind is given by Gabg/C131 2(gab;g/C27gag;b/C27cabg/C27cagb/C28cbga); (10) where the gs are the METRIC TENSORS , the cs are COMMUTATION COEFFICIENTS , and the commas indi- cate the COMMA DERIVATIVE .I na n ORTHONORMAL BASIS ,gab;g/C300 and gmg/C30dmg;so Gabg/C30Gm abgmg/C30Gmab/C301 2(cabg/C27cagb/C28cbga) (11) and Gijk/C300 for i"j"k (12) Giik/C30/C281 2@gii @xkfori"k (13) Giji/C30Gjii/C301 2@gii @xj(14) Gk ij/C300 for i"j"k (15) Gkii/C30/C281 2gkk@gii @xkfori"k (16) Giij/C30Giji/C301 2gii@gii @xj/C301 2@lngii @xj: (17) For TENSORS ofRANK 3, the Christoffel symbols of the second kind may be concisely summarized in MATRIX form: Gu/C13Gu rrGuruGurf GuurGuuuGuuf GufrGufuGuff2 643 75: (18) The Christoffel symbols are given in terms of the coefficients of the FIRST FUNDAMENTAL FORM E,F, andGby G1 11/C30GEu/C282FFu/C27FEv 2(EG/C28F2)(19) G112/C30GEv/C28FGu 2(EG/C28F2)(20) G122/C302GFv/C28GGu/C28FGv 2(EG/C28F2)(21) G211/C302EFu/C28EEv/C28FEu 2(EG/C28F2)(22) G212/C30EGu/C28FEv 2(EG/C28F2)(23) G222/C30EGv/C282FFv/C27FGu 2(EG/C28F2); (24) andG121/C30G112andG221/C30G212:IfF/C300, the Christoffel symbols of the second kind simplify toG111/C30Eu 2E(25) G112/C30Ev 2E(26) G122/C30/C28Gu 2E(27) G211/C30/C28Ev 2G(28) G212/C30Gu 2G(29) G222/C30Gv 2G(30) (Gray 1997). The following relationships hold between the Chris- toffel symbols of the second kind and coefficients of the first FUNDAMENTAL FORM , G1 11E/C27G211F/C301 2Eu (31) G1 12E/C27G212F/C301 2Ev (32) G1 22E/C27G222F/C30Fv/C281 2Gu (33) G1 11F/C27G211G/C30Fu/C281 2Ev (34) G1 12F/C27G212G/C301 2Gu (35) G1 22F/C27G222G/C301 2Gv (36) G1 11/C27G212/C30(lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG/C28F2p )u (37) G1 12/C27G222/C30(lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG/C28F2p )v (38) (Gray 1997). For a surface given in M ONGE’S FORM z/C30F(x;y); Gk ij/C30zijzk 1/C27z2 1/C27z22: (39) Christoffel symbols of the second kind arise in the computation of GEODESICS . The GEODESIC EQUATION of free motion is dt2/C30/C28habdjadjb; (40) or d2ja dt2/C300: (41) Expanding, d dt@ja @xmdxm dt ! /C30@ja @xmd2xm dt2/C27@2ja @xm@xndxm dtdxn dt/C300 (42) @ ja @xmd2xm dt2@xl @ ja /C27@2 ja @xm @xndxm d tdxn dt@xl @ ja /C300: (43) But @ ja @xn@x l @ ja /C30 dl m ; (44) so dlmd2xm dt2 /C27@2 ja @xm @xn@xl @ ja ! dx m dtdxn dt /C30d2xl dt2 /C27G l mndx m dtdxn dt; (45) where Glmn /C13@2ja @xm@xn@xl @ja: (46) See also CARTAN TORSION COEFFICIENT ,CHRISTOFFEL SYMBOL ,CHRISTOFFEL SYMBOL OF THE FIRST KIND, COMMA DERIVATIVE ,C OMMUTATION COEFFICIENT , CONNECTION COEFFICIENT ,GAUSS EQUATIONS ,SEMI- COLON DERIVATIVE ,TENSOR References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 160 /C1/67, 1985. Gray, A. "Christoffel Symbols." §22.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 509 /C1/13, 1997. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 47 /C1/8, 1953. Sternberg, S. Differential Geometry. New York: Chelsea, p. 354, 1983. Christoffel-Darboux Formula For three consecutive orders of an ORTHOGONAL POLYNOMIAL , the following relationship holds for n/C302, 3, ..., pn(x)/C30(Anx/C27Bn)pn/C281(x)/C28Cnpn/C282(x); (1) where An>0;Bn;andCn>0 are constants. Denoting the highest COEFFICIENT ofpn(x)b ykn; An/C30kn kn/C281(2) Cn/C30An An/C281/C30knkn/C282 k2 n/C281: (3) Then p0(x)p0(y)/C27.../C27pn(x)pn(y) /C30kn kn/C271pn/C271(x)pn(y)/C28pn(x)pn/C271(y) x/C28y: (4) In the special case of x/C30y, (4) gives[p0(x)]2/C27.../C27[pn(x)]2 /C30kn kn/C271[p?n/C271(x)pn(x)/C28p?n(x)pn/C271(x)]: (5) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 785, 1972. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 42 /C1/44, 1975. Christoffel-Darboux Identity X/C12 k/C300fk(x)fk(y) gk /C30fm/C271(x)fm(y)/C28fm(x)fm/C271(y) amgm(x/C28y);(1) where fk(x) are ORTHOGONAL POLYNOMIALS with WEIGHTING FUNCTION W(x); gm/C13g[fm(x)]2W(x)dx; (2) and ak/C13Ak/C271 Ak(3) where Akis the COEFFICIENT ofxkinfk(x):/ References Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, p. 322, 1956. Chromatic Number The fewest number of colors g(G) necessary to color the vertices of GRAPH or regions of a SURFACE (Skiena 1990, p. 210). The chromatic number is the smallest positive integer zsuch that the CHROMATIC POLYNO- MIAL pG(z)>0:Calculating the chromatic number of aGRAPH is an NP -COMPLETE PROBLEM (Skiena 1990, pp. 211 /C1/12). For any two positive integers gandk, there exists a graph of girth at least gand chromatic number at least k(Erdos 1961, Lova ´sz 1968; Skiena 1990, p. 215). The chromatic number of a surface of GENUS gis given by the H EAWOOD CONJECTURE , g(g)/C301 2(7/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 48g/C271p )jk ; where xbcis the FLOOR FUNCTION .g(g) is sometimes also denoted x(g) (which is unfortunate, since x(g)/C30 2/C282gcommonly refers to the E ULER CHARACTERIS- TIC). For g/C300, 1, ..., the first few values of x(g) are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, ... (Sloane’s A000934). Erdos (1959) proved that there are graphs with arbitrarily large GIRTH and CHROMATIC NUMBER (Bolloba ´s and West 2000). See also BETTI NUMBER ,BRELAZ’S HEURISTIC ALGO- RITHM ,BROOKS’ THEOREM ,CHROMATIC POLYNOMIAL , EDGE CHROMATIC NUMBER ,EDGE COLORING ,EULER CHARACTERISTIC ,GENUS (SURFACE ), HEAWOOD CON- JECTURE ,M AP COLORING ,PERFECT GRAPH ,TORUS COLORING References Bolloba ´s, B. and West, D. B. "A Note on Generalized Chromatic Number and Generalized Girth." Discr. Math. 213,29/C1/4, 2000. Chartrand, G. "A Scheduling Problem: An Introduction to Chromatic Numbers." §9.2 in Introductory Graph Theory. New York: Dover, pp. 202 /C1/09, 1985. Eppstein, D. "The Chromatic Number of the Plane." http:// www.ics.uci.edu/~eppstein/junkyard/plane-color/. Erdos, P. "Graph Theory and Probability." Canad. J. Math. 11,34/C1/8, 1959. Erdos, P. "Graph Theory and Probability II." Canad. J. Math. 13, 346 /C1/52, 1961. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 9, 1984. Lova´sz, L. "On Chromatic Number of Finite Set-Systems.’ Acta Math. Acad. Sci. Hungar. 19,59/C1/7, 1968. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A000934/M3292 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Chromatic Polynomial A POLYNOMIAL pG(z)ofa GRAPH G which counts the number of ways to color g with exactly z colors. For example, the CUBICAL GRAPH has chromatic polyno- mial pG(z) /C30z8 /C2812z7 /C2766z6 /C28214z5 /C27441z4 /C28572z3 /C27423z2 /C28133z ; (1) so the number of 1-, 2-, ... colorings are 0, 2, 114, 2652, 29660, 198030, .... The chromatic polynomial of a graph g in the variable z can be determined using ChromaticPolynomial [g, z] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The chromatic polynomial of a DISCONNECTED GRAPH is the product of the chromatic polynomials of its CONNECTED COMPONENTS . The chromatic polynomial of a graph of order n has degree n, with leading coefficient 1 and constant term 0. Furthermore, the coefficients alternate signs, and the coefficient of the (n /C281)/st term is /C28e ; where e is the number of edges.Interestingly, pG(/C281) is equal to the number of acyclic orientations of G (Stanley 1973). Except for special cases (such as TREES ), the calcula- tion of PG/(z) is exponential in the minimum number of edges in G and the COMPLEMENT GRAPH ¯G (Skiena 1990, p. 211), and calculating the chromatic polyno- mial of a GRAPH is at least an NP-COMPLETE PROBLEM (Skiena 1990, pp. 211 /C1/12). Tutte (1970) showed that the chromatic polynomial of a planar triangulation possess a ROOT close to f2 /C30 f /C271 /C302:618033... ; where f is the GOLDEN MEAN . More precisely, if n is the number of VERTICES of G, then PG(f2)5f5/C28n(2) (Tutte 1970, Le Lionnais 1983).Read (1968) conjectured that, for any chromatic polynomial c nzn/C27.../C27c1z; (3) there does not exist a 1 5p5q5r5nsuch that ½cp½> ½cq½and½cq½B½cr½(Skiena 1990, p. 221). The CHROMATIC NUMBER of a graph gives the smallest number of colors with which a graph can be colored,and so is the smallest positive integer zsuch that p G(z)>0 (Skiena 1990, p. 211). See also CHROMATIC NUMBER , K-COLORING References Berman, G. and Tutte, W. T. "The Golden Root of a Chromatic Polynomial." J. Combin. Th. 6, 301/C1/02, 1969. Birkhoff, G. D. "A Determinant Formula for the Number of Ways of Coloring a Map." Ann. Math. 14,4 2/C1/6, 1912. Birkhoff, G. D. and Lewis, D. C. "Chromatic Polynomials." Trans. Amer. Math. Soc. 60, 355/C1/51, 1946. Chva´tal, V. "A Note on Coefficients of Chromatic Polyno- mials." J. Combin. Th. 9,9 5/C1/6, 1970. Erdos, P. and Hajnal, A. "On Chromatic Numbers of Graphs and Set-Systems." Acta Math. Acad. Sci. Hungar. 17,6 1/C1/ 9, 1966. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983. Read, R. C. "An Introduction to Chromatic Polynomials." J. Combin. Th. 4,5 2/C1/1, 1968. Saaty, T. L. and Kainen, P. C. "Chromatic Numbers and Chromatic Polynomials." Ch. 6 in The Four-Color Pro- blem: Assaults and Conquest. New York: Dover, pp. 134 /C1/ 63 1986. Skiena, S. "Chromatic Polynomials." §5.5.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 210 /C1/12, 1990. Stanley, R. P. "Acyclic Orientations of Graphs." Disc. Math. 5, 171/C1/78, 1973. Tutte, W. T. "On Chromatic Polynomials and the Golden Ratio." J. Combin. Th. 9, 289/C1/96, 1970. Chu Identity CHU-VANDERMONDE IDENTITY Chu Space A Chu space is a BINARY RELATION from a SET A to an ANTISET X which is defined as a SET which transforms via converse functions. See also ANTISET References Stanford Concurrency Group. "Guide to Papers on Chu Spaces." http://boole.stanford.edu/chuguide.html. Church’s Theorem No decision procedure exists for ARITHMETIC . Church’s Thesis CHURCH- TURING THESIS Church-Turing Thesis The TURING MACHINE concept defines what is meant mathematically by an algorithmic procedure. Stated another way, a function f is effectively COMPUTABLE IFF it can be computed by a TURING MACHINE . See also ALGORITHM ,COMPUTABLE FUNCTION ,DECID- ABLE ,TURING MACHINE References Penrose, R. The Emperor’s New Mind: Concerning Compu- ters, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 47 /C1/9, 1989. Pour-El, M. B. "The Structure of Computability in Analysis and Physical Theory: An Extension of Church’s Thesis." Ch. 13 in Handbook of Computability Theory (Ed. E. R. Griffor). Amsterdam, Netherlands: Elsevier, pp. 449 /C1/70, 1999. Chu-Vandermonde Identity A special case of GAUSS’S THEOREM , with a being a NEGATIVE INTEGER /C28n : 2F1(/C28n; b; c;1)/C30(c /C28 b)n (c)n; where2F1(a ; b; c; z)isa HYPERGEOMETRIC FUNC- TION and (a)n is a POCHHAMMER SYMBOL (Bailey 1935, p. 3; Koepf 1998, p. 32). The identity is sometimes also called Vandermonde’s theorem. The identity (x /C27a)n /C30X/C12 k /C300n kl11sl11n (x)k(a)n/C28k (Koepf 1998, p. 42), wheren kl1ml11 is a BINOMIAL COEFFI- CIENT and (a)n /C13a(a /C281) /C1/C1/C1(a /C28n /C271) is the POCH- HAMMER SYMBOL is sometimes also known as the Chu-Vandermonde identity. (0) can be written as x /C27a nl11sl11n /C30Xn k /C300x kl11sl11n a n /C28kl11sl11n ;which is sometimes known as VANDERMONDE’S CON- VOLUTION FORMULA (Roman 1984). A special case gives the identity Xmax( k; n) l/C300m k /C28ll11sl11n n ll11sl11n /C30m /C27n kl11sl11n : The identities Xn k/C300a kl11sl11n b n /C28kl11sl11n /C30a /C27b nl11sl11n (1) Xn k /C300n kl11sl11n s t /C28kl11sl11n /C30n /C27s tl11sl11n (2) Xn k /C300n kl11sl11n s t /C27kl11sl11n /C30n /C27s n /C27tl11sl11n (3) are all special instances of the Chu-Vandermonde identity (Koepf 1998, p. 41). See also BINOMIAL THEOREM ,G AUSS’S HYPERGEO- METRIC THEOREM , Q-CHU-VANDERMONDE IDENTITY , UMBRAL CALCULUS References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, 1935. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, pp. 130 and 181 /C1/82, 1996. Roman, S. The Umbral Calculus. New York: Academic Press, p. 29, 1984. Chva ´tal Graph Gru¨nbaum conjectured that for every m /C211, n /C212, there exists an m-regular, m-chromatic graph of GIRTH at least n. This result is trivial for n /C302 and m /C302; 3; but only two other such graphs are known: the Chva ´tal graph illustrated above, and the G RU¨N- BAUM GRAPH . See also GRU¨ NBAUM GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 241, 1976. Gru¨nbaum, B. "A Problem in Graph Coloring." Amer. Math. Monthly 77, 1088 /C1/092, 1970. Chva ´tal’s Art Gallery Theorem ART GALLERY THEOREM Chva ´tal’s Theorem Let a GRAPH G have VERTICES with VERTEX DEGREES d1 5/C1/C1/C15dm : If for every i Bn=2 we have either di ] i /C271or dn/C28i ]n /C28i ; then the GRAPH is HAMILTONIAN . See also HAMILTONIAN GRAPH References Chva´tal, V. "On Hamilton’s Ideals." J. Combin. Th. 12, 163 /C1/ 68, 1972. ci COSINE INTEGRAL Ci COSINE INTEGRAL Cigarettes It is possible to place 7 cigarettes in such a way that each touches the other if l=d > 7ffiffiffi 3p =2 (Gardner 1959, p. 115). References Gardner, M. The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, 1959. Cin COSINE INTEGRAL C-Infinity Function A C/C12 function is a function that is DIFFERENTIABLE for all degrees of differentiation. For instance, f(x) /C30 e2xis C/C12because its nth derivative f(n)(x) /C302ne2x exists and is CONTINUOUS . All polynomials are C /C12: The reason for the notation is that /Ck FUNCTIONS have k continuous derivatives. /C /C12 functions are also called "smooth" because neither they nor their derivatives have "corners," which would make their graph look somewhat rough. For example, f(x) /C30½x3 ½ is not smooth.There are special C/C12 functions which are very useful in analysis and geometry. For example, there are smooth functions called BUMP FUNCTIONS , which are smooth approximations to a CHARACTERISTIC FUNC- TION . Typically, these functions require some CALCU- LUS to show that they are indeed C /C12:/ Any ANALYTIC FUNCTION is smooth. But a smooth function is not necessarily analytic. For instance, an analytic function cannot be a BUMP FUNCTION . Con- sider the following function, whose TAYLOR SERIES at 0 is identically zero, yet the function is not zero: f(x) /C300 for x 50 e /C281=xfor x > 0:l12) The function f goes to zero very quickly. One property of smooth functions is that they can look very different at different scales. The set of smooth functions cannot be made into a BANACH SPACE , which makes some problems hard, but instead has the weaker structure of a F RE´CHET SPACE . See also C-K FUNCTION ,C-INFINITY TOPOLOGY ,CAL- CULUS ,D IFFERENTIAL TOPOLOGY ,F RE´ CHET SPACE , PARTITION OF UNITY,SARD’S THEOREM Circle A circle is the set of points equidistant from a givenpoint O. The distance rfrom the CENTER is called the RADIUS , and the point Ois called the CENTER . Twice the RADIUS is known as the DIAMETER d/C302r:The PERIMETER Cof a circle is called the CIRCUMFERENCE , and is given by C/C30pd/C302pr: (1) The angle a circle subtends from its center is a FULL ANGLE , equal to 360 8or 2pRADIANS . The circle is a CONIC SECTION obtained by the intersection of a CONE with a PLANE PERPENDICULAR to the CONE ’s symmetry axis. A circle is the degen- erate case of an ELLIPSE with equal semimajor and semiminor axes (i.e., with ECCENTRICITY 0). The interior of a circle is called a DISK. The generalization of a circle to 3-D is called a SPHERE , and to n-D for n]4a HYPERSPHERE . The region of intersection of two circles is called a LENS . The region of intersection of three symmetri- cally placed circles (as in a V ENN DIAGRAM ), in the special case of the center of each being located at the intersection of the other two, is called a R EULEAUX TRIANGLE . The PARAMETRIC EQUATIONS for a circle of RADIUS a are x/C30acost (2) y/C30asint: (3) For a body moving uniformly around the circle, x?/C30/C28 asint (4) y?/C30acost; (5) and xƒ/C30/C28acost (6) yƒ/C30/C28asint: (7) When normalized, the former gives the equation for the unit TANGENT VECTOR of the circle, ( /C28sint;cost): The circle can also be parameterized by the rational functions x/C301/C28t2 1/C27t2(8) y/C302t 1/C27t2; (9) but an ELLIPTIC CURVE cannot. The following plots show a sequence of NORMAL and TANGENT VECTORS for the circle. The ARC LENGTH s,CURVATURE k;and TANGENTIAL ANGLE fof the circle ares(t)/C30gds/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2/C27y?2q dt/C30at (10) k(t)/C30x?yƒ/C28y?xƒ (x?2/C27y?2)3=2/C301 a(11) f(t)/C30gk(t)dt/C30t a: (12) The C ESA`RO EQUATION is k/C301 a: (13) InPOLAR COORDINATES , the equation of the circle has a particularly simple form. r/C30a (14) is a circle of RADIUS acentered at ORIGIN , r/C302acosu (15) is circle of RADIUS acentered at ( a;0);and r/C302asinu (16) is a circle of RADIUS acentered on (0 ;a):In C ARTE- SIAN COORDINATES , the equation of a circle of RADIUS acentered on ( x0;y0)i s (x/C28x0)2/C27(y/C28y0)2/C30a2: (17) InPEDAL COORDINATES with the PEDAL POINT at the center, the equation is pa/C30r2(18) The circle having P1P2as a diameter is given by (x/C28x1)(x/C28x2)/C27(y/C28y1)(y/C28y2)/C300: (19) The equation of a circle passing through the three points ( xi;yi) for i/C301, 2, 3 (the CIRCUMCIRCLE of the TRIANGLE determined by the points) is x2/C27y2xy 1 x2 1/C27y21x1y11 x22/C27y22x2y21 x23/C27y23x3y31l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112/C300: (20) The CENTER and RADIUS of this circle can be identified by assigning coefficients of a QUADRATIC CURVE ax2/C27cy2/C27dx/C27ey/C27f/C300; (21) where a/C30cand b/C300 (since there is no xycross term). COMPLETING THE SQUARE gives ax/C27d 2a !2 /C27ay/C27e 2a !2 /C27f/C28d2/C27e2 4a/C300: (22) The CENTER can then be identified as x0 /C30/C28d 2a (23) y0 /C30/C28e 2a (24) and the RADIUS as r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 /C27 e2 4a2/C28f as ; (25) where a /C30x1y11 x2y21 x3y31l112l112l112l112l112l112l112l112l112l112l112l112(26) d /C30/C28x 2 1 /C27y21y11 x22 /C27y22y21 x2 3 /C27y23y31l112l112l112l112l112l112l112l112l112l112l112l112(27) e /C30x 2 1 /C27y21x11 x22 /C27y22x21 x23 /C27y23x31l112l112l112l112l112l112l112l112l112l112l112l112(28) f /C30/C28x 2 1 /C27y21x1y1 x2 2 /C27y22x2y2 x2 3 /C27y23x3y3l112l112l112l112l112l112l112l112l112l112l112l112(29) Four or more points which lie on a circle are said to be CONCYCLIC . Three points are trivially concyclic since three noncollinear points determine a circle. The CIRCUMFERENCE -to-DIAMETER ratio C =d for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor s increases its PERIMETER by s), and d also scales by s. This ratio is denoted p (PI), and has been proved TRANSCENDENTAL . With d the DIAMETER and r the RADIUS , C /C30 pd /C302pr : (30) Knowing C=d ; we can then compute the AREA of the circle either geometrically or using CALCULUS . From CALCULUS , A /C30g2 p 0dugr 0rdr/C30(2p)1 2 r2l11)l117 /C30 pr2 : (31) Now for a few geometrical derivations. Using con- centric strips, we have As the number of strips increases to infinity, we are left with a TRIANGLE on the right, soA /C3012(2pr)r /C30 pr2 : (32) This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC ). If we cut the circle instead into wedges, As the number of wedges increases to infinity, we are left with a RECTANGLE ,so A/C30(pr)r/C30pr2: (33) See also ADAMS’ CIRCLE ,ARC,BLASCHKE’S THEOREM , BRAHMAGUPTA’S FORMULA ,BROCARD CIRCLE ,CASEY’S THEOREM ,CEVIAN CIRCLE ,CHORD ,CIRCLE INSCRIB- ING,CIRCLE- LINE INTERSECTION ,CIRCUMCIRCLE ,CIR- CUMFERENCE ,CLIFFORD’S CIRCLE THEOREM ,CLOSED DISK,CONCENTRIC CIRCLES ,COSINE CIRCLE ,COTES CIRCLE PROPERTY ,D IAMETER ,D ISK,D ROZ-FARNY CIRCLES ,EULER TRIANGLE FORMULA ,EXCIRCLE ,EX- COSINE CIRCLE ,E YEBALL THEOREM ,F EUERBACH’S THEOREM ,F IVE CIRCLES THEOREM ,F IVE DISKS PROBLEM ,FLOWER OF LIFE,FORD CIRCLE ,FUHRMANN CIRCLE ,GERGORIN CIRCLE THEOREM ,HART CIRCLE , HOPF CIRCLE ,INCIRCLE ,INVERSIVE DISTANCE ,JOHN- SON CIRCLE ,KINNEY’S SET,LEMOINE CIRCLE ,LENS, LESTER CIRCLE ,M AGIC CIRCLES ,M ALFATTI CIRCLES , MCCAY CIRCLE ,M IDCIRCLE ,M ONGE’S THEOREM , NEUBERG CIRCLE ,N INE-POINT CIRCLE ,O PEN DISK, P-CIRCLE ,PARRY CIRCLE ,PI,POINT CIRCLE ,POLAR CIRCLE ,POWER (CIRCLE ), PRIME CIRCLE ,PSEUDOCIR- CLE,PTOLEMY’S THEOREM ,PURSER’S THEOREM ,RADI- CAL AXIS,R ADIUS ,R EULEAUX TRIANGLE ,SEED OF LIFE,SEIFERT CIRCLE ,SEMICIRCLE ,SEVEN CIRCLES THEOREM ,SIMILITUDE CIRCLE ,SIX CIRCLES THEO- REM,S ODDY CIRCLES ,S PHERE ,T AYLOR CIRCLE , TRIPLICATE- RATIO CIRCLE ,T UCKER CIRCLES ,U NIT CIRCLE ,VENN DIAGRAM ,VILLARCEAU CIRCLES ,YIN- YANG References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 125 and 197, 1987. Casey, J. "The Circle." Ch. 3 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, withNumerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 96 /C1 /50, 1893. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 74 /C1/5, 1996. Coxeter, H. S. M. and Greitzer, S. L. "Some Properties of Circles." Ch. 2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 27 /C1/0, 1967. Dunham, W. "Archimedes’ Determination of Circular Area." Ch. 4 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 84 /C1/12, 1990. Eppstein, D. "Circles and Spheres." http://www.ics.uci.edu/ ~eppstein/junkyard/sphere.html. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 1, 1999. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948. Lachlan, R. "The Circle." Ch. 10 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 148 /C1/ 73, 1893. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 65 /C1/6, 1972. MacTutor History of Mathematics Archive. "Circle." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Cir- cle.html. Pappas, T. "Infinity & the Circle" and "Japanese Calculus." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 68 and 139, 1989. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Yates, R. C. "The Circle." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 21 /C1/5, 1952. Circle Bundle A circle bundle p : E 0 M is a FIBER BUNDLE whose FIBERS p/C281(x) are circles. It may also have the structure of a PRINCIPAL BUNDLE if there is an action of SO(2) that preserves the fibers, and is locally trivial. That is, if every point has a TRIVIALIZATION U /C29S1 such that the action of SO(2) on S1 is the usual one. See also BUNDLE ,GROUP ACTION ,PRINCIPAL BUNDLE Circle Caustic Consider a point light source located at a point ( m; 0): The CATACAUSTIC of a unit CIRCLE for the light at m /C30 /C12 is the NEPHROID x /C301 4[3 cot t /C28cos(3 t)] (1) y /C3014[3 sin t /C28sin(3 t)]: (2) The CATACAUSTIC for the light at a finite distance m >1 is the curve x /C30m(1 /C28 3m cos t /C27 2m cos3 t) /C28(1 /C27 2m2) /C27 3 m cos t (3) y /C302m2 sin3 t 1 /C27 2m2 /C28 3m cos t ; (4) and for the light on the CIRCUMFERENCE of the CIRCLE m /C301 is the CARDIOID x /C3023cos t(1 /C27cos t) /C2813 (5) y /C3023sin t(1 /C27cos t): (6) If the point is inside the circle, the catacaustic is a discontinuous two-part curve. These four cases are illustrated below. The CATACAUSTIC for PARALLEL rays crossing a CIRCLE is a CARDIOID . See also CATACAUSTIC ,CAUSTIC Circle Chord Picking CIRCLE LINE PICKING Circle Covering An arrangement of overlapping circles which cover the entire plane. A lower bound for a covering using equivalent circles is 2 p=ffiffiffiffiffiffi 27p (Williams 1979, p. 51). See also CIRCLE PACKING ,DISK COVERING PROBLEM , FIVE DISKS PROBLEM ,FLOWER OF LIFE,SEED OF LIFE References Williams, R. "Circle Coverings." §2/C1/inThe Geometrical Foundation of Natural Structure: A Source Book of De- sign. New York: Dover, pp. 51 /C1/2, 1979. Circle Covering by Arcs The probability P(a ; n) that n random arcs of angular size a cover the circumference of a circle completely (for a circle with unit circumference) is P(a; n) /C30X1=abc k /C300(/C281)k n kl11sl11n (1 /C28ka)n/C281 ; where xbcis the FLOOR FUNCTION (Solomon 1978, p. 75). This was first given correctly by Stevens (1939), although partial results were obtains by Whitworth (1897), Baticle (1935), Garwood (1940), Darling (1953), and Shepp (1972). The probability that n arcs leave exactly l gaps is given by Pl gaps(a ; n) /C30n ll11sl11nXk j/C301(/C281)j/C28l n /C28l j /C28ll11sl11n (1 /C28ja)n /C281 (Stevens 1939; Solomon 1978, p. 76). See also CIRCLE POINT PICKING ,CIRCLE LINE PICKING References Baticle, M. "Le proble `me des re´partitions." C. R. Acad. Sci. Paris 201, 862 /C1/64, 1935. Fisher, R. A. "Tests of Significance in Harmonic Analysis." Proc. Roy. Soc. London Ser. A 125,54/C1/9, 1929. Fisher, R. A. "On the Similarity of the Distributions Found for the Test of Significance in Harmonic Analysis, and in Stevens’s Problem in Geometric Probability." Eugenics 10, 14 /C1/7, 1940. Darling, D. A. "On a Class of Problems Related to the Random Division of an Interval." Ann. Math. Stat. 24, 239 /C1/53, 1953. Garwood, F. "An Application to the Theory of Probability of the Operation of Vehicular-Controlled Traffic Signals." J. Roy. Stat. Soc. Suppl. 7,65/C1/7, 1940. Shepp, L. A. "Covering the Circle with Random Arcs." Israel J. Math. 11, 328 /C1/45, 1972. Siegel, A. F. Random Coverage Problems in Geometric Probability with an Application to Time Series Analysis. Ph.D. thesis. Stanford, CA: Stanford University, 1977. Solomon, H. "Covering a Circle Circumference and a Sphere Surface." Ch. 4 in Geometric Probability. Philadelphia, PA: SIAM, pp. 75 /C1/6, 1978. Stevens, W. L. "Solution to a Geometrical Problem in Probability." Ann. Eugenics 9, 315 /C1/20, 1939. Whitworth, W. A. DCC Exercises in Choice and Chance. 1897. Reprinted New York: Hafner, 1965. Circle Cutting CIRCLE DIVISION BY CHORDS ,C IRCLE DIVISION BY LINESCircle Division by Chords A related problem, sometimes called Moser’s circle problem, is to find the number of pieces into which a CIRCLE is divided if n points on its CIRCUMFERENCE are joined by CHORDS with no three CONCURRENT . The answer is g(n) /C30n 4l11sl11n /C27n 2l11sl11n /C271 (1) /C301 24(n4 /C286n3 /C2723n2 /C2818n /C2724) ; (2) (Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), wheren ml1ml11 is a BINOMIAL COEFFICIENT . The first few values are 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (Sloane’s A000127). This sequence demonstrates the danger in making as- sumptions based on limited trials. While the series starts off like 2n /C281 ; it begins differing from this GEOMETRIC SERIES atn/C306. See also CAKE CUTTING ,CIRCLE DIVISION BY LINES, CYLINDER CUTTING ,HAM SANDWICH THEOREM ,PAN- CAKE THEOREM ,PIZZA THEOREM ,PLANE DIVISION BY CIRCLES ,PLANE DIVISION BY ELLIPSES ,PLANE DIVI- SION BY LINES,SQUARE DIVISION BY LINES,TORUS CUTTING References Conway, J. H. and Guy, R. K. "How Many Regions." In The Book of Numbers. New York: Springer-Verlag, pp. 76 /C1/9, 1996. Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697/C1/12, 1988. Noy, M. "A Short Solution of a Problem in Combinatorial Geometry." Math. Mag. 69,5 2/C1/3, 1996. Sloane, N. J. A. Sequences A000127/M1119 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Yaglom, A. M. and Yaglom, I. M. Problem 47 in Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, 1987. Circle Division by Lines Determining the maximum number of pieces in which it is possible to divide a CIRCLE for a given number of cuts is called the circle cutting, or sometimes PAN- CAKE CUTTING , problem. The minimum number is always n/C271;where nis the number of cuts, and it is always possible to obtain any number of pieces between the minimum and maximum. The first cut creates 2 regions, and the nth cut creates n new regions, so f(1) /C302 (1) f(2) /C302 /C27f(1) (2) f(n) /C30n /C27f(n /C281): (3) Therefore, f(n) /C30n /C27[(n /C281) /C27f(n /C282)] /C30n /C27(n /C281) /C27.../C272 /C27f(1) /C30f(1) /C27Xn k/C302kf(1) /C302 /C271 2(n /C272)(n /C281) /C3012(n2 /C27n /C272): (4) Evaluating for n /C301, 2, ... gives 2, 4, 7, 11, 16, 22, ... (Sloane’s A000124). This is equivalent to the maximal number of regions into which a PLANE can be cut by n lines. See also CIRCLE DIVISION BY CHORDS ,PLANE DIVI- SION BY CIRCLES ,SPACE DIVISION BY PLANES ,SPACE DIVISION BY SPHERES ,SQUARE DIVISION BY LINES References Sloane, N. J. A. Sequences A000124/M1041 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M1041 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Yaglom, A. M. and Yaglom, I. M. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, pp. 102 /C1/06, 1987. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 31, 1986. Circle Evolute x /C30cos tx?/C30/C28 sin txƒ/C30/C28cos t (1) y /C30sin ty?/C30cos tyƒ/C30/C28sin t; (2) so the RADIUS OF CURVATURE is R /C30(x?2 /C27 y?2)3 =2 yƒx?/C28xƒy?/C30(sin2 t /C27 cos2 t)3=2 ( /C28sin t)(/C28sin t) /C28 (/C28cos t) cos t /C301; (3) and the TANGENT VECTOR is ˆT /C30/C28sin t cos tl12ml121 : (4) Therefore, cos t /C30 ˆT /C215 ˆx /C30/C28sin t (5) sin t /C30 ˆT /C215 ˆy /C30cos t; (6)so j(t) /C30x /C28R sin t /C30cos t /C281 /C215 cos t /C300 (7) h(t) /C30y /C27R cos t /C30sin t /C271 /C215 (/C28sin t) /C300 ; (8) and the EVOLUTE degenerates to a POINT at the ORIGIN . See also CIRCLE INVOLUTE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 99, 1997. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 55 /C1/9, 1991. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 137, 1999. Circle Inscribing Ifris the INRADIUS of a CIRCLE inscribed in a RIGHT TRIANGLE with sides aandband HYPOTENUSE c, then r/C301 2(a/C27b/C28c): AS ANGAKU PROBLEM dated 1803 from the Gumma Prefecture asks to construct the figure consisting of a circle centered at O, a second smaller circle centered atO2tangent to the first, and an ISOSCELES TRIANGLE whose base ABcompletes the diameter of the larger circle through the smaller XB. Now inscribe a third circle with center O3inside the large circle, outside the small one, and on the side of a leg of the triangle.It then follows that the line O 3A/C222XB:To find the explicit position and size of the circle, let the circle O have radius 1/2 and be centered at (0 ;0) and let the circle O2have diameter 0 BrB1:Then solving the simultaneous equations 1 2r/C27al11)l1172 /C3012rl11)l1172 /C27y2(1) 12/C28al11)l1172 /C30r/C2812l11)l1172 /C27y2(2) for a and y gives a /C30r(1 /C28 r) 1 /C27 r (3) y /C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(1 /C28 r)p 1 /C27 r: (4) See also INCIRCLE ,INSCRIBED ,POLYGON References Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/1, May 1998. Circle Involute First studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a CYCLOID . For a CIRCLE with a /C301, the PARA- METRIC EQUATIONS of the circle and their derivatives are given by x /C30cos tx?/C30/C28 sin txƒ/C30/C28cos t (1) y /C30sin ty?/C30cos tyƒ/C30/C28sin t: (2) The TANGENT VECTOR is ˆT /C30/C28sin t cos tl12ml121 (3) and the ARC LENGTH along the circle is s /C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2 /C27y?2q dt /C30g dt /C30t; (4) so the involute is given by ri /C30r /C28s ˆT /C30 cos t sin tl12ml121 /C28t /C28sin t cos tl12ml121 /C30cos t /C27t sin t sin t /C28t cos tl12ml121 ; (5)or x /C30a(cos t /C27t sin t) (6) y /C30a(sin t /C28t cos t) : (7) The ARC LENGTH , CURVATURE , and TANGENTIAL ANGLE are s/C30gds/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2/C27y?2q dt/C301 2at2(8) k/C301 at(9) f/C30t: (10) The C ESA`RO EQUATION is k/C301ffiffiffiffiffiasp : (11) See also CIRCLE ,C IRCLE EVOLUTE ,E LLIPSE INVO- LUTE ,INVOLUTE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 220, 1987. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 105, 1997. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 6 /C1/, 1999. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 190 /C1/91, 1972. MacTutor History of Mathematics Archive. "Involute of a Circle." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Involute.html. Circle Involute Pedal Curve The PEDAL CURVE ofCIRCLE INVOLUTE f/C30cost/C27tsint g/C30sint/C28tcost with the center as the PEDAL POINT is the A RCHI- MEDES’ SPIRAL x/C30tsint y/C30/C28tcost: Circle Lattice Points For every POSITIVE INTEGER n, there exists a CIRCLE which contains exactly nlattice points in its interior. H. Steinhaus proved that for every POSITIVE INTEGER n, there exists a CIRCLE ofAREA nwhich contains exactly nlattice points in its interior. SCHINZEL’S THEOREM shows that for every POSITIVE INTEGER n, there exists a CIRCLE in the PLANE having exactly nLATTICE POINTS on its CIRCUMFERENCE . The theorem also explicitly identifies such "S CHINZEL CIRCLES "a s x/C281 2l11)l1172 /C27y2/C30145k/C281forn/C302k x/C281 3l11)l1172 /C27y2/C301952kforn/C302k/C271:8 >< >:(1) Note, however, that these solutions do not necessarily have the smallest possible RADIUS . For example, while the S CHINZEL CIRCLE centered at (1/3, 0) and with RADIUS 625/3 has nine lattice points on its CIRCUMFERENCE , so does the CIRCLE centered at (1/ 3, 0) with RADIUS 65/3. Letrbe the smallest INTEGER RADIUS of a CIRCLE centered at the ORIGIN (0, 0) with L(r)LATTICE POINTS . In order to find the number of lattice points of the CIRCLE , it is only necessary to find the number in the first octant, i.e., those with 0 5y5r=ffiffiffi 2pl1=l1; ;where zbc is the FLOOR FUNCTION . Calling this N(r);then for r] 1;L(r)/C308N(r)/C284;soL(r)/C134 (mod 8) :The multi- plication by eight counts all octants, and the subtrac- tion by four eliminates points on the axes which themultiplication counts twice. (Sinceffiffiffi 2p is IRRATIONAL , a mid-arc point is never a LATTICE POINT .) GAUSS’S CIRCLE PROBLEM asks for the number of lattice points within aCIRCLE ofRADIUS r N(r)/C301/C274rbc/C274Xrbc i/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28i2pjk : (2) Gauss showed that N(r)/C30pr2/C27E(r); (3) where ½E(r)½52ffiffiffi 2p pr: (4) The number of lattice points on the CIRCUMFERENCE of circles centered at (0, 0) with radii 0, 1, 2, ... are 1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, ... (Sloane’s A046109). The following table gives the smallest RADIUS r5 390;800 for a circle centered at (0, 0) having a given number of LATTICE POINTS L(r) (Sloane’s A046112). Note that the high-water mark radii are alwaysmultiples of five. /L(r)/ r /L(r)/ r 1 0 108 1,105 4 1 132 40,625 12 5 140 21,12520 25 156 203,12528 125 180 5,52536 65 196 274,625 44 3,125 252 27,625 52 15,625 300 71,82560 325 324 32,04568 390,625 420 359,125 76 /51;953;125 /540 160,225 84 1,625 92 /548;828;125 / 100 4,225 If the CIRCLE is instead centered at (1/2, 0), then the CIRCLES ofRADII 1/2, 3/2, 5/2, ... have 2, 2, 6, 2, 2, 2, 6, 6, 6, 2, 2, 2, 10, 2, ... (Sloane’s A046110) on their CIRCUMFERENCES . If the CIRCLE is instead centered at (1/3, 0), then the number of lattice points on the CIRCUMFERENCE of the CIRCLES of RADIUS 1/3, 2/3, 4/3, 5/3, 7/3, 8/3, ... are 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 1, 5, 3, ... (Sloane’s A046111). Let 1. an be the RADIUS of the CIRCLE centered at (0, 0) having 8n /C274 lattice points on its CIRCUMFERENCE , 2. bn =2 be the RADIUS of the CIRCLE centered at (1/ 2, 0) having 4n /C272 lattice points on its CIRCUM- FERENCE , 3. cn =3 be the RADIUS of CIRCLE centered at (1/3, 0) having 2n /C271 lattice points on its CIRCUMFERENCE . Then the sequences fan g;fbn g; and fcn g are equal, with the exception that bn /C300if2½n and cn /C300if3½n: However, the sequences of smallest radii having the above numbers of lattice points are equal in the three cases and given by 1, 5, 25, 125, 65, 3125, 15625, 325, ... (Sloane’s A046112). KULIKOWSKI’S THEOREM states that for every POSITIVE INTEGER n, there exists a 3-D SPHERE which has exactly n LATTICE POINTS on its surface. The SPHERE is given by the equation (x/C28a)2/C27(y/C28b)2/C27(z/C28ffiffiffi 2p )2/C30c2/C272; where aandbare the coordinates of the center of the so-called S CHINZEL CIRCLE and cis its RADIUS (Honsberger 1973). See also CIRCLE ,CIRCUMFERENCE ,G AUSS’S CIRCLE PROBLEM ,KULIKOWSKI’S THEOREM ,LATTICE POINT , SCHINZEL CIRCLE ,SCHINZEL’S THEOREM References Honsberger, R. "Circles, Squares, and Lattice Points." Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117 /C1/27, 1973. Kulikowski, T. "Sur l’existence d’une sphe `re passant par un nombre donne ´aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 5,8 9/C1/0, 1959. Schinzel, A. "Sur l’existence d’un cercle passant par un nombre donne ´de points aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 4,7 1/C1/2, 1958. Sierpinski, W. "Sur quelques proble `mes concernant les points aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 4,2 5/C1/1, 1958. Sierpinski, W. "Sur un proble `me de H. Steinhaus concernant les ensembles de points sur le plan." Fund. Math. 46, 191/C1/94, 1959. Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964. Weisstein, E. W. "Circle Lattice Points." M ATHEMATICA NOTEBOOK CIRCLE LATTICE POINTS.M . Circle Lattice Theorem GAUSS’S CIRCLE PROBLEMCircle Line Picking Given a UNIT CIRCLE , pick two points at random on its circumference, forming a CHORD . Without loss of generality, the first point can be taken as (1 ;0);and the second by (cos u;sinu);with u/C23[0;p] (by sym- metry, the range can be limited to pinstead of 2 p): The distance sbetween the two points is then s(u)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C282 cos up /C302½sin(1 2u)½: (1) The average distance is then given by ¯s/C30gp 0s(u)du gp 0du/C304 p: (2) The probability function Psis obtained from Ps/C30du dsl112l112l112l112l112l112l112l112l112l112P u/C301 p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28(1 2s)2q : (3) The RAW MOMENTS are then m?n/C30gp 0[2 sin(1 2u)]ndu gp 0du(4) /C30g2 0sn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28(1 2s)2q (5) /C302nG(12(1/C27n)) ffiffiffippG(1/C271 2n); (6) giving the first few as m?2 /C302 (7) m?3 /C3032 3p (8) m ?4 /C306: (9) The CENTRAL MOMENTS are m2 /C302 /C2816 p2 (10) m3 /C308(48 /C28 5 p2) 3 p3 (11) m4 /C306 /C2764(p2 /C28 36) 3p4; (12) giving the SKEWNESS and KURTOSIS as g1 /C302ffiffiffi 2p (48 /C28 5p2) 3(p2 /C28 8)3 =2 (13) g2 /C30/C289p4 /C27 320p2 /C28 2304 6(p2 /C28 8)2 : (14) BERTRAND’S PROBLEM asks for the PROBABILITY that a CHORD drawn at random on a CIRCLE of RADIUS r has length ]r :/ See also BALL LINE PICKING ,BERTRAND’S PROBLEM , CIRCLE COVERING BY ARCS,CIRCLE TRIANGLE PICK- ING,DISK LINE PICKING Circle Map A 1-D MAP which maps a CIRCLE onto itself un/C271 /C30 un /C27V/C28K 2psin(2pun) ; (1) where un/C271is computed mod 1 and K is a constant. Note that the circle map has two parameters: V and K. V can be interpreted as an externally applied frequency, and K as a strength of nonlinearity. The 1- DJ ACOBIAN is @ un/C271 @ un/C301 /C28K cos(2 pun) ; (2) so the circle map is not AREA-PRESERVING . It is related to the STANDARD MAP In/C271 /C30In /C27K 2psin(2pun) (3) un/C271 /C30 un /C27In /C271 ; (4) for I and u computed mod 1. Writing un/C271 as un/C271 /C30 un /C27In /C27K 2 psin(2 pun) (5)gives the circle map with In /C30V and K /C30/C28K : The unperturbed circle map has the form un/C271 /C30 un /C27V: (6) If V is RATIONAL , then it is known as the map WINDING NUMBER , defined by V/C30W /C13p q ; (7) and implies a periodic trajectory, since un will return to the same point (at most) every q ORBITS .IfV is IRRATIONAL , then the motion is quasiperiodic. If K is NONZERO , then the motion may be periodic in some finite region surrounding each RATIONAL V: This execution of periodic motion in response to an IRRA- TIONAL forcing is known as MODE LOCKING . If a plot is made of K vs. V with the regions of periodic MODE-LOCKED parameter space plotted around RA- TIONAL V values (WINDING NUMBERS ), then the re- gions are seen to widen upward from 0 at K /C300to some finite width at K /C301. The region surrounding each RATIONAL NUMBER is known as an ARNOLD TONGUE .At K /C300, the ARNOLD TONGUES are an isolated set of MEASURE zero. At K /C301, they form a CANTOR SET of DIMENSION d :0:08700 : For K /C211, the tongues overlap, and the circle map becomes non- invertible. Let Vn be the parameter value of the circle map for a cycle with WINDING NUMBER Wn /C30Fn =Fn /C271passing with an angle u /C300 ; where Fnis a F IBONACCI NUMBER . Then the parameter values Vnaccumulate at the rate d/C13lim n0/C12Vn/C28Vn/C281 Vn/C271/C28Vn/C30/C282:833 (8) (Feigenbaum et al. 1982). See also ARNOLD TONGUE ,DEVIL’S STAIRCASE ,MODE LOCKING ,W INDING NUMBER (MAP) References Devaney, R. L. An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, pp. 108 /C1/ 11, 1987. Feigenbaum, M. J.; Kadanoff, L. P.; and Shenker, S. J. "Quasiperiodicity in Dissipative Systems: A Renormaliza- tion Group Analysis." Physica D 5, 370/C1/86, 1982. Rasband, S. N. "The Circle Map and the Devil’s Staircase." §6.5 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 128 /C1/32, 1990. Circle Method A method employed by Hardy, Ramanujan, and Littlewood to solve many asymptotic problems in ADDITIVE NUMBER THEORY , particularly in deriving an asymptotic formula for the PARTITION FUNCTION P. The circle method proceeds by choosing a circular CONTOUR satisfying certain technical properties (Apostol 1997). The method was modified by Rade- macher using a different contour in his derivative of the exact convergent formula for the PARTITION FUNCTION P. See also PARTITION FUNCTION P References Apostol, T. M. "The Plan of the Proof." §5.2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 95 /C1/6, 1997. Circle Negative Pedal Curve The NEGATIVE PEDAL CURVE of a circle is an ELLIPSE if the PEDAL POINT is inside the CIRCLE , and a HYPER- BOLA if the PEDAL POINT is outside the CIRCLE . Circle Notation A NOTATION for LARGE NUMBERS due to Steinhaus (1983). In circle notation, is defined as n in n SQUARES , where numbers written inside squares (and triangles) are interpreted in terms of STEINHAUS- MOSER NOTATION . The particular number known as the MEGA is then defined as follows (correcting the typographical error of Steinhaus). See also MEGA,M EGISTRON ,STEINHAUS- MOSER NO- TATION References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 28 /C1/9, 1999. Circle Order A POSET P is a circle order if it is ISOMORPHIC to a SET ofDISKS ordered by containment. See also ISOMORPHIC POSETS ,PARTIALLY ORDERED SET Circle Orthotomic The ORTHOTOMIC of the CIRCLE represented by x/C30cost (1) y/C30sint (2)with a source at ( x, y)i s x/C30xcos(2 t)/C28ysin(2 t)/C272 sin t (3) y/C30/C28xsin(2 t)/C28ycos(2 t)/C272 cos t: (4) Circle Packing A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (orall) of them are mutually tangent. The generalization to spheres is called a SPHERE PACKING .TESSELLA- TIONS of regular polygons correspond to particular circle packings (Williams 1979, pp. 35 /C1/1). There is a well developed theory of circle packing in the context of discrete conformal mapping (Stephenson). The densest packing of circles in the PLANE is the hexagonal lattice of the bee’s honeycomb (right figure; Steinhaus 1983, p. 202), which has a PACKING DEN- SITY of hh/C301 6pffiffiffi 3p :0:9068996821 (1) (Wells 1986, p. 30). Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fejes To ´th proved that the hexagonal lattice is indeed the densest of allpossible plane packings. Wells (1991, pp. 30 /C1/1) considers the maximum size possible for nidentical circles packed on the surface of a UNIT SPHERE . Using discrete conformal mapping, the radii of thecircles in the above packing inside a UNIT CIRCLE can be determined as roots of the polynomial equations a6/C27378a5/C273411 a4/C288964 a3/C2810233 a2/C273402 a/C2827 /C300 (2) 169b6/C2724978 b5/C272307 b4/C2814580 b3/C273375 b2/C27162b /C2827/C300 (3) c6/C27438c5/C2719077 c4/C2815840 c3/C28360c2/C272592 c/C28432 /C300 (4) with a:0:266746 (5) b:0:321596 (6) c:0:223138 : (7) The following table gives the packing densities hfor the circle packings corresponding to the regular and semiregular plane tessellations (Williams 1979, p. 49). TESSELLATION /hexact /happrox. /f3;6g//1 12ffiffiffiffiffiffi 12p p/ 0.9069 /f4;4g//1 4p/ 0.7854 /f6;3g//19ffiffiffi 3p p/ 0.6046 /32:42// (2/C28ffiffiffi3p )p /0.8418 /32:4:3:4// (2/C28ffiffiffi3p )p /0.8418 /3:6:3:6//1 8ffiffiffi 3p p/ 0.6802 /34:6//1 7ffiffiffi 2p p/ 0.7773 3.122 /(7ffiffiffi 3p /C2812)p/0.3907 4.82 /(3/C282ffiffiffi 2p )p/0.5390 /3:4:6:4//1 3(2ffiffiffi 3p /C283)p/0.7290 /3:4:6:4//1 3(2ffiffiffi 3p /C283)p/0.4860 Solutions for the smallest diameter CIRCLES into which nUNIT CIRCLES can be packed have been proved optimal for n/C301 through 10 (Kravitz 1967). The best known results are summarized in the following table, and the first few cases are illustrated above (Friedman).nd exact dapprox. 1 1 1.00000 2 2 2.00000 3 /1/C272 3ffiffiffi 3p / 2.15470... 4 /1/C27ffiffiffi 2p / 2.41421... 5 /1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(1/C271=ffiffiffi 5p )q /2.70130... 6 3 3.00000 7 3 3.000008 /1/C27csc(p=7)/ 3.30476... 9 /1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(2/C27ffiffiffi 2p )p /3.61312... 10 3.82... 1112 4.02... The following table gives the diameters dof circles giving the densest known packings of nequal circles packed inside a UNIT SQUARE , the first few of which are illustrated above (Friedman). All n/C301t o2 0 solutions (in addition to all solutions n/C30k2) have been proved optimal (Friedman). Peikert (1994) uses a normalization in which the centers ofncircles of diameter mare packed into a square of side length 1. Friedman lets the circles have unit radius and givesthe smallest square side length s. A tabulation of analytic sand diagrams for n/C301 to 25 circles is given by Friedman. Coordinates for optimal packings are given by Nurmela and O ¨sterga ˚rd. nd /:d/ m /:m/ 1 1 1.000000 22 2/C27ffiffiffi 2p0.585786 /ffiffiffi 2p / 1.414214 34 4/C27ffiffiffi 2p /C27ffiffiffi 6p0.508666 /ffiffiffi6p /C28ffiffiffi 2p /1.035276 4 /1 2/ 0.500000 1 1.000000 5 /ffiffiffi 2p /C281/ 0.414214 /1 2ffiffiffi 2p / 0.707107 6 /1 23(6ffiffiffiffiffiffi13p /C2813) / 0.375361 /1 6ffiffiffiffiffiffi 13p / 0.600925 7 /2 13(4 /C28ffiffiffi3p ) / 0.348915 /4 /C282ffiffiffi3p / 0.535898 82 2 /C27ffiffiffi2p /C27ffiffiffi6p0.341081 /1 2(ffiffiffi 6p /C28ffiffiffi 2p )/ 0.517638 9 /1 3/ 0.333333 /12/ 0.500000 10 0.296408 0.421280 The smallest SQUARE into which two UNIT CIRCLES , one of which is split into two pieces by a chord, can be packed is not known (Goldberg 1968, Ogilvy 1990). The best known packings of circles into an equilateral triangle are shown above for the first few cases (Friedman). A rigid packing of circles can be obtained from a hexagonal tessellation by removing the centers of a hexagonal web, then replacing each remaining circle with three equal inscribed circles (appropriately oriented), as illustrated above (Meschkowski 1966, Wells 1991). If the original circles have unit radius, the lengths r, y/C28; and y/C27 can be obtained by solving r /C30y/C28 cos 30 /C14; (8) r /C27y/C28/C301 (9) y/C27/C30r tan 30 /C14; (10) givingr /C302ffiffiffi 3p /C283 (11) y/C28/C304 /C282ffiffiffi3p (12) y /C27/C302 /C28ffiffiffi3p : (13) The resulting circles cover a fraction h/C30h h2 33pr2 p12 ! /C30(7ffiffiffi 3p /C2812)p:0:390675 (14) of the plane, believed to be the smallest possible for a rigid packing of circles (Wells 1991). See also CIRCLE COVERING ,DESCARTES CIRCLE THE- OREM ,FOUR COINS PROBLEM ,H YPERSPHERE PACK- ING,M ALFATTI’S RIGHT TRIANGLE PROBLEM , MERGELYAN- WESLER THEOREM ,SANGAKU PROBLEM , SODDY CIRCLES ,SPHERE PACKING ,SQUARE PACKING , TANGENT CIRCLES ,TRIANGLE PACKING ,UNIT CELL References Boll, D. "Packing Results." http://www.frii.com/~dboll/pack- ing.html. Bowers, P. L. and Stephenson, K. "Uniformizing Dessins and Bely /Maps via Circle Packing." Preprint. Casado, L. G.and Szabo ´, P. G. "Equal Circle Packing in a Square." http://www.inf.u-szeged.hu/~pszabo/Packing_cir- cles.html. Collins, C. R. and Stephenson, K. "A Circle Packing Algo- rithm." Preprint. Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lat- tices, and Groups, 2nd ed. New York: Springer-Verlag, 1992. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1991. Donovan, J. "Packing Circles in Squares and Circles Page." http://home.att.net/~donovanhse/Packing/. Eppstein, D. "Covering and Packing." http://www.ics.u- ci.edu/~eppstein/junkyard/cover.html. Fejes To ´th, L. Lagerungen in der Ebene auf der Kugel und im Raum. Berlin: Springer-Verlag, 1953. Fejes To ´th, L. "On the Stability of a Circle Packing." Ann. Univ. Sci. Budapestinensis, Sect. Math. 3/C1/,6 3/C1/6, 1960/ 1961. Folkman, J. H. and Graham, R. "A Packing Inequality for Compact Convex Subsets of the Plane." Canad. Math. Bull. 12, 745/C1/52, 1969. Friedman, E. "Circles in Circles." http://www.stetson.edu/ ~efriedma/cirincir/. Friedman, E. "Squares in Circles." http://www.stetson.edu/ ~efriedma/squincir/. Friedman, E. "Triangles in Circles." http://www.stetson.edu/ ~efriedma/triincir/. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,1 8/C1/8, Jan. 1979. Gardner, M. "Tangent Circles." Ch. 10 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 149 /C1/66, 1992. Goldberg, M. "Problem E1924." Amer. Math. Monthly 75, 195, 1968. Goldberg, M. "The Packing of Equal Circles in a Square." Math. Mag. 43,2 4/C1/0, 1970. Goldberg, M. "Packing of 14, 16, 17, and 20 Circles in a Circle." Math. Mag. 44, 134/C1/39, 1971. Graham, R. L. and Luboachevsky, B. D. "Repeated Patterns of Dense Packings of Equal Disks in a Square." Electronic J. Combinatorics 3, R16 1 /C1/7, 1996. http://www.combina- torics.org/Volume_3/volume3.html#R16. Graham, R. L.; Luboachevsky, B. D.; Nurmela, K. J.; and O¨ sterga ˚rd, P. R. J. "Dense Packings of Congruent Circles in a Circle." Discrete Mat. 181, 139 /C1/54, 1998. Kravitz, S. "Packing Cylinders into Cylindrical Containers." Math. Mag. 40,65/C1/0, 1967. Likos, C. N. and Henley, C. L. "Complex Alloy Phases for Binary Hard-Disc Mixtures." Philos. Mag. B 68,85/C1/13, 1993. Maranas, C. D.; Floudas, C. A.; and Pardalos, P. M. "New Results in the Packing of Equal Circles in a Square." Disc. Math. 142, 287 /C1/93, 1995. McCaughan, F. "Circle Packings." http://www.pmms.cam.a- c.uk/~gjm11/cpacking/info.html. Meschkowski, H. Unsolved and Unsolvable Problems in Geometry. London: Oliver & Boyd, 1966. Molland, M. and Payan, Charles. "A Better Packing of Ten Equal Circles in a Square." Discrete Math. 84, 303 /C1/05, 1990. Nurmela, K. J. and O¨ sterga ˚rd, P. R. J. "Packing Up to 50 Equal Circles in a Square." Disc. Comput. Geom. 18, 111 /C1/ 20, 1997. Nurmela, K. J. and O¨ sterga ˚rd, P. R. J.packings/square/ . http://www.tcs.hut.fi/packings/square/. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 145, 1990. Peikert, R. "Dichteste Packungen von gleichen Kreisen in einem Quadrat." Elem. Math. 49,16/C1/6, 1994. Peikert, R.; Wu¨rtz, D.; Monagan, M.; and de Groot, C. "Packing Circles in a Square: A Review and New Results." In System Modelling and Optimization, Proceedings of the Fifteenth IFIP Conference Held at the University of Zu¨rich, September 2 /C1/, 1991 (Ed. P. Kall). Berlin: Springer-Ver- lag, pp. 45 /C1/4, 1992. Peikert, R. "Packing of Equal Circles in a Square." http:// www.inf.ethz.ch/~peikert/personal/CirclePackings/. Reis, G. E. "Dense Packing of Equal Circle within a Circle." Math. Mag. 48,33/C1/7, 1975. Schaer, J. "The Densest Packing of Nine Circles in a Square." Can. Math. Bul. 8, 273 /C1/77, 1965. Schaer, J. "The Densest Packing of Ten Equal Circles in a Square." Math. Mag. 44, 139 /C1/40, 1971. Specht, E. "The Best Known Packings of Equal Circles in the Unit Square." http://hydra.nat.uni-magdeburg.de/pack- ing/csq.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 202, 1999. Stephenson, K. "Circle Packing." http://www.math.utk.edu/ ~kens/#Packing. Stephenson, K. "Circle Packing Bibliography as of April 1999." http://www.math.utk.edu/~kens/CP-bib.ps. Stephenson, K. "Circle Packings in the Approximation of Conformal Mappings." Bull. Amer. Math. Soc. 23, 407 /C1/16, 1990. Stephenson, K. "A Probabilistic Proof of Thurston’s Con- jecture on Circle Packings." Rend. Sem. Math. Fis. Milano 66, 201 /C1/91, 1998. Valette, G. "A Better Packing of Ten Equal Circles in a Square." Discrete Math. 76,57/C1/9, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 30, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 30 /C1/1, 1991. Williams, R. "Circle Packings, Plane Tessellations, and Networks." §2.3 in The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, pp. 34 /C1/7, 1979.Circle Pedal Curve The PEDAL CURVE of a CIRCLE is a CARDIOID if the PEDAL POINT is taken on the CIRCUMFERENCE , and otherwise a LIMAC ¸ ON. Circle Point Picking A uniform distribution of points on the CIRCUMFER- ENCE of a UNIT CIRCLE can be obtained by picking two numbers x1;x2from a UNIFORM DISTRIBUTION on (/C281;1);and rejecting pairs with x2 1/C27x22]1:From the remaining points, the DOUBLE-ANGLE FORMULAS then imply that the points with C ARTESIAN COORDI- NATES x/C30x2 1/C28x22 x2 1/C27x22 y/C302x1x2 x2 1/C27x22 have the desired distribution (von Neumann 1951, Cook 1957). This method can also be extended to SPHERE POINT PICKING (Cook 1957). The plots above show the distribution of points for 50, 100, and 500initial points (where the counts refer to the number ofpoints before throwing away). See also C IRCLE COVERING BY ARCS,D ISK POINT PICKING ,SPHERE POINT PICKING References Cook, J. M. "Technical Notes and Short Papers: Rational Formulae for the Production of a Spherically Symmetric Probability Distribution." Math. Tables Aids Comput. 11, 81 /C1/2, 1957. von Neumann, J. "Various Techniques Used in Connection with Random Digits." NBS Appl. Math. Ser., No. 12. Washington, DC: U.S. Government Printing Office, pp. 36 /C1/8, 1951. Watson, G. S. and Williams, E. J. "On the Construction of Significance Tests on the Circle and Sphere." Biometrika 43, 344 /C1/52, 1956. Circle Quadrature CIRCLE SQUARING Circle Radial Curve The RADIAL CURVE of a unit CIRCLE from a RADIAL POINT (x; 0) is another CIRCLE with PARAMETRIC EQUATIONS x(t) /C30x /C28cos t y(t) /C30/C28sin t: Circle Squaring Construct a SQUARE equal in AREA to a CIRCLE using only a STRAIGHTEDGE and COMPASS . This was one of the three GEOMETRIC PROBLEMS OF ANTIQUITY , and was perhaps first attempted by Anaxagoras. It was finally proved to be an impossible problem when PI was proven to be TRANSCENDENTAL by Lindemann in 1882.’ However, approximations to circle squaring are given by constructing lengths close to p /C303 :1415926 ... : Ramanujan (1913 /C1/4), Olds (1963), Gardner (1966, pp. 92 /C1/3), and (Bold 1982, p. 45) give geometric constructions for 355=113 /C303 :1415929... : Dixon (1991) gives constructions for 6=5(1 /C27 f) /C30 3:141640... andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40=3/C282ffiffiffi3pq/C303:141533 . . . (K OCHANS- KY’S APPROXIMATION ). While the circle cannot be squared in E UCLIDEAN SPACE ,i tcan in G AUSS- BOLYAI- LOBACHEVSKY SPACE (Gray 1989). See also BANACH- TARSKI PARADOX ,GEOMETRIC CON- STRUCTION ,KOCHANSKY’S APPROXIMATION ,QUADRA- TURE ,SQUARINGReferences Bold, B. "The Problem of Squaring the Circle." Ch. 6 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 39 /C1/8, 1982. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190 /C1/91, 1996. Dixon, R. Mathographics. New York: Dover, pp. 44 /C1/9 and 52/C1/3, 1991. Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 inJourney through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 20 /C1/6, 1990. Gardner, M. "The Transcendental Number Pi." Ch. 8 in Martin Gardner’s New Mathematical Diversions fromScientific American. New York: Simon and Schuster, pp. 91 /C1 /02, 1966. Gray, J. Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd ed. Oxford, England: Oxford University Press, 1989. Hertel, E. "On the Set-Theoretical Circle-Squaring Pro- blem." http://www.minet.uni-jena.de/Math-Net/reports/ sources/2000/00 /C1/6report.ps. Jesseph, D. M. Squaring the Circle: The War Between Hobbes and Wallis. Chicago: University of Chicago Press, 1999. Klein, F. "Transcendental Numbers and the Quadrature of the Circle." Part II in "Famous Problems of ElementaryGeometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 49 /C1 /0, 1980. Meyers, L. F. "Update on William Wernick’s ‘Triangle Constructions with Three Located Points."’ Math. Mag. 69,4 6/C1/9, 1996. Olds, C. D. Continued Fractions. New York: Random House, pp. 59 /C1/0, 1963. Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/72, 1913 /C1/914. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 48, 1986. Circle Strophoid The STROPHOID of a CIRCLE with pole at the center and fixed point on the CIRCUMFERENCE is a F REETH’S NEPHROID . Circle Tangents Given the above figure, GE/C30FH, since AB/C30AG/C27GB/C30GE/C27GF/C30GE/C27(GE/C27EF) /C302GE/C27EF CD/C30CH/C27HD/C30EH/C27FH/C30FH/C27(FH/C27EF) /C30EF/C272FH: Because AB /C30CD, it follows that GE /C30FH. The line tangent to a CIRCLE of RADIUS a centered at (x, y) x?/C30x /C27a cos t y?/C30y /C27a sin t through (0, 0) can be found by solving the equation x /C27a cos t y /C27a sin tl12ml121 /C215a cos t a sin tl12ml121 /C300; giving t /C309cos/C281/C28ax 9 yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2 /C28 a2p x2 /C27 y2 ! : Two of these four solutions give tangent lines, as illustrated above, and the lengths of these lines are equal (Casey 1888, p. 29). A line tangent to two given circles at centers r1 and r2 of radii a1and a2 Ba1may be constructed by constructing the tangent to the single circle of radius a1 /C28a2centered at r1and through r2 ; then translat- ing this line along the radius through r1a distance a2 until it falls on the original two circles (Casey 1888, pp. 31 /C1/2). See also KISSING CIRCLES PROBLEM ,M IQUEL POINT , MONGE’S PROBLEM ,NINE-POINT CIRCLE ,PEDAL CIR- CLE,TANGENT CIRCLES ,TANGENT LINE,TRIANGLEReferences Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., 1888. Dixon, R. Mathographics. New York: Dover, p. 21, 1991. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 4 /C1 /, 1991. Circle Triangle Picking Select three points at random on a unit CIRCLE . Find the distribution of possible areas. The first point can be assigned coordinates (1 ;0) without loss of generality. Call the central angles from the first point to the second and third u1andu2: The range of u1can be restricted to [0 ;p] because of symmetry, but u2can range from [0 ;2p):Then A(u1;u2)/C302½sin(1 2u1) sin(12u2) sin[12(u1/C28u2)]½;(1) so ¯A/C30gp 0g2p 0A(u1;u2)du2du1 C; (2) where C/C13gp 0g2p 0du2du1/C302p2: (3) Therefore, ¯A/C302 2p2gp 0g2p 0½sin(1 2u1) sin(12u2) sin[12(u1/C28u2)]½du2du1 /C301 p2gp 0sin(1 2u1)g2p 0sin(12u2)½sin[12(u2/C28u1)]½du22 435du 1 /C301 p2gp 0g2p 0 u2/C28u1>0sin(1 2u1) sin(12u2) sin[12(u1/C28u2)]du2du1 /C271 p2gp 0g2p 0 u2/C28u1B0sin(1 2u1) sin(12u2) sin[12(u1/C28u2)]du2du1 /C301 p2gp 0sin(12u1)g2p u1sin(12u2) sin[12(u2/C28u1)]du22 435du 1 /C271 p2 g p 0sin(1 2 u1) /C2g u1 0sin(12 u2) sin[12(u2 /C28 u1)] du2"# du1 : (4) But g(12 u2)sin[12( u2 /C28 u1)] d u2 /C30g sin(12 u2)[sin(12 u2)cos(12 u2) /C28sin(12 u1) cos(12 u2)] du2 /C30cos(1 2 u1)g sin2(12 u2) du2 /C28sin(12 u1)g sin(12 u1) cos(12 u2) d u2 /C301 2cos(12 u1)g(1 /C28cos u2) du2 /C2812sin(12 u2)g sin u2 du2 (5) Write (4) as ¯A /C301 p2 g p 0sin(12 u1)I1 du1 /C27g p 0sin(12 u1)I2 du1l12ml121 ; (6) then I1 /C13g2p 0sin(12 u2) sin[12( u2 /C28 u1)] du2 ; (7) and I2 /C13g u1 0sin(12 u2) sin[12( u1 /C28 u2)] du2 : (8) From (6), I1 /C301 2cos(12 u2)[u2 /C28sin u2]2 p u1/C2712sin(12 u1)[cos u2]2 p u1 /C3012cos(12 u1)(2p /C28 u1 /C27sin u1) /C2712sin(12 u1)(1 /C28cos u1) /C30 p cos(12 u1) /C2812 u1 cos(12 u1) /C2712[cos(12 u1) sin u1 /C28cos u1sin(12 u1)] /C2712sin(12 u1) /C30 p cos(12 u1) /C2812 u1 cos(12 u1) /C2712 /C2712sin( u1 /C2812 u1) /C2712sin(12 u1) /C30 p cos(12 u1) /C2812 u1 cos(12 u1) /C27sin(12 u1) ; (9) so g p 0I1 sin(1 2 u1) d u1 /C3054 p: (10) Also,I2 /C301 2cos(12 u1)[sin u2 /C28 u2]u1 0/C2812sin(12 u1)[cos u]u1 0 /C3012cos(12 u2)(sin u1 /C28 u1) /C2812sin(12 u1)(cos u1 /C281) /C30/C2812 u1 cos(12 u1) /C2712[sin u1 cos(12 u1) /C28cos u1 sin(1 2 u2)] /C2712sin(12 u1) /C30/C281 2 u1 cos(12 u1) /C27sin(12 u1); (11) so g p 0I2 sin(1 2 u1) du1 /C3014p: (12) Combining (10) and (12) gives ¯A/C301 p25p 4/C27p 4 ! /C303 2p:0:4775 : (13) The first few moments are m?2/C3038 (14) m?3/C3041 32p(15) m?4/C3045 128; (16) so the VARIANCE is s2 A/C30/C142A/C1432/C28/C142A2/C143/C303(p2/C286) 8p2:0:1470 : (17) See also CIRCLE LINE PICKING ,D ISK TRIANGLE PICKING ,POINT- POINT DISTANCE–1- D, SPHERE POINT PICKING Circle-Circle Intersection Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points. Let two CIRCLES ofRADII Rand rand centered at (0;0) and ( d;0) intersect in a LENS -shaped region. The equations of the two circles are x2 /C27y2 /C30R2 (1) (x /C28d)2 /C27y2 /C30r2 : (2) Combining (1) and (2) gives (x /C28d)2 /C27(R2 /C27x2) /C30r2 : (3) Multiplying through and rearranging gives x2 /C282 dx /C27d2 /C28x2 /C30r2 /C28R2 : (4) Solving for x results in x /C30d2 /C28 r2 /C27 R2 2d: (5) The line connecting the cusps of the LENS therefore has half-length given by plugging x back in to obtain y2 /C30R2 /C28x2 /C30R2 /C28d2 /C28 r2 /C27 R2 2d !2 /C304d2R2 /C28 (d2 /C28 r2 /C27 R2)2 4d2 ; (6) giving a half-height y /C30a =2of a /C301 dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4d2R2 /C28(d2 /C28r2 /C27R2)2q /C301 d[(/C28d /C27r /C28R)(/C28d /C28r /C27R)(/C28d /C27r /C27R)(d /C27r /C27R)]1=2 : (7) This same formulation applies directly to the SPHERE- SPHERE INTERSECTION problem. To find the AREA of the asymmetric "LENS " in which the CIRCLES intersect, simply use the formula for the circular SEGMENT of radius R?/and triangular height d? A(R?; d?) /C30R?2 cos/C281d? R? ! /C28d?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R?2 /C28d?2p (8) twice, one for each half of the "LENS ." Noting that the heights of the two segment triangles are d1 /C30x /C30d2 /C28 r2 /C27 R2 2d (9) d2 /C30d /C28x /C30d2 /C27 r2 /C28 R2 2d: (10) The result is A /C30A(R; d1) /C27A(r ; d2) /C30r2 cos/C281d2 /C27 r2 /C28 R2 2dr ! /C27R2 cos/C281d2 /C27 R2 /C28 r2 2dR ! /C281 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (/C28d /C27r /C27R)(d /C27r /C28R)(d /C28r /C27R)(d /C27r /C27R)p : (11) The limiting cases of this expression can be checkedto give 0 when d /C30R /C27r and A /C302R2 cos/C281d 2R ! /C281 2 dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4R2 /C28d2p (12) /C302A1 2 d; Rl11)l117 (13) when r /C30R, as expected. In order for half the area of two UNIT DISKS (R /C301) to overlap, set A /C30 pR2 =2 /C30 p=2 in the above equation 12 p /C302 cos /C28112 dl11)l117 /C2812 dffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C28d2p (14) and solve numerically, yielding d :0:807946 :/ If three symmetrically placed equal circles intersect in a single point, as illustrated above, the total area of the three lens-shaped regions formed by the pairwise intersection of circles is given by A/C30p/C283 2ffiffiffi 3p : (15) Similarly, the total area of the four lens-shaped regions formed by the pairwise intersection of circles is given by A/C302(p/C282): (16) See also BORROMEAN RINGS,BROCARD TRIANGLES , CIRCLE- ELLIPSE INTERSECTION ,C IRCLE- LINE INTER- SECTION ,C IRCULAR TRIANGLE ,D OUBLE BUBBLE , GOAT PROBLEM ,LENS,R EULEAUX TRIANGLE ,SEG- MENT ,S PHERE- SPHERE INTERSECTION ,T RIQUETRA , VENN DIAGRAM Circle-Ellipse Intersection An ellipse intersects a circle in 0, 1, 2, 3, or 4 points. The points of intersection of a circle of center (x0 ; y0) and radius r with an ellipse of semi-major and semi- minor axes a and b, respectively and center (xe ; ye) can be determined by simultaneously solving (x /C28x0)2 /C27(y /C28y0)2 /C30r2 (1) (x /C28 xe)2 a2/C27(y /C28 ye)2 b2/C301: (2) If (x0 ; y0) /C30(xe ; ye) /C30(0; 0); then the solution takes on the particularly simple form x/C309affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28b2 a2/C28b2s (3) y/C309bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28r2 a2/C28b2s : (4) See also CIRCLE ,CIRCLE- CIRCLE INTERSECTION ,EL- LIPSE Circle-Line Intersection ALINE determined by two points ( x1;y1) and ( x2;y2) may intersect a CIRCLE ofRADIUS rand center (0, 0) in two imaginary points, a degenerate single point (corresponding to the line being tangent to the circle),or two real points. Defining d x/C30x2/C28x1 (1) dy/C30y2/C28y1 (2) dr/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 x/C27d2yq (3) D/C30x1x2 y1y2l112l112l112l112l112l112l112l112/C30x 1y2/C28x2y1 (4)gives the points of intersection as x/C30/C28Ddy9sgn/C31(dy)dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2d2 r/C28D2p d2 r; (5) y/C30/C28Ddx9½dy½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2d2 r/C28D2p d2r; (6) where the function sgn /C31is defined as sgn/C31(x)/C13/C281 for xB0 1 otherwise :l12) (7) The discriminant D/C13r2d2 r/C28D2(8) therefore determines the incidence of the line and circle as summarized in the following table. /D/ Incidence /DB0/no intersection /D/C300/tangent /D>0/intersection Circle-Point Midpoint Theorem Taking the locus of MIDPOINTS from a fixed point to a circle of radius rresults in a circle of radius r=2:This follows trivially from r( u) /C30/C28x 0l12ml121 /C271 2r cos u r sin ul12ml121 /C28/C28x 0l12ml121 l11sl11n /C301 2 r cos u /C2812 x 12sin u"# : References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 17, 1929. Circles-and-Squares Fractal A FRACTAL produced by iteration of the equation zn/C271 /C30z2 n (mod m) which results in a MøIRE´ -like pattern. See also FRACTAL ,MøIRE´ PATTERN Circuit GRAPH CYCLE Circuit Rank Also known as the CYCLOMATIC NUMBER . The circuit rank is the smallest number of EDGES g which must be removed from a GRAPH of N EDGES and n nodes such that no CIRCUIT remains. g /C30N /C28n /C271 : Circulant Determinant Gradshteyn and Ryzhik (2000) define circulants by x1 x2x3/C1/C1/C1 xn xn x1x2/C1/C1/C1 xn/C281 xn/C281xnx1/C1/C1/C1 xn/C282 nnn::: n x2 x3x4/C1/C1/C1 x1l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112 /C30Y j/C301(x1 /C27x2 vj /C27x3 v2 j /C27.../C27xn vn/C281 j) (1) where vj is the nth ROOT OF UNITY . The second-ordercirculant determinant is x1x2 x2x1l112l112l112l112l112l112l112l112/C30(x 1 /C27x2)(x1 /C28x2) ; (2) and the third order is x1x2x3 x3x1x2 x2x3x1l112l112l112l112l112l112l112l112l112l112l112l112/C30(x 1 /C27x2 /C27x3)(x1 /C27 vx2 /C27 v2x3) /C2(x1 /C27 v2x2 /C27 vx3) ; (3) where v and v2 are the COMPLEX CUBE ROOTS of UNITY . The EIGENVALUES l of the corresponding n /C29n CIR- CULANT MATRIX are lj /C30x1 /C27x2 vj /C27x3 v2 j /C27.../C27xn vn/C281 j: (4) See also CIRCULANT MATRIX References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1111 /C1/112, 2000. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 114, 1991. Circulant Graph AGRAPH ofnVERTICES in which the ithVERTEX is adjacent to the ( i/C27j)/th and ( i/C28j)/thVERTICES for each jin a list l. The circulant graph Ci1;2;...;n=2bc(n) gives the COMPLETE GRAPH Knand the graph Ci1(n) gives the CYCLIC GRAPH Cn:/ The number of circulant graphs on n/C301, 2, ... nodes (counting empty graphs) are given by 1, 2, 2, 4, 3, 8, 4, 12, ... (Sloane’s A049287). Note that these numbers cannot be counted simply by enumerating the num- ber of nonempty subsets of f1 ; 2 ; ...; n=2bc g since, for example, Ci1(5) /C30Ci2(5) /C30C5 : There is an easy for- mula for prime orders, and formulas are known for squarefree and prime-squared orders. Special cases are summarized in the table below. Graph Symbol OCTAHEDRAL GRAPH /Ci1; 2(6) / 16-CELL /Ci1; 2; 3(8) / See also 16-CELL,OCTAHEDRAL GRAPH References Buckley, F. and Harary, F. Distances in Graphs. Redwood City, CA: Addison-Wesley, 1990. Liskovets, V. A.; and Po¨schel, R. "On the Enumeration of Circulant Graphs of Prime-Power and Square-Free Or- ders." Preprint. MATH-AL-8 /C1/996, TU-Dresden. Klin, M.; Liskovets, V.; and Po¨schel, R. "Analytical Enu- meration of Circulant Graphs with Prime-Squared Num- ber of Vertices." Se´m. Lothar. Combin. 36, Art. B36d, 1996. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 99 and 140, 1990. Zhou, A. and Zhang, X. D. "Enumeration of Circulant Graphs with Order n and Degree 4 or 5/" [Chinese]. Dianzi Keji Daxue Xuebao 25, 272 /C1/76, 1996. Circulant Matrix An n /C29n MATRIX C defined as follows, Cn/C301 n 1ðÞn2ðÞ /C1/C1/C1n n/C281ðÞ n n/C281ðÞ 1n1ðÞ/C1/C1/C1n n/C282ðÞ nnn::: n n 1ðÞn2ðÞn3ðÞ/C1/C1/C1 12 6643 775; where n kl1ml11 is a BINOMIAL COEFFICIENT . The DETERMI- NANT of Cn is given by the beautiful formula Cn/C30Yn/C281 j/C300[( 1/C27 vj)n/C281]; where v0 /C131; v1 ; ..., vn /C281are the nth ROOTS OF UNITY . The determinants for n /C301, 2, ..., are given by 1, /C283, 28, /C28375, 3751, 0, 6835648, /C281343091375, 364668913756, ... (Sloane’s A048954), which is 0 when n /C130 (mod 6):/ Circulant matrices are examples of LATIN SQUARES . See also CIRCULANT DETERMINANT References Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 1994. Sloane, N. J. A. Sequences A048954 and A049287 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes’ Folium." Math. Mag. 61, 172 /C1/87, 1988. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 114, 1991. Circular Chessboard A circular pattern obtained by superposing parallel equally spaced lines on a set of concentric circles of increasing radii, then coloring the regions in chess- board fashion. The pattern appeared on the cover of early editions of Scripta Mathematica. See also CHESSBOARD References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 243 /C1/45 and 249 /C1/51, 1984. Circular Cylinder CYLINDER Circular Cylindrical Coordinates CYLINDRICAL COORDINATES Circular Functions The functions describing the horizontal and vertical positions of a point on a CIRCLE as a function of ANGLE (COSINE and SINE) and those functions derived from them: cotx/C131 tanx/C30cosx sinx(1) cscx/C131 sinx(2) secx/C131 cosx(3) tanx/C13sinx cosx: (4) Circular functions are also called TRIGONOMETRIC FUNCTIONS , and the study of circular functions is called TRIGONOMETRY . See also COSECANT ,COSINE ,COTANGENT ,ELLIPTIC FUNCTION ,G ENERALIZED HYPERBOLIC FUNCTIONS , HYPERBOLIC FUNCTIONS ,SECANT ,SINE,T ANGENT , TRIGONOMETRIC FUNCTIONS ,TRIGONOMETRY References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/9, 1972. Circular Permutation The number of ways to arrange n distinct objects along a FIXED (i.e., cannot be picked up out of the plane and turned over) CIRCLE is Pn /C30(n /C281)!: The number is (n /C281)! instead of the usual FACTORIAL n! since all CYCLIC PERMUTATIONS of objects are equivalent because the CIRCLE can be rotated. For example, of the 3! /C306 permutations of three objects, the (3 /C281)! /C302 distinct circular permutations are f1 ; 2 ; 3g and f1; 3; 2g: Similarly, of the 4! /C3024 permutations of four objects, the (3 /C281)! /C306 distinct circular permutations are f1; 2; 3; 4g;f1 ; 2; 4; 3g; f1; 3; 2; 4g; f1; 3; 4; 2g; f1; 4 ; 2 ; 3 g; and f1; 4; 3; 2g: Of these, there are only three FREE permutations (i.e., inequivalent when flipping the circle is allowed): f1; 2; 3; 4g;f1 ; 2 ; 4 ; 3 g; and f1; 3; 2; 4g: The number of free circular permuta- tions of order n is P ?n /C301 for n /C301, 2, and P?n /C301 2(n /C281)! for n ]3; giving the sequence 1, 1, 1, 3, 12, 60, 360, 2520, ... (Sloane’s A001710). See also CYCLIC PERMUTATION ,FACTORIAL ,PERMUTA- TION ,PRIME CIRCLE References Sloane, N. J. A. Sequences A001710/M2933 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Circular Reciprocation RECIPROCATIONCircular Triangle A triangle ABC formed by three circular ARCS .By extending the arcs into complete circles, the points of intersection A?; B ?; and C ? are obtained. This gives the three circular triangles, A?B ?C ?; AB ?C?; A?BC?; and A?B?C ; which are called the ASSOCIATED TRIANGLES to ABC . In addition, circular triangles A?B?C ?; AB ?C ?; A?BC?; and A?B ?C can also be drawn. The circular triangle and its associated circles have a total of eight INCIRCLES and six CIRCUMCIRCLE . These systems of circles have some remarkable properties, including the HART CIRCLE , which is an analog of the NINE-POINT CIRCLE in FEUERBACH’S THEOREM . See also APOLLONIUS’ PROBLEM ,A RC,A SSOCIATED TRIANGLES ,CIRCLE- CIRCLE INTERSECTION ,FEUERBA- CH’S THEOREM ,H ART CIRCLE ,H ARUKI’S THEOREM , NINE-POINT CIRCLE ,SPHERICAL TRIANGLE ,TRIQUE- TRA References Lachlan, R. "Properties of a Circular Triangle." §397/C1/04 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 251 /C1/57, 1893. Circular-Cylinder Coordinates CYLINDRICAL COORDINATES Circumcenter The center O of a TRIANGLE’S CIRCUMCIRCLE . It can be found as the intersection of the PERPENDICULAR BISECTORS . If the TRIANGLE is ACUTE , the circumcen- ter is in the interior of the TRIANGLE .Ina RIGHT TRIANGLE , the circumcenter is the MIDPOINT of the HYPOTENUSE . OO1 /C27OO2 /C27OO3 /C30R /C27r ; (1) where Oiare the MIDPOINTS of sides Ai ; R is the CIRCUMRADIUS , and r is the INRADIUS (Johnson 1929, p. 190). The TRILINEAR COORDINATES of the circum- center are cos A : cos B : cos C ; (2) and the exact trilinears are therefore R cos A : R cos B : R cos C : (3) The AREAL COORDINATES are (1 2 a cot A;12 b cot B ;12 c cot C) : (4) The distance between the INCENTER and circumcenter isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R(R /C282r)p : Given an interior point, the distances to the VERTICES are equal IFF this point is the circumcenter. It lies on the BROCARD AXIS. The circumcenter O and ORTHOCENTER H are ISO- GONAL CONJUGATES . The ORTHOCENTER H of the PEDAL TRIANGLE DO1O2O3formed by the CIRCUMCENTER Oconcurs with the circumcenter Oitself, as illustrated above. The circumcenter also lies on the E ULER LINE . See also BROCARD DIAMETER ,C ARNOT’S THEOREM , CENTROID (TRIANGLE ), CIRCLE ,EULER LINE,INCEN- TER,LESTER CIRCLE ,ORTHOCENTER References Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 623, 1970. Dixon, R. Mathographics. New York: Dover, p. 55, 1991. Eppstein, D. "Circumcenters of Triangles." http://www.ics.u- ci.edu/~eppstein/junkyard/circumcenter.html. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Kimberling, C. "Circumcenter." http://cedar.evansville.edu/ ~ck6/tcenters/class/ccenter.html. Circumcircle ATRIANGLE’S circumscribed circle. Its center Ois called the CIRCUMCENTER , and its RADIUS Rthe CIRCUMRADIUS . The circumcircle can be specified using TRILINEAR COORDINATES as bga/C27gab/C27abc/C300: (1) The S TEINER POINT Sand T ARRY POINT Tlie on the circumcircle. When an arbitrary point P is taken on the circum- circle, then the feet P1 ; P2 ; and P3of the perpendi- culars from P to the sides (or their extensions) of the TRIANGLE are COLLINEAR on a line called the SIMSON LINE. Furthermore, the reflections PA ; PB ; PCof any point P on the CIRCUMCIRCLE taken with respect to the sides BC, AC, AB of the triangle are COLLINEAR , not only with each other but also with the ORTHO- CENTER H (Honsberger 1995, pp. 44 /C1/7). The tangent to a triangle’s circumcircle at a vertex is ANTIPARALLEL to the opposite side, the sides of the ORTHIC TRIANGLE are parallel to the tangents to the circumcircle at the vertices, and the radius of the circumcircle at a vertex is perpendicular to all lines ANTIPARALLEL to the opposite sides (Johnson 1929, pp. 172 /C1/73). A GEOMETRIC CONSTRUCTION for the circumcircle is given by Pedoe (1995, pp. xii-xiii). The equation for the circumcircle of the TRIANGLE with VERTICES (xi ; yi) for i /C301, 2, 3 is x2 /C27y2xy 1 x2 1 /C27y21x1y11 x22 /C27y22x2y21 x23 /C27y23x3y31l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112/C300 : (2) Expanding the DETERMINANT , a(x2 /C27y2) /C272dx /C272fy /C27g /C300; (3) where a /C30x1y11 x2y21 x3y31l112l112l112l112l112l112l112l112l112l112l112l112(4) d /C30/C28 1 2x2 1 /C27y21y11 x22 /C27y22y21 x23 /C27y23y31l112l112l112l112l112l112l112l112l112l112l112l112(5) f /C30 1 2x2 1 /C27y21x11 x22 /C27y22x21 x23 /C27y23x31l112l112l112l112l112l112l112l112l112l112l112l112(6) g /C30/C28x 2 1 /C27y21x1y1 x2 2 /C27y22x2y2 x2 3 /C27y23x3y3l112l112l112l112l112l112l112l112l112l112l112l112: (7) COMPLETING THE SQUARE givesax/C27d a !2 /C27ay/C27f a !2 /C28d2 a/C28f2 a/C27g /C300 (8) which is a CIRCLE OF THE FORM (x /C28x0)2 /C27(y /C28y0)2 /C30r2 ; (9) with CIRCUMCENTER x0 /C30/C28d a (10) y0 /C30/C28f a (11) and CIRCUMRADIUS r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f2 /C27 d2 a2/C28g as : (12) If a polygon with side lengths a, b, c, ... and standard trilinear equations a /C300; b /C300 ; g /C300; ... has a cir- cumcircle, then for any point of the circle, a a /C27b b /C27c g /C27.../C300 (13) (Casey 1878, 1893). See also CIRCLE ,C IRCUMCENTER ,C IRCUMRADIUS , EXCIRCLE ,INCIRCLE ,PARRY POINT ,PIVOT THEOREM , PURSER’S THEOREM ,SIMSON LINE,STEINER POINTS , TARRY POINT References Casey, J. Trans. Roy. Irish Acad. 26, 527 /C1/10, 1878. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 128 /C1/29, 1893. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 7, 1967. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., 1995. Lachlan, R. "The Circumcircle." §118 /C1/22 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 66 /C1/0, 1893. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Circumference The PERIMETER of a CIRCLE . For RADIUS rorDIAMETER d/C302r; C/C302pr/C30pd; where pisPI. See also CIRCLE ,DIAMETER ,GRAPH CIRCUMFERENCE , PERIMETER ,PI,RADIUS Circumflex HAT Circuminscribed Given two CLOSED CURVES , the circuminscribed curve is simultaneously INSCRIBED in the outer one and CIRCUMSCRIBED on the inner one. See also PONCELET’S PORISM ,STEINER CHAIN Circumradius The radius of a TRIANGLE’S CIRCUMCIRCLE or of a POLYHEDRON ’s CIRCUMSPHERE , denoted R. For a TRIANGLE , R /C30abcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a /C27 b /C27 c)(b /C27 c /C28 a)(c /C27 a /C28 b)(a /C27 b /C28 c)p (1) where the side lengths of the TRIANGLE are / a ; b; and c/. This equation can also be expressed in terms of the RADII of the three mutually tangent CIRCLES centered at the TRIANGLE’S VERTICES . Relabeling the diagram for the SODDY CIRCLES with VERTICES O1 ; O2 ; and O3 and the radii r1 ; r2 ; and r3 ; and using a /C30r1 /C27r2 (2) b /C30r2 /C27r3 (3) c /C30r1 /C27r3 (4)then gives R /C30(r1 /C27 r2)(r1 /C27 r3)(r2 /C27 r3) 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1r2r3(r1 /C27 r2 /C27 r3)p : (5) If O is the CIRCUMCENTER and M is the triangle CENTROID , then OM2 /C30R2 /C281 9(a2 /C27b2 /C27c2): (6) Rr /C30abc 4s (7) cos a1 /C27cos a2 /C27cos a3 /C301 /C27r R (8) r /C302R cos a1 cos a2 cos a3 (9) (Johnson 1929, pp. 189 /C1/91). Let d be the distance between INRADIUS r and circumradius R, d /C30rR : Then R2 /C28d2 /C302Rr (10) 1 R /C28 d /C271 R /C27 d /C301 r (11) (Mackay 1886 /C1/7; Casey 1888, pp. 74 /C1/5). These and many other identities are given in Johnson (1929, pp. 186 /C1/90). The HYPOTENUSE of a RIGHT TRIANGLE is a DIAMETER of the triangle’s CIRCUMCIRCLE , so the circumradius is given by R /C3012 c ; (12) where cis the HYPOTENUSE . For an A RCHIMEDEAN SOLID , expressing the circum- radius in terms of the INRADIUS rand MIDRADIUS r gives R/C301 2(r/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C27a2)p (13) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C271 4a2q (14) for an A RCHIMEDEAN SOLID . See also CARNOT’S THEOREM ,CIRCUMCIRCLE ,CIRCUM- SPHERE ,INCIRCLE ,INRADIUS References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Mackay, J. S. "Historical Notes on a Geometrical Theorem and its Developments [18th Century]." Proc. Edinburgh Math. Soc. 5,6 2/C1/8, 1886 /C1/887. Circumscribed A geometric figure which touches only the vertices (or other extremities) of another figure. See also CIRCUMCENTER ,CIRCUMCIRCLE ,CIRCUMIN- SCRIBED ,CIRCUMRADIUS ,INSCRIBED Circumsphere A SPHERE circumscribed in a given solid. Its radius is called the CIRCUMRADIUS . The figures above depict the circumspheres of the Platonic solids. See also INSPHERE ,MIDSPHERE Cis Another name for the complex exponential, Cis x /C13eix /C30cos x /C27i sin x: See also EXPONENTIAL FUNCTION ,PHASOR Cissoid Given two curves C1 and C2 and a fixed point O, let a line from O cut C1 at Q and C2 at R. Then the LOCUS of a point P such that OP /C30QR is the cissoid. The word cissoid means "ivy shaped." Curve 1Curve2Pole Cissoid LINE PARALLEL LINEany point line LINE CIRCLE center CONCHOID OF NICOMEDES CIRCLE tangent lineon CIRCUM- FERENCEobliquecissoid CIRCLE tangentlineon CIRCUM- FERENCE opp. tan- gentCISSOID OF DIOCLES CIRCLE radial line on CIRCUM- FERENCEstrophoid CIRCLE concentric CIRCLEcenter CIRCLE CIRCLE same CIRCLE/(affiffiffi 2p ;0)/ LEMNISCATE See also CISSOID OF DIOCLESReferences Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 53 /C1/6 and 205, 1972. Lockwood, E. H. "Cissoids." Ch. 15 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 130 /C1/33, 1967. Yates, R. C. "Cissoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 26 /C1/0, 1952. Cissoid of Diocles A curve invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube bygeometrical methods. The name "cissoid" first ap-pears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found, in 1658, that the AREA between the curve and its asymptote was 3 a (MacTutor Archive). From a given point there areeither one or three TANGENTS to the cissoid. Given an origin Oand a point Pon the curve, let Sbe the point where the extension of the line OPinter- sects the line x/C302aandRbe the intersection of the CIRCLE of RADIUS aand center ( a;0) with the extension of OP. Then the cissoid of Diocles is the curve which satisfies OP/C30RS. The cissoid of Diocles is the ROULETTE of the VERTEX of a PARABOLA rolling on an equal PARABOLA . Newton gave a method of drawing the cissoid of Diocles using two line segments of equal length at RIGHT ANGLES .I f they are moved so that one line always passes through a fixed point and the end of the other line segment slides along a straight line, then the MID- POINT of the sliding line segment traces out a cissoid of Diocles. The cissoid of Diocles is given by the PARAMETRIC EQUATIONS x/C302asin2u (1) y/C302asin3u cosu: (2) Converting these to POLAR COORDINATES gives r2 /C30x2 /C27y2 /C304a2sin4 u /C27sin6 u cos2 u ! /C304a2 sin4 u(1 /C27tan2 u) /C304a2 sin4 u sec2 u; (3) so r /C302a sin2 u sec u /C302a sin u tan u: (4) In CARTESIAN COORDINATES , x3 2a /C28 x /C308a3 sin6 u 2a /C28 2a sin2 u /C304a2sin6 u 1 /C28 sin2 u /C304a2sin6 u cos2 u /C30y2 : (5) An equivalent form is x(x2 /C27y2) /C302ay2 : (6) Using the alternative parametric form x(t) /C302at2 1 /C27 t2 (7) y(t) /C302at3 1 /C27 t2 (8) (Gray 1997), gives the CURVATURE as k(t) /C303 a ½t½(t2 /C27 4)3=2 : (9) References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 214, 1987. Gray, A. "The Cissoid of Diocles." §3.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 57 /C1/1, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 98 /C1/00, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 130 /C1/33, 1967. MacTutor History of Mathematics Archive. "Cissoid of Diocles." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Cissoid.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 34, 1986. Yates, R. C. "Cissoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 26 /C1/0, 1952. Cissoid of Diocles Caustic The CAUSTIC of the cissoid where the RADIANT POINT is taken as (8a; 0) is a CARDIOID . Cissoid of Diocles Inverse Curve If the cusp of the CISSOID OF DIOCLES is taken as the INVERSION CENTER , then the cissoid inverts to a PARABOLA .Cissoid of Diocles Pedal Curve The PEDAL CURVE of the cissoid, when the PEDAL POINT is on the axis beyond the ASYMPTOTE at a distance from the cusp which is four times that of the ASYMPTOTE is a CARDIOID . C-k Function A function with k CONTINUOUS DERIVATIVES is called a Ck function. In order to specify a Ck function on a domain X, the notation Ck(X) is used. The most common Ck space is C0 ; the space of CONTINUOUS FUNCTIONS , whereas C1 is the space of CONTINUOUSLY DIFFERENTIABLE FUNCTIONS . Cartan (1977, p. 327) writes humorously that "by ‘differentiable,’ we mean of class Ck ; with k being as large as necessary." Of course, any SMOOTH FUNCTION is Ck ; and when l /C21k, then any Cl function is Ck : It is natural to think of a Ck function as being a little bit rough, but the graph of a C3function "looks" smooth. Examples of Ckfunctions are ½x½k/C271(forkeven) and xk/C271sin(1 =x);which do not have a ( k/C271)/st derivative at 0. The notion of Ckfunction may be restricted to those whose first kderivatives are BOUNDED functions. The reason for this restriction is that the set of Ck functions has a NORM which makes it a B ANACH SPACE , ½½f½½Ck(X)/C30Xk n/C300sup x/C23X½f(n)(x)½: See also BANACH SPACE ,C -INFINITY FUNCTION , CALCULUS ,C ONTINUOUSLY DIFFERENTIABLE FUNC- TION ,CONTINUOUS FUNCTION ,DIFFERENTIAL EQUA- TION ,REGULARITY (PDE) References Cartan, H. Cours de calcul diffe´rentiel. Paris: Hermann, 1977. Krantz, S. G. "Continuously Differentiable and Ck Func- tions." §1.3.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 12 /C1/3, 1999. Clairaut’s Difference Equation This entry contributed by RONALD M. AARTS Clairaut’s difference equation is a special case of Lagrange’s equation (Sokolnikoff and Redheffer 1958) defined by uk /C30kDuk /C27F( Duk) ; or in "x notation," y /C30xDy Dx /C27FDy Dx ! (Spiegel 1970). It is so named by analogy with CLAIRAUT’S DIFFERENTIAL EQUATION y /C30xdy dx /C27Fdydx ! : See also C LAIRAUT’S DIFFERENTIAL EQUATION References Sokolnikoff, I. S. and Redheffer, R. M. Mathematics of Physics and Modern Engineering. New York: McGraw- Hill, 1958. Spiegel, M. R. Schaum’s Outline of Theory and Problems of Calculus of Finite Differences and Difference Equations. New York: McGraw-Hill, 1970. Clairaut’s Differential Equation y /C30xdy dx /C27fdydx ! (1) or y /C30px /C27f(p); (2) where f is a FUNCTION of one variable and p /C13dy=dx: The general solution is y /C30cx /C27f(c) : (3) The singular solution ENVELOPES are x /C30/C28f ?(c) and y /C30f(c) /C28cf ?(c):/ A PARTIAL DIFFERENTIAL EQUATION known as Clair- aut’s equation is given by u /C30xux /C27yuy /C27f(ux ; uy) (4)(Iyanaga and Kawada 1980, p. 1446; Zwillinger 1997, p. 132). See also CLAIRAUT’S DIFFERENCE EQUATION , D’ALEM- BERT’S EQUATION References Boyer, C. B. A History of Mathematics. New York: Wiley, p. 494, 1968. Ford, L. R. Differential Equations. New York: McGraw-Hill, p. 16, 1955. Ince, E. L. Ordinary Differential Equations. New York: Dover, pp. 39 /C1/0, 1956. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1446, 1980. Zwillinger, D. "Clairaut’s Equation." §II.A.38 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 120 and 158 /C1/60, 1997. Clarity The RATIO of a measure of the size of a "fit" to the size of a "residual." References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 667, 1977. Clark’s Triangle ANUMBER TRIANGLE created by setting the vertex equal to 0, filling one diagonal with 1s, the other diagonal with multiples of an INTEGER f, and filling in the remaining entries by summing the elements on either side from one row above. Call the first column n/C300 and the last column m/C30nso that c(m;0)/C30fm (1) c(m;m)/C301 (2) then use the RECURRENCE RELATION c(m;n)/C30c(m/C281;n/C281)/C27c(m/C281;n) (3) to compute the rest of the entries. For n/C301, we have c(m;1)/C30c(m/C281;0)/C27c(m/C281;1) (4) c(m;1)/C28c(m/C281;1)/C30c(m/C281;0)/C30f(m/C281):(5) For arbitrary m, the value can be computed by SUMMING this RECURRENCE , c(m; 1) /C30fXm/C281 k/C301k ! /C271 /C301 2 fm(m /C281) /C271: (6) Now, for n /C302 we have c(m; 2) /C30c(m /C281; 1) /C27c(m /C281; 2) (7) c(m; 2) /C28c(m /C281; 2) /C30c(m /C281 ; 1) /C3012 f(m /C281)m /C271 ; (8) so SUMMING the RECURRENCE gives c(m; 2) /C30Xm/C281 k /C301[1 2 fk(k /C281) /C271] /C30Xm k/C301(12 fk2 /C2812 fk /C271) /C301 2 f[16 m(m /C271)(2m /C271)] /C2812 f[12 m(m /C271)] /C27m /C3016(m /C281)(fm2 /C282fm /C276): (9) Similarly, for n /C303 we have c(m; 3) /C28c(m /C281 ; 3) /C30c(m /C281; 2) /C3016 fm3 /C28fm2 /C27(11 6f /C271)m /C28(f /C272): (10) Taking the SUM, c(m; 3) /C30Xm k /C30216 fk3 /C28fk2 /C27(11 6f /C271)k /C28(f /C272): (11) Evaluating the SUM gives c(m; 3) /C301 24(m /C281)(m /C282)(fm2 /C283fm /C2712): (12) So far, this has just been relatively boring ALGEBRA . But the amazing part is that if f /C306 is chosen as the INTEGER , then c(m; 2) and c(m; 3) simplify to c(m; 2) /C3016(m /C281)(6m2 /C2812m /C276) /C30(m /C281)3(13) c(m; 3) /C3014(m /C281)2(m /C282)2 ; (14) which are consecutive CUBES (m /C281)3 and nonconse- cutive SQUARES n2 /C30[(m /C281)(m /C282)=2]2 :/ See also BELL TRIANGLE ,C ATALAN’S TRIANGLE , EULER’S TRIANGLE ,L EIBNIZ HARMONIC TRIANGLE , LOSSNITSCH’S TRIANGLE ,N UMBER TRIANGLE ,P AS- CAL’S TRIANGLE ,SEIDEL- ENTRINGER- ARNOLD TRIAN- GLE,SUM References Clark, J. E. "Clark’s Triangle." Math. Student 26, No. 2, p. 4, Nov. 1978. Class The word "class" has many specialized meanings in mathematics in which it refers to a group of objects with some common property (e.g., CHARACTERISTIC CLASS or CONJUGACY CLASS .) In statistics, a class is a grouping of values by which data is binned for computation of a FREQUENCY DISTRIBUTION (Kenney and Keeping 1962, p. 14). The range of values of a given class is called a CLASS INTERVAL , the boundaries of an interval are called CLASS LIMITS , and the middle of a CLASS INTERVAL is called the CLASS MARK . class intervalclassmarkabsolutefrequencyrelativefrequencycumulativeabsolutefrequencyrelativecumulativefrequency 0.00 /C1 / 9.995 1 0.01 1 0.01 10.00 /C1/ 9.9915 3 0.03 4 0.04 20.00 /C1/ 9.9925 8 0.08 12 0.12 30.00 /C1/ 9.9935 18 0.18 30 0.30 40.00 /C1/ 9.9945 24 0.24 54 0.54 50.00/C1/ 9.9955 22 0.22 76 0.76 60.00/C1/ 9.9965 15 0.15 91 0.91 70.00/C1/ 9.9975 8 0.08 99 0.99 80.00/C1/ 9.9985 0 0.00 99 0.99 90.00/C1/ 9.9995 1 0.01 100 1.00 See also CHARACTERISTIC CLASS,CLASS BOUNDARIES , CLASS GROUP FACTORIZATION METHOD ,CLASS INTER- VAL,CLASS LIMITS ,CLASS MARK,CLASS (MULTIPLY PERFECT NUMBER ), CLASS NUMBER ,C LASS (SET), CONJUGACY CLASS ,FREQUENCY DISTRIBUTION Class (Group) CONJUGACY CLASS Class (Map) A MAP u : Rn 0 Rn from a DOMAIN G is called a map of class Cr if each component of u(x) /C30(u1(x1 ; ...; xn) ; ...; um(x1 ; ...; xn)) is of class Cr (0 5r 5/C12 or r /C30 v)in G, where Cd denotes a continuous function which is differentiable d times. Class (Multiply Perfect Number) The number k in the expression s(n) /C30kn for a MULTIPLY PERFECT NUMBER is called its class. See also MULTIPLY PERFECT NUMBER Class (Set) A class is a generalized set invented to get around RUSSELL’S PARADOX while retaining the arbitrary criteria for membership which leads to difficulty for SETS . The members of classes are SETS , but it is possible to have the class C of "all SETS which are not members of themselves" without producing a PARA- DOX (since C is a PROPER CLASS (and not a SET), it is not a candidate for membership in C). The distinction between classes and sets is a concept from VON NEUMANN- BERNAYS- GO¨ DEL SET THEORY . See also AGGREGATE ,P ROPER CLASS ,R USSELL’S PARADOX ,SET,TYPE, VON NEUMANN- BERNAYS- GO¨ DEL SET THEORY References Gonseth, F. "Faiblesse des ide´es ge´ne´rales de classe et d’attribut." §108 in Les mathe ´matiques et la re´alite´: Essai sur la me´thode axiomatique. Paris: Fe´lix Alcan, pp. 259 /C1/ 61, 1936. Class Boundaries Because of rounding, the stated CLASS LIMITS do not correspond to the actual ranges of data falling in them. For example, if the CLASS LIMITS are 1.00 and 2.00, then all values between 0.95 and 2.05 would actually fall in the given CLASS , so the class bound- aries are 0.95 and 2.05 (Kenney and Keeping 1962, p. 17). See also CLASS LIMITS References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 17, 1962. Class Field See also CLASS FIELD THEORY Class Field Theory See also CLASS FIELD,CLASS NUMBER ,RECIPROCITYLAW References Garbanati, D. "Class Field Theory Summarized." Rocky Mtn. J. Math. 11, 195 /C1/25, 1981. Hazewinkel, M. "Local Class Field Theory is Easy." Adv. Math. 18, 148 /C1/81, 1975. Class Group Factorization Method A PRIME FACTORIZATION ALGORITHM . References Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673 /C1/15, 1990. Class Interval One of the ranges into which data in a FREQUENCY DISTRIBUTION table (or HISTOGRAM ) are BINNED . The ends of a class interval are called CLASS LIMITS , and the middle of an interval is called a CLASS MARK . See also BIN,C LASS BOUNDARIES ,C LASS LIMITS , CLASS MARK,HISTOGRAM ,SHEPPARD’S CORRECTION References Kenney, J. F. and Keeping, E. S. "Class Intervals." §1.9 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 15 /C1/7, 1962. Class Limits The end values which specify a CLASS INTERVAL . See also CLASS BOUNDARIES ,CLASS INTERVAL References Kenney, J. F. and Keeping, E. S. "Class Limits and Class Boundaries." §1.10 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 17, 1962. Class Mark The average of the values of the CLASS LIMITS for a given class. A class mark is also called a midvalue or central value (Kenney and Keeping 1962, p. 14), and is commonly denoted xc :/ See also CLASS INTERVAL ,CLASS LIMITS References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 14, 1962. Class Number For any IDEAL I, there is an IDEAL Iisuch that IIi/C30z; (1) where zis a PRINCIPAL IDEAL , (i.e., an IDEAL of rank 1). Moreover, there is a finite list of ideals Iisuch that this equation may be satisfied for every I. The size of this list is known as the class number. When the class number is 1, the RING corresponding to a given IDEAL has unique factorization and, in a sense, the class number is a measure of the failure of unique factorization in the original number ring. A finite series giving exactly the class number of a RING is known as a CLASS NUMBER FORMULA .A CLASS NUMBER FORMULA is known for the full ring of cyclotomic integers, as well as for any subring of the cyclotomic integers. Finding the class number is acomputationally difficult problem. Leth(d) denote the class number of a quadratic ring, corresponding to the BINARY QUADRATIC FORM ax2/C27bxy/C27cy2; (2) with DISCRIMINANT d/C13b2/C284ac: (3) Then the class number h(d) for DISCRIMINANT dgives the number of possible factorizations of ax2/C27bxy/C27 cy2in the QUADRATIC FIELD Q(ffiffiffi dp ):Here, the factors are of the form x/C27yffiffiffidp ;with xandyhalf INTEGERS . Some fairly sophisticated mathematics shows that the class number for discriminant dcan be given by the CLASS NUMBER FORMULA h(d)/C13/C281 2l nh(d)Xd/C281 r/C301(d=r)ln sinpr d ! ford>0 /C28w(d) 2½d½X½d½/C281 r/C301(d=r)r fordB0;8 >>>>< >>>>: (4) where ( d=r) is the K RONECKER SYMBOL ,h(d) is the FUNDAMENTAL UNIT ,w(d) is the number of substitu- tions which leave the BINARY QUADRATIC FORM un- changed w(d)/C306 for d/C30/C283 4 for d/C30/C284 2 otherwise ;8 < :(5) and the sums are taken over all terms where the KRONECKER SYMBOL is defined (Cohn 1980). The class number for d/C210 can also be written h2h(d)/C30Yd/C281 r/C301sin/C28(d=r)pr d ! (6) ford/C210, where the PRODUCT is taken over terms for which the K RONECKER SYMBOL is defined. The class number h(d) is related to the D IRICHLET L- SERIES by h(d)/C30Ld(1) k(d); (7) where k(d) is the D IRICHLET STRUCTURE CONSTANT .Oesterle ´(1985) showed that class number h(/C28d) satisfies the INEQUALITY h(/C28d)>1 7000Y p½d/C311/C282ffiffiffippl1=l1; p/C271 ! lnd; (8) for/C28dB0;where xbcis the FLOOR FUNCTION , the product is over PRIMES dividing d, and the +indicates that the GREATEST PRIME FACTOR ofdis omitted from the product. It is also known that if disRELATIVELY PRIME to 5077, then the denominator 7000 in (8) can be replaced by 55. The Mathematica function NumberTheory‘Num- berTheoryFunctions‘ClassNumber[ n] gives the class number h(d) for daNEGATIVE SQUAREFREE number OF THE FORM 4k/C271:/ GAUSS’S CLASS NUMBER PROBLEM asks to determine a complete list of fundamental DISCRIMINANTS /C28dsuch that the CLASS NUMBER is given by h(/C28d)/C30nfor a given n. This problem has been solved for n57 and ODD n523:Gauss conjectured that the class number h(/C28d)o fa n IMAGINARY QUADRATIC FIELD with DIS- CRIMINANT /C28dtends to infinity with d, an assertion now known as G AUSS’S CLASS NUMBER CONJECTURE . The discriminants dhaving h(/C28d)/C301;2, 3, 4, 5, ... are Sloane’s A014602 (Cohen 1993, p. 229; Cox 1997, p. 271), Sloane’s A014603 (Cohen 1993, p. 229), Sloa-ne’s A006203 (Cohen 1993, p. 504), Sloane’s A013658 (Cohen 1993, p. 229), Sloane’s A046002, Sloane’s A046003, .... The complete set of negative discrimi-nants having class numbers 1 /C1 /and ODD 7/C1/3 are known. Buell (1977) gives the smallest and largestfundamental class numbers for dB4;000;000;par- titioned into EVEN discriminants, discriminants 1 (mod 8), and discriminants 5 (mod 8). Arno et al. (1993) give complete lists of values of dwith h(/C28d)/C30k forODD k/C305, 7, 9, ..., 23. Wagner gives complete lists of values for k/C305, 6, and 7. Lists of NEGATIVE discriminants corresponding to IMAGINARY QUADRATIC FIELDS Q(ffiffiffiffiffiffiffiffiffiffiffiffiffi /C28d(n)p ) having small class numbers h(/C28d) are given in the table below. In the table, Nis the number of "fundamental" values of /C28dwith a given class number h(/C28d);where "fundamental" means that /C28dis not divisible by any SQUARE NUMBER s2such that h(/C28d=s2)Bh(/C28d):For example, although h(/C2863)/C302;-63 is not a funda- mental discriminant since 63 /C3032/C2157 and h(/C2863=32)/C30h(/C287)/C301Bh(/C2863):EVEN values 8 5 h(/C28d)524 have been computed by Weisstein. The number of negative discriminants having class num-ber 1, 2, 3, ... are 9, 18, 16, 54, 25, 51, 31, ... (Sloane’sA046125). The largest negative discriminants having class numbers 1, 2, 3, ... are 163, 427, 907, 1555, 2683, ... (Sloane’s A038552). The following table lists the numbers having class numbers h525:The search was terminated at 50000, 70000, 90000, and 90000 for class numbers 18, 20, 22, and 24, respectively. As far as I know, analytic upper bounds are not currently known for these cases. /h(/C28d)/N Sloane d 1 9 A014602 3, 4, 7, 8, 11, 19, 43, 67, 163 2 18 A014603 15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427 3 16 A006203 23, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907 4 54 A013658 39, 55, 56, 68, 84, 120, 132, 136, 155, 168, 184, 195, 203, 219, 228, 259, 280, 291, 292, 312, 323, 328, 340, 355, 372, 388, 408, 435, 483, 520, 532, 555, 568, 595, 627, 667, 708, 715, 723, 760,763, 772, 795, 955, 1003, 1012, 1027, 1227, 1243, 1387, 1411,1435, 1507, 1555 5 25 A046002 47, 79, 103, 127, 131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947, 1051, 1123, 1723, 1747, 1867, 2203, 2347,2683 6 51 A046003 87, 104, 116, 152, 212, 244, 247, 339, 411, 424, 436, 451, 472, 515, 628, 707, 771, 808, 835, 843, 856, 1048, 1059, 1099, 1108,1147, 1192, 1203, 1219, 1267, 1315, 1347, 1363, 1432, 1563,1588, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563,2787, 2923, 3235, 3427, 3523, 3763 7 31 A046004 71, 151, 223, 251, 463, 467, 487, 587, 811, 827, 859, 1163, 1171, 1483, 1523, 1627, 1787, 1987, 2011, 2083, 2179, 2251, 2467,2707, 3019, 3067, 3187, 3907, 4603, 5107, 5923 8 131 A046005 95, 111, 164, 183, 248, 260, 264, 276, 295, 299, 308, 371, 376, 395, 420, 452, 456, 548, 552, 564, 579, 580, 583, 616, 632, 651,660, 712, 820, 840, 852, 868, 904, 915, 939, 952, 979, 987, 995,1032, 1043, 1060, 1092, 1128, 1131, 1155, 1195, 1204, 1240,1252, 1288, 1299, 1320, 1339, 1348, 1380, 1428, 1443, 1528,1540, 1635, 1651, 1659, 1672, 1731, 1752, 1768, 1771, 1780,1795, 1803, 1828, 1848, 1864, 1912, 1939, 1947, 1992, 1995,2020, 2035, 2059, 2067, 2139, 2163, 2212, 2248, 2307, 2308,2323, 2392, 2395, 2419, 2451, 2587, 2611, 2632, 2667, 2715,2755, 2788, 2827, 2947, 2968, 2995, 3003, 3172, 3243, 3315,3355, 3403, 3448, 3507, 3595, 3787, 3883, 3963, 4123, 4195,4267, 4323, 4387, 4747, 4843, 4867, 5083, 5467, 5587, 5707,5947, 6307 9 34 A046006 199, 367, 419, 491, 563, 823, 1087, 1187, 1291, 1423, 1579, 2003, 2803, 3163, 3259, 3307, 3547, 3643, 4027, 4243, 4363,4483, 4723, 4987, 5443, 6043, 6427, 6763, 6883, 7723, 8563,8803, 9067, 10627 10 87 A046007 119, 143, 159, 296, 303, 319, 344, 415, 488, 611, 635, 664, 699, 724, 779, 788, 803, 851, 872, 916, 923, 1115, 1268, 1384, 1492,1576, 1643, 1684, 1688, 1707, 1779, 1819, 1835, 1891, 1923,2152, 2164, 2363, 2452, 2643, 2776, 2836, 2899, 3028, 3091,3139, 3147, 3291, 3412, 3508, 3635, 3667, 3683, 3811, 3859,3928, 4083, 4227, 4372, 4435, 4579, 4627, 4852, 4915, 5131,5163, 5272, 5515, 5611, 5667, 5803, 6115, 6259, 6403, 6667,7123, 7363, 7387, 7435, 7483, 7627, 8227, 8947, 9307, 10147,10483, 13843 11 41 A046008 167, 271, 659, 967, 1283, 1303, 1307, 1459, 1531, 1699, 2027, 2267, 2539, 2731, 2851, 2971, 3203, 3347, 3499, 3739, 3931,4051, 5179, 5683, 6163, 6547, 7027, 7507, 7603, 7867, 8443,9283, 9403, 9643, 9787, 10987, 13003, 13267, 14107, 14683,15667 12 206 A046009 231, 255, 327, 356, 440, 516, 543, 655, 680, 687, 696, 728, 731, 744, 755, 804, 888, 932, 948, 964, 984, 996, 1011, 1067, 1096,1144, 1208, 1235, 1236, 1255, 1272, 1336, 1355, 1371, 1419,1464, 1480, 1491, 1515, 1547, 1572, 1668, 1720, 1732, 1763,1807, 1812, 1892, 1955, 1972, 2068, 2091, 2104, 2132, 2148,2155, 2235, 2260, 2355, 2387, 2388, 2424, 2440, 2468, 2472,2488, 2491, 2555, 2595, 2627, 2635, 2676, 2680, 2692, 2723,2728, 2740, 2795, 2867, 2872, 2920, 2955, 3012, 3027, 3043,3048, 3115, 3208, 3252, 3256, 3268, 3304, 3387, 3451, 3459,3592, 3619, 3652, 3723, 3747, 3768, 3796, 3835, 3880, 3892, 3955, 3972, 4035, 4120, 4132, 4147, 4152, 4155, 4168, 4291, 4360, 4411, 4467, 4531, 4552, 4555, 4587, 4648, 4699, 4708,4755, 4771, 4792, 4795, 4827, 4888, 4907, 4947, 4963, 5032,5035, 5128, 5140, 5155, 5188, 5259, 5299, 5307, 5371, 5395,5523, 5595, 5755, 5763, 5811, 5835, 6187, 6232, 6235, 6267,6283, 6472, 6483, 6603, 6643, 6715, 6787, 6843, 6931, 6955,6963, 6987, 7107, 7291, 7492, 7555, 7683, 7891, 7912, 8068,8131, 8155, 8248, 8323, 8347, 8395, 8787, 8827, 9003, 9139,9355, 9523, 9667, 9843, 10003, 10603, 10707, 10747, 10795,10915, 11155, 11347, 11707, 11803, 12307, 12643, 14443,15163, 15283, 16003, 17803 13 37 A046010 191, 263, 607, 631, 727, 1019, 1451, 1499, 1667, 1907, 2131, 2143, 2371, 2659, 2963, 3083, 3691, 4003, 4507, 4643, 5347,5419, 5779, 6619, 7243, 7963, 9547, 9739, 11467, 11587, 11827,11923, 12043, 14347, 15787, 16963, 20563 14 96 A046011 215, 287, 391, 404, 447, 511, 535, 536, 596, 692, 703, 807, 899, 1112, 1211, 1396, 1403, 1527, 1816, 1851, 1883, 2008, 2123,2147, 2171, 2335, 2427, 2507, 2536, 2571, 2612, 2779, 2931, 2932, 3112, 3227, 3352, 3579, 3707, 3715, 3867, 3988, 4187, 4315, 4443, 4468, 4659, 4803, 4948, 5027, 5091, 5251, 5267,5608, 5723, 5812, 5971, 6388, 6499, 6523, 6568, 6979, 7067,7099, 7147, 7915, 8035, 8187, 8611, 8899, 9115, 9172, 9235,9427, 10123, 10315, 10363, 10411, 11227, 12147, 12667, 12787,13027, 13435, 13483, 13603, 14203, 16867, 18187, 18547,18643, 20227, 21547, 23083, 23692, 30067 15 68 A046012 239, 439, 751, 971, 1259, 1327, 1427, 1567, 1619, 2243, 2647, 2699, 2843, 3331, 3571, 3803, 4099, 4219, 5003, 5227, 5323,5563, 5827, 5987, 6067, 6091, 6211, 6571, 7219, 7459, 7547,8467, 8707, 8779, 9043, 9907, 10243, 10267, 10459, 10651,10723, 11083, 11971, 12163, 12763, 13147, 13963, 14323,14827, 14851, 15187, 15643, 15907, 16603, 16843, 17467,17923, 18043, 18523, 19387, 19867, 20707, 22003, 26203,27883, 29947, 32323, 34483 16 322 A046013 399, 407, 471, 559, 584, 644, 663, 740, 799, 884, 895, 903, 943, 1015, 1016, 1023, 1028, 1047, 1139, 1140, 1159, 1220, 1379,1412, 1416, 1508, 1560, 1595, 1608, 1624, 1636, 1640, 1716,1860, 1876, 1924, 1983, 2004, 2019, 2040, 2056, 2072, 2095,2195, 2211, 2244, 2280, 2292, 2296, 2328, 2356, 2379, 2436,2568, 2580, 2584, 2739, 2760, 2811, 2868, 2884, 2980, 3063,3108, 3140, 3144, 3160, 3171, 3192, 3220, 3336, 3363, 3379,3432, 3435, 3443, 3460, 3480, 3531, 3556, 3588, 3603, 3640,3732, 3752, 3784, 3795, 3819, 3828, 3832, 3939, 3976, 4008,4020, 4043, 4171, 4179, 4180, 4216, 4228, 4251, 4260, 4324,4379, 4420, 4427, 4440, 4452, 4488, 4515, 4516, 4596, 4612,4683, 4687, 4712, 4740, 4804, 4899, 4939, 4971, 4984, 5115,5160, 5187, 5195, 5208, 5363, 5380, 5403, 5412, 5428, 5460,5572, 5668, 5752, 5848, 5860, 5883, 5896, 5907, 5908, 5992,5995, 6040, 6052, 6099, 6123, 6148, 6195, 6312, 6315, 6328,6355, 6395, 6420, 6532, 6580, 6595, 6612, 6628, 6708, 6747,6771, 6792, 6820, 6868, 6923, 6952, 7003, 7035, 7051, 7195,7288, 7315, 7347, 7368, 7395, 7480, 7491, 7540, 7579, 7588,7672, 7707, 7747, 7755, 7780, 7795, 7819, 7828, 7843, 7923,7995, 8008, 8043, 8052, 8083, 8283, 8299, 8308, 8452, 8515,8547, 8548, 8635, 8643, 8680, 8683, 8715, 8835, 8859, 8932,8968, 9208, 9219, 9412, 9483, 9507, 9508, 9595, 9640, 9763,9835, 9867, 9955, 10132, 10168, 10195, 10203, 10227, 10312,10387, 10420, 10563, 10587, 10635, 10803, 10843, 10948,10963, 11067, 11092, 11107, 11179, 11203, 11512, 11523,11563, 11572, 11635, 11715, 11848, 11995, 12027, 12259,12387, 12523, 12595, 12747, 12772, 12835, 12859, 12868,13123, 13192, 13195, 13288, 13323, 13363, 13507, 13795,13819, 13827, 14008, 14155, 14371, 14403, 14547, 14707,14763, 14995, 15067, 15387, 15403, 15547, 15715, 16027,16195, 16347, 16531, 16555, 16723, 17227, 17323, 17347,17427, 17515, 18403, 18715, 18883, 18907, 19147, 19195,19947, 19987, 20155, 20395, 21403, 21715, 21835, 22243,22843, 23395, 23587, 24403, 25027, 25267, 27307, 27787,28963, 31243 17 45 A046014 383, 991, 1091, 1571, 1663, 1783, 2531, 3323, 3947, 4339, 4447, 4547, 4651, 5483, 6203, 6379, 6451, 6827, 6907, 7883, 8539,8731, 9883, 11251, 11443, 12907, 13627, 14083, 14779, 14947,16699, 17827, 18307, 19963, 21067, 23563, 24907, 25243,26083, 26107, 27763, 31627, 33427, 36523, 37123 18 150 A046015 335, 519, 527, 679, 1135, 1172, 1207, 1383, 1448, 1687, 1691, 1927, 2047, 2051, 2167, 2228, 2291, 2315, 2344, 2644, 2747,2859, 3035, 3107, 3543, 3544, 3651, 3688, 4072, 4299, 4307,4568, 4819, 4883, 5224, 5315, 5464, 5492, 5539, 5899, 6196,6227, 6331, 6387, 6484, 6739, 6835, 7323, 7339, 7528, 7571,7715, 7732, 7771, 7827, 8152, 8203, 8212, 8331, 8403, 8488,8507, 8587, 8884, 9123, 9211, 9563, 9627, 9683, 9748, 9832,10228, 10264, 10347, 10523, 11188, 11419, 11608, 11643,11683, 11851, 11992, 12067, 12148, 12187, 12235, 12283,12651, 12723, 12811, 12952, 13227, 13315, 13387, 13747,13947, 13987, 14163, 14227, 14515, 14667, 14932, 15115,15243, 16123, 16171, 16387, 16627, 17035, 17131, 17403,17635, 18283, 18712, 19027, 19123, 19651, 20035, 20827,21043, 21652, 21667, 21907, 22267, 22443, 22507, 22947,23347, 23467, 23683, 23923, 24067, 24523, 24667, 24787,25435, 26587, 26707, 28147, 29467, 32827, 33763, 34027,34507, 36667, 39307, 40987, 41827, 43387, 48427 19 47 A046016 311, 359, 919, 1063, 1543, 1831, 2099, 2339, 2459, 3343, 3463, 3467, 3607, 4019, 4139, 4327, 5059, 5147, 5527, 5659, 6803, 8419, 8923, 8971, 9619, 10891, 11299, 15091, 15331, 16363, 16747, 17011, 17299, 17539, 17683, 19507, 21187, 21211, 21283, 23203, 24763, 26227, 27043, 29803, 31123, 37507, 38707 20 350 A046017 455, 615, 776, 824, 836, 920, 1064, 1124, 1160, 1263, 1284, 1460, 1495, 1524, 1544, 1592, 1604, 1652, 1695, 1739, 1748, 1796, 1880, 1887, 1896, 1928, 1940, 1956, 2136, 2247, 2360, 2404, 2407, 2483, 2487, 2532, 2552, 2596, 2603, 2712, 2724, 2743, 2948, 2983, 2987, 3007, 3016, 3076, 3099, 3103, 3124, 3131, 3155, 3219, 3288, 3320, 3367, 3395, 3496, 3512, 3515, 3567, 3655, 3668, 3684, 3748, 3755, 3908, 3979, 4011, 4015, 4024, 4036, 4148, 4264, 4355, 4371, 4395, 4403, 4408, 4539, 4548, 4660, 4728, 4731, 4756, 4763, 4855, 4891, 5019, 5028, 5044, 5080, 5092, 5268, 5331, 5332, 5352, 5368, 5512, 5560, 5592, 5731, 5944, 5955, 5956, 5988, 6051, 6088, 6136, 6139, 6168, 6280, 6339, 6467, 6504, 6648, 6712, 6755, 6808, 6856, 7012, 7032, 7044, 7060, 7096, 7131, 7144, 7163, 7171, 7192, 7240, 7428, 7432, 7467, 7572, 7611, 7624, 7635, 7651, 7667, 7720, 7851, 7876, 7924, 7939, 8067, 8251, 8292, 8296, 8355, 8404, 8472, 8491, 8632, 8692, 8755, 8808, 8920, 8995, 9051, 9124, 9147, 9160, 9195, 9331, 9339, 9363, 9443, 9571, 9592, 9688, 9691, 9732, 9755, 9795, 9892, 9976, 9979, 10027, 10083, 10155, 10171, 10291, 10299, 10308, 10507, 10515, 10552, 10564, 10819, 10888, 11272, 11320, 11355, 11379, 11395, 11427, 11428, 11539, 11659, 11755, 11860, 11883, 11947, 11955, 12019, 12139, 12280, 12315, 12328, 12331, 12355, 12363, 12467, 12468, 12472, 12499, 12532, 12587, 12603, 12712, 12883, 12931, 12955, 12963, 13155, 13243, 13528, 13555, 13588, 13651, 13803, 13960, 14307, 14331, 14467, 14491, 14659, 14755, 14788, 15235, 15268, 15355, 15603, 15688, 15691, 15763, 15883, 15892, 15955, 16147, 16228, 16395, 16408, 16435, 16483, 16507, 16612, 16648, 16683, 16707, 16915, 16923, 17067, 17187, 17368, 17563, 17643, 17763, 17907, 18067, 18163, 18195, 18232, 18355, 18363, 19083, 19443, 19492, 19555, 19923, 20083, 20203, 20587, 20683, 20755, 20883, 21091, 21235, 21268, 21307, 21387, 21508, 21595, 21723, 21763, 21883, 22387, 22467, 22555, 22603, 22723, 23443, 23947, 24283, 24355, 24747, 24963, 25123, 25363, 26635, 26755, 26827, 26923, 27003, 27955, 27987, 28483, 28555, 29107, 29203, 30283, 30787, 31003, 31483, 31747, 31987, 32923, 33163, 34435, 35683, 35995, 36283, 37627, 37843, 37867, 38347, 39187, 39403, 40243, 40363, 40555, 40723, 43747, 47083, 48283, 51643, 54763, 58507 21 85 A046018 431, 503, 743, 863, 1931, 2503, 2579, 2767, 2819, 3011, 3371, 4283, 4523, 4691, 5011, 5647, 5851, 5867, 6323, 6691, 7907, 8059, 8123, 8171, 8243, 8387, 8627, 8747, 9091, 9187, 9811, 9859, 10067, 10771, 11731, 12107, 12547, 13171, 13291, 13339, 13723, 14419, 14563, 15427, 16339, 16987, 17107, 17707, 17971, 18427, 18979, 19483, 19531, 19819, 20947, 21379, 22027, 22483, 22963, 23227, 23827, 25603, 26683, 27427, 28387, 28723, 28867, 31963, 32803, 34147, 34963, 35323, 36067, 36187, 39043, 40483, 44683, 46027, 49603, 51283, 52627, 55603, 58963, 59467, 61483 22 139 A046019 591, 623, 767, 871, 879, 1076, 1111, 1167, 1304, 1556, 1591, 1639, 1903, 2215, 2216, 2263, 2435, 2623, 2648, 2815, 2863, 2935, 3032, 3151, 3316, 3563, 3587, 3827, 4084, 4115, 4163, 4328, 4456, 4504, 4667, 4811, 5383, 5416, 5603, 5716, 5739, 5972, 6019, 6127, 6243, 6616, 6772, 6819, 7179, 7235, 7403, 7763, 7768, 7899, 8023, 8143, 8371, 8659, 8728, 8851, 8907, 8915, 9267, 9304, 9496, 10435, 10579, 10708, 10851, 11035, 11283, 11363, 11668, 12091, 12115, 12403, 12867, 13672, 14019, 14059, 14179, 14548, 14587, 14635, 15208, 15563, 15832, 16243, 16251, 16283, 16291, 16459, 17147, 17587, 17779, 17947, 18115, 18267, 18835, 18987, 19243, 19315, 19672, 20308, 20392, 22579, 22587, 22987, 24243, 24427, 25387, 25507, 25843, 25963, 26323, 26548, 27619, 28267, 29227, 29635, 29827, 30235, 30867, 31315, 33643, 33667, 34003, 34387, 35347, 41083, 43723, 44923, 46363, 47587, 47923, 49723, 53827, 77683, 85507 23 68 A046020 647, 1039, 1103, 1279, 1447, 1471, 1811, 1979, 2411, 2671, 3491, 3539, 3847, 3923, 4211, 4783, 5387, 5507, 5531, 6563, 6659, 6703, 7043, 9587, 9931, 10867, 10883, 12203, 12739, 13099, 13187, 15307, 15451, 16267, 17203, 17851, 18379, 20323, 20443, 20899, 21019, 21163, 22171, 22531, 24043, 25147, 25579, 25939, 26251, 26947, 27283, 28843, 30187, 31147, 31267, 32467, 34843, 35107, 37003, 40627, 40867, 41203, 42667, 43003, 45427, 45523, 47947, 90787 24 511 A048925 695, 759, 1191, 1316, 1351, 1407, 1615, 1704, 1736, 1743, 1988, 2168, 2184, 2219, 2372, 2408, 2479, 2660, 2696, 2820, 2824, 2852, 2856, 2915, 2964, 3059, 3064, 3127, 3128, 3444, 3540, 3560, 3604, 3620, 3720, 3864, 3876, 3891, 3899, 3912, 3940, 4063, 4292, 4308, 4503, 4564, 4580, 4595, 4632, 4692, 4715, 4744, 4808, 4872, 4920, 4936, 5016, 5124, 5172, 5219, 5235, 5236, 5252, 5284, 5320, 5348, 5379, 5432, 5448, 5555, 5588, 5620, 5691, 5699, 5747, 5748, 5768, 5828, 5928, 5963, 5979, 6004, 6008, 6024, 6072, 6083, 6132, 6180, 6216, 6251, 6295, 6340, 6411, 6531, 6555, 6699, 6888, 6904, 6916, 7048, 7108,7188, 7320, 7332, 7348, 7419, 7512, 7531, 7563, 7620, 7764, 7779, 7928, 7960, 7972, 8088, 8115, 8148, 8211, 8260, 8328, 8344, 8392, 8499, 8603, 8628, 8740, 8760, 8763, 8772, 8979, 9028, 9048, 9083, 9112, 9220, 9259, 9268, 9347, 9352, 9379, 9384, 9395, 9451, 9480, 9492, 9652, 9672, 9715, 9723, 9823, 9915, 9928, 9940, 10011, 10059, 10068, 10120, 10180, 10187, 10212, 10248, 10283, 10355, 10360, 10372, 10392, 10452, 10488, 10516, 10612, 10632, 10699, 10740, 10756, 10788, 10792, 10840, 10852, 10923, 11019, 11032, 11139, 11176, 11208, 11211, 11235, 11267, 11307, 11603, 11620, 11627, 11656, 11667, 11748, 11752, 11811, 11812, 11908, 11928, 12072, 12083, 12243, 12292, 12376, 12408, 12435, 12507, 12552, 12628, 12760, 12808, 12820, 12891, 13035, 13060, 13080, 13252, 13348, 13395, 13427, 13444, 13512, 13531, 13539, 13540, 13587, 13611, 13668, 13699, 13732, 13780, 13912, 14035, 14043, 14212, 14235, 14260, 14392, 14523, 14532, 14536, 14539, 14555, 14595, 14611, 14632, 14835, 14907, 14952, 14968, 14980, 15019, 15112, 15267, 15339, 15411, 15460, 15483, 15528, 15555, 15595, 15640, 15652, 15747, 15748, 15828, 15843, 15931, 15940, 15988, 16107, 16132, 16315, 16360, 16468, 16563, 16795, 16827, 16872, 16888, 16907, 16948, 17032, 17043, 17059, 17092, 17283, 17560, 17572, 17620, 17668, 17752, 17812, 17843, 18040, 18052, 18088, 18132, 18148, 18340, 18507, 18568, 18579, 18595, 18627, 18628, 18667, 18763, 18795, 18811, 18867, 18868, 18915, 19203, 19528, 19579, 19587, 19627, 19768, 19803, 19912, 19915, 20260, 20307, 20355, 20427, 20491, 20659, 20692, 20728, 20803, 20932, 20955, 20980, 20995, 21112, 21172, 21352, 21443, 21448, 21603, 21747, 21963, 21988, 22072, 22107, 22180, 22323, 22339, 22803, 22852, 22867, 22939, 23032, 23035, 23107, 23115, 23188, 23235, 23307, 23368, 23752, 23907, 23995, 24115, 24123, 24292, 24315, 24388, 24595, 24627, 24628, 24643, 24915, 24952, 24955, 25048, 25195, 25347, 25467, 25683, 25707, 25732, 25755, 25795, 25915, 25923, 25972, 25987, 26035, 26187, 26395, 26427, 26467, 26643, 26728, 26995, 27115, 27163, 27267, 27435, 27448, 27523, 27643, 27652, 27907, 28243, 28315, 28347, 28372, 28459, 28747, 28891, 29128, 29283, 29323, 29395, 29563, 29659, 29668, 29755, 29923, 30088, 30163, 30363, 30387, 30523, 30667, 30739, 30907, 30955, 30979, 31252, 31348, 31579, 31683, 31795, 31915, 32008, 32043, 32155, 32547, 32635, 32883, 33067, 33187, 33883, 34203, 34363, 34827, 34923, 36003, 36043, 36547, 36723, 36763, 36883, 37227, 37555, 37563, 38227, 38443, 38467, 39603, 39643, 39787, 40147, 40195, 40747, 41035, 41563, 42067, 42163, 42267, 42387, 42427, 42835, 43483, 44947, 45115, 45787, 46195, 46243, 46267, 47203, 47443, 47707, 48547, 49107, 49267, 49387, 49987, 50395, 52123, 52915, 54307, 55867, 56947, 57523, 60523, 60883, 61147, 62155,62203, 63043, 64267, 79363, 84043, 84547, 111763 25 95 A056987 479, 599, 1367, 2887, 3851, 4787, 5023, 5503, 5843, 7187, 7283, 7307, 7411, 8011, 8179, 9227, 9923, 10099, 11059, 11131,11243, 11867, 12211, 12379, 12451, 12979, 14011, 14923,15619, 17483, 18211, 19267, 19699, 19891, 20347, 21107,21323, 21499, 21523, 21739, 21787, 21859, 24091, 24571,25747, 26371, 27067, 27091, 28123, 28603, 28627, 28771,29443, 30307, 30403, 30427, 30643, 32203, 32443, 32563,32587, 33091, 34123, 34171, 34651, 34939, 36307, 37363,37747, 37963, 38803, 39163, 44563, 45763, 48787, 49123,50227, 51907, 54667, 55147, 57283, 57667, 57787, 59707,61027, 62563, 63067, 64747, 66763, 68443, 69763, 80347,85243, 89083, 93307 The table below gives lists of POSITIVE fundamental discriminants dhaving small class numbers h(d); corresponding to REAL QUADRATIC FIELDS . All POSI- TIVE SQUAREFREE values of d597 (for which the KRONECKER SYMBOL is defined) are included. /h(d)/d 1 5, 13, 17, 21, 29, 37, 41, 53, 57, 61, 69, 73, 77 26 5 The POSITIVE dfor which h(d/C301) is given by Sloane’s A014539. See also CLASS FIELD THEORY ,C LASS NUMBER FORMULA ,DIRICHLET L-SERIES ,DISCRIMINANT (BIN- ARY QUADRATIC FORM), GAUSS’S CLASS NUMBER CONJECTURE ,G AUSS’S CLASS NUMBER PROBLEM , HEEGNER NUMBER ,IDEAL , J-FUNCTION ,RING References Arno, S. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith. 40, 321 /C1/34, 1992. Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." http:// www.math.uiuc.edu/Algebraic-Number-Theory/0009/. Buell, D. A. "Small Class Numbers and Extreme Values of L-Functions of Quadratic Fields." Math. Comput. 139, 786 /C1/96, 1977. Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993. Cohn, H. Advanced Number Theory. New York: Dover, pp. 163 and 234, 1980. Cox, D. A. Primes of the Form x2 /C27ny2 : Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997. Davenport, H. "Dirichlet’s Class Number Formula." Ch. 6 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 43 /C1/3, 1980. Himmetoglu, S. Berechnung von Klassenzahlen Imaginaer- Quadratischer Zahlko ¨rper. Diplomarbeit. Heidelberg, Germany: University of Heidelberg Faculty for Mathe- matics, March 1986. Iyanaga, S. and Kawada, Y. (Eds.). "Class Numbers of Algebraic Number Fields." Appendix B, Table 4 in En- cyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1494 /C1/496, 1980. Montgomery, H. and Weinberger, P. "Notes on Small Class Numbers." Acta. Arith. 24, 529 /C1/42, 1974. Mu¨ller, H. "A Calculation of Class-Numbers of Imaginary Quadratic Numberfields." Tamkang J. Math. 9, 121 /C1/28, 1978. Oesterle ´, J. "Nombres de classes des corps quadratiques imaginaires." Aste´rique 121 /C1/22, 309 /C1/23, 1985. Sloane, N. J. A. Sequences A003657/M2332, A006203/ M5131, A013658, A014539, A014602, A014603, A038552, A046002, A046003, A046125, A048925, and A056987 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Stark, H. M. "A Complete Determination of the Complex Quadratic Fields of Class Number One." Michigan Math. J. 14,1/C1/7, 1967. Stark, H. M. "On Complex Quadratic Fields with Class Number Two." Math. Comput. 29, 289 /C1/02, 1975. Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65, 785 /C1/00, 1996. Weisstein, E. W. "Class Numbers." MATHEMATICA NOTE- BOOK CLASS NUMBERS.M . Class Number Formula A class number formula is a finite series giving exactly the CLASS NUMBER of a RING . For a RING of quadratic integers, the class number is denoted h(d); where d is the discriminant. A class number formula is known for the full ring of cyclotomic integers, as well as for any subring of the cyclotomic integers. This formula includes the quadratic case as well as many cubic and higher-order RINGS . See also CLASS NUMBER ,RINGClass Representative A set of class representatives is a SUBSET of X which contains exactly one element from each EQUIVALENCE CLASS . See also EQUIVALENCE CLASS Classical Algebraic Geometry Classical algebraic geometry is the study of ALGE- BRAIC VARIETIES , both AFFINE VARIETIES in Cn and PROJECTIVE VARIETIES in C ’n /. The original motivation was to study systems of polynomials and their roots. See also ALGEBRAIC GEOMETRY ,ALGEBRAIC VARIETY , POLYNOMIAL Classical Canonical Form JORDAN CANONICAL FORM Classical Groups The four following types of GROUPS , 1. LINEAR GROUPS , 2. ORTHOGONAL GROUPS , 3. SYMPLECTIC GROUPS , and 4. UNITARY GROUPS , which were studied before more exotic types of groups (such as the SPORADIC GROUPS ) were discovered. See also GROUP ,G ROUP THEORY ,L INEAR GROUP , ORTHOGONAL GROUP ,SIMPLE GROUP ,SYMPLECTIC GROUP ,UNITARY GROUP Classification The classification of a collection of objects generally means that a list has been constructed with exactly one member from each ISOMORPHISM type among the objects, and that tools and techniques can effectively be used to identify any combinatorially given object with its unique representative in the list. Examples of mathematical objects which have been classified include the finite SIMPLE GROUPS and 2-MANIFOLDS but not, for example, KNOTS . See also ENUMERATION PROBLEM Classification Theorem CLASSIFICATION THEOREM OF FINITE GROUPS ,CLAS- SIFICATION THEOREM OF SURFACES Classification Theorem of Finite Groups The classification theorem of FINITE SIMPLE GROUPS , also known as the ENORMOUS THEOREM , which states that the FINITE SIMPLE GROUPS can be classified completely into 1. CYCLIC GROUPS Zp of PRIME ORDER , 2. ALTERNATING GROUPS An of degree at least five, 3. LIE-TYPE CHEVALLEY GROUPS PSL(n; q); PSU (n; q); PsP(2n; q); and PV e(n; q);/ 4. LIE-TYPE (TWISTED CHEVALLEY GROUPS or the TITS GROUP )3D4(q); E6(q) ; E7(q) ; E8(q) ; F4(q); 2F4(2n) ?; G2(q) ; 2G2(3n) ; 2B(2n) ;/ 5. SPORADIC GROUPS M11 ; M12 ; M22 ; M23 ; M24 ; J2 /C30 HJ ; Suz, HS, McL , Co3 ; Co2 ; Co1 ; He, Fi22 ; Fi23 ; Fi?24 ; HN, Th, B, M, J1 ; O’N, J3 ; Ly, Ru, J4 :/ The "PROOF " of this theorem is spread throughout the mathematical literature and is estimated to be approximately 15,000 pages in length. See also FINITE GROUP ,GROUP , J-FUNCTION ,SIMPLE GROUP References Cartwright, M. "Ten Thousand Pages to Prove Simplicity." New Scientist 109,26/C1/0, 1985. Cipra, B. "Are Group Theorists Simpleminded?" What’s Happening in the Mathematical Sciences, 1995 /C1/996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 82 /C1/9, 1996. Cipra, B. "Slimming an Outsized Theorem." Science 267, 794 /C1/95, 1995. Gorenstein, D. "The Enormous Theorem." Sci. Amer. 253, 104 /C1/15, Dec. 1985. Solomon, R. "On Finite Simple Groups and Their Classifica- tion." Not. Amer. Math. Soc. 42, 231 /C1/39, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 57, 1986. Classification Theorem of Surfaces All closed surfaces, despite their seemingly diverse forms, are topologically equivalent to SPHERES with some number of HANDLES or CROSS-CAPS . The tradi- tional proof follows Seifert and Threlfall (1980), but Conway’s so-called "zero-irrelevancy" ("ZIP") provides a more streamlined approach (Francis and Weeks 1999). See also CROSS- CAP,HANDLE References Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/99, 1999. Seifert, H. and Threlfall, W. A Textbook of Topology. New York: Academic Press, 1980. Clausen Formula Clausen’s4F3 identity 4F3a ; b; c; d e; f ; g;1l11sl11n /C30(2a)½d½(a /C27 b)½d½(2b)½d½ (2a /C27 2b)½d½a ½d½b½d½; (1) holds for a /C27b /C27c /C28d /C301=2 ; e /C30a /C27b /C271=2; a /C27f /C30 d /C271 /C30b /C27g ; where d a nonpositive integer and (a)n is the POCHHAMMER SYMBOL (Petkovsek et al. 1996). Closely related identities include4F3 /C301 2 a;12(a /C271); b /C27n;/C28n 12 b ;12(b /C271); a /C271;1"# /C30(b /C28 a)n (b)n(2) and 4F312 a;12(a /C271); b /C27n ;/C28n 12(b /C271);12(b /C272); a;1 # /C30(b /C28 a /C27 1)n (b /C27 1)n/C281(b /C27 2n) (3) (Bailey 1935; Slater 1966, p. 245; Andrews and Burge 1993) Another identity ascribed to Clausen which involves the HYPERGEOMETRIC FUNCTION2F1(a; b; c; z) and the GENERALIZED HYPERGEOMETRIC FUNCTION 3F2(a ; b; c; d; e; z) is given by 2F1a ; b a /C27b /C271 2; xl11sl11nl12ml121 2 /C303 F22a; a /C27b; 2b a /C27b /C2712; 2a /C272b; xl11sl11n (4) (Clausen 1828; Bailey 1935, p. 86; Hardy 1999, p. 106). See also GENERALIZED HYPERGEOMETRIC FUNCTION , HYPERGEOMETRIC FUNCTION References Andrews, G. E. and Burge, W. H. "Determinant Identities." Pacific J. Math. 158,1/C1/4, 1993. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, 1935. Clausen, T. "Ueber die Falle wenn die Reihe y/C301/C27a/C215b 1/C215gx/C27... ein quadrat von der Form x/C301/C27a?b?g? 1 /C215d?e?x/C27. . . hat." J. fu¨r Math. 3,8 9/C1/5, 1828. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, pp. 43 and 127, 1996. Slater, L. J. Generalized Hypergeometric Functions. Cam- bridge, England: Cambridge University Press, 1966. Clausen Function Define Sn(x)/C13X/C12 k/C301sin(kx) kn(1) Cn(x)/C13X/C12 k/C301cos(kx) kn; (2) and write Cln(x) /C13Sn(x) /C30X/C12 k /C301sin(kx) knn even Cn(x) /C30X/C12 k /C301cos(kx) knn odd:8 >>>>< >>>>:(3) Then the Clausen function Cl n(x) can be given symbolically in terms of the POLYLOGARITHM as Cln(x) /C301 2 i[Lin(e /C28ix) /C28Lin(eix)] n even 1 2[Lin(e /C28ix) /C27Lin(eix)] n odd:( (4) For n /C301, the function takes on the special form Cl1(x) /C30C1(x) /C30/C28ln½2 sin(1 2 x)½ (5) and for n /C302, it becomes CLAUSEN’S INTEGRAL Cl2(x) /C30S2(x) /C30/C28gx 0ln[2 sin(12 t)] dt: (6) The symbolic sums of opposite parity are summable symbolically, and the first few are given by C2(x) /C3016 p2 /C2812 px /C2714 x2 (7) C4(x) /C301 90 /C281 12 p2x2 /C271 12 px3 /C281 48 x4 (8) S1(x) /C301 2( p /C28x) (9) S3(x) /C301 6 p2x /C2814 px2 /C271 12 x3 (10) S5(x) /C301 90 p4x /C281 36 p2x3 /C271 48 px4 /C281 240 x5 (11) for 0 5x 52p (Abramowitz and Stegun 1972). See also CLAUSEN’S INTEGRAL ,POLYGAMMA FUNC- TION ,POLYLOGARITHM References Abramowitz, M. and Stegun, C. A. (Eds.). "Clausen’s Inte- gral and Related Summations" §27.8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005 /C1/006, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 783, 1985. Clausen, R. "U¨ ber die Zerlegung reeller gebrochener Funk- tionen." J. reine angew. Math. 8, 298 /C1/00, 1832. Grosjean, C. C. "Formulae Concerning the Computation of the Clausen Integral Cl2( a) :/" J. Comput. Appl. Math. 11, 331 /C1/42, 1984. Jolley, L. B. W. Summation of Series. London: Chapman, 1925. Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, pp. 170 /C1/80, 1958. Wheelon, A. D. A Short Table of Summable Series. Report No. SM-14642. Santa Monica, CA: Douglas Aircraft Co., 1953.Clausen’s Integral The n /C302 case of the S2 CLAUSEN FUNCTION Cl2(u)/C30/C28gu 0ln[2 sin(12t)]dt: See also CLAUSEN FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005 /C1/006, 1972. Ashour, A. and Sabri, A. "Tabulation of the Function c(u)/C30a/C12 n/C301sin(nu) n2:/"Math. Tables Aids Comp. 10, 54 and 57/C1/5, 1956. Clausen, R. "U ¨ber die Zerlegung reeller gebrochener Funk- tionen." J. reine angew. Math. 8, 298/C1/00, 1832. Lewin, L. "Clausen’s Integral." Ch. 4 in Dilogarithms and Associated Functions. London: Macdonald, pp. 91 /C1/05, 1958. Clausen’s Product Identity 2F1(1 4/C27a;14/C27b;q/C27a/C27b;x)2F1(14/C28a;14/C28b;1/C28a /C28b;x) /C303F2(12;12/C27a/C28b;12/C28a/C27b;1/C27a/C27b;1/C28a /C28b;x); where2F1(a;b;c;x)i sa HYPERGEOMETRIC FUNC- TION . Koepf, W. Hypergeometric Summation: An Algorith- mic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 118, 1998. Cleavance Center The point of concurrence Sof a triangle’s CLEAVERS M1C1 ; M2C2 ; and M3C3 ; which is simply the SPIEKER CENTER , i.e., the INCENTER of the MEDIAL TRIANGLE (Honsberger 1995, p. 2). See also CLEAVANCE CENTER ,M EDIAL TRIANGLE , NAGEL POINT ,SPIEKER CENTER References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 2, 1995. Cleaver A PERIMETER -bisecting segment of a polygon originat- ing from the MIDPOINT of one side. Each cleaver M1C1 ; M2C2 ; and M3C3 in a TRIANGLE DA1A2A3 is parallel to an ANGLE BISECTOR of the triangle (shown as dashed lines above). In addition, the three cleavers CONCUR in a point S known as the CLEAVANCE CENTER , which is the SPIEKER CENTER , i.e., INCENTER of the MEDIAL TRIANGLE (Honsberger 1995, p. 2). See also B-LINE,CLEAVANCE CENTER ,M EDIAL TRI- ANGLE ,MIDPOINT ,SPLITTER References Avishalom, D. "Perimeter-Bisectors in a Triangle" [Hebrew]. Riveon Lematematika 13,46/C1/9, 1959. Avishalom, D. "The Perimetric Bisection of Triangles." Math. Mag. 36,60/C1/2, 1963. Honsberger, R. "Cleavers and Splitters." Episodes in Nine- teenth and Twentieth Century Euclidean Geometry. Wa- shington, DC: Math. Assoc. Amer., pp. 1 /C1/4, 1995. Jarden, D. "Synthetical Proof for the Theorem on the Center of Perimeter-Bisectors in a Triangle" [Hebrew]. Riveon Lematematika 13, 50, 1959. Clebsch Diagonal Cubic A CUBIC ALGEBRAIC SURFACE given by the equation x3 0 /C27x31 /C27x32 /C27x33 /C27x34 /C300; (1) with the added constraint x0 /C27x1 /C27x2 /C27x3 /C27x4 /C300: (2) The implicit equation obtained by taking the plane at infinity as x0 /C27x1 /C27x2 /C27x3 =2is 81(x3 /C27y3 /C27z3) /C28189(x2y /C27x2z /C27y2x /C27y2z /C27z2x /C27z2y) /C2754xyz /C27126(xy /C27xz /C27yz) /C289(x2 /C27y2 /C27z2) /C289(x /C27y /C27z) /C271 /C300 (3) (Hunt, Nordstrand). On Clebsch’s diagonal surface, all 27 of the complex lines (SOLOMON’S SEAL LINES ) present on a general smooth CUBIC SURFACE are real. In addition, there are 10 points on the surface where 3 of the 27 lines meet. These points are called ECKARDT POINTS (Fischer 1986, Hunt), and the Clebsch diagonal surface is the unique CUBIC SUR- FACE containing 10 such points (Hunt). If one of the variables describing Clebsch’s diagonal surface is dropped, leaving the equations x30/C27x31/C27x32/C27x33/C300; (4) x0/C27x1/C27x2/C27x3/C300; (5) the equations degenerate into two intersecting PLANES given by the equation (x/C27y)(x/C27z)(y/C27z)/C300: (6) See also CUBIC SURFACE ,ECKARDT POINT References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 9 /C1/1, 1986. Fischer, G. (Ed.). Plates 10 /C1/2i n Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 13 /C1/5, 1986. Hunt, B. The Geometry of Some Special Arithmetic Quoti- ents. New York: Springer-Verlag, pp. 122 /C1/28, 1996. Nordstrand, T. "Clebsch Diagonal Surface." http:// www.uib.no/people/nfytn/clebtxt.htm. Clebsch-Aronhold Notation A notation used to describe curves. The fundamental principle of Clebsch-Aronhold notation states that if each of a number of forms be replaced by a POWER of a linear form in the same number of variables equal to the order of the given form, and if a sufficient numberof equivalent symbols are introduced by the A RON- HOLD PROCESS so that no actual COEFFICIENT appears except to the first degree, then every identicalrelation holding for the new specialized forms holds for the general ones. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 79, 1959. ClebschGordan CLEBSCH- GORDAN COEFFICIENT Clebsch-Gordan Coefficient A mathematical symbol used to integrate products of three SPHERICAL HARMONICS . Clebsch-Gordan coeffi- cients commonly arise in applications involving the addition of angular momentum in quantum me- chanics. If products of more than three SPHERICAL HARMONICS are desired, then a generalization known as WIGNER 6J-SYMBOLS or WIGNER 9J-SYMBOLS is used. The Clebsch-Gordan coefficients are written Cj m1m2/C30(j1j2m1m2 ½j1j2jm) (1) and are defined by CJM /C30X M /C30M1 /C27M2CJM 1M2CM1M2; (2) where J /C13J1 /C27J2 :/ The coefficients are subject to the restrictions that (j1 ; j2 ; j) be positive integers or half-integers, j1 /C27j2 /C27 j is an integer, (m1 ; m2 ; m) are positive or negative integers or half integers, j1 /C27j2 /C28j ]0 (3) j1 /C28j2 /C27j ]0 (4) /C28j1 /C27j2 /C27j ]0; (5) and /C28½j1 ½5m1 5½j1 ½;/C28½j2 ½5m2 5½j2 ½; and /C28½j ½5m 5½j ½ (Abramowitz and Stegun 1972, p. 1006). In addition, by use of symmetry relations, coefficients may always be put in the standard form j1 Bj2 Bj and m ]0:/ The Clebsch-Gordan coefficients are implemented in Mathematica asClebschGordan [{j1, m1}, {j2, m2}, {j, m}] (assumed to be in standard form) and satisfy (j1j2m1m2 ½j1j2jm) /C300 for m1 /C27m2 "m (6) and are The Clebsch-Gordan coefficients are sometimes ex- pressed using the related RACAH V-COEFFICIENTS , V(j1j2j; m1m2m) (7) or WIGNER 3J-SYMBOLS . Connections among the three are (j1j2m1m2 ½j1j2jm) /C30(/C281)m/C27j1/C28j2ffiffiffiffiffiffiffiffiffiffiffiffiffi 2j /C271pj1 j2 j m1m2/C28ml11sl11n (8) (j1j2m1m2 ½j1j2jm) /C30(/C281)j/C27mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2j /C271Vp (j 1j2j; m1m2 /C28m) (9)V(j1j2j; m1m2m) /C30(/C281)/C28j1/C27j2/C27j j1 j2 j1 m1m2m2l11sl11n : (10) They have the symmetry (j1j2m1m2½j1j2jm)/C30(/C281)j1/C27j2/C28j(j2j1m2m1½j2j1jm);(11) and obey the orthogonality relationships X j;m(j1j2m1m2½j1j2jm)(j1j2jm½j1j2m?1m?2) /C30dm1m?1dm2m?2(12) X m1;m2(j1j2m1m2½j1j2jm)(j1j2j?m?½j1j2m1m2) /C30djj?dmm?: (13) See also RACAH V-COEFFICIENT ,R ACAH W-COEFFI- CIENT ,W IGNER 3J-SYMBOL ,W IGNER 6J-SYMBOL , WIGNER 9J-SYMBOL References Abramowitz, M. and Stegun, C. A. (Eds.). "Vector-Addition Coefficients." §27.9 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1006 /C1/010, 1972. Cohen-Tannoudji, C.; Diu, B.; and Laloe ¨, F. "Clebsch-Gordan Coefficients." Complement BXinQuantum Mechanics, Vol. 2. New York: Wiley, pp. 1035 /C1/047, 1977. Condon, E. U. and Shortley, G. §3.6/C1/.14 in The Theory of Atomic Spectra. Cambridge, England: Cambridge Univer- sity Press, pp. 56 /C1/8, 1951. Fano, U. and Fano, L. Basic Physics of Atoms and Molecules. New York: Wiley, p. 240, 1959. Messiah, A. "Clebsch-Gordan (C.-G.) Coefficients and ‘3 j’ Symbols." Appendix C.I in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 1054 /C1/060, 1962. Rose, M. E. Elementary Theory of Angular Momentum. New York: Dover, 1995. Shore, B. W. and Menzel, D. H. "Coupling and Clebsch- Gordan Coefficients." §6.2 in Principles of Atomic Spectra. New York: Wiley, pp. 268 /C1/76, 1968. Sobel’man, I. I. "Angular Momenta." Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer- Verlag, 1992. Clement Matrix KACMATRIX Clenshaw Recurrence Formula The downward Clenshaw recurrence formula evalu- ates a sum of products of indexed COEFFICIENTS by functions which obey a RECURRENCE RELATION .I f f(x)/C30XN k/C300ckFk(x) and Fn/C271(x)/C30a(n;x)Fn(x)/C27b(n;x)Fn/C281(x); where the ck/s are known, then define yN /C272 /C30yN /C271 /C300 yk /C30 a(k; x)yk/C271 /C27 b(k /C271; x)yk/C272 /C27ck for k /C30N ; N /C281; ... and solve backwards to obtain y2 and y1 : ck /C30yk /C28 a(k; x)yk/C271 /C28 b(k /C271; x)yk /C272 f(x) /C30XN k /C300ckFk(x) /C30c0F0(x) /C27[y1 /C28 a(1; x)y2 /C28 b(2; x)y3]F1(x) /C27[y2 /C28 a(2; x)y3 /C28 b(3; x)y4]F2(x) /C27[y3 /C28 a(3; x)y4 /C28 b(4; x)y5]F3(x) /C27[y4 /C28 a(4; x)y5 /C28 b(5; x)y6]F4(x) /C27... /C30c0F0(x) /C27y1F1(x) /C27y2[F2(x) /C28 a(1; x)F1(x)] /C27y3[F3(x) /C28 a(2; x)F2(x) /C28 b(2; x)] /C27y4[F4(x) /C28 a(3; x)F3(x) /C28 b(3; x)] /C27... /C30c0F0(x) /C27y2[ fa(1; x)F1(x) /C27 b(1; x)F0(x) g /C28a(1; x)F1(x)] /C27y1F1(x) /C30c0F0(x) /C27y1F1(x) /C27 b(1; x)F0(x)y2 : The upward Clenshaw recurrence formula is y/C282 /C30y/C281 /C300 yk /C301 b(k /C27 1 ; x) [yk /C282 /C28 a(k ; x)yk /C281 /C28ck] for k /C300 ; 1 ; ...; N /C281: f(x) /C30cNFN(x) /C28 b(N ; x)FN /C281(x)yN /C281 /C28FN(x)yN /C282 : References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Recurrence Relations and Clenshaw’s Recur- rence Formula." §5.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 172 /C1/78, 1992. Cliff Random Number Generator A RANDOM NUMBER generator produced by iterating Xn/C271 /C30 100 ln Xn (mod1) jj for a SEED X0 /C300:1: This simple generator passes the NOISE SPHERE test for randomness by showing no structure. See also RANDOM NUMBER ,SEED References Pickover, C. A. "Computers, Randomness, Mind, and In- finity." Ch. 31 in Keys to Infinity. New York: W. H. Freeman, pp. 233 /C1/47, 1995.Clifford Algebra Let V be an n-D linear SPACE over a FIELD K, and let Q be a QUADRATIC FORM on V. A Clifford algebra is then defined over the T(V) =I(Q) ; where T(V) is the tensor algebra over V and I is a particular IDEAL of T(V) :/ Clifford algebraists call their higher dimensional numbers HYPERCOMPLEX even though they do not share all the properties of complex numbers and no classical function theory can be constructed over them.See also H YPERCOMPLEX NUMBER ,QUATERNION References Ab //amowicz, R. Hecke Algebra, SVD, and Other Computa- tional Examples with CLIFFORD. 14 Oct 1999. http:// xxx.lanl.gov/abs/math.RA/9910069/. Ablamowicz, R.; Lounesto, P.; and Parra, J. M. Clifford Algebras with Numeric and Symbolic Computations. Boston, MA: Birkha ¨user, 1996. Huang, J.-S. "The Clifford Algebra." §6.2 in Lectures on Representation Theory. Singapore: World Scientific, pp. 63 /C1/5, 1999. Iyanaga, S. and Kawada, Y. (Eds.). "Clifford Algebras." §64 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 220 /C1/22, 1980. Lounesto, P. "Counterexamples to Theorems Published and Proved in Recent Literature on Clifford Algebras, Spinors, Spin Groups, and the Exterior Algebra." http://www.hit.fi/ ~lounesto/counterexamples.htm. Clifford’s Circle Theorem Let C1 ; C2 ; C3 ; and C4be four CIRCLES of GENERAL POSITION through a point P. Let Pijbe the second intersection of the CIRCLES Ciand Cj : Let Cijkbe the CIRCLE PijPikPjk : Then the four CIRCLES C234 ; C134 ; C124 ; and C123all pass through the point P1234: Similarly, let C5be a fifth CIRCLE through P. Then the five points P2345;P1345;P1245;P1235andP1234all lie on one CIRCLE C12345 :And so on. See also CIRCLE ,COX’S THEOREM References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 32 /C1/3, 1991. Clifford’s Curve Theorem The dimension of a special series can never exceed half its order. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 263, 1959. Clique A clique of a GRAPH is its maximal COMPLETE SUBGRAPH (Harary 1994, p. 20), although some authors define a clique as any COMPLETE SUBGRAPH and then refer to "maximum cliques" (Skiena 1990, p. 217). The problem of finding the size of a clique for a given GRAPH is an NP-COMPLETE PROBLEM (Skiena 1997). Cliques arise in a number of areas of GRAPH THEORY and combinatorics, including the theory of ERROR- CORRECTING CODES . The command MaximumCli- que[g] in the Mathematica add-on packageDiscre- teMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ) finds the size of the largest clique in a given GRAPH . The number of graphs on n nodes having 3 cliques are 0, 0, 1, 4, 12, 31, 67, ... (Sloane’s A005289). A COMPLETE K-PARTITE GRAPH has maximum clique size k. The largest order n graph which does not contain the COMPLETE GRAPH Kpas a SUBGRAPH is called the TURA´ N’S GRAPH Tn;p (Skiena 1990, p. 218). See also CLIQUE GRAPH ,CLIQUE NUMBER ,COMPLETE GRAPH ,INDUCED SUBGRAPH ,PARTY PROBLEM ,PER- FECT GRAPH ,RAMSEY NUMBER ,TURA´ N’S THEOREM References Bellare, M.; Goldreich, O.; and Sudan, M. "Free Bits, PCPs, and Non-Approximability--Towards Tight Results." SIAM J. Comput. 27, 804 /C1/15, 1998. Cormen, T.; Leiserson, C.; and Rivest, R. Introduction to Algorithms. Cambridge, MA: MIT Press, 1990. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Karp, R. M. "Reducibility Among Combinatorial Problems." In Complexity of Computer Calculations (Ed. R. Miller and J. Thatcher). New York: Plenum, pp. 85 /C1/03, 1972. Garey, M. R. and Johnson, D. S. Computers and Intract- ability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman, 1983.Manber, U. Introduction to Algorithms: A Creative Ap- proach. Reading, MA: Addison-Wesley, 1989. Skiena, S. "Maximum Cliques." §5.6.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 215 and 217 /C1/18, 1990. Skiena, S. S. "Clique and Independent Set" and "Clique." §6.2.3 and 8.5.1 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 144 and 312 /C1/14, 1997. Sloane, N. J. A. Sequences A005289/M3440 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Clique Graph The clique graph of a given GRAPH G is the GRAPH INTERSECTION of the family of CLIQUES of G.A GRAPH G is a clique graph IFF it contains a family F of COMPLETE SUBGRAPHS whose GRAPH UNION is G, such that whenever every pair of such complete graphs in some subfamily F ? has a nonempty graph intersec- tion, the intersection of all members of F ? is not empty (Harary 1994, p. 20). See also CLIQUE ,CLIQUE NUMBER ,COMPLETE GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Clique Number The number of VERTICES in the largest CLIQUE ofG, denoted v(G):For an arbitrary GRAPH , v(G)]Xn i/C3011 n/C28di; where diis the DEGREE ofVERTEX i. The following table gives the number Nk(n)o fn-node graphs having clique number kfor small k. kSloane /Nk(n)/ 1 1 ,1 ,1 ,1 ,1 ,1 ,. . . 2 A052450 0, 1, 2, 6, 13, 37, 106, ... 3 A052451 0, 0, 1, 3, 15, 82, 578, ... 4 A052452 0, 0, 0, 1, 4, 30, 301, ... 5 0,0,0,0,1,5,51,... 6 0,0,0,0,0,1,6,... See also CLIQUE ,CLIQUE GRAPH References Aigner, M. "Tura ´n’s Graph Theorem." Amer. Math. Monthly 102, 808 /C1/16, 1995. Sloane, N. J. A. Sequences A052450, 052451, and A052452 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Clock Arithmetic CONGRUENCE Clock Prime A prime number obtained by reading digits around an analog clock. In a clockwise directions, the primes are 2, 3, 5, 7, 11, 23, 67, 89, 4567, 23456789, 23456789101112123, ... (Sloane’s A036342). In a counterclockwise direction, the primes are 2, 3, 5, 7, 11, 43, 109, 10987, 76543, 6543211211, 4321121110987, ... (Sloane’s A036342). In either direction, the primes are 2, 3, 5, 7, 11, 23, 43, 67, 89, 109, 4567, 10987, 76543, 23456789, 6543211211, ... (Sloane’s A036344). On a 24-hour digital clock, there are 211 possible prime values: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 101, ... (Sloane’s A050246). References Rivera, C. "Problems & Puzzles: Puzzle Primes on a Clock.- 019." http://www.primepuzzles.net/puzzles/puzz_019.htm. Sloane, N. J. A. Sequences A036342, A036343, A036344, and A050246 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Clock Solitaire A solitaire game played with CARDS . The chance of winning is 1/13, and the AVERAGE number of CARDS turned up is 42.4. References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 244 /C1/47, 1978. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, pp. 377 and 577, 1997.Moyse, A. Jr. 150 Ways to Play Solitaire. Chicago: Whitman, 1950. Close Packing SPHERE PACKING Closed A mathematical structure A is said to be closed under an operation /C27 if, whenever a and b are both elements of A, then so is a /C27b:/ A mathematical object taken together with its bound- ary is also called closed. For example, while the interior of a SPHERE is an OPEN BALL , the interior together with the sphere itself is a CLOSED BALL . See also CLOSED BALL,CLOSED CURVE ,CLOSED DISK, CLOSED FORM,CLOSURE (TOPOLOGY ) Closed Ball The closed ball with center x and radius r is defined by Br(x) /C30fy : ½y /C28x½5r g: See also BALL,CLOSED DISK,OPEN BALL References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 1, 1991. Closed Curve In the plane, a closed curve is a CURVE with no endpoints and which completely encloses an AREA . See also CURVE ,JORDAN CURVE ,SIMPLE CURVE References Krantz, S. G. "Closed Curves." §2.1.2 in Handbook of Com- plex Analysis. Boston, MA: Birkha ¨user, pp. 19 /C1/0, 1999. Closed Curve Problem Find NECESSARY and SUFFICIENT conditions that determine when the integral curve of two periodic functions k(s) and t(s) with the same period Lis a CLOSED CURVE . Closed Disk An n-D closed disk of RADIUS r is the collection of points of distance 5r from a fixed point in EUCLIDEAN n-space. Krantz (1999, p. 3) uses the symbol ¯D(x; r) to denote the closed disk, and ¯D /C30 ¯D(0 ; 1) to denote the unit closed disk centered at the origin See also DISK,OPEN DISK References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 3, 1999. Closed Form A discrete FUNCTION A(n; k) is called closed form (or sometimes "hypergeometric") in two variables if the ratios A(n /C271; k) =A(n; k) and A(n; k /C271)=A(n; k) are both RATIONAL FUNCTIONS . A pair of closed form functions (F, G) is said to be a WILF-ZEILBERGER PAIR if F(n /C271; k) /C28F(n ; k) /C30G(n; k /C271) /C28G(n; k) : See also ELEMENTARY NUMBER ,LIOUVILLIAN NUM- BER,RATIONAL FUNCTION ,W ILF-ZEILBERGER PAIR References Chow, T. Y. "What is a Closed-Form Number?" Amer. Math. Monthly 106, 440 /C1/48, 1999. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 141, 1996. Zeilberger, D. "Closed Form (Pun Intended!)." Contemporary Math. 143, 579 /C1/07, 1993. Closed Graph Theorem A linear OPERATOR between two BANACH SPACES is continuous IFF it has a "closed" graph. See also BANACH SPACE References Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.Closed Interval An INTERVAL which includes its LIMIT POINTS . If the endpoints of the interval are FINITE numbers a and b, then the INTERVAL is denoted [a, b]. If one of the endpoints is 9/C12 ; then the interval still contains all of its LIMIT POINTS ,so[a;/C12) and (/C28/C12; b] are also closed intervals. See also CLOSED BALL,CLOSED DISK,CLOSED SET, HALF-CLOSED INTERVAL ,INTERVAL ,OPEN INTERVAL References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 1, 1991. Closed Manifold A COMPACT MANIFOLD without boundary. See also OPEN MANIFOLD Closed Set There are several equivalent definitions of a closed SET.A SETSis closed if 1. The COMPLEMENT ofSis an OPEN SET , 2.Sis its own CLOSURE , 3. Sequences/nets/filters in Swhich converge do so within S, 4. Every point outside Shas a NEIGHBORHOOD disjoint from S. The POINT-SET TOPOLOGICAL definition of a closed set is a set which contains all of its LIMIT POINTS . Therefore, a closed set Cis one for which, whatever point xis picked outside of C,xcan always be isolated in some OPEN SET which doesn’t touch C. The most commonly encountered closed sets are the CLOSED INTERVAL , closed path, CLOSED DISK , interior of a closed path together with the path itself, and CLOSED BALL . The C ANTOR SET is an unusual closed set in the sense that it consists entirely of BOUNDARY POINTS (and is nowhere DENSE , so it has L EBESGUE MEASURE 0). It is possible for a set to be neither OPEN nor closed, e.g., the HALF-CLOSED INTERVAL (0; 1]:/ See also BOREL SET,BOUNDARY POINT ,CANTOR SET, CLOSED BALL,C LOSED INTERVAL ,C LOSED DISK, COMPACT SET,HALF-CLOSED INTERVAL ,OPEN SET References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 3, 1999. Closed Star The CLOSURE St y of a STAR St y at a vertex v of a SIMPLICIAL COMPLEX K. See also LINK (SIMPLICIAL COMPLEX ), STAR References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Closed Subgroup A SUBSET of a TOPOLOGICAL GROUP which is CLOSED as a SUBSET and also a SUBGROUP . See also EFFECTIVE ACTION ,FREE ACTION ,GROUP , ISOTROPY GROUP ,M ATRIX GROUP ,O RBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION , TOPOLOGICAL GROUP ,TRANSITIVE Closure (Set) A SET S and a BINARY OPERATOR /C31 are said to exhibit closure if applying the BINARY OPERATOR to two elements S returns a value which is itself a member of S. The term "closure" is also used to refer to a "closed" version of a given set. The closure of a SET can be defined in several equivalent ways, including 1. The SET plus its LIMIT POINTS , also called "boundary" points, the union of which is also called the "frontier." 2. The unique smallest CLOSED SET containing the given SET. 3. The COMPLEMENT of the interior of the COMPLE- MENT of the set. 4. The collection of all points such that every NEIGHBORHOOD of these points intersects the original SET in a nonempty SET. In topologies where the T2-SEPARATION AXIOM is assumed, the closure of a finite SET S is S itself. See also BINARY OPERATOR ,BOUNDARY SET,CLOSURE (TOPOLOGY ), CONNECTED SET,EXISTENTIAL CLOSURE , REFLEXIVE CLOSURE ,T IGHT CLOSURE ,T RANSITIVE CLOSUREReferences Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Closure (Topology) The closure of a set A is the smallest closed set containing A. Closed sets are CLOSED under arbitrary intersection, so it is also the intersection of all closed sets containing A. Typically, it is just A with all of its ACCUMULATION POINTS . See also CLOSED SET,C LOSURE (SET), SEQUENCE , TOPOLOGY Closure Relation d(x /C28t) /C30X/C12 n/C300fn(x) fn(t) ; where d(x) is the DELTA FUNCTION . Clothoid CORNU SPIRAL Clove Hitch A HITCH also called the BOATMAN’S KNOT or PEG KNOT . References Owen, P. Knots. Philadelphia, PA: Courage, pp. 24 /C1/7, 1993. Club SPHINX Clump RUN Cluster Given a POINT LATTICE , a cluster is a group of filled cells which are all connected to their neighbors vertically or horizontally. See also CLUSTER PERIMETER ,PERCOLATION THEORY , S-CLUSTER , S-RUN References Stauffer, D. and Aharony, A. Introduction to Percolation Theory, 2nd ed. London: Taylor & Francis, 1992. Cluster Perimeter The number of empty neighbors of a CLUSTER . See also PERIMETER POLYNOMIAL Cluster Prime An ODD PRIME p is called a cluster prime if every EVEN positive integer less than p /C282 can be written as a difference of two primes q /C28q?; where q; q?5p : The first 23 odd primes 3, 5, 7, ..., 89 are all cluster primes. The first few odd primes that are not cluster primes are 97, 127, 149, 191, 211, ... (Sloane’s A038133). The numbers of cluster primes less than 101,102, ... are 23, 99, 420, 1807, ... (Sloane’s A039506), and the corresponding numbers of noncluster primes are 0, 1, 68, 808, 7784, ... (Sloane’s A039507). It is not known if there are infinitely many cluster primes, but Bleck- smith et al. (1999) show that for every positive integer s, there is a bound x0 /C30xx(s) such that if x ] x0 ; then pc(x) Bx (ln x)s ; where pc(x) is the number of cluster primes not exceeding x. Blecksmith et al. (1999) also show that the sum of the reciprocals of the cluster primes is finite. See also PRIME CONSTELLATION References Blecksmith, R.; Erdos, P.; and Selfridge, J. L. "Cluster Primes." Amer. Math. Monthly 106,43/C1/8, 1999. Sloane, N. J. A. Sequences A038133, A039506, and A039507 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. C-Matrix Any SYMMETRIC MATRIX ( CT /C30C)or SKEW SYMMETRIC MATRIX (/CT /C30/C28C) Cnwith diagonal elements 0 and others 9 1 satisfying CCT /C30( n/C281)I; where I is the IDENTITY MATRIX , is known as a C- matrix (Ball and Coxeter 1987). There are two symmetric C-matrices of order 2, 0 /C281 /C2810l12ml121 ;01 10l12ml121 and two antisymmetric C-matrices of order 2, 01 /C2810l12ml121 ;01 /C2810l12ml121 : Further examples include C4 /C300 /C27/C27/C27 /C28 0 /C28/C27 /C28/C27 0 /C28 /C28/C28/C27 02 6643 775C6 /C300 /C27/C27/C27/C27/C27 /C27 0 /C27/C28/C28/C27 /C27/C27 0 /C27/C28/C28 /C27/C28/C27 0 /C27/C28 /C27/C28/C28/C27 0 /C27 /C27/C27/C28/C28/C27 02 66666643 7777775 There are no symmetric C-matrices of order 4 or 22 (Ball and Coxeter 1987, p. 309). The following table gives the number of C-matrices of orders n /C301, 2, .... Type Sloane Numbers symmetric 0, 2, 0, 0, 0, 384, 0, 0, ... antisymmetric 0, 2, 0, 16, 0, 0, 0, 30720, ... total 0, 4, 0, 16, 0, 384, 0, 30720, ... A C-matrix of an odd prime power order may be constructed using a general method due to Paley (Paley 1933, Ball and Coxeter 1987). References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 308 /C1/09, 1987. Belevitch, V. Ann. de la Socie´te´ scientifique de Bruxelles 82, 13 /C1/2, 1968. Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626 /C1/30, 1972. Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 689, 1996. Paley, R. E. A. C. "On Orthogonal Matrices." J. Math. Phys. 12, 311/C1/20, 1933. Raghavarao, D. Constructions and Combinatorial Problems in Design of Experiments. New York: Dover, 1988. Coanalytic Set ADEFINABLE SET which is the complement of an ANALYTIC SET . See also ANALYTIC SET Coastline Paradox Determining the length of a country’s coastline is not as simple as it first appears, as first considered byL. F. Richardson (1881 /C1 /953). In fact, the answer depends on the length of the RULER you use for the measurements. A shorter RULER measures more of the sinuosity of bays and inlets than a larger one, sothe estimated length continues to increase as the RULER length decreases. In fact, a coastline is an example of a FRACTAL , and plotting the length of the RULER versus the measured length of the coastline on a log-log plot gives a straight line, the slope of which is the FRACTAL DIMENSION of the coastline (and will be a number between 1 and 2). See also LONGIMETER References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 29 /C1/1, 1991. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 109 /C1/10, 1999. Coates-Wiles Theorem In 1976, Coates and Wiles showed that ELLIPTIC CURVES with COMPLEX MULTIPLICATION having an infinite number of solutions have L-functions which are zero at the relevant fixed point. This is a special case of the SWINNERTON- DYER CONJECTURE . References Cipra, B. "Fermat Prover Points to Next Challenges." Science 271, 1668 /C1/669, 1996. Coaxal Circles CIRCLES which share a RADICAL LINE with a given circle are said to be coaxal. The centers of coaxal circles are COLLINEAR , and the collection of all coaxal circles is called a pencil of coaxal circles (Coxeter and Greitzer 1967, p. 35). It is possible to combine the two types of coaxal systems illustrated above such that the sets are orthogonal. Members of a COAXAL SYSTEM satisfy x2 /C27y2 /C272lx /C27c /C30(x /C27 l)2 /C27y2 /C27c /C28 l2 /C300 for values of l: Picking /l2 /C30c/ then gives the two circles (x 9ffiffifficp)2 /C27y2 /C300 of zero RADIUS , known as POINT CIRCLES . The twopoint circles /(9ffiffifficp; 0)/, real or imaginary, are called the LIMITING POINTS . See also CIRCLE ,COAXALOID SYSTEM ,GAUSS- BODEN- MILLER THEOREM ,LIMITING POINT ,POINT CIRCLE , RADICAL LINE References Casey, J. "Coaxal Circles." §6.5 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Exam- ples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 113 /C1/26, 1888. Coolidge, J. L. "Coaxal Circles." §1.7 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 95 /C1/13, 1971. Coxeter, H. S. M. and Greitzer, S. L. "Coaxal Circles." §2.3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 35 /C1/6 and 122, 1967. Dixon, R. Mathographics. New York: Dover, pp. 68 /C1/2, 1991. Durell, C. V. "Coaxal Circles." Ch. 11 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 121 /C1/25, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 34 /C1/7, 199, and 279, 1929. Lachlan, R. "Coaxal Circles." Ch. 13 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 199 /C1/17, 1893. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 143 /C1/44, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 33 /C1/4, 1991. Coaxal Planes SHEAF OF PLANES Coaxal System A system of COAXAL CIRCLES . See also COAXAL CIRCLES ,PONCELET’S COAXAL THE- OREM Coaxaloid System A system of circles obtained by multiplying each RADIUS in a COAXAL SYSTEM by a constant. References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 276 /C1/77, 1929. Coaxial Circles COAXAL CIRCLES Cobordant Manifold Two open MANIFOLDS M and M ? are cobordant if there exists a MANIFOLD with boundary Wn/C271 such that an acceptable restrictive relationship holds. See also COBORDISM , H-COBORDISM THEOREM ,MORSE THEORY Cobordism BORDISM , H-COBORDISM Cobordism Group BORDISM GROUP Cobordism Ring BORDISM GROUP Cobweb Equation This entry contributed by RONALD M. AARTS The simple first-order DIFFERENCE EQUATION yt /C271 /C28Ayt /C30B ; (1) where A /C30/C28ms md(2) B /C30bd /C28 bs md(3) and Dt /C30/C28mdpt /C27bd (4) St/C271 /C30mspt /C27bs (5) are the price-demand and price-supply curves, where /C28mdand bdrepresent the slope and D-intercept, respectively, for the demand curve, and msand bs represent the corresponding constants for the supply curve (Ezekiel 1938, Goldberg 1986). A class of behaviors related to this equation is known as "Cobweb phenomena" in economics. See also DIFFERENCE EQUATION References Ezekiel, M. "The Cobweb Theorem." Quart. J. Econ. 52, 255 /C1/80, 1938. Goldberg, S. Introduction to Difference Equations, with Illustrative Examples from Economics, Psychology, and Sociology. New York: Dover, 1986. Cochleoid The cochleoid, whose name means "snail-form" inLatin, was first discussed by J. Peck in 1700 (Mac- Tutor Archive). It has also been called the oui-ja board curve (Beyer 1987, p. 215). The points of contact of PARALLEL TANGENTS to the cochleoid lie on a STROPHOID . In POLAR COORDINATES , r /C30a sin u u: (1) In CARTESIAN COORDINATES , (x2 /C27y2) tan/C281y x ! /C30ay : (2) The CURVATURE is k /C302ffiffiffi 2p u3[2u /C28 sin(2u)] [1 /C27 2u2 /C28 cos(2 u) /C28 2u sin(2u)]3 =2 : (3) See also QUADRATRIX OF HIPPIAS References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 196, 1972. MacTutor History of Mathematics Archive. "Cochleoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Co- chleoid.html. Cochleoid Inverse Curve The INVERSE CURVE of the COCHLEOID r/C30sinu u(1) with INVERSION CENTER at the ORIGIN and inversion radius k is the QUADRATRIX OF HIPPIAS . x /C30kt cot u (2) y /C30kt: (3) Cochloid CONCHOID OF NICOMEDES Cochran’s Theorem The converse of FISHER’S THEOREM . Cocked Hat Curve The PLANE CURVE (x2 /C272ay /C28a2)2 /C30y2(a2 /C28x2) ; which is similar to the BICORN . References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Cocktail Party Graph A GRAPH consisting of two rows of paired nodes in which all nodes but the paired ones are connected with an EDGE . It is the complement of the LADDER GRAPH . See also LADDER GRAPH Coconut MONKEY AND COCONUT PROBLEM Codazzi Equations MAINARDI- CODAZZI EQUATIONS Code A code is a set of n-tuples of elements ("WORDS ") taken from an ALPHABET .See also ALPHABET ,C ODING THEORY ,E NCODING , ERROR- CORRECTING CODE,G RAY CODE,H UFFMAN CODING , ISBN, LINEAR CODE, UPC, WORD Codimension The minimum number of parameters needed to fully describe all possible behaviors near a nonstructurally stable element. See also BIFURCATION Coding Theory Coding theory, sometimes called ALGEBRAIC CODING THEORY , deals with the design of ERROR-CORRECTING CODES for the reliable transmission of information across noisy channels. It makes use of classical and modern algebraic techniques involving FINITE FIELDS , GROUP THEORY , and polynomial algebra. It has con- nections with other areas of DISCRETE MATHEMATICS , especially NUMBER THEORY and the theory of experi- mental designs. See also ENCODING ,E RROR- CORRECTING CODE,FI- NITE FIELD,HADAMARD MATRIX References Alexander, B. "At the Dawn of the Theory of Codes." Math. Intel. 15,20/C1/6, 1993. Berlekamp, E. R. Algebraic Coding Theory, rev. ed. New York: McGraw-Hill, 1968. Golomb, S. W.; Peile, R. E.; and Scholtz, R. A. Basic Con- cepts in Information Theory and Coding: The Adventures of Secret Agent 00111. New York: Plenum, 1994. Hill, R. First Course in Coding Theory. Oxford, England: Oxford University Press, 1986. Humphreys, O. F. and Prest, M. Y. Numbers, Groups, and Codes. New York: Cambridge University Press, 1990. MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error- Correcting Codes. New York: Elsevier, 1978. Roman, S. Coding and Information Theory. New York: Springer-Verlag, 1992. Stepanov, S. A. Codes on Algebraic Curves. New York: Kluwer, 1999. Vermani, L. R. Elements of Algebraic Coding Theory. Boca Raton, FL: CRC Press, 1996. Weisstein, E. W. "Books about Coding Theory." http:// www.treasure-troves.com/books/CodingTheory.html. Codomain ASETwithin which the values of a function lie (as opposed to the RANGE , which is the set of values that the function actually takes). See also DOMAIN ,RANGE (IMAGE ) References Borowski, E. J. and Borwein, J. M. (Eds.). The HarperCol- lins Dictionary of Mathematics. New York: HarperCollins, p. 89, 1991. Griffel, D. H. Applied Functional Analysis. New York: Wiley, p. 116, 1984. Coefficient A multiplicative factor (usually indexed) such as one of the constants ai in the POLYNOMIAL anxn /C27an /C281xn/C281 /C27... /C27a2x2 /C27a1x /C27a0 :/ See also BINOMIAL COEFFICIENT ,C ARTAN TORSION COEFFICIENT ,C ENTRAL BINOMIAL COEFFICIENT , CLEBSCH- GORDAN COEFFICIENT ,COEFFICIENT FIELD, COEFFICIENT NOTATION ,COMMUTATION COEFFICIENT , CONNECTION COEFFICIENT ,C ORRELATION COEFFI- CIENT ,C ROSS- CORRELATION COEFFICIENT ,E XCESS COEFFICIENT ,G AUSSIAN COEFFICIENT ,L AGRANGIAN COEFFICIENT ,MULTINOMIAL COEFFICIENT ,PEARSON’S SKEWNESS COEFFICIENTS ,PRODUCT- MOMENT COEFFI- CIENT OF CORRELATION ,QUARTILE SKEWNESS COEF- FICIENT ,QUARTILE VARIATION COEFFICIENT ,RACAH V-COEFFICIENT ,RACAH W-COEFFICIENT ,REGRESSION COEFFICIENT ,ROMAN COEFFICIENT ,TRIANGLE COEF- FICIENT ,U NDETERMINED COEFFICIENTS METHOD , VARIATION COEFFICIENT Coefficient Field Let V be a VECTOR SPACE over a FIELD K, and let A be a nonempty SET. For an appropriately defined AFFINE SPACE A, K is called the coefficient field. Coefficient Notation Given a SERIES OF THE FORM A(z) /C30X kakzk ; the notation [zk](A(z)) is used to indicate the coeffi- cient ak(Sedgewick and Flajolet 1996). This corre- sponds to the Mathematica functions Coefficient [A[z], z, k] and SeriesCoeffi- cient [series , k]. References Sedgewick, R. and Flajolet, P. An Introduction to the Analysis of Algorithms. Reading, MA: Addison-Wesley, 1996. Coercive Functional A bilinear FUNCTIONAL f on a normed SPACE E is called coercive (or sometimes ELLIPTIC ) if there exists a POSITIVE constant K such that f(x; x) ]K ½½x½½2 for all x /C23 E:/ See also LAX-MILGRAM THEOREM References Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990.Cofactor The signed version Cij of a MINOR Mij of a MATRIX Cij /C13(/C281)i/C27jMij used in the computation of the matrix’s DETERMINANT det(A) /C30X iaiCij: The cofactor can be computed in Mathematica using Cofactor[m_List,{i_Integer,j_Integer}]: /C30 (-1)^(i /C27j)Drop[Transpose[Drop[Transpose[m], {j}]],{i}] See also DETERMINANT ,DETERMINANT EXPANSION BY MINORS ,MINOR References Muir, T. A Treatise on the Theory of Determinants. New York: Dover, p. 54, 1960. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 235, 1990. Cofactor Expansion DETERMINANT EXPANSION BY MINORS Cofinite Filter This entry contributed by VIKTOR BENGTSSON If S is an infinite set, then the collection FS /C30fA ⁄ S : S /C28A is finite g is a FILTER called the cofinite (or Fre´chet) filter on S. See also FILTER ,ULTRAFILTER Cohen-Kung Theorem Guarantees that the trajectory of LANGTON’S ANT is unbounded. Cohomology Cohomology is an invariant of a TOPOLOGICAL SPACE , formally "dual" to HOMOLOGY , and so it detects "holes" in a SPACE . Cohomology has more algebraic structure than HOMOLOGY , making it into a GRADED RING (with multiplication given by the so-called "CUP PRODUCT "), whereas HOMOLOGY is just a graded ABELIAN GROUP invariant of a SPACE . A generalized homology or cohomology theory must satisfy all of the EILENBERG- STEENROD AXIOMS with the exception of the dimension axiom. See also ALEKSANDROV- CECH COHOMOLOGY ,ALEXAN- DER-SPANIER COHOMOLOGY ,CECH COHOMOLOGY ,CUP PRODUCT , DE RHAM COHOMOLOGY ,DOLBEAULT CO- HOMOLOGY ,G RADED ALGEBRA ,H OMOLOGY (TOPOL- OGY) Cohomology Class See also INTEGRAL COHOMOLOGY CLASS Cohomotopy Group Cohomotopy groups are similar to HOMOTOPY GROUPS . A cohomotopy group is a GROUP related to the HOMOTOPY classes of MAPS from a SPACE X into a SPHERE Sn :/ See also HOMOTOPY GROUP Coin A flat disk which acts as a two-sided DIE. See also BERNOULLI TRIAL,CARDS ,COIN PARADOX , COIN TOSSING ,D ICE,FELLER’S COIN-TOSSING CON- STANTS ,FOUR COINS PROBLEM ,GAMBLER’S RUIN References Brooke, M. Fun for the Money. New York: Scribner’s, 1963. Coin Flipping COIN TOSSING Coin Paradox After a half rotation of the coin on the left around the central coin (of the same RADIUS ), the coin undergoes a complete rotation. In other words, a coin makes two complete rotations when rolled around the boundary of an identical coin. This fact is readily apparent in the generation of the CARDIOID as one disk rolling on another. See also CARDIOID References Pappas, T. "The Coin Paradox." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 220, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 145, 1999. Coin Problem Let there be n ]2 INTEGERS 0 Ba1 B...Banwith (a1 ; a2 ; ...; an) /C301 (all RELATIVELY PRIME ). For large enough N /C30an i/C301 aixi ; there is a solution in NONNEGA- TIVE INTEGERS xi : The greatest N /C30g(a1 ; a2 ; ...; an)for which there is no solution is called the coin problem. Sylvester showed g(a1 ; a2) /C30(a1 /C281)(a2 /C281) /C281 ; and an explicit solution is known for n /C303, but no closed form solution is known for larger N. References Guy, R. K. "The Money-Changing Problem." §C7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 113 /C1/14, 1994. Coin Tossing An idealized coin consists of a circular disk of zero thickness which, when thrown in the air and allowed to fall, will rest with either side face up ("heads" H or "tails" T) with equal probability. A coin is therefore a two-sided DIE. Despite slight differences between the sides and NONZERO thickness of actual coins, the distribution of their tosses makes a good approxima- tion to a p /C301 =2B ERNOULLI DISTRIBUTION . There are, however, some rather counterintuitive properties of coin tossing. For example, it is twice as likely that the triple TTH will be encountered before THT than after it, and three times as likely that THH will precede HHT . Furthermore, it is six times as likely that HTT will be the first of HTT , TTH , and TTT to occur (Honsberger 1979). There are also strings S of Hs and Ts that have the property that the expected wait W(S1) to see string S1is less than the expected wait W(S2) to see S2 ; but the probability of seeing S1before seeing S2is less than 1/2 (Berlekamp et al. 1982; Gardner 1988). Examples include 1. THTH and HTHH , for which W(THTH ) /C3020 and W(HTHH ) /C3018 ; but for which the probability that THTH occurs before HTHH is 9/14 (Gardner 1988, p. 64), 2. W(TTHH ) /C30W(THHH ) /C3016 ; W(HHH ) ; but for which the probability that TTHH occurs before HHH is 7/12, and for which the probability that THHH occurs before HHH is 7/8 (Penney 1969; Gardner 1988, p. 66). More amazingly still, spinning a penny instead of tossing it results in heads only about 30% of the time (Paulos 1995). The study of RUNS of two or more identical tosses is well-developed, but a detailed treatment is surpris- ingly complicated given the simple nature of theunderlying process. See also B ERNOULLI DISTRIBUTION ,BERNOULLI TRIAL, CARDS ,COIN,DICE,GAMBLER’S RUIN,M ARTINGALE , RUN,SAINT PETERSBURG PARADOX References Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Games in General. London: Academic Press, p. 777, 1982. Ford, J. "How Random is a Coin Toss?" Physics Today 36, 40 /C1/7, 1983. Gardner, M. "Nontransitive Paradoxes." Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 64 /C1/6, 1988. Honsberger, R. "Some Surprises in Probability." Ch. 5 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 100 /C1/03, 1979. Keller, J. B. "The Probability of Heads." Amer. Math. Monthly 93, 191 /C1/97, 1986. Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, p. 75, 1995. Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 238 /C1/39, 1990. Penney, W. "Problem 95. Penney-Ante." J. Recr. Math. 2, 241, 1969. Sloane, N. J. A. Sequences A000225/M2655 and A050227 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Spencer, J. "Combinatorics by Coin Flipping." Coll. Math. J., 17, 407 /C1/12, 1986. Whittaker, E. T. and Robinson, G. "The Frequency Distribu- tion of Tosses of a Coin." §90 in The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 176 /C1/77, 1967. Coincidence A coincidence is a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection (Diaconis and Mosteller 1989). Given a large number events, extremely unlikely coincidences are possible–and perhaps even common. To quote Sherlock Holmes, "Amid the action and reaction of so dense a swarm of humanity, every possible combination of events may be expected to take place, and many a little problem will be presented which may be striking and bizarre..." (Conan Doyle 1988, p. 245). See also BIRTHDAY PROBLEM ,LAW OF TRULY LARGE NUMBERS ,O DDS,P ROBABILITY ,R ANDOM NUMBER , SIGNIFICANCE References Bogomolny, A. "Coincidence." http://www.cut-the-knot.com/ do_you_know/coincidence.html. Conan Doyle, A. "The Adventure of the Blue Carbuncle." In The Complete Sherlock Holmes. New York: Doubleday, pp. 244 /C1/57, 1988. Falk, R. "On Coincidences." Skeptical Inquirer 6,18/C1/1, 1981 /C1/2. Falk, R. "The Judgment of Coincidences: Mine Versus Yours." Amer. J. Psych. 102, 477 /C1/93, 1989. Falk, R. and MacGregor, D. "The Surprisingness of Coin- cidences." In Analysing and Aiding Decision Processes (Ed. P. Humphreys, O. Svenson, and A. Va´ri). New York: Elsevier, pp. 489 /C1/02, 1984. Diaconis, P. and Mosteller, F. "Methods of Studying Coin- cidences." J. Amer. Statist. Assoc. 84, 853 /C1/61, 1989. Jung, C. G. Synchronicity: An Acausal Connecting Princi- ple. Princeton, NJ: Princeton University Press, 1973.Kammerer, P. Das Gesetz der Serie: Eine Lehre von den Wiederholungen im Lebens--und im Weltgeschehen. Stutt- gart, Germany: Deutsche Verlags-Anstahlt, 1919. Stewart, I. "What a Coincidence!" Sci. Amer. 278,95/C1/6, June 1998. Coincident Two LINES or plane CONGRUENT geometric figures which lie on top of each other are said to be coincident. See also CONGRUENT ,HOMOTHETIC ,SIMILAR Colatitude The polar angle on a SPHERE measured from the North Pole instead of the equator. The angle f in SPHERICAL COORDINATES is the COLATITUDE .Itis related to the LATITUDE d by f /C3090 /C14/C28 d :/ See also LATITUDE ,LONGITUDE ,SPHERICAL COORDI- NATES Colinear COLLINEAR Collapsoid The collapsoids are a class of non-convex collapsible polyhedra. They can be constructed by replacing each edge of a DODECAHEDRON orICOSAHEDRON by the diagonal of a pyramid (with base removed). Thirty such pyramids are then fitted together using tabs. References Pedersen, J. "Collapsoids." Math. Gaz. 59,8 1/C1/4, 1975. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 34, 1991. Collatz Problem A problem posed by L. Collatz in 1937, also called the 3X/C271 MAPPING ,HASSE’S ALGORITHM ,KAKUTANI’S PRO- BLEM ,S YRACUSE ALGORITHM ,S YRACUSE PROBLEM , THWAITES CONJECTURE , and ULAM’S PROBLEM (Lagar- ias 1985). Thwaites (1996) has offered a £1000 reward for resolving the CONJECTURE . Let a0be an INTEGER . Then the Collatz problem asks if iterating an/C301 2an/C281 foran/C281even 3an/C281/C271 for an/C281oddl12) (1) always returns to 1 for POSITIVE a0:This question has been tested and found to be true for all numbers /53/C215253:2:702/C291016(Oliveira e Silva 1999), im- proving the earlier results of 1015(Vardi 1991, p. 129) and 5 :6/C291013(Leavens and Vermeulen 1992). The members of the SEQUENCE produced by the Collatz are sometimes known as HAILSTONE NUMBERS . Be- cause of the difficulty in solving this problem, Erdos commented that "mathematics is not yet ready for such problems" (Lagarias 1985). If NEGATIVE numbers are included, there are four known cycles (excluding the trivial 0 cycle): (4, 2, 1), ( /C282,/C281), (/C285,/C287,/C2810), and (/C2817,/C2825,/C2837,/C2855,/C2882,/C2841,/C2861,/C2891, /C28136,/C2868,/C2834). The number of tripling steps needed to reach 1 for n/C301, 2, ... are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane’s A006667). The Collatz problem was modified by Terras (1976, 1979), who asked if iterating tn/C301 2tn/C281 fortn/C281even 12(3tn/C281/C271) for tn/C281odd( (2) always returns to 1 for initial integer value t0:If NEGATIVE numbers are included, there are 4 known cycles: (1, 2), ( /C281), (/C285,/C287,/C2810), and ( /C2817,/C2825, /C2837,/C2855,/C2882,/C2841,/C2861,/C2891,/C28136,/C2868,/C2834). It is a special case of the "generalized Collatz problem" with d/C302,m0/C301;m1/C303;r0/C300;and r1/C30 /C281:Terras (1976, 1979) also proved that the set of INTEGERS Sk/C13fn:nhas stopping time 5kghas a limiting asymptotic density F(k);such that if Nx(k) is the number of nsuch that n5xands(n)5k;then the limit F(k)/C30lim x0/C12Nx(k) x; (3) exists. Furthermore, F(k)01a sk0/C12;so almost all INTEGERS have a finite stopping time. Finally, for all k]1; 1/C28F(k)/C30lim x0/C12Nx(k) x52/C28nk; (4) where H(x)/C30/C28xlgx/C28(1/C28x) lg(1/C28x) (5) u/C301 lg 3(6) h/C301/C28H(u)/C300:05004 . . . (7) (Lagarias 1985). Conway proved that the original Collatz problem has no nontrivial cycles of length B400:Lagarias (1985) showed that there are no nontrivial cycles with length B275;000:Conway (1972) also proved that Collatz- type problems can be formally UNDECIDABLE . A generalization of the C OLLATZ PROBLEM letsd]2 be a POSITIVE INTEGER andm0;...,md/C281beNONZERO INTEGERS . Also let ri/C23Zsatisfyri/C13imi(mod d): (8) Then T(x)/C30mix/C28ri d(9) forx/C13i(mod d) defines a generalized Collatz map- ping. An equivalent form is T(x)/C30mix d$% /C27Xi (10) forx/C13i(mod d) where X0;...,Xd/C281are INTEGERS and rbcis the FLOOR FUNCTION . The problem is connected with ERGODIC THEORY and M ARKOV CHAINS (Mat- thews 1995). Matthews (1995) obtained the followingtable for the mapping T k(x)/C301 2x forx/C130 (mod 2) 12(3x/C27k) for x/C131 (mod 2) ;( (11) where k/C30T5k:/ k# Cycles Max. Cycle Length 05 2 7 11 0 3 42 13 118 3 17 118 4 19 1185 21 1656 23 433 Matthews and Watts (1984) proposed the following conjectures. 1. If m 0/C1/C1/C1md/C281 jj Bdd;then all trajectories fTK(n)gforn/C23Zeventually cycle. 2. If m0/C1/C1/C1md/C281 jj >dd;then almost all trajectories fTK(n)gforn/C23Zare divergent, except for an exceptional set of INTEGERS nsatisfying #fn/C23S/C28X5nBXg/C30o(X): j 3. The number of cycles is finite.4. If the trajectory fT K(n)gforn/C23Zis not even- tually cyclic, then the iterates are uniformlydistribution mod d afor each a]1;with limN0/C121 N/C271card fK5NTK(n)/C13j(mod da)gl112l112 /C30d/C28 a (12) for 0 5j 5d a /C281:/ Matthews believes that the map T(x) /C307x /C273 for x /C130 (mod 3) 1 3(7x /C272) for x /C131 (mod 3) 13(x /C282) for x /C132 (mod 3)8 >< >:(13) will either reach 0 (mod 3) or will enter one of the cycles (/C281) or (/C282;/C284); and offers a $100 (Austra- lian?) prize for a proof. See also HAILSTONE NUMBER References Applegate, D. and Lagarias, J. C. "Density Bounds for the 3x /C271 Problem 1. Tree-Search Method." Math. Comput. 64, 411 /C1/26, 1995. Applegate, D. and Lagarias, J. C. "Density Bounds for the 3x /C271 Problem 2. Krasikov Inequalities." Math. Comput. 64, 427 /C1/38, 1995. Burckel, S. "Functional Equations Associated with Con- gruential Functions." Theor. Comp. Sci. 123, 397 /C1/06, 1994. Conway, J. H. "Unpredictable Iterations." Proc. 1972 Num- ber Th. Conf. , University of Colorado, Boulder, Colorado, pp. 49 /C1/2, 1972. Crandall, R. "On the ‘/3x /C271/’ Problem." Math. Comput. 32, 1281 /C1/292, 1978. Everett, C. "Iteration of the Number Theoretic Function f(2n) /C30n; f(2n /C271) /C30f(3n /C272):/" Adv. Math. 25,42/C1/5, 1977. Guy, R. K. "Collatz’s Sequence." §E16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215 /C1/18, 1994. Lagarias, J. C. "The 3x /C271 Problem and Its Generaliza- tions." Amer. Math. Monthly 92,3/C1/3, 1985. http:// www.cecm.sfu.ca/organics/papers/lagarias/. Leavens, G. T. and Vermeulen, M. "/3x /C271 Search Pro- grams." Comput. Math. Appl. 24,79/C1/9, 1992. Margenstern, M. and Matiyasevich, Y. "A Binomial Repre- sentation of the 3x /C271 Problem." Acta Arith. 91, 367 /C1/78, 1999. Matthews, K. R. "The Generalized 3x /C271 Mapping." http:// www.maths.uq.oz.au/~krm/survey.ps. Rev. Mar. 30, 1999. Matthews, K. R. and Watts, A. M. "A Generalization of Hasses’s Generalization of the Syracuse Algorithm." Acta Arith. 43, 167 /C1/75, 1984. Oliveira e Silva, T. "Maximum Excursion and Stopping Time Record-Holders for the 3x /C271 Problem: Computational Results." Math. Comput. 68, 371 /C1/84, 1999. Schroeppel, R.; Gosper, R. W.; Henneman, W.; and Banks, R. Item 133 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 64, Feb. 1972. Sloane, N. J. A. Sequences A006667/M0019 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Terras, R. "A Stopping Time Problem on the Positive Integers." Acta Arith. 30, 241 /C1/52, 1976. Terras, R. "On the Existence of a Density." Acta Arith. 35, 101 /C1/02, 1979. Thwaites, B. "Two Conjectures, or How to Win £1100." Math.Gaz. 80,35/C1/6, 1996. Vardi, I. "The 3x /C271 Problem." Ch. 7 in Computational Recreations in Mathematica. Redwood City, CA: Addi- son-Wesley, pp. 129 /C1/37, 1991.Collinear Three or more points P1 ; P2 ; P3 ; ..., are said to be collinear if they lie on a single straight LINE L. A line on which points lie, especially if it is related to a geometric figure such as a TRIANGLE , is sometimes called an AXIS. Three points are collinear IFF the ratios of distances satisfy x2 /C28x1 : y2 /C28y1 : z2 /C28z1 /C30x3 /C28x1 : y3 /C28y1 : z3 /C28z1 : Two points are trivially collinear since two points determine a LINE. Let points P1 ; P2 ; and P3 lie, one each, on the sides of a triangle DA1A2A3or their extensions, and reflect these points about the midpoints of the triangle sides to obtain P?1 ; P?2 ; and P ?3 : Then P ?1 ; P ?2 ; and P?3are collinear IFF P1 ; P2 ; and P3 are (Honsberger 1995). See also AXIS,CONCYCLIC ,CONFIGURATION ,DIRECTED ANGLE ,DROZ-FARNY THEOREM ,GENERAL POSITION , LINE,N-CLUSTER ,SYLVESTER’S LINE PROBLEM References Coxeter, H. S. M. and Greitzer, S. L. "Collinearity and Concurrence." Ch. 3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 51 /C1/9, 1967. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 153 /C1/54, 1995. Collineation A transformation of the plane which transforms COLLINEAR points into COLLINEAR points. A projective collineation transforms every 1-D form projectively, and a perspective collineation is a collineation which leaves all lines through a point and points through a line invariant. In an ELATION , the center and axis are incident; in a HOMOLOGY they are not. For further discussion, see Coxeter (1969, p. 248). See also AFFINITY ,CORRELATION ,ELATION ,EQUIAF- FINITY ,HOMOLOGY (GEOMETRY ), PERSPECTIVE COLLI- NEATION ,PROJECTIVE COLLINEATION References Coxeter, H. S. M. "Collineations and Correlations." §14.6 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 247 /C1/51, 1969. Collision-Free Hash Function A function Hthat maps an arbitrary length message Mto a fixed length message digest MDis a collision- free hash function if 1. It is a ONE-WAY HASH FUNCTION . 2. It is hard to find two distinct messages ( M?;M) that hash to the same result H(M?)/C30H(M):More precisely, any efficient algorithm (solving a P- PROBLEM ) succeeds in finding such a collision with negligible probability (Russell 1992). See also HASH FUNCTION References Bakhtiari, S.; Safavi-Naini, R.; and Pieprzyk, J. Crypto- graphic Hash Functions: A Survey. Technical Report 95 /C1/ 9, Department of Computer Science, University of Wol- longong, July 1995. ftp://ftp.cs.uow.edu.au/pub/papers/ 1995/tr-95 /C1/9.ps.Z. Russell, A. "Necessary and Sufficient Conditions for Colli- sion-Free Hashing." In Abstracts of Crypto 92. pp. 10 /C1/2 /C1/ 0 /C1/7. ftp://theory.lcs.mit.edu/pub/people/acr/hash.ps. Collocation Method A method of determining coefficients al in an expan- sion y(x) /C30y0(x) /C27Xq l /C301alyl(x) so as to nullify the values of an ORDINARY DIFFER- ENTIAL EQUATION L[y(x)] /C300 at prescribed points. References Itoˆ, K. (Ed.). "Methods Other than Difference Methods." §303I in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 1139, 1980. Cologarithm The LOGARITHM of the RECIPROCAL of a number, equal to the NEGATIVE of the LOGARITHM of the number itself, colog x /C13log1 x ! /C30/C28log x: See also ANTILOGARITHM ,LOGARITHM Colon Product Let AB and CD be DYADS . Their colon product is defined by AB : CD /C13C /C215 AB /C215 D /C30(A /C215 C)(B /C215 D): See also DYAD Colorable Color each segment of a KNOT DIAGRAM using one of three colors. If1. At any crossing, either the colors are all different or all the same, and 2. At least two colors are used, then a KNOT is said to be colorable (or more specifi- cally, THREE-COLORABLE ). Colorability is invariant under REIDEMEISTER MOVES , and can be generalized. For instance, for five colors 0, 1, 2, 3, and 4, a KNOT is five-colorable if 1. at any crossing, three segments meet. If the overpass is numbered a and the two underpasses B and C, then 2a /C13b /C27c (mod 5); and 2. at least two colors are used. Colorability cannot always distinguish HANDEDNESS . For instance, three-colorability can distinguish the mirror images of the TREFOIL KNOT but not the FIGURE-OF-EIGHT KNOT . Five-colorability, on the other hand, distinguishes the MIRROR IMAGES of the FIGURE- OF-EIGHT KNOT but not the TREFOIL KNOT . See also COLORING ,W ORM Coloring A coloring of plane regions, LINK segments, etc., is an assignment of a distinct labeling (which could be a number, letter, color, etc.) to each component. Color- ing problems generally involve TOPOLOGICAL consid- erations (i.e., they depend on the abstract study of the arrangement of objects), and theorems about color- ings, such as the famous FOUR-COLOR THEOREM , can be extremely difficult to prove. See also COLORABLE ,EDGE COLORING ,FOUR- COLOR THEOREM , K-COLORING ,LOVA´ SZ NUMBER ,POLYHE- DRON COLORING ,SIX-COLOR THEOREM ,THREE- COLOR- ABLE ,VERTEX COLORING References Eppstein, D. "Coloring." http://www.ics.uci.edu/~eppstein/ junkyard/color.html. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986. Columbian Number SELF NUMBER Column Space See also ROW SPACE Column Vector An m /C291 MATRIX a11 a21 n am12 6643 775: See also MATRIX ,ROW VECTOR ,VECTOR Column-Convex Polyomino A column-convex polyomino is a self-avoiding CONVEX POLYOMINO such that the intersection of any vertical line with the polyomino has at most two connected components. Column-convex polyominos are also called vertically convex polyominoes. A ROW-CONVEX POLYOMINO is similarly defined. The number a(n)of column-convex n-polyominoes are given by the third- order RECURRENCE RELATION a(n) /C305a(n /C281) /C287a(n /C282) /C274a(n /C283) with a(1) /C301 ; a(2) /C302 ; a(3) /C306; and a(4) /C3019 (Hick- erson 1999). The first few are 1, 2, 6, 19, 61, 196, 629, 2017, ... (Sloane’s A001169). a(n) has GENERATING FUNCTION f(x)x(1 /C28 x)3 1 /C28 5x /C27 7x2 /C28 4x3 /C30x /C272x2 /C276x3 /C2719x4 /C27...: See also CONVEX POLYOMINO ,P OLYOMINO ,R OW- CONVEX POLYOMINO References Enting, I. G. and Guttmann, A. J. "On the Area of Square Lattice Polygons." J. Statist. Phys. 58, 475 /C1/84, 1990. Phys. Rev. Ser. 2 103,1/C1/6, 1956. Hickerson, D.. "Counting Horizontally Convex Polyomi- noes." J. Integer Sequences 2, No. 99.1.8, 1999. http:// www.research.att.com/~njas/sequences/JIS/HICK2/ chcp.html. Klarner, D. A. "Some Results Concerning Polyominoes." Fib. Quart. 3,9/C1/0, 1965. Klarner, D. A. "Cell Growth Problems." Canad. J. Math. 19, 851 /C1/63, 1967. Klarner, D. A. "The Number of Graded Partially Ordered Sets." J. Combin. Th. 6,12/C1/9, 1969. Lunnon, W. F. "Counting Polyominoes." In Computers in Number Theory, Proc. Science Research Council Atlas Symposium No. 2 held at Oxford, from 18 /C1/3 August, 1969 (Ed. A. O. L. Atkin and B. J. Birch). London: Aca- demic Press, pp. 347 /C1/72, 1971.Po´lya, G. "On the Number of Certain Lattice Polygons." J. Combin. Th. 6, 102 /C1/05, 1969. Sloane, N. J. A. Sequences A001169/M1636 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stanley, R. P. "Generating Functions." In Studies in Combi- natorics (Ed. G.-C. Rota). Washington, DC: Amer. Math. Soc., pp. 100 /C1/41, 1978. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, p. 259, 1999. Temperley, H. N. V. "Combinatorial Problems Suggested By the Statistical Mechanics of Domains and of Rubber-Like Molecules." Colunar Triangle Given a SCHWARZ TRIANGLE (pqr) ; replacing each VERTEX with its antipodes gives the three colunar SPHERICAL TRIANGLES (pq ?r ?) ; (p ?qr?); (p?q ?r) where 1 p /C271 p?/C301 1 q /C271 q?/C301 1 r /C271 r?/C301: See also SCHWARZ TRIANGLE ,SPHERICAL TRIANGLE References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 112, 1973. Comass The comass of a DIFFERENTIAL P-FORM f is the largest value of f on a p vector of p-volume one, sup v/C23LpTM;vjj/C301f(v) jj : See also CALIBRATION FORM Comb Function SHAH FUNCTION Combination The number of ways of picking k unordered outcomes from npossibilities. Also known as the BINOMIAL COEFFICIENT orCHOICE NUMBER and read " nchoose r." nCk/C13n kl11sl11n /C13n! k!(n/C28k)!; where n!isa FACTORIAL (Uspensky 1937, p. 18). For example, there are4 2l1ml11 /C306 combinations on f1; 2; 3; 4g; namely f1 ; 2g;f1 ; 3g;f1; 4g;f2; 3g; f2; 4g; and f3 ; 4 g: These combinations are known as K-SUBSETS . Muir (1960, p. 7) uses the nonstandard notations (n)k /C30 n kl1ml11 and (¯n)k /C30 n/C28k kl1ml11 :/ See also BINOMIAL COEFFICIENT ,D ERANGEMENT , FACTORIAL , K-SUBSET ,PERMUTATION ,SUBFACTORIAL References Conway, J. H. and Guy, R. K. "Choice Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 67 /C1/8, 1996. Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Ruskey, F. "Information on Combinations of a Set." http:// www.theory.csc.uvic.ca/~cos/inf/comb/CombinationsIn- fo.html. Skiena, S. "Combinations." §1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 40 /C1/6, 1990. Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, p. 18, 1937. Combination Lock Let a combination of n buttons be a SEQUENCE of disjoint nonempty SUBSETS of the SET f1; 2; ...; ng: If the number of possible combinations is denoted an ; then an satisfies the RECURRENCE RELATION an /C30Xn/C281 i/C300n n /C28il11sl11n ai ; (1) with a0 /C301: This can also be written an /C30dn dxn1 2 /C28 ex ! j x /C300/C301 2X/C12 k /C300kn 2k ; (2) where the definition 00 /C301 has been used. Further- more, anXn k /C301An; k2n/C28k /C30Xn k/C301An; k2k /C281 ; (3) where An ; kare EULERIAN NUMBERS . In terms of the STIRLING NUMBERS OF THE SECOND KIND s(n; k) ; an /C30Xn k/C301k!s(n; k) : (4) /an can also be given in closed form as an /C301 2 Li /C28n(12) ; (5) where Lin(z) is the POLYLOGARITHM . The first few values of an for n /C301, 2, ... are 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... (Sloane’s A000670).The quantity bn /C13an n! (6) satisfies the inequality 1 2(ln 2)n 5bn 51 (ln 2)n : (7) References Sloane, N. J. A. Sequences A000670/M2952 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Velleman, D. J. and Call, G. S. "Permutations and Combina- tion Locks." Math. Mag. 68, 243 /C1/53, 1995. Combinatorial Composition COMPOSITION Combinatorial Design References Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, 1996. Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. Combinatorial Dual Graph Let m(G) be the cycle rank of a graph G, m/C31(G) be the cocycle rank, and the relative complement G /C28H of a SUBGRAPH H of G be defined as that subgraph obtained by deleting the lines of H. Then a graph G /C31 is a combinatorial dual of G if there is a one-to-one correspondence between their sets of lines such that for any choice Yand Y/C31of corresponding subsets of lines, M/C31(G/C28Y)/C30m/C31(G)/C28m(/C142Y/C31/C143); where /C142Y/C31/C143is the subgraph of G/C31with the line set Y/C31:/ Whitney showed that the GEOMETRIC DUAL GRAPH and combinatorial dual graph are equivalent (Harary 1994, p. 115), and so may simply be called "the" DUAL GRAPH . Also, a graph is PLANAR IFF it has a combina- torial dual (Harary 1994, p. 115). See also DUAL GRAPH ,G EOMETRIC DUAL GRAPH , PLANAR GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 113 /C1/15, 1994. Combinatorial Geometry See also MATROID References Friedman, E. "Erich’s Combinatorial Geometry Page." http:// www.stetson.edu/~efriedma/comb.html. Pach, J. and Agarwal, P. K. Combinatorial Geometry. New York: Wiley, 1995. Combinatorial Number BINOMIAL COEFFICIENT Combinatorial Optimization References Ausiello, G.; Crescenzi, P.; Gambois, G.; Kann, V.; March- etti-Spaccamela, A.; and Protasi, M. Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Berlin: Springer- Verlag, 1999. Du, D.-Z. and Pardalos, P. M. (Eds.). Handbook of Combi- natorial Optimization, Vols. 1 /C1/. Amsterdam, Nether- lands: Kluwer, 1998. Combinatorial Species SPECIES Combinatorial Topology Combinatorial topology is a special type of ALGEBRAIC TOPOLOGY that uses COMBINATORIAL methods. For example, SIMPLICIAL HOMOLOGY is a combinatorial construction in ALGEBRAIC TOPOLOGY , so it belongs to combinatorial topology. See also ALGEBRAIC TOPOLOGY ,SIMPLICIAL HOMOL- OGY,TOPOLOGY References Alexandrov, P. S. Combinatorial Topology. New York: Do- ver, 1998. Pontryagin, L. S. Foundations of Combinatorial Topology. New York: Dover, 1999. Combinatorics The branch of mathematics studying the enumera- tion, combination, and permutation of sets of ele- ments and the mathematical relations which characterize these properties. See also ALGEBRAIC COMBINATORICS ,A NTICHAIN , CHAIN ,D ILWORTH’S LEMMA ,D IVERSITY CONDITION , ENUMERATION PROBLEM ,ERDOS- SZEKERES THEOREM , INCLUSION- EXCLUSION PRINCIPLE ,K IRKMAN’S SCHOOLGIRL PROBLEM ,K IRKMAN TRIPLE SYSTEM , LENGTH (PARTIAL ORDER ), PARTIAL ORDER ,PIGEON- HOLE PRINCIPLE ,R AMSEY’S THEOREM ,S CHRO ¨ DER- BERNSTEIN THEOREM ,SCHUR’S LEMMA ,SPERNER’S THEOREM ,TOTAL ORDER ,U MBRAL CALCULUS , VANDER WAERDEN’S THEOREM ,W IDTH (PARTIAL ORDER ) References Abramowitz, M. and Stegun, C. A. (Eds.). "Combinatorial Analysis." Ch. 24 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 821 /C1/827, 1972. Aigner, M. Combinatorial Theory. New York: Springer- Verlag, 1997. Bellman, R. and Hall, M. Combinatorial Analysis. Amer. Math. Soc., 1979. Berge, C. Principles of Combinatorics. New York: Academic Press, 1971. Bergeron, F.; Labelle, G.; and Leroux, P. Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, 1998. Biggs, N. L. "The Roots of Combinatorics." Historia Mathe- matica 6, 109/C1/36, 1979. Bose, R. C. and Manvel, B. Introduction to Combinatorial Theory. New York: Wiley, 1984. Brown, K. S. "Combinatorics." http://www.seanet.com/ ~ksbrown/icombina.htm. Cameron, P. J. Combinatorics: Topics, Techniques, Algo- rithms. New York: Cambridge University Press, 1994. Cohen, D. Basic Techniques of Combinatorial Theory. New York: Wiley, 1978. Cohen, D. E. Combinatorial Group Theory: A Topological Approach. New York: Cambridge University Press, 1989. Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, 1996. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Dinitz, J. H. and Stinson, D. R. (Eds.). Contemporary Design Theory: A Collection of Surveys. New York: Wiley, 1992. Eisen, M. Elementary Combinatorial Analysis. New York: Gordon and Breach, 1969. Electronic Journal of Combinatorics. http://www.combina- torics.org/previous_volumes.html. Eppstein, D. "Combinatorial Geometry." http://www.ics.u- ci.edu/~eppstein/junkyard/combinatorial.html. Erdos, P. and Spencer, J. Probabilistic Methods in Combi- natorics. New York: Academic Press, 1974. Erickson, M. J. Introduction to Combinatorics. New York: Wiley, 1996. Fields, J. "On-Line Dictionary of Combinatorics." http:// www.math.uic.edu/~fields/comb_dic/. Gardner, M. "Combinatorial Theory." Ch. 3 in The Sixth Book of Mathematical Games from Scientific American.Chicago, IL: University of Chicago Press, pp. 19 /C1 /8, 1984. Godsil, C. D. "Problems in Algebraic Combinatorics." Elec- tronic J. Combinatorics 2,F 11 /C1/0, 1995. http://www.com- binatorics.org/Volume_2/volume2.html#F1. Graham, R. L.; Gro ¨tschel, M.; and Lova ´sz, L. (Eds.). Hand- book of Combinatorics, 2 vols. Cambridge, MA: MIT Press, 1996. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed.Reading, MA: Addison-Wesley, 1994. Grimaldi, R. P. Discrete and Combinatorial Mathematics: An Applied Introduction, 4th ed. Longman, 1998. Hall, M. Jr. Combinatorial Theory, 2nd ed. New York: Wiley, 1986. Harary, F. Applied Combinatorial Mathematics. New York: Wiley, 1964. Knuth, D. E. (Ed.). Stable Marriage and Its Relation to Other Combinatorial Problems. Providence, RI: Amer. Math. Soc., 1997. Kreher, D. L. and Stinson, D. Combinatorial Algorithms: Generation, Enumeration, and Search. Boca Raton, FL: CRC Press, 1999. Kucera, L. Combinatorial Algorithms. Bristol, England: Adam Hilger, 1989. Liu, C. L. Introduction to Combinatorial Mathematics. New York: McGraw-Hill, 1968. MacMahon, P. A. Combinatory Analysis, 2 vols. New York: Chelsea, 1960. Marcus, D. Combinatorics: A Problem Oriented Approach. Washington, DC: Math. Assoc. Amer., 1998. Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Petit, S. "Encyclopedia of Combinatorial Structures." http:// algo.inria.fr/encyclopedia/. Raghavarao, D. Constructions and Combinatorial Problems in Design of Experiments. New York: Dover, 1988. Riordan, J. Combinatorial Identities, reprint ed. with correc- tions. Huntington, NY: Krieger, 1979. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980. Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Rosen, K. H. (Ed.). Handbook of Discrete and Combinatorial Mathematics. Boca Raton, FL: CRC Press, 2000. Rota, G.-C. (Ed.). Studies in Combinatorics. Providence, RI: Math. Assoc. Amer., 1978. Ruskey, F. "The (Combinatorial) Object Server." http:// www.theory.csc.uvic.ca/~cos/. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Slomson, A. Introduction to Combinatorics. Boca Raton, FL: Chapman and Hall, 1997. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, 1999. Stanley, R. P. Enumerative Combinatorics, Vol. 2. Cam- bridge, England: Cambridge University Press, 1999. Street, A. P. and Wallis, W. D. Combinatorial Theory: An Introduction. Winnipeg, Manitoba: Charles Babbage Re- search Center, 1977. Tucker, A. Applied Combinatorics, 3rd ed. New York: Wiley, 1995. van Lint, J. H. and Wilson, R. M. A Course in Combinato- rics. New York: Cambridge University Press, 1992. Weisstein, E. W. "Books about Combinatorics." http:// www.treasure-troves.com/books/Combinatorics.html. Wilf, H. S. Combinatorial Algorithms: An Update. Philadel- phia, PA: SIAM, 1989. Comedian Triangles Two triangles having the same MEDIAN are said to be comedian triangles. See also COSYMMEDIAN TRIANGLES ,M EDIAN (TRIAN- GLE) References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 63, 1893.Comma A typesetting symbol which has several distinct meanings in mathematics. It is used for a number of purposes. 1. To denote Boundaries between elements in a list, as in f1; 2; 3; ...g:/ 2. To delimit indices in the element of a MATRIX ,as in ai; j(although it is frequently omitted when implied by context). 3. To indicate the COMMA DERIVATIVE of a TENSOR . 4. In place of a DECIMAL POINT in continental Europe, e.g., 3,14159. See also COMMA DERIVATIVE ,DECIMAL POINT References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 275, 1997. Comma Derivative For A a TENSOR , A;k /C13@A @xk /C13@kA Ak ;k /C131 gk@Ak @xk /C13@kAk : Schmutzer (1968, p. 70) uses the older notation Akj /. See also COVARIANT DERIVATIVE ,TENSOR References Schmutzer, E. Relativistische Physik (Klassische Theorie). Leipzig, Germany: Akademische Verlagsgesellschaft, 1968. Comma of Didymus The musical interval by which four fifths exceed a seventeenth (i.e., two octaves and a major third), @A @xk /C13@kA also called a SYNTONIC COMMA . See also COMMA OF PYTHAGORAS ,DIESIS ,SCHISMA Comma of Pythagoras The musical interval by which twelve fifths exceed seven octaves, Ak ;k Successive CONTINUED FRACTION CONVERGENTS to 1 gk@Ak @xk/C13@kAk give increasingly close approximations Akjofmfifths by n octaves as 1, 2, 5/3, 12/7, 41/24, 53/31, 306/179, 665/389, ... (Sloane’s A005664 and A046102; Jeans 1968, p. 188), shown in bold in the table below. All near-equalities of m fifths and n octaves having with are given in the following table. mn Ratio mn Ratio 12 7 1.013643265 265 155 1.010495356 41 24 0.9886025477 294 172 0.9855324037 53 31 1.002090314 306 179 0.9989782832 65 38 1.015762098 318 186 1.012607608 94 55 0.9906690375 347 203 0.9875924759 106 62 1.004184997 359 210 1.001066462 118 69 1.017885359 371 217 1.014724276 147 86 0.9927398469 400 234 0.9896568543 159 93 1.006284059 412 241 1.003159005 188 110 0.9814251419 424 248 1.016845369 200 117 0.994814985 453 265 0.9917255479 212 124 1.008387509 465 272 1.005255922 241 141 0.9834766286 477 279 1.018970895 253 148 0.9968944607 494 289 0.9804224033 See also COMMA OF DIDYMUS ,DIESIS ,SCHISMA References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 257, 1995. Guy, R. K. "Small Differences Between Powers of 2 and 3." §F23 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 261, 1994. Sloane, N. J. A. Sequences A005664/M1428 and A046102 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html.Commandino’s Theorem The four medians of a TETRAHEDRON CONCUR in a point which divides each MEDIAN in the ratio 1:3, the longer segment being on the side of the vertex of the TETRAHEDRON . See also BIMEDIAN ,M EDIAN (TETRAHEDRON ), TETRA- HEDRON References Altshiller-Court, N. "Commandino’s Theorem." §170 in Mod- ern Pure Solid Geometry. New York: Chelsea, pp. 51 /C1/2, 1979. Commandino, F. Prop. 17 in De centro gravitatis solidorum . p. 21, 1565. Common Cycloid CYCLOID Common Fraction A FRACTION in which NUMERATOR and DENOMINATOR are both integers, as opposed to fractions. Common fractions are sometimes also called vulgar fractions. See also COMPLEX FRACTION ,FRACTION Common Logarithm The LOGARITHM inBASE 10. The notation log xis used by physicists, engineers, and calculator keypads to denote the common logarithm. However, mathemati- cians generally use the same symbol to mean the NATURAL LOGARITHM LN ,l nx:Worse still, in Russian literature the notation lg xis used to denote a base-10 logarithm, which conflicts with the use of the symbol LGto indicate the logarithm to base 2. To avoid all ambiguity, it is best to explicitly specify log10xwhen the logarithm to base 10 is intended. In this work, logx/C30log10x;lnx/C30logexis used for the NATURAL LOGARITHM , and lg x /C30log2 x is the logarithm to the base 2. Hardy and Wright (1979, p. 8) assert that the common logarithm has "no mathematical interest." Common and natural logarithms can be expressed in terms of each other as ln x /C30log10 x log10 e log10 x /C30ln x ln 10 : See also LG,LN,LOGARITHM ,NATURAL LOGARITHM References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Common Residue The value of b, where a /C13b (mod m) ; taken to be NONNEGATIVE and smaller than m. See also MINIMAL RESIDUE ,RESIDUE (CONGRUENCE ) Commutation Coefficient A TENSOR -like coefficient which gives the difference between PARTIAL DERIVATIVES of two coordinates with respect to the other coordinate, c m ab /C0em /C30[ /C0ea ; /C0e b] /C309a /C0eb /C289b /C0e a : See also CONNECTION COEFFICIENT ,PARTIAL DERIVA- TIVE Commutative Two elements x and y of a set S are said to be commutative under a binary operation + if they satisfy x + y /C30y + x: Real numbers are commutative under addition x /C27y /C30y /C27x and multiplication x /C215 y /C30y /C215 x: See also ASSOCIATIVE ,C OMMUTE ,C OMMUTATIVE ALGEBRA ,C OMMUTATIVE MATRICES ,C OMMUTATIVE RING,DISTRIBUTIVE ,TRANSITIVECommutative Algebra Let A denote an R/-algebra, so that A is a VECTOR SPACE over R and A /C29A 0 A (1) (x; y) /C2x /C215 y : (2) Now define Z /C13fx /C23 A : x /C215 y /C300 for some y /C23 A "0 g; (3) where 0 /C23 Z : An ASSOCIATIVE R/-algebra is commuta- tive if x /C215 y /C30y /C215 x for all x; y /C23 A: Similarly, a RING is commutative if the MULTIPLICATION operation is commutative, and a LIE ALGEBRA is commutative if the COMMUTATOR [A, B] is 0 for every A and B in the LIE ALGEBRA . See also ABELIAN GROUP ,COMMUTATIVE References Atiyah, M. F. and MacDonald, I. G. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, pp. 9 /C1/0, 1969. Cox, D.; Little, J.; and O’Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer- Verlag, 1996. Eisenbud, D. (Ed.). Commutative Algebra, Algebraic Geome- try, and Computational Methods. Singapore: Springer- Verlag, 1999. Finch, S. "Zero Structures in Real Algebras." http:// www.mathsoft.com/asolve/zerodiv/zerodiv.html. MacDonald, I. G. and Atiyah, M. F. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, 1969. Samuel, P. and Zariski, O. Commutative Algebra, Vol. 2. New York: Springer-Verlag, 1997. Zariski, O. and Samuel, P. Commutative Algebra I. New York: Springer-Verlag, 1958. Commutative Group ABELIAN GROUP Commutative Matrices COMMUTING MATRICES Commutative Ring A RING is commutative if the MULTIPLICATION opera- tion is COMMUTATIVE . See also COMMUTATIVE ,RING Commutator Let ˜A;˜B;...be OPERATORS . Then the commutator of ˜A and ˜Bis defined as [˜A;˜B]/C13˜A˜B/C28˜B˜A: (1) Leta,b, ... be constants. Identities include [f(x);x]/C300 (2) [˜A;˜A]/C300 (3) [ ˜A; ˜B] /C30/C28[ ˜B ; ˜A] (4) [ ˜A; ˜B ˜C] /C30[ ˜A; ˜B] ˜C /C27 ˜B[ ˜A; ˜C] (5) [ ˜A ˜B ; ˜C] /C30[ ˜A; ˜C] ˜B /C27 ˜A[ ˜B ; ˜C] (6) [a /C27 ˜A; b /C27 ˜B] /C30[ ˜A; ˜B] (7) [ ˜A /C27 ˜B ; ˜C /C27 ˜D] /C30[ ˜A; ˜C] /C27[ ˜A; ˜D] /C27[ ˜B ; ˜C] /C27[ ˜B; ˜D] : (8) Let A and B be TENSORS . Then [A; B] /C139AB /C289BA: (9) There is a related notion of commutator in the theory of groups. The commutator of two GROUP elements A and B is ABA /C281B /C281 ; and two elements A and B are said to COMMUTE when their commutator is the IDENTITY ELEMENT . When the group is a LIE GROUP , the LIE BRACKET in its LIE ALGEBRA is an infinitesi- mal version of the group commutator. For instance, let A and B be square matrices, and let a(s) and b(t)be paths in the LIE GROUP of INVERTIBLE MATRICES which satisfy a(0) /C30 b(0) /C301 (10) @x @s j s/C300/C30A (11) @ b @s j s /C300/C30B; (12) then @ @s@ @ta(s) b(t) a/C281(s) b/C281(t)j (s/C300 ; t /C300)/C302[A; B] : (13) See also AD, AD,A NTICOMMUTATOR ,C OMMUTATOR SUBGROUP ,JACOBI IDENTITIES References Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 13, 1996. Commutator Series (Lie Algebra) The commutator series of a LIE ALGEBRA g; sometimes called the derived series, is the sequence of subalge- bras recursively defined by gk /C271 /C30[ gk ;gk]; with g0 /C30g: The sequence of subspaces is always decreasing with respect to inclusion or dimension, and becomes stable when g is finite dimensional. The notation [ a;b] means the linear span of elements of the form [A, B], where A /C23a and B /C23b:/ When the commutator series ends in the zero sub- space, the Lie algebra is called SOLVABLE . For example, consider the LIE ALGEBRA of strictly UPPERTRIANGULAR MATRICES , then g0 /C300 a12a13a14a15 00 a23a24a25 00 0a34a35 0000 a45 000002 666643 77775(1) g 1 /C3000 a13a14a15 00 0 a24a25 00 0 0 a35 00000 000002 666643 77775(2) g 2 /C300000 a15 0000 0 0000 0 0000 0 0000 02 666643 77775; (3) and g 3 /C300: By definition, gk ƒgk where gk is the term in the LOWER CENTRAL SERIES , as can be seen by the example above. In contrast to the SOLVABLE LIE ALGEBRAS , the SEMISIMPLE LIE ALGEBRAS have a constant commu- tator series. Others are in between, e.g., [ gln ;gln] /C30sln ; (4) which is semisimple, because the TRACE satisfies Tr(AB) /C30Tr(BA) : (5) Here, gln is a general linear Lie algebra and sln is the SPECIAL LINEAR LIE ALGEBRA . Here are some Mathematica functions for determin- ing the commutator series, given a list of matrices which is a basis for g:/ MatrixBasis[a_List]: /C30 Partition[#1,Length[a[[1]]]]&/@ LatticeReduce[Flatten/@a] LieCommutator[a_,b_]: /C30a.b-b.a NextDerived[{}] /C30{}; NextDerived[g_List]: /C30 MatrixBasis[Flatten[Outer[LieCommutator,g,g,1] ,1]] kthDerived[g_List,k_Integer]: /C30 Nest[NextDerived,g,k] For example, gl5/C30Flatten[Table[ReplacePart [Table [0,{i,5},{j,5}],1,{k,l}],{k,5},{l,5}],1];sl5 /C30 kthDerived[gl5, 1] See also BOREL SUBALGEBRA ,COMMUTATOR SERIES (GROUP ), LIE ALGEBRA ,LIE GROUP ,NILPOTENT LIE GROUP ,N ILPOTENT LIE ALGEBRA ,REPRESENTATION (LIE ALGEBRA ), REPRESENTATION (SOLVABLE LIE GROUP ), SOLVABLE LIE GROUP ,SPLIT SOLVABLE LIE ALGEBRA Commutator Subgroup The commutator subgroup of a GROUP G is the SUBGROUP generated by the COMMUTATORS of its elements, and is denoted [G, G]. It is always a NORMAL SUBGROUP . It can range from the identity subgroup (in the case of an ABELIAN GROUP ), to the whole group. For instance, in the QUATERNION group f91;9i ;9j ;9kg with eight elements, the commuta- tors form the subgroup f1;/C281g: The commutator subgroup of the SYMMETRIC GROUP is the ALTERNAT- ING GROUP . The commutator subgroup of the ALTER- NATING GROUP An is the whole group An : When n ]5; Anis a SIMPLE GROUP and its only nontrivial normal subgroup is itself. Since [An ; An] is a nontrivial normal subgroup, it must be An :/ The first homology of a group G is the ABELIANIZA- TION H1(G) /C30G=[G ; G] : See also ABELIAN GROUP ,ABELIANIZATION ,COMMU- TATOR ,GROUP ,GROUP COHOMOLOGY ,NORMAL SUB- GROUP Commute Two algebraic objects that are COMMUTATIVE , i.e., A and B such that A + B /C30B + A for some operation +; are said to commute with each other. See also COMMUTATIVE ,COMMUTATOR Commuting Matrices This entry contributed by RONALD M. AARTS Two matrices A and B which satisfy AB /C30BA under MATRIX MULTIPLICATION are said to be com- muting. In general, MATRIX MULTIPLICATION is not COMMU- TATIVE . Furthermore, in general there is no MATRIX INVERSE A /C281 even when A "0: Finally, AB can be zero even without A /C300 or B /C300: And when AB /C300 ; we may still have BA "0; a simple example of which is provided by A /C3001 00l12ml121 B /C301000l12ml121 ; for whichAB /C300 ; but BA /C3001 00l12ml121 /C30A (Taussky 1957). See also C OMMUTATIVE References Gantmacher, F. R. Ch. 8 in The Theory of Matrices, Vol. 1. Providence, RI: Amer. Math. Soc., 1998. Taussky, O. "Commutativity in Finite Matrices." Amer. Math. Monthly 64, 229 /C1/35, 1957. Co-Monotone Approximation COMONOTONE APPROXIMATION Comonotone Approximation This entry contributed by RONALD M. AARTS The approximation of a piecewise MONOTONIC FUNC- TION f by a polynomial with the same monotonicity. Such comonotonic approximations can always be accomplished with nth degree polynomials, and have an error of Av(f;1=n) (Passow and Raymon 1974, Passow et al. 1974, Newman 1979). References Newman, D. J. "Efficient Co-Monotone Approximation." J. Approx. Th. 25, 189 /C1/92, 1979. Passow, E. and Raymon, L. "Monotone and Comonotone Approximation." Proc. Amer. Math. Soc. 42, 340 /C1/49, 1974. Passow, E.; Raymon, L.; and Roulier, J. A. "Comonotone Polynomial Approximation." J. Approx. Th. 11, 221 /C1/24, 1974. Compact Closure A set U has compact closure if its CLOSURE is COMPACT . Typically, compact closure is equivalent to the condition that U is BOUNDED . See also BOUNDED ,COMPACT SET,TOPOLOGY Compact Group COMPACT LIE GROUP Compact Lie Group If the parameters of a LIE GROUP vary over a CLOSED INTERVAL , them the LIE GROUP is said to be compact. Every representation of a compact group is equiva- lent to a UNITARY representation. See also LIE GROUP References Huang, J.-S. "Compact Lie Groups." Part 3 in Lectures on Representation Theory. Singapore: World Scientific, pp. 71 /C1/28, 1999. Compact Manifold A compact manifold is a MANIFOLD which is compact as a TOPOLOGICAL SPACE . Examples are the CIRCLE (the only 1-D compact manifold) and the n-dimen- sional sphere and torus. Compact manifolds in two dimensions are completely classified by their orienta- tion and the number of holes (GENUS ). For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many CHARTS , and that any continuous real-valued function is bounded on a compact manifold. However, it is an open question if the known compact manifolds in 3-D are complete, and it is not even known what a complete list in 4-D should look like. The following terse table therefore summarizes current knowledge about the number of compact manifolds N(D)ofD dimensions. D /N(D)/ 11 22 See also MANIFOLD ,SPHERE ,TOPOLOGICAL SPACE , TORUS ,TYCHONOF COMPACTNESS THEOREM Compact Set The SET S is compact if, from any SEQUENCE of elements X1 ; X2 ; ...of S, a subsequence can always be extracted which tends to some limit element X of S. Compact sets are therefore sets which are both CLOSED and BOUNDED . See also BOUNDED SET,CLOSED SET References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Compact Space A TOPOLOGICAL SPACE is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose union is X. A subset A of a TOPOLOGICAL SPACE X is compact if it is compact as a TOPOLOGICAL SPACE with the relative topology (i.e., every family of open sets of X whose union contains A has a finite subfamily whose union contains A). Compact Support A function has compact support if it is zero outside of a COMPACT SET. A function with compact support isonly interesting in a BOUNDED domain. Alternatively, one can say that a function has compact support if its SUPPORT is a COMPACT SET. For example, the function f : x 0 x2 in its entire domain (i.e., f : R 0 R /C27) does not have compact support, while any BUMP FUNCTION does have compact support. See also BUMP FUNCTION ,COMPACT SET,SUPPORT Compact Surface A compact surface is a SURFACE which is also a COMPACT SET. A compact surface has a TRIANGULA- TION with a finite number of triangles. The SPHERE and TORUS are compact. See also COMPACT SET,TRIANGULATION Compactification A compactification of a TOPOLOGICAL SPACE X is a larger space Y containing X which is also compact. The smallest compactification is the ONE-POINT COM- PACTIFICATION . For example, the real line is not compact. It is contained in the circle, which is obtained by adding a point at infinity. Similarly, the plane is compactified by adding one point at infinity, giving the SPHERE . See also COMPACT SET,STEREOGRAPHIC PROJECTION , TOPOLOGICAL SPACE Compactness Theorem Inside a BALL BinR3; frectifiable currents SinBLarea S5c;length @S5cg is compact under the FLAT NORM . References Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103, 369/C1/76, 1996. Compact-Open Topology The compact-open topology is a common topology used on FUNCTION SPACES . Suppose Xand Yare TOPOLOGICAL SPACES and C(X;Y) is the set of con- tinuous maps from f:X0Y:The compact-open topology on C(X;Y) is generated by subsets of the following form, B(K;U)/C30ff½f(K)ƒUg; where Kis compact in XandUis open in Y. (Hence the terminology "compact-open.") It is important tonote that these sets are not CLOSED under intersec- tion, and do not form a BASIS . Instead, the sets B(K;U) form a SUBBASIS for the compact-open topology. That is, the open sets in the compact-opentopology are the arbitrary unions of finite intersec- tions of B(K;U): / The simplest FUNCTION SPACE to compare topologies is the space of real-valued continuous functions f : R 0 R: A sequence of functions fnconverges to f /C300 IFF for every B(K ; U) containing f contains all but a finite number of the fn : Hence, for all K /C210 and all e > 0; there exists an N such that for all n /C21N, fn(x) jjBe for all ½x½5K : For example, the sequence of functions fn /C30 sin(nx=2)=(n /C271) /C27x2n =e /C28n2 =2converges to the zero function, although each function is unbounded. When Y is a METRIC SPACE , the compact-open topol- ogy is the same as the topology of COMPACT CONVER- GENCE .IfX is a LOCALLY COMPACT HAUSDORFF space, a fairly weak condition, then the evaluation map e : X /C29C(X ; Y) 0 Y defined by e(x; f) /C30f(x)is CONTINUOUS . Similarly, H : X /C29Z 0 Y is CONTINUOUS IFF the map ˜H : Z 0 C(X ; Y) ; given by H(x; z) /C30 ˜H(z)(x) ; is CONTINUOUS . Hence, the compact-open topology is the right topol- ogy to use in HOMOTOPY theory. See also ALGEBRAIC TOPOLOGY ,COMPACT CONVER- GENCE ,HOMOTOPY THEORY ,TOPOLOGICAL SPACE References Munkres, J. Topology. Englewood Cliffs, NJ: Prentice Hall, pp. 285 /C1/89, 1975. Companion Knot Let K1 be a knot inside a TORUS . Now knot the TORUS in the shape of a second knot (called the companion knot) K2 : Then the new knot resulting from K1is called the SATELLITE KNOT K3 :/ References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 115 /C1/18, 1994. Companion Matrix The companion matrix to a MONIC POLYNOMIAL a(x) /C30a0 /C27a1x /C27.../C27an/C281xn/C281 /C27xn (1) is the n /C29n SQUARE MATRIXA /C3000 /C1/C1/C1 0 /C28a0 10 /C1/C1/C1 0 /C28a1 01 /C1/C1/C1 0 /C28a2 nn :::::: n 00 /C1/C1/C1 1 /C28an/C2812 666643 77775(2) with ones on the SUBDIAGONAL and the last column given by the coefficients of a(x) : Note that in the literature, the companion matrix is sometimes de- fined with the rows and columns switched, i.e., the TRANSPOSE of the above matrix. When eiis the STANDARD BASIS , a companion matrix satisfies Aei/C30ei/C271 (3) for i Bn, as well as Aen/C30X /C28aiei; (4) including A ne1/C30X /C28aiA ie1: (5) The MINIMAL POLYNOMIAL of the companion matrix is therefore a(x); which is also its CHARACTERISTIC POLYNOMIAL . Companion matrices are used to write a matrix in RATIONAL CANONICAL FORM . In fact, any n /C29n matrix whose MINIMAL POLYNOMIAL p(x) has DEGREE n is SIMILAR to the companion matrix for p(x) : The RATIONAL CANONICAL FORM is more interesting when the degree of p(x) is less than n. The following Mathematica command gives the com- panion matrix for a polynomial pin the variable x. CompanionMatrix[p_,x_]: /C30 Module[{rnk /C30Exponent[p,x], v/C30CoefficientList[p,x],w}, w/C30Drop[v/Last[v],-1]; If[rnk /C30/C301,{-w}, Transpose[Append[(Prepend[#1,0]&/@IdentityMa- trix[rnk-1]),-w]]]] See also MATRIX ,M INIMAL POLYNOMIAL (MATRIX ), RATIONAL CANONICAL FORM References Dummit, D. and Foote, R. Abstract Algebra. Englewood Cliffs, NJ: Prentice Hall, 1991. Herstein, I. §6.7 in Topics in Algebra, 2nd ed. New York: Wiley, 1975. Jacobson, N. §3.10 in Basic Algebra I. New York: W. H. Freeman, 1985. Comparability Graph The comparability graph of a POSET P/C30(X;5) is the GRAPH with vertex set Xfor which vertices xand y are adjacent IFFeither x5yory5xinP. See also INTERVAL GRAPH ,PARTIALLY ORDERED SET Comparison Test Let a akand a bkbe a SERIES with POSITIVE terms and suppose a1 5b1 ; a2 5b2 ; .... 1. If the bigger series CONVERGES , then the smaller series also CONVERGES . 2. If the smaller series DIVERGES , then the bigger series also DIVERGES . See also CONVERGENCE TESTS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 280 /C1/81, 1985. Compass A tool with two arms joined at their ends which can be used to draw CIRCLES .In GEOMETRIC CONSTRUCTIONS , the classical Greek rules stipulate that the compass cannot be used to mark off distances, so it must "collapse" whenever one of its arms is removed from the page. This results in significant complication in the complexity of GEOMETRIC CONSTRUCTIONS . See also CONSTRUCTIBLE POLYGON ,GEOMETRIC CON- STRUCTION ,G EOMETROGRAPHY ,M ASCHERONI CON- STRUCTION ,PLANE GEOMETRY ,POLYGON ,PONCELET- STEINER THEOREM ,R ULER ,S IMPLICITY ,S TEINER CONSTRUCTION ,STRAIGHTEDGE References Dixon, R. "Compass Drawings." Ch. 1 in Mathographics. New York: Dover, pp. 1 /C1/8, 1991. Compatible Let Akk be the MATRIX NORM associated with the MATRIX A and xkk be the VECTOR NORM associated with a VECTOR x. Let the product Ax be defined, then Akkand xkkare said to be compatible if Axkk5 Akkxkk: References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 2000, 1980. Complement In general, the word "complement" refers to that subset F ? of some set S which excludes a given subset F. Taking F and its complement F ? together then gives the whole of the original set. The notations F ? and ¯F are commonly used to denote the complement of a set F.This concept is commonly used and made precise in the particular cases of a GRAPH COMPLEMENT , KNOT COMPLEMENT , and COMPLEMENT SET. The word "com- plementary" is also used in the same way, so combin- ing an angle and its COMPLEMENTARY ANGLE gives a RIGHT ANGLE and a complementary error function ERFC and the usual error function ERF give unity when added together, erfcx/C27erfx/C301: See also COMPLEMENT SET,COMPLEMENTARY ANGLE , ERFC,GRAPH COMPLEMENT ,KNOT COMPLEMENT References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 23, 1984. Complement Graph GRAPH COMPLEMENT Complement Knot KNOT COMPLEMENT Complement Set Given a set Swith a subset E, the complement of Eis defined as E?/C13fF:F/C23S;FQEg: (1) Using SET DIFFERENCE notation, the complement is defined by E?/C30S_E: (2) IfE/C30S, then E?/C13S?/C30¥; (3) where ¥is the EMPTY SET . The complement is implemented in Mathematica asComplement [l,l1, ...]. Given a single SET, the second PROBABILITY AXIOM gives 1/C30P(S)/C30P(E@E?): (4) Using the fact that ESE?/C30¥; 1/C30P(E)/C27P(E?) (5) P(E?)/C301/C28P(E): (6) This demonstrates that P(S?)/C30P(¥)/C301/C28P(S)/C301/C281/C300: (7) Given two SETS, P(ESF?)/C30P(E)/C28P(ESF) (8) P(E ?S F ?) /C301 /C28P(E) /C28P(F) /C27P(E S F): (9) See also INTERSECTION ,SET DIFFERENCE ,SYMMETRIC DIFFERENCE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Complementary Angle Two ANGLES a and p=2 /C28 a are said to be complemen- tary. See also ANGLE ,R IGHT ANGLE ,S UPPLEMENTARY ANGLE Complementary Error Function ERFC Complementary Modulus If k is the MODULUS of an ELLIPTIC INTEGRAL or ELLIPTIC FUNCTION , then k ?/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2p is called the complementary modulus. Complete elliptic integrals with respect to the complementary modulus are often denoted K ?(k) /C13K(k?) /C30K(ffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28k 2p ) and E ?(k) /C13E(k ?) /C30E(ffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28k 2p ): See also MODULUS (ELLIPTIC INTEGRAL ) References To¨lke, F. "Parameterfunktionen." Ch. 3 in Praktische Funk- tionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83 /C1/15, 1966. Complementation The process of taking the COMPLEMENT of a set or truth function. In the latter case, complementation is equivalent to the NOT operation. See also COMPLEMENT , NOT Complete COMPLETE AXIOMATIC THEORY ,COMPLETE BIGRAPH , COMPLETE GRAPH ,C OMPLETE QUADRANGLE ,C OM- PLETE QUADRILATERAL ,COMPLETE SEQUENCE ,COM- PLETE SET OF FUNCTIONS ,C OMPLETE SPACE ,COMPLETENESS PROPERTY ,W EAKLY COMPLETE SE- QUENCE Complete Axiomatic Theory An axiomatic theory (such as a GEOMETRY ) is said to be complete if each valid statement in the theory is capable of being proven true or false. See also CONSISTENCY Complete Beta Function BETA FUNCTION ,INCOMPLETE BETA FUNCTION Complete Bigraph COMPLETE BIPARTITE GRAPH Complete Binary Tree A labeled BINARY TREE containing the labels 1 to n with root 1, branches leading to nodes labeled 2 and 3, branches from these leading to 4, 5 and 6, 7, respectively, and so on (Knuth 1997, p. 401). See also BINARY TREE,COMPLETE TREE,COMPLETE TERNARY TREE,HEAP References Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, p. 144, 1998. Complete Bipartite Graph ABIPARTITE GRAPH (i.e., a set of VERTICES decomposed into two disjoint sets such that there are no two VERTICES within the same set are adjacent) such that every pair of VERTICES in the two sets are adjacent. If there are p and q VERTICES in the two sets, the complete bipartite graph (sometimes also called a COMPLETE BIGRAPH ) is denoted Kp ; q : The above figures show K3 ; 2and K2 ; 5 : K3 ; 3is also known as the UTILITY GRAPH , and is the unique 4-CAGE GRAPH . A complete bipartite graph Kn ; nis a CIRCULANT GRAPH (Skiena 1990, p. 99). The complete bipartite graph K18 ; 18 illustrated above plays an important role in the novel by Eco (1989, p. 473; Skiena 1990, p. 143). See also BIPARTITE GRAPH ,CAGE GRAPH ,COMPLETE GRAPH ,C OMPLETE K -PARTITE GRAPH , K-PARTITE GRAPH ,THOMASSEN GRAPH ,UTILITY GRAPH References Eco, U. Foucault’s Pendulum. San Diego: Harcourt Brace Jovanovich, p. 473, 1989. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 12, 1986. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Complete Convex Function This entry contributed by RONALD M. AARTS A function f(x) is completely convex in an OPEN INTERVAL (a, b) if it has DERIVATIVES of all orders there and if (/C281)kf(2k)(x) ]0 for k /C300, 1, 2, ... in that interval (Widder 1945, p. 177). For example, the functions sin x and cos x are completely convex in the intervals (0; p) and (/C28p=2; p=2) respectively. See also COMPLETELY MONOTONIC FUNCTION References Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941.Complete Digraph Complete digraphs are digraphs in which every pair of nodes is connected by a bidirectional edge. See also COMPLETE GRAPH ,D IGRAPH ,R AMSEY’S THEOREM Complete Direct Sum RINGDIRECT PRODUCT Complete Functions COMPLETE SET OF FUNCTIONS Complete Gamma Function GAMMA FUNCTION ,INCOMPLETE GAMMA FUNCTION Complete Graph AGRAPH in which each pair of VERTICES is connected by an EDGE . The complete graph with nVERTICES is denoted Kn;and hasn 2l1ml11 undirected edges, wheren kl1ml11 is aBINOMIAL COEFFICIENT . In older literature, complete GRAPHS are called UNIVERSAL GRAPHS . The number of EDGES inKvisv(v/C281)=2 (the trian- gular numbers), and the GENUS is (v/C283)(v/C284)=12 for v]3:The ADJACENCY MATRIX Aof the complete graph Gtakes the particularly simple form of all 1s with 0s on the diagonal, i.e., the UNIT MATRIX minus the IDENTITY MATRIX , A/C30J/C28I: (1) /K3is the CYCLE GRAPH C3;as well as the ODD GRAPH O2(Skiena 1990, p. 162). K4is the TETRAHEDRAL GRAPH , as well as the WHEEL GRAPH W4;and is also a PLANAR GRAPH .K5is nonplanar. Conway and Gordon (1983) proved that every embedding of K6isINTRIN- SICALLY LINKED with at least one pair of linked triangles. They also showed that any embedding of K7 contains a knotted HAMILTONIAN CYCLE . The CHROMATIC POLYNOMIAL pKn(z)ofKn is given by the FALLING FACTORIAL (z)n ; and the CHROMATIC NUMBER by n. It is not known in general if a set of TREES with 1, 2, ..., n /C281 EDGES can always be packed into Kn : However, if the choice of TREES is restricted to either the path or star from each family, then the packing can always be done (Zaks and Liu 1977, Honsberger 1985). See also CLIQUE ,COMPLETE BIPARTITE GRAPH ,COM- PLETE DIGRAPH ,COMPLETE K-PARTITE GRAPH ,EMPTY GRAPH ,GRAPH COMPLEMENT ,ODD GRAPH References Chartrand, G. Introductory Graph Theory. New York: Dover, pp. 29 /C1/0, 1985. Conway, J. H. and Gordon, C. M. "Knots and Links in Spatial Graphs." J. Graph Th. 7, 445 /C1/53, 1983. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 60 /C1/3, 1985. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 12, 1986. Skiena, S. "Complete Graphs." §4.2.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 82 and 140 /C1/41, 1990. Zaks, S. and Liu, C. L. "Decomposition of Graphs into Trees." In Proceedings of the Eighth Southeastern Con- ference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977 (Ed. F. Hoffman, L. Lesniak-Foster, D. McCarthy, R. C. Mul- lin, K. B. Reid, and R. G. Stanton). Congr. Numerantum 19, 643 /C1/54, 1977. Complete k-Partite Graph A K-PARTITE GRAPH (i.e., a set of VERTICES decomposed into k disjoint sets such that no two VERTICES within the same set are adjacent) such that every pair of VERTICES in the k sets are adjacent. If there are p, q, ..., r VERTICES in the k sets, the complete k-partite graph is denoted /Kp;q ;:::;r :/ The above figure shows K2 ; 3 ; 5 :/ See also COMPLETE GRAPH ,C OMPLETE K-PARTITE GRAPH , K-PARTITE GRAPHReferences Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 23, 1994. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 12, 1986. Skiena, S. "Complete k-Partite Graphs." §4.2.2 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 142 /C1/44, 1990. Complete Metric Space A complete metric space is a METRIC SPACE in which every CAUCHY SEQUENCE is CONVERGENT . Examples include the REAL NUMBERS with the usual metric and the P-ADIC NUMBERS . Complete Minimal Surface A surface which is simultaneously COMPLETE and MINIMAL . There have been a large number of funda- mental breakthroughs in the study of such surfaces in recent years, and they remain the focus of intensive current research. Until the COSTA MINIMAL SURFACE was discovered in 1984, the only other known complete minimal em- beddable surfaces in R3with no self-intersections were the PLANE , CATENOID , and HELICOID . The plane is genus 0 and the catenoid and the helicoid are genus 0 with two punctures, but the Costa minimal surface is genus 1 with three punctures (Schwalbe and Wagon 1999). See also COMPLETE SURFACE ,COSTA MINIMAL SUR- FACE ,MINIMAL SURFACE ,NIRENBERG’S CONJECTURE References Schwalbe, D. and Wagon, S. "The Costa Surface, in Show and Mathematica ."Mathematica in Educ. Res. 8,5 6/C1/3, 1999. Complete Permutation DERANGEMENT Complete Product The complete products of a B OOLEAN ALGEBRA of subsets generated by a set fAkgp k/C301ofCARDINALITY p are the 2pBOOLEAN FUNCTIONS B1B2/C1/C1/C1Bp/C13B1SB2S/C1/C1/C1SBp; where each Bkmay equal Akor its complement ¯Ak: For example, the 23/C308 complete products of A/C30 fA1;A2;A3gare A1A2A3;A1A2¯A3;A1¯A2A3;¯A1A2A3; A1¯A2¯A3;¯A1A2¯A3;¯A1¯A2A3;¯A1¯A2¯A3: Each B OOLEAN FUNCTION has a unique representa- tion (up to order) as a union of complete products. For example, A1A2 @ ¯A3 /C30(A1A2A3 @ A1A2¯A3) @ (A1A2¯A3 @ ¯A1A2¯A3 @ A1¯A2¯A3 @ ¯A1¯A2¯A3) /C30A1A2A3 @ a1A2¯A3 @ ¯A1A2¯A3 @ A1¯A2¯A3 @ ¯A1¯A2¯A3 /C30A1A2A3 /C27A1A2¯A3 /C27 ¯A1¯A2¯A3 (Comtet 1974, p. 186). See also BOOLEAN FUNCTION ,CONJUNCTION References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 186, 1974. Complete Quadrangle If the four points making up a QUADRILATERAL are joined pairwise by six distinct lines, a figure known as a complete quadrangle results. A complete quadran- gle is therefore a set of four points, no three collinear, and the six lines which join them. Note that a complete quadrilateral is different from a COMPLETE QUADRANGLE . The midpoints of the sides of any complete quad- rangle and the three diagonal points all lie on a CONIC known as the NINE-POINT CONIC .Ifitisan ORTHO- CENTRIC QUADRILATERAL , the CONIC reduces to a CIRCLE . The ORTHOCENTERS of the four TRIANGLES of a complete quadrangle are COLLINEAR on the RADICAL LINE of the CIRCLES on the diameters of a QUAD- RILATERAL . See also COMPLETE QUADRANGLE ,PTOLEMY’S THEO- REM References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 230 /C1/31, 1969. Demir, H. "The Compleat [sic] Cyclic Quadrilateral." Amer. Math. Monthly 79, 777 /C1/78, 1972. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 80, 1928. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 25, 1930. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 61 /C1/2, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 101 /C1/04, 1990.Complete Quadrilateral The figure determined by four lines, no three of which are concurrent, and their six points of intersection (Johnson 1929, pp. 61 /C1/2). Note that this figure is different from a COMPLETE QUADRANGLE . A complete quadrilateral has three diagonals (compared to two for an ordinary QUADRILATERAL ). The MIDPOINTS of the diagonals of a complete quadrilateral are COLLI- NEAR on a line M (Johnson 1929, pp. 152 /C1/53). A theorem due to Steiner (Mention 1862, Johnson 1929, Steiner 1971) states that in a complete quad- rilateral, the bisectors of angles are CONCURRENT at 16 points which are the incenters and EXCENTERS of the four TRIANGLES . Furthermore, these points are the intersections of two sets of four CIRCLES each of which is a member of a conjugate coaxal system. The axes of these systems intersect at the point common to the CIRCUMCIRCLES of the quadrilateral. Newton proved that, if a CONIC SECTION is inscribed in a complete quadrilateral, then its center lies on M (Wells 1991). In addition, the ORTHOCENTERS of the four triangles formed by a complete quadrilateral lie on a line which is perpendicular to M. Plu¨cker proved that the circles having the three diagonals as dia- meters have two common points which lie on the line joining the four triangles’ ORTHOCENTERS (Wells 1991). See also COMPLETE QUADRANGLE ,GAUSS- BODENMIL- LER THEOREM ,M IDPOINT ,O RTHOCENTER ,P OLAR CIRCLE ,QUADRILATERAL References Carnot, L. N. M. De la corre ´lation des figures de ge ´ome´trie. Paris: l’Imprimerie de Crapelet, p. 122, 1801. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 230 /C1/31, 1969. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 81, 1928. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 25, 1930. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 61 /C1/2, 149, 152 /C1/53, and 255 /C1/56, 1929. Mention, M. J. "De ´monstration d’un The ´ore`me de M. Stei- ner." Nouv. Ann. Math., 2nd Ser. 1,1 6/C1/0, 1862. Mention, M. J. "De ´monstration d’un The ´ore`me de M. Stei- ner." Nouv. Ann. Math., 2nd Ser. 1,6 5/C1/7, 1862. Steiner, J. Gesammelte Werke, 2nd ed, Vol. 1. New York: Chelsea, p. 223, 1971. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 35, 1991. Complete Residue System A set of numbers a0 ; a1 ; ..., am/C281(mod m) form a complete set of residues, also called a covering system, if they satisfy ai /C13i (mod m) for i /C300, 1, ..., m /C281: For example, a complete system of residues is formed by a base b and a modulus m if the residues riin bi /C13ri (mod m) for i /C301, ..., m /C281 run through the values 1, 2, ..., m /C281:/ See also CONGRUENCE ,E XACT COVERING SYSTEM , HAUPT- EXPONENT ,ORDER (MODULO ), REDUCED RESI- DUE SYSTEM ,RESIDUE CLASS References Guy, R. K. "Covering Systems of Congruences." §F13 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 251 /C1/53, 1994. Nagell, T. "Residue Classes and Residue Systems." §20 in Introduction to Number Theory. New York: Wiley, pp. 69 /C1/ 1, 1951. Complete Sequence A SEQUENCE of numbers V /C30fnn g is complete if every POSITIVE INTEGER n is the sum of some subsequence of V, i.e., there exist ai /C300 or 1 such that n /C30X/C12 i/C301ai ni (Honsberger 1985, pp. 123 /C1/26). The FIBONACCI NUM- BERS are complete. In fact, dropping one number still leaves a complete sequence, although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). The SEQUENCE of PRIMES with the element f1g prepended, f1; 2; 3; 5 ; 7 ; 11 ; 13; 17; 19 ; 23 ; ...g is complete, even if any number of PRIMES each > 7 are dropped, as long as the dropped terms do not include two consecutive PRIMES (Honsberger 1985, pp. 127 /C1/28). This is a consequence of BERTRAND’S POSTULATE . See also BERTRAND’S POSTULATE ,BROWN’S CRITER- ION,FIBONACCI DUAL THEOREM ,GREEDY ALGORITHM , WEAKLY COMPLETE SEQUENCE ,ZECKENDORF’S THEO- REM References Brown, J. L. Jr. "Unique Representations of Integers as Sums of Distinct Lucas Numbers." Fib. Quart. 7, 243 /C1/ 52, 1969. Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer for Fibonacci Numbers. XII." Fib. Quart. 11, 317 /C1/31, 1973. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.Complete Set of Functions A set of ORTHONORMAL FUNCTIONS ffn(x) g is termed complete in the CLOSED INTERVAL x /C23 [a ; b] if, for every PIECEWISE CONTINUOUS function f(x) in the interval, the minimum square error En /C13½½f /C28(c1 f1 /C27.../C27cn fn) ½½2 (where ½½f ½½ denotes the L2-NORM with respect to a WEIGHTING FUNCTION w(x)) converges to zero as n becomes infinite. Symbolically, a set of functions is complete if lim m0/C12gb af(x) /C28Xm n/C300an fn(x)"# 2 w(x) dx /C300; where the above integral is a LEBESGUE INTEGRAL . See also BESSEL’S INEQUALITY ,H ILBERT SPACE , L2- NORM References Arfken, G. "Completeness of Eigenfunctions." §9.4 in Math- ematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 523 /C1/38, 1985. Complete Space A SPACE of COMPLETE FUNCTIONS . See also COMPLETE METRIC SPACE Complete Surface A surface which has no edges. See also COMPLETE MINIMAL SURFACE ,EMBEDDED SURFACE ,MINIMAL SURFACE Complete Ternary Tree A labeled TERNARY TREE containing the labels 1 to n with root 1, branches leading to nodes labeled 2, 3, 4, branches from these leading to 5, 6, 7 and 8, 9, 10 respectively, and so on (Knuth 1997, p. 401). See also COMPLETE BINARY TREE,COMPLETE TREE, TERNARY TREE References Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Complete Tree See also COMPLETE BINARY TREE,COMPLETE TERN- ARY TREE Complete Vector Space A VECTOR SPACE is complete if every CAUCHY SE- QUENCE in the space converges to an element in the space. For example, the rationals are not complete, whereas the real numbers are. See also VECTOR SPACE Completely Monotonic Function This entry contributed by RONALD M. AARTS A completely monotonic function is a function f(x) such that (/C281)/C28nf(n)(x) ]0 for n /C300, 1, 2, .... Such functions occur in areas such as probability theory (Feller 1971), numerical analy- sis, and elasticity (Ismail et al. 1986). See also COMPLETE CONVEX FUNCTION ,M ONOTONIC FUNCTION References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971. Ismail, M. E. H.; Lorch, L.; and Muldon, M. E. "Completely Monotonic Functions Associated with the Gamma Func- tion and Its q-Analogues." J. Math. Anal. Appl. 116,1/C1/, 1986. Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Completely Multiplicative Function A real valued arithmetical function f(n) is called completely multiplicative if f(mn) /C30f(m)f(n) holds for each pair of integers (m, n). See also MULTIPLICATIVE FUNCTION References Ka´tai, I. and Kova´cs, B. "Multiplicative Functions with Nearly Integer Values." Acta Sci. Math. 48, 221 /C1/25, 1985. Completely Regular Graph A POLYHEDRAL GRAPH is completely regular if the DUAL GRAPH is also REGULAR . There are only five types. Let r be the number of EDGES at each node, r/C31 the number of EDGES at each node of the DUAL GRAPH , V the number of VERTICES , E the number of EDGES , and F the number of faces in the PLATONIC SOLID corresponding to the given graph. The following table summarizes the completely regular graphs.Type /r//r/C31/ VEF Tetrahedral 3 3464 Cubical 3 4 8 12 6 Dodecahedral 3 5 20 39 12 Octahedral 4 3 6 12 8 Icosahedral 5 3 12 30 20 Completeness Property All lengths can be expressed as REAL NUMBERS . Completing the Square The conversion of an equation OF THE FORM ax2 /C27 bx /C27c to the form ax/C27b 2a !2 /C27 c /C28b2 4a ! ; which, defining B /C13b =2a and C /C13c /C28b2 =4a ; simplifies to a(x /C27B)2 /C27C: Completion A METRIC SPACE X which is not complete has a CAUCHY SEQUENCE which does not CONVERGE . The completion of X is obtained by adding the limits to the Cauchy sequences. The completion is always COM- PLETE . For example, the rational numbers, with the distance metric, are not complete because there exist CAUCHY SEQUENCES that do not converge, e.g., 1, 1.4, 1.41, 1.414, ... does not converge becauseffiffiffi 2p is not rational. The completion of the rationals is the real numbers. Note that the completion depends on the METRIC . For instance, for any PRIME p, the rationals have a METRIC given by the P-ADIC NORM , and then the completion of the rationals is the set of P-ADIC NUMBERS . Another common example of a completion is the space of L2-FUNCTIONS . Technically speaking, the completion of Xis the set of CAUCHY SEQUENCES and Xis contained in this set, ISOMETRICALLY , as the constant sequences. See also CAUCHY SEQUENCE , L2-SPACE ,LOCAL FIELD, METRIC SPACE , P-ADIC NUMBER ,REAL NUMBER Complex CW -COMPLEX ,SIMPLICIAL COMPLEX Complex Addition Two COMPLEX NUMBERS z /C30x /C27iy and z ?/C30x?/C27iy ? are added together componentwise, z /C27z?/C30(x /C27x ?) /C27i(y /C27y?) : In component form, (x; y) /C27(x?; y?) /C30(x /C27x?; y /C27y?) (Krantz 1999, p. 1). See also COMPLEX DIVISION ,COMPLEX MULTIPLICA- TION ,COMPLEX NUMBER ,VECTOR ADDITION References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 1, 1999. Complex Analysis The study of COMPLEX NUMBERS , their DERIVATIVES , manipulation, and other properties. Complex analy- sis is an extremely powerful tool with an unexpect- edly large number of practical applications to the solution of physical problems. CONTOUR INTEGRA- TION , for example, provides a method of computing difficult INTEGRALS by investigating the singularities of the function in regions of the COMPLEX PLANE near and between the limits of integration. The most fundamental result of complex analysis is the CAUCHY- RIEMANN EQUATIONS , which give the conditions a FUNCTION must satisfy in order for a complex generalization of the DERIVATIVE , the so- called COMPLEX DERIVATIVE , to exist. When the COM- PLEX DERIVATIVE is defined "everywhere," the func- tion is said to be ANALYTIC . A single example of the unexpected power of complex analysis is PICARD’S THEOREM , which states that an ANALYTIC FUNCTION assumes every COMPLEX NUMBER , with possibly one exception, infinitely often in any NEIGHBORHOOD of an ESSENTIAL SINGULARITY ! See also ANALYTIC CONTINUATION ,ARGUMENT PRIN- CIPLE ,BRANCH CUT,BRANCH POINT ,CAUCHY INTE- GRAL FORMULA ,C AUCHY INTEGRAL THEOREM , CAUCHY PRINCIPAL VALUE ,CAUCHY- RIEMANN EQUA- TIONS ,C OMPLEX NUMBER ,C ONFORMAL MAPPING , CONTOUR INTEGRATION , DE MOIVRE’S IDENTITY ,EU- LER FORMULA ,INSIDE- OUTSIDE THEOREM ,JORDAN’S LEMMA ,LAURENT SERIES ,LIOUVILLE’S CONFORMAL- ITY THEOREM ,M ONOGENIC FUNCTION ,M ORERA’S THEOREM ,P ERMANENCE OF ALGEBRAIC FORM,P I- CARD’S THEOREM ,POLE,POLYGENIC FUNCTION ,RE- SIDUE (COMPLEX ANALYSIS ) References Arfken, G. "Functions of a Complex Variable I: Analytic Properties, Mapping" and "Functions of a Complex Vari- able II: Calculus of Residues." Chs. 6 /C1/inMathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 352 /C1/95 and 396 /C1/36, 1985.Boas, R. P. Invitation to Complex Analysis. New York: Random House, 1987. Churchill, R. V. and Brown, J. W. Complex Variables and Applications, 6th ed. New York: McGraw-Hill, 1995. Conway, J. B. Functions of One Complex Variable, 2nd ed. New York: Springer-Verlag, 1995. Forsyth, A. R. Theory of Functions of a Complex Variable, 3rd ed. Cambridge, England: Cambridge University Press, 1918. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, 1996. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, 1999. Lang, S. Complex Analysis, 3rd ed. New York: Springer- Verlag, 1993. Morse, P. M. and Feshbach, H. "Functions of a Complex Variable" and "Tabulation of Properties of Functions ofComplex Variables." Ch. 4 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 348 /C1 /91 and 480 /C1/85, 1953. Needham, T. Visual Complex Analysis. New York: Claren- don Press, 2000. Silverman, R. A. Introductory Complex Analysis. New York: Dover, 1984. Weisstein, E. W. "Books about Complex Analysis." http:// www.treasure-troves.com/books/ComplexAnalysis.html. Complex Conjugate The complex conjugate of a COMPLEX NUMBER z/C13 a/C27biis defined to be ¯z/C13a/C28bi: (1) Note that there are several notations in common use for the complex conjugate. Older physics and engi- neering texts tend to prefer z/C31(Bekefi and Barrett 1987, p. 616; Arfken 1985, p. 356; Harris and Stocker1998, p. 21; Hecht 1998, p. 18; Herkommer 1999, p. 262), while many modern math and physics texts favor ¯z(Abramowitz and Stegun 1972, p. 16; Kaplan 1981, p. 28; Roman 1987, p. 534; Kreyszig 1988, p. 568; Kaplan 1992, p. 572; Harris and Stocker 1998, p. 21; Krantz 1999, p. 2; Anton 2000, p. 528).In the latter case, the notation z/C31is then reserved to denote the ADJOINT operator, which is denoted z$in many older physics texts. In this work, ¯zis used to denote the complex conjugate, and z/C31is used to denote the ADJOINT . The CONJUGATE MATRIX of a MATRIX A/C30(aij) is the MATRIX obtained by replacing each element aijwith its complex conjugate, ¯A/C30(¯aij) (Arfken 1985, p. 210). The complex conjugate is implemented in Mathema- tica asConjugate [z]. The common notational conventions are summarized in the table below. convention complex conjugate ADJOINT mathematics /¯A// A/C31/ engineering /A /C31// A $/ By definition, the complex conjugate satisfies ¯¯z /C30z: (2) The complex conjugate is DISTRIBUTIVE under COM- PLEX ADDITION , z1 /C27z2 /C30z1 /C27z2 ; (3) since (a1 /C27ib1) /C27(a2 /C27ib2) /C30(a1 /C27a2) /C27i(b1 /C27b2) /C30(a1 /C27a2) /C28i(b1 /C27b2) /C30(a1 /C28ib1) /C27(a2 /C28ib2) /C30a1 /C27ib1 /C27a2 /C27ib2 ; and DISTRIBUTIVE over COMPLEX MULTIPLICATION , z1z2 /C30 ¯z1 ¯z2 ; (4) since (a1 /C27b1i)(a2 /C27b2i) /C30(a1a2 /C28b1b2) /C27i(a1b2 /C27a2b1) /C30(a1a2 /C28b1b2) /C28i(a1b2 /C27a2b1) /C30(a1 /C28ib1)(a2 /C28ib2) /C30a1 /C27ib1a2 /C27ib2 : See also ADJOINT MATRIX ,COMPLEX ANALYSIS ,COM- PLEX DIVISION ,COMPLEX NUMBER ,CONJUGATE MA- TRIX,MODULUS (COMPLEX NUMBER ) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Anton, H. Elementary Linear Algebra, 8th ed. New York: Wiley, 2000. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 355 /C1/56, 1985. Bekefi, G. and Barrett, A. H. Electromagnetic Vibrations, Waves, and Radiation. Cambridge, MA: MIT Press, p. 616, 1987. Hecht, E. Optics, 3rd ed. Reading, MA: Addison-Wesley, p. 18, 1998. Herkommer, M. A. Number Theory: A Programmer’s Guide. New York: McGraw-Hill, p. 262, 1999. Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 21, 1998. Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1992. Kaplan, W. Advanced Mathematics for Engineers. Reading, MA: Addison-Wesley, 1981. Krantz, S. G. "Complex Conjugate." §1.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 2, 1999. Kreyszig, E. Advanced Engineering Mathematics, 6th ed. New York: Wiley, p. 568, 1988. Roman, S. "The Conjugate of a Complex Number and Complex Division." §11.2 in College Algebra and Trigono- metry. San Diego, CA: Harcourt, Brace, Jovanovich, pp. 534 /C1/41, 1987.Complex Derivative A DERIVATIVE of a COMPLEX function, which must satisfy the CAUCHY- RIEMANN EQUATIONS in order to be COMPLEX DIFFERENTIABLE . See also CAUCHY- RIEMANN EQUATIONS ,C OMPLEX DIFFERENTIABLE ,DERIVATIVE References Krantz, S. G. "The Complex Derivative." §1.3.5 and 2.2.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 15 /C1/6 and 24, 1999. Complex Differentiable Let z /C30x /C27iy and f(z) /C30u(x; y) /C27iv(x; y) on some region G containing the point z0 : If f(z) satisfies the CAUCHY- RIEMANN EQUATIONS and has continuous first PARTIAL DERIVATIVES at z0 ; then f ?(z0) exists and is given by f ?(z0) /C30lim z0z0f(z) /C28 f(z0) z /C28 z0; and the function is said to be COMPLEX DIFFERENTI- ABLE (or, equivalently, ANALYTIC , HOLOMORPHIC ,or regular). A function f : C 0 C can be thought of as a map from the plane to the plane, f : R2 0 R2 : Then f is complex differentiable iff its JACOBIAN is of the form a /C28b bal12ml121 at every point. That is, its derivative is given by the multiplication of a COMPLEX NUMBER a /C27bi: For instance, the function f(z)/C30¯z;where ¯zis the COMPLEX CONJUGATE ,i snotcomplex differentiable. See also ANALYTIC FUNCTION ,C AUCHY- RIEMANN EQUATIONS ,COMPLEX DERIVATIVE ,DIFFERENTIABLE , ENTIRE FUNCTION ,H OLOMORPHIC FUNCTION ,PSEU- DOANALYTIC FUNCTION References Krantz, S. G. "Alternative Terminology for Holomorphic Functions." §1.3.6 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 16, 1999. Complex Division The division of two COMPLEX NUMBERS can be accom- plished by multiplying the NUMERATOR and DENOMI- NATOR by the COMPLEX CONJUGATE of the DENOMINATOR , for example, with z1/C30a/C27biandz2/C30 c/C27di;z/C30z1=z2is given by z/C30a/C27bi c/C27di/C30(a/C27bi)c/C27di (c/C27di)c/C27di/C30(a/C27bi)(c/C28di) (c/C27di)(c/C28di) /C30(ac/C27bd)/C27i(bc/C28ad) c2/C27d2; where ¯z denotes the COMPLEX CONJUGATE . In compo- nent notation, (x; y) (x?; y?) /C30xx?/C27yy?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2 /C27 y?2p ;yx ?/C28xy?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix?2 /C27 y?2p ! : See also COMPLEX ADDITION ,COMPLEX MULTIPLICA- TION ,COMPLEX NUMBER ,DIVISION Complex Form (Type) The DIFFERENTIAL FORMS on Cn decompose into forms of type (p, q). For example, on C ; the EXTERIOR ALGEBRA decomposes into four types: fflC /C30ffl0 /C154ffl1;0 /C154ffl0 ;1 /C154ffl1 ;1 /C30/C1421/C143/C154/C142 dz /C143/C154/C142 d¯z /C143/C154/C142 dz ffld¯z/C143; (1) where dz /C30dx /C27idy; d¯z /C30dx /C28idy ; and /C156/ denotes the DIRECT SUM. In general, a (p, q)-form is the sum of terms with pdz s and qd¯z/s. A k-form decomposes into a sum of (p, q)-forms, where k /C30p /C27q :/ For example, the 2-forms on C2 decompose as ffl2 C2 /C30ffl2;0 /C154ffl1 ;1 /C154ffl0 ;2 (2) /C30/C142dz1 ffl dz2 /C143/C154/C142 dz1 ffl d¯z1 ; dz1 ffl d¯z2 ; dz2 ffl d¯z1 ; dz2 ffl d¯z2 /C143/C154/C142 d¯z1 ffl d¯z2 /C143: (3) The decomposition into forms of type (p, q)is preserved by HOLOMORPHIC MAPS . More precisely, when f : X 0 Y is holomorphic and a is a (p, q)- form on Y, then the PULLBACK f /C31a is a (p, q)-form on X. Recall that the EXTERIOR ALGEBRA is generated by the ONE-FORMS ,by WEDGE PRODUCT and addition. Then the forms of type (p, q) are generated by Lp( L1; 0) fflLq( L0 ; 1) : (4) The SUBSPACE L1 ; 0 of the complex one-forms can be identified as the /C27i/-EIGENSPACE of the ALMOST COM- PLEX STRUCTURE J, which satisfies J2 /C30/C28I : Similarly, the /C28i/-EIGENSPACE is the SUBSPACE ffl0 ; 1 : In fact, the decomposition of TX /C156C /C30TX1 ; 0 /C154TX0; 1 determines the ALMOST COMPLEX STRUCTURE J on TX. More abstractly, the forms into type (p, q) are a REPRESENTATION of C /C31; where l acts by multiplication by lp ¯lq :/ See also ALMOST COMPLEX STRUCTURE ,C OMPLEX MANIFOLD ,DEL BAR OPERATOR ,DOLBEAULT COHO- MOLOGY References Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, pp. 106 /C1/26, 1994. Weil, A. Introduction a` l’e´tude des varie´te`sKa¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer-site´ de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, 1980. Complex Fraction A FRACTION in which NUMERATOR and DENOMINATOR are themselves fractions. See also COMMON FRACTION ,FRACTION Complex Function A FUNCTION whose RANGE is in the COMPLEX NUMBERS is said to be a complex function, or a complex-valued function. See also REAL FUNCTION ,SCALAR FUNCTION ,VECTOR FUNCTION Complex Infinity An infinite number in the COMPLEX PLANE whose ARGUMENT is unknown. See also C*,DIVISION BY ZERO,EXTENDED COMPLEX PLANE ,INFINITY ,P OINT AT INFINITY ,R IEMANN SPHERE Complex Line Integral LINE INTEGRAL Complex Manifold A complex manifold is a MANIFOLD M whose COORDI- NATE CHARTS are open subsets of Cn and the TRANSI- TION FUNCTIONS between charts are HOLOMORPHIC FUNCTIONS . Naturally, a complex manifold of dimen- sion n also has the structure of a REAL SMOOTH MANIFOLD of dimension 2n :/ A function f : M 0 C is HOLOMORPHIC if it is HOLO- MORPHIC in every COORDINATE CHART . Similarly, a map f : M 0 N is HOLOMORPHIC if its restrictions to coordinate charts on N are holomorphic. Two complex manifolds M and N are considered equivalent if there is a map f : M 0 N which is a DIFFEOMORPHISM and whose inverse is HOLOMORPHIC . See also ALGEBRAIC VARIETY ,CONFORMAL MAPPING , HOLOMORPHIC FUNCTION ,M ANIFOLD ,RIEMANN SUR- FACE ,STEIN MANIFOLD Complex Matrix AMATRIX whose elements may contain COMPLEX NUMBERS . The MATRIX PRODUCT of two 2 /C292 complex matrices is given by x11/C27y11ix12/C27y12i x21/C27y21ix22/C27y22il12ml121 u11/C27v11iu12/C27v12i u21/C27v21iu22/C27v22il12ml121 /C30R11R12 R21R22l12ml121 /C27iI11I12 I21I22l12ml121 ; where R11 /C30u11x11 /C27u21x21 /C28v11y11 /C28v21y12 R12 /C30u12x11 /C27u22x12 /C28v11y11 /C28v22y12 R21 /C30u11x21 /C27u21x22 /C28v11y21 /C28v21y22 R22 /C30u12x21 /C27u22x22 /C28v12y21 /C28v22y22 I11 /C30v11x11 /C27v21x21 /C27u11y11 /C27u21y12 I12 /C30v12x11 /C27v22x12 /C27u12y11 /C27u22y12 I21 /C30v11x21 /C27u21x22 /C27u11y21 /C27u21y22 I22 /C30v12x21 /C27v22x22 /C27u12y21 /C27u22y22 : Hadamard (1893) proved that the DETERMINANT of any complex n /C29n matrix A with entries in the closed UNIT DISK ½aij ½51 satisfies ½det A½5nn=2 (1) (HADAMARD’S MAXIMUM DETERMINANT PROBLEM ), with equality attained by the VANDERMONDE MATRIX of the n ROOTS OF UNITY (Faddeev and Sominskii 1965, p. 331; Brenner 1972). The first few values for n /C301, 2, ... are 1, 2, 3ffiffiffi 3p ; 16, 25ffiffiffi5p ; 216, .... Studying the maximum possible eigenvalue norms for random complex n /C29n matrices is computationally intractable. Although average properties of the dis- tribution of ½l ½ can be determined, finding the max- imum value corresponds to determining if the set of matrices contains a SINGULAR MATRIX , which has been proven to be an NP-COMPLETE PROBLEM (Poljak and Rohn 1993, Kaltofen 1999). The above plots show the distributions for 2 /C292; 3 /C293; and 4 /C294 matrix eigenvalue norms for elements uniformly distributed inside the unit disk ½z½51: Similar plots are obtain- ed for elements uniformly distributed inside ½R[z] ½;½I[z]½51: The exact distribution of eigenvalues for complex matrices with both real and imaginary parts distributed as independent standard normal variates is given by Ginibre (1965), Hwang (1986), and Mehta (1991). See also COMPLEX VECTOR ,H ADAMARD’S MAXIMUM DETERMINANT PROBLEM ,INTEGER MATRIX , K-MATRIX , MATRIX ,REAL MATRIXReferences Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626 /C1/30, 1972. Edelman, A. "The Probability that a Random Real Gaussian Matrix has k Real Eigenvalues, Related Distributions, and the Circular Law." J. Multivariate Anal. 60, 203 /C1/32, 1997. Faddeev, D. K. and Sominskii, I. S. Problems in Higher Algebra. San Francisco: W. H. Freeman, 1965. Ginibre, J. "Statistical Ensembles of Complex, Quaternion, and Real Matrices." J. Math. Phys. 6, 440 /C1/49, 1965. Hadamard, J. "Re´solution d’une question relative aux de´terminants." Bull. Sci. Math. 17,30/C1/1, 1893. Hwang, C. R. "A Brief Survey on the Spectral Radius and the Spectral Distribution of Large Random Matrices with i.i.d. Entries." In Random Matrices and Their Applica- tions . Providence, RI: Amer. Math. Soc., pp. 145 /C1/52, 1986. Kaltofen, E. "Challenges of Symbolic Computation: My Favorite Open Problems." Submitted to J. Symb. Comput. Mehta, M. L. Random Matrices, 2nd rev. enl. ed. New York: Academic Press, 1991. Poljak, S. and Rohn, J. "Checking Robust Nonsingularity is NP-Hard." Math. Control Signals Systems 6,1/C1/, 1993. Complex Measure A MEASURE which takes values in the COMPLEX NUMBERS . The set of complex measures on a MEASURE SPACE X forms a VECTOR SPACE . Note that this is not the case for the more common POSITIVE MEASURES . Also, the space of finite measures (/½ m(X) ½B/C12) has a norm given by the TOTAL VARIATION MEASURE ½½m ½½/C30 ½ m½(X) ½; which makes it a BANACH SPACE . Using the POLAR REPRESENTATION of m; it is possible to define the LEBESGUE INTEGRAL using a complex measure, g fdm /C30g eiufd½ m½: Sometimes, the term "complex measure" is used to indicate an arbitrary measure. The definitions for measure can be extended to measures which take values in any VECTOR SPACE . For instance in SPEC- TRAL THEORY , measures on C ; which take values in the bounded linear maps from a HILBERT SPACE to itself, represent the SPECTRUM of an operator. See also BANACH SPACE ,LEBESGUE INTEGRAL ,M EA- SURE ,M EASURE SPACE ,P OLAR REPRESENTATION (MEASURE ), SPECTRAL THEORY References Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 116 /C1/32, 1987. Complex Modulus MODULUS (COMPLEX NUMBER ) Complex Multiplication Two COMPLEX NUMBERS x/C30a/C27iband y¼c/C27idare multiplied as follows: xy /C30(a /C27ib)(c /C27id) /C30ac /C27ibc /C27iad /C28bd /C30(ac /C28bd) /C27i(ad /C27bc) : In component form, (x; y)(x?; y?) /C30(xx?/C28yy?; xy ?/C27yx?) (1) (Krantz 1999, p. 1). The special case of a COMPLEX NUMBER multiplied by a SCALAR a is then given by (x; y)(x?; y?) /C30(a ; 0)(x; y) /C30(ax ; ay) : (2) Surprisingly, complex multiplication can be carried out using only three REAL multiplications, ac, bd, and (a /C27b)(c /C27d)as R[(a /C27ib)(c /C27id)] /C30ac /C28bd J[(a /C27ib)(c /C27id)] /C30(a /C27b)(c /C27d) /C28ac /C28bd: Complex multiplication has a special meaning for ELLIPTIC CURVES . See also COMPLEX ADDITION ,C OMPLEX DIVISION , COMPLEX NUMBE R,E LLIPTIC CURVE ,IMAGINARY PART,MULTIPLICATION ,REAL PART References Cox, D. A. Primes of the Form x2 /C27ny2 : Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 1, 1999. Complex Number The complex numbers are the FIELD C of numbers OF THE FORM x /C27iy ; where x and y are REAL NUMBERS and I is the IMAGINARY UNIT equal to the SQUARE ROOT of /C281,ffiffiffiffiffiffi /C281p : When a single letter z /C30x /C27iy is used to denote a complex number, it is sometimes called an "AFFIX ." In component notation, z /C30x /C27iy can be written (x, y). The FIELD of complex numbers includes the FIELD of REAL NUMBERS as a SUBFIELD . The set of complex numbers is implemented in Mathematica as Complexes . A number x can then be tested to see if it is complex using the command Element[ x, Complexes]. Through the EULER FORMULA , a complex number z /C30x /C27iy (1) may be written in "PHASOR " form z /C30½z ½(cos u /C27i sin u) /C30½z ½ei u : (2) Here, ½z ½ is known as the MODULUS and u is known as the ARGUMENT or PHASE . The ABSOLUTE SQUARE of z is defined by ½z½2 /C30z¯z; with ¯z the COMPLEX CONJUGATE , and the argument may be computed from arg(z) /C30 u /C30tan/C281y x ! : (3)DE MOIVRE’S IDENTITY relates POWERS of complex numbers zn /C30½z½n[cos(n u) /C27i sin(nu)] : (4) COMPLEX DIVISION and COMPLEX MULTIPLICATION can also be defined for complex numbers. Finally, the REAL R(z) and IMAGINARY PARTS I(z) are given by R(z) /C301 2(z /C27 ¯z) (5) J(z) /C30z /C28 ¯z 2i/C30/C281 2i(z /C28 ¯z) /C3012i(¯z /C28z) : (6) The POWERS of complex numbers can be written in closed form as follows: zn/C30xn/C28n 2l11sl11n xn/C282y2/C27n 4l11sl11n xn/C284y4/C28...l12ml121 /C27in 1l11sl11n xn/C281y/C28n 3l11sl11n xn/C283y3/C27...l12ml121 : (7) The first few are explicitly z2/C30(x2/C28y2)/C27i(2xy) (8) z3/C30(x3/C283xy2)/C27i(3x2y/C28y) (9) z4/C30(x4/C286x2y2/C27y4)/C27i(4x3y/C284xy3) (10) z5/C30(x5/C2810x3y2/C275xy4)/C27i(5x4y/C2810x2y3/C27y5)ð11Þ (Abramowitz and Stegun 1972). See also ABSOLUTE SQUARE ,A RGUMENT (COMPLEX NUMBER ), COMPLEX DIVISION ,COMPLEX MULTIPLICA- TION ,COMPLEX PLANE , I,IMAGINARY NUMBER ,M OD- ULUS (COMPLEX NUMBER ), PHASE ,P HASOR ,R EAL NUMBER ,SURREAL NUMBER References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 16 /C1/7, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 353 /C1/57, 1985. Bold, B. "Complex Numbers." Ch. 3 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 19 /C1/7, 1982. Courant, R. and Robbins, H. "Complex Numbers." §2.5 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 88 /C1/03, 1996. Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A; Koecher, M.; Mainzer, M.; and Remmert, R. Numbers. New York: Springer-Verlag, 1990. Krantz, S. G. "Complex Arithmetic." §1.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 1 /C1/, 1999. Morse, P. M. and Feshbach, H. "Complex Numbers and Variables." §4.1 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 349 /C1/56, 1953. Nahin, P. J. An Imaginary Tale: The Story offfiffiffiffiffiffi /C281p :/Prince- ton, NJ: Princeton University Press, 1998. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Complex Arithmetic." §5.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 171 /C1/72, 1992. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 21 /C1/3, 1986. Complex Plane The plane of COMPLEX NUMBERS spanned by the vectors 1 and i, where i is the IMAGINARY NUMBER . Every COMPLEX NUMBER corresponds to a unique POINT in the complex plane. The LINE in the plane with i /C300 is the REAL LINE. The complex plane is sometimes called the ARGAND PLANE or GAUSS PLANE , and a plot of COMPLEX NUMBERS in the plane is sometimes called an ARGAND DIAGRAM . See also AFFINE COMPLEX PLANE ,ARGAND DIAGRAM , ARGAND PLANE ,B ERGMAN SPACE ,C *,C OMPLEX PROJECTIVE PLANE ,E XTENDED COMPLEX PLANE , ISOTROPIC LINE,L EFT HALF-PLANE ,L OWER HALF- DISK,L OWER HALF-PLANE ,R IGHT HALF-PLANE , UPPER HALF-DISK,UPPER HALF-PLANE References Courant, R. and Robbins, H. "The Geometric Interpretation of Complex Numbers." §5.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 92 /C1/7, 1996. Krantz, S. G. "The Topology of the Complex Plane." §1.1.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 3 /C1/, 1999. Complex Projective Plane The set P2 is the set of all EQUIVALENCE CLASSES [a; b; c] of ordered triples (a ; b; c) /C23C3_(0; 0; 0) un- der the equivalence relation (a; b; c) /C2(a?; b?; c?)if (a; b; c) /C30( la?; lb ?; lc ?) for some NONZERO COMPLEX NUMBER l :/ See also COMPLEX PROJECTIVE PLANE Complex Projective SpaceSee also COMPLEX SPACE ,REAL PROJECTIVE SPACE Complex Representation PHASOR Complex Space See also COMPLEX PROJECTIVE SPACE ,REAL SPACE , TWISTOR SPACE Complex Structure The complex structure of a point x /C30x1 ; x2in the PLANE is defined by the linear MAP J : R2 0 R2 J(x1 ; x2) /C30(/C28x2 ; x1) ; and corresponds to a clockwise rotation by p=2 : This map satisfies J2 /C30/C28I (Jx) /C215(Jy) /C30x /C215 y (Jx) /C215 x /C300; where I is the IDENTITY MAP. More generally, if V is a 2-D VECTOR SPACE , a linear map J : V 0 V such that J2 /C30/C28I is called a complex structure on V.If V /C30R2 ; this collapses to the previous definition. See also MODULI SPACE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 4 and 247, 1997. Complex System References Goles, E. and Martı ´nez, S. (Eds.). Cellular Automata and Complex Systems. Amsterdam, Netherlands: Kluwer, 1999. Complex Vector A VECTOR whose elements are COMPLEX NUMBERS . See also COMPLEX NUMBER ,REAL VECTOR ,VECTOR Complex Vector Bundle A complex vector bundle is a VECTOR BUNDLE p:E0 Mwhose FIBER p/C281(x)i sa COMPLEX VECTOR SPACE .I t is not necessarily a COMPLEX MANIFOLD , even if its BASE MANIFOLD Mis a COMPLEX MANIFOLD .I fa complex vector bundle also has the structure of a COMPLEX MANIFOLD , and pisHOLOMORPHIC , then it is called a HOLOMORPHIC VECTOR BUNDLE . See also BUNDLE ,COMPLEX VECTOR SPACE ,H OLO- MORPHIC VECTOR BUNDLE ,MANIFOLD ,VECTOR SPACE Complex Vector Space A complex vector space is a VECTOR SPACE whose FIELD of scalars is the COMPLEX numbers. A linear transformation between complex vector spaces is given by a matrix with complex entries (i.e., a COMPLEX MATRIX ). See also BASIS (VECTOR SPACE ), COMPLEX STRUC- TURE ,L INEAR TRANSFORMATION ,R EAL VECTOR SPACE ,VECTOR SPACE Complexes COMPLEX NUMBER Complexity (Number) The number of 1s needed to represent an INTEGER using only additions, multiplications, and parenth- eses are called the integer’s complexity. For example, 1 /C301 2 /C301 /C271 3 /C301 /C271 /C271 4 /C30(1 /C271)(1 /C271) /C301 /C271 /C271 /C271 5 /C30(1 /C271)(1 /C271) /C271 /C301 /C271 /C271 /C271 /C271 6 /C30(1 /C271)(1 /C271 /C271) 7 /C30(1 /C271)(1 /C271 /C271) /C271 8 /C30(1 /C271)(1 /C271)(1 /C271) 9 /C30(1 /C271 /C271)(1 /C271 /C271) 10 /C30(1 /C271 /C271)(1 /C271 /C271) /C271 /C30(1 /C271)(1 /C271 /C271 /C271 /C271) So, for the first few n, the complexity is 1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, ... (Sloane’s A005245). References Guy, R. K. "Expressing Numbers Using Just Ones." §F26 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 263, 1994. Guy, R. K. "Some Suspiciously Simple Sequences." Amer. Math. Monthly 93, 186 /C1/90, 1986. Guy, R. K. "Monthly Unsolved Problems, 1969 /C1/987." Amer. Math. Monthly 94, 961 /C1/70, 1987. Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/09, 1989. Rawsthorne, D. A. "How Many 1’s are Needed?" Fib. Quart. 27,14/C1/7, 1989. Sloane, N. J. A. Sequences A005245/M0457 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Complexity (Sequence) BLOCK GROWTH Complexity Theory The theory of classifying problems based on how difficult they are to solve. A problem is assigned to the P-PROBLEM (polynomial time) class if the number of steps needed to solve it is bounded by some POWER of the problem’s size. A problem is assigned to the NP- PROBLEM (nondeterministic polynomial time) class if it permits a nondeterministic solution and the num- ber of steps of the solution is bounded by some power of the problem’s size. The class of P-PROBLEMS is a subset of the class of NP-PROBLEMS , but there also exist problems which are not NP. If a solution is known to an NP-PROBLEM , it can be reduced to a single period verification. A problem is NP-COMPLETE if an ALGORITHM for solving it can be translated into one for solving any other NP -PRO- BLEM . Examples of NP -COMPLETE PROBLEMS include the HAMILTONIAN CYCLE and TRAVELING SALESMAN PROBLEMS .LINEAR PROGRAMMING , thought to be an NP-PROBLEM , was shown to actually be a P -PROBLEM by L. Khachian in 1979. It is not known if all apparently NP -PROBLEMS are actually P -PROBLEMS . See also BIT COMPLEXITY ,NP -COMPLETE PROBLEM , NP-PROBLEM ,P-PROBLEM References Bridges, D. S. Computability. New York: Springer-Verlag, 1994. Brookshear, J. G. Theory of Computation: Formal Lan- guages, Automata, and Complexity. Redwood City, CA: Benjamin/Cummings, 1989. Cooper, S. B.; Slaman, T. A.; and Wainer, S. S. (Eds.). Computability, Enumerability, Unsolvability: Directions in Recursion Theory. New York: Cambridge University Press, 1996. Davis, M. Computability and Unsolvability. New York: Dover, 1982. Du, D.-Z. and Ko, K.-I. Theory of Computational Complexity. New York; Wiley, 2000. Garey, M. R. and Johnson, D. S. Computers and Intract- ability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman, 1983. Goetz, P. "Phil Goetz’s Complexity Dictionary." http:// www.cs.buffalo.edu/~goetz/dict.html. Griffor, E. R. (Ed.). Handbook of Computability Theory. Amsterdam, Netherlands: Elsevier, 1999. Hopcroft, J. E. and Ullman, J. D. Introduction to Automated Theory, Languages, and Computation. Reading, MA: Addison-Wesley, 1979. Lewis, H. R. and Papadimitriou, C. H. Elements of the Theory of Computation, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997. Sudkamp, T. A. Language and Machines: An Introduction to the Theory of Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1996. Weisstein, E. W. "Books about Computational Complexity." http://www.treasure-troves.com/books/Computational- Complexity.html. Welsh, D. J. A. Complexity: Knots, Colourings and Count- ing. New York: Cambridge University Press, 1993. Complex-Valued Function COMPLEX FUNCTION Component A GROUP L is a component of H if L is a QUASISIMPLE GROUP which is a SUBNORMAL SUBGROUP of H. See also GROUP ,Q UASISIMPLE GROUP ,SUBGROUP , SUBNORMAL SUBGROUP Component Graph An n-component of a GRAPH G is a maximal n- connected SUBGRAPH . References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Composite Knot A KNOT which is not a PRIME KNOT . Composite knots are special cases of SATELLITE KNOTS . See also KNOT,PRIME KNOT,SATELLITE KNOT Composite Number A composite number n is a POSITIVE INTEGER n /C211 which is not PRIME (i.e., which has FACTORS other than 1 and itself). The first few composite numbers (sometimes called "composites" for short) are 4, 6, 8, 9, 10, 12, 14, 15, 16, ... (Sloane’s A002808), which can be written 22,2 /C215 3; 23,32,2 /C215 5; 22 /C215 3; 2 /C215 7 ; 3 /C215 5; and 24, respectively. The number 1 is a special case which is considered to be neither composite nor PRIME . A composite number C can always be written as a PRODUCT in at least two ways (since 1 /C215 C is always possible). Call these two products C /C30ab /C30cd ; (1) then it is obviously the case that C ½ab (C divides ab). Set c /C30mn; (2) where m is the part of C which divides a, and n is the part of C which divides b. Then there are p and q such that a /C30mp (3) b /C30nq : (4) Solving ab /C30cd for d gives d /C30ab c/C30(mp)(nq) mn/C30pq : (5) It then follows that S /C13a2 /C27b2 /C27c2 /C27d2 /C30m2p2 /C27n2q2 /C27m2n2 /C27p2q2 /C30(m2 /C27q2)(n2 /C27p2) : (6)It therefore follows that a2 /C27b2 /C27c2 /C27d2is never PRIME ! In fact, the more general result that S /C13ak /C27bk /C27ck /C27dk (7) is never PRIME for k an INTEGER ]0 also holds (Honsberger 1991). See also AMENABLE NUMBER ,GRIMM’S CONJECTURE , HIGHLY COMPOSITE NUMBER ,PRIME FACTORIZATION PRIME GAPS,PRIME NUMBER ,W EAKLY PRIME References Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 19 /C1/0, 1991. Sloane, N. J. A. Sequences A002808/M3272 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Composite Runs PRIME GAPS Compositeness Certificate A compositeness certificate is a piece of information which guarantees that a given number p is COMPO- SITE. Possible certificates consist of a FACTOR of a number (which, in general, is much quicker to check by direct division than to determine initially), or of the determination that either ap/C281 f1 (mod p); (i.e., p violates FERMAT’S LITTLE THEOREM ), or a "/C281; 1 and a2 /C131 (mod p) : A quantity a satisfying either property is said to be a WITNESS to p’s compositeness. See also ADLEMAN- POMERANCE- RUMELY PRIMALITY TEST,FERMAT’S LITTLE THEOREM ,M ILLER’S PRIMAL- ITY TEST,PRIMALITY CERTIFICATE ,W ITNESS Compositeness Test A test which always identifies PRIME NUMBER s cor- rectly, but may incorrectly identify a COMPOSITE NUMBER as a PRIME . See also PRIMALITY TEST Composition The combination of two FUNCTIONS to form a single new FUNCTION . The composition of two functions f andgis denoted f(gand is defined by f(g/C30f(g(x)); (1) where fis a function whose domain includes the range of g. The notation f(g(x)/C30f(g(x)); (2) is sometimes used to explicitly indicate the symbol used for the variable. Composition is associative, so that f((g(h) /C30(f(g)(h: (3) If the functions g is continuous at x0and f is continuous at g(x0) ; then f(g is also continuous at x0 :/ A combinatorial composition is defined as an unor- dered arrangement of k nonnegative integers which sum to n (Skiena 1990, p. 60). The compositions of n into k parts is given by Compositions [n, k] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ), and the number Ck(n) of compositions of a number n of length k is given by the formula Ck(n) /C30n /C27k /C281 k /C281l11sl11n /C30(n /C27 k /C28 1)! n!(k /C28 1)!; (4) implemented as NumberOfCompositions [n, k]in the Mathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The following table gives Ck(n) for n /C301, 2, ... and small k. k Sloane /Ck(1); Ck(2) ; ... 2 Sloane’s A0000272, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ... 3 Sloane’sA0002173, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, ... 4 Sloane’sA0002924, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, ... 5 Sloane’sA0003325, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, ... 6 Sloane’sA0003896, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, ... 7 Sloane’sA0005797, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, ... 8 Sloane’sA0005808, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, ... 9 Sloane’sA0005819, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, ... An operation called composition is also defined on BINARY QUADRATIC FORMS . For two numbers repre- sented by two forms, the product can then be represented by the composition. For example, the composition OF THE FORM s2x2 /C2715y2 and 3x2 /C2710y2is given by 6x2 /C275y2 ; and in this case, the product of 17 and 13 would be REPRESENTED AS ((6 /C215 36 /C275 /C215 1 /C30 221)) : There are several algorithms for computing binary quadratic form composition, which is the basis for some factoring methods. See also ADEM RELATIONS ,B HARGAVA’S THEOREM , BINARY OPERATOR ,BINARY QUADRATIC FORM,RAN- DOM COMPOSITION References Apostol, T. M. "Composite Functions and Continuity." §3.7 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 140 /C1/41, 1967. Klingsberg, P. "A Gray Code for Compositions." J. Algo- rithms 3,41/C1/4, 1982. Skiena, S. "Compositions." §2.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 60 /C1/2, 1990. Composition Series Every FINITE GROUP G of order greater than one possesses a finite series of SUBGROUPS , called a composition series, such that I1Hs1 ...1H21H11G ; where Hi/C271is a maximal subgroup of Hiand H1G means that H is a NORMAL SUBGROUP of G.A composition series is therefore a NORMAL SERIES without repetition whose factors are all simple (Scott 1987, p. 36). The QUOTIENT GROUPS G=H1 ; H1 =H2 ; ..., Hs/C281 =Hs ; Hs are called composition quotient groups. See also FINITE GROUP ,INVARIANT SUBGROUP ,JOR- DAN- HO¨ LDER THEOREM ,N ORMAL SERIES ,N ORMAL SUBGROUP ,QUOTIENT GROUP ,SUBGROUP References Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993. Scott, W. R. "Composition Series." §2.5 in Group Theory. New York: Dover, pp. 36 /C1/8, 1987. Composition Theorem Given a QUADRATIC FORM Q(x; y) /C13x2 /C27y2 ; then Q(x; y)Q(x?; y?) /C30Q(xx ?/C28yy?; x?y /C27x?y) ; since (x2 /C27y2)(x?2 /C27y?2) /C30(xx ?/C28yy?)2 /C27(xy?/C27x?y)2 /C30x2x?2/C27y2y?2/C27x?2y2/C27x2y?2: See also GENUS THEOREM ,QUADRATIC FORM Compound Interest Let P be the PRINCIPAL (initial investment), r be the annual compounded rate, i(n) the "nominal rate," n be the number of times INTEREST is compounded per year (i.e., the year is divided into n CONVERSION PERIODS ), and t be the number of years (the "term"). The INTEREST rate per CONVERSION PERIOD is then r /C13i(n) n: (1) If interest is compounded n times at an annual rate of r (where, for example, 10% corresponds to r /C300:10); then the effective rate over 1=n the time (what an investor would earn if he did not redeposit his interest after each compounding) is (1 /C27r)1=n : (2) The total amount of holdings A after a time t when interest is re-invested is then A /C30P 1 /C27i(n) n !nt /C30P(1 /C27r)nt : (3) Note that even if interest is compounded continu- ously, the return is still finite since lim n 0/C121 /C271 n !n /C30e; (4) where E is the base of the NATURAL LOGARITHM . The time required for a given PRINCIPAL to double (assuming n /C301 CONVERSION PERIOD ) is given by solving 2P /C30P(1 /C27r)t ; (5) or t /C30ln 2 ln(1 /C27 r) ; (6) where LN is the NATURAL LOGARITHM . This function can be approximated by the so-called RULE OF 72: t :0:72 r: (7) See also E,INTEREST ,L N,N ATURAL LOGARITHM , PRINCIPAL ,RULE OF 72,SIMPLE INTEREST References Kellison, S. G. The Theory of Interest, 2nd ed. Burr Ridge, IL: Richard D. Irwin, pp. 14 /C1/6, 1991. Milanfar, P. "A Persian Folk Method of Figuring Interest." Math. Mag. 69, 376, 1996. Compound Polyhedron POLYHEDRON COMPOUNDCompressible Surface Let L be a LINK in R3 and let there be a DISK D in the LINK COMPLEMENT R3 /C28L : Then a surface F such that D intersects F exactly in its boundary and its boundary does not bound another disk on F is called a compressible surface (Adams 1994, p. 86). See also KNOT COMPLEMENT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. Compression See also INFORMATION THEORY References Hankerson, D.; Harris, G. A.; and Johnson, P. D. Jr. Intro- duction to Information Theory and Data Compression. Boca Raton, FL: CRC Press, 1998. Computability COMPLEXITY THEORY Computable Function Any computable function can be incorporated into a PROGRAM using while-loops (i.e., "while something is true, do something else"). For-loops (which have a fixed iteration limit) are a special case of while-loops, so computable functions could also be coded using a combination of for- and while-loops. The ACKERMANN FUNCTION is the simplest example of a WELL DEFINED TOTAL FUNCTION which is computable but not PRIMI- TIVE RECURSIVE , providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive (Do¨tzel 1991). See also ACKERMANN FUNCTION ,CHURCH’S THESIS , COMPUTABLE NUMBER ,PRIMITIVE RECURSIVE FUNC- TION ,TURING MACHINE References Do¨tzel, G. "A Function to End All Functions." Algorithm: Recreational Programming 2,1 6/C1/7, 1991. Computable Number A number which can be computed to any number of DIGITS desired by a T URING MACHINE . Surprisingly, most IRRATIONALS are not computable numbers! References Penrose, R. The Emperor’s New Mind: Concerning Compu- ters, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, 1989. Turing, A. M. "On Computable Numbers with an Applica- tion to the Entscheidungsproblem." Proc. London Math. Soc. 42, 230 /C1/65, 1936. Computational Complexity COMPLEXITY THEORY Computational Geometry The study of efficient algorithms for solving geometric problems. Examples of problems treated by computa- tional geometry include determination of the CONVEX HULL and VORONOI DIAGRAM for a set of points, TRIANGULATION of points in a plane or in space, and other related problems. See also CONVEX HULL,DELAUNAY TRIANGULATION , DISCRETE GEOMETRY ,GEOMETRIC PROBABILITY ,HAP- PY END PROBLEM ,INTERSECTION DETECTION ,M IN- KOWSKI SUM,N EAREST NEIGHBOR PROBLEM , POLYHEDRON PACKING ,SPAN (GEOMETRY ), SYLVES- TER’S FOUR- POINT PROBLEM ,TESSELLATION ,TRIAN- GULATION ,VERTEX ENUMERATION ,VORONOI DIAGRAM References de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. Computational Geometry: Algorithms and Appli- cations, 2nd rev. ed. Berlin: Springer-Verlag, 2000. Goodman, J. E. and O’Rourke, J. Handbook of Discrete and Computational Geometry. Boca Raton, FL: CRC Press, 1997. O’Rourke, J. Computational Geometry in C, 2nd ed. Cam- bridge, England: Cambridge University Press, 1998. Preparata, F. R. and Shamos, M. I. Computational Geome- try: An Introduction. New York: Springer-Verlag, 1985. Sack, J.-R. and Urrutia, J. (Eds.) Handbook of Computa- tional Geometry. Amsterdam, Netherlands: North-Hol- land, 2000. Skiena, S. S. "Computational Geometry." §8.6 in The Algo- rithm Design Manual. New York: Springer-Verlag, pp. 345 /C1/96, 1997. Concatenated Number Sequences CONSECUTIVE NUMBER SEQUENCES Concatenation The concatenation of two strings a and b is the string ab formed by joining a and b. Thus the concatenation of the strings "book" and "case" is the string "book- case". The concatenation of two strings a and b is often denoted ab, a½½b; or, in Mathematica , a B/C21 b: Concatenation is an associative operation, so that the concatenation of three or more strings, for example abc, abcd , etc., is WELL DEFINED . The concatenation of two or more numbers is the number formed by concatenating their numerals. For example, the concatenation of 1, 234, and 5678 is 12345678. The value of the result depends on the numeric base, which is typically understood from context.The formula for the concatenation of numbers p and q in base b is p ½½q /C30pbl(q) /C27q; where l(q) /C30 logb q bc /C271 is the LENGTH of q in base b and xbcis the FLOOR FUNCTION . See also CONSECUTIVE NUMBER SEQUENCES ,LENGTH (NUMBER ), SMARANDACHE SEQUENCES Concave A SET in Rd is concave if it does not contain all the LINE SEGMENTS connecting any pair of its points. If the SET does contain all the LINE SEGMENTS ,itis called CONVEX . See also CONNECTED SET,CONVEX FUNCTION ,CON- VEX HULL,CONVEX OPTIMIZATION THEORY ,CONVEX POLYGON ,DELAUNAY TRIANGULATION ,SIMPLY CON- NECTED Concave Function A function f(x) is said to be concave on an interval [a, b] if, for any points x1and x2in [a, b], the function /C28f(x)is CONVEX on that interval (Gradshteyn and Ryzhik 2000). See also CONVEX FUNCTION References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1132, 2000. Concentrated Letmbe a POSITIVE MEASURE on a SIGMA ALGEBRA M, and let lbe an arbitrary (real or complex) MEASURE onM. If there is a SETA/C23Msuch that l(E)/C30l(ASE) for every E/C23M;then lis said to be concentrated on A. This is equivalent to requiring that l(E) /C300 whenever E S A /C30¥:/ See also ABSOLUTELY CONTINUOUS ,M UTUALLY SIN- GULAR References Rudin, W. Functional Analysis, 2nd ed. New York: McGraw- Hill, p. 121, 1991. Concentric Two geometric figures are said to be concentric if their CENTERS coincide. The region between two concentric CIRCLES is called an ANNULUS . See also ANNULUS ,CONCENTRIC CIRCLES ,CONCYCLIC , ECCENTRIC Concentric Circles Concentric circles are circles with a common center. The region between two CONCENTRIC circles of differ- ent RADII is called an ANNULUS . Any two circles can be made concentric by INVERSION by picking the INVER- SION CENTER as one of the LIMITING POINTS . Given two concentric circles with RADII R and 2R; what is the probability that a chord chosen at random from the outer circle will cut across the inner circle? Depending on how the "random" CHORD is chosen, 1/2, 1/3, or 1/4 could all be correct answers. 1. Picking any two points on the outer circle and connecting them gives 1/3. 2. Picking any random point on a diagonal and then picking the CHORD that perpendicularly bisects it gives 1/2. 3. Picking any point on the large circle, drawing a line to the center, and then drawing the perpendi- cularly bisected CHORD gives 1/4. So some care is obviously needed in specifying what is meant by "random" in this problem. Given an arbitrary CHORD BB? to the larger of two concentric CIRCLES centered on O, the distance between inner and outer intersections is equal on both sides (AB /C30A?B?) : To prove this, take the PERPENDICULAR to BB? passing through O and cross- ing at P. By symmetry, it must be true that PA and PA? are equal. Similarly, PB and PB? must be equal. Therefore, PB /C28PA /C30AB equals PB?/C28PA ?/C30A?B ?: In- cidentally, this is also true for HOMEOIDS , but the proof is nontrivial. See also ANNULUS ,LIMITING POINT Conchoid A curve whose name means "shell form." Let C be a curve and O a fixed point. Let P and P? be points on a line from O to C meeting it at Q, where P?Q /C30QP /C30k; with k a given constant. For example, if C is a CIRCLE and O is on C, then the conchoid is a LIMAC ¸ ON, while in the special case that k is the DIAMETER of C, then the conchoid is a CARDIOID . The equation for a parametrically represented curve (f(t) ; g(t)) with O /C30 (x0 ; y0)is x /C30f 9k(f /C28 x0)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (f /C28 x0)2/C27(g/C28y0)2q y/C30g9k(g/C28y0)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(f/C28x 0)2/C27(g/C28y0)2q : See also CONCHO- SPIRAL ,CONCHOID OF DE SLUZE , CONCHOID OF NICOMEDES ,CONICAL SPIRAL ,DU¨ RER’S CONCHOID References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 49 /C1/1, 1972. Lockwood, E. H. "Conchoids." Ch. 14 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 126 /C1/29, 1967. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 38 /C1/9, 1991. Yates, R. C. "Conchoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 31 /C1/3, 1952. Conchoid of de Sluze A curve first constructed by Rene´ de Sluze in 1662. In CARTESIAN COORDINATES , a(x /C28a)(x2 /C27y2) /C30k2x2 ; and in POLAR COORDINATES , r /C30k2 cos u a/C27a sec u: The above curve has k2 =a /C301; a /C30/C280:5:/ Conchoid of Nicomedes A curve studied by the Greek mathematician Nico- medes in about 200 BC , also called the COCHLOID .It is the LOCUS of points a fixed distance away from a line as measured along a line from the FOCUS point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family. This curve was a favorite with 17th century mathematicians and could be used to solve the problems of CUBE DUPLICATION , ANGLE TRISECTION , HEPTAGON construction, and other NEUSIS CONSTRUCTIONS (Johnson 1975). In POLAR COORDINATES , r /C30b /C27a sec u: (1) In CARTESIAN COORDINATES , (x /C28a)2(x2 /C27y2) /C30b2x2 : (2) The conchoid has x /C30a as an asymptote and the AREA between either branch and the ASYMPTOTE is infinite. The AREA of the loop isA /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28a2p /C282ab lnb /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 a2p a ! /C27b2 cos/C281a b ! : (3) See also CONCHOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215, 1987. Johnson, C. "A Construction for a Regular Heptagon." Math. Gaz. 59,17/C1/1, 1975. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 135 /C1/39, 1972. MacTutor History of Mathematics Archive. "Conchoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Con- choid.html. Pappas, T. "Conchoid of Nicomedes." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 94 /C1/ 5, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 154 /C1/55, 1999. Szmulowicz, F. "Conchoid of Nicomedes from Reflections and Refractions in a Cone." Amer. J. Phys. 64, 467 /C1/71, Apr. 1996. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 34, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 38 /C1/9, 1991. Yates, R. C. "Conchoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 31 /C1/3, 1952. Concho-Spiral The SPACE CURVE with PARAMETRIC EQUATIONS r/C30mua u/C30u z/C30muc: See also CONICAL SPIRAL ,SPIRAL Concordant Form A concordant form is an integer TRIPLE (a;b;N) where a2 /C27b2 /C30c2 a2 /C27Nb2 /C30d2 ;l12) with c and d integers. Examples include 146632 /C271113842 /C301123452 146632 /C2747 /C215 1113842 /C307637512l12) 11412 /C27132602 /C30133092 11412 /C2753 /C215 132602 /C30965412l12) 28731612 /C2724010802 /C3037443612 28731612 /C2783 /C215 24010802 /C30220627612 :l12) Dickson (1962) states that C. H. Brooks and S. Wat- son found in The Ladies’ and Gentlemen’s Diary (1857) that x2 /C27y2 and x2 /C27Ny2 can be simultaneously squares for N B100 only for 1, 7, 10, 11, 17, 20, 22, 23, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 59, 60, 61, 68, 71, 72, 74, 76, 77, 79, 82, 85, 86, 90, 92, 93, 94, 97, 99, and 100 (which evidently omits 47, 53, and 83 from above). The list of concordant primes less than 1000 is now complete with the possible exception of the 16 primes 103, 131, 191, 223, 271, 311, 431, 439, 443, 593, 607, 641, 743, 821, 929, and 971 (Brown). See also CONGRUUM References Brown, K. S. "Concordant Forms." http://www.seanet.com/ ~ksbrown/kmath286.htm. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 475, 1952. Concur Two or more lines which intersect in a POINT are said to concur. See also CONCURRENT References Coxeter, H. S. M. and Greitzer, S. L. "Collinearity and Concurrence." Ch. 3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 51 /C1/9, 1967. Concurrency Principle See also CONCURRENT RELATION Concurrent Two or more LINES are said to be concurrent if they intersect in a single point. Two LINES concur if their TRILINEAR COORDINATES satisfy l1m1n1 l2m2n2 l3m3n3l112l112l112l112l112l112l112l112l112l112l112l112/C300: (1) Three LINES concur if their TRILINEAR COORDINATESsatisfy l1 a /C27m1 b /C27n1 g /C300 (2) l2 a /C27m2 b /C27n2 g /C300 (3) l3 a /C27m3 b /C27n3 g /C300; (4) in which case the point is m2n3 /C28n2m3 : n2l3 /C28l2n3 : l2m3 /C28m2l3 : (5) Three lines A1x /C27B1y /C27C1 /C300 (6) A2x /C27B2y /C27C2 /C300 (7) A3x /C27B3y /C27C3 /C300 (8) are concurrent if their COEFFICIENTS satisfy A1B1C1 A2B2C2 A3B3C3l112l112l112l112l112l112l112l112l112l112l112l112/C300: (9) See also C ONCYCLIC ,POINT Concurrent Normals Conjecture It is conjectured that any convex body in Euclidean n- space has an interior lying on normals through 2 n distinct boundary points (Croft et al. 1991). This has been proved for n/C302 and 3 by Heil (1979ab, 1985). It is known that higher dimensions always contain at least a 6-normal point, but the general conjectureremains open. References Coxeter, H. S. M. and Greitzer, S. L. "Collinearity and Concurrence." Ch. 3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 51 /C1/9, 1967. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Concurrent Normals." §A3 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 14 /C1/5, 1991. Heil, E. "Existenz eines 6-Normalenpunktes in einem konvexen Ko ¨rper." Arch. Math. (Basel) 32, 412/C1/16, 1979a. Heil, E. "Correction to ‘Existenz eines 6-Normalenpunktes in einem konvexen Ko ¨rper."’ Arch. Math. (Basel) 33, 496, 1979b. Heil, E. "Concurrent Normals and Critical Points under Weak Smoothness Assumptions." In Discrete Geometry and Convexity (Ed. J. E. Goodman, E. Lutwak, J. Malke- vitch, and R. Pollack). Ann. New York Acad. Sci. 440, pp. 170 /C1/78, 1985. Concurrent Relation LetXandYbe sets, and let R⁄X/C29Ybe a relation onX/C29Y:Then Ris a concurrent relation if and only if for any finite subset FofX, there exists a single element pofYsuch that if a/C23F;then aRp. Examples of concurrent relations include the following: 1. The relation B on either the natural numbers, the integers, the rational numbers, or the real numbers. 2. The relation R between elements of an exten- sion E of a field F; defined by R /C30 (a ; b) /C23E /C29E : b is algebraic over F and f x is in the extension of F by yg: 3. The containment relation ⁄ between open neighborhoods of a given point p of a TOPOLOGICAL SPACE X. See also CONCURRENCY PRINCIPLE References Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985. Robinson, A. "Germs." In Applications of Model Theory to Algebra, Analysis and Probability (International Sympos., Pasadena, Calif., 1967). New York: Holt, Rinehart and Winston, pp. 138 /C1/49, 1969. Insall, M. "Hyperalgebraic Primitive Elements for Rela- tional Algebraic and Topological Algebraic Models." Stu- dia Logica 57, 409 /C1/18, 1996. Concyclic Four or more points P1 ; P2 ; P3 ; P4 ; ... which lie on a CIRCLE C are said to be concyclic. Three points are trivially concyclic since three noncollinear points determine a CIRCLE . The number of the n2 LATTICE POINTS x; y /C23 [1; n] which can be picked with no four concyclic is i(n2 =3 /C28 e) (Guy 1994). A theorem states that if any four consecutive points of a POLYGON are not concyclic, then its AREA can be increased by making them concyclic. This fact arises in some PROOFS that the solution to the ISOPERI- METRIC PROBLEM is the CIRCLE . See also ANTIPARALLEL ,C IRCLE ,C OLLINEAR ,C ON- CENTRIC ,CYCLIC HEXAGON ,CYCLIC PENTAGON ,CYC- LIC QUADRILATERAL ,ECCENTRIC ,N-CLUSTER References Coolidge, J. L. "Concurrent Circles and Concyclic Points." §1.6 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 85 /C1/5, 1971. Guy, R. K. "Lattice Points, No Four on a Circle." §F3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 241, 1994.Condensation A method of computing the DETERMINANT of a SQUARE MATRIX due to Charles Dodgson (1866) (who is more famous under his pseudonym Lewis Carroll). The method is useful for hand calculations because, for an INTEGER MATRIX , all entries in submatrices computed along the way must also be integers. The method is also implemented efficiently in a parallel computa- tion. Condensation is also known as the method of contractants (Macmillan 1955, Lotkin 1959). Given an n /C29n matrix, condensation successively computes an (n /C281) /C29(n /C281) matrix, an (n /C282) /C29(n /C28 2) matrix, etc., until arriving at a 1 /C291 matrix whose only entry ends up being the DETERMINANT of the original matrix. To compute the k /C29k matrix (/n /C281 ] k ]1); take the k2 2 /C292 connected subdeterminants of the (k /C271) /C29(k /C271) matrix and divide them by the k2 central entries of the (k /C272) /C29(k /C272) matrix, with no divisions performed for k /C30n /C281: The k /C29k matrices arrived at in this manner are the matrices of determinants of the k2(n /C28k /C271) /C29(n /C28k /C271) con- nected submatrices of the original matrices. For example, the first condensation of the 3 /C293 matrix abc def ghi2 435 yields the matrix ae /C28bd bf /C28ce dh /C28eg ei /C28fhl12ml121 ; and the second condensation yields [((ae 2i /C28aefh /C28bdei /C27bdfh ) /C28(bdfh /C28befg /C28cdeh /C27ce2g))=e] which is the determinant of the original matrix. Collecting terms gives (1)aei /C27(/C281)afh /C27(/C281)bdi /C27(0)bde /C281fh /C27(1)bfg /C27(1)cdh /C27(/C281)ceg; of which the nonzero terms correspond to the PERMU- TATION MATRICES . In the 4 /C294 case, 24 nonzero terms are obtained together with 18 vanishing ones. These 42 terms correspond to the ALTERNATING SIGN MA- TRICES for which any /C281s in a row or column must have a /C271 "outside" it (i.e., all /C281s are "bordered" by /C271/s). See also ALTERNATING SIGN MATRIX ,DETERMINANT , DETERMINANT EXPANSION BY MINORS References Bareiss, E. H. "Sylvester’s Identity and Multistep Integer- Preserving Gaussian Elimination." Math. Comput. 22, 565/C1/78, 1968. Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/46. Dodgson, C. L. "Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetic Values." Proc. Roy. Soc. Ser. A 15, 150 /C1/55, 1866. Lotkin, M. "Note on the Method of Contractants." Amer. Math. Soc. 55, 476 /C1/79, 1959. Macmillan, R. H. A New Method for the Numerical Evalua- tion of Determinants." J. Roy. Aeronaut. Soc. 59, 772, 1955. Robbins, D. P. and Rumsey, H. Jr. "Determinants and Alternating Sign Matrices." Adv. Math. 62, 169 /C1/84, 1986. Condition A requirement NECESSARY for a given statement or theorem to hold. Also called a CRITERION . See also BOUNDARY CONDITIONS ,CARMICHAEL CON- DITION ,CAUCHY BOUNDARY CONDITIONS ,CONDITION NUMBER ,DIRICHLET BOUNDARY CONDITIONS ,DIVER- SITY CONDITION ,FELLER- LE´ VY CONDITION ,H O¨ LDER CONDITION ,LICHNEROWICZ CONDITIONS ,LINDEBERG CONDITION ,LIPSCHITZ CONDITION ,LYAPUNOV CONDI- TION ,NEUMANN BOUNDARY CONDITIONS ,ROBERTSON CONDITION ,ROBIN BOUNDARY CONDITIONS ,TAYLOR’S CONDITION ,TRIANGLE CONDITION ,WEIERSTRASS- ERD- MAN CORNER CONDITION ,W INKLER CONDITIONS Condition Number The ratio of the largest to smallest SINGULAR VALUE of a MATRIX . A system is said to be SINGULAR if the condition number is INFINITE , and ILL-CONDITIONED if it is too large. The p-norm condition number of a matrix can be computed using MatrixCondition- Number [m, p] in the Mathematica add-on package LinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ) for p /C30 1, 2, or /C12; where omitting the p is equivalent to specifying Infinity . See also ILL-CONDITIONED MATRIX ,SINGULAR MA- TRIX,SINGULAR VALUE DECOMPOSITION Conditional The formal term in PROPOSITIONAL CALCULUS for the CONNECTIVE IMPLIES . See also BICONDITIONAL ,IMPLIES References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 13, 1997. Conditional Convergence If the SERIES X/C12 n/C300un CONVERGES , butX/C12 n/C300½un ½ does not, where ½x½ is the ABSOLUTE VALUE , then the SERIES is said to be conditionally CONVERGENT . The RIEMANN SERIES THEOREM states that, by a suitable rearrangement of terms, a conditionally convergent SERIES may be made to converge to any desired value, or to DIVERGE . See also ABSOLUTE CONVERGENCE ,C ONVERGENCE TESTS ,D IVERGENT SERIES ,RIEMANN SERIES THEO- REM,SERIES References Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 170 /C1/71, 1984. Hardy, G. H. Divergent Series. New York: Oxford University Press, 1949. Conditional Probability The conditional probability of an EVENT A assuming that B has occurred, denoted P(A½B); equals P(A½B) /C30P(A S B) P(B); (1) which can be proven directly using a VENN DIAGRAM . Multiplying through, this becomes P(A½B)P(B) /C30P(A S B) ; (2) which can be generalized to P(A S B S C) /C30P(A)P(B½A)P(C½A S B) : (3) Rearranging (1) gives P(B ½A) /C30P(BSA) P(A): (4) Solving (4) for /P(BSA)/C30P(ASB)/and plugging in to (1) gives P(A½B)/C30P(A)P(B½A) P(B): (5) See also BAYES’ FORMULA ,FERMAT’S PRINCIPLE OF CONJUNCTIVE PROBABILITY ,TOTAL PROBABILITY THE- OREM References Papoulis, A. "Conditional Probabilities and Independent Sets." §2/C1/inProbability, Random Variables, and Stochas- tic Processes, 2nd ed. New York: McGraw-Hill, pp. 33 /C1/5, 1984. Condom Problem GLOVE PROBLEM Condon-Shortley Phase The (/C281)mphase factor in some definitions (e.g., Arfken 1985) of the SPHERICAL HARMONICS and associated LEGENDRE POLYNOMIALS . Using the Con- don-Shortley convention gives Ym l( u; f) /C30(/C281)mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l /C27 1 4p(l /C28 m)! (l /C27 m)!s Pm l(cos u)eimf : The Condon-Shortley phase is not necessary in the definition of the SPHERICAL HARMONICS , but including it simplifies the treatment of angular moment in quantum mechanics. In particular, they are a con- sequence of the ladder operators L/C28andL/C27(Arfken 1985, p. 693). See also LEGENDRE POLYNOMIAL ,SPHERICAL HARMO- NIC References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 682 and 692, 1985. Condon, E. U. and Shortley, G. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 1951. Shore, B. W. and Menzel, D. H. Principles of Atomic Spec- tra. New York: Wiley, p. 158, 1968. Conductor J-CONDUCTOR Cone A cone is a PYRAMID with a circular CROSS SECTION , and a right cone is a cone with its vertex above the center of its base. However, in discussions of CONIC SECTIONS , the word "cone" is taken mean " DOUBLE CONE ," consisting of two cones placed apex to apex. This is a QUADRATIC SURFACE , and each single cone is called a " NAPPE ." The HYPERBOLA can then be defined as the intersection of a PLANE with both NAPPES of the cone. A right cone of height hcan be described by the PARAMETRIC EQUATIONS x/C30h/C28z hrcosu (1) y/C30h/C28z hrsinu (2) z/C30z (3) forz/C23[0;h] and u/C23[0;2p):The VOLUME of a cone is therefore V/C301 3Abh; (4) where Abis the base AREA andhis the height. If the base is circular, then V/C301 3pr2h: (5) This amazing fact was first discovered by Eudoxus, and other proofs were subsequently found by Archi-medes in On the Sphere and Cylinder (ca. 225 BC ) and Euclid in Proposition XII.10 of his E LEMENTS (Dunham 1990). The CENTROID can be obtained by setting R2/C300 in the equation for the centroid of the CONICAL FRUSTUM , ¯z/C30/C142z/C143 V/C30h(R2 1/C272R1R2/C273R22) 4(R2 1/C27R1R2/C27R22); (6) (Eshbach 1975, p. 453; Beyer 1987, p. 133) yielding ¯z/C301 4h: (7) For a right circular cone, the SLANT HEIGHT sis s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C27h2p (8) and the surface AREA (not including the base) is S/C30prs/C30prffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir 2/C27h2p : (9) The LOCUS of the apex of a variable cone containing anELLIPSE fixed in 3-space is a HYPERBOLA through the FOCI of the ELLIPSE . In addition, the LOCUS of the apex of a cone containing that HYPERBOLA is the original ELLIPSE . Furthermore, the ECCENTRICITIES of the ELLIPSE and HYPERBOLA are reciprocals. There are three ways in which a grid can be mapped onto a cone so that it forms a CONE NET (Steinhaus 1983, pp. 225 /C1/27). Using the parameterization x /C30h /C28 u hr cos v (10) y /C30h /C28 u hr sin v (11) z /C30u (12) gives coefficients of the FIRST FUNDAMENTAL FORM E /C301 /C27r2 h2 (13) F /C300 (14) G /C30r2(h /C28 u)2 h2; (15) SECOND FUNDAMENTAL FORM coefficients e /C300 (16) f /C300 (17) g /C30r(h /C28 u)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C27 r2p ; (18) AREA ELEMENT dS /C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C27 r2p h2(h /C28u) ; (19) GAUSSIAN CURVATURE K /C300; (20) and MEAN CURVATURE M /C30h2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C27 r2p (2hr/C282ru): (21) Note that writing z/C30vinstead of z/C30uwould give a HELICOID instead of a CONE . See also BICONE ,CONE NET,CONIC SECTION ,CONICAL FRUSTUM ,C YLINDER ,D OUBLE CONE,G ENERALIZED CONE,H ELICOID ,NAPPE ,PYRAMID ,SPHERE ,SPHER- ICON References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 129 and 133, 1987. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 76 /C1/7, 1990. Eshbach, O. W. Handbook of Engineering Fundamentals. New York: Wiley, 1975.Harris, J. W. and Stocker, H. "Cone." §4.7 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 104 /C1/05, 1998. Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 inGeometry and the Imagination. New York: Chelsea, pp. 7/C1/1, 1999. Kern, W. F. and Bland, J. R. "Cone" and "Right Circular Cone." §24/C1/5i n Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 57 /C1/4, 1948. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Yates, R. C. "Cones." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 34 /C1/5, 1952. Cone (Space) The JOIN of a TOPOLOGICAL SPACE Xand a point P,/ C(X)/C30X+P/. References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 6, 1976. Cone Graph AGRAPH Cn/C27Km;where Cnis a CYCLIC GRAPH andKm is a COMPLETE GRAPH . Cone Net The mapping of a grid of regularly ruled squares onto aCONE with no overlap or misalignment. Cone nets are possible for vertex angles of 90 8, 1808, and 270 8, where the dark edges in the upper diagrams aboveare joined. Beautiful photographs of cone net models(lower diagrams above) are presented in Steinhaus (1983). The transformation from a point ( x, y) in the grid plane to a point /(x?; y?; z ?)/ on the cone is given by x?/C30rn cosu n ! (1) y?/C30rn sinu n ! (2) z ?/C30(1 /C28r)h; (3) where n /C301/4, 1/2, or 3/4 is the fraction of a circle forming the base, and h /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28n2p (4) u /C30tan/C281y x ! (5) r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p : (6) See also CONE,SPHERICON References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 224 /C1/28, 1999. Cone-Plane Intersection CONIC SECTION Cone-Sphere Intersection Let a CONE of opening parameter c and vertex at / (0; 0; 0)/ intersect a SPHERE of RADIUS r centered at / (x0 ; y0 ; z0)/, with the CONE oriented such that its axis does not pass through the center of the SPHERE . Then the equations of the curve of intersection are x2 /C27 y2 c2/C30z2 (1) (x /C28x0)2 /C27(y /C28y0)2 /C27(z /C28z0)2 /C30r2 : (2) Combining (1) and (2) gives (x /C28x0)2 /C27(y /C28y0)2 /C27x2 /C27 y2 c2/C282z0 cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p /C27z2 0 /C30r2(3)x21 /C271 c2 ! /C282x0x /C27y2 1 /C271 c2 ! /C282y0y /C27(x20 /C27y20 /C27z20 /C28r2) /C282z0 cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p /C300 : (4) Therefore, x and y are connected by a complicated QUARTIC EQUATION , and x, y, and z by a QUADRATIC EQUATION . If the CONE -SPHERE intersection is on-axis so that a CONE of opening parameter c and vertex at /(0; 0; z0)/ is oriented with its AXIS along a radial of the SPHERE of radius r centered at /(0; 0; 0)/, then the equations of the curve of intersection are (z /C28z0)2 /C30x2 /C27 y2 c2 (5) x2 /C27y2 /C27z2 /C30r2 : (6) Combining (5) and (6) gives c2(z /C28z0)2 /C27z2 /C30r2 (7) c2(z2 /C282z0z /C27z2 0) /C27z2 /C30r2 (8) z2(c2 /C271) /C282c2z0z /C27(z20c2 /C28r2) /C300: (9) Using the QUADRATIC EQUATION gives z /C302c2z0 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4c4z2 0 /C28 4(c2 /C27 1)(z20c2 /C28 r2)p 2(c2 /C27 1) /C30c2z09ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2(r2/C28z2 0)/C27r2p c2/C271: (10) So the curve of intersection is planar. Plugging (10) into (5) shows that the curve is actually a CIRCLE , with RADIUS given by a/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p : (11) See also CONE,SPHERE References Kenison, E. and Bradley, H. C. Descriptive Geometry. New York: Macmillan, pp. 282 /C1/83, 1935. Confidence Interval The probability that a measurement will fall within a given CLOSED INTERVAL [a, b]. For a CONTINUOUS DISTRIBUTION , CI(a;b)/C13ga bP(x)dx; (1) where P(x) is the PROBABILITY DISTRIBUTION FUNC- TION . Usually, the confidence interval of interest is symmetrically placed around the mean, so CI(x)/C13CI(m/C28x;m/C27x)/C30gm/C27x m/C28xP(x)dx; (2) where mis the MEAN . For a G AUSSIAN DISTRIBUTION , the probability that a measurement falls within /ns/of the mean mis CI(ns)/C131 sffiffiffiffiffiffi 2ppgm/C27ns m/C28nse/C28(x/C28m)2=2s2dx /C302 sffiffiffiffiffiffi 2ppgm/C27ns 0e/C28(x/C28m)2=2s2dx: (3) Now let /u/C13(x/C28m)=ffiffiffi2p s /,s o /du/C30dx=ffiffiffi2p s /. Then CI(ns)/C302 sffiffiffiffiffiffi 2ppffiffiffi 2p sgn=ffiffi 2p 0e/C28u2du/C302ffiffiffippgn=ffiffi 2p 0e/C28u2du /C30erfnffiffiffi 2p ! (4) where erf( x) is the so-called ERFfunction. The variate value producing a confidence interval CI is often denoted /xCI/,s o xCI/C30ffiffiffi 2p erf/C281(CI) : (5) range CI s 0.6826895 2s0.9544997 3s0.9973002 4s0.9999366 5s0.9999994 To find the standard deviation range corresponding to a given confidence interval, solve (4) for n. n/C30ffiffiffi 2p erf/C281(CI) (6) CI range 0.80091.28155 s 0.90091.64485 s 0.95091.95996 s 0.99092.57583 s 0.99592.80703 s 0.99993.29053 sConfiguration The word configuration is sometimes used to describe a finite collection of points /p/C30(p1;...;pn)/,/pi/C23Rd /, where Rdis a E UCLIDEAN SPACE . The term "configuration" also is used to describe afinite incidence structure /(vr;bk)/with the following properties (Gropp 1992). 1. There are vpoints and blines. 2. There are kpoints on each line and rlines through each point. 3. Two different lines intersect each other at most once and two different points are connected by aline at most once. The conditions vr/C30bk v]r(k/C281)/C271 are NECESSARY for the existence of a configuration. Fork/C303, these conditions are also SUFFICIENT , and fork/C304 this is probably also the case (Gropp 1992). The necessary conditions hold, but there is no 22 5. For k/C306 and 7, the above conditions are not SUFFICIENT , as illustrated by the affine projective plane of order 6 (36 7,4 2 6) and the projective plane (437,4 3 7). Configurations are among the oldest combinatorialstructures, having been defined by T. Reye in 1876. Anr- REGULAR GRAPH can be regarded as a configura- tion /(vr;b2)/by associating nodes with the points, and edges with the lines. The following table summarizesthe number of different configurations for some special values (Gropp 1992). configuration distinct (12 2,83)5 (152,1 0 3)1 8 A symmetric configuration /nk/C30(nk;nk)/consists of n lines and npoints arranged such that klines pass through each point and there are kpoints on each line. All symmetric /n3/configurations are known for / n514/. The number of 7 3,8 3,9 3. . . configurations are 1, 1, 3, 10, 31, 229, 2036, 21399, 245342, ..., correcting an error of von Sterneck for 12 3(Sloane’s A001403; Sterneck 1894, 1895; Wells 1991, p. 72; Colbourn and Dinitz 1996; Gropp 1997; Hilbert and Cohn-Vossen 1999). The F ANO PLANE , in which the central point corre- sponds to the POINT AT INFINITY , is the unique 7 3 configuration. There are no 7 3configurations using points all at finite distances (Wells 1986, p. 75). There are no 8 3configurations using points all at finite distances (Wells 1986, p. 75), but a single configuration exists with a POINT AT INFINITY . There are three 9 3configurations, of which P APPUS’S HEXAGON THEOREM (left figure) is one (Wells 1985, p. 75). The other two consist of embedded EQUILAT- ERAL TRIANGLES (Wells 1991, pp. 159 /C1/60). In the second 9 3configuration, the angle ucan be computed using the above figure. For the top triangle,trigonometry gives tan(30 /C14/C28u)/C30x 1 4ffiffiffi 3p: (1) Solving for xand plugging into the trigonometric equation from the bottom triangle gives tanu/C301 4ffiffiffi 3p 1 2/C27x/C30ffiffiffi 3p 2/C27ffiffiffi3p tan(30/C14/C28u): (2) Now using the identity tan(a/C28b)/C30tana/C28tanb 1/C27tanatanb(3) with /a/C30u;b/C3030/C14 /givestan(u/C2830/C14)/C30tanu/C281ffiffiffi3p 1/C271ffiffiffi3ptanu/C30ffiffiffi 3p tanu/C281ffiffiffi 3p /C27tanu:(4) Plugging in gives tanu2/C27ffiffiffi 3pffiffiffi 3p tanu/C281ffiffiffi3p /C27tanu ! /C30ffiffiffi 3p ; (5) which simplifies to tan2u/C30sec2u/C281/C303 5(6) sec2u/C308 5(7) cos2u/C301 2[1/C27cos(2 u)]/C3058 (8) 1 4/C30cos(2 u) (9) u/C301 2cos/C28114l11)l117 :0:659058 rad : (10) Some additional trigonometry then gives the posi- tions of the three innermost EQUILATERAL TRIANGLE vertices, P1/C301 8(5/C28ffiffiffi 5p );1 8(ffiffiffiffiffiffi 15p /C28ffiffiffi3p )l11)l117 (11) P 2/C301 4ffiffiffi 5p ;1 4ffiffiffi 3pl11)l117 (12) P3/C301 8(7/C28ffiffiffi 5p );1 8(3ffiffiffi 3p /C28ffiffiffiffiffiffi15p )l11)l117 : (13) For the third 9 3configuration, solving the five simultaneous equations tan(u/C2830/C14)/C30x h1(14) tan(60/C14/C28u)/C30h2 1 2(15) h1/C27xffiffiffi 3p /C27h2/C301 2ffiffiffi 3p (16) tan(60/C14/C28u)/C30ffiffiffi 3p /C28tanu 1/C27ffiffiffi3p tanu/C30h2/C27xffiffiffi3p l/C271 2(17) tan 60/C14/C30ffiffiffi 3p /C30h2/C27xffiffiffi 3p 1 2/C28l(18) gives u /C301 2cos/C28114l11)l117 (19) x /C3014(7 /C283ffiffiffi 5p ) (20) l /C301 4(ffiffiffi 5p /C281) (21) h1 /C301 4(ffiffiffiffiffiffi 15p /C28ffiffiffi3p ) (22) h 2 /C301 2(ffiffiffiffiffiffi 15p /C282ffiffiffi3p ) : (23) The six points are then given by P 1 /C301 4(3ffiffiffi 5p /C285);1 4(3ffiffiffi 3p /C28ffiffiffiffiffiffi15p )l11)l117 (24) P 2 /C301 2 ;12(ffiffiffiffiffiffi 15p /C282ffiffiffi3p )l11)l117 (25) P 3 /C303 4(3 /C28ffiffiffi 5p );1 4(3ffiffiffi 3p /C28ffiffiffiffiffiffi15p )l11)l117 (26) P 4 /C301 2(ffiffiffi 5p /C281); 0l11)l117 (27) P5 /C301 4(5 /C28ffiffiffi 5p ) ;1 4(ffiffiffiffiffiffi 15p /C28ffiffiffi3p )l11)l117 (28) P 6 /C301 4(3 /C28ffiffiffi 5p ) ;1 4(3ffiffiffi 3p /C28ffiffiffiffiffiffi15p )l11)l117 : (29) The DESARGUES CONFIGURATION , illustrated above, is one of the ten 103 configurations. Page and Dorwart (1984) discuss the 31 113 configurations (Wells 1991, p. 63). The CREMONA- RICHMOND CONFIGURATION , illustrated above, is one of the 245342 153 configurations.See also BAR (EDGE), CREMONA- RICHMOND CONFIG- URATION ,D ESARGUES CONFIGURATION ,D OUBLE SIXES,E QUILATERAL TRIANGLE ,E UCLIDEAN SPACE , FANO PLANE ,FRAMEWORK ,ORCHARD- PLANTING PRO- BLEM ,ORIENTED MATROID ,PAPPUS’S HEXAGON THE- OREM ,PROJECTIVE PLANE ,REGULAR GRAPH ,REYE’S CONFIGURATION ,RIGID GRAPH ,TENSEGRITY ,TESSER- ACT References Bokowski, J. and Sturmfels, B. Computational Synthetic Geometry. Berlin: Springer-Verlag, p. 41, 1988. Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 255, 1996. Gropp, H. "Configurations and the Tutte Conjecture." Ars. Combin. A 29, 171 /C1/77, 1990. Gropp, H. "On the History of Configurations." Conference San Sebastien (Spain). Sept. 1990. Gropp, H. "Enumeration of Regular Graphs 100 Years Ago." Discrete Math. 101,73/C1/5, 1992. Gropp, H. "Non-Symmetric Configurations with Deficiencies 1 and 2." Combinatorics ’90. Recent Trends and Applica- tions. Proceedings of the International Conference Held in Gaeta, May 20 /C1/7, 1990 (Ed. A. Barlotti, A. Bichera, P. V. Ceccherini, and G. Tallini). Amsterdam, Nether- lands: North-Holland, pp. 227 /C1/39, 1992. Gropp, H. "Configurations and Their Realization." Discr. Math. 174, 137 /C1/51, 1997. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Page, W. and Dorwart, H. L. "Numerical Patterns and Geometrical Configurations." Math. Mag. 57,82/C1/2, 1984. Sloane, N. J. A. Sequences A001403 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sterneck, R. D. von. "Die Configuration 113." Monatshefte f. Math. Phys. 5, 325 /C1/31, 1894. Sterneck, R. D. von. "Die Configuration 123." Monatshefte f. Math. Phys. 6, 223 /C1/55, 1895. Sturmfels, B. and White, N. "All 113 and 123 Configurations are Rational." Aeq. Math. 39, 254 /C1/60, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 75, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 63 and 159 /C1/60, 1991. Confluent Hypergeometric Differential Equation The second-order ordinary differential equation xyƒ/C27(c /C28x)y?/C28ay /C300; (1) sometimes also called Kummer’s differential equation (Zwillinger 1997, p. 124). It has a REGULAR SINGULAR POINT at 0 and an irregular singularity at /C12:The solutions y/C30b11F1(a;c;x)/C27b2U(a;c;x) (2) are called CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST and SECOND KINDS , respectively. Note that the CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND is also denoted /M(a;c;x)/or /F(a;c;z)/. See also CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND,C ONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND,GENERAL CONFLU- ENT HYPERGEOMETRIC DIFFERENTIAL EQUATION ,HY- PERGEOMETRIC DIFFERENTIAL EQUATION ,WHITTAKER DIFFERENTIAL EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 504, 1972. Arfken, G. "Confluent Hypergeometric Functions." §13.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 753 /C1/58, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 551 /C1/55, 1953. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 123 /C1/24, 1997. Confluent Hypergeometric Function CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND,CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND,CONFLUENT HYPERGEOMETRIC LIMIT FUNCTION Confluent Hypergeometric Function of the First Kind The confluent hypergeometric function is a degener- ate form the HYPERGEOMETRIC FUNCTION 2F1(a ; b; c; z) which arises as a solution the CON- FLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATION .It is commonly denoted1F1(a; b; z)/, /M(a ; b ; z)/,or / F(a; b; z)/, and is also known as KUMMER’S FUNCTION of the first kind. An alternate form of the solution to the CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATION is known as the WHITTAKER FUNCTION . The confluent hypergeometric function has a HYPER- GEOMETRIC SERIES given by 1F1(a; b; z) /C301 /C27a bz /C27a(a /C27 1) b(b /C27 1)z2 2! /C27... /C30X/C12 k/C300(a)k (b)kzk k! ; (1) where /(a)k/ and /(b)k/ are POCHHAMMER SYMBOLS .Ifa and b are INTEGERS , a B0, and either b /C210or b Ba, then the series yields a POLYNOMIAL with a finite number of terms. If b is an INTEGER 50, then 1F1(a; b; z) is undefined. The confluent hypergeo- metric function is given in terms of the LAGUERRE POLYNOMIAL by Lm n (x) /C30(m /C27 n)! m!n!1 F1(/C28n; m /C271; x) ; (2) (Arfken 1985, p. 755), and also has an integral representation1F1(a; b; z) /C30G(b) G(b /C28 a) G(a) g1 0eztta /C281(1 /C28t)b/C28a /C281 dt (3) (Abramowitz and Stegun 1972, p. 505). BESSEL FUNCTIONS , the ERROR FUNCTION , the incom- plete GAMMA FUNCTION , HERMITE POLYNOMIAL ,LA- GUERRE POLYNOMIAL , as well as other are all special cases of this function (Abramowitz and Stegun 1972, p. 509). Kummer showed that ex 1F1(a;b;/C28x)/C301F1(b/C28a;b;x) (4) (Koepf 1998, p. 42). KUMMER’S SECOND FORMULA gives 1F11 2/C27m;2m/C271;zl11)l117 /C30M0;m(z) /C30zm/C271=21/C27X/C12 p/C301z2p 24pp!(m/C271)(m/C272)/C1/C1/C1(m/C27p)"# ; (5) where1F1(a;b;z) is the CONFLUENT HYPERGEO- METRIC FUNCTION and /m"/C281=2;/C281;/C283=2/, .... See also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATION ,CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND,CONFLUENT HYPERGEOMETRIC LIMIT FUNCTION ,G ENERALIZED HYPERGEOMETRIC FUNCTION ,HEINE HYPERGEOMETRIC SERIES ,HYPER- GEOMETRIC FUNCTION ,H YPERGEOMETRIC SERIES , KUMMER’S FORMULAS ,W EBER- SONINE FORMULA , WHITTAKER FUNCTION References Abad, J. and Sesma, J. "Computation of the Regular Confluent Hypergeometric Function." Mathematica J. 5, 74/C1/6, 1995. Abramowitz, M. and Stegun, C. A. (Eds.). "Confluent Hy- pergeometric Functions." Ch. 13 in Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 503 /C1/15, 1972. Arfken, G. "Confluent Hypergeometric Functions." §13.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 753 /C1/58, 1985. Buchholz, H. The Confluent Hypergeometric Function with Special Emphasis on its Applications. New York: Springer-Verlag, 1969. Iyanaga, S. and Kawada, Y. (Eds.). "Hypergeometric Func- tion of Confluent Type." Appendix A, Table 19.I inEncyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1469, 1980. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities.Braunschweig, Germany: Vieweg, 1998. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 551 /C1 /54 and 604 /C1/05, 1953. Slater, L. J. Confluent Hypergeometric Functions. Cam- bridge, England: Cambridge University Press, 1960. Spanier, J. and Oldham, K. B. "The Kummer Function / M(a;c;x)/." Ch. 47 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 459 /C1/69, 1987. Tricomi, F. G. Fonctions hyperge ´ome´triques confluentes. Paris: Gauthier-Villars, 1960. Confluent Hypergeometric Function of the Second Kind Gives the second linearly independent solution to the CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUA- TION . It is also known as the KUMMER’S FUNCTION of the second kind, the TRICOMI FUNCTION , or the GORDON FUNCTION . It is denoted /U(a ; b; z)/ and has an integral representation U(a; b; z) /C301 G(a) g/C12 0e /C28ztta /C281(1 /C27t)b /C28a/C281 dt (Abramowitz and Stegun 1972, p. 505). The WHIT- TAKER FUNCTIONS give an alternative form of the solution. For small z, the function behaves as /z1 /C28b /. See also BATEMAN FUNCTION ,CONFLUENT HYPERGEO- METRIC FUNCTION OF THE FIRST KIND,CONFLUENT HYPERGEOMETRIC LIMIT FUNCTION ,COULOMB WAVE FUNCTION ,CUNNINGHAM FUNCTION ,GORDON FUNC- TION ,H YPERGEOMETRIC FUNCTION ,P OISSON- CHAR- LIER POLYNOMIAL ,T ORONTO FUNCTION ,W EBER FUNCTIONS ,W HITTAKER FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Confluent Hy- pergeometric Functions." Ch. 13 in Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 503 /C1/15, 1972. Arfken, G. "Confluent Hypergeometric Functions." §13.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 753 /C1/58, 1985. Buchholz, H. The Confluent Hypergeometric Function with Special Emphasis on its Applications. New York: Springer-Verlag, 1969. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 671 /C1/72, 1953. Spanier, J. and Oldham, K. B. "The Tricomi Function / U(a;c;x)/." Ch. 48 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 471 /C1/77, 1987. Confluent Hypergeometric Limit Function 0F1(; a; z) /C13 lim q0/C121 F1q; a;z q ! : (1) It has a series expansion 0F1(;a; z) /C30X/C12 n/C300zn (a)nn! (2) and satisfies zd2y dz2 /C27ady dz /C28y /C300: (3) AB ESSEL FUNCTION OF THE FIRST KIND can beexpressed in terms of this function by Jn(x) /C301 2 xl11)l117n n!0 F1(; n /C271; /C2814 x2) (4) (Petkovsek et al. 1996). See also CONFLUENT HYPERGEOMETRIC FUNCTION , GENERALIZED HYPERGEOMETRIC FUNCTION ,H YPER- GEOMETRIC FUNCTION References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 38, 1996. Confocal Conics Confocal conics are CONIC SECTIONS sharing a com- mon FOCUS . Any two confocal CENTRAL CONICS are orthogonal (Ogilvy 1990, p. 77). See also CONFOCAL ELLIPSES ,CONFOCAL ELLIPSOIDAL COORDINATES ,C ONFOCAL HYPERBOLAS ,C ONFOCAL PARABOLAS ,C ONFOCAL QUADRICS ,C ONIC SECTION , FOCUS References Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 77 /C1/8, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 39 /C1/0, 1991. Confocal Ellipses ELLIPSES sharing common FOCI (left figure). The family of confocal ellipses covers the plane simply, in the sense that there is a unique ellipse passing through each point in the plane (Hilbert and Cohn-Vossen 1999, p. 5). The figure on the right shows confocal ellipses superimposed on CONFOCAL HYPER- BOLAS , which form an orthogonal net of curves (Hilbert and Cohn-Vossen 1999, pp. 5 /C1/). See also CONFOCAL CONICS ,CONFOCAL HYPERBOLAS , CONFOCAL PARABOLAS ,ELLIPSE References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Confocal Ellipsoidal Coordinates The confocal ellipsoidal coordinates, called simply "ellipsoidal coordinates" by Morse and Feshbach (1953) and "elliptic coordinates" by Hilbert andCohn-Vossen (1999, p. 22), are given by the equations x2 a2/C27j/C27y2 b2/C27j/C27z2 c2/C27j/C301 (1) x2 a2/C27h/C27y2 b2/C27h/C27z2 c2/C27h/C301 (2) x2 a2/C27z/C27y2 b2/C27z/C27z2 c2/C27z/C301; (3) where //C28cBjB/C12 /,//C28b2BhB/C28c2/, and //C28a2BzB/C28b2/. These coordinates correspond to three CONFOCAL QUADRICS all sharing the same pair of foci. Surfaces of constant /j/are confocal ELLIPSOIDS , surfaces of constant hare one-sheeted HYPERBOLOIDS , and sur- faces of constant /z/are two-sheeted HYPERBOLOIDS (Hilbert and Cohn-Vossen 1999, pp. 22 /C1/3). For every / (x;y;z)/, there is a unique set of ellipsoidal coordi- nates. However, /(j;h;z)/specifies eight points sym- metrically located in OCTANTS . Solving for x,y, and zgives x2/C30(a2/C27j)(a2/C27h)(a2/C27z) (b2/C28a2)(c2/C28a2)(4) y2/C30(b2/C27j)(b2/C27h)(b2/C27z) (a2/C28b2)(c2/C28b2)(5) z2/C30(c2/C27j)(c2/C27h)(c2/C27z) (a2/C28c2)(b2/C28c2): (6) The L APLACIAN is92C/C30(h/C28z)f(j)@ @jf(j)@C @j"# /C27(z/C28j)f(h)@ @h /C2f(h)@C @h"# /C27(j/C28h)f(z)@ @zf(z)@C @z"# ;(7) where f(x)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C27a2)(x/C27b2)(x/C27c2)p : (8) Another definition is x2 a2/C28l/C27y2 b2/C28l/C27z2 c2/C28l/C301 (9) x2 a2/C28m/C27y2 b2/C28m/C27z2 c2/C28m/C301 (10) x2 a2/C28n/C27y2 b2/C28n/C27z2 c2/C28n/C301; (11) where lBc2BmBb2BnBa2(12) (Arfken 1970, pp. 117 /C1/18). Byerly (1959, p. 251) uses a slightly different definition in which the Greek variables are replaced by their squares, and a/C300. Equation (9) represents an ELLIPSOID , (10) represents a one-sheeted HYPERBOLOID , and (11) represents a two-sheeted HYPERBOLOID . In terms of C ARTESIAN COORDINATES , x2/C30(a2/C28l)(a2/C28m)(a2/C28n) (a2/C28b2)(a2/C28c2)(13) y2/C30(b2/C28l)(b2/C28m)(b2/C28n) (b2/C28a2)(b2/C28c2)(14) z2/C30(c2/C28l)(c2/C28m)(c2/C28n) (c2/C28a2)(c2/C28b2): (15) The SCALE FACTORS are hl/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (m/C28l)(n/C28l) 4(a2/C28l)(b2/C28l)(c2/C28l)s (16) hm/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n/C28m)(l/C28m) 4(a2/C28m)(b2/C28m)(c2/C28m)s (17) hn/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (l/C28n)(m/C28n) 4(a2/C28n)(b2/C28n)(c2/C28n)s : (18) The L APLACIAN is 92 /C302a2b2 /C27 a2c2 /C27 b2c2 /C28 2n(a2 /C27 b2 /C27 c2) /C27 3 n2 ( m /C28 n)(n /C28 l)@ @ n /C274(a2 /C28 n)(b2 /C28 n)(c2 /C28 n) (m /C28 n)(n /C28 l)@2 @ n2 /C272a2b2 /C27 a2c2 /C27 b2c2 /C28 2m(a2 /C27 b2 /C27 c2) /C27 3 m2 ( n /C28 m)( m /C28 l)@ @ m /C274(a2 /C28 m)(b2 /C28 m)(c2 /C28 m) (m /C28 l)( n /C28 m)@2 @ m2 /C272/C28(a2b2 /C27 a2c2 /C27 b2c2) /C27 2 l(a2 /C27 b2 /C27 c2) /C28 3l2 ( m /C28 l)(n /C28 l)@ @ l (19) Using the NOTATION of Byerly (1959, pp. 252 /C1/53), this can be reduced to 92 /C30( m2 /C28 n2)@2 @ a2 /C27(l2 /C28 n2)@2 @ b2 /C27( l2 /C28 m2)@2 @ g2 ; (20) where a /C30cg l cdlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( l2 /C28 b2)( l2 /C28 c2)p /C30Fb c;p 2 ! /C28Fb c; sin/C281c l ! ! (21) b /C30cg m bdmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (c2 /C28 m2)( m2 /C28 b2)p /C30Fffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28b2 c2s ; sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28b2 m2 1 /C28b2 c2vuuuuuut0 BBBB@1 CCCCA2 666643 77775(22) g /C30c g n 0dnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b2 /C28 n2)(c2 /C28 n2)p /C30Fb c; sin/C281n b ! ! : (23) Here, F is an ELLIPTIC INTEGRAL OF THE FIRST KIND . In terms of a; b; and g; l /C30c dc a;b c ! (24) m /C30b nd b;ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28b2 c2s ! (25) n /C30b sn g ;b c ! ; (26) where dc, nd, and sn are JACOBI ELLIPTIC FUNCTIONS . The HELMHOLTZ DIFFERENTIAL EQUATION is separable in confocal ellipsoidal coordinates. See also HELMHOLTZ DIFFERENTIAL EQUATION– CON-FOCAL ELLIPSOIDAL COORDINATES References Abramowitz, M. and Stegun, C. A. (Eds.). "Definition of Elliptical Coordinates." §21.1 in Handbook of Mathema- tical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972. Arfken, G. "Confocal Ellipsoidal Coordinates /( j1 ; j2 ; j3)/." §2.15 in Mathematical Methods for Physicists, 2nd ed. New York: Academic Press, pp. 117 /C1/18, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 251 /C1/52, 1959. Hilbert, D. and Cohn-Vossen, S. "The Thread Construction of the Ellipsoid, and Confocal Quadrics." §4in Geometry and the Imagination. New York: Chelsea, pp. 19 /C1/5, 1999. Moon, P. and Spencer, D. E. "Ellipsoidal Coordinates / ( h; u ; l)/." Table 1.10 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 40 /C1/4, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953. Confocal Hyperbolas HYPERBOLAS sharing common FOCI (left figure). The family of confocal hyperbolas covers the plane simply, in the sense that there is a unique hyperbola passing through each point in the plane (Hilbert and Cohn- Vossen 1999, p. 5). The figure on the right shows confocal hyperbolas superimposed on CONFOCAL EL- LIPSES , which form an orthogonal net of curves (Hilbert and Cohn-Vossen 1999, pp. 5 /C1/). See also CONFOCAL CONICS ,C ONFOCAL ELLIPSES , CONFOCAL PARABOLAS ,ELLIPSE References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 5, 1999. Confocal Parabolas PARABOLAS sharing a common FOCUS . See also CONFOCAL CONICS ,C ONFOCAL ELLIPSES , CONFOCAL HYPERBOLAS ,PARABOLA References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 5, 1999. Confocal Parabolic Coordinates CONFOCAL PARABOLOIDAL COORDINATES Confocal Paraboloidal Coordinates x2 a2 /C28 l /C27y2 b2 /C28 l /C30z /C28 l (1) x2 a2 /C28 m /C27y2 b2 /C28 m /C30z /C28 m (2) x2 a2 /C28 n /C27y2 b2 /C28 n /C30z /C28 n ; (3) where /l /C23 (/C28/C12; b2)/,/m /C23 (b2 ; a2)/, and / n /C23 (a2 ;/C12)/. x2 /C30(a2 /C28 l)(a2 /C28 m)(a2 /C28 n) (b2 /C28 a2) (4) y2 /C30(b2 /C28 l)(b2 /C28 m)(b2 /C28 n) (a2 /C28 b2) (5) z /C30 l /C27 m /C27 n /C28a2 /C28b2 : (6) The SCALE FACTORS are hl /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( m /C28 l)( n /C28 l) 4(a2 /C28 l)(b2 /C28 l)s (7) hm /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( n /C28 m)( l /C28 m) 4(a2 /C28 m)(b2 /C28 m)s (8) hn /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (l /C28 n)( m /C28 n) 16(a2 /C28 n)(b2 /C28 n)s : (9)The LAPLACIAN is 92 /C302(a2 /C27 b2 /C28 2 n) (m /C28 n)( n /C28 l)@ @ n /C274(a2 /C28 n)( n /C28 b2) ( m /C28 n)( n /C28 l)@2 @ n2 /C272(a2 /C27 b2 /C28 2m) ( m /C28 l)(n /C28 m)@ @ m /C274(a2 /C28 m)( m /C28 b2) (m /C28 l)( n /C28 m)@2 @ m2 /C272(2l /C28 a2 /C28 b2) ( m /C28 l)(n /C28 l)@ @ l /C274(l /C28 a2)( l /C28 b2) ( m /C28 l)(n /C28 l)@2 @ l2 : (10) The HELMHOLTZ DIFFERENTIAL EQUATION is SEPAR- ABLE . See also HELMHOLTZ DIFFERENTIAL EQUATION– CON- FOCAL PARABOLOIDAL COORDINATES References Arfken, G. "Confocal Parabolic Coordinates (/j1 ; j2 ; j3):/" §2.17 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 119 /C1/20, 1970. Moon, P. and Spencer, D. E. "Paraboloidal Coordinates / ( m; n ; l)/." Table 1.11 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 44 /C1/8, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 664, 1953. Confocal Quadrics A set of QUADRATIC SURFACES which share FOCI. Ellipsoids and one- and two-sheeted hyperboloids can be confocal. These three types of surfaces can be combined to form an orthogonal coordinate system known as CONFOCAL ELLIPSOIDAL COORDINATES (Hil- bert and Cohn-Vossen 1991, pp. 22 /C1/3). The planes of symmetry of the tangent cone from any point P in space to any surface of the confocal system which does not enclose P are the tangent planes at P to the three surfaces of the system that pass through P. As a limiting case, this result means that every surface of the confocal system when viewed from a point lying on a focal curve and not enclosed by the surface looks like a circle with its center on the line of sight, provided that the line of sight is tangent to the focal curve (Hilbert and Cohn-Vossen 1999, p. 24). See also CONFOCAL ELLIPSOIDAL COORDINATES ,EL- LIPSOID ,HYPERBOLOID ,QUADRATIC SURFACE References Hilbert, D. and Cohn-Vossen, S. "The Thread Construction of the Ellipsoid, and Confocal Quadrics." §4in Geometry and the Imagination. New York: Chelsea, pp. 19 /C1/5, 1999. Confoliation A topological structure which interpolates between contact structures and codimension-one FOLIATIONS . See also FOLIATION References Eliashberg, Y. M. and Thurston, W. P. Confolations. Provi- dence, RI: Amer. Math. Soc., 1998. Conformal Latitude An AUXILIARY LATITUDE defined by x /C132 tan /C281tan(1 4 p /C2712 f)1 /C28 e sin f 1 /C27 e sin f"#e=28 < :9 = ;/C281 2 p /C302 tan /C2811 /C27 sin f 1 /C28 sin f1 /C28 e sin f 1 /C27 e sin f !e () 1=2 /C281 2 p /C30 f /C28(1 2 e2 /C275 24 e4 /C273 32 e6 /C27281 5760 e8 /C27...) sin(2f) /C27(5 48 e4 /C277 80 e6 /C27697 11520 e8 /C27...) sin(4f) /C28(13 480 e6 /C27461 13440 /C27...) sin(6f) /C27(1237 161280 e8 /C27...) sin(8f) /C27... The inverse is obtained by iterating the equation f /C302 tan/C281tan(1 4 p /C2712 x)1 /C27 e sin f 1 /C28 e sin f !e=22 435/C28 1 2 p using f /C30 x as the first trial. A series form is f /C30 x /C27(1 2 e2 /C275 24 e4 /C271 12 e6 /C2713 360 e8 /C27...) sin(2x) /C27(7 48 e4 /C2729 240 e6 /C27811 11520 e8 /C27...) sin(4x) /C27(7 120 e6 /C2781 1120 e8 /C27...) sin(6x) /C27(4279 161280e8 /C27...) sin(8x) /C27... The conformal latitude was called the ISOMETRIC LATITUDE by Adams (1921), but this term is now used to refer to a different quantity. See also AUXILIARY LATITUDE ,LATITUDE References Adams, O. S. "Latitude Developments Connected with Geo- desy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridianal Projections." Spec. Pub.No. 67. U. S. Coast and Geodetic Survey, pp. 18 and 84 /C1 /5, 1921. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington,DC: U. S. Government Printing Office, pp. 15 /C1 /6, 1987.Conformal Map CONFORMAL MAPPING Conformal Mapping A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transfor- mation, or biholomorphic map, is a TRANSFORMATION w/C30f(z) that preserves local ANGLES .A n ANALYTIC FUNCTION is conformal at any point where it has a NONZERO DERIVATIVE . Conversely, any conformal mapping of a complex variable which has continuous partial derivatives is analytic. Conformal mapping isextremely important in COMPLEX ANALYSIS , as well as in many areas of physics and engineering. Several conformal transformations of regular gridsare illustrated in the first figure above, and areimplemented as ComplexMap in the Mathematica add-on package Graphics‘ComplexMap‘ (which can be loaded with the command BBGraphics‘ ). In the second figure above, contours of constant ½z½are shown together with their corresponding contours after the transformation. Moon and Spencer (1988) and Krantz (1999, pp. 183 /C1 /94) give tables of confor- mal mappings. Letuandfbe the tangents to the curves gandf(g)a t z0andw0in the COMPLEX PLANE , w/C28w0/C13f(z)/C28f(z0)/C30f(z)/C28f(z0) z/C28z0(z/C28z0) (1) arg(w/C28w0)/C30argf(z)/C28f(z0) z/C28z0"# /C27arg(z/C28z0):(2) Then as w0w0andz0z0; f/C30argf?(z0)/C27u (3) ½w½/C30½f?(z0)½½z½: (4) A function f:C0Cis conformal IFF there are complex numbers a"0 and bsuch that f(z)/C30az/C27b (5) forz/C23C(Krantz 1999, p. 80). Furthermore, if h:C0 Cis an analytic function such that lim ½z½0/C27/C12½h(z)½/C30/C27/C12; (6) then his a polynomial in z(Greene and Krantz 1997; Krantz 1999, p. 80). Conformal transformations can prove extremely use- ful in solving physical problems. By letting w/C13f(z); the REAL and IMAGINARY PARTS ofw(z) must satisfy the C AUCHY- RIEMANN EQUATIONS and L APLACE’S EQUATION , so they automatically provide a scalar POTENTIAL and a so-called stream function. If a physical problem can be found for which the solutionis valid, we obtain a solution–which may have been very difficult to obtain directly–by working back- wards. For example, let w(z)/C30Az n/C30Arneinu; (7) the REAL and IMAGINARY PARTS then give f/C30Arncos(nu) (8) c/C30Arnsin(nu): (9) Forn/C30/C28 2, f/C30A r2cos(2 u) (10) c/C30/C28A r2sin(2u); (11) which is a double system of LEMNISCATES (Lamb 1945, p. 69). Forn/C30/C28 1,f/C30A rcosu (12) c/C30A rsinu: (13) This solution consists of two systems of CIRCLES , and fis the POTENTIAL FUNCTION for two PARALLEL opposite charged line charges (Feynman et al. 1989, §7/C1/; Lamb 1945, p. 69). Forn/C301=2; f/C30Ar1=2cosu 2 ! /C30Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p /C27x 2s (14) c/C30Ar1=2sinu 2 ! /C30Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p /C28x 2s : (15) /fgives the field near the edge of a thin plate (Feynman et al. 1989, §7/C1/). Forn/C301, f/C30Arcosu/C30Ax (16) c/C30Arsinu/C30Ay; (17) giving two straight lines (Lamb 1945, p. 68). Forn/C303=2; w/C30Ar3=2e3iu=2: (18) /fgives the field near the outside of a rectangular corner (Feynman et al. 1989, §7/C1/). For n /C302, w /C30A(x /C27iy)2 /C30A[(x2 /C28y2) /C272ixy] (19) f /C30A(x2 /C28y2) /C30Ar2 cos(2 u) (20) c /C302Axy /C30Ar2 sin(2u) : (21) These are two PERPENDICULAR HYPERBOLAS , and f is the POTENTIAL FUNCTION near the middle of two point charges or the field on the opening side of a charged RIGHT ANGLE conductor (Feynman 1989, §7 /C1/). See also ANALYTIC FUNCTION ,C AUCHY- RIEMANN EQUATIONS ,CAYLEY TRANSFORM ,CONFORMAL PRO- JECTION ,HARMONIC FUNCTION ,LAPLACE’S EQUATION , MO¨ BIUS TRANSFORMATION ,Q UASICONFORMAL MAP, SCHWARZ- CHRISTOFFEL MAPPING ,SIMILAR References Arfken, G. "Conformal Mapping." §6.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 392 /C1/94, 1985. Bergman, S. The Kernel Function and Conformal Mapping. New York: Amer. Math. Soc., 1950. Carathe ´odory, C. Conformal Representation. New York: Dover, 1998. Carrier, G.; Crook, M.; and Pearson, C. E. Functions of a Complex Variable: Theory and Technique. New York: McGraw-Hill, 1966. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 80, 1967. Feynman, R. P.; Leighton, R. B.; and Sands, M. The Feyn- man Lectures on Physics, Vol. 1. Redwood City, CA: Addison-Wesley, 1989. Greene, R. E. and Krantz, S. G. Function Theory of One Complex Variable. New York: Wiley, 1997. Katznelson, Y. An Introduction to Harmonic Analysis. New York: Dover, 1976. Kober, H. Dictionary of Conformal Representations. New York: Dover, 1957. Krantz, S. G. "Conformality," "The Geometric Theory of Holomorphic Functions," "Applications That Depend on Conformal Mapping," and "A Pictorial Catalog of Con- formal Maps." §2.2.5, Ch. 6, Ch. 14, and Appendix to Ch. 14 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 25, 79 /C1/8, and 163 /C1/94, 1999. Kythe, P. K. Computational Conformal Mapping. Boston, MA: Birkha ¨user, 1998. Lamb, H. Hydrodynamics, 6th ed. New York: Dover, 1945. Mathews, J. "Conformal Mappings." http://www.ecs.fullerto- n.edu/~mathews/fofz/cmaps.html. Moon, P. and Spencer, D. E. "Conformal Transformations." §2.01 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 49 /C1/6, 1988.Morse, P. M. and Feshbach, H. "Conformal Mapping." §4.7 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 358 /C1/62 and 443 /C1/53, 1953. Nehari, Z. Conformal Map. New York: Dover, 1982. Conformal Projection A MAP PROJECTION which is a CONFORMAL MAPPING , i.e., one for which local (infinitesimal) angles on a sphere are mapped to the same angles in the projec- tion. On maps of an entire sphere, however, there are usually singular points at which local angles are distorted. The term conformal was applied to map projections by Gauss in 1825, and eventually supplanted the alter- native terms "orthomorphic" (Germain 1865, Lee 1944; Snyder 1987, p. 4) and "autogonal" (Tissot1881, Lee 1944). No projection can be both EQUAL-AREA and conform, and projections which are neither EQUAL-AREA nor conformal are sometimes called APHYLACTIC (Lee 1944; Snyder 1987, p. 4). See also CONFORMAL MAPPING ,EQUIDISTANT PROJEC- TION ,LAMBERT CONFORMAL CONIC PROJECTION ,MAP PROJECTION References Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190/C1/00, 1944. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987. Thomas, P. S. Conformal Projections in Geodesy and Carto- graphy. Washington, DC: U. S. Coast and Geodetic Survey Spec. Pub. 251, 1952. Tissot, A. Me´moir sur la repre ´sentation des surfaces et les projections des cartes ge ´ographiques. Paris: Gauthier- Villars, 1881. Conformal Tensor WEYLTENSOR Conformal Transformation CONFORMAL MAPPING Congruence If two numbers bandchave the property that their difference b/C28cis integrally divisible by a number m (i.e., b/C28c=mis an integer), then bandcare said to be "congruent modulo m." The number mis called the MODULUS , and the statement " bis congruent to c (modulo m)" is written mathematically as b/C13c(mod m): (1) Ifb/C28cisnotintegrally divisible by m, then we say " b isnotcongruent to c(modulo m)," which is written bfc(mod m): (2) The explicit "(mod m)" is sometimes omitted when the MODULUS mis understood by context, so in such cases, care must be taken not to confuse the symbol /C13 with the EQUIVALENCE sign. The quantity bis sometimes called the "base," and the quantity cis called the RESIDUE orREMAINDER . There are several types of residues. The COMMON RESIDUE defined to be NONNEGATIVE and smaller than m, while the MINIMAL RESIDUE iscorc/C28m;which- ever is smaller in ABSOLUTE VALUE . In many compu- ter languages (such as FORTRAN orMathematica ), the COMMON RESIDUE ofb(mod m) is written mod( b,m) (FORTRAN )o rMod[ b,m](Mathematica ). Congruence arithmetic is perhaps most familiar as a generalization of the arithmetic of the clock. Sincethere are 60 minutes in an hour, "minute arithmetic"uses a modulus of m/C3060. If one starts at 40 minutes past the hour and then waits another 35 minutes,40/C2735/C1315 (mod 60) ;so the current time would be 15 minutes past the (next) hour. Similarly, "hour arithmetic" on a 12-hour clock uses amodulus of m/C3012, so 10 o’clock (a.m.) plus five hours gives 10 /C275/C133 (mod 12) ;or 3 o’clock (p.m.) Congruences satisfy a number of important proper-ties, and are extremely useful in many areas of NUMBER THEORY . Using congruences, simple DIVISI- BILITY TESTS to check whether a given number is divisible by another number can sometimes bederived. For example, if the sum of a number’s digitsis divisible by 3 (9), then the original number isdivisible by 3 (9). Congruences also have their limitations. For exam- ple, if a/C13band c/C13d(mod n);then it follows that a x/C13bx;but usually not that xc/C13xdorac/C13bd:Inaddition, by "rolling over," congruences discard abso- lute information. For example, knowing the number of minutes past the hour is useful, but knowing the hour the minutes are past is often more useful still. Leta/C13a?(mod m) and b/C13b?(mod m);then impor- tant properties of congruences include the following, where [means " IMPLIES ": 1. Equivalence: a/C13b(mod 0) [a/C30b(which can be regarded as a definition). 2. Determination: either a/C13b(mod m)o r afb(mod m):/ 3. Reflexivity: a/C13a(mod m):/ 4. Symmetry: a/C13b(mod m)[b/C13a(mod m):/ 5. Transitivity: a/C13b(mod m) and b/C13c(mod m)/ /[a/C13c(mod m):/ 6.a/C27b/C13a?/C27b?(mod m):/ 7.a/C28b/C13a?/C28b?(mod m):/ 8.ab/C13a?b?(mod m):/ 9.a/C13b(mod m)[ka/C13kb(mod m):/ 10.a/C13b(mod m)[an/C13bn(mod m):/ 11. /a/C13b(mod m1)/and /a/C13b(mod m2)[a/C13 b(mod[ m1;m2]);where [ m1;m2] is the LEAST COMMON MULTIPLE . 12.ak/C13bk(mod m)[a/C13bmodm (k;m)l11)l117 ;where ( k, m) is the GREATEST COMMON DIVISOR . 13. If a/C13b(mod m);then P(a)/C13P(b) (mod m);for P(x)aPOLYNOMIAL . Properties (6 /C1/) can be proved simply by defining a/C13a?/C27rd (3) b/C13b?/C27sd; (4) where randsare INTEGERS . Then a/C27b/C30a?/C27b?/C27(r/C27s)d (5) a/C28b/C30a?/C28b?/C27(r/C28s)d (6) ab/C30a?b?/C27(a?s/C27b?r/C27rsd)d; (7) so the properties are true. Congruences also apply to FRACTIONS . For example, note that 2/C294/C1313/C293/C1326/C296/C131 (mod 7) ; (8) so 1 2/C13414/C13223/C13316/C136 (mod 7) : (9) To find p=q(mod m), use an ALGORITHM similar to the GREEDY ALGORITHM . Let q0/C13qand find p0/C30m q0&’ ; (10) where //C26x/C27/is the CEILING FUNCTION , then compute q1 /C13q0p0(mod m) : (11) Iterate until qn /C301; then p q /C13pYn/C281 i/C300pi(mod m): (12) This method always works for m PRIME , and some- times even for m COMPOSITE . However, for a COMPO- SITE m, the method can fail by reaching 0 (Conway and Guy 1996). Finding a fractional congruence is equivalent to solving a corresponding LINEAR CON- GRUENCE EQUATION ax /C13b (mod m): (13) See also ALGEBRAIC CONGRUENCE ,C ANCELLATION LAW,CHINESE REMAINDER THEOREM ,COMMON RE- SIDUE ,C ONGRUENCE AXIOMS ,C ONGRUENCE EQUA- TION ,DIVISIBILITY TESTS,FUNCTIONAL CONGRUENCE , GREATEST COMMON DIVISOR ,LEAST COMMON MULTI- PLE,LINEAR CONGRUENCE EQUATION ,M INIMAL RE- SIDUE ,M ODULUS (CONGRUENCE ), QUADRATIC CONGRUENCE EQUATION ,Q UADRATIC RECIPROCITY LAW,RESIDUE (CONGRUENCE ), RSA ENCRYPTION References Burton, D. M. "The Theory of Congruences." Ch. 4 in Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, pp. 80 /C1/05, 1989. Conway, J. H. and Guy, R. K. "Arithmetic Modulo p." In The Book of Numbers. New York: Springer-Verlag, pp. 130 /C1/ 32, 1996. Courant, R. and Robbins, H. "Congruences." §2 in Supple- ment to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 31 /C1/0, 1996. Hardy, G. H. and Wright, E. M. "Congruences and Classes of Residues," "Elementary Properties of Congruences," "Linear Congruences," "General Properties of Con- gruences," and "Congruences to Composite Moduli." §5.2 /C1/.4 and Chs. 7 /C1/ in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 49 /C1/2 and 82 /C1/06, 1979. Hilton, P.; Holton, D.; and Pedersen, J. "A Far Nicer Arithmetic." Ch. 2 in Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 25 /C1/0, 1997. Nagell, T. "Theory of Congruences." Ch. 3 in Introduction to Number Theory. New York: Wiley, pp. 68 /C1/31, 1951. Se´roul, R. "Congruences." §2.5 in Programming for Mathe- maticians. Berlin: Springer-Verlag, pp. 11 /C1/2, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 55, 1993. Weisstein, E. W. "Fractional Congruences." MATHEMATICA NOTEBOOK MODFRACTION.M . Congruence Arithmetic CONGRUENCECongruence Axioms The five of HILBERT’S AXIOMS which concern geo- metric equivalence. See also CONGRUENCE AXIOMS ,CONTINUITY AXIOMS , HILBERT’S AXIOMS ,INCIDENCE AXIOMS ,O RDERING AXIOMS ,PARALLEL POSTULATE References Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: Open Court, 1980. Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert’s System of Axioms." §163B in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 544 /C1/45, 1980. Congruence Equation An equation OF THE FORM f(x) /C13b (mod m) ; (1) where the values of 0 5x Bm for which the equation holds are sought. Such an equation may have none, one, or many solutions. There is a general method for solving both the general LINEAR CONGRUENCE EQUA- TION ax /C13b (mod m) (2) and the general QUADRATIC CONGRUENCE EQUATION a2x2 /C27a1x /C27a0 /C130 (mod n) : (3) However, solution of the general polynomial congru- ence amxm /C27... /C27a2x2 /C27a1x /C27a0 /C130 (mod n) (4) is intractable. Note that any polynomial congruence will give congruent results when congruent values are substituted. Two or more simultaneous congruences x /C13a (mod m) (5) x /C13b (mod n) (6) are solvable using the CHINESE REMAINDER THEOREM . See also CHINESE REMAINDER THEOREM ,C ONGRU- ENCE ,LINEAR CONGRUENCE EQUATION ,Q UADRATIC CONGRUENCE EQUATION Congruence Transformation A transformation OF THE FORM g /C30DT hD ; where det(D) "0 and det(D) is the DETERMINANT .ISOME- TRIES are also called congruence transformations. See also SYLVESTER’S INERTIA LAW Congruent There are at least two meanings on the word congruent in mathematics. Two geometric figures are said to be congruent if they are equivalent to within ROTATION and TRANSLATION (i.e., IFF one can be transformed into the other by an ISOMETRY ). This relationship is written A $B: Unfortunately, the symbol $is also used to denote an ISOMORPHISM . A number a is said to be congruent to b modulo m if m½a /C28b (m DIVIDES a /C28b) :/ See also COINCIDENT ,C ONGRUENCE ,H OMOTHETIC , ISOMETRY ,ROTATION ,SIMILAR ,TRANSLATION References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 80, 1967. Congruent Incircles Point The point Y for which TRIANGLES BYC , CYA , and AYB have congruent INCIRCLES . It is a special case of an ELKIES POINT . References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Congruent Isoscelizers Point In 1989, P. Yff proved there is a unique configuration of ISOSCELIZERS for a given TRIANGLE such that all three have the same length. Furthermore, these ISOSCELIZERS meet in a point called the congruent isoscelizers point, which has TRIANGLE CENTER FUNC- TION a /C30cos(1 2 B) /C27cos(12 C) /C28cos(12 A): See also ISOSCELIZER References Kimberling, C. "Congruent Isoscelizers Point." http://cedar.- evansville.edu/~ck6/tcenters/recent/conisos.html. Congruent Matrices Two SQUARE MATRICES A and B are called congruent if there exists a nonsingular matrix P such thatB /C30PTAP ; where PT is the TRANSPOSE . See also TRANSPOSE References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 115 1962. Congruent Numbers A set of numbers (a ; x; y; t) such that x2 /C27ay2 /C30z2 x2 /C28ay2 /C30t2 :l12) They are a generalization of the CONGRUUM PROBLEM , which is the case y /C301. For a /C30101, the smallest solution is x /C302015242462949760001961 y /C30118171431852779451900 z /C302339148435306225006961 t /C301628124370727269996961 : See also CONGRUUM References Guy, R. K. "Congruent Number." §D76 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 195 /C1/97, 1994. Congruum A number h which satisfies the conditions of the CONGRUUM PROBLEM : x2 /C27h /C30a2 and x2 /C28h /C30b2 ; where x; h; a; b are integers. The list of congrua is given by 24, 96, 120, 240, 336, 384, 480, 720, ... (Sloane’s A057102). See also CONCORDANT FORM,CONGRUUM PROBLEM References Sloane, N. J. A. Sequences A057102 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Congruum Problem Find a SQUARE NUMBER x2such that, when a given integer his added or subtracted, new SQUARE NUMBERS are obtained so that x2/C27h/C30a2(1) and x2 /C28h /C30b2 : (2) This problem was posed by the mathematicians The´odore and Jean de Palerma in a mathematical tournament organized by Frederick II in Pisa in 1225. The solution (Ore 1988, pp. 188 /C1/91) is x /C30m2 /C27n2 (3) h /C304mn(m2 /C28n2) ; (4) where m and n are INTEGERS . a and b are then given by a /C30m2 /C272mn /C28n2 (5) b /C30n2 /C272mn /C28m2 (6) Fibonacci proved that all numbers h (the CONGRUA ) are divisible by 24. FERMAT’S RIGHT TRIANGLE THEO- REM is equivalent to the result that a congruum cannot be a SQUARE NUMBER . A table for small m and n is given in Ore (1988, p. 191), and a larger one (for h 51000) by Lagrange (1977). The first mn h x a b Sloane A057103 A055096 A057104 A057105 21 2 4571 3 1 96 10 14 2 3 2 120 13 17 7 4 1 240 17 23 7 4 2 384 20 28 4 4 3 336 25 31 17 See also CONCORDANT FORM,CONGRUENT NUMBERS , CONGRUUM ,SQUARE NUMBER References Alter, R. and Curtz, T. B. "A Note on Congruent Numbers." Math. Comput. 28, 303 /C1/05, 1974. Alter, R.; Curtz, T. B.; and Kubota, K. K. "Remarks and Results on Congruent Numbers." In Proc. Third South- eastern Conference on Combinatorics, Graph Theory, and Computing, 1972, Boca Raton, FL. Boca Raton, FL: Florida Atlantic University, pp. 27 /C1/5, 1972. Bastien, L. "Nombres congruents." Interme ´d. des Math. 22, 231 /C1/32, 1915. Ge´rardin, A. "Nombres congruents." Interme ´d. des Math. 22, 52 /C1/3, 1915. Lagrange, J. "Construction d’une table de nombres congru- ents." Calculateurs en Math., Bull. Soc. math. France. , Me´moire 49 /C1/0, 125 /C1/30, 1977. Ore, Ø. Number Theory and Its History. New York: Dover, 1988.Sloane, N. J. A. Sequences A055096, A057103, A057104, and A057105 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Conic CONIC SECTION Conic Constant K /C13/C28e2 ; where e is the ECCENTRICITY of a CONIC SECTION . See also CONIC SECTION ,ECCENTRICITY Conic Double Point ISOLATED SINGULARITY Conic Equidistant Projection AMAP PROJECTION with transformation equations x/C30rsinu (1) y/C30r0/C28rcosu; (2) where r/C30(G/C28f) (3) u/C30n(l/C28l0) (4) r0/C30(G/C28u0) (5) G/C30cosf1 n/C27f1 (6) n/C30cosf1/C28cosf2 f2/C28f1: (7) The inverse FORMULAS are given by f/C30G/C28r (8) l/C30l0/C27u n; (9) where r /C30sgn(n)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27( r0 /C28y)2q (10) u /C30tan/C281 x r0 /C28 y ! : (11) See also EQUIDISTANT PROJECTION Conic Projection A conic projection of points on a unit sphere centered at O consists of extending the line OS for each point S until it intersects a cone with apex A which tangent to the sphere along a circle passing through a point T in a point C. For a cone with apex a height h above O, the angle from the Z-AXIS at which the cone is tangent is given by u /C30sec /C281 h; (1) and the radius of the circle of tangency and height above O at which it is located are given by r /C30sin u /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C28 1p h (2) z /C30cos u /C301 h : (3) Letting f?/C30p=2 /C28 f be the colatitude of a point S on asphere, the length of the vector OC along OS is l /C30sec(u /C28 f?) /C30sec(sec /C281 h /C28 f?) /C30csc( f /C27sec/C281 h) : (4) The left figure above shows the result of re-projecting onto a plane perpendicular to the Z-AXIS (equivalent to looking at the cone from above the apex), while the figure on the right shows the cone cut along the solid line and flattened out. The equations transforming a point on a sphere ( f; l) to a point on the flattened cone are x /C30csc(sec /C281 h /C27 f) cos f sinlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C28 1p ! (5) y /C30csc(sec /C281 h /C27 f) cos f coslffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2 /C28 1p ! : (6) This form of the projection, however, is seldom used in practice, and the term "conic projection" is used instead to refer to any projection in which lines of latitude are mapped to equally spaced radial lines and lines of latitude (parallels) are mapped tocircumferential lines with arbitrary mathematically spaced separations (Snyder 1987, p. 5). See also A LBERS EQUAL- AREA CONIC PROJECTION , CONIC EQUIDISTANT PROJECTION ,CYLINDRICAL PRO- JECTION ,LAMBERT AZIMUTHAL EQUAL- AREA PROJEC- TION ,POLYCONIC PROJECTION References Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190/C1/00, 1944. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 5, 1987. Conic Section The conic sections are the nondegenerate curves generated by the intersections of a PLANE with one or two NAPPES of a CONE . For a PLANE perpendicular to the axis of the CONE , a circle is produced. For a PLANE which is not perpendicular to the axis and which intersects only a single nappe, the curveproduced is either an ELLIPSE or a PARABOLA (Hilbert and Cohn-Vossen 1999, p. 8). The curve produced by aPLANE intersecting both NAPPES is a HYPERBOLA (Hilbert and Cohn-Vossen 1999, pp. 8 /C1/). The ELLIPSE and HYPERBOLA are known as CENTRAL CONICS . Because of this simple geometric interpretation, the conic sections were studied by the Greeks long beforetheir application to inverse square law orbits wasknown. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ELLIPSES , and Newton was then able to derive the shape of orbitsmathematically using CALCULUS , under the assump- tion that gravitational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes which are any of the four types of conic sections are possible. A conic section may more formally be defined as the locus of a point Pthat moves in the PLANE of a fixed point Fcalled the FOCUS and a fixed line dcalled the DIRECTRIX (with Fnot on d) such that the ratio of the distance of Pfrom Fto its distance from dis a constant ecalled the ECCENTRICITY .I fe/C300, the conic is a CIRCLE ,i f0BeB1;the conic is an ELLIPSE ,i f e/C301, the conic is a PARABOLA , and if e/C211, it is a HYPERBOLA . A conic section with DIRECTRIX atx/C300, focus at (p;0);and ECCENTRICITY e/C210 has Cartesian equa- tion y2/C27(1/C28e2)x2/C282px/C27p2/C300 (1) (Yates 1952, p. 36), where pis called the FOCAL PARAMETER . Plugging in pfor an ELLIPSE givesy2/C27(1/C28e2)x2/C282a(1/C28e2) ex/C27a2(1/C28e2)2 e2/C300;(2) for a PARABOLA (1) simplifies to y2/C304p(x/C28p); (3) and for a HYPERBOLA , (1) simplifies to y2/C27(1/C28e2)x2/C282a(e2/C281) ex/C27a2(e2/C281)2 e2/C300:(4) The polar equation of a conic section with FOCAL PARAMETER pis given by r/C30ep 1/C27ecosu: (5) The PEDAL CURVE of a conic section with PEDAL POINT at a FOCUS is either a CIRCLE or a LINE. In particular the ELLIPSE PEDAL CURVE and HYPERBOLA PEDAL CURVE are both CIRCLES , while the PARABOLA PEDAL CURVE is a LINE (Hilbert and Cohn-Vossen 1999, pp. 25 /C1/7). Five points in a plane determine a conic (Coxeter andGreitzer 1967, p. 76; Le Lionnais 1983, p. 56; Wells 1991), as do five tangent lines in a plane (Wells 1991). This follows from the fact that a conic section is a QUADRATIC CURVE , which has general form ax2/C272bxy/C27cy2/C27dx/C27fy/C27g/C300; (6) so dividing through by ato obtain x2/C272b?xy/C27c?y2/C27d?x/C27f?y/C27g?/C300 (7) leaves five constants. Five points, ( xi;yi) for i/C301, ..., 5, therefore determine the constants uniquely. The GEOMETRIC CONSTRUCTION of a conic section from five points lying on it is called the B RAIKENRIDGE- MA- CLAURIN CONSTRUCTION . Two conics that do not coincide or have an entirestraight line in common cannot meet at more thanfour points (Hilbert and Cohn-Vossen 1999, pp. 24 and 160). There is an infinite family of conics touching four lines. However, of the eleven regionsinto which plane division cuts the plane, only five can contain a conic section which is tangent to all four lines. Parabolas can occur in one region only (which also contains ellipses and one branch of hyperbolas), and the only closed region contains only ellipses. Let a polygon of 2n sides be inscribed in a given conic, with the sides of the polygon being termed alternately "odd" and "even" according to some definite conven- tion. Then the n(n /C282) points where an odd side meet a nonadjacent even side lie on a curve of order n /C282 (Evelyn et al. 1974, p. 30). See also BRAIKENRIDGE- MACLAURIN CONSTRUCTION , BRIANCHON’S THEOREM ,C ENTRAL CONIC ,C IRCLE , CONE,C YLINDRICAL SECTION ,E CCENTRICITY ,E L- LIPSE ,F ERMAT CONIC ,F OCAL PARAMETER ,F OUR CONICS THEOREM ,FRE´ GIER’S THEOREM ,HYPERBOLA , NAPPE ,PARABOLA ,PASCAL’S THEOREM ,PLANE DIVI- SION BY ELLIPSES ,QUADRATIC CURVE ,SEYDEWITZ’S THEOREM ,SKEW CONIC ,STEINER’S THEOREM ,THREE CONICS THEOREM References Besant, W. H. Conic Sections, Treated Geometrically, 8th ed. rev. Cambridge, England: Deighton, Bell, 1890. Casey, J. "Special Relations of Conic Sections" and "Invar- iant Theory of Conics." Chs. 9 and 15 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most RecentExtensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 307 /C1 /32 and 462 /C1/45, 1893. Chasles, M. Traite ´des sections coniques. Paris, 1865. Coolidge, J. L. A History of the Conic Sections and Quadric Surfaces. New York: Dover, 1968. Coxeter, H. S. M. "Conics" §8.4 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115 /C1/19, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 138 /C1/41, 1967. Downs, J. W. Practical Conic Sections. Palo Alto, CA: Dale Seymour, 1993. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. The Seven Circles Theorem and Other New Theorems. London: Stacey International, p. 30, 1974. Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 inGeometry and the Imagination. New York: Chelsea, pp. 7/C1/1, 1999. Iyanaga, S. and Kawada, Y. (Eds.). "Conic Sections." §80 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 271 /C1/76, 1980. Klein, F. "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, andthe Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 42 /C1 /4, 1980. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Lebesgue, H. Les Coniques. Paris: Gauthier-Villars, 1955. Ogilvy, C. S. "The Conic Sections." Ch. 6 in Excursions in Geometry. New York: Dover, pp. 73 /C1/5, 1990. Pappas, T. "Conic Sections." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 196 /C1/97, 1989. Salmon, G. Conic Sections, 6th ed. New York: Chelsea, 1960. Smith, C. Geometric Conics. London: MacMillan, 1894.Sommerville, D. M. Y. Analytical Conics, 3rd ed. London: G. Bell and Sons, 1961. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 238 /C1/40, 1999. Weisstein, E. W. "Books about Conic Sections." http:// www.treasure-troves.com/books/ConicSections.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 175, 1991. Yates, R. C. "Conics." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 36 /C1/6, 1952. Conic Section Tangent Given a CONIC SECTION x2/C27y2/C272gx/C272fy/C27c/C300; the tangent at /(x1;y1)/is given by the equation xx1/C27yy1/C27g(x/C27x1)/C27f(y/C27y1)/C27c/C300: Conical Coordinates There are several different definitions of conical coordinates defined by Morse and Feshbach (1953), Byerly (1959), Arfken (1970), and Moon and Spencer(1988). The ( l;m;n) system defined in Mathematica is x/C30lmn ab(1) y/C30l affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (m2/C28a2)(n2/C28a2) a2/C28b2s (2) z/C30l bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (m2/C28b2)(n2/C28b2) b2/C28a2s ; (3) where b2>m2>c2>n2:Byerly (1959) uses a ( r;m;n) system which is essentially the same coordinatesystem as above, but replacing lwith r,awith b, and bwith c. Moon and Spencer (1988) use ( r;u;l) instead of ( l;m;n): / The above equations give x2 /C27y2 /C27z2 /C30 l2 (4) x2 m2 /C27y2 m2 /C28 a2 /C27z2 m2 /C28 b2 /C300 (5) x2 n2 /C27y2 n2 /C28 a2 /C27z2 n2 /C28 b2 /C300: (6) The SCALE FACTORS are hl /C301 (7) hm /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2( m2 /C28 n2) ( m2 /C28 a2)(b2 /C28 m2)s (8) hn /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2( m2 /C28 n2) ( n2 /C28 a2)(n2 /C28 b2)s : (9) The LAPLACIAN is 92 /C30n(2n2 /C28 a2 /C28 b2) ( m /C28 n)(m /C27 n)l2@ @ n /C27(a /C28 n)(a /C27 n)(n /C28 b)(n /C27 b) ( n /C28 m)( n /C27 m) l2@2 @ n2 /C27m(2m2 /C28 a2 /C28 b2) ( m /C28 n)(m /C27 n) l2@ @ m /C27( m /C28 b)(m /C27 b)( m /C28 a)(m /C27 a) ( n /C28 m)( n /C27 m) l2@2 @ m2 /C272 l@ @ l /C27@2 @ l2 : (10) The HELMHOLTZ DIFFERENTIAL EQUATION is separable in conical coordinates. See also HELMHOLTZ DIFFERENTIAL EQUATION– CON- ICAL COORDINATES References Arfken, G. "Conical Coordinates (/j1 ; j2 ; j3) :/" §2.16 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 118 /C1/19, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 263, 1959. Moon, P. and Spencer, D. E. "Conical Coordinates (r ; u ; l):/" Table 1.09 in Field Theory Handbook, Including Coordi- nate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 37 /C1/0, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 659, 1953. Spence, R. D. "Angular Momentum in Sphero-Conal Coordi- nates." Amer. J. Phys. 27, 329 /C1/35, 1959.Conical Frustum A conical frustum is a FRUSTUM created by slicing the top off a CONE (with the cut made parallel to the base). For a right circular CONE , let s be the slant height and R1 and R2 the top and bottom RADII . Then s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (R1 /C28R2)2 /C27h2 :q (1) The SURFACE AREA , not including the top and bottom CIRCLES ,is A /C30 p(R1 /C27R2)s /C30 p(R1 /C27R2)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(R 1 /C28R2)2 /C27h2 :q (2) The VOLUME of the frustum is given by V /C30 pgh 0[r(z)]2 dz: (3) But r(z) /C30R1 /C27(R2 /C28R1)z h ; (4) so V /C30 pgh 0[r(z)]2 dz /C30 pgh 0R1 /C27(R2 /C28R1)z h"#2 dz /C301 3 ph(R2 1 /C27R1R2 /C27R22) : (5) This formula can be generalized to any PYRAMID by letting Ai be the base AREAS of the top and bottom of the frustum. Then the VOLUME can be written as V /C301 3h(A1 /C27A2 /C27ffiffiffiffiffiffiffiffiffiffiffi A1A2p ): (6) The area-weighted integral of z over the frustum is zhi/C30 pgh 0z[r(z)]2dz/C301 12ph2(R2 1/C272R1R2/C273R22);(7) so the CENTROID is located along the Z-AXIS at a height ¯z/C30zhi V/C30h(R2 1/C272R1R2/C273R22) 4(R2 1/C27R1R2/C27R22)(8) (Eshbach 1975, p. 453; Beyer 1987, p. 133; Harris and Stocker 1998, p. 105). The special case of the CONE is given by taking R2/C300;yielding ¯z/C30h=4:/ See also CONE,F RUSTUM ,P YRAMIDAL FRUSTUM , SPHERICAL SEGMENT References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 129 /C1/30 and 133, 1987. Eshbach, O. W. Handbook of Engineering Fundamentals. New York: Wiley, 1975. Harris, J. W. and Stocker, H. "Frustum of a Right Circular Cone." §4.7.2 in Handbook of Mathematics and Computa- tional Science. New York: Springer-Verlag, p. 105, 1998. Kern, W. F. and Bland, J. R. "Frustum of Right Circular Cone." §29 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 71 /C1/5, 1948. Conical Function Functions which can be expressed in terms of LE- GENDRE FUNCTIONS OF THE FIRST and SECOND KINDS . See Abramowitz and Stegun (1972, p. 337). Pm /C281 =2 /C27ip(cos u) /C301 /C274p2 /C27 12 22sin2(1 2 u) /C27(4p2 /C27 12)(4p2 /C27 32) 2242 sin4(1 2 u) /C27... /C302 p g u 0cosh( pt)dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(cos t /C28 cos u)p Q m /C281=2 /C14ip(cos u) /C309i sinh( p p)g/C12 0cos(pt)dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(cosh t /C27 cos u)p /C27g/C12 0cosh( pt)dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(cos t /C28 cos u)p : See also T OROIDAL FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Conical Func- tions." §8.12 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, p. 337, 1972. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1464, 1980. Conical Projection CONIC PROJECTION Conical Spiral A SPACE CURVE given by the PARAMETRIC EQUATIONS x /C30h /C28 z hr cos(az) y /C30h /C28 z hr sin(az) z /C30z for h the height of the cone, r its radius, and a a constant. See also CONE,SEASHELL Conical Surface GENERALIZED CONE Conical Wedge The SURFACE also called the CONOCUNEUS OF WALLIS and given by the parametric equation x /C30u cos v y /C30u sin v z /C30c(1 /C282 cos2 v) : See also CYLINDRICAL WEDGE ,W EDGE References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 302, 1993. Conjecture A proposition which is consistent with known data, but has neither been verified nor shown to be false. It is synonymous with HYPOTHESIS . References Rivera, C. "Problems & Puzzles: Conjectures." http:// www.primepuzzles.net/conjectures/. Conjugacy Class A complete set of mutually conjugate GROUP ele- ments. Each element in a GROUP belongs to exactly one class, and the IDENTITY ELEMENT (I/C301) is always in its own class. The ORDERS of all classes must be integral FACTORS of the ORDER of the GROUP . From the last two statements, a GROUP ofPRIME order has one class for each element. More generally, in an A BELIAN GROUP , each element is in a conjugacy class by itself. Two operations belong to the same class when one may be replaced by the other in a new COORDINATE SYSTEM which is accessible by a symmetry operation (Cotton 1990, p. 52). These sets correspond directly tothe sets of equivalent operations. To see how to compute conjugacy classes, consider the FINITE GROUP D3, which has the following MULTI- PLICATION TABLE . /D3/ 1 ABCDE 11 ABCDE AA 1 DEBC BBE 1 DCA CCDE 1 AB DDCABE 1 EEBCA 1 D /f1g is always in a conjugacy class of its own. To find another conjugacy class take some element, say A, and find the results of all similarity transformations X /C281AX /C30X /C281(AX)on A. For example, for X /C30A, the product of A by A can be read of as the element at the intersection of the row containing A (the first multi- plicand) with the column containing A (the second multiplicand), giving A/C281AA /C30A/C2811: Now, we want to find Z where A/C2811 /C30Z ; so pre-multiply both sides by A to obtain (AA/C281)1 /C301 /C30AZ; so Z is the element whose column intersects row A in 1, i.e., A. Thus, A/C281AA /C30A: Similarly, B /C281AB /C30C; and continuing the process for all elements gives A/C281AA /C30A (1) B/C281AB /C30C (2) C/C281AC /C30B (3) D/C281AD /C30C (4) E/C281AE /C30B (5) The possible outcomes are A, B,orC,so fA; B ; C g forms a conjugacy class. To find the next conjugacy class, take one of the elements not belonging to an existing class, say D. Applying a similarity transfor- mation gives A/C281DA /C30E (6) B /C281DB /C30D ; (7) so we need proceed no further since D and E both appear, meaning fD; E g form a conjugacy class and we have exhausted all elements of the group. Let G be a FINITE GROUP of ORDER ½G½; and let s be the number of conjugacy classes of G.If½G ½ is ODD, then ½G ½/C13s (mod 16) (Burnside 1955, p. 295). Furthermore, if every PRIME pi DIVIDING ½G½ satisfies pi /C131 (mod 4); then½G½/C13s (mod 32) (Burnside 1955, p. 320). Poonen (1995) showed that if every PRIME piDIVIDING ½G½ satisfies pi /C131 (mod m) for m ]2; then ½G ½/C13s (mod 2m2) : References Burnside, W. Theory of Groups of Finite Order, 2nd ed. New York: Dover, 1955. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990. Poonen, B. "Congruences Relating the Order of a Group to the Number of Conjugacy Classes." Amer. Math. Monthly 102, 440 /C1/42, 1995. Conjugate COMPLEX CONJUGATE ,CONJUGATE ELEMENT ,CONJU- GATE GRADIENT METHOD ,C ONJUGATE MATRIX , CONJUGATE POINTS ,CONJUGATE SUBGROUP ,CONJU- GATION MOVE Conjugate Element Given a GROUP with elements A and X, there must be an element B which is a SIMILARITY TRANSFORMATION of A; B /C30X /C281AX so A and B are conjugate with respect to X. Conjugate elements have the following properties: 1. Every element is conjugate with itself. 2. If A is conjugate with B with respect to X, then B is conjugate to A with respect to X. 3. If A is conjugate with B and C, then B and C are conjugate with each other. See also CONJUGACY CLASS,CONJUGATE SUBGROUP Conjugate Gradient Method An ALGORITHM for finding the nearest LOCAL MINI- MUM of a function of n variables which presupposes that the GRADIENT of the function can be computed. It uses conjugate directions instead of the local GRADI- ENT for going downhill. If the vicinity of the MINIMUM has the shape of a long, narrow valley, the minimum is reached in far fewer steps than would be the case using the STEEPEST DESCENT METHOD . See also GRADIENT ,L OCAL MINIMUM ,M INIMUM , STEEPEST DESCENT METHOD References Brodie, K. W. §3.1.7 in The State of the Art in Numerical Analysis (Ed. D. A. E. Jacobs). London: Academic Press, 1977. Bulirsch, R. and Stoer, J. §8.7 in Introduction to Numerical Analysis. New York: Springer-Verlag, 1991. Polak, E. §2.3 in Computational Methods in Optimization. New York: Academic Press, 1971. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 413 /C1/17, 1992. Conjugate Matrix The matrix ¯A obtained from a given matrix A by taking the COMPLEX CONJUGATE of each element of A (Courant and Hilbert 1989, p. 9). The notation A/C31 is sometimes also used, which can lead to confusion since this symbol is also used to denote the ADJOINT MATRIX . See also ADJOINT MATRIX ,COMPLEX CONJUGATE References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 355 /C1/56, 1985. Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 12 /C1/3, 1962. Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989. Conjugate Partition Pairs of partitions for a single number whose FER- RERS DIAGRAMS transform into each other when reflected about the line y /C30/C28x; with the coordinates of the upper left dot taken as (0, 0), are called conjugate (or transpose) partitions. For example, the conjugate partitions illustrated above correspond to the partitions 6 /C273 /C273 /C272 /C271 and 5 /C274 /C273 /C271 /C271 /C27 1 of 15. A partition that is conjugate to itself is said to be a SELF-CONJUGATE PARTITION . The conjugate partition of a given partition l can be implemented in Mathematica as follows. ConjugatePartition[l_List]: /C30 Module[{i,r /C30 Reverse[l],n /C30 Length[l]}, Table[n /C271 /C28Position[r,_?(# /C21/C30 i&), Infinity,1][[1,1]], {i,l[[1]]} ]] A similar implementation is given as Transpose- Partition [l] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also DURFEE SQUARE ,FERRERS DIAGRAM ,PARTI- TION FUNCTION P,SELF-CONJUGATE PARTITION References Andrews, G. E. The Theory of Partitions. Cambridge, Eng- land: Cambridge University Press, pp. 7 /C1/, 1998.Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 55 /C1/6, 1990. Conjugate Permutation INVERSE PERMUTATION Conjugate Points HARMONIC CONJUGATE POINTS ,INVERSE POINTS ,ISO- GONAL CONJUGATE ,ISOTOMIC CONJUGATE POINT Conjugate Subgroup A SUBGROUP H of an original GROUP G has elements hi : Let x be a fixed element of the original GROUP G which is not a member of H. Then the transformation xhix/C281 ; (i /C301, 2, ...) generates the so-called conjugate subgroup xHx/C281 : If, for all x, xHx/C281 /C30H ; then H is a SELF-CONJUGATE (also called "invariant" or "normal") SUBGROUP . All SUBGROUPS of an ABELIAN GROUP are SELF-CON- JUGATE . See also SELF-CONJUGATE SUBGROUP ,S UBGROUP , SYLOW THEOREMS Conjugate Transpose Matrix ADJOINT MATRIX Conjugation The process of taking a COMPLEX CONJUGATE of a COMPLEX NUMBER , COMPLEX MATRIX , etc., or of per- forming a CONJUGATION MOVE on a KNOT . See also COMPLEX CONJUGATE ,C OMPLEX MATRIX , COMPLEX NUMBER ,C ONJUGATE MATRIX ,C ONJUGA- TION MOVE Conjugation Move A type I MARKOV MOVE . See also MARKOV MOVES ,STABILIZATION Conjunction A product of ANDs, denoted ffln k/C301Ak: The conjunctions of a BOOLEAN ALGEBRA A of subsets of cardinality p are the 2p functions Al /C30@ i /C23 lAi ; where l ƒf1 ; 2 ; ...; p g: For example, the 8 conjunc- tions of A /C30fA1 ; A2 ; A3 g are ¥; A1 ; A2 ; A3 ; A1A2 ; A2A3 ; A3A1 ; and A1A2A3 (Comtet 1974, p. 186). See also AND, BOOLEAN ALGEBRA ,BOOLEAN FUNC- TION ,COMPLETE PRODUCT ,DISJUNCTION , NOT, OR References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 186, 1974. Conjunctive Normal Form A statement is in conjunctive normal form if it is a CONJUNCTION (sequence of ANDs) consisting of one or more conjuncts, each of which is a DISJUNCTION (OR) of one or more statement letters and negations of statement letters. Examples of disjunctive normal forms include A (1) (A /C150B) ffl(!A /C150C) (2) (A /C150B /C150!A) ffl(C /C150!B) ffl(A /C150!C) (3) A /C150B (4) A ffl(B /C150C) ; (5) where /C150 denotes OR, ffl denotes AND, and ! denotes NOT. Every statement in logic consisting of a combi- nation of multiple ffl;/C150; and !/s can be written in conjunctive normal form. See also DISJUNCTIVE NORMAL FORM References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 27, 1997. Connected Component A TOPOLOGICAL SPACE decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity, i.e., if a /C2b and b /C2c then a /C2c : Hence, being in the same component is an EQUIVALENCE RELATION , and the equivalence classes are the connected components. Using PATH-CONNECTEDNESS , the path-connected component containing x /C23 X is the set of all y path- connected to x. That is, it is the set of y such that there is a continuous path from x to y. Technically speaking, in some TOPOLOGICAL SPACES , path-connected is not the same as connected. A subset Y of X is connected if there is no way to write Y /C30U @ V with U and V disjoint OPEN SETS. Every TOPOLOGICAL SPACE decomposes into a disjoint union X /C30@ Yi where the Yi are connected. The Yi are called the connected components of X. See also CONNECTED SET,PATH-CONNECTED ,TOPO- LOGICAL SPACE Connected Digraph There are two distinct notions of connectivity in a DIGRAPH .ADIGRAPH is WEAKLY CONNECTED if there is an undirected path between any pair of vertices, and STRONGLY CONNECTED if there is a directed path between every pair of vertices (Skiena 1990, p. 173). The following tables summarized the number of weakly and strongly connected digraphs on n /C301, 2, ... nodes. The 8 weakly but not strongly connected digraphs on three nodes are illustrated above. connectivity Sloane counts weakly connected A003085 1, 2, 13, 199, 9364, ... strongly connectedA035512 1, 1, 5, 83, 5048, 1047008, ... weakly but notstronglyA056988 0, 1, 8, 116, 4316, 483835, ... See also C ONNECTED GRAPH ,D IGRAPH ,STRONGLY CONNECTED DIGRAPH ,W EAKLY CONNECTED DIGRAPH References Skiena, S. "Strong and Weak Connectivity." §5.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 172 /C1/74, 1990. Sloane, N. J. A. Sequences A003085/M2067, A035512, and A056988 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Connected Graph A GRAPH which is connected in the sense of a TOPOLOGICAL SPACE , i.e., there is a path from any point to any other point in the GRAPH . The number of n-node connected unlabeled graphs for n /C301, 2, ... are 1, 1, 2, 6, 21, 112, 853, 11117, ... (Sloane’s A001349). The total number of (not necessarily connected) unlabeled n-node graphs is given by the EULER TRANSFORM of the preceding sequence, 1, 2, 4, 11, 34, 156, 1044, 12346, ... (Sloane’s A000088; Sloane and Plouffe 1995, p. 20). The numbers of connected labeled graphs on n-nodes are 1, 1, 4, 38, 728, 26704, ... (Sloane’s A001187), and the total number of (not necessarily connected) labeled n-node graphs is given by the EXPONENTIAL TRANSFORM of the preceding sequence: 1, 2, 8, 64, 1024, 32768, ... (Sloane’s A006125; Sloane and Plouffe 1995, p. 19). If an is the number of unlabeled connected graphs on n nodes satisfying some property, than the EULER TRANSFORM bnis the total number of unlabeled graphs (connected or not) with the same property. This application of the EULER TRANSFORM is called RIDDELL’S FORMULA . If G is DISCONNECTED , then its complement ¯G is connected (Skiena 1990, p. 171; Bolloba ´s 1998). How- ever, the converse is not true, as can be seen using the example of the CYCLE GRAPH C5which is connected and isomorphic to its complement. One can also speak of connected graphs in which each vertex has degree at least k (i.e., the minimum of the DEGREE SEQUENCE is ]k) : The usual CONNECTED GRAPH is therefore connected with minimal degree /]1:The following table gives the number of con- nected graphs with minimal degree ]konnvertices for small k. kSloane sequence 1 A001349 1, 1, 2, 6, 21, 112, 853, 11117, ... 2 A004108 0, 0, 1, 3, 11, 61, 507, 7442, ...3 A007112 0, 0, 0, 1, 3, 19, 150, 2589, ... See also A LGEBRAIC CONNECTIVITY ,B ICONNECTED GRAPH ,DEGREE SEQUENCE ,DISCONNECTED GRAPH , EULER TRANSFORM ,P LANAR CONNECTED GRAPH , POLYHEDRAL GRAPH ,POLYNEMA ,R EGULAR GRAPH , RIDDELL’S FORMULA ,SEQUENTIAL GRAPH ,STEINITZ’S THEOREM ,TAIT’S HAMILTONIAN GRAPH CONJECTURE References Bolloba ´s, B. Modern Graph Theory. New York: Springer- Verlag, 1998. Cadogan, C. C. "The Mo ¨bius Function and Connected Graphs." J. Combin. Th. B 11, 193/C1/00, 1971. Chartrand, G. "Connected Graphs." §2.3 in Introductory Graph Theory. New York: Dover, pp. 41 /C1/5, 1985. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 13, 1994. Skiena, S. "Connectivity." §5.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 171 /C1/ 80, 1990. Sloane, N. J. A. Sequences A000088/M1253, A001187/ M3671, A001349/M1657, A004108/M2910, A006125/ M1897, and A007112/M3059 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Tutte, W. T. The Connectivity of Graphs. Toronto, Canada: Toronto University Press, 1967. Connected Set A connected set is a SET which cannot be partitioned into two nonempty SUBSETS which are open in the relative topology induced on the SET. Equivalently, it is a SET which cannot be partitioned into two none- mpty SUBSETS such that each SUBSET has no points in common with the CLOSURE of the other. The REAL NUMBERS are a connected set. See also CLOSED SET,CLOSURE (SET), EMPTY SET, OPEN SET,SET,SIMPLY CONNECTED ,SUBSET References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 3, 1999. Connected Space A SPACE D is connected if any two points in D can be connected by a curve lying wholly within D.A SPACE is 0-connected (a.k.a. PATHWISE-CONNECTED ) if every MAP from a 0-SPHERE to the SPACE extends continu- ously to the 1-DISK. Since the 0-SPHERE is the two endpoints of an interval (1-DISK), every two points have a path between them. A space is 1-connected (a.k.a. SIMPLY CONNECTED ) if it is 0-connected and if every MAP from the 1-SPHERE to it extends continu- ously to a MAP from the 2-DISK. In other words, every loop in the SPACE is CONTRACTIBLE .A SPACE is n- MULTIPLY CONNECTED if it is (n /C281)/-connected and if every MAP from the n-SPHERE into it extends con- tinuously over the (n /C271)/-DISK. A theorem of Whitehead says that a SPACE is infinitely connected IFF it is CONTRACTIBLE . See also CONNECTIVITY ,C ONTRACTIBLE ,L OCALLY PATHWISE- CONNECTED ,MULTIPLY CONNECTED ,PATH- WISE- CONNECTED ,SIMPLY CONNECTED Connected Sum The connected sum M1#M2 of n-manifolds M1 and M2 is formed by deleting the interiors of n-BALLS bn iin mni and attaching the resulting punctured MANIFOLDS Mi /C28 ˙Bi to each other by a HOMEOMORPHISM h : @B2 0 @B1 ; so M1#M2 /C30(M1 /C28 ˙B1)@ h(M2 /C28 ˙B2): /Bi is required to be interior to Mi and @Bi bicollared in Mi to ensure that the connected sum is a MANIFOLD . The connected sum of two KNOTS is called a KNOT SUM. See also KNOT SUMReferences Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 39, 1976. Connected Sum Decomposition Every COMPACT 3-MANIFOLD is the CONNECTED SUM of a unique collection of PRIME 3-MANIFOLDS . See also JACO-SHALEN- JOHANNSON TORUS DECOMPO- SITION Connection See also CONNECTION COEFFICIENT ,C ONNECTION (VECTOR BUNDLE ), GAUSS- MANIN CONNECTION Connection (Vector Bundle) A connection on a VECTOR BUNDLE p:E0Mis a way to "differentiate" SECTIONS , in a way that is analogous to the EXTERIOR DERIVATIVE dfof a function f.I n particular, a connection 9is a function from smooth sections G(M;E) to smooth sections of ETENSOR with ONE-FORMS G(M;E/C156T/C31M) that satisfies the follow- ing conditions. 1.9fs/C30s/C156df/C27f9s(Leibniz rule), and 2.9s1/C27s2/C309s1/C279s2:/ Alternatively, a connection can be considered as a linear map from SECTIONS ofE/C156TM;i.e., a section of Ewith a VECTOR FIELD X, to sections of E, in analogy to the DIRECTIONAL DERIVATIVE . The DIRECTIONAL DERIVATIVE of a function f, in the direction of a vector field X, is given by df(X):The connection, along with a vector field X, may be applied to a section sofEto get the section 9Xs:From this perspective, connec- tions must also satisfy 9fXs/C30f9Xs (1) for any smooth function f. This property follows from the first definition. For example, the TRIVIAL BUNDLE E/C30M/C29Rkadmits aFLAT CONNECTION since any SECTION scorresponds to a function ˜s:M0Rk:Then setting 9s/C30dsgives the connection. Any connection on the TRIVIAL BUN- DLEis of the form 9s/C30ds/C27s/C156a;where ais any ONE- FORM with values in Hom( E;E)/C30E/C31/C156E;i.e.,ais a matrix of ONE-FORMS . The matrix of ONE-FORMS a/C30dx 2xd y 0 0 dx/C283dy 0 xy dx 0 y2dx/C27dy2 435 (2) determines a connection 9on the rank-3 bundle over R2 : It acts on a section s /C30(s1 ; s2 ; s3) by the following. 9@=@xs /C30sx /C27 a( @=@x)s /C30sx /C27100 010 xy 0 y22 435s /C30( @s 1 =@x /C27s1 ;@s2 =@x /C27s2 ;@s3 =@x /C27xys1 /C27y2s3) (3) 9@=@ys /C30sy /C27 a( @=@y)s /C30sy /C2702 x 0 0 /C2830 0012 435s /C30( @s 1 =@x /C272xs2 ;@s2 =@x /C283s2 ;@s3 =@x /C27s3): (4) In any TRIVIALIZATION , a connection can be described just as in the case of a TRIVIAL BUNDLE . However, if the bundle E is not TRIVIAL , then the EXTERIOR DERIVATIVE ds is not WELL DEFINED (globally) for a SECTION s. Still, the difference between any two connections must be ONE-FORMS with values in ENDOMORPHISMS of E, i.e., matrices of one forms. So the space of connections forms an AFFINE SPACE . The CURVATURE of the bundle is given by the formula V/C309(9: In coordinates, V/C30 a ffl a is matrix of TWO- FORMS . For instance, in the example above, V/C3002 xdxffldy 0 0 /C283x ffldy 0 02x3ydxffldy y2 dx ffldy2 435 (5) is the curvature. Another way of describing a connection is as a splitting of the TANGENT BUNDLE TE of E as TM /C154 E : The vertical part of TE corresponds to tangent vectors along the fibers, and is the kernel of dp : TE 0 TM : The horizontal part is not WELL DEFINED a priori. A connection defines a subspace of TE(x; v) which is isomorphic to TMx : It defines k FLAT SECTIONS sisuch that 9si /C300; which are a BASIS for the FIBERS of E, at least nearby x. These flat sections determine the horizontal part of TE near x. Also, a connection on a vector bundle can be defined by a CONNECTION on the ASSOCIATED PRINCIPAL BUNDLE . In some settings there is a canonical connection. For example, a RIEMANNIAN MANIFOLD has the LEVI- CIVITA CONNECTION , given by the CHRISTOFFEL SYM- BOLS OF THE FIRST and SECOND KINDS , which is the unique torsion-free connection compatible with the metric. A HOLOMORPHIC VECTOR BUNDLE with a HERMITIAN METRIC has a unique connection which is compatible with both metric and the COMPLEX STRUCTURE . See also CONNECTION (PRINCIPAL BUNDLE ), CURVA- TURE ,C URVATURE (BUNDLE ), HERMITIAN METRIC , LEVI-CIVITA CONNECTION ,P ARALLEL TRANSPORT , PRINCIPAL BUNDLE ,SECOND FUNDAMENTAL FORM, SECTION (BUNDLE ), TORSION (BUNDLE )Connective A function, or the symbol representing a function, which corresponds to English conjunctions such as "and," "or," "not," etc. that takes one or more truth values as input and returns a single truth value as output. The terms "logical connective" and "proposi- tional connective" are also used. The following table summarizes some common connectives and their notations. connective symbol AND /A fflB ; A /C215 B; A:B; AB, A&B ; A&&B/ EQUIVALENT /A /C13B ; A UB ; A XB/ IMPLIES /A [B ; A ‡B ; A 0 B/ NAND /A¯fflB ; A½B ;A /C215 B/ NONEQUIVALENT /A fB ; A UB ; A uXB/ NOR /A¯/C150B ; A ¡B ;A /C27B/ NOT /!A;/C15A;¯A;/C2A/ OR /A/C150B;A/C27B;A½B;A½½B/ XNOR AXNOR B XOR /A¯/C150B;A/C154B/ See also AND, BINARY OPERATOR ,E QUIVALENT , IMPLIES , OR, NAND, NONEQUIVALENT , NOR, NOT, PROPOSITIONAL CALCULUS ,T RUTH TABLE ,XNOR, XOR References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997. Connective Constant SELF-AVOIDING WALKCONNECTIVE CONSTANT Connectivity CONNECTED SPACE ,E DGE CONNECTIVITY ,V ERTEX CONNECTIVITY Connectivity Pair An ordered pair ( a, b) of nonnegative integers such that there is some set of apoints and bedges whose removal disconnects the graph and there is no set of a/C281 nodes and bedges or anodes and b/C281 edges with this property. References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Connes Function The APODIZATION FUNCTION A(x) /C30 1 /C28x2 a2 !2 : Its FULL WIDTH AT HALF MAXIMUM isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C282ffiffiffi 2pp a ; and its INSTRUMENT FUNCTION is I(x) /C308affiffiffiffiffiffi 2pp J5 =2(2pka) (2pka)5 =2 ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . See also APODIZATION FUNCTION Conocuneus of Wallis CONICAL WEDGE Conoid PLU¨ CKER’S CONOID ,RIGHT CONOID Consecutive Number Sequences Consecutive number sequences are sequences con- structed by concatenating numbers of a given type. Many of these sequences were considered by Smar- andache, so they are sometimes known as SMARAN- DACHE SEQUENCES . The nth term of the consecutive integer sequence consists of the concatenation of the first n POSITIVE INTEGERS : 1, 12, 123, 1234, ... (Sloane’s A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998). This se- quence gives the digits of the CHAMPERNOWNE CON- STANT and contains no PRIMES in the first 7,746 terms (Weisstein, Jan. 23, 2000). Fleuren (1999) has ver- ified the absence of primes up to n /C30200. This is roughly consistent with simple arguments based on the distribution of primes which suggest that only a single prime is expected in the first 15,000 or so terms. The number of digits of the n term can be computed by noticing the pattern in the following table, where d /C30[log10 n] /C271 is the number of digits in n. dn Range Digits 11/C1/ n 210/C1/9 /9 /C272(n /C289)/ 3 100 /C1/99 /9 /C2790 /C215 2 /C273(n /C2899) / 4 1000 /C1/ 999/9 /C2790 /C215 2 /C27900 /C215 3 /C274(n /C28999) /Therefore, the number of digits D(n) in the nth term can be written D(n) /C30d(n /C271 /C2810d/C281) /C27Xd/C281 k /C3019k /C215 10k /C281 /C30(n /C271)d /C2810d /C28 1 9; where the second term is the REPUNIT Rd :/ The nth term of the reverse integer sequence consists of the concatenation of the first n POSITIVE INTEGERS written backwards: 1, 21, 321, 4321, ... (Sloane’s A000422; Smarandache 1993, Dumitrescu and Se- leacu 1994, Stephan 1998). The only PRIME in the first 7,287 terms (Weisstein, Jan. 23, 2000) of this se- quence is the 82nd term 828180...321 (Stephan 1998, Fleuren 1999), which has 155 digits. This is roughly consistent with simple arguments based on the distribution of prime which suggest that a single prime is expected in the first 15,000 or so terms. The terms of the reverse integer sequence have the same number of digits as do the consecutive integer sequence. The concatenation of the first n PRIMES gives 2, 23, 235, 2357, 235711, ... (Sloane’s A019518; Smith 1996, Mudge 1997). This sequence converges to the digits of the COPELAND- ERDOS CONSTANT and is PRIME for terms 1, 2, 4, 128, 174, 342, 435, 1429, ... (Sloane’s A046035; Ibstedt 1998, pp. 78 /C1/9), with no others less than 4,706 (Weisstein, Jan. 23, 2000). The concatenation of the first n ODD NUMBERS gives 1, 13, 135, 1357, 13579, ... (Sloane’s A019519; Smith 1996, Marimutha 1997, Mudge 1997). This sequence is PRIME for terms 2, 10, 16, 34, 49, 2570, ... (Sloane’s A046036; Weisstein, Ibstedt 1998, pp. 75 /C1/6), with no others less than 4,354 (Weisstein, Jan. 1, 2000). The 2570th term, given by 1 3 5 7...5137 5139, has 9725 digits and was discovered by Weisstein in Aug. 1998. The concatenation of the first n EVEN NUMBERS gives 2, 24, 246, 2468, 246810, ... (Sloane’s A019520; Smith 1996; Marimutha 1997; Mudge 1997; Ibstedt 1998,pp. 77 /C1 /8). The concatenation of the first nSQUARE NUMBERS gives 1, 14, 149, 14916, ... (Sloane’s A019521; Mar- imutha 1997). The only PRIME in the first 2,822 terms is the third term, 149, (Weisstein). The concatenation of the first nCUBIC NUMBERS gives 1, 18, 1827, 182764, ... (Sloane’s A019522; Marimutha 1997). There are no PRIMES in the first 2,652 terms (Weisstein). See also CHAMPERNOWNE CONSTANT ,C ONCATENA- TION ,COPELAND- ERDOS CONSTANT ,CUBIC NUMBER , DEMLO NUMBER ,E VEN NUMBER ,O DD NUMBER , SMARANDACHE SEQUENCES ,SQUARE NUMBER References Dumitrescu, C. and Seleacu, V. (Eds.). Some Notions and Questions in Number Theory. Glendale, AZ: Erhus Uni- versity Press, 1994. Fleuren, M. "Smarandache Factors and Reverse Factors." Smarandache Notions J. 10,5/C1/8, 1999. Ibstedt, H. "Smarandache Concatenated Sequences." Ch. 5 in Computer Analysis of Number Sequences. Lupton, AZ: American Research Press, pp. 75 /C1/9, 1998. Marimutha, H. "Smarandache Concatenate Type Se- quences." Bull. Pure Appl. Sci. 16E, 225 /C1/26, 1997. Mudge, M. "Top of the Class." Personal Computer World, 674 /C1/75, June 1995. Mudge, M. "Not Numerology but Numeralogy!" Personal Computer World, 279 /C1/80, 1997. Rivera, C. "Problems & Puzzles: Puzzle Primes by Listing.- 008." http://www.primepuzzles.net/puzzles/puzz_008.htm. Sloane, N. J. A. Sequences A000422, A007908, A019518, A019519, A019520, A019521, A019522, A046035, and A046036 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993. Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101 /C1/07, 1996. Stephan, R. W. "Factors and Primes in Two Smarandache Sequences." Smarandache Notions J. 9,4/C1/0, 1998. Conservation of Number Principle A generalization of Poncelet’s CONTINUITY PRINCIPLE made by H. Schubert in 1874 /C1/9. The conservation of number principle asserts that the number of solutions of any determinate algebraic problem in any number of parameters under variation of the parameters is invariant in such a manner that no solutions become INFINITE . Schubert called the application of this technique the CALCULUS of ENUMERATIVE GEOMETRY . See also CONTINUITY PRINCIPLE ,DUALITY PRINCIPLE , HILBERT’S PROBLEMS References Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 340, 1945. Conservative Field The following conditions are equivalent for a con- servative VECTOR FIELD : 1. For any oriented simple closed curve C, the LINE INTEGRAL FC/ F /C215 ds /C300 :/ 2. For any two oriented simple curves C1and C2 with the same endpoints, fC1F /C215 ds /C30fC2F /C215 ds :/ 3. There exists a SCALAR POTENTIAL FUNCTION f such that F /C309f ; where 9 is the GRADIENT . 4. The CURL 9/C29F /C300:/ See also CURL,G RADIENT ,L INE INTEGRAL ,P OIN- CARE ´ ’S THEOREM ,P OTENTIAL FUNCTION ,V ECTOR FIELDConsistency The absence of CONTRADICTION (i.e., the ability to prove that a statement and its negative are both true) in an AXIOMATIC SYSTEM is known as consistency. See also AXIOMATIC SET THEORY ,AXIOMATIC SYSTEM , COMPLETE AXIOMATIC THEORY ,C ONSISTENCY STRENGTH ,GO¨ DEL’S INCOMPLETENESS THEOREM Consistency Strength If the CONSISTENCY of one of two propositions implies the CONSISTENCY of the other, the first is said to have greater consistency strength. Constant Any REAL NUMBER which is "significant" (or interest- ing) in some way. In this work, the term "constant" is generally reserved for REAL nonintegral numbers of interest, while "NUMBER " is reserved for interesting INTEGERS (e.g., BRUN’S CONSTANT , but BEAST NUM- BER). In contexts like LINEAR COMBINATION , the term "constant" is generally used to mean "SCALAR "or "REAL NUMBER ," and need not exclude integer values. Certain constants are known to many DECIMAL DIGITS and recur throughout many diverse areas of mathe- matics, often in unexpected and surprising places (e.g., PI, E, and to some extent, the EULER- MASCHER- ONI CONSTANT g): Other constants are more specia- lized and may be known to only a few DIGITS . S. Plouffe maintains a site about the computation and identification of numerical constants. Plouffe’s site also contains a page giving the largest number of DIGITS computed for the most common constants. S. Finch maintains a delightful, more expository site containing detailed essays and references on con- stants both common and obscure. The mathematician Glaisher remarked, "No doubt the desire to obtain the values of these quantities to a great many figures is also partly due to the fact that most of them are interesting in themselves; for e, p;g; 1n 2 ;and many other numerical quantities occupy a curious, and some of them almost a mysterious, place in mathematics, so that there is a natural tendency to do all that can be done towards their precise deter- mination" (Gourdon and Sebah). See also COEFFICIENT ,N UMBER ,R EAL NUMBER , SCALAR References Bailey, D. H. and Crandall, R. E. "On the Random Char- acter of Fundamental Constant Expansions." Manuscript, Mar. 2000. http://www.nersc.gov/~dhbailey/dhbpapers/dhbpapers.html. Borwein, J. and Borwein, P. A Dictionary of Real Numbers. London: Chapman & Hall, 1990. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/constant.html. Gourdon, X. and Sebah, P. "Mathematical Constants and Computation." http://xavier.gourdon.free.fr/Constants/ constants.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Plouffe, S. "Plouffe’s Inverter." http://www.lacim.uqam.ca/pi/ . Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u- qam.ca/pi/records.html. Robinson, H. P. and Potter, E. Mathematical Constants. Report UCRL-20418. Berkeley, CA: University of Califor- nia, 1971. Wells, D. W. The Penguin Dictionary of Curious and Inter- esting Numbers. Harmondsworth, England: Penguin Books, 1986. Constant Function A FUNCTION f(x) /C30c which does not change as its parameters vary. The GRAPH of a 1-D constant FUNCTION is a straight LINE. The DERIVATIVE of a constant FUNCTION c is d dx c /C300; (1) and the INTEGRAL is g cdx/C30cx : (2) The FOURIER TRANSFORM of the constant function f(x) /C301 is given by F[1] /C30g/C12 /C28/C12e /C282 pikx dx /C30 d(k) ; (3) where d(k) is the DELTA FUNCTION . See also FOURIER TRANSFORM–1 References Spanier, J. and Oldham, K. B. "The Constant Function c." Ch. 1 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 11 /C1/4, 1987. Constant Precession Curve CURVE OF CONSTANT PRECESSION Constant Problem Given an expression involving known constants, integration in finite terms, computation of limits, etc., determine if the expression is equal to ZERO . The constant problem is a very difficult unsolved problem in transcendental NUMBER THEORY . However, it isknown that the problem is UNDECIDABLE if the expression involves oscillatory functions such as SINE. However, the FERGUSON- FORCADE ALGORITHM is a practical algorithm for determining if there exist integers ai for given real numbers xi such that a1x1 /C27a2x2 /C27:::/C27anxn /C300 ; or else establish bounds within which no relation can exist (Bailey 1988). See also FERGUSON- FORCADE ALGORITHM ,HERMITE- LINDEMANN THEOREM ,INTEGER RELATION ,S CHA- NUEL’S CONJECTURE References Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving p;e, and Euler’s Constant." Math. Comput. 50, 275/C1/81, 1988. Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440/C1/48, 1999. Chen, Z.-Z. and Kao, M.-Y. Reducing Randomness via Irrational Numbers. 7 Jul 1999. http://xxx.lanl.gov/abs/ cs.DS/9907011/. Richardson, D. "The Elementary Constant Problem." In Proc. Internat. Symp. on Symbolic and Algebraic Compu-tation, Berkeley, July 27 /C1 /9, 1992 (Ed. P. S. Wang). ACM Press, 1992. Richardson, D. "How to Recognize Zero." J. Symb. Comp. 24, 627/C1/45, 1997. Sackell, J. "Zero-Equivalence in Function Fields Defined by Algebraic Differential Equations." Trans. Amer. Math. Soc. 336, 151/C1/71, 1993. Constant Width Curve CURVE OF CONSTANT WIDTH Constructible Number A number which can be represented by a FINITE number of ADDITIONS ,SUBTRACTIONS ,MULTIPLICA- TIONS ,DIVISIONS , and FINITE SQUARE ROOT extrac- tions of integers. Such numbers correspond to LINE SEGMENTS which can be constructed using only STRAIGHTEDGE and COMPASS . All RATIONAL NUMBERS are constructible, and all constructible numbers are ALGEBRAIC NUMBERS (Courant and Robbins 1996, p. 133). If a CUBIC EQUATION with rational coefficients has no rational root, then none of its roots is constructible (Courant and Robbins, p. 136). In particular, let F0be the FIELD of RATIONAL NUMBERS . Now construct an extension field F1of constructible numbers by the adjunction offfiffiffiffiffi k0p ; where k0is in F0;butffiffiffiffiffi k0p is not, consisting of all numbers OF THE FORM a0/C27b0ffiffiffiffiffi k0p ;where a0;b0/C23F0: Next, construct an extension field F2ofF1by the adjunction offfiffiffiffiffiffi K1p ;defined as the numbers a1/C27 b1ffiffiffiffiffi k1p ;where a1;b1/C23F1;and k1is a number in F1 for whichffiffiffiffiffiffi K1p does not lie in F1:Continue the process ntimes. Then constructible numbers are precisely those which can be reached by such a sequence of extension fields Fn;where nis a measure of the "complexity" of the construction (Courant and Rob- bins 1996). See also ALGEBRAIC NUMBER ,COMPASS ,CONSTRUC- TIBLE POLYGON ,E UCLIDEAN NUMBER ,R ATIONAL NUMBER ,STRAIGHTEDGE References Bold, B. "Achievement of the Ancient Greeks" and "An Analytic Criterion for Contractibility." Chs. 1 /C1/ in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 1 /C1/7, 1982. Courant, R. and Robbins, H. "Constructible Numbers and Number Fields." §3.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 127 /C1/34, 1996. Constructible Polygon COMPASS and STRAIGHTEDGE constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, ..., sides. However, this listing is not a complete enumeration of "constructible" polygons. A regular n- gon (/n ]3) can be constructed by STRAIGHTEDGE and COMPASS IFF n /C302kp1p2 /C1/C1/C1ps ; where k is in INTEGER ]0 and the piare distinct FERMAT PRIMES .FERMAT NUMBERS are OF THE FORM Fm /C3022m /C271 ; where m is an INTEGER ]0: The only known PRIMES of this form are 3, 5, 17, 257, and 65537. The fact that this condition was SUFFICIENT was first proved by Gauss in 1796 when he was 19 years old. That this condition was also NECESSARY was not explicitly proven by Gauss, and the first proof of this fact is credited to Wantzel (1836). See also COMPASS ,CONSTRUCTIBLE NUMBER ,CYCLO- TOMIC POLYNOMIAL ,FERMAT NUMBER ,G EOMETRIC CONSTRUCTION ,GEOMETROGRAPHY ,H EPTADECAGON , HEXAGON ,OCTAGON ,PENTAGON ,POLYGON ,SQUARE , STRAIGHTEDGE ,TRIANGLEReferences Bachmann, P. Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie. Leipzig, Germany: Teub- ner, 1872. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 94 /C1/6, 1987. Bold, B. "The Problem of Constructing Regular Polygons." Ch. 7 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 49 /C1/1, 1982. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 119, 1996. De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97 /C1/08, 1991. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352 /C1/86, 1955. Dixon, R. "Compass Drawings." Ch. 1 in Mathographics. New York: Dover, pp. 1 /C1/8, 1991. Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. Translated by A. A. Clarke. New Haven, CT: Yale University Press, 1965. Kazarinoff, N. D. "On Who First Proved the Impossibility of Constructing Certain Regular Polygons with Ruler and Compass Alone." Amer. Math. Monthly 75, 647 /C1/48, 1968. Klein, F. "The Division of the Circle into Equal Parts." Part I, Ch. 3 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 16 /C1/3, 1980. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 137 /C1/38, 1990. Wantzel, M. L. "Recherches sur les moyens de reconnaı ˆtre si un proble `me de ge´ome´trie peut se re´soudre avec la re`gle et le compas." J. Math. pures appliq. 1, 366 /C1/72, 1836. Construction BRAIKENRIDGE- MACLAURIN CONSTRUCTION ,C ON- STRUCTIBLE NUMBER ,C ONSTRUCTIBLE POLYGON , CONSTRUCTIVE DILEMMA ,G EOMETRIC CONSTRUC- TION ,HAUY CONSTRUCTION ,MASCHERONI CONSTRUC- TION ,M ATCHSTICK CONSTRUCTION ,N EUSIS CONSTRUCTION ,PALEY CONSTRUCTION ,STEINER CON- STRUCTION ,W YTHOFF CONSTRUCTION Constructive Dilemma A formal argument in LOGIC in which it is stated that (1)P[Qand R[S(where [means " IMPLIES "), and (2) either PorRis true, from which two statements it follows that either QorSis true. See also DESTRUCTIVE DILEMMA ,DILEMMA Contact Angle The ANGLE a between the normal vector of a SPHERE (or other geometric object) at a point where a PLANE is tangent to it and the normal vector of the plane. In the above figure, a /C30cos/C281a R ! /C30sin/C281R /C28 h R ! /C215 See also SPHERICAL CAP Contact Number KISSING NUMBER Contact Triangle The TRIANGLE formed by the points of intersection of a TRIANGLE T’s INCIRCLE with T. This is the PEDAL TRIANGLE of T with the INCENTER as the PEDAL POINT (cf., TANGENTIAL TRIANGLE ). The lines from the vertices of the contact triangle to the vertices of the original triangle CONCUR in the GERGONNE POINT . Furthermore, the contact triangle and TANGENTIAL TRIANGLE are perspective from the GERGONNE POINT . See also ADAMS’ CIRCLE ,G ERGONNE POINT ,PEDAL TRIANGLE ,SEVEN CIRCLES THEOREM ,T ANGENTIAL TRIANGLE References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/29, 1996.Contained Partition A PARTITION p is said to contain another partition q if the FERRERS DIAGRAM of p contains the FERRERS DIAGRAM of q. For example, f3; 3; 2g (left figure) contains both f3; 3; 1g and f3; 3; 2g (right figures). YOUNG’S LATTICE YPis the PARTIAL ORDER of parti- tions contained within p ordered by containment (Skiena 1990, p. 77). See also PARTITION ,YOUNG’S LATTICE References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Contained Pattern A subset t /C23 Sn of a permutation f1 ; ...; ng is said to contain a /C23 Skif there exist 1 5i1 B...Bik 5n such that t /C30( ti ; ... ; tk)is ORDER ISOMORPHIC to a /C30 ( a1 ; ...; ak): Here, Snis the SYMMETRIC GROUP on n elements. In other words, t contains a IFF any K-SUBSET of t is ORDER ISOMORPHIC to a:/ See also AVOIDED PATTERN ,O RDER ISOMORPHIC , PERMUTATION PATTERN ,W ILF CLASS,W ILF EQUIVA- LENT References Mansour, T. Permutations Avoiding a Pattern from Skand at Least Two Patterns from S3 : 31 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007194/. Content The content of a POLYTOPE or other n-dimensional object is its generalized VOLUME (i.e., its "hypervo- lume"). Just as a three-dimensional object has VO- LUME , SURFACE AREA , and GENERALIZED DIAMETER ,an n-dimensional object has "measures" of order 1, 2, ..., n. The content of an integer polynomial P /C23Z(x) ; denoted cont( P) ; is the largest integer k ]1 such that P=k also has integer coefficients. Gauss’s lemma for contents states that if P and Q are two polynomials with integer coefficients, then cont( PQ)/C30cont( P)cont( Q) (Se´roul 2000, p. 287). See also POLYNOMIAL ,VOLUME References Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 287, 2000. Contests MATHEMATICS CONTESTS Contiguous Function A HYPERGEOMETRIC FUNCTION in which one para- meter changes by /C271or/C281 is said to be contiguous. There are 26 functions contiguous to2F1(a ; b; c; x) taking one pair at a time. There are 325 taking two or more pairs at a time. See Abramowitz and Stegun (1972, pp. 557 /C1/58). See also HYPERGEOMETRIC FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Contingency A SENTENCE is called a contingency if its TRUTH TABLE contains at least one ‘T’ and at least one ‘F.’ See also CONTRADICTION ,TAUTOLOGY ,TRUTH TABLE References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 13, 1958. Continued Fraction A "general" continued fraction representation of a REAL NUMBER xisOF THE FORM x/C30a0/C27b1 a1/C27b2 a2/C27b3 a3/C27...; (1) which can be written x/C30a0/C27b1 a1/C27b2 a2/C27/C1/C1/C1 /C215 (2) An archaic word for a continued fraction is ANTHY- PHAIRETIC RATIO . The SIMPLE CONTINUED FRACTION representation of a number x(which is usually what is meant when the term "continued fraction" is used without qualifica- tion) is given by x/C30a0/C271 a1/C271 a2/C271 a3/C27...; (3) which can be written in a compact abbreviatedNOTATION as x/C30[a0;a1;a2;a3;... ]: (4) Some care is needed, since some authors begin indexing the terms at a1instead of a0;causing the parity of certain fundamental results in continued fraction theory to be reversed. Starting the indexing with a0; a0/C30xbc (5) is the integral part of x, where xbcis the FLOOR FUNCTION , a1/C301 x/C28a0$% (6) is the integral part of the RECIPROCAL ofx/C28a0; a2/C301 1 x/C28a0/C28a166666647777775(7) is the integral part of the reciprocal of the remainder, etc. Writing the remainders according to the RECUR- RENCE RELATION r0/C30x (8) rn/C301 rn/C281/C28an/C281(9) gives the concise formula an/C30rnbc : (10) The quantities anare called PARTIAL QUOTIENTS , and the quantity obtained by including nterms of the continued fraction cn/C30pn qn/C30[a0;a1;...;an] /C30a0/C271 a1/C271 a2/C271 .../C271 an(11) is called the nthCONVERGENT . For example, consider the computation of the continued fraction of p;given byp/C30[3;7;15;1;292;1;1;... ]:/ Term Value PQs Convergent Value /a0// pbc/C303// [3]// 3/ 3.00000 /a1//1 p/C283jk /C307// [3;7]//22 7/ 3.14286 /a2///C281 1 p/C283/C287/C29/C3015//[3;7;15] //333 106/ 3.14151 Continued fractions provide, in some sense, a series of "best" estimates for an IRRATIONAL NUMBER . Func- tions can also be written as continued fractions, providing a series of better and better rational approximations. Continued fractions have also proved useful in the proof of certain properties ofnumbers such as Eandp(PI). Because irrationals which are square roots of RATIONAL NUMBERS have periodic continued fractions, an exact representationfor a tabulated numerical value (i.e., 1.414... forP YTHAGORAS’S CONSTANT ,ffiffiffi 2p ) can sometimes be found if it is suspected to represent an unknown QUADRATIC SURD . Continued fractions are also useful for finding near commensurabilities between events with different periods. For example, the Metonic cycle used for calendrical purposes by the Greeks consists of 235lunar months which very nearly equal 19 solar years, and 235/19 is the sixth CONVERGENT of the ratio of the lunar phase (synodic) period and solar period (365.2425/29.53059). Continued fractions can also be used to calculate gear ratios, and were used for this purpose by the ancient Greeks (Guy 1990). LetPn=Qnbe convergents of a nonsimple continued fraction. Then P/C281/C131Q/C281/C130 (12) P0/C13a0Q0/C131 (13) and subsequent terms are calculated from the RECUR- RENCE RELATIONS Pj/C30ajPj/C281/C27bjPj/C282 (14) Qj/C30ajQj/C281/C27bjQj/C282 (15) forj/C301, 2, ..., n. It is also true that PnQn/C281/C28Pn/C281Qn/C30(/C281)n/C281Yn k/C301bk: (16) The error in approximating a number by a given CONVERGENT is roughly the MULTIPLICATIVE INVERSE of the square of the DENOMINATOR of the first neglected term.Afinite simple continued fraction representation terminates after a finite number of terms. To "round" a continued fraction, truncate the last term unless itis91, in which case it should be added to the previous term (Gosper 1972, Item 101A). To takeone over a continued fraction, add (or possibly delete)an initial 0 term. To negate, take the NEGATIVE of all terms, optionally using the identity [/C28a;/C28b;/C28c;/C28d;... ] /C30[/C28a/C281;1;b/C281;c;d;... ]: (17) A particularly beautiful identity involving the termsof the continued fraction is [a0;a1;...;an] [a0;a1;...;an/C281]/C30[an;an/C281;...;a1;a0] [an;an/C281;...;a1]/C215 (18) Finite simple fractions represent rational numbersand all rational numbers are represented by finitecontinued fractions. There are two possible represen- tations for a finite simple fraction: [a 0;...;an] /C30[a0;...;an/C281;an/C281;1] for an/C211 [a0;...;an/C282;an/C281/C271] for an/C301l12) (19) On the other hand, an infinite simple fraction represents a unique IRRATIONAL NUMBER , and each IRRATIONAL NUMBER has a unique infinite continued fraction. Consider the CONVERGENTS cn/C30pn=qnof a simple continued fraction, and define p/C282/C130q/C282/C131 (20) p/C281/C131q/C281/C130 (21) p0/C13a0q0/C131: (22) Then subsequent terms can be calculated from the RECURRENCE RELATIONS pn/C30anpn/C281/C27pn/C282 (23) qn/C30anqn/C281/C27qn/C282: (24) The CONTINUED FRACTION FUNDAMENTAL RECUR- RENCE RELATION forsimple continued fractions is pnqn/C281/C28pn/C281qn/C30(/C281)n/C271: (25) It is also true that if a0"0; pn pn/C281/C30[an;an/C281;...;a0] (26) qn qn/C281/C30[an;...;a1]: (27) Furthermore, pn qn/C30pn/C271/C28pn/C281 qn/C271/C28qn/C281: (28) Also, if a convergent cn/C30pn=qn>1;then qn pn/C30[0;a0;a1;...;an]: (29) Similarly, if cn/C30pn=qnB1;then a0/C300 and qn pn/C30[a1;...;an]: (30) The convergents cn/C30pn=qnalso satisfy cn/C28cn/C281/C30(/C281)n/C271 qnqn/C281(31) cn/C28cn/C282/C30an(/C281)n qnqn/C282: (32) Plotted above on semilog scales are cn/C28p(neven; left figure) and p/C28cn(nodd; right figure) as a function of nfor the convergents of p:In general, the EVEN convergents c2n/C271of an infinite simple continued fraction for a number xform an INCREASING SE- QUENCE , and the ODD convergents c2nform a DE- CREASING SEQUENCE (so any EVEN convergent is less than any ODD convergent). Summarizing, c0Bc2Bc4B/C1/C1/C1Bc2n/C282Bc2nB/C1/C1/C1Bx (33) xB/C1/C1/C1Bc2n/C271Bc2n/C281Bc5Bc3Bc1: (34) Furthermore, each convergent for n]3 lies between the two preceding ones. Each convergent is nearer to the value of the infinite continued fraction than theprevious one. In addition, for a number x/C30 [a 0;a1;... ]; 1 (an/C271/C272)q2 nBx/C28pn qnl112l112l112l112l112l112l112l112l112l112B 1 an/C271q2 n: (35) The SQUARE ROOT of a SQUAREFREE INTEGER has a periodic continued fraction OF THE FORM ffiffiffinp/C30[a0;a1;...;an;2a0] (36) (Rose 1994, p. 130). Furthermore, if Dis not a SQUARE NUMBER , then the terms of the continued fraction offfiffiffiffi Dp satisfy 0BanB2ffiffiffiffi Dp : (37) In particular, [¯a]/C30a/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C274p 2(38) [1;¯a]/C30/C281/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274ap 2(39) [a;2a]/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C271p (40) [a;b]/C30abffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ab/C27(ab(ab/C274)p 2b(41)[a1;...;an] /C30/C28(qn/C281/C28pn)/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (qn/C281/C28pn)2/C274qnpn/C281q 2qn(42) [a0;b1;...;bn]/C30a0/C271 [b1;...;bn](43) [b1;...;bn]/C30[b1;...;bn]pn/C27pn/C281 [b1;...;bn]qn/C27qn/C281: (44) The first follows from a/C30n/C271 n/C271 n/C271 n/C27... /C30n/C271 n/C271 n/C271 n/C27...0 BBB@1 CCCA: (45) Therefore, a/C28n/C301 n/C271 n/C271 n/C27...; (46) so plugging (46) into (45) gives a/C30n/C271 n/C27(a/C28n)/C30n/C271 a: (47) Expanding a2/C28na/C281/C300; (48) and solving using the QUADRATIC FORMULA gives a/C30n/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2/C274p 2: (49) The analog of this treatment in the general case gives a/C30apn/C27pn/C281 aqn/C27qn/C281: (50) The following table gives the repeating simple con- tinued fractions for the square roots of the first few integers (excluding the trivial SQUARE NUMBERS ). N /affiffiffiffiffi Np / N /affiffiffiffiffiNp / 2 /[1;¯2]/ 22 /[4;1;2;4;2;1;8]/ 3 /[1;1;2]/ 23 /[4;1;3;1;8]/ 5 /[2;4]/ 24 /[4;1;8]/ 6 /[2;2;4]/ 26 /[5;10] / 7 /[2;1;1;1;4]/ 27 /[5;5;10] / 8 /[2;1;4]/ 28 /[5;3;2;3;10] / 10 /[3;¯6]/ 29 /[5;2;1;1;2;10] / 11 /[3;3;6]/ 30 /[5;2;10] / 12 /[3;2;6]/ 31 /[5;1;1;3;5;3;1;1;10] / 13 /[3;1;1;1;1;6]/ 32 /[5;1;1;1;10] / 14 /[3;1;2;1;6]/ 33 /[5;1;2;1;10] / 15 /[3;1;6]/ 34 /[5;1;4;1;10] / 17 /[4;¯8]/ 35 /[5;1;10] / 18 /[4;4;8]/ 37 /[6;12] / 19 /[4;2;1;3;1;2;8]/38 /[6;6;12] / 20 /[4;2;8]/ 39 /[6;4;12] / 21 /[4;1;1;2;1;1;8]/40 /[6;3;12] / The periods of the continued fractions of the square roots of the first few nonsquare integers 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, ... (Sloane’s A000037) are 1, 2, 1, 2, 4, 2, 1, 2, 2, 5, ... (Sloane’s A013943; Williams 1981,Jacobson et al. 1995). An upper bound for the length is roughly O(lnDffiffiffiffi Dp ): / An even stronger result is that a continued fraction is periodic IFFit is a ROOT of a quadratic POLYNOMIAL . Calling the portion of a number xremaining after a given convergent the "tail," it must be true that therelationship between the number xand terms in its tail is OF THE FORM x/C30ax/C27b cd/C27d; (51) which can only lead to a QUADRATIC EQUATION . LOGARITHMS logb0b1can be computed by defining b2; ... and the POSITIVE INTEGER n1;...such that bn1 1Bb0Bbn1/C271 1 b2/C30b0 bn1 1(52) bn2 2Bb1Bbn2/C271 2 b3/C30b1 bn2 2(53) and so on. Then logb0b1/C30[n1;n2;n3; :::]: (54) A geometric interpretation for a reduced FRACTION y=xconsists of a string through a LATTICE of points with ends at (1 ;0) and ( x, y) (Klein 1907, 1932; Steinhaus 1983, p. 40; Gardner 1984, pp. 210 /C1/11, Ball and Coxeter 1987, pp. 86 /C1/7; Davenport 1992). This interpretation is closely related to a similar one for the GREATEST COMMON DIVISOR . The pegs it presses against ( xi;yi) give alternate CONVERGENTS yi=xi;while the other CONVERGENTS are obtained from the pegs it presses against with the initial end at (0;1):The above plot is for e/C282;which has con- vergents 0, 1, 2/3, 3/4, 5/7, .... Let the continued fraction for xbe written [a0;a1; :::; an]:Then the limiting value is almost always KHINTCHINE’S CONSTANT K/C13lim n0/C12(a1a2...an)1=n/C302:68545 . . . : (55) Continued fractions can be used to express the POSITIVE ROOTS of any POLYNOMIAL equation. Con- tinued fractions can also be used to solve linear DIOPHANTINE EQUATIONS and the P ELL EQUATION . Euler showed that if a CONVERGENT SERIES can be written in the form c1/C27c1c2/C27c1c2c3/C27...; (56) then it is equal to the continued fraction c1 1/C28c2 1/C27c2/C28c3 1/C27c3/C28...: (57) Gosper has invented an ALGORITHM for performing analytic ADDITION ,SUBTRACTION ,MULTIPLICATION , and DIVISION using continued fractions. It requires keeping track of eight INTEGERS which are concep- tually arranged at the VERTICES of a CUBE . Although this ALGORITHM has not appeared in print, similar algorithms have been constructed by Vuillemin(1987) and Liardet and Stambul (1998). Gosper’s algorithm for computing the continued frac- tion for ( ax/C27b)=(cx/C27d) from the continued fraction for x is described by Gosper (1972), Knuth (1981, Exercise 4.5.3.15, pp. 360 and 601), and Fowler (1999). (In line 9 of Knuth’s solution, Xk 1 A=C bc should be replaced by Xk 1 min A=C bc ; ð / / (A /C27B)=(C /C27D) bc Þ :/) Gosper (1972) and Knuth (1981) also mention the bivariate case (axy /C27bx/ //C27cy /C27d) =(Axy /C27Bx /C27Cy /C27D) :/ Ramanujan developed a number of interesting closed- form expressions for continued fractions, including 1 1 /C27e /C282 p 1 /C27e /C284 p 1 /C27 .../C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffi 5p 2s /C28ffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27 1p 2"# e2 p =5(58) 1 1/C27e /C282 pffiffi 5p 1 /C27e /C284 pffiffi 5p 1 /C27 ... /C30ffiffiffi 5p 1 /C27 53 =4ffiffiffi5p /C28 1 2 ! 5 =2 /C2812 435/C28ffiffiffi 5p /C27 1 28 >>>>>< >>>>>:9 >>>>>= >>>>>;e 2 p =ffiffi 5p (59) and 4g/C12 0xe /C282ffiffi 5p cosh xdx /C301 2[ z(2;14(1 /C27ffiffiffi 5p )) /C28 z(2;1 4(3/C27ffiffiffi 5p )] /C301 1/C2712 1/C2712 1/C2722 1/C2722 1/C2732 1/C2732 1/C27(60) (Watson 1929; Preece 1931; Watson 1931; Hardy 1999, p. 8). See also GAUSSIAN BRACKETS ,HURWITZ’S IRRATIONAL NUMBER THEOREM ,K HINTCHINE’S CONSTANT ,L A- GRANGE’S CONTINUED FRACTION THEOREM ,L AME´ ’S THEOREM ,L EHMER CONTINUED FRACTION ,L E´ VY CONSTANT ,L OCHS THEOREM ,P ADE´APPROXIMANT , PARTIAL QUOTIENT ,PI,QUADRATIC IRRATIONAL NUM- BER,Q UOTIENT- DIFFERENCE ALGORITHM ,R OGERS- RAMANUJAN CONTINUED FRACTION ,SEGRE’S THEO- REM,TROTT’S CONSTANT References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 19, 1972. Acton, F. S. "Power Series, Continued Fractions, and Ra- tional Approximations." Ch. 11 in Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., 1990. Adamchik, V. "Limits of Continued Fractions and Nested Radicals." Mathematica J. 2,5 4/C1/7, 1992. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 54 /C1/7 and 86/C1/7, 1987. Berndt, B. C. and Gesztesy, F. (Eds.). Continued Fractions: From Analytic Number Theory to Constructive Approxi-mation, A Volume in Honor of L.J. Lange. Providence, RI: Amer. Math. Soc., 1999. Beskin, N. M. Fascinating Fractions. Moscow: Mir Publish- ers, 1980. Brezinski, C. History of Continued Fractions and Pade ´ Approximants. New York: Springer-Verlag, 1980. Conway, J. H. and Guy, R. K. "Continued Fractions." In The Book of Numbers. New York: Springer-Verlag, pp. 176 /C1/ 79, 1996. Courant, R. and Robbins, H. "Continued Fractions. Dio- phantine Equations." §2.4 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 49 /C1/1, 1996. Davenport, H. §IV.12 in The Higher Arithmetic: An Intro- duction to the Theory of Numbers, 6th ed. New York: Cambridge University Press, 1992. Dunne, E. and McConnell, M. "Pianos and Continued Fractions." Math. Mag. 72, 104/C1/15, 1999. Euler, L. Introduction to Analysis of the Infinite, Book I. New York: Springer-Verlag, 1980. Fowler, D. H. The Mathematics of Plato’s Academy: A New Reconstruction, 2nd ed. Oxford, England: Oxford Univer- sity Press, 1999. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 210 /C1/11, 1984. Gosper, R. W. Item 101a in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 37 /C1/9, Feb. 1972. Gosper, R. W. Item 101b in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 39 /C1/4, Feb. 1972. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Conti- nuants." §6.7 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 301 /C1/09, 1994. Guy, R. K. "Continued Fractions" §F20 in Unsolved Pro- blems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 259, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Jacobson, M. J. Jr.; Lukes, R. F.; and Williams, H. C. "An Investigation of Bounds for the Regulator of QuadraticFields." Experiment. Math. 4, 211/C1 /25, 1995. Khinchin, A. Ya. Continued Fractions. New York: Dover, 1997. Kimberling, C. "Continued Fractions." http://cedar.evansvil- le.edu/~ck6/integer/contfr.html. Klein, F. Ausgewa ¨hlte Kapitel der Zahlentheorie I. Go¨ttin- gen, Germany: n.p., 1896. Klein, F. Elementary Number Theory. New York, p. 44, 1932. Kline, M. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, p. 316, 1998. Liardet, P. and Stambul, P. "Algebraic Computation with Continued Fractions." J. Number Th. 73,9 2/C1/21, 1998. Lorentzen, L. and Waadeland, H. Continued Fractions with Applications. Amsterdam, Netherlands: North-Holland, 1992. Moore, C. D. An Introduction to Continued Fractions. Washington, DC: National Council of Teachers of Mathe- matics, 1964. Olds, C. D. Continued Fractions. New York: Random House, 1963. Perron, O. Die Lehre von Kettenbru ¨chen, 3. verb. und erweiterte Aufl. Stuttgart, Germany: Teubner, 1954 /C1/7. Pettofrezzo, A. J. and Bykrit, D. R. Elements of Number Theory. Englewood Cliffs, NJ: Prentice-Hall, 1970. Preece, C. T. "Theorems Stated by Ramanujan (X)." J. London Math. Soc. 6,22/C1/2, 1931. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Evaluation of Continued Fractions." §5.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 163 /C1/67, 1992. Riesel, H. "Continued Fractions." Appendix 8 in Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 327 /C1/42, 1994. Rockett, A. M. and Szu¨sz, P. Continued Fractions. New York: World Scientific, 1992. Rose, H. E. A Course in Number Theory, 2nd ed. Oxford, England: Oxford University Press, 1994. Rosen, K. H. Elementary Number Theory and Its Applica- tions. New York: Addison-Wesley, 1980. Schur, I. "Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbru ¨che." Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Klasse , pp. 302 /C1/21, 1917. Sloane, N. J. A. Sequences A000037/M0613 and A013943 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 39 /C1/2, 1999. Van Tuyl, A. L. "Continued Fractions." http://www.calvi- n.edu/academic/math/confrac/. Vuillemin, J. "Exact Real Computer Arithmetic with Con- tinued Fractions." INRIA Report 760. Le Chasny, France: INRIA, Nov. 1987. http://www.inria.fr/RRRT/RR- 0760.html. Wagon, S. "Continued Fractions." §8.5 in Mathematica in Action. New York: W. H. Freeman, pp. 263 /C1/71, 1991. Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948. Watson, G. N. "Theorems Stated by Ramanujan (VII): Theorems on a Continued Fraction." J. London Math. Soc. 4,39/C1/8, 1929. Watson, G. N. "Theorems Stated by Ramanujan (IX): Two Continued Fractions." J. London Math. Soc. 4, 231 /C1/37, 1929. Weisstein, E. W. "Books about Continued Fractions." http:// www.treasure-troves.com/books/ContinuedFrac- tions.html. Williams, H. C. "A Numerical Investigation into the Length of the Period of the Continued Fraction Expansion offfiffiffiffi Dp :/" Math. Comp. 36, 593 /C1/01, 1981. Continued Fraction Constant A continued fraction with partial quotients which increase in ARITHMETIC PROGRESSION is [A /C27D; A /C272D ; A /C273D ; ...]/C30IA =D2 D ! I1 /C27A =D2 D ! ; where In(x)isa MODIFIED BESSEL FUNCTION OF THE FIRST KIND (Schroeppel 1972). A special case isC /C300 /C271 1 /C271 2 /C271 3 /C271 4 /C271 5 /C27 ...; which has the value C /C30I1(2) I0(2) /C300 :697774658... (Lehmer 1973, Rabinowitz 1990). See also E,GOLDEN RATIO,M ODIFIED BESSEL FUNC- TION OF THE FIRST KIND,P I,R ABBIT CONSTANT , THUE- MORSE CONSTANT References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html. Guy, R. K. "Review: The Mathematics of Plato’s Academy." Amer. Math. Monthly 97, 440 /C1/43, 1990. Lehmer, D. H. "Continued Fractions Containing Arithmetic Progressions." Scripta Math. 29,17/C1/4, 1973. Rabinowitz, S. Problem E3264. "Asymptotic Estimates from Convergents of a Continued Fraction." Amer. Math. Monthly 97, 157 /C1/59, 1990. Schroeppel, R. Item 99 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 36, Feb. 1972. Continued Fraction Factorization Algorithm A PRIME FACTORIZATION ALGORITHM which uses RE- SIDUES produced in the CONTINUED FRACTION offfiffiffiffiffiffiffiffiffi mNp for some suitably chosen m to obtain a SQUARE NUMBER . The ALGORITHM solves x2 /C13y2 (mod n) by finding an m for which m2(mod n) has the smallest upper bound. The method requires (by conjecture) about exp(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lnn ln ln np ) steps, and was the fastest PRIME FACTORIZATION ALGORITHM in use before the QUADRATIC SIEVE , which eliminates the 2 under the SQUARE ROOT (Pomerance 1996), was developed. See also EXPONENT VECTOR ,PRIME FACTORIZATION ALGORITHMS References Morrison, M. A. and Brillhart, J. "A Method of Factoring and the Factorization of F7:/"Math. Comput. 29, 183/C1/05, 1975. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/485, 1996. Continued Fraction Fundamental Recurrence Relation For a SIMPLE CONTINUED FRACTION x/C30[a0;a1;... ] with CONVERGENTS pn=qn;the fundamental RECUR- RENCE RELATION is given by pnqn/C281 /C28pn /C281qn /C30(/C281)n/C271 : See also SIMPLE CONTINUED FRACTION ,CONTINUED FRACTION References Olds, C. D. Continued Fractions. New York: Random House, p. 27, 1963. Continued Fraction Map f(x) /C301 x /C281 x$% for x /C23 [0; 1]; where xbcis the FLOOR FUNCTION . The NATURAL INVARIANT of the map is r(y) /C301 (1 /C27 y)ln2: References Beck, C. and Schlo¨gl, F. Thermodynamics of Chaotic Systems. Cambridge, England: Cambridge University Press, pp. 194 /C1/95, 1995. Continued Fraction Unit Fraction Algorithm An algorithm for computing a UNIT FRACTION , called the FAREY SEQUENCE method by Bleicher (1972).References Bleicher, M. N. "A New Algorithm for the Expansion of Continued Fractions." J. Number Th. 4, 342 /C1/82, 1972. Eppstein, D. Egypt.ma Mathematica notebook. http:// www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma. Continued Square Root NESTED RADICAL Continued Vector Product VECTOR TRIPLE PRODUCT Continuity The property of being CONTINUOUS . See also CONTINUITY AXIOMS ,CONTINUITY CORREC- TION ,CONTINUITY PRINCIPLE ,CONTINUOUS DISTRIBU- TION ,CONTINUOUS FUNCTION ,CONTINUOUS SPACE , FUNDAMENTAL CONTINUITY THEOREM ,LIMIT References Kaplan, W. "Limits and Continuity." §2.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 82 /C1/ 6, 1992. Smith, W. K. Limits and Continuity. New York: Macmillan, 1964. Continuity Axioms "The" continuity axiom is an additional AXIOM which must be added to those of Euclid’s ELEMENTS in order to guarantee that two equal CIRCLES of RADIUS r intersect each other if the separation of their centers is less than 2r (Dunham 1990). The continuity axioms are the three of HILBERT’S AXIOMS which concern geometric equivalence. ARCHIMEDES’ LEMMA is sometimes also known as "the continuity axiom." See also CONGRUENCE AXIOMS ,H ILBERT’S AXIOMS , INCIDENCE AXIOMS ,O RDERING AXIOMS ,P ARALLEL POSTULATE References Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 38, 1990. Hilbert, D. The Foundations of Geometry. Chicago, IL: Open Court, 1980. Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert’s System of Axioms." §163B in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 544 /C1/45, 1980. Continuity Correction A correction to a discrete BINOMIAL DISTRIBUTION to approximate a continuous distribution. P(a5X5b):Pa/C281 2/C28npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np(1/C28p)p 5z5b/C271 2/C28npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np(1/C28p)p ! ; where z /C13(x /C28 m) s is a continuous variate with a NORMAL DISTRIBUTION and X is a variate of a BINOMIAL DISTRIBUTION . See also BINOMIAL DISTRIBUTION ,NORMAL DISTRIBU- TION References Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, p. 87, 1993. Continuity Principle The metric properties discovered for a primitive figure remain applicable, without modifications other than changes of signs, to all correlative figures which can be considered to arise from the first. As stated by Lachlan (1893), the principle states that if, from the nature of a particular problem, a certain number of solutions are expected (and are, in fact, found in any one case), then there will be the same number of solutions in all cases, although some solutions may be imaginary. For example, two circles intersect in two points, so it can be stated that every two circles intersect in two points, although the points may be imaginary or may coincide. The principle is extremely powerful (if somewhat difficult to state precisely), and allows immediate derivation of some geometric propositions from other propositions which may appear simpler and may be substantially easier to prove. The continuity principle was first enunciated by Kepler and thereafter enunciated by Boscovich. How- ever, it was not generally accepted until formulated by Poncelet in 1822. Formally, it amounts to the statement that if an analytic identity in any finite number of variables holds for all real values of the variables, then it also holds by ANALYTIC CONTINUA- TION for all complex values (Bell 1945). This principle is also called "Poncelet’s continuity principle," or sometimes the "permanence of mathematical rela- tions principle" (Bell 1945). See also ANALYTIC CONTINUATION ,CONSERVATION OF NUMBER PRINCIPLE ,D UALITY PRINCIPLE ,P ERMA- NENCE OF ALGEBRAIC FORM References Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 340, 1945. Lachlan, R. "The Principle of Continuity." §8in An Elemen- tary Treatise on Modern Pure Geometry. London: Macmil- lian, pp. 4 /C1/, 1893. Poncelet, J.-V. Traite ´ des Proprie ´te´s Projectives. 1822.Continuous A general mathematical property obeyed by mathe- matical objects in which all elements are within a NEIGHBORHOOD of nearby points. The continuous maps between TOPOLOGICAL SPACES form a CATE- GORY . The designation "continuous" is sometimes used to indicate membership in this category. See also ABSOLUTELY CONTINUOUS ,C ONTINUOUS DISTRIBUTION ,CONTINUITY ,CONTINUOUS FUNCTION , CONTINUOUS SPACE ,DIFFERENTIABLE ,JUMP,PIECE- WISE CONTINUOUS References Jeffreys, H. and Jeffreys, B. S. "Limits of Functions: Con- tinuity." §1.06 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 17 /C1/3, 1988. Continuous Distribution A STATISTICAL DISTRIBUTION for which the variables may take on a continuous range of values. Abramo- witz and Stegun (1972, p. 930) give a table of the parameters of most common continuous distributions. See also BETA DISTRIBUTION ,BIVARIATE DISTRIBU- TION ,CAUCHY DISTRIBUTION ,CHI DISTRIBUTION ,CHI- SQUARED DISTRIBUTION ,CORRELATION COEFFICIENT , DISCRETE DISTRIBUTION ,DOUBLE EXPONENTIAL DIS- TRIBUTION ,E QUALLY LIKELY OUTCOMES DISTRIBU- TION ,EXPONENTIAL DISTRIBUTION ,EXTREME VALUE DISTRIBUTION , F-DISTRIBUTION ,FERMI- DIRAC DISTRI- BUTION ,F ISHER’S Z-DISTRIBUTION ,F ISHER- TIPPETT DISTRIBUTION ,GAMMA DISTRIBUTION ,GAUSSIAN DIS- TRIBUTION ,H ALF-NORMAL DISTRIBUTION ,L APLACE DISTRIBUTION ,LATTICE DISTRIBUTION ,LE´ VY DISTRI- BUTION ,L OGARITHMIC DISTRIBUTION ,L OG-SERIES DISTRIBUTION ,LOGISTIC DISTRIBUTION ,LORENTZIAN DISTRIBUTION ,MAXWELL DISTRIBUTION ,NORMAL DIS- TRIBUTION ,PARETO DISTRIBUTION ,PASCAL DISTRIBU- TION ,P EARSON TYPE III DISTRIBUTION ,P OISSON DISTRIBUTION ,PO´ LYA DISTRIBUTION ,RATIO DISTRIBU- TION ,RAYLEIGH DISTRIBUTION ,RICE DISTRIBUTION , SNEDECOR’S F-DISTRIBUTION ,STUDENT’S T-DISTRIBU- TION ,STUDENT’S Z-DISTRIBUTION ,UNIFORM DISTRIBU- TION ,W EIBULL DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 927 and 930, 1972. Evans, M.; Hastings, N.; and Peacock, B. Statistical Dis- tributions, 3rd ed. New York: Wiley, 2000. Kotz, S.; Balakrishnan, N.; and Johnson, N. L. Continuous Multivariate Distributions, Vol. 1: Models and Applica-tions, 2nd ed. New York: Wiley, 2000. McLaughlin, M. "Common Probability Distributions." http:// www.geocities.com/~mikemclaughlin/math_stat/Dists/Compendium.html. Continuous Function There are several commonly used methods of defining the slippery, but extremely important, concept of a continuous function. The space of continuous func- tions is denoted C0 ; and corresponds to the k /C300 case of a C-K FUNCTION . A continuous function can be formally defined as a FUNCTION f : X 0 Y where the pre-image of every OPEN SET in Y is OPEN in X. More concretely, a function f(x) in a single variable x is said to be continuous at point x0 if 1. f(x0) is defined, so that x0 is in the DOMAIN of f. 2. limx 0x0f(x) exists for x in the DOMAIN of f. 3. limx 0x0f(x) /C30f(x0) ;/ where lim denotes a LIMIT . Many mathematicians prefer to define the continuity of a function via a so-called EPSILON-DELTA DEFINI- TION of a LIMIT . In this formalism, a LIMIT c of function f(x)asx approaches a point x0 ; lim x0x0f(x) /C30c ; (1) is defined when, given any e > 0; a d > 0 can be found such that for every x in some domain D and within the neighborhood of x0 of radius d (except possibly x0 itself), f(x) /C28c jj B e: (2) Then if x0 is in D and lim x 0x0f(x) /C30f(x0) /C30c ; (3) /f(x) is said to be continuous at x0 :/ If f is DIFFERENTIABLE at point x0 ; then it is also continuous at x0 : If two functions f and g are continuous at x0 ; then 1. f /C27g is continuous at x0 :/ 2. f /C28g is continuous at x0 :/ 3. f /C29g is continuous at x0 :/ 4. f }g is continuous at x0if g(x0) "0 and is discontinuous at x0 if g(x0) /C300:/ 5. f(g is continuous at x0 ; where f(g denotes f(g(x)); the COMPOSITION of the functions f and g. The notion of continuity for a function in two vari- ables is slightly trickier, as illustrated above by the plot of the function z /C30x2 /C28 y2 x2 /C27 y2 : (4) This function is discontinuous at the origin, but has limit 0 along the line x /C30y, limit 1 along the X-AXIS , and limit /C281 along the Y-AXIS (Kaplan 1992, p. 83). See also C-K FUNCTION ,CONTINUOUSLY DIFFERENTI- ABLE FUNCTION ,CRITICAL POINT ,D IFFERENTIABLE , LIMIT,NEIGHBORHOOD ,PIECEWISE CONTINUOUS ,STA- TIONARY POINT References Bartle, R. G. and Sherbert, D. Introduction to Real Analysis. New York: Wiley, p. 141, 1991. Kaplan, W. "Limits and Continuity." §2.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 82 /C1/ 6, 1992. Continuous Group A GROUP having CONTINUOUS group operations. A continuous group is necessarily infinite, since an INFINITE GROUP just has to contain an infinite number of elements. But some infinite groups, such as the integers or rationals, are not continuous groups. See also DISCRETE GROUP ,FINITE GROUP ,INFINITE GROUP Continuous Space A TOPOLOGICAL SPACE . See also NET Continuous Transformation HOMEOMORPHISM Continuous Vector Bundle A continuous vector bundle is a VECTOR BUNDLE p: E0Mwith only the structure of a TOPOLOGICAL MANIFOLD . The map p is CONTINUOUS . It has no SMOOTH STRUCTURE or METRIC . See also BUNDLE ,M ANIFOLD ,M ETRIC (BUNDLE ), VECTOR BUNDLE Continuously Differentiable Function The space of continuously differentiable functions is denoted C1 ; and corresponds to the k /C301 case of a C-K FUNCTION . See also C-K FUNCTION ,CONTINUOUS FUNCTION References Krantz, S. G. "Continuously Differential and Ck Functions" and "Differentiable and Ck Curves." §1.3.1 and 2.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 12 /C1/3 and 21, 1999. Continuum The nondenumerable set of REAL NUMBERS , denoted C. It satisfies /C210/C27C /C30C (1) and Cr /C30C ; (2) where /C2100 is ALEPH-0 . It is also true that /C210/C2100 0/C30C : (3) However, CC /C30F (4) is a SET larger than the continuum. Paradoxically, there are exactly as many points C on a LINE (or LINE SEGMENT )asina PLANE , a 3-D SPACE , or finite HYPERSPACE , since all these SETS can be put into a ONE-TO-ONE correspondence with each other. The CONTINUUM HYPOTHESIS , first proposed by Georg Cantor, holds that the CARDINAL NUMBER of the continuum is the same as that of ALEPH-1 . The surprising truth is that this proposition is UNDECID- ABLE , since neither it nor its converse contradicts the tenets of SET THEORY . See also ALEPH-0 ,ALEPH-1 ,CONTINUUM HYPOTHESIS , DENUMERABLE SET Continuum Hypothesis Portions of this entry contributed by MATTHEW SZUD- ZIK The proposal originally made by Georg Cantor that there is no infinite set with a CARDINAL NUMBER between that of the "small" infinite set of INTEGERS /C2100 and the "large" infinite set of REAL NUMBERS C (the "CONTINUUM "). Symbolically, the continuum hypoth- esis is that /C2101 /C30C:/Go¨del showed that no CONTRADICTION would arise if the continuum hypothesis were added to conventional ZERMELO- FRAENKEL SET THEORY . However, using a technique called FORCING , Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to SET THEORY . Together, Go¨del’s and Cohen’s results established that the validity of the continuum hy- pothesis depends on the version of SET THEORY being used, and is therefore UNDECIDABLE (assuming the ZERMELO- FRAENKEL AXIOMS together with the AXIOM OF CHOICE ). Conway and Guy (1996, p. 282) recount a generalized version of the continuum hypothesis originally due to Hausdorff in 1908 which is also UNDECIDABLE :is 2/C210 a /C30/C210a/C271for every a/? The continuum hypothesis follows from generalized continuum hypothesis, so ZF /C27GCH /C159CH :/ In 2000, H. Woodin formulated a new plausible "axiom" whose adoption (in addition to the ZER- MELO- FRAENKEL AXIOMS and AXIOM OF CHOICE ) would imply that the Continuum Hypothesis is false. Since set theoreticians have felt for some time that the Continuum Hypothesis should be false, if Woodin’s axiom proves to be particularly elegant, useful, or intuitive, it may catch on. It is interesting to compare this to a situation with Euclid’s PARALLEL POSTULATE more than 300 years ago, when Wallis proposed an additional axiom that would imply the PARALLEL POSTULATE (Greenberg 1994, pp. 152 /C1/53). See also ALEPH-0 ,A LEPH-1 ,A XIOM OF CHOICE , CARDINAL NUMBER ,C ONTINUUM ,D ENUMERABLE SET,FORCING ,HILBERT’S PROBLEMS ,LEBESGUE MEA- SURABILITY PROBLEM ,U NDECIDABLE ,Z ERMELO- FRAENKEL AXIOMS ,ZERMELO- FRAENKEL SET THEORY References Cohen, P. J. "The Independence of the Continuum Hypoth- esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143 /C1/148, 1963. Cohen, P. J. "The Independence of the Continuum Hypoth- esis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105/C1/10, 1964. Cohen, P. J. Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 282, 1996. Ferreiro ´s, J. "The Notion of Cardinality and the Continuum Hypothesis." Ch. 6 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 171 /C1/14, 1999. Go¨del, K. The Consistency of the Continuum-Hypothesis. Princeton, NJ: Princeton University Press, 1940. Greenberg, M. J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W. H. Freeman, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 225 /C1/26, 1998. Jech, T. J. Set Theory, 2nd ed. Berlin: Springer-Verlag, 1997. McGough, N. "The Continuum Hypothesis." http://www.ii.- com/math/ch/. Contour A path in the COMPLEX PLANE over which CONTOUR INTEGRATION is performed to compute a CONTOUR INTEGRAL . When choosing a contour to evaluate an integral on the REAL LINE, a contour is generally chosen based on the range of integration and the position of POLES in the COMPLEX PLANE . For example, for an integral from /C28/C12 to /C27/C12 along the real axis, the contour at left could be chosen if the function f had no POLES on the REAL LINE, and the middle contour could be chosen if it had a POLE at the origin. To perform an integral over the positive real axis from 0 to /C27/C12 for a function with a POLE at 0, the contour at right could be chosen. See also CONTOUR INTEGRAL ,CONTOUR INTEGRATION , HANKEL CONTOUR ,INSIDE- OUTSIDE THEOREM ,POLE, RESIDUE (COMPLEX ANALYSIS ) Contour Integral An integral obtained by CONTOUR INTEGRATION . The particular path in the COMPLEX PLANE used to compute the integral is called a CONTOUR . Watson (1966 p. 20) uses the notation f(a /C27) f(z) dz to denote the contour integral of f(z) with CONTOUR encircling the point aonce in a counterclockwise direction. See also CONTOUR ,CONTOUR INTEGRATION References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Contour Integration Contour integration is the process of calculating the values of a CONTOUR INTEGRAL around a given CONTOUR in the COMPLEX PLANE . As a result of a truly amazing property of HOLOMORPHIC FUNCTIONS , such integrals can be computed easily simply bysumming the values of the RESIDUES inside the CONTOUR . LetP(x) and Q(x)b e POLYNOMIALS ofDEGREES nandmwith COEFFICIENTS bn;...,b0andcm;...,c0:Take the CONTOUR in the UPPER HALF-PLANE , replace xbyz, and write z/C13Reiu:Then g/C12 /C28/C12P(z)dz Q(z)/C30lim R0/C12gR /C28RP(z)dz Q(z): (1) Define a path gRwhich is straight along the REAL axis from/C28RtoRand make a circular half-arc to connect the two ends in the upper half of the COMPLEX PLANE . The RESIDUE THEOREM then gives lim R/C28/C12ggRP(z)dz Q(z) /C30lim R/C28/C12gR /C28RP(z)dz Q(z)/C27lim R0/C12gp 0P(Reiu) Q(Reiu)iReiudu /C302piX I[z]>0ResP(z) Q(z)"# ; (2) where Res denotes the RESIDUES . Solving, lim R0/C12gR /C28RP(z)dz Q(z) /C302piX I[z]>0ResP(z) Q(z)/C28lim R0/C12gp 0P(Reiu) Q(Reiu)iReiudu Define Ir/C13lim R0/C12gp 0P(Reiu) Q(Reiu)iReiudu /C30lim R0/C12gp 0bn(Reiu)n/C27bn/C281(Reiu)n/C281/C27.../C27b0 cm(Reiu)m/C27cm/C281(Reiu)m/C281/C27.../C27c0iR du /C30lim R0/C12gp 0bn cm(Reiu)n/C28miR du /C30lim R0/C12gp 0bn cmRn/C271/C28mi(eiu)n/C28mdu (3) and set e/C13/C28(n/C271/C28m); (4) then equation (3) becomes Ir/C13lim R0/C12i Rebr cmgp 0ei(n/C28m)udu: (5) Now, lim R0/C12R/C28e/C300 (6) foro>0:That means that for /C28n/C281/C27m]1;orm] n/C272;IR/C300;so g/C12 /C28/C12P(z)dz Q(z)/C302piX I[z]>0ResP(z) Q(z)"# (7) for m ]n /C272: Apply JORDAN’S LEMMA with f(x) /C13 P(x) =Q(x) : We must have lim x0/C12f(x) /C300 ; (8) so we require m ]n /C271: Then g/C12 /C28/C12P(z) Q(z)eiaz dz /C302piX I[z]>0ResP(z) Q(z)eiaz"# (9) for m ]n /C271 :/ Since this must hold separately for REAL and IMAGIN- ARY PARTS , this result can be extended to g/C12 /C28/C12P(x) Q(x)cos(ax) dx /C302pRX I[z]>0ResP(z) Q(z)eiaz"#() (10) g/C12 /C28/C12P(x) Q(x)sin(ax) dx /C302pIX I[z]>0ResP(z) Q(z)eiaz"#() : (11) It is also true that g/C12 /C28/C12P(z) Q(z)ln(az) dz /C300 : (12) See also CAUCHY INTEGRAL FORMULA ,CAUCHY INTE- GRAL THEOREM ,CONTOUR ,CONTOUR INTEGRAL ,IN- SIDE- OUTSIDE THEOREM ,JORDAN’S LEMMA ,RESIDUE (COMPLEX ANALYSIS ), SINE INTEGRAL References Krantz, S. G. "Applications to the Calculation of Definite Integrals and Sums." §4.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 51 /C1/3, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 353 /C1/56, 1953. Contour Plot A plot of EQUIPOTENTIAL CURVES . If desired, the regions between contours can be shaded or coloredto indicate their magnitude. Contour plots are im- plemented in Mathematica as ContourPlot [f,{x, xmin , xmin }, {y, ymin , ymax }]. See also EQUIPOTENTIAL CURVE ,LEVEL CURVE ,LEVEL SET,LEVEL SURFACE Contractant CONDENSATION Contracted Cycloid CURTATE CYCLOID Contraction (Geometry) An AFFINE TRANSFORMATION in which the scale is reduced. See also EXPANSION Contraction (Graph) The merging of nodes in a GRAPH by eliminating segments between two nodes. Contraction (Ideal) When f : A 0 B is a ring HOMOMORPHISM and b is an IDEAL in B, then f /C281(b) is an ideal in A, called the contraction of b and sometimes denoted bc :/ The contraction of a PRIME IDEAL is always prime. For example, consider f : Z 0 Z[ffiffiffi 2p ] : Then the contrac- tion offfiffiffi 2pl11ml111 is the ideal of even integers. See also ALGEBRAIC NUMBER THEORY ,E XTENSION (IDEAL ), IDEAL ,PRIME IDEAL ,RING References Atiyah, M. F. and MacDonald, I. G. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, pp. 9 /C1/0, 1969. Contraction (Tensor) The contraction of a TENSOR is obtained by setting unlike indices equal and summing according to the EINSTEIN SUMMATION convention. Contraction re- duces the RANK of a TENSOR by 2. For a second RANK TENSOR , contr( B?ji)/C13B?ii B?ii/C30@x?i @xk@xl @x?iBk l/C30@xl @xkBkl/C30dl kBk l/C30Bkk: Therefore, the contraction is invariant, and must be a SCALAR . In fact, this SCALAR is known as the TRACE of aMATRIX inMATRIX theory. References Arfken, G. "Contraction, Direct Product." §3.2 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 124 /C1/26, 1985. Jeffreys, H. and Jeffreys, B. S. "Transformation of Coordi- nates." §3.02 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 86 /C1/7, 1988. Contradiction A SENTENCE is called a contradiction if its TRUTH TABLE contains only ‘F.’ See also CONSISTENCY STRENGTH ,C ONTINGENCY , TAUTOLOGY ,TRUTH TABLE References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 13, 1958. Contradiction Law No A is not-A. See also NOT Contravariant Tensor A contravariant tensor is a TENSOR having specific transformation properties (cf., a COVARIANT TENSOR ). To examine the transformation properties of a contra- variant tensor, first consider a TENSOR of RANK 1(a VECTOR ) dr /C30dx1 ˆx1 /C27dx2 ˆx2 /C27dx3 ˆx3 ; (1) for which dx?i /C30@x?i @xjdxj : (2) Now let Ai /C13dxi ; then any set of quantities Ajwhich transform according to A?i /C30@x?i @xjAj ; (3) or, defining aij /C13@x?i @xj; (4) according to A?i /C30aijAj (5) is a contravariant tensor. Contravariant tensors are indicated with raised indices, i.e., a m :/ COVARIANT TENSORS are a type of TENSOR with differing transformation properties, denoted an : How- ever, in 3-D CARTESIAN COORDINATES , @xj @x?i/C30@x?i @xj/C13aij (6) for i ; j /C301 ; 2, 3, meaning that contravariant and covariant tensors are equivalent. The two types of tensors do differ in higher dimensions, however.Contravariant FOUR-VECTORS satisfy a m /C30L m n an ; (7) where L is a LORENTZ TENSOR . To turn a COVARIANT TENSOR aninto a contravariant tensor am (INDEX RAISING ), use the METRIC TENSOR gmn to write g mnan /C30am : (8) Covariant and contravariant indices can be used simultaneously in a MIXED TENSOR . See also CONTRAVARIANT VECTOR ,COVARIANT TEN- SOR,FOUR- VECTOR ,INDEX RAISING ,LORENTZ TENSOR , METRIC TENSOR ,MIXED TENSOR ,TENSOR References Arfken, G. "Noncartesian Tensors, Covariant Differentia- tion." §3.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 158 /C1/64, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44 /C1/6, 1953. Contravariant Vector The usual type of VECTOR , which can be viewed as a CONTRAVARIANT TENSOR ("KET") of RANK 1. Contra- variant vectors are dual to ONE-FORMS ("BRAS ," a.k.a. COVARIANT VECTORS ). See also BRA,COVARIANT VECTOR ,CONTRAVARIANT TENSOR ,KET,ONE-FORM,VECTOR Control Theory The mathematical study of how to manipulate the parameters affecting the behavior of a system to produce the desired or optimal outcome. See also KALMAN FILTER ,LINEAR ALGEBRA ,PONTRYA- GIN MAXIMUM PRINCIPLE References Zabczyk, J. Mathematical Control Theory: An Introduction. Boston, MA: Birkha ¨user, 1993. Convective Acceleration The acceleration of an element of fluid, given by the CONVECTIVE DERIVATIVE of the VELOCITY v, Dv Dt/C30@v @t /C27v /C2159v; where 9 is the GRADIENT operator. See also ACCELERATION ,C ONVECTIVE DERIVATIVE , CONVECTIVE OPERATOR References Batchelor, G K. An Introduction to Fluid Dynamics. Cam- bridge, England: Cambridge University Press, p. 73, 1977. Convective Derivative A DERIVATIVE taken with respect to a moving coordi- nate system, also called a LAGRANGIAN DERIVATIVE .It is given by D Dt /C30@ @t /C27v /C2159; where 9 is the GRADIENT operator and v is the VELOCITY of the fluid. This type of derivative is especially useful in the study of fluid mechanics. When applied to v, Dv Dt/C30@v @t /C27( 9/C29v) /C29v /C279(1 2 v2) : See also CONVECTIVE OPERATOR ,DERIVATIVE ,VELO- CITY References Batchelor, G K. An Introduction to Fluid Dynamics. Cam- bridge, England: Cambridge University Press, p. 73, 1977. Convective Operator Defined for a VECTOR FIELD A by (A /C2159) ; where 9 is the GRADIENT operator. Applied in arbitrary orthogonal 3-D coordinates to a VECTOR FIELD B, the convective operator becomes [(A /C2159)B]j /C30X3 k/C301Ak hk@Bj @qk/C27Bk hkhjAj@hj @qk/C28Ak@hk @qj ! "# ; (1) where the hi/s are related to the METRIC TENSORS by hi /C30ffiffiffiffiffigiip: In CARTESIAN COORDINATES , (A /C2159)B /C30 Ax@Bx @x/C27Ay@Bx @y/C27Az@Bx @z ! ˆx /C27 Ax@By @x/C27Ay@Ky @y/C27Az@By @z ! ˆy /C27 Ax@Bz @x/C27Ay@Bz @y/C27Az@Bz @z ! ˆz : (2) In CYLINDRICAL COORDINATES , (A /C2159)B /C30 Ar@Br @r/C27Af r@Br @ f/C27Az@Br @z/C28AfB f r ! ˆr /C27 Ar@Bf @r/C27Af r@Bf @ f/C27Az@Bf @z/C27AfBr r ! ˆf /C27 Ar@Bz @r/C27Af r@Bz @ f/C27Az@Bz @z ! ˆz : (3) In SPHERICAL COORDINATES ,(A /C2159)B /C30 Ar@Br @r/C27Af r@Br @ u/C27Af r sin u@Br @ f/C28AuBu /C27 AfBf r ! ˆr /C27 Ar@Bu @r/C27Au r@Bu @ u/C27Af r sin u@Bu @ f/C27AuBr r/C28AfBf cot u r ! ˆu /C27 Ar@Bf @r/C27Au r@Bf @ u/C27Af r sin u@Bf @ f/C27AfBr r/C27AfBu cot u r ! ˆf: (4) See also CONVECTIVE ACCELERATION ,C ONVECTIVE DERIVATIVE ,CURVILINEAR COORDINATES ,GRADIENT Convergence ALMOST EVERYWHERE CONVERGENCE ,CONVERGENCE IMPROVEMENT ,CONVERGENCE TESTS ,CONVERGENT , CONVERGENT SEQUENCE ,C ONVERGENT SERIES , POINTWISE CONVERGENCE Convergence Acceleration CONVERGENCE IMPROVEMENT Convergence Improvement The improvement of the convergence properties of a SERIES , also called CONVERGENCE ACCELERATION , such that a SERIES reaches its limit to within some accuracy with fewer terms than required before. Convergence improvement can be effected by forming aLINEAR COMBINATION with a SERIES whose sum is known. Useful sums include X/C12 n/C3011 n(n/C271)/C301 (1) X/C12 n/C3011 n(n/C271)(n/C272)/C301 4(2) X/C12 n/C3011 n(n/C271)(n/C272)(n/C273)/C301 18(3) X/C12 n/C3011 n(n/C271)/C1/C1/C1(n/C27p)/C301 p /C215p!: (4) Kummer’s transformation takes a convergent series s/C30X/C12 k/C300ak (5) and another convergent series c/C30X/C12 k/C300ck (6) with known csuch that lim k 0/C12ak ck/C30 l "0: (7) Then a series with more rapid convergence to the same value is given by s /C30 lc /C27X/C12 k/C3001 /C28 lck ak ! ak (8) (Abramowitz and Stegun 1972). The EULER TRANSFORM takes a convergent alternat- ing series X/C12 k /C300(/C281)kak /C30a0 /C28a1 /C27a2 ... (9) into a series with more rapid convergence to the same value to s /C30X/C12 k /C300(/C281)k Dka0 2k/C271; (10) where Dka0 /C30Xk m/C300/C13(/C281)m k ml11sl11n ak /C28m (11) (Abramowitz and Stegun 1972; Beeler et al. 1972). Given a series OF THE FORM S /C30X/C12 n /C301f1 n ! ; (12) where f(z)isan ANALYTIC at 0 and on the closed unit DISK, and f(z)½z00 /C30O(z2) ; (13) then the series can be rearranged to S /C30X/C12 n/C301X/C12 m/C302fm1 n !m /C30X/C12 m/C302X/C12 n/C301fm1 n !m /C30X/C12 m/C302fm z(m); (14) where f(z) /C30X/C12 m/C302fmzm (15) is the MACLAURIN SERIES of f and z is the RIEMANN ZETA FUNCTION (Flajolet and Vardi 1996). The trans- formed series exhibits geometric convergence. Simi- larly, if f(z)is ANALYTIC in ½z ½51=n0 for some POSITIVE INTEGER n0 ; thenS /C30Xn0 /C281 n/C301f1 n ! /C27X/C12 m/C302fmz(m) /C281 1m /C28.../C281 (n0 /C28 1)m"# ; (16) which converges geometrically (Flajolet and Vardi 1996). (16) can also be used to further accelerate the convergence of series (14). See also EULER TRANSFORM ,W ILF-ZEILBERGER PAIR References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 288 /C1/89, 1985. Beeler et al. Item 120 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 55, Feb. 1972. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Convergence Tests A test to determine if a given SERIES CONVERGES or DIVERGES . See also ABEL’S UNIFORM CONVERGENCE TEST,BER- TRAND’S TEST, D’ALEMBERT RATIO TEST,DIVERGENCE TESTS ,ERMAKOFF’S TEST,G AUSS’S TEST,INTEGRAL TEST,K UMMER’S TEST,L IMIT COMPARISON TEST, LIMIT TEST,RAABE’S TEST,RADIUS OF CONVERGENCE , RATIO TEST,RIEMANN SERIES THEOREM ,ROOT TEST References Arfken, G. "Convergence Tests." §5.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 280 /C1/93, 1985. Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 55 /C1/7, 1991. Convergent The RATIONAL NUMBER obtained by keeping only a limited number of terms in a CONTINUED FRACTION is called a convergent. For example, in the SIMPLE CONTINUED FRACTION for the GOLDEN RATIO , f/C301/C271 1/C271 1/C27...; the convergents are 1;1/C271 1/C302;1/C271 1/C271 1/C303 2;... The word convergent is also used to describe a CONVERGENT SEQUENCE orCONVERGENT SERIES . See also CONTINUED FRACTION ,C ONVERGENT SE- QUENCE ,C ONVERGENT SERIES ,PARTIAL QUOTIENT , SIMPLE CONTINUED FRACTION Convergent Sequence A SEQUENCE Sn converges to the limit S lim n0/C12Sn /C30S if, for any e > 0; there exists an N such that ½Sn /C28S½B e for n /C21N.IfSndoes not converge, it is said to DIVERGE . This condition can also be written as lim n0/C12Sn /C30lim n 0/C12Sn /C30S: Every bounded MONOTONIC SEQUENCE converges. Every unbounded SEQUENCE diverges. See also CONDITIONAL CONVERGENCE ,STRONG CON- VERGENCE ,W EAK CONVERGENCE References Jeffreys, H. and Jeffreys, B. S. "Bounded, Unbounded, Convergent, Oscillatory." §1.041 in Methods of Mathema- tical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 11 /C1/2, 1988. Convergent Series The infinite SERIES a/C12 n/C301 anis convergent if the SEQUENCE of partial sums Sn /C30Xn k/C301ak is convergent. Conversely, a SERIES is divergent if the SEQUENCE of partial sums is divergent. If auk and avk are convergent SERIES , then a(uk /C27vk) and a(uk /C28vk) are convergent. If c "0 ; then aukand c aukboth converge or both diverge. Convergence and diver- gence are unaffected by deleting a finite number of terms from the beginning of a series. Constant terms in the denominator of a sequence can usually be deleted without affecting convergence. All but the highest POWER terms in POLYNOMIALS can usually be deleted in both NUMERATOR and DENOMINATOR of a SERIES without affecting convergence. If a SERIES converges absolutely, then it converges. See also CONVERGENCE TESTS,RADIUS OF CONVER- GENCE References Bromwich, T. J. I’a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991. Conversion Period The period of time between INTEREST payments. See also COMPOUND INTEREST ,INTEREST ,S IMPLE INTERESTConvex A SET in EUCLIDEAN SPACE Rd is a CONVEX SET if it contains all the LINE SEGMENTS connecting any pair of its points. If the SET does not contain all the LINE SEGMENTS , it is called CONCAVE . See also CONNECTED SET,CONVEX FUNCTION ,CON- VEX HULL,CONVEX OPTIMIZATION THEORY ,CONVEX POLYGON ,CONVEX SET,DELAUNAY TRIANGULATION , MINKOWSKI CONVEX BODY THEOREM ,SIMPLY CON- NECTED References Benson, R. V. Euclidean Geometry and Convexity. New York: McGraw-Hill, 1966. Busemann, H. Convex Surfaces. New York: Interscience, 1958. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Convexity." Ch. A in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 6 /C1/7, 1994. Eggleston, H. G. Problems in Euclidean Space: Applications of Convexity. New York: Pergamon Press, 1957. Gruber, P. M. "Seven Small Pearls from Convexity." Math. Intell. 5,1 6/C1/9, 1983. Gruber, P. M. "Aspects of Convexity and Its Applications." Expos. Math. 2,4 7/C1/3, 1984. Guggenheimer, H. Applicable Geometry--Global and Local Convexity. New York: Krieger, 1977. Kelly, P. J. and Weiss, M. L. Geometry and Convexity: A Study of Mathematical Methods. New York: Wiley, 1979. Webster, R. Convexity. Oxford, England: Oxford University Press, 1995. Convex Function A function whose value at the MIDPOINT of every INTERVAL in its DOMAIN does not exceed the AVERAGE of its values at the ends of the INTERVAL . In other words, a function f(x) is convex on an INTERVAL [a, b] if for any two points x1andx2in [a, b], f[1 2(x1/C27x2)]512[f(x1)/C27f(x2)] (Gradshteyn and Ryzhik 2000). If f(x) has a second DERIVATIVE in [a, b], then a NECESSARY and SUFFI- CIENT condition for it to be convex on that INTERVAL is that the second DERIVATIVE f ƒ(x) > 0 for all x in [a, b]. If the inequality above is STRICT for all x1 and x2 ; then f(x) is called strictly convex. Examples of convex functions include xp for p ]1; x ln x for x /C210, and ½x½ for all x. If the sign of the inequality is reversed, the function is called CONCAVE . See also CONCAVE FUNCTION ,L OGARITHMICALLY CONVEX FUNCTION References Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How Likely is a Function to be Convex?" Math. Mag. 61, 211 /C1/ 19, 1988. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1132, 2000. Webster, R. Convexity. Oxford, England: Oxford University Press, 1995. Convex Hull The convex hull of a set of points S in n-D is the INTERSECTION of all convex sets containing S. For N points p1 ; ..., pN ; the convex hull C is then given by the expression C /C13XN j/C301ljpj : lj ]0 for all j andXN j/C301lj /C301() : Computing the convex hull is a problem in COMPUTA- TIONAL GEOMETRY . The indices of the points specify- ing the convex hull of a set of points in two dimensions is given by the command Convex- Hull [pts] in the Mathematica add-on packageDis- creteMath‘ComputationalGeometry‘ (which can be loaded with the command BBDiscreteMath‘ ). Future versions of Mathematica will support n- dimensional convex hulls. In d dimensions, the "gift wrapping" algorithm, which has complexity O(n d=2bc/C271); where xbcis the FLOOR FUNCTION , can be used (Skiena 1997, p. 352). In 2- and 3-D, however, specialized algorithms exist with complexity O(n ln n) (Skiena 1997, pp. 351 /C1/52). Yao (1981) has proved that any decision-tree algo- rithm for the 2-D case requires quadratic or higher- order tests, and that any algorithm using quadratictests (which includes all currently known algorithms) cannot be done with lower complexity than O(n ln n): However, it remains an open problem whether better complexity can be obtained using higher-order poly- nomial tests (Yao 1981). O’Rourke (1997) gives a robust 2-D implementation as well as an O(n2) 3-D implementation. Qhull works efficiently in 2 to 8 dimensions (Barber et al. 1997). The DUAL POLYHEDRON of any non-convex UNIFORM POLYHEDRON is a stellated form of the CONVEX HULL of the given polyhedron (Wenninger 1983, pp. 3 /C1/and 40). See also CARATHE ´ ODORY’S FUNDAMENTAL THEOREM , COMPUTATIONAL GEOMETRY ,CROSS POLYTOPE ,GROE- MER PACKING ,G ROEMER THEOREM ,H APPY END PROBLEM ,RADON’S THEOREM ,SAUSAGE CONJECTURE , SPAN (GEOMETRY ), SYLVESTER’S FOUR- POINT PRO- BLEM ,TEMPERATURE References Barber, C.; Dobkin, D.; and Huhdanpaa, H. "The Quickhull Algorithm for Convex Hulls." ACM Trans. Mathematical Software 22, 469/C1/83, 1997. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 8, 1991. de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. "Convex Hulls: Mixing Things." Ch. 11 in Com- putational Geometry: Algorithms and Applications, 2nd rev. ed. Berlin: Springer-Verlag, pp. 235 /C1/50, 2000. Edelsbrunner, H. and Mu ¨cke, E. P. "Three-Dimensional Alpha Shapes." ACM Trans. Graphics 13,4 3/C1/2, 1994. O’Rourke, J. Computational Geometry in C, 2nd ed. Cam- bridge, England: Cambridge University Press, 1998. Preparata, F. R. and Shamos, M. I. Computational Geome- try: An Introduction. New York: Springer-Verlag, 1985. Santalo ´,L .A . Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Seidel, R. "Convex Hull Computations." Ch. 19 in Handbook of Discrete and Computational Geometry (Ed. J. E. Good- man and J. O’Rourke). Boca Raton, FL: CRC Press, pp. 361 /C1/75, 1997. Skiena, S. S. "Convex Hull." §8.6.2 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 351 /C1/54, 1997. Weisstein, E. W. "Convex Hull 3D." M ATHEMATICA NOTE- BOOK CONVEX HULL.M . Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Yao, A. C.-C. "A Lower Bound to Finding Convex Hulls." J. ACM 28, 780/C1/87, 1981. Convex Optimization Theory The problem of maximizing a linear function over a CONVEX POLYHEDRON , also known as OPERATIONS RESEARCH or OPTIMIZATION THEORY . The general problem of convex optimization is to find the mini- mum of a convex (or quasiconvex) function fon a FINITE -dimensional convex body A. Methods of solu- tion include Levin’s algorithm and the method ofcircumscribed ELLIPSOIDS , also called the Nemir- ovsky-Yudin-Shor method. References Tokhomirov, V. M. "The Evolution of Methods of Convex Optimization." Amer. Math. Monthly 103,65/C1/1, 1996. Convex Polygon A POLYGON is CONVEX if it contains all the LINE SEGMENTS connecting any pair of its points. Let f(n) be the smallest number such that when W is a set of more than f(n) points in GENERAL POSITION (with no three points COLLINEAR ) in the plane, all of the VERTICES of some convex n-gon are contained in W. The answers for n /C302, 3, and 4 are 2, 4, and 8. It is conjectured that f(n) /C302n/C282 ; but only proven that 2n /C282 5f(n) 52n /C284 n /C282l11sl11n ; wheren kl1ml11 is a BINOMIAL COEFFICIENT . See also CONVEX POLYOMINO ,CONVEX POLYHEDRON , CONVEX POLYOMINO ,CONVEX POLYTOPE ,HAPPY END PROBLEM ,LATTICE POLYGON ,POLYGON Convex Polyhedron A convex polyhedron can be defined algebraically as the set of solutions to a system of linear inequalities mx5b; (1) where m is a real s /C293 MATRIX and b is a real s- VECTOR . Although usage varies, most authors addi- tionally require that a solution be bounded for it to define a CONVEX POLYHEDRON . An example of a convex polyhedron is illustrated above. The more simple DODECAHEDRON is given by a system with s /C3012. Explicit examples are given in the following table. convex polyhedron s /m/ b TETRAHEDRON 4 111 1 /C281 /C281 /C2811 /C281 /C281 /C28112 6643 7752 00 02 6643 775CUBE 6 100 /C28100 0100 /C2810 001 00 /C2812 66666643 77777751 1 111 12 66666643 7777775 OCTAHEDRON 8 111 11 /C281 1 /C2811 1 /C281 /C281 /C28111 /C2811 /C281 /C281 /C2811 /C281 /C281 /C2812 666666666643 777777777751 1 1 1111 12 666666666643 77777777775 In general, given the MATRICES , the VERTICES (and FACES ) can be found using an algorithmic procedure known as VERTEX ENUMERATION . Geometrically, a convex polyhedron can be defined as a POLYHEDRON for which a line connecting any two (noncoplanar) points on the surface always lies in the interior of the polyhedron. The 92 convex polyhedra having only REGULAR POLYGONS as faces are called the JOHNSON SOLIDS , which include the PLATONIC SOLIDS and ARCHIMEDEAN SOLIDS . No method is known for computing the VOLUME of a general convex polyhedron (Ogilvy 1990, p. 173). Every convex polyhedron can be represented in the plane or on the surface of a sphere by a 3-connected PLANAR GRAPH (called a POLYHEDRAL GRAPH ). Con- versely, by a theorem of Steinitz as restated by Gru¨nbaum, every 3-connected PLANAR GRAPH can be realized as a convex polyhedron (Duijvestijn and Federico 1981). The numbers of vertices V, edges E, and faces Fof a convex polyhedron are related by the POLYHEDRAL FORMULA V/C27F/C28E/C302: See also ARCHIMEDEAN SOLID ,C ONVEX POLYGON , CONVEX POLYOMINO ,CONVEX POLYTOPE ,D ELTAHE- DRON ,JOHNSON SOLID ,KEPLER- POINSOT SOLID ,PLA- TONIC SOLID,POLYHEDRAL FORMULA ,POLYHEDRAL GRAPH ,POLYHEDRON ,REGULAR POLYHEDRON ,VER- TEX ENUMERATION References Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral ( /3/-Connected Planar) Graphs." Math. Comput. 37, 523/C1/32, 1981. Ogilvy, C. S. Excursions in Geometry. New York: Dover, 1990. Lyusternik, L. A. Convex Figures and Polyhedra. New York: Dover, 1963. Yaglom, I. M. and Boltianskii, V. G. Convex Figures. New York: Holt, Rinehart and Winston, 1961. Convex Polyomino A convex polyomino (sometimes called a "convex polygon") is a polyomino whose PERIMETER is equal to that of its minimal bounding box (Bousquet-Me ´lou et al. 1999). Furthermore, if it contains at least one corner of its minimal bounding box, it is said to be a DIRECTED CONVEX POLYOMINO .A COLUMN-CONVEX POLYOMINO is a self-avoiding polyomino such that the intersection of any vertical line with the poly-omino has at most two connected components, and a ROW-CONVEX POLYOMINO is similarly defined. The anisotropic perimeter and area generating func-tion G(x;y;q)/C30X m]1X n]1X a]1C(m;n;a)xmynqa; (1) where C(m;n;a) is the number of polygons with 2 m horizonal bonds, 2 nvertical bonds, and area ais given by G(x;y;q)/C302X m]1ym/C272 (xq)2 mN(xqm/C281)N(xqm) /C2[Tm/C271S(xqm)/C28yTmS(xqm/C271)]2 /C27X m]1xymqm(Tm)2 (xq)m/C281(xq)m; (2) where N(x)/C30X n]0(/C281)nxnqn/C271 2ðÞ (q)n(yq)n(3) S(x)/C30X n]1xnqn (yq)nXn/C281 j/C300(/C281)jqj 2ðÞ (q)j(yqj/C271)n/C28j"# (4) andTn(x) is the polynomial RECURRENCE RELATION Tn(x)/C302Tn/C281(x)/C27(xqn/C281/C281)Tn/C282(x) (5)with T0(x)/C301 and T1(x)/C301 (Bousquet-Me ´lou 1992b). The first few of these polynomials are given by T2(x)/C301/C27qx T3(x)/C301/C27(2q/C27q2)x T4(x)/C301/C27(3q/C272q2/C27q3)x/C27q4x2 T5(x)/C301/C27(4q/C273q2/C272q3/C27q4)x/C27(2q4/C272q5/C27q6)x2: Expanding the generating function shows that the number of convex polyominoes having PERIMETER 2n/C278 is given by (2n/C2711)4n/C284(2n/C271)2n nl11sl11n ; (6) wheren kl1ml11 is a BINOMIAL COEFFICIENT (Delest and Viennot 1984, Bousquet-Me ´lou 1992). This function has been computed exactly for the column-convex and directed column-convex polyomi- noes (Bousquet-Me ´lou 1996, Bousquet-Me ´louet al. 1999). G(1;1;q)i sa Q-SERIES , but becomes algebraic for column-convex polyominoes. However, G(x;y;q) for column-convex polyominoes again involves Q- SERIES (Temperley 1956, Bousquet-Me ´lou et al. 1999). /G(x;y)/C30G(x;y;1) is an algebraic function of xandy (called the "fugacities") given by G(x;y)/C30X x]1X y]1C(m;n)xmyn /C30R(x;y)xy [D(x;y)]2/C284x2y2 D3=2; (7) where R(x;y)/C301/C283x/C283y/C273x2/C273y2/C275xy/C28x3/C28y3/C28x2y /C28xy2/C28xy(x/C28y)2(8) D(x;y)/C301/C282x/C282y/C282xy/C27x2/C27y2 /C30(1/C28y)21/C28x(2/C272y/C28x) (1/C28y)2"# (9) (Lin and Chang 1988, Bousquet-Me ´lou 1992). This can be solved to explicitly give C(m;n)/C30mn/C281 m/C27n/C2822m/C272n/C284 2m/C282l11sl11n /C282(m/C27n/C282)m/C27n/C283 m/C281l11sl11n m/C27n/C283 n/C281l11sl11n (10) (Gessel 1990, Bousquet-Me ´lou 1992). /G(x;y) satisfies the inversion relation G(x;y)/C27y3G(x=y;1=y)/C30xy/C28x3y@ @x1/C28x/C27y D(x;y);(11) where D(x; y) /C301 /C282x /C282y /C282xy /C27x2 /C27y2 /C30(1 /C28y)2 1 /C28x(2 /C27 2y /C28 x) (1 /C28 y)2"# (12) (Lin and Chang 1988, Bousquet-Me ´lou et al. 1999). The half-vertical perimeter and area generating function for column-convex polyominos of width 3 is given by the special case H3(y; q) /C30yq3 (1 /C28 yq)4(1 /C28 yq2)2(1 /C28 yq3) /C2(y6q8 /C274y5q7 /C272y5q6 /C27y4q6 /C28y4q4 /C284y3q5 /C286y3q4 /C284y3q3 /C28y2q4 /C27y2q2 /C272yq2 /C274yq /C271) (13) of the general rational function (Bousquet-Me ´lou et al. 1999), which satisfies the reciprocity relation H3(1=y; 1=q) /C30/C281 yq3H3(y; q) : (14) The anisotropic area and perimeter generating func- tion G(x; y; q) and partial generating functions Hm(y; q) ; connected by G(x; y; q) /C30X m]1Hm(y; q)xm ; (15) satisfy the self-reciprocity and inversion relations Hm(1=y; 1=q) /C30/C281 yqmHm(y; q) (16) and G(x; y; q) /C27yG(xq; 1=y; 1=q) /C300 (Bousquet-Me ´lou et al. 1999). See also COLUMN- CONVEX POLYO MINO ,D IRECTED CONVEX POLYOMINO ,POLYOMINO References Bousquet-Me ´lou, M. "Convex Polyominoes and Heaps of Segments." J. Phys. A: Math. Gen. 25, 1925 /C1/934, 1992a. Bousquet-Me ´lou, M. "Convex Polyominoes and Algebraic Languages." J. Phys. A: Math. Gen. 25, 1935 /C1/944, 1992b. Bousquet-Me ´lou, M. "A Method for Enumeration of Various Classes of Column-Convex Polygons." Disc. Math. 154,1/C1/ 5, 1996. Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/ 9908123/. Delest, M.-P. and Viennot, G. "Algebraic Languages and Polyominoes [sic] Enumeration." Theoret. Comput. Sci. 34, 169 /C1/06, 1984. Gessel, I. M. "On the Number of Convex Polyominoes." Preprint. 1990. Lin, K. Y. and Chang, S. J. "Rigorous Results for the Number of Convex Polygons on the Square and Honey- comb Lattices." J. Phys. A: Math. Gen. 21, 2635 /C1/642, 1988.Temperley, H. N. V. "Combinatorial Problems Suggested by the Statistical Mechanics of Domains and of Rubber-Like Molecules." Phys. Rev. 103,1/C1/6, 1956. Convex Polytope See also CONVEX POLYGON ,CONVEX POLYHEDRON , POLYTOPE Convex Set A SET S in n-dimensional space is called a convex set if the line segment joining any pair of points of S lies entirely in S. See also CONVEX References Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Convexity." Ch. A in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 6 /C1/7, 1994. Klee, V. "What is a Convex Set?" Amer. Math. Monthly 78, 616/C1/31, 1971. Lay, S. R. Convex Sets and Their Applications. New York: Wiley, 1979. Valentine, F. A. Convex Sets. New York: McGraw-Hill, 1964. Convolution A convolution is an integral which expresses the amount of overlap of one function g(t) as it is shifted over another function f(t):It therefore "blends" one function with another. For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam (theF OURIER TRANSFORM of the sampling distribution). The convolution is sometimes also known by itsGerman name, faltung ("folding"). A convolution over a finite range [0 ;t] is given by f(t)+g(t)/C13gt 0f(t)g(t/C28t)dt; (1) where the symbol f+g(occasionally also written as f/C156g) denotes convolution of fandg. Convolution is more often taken over an infinite range, f(t)+g(t)/C13g/C12 /C28/C12f(t)g(t/C28t)dt /C30g/C12 /C28/C12g(t)f(t/C28t)dt: (2) Let f,g, and hbe arbitrary functions and aa constant. Convolution the satisfies the followingproperties, f+g/C30g+f (3) f+(g+h)/C30(f+g)+h (4) f+(g/C27h)/C30(f+g)/C27(f+h) (5) (Bracewell 1999, p. 27), as well as a(f + g) /C30(af) + g /C30f + (ag) : (6) Taking the DERIVATIVE of a convolution gives d dx (f + g) /C30df dx+ g /C30f +dg dx : (7) The AREA under a convolution is the product of areas under the factors, g/C12 /C28/C12(f + g) dx /C30g/C12 /C28/C12g/C12 /C28/C12f(u)g(x /C28u) dul12ml121 dx /C30g/C12 /C28/C12f(u)g/C12 /C28/C12g(x /C28u) dxl12ml121 du /C30g/C12 /C28/C12f(u) dul12ml121g/C12 /C28/C12g(x) dxl12ml121 : (8) The horizontal CENTROIDS add x(f + g) hi /C30 xfhi/C27 xghi ; (9) as do the VARIANCES x2(f + g)l11ml111 /C30 x2fl11ml111 /C27 x2gl11ml111 ; (10) where xnfhi/C13g/C12 /C28/C12xnf(x) dx g/C12 /C28/C12f(x) dx: (11) There is also a definition of the convolution which arises in probability theory and is given by F(t) + G(t) /C30g F(t /C28x) dG(x); (12) where f F(t /C28x) dG(x)isaS TIELTJES INTEGRAL . See also AUTOCORRELATION ,CAUCHY PRODUCT ,CON- VOLUTION THEOREM ,CROSS- CORRELATION ,W IENER- KHINTCHINE THEOREM References Bracewell, R. "Convolution" and "Two-Dimensional Convo- lution." Ch. 3 in The Fourier Transform and Its Applica- tions, 3rd ed. New York: McGraw-Hill, pp. 25 /C1/0 and 243 /C1/ 44, 1999. Hirschman, I. I. and Widder, D. V. The Convolution Trans- form. Princeton, NJ: Princeton University Press, 1955. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 464 /C1/65, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Convolution and Deconvolution Using the FFT." §13.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 531 /C1/37, 1992. Weisstein, E. W. "Books about Convolution." http:// www.treasure-troves.com/books/Convolution.html. Convolution Theorem Let f(t) and g(t) be arbitrary functions of time t with FOURIER TRANSFORMS . Takef(t) /C30F/C281[F( n)] /C30g/C12 /C28/C12F( n)e2 pi nt d n (1) g(t) /C30F/C281[G( n)] /C30g/C12 /C28/C12G( n)e2 pi nt d n; (2) where F/C281 denotes the inverse FOURIER TRANSFORM (where the transform pair is defined to have con- stants A /C301 and B /C30/C282 p): Then the CONVOLUTION is f + g /C13g/C12 /C28/C12g(t?)f(t /C28t?) dt? /C30g/C12 /C28/C12g(t?)g/C12 /C28/C12F( n)e2 pin(t/C28t?) d nl12ml121 dt?: (3) Interchange the order of integration, f + g /C30g/C12 /C28/C12F( n)g/C12 /C28/C12g(t?)e /C282pi nt? dt?l12ml121 e2 pint dn /C30g/C12 /C28/C12F( n)G( n)e2 pi nt d n /C30F/C281[F( n)G( n)] : (4) So, applying a FOURIER TRANSFORM to each side, we have F[f + g] /C30F[f]F[g]: (5) The convolution theorem also takes the alternate forms F[fg] /C30F[f] + F[g] (6) F/C281(F[f]F[g]) /C30f + g (7) F/C281(F[f] + F[g]) /C30fg: (8) See also AUTOCORRELATION ,CONVOLUTION ,FOURIER TRANSFORM ,W IENER- KHINTCHINE THEOREM References Arfken, G. "Convolution Theorem." §15.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 810 /C1/14, 1985. Bracewell, R. "Convolution Theorem." The Fourier Trans- form and Its Applications, 3rd ed. New York: McGraw- Hill, pp. 108 /C1/12, 1999. Conway Groups The AUTOMORPHISM GROUP Co1 of the LEECH LATTICE modulo a center of order two is called "the" Conway group. There are 15 exceptional CONJUGACY CLASSES of the Conway group. This group, combined with the GROUPS Co2and Co3obtained similarly from the LEECH LATTICE by stabilization of the 1-D and 2-D sublattices, are collectively called Conway groups. The Conway groups are SPORADIC GROUPS . See also LEECH LATTICE ,SPORADIC GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#spo. Conway Notation CONWAY’S KNOT NOTATION ,C ONWAY POLYHEDRON NOTATION Conway Polyhedron Notation A NOTATION for POLYHEDRA which begins by specify- ing a "seed" polyhedron using a capital letter. The PLATONIC SOLIDS are denoted T (TETRAHEDRON ), O (OCTAHEDRON ), C (CUBE ), I (ICOSAHEDRON ), and D (DODECAHEDRON ), according to their first letter. Other polyhedra include the PRISMS ,Pn, ANTIPRISMS , An, and PYRAMIDS ,Yn, where n ]3 specifies the number of sides of the polyhedron’s base. Operations to be performed on the polyhedron are then specified with lower-case letters preceding the capital letter. See also POLYHEDRON ,SCHLA ¨ FLI SYMBOL ,W YTHOFF SYMBOL References Hart, G. "Conway Notation for Polyhedra." http://www.geor- gehart.com/virtual-polyhedra/conway_notation.html. Conway Polynomial ALEXANDER POLYNOMIAL Conway Puzzle Construct a 5 /C295 /C295 cube from thirteen 1 /C292 /C294 blocks, one 2 /C292 /C292 block, one 1 /C292 /C292; and three 1 /C291 /C293 blocks. See also BOX-PACKING THEOREM ,CUBE DISSECTION , DE BRUIJN’S THEOREM ,KLARNER’S THEOREM ,POLY- CUBE ,SLOTHOUBER- GRAATSMA PUZZLE References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 77 /C1/0, 1976. Conway Sequence The LOOK AND SAY SEQUENCE generated from a starting DIGIT of 3, as given by Vardi (1991). See also CONWAY’S CONSTANT ,COSMOLOGICAL THEO- REM,LOOK AND SAY SEQUENCE References Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 13 /C1/4, 1991.Conway Sphere A sphere with four punctures occurring where a KNOT passes through the surface. References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 94, 1994. Conway-Alexander Polynomial ALEXANDER POLYNOMIAL Conway’s Constant The constant l/C301:303577269034296 . . . (Sloane’s A014715) giving the asymptotic rate of growth Clnof the number of DIGITS in the nth term of the LOOK AND SAY SEQUENCE , given by the unique positive real root of the POLYNOMIAL 0/C30x71/C28x69/C282x68/C28x67/C272x66/C272x65/C27x64/C28x63/C28x62 /C28x61/C28x60/C28x59/C272x58/C275x57/C273x56/C282x55/C2810x54 /C283x53/C282x52/C276x51/C276x50/C27x49/C279x48/C283x47 /C287x46/C288x45/C288x44/C2710x43/C276x42/C278x41/C284x40 /C2812x39/C277x38/C287x37/C277x36/C27x35/C283x34/C2710x33 /C27x32/C286x31/C282x30/C2810x29/C283x28/C272x27/C279x26 /C283x25/C2714x24/C288x23/C287x21/C279x20/C283x19/C284x18 /C2810x17/C287x16/C2712x15/C277x14/C272x13/C2812x12/C284x11 /C282x10/C285x9/C27x7/C287x6/C277x5/C284x4/C2712x3/C286x2 /C273x/C286; (1) illustrated in the figure above. Note that the POLY- NOMIAL given in Conway (1987, p. 188) contains a misprint. The CONTINUED FRACTION forlis 1, 3, 3, 2, 2, 54, 5, 2, 1, 16, 1, 30, 1, 1, 1, 2, 2, 1, 14, 1, ... (Sloane’s A014967). See also CONWAY SEQUENCE ,COSMOLOGICAL THEO- REM,LOOK AND SAY SEQUENCE References Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay." §5.11 in Open Problems in Commu- nications and Computation (Ed. T. M. Cover and B. Go- pinath). New York: Springer-Verlag, pp. 173 /C1/88, 1987. Conway, J. H. and Guy, R. K. "The Look and Say Sequence." In The Book of Numbers. New York: Springer-Verlag, pp. 208 /C1/09, 1996. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cnwy/cnwy.html. Hilgemeier, M. "Die Gleichniszahlen-Reihe." Bild der Wis- sensch. , pp. 194 /C1/96, Dec. 1986. Hilgemeier, M. "‘One Metaphor Fits All’: A Fractal Voyage with Conway’s Audioactive Decay." Ch. 7 in Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York: St. Martin’s Press, 1996. Sloane, N. J. A. Sequences A014715 and A014967 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 13 /C1/4, 1991. Conway’s Game of Life LIFE Conway’s Knot The KNOT with BRAID WORD s3 2 s1 s/C281 3s/C282 2s1 s/C281 2s1 s /C281 3: The JONES POLYNOMIAL of Conway’s knot is t/C284(/C281 /C272t /C282t2 /C272t3 /C27t6 /C282t7 /C272t8 /C282t9 /C27t10) ; the same as for the KINOSHITA- TERASAKA KNOT . Conway’s Knot Notation A concise NOTATION based on the concept of the TANGLE used by Conway (1967) to enumerate KNOTS up to 11 crossings. An ALGEBRAIC KNOT containing no NEGATIVE signs in its Conway knot NOTATION is an ALTERNATING KNOT . References Conway, J. H. "An Enumeration of Knots and Links, and Some of Their Algebraic Properties." In Computation Problems in Abstract Algebra (Ed. J. Leech). Oxford, England: Pergamon Press, pp. 329 /C1/58, 1967. Conway’s Life LIFE Cookie-Cutter Problem Maximize the number of cookies you can cut from a given expanse of dough (Hoffman 1998, p. 173). See also BIN-PACKING PROBLEM ,TILING PROBLEMReferences Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Coordinate Chart A coordinate chart is a way of expressing the points of a small NEIGHBORHOOD , usually on a MANIFOLD M,as coordinates in EUCLIDEAN SPACE . An example from geography is the coordinate chart given by the functions of LATITUDE and LONGITUDE . This coordi- nate chart is not valid on the whole globe, since it doesn’t give unique coordinates at the north or south pole (which way is east from the north pole?). Technically, a coordinate chart is a map f : U 0 V where U is an open set in M, V is an open set in Rn and n is the dimension of the manifold. Often, through notational abuse, the open set U is equated with V, and calculations on the manifold are done in the coordinate chart. This technique has the draw- back that it must be checked whether a change of coordinates affects the result of a calculation. The map f must be one-to-one, and in fact must be a HOMEOMORPHISM .Ona SMOOTH MANIFOLD , it must be a DIFFEOMORPHISM , although if the chart defines the smooth structure then this is a tautology. Simi- larly, on a complex manifold, the map f is holo- morphic. If there are two neighborhoods U1and U2with coordinate charts f1and f2 ; the TRANSITION FUNC- TION f2( f/C281 1is WELL DEFINED since coordinate charts are one-to-one. See also ATLAS ,CHART ,COMPLEX MANIFOLD ,EUCLI- DEAN SPACE ,MANIFOLD ,SMOOTH MANIFOLD ,TRANSI- TION FUNCTION Coordinate Geometry ANALYTIC GEOMETRY ,CARTESIAN GEOMETRY Coordinate System A system for specifying points using COORDINATES measured in some specified way. The simplest co- ordinate system consists of coordinate axes oriented perpendicularly to each other, known as CARTESIAN COORDINATES . Depending on the type of problem under consideration, coordinate systems possessing special properties may allow particularly simple solution. See also CURVILINEAR COORDINATES ,CYCLIDIC CO- ORDINATES ,SKEW COORDINATE SYSTEM ,O RTHOGO- NAL COORDINATE SYSTEM Coordinates A set of n variables which fix a geometric object. If the coordinates are distances measured along PERPENDI- CULAR axes, they are known as CARTESIAN COORDI- NATES . The study of GEOMETRY using one or more coordinate systems is known as ANALYTIC GEOMETRY . See also AREAL COORDINATES ,BARYCENTRIC COORDI- NATES ,BIPOLAR COORDINATES ,BIPOLAR CYLINDRICAL COORDINATES ,B ISPHERICAL COORDINATES ,C ARTE- SIAN COORDINATES ,CHOW COORDINATES ,CIRCULAR CYLINDRICAL COORDINATES ,CONFOCAL ELLIPSOIDAL COORDINATES ,C ONFOCAL PARABOLOIDAL COORDI- NATES ,CONICAL COORDINATES ,CURVILINEAR COORDI- NATES ,C YCLIDIC COORDINATES ,C YLINDRICAL COORDINATES ,ELLIPSOIDAL COORDINATES ,ELLIPTIC CYLINDRICAL COORDINATES ,G AUSSIAN COORDINATE SYSTEM ,G RASSMANN COORDINATES ,H ARMONIC CO- ORDINATES ,H OMOGENEOUS COORDINATES ,O BLATE SPHEROIDAL COORDINATES ,O RTHOCENTRIC COORDI- NATES ,PARABOLIC COORDINATES ,PARABOLIC CYLIND- RICAL COORDINATES ,P ARABOLOIDAL COORDINATES , PEDAL COORDINATES ,POLAR COORDINATES ,PROLATE SPHEROIDAL COORDINATES ,Q UADRIPLANAR COORDI- NATES ,RECTANGULAR COORDINATES ,SPHERICAL CO- ORDINATES ,T OROIDAL COORDINATES ,T RILINEAR COORDINATES References Arfken, G. "Coordinate Systems." Ch. 2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 85 /C1/17, 1985. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 1, 1961. Coordination Number KISSING NUMBER Copeland-Erdos Constant The decimal 0.23571113171923... (Sloane’s A033308) obtained by concatenating the PRIMES : 2, 23, 235, 2357, 235711, ... (Sloane’s A019518; one of the SMARANDACHE SEQUENCES ). Copeland and Erdos (1946) showed that it is a NORMAL NUMBER in base 10. The first few digits of the CONTINUED FRACTION of the Copeland-Erdos constant are 0, 4, 4, 8, 16, 18, 5, 1, ... (Sloane’s A030168). The positions of the first occur- rence of n in the CONTINUED FRACTION are 8, 16, 20, 2, 7, 15, 12, 4, 17, 254, ... (Sloane’s A033309). The incrementally largest terms are 4, 8, 16, 18, 58, 87, 484, ... (Sloane’s A033310), which occur at positions 2, 4, 5, 6, 18, 36, 82, 89, ... (Sloane’s A033311). See also CHAMPERNOWNE CONSTANT ,PRIME NUMBER References Champernowne, D. G. "The Construction of Decimals Nor- mal in the Scale of Ten." J. London Math. Soc. 8, 1933.Copeland, A. H. and Erdos, P. "Note on Normal Numbers." Bull. Amer. Math. Soc. 52, 857 /C1/60, 1946. Sloane, N. J. A. Sequences A019518, A030168, A033308, A033309, A033310, and A033311 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Coplanar Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar IFF the volume of the TETRAHEDRON defined by them is 0, x1y1z11 x2y2z21 x3y3z31 x4y4z41l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112/C300: See also P LANE Copolar Triangles PERSPECTIVE TRIANGLES Coprime RELATIVELY PRIME Coproduct Denoted‘:/ Copson-de Bruijn Constant DEBRUIJN CONSTANT Copson’s Inequality Let fangbe a NONNEGATIVE SEQUENCE and f(x)a NONNEGATIVE integrable function. Define An/C30Xn k/C301ak (1) Bn/C30X/C12 k/C30nak (2) and F(x)/C30gx 0f(t)dt (3) G(x)/C30g/C12 xf(t)dt; (4) and take 0 BpB1:For integrals, g/C12 0G(x) x"#p dx>p p/C281 !p g/C12 0[f(x)]pdx (5) (unless fis identically 0). For sums, 1 /C271 p /C28 1 ! Bp 1 /C27X/C12 n/C302Bn n !p >p p /C28 1 !pX/C12 n/C301ap n(6) (unless all an /C300):/ References Beesack, P. R. "On Some Integral Inequalities of E. T. Cop- son." In General Inequalities 2: Proceedings of the Second International Conference on General Inequalities, held in the Mathematical Research Institut at Oberwolfach, Black Forest, July 30-August 5, 1978 (Ed. E. F. Beckenbach). Basel: Birkha ¨user, 1980. Copson, E. T. "Some Integral Inequalities." Proc. Royal Soc. Edinburgh 75A, 157 /C1/64, 1975 /C1/976. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. Theorems 326 /C1/27, 337 /C1/38, and 345 in Inequalities. Cambridge, England: Cambridge University Press, 1934. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, 1991. Copula A function that joins univariate distribution functions to form multivariate distribution functions. A 2-D copula is a function C : I2 0 I such that C(0; t) /C30C(t; 0) /C300 and C(1; t) /C30C(t; 1) /C30t for all t /C23 I ; and C(u2 ; v2) /C28C(u1 ; v2) /C28C(u2 ; v1) /C27C(u1 ; v1) ]0 for all u1 ; u2 ; v1 ; v2 /C23 I such that u1 5u2 and v1 5v2 :/ See also SKLAR’S THEOREM Cordial Graph A GRAPH is called cordial if it is possible to label its vertices with 0s and 1s so that when the edges are labeled with the difference of the labels at their endpoints, the number of vertices (edges) labeled with ones and zeros differ at most by one. Cordial labelings were introduced by Cahit (1987) as a weakened version of GRACEFUL and HARMONIOUS . An EULER GRAPH is not cordial if the number of its vertices is multiple of four. For example, all TREES are cordial, CYCLE GRAPHS of length n are cordial if n is not a multiple of four, COMPLETE GRAPHS on n vertices are cordial if n B4, and the WHEEL GRAPH on n /C271 vertices is cordial IFF n is not congruent to 3 modulo 4. See also GRACEFUL GRAPH ,H ARMONIOUS GRAPH , LABELED GRAPH References Cahit, I. "Cordial Graphs: A Weaker Version of Graceful and Harmonious Graphs." Ars Combin. 23, 201 /C1/08, 1987.Cordiform Projection WERNER PROJECTION Cork Plug A 3-D SOLID which can stopper a SQUARE , TRIANGU- LAR,or CIRCULAR HOLE . There is an infinite family of such shapes. The one with smallest VOLUME has TRIANGULAR CROSS SECTIONS and V /C30 pr3; that with the largest VOLUME is made using two cuts from the top diameter to the EDGE and has VOLUME V /C304 pr3 =3:/ See also CROSS SECTION ,STEREOLOGY ,TRIP-LET Corkscrew Surface A surface also called the TWISTED SPHERE . References Gray, A. "The Corkscrew Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 477 /C1/78, 1997. Cornish-Fisher Asymptotic Expansion y:m/C27sw; where w/C30x/C27[g1h1(x)]/C27[g2h2(x)/C27g2 1h11(x)] /C27[g3h3(x)/C27g1g2h12(x)/C27g31h111(x)] /C27[g4h4(x)/C27g22h22(x)/C27g1g3h13(x)]/C27g21g2h112(x) /C27g41h1111(x)]/C27...; where h1(x) /C301 6 He2(x) h2(x) /C301 24 He3(x) h11(x) /C30/C281 36[2He3(x) /C27He1(x)] h3(x) /C301 120 He4(x) h12(x) /C30/C281 24[He4(x) /C27He2(x)] h111(x) /C301 324[12He4(x) /C2719He2(x)] h4(x) /C301 720 He5(x) h22(x) /C30/C281 384[3He5(x) /C276He3(x) /C272He1(x)] h13(x) /C30/C281 180[2He5 /C273He3(x)] h112(x) /C301 288[14He5(x) /C2737He3(x) /C278He1(x)] h1111(x) /C30/C281 7776[252He5(x) /C27832He3(x) /C27227He1(x)] : See also CHARLIER SERIES ,EDGEWORTH SERIES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 935, 1972. Cornish, E. A. and Fisher, R. A. "Moments and Cumulants in the Specification of Distributions." Extrait de la Revue de l’Institute International de Statistique 4,1/C1/4, 1937. Reprinted in Fisher, R. A. Contributions to Mathematical Statistics. New York: Wiley, 1950. Wallace, D. L. "Asymptotic Approximations to Distribu- tions." Ann. Math. Stat. 29, 635/C1/54, 1958. Wasow, W. "On the Asymptotic Transformation of Certain Distributions into the Normal Distribution." Proceedings of Symposia in Applied Mathematica VI, Numerical Analysis . New York: McGraw-Hill, pp. 251 /C1/59, 1956. Cornu Spiral A plot in the COMPLEX PLANE of the points B(t)/C30S(t)/C27iC(t); (1) where S(t) and C(t) are the F RESNEL INTEGRALS (von Seggern 1993, p. 210; Gray 1997, p. 65). The Cornu spiral is also known as the CLOTHOID or E ULER’SSPIRAL . It was probably first studied by Johann Bernoulli around 1696 (Bernoulli 1967, pp. 1084 /C1/ 086). A Cornu spiral describes diffraction from the edge of a HALF-PLANE . The quantities C(t)=S(t) and S(t)=C(t) are plotted above. The SLOPE of the curve’s TANGENT VECTOR (above right figure) is mT(t)/C30S?(t) C?(t)/C30tan1 2pt2l11)l117 ; (2) plotted below. The C ESA`RO EQUATION for a Cornu spiral is r/C30c2=s; where ris the RADIUS OF CURVATURE and sthe ARC LENGTH . The TORSION ist/C300:/ Gray (1997) defines a generalization of the Cornu spiral given by PARAMETRIC EQUATIONS x(t) /C30agt 0sinun/C271 n /C27 1 ! du (3) /C30atn/C272 (n /C27 1)(n /C27 2) /C21F21 2 /C271 2(n /C27 1);32 ;32 /C271 2(n /C27 1); /C28t2(n/C271) 4(n /C27 1)2 ! (4) y(t) /C30agt 0cosun/C271 n /C27 1 ! du (5) /C30at1F21 2(n /C27 1);12 ; 1 /C271 2(n /C27 1); /C28t2(n/C271) 4(n /C27 1)2 ! ; (6) where1F2(a; b; c; x)isa GENERALIZED HYPERGEO- METRIC FUNCTION . The ARC LENGTH , CURVATURE , and TANGENTIAL ANGLE of this curve are s(t) /C30at (7) k(t) /C30/C28tn a (8) f(t) /C30/C28tn/C271 n /C27 1 : (9) The CESA` RO EQUATION is k /C30/C28sn an/C271 : (10) Dillen (1990) describes a class of "polynomial spirals"for which the CURVATURE is a polynomial function of the ARC LENGTH . These spirals are a further general- ization of the Cornu spiral. The curves plotted above correspond to k /C30s ; k /C30s2 ; k /C30s2 /C282:19 ; k /C30s2 /C284; k /C30s2 /C271; and k /C305s4 /C2818s2 /C275 ; respectively. See also FRESNEL INTEGRALS ,NIELSEN’S SPIRAL References Bernoulli, J. Opera, Tomus Secundus. Brussels, Belgium: Culture er Civilisation, 1967. Dillen, F. "The Classification of Hypersurfaces of a Eucli- dean Space with Parallel Higher Fundamental Form." Math. Z. 203, 635 /C1/43, 1990. Gray, A. "Clothoids." §3.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 64 /C1/6, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 190 /C1/91, 1972. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, 1993. Cornucopia The SURFACE given by the PARAMETRIC EQUATIONS x /C30ebv cos v /C27eav cos u cos v y /C30ebv sin v /C27eav cos u sin v z /C30eav sin u: References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 304, 1993. Corollary An immediate consequence of a result already proved. Corollaries usually state more complicated THEOREMS in a language simpler to use and apply. See also LEMMA ,PORISM ,THEOREM Corona (Polyhedron) AUGMENTED SPHENOCORONA ,HEBESPHENOMEGACOR- ONA,SPHENOCORONA ,SPHENOMEGACORONA Corona (Tiling) The first corona of a TILE is the set of all tiles that have a common boundary point with that tile (includ- ing the original tile itself). The second corona is the set of tiles that share a point with something in the first corona, and so on. References Eppstein, D. "Heesch’s Problem." http://www.ics.uci.edu/ ~eppstein/junkyard/heesch/. Correlation The degree of association between two or more quantities. In a 2-D plot, the degree of correlation between the values on the two axes is quantified by the so-called CORRELATION COEFFICIENT . See also AUTOCORRELATION ,C ORRELATION COEFFI- CIENT ,C ORRELATION (GEOMETRIC ), CORRELATION (STATISTICAL ), CROSS- CORRELATION References Kenney, J. F. and Keeping, E. S. "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252 /C1/85, 1962. Whittaker, E. T. and Robinson, G. "Correlation." Ch. 12 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 317 /C1/42, 1967. Correlation (Geometric) A point-to-line and line-to-point TRANSFORMATION which transforms points A into lines a ? and lines b into points B?such that a?passes through B?IFFA? lies on b. See also LINE,POINT ,POLARITY ,PROJECTIVE CORRE- LATION References Coxeter, H. S. M. "Collineations and Correlations." §14.6 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 247 /C1/52, 1969. Correlation (Statistical) For two variables xandy, the correlation is defined by cor(x;y)/C13cov(x;y) sxsy; (1) where sxdenotes STANDARD DEVIATION and cov( x;y) is the COVARIANCE of these two variables. For the general case of variables xiandxj;where i;j/C301;2, ..., n, cor(xi;xj)/C30cov(xi;yj)ffiffiffiffiffiffiffiffiffiffiffiffiViiVjjp ; (2) where Viiare elements of the COVARIANCE MATRIX .I n general, a correlation gives the strength of the relationship between variables. For i/C30j,cor(xi;xi)/C30cov(xi;xi) si/C30sii si/C30s2 i si/C30si: (3) The variance of any quantity is always NONNEGATIVE by definition, so varx sx/C27y sy ! ]0: (4) From a property of VARIANCES , the sum can be expanded varx sx ! /C27vary sy ! /C272covx sx;y sy ! ]0 (5) 1 s2 xvar(x)/C271 s2yvar(y)/C272 sxsycov(x;y)]0 (6) 1/C271/C272 sxsycov(x;y)/C302/C272 sxsycov(x;y)]0:(7) Therefore, cor(x;y)/C30cov(x;y) sxsy]/C281: (8) Similarly, varx sx ! /C28y sy ! ]0 (9) varx sx ! /C27var/C28y sy ! /C272 covx sx;/C28y sy ! ]0 (10) 1 s2 xvar(x)/C271 s2yvar(y)/C282 sxsycov(x;y)]0 (11) 1/C271/C282 sxsycov(x;y)/C302/C282 sxsycov(x;y)]0:(12) Therefore, cor(x;y)/C30cov(x;y) sxsy51; (13) so/C2815cor(x;y)51:For a LINEAR COMBINATION of two variables, var(y/C28bx)/C30var(y)/C27var(/C28bx)/C302 cov( y;/C28bx) /C30var(y)/C27b2var(x)/C282bcov(x;y) /C30s2 y/C27s2x/C282bcov(x;y): (14) Examine the cases where cor( x;y)/C3091; cor(x;y)/C13cov(x;y) sxsy/C3091 (15) var(y/C28bx)/C30b2s2x/C27s2y/C142bsxsy/C30(bsx/C14sy)2:(16) The VARIANCE will be zero if b/C139sy=sx;which requires that the argument of the VARIANCE is a constant. Therefore, y /C28bx /C30a; so y /C30a /C27bx : If cor(x ; y) /C3091; y is either perfectly correlated (b /C210) or perfectly anticorrelated (b B0) with x. See also COVARIANCE ,C OVARIANCE MATRIX ,V AR- IANCE Correlation Coefficient The correlation coefficient is a quantity which gives the quality of a LEAST SQUARES FITTING to the original data. To define the correlation coefficient, first con- sider the sum of squared values ssxx;ssxy;and ssyyof a set of ndata points ( xi;yi) about their respective means, ssxx/C13X (xi/C28¯x)2(1) /C30X x2/C282¯xX x/C27X ¯x2 /C30X x2/C282n¯x2/C27n¯x2/C30X x2/C28n¯x2(2) ssyy/C13X (yi/C28¯y)2(3) /C30X y2/C282¯yX y/C27X ¯y2 /C30X y2/C282n¯y2/C27n¯y2/C30X y2/C28n¯y2(4) ssxy/C13X (xi/C28¯x)(yi/C28¯y) (5) /C30X (xiyi/C28¯xyi/C28xi¯y/C27¯x¯y) /C30X xy/C28n¯x¯y/C28n¯x¯y/C27n¯x¯y/C30X xy/C28n¯x¯y: (6) For linear LEAST SQUARES FITTING , the COEFFICIENT b in y/C30a/C27bx (7) is given by b/C30nPxy/C28PxPy nPx2/C28(Px)2/C30ssxy ssxx; (8) and the COEFFICIENT b?in x/C30a?/C27b?y (9) is given by b?/C30nPxy/C28PxPy nPy2/C28(Py)2: (10) The correlation coefficient r2(sometimes also denoted R2) is then defined by r/C13ffiffiffiffiffiffiffi bb?p /C30nPxy/C28PxPyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [nPx2/C28(Px)2][nPy2/C28(Py)2]q ;(11) which can be written more simply as r2/C30ss2 xy ssxxssyy: (12) The correlation coefficient is also known as the PRODUCT-MOMENT COEFFICIENT OF CORRELATION or PEARSON’S CORRELATION . The correlation coefficients for linear fits to increasingly noisy data are shown above. The correlation coefficient has an important physical interpretation. To see this, define A/C13X x2/C28n¯x2hi/C281 (13) and denote the "expected" value for yiasˆyi:Sums of ˆyi are then ˆyi/C30a/C27bxi/C30¯y/C28b¯x/C27bxi/C30¯x/C27b(xi/C28¯x) /C30A(¯yX x2/C28¯xX xy/C27xiX xy/C28n¯x¯yxi) /C30A[¯yX x2/C27(xi/C28¯x)X xy/C28n¯x¯yxi] (14) X ˆyi/C30A(n¯yX x2/C28n2¯x2¯y) (15) X ˆy2 i/C30A2[n¯y2(X x2)2/C28n2¯x2¯y2(X x2) /C282n¯x¯y(X xy)(X x2)/C272n2¯x3¯y(X xy) /C27(X x2)(X xy)2/C28n¯x2(X xy)] (16) X yiˆyi/C30AX [yi¯yX x2/C27yi(xi/C28¯x) /C2X xy/C28n¯x¯yxiyi] /C30A[n¯y2X x2/C27(X xy)2/C28n¯x¯y /C2X xy/C28n¯x¯y(X xy)] /C30A[n¯y2X x2/C27(X xy)2/C282n¯x¯yX xy]: (17) The sum of squared residuals is then SSR/C13X (ˆyi/C28¯y)2/C30X (ˆy2i/C282¯yˆyi/C27¯y2) /C30A2(X xy/C28n¯x¯y)2(X x2/C28n¯x2)/C30(Pxy/C28n¯x¯y)2 Px2/C28n¯x2 /C30bssxy/C30ss2 xy ssxx/C30ssyyr2/C30b2ssxx; (18) and the sum of squared errors is SSE /C13X (yi /C28 ˆyi)2 /C30X (yi /C28 ¯y /C28b¯x /C28bxi)2 /C30X [yi /C28 ¯y /C28b(xi /C28 ¯x)]2 /C30X (yi /C28 ¯y)2 /C27b2X (xi /C28 ¯x)2 /C282b /C2X (xi /C28 ¯x)(yi /C28 ¯y) /C30ssyy /C27b2 ssxx /C282bssxy : (19) But b /C30ssxy ssxx(20) r2 /C30ss2 xy ssxxssyy; (21) so SSE /C30ssyy /C27ss2xy ss2 xxssxx /C282ssxy ssxxssxy (22) /C30ssyy /C28ss2 xy ssxx(23) /C30ssyy1 /C28ss2xy ssxxssyy ! (24) /C30ssyy(1 /C28r2); (25) and SSE /C27SSR /C30ssyy(1 /C28r2) /C27ssyyr2 /C30ssyy : (26) The square of the correlation coefficient r2 is there- fore given by r2 /C13SSR ssyy/C30ss2xy ssxxssyy/C30(P xy /C28 n¯x¯y)2 (P x2 /C28 n¯x2)(P y2 /C28 n¯y2) : (27) In other words, r2 is the proportion of ssyywhich is accounted for by the regression. If there is complete correlation, then the lines obtained by solving for best-fit (a, b) and (a ?; b ?) coincide (since all data points lie on them), so solving (9) for y and equating to (7) gives y /C30/C28a ? b?/C27x b?/C30a /C27bx: (28) Therefore, a /C30/C28a?=b? and b /C301=b?; giving r2/C30bb?/C301: (29) The correlation coefficient is independent of both origin and scale, so r(u;v)/C30r(x;y); (30) where u/C13x/C28x0 h(31)v/C13y/C28y0 h: (32) See also CORRELATION INDEX ,CORRELATION COEFFI- CIENT– GAUSSIAN BIVARIATE DISTRIBUTION ,CORRELA- TION RATIO,LEAST SQUARES FITTING ,REGRESSION COEFFICIENT ,SPEARMAN RANK CORRELATION COEFFI- CIENT References Acton, F. S. Analysis of Straight-Line Data. New York: Dover, 1966. Kenney, J. F. and Keeping, E. S. "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252 /C1/85, 1962. Gonick, L. and Smith, W. "Regression." Ch. 11 in The Cartoon Guide to Statistics. New York: Harper Perennial, pp. 187 /C1/10, 1993. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Linear Correlation." §14.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 630 /C1/33, 1992. Whittaker, E. T. and Robinson, G. "The Coefficient of Correlation for Frequency Distributions which are notNormal." §166 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 334 /C1 /36, 1967. Correlation Coefficient * /Gaussian Bivariate Distribution For a G AUSSIAN BIVARIATE DISTRIBUTION , the distri- bution of correlation COEFFICIENTS is given by P(r)/C301 p(N/C282)(1/C28r2)(N/C284)=2 /C2(1/C28r2)(N/C281)=2g/C12 0db (cosh b/C28rr)N/C281 /C301 p(N/C282)(1/C28r2)(N/C284)=2(1/C28r2)(N/C281)=2ffiffiffi p 2s G(N/C281) GN/C281 2l11)l117 /C29(1/C28rr)/C28(N/C283=2) 2F11 2;12;2N/C281 2;rr/C271 2 ! /C30(N/C282)G(N/C281)(1/C28r2)(N/C281)=2(1/C28r2)(N/C284)=2 ffiffiffiffiffiffi 2pp GN/C281 2l11)l117 (1/C28rr)N/C283=2 /C21/C271 4rr/C271 2N/C281/C279 16(rr/C271)2 (2N/C281)(2N/C271)/C27/C1/C1/C1"# ; (1) where ris the population correlation COEFFICIENT , 2F1(a;b;c;x)i sa HYPERGEOMETRIC FUNCTION , and G(z) is the GAMMA FUNCTION (Kenney and Keeping 1951, pp. 217 /C1/21). The MOMENTS are /C142r/C143/C30r/C28r(1/C28r2) 2n(2) var(r)/C30(1/C28r2)2 n1/C2711r2 2n/C27/C1/C1/C1 ! (3) g1/C306rffiffiffinp 1/C2777r2/C2830 12n/C27/C1/C1/C1 ! g2/C306 n(12r2/C281)/C27...; (4) where n/C13n/C281:If the variates are uncorrelated, then r/C300 and 2f11 2;12;2n/C281 2;rr/C271 2 ! /C302F112;12;2N/C281 2;12 ! /C30GN/C281 2l11)l117 23=2/C28Nffiffiffipp GN 2 !"#2 ; (5) so P(r)/C30(N/C282)G(N/C281)ffiffiffiffiffiffi 2pp GN/C281 2l11)l117 /C2(1/C28r2)(N/C284)=2GN/C2812l11)l117 23=2/C28Nffiffiffipp GN 2 !"#2 /C3021/C28N(N/C282)G(N/C281) GN 2 !"#2 (1/C28r2)(N/C284=2): (6) But from the L EGENDRE DUPLICATION FORMULA , ffiffiffippG(N/C281)/C302N/C282GN 2 ! GN/C281 2 ! ; (7) so P(r)/C30(21/C28N)(2N/C282)(N/C282)GN 2 ! GN/C281 2 ! ffiffiffippGN 2 !"#2 /C2(1/C28r2)(N/C284)=2/C30(N/C282)GN/C281 2 ! 2ffiffiffippGN 2 ! (1/C28r2)(N/C284)=2 /C301ffiffiffippn 2Gn/C271 2 ! Gn 2/C271 ! (1/C28r2)(n/C282)=2 /C301ffiffiffippGn/C271 2 ! Gn 2 ! (1/C28r2)(n/C282)=2: (8) The uncorrelated case can be derived more simply by letting bbe the true slope, so that h/C30a/C27bx:Then t/C13(b/C28b)Sx Syffiffiffiffiffiffiffiffiffiffiffiffiffiffi N/C282 1/C28r2s /C30(b/C28b)r bffiffiffiffiffiffiffiffiffiffiffiffiffiffi N/C282 1/C28r2s (9) is distributed as S TUDENT’S Twith n/C13N/C282DEGREES OF FREEDOM . Let the population regression COEFFI- CIENT rbe 0, then b/C300;so t/C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1/C28r2s ; (10) and the distribution is P(t)dt/C301ffiffiffiffiffinppGn/C271 2 ! Gn 2 ! 1/C27t2 n !(n/C271)=2dt: (11) Plugging in for tand using dt/C30ffiffiffinpffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p /C28r1 2l11)l117 (/C282r)(1/C28r2)/C281=2 1/C28r22 435dr /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1/C28r2s 1/C28r2/C27r2 1/C28r2 ! dr/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n (1/C28r)3s dr (12) gives P(t)dt/C301ffiffiffiffiffinppGn/C271 2 ! Gn 2 ! 1/C27r2n (1/C28r2)n"#(n/C271)=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n (1/C28r)3s dr /C30(1/C28r2)/C283=2 ffiffiffippGn/C271 2l11)l117 Gn 2l11)l117 1 1/C28r2l11)l117(n/C271)=2dr /C301ffiffiffippGn/C271 2 ! Gn 2 ! (1/C28r2)/C283=2(1/C28r2)(n/C271)=2dr /C301ffiffiffippGn/C271 2 ! Gn 2 ! (1/C28r2)(n/C282)=2dr; (13) so P(r)/C301ffiffiffippGn/C271 2l11)l117 Gn 2l11)l117 (1/C28r2)(n/C282)=2(14) as before. See Bevington (1969, pp. 122 /C1/23) or Pugh and Winslow (1966, §12/C1/). If we are interested instead in the probability that a correlation COEFFI- CIENT would be obtained ]½r½;where ris the observed COEFFICIENT , then 392 Let I/C131 2(n/C282):For EVEN n;the exponent Iis an INTEGER so, by the BINOMIAL THEOREM , (1/C28r2)I/C30XI k/C300I kl11sl11n (/C28r2)k(17) and Pc(r)/C301/C282ffiffiffippGn/C271 2 ! Gn 2 ! /C2(/C281)k I! (I/C28k)!k!grjj 0XI k/C300r?2kdr? /C301/C282ffiffiffippGn/C271 2 ! Gn 2 ! /C2XI k/C300(/C281)k I! (I/C28k)!k!½r½2k/C271 2k/C271"# : (18) For ODDn;the integral is Pc(r)/C301/C282g½r½ 0P(r?)dr? /C301/C282ffiffiffippGn/C271 2 ! Gn 2 !g½r½ 0(ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p )n/C282dr:(19) Letr/C13sinxsodr/C30cosxd x ;thenPc(r)/C301/C282ffiffiffippGn/C271 2 ! Gn 2 !gsin/C281rjj 0cosn/C282xcosxd x /C301/C282ffiffiffippGn/C271 2 ! Gn 2 ! /C27gsin/C281rjj 0cosn/C281xd x : (20) ButnisODD,s on/C281/C132nisEVEN . Therefore 2ffiffiffippGn/C271 2 ! Gn 2 ! /C302ffiffiffippG(n/C271) Gn/C271 2l11)l117 /C302ffiffiffippn! (2n/C281)!!ffiffiffipp 2n /C302 p2nn! p(2n/C281)!!/C302 p(2n)!! (2n/C281)!!: (21) Combining with the result from the COSINE INTEGRAL gives Pc(r)/C301/C282 p(2n)!!(2n/C281)!! (2n/C281)!!(2 n)!! /C2sinxXn/C281 k/C300(2k)!! (2k/C271)!!cos2k/C271x/C27x"# sin/C281rjj 0: (22) Use cos2k/C281x/C30(1/C28r2)(2k/C281)=2/C30(1/C28r2)(k/C281=2); (23) and define J/C13n/C281/C30(n/C283)=2;then Pc(r)/C301/C282 p /C2sin/C281rjj/C27rjjXJ k/C300(2k)!! (2k/C271)!!(1/C28r2)k/C271=2"# : (24) (In Bevington 1969, this is given incorrectly.) Com- bining the correct solutions Pc(r)/C301/C282ffiffiffippG[(n/C271)=2] G(n=2)XI k/C300(/C281)k I! (1/C28k)!k!rjj2k/C271 2k/C271"# forneven 1/C282 psin/C281rjj/C27rjjXJ k/C300(2k)!! (2k/C271)!!(1/C28r2)k/C271=2"# fornodd8 >>>>>>>>< >>>>>>>>: (25) Ifr"0;a skew distribution is obtained, but the variable zdefined by z/C13tanh/C281r (26) is approximately normal with mz /C30tanh/C281 r (27) s2 z /C301 N /C28 3 (28) (Kenney and Keeping 1962, p. 266). Let bj be the slope of a best-fit line, then the multiple correlation COEFFICIENT is R2 /C13Xn j/C301bjs2 jy s2 y ! /C30Xn j/C301bjsj syrjy ! ; (29) where sjy is the sample VARIANCE . On the surface of a SPHERE , r /C13g fg dV g fdVg gdV; (30) where dV is a differential SOLID ANGLE . This defini- tion guarantees that /C281 Br B1: If f and g are expanded in REAL SPHERICAL HARMONICS , f( u; f) /C13X/C12 l/C300Xl m/C300[Cm lYmc l( u; f) sin(mf) /C27SmlYms l(u ; f)] (31) g(u ; f) /C13X/C12 t/C300Xl m/C300[AmlYmc l( u; f)sin(mf) /C27BmlYms l( u; f)] : (32) Then r1 /C30Pl m/C300(Cm lAml/C27 SmlBml)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPl m/C300(Cm2 l/C27 Sm2 l)qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPlm/C300(Am2 l/C27 Bm2 l)q : (33) The confidence levels are then given by G1(r) /C30r G2(r) /C30r 1 /C271 2 s2l11)l117 /C3012 r(3 /C28r2) G3(r) /C30r 1 /C271 2 s2 1 /C2734 s2l11)l117hi /C3018 r(15 /C2810r2 /C273r4) G4(r) /C30r 1 /C271 2 s2 1 /C2734 s2 1 /C2756 s2l11)l117hino /C301 16 r(35 /C2835r2 /C2721r4 /C285r6) ; where s /C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28r2p (34) (Eckhardt 1984). See also FISHER’S Z’-TRANSFORMATION ,S PEARMAN RANK CORRELATION COEFFICIENT ,SPHERICAL HAR- MONICReferences Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1969. Eckhardt, D. H. "Correlations Between Global Features of Terrestrial Fields." Math. Geology 16, 155 /C1/71, 1984. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Pugh, E. M. and Winslow, G. H. The Analysis of Physical Measurements. Reading, MA: Addison-Wesley, 1966. Correlation Dimension Define the correlation integral as C(e) /C13 lim n0/C121 N2X/C12 i; j/C301 i"jH( e /C28 xi /C28xjl119l119l119l119) ; (1) where H is the HEAVISIDE STEP FUNCTION . When the below limit exists, the correlation dimension is then defined as D2 /C13dcor /C13 lim e; e?00 /C27lnC( e) C(e?)"# lne e ? ! : (2) If n is the CORRELATION EXPONENT , then lim e00n 0 D2 : (3) It satisfies dcor 5dinf 5dcap /C30?dLya : (4) To estimate the correlation dimension of an M- dimensional system with accuracy (1 /C28Q) requires Nmin data points, where Nmin ]R(2 /C28 Q) 2(1 /C28 Q)"#M ; (5) where R ]1 is the length of the "plateau region." If an ATTRACTOR exists, then an estimate of D2saturates above some M given by M ]2D /C271; (6) which is sometimes known as the fractal Whitney embedding prevalence theorem. See also CORRELATION EXPONENT , Q-DIMENSION References Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 547 /C1/48, 1995. Correlation Exponent A measure n of a STRANGE ATTRACTOR which allows the presence of CHAOS to be distinguished from random noise. It is related to the CAPACITY DIMENSION D and INFORMATION DIMENSION s; satisfying n 5 s 5D: (1) It satisfies n 5DKY ; (2) where DKYis the KAPLAN- YORKE DIMENSION . As the cell size goes to zero, lim e 00n 0 D2 ; (3) where D2 is the CORRELATION DIMENSION . See also CORRELATION DIMENSION ,INFORMATION DIMENSION ,KAPLAN- YORKE DIMENSION References Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/08, 1983. Correlation Index Given a curved regression, the correlation index is defined by rc /C13syˆy sysˆy; where syand sˆyare the standard deviations of the data points y and the estimates ˆy given by the regression line, and the quantity syˆyis not defined by Kenney and Keeping 1962. Then r2 c /C30s2 ˆy s2 y/C301 /C28s2 ey s2 y; where s2 ey is the variance of the observed ys about the best-fitting curved line (Kenney and Keeping 1962, p. 293). See also CORRELATION COEFFICIENT ,REGRESSION References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Correlation Integral Consider a set of points /Xi/ on an ATTRACTOR , then the correlation integral is C(l) /C13 lim N 0/C121 N2f ; where f is the number of pairs (i, j) whose distance Xi /C28Xjl112l112l112l112B l: For small l,C(l) /C2ln ; where n is the CORRELATION EXPONENT . References Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/08, 1983. Correlation Ratio Let there be Ni observations of the ith phenomenon, where i /C301, ..., p and N /C13X Ni (1) ¯yi /C131 NiX ayia (2) ¯y /C131 NX iX ayia : (3) Then E2 yx /C13P iNi(¯yi /C28 ¯y)2 P iP a(yia /C28 ¯y)2 : (4) Let hyxbe the population correlation ratio. If Ni /C30Nj for i "j; then f(E2) /C30e /C28 l(E2)a /C281(1 /C28 E2)b /C281 1F1(a ; b; lE2) B(a ; b) ; (5) where l /C13N h2 2(1 /C28 h2) (6) a/C13n1 2(7) b/C13n2 2(8) and1F1(a;b;z) is the CONFLUENT HYPERGEOMETRIC LIMIT FUNCTION .I fl/C300;then f(E2)/C30b(a;b) (9) (Kenney and Keeping 1951, pp. 323 /C1/24). See also CORRELATION COEFFICIENT ,R EGRESSION COEFFICIENT References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Cos COSINE Cosecant The function defined by csc x /C131 =sin x; where sin x is the SINE. The MACLAURIN SERIES of the cosecant function is csc x /C301 x /C271 6 x /C277 360 x3 /C2731 15120 x5 /C27... /C27(/C281)n/C2712(22n/C281 /C28 1)B2n (2n)! x2n/C281 /C27... ; where B2n is a BERNOULLI NUMBER . See also INVERSE COSECANT ,SECANT ,SINE References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/9, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215, 1987. Spanier, J. and Oldham, K. B. "The Secant sec(x) and Cosecant csc(x) Functions." Ch. 33 in An Atlas of Func- tions. Washington, DC: Hemisphere, pp. 311 /C1/18, 1987. Coset This entry contributed by NICOLAS BRAY For a SUBGROUP H of a GROUP G and an element x of G, define xH /to be the set fxh : h /C23 H g and Hx to be the set fhx : h /C23 H g: A SUBSET of G of the form xH for some x /C23 G is said to be a LEFT COSET of H and a subset of the form Hx is said to be a RIGHT COSET of H. For any SUBGROUP H, we can define an EQUIVALENCE RELATION /C2 by x /C2y if x /C30yh for some h /C23 H : The EQUIVALENCE CLASSES of this EQUIVALENCE RELATIONare exactly the LEFT COSETS of H, and an element x of G is in the EQUIVALENCE CLASS xH. Thus the LEFT COSETS of H form a partition of G. It is also true that any two LEFT COSETS of H have the same CARDINALITY , and in particular, every coset of H has the same CARDINALITY as eH /C30H, where e is the IDENTITY ELEMENT . Thus, the CARDINALITY of any LEFT COSET of H has CARDINALITY the order of H. The same results are true of the RIGHT COSETS of G as well and, in fact, one can prove that the set of LEFT COSETS of H has the same CARDINALITY as the set of RIGHT COSETS ofH. See also EQUIVALENCE CLASS ,GROUP ,LEFT COSET , QUOTIENT GROUP ,RIGHT COSET ,SUBGROUP Cosh HYPERBOLIC COSINE CoshIntegral CHI Cosine One of the basic TRIGONOMETRIC FUNCTIONS encoun- tered in TRIGONOMETRY . Let ube an ANGLE measured counterclockwise from the X-AXIS along the arc of the unit CIRCLE . Then cos uis the horizontal coordinate of the arc endpoint. As a result of this definition, the cosine function is periodic with period 2 p:/ The definition of the cosine function can be extendedto complex arguments zusing the definition cosz/C30 1 2(eiz/C27e/C28iz); (1) where eis the base of the NATURAL LOGARITHM andi is the IMAGINARY NUMBER . A related function known as the HYPERBOLIC COSINE is similarly defined, cosh z /C301 2(ez /C27e /C28z) : (2) The cosine function has a FIXED POINT at 0.739085. The cosine function can be defined algebraically using the infinite sum cos x /C13X/C12 n/C300(/C281)nx2n (2n)!/C301 /C28x2 2! /C27x4 4! /C28x6 6! /C27...; (3) or the INFINITE PRODUCT cos x /C30Y/C12 n/C3011 /C284x2 p2(2n /C28 1)2"# : (4) A close approximation to cos(x) for x /C23 [0; p=2] is cosp 2x ! :1 /C28x2 x /C27 (1 /C28 x)ffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28 x 3s (5) (Hardy 1959). The difference between cos x and Hardy’s approximation is plotted below. The cosine obeys the identity cos(nu) /C302 cos u cos[(n /C281)u] /C28cos[(n /C282)u] (6) and the MULTIPLE-ANGLE FORMULA cos(nx) /C30Xn k /C300n kl11sl11n cosk x sinn /C28k x cos[12(n /C28k) p]; (7) wheren kl1ml11 is a BINOMIAL COEFFICIENT . Summing the COSINE of a multiple angle from n /C300to N /C281 can be done in closed form using XN /C281 n/C300cos(nx) /C30RXN /C281 n/C300einx"# ; (8) where R[z] is the REAL PART of z. The EXPONENTIAL SUM FORMULAS give XN n/C301cos(nx) /C30Rsin(12 Nx) sin(1 2 x)ei(N /C271)x =2"# /C30sin(1 2 Nx) sin(1 2 x)cos[1 2 x(N /C271)] : (9)Similarly, X/C12 n/C300pn cos(nx) /C30RX/C12 n/C300pnein x"# ; (10) where ½p ½B1: The EXPONENTIAL SUM FORMULA gives X/C12 n/C300pn cos(nx) /C30R1 /C28 pe/C28ix 1 /C28 2p cos x /C27 p2"# /C301 /C28 p cos x 1 /C28 2p cos x /C27 p2 : (11) The sum of cos2(kx) can also be done in closed form, XN k /C300cos2(kx) /C3014f3 /C272N /C27csc x sin[x(1 /C272N)] g: (12) The FOURIER TRANSFORM of cos(2 pk0x) is given by F[cos(2 pk0x)] /C30g/C12 /C28/C12e/C282 pikx cos(2 pk0x) dx /C3012[ d(k /C28k0) /C27 d(k /C27k0)]; (13) where d(k) is the DELTA FUNCTION . Cvijovic and Klinowski (1995) note that the following series Cn(a)/C30X/C12 k/C300cos(2 k/C271)a (2k/C271)n(14) has closed form for n/C302n; C2n(a)/C30(/C281)n 4(2n/C281)!p2nE2n/C281a p ! ; (15) where En(x)i sa nE ULER POLYNOMIAL . See also EULER POLYNOMIAL ,E XPONENTIAL SUM FORMULAS ,F OURIER TRANSFORM– COSINE ,H YPER- BOLIC COSINE ,S INE,T ANGENT ,T RIGONOMETRIC FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/9, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215, 1987. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 68, 1959. Cvijovic, D. and Klinowski, J. "Closed-Form Summation of Some Trigonometric Series." Math. Comput. 64, 205/C1/10, 1995. Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975. Project Mathematics . "Sines and Cosines, Parts I-III." Videotape. http://www.projmath.caltech.edu/sincos1.htm. Spanier, J. and Oldham, K. B. "The Sine /sin(x)/ and Cosine cos(x) Functions." Ch. 32 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 295 /C1/10, 1987. Cosine Apodization Function The APODIZATION FUNCTION A(x) /C30cospx 2a ! : Its FULL WIDTH AT HALF MAXIMUM is 4a =3: Its INSTRUMENT FUNCTION is I(k) /C304a cos(2 pak) p(1 /C28 16a2k2) : See also APODIZATION FUNCTION Cosine Circle Draw ANTIPARALLELS through the SYMMEDIAN POINT K. The points where these lines intersect the sides then lie on a CIRCLE , known as the cosine circle (or sometimes the second LEMOINE CIRCLE ), which has center at K. The CHORDS P2Q3 ; P3Q1 ; and P1Q2are proportional to the COSINES of the ANGLES of DA1A2A3 ; giving the circle its name. The center of the cosine circle is the CIRCUMCENTER O of DABC :/ TRIANGLES P1P2P3and DA1A2A3 are directly similar, and TRIANGLES DQ1Q2Q3 and A1A2A3 are similar. The MIQUEL POINT of DP1P2P3 is at the BROCARD POINT V of DP1P2P3 :/ The cosine circle is a special case of a TUCKER CIRCLE . See also BROCARD POINTS ,E XCOSINE CIRCLE ,LE- MOINE CIRCLE ,M IQUEL POINT ,T AYLOR CIRCLE , TUCKER CIRCLESReferences Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 66, 1971. Honsberger, R. "The Lemoine Circles." §9.2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 88 /C1/9, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 271 /C1/73, 1929. Lachlan, R. "The Cosine Circle." §129 /C1/30 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 75, 1893. Cosine Hexagon The closed cyclic self-intersecting hexagon formed by joining the adjacent ANTIPARALLELS in the construc- tion of the COSINE CIRCLE . The sides of this hexagon have the property that, in addition to P1Q2 ; P2Q3 ; and P3Q1being ANTIPARALLEL to /A1A2 ; A2A3 ; A1A3/, the remaining sides P1Q1 ½½A2A3 ; P2Q2 ½½A1A3 ; and P3Q3 ½½A1A2 : The cosine hexagon is a special case of a TUCKER HEXAGON . See also COSINE CIRCLE ,LEMOINE HEXAGON ,TUCKER HEXAGON Cosine Integral There are (at least) three types of "cosine integrals," denoted ci( x);Ci(x);and Cin( x): ci(x)/C13/C28g/C12 xcostd t t(1) /C301 2[ei(ix)/C27ei(/C28ix)] (2) /C30/C2812[E1(ix)/C27E1(/C28ix)]; (3) Ci(x)/C13g/C27lnz/C27gz 0cost/C281 tdt (4) Cin(x)/C13gz 0(1/C28cost)dt t(5) /C30/C28Ci(x)/C27lnx/C27g: (6) Here, ei( x) is the EXPONENTIAL INTEGRAL ,En(x) is the EN-FUNCTION , and gis the E ULER- MASCHERONI CON- STANT . ci(x) is the function returned by the Mathe- matica command CosIntegral [x] and displayed above. /ci(x) has zeros at 0.616505, 3.38418, 6.42705, .... Extrema occur when ci?(x)/C30cosx x/C300; (7) or cos x/C300;orp=2;3p=2;5p=2;..., which are alter- nately maxima and minima. At these points, ci( x) equals 0.472001, /C280:198408 ;0.123772, .... Inflection points occur when ciƒ(x)/C30/C28cosx x2/C28sinx x/C300; (8) which simplifies to 1/C27xtanx/C300; (9) which has solutions 2.79839, 6.12125, 9.31787, .... To compute the integral of an EVEN power times a cosine, I/C13gx2ncos(mx)dx; (10) use INTEGRATION BY PARTS . Let u/C30x2ndv/C30cos(mx)dx (11) du/C302nx2n/C281dx v/C301 msin(mx); (12)so I/C301 mx2nsin(mx)/C282n mgx2n/C281sin(mx)dx: (13) Using INTEGRATION BY PARTS again, u/C30x2n/C281dv/C30sin(mx)dx (14) du/C30(2n/C281)x2n/C282dx v/C30/C281 mcos(mx); (15) and gx2ncos(mx)dx /C301 mx2nsin(mx)/C282n m /C2/C281 mx2n/C281cos(mx)/C272n/C281 mgx2n/C282cos(mx)dx"# /C301 mx2nsin(mx)/C272n m2x2n/C281cos(mx) /C28(2n)(2n/C281) m2gx2n/C282cos(mx)dx /C301 mx2nsin(mx)/C272n m2x2n/C281cos(mx)/C27... /C27(2n)! m2ngx0cos(mx)dx /C301 mx2nsin(mx)/C272n m2x2n/C281cos(mx)/C27... /C27(2n)! m2n/C271sin(mx) /C30sin(mx)Xn k/C300(/C281)k/C271 (2n)! (2n/C282k)!m2k/C271x2n/C282k /C27cos(mx)Xn k/C301(/C281)k/C271 (2n)! (2k/C282n/C281)!m2kx2n/C282k/C271: (16) Letting k?/C13n/C28k;/ gx2ncos(mx)dx /C30sin(mx)Xn k/C300(/C281)n/C28k/C271 (2n)! (2k)!m2n/C282k/C271x2k /C27cos(mx)Xn/C281 k/C300(/C281)n/C28k/C271 (2n)! (2k/C281)!m2n/C282kx2k/C271 /C30(/C281)n/C271(2n)! sin( mx)Xn/C281 k/C300(/C281)k (2k)!m2n/C282k/C271x2k" /C27cos(mx)Xn k/C301(/C281)k/C271 (2k/C283)!m2n/C282k/C272x2k/C271l121 : (17) To find a closed form for an integral power of a cosine function, I/C13gcosmxd x ; (18) perform an INTEGRATION BY PARTS so that u/C30cosm/C281xd v/C30cosxd x (19) du/C30/C28(m/C281) cosm/C282xsinxd x v /C30sinx: (20) Therefore I/C30sinxcosm/C281x/C27(m/C281)gcosm/C282xsin2xd x /C30sinxcosm/C281x/C27(m/C281) /C2gcosm/C282xd x/C28gcosmxd xl12ml121 /C30sinxcosm/C281x/C27(m/C281)gcosm/C282xd x/C28Il12ml121 ;(21) so I1/C27(m/C281) ½/C138 /C30sinxcosm/C281x/C27(m/C281)gcosm/C282xd x (22) I/C30gcosmxd x /C30sinxcosm/C281x m/C27m/C281 mgcosm/C282xd x : (23) Now, if misEVEN som/C132n;thengcos2nxd x/C30sinxcos2n/C281x 2n/C272n/C281 2ngcos2n/C282xd x /C30sinxcos2n/C281x 2n /C272n/C281 2nsinxcos2n/C283x 2n/C282/C272n/C283 2n/C282gcos2n/C284xd x"# /C30sinx1 2ncos2n/C281x/C272n/C281 (2n)(2n/C282)cos2n/C283x"# /C27(2n/C281)(2n/C283) (2n)(2n/C282)gcos2n/C284xd x /C30sinx1 2ncos2n/C281x/C272n/C281 (2n)(2n/C282)cos2n/C283x/C27..."# /C27(2n/C281)(2n/C283)/C1/C1/C11 (2n)(2n/C282)/C1/C1/C12gcos0xd x /C30sinxXn k/C301(2n/C282k)!! (2n)!!(2n/C281)!! (2n/C282k/C271)!!cos2n/C282k/C271x /C27(2n/C281)!! (2n)!!x: (24) Now let k?/C13n/C28k/C271;son/C28k/C30k?/C281;/ gcos2nxd x /C30sinxXn k/C301(2k/C282)!! (2n)!!(2n/C281)!! (2k/C281)!!cos2k/C281x /C27(2n/C281)!! (2n)!!x /C30(2n/C281)!! (2n)!! /C2sinxXn/C281 k/C300(2k)!! (2k/C271)!!cos2k/C271x/C27x"# : (25) Now if misODD som/C132n/C271;then gcos2n/C271xd x/C30sinxcos2nx 2n/C271/C272n 2n/C271gcos2n/C281xd x /C30sinxcos2nx 2n/C271/C272n 2n/C271 /C2sinxcos2n/C282x 2n/C281/C272n/C282 2n/C281gcos2n/C283xd x"# /C30sinx1 2n/C271cos2nx/C272n (2n/C271)(2n/C281)cos2n/C282x"# /C27(2n)(2n/C282) (2n/C271)(2n/C281)gcos2n/C283xd x /C30sin x1 2n /C27 1cos2n x /C272n (2n /C27 1)(2n /C28 1)cos2n /C282 x" /C27...l121 /C27(2n)(2n /C28 2) /C1/C1/C12 (2n /C27 1)(2n /C28 1) /C1/C1/C13 g cos xdx /C30sin xXn k /C300(2n/C282k/C281)!! (2n/C271)!!(2n)!! (2n/C282k)!! cos2n/C282k x: (26) Now let k?/C13n /C28k; g cos2n xdx /C30(2n)!! (2n /C27 1)!!sin xXn k /C300(2k /C28 1)!! (2k)!!cos2k x: (27) The general result is then g cosm xdx /C30(2n /C28 1)!! (2n)!!sin xXn/C281 k/C300(2k)!! (2k /C27 1)!!cos2k /C271 x /C27x"# for m /C302n (2n)!! (2n /C27 1)!!sin xXn k /C300(2k /C28 1)!! (2k)!!cos2k x for m /C302n /C271:8 >>>>>>>>< >>>>>>>>: (28) The infinite integral of a cosine times a Gaussian can also be done in closed form, g/C12 /C28/C12e/C28ax2cos(kx)dx/C30ffiffiffi p as e/C28k2=4a: (29) See also CHI,D AMPED EXPONENTIAL COSINE INTE- GRAL ,N IELSEN’S SPIRAL ,S HI,S ICI SPIRAL ,S INE INTEGRAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Cosine Integrals." §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231 /C1/33, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 342 /C1/43, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Fresnel Integrals, Cosine and Sine Integrals."§6.79 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 248 /C1 /52, 1992. Spanier, J. and Oldham, K. B. "The Cosine and Sine Integrals." Ch. 38 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 361 /C1/72, 1987.Cosines Law LAW OF COSINES CosIntegral COSINE INTEGRAL Cosmic Figure PLATONIC SOLID Cosmological Theorem There exists an INTEGER Nsuch that every string in the LOOK AND SAY SEQUENCE "decays" in at most N days to a compound of "common" and "transuranic elements." The table below gives the periodic table of atoms associated with the LOOK AND SAY SEQUENCE as named by Conway (1987). The "abundance" is theaverage number of occurrences for long strings out ofevery million atoms. The asymptotic abundances are zero for transuranic elements, and 27.246... for arsenic (As), the next rarest element. The mostcommon element is hydrogen (H), having an abun-dance of 91,970.383.... The starting element is U, represented by the string "3," and subsequent terms are those giving a description of the current term: onethree (13); one one, one three (1113); three ones, one three (3113), etc. Abundance n /En//Enis the derivate of /En/C271/ 102.56285249 92 U 3 9883.5986392 91 Pa 137581.9047125 90 Th 11136926.9352045 89 Ac 31135313.7894999 88 Ra 1321134076.3134078 87 Fr 11131221133127.0209328 86 Rn 3113112221132398.7998311 85 At Ho.13221131840.1669683 84 Po 11132221131411.6286100 83 Bi 31133221131082.8883285 82 Pb Pm.123222113830.70513293 81 Tl 111213322113637.25039755 80 Hg 31121123222113488.84742982 79 Au 132112211213322113375.00456738 78 Pt 111312212221121123222113287.67344775 77 Ir 3113112211322112211213322113220.68001229 76 Os 1321132122211322212221121123222113169.28801808 75 Re 11312211312113221133211322112211213322113315.56655252 74 W Ge.Ca.312211322212221121123222113 242.07736666 73 Ta 13112221133211322112211213322113 2669.0970363 72 Hf 11132.Pa.H.Ca.W 2047.5173200 71 Lu 311312 1570.6911808 70 Yb 1321131112 1204.9083841 69 Tm 11131221133112 1098.5955997 68 Er 311311222.Ca.Co 47987.529438 67 Ho 1321132.Pm 36812.186418 66 Dy 111312211312 28239.358949 65 Tb 3113112221131112 21662.972821 64 Gd Ho.13221133112 20085.668709 63 Eu 1113222.Ca.Co 15408.115182 62 Sm 311332 29820.456167 61 Pm 132.Ca.Zn 22875.863883 60 Nd 111312 17548.529287 59 Pr 31131112 13461.825166 58 Ce 1321133112 10326.833312 57 La 11131.H.Ca.Co 7921.9188284 56 Ba 311311 6077.0611889 55 Cs 13211321 4661.8342720 54 Xe 11131221131211 3576.1856107 53 I 311311222113111221 2743.3629718 52 Te Ho.1322113312211 2104.4881933 51 Sb Eu.Ca.3112221 1614.3946687 50 Sn Pm.13211 1238.4341972 49 In 11131221 950.02745646 48 Cd 3113112211 728.78492056 47 Ag 132113212221 559.06537946 46 Pd 111312211312113211 428.87015041 45 Rh 311311222113111221131221 328.99480576 44 Ru Ho.132211331222113112211 386.07704943 43 Tc Eu.Ca.311322113212221 296.16736852 42 Mo 13211322211312113211 227.19586752 41 Nb 1113122113322113111221131221 174.28645997 40 Zr Er.12322211331222113112211 133.69860315 39 Y 1112133.H.Ca.Tc 102.56285249 38 Sr 3112112.U 78.678000089 37 Rb 1321122112 60.355455682 36 Kr 11131221222112 46.299868152 35 Br 3113112211322112 35.517547944 34 Se 13211321222113222112 27.246216076 33 As 11131221131211322113322112 1887.4372276 32 Ge 31131122211311122113222.Na 1447.8905642 31 Ga Ho.13221133122211332 23571.391336 30 Zn Eu.Ca.Ac.H.Ca.312 18082.082203 29 Cu 131112 13871.123200 28 Ni 11133112 45645.877256 27 Co Zn.32112 35015.858546 26 Fe 1312211226861.360180 25 Mn 111311222112 20605.882611 24 Cr 31132.Si 15807.181592 23 V 13211312 12126.002783 22 Ti 11131221131112 9302.0974443 21 Sc 3113112221133112 56072.543129 20 Ca Ho.Pa.H.12.Co 43014.360913 19 K 1112 32997.170122 18 Ar 3112 25312.784218 17 Cl 132112 19417.939250 16 S 1113122112 14895.886658 15 P 311311222112 32032.812960 14 Si Ho.1322112 24573.006696 13 Al 1113222112 18850.441228 12 Mg 3113322112 14481.448773 11 Na Pm.123222112 11109.006696 10 Ne 111213322112 8521.9396539 9 F 31121123222112 6537.3490750 8 O 132112211213322112 5014.9302464 7 N 111312212221121123222112 3847.0525419 6 C 3113112211322112211213322112 2951.1503716 5 B 1321132122211322212221121123222112 2263.8860325 4 Be 111312211312113221133211322112211213322112 4220.0665982 3 Li Ge.Ca.3122113222122211211232221223237.2968588 2 He 1311222113321132211221121332211291790.383216 1 H Hf.Pa.22.Ca.Li See also CONWAY’S CONSTANT ,LOOK AND SAY SE- QUENCE References Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay." §5.11 in Open Problems in Commu- nication and Computation (Ed. T. M. Cover and B. Gopi- nath). New York: Springer-Verlag, pp. 173 /C1/88, 1987. Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay." Eureka, 5/C1/8, 1985. Ekhad, S. B. and Zeilberger, D. "Proof of Conway’s Lost Cosmological Theorem." Electronic Research Announce- ment of the Amer. Math. Soc. 3,7 8/C1/2, 1997. http:// www.math.temple.edu/~zeilberg/mamarim/mamar- imhtml/horton.html. Hilgemeier, M. "Die Gleichniszahlen-Reihe." Bild der Wis- sensch. 12, 19, 1986. Hilgemeier, M. "‘One Metaphor Fits All’: A Fractal Voyage with Conway’s Audioactive Decay." Ch. 7 in Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York: St. Martin’s Press, 1996. Costa Minimal Surface A COMPLETE MINIMAL EMBEDDABLE SURFACE of finite topology (i.e., it has no BOUNDARY and does not intersect itself). Until this surface was discovered by Costa (1984), the only other known complete minimal embeddable surfaces in R3 with no self-intersections were the PLANE , CATENOID , and HELICOID . The plane is genus 0 and the catenoid and the helicoid are genus 0 with two punctures, but the Costa minimal surface is genus 1 with three punctures (Schwalbe and Wagon 1999). In addition, and rather amazingly, the Costa surface belongs to the D4 DIHEDRAL GROUP of symmetries. An animation by S. Dickson illus- trates the homotopy of the TORUS into a Costa surface (Wolfram Research). As discovered by Gray (Ferguson et al. 1996, Gray 1997), the Costa surface can be represented parame- trically explicitly by x /C301 2 R/C28z(u /C27iv) /C27 pu /C27p2 4e1/C27p 2e1[z(u /C27iv /C2812) /C28 z(u /C27iv /C2812 i)]() y /C301 2 R/C28iz(u /C27iv) /C27 pv /C27p2 4e1/C28p 2e1[i z(u /C27iv /C2812) /C28iz(u /C27iv /C2812 i)]() z /C3014ffiffiffiffiffiffi 2pp ln/C212(u /C27 iv) /C28 e1 /C212(u /C27 iv) /C27 e1l112l112l112l112l112l112l112l112l112l112; where z(z) is the WEIERSTRASS ZETA FUNCTION , /C212(g2 ; g3; z) is the WEIERSTRASS ELLIPTIC FUNCTION with (g2 ; g3) /C30(189 :072772... ; 0) the invariants cor- responding to the half-periods 1/2 and i =2; and first root e1 /C30/C212(1 2;0; g3) /C30/C212(12½12 ;12 i) :6:87519 ; where /C212(z; g2 ; g3) /C30/C212(z½ v1 ; v2) is the WEIERSTRASS ELLIPTIC FUNCTION . See also COMPLETE MINIMAL SURFACE ,M INIMAL SURFACE ,W EIERSTRASS ELLIPTIC FUNCTION ,W EIER- STRASS ZETA FUNCTION References Costa, A. "Examples of a Complete Minimal Immersion in R3 of Genus One and Three Embedded Ends." Bil. Soc. Bras. Mat. 15,47/C1/4, 1984. do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 43, 1986.Ferguson, H.; Gray, A.; and Markvorsen, S. "Costa’s Mini- mal Surface via Mathematica ." Mathematica in Educ. Res. 5,5/C1/0, 1996. Ferguson, H.; Ferguson, C.; Nemeth, R.; Schwalbe, D.; and Wagon, S. "Invisible Handshake." Math. Intell. 21, 1999. To appear. Gray, A. "Costa’s Minimal Surface." §32.5 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 747 /C1/57, 1997. Hoffman, D. and Meeks, W. H. III. "A Complete Embedded Minimal Surfaces in R3 with Genus One and Three Ends." J. Diff. Geom. 21, 109 /C1/27, 1985. Nordstrand, T. "Costa-Hoffman-Meeks Minimal Surface." http://www.uib.no/people/nfytn/costatxt.htm. Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 149 /C1/50, 1986. Peterson, I. "Three Bites in a Doughnut: Computer-Gener- ated Pictures Contribute to the Discovery of a New Minimal Surface." Sci. News 127, 161 /C1/76, 1985. Peterson, I. "The Song in the Stone: Developing the Art of Telecarving a Minimal Surface." Sci. News 149, 110 /C1/11, Feb. 17, 1996. Schwalbe, D. and Wagon, S. "The Costa Surface, in Show and Mathematica ." Mathematica in Educ. Res. 8,56/C1/3, 1999. Wolfram Research, Inc. "3-D Zoetrope at SIGGRAPH 2000." http://www.wolfram.com/news/zoetrope.html. Costa-Hoffman-Meeks Minimal Surface COSTA MINIMAL SURFACE Cosymmedian Triangles Extend the SYMMEDIANS of a TRIANGLE DA1A2A3to meet the CIRCUMCIRCLE at P1 ; P2 ; P3 : Then the SYMMEDIAN POINT K of DA1A2A3is also the SYMME- DIAN POINT of DP1P2P3 : The TRIANGLES DA1A2A3 and DP1P2P3are cosymmedian triangles, and have the same BROCARD CIRCLE , second BROCARD TRIANGLE , BROCARD ANGLE ,BROCARD POINTS , and CIRCUMCIR- CLE. See also BROCARD ANGLE ,BROCARD CIRCLE ,BROCARD POINTS ,BROCARD TRIANGLES ,CIRCUMCIRCLE ,COME- DIAN TRIANGLES ,SYMMEDIAN ,SYMMEDIAN POINT References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 63, 1893. Cot COTANGENT Cotangent The function defined by cot x /C131=tan x; where tan x is the TANGENT . The notations ctn x (Erde ´lyi et al. 1981, p. 7) and ctg x (Gradshteyn and Ryzhik 2000, p. xxix) are sometimes used in place of cot x:/ The MACLAURIN SERIES for cot x is cot x /C301 x /C281 3 x /C281 45 x3 /C282 945 x5 /C281 4725 x7 /C28... /C28( /C281)n/C27122nB2n (2n)!/C28...; where Bn is a BERNOULLI NUMBER . p cot( px) /C301 x /C272xX/C12 n/C3011 x2 /C28 n2 : It is known that, for n ]3; cot( p=n) is rational only for n /C304. See also HYPERBOLIC COTANGENT ,INVERSE COTAN- GENT ,LEHMER’S CONSTANT ,TANGENT References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/9, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215, 1987. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 6, 1981. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Spanier, J. and Oldham, K. B. "The Tangent tan(x) and Cotangent cot(x) Functions." Ch. 34 in An Atlas of Func- tions. Washington, DC: Hemisphere, pp. 319 /C1/30, 1987. Cotangent Bundle The cotangent bundle of a MANIFOLD is similar to the TANGENT BUNDLE , except that it is the set (x, f) where x /C23 M and f is a dual vector in the TANGENT SPACE to x /C23 M : The cotangent bundle is denoted T /C31M :/ See also TANGENT BUNDLE Cotes Circle Property x2n /C271 /C30 x2 /C282x cosp 2n ! /C271"# /C29 x2 /C282x cos3p 2n ! /C271"# /C29/C1/C1/C1/C29 /C29 x2 /C282x cos(2n /C28 1)p 2n ! /C271"# : See also COSINE ,TRIGONOMETRIC FUNCTIONS Cotes Number The numbers lnnin the GAUSSIAN QUADRATURE formula Qn(f) /C30Xn n/C301lnnf(xnn) : See also CHRISTOFFEL NUMBER ,GAUSSIAN QUADRA- TURE References Cajori, F. A History of Mathematical Notations, Vols. 1 /C1/. New York: Dover, p. 42, 1993. Cotes’ Spiral The planar orbit of a particle under a r/C283 force field. It is an EPISPIRAL . See also EPISPIRAL Coth HYPERBOLIC COTANGENT . Cotree The cotree T /C31 of a spanning tree T in a CONNECTED GRAPH G is the spacing SUBGRAPH of G containing exactly those edges of G which are not in T (Harary 1994, p. 39). See also TWIG References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Coulomb Wave Function A special case of the CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND . It gives the solution to the radial Schro ¨dinger equation in the Coulomb potential /(1=r) of a point nucleus d2W dr2 /C27 1 /C282h r/C28L(L /C27 1) r2"# W /C300 (1) (Abramowitz and Stegun 1972; Zwillinger 1997, p. 122). The complete solution is W /C30C1FL( h; r) /C27C2GL(h ; r) : (2) The Coulomb function of the first kind is FL( h; r) /C30CL(h) rL /C271e /C28ip 1F1(L /C271 /C28i h;2L /C272; 2i r) ; (3) where CL( h) /C132Le/C28 ph =2 ½G(L /C27 1 /C27 i h) ½ G(2L /C27 2); (4) /1F1(a; b; z) is the CONFLUENT HYPERGEOMETRIC FUNCTION , G(z) is the GAMMA FUNCTION , and the Coulomb function of the second kind is GL( h; r) /C302h C2 0( h)FL( h ; r) ln(2 r) /C27qL(h) pL( h)"# /C271 (2L /C27 1)CL( h)r /C28LX/C12 K /C30/C28LaL k ( h) rK /C27L ; (5) where qL ; pL ; and aL kare defined in Abramowitz and Stegun (1972, p. 538). See also CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND References Abramowitz, M. and Antosiewicz, H. A. "Coulomb Wave Functions in the Transition Region." Phys. Rev. 96,75/C1/7, 1954. Abramowitz, M. and Rabinowitz, P. "Evaluation of Coulomb Wave Functions along the Transition Line." Phys. Rev. 96, 77 /C1/9, 1954. Abramowitz, M. and Stegun, C. A. (Eds.). "Coulomb Wave Functions." Ch. 14 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 537 /C1/44, 1972. Biedenharn, L. C.; Gluckstern, R. L.; Hull, M. H. Jr.; and Breit, G. "Coulomb Wave Functions for Large Charges and Small Velocities." Phys. Rev. 97, 542 /C1/54, 1955. Bloch, I.; Hull, M. H. Jr.; Broyles, A. A.; Bouricius, W. G.; Freeman, B. E.; and Breit, G. "Coulomb Functions for Reactions of Protons and Alpha-Particles with the Lighter Nuclei." Rev. Mod. Phys. 23, 147 /C1/82, 1951.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 631 /C1/33, 1953. National Bureau of Standards. Tables of Coulomb Wave Functions, Vol. 1, Applied Math Series 17. Washington, DC: U.S. Government Printing Office, 1952. Stegun, I. A. and Abramowitz, M. "Generation of Coulomb Wave Functions by Means of Recurrence Relations." Phys. Rev. 98, 1851 /C1/852, 1955. Count The largest n such that ½zn ½B4inaM ANDELBROT SET. Points of different count are often assigned different colors. Countable Additivity Probability Axiom For a COUNTABLE SET of n disjoint events E1 ; E2 ; ..., En P @n i/C301Eil11sl11n /C30Xn i/C301P(Ei): See also COUNTABLE SET Countable Set A SET which is either FINITE or DENUMERABLE . However, some author (Ciesielski 1997, p. 64) use the definition "equipollent to the finite ordinals," commonly used to define a DENUMERABLE SET,to define a countable set. See also ALEPH-0 ,A LEPH-1 ,C OUNTABLY INFINITE , DENUMERABLE SET,FINITE ,INFINITE ,UNCOUNTABLY INFINITE References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Countable Space FIRST- COUNTABLE SPACE Countably Infinite Any SETwhich can be put in a ONE-TO-ONE correspon- dence with the NATURAL NUMBERS (or INTEGERS )s o that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set. Once one countable set Sis given, any other set which can be put into a ONE- TO-ONE correspondence with Sis also countable. Countably infinite sets have CARDINAL NUMBER ALEPH-0 . Examples of countable sets include the INTEGERS , ALGEBRAIC NUMBERS , and RATIONAL NUMBERS . Georg Cantor showed that the number of REAL NUMBERS is rigorously larger than a countably infinite set, and the postulate that this number, the so-called "CON- TINUUM ," is equal to ALEPH-1 is called the CONTINUUM HYPOTHESIS . Examples of nondenumerable sets in- clude the REAL , COMPLEX , IRRATIONAL , and TRANS- CENDENTAL NUMBERS . See also ALEPH-0 ,ALEPH-1 ,CANTOR DIAGONAL SLASH , CARDINAL NUMBER ,C ONTINUUM ,C ONTINUUM HY- POTHESIS ,C OUNTABLE SET,H ILBERT HOTEL ,U N- COUNTABLY INFINITE References Courant, R. and Robbins, H. "The Denumerability of the Rational Number and the Non-Denumerability of the Continuum." §2.4.2 in What is Mathematics?: An Elemen- tary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 79 /C1/3, 1996. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, p. 10, 1988. Counterfeit Coin Problem WEIGHING Counting Generalized Principle If r experiments are performed with nipossible outcomes for each experiment i /C301; 2 ; ...; r; then there are a total ofQr i/C301 ni possible outcomes. Counting Number A POSITIVE INTEGER : 1, 2, 3, 4, ... (Sloane’s A000027), also called a NATURAL NUMBER . However, zero (0) is sometimes also included in the list of counting numbers. Due to lack of standard terminology, the following terms are recommended in preference to "counting number," "NATURAL NUMBER ," and "WHOLE NUMBER ." set name symbol ..., -2, -1, 0, 1, 2, ...INTEGERS Z 1, 2, 3, 4, ... POSITIVE INTEGERS Z/C27 0, 1, 2, 3, 4, ... NONNEGATIVE INTE- GERSZ* 0, -1, -2, -3, -4, ... NONPOSITIVE INTE- GERS -1, -2, -3, -4, ... NEGATIVE INTEGERS Z- See also NATURAL NUMBER ,W HOLE NUMBER ,Z,Z -, Z/C27,Z* References Sloane, N. J. A. Sequences A000027/M0472 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Coupon Collector’s Problem Let n objects be picked repeatedly with probability pi that object i is picked on a given try, with X ipi /C301: Find the earliest time at which all n objects have been picked at least once. References Hildebrand, M. V. "The Birthday Problem." Amer. Math. Monthly 100, 643, 1993. Cousin Primes Pairs of PRIMES OF THE FORM (p, p /C274) are called cousin primes. The first few are (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), ... (Sloane’s A023200 and A046132). According to the first FIRST HARDY- LITTLEWOOD CONJECTURE , the cousin primes have the same asymptotic density as the TWIN PRIMES , Px(p; p /C274) /C22Y p ]3p(p /C28 2) (p /C28 1)2 gx 2dx ? (ln x?)2 /C301:320323632 gx 2dx? (ln x?)2 whereQ 2 /C301 :320323632 is the TWIN PRIMES CON- STANT . An analogy to BRUN’S CONSTANT , the constant B4 /C13(1 7 /C271 11) /C27(1 13 /C271 17) /C27(1 19 /C271 23) /C27(1 37 /C271 41) /C27...; (omitting the initial term 1=3 /C271=7) can be defined. Using cousin primes up to 242, the value of B4is estimated as B4:1:1970449 (Wolf 1996). See also BRUN’S CONSTANT ,PRIME CONSTELLATION , SEXY PRIMES ,TWIN PRIMES ,TWIN PRIMES CONSTANT References Sloane, N. J. A. Sequences A023200 and A046132 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/eisonline.html. Covariance Given nsets of variates denoted fx1g;...,fxng;the covariance sij/C13cov(xi;xj)o fxiandxjis defined by cov(xi;xj)/C13/C142(xi/C28mi)(xj/C28mj)/C143 (1) /C30/C142xixj/C143/C28/C142xi/C143/C142xj/C143; (2) where mi/C30/C142xi/C143andmj/C30/C142xj/C143are the MEANS ofxiand xj;respectively. The matrix ( Vij) of the quantities Vij /C30cov(xi; xj) is called the COVARIANCE MATRIX .In the special case i /C30j, cov(xi ; xi) /C30/C142x2 i /C143/C28/C142xi /C1432 /C30 s2i ; (3) giving the usual VARIANCE sii /C30 s2 i /C30var(xi) ;:/ The covariance of two variates xiand xjprovides a measure of how strongly correlated these variables are, and the derived quantity cor(xi ; xj) /C13cov(xi ; xj) si sj; (4) where si ; sjare the STANDARD DEVIATIONS , is called CORRELATION of xiand xj : The covariance is sym- metric since cov(x ; y) /C30cov(y; x): (5) For two variables, the covariance is related to the VARIANCE by var(x /C27y) /C30var(x) /C27var(y) /C272 cov(x; y) : (6) For two independent variates x /C30xi and y /C30xj ; cov(x; y) /C30/C142xy/C143/C28 mx my /C30/C142x/C143/C142y/C143/C28 mx my /C300 ; (7) so the covariance is zero. However, if the variables are correlated in some way, then their covariance will be NONZERO . In fact, if cov(x; y) > 0; then y tends to increase as x increases. If cov(x; y) B0 ; then y tends to decrease as x increases. The covariance obeys the identity cov(x /C27z ; y) /C30/C142(x /C27z)y /C28(x /C27z)(y) /C143 /C30/C142xy /C143/C27/C142zy /C143/C28( /C142x/C143/C27/C142z /C143) /C142y/C143 /C30/C142xy /C143/C28/C142x/C143/C142y/C143/C27/C142zy/C143/C28/C142z /C143/C142y/C143 /C30cov(x; y) /C27cov(z ;y) : (8) By induction, it therefore follows that covXn i/C301xi ; y ! /C30Xn i/C301cov(xi ; y) (9) covXn i/C301xi ;Xm j /C301yj ! /C30Xn i /C301cov xiXm j/C301yj ! (10) /C30Xn i/C301covXm j/C301yj ; xi ! (11) /C30Xn i /C301Xm j/C301cov(yj ; xi) (12) /C30Xn i/C301Xn j/C301cov(xi ; yj) : (13)See also CORRELATION (STATISTICAL ), COVARIANCE MATRIX ,VARIANCE Covariance Matrix Given n sets of variates denoted fx1 g; ..., fxn g , the first-order covariance matrix is defined by Vij /C30cov(xi ; xj) /C13/C142(xi /C28 mi)(xj /C28 mj) /C143; where mi is the MEAN . Higher order matrices are given by Vmn ij/C30/C142(xi /C28 mi)m(xj /C28 mj)n /C143: An individual matrix element Vij /C30cov(xi ; xj) is called the COVARIANCE of xi and xj :/ See also CORRELATION (STATISTICAL ), COVARIANCE , ERROR PROPAGATION ,VARIANCE Covariant Derivative The covariant derivative of a CONTRAVARIANT TENSOR Aa (also called the "semicolon derivative" since its symbol is a semicolon) is given by 9 /C215 A /C13Aa ; b /C30Aa ; b /C27Ga bkAk ; (1) where Ak ;k is a COMMA DERIVATIVE and 9/C215is a general- ization of the symbol commonly used to denote the DIVERGENCE of a vector function in 3-D, Gk ijis a CONNECTION COEFFICIENT , and EINSTEIN SUMMATION has been used in the last term. The covariant derivative of a COVARIANT TENSOR Aa is Aa;b/C301 gbb@Aa @xb/C28GkabAk; (2) Schmutzer (1968, p. 72) uses the older notation Aj ½½kor Aj½½k:/ See also COMMA DERIVATIVE ,CONNECTION COEFFI- CIENT ,COVARIANT TENSOR ,DIVERGENCE ,LEVI-CIVITA CONNECTION References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 48 /C1/0, 1953. Schmutzer, E. Relativistische Physik (Klassische Theorie). Leipzig, Germany: Akademische Verlagsgesellschaft, 1968. Covariant Tensor A covariant tensor is a TENSOR having specific transformation properties (cf., a CONTRAVARIANT TEN- SOR). To examine the transformation properties of a covariant tensor, first consider the GRADIENT 9f/C30@f @x1ˆx1/C27@f @x2ˆx2/C27@f @x3ˆx3; (1) for which @ f? @x?i/C30@ f @xj@xj @x ?i; (2) where f(x1 ; x2 ; x3) /C30 f?(x?1 ; x?2 ; x?3) : Now let Ai /C13@ f @xi; (3) then any set of quantities Aj which transform accord- ing to A?i /C30@xj @x?iAj (4) or, defining aij /C13@xj @x?i; (5) according to A?i /C30aijAj (6) is a covariant tensor. Covariant tensors are indicated with lowered indices, i.e., am :/ CONTRAVARIANT TENSORS are a type of TENSOR with differing transformation properties, denoted a n : How- ever, in 3-D CARTESIAN COORDINATES , @xj @x?i/C30@x?i @xj/C13aij (7) for i ; j /C301 ; 2, 3, meaning that contravariant and covariant tensors are equivalent. The two types of tensors do differ in higher dimensions, however. Covariant FOUR-VECTORS satisfy am /C30L n man ; (8) where L is a LORENTZ TENSOR . To turn a CONTRAVARIANT TENSOR an into a covariant tensor am(INDEX LOWERING ), use the METRIC TENSOR gmn to write gmnan /C30a m : (9) Covariant and contravariant indices can be used simultaneously in a MIXED TENSOR . See also CONTRAVARIANT TENSOR ,F OUR- VECTOR , INDEX LOWERING ,LORENTZ TENSOR ,METRIC TENSOR , MIXED TENSOR ,TENSOR References Arfken, G. "Noncartesian Tensors, Covariant Differentia- tion." §3.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 158 /C1/64, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44 /C1/6, 1953. Covariant Vector A COVARIANT TENSOR of RANK 1, more commonly called a ONE-FORM (or "BRA").See also BRA,CONTRAVARIANT VECTOR ,CONTRAVAR- IANT TENSOR ,KET,ONE-FORM,VECTOR Cover A family g of nonempty SUBSETS of X whose UNION contains the given set X (and which contains no duplicated subsets) is called a cover (or covering) of X. For example, there is only a single cover of f1g; namely f1 g itself. However, there are five covers of f1; 2g; namely ff1g;f2gg;ff1; 2 gg;ff1 g;f1; 2gg; ff2g;f1; 2gg; and ff1g;f2g;f1; 2gg:/ A MINIMAL COVER is a cover for which removal of one member destroys the covering property. For example, of the five covers of f1; 2g; only ff1g;f2gg and ff1; 2gg are minimal covers. There are various other types of specialized covers, including PROPER COVERS , antichain covers, k-covers, and k /C31/-covers (Macula 1994). The number of possible covers for a set of N elements are ½C(N)½/C301 2XN k /C300(/C281)k N kl11sl11n 22N /C28k ; the first few of which are 1, 5, 109, 32297, 2147321017, 9223372023970362989, ... (Sloane’s A003465). See also MINIMAL COVER ,PROPER COVER References Eppstein, D. "Covering and Packing." http://www.ics.u- ci.edu/~eppstein/junkyard/cover.html. Macula, A. J. "Covers of a Finite Set." Math. Mag. 67, 141 /C1/ 44, 1994. Sloane, N. J. A. Sequences A003465/M4024 and A055621 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Cover Relation The transitive reflexive reduction of a PARTIAL ORDER . An element z of a POSET (X ;5) covers another element x provided that there exists no third element y in the poset for which x 5y 5z: In this case, z is called an "upper cover" of xandxa "lower cover" of z. See also PARTIAL ORDER Covering COVER ,COVERING MAP,PACKING Covering Dimension LEBESGUE COVERING DIMENSION Covering Map A covering map is a SURJECTIVE OPEN MAP f:X0Y whose preimages f/C281(y) are a DISCRETE SET inX. For example, the map f(z)/C30z2;as a map f:C/C2800C/C28 0; is a covering. Note that f /C281(w) always consists of two points. In general, the cardinality of f /C281(y)is independent of y /C23 Y :/ Another example is p : C 0 C=G#T; where G/C30f(a /C27 bI) ½a; b /C23Zg: The map p is actually the UNIVERSAL COVER of the torus T: If f : X 0 T is any covering of the torus, then there exists a covering ˜p : C 0 X such that p factors through ˜p; i.e., p /C30f(˜p:/ See also SIMPLY CONNECTED ,TOPOLOGICAL SPACE , UNIVERSAL COVER Covering System COMPLETE RESIDUE SYSTEM Coversine covers A /C131 /C28sin A; where sin A is the SINE. See also EXSECANT ,HAVERSINE ,SINE,VERSINE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 78, 1972. Coxeter Diagram COXETER- DYNKIN DIAGRAM Coxeter Graph A non-Hamiltonian graph with a high degree of symmetry such that there is a GRAPH AUTOMORPHISM taking any path of length three into any other. See also COXETER- DYNKIN DIAGRAM ,LEVI GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 241, 1976. Tutte, W. T. "A Non-Hamiltonian Graph." Canad. Math. Bull. 3,1/C1/, 1960. Coxeter Group A group generated by the elements Pifor i /C301, ..., n subject to(PiPj)Mij /C301 ; where Mijare the elements of a COXETER MATRIX . Coxeter used the NOTATION [3p; q; r] for the Coxeter group generated by the nodes of a Y-shaped COXETER- DYNKIN DIAGRAM whose three arms have p, q, and r EDGES . A Coxeter group of this form is finite IFF 1 p /C27 1 /C271 q /C27 1 /C271 r /C27 1> 1: See also BIMONSTER ,B UILDING ,C OXETER- DYNKIN DIAGRAM References Arnold, V. I. "Snake Calculus and Combinatorics of Ber- noulli, Euler, and Springer Numbers for Coxeter Groups." Russian Math. Surveys 47,3/C1/5, 1992. Garrett, P. Buildings and Classical Groups. Boca Raton, FL: Chapman and Hall, 1997. Hsiang, W. Y. "Coxeter Groups, Weyl Reduction, and Weyl Formulas." Lec. 4 in Lectures on Lie Groups. Singapore: World Scientific, pp. 58 /C1/7, 2000. Coxeter Matrix An n /C29n SQUARE MATRIX M with Mii /C301 Mij /C30Mji > 1 for all i ; j /C301; ..., n. See also COXETER GROUP Coxeter-Dynkin Diagram A LABELED GRAPH whose nodes are indexed by the generators of a COXETER GROUP having (Pi ; Pj)asan EDGE labeled by Mij whenever Mij > 2 ; where Mij is an element of the COXETER MATRIX . Coxeter-Dynkin diagrams are used to visualize COXETER GROUPS .A Coxeter-Dynkin diagram is associated with each RATIONAL DOUBLE POINT (Fischer 1986), and a Cox- eter diagram is sufficient to characterize the algebra of the group. See also COXETER GROUP ,D YNKIN DIAGRAM ,R A- TIONAL DOUBLE POINT References Arnold, V. I. "Critical Points of Smooth Functions." Proc. Int. Congr. Math. 1,1 9/C1/9, 1974. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 12 /C1/3, 1986. Coxeter’s Loxodromic Sequence of Tangent Circles An infinite sequence of CIRCLES such that every four consecutive CIRCLES are mutually tangent, and the CIRCLES ’RADII ...,R/C28n;...,R/C281;R0;R1;R2;R3;R4;..., Rn ; Rn /C271 ; ..., are in GEOMETRIC PROGRESSION with ratio k /C13Rn/C271 Rn/C30 f /C27ffiffiffiffi fp ; where f is the GOLDEN RATIO (Gardner 1979ab). Coxeter (1968) generalized the sequence to SPHERES . See also ARBELOS ,B OWL OF INTEGERS ,G OLDEN RATIO,HEXLET ,PAPPUS CHAIN ,STEINER CHAIN References Coxeter, D. "Coxeter on ‘Firmament."’ http://www.bangor.- ac.uk/SculMath/image/donald.htm. Coxeter, H. S. M. "Loxodromic Sequences of Tangent Spheres." Aequationes Math. 1, 112 /C1/17, 1968. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,18/C1/8, Jan. 1979a. Gardner, M. "Mathematical Games: How to be a Psychic, Even if You are a Horse or Some Other Animal." Sci. Amer. 240,18/C1/5, May 1979b. Coxeter-Todd Lattice The complex LATTICE L v 6corresponding to real lattice K12having the densest HYPERSPHERE PACKING (KIS- SING NUMBER ) in 12-D. The associated AUTOMORPH- ISM GROUP G0 was discovered by Mitchell (1914). The order of G0 is given by ½Aut( L v 6 ) ½/C3029 /C215 37 /C215 5 /C215 7 /C3039; 191; 040: The order of the AUTOMORPHISM GROUP of K12 is given by ½Aut(K12) ½/C30210 /C215 37 /C215 5 /C215 7 (Conway and Sloane 1983). See also BARNES- WALL LATTICE ,LEECH LATTICE References Conway, J. H. and Sloane, N. J. A. "The Coxeter-Todd Lattice, the Mitchell Group and Related Sphere Packings." Math. Proc. Camb. Phil. Soc. 93, 421 /C1/40, 1983. Conway, J. H. and Sloane, N. J. A. "The 12-Dimensional Coxeter-Todd Lattice K12 :/" §4.9 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 127 /C1/29, 1993. Coxeter, H. S. M. and Todd, J. A. "As Extreme Duodenary Form." Canad. J. Math. 5, 384 /C1/92, 1953. Mitchell, H. H. "Determination of All Primitive Collineation Groups in More than Four Variables." Amer. J. Math. 36, 1 /C1/2, 1914. Todd, J. A. "The Characters of a Collineation Group in Five Dimensions." Proc. Roy. Soc. London Ser. A 200, 320 /C1/36, 1950. Cox’s Theorem Let s1 ; ..., s4be four PLANES in GENERAL POSITION through a point P and let Pijbe a point on the LINE si/C215 sj : Let sijkdenote the PLANE PijPikPjk : Then the four PLANES s234 ; s134 ; s124 ; s123all pass through one point P1234 : Similarly, let s1 ; ..., s5be five PLANES inGENERAL POSITION through P. Then the five points P2345 ; P1345 ; P1245 ; P1235 ; and P1234 all lie in one PLANE . And so on. See also CLIFFORD’S CIRCLE THEOREM ,PLANE Crame ´r Conjecture The unproven CONJECTURE that lim n 0/C12pn /C271 /C28 pn (ln pn)2 /C301; where pn is the nth PRIME . References Crame ´r, H. "On the Order of Magnitude of the Difference Between Consecutive Prime Numbers." Acta Arith. 2,23/C1/ 6, 1936. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 7, 1994. Riesel, H. "The Crame ´r Conjecture." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 79 /C1/2, 1994. Rivera, C. "Problems & Puzzles: Conjecture The Cramer’s Conjecture.-007." http://www.primepuzzles.net/conjec- tures/conj_007.htm. Crame ´r-Euler Paradox A curve of order n is generally determined by n(n /C27 3)=2 points. So a CONIC SECTION is determined by five points and a CUBIC CURVE should require nine. But the MACLAURIN- BE´ ZOUT THEOREM says that two curves of degree n intersect in n2 points, so two CUBICS intersect in nine points. This means that n(n/C273)=2 points do not always uniquely determine a single curve of order n. The paradox was publicized by Stirling, and explained by Plu ¨cker. See also CUBIC CURVE ,MACLAURIN- BE´ ZOUT THEOREM Cramer’s Rule Given a set of linear equations a1x/C27b1y/C27c1z/C30d1 a2x/C27b2y/C27c2z/C30d2 a3x/C27b3y/C27c3z/C30d3;8 < :(1) consider the DETERMINANT D/C13a1b1c1 a2b2c2 a3b3c3l112l112l112l112l112l112l112l112l112l112l112l112: (2) Now multiply Dbyx, and use the property of DETERMINANTS that MULTIPLICATION by a constant is equivalent to MULTIPLICATION of each entry in a given row by that constant xa1b1c1 a2b2c2 a3b3c3l112l112l112l112l112l112l112l112l112l112l112l112/C30a 1xb1c1 a2xb2c2 a3xb3c3l112l112l112l112l112l112l112l112l112l112l112l112: (3) Another property of DETERMINANTS enables us to add a constant times any column to any column and obtain the same DETERMINANT , so add y times column 2 and z times column 3 to column 1, xD /C30a1x /C27b1y /C27c1zb1c1 a2x /C27b2y /C27c2zb2c2 a3x /C27b3x /C27c3zb3c3l112l112l112l112l112l112l112l112l112l112l112l112/C30d 1b1c1 d2b2c2 d3b3c3l112l112l112l112l112l112l112l112l112l112l112l112: (4) If d /C300 ; then (4) reduces to xD /C300, so the system has nondegenerate solutions (i.e., solutions other than (0, 0, 0)) only if D /C300 (in which case there is a family of solutions). If d "0 and D /C300, the system has no unique solution. If instead d "0 and D "0; then solutions are given by x /C30d1b1c1 d2b2c2 d3b3c3l112l112l112l112l112l112l112l112l112l112l112l112 D; (5) and similarly for y /C30a1d1c1 a2d2c2 a3d3c3l112l112l112l112l112l112l112l112l112l112l112l112 D (6) z /C30a1b1d1 a2b2d2 a3b3d3l112l112l112l112l112l112l112l112l112l112l112l112 D (7) This procedure can be generalized to a set of n equations so, given a system of n linear equations a 11a12 /C1/C1/C1 a1n nn::: n a1n1an2/C1/C1/C1 ann2 435x 1 n xn2435/C30d 1 n dn2435; (8) let D /C13a 11a12 /C1/C1/C1 a1n nn::: n a1n1an2/C1/C1/C1 annl112l112l112l112l112l112l112l112l112l112l112l112: (9) If d /C300; then nondegenerate solutions exist only if D /C300. If d "0 and D /C300, the system has no unique solution. Otherwise, compute D k /C13a11 /C1/C1/C1 a1(k /C281)d1a1(k /C271)/C1/C1/C1 a1n n::: nnn::: n an1/C1/C1/C1 an(k /C281)dnan(k /C271)/C1/C1/C1 annl112l112l112l112l112l112l112l112l112l112l112l112: (10) Then x k /C30Dk =D for 1 5k 5n: In the 3-D case, the VECTOR analog of Cramer’s rule is (A /C29B) /C29(C /C29D) /C30(A /C215 B /C29D)C /C28(A /C215 B /C29C)D: (11) See also DETERMINANT ,LINEAR ALGEBRA ,M ATRIX , SYSTEM OF EQUATIONS ,VECTORReferences Cramer, G. "Intr. a` l’analyse de lignes courbes alge´briques." Geneva, 657 /C1/59, 1750. Muir, T. The Theory of Determinants in the Historical Order of Development, Vol. 1. New York: Dover, pp. 11 /C1/4, 1960. Crame ´r’s Theorem If X and Y are INDEPENDENT variates and X /C27Y is a GAUSSIAN DISTRIBUTION , then both X and Y must have GAUSSIAN DISTRIBUTIONS . This was proved by Crame ´r in 1936. Craps A game played with two DICE. If the total is 7 or 11 (a "natural"), the thrower wins and retains the DICE for another throw. If the total is 2, 3, or 12 ("craps"), the thrower loses but retains the DICE. If the total is any other number (called the thrower’s "point"), the thrower must continue throwing and roll the "point" value again before throwing a 7. If he succeeds, he wins and retains the DICE, but if a 7 appears first, the player loses and passes the DICE. The following table summarizes the probabilities of winning on a roll-by-roll basis, where P(p /C30n) is the probability of rolling a point n. For rolls that are not naturals (W) or craps (L), the probability that the point p /C30 n will be rolled first is found from P(win½p /C30n) /C30P(p /C30 n) P(p /C30 7) /C27 P(p /C30 n) /C30P(p /C30 n) 1 6 36 /C27 P(p /C30 n) : n /P(p /C30n)/ W/L /P(win½p /C30n)/ 2 /1 36/ L0 3 /2 36/ L0 4 /3 36//3 9/ 5 /4 36//4 10/ 6 /5 36//5 11/ 7 /6 36/ W1 8 /5 36//5 11/ 9 /4 36//4 10/ 10 /3 36//39/ 11 /2 36/ W1 12 /1 36/ L0 Summing P(p/C30n) from n/C301 to 12 then gives the probability of winning as 244 =495:0:492929 (Kraitchik 1942), just under 50%. See also DICE References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 12 /C1/3, 1951. Kraitchik, M. "Craps." §6.5 in Mathematical Recreations. New York: W. W. Norton, pp. 123 /C1/26, 1942. CRC CYCLIC REDUNDANCY CHECK Creative Telescoping TELESCOPING SUM,ZEILBERGER’S ALGORITHM Cremona Transformation An entire Cremona transformation is a BIRATIONAL TRANSFORMATION of the PLANE . Cremona transforma- tions are MAPS OF THE FORM xi /C271 /C30f(xi ; yi) ; yi /C271 /C30g(xi ; yi) ; in which f and g are POLYNOMIALS . A quadratic Cremona transformation is always factorable. See also NOETHER’S TRANSFORMATION THEOREM References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 203 /C1/04, 1959. Cremona-Richmond Configuration A153configuration of 15 lines and 15 points, with three lines through three points, three points on every line, and containing no triangles. See also CONFIGURATION References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 40, 1991. Cribbage Cribbage is a game in which each of two players is dealt a hand of six CARDS . Each player then discards two of his six cards to a four-card "crib" which alternates between players. After the discard, thetop card in the remaining deck is turned up. Cards are then alternately played out by the two players, with points being scored for pairs, runs, cumulative total of 15 and 31, and playing the last possible card ("go") not giving a total over 31. All face cards are counted as 10 for the purpose of playing out, but the normal values of Jack /C3011 ; Queen /C3012 ; King /C3013 are used to determine runs. Aces are always low (/ace /C301): After all cards have been played, each player counts the four cards in his hand taken in conjunction with the single top card. Points are awarded for pairs, flushes, runs, and combinations of cards giving 15. A Jack having the same suit as a top card is awarded an additional point for "nobbs." The crib is then also counted and scored. The winner is the first person to "peg" a certain score, as recorded on a "cribbage board." The best possible score in a hand is 29, corresponding to three 5s and a Jack with a top 5 the same suit as the Jack. Hands with scores of 19, 25, 26, and 27 are not possible. A hand scoring zero points is therefore sometimes humorously referred to as a "19-point" hand. See also BRIDGE CARD GAME,CARDS ,POKER Criss-Cross Method A standard form of the LINEAR PROGRAMMING problem of maximizing a linear function over a CONVEX POLYHEDRON is to maximize c /C215 x subject to mx 5b and x ]0; where m is a given s /C29d matrix, c and b are given d-vector and s-vectors, respectively. The Criss-cross method always finds a VERTEX solution if an optimal solution exists. See also CONVEX POLYHEDRON ,LINEAR PROGRAM- MING ,VERTEX (POLYHEDRON ) Criterion A requirement NECESSARY for a given statement or theorem to hold. Also called a CONDITION . See also BROWN’S CRITERION ,C AUCHY CRITERION , EULER’S CRITERION ,G AUSS’S CRITERION ,K ORSELT’S CRITERION ,LEIBNIZ CRITERION ,POCKLINGTON’S CRI- TERION ,VANDIVER’S CRITERIA ,W EYL’S CRITERION Critical Damping DAMPED SIMPLE HARMONIC MOTION– CRITICAL DAMP- ING Critical Index LetFbe the M ACLAURIN SERIES of a MEROMORPHIC FUNCTION fwith a finite or infinite number of POLES at points zk;indexed so that 0 B½z1 ½5½z2 ½5½z3 ½5...; then a POLE will occur as many times in the sequence fzk g as indicated by its order. Any index such that ½zm ½B½zm/C271 ½ holds is then called a critical index of f (Henrici 1988, pp. 641 /C1/42). References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, pp. 641 /C1/42, 1988. Critical Line The LINE R(s) /C301 =2 in the COMPLEX PLANE on which the RIEMANN HYPOTHESIS asserts that all nontrivial (COMPLEX ) ROOTS of the RIEMANN ZETA FUNCTION z(s) lie. Although it is known that an INFINITE number of zeros lie on the critical line and that these comprise at least 40% of all zeros, the RIEMANN HYPOTHESIS is still unproven. See also CRITICAL STRIP,R IEMANN HYPOTHESIS , RIEMANN ZETA FUNCTION References Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361 /C1/372, 1979. Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. II." Math. Comput. 39, 681 /C1/88, 1982. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 142, 1991. Critical Point A FUNCTION y /C30f(x) has critical points at all points x0 where f ?(x0) /C300or f(x) is not DIFFERENTIABLE .A FUNCTION z /C30f(x; y) has critical points where the GRADIENT 9f /C300or @f =@x or the PARTIAL DERIVATIVE @f =@y is not defined. See also FIXED POINT ,INFLECTION POINT ,O NLY CRITICAL POINT IN TOWN TEST,STATIONARY POINTCritical Strip The region /0 B s B1/, where s is defined as the REAL PART of a COMPLEX NUMBER s /C30 s /C27it : All nontrivial zeros (i.e., those at negative integer) of the RIEMANN ZETA FUNCTION lie inside this strip. See also CRITICAL LINE,R IEMANN HYPOTHESIS , RIEMANN ZETA FUNCTION References Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361 /C1/372, 1979. Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. II." Math. Comput. 39, 681 /C1/88, 1982. Crofton Cell A RANDOM POLYGON containing the origin (Kovalenko 1999). See also RANDOM POLYGON References Kovalenko, I. N. "A Simplified Proof of a Conjecture of D. G. Kendall Concerning Shapes of Random Polygons." J. Appl. Math. Stoch. Anal. 12, 301 /C1/10, 1999. Crofton’s Formula Let n points j1 ; ..., jnbe randomly distributed on a domain S, and let H be some event that depends on the positions of the n points. Let S ? be a domain slightly smaller than S but contained within it, and let dS be the part of S not in S?: Let P[H] be the probability of event H, s be the measure of S, and dS the measure of dS; then Crofton’s formula states that dP[H] /C30n(P[H j1 /C23 dS] /C28P[H])s/C281 ds (Solomon 1978, p. 99). See also CROFTON’S INTEGRALS References Ruben, H. and Reed, W. J. "A More General Form of the Theory of Crofton." J. Appl. Prob. 10, 479/C1/82, 1973. Solomon, H. "Crofton’s Theorem and Sylvester’s Problem in Two and Three Dimensions." Ch. 5 in Geometric Prob- ability. Philadelphia, PA: SIAM, pp. 97 /C1/25, 1978. Crofton’s Integrals Consider a convex plane curve K with PERIMETER L, and the set of points P exterior to K. Further, let t1 and t2be the perpendicular distances from P to K (with corresponding tangent points A1 and A2 on K), and let v /C30/C218A1PA2 : Then gP ext : to Ksin v t1t2dP /C302p2 (1) (Crofton 1885; Solomon 1978, p. 28). If K has a continuous RADIUS OF CURVATURE and the radii of curvature at points A1and A2are r1and r2 ; then gP ext : to Ksin v t1t2r1 r2 dP /C301 2 L2 (2) (Solomon 1978, p. 28), and furthermore gP ext : to Ksin v t1t2(r1 /C27 r2) dP /C302pL (3) (Santalo ´ 1953; Solomon 1978, p. 28). See also CROFTON’S FORMULA References Crofton, M. W. "Probability." Encyclopaedia Britannica, 9th ed., Vol. 19. Philadelphia, PA: J. M. Stoddart, pp. 768 /C1/ 88, 1885. Santalo ´,L.Introduction to Integral Geometry. Paris: Her- mann, 1953. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978. Crofton’s Theorem CROFTON’S FORMULA Crook A6- POLYIAMOND . See also POLYIAMOND References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Crookedness Let a KNOT K be parameterized by a VECTOR FUNC- TION v(t) with t /C23S1 ; and let w be a fixed UNIT VECTOR in R3 : Count the number of RELATIVE MINIMA of the projection function w /C215 v(t) : Then the MINIMUM suchnumber over all directions w and all K of the given type is called the crookedness m(K): Milnor (1950) showed that 2pm(K) is the INFIMUM of the total curvature of K. For any TAME KNOT K in R3 ; m(K) /C30 b(K) where b(K) is the BRIDGE INDEX . See also BRIDGE INDEX References Milnor, J. W. "On the Total Curvature of Knots." Ann. Math. 52, 248 /C1/57, 1950. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 115, 1976. Cross In general, a cross is a figure formed by two inter- secting LINE SEGMENTS .In LINEAR ALGEBRA , a cross is defined as a set of n mutually PERPENDICULAR pairs of VECTORS of equal magnitude from a fixed origin in EUCLIDEAN n-SPACE . The word "cross" is also used to denote the operation of the CROSS PRODUCT ,soa /C29b would be pronounced "across b." See also CROSS PRODUCT ,D OT,E UTACTIC STAR, GAULLIST CROSS ,GREEK CROSS ,LATIN CROSS ,M AL- TESE CROSS ,PAPAL CROSS ,SAINT ANDREW’S CROSS , SAINT ANTHONY’S CROSS ,STAR Cross Curve CRUCIFORM Cross Fractal CANTOR SQUARE FRACTAL Cross of Lorraine GAULLIST CROSS Cross Polytope A regular POLYTOPE inn-D corresponding to the CONVEX HULL of the points formed by permuting the coordinates ( 91, 0, 0, ..., 0). A cross-polytope (also called an orthoplex) is denoted ? missing and has 2 n vertices and S CHLA ¨FLI SYMBOL f3;...;3|fflfflfflfflffl{zfflfflfflfflffl} n/C282;4g: The cross polytope is named because its 2 nvertices are located equidistant from the origin along the Cartesian axes in n-space, which each such axis perpendicular to all others. A cross polytope isbounded by 2 n(n/C281)/-simplexes, and is a dipyramid erected (in both directions) into the nth dimension, with an ( n/C281)/-dimensional cross polytope as its base. In 1-D, the cross polytope is the LINE SEGMENT [/C281; 1]: In 2-D, the cross polytope f4g is the filled SQUARE with vertices (/C281 ; 0); (0;/C281); (1; 0); (0; 1): In 3-D, the cross polytope (3; 4) is the convex hull of the OCTAHEDRON with vertices (/C281; 0; 0); (0;/C281 ; 0); (0; 0;/C281); (1; 0; 0); (0; 1 ; 0); (0; 0; 1): In 4-D, the cross polytope f3 ; 3 ; 4g is the 16-CELL , depicted in the above figure by projecting onto one of the four mutually perpendicular 3-spaces within the 4-space obtained by dropping one of the four vertex compo- nents (R. Towle). The graph of bn missing is isomorphic with the CIRCULANT GRAPH Ci1 ; 2 ;... ;(n/C281)(2n) :/ See also 16-CELL,HYPERCUBE ,POLYTOPE ,SIMPLEX Cross Product For VECTORS u and v, the cross product is defined by u /C29v /C30ˆx(uyvz /C28uzvy) /C28ˆy(uxvz /C28uzvx) /C27ˆz(uxvy /C28uyvx) : (1) This can be written in a shorthand NOTATION which takes the form of a DETERMINANT u /C29v /C30ˆx ˆy ˆz uxuyuz vxvyvzl112l112l112l112l112l112l112l112l112l112l112l112: (2) Here, /u /C29v/ is always PERPENDICULAR to both u and v, with the orientation determinant by the RIGHT-HAND RULE . It is also true that u /C29v jj /C30ujjvjjsin u; (3) /C30 ujjvjjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28(ˆu /C215 ˆv)2q ; (4) where u is the angle between u and v, given by the DOT PRODUCT cos u /C13ˆu /C215 ˆv: (5) Jeffreys and Jeffreys (1988) use the notation u fflv to denote the cross product. The cross product is implemented in Mathematica 3.0 and higher asCross [a, b]. Identities involving the cross product include d dt[r1(t) /C29r2(t)] /C30r1(t) /C29dr2 dt/C27dr1 dt/C29r2(t) (6) A /C29B /C30/C28B /C29A (7) A /C29(B /C27C) /C30A /C29B /C27A /C29C (8) (tA) /C29B /C30t(A /C29B) : (9) For a proof that A /C29B is a PSEUDOVECTOR , see Arfken (1985, pp. 22 /C1/3). In TENSOR notation, A /C29B /C30 eijkAjBk ; (10) where eijk is the PERMUTATION SYMBOL . See also CARTESIAN PRODUCT ,DOT PRODUCT ,PERMU- TATION SYMBOL ,RIGHT- HAND RULE,SCALAR TRIPLE PRODUCT ,VECTOR ,VECTOR DIRECT PRODUCT ,VEC- TOR MULTIPLICATION References Arfken, G. "Vector or Cross Product." §1.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 18 /C1/6, 1985. Jeffreys, H. and Jeffreys, B. S. "Vector Product." §2.07 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 67 /C1/3, 1988. Cross Section The cross section of a SOLID is a plane figure obtained by its intersection with a PLANE . The cross section of an object therefore represents an infinitesimal "slice" of a solid, and may be different depending on the orientation of the slicing plane. While the cross section of a SPHERE is always a DISK, the cross section of a CUBE may be a SQUARE ,HEXAGON , or other shape. See also AXONOMETRY ,CAVALIERI’S PRINCIPLE ,INNER QUERMASS ,L AMINA ,P LANE ,P ROJECTION ,R ADON TRANSFORM ,STEREOLOGY Cross Sequence A sequence s(l) n(x)/C30[h(t)]lsn(x); where /sn(x)/ is a SHEFFER SEQUENCE ,/h(t)/ is invertible, and l ranges over the real numbers is called a STEFFENSEN SEQUENCE .If /sn(x)/ is an associated SHEFFER SEQUENCE , then /s(l) n / is called a cross se- quence. Examples include the ACTUARIAL POLYNOMIAL and POISSON- CHARLIER POLYNOMIAL . See also APPELL CROSS SEQUENCE ,S HEFFER SE- QUENCE ,STEFFENSEN SEQUENCE References Roman, S. "Cross Sequences and Steffensen Sequences." §5.3 in The Umbral Calculus. New York: Academic Press, pp. 140 /C1/43, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Cross Surface A SPHERE with a single CROSS-CAP . This term is more appropriate in purely topological applications than the more common term REAL PROJECTIVE PLANE , which implies the presence of an affine structure (Francis and Weeks 1999). The double cross surface is the KLEIN BOTTLE and the triple cross surface is called D YCK’S SURFACE (Francis and Collins 1993, Francis and Weeks 1999). See also CROSS- CAP,REAL PROJECTIVE PLANE References Francis, G. and Collins, B. "On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art." Ch. 11 in The Visual Mind: Art and Mathematics (Ed. M. Emmer). Cambridge, MA: MIT Press, 1993. Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393/C1/99, 1999. Cross-Cap The self-intersection of a one-sided SURFACE . "Cross- cap" is sometimes also written without the hyphen as the single word "crosscap." The cross-cap can be thought of as the object produced by puncturing a surface a single time, attaching two ZIPS around the puncture in the same direction, distorting the hole so that the zips line up, requiring that the surfaceintersect itself, and then zipping up. The cross-cap can also be described as a circular HOLE which, when entered, exits from its opposite point (from a topolo-gical viewpoint, both singular points on the cross-capare equivalent). The cross-cap has a segment of double points which terminates at two " PINCH POINTS " known as W HITNEY SINGULARITIES .A CROSS-HANDLE is homeomorphic to two cross-caps (Francis and Weeks 1999). ASPHERE with one cross-cap has traditionally been called a REAL PROJECTIVE PLANE . While this is appro- priate in the study of PROJECTIVE GEOMETRY when an affine structure is present, J. H. Conway advocates use of the term CROSS SURFACE in a purely topological interpretation (Francis and Weeks 1999). The cross- cap is one of the three possible SURFACES obtained by sewing a M O¨BIUS STRIP to the edge of a DISK. The other two are the B OY SURFACE and R OMAN SURFACE . The cross-cap can be generated using the generalmethod for NONORIENTABLE SURFACES using the polynomial function f(x;y;z)/C30(xz;yz;1 2(z2/C28x2)) (1) (Pinkall 1986). Transforming to SPHERICAL COORDI- NATES gives x(u;v)/C301 2cosusin(2 v) (2) y(u;v)/C3012sinusin(2 v) (3) z(u;v)/C301 2(cos2v/C28cos2usin2v) (4) foru/C23[0;2p) and v/C23[0;p=2]:To make the equations slightly simpler, all three equations are normally multiplied by a factor of 2 to clear the arbitrary scaling constant. Three views of the cross-cap gener- ated using this equation are shown above. Note thatthe middle one looks suspiciously like B OUR’S MINI- MAL SURFACE . Another representation is f(x;y;z)/C30(yz;2xy;x2/C28y2); (5) (Gray 1997), giving PARAMETRIC EQUATIONS x /C301 2sin u sin(2 v) (6) y /C30sin(2 u) sin2 v (7) z /C30cos(2 u) sin2 v ; (8) (Geometry Center) where, for aesthetic reasons, the y- and z-coordinates have been multiplied by 2 to produce a squashed, but topologically equivalent, surface. Nordstrand gives the implicit equation 4x2(x2 /C27y2 /C27z2 /C27z) /C27y2(y2 /C27z2 /C281) /C300 (9) which can be solved for z to yield z /C30/C282x2 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (y2 /C27 2x2)(1 /C28 4x2 /C28 y2)p 4x2 /C27 y2 : (10) Taking the inversion of a cross-cap such that (0, 0, /C281=2) is sent to /C12 gives a CYLINDROID , shown above (Pinkall 1986). See also BOY SURFACE ,CAP,CLASSIFICATION THEO- REM OF SURFACES ,CROSS- HANDLE ,CROSS SURFACE , HANDLE ,M O¨ BIUS STRIP,N ONORIENTABLE SURFACE , PROJECTIVE PLANE ,ROMAN SURFACE References Fischer, G. (Ed.). Plate 107 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 108, 1986. Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/99, 1999. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 15, 1984. Gray, A. "The Cross Cap." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 333 /C1/35, 1997. Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 64, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 197, 1991. Cross-Correlation The cross-correlation of two COMPLEX FUNCTIONS f(t) and g(t) of a real variable t, denoted f w g is defined by f w g /C13 ¯f(/C28t) + g(t) ; (1) where + denotes CONVOLUTION and ¯f(t) is the COM-PLEX CONJUGATE of f(t) : Since CONVOLUTION is de- fined by f(t) + g(t) /C30g/C12 /C28/C12f( t)g(t /C28 t) dt ; (2) it follows that f w g /C13g/C12 /C28/C12¯f(/C28t)g(t /C28 t) dt : (3) Letting t ?/C30/C28t; dt ?/C30/C28 d t so (3) is equivalent to fwg/C30g/C28/C12 /C12¯f(t?)g(t/C27t?)(/C28dt?) /C30g/C12 /C28/C12¯f(t)g(t/C27t)dt: (4) The cross-correlation satisfies the identity (gwh)w(gwh)/C30(gwg)w(hwh): (5) IfforgisEVEN , then fwg/C30f+g; (6) where +again denotes CONVOLUTION . See also AUTOCORRELATION ,CONVOLUTION ,CROSS- CORRELATION THEOREM ,FOURIER TRANSFORM References Bracewell, R. "Pentagram Notation for Cross Correlation." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 46 and 243, 1999. Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 244 /C1/45 and 252 /C1/53, 1962. Cross-Correlation Coefficient The COEFFICIENT rin a G AUSSIAN BIVARIATE DISTRI- BUTION . Cross-Correlation Theorem Letfwgdenote the CROSS-CORRELATION of functions f(t) and g(t):Then f w g /C30g/C12 /C28/C12¯f( t)g(t /C27 t) dt /C30g/C12 /C28/C12g/C12 /C28/C12¯F( n)e2 pint dng/C12 /C28/C12G( n ?)e /C282 pin ?(t/C27 t) dn ?"# dt /C30g/C12 /C28/C12g/C12 /C28/C12g/C12 /C28/C12¯F( n)G(n ?)e /C282pi t(n ?/C28n) e/C282pi n?t dt dn d n? /C30g/C12 /C28/C12g/C12 /C28/C12¯F( n)G( n?)e /C282pin ?tg/C12 /C28/C12e/C282pit( n?/C28n) dt"# dn dn ? /C30g/C12 /C28/C12g/C12 /C28/C12¯F( n)G( n?)e /C282pin ?t d( n?/C28n) dn? dn /C30g/C12 /C28/C12¯F( n)G( n)e /C282 pint dn (1) where F denotes the FOURIER TRANSFORM , ¯z is the COMPLEX CONJUGATE , and f(t) /C13F[F(n)] /C30g/C12 /C28/C12F(n)e /C282pint dt (2) g(t) /C13F[G( n)] /C30g/C12 /C28/C12G( n)e /C282 pi nt dt: (3) Applying a FOURIER TRANSFORM on each side gives the cross-correlation theorem, f w g /C30F[ ¯F(n)G( n)]: (4) If F /C30G, then the cross-correlation theorem reduces to the WIENER- KHINTCHINE THEOREM . See also FOURIER TRANSFORM ,W IENER- KHINTCHINE THEOREM Crosscram DOMINEERING Crossed Hyperbolic Rotation Exchanges branches of the HYPERBOLA x ?y?/C30xy: x?/C30 m/C281x y?/C30/C28my: See also HYPERBOLIC ROTATIONCrossed Ladders Problem Given two crossed LADDERS resting against two buildings, what is the distance between the build- ings? Let the height at which they cross be h and the lengths of the LADDERS l1 and l2 : The height at which l2touches the building h2is then obtained by simultaneously solving the equations l2 1 /C30h21 /C27d2 (1) l22 /C30h22 /C27d2 (2) and 1 h /C301 h1/C271 h2; (3) the latter of which follows either immediately from the CROSSED LADDERS THEOREM or from similar triangles with d1 /C30dh=h2 ; d2 /C30dh=h1 ; and d /C30d1 /C27 d2 : Eliminating d gives the equations h41 /C282hh31 /C27(h /C28h1)2(l22 /C28l21) /C300: (4) h42 /C282hh32 /C27(h /C28h2)2(l21 /C28l22) /C300: (5) These quartic equations can be solved for h1and h2 given known values of h, l1 ; and l2 :/ There are solutions in which not only l1 ; l2 ; h1 ; h2 ; and h are all integers, but so are d1 ; and d2 : One example is h1 /C30119; h1 /C3070; h /C30 30, d1 /C3040; d2 /C3016 :/ The problem can also be generalized to the situation in which the ends of the ladders are not pinned against the buildings, but propped fixed distances d1 andd2away. See also CROSSED LADDERS THEOREM ,LADDER References Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 62 /C1/4, 1979. Crossed Ladders Theorem In the above figure, let E be the intersection of AD and BC and specify that ABIEFICD: Then 1 AB /C271 CD /C301 EF: A beautiful related theorem due to H. Stengel can be stated as follows. In the above figure, let E lie on the side AB and D lie on the side BC. Now let EC intersect the line AD at a point F, and construct points H, I, and J so that EIIDHIFJIBG : Then 1 EI /C271 DH /C301 FJ /C271 BG : See also CROSSED LADDERS PROBLEM Crossed Trough The SURFACEz /C30cx2y2 : See also MONKEY SADDLE References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 286, 1993. Cross-Handle A cross-handle is a topological structure which can be thought of as the object produced by puncturing a surface twice, attaching a ZIP around each puncture travelling in the same direction, pulling the edges of the zips together after one tube first passes through itself it order for the direction of the zips to match up, and then zipping up. In 3-space, the cross-handle contains a line of self-intersection. A cross-handle is homeomorphic to two CROSS-CAPS (Francis and Weeks 1999). DYCK’S THEOREM states that HANDLES and cross-handles are equivalent in the presence of a CROSS-CAP . See also CAP,CROSS- CAP,DYCK’S THEOREM ,HANDLE References Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393/C1/99, 1999. Crossing Number (Graph) Given a "good" GRAPH G(i.e., one for which all intersecting EDGES intersect in a single point and arise from four distinct VERTICES ), the crossing number n(G) is the minimum possible number of crossings with which the GRAPH can be drawn. A GRAPH with crossing number 0 is a PLANAR GRAPH . Garey and Johnson (1983) showed that determining the crossing number is an NP -COMPLETE PROBLEM . GUY’S CONJECTURE suggests that the crossing number for the COMPLETE GRAPH Knis n(Kn)/C301 4n 2$% n/C281 2$% n/C282 2$% n/C283 2$% ; (1) which can be rewritten n(Kn) /C301 64 n(n /C282)2(n /C284) for n even 1 64(n /C281)2(n /C283)2for n odd:( (2) The values of (2) for n /C301, 2, ... are then given by 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588, ... (Sloane’s A000241), although it has not been proven that these agree with the actual crossing numbers for n ]11 :/ ZARANKIEWICZ’S CONJECTURE asserts that the cross- ing number for a COMPLETE BIGRAPH is n(Km; n) /C30n 2$% n /C28 1 2$% m 2$% m /C28 1 2$% : (3) It has been checked up to m; n /C307 ; and Zarankiewicz has shown that, in general, the FORMULA provides an upper bound to the actual number. The table below gives known results. When the number is not known exactly, the prediction of ZARANKIEWICZ’S CONJEC- TURE is given in parentheses. 1234 5 6 7 10000 0 0 0 2 00000 0 3 1246 9 44 8 1 2 1 8 51 6 2 4 36 63 6 5 4 7 77, 79, or (81) Kleitman (1970, 1976) computed the exact crossing numbers n(K5 ; n) for all positive n. See also GUY’S CONJECTURE ,RECTILINEAR CROSSING NUMBER ,TOROIDAL CROSSING NUMBER ,ZARANKIE- WICZ’S CONJECTURE References Erdos, P. and Guy, R. K. "Crossing Number Problems." Amer. Math. Monthly 80,52/C1/7, 1973. Gardner, M. "Crossing Numbers." Ch. 11 in Knotted Dough- nuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 133 /C1/44, 1986. Garey, M. R. and Johnson, D. S. "Crossing Number is NP- Complete." SIAM J. Alg. Discr. Meth. 4, 312 /C1/16, 1983. Guy, R. K. "The Crossing Number of the Complete Graph." Bull. Malayan Math. Soc. 7,68/C1/2, 1960. Guy, R. K. "Latest Results on Crossing Numbers." In Recent Trends in Graph Theory, Proc. New York City Graph Theory Conference, 1st, 1970. (Ed. New York City Graph Theory Conference Staff). New York: Springer-Verlag, 1971. Guy, R. K. "Crossing Numbers of Graphs." In Graph Theory and Applications: Proceedings of the Conference at Wes- tern Michigan University, Kalamazoo, Mich., May 10 /C1/3,1972 (Ed. Y. Alavi, D. R. Lick, and A. T. White). New York: Springer-Verlag, pp. 111 /C1/24, 1972. Kleitman, D. J. "The Crossing Number of ." J. Combin. Th. 9, 315 /C1/23, 1970. Kleitman, D. J. "A Note on the Parity of the Numbers of Crossings of a Graph." J. Combin. Th., Ser. B 21,88/C1/9, 1976. Koman, M. "Extremal Crossing Numbers of Complete k- Chromatic Graphs." Mat. Casopis Sloven. Akad. Vied. 20, 315 /C1/25, 1970. Kovari, T.; So´s, V. T.; and Tura´n, P. "On a Problem of K. Zarankiewicz." Colloq. Math. 3,50/C1/7, 1954. Moon, J. W. "On the Distribution of Crossings in Random Complete Graphs." SIAM J. 13, 506 /C1/10, 1965. Owens, A. "On the Biplanar Crossing Number." IEEE Trans. Circuit Th. 18, 277 /C1/80, 1971. Pach, J. and To´th, G. "Thirteen Problems on Crossing Numbers." Geocombin. 9, 195 /C1/07, 2000. Richter, R. B. and Thomassen, C. "Relations Between Cross- ing Numbers of Complete and Complete Bipartite Graphs." Amer. Math. Monthly 104, 131 /C1/37, 1997. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 251, 1990. Sloane, N. J. A. Sequences A014540 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Thomassen, C. "Embeddings and Minors." In Handbook of Combinatorics, 2 vols. (Ed. R. L. Graham, M. Gro¨tschel, and L. Lova´sz.) Cambridge, MA: MIT Press, p. 314, 1996. Tutte, W. T. "Toward a Theory of Crossing Numbers." J. Comb. Th. 8,45/C1/3, 1970. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Paul Erdos’s 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thomason). Cambridge, England: Cambridge Univer- sity Press, pp. 557 /C1/62, 1997. Crossing Number (Link) The least number of crossings that occur in any projection of a LINK . In general, it is difficult to find the crossing number of a given LINK . Knots and links are generally tabulated based on their crossingnumbers. See also K NOT,LINK References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 67 /C1/9, 1994. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,3 3/C1/8, Fall 1998. Cross-Ratio [a;b;c;d]/C13(a/C28b)(c/C28d) (a/C28d)(c/C28b): (1) For a M O¨BIUS TRANSFORMATION f, [a;b;c;d]/C30[f(a);f(b);f(c);f(d)]: (2) There are six different values which the cross-ratio may take, depending on the order in which the points are chosen. Let l /C13[a ; b ; c ; d] : Possible values of the cross-ratio are then l ; 1 /C28 l; 1=l ; ( l /C281)= l; 1 =(1 /C28 l); and l =(l /C281):/ Given lines a, b, c, and d which intersect in a point O, let the lines be cut by a line l, and denote the points of intersection of l with each line by A, B, C, and D. Let the distance between points A and B be denoted AB, etc. Then the cross-ratio [AB; CD] /C13(AB)(CD) (BC)(AD) (3) is the same for any position of the l (Coxeter and Greitzer 1967). Note that the definitions / (AB=AD) =(BC =CD)/ and /(CA=CB)=(DA =DB)/ are used instead by Kline (1990) and Courant and Robbins (1966), respectively. The identity [AD; BC] /C27[AB ; DC] /C301 (4) holds IFF /AC ==BD /, where / ==/ denotes SEPARATION . The cross-ratio of four points on a radial line of an INVERSION CIRCLE is preserved under INVERSION (Ogilvy 1990, p. 40). See also BIVALENT RANGE ,E QUICROSS ,H ARMONIC RANGE ,H OMOGRAPHIC ,M O¨ BIUS TRANSFORMATION , SEPARATION References Anderson, J. W. "The Cross Ratio." §2.3 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 30 /C1/6, 1999. Casey, J. "Theory of Anharmonic Section." §6.6 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 126 /C1/40, 1888. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, 1996. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 107 /C1/08, 1967. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 73 /C1/6, 1928. Graustein, W. C. "Cross Ratio." Ch. 6 in Introduction to Higher Geometry. New York: Macmillan, pp. 72 /C1/3, 1930. Kline, M. Mathematical Thought from Ancient to Modern Times, Vol. 1. Oxford, England: Oxford University Press, 1990. Lachlan, R. "Theory of Cross Ratio." Ch. 16 in An Elemen- tary Treatise on Modern Pure Geometry. London: Macmil- lian, pp. 266 /C1/82, 1893. Mo¨bius, A. F. Ch. 5 in Der barycentrische Calcul: Ein neues Hu¨lfsmittel zur analytischen Behandlung der Geometrie, dargestellt und insbesondere auf die Bildung neuer Clas- sen von Aufgaben und die Entwickelung mehrerer Ei- genschaften der Kegelschnitte angewendet. Leipzig, Germany: J. A. Barth, 1827. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 39 /C1/1, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 41, 1991.Cross-Stitch Curve A fractal curve of infinite length which bounds an area twice that of the original square. See also BOX FRACTAL ,CANTOR SQUARE FRACTAL , FRACTAL ,SIERPINSKI CURVE References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 228 /C1/29, 1984. Crout’s Method A ROOT finding technique used in LU DECOMPOSITION . It solves the /N2/ equations i Bjli1u1j /C27li2u2j /C27/C1/C1/C1/C27liiujj /C30aij i /C30jli1u1j /C27li2u2j /C27/C1/C1/C1/C27liiujj /C30aij i /C21jli1u1j /C27li2u2j /C27/C1/C1/C1/C27liiujj /C30aij for the /N2 /C27N/ unknowns /lij/ and /uij/. See also LU DECOMPOSITION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 36 /C1/8, 1992. Crowd A group of SOCIABLE NUMBERS of order 3. Crown A6 - POLYIAMOND . See also POLYIAMOND References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Crucial Point The HOMOTHETIC CENTER of the ORTHIC TRIANGLE and the triangular hull of the three EXCIRCLES . It has TRIANGLE CENTER FUNCTION a /C30tan A /C30sin(2 B) /C27sin(2 C) /C28sin(2 A) : References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Lyness, R. and Veldkamp, G. R. Problem 682 and Solution. Crux Math. 9,23/C1/4, 1983. Cruciform A plane curve also called the CROSS CURVE and POLICEMAN ON POINT DUTY CURVE (Cundy and Rollett 1989). It is given by the equation x2y2 /C28a2x2 /C28b2y2 /C300; (1) which is equivalent to 1 /C28a2 x2 /C28b2 y2 /C300 (2) a2 x2 /C27b2 y2 /C301; (3) or, rewriting, y2 /C30b2x2 x2 /C28 a2 : (4) In parametric form, x /C30a sec t (5) y /C30b csc t: (6)The CURVATURE is k /C303ab csc2 t sec2 t (b2 cos2 t cos2 ta2 sec2 t tan2 t)3 =2 : (7) References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 127 and 130 /C1/31, 1972. Crunode A point where a curve intersects itself so that two branches of the curve have distinct tangent lines. The MACLAURIN TRISECTRIX , shown above, has a crunode at the origin. See also ACNODE ,SPINODE ,TACNODE Cryptarithm CRYPTARITHMETIC Cryptarithmetic A number PUZZLE in which a group of arithmetical operations has some or all of its DIGITS replaced by letters or symbols, and where the original DIGITS must be found. In such a puzzle, each letter repre- sents a unique digit. See also ALPHAMETIC ,DIGIMETIC ,SKELETON DIVISION References Bogomolny, A. "Cryptarithms." http://www.cut-the-knot.- com/st_crypto.html. Brooke, M. One Hundred & Fifty Puzzles in Crypt-Arith- metic. New York: Dover, 1963. Kraitchik, M. "Cryptarithmetic." §3.11 in Mathematical Recreations. New York: W. W. Norton, pp. 79 /C1/3, 1942. Marks, R. W. The New Mathematics Dictionary and Hand- book. New York: Bantam Books, 1964. Cryptographic Hash Function A cryptographic hash function is most commonly one of the following: a ONE-WAY HASH FUNCTION ,aCOLLI- SION-FREE HASH FUNCTION ,a TRAPDOOR ONE-WAY HASH FUNCTION , or a function from a class of UNIVERSAL HASH FUNCTIONS . See also BIRTHDAY ATTACK ,COLLISION- FREE HASH FUNCTION ,H ASH FUNCTION ,ONE-WAY HASH FUNC- TION ,TRAPDOOR ONE-WAY HASH FUNCTION ,UNIVER- SAL HASH FUNCTION References Bakhtiari, S.; Safavi-Naini, R.; and Pieprzyk, J. Crypto- graphic Hash Functions: A Survey. Technical Report 95 /C1/ 9, Department of Computer Science, University of Wol- longong, July 1995. ftp://ftp.cs.uow.edu.au/pub/papers/ 1995/tr-95 /C1/9.ps.Z. Cryptography The science of adversarial information protection. See also CODING THEORY ,C RYPTARITHM ,C RYPTO- GRAPHIC HASH FUNCTION ,KNAPSACK PROBLEM ,PUB- LIC-KEY CRYPTOGRAPHY ,T RAPDOOR ONE-WAY FUNCTION References Davies, D. W. The Security of Data in Networks. Los Angeles, CA: IEEE Computer Soc., 1981. Diffie, W. and Hellman, M. "New Directions in Cryptogra- phy." IEEE Trans. Info. Th. 22, 644 /C1/54, 1976. Honsberger, R. "Four Clever Schemes in Cryptography." Ch. 10 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 151 /C1/73, 1985. Simmons, G. J. "Cryptology, The Mathematics of Secure Communications." Math. Intel. 1, 233 /C1/46, 1979. van Tilborg, H. C. A. Fundamentals of Cryptography: A Professional Reference and Interactive Tutorial. Norwell, MA: Kluwer, 1999. Crystallographic Point Groups The crystallographic point groups are the POINT GROUPS in which translational periodicity is required (the so-called CRYSTALLOGRAPHY RESTRICTION ). There are 32 such groups, summarized in the following table which organized them by SCHO¨ NFLIES SYMBOL type. type point groups nonaxial /Ci ; Cs/ cyclic /C1 ; C2 ; C3 ; C4 ; C6/ cyclic with horizontal planes /C2h ; C3h ; C4h ; C6h/ cyclic with vertical planes /C2v ; C3v ; C4v ; C6v/ dihedral /D2 ; D3 ; D4 ; D6/ dihedral with horizontal planes/D2h ; D3h ; D4h ; D6h/ dihedral with planes between axes/D2d ; D3d/ improper rotation /S4 ; S6/ cubic groups /T ; Th ; Td ; O; Oh/Note that while the TETRAHEDRAL /Td/ and OCTAHEDRAL /Oh/ POINT GROUPS are also crystallographic point groups, the ICOSAHEDRAL GROUP /Ih/ is not. The orders, classes, and group operations for these groups can be concisely summarized in their CHARACTER TABLES . See also CHARACTER TABLE ,CRYSTALLOGRAPHY RE- STRICTION ,DIHEDRAL GROUP ,GROUP ,GROUP THEO- RY,H ERMANN- MAUGUIN SYMBOL ,LATTICE GROUPS , OCTAHEDRAL GROUP ,P OINT GROUPS ,S CHO¨ NFLIES SYMBOL ,SPACE GROUPS ,TETRAHEDRAL GROUP References Arfken, G. "Crystallographic Point and Space Groups." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 248 /C1/49, 1985. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 379, 1990. Hahn, T. (Ed.). International Tables for Crystallography, vol. A, 4th ed. Dordrecht, Netherlands: Kluwer, p. 752, 1995. Lomont, J. S. "Crystallographic Point Groups." §4.4 in Applications of Finite Groups. New York: Dover, pp. 132 /C1/46, 1993. Yale, P. B. "Crystallographic Point Groups." §3.4 in Geome- try and Symmetry. New York: Dover, pp. 103 /C1/08, 1988. Crystallography Restriction If a discrete GROUP of displacements in the plane has more than one center of rotation, then the only rotations that can occur are by 2, 3, 4, and 6. This can be shown as follows. It must be true that the sum of the interior angles divided by the number of sides is a divisor of 3608. 180/C14(n /C28 2) n/C30360/C14 m; where m is an INTEGER . Therefore, symmetry will be possible only for 2n n /C28 2 /C30m; where m is an INTEGER . This will hold for 1-, 2-, 3-, 4-, and 6-fold symmetry. That it does not hold for n /C216is seen by noting that n /C306 corresponds to m /C303. The m /C302 case requires that /n /C30n /C282/ (impossible), and them/C301 case requires that n/C30/C28 2 (also impossible). The POINT GROUPS that satisfy the crystallographic restriction are called CRYSTALLOGRAPHIC POINT GROUPS . See also CRYSTALLOGRAPHIC POINT GROUPS ,POINT GROUPS ,SYMMETRY References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 5, 1999. Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., p. 5, 1999. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 304, 1999. Yale, P. B. Geometry and Symmetry. New York: Dover, p. 104, 1988. Csa´sza´r Polyhedron A POLYHEDRON topologically equivalent to a TORUS which was discovered in the late 1940s by A´ kos Csa´sza´r (Gardner 1975). It has 7 VERTICES , 14 faces, and 21 EDGES , and is the DUAL POLYHEDRON of the SZILASSI POLYHEDRON . The SKELETON of the Csa´sza´r polyhedron, illustrated above, is ISOMORPHIC to the COMPLETE GRAPH K7. Rather surprisingly, the graph of the Csa´sza´r poly- hedron’s skeleton and its DUAL GRAPH can be used to find STEINER TRIPLE SYSTEMS (Gardner 1975). The figure above shows how to construct the Csa´sza´r polyhedron. See also SZILASSI POLYHEDRON ,TOROIDAL POLYHE- DRON References Csa´sza´r, A´ . "A Polyhedron without Diagonals." Acta Sci. Math. 13, 140 /C1/42, 1949 /C1/950. Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102 /C1/07, May 1975.Gardner, M. "The Csa´sza´r Polyhedron." Ch. 11 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 139 /C1/52, 1988. Gardner, M. Fractal Music, Hypercards, and More: Mathe- matical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 118 /C1/20, 1992. Hart, G. "Toroidal Polyhedra." http://www.georgehart.com/ virtual-polyhedra/toroidal.html. Csc COSECANT Csch HYPERBOLIC COSECANT C-Table C-DETERMINANT Ctg COTANGENT Cth HYPERBOLIC COTANGENT Ctn COTANGENT Cubature Ueberhuber (1997, p. 71) and Krommer and Ueber- huber (1998, pp. 49 and 155 /C1/65) use the word "QUADRATURE " to mean numerical computation of a univariate INTEGRAL , and "cubature" to mean numer- ical computation of a MULTIPLE INTEGRAL . Cubature techniques available in Mathematica include MONTE CARLO INTEGRATION , implemented asNIntegrate [f, ..., Method- /C21MonteCarlo ]orNIntegrate [f, ..., Method- /C21QuasiMonteCarlo ], and the adaptive Genz-Malik algorithm, implemented as NIntegra- te[f, ...,Method- /C21MultiDimensional ]. See also MONTE CARLO INTEGRATION ,N UMERICAL INTEGRATION ,QUADRATURE References Cools, R. "Monomial Cubature Rules Since "Stroud": A Compilation--Part 2." J. Comput. Appl. Math. 112,2 1/C1/7, 1999. Cools, R. "Encyclopaedia of Cubature Formulas." http:// www.cs.kuleuven.ac.be/~nines/research/ecf/ecf.html. Cools, R. and Rabinowitz, P. "Monomial Cubature Rules Since "Stroud": A Compilation." J. Comput. Appl. Math. 48, 309/C1/26, 1993. Krommer, A. R. and Ueberhuber, C. W. "Construction of Cubature Formulas." §6.1 in Computational Integration. Philadelphia, PA: SIAM, pp. 155 /C1/65, 1998. Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, 1997. Cube The three-dimensional P LATONIC SOLID P3which is also called the HEXAHEDRON . The cube is composed of six SQUARE faces, 6 f4g;which meet each other at RIGHT ANGLES , and has eight VERTICES and 12 EDGES . It is also the UNIFORM POLYHEDRON U6and Wennin- ger model W3:It is described by the S CHLA ¨FLI SYMBOL f4;3gand W YTHOFF SYMBOL 3½24:/ The DUAL POLYHEDRON of the cube is the OCTAHE- DRON . It has the OhOCTAHEDRAL GROUP of symme- tries, and is a ZONOHEDRON . The connectivity of the vertices is given by the CUBICAL GRAPH . Because the VOLUME of a cube of side length nis given by n3;a number OF THE FORM n3is called a CUBIC NUMBER (or sometimes simply "a cube"). Simi- larly, the operation of taking a number to the third POWER is called CUBING . Sodium chloride (NaCl; common table salt) naturally forms cubic crystals. Using so-called "wallet hinges," a ring of six cubes can be rotated continuously (Wells 1975; Wells 1991, pp. 218 /C1/19). The cube cannot be STELLATED .A PLANE passing through the MIDPOINTS of opposite sides (perpendi- cular to a C3axis) cuts the cube in a regular HEXAGONAL CROSS SECTION (Gardner 1960; Steinhaus 1983, p. 170; Cundy and Rollett 1989, p. 157; Holden1991, pp. 22 /C1 /3). Since there are four such axes, there are four possible HEXAGONAL CROSS SECTIONS . If the vertices of the cube are ( 91;91;91);then the vertices of the inscribed HEXAGON are (0 ;/C281;/C281); (1;0;/C281);(1;1;0);(0;1;1);(/C281;0;1);and (/C281;/C281;0):AHEXAGON is also obtained when the cube is viewed from above a corner along the exten-sion of a space diagonal (Steinhaus 1983, p. 170). A HYPERBOLOID of one sheet is obtained as the envelope of a cube rotated about a space diagonal (Steinhaus1983, pp. 171 /C1 /72). The centers of the faces of an OCTAHEDRON form a cube, and the centers of the faces of a cube form an OCTAHEDRON (Steinhaus 1983, pp. 194 /C1/95). The lar- gest SQUARE which will fit inside a cube of side ahas each corner a distance 1/4 from a corner of a cube. Theresulting SQUARE has side length 3ffiffiffi 2p a=4;and the cube containing that side is called P RINCE RUPERT’S CUBE . The solid formed by the faces having the sides of the STELLA OCTANGULA (left figure) as DIAGONALS is a cube (right figure; Ball and Coxeter 1987). Affixing a SQUARE PYRAMID of height 1/2 on each face of a cube having unit edge length results in a RHOMBIC DODE- CAHEDRON (Bru¨ckner 1900, p. 130; Steinhaus 1983, p. 185). The cube can be constructed by CUMULATION of a unit edge-length TETRAHEDRON by a pyramid with height 1 6ffiffiffi 6p :The following table gives polyhedra which can be constructed by CUMULATION of acube by pyramids of given heights h. h /(r /C27h)=h/ Result /1 6// 4=3/ TETRAKIS HEXAHEDRON /1 2/ 2 RHOMBIC DODECAHEDRON /12ffiffiffi 2p //1 /C27ffiffiffi2p / 24-faced star DELTAHEDRON The VERTICES of a cube of side length 2 with face- centered axes are given by (91;91 ;91): If the cube is oriented with a space diagonal along the Z-AXIS , the coordinates are (0, 0,ffiffiffi 3p ) ; (0, 2ffiffiffiffiffiffiffiffi 2=3p ; 1 =ffiffiffi 3p ) ; (/ffiffiffi2p ;ffiffiffiffiffiffiffiffi 2=3p ;/C281=ffiffiffi 3p ); ( /ffiffiffi 2p ;/C28ffiffiffiffiffiffiffiffi 2 =3p ; 1=ffiffiffi 3p ) ; (0, /C282ffiffiffiffiffiffiffiffi 2=3p ; /C281=ffiffiffi 3p ); (//C28ffiffiffi 2p ;/C28ffiffiffiffiffiffiffiffi 2=3p ; 1=ffiffiffi 3p ) ; (//C28ffiffiffi 2p ;ffiffiffiffiffiffiffiffi 2=3p ;/C281 =ffiffiffi 3p ); and the negatives of these vectors. A FACETED version is the GREAT CUBICUBOCTAHEDRON . A cube of side length 1 has INRADIUS , MIDRADIUS , and CIRCUMRADIUS of r /C301 2 /C300:5 (1) r /C3012ffiffiffi 2p :0:70710 (2) R /C301 2ffiffiffi 3p :0:86602 : (3) The cube has a DIHEDRAL ANGLE of a /C301 2 p: (4) The SURFACE AREA and VOLUME of the cube are S /C306a2 (5) V /C30a3 : (6) See also AUGMENTED TRUNCATED CUBE,BIAUGMEN- TED TRUNCATED CUBE,BIDIAKIS CUBE,BISLIT CUBE, BROWKIN’S THEOREM ,CUBE DISSECTION ,CUBE DOVE- TAILING PROBLEM ,CUBE DUPLICATION ,CUBIC NUM- BER,CUBICAL GRAPH ,CUBOID ,GOURSAT’S SURFACE , HADWIGER PROBLEM ,HYPERCUBE ,KELLER’S CONJEC- TURE ,P LATONIC SOLID ,P RINCE RUPERT’S CUBE, PRISM ,RUBIK’S CUBE,SOMA CUBE,STELLA OCTANGU- LA,TESSERACT ,UNIT CUBE References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127 and 228, 1987. Bru¨ckner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900. Cundy, H. and Rollett, A. "Cube. 43" and "Hexagonal Section of a Cube." §3.5.2 and 3.15.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 85, 1989. Davie, T. "The Cube (Hexahedron)." http://www.dcs.st-an- d.ac.uk/~ad/mathrecs/polyhedra/cube.html. Eppstein, D. "Rectilinear Geometry." http://www.ics.uci.edu/ ~eppstein/junkyard/rect.html.Gardner, M. "Mathematical Games: More About the Shapes that Can Be Made with Complex Dominoes." Sci. Amer. 203, 186 /C1/98, Nov. 1960. Harris, J. W. and Stocker, H. "Cube" and "Cube (Hexahe- dron)." §4.2.4 and 4.4.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 97 /C1/8 and 100, 1998. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Kern, W. F. and Bland, J. R. "Cube." §9in Solid Mensura- tion with Proofs, 2nd ed. New York: Wiley, pp. 19 /C1/0, 1948. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 170 /C1/72 and 192, 1999. Wells, D. "Puzzle Page." Games and Puzzles. Sep. 1975. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 41 /C1/2 and 218 /C1/19, 1991. Wenninger, M. J. "The Hexahedron (Cube)." Model 3 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 16, 1989. Cube 2-Compound A POLYHEDRON COMPOUND obtained by allowing two CUBES to share opposite VERTICES , then rotating one a sixth of a turn (Holden 1971, p. 34). See also CUBE,C UBE 3-COMPOUND ,C UBE 4-COM- POUND ,CUBE 5-COMPOUND ,POLYHEDRON COMPOUND References Hart, G. "Compound of Two Cubes." http://www.georgehart.- com/virtual-polyhedra/vrml/cubes_D6_D3.wrl. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 213, 1999. Weisstein, E. W. "Polyhedra." M ATHEMATICA NOTEBOOK POLYHEDRA.M . Cube 3-Compound A compound with the symmetry of the CUBE which arises by joining three CUBES such that each shares two C2 axes (Holden 1971, p. 35). The solid is depicted atop the left pedestle in M. C. Escher’s woodcut Waterfall. See also CUBE,C UBE 2-COMPOUND ,C UBE 4-COM- POUND ,CUBE 5-COMPOUND ,ESCHER’S SOLID,POLY- HEDRON COMPOUND References Hart, G. "The Compound of Three Cubes." http://www.geor- gehart.com/virtual-polyhedra/vrml/cubes_S4_D4.wrl. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Cube 4-Compound A compound also called BAKOS’ COMPOUND having the symmetry of the CUBE which arises by joining four CUBES such that each C3 axis falls along the C3 axis of one of the other CUBES (Bakos 1959; Holden 1971, p. 35). Let the first cube c1consists of a cube in standard position rotated by p=3 radians around the (1; 1; 1)/-axis, then the other three cubes are obtained by rotating c1around the (0 ;0;1)/-axis ( Z-AXIS )b y p=2;/C28p=2;andpradians, respectively. See also CUBE,C UBE 2-COMPOUND ,C UBE 3-COM- POUND ,CUBE 5-COMPOUND ,POLYHEDRON COMPOUND References Bakos, T. "Octahedra Inscribed in a Cube." Math. Gaz. 43, 17/C1/0, 1959. Hart, G. "The Compound of Four Cubes." http://www.geor- gehart.com/virtual-polyhedra/vrml/cubes_S4_D3.wrl. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Cube 5-Compound APOLYHEDRON COMPOUND consisting of the arrange- ment of five CUBES in the VERTICES of a DODECAHE- DRON (or the centers of the faces of the ICOSAHEDRON ). The cube 5-compound is the dual of the OCTAHEDRON 5-COMPOUND . In the above figure, let a/C301 be the length of a CUBE EDGE . Then x/C301 2(3/C28ffiffiffi 5p ) u/C30tan/C2813/C28ffiffiffi 5p 2 ! :20/C1454? f/C30tan/C281ffiffiffi5p /C281 2 ! :31/C1443? c/C3090/C14/C28f:58/C1417? a/C3090/C14/C28u:69/C1406?: The compound is most easily constructed using pieces like the ones in the above line diagram. The cube 5- compound has the 30 facial planes of the RHOMBIC TRIACONTAHEDRON (Steinhaus 1983, pp. 199 and 209; Ball and Coxeter 1987). For cubes of unit edge lengths, the resulting com- pound has edge lengths s1/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(65/C2829ffiffiffi 5p )q (1) s2/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 27/C2812ffiffiffi 5pq (2) s3/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(25/C2811ffiffiffi 5p )q (3) s4/C30ffiffiffi5p /C282 (4) s 5/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32(7/C283ffiffiffi 5p )q (5) s6/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 5pq (6) s7/C301 2(3/C28ffiffiffi 5p ): (7) The CIRCUMRADIUS is R/C301 2ffiffiffi 3p ; (8) and the SURFACE AREA and VOLUME are S/C30165ffiffiffi5p /C28360 (9) V /C301 2(55ffiffiffi 5p /C28120) : (10) See also CUBE,C UBE 2-COMPOUND ,C UBE 3-COM- POUND ,CUBE 4-COMPOUND ,CUBE 5-COMPOUND– OC- TAHEDRON 5-COMPOUND ,C UBE 20-COMPOUND , DODECAHEDRON ,O CTAHEDRON 5-COMPOUND ,POLY- HEDRON COMPOUND ,RHOMBIC TRIACONTAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 135 and 137, 1987. Cundy, H. and Rollett, A. "Five Cubes in a Dodecahedron." §3.10.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 135 /C1/36, 1989. Hart, G. "Standard Compound of Five Cubes." http:// www.georgehart.com/virtual-polyhedra/vrml/compoun- d_of_5_cubes_(5_colors).wrl. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Cube 20-Compound See also CUBE,C UBE 2-COMPOUND ,C UBE 3-COM- POUND ,CUBE 4-COMPOUND ,CUBE 5-COMPOUND ,POLY- HEDRON COMPOUND References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 139 /C1/40, 1983. Cube 5-Compound /C1/Octahedron 5- Compound The compound of the CUBE 5-COMPOUND and its dual, the OCTAHEDRON 5-COMPOUND . See also CUBE 5-COMPOUND ,O CTAHEDRON 5-COM- POUND Cube Dissection A CUBE can be divided into n subcubes for only n /C301, 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and n ]48 (Sloane’s A014544). The seven pieces used to construct the 3 /C293 /C293 cube dissection known as the SOMA CUBE are one 3- POLYCUBE and six 4-POLYCUBES (1 /C215 3 /C276 /C215 4 /C3027); illustrated above. Another 3 /C293 /C293 cube dissection due to Steinhaus (1983) uses three 5-POLYCUBES and three 4-POLY- CUBES (3 /C215 5 /C273 /C215 4 /C3027); illustrated above. There are two solutions. It is possible to cut a 1 /C293 RECTANGLE into two identical pieces which will form a CUBE (without overlapping) when folded and joined. In fact, an INFINITE number of solutions to this problem were discovered by C. L. Baker (Hunter and Madachy 1975). Lonke (2000) has considered the number f(j ; k; n)of j-dimensional faces of a random k-dimensional cen- tral section of the n-cube Bn /C12/C30[/C281 ; 1]n ; and gives the special result f(0;k;n)/C302kn kl11sl11nffiffiffiffiffiffi 2k ps g/C12 0e/C28kt2=2gn/C28k(tBn/C28k /C12)dt; where gn/C28kis the ( n/C28k)/-dimensional Gaussian prob- ability measure. See also CONWAY PUZZLE ,D ISSECTION ,H ADWIGER PROBLEM ,POLYCUBE ,SLOTHOUBER- GRAATSMA PUZ- ZLE,SOMA CUBE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 112 /C1/13, 1987. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 203 /C1/05, 1989. Gardner, M. "Block Packing." Ch. 18 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 227 /C1/39, 1988. Gardner, M. Fractal Music, Hypercards, and More: Mathe- matical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 297 /C1/98, 1992. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 75 /C1/0, 1976. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 69 /C1/0, 1975. Lonke, Y. "On Random Sections of the Cube." Discr. Comput. Geom. 23, 157 /C1/69, 2000. Sloane, N. J. A. Sequences A014544 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 168 /C1/69, 1999. Cube Division by Planes What is the average number of regions into which n randomly chosen planes divide a cube? See also CYLINDER CUTTING ,S PACE DIVISION BY PLANES Cube Dovetailing Problem Given the figure on the left (without looking at the solution on the right), determine how to disengage the two slotted CUBE halves without cutting, break- ing, or distorting. References Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 145 and 249, 1958. Ogilvy, C. S. Excursions in Mathematics. New York: Dover, pp. 57, 59, and 143, 1994. Cube Duplication Also called the DELIAN PROBLEM or DUPLICATION OF THE CUBE . A classical problem of antiquity which, given the EDGE of a CUBE , requires a second CUBE to be constructed having double the VOLUME of the first using only a STRAIGHTEDGE and COMPASS . Under these restrictions, the problem cannot be solved because the DELIAN CONSTANT 21 =3 (the re- quired RATIO of sides of the original CUBE and that tobe constructed) is not a EUCLIDEAN NUMBER . The problem can be solved, however, using a NEUSIS CONSTRUCTION . See also ALHAZEN’S BILLIARD PROBLEM ,C OMPASS , CUBE,DELIAN CONSTANT ,GEOMETRIC PROBLEMS OF ANTIQUITY ,NEUSIS CONSTRUCTION ,STRAIGHTEDGE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 93 /C1/4, 1987. Bold, B. "The Delian Problem." Ch. 4 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 29 /C1/1, 1982. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190 /C1/91, 1996. Courant, R. and Robbins, H. "Doubling the Cube" and "A Classical Construction for Doubling the Cube." §3.3.1 and 3.5.1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 134 /C1/35 and 146, 1996. Do¨rrie, H. "The Delian Cube-Doubling Problem." §35 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 170 /C1/72, 1965. Klein, F. "The Delian Problem and the Trisection of the Angle." Ch. 2 in "Famous Problems of Elementary Geo-metry: The Duplication of the Cube, the Trisection of theAngle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 13 /C1 /5, 1980. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 33 /C1/4, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 49 /C1/0, 1991. Cube Line Picking The average DISTANCE between two points chosen at random inside a unit cube (the n/C303 case of HYPER- CUBE LINE PICKING )i s D(3)/C301 105[4/C2717ffiffiffi 2p /C286ffiffiffi3p /C2721 ln(1 /C27ffiffiffi2p ) /C2742 ln(2 /C27ffiffiffi3p )/C287p] (Robbins 1978, Le Lionnais 1983). Pick npoints on a CUBE , and space them as far apart as possible. The best value known for the minimum straight LINE distance between any two points is given in the following table. n /d(n)/ 5 1.1180339887498 6 1.0606601482100 71819 0.86602540378463 10 0.74999998333331 11 0.70961617562351 12 0.70710678118660 13 0.70710678118660 14 0.70710678118660 15 0.625 See also CUBE POINT PICKING ,C UBE TRIANGLE PICKING ,DISCREPANCY THEOREM ,H YPERCUBE LINE PICKING ,POINT PICKING ,POINT- POINT DISTANCE–1- D References Bolis, T. S. Solution to Problem E2629. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 277 /C1/78, 1978. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/geom/geom.html. Ghosh, B. "Random Distances within a Rectangle and between Two Rectangles." Bull. Calcutta Math. Soc. 43, 17 /C1/4, 1951. Holshouser, A. L.; King, L. R.; and Klein, B. G. Solution to Problem E3217, "Minimum Average Distance between Points in a Rectangle." Amer. Math. Monthly 96,64/C1/5, 1989. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983. Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978. Santalo ´,L.A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Cube Packing References Friedman, E. "Cubes in Cubes." http://www.stetson.edu/ ~efriedma/cubincub/. Cube Point Picking Pick N points p1 ; ..., pNrandomly in a unit n-cube. Let C be the CONVEX HULL ,so C /C13XN j/C301ljpj : lj ]0 for all j andXN j/C301lj /C301() : (1) Let V(n; N) be the expected n-D VOLUME (the CON- TENT )ofC, S(n; N) be the expected (n /C281)/-D SURFACE AREA of C, and P(n ; N) the expected number of VERTICES on the POLYGONAL boundary of C. Then lim N 0/C12N[1 /C28 V(2; N)] ln N/C308 3 lim N 0/C12ffiffiffiffiffi Np [4 /C28S(2; N)] /C30ffiffiffiffiffiffi 2pp 2 /C28g1 0(ffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27t2p /C281)t/C283=2 dt"# /C304:2472965... ; (2)lim N 0/C12P(2; N) /C288 3 ln N /C3083( g /C28ln 2) /C30/C280:309150708... (3) (Re´nyi and Sulanke 1963, 1964). See also BALL POINT PICKING ,CUBE LINE PICKING , SPHERE POINT PICKING References Re´nyi, A. and Sulanke, R. "U¨ ber die konvexe Hu¨lle von n zufa¨llig gewa¨hlten Punkten, I." Z. Wahrscheinlichkeits 2, 75 /C1/4, 1963. Re´nyi, A. and Sulanke, R. "U¨ ber die konvexe Hu¨lle von n zufa¨llig gewa¨hlten Punkten, II." Z. Wahrscheinlichkeits 3, 138 /C1/47, 1964. Cube Power A number raised to the third POWER . x3 is read as "x cubed." See also CUBIC NUMBER Cube Root Given a number z, the cube root of z, denotedffiffiffiz3por z1=3(zto the 1/3 POWER ), is a number asuch that a3/C30z:There are three (not necessarily distinct) cube roots for any number. For real arguments, the cube root is an INCREASING FUNCTION , although the usual derivative test cannot be used to establish this fact at the ORIGIN since the derivative approaches infinity there (as illustrated above). See also CUBE DUPLICATION ,CUBED ,D ELIAN CON- STANT ,G EOMETRIC PROBLEMS OF ANTIQUITY , K- MATRIX ,SQUARE ROOT Cube Tetrahedron Picking Given four points chosen at random inside a UNIT CUBE , the average VOLUME of the TETRAHEDRON determined by these points is given by ¯V /C30g1 0/C1/C1/C1g10 ½V(xi) ½dx1 /C1/C1/C1dx4dy1 /C1/C1/C1dy4dz1 /C1/C1/C1dz4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 12 g10 /C1/C1/C1g10 dx1 /C1/C1/C1dx4dy1 /C1/C1/C1dy4dz1 /C1/C1/C1dz4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 12 where the VERTICES are located at (xi ;yi ;zi) where i /C301, ..., 4, and the (signed) VOLUME is given by the DETERMINANT V /C301 3!x1y1z11 x2y2z21 x3y3z31 x4y4z41l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112: The integral is extremely difficult to compute. The analytic result is not known, but numerically is given by ¯V :0:0138 : (Note that the result quoted in the reply to Seidov 2000 actually refers to the average volume for TETRAHEDRON TETRAHEDRON PICKING .) See also CUBE,POINT PICKING ,SPHERE TETRAHE- DRON PICKING ,SQUARE TRIANGLE PICKING ,TETRA- HEDRON References Seidov, Z. F. "Letters: Random Triangle." Mathematica J. 7, 414, 2000. Cube Triangle Picking Pick 3 points at random in the unit n-HYPERCUBE . Denote the probability that the three points form an OBTUSE TRIANGLEQ(n): Langford (1969) proved F(2) /C3097 150 /C271 40 p /C300 :725206483...See also BALL TRIANGLE PICKING ,C UBE POINT PICKING References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/geom/geom.html. Langford, E. "The Probability that a Random Triangle is Obtuse." Biometrika 56, 689 /C1/90, 1969. Santalo ´,L.A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Cubed A number to the POWER 3 is said to be cubed, so that x3 is called "x cubed." See also CUBE ROOT,SQUARED Cubefree A number is said to be cubefree if its PRIME FACTOR- IZATION contains no tripled factors. All PRIMES are therefore trivially cubefree. The cubefree numbers are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ... (Sloane’s A004709). The cubeful numbers (i.e., those that contain at least one cube) are 8, 16, 24, 27, 32, 40, 48, 54, ... (Sloane’s A046099). The number of cubefree numbers less than 10, 100, 1000, ... are 9, 85, 833, 8319, 83190, 831910, ..., and their asymptotic density is 1=z(3) :0:831907 ; where z(n) is the RIEMANN ZETA FUNCTION . See also BIQUADRATEFREE ,CUBEFREE PART,PRIME NUMBER ,RIEMANN ZETA FUNCTION ,SQUAREFREE References Sloane, N. J. A. Sequences A004709 and A046099 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Cubefree Part That part of a POSITIVE INTEGER left after all cubic factors are divided out. For example, the cubefree part of 24 /C3023/C2153 is 3. For n/C301, 2, ..., the first few are 1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, ... (Sloane’s A050985). The squarefree part function can be implemented in Mathematica as SquarefreePart[n_Integer?Positive] : /C30 Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]) See also CUBEFREE ,CUBIC PART,SQUAREFREE PART References Sloane, N. J. A. Sequences A050985 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Cubefree Word A cubefree word contains no cubed words as sub- words. The number of binary cubefree words of length n /C301, 2, ... are 2, 4, 6, 10, 16, 24, 36, 56, 80, 118, ... (Sloane’s A028445). Binary cubefree words satisfy 2 /C215 1:080n 5c(n) 52 /C215 1:522n : (1) The number of ternary cubefree words of length n /C301, 2, ... are 3, 9, 24, 66, 180, 486, 1314, ... (Sloane’s A051042). The number of quaternary cubefree words of length n /C301, 2, ... are 4, 16, 60, 228, 864, 3264, 12336, ... (Sloane’s A051043). See also OVERLAPFREE WORD,SQUAREFREE WORD, WORD References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/words/words.html. Sloane, N. J. A. Sequences A028445, A051042, and A051043 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Cube-Octahedron Compound A POLYHEDRON COMPOUND composed of a CUBE and its DUAL POLYHEDRON , the OCTAHEDRON . For a CUBE of edge length 1, the 14 vertices are located at (/91=2; 91=2 ;91=2); ( 9 1, 0, 0), (0, 9 1, 0), (0, 0, 9 1). Since the edges of the cube and octahedron bisect each other, the resulting solid has side lengths 1/2 andffiffiffiffiffiffiffiffiffiffi 2=2p ; and SURFACE AREA and VOLUME given byS /C303(1 /C27ffiffiffiffiffi 3)p V /C303 2 : The CONVEX HULL of the cube-octahedron compound is a RHOMBIC DODECAHEDRON . The solid common to both the CUBE and OCTAHEDRON (left figure) in a cube-octahedron compound is a CUBOCTAHEDRON (middle figure). The edges intersect- ing in the points plotted above are the diagonals of RHOMBUSES , and the 12 RHOMBUSES form a RHOMBIC DODECAHEDRON (right figure; Ball and Coxeter 1987). See also CUBE,C UBOCTAHEDRON ,O CTAHEDRON , POLYHEDRON COMPOUND References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 137, 1987. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 158, 1969. Cundy, H. and Rollett, A. "Cube Plus Octahedron." §3.10.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 130, 1989. Weisstein, E. W. "Polyhedra." M ATHEMATICA NOTEBOOK POLYHEDRA.M . Wenninger, M. J. "Compound of a Cube and Octahedron." §43 in Polyhedron Models. New York: Cambridge Uni- versity Press, p. 68, 1989. Cubic Close Packing SPHERE PACKING Cubic Curve A cubic curve is an ALGEBRAIC CURVE of degree 3. An algebraic curve over a FIELD Kis an equation f(X;Y)/C300;where f(X;Y)i sa POLYNOMIAL inXand Ywith COEFFICIENTS inK, and the degree of fis the MAXIMUM degree of each of its terms ( MONOMIALS ). Newton showed that all cubics can be generated by the projection of the five divergent cubic parabolas. Newton’s classification of cubic curves appeared in the chapter "Curves" in Lexicon Technicum by John Harris published in London in 1710. Newton also classified all cubics into 72 types, missing six of them. In addition, he showed that any cubic can be obtainedby a suitable projection of the ELLIPTIC CURVE y2/C30ax3/C27bx2/C27cx/C27d; (1) where the projection is a BIRATIONAL TRANSFORMA- TION , and the general cubic can also be written as y2 /C30x3 /C27ax /C27b: (2) Newton’s first class is equations OF THE FORM xy2 /C27ey /C30ax3 /C27bx2 /C27cx /C27d: (3) This is the hardest case and includes the SERPENTINE CURVE as one of the subcases. The third class was ay2 /C30x(x2 /C282bx /C27c) ; (4) which is called NEWTON’S DIVERGING PARABOLAS . Newton’s 66th curve was the TRIDENT OF NEWTON . Newton’s classification of cubics was criticized by Euler because it lacked generality. Plu¨cker later gave a more detailed classification with 219 types. The NINE ASSOCIATED POINTS THEOREM states that Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth (Evelyn et al. 1974, p. 15). Pick a point P, and draw the tangent to the curve at P. Call the point where this tangent intersects the curve Q. Draw another tangent and call the point of intersection with the curve R. Every curve of third degree has the property that, with the areas in the above labeled figure, B/C3016A (5) (Honsberger 1991). See also CAYLEY- BACHARACH THEOREM ,CUBIC EQUA- TION ,E LLIPTIC CURVE ,N INE ASSOCIATED POINTS THEOREM ,TRIANGLE CUBIC CURVE References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. The Seven Circles Theorem and Other New Theorems. London: Stacey International, p. 15, 1974. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 114 /C1/18, 1991. Newton, I. Mathematical Works, Vol. 2. New York: Johnson Reprint Corp., pp. 135 /C1/61, 1967. Wall, C. T. C. "Affine Cubic Functions III." Math. Proc. Cambridge Phil. Soc. 87,1/C1/4, 1980. Westfall, R. S. Never at Rest: A Biography of Isaac Newton. New York: Cambridge University Press, 1988.Yates, R. C. "Cubic Parabola." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 56 /C1/ 9, 1952. Cubic Equation A cubic equation is a POLYNOMIAL equation of degree three. Given a general cubic equation z3/C27a2z2/C27a1z/C27a0/C300 (1) (the COEFFICIENT a3ofz3may be taken as 1 without loss of generality by dividing the entire equation through by a3);first attempt to eliminate the a2term by making a substitution OF THE FORM z/C13x/C28l: (2) Then (x/C28l)3/C27a2(x/C28l)2/C27a1(x/C28l)/C27a0/C300 (3) (x3/C283lx2/C273l2x/C28l3)/C27a2(x2/C282lx/C27l2) /C27a1(x/C28l)/C27a0/C300 (4) x3/C27(a2/C283l)x2/C27(a1/C282a2l/C273l2)x /C27(a0/C28a1l/C27a2l2/C28l3)/C300: (5) The x2is eliminated by letting l/C30a2=3;so z/C13x/C281 3a2: (6) Then z3/C30(x/C281 3a2)3/C30x3/C28a2x2/C2713a2 2x/C281 27a32: (7) a2z2/C30a2(x/C281 3a2)2/C30a2x2/C2823a2 2x/C271 9a3 2 (8) a1z/C30a1(x/C281 3a2)/C30a1x/C2813a1a2; (9) so equation (1) becomes x3/C27(/C28a2/C27a2)x2/C27(13a2 2/C282 3a2 2/C27a1)x /C28(1 27a32/C281 9a3 2/C271 3a1a2/C28a0)/C300 (10) x3/C27(a1/C2813a2 2)x/C28(1 3a1a2/C282 27a3 2/C28a0)/C300 (11) x3/C273/C2153a1/C28a2 2 9x/C282/C2159a1a2/C2827a0/C282a32 54/C300:(12) Defining p/C133a1/C28a22 3(13) q/C139a1a2/C2827a0/C282a32 27(14) then allows (12) to be written in the standard form x3/C27px/C30q: (15) The simplest way to proceed is to make V IETA’S SUBSTITUTION x/C30w/C28p 3w; (16) which reduces the cubic to the equation w3/C28p3 27w3/C28q/C300; (17) which is easily turned into a QUADRATIC EQUATION in w3by multiplying through by w3to obtain (w3)2/C28q(w3)/C281 27p3/C300 (18) (Birkhoff and Mac Lane 1996, p. 106). The result from the QUADRATIC EQUATION is w3/C301 2q9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2/C274 27p3ql11)l117 /C3012q9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14q2/C271 27p3q /C30R9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2/C27Q3p ; (19) where Qand Rare sometimes more useful to deal with than are pand q. There are therefore six solutions for w(two corresponding to each sign for each ROOT ofw3):Plugging wback in to (17) gives three pairs of solutions, but each pair is equal, so there are three solutions to the cubic equation. Equation (12) may also be explicitly factored by attempting to pull out a term OF THE FORM (x/C28B) from the cubic equation, leaving behind a quadratic equation which can then be factored using the QUADRATIC FORMULA . This process is equivalent to making V IETA’S SUBSTITUTION , but does a slightly better job of motivating Vieta’s "magic" substitution, and also at producing the explicit formulas for the solutions. First, define the intermediate variables Q/C133a1/C28a2 2 9(20) R/C139a2a1/C2827a0/C282a32 54(21) (which are identical to pand qup to a constant factor). The general cubic equation (12) then becomes x3/C273Qx/C282R/C300: (22) LetBandCbe, for the moment, arbitrary constants. An identity satisfied by PERFECT CUBIC POLYNOMIAL equations is that x3/C28B3/C30(x/C28B)(x2/C27Bx/C27B2): (23) The general cubic would therefore be directly factor- able if it did not have an xterm (i.e., if Q/C300). However, since in general Q"0;add a multiple of (x/C28B)/*/sayC(x/C28B)/*/to both sides of (23) to give the slightly messy identity(x3/C28B3)/C27C(x/C28B)/C30(x/C28B)(x2/C27Bx/C27B2/C27C) /C300; (24) which, after regrouping terms, is x3/C27Cx/C28(B3/C27BC)/C30(x/C28B)[x2/C27Bx/C27(B2/C27C)] /C300: (25) We would now like to match the COEFFICIENTS Cand /C28(B3/C27BC) with those of equation (22), so we must have C/C303Q (26) B3/C27BC/C302R: (27) Plugging the former into the latter then gives B3/C273QB/C302R: (28) Therefore, if we can find a value of Bsatisfying the above identity, we have factored a linear term from the cubic, thus reducing it to a QUADRATIC EQUATION . The trial solution accomplishing this miracle turnsout to be the symmetrical expression B/C30[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 3/C27R2p ]1=3/C27[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]1=3: (29) Taking the second and third POWERS ofBgives B2/C30[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q3/C27R2p ]2=3/C272[R2/C28(Q3/C27R2)]1=3 /C27[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q3/C27R2p ]2=3 /C30[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]2=3/C27[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]2=3/C282Q(30) B3/C30/C282QB /C27[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q3/C27R2p ]1=3/C27[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]1=3no /C29[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q3/C27R2p ]2=3/C27[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]2=3no /C30[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]/C27[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ] /C27[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]1=3[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]2=3 /C27[R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q3/C27R2p ]2=3[R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p ]1=3/C282QB /C30/C282QB/C272R/C27[R2/C28(Q3/C27R2)]1=3 /C29R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q3/C27R2pl11)l1171=3 /C27R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q3/C28R2pl11)l1171=3l12ml121 /C30/C282QB/C272R/C28QB/C30/C283QB/C272R: (31) Plugging B3andBinto the left side of (28) gives (/C283QB/C272R)/C273QB/C302R; (32) so we have indeed found the factor ( x/C28B) of (22), and we need now only factor the quadratic part. Plugging C/C303Qinto the quadratic part of (25) and solving the resulting x2/C27Bx/C27(B2/C273Q)/C300 (33) then gives the solutions x/C301 2/C28B9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2/C284(B2/C273Q)phi /C30/C2812B912ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C283B2/C2812Qp /C30/C281 2B912ffiffiffiffiffi 3ipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2/C274Qp : (34) These can be simplified by defining A/C13R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2phi1=3 /C28R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2phi1=3 (35) A2/C30R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2phi2=3 /C282R2/C28(Q3/C27R2)l12l191=3 /C27R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2phi2=3 /C30R/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2phi2=3 /C27R/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 3/C27R2p )hi2=3 /C272Q /C30B2/C274Q; (36) so that the solutions to the quadratic part can be written x/C30/C281 2B912ffiffiffi 3p iA: (37) Defining D/C13Q3/C27R2(38) S/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R/C27ffiffiffiffi Dpq (39) T/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R/C28ffiffiffiffi Dp ;q (40) where Dis the DISCRIMINANT (which is defined slightly differently, including the opposite SIGN,b y Birkhoff and Mac Lane 1996) then gives very simple expressions for AandB, namely B/C30S/C27T (41) A/C30S/C28T: (42) Therefore, at last, the ROOTS of the original equation inzare then given by z1/C30/C281 3a2/C27(S/C27T) (43) z2/C30/C2813a2/C2812(S/C27T)/C2712iffiffiffiffiffi 3p (S/C28T) (44) z3/C30/C2813a2/C2812(S/C27T)/C2812iffiffiffiffiffi 3p (S/C28T); (45) with a2the COEFFICIENT ofz2in the original equation, andSandTas defined above. These three equations giving the three ROOTS of the cubic equation are sometimes known as C ARDANO’S FORMULA . Note that if the equation is in the standard form of Vieta x3/C27px/C30q; (46) in the variable x, then a2/C300;a1/C30p;anda0/C30/C28q;and the intermediate variables have the simple form (cf. Beyer 1987)Q/C301 3p (47) R/C3012q (48) D/C13Q3/C27R2/C30p 3 !2 /C27q 2 !2 : (49) The solutions satisfy N EWTON’S RELATIONS z1/C27z2/C27z3/C30/C28a2 (50) z1z2/C27z2z3/C27z1z3/C30a1 (51) z1z2z3/C30/C28a0: (52) In standard form (46), a2/C300;a1/C30p;and a0/C30/C28q;so eliminating qgives p/C30/C28(z2 i/C27zizj/C27z2j) (53) fori"j;and eliminating pgives q/C30/C28zizj(zi/C27zj) (54) fori"j:In addition, the properties of the SYMMETRIC POLYNOMIALS appearing in N EWTON’S RELATIONS give z21/C27z22/C27z23/C30/C282p (55) z31/C27z32/C27z33/C303q (56) z41/C27z42/C27z43/C302p2(57) z51/C27z52/C27z53/C30/C285pq: (58) The equation for z1in C ARDANO’S FORMULA does not have an iappearing in it explicitly while z2andz3do, but this does not say anything about the number of REAL and COMPLEX ROOTS (since Sand Tare themselves, in general, COMPLEX ). However, deter- mining which ROOTS are REAL and which are COM- PLEX can be accomplished by noting that if the DISCRIMINANT D/C210, one ROOT isREAL and two are COMPLEX CONJUGATES ;i fD/C300, all ROOTS are REAL and at least two are equal; and if DB0, all ROOTS are REAL and unequal. If DB0, define u/C13cos/C281 Rffiffiffiffiffiffiffiffiffiffiffi /C28Q3p ! : (59) Then the REAL solutions are OF THE FORM z1/C302ffiffiffiffiffiffiffiffi /C28Qp cosu 3 ! /C281 3a2 (60) z2/C302ffiffiffiffiffiffiffiffi /C28Qp cosu/C272p 3 ! /C2813a2 (61) z3/C302ffiffiffiffiffiffiffiffi /C28Qp cosu/C274p 3 ! /C2813a2: (62) This procedure can be generalized to find the REAL ROOTS for any equation in the standard form (46) by using the identity sin3 u /C283 4sin u /C2714sin(3u) /C300 (63) (Dickson 1914) and setting x /C13ffiffiffiffiffiffiffiffi 4 ½p ½ 3s y (64) (Birkhoff and Mac Lane 1996, pp. 90 /C1/1), then 4 pjj 3 !3 =2 y3 /C27pffiffiffiffiffiffiffiffiffi 4 pjj 3s y /C30q (65) y3 /C2734p pjjy /C303 4 pjj !3 =2 q (66) 4y3 /C273 sgn(p)y /C301 2 q3 pjj !3=2 /C13C : (67) If p /C210, then use sinh(3 u) /C304 sinh3 u /C273 sinh u (68) to obtain y /C30sinh(1 3sinh/C281 C): (69) If p B0 and Cjj]1; use cosh(3 u) /C304 cosh3 u /C283 cosh u; (70) and if p B0 and Cjj51; use cos(3 u) /C304 cos3 u /C283 cos u; (71) to obtain y /C30cosh(13cosh/C281 C) for C ]1 /C28cosh(13cosh/C281 Cjj) for C 5/C281 cos(1 3cos/C281 C) for CjjB1:8 >>< >>:(72) The solutions to the original equation are then xi /C302ffiffiffiffiffiffi pjj 3s yi /C281 3 a2 : (73) An alternate approach to solving the cubic equation is to use LAGRANGE RESOLVENTS (Faucette 1996). Let v /C13e2pi=3 ; define (1; x1) /C30x1 /C27x2 /C27x3 (74) ( v; x1) /C30x1 /C27 vx2 /C27 v2x3 (75) (v2 ; x1) /C30x1 /C27 v2x2 /C27 vx3 ; (76) where xi are the ROOTS of x3 /C27px /C28q /C300; (77) and consider the equation[x /C28(u1 /C27u2)][x /C28( vu1 /C27 v2u2)][x /C28( v2u1 /C27 vu2)] /C300; (78) where u1and u2are COMPLEX NUMBERS . The ROOTS are then xj /C30 vju1 /C27 v2ju2 (79) for j /C300, 1, 2. Multiplying through gives x3/C283u1u2x/C28(u3 1/C27u32)/C300; (80) which can be written in the form (77), where u31/C27u32/C30q (81) u31u32/C30/C28p 3 !3 : (82) Some curious identities involving the roots of a cubic equation due to Ramanujan are given by Berndt (1994). See also CASUS IRREDUCIBILUS ,DISCRIMINANT (POLY- NOMIAL ), PERFECT CUBIC POLYNOMIAL ,Q UADRATIC EQUATION ,QUARTIC EQUATION ,QUINTIC EQUATION , SEXTIC EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972. Berger, M. §16.4.1 /C1/6.4.11.1 in Geometry I. New York: Springer-Verlag, 1994. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 22 /C1/3, 1994. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 9 /C1/1, 1987. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 90 /C1/1, 106 /C1/07, and 414/C1/17, 1996. Borwein, P. and Erde ´lyi, T. "Cubic Equations." §1.1.E.1b in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 4, 1995. Dickson, L. E. "A New Solution of the Cubic Equation." Amer. Math. Monthly 5,3 8/C1/9, 1898. Dickson, L. E. Elementary Theory of Equations. New York: Wiley, pp. 36 /C1/7, 1914. Dunham, W. "Cardano and the Solution of the Cubic." Ch. 6 inJourney through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 133 /C1/54, 1990. Ehrlich, G. §4.16 in Fundamental Concepts of Abstract Algebra. Boston, MA: PWS-Kent, 1991. Faucette, W. M. "A Geometric Interpretation of the Solution of the General Quartic Polynomial." Amer. Math. Monthly 103,5 1/C1/7, 1996. Jones, J. "Omar Khayya ´m and a Geometric Solution of the Cubic." http://jwilson.coe.uga.edu/emt669/Student.- Folders/Jones.June/omar/omarpaper.html. Kennedy, E. C. "A Note on the Roots of a Cubic." Amer. Math. Monthly 40, 411/C1/12, 1933. King, R. B. Beyond the Quartic Equation. Boston, MA: Birkha ¨user, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Quadratic and Cubic Equations." §5.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 178 /C1/80, 1992. Spanier, J. and Oldham, K. B. "The Cubic Function x3 /C27 ax2 /C27bx /C27c and Higher Polynomials." Ch. 17 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 131 /C1/47, 1987. van der Waerden, B. L. §64 in Algebra. New York: Frederick Ungar, 1970. Whittaker, E. T. and Robinson, G. "The Solution of the Cubic." §62 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 124 /C1/26, 1967. Cubic Graph Cubic graphs, also called trivalent graphs, are graphs all of whose nodes have degree 3 (i.e., 3-REGULAR GRAPHS ). Cubic graphs on n nodes exists only for even n (Harary 1994, p. 15). The numbers of cubic graphs on 2, 4, 6, ... nodes are 0, 1, 2, 6, 21, 94, 540, 4207, ... (Sloane’s A005638). The unique 4-node cubic graph is the COMPLETE GRAPH k4 : The two 6-node cubic graphs are the UTILITY GRAPH K3; 3 and the CIRCULANT GRAPH Ci1 ; 3(6) : The connected 3-regular graphs have been determined by Brinkmann (1996) up to 24 nodes. /(3; g)/-CAGE GRAPHS and UNITRANSITIVE GRAPHS are cubic. In addition, the following tables gives polyhe- dra whose SKELETONS are cubic. POLYHEDRON nodes TETRAHEDRON 4 CUBE 8 TRUNCATED TETRAHEDRON 12 DODECAHEDRON 20 TRUNCATED CUBE 24 TRUNCATED OCTAHEDRON 24 GREAT RHOMBICUBOCTAHEDRON (ARCHIMEDEAN )48 TRUNCATED ICOSAHEDRON 60 GREAT RHOMBICOSIDODECAHEDRON (ARCHIMEDEAN )120 See also BARNETTE’S CONJECTURE ,BICUBIC GRAPH , CAGE GRAPH ,C UBICAL GRAPH ,F RUCHT GRAPH ,QUARTIC GRAPH ,QUINTIC GRAPH ,REGULAR GRAPH , TAIT’S HAMILTONIAN GRAPH CONJECTURE ,T UTTE CONJECTURE ,UNITRANSITIVE GRAPH References Brinkmann, G. "Fast Generation of Cubic Graphs." J. Graph Th.23, 139/C1/49, 1996. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Robinson, R. W.; Wormald, N. C. "Number of Cubic Graphs." J. Graph. Th. 7, 463/C1/67, 1983. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 177, 1990. Sloane, N. J. A. Sequences A005638/M1656 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Tutte, W. T. "A Family of Cubical Graphs." Proc. Cambridge Philos. Soc. , 459/C1/74, 1947. Tutte, W. T. "A Theory of 3-Connected Graphs." Indag. Math. 23, 441/C1/55, 1961. Cubic Number AFIGURATE NUMBER OF THE FORM n3;fornaPOSITIVE INTEGER . The first few are 1, 8, 27, 64, ... (Sloane’s A000578). The GENERATING FUNCTION giving the cubic numbers is x(x2/C274x/C271) (x/C281)4/C30x/C278x2/C2727x3/C27... ( 1 ) The HEX PYRAMIDAL NUMBERS are equivalent to the cubic numbers (Conway and Guy 1996). As a part of the study of W ARING’S PROBLEM ,i ti s known that every positive integer is a sum of no more than 9 positive cubes ( /g(3)/C309;proved by Dickson, Pillai, and Niven in the early twentieth century), thatevery "sufficiently large" integer is a sum of no morethan 7 positive cubes ( /G(3)57):However, it is not known if 7 can be reduced (Wells 1986, p. 70). The number of positive cubes needed to represent the numbers 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, ...(Sloane’s A002376), and the number of distinct ways to represent the numbers 1, 2, 3, ... in terms of positive cubes are 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, ... (Sloane’s A003108). In 1939, Dickson proved that the only INTEGERS requiring nine positive cubes are 23 and 239. Wiefer- ich proved that only 15 INTEGERS require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 213, 238, 303, 364,420, 428, and 454 (Sloane’s A018889). The quantityG(3) in W ARING’S PROBLEM therefore satisfies G(3)5 7;and the largest number known requiring seven cubes is 8042. Deshouillers et al. (1999) conjectured that 7,373,170,279,850 is the largest integer that cannot be expressed as the sum of four nonnegative cubes. The following table gives the first few numbers which require at least N /C301, 2, 3, ..., 9 (i.e., Nor more) positive cubes to represent them as a sum. N Sloane Numbers 1 Sloane’s A0005781, 8, 27, 64, 125, 216, 343,512, ... 2 Sloane’s A0033252, 9, 16, 28, 35, 54, 65, 72,91, ... 3 Sloane’s A0030723, 10, 17, 24, 29, 36, 43, 55, 62, ... 4 Sloane’s A0033274, 11, 18, 25, 30, 32, 37, 44,51, ... 5 Sloane’s A0033285, 12, 19, 26, 31, 33, 38, 40, 45, ... 6 Sloane’s A0033296, 13, 20, 34, 39, 41, 46, 48, 53, ... 7 Sloane’s A0188907, 14, 21, 42, 47, 49, 61, 77, ... 8 Sloane’s A01888915, 22, 50, 114, 167, 175, 186, ... 9 Sloane’s A01888823, 239 There is a finite set of numbers which cannot be expressed as the sum of distinct positive cubes: 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,23, 24, 25, 26, ...(Sloane’s A001476). It is known that every integer is a sum of at most 5 signed cubes ( /eg(3)55i nW ARING’S PROBLEM ). It is believed that 5 can be reduced to 4, so that N/C30A3/C27B3/C27C3/C27D3(2) for any number N, although this has not been proved for numbers OF THE FORM 9n94:However, every multiple of 6 can be REPRESENTED AS a sum of foursigned cubes as a result of the algebraic identity 6x/C30(x/C271)3/C27(x/C281)3/C28x3/C28x3: (3) In fact, all numbers NB1000 and not OF THE FORM 9n94 are known to be expressible as the SUM N/C30A3/C27B3/C27C3(4) ofthree (positive or negative) cubes with the excep- tion of N/C3030, 33, 42, 52, 74, 110, 114, 156, 165, 195, 290, 318, 366, 390, 420, 444, 452, 478, 501, 530, 534,564, 579, 588, 600, 606, 609, 618, 627, 633, 732, 735,758, 767, 786, 789, 795, 830, 834, 861, 894, 903, 906, 912, 921, 933, 948, 964, 969, and 975 (Sloane’s A046041; Miller and Woollett 1955; Gardiner et al. 1964; Guy 1994, p. 151). While it is known that (4) has no solutions for Nof the form 9 n94 (Hardy and Wright 1979, p. 327), there is known reason for excluding the above integers (Gardiner et al. 1964). Mahler proved that 1 has infinitely-many representa-tions as 3 signed cubes. The following table gives the numbers which can be represented in exactly W different ways as a sum of Npositive cubes. (Combining all Ws for a given N then gives the sequences in the previous table.) Forexample, 157/C304 3/C2743/C2733/C2713/C2713/C3053/C2723/C2723/C2723/C2723(5) can be represented in W/C302 ways by N/C305 cubes. The smallest number representable in W/C302 ways as a sum of N/C302 cubes, 1729/C3013/C27123/C3093/C27103; (6) is called the H ARDY- RAMANUJAN NUMBER and has special significance in the history of mathematics as aresult of a story told by Hardy about Ramanujan. Note that Sloane’s A001235 is defined as the se- quence of numbers which are the sum of cubes in two or more ways, and so appears identical in the first few terms to the ( N/C302;W/C302) series given below. NW Sloane numbers 1 0 A007412 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ... 1 1 A000578 1, 8, 27, 64, 125, 216, 343, 512, ... 2 0 A057903 1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ... 2 1 2, 9, 16, 28, 35, 54, 65, 72, 91, ... 2 2 A018850 1729, 4104, 13832, 20683, 32832, ... 2 3 A003825 87539319, 119824488, 143604279, ... 2 4 A003826 6963472309248, 12625136269928, ... 2 5 48988659276962496, ... 2 6 8230545258248091551205888, ... 3 0 A057904 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ... 3 1 A025395 3, 10, 17, 24, 29, 36, 43, 55, 62, ... 3 2 251, ... 4 0 A057905 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ... 4 1 A025403 4, 11, 18, 25, 30, 32, 37, 44, 51, ... 4 2 A025404 219, 252, 259, 278, 315, 376, 467, ... 5 0 A057906 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15, ... 5 1 A048926 5, 12, 19, 26, 31, 33, 38, 40, 45, ... 5 2 A048927 157, 220, 227, 246, 253, 260, 267, ... 6 0 A057907 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, ... 6 1 A048929 6, 13, 20, 27, 32, 34, 39, 41, 46, ... 6 2 A048930 158, 165, 184, 221, 228, 235, 247, ... 6 3 A048931 221, 254, 369, 411, 443, 469, 495, ... The following table gives the possible residues (mod n) for cubic numbers for n/C301 to 20, as well as the number of distinct residues s(n):/ n /s(n)//x3(mod n)/ 2 2 0, 1 3 3 0, 1, 24 3 0, 1, 3 5 5 0, 1, 2, 3, 4 6 6 0, 1, 2, 3, 4, 57 3 0, 1, 68 5 0, 1, 3, 5, 7 9 3 0, 1, 810 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 11 11 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 101 2 9 0 ,1 ,3 ,4 ,5 ,7 ,8 ,9 ,1 11 3 5 0 ,1 ,5 ,8 ,1 2 1 4 6 0 ,1 ,6 ,7 ,8 ,1 3 15 15 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1416 10 0, 1, 3, 5, 7, 8, 9, 11, 13, 1517 17 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 18 6 0, 1, 8, 9, 10, 17 19 7 0, 1, 7, 8, 11, 12, 18 20 15 0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19 Dudeney found two RATIONAL NUMBERS other than 1 and 2 whose cubes sum to 9, 415280564497 348671682660and676702467503348671682660(7) (Gardner 1958). The problem of finding two RATIONAL NUMBERS whose cubes sum to six was "proved" impossible by Legendre. However, Dudeney found the simple solutions 17/21 and 37/21. The only three consecutive INTEGERS whose cubes sum to a cube are given by the D IOPHANTINE EQUATION 33/C2743/C2753/C3063: (8) CATALAN’S CONJECTURE states that 8 and 9 (23and 32) are the only consecutive POWERS (excluding 0 and 1), i.e., the only solution to C ATALAN’S DIOPHANTINE PROBLEM . This CONJECTURE has not yet been proved or refuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the CONJECTURE not hold. It is also known that 8 and 9 are the only consecutive cubic and SQUARE NUMBERS (in either order). There are six POSITIVE INTEGERS equal to the sum of the DIGITS of their cubes: 1, 8, 17, 18, 26, and 27 (Sloane’s A046459; Moret Blanc 1879). There are four POSITIVE INTEGERS equal to the sums of the cubes of their digits: 153/C3013/C2753/C2733(9) 370/C3033/C2773/C2703(10) 371/C3033/C2773/C2713(11) 407/C3043/C2703/C2773(12) (Ball and Coxeter 1987). There are two SQUARE NUMBERS OF THE FORM n3 /C284:4/C3023 /C284 and 121 /C30 53 /C284 (Le Lionnais 1983). A cube cannot be the concatenation of two cubes, since if c3 is the con- catenation of a3 and b3 ; then c3 /C3010ka3 /C27b3 ; where k is the number of digits in b3 : After shifting any powers of 1000 in 10k into a3 ; the original problem is equivalent to finding a solution to one of the DIOPHANTINE EQUATIONS c3 /C28b3 /C30a3 (13) c3 /C28b3 /C3010a3 (14) c3 /C28b3 /C30100a3 : (15) None of these have solutions in integers, as proved independently by Sylvester, Lucas, and Pepin (Dick- son 1966, pp. 572 /C1/78). See also BIQUADRATIC NUMBER ,C ENTERED CUBE NUMBER ,C LARK’S TRIANGLE ,D IOPHANTINE EQUA- TION–3RD POWERS ,HARDY- RAMANUJAN NUMBER ,PAR- TITION ,SQUARE NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 14, 1987. Bertault, F.; Ramare ´, O.; and Zimmermann, P. "On Sums of Seven Cubes." Math. Comput. 68, 1303 /C1/310, 1999. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 42 /C1/4, 1996. Davenport, H. "On Waring’s Problem for Cubes." Acta Math. 71, 123 /C1/43, 1939. Deshouillers, J.-M.; Hennecart, F.; and Landreau, B. "7 373 170 279 850." Math. Comput. 69, 421 /C1/39, 1999. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966. Gardiner, V. L.; Lazarus, R. B.; and Stein, P. R. "Solutions of the Diophantine Equation x3 /C27y3 /C30z3 /C28d:/" Math. Com- put. 18, 408 /C1/13, 1964. Gardner, M. "Mathematical Games: About Henry Ernest Dudeney, A Brilliant Creator of Puzzles." Sci. Amer. 198, 108 /C1/12, Jun. 1958. Guy, R. K. "Sum of Four Cubes." §D5 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 151 /C1/52, 1994. Hardy, G. H. and Wright, E. M. "Representation by Cubes and Higher Powers." Ch. 21 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 317 /C1/39, 1979. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 53, 1983. Miller, J. C. P. and Woollett, M. F. C. "Solutions of the Diophantine Equation x3 /C27y3 /C27z3 /C30k:/" J. London Math. Soc. 30, 101 /C1/10, 1955. Sloane, N. J. A. Sequences A000578/M4499, A001235, A001476, A002376/M0466, A003108/M0209, A003072, A003325, A003327, A003328, A003825, A003826, A007412/M0493, A011541, A018850, A018888, A018889, A018890, A025395, A046040, A046459, A048926, A048927, A048928, A048929, A048930, A048931, A048932, A057903, A057904, A057905, A057906, and A057907 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 70, 1986.Cubic Part The largest cube dividing a POSITIVE INTEGER n. For n /C301, 2, ..., the first few are 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, ... (Sloane’s A008834). See also CUBEFREE PART,CUBIC NUMBER ,SQUARE PART References Sloane, N. J. A. Sequences A008834 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Cubic Reciprocity Theorem A RECIPROCITY THEOREM for the case n /C303 solved by Gauss using "INTEGERS " OF THE FORM a /C27br ; when r is a root of x2 /C27x /C271 /C300 (i.e., r equals /C28(/C281)1 =3 or (/C281)2=3) and a, b are INTEGERS . See also CUBIC RESIDUE ,RECIPROCITY THEOREM References Ireland, K. and Rosen, M. "Cubic and Biquadratic Recipro- city." Ch. 9 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108 /C1/37, 1990. Cubic Residue If there is an INTEGER x such that x3 /C13q (mod p) ; (1) then q is said to be a cubic residue (mod p). If not, q is said to be a cubic nonresidue (mod p). See also CUBIC RECIPROCITY THEOREM ,QUADRATIC RESIDUE References Nagell, T. Introduction to Number Theory. New York: Wiley, p. 115, 1951. Cubic Spline A cubic spline is a SPLINE constructed of piecewise third-order POLYNOMIALS which pass through a set of control points. The second DERIVATIVE of each POLY- NOMIAL is commonly set to zero at the endpoints, since this provides a boundary condition that com- pletes the system of n /C282 equations, leading to a simple 3-diagonal system which can be solved easily to give the coefficients of the polynomials. However, this choice is not the only one possible, and other boundary conditions can be used instead. See also SPLINE ,THIN PLATE SPLINE References Burden, R. L.; Faires, J. D.; and Reynolds, A. C. Numerical Analysis, 6th ed. Boston, MA: Brooks/Cole, pp. 120 /C1/21, 1997. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Cubic Spline Interpolation." §3.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 107 /C1/10, 1992. Cubic Surface An ALGEBRAIC SURFACE of ORDER 3. Schla ¨fli and Cayley classified the singular cubic surfaces. On the general cubic, there exists a curious geometrical structure called DOUBLE SIXES , and also a particular arrangement of 27 (possibly complex) lines, as dis- covered by Schla ¨fli (Salmon 1965, Fischer 1986) and sometimes called SOLOMON’S SEAL LINES . A nonregu- lar cubic surface can contain 3, 7, 15, or 27 real lines (Segre 1942, Le Lionnais 1983). The CLEBSCH DIAG- ONAL CUBIC contains all possible 27. The maximum number of ORDINARY DOUBLE POINTS on a cubic surface is four, and the unique cubic surface having four ORDINARY DOUBLE POINTS is the CAYLEY CUBIC . Schoutte (1910) showed that the 27 lines can be put into a ONE-TO-ONE correspondence with the vertices of a particular POLYTOPE in 6-D space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the POLYTOPE and conversely (Du Val 1931). A similar correspondence can be made between the 28 bitangents of the general plane QUARTIC CURVE and a 7-D POLYTOPE (Coxeter 1928) and between the tritangent planes of the canonical curve of genus 4 and an 8-D POLYTOPE (Du Val 1933). A smooth cubic surface contains 45 TRITANGENTS (Hunt). The Hessian of smooth cubic surface contains at least 10 ORDINARY DOUBLE POINTS , although the Hessian of the CAYLEY CUBIC contains 14 (Hunt). See also CAYLEY CUBIC ,CLEBSCH DIAGONAL CUBIC , DOUBLE SIXES,ECKARDT POINT ,ISOLATED SINGULAR- ITY,NORDSTRAND’S WEIRD SURFACE ,SOLOMON’S SEAL LINES,TRITANGENT References Bruce, J. and Wall, C. T. C. "On the Classification of Cubic Surfaces." J. London Math. Soc. 19, 245 /C1/56, 1979. Cayley, A. "A Memoir on Cubic Surfaces." Phil. Trans. Roy. Soc. 159, 231 /C1/26, 1869. Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc. 24, 7 /C1/, 1928. Du Val, P. "On the Directrices of a Set of Points in a Plane." Proc. London Math. Soc. Ser. 2 35,23/C1/4, 1933. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 9 /C1/4, 1986. Fladt, K. and Baur, A. Analytische Geometrie spezieler Fla¨chen und Raumkurven. Braunschweig, Germany: Vieweg, pp. 248 /C1/55, 1975. Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Hunt, B. "The 27 Lines on a Cubic Surface" and "Cubic Surfaces." Ch. 4 and Appendix B.4 in The Geometry of Some Special Arithmetic Quotients. New York: Springer- Verlag, pp. 108 /C1/67 and 302 /C1/10, 1996.Klein, F. "U¨ ber Fla¨chen dritter Ordnung." Gesammelte Abhandlungen, Band II. Berlin: Springer-Verlag, pp. 11 /C1/2, 1973. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 49, 1983. Rodenberg, C. "Zur Classification der Fla¨chen dritter Ord- nung." Math. Ann. 14,46/C1/10, 1878. Salmon, G. Analytic Geometry of Three Dimensions. New York: Chelsea, 1965. Schla¨fli, L. "On the Distribution of Surface of Third Order into Species." Phil. Trans. Roy. Soc. 153, 193 /C1/47, 1864. Schoutte, P. H. "On the Relation Between the Vertices of a Definite Sixdimensional Polytope and the Lines of a Cubic Surface." Proc. Roy. Acad. Amsterdam 13, 375 /C1/83, 1910. Segre, B. The Nonsingular Cubic Surface. Oxford, England: Clarendon Press, 1942. Cubical Conic Section CUBICAL ELLIPSE ,C UBICAL HYPERBOLA ,C UBICAL PARABOLA ,SKEW CONIC Cubical Ellipse An equation OF THE FORM y/C30ax3/C27bx2/C27cx/C27d where only one ROOT is real. See also CUBICAL CONIC SECTION ,CUBICAL HYPERBO- LA,CUBICAL PARABOLA ,CUBICAL PARABOLIC HYPER- BOLA ,ELLIPSE ,SKEW CONIC Cubical Graph The PLATONIC GRAPH corresponding to the connectiv- ity of the CUBE . Several symmetrical circular embed- dings of this graph are illustrated in the second figure above. The cubical graph has 8 nodes, 12 edges, VERTEX CONNECTIVITY 3, and EDGE CONNECTIVITY 3, GRAPH DIAMETER 3, GRAPH RADIUS 3, and GIRTH 4. The cubical graph’s CHROMATIC POLYNOMIAL is pG(z) /C30z8 /C2812z7 /C2766z6 /C28214z5 /C27441z4 /C28572z3 /C27423z2 /C28133z ; and the CHROMATIC NUMBER is x(G) /C302:/ The maximum number of nodes in a cubical graph which induce a cycle is six (Danzer and Klee 1967; Skiena 1990, p. 149). See also BIDIAKIS CUBE,BISLIT CUBE,CUBE,DODE- CAHEDRAL GRAPH ,ICOSAHEDRAL GRAPH ,O CTAHE- DRAL GRAPH ,PLATONIC GRAPH ,TETRAHEDRAL GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976. Danzer, L. and Klee, V. "Lengths of Snakes in Boxes." J. Combin. Th. 2, 258 /C1/65, 1967. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Cubical Hyperbola An equation OF THE FORM y /C30ax3 /C27bx2 /C27cx /C27d;where the three ROOTS are REAL and distinct, i.e., y /C30a(x /C28r1)(x /C28r2)(x /C28r3) /C30a[x3 /C28(r1 /C27r2 /C27r3)x2 /C27(r1r2 /C27r1r3 /C27r2r3)x /C28r1r2r3] : See also CUBICAL CONIC SECTION ,CUBICAL ELLIPSE , CUBICAL HYPERBOLA ,CUBICAL PARABOLA ,HYPERBO- LA Cubical Parabola An equation OF THE FORM y /C30ax3 /C27bx2 /C27cx /C27d; where the three ROOTS of the equation coincide (and are therefore real), i.e., y /C30a(x /C28r)3 /C30a(x3 /C283rx2 /C283r2x /C28r3) : See also CUBICAL CONIC SECTION ,CUBICAL ELLIPSE , CUBICAL HYPERBOLA ,CUBICAL PARABOLIC HYPERBO- LA,PARABOLA ,SEMICUBICAL PARABOLA References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 215 and 223, 1987. Cubical Parabolic Hyperbola An equation OF THE FORM y/C30ax3/C27bx2/C27cx/C27d; where two of the ROOTS of the equation coincide (and all three are therefore real), i.e., y /C30a(x /C28r1)2(x /C28r2) /C30a[x3 /C28(2r1 /C27r2)x2 /C27r1(r1 /C272r2)x /C28r2 1r2] : See also CUBICAL CONIC SECTION ,CUBICAL ELLIPSE , CUBICAL HYPERBOLA ,CUBICAL PARABOLA ,HYPERBO- LA Cubicuboctahedron GREAT CUBICUBOCTAHEDRON ,SMALL CUBICUBOCTA- HEDRON Cubique d’Agnesi WITCH OF AGNESI Cubitruncated Cuboctahedron The UNIFORM POLYHEDRON U16whose DUAL is the TETRADYAKIS HEXAHEDRON . It has W YTHOFF SYMBOL 34 34½:Its faces are 8 f6g/C276f8g/C276f83g:It is a FACETED OCTAHEDRON . the CIRCUMRADIUS for a cubitruncated cuboctahedron of unit edge length is r/C3012ffiffiffiffiffi 7p : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 113 /C1/14, 1971. Cuboctahedron The A RCHIMEDEAN SOLID A1(also called the DYMAX- ION orHEPTAPARALLELOHEDRON ) with faces /8f3g/C27 6f4g:It is one of the two convex QUASIREGULAR POLYHEDRA .I ti s UNIFORM POLYHEDRON U7and Wenninger model W11:It has S CHLA ¨FLI SYMBOL /3 4l1sl1n / and W YTHOFF SYMBOL 2|34. The DUAL POLYHEDRON is the RHOMBIC DODECAHE- DRON . The cuboctahedron has the OhOCTAHEDRAL GROUP of symmetries. According to Heron, Archi- medes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The VERTICES of a cuboc- tahedron with EDGE length offfiffiffi 2p are (0,91,91), (91, 0,91), and ( 91,91, 0). The INRADIUS rof the dual, MIDRADIUS rof the solid and dual, and CIRCUMRADIUS Rof the solid for a/C301 are r/C303 4/C300:75 (1) r/C301 2ffiffiffiffiffi 3p :0:86602 (2) R/C301: (3) The distances from the center of the solid to the centroids of the triangular and square faces are r3/C301 3ffiffiffi 6p (4) r4/C301 2ffiffiffi 2p : (5) The SURFACE AREA and VOLUME are S/C306/C272ffiffiffiffiffi 3p (6) V/C305 3ffiffiffi 2p : (7) FACETED versions of the cuboctahedron include the CUBOHEMIOCTAHEDRON and OCTAHEMIOCTAHEDRON . The solid common to both the CUBE and OCTAHEDRON (left figure) in a CUBE-OCTAHEDRON COMPOUND is a CUBOCTAHEDRON (right figure; Ball and Coxeter 1987). The mineral argentite (Ag 2S) forms cuboctahe- dral crystals (Steinhaus 1983, p. 203). The cubocta- hedron can be inscribed in the RHOMBIC DODECAHEDRON (Steinhaus 1983, p. 206). Wenninger (1989) lists four of the possible STELLA- TIONS of the cuboctahedron: the CUBE-OCTAHEDRON COMPOUND , a truncated form of the STELLA OCTAN- GULA , a sort of compound of six intersecting square pyramids, and an attractive concave solid formed of rhombi meeting four at a time. If a cuboctahedron is oriented with triangles on top and bottom, the two halves may be rotated one sixth of a turn with respect to each other to obtain JOHNSON SOLID J27, the TRIANGULAR ORTHOBICUPOLA . In cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connect- ing the centers of the external 12 spheres gives a cuboctahedron (Steinhaus 1983, pp. 203 /C1/07), which is therefore also a SPACE-FILLING POLYHEDRON . See also ARCHIMEDEAN SOLID ,CUBE,CUBE-OCTAHE- DRON COMPOUND ,CUBOHEMIOCTAHEDRON ,OCTAHE- DRON ,O CTAHEMIOCTAHEDRON ,Q UASIREGULAR POLYHEDRON ,R HOMBIC DODECAHEDRON ,R HOMBICDODECAHEDRON STELLATIONS ,R HOMBUS ,S PACE- FILLING POLYHEDRON ,SPHERE PACKING ,STELLATION , TRIANGULAR ORTHOBICUPOLA References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 137, 1987. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Cundy, H. and Rollett, A. "Cuboctahedron. /(3:4)2 /."§3.7.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 102, 1989. Ghyka, M. The Geometry of Art and Life. New York: Dover, p. 54, 1977. Heath, T. L. A History of Greek Mathematics, Vol. 1: From Thales to Euclid. New York: Dover, 1981. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 203 /C1/05, 1999. Wenninger, M. J. "The Cuboctahedron." Model 11 in Poly- hedron Models. Cambridge, England: Cambridge Univer- sity Press, p. 25, 1989. Wenninger, M. J. "Commentary on the Stellation of the Archimedean Solids." In Polyhedron Models. New York: Cambridge University Press, pp. 66 /C1/2, 1989. Cuboctahedron-Rhombic Dodecahedron Compound The POLYHEDRON COMPOUND consisting of the CUBOC- TAHEDRON and its dual, the RHOMBIC DODECAHEDRON , illustrated in the left figure above. The right figure shows the solid common to the two polyhedra. If the CUBOCTAHEDRON has unit edge length, the compound can be constructed by midpoint CUMULATION with heights h3/C301 4ffiffiffi 6p (1) h4/C301 2ffiffiffi 2p : (2) The resulting compound has side lengths s1/C301 8ffiffiffi 6p (3) s2/C301 2(4) s3/C3014ffiffiffi 6p (5) s4 /C301 2ffiffiffi 2p ; (6) and SURFACE AREA and VOLUME S /C303 4(4 /C275ffiffiffi 2p /C272ffiffiffi3p ) (7) V /C3031 16ffiffiffi 2p : (8) See also CUBOCTAHEDRON ,POLYHEDRON COMPOUND , POLYHEDRON DUAL,RHOMBIC DODECAHEDRON Cuboctatruncated Cuboctahedron CUBITRUNCATED CUBOCTAHEDRON Cubocycloid ASTROID Cubohemioctahedron The UNIFORM POLYHEDRON U15whose DUAL is the HEXAHEMIOCTACRON . It has WYTHOFF SYMBOL4 34|3. Its faces are 4{6}/C276{4}. It is a FACETED version of the CUBOCTAHEDRON . Its CIRCUMRADIUS for unit edge length is R /C301. References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 121 /C1/22, 1971. Cuboid A rectangular PARALLELEPIPED , sometimes also called a brick. A cuboid of side lengths a, b, and c has VOLUME V /C30abc (1) and SURFACE AREAS /C302(ab /C27ac /C27bc) : (2) The face diagonals are dab /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2p (3) dac /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2 /C27c2p (4) dbc /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 /C27c2p (5) and the body diagonal is dabc /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2 /C27b2 /C27c2p : (6) A cuboid with all sides equal is called a CUBE . See also CUBE,E ULER BRICK ,P ARALLELEPIPED , PRISM ,SPIDER AND FLY PROBLEM References Harris, J. W. and Stocker, H. "Cuboid." §4.2.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 97, 1998. Cullen Number A number OF THE FORM Cn /C302nn /C271 : The first few are 3, 9, 25, 65, 161, 385, ... (Sloane’s A002064). Cullen numbers are DIVISIBLE by /p /C302n /C281/ if p is a PRIME OF THE FORM /8k 93/. The only Cullen numbers Cnfor /n B300;000 / which are PRIME are for n /C301, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, ... (Sloane’s A005849; Ballinger). The largest PRIME Cullen num- ber known is for n/C30361275, but the range 335000 /C1/ 45000 has not yet been fully checked. See also CUNNINGHAM NUMBER ,FERMAT NUMBER , SIERPINSKI NUMBER OF THE FIRST KIND,W OODALL NUMBER References Ballinger, R. "Cullen Primes: Definition and Status." http:// vamri.xray.ufl.edu/proths/cullen.html. Caldwell, C. K. "The Top Twenty: Cullen Primes." http:// www.utm.edu/research/primes/lists/top20/Cullen.html. Guy, R. K. "Cullen Numbers." §B20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 77, 1994. Keller, W. "New Cullen Primes." Math. Comput. 64, 1733 /C1/ 741, 1995. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/cullen/. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 360 /C1/61, 1996. Sloane, N. J. A. Sequences A002064/M2795 and A0058495401 in "An On-Line Version of the Encyclopediaof Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Cumulant Let /f(t)/be the CHARACTERISTIC FUNCTION , defined as the F OURIER TRANSFORM of the PROBABILITY DENSITY FUNCTION (using FOURIER TRANSFORM parameters / a /C30b /C301/), f(t) /C30F[P(x)] /C30g/C12 /C28/C12eitxP(x) dx: (1) Then the cumulants / kn/ are then defined by ln f(t) /C13X/C12 n/C300kn(it)n n! (2) (Abramowitz and Stegun 1972, p. 928). Taking the MACLAURIN SERIES gives ln f(t) /C30(it)m?1 /C271 2 (it)2(m?2 /C28 m?12) /C271 3! (it)3 /C2(2m?13/C283m?1 m?2 /C27 m?3) /C271 4!(it)4 /C2(/C286m?14/C2712m?12m ?2 /C283 m?22/C284m ?1 m?3 /C27 m?4) /C271 5! /C2(it)5 /C2[24m ?15/C2860m ?13m?2 /C2720m ?12m?3 /C2810m?2 m?3 /C275 m?1(6m?22/C28 m?4) /C27 m?5] /C27...; (3) where /mn ?/ are RAW MOMENTS ,so k1 /C30 m?1 (4) k2 /C30 m ?2 /C28 m ?1 (5) k3 /C302m ?13/C283m?1 m?2 /C27 m?3 (6) k4 /C30/C286m ?14/C2712m ?12m?2 /C283m?22/C284m?1 m?3 /C27 m?4 (7) k5 /C30/C2824 m?15/C2860 m?13m?2 /C2720 m?12m?3 /C2810m ?2 m?3 /C275 m?1(6m?22/C28 m?4) /C27 m?5 : (8) In terms of the CENTRAL MOMENTS mn ; k1 /C30 m (9) k2 /C30 m2 /C30 s2 (10) k3 /C30 m3 (11) k4 /C30 m4 /C283m2 2 (12) k5 /C30 m5 /C2810 m2 m3 ; (13) where m is the MEAN and s2 /C13 m2 is the VARIANCE . The K-STATISTIC are UNBIASED ESTIMATORS of the cumulants. See also CHARACTERISTIC FUNCTION (PROBABILITY ), CUMULANT- GENERATING FUNCTION , K -STATISTIC , KURTOSIS ,MEAN,MOMENT ,SHEPPARD’S CORRECTION , SKEWNESS ,UNBIASED ESTIMATOR ,VARIANCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. New York: Dover, p. 928, 1972. Kenney, J. F. and Keeping, E. S. "Cumulants and the Cumulant-Generating Function," "Additive Property of Cumulants," and "Sheppard’s Correction." §4.10 /C1/.12 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 77 /C1/2, 1951. Cumulant-Generating Function Let /M(h)/ be the MOMENT-GENERATING FUNCTION , then K(h) /C13ln M(h) /C30 k1h /C271 2!h2 k2 /C271 3!h3 k3 /C27...; (1) where /k1 ; k2/, ..., are the CUMULANTS . If L /C30XN j/C301cjxj (2) is a function of N independent variables, then the cumulant-generating function for L is given by K(h)/C30XN j/C301Kj(cjh): (3) See also CUMULANT ,MOMENT- GENERATING FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972. Kenney, J. F. and Keeping, E. S. "Cumulants and the Cumulant-Generating Function" and "Additive Propertyof Cumulants." §4.10/C1 /.11 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 77 /C1/0, 1951. Cumulation The dual operation of TRUNCATION which replaces the faces of a POLYHEDRON with PYRAMIDS of height h (where hmay be positive, zero, or negative) having the face as the base. This operation is implemented in Mathematica under the misnomer Stellate [poly, ratio ] in the Mathematica add-on package Graphic- s‘Polyhedra‘ (which can be loaded with the com- mandBBGraphics‘ ). The operation is sometimes also called accretion, or sometimes akisation (since it transforms a regular polygon to an n-akis polyhe- dron, i.e., quadruples the number of faces). The following plots show cumulation series for the TETRAHEDRON ,CUBE ,OCTAHEDRON ,DODECAHEDRON , and ICOSAHEDRON . Cumulation with h /C300 gives a triangulated version of the original solid. The following table gives special solids formed by cumulation of given heights on simple solids. In this table, r is the INRADIUS , and (r /C27 h) =h is the "stellation ratio" as defined in Mathema- tica. Original h /(r/C27h)=h/ Result CUBE /1 6// 4=3/ TETRAKIS HEXAHE- DRON CUBE /1 2/ 2 RHOMBIC DODECAHE- DRON CUBE /12ffiffiffi 2p // 1/C27ffiffiffi2p / 24-faced star DELTA- HEDRON DODECAHEDRON /1 19ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5(65/C2722ffiffiffi 5p )q //3 19(10/C28ffiffiffi5p ) / PENTAKIS DODECAHE- DRON DODECAHEDRON /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10(5/C28ffiffiffi 5p )q // 2ffiffiffi5p /C283 / 60-faced star DELTA- HEDRON DODECAHEDRON /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5(5/C272ffiffiffi 5p )q //ffiffiffi5p / SMALL STELLATED DODECAHEDRON ICOSAHEDRON /1 6ffiffiffi 3p (ffiffiffi5p /C283) // 3(ffiffiffi5p /C282) / GREAT DODECAHE- DRON ICOSAHEDRON /1 15ffiffiffiffiffiffi15p //1 5(10/C283ffiffiffi 5p )/ SMALL TRIAMBIC ICOSAHEDRON ICOSAHEDRON /1 3ffiffiffi 6p // 1/C283ffiffiffi2p /C27ffiffiffiffiffiffi10p / 60-faced star DELTA- HEDRON ICOSAHEDRON /1 6ffiffiffi 3p (3/C27ffiffiffi5p ) / 3 GREAT STELLATED DODECAHEDRON OCTAHEDRON /ffiffiffi3p /C282 3ffiffiffi 6p // 5/C283ffiffiffi 2p / SMALL TRIAKIS OCTAHEDRON OCTAHEDRON /1 3ffiffiffi 6p / 3 STELLA OCTANGULA TETRAHEDRON /1 15ffiffiffi6p //7 5/ TRIAKIS TETRAHE- DRON TETRAHEDRON /16ffiffiffi 6p / 2 CUBE TETRAHEDRON /1 3ffiffiffi 6p / 3 9-faced star DELTA- HEDRON Another type of cumulation (which I call "midpoint cumulation") replaces each facial polygon with trian- gular polygons joining vertices with the neighboring edge midpoints, and then constructs a pyramid with base determined by the face’s midpoints. Midpointcumulation allow compounds of Archimedean solids and their duals to be easily constructed. ARCHIMEDEAN SOLIDdual face 1 face 2 CUBOCTAHE- DRONRHOMBIC DO- DECAHEDRON3:1 4ffiffiffi 6p / 4:1 2ffiffiffi 2p / ICOSIDODECA- HEDRONRHOMBIC TRIA- CONTAHE- DRON3:1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C283ffiffiffi 5pp )q /1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15(5/C272ffiffiffi 5p )q / SMALL RHOM- BICUBOCTAHE- DRONDELTOIDAL ICOSITETRAHE-DRON3:1 42ffiffiffi 3p (3/C28ffiffiffi 2p )// 4:1 2(ffiffiffi 2p /C281)/ TRUNCATED CUBESMALL TRIAKIS OCTAHEDRON/3:1 6ffiffiffiffiffi 3p (3/C282ffiffiffiffiffi 2p )// 8:12(1/C27ffiffiffiffiffi 2p )/ TRUNCATED DODECAHE- DRONTRIAKIS ICOSA- HEDRON/3:1 372ffiffiffiffiffi 3p (1/C275ffiffiffiffiffi 5p )//1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(6/C27ffiffiffiffiffi 5p )q / TRUNCATED ICOSAHEDRONPENTAKIS DO- DECAHEDRON/1 38ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10(305/C27131ffiffiffi 5p )q // 6:1 4ffiffiffi 3p (ffiffiffi5p /C283) / TRUNCATED OCTAHEDRONTETRAKIS HEX- AHEDRON/4:1 8ffiffiffiffiffi 2p // 3:14ffiffiffi 6p / TRUNCATED TETRAHEDRONTRIAKIS TET- RAHEDRON/3:1 30ffiffiffiffiffi 6p // 6:1 2ffiffiffiffiffi 6p / See also ELEVATUM ,ESCHER’S SOLID ,INVAGINATUM , PYRAMID ,STELLATION ,TRUNCATION References Graziotti, U. Polyhedra, the Realm of Geometric Beauty. San Francisco, CA: 1962. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Cumulative Distribution Function DISTRIBUTION FUNCTION Cumulative Frequency Let the ABSOLUTE FREQUENCIES of occurrence of an event in a number of CLASS INTERVALS be denoted f1 ; f2 ; .... The cumulative frequency corresponding to the upper boundary of any CLASS INTERVAL ciin a FREQUENCY DISTRIBUTION is the total absolute fre- quency of all values less than that boundary, denoted FB/C13X i5nfi: See also ABSOLUTE FREQUENCY ,C LASS INTERVAL , CUMULATIVE FREQUENCY POLYGON ,FREQUENCY DIS- TRIBUTION ,RELATIVE CUMULATIVE FREQUENCY ,RE- LATIVE FREQUENCY References Kenney, J. F. and Keeping, E. S. "Cumulative Frequencies." §1.11 in Mathematics of Statistics, Pt. 1, 3rd ed. Prince- ton, NJ: Van Nostrand, pp. 17 /C1/9, 1962. Cumulative Frequency Polygon A plot of the cumulative frequency against the upper class boundary with the points joined by line seg- ments. Any continuous cumulative frequency curve, including a cumulative frequency polygon, is called an OGIVE . See also ABSOLUTE FREQUENCY ,C LASS INTERVAL , FREQUENCY DISTRIBUTION ,F REQUENCY POLYGON , OGIVE,RELATIVE CUMULATIVE FREQUENCY ,RELATIVE FREQUENCY References Kenney, J. F. and Keeping, E. S. "Cumulative Frequency Polygons." §2.6 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 28 /C1/9, 1962. Cundy and Rollett’s Egg An OVAL dissected into pieces which are to used to create pictures. The resulting figures resemble those constructed out of TANGRAMS . See also DISSECTION ,EGG,OVAL,TANGRAM References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 19 /C1/1, 1989.Dixon, R. Mathographics. New York: Dover, p. 11, 1991. Cunningham Chain A SEQUENCE of PRIMES q1 Bq2 B...Bqk is a Cunning- ham chain of the first kind (second kind) of length k if q1 /C271 /C302qi /C271(q1 /C271 /C302qi /C281) for i /C30 1, ..., k /C281: Cunningham PRIMES of the first kind are SOPHIE GERMAIN PRIMES . The two largest known Cunningham chains (of the first kind) of length three are ( 384205437 /C215 24000 /C281; 384205437 /C215 24001 /C281 ; 384205437 /C215 24002 /C281) and (/651358155 /C215 23291 /C281; 651358155 /C215 23292 /C281; 651358155 /C215 23293 /C281); both discovered by W. Roon- guthai in 1998. See also BITWIN CHAIN ,PRIME ARITHMETIC PROGRES- SION,PRIME CLUSTER References Forbes, T. "Prime Clusters and Cunningham Chains." Math. Comput. 68, 1739 /C1/748, 1999. Guy, R. K. "Cunningham Chains." §A7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 18 /C1/9, 1994. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 333, 1996. Roonguthai, W. "Yves Gallot’s Proth and Cunningham Chains." http://ksc9.th.com/warut/cunningham.html. Cunningham Function Sometimes also called the PEARSON- CUNNINGHAM FUNCTION . It can be expressed using WHITTAKER FUNCTIONS (Whittaker and Watson 1990, p. 353). vn;m(x) /C13e pi(m=2 /C28n) /C27x G(1 /C27 n /C281 2m) U(1 2m /C28n; 1 /C27m; x) ; where U(a ; b ; z)isa CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND (Abramowitz and Stegun 1972, p. 510). See also CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND,W HITTAKER FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Cunningham Number ABINOMIAL NUMBER OF THE FORM C9(b;n)/C13bn91: Bases bkwhich are themselves powers need not be considered since they correspond to ( bk)n91/C30bkn91: PRIME NUMBERS OF THE FORM C9(b;n) are very rare. ANECESSARY (but not SUFFICIENT ) condition for C/C27(2;n)/C302n/C271t ob e PRIME is that nbeOF THE FORM n/C302m:Numbers OF THE FORM Fm/C30 C /C27(2; 2m) /C3022m /C271 are called FERMAT NUMBERS , and the only known PRIMES occur for /C /C27(2; 1) /C303/, C /C27(2; 2) /C305; C /C27(2; 4) /C3017 ; C /C27(2; 8) /C30257; and C /C27(2; 16) /C3065537 (i.e., n /C300, 1, 2, 3, 4). The only other PRIMES C /C27(b; n) for nontrivial b 511 and 2 5 n 51000 are C /C27(6; 2) /C3037 ; C /C27(6; 4) /C301297 ; and C /C27(10 ; 2) /C30101:/ PRIMES OF THE FORM C /C28(b; n) are also very rare. The MERSENNE NUMBERS Mn /C30C /C28(2; n) /C302n /C281 are known to be prime only for 37 values, the first few of which are n /C302, 3, 5, 7, 13, 17, 19, ... (Sloane’s A000043). There are no other PRIMES C/C28(b ; n) for nontrivial b 520 and 2 5n 51000 :/ In 1925, Cunningham and Woodall (1925) gathered together all that was known about the PRIMALITY and factorization of the numbers C 9(b ; n) and published a small book of tables. These tables collected from scattered sources the known prime factors for the bases 2 and 10 and also presented the authors’ results of 30 years’ work with these and other bases. Since 1925, many people have worked on filling in these tables. D. H. Lehmer, a well-known mathema- tician who died in 1991, was for many years a leader of these efforts. Lehmer was a mathematician who was at the forefront of computing as modern electro- nic computers became a reality. He was also known as the inventor of some ingenious pre-electronic computing devices specifically designed for factoring numbers. Updated factorizations were published in Brillhart et al. (1988). The current archive of Cunningham number factorizations for b /C30 1, ..., 9 12 is kept on ftp://sable.ox.ac.uk/pub/math/cunningham/. The tables have been extended by Brent and te Riele (1992) to b /C30 13, ..., 100 with m B255 for b B30 and m B100 for b ]30 : All numbers with exponent 58 and smaller, and all composites with 590 digits have now been factored. See also BINOMIAL NUMBER ,CULLEN NUMBER ,FER- MAT NUMBER ,M ERSENNE NUMBER ,REPUNIT ,RIESEL NUMBER ,SIERPINSKI NUMBER OF THE FIRST KIND, WOODALL NUMBER References Brent, R. P. and te Riele, H. J. J. "Factorizations of an 91; 13 5a B100 /" Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, June 1992. ftp://sable.ox.a- c.uk/pub/math/factors/. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn 91 ; b /C302, 3; 5; 6; 7; 10; 11; 12 Up to High Powers, rev. ed. Provi- dence, RI: Amer. Math. Soc., 1988. Updates are available electronically from ftp://sable.ox.ac.uk/pub/math/cunning- ham/. Cunningham, A. J. C. and Woodall, H. J. Factorisation of yn /C141 ; y /C30 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers (n). London: Hodgson, 1925. Mudge, M. "Not Numerology but Numeralogy!" Personal Computer World, 279 /C1/80, 1997.Ribenboim, P. "Numbers k /C292n 91:/" §5.7 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 355 /C1/60, 1996. Sloane, N. J. A. Sequences A000043/M0672 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Cunningham Project CUNNINGHAM NUMBER Cup See also CAP,CUP PRODUCT References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971. Cup Product The cup product is a product on COHOMOLOGY CLASSES . In the case of DE RHAM COHOMOLOGY ,a COHOMOLOGY CLASS can be represented by a CLOSED FORM . The cup product of [ a] and [ b] is represented by the CLOSED FORM [a ffl b] ; where ffl is the WEDGE PRODUCT of DIFFERENTIAL K-FORMS . It is the dual operation to intersection in HOMOLOGY . In general, the cup product is a map /C150: Hp /C29Hq 0 Hp /C27q which satisfies a /C150b /C30(/C281)pqb /C150a :/ See also COHOMOLOGY ,CUP, DE RHAM COHOMOLOGY , HOMOLOGY References Hazewinkel, M. (Managing Ed.). §200.K, 201.I, and 237.D in Encyclopaedia of Mathematics: An Updated and Anno- tated Translation of the Soviet "Mathematical Encyclopae-dia," Vol. 2. Dordrecht, Netherlands: Reidel, pp. 756, 766/C1 /67, and 879, 1988. Cupola Ann-gonal cupola Qnis a POLYHEDRON having n obliquely oriented TRIANGULAR and nrectangular faces separating an fngand a f2ngREGULAR POLY- GON, each oriented horizontally. The coordinates of the base VERTICES are Rcosp(2k/C271) 2n"# ;Rsinp(2k/C271) 2n"# ;0 ! ; (1) and the coordinates of the top VERTICES are rcos2kp n"# ;rsin2kp n"# ;z ! ; (2) where Randrare the CIRCUMRADII of the base and top R /C301 2a cscp 2n ! (3) r /C3012a cscp n ! ; (4) and z is the height. A cupola with all unit edge lengths (in which case the triangles become unit equilateral triangles and the rectangles become unit squares) is possible only for n /C303, 4, 5, in which case the height z can be obtained by letting k /C30 0 in the equations (1) and (2) to obtain the coordinates of neighboring bottom and top VER- TICES , b /C30R cosp 2n ! R sinp 2n ! 02 66666643 7777775(5) t /C30r 0 z2 435: (6) Since all side lengths are a, ½b /C28t ½ 2 /C30a2 : (7) Solving for z then gives R cosp 2n ! /C28r"#2 /C27R2 sin2p 2n ! /C27z2 /C30a2 (8) z2 /C27R2 /C27r2 /C282rR cosp 2n ! /C30a2 (9) z /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C282rR cosp 2n ! /C28r2 /C28R2vuut /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C281 4csc2p n !vuut(10) See also BICUPOLA ,ELONGATED CUPOLA ,GYROELON- GATED CUPOLA ,P ENTAGONAL CUPOLA ,R OTUNDA , SQUARE CUPOLA ,TRIANGULAR CUPOLAReferences Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169 /C1/00, 1966. Cupolarotunda A CUPOLA adjoined to a ROTUNDA . See also GYROCUPOLAROTUNDA ,ORTHOCUPOLAROTUN- DA Curl The curl of a TENSOR field is given by (9/C29A) a /C30 eamnAv: m ; (1) where eijkis the LEVI-CIVITA TENSOR and ";" is the COVARIANT DERIVATIVE . For a VECTOR FIELD , the curl is denoted curl(F) /C139/C29F; (2) and 9/C29F is normal to the PLANE in which the "circulation" is MAXIMUM . Its magnitude is the limit- ing value of circulation per unit AREA , (9/C29F) /C215 ˆn /C13lim A00GCF /C215 ds A: (3) Let F /C13F1 ˆu1 /C27F2 ˆu2 /C27F3 ˆu3 (4) and hi /C13@r @uil112l112l112l112l112l112l112l112l112l112; (5) then 9/C29F /C13 1 h1h2h3h1 ˆu1h2 ˆu2h3 ˆu3 @ @u1@ @u2@ @u3 h1F1h2F2h2F2l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112 /C30 1 h2h3@ @u2(h3F3) /C28@ @u3(h2F2)"# ˆu1 /C271 h1h3@ @u3(h1F1) /C28@ @u1(h3F3)"# ˆu2 /C271 h1h2@ @u1(h2F2) /C28@ @u2(h1F1)"# ˆu3 : (6) Special cases of the curl formulas above can be given for CURVILINEAR COORDINATES . See also CURL THEOREM ,CURVILINEAR COORDINATES , DIVERGENCE ,GRADIENT ,VECTOR DERIVATIVE References Arfken, G. "Curl, 9/C29:/" §1.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 42 /C1/ 7, 1985. Curl Theorem A special case of STOKES’ THEOREM in which F is a VECTOR FIELD and M is an oriented, compact em- bedded 2-MANIFOLD with boundary in /R2 /, given by gS( 9/C29F) /C215 da /C30g@SF /C215 ds: (1) There are also alternate forms. If F /C13cF ; (2) then gSda /C299F /C30gCFds: (3) and if F /C13c /C29P ; (4) then gS(da /C299) /C29P /C30gCds /C29P: (5) See also CHANGE OF VARIABLES THEOREM ,C URL, STOKES’ THEOREM References Arfken, G. "Stokes’s Theorem." §1.12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 61 /C1/4, 1985. Curlicue Fractal The curlicue fractal is a figure obtained by the following procedure. Let s be an IRRATIONAL NUMBER . Begin with a line segment of unit length, which makes an ANGLE f0 /C130 to the horizontal. Then define un iteratively by un/C271 /C30( un /C272 ps)(mod 2p) ;with u0 /C300: To the end of the previous line segment, draw a line segment of unit length which makes an angle fn /C271 /C30 un /C27 fn(mod 2 p) ; to the horizontal (Pickover 1995). The result is a FRACTAL , and the above figures correspond to the curlicue fractals with 10,000 points for the GOLDEN RATIO f ; ln 2 ; e,ffiffiffi 2p ; the EULER- MASCHERONI CON- STANT g ; p; and FEIGENBAUM CONSTANT d :/ The TEMPERATURE of these curves is given in the following table. Constant Temperature GOLDEN RATIO f/ 46 /ln 2/ 51 e 58 /ffiffiffi 2p / 58 EULER- MASCHERONI CONSTANT g/ 63 /p/ 90 FEIGENBAUM CONSTANT a/ 92 References Berry, M. and Goldberg, J. "Renormalization of Curlicues." Nonlinearity 1,1/C1/6, 1988. Moore, R. and van der Poorten, A. "On the Thermodynamics of Curves and Other Curlicues." McQuarie Univ. Math. Rep. 89 /C1/031, April 1989. Pickover, C. A. "The Fractal Golden Curlicue is Cool." Ch. 21 in Keys to Infinity. New York: W. H. Freeman, pp. 163 /C1/ 67, 1995. Pickover, C. A. Mazes for the Mind: Computers and the Unexpected. New York: St. Martin’s Press, 1993. Sedgewick, R. Algorithms in C, 3rd ed. Reading, MA: Addison-Wesley, 1998. Stewart, I. Another Fine Math You’ve Got Me Into.... New York: W. H. Freeman, 1992. Stoschek, E. "Module 35: Curlicue Variations: Polygon Patterns in the Gauss Plane of Complex Numbers." http://marvin.sn.schule.de/~inftreff/modul35/tas- k35_e.htm. Stoschek, E. "Module 36: The Feigenbaum-Constant din the Gauss Plane." http://marvin.sn.schule.de/~inftreff/modul36/task36_e.htm. Curly Brace BRACE Current A linear FUNCTIONAL on a smooth differential form. See also FLAT NORM,INTEGRAL CURRENT ,RECTIFI- ABLE CURRENT Curtate Cycloid The path traced out by a fixed point at a RADIUS b B a, where a is the RADIUS of a rolling CIRCLE , some- times also called a CONTRACTED CYCLOID . x /C30af /C28b sin f (1) y /C30a /C28b cos f : (2) The ARC LENGTH from f /C300is s /C302(a /C27b)E(u) ; (3) where sin(1 2 f) /C30sn u (4) k2 /C304ab (a /C27 c)2 ; (5) and E(u) is a complete ELLIPTIC INTEGRAL OF THE SECOND KIND and sn uis a J ACOBI ELLIPTIC FUNC- TION . See also CYCLOID ,PROLATE CYCLOID ,TROCHOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 216, 1987. Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 325, 1998. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 194 /C1/97, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 146, 1967. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 147 /C1/48, 1999. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 292, 1995. Curtate Cycloid Evolute The EVOLUTE of the CURTATE CYCLOID x/C30af/C28bsinf (1) y/C30a/C28bcosf: (2) is given by x/C30a[/C282bf/C272afcosf/C282asinf/C27bsin(2f)] 2(acosf/C28b)(3) y/C30a(a/C28bcosf)2 b(acosf/C28b): (4)Curvature In general, there are two important types of curva- ture: EXTRINSIC CURVATURE and INTRINSIC CURVA- TURE . The EXTRINSIC CURVATURE of curves in 2- and 3-space was the first type of curvature to be studiedhistorically, culminating in the F RENET FORMULAS , which describe a SPACE CURVE entirely in terms of its "curvature," TORSION , and the initial starting point and direction. After the curvature of 2- and 3-d curves was studied, attention turned to the curvature of surfaces in 3- space. The main curvatures which emerged from this scrutiny are the MEAN CURVATURE ,GAUSSIAN CURVA- TURE , and the W EINGARTEN MAP .M EAN CURVATURE was the most important for applications at the timeand was the most studied, but Gauss was the first torecognize the importance of the G AUSSIAN CURVA- TURE . Because G AUSSIAN CURVATURE is "intrinsic," it is detectable to 2-dimensional "inhabitants" of the sur-face, whereas MEAN CURVATURE and the W EINGARTEN MAP are not detectable to someone who can’t study the 3-dimensional space surrounding the surface onwhich he resides. The importance of G AUSSIAN CUR- VATURE to an inhabitant is that it controls the surface AREA ofSPHERES around the inhabitant. Riemann and many others generalized the concept ofcurvature to SECTIONAL CURVATURE ,SCALAR CURVA- TURE , the R IEMANN TENSOR ,RICCI CURVATURE , and a host of other INTRINSIC and EXTRINSIC CURVATURES . General curvatures no longer need to be numbers,and can take the form of a MAP,GROUP ,GROUPOID , tensor field, etc. The simplest form of curvature and that usually first encountered in CALCULUS is an EXTRINSIC CURVA- TURE . In 2-D, let a PLANE CURVE be given by CARTESIAN PARAMETRIC EQUATIONS x/C30x(t) and y/C30 y(t):Then the curvature kis defined by k/C13df ds/C30df dt ds dt/C30df dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx dt !2 /C27dy dt !2vuut/C30df dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2/C27y?2p ;(1) where fis the TANGENTIAL ANGLE and sis the ARC LENGTH . As can readily be seen from the definition, curvature therefore has units of inverse distance. Thedf dtderivative in the above equation can be found using the identity tanf/C30dy dx/C30dy=dt dx=dt/C30y? x?; (2) so d dt(tanf)/C30sec2fdf dt/C30x?yƒ/C28y?xƒ x?2(3) and df dt/C301 sec2fd dt(tanf)/C301 1/C27tan2fx?yƒ/C28y?xƒ x?2 /C301 1/C27y?2 x?2x?yƒ/C28y?xƒ x?2/C30x?yƒ/C28y?xƒ x?2/C27y?2: (4) Combining (1), (2), and (4) then gives k/C30x?yƒ/C28y?xƒ (x?2/C27y?2)3=2: (5) For a 2-D curve written in the form y/C30f(x);the equation of curvature becomes k/C30d2y dx2 1/C27(dy dx)2hi3=2: (6) If the 2-D curve is instead parameterized in POLAR COORDINATES , then k/C30r2/C272r2 u/C28rruu (r2/C27r2 u)3=2; (7) where ru/C13@r=@u(Gray 1997, p. 89). In PEDAL CO- ORDINATES , the curvature is given by k/C301 rdp dr: (8) The curvature for a 2-D curve given implicitly by g(x;y)/C300 is given by k/C30gxxg2 y/C282gxygxgy/C27gyyg2x (g2 x/C27g2y)3=2(9) (Gray 1997). Now consider a parameterized SPACE CURVE r(t)i n3 - D for which the TANGENT VECTOR ˆTis defined as ˆT/C13dr dt dr dtl112l112l112l112l112l112l112l112l112l112/C30dr dt ds dt: (10) Therefore, dr dt/C30ds dtˆT (11) d2r dt2/C30d2s dt2ˆT/C27ds dtdˆT dt/C30ds2 dt2ˆT/C27kˆNds dt !2 ; (12) where ˆNis the NORMAL VECTOR . But dr dt/C29d2r dt2/C30ds dtd2s dt2(ˆT/C29ˆT)/C27kds dt !3 (ˆT/C29ˆN)/C30kds dt !3 (ˆT/C29ˆN) (13) dr dt/C29d2r dt2l112l112l112l112l112l112l112l112l112l112/C30k ds dt !3 /C30kdr dtl112l112l112l112l112l112l112l112l112l1123 ; (14) so k/C30dˆT dsl112l112l112l112l112l112l112l112l112l112/C30 dr dt/C29d2r dt2l112l112l112l112l112l112 dr dtl112l112l112l112l112l112 3: (15) The curvature of a 2-D curve is related to the RADIUS OF CURVATURE of the curve’s OSCULATING CIRCLE . Consider a CIRCLE specified parametrically by x/C30acost (16) y/C30asint (17) which is tangent to the curve at a given point. The curvature is then k/C30x?yƒ/C28y?xƒ (x?2/C27y?2)3=2/C30a2 a3/C301 a; (18) or one over the RADIUS OF CURVATURE . The curvature of a CIRCLE can also be repeated in vector notation. For the CIRCLE with 05tB2p;the ARC LENGTH is s(t)/C30gt 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx dt !2 /C27dy dt !2vuutdt /C30gt 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2cos2t/C27a2sin2tp dt/C30at; (19) sot/C30s=aand the equations of the CIRCLE can be rewritten as x/C30acoss a ! (20) y/C30asins a ! : (21) The POSITION VECTOR is then given by r(s)/C30acoss a ! ˆx/C27asins a ! ˆy; (22) and the TANGENT VECTOR is ˆT/C30dr ds/C30/C28sins a ! ˆx/C27coss a ! ˆy; (23) so the curvature is related to the RADIUS OF CURVA- TURE aby k/C30dˆT dsl112l112l112l112l112l112l112l112l112l112/C30 j/C281 acoss a ! ˆx/C281 asins a ! ˆyj /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2s a ! /C27 sin2s a ! a2vuuuut/C301 a ; (24) as expected. Four very important derivative relations in differen- tial geometry related to the FRENET FORMULAS are ˙r /C30T (25) ¨r /C30 kN (26) /C5r /C30 ˙kN /C27 k(tB /C28 kT) (27) [˙r ; ¨r; /C5r] /C30 k2 t ; (28) where T is the TANGENT VECTOR , N is the NORMAL VECTOR , B is the BINORMAL VECTOR , and t is the TORSION (Coxeter 1969, p. 322). The curvature at a point on a surface takes on a variety of values as the PLANE through the normal varies. As k varies, it achieves a minimum and a maximum (which are in perpendicular directions) known as the PRINCIPAL CURVATURES . As shown in Coxeter (1969, pp. 352 /C1/53), k2 /C28X bi i k /C27det(bj i) /C300 (29) k2 /C282H k /C27K /C300; (30) where K is the GAUSSIAN CURVATURE , H is the MEAN CURVATURE , and det denotes the DETERMINANT . The curvature k is sometimes called the FIRST CURVATURE and the TORSION t the SECOND CURVA- TURE . In addition, a THIRD CURVATURE (sometimes called TOTAL CURVATURE ) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds2 T /C27ds2Bq (31) is also defined. A signed version of the curvature of a CIRCLE appearing in the D ESCARTES CIRCLE THEOREM for the radius of the fourth of four mutually tangent circles is called the BEND . See also BEND (CURVATURE ), CURVATURE CENTER , CURVATURE SCALAR ,E XTRINSIC CURVATURE ,FIRST CURVATURE ,FOUR- VERTEX THEOREM ,GAUSSIAN CUR- VATURE ,INTRINSIC CURVATURE ,LANCRET EQUATION , LINE OF CURVATURE ,M EAN CURVATURE ,N ORMAL CURVATURE ,P RINCIPAL CURVATURES ,R ADIUS OF CURVATURE ,R ICCI CURVATURE ,R IEMANN TENSOR , SECOND CURVATURE ,SECTIONAL CURVATURE ,SODDY CIRCLES ,THIRD CURVATURE ,TORSION (DIFFERENTIAL GEOMETRY ), WEINGARTEN MAP References Casey, J. Exploring Curvature. Wiesbaden, Germany: Vieweg, 1996. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Fischer, G. (Ed.). Plates 79 /C1/5i n Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 74 /C1/1, 1986. Gray, A. "Curvature of Curves in the Plane," "Drawing Plane Curves with Assigned Curvature," and "DrawingSpace Curves with Assigned Curvature." §1.5, 6.4, and 10.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 14 /C1 /7, 140 /C1/46, and 222 /C1/24, 1997. Kreyszig, E. "Principal Normal, Curvature, Osculating Circle." §12 in Differential Geometry. New York: Dover, pp. 34 /C1/6, 1991. Yates, R. C. "Curvature." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 60 /C1/4, 1952. Curvature Center The point on the POSITIVE RAY of the NORMAL VECTOR at a distance r(s);where ris the RADIUS OF CURVA- TURE . It is given by z/C30x/C27rN/C30x/C27r2T ds; (1) where Nis the NORMAL VECTOR andTis the TANGENT VECTOR . It can be written in terms of xexplicitly as z/C30x/C27xƒ(x?/C215x?)2/C28x?(x?/C215x?)(x?/C215xƒ) (x?/C215x?)(xƒ/C215xƒ)/C28(x?/C215xƒ)2: (2) For a CURVE represented parametrically by (f(t);g(t)); a/C30f/C28(f?2/C28g?2)g? f?gƒ/C28fƒg?(3) b/C30g/C27(f?2/C28g?2)f? f?gƒ/C28fƒg?(4) References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997. Curvature Scalar SCALAR CURVATURE Curvature Vector K/C13dT ds; where T is the TANGENT VECTOR defined by T /C13dx ds dx dsl112l112l112l112l112l112l112l112l112l112: Curve A CONTINUOUS MAP from a 1-D SPACE to an n-D SPACE . Loosely speaking, the word "curve" is often used to mean the GRAPH of a 2- or 3-D curve. The simplest curves can be represented parametrically in n-D SPACE as x1 /C30f1(t) x2 /C30f2(t) n xn /C30fn(t) : Other simple curves can be simply defined only implicitly, i.e., in the form f(x1;x2;... )/C300: See also PLANE CURVE ,SPACE CURVE ,SPHERICAL CURVE References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 71 /C1/5, 1989. "Geometry." The New Encyclopædia Britannica, 15th ed. 19, pp. 946 /C1/51, 1990. Gallier, J. H. Curves and Surfaces for Geometric Design: Theory and Algorithms. New York: Academic Press, 1999. Oakley, C. O. Analytic Geometry. New York: Barnes and Noble, 1957. Rutter, J. W. Geometry of Curves. Boca Raton, FL: Chap- man and Hall/CRC, 2000. Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, 1995. Seggern, D. von CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, 1993. Smith, P. F.; Gale, A. S.; and Neelley, J. H. New Analytic Geometry, Alternate Edition. Boston, MA: Ginn and Company, 1938. Walker, R. J. Algebraic Curves. New York: Springer-Verlag, 1978. Weisstein, E. W. "Books about Curves." http://www.trea- sure-troves.com/books/Curves.html. Yates, R. C. The Trisection Problem. Reston, VA: National Council of Teachers of Mathematics, 1971. Zwillinger, D. (Ed.). "Algebraic Curves." §8.1 in CRC Stan- dard Mathematical Tables and Formulae, 3rd ed. Boca Raton, FL: CRC Press, 1996. Curve of Constant Breadth CURVE OF CONSTANT WIDTHCurve of Constant Precession A curve whose CENTRODE revolves about a fixed axis with constant ANGLE and SPEED when the curve is traversed with unit SPEED . The TANGENT INDICATRIX of a curve of constant precession is a SPHERICAL HELIX .A n ARC LENGTH parameterization of a curve of constant precession with NATURAL EQUATIONS k(s)/C30/C28vsin(ms) (1) t(s)/C30/C28vcos(ms) (2) is x(s)/C30a/C27m 2asin[(a/C28m)s] a/C28m/C28a/C28m 2asin[(a/C27m)s] a/C27m(3) y(s)/C30a/C27m 2asin[(a/C28m)s] a/C28m/C27a/C28m 2acos[(a/C27m)s] a/C27m(4) z(s)/C30v masin(ms); (5) where a/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2/C27m2p (6) and v;and mare constant. This curve lies on a circular one-sheeted HYPERBOLOID x2/C27y2/C28m2 v2z2/C304m2 v4: (7) The curve is closed IFFm=aisRATIONAL . References Scofield, P. D. "Curves of Constant Precession." Amer. Math. Monthly 102, 531/C1/37, 1995. Curve of Constant Slope GENERALIZED HELIX Curve of Constant Width Curves which, when rotated in a square, make contact with all four sides. Such curves are sometimesalso known as ROLLERS . The "width" of a closed convex curve is defined as the distance between parallel lines bounding it ("support- ing lines"). Every curve of constant width is convex.Curves of constant width have the same "width" regardless of their orientation between the parallel lines. In fact, they also share the same PERIMETER (BARBIER’S THEOREM ). Examples include the CIRCLE (with largest AREA ), and R EULEAUX TRIANGLE (with smallest AREA ) but there are an infinite number. A curve of constant width can be used in a special drillchuck to cut square " HOLES ." A generalization gives solids of constant width. These do not have the same surface AREA for a given width, but their shadows are curves of constant width with the same width! See also DELTA CURVE ,KAKEYA NEEDLE PROBLEM , REULEAUX TRIANGLE References Blaschke, W. "Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts." Math. Ann. 76, 504 /C1/13, 1915. Bogomolny, A. "Shapes of Constant Width." http://www.cut- the-knot.com/do_you_know/cwidth.html. Bo¨hm, J. "Convex Bodies of Constant Width." Ch. 4 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 96 /C1/00, 1986. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 7, 1991. Fischer, G. (Ed.). Plates 98 /C1/02 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 89 and 96, 1986. Gardner, M. "Mathematical Games: Curves of Constant Width, One of which Makes it Possible to Drill Square Holes." Sci. Amer. 208, 148 /C1/56, Feb. 1963. Gardner, M. "Curves of Constant Width." Ch. 18 in The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 212 /C1/21, 1991. Goldberg, M. "Circular-Arc Rotors in Regular Polygons." Amer. Math. Monthly 55, 393 /C1/02, 1948. Kelly, P. Convex Figures. New York: Harcourt Brace, 1995. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, 1957. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 150 /C1/51, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 219 /C1/20, 1991. Yaglom, I. M. and Boltyanski, V. G. Convex Figures. New York: Holt, Rinehart, and Winston, 1961. Curvilinear Coordinates A COORDINATE SYSTEM composed of intersecting surfaces. If the intersections are all at right angles, then the curvilinear coordinates are said to form an ORTHOGONAL COORDINATE SYSTEM . If not, they form a SKEW COORDINATE SYSTEM . A general METRIC gmn has a LINE ELEMENT ds2 /C30gmndu mdu n ; (1) where EINSTEIN SUMMATION is being used. Curvi- linear coordinates are defined as those with a diag- onal METRIC so that gmn /C13 dm n h2 m ; (2) where dm nis the KRONECKER DELTA . Curvilinear coordinates therefore have a simple LINE ELEMENT ds2 /C30 dmn h2 mdumdun /C30h2mdum2 ; (3) which is just the PYTHAGOREAN THEOREM , so the differential VECTOR is dr /C30hm dumˆum ; (4)or dr /C30@r @u1du1 /C27@r @u2du2 /C27@r @u3du3 ; (5) where the SCALE FACTORS are hi /C13@r @uil112l112l112l112l112l112l112l112l112l112 (6) and ˆu i /C13@r @ui ½@r @ui½/C301 hi@r @ui: (7) Equation (5) may therefore be re-expressed as dr /C30h1 du1 ˆu1 /C27h2 du2 ˆu2 /C27h3 du3 ˆu3 : (8) The GRADIENT is grad( f)/C139f /C301 h1@f @u1ˆu1/C271 h2@f @u2ˆu2/C271 h3@f @u3ˆu3;(9) the DIVERGENCE is div(F)/C139 /C215F/C131 h1h2h3 /C2@ @u1(h2h3F1)/C27@ @u2(h3h1F2)/C27@ @u3(h1h2F3)"# ;(10) and the CURL is 9/C29F/C131 h1h2h3h1ˆu1h2ˆu2h3ˆu3 @ @u1@ @u2@ @u3 h1F1h2F2h2F2l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112l112 /C30 1 h2h3@ @u2(h3F3)/C28@ @u3(h2F2)"# ˆu1 /C271 h1h3@ @u3(h1F1)/C28@ @u1(h3F3)"# ˆu2 /C271 h1h2@ @u1(h2F2)/C28@ @u2(h1F1)"# ˆu3: (11) See also ORTHOGONAL COORDINATE SYSTEM ,SKEW COORDINATE SYSTEM References Byerly, W. E. "Orthogonal Curvilinear Coo ¨rdinates." §130 in An Elementary Treatise on Fourier’s Series, and Spheri- cal, Cylindrical, and Ellipsoidal Harmonics, with Appli-cations to Problems in Mathematical Physics. New York: Dover, pp. 238 /C1 /39, 1959. Moon, P. and Spencer, D. E. Foundations of Electrody- namics. Princeton, NJ: Van Nostrand, 1960. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1 /C1/, 1988. Cushion The QUARTIC SURFACE resembling a squashed round cushion on a barroom stool and given by the equation z2x2 /C28z4 /C282zx2 /C272z3 /C27x2 /C28z2 /C28(x2 /C28z)2 /C28y4 /C282x2y2 /C28y2z2 /C272y2z /C27y2 /C300 : See also QUARTIC SURFACE References Nordstrand, T. "Surfaces." http://www.uib.no/people/nfytn/ surfaces.htm. Cusp A cusp is a point on a continuous curve where the tangent vector reverses sign as the curve is traversed. A cusp is a type of DOUBLE POINT . The above plot shows the curve x3 /C28y2 /C300; which has a cusp at the ORIGIN . See also CRUNODE ,D OUBLE CUSP,D OUBLE POINT , ORDINARY DOUBLE POINT ,RAMPHOID CUSP,SALIENT POINT ,SPINODE ,TACNODE References Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 57 /C1/8, 1978.Cusp Catastrophe A CATASTROPHE which can occur for two control factors and one behavior axis. The cusp catastrophe is the universal unfolding of the singularity f(x) /C30x4 and has the equation F(x; u; v) /C30x4 /C27ux2 /C27vx: The equation y /C30x2 =3 also has a cusp catastrophe. See also CATASTROPHE THEORY References Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 28, 1993. Cusp Form A cusp form is a MODULAR FORM for which the coefficient c(0) /C300 in the FOURIER SERIES f( t) /C30X/C12 n/C300c(n)e2 pint (Apostol 1997, p. 114). The only entire cusp form of weight k B 12 is the zero function (Apostol 1997, p. 116). The set of all cusp forms in Mk (all MODULAR FORMS of weight k) is a linear subspace of Mk which is denoted Mk ; 0 : The dimension of Mk ; 0 is 1 for k /C30 12, 16, 18, 20, 22, and 26 (Apostol 1997, p. 119). For a cusp form f /C23 M2k; 0 ; c(n) /C30O(nk) (1) (Apostol 1997, p. 135) or, more precisely, c(n) /C30O(nk /C281 =4/C27 e) (2) for every e>0 (Selberg 1965; Apostol 1997, p. 136). It is conjectured that the /C281=4 in the exponent can be reduced to /C281=2 (Apostol 1997, p. 136). See also MODULAR FORM References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 114 and 116, 1997. Selberg, A. "On the Estimate of Coefficients of Modular Forms." Proc. Sympos. Pure Math. 8,1/C1/5, 1965. Cusp Map The function f(x) /C301 /C282½x½1 =2 for x /C23 [/C281 ; 1]: The INVARIANT DENSITY is r(y) /C301 2(1 /C28y): References Beck, C. and Schlo¨gl, F. Thermodynamics of Chaotic Systems. Cambridge, England: Cambridge University Press, p. 195, 1995. Cusp Point CUSP Cut Given a weighted, UNDIRECTED GRAPH G /C30(V ; E) and a GRAPHICAL PARTITION of V into two sets A and B, the cut of G with respect to A and B is defined as cut(A; B) /C30X i /C23A; j /C23BW(i ; j); where W(i ; j) denotes the weight for the edge con- necting vertices i and j. See also BRANCH CUT,CUT SET References Demmel, J. "CS 267: Lectures 20 and 21, Mar 21, 1996 and Apr 2, 1999. Graph Partitioning, Part 1." http:// www.cs.berkeley.edu/~demmel/cs267/lecture18/lec- ture18.html. Cut Set A set of edges of a GRAPH which, if removed (or "cut"), disconnects the graph (i.e., forms a DISCONNECTED GRAPH ). See also ARTICULATION VERTEX ,D ISCONNECTED GRAPHReferences Skiena, S. "Reconstructing Graphs from Cut-Set Sizes." Info. Proc. Lett. 32, 123 /C1/27, 1989. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Cutpoint ARTICULATION VERTEX Cutting The slicing of a 3-D object by a plane (or more general slice). See also ARCHIMEDES’ HAT-BOX THEOREM ,ARRANGE- MENT ,CAKE CUTTING ,CYLINDER CUTTING ,DIVISION , HADWIGER PROBLEM ,H AM SANDWICH THEOREM , PANCAKE CUTTING ,PIE CUTTING ,SQUARE DIVISION BY LINES,TORUS CUTTING Cut-Vertex ARTICULATION VERTEX CW-Approximation Theorem If X is any SPACE , then there is a CW -COMPLEX Y and a MAP f : Y 0 X inducing ISOMORPHISMS on all HOMOTOPY , HOMOLOGY , and COHOMOLOGY groups. CW-Complex A CW-complex is a homotopy-theoretic generalization of the notion of a SIMPLICIAL COMPLEX . A CW-complex is any SPACE X which can be built by starting off with a discrete collection of points called X0 ; then attach- ing 1-D DISKS D1 to X0 along their boundaries S0 ; writing X1 for the object obtained by attaching the D1 / stoX0 ; then attaching 2-D DISKS D2 to X1 along their boundaries S1 ; writing X2 for the new SPACE , and so on, giving spaces Xn for every n. A CW-complex is any SPACE that has this sort of decomposition into SUBSPACES Xn built up in such a hierarchical fashion (so the Xn/s must exhaust all of X). In particular, Xn may be built from Xn/C281 by attaching infinitely many n-DISKS , and the attaching MAPS Sn/C281 0 Xn /C281 may be any continuous MAPS . The main importance of CW-complexes is that, for the sake of HOMOTOPY , HOMOLOGY , and COHOMOLOGY groups, every SPACE is a CW-complex. This is called the CW -APPROXIMATION THEOREM . Another is WHITE- HEAD’S THEOREM , which says that MAPS between CW- complexes that induce ISOMORPHISMS on all HOMO- TOPY GROUPS are actually HOMOTOPY equivalences. See also COHOMOLOGY ,CW -APPROXIMATION THEO- REM,HOMOLOGY GROUP ,HOMOTOPY GROUP ,SIMPLI- CIAL COMPLEX ,S PACE ,S UBSPACE ,W HITEHEAD’S THEOREM Cycle (Circle) A CIRCLE with an arrow indicating a direction. Cycle (Map) An n-cycle is a finite sequence of points Y0 ; ..., Yn/C281 such that, under a MAP G, Y1 /C30G(Y0) Y2 /C30G(Y1) Yn/C281 /C30G(Yn/C282) Y0 /C30G(Yn/C281) : In other words, it is a periodic trajectory which comes back to the same point after n iterations of the cycle. Every point Yj of the cycle satisfies Yj /C30Gn(Yj) and is therefore a FIXED POINT of the mapping Gn : A fixed point of G is simply a CYCLE of period 1. Cycle (Permutation) A SUBSET of a PERMUTATION whose elements trade places with one another. Permutations cycles are called "orbits" by Comtet (1974, p. 256). For example, in the PERMUTATION GROUP f4 ; 2; 1; 3g;f1; 3; 4g is a 3-cycle (/1 0 3 ; 3 0 4 ; and 4 0 1) and f2 g is a 1-cycle / (2 0 2): There is a great deal of freedom in picking the representation of a cyclic decomposition since (1) the cycles are disjoint and can therefore be specified in any order, and (2) any rotation of a given cycle specifies the same cycle (Skiena 1990, p. 20). There- fore, (431)(2), (314)(2), (143)(2), (2)(431), (2)(314), and (2)(143) all describe the same cycle. The cyclic decomposition of a PERMUTATION can be computed in Mathematica with the function ToCy- cles [p] in the Mathematica add-on package Dis- creteMath‘Permutations‘ (which can be loaded with the command BBDiscreteMath‘ ) and the PERMUTATION corresponding to a cyclic decomposition can be computed withFromCycles [c1, ..., cn] in the Mathematica add-on package DiscreteMath‘Per- mutations‘ (which can be loaded with the command BBDiscreteMath‘ ). According to Vardi (1991), the Mathematica code for ToCycles is one of the most obscure ever written. Every PERMUTATION GROUP on n symbols can be uniquely expressed as a product of disjoint cycles (Skiena 1990, p. 20). A cycle decomposition of a PERMUTATION can be viewed as a CLASS of a PERMUTA- TION GROUP . The number d1(n; k)of k-cycles in a PERMUTATION GROUP of order n is given by d1(n; k) /C30(/C281)n/C28kS1(n; k) /C30½S1(n; k)½; (1) where S1(n; m) are the STIRLING NUMBERS OF THE FIRST KIND . More generally, let dr(n; k) be the number of permutations of n having exactly k cyclesall of which are of length ]r : d2(n; k) are sometimes called the associated STIRLING NUMBERS OF THE FIRST KIND (Comtet 1974, p. 256). The quantities d3(n ; k) appear in a closed-form expression for the coefficients of in STIRLING’S SERIES (Comtet 1974, p. 257 and 267). The following table gives the triangles for dr(n; k) :/ r Sloane /dr(n; k)/ 1 A008275 1; 1, 1; 2, 3, 1; 6, 11, 6, 1; 24, 50, 35, 10, 1; ... 2 A008306 1; 2; 6, 3; 24, 20; 120, 130, 15; 720, 924, 210; ... 3 A050211 2; 6; 24; 120, 40; 720, 420; 5040, 3948; 40320, ... 4 A050212 6; 24; 120; 720; 5040, 1260; 40320, 18144; ... 5 A050213 24; 120; 720; 5040; 40320; 362880, 72576; ... The functions dr(n;k) are given by the RECURRENCE RELATION dr(n;k)/C30(n/C281)dr(n/C281;k) /C27(n/C281)r/C281dr(n/C28r;k/C281); (2) where ( n)kis the FALLING FACTORIAL , combined with the initial conditions dr(n;k)/C300 for n5kr/C281 (3) dr(n;1)/C30(n/C281)! (4) (Riordan 1958, p. 85; Comtet 1974, p. 257). See also GOLOMB- DICKMAN CONSTANT ,PERMUTATION , PERMUTATION GROUP ,S TIRLING NUMBER OF THE FIRST KIND,STIRLING’S SERIES ,SUBSET References Biggs, N. Discrete Mathematics, rev. ed. Oxford, England: Clarendon Press, 1993. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 257, 1974. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Riordan, J. Combinatorial Identities. New York: Wiley, 1958. Skiena, S. "The Cycle Structure of Permutations." §1.2.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 20 /C1/4, 1990. Sloane, N. J. A. Sequences A008275, A008306, A050211, A050212, A050213 in "An On-Line Version of the En- cyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Stanton, D. and White, D. Constructive Combinatorics. New York: Springer-Verlag, 1986. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, p. 223, 1991. Cycle Decomposition CYCLE (PERMUTATION ) Cycle Graph A cycle graph Cnis a graph on n nodes containing a single cycle through all nodes. Cycle graphs can be generated using Cycle [n] in the Mathematica add- on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The CHROMATIC NUMBER of Cn is given by x(Cn) /C303 for n odd 2 for n even :l12) A cycle graph of a GROUP is a GRAPH which shows cycles of a GROUP as well as the connectivity between the cycles. Several examples are shown above. For Z4, the group elements Aisatisfy A4 i /C301; where 1 is the IDENTITY ELEMENT , and two elements satisfy A21 /C30A23 /C301:/ For a CYCLIC GROUP of COMPOSITE ORDER n (e.g., Z4, Z6, Z8), the degenerate subcycles corresponding to factors dividing n are often not shown explicitly since their presence is implied. See also CHAIN (GRAPH ), CHARACTERISTIC FACTOR , CYCLIC GRAPH ,CYCLIC GROUP ,G RAPH CYCLE ,H A- MILTONIAN CYCLE ,SQUARE GRAPH ,TRIANGLE GRAPH , WALKReferences Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 83 /C1/8, 1993. Skiena, S. "Cycles, Stars, and Wheels." §4.2.3 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 144 /C1/47, 1990. Cyclic Graph A GRAPH of n nodes and n edges such that node i is connected to the two adjacent nodes i /C271 and i /C281 (mod n), where the nodes are numbered 0, 1, ..., n /C281:/ See also CYCLE GRAPH ,FOREST ,GRAPH CYCLE ,STAR GRAPH ,W HEEL GRAPH References Balaban, A. T. "Enumeration of Cyclic Graphs." In Chemical Applications of Graph Theory (Ed. A. T. Balaban). Lon- don: Academic Press, pp. 63 /C1/05, 1976. Cyclic Group A cyclic group Zn(also commonly denoted Znor Cn; Shanks 1993, p. 75) of ORDER n is a GROUP defined by the element X (the GENERATOR ) and its n POWERS up to Xn /C30I ; where I is the IDENTITY ELEMENT . Cyclic groups are ABELIAN . There exists a unique cyclic group of every order n ]2; so cyclic groups of the same order are always isomorphic (Scott 1987, p. 34; Shanks 1993, p. 74). Furthermore, subgroups of cyclic groups are cyclic, and all GROUPS of PRIME ORDER are cyclic. In fact, the only SIMPLE ABELIAN GROUPS are the cyclic groups of order n /C30 1ora n a prime (Scott 1987, p. 35). Examples of cyclic groups include Z2 ; Z3 ; Z4 ; and the MODULO MULTIPLICATION GROUPS Mm such that m /C30 2, 4, pn ; or 2pn ; for p an ODD PRIME and n ]1 (Shanks 1993, p. 92). By computing the CHARACTERISTIC FAC- TORS , any A BELIAN GROUP can be expressed as a GROUP DIRECT PRODUCT of cyclic SUBGROUPS , for example, Z2/C156Z4orZ2/C156Z2/C156Z2. See also ABELIAN GROUP ,CHARACTERISTIC FACTOR , FINITE GROUP Z2,FINITE GROUP Z3,FINITE GROUP Z4, FINITE GROUP Z5,FINITE GROUP Z6,M ETACYCLIC GROUP ,M ODULO MULTIPLICATION GROUP ,S IMPLE GROUP References Lomont, J. S. "Cyclic Groups." §3.10.A in Applications of Finite Groups. New York: Dover, p. 78, 1987. Scott, W. R. "Cyclic Groups." §2.4 in Group Theory. New York: Dover, pp. 34 /C1/5, 1987. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993. Cyclic Hexagon A hexagon (not necessarily regular) on whose VER- TICES a CIRCLE may be CIRCUMSCRIBED . Let si /C13Y i(a2 1 ; a22 ; a23 ; a24 ; a25 ; a26) (1) denote the ith-order SYMMETRIC POLYNOMIAL on the six variables consisting of the squares a2 iof the hexagon side lengths ai ; so s1 /C30a2 1 /C27a22 /C27a23 /C27a24 /C27a25 /C27a26 (2) s2 /C30a21a22 /C27a21a23 /C27a21a24 /C27a21a25 /C27a21a26 /C27a22a23 /C27a22a24 /C27a22a25 /C27a22a26 /C27a23a24 /C27a23a25 /C27a23a26 /C27a24a25 /C27a24a26 /C27a25a26 (3) s3 /C30a21a22a23 /C27a21a22a24 /C27a21a22a25 /C27a21a22a26 /C27a22a23a24 /C27a22a23a25 /C27a22a23a26 /C27a23a24a25 /C27a23a24a26 /C27a24a25a26 (4) s4 /C30a21a22a23a24 /C27a21a22a23a25 /C27a21a22a23a26 /C27a21a23a24a25 /C27a21a23a24a26 /C27a21a23a25a26 /C27a21a24a25a26 /C27a22a23a24a25 /C27a22a23a24a26 /C27a22a23a25a26 /C27a22a24a25a26 /C27a23a24a25a26 (5) s5 /C30a21a22a23a24a25 /C27a21a22a23a24a26 /C27a21a22a23a25a26 /C27a21a22a24a25a26 /C27a21a23a24a25a26 /C27a22a23a24a25a26 (6) s6 /C30a21a22a23a24a25a26 : (7) Then let K be the AREA of the hexagon and define u /C3016K2 (8) t2 /C30u /C284s2 /C27 s21 (9) t3 /C308s3 /C27 s1t2 /C2816ffiffiffiffiffis6p(10) t4 /C30t2 2 /C2864s4 /C2764 s1ffiffiffiffiffis6p(11) t5 /C30128s5 /C2732t2ffiffiffiffiffis 6p: (12) The AREA of the hexagon then satisfiesut3 4 /C27t23t24 /C2816t33t5 /C2818ut3t4t5 /C2827u2t25 /C300; (13) or this equation withffiffiffiffiffis6preplaced by /C28ffiffiffiffiffis 6p; a seventh order POLYNOMIAL in u. This is 1=(4u2) times the DISCRIMINANT of the CUBIC EQUATION z3 /C272t3z2 /C28ut4z /C272u2t5 : (14) See also CONCYCLIC ,C YCLIC PENTAGON ,C YCLIC POLYGON ,FUHRMANN’S THEOREM References Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Discr. Comput. Geom. 12, 223 /C1/36, 1994. Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523 /C1/30, 1995. Cyclic Number A number having n /C281 DIGITS which, when MULTI- PLIED by 1, 2, 3, ..., n /C281 ; produces the same digits in a different order. Cyclic numbers are generated by the UNIT FRACTIONS 1=n which have maximal period DECIMAL EXPANSIONS (which means n must be PRIME ). The first few numbers which generate cyclic numbers are 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (Sloane’s A001913). A much larger generator is 17389. It has been conjectured, but not yet proven, that an INFINITE number of cyclic numbers exist. In fact, the FRACTION of PRIMES which generate cyclic numbers seems to be approximately 3/8. See Yates (1973) for a table of PRIME period lengths for PRIMES B1;370;471: When a cyclic number is multiplied by its generator, the result is a string of 9s. This is a special case of MIDY’S THEOREM . 07 /C300.142857 17 /C300.0588235294117647 19 /C300.052631578947368421 23 /C300.0434782608695652173913 29 /C300.0344827586206896551724137931 47 /C300.02127659574468085106382978723404255319- 0.021276595744680851063829787234042553191489- 3617 59 /C300.01694915254237288135593220338983050847- 0.016949152542372881355932203389830508474576- 2711864406779661 61 /C300.01639344262295081967213114754098360655- 0.016393442622950819672131147540983606557377- 049180327868852459 97/C300.01030927835051546391752577319587628865- 0.010309278350515463917525773195876288659793- 81443298969072164948453608247422680412371134- 0206185567 See also DECIMAL EXPANSION ,FULL REPTEND PRIME , MIDY’S THEOREM References Gardner, M. "Cyclic Numbers." Ch. 10 in Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 111 /C1/22, 1979. Guttman, S. "On Cyclic Numbers." Amer. Math. Monthly 44, 159 /C1/66, 1934. Kraitchik, M. "Cyclic Numbers." §3.7 in Mathematical Recreations. New York: W. W. Norton, pp. 75 /C1/6, 1942. Rao, K. S. "A Note on the Recurring Period of the Reciprocal of an Odd Number." Amer. Math. Monthly 62, 484 /C1/87, 1955. Rivera, C. "Problems & Puzzles: Puzzle Period Length of/1 =p/ .-012." http://www.primepuzzles.net/puzzles/ puzz_012.htm. Sloane, N. J. A. Sequences A001913/M4353 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Yates, S. Primes with Given Period Length. Trondheim, Norway: Universitetsforlaget, 1973. Cyclic Pentagon A cyclic pentagon is a not necessarily regular PENTA- GON on whose VERTICES a CIRCLE may be CIRCUM- SCRIBED . Let such a pentagon have edge lengths a1 ; ..., a5 ; and AREA K, and let si /C13Pi(a2 1 ; a22 ; a23 ; a24 ; a25) (1) denote the ith-order SYMMETRIC POLYNOMIAL on the five variables consisting of the squares a2 iof the pentagon side lengths ai ; so s1 /C30a2 1 /C27a22 /C27a23 /C27a24 /C27a25 (2) s2 /C30a21a22 /C27a21a23 /C27a21a24 /C27a21a25 /C27a22a23 /C27a22a24 /C27a22a25 /C27a23a24 /C27a23a25 /C27a24a25 (3) s3 /C30a21a22a23 /C27a21a22a24 /C27a21a22a25 /C27a22a23a24 /C27a22a23a25 /C27a23a24a25 (4) s4 /C30a21a22a23a24 /C27a21a22a23a25 /C27a21a23a24a25 /C27a21a22a24a25 /C27a22a23a24a25 (5) s5 /C30a21a22a23a24a25 : (6) In addition, also define u /C3016K2 (7) t2 /C30u /C284s2 /C27 s21 (8) t3 /C308s3 /C27 s1t2 (9) t4 /C30/C2864 s4 /C27t22 (10) t5 /C30128s5 : (11) Then the AREA of the pentagon satisfies ut34 /C27t23t24 /C2816t33t5 /C2818ut3t4t5 /C2827u2t25 /C300; (12)a seventh order POLYNOMIAL in u (Robbins 1995). This is also 1=(4u2) times the DISCRIMINANT of the CUBIC EQUATION z3 /C272t3z2 /C28ut4z /C272u2t5 (13) (Robbins 1995). See also CONCYCLIC ,CYCLIC HEXAGON ,CYCLIC POLY- GON References Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Discr. Comput. Geom. 12, 223 /C1/36, 1994. Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523 /C1/30, 1995. Cyclic Permutation A PERMUTATION which shifts all elements of a SET by a fixed offset, with the elements shifted off the end inserted back at the beginning. For a SET with elements a0 ; a1 ; ..., an /C281 ; a cyclic permutation of one place to the left would yield a1 ; ..., an/C281 ; a0 ; and a cyclic permutation of one place to the right would yield an /C281 ; a0 ; a1 ; .... The mapping can be written as ai 0 ai/C27k(mod n)for a shift of k places. A shift of k places to the left is implemented in Mathematica asRotateLeft [list, k], while a shift of kplaces to the right is implemen- ted asRotateRight [list,k]. See also PERMUTATION Cyclic Polygon A cyclic polygon is a POLYGON with VERTICES upon which a CIRCLE can be CIRCUMSCRIBED . Since every TRIANGLE has a CIRCUMCIRCLE , every TRIANGLE is cyclic. It is conjectured that for a cyclic polygon of 2m/C271 sides, 16 K2(where Kis the AREA ) satisfies a MONIC POLYNOMIAL of degree Dm;where Dm/C30Xm/C281 k/C300(m/C28k)2m/C271 kl11sl11n (1) /C301 2(2m/C271)2m ml11sl11n /C2822ml12ml121 (2) (Robbins 1995). It is also conjectured that a cyclic polygon with 2 m/C272 sides satisfies one of two POLY- NOMIALS of degree Dm:The first few values of Dmare 1, 7, 38, 187, 874, ... (Sloane’s A000531). For TRIANGLES n/C303/C302/C2151/C271;the POLYNOMIAL is HERON’S FORMULA , which may be written 16K2/C302a2b2/C272a2c2/C272b2c2/C28a4/C28b4/C28c4; (3) and which is of order D1/C301i n1 6 K2:For a CYCLIC QUADRILATERAL , the POLYNOMIAL is B RAHMAGUPTA’S FORMULA , which may be written 16K2 /C30/C28a4 /C272a2b2 /C28b4 /C272a2c2 /C272b2c2 /C28c4 /C278abcd /C272a2d2 /C272b2d2 /C272c2d2 /C28d4 ; (4) which is of order D1 /C301in16 K2 : Robbins (1995) gives the corresponding FORMULAS for the CYCLIC PENTA- GON and CYCLIC HEXAGON . See also CONCYCLIC ,CYCLIC HEXAGON ,CYCLIC PEN- TAGON ,C YCLIC QUADRANGLE ,C YCLIC QUADRILAT- ERAL ,JAPANESE THEOREM References Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Discr. Comput. Geom. 12, 223 /C1/36, 1994. Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523 /C1/30, 1995. Sloane, N. J. A. Sequences A000531 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Cyclic Quadrangle Let A1 ; A2 ; A3 ; and A4 be four POINTS on a CIRCLE , and H1 ; H2 ; H3 ; H4the ORTHOCENTERS of TRIANGLES DA2A3A4 ; etc. If, from the eight POINTS , four with different subscripts are chosen such that three are from one set and the fourth from the other, these POINTS form an ORTHOCENTRIC SYSTEM . There are eight such systems, which are analogous to the six sets of ORTHOCENTRIC SYSTEMS obtained using the feet of the ANGLE BISECTORS , ORTHOCENTER , and VERTICES of a generic TRIANGLE . On the other hand, if all the POINTS are chosen from one set, or two from each set, with all different subscripts, the four POINTS lie on a CIRCLE . There are four pairs of such CIRCLES , and eight POINTS lie by fours on eight equal CIRCLES . The S IMSON LINE ofA4with regard to TRIANGLE DA1A2A3is the same as that of H4with regard to the TRIANGLE DH1A2A3:/ See also ANGLE BISECTOR ,CONCYCLIC ,CYCLIC POLY- GON,C YCLIC QUADRILATERAL ,O RTHOCENTRIC SYS- TEM References Coxeter, H. S. M. and Greitzer, S. L. "Cyclic Quadrangles; Brahmagupta’s Formula." §3.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56 /C1/0, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 251 /C1/53, 1929.Cyclic Quadrilateral AQUADRILATERAL for which a CIRCLE can be circum- scribed so that it touches each VERTEX . The AREA is then given by a special case of B RETSCHNEIDER’S FORMULA . Let the sides have lengths a,b,c, and d, letsbe the SEMIPERIMETER s/C131 2(a/C27b/C27c/C27d); (1) and let Rbe the CIRCUMRADIUS . Then A/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s/C28a)(s/C28b)(s/C28c)(s/C28d)p (2) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ac/C27bd)(ad/C27bc)(ab/C27cd)p 4R: (3) Solving for the CIRCUMRADIUS gives R/C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ac/C27bd)(ad/C27bc)(ab/C27cd) (s/C28a)(s/C28b)(s/C28c)(s/C28d)s : (4) The DIAGONALS of a cyclic quadrilateral have lengths p/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ab/C27cd)(ac/C27bd) ad/C27bcs (5) q/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ac/C27bd)(ad/C27bc) ab/C27cds ; (6) so that pq/C30ac/C27bd:/ In general, there are three essentially distinct cyclic quadrilaterals (modulo ROTATION and REFLECTION ) whose edges are permutations of the lengths a,b,c, and d. Of the six corresponding DIAGONAL lengths, three are distinct. In addition to pand q, there is therefore a "third" DIAGONAL which can be denoted r. It is given by the equation r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ad/C27bc)(ab/C27cd) ac/C27bds : (7) This allows the AREA formula to be written in the particularly beautiful and simple form A/C30pqr 4R: (8) The DIAGONALS are sometimes also denoted p,q, and r. The AREA of a cyclic quadrilateral is the MAXIMUM possible for any QUADRILATERAL with the given side lengths. Also, the opposite ANGLES of a cyclic quad- rilateral sum to pRADIANS (Dunham 1990). There exists a closed BILLIARDS path inside a cyclic quad- rilateral if its CIRCUMCENTER lies inside the quad- rilateral (Wells 1991, p. 11). The INCENTERS of the four triangles composing the cyclic quadrilateral form a RECTANGLE . Furthermore, the sides of the RECTANGLE are PARALLEL to the lines connecting the MID-ARC POINTS between each pair of vertices (left figure above; Fuhrmann 1890, p. 50; Johnson 1929, pp. 254 /C1/55; Wells 1991). If the EX- CENTERS of the triangles constituting the quadrilat- eral are added to the INCENTERS ,a4/C294 rectangular grid is obtained (right figure; Johnson 1929, p. 255;Wells 1991). Consider again the four triangles contained in a cyclic quadrilateral. Amazingly, the CENTROIDS Mi;NINE- POINT CENTERS Ni;and ORTHOCENTERS Hiformed by these triangles are similar to the original quadrilat- eral. In fact, the triangle formed by the ORTHOCEN- TERS is congruent to it (Wells 1991, p. 44). A cyclic quadrilateral with RATIONAL sides a,b,c, and d,DIAGONALS pand q,CIRCUMRADIUS r, and AREA ais given by a/C3025,b/C3033,c/C3039,d/C3065, p/C3060,q/C3052,r/C3065=2;anda/C301344.Letahbo be a QUADRILATERAL such that the angles /C218hab and/C218hob are RIGHT ANGLES , then ahbo is a cyclic quadrilateral (Dunham 1990). This is a COR- OLLARY of the theorem that, in a RIGHT TRIANGLE , the MIDPOINT of the HYPOTENUSE is equidistant from the three VERTICES . Since Mis the MIDPOINT of both RIGHT TRIANGLES DAHB andDBOH ;it is equidistant from all four VERTICES ,s oa CIRCLE centered at M may be drawn through them. This theorem is one ofthe building blocks of Heron’s derivation of H ERON’S FORMULA . An application of B RAHMAGUPTA’S THEOREM gives the pretty result that, for a cyclic quadrilateral withperpendicular diagonals, the distance from the CIR- CUMCENTER Oto a side is half the length of the opposite side, so in the above figure, OMAB/C301 2CD/C30CMCD/C30DMCD; (9) and so on (Honsberger 1995, pp. 37 /C1/8). LetMACandMBDbe the MIDPOINTS of the diagonals of a cyclic quadrilateral ABCD , and let Pbe the intersection of the diagonals. Then the ORTHOCENTER ofTRIANGLE DPMACMBDis the ANTICENTER Tof ABCD (Honsberger 1995, p. 39). Place four equal CIRCLES so that they intersect in a point. The quadrilateral ABCD is then a cyclic quadrilateral (Honsberger 1991). For a CONVEX cyclic quadrilateral Q, consider the set of CONVEX cyclic quadrilaterals Q½½whose sides are PARALLEL to Q. Then the Q½½of maximal AREA is the one whose DIAGONALS are PERPENDICULAR (Gu¨rel 1996). See also BICENTRIC QUADRILATERAL ,BRAHMAGUPTA’S THEOREM ,B RETSCHNEIDER’S FORMULA ,B UTTERFLY THEOREM ,CENTROID (TRIANGLE ), CONCYCLIC ,CYCLIC POLYGON ,CYCLIC QUADRANGLE ,EULER BRICK,HER- ON’S FORMULA ,M ALTITUDE ,M ID-ARC POINTS ,NINE- POINT CENTER ,O RTHOCENTER ,P ONCELET TRANS- VERSE ,PTOLEMY’S THEOREM ,Q UADRILATERAL ,TAN- GENTIAL QUADRILATERAL References Andreescu, T. and Gelca, R. "Cyclic Quadrilaterals." §1.2 in Mathematical Olympiad Challenges. Boston, MA: Bir- kha¨user, pp. 6 /C1/, 2000. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 121, 1990. Fuhrmann, W. Synthetische Beweise Planimetrischer Sa¨tze. Berlin, 1890. Gu¨rel, E. Solution to Problem 1472. "Maximal Area of Quadrilaterals." Math. Mag. 69, 149, 1996. Harris, J. W. and Stocker, H. "Quadrilateral of Chords." §3.6.7 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 85, 1998. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 36 /C1/7, 1991. Honsberger, R. "Cyclic Quadrilaterals." §4.2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 35 /C1/0, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 182 /C1/94, 1929. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 11 and 43 /C1/4, 1991. Cyclic Redundancy Check A sophisticated CHECKSUM (often abbreviated CRC), which is based on the algebra of polynomials over the integers (mod 2). It is substantially more reliable in detecting transmission errors, and is one common error-checking protocol used in modems. The CRC is a form of HASH FUNCTION .To compare large data blocks using the CRC, first precalculate the CRCs for each block. Two blocks can then be rapidly compared by seeing if their CRCs are equal, saving a great deal of calculation time in most cases. The method is not infallible since for an N-bit checksum, 1 =2N of random blocks will have the same checksum for inequivalent data blocks. However, if N is large, the probability that two inequivalent blocks have the same CRC can be made very small. See also CHECKSUM ,ERROR- CORRECTING CODE,HASH FUNCTION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Cyclic Redundancy and Other Checksums." Ch. 20.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 888 /C1/95, 1992. Cyclic Triple The 3-node TOURNAMENT (and DIRECTED GRAPH ) illustrated above (Harary 1994, p. 205). See also TOURNAMENT ,TRANSITIVE TRIPLE References Harary, F. "Tournaments." Graph Theory. Reading, MA: Addison-Wesley, 1994. Cyclically Symmetric Plane Partition A PLANE PARTITION whose solid Young diagram is invariant under the rotation which cyclically per- mutes the x-, y-, and z-axes. MACDONALD’S PLANE PARTITION CONJECTURE gives a formula for the num- ber of cyclically symmetric plane partitions (CSPPs) of a given integer whose YOUNG DIAGRAMS fit inside an n /C29n /C29n box. Macdonald gave a product repre- sentation for the power series whose coefficients qn were the number of such partitions of n. See also MACDONALD’S PLANE PARTITION CONJEC- TURE ,MAGOG TRIANGLE ,PLANE PARTITION References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637/C1/46. Cyclic-Inscriptable Quadrilateral BICENTRIC QUADRILATERAL Cyclid CYCLIDE Cyclide A pair of focal conics which are the envelopes of two one-parameter families of spheres, sometimes also called a CYCLID . The cyclide is a QUARTIC SURFACE , and the lines of curvature on a cyclide are all straight lines or circular arcs (Pinkall 1986). The STANDARD TORI and their INVERSIONS in an INVERSION SPHERE S centered at a point x0 and of RADIUS r, given by I(x0 ; r) /C30x0 /C27x /C28 x0r2 ½x /C28 x0 ½2 ; are both cyclides (Pinkall 1986). Illustrated above are RING CYCLIDES , HORN CYCLIDES , and SPINDLE CY- CLIDES . The figures on the right correspond to x0 lying on the torus itself, and are called the PARABOLIC RING CYCLIDE , PARABOLIC HORN CYCLIDE , and PARA- BOLIC SPINDLE CYCLIDE , respectively. See also CYCLIDIC COORDINATES ,H ORN CYCLIDE , INVERSION ,INVERSION SPHERE ,P ARABOLIC HORN CYCLIDE ,PARABOLIC RING CYCLIDE ,RING CYCLIDE , SPINDLE CYCLIDE ,STANDARD TORI References Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 273, 1959. Eisenhart, L. P. "Cyclides of Dupin." §133 in A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, pp. 312 /C1/14, 1960. Fischer, G. (Ed.). Plates 71 /C1/7in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 66 /C1/2, 1986. JavaView. "Classic Surfaces from Differential Geometry: Dupin Cycloid." http://www-sfb288.math.tu-berlin.de/vgp/ javaview/demo/surface/common/PaSurface_DupinCy- cloid.html. Marsan, A. "Cyclides." http://www.engin.umich.edu/dept/ meam/deslab/cadcam/Cyclides/cyclide.html. Nordstrand, T. "Dupin Cyclide." http://www.uib.no/people/ nfytn/dupintxt.htm. Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28 /C1/0, 1986.Salmon, G. Analytic Geometry of Three Dimensions. New York: Chelsea, p. 527, 1979. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 62, 1991. Cyclidic Coordinates A general system of fourth-order CURVILINEAR CO- ORDINATES based on the CYCLIDE in which LAPLACE’S EQUATION is SEPARABLE (either simply separable or R-separable). Boˆcher (1894) treated all possible sys- tems of this class (Moon and Spencer 1988, p. 49). See also BICYCLIDE COORDINATES ,CAP-CYCLIDE CO- ORDINATES ,D ISK-CYCLIDE COORDINATES ,O RTHOGO- NAL COORDINATE SYSTEM References Boˆcher, M. U¨ber die Reihenentwicklungen der Potentialthe- orie. Leipzig, Germany: Teubner, 1894. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics,with Applications to Problems in Mathematical Physics.New York: Dover, p. 273, 1959. Casey, J. "On Cyclides and Sphero-Quartics." Philos. Trans. Roy. Soc. London 161, 585/C1 /21, 1871. Darboux, G. "Remarques sur la the ´orie des surfaces ortho- gonales." Comptes Rendus 59, 240/C1/42, 1864. Darboux, G. "Sur l’application des me ´thodes de la physique mathe ´matique a `l’e´tude de corps termine ´s par des cyclides." Comptes Rendus 83, 1037 /C1/039, 1864. Klein, F. U¨ber lineare Differentialgleichungen der zweiter Ordnung; Vorlesungen gehalten im Sommersemester1894. Go¨ttingen, Germany: 1894. Maxwell, J. C. "On the Cyclide." Quart. J. Pure Appl. Math. 9, 111/C1 /26, 1868. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations,and Their Solutions, 2nd ed. New York: Springer-Verlag, 1988. Wangerin. Preisschriften der Jablanowski’schen Ge- sellschaft, No. 18, 1875 /C1 /876. Wangerin. Crelle’s J. 82, 1875 /C1/876. Wangerin. Berliner Monatsber. 1878. Cycloid The cycloid is the locus of a point on the rim of a CIRCLE ofRADIUS arolling along a straight LINE.I t was studied and named by Galileo in 1599. Galileo attempted to find the AREA by weighing pieces of metal cut into the shape of the cycloid. Torricelli,Fermat, and Descartes all found the AREA . The cycloid was also studied by Roberval in 1634, Wren in 1658,Huygens in 1673, and Johann Bernoulli in 1696.Roberval and Wren found the ARC LENGTH (MacTutor Archive). Gear teeth were also made out of cycloids, as first proposed by Desargues in the 1630s (Cundy and Rollett 1989). In 1696, Johann Bernoulli challenged other mathe- maticians to find the curve which solves the BRACHIS- TOCHRONE PROBLEM , knowing the solution to be a cycloid. Leibniz, Newton, Jakob Bernoulli and L’Hos-pital all solved Bernoulli’s challenge. The cycloid alsosolves the TAUTOCHRONE PROBLEM , as alluded to in the following passage from Moby Dick : "[The try-pot] is also a place for profound mathematical meditation.It was in the left-hand try-pot of the Pequod , with the soapstone diligently circling round me, that I wasfirst indirectly struck by the remarkable fact, that ingeometry all bodies gliding along a cycloid, mysoapstone, for example, will descend from any point in precisely the same time" (Melville 1851). Because of the frequency with which it provoked quarrelsamong mathematicians in the 17th century, the cycloid became known as the "Helen of Geometers" (Boyer 1968, p. 389). The cycloid is the CATACAUSTIC of a CIRCLE for a RADIANT POINT on the circumference, as shown by Jakob and Johann Bernoulli in 1692. The CAUSTIC of the cycloid when the rays are parallel to the Y-AXIS is a cycloid with twice as many arches. The RADIAL CURVE of a CYCLOID is a CIRCLE . The EVOLUTE and INVOLUTE of a cycloid are identical cycloids. If the cycloid has a CUSP at the ORIGIN , its equation in CARTESIAN COORDINATES is x/C30acos/C281a/C28y a ! /C14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ay/C28y2p : (1) In parametric form, this becomes x/C30a(t/C28sint) (2) y/C30a(1/C28cost): (3) If the cycloid is upside-down with a cusp at (0 ;a);(2) and (3) become x/C302asin/C281y 2a ! /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ay/C28y 2p (4) or x/C30a(t/C27sint) (5) y/C30a(1/C28cost) (6) (sign of sin tflipped for x). The DERIVATIVES of the parametric representation (2) and (3) are x?/C30a(1/C28cost) (7) y?/C30asint (8)dy dx/C30y? x?/C30asint a(1/C28cost)/C30sint 1/C28cost/C302 sin(1 2t)cos(12t) 2 sin2(1 2t) /C30cot(12t) (9) The squares of the derivatives are x?2/C30a2(1/C282 cos t/C27cos2t) (10) y?2/C30a2sin2t; (11) so the ARC LENGTH of a single cycle is L/C30gds/C30g2p 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2/C27y?2q dt /C30ag2p 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C282 cos t/C27cos2t)/C27sin2tq dt /C30affiffiffi 2pg2p 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28costp dt/C302ag2p 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28cost 2s dt /C302ag2p 0sin(1 2t)l112l112l112l112l112l112dt: (12) Now let u/C13t=2s o du/C30dt=2:Then L/C304agp 0sinud u/C304a[/C28cosu]p 0 /C30/C284a[(/C281)/C281]/C308a: (13) The ARC LENGTH ,CURVATURE , and TANGENTIAL ANGLE are s/C308asin2(1 4t) (14) k/C30/C2814acsc(12t) (15) f/C30/C281 2at: (16) The AREA under a single cycle is A/C30g2p 0yd x/C30a2g2p 0(1/C28cosf)(1/C28cosf)df /C30a2g2p 0(1/C28cosf)2df /C30a2g2p 0(1/C282 cos f/C27cos2f)df /C30a2g2p 0f1/C282 cos f/C271 2[1/C27cos(2 f)]gdf /C30a2g2p 0[32/C282 cos f/C2712cos(2 f)]df /C30a2[3 2f/C282 sin f/C2714sin(2f)]2p 0 /C30a2322p/C303pa2: (17) The NORMAL is ˆT /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28 2 cos tp 1 /C28cos t sin tl12ml121 : (18) See also BRACHISTOCHRONE PROBLEM ,CURTATE CY- CLOID ,CYCLIDE ,CYCLOID EVOLUTE ,CYCLOID INVO- LUTE ,EPICYCLOID ,HYPOCYCLOID ,PROLATE CYCLOID , TAUTOCHRONE PROBLEM ,TROCHOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 216, 1987. Bogomolny, A. "Cycloids." http://www.cut-the-knot.com/ pythagoras/cycloids.html. Boyer, C. B. A History of Mathematics. New York: Wiley, 1968. Cundy, H. and Rollett, A. "Cycloid." §5.1.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 215 /C1/16, 1989. Gardner, M. "The Cycloid: Helen of Geometers." Ch. 13 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 127 /C1/34, 1984. Gray, A. "Cycloids." §3.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 50 /C1/2, 1997. Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 325, 1998. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 197, 1972. Lockwood, E. H. "The Cycloid." Ch. 9 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 80 /C1/9, 1967. MacTutor History of Mathematics Archive. "Cycloid." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Cy-cloid.html. Melville, H. "The Tryworks." Ch. 96 in Moby Dick. New York: Bantam, 1981. Originally published in 1851. Muterspaugh, J.; Driver, T.; and Dick, J. E. "The Cycloid and Tautochronism." http://php.indiana.edu/~jedick/pro- ject/intro.html. Pappas, T. "The Cycloid--The Helen of Geometry." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 6/C1 /, 1989. Phillips, J. P. "Brachistochrone, Tautochrone, Cycloid--Ap- ple of Discord." Math. Teacher 60, 506/C1/08, 1967. Proctor, R. A. A Treatise on the Cycloid. London: Longmans, Green, 1878. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 147, 1999. Wagon, S. "Rolling Circles." Ch. 2 in Mathematica in Action. New York: W. H. Freeman, pp. 39 /C1/6, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 44 /C1/7, 1991. Whitman, E. A. "Some Historical Notes on the Cycloid." Amer. Math. Monthly 50, 309/C1/15, 1948. Yates, R. C. "Cycloid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 65 /C1/0, 1952. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 291 /C1/92, 1995.Cycloid Evolute The EVOLUTE of the CYCLOID x(t)/C30a(t/C28sint) y(t)/C30a(1/C28cost) is given by x(t)/C30a(t/C27sint) y(t)/C30a(cost/C281): As can be seen in the above figure, the EVOLUTE is simply a shifted copy of the original CYCLOID , so the CYCLOID is its own EVOLUTE . Cycloid Involute The INVOLUTE of the CYCLOID x(t)/C30a(t/C28sint) y(t)/C30a(1/C28cost) is given by x(t)/C30a(t/C27sint) y(t)/C30a(3/C27cost): As can be seen in the above figure, the INVOLUTE is simply a shifted copy of the original CYCLOID , so the CYCLOID is its own INVOLUTE ! Cycloid Radial Curve The RADIAL CURVE of the CYCLOID is the CIRCLE x/C30x0/C272asinf y /C30/C282a /C27y0 /C272a cos f: Cyclomatic Number CIRCUIT RANK Cyclotomic CYCLOTOMIC POLYNOMIAL Cyclotomic Equation The equation xp /C301 ; where solutions zk /C30e2 pik=p are the ROOTS OF UNITY sometimes called DE MOIVRE NUMBERS . Gauss showed that the cyclotomic equation can be reduced to solving a series of QUADRATIC EQUATIONS whenever p is a FERMAT PRIME . Wantzel (1836) subsequently showed that this condition is not only SUFFICIENT , but also NECESSARY . An "irreducible" cyclotomic equation is an expression OF THE FORM xp /C28 1 x/C281/C30xp/C281/C27xp/C282/C27.../C271/C300; where pisPRIME . Its ROOTS zisatisfy zijj/C301:/ See also CYCLOTOMIC POLYNOMIAL , DE MOIVRE NUMBER ,POLYGON ,PRIMITIVE ROOT OF UNITY References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 99 /C1/00, 1996. Scott, C. A. "The Binomial Equation xp/C281/C300:/"Amer. J. Math. 8, 261/C1/64, 1886. Wantzel, M. L. "Recherches sur les moyens de reconnaı ˆtre si un Proble `me de Ge ´ome´trie peut se re ´soudre avec la re `gle et le compas." J. Math. pures appliq. 1, 366/C1/72, 1836. Cyclotomic Factorization zp/C28yp/C30(z/C28y)(z/C28zy)/C1/C1/C1(z/C28zp/C281y); where z/C13e2pi=p(aDEMOIVRE NUMBER ) and pis a PRIME . Cyclotomic Field The smallest field containing m/C23Z]1 with zaPRIME ROOT OF UNITY is denoted Rm(z); xp/C27yp/C30Yp k/C301(x/C27zky): Specific cases are R3/C30Q(ffiffiffiffiffiffi /C283p ) R4/C30Q(ffiffiffiffiffiffi /C281p )R6/C30Q(ffiffiffiffiffiffi /C283p ); where Qdenotes a QUADRATIC FIELD . References Koch, H. "Cyclotomic Fields." §6.4 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 180 /C1/84, 2000. Weiss, E. Algebraic Number Theory. New York: Dover, 1998. Cyclotomic Integer A number OF THE FORM a0/C27a1z/C27.../C27ap/C281zp/C281; where z/C13e2pi=p is a DEMOIVRE NUMBER and pis a PRIME NUMBER . Unique factorizations of cyclotomic INTEGERS fail for p/C2123. Cyclotomic Invariant Letpbe an ODD PRIME andFnthe CYCLOTOMIC FIELD ofpn/C271/thROOTS of unity over the rational FIELD . Now letpe(n)be the POWER ofpwhich divides the CLASS NUMBER hnofFn:Then there exist INTEGERS mp;lp] 0 and npsuch that e(n)/C30mppn/C27lpn/C27np for all sufficiently large n. For REGULAR PRIMES , mp/C30lp/C30np/C300:/ References Johnson, W. "Irregular Primes and Cyclotomic Invariants." Math. Comput. 29, 113/C1/20, 1975. Cyclotomic Number DEMOIVRE NUMBER ,SYLVESTER CYCLOTOMIC NUM- BER Cyclotomic Polynomial A polynomial given by Fn(x)/C30Y ?n k/C301(x/C28zk); (1) where zkare the ROOTS OF UNITY inCgiven by zk/C13e2pik=n(2) andkruns over integers RELATIVELY PRIME ton. The prime may be dropped if the product is instead taken over PRIMITIVE ROOTS OF UNITY , so that Fn(x)/C30Yn k/C301primitive zk(x/C28zk): (3) The notation Fn(x) is also frequently encountered. Dickson et al. (1923) and Apostol (1975) give exten- sive bibliographies for cyclotomic polynomials. /Fn(x)i sa n INTEGER POLYNOMIAL and an IRREDUCIBLE POLYNOMIAL with DEGREE f(n);where f(n) is the TOTIENT FUNCTION . Cyclotomic polynomials are re- turned by the Mathematica command Cycloto- mic[n,x]. The roots of cyclotomic polynomials lie on the UNIT CIRCLE in the COMPLEX PLANE , as illustrated above for the first few cyclotomic polynomials. The first few cyclotomic POLYNOMIALS are F1(x)/C30x/C281 F2(x)/C30x/C271 F3(x)/C30x2/C27x/C271 F4(x)/C30x2/C271 F5(x)/C30x4/C27x3/C27x2/C27x/C271 F6(x)/C30x2/C28x/C271 F7(x)/C30x6/C27x5/C27x4/C27x3/C27x2/C27x/C271 F8(x)/C30x4/C271 F9(x)/C30x6/C27x3/C271 F10(x)/C30x4/C28x3/C27x2/C28x/C271: Ifpis an ODD PRIME , then Fp(x)/C30xp/C281 x/C281/C30xp/C281/C27xp/C282/C27.../C27x/C271 (4) F2p(x)/C30x2p/C281 xp/C281x/C281 x2/C281/C30xp/C281/C28xp/C282/C27.../C28x/C271 (5)F4p(x)/C30x4p/C281 x2p/C281x2/C281 x4/C281 /C30x2p/C282/C28x2p/C284/C27.../C28x2/C271 (6) (Riesel 1994, p. 306). Similarly, for pagain an ODD PRIME , xp/C281/C30F1(x)Fp(x) (7) x2p/C281/C30F1(x)F2(x)Fp(x)F2p(x) (8) x4p/C281/C30F1(x)F4(x)F2(x)Fp(x)F2p(x)F4p(x): (9) For the first few remaining values of n, x/C281/C30F1(x) (10) x2/C281/C30F1(x)F2(x) (11) x4/C281/C30F1(x)F2(x)F4(x) (12) x8/C281/C30F1(x)F2(x)F4(x)F8(x) (13) x9/C281/C30F1(x)F3(x)F9(x) (14) x15/C281/C30F1(x)F3(x)F5(x)F15(x) (15) x16/C281/C30F1(x)F2(x)F4(x)F8(x)F16(x) (16) x18/C281/C30F1(x)F2(x)F3(x)6(x)F9(x)F18(x) (17) (Riesel 1994, p. 307). ForpaPRIME relatively prime to n, Fnp(x)/C30Fn(xp) Fn(x); (18) but if p½n; Fnp(x)/C30Fn(xp) (19) (Nagell 1951, p. 160).An explicit equation for F n(x) for SQUAREFREE nis given by Fn(x)/C30Xf(n) j/C300anjzf(n)/C28j; (20) where Anjis calculated using the RECURRENCE RELA- TION anj/C30 /C28m(n) jXj/C281 m/C300anmm(GCD( n;j/C28m))f(GCD( n;j/C28m));(21) with an0/C301;where mnis the M O¨BIUS FUNCTION and GCD( m;n) is the GREATEST COMMON DENOMINATOR ofmandn. The POLYNOMIAL xn/C281can be factored as xn/C281/C30Y d½nFd(x); (22) where Fd(x)i sa CYCLOTOMIC POLYNOMIAL . Further- more, xn/C271/C30x2n/C281 xn/C281/C30Q d½2nFd(x)Q d½nFd(x): (23) The COEFFICIENTS of the inverse of the cyclotomic POLYNOMIAL 1 1/C27x/C27x2/C301/C28x/C27x3/C28x4/C27x6/C28x7/C27x9/C28x10/C27... /C13X/C12 n/C300cnxn(24) can also be computed from cn/C301/C2821 3(n/C272)jk /C2713(n/C271)jk /C2713njk (25) /C301/C2831 3(n/C272)jk /C27nbc (26) /C302ffiffiffi 3psin[2 3p(n/C271)]; (27) where /C28x/C29is the FLOOR FUNCTION . The LOGARITHM of the cyclotomic polynomial Fn(x)/C30Y djn(1/C28xn=d)m(d)(28) is the M O¨BIUS INVERSION FORMULA (Vardi 1991, p. 225). ForpPRIME , Fp(x)/C30Xp/C281 k/C300xk; (29) i.e., the coefficients are all 1. The first cyclotomic polynomial to have a coefficient other than 91 and 0 isF105(x);which has coefficients of /C282 for x7andx41: This is true because 105 is the first number to havethree distinct ODD PRIME factors, i.e., Td(McClellan and Rader 1979, Schroeder 1997). The smallest values of nfor which Fn(x) has one or more coeffi- cients91,92,93, ... are 0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465,10465, 11305, ... (Sloane’s A013594). It appears to be true that, for m;n>1;ifF m(x)/C27 Fn(x) factors, then the factors contain a cyclotomic polynomial. For example, F7(x)/C27F22(x)/C30(x2/C271)(x8/C28x7/C272x4/C272) /C30F4(x)(x8/C28x7/C272x4/C272): (30) This observation has been checked up to m;n/C30150 (C. Nicol). If mand nare prime, then Cm/C27Cnis irreducible.Migotti (1883) showed that COEFFICIENTS ofFpq(x) for pandqdistinct PRIMES can be only 0, 91. Lam andLeung (1996) considered Fpq(x)/C13Xpq/C281 k/C300akxk(31) forp, q PRIME . Write the TOTIENT FUNCTION as f(pq)/C30(p/C281)(q/C281)/C30rp/C27sq (32) and let 05k5(p/C281)(q/C281); (33) then 1.ak/C301IFFk/C30ip/C27jqfor some i/C23[0;r] and j/C23[0;s];/ 2.ak/C30/C281IFFk/C27pq/C30ip/C27jpfori/C23[r/C271;q/C281] andj/C23[s/C271;p/C281];/ 3. otherwise ak/C300:/ The number of terms having ak/C301i s( r/C271)(s/C271); and the number of terms having ak/C30/C281i s( p/C28s/C28 1)(q/C28r/C281):Furthermore, assume q/C21p, then the middle COEFFICIENT ofFpqis (/C281)r:/ Resultants of cyclotomic polynomials have been com- puted by Lehmer (1930), Diederichsen (1940), andApostol (1970). It is known that r(F k(x);Fn(x))/C301i f (m;n)/C301;i.e.,mandnare relatively prime (Apostol 1975). Apostol (1975) showed that for positive inte-gers mand nand arbitrary nonzero complex num- bers aandb, r(F m(ax);Fn(bx)) /C30bf(m)f(n)Y d½nFm=dad bd !"#m(n=d)f(m)=f(m=d) ; (34) where d/C30GCD( m;d) is the GREATEST COMMON DIVI- SORofmandd,f(n) is the TOTIENT FUNCTION ,m(n)i s the M O¨BIUS FUNCTION , and the product is over the divisors of n.I fmandnare distinct primes pandq, then (34) simplifies to r(Fq(ax);Fp(bx)) /C30apq/C28bpq ap/C28bpa/C28b aq/C28bqfora"b a(p/C281)(q/C281)fora/C30b:8 < :(35) The following table gives the RESULTANTS r(Fk(x);Fn(x)) (Sloane’s A054372). /k_n/1234567 10 2203310 42210 551110 613411077111110 See also A URIFEUILLEAN FACTORIZATION ,G AUSS’S CYCLOTOMIC FORMULA ,LUCAS’S THEOREM ,M O¨ BIUS INVERSION FORMULA ,P RIMITIVE ROOT OF UNITY, ROOT OF UNITY References Apostol, T. M. "Resultants of Cyclotomic Polynomials." Proc. Amer. Math. Soc. 24, 457/C1/62, 1970. Apostol, T. M. "The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx):/"Math. Comput. 29,1/C1/, 1975. Beiter, M. "The Midterm Coefficient of the Cyclotomic Polynomial Fpq(x):/"Amer. Math. Monthly 71, 769/C1/70, 1964. Beiter, M. "Magnitude of the Coefficients of the Cyclotomic Polynomial Fpq:/"Amer. Math. Monthly 75, 370/C1/72, 1968. Bloom, D. M. "On the Coefficients of the Cyclotomic Poly- nomials." Amer. Math. Monthly 75, 372/C1/77, 1968. Brent, R. P. "On Computing Factors of Cyclotomic Polyno- mials." Math. Comput. 61, 131/C1/49, 1993. Carlitz, L. "The Number of Terms in the Cyclotomic Polynomial Fpq(x):/"Amer. Math. Monthly 73, 979/C1/81, 1966. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996. de Bruijn, N. G. "On the Factorization of Cyclic Groups." Indag. Math. 15, 370/C1/77, 1953. Dickson, L. E.; Mitchell, H. H.; Vandiver, H. S.; and Wahlin, G. E. Algebraic Numbers. Bull Nat. Res. Council, Vol. 5, Part 3, No. 28. Washington, DC: National Acad. Sci., 1923. Diederichsen, F.-E. "U ¨ber die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer A ¨quivalenz." Abh. Math. Sem. Hanisches Univ. 13, 357/C1/12, 1940. Lam, T. Y. and Leung, K. H. "On the Cyclotomic Polynomial Fpq(X):/"Amer. Math. Monthly 103, 562/C1/64, 1996. Lehmer, E. "On the Magnitude of the Coefficients of the Cyclotomic Polynomial." Bull. Amer. Math. Soc. 42, 389/C1/ 92, 1936. Lehmer, E. "On the Magnitude of Coefficients of the Cyclotomic Polynomials." Bull. Amer. Math. Soc. 42, 389/C1/92, 1936. McClellan, J. H. and Rader, C. Number Theory in Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1979. Migotti, A. "Zur Theorie der Kreisteilungsgleichung." Sitz- ber. Math.-Naturwiss. Classe der Kaiser. Akad. der Wiss., Wien 87,7/C1/4, 1883. Nagell, T. "The Cyclotomic Polynomials" and "The Prime Divisors of the Cyclotomic Polynomial." §46 and 48 in Introduction to Number Theory. New York: Wiley, pp. 158 /C1/60 and 164 /C1/68, 1951. Riesel, H. "The Cyclotomic Polynomials" in Appendix 6. Prime Numbers and Computer Methods for Factorization,2nd ed. Boston, MA: Birkha ¨user, pp. 305 /C1 /08, 1994. Schroeder, M. R. Number Theory in Science and Commu- nication, with Applications in Cryptography, Physics,Digital Information, Computing, and Self-Similarity, 3rded.New York: Springer-Verlag, p. 245, 1997. Se´roul, R. "Cyclotomic Polynomials." §10.8 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 265 /C1 /69, 2000. Sloane, N. J. A. Sequences A013594 and A054372 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 8 and 224 /C1 /25, 1991. Cylinder In common usage, the term "cylinder" refers to a SOLID of circular CROSS SECTION in which the centers of the CIRCLES all lie on a single LINE (i.e., a right circular cylinder). In mathematical usage, "cylinder" is commonly taken to refer to only the lateral sides of this solid, excluding the top and bottom caps. If a plane inclined with respect to the caps intersects acylinder, it does so in an ELLIPSE . The cylinder was extensively studied by Archimedes in his two-volumework On the Sphere and Cylinder in ca. 225 BC. A cylinder is called a right cylinder if it is "straight" inthe sense that its CROSS SECTIONS lie directly on top of each other; otherwise, the cylinder is called oblique. The lateral surface of a cylinder of height hand RADIUS rcan be described parametrically by x/C30rcosu (1) y/C30rsinu (2) z/C30z; (3) forz/C23[0;h] and u/C23[0;2p):These are the basis for CYLINDRICAL COORDINATES . The SURFACE AREA (of the sides) and VOLUME of the cylinder of height hand RADIUS rare S/C302prh (4) V/C30pr2h: (5) Therefore, if top and bottom caps are added, thevolume-to-surface area ratio for a cylindrical solid is V S /C30pr2h 2prh /C27 2 pr2 /C301 21 r /C271 h !/C281 ; (6) which is related to the HARMONIC MEAN of the radius r and height h. The fact that Vsphere Vcircumscribed cylinder /C28 Vsphere/C304 3 2 /C284 3/C304 3 2 3/C302 (7) was known to Archimedes (Steinhaus 1983, p. 223). Using the parametrization x(u; v) /C30a cos v (8) y(u ; v) /C30a sin v (9) z(u; v) /C30u (10) gives coefficients of the FIRST FUNDAMENTAL FORM E /C301 (11) F /C300 (12) G /C30a2 ; (13) the coefficients of the SECOND FUNDAMENTAL FORM e /C300 (14) f /C300 (15) g /C30a (16) AREA ELEMENT dS /C30aduffl dv; (17) GAUSSIAN CURVATURE K /C300; (18) and MEAN CURVATURE H /C301 2a : (19) See also ARCHIMEDES’ HAT-BOX THEOREM ,BARREL , CONE,CYLINDER DISSECTION ,CYLINDER- SPHERE IN- TERSECTION ,C YLINDRICAL SEGMENT ,C YLINDRICAL WEDGE ,ELLIPTIC CYLINDER ,GENERALIZED CYLINDER , SPHERE ,STEINMETZ SOLID ,VIVIANI’S CURVE References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 129, 1987. Harris, J. W. and Stocker, H. "Cylinder." §4.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 102 /C1/04, 1998. Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 in Geometry and the Imagination. New York: Chelsea, pp. 7 /C1/1, 1999.JavaView. "Classic Surfaces from Differential Geometry: Cylinder." http://www-sfb288.math.tu-berlin.de/vgp/java- view/demo/surface/common/PaSurface_Cylinder.html. Kern, W. F. and Bland, J. R. "Circular Cylinder" and "Right Circular Cylinder." §16 /C1/7in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 36 /C1/2, 1948. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Cylinder Cutting The maximum number of pieces into which a cylinder can be divided by n oblique cuts is given by f(n) /C30n /C271 3l11sl11n /C27n /C271 /C301 6(n /C272)(n /C273); wherea bl1ml11 is a BINOMIAL COEFFICIENT . This problem is sometimes also called cake cutting or pie cutting, and has the same solution as SPACE DIVISION BY PLANES . For n /C301, 2, ... cuts, the maximum number of pieces is 2, 4, 8, 15, 26, 42, ... (Sloane’s A000125). See also CIRCLE DIVISION BY LINES,CUBE DIVISION BY PLANES ,HAM SANDWICH THEOREM ,PANCAKE THEO- REM,SPACE DIVISION BY PLANES ,TORUS CUTTING References Bogomolny, A. "Can You Cut a Cake into 8 Pieces with Three Movements." http://www.cut-the-knot.com/do_you_know/ cake.html. Sloane, N. J. A. Sequences A000125/M1100 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Cylinder Dissection A cylinder can be dissected into unequal squares, with nine squares required at a minimum. Trivial squarings can be constructed by taking rectangle dissections and matching edges, but there are two nontrivial nine-square tilings (Stewart 1997). See also MO¨ BIUS STRIP DISSECTION ,PERFECT SQUARE DISSECTION ,TORUS DISSECTION References Stewart, I. "Squaring the Square." Sci. Amer. 277,9 4/C1/6, July 1997. Cylinder Function The cylinder function is defined as C(x; y) /C131 forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p 5a 0 forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p > a :l12) (1) The BESSEL FUNCTIONS are sometimes also called cylinder functions. To find the FOURIER TRANSFORM of the cylinder function, let kx /C30k cos a (2) ky /C30k sin a (3) x /C30r cos u (4) y /C30r sin u: (5) Then F(k; a) /C30F(C(x; y)) /C30g2 p 0ga 0ei(k cos a r cos u/C27k sin a r sin u)rdrd u /C30g2 p 0ga 0eikr cos( u /C28a)rdrd u: (6) Let b /C30 u /C28 a; so db /C30du: Then F(k; a) /C30g2p /C28 a /C28 aga 0eikr cos brdrd u /C30g2p 0ga 0eikr cos brdrd u /C302pga 0J0(kr)rdr ; (7) where J0(x) is a zeroth order BESSEL FUNCTION OF THE FIRST KIND . Let u /C13kr; so du /C30kdr ; thenF(k ; a) /C302 p k2 gka 0J0(u)udu/C302p k2 [uJ1(u)]ka 0 /C302 pa kJ1(ka) /C302pa2J1(ka) ka: (8) As defined by Watson (1966), a "cylinder function" is any function which satisfies the RECURRENCE RELA- TIONS Cn/C281(z)/C27Cn/C271(z)/C302n zCn(z) (9) Cn/C281(z)/C28Cn/C271(z)/C302C?n(z): (10) This class of functions can be expressed in terms of BESSEL FUNCTIONS . See also BESSEL FUNCTION OF THE FIRST KIND, CYLINDER FUNCTION ,CYLINDRICAL FUNCTION ,HEMI- SPHERICAL FUNCTION References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Cylinder-Cylinder Intersection STEINMETZ SOLID Cylinder-Plane Intersection CYLINDRICAL SECTION Cylinder-Sphere Intersection The curve formed by the intersection of a CYLINDER and a SPHERE is known as V IVIANI’S CURVE . The problem of finding the lateral SURFACE AREA of a CYLINDER of radius rinternally tangent to a SPHERE of radius Rwas given in a S ANGAKU PROBLEM from 1825. The easiest way to determine the solution is to solve the simultaneous equations x2 /C27y2 /C27z2 /C30R2 (1) y2 /C27[z /C28(R /C28r)]2 /C30r2 (2) for x and y, x /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(R /C28r)(R /C28z)p (3) y /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(R /C28z)(2r /C28R /C28z)p : (4) These give the PARAMETRIC EQUATIONS for VIVIANI’S CURVE in this case (left figure). The SURFACE AREA can the be found by constructing a series of curved segments (right figure). The arc length element around the surface of the cylinder at a height z is given by ds /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27dy dz !2vuutdz /C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (R /C28 z)(2r /C28 R /C27 z)p : (5) The SURFACE AREA of one quarter of the surface is then S1 =4 /C30g x(z) ds /C30gR R/C282rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(R /C28r)(R /C28z)p rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (R /C28 z)(2r /C28 R /C28 z)p dz /C30gR R/C282rrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(R /C28 r) 2r /C28 R /C27 zs dz; (6) where some care is needed treating the lower limit, S1 =4 /C30 lim r?0r/C284r[ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(R /C28r)p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(R /C28r)(r /C28r ?)p ] /C304r3 =2ffiffiffiffiffiffiffiffiffiffiffiffi R /C28rp : (7) The total SURFACE AREA is then S /C304S1 =4 /C3016r3=2ffiffiffiffiffiffiffiffiffiffiffiffiR /C28rp (8) a result obtained in a more roundabout geometric arguments by Rothman (1998). (Note that the answer printed in the original Rothman article was incorrect;the corrected answer has been posted on the Internet version of the article.) See also C YLINDER ,SPHERE ,VIVIANI’S CURVE References Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/1, May 1998. Cylindrical Algebraic Decomposition This entry contributed by ADAM STRZEBONSKI Define a cell in R1 as an open interval or a point. A cell in Rk /C271 then has one of two forms, f(x; y):x /C23 C ; and f(x) By Bg(x) g or f(x; y):x /C23 C ; and y /C30f(x) g; where x /C30fx1 ; ...; xk g; C is a cell in Rk ; f and g are either (1) continuous functions on C such that for some polynomials F and G, F(x; f(x)) /C300 and G(x; g(x)) /C300 ; or (2) 9/C12 ; and f(x) Bg(x) for all x /C23 C :/ A cylindrical algebraic decomposition of S ƒRn is a representation of S as a finite union of disjoint cells. Let F be finite set of polynomials in n variables. A cylindrical algebraic decomposition of S ƒRn is said to be F-invariant if each of the polynomials from F has a constant sign on each cell of the decomposition. The cylindrical algebraic decomposition (CAD) algo- rithm, given a finite set F of polynomials in n variables, computes an F-invariant cylindrical alge- braic decomposition of Rn : Given a logical combina- tion of polynomial equations and inequalities in n real unknowns, one can use the CAD algorithm to find a cylindrical algebraic decomposition of its solution set. For example, the decomposition of x2 /C27y2 /C27z2 B1 is given by /C281 Bx B1 1 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p By Bffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28x2p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2 /C28y2p Bz Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x2/C28y2p :8 < : Mathematica 4.0 contains the function Cylindri- calAlgebraicDecomposition which performs cy- lindrical algebraic decompositions. Although the process is algorithmic, it becomes computationally infeasible for complicated inequalities. See also CYLINDRICAL PARTS ,GENERIC CYLINDRICAL ALGEBRAIC DECOMPOSITION ,Q UANTIFIER ELIMINA- TION ,TARSKI’S THEOREM References Caviness, B. F. and Johnson, J. R. (Eds.). Quantifier Elim- ination and Cylindrical Algebraic Decomposition. New York: Springer-Verlag, 1998. Collins, G. E. "Quantifier Elimination for the Elementary Theory of Real Closed Fields by Cylindrical Algebraic Decomposition." Lect. Notes Comput. Sci. 33, 134/C1/83, 1975. Collins, G. E. "Quantifier Elimination by Cylindrical Alge- braic Decomposition--Twenty Years of Progress." In Quan- tifier Elimination and Cylindrical AlgebraicDecomposition (Ed. B. F. Caviness and J. R. Johnson). New York: Springer-Verlag, pp. 8 /C1 /3, 1998. Collins, G. E. and Hong, H. "Partial Cylindrical Algebraic Decomposition for Quantifier Elimination." J. Symb. Comput. 12, 299/C1/28, 1991. Dolzmann, A. and Sturm, T. "Simplification of Quantifier- Free Formulae Over Ordered Fields." J. Symb. Comput. 24, 209/C1/31, 1997. Faugere, J. C.; Gianni, P.; Lazard, D.; and Mora, T. "Efficient Computation of Zero-Dimensional GroebnerBases by Change of Ordering." J. Symb. Comput. 16, 329/C1 /44, 1993. Hong, H. "An Improvement of the Projection Operator in Cylindrical Algebraic Decomposition." In ISSAC ’90: Proceedings of the International Symposium on Symbolicand Algebraic Computation, August 20 /C1 /4, 1990, Tokyo, Japan (Ed. S. Watanabe and M. Nagata). New York: ACM Press, pp. 261 /C1/64, 1990. Loos, R. and Weispfenning, V. "Applying Lattice Quantifier Elimination." Comput. J. 36, 450/C1/61, 1993. McCallum, S. "Solving Polynomial Strict Inequalities Using Cylindrical Algebraic Decomposition." Comput. J. 36, 432/C1/38, 1993. McCallum, S. "An Improved Projection for Cylindrical Algebraic Decomposition of Three Dimensional Space." J. Symb. Comput. 5, 141/C1/61, 1988. McCallum, S. "An Improved Projection for Cylindrical Algebraic Decomposition." In Quantifier Elimination and Cylindrical Algebraic Decomposition (Ed. B. F. Caviness and J. R. Johnson). New York: Springer-Verlag, pp. 242 /C1/ 68, 1998. Strzebonski, A. "An Algorithm for Systems of Strong Poly- nomial Inequalities." Mathematica J. 4,7 4/C1/7, 1994. Strzebonski, A. "A Real Polynomial Decision Algorithm Using Arbitrary-Precision Floating Point Arithmetic." Reliable Comput. 5, 337/C1/46, 1999. Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525/C1/41, 2000. Cylindrical Coordinates Cylindrical coordinates are a generalization of 2-D POLAR COORDINATES to 3-D by superposing a height (z) axis. Unfortunately, there are a number of different notations used for the other two coordinates. Either rorris used to refer to the radial coordinate and either foruto the azimuthal coordinates. Arfken (1985), for instance, uses ( r;f;z);whileBeyer (1987) uses ( r;u;z):In this work, the NOTA- TION (r;u;z) is used. r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p (1) u/C30tan/C281y x ! (2) z/C30z; (3) where r/C23[0;/C12);u/C23[0;2p);andz/C23(/C28/C12;/C12):In terms ofx,y, and z x/C30rcosu (4) y/C30rsinu (5) z/C30z: (6) Morse and Feshbach (1953) define the cylindrical coordinates by x/C30j1j2 (7) y/C30j1ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28j2 2q (8) z/C30j3; (9) where j1/C30randj2/C30cosu:The METRIC elements of the cylindrical coordinates are grr/C301 (10) guu/C30r2(11) gzz/C301; (12) so the SCALE FACTORS are gr/C301 (13) gu/C30r (14) gz/C301: (15) The LINE ELEMENT is ds/C30drˆr/C27rduˆu/C27dzˆz; (16) and the VOLUME ELEMENT is dV/C30rd rd udz: (17) The J ACOBIAN is @(x;y;z) @(r;u;z)l112l112l112l112l112l112l112l112l112l112/C30r: (18) AC ARTESIAN VECTOR is given in CYLINDRICAL COOR- DINATES by r/C30rcosu rsinu z2 435: (19) To find the UNIT VECTORS , ˆr/C13dr dr dr drl112l112l112l112l112l112l112l112l112l112/C30cosu sinu 02 435 (20) ˆu/C13 dr du dr dul112l112l112l112l112l112l112l112l112l112/C30/C28sinu cosu 02 435 (21) ˆz/C13 dr dz dr dzl112l112l112l112l112l112l112l112l112l112/C300 012 435: (22) Derivatives of unit VECTORS with respect to the coordinates are @ˆr @r/C300 (23) @ˆr @u/C30/C28sinu cosu 02435/C30ˆu (24) @ˆr @z/C300 (25) @ˆu @r/C300 (26) @ˆu @u/C30/C28cosu /C28sinu 02435/C30/C28ˆr (27) @ˆu @z/C300 (28) @ˆz @r/C300 (29) @ˆz @u/C300 (30) @ˆz @z/C300: (31) The GRADIENT of a VECTOR FIELD in cylindrical coordinates is given by 9/C13ˆr@ @r/C27ˆu1 r@ @u/C27ˆz@ @z; (32) so the GRADIENT components become9rˆr/C300 (33) 9uˆr/C301 rˆu (34) 9zˆr/C300 (35) 9rˆu/C300 (36) 9uˆu/C30/C281 rˆr (37) 9zˆu/C300 (38) 9rˆz/C300 (39) 9uˆz/C300 (40) 9zˆz/C300: (41) Now, since the CONNECTION COEFFICIENTS are defined by Gi jk/C30ˆxi/C215(9kˆxj); (42) Gr/C30000 0/C281 r0 0002 6643 775(43) Gu/C3001 r0 000 0002 6643 775(44) Gz/C30000 000 0002 435; (45) the COVARIANT DERIVATIVES , given by Aj;k/C301 gkk@Aj @xk/C28Gi jkAi; (46) are Ar;r/C30@Ar @r/C28GirrAi/C30@Ar @r(47) Ar;u/C301 r@Ar @u/C28GiruAi/C301 r@Au @r/C28GuruAu /C301 r@Ar @u/C28Au r(48) Ar;z/C30@Ar @z/C28GirzAi/C30@Ar @z(49) Au;r/C30@Au @rGi urAi/C30@Au @r(50) Au;u/C301 r@Au @u/C28Gi uuAi/C301 r@Au @u/C28GruuAr /C301 r@Au @u/C27Ar r(51) Au;z/C30@Au @z/C28Gi uzAi/C30@Au @z(52) Az;r/C30@Az @r/C28Gi zrAi/C30@Az @r(53) Az;u/C30@Az @u/C28GizuAi/C301 r@Az @u(54) Az;z/C30@Az @z/C28GizzAi/C30@Az @z: (55) CROSS PRODUCTS of the coordinate axes are ˆr/C29ˆz/C30/C28 ˆu (56) ˆu/C29ˆz/C30ˆr (57) ˆr/C29ˆu/C30ˆz: (58) The COMMUTATION COEFFICIENTS are given by cm ab /C0em/C30[ /C0ea; /C0eb]/C309a /C0eb/C289b /C0ea; (59) But [ˆr;ˆr]/C30[ˆu;ˆu]/C30[ˆf;ˆf]/C300; (60) soca rr/C30cauu/C30caff/C300;where a/C30r;u;f:Also [ˆr;ˆu]/C30/C28[ˆu;ˆr]/C309rˆu/C289uˆr/C300/C281 rˆu/C30/C281 rˆu;(61) socuru/C30/C28cuur/C30/C281 r;crru/C30cf ru/C300:Finally, [ˆr;ˆf]/C30[ˆu;ˆf]/C300: (62) Summarizing, cr/C30000 0000002 435 (63) c u/C300/C281 r0 1 r00 0002 6666643 777775(64) c f/C30000 0000002 435: (65) Time DERIVATIVES of the VECTOR are ˙r/C30cosu˙r/C28rsinu˙u sinu˙r/C27rcosu˙u ˙z2435/C30˙rˆr/C27r˙uˆu/C27˙zˆz (66)¨r/C30 /C28sinu˙r˙u/C27cosu¨r/C28sinu˙r˙u/C28rcosu˙u 2/C28rsinu¨u cosu˙r˙u/C27sinu¨r/C27cosu˙r˙u/C28rsinu˙u2/C27rcosu¨u ¨z2435 /C30/C282 sin u˙r˙u/C27cosu¨r/C28rcosu˙u 2/C28rsinu¨u 2 cos u˙r˙u/C27sinu¨r/C28rsinu˙u2/C27rcosu¨u ¨z2 435 /C30(¨r/C28r˙u 2)ˆr/C27(2˙r˙u/C27r¨u)ˆu/C27˙zˆz: (67) SPEED is given by v/C13½˙r½/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˙r2/C27r2˙u2/C27˙z2p : (68) Time derivatives of the UNIT VECTORS are ˙ˆr/C30/C28sinu˙u cosu˙u 02 435/C30˙uˆu (69) ˙ˆu/C30/C28cosu˙u /C28sinu˙u 02 435/C30/C28 ˙uˆr (70) ˙ˆz/C300 002 435/C300: (71) C ROSS PRODUCTS of the axes are ˆr/C29ˆz/C30/C28 ˆu (72) ˆu/C29ˆz/C30ˆr (73) ˆr/C29ˆu/C30ˆz: (74) The CONVECTIVE DERIVATIVE is D˙r Dt/C13@ @t/C27˙r/C2159 ! ˙r/C30@˙r @t/C27˙r/C2159˙r: (75) To rewrite this, use the identity 9(A /C215B)/C30A/C29(9/C29B)/C27B/C29(9/C29A)/C27(A /C2159)B /C27(B /C2159)A (76) and set A/C30B, to obtain 9(A /C215A)/C302A/C29(9/C29A)/C272(A /C2159)A; (77) so (A /C2159)A/C30(1 2A2)/C28A/C29(9/C29A): (78) Then D˙r Dt/C30¨r/C279(1 2˙r2)/C28˙r/C29(9/C29˙r) /C30¨r/C27(9/C29˙r)/C29˙r/C279(12˙r2): (79) The CURL in the above expression gives 9/C29˙r/C301 r@ @r(r2˙u)ˆz/C302˙uˆz; (80) so /C28˙r /C29( 9/C29˙r) /C30/C282 ˙u(˙rˆr /C29ˆz /C27r ˙u ˆu /C29ˆz) /C30/C282 ˙u(/C28˙r ˆu /C27r ˙uˆr) /C302˙r ˙u ˆu /C282r ˙u2 ˆr: (81) We expect the gradient term to vanish since SPEED does not depend on position. Check this using the identity 9(f2) /C302f 9f ; 9(1 2˙r2) /C3012 9(˙r2 /C27r2 ˙u2 /C27 ˙z2) /C30 ˙r 9˙r /C27r ˙u9(r ˙u) /C27 ˙z 9˙z : (82) Examining this term by term, ˙r 9˙r /C30 ˙r@ @t9r /C30 ˙r@ @tˆr /C30 ˙r˙ˆr /C30 ˙r ˙u ˆu (83) r ˙u 9(r ˙u) /C30r ˙u r@ @t9u /C27 ˙u9r"# /C30r ˙u r@ @t1 rˆu ! /C27 ˙uˆr"# /C30r ˙u r /C281 r2˙r ˆu /C271 r˙ˆu ! /C27 ˙uˆr"# /C30/C28 ˙u˙r ˆu /C27r ˙u(/C28˙uˆr) /C27r ˙u2 ˆr /C30/C28 ˙u˙r ˆu (84) ˙z 9˙z /C30 ˙z@ @t9z /C30 ˙z@ @tˆz /C30 ˙z˙ˆz /C300 ; (85) so, as expected, 9(1 2˙r2) /C300 : (86) We have already computed , so combining all three pieces gives D˙r Dt /C30(¨r /C28r ˙u2 /C282r ˙u2)ˆr /C27(2˙r ˙u /C272˙r ˙u /C27r ¨u) ˆu /C27 ¨zˆz /C30(¨r /C283r ˙u2)ˆr /C27(4˙r ˙u /C27r ¨u) ˆu /C27 ¨zˆz : (87) The DIVERGENCE is 9 /C215 A /C30Ar ;r /C30Ar;r /C27( Gr rrAt /C27GrurAu /C27GrzrAz) /C27A u ;u /C27( Gu ruAr /C27GuuuAu /C27G uzuAz) /C27Az ;z /C27( Gz rzAr /C27GzuzAu /C27GzzzAz) /C30Ar ;r /C27Au; u /C27Az;z /C27(0 /C270 /C270) /C271 r /C270 /C270 ! /C27(0 /C270 /C270) /C301 gr@ @rAr /C271 gu@ @ uAu /C271 gz@ @zAz /C271 rAr /C30@ @r /C271 r ! Ar /C271 r@ @ uAu /C27@ @zAz ; (88) or, in VECTOR notation 9 /C215 F /C301 r@ @r(rFr) /C271 r@Fu @ u/C27@Fz @z: (89)The CROSS PRODUCT is 9/C29F /C301 r@Fz @ u/C28@Fu @z ! ˆr /C27@Fr @z/C28@Fz @r ! ˆu /C271 r@ @r (rFu) /C28@Fr @ u"# ˆz : (90) The scalar LAPLACIAN is 92f /C131 r@ @rr@f @r ! /C271 r2@2f @ u2 /C27@2f @z2 /C30@2f @r2 /C271 r@f @r /C271 r2@2f @ u2 /C27@2f @z2 : (91) The vector LAPLACIAN is 92v /C30@2vr @r2 /C271 r2@2vr @ f2 /C27@2vr z2/C271 r@vr @r/C282 r2@vf @ f/C28vr r2 @2 @r2 /C271 r2@2vf @ f2 /C27@2vf @z2 /C271 r@vf @r/C272 r2@vr @f/C28vf r2 @2vz @r2/C271 r2@2vz @f2/C27@2vz @z2/C271 r@vz @r2 6666666643 777777775: (92) The H ELMHOLTZ DIFFERENTIAL EQUATION is separable in cylindrical coordinates and has STA¨CKEL DETERMI- NANT S/C301 (for r,u;z)o rS/C301=(1/C28j2 2) (for Morse and Feshbach’s j1;j2;j3):/ See also ELLIPTIC CYLINDRICAL COORDINATES ,HELM- HOLTZ DIFFERENTIAL EQUATION– CIRCULAR CYLINDRI- CAL COORDINATES ,POLAR COORDINATES ,SPHERICAL COORDINATES References Arfken, G. "Circular Cylindrical Coordinates." §2.4 in Math- ematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 95 /C1/01, 1985. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 212, 1987. Moon, P. and Spencer, D. E. "Circular-Cylinder Coordinates (r;c;z):/" Table 1.02 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 12 /C1/7, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 657, 1953. Cylindrical Equal-Area Projection The MAP PROJECTION having transformation equa- tions x /C30( l /C28 l0)cos fs (1) y /C30sin f sec fs (2) for the normal aspect, where l is the LONGITUDE , l0 is the standard LONGITUDE (horizontal center of the projection), f is the LATITUDE , and fs is the so-called "standard latitude." The inverse transformation equations for the normal aspect are f /C30sin/C281(y cos fs) (3) l /C30x sec fs /C27 l0 : (4) Special cases of cylindrical equal-area projections are summarized in the following table (Maling 1992). / cs/ MAP PROJECTION 08 LAMBERT CYLINDRICAL EQUAL-AREA PROJECTION 308 BEHRMANN CYLINDRICAL EQUAL-AREA PROJECTION 37.383 8 TRISTAN EDWARDS PROJECTION 44.138 8 PETERS PROJECTION 458 GALL ORTHOGRAPHIC PROJECTION 508 BALTHASART PROJECTION An oblique form of the cylindrical equal-area projec- tion is given by the equations lp /C30 tan/C281cos f1 sin f2 cos l1 /C28 sin f1 cos f2 cos l2 sin f1 cos f2 sinl2 /C28 cos f1 sin f2 sin l1 ! (5) fp /C30tan/C281 /C28cos(lp /C28 l1) tan f1"# ; (6)and the inverse FORMULAS are f /C30sin/C281(y sin fp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28y2p cos fp sin x) (7) l /C30 l0 /C27tan/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 y2p sin fp sin x /C28 y cos fpffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28 y2p cos x ! : (8) A transverse form of the cylindrical equal-area projection is given by the equations x/C30cosfsin(l/C28l0) (9) y/C30tan/C281 tanf cos(l/C28l0)"# /C28f0; (10) and the inverse FORMULAS are f/C30sin/C281[ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p sin(y/C27f0)] (11) l/C30l0/C27tan/C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p cos(y/C27f0)"# : (12) See also BALTHASART PROJECTION ,BEHRMANN CY- LINDRICAL EQUAL- AREA PROJECTION ,C YLINDRICAL EQUIDISTANT PROJECTION ,EQUAL- AREA PROJECTION , GALL ORTHOGRAPHIC PROJECTION ,LAMBERT CYLIND- RICAL EQUAL- AREA PROJECTION ,PETERS PROJECTION TRISTAN EDWARDS PROJECTION References Maling, D. H. Coordinate Systems and Map Projections, 2nd ed, rev. Woburn, MA: Butterworth-Heinemann, 1993. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 76 /C1/5, 1987. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 221 /C1/22, 1999. Cylindrical Equidistant Projection The MAP PROJECTION having transformation equa- tions x /C30( l /C28 l0)cos f1 (1) y /C30 f; (2) and the inverse FORMULAS are f /C30y (3) l /C30 l0 /C27x sec f1 ; (4) The following table gives special cases of the cylind- rical equidistant projection. /f1/ projection name 08 EQUIRECTANGULAR PROJECTION /37 /C1430?/ MILLER EQUIDISTANT PROJECTION 43 8 MILLER EQUIDISTANT PROJECTION 45 8 GALL ISOGRAPHIC PROJECTION /50 /C1428?/ MILLER EQUIDISTANT PROJECTION See also CYLINDRICAL EQUAL- AREA PROJECTION , EQUIDISTANT PROJECTION ,E QUIRECTANGULAR PRO- JECTION ,G ALL ISOGRAPHIC PROJECTION ,M ILLER EQUIDISTANT PROJECTION References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 90 /C1/1, 1987. Snyder, J. P. Flattening the Earth: Two Thousand Years of Map Projections. Chicago, IL: University of Chicago Press, 1993. Cylindrical Equirectangular Projection CYLINDRICAL EQUIDISTANT PROJECTIONCylindrical Function Rm(x; y) /C13J ?m(x)Y ?m(y) /C28 J ?m(y)Y ?m(x) Jm(x)Y ?m(y) /C28 J ?m(y)Ym(x) Sm(x; y) /C13J ?m(x)Ym(y) /C28 Jm(y)Y ?m(x) Jm(x)Ym(y) /C28 Jm(y)Ym(x) : See also CYLINDER FUNCTION ,HEMISPHERICAL FUNC- TION Cylindrical Harmonics BESSEL FUNCTION OF THE FIRST KIND Cylindrical Hoof CYLINDRICAL WEDGE Cylindrical Parts The cylindrical parts of a system of real algebraic equations and inequalities in variables fx1 ;...; xn g are the terms f1 5x1 5g1 f2(x1) 5x2 5g2(x1) n fn(x1 ; x2 ; ...; xn) 5xn 5gn(x1 ; ... ; xn/C281) ; where ‘/5/’ is one of B;5; or /C30; and fi and gi are 9/C12 or algebraic expressions in variables fx1 ; ...; xi /C281 g that are real-valued for all (i /C281)/-tuples of real numbers fa1 ; ...; ai /C281 g satisfying f1 5a1 5g1 f2(a1) 5a2 5g2(a2) n fi/C281(a1 ; ...; ai/C282) 5ai/C281 5gi/C281(a1 ; ...; ai/C282) : The CONJUNCTION of a finite number of disjoint cylindrical parts is called a CYLINDRICAL ALGEBRAIC DECOMPOSITION . See also CYLINDRICAL ALGEBRAIC DECOMPOSITION References Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525/C1/41, 2000. Cylindrical Projection A cylindrical projection of points on a unit sphere centered at O consists of extending the line OS for each point S until it intersects a cylinder tangent to the sphere at its equator at a corresponding point C. If the sphere is tangent to the cylinder at longitude l0 ; then a point on the sphere with latitude f and longitude l is mapped to a point on the cylinder with height tan f:/ Unwrapping and flattening out the cylinder then gives the Cartesian coordinates x /C30 l /C28 l0 (1) y /C30tan f : (2) The cylindrical projection of the Earth is illustrated above. This form of the projection, however, is seldom used in practice, and the term "cylindrical projection" is used instead to refer to any projection in which lines of longitude are mapped to equally spaced parallel lines and lines of latitude (parallels) are mapped to parallel lines with arbitrary mathematically spaced separations (Snyder 1987, p. 5). For example, the common MERCATOR PROJECTION uses the complicated transformation y /C30ln[tan(1 4 p /C2712 f)] (3) instead of tan f in order to achieve certain desirable properties in the projection. Craig (1882) used the term "cylindric" instead of "cylindrical" (Lee 1944), but this convention did notcatch on. See also B EHRMANN CYLINDRICAL EQUAL- AREA PRO- JECTION ,CYLINDRICAL EQUAL- AREA PROJECTION ,CY- LINDRICAL EQUIDISTANT PROJECTION ,G ALL ORTHOGRAPHIC PROJECTION ,MERCATOR PROJECTION , MILLER CYLINDRICAL PROJECTION ,PETERS PROJEC- TION ,PSEUDOCYLINDRICAL PROJECTION References Craig, T. A Treatise on Projections. Washington, DC: U. S. Government Printing Office, 1882. Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190/C1/00, 1944. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987. Cylindrical Section The intersection of a PLANE with a right circular CYLINDER is a CIRCLE (if the plane is at a right angle to the axis), an ELLIPSE , or, if the plane is parallel to the axis, a single line (if the plane is tangent to the cylinder), pair of parallel lines bounding an infiniterectangle (if the plane cuts the cylinder), or no intersection at all (if the plane missed the cylinder entirely; Hilbert and Cohn-Vossen 1999, pp. 7 /C1 /). The volume of the cylindrical section can be obtainedinstantly by noting that two such sections can be fitted together to form a cylinder of radius Rand height h1 /C27h2 ; so the volume of the original wedge is half that of the cylinder of height h1 /C27h2 : The volume can be found directly through integration by noting that the height in polar and Cartesian coordinates is given by h(r ; u) /C30h1 /C271 21 /C27r Rcos u ! (h2 /C28h1) (1) h(x; y) /C30h1 /C27121 /C27x R ! (h2 /C28h1); (2) so V /C30gR 0g2 p 0gh(r; u) 0rdrd u dz (3) /C30gR /C28Rgffiffiffiffiffiffiffiffiffiffi R2 /C28x2p /C28ffiffiffiffiffiffiffiffiffiffi R2 /C28x2pgh(x; y) 0dx dy dz ; (4) giving (1). Similarly, the volume-weighted coordi- nates are given by xhi/C301 8 pR3(h2 /C28h1) (5) yhi/C300 (6) zhi/C301 32 pR2(5h2 1 /C276h1h2 /C275h22); (7) so the centroids are given by ¯x /C30xhi V/C30R(h2 /C28 h1) 4(h1 /C27 h2) (8) ¯y /C30yhi V/C300 (9) ¯z /C30zhi V/C305h2 1 /C27 6h1h2 /C27 5h22 16(h1 /C27 h2); (10) (cf. the strange parameterization used by Harris and Stocker 1998, p. 103). See also CONIC SECTION ,C YLINDER ,C YLINDRICAL SEGMENT ,CYLINDRICAL WEDGE ,ELLIPSE References Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 in Geometry and the Imagination. New York: Chelsea, pp. 7 /C1/1, 1999.Cylindrical Segment The solid portion of a CYLINDER below a cutting PLANE which is oriented PARALLEL to the CYLINDER ’s axis of symmetry (i.e., a portion of a horizontal cylindrical tank which is partially filled with fluid). The solid cut from a circular CYLINDER by a tilted PLANE which does not cut the base (sometimes called a truncated cylinder) has VOLUME V /C301 2 pR2(h1 /C27h2) ; (1) lateral SURFACE AREA SL /C30 pR(h1 /C27h2) ; (2) and top SURFACE AREA ST /C30 pRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C271 4(h2 /C28h1)2q (3) (Harris and Stocker 1998, p. 103). For a CYLINDER of RADIUS r and length L, the VOLUME V(L; r ; h) of the cylindrical segment is given by multiplying the AREA of a circular SEGMENT of height h by L, V(L ; r ; h) /C30Lr2 cos /C281r/C28h r ! /C28(r/C28h)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2rh/C28h2p"# ; plotted above. Note that the above equation gives V(h/C300)/C300;V(h/C30r)/C30pr2L=2;and V(h/C302r)/C30pr2L; as it must. See also CYLINDRICAL WEDGE ,SECTOR ,SEGMENT , SPHERICAL SEGMENT Cylindrical Surface GENERALIZED CYLINDER Cylindrical Wedge A wedge is cut from a CYLINDER by slicing with a plane that intersects the base of the cylinder. The VOLUME of a cylindrical wedge can be found by noting that the plane cutting the cylinder passes through the three points illustrated above (with c/C30a/C28b);so the three-point form of the plane gives the equation /C28hx/C27bz/C27(a/C28b)h/C300: (1) Solving for zgives z/C30h(x/C28a/C27b) b: (2) The volume is therefore V/C302gffiffiffiffiffiffiffiffiffiffi a2/C28x2p 0ga a/C28bh(x/C27b/C28a) bdx dy (3) /C30h 6b2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2a/C28b)bp (3a2/C282ab/C27b2)/C283pa2(a/C28b)h /C276a2(a/C28b)tan/C281 a/C28bffiffiffiffiffiffiffiffiffiffiffiffiffi (2a/C28b)bpl11)l117 /C138; (4) and the lateral SURFACE AREA SL/C302h b /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2a/C28b)bp /C28a(a/C28b)cot/C281 a/C28bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2a/C28b)bp ! "# ; (5) (apparently given incorrectly by Harris and Stocker 1998, p. 104). A special case of the cylindrical wedge, also called a cylindrical hoof, is a wedge passing through a DIAMETER of the base (so that a/C30b). Let the height of the wedge be hand the radius of the CYLINDER from which it is cut r. Then plugging the points(0;/C28r;0);(0;r;0);and ( r;0;h) into the 3-point equation for a PLANE gives the equation for the plane as hx/C28rz/C300: (6) combining with the equation of the CIRCLE which describes the curved part remaining of the cylinder (and writing t/C30xthen gives the PARAMETRIC EQUA- TIONS of the "tongue" of the wedge as x/C30t (7) y/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28t2p (8) z/C30ht r(9) fort/C23[0;r]:To examine the form of the tongue, it needs to be rotated into a convenient plane. This can be accomplished by first rotating the plane of the curve by 90 8about the X-AXIS using the ROTATION MATRIX Rx(90/C14) and then by the ANGLE u/C30tan/C281h r ! (10) above the Z-AXIS . The transformed plane now rests in thexz-plane and has PARAMETRIC EQUATIONS x/C30tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C27r2p r(11) z/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28t2p (12) and is shown below. The length of the tongue (measured down its middle) is obtained by plugging t/C30rinto the above equation forx, which becomes L/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C27r2p (13) (and which follows immediately from the P YTHAGOR- EAN THEOREM ). The VOLUME of the wedge is given by V/C302 3r2h (14) and the lateral SURFACE AREA by SL/C302rh: (15) While the centroid of the general cylindrical wedge is complicated for a"b;for the cylindrical hoof, the centroid is given by ¯x /C30ga a /C28bgffiffiffiffiffiffiffiffiffiffi a2 /C28x2p /C28ffiffiffiffiffiffiffiffiffiffi a2 /C28x2pgh(b/C28a /C27x)=b 0x dz dy dz ; (16) giving xhi/C303 16 pr (17) yhi/C300 (18) zhi/C303 32 ph: (19) See also CONICAL WEDGE ,C YLINDRICAL SECTION , CYLINDRICAL SEGMENT ,W EDGE References Harris, J. W. and Stocker, H. "Obliquely Cut Circular Cylinder" and "Segment of a Cylinder." §4.6.3 /C1/.6.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 103 /C1/04, 1998.Kern, W. F. and Bland, J. R. "Truncated Prism (or Cylin- der)." §31 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 81 /C1/3 and 127, 1948. Cylindroid PLU¨ CKER’S CONOID C* The RIEMANN SPHERE C /C31/C30C @f/C12g; also called the EXTENDED COMPLEX PLANE . The notation ˆC is some- times also used (Krantz 1999, p. 82). The notation C /C31 also stands for C /C28f0 g; the punctu- red plane, which is both a L IE GROUP and an A BELIAN VARIETY . See also C, COMPLEX NUMBER ,C OMPLEX PLANE , EXTENDED COMPLEX PLANE ,Q,R,R IEMANN SPHERE , Z References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 82, 1999. D d’Alembert Ratio Test RATIO TEST d’Alembert’s Equation The ORDINARY DIFFERENTIAL EQUATION y /C30xf(y ?) /C27g(y?) ; where y /C13dy=dx and f and g are given functions. This equation is sometimes also known as LAGRANGE’S EQUATION (Zwillinger 1997). See also LAGRANGE’S EQUATION References Ince, E. L. Ordinary Differential Equations. New York: Dover, pp. 38 /C1/9, 1956. Murphy, G. M. Ordinary Differential Equations and Their Solution. Princeton, NJ: Van Nostrand, pp. 65 /C1/6, 1960. Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Brookline, MA: Math. Sci. Press, pp. 217 /C1/18, 1950. Zwillinger, D. "Lagrange’s Equation." §II.A.69 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 120 and 265 /C1/68, 1997. d’Alembert’s Solution The method of d’Alembert provides a solution to the one-dimensional WAVE EQUATION @2y @x2 /C301 c2@2y @t2 (1) that models vibrations of a string. The general solution can be obtained by introducing new variables j /C30x /C28ct and h /C30x /C27ct ; and applying the CHAIN RULE to obtain @ @x /C30@ j @x@ @ j /C27@ h @x@ @ h (2) /C30@ @ j /C27@ @ h (3) @ @t /C30@ j @t@ @ j /C27@ h @t@ @ h (4) /C30/C28c@ @ j /C27c@ @ h : (5) Using (3) and (5) to compute the left and right sides of (1) then gives @2y @x2 /C30@ @ j /C27@ @ h ! @y @ j /C27@y @ h ! /C30@2y @ j2 /C272@2y @ j@ h /C27@2y @ h2(6)@2y @t2 /C30/C28 c@ @ j /C27c@ @ h ! /C28c@y @ j /C27c@y @ h ! /C30c2@2y @ j2 /C282c2@2y @ j@ h /C27c2@2y @ h2 : (7) respectively, so plugging in and expanding then gives @2y @ j@ h /C300: (8) This partial differential equation has general solution /C30f(j) /C27g( h) (9) /C30f(x /C28ct) /C27g(x /C27ct) : (10) where f and g are arbitrary functions, with f representing a right-traveling wave and g a left- traveling wave. See also WAVE EQUATION References Bekefi, G. and Barrett, A. H. Electromagnetic Vibra- tions, Waves, and Radiation. Cambridge, MA: MIT Press, pp. 161 /C1/63, 1987. d’Alembert’s Theorem If three CIRCLES A, B, and C are taken in pairs, the external SIMILARITY POINTS of the three pairs lie on a straight LINE. Similarly, the external SIMILARITY POINT of one pair and the two internal SIMILARITY POINTS of the other two pairs lie upon a straight LINE, forming a SIMILARITY AXIS of the three CIRCLES . See also SIMILARITY POINT References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 155, 1965. d’Alembertian Written in the NOTATION of PARTIAL DERIVATIVES , the d’Alembertian I2 is defined by I2 /C1392 /C281 c2@2 @t2 ; where c is the speed of light. Writing in TENSOR notation, I2f/C13glkf;l0CB0C@ ;k/C30glk@2f @xl@xk/C28Gl@f @xl: See also GRADIENT FOUR- VECTOR ,HARMONIC COOR- DINATES ,LAPLACIAN ,W AVE EQUATION d’Alembertian Operator Written in the NOTATION of PARTIAL DERIVATIVES , I2 where c is the speed of light. Writing in TENSOR notation, I2 /C1392 /C281 c2@2 @t2 ; See also HARMONIC COORDINATES d’Ocagne’s Identity FmFnþ1 /C28FnFmþ1 ¼ð/C281 ÞnFm/C28n ; where Fn is a FIBONACCI NUMBER . See also CASSINI’S IDENTITY ,C ATALAN’S IDENTITY , FIBONACCI NUMBER # 1999 /C1/001 Wolfram Research, Inc. d’Octagne’s Identity # 1999 /C1/001 Wolfram Research, Inc. DAG ACYCLIC DIGRAPH Dagger The symbol $ most commonly used in older physics texts to denote the ADJOINT operator. The dagger is also known as the obelisk, obelus, or long cross (Bringhurst 1997, p. 275). See also ADJOINT ,DOUBLE DAGGER References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, 1997. Daisy A figure resembling a daisy or sunflower in which copies of a geometric figure of increasing size are placed at regular intervals along a spiral. The result-ing figure appears to have multiple spirals spreading out from the center. See also HEXLET ,P HYLLOTAXIS ,S PIRAL ,S WIRL , WHIRL References Dixon, R. "On Drawing a Daisy." §5.1 in Mathographics. New York: Dover, pp. 122 /C1/43, 1991. Damped Exponential Cosine Integral g/C12 0e/C28wTcos(vt)dv: (1) Integrate by parts with u/C13e/C28vTdv/C30cos(vt)dv (2) du/C13/C28Te/C28vTdvv/C301 tsin(vt); (3) so ge/C28vTcos(vt)dv /C301 te/C28wtsin(vt)/C27T tge/C28wTsin(vt)dv: (4) Now integrate ge/C28vTsin(vt)dv (5) by parts. Let u/C30e/C28vTdv/C30sin(vt)dv (6) du/C30/C28Te/C28vTdvv/C30/C281 tcos(vt); (7) so ge/C28vtsin(vt)dv /C30/C281 tcos(vt)/C28T tge/C28vTcos(vt)dv (8) and ge vT cos(vt)d v /C301 te /C28vt sin( vt) /C28T t2 e /C28 vt cos(vt) /C28T2 t2 ge/C28 vT cos(vt)dv (9) 1 /C27T2 t2 ! ge /C28 vT cos(vt)dv /C30e /C28 vT1 tsin( vt) /C28T t2cos(vt)"# (10) t2 /C27 T2 t2 ge /C28vT cos(vt)dv /C30e /C28 vt t2t sin( vT) /C28T cos(vt) ½/C138 (11) ge /C28vT cos( vt)dv /C30e /C28 vT t2 /C27 T2t sin(vt) /C28T cos(vT) ½/C138 : (12) Therefore, g/C12 0e /C28 vT cos(vt)dv /C300 /C27T t2 /C27 T2 /C30T t2 /C27 T2 : (13) See also COSINE INTEGRAL ,F OURIER TRANSFORM– LORENTZIAN FUNCTION ,LORENTZIAN FUNCTION Damped Simple Harmonic Motion Adding a damping force proportional to ˙x to the equation of SIMPLE HARMONIC MOTION , the first derivative of x with respect to time, the equation of motion for damped simple harmonic motion is ¨x /C27 b˙x /C27 v2 0x /C300 ; (1) where b is the damping constant. This equation arises, for example, in the analysis of the flow of current in an electronic CLR circuit, (which contains a capacitor, an inductor, and a resistor ). The curve produced by two damped harmonic oscillators at right angles to each other is called a HARMONOGRAPH , and simplifies to a LISSAJOUS CURVE if b1 /C30 b2 /C300:/ The damped harmonic oscillator can be solved by looking for trial solutions OF THE FORM x /C30ert : Plug- ging this into (1) gives r2 /C27 br /C27 v200CB0C@ ert /C300 (2) r2 /C27 br /C27 v20 /C300 : (3) This is a QUADRATIC EQUATION with solutionsr /C301 2/C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284 v2 0q0C@80C@9 : (4) There are therefore three solution regimes depending on the SIGN of the quantity inside the SQUARE ROOT , a /C13 b2 /C284v2 0 : (5) The three regimes are summarized in the following table. /a/ regime /a B0/ UNDERDAMPING /a /C300/ CRITICAL DAMPING /a > 0/ OVERDAMPING If a periodic (sinusoidal) forcing term is added at angular frequency v; the same three solution regimes are again obtained. Surprisingly, the resulting mo- tion is still periodic (after an initial transient re- sponse, corresponding to the solution to the unforced case, has died out), but it has an amplitude different from the forcing amplitude. The "particular" solution xp(t) to the forced second- order nonhomogeneous ORDINARY DIFFERENTIAL EQUATION ¨x /C27p(t)˙x /C27q(t)x /C30A cos(vt) (6) due to forcing is given by the equation xp(t) /C30/C28x1(t)gx2(t)g(t) W(t)dt /C27x2(t)gx1(t)g(t) W(t)dt; (7) where x1andx2are the homogeneous solutions to the unforced equation ¨x/C27p(t)˙x/C27q(t)x/C300 (8) and W(t) is the W RONSKIAN of these two functions. Once the sinusoidal case of forcing is solved, it can be generalized to any periodic function by expressing the periodic function in a F OURIER SERIES . See also DAMPED SIMPLE HARMONIC MOTION , DAMPED SIMPLE HARMONIC MOTION– CRITICAL DAMP- ING,DAMPED SIMPLE HARMONIC MOTION– OVERDAMP- ING,D AMPED SIMPLE HARMONIC MOTION– UNDERDAMPING ,HARMONOGRAPH ,LISSAJOUS CURVE , SIMPLE HARMONIC MOTION References Papoulis, A. "Motion of a Harmonically Bound Particle." §15/C1/inProbability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 524 /C1/28, 1984. Damped Simple Harmonic Motion * / Critical Damping Critical damping is a special case of damped simple harmonic motion in which a /C13 b2 /C284 v2 0 /C300; (1) so b /C302v0 : (2) In this case, a /C300 so the solutions OF THE FORM x /C30ert satisfy r9/C301 2(/C28b) /C30/C2812 b /C30/C28v 0 : (3) One of the solutions is therefore x1 /C30e/C28 v0t : (4) In order to find the other linearly independent solution, we can make use of the identity x2(t) /C30x1(t)ge/C28gp(t)dt x1(t) ½/C1382dt: (5) Since we have p(t) /C302v0 ; e /C28fp(t)dt simplifies to e /C282 v0t : Equation (5) therefore becomes x2(t) /C30e /C28 v0tge/C282 v0t e /C28 v0t ½/C1382 dt /C30e/C28 v0tgdt /C30te /C28 v0t : (6) The general solution is therefore x /C30(A /C27Bt)e /C28 v0t : (7) In terms of the constants A and B, the initial values are x(0) /C30A (8) x?(0) /C30B /C28Av; (9) so A /C30x(0) (10) B ¼ x?ð0Þþv0xð0Þ: (11)The above plot shows a critically damped simple harmonic oscillator with v /C300:3; b /C300 :15 for a vari- ety of initial conditions (A, B). For sinusoidally forced simple harmonic motion with critical damping, the equation of motion is ¨x /C272 v0 ˙x /C27 v2 0x /C30A cos(vt) ; (12) and the WRONSKIAN is W(t) /C13x1 ˙x2 /C28 ˙x1x2 /C30e /C282 v0t : (13) Plugging this into the equation for the particular solution gives xp(t) /C30/C28e /C28v0tgte /C28 v0tA cos vtðÞ e /C282v0tdt /C27te /C28 v0tge /C28 v0tA cos(vt) e /C282 v0tdt /C30A v2 /C27 v2 0 ðÞv2 0 /C28 v20CB0C@ cos(vt) /C272vv0 sin( vt)0C10CC : (14) In order to put this in the desired form, note that we want to equate C cos u /C27S sin u /C30Q cos(u /C27 d) /C30Q( cos u cos d /C28sin u sin d) : (15) This means C /C13Q cos d /C30 v20 /C28 v2 (16) S /C13/C28Q sin d /C302 vv0 ; (17) so Q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 /C27S2p (18) d/C30tan/C281/C28S C ! : (19) Plugging in, Q/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v4 0/C282v20v2/C27v4/C274v20v2q /C30v2 0v2: (20) d/C30tan/C2812vv0 v2/C28v2 0 ! : (21) The solution in the requested form is therefore xp/C30A v2/C27v20 ðÞ2v2 0/C27v20CB0C@ cos(vt/C27d) A v2/C27v2 0cosvt/C27d ðÞ ; (22) where dis defined by (21). See also DAMPED SIMPLE HARMONIC MOTION , DAMPED SIMPLE HARMONIC MOTION– OVERDAMPING , DAMPED SIMPLE HARMONIC MOTION– UNDERDAMPING , SIMPLE HARMONIC MOTION References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 528, 1984. Damped Simple Harmonic Motion * / Overdamping Overdamped simple harmonic motion occurs when b2 /C284v2 0 > 0; (1) so a /C13 b2 /C284v20 > 0: (2) x1 /C30er/C28t (3) x2 /C30er/C27t ; (4) where r9/C131 2/C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284v2 0q0C@80C@9 : (5) The general solution is therefore x /C30Aer/C28t /C27Ber/C27t ; (6) where A and B are constants. The initial values are x(0) /C30A /C27B (7) x?(0) /C30Ar/C28/C27Br/C27; (8) so A /C30x(0) /C27r/C27x(0) /C28 x?(0) r/C28/C28 r/C27(9) B /C30/C28r/C27x(0) /C28 x?(0) r/C28/C28 r/C27: (10) The above plot shows an overdamped simple harmo- nic oscillator with v /C300:3; b /C300:075 and three differ- ent initial conditions (A, B). For a cosinusoidally forced overdamped oscillator with forcing function g(t) /C30C cos(vt) ; the particular solutions arey1(t) /C30er1t (11) y2(t) /C30er2t ; (12) where r1 /C131 2/C28b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284v2 0q0C@80C@9 (13) r2 /C131 2/C28b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284 v2 0q0C@80C@9 : (14) These give the identities r1 /C27r2 /C30/C28b (15) r1 /C28r2 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284 v2 0q (16) and v2 0 /C301 4b /C28 r1 /C28r2 ðÞ2hi /C30r1r2 : (17) The WRONSKIAN is W(t) /C30y1y?2 /C28y ?1y2 /C30er1tr2er2t /C28r1er1ter2t /C30 r2 /C28r1 ðÞ e r1/C27r2 ðÞ t: (18) The particular solution is yp /C30/C28y1v1 /C27y2v2 ; (19) where v1 /C13gy2g(t) W(t)/C30C r2 /C28 r1v sin( vt) /C28 r2 cos(vt) er2t r2 2 /C27 v2 ðÞ(20) v2 /C13gy2g(t) W(t)/C30C r2 /C28 r1v sin vtðÞ/C28 r1 cos vtðÞ er1t r22 /C27 v2 ðÞ:ð21Þ Therefore, yp/C30Ccos(vt)r1r2/C28v2ðÞ /C28sin(vt)vr1/C27r2 ðÞ r21/C27v2 ðÞ r22/C27v2 ðÞ /C30Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2v2/C27v2/C28v2 0 ðÞ2q cosvt/C27d ðÞ ; (22) where d/C30tan/C281 bv v2/C28v20 ! : (23) See also DAMPED SIMPLE HARMONIC MOTION , DAMPED SIMPLE HARMONIC MOTION– CRITICAL DAMP- ING,D AMPED SIMPLE HARMONIC MOTION– UNDER - DAMPING ,SIMPLE HARMONIC MOTION References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 527 /C1/28, 1984. Damped Simple Harmonic Motion * / Underdamping Underdamped simple harmonic motion occurs when b2/C284v2 0B0; (1) so a/C13b2/C284v20B0: (2) Define g/C13ffiffiffiffiffiffi/C28ap/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4v2 0/C28b2q ; (3) then solutions satisfy r9/C30/C281 2b9ig; (4) where r9/C1312/C28b9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2/C284v2 0q0C@80C@9 ; (5) and are OF THE FORM x/C30e/C28b=29ig ðÞ t: (6) Using the E ULER FORMULA eix/C30cosx/C27isinx; (7) this can be rewritten x/C30e/C28b=2ðÞ tcosgtðÞ9isingtðÞ ½/C138 : (8) We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linear sums of LINEARLY INDEPENDENT solutions are also solutions. Since we have a sum of such solutions in (8), it follows that the IMAGINARY and REAL PARTS separately satisfy the ODE and are therefore the solutions we seek. The constant in front of the sineterm is arbitrary, so we can identify the solutions as x1/C30e/C28b=2ðÞ tcos(gt) (9) x2/C30e/C28b=2ðÞ tsin(gt); (10) so the general solution is x/C30e/C28b=2ðÞ t[Acos(gt)/C27Bsin(gt)]: (11) The initial values are x(0)/C30A (12) x?(0)/C30/C281 2bA/C27B;g (13) soAand Bcan be expressed in terms of the initial conditions by A/C30x(0) (14) B/C30bx(0) 2g/C27x?(0)g: (15) The above plot shows an underdamped simple har- monic oscillator with v/C300:3;b/C300:4 for a variety of initial conditions ( A, B ). For a cosinusoidally forced underdamped oscillatorwith forcing function g(t)/C30Ccos(vt);use g/C13 1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4v2 0/C28b2q (16) a/C131 2b (17) to obtain 4v2 0/C28b2/C304g2(18) v20/C30g2/C271 4b2/C30g2/C27a2(19) b/C302a: (20) The particular solutions are y1(t)/C30e/C28atcos(gt) (21) y2(t)/C30e/C28atsin(gt): (22) The W RONSKIAN is W(t)/C13y1y?2/C28y?1y2 /C30e/C28atcos(gt)/C28ae/C28atsin(gt)/C27e/C28atgcos(gt) ½/C138 /C28e/C28atsin(gt)/C28ae/C28atcos(gt)/C28e/C28atgsin(gt) ½/C138 /C30e/C282ata[/C28sin (gt) cos( gt)/C27sin (gt) cos( gt)] f /C27g[cos2(gt)/C27sin2(gt)]g /C30ge/C282at: (23) The particular solution is given by yp /C30/C28y1v1 /C27y2v2 ; (24) where v1 /C30gy2g(t) W(t)/C30C g ge at cos(gt) cos(vt)dt (25) v2 /C30gy2g(t) W(t)/C30C g ge at cos(gt) cos( vt)dt: (26) Using computer algebra to perform the algebra, the particular solution is yp(t) /C30Ca2 /C27 g2 /C28 v2ðÞ cos( vt) /C27 2av sin( vt) a2 /C27 ( g /C28 v)2hi a2 /C27 ( g /C27 v)2hi /C30Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 0 /C28 v2 ðÞ2/C27b2 v2q v20 /C28 v2 ðÞ2/C28v2 4v20 /C28 b20CB0C@ cos(vt /C27 d) ; ð27Þ where d /C30tan/C281 bv v2 /C28 v20 ! : (28) If the forcing function is sinusoidal instead of cosinu- soidal, then d?/C30d/C281 2p/C30tan/C281x/C2812p/C30tan /C281/C281 x ! ; (29) so d?/C30tan/C281v2 0/C28v2 bv ! : (30) See also DAMPED SIMPLE HARMONIC MOTION , DAMPED SIMPLE HARMONIC MOTION– CRITICAL DAMP- ING,DAMPED SIMPLE HARMONIC MOTION– OVERDAMP- ING,SIMPLE HARMONIC MOTION References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 525 /C1/27, 1984. d-Analog N.B. A detailed online essay by S. Finch was the starting point for this entry. The d-analog of a COMPLEX NUMBER sis defined as [s]d/C301/C282d sd(1) (Flajolet et al. 1995). For integer n, [2]!/C131 and [n]d!/C30[3][4] /C1/C1/C1[n]/C301/C282d 3d ! 1/C282d 4d ! /C1/C1/C11/C282d nd ! : (2) It can then be extended to complex values via [s]d!/C30Y/C12 j/C301[j/C272] [j/C27s](3) (Flajolet et al. 1995). It satisfies the basic functional identity [s]d!/C30[s]d[s/C281]d!: (4) The d-analog of the POLYGAMMA FUNCTION is [c]d(s/C271)/C30d dsln[s]d! /C30/C28d/C2152dX/C12 m/C3011 (m/C27s)(m/C27s)d/C282dhi : (5) The first few values are [c]1(s)/C303/C282s s2/C283s/C272(6) [c]2(s)/C30c0(s/C282)/C282c0(s)/C27c0(s/C272); (7) where c0(x) is the DIGAMMA FUNCTION . The d-analog of the E ULER- MASCHERONI CONSTANT g is [g]d/C30/C28[c]d(3)/C30d/C2152dX/C12 m/C3031 mmd/C282d ðÞ(8) (Flajolet et al. 1995). The first few values are [g]1/C303 2(9) [g]2/C3011 12(10) [g]3/C309 2/C28H3/C28iffiffi 3p/C28H3/C27iffiffi 3p (11) [g]4/C304712/C28H 2/C282i/C28H2/C272i; (12) where Hnis a HARMONIC NUMBER . The d-analog of the HARMONIC NUMBERS isH2½/C138d/C300 and Hn½/C138d/C30d/C2152d 1 3d/C271[3]/C271 4d/C271[4]/C27.../C271 nd/C271[n] ! (13) /C30[c]d(n/C271)/C27[g]d (14) (Flajolet et al. 1995). The d-analog of INFINITY FACTORIAL is given by [ /C12!]d /C30Y/C12 n/C3031 /C282d nd ! : (15) This INFINITE PRODUCT can be evaluated in closed form in terms of p; the HYPERBOLIC SINE sinh x; and GAMMA FUNCTIONS G(x) involving roots of unity zk n /C13 (/C281)k=n ; d1 /C300 (16) d2 /C301 6 (17) d3 ¼sinh ðpffiffiffi 3p Þ 42pffiffiffi3p (18) d 4 /C30cosh p sinh p 60p (19) d5 /C301 1240 G 2z1 50CB0C@ G/C282z250CB0C@0C@10C@10C@10C@12 (20) d6 /C30sinh2( pffiffiffi 3p ) 1512p2 (21) d7 /C301 28448 G 2 z1 70CB0C@ G/C282 z270CB0C@ G 2 z370CB0C@ 0C@10C@10C@10C@12 (22) d8 /C30sinh 2pðÞ sinh 2z1 40CB0C@0C@10C@10C@10C@12 16320 p3 (23) d9 /C30sinh pffiffiffi 3p0CB0C@ 588672 pffiffiffi 3p G 2z1 90CB0C@ G/C282z290CB0C@ G/C282z490CB0C@ 0C@10C@10C@10C@12 : (24) These are all special cases of a general result for INFINITE PRODUCTS . See also INFINITE PRODUCT , Q-ANALOG References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/infprd/infprd.html. Flajolet, P.; Labelle, G.; Laforest, L.; and Salvy, B. "Hyper- geometrics and the Cost Structure of Quadtrees." Random Structure Alg. 7, 117 /C1/44, 1995. http://pauillac.inria.fr/ algo/flajolet/Publications/publist.html. Kahovec, H. "Basic Infinite Products." http://www.mathsoft.- com/asolve/constant/infprd/kahovec/ip.html. Kahovec, H. "Proof of the Infinite Product Formulas." http:// www.mathsoft.com/asolve/constant/infprd/kahovec/ proof01.html.Dandelin Spheres The inner and outer SPHERES TANGENT internally to a CONE and also to a PLANE intersecting the CONE are called Dandelin spheres. The SPHERES can be used to show that the intersec- tion of the PLANE with the CONE is an ELLIPSE . Let p be a PLANE intersecting a right circular CONE with vertex O in the curve E. Call the SPHERES TANGENT to the CONE and the PLANE S1and S2 ; and the CIRCLES on which the SPHERES are TANGENT to the CONE R1 and R2 : Pick a line along the CONE which intersects R1at Q, E at P, and R2at T. Call the points on the PLANE where the CIRCLES are TANGENT F1and F2: Because intersecting tangents have the same length, F1P/C30QP F2P/C30TP: Therefore, PF1/C27PF2/C30QP/C27PT/C30QT; which is a constant independent of P,s o Eis an ELLIPSE with a/C30QT=2:/ See also CONE,SPHERE References Honsberger, R. "Kepler’s Conics." Ch. 9 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., p. 170, 1979. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 40 /C1/4, 1991. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 80 /C1/1, 1990. Ogilvy, C. S. Excursions in Mathematics. New York: Dover, pp. 68 /C1/9, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 48, 1991. Danielson-Lanczos Lemma The DISCRETE FOURIER TRANSFORM of length N (where N is EVEN ) can be rewritten as the sum of two DISCRETE FOURIER TRANSFORMS , each of length N =2 : One is formed from the EVEN -numbered points; the other from the ODD-numbered points. Denote the kth point of the DISCRETE FOURIER TRANSFORM by Fn : Then Fn /C30XN /C281 k/C300fke /C282pink=N /C30XN =2 /C281 k/C300e /C282 pikn=(N =2)f2k /C27WnXN =2 /C281 k /C300e/C282 pikn= N =2 ðÞf2k /C271 /C30Fe n /C27WnFo n ; where W /C13e/C282 pi=N and n /C300; ... ; N : This procedure can be applied recursively to break up the N =2 even and ODD points to their N =4 EVEN and ODD points. If N is a POWER of 2, this procedure breaks up the original transform into 1gN transforms of length 1. Each transform of an individual point has Feeo /C1/C1/C1 n/C30fk for some k. By reversing the patterns of evens and odds, then letting e /C300 and o /C301, the value of k in BINARY is produced. This is the basis for the FAST FOURIER TRANSFORM . See also DISCRETE FOURIER TRANSFORM ,FAST FOUR- IER TRANSFORM ,FOURIER TRANSFORM References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, pp. 407 /C1/11, 1989. Darboux Integral A variant of the RIEMANN INTEGRAL defined when the UPPER and LOWER INTEGRALS , taken as limits of the LOWER SUM Lf; f;N ðÞ /C30Xn r/C301Mf; dr ðÞ /C28 f xr /C281 ðÞ and UPPER SUM Uf; f;N ðÞ /C30Xn r/C301Mf; dr ðÞ /C28 f xr/C281 ðÞ ; are equal. Here, f(x)isa REAL FUNCTION , f(x)isa monotonic increasing function with respect to which the sum is taken, m(f;S) denotes the lower bound of f(x) over the interval S, and M(f;S) denotes the upper bound.See also LOWER INTEGRAL ,LOWER SUM,R IEMANN INTEGRAL ,UPPER INTEGRAL ,UPPER SUM References Kestelman, H. Modern Theories of Integration, 2nd rev. ed. New York: Dover, p. 250, 1960. Darboux Problem GOURSAT PROBLEM Darboux Vector The rotation VECTOR of the TRIHEDRON of a curve with CURVATURE k "0 when a point moves along a curve with unit SPEED . It is given by D /C30 tT /C27 kB ; (1) where t is the TORSION , T the TANGENT VECTOR , and B the BINORMAL VECTOR . The Darboux vector field satisfies ˙T /C30D /C29T (2) ˙N /C30D /C29N (3) ˙B/C30D/C29B: (4) See also BINORMAL VECTOR ,CURVATURE ,TANGENT VECTOR ,TORSION (DIFFERENTIAL GEOMETRY ) References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 205, 1997. Darboux’s Formula Darboux’s formula is a theorem on the expansion of functions in infinite series. T AYLOR SERIES may be obtained as a special case of the formula, which maybe stated as follows. Letf(z) be analytic at all points of the line joining ato z, and let f(t) be any POLYNOMIAL of degree nint. Then if 0 5t51;differentiation gives d dtX/C12 m/C301(/C281)m(z/C28a)mB(n/C28m)(t)f(m)(a/C27t(a/C28z)) /C30/C28(z/C28a)f(n)(t)f?(a/C27t(z/C28a)) /C27(/C281)n(z/C28a)n/C271f(t)f(n/C271)(a/C27t(z/C28a)): (1) Butf(n)(t)/C30f(n)(0);so integrating tover the interval 0 to 1 gives f(n)(0)[f(z)/C28f(a)] /C30Xn m/C301(/C281)m/C281(z/C28a)m[f(n/C28m)(1)f(m)(z) /C28f(n/C28m)(0)f(m)(a)] /C27(/C281)n(z /C28a)n/C271g1 0f(t)f(n /C271)(a /C27t(z /C28a))dt: (2) The TAYLOR SERIES follows by letting f(t) /C30(t /C281)n and letting n 0/C12 (Whittaker and Watson 1990, p. 125). See also BU¨ RMANN’S THEOREM ,E ULER- MACLAURIN INTEGRATION FORMULAS ,MACLAURIN SERIES ,TAYLOR SERIES References Whittaker, E. T. and Watson, G. N. "A Formula Due to Darboux." §7.1 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 125, 1990. Darboux-Stieltjes Integral DARBOUX INTEGRAL Darling’s Products A generalization of the HYPERGEOMETRIC FUNCTION identity 2F1( a; b; g;z)2F1(1 /C28 a;1 /C28 b;2/C28 g;z) /C302 F1(a /C271 /C28 g ; b /C271 /C28 g;2/C28 g;z)2F1( g /C28 a; g /C28 b; g;z) (1) to the GENERALIZED HYPERGEOMETRIC FUNCTION 3F2(a ;b;c;d;e;x) : Darling’s products are 3F2a; b; g;z d; o0C1B0C1@ 3F21 /C28 a;1 /C28 b;1 /C28 g;z 2 /C28 d; 2 /C28 o0C1B0C1@ /C30o /C28 1 o /C28 d3F2a /C271 /C28 d; b /C271 /C28 d; g /C271 /C28 d;z 2 /C28 d ; o /C271 /C28 d0C1B /C23F2d /C28 a; d /C28 b; d /C28 g;z d; d /C271 /C28 o0C1B0C1@ /C27d /C28 1 d /C28 o 3F2a /C271 /C28 o ; b /C271 /C28 o ; g /C271 /C28 o;z 2 /C28 o ; d /C271 /C28 o0C1B0C1@ /C23F2o /C28 a; o /C28 b; o /C28 g;z o ; o /C271 /C28 d0C1B0C1@ (2) and (1 /C28z)a /C27 b/C27 g/C28 d/C28 o 3F2a; b; g;z d; o0C1B0C1@ /C30o /C28 1 o /C28 d3F2d /C28 a; d /C28 b; d /C28 g;z d; d /C271 /C28 o0C1B0C1@ /C23F2o /C28 a; o /C28 b; o /C28 g;z o /C281; o /C271 /C28 d0C1B0C1@ /C27d /C28 1 d /C28 o 3F2o /C28 a; o /C28 b; o /C28 g;z o ; o /C271 /C28 d0C1B0C1@/C23F2d /C28 a; d /C28 b; d /C28 g;z d /C281; d /C271 /C28 o0C1B0C1@ ; (3) which reduce to (1) when g /C30 o 0/C12:/ See also GENERALIZED HYPERGEOMETRIC FUNCTION References Bailey, W. N. "Darling’s Theorems of Products." §10.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 88 /C1/2, 1935. Dart PENROSE TILES Darwin’s Expansions Series expansions of the PARABOLIC CYLINDER FUNC- TIONS U(a;x) and W(a;x) : The formulas can be found in Abramowitz and Stegun (1972). See also PARABOLIC CYLINDER FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 689 /C1/90 and 694 /C1/95, 1972. Darwin-de Sitter Spheroid A SURFACE OF REVOLUTION OF THE FORM r( f) /C30a 1 /C28e sin2 f /C283 8 e2 /C27k ! sin2(2f)"# ; where k is a second-order correction to the figure of a rotating fluid. See also OBLATE SPHEROID ,P ROLATE SPHEROID , SPHEROID References Zharkov, V. N. and Trubitsyn, V. P. Physics of Planetary Interiors. Tucson, AZ: Pachart Publ. House, 1978. Data Cube A 3-D data set consisting of stacked 2-D data slices as a function of a third coordinate. See also GRAPH (FUNCTION ) Data Structure A formal structure for the organization of informa- tion. Examples of data structures include the LIST, QUEUE ,STACK , and TREE . References Tarjan, R. E. Data Structures and Network Algorithms. Philadelphia, PA: SIAM Press, 1983. Wood, D. Data Structures, Algorithms, and Performance. Reading, MA: Addison-Wesley, 1993. Database A database can be roughly defined as a structure consisting of 1. A collection of information (the data), 2. A collection of queries that can be submitted, and 3. A collection of algorithms by which the structure responds to queries, searches the data, and re- turns the results. References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 48, 1996. Daubechies Wavelet Filter A WAVELET used for filtering signals. Daubechies (1988, p. 980) has tabulated the numerical values up to order p/C3010. See also WAVELET References Daubechies, I. "Orthonormal Bases of Compactly Supported Wavelets." Comm. Pure Appl. Math. 41, 909/C1/96, 1988. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Interpolation and Extrapolation." Ch. 3 in Numerical Recipes in FORTRAN: The Art of ScientificComputing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 584 /C1 /86, 1992. Davenport-Schinzel Sequence Form a sequence from an ALPHABET of letters [1 ;n] such that there are no consecutive letters and no alternating subsequences of length greater than d. Then the sequence is a Davenport-Schinzel sequenceif it has maximal length N d(n):The value of N1(n)i s the trivial sequence of 1s: 1, 1, 1, ... (Sloane’sA000012). The values of N 2(n) are the POSITIVE INTEGERS 1, 2, 3, 4, ... (Sloane’s A000027). The values ofN3(n) are the ODD INTEGERS 1, 3, 5, 7, ... (Sloane’s A005408). The first nontrivial Davenport-Schinzelsequence N 4(n) is given by 1, 4, 8, 12, 17, 22, 27, 32, ... (Sloane’s A002004). Additional sequences are givenby Guy (1994, p. 221) and Sloane. References Agarwal, P. K. and Sharir, M. "Davenport-Schinzel Se- quences and Their Geometric Applications." Ch. 1 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Hol-land, pp. 1 /C1 /7, 2000. Davenport, H. and Schinzel, A. "A Combinatorial Problem Connected with Differential Equations." Amer. J. Math. 87, 684/C1/90, 1965. Guy, R. K. "Davenport-Schinzel Sequences." §E20 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 220 /C1/22, 1994. Roselle, D. P. and Stanton, R. G. "Results of Davenport- Schinzel Sequences." In Proc. Louisiana Conference onCombinatorics, Graph Theory, and Computing. LouisianaState University, Baton Rouge, March 1 /C1 /, 1970 (Ed. R. C. Mullin, K. B. Reid, and D. P. Roselle). Winnipeg, Mani- toba: Utilitas Mathematica, pp. 249 /C1/67, 1960. Sharir, M. and Agarwal, P. Davenport-Schinzel Sequences and Their Geometric Applications. New York: Cambridge University Press, 1995. Sloane, N. J. A. Sequences A000012/M0003, A000027/ M0472, and A002004/M3328 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Davey-Stewartson Equations The system of PARTIAL DIFFERENTIAL EQUATIONS iut/C27uxx/C27auyy/C27buujj2/C28uv/C300 vxx/C27gvyy/C27dujj20C@n0C@o yy/C300: References Champagne, B. and Winternitz, P. "On the Infinite-Dimen- sional Group of the Davey-Stewartson Equations." J. Math. Phys. 29,1/C1/, 1988. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Dawson’s Integral AnINTEGRAL which arises in computation of the Voigt lineshape: D(x)/C13e/C28x2gx 0ey2dy: (1) It is sometimes generalized such that D9(x)/C13e/C14x2gx 0e9y2dy; (2) giving D/C27(x)/C301 2ffiffiffippe/C28x2erfi(x) (3) D/C28(x)/C301 2ffiffiffippex2erf(x); (4) where erf( z) is the ERF function and erfi( z) is the imaginary error function ERFI.D/C27(x) is illustrated in the left figure above, and D/C28(x) in the right figure. D/C27(x) has an ASYMPTOTIC SERIES D/C27(x) /C21 2x /C271 4x3 /C27... (5) The plots above show the behavior of D/C27(z) in the COMPLEX PLANE . The plots above show the behavior of D/C28(z) in the COMPLEX PLANE . /D/C27 has a maximum at D? /C27(x) /C300; or 1 /C28ffiffiffippe /C28x2 x2 erfi(x) /C300; (6) giving D/C27(0:9241388730) /C300:5410442246 ; (7) and an inflection at Dƒ/C27(x) /C300; or /C282x /C27ffiffiffippe /C28x22x2 /C2810CB0C@ erfi(x) /C300; (8) giving D/C27(1:5019752683) /C300:4276866160 : (9) See also ERFI,GAUSSIAN FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 298, 1972. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Dawson’s Integrals." §6.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing,2nd ed. Cambridge, England: Cambridge University Press, pp. 252 /C1/54, 1992. Spanier, J. and Oldham, K. B. "Dawson’s Integral." Ch. 42 inAn Atlas of Functions. Washington, DC: Hemisphere, pp. 405 /C1/10, 1987. dc JACOBI ELLIPTIC FUNCTIONS #1999/C1/001 Wolfram Research, Inc. de Bruijn Constant Also called the C OPSON-DE BRUIJN CONSTANT . It is the minimal constant c/C301:0164957714 . . . such that the inequality X/C12 n/C301an5cX/C12 n/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 n/C27a2 n/C271/C27a2n/C272/C27... ns always holds. References Copson, E. T. "Note on Series of Positive Terms." J. London Math. Soc. 2,9/C1/2, 1927. Copson, E. T. "Note on Series of Positive Terms." J. London Math. Soc. 3,4 9/C1/1, 1928. de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, 1981. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/copson/copson.html. de Bruijn Diagram DEBRUIJN GRAPH de Bruijn Graph A graph whose nodes are sequences of symbols from some ALPHABET and whose edges indicate the se- quences which might overlap. References Golomb, S. W. Shift Register Sequences. San Francisco, CA: Holden-Day, 1967. Ralston, A. "de Bruijn Sequences--A Model Example of the Interaction of Discrete Mathematics and Computer Science." Math. Mag. 55, 131/C1/43, 1982. de Bruijn Sequence The shortest circular sequence of length sasuch that every string of length non the ALPHABET aof size s occurs as a contiguous subrange of the sequence described by a. A de Bruijn sequence can be gener- ated using DeBruijnSequence [a,n] in the Mathe- matica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). For example, a de Bruijn sequence of order non the alphabet fa;b;cgis given by fa;a;c;b;b;c;c;a;bg:/ Every de Bruijn sequence corresponds to an EULER- IAN CYCLE on a DE BRUIJN GRAPH . Surprisingly, it turns out that the lexicographic sequence of LYNDON WORDS of lengths DIVISIBLE by n gives the lexicogra- phically smallest de Bruijn sequence (Ruskey). de Bruijn sequences can be generated by feedback shift registers (Golomb 1966; Ronse 1984; Skiena 1990, p. 196). See also DE BRUIJN GRAPH ,LYNDON WORD References de Bruijn, N. G. "A Combinatorial Problem." Koninklijke Nederlandse Akademie v. Wetenschappen 49, 758 /C1/64, 1946. Golomb, S. W. Shift Register Sequences. San Francisco, CA: Holden-Day, 1967. Good, I. J. "Normal Recurring Decimals." J. London Math. Soc. 21, 167 /C1/72, 1946. Knuth, D. E. "Oriented Subtrees of an Arc Digraph." J. Combin. Th. 3, 309 /C1/14, 1967. Ronse, C. Feedback Shift Registers. Berlin: Springer-Verlag, 1984. Ruskey, F. "Information on Necklaces, Lyndon Words, de Bruijn Sequences." http://www.theory.csc.uvic.ca/~cos/inf/ neck/NecklaceInfo.html. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 195 /C1/96, 1990. de Bruijn’s Theorem A box can be packed with a HARMONIC BRICK a /C29ab /C29 abc IFF the box has dimensions ap /C29abq /C29abcr for some natural numbers p, q, r (i.e., the box is a multiple of the brick). See also BOX-PACKING THEOREM ,CONWAY PUZZLE , KLARNER’S THEOREM References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 69 /C1/2, 1976. de Bruijn-Newman Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let J be the XI FUNCTION defined by J(iz) /C301 2z2 /C2814 ! p /C28z=2 /C281 4G1 2 z /C2714 ! z z /C2712 ! : (1) /J(z =2)=8 can be viewed as the FOURIER TRANSFORM of the signal F(t) /C30X/C12 n/C3012p2n4e9t /C283pn2e5t0CB0C@ e/C28pn2e4t (2) for t /C23R ]0: Then denote the FOURIER TRANSFORM of F(t)e lt2 as H( l; z) ; F F(t)e lt2hi /C30H( l;z) : (3)de Bruijn (1950) proved that H has only REAL zeros for l ]1=2: C. M. Newman (1976) proved that there exists a constant L such that H has only REAL zeros IFF l ]L: The best current lower bound (Csordas et al. 1993, 1994) is L>/C285:895 /C2910 /C289 : The RIEMANN HYPOTHESIS is equivalent to the conjecture that L50:/ See also XI FUNCTION References Csordas, G.; Odlyzko, A.; Smith, W.; and Varga, R. S. "A New Lehmer Pair of Zeros and a New Lower Bound for the de Bruijn-Newman Constant." Elec. Trans. Numer. Ana- lysis 1, 104 /C1/11, 1993. Csordas, G.; Smith, W.; and Varga, R. S. "Lehmer Pairs of Zeros, the de Bruijn-Newman Constant and the Riemann Hypothesis." Constr. Approx. 10, 107 /C1/29, 1994. de Bruijn, N. G. "The Roots of Trigonometric Integrals." Duke Math. J. 17, 197 /C1/26, 1950. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/dbnwm/dbnwm.html. Newman, C. M. "Fourier Transforms with only Real Zeros." Proc. Amer. Math. Soc. 61, 245 /C1/51, 1976. de Gua’s Theorem The square of the AREA of the base (i.e., the face opposite the right TRIHEDRAL ANGLE )ofa TRIRECTAN- GULAR TETRAHEDRON is equal to the sum of the squares of the AREAS of its other three faces. This theorem was presented to the Paris Academy of Sciences in 1783 by J. P. de Gua de Malves (1712 /C1/ 785), although it was known to Descartes (1859) and to Faulhaber (Altshiller-Court 1979, p. 300). It is a special case of a general theorem presented by Tinseau to the Paris Academy in 1774 (Osgood andGraustein 1930, p. 517; Altshiller-Court 1979). See also P YTHAGOREAN THEOREM ,TRIRECTANGULAR TETRAHEDRON References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, pp. 92 and 300, 1979. Descartes, R. Oeuvres ine ´dites de Descartes. Paris, 1859. Osgood, W. F. and Graustein, W. C. Plane and Solid Analytic Geometry. New York: Macmillan, Th. 2, p. 517, 1930. #1999/C1/001 Wolfram Research, Inc. de Jonquie `res Theorem For an algebraic curve, the total number of groups of agr Nconsisting in a point of multiplicity k1;one of multiplicity k2;..., one of multiplicity kp;where X ki/C30N (1) X (ki/C281)/C30r; (2) and where a1points have one multiplicity, a2another, etc., and Y /C30k1k2...kp (3) is Qp(p /C28 1)...( p /C28 r) a1!a2! /C1/C1/C1 /C2P p /C28 r /C28P i@P @ki p /C28 r /C27 1 /C27P ij@2 P @ki @kj p /C28 r /C27 2 /C27...2 66643 7775: (4) References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 288, 1959. de Jonquie `res Transformation A transformation of an algebraic curve which is of the same type as its inverse. A de Jonquie `res transforma- tion is always factorable. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 203 /C1/04, 1959. de la Loubere’s Method A method for constructing MAGIC SQUARES of ODD order, also called the SIAMESE METHOD . See also MAGIC SQUARE de Longchamps Point The reflection of the ORTHOCENTER about the CIRCUM- CENTER of a TRIANGLE . This point is also the ORTHO- CENTER of the ANTICOMPLEMENTARY TRIANGLE . It has TRIANGLE CENTER FUNCTION a /C30cos A /C28cos B cosC : The SODDY LINE intersects the EULER LINE in the de Longchamps point (Oldknow 1996). See also CIRCUMCENTER ,EULER LINE,ORTHOCENTER , SODDY LINE References Altshiller-Court, N. "On the de Longchamps Circle of the Triangle." Amer. Math. Monthly 33, 368 /C1/75, 1926. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/29, 1996. Vandeghen, A. "Soddy’s Circles and the de Longchamps Point of a Triangle." Amer. Math. Monthly 71, 176 /C1/79, 1964. de Me´re´’s Problem The probability of getting at least one "6" in four rolls of a single 6-sided DIE is1 /C285 6 !4 :0:5177 ; (1) which is slightly higher than the probability of at least one double-six in 24 throws of two dice, 1 /C283536 ! 24 :0 :4914 : (2) The French nobleman and gambler Chevalier de Me´re´ suspected that (1) was higher than (2), but his mathematical skills were not great enough to demon- strate why this should be so. He posed the question to Pascal, who solved the problem and proved de Me´re´ correct. In fact, de Me´re´’s observation remains true even if two dice are thrown 25 times, since the probability of throwing at least one double-six is then 1/C283536 ! 25:0:5055 : (3) See also B OXCARS ,DICE References Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, pp. 28 /C1/9 and 44 /C1/5, 1993. Kraitchik, M. "A Dice Problem." §6.2 in Mathematical Recreations. New York: W. W. Norton, pp. 118 /C1/19, 1942. Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 21 /C1/2, 1937. de Moivre Number A solution /zk¼e2pik=d /to the CYCLOTOMIC EQUATION xd¼1: The de Moivre numbers give the coordinates in the COMPLEX PLANE of the VERTICES of a REGULAR POLY- GON with dsides and unit RADIUS . nde Moivre Number 291 31 ,1 2/C2819iffiffiffi 3p0C@n0C@o / 4 /91;9i/ 51,1 4/C281 /C27ffiffiffi 5p 9iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C272ffiffiffi 5pq 0C@80C@9 ; 1 4/C281 /C28ffiffiffi 5p 9iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C282ffiffiffi 5pq 0C@80C@9 / 6 /91;91 291 /C27iffiffiffi 3p0C@n0C@o / See also CYCLOTOMIC EQUATION ,CYCLOTOMIC POLY- NOMIAL ,EUCLIDEAN NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996. de Moivre’s Identity ei(nu) /C30 ei u0CB0C@ n: (1) From the EULER FORMULA it follows that cos(nu) /C27i sin(nu) /C30(cos u /C27i sin u)n : (2) A similar identity holds for the HYPERBOLIC FUNC- TIONS , (cosh z /C27sinh z)n /C30cosh( nz) /C27sinh( nz): (3) See also EULER FORMULA References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 356 /C1/57, 1985. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 96 /C1/00, 1996. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 156, 1951. de Moivre’s Quintic AQUINTIC EQUATION OF THE FORM x5/C27ax3/C271 5a2x/C27b/C300: See also QUINTIC EQUATION de Moivre-Laplace Theorem The asymptotic form of the n-step B ERNOULLI DIS- TRIBUTION with parameters pand q/C301/C28pis given by Pn(k)/C30n k0C@80C@9 pkqn/C28k/C21ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pnpqp e/C28(k/C28np)2=(2npq)(1) (Papoulis 1984, p. 66).Uspensky (1937) defines the de Moivre-Laplace the- orem as the fact that the sum of those terms of the BINOMIAL SERIES of (p/C27q)nfor which the number of successes xfalls between d1andd2is approximately Q:1ffiffiffiffiffiffi 2ppgt2 t1e/C28t2=2dt; (2) where t1/C13d1/C281 2/C28np s(3) t2/C13d2/C2712/C28np s(4) s/C13ffiffiffiffiffiffiffiffiffiffinpq :p(5) More specifically, Uspensky (1937, p. 129) showed that Q/C301ffiffiffiffiffiffi 2ppgt2 t1e/C28t2=2dt/C27q/C28p 6ffiffiffiffiffiffiffiffiffi2psp 1/C28t20CB0C@ e/C28t2=2hit2 t1/C27V;(6) where the error term satisfies ½V½B0:13/C270:18½p-q½ s2/C27e/C283s=2(7) fors]5 (Uspensky 1937, p. 129; Kenney and Keep- ing 1958, pp. 36 /C1/7). Note that Kenney and Keeping (1958, p. 37) give the slightly smaller DENOMINATOR 0:12/C270:18½p/C28q½:/ ACOROLLARY states that the probability that x successes in ntrials will differ from the expected value npby more than disPd¼1/C28Qd;where Qd/C302ffiffiffiffiffiffi2ppgd 0e/C28t2=2dt; (8) with d/C13d/C271 2 s(9) (Kenney and Keeping 1958, p. 39). Uspensky (1937, p. 130) showed that Qd1/C13P(x/C28np jj5d) is given by Qd1/C302ffiffiffiffiffiffi 2ppgd1 0e/C28u2=2du/C271/C28u1/C28u2ffiffiffiffiffiffiffiffiffi2psp e/C28d2 1=2/C27V1;(10) where d1/C13d d(11) u1 /C13ðnq þ dÞ/C0/C28nq þ d /C29 ð12Þ u2 /C13ðnp þ dÞ/C0/C28np þ d /C29; ð13Þ and the error term satisfies jV1 jB0 :20 þ 0 :25 jp /C0 qj s2 þ e /C03 s=2 ; ð14Þ for s ]5 (Uspensky 1937, p. 130; Kenney and Keep- ing 1958, pp. 40 /C1/1). See also BERNOULLI DISTRIBUTION ,BINOMIAL SERIES , GAUSSIAN DISTRIBUTION ,N ORMAL DISTRIBUTION , WEAK LAW OF LARGE NUMBERS References de la Valle´e-Poussin, C. "Demonstration nouvelle du the´o- re`me de Bernoulli." Ann. Soc. Sci. Bruxelles 31, 219 /C1/36, 1907. de Moivre, A. Miscellanea analytica. Lib. 5, 1730. de Moivre, A. The Doctrine of Chances, or, a Method of Calculating the Probabilities of Events in Play, 3rd ed. New York: Chelsea, 2000. Reprint of 1756 3rd ed. Original ed. published 1716. Kenney, J. F. and Keeping, E. S. "The DeMoivre-Laplace Theorem" and "Simple Sampling of Attributes." §2.10 and 2.11 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 36 /C1/1, 1951. Laplace, P. The´orie analytiques de probabilite ´s, 3e`me e´d., revue et augmente ´e par l’auteur. Paris: Courcier, 1820. Reprinted in uvres comple `tes de Laplace, tome 7. Paris: Gauthier-Villars, pp. 280 /C1/85, 1886. Mirimanoff, D. "Le jeu de pile ou face et les formules de Laplace et de J. Eggenberger." Commentarii Mathematici Helvetici 2, 133 /C1/68, 1930. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Uspensky, J. V. "Approximate Evaluation of Probabilities in Bernoullian Case." Ch. 7 in Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 119 /C1/38, 1937. de Morgan’s and Bertrand’s Test BERTRAND’S TEST de Morgan’s Duality Law For every proposition involving logical addition and multiplication ("or" and "and"), there is a correspond- ing proposition in which the words "addition" and "multiplication" are interchanged. de Morgan’s Laws Let @ represent "or", S represent "and", and ? repre- sent "not." Then, for two logical units E and F, (E @ F) ?/C30E ?S F ? (E S F)?/C30E ?@ F ?: These laws also apply in the more general context of BOOLEAN ALGEBRA and, in particular, in the BOOLEAN ALGEBRA of SET THEORY , in which case @would denoteUNION , S INTERSECTION , and ? complementation with respect to any superset of E and F. References Dugundji, J. Topology. Englewood Cliffs, NJ: Prentice-Hall, 1965. Halmos, P. R. Naive Set Theory. New York: Springer- Verlag, 1974. Kelley, J. L. General Topology. New York: Springer-Verlag, 1975. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 23, 1984. Simpson, R. E. Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 540 /C1/41, 1987. de Polignac’s Conjecture Every EVEN NUMBER is the difference of two consecu- tive PRIMES in infinitely many ways (Dickson 1952, p. 424). If true, taking the difference 2, this conjec- ture implies that there are infinitely many TWIN PRIMES (Ball and Coxeter 1987). The CONJECTURE has never been proven true or refuted. See also EVEN NUMBER ,G OLDBACH CONJECTURE , TWIN PRIMES References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 64, 1987. Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, p. 76, 1989. de Polignac, A. "Six propositions arithmologiques de ´duites de crible d’E ´ratosthe `ne." Nouv. Ann. Math. 8, 423/C1/29, 1849. de Polignac, A. Comptes Rendus Paris 29, 400 and 738 /C1/39, 1849. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, 1952. de Rham Cohomology de Rham cohomology is a formal set-up for the analytic problem: If you have a DIFFERENTIAL K- FORM von a MANIFOLD M, is it the EXTERIOR DERIVATIVE of another DIFFERENTIAL K-FORM v?/? Formally, if v/C30dv?then dv/C300::This is more commonly stated as d(d/C300;meaning that if vis to be the EXTERIOR DERIVATIVE of a DIFFERENTIAL K- FORM ,a NECESSARY condition that vmust satisfy is that its EXTERIOR DERIVATIVE is zero. de Rham cohomology gives a formalism that aims to answer the question, "Are all differential k-forms on a MANIFOLD with zero EXTERIOR DERIVATIVE the EXTER- IOR DERIVATIVES of (k/C281)/-forms?" In particular, the kth de Rham cohomology vector space is defined to be the space of all k-forms with EXTERIOR DERIVATIVE 0, modulo the space of all boundaries of ( k/C281)/-forms. This is the trivial VECTOR SPACE IFF the answer to our question is yes. The fundamental result about de Rham cohomology is that it is a topological invariant of the MANIFOLD , namely: the kth de Rham cohomology VECTOR SPACE of a MANIFOLD M is canonically isomorphic to the ALEXANDER- SPANIER COHOMOLOGY VECTOR SPACE Hk(M;R) (also called cohomology with compact sup- port). In the case that M is compact, ALEXANDER- SPANIER COHOMOLOGY is exactly singular cohomol- ogy. See also ALEXANDER- SPANIER COHOMOLOGY ,CHANGE OF VARIABLES THEOREM ,C OHOMOLOGY ,D IFFEREN- TIAL K-FORM,EXTERIOR DERIVATIVE ,VECTOR SPACE de Sluze Conchoid CONCHOID OF DE SLUZE de Sluze Pearls PEARLS OF SLUZE Dead Variable DUMMY VARIABLE Debye Functions gx 0tndt et /C28 1 /C30xn1 n /C28x 2(n /C27 1) /C27X/C12 k /C301B2kx2k (2k /C27 n)(2k!)"# ; (1) where xjjB2p and Bn are BERNOULLI NUMBERS . g/C12 xtndt et /C28 1 /C30X/C12 k /C301e /C28kxxn k/C27nxn/C281 k2/C27n(n /C28 1)xn/C282 k3/C27.../C27n! kn/C271"# ; (2) where x /C210. The sum of these two integrals is g/C12 0tndt et /C28 1 /C30n!z(n /C271); (3) where z(z) is the RIEMANN ZETA FUNCTION . References Abramowitz, M. and Stegun, C. A. (Eds.). "Debye Func- tions." §27.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, p. 998, 1972. Debye’s Asymptotic Representation An asymptotic expansion for a HANKEL FUNCTION OF THE FIRST KIND H(1) n(x) /C21ffiffiffipp exp fix[cos a /C27( a /C28p=2) sin a] g/C29eip=4 X/C271 8 /C275 24tan2 a ! 3e3pi=4 2X3" /C273 128 /C2777 576tana /C27385 3456 tan4 a ! 3 /C215 e5 pi=4 22X5/C27.../C138; where n x /C30sin a; 1 /C28n x>3 x n1 =2 ; and X /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28x cos12 a ! :vuut See also H ANKEL FUNCTION OF THE FIRST KIND References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1475, 1980. Decade A power of 10. See also OCTAVE Decagon The constructible regular 10-sided POLYGON with SCHLA ¨FLI SYMBOL f10g:The INRADIUS r,CIRCUMRA- DIUS R, and AREA can be computed directly from the formulas for a general REGULAR POLYGON with side length s and n /C3010 sides, r /C301 2s cotp 10 ! /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 /C2810ffiffiffi 5p sq (1) R /C301 2 s cscp 10 ! /C30121 /C27ffiffiffi 5p0C@n0C@o s /C30 fs (2) A /C301 4 ns2 cotp 10 ! /C3052ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq s 2 : (3) Here, f is the GOLDEN MEAN . See also DECAGRAM ,D ODECAGON ,T RIGONOMETRY VALUES PI/10,UNDECAGON References Dixon, R. Mathographics. New York: Dover, p. 18, 1991. Decagonal Number A FIGURATE NUMBER OF THE FORM 4n2 /C283n : The first few are 1, 10, 27, 52, 85, ... (Sloane’s A001107). The GENERATING FUNCTION giving the decagonal numbers is x(7x /C27 1) (1 /C28 x)3 /C30x /C2710x2 /C2727x3 /C2752x4 /C27... The first few odd decagonal numbers are 1, 27, 85, 175, 297, ... (Sloane’s A028993), and the first few even decagonal numbers are 10, 52, 126, 232, 360, 540, ... (Sloane’s A028994). See also DECAGON ,FIGURATE NUMBER References Sloane, N. J. A. Sequences A001107/M4690, A028993, and A028994 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html.Decagram The STAR POLYGON f10 =3g:/ See also DECAGON ,STAR POLYGON Decahedral Graph A POLYHEDRAL GRAPH having 10 vertices. There are 32,300 nonisomorphic nonahedral graphs, as first enumerated by Duijvestijn and Federico (1981). See also POLYHEDRAL GRAPH References Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (/3/-Connected Planar) Graphs." Math. Comput. 37, 523 /C1/32, 1981. Decic Surface An ALGEBRAIC SURFACE which can be represented implicitly by a POLYNOMIAL of degree 10 in x, y, and z. An example is the BARTH DECIC . See also ALGEBRAIC SURFACE ,BARTH DECIC,CUBIC SURFACE ,QUADRATIC SURFACE ,QUARTIC SURFACE Decidable A THEORY is decidable IFF there is an algorithm which can determine whether or not any SENTENCE r is a member of the THEORY . See also CHURCH- TURING THESIS ,D ETERMINISTIC , GO¨ DEL’S COMPLETENESS THEOREM ,GO¨ DEL’S INCOM- PLETENESS THEOREM ,K REISEL CONJECTURE ,S EN- TENCE ,TARSKI’S THEOREM ,THEORY ,UNDECIDABLE References Enderton, H. B. Elements of Set Theory. New York: Aca- demic Press, 1977. Kemeny, J. G. "Undecidable Problems of Elementary Num- ber Theory." Math. Ann. 135, 160 /C1/69, 1958. Decillion In the American system, 1033. See also LARGE NUMBER Decimal The BASE -10 notational system for representing REAL NUMBERS . The expression of a number in the decimal system is called its DECIMAL EXPANSION , examples of which are 1, 13, 2028, 12.1, and 3.14159. Each number is called a decimal DIGIT , and the period placed to the right of the units place in a decimal number is called the DECIMAL POINT . See also 10,BASE (NUMBER ), BINARY ,DECIMAL POINT , HEXADECIMAL ,NEGADECIMAL ,OCTAL References Pappas, T. "The Evolution of Base Ten." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 2 /C1/, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 78 /C1/0, 1986. Decimal Comma The symbol used in continental Europe to denote a DECIMAL POINT , point example 3,14159.... See also DECIMAL POINT Decimal Expansion The decimal expansion of a number is its representa- tion in base 10. For example, the decimal expansion of 252is 625, of pis 3.14159..., and of 1 =9 is 0.1111.... Ifr/C30p=qhas a finite decimal expansion, then r/C30a1 10/C27a2 102/C27.../C27an 10n /C30a110n/C281/C27a210n/C282/C27.../C27an 10n /C30a110n/C281/C27a210n/C282/C27.../C27an 2n/C2155n: (1) FACTORING possible common multiples gives r/C30p 2a5b; (2) where pf0 (mod 2, 5). Therefore, the numbers with finite decimal expansions are fractions of this form. The number of decimals is given by max( a;b) (Wells 1986, p. 60). Numbers which have a finite decimal expansion are called REGULAR NUMBERS . Any NONREGULAR fraction m=nis periodic, and has a period l(n) independent of m, which is at most n/C281 DIGITS long. If nisRELATIVELY PRIME to 10, then the period l(n)o fm=nis a divisor of f(n) and has at most f(n)DIGITS , where fis the TOTIENT FUNCTION .I t turns out that l(n) is the HAUPT-EXPONENT of 10 (mod n) (Glaisher 1878, Lehmer 1941). When a rational number m=nwith ( m;n)/C301 is expanded, the period begins after sterms and has length t, where sandtare the smallest numbers satisfying 102/C1310s/C27t(mod n): (3) When nf0 (mod 2, 5), s/C300, and this becomes a purely periodic decimal with 10t/C131 (mod n): (4) As an example, consider n/C3084. 100/C1311 01/C1310 102/C1316 103/C13/C288 104/C1341 05/C1340 106/C13/C2820 107/C13/C2832; 108/C1316 sos/C302,t/C306. The decimal representation is 1 =84/C30 0:011910476 :When the DENOMINATOR of a fraction m=nhas the form n/C30n02a5bwith ( n0;10)/C301;then the period begins after max( a;b) terms and the length of the period is the exponent to which 10 belongs (modn 0);i.e., the number xsuch that 10x/C131 mod n0 ðÞ :Ifq isPRIME andl(q)i s EVEN , then breaking the repeat- ing DIGITS into two equal halves and adding gives all 9s. For example, 1 =7/C300:142857 ;and 142 /C27857/C30999. For 1 =qwith a PRIME DENOMINATOR other than 2 or 5, all cycles n=qhave the same length (Conway and Guy 1996). Ifnis a PRIME and 10 is a PRIMITIVE ROOT ofn, then the period l(n) of the repeating decimal 1 =nis given by l(n)/C30f(n); (5) where f(n) is the TOTIENT FUNCTION . Furthermore, the decimal expansions for p=n;with p/C301, 2, ..., n/C281 have periods of length n/C281 and differ only by a cyclic permutation. Such numbers are called LONG PRIMES by conway and guy (1996). an equivalent definition is that 10i/C131(mod n) (6) fori/C30n/C281 and no iless than this. In other words, a NECESSARY (but not SUFFICIENT ) condition is that the number 9 Rn/C281(where Rnis a REPUNIT )i s DIVISIBLE by n, which means that RnisDIVISIBLE byn. The first few numbers with maximal decimal expan-sions, called FULL REPTEND PRIMES , are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ... (Sloane’s A001913). The decimals corresponding to these are called CYCLIC NUMBERS . No general method is known for finding FULL REPTEND PRIMES . Artin conjectured that A RTIN’S CONSTANT C/C300:3739558136 . . . is the fraction of PRIMES pfor with 1 =phas decimal maximal period (Conway and Guy 1996). D. Lehmerhas generalized this conjecture to other bases, obtain- ing values which are small rational multiples of C. To find DENOMINATORS with short periods, note that 101/C281/C3032 102/C281/C3032/C21511 103 /C281 /C3033 /C21537 104 /C281 /C3032 /C21511 /C215101 105 /C281 /C3032 /C21541 /C215271 106 /C281 /C3033 /C2157 /C21511 /C21513 /C21537 107 /C281 /C3032 /C215239 /C2154649 108 /C281 /C3032 /C21511 /C21573 /C215101 /C215137 109 /C281 /C3034 /C21537 /C215333667 1010 /C281 /C3032 /C21511 /C21541 /C215271 /C2159091 1011 /C281 /C3032 /C21521649 /C215513239 1012 /C281 /C3033 /C2157 /C21511 /C21513 /C21537 /C215101 /C2159901 : The period of a fraction with DENOMINATOR equal to a PRIME FACTOR above is therefore the POWER of 10 in which the factor first appears. For example, 37 appears in the factorization of 103 /C281 and 109 /C281; so its period is 3. Multiplication of any FACTOR by a 2a5b still gives the same period as the FACTOR alone. A DENOMINATOR obtained by a multiplication of two FACTORS has a period equal to the first POWER of 10 in which both FACTORS appear. The following table gives the PRIMES having small periods (Sloane’s A046106, A046107, and A046108; Ogilvy and Anderson 1988). period primes 13 2113374 101 5 41, 271 67,137 239, 4649 8 73, 137 9 333667 10 9091 11 21649, 513239 12 9901 13 53, 79, 265371653 14 909091 15 31, 2906161 16 17, 5882353 17 2071723, 5363222357 18 19, 52579 19 1111111111111111111 20 3541, 27961A table of the periods e of small PRIMES other than the special p /C305, for which the decimal expansion is not periodic, follows (Sloane’s A002371). pepe pe 3 1 31 15 67 33 76 3 737 1 3 5 11 2 41 5 73 8 1 3 64 32 1 7 91 3 17 16 47 46 83 41 19 18 53 13 89 4423 22 59 58 97 9629 28 61 60 101 4 Shanks (1873ab) computed the periods for all PRIMES up to 120,000 and published those up to 29,989. See also DECIMAL ,DECIMAL POINT ,FRACTION ,HAUPT- EXPONENT ,MIDY’S THEOREM ,REPEATING DECIMAL References Conway, J. H. and Guy, R. K. "Fractions Cycle into Deci- mals." In The Book of Numbers. New York: Springer- Verlag, pp. 157 /C1/63 and 166 /C1/71, 1996. Das, R. C. "On Bose Numbers." Amer. Math. Monthly 56, 87/C1/9, 1949. de Polignac, A. "Note sur la divisibilite ´des nombres." Nouv. Ann. Math. 14, 118/C1/20, 1855. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 159 /C1/ 79, 1952. Glaisher, J. W. L. "Periods of Reciprocals of Integers Prime to 10." Proc. Cambridge Philos. Soc. 3, 185/C1/06, 1878. Lehmer, D. H. "Guide to Tables in the Theory of Numbers." Bulletin No. 105. Washington, DC: National Research Council, pp. 7 /C1/2, 1941. Lehmer, D. H. "A Note on Primitive Roots." Scripta Math. 26, 117/C1/19, 1963. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, p. 60, 1988. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur.Princeton, NJ: Princeton University Press, pp. 147 /C1 /63, 1957. Rao, K. S. "A Note on the Recurring Period of the Reciprocal of an Odd Number." Amer. Math. Monthly 62, 484/C1/87, 1955. Shanks, W. "On the Number of Figures in the Period of the Reciprocal of Every Prime Number Below 20,000." Proc. Roy. Soc. London 22, 200, 1873a. Shanks, W. "On the Number of Figures in the Period of the Reciprocal of Every Prime Number Between 20,000 and30,000." Proc. Roy. Soc. London 22, 384, 1873b. Shiller, J. K. "A Theorem in the Decimal Representation of Rationals." Amer. Math. Monthly 66, 797/C1 /98, 1959. Sloane, N. J. A. Sequences A001913/M4353, A002329/ M4045, A002371/M4050, A046106, A046107, andA046108 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 60, 1986. Decimal Period DECIMAL COMMA ,D ECIMAL EXPANSION ,D ECIMAL POINT Decimal Point The symbol uses to separate the integer part of a decimal number from its fractional part is called the decimal point. In the United States, the decimal point is denoted with a period (e.g., 3.1415), whereas a raised period is used in Britain (e.g., 3:1415) ; and a DECIMAL COMMA is used in continental Europe (e.g., 3,1415). The number 3.1415 is voiced "three point one four one five," while in continental Europe, 3,1415 would be voiced "three comma one four one five." See also COMMA ,DECIMAL ,DECIMAL COMMA ,DECI- MAL EXPANSION Decision Problem Does there exist an ALGORITHM for deciding whether or not a specific mathematical assertion does or does not have a proof? The decision problem is also known as the ENTSCHEIDUNGSPROBLEM (which, not so coin- cidentally, is German for "decision problem"rpar;. Using the concept of the TURING MACHINE , Turing showed the answer to be NEGATIVE for elementary NUMBER THEORY . J. Robinson and Tarski showed the decision problem is undecidable for arbitrary FIELDS . Decision Theory A branch of GAME THEORY dealing with strategies to maximize the outcome of a given process in the face of uncertain conditions. See also NEWCOMB’S PARADOX ,O PERATIONS RE- SEARCH ,PRISONER’S DILEMMA Deck Transformation The deck transformations of a UNIVERSAL COVER ˜X form a group G; which is the FUNDAMENTAL GROUP of the QUOTIENT SPACE X /C30 ˜X =G: Deck transformations are also called covering trans- formations, and are defined for any COVER p : A 0 X : They act on A by homeomorphisms which preserve the projection p. The UNIVERSAL COVER of X, denoted ˜X ; is a SIMPLY CONNECTED space and is a COVERING of p : ˜X 0 X : Every loop in X, say a function f on the unit interval with f(0) /C30f(1) /C30p ; lifts to a path ˜f /C23 ˜X ; which only depends on the choice of ˜f /C23p/C281(p); i.e., the startingpoint in the PREIMAGE of p: Moreover, the endpoint ˜f(1) depends only on the HOMOTOPY CLASS of f and ˜f(0): Given a point q /C23 ˜X ; and a; a member of the FUNDAMENTAL GROUP of X, a point a /C215q is defined to be the endpoint of a LIFT of a path f which represents a:/ For example, when X is the SQUARE TORUS then ˜X is the plane and the preimage p/C281(p) is a translation of the integer lattice f(n ;m)gƒR2 : Any loop in the torus lifts to a path in the plane, with the endpoints lying in the integer lattice. These translated integer lattices are the ORBITS of the action of Z /C29Z on R2 by addition. The above animation shows the action of some deck transformations on some disks in the plane. The spaces are the torus and its UNIVERSAL COVER , the plane. An element of the fundamental group, shown as the path in blue, defines a deck transformation of the universal cover. It moves around the points in the universal cover. The points moved to have the same projection in the torus. The blue path is a loop in the torus, and all of its preimages are shown. See also COVER ,F UNDAMENTAL GROUP ,G ROUP ACTION ,UNIVERSAL COVER References Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, pp. 163 /C1/64, 1995. Massey, W. S. A Basic Course in Algebraic Topology. New York: Springer-Verlag, pp. 130 /C1/40, 1991. Decomposable ADIFFERENTIAL K-FORM vof degree pin an EXTERIOR ALGEBRA fflVis decomposable if there exist pONE- FORMS aisuch that v/C30a1ffl...fflapi; (1) where afflbdenotes a WEDGE PRODUCT . Forms of degree 0, 1, dim V/C281;and dim Vare always decom- posable. Hence the first instance of indecomposable forms occurs in R4;in which case e1ffle2/C27e3ffle4is indecomposable. If ap-form vhas an ENVELOPE of dimension pthen it is decomposable. In fact, the ONE-FORMS in the (dual) basis to the envelope can be used as the aiabove. The P LU¨CKER RELATIONS form a system of quadratic equations on the aIin v/C30X aIei1ffl...ffleip; (2) which is equivalent to v being decomposable. Since a decomposable p-form corresponds to a p-dimensional subspace, these quadratic equations show that the GRASSMANNIAN is a PROJECTIVE VARIETY . In particu- lar, v is decomposable if for every b /C23fflp /C271 V +; i(i(b) v) v /C300 ; (3) where i denotes CONTRACTION and V + is the DUAL SPACE to V. Here is a Mathematica function which tests whether the ANTISYMMETRIC TENSOR w is decomposable. BBDiscreteMath‘Combinatorica‘; ContractAll[a_List, b_List] : /C30 Module[{k /C30 TensorRank[a] - TensorRank[b]}, If[k /C21/C30 0, Map[Flatten[#1].Flatten[b] &, a, {k}], ContractAll[b, a] ] ] Envelope[a_List?VectorQ] : /C30 Select[{a}, #1 ! /C30 Table[0, {Length[a]}] &] Envelope[a_List] : /C30 Module[ { z, inds, vects, d /C30 Dimensions[a][[1]], r /C30 TensorRank[a] }, z /C30 Table[0, ##1] & @@ Table[{d}, {r - 1}]; inds /C30 KSubsets[Range[d], r - 1]; vects /C30 Map[ContractAll[a, ReplacePart[z, 1, #1]] &, inds]; Select[RowReduce[vects], #1 ! /C30 Table[0, {d}] &] ] DecomposableQ[a_?ListQ] : /C30 (Length[Envelope[a]] /C30/C30 TensorRank[a]) See also CONTRACTION (TENSOR ), EXTERIOR ALGEBRA , GRASSMANNIAN ,PLU¨ CKER RELATIONS ,VECTOR SPACE , WEDGE PRODUCT References Sternberg, S. Differential Geometry. New York: Chelsea, pp. 14 /C1/0, 1983. Decomposition A rewriting of a given quantity (e.g., a MATRIX )in terms of a combination of "simpler" quantities. See also CHOLESKY DECOMPOSITION ,COMPOSITION , CONNECTED SUM DECOMPOSITION ,JACO-SHALEN- JO- HANNSON TORUS DECOMPOSITION ,LUD ECOMPOSI- TION ,P RIME FACTORIZATION ,QRD ECOMPOSITION , SINGULAR VALUE DECOMPOSITION Decomposition Group References Koch, H. "Decomposition Group and Ramification Group." §6.1 in Number Theory: Algebraic Numbers and Func- tions. Providence, RI: Amer. Math. Soc., pp. 172 /C1/76, 2000.Deconvolution The inversion of a CONVOLUTION equation, i.e., the solution for f of an equation OF THE FORM f + g ¼ h þ e; given g and h, where o is the NOISE and + denotes the CONVOLUTION . Deconvolution is ill-posed and will usually not have a unique solution even in the absence of NOISE . Linear deconvolution ALGORITHMS include INVERSE FILTERING and WIENER FILTERING . Nonlinear ALGO- RITHMS include the CLEAN algorithm, MAXIMUM ENTROPY METHOD , and LUCY. See also CONVOLUTION , LUCY, MAXIMUM ENTROPY METHOD ,W IENER FILTER References Cornwell, T. and Braun, R. "Deconvolution." Ch. 8 in Synthesis Imaging in Radio Astronomy: Third NRAO Summer School, 1988 (Ed. R. A. Perley, F. R. Schwab, and A. H. Bridle). San Francisco, CA: Astronomical So- ciety of the Pacific, pp. 167 /C1/83, 1989. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Convolution and Deconvolution Using the FFT." §13.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 531 /C1/37, 1992. Decreasing Function A function f(x) decreases on an INTERVAL I if fbðÞB faðÞfor all b /C21a, where a; b /C23 I : Conversely, a function f(x) increases on an INTERVAL I if fbðÞ> faðÞfor all b /C21a with a ;b /C23 I :/ If the DERIVATIVE f ?(x)ofa CONTINUOUS FUNCTION f(x) satisfies f ?(x) B0onan OPEN INTERVAL (a, b), then f(x) is decreasing on (a, b). However, a function may decrease on an interval without having a derivative defined at all points. For example, the function /C28x1=3 is decreasing everywhere, including the origin x /C300, despite the fact that the DERIVATIVE is not defined at that point. See also DERIVATIVE ,INCREASING FUNCTION ,N ON- DECREASING FUNCTION ,NONINCREASING FUNCTION References Jeffreys, H. and Jeffreys, B. S. "Increasing and Decreasing Functions." §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988. Decreasing Sequence A SEQUENCE a1 ;a2 ::: fg for which a1]a2]...:/ See also INCREASING SEQUENCE ,SEQUENCE Decreasing Series ASERIES s1;s2;. . . for which s1]s2]...:/ Dedekind Cut A set partition of the RATIONAL NUMBERS into two nonempty subsets S1 and S2 such that all members of S1are less than those of S2and such that S1has no greatest member. REAL NUMBERS can be defined using either Dedekind cuts or CAUCHY SEQUENCES . See also CANTOR- DEDEKIND AXIOM ,C AUCHY SE- QUENCE References Courant, R. and Robbins, H. "Alternative Methods of Defining Irrational Numbers. Dedekind Cuts." §2.2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 71 /C1/2, 1996. Jeffreys, H. and Jeffreys, B. S. "Nests of Intervals: Dedekind Section." §1.031 in Methods of Mathematical Physics, 3rd ed.Cambridge, England: Cambridge University Press, pp. 6/C1/, 1988. Dedekind Eta DEDEKIND ETAFUNCTION Dedekind Eta Function Let q/C30e2pit; (1) then the Dedekind eta function is defined over the UPPER HALF-PLANE H/C30t:I½t/C138>0 fg by h(t)/C13q1=24Y/C12 n/C3011/C28qnðÞ /C30q;qðÞ/C12; (2)which can be written as h(t)/C30q1=241/C27X/C12 n/C301(/C281)nqn3n/C281 ðÞ =2/C27qn3n/C271 ðÞ =20C10CC() (3) (Weber 1902, pp. 85 and 112; Atkin and Morain 1993). h(t)i sa MODULAR FORM first introduced by Dedekind in 1877, and is related to the MODULAR DISCRIMINANT of the W EIERSTRASS ELLIPTIC FUNCTION by D(t)/C30(2p)12h(p)½/C13824(4) (Apostol 1997, p. 47). The derivative of h(t) satisfies /C284pid dtlnh(t)½/C138/C30G2(t) (5) d dtln/C281 t"# /C30d dtlnh(t)½/C138/C271 2d drln(/C28it); (6) where G2(t)i sa n EISENSTEIN SERIES . Letting z24/C30e2pi=24/C30epi=12be a ROOT OF UNITY ,h(t) satisfies h(t/C271)/C30epi=12h(t) (7) h(t/C27n)/C30epin=12h(t) (8) h/C281 t ! /C30ffiffiffiffiffiffiffiffi /C28itp h(t) (9) where nis an integer (Weber 1902, p. 113; Atkin and Morain 1993; Apostol 1997, p. 47). The Dedekind eta function is related to the J ACOBI THETA FUNCTION q3 by q30;epit0CB0C@ /C30h21 2t/C271 ðÞ ! h(t/C271)(10) (Apostol 1997, p. 91). Macdonald (1972) has related most expansions OF THE FORM q;qðÞc /C12to affine ROOT SYSTEMS . Exceptions not included in Macdonald’s treatment include c/C302, found by Hecke and Rogers, c/C304, found by Ramanu- jan , and c/C3026, found by Atkin (Leininger and Milne 1997). Using the Dedekind eta function, the J ACOBI TRIPLE PRODUCT identity is written q;qðÞ3/C12/C30X/C12 n/C300(/C281)n(2n/C271)qnn/C271 ðÞ =2(11) (Jacobi 1829, Hardy and Wright 1979, Leininger and Milne 1997, Hirschhorn 1999). Dedekind’s functional equation states that ifab cd0C10CC /C23G; where Gis the MODULAR GROUP GAMMA ,c/C210, and t/C23H;then ha t /C27 b c t /C27 d ! /C30 e(a ;b;c ;d) /C28iffiffiffiffiffiffiffiffiffiffiffiffiffi c t /C27dphi h( t); (12) where e(a; b;c ;d) /C30exp pia /C27 d 12c/C27s /C28d;c ðÞ !"# ; (13) and sh;kðÞ/C30Xk /C281 r/C301r khr k/C28hr k"# /C281 2 ! (14) is a DEDEKIND SUM (Apostol 1997, pp. 52 /C1/7), with xbc the FLOOR FUNCTION . See also DIRICHLET ETA FUNCTION ,DEDEKIND SUM, ELLIPTIC LAMBDA FUNCTION ,INFINITE PRODUCT , INVARIANT (ELLIPTIC FUNCTION ), JACOBI THETA FUNCTIONS ,KLEIN’S ABSOLUTE INVARIANT , Q-SERIES , TAU FUNCTION ,W EBER FUNCTIONS References Apostol, T. M. "The Dedekind Eta Function." Ch. 3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 47 /C1/3, 1997. Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Math. Comput. 61,29/C1/8, 1993. Bhargava, S. and Somashekara, D. "Some Eta-Function Identities Deducible from Ramanujan’s 1c1 Summation." J. Math. Anal. Appl. 176, 554 /C1/60, 1993. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hirschhorn, M. D. "Another Short Proof of Ramanujan’s Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580 /C1/83, 1999. Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum Borntrae- ger, p. 90, 1829. Leininger, V. E. and Milne, S. C. "Some New Infinite Families of Eta Function Identities." Preprint. http:// www.math.ohio-state.edu/~milne/preprints.html. Leininger, V. E. and Milne, S. C. "Expansions for qðÞn2/C27n /C12and Basic Hypergeometric Series in U(n) :/" Preprint. http:// www.math.ohio-state.edu/~milne/preprints.html. Ko¨hler, G. "Some Eta-Identities Arising from Theta Series." Math. Scand. 66, 147 /C1/54, 1990. Macdonald, I. G. "Affine Root Systems and Dedekind’s h/- Function." Invent. Math. 15,91/C1/43, 1972. Ramanujan, S. "On Certain Arithmetical Functions." Trans. Cambridge Philos. Soc. 22, 159 /C1/84, 1916. Siegel, C. L. "A Simple Proof of h /C281=t ðÞ /C30 htðÞffiffiffiffiffiffiffi t =ip :/" Math- ematika 1, 4, 1954. Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1902. Dedekind Function c(n) /C30nY distinct prime factors p of n1 /C27p /C2810CB0C@ where the PRODUCT is over the distinct PRIME FAC-TORS of n. The first few values are 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, ... (Sloane’s A001615). See also DEDEKIND ETA FUNCTION ,EULER PRODUCT , TOTIENT FUNCTION References Cox, D. A. Primes of the Form x2/C27ny2:Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, p. 228, 1997. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 96, 1994. Sloane, N. J. A. Sequences A001615/M2315 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Dedekind Number ANTICHAIN Dedekind Ring A abstract commutative RING in which every NON- ZERO IDEAL is a unique product of PRIME IDEALS . References Noether, E. "Abstract Development of Ideal Theory in Algebraic Number Fields and Function Fields." Math. Ann. 96,2 6/C1/1, 1927. Dedekind Section DEDEKIND CUT Dedekind Sum Given RELATIVELY PRIME INTEGERS pand q(i.e., (p;q)/C301);the Dedekind sum is defined by sp;qðÞ/C13Xq i/C301i q ! ! pi q ! ! ; (1) where (x)ðÞ/C13x/C28xbc/C281 2xQZ 0 x/C23Z;8 < :(2) with xbcthe FLOOR FUNCTION .(x)ðÞ is an ODD FUNC- TION since ( x)ðÞ/C30/C28 (x)ðÞ and is periodic with period 1. The Dedekind sum is meaningful even if ( p;q)"1;so the relatively prime restriction is sometimes dropped (Apostol 1997, p. 72). The symbol s(p;q) is sometimes used instead of s(p;a) (Beck 2000). The Dedekind sum can also be expressed in the form s(p;q)/C301 4qXq/C281 r/C301cotppr k ! cotpr q ! : (3) If 0BhBk;letr0;r1;...,rn/C271denote the remainders in the E UCLIDEAN ALGORITHM given by r0/C30k (4) r1/C30h (5) rj þ1 /C13rj/C01 ðmod rj Þ (6) for 1 5rj/C271 Brj and rn /C271 /C301: Then sh;kðÞ/C301 12Xn/C271 j/C301/C281ðÞj /C271r2 j/C27 r2j/C281 /C27 1 rjrj/C281() /C28/C281ðÞn/C271 8 (7) (Apostol 1997, pp. 72 /C1/3). In general, there is no simple formula for closed-form evaluation of s(p ;q) ; but some special cases are s(1;q) /C30(q /C28 1)(q /C28 2) 12q (8) s 2 ;q odd ðÞ /C30(q /C28 1)(q /C28 2) 24q (9) (Apostol 1997, p. 62). Apostol (1997, p. 73) gives the additional special cases 12hks h;kðÞ/C30 k /C281 ðÞ k /C28h2 /C2810CB0C@ for k /C131 (mod h)(10) 12hks h ;kðÞ/C30 k /C282 ðÞ k /C281 2h2 /C2710CB0C@"# for k /C132 (mod h)(11) 12hks h; kðÞ/C30k2 /C27 h2 /C286h /C2720CB0C@ k /C27h2 /C271 for k /C13/C281 (mod h)(12) 12hks h;kðÞ/C30k2 /C28h2 /C28 tr/C28 1 ðÞ r /C28 2 ðÞ h /C27 r2 /C27 1 r k /C27h2 /C271 (13) for k /C13r modh ðÞ and h /C13t (mod r) ; where r ]1 and t ¼ 91: Finally, 12hks(h;k) /C30k2 /C28h2 /C27 4r(t /C28 2)(t /C27 2)h /C27 26 5 k /C27h2 /C271 (14) for k /C135 (mod h) and h /C13t (mod5) ; where t ¼91or 9 2. Dedekind sums obey 2-term s(p;q) /C27s(q ;p) /C30/C281 4 /C271 12p q /C27q p /C271 pq ! (15) (Dedekind 1953; Rademacher and Grosswald 1972; Pommersheim 1993; Apostol 1997, pp. 62 /C1/4) and 3- term sbc?;a ðÞ /C27sca?;b ðÞ /C27sab?;c ðÞ /C30/C281 4 /C271 12a bc /C27b ca /C27c ab ! (16)(Rademacher 1954), reciprocity laws, where a, a ?; b, b?; and c, c? are pairwise COPRIME , and aa ?/C131 (mod b) (17) bb ?/C131 (mod c) (18) cc0/C131 (mod a) (19) (Pommersheim 1993). /6ps(p ;q) is an integer, and if u /C30(3;q) ; then 12pqs(p ;q) /C130 (mod up) (20) and 12pqs(q;p) /C13q2 /C271 (mod up): (21) In addition, s(p ;q) satisfies the congruence 12qs(p ;q) /C13(q /C281)(q /C272) /C284p(q /C281) /C274X rBq =22pr q$% (mod 8); (22) which, if q is odd, becomes 12qs(p;q) /C13q /C281 /C274X rBq=22pr q$% (mod 8) (23) (Apostol 1997, pp. 65 /C1/6). If q /C303, 5, 7, or 13, let r /C30 24 =(q /C281); let integers a, b, c, d be given with ad /C28 bc /C301 such that c /C30c1q and c1 > 0; and let d /C30 s(a; c) /C28a /C27 d 12c() /C28 s(a1;c1) /C28a /C27 d 12c1() : (24) Then rd is an even integer (Apostol 1997, pp. 66 /C1/9). Let p, q, u, v /C23N with (p;q) /C30(u;v) /C301 (i.e., are pairwise RELATIVELY PRIME ), then the Dedekind sums also satisfy s(p;q)/C27s(u;v) /C30s(pu?/C28qv?;pv/C27qu)/C281 4/C271 12q vt/C27v tq/C27t qv ! ;(25) where t/C30pv/C27qu;and u?;v?are any INTEGERS such that uu?/C27vv?/C301 (Pommersheim 1993). Ifpis prime, then (p/C271)s(h;k)/C30s(ph;k)/C27Xp/C281 m/C300s(h/C27mk;pk) (26) (Dedekind 1953; Apostol 1997, p. 73). Moreover, it has been beautifully generalized by Knopp (1980). See also DEDEKIND ETA FUNCTION ,ISEKI’S FORMULA References Apostol, T. M. "Properties of Dedekind Sums," "The Reci- procity Law for Dedekind Sums," and "Congruence Prop- erties of Dedekind Sums." §3.7/C1/.9 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 52 and 61 /C1/9, 1997. Apostol, T. M. Ch. 12 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976. Beck, M. "Dedekind Cotangent Sums." Submitted. Dedekind, R. "Erlauterungen zu den Fragmenten, XXVIII." In Collected Works of Bernhard Riemann. New York: Dover, pp. 466 /C1/78, 1953. Iseki, S. "The Transformation Formula for the Dedekind Modular Function and Related Functional Equations." Duke Math. J. 24, 653 /C1/62, 1957. Knopp, M. I. "Hecke Operators and an Identity for Dedekind Sums." J. Number Th. 12,2/C1/, 1980. Pommersheim, J. "Toric Varieties, Lattice Points, and Dedekind Sums." Math. Ann. 295,1/C1/4, 1993. Rademacher, H. "Generalization of the Reciprocity Formula for Dedekind Sums." Duke Math. J. 21, 391 /C1/98, 1954. Rademacher, H. and Grosswald, E. Dedekind Sums. Wa- shington, DC: Math. Assoc. Amer., 1972. Rademacher, H. and Whiteman, A. L. "Theorems on Dede- kind Sums." Amer. J. Math. 63, 377 /C1/07, 1941. Dedekind’s Axiom For every partition of all the points on a line into two nonempty SETS such that no point of either lies between two points of the other, there is a point of one SET which lies between every other point of that SET and every point of the other SET. Dedekind’s Problem The determination of the number of monotone BOO- LEAN FUNCTIONS of n variables (equivalent to the number of ANTICHAINS on the n-set 1; 2;:::; n fg )is called Dedekind’s problem. See also ANTICHAIN ,BOOLEAN FUNCTION References Dedekind, R. "U¨ ber Zerlegungen von Zahlen durch ihre gro¨ssten gemeinsammen Teiler." In Gesammelte Werke, Bd. 1. pp. 103 /C1/48, 1897. Kleitman, D. "On Dedekind’s Problem: The Number of Monotone Boolean Functions." Proc. Amer. Math. Soc. 21, 677 /C1/82, 1969 677 /C1/82. Kleitman, D. and Markowsky, G. "On Dedekind’s Problem: The Number of Isotone Boolean Functions. II." Trans. Amer. Math. Soc. 213, 373 /C1/90, 1975. Deducible If q is logically deducible from p, this is written p /C159 q:/ Deep Theorem Qualitatively, a deep theorem is a theorem whose proof is long, complicated, difficult, or appears to involve branches of mathematics which are not obviously related to the theorem itself (Shanks 1993). Shanks (1993) cites the QUADRATIC RECIPRO- CITY THEOREM as an example of a deep theorem. See also THEOREM ,TRIVIALReferences Shanks, D. "Is the Quadratic Reciprocity Law a Deep Theorem?" §2.25 in Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 64 /C1/6, 1993. Defective Matrix A MATRIX whose EIGENVECTORS are not COMPLETE . Defective Number DEFICIENT NUMBER Deficiency Given BINOMIAL COEFFICIENTN k0CB0C@ ; write N /C28k /C27i /C30aibi ; for 1 5i 5k; where bicontains only those prime factors > k: Then the number of i for which bi /C301 (i.e., for which all the factors of N /C28k /C27i are 5k is called the deficiency ofN k0CB0C@ (Erdos et al. 1993, Guy 1994). The following table gives the GOOD BINOMIAL COEFFICIENTS (i.e., those with 1 pf N k0CB0C@ > kÞ) having deficiency d ]1 (Erdos et al. 1993), and Erdos et al. (1993) conjecture that there are no other with d /C211. d Good Binomial Coefficients 1 /3 20C@80C@9 ;730C@80C@9 ;13 40C@80C@9 ;14 40C@80C@9 ;23 50C@80C@9 ;62 60C@80C@9 ;89 80C@80C@9 ; ... 2 /740C@80C@9 ;44 80C@80C@9 ;74100C@80C@9 ;174 120C@80C@9 ;239 140C@80C@9 ;5179 270C@80C@9 ; / /8413 280C@80C@9 ;96622 420C@80C@9 / 3 /46100C@80C@9 ;47100C@80C@9 ;241 160C@80C@9 ;2105 250C@80C@9 ;1119 270C@80C@9 ; 6459 330C@80C@9 / 4 /47110C@80C@9 / 9 /284 280C@80C@9 / See also ABUNDANCE ,GOOD BINOMIAL COEFFICIENT References Erdos, P.; Lacampagne, C. B.; and Selfridge, J. L. "Esti- mates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215/C1/24, 1993. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 84 /C1/5, 1994. Deficient Number Numbers which are not PERFECT and for which s(N) /C13 s(N) /C28N BN ; or equivalently s(n) B2n; where s(N) is the DIVISOR FUNCTION . Deficient num- bers are sometimes called DEFECTIVE NUMBERS (Singh 1997). PRIMES , PRIME POWERS , and any divisors of a PERFECT or deficient number are all deficient. The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, ... (Sloane’s A005100). See also ABUNDANT NUMBER ,L EAST DEFICIENT NUMBER ,PERFECT NUMBER References Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3 /C1/3, 1952. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 45, 1994. Singh, S. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker, p. 11, 1997. Sloane, N. J. A. Sequences A005100/M0514 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Souissi, M. Un Texte Manuscrit d’Ibn Al-Banna’ Al-Marra- kusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975. Definable Set An ANALYTIC ,BOREL ,or COANALYTIC SET. Defined If A and B are equal by definition (i.e., A is defined as B), then this is written symbolically as A /C13B; A:/C30B; or sometimes ‹:/ Definite Integral An INTEGRAL gb af(x)dx with upper and lower limits. The first FUNDAMENTAL THEOREM OF CALCULUS allows definite integrals to be computed in terms of INDEFINITE INTEGRALS , since if F is the INDEFINITE INTEGRAL for f(x) ; then gb af(x)dx /C30F(b) /C28F(a) : See also CALCULUS ,F UNDAMENTAL THEOREMS OF CALCULUS ,INDEFINITE INTEGRAL ,INTEGRALDegen’s Eight-Square Identity See also EULER FOUR- SQUARE IDENTITY ,FIBONACCI IDENTITY Degeneracy The property of being DEGENERATE . See also DEGENERATE Degenerate A limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. For example, the POINT is a degenerate case of the CIRCLE as the RADIUS approaches 0, and the CIRCLE is a degenerate form of an ELLIPSE as the ECCENTRICITY approaches 0. Another example is the two identical ROOTS of the second-order POLYNOMIAL (x /C281)2 : Since the n ROOTS of an nth degree POLY- NOMIAL are usually distinct, ROOTS which coincide are said to be degenerate. Degenerate cases often require special treatment in numerical and analytical solu- tions. For example, a simple search for both ROOTS of the above equation would find only a single one: 1. The word degenerate also has several very specific and technical meanings in different branches of mathematics. See also TRIVIAL References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 513 /C1/14, 1985. Degree The word "degree" has many meanings in mathe- matics. The most common meaning is the unit of ANGLE measure defined such that an entire rotation is 3608. This unit harks back to the Babylonians, who used a base 60 number system. 3608 likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The degree is subdivided into 60 MINUTES per degree, and 60 SECONDS per MINUTE . The word "degree" is also used in many contexts where it is synonymous with "order," as applied for example to polynomials. See also ARC MINUTE ,ARC SECOND ,DEGREE (EXTEN- SION FIELD), DEGREE OF FREEDOM ,DEGREE (MAP), DEGREE (POLYNOMIAL ), DEGREE (VERTEX ), INDEGREE , LOCAL DEGREE ,OUTDEGREE References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 276, 1997. Degree (Algebraic Surface) ORDER (ALGEBRAIC SURFACE ) Degree (Extension Field) The degree (or relative degree, or index) of an EXTENSION FIELD K =F ; denoted K : F ½/C138 ; is the dimen- sion of K as a VECTOR SPACE over F, i.e., K : F ½/C138 /C30dimFK : If K : F ½/C138 is finite, then the extension is said to be finite; otherwise, it is said to be infinite. See also EXTENSION FIELD References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 424, 1998. Degree (Map) Let f : M /C2N be a MAP between two compact, con- nected, oriented n-D MANIFOLDS without boundary. Then f induces a HOMOMORPHISM f/C31 from the HOMOL- OGY GROUPS Hn(M)to Hn(N); both canonically iso- morphic to the INTEGERS , and so f/C31 can be thought of as a HOMOMORPHISM of the INTEGERS . The INTEGER d(f) to which the number 1 gets sent is called the degree of the MAP f. There is an easy way to compute d(f) if the MANI- FOLDS involved are smooth. Let x /C23N; and approx- imate f by a smooth map HOMOTOPIC to f such that x is a "regular value" of f (which exist and are every- where by SARD’S THEOREM ). By the IMPLICIT FUNCTION THEOREM , each point in f /C281(x) has a NEIGHBORHOOD such that f restricted to it is a DIFFEOMORPHISM .If the DIFFEOMORPHISM is orientation preserving, as- sign it the number /C271 ; and if it is orientation reversing, assign it the number /C281. Add up all the numbers for all the points in f /C281(x) ; and that is the d(f); the degree of f. One reason why the degree of a map is important is because it is a HOMOTOPY invariant. A sharper result states that two self- maps of the n-sphere are homotopic IFF they have the same degree. This is equivalent to the result that the nth HOMOTOPY GROUP of the n-SPHERE is the set Z of INTEGERS . The ISOMORPHISM is given by taking the degree of any representation. One important application of the degree concept is that homotopy classes of maps from n-spheres to n- spheres are classified by their degree (there is exactly one homotopy class of maps for every INTEGER n, and n is the degree of those maps). Degree (Polynomial) The highest POWER in a UNIVARIATE POLYNOMIAL is known as its degree, or sometimes "order." For example, the POLYNOMIALP(x) /C30anxn /C27.../C27a2x2 /C27a1x /C27a0 is of degree n, denoted P(x) /C30n: The degree of a polynomial is implemented in Mathematica as Ex- ponent [poly, x]. See also ORDER (POLYNOMIAL ) Degree (Vertex) VERTEX DEGREE Degree Matrix A DIAGONAL MATRIX corresponding to a GRAPH that has the VERTEX DEGREE of viin the ith position (Skiena 1990, p. 235). See also VERTEX DEGREE References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Degree of Freedom The number of degrees of freedom in a problem, distribution, etc., is the number of parameters which may be independently varied. See also LIKELIHOOD RATIO Degree Sequence Given an UNDIRECTED GRAPH , a degree sequence is a monotonic nonincreasing sequence of the VERTEX DEGREES (valencies) of its VERTICES . The number of degree sequences for a graph of a given order is closely related to GRAPHICAL PARTITIONS . The mini- mum vertex degree in a GRAPH Gis denoted d(G);and the maximum degree is denoted D(G) (Skiena 1990, p. 157). A GRAPH whose degree sequence contains multiple copies of a single integer is called a REGULAR GRAPH . A graph corresponding to a given degree sequence can be constructed using RealizeDegree- Sequence [d] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). It is possible for two topologically distinct graphs to have the same DEGREE SEQUENCE . The number of distinct degree sequences for graphs of n /C301, 2, ... nodes are given by 1, 2, 4, 11, 31, 102, 342, ... (Sloane’s A004251), compared with the total num- ber of nonisomorphic simple undirected graphs with n NODES of 1, 2, 4, 11, 34, 156, 1044, ... (Sloane’s A000088). The first order having fewer degree se- quences than number of nonisomorphic graphs is therefore n /C305. For the graphs illustrated above, the degree sequences are given in the following table. 1 / f0g/ 2 / f0;0g;f1; 1g/ 3 / f0;0; 0g;f1;1;0 g;f2 ;1;1 g;f2 ;2;2 g/ 4 / f0;0; 0;0g;f1;1 ;0;0 g;f2 ;1;1 ;0g;f2;2 ;2;0 g;/ / f3;2; 2;1g;f3;3 ;2;2 g;f3 ;3;3 ;3g;f1;1 ;1;1 g;/ / f2;2; 1;1g;f2;2 ;2;2 g;f3 ;1;1 ;1g/ The possible sums of elements for a degree sequence of order n are 0, 2, 4, 6, ..., n(n /C281):/ A degree sequence is said to be k-connected if there exists some k-CONNECTED GRAPH corresponding to the degree sequence. For example, while the degree sequence f1;2; 1g is 1- but not 2-connected, f2;2; 2g is 2-connected. See also DEGREE SET,D EGREE (VERTEX ), GRAPHIC SEQUENCE ,G RAPHICAL PARTITION , K -CONNECTED GRAPH ,REGULAR GRAPH References Ruskey, F. "Information on Degree Sequences." http:// www.theory.csc.uvic.ca/~cos/inf/nump/DegreeSequen- ces.html. Ruskey, F.; Cohen, R.; Eades, P.; and Scott, A. "Alley CATs in Search of Good Homes." Congres. Numer. 102,97/C1/10, 1994.Skiena, S. "Realizing Degree Sequences." §4.4.2 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 157 /C1/60, 1990. Sloane, N. J. A. Sequences A004251/M1250 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Degree Set The set of integers which make up a DEGREE SEQUENCE . Any set of positive integers is the degree set for some graph. See also DEGREE SEQUENCE References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 167, 1990. Dehn Invariant An invariant defined using the angles of a 3-D POLYHEDRON . It remains constant under solid DISSEC- TION and reassembly. Solids with the same VOLUME can have different Dehn invariants. Two POLYHEDRA can be dissected into each other only if they have the same volume and the same Dehn invariant. In 1902, Dehn showed that two interdis- sectable polyhedra must have equal Dehn invariants, settling the third of HILBERT’S PROBLEMS , and Sydler (1965) showed that two polyhedra with the same Dehn invariants are interdissectable. See also DISSECTION ,E HRHART POLYNOMIAL ,H IL- BERT’S PROBLEMS References Sydler, J.-P. "Conditions ne´cessaires et suffisantes pour l’e´quivalence des polye`dres de l’espace euclidean a` trois dimensions." Comment. Math. Helv. 40,43/C1/0, 1965. Dehn Surgery The operation of drilling a TUBULAR NEIGHBORHOOD of a KNOT K in S3 and then gluing in a solid TORUS so that its meridian curve goes to a (p, q)-curve on the TORUS boundary of the KNOT exterior. Every compact connected 3- MANIFOLD comes from Dehn surgery on a LINK inS3:/ See also KIRBY CALCULUS ,TUBULAR NEIGHBORHOOD References Adams, C. C. "The Poincare ´Conjecture, Dehn Surgery, and the Gordon-Luecke Theorem." §9.3 in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 257 /C1/63, 1994. Dehn’s Lemma An embedding of a 1-SPHERE in a 3-MANIFOLD which exists continuously over the 2-DISK also extends over the DISK as an embedding. This theorem was pro- posed by Dehn in 1910, but a correct proof was not obtained until the work of Papakyriakopoulos (1957ab). References Hempel, J. 3-Manifolds. Princeton, NJ: Princeton Univer- sity Press, 1976. Papakyriakopoulos, C. D. "On Dehn’s Lemma and the Asphericity of Knots." Proc. Nat. Acad. Sci. USA 43, 169 /C1/72, 1957a. Papakyriakopoulos, C. D. "On Dehn’s Lemma and the Asphericity of Knots." Ann. Math. 66,1/C1/6, 1957b. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 100 /C1/01, 1976. Del GRADIENT Del Bar Operator The operator ¯@ is defined on a COMPLEX MANIFOLD , and is called the ‘del bar operator.’ The EXTERIOR DERIVATIVE d takes a function and yields a ONE-FORM . It decomposes as d /C30@/C27 ¯@; (1) as complex ONE-FORMS decompose into TYPE L1 ¼L1;0 /C156L0 ;1 (2) where /C156denotes the DIRECT SUM. More concretely, in coordinates zk /C30xk /C27iyk ; @f /C30X @f @xk/C28i@f @yk ! dzk (3) and ¯@f /C30X @f @xk/C27i@f @yk ! d¯zk : (4) These operators extend naturally to forms of higher degree. In general, if a is a (p, q)-FORM , then @ a is a (p /C271;q)/-form and ¯@ a is a (p ;q /C271)/-form. The equation ¯@f /C300 expresses the condition of f being a HOLO- MORPHIC FUNCTION . More generally, a (p ;0)/-FORM a is called HOLOMORPHIC if¯@ a /C300 ; in which case its coefficients, as written in a COORDINATE CHART , are HOLOMORPHIC FUNCTIONS . The del bar operator is also well-defined on SECTIONS of a HOLOMORPHIC VECTOR BUNDLE . The reason is because a change in coordinates or trivializations is HOLOMORPHIC . See also ALMOST COMPLEX STRUCTURE ,A NALYTIC FUNCTION ,CAUCHY- RIEMANN EQUATIONS ,COMPLEX MANIFOLD ,COMPLEX FORM (TYPE), DIFFERENTIAL K-FORM,DOLBEAULT COHOMOLOGY ,DOLBEAULT OPERA- TORS ,HOLOMORPHIC FUNCTION ,HOLOMORPHIC VEC- TOR BUNDLE References Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, 1994. Weil, A. Introduction a `l’e´tude des varie ´te`sK a¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer- site´de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, pp. 27 /C1/5, 1980. Del Pezzo Surface ASURFACE which is related to C AYLEY NUMBERS . References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 211, 1973. Hunt, B. "Del Pezzo Surfaces." §4.1.4 in The Geometry of Some Special Arithmetic Quotients. New York: Springer- Verlag, pp. 128 /C1/29, 1996. Delambre’s Analogies GAUSS’S FORMULAS Delannoy Number The Delannoy numbers are the number of lattice paths from (0 ;0) to ( b, a) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (i.e, 0;/C160;andP):They are given by the RECURRENCE RELATION D(a;b)/C30D(a/C281;b)/C27D(a;b/C281)/C27D(a/C281;b/C281);(1) with D(0;0)/C301:They have the GENERATING FUNC- TION X/C12 p;q/C301D(p;q)xpyq/C30(1/C28x/C28y/C28xy)/C281(2) (Comtet 1974, p. 81). For n /C13a /C30b; the Delannoy numbers are the number of "king walks" D(n;n) /C30Pn(3) ; where Pn(x)isaL EGENDRE POLYNOMIAL (Moser 1955; Comtet 1974, p. 81; Vardi 1991). Another expression is D(n;n) /C30Xn k /C300n k0C@80C@9 n /C27k k0C@80C@9 /C302F1(/C28n ;n /C271;1 ;/C281); (3) wherea b0CB0C@ is a BINOMIAL COEFFICIENT and 2F1(a ;b;c;z)isa HYPERGEOMETRIC FUNCTION . The values of D(n ;n) for n /C301, 2, ... are 3, 13, 63, 321, 1683, 8989, 48639, ... (Sloane’s A001850). The SCHRO ¨ DER NUMBERS bear the same relation to the Delannoy numbers as the CATALAN NUMBERS do to the BINOMIAL COEFFICIENTS . See also BINOMIAL COEFFICIENT ,CATALAN NUMBER , MOTZKIN NUMBER ,SCHRO ¨ DER NUMBER References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 80 /C1/1, 1974. Dickau, R. M. "Delannoy and Motzkin Numbers." http:// www.prairienet.org/~pops/delannoy.html. Goodman and Narayana. "Lattice Paths with Diagonal Steps." U. Alberta. No. 39, 1967. Moser, L. "King Paths on a Chessboard." Math. Gaz. 39, 54, 1955. Moser, L. and Zayachkowski, H. S. "Lattice Paths with Diagonal Steps." Scripta Math. 26, 223 /C1/29, 1963. Sloane, N. J. A. Sequences A001850/M2942 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stocks, D. R. Jr. "Lattice Paths in E3 with Diagonal Steps." Canad. Math. Bull. 10, 653 /C1/58, 1967. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, 1991. Delaunay Triangulation The Delaunay triangulation is a TRIANGULATION which is equivalent to the NERVE of the cells in a VORONOI DIAGRAM , i.e., that triangulation of the CONVEX HULL of the points in the diagram in which every CIRCUMCIRCLE of a TRIANGLE is an empty circle (Okabe et al. 1992, p. 94). The Mathematica com- mand PlanarGraphPlot [pts] in the Mathematica add-on package DiscreteMath‘Computational- Geometry‘ (which can be loaded with the commandBBDiscreteMath‘ ) plots the Delaunay triangula- tion of the given list of points. The Delaunay triangulation and VORONOI DIAGRAM in R2 are dual to each other. See also TRIANGULATION ,VORONOI DIAGRAM References Lee, D. T. and Schachter, B. J. "Two Algorithms for Con- structing a Delaunay Triangulation." Int. J. Computer Information Sci. 9, 219 /C1/42, 1980. Okabe, A.; Boots, B.; and Sugihara, K. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. New York: Wiley, 1992. Preparata, F. R. and Shamos, M. I. Computational Geome- try: An Introduction. New York: Springer-Verlag, 1985. Delian Constant The number 21 =3 (the CUBE ROOT of 2) which is to be constructed in the CUBE DUPLICATION problem. This number is not a EUCLIDEAN NUMBER although it is an ALGEBRAIC of third degree. See also CUBE,C UBE DUPLICATION ,C UBE ROOT, GEOMETRIC CONSTRUCTION ,G EOMETRIC PROBLEMS OF ANTIQUITY References Conway, J. H. and Guy, R. K. "Three Greek Problems." In The Book of Numbers. New York: Springer-Verlag, pp. 192 /C1/94, 1996. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 33 /C1/4, 1986. Delian Problem CUBE DUPLICATION ,DELIAN CONSTANT Delta Amplitude Given an AMPLITUDE f and a MODULUS m in an ELLIPTIC INTEGRAL , D(f)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28msin2f:q See also AMPLITUDE ,ELLIPTIC INTEGRAL ,M ODULUS (ELLIPTIC INTEGRAL ) Delta Curve A curve which can be turned continuously inside an EQUILATERAL TRIANGLE . There are an infinite number of delta curves, but the simplest are the CIRCLE and lens-shaped D/-biangle. All the Dcurves of height h have the same PERIMETER 2ph=3:Also, at each position of a Dcurve turning in an EQUILATERAL TRIANGLE , the perpendiculars to the sides at the points of contact are CONCURRENT at the instanta- neous center of rotation. See also EQUILATERAL TRIANGLE ,LENS,REULEAUX POLYGON ,REULEAUX TRIANGLE ,ROTOR References Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 56 /C1/9, 1973. Delta Function AGENERALIZED FUNCTION which can be defined as the limit of a class of DELTA SEQUENCES . The delta function is sometimes called "Dirac’s delta function" or the "impulse symbol" (Bracewell 1999). Formally, d is a LINEAR FUNCTIONAL from a space (commonly taken as a S CHWARZ SPACE Sor the space of all smooth functions of compact support D) of test functions f. The action of donf, commonly denoted d[f]o r d;fhi ;then gives the value at 0 of ffor any function f. In engineering contexts, the functional nature of the delta function is often suppressed, and dis instead viewed as a "special kind" of function, resulting in the useful (but unfortunately deceptive) notation d(x):In addition, it is possible to define the delta function asan integral satisfying certain properties at infinity(although this is often not explicitly stated), andcommonly used (equivalent) definitions of this type include d(x)/C30 1 plim e00e x2/C27e2; (1) /C30lim e00exjje/C281(2) /C30lim e00/C271 2ffiffiffiffiffipepe/C28x2=(4e)(3) /C30lim e001 pxsinx e ! (4) /C30lim e001 eAix e ! (5)/C30lim e001 eJ1=ex/C271 e ! (6) /C30lim e001 ee/C28x2=eLn2x e !0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1(7) /C30lim n0/C121 2psin n/C271 2 ! x"# sin12x ! : (8) Here, Ai( x)i sa nA IRY FUNCTION ,Jn(x)i saB ESSEL FUNCTION OF THE FIRST KIND , and Ln(x)i saL A- GUERRE POLYNOMIAL of arbitrary positive integer order. (8) is sometimes called the D IRICHLET KERNEL . The fundamental equation that defines derivatives of the delta function d(x)i s gfðxÞdðnÞðxÞdx/C13/C28g@f @xdðn/C281ÞðxÞdx: ð9Þ Letting f(x)/C30xg(x) in this definition, it follows that gxg(x)d?(x)dx/C30/C28gd(x)@ @x[xg(x)]dx; /C30/C28gd(x)g(x)/C27xg?(x) ½/C138 dx /C30/C28gg(x)d(x)dx; (10) where the second term can be dropped sincefxg?(x)d(x)dx/C300;so (10) implies xd?(x)/C30/C28d(x): (11) In general, the same procedure gives gxnf(x) ½/C138 d(n)(x)dx/C30(/C281)ng@nxnf(x) ½/C138 @xnd(x)dx; (12) but since any power of xtimes d(x) integrates to 0, it follows that only the constant term contributes.Therefore, all terms multiplied by derivatives of f(x) vanish, leaving n!f(x);so gxnf(x) ½/C138 dnðÞ(x)dx/C30(/C281)nn!gf(x)d(x)dx; (13) which implies xnd(n)(x)/C30(/C281)nn!d(x): (14) Other identities involving the derivative of the delta function include d?(/C28x)/C30/C28d?(x) (15) g/C12 /C28/C12f(x)d?(x/C28a)dx/C30/C28f?(a) (16) (d?+f)(a)/C30g/C12 /C28/C12d?(a/C28x)f(x)dx/C30f?(a) (17) where +denotes CONVOLUTION , g/C12 /C28/C12d?(x) jj dx/C30/C12; (18) and x2d?(x)/C300: (19) The delta function can also be viewed as the DERIVA- TIVE of the H EAVISIDE STEP FUNCTION , d dxH(x) ½/C138/C30d(x) (20) (Bracewell 1999, p. 94). Additional identities include d(x/C28a)/C300 (21) forx"a; ga/C27o a/C28od(x/C28a)dx/C301; (22) where ois any POSITIVE number, and g/C12 /C28/C12f(x)d(x/C28a)dx/C30f(a) (23) d(ax)/C301 ajjd(x) (24) dx2/C28a20CB0C@ /C301 2ajjd(x/C27a)/C27d(x/C28a) ½/C138 (25) More generally, the delta function of a function is given by d[g(x)]/C30X id(x/C28xi) g?(xi) jj; (26) where the xi/s are the ROOTS ofg. For example, examine d(x2/C27x/C282)/C30d[(x/C281)(x/C272)]: (27) Then g?(x)/C302x/C271;sog?(x1)/C30g?(1)/C303 and g?(x2)/C30 g?(/C282)/C30/C283;and we have d(x2/C27x/C282)/C301 3d(x/C281)/C2713d(x/C272): (28) AF OURIER SERIES expansion of d(x/C28a) gives an/C301pgp /C28pd(x/C28a) cos( nx)dx/C301pcos(na) (29) b n/C301pgp /C28pd(x/C28a) sin( nx)dx/C301psin(na); (30)so d(x/C28a)/C301 2p/C271 p /C2X/C12 n/C301[cos(n a) cos(n x)/C27sin(n a) sin(n x)] /C301 2p/C271pX /C12 n/C301cos[n(x/C28a)]: (31) The delta function is given as a F OURIER TRANSFORM as d(x)/C30F1½/C138/C30g/C12 /C28/C12e/C282pikxdk: (32) Similarly, F/C281[d(x)]/C30g/C12 /C28/C12dxðÞe2pikxdx/C301 (33) (Bracewell 1999, p. 95). More generally, the F OURIER TRANSFORM of the delta function is Fd(x/C28x0) ½/C138 /C30g/C12 /C28/C12e/C282pikxd(x/C28x0)dx/C30e2pikx0:(34) Delta functions can also be defined in 2-D, so that in 2-D C ARTESIAN COORDINATES d2(x;y)/C300 /C12x2/C27y2"0 x2/C27y2/C300;0C1n (35) g/C12 /C28/C12g/C12 /C28/C12d2(x;y)dxdy/C301 (36) d2(ax;by)/C301 ½ab½d2(x;y); (37) and d2(x;y)/C30d(x)d(y): (38) Similarly, in POLAR COORDINATES , d2(x;y)/C30d(r) p½r½(39) (Bracewell 1999, p. 85). In 3-D C ARTESIAN COORDINATES d3(x;y;z)/C30d3(x)/C300 /C12x2/C27y2/C27z2"0 x2/C27y2/C27z2/C3000C1n (40) g/C12 /C28/C12g/C12 /C28/C12g/C12 /C28/C12d3(x;y;z)dxdydz /C301 (41) and d(x)d(y)d(z): (42) inCYLINDRICAL COORDINATES (r;u;z); d3(r ; u;z) /C30d(r) d(z) pr: (43) In SPHERICAL COORDINATES (r; u; f) ; d3(r ; u ; f) /C30d(r) 2 pr2 (44) (Bracewell 1999, p. 85). A series expansion in CYLINDRICAL COORDINATES gives d3 r1 /C28r2 ðÞ /C301 r1d r1 /C28r2 ðÞ du1 /C28 u2 ðÞ d z1 /C28z2 ðÞ /C301 r1d r1 /C28r2 ðÞ1 2pX/C12 m/C30/C28/C12eim u1/C28u2 ðÞ 1 2p g/C12 /C28/C12eik z1/C28z2 ðÞdk : (45) The delta function also obeys the so-called SIFTING PROPERTY gf(x) d(x /C28x0)dx /C30f(x0) (46) (Bracewell 1999, pp. 74 /C1/5). See also DELTA SEQUENCE ,D OUBLET FUNCTION , FOURIER TRANSFORM– DELTA FUNCTION ,G ENERAL- IZED FUNCTION ,IMPULSE SYMBOL ,P OINCARE ´ -BER- TRAND THEOREM ,S HAH FUNCTION ,S OKHOTSKII’S FORMULA References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 481 /C1/85, 1985. Bracewell, R. "The Impulse Symbol." Ch. 5 in The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 69 /C1/7, 1999. Dirac, P. A. M. Quantum Mechanics, 4th ed. London: Oxford University Press, 1958. Gasiorowicz, S. Quantum Physics. New York: Wiley, pp. 491 /C1/94, 1974. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 97 /C1/8, 1984. Spanier, J. and Oldham, K. B. "The Dirac Delta Function d(x /C28a) :/" Ch. 10 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 79 /C1/2, 1987. van der Pol, B. and Bremmer, H. Operational Calculus Based on the Two-Sided Laplace Integral. Cambridge, England: Cambridge University Press, 1955. Delta Operator A SHIFT-INVARIANT OPERATOR Q for which Qx is a NONZERO constant.1. Qa /C300 for every constant a. 2. If p(x)isa POLYNOMIAL of degree n, Qp(x)isa POLYNOMIAL of degree n /C281 :/ 3. Every delta sequence has a unique BASIC POLYNOMIAL SEQUENCE . See also BASIC POLYNOMIAL SEQUENCE ,SHIFT- INVAR- IANT OPERATOR ,UMBRAL CALCULUS References Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Delta Sequence A SEQUENCE of strongly peaked functions for which lim n0/C12g/C12 /C28/C12dn(x)f(x) dx /C30f(0) (1) so that in the limit as /n 0/C12/, the sequences become DELTA FUNCTIONS . Examples include dn(x) /C300 x B/C281 2n n /C281 2n Bx B1 2n 0 x /C211 2n8 >< >:ð2Þ /C30nffiffiffipp e /C28n2x2 ð3Þ /C30n psinc( ax) /C13sin(nx) px ð4Þ /C301 pxeinx /C28 e /C28inx 2i ð5Þ /C301 2pix[eixt]n /C28n ð6Þ /C301 2 p gn /C28neixt dt ð7Þ /C301 2psin[(n/C271 2)x] sin(1 2x); ð8Þ where (8) is known as the D IRICHLET KERNEL . See also DELTA FUNCTION Delta Variation VARIATION Deltahedron APOLYHEDRON whose faces are CONGRUENT EQUILAT- ERAL TRIANGLES (Wells 1986, p. 73). There are an infinite number of deltahedra, but only eight convex ones (Freudenthal and van der Waerden 1947). Among this list of eight, faces composed of coplanar equilateral triangles sharing an edge (such as the RHOMBIC DODECAHEDRON ) are not allowed. The eight convex deltahedra have n /C304, 6, 8, 10, 12, 14, 16, and 20 faces. These are summarized in the table below, and illustrated in the following figures. n Name 4 TETRAHEDRON 6 TRIANGULAR DIPYRAMID 8 OCTAHEDRON 10 PENTAGONAL DIPYRAMID 12 SNUB DISPHENOID 14 TRIAUGMENTED TRIANGULAR PRISM 16 GYROELONGATED SQUARE DIPYRAMID 20 ICOSAHEDRON The 24-faced deltahedra formed by (1) CUMULATION of the CUBE and (2) STELLA OCTANGULA are both con- cave. The "caved in" CUMULATED DODECAHEDRON is a deltahedron with 60 faces. It is ICOSAHEDRON STELLA- TION I20(Wells 1991, p. 78). Cundy (1952) identified 17 concave deltahedra with two kinds of VERTICES . See also CUMULATION ,G YROELONGATED SQUARE DIPYRAMID ,ICOSAHEDRON ,O CTAHEDRON ,PENTAGO- NAL DIPYRAMID ,SNUB DISPHENOID TETRAHEDRON , TRIANGULAR DIPYRAMID ,T RIAUGMENTED TRIANGU- LAR PRISM References Cundy, H. M. "Deltahedra." Math. Gaz. 36, 263/C1/66, 1952. Cundy, H. and Rollett, A. "Deltahedra." §3.11 in Mathema- tical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 142 /C1/44, 1989. Freudenthal, H. and van der Waerden, B. L. "On an Assertion of Euclid." Simon Stevin 25, 115/C1/21, 1947. Gardner, M. Fractal Music, Hypercards, and More: Mathe- matical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 40, 53, and 58 /C1/0, 1992. Pugh, A. Polyhedra: A Visual Approach. Berkeley, CA: University of California Press, pp. 35 /C1/6, 1976. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 73, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 51 and 78, 1991. Deltohedron TRAPEZOHEDRON Deltoid A 3-cusped HYPOCYCLOID , also called a tricuspoid. The deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner in 1856 and is sometimes called Steiner’s hypocycloid (Lockwood 1967; Coxeter and Greitzer 1967, p. 44; MacTutor Archive). The equation of the deltoid is obtained by setting n /C13 a =b /C303 in the equation of the HYPOCYCLOID , where a is the RADIUS of the large fixed CIRCLE and b is the RADIUS of the small rolling CIRCLE , yielding the parametric equations x /C302 3cos f /C2813cos(2 f)"# a /C302b cos f /C27b cos(2 f) (1) y /C3023sinf /C2713sin(2f)"# a /C302b sinf /C28b sin(2f) : (2) The ARC LENGTH , CURVATURE , and TANGENTIAL ANGLE are s(t) /C304gt 0½ sin3 2 t? ! dt?/C3016 3sin234 t ! (3) k(t) /C30/C281 8csc32 t ! (4) f(t) /C30/C281 2 t: (5) As usual, care must be taken in the evaluation of stðÞ for t > 2 p=3: Since the form given above comes from an integral involving the ABSOLUTE VALUE of a function, it must be monotonic increasing. Each branch can be treated correctly by defining n /C303t 2p"# /C271; (6) where xbcis the FLOOR FUNCTION , giving the formula s(t) /C30(/C281)1 /C27[n (mod2)] 16 3sin23 4 t ! /C2732 312 n"# : (7) The total ARC LENGTH is computed from the general HYPOCYCLOID equation sn /C308a(n /C28 1) n: (8) With n /C303, this gives s3 /C3016 3a : (9) The AREA is given by An /C30(n /C28 a)(n /C28 2) n2 pa2 (10)with n /C303 A3 /C3029 pa 2 : (11) The length of the tangent to the tricuspoid, measured between the two points P,Qin which it cuts the curve again, is constant and equal to 4 a:If you draw TANGENTS atPandQ, they are at R IGHT ANGLES . See also ASTROID ,HYPOCYCLOID ,SIMSON LINE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 219, 1987. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 44, 1967. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 70, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 131 /C1/35, 1972. Lockwood, E. H. "The Deltoid." Ch. 8 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 72 /C1/9, 1967. MacBeath, A. M. "The Deltoid." Eureka 10,2 0/C1/3, 1948. MacBeath, A. M. "The Deltoid, II." Eureka 11,2 6/C1/9, 1949. MacBeath, A. M. "The Deltoid, III." Eureka 12,5/C1/, 1950. MacTutor History of Mathematics Archive. "Tricuspoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Tri-cuspoid.html. Patterson, B. C. "The Triangle: Its Deltoids and Foliates." Amer. Math. Monthly 47,1 1/C1 /8, 1940. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 52, 1991. Yates, R. C. "Deltoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 71 /C1/4, 1952. Deltoid Caustic The caustic of the D ELTOID when the rays are PARALLEL in any direction is an ASTROID . Deltoid Evolute AHYPOCYCLOID EVOLUTE forn/C303 is another D EL- TOID scaled by a factor n=(n /C282) /C303=1 /C303 and rotated 1=(2 /C2153) /C301=6 of a turn. Deltoid Involute A HYPOCYCLOID INVOLUTE for n /C303 is another DEL- TOID scaled by a factor (n /C282)=n /C301 =3 and rotated 1=(2 /C2153) /C301=6 of a turn. Deltoid Pedal Curve The PEDAL CURVE for a DELTOID with the PEDAL POINT at the CUSP is a FOLIUM . For the PEDAL POINT at the CUSP (NEGATIVE x-intercept), it is a BIFOLIUM . At the center, or anywhere on the inscribed EQUILATERAL TRIANGLE ,itisa TRIFOLIUM . Deltoid Radial Curve The TRIFOLIUM x /C30x0 /C274a cos f /C284a cos(2 f) y /C30y0 /C274a sinf /C274a sin(2f) : Deltoidal Hexecontahedron The 60-faced DUAL POLYHEDRON of the SMALL RHOM- BICOSIDODECAHEDRON A5 and Wenninger dual W14 : It is sometimes also called the trapezoidal hexecontahe- dron or strombic hexecontahedron. See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , HEXECONTAHEDRON ,SMALL RHOMBICOSIDODECAHE- DRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 24, 1983. Deltoidal Icositetrahedron The 24-faced DUAL POLYHEDRON of the SMALL RHOM- BICUBOCTAHEDRON A6 and Wenninger dual W13 : It is also called the TRAPEZOIDAL ICOSITETRAHEDRON . For a SMALL RHOMBICUBOCTAHEDRON with unit edge length, the deltoidal icositetrahedron has edge lengths s1 /C302 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C28ffiffiffi 2pq (1) s2 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C282ffiffiffi 2pq (2) and INRADIUS r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 177 /C274ffiffiffi2p0C@n0C@os : (3) Normalizing so the smallest edge has unit edge length s 1 /C301 gives a deltoidal icositetrahedron with SURFACE AREA and VOLUME S /C306ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 29 /C282ffiffiffi 2p :q (4) V /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 122 /C2771ffiffiffi 2p :q (5) See also ARCHIMEDEAN SOLID ,D ELTOIDAL ICOSITE- TRAHEDRON STELLATIONS ,DELTOIDAL ICOSITETRAHE- DRON STELLATIONS ,ICOSITETRAHEDRON ,S MALL RHOMBICUBOCTAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 23, 1983.Deltoidal Icositetrahedron Stellations The CONVEX HULLS of the SMALL CUBICUBOCTAHE- DRON U13 ; SMALL RHOMBIHEXAHEDRON U18 ; and STEL- LATED TRUNCATED HEXAHEDRON U19are all the Archimedean SMALL RHOMBICUBOCTAHEDRON A6 ; whose dual is the deltoidal icositetrahedron, so the duals of these solids (i.e., the SMALL HEXACRONIC ICOSITETRAHEDRON , SMALL RHOMBIHEXAHEDRON , and GREAT TRIAKIS OCTAHEDRON ) are all stellations of the deltoidal icositetrahedron (Wenninger 1983, p. 57). See also ARCHIMEDEAN SOLID,ICOSITETRAHEDRON , SMALL RHOMBICUBOCTAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Demiregular Tessellation TESSELLATION Demlo Number The initially PALINDROMIC NUMBERS 1, 121, 12321, 1234321, 123454321, ... (Sloane’s A002477). For the first through ninth terms, the sequence is given by the GENERATING FUNCTION /C2810x /C27 1 (x /C28 1)(10 x /C28 1)(100 x /C28 1) /C301 /C27121x /C2712321 x2 /C271234321 x3 /C27::: (Plouffe 1992, Sloane and Plouffe 1995). The defini- tion of this sequence is slightly ambiguous from the tenth term on. See also CONSECUTIVE NUMBER SEQUENCES ,PALIN- DROMIC NUMBER References Kaprekar, D. R. "On Wonderful Demlo Numbers." Math. Student 6,6 8/C1/0, 1938. Plouffe, S. "Approximations de Se ´ries Ge ´ne´ratrices et quel- ques conjectures." Montre ´al, Canada: Universite ´du Que ´- bec a`Montre ´al, Me ´moire de Maı ˆtrise, UQAM, 1992. Sloane, N. J. A. Sequences A002477/M5386 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Dendrite A system of line segments connecting a given set of points. See also PLATEAU’S PROBLEM ,TRAVELING SALESMAN PROBLEM References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 120 /C1/25, 1999. Dendrite Fractal AJ ULIA SET with constant c chosen at the boundary of the MANDELBROT SET (Branner 1989; Dufner et al. 1998, p. 225). The image above was computed using c /C30i. See also JULIA SET References Branner, B. "The Mandelbrot Set." In Chaos and Fractals: The Mathematics behind the Computer Graphics (Ed. R. L. Devaney and L. Keen). Providence, RI: Amer. Math. Soc., pp. 75 /C1/05, 1989. Dufner, J.; Roser, A.; and Unseld, F. Fraktale und Julia- Mengen. Harri Deutsch, p. 225, 1998. Denjoy Integral A type of INTEGRAL which is an extension of both the RIEMANN INTEGRAL and the LEBESGUE INTEGRAL . The original Denjoy integral is now called a Denjoy integral "in the restricted sense," and a more general type is now called a Denjoy integral "in the wider sense." The independently discovered PERRON INTE- GRAL turns out to be equivalent to the Denjoy integral "in the restricted sense." See also INTEGRAL ,L EBESGUE INTEGRAL ,P ERRON INTEGRAL ,RIEMANN INTEGRAL References Iyanaga, S. and Kawada, Y. (Eds.). "Denjoy Integrals." §103 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 337 /C1/40, 1980. Kestelman, H. "General Denjoy Integral." §9.2 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 217 /C1/27, 1960.Denominator The number q in a FRACTION p =q:/ See also FRACTION ,N UMERATOR ,RATIO,RATIONAL NUMBER Dense A set A in a FIRST-COUNTABLE SPACE is dense in B if B /C30A @ L; where L is the limit of sequences of elements of A. For example, the rational numbers are dense in the reals. In general, a SUBSET A of X is dense if its CLOSURE cl(A) /C30X :/ See also CLOSURE (SET), DENSITY ,D ERIVED SET, NOWHERE DENSE ,PERFECT SET Density DENSITY (POLYGON ), DENSITY (SEQUENCE ), NATURAL DENSITY Density (Polygon) The number q in a STAR POLYGON fp=qg:/ See also STAR POLYGON Density (Sequence) Let a SEQUENCE aifg/C12 i/C301be strictly increasing and composed of NONNEGATIVE INTEGERS . Call A(x) the number of terms not exceeding x. Then the density is given by limx 0/C12A(x)=x if the LIMIT exists. References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 199, 1994. Density Function PROBABILITY FUNCTION Denumerable Set A SET is denumerable IFF it is EQUIPOLLENT to the finite ORDINAL NUMBERS . (Moore 1982, p. 6; Rubin 1967, p. 107; Suppes 1972, pp. 151 /C1/52). However, Ciesielski (1997, p. 64) calls this property "counta- ble." The set ALEPH-0 is most commonly called "denumerable" to "COUNTABLY INFINITE ". See also COUNTABLE SET,COUNTABLY INFINITE References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Ferreiro ´s, J. "Non-Denumerability of R:/"§6.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 177 /C1/ 83, 1999. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Denumerably Infinite COUNTABLY INFINITE Depth (Graph) GRAPH THICKNESS Depth (Size) The depth of a box is the horizontal DISTANCE from front to back (usually not necessarily defined to be smaller than the WIDTH , the horizontal DISTANCE from side to side). See also HEIGHT ,W IDTH (SIZE) Depth (Statistics) The smallest RANK (either up or down) of a set of data. See also RANK (STATISTICS ) References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 30, 1977. Depth (Tree) The depth of a RESOLVING TREE is the number of levels of links, not including the top. The depth of the link is the minimal depth for any RESOLVING TREE of that link. The only links of length 0 are the trivial links. A KNOT of length 1 is always a trivial KNOT and links of depth one are always HOPF LINKS , possibly with a few additional trivial components (Bleiler and Scharlemann 1988). The LINKS of depth two have also been classified (Scharlemann and Thompson 1991). References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 169, 1994. Bleiler, S. and Scharlemann, M. "A Projective Plane in R4 with Three Critical Points is Standard. Strongly Invertible Knots have Property P." Topology 27, 519 /C1/40, 1988. Scharlemann, M. and Thompson, A. "Detecting Unknotted Graphs in 3/-Space." J. Diff. Geom. 34, 539 /C1/60, 1991. Depth-First Traversal A search algorithm of a GRAPH which explores the first son of a node before visiting its brothers. Tarjan (1972) and Hopcroft and Tarjan (1973) showed that depth-first search gives linear time algorithms for many problems in graph theory (Skiena 1990). See also BREADTH- FIRST TRAVERSALReferences Hopcroft, J. and Tarjan, R. "Algorithm 447: Efficient Algo- rithms for Graph Manipulation." Comm. ACM 16, 372/C1/ 78, 1973. Skiena, S. "Breadth-First and Depth-First Search." §3.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 95 /C1/7, 1990. Tarjan, R. E. "Depth-First Search and Linear Graph Algo- rithms." SIAM J. Comput. 1, 146/C1/60, 1972. Derangement A derangement of nordered objects, denoted ! n;is a PERMUTATION in which none of the objects appear in their "natural" (i.e., ordered) place. For example, the only derangements of f1;2;3gare f2;3;1gand f3;1;2g;so !3/C302:Similarly, the derangements of f1;2;3;4gare f2;1;4;3g;f2;3;4;1g;f2;4;1;3g; f3;1;4;2g; f3;4;1;2g; f3;4;2;1g; f4;1;2;3g; f4;3;1;2g;and f4;3;2;1g:Derangements are permu- tations without fixed points (i.e., having no cycles oflength one). The derangements of a list of nelements can be computed using Derangments [n] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mandBBDiscreteMath‘ ). The problem was formulated by P. R. de Montmort in 1708, and solved by him in 1713 (de Montmort 1713 /C1 / 714). Nicholas Bernoulli also solved the problem using the INCLUSION-EXCLUSION PRINCIPLE (de Mon- tmort 1713 /C1/714, p. 301; Bhatnagar, p. 8). The function giving the number of distinct derange-ments on nelements is called the SUBFACTORIAL !n and is equal to !n/C13n!Xn k/C300(/C281)k k!(1) (Bhatnagar, pp. 8 /C1/)o r !n/C13n! e"# ; (2) where k! is the usual FACTORIAL and [ x] is the NEAREST INTEGER FUNCTION . These are also called RENCONTRES NUMBERS (named after rencontres soli- taire), or COMPLETE PERMUTATIONS , or derangements. The number of derangements ! n/C30d(n) of length n satisfy the RECURRENCE RELATIONS d(n)/C30(n/C281)[d(n/C281)/C27d(n/C282)] (3) and d(n)/C30nd(n/C281)/C27(/C281)n; (4) with d(1)/C300 and d(2)/C301 (Skiena 1990, p. 33). The first few are 0, 1, 2, 9, 44, 265, 1854, ... (Sloane’sA000166). This sequence cannot be expressed as afixed number of hypergeometric terms (Petkovsek et al.1996, pp. 157 /C1 /60). See also MARRIED COUPLES PROBLEM ,PERMUTATION , ROOT,SUBFACTORIAL References Aitken, A. C. Determinants and Matrices. Westport, CT: Greenwood Pub., p. 135, 1983. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 46 /C1/7, 1987. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, 1995. Comtet, L. "The ‘Proble `me des Recontres’." §4.2 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 180 /C1/83, 1974. Coolidge, J. L. An Introduction to Mathematical Probability. Oxford, England: Oxford University Press, p. 24, 1925. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 115 /C1/16, 1996. de Montmort, P. R. Essai d’analyse sur les jeux de hasard. Paris, 1708. Second edition published 1713 /C1/714. Third edition reprinted in New York: Chelsea, pp. 131 /C1/38, 1980. Dickau, R. M. "Derangements." http://forum.swarthmor- e.edu/advanced/robertd/derangements.html. Durell, C. V. and Robson, A. Advanced Algebra. London, p. 459, 1937. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Ruskey, F. "Information on Derangements." http:// www.theory.csc.uvic.ca/~cos/inf/perm/Derange- ments.html. Skiena, S. "Derangements." §1.4.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 33 /C1/4, 1990. Sloane, N. J. A. Sequences A000166/M1937 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stanley, R. P. Enumerative Combinatorics, Vol. 1. New York: Cambridge University Press, p. 67, 1986. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 123, 1991. Derivation A derivation is a sequence of steps, logical or computational, from one result to another. The word derivation comes from the word "derive." "Derivation" can also refer to a particular type of operator used to define a DERIVATION ALGEBRA on a ring or algebra. See also DERIVATION ALGEBRA Derivation Algebra Let A be any algebra over a FIELD F, and define a derivation of A as a linear operator D on A satisfying (xy)D /C30(xD)y /C27x(yD)for all x;y /C23 A: Then the set D(A) of all derivations of A in a SUBSPACE of the associative algebra of all linear operators on A is a LIE ALGEBRA , called the derivation algebra. See also LIE ALGEBRA References Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, pp. 3 /C1/, 1996. Derivative The derivative of a FUNCTION represents an infinite- simal change in the function with respect to whatever parameters it may have. The "simple" derivative of a function fwith respect to xis denoted either f?(x)o r df dx(1) (and often written in-line as df=dx):When derivatives are taken with respect to time, they are often denoted using Newton’s OVERDOT notation for FLUXIONS , dx dt/C30˙x: (2) When a derivative is taken ntimes, the notation x(n) or dnf dxn(3) is used, with ˙x;¨x; /C5x;etc: (4) the corresponding FLUXION notation. When a function f(x;y;. . .) depends on more than one variable, a PARTIAL DERIVATIVE @f @x;@2f @x@y;etc: (5) can be used to specify the derivative with respect toone or more variables. The derivative of a function f(x) with respect to the variable xis defined as f?(x)/C13lim h00f(x/C27h)/C28f(x) h: (6) Note that in order for the limit to exist, both limh00/C27 and limh00/C28must exist and be equal, so the FUNCTION must be continuous. However, continuity is a NECES- SARY butnot SUFFICIENT condition for differentiabil- ity. Since some DISCONTINUOUS functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives." A 3-D generalization of the derivative to an arbitrary direction is known as the DIRECTIONAL DERIVATIVE .I n general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into "TANGENT MAPS ." Simple derivatives of some simple functions follow. d dxxn/C30nxn/C281(7) d dxln½x½/C301 x(8) d dxsinx/C30cosx (9) d dxcosx/C30/C28sinx (10) d dxtanx/C30d dxsinx cosx ! /C30cosxcosx/C28sinx(/C28sinx) cos2x /C301 cos2x/C30sec2x (11) d dxcscx/C30d dx(sinx)/C281/C30/C28(sinx)/C282cosx/C30/C28cosx sin2x /C30/C28cscxcotx (12) d dxsecx/C30d dx(cosx)/C281/C30/C28(cosx)/C282(/C28sinx)/C30sinx cos2x /C30secxtanx (13) d dxcotx/C30d dxcosx sinx ! /C30sinx(/C28sinx)/C28cosxcosx sin2x /C30/C281 sin2x/C30/C28csc2x (14) d dxex/C30ex(15) d dxax/C30d dxelnax/C30d dxexlna/C30(lna)exlna/C30(lna)ax(16) d dxsin/C281x/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p (17) d dxcos/C281x/C30/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p (18) d dxtan/C281x/C301 1/C27x2(19)d dxcot/C281x/C30/C281 1/C27x2(20) d dxsec/C281x/C301 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C281p (21) d dxcsc/C281x/C30/C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffix2/C281p (22) d dxsinhx/C30coshx (23) d dxcoshx/C30sinhx (24) d dxtanh x/C30sech2x (25) d dxcothx/C30/C28csch2x (26) d dxsechx/C30/C28sechxtanh x (27) d dxcschx/C30/C28cschxcothx (28) d dxsnx/C30cnxdnx (29) d dxcnx/C30/C28snxdnx (30) d dxdnx/C30/C28k2snxcnx: (31) where sn( x)/C13sn(x;k);cn(x)/C13cn(x;k);etc. are J ACOBI ELLIPTIC FUNCTIONS , and the PRODUCT RULE and QUOTIENT RULE have been used extensively to expand the derivatives. There are a number of important rules for computing derivatives of certain combinations of functions.Derivatives of sums are equal to the sum of deriva- tives so that f(x)/C27/C1/C1/C1/C27h(x) ½/C138 ?/C30f?(x)/C27/C1/C1/C1/C27h?(x): (32) In addition, if cis a constant, d dxcf(x) ½/C138/C30cf?(x): (33) The PRODUCT RULE for differentiation states d dxf(x)g(x) ½/C138 /C30f(x)g?(x)/C27f?(x)g(x); (34) where f?denotes the DERIVATIVE offwith respect to x. This derivative rule can be applied iteratively to yield derivate rules for products of three or more functions, for example, [fgh] ?/C30(fg)h?/C27(fg) ?h /C30fgh?/C27(fg?/C27f ?g)h /C30f ?gh /C27fg ?h /C27fgh?: (35) The QUOTIENT RULE for derivatives states that d dxf(x) g(x)"# /C30g(x)f ?(x) /C28 f(x)g?(x) g(x)½/C1382 (36) while the POWER RULE gives d dxxnðÞ/C30nxn/C281 (37) Other very important rule for computing derivatives is the CHAIN RULE , which states that dy dx /C30dy du /C215du dx; (38) or more generally, dz dt /C30@z @xdx dt /C27@z @ydy dt; (39) were /@z=@x/ denotes a PARTIAL DERIVATIVE . Miscellaneous other derivative identities include dy dx /C30dy dt dx dt(40) dydx /C301 dx dy: (41) If F(x;y) /C30C ; where C is a constant, then dF /C30@F @ydy /C27@F @xdx /C300; (42) so dy dx /C30/C28@F @x @F @y: (43) A vector derivative of a vector function X(t) /C13x1(t) x2(t) n xk(t)2 6643 775 (44) can be defined bydX dt/C30dx1 dt dx2 dt n dtk dt2 66666666643 7777777775(45) The nth derivatives of x nf(x) for n /C301, 2, ... are d dx [xf(x)] /C30f(x) /C27xf ?(x) (46) d2 dx2x2f(x)0C10CC /C302f(x)/C274xf?(x)/C27x2fƒ(x) (47) d3 dx3x3f(x)0C10CC /C306f(x)/C2718xf?(x)/C279x2fƒ(x)/C27x3f§(x):(48) See also BLANCMANGE FUNCTION ,C ARATHE ´ ODORY DERIVATIVE ,CHAIN RULE,COMMA DERIVATIVE ,CON- VECTIVE DERIVATIVE ,COVARIANT DERIVATIVE ,DIREC- TIONAL DERIVATIVE ,E ULER- LAGRANGE DERIVATIVE , FLUXION ,FRACTIONAL CALCULUS ,FRE´ CHET DERIVA- TIVE,L AGRANGIAN DERIVATIVE ,L IE DERIVATIVE , LOGARITHMIC DERIVATIVE ,PINCHERLE DERIVATIVE , POWER RULE,PRODUCT RULE, Q-SERIES ,QUOTIENT RULE,SCHWARZIAN DERIVATIVE ,SEMICOLON DERIVA- TIVE,W EIERSTRASS FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999. Beyer, W. H. "Derivatives." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 229 /C1/32, 1987. Griewank, A. Principles and Techniques of Algorithmic Differentiation. Philadelphia, PA: SIAM, 2000. Derivative Test FIRST DERIVATIVE TEST,SECOND DERIVATIVE TEST Derived Polygon Given a POLYGON with an EVEN NUMBER of sides, the derived polygon is obtained by joining the points which are a fractional distance r along each side. If r /C301=2; then the derived polygons are called MID- POINT POLYGONS and tend to a shape with opposite sides parallel and equal in length. Furthermore, alternate polygons have approximately the same length, and the original and all derived polygons have the same centroid. Amazingly, if r "1; the derived polygons still ap- proach a shape with opposite sides parallel and equal in length, and all have the same centroid. The above illustrations show 20 derived polygons for ratios r /C30 0:3; 0.5, 0.7, and 0.9. More amazingly still, if the original polygon is skew, a plane polygonal is ap- proached which has these same properties. See also MIDPOINT POLYGON ,W HIRL References Cadwell, J. H. Topics in Recreational Mathematics. Cam- bridge, England: Cambridge University Press, 1966. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 53 /C1/4, 1991. Derived Set The LIMIT POINTS of a SETP, denoted P?:/ See also DENSE ,LIMIT POINT ,PERFECT SET References Ferreiro ´s, J. "Cantor’s Derived Sets" and "Derived Sets and Cardinalities." §4.4.3 and 6.6 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathe- matics. Basel, Switzerland: Birkha ¨user, pp. 141 /C1/44 and 202/C1/08, 1999.Dervish AQUINTIC SURFACE having the maximum possible number of ORDINARY DOUBLE POINTS (31), which was constructed by W. Barth in 1994 (Endraß). The implicit equation of the surface is 64(x/C28w)x4/C284x3w/C2810x2y2/C284x2w20C1 /C2716xw3/C2820xy2w/C275y4/C2716w4/C2820y2w2/C138 /C285ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C28ffiffiffi 5pq 2z/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C28ffiffiffi 5pq w0C@80C@9 /C24x2/C27y2/C27z20CB0C@ /C27(1/C273ffiffiffi 5p )w2hi2 ; (1) where wis a parameter (Endraß). The surface can also be described by the equation aF/C27q/C300; (2) where F/C30h1h2h3h4h5; (3) h1¼x/C0z ð4Þ h2/C30cos2p 5 ! x/C28sin2p 5 ! y/C28z (5) h3/C30cos4p 5 ! x/C28sin4p 5 ! y/C28z (6) h4/C30cos6p 5 ! x/C28sin6p 5 ! y/C28z (7) h5/C30cos8p 5 ! x/C28sin8p 5 ! y/C28z (8) q/C30(1/C28cz)x2/C27y2/C281/C27rz20CB0C@2; (9) and r /C301 41 /C27ffiffiffi 5p0C@n0C@o (10) a /C30/C288 51 /C271ffiffiffi 5p !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5pq (11) c /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5pq (12) (Nordstrand). The dervish is invariant under the GROUP D5and contains exactly 15 lines. Five of these are the intersection of the surface with a D5/-invariant cone containing 16 nodes, five are the intersection of the surface with a D5/-invariant plane containing 10 nodes, and the last five are the intersection of the surface with a second D5/-invariant plane containing no nodes (Endraß). See also ALGEBRAIC SURFACE ,QUINTIC SURFACE References Endraß, S. "Togliatti Surfaces." http://enriques.mathemati- k.uni-mainz.de/kon/docs/Etogliatti.shtml. Endraß, S. "Fla¨chen mit vielen Doppelpunkten." DMV- Mitteilungen 4,17/C1/0, 4/1995. Endraß, S. Symmetrische Fla¨che mit vielen gewo¨hnlichen Doppelpunkten. Ph.D. thesis. Erlangen, Germany, 1996. Nordstrand, T. "Dervish." http://www.uib.no/people/nfytn/ dervtxt.htm. Desargues’ Configuration The 103CONFIGURATION of ten lines intersecting three at a time in 10 points which arises in DESAR- GUES’ THEOREM . See also CONFIGURATION ,DESARGUES’ THEOREMDesargues’ Theorem If the three straight LINES joining the corresponding VERTICES of two TRIANGLES ABC and A?B ?Cƒ all meet in a point (the PERSPECTIVE CENTER ), then the three intersections of pairs of corresponding sides lie on a straight LINE (the PERSPECTIVE AXIS). Equivalently, if two TRIANGLES are PERSPECTIVE from a POINT , they are PERSPECTIVE from a LINE. The 10 lines and 10 3-line intersections form a 103 CONFIGURATION sometimes called DESARGUES’ CON- FIGURATION . Desargues’ theorem is SELF-DUAL upon application of the DUALITY PRINCIPLE of PROJECTIVE GEOMETRY . See also DESARGUES’ CONFIGURATION ,DUALITY PRIN- CIPLE ,PAPPUS’S HEXAGON THEOREM ,PASCAL LINES, PASCAL’S THEOREM ,PERSPECTIVE AXIS,PERSPECTIVE CENTER ,PERSPECTIVE TRIANGLES ,SELF-DUAL References Coxeter, H. S. M. and Greitzer, S. L. "Perspective Triangles; Desargues’s Theorem." §3.6 in Geometry Revisited. Wa- shington, DC: Math. Assoc. Amer., pp. 70 /C1/2, 1967. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 44, 1928. Eves, H. "Desargues’ Two-Triangle Theorem." §6.2.5 in A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, pp. 249 /C1/51, 1965. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 23 /C1/5, 1930. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 89 /C1/2, 1990. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 231, 1929. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 77, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 54 /C1/5, 1991. Descartes Circle Theorem A special case of A POLLONIUS’ PROBLEM requiring the determination of a CIRCLE touching three mutually TANGENT CIRCLES (also called the KISSING CIRCLES PROBLEM ). There are two solutions: a small circle surrounded by the three original CIRCLES , and a large circle surrounding the original three. Frederick Soddy gave the FORMULA for finding the RADIUS of the so-called inner and outer SODDY CIRCLES given the RADII of the other three. The relationship is 2 k2 1 /C27 k22 /C27 k23 /C27 k240CB0C@ /C30 k1 /C27 k2 /C27 k3 /C27 k4 ðÞ2; where kiare the CURVATURES of the CIRCLES . Here, the NEGATIVE solution corresponds to the outer SODDY CIRCLE and the POSITIVE solution to the inner SODDY CIRCLE . This formula was known to Descartes and Vie`te (Boyer and Merzbach 1991, p. 159), but Soddy ex- tended it to SPHERES .In n-D space, n /C272 mutually touching n-SPHERES can always be found, and the relationship of their CURVATURES is nXn/C272 i/C301k2i ! /C30Xn/C272 i/C301ki ! 2 : See also APOLLONIUS’ PROBLEM ,FOUR COINS PRO- BLEM ,SANGAKU PROBLEM ,SODDY CIRCLES ,SPHERE PACKING ,TANGENT CIRCLES References Boyer, C. B. and Merzbach, U. C. A History of Mathematics, 2nd ed. New York: Wiley, 1991. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 13 /C1/6, 1969. Fukagawa, H. and Pedoe, D. "The Descartes Circle Theo- rem." §1.7 in Japanese Temple Geometry Problems. Win- nipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 16 /C1/7 and 92, 1989. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/1, May 1998. Wilker, J. B. "Four Proofs of a Generalization of the Descartes Circle Theorem." Amer. Math. Monthly 76, 278 /C1/82, 1969. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, pp. 50 /C1/ 1, 1979. Descartes Folium FOLIUM OF DESCARTES Descartes Ovals CARTESIAN OVALS Descartes Total Angular Defect The total angular defect is the sum of the ANGULAR DEFECTS over all VERTICES of a POLYHEDRON , where the ANGULAR DEFECT d at a given VERTEX is the difference between the sum of face angles and 2p: For any convex POLYHEDRON , the Descartes total angular defect is D/C30X idi /C304p: (1) This is equivalent to the POLYHEDRAL FORMULA for a closed rectilinear surface, which satisfiesD/C302p(V /C28E /C27F) : (2) A POLYHEDRON with N0 equivalent VERTICES is called aP LATONIC SOLID and can be assigned a SCHLA ¨ FLI SYMBOL fp; qg: It then satisfies N0 /C304p d (3) and d /C302p/C28q 1 /C282 p ! p; (4) so N0 /C304p 2p /C27 2q /C28 pq : (5) See also ANGULAR DEFECT ,PLATONIC SOLID,POLY- HEDRAL FORMULA ,POLYHEDRON Descartes’ Formula DESCARTES TOTAL ANGULAR DEFECT Descartes’ Sign Rule A method of determining the maximum number of POSITIVE and NEGATIVE REAL ROOTS of a POLYNOMIAL . For POSITIVE ROOTS , start with the SIGN of the COEFFICIENT of the lowest (or highest) POWER . Count the number of SIGN changes n as you proceed from the lowest to the highest POWER (ignoring POWERS which do not appear). Then n is the maximum number of POSITIVE ROOTS . Furthermore, the number of allowable ROOTS is n, n /C282; n /C284; .... For example, consider the POLYNOMIAL f(x) /C30x7 /C27x6 /C28x4 /C28x3 /C28x2 /C27x /C281: (1) Since there are three SIGN changes, there are a maximum of three possible POSITIVE ROOTS . For NEGATIVE ROOTS , starting with a POLYNOMIAL f(x) ; write a new POLYNOMIAL f(/C28x) with the SIGNS of all ODD POWERS reversed, while leaving the SIGNS of the EVEN POWERS unchanged. Then proceed as before to count the number of SIGN changes n. Then n is the maximum number of NEGATIVE ROOTS . For example, consider the POLYNOMIAL f(x) /C30x7 /C27x6 /C28x4 /C28x3 /C28x2 /C27x /C281; (2) and compute the new POLYNOMIAL f(/C28x)/C30/C28x7/C27x6/C28x4/C27x3/C28x2/C28x/C281: (3) In this example, there are four SIGN changes, so there are a maximum of four NEGATIVE ROOTS . See also BOUND ,ROOT,STURM FUNCTION References Anderson, B.; Jackson, J.; and Sitharam, M. "Descartes’ Rule of Signs Revisited." Amer. Math. Monthly 105, 447 /C1/ 51, 1998. Grabiner, D. J. "Descartes’ Rule of Signs: Another Construc- tion." Amer. Math. Monthly 106, 854 /C1/55, 1999. Hall, H. S. and Knight, S. R. Higher Algebra: A Sequel to Elementary Algebra for Schools. London: Macmillan, pp. 459 /C1/60, 1950. Henrici, P. "Sign Changes. The Rule of Descartes." §6.2 in Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, pp. 439 /C1/43, 1988. Itenberg, U. and Roy, M. F. "Multivariate Descartes’ Rule." Beitra ¨ge Algebra Geom. 37, 337 /C1/46, 1996. Struik, D. J. (Ed.). A Source Book in Mathematics 1200 /C1/ 800. Princeton, NJ: Princeton University Press, pp. 89 /C1/3, 1986. Descartes-Euler Polyhedral Formula POLYHEDRAL FORMULA Descending Plane Partition 776631 6542 33 2 A descending plane partition of order n is a 2-D array (possibly empty) of positive integers less than or equal to n such that the left-hand edges are succes- sively indented, rows are nonincreasing across, col- umns are decreasing downwards, and the number of entries in each row is strictly less than the largest entry in that row. Implicit in this definition are the requirements that no "holes" are allowed in the array, all rows are flush against the top, and the diagonal element must be filled if any element of its row is filled. The above example shows a decreasing plane partition of order seven. 33 33323132 f 2 The sole descending plane partition of order one is the empty one ¥; the two of order two are "2" and f; and the seven of order three are illustrated above. In general, the number of descending plane partitions of order n is equal to the number of /C271/-bordered ALTERNATING SIGN MATRICES : 1, 2, 7, 42, 429, ... (Sloane’s A005130). See also ALTERNATING SIGN MATRIX ,PLANE PARTI- TION References Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math. 53, 193 /C1/25, 1979. Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/46.Sloane, N. J. A. Sequences A005130/M1808 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Descriptive Geometry PROJECTIVE GEOMETRY Descriptive Set Theory The study of DEFINABLE SETS and functions in POLISH SPACES . References Becker, H. and Kechris, A. S. The Descriptive Set Theory of Polish Group Actions. New York: Cambridge University Press, 1996. Design A formal description of the constraints on the possible configurations of an experiment which is subject to given conditions. A design is sometimes called an EXPERIMENTAL DESIGN . See also BLOCK DESIGN ,C OMBINATORICS ,D ESIGN THEORY ,HADAMARD DESIGN ,HOWELL DESIGN ,SPHE- RICAL DESIGN ,SYMMETRIC BLOCK DESIGN ,TRANS- VERSAL DESIGN Design Theory The study of DESIGNS and, in particular, NECESSARY and SUFFICIENT conditions for the existence of a BLOCK DESIGN . See also BLOCK DESIGN ,BRUCK- RYSER- CHOWLA THE- OREM ,DESIGN ,FISHER’S BLOCK DESIGN INEQUALITY References Assmus, E. F. Jr. and Key, J. D. Designs and Their Codes. New York: Cambridge University Press, 1993. Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, 1996. Dinitz, J. H. and Stinson, D. R. (Eds.). "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys. New York: Wiley, pp. 1 /C1/2, 1992. Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. Desmic Surface LetD1;D2;andD3be tetrahedra in projective 3-space P3:Then the tetrahedra are said to be desmically related if there exist constants a;b;andgsuch that aD1/C27bD2/C27gD3/C300: A desmic surface is then defined as a QUARTIC SUR- FACE which can be written as aD1/C27bD2/C27cD3/C300 for desmically related tetrahedra D1;D2;andD3: Desmic surfaces have 12 ORDINARY DOUBLE POINTS , which are the vertices of three tetrahedra in 3-space (Hunt). See also QUARTIC SURFACE References Hunt, B. "Desmic Surfaces." §B.5.2 in The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 311 /C1/15, 1996. Jessop, C. §13 in Quartic Surfaces with Singular Points. Cambridge, England: Cambridge University Press, 1916. Destructive Dilemma A formal argument in LOGIC in which it is stated that 1. P [Q and R [S (where [means "IMPLIES "), and 2. Either not-Q or not-S is true, from which two statements it follows that either not- Por not- Ris true. See also CONSTRUCTIVE DILEMMA ,DILEMMA Determinant Determinants are mathematical objects which are very useful in the analysis and solution of SYSTEMS OF LINEAR EQUATIONS . As shown by C RAMER’S RULE ,a nonhomogeneous system of linear equations has anontrivial solution IFFthe determinant of the sys- tem’s MATRIX isNONZERO (i.e., the MATRIX is non- singular). For example, eliminating x,y, and zfrom the equations a1x/C27a2y/C27a3z/C300 (1) b1x/C27b2y/C27b3z/C300 (2) c1x/C27c2y/C27c3z/C300 (3) gives the expression a1b2c3/C28a1b3c2/C27a2b3c1/C28a2b1c3/C27a3b1c2/C28a3b2c1 /C300; (4) which is called the determinant for this system of equation. Determinants are defined only for SQUARE MATRICES . If the determinant of a MATRIX is 0, the MATRIX is said to be a SINGULAR MATRIX . The determinant of a MATRIX A; a1a2/C1/C1/C1 an b1b2/C1/C1/C1 bn nn:::n z1z2/C1/C1/C1 zn0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1(5) is commonly denoted det A;Ajj;or in component notation as a9a 1b2c3/C1/C1/C1 ðÞ ;Da1b2c3/C1/C1/C1 ðÞ ;ora1b2c3/C1/C1/C1 jj (Muir 1960, p. 17). A2/C292 determinant is defined to be detab cd0C1B0C1@ /C13ab cd0C@10C@10C@10C@10C@10C@10C@10C@1/C13ad/C28bc: (6) Ak/C29kdeterminant can be expanded "by MINORS "t o obtaina11 a21 n ak1a12 a22 n ak2a13 a23 n ak3/C1/C1/C1 /C1/C1/C1::: /C1/C1/C1a1k a2k n akk0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30a 11a22 n ak2a23 n ak3/C1/C1/C1::: /C1/C1/C1a2k n akk0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C28a 12a21 n ak1a23 n ak3/C1/C1/C1::: /C1/C1/C1a2k n akk0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C27/C1/C1/C1 9a 1ka21 n ak1a22 n ak2/C1/C1/C1::: /C1/C1/C1a2(k/C281) n ak(k/C281)0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (7) A general determinant for a MATRIX Ahas a value Ajj/C30X iaijaij; (8) with no implied summation over jand where aijis the COFACTOR ofaijdefined by aij/C13(/C281)i/C27jCij: (9) Here, Cis the ( n/C281)/C29(n/C281)MATRIX formed by eliminating row iand column jfrom A:This process is called DETERMINANT EXPANSION BY MINORS (or "Laplacian expansion by minors," sometimes further shortened to simply "Laplacian expansion"). A determinant can also be computed by writing down allPERMUTATIONS off1;...;ng;taking each permuta- tion as the subscripts of the letters a,b, ..., and summing with signs determined by ep/C30(/C281)i(p); where i(p) is the number of PERMUTATION INVERSIONS in permutation p(Muir 1960, p. 16), and en1n2... i s the PERMUTATION SYMBOL . For example, with n/C303, the permutations and the number of inversions theycontain are 123 (0), 132 (1), 213 (1), 231 (2), 312 (2),and 321 (3), so the determinant is given by a 1a2a3 b1b2b3 c1c2c30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C30a 1b2c3/C28a1b3c2/C28a2b1c3/C27a2b3c1/C27a3b1c2 /C28a3b2c1: ð10Þ Ifcis a constant and Aann/C29nSQUARE MATRIX , then aAjj/C30anAjj: (11) Given an n/C29ndeterminant, the additive inverse is /C28Ajj/C30(/C281)nAjj: (12) Determinants are also DISTRIBUTIVE ,s o ABjj/C30AjjBjj: (13) This means that the determinant of a MATRIX INVERSE can be found as follows: Ijj/C30AA/C2810C@10C@10C@10C@1/C30AjjA /C2810C@10C@10C@10C@1/C301; (14) where Iis the IDENTITY MATRIX ,s o Ajj/C301 A/C2810C@10C@10C@10C@1: (15) Determinants areMULTILINEAR in rows and columns, since a1a2a3 a4a5a6 a7a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30a 100 a4a5a6 a7a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C270a 20 a4a5a6 a7a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C2700 a 3 a4a5a6 a7a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 (16) and a 1a2a3 a4a5a6 a7a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30a 1a2a3 0aa6 0a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C270a 2a3 a4a5a6 0a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C270a 2a3 0a5a6 a7a8a90C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (17) The determinant of the SIMILARITY TRANSFORMATION of a matrix is equal to the determinant of the original MATRIX BAB/C2810C@10C@10C@10C@1/C30BjjAjjB/C2810C@10C@10C@10C@1/C30BjjAjj1 Bjj/C30Ajj: (18) The determinant of a similarity transformation minus a multiple of the unit MATRIX is given by B/C281AB/C28lI0C@10C@10C@10C@1/C30B/C281AB/C28B/C281lIB0C@10C@10C@10C@1/C30B /C281(A/C28lI)B0C@10C@10C@10C@1 /C30B /C2810C@10C@10C@10C@1A/C28lI jj Bjj/C30A/C28lI jj : (19) The determinant of a MATRIX TRANSPOSE equals the determinant of the original MATRIX , Ajj/C30AT0C@10C@10C@10C@1; (20) and the determinant of a COMPLEX CONJUGATE is equal to the COMPLEX CONJUGATE of the determinant ¯A0C@10C@10C@10C@1/C30Ajj: (21) Letobe a small number. Then I/C27eA jj /C301/C27eTr(A)/C27Oe20CB0C@ ; (22) where Tr( A) is the TRACE ofA:The determinant takes on a particularly simple form for a TRIANGULAR MATRIX a11a21/C1/C1/C1 ak1 0a22/C1/C1/C1 ak2 nn:::n 00 nakk0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30Y k n/C301ann: (23) Important properties of the determinant include the following, which include invariance under ELEMEN- TARY ROW AND COLUMN OPERATIONS .1. Switching two rows or columns changes thesign. 2. Scalars can be factored out from rows and columns.3. Multiples of rows and columns can be added together without changing the determinant’s va- lue.4. Scalar multiplication of a row by a constant c multiplies the determinant by c. 5. A determinant with a row or column of zeros hasvalue 0. 6. Any determinant with two rows or columns equal has value 0. Property 1 can be established by induction. For a 2 /C29 2 MATRIX , the determinant is a1b1 a2b20C@10C@10C@10C@10C@10C@10C@10C@1/C30a 1b2/C28b1a2/C30/C28 b1a2/C28a1b2 ðÞ /C30/C28b1a1 b2a20C@10C@10C@10C@10C@10C@10C@10C@1(24) For a 3 /C293 MATRIX , the determinant is a1b1c1 a2b2c2 a3b3c30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30a 1b2c2 b3c30C@10C@10C@10C@10C@10C@10C@10C@1/C28b 1a2c2 a3c30C@10C@10C@10C@10C@10C@10C@10C@1/C27c 1a2b2 a3b30C@10C@10C@10C@10C@10C@10C@10C@1 /C30/C28 a 1c2b2 c3b30C@10C@10C@10C@10C@10C@10C@10C@1/C27b 1c2a2 c3a30C@10C@10C@10C@10C@10C@10C@10C@1/C28c 1a2b2 a3b30C@10C@10C@10C@10C@10C@10C@10C@10C@80C@9 /C30/C28a 1c1b1 a2c2b2 a3c3b30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C30/C28 /C28 a 1b2c2 b3c30C@10C@10C@10C@10C@10C@10C@10C@1/C27b 1a2c2 a3c30C@10C@10C@10C@10C@10C@10C@10C@1/C27c1b2a2 b3a30C@10C@10C@10C@10C@10C@10C@10C@10C@80C@9 /C30/C28b1a1c1 b2a2c2 b3a3c30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C30/C28 /C28 a 1c2b2 c3b30C@10C@10C@10C@10C@10C@10C@10C@1/C28b 1a2c2 a3c30C@10C@10C@10C@10C@10C@10C@10C@1/C27c 1b2a2 b3a30C@10C@10C@10C@10C@10C@10C@10C@10C@80C@9 /C30/C28c 1b1a1 c2b2a2 c3b3a30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (25) Property 2 follows likewise. For 2 /C292 and 3 /C293 matrices, ka 1b1 ka2b20C@10C@10C@10C@10C@10C@10C@10C@1/C30ka 1b2 ðÞ /C28kb1a2 ðÞ /C30ka1b1 a2b20C@10C@10C@10C@10C@10C@10C@10C@1(26) and ka 1b1c1 ka2b2c2 ka3b3c30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30ka 1b2c2 b3c30C@10C@10C@10C@10C@10C@10C@10C@1/C28b 1ka2c2 ka3c30C@10C@10C@10C@10C@10C@10C@10C@1 /C27c1ka2b2 ka3b30C@10C@10C@10C@10C@10C@10C@10C@1/C30ka 1b1c1 a2b2c2 a3b3c30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (27) Property 3 follows from the identity a 1 /C27kb1b1c1 a2 /C27kb2b2c2 a3 /C27kb3b3c30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C30 a 1 /C27kb1 ðÞ /C2b2c2 b3c30C@10C@10C@10C@10C@10C@10C@10C@1/C28b 1a /C27kb2c2 a3 /C27kb3c30C@10C@10C@10C@10C@10C@10C@10C@1/C27c 1a2 /C27kb2b2 a3 /C27kb3b30C@10C@10C@10C@10C@10C@10C@10C@1: (28) If a ij is an n /C29n MATRIX with aij REAL NUMBERS , then det[aij] has the interpretation as the oriented n- dimensional CONTENT of the PARALLELEPIPED spanned by the column vectors [ai;1] ; ..., [ai ;n]inRn :: Here, "oriented" means that, up to a change of /C27or /C28 SIGN, the number is the n-dimensional CONTENT , but the SIGN depends on the "orientation" of the column vectors involved. If they agree with the standard orientation, there is a /C27SIGN; if not, there is a /C28SIGN. The PARALLELEPIPED spanned by the n-D vectors v1 through vi is the collection of points t1v1 /C27.../C27tivi ; (29) where tjis a REAL NUMBER in the CLOSED INTERVAL [0;1]::/ Several accounts state that Lewis Carroll (Charles Dodgson ) sent Queen Victoria a copy of one of his mathematical works, in one account, An Elementary Treatise on Determinants . Heath (1974) states, "A well-known story tells how Queen Victoria, charmed by Alice in Wonderland , expressed a desire to receive the author’s next work, and was presented, in due course, with a loyally inscribed copy of An Elementary Treatise on Determinants ," while Gattegno (1974) asserts "Queen Victoria, having enjoyed Alice so much, made known her wish to receive the author’s other books, and was sent one of Dodgson’s mathe- matical works." However, in Symbolic Logic (1896), Carroll stated, "I take this opportunity of giving what publicity I can to my contradiction of a silly story, which has been going the round of the papers, about my having presented certain books to Her Majesty the Queen. It is so constantly repeated, and is such absolute fiction, that I think it worth while to state, once for all, that it is utterly false in every particular: nothing even resembling it has occurred" (Mikkelson and Mikkelson). Hadamard (1893) showed that the absolute value ofthe determinant of a COMPLEX n /C29n matrix with entries in the UNIT DISK satisfies det A jj5nn=2 (30) (Brenner 1972). The plots above show the distribution of determinants for random n /C29n complex matrices with entries satisfying aij0C@10C@10C@10C@1B1 for n/C302, 3, and 4. There are an infinite number of 3 /C293 determinants with no 0 or 91 entries having unity determinant. One parametric family is /C288n2/C288n 2n/C2714 n /C284n2/C284nn /C2712 n/C271 /C284n2/C284n/C281 n 2n/C2810C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (31) Specific examples having small entries include 232 4239670C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1;235 3239570C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1;236 323 17 11 160C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1;. . . (32) (Guy 1989, 1994). See also C AYLEY- MENGER DETERMINANT ,CIRCULANT DETERMINANT ,COFACTOR ,CONDENSATION ,CRAMER’S RULE,DETERMINANT EXPANSION BY MINORS ,DETER- MINANT IDENTITIES ,ELEMENTARY ROW AND COLUMN OPERATIONS ,H ADAMARD’S MAXIMUM DETERMINANT PROBLEM ,H ESSIAN DETERMINANT ,H YPERDETERMI- NANT ,IMMANANT ,JACOBIAN ,K NOT DETERMINANT , MATRIX ,M INOR ,PERMANENT ,PFAFFIAN ,SINGULAR MATRIX ,SYLVESTER’S DETERMINANT IDENTITY ,SYL- VESTER MATRIX ,SYSTEM OF EQUATIONS ,V ANDER- MONDE DETERMINANT ,W RONSKIAN References Andrews, G. E. and Burge, W. H. "Determinant Identities." Pacific J. Math. 158,1/C1/4, 1993. Arfken, G. "Determinants." §4.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 168 /C1/76, 1985. Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626/C1/30, 1972. Dostor, G. Ele´ments de la the ´orie des de ´terminants, avec application a `l’alge`bre, la trigonome ´trie et la ge ´ome´trie analytique dans le plan et l’espace, 2e `me ed. Paris: Gauthier-Villars, 1905. Gattegno, J. Lewis Carroll: Fragments of a Looking-Glass. New York: Crowell, 1974. Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903/C1/09, 1989. Guy, R. K. "A Determinant of Value One." §F28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 265 /C1/66, 1994. Hadamard, J. "Re ´solution d’une question relative aux de´terminants." Bull. Sci. Math. 17,3 0/C1/1, 1893. Heath, P. The Philosopher’s Alice: Alice’s Adventures in Wonderland and Through the Looking-Glass. New York: St. Martin’s Press, 1974. Kowalewski, G. Einfu ¨hrung in die Determinantentheorie. New York: Chelsea, 1948. Mikkelson, D. P. and Mikkelson, B. "Fit for a Queen." http:// www.snopes.com/errata/carroll.htm. Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Whittaker, E. T. and Robinson, G. "Determinants and Linear Equations." Ch. 5 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 71 /C1/7, 1967. Yvinec, Y. "Geometric Computing: Exact Sign of a Determi- nant." http://www-sop.inria.fr/prisme/personnel/yvinec/ Determinants/english.html. Determinant (Binary Quadratic Form) The determinant of a BINARY QUADRATIC FORM Au2 /C272Buv /C27Cv2 is D /C13B2 /C28AC: It is equal to 1/4 of the corresponding DISCRIMINANT . Determinant (Knot) KNOT DETERMINANT Determinant Expansion by Minors Also known as "Laplacian" determinant expansion by minors, expansion by minors is a technique for computing the DETERMINANT of a given SQUARE MATRIX M : Although efficient for small matrices, techniques such as GAUSSIAN ELIMINATION are much more efficient when the matrix size becomes large. Let Mjjdenote the DETERMINANT of a MATRIX M ; then Mjj/C30Xk i/C301/C281ðÞi/C27jaijMij ; (1) where Mijis a so-called MINOR of M ; obtained by taking the determinant of M with row i and column j "crossed out." For example, for a 3 /C293 matrix, the above formula gives a11a12a13 a21a22a23 a31a32a330C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C30a 11a22a23 a32a330C@10C@10C@10C@10C@10C@10C@10C@1/C28a 12a21a23 a31a330C@10C@10C@10C@10C@10C@10C@10C@1/C27a 13a21a22 a31a320C@10C@10C@10C@10C@10C@10C@10C@1: (2) The procedure can then be iteratively applied to calculate the minors in terms of subminors, etc. The factor (/C281) i/C27j is sometimes absorbed into the minor as Mjj/C30Xk i/C301aijCij ; (3) in which case Cij is called a COFACTOR . The equation for the determinant can also be formally written as Ajj/C30X p(/C281)I(p)Yn i/C301ai;p(i); (4)where pranges over all permutations of 1 ;2; :::;n fg and I( p) is the INVERSION NUMBER ofp(Bressoud and Propp 1999). See also COFACTOR ,CONDENSATION ,DETERMINANT , GAUSSIAN ELIMINATION References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 169 /C1/70, 1985. Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637/C1/46. Muir, T. "Minors and Expansions." Ch. 4 in A Treatise on the Theory of Determinants. New York: Dover, pp. 53 /C1/37, 1960. Determinant Identities Interesting DETERMINANT identities include 1ab/C27c 1bc/C27a 1ca/C27b0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C300 (1) (Muir 1960, p. 39), a/C27b/C27c/C27dbcd b/C27c/C27d/C27acda c/C27d/C27a/C27bdac d/C27a/C27b/C27cabc0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C301bcd 1cda 1dab 1abc0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C2(a/C27b/C27c/C27d) (2) (Muir 1960, p. 41), 1aa 2a3 1bb2b3 1cc2c3 1dd2d30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30(b/C28a)(c/C28a)(c/C28b)(d/C28a)(d/C28b) /C2(d/C28c) (3) (Muir 1960, p. 42), bcd a a 2a3 cda b b2b3 dab c c2c3 abc d d2d30C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C301a 2a3a4 1b2b3b4 1c2c3c4 1d2d3d40C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1(4) (Muir 1960, p. 47), 0a 2b2c2 a20g2b2 b2g20a2 c2b2a200C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C300aabbcg aa0cgaa bbcg0aa cgbbaa00C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1(5) (Muir 1960, p. 42), 11 1 1 11/C27x 11 111 /C27y 1 11 11 /C27z0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30xyz (6) (Muir 1960, p. 44), and the C AYLEY- MENGER DETER- MINANT 0 abc a 0 cb bc 0 a cba 00C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C3001 1 1 10 c 2b2 1 c20 a2 1 b2a200C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1(7) (Muir 1960, p. 46), which is closely related to H ERON’S FORMULA . See also DETERMINANT References Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Determinant Theorem Given a MATRIX M ; the following are equivalent: 1. Mjj"0:/ 2. The columns of M are linearly independent. 3. The rows of M are linearly independent. 4. Range( /M) /C30 Rn ::/ 5. Null( /M) /C30f0g:/ 6. M has a MATRIX INVERSE . See also DETERMINANT ,MATRIX INVERSE ,NULLSPACE , RANGE (IMAGE ) Deterministic AT URING MACHINE is called deterministic if there is always at most one instruction associated with a given present internal state/tape state pair (q, s). Otherwise, it is called nondeterministic (Itoˆ 1987, p. 137). In prediction theory, let fXt g be a weakly stationary process, and let Mt(X) be a subspace spanned by the Xs(with s 5t) : If Mt(X) is independent of t so that Mt(X) /C30M(X) for every t, then fXt g is said to be deterministic (Itoˆ 1987, p. 1463). See also TURING MACHINE References Itoˆ, K. (Ed.). "Turing Machines." §31B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 136 /C1/37, 1987. Itoˆ, K. (Ed.). §395D in Encyclopedic Dictionary of Mathe- matics, 2nd ed., Vol. 3. Cambridge, MA: MIT Press, p. 1463, 1987. Developable Surface A surface on which the GAUSSIAN CURVATURE K is everywhere 0. See also BINORMAL DEVELOPABLE ,GAUSSIAN CURVA- TURE ,NORMAL DEVELOPABLE ,SYNCLASTIC ,TANGENT DEVELOPABLE References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 5, 1987.Deviation The DIFFERENCE of a quantity from some fixed value, usually the "correct" or "expected" one. See also ABSOLUTE DEVIATION ,AVERAGE ABSOLUTE DEVIATION ,D IFFERENCE ,D ISPERSION (STATISTICS ), MEAN DEVIATION ,S IGNED DEVIATION ,S TANDARD DEVIATION References Kenney, J. F. and Keeping, E. S. "Deviations." §6.3 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 76 1962. Devil on Two Sticks DEVIL’S CURVE Devil’s Curve The devil’s curve was studied by G. Cramer in 1750 and Lacroix in 1810 (MacTutor Archive). It appearedinNouvelles Annales in 1858. The Cartesian equation is y 4/C28a2y2/C30x4/C28b2x2; (1) equivalent to y2y2/C28a20CB0C@ /C30x2x2/C28b20CB0C@ ; (2) the polar equation is r2sin2u/C28cos2u0CB0C@ /C30a2sin2u/C28b2cos2u; (3) and the PARAMETRIC EQUATIONS are x/C30costffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2sin2t/C28b2cos2t sin2t/C28cos2ts ð4Þ y/C30sintffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2sin2t/C28b2cos2t sin2t/C28cos2ts : (5) The curve illustrated above corresponds to para- meters a2 /C301 and b2 /C302 :/ A special case of the Devil’s curve is the so-called "electric motor curve": y2 y2 /C28960CB0C@ /C30x2 x2 /C281000CB0C@ (6) (Cundy and Rollett 1989). References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 92 /C1/3, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 151 /C1/52, 1972. MacTutor History of Mathematics Archive. "Devil’s Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Dev- ils.html. Devil’s Needle Puzzle BAGUENAUDIER Devil’s Staircase A plot of the WINDING NUMBER W resulting from MODE LOCKING as a function of V for the CIRCLE MAP un/C271 /C30 un /C27V/C28K 2psin(2pun) with K /C301. (Since the CIRCLE MAP becomes MODE-LOCKED , the WINDING NUMBER is independent of the initial starting argument u0 :/) At each value of V; the WINDING NUMBER is some RATIONAL NUMBER . The result is a monotonic increasing "staircase" for which the simplest RATIONAL NUMBERS have the largest steps. The Devil’s staircase continuously maps the interval [0; 1] onto [0;1]; but is constant almost everywhere (i.e., except on a CANTOR SET). For K /C301, the MEASURE of quasiperiodic states (/ V IRRATIONAL ) on the V/-axis has become zero, and the measure of MODE-LOCKED state has become 1. The DIMENSION of the Devil’s staircase :0:8700 93 :7 /C2910 /C284 :/ See also CANTOR FUNCTION ,CIRCLE MAP,M INKOWS- KI’S QUESTION MARK FUNCTION ,W INDING NUMBER (MAP) References Devaney, R. L. An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, pp. 109 /C1/ 10, 1987. Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993. Rasband, S. N. "The Circle Map and the Devil’s Staircase." §6.5 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 128 /C1/32, 1990. Diabolic Square The term used by Hunter and Madachy (1975, p. 24) and Madachy (1979, p. 87) to refer to a PANMAGIC SQUARE . See also PANMAGIC SQUARE References Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, 1975. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, 1979. Diabolical Cube A 6-piece POLYCUBE DISSECTION of the 3 /C293 CUBE . See also CUBE DISSECTION ,SOMA CUBE References Gardner, M. "Polycubes." Ch. 3 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 29 /C1/0, 1986. Diabolical Square DIABOLIC SQUARE Diabolo One of the three 2-POLYABOLOES . See also POLYABOLO Diacaustic The ENVELOPE of refracted rays for a given curve. See also CATACAUSTIC ,CAUSTIC References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 60, 1972. Diagonal A diagonal of a SQUARE MATRIX which is traversed in the "southeast" direction. "The" diagonal (or "main diagonal" or "principal diagonal"rpar; of an n /C29n square matrix is the diagonal from a11 to ann :/ See also DIAGONAL MATRIX ,D IAGONAL METRIC , DIAGONAL (POLYGON ), DIAGONAL (POLYHEDRON ), DI- AGONAL RAMSEY NUMBER ,DIAGONAL SLASH ,DIAGO- NAL TRIANGLE ,D IAGONALIZABLE MATRIX ,SHALLOW DIAGONAL ,SKEW DIAGONAL ,SUBDIAGONAL ,SUPER- DIAGONAL ,TRIDIAGONAL MATRIX Diagonal (Polygon) A LINE SEGMENT connecting two nonadjacent VER- TICES of a POLYGON . The number of ways a fixed convex n-gon can be divided into TRIANGLES by nonintersecting diagonals is Cn/C282(with Cn/C283diag-onals), where Cnis a CATALAN NUMBER . This is EULER’S POLYGON DIVISION PROBLEM . Counting the number of regions determined by drawing the diag- onals of a regular n-gon is a more difficult problem, as is determining the number of n-tuples of CONCUR- RENT diagonals (Kok 1972). The number of regions which the diagonals of a CONVEX POLYGON divide its center if no three are concurrent in its interior is N /C30n 40C@80C@9 /C27n /C281 40C@80C@9 /C301 24(n /C281)(n /C282) n2 /C283n /C27120CB0C@ : The first few values are 0, 0, 1, 4, 11, 25, 50, 91, 154, 246, ... (Sloane’s A006522). See also CATALAN NUMBER ,D IAGONAL (POLYHE- DRON ), EULER’S POLYGON DIVISION PROBLEM ,POLY- GON,VERTEX (POLYGON ) References Kok, J. Item 2 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 3, Feb. 1972. Sloane, N. J. A. Sequences A006522/M3413 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Diagonal (Polyhedron) A LINE SEGMENT connecting two nonadjacent sides of a POLYHEDRON . Any polyhedron having no diagonals must have a SKELETON which is a COMPLETE GRAPH (Gardner 1975). The only SIMPLE POLYHEDRON with no diagonals is the TETRAHEDRON . The only known TOROIDAL POLYHEDRON with no diagonals is the CSA´ SZA´ R POLYHEDRON . See also CSA´ SZA´ R POLYHEDRON , TETRAHEDRON References Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102 /C1/07, May 1975. See also CSA´ SZA´ R POLYHEDRON ,D IAGONAL (POLY- GON), EULER BRICK,POLYHEDRON ,SPACE DIAGONAL , TETRAHEDRON Diagonal (Solidus) SOLIDUS Diagonal Block Matrix BLOCK DIAGONAL MATRIX Diagonal Matrix A diagonal matrix is a SQUARE MATRIX AOF THE FORM aij/C30cidij; (1) where dijis the K RONECKER DELTA ,ciare constants, and i; j /C301; 2, ..., n, with is no implied summation over indices. The general diagonal matrix is therefore OF THE FORM c10 /C1/C1/C1 0 0 c2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 cn2 6643 775 (2) often denoted diag c 1 ;c2 ;...;cn ðÞ : The diagonal matrix with elements l /C30 c1 ; ... ;cn fg can be computed in Mathematica usingDiagonalMatrix [l]. Given a MATRIX EQUATION OF THE FORM a11/C1/C1/C1 a1n n::: n an1/C1/C1/C1 ann2 435l 1/C1/C1/C1 0 n::: n 0 /C1/C1/C1 ln2435 /C30l 1/C1/C1/C1 0 n::: n 0 /C1/C1/C1 ln2 435a 11/C1/C1/C1 a1n n::: n an1/C1/C1/C1 ann2435; (3) multiply through to obtain a 11 l1/C1/C1/C1 a1n ln n::: n an1 l1/C1/C1/C1 ann ln2 435/C30a 11 l1/C1/C1/C1 a1n l1 n::: n an1 ln/C1/C1/C1 ann ln2435: (4) Since in general, l i " lj for i "j; this can be true only if off-diagonal components vanish. Therefore, A must be diagonal. Given a diagonal matrix T; the MATRIX POWER can be computed simply by taking each element to the power in question, Tn /C30t10 /C1/C1/C1 0 0 t2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 tk26643 775n /C30tn 10 /C1/C1/C1 0 0 tn2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 tn k2 6643 775: (5) Similarly, a MATRIX EXPONENTIAL can be performed simply by exponentiating each of the diagonal ele- ments, exp(A) /C30et1 0 /C1/C1/C1 0 0 et2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 etk2 6643 775: (6) See also CANONICAL BOX MATRIX, DIAGONAL ,DIAG- ONALIZABLE MATRIX ,EXPONENTIAL MATRIX ,M ATRIX , NORMAL MATRIX ,P ERSYMMETRIC MATRIX ,S KEW SYMMETRIC MATRIX ,SYMMETRIC MATRIX ,TRIANGU- LAR MATRIX ,TRIDIAGONAL MATRIX References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 181 /C1/84 and 217 /C1/29, 1985.Diagonal Metric A METRIC gij which is zero for i "j:/ See also METRIC Diagonal Quadratic Form If A /C30(aij)isa DIAGONAL MATRIX , a special case of a SYMMETRIC MATRIX , then Q( y) /C30vTAv /C30X aiiv2 i is a diagonal quadratic form, and Q(v;w) /C30vTAw is its associated diagonal SYMMETRIC BILINEAR FORM . For a general SYMMETRIC MATRIX A ; a SYMMETRIC BILINEAR FORM Q may be diagonalized by a nonde- generate n /C29n matrix C such that Q(C y;Cw)isa diagonal form. That is, CTAC is a DIAGONAL MATRIX . Note that C may not be an ORTHOGONAL MATRIX . Here is a Mathematica function to find a matrix C which will diagonalize a symmetric bilinear form, given a SYMMETRIC MATRIX . DiagonalizerMatrix[a_List?MatrixQ] : /C30 Module[ { q, ctr, t2, v1 /C30 Prepend[Table[0, {Length[a] - 1}], 1] }, q[v_] : /C30 v.a.v; If[(t2 /C30 q[v1]) ! /C30 0, v1 / /C30 Sqrt[Abs[t2]]]; ctr /C30 {v1}; Do[ v1 /C30 NullSpace[ctr.a][[1]]; If[(t2 /C30 q[v1]) ! /C30 0, v1 / /C30 Sqrt[Abs[t2]]]; AppendTo[ctr, v1], {Length[a] - 1} ];Transpose[Sort[ctr, q[#1] /C21 q[#2] &]] ] For example, consider A/C3012 230C1B0C1@ : Then taking C/C301/C282 010C1B0C1@ gives CTAC/C3010 0/C2810C1B0C1@ ; soAhas SIGNATURE (1;1):/ See also QUADRATIC FORM,S IGNATURE (MATRIX ), SYMMETRIC BILINEAR FORM,VECTOR SPACE Diagonal Ramsey Number AR AMSEY NUMBER OF THE FORM Rðk ;k;2Þ:/ See also RAMSEY NUMBER Diagonal Slash CANTOR DIAGONAL METHOD Diagonal Triangle The TRIANGLE determined by the intersections of the sides and diagonals of a CYCLIC QUADRILATERAL . Each vertex is the POLE of the opposite side with respect to the CIRCLE See also CYCLIC QUADRILATERAL ,POLE (INVERSION ), TRIANGLE References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 44, 1991. Diagonalizable Matrix This entry contributed by VIKTOR BENGTSSON An n /C29n/-matrix A is said to be diagonalizable if it can be written on the form A /C30PDP /C281 ; where D is a DIAGONAL n /C29n matrix with the EIGENVALUES of A as its entries and P is an INVER- TIBLE n /C29n matrix consisting of the EIGENVECTORS corresponding to the EIGENVALUES in D:/ The diagonalization theorem states that a quadratic matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. Diagonalization (and most other forms of matrix factorisation) are particularly useful when studying linear transforma- tions, discrete dynamical systems, continuous sys- tems, and so on. See also CANTOR DIAGONAL ARGUMENT ,D IAGONAL MATRIX ,D IAGONAL QUADRATIC FORM,INVERTIBLE MATRIXDiagonalization MATRIX DIAGONALIZATION Diagonals Problem EULER BRICK Diagram A schematic mathematical illustration showing the relationships between or properties of mathematical objects. See also ALTERNATING KNOT DIAGRAM ,A RGAND DIAGRAM ,C OXETER- DYNKIN DIAGRAM , DE BRUIJN DIAGRAM ,D YNKIN DIAGRAM ,F ERRERS DIAGRAM , HASSE DIAGRAM ,H EEGAARD DIAGRAM ,K NOT DIA- GRAM ,L INK DIAGRAM ,P LOT,STEM-AND- LEAF DIA- GRAM ,VENN DIAGRAM ,VORONOI DIAGRAM ,YOUNG DIAGRAM Diagrammatic Move KNOT MOVE Diameter The diameter of a CIRCLE is the DISTANCE from a point on the CIRCLE to a point p RADIANS away, and is the maximum distance from one point on a circle to another. The diameter of a SPHERE is the maximum distance between two ANTIPODAL POINTS on the sur- face of the sphere. If r is the RADIUS of a CIRCLE or SPHERE , then d /C302r: The ratio of the CIRCUMFERENCE C of a CIRCLE or GREAT CIRCLE of a SPHERE to the diameter disPI, p/C30C d: See also BROCARD DIAMETER ,CIRCUMFERENCE ,GEN- ERALIZED DIAMETER ,GRAPH DIAMETER ,PI,RADIUS , SPHERE ,TRANSFINITE DIAMETER Diamond Another word for a RHOMBUS . The diamond is also the name given to the unique 2-POLYIAMOND . See also KITE,LOZENGE ,PARALLELOGRAM ,POLYIA- MOND ,QUADRILATERAL ,RHOMBUS Dice A die (plural "dice") is a SOLID with markings on each of its faces. The faces are usually all the same shape, making P LATONIC SOLIDS and A RCHIMEDEAN SOLID DUALS the obvious choices. The die can be "rolled" by throwing it in the air and allowing it to come to reston one of its faces. Dice are used in many games ofchance as a way of picking RANDOM NUMBERS on which to bet, and are used in board or role-playinggames to determine the number of spaces to move,results of a conflict, etc. A COIN can be viewed as a degenerate 2-sided case of a die. The most common type of die is a six-sided CUBE with the numbers 1 /C1/placed on the faces. The value of the roll is indicated by the number of "spots" showing on the top. For the six-sided die, opposite faces are arranged to always sum to seven. This gives twopossible MIRROR IMAGE arrangements in which the numbers 1, 2, and 3 may be arranged in a clockwise orcounterclockwise order about a corner. Commercialdice may, in fact, have either orientation. Theillustrations below show 6-sided dice with counter- clockwise and clockwise arrangements, respectively. The CUBE has the nice property that there is an upward-pointing face opposite the bottom face from which the value of the "roll" can easily be read. This would not be true, for instance, for a TETRAHEDRAL die, which would have to be picked up and turned over to reveal the number underneath (although it could be determined by noting which number 1 /C1/was not visible on one of the upper three faces). The arrangement of spots corresponding to a roll of 5 on a six-sided die is called the QUINCUNX . There are also special names for certain rolls of two six-sided dice: two 1s are called SNAKE EYES and two 6s are called B OXCARS . Shapes of dice other than the usual 6-sided CUBE are commercially available from companies such as Dice& Games, Ltd. Diaconis and Keller (1989) show that there exist "fair" dice other than the usual P LATONIC SOLIDS and duals of the A RCHIMEDEAN SOLIDS , wherea fair die is one for which its symmetry group actstransitively on its faces (i.e., ISOHEDRA ). There are 30 isohedra. The probability of obtaining ppoints (a roll of p)o nn s-sided dice can be computed as follows. The number of ways in which pcan be obtained is the COEFFICIENT ofxpin f(x)/C30x/C27x2/C27.../C27xs0CB0C@n(1) since each possible arrangement contributes one term. f(x) can be written as a MULTINOMIAL SERIES f(x)/C30xnXs/C281 i/C300xi ! n /C30xn1/C28xs 1/C28x !n ; (2) so the desired number cis the COEFFICIENT ofxpin xn1/C28xsðÞn1/C28x ðÞ/C28n: (3) Expanding, xnXn k/C300/C281ðÞkn k0C@80C@9 xskX/C12 l/C300n/C27l/C281 l0C@80C@9 xl; (4) so in order to get the COEFFICIENT ofxp;include all terms with p/C30n/C27sk/C27l: (5) cis therefore c/C30Xn k/C300(/C281)kn k0C@80C@9 p/C28sk/C281 p/C28sk/C28n0C@80C@9 : (6) But p/C28sk/C28n>0 only when kB(p/C28n)=s;so the other terms do not contribute. Furthermore, p/C28sk/C281 p/C28sk/C28n0C@80C@9 /C30p/C28sk/C281 n/C2810C@80C@9 ; (7) so c/C30X(p/C28n)=s bc k/C300(/C281)kn k0C@80C@9 p/C28sk/C281 n/C2810C@80C@9 ; (8) where xbcis the FLOOR FUNCTION , and P(p;n;s)/C301 snX(p/C28n)=s bc k/C300(/C281)kn k0C@80C@9 p/C28sk/C281 n/C2810C@80C@9 (9) (Uspensky 1937, pp. 23 /C1/4). Consider now s/C306. For n/C302 six-sided dice, kmax/C13p/C282 6$% /C300 for 2 5p57 1 for 12 5p58;0C1n (10) and P(p;2;6)/C301 62Xkmax k/C300(/C281)k2 k0C@80C@9 p/C286k/C281 10C@80C@9 /C301 62Xkmax k/C300(/C281)k 2! k!(2/C28k)!(p/C286k/C281) /C301 36Xkmax k/C300(1/C282k)(k/C271)(p/C286k/C281) 1 36p/C281 13/C28pfor 25p57 for 85p5120C1n /C306/C28p/C287 jj 36for 25p512: (11) The most common roll is therefore seen to be a 7, with probability 6 =36/C301=6;and the least common rolls are 2 and 12, both with probability 1/36. Forn/C303 six-sided dice, kmax¼np/C283 6$% ¼0 for 3 5p58 1 for9 5p514 2 for15 5p518;8 < :(12) and P(p;3;6) /C301 63Xkmax k/C300(/C281)k3 k0C@80C@9 p/C286k/C281 20C@80C@9 /C301 63Xkmax k/C300(/C281)k 3! k!(3/C28k)!(p/C286k/C281)(p/C286k/C282) 2 /C301 216 /C2(p/C281)(p/C282) 2 for 35p58 (p/C281)(p/C282) 2/C283(p/C287)(p/C288) 2 for 95p514 (p/C281)(p/C282) 2/C283(p/C287)(p/C288) 2/C273(p/C2813)(p/C2814) 2 for 155p518:8 >>>>>>>>>>>>>< >>>>>>>>>>>>>: /C30 1 2161 2(p/C281)(p/C282) for 3 5p58 /C28p2/C2721p/C2883 for 9 5p514 12(19/C28p)(20/C28p) for 15 5p518:8 >>>>>< >>>>>:(13) For three six-sided dice, the most common rolls are 10 and 11, both with probability 1/8; and the leastcommon rolls are 3 and 18, both with probability 1/ 216. For four six-sided dice, the most common roll is 14, with probability 73/648; and the least common rollsare 4 and 24, both with probability 1/1296. In general, the likeliest roll /pL/forns-sided dice is given bypL(n;s)/C301 2n(s/C271)$% ; (14) which can be written explicitly as pL(n;s)/C3012n(s/C271) for neven 12n(s/C271)/C281 ½/C138 fornodd;seven 1 2n(s/C271) for nodd;sodd:8 >>>>>>>>< >>>>>>>>:(15) For 6-sided dice, the likeliest rolls are given by p L(n;6)/C307 2n$% /C3072n for n even 12(7n/C281) for n odd ;8 >>>< >>>:(16) or 7, 10, 14, 17, 21, 24, 28, 31, 35, ... for n/C302, 3, ... (Sloane’s A030123) dice. The probabilities corre- sponding to the most likely rolls can be computed byplugging p/C30p Linto the general formula together with kL(n;s)/C301 2n forneven n(s/C281)/C281 2s$% fornodd;seven n(s/C281) 2s$% fornodd;sodd:8 >>>>>>>>>< >>>>>>>>>:(17) Unfortunately, P(p L;n;s) does not have a simple closed-form expression in terms of sandn. However, the probabilities of obtaining the likeliest roll totals can be found explicitly for a particular s. For n6- sided dice, the probabilities are 1/6, 1/8, 73/648, 65/648, 361/3888, 24017/279936, 7553/93312, ... forn/C302, 3, .... The probabilities for obtaining a given total using n6- sided dice are shown above for n/C301, 2, 3, and 4 dice. They can be seen to approach a G AUSSIAN DISTRIBU- TION as the number of dice is increased. See also BOXCARS ,COIN TOSSING ,CRAPS , DE ME´ RE´ ’S PROBLEM ,EFRON’S DICE,ISOHEDRON ,POKER ,QUIN- CUNX ,SICHERMAN DICE,SNAKE EYES,YAHTZEE References Culin, S. "Tjou-sa-a--Dice." §72 in Games of the Orient: Korea, China, Japan. Rutland, VT: Charles E. Tuttle, pp. 78 /C1/9, 1965. Diaconis, P. and Keller, J. B. "Fair Dice." Amer. Math. Monthly 96, 337 /C1/39, 1989. Dice & Games, Ltd. "Dice & Games Hobby Games Acces- sories." http://www.dice.co.uk/hob.htm. Gardner, M. "Dice." Ch. 18 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 251 /C1/62, 1978. Pegg, E. Jr. "Fair Dice." http://www.mathpuzzle.com/Fair- dice.htm. Robertson, L. C.; Shortt, R. M.; Landry, S. G. "Dice with Fair Sums." Amer. Math. Monthly 95, 316 /C1/28, 1988. Sloane, N. J. A. Sequences A030123 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 23 /C1/4, 1937. Dichroic Polynomial A POLYNOMIAL ZG(q ;v) in two variables for abstract GRAPHS .A GRAPH with one VERTEX has Z /C30q. Adding a VERTEX not attached by any EDGES multiplies the Z by q. Picking a particular EDGE of a GRAPH G, the POLYNOMIAL for G is defined by adding the POLY- NOMIAL of the GRAPH with that EDGE deleted to v times the POLYNOMIAL of the graph with that EDGE collapsed to a point. Setting v /C30/C28 1 gives the number of distinct VERTEX colorings of the GRAPH . The dichroic POLYNOMIAL of a PLANAR GRAPH can be expressed as the SQUARE BRACKET POLYNOMIAL of the corresponding ALTERNATING LINK by ZG(q;v) /C30qN =2BL(G) ; where N is the number of VERTICES in G. Dichroic POLYNOMIALS for some simple GRAPHS are ZK1/C30q ZK2/C30q2 /C27vq ZK3/C30q3 /C273vq2 /C273v2q /C27v3 : References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 231 /C1/35, 1994. Dickman Function The probability that a random integer between 1 and x will have its GREATEST PRIME FACTOR 5xaap- proaches a limiting value F(a)a s x0/C12;where F(a)/C301 for a>1 andF(a)/C30ga 0Ft 1/C28t ! dt t for 05a51 (Dickman 1930, Knuth 1997). Similarly, the second-largest prime factor will be 5xbwith approximate probability G(b);where G(b)/C301 for b] 1=2 and G(b)/C30gb 0Gt 1/C28t ! /C28Ft 1/C28t ! "# dt t for 05b51=2::/ See also GREATEST PRIME FACTOR ,PRIME FACTORS References Dickman, K. Arkiv fo ¨r Mat., Astron. och Fys. 22A,1/C1/4, 1930. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, pp. 382 /C1/84, 1998. Norton, K. K. Numbers with Small Prime Factors, and the Least k th Power Non-Residue. Providence, RI: Amer. Math. Soc., 1971. Ramaswami, V. "On the Number of Positive Integers Less than xand Free of Prime Divisors Greater than xc:/"Bull. Amer. Math. Soc. 55, 1122 /C1/127, 1949. Ramaswami, V. "The Number of Positive Integers 5Xand Free of Prime Divisors >xG;and a Problem of S. S. Pillai." Duke Math. J. 16,9 9/C1/09, 1949. Dicone BICONE Dictionary Order LEXICOGRAPHIC ORDER Dido’s Problem Find the figure bounded by a line which has the maximum AREA for a given PERIMETER . The solution is a SEMICIRCLE . The problem is based on a passage from Virgil’s Aeneid : "The Kingdom you see is Carthage, the Tyrians, the town of Agenor; But the country around is Libya, no folk to meet in war. Dido, who left the city of Tyre to escape her brother,Rules here–a long a labyrinthine tale of wrongIs hers, but I will touch on its salient points in order.... Dido, in great disquiet, organised her friends for escape. They met together, all those who harshly hated the tyrant Or keenly feared him: they seized some ships which chanced to be ready... They came to this spot, where to-day you can behold the mighty Battlements and the rising citadel of New Carthage, And purchased a site, which was named ‘Bull’s Hide’ after the bargain By which they should get as much land as they could enclose with a bull’s hide." See also ISOPERIMETRIC PROBLEM ,ISOVOLUME PRO- BLEM ,PERIMETER ,SEMICIRCLE References Thomas, I. Greek Mathematical Works, Vol. 2: From Aris- tarchus to Pappus. London: Heinemann, 1980. Tikhomirov, V. M. Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., pp. 9 /C1/8, 1991. Virgil. Translated by C. D. Lewis. Book I, lines 307 /C1/72 in The Aeneid. New York: Doubleday, pp. 22 /C1/3, 1953. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 122 /C1/24, 1991. Diesis The symbol %; also called the DOUBLE DAGGER (Bring- hurst 1997, p. 277). References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, 1997. Diffeomorphic See also DIFFEOMORPHISM Diffeomorphism A diffeomorphism is a MAP between MANIFOLDS which is DIFFERENTIABLE and has a DIFFERENTIABLE in- verse. See also ANOSOV DIFFEOMORPHISM ,AXIOM AD IFFEO- MORPHISM ,D IFFEOMORPHIC ,P ESIN THEORY ,S YM- PLECTIC DIFFEOMORPHISM ,TANGENT MAP Difference The difference of two numbers n1and n2is n1 /C28n2 ; where the MINUS sign denotes SUBTRACTION . See also BACKWARD DIFFERENCE ,F INITE DIFFER- ENCE ,FORWARD DIFFERENCE ,M INUS ,SUBTRACTION , SYMMETRIC DIFFERENCE Difference Equation A difference equation is the discrete analog of a DIFFERENTIAL EQUATION . A difference equation in- volves a FUNCTION with INTEGER -valued arguments f(n) in a form like f(n) /C28f(n /C281) /C30g(n) ; (1) where g is some FUNCTION . The above equation is the discrete analog of the first-order ORDINARY DIFFER- ENTIAL EQUATION f ?(x) /C30g(x) (2) Examples of difference equations often arise inDYNAMICAL SYSTEMS . Examples include the iteration involved in the MANDELBROT and JULIA SET defini- tions, f(n /C271) /C30f(n)2 /C27c ; (3) with c a constant, as well as the LOGISTIC EQUATION f(n /C271) /C30rf(n)1/C28f(n) ½/C138 ; (4) with ra constant. See also FINITE DIFFERENCE ,O RDINARY DIFFEREN- TIAL EQUATION ,RECURRENCE RELATION References Agarwal, R. P. Difference Equations and Inequality: Theory, Methods, and Applications, 2nd ed., rev. exp. New York: Dekker, 2000. Batchelder, P. M. An Introduction to Linear Difference Equations. New York: Dover, 1967. Bellman, R. E. and Cooke, K. L. Differential-Difference Equations. New York: Academic Press, 1963. Beyer, W. H. "Finite Differences." CRC Standard Mathema- tical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 429 /C1/60, 1988. Brand, L. Differential and Difference Equations. New York: Wiley, 1966. Fulford, G.; Forrester, P.; and Jones, A. Modelling with Differential and Difference Equations. New York: Cam- bridge University Press, 1997. Goldberg, S. Introduction to Difference Equations, with Illustrative Examples from Economics, Psychology, and Sociology. New York: Dover, 1986. Levy, H. and Lessman, F. Finite Difference Equations. New York: Dover, 1992. Richtmyer, R. D. and Morton, K. W. Difference Methods for Initial-Value Problems, 2nd ed. New York: Interscience Publishers, 1967. Weisstein, E. W. "Books about Difference Equations." http:// www.treasure-troves.com/books/DifferenceEqua- tions.html. Difference of Successes Ifx1=n1andx2=n2are the observed proportions from standard NORMALLY DISTRIBUTED samples with pro- portion of success u;then the probability that w/C13x1 n1/C28x2 n2(1) will be as great as observed is Pd/C301/C282gdjj 0f(t)dt (2) where d/C13w sw(3) sw/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆu1/C28ˆu0CB0C@ 1 n1/C271 n2 !vuut(4) ˆu /C13x1 /C27 x2 n1 /C27 n2: (5) Here, ˆu is the UNBIASED ESTIMATOR . The SKEWNESS and KURTOSIS of this distribution are g2 1 /C30n1 /C28 n2 ðÞ2 n1n2n1 /C27 n2 ðÞ1 /C28 4 ˆu(1 /C28 ˆu) ˆu(1 /C28 ˆu) (6) g2 /C30n2 1 /C28 n1n2 /C27 n22 n1n2n1 /C27 n2 ðÞ1 /C28 6 ˆu 1 /C28 ˆu0CB0C@ ˆu 1 /C28 ˆu0CB0C@ : (7) Difference Operator BACKWARD DIFFERENCE ,FORWARD DIFFERENCE Difference Quotient Dhf(x) /C13f(x /C27 h) /C28 f(x) h/C30Df h: It gives the slope of the SECANT LINE passing through f(x) and f(x /C27h) : In the limit n 0 0; the difference quotient becomes the PARTIAL DERIVATIVE lim h01Dx(h)f(x; y) /C30@f @x : Difference Set Let G be a GROUP of ORDER h and D be a set of k elements of G. If the set of differences di /C28dj contains every NONZERO element of G exactly l times, then D is a (h;k; l)/-difference set in G of ORDER n /C30k /C28 l : If l /C301 ; the difference set is called planar. The quad- ratic residues in the FINITE FIELD GF(11) form a difference set. If there is a difference set of size k in a group G, then 2 k 20CB0C@ must be a multiple of Gjj/C281 ; where k20CB0C@ is a BINOMIAL COEFFICIENT . See also BRUCK- RYSER- CHOWLA THEOREM ,F IRST MULTIPLIER THEOREM ,PRIME POWER CONJECTURE References Gordon, D. M. "The Prime Power Conjecture is True for n B2 ;000; 000:/" Electronic J. Combinatorics 1,R61 /C1/, 1994. http://www.combinatorics.org/Volume_1/volu- me1.html#R6. Difference Table A table made by subtracting adjacent entries in a sequence, then repeating the process with those numbers. See also DIVIDED DIFFERENCE ,FINITE DIFFERENCE , INTERPOLATION ,QUOTIENT- DIFFERENCE TABLEReferences Sloane, N. J. A. and Plouffe, S. "Analysis of Differences." §2.5 in The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 10 /C1/3, 1995. Whittaker, E. T. and Robinson, G. "Difference Table." §2in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 2 /C1/, 1967. Different Two quantities are said to be different (or "unequal") if they are not EQUAL . The term "different" also has a technical usage related to MODULES . Let a MODULE M in an INTEGRAL DOMAIN D1forRffiffiffiffi Dp0C@n0C@o be expressed using a two- element basis as M/C30j1;j2 ½/C138 ; where j1andj2are in D1:Then the different of the MODULE is defined as D/C30D(M)/C30j1j2 j? 1j?20C@10C@10C@10C@10C@10C@10C@10C@1/C30j 1j? 2/C28j?1j2: The different D"0IFFj1and j2are linearly independent. The DISCRIMINANT is defined as the square of the different. See also DISCRIMINANT (MODULE ), EQUAL ,MODULE References Cohn, H. Advanced Number Theory. New York: Dover, pp. 72 /C1/3, 1980. Different Prime Factors DISTINCT PRIME FACTORS Differentiable AREAL FUNCTION is said to be differentiable at a point if its DERIVATIVE exists at that point. The notion of differentiability can also be extended to COMPLEX FUNCTIONS (leading to the C AUCHY- RIEMANN EQUA- TIONS and the theory of HOLOMORPHIC FUNCTIONS ), although a few additional subtleties arise in COMPLEX DIFFERENTIABILITY that are not present in the real case. Amazingly, there exist CONTINUOUS FUNCTIONS which are nowhere differentiable. Two examples are the BLANCMANGE FUNCTION and W EIERSTRASS FUNCTION . See also ANALYTIC FUNCTION ,BLANCMANGE FUNC- TION ,CAUCHY- RIEMANN EQUATIONS ,COMPLEX DIF- FERENTIABLE ,C ONTINUOUS FUNCTION ,D ERIVATIVE , HOLOMORPHIC FUNCTION ,P ARTIAL DERIVATIVE , WEAKLY DIFFERENTIABLE ,W EIERSTRASS FUNCTION References Krantz, S. G. "Alternative Terminology for Holomorphic Functions" and "Differentiable and CkCurves." §1.3.6 and 2.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 16 and 21, 1999. Differentiable Manifold SMOOTH MANIFOLD Differential A ONE-FORM . See also DIFFERENTIAL K-FORM,E XACT DIFFEREN- TIAL,INEXACT DIFFERENTIAL ,ONE-FORM Differential Calculus That portion of "the" CALCULUS dealing with DERIVA- TIVES . See also INTEGRAL CALCULUS Differential Equation An equation which involves the DERIVATIVES of a function as well as the function itself. If PARTIAL DERIVATIVES are involved, the equation is called a PARTIAL DIFFERENTIAL EQUATION ; if only ordinary DERIVATIVES are present, the equation is called an ORDINARY DIFFERENTIAL EQUATION . Differential equa- tions play an extremely important and useful role in applied math, engineering, and physics, and much mathematical and numerical machinery has been developed for the solution of differential equations. See also ADAMS’ METHOD ,D IFFERENCE EQUATION , INTEGRAL EQUATION ,ORDINARY DIFFERENTIAL EQUA- TION ,PARTIAL DIFFERENTIAL EQUATION References Arfken, G. "Differential Equations." Ch. 8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 437 /C1/96, 1985. Dormand, J. R. Numerical Methods for Differential Equa- tions: A Computational Approach. Boca Raton, FL: CRC Press, 1996. Differential Evolution A simple EVOLUTION STRATEGY which is fairly fast and reasonably robust. See also EVOLUTION STRATEGIES ,G ENETIC ALGO- RITHM ,OPTIMIZATION THEORY References Price, K. and Storn, R. "Differential Evolution." Dr. Dobb’s J., No. 264, 18 /C1/8, Apr. 1997. Differential Form DIFFERENTIAL K-FORM Differential Geometry Differential geometry is the study of RIEMANNIAN MANIFOLDS . Differential geometry deals with metrical notions on MANIFOLDS , while DIFFERENTIAL TOPOLOGY deals with those nonmetrical notions of MANIFOLDS . See also DIFFERENTIAL TOPOLOGYReferences Dillen, F. J. E. and Verstraelen, L. C.A. (Eds.). Handbook of Differential Geometry, Vol. 1. Amsterdam, Netherlands: North-Holland, 2000. Eisenhart, L. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, 1960. Graustein, W. C. Differential Geometry. New York: Dover, 1966. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997. Kreyszig, E. Differential Geometry. New York: Dover, 1991. Lipschutz, M. M. Theory and Problems of Differential Geometry. New York: McGraw-Hill, 1969. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 2nd ed. Berkeley, CA: Publish or Perish Press, 1979. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 2, 2nd ed. Berkeley, CA: Publish or Perish Press, 1990. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 3, 2nd ed. Berkeley, CA: Publish or Perish Press, 1990. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 4, 2nd ed. Berkeley, CA: Publish or Perish Press, 1979. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 5, 2nd ed. Berkeley, CA: Publish or Perish Press, 1979. Struik, D. J. Lectures on Classical Differential Geometry. New York: Dover, 1988. Weatherburn, C. E. Differential Geometry of Three Dimen- sions, 2 vols. Cambridge, England: Cambridge University Press, 1961. Weisstein, E. W. "Books about Differential Geometry." http://www.treasure-troves.com/books/DifferentialGeome- try.html. Differential Ideal A differential ideal J on a MANIFOLD M is an IDEAL in the EXTERIOR ALGEBRA of DIFFERENTIAL K-FORMS on M which is also CLOSED under the EXTERIOR DERIVA- TIVE d. That is, for any differential form a and any form b /C23I; then 1. a ffl b /C23I; and 2. d b /C23I/ For example, I/C30 xdy;dx ffldy hi is a differential ideal on M /C30R2 :/ A smooth map f : X 0 M is called an integral of J if the PULLBACK MAP of all forms in J vanish on X, i.e., f+(I)/C300:/ See also DIFFERENTIAL FORM,E NVELOPE (FORM), INTEGRABLE (DIFFERENTIAL IDEAL ), MANIFOLD Differential k-Form A differential k-form is a TENSOR ofRANK kwhich is antisymmetric under exchange of any pair of indices. The number of ALGEBRAICALLY INDEPENDENT compo- nents in n-D is given by the BINOMIAL COEFFICIENT n k0CB0C@ :In particular, a ONE-FORM v1(often simply called a "differential") is a quantity v1 /C30b1dx1 /C27b2dx2 /C27.../C27bndxn ; (1) where b1 /C30b1x1 ;x2 ;...;xn ðÞ and b2 /C30b2x1 ;x1 ; ... ;xn ðÞ are the components of a COVARIANT TENSOR . Chan- ging variables from x to y gives v1 /C30Xn i/C301bidxi /C30Xn i/C301Xn j/C301bi@xi @yjdyj /C30Xn j/C301bjdyj ; (2) where ¯bj /C13Xn i/C301bj@xi @yj; (3) which is the covariant transformation law. A p-ALTERNATING MULTILINEAR FORM on a VECTOR SPACE V corresponds to an element of fflp V +; the pth EXTERIOR POWER of the DUAL SPACE to V. A differ- ential p-form on a MANIFOLD is a SECTION of the VECTOR BUNDLE fflp T +M ; the pth EXTERIOR POWER of the COTANGENT BUNDLE . Hence, it is possible to write a p-form in coordinates by X Ijj/C30paIdxi1ffl...ffldxip(4) where I ranges over all increasing subsets of p elements from 1 ;...;n fg ; and the aIare functions. An important operation on differential forms, the EXTERIOR DERIVATIVE , is used in the celebrated STOKES’ THEOREM . The EXTERIOR DERIVATIVE d of a p form is a (p /C271)/-form. In fact, by definition, if xi is the coordinate function, thought of as a ZERO-FORM , then dxiðÞ/C30dxi :/ Another important operation on forms is the WEDGE PRODUCT , or exterior product. If a is a p-form and b is q-form, then a ffl b is a p /C27q form. Also, a p-form can be CONTRACTED with an r-vector, i.e., a SECTION of fflr TM ; to give a (p /C28r)/-form, or if r /C21p,an( r /C28p)/- vector. If the manifold has a METRIC , then there is an operation dual to the exterior product, called the INTERIOR PRODUCT . In higher dimensions, there are more kinds of differential forms. For instance, on the TANGENT SPACE to R2 there is the ZERO-FORM 1, two ONE-FORMS dx and dy, and one TWO-FORM dx ffldy: A ONE-FORM can be written uniquely as fdx /C27gdy : In four dimen- sions, dx1 ffldx2 /C27dx3 ffldx4is a TWO-FORM which cannot be written as a fflb:/ The minimum number of terms necessary to write a form is sometimes called the rank of the form, usually in the case of a TWO-FORM . When a form has rank one, it is called DECOMPOSABLE . Another meaning for rank of a form is its rank as a TENSOR , in which case a p- form can be described as an ANTISYMMETRIC TENSOR of rank p, in fact of type (0;p) : The rank of a form can also mean the dimension of its ENVELOPE , in which case the rank is an integer-valued function. With thelatter definition of rank, a p-form is decomposable IFF it has rank p. When n is the dimension of a MANIFOLD M, then n is also the dimension of the TANGENT SPACE TMx : Consequently, an n-form always has rank one, and for p /C21n,ap-form must be zero. Hence, an n-form is called a TOP-DIMENSIONAL FORM .A TOP-DIMENSIONAL FORM can be INTEGRATED without using a METRIC . Consequently, a p-form can be integrated on a p- dimensional SUBMANIFOLD . Differential forms are a VECTOR SPACE (with a C-INFINITY TOPOLOGY ) and therefore have a dual space. Submanifolds represent an element of the dual via integration, so it is common to say that they are in the dual space of forms, which is the space of CURRENTS . With a METRIC , the H ODGE STAR operator +defines a map from p-forms to ( n/C28p)/-forms such that /C31/C31/C30 (/C281)p(n/C28p):/ When f:M0Nis a SMOOTH MAP , it pushes forward TANGENT VECTORS from TM toTNaccording to the JACOBIAN f/C31:Hence, a differential form on Npulls back to a differential form on M. f/C31ay1ffl...fflyp0CB0C@ /C30af/C31y1ffl...fflf/C31yp0CB0C@ (5) The PULLBACK MAP is a linear map which commutes with the EXTERIOR DERIVATIVE , f+(da)/C30df+(a): (6) See also ANGLE BRACKET ,BRA,COVARIANT TENSOR , EXTERIOR ALGEBRA ,EXTERIOR DERIVATIVE ,H ODGE STAR,INTEGRATION (FORM), JACOBIAN ,K ET,M ANI- FOLD ,O NE-FORM,S TOKES’ THEOREM ,S YMPLECTIC FORM,T ANGENT BUNDLE ,T ENSOR ,T WO-FORM, WEDGE PRODUCT ,ZERO-FORM References Berger, M. Differential Geometry. New York: Springer- Verlag, pp. 146 /C1/37, 1988. Flanders, H. Differential Forms with Applications to the Physical Sciences. New York: Academic Press, 1963. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 2nd ed. Houston, TX: Publish or Perish, pp. 273 /C1/83, 1999. Sternberg, S. Differential Geometry. New York: Chelsea, pp. 14 /C1/0, 1983. Weintraub, S. H. Differential Forms: A Complement to Vector Calculus. San Diego, CA: Academic Press, 1996. Differential Operator The OPERATOR representing the computation of a DERIVATIVE , ˜D/C13d dx: (1) The second derivative is then denoted ˜D2;the third ˜D3;etc. The INTEGRAL is denoted ˜D/C281:/ The differential operator satisfies the identity x /C28d dx /C30/C28ex2 =2d dx e /C28x2 =2 (2) (Arfken 1985, p. 720). Furthermore, 2x /C28d dx !n 1 /C30Hn(x) ; (3) where Hn(x)isaH ERMITE POLYNOMIAL . The symbol q can be used to denote the operator q/C13zd dz (4) (Bailey 1935, p. 8). See also CONVECTIVE DERIVATIVE ,DERIVATIVE ,FRAC- TIONAL DERIVATIVE ,GRADIENT References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: University Press, 1935. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985. Differential Structure EXOTIC R4,EXOTIC SPHERE Differential Topology The motivating force of TOPOLOGY , consisting of the study of smooth (differentiable) MANIFOLDS . Differ- ential topology deals with nonmetrical notions of MANIFOLDS , while DIFFERENTIAL GEOMETRY deals with metrical notions of MANIFOLDS . See also DIFFERENTIAL GEOMETRY References Dieudonne ´,J. A History of Algebraic and Differential Topology: 1900 /C1/960. Boston, MA: Birkha ¨user, 1989. Munkres, J. R. Elementary Differential Topology. Princeton, NJ: Princeton University Press, 1963. Differentiating Under the Integral Sign INTEGRATION UNDER THE INTEGRAL SIGN,LEIBNIZ INTEGRAL RULE Differentiation The computation of a DERIVATIVE . See also CALCULUS ,DERIVATIVE ,INTEGRAL ,INTEGRA- TION References Griewank, A. Principles and Techniques of Algorithmic Differentiation. Philadelphia, PA: SIAM, 2000.Digamma Function ASPECIAL FUNCTION which is given by the LOGARITH- MIC DERIVATIVE of the GAMMA FUNCTION (or, depend- ing on the definition, the LOGARITHMIC DERIVATIVE of the FACTORIAL ). Because of this ambiguity, two different notations are sometimes (but not always) used, with C(z)/C13d dzlnG(z)/C30G?(z) G(z)(1) defined as the LOGARITHMIC DERIVATIVE of the GAMMA FUNCTION G(z);and F(z)/C13d dzlnz! (2) defined as the LOGARITHMIC DERIVATIVE of the FAC- TORIAL function. The two are connected by the relationship F(z)/C30C(z/C271): (3) ThenthDERIVATIVE ofC(z) is called the POLYGAMMA FUNCTION , denoted cn(z):The notation c0(z)/C30C(z)i s therefore frequently used for the digamma functionitself, and Erde ´lyiet al. (1981) use the notation c(z) forC(z):The function C(z)/C30c 0(z) is returned by the functionPolyGamma [z]o rPolyGamma [0,z]i nMath- ematica . From a series expansion of the FACTORIAL function, c0(z/C271)/C30d dz /C2lim n0/C12[lnn!/C27zlnn/C28ln(z/C271)/C28ln(z/C272) /C28.../C28ln(z/C27n) (4) /C30 lim n0/C12lnn /C281 z /C27 1 /C281 z /C27 2 /C28.../C281 z /C27 n ! (5) /C30/C28g /C28X/C12 n /C3011 z /C27 1 /C281 n ! (6) /C30/C28g /C27X/C12 n /C301z n(n /C27 z) (7) /C30lnz /C271 2z /C28X/C12 n /C301B2n 2nz2n ; (8) where g is the EULER- MASCHERONI CONSTANT and B2n are BERNOULLI NUMBERS . The digamma function satisfies c0(z) /C30g/C12 0e /C28t t/C28e/C28zt 1 /C28 e /C28t ! dt : (9) For integral z /C13n; c0(n) /C30/C28g /C27Xn/C281 k /C3011 k /C30/C28g /C27Hn /C281 ; (10) where g is the EULER- MASCHERONI CONSTANT and Hn is a HARMONIC NUMBER . Other identities include dc0 dz/C30X/C12 n/C3001 (z /C27 n)2 (11) c0(1 /C28z) /C28 c0(z) /C30p cot( pz) (12) c0(z /C271) /C30 c0(z) /C271 z (13) c0(2z) /C301 2 c0(z) /C2712 c 0 z /C2712 ! /C27ln2 : (14) Special values are c 012 ! /C30/C28g /C282 ln2 (15) c 0(1) /C30/C28g : (16) At integral values, c0(n /C271) /C30/C28g /C27Xn k /C3011k ; (17) and at half-integral values, c 012 /C27n ! /C30/C28g /C282 ln2 /C272X n k /C3011 2k /C28 1 /C30/C28g /C27Hn/C281 =2 ; (18) where Hnis a HARMONIC NUMBER . At rational argu-ments, c0(p=q) is given by GAUSS’S DIGAMMA THEO- REM. Sums and differences of c1(r =s) for small integral r and s can be expressed in terms of CATALAN’S CONSTANT and p:/ See also BARNES’ G-FUNCTION , G-FUNCTION ,GAMMA FUNCTION ,GAUSS’S DIGAMMA THEOREM ,H ARMONIC NUMBER ,H URWITZ ZETA FUNCTION ,L OGARITHMIC DERIVATIVE ,M ELLIN’S FORMULA ,POLYGAMMA FUNC- TION ,RAMANUJAN FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Psi (Digamma) Function." §6.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 258 /C1/59, 1972. Arfken, G. "Digamma and Polygamma Functions." §10.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 549 /C1/55, 1985. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The c Function." §1.7 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 15 /C1/0, 1981. Jeffreys, H. and Jeffreys, B. S. "The Digamma (/F) and Trigamma (/F?) Functions." Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 465 /C1/66, 1988. Spanier, J. and Oldham, K. B. "The Digamma Function c(x):/ " Ch. 44 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 423 /C1/34, 1987. Digimetic ACRYPTARITHM in which DIGITS are used to represent other DIGITS . See also CRYPTARITHM Digit The number of digits Din an INTEGER nis the number of numbers in some base (usually 10) re- quired to represent it. The numbers 1 to 9 aretherefore single digits, while the numbers 10 to 99 are double digits. Terms such as "double-digit infla- tion" are occasionally encountered, although thisparticular usage has thankfully not been needed inthe U.S. for some time. The number of (base 10) digits in a number ncan be calculated as D/C301/C27log 10njj bc ; where xbcis the FLOOR FUNCTION . The number of digits din the number nrepresented in base bis given by the Mathematica function DigitCount [n,b,d], withDigitCount [n,b] giving a list of the numbers of each digit in n. Numbers in base-10 which are divisible by their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44,48, 55, 66, 77, 88, 99, 111, 112, 115, 122, ... (Sloane’s A034838). Numbers which are divisible by the sum of their digits are called H ARSHAD NUMBERS :1 ,2 ,3 ,4 ,5 , 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (Sloane’s A005349). Numbers which are divisible by both their digits and the sum of their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 48, 111, 112, 126, 132, 135, 144, ... (Sloane’s A050104). Numbers which are equal to (i.e., not just divisible by) the product of their divisors and the sum of their divisors are called SUM-PRODUCT NUMBERS and are given by 1, 135, 144, ... (Sloane’s A038369). b order Sloane Numbers (/]b)/ 2 increasing 2 nondecreasing A000225 3, 7, 15, 31, 63, 127, 255, 511, 1023, ... 2 nonincreasing A031997 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, ... 2 decreasing 2 10 increasing A009993 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, ... 10 nondecreasing A009994 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, ... 10 nonincreasing A009996 10, 11, 20, 21, 22, 30, 31, 32, 33, 40, 41, 42, ... 10 decreasing A009995 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, ... 16 increasing A023784 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, ... 16 nondecreasing A023757 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, ... 16 nonincreasing A023771 17, 32, 33, 34, 48, 49, 50, 51, 64, 65, 66, 67, ... 16 decreasing A023797 32, 33, 48, 49, 50, 64, 65, 66, 67, 80, 81, 82, ... In HEXADECIMAL , numbers with increasing digits are called METADROMES , those with nondecreasing digits are called PLAINDRONES , those with nonincreasing digits are called NIALPDROMES , and those with de- creasing digits are called KATADROMES . The count of numbers with strictly increasing digits in base- bis 2b/C281;and the number with strictly decreasing digits is 2b/C281:/See also 196-ALGORITHM ,A DDITIVE PERSISTENCE , DIGIT PRODUCT ,DIGIT SERIES ,DIGIT-SHIFTING CON- STANTS ,DIGITADDITION ,DIGITAL ROOT,FACTORION , FIGURES ,H ARSHAD NUMBER ,K ATADROME ,LENGTH (NUMBER ), METADROME ,M ULTIPLICATIVE PERSIS- TENCE ,NARCISSISTIC NUMBER ,NIALPDROME ,PLAIN- DROME ,SCIENTIFIC NOTATION ,SIGNIFICANT DIGITS , SMITH NUMBER ,SUM-PRODUCT NUMBER References Bailey, D. H. and Crandall, R. E. "On the Random Char- acter of Fundamental Constant Expansions." Manuscript, Mar. 2000. http://www.nersc.gov/~dhbailey/dhbpapers/ dhbpapers.html. Sloane, N. J. A. Sequences A0053490481, A034838, A038369, and A050104 in "An On-Line Version of theEncyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Digit Block Let uB(n) be the number of DIGIT BLOCKS of a sequence Bin the base- bexpansion of n, which can be implemented in Mathematica as u[n_Integer, b_Integer, block_List] : /C30 Count[Partition[IntegerDigits[n, b], Length[block], 1], block] The following table gives the sequence uB(n) fg for a number of blocks B. BSloane sequence 00 A056973 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, ... 01 A037800 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, ... 10 A033264 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, ... 11 A014081 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, 0, ... 000 A056974 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, ... 001 A056975 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ... 010 A056976 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, ... 011 A056977 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, ... 100 A056978 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, ... 101 A056979 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, ... 110 A056980 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, ... 111 A014082 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, ... See also DIGIT SERIES ,RUDIN- SHAPIRO SEQUENCE References Sloane, N. J. A. Sequences A014081, A014082, A033264, A037800, A056973, A056974, A056975, A056976, A056977, A056978, A056979, and A056980 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Digit Product Let sb(n) be the sum of the base- b digits of n, and e(n) /C30(/C281)S2(n) the THUE- MORSE SEQUENCE , then Y/C12 n/C3002n /C27 1 2n /C27 2 !e(n) /C301 2ffiffiffi 2p : (1) See also DIGIT,DIGIT SERIES References Allouche, J.-P. "Series and Infinite Products Related to Binary Expansions of Integers." http://algo.inria.fr/semi- nars/sem92 /C1/3/allouche.ps. Shallit, J. O. "On Infinite Products Associated with Sums of Digits." J. Number Th. 21, 128 /C1/34, 1985. Digit Series Let sb(n) be the sum of the base- b digits of n, which can be implemented in Mathematica as s[n_, b_] : /C30 Plus @@ IntegerDigits[n, b] Then X/C12 n/C301sb(n) n(n /C27 1) /C30b b /C28 1 lnb ; (1) the b /C302 case of which was given in the 1981 Putnam competition (Allouche 1992). In addition, X/C12 n/C301s22n /C27 1 n2(n /C27 1)2 /C30p2 9 (2) X/C12 n/C302s2(n) ½/C13828n3 /C27 4n2 /C27 n /C28 1 4nn2 /C28 1 ðÞ 4n2 /C28 1 ðÞ/C3017 24 /C27ln2 (3) (Allouche 1992, Allouche and Shallit 1992). Let u(n) be the number of DIGIT BLOCKS of 11 in the binary expansion of n, thenX/C12 n/C301u(n) n(n /C27 1) /C3032ln2/C2814p (4) (Allouche 1992). See also D IGIT,DIGIT BLOCK ,DIGIT PRODUCT References Allouche, J.-P. "Series and Infinite Products Related to Binary Expansions of Integers." 1992. http://algo.inria.fr/ seminars/sem92 /C1/3/allouche.ps. Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences." Theor. Comput. Sci. 98, 163/C1/97, 1992. Shallit, J. O. "On Infinite Products Associated with Sums of Digits." J. Number Th. 21, 128/C1/34, 1985. Digitaddition Start with an INTEGER n, known as the GENERATOR . Add the SUM of the GENERATOR ’s digits to obtain the digitaddition n?:A number can have more than one GENERATOR . If a number has no GENERATOR ,i ti s called a SELF NUMBER . The sum of all numbers in a digitaddition series is given by the last term minus the first plus the sum of the DIGITS of the last. If the digitaddition process is performed on n?to yield itsdigitaddition nƒ;onnƒto yield n§;etc., a single- digit number, known as the DIGITAL ROOT ofn,i s eventually obtained. The digital roots of the first fewintegers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (Sloane’s A010888). If the process is generalized so that the kth (instead of first) powers of the digits of a number are repeatedly added, a periodic sequence of numbers iseventually obtained for any given starting number n. If the original number nis equal to the sum of the kth powers of its digits, it is called a N ARCISSISTIC NUMBER . If the original number is the smallest number in the eventually periodic sequence of num-bers in the repeated k-digitadditions, it is called a RECURRING DIGITAL INVARIANT . Both N ARCISSISTIC NUMBERS and RECURRING DIGITAL INVARIANTS are relatively rare. The only possible periods for repeated 2-digitaddi- tions are 1 and 8, and the periods of the first few positive integers are 1, 8, 8, 8, 8, 8, 1, 8, 8, 1, .... The possible periods pforn-digitadditions are summar- ized in the following table, together with digitaddi-tions for the first few integers and the corresponding sequence numbers. Some periods do not show up for a long time. For example, a period-6 10-digitadditiondoes not occur until the number 266. n Sloane ps n-Digitadditions 2 Sloane’s A0311761 ,8 1 ,8 ,8 ,8 ,8 ,8 ,1 ,8 ,8 , 1, ... 3 Sloane’s A0311781 ,2 ,3 1 ,1 ,1 ,3 ,1 ,1 ,1 ,1 ,1 , 1, 1, 1, 3, ... 4 Sloane’s A0311821 ,2 ,7 1 ,7 ,7 ,7 ,7 ,7 ,7 ,7 ,7 , 1, 7, 1, 7, 7, ... 5 Sloane’s A0311861, 2, 4, 6, 10, 12, 22, 281, 12, 22, 4, 10, 22, 28, 10, 22, 1, ... 6 Sloane’s A0311951, 2, 3, 4, 10, 301, 10, 30, 30, 30, 10,10, 10, 3, 1, 10, ... 7 Sloane’s A0312001, 2, 3, 6,12, 14, 21,27, 30, 56, 921, 92, 14, 30, 92, 56, 6, 92, 56, 1, 92, 27, ... 8 Sloane’s A0312111, 25, 154 1, 25, 154, 154, 154, 154, 25, 154, 154, 1,25, 154, 154, 1, ... 9 Sloane’s A0312121, 2, 3, 4,8, 10, 19,24, 28, 30, 80, 931, 30, 93, 1, 19, 80, 4, 30, 80, 1, 30, 93, 4, 10,... 10 Sloane’s A0312131, 6, 7, 17, 81, 1231, 17, 123, 17, 17, 123,123, 123, 123, 1, 17, 123, 17 ... The numbers having period-1 2-digitadded sequencesare also called HAPPY NUMBERS . The first few num- bers having period pn-digitadditions are summar- ized in the following table, together with theirsequence numbers. np Sloane Members 2 1 Sloane’s A0077701, 7, 10, 13, 19, 23, 28,31, 32, ... 2 8 Sloane’s A0311772, 3, 4, 5, 6, 8, 9, 11, 12,14, 15, ... 3 1 Sloane’s A0311791, 2, 3, 5, 6, 7, 8, 9, 10,11, 12, ... 3 2 Sloane’s A03118049, 94, 136, 163, 199,244, 316, ... 3 3 Sloane’s A0311814, 13, 16, 22, 25, 28, 31,40, 46, ... 4 1 Sloane’s A0311831, 10, 12, 17, 21, 46, 64,71, 100, ... 4 2 Sloane’s A03118466, 127, 172, 217, 228,271, 282, ... 4 7 Sloane’s A0311852, 3, 4, 5, 6, 7, 8, 9, 11,13, 14, ... 5 1 Sloane’s A0311871, 10, 100, 145, 154, 247,274, ...5 2 Sloane’s A031188133, 139, 193, 199, 226, 262, ... 5 4 Sloane’s A0311894, 37, 40, 55, 73, 124,142, ... 5 6 Sloane’s A03119016, 61, 106, 160, 601, 610, 778, ... 5 10 Sloane’s A0311915, 8, 17, 26, 35, 44, 47, 50, 53, ... 5 12 Sloane’s A0311922, 11, 14, 20, 23, 29, 32,38, 41, ... 5 22 Sloane’s A0311933, 6, 9, 12, 15, 18, 21, 24,27, ... 5 28 Sloane’s A0311947, 13, 19, 22, 25, 28, 31,34, 43, ... 6 1 Sloane’s A0115571, 10, 100, 1000, 10000,100000, ... 6 2 Sloane’s A0313573468, 3486, 3648, 3684,3846, ... 6 3 Sloane’s A0311969, 13, 31, 37, 39, 49, 57,73, 75, ... 6 4 Sloane’s A031197255, 466, 525, 552, 646,664, ... 6 10 Sloane’s A0311982, 6, 7, 8, 11, 12, 14, 15,17, 19, ... 6 30 Sloane’s A0311993, 4, 5, 16, 18, 22, 29, 30,33, ... 7 1 Sloane’s A0312011, 10, 100, 1000, 1259,1295, ... 7 2 Sloane’s A03120222, 202, 220, 256, 265,526, 562, ... 7 3 Sloane’s A031203124, 142, 148, 184, 214,241, 259, ... 7 6 7, 70, 700, 7000, 70000, 700000, ... 7 12 Sloane’s A03120417, 26, 47, 59, 62, 71, 74, 77, 89, ... 7 14 Sloane’s A0312053, 30, 111, 156, 165, 249, 294, ... 7 21 Sloane’s A03120619, 34, 43, 91, 109, 127, 172, 190, ... 7 27 Sloane’s A03120712, 18, 21, 24, 39, 42, 45,54, 78, ... 7 30 Sloane’s A0312084, 13, 16, 25, 28, 31, 37, 40, 46, ... 7 56 Sloane’s A0312096, 9, 15, 27, 33, 36, 48, 51, 57, ... 7 92 Sloane’s A0312102, 5, 8, 11, 14, 20, 23, 29, 32, 35, ... 8 1 1, 10, 14, 17, 29, 37, 41, 71, 73, ... 8 25 2, 7, 11, 15, 16, 20, 23, 27, 32, ... 8 154 3, 4, 5, 6, 8, 9, 12, 13, 18, 19, ... 9 1 1, 4, 10, 40, 100, 400, 1000, 1111, ... 9 2 127, 172, 217, 235, 253, 271, 325, ... 9 3 444, 4044, 4404, 4440, 4558, ... 9 4 7, 13, 31, 67, 70, 76, 103, 130, ... 9 8 22, 28, 34, 37, 43, 55, 58, 73, 79, ... 9 10 14, 38, 41, 44, 83, 104, 128, 140, ... 9 19 5, 26, 50, 62, 89, 98, 155, 206, ... 9 24 16, 61, 106, 160, 337, 373, 445, ... 9 28 19, 25, 46, 49, 52, 64, 91, 94, ... 9 30 2, 8, 11, 17, 20, 23, 29, 32, 35, ... 9 80 6, 9, 15, 18, 24, 33, 42, 48, 51, ... 9 93 3, 12, 21, 27, 30, 36, 39, 45, 54, ... 10 1 Sloane’s A0115571, 10, 100, 1000, 10000, 100000, ... 10 6 266, 626, 662, 1159, 1195, 1519, ... 10 7 46, 58, 64, 85, 122, 123, 132, ... 10 17 2, 4, 5, 11, 13, 20, 31, 38, 40, ... 10 81 17, 18, 37, 71, 73, 81, 107, 108, ... 10 123 3, 6, 7, 8, 9, 12, 14, 15, 16, 19, ... See also 196-ALGORITHM ,A DDITIVE PERSISTENCE ,DIGIT,DIGITAL ROOT,M ULTIPLICATIVE PERSISTENCE , NARCISSISTIC NUMBER ,RECURRING DIGITAL INVAR- IANT References Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Digital Root Consider the process of taking a number, adding its DIGITS , then adding the DIGITS of numbers derived from it, etc., until the remaining number has only one DIGIT . The number of additions required to obtain a single DIGIT from a number n is called the ADDITIVE PERSISTENCE of n, and the DIGIT obtained is called the digital root of n. For example, the sequence obtained from the starting number 9876 is (9876, 30, 3), so 9876 has an ADDITIVE PERSISTENCE of 2 and a digital root of 3. The digital roots of the first few integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 9, 1, ... (Sloane’s A010888). The digital root of an INTEGER n can therefore be com- puted without actually performing the iteration using the simple congruence formula n (mod 9) n f0 (mod 9) 9 n /C130 (mod 9):0C1n See also ADDITIVE PERSISTENCE ,D IGITADDITION , KAPREKAR NUMBER ,M ULTIPLICATIVE DIGITAL ROOT, MULTIPLICATIVE PERSISTENCE ,N ARCISSISTIC NUM- BER,RECURRING DIGITAL INVARIANT ,SELF NUMBER References Sloane, N. J. A. Sequences A007612/M1114 and A010888 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Digit-Extraction Algorithm An algorithm which allows digits of a given number to be calculated without requiring the computation of earlier digits. The BAILEY- BORWEIN-PLOUFFE ALGO- RITHM for PI is the best-known such algorithm, but an algorithm also exists for E. See also BAILEY- BORWEIN- PLOUFFE ALGORITHM Digit-Shifting Constants Given a REAL NUMBER x, find the powers of a base b that will shift the digits of xa number of places nto the left. This is equivalent to solving bx/C30bnx (1) or x /C30n /C27logbx: (2) The solution is given by x /C30/C28W /C28b/C28nlnb ðÞ lnb; (3) where W(x)isL AMBERT’S W-FUNCTION . The above plot shows logbx /C27n /C28x for b /C3010 and small values of n. As can be seen, there are two distinct solutions, corresponding to two different BRANCHES of W(x) in (3). For n /C301, 2, ..., these solutions are approximately given by 0.137129, 0.0102386, 0.00100231, 0.000100023, 0.0000100002, ..., and 1, 2.37581, 3.55026, 4.66925, 5.76046, ..., respectively. For example, 100 :0102385... /C301 :02385... (4) and 102 :37581... /C30237:581 ... (5) See also BASE (NUMBER ), DIGIT,LOGARITHM Digon The DEGENERATE POLYGON (corresponding to a LINE SEGMENT ) with SCHLA ¨ FLI SYMBOL {2}. See also LINE SEGMENT ,POLYGON ,TRIGONOMETRY VALUES PI/2 Digraph DIRECTED GRAPHDihedral Angle The ANGLE u between two PLANES . The dihedral angle between the planes A1x /C27B1y /C27C1z /C27D1 /C300 (1) A2x /C27B2y /C27C2z /C27D2 /C300 (2) which have normal vectors N1 /C30 A1 ;B1 ;C1 ðÞ and N2 /C30 A2 ; B2 ;C2 ðÞ is simply given via the DOT PRODUCT of the normals, cos u /C30N1 /C215N2 /C30A1A2 /C27 B1B2 /C27 C1C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 1 /C27 B21 /C27 C21pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A22 /C27 B22 /C27 C22p : (3) The dihedral angle between planes in a general TETRAHEDRON is closely connected with the face areas via a generalization of the LAW OF COSINES . See also ANGLE ,PLANE ,TETRAHEDRON ,TRIHEDRON , VERTEX ANGLE References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 15, 1948. Dihedral Group A GROUP of symmetries for an n-sided REGULAR POLYGON , denoted Dn : The ORDER of Dn is 2n:/ See also FINITE GROUP D3,FINITE GROUP D4 References Arfken, G. "Dihedral Groups, Dn:/"Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 248, 1985. Lomont, J. S. "Dihedral Groups." §3.10.B in Applications of Finite Groups. New York: Dover, pp. 78 /C1/0, 1987. Dihedral Prime A number nsuch that the "LED representation" of n (i.e., the arrangement of horizonal and vertical lines seen on a digital clock or pocket calculator), nupside down, nin a mirror, and nupside-down-and-in-a- mirror are all primes. The digits of nare therefore restricted to 0, 1, 2, 5, and 8. The first few dihedral primes are 2, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, ... (Sloane’s A038136). References Rivera, C. "Problems & Puzzles: Puzzle The Mirrorable Numbers (by Mike Keith).-039." http://www.primepuz- zles.net/puzzles/puzz_039.htm. Sloane, N. J. A. Sequences A038136 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Dijkstra Tree The shortest path-spanning TREE from a VERTEX of a GRAPH . Dijkstra’s Algorithm An ALGORITHM for finding a GRAPH GEODESIC , i.e., the shortest path between two VERTICES in a GRAPH .It functions by constructing a shortest-path tree from the initial vertex to every other vertex in the graph. The algorithm is implemented asDijkstra [g] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). See also FLOYD’S ALGORITHM ,GRAPH GEODESIC References Dijkstra, E. W. "A Note on Two Problems in Connection with Graphs." Numerische Math. 1, 269 /C1/71, 1959. Skiena, S. "Dijkstra’s Algorithm." §6.1.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 225 /C1/27, 1990. Whiting, P. D. and Hillier, J. A. "A Method for Finding the Shortest Route through a Road Network." Operational Res. Quart. 11,37/C1/0, 1960. Dilation A SIMILARITY TRANSFORMATION which transforms each line to a PARALLEL line whose length is a fixed multiple of the length of the original line. The simplest dilation is therefore a TRANSLATION , and any dilation that is not merely a TRANSLATION is called a CENTRAL DILATION . Two triangles related by a CENTRAL DILATION are said to be PERSPECTIVE TRIAN- GLES because the lines joining corresponding vertices CONCUR . A dilation corresponds to an EXPANSION plus a TRANSLATION .See also EXPANSION ,PARALLEL ,PERSPECTIVE TRIAN- GLES ,TRANSLATION References Coxeter, H. S. M. and Greitzer, S. L. "Dilation." §4.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 94 /C1/5, 1967. Dilative Rotation SPIRAL SIMILARITY Dilcher’s Formula X 15k5nn k0C@80C@9(/C281)k /C281 km /C30X 15i1 5i2 5...5im 5n1 i1i2 /C1/C1/C1im; (1) wheren k0CB0C@ is a BINOMIAL COEFFICIENT (Dilcher 1995, Flajolet and Sedgewick 1995, Prodinger 2000). An inverted version is given by X 1 5k 5nn k0C@80C@9 (/C281)k /C281X 1 5i15i25...5im/C30k1 i1i2 /C1/C1/C1im /C30X 1 5k 5n1 km /C30H(m) n ; (2) where H(k) nis a HARMONIC NUMBER of order m (Herna ´ndez 1999, Prodinger 2000). A Q-ANALOG of (1) is given by X 1 5k 5nn k0C1B0C1@ q(/C281)k/C281qk /C27 1 20C@80C@9 /C27 (m /C28 1)k 1 /C28 qk ðÞm /C30X 1 5i15i25...5im5nqi1 1 /C28 qi1/C1/C1/C1qim 1 /C28 qim; (3) where n k0C1B0C1@ q/C30(q;q)n (q;q)k(q;q)n/C28k(4) is a G AUSSIAN POLYNOMIAL (Prodinger 2000). See also BINOMIAL IDENTITY References Dilcher, K. "Some q-Series Identities Related to Divisor Functions." Disc. Math. 145,8 3/C1/3, 1995. Flajolet, P. and Sedgewick, R. "Mellin Transforms and Asymptotics: Finite Differences and Rice’s Integrals." Theor. Comput. Sci. 144, 101/C1/24, 1995. Herna ´ndez, V. "Solution IV of Problem 10490: A Reciprocal Summation Identity." Amer. Math. Monthly 106, 589/C1/90, 1999. Prodinger, H. "A q-Analogue of a Formula of Hernandez Obtained by Inverting a Result of Dilcher." Austral. J. Combin. 21, 271/C1/74, 2000. Dilemma Informally, a situation in which a decision must be made from several alternatives, none of which is obviously the optimal one. In formal LOGIC , a di- lemma is a specific type of argument using two conditional statements which may take the form of a CONSTRUCTIVE DILEMMA or a DESTRUCTIVE DI- LEMMA . See also CONSTRUCTIVE DILEMMA ,D ESTRUCTIVE DILEMMA ,MONTY HALL PROBLEM ,PARADOX ,PRISON- ER’S DILEMMA Dilogarithm A special case of the POLYLOGARITHM Lin(z) for n/C302. It is denoted Li2(z);or sometimes L2(z):The notation Li2(x) for the dilogarithm is unfortunately similar to that for the LOGARITHMIC INTEGRAL Li(x):The diloga- rithm can be defined by the sum Li2(z)/C30X/C12 k/C301zk k2(1) or the integral Li2(z)/C13g0 zln(1/C28t)dt t: (2) There are also two different commonly encountered normalizations for the Li2(z) function, both denoted L(z);and one of which is known as the R OGERS L- FUNCTION . The major functional equations for the dilogarithm are given byLi2(x)/C27Li2(/C28x)/C301 2Li2x20CB0C@ (3) Li2(1/C28x)/C27Li21/C28x/C2810CB0C@ /C30/C2812(lnx) 2(4) Li2(x)/C27Li2(1/C28x)/C301 6p2/C28(lnx) ln(1/C28x) (5) Li2(/C28x)/C28Li2(1/C28x)/C2712Li 21/C28x20CB0C@ /C30/C281 12p2/C28(lnx) ln(x/C271): (6) A complete list of Li2(x) which can be evaluated in closed form is given by Li2(/C281)/C30/C281 12p2(7) Li2(0)/C300 (8) Li212 ! /C301 12p2/C2812(ln 2) 2(9) Li2(1)/C3016p 2(10) Li2(/C28f)/C30/C281 10p2/C28(lnf)2(11) /C30/C281 10p2/C28csch/C28120CB0C@ 2(12) Li2(/C28f/C281)/C30/C281 15p2/C271 2(lnf)2(13) /C30/C281 15p2/C271 2csch/C28120CB0C@ 2(14) Lif/C2820CB0C@ /C301 15p2/C28(lnf)2(15) /C301 15p2/C28csch/C28120CB0C@ 2(16) Lif/C2810CB0C@ /C301 10p2/C28(lnf)2(17) /C301 10p2/C28csch/C28120CB0C@ 2; (18) where fis the GOLDEN RATIO (Lewin 1981, Borwein et al.1998). There are several remarkable identities involving the DILOGARITHM function. Ramanujan gave the identi- ties Li21 3 ! /C2816 Li 219 ! /C301 18 p2 /C2816(ln 3) 2 (19) Li2/C2812 ! /C2715Li 219 ! /C30/C281 18 p2 /C27ln 2 ln 3 /C2812(ln 2) 2 /C2813(ln 3) 2(20) Li21 4 ! /C2713Li 219 ! /C301 18 p2 /C272 ln 2 ln 3 /C282(ln 2)2 /C282 3(ln 3)2ð21Þ Li2/C2813 ! /C2813 Li 219 ! /C30/C281 18 p2 /C2716(ln 3) 2(22) Li2/C2818 ! /C27Li 219 ! /C30/C2812ln98 ! 2 (23) Li21 2ffiffiffi 5p /C2810C@n0C@o ! /C301 10 p2 /C28 ln1 21 /C27ffiffiffi 5p0C@n0C@o !"#2 (24) (Berndt 1994, Gordon and McIntosh 1997), and Bailey et al. show that p2 /C3036Li21 2 ! /C2836Li214 ! /C2812Li 218 ! /C276Li 21 64 ! (25) 12Li212 ! /C30p 2 /C286ln2ðÞ2(26) See also ABEL’S DUPLICATION FORMU LA,A BEL’S FUNCTIONAL EQUATION ,C LAUSEN FUNCTION ,IN- VERSE TANGENT INTEGRAL , L-ALGEBRAIC NUMBER , LEGENDRE’S CHI-FUNCTION ,LOGARITHM ,POLYLOGA- RITHM ,R OGERS L-FUNCTION ,SPENCE’S FUNCTION , SPENCE’S INTEGRAL ,TRILOGARITHM ,W ATSON IDENTI- TIES References Abramowitz, M. and Stegun, C. A. (Eds.). "Dilogarithm." §27.7 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1004 /C1/005, 1972. Andrews, G. E.; Askey, R.; and Roy, R. Special Functions. Cambridge, England: Cambridge University Press, 1999. Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 323 /C1/26, 1994. Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Losinek, P. "Special Values of Multidimensional Polyloga- rithms." CECM-98:106, 14 May 1998. http://www.cecm.s- fu.ca/preprints/1998pp.html#98:106.Bytsko, A. G. J. Physics A 32, 8045, 1999. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Euler’s Dilogarithm." §1.11.1 in Higher Transcen- dental Functions, Vol. 1. New York: Krieger, pp. 31 /C1/2, 1981. Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431 /C1/48, 1997. Kirillov, A. N. "Dilogarithm Identities." Progr. Theor. Phys. Suppl. 118,61/C1/42, 1995. Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, 1958. Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981. Lewin, L. "The Dilogarithm in Algebraic Fields." J. Austral. Soc. Ser. A 33, 302 /C1/30, 1982. Watson, G. N. Quart. J. Math. Oxford Ser. 8, 39, 1937. Dilworth’s Lemma The WIDTH of a set P is equal to the minimum number of CHAINS needed to COVER P. Equivalently, if a set P of ab /C271 elements is PARTIALLY ORDERED , then P contains a CHAIN of size a /C271oran ANTICHAIN of size b /C271 : Letting N be the CARDINALITY of P, W the WIDTH , and L the LENGTH , this last statement says N 5LW : Dilworth’s lemma is a generalization of the ERDOS-SZEKERES THEOREM .RAMSEY’S THEOREM gen- eralizes Dilworth’s lemma. See also ANTICHAIN ,CHAIN ,COMBINATORICS ,ERDOS- SZEKERES THEOREM ,RAMSEY’S THEOREM References Dilworth, R. P. "A Decomposition Theorem for Partially Ordered Sets." Ann. Math. 51, 161/C1/66, 1950. Skiena, S. "Dilworth’s Lemma." §6.4.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 241 /C1/43, 1990. Dilworth’s Theorem DILWORTH’S LEMMA Dimension The dimension of an object is a topological measure of the size of its covering properties. Roughly speaking, it is the number of coordinates needed to specify a point on the object. For example, a RECTANGLE is two- dimensional, while a CUBE is three-dimensional. The dimension of an object is sometimes also called its"dimensionality." The prefix "hyper-" is usually used to refer to the 4- (and higher-) dimensional analogs of 3-dimensionalobjects, e.g. HYPERCUBE ,HYPERPLANE . The notion of dimension is important in mathematicsbecause it gives a precise parameterization of theconceptual or visual complexity of any geometricobject. In fact, the concept can even be applied to abstract objects which cannot be directly visualized. For example, the notion of time can be considered asone-dimensional, since it can be thought of as con- sisting of only "now," "before" and "after." Since "before" and "after," regardless of how far back or how far into the future they are, are extensions, time is like a line, a 1-dimensional object. To see how lower and higher dimensions relate to each other, take any geometric object (like a POINT , LINE, CIRCLE , PLANE , etc.), and "drag" it in an opposing direction (drag a POINT to trace out a LINE, a LINE to trace out a box, a CIRCLE to trace out a CYLINDER ,aDISK to a solid CYLINDER , etc.). The result is an object which is qualitatively "larger" than the previous object, "qualitative" in the sense that, regardless of how you drag the original object, you always trace out an object of the same "qualitative size." The POINT could be made into a straight LINE,a CIRCLE ,aHELIX , or some other CURVE , but all of these objects are qualitatively of the same dimension. The notion of dimension was invented for the purpose of measuring this "qualitative" topological property. Finite collections of objects (e.g., points in space) are considered 0-dimensional. Objects that are "dragged" versions of 0-dimensional objects are then called 1- dimensional. Similarly, objects which are dragged 1- dimensional objects are 2-dimensional, and so on. Dimension is formalized in mathematics as the intrinsic dimension of a TOPOLOGICAL SPACE . This dimension is called the LEBESGUE COVERING DIMEN- SION (also known simply as the TOPOLOGICAL DIMEN- SION). The archetypal example is EUCLIDEAN n-space Rn ; which has topological dimension n. The basic ideas leading up to this result (including the DIMEN- SION INVARIANCE THEOREM , DOMAIN INVARIANCE THE- OREM , and LEBESGUE COVERING DIMENSION ) were developed by Poincare ´, Brouwer, Lebesgue, Urysohn, and Menger. There are several branchings and extensions of the notion of topological dimension. Implicit in the notion of the LEBESGUE COVERING DIMENSION is that dimen- sion, in a sense, is a measure of how an object fills space. If it takes up a lot of room, it is higher dimensional, and if it takes up less room, it is lower dimensional. HAUSDORFF DIMENSION (also called FRACTAL DIMENSION ) is a fine tuning of this definition that allows notions of objects with dimensions other than INTEGERS .FRACTALS are objects whose HAUS- DORFF DIMENSION is different from their TOPOLOGICAL DIMENSION . The concept of dimension is also used in ALGEBRA , primarily as the dimension of a VECTOR SPACE over a FIELD . This usage stems from the fact that VECTOR SPACES over the reals were the first VECTOR SPACES to be studied, and for them, their topological dimension can be calculated by purely algebraic means as the CARDINALITY of a maximal linearly independent sub- set. In particular, the dimension of a SUBSPACE of Rn is equal to the number of LINEARLY INDEPENDENT VECTORS needed to generate it (i.e., the number of VECTORS in its BASIS ). Given a transformation A of Rn ;dim[Range( A)] /C27dim[Null( A)] /C30dim(Rn) : See also 4-DIMENSIONAL GEOMETRY ,BASIS (VECTOR SPACE ), CAPACITY DIMENSION ,CODIMENSION ,CORRE- LATION DIMENSION ,EXTERIOR DIMENSION ,FRACTAL DIMENSION ,HAUSDORFF DIMENSION ,HAUSDORFF- BE- SICOVITCH DIMENSION ,K APLAN- YORKE DIMENSION , KRULL DIMENSION ,LEBESGUE COVERING DIMENSION , LEBESGUE DIMENSION ,LYAPUNOV DIMENSION ,POSET DIMENSION , Q-DIMENSION ,S IMILARITY DIMENSION , TOPOLOGICAL DIMENSION References Abbott, E. A. Flatland: A Romance of Many Dimensions. New York: Dover, 1992. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 8, 1991. Czyz, J. Paradoxes of Measures and Dimensions Originating in Felix Hausdorff’s Ideas. Singapore: World Scientific, 1994. Hinton, C. H. The Fourth Dimension. Pomeroy, WA: Health Research, 1993. Manning, H. The Fourth Dimension Simply Explained. Magnolia, MA: Peter Smith, 1990. Manning, H. Geometry of Four Dimensions. New York: Dover, 1956. Neville, E. H. The Fourth Dimension. Cambridge, England: Cambridge University Press, 1921. Rucker, R. von Bitter. The Fourth Dimension: A Guided Tour of the Higher Universes. Boston, MA: Houghton Mifflin, 1984. Sommerville, D. M. Y. An Introduction to the Geometry of N Dimensions. New York: Dover, 1958. Weisstein, E. W. "Books about Dimensions." http:// www.treasure-troves.com/books/Dimensions.html. Dimension Axiom One of the EILENBERG- STEENROD AXIOMS . Let X be a single point space. Hn(X) /C300 unless n /C300, in which case H0(X) /C300 where G are some GROUPS . The H0 are called the COEFFICIENTS of the HOMOLOGY THEORY H( /C215) :/ See also EILENBERG- STEENROD AXIOMS ,H OMOLOGY (TOPOLOGY ) Dimension Invariance Theorem /Rn is HOMEOMORPHIC to Rm IFF n /C30m. This theorem was first proved by Brouwer. See also DOMAIN INVARIANCE THEOREM Dimensionality DIMENSION Dimensionality Theorem For a FINITE GROUP ofhelements with an ni/th dimensional ith irreducible representation, X in2 i /C30h Diminished Polyhedron A UNIFORM POLYHEDRON with pieces removed. Diminished Rhombicosidodecahedron JOHNSON SOLID J76 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Dini Expansion An expansion based on the ROOTS of x/C28n xJt n(x) /C27HJn(x) ½/C138 /C300; where Jn(x)isaB ESSEL FUNCTION OF THE FIRST KIND , is called a Dini expansion. See also BESSEL FUNCTION FOURIER EXPANSION References Bowman, F. Introduction to Bessel Functions. New York: Dover, p. 109, 1958. Dini’s Surface A surface of constant NEGATIVE CURVATURE obtainedby twisting a PSEUDOSPHERE and given by the PARA- METRIC EQUATIONS x /C30a cos u sin v (1) y /C30a sin u sin v (2) z /C30a cos v /C27ln tan1 2 v !"#() /C27bu : (3) The above figure corresponds to a /C301, b /C300:2; u /C23 [0;4p]; and v /C23 (0; 2]:/ The coefficients of the FIRST FUNDAMENTAL FORM are E /C301 2a2 /C272b2 /C28a2 cos(2 v)0C10CC (4) F /C30ab cos v cot v (5) G /C30a2 cot2 v; (6) the coefficients of the SECOND FUNDAMENTAL FORM are e /C30/C28a2 cos v sin vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2p (7) f /C30ab cos vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 /C27 b2p (8) g /C30a2 cot vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2p ; (9) and the AREA ELEMENT is dA /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2p cosv: (10) The G AUSSIAN and MEAN CURVATURES are given by K/C30/C281 a2/C27b2(11) H/C30/C28cot(2 v)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2p : (12) See also PSEUDOSPHERE References Gray, A. "Dini’s Surface." §21.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 493 /C1/95, 1997. Nordstrand, T. "Dini’s Surface." http://www.uib.no/people/ nfytn/dintxt.htm. Dini’s Test A test for the convergence of F OURIER SERIES . Let fx(t)/C13f(x/C27t)/C27f(x/C28t)/C282f(x); then if gp 0fx(t) jj dt t is FINITE , the FOURIER SERIES converges to f(x)atx. See also FOURIER SERIES References Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 65 /C1/8, 1991. Dinitz Problem Given any assignment of n-element sets to the n2 locations of a square n /C29n array, is it always possible to find a PARTIAL LATIN SQUARE ? The fact that such a PARTIAL LATIN SQUARE can always be found for a 2 /C292 array can be proven analytically, and techniques were developed which also proved the existence for 4 /C294 and 6 /C296 arrays. However, the general problem eluded solution until it was answered in the affirma- tive by Galvin in 1993 using results of Janssen (1993ab) and F. Maffray. See also PARTIAL LATIN SQUARE References Chetwynd, A. and Ha ¨ggkvist, R. "A Note on List-Colorings." J. Graph Th. 13,8 7/C1/5, 1989. Cipra, B. "Quite Easily Done." In What’s Happening in the Mathematical Sciences 2, pp. 41 /C1/6, 1994. Erdos, P.; Rubin, A.; and Taylor, H. "Choosability in Graphs." Congr. Numer. 26, 125/C1/57, 1979. Ha¨ggkvist, R. "Towards a Solution of the Dinitz Problem?" Disc. Math. 75, 247/C1/51, 1989. Janssen, J. C. M. "The Dinitz Problem Solved for Rectan- gles." Bull. Amer. Math. Soc. 29, 243/C1/49, 1993a. Janssen, J. C. M. Even and Odd Latin Squares. Ph.D. thesis. Lehigh University, 1993b. Kahn, J. "Recent Results on Some Not-So-Recent Hyper- graph Matching and Covering Problems." Proceedings of the Conference on Extremal Problems for Finite Sets. Visegra `d, Hungary, 1991. Kahn, J. "Coloring Nearly-Disjoint Hypergraphs with / nþoðnÞ/Colors." J. Combin. Th. Ser. A 59,3 1/C1/9, 1992. Diocles’s Cissoid CISSOID OF DIOCLES Diophantine Equation An equation in which only INTEGER solutions are allowed. H ILBERT’S 10TH PROBLEM asked if a techni- que for solving a general Diophantine existed. A general method exists for the solution of first degreeDiophantine equations. However, the impossibility of obtaining a general solution was proven by Julia Robinson and Martin Davis in 1970, following proof ofthe result that the relation n/C30F 2m(where F2mis a FIBONACCI NUMBER ) is Diophantine by Yuri Matiya- sevich (Matiyasevich 1970, Davis 1973, Davis andHersh 1973, Davis 1982, Matiyasevich 1993). Morespecifically, Matiyasevich showed that there is a polynomial Pinn,m, and a number of othervariables x,y,z, ... having the property that n/C30 F 2mIFFthere exist integers x,y,z, ... such that P(n;m;x;y;z;... )/C300::/ Jones and Matiyasevich (1982) proved that no ALGO- RITHMS can exist to determine if an arbitrary Dio- phantine equation in nine variables has solutions. As a consequence of this result, it can be proved that there does not exists a general algorithm for solving a QUARTIC DIOPHANTINE EQUATION , although the algo- rithm for constructing such an unsolvable quarticDiophantine equation can require arbitrarily manyvariables (Matiyasevich 1993). Ogilvy and Anderson (1988) give a number of Dio- phantine equations with known and unknown solu- tions. A linear Diophantine equation (in two variables) is an equation of the general form ax/C27by/C30c; (1) where solutions are sought with a,b, and c INTEGERS . Such equations can be solved completely, and the firstknown solution was constructed by Brahmagupta. Consider the equation ax/C27by/C301: (2) Now use a variation of the E UCLIDEAN ALGORITHM , letting a/C30r1andb/C30r2 r1/C30q1r2/C27r3 (3) r2/C30q2r3/C27r4 (4) rn/C283/C30qn/C283rn/C282/C27rn/C281 (5) rn/C282/C30qn/C282rn/C281/C271: (6) Starting from the bottom gives 1/C30rn/C282/C28qn/C282rn/C281 (7) rn/C281/C30rn/C283/C28qn/C283rn/C282; (8) so 1/C30rn/C282/C28qn/C282(rn/C283/C28qn/C283rn/C282) /C30/C28qn/C282rn/C283/C27(1/C28qn/C282qn/C283)rn/C282: (9) Continue this procedure all the way back to the top. Take as an example the equation 1027 x/C27712y/C301: (10) Proceed as follows. 1027/C30712 /C2151/C27315½1/C30/C28165 /C2151027/C27238 /C215712/C160 712/C30315 /C2152/C2782½1/C3073 /C215712/C28165 /C215315½ 315/C3082 /C2153/C2769½1/C30/C2819 /C215315/C2773 /C21582½ 82/C3069 /C2151/C2713½1/C3016 /C215 82/C2819 /C21569½ 69/C3013 /C2155/C274½1/C30/C283/C215 69/C2716 /C21513½ 13/C30 4/C2153/C271¡1/C301/C215 13/C283/C2154½ 1/C300/C215 4/C271/C2151½ The solution is therefore x/C30/C28165, y/C30238. The above procedure can be simplified by noting that the two left-most columns are offset by one entry and alternate signs, as they must since 1 /C30/C28Ai /C271ri /C27Airi/C271 (11) ri/C271 /C30ri/C281 /C28riqi/C281 (12) 1 /C30Airi /C281 /C28 Aiqi /C281 /C27Ai/C2710CB0C@ ; (13) so the COEFFICIENTS of ri/C281 and ri/C271 are the same and Ai/C281 /C30/C28(Aiqi/C281 /C27Ai/C271) : (14) Repeating the above example using this information therefore gives 1027 /C30 712 /C2151/C27315½1 /C30/C28165 /C215 1027/C27 238 /C215712/C160 712 /C30 315 /C2152/C27 82 ½1 /C30 73 /C215 712/C28165 /C215315½ 315 /C30 82 /C2153/C2769 ½1 /C30/C2819 /C215 315/C27 73 /C215 82 ½ 82 /C30 69 /C2151/C2713 ½1 /C30 16 /C215 82/C28 19 /C215 69 ½ 69 /C30 13 /C2155/C27 4 ½1 /C30/C283 /C215 69/C27 16 /C215 13 ½ 13 /C30 4 /C2153/C27 1 ¡1 /C30 1 /C215 13/C28 3 /C215 4 ½ 1 /C30 0 /C215 4/C27 1 /C215 1 ½ and we recover the above solution. Call the solutions to ax /C27by /C301 (15) /x0and y0 : If the signs in front of ax or by are NEGATIVE , then solve the above equation and take the signs of the solutions from the following table: equation xy /ax /C27by /C301//x0//y0/ /ax /C28by /C301//x0///C28y0/ //C28ax /C27by /C301///C28x0//y0/ //C28ax /C28by /C301///C28x0///C28y0/ In fact, the solution to the equation ax /C28by /C301 (16) is equivalent to finding the CONTINUED FRACTION for a =b; with a and b RELATIVELY PRIME (Olds 1963). If there are n terms in the fraction, take the (n /C281)/th convergent pn/C281 =qn/C281 : But pnqn/C281 /C28pn/C281qn /C30(/C281)n ; (17) so one solution is x0 /C30(/C281)nqn/C281 ; y0 /C30(/C281)npn/C281 ; with a general solution x /C30x0 /C27kb (18) y /C30y0 /C27ka (19) with k an arbitrary INTEGER . The solution in terms of smallest POSITIVE INTEGERS is given by choosing an appropriate k.Now consider the general first-order equation OF THE FORM ax /C27by /C30c : (20) The GREATEST COMMON DIVISOR d /C13GCD( a;b) can be divided through yielding a?x /C27b?y /C30c ?; (21) where a?/C13a=d; b?/C13b=d; and c ?/C13c=d : If d¶c ; then c? is not an INTEGER and the equation cannot have a solution in INTEGERS . A necessary and sufficient condition for the general first-order equation to have solutions in INTEGERS is therefore that d½c: If this is the case, then solve a?x /C27b ?y /C301 (22) and multiply the solutions by c?; since a ?(c ?x) /C27b?(c?y) /C30c?: (23) D. Wilson has compiled a list of the smallest nth POWERS which are the sums of n distinct smaller nth POWERS . The first few are 3, 5, 6, 15, 12, 25, 40, ...(Sloane’s A030052): 31 /C3011 /C2721 52 /C3032 /C2742 63 /C3033 /C2743 /C2753 154 /C3044 /C2764 /C2784 /C2794 /C27144 125 /C3045 /C2755 /C2765 /C2775 /C2795 /C27115 256 /C3016 /C2726 /C2736 /C2756 /C2766 /C2776 /C2786 /C2796 /C27106 /C27126 /C27136 /C27156 /C27166 /C27176 /C27186 /C27236 407 /C3017 /C2737 /C2757 /C2797 /C27127 /C27147 /C27167 /C27177 /C27187/C27207/C27217/C27227/C27257/C27287/C27397 848/C3018/C2728/C2738/C2758/C2778/C2798/C27108/C27118 /C27128/C27138/C27148/C27158/C27168/C27178/C27188 /C27198/C27218/C27238/C27248/C27258/C27268/C27278 /C27298/C27328/C27338/C27358/C27378/C27388/C27398 /C27418/C27428/C27438/C27458/C27468/C27478/C27488 /C27498/C27518/C27528/C27538/C27578/C27588/C27598 /C27618/C27638/C27698/C27738 479/C3019/C2729/C2749/C2779/C27119/C27149/C27159/C27189 /C27269/C27279/C27309/C27319/C27329/C27339 /C27369/C27389/C27399/C27439 6310/C30110/C27210/C27410/C27510/C27610/C27810/C271210 /C271510/C271610/C271710/C272010/C272110/C272510 /C272610/C272710/C272810/C273010/C273610/C273710 /C273810/C274010/C275110/C276210: See also ABC CONJECTURE ,A RCHIMEDES’ CATTLE PROBLEM ,BACHET EQUATION ,BRAHMAGUPTA’S PRO- BLEM ,CANNONBALL PROBLEM ,CATALAN’S PROBLEM , DIOPHANTINE EQUATION–2ND POWERS ,DIOPHANTINE EQUATION–3RD POWERS ,DIOPHANTINE EQUATION–4TH POWERS ,DIOPHANTINE EQUATION–5TH POWERS ,DIO- PHANTINE EQUATION–6TH POWERS ,D IOPHANTINE EQUATION–7TH POWERS ,D IOPHANTINE EQUATION– 8TH POWERS ,DIOPHANTINE EQUATION–9TH POWERS , DIOPHANTINE EQUATION–10TH POWERS ,DIOPHANTINE EQUATION NTH POWERS ,D IOPHANTUS PROPERTY , EULER BRICK,E ULER QUARTIC CONJECTURE ,F ER- MAT’S LAST THEOREM ,F ERMAT ELLIPTIC CURVE THEOREM ,G ENUS THEOREM ,H URWITZ EQUATION , MARKOV NUMBER ,MONKEY AND COCONUT PROBLEM , MULTIGRADE EQUATION , P-ADIC NUMBER ,PELL EQUA- TION ,PYTHAGOREAN QUADRUPLE ,PYTHAGOREAN TRI- PLE,THUE EQUATION References Bashmakova, I. G. Diophantus and Diophantine Equations. Washington, DC: Math. Assoc. Amer., 1997. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Carmichael, R. D. The Theory of Numbers, and Diophantine Analysis. New York: Dover, 1959. Chen, S. "Equal Sums of Like Powers: On the Integer Solution of the Diophantine System." http://www.nease.- net/~chin/eslp/. Chen, S. "References." http://www.nease.net/~chin/eslp/re- ferenc.htm. Courant, R. and Robbins, H. "Continued Fractions. Dio- phantine Equations." §2.4 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas andMethods, 2nd ed. Oxford, England: Oxford University Press, pp. 49 /C1 /1, 1996. Davis, M. "Hilbert’s Tenth Problem is Unsolvable." Amer. Math. Monthly 80, 233/C1/69, 1973. Davis, M. and Hersh, R. "Hilbert’s 10th Problem." Sci. Amer. 229,8 4/C1/1, Nov. 1973. Davis, M. "Hilbert’s Tenth Problem is Unsolvable." Appen- dix 2 in Computability and Unsolvability. New York: Dover, 1999 /C1/35, 1982. Dickson, L. E. "Linear Diophantine Equations and Con- gruences." Ch. 2 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 41 /C1/ 9, 1952. dmoz. "Equal Sums of Like Powers." http://dmoz.org/Science/ Math/Number_Theory/Diophantine_Equations/Equal_- Sums_of_Like_Powers/. Do¨rrie, H. "The Fermat-Gauss Impossibility Theorem." §21 in100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 96 /C1/04, 1965. Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Guy, R. K. "Diophantine Equations." Ch. D in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/98, 1994. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hunter, J. A. H. and Madachy, J. S. "Diophantos and All That." Ch. 6 in Mathematical Diversions. New York: Dover, pp. 52 /C1/4, 1975. Ireland, K. and Rosen, M. "Diophantine Equations." Ch. 17 inA Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 269 /C1/96, 1990. Jones, J. P. and Matiyasevich, Yu. V. "Exponential Dio- phantine Representation of Recursively Enumerable Sets." Proceedings of the Herbrand Symposium, Mar- seilles, 1981. Amsterdam, Netherlands: North-Holland, pp. 159 /C1/77, 1982. Lang, S. Introduction to Diophantine Approximations, 2nd ed.New York: Springer-Verlag, 1995. Matiyasevich, Yu. V. "Solution of the Tenth Problem of Hilbert." Mat. Lapok 21,8 3/C1/7, 1970.Matiyasevich, Yu. V. Hilbert’s Tenth Problem. Cambridge, MA: MIT Press, 1993. http://www.informatik.uni-stutt-gart.de/ifi/ti/personen/Matiyasevich/H10Pbook/. Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." http://euler.free.fr/. Mordell, L. J. Diophantine Equations. New York: Academic Press, 1969. Nagell, T. "Diophantine Equations of First Degree." §10 in Introduction to Number Theory. New York: Wiley, pp. 29 /C1 / 2, 1951. Ogilvy, C. S. and Anderson, J. T. "Diophantine Equations." Ch. 6 in Excursions in Number Theory. New York: Dover, pp. 65 /C1/3, 1988. Olds, C. D. Ch. 2 in Continued Fractions. New York: Random House, 1963. Sloane, N. J. A. Sequences A030052 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . Weisstein, E. W. "Books about Diophantine Equations." http://www.treasure-troves.com/books/DiophantineEqua-tions.html. Diophantine Equation * /10th Powers The 10.1.2 equation A10/C30B10/C27C10(1) is a special case of F ERMAT’S LAST THEOREM with n/C3010, and so has no solution. The smallest 10.1.15 solution is 10010/C279410/C279110/C272/C2157710/C277610/C276310/C276210/C275210 /C274510/C273510/C273310/C271610/C271010/C27110/C3010810(2) (J.-C. Meyrignac 1999, PowerSum). The smallest 10.1.22 solution is 3310/C302/C2153010/C272/C2152610/C272310/C272110/C271910/C271810 /C272/C2151310/C272/C2151210/C275/C2151010/C272/C215910/C27710/C27610/C27310(3) (Ekl 1998). The smallest 10.1.23 solution is 5/C215110/C27210/C27310/C27610/C276/C215710/C274/C215910 /C271010/C272/C2151210/C271310/C271410/C301510(4) (Lander et al. 1967). The smallest 10.2.13 solution is 5110/C273210/C304910/C274310/C274110/C273710/C272810/C272610 /C272510/C271510/C271010/C2710910/C27510/C27310: (5) The smallest 10.2.15 solution is 3510/C27310/C303310/C273210/C272410/C272110/C272/C2152010 /C273/C2151310/C271210/C271110/C27910/C27710/C272/C215110(6) (Ekl 1998). The smallest 10.2.19 solution is 5/C215210/C27510/C27610/C271010/C276/C2151110 /C272/C2151210/C273/C2151510/C30910/C271710(7) (Lander et al. 1967). The smallest 10.3.13 solution is 4610/C273210/C272210 /C304310/C274310/C272710/C272610/C271710/C271610 /C271210/C27910/C27910/C27610/C27410/C27310/C27310: (8) The smallest 10.3.14 solution is 3010/C272810/C27410/C303110/C272310/C272/C2152010/C272/C2151710 /C271610/C271010/C273/C215910/C27510/C272/C215210(9) (Ekl 1998). The smallest 10.3.24 solution is 110/C27210/C27310/C2710 /C215410/C27710/C277/C215810 /C271010/C271210/C271610/C301110/C272/C2151510(10) (Lander et al. 1967). The 10.4.12 equation has solution 5110/C274910/C274310/C273910/C272910/C272810/C272/C2151710 /C271610/C271310/C27710/C27410/C305310/C2724410/C272210(11) (E. Bainville 1999, PowerSum). The smallest 10.4.15 solution is 4/C2152310/C302610/C275/C2151810/C273/C2151710/C271510/C271210/C27610 /C273/C215410(12) (Ekl 1998). The smallest 10.4.23 solution is 5/C215110/C272/C215210/C272/C215310/C27410/C274/C215610/C273:710/C27810 /C272/C2151010/C272/C2151410/C271510/C303/C2151110/C271610(13) (Lander et al. 1967). The smallest 10.5.16 solutions are 4/C215110/C27210/C272/C215410/C27610/C272/C2151210 /C275/C2151310/C271510/C302/C215310/C27810/C271410/C271610(14) 2010/C271110/C27810/C27310/C27110/C302/C2151810/C271710 /C271610/C271010/C272/C215710/C276/C215410/C272/C215210(15) (Lander et al. 1967, Ekl 1998). The smallest 10.6.6 solution is 9510/C277110/C273210/C272810/C272510/C271610 /C309210/C278510/C273410/C273410/C272310/C27510: (16) The smallest 10.6.16 solution is 1810/C271210/C271110/C271010/C27310/C27210 /C301710/C271610/C274/C2151310/C274/C215710/C274/C215610/C27510/C27410(17) (Ekl 1998). The smallest 10.6.27 solution is110/C274/C215310/C272/C215410/C272/C215510/C277/C215610 /C279/C215710/C271010/C271310/C302/C215210/C27810/C271110/C272/C2151210(18) (Lander et al. 1967). The smallest 10.7.7 solutions are 3810/C273310/C272610/C272610/C271510/C27810/C27110 /C303610/C273510/C273210/C272910/C272410/C272310/C272210(19) 6810/C276110/C275510/C273210/C273110/C272810/C27110 /C306710/C276410/C274910/C274410/C272310/C272010/C271710(20) (Lander et al. 1967, Ekl 1998). References Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. PowerSum. "Index of Equal Sums of Like Powers." http:// www.chez.com/powersum/. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . Diophantine Equation * /2nd Powers A general quadratic Diophantine equation in two variables xandyis given by ax2/C27cy2/C30k; (1) where a,c, and kare specified (positive or negative) integers and xandyare unknown integers satisfying the equation whose values are sought. The slightlymore general second-order equation ax 2/C27bxy/C27cy2/C30k (2) is one of the principal topics in Gauss’s Disquisitiones arithmeticae . According to Ito ˆ(1987), equation (2) can be solved completely using solutions to the P ELL EQUATION . In particular, all solutions of ax2/C27bxy/C27cy2/C301 (3) are among the CONVERGENTS of the CONTINUED FRACTIONS of the roots of ax2/C27bx/C27c:InMathema- tica 5.0, solution to the general bivariate quadratic Diophantine equation will be implemented as Re- duce [eqn&&Element [x|y,Integers ], {x,y}]. For quadratic Diophantine equations in more than two variables, there exist additional deep results due to C. L. Siegel. An equation OF THE FORM x2/C28Dy2/C301; (4) where Dis an INTEGER is a very special type of equation called a P ELL EQUATION . Pell equations, as well as the analogous equation with a minus sign on the right, can be solved by finding the CONTINUED FRACTION forffiffiffiffi Dp :The more complicated equation x2/C28Dy2/C30c (5) can also be solved for certain values of candD, but the procedure is more complicated (Chrystal 1961). However, if a single solution to (5) is known, other solutions can be found using the standard techniquefor the P ELL EQUATION . The following table summarizes possible representa- tion of primes pof given forms, where xand yare positive integers. No odd primes other than those indicated share these properties (Nagell 1951, p. 188). form congruence for p /x2/C27y2///C131 (mod 4) /x2/C272y2///C131;3 (mod 8) /x2/C273y2 ///C131 (mod 6) /x2/C277y2///C131;9;11 (mod 14) /2x2/C273y2///C135;11 (mod 24) As a part of the study of W ARING’S PROBLEM ,i ti s known that every positive integer is a sum of no more than 4 positive squares ( /g(2)/C304; L AGRANGE’S FOUR- SQUARE THEOREM ), that every "sufficiently large" integer is a sum of no more than 4 positive squares( /G(2)/C304);and that every integer is a sum of at most 3 signed squares ( eg(2)/C303):If zero is counted as a square, both POSITIVE and NEGATIVE numbers are included, and the order of the two squares is distin-guished, Jacobi showed that the number of ways anumber can be written as the sum of two squares (the r 2(n) function) is four times the excess of the number ofDIVISORS of the form 4 x/C271 over the number of DIVISORS OF THE FORM 4x/C281:/ In 1769 Euler (1862) noted the identity ab apr9bqs ðÞ2/C27abaps/C14bqr ðÞ2 /C30aap2/C27bbq20CB0C@ abr2/C27abs20CB0C@ ; (6) which gives a parametric solution to the equation Ax2/C27By2/C30C (7) for integers A;B;C;x;ywith Ccomposite (Dickson 1957, p. 407). Call a Diophantine equation consisting of finding a sum of mkthPOWERS which is equal to a sum of n kthPOWERS a" /k:m:nequation." The 2.1.2 quadratic Diophantine equationA2/C30B2/C27C2; (8) corresponds to finding a P YTHAGOREAN TRIPLE (A,B, C) has a well-known general solution (Dickson 1966, pp. 165 /C1/70). To solve the equation, note that every PRIME OF THE FORM 4x/C271 can be expressed as the sum of two RELATIVELY PRIME squares in exactly one way. A set of INTEGERS satisfying the 2.1.3 equation A2/C30B2/C27C2/C27D2(9) is called a P YTHAGOREAN QUADRUPLE . Parametric solutions to the 2.2.2 equation A2/C27B2/C30C2/C27D2(10) are known (Dickson 1966; Guy 1994, p. 140). To find in how many ways a general number mcan be expressed as a sum of two squares, factor it as follows m/C302a0p2a1 1/C1/C1/C1p2annqb1 1/C1/C1/C1qbrr; (11) where the ps are primes OF THE FORM 4x/C281 and the qs are primes OF THE FORM x/C271:If the as are integral, then define B/C132b1/C271 ðÞ 2b2/C271 ð Þ/C1/C1/C1 2br/C271 ðÞ /C281: (12) Then mis a sum of two unequal squares in N(m)/C300 for any aihalf -integral 1 2b1/C271 ðÞ b2/C271 ð Þ/C1/C1/C1 br/C271 ðÞ for all aiintegral ;Bodd 1 2b1/C271 ðÞ b2/C272 ð Þ/C1/C1/C1 br/C271 ðÞ /C2812 for all a iintegral ;Beven :8 >>>>>>>>>>>< >>>>>>>>>>>:(13) Solutions to an equation OF THE FORM A2/C27B20CB0C@ C2/C27D20CB0C@ /C30E2/C27F2(14) are given by the F IBONACCI IDENTITY a2/C27b20CB0C@ c2/C27d20CB0C@ /C30(ac9bd)2/C27(bc/C14ad)2 /C13e2/C27f2: (15) Another similar identity is the E ULER FOUR-SQUARE IDENTITY a2 1/C27a220CB0C@ b21/C27b220CB0C@ c21/C27c220CB0C@ d21/C27d220CB0C@ /C30e21/C27e22/C27e23/C27e24 (16) a21/C27a22/C27a23/C27a240CB0C@ b21/C27b22/C27b23/C27b240CB0C@ /C30a1b1/C28a2b2/C28a3b3/C28a4b4 ðÞ2 /C27a1b2/C27a2b1/C27a3b4/C28a4b3 ðÞ2 /C27a1b3/C28a2b4/C27a3b1/C27a4b2 ðÞ2 /C27a1b4/C27a2b3/C28a3b2/C27a4b1 ðÞ2: (17) Degen’s eight-square identity holds for eight squares, but no other number, as proved by Cayley. The two- square identity underlies much of TRIGONOMETRY , the four-square identity some of QUATERNIONS , and the eight-square identity, the C AYLEY ALGEBRA (a non- commutative nonassociative algebra; Bell 1945). Chen Shuwen found the 2.6.6 equation 872/C272332/C272642/C273962/C274962/C275402 /C30902/C272062/C273092/C273662/C275222/C275232: (18) RAMANUJAN’S SQUARE EQUATION 2n/C287/C30x2(19) has been proved to have only solutions n/C303, 4, 5, 7, and 15 (Schroeppel 1972). See also ALGEBRA ,CANNONBALL PROBLEM ,CONTIN- UED FRACTION ,EULER FOUR- SQUARE IDENTITY ,FER- MAT DIFFERENCE EQUATION ,G ENUS THEOREM , HILBERT SYMBOL ,LAGRANGE NUMBER (DIOPHANTINE EQUATION ), LEBESGUE IDENTITY ,P ELL EQUATION , PYTHAGOREAN QUADRUPLE ,P YTHAGOREAN TRIPLE , QUADRATIC RESIDUE ,S QUARE NUMBER ,S UM OF SQUARES FUNCTION ,W ARING’S PROBLEM References Beiler, A. H. "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Enter- tains. New York: Dover, pp. 248 /C1/68, 1966. Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 159, 1945. Chrystal, G. Textbook of Algebra, 2 vols. New York: Chelsea, 1961. Degan, C. F. Canon Pellianus. Copenhagen, Denmark, 1817. Dickson, L. E. "Number of Representations as a Sum of 5, 6, 7, or 8 Squares." Ch. 13 in Studies in the Theory of Numbers. Chicago, IL: University of Chicago Press, 1930. Dickson, L. E. "Pell Equation; ax2/C27bx/C27cMade a Square" and "Further Single Equations of the Second Degree."Chs. 12 /C1 /3i n History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 341 /C1/34, 1966. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Itoˆ, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed, Vol. 1. Cambridge, MA: MIT Press, p. 450, 1987. Lam, T. Y. The Algebraic Theory of Quadratic Forms. Reading, MA: W. A. Benjamin, 1973. Nagell, T. "Diophantine Equations of the Second Degree." Ch. 6 in Introduction to Number Theory. New York: Wiley, pp. 188 /C1/26, 1951. Rajwade, A. R. Squares. Cambridge, England: Cambridge University Press, 1993. Scharlau, W. Quadratic and Hermitian Forms. Berlin: Springer-Verlag, 1985. Schroeppel, R. Item 31 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 14, Feb. 1972. Shapiro, D. B. "Products of Sums and Squares." Expo. Math. 2, 235/C1/61, 1984. Smarandache, F. "Un metodo de resolucion de la ecuacion diofantica." Gaz. Math. 1, 151/C1/57, 1988.Smarandache, F. "Method to Solve the Diophantine Equa- tionax2/C28by2/C27c/C300:/"I nCollected Papers, Vol. 1. Buchar- est, Romania: Tempus, 1996. Taussky, O. "Sums of Squares." Amer. Math. Monthly 77, 805/C1/30, 1970. Whitford, E. E. Pell Equation. New York: Columbia Uni- versity Press, 1912. #1999/C1/001 Wolfram Research, Inc. Diophantine Equation * /3rd Powers As a part of the study of W ARING’S PROBLEM ,i ti s known that every positive integer is a sum of no more than 9 positive cubes ( /g(3)/C309);that every "suffi- ciently large" integer is a sum of no more than 7 positive cubes ( /G(3)57; although it is not known if 7 can be reduced), and that every integer is a sum of atmost 5 signed cubes ( eg(3)55; although it is not known if 5 can be reduced to 4). It is known that every ncan be written is the form n/C30A 2/C27B2/C28C3: (1) The 3.1.2 equation A3/C30B3/C27C3(2) is a case of F ERMAT’S LAST THEOREM with n/C303. In fact, this particular case was known not to have any solutions long before the general validity of F ERMAT’S LAST THEOREM was established. Thue showed that a Diophantine equation OF THE FORM AX3/C28BY3/C301 (3) forA,B, and lintegers, has only finite many solutions (Hardy 1999, pp. 78 /C1/9). Miller and Woollett (1955) and Gardiner et al. (1964) investigated integer solutions of A3/C27B3/C27C3/C30D (4) i.e., numbers representable as the sum of three (positive or negative) CUBIC NUMBERS . The general rational solution to the 3.1.3 equation A3/C30B3/C27C3/C27D3(5) was found by Euler and Vieta (Dickson 1966, pp. 550 /C1/54; Hardy 1999, pp. 20 /C1/1). Hardy and Wright (1979, pp. 199 /C1/01) give a solution which can be based on the identities a3a3/C27b30CB0C@3 /C30b3a3/C27b30CB0C@3/C27a3a3/C282b30CB0C@3/C27b32a3/C27b30CB0C@3(6) a3a3/C272b30CB0C@3 /C30a3a3/C28b30CB0C@3/C27b3a3/C28b30CB0C@3/C272a3/C27b30CB0C@3: (7) This is equivalent to the general 3.2.2 solution found by Ramanujan (Dickson 1966, pp. 500 and 554;Berndt 1994, pp. 54 and 107; Hardy 1999, p. 11, 68, and 237). The smallest integer solutions are 33/C2743/C2753/C3063(8) 13/C2763/C2783/C3093(9) 73/C27143/C27173/C30203(10) 113/C27153/C27273/C30293(11) 283/C27533/C27753/C30843(12) 263/C27553/C27783/C30873(13) 333/C27703/C27923/C301053(14) (Fredkin 1972; Madachy 1979, pp. 124 and 141). Other general solutions have been found by Binet(1841) and Schwering (1902), although Ramanujan’sformulation is the simplest. No general solution giving all POSITIVE integral solutions is known (Dickson 1966, pp. 550 /C1/61). Y. Kohmoto has found a 3.1.39solution, 21000003/C3020460003/C278820003/C272160003 /C3019796003/C2711454003/C27850003 /C3020811003/C276281103/C2718903 /C3020431503/C279012003/C27304503 /C3020022803/C2710724803/C27303603 /C3019604803/C2711995203/C27152003 /C3019488003/C2712297603/C27302403 /C3020781603/C276588123/C27131883 /C3020091123/C2710480403/C27138883: (15) 3.1.4 equations include 113/C27123/C27133/C27143/C30203(16) 53/C2773/C2793/C27103/C30133: (17) 3.1.5 equations include 13/C2733/C2743/C2753/C2783/C3093(18) 33/C2743/C2753/C2783/C27103/C30123; (19) and a 3.1.6 equation is given by 13/C2753/C2763/C2773/C2783/C27103/C30133: (20) The 3.2.2 equation A3/C27B3/C30C3/C27D3(21) has a known parametric solution (Dickson 1966, pp. 550 /C1/54; Guy 1994, p. 140), and 10 solutions with sum B105, 1729/C3013/C27123/C27/C3093/C27103(22) 4104/C3023/C27163/C3093/C27153(23) 13832 /C3023/C27243/C30183/C27203(24) 20683 /C30103/C27273/C30193/C27243(25)32832 /C3043/C27323/C30183/C27303(26) 39312 /C3023/C27343/C30153/C27333(27) 40033 /C3093/C27343/C30163/C27333(28) 46683 /C3033/C27363/C30273/C27303(29) 64232 /C30173/C27393/C30263/C27363(30) 65728 /C30123/C27403/C30313/C27333(31) (Sloane’s A001235; Moreau 1898). The first number(Madachy 1979, pp. 124 and 141) in this sequence,the so-called H ARDY- RAMANUJAN NUMBER , is asso- ciated with a story told about Ramanujan byG. H. Hardy, but was known as early as 1657 (Berndtand Bhargava 1993). The smallest number represen- table in nways as a sum of cubes is called the nth TAXICAB NUMBER . Ramanujan gave a general solution to the 3.2.2 equation as a/C27l2g0CB0C@ 3/C27lb/C27g ðÞ3/C30la/C27g ðÞ3/C27b/C27l2g0CB0C@ 3(32) where a2/C27ab/C27b2/C303lg2(33) (Berndt 1994, p. 107). Another form due to Ramanu-jan is A 2/C277AB/C289B20CB0C@3/C272A2/C284AB/C2712B20CB0C@3 /C302A2/C2710B20CB0C@3/C27A2/C289AB/C28B20CB0C@3: (34) Hardy and Wright (1979, Theorem 412) prove thatthere are numbers that are expressible as the sum oftwo cubes in nways for any n(Guy 1994, pp. 140 /C1 / 41). The proof is constructive, providing a method forcomputing such numbers: given RATIONALS NUMBERS rands, compute t/C30rr3/C272s3ðÞ r3/C28s3(35) u/C30s2r3/C27s3ðÞ r3/C28s3(36) v/C30tt3/C282u3ðÞ t3/C27u3(37) w/C30u2t3/C28u3ðÞ t3/C27u3: (38) Then r3/C27s3/C30t3/C28u3/C30v3/C27w3(39) The DENOMINATORS can now be cleared to produce an integer solution. If r=sis picked to be large enough, thevandwwill be POSITIVE .I fr=sis still larger, the v=wwill be large enough for vandwto be used as the inputs to produce a third pair, etc. However, the resulting integers may be quite large, even for n/C302. E.g., starting with 33/C2713/C3028;the algorithm finds 28/C3028340511 21446828 !3 /C276328470521446828 ! 3 ; (40) giving 28 /C215214468283/C303/C21521446828ðÞ3/C27214468283(41) /C30283405113/C27632847053: (42) The numbers representable in three ways as a sum of two cubes (a 3.23equation) are 87539319 /C301673/C274363/C302283/C274233 /C302553/C274143ð43Þ 119824488 /C30113þ4933¼903þ4923 ¼3463þ4283ð44Þ 143604279 /C301113/C275223/C303593/C274603 /C304083/C274233ð45Þ 175959000 /C30703/C275603/C301983/C275523 /C303153/C275253ð46Þ 327763000 /C303003/C276703/C303393/C276613 /C305103/C275803ð47Þ (Guy 1994, Sloane’s A003825). Wilson (1997) found 32 numbers representable in four ways as the sum of twocubes (a 3.2 4equation). The first is 6963472309248 /C3024213/C27190833/C3054362/C27189483 /C30102003/C27180723/C30133223/C27166303: (48) The smallest known numbers so representable are 6963472309248, 12625136269928, 21131226514944, 26059452841000, ... (Sloane’s A003826). Wilson alsofound six five-way sums, 48988659276962496 /C3038787 3/C273657573 /C301078393/C273627533 /C302052923/C273429523 /C302214243/C273365883 /C302315183/C273319543(49) 490593422681271000 /C30483693/C277886313 /C302337753/C277817853 /C302851203/C277760703 /C305431453/C276912953/C305792403/C276666303(50) 6355491080314102272 /C301031133/C2718522153 /C305804883/C2718331203 /C307887243/C2718033723 /C3011507923/C2716905443 /C3014620503/C2714782383(51) 27365551142421413376 /C301677513/C2730133053 /C302653923/C2730127923 /C309443763/C2729822403 /C3012831483/C2729338443 /C3018721843/C2727502883(52) 1199962860219870469632 /C305915433/C27106258653 /C309358563/C27106240563 33301683/C27105163203 /C3066019123/C2796983843 /C3083875503/C2784804183(53) 111549833098123426841016 /C3010740733/C27481379993 /C3087878703/C27480403563 /C30139509723/C27477443823 /C30244501923/C27459364623 /C30337844783/C27417912043; (54) and a single six-way sum 8230545258248091551205888 /C30112393173/C272018914353 /C30177812643/C272018570643 /C30632731923/C271998100803 /C30859709163/C271965675483 /C301254363283/C271842692963 /C301593634503/C271611279423: (55) A solution to the 3.4.4 equation is 23/C2733/C27103/C27113/C3013/C2753/C2783/C27123(56) (Madachy 1979, pp. 118 and 133). 3.6.6 equations also exist: 13/C2723/C2743/C2783/C2793/C27123 /C3033/C2753/C2763/C2773/C27103/C27113(57) 873 /C272333 /C272643 /C273963 /C274963 /C275403 /C30903 /C272063 /C273093 /C273663 /C275223 /C275233 : (58) (Madachy 1979, p. 142; Chen Shuwen). Euler gave the general solution to A3 /C27B3 /C30C2 (59) as A /C303n2 /C276n2 /C28n (60) B /C30/C283n3 /C276n2 /C27n (61) C/C306n23n2/C2710CB0C@ : (62) See also CANNONBALL PROBLEM ,C UBIC NUMBER , HARDY- RAMANUJAN NUMBER ,M ULTIGRADE EQUA- TION ,S UPER- D NUMBER ,T AXICAB NUMBER ,T RI- MORPHIC NUMBER ,W ARING’S PROBLEM References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Berndt, B. C. and Bhargava, S. "Ramanujan--For Low- brows." Amer. Math. Monthly 100, 645/C1/56, 1993. Binet, J. P. M. "Note sur une question relative a `la the ´orie des nombres." C. R. Acad. Sci. (Paris) 12, 248/C1/50, 1841. Chen, S. "Equal Sums of Like Powers: On the Integer Solution of the Diophantine System." http://www.nease.- net/~chin/eslp/ Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966. Fredkin, E. Item 58 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 23, Feb. 1972. Gardiner, V. L.; Lazarus, R. B.; and Stein, P. R. "Solutions of the Diophantine Equation x3/C27y3/C30z3/C28d:/"Math. Com- put. 18, 408/C1/13, 1964. Guy, R. K. "Sums of Like Powers. Euler’s Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/44, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Koyama, K.; Tsuruoka, Y.; and Sekigawa, S. "On Searching for Solutions of the Diophantine Equation x3/C27y3/C27z3/C30n:/" Math. Comput. 66, 841/C1/51, 1997. Kraus, A. "Sur l’e ´quation a3/C27b3/C30cp:/"Experim. Math. 7,1/C1/ 3, 1998. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, 1979. Miller, J. C. P. and Woollett, M. F. C. "Solutions of the Diophantine Equation x3/C27y3/C27z3/C30k:/"J. London Math. Soc. 30, 101/C1/10, 1955. Moreau, C. "Plus petit nombre e ´gal a`la somme de deux cubes de deux fac ¸ons." L’Intermediaire Math. 5, 66, 1898. Nagell, T. "The Diophantine Equation j3/C27h3/C27z3and Analogous Equations" and "Diophantine Equations ofthe Third Degree with an Infinity of Solutions." §65 and 66 in Introduction to Number Theory. New York: Wiley, pp. 241 /C1 /48, 1951.Rivera, C. "Problems & Puzzles: Puzzle p3/C30a3/C27b3/C27c3; pa;b;cPrime.-048." http://www.primepuzzles.net/puzzles/ puzz_048.htm. Schwering, K. "Vereinfachte Lo ¨sungen des Eulerschen Aufgabe: x3/C27y3/C27z3/C27v3/C300::/"Arch. Math. Phys. 2, 280/C1/ 84, 1902. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 157, 1993. Sloane, N. J. A. Sequences A001235 and A003825 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . Wilson, D. Personal communication, Apr. 17, 1997. #1999/C1/001 Wolfram Research, Inc. Diophantine Equation * /4th Powers As a consequence of Matiyasevich’s refutation of Hilbert’s 10th problem, it can be proved that there does not exists a general algorithm for solving a general quartic Diophantine equation. However, thealgorithm for constructing such an unsolvable quarticDiophantine equation can require arbitrarily many variables (Matiyasevich 1993). As a part of the study of W ARING’S PROBLEM ,i ti s known that every positive integer is a sum of no more than 19 positive biquadrates /g(4)/C3019 ðÞ ;that every "sufficiently large" integer is a sum of no more than16 positive biquadrates /G(4)/C3016 ðÞ ;and that every integer is a sum of at most 10 signed biquadrates ( eg(4)510; although it is not known if 10 can be reduced to 9). The first few numbers nwhich are a sum of four fourth POWERS (/m/C281 equations) are 353, 651, 2487, 2501, 2829, ... (Sloane’s A003294). The 4.1.2 equation x4/C30y4/C27z4(1) is a case of F ERMAT’S LAST THEOREM with n/C304 and therefore has no solutions. In fact, the equations x49y4/C28z2(2) also have no solutions in INTEGERS (Nagell 1951, pp. 227 and 229). The equation x4/C28y4/C302z2(3) has no solutions in integers (Nagell 1951, p. 230). The only number OF THE FORM 4x4/C27y4(4) which is PRIME is 5 (Baudran 1885, Le Lionnais 1983). Let the notation p:m:nstand for the equation consisting of a sum of mpth powers being equal to a sum of npth powers. In 1772, Euler proposed that the 4.1.3 equation A4/C27B4/C27C4/C30D4(5) had no solutions in INTEGERS (Lander et al. 1967). This assertion is known as the E ULER QUARTIC CONJECTURE . Ward (1948) showed there were no solutions for D510;000;which was subsequently improved to D5220;000 by Lander et al. (1967). However, the E ULER QUARTIC CONJECTURE was dis- proved in 1987 by N. Elkies, who, using a geometric construction, found 2;682;4404/C2715;365;6394/C2718;796;7604 /C3020;615;6734(6) and showed that infinitely many solutions existed (Guy 1994, p. 140). In 1988, Roger Frye found 95;8004/C27217;5194/C27414;5604/C30422;4814(7) and proved that there are no solutions in smaller INTEGERS (Guy 1994, p. 140). Another solution was found by Allan MacLeod in 1997, 638;523;2494/C30630;662;6244 /C27275;156;2404/C27219;076;4654(8) (Ekl 1998). It is not known if there is a parametric solution. In contrast, there are many solutions to the equation A4/C27B4/C27C4/C302D4(9) (see below). The 4.1.4 equation A4/C27B4/C27C4/C27D4/C30E4(10) has solutions 304/C271204/C272724/C273154/C303534(11) 2404/C273404/C274304/C275994/C306514(12) 4354/C277104/C2713845/C2724204/C3024874(13) 11304/C2711904/C2714324/C2723654/C3025014(14) 8504/C2710104/C2715464/C2727454/C3028294(15) 22704/C2723454/C2724604/C2731524/C3037234(16) 3504/C2716524/C2732304/C2733954/C3039734(17) 2054/C2710604/C2726504/C2740944/C3042674(18) 13944/C2717504/C2735454/C2736704/C3043334(19) 6994/C277004/C2728404/C2742504/C3044494(20) 3804/C2716604/C2718804/C2749074/C3049494(21) 10004/C2711204/C2732334/C2750804/C3052814(22) 4104/C2714124/C2739104/C2750554/C3054634(23) 9554/C2717704/C2726344/C2754004/C3054914(24) 304/C2716804/C2730434/C2754004/C3055434(25)13544/C2718104/C2743554/C2751504/C3057294(26) 5424/C2727704/C2742804/C2756954/C3061674(27) 504/C278854/C2750004/C2759844/C3066094(28) 14904/C2734684/C2747904/C2761854/C3068014(29) 13904/C2728504/C2753654/C2763684/C3071014(30) 1604/C2713454/C2727904/C2771664/C3072094(31) 8004/C2730524/C2754404/C2766354/C3073394(32) 22304/C2731964/C2756204/C2769954/C3077034(33) (Norrie 1911, Patterson 1942, Leech 1958, Brudno 1964, Lander et al. 1967), but it is not known if there is a parametric solution (Guy 1994, p. 139). There are an infinite number of solutions to the 4.1.5 equation A4/C30B4/C27C4/C27D4/C27E4/C27F4: (34) Some of the smallest are 24/C2724/C2734/C2744/C2744/C3054(35) 44/C2764/C2784/C2794/C27144/C30154(36) 44/C27214/C27224/C27264/C27284/C30354(37) 14/C2724/C27124/C27244/C27444/C30454(38) 14/C2784/C27124/C27324/C27644/C30654(39) 24/C27394/C27444/C27464/C27524/C30654(40) 224/C27524/C27574/C27744/C27764/C30954(41) 224/C27284/C27634/C27724/C27944/C301054(42) (Berndt 1994). Berndt and Bhargava (1993) andBerndt (1994, pp. 94 /C1 /6) give Ramanujan’s solutions for arbitrary s,t,m, and n, 8s2/C2740st/C2824t20CB0C@4/C276s2/C2844st/C2818t20CB0C@4 /C2714s2/C284st/C2842t20CB0C@4/C279s2/C2727t20CB0C@4/C274s2/C2712t20CB0C@4 /C3015s2/C2745t20CB0C@4; (43) and 4m2/C2812n20CB0C@4/C273m2/C279n20CB0C@4/C272m2/C2812mn/C286n20CB0C@4 /C274m2/C2712n20CB0C@4/C272m2/C2712mn/C286n20CB0C@4 /C305m2/C2715n20CB0C@4: ð44Þ These are also given by Dickson (1966, p. 649), andtwo general FORMULAS are given by Beiler (1966, p. 290). Other solutions are given by Fauquembergue (1898), Haldeman (1904), and Martin (1910). Parametric solutions to the 4.2.2 equation A4/C27B4/C30C4/C27D4(45) are known (Euler 1802; Ge ´rardin 1917; Guy 1994, pp. 140 /C1/41), but no "general" solution is known (Hardy 1999, p. 21). A few specific solutions are 594/C271584/C301334/C271344/C30635;318;657 (46) 74/C272394/C301574/C272274/C303;262;811;042 (47) 1934/C272924/C302564/C272574/C308;657;437;697 (48) 2984/C274974/C302714/C275024/C3068;899;596;497 (49) 5144/C273594/C301034/C275424/C3086;409;838;577 (50) 2224/C276314/C305034/C275584/C30160;961;094;577 (51) 214/C277174/C304714/C276814/C30264;287;694;402 (52) 764/C2712034/C306534/C2711764 /C302;094;447;251;857 ð53Þ 9974/C2713424/C308784/C2713814 /C304;231;525;221;377 ð54Þ (Sloane’s A003824 and A018786; Richmond 1920; Dickson, pp. 60 /C1/2; Dickson 1966, pp. 644 /C1/47; Leech 1957; Berndt 1994, p. 107; Ekl 1998 [with typo]), thesmallest of which is due to Euler (Hardy 1999, p. 21).Lander et al. (1967) give a list of 25 primitive 4.2.2 solutions. General (but incomplete) solutions aregiven by x/C30a/C27b (55) y/C30c/C28d (56) u/C30a/C28b (57) v/C30c/C27d; (58) where a/C30nm 2/C27n20CB0C@ /C28m4/C2718m2n2/C28n40CB0C@ (59) b/C302mm6/C2710m4n4/C27m2n4/C274n60CB0C@ (60) c/C302n4m6/C27m4n2/C2710m2n4/C27n60CB0C@ (61) d/C30mm2/C27n20CB0C@ /C28m4/C2718m2n2/C28n40CB0C@ (62) (Hardy and Wright 1979). Parametric solutions to the 4.2.3 equation A4/C27B4/C30C4/C27D4/C27E4(63) are known (Ge ´rardin 1910, Ferrari 1913). The smal- lest solution is 34/C2754/C2784/C3074/C2774(64) (Lander et al. 1967).Ramanujan gave the 4.2.4 equation 34/C2794/C3054/C2754/C2764/C2784: (65) Ramanujan gave the 4.3.3 equations 24/C2744/C2774/C3034/C2764/C2764(66) 34/C2774/C2784/C3014/C2724/C2794(67) 64/C2794/C27124/C3024/C2724/C27134(68) (Berndt 1994, p. 101). Similar examples can be found in Martin (1896). Parametric solutions were given byGe´rardin (1911). Ramanujan also gave the general expression 3 4/C272x4/C2810CB0C@4/C274x5/C27x0CB0C@4 /C304x4/C2710CB0C@4/C276x4/C2830CB0C@4/C274x5/C285x0CB0C@4(69) (Berndt 1994, p. 106). Dickson (1966, pp. 653 /C1/55) cites several FORMULAS giving solutions to the 4.3.3 equation, and Haldeman (1904) gives a general FORMULA . Ramanujan gave the 4.3.4 identities 24/C2724/C2774/C3044/C2744/C2754/C2764(70) 34/C2794/C27144/C3074/C2784/C27104/C27134(71) 74/C27104/C27134/C3054/C2754/C2764/C27144(72) (Berndt 1994, p. 101). Haldeman (1904) gives general FORMULAS for 4/C1/and 4 /C1/equations. Ramanujan gave 2ab/C27ac/C27bc ðÞ2/C30a4/C27b4/C27c4(73) 2ab/C27ac/C27bc ðÞ4/C30a4b/C28c ðÞ4/C27b4c/C28a ðÞ4/C27c4a/C28b ðÞ4(74) 2ab/C27ac/C27bc/C27 ðÞ6 /C30a2b/C27b2c/C27c2a0CB0C@4/C27ab2/C27bc2/C27ca20CB0C@4/C273(abc)4 (75) 2ab/C27ac/C27bc ðÞ8/C30a3/C272abc0CB0C@4b/C28c ðÞ4 /C27b3/C272abc0CB0C@4c/C28a ðÞ4/C27c3/C272abc0CB0C@4a/C28b ðÞ4; (76) where a/C27b/C27c/C300 (77) (Berndt 1994, pp. 96 /C1/7). F ORMULA (74) is equivalent to F ERRARI’S IDENTITY a2/C272ac/C282bc/C28b20CB0C@4/C27b2/C282ab/C282ac/C28c20CB0C@4 /C27c2/C272ab/C272bc/C28a20CB0C@4 /C302a2/C27b2/C27c2/C28ab/C27ac/C27bc0CB0C@4: (78) BHARGAVA’S THEOREM is a general identity which gives the above equations as a special case, and may have been the route by which Ramanujan proceeded. Another identity due to Ramanujan is a /C27b /C27c ðÞ4/C27 b /C27c /C27d ðÞ4/C27 a /C28d ðÞ4 /C30 c /C27d /C27a ðÞ4/C27 d /C27a /C27b ðÞ4/C27 b /C28c ðÞ4; (79) where a =b /C30c =d; and 4 may also be replaced by 2 (Ramanujan 1957, Hirschhorn 1998). V. Kyrtatas noticed that a /C303, b /C307, c /C3020, d /C3025, e /C3038, and f /C3039 satisfy a4 /C27 b4 /C27 c4 d4 /C27 e4 /C27 f4 /C30a/C27b/C27c d/C27e/C27f(80) and asks if there are any other distinct integer solutions. See also BHARGAVA’S THEOREM ,BIQUADRATIC NUM- BER,FORD’S THEOREM ,MULTIGRADE EQUATION ,WAR- ING’S PROBLEM References Barbette, E. Les sommes de p -ie´mes puissances distinctes e´gales a `une p-ie ´me puissance. Doctoral Dissertation, Liege, Belgium. Paris: Gauthier-Villars, 1910. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Berndt, B. C. and Bhargava, S. "Ramanujan--For Low- brows." Am. Math. Monthly 100, 645/C1/56, 1993. Bhargava, S. "On a Family of Ramanujan’s Formulas for Sums of Fourth Powers." Ganita 43,6 3/C1/7, 1992. Brudno, S. "A Further Example of A4/C27B4/C27C4/C27D4/C30E4:/" Proc. Cambridge Phil. Soc. 60, 1027 /C1/028, 1964. Chen, S. "Equal Sums of Like Powers: On the Integer Solution of the Diophantine System." http://www.nease.- net/~chin/eslp/ Dickson, L. E. Introduction to the Theory of Numbers. New York: Dover. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966. Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Euler, L. Nova Acta Acad. Petrop. as annos 1795 /C1/79613, 45, 1802. Fauquembergue, E. L’interme ´diaire des Math. 5, 33, 1898. Ferrari, F. L’interme ´diaire des Math. 20, 105/C1/06, 1913. Guy, R. K. "Sums of Like Powers. Euler’s Conjecture" and "Some Quartic Equations." §D1 and D23 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/44 and 192 /C1/93, 1994. Haldeman, C. B. "On Biquadrate Numbers." Math. Mag. 2, 285/C1/96, 1904. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. §13.7 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clar- endon Press, 1979. Hirschhorn, M. D. "Two or Three Identities of Ramanujan." Amer. Math. Monthly 105,5 2/C1/5, 1998. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Leech, J. "Some Solutions of Diophantine Equations." Proc. Cambridge Phil. Soc. 53, 778/C1/80, 1957. Leech, J. "On A4/C27B4/C27C4/C27D4/C30E4:/"Proc. Cambridge Phil. Soc. 54, 554/C1/55, 1958. Martin, A. "About Biquadrate Numbers whose Sum is a Biquadrate." Math. Mag. 2, 173/C1/84, 1896. Martin, A. "About Biquadrate Numbers whose Sum is a Biquadrate--II." Math. Mag. 2, 325/C1/52, 1904. Nagell, T. "Some Diophantine Equations of the Fourth Degree with Three Unknowns" and "The DiophantineEquation 2 x 4/C28y4/C30z2:/"§62 and 63 in Introduction to Number Theory. New York: Wiley, pp. 227 /C1/35, 1951. Norrie, R. University of St. Andrews 500th Anniversary Memorial Volume. Edinburgh, Scotland: pp. 87 /C1/9, 1911. Patterson, J. O. "A Note on the Diophantine Problem of Finding Four Biquadrates whose Sum is a Biquadrate."Bull. Amer. Math. Soc. 48, 736/C1 /37, 1942. Ramanujan, S. Notebooks. New York: Springer-Verlag, pp. 385 /C1/86, 1987. Richmond, H. W. "On Integers Which Satisfy the Equation t39x39y39z3/C300:/"Trans. Cambridge Phil. Soc. 22, 389/C1/ 03, 1920. Rivera, C. "Problems & Puzzles: Puzzle p4/C30a4/C27b4/C27c4/C27d4; a;b;c;d>0:/-047." http://www.primepuzzles.net/puzzles/ puzz_047.htm. Sloane, N. J. A. Sequences A003294/M5446, A003824, and A018786 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Ward, M. "Euler’s Problem on Sums of Three Fourth Powers." Duke Math. J. 15, 827/C1/37, 1948. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . #1999/C1/001 Wolfram Research, Inc. Diophantine Equation * /5th Powers The 5.1.2 fifth-order Diophantine equation A5/C30B5/C27C5(1) is a special case of F ERMAT’S LAST THEOREM with n/C305, and so has no solution. improving on the results on Lander et al. (1967), who checked up to 2:8/C291014:(In fact, no solutions are known for POWERS of 6 or 7 either.) No solutions to the 5.1.3 equation A5/C27B5/C27C5/C30D5(2) are known (Lander et al. 1967). For 4 fifth POWERS , we have the 5.1.4 equation 275/C27845/C271105/C271335/C301445(3) (Lander and Parkin 1967, Lander et al. 1967, Ekl 1998), but it is not known if there is a parametric solution (Guy 1994, p. 140). Sastry (1934) found a 2-parameter solution for 5.1.5 equations (75v 5/C28u5)5/C27(u5/C2725v5)5/C27(u5/C2825v5)5 /C27(10u3v2)5/C27(50uv4)5/C30(u5/C2775v5)5(4) (quoted in Lander and Parkin 1967), and Lander andParkin (1967) found the smallest numerical solutions.Lander et al. (1967) give a list of the smallest solutions, the first few being 195/C27435/C27465/C27475/C27675/C30725(5) 215/C27235/C27375/C27795/C27845/C30945(6) 75/C27435/C27575/C27805/C271005/C301075(7) 785/C271205/C271915/C272595/C273475/C303655(8) 795/C272025/C272585/C272615/C273955/C304155(9) 45/C27265/C271395/C272965/C274125/C304275(10) 315/C271055/C271395/C273145/C274165/C304355(11) 545/C27915/C271015/C274045/C274305/C304805(12) 195þ2015þ3475þ388 þ4485¼5035ð13Þ 1595/C271725/C272005/C273565/C275135/C305305(14) 2185/C272765/C273855/C274095/C274955/C305535(15) 25/C272985/C273515/C274745/C275005/C305755(16) (Lander and Parkin 1967, Lander et al. 1967). The 5.1.6 equation has solutions 45/C2755/C2765/C2775/C2795/C27115/C30125(17) 55/C27105/C27115/C27165/C27195/C27295/C30305(18) 155/C27165/C27175/C27225/C27245/C27285/C30325(19) 135/C27185/C27235/C27315/C27365/C27665/C30675(20) 75/C27205/C27295/C27315/C27345/C27665/C30675(21) 225/C27355/C27485/C27585/C27615/C27645/C30785(22) 45/C27135/C27195/C27205/C27675/C27965/C30995(23) 65/C27175/C27605/C27645/C27735/C27895/C30995(24) (Martin 1887, 1888, Lander and Parkin 1967, Lander et al. 1967). The smallest 5.1.7 solution is 15/C2775/C2785/C27145/C27155/C27185/C27205/C30235(25) (Lander et al. 1967). No solutions to the 5.2.2 equation A5/C27B5/C30C5/C27D5(26) are known, despite the fact that sums up to 1 :026/C29 1026have been checked (Guy 1994, p. 140). The smallest 5.2.3 solution is 141325/C272205/C30140685/C2762375/C2750275(27) (B. Scher and E. Seidl 1996, Ekl 1998). Sastry’s(1934) 5.1.5 solution gives some 5.2.4 solutions. The smallest primitive 5.2.4 solutions are 4 5/C27105/C27205/C27285/C3035/C27295(28) 55/C27135/C27255/C27375/C30125/C27385(29)265/C27295/C27355/C27505/C30285/C27525(30) 55/C27255/C27625/C27635/C30615/C27645(31) 65/C27505/C27535/C27825/C30165/C27855(32) 565/C27635/C27725/C27865/C30315/C27965(33) 445/C27585/C27675/C27945/C30145/C27995(34) 115/C27135/C27375/C27995/C30635/C27975(35) 485/C27575/C27765/C271005/C30255/C271065(36) 585/C27765/C27795/C271025/C30545/C271115(37) (Rao 1934, Moessner 1948, Lander et al. 1967). The smallest primitive 5.2.5 solutions are 45/C2755/C2775/C27165/C27215/C3015/C27225(38) 95/C27115/C27145/C27185/C27305/C30235/C27295(39) 105/C27145/C27265/C27315/C27335/C30165/C27385(40) 45/C27225/C27295/C27355/C27365/C30245/C27425(41) 85/C27155/C27175/C27195/C27455/C30305/C27445(42) 55/C2765/C27265/C27275/C27445/C30365/C27425(43) (Rao 1934, Lander et al. 1967). Parametric solutions are known for the 5.3.3 (Sastry and Lander 1934; Moessner 1951; Swinnerton-Dyer1952; Lander 1968; Bremmer 1981; Guy 1994, pp. 140 and 142; Choudhry 1999). Swinnerton-Dyer (1952) gave two parametric solutions to the 5.3.3equation but, forty years later, W. Gosper discovered that the second scheme has an unfixable bug. Choudhry (1999) gave a parametric solution to themore general equation ax 5/C27by5/C27cx5/C30au5/C27bv5/C27cw5(44) with a/C27b/C27c/C300:The smallest primitive solutions to the 5.3.3 equation with unit coefficients are 245/C27285/C27675/C3035/C27545/C27625(45) 185/C27445/C27665/C30135/C27515/C27645(46) 215/C27435/C27745/C3085/C27625/C27685(47) 565/C27675/C27835/C30535/C27725/C27815(48) 495/C27755/C271075/C30395/C27925/C271005(49) (Moessner 1939, Moessner 1948, Lander et al. 1967, Ekl 1998). A two-parameter solution to the 5.3.4 equation was given by Xeroudakes and Moessner (1958). Gloden(1949) also gave a parametric solution. The smallest solution is 15 /C2785 /C27145 /C27275 /C3035 /C27225 /C27255 (50) (Rao 1934, Lander et al. 1967). Several parametric solutions to the 5.4.4 equation were found by Xeroudakes and Moessner (1958). The smallest 5.4.4 solution is 55 /C2765 /C2765 /C2785 /C3045 /C2775 /C2775 /C2775 (51) (Rao 1934, Lander et al. 1967). The first 5.4.4.4 equation is 35 /C27485 /C27525 /C27615 /C30135 /C27365 /C27515 /C27645 /C30185 /C27365 /C27445 /C27665 (52) (Lander et al. 1967). Moessner and Gloden (1944) give the 5.5.6 solution 225 /C27175 /C27165 /C2765 /C2755 /C30215 /C27205 /C27125 /C27105 /C2725 /C2715 : (53) Chen Shuwen found the 5.6.6 solution 875/C272335/C272645/C273965/C274965/C275405 /C30905/C272065/C273095/C273665/C275225/C275235: (54) See also MULTIGRADE EQUATION References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, p. 95, 1994. Bremner, A. "A Geometric Approach to Equal Sums of Fifth Powers." J. Number Th. 13, 337/C1/54, 1981. Chen, S. "Equal Sums of Like Powers: On the Integer Solution of the Diophantine System." http://www.nease.- net/~chin/eslp/ Choudhry, A. "The Diophantine Equation ax5/C27by5/C27cz5/C27/C30au5/C27bv5/C27cw5:/"Rocky Mtn. J. Math. 29, 459/C1/62, 1999. Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Gloden, A. "Uuml;ber mehrgeradige Gleichungen." Arch. Math. 1, 482/C1/83, 1949. Guy, R. K. "Sums of Like Powers. Euler’s Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/44, 1994. Lander, L. J. and Parkin, T. R. "A Counterexample to Euler’s Sum of Powers Conjecture." Math. Comput. 21, 101/C1/03, 1967. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. Lander, L. J. "Geometric Aspects of Diophantine Equations Involving Equal Sums of Like Power." Amer. Math. Monthly 75, 1061 /C1/073, 1968. Martin, A. "Methods of Finding nth-Power Numbers Whose Sum is an nth Power; With Examples." Bull. Philos. Soc. Washington 10, 107/C1/10, 1887. Martin, A. Smithsonian Misc. Coll. 33, 1888. Martin, A. "About Fifth-Power Numbers whose Sum is a Fifth Power." Math. Mag. 2, 201/C1/08, 1896. Moessner, A. "Einige numerische Identita ¨ten." Proc. Indian Acad. Sci. Sect. A 10, 296/C1/06, 1939.Moessner, A. "Alcune richerche di teoria dei numeri e problemi diofantei." Bol. Soc. Mat. Mexicana 2,3 6/C1/9, 1948. Moessner, A. "Due Sistemi Diofantei." Boll. Un. Mat. Ital. 6, 117/C1/18, 1951. Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. E ´cole Polytech. de Timisoara 11, 196/C1/19, 1944. Rao, K. S. "On Sums of Fifth Powers." J. London Math. Soc. 9, 170/C1/71, 1934. Sastry, S. and Chowla, S. "On Sums of Powers." J. London Math. Soc. 9, 242/C1/46, 1934. Swinnerton-Dyer, H. P. F. "A Solution of A5/C27B5/C27C5/C30D5/C27E5/C27F5:/"Proc. Cambridge Phil. Soc. 48, 516/C1/18, 1952. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . Xeroudakes, G. and Moessner, A. "On Equal Sums of Like Powers." Proc. Indian Acad. Sci. Sect. A 48, 245/C1/55, 1958. Diophantine Equation * /6th Powers The 6.1.2 equation A6/C30B6/C27C6(1) is a special case of F ERMAT’S LAST THEOREM with n/C306, and so has no solution. No 6.1. nsolutions are known for n56 (Lander et al. 1967; Guy 1994, p. 140). The smallest 6.1.7 solution is 746/C272346/C274026/C274746/C277026/C278946/C2710176 /C3011416(2) (Lander et al. 1967; Ekl 1998). The smallest primitive 6.1.8 solutions are 86/C27126/C27306/C27786/C271026/C271386/C271656/C272466 /C302516(3) 486/C271116/C271566/C271866/C271886/C272286/C272406/C274266 /C304316(4) 936/C27936/C271956/C271976/C273036/C273036/C273036/C274116 /C304406(5) 2196/C272556/C272616/C272676/C272896/C273516/C273516/C273516 /C304406(6) 126/C27666/C271386/C271746/C272126/C272886/C273066/C274416 /C304556(7) 126/C27486/C272226/C272366/C273336/C273846/C273906/C274266 /C304936(8) 666/C27786/C271446/C272286/C272566/C272886/C274356/C274446 /C304996(9) 166/C27246/C27606/C271566/C272046/C272766/C273306/C274926 /C305026(10) 616/C27966/C271566/C272286/C272766/C273186/C273546/C275346 /C305476(11) 1706/C271776/C272766/C273126/C273126/C274086/C274506/C274986 /C305596(12) 606/C271026/C271266/C272616/C272706/C273386/C273546/C275706 /C305816(13) 576/C271466/C271506/C273606/C273906/C274026/C274446/C275286 /C305836(14) 336/C27726/C271226/C271926/C272046/C273906/C275346/C275346 /C306076(15) 126/C27906/C271146/C271146/C272736/C273066/C274926/C275926 /C306236(16) (Lander et al. 1967). The smallest 6.1.9 solution is 16/C27176/C27196/C27226/C27316/C27376/C27376/C27416/C27496 /C30546(17) (Lander et al. 1967). The smallest 6.1.10 solution is 26/C2746/C2776/C27146/C27166/C27266/C27266/C27306/C27326/C27326 /C30396(18) (Lander et al. 1967). The smallest 6.1.11 solution is 26/C2756/C2756/C2756/C2776/C2776/C2796/C2796/C27106/C27146/C27176 /C30186(19) (Lander et al. 1967). There is also at least one 6.1.16 identity, 16/C2726/C2746/C2756/C2766/C2776/C2796/C27126/C27136/C27156 /C27166/C27186/C27206/C27216/C27226/C27236/C30286(20) (Martin 1893). Moessner (1959) gave solutions for 6.1.16, 6.1.18, 6.1.20, and 6.1.23 equations. Ekl (1996) has searched and found no solutions to the 6.2.2 A6/C27B6/C30C6/C27D6(21) with sums less than 7 :25/C291026:No solutions are known to the 6.2.3 or 6.2.4 equations. The smallest primitive 6.2.5 equations are 10926/C278616/C276026/C272126/C27846/C3011176/C277706(22) 18936/C2714686/C2714076/C2713026/C2712466 /C3020416/C276916(23) 21846/C2720966/C2714846/C2712666/C2712396 /C3024416/C277526(24) 26536/C2729626/C2714886/C2712816/C273906 /C3028276/C271516(25) 29546/C2724816/C278506/C277986/C274206 /C3029596/C2724706(26) (E. Brisse 1999 Resta 1999, PowerSum). The smallest 6.2.6 equation is2416/C27176/C302186/C272106/C271186/C272:636/C27426(27) (Ekl 1998). The smallest 6.2.7 solution is 186/C27226/C27366/C27586/C27696/C27786/C27786 /C30566/C27916(28) (Lander et al. 1967). The smallest 6.2.8 solution is 86/C27106/C27126/C27156/C27246/C27306/C27336/C27366 /C30356/C27376(29) (Lander et al. 1967). The smallest 6.2.9 solution is 16/C2756/C2756/C2776/C27136/C27136/C27136/C27176/C27196 /C3066/C27216(30) (Lander et al. 1967). The smallest 6.2.10 solution is 16/C2716/C2716/C2746/C2746/C2776/C2796/C27116/C27116/C27116 /C30126/C27126(31) (Lander et al. 1967). Parametric solutions are known for the 6.3.3 equation A6/C27B6/C27C6/C30D6/C27E6/C27F6(32) (Guy 1994, pp. 140 and 142). Known solutions are 36/C27196/C27226/C30106/C27156/C27236(33) 366/C27376/C27676/C30156/C27526/C27656(34) 336/C27476/C27746/C30236/C27546/C27736(35) 326/C27436/C27816/C3036/C27556/C27806(36) 376/C27506/C27816/C30116/C27656/C27786(37) 256/C27626/C271386/C30826/C27926/C271356(38) 516/C271136/C271366/C30406/C271256/C271296(39) 716/C27926/C271476/C3016/C271326/C271336(40) 1116/C271216/C272306/C30266/C271696/C272256(41) 756/C271426/C272456/C30146/C271636/C272436(42) (Rao 1934, Lander et al. 1967, Ekl 1998). Ekl (1998) mentions but does not list the 87 smallest solutions to the 6.2.6 equation. The smallest primitive 6.3.4 solutions are 736/C27586/C27416/C30706/C27656/C27326/C27156(43) 856/C27626/C27616/C30836/C27696/C27566/C27526(44) 856/C27746/C27616/C30876/C27716/C27566/C27266(45) 906/C27886/C27116/C30926/C27786/C27746/C27216(46) 956/C27836/C27266/C301016/C27286/C27246/C27236(47) 1306/C27446/C27236/C301196/C271086/C27866/C27386(48) 1256/C271146/C27386/C301266/C271046/C27936/C27686(49) 2056/C271136/C27186/C301986/C271486/C271336/C27396(50) 2116/C271236/C27346/C302106/C271346/C27736/C27396(51) 2126/C271646/C271036/C302176/C271306/C271146/C2786(52) 2226/C27346/C27256/C302176/C271566/C27966/C27686(53) 2186/C271676/C27296/C302246/C271076/C271026/C27656(54) 2266/C271106/C27176/C302246/C271436/C27726/C27346(55) 2446/C271236/C271126/C302386/C271806/C27916/C27726(56) 2416/C271726/C271566/C302466/C271456/C271326/C27566(57) 2576/C271556/C2766/C302526/C271816/C271436/C271146(58) 2656/C271476/C27126/C302316/C272216/C272106/C271146(59) 2606/C272186/C271856/C302766/C271526/C271126/C27256(60) 3056/C27856/C27666/C302736/C272676/C271726/C271226(61) 3126/C272416/C27336/C303156/C272286/C27996/C2726(62) 3316/C272346/C27596/C303066/C272946/C271516/C27956(63) 3326/C272436/C27436/C303386/C271776/C271686/C27956(64) 3516/C272656/C272216/C303366/C273096/C271696/C27736(65) 3656/C271376/C271266/C303606/C272346/C271756/C271336(66) 3606/C272656/C272006/C303366/C273186/C272126/C271696(67) 3486/C273256/C27366/C303576/C272766/C272766/C27826(68) 3736/C272886/C271046/C303636/C272926/C272666/C271206(69) 3866/C271136/C27626/C303786/C272606/C272096/C27886(70) (Lander et al. 1967, Ekl 1998). Moessner (1947) gave three parametric solutions to the 6.4.4 equation. The smallest 6.4.4 solution is 26/C2726/C2796/C2796/C3036/C2756/C2766/C27106(71) (Rao 1934, Lander et al. 1967). The smallest 6.4.4.4 solution is 16/C27346/C27496/C271116/C3076/C27436/C27696/C271106 /C30186/C27256/C27776/C271096(72) (Lander et al. 1967). Moessner and Gloden (1944) give the 6.7.8 solution 326/C27316/C27236/C27226/C27136/C2766/C2756 /C30336/C27286/C27276/C27206/C27116/C27106/C2726/C2716:(73) References Ekl, R. L. "Equal Sums of Four Seventh Powers." Math. Comput. 65, 1755 /C1/756, 1996.Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Guy, R. K. "Sums of Like Powers. Euler’s Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/44, 1994. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. Martin, A. "On Powers of Numbers Whose Sum is the Same Power of Some Number." Quart. J. Math. 26, 225/C1/27, 1893. Moessner, A. "On Equal Sums of Like Powers." Math. Student 15,8 3/C1/8, 1947. Moessner, A. "Einige zahlentheoretische Untersuchungen und diophantische Probleme." Glasnik Mat.-Fiz. Astron. Drustvo Mat. Fiz. Hrvatske Ser. 2 14, 177/C1/82, 1959. Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. E ´cole Polytech. de Timisoara 11, 196/C1/19, 1944. PowerSum. "Index of Equal Sums of Like Powers." http:// www.chez.com/powersum/. Rao, S. K. "On Sums of Sixth Powers." J. London Math. Soc. 9, 172/C1/73, 1934. Resta, G. "New Results on Equal Sums of Sixth Powers." Instituto di Matematica Computazionale, Pisa, Italy. April 1999. http://www.chez.com/powersum/Tr-b4 /C1/8.zip. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . Diophantine Equation * /7th Powers The 7.1.2 equation A7/C27B7/C30C7(1) is a special case of F ERMAT’S LAST THEOREM with n/C307, and so has no solution. No solutions to the 7.1.3, 7.1.4, 7.1.5, 7.1.6 equations are known. There is now a known solutions to the 7.1.7 equation, 5687/C305257/C274397/C274307/C274137/C272667/C272587/C271277 (2) (M. Dodrill 1999, PowerSum), requiring an update by Guy (1994, p. 140). The smallest 7.1.8 solution is 127/C27357/C27537/C27587/C27647/C27837/C27857/C27907 /C301027(3) (Lander et al. 1967, Ekl 1998). The smallest 7.1.9 solution is 67/C27147/C27207/C27227/C27277/C27337/C27417/C27507/C27597 /C30627(4) (Lander et al. 1967). No solutions to the 7.2.2, 7.2.3, 7.2.4, or 7.2.5 equations are known. The smallest 7.2.6 equation is 1257/C27247/C301217/C27947/C27837/C27617/C27577/C27277(5) (Meyrignac). The smallest 7.2.8 solution is 57/C2767/C2777/C27157/C27157/C27207/C27287/C27317 /C30107/C27337(6) (Lander et al. 1967, Ekl 1998). A 7.2.10.10 solution is 27/C27277/C3047/C2787/C27137/C27147/C27147/C27167/C27187/C27227 /C27237/C27237 /C3077/C2777/C2797/C27137/C27147/C27187/C27207/C27227 /C27227/C27237(7) (Lander et al. 1967). No solutions to the 7.3.3 equation are known (Ekl 1996), nor are any to 7.3.4. The smallest 7.3.5equations are 96 7/C27417/C27177/C30877/C272/C215777/C27687/C27567(8) 1537/C27437/C27147/C301407/C271377/C27597/C27427/C27427:(9) No solutions are known to the 7.3.6 equation. The smallest 7.3.7 solution is 77/C2777/C27127/C27167/C27277/C27287/C27317 /C30267/C27307/C27307(10) (Lander et al. 1967). Guy (1994, p. 140) asked if a 7.4.4 equation exists. The following solution provide an affirmative answer 1497/C271237/C27147/C27107/C301467/C271297/C27907/C27157(11) 1947/C271507/C271057/C27237 /C301927/C271527/C271327/C27387(12) 3547/C271127/C27527/C27197/C303437/C272817/C27467/C27357(13) (Ekl 1996, Elk 1998, M. Lau 1999, PowerSum).Numerical solutions to the 7.4.5 equation are givenby Gloden (1948). The smallest primitive 7.4.5 solu-tions are 50 7/C27437/C27167/C27127/C30527/C27297/C27267/C27117/C2737(14) 817/C27587/C27197/C2797/C30777/C27687/C27567/C27487/C2727(15) 877þ747þ697þ407 /C30827þ797þ757þ257þ97ð16Þ 997/C27767/C27327/C27297 /C30937/C27887/C27667/C27367/C27357(17) 987/C27827/C27587/C27347 /C30997/C27757/C27697/C27167/C27137(18) 1047/C27967/C27607/C27147 /C301027/C27957/C27817/C27577/C27237(19) 1117/C271027/C27407/C27297 /C301127/C27967/C27827/C27557/C27217(20) 1137/C271027/C27867/C27237 /C301207/C27817/C27587/C27557/C27107(21) (Lander et al. 1967, Ekl 1998). Gloden (1949) gives parametric solutions to the 7.5.5 equation. The first few 7.5.5 solutions are87/C2787/C27137/C27167/C27197 /C3027/C27127/C27157/C27177/C27187(22) 47/C2787/C27147/C27167/C27237 /C3077/C2777/C2797/C27207/C27227(23) 117/C27127/C27187/C27217/C27267 /C3097/C27107/C27227/C27237/C27247(24) 67/C27127/C27207/C27227/C27277 /C30107/C27137/C27137/C27257/C27267(25) 37/C27137/C27177/C27247/C27387 /C30147/C27267/C27327/C27327/C27337(26) (Lander et al. 1967). Ekl (1998) mentions but does not list 107 primitive solutions to 7.5.5. A parametric solution to the 7.6.6 equation was given by Sastry and Rai (1948). The smallest is 27/C2737/C2767/C2767/C27107/C27137 /C3017/C2717/C2777/C2777/C27127/C27127(27) (Lander et al. 1967). Another found by Chen Shuwen is 877/C272337/C272647/C273967/C274967/C275407 /C30907/C272067/C273097/C273667/C275227/C275237: (28) Moessner and Gloden (1944) gave the 7.9.10 solution 427/C27377/C27367/C27297/C27237/C27197/C27137/C2767/C2757 /C30417/C27407/C27337/C27287/C27277/C27157/C27147/C2797/C2727 /C2717: (29) References Ekl, R. L. "Equal Sums of Four Seventh Powers." Math. Comput. 65, 1755 /C1/756, 1996. Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Gloden, A. "Zwei Parameterlo ¨sungen einer mehrgeradigen Gleichung." Arch. Math. 1, 480/C1/82, 1949. Guy, R. K. "Sums of Like Powers. Euler’s Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/44, 1994. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. E ´cole Polytech. de Timisoara 11, 196/C1/19, 1944. Nagell, T. "The Diophantine Equation /x7þy7þz7¼0/."§67 inIntroduction to Number Theory. New York: Wiley, pp. 248 /C1/51, 1951. PowerSum. "Index of Equal Sums of Like Powers." http:// www.chez.com/powersum/. Sastry, S. and Rai, T. "On Equal Sums of Like Powers." Math. Student 16,1 8/C1/9, 1948. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . Diophantine Equation * /8th Powers The 8.1.2 equation A8/C27B8/C30C8(1) is a special case of F ERMAT’S LAST THEOREM with n/C308, and so has no solution. No 8.1.3, 8.1.4, 8.1.5, 8.1.6, 8.1.7, or 8.1.8 solutions are known. The smal- lest 8.1.9 is 11678/C3010948 /C2710408/C275608/C275588 /C273668/C273488/C272848/C272718/C271908(2) (N. Kuosa). The smallest 8.1.10 is 2358/C302268/C271848/C271718/C271528/C271428 /C27668/C27588/C27348/C27168/C2768(3) (N. Kuosa, PowerSum). The smallest 8.1.11 solution is 148/C27188/C27448/C27448/C27668/C27708/C27928/C27938 /C27968/C271068/C271128/C301258(4) (Lander et al. 1967, Ekl 1998). The smallest 8.1.12 solution is 88/C2788/C27108/C27248/C27248/C27248/C27268/C27308/C27348 /C27448/C27528/C27638/C30658(5) (Lander et al. 1967). The general identity 28k/C274/C2710CB0C@ 8/C3028k/C274/C2810CB0C@ 8/C2727k/C2740CB0C@ 8/C272k/C2710CB0C@ 8 /C27725k/C2730CB0C@ 8/C2723k/C2720CB0C@ 8hi (6) gives a solution to the 8.1.17 equation (Lander et al. 1967). No 8.2.2, 8.2.3, 8.2.4, 8.2.5, 8.2.6, or 8.2.7 solutions are known. The smallest 8.2.8 solution is 1298/C27958/C301288/C27928/C27868/C27828/C27748/C27578/C27558 /C27208: (7) The smallest 8.2.9 solution is 28/C2778/C2788/C27168/C27178/C27208/C27208/C27248/C27248 /C30118/C27278(8) (Lander et al. 1967, Ekl 1998). No 8.3.3, 8.3.4, 8.3.5, or 8.3.6 solutions are known. The smallest 8.3.7 solution is 1088/C27688/C2758 /C301028/C27888/C27888/C27528/C27378/C27268/C2768: (9) The smallest 8.3.8 solution is68/C27128/C27168/C27168/C27388/C27388/C27408/C27478 /C3088/C27178/C27508(10) (Lander et al. 1967, Ekl 1998). No 8.4.4 solutions is known. The smallest 8.4.5solution is 221 8/C271088/C27948/C27948 /C301958/C271948/C271888/C271268/C27388: (11) The smallest 8.4.6 solution is 478/C27298/C27128/C2758 /C30458/C27408/C27308/C27268/C27238/C2738(12) (Ekl 1998). The smallest 8.4.7 solution is 78/C2798/C27168/C27228/C27228/C27288/C27348 /C3068/C27118/C27208/C27358(13) (Lander et al. 1967). The smallest 8.5.5 solutions are 438/C27208/C27118/C27108/C2718 /C30418/C27358/C27328/C27288/C2758(14) 428/C27418/C27358/C2798/C2768 /C30458/C27368/C27278/C27138/C2788(15) 638/C27638/C27318/C27158/C2768 /C30658/C27598/C27488/C27378/C2778(16) 758/C27478/C27398/C27268/C2768 /C30678/C27678/C27628/C27208/C27118(17) 778/C27768/C27718/C27428/C27288 /C30868/C27418/C27368/C27328/C27298(18) 908/C27818/C27108/C2748/C2738 /C30928/C27748/C27558/C27508/C27378(19) 938/C27658/C27658/C27418/C27138 /C30818/C27818/C27798/C27758/C27458(20) 898/C27878/C27288/C27148/C27148 /C30968/C27368/C27338/C27318/C27248(21) 938/C27908/C27328/C27188/C2798 /C30948/C27868/C27718/C27608/C27198(22) 1048/C27738/C27368/C27178/C2738 /C301038/C27788/C27688/C27118/C2798(23) 1038/C27868/C27588/C27118/C2788 /C301048/C27788/C27698/C27628/C2798(24) 1088/C271018/C27888/C27458/C2718 /C301168/C27598/C27468/C27158/C2738(25) 1168þ928þ798þ338þ258 ¼1138þ1038þ608þ448þ318(26) 1238/C27978/C27718/C27108/C2728 /C301258/C27778/C27488/C27378/C27268(27) 1218/C271098/C27718/C27708/C27408 /C301208/C271048/C27998/C27758/C27618(28) 1278/C27438/C27268/C27108/C2738 /C301238/C271058/C27698/C27428/C27148(29) (Letac 1942, Lander et al. 1967, Ekl 1998). The smallest 8.5.6 solutions are 368þ368þ338þ258þ218 /C30388þ348þ328þ158þ158þ138ð30Þ 398/C27338/C27328/C27258/C27198 /C30378/C27358/C27358/C27178/C27168/C2728(31) 418þ218þ208þ198þ168 /C30408þ318þ308þ178þ98þ88ð32Þ 438/C27348/C27248/C2788/C2718 /C30428/C27378/C27288/C27168/C27168/C27158(33) 448/C27428/C27248/C27178/C2748 /C30478/C27208/C27188/C2788/C2768/C2768(34) 498/C27298/C27228/C2718/C2718 /C30478/C27428/C27268/C27238/C27178/C2758(35) 468/C27468/C27338/C27308/C2798 /C30458/C27458/C27368/C27368/C27348/C27328(36) 518/C27488/C27398/C27218/C27108 /C30538/C27458/C27258/C27228/C27228/C2768(37) 558þ378þ198þ198þ188 /C30518þ508þ358þ268þ118þ98ð38Þ 588/C27178/C27138/C27108/C2778 /C30568/C27458/C27418/C27408/C2788/C2718(39) 558/C27538/C27248/C27218/C2728 /C30528/C27528/C27508/C27258/C27178/C2778(40) 588/C27518/C27178/C27118/C27118 /C30608/C27378/C27348/C27298/C27238/C2738(41) 548/C27518/C27518/C27438/C2748 /C30598/C27468/C27418/C27308/C27178/C2728(42) 588/C27538/C27358/C27198/C27178 /C30618/C27308/C27258/C27238/C27168/C2718(43) 618/C27298/C27288/C27278/C27268 /C30578/C27528/C27488/C27178/C27148/C2758(44)588/C27518/C27498/C2788/C2768 /C30618/C27448/C27328/C27268/C27108/C2718(45) 628/C27538/C27388/C27328/C27238 /C30618/C27528/C27508/C27348/C27248/C2718(46) 598/C27578/C27478/C27408/C2788 /C30628/C27528/C27458/C27178/C27158/C2728(47) 638/C27628/C27558/C27438/C27278 /C30658/C27598/C27568/C27178/C27138/C27108(48) (Ekl 1998). Moessner and Gloden (1944) found solutions to the 8.6.6 equation. The smallest 8.6.6 solution is 38/C2768/C2788/C27108/C27158/C27238 /C3058/C2798/C2798/C27128/C27208/C27228(49) (Lander et al. 1967). Ekl (1998) mentions but does not list 204 primitive solutions to the 8.6.6 equation. Moessner and Gloden (1944) found solutions to the 8.6.7 equation. Parametric solutions to the 8.7.7 equation were given by Moessner (1947) and Gloden (1948). The smallest 8.7.7 solution is 18/C2738/C2758/C2768/C2768/C2788/C27138 /C3048/C2778/C2798/C2798/C27108/C27118/C27128(50) (Lander et al. 1967). Sastry (1934) used the smallest 17 /C1/solution to give a parametric 8.8.8 solution. The smallest 8.8.8 solutionis 1 8/C2738/C2778/C2778/C2778/C27108/C27108/C27128 /C3048/C2758/C2758/C2768/C2768/C27118/C27118/C27118(51) (Lander et al. 1967). Letac (1942) found solutions to the 8.9.9 equation. Moessner and Gloden (1944) found the 8.9.10 solution 548/C27538/C27468/C27378/C27298/C27238/C27228/C2768/C2758 /C30558/C27/C27508/C27498/C27338/C27328/C27268/C27188/C2798/C2728 /C2718: (52) References Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Gloden, A. "Parametric Solutions of Two Multi-Degreed Equalities." Amer. Math. Monthly 55,8 6/C1/8, 1948. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. Letac, A. Gazetta Mathematica 48,6 8/C1/9, 1942. Moessner, A. "On Equal Sums of Like Powers." Math. Student 15,8 3/C1/8, 1947. Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. E ´cole Polytech. de Timisoara 11, 196/C1/19, 1944. Sastry, S. "On Sums of Powers." J. London Math. Soc. 9, 242/C1/46, 1934. Weisstein, E. W. "Like Powers." M ATHEMATICA NOTEBOOK LIKEPOWERS.M . Diophantine Equation * /9th Powers The 9.1.2 equation A9/C30B9/C27C9(1) is a special case of F ERMAT’S LAST THEOREM with n/C309, and so has no solution. No 9.1.3, 9.1.4, 9.1.5, 9.1.6, 9.1.7, 9.1.8, 9.1.9, 9.1.10, or 9.1.11 solutions are known. The smallest 9.1.12 solution is 1039/C30919/C27919/C27899/C27719/C27689/C27659 /C27439/C27429/C27199/C27169/C27139/C2759: (2) To 9.1.13 solution is known. The smallest 9.1.14 solution is 669/C30639/C27549/C27519/C27499/C27389/C27359/C27299 /C27249/C27219/C27129/C27109/C2779/C2729/C2719(3) (Ekl 1998). No 9.2.2, 9.2.3, 9.2.4,. 9.2.5, 9.2.6, 9.2.7, 9.2.8, or 9.2.9 solutions are known. A 9.2.10 solution is given by 1219/C272/C2151169/C271159/C27899/C27529/C27289 /C27269/C27149/C2799/C301379/C27699(4) (L. Morelli 1999, PowerSum). No 9.2.11 solutions are known. The smallest 9.2.12 solution is 4/C21529/C272/C21539/C2749/C2779/C27169/C27179/C272/C215199 /C30159/C27219(5) (Lander et al. 1967, Ekl 1998). There are no known 9.1.13 or 9.1.14 solutions. The smallest 9.1.15 solu-tion is 2 9/C2729/C2749/C2769/C2769/C2779/C2799/C2799/C27109/C27159 /C27189/C27219/C27219/C27239/C27239/C30269(6) (Lander et al. 1967). There are no known 9.3.3, 9.3.4, 9.3.5, 9.3.6, 9.3.7, or9.3.8 solutions. The smallest 9.3.9 solution is 2/C21538 9/C2739/C30419/C27239/C272/C215209/C27189/C272/C215139/C27129/C2799 (7) (Ekl 1998). There is no known 9.3.10 solution. The smallest 9.3.11 solution is 29/C2739/C2769/C2779/C2799/C2799/C27199/C27199/C27219/C27259 /C27299/C30139/C27169/C27309(8) (Lander et al. 1967). There are no known 9.4.4 or 9.4.5 solutions are known. The smallest 9.4.6 solution is909/C27649/C27359/C27359 /C30869/C27809/C27629/C27439/C27279/C27169: (9) There are no known 9.4.7 or 9.4.8 solutions. The smallest 9.4.9 solution is 389/C27319/C27129/C2729 /C30369/C272/C215329/C27309/C27159/C27139/C2789/C2749/C2739(10) (Ekl 1998). The smallest 9.4.10 solutions are 29/C2769/C2769/C2799/C27109/C27119/C27149/C27189/C27199/C27199 /C3059/C27129/C27169/C27219(11) (Lander et al. 1967). The smallest 9.5.5 solution is 1929/C271019/C27919/C27309/C27269 /C301809/C271759/C271169/C27179/C27129: (12) There is no known 9.5.6 solution. The smallest 9.5.7 solution is 359/C27269/C272/C215159/C27129 /C30339/C27329/C27249/C27169/C27149/C2789/C2769(13) (Ekl 1998). There are no known 9.5.8, 9.5.9, or 9.5.10solutions. The smallest 9.5.11 solution is 3 9/C2759/C2759/C2799/C2799/C27129/C27159/C27159/C27169/C27219 /C27219/C3079/C2789/C27149/C27209/C27229(14) (Lander et al. 1967). The smallest 9.6.6 solutions are 239/C27189/C27149/C27139/C27139/C2719 /C30229/C27219/C27159/C27109/C2799/C2759(15) 319/C27239/C27219/C27149/C2799/C2729 /C30299/C27299/C27159/C27119/C27109/C2769(16) 469/C27449/C27279/C27279/C27279/C2799 /C30489/C27399/C27239/C27159/C27139/C27129(17) 479/C27479/C27229/C27229/C27129/C2749 /C30509/C27399/C27359/C27139/C27109/C2779(18) 549/C27529/C27489/C27479/C27469/C27149 /C30609/C27189/C27179/C27169/C27159/C27159(19) 709/C27449/C27369/C27339/C27199/C2749 /C30649/C27639/C27579/C27479/C27229/C27139(20) 689/C27589/C27509/C27469/C27419/C2779 /C30709/C27489/C27269/C27259/C27239/C27189(21) (Lander et al. 1967, Ekl 1998). Ekl (1998) mentions but does not list nine primitive solutions to the 9.7.7 equation. Moessner (1947) gives a parametric solution to the 9.10.10 equation. Palama ´ (1953) gave a solution to the 9.11.11 equation. Moessner and Gloden (1944) give the 9.11.12 solution 729 /C27679 /C27669 /C27539 /C27439 /C27379 /C27359 /C27299 /C27199 /C2769 /C2759 /C30719 /C27709 /C27639 /C27559 /C27409 /C27399 /C27339 /C27329 /C27179 /C2799 /C2729 /C2719 : (22) References Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446 /C1/59, 1967. Moessner, A. "On Equal Sums of Like Powers." Math. Student 15,83/C1/8, 1947. Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. E´ cole Polytech. de Timisoara 11, 196 /C1/19, 1944. Palama ´, G. "Diophantine Systems of the Type ap i /C301ak i /C30 ap i /C301bk i(k /C301, 2, ..., n, n /C272 ; n /C274 ; ..., n /C272r):/" Scripta Math. 19, 132 /C1/34, 1953. PowerSum. "Index of Equal Sums of Like Powers." http:// www.chez.com/powersum/. Weisstein, E. W. "Like Powers." MATHEMATICA NOTEBOOK LIKEPOWERS.M . Diophantine Equation * /nth Powers The 2 /C1/ equation An /C27Bn/C27/C30Cn (1) is a special case of FERMAT’S LAST THEOREM and so has no solutions for n ]3: Lander et al. (1967) give a table showing the smallest n for which a solution to xk 1 /C27xk2 /C27.../C27xkm /C30yk1 /C27yk2 /C27.../C27ykn ; (2) with 1 5m 5n is known. An updated table is given below; a more extensive table may be found at the PowerSum web site. k m 234567 8 910 1233478111523 2222478 91219 3 337 81124 44 7 1 0 2 3 55 5 1 1 1 6 66 2 777Take the results from the R AMANUJAN 6 /C1/0 /C1/ IDENTITY that for ad /C30bc, with F2m(a ;b;c ;d) /C30(a /C27b /C27c)2m /C27(b /C27c /C27d)2m /C28(c /C27d /C27a)2m /C28(d/C27a/C27b)2m/C27(a/C28d)2m/C28(b/C28c)2m(3) and f2m(x;y)/C30(1/C27x/C27y)2m/C27(x/C27y/C27xy)2m/C28(y/C27xy/C271)2m /C28(xy/C271/C27x)2m/C27(1/C28xy)2m/C28(x/C28y)2m;(4) then F2m(a;b;c;d)/C30a2mf2m(x;y): (5) Using f2(x;y)/C300 (6) f4(x;y)/C300 (7) now gives (a/C27b/C27c)n/C27(b/C27c/C27d)n/C27(a/C28d)n /C30(c/C27d/C27a)n/C27(d/C27a/C27b)n/C27(b/C28c)n(8) forn/C302o r4 . See also DIOPHANTINE EQUATION ,RAMANUJAN 6 -10-8 IDENTITY References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, p. 101, 1994. Berndt, B. C. and Bhargava, S. "Ramanujan--For Low- brows." Amer. Math. Monthly 100, 644/C1/56, 1993. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 653 /C1/57, 1966. Gloden, A. Mehrgradige Gleichungen. Groningen, Nether- lands: P. Noordhoff, 1944. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. PowerSum. "Index of Equal Sums of Like Powers." http:// www.chez.com/powersum/. Reznick, B. Sums of Even Powers of Real Linear Forms. Providence, RI: Amer. Math. Soc., 1992. Sekigawa, H. and Koyama, K. "Nonexistence Conditions of a Solution for the Congruence xk 1/C27.../C27xks/C13Nmod pnðÞ :/" Math. Comput. 68, 1283 /C1/297, 1999. Diophantine Quadruple DIOPHANTINE SET Diophantine Set A set SofPOSITIVE INTEGERS is said to be Diophan- tine IFFthere exists a POLYNOMIAL Qwith integral coefficients in m]1 indeterminates such that S/C30Qx1; :::;xm ðÞ ]1:x1]1; :::;xm]1 fg : It has been proved that the set of PRIME NUMBERS is a Diophantine set. References Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 189 /C1/92, 1995. Diophantus Property A set of mdistinct POSITIVE INTEGERS S/C30a1; :::;am fg satisfies the Diophantus property DnðÞof order n(a positive integer) if, for all i;j/C301;...,mwith i"j; aiaj/C27n/C30b2 ij; (1) thebij/s are INTEGERS . The set Sis called a Diophan- tine n-tuple. Diophantine 1-doubles are abundant: (1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (1, 8), (3, 8), (6, 8), (7, 9), (8, 10), (9, 11), ... (Sloane’s A050269 and A050270). Diophantine 1-triples are less abundant: (1, 3, 8), (2, 4, 12), (1, 8, 15), (3, 5, 16), (4, 6, 20), ... (Sloane’s A050273, A050274, and A050275). Fermat found the smallest Diophantine 1-quadruple: 1;3;8;120 fg (Davenport and Baker 1969, Jones 1976). There are no others with largest term 5200; and Davenport and Baker (1969) showed that if c/C271; 3c/C271;and 8 c/C271 are all squares, then c/C30120. Jones (1976) derived an infinite sequence of polynomials S/C30x;x/C272;c 1xðÞ;c2xðÞ; ::: fg such that the product of any two, increased by 1, is the square of a polynomial.Letting c /C281xðÞ/C30c0xðÞ/C300;then the general ckxðÞis given by the RECURRENCE RELATION ck/C304x2/C278x/C2720CB0C@ ck/C281/C28ck/C282/C274x/C271 ðÞ : (2) The first few ckare c1/C304x/C271 ðÞ c2/C3043/C2711x/C2712x2/C274x30CB0C@ c3/C3083/C2723x/C2762x2/C2774x3/C2740x4/C278x50CB0C@ : Letting x/C301 gives the sequence sn/C301;3, 8, 120, 1680, 23408, 326040, ... (Sloane’s A051047), for whichffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi snsn/C271/C271p is 2, 5, 31, 449, 6271, 87361, ... (Sloane’s A051048). General D1ðÞquadruples are F2n;F2n/C272;F2n/C274;4F2n/C271F2n/C272F2n/C273;0C80C9 (3) where Fnare F IBONACCI NUMBERS , and n;n/C272;4n/C274;4n/C271 ðÞ ;2n/C271 ðÞ 2n/C273 ðÞ fg : (4) The quadruplet 2Fn/C281;2Fn/C271;2F3 nFn/C271Fn/C272;0C82Fn/C271Fn/C272Fn/C2732F2 n/C271/C28F2 n0CB0C@ g (5) isDF2 nðÞ (Dujella 1996). Dujella (1993) showed there exist no Diophantine quadruples D4k/C272 ðÞ :/ References Brown, E. "Sets in Which xy/C27kis Always a Square." Math. Comput. 45, 613/C1/20, 1985. Davenport, H. and Baker, A. "The Equations 3 x2/C282/C30y2and 8x2/C287/C30z2:/"Quart. J. Math. (Oxford) Ser. 2 20, 129/C1/37, 1969. Diofant Aleksandri /˘1/ski /˘1:Arifmetika i kniga o mnogou- gol’nyh chislakh [Russian]. Moscow: Nauka, 1974. Dujella, A. "Generalization of a Problem of Diophantus." Acta Arith. 65,1 5/C1/7, 1993. Dujella, A. "Diophantine Quadruples for Squares of Fibo- nacci and Lucas Numbers." Portugaliae Math. 52, 305/C1/ 18, 1995. Dujella, A. "Generalized Fibonacci Numbers and the Pro- blem of Diophantus." Fib. Quart. 34, 164/C1/75, 1996. Hoggatt, V. E. Jr. and Bergum, G. E. "A Problem of Fermat and the Fibonacci Sequence." Fib. Quart. 15, 323/C1/30, 1977. Jones, B. W. "A Variation of a Problem of Davenport and Diophantus." Quart. J. Math. (Oxford) Ser. (2) 27, 349/C1/ 53, 1976. Morgado, J. "Generalization of a Result of Hoggatt and Bergum on Fibonacci Numbers." Portugaliae Math. 42, 441/C1/45, 1983 /C1/984. Sloane, N. J. A. Sequences A050269, A050269, A050273, A050274, A050275, A051047, and A051048 in "An On- Line Version of the Encyclopedia of Integer Sequences."http://www.research.att.com/~njas/sequences/eisonli- ne.html. Diophantus’s Riddle "Diophantus’s youth lasts 1/6 of his life. He grew a beard after 1/12 more of his life. After 1/7 more of hislife, Diophantus married. Five years later, he had a son. The son lived exactly half as long as his father, and Diophantus died just four years after his son’sdeath. All of this totals the years Diophantus lived." LetDbe the number of years Diophantus lived, and letSbe the number of years his son lived. Then the above word problem gives the two equations D/C30 1 6/C271 12/C2717 ! D/C275/C27S/C274 S/C3012D: Solving this simultaneously gives S/C3042 as the age of the son and D/C3084 as the age of Diophantus. References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 186 /C1/87, 1998. Pappas, T. "Diophantus’ Riddle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 123 and 232, 1989. #1999/C1/001 Wolfram Research, Inc. Dipyramid Two PYRAMIDS symmetrically placed base-to-base, also called a BIPYRAMID . The dipyramids are DUALS of the regular PRISMS . Consider the dipyramids generated by taking the duals of the n-PRISMS . The edge lengths of the base Sb n and slant edges Ssn ; half-height (half the distance from peak to peak) hn ; surface areas Snand volumes Vn (after scaling so that the smallest edge length is 1) are given by Sb 3 ;Ss3 /C302;4 3 (1) h3 /C3023 (2) S 3 /C3098ffiffiffi 7p (3) V 3 /C303 16ffiffiffi3p (4) sb 4 ;ss4 ¼ffiffiffi 2p ;ffiffiffi2p ð5Þ h 4 ¼ 1 ð6Þ S4 /C302ffiffiffi 3p (7) V4 /C301 3ffiffiffi 2p (8) Sb 4 ;Ss4 /C30ffiffiffi 5p /C281;4 5ffiffiffi 5p (9) h4 /C301 2ffiffiffi 2p (10) S4 /C302ffiffiffi3p (11)V 4 /C301 3ffiffiffi 2p (12) Sb 5 ;Ss5 /C30ffiffiffi 5p /C281 ;4 5ffiffiffi 5p (13) h5 ¼1 5 ð5 þffiffiffi 5p Þð 14Þ S5 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 95 /C2740ffiffiffi 5pq (15) V5 /C301 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1265 /C2729ffiffiffi 5p0C@n0C@os (16) S b 6 ; Ss6 /C302 3ffiffiffi 3p ;4 3ffiffiffi 3p (17) h6 /C302 (18) S6 /C303ffiffiffiffiffiffi15p (19) V 6 /C303 (20) Sb 8 ;Ss8 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 22/C28ffiffiffi 2p0C@n0C@or ;2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq (21) h8 /C302 /C27ffiffiffi 2p (22) S8 /C304ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23 /C2716ffiffiffi 2pq (23) V8 /C302 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi258/C2741ffiffiffi 2p0C@n0C@or (24) S b 10 ;Ss10 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 55 /C28ffiffiffi 5p0C@n0C@os ;4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 55 /C272ffiffiffi 5p0C@n0C@os (25) h10 /C303 /C27ffiffiffi 5p (26) S10 /C305ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 55 /C2724ffiffiffi 5pq (27) V10 /C305 6 /C27 15 /C277ffiffiffi 5p0C@n0C@o : (28) JOHNSON SOLID J12is a triangular dipyramid, the OCTAHEDRON is a square dipyramid, and J OHNSON SOLID J13is a pentagonal dipyramid. See also DELTAHEDRON ,E LONGATED DIPYRAMID , JOHNSON SOLID,OCTAHEDRON ,PENTAGONAL DIPYR- AMID ,PRISM ,PYRAMID ,TRAPEZOHEDRON ,TRIANGU- LAR DIPYRAMID ,TRIGONAL DIPYRAMID References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 117, 1989. Pedagoguery Software. Poly . http://www.peda.com/poly/. Dirac Delta Function DELTA FUNCTION Dirac Distribution DELTA FUNCTION # 1999 /C1/001 Wolfram Research, Inc. Dirac Equation The quantum electrodynamical law which applies to spin-1/2 particles and is the relativistic generalization of the SCHRO ¨ DINGER EQUATION .In3 /C271 dimensions (three space dimensions and one time dimension), it is given by ih c@ c @t/C30 axpx /C27 aypy /C27 azpz /C27 a4mcðÞ0C10CC c; (1) where h is h-bar, c is the speed of light, c is the wavefunction , m is the mass of the particle, ai are the DIRAC MATRICES , si are PAULI SPIN MATRICES , and pi /C30pi000 0 pi00 00 pi0 000 pi2 6643 775: (2) In 1/C271 dimensions, the Dirac equation is the system of PARTIAL DIFFERENTIAL EQUATIONS ut /C27vx /C27imu /C272i l ujj2/C28vjj20C@n0C@o u/C300 (3) vt/C27ux/C27imv/C272ilvjj2/C28ujj20C@n0C@o v/C300 (4) (Alvarez et al. 1982; Zwillinger 1997, p. 137); See also SCHRO ¨ DINGER EQUATION References Alvarez, A.; Pen-Yu, K.; and Vazquez, L. "The Numerical Study of a Nonlinear One-Dimensional Dirac Equation." Appl. Math. Comput. 18,1/C1/5, 1983. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Dirac Gamma Matrices DIRAC MATRICES Dirac Matrices The Dirac matrices are a class of 4 /C294 matrices which arise in quantum electrodynamics. There are a variety of different symbols used, and Dirac matrices are also known as gamma matrices or Dirac gammamatrices. The Dirac matrices are defined as the 4 /C294 matrices s i/C30I2/C156si;Pauli (1) ri/C30si;Pauli/C156I2; (2)where si;Rauli are the /2/C292 ðÞ PAULI MATRICES ,2is the 2 /C292 ðÞ IDENTITY MATRIX ,i/C301, 2, 3, and A/C156Bis the MATRIX DIRECT PRODUCT . Explicitly, this set of Dirac matrices is then given by I/C301000 0100001000012 6643 775(3) s 1/C300100 1000000100102 6643 775(4) s 2/C300/C28i00 i000 000 /C28i 00 i02 6643 775(5) s3/C301000 0/C2810 0 0010 000 /C2812 6643 775(6) r1/C300010 0001 1000 01002 6643 775(7) r2/C3000 /C28i0 00 0 /C28i i00 0 0i002 6643 775(8) r 3/C3010 0 0 01 0 000 /C2810 00 0 /C2812 6643 775(9) These matrices satisfy the anticommutation identi- ties s isj/C27sjsi/C302dijI (10) rirj/C27rjri/C302dijI; (11) where dijis the K RONECKER DELTA , the commutation identity si;rj0C10CC /C30sirj/C27sjri/C300; (12) and are cyclic under permutations of indices sisi/C30isk (13) riri/C30irk: (14) A total of 16 Dirac matrices can be defined via Eij/C30sirj (15) fori;j/C300;1, 2, 3 and where s0/C30r0/C13I:These matrices satisfy 1. Eij0C@10C@10C@10C@1/C301; where |A| is the DETERMINANT , 2. E2 ij /C30I ;/ 3. Eij /C30E/C31ij ; where A + denotes the ADJOINT MATRIX , making them Hermitian, and therefore unitary, 4. Tr Eij0CB0C@ /C300 ; except Tr E00ðÞ/C304 ;/ 5. Any two Eijmultiplied together yield a Dirac matrix to within a multiplicative factor of /C281or 9i ;/ 6. The Eij are linearly independent, 7. The Eijform a complete set, i.e., any 4 /C294 constant matrix may be written as A /C30X3 i;j/C300cijEij ; (16) where the cij are real or complex and are given by cmn /C301 4 Tr AEmnðÞ (17) (Arfken 1985). Dirac’s original matrices were written aiand were defined by ai /C30E1i /C30 r1 si (18) a4 /C30E30 /C30 r3 ; (19) for i /C301, 2, 3, giving a1 /C30E11 /C300001 0010010010002 6643 775 (20) a 2 /C30E12 /C30000 /C28i 00 i 0 0 /C28i 00 i 0002 6643 775 (21) a3 /C30E13 /C300010 000 /C281 1000 0 /C2810 02 6643 775 (22) a4 /C30E30 /C3010 0 1 01 0 0 00 /C2810 00 0 /C2812 6643 775: (23) The additional matrix a 5 /C30E20 /C30 r2 /C3000 /C28i 0 00 0 /C28i i 00 0 0 i 002 6643 775 (24) is sometimes defined.A closely related set of Dirac matrices is defined by g i /C300 si /C28si00C1B0C1@ (25) g4 /C30I 0 2I /C28I0C1B0C1@ (26) for i /C301, 2, 3 (Goldstein 1980). Instead of g4 ;g0 ; is commonly used. Unfortunately, there are two differ- ent conventions for its definition, the "chiral basis" g0 /C300 I I 0 : (27) and the "Dirac basis" g0 /C30 I 0 0 /C28I : (28) Other sets of Dirac matrices are sometimes defined as yi /C30E2i (29) y4 /C30E30 (30) y5 /C30/C28E10 (31) and di /C30E3i (32) for i /C301, 2, 3 (Arfken 1985). Any of the 15 Dirac matrices (excluding the identity matrix) commute with eight Dirac matrices and anticommute with the other eight. Let M /C13 1 21 /C27Eij0CB0C@ ; then M2 /C30M (33) (Arfken 1985, p. 216). In addition a1 a2 a32 435/C29a 1 a2 a32435/C302is: (34) The products of a iandyisatisfy a1a2a3a4a5/C301 (35) y1y2y3y4y5/C301: (36) The 16 Dirac matrices form six anticommuting sets of five matrices each: 1.a1;a2;a3;a4;a5;/ 2.y1;y2;y3;y4;y5;/ 3.d1;d2;d3;r1;r2;/ 4.a1;y1;d1;s2;s3;/ 5.a2;y2;d2;s1;s3;/ 6.a3;y3;d3;s1;s2;:/ See also PAULI MATRICES References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211 /C1/17, 1985. Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 580, 1980. Dirac Notation A notation invented by Dirac which is very useful in quantum mechanics. The notation defines the "KET" vector, denoted c/C143;j and its transpose, called the "BRA" vector and denoted /C142cj:: The "bracket" is then defined by /C142 fj c/C143:: Dirac notation satisfies the iden- tities /C142 f ˜O0C@10C@10C@10C@1c/C143/C13/C142fj ˜O c/C143 /C142f j c/C143/C13g/C12 /C28/C12¯fcdx; where ¯c is the COMPLEX CONJUGATE . See also ANGLE BRACKET ,B RA,D IFFERENTIAL K- FORM,KET,L2-SPACE ,ONE-FORM Dirac Operator The operator D /C30/C28id/C27d/C31ðÞ ; where d/C31 is the ADJOINT . Dirac’s Theorem A GRAPH with n ]3 VERTICES in which each VERTEX has VERTEX DEGREE ]n=2 has a HAMILTONIAN CIR- CUIT. See also HAMILTONIAN CIRCUIT Direct Analytic Continuation If (f, U) and (g, V) are FUNCTIONS ELEMENTS , then (g, V) is a direct analytic continuation of (f, U)ifU S V "0¥ and f and G are equal on U S V ::/ See also ANALYTIC CONTINUATION ,GLOBAL ANALYTIC CONTINUATION References Krantz, S. G. "Direct Analytic Continuation." §10.1.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 128, 1999. Direct Product The direct product is defined for a number of classes of algebraic objects, including GROUPS , RINGS , and MODULES . In each case, the direct product of an algebraic object is given by the CARTESIAN PRODUCT of its elements, considered as sets, and its algebraic operations are defined componentwise. For instance, the direct product of two VECTOR SPACES of DIMEN- SIONS n and m is a VECTOR SPACE of DIMENSION n /C27m:/Direct products satisfy the property that, given maps a : S 0 A and b : S 0 B; there exists a unique map S 0 A /C29B given by a(s) ; b(s) ðÞ :: The notion of map is determined by the CATEGORY , and this definition extends to other CATEGORIES such as TOPOLOGICAL SPACES . Note that no notion of commutativity is necessary, in contrast to the case for the COPRODUCT . In fact, when A and B are ABELIAN , as in the cases of MODULES (e.g., VECTOR SPACES )orA BELIAN GROUPS ) (which are MODULES over the integers), then the DIRECT SUM A /C154B is well-defined and is the same as the direct product. Although the terminology is slightly confusing because of the distinction between the elementary operations of addition and multiplica- tion, the term "direct sum" is used in these cases instead of "direct product" because of the implicit connotation that addition is always commutative. Note that direct products and DIRECT SUMS differ for infinite indices. An element of the DIRECT SUM is zero for all but a finite number of entries, while an element of the direct product can have all nonzero entries. Some other unrelated objects are sometimes also called a direct product. For example, the TENSOR DIRECT PRODUCT is the same as the TENSOR PRODUCT , in which case the dimensions multiply instead of add. Here, "direct" may be used to distinguish it from the EXTERNAL TENSOR PRODUCT . See also CARTESIAN PRODUCT ,C ATEGORY THEORY , COPRODUCT ,DIRECT SUM,GROUP DIRECT PRODUCT , MATRIX DIRECT PRODUCT ,PRODUCT (CATEGORY THE- ORY), RING DIRECT PRODUCT ,SET DIRECT PRODUCT , TENSOR DIRECT PRODUCT ,TENSOR PRODUCT (VECTOR SPACE ) Direct Proportion DIRECTLY PROPORTIONAL Direct Search Factorization Direct search factorization is the simplest (and most simple-minded) PRIME FACTORIZATION ALGORITHM .I t consists of searching for factors of a number bysystematically performing TRIAL DIVISIONS , usually using a sequence of increasing numbers. Multiples of small PRIMES are commonly excluded to reduce the number of trial DIVISORS , but just including them is sometimes faster than the time required to exclude them. Direct search factorization is very inefficient, and can be used only with fairly small numbers. When using this method on a number n, only DIVISORS up toffiffiffinpbc (where xbcis the FLOOR FUNC- TION ) need to be tested. This is true since if all INTEGERS less than this had been tried, then nffiffiffinpbc/C27 1 Bffiffiffinp: (1) In other words, all possible FACTORS have had their COFACTORS already tested. It is also true that, when the smallest PRIME FACTOR p of n is >ffiffiffin3p; then its COFACTOR m (such that n /C30pm) must be PRIME .To prove this, suppose that the smallest p is >ffiffiffin3p;: If m /C30ab, then the smallest value a and b could assume is p. But then n /C30pm /C30pab /C30p3 > n; (2) which cannot be true. Therefore, m must be PRIME ,so n ¼ p1p2 (3) See also PRIME FACTORIZATION ALGORITHMS ,TRIAL DIVISION Direct Sum The direct sum A /C154B of two sets of integers A and B consists of the set a /C27b : a /C23 A ;b /C23 B fg ; and can be generalized to an arbitrary number of sets A /C154B /C154/C1/C1/C1 in the obvious way. For example, the direct sum of A /C30f1;2 g; B /C30f1;2 g; and C /C30f2 ;3g is A /C154B /C154C /C30 f4; 5;5; 6;5;6 ;6;7 g:: The direct sum of a sequence of sets l can be implemented in Mathematica as follows. DirectSum[l__] : /C30 Flatten[Outer[Plus, l]] The significant property of the direct sum is that it is the COPRODUCT in the CATEGORY of MODULES (i.e., a MODULE DIRECT SUM). This general definition gives as a consequence the definition of the direct sum A /C154B of ABELIAN GROUPS A and B (since they are Z/- modules, i.e., MODULES over the INTEGERS ) and the direct sum of VECTOR SPACES (since they are MODULES over a FIELD ). Note that the direct sum of Abelian groups is the same as the GROUP DIRECT PRODUCT , but that the term direct sum is not used for groups which are NON- ABELIAN . Note that DIRECT PRODUCTS and direct sums differ for infinite indices. An element of the direct sum is zero for all but a finite number of entries, while an element of the DIRECT PRODUCT can have all nonzero entries. See also ABELIAN GROUP ,DIRECT PRODUCT ,GROUP DIRECT PRODUCT ,M ATRIX DIRECT SUM,M ODULE , MODULE DIRECT SUM Direct Variation DIRECTLY PROPORTIONAL Directed Acyclic Graph ACYCLIC DIGRAPHDirected Angle The symbol /C140ABC denotes the directed angle from AB to BC, which is the signed angle through which AB must be rotated about B to coincide with BC. Four points ABCD lie on a CIRCLE (i.e., are CON- CYCLIC ) IFF /C140ABC /C30/C140ADC :: It is also true that /C140l1l2 /C27/C140l2l1 /C300/C14 or 360/C14: Three points A, B, and C are COLLINEAR IFF /C140ABC /C30 0or180: or 1808. For any four points, A, B, C, and D, /C140ABC/C27/C140CDA/C30/C140BAD/C27/C140DCB : See also ANGLE ,C OLLINEAR ,C ONCYCLIC ,M IQUEL EQUATION References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 11 /C1/5, 1929. Directed Convex Polyomino ACONVEX POLYOMINO containing at least one edge of its minimal bounding rectangle. The perimeter and area generating function for directed polygons of width m, height n, and area qis given by Gðx;y;qÞ¼X x]1X y]1X q]1Cðm;n;aÞxmynqn /C30yRðxÞ/C28ˆNðxÞ NðxÞð1Þ where N(x) /C30X n]0( /C281)nxnqn /C271 2ðÞ (q)n(yq)n(2) ˆN(x) /C30X n]1( /C281)nxnqn /C271 2ðÞ (q)n/C281(yq)n(3) R(x) /C30yX n]2xnqn (yq)nPn/C282 m/C300( /C281)m qm /C27 2 20C@80C@9 (q)m(yqm/C271)n/C28m/C2810 BBB@1 CCCA2 66643 7775(4) (Bousquet-Me ´lou 1992). The anisotropic perimeter generating function for directed convex polygons of width x and height y is given by G(x;y) /C30X x]1X y]1C(m;n)xmyn /C30xyffiffiffiffiffiffiffiffiffiffiffiffiffiffi D(x;y)p ; (5) where D(x ;y) /C301 /C282x /C282y /C282xy /C27x2 /C27y2 /C30(1 /C28y)2 1 /C28x(2 /C27 2y /C28 x) (1 /C28 y)2"# (6) (Lin and Chang 1988, Bousquet 1992, Bousquet- Me´lou et al. 1999). This can be solved to explicitly give C(m;n) /C30m /C27n /C282 m /C2810C@80C@9 m /C27n /C282 n /C2810C@80C@9 (7) (Bousquet-Me ´lou 1992). Expanding the generating function gives G(x;y) /C30X m]1Hm(y)xm (8) /C30y 1 /C28 y x /C27y(1 /C27 y) (1 /C28 y)3 x2 /C27y(1 /C27 4y /C27 y2) (1 /C28 y)5x3 /C27... (9) /C30(y /C27y2 /C27y3 /C27y4 /C27y5 /C27...)x /C27(y /C274y2 /C279y3 /C2716y4 /C2725y5 /C27...)x2 /C27(y /C279y2 /C2736y3 /C27100y4 /C27225y5 /C27...)x3 /C27(y /C2716y2 /C27100y3 /C27400y4 /C271225 y5 /C27...)x4 /C27... (10) An explicit formula of Hm(y) is given by Bousquet- Me´lou (1992). These functions satisfy the reciprocity relations Hm(1=y) /C30/C28ym/C282Hm(y) (11) G(x;y) /C27y2G(x=y;1 =y) /C300 (12) (Bousquet-Me ´lou et al. 1999).The anisotropic area and horizontal perimeter gen- erating function G(x;q) and partial generating func- tions Hm(q);connected by G(x;q)/C30X m]1Hm(q)xm; satisfy the self-reciprocity and inversion relations Hm(1=q)/C30/C281 qHm(q) and G(x;q)/C27qG(x;1=q)/C300 (Bousquet-Me ´louet al. 1999). See also CONVEX POLYOMINO ,LATTICE POLYGON References Bousquet-Me ´lou, M. "Convex Polyominoes and Heaps of Segments." J. Phys. A: Math. Gen. 25, 1925 /C1/934, 1992. Bousquet-Me ´lou, M. "Convex Polyominoes and Algebraic Languages." J. Phys. A: Math. Gen. 25, 1935 /C1/944, 1992. Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/9908123/. Lin, K. Y. and Chang, S. J. "Rigorous Results for the Number of Convex Polygons on the Square and Honey-comb Lattices." J. Phys. A: Math. Gen. 21, 2635 /C1 /642, 1988. Directed Graph AGRAPH in which each EDGE is replaced by a directed EDGE , also called a digraph or reflexive graph. A COMPLETE directed graph is called a TOURNAMENT .A directed graph having no symmetric pair of directed edges is called an ORIENTED GRAPH . IfGis an undirected connected GRAPH , then one can always direct the circuit EDGES ofGand leave the SEPARATING EDGES undirected so that there is a directed path from any node to another. Such a GRAPH is said to be transitive if the adjacency relation is transitive. The number of directed graphs of n nodes for n /C301, 2, ... are 1, 3, 16, 218, 9608, ... (Sloane’s A000273). See also ACYCLIC DIGRAPH ,ARBORESCENCE ,CAYLEY GRAPH ,G RAPH ,INDEGREE ,N ETWORK ,O RIENTED GRAPH ,O UTDEGREE ,S INK (DIRECTED GRAPH ), SOURCE ,STRONGLY CONNECTED DIGRAPH ,TOPOLOGY (DIGRAPH) ,TOURNAMENT ,W EAKLY CONNECTED DI- GRAPH References Chartrand, G. "Directed Graphs as Mathematical Models." §1.5 in Introductory Graph Theory. New York: Dover, pp. 16 /C1/9, 1985. Harary, F. "Digraphs." Ch. 16 in Graph Theory. Reading, MA: Addison-Wesley, pp. 10 and 198 /C1/11, 1994. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 122, 1986. Sloane, N. J. A. Sequences A000273/M3032 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Directed Set A set S together with a RELATION ] which is both transitive and reflexive such that for any two ele- ments a;b /C23 S; there exists another element c /C23 S with a]c]b:In this case, the relation ]is said to "direct" the set. See also NET Direction Cosine Letabe the ANGLE between vandx,bthe ANGLE between vandy, and cthe ANGLE between vandz. Then the direction cosines are equivalent to the (x;y;z) coordinates of a UNIT VECTOR ˆv; a/C13cosa/C13v/C215ˆx vjj(1) b/C13cosb/C13v/C215ˆy vjj(2)g/C13cosc/C13v/C215ˆz vjj: (3) From these definitions, it follows that a2/C27b2/C27g2/C301: (4) To find the J ACOBIAN when performing integrals over direction cosines, use u/C30sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2q0C@80C@9 (5) f/C30tan/C281b a ! (6) g/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28a 2/C28b2q : (7) The J ACOBIAN is @(u;f) @(a;b)0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30@u @a@u @b @f @a@f @b0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (8) Using d dxsin/C281x0CB0C@ /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p (9) d dxtan/C281x0CB0C@ /C301 1/C27x2; (10) @(u;f) @(a;b)0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C301 2a2/C27b20CB0C@ /C281=22a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28a2/C28b2q1 2a2/C27b20CB0C@ /C281=22b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28a2/C28b2q /C28a/C282b 1/C27b2 a2a/C281 1/C27b2 a20C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1 /C30 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28a2/C28b2qa2/C27b20CB0C@ /C281=2 1/C27b2 a21/C27b2 a2 ! /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b20CB0C@ 1/C28a2/C28b20CB0C@q ; (11) so dV/C30sinudfdu/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2q@(u;f) @(a;b)0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1dadb /C30 dadbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28a2/C28b2q /C30dadb g: (12) Direction cosines can also be defined between two sets of C ARTESIAN COORDINATES , a1 /C13ˆx?/C215ˆx (13) a2 /C13ˆx?/C215ˆy (14) a3 /C13ˆx?/C215ˆz (15) b1 /C13ˆy?/C215ˆx (16) b2 /C13ˆy?/C215ˆy (17) b3 /C13ˆy ?/C215ˆz (18) g1 /C13ˆz ?/C215ˆx (19) g2 /C13ˆz ?/C215ˆy (20) g3 /C13ˆz?/C215ˆz : (21) Projections of the unprimed coordinates onto the primed coordinates yield ˆx?/C30 ˆx?/C215ˆxðÞ ˆx /C27 ˆx ?/C215ˆyðÞ ˆy /C27 ˆx ?/C215ˆzðÞ ˆz /C30 a1 ˆx /C27 a2 ˆy /C27 a3 ˆz ð22Þ ˆy ?/C30 ˆy?/C215ˆxðÞ ˆx /C27 ˆy?/C215ˆyðÞ ˆy /C27 ˆy?/C215ˆzðÞ ˆz /C30 b1 ˆx /C27 b2 ˆy /C27 b3 ˆz ð23Þ ˆz ?/C30 ˆz ?/C215ˆxðÞ ˆx /C27 ˆz?/C215ˆyðÞ ˆy /C27 ˆz ?/C215ˆzðÞ ˆz /C30 g1 ˆx /C27 g2 ˆy /C27 g3 ˆz ;ð24Þ and x?/C30r /C215ˆx?/C30a1x /C27 a2y /C27 a3z (25) y?/C30r /C215ˆy ?/C30b1x /C27 b2y /C27 b3z (26) z?/C30r /C215ˆz?/C30 g1x /C27 g2y /C27 g3z : (27) Projections of the primed coordinates onto the un- primed coordinates yield ˆx /C30 ˆx /C215ˆx? ðÞ ˆx?/C27 ˆx /C215ˆy ? ðÞ ˆy ?/C27 ˆx /C215ˆz ? ðÞ ˆz? /C30 a1 ˆx?/C27b1 ˆy?/C27g1 ˆz? (28) ˆy /C30 ˆy /C215ˆx? ðÞ ˆx?/C27 ˆy /C215ˆy ? ðÞ ˆy ?/C27 ˆy /C215ˆz ? ðÞ ˆz? /C30 a2 ˆx ?/C27 b2 ˆy?/C27g2 ˆz? (29) ˆz /C30 ˆz /C215ˆx? ðÞ ˆx?/C27 ˆz /C215ˆx? ðÞ ˆy?/C27 ˆz /C215ˆz ? ðÞ ˆz? /C30 a3 ˆx?/C27 b3 ˆy ?/C27 g3 ˆz ?; (30) and x /C30r /C215ˆx /C30 a1x /C27 b1y /C27 g1z (31) y /C30r /C215ˆy /C30 a2x /C27 b2y /C27 g2z (32) z /C30r /C215ˆz /C30 a3x /C27 b3y /C27 g3z : (33) Using the orthogonality of the coordinate system, it must be true that ˆx /C215ˆy /C30ˆy /C215ˆz /C30ˆz /C215ˆx /C300 (34) ˆx /C215ˆx /C30ˆy /C215ˆy /C30ˆz /C215ˆz /C301; (35) giving the identities al am /C27 bl bm /C27 gl gm /C300 (36) for l ;m /C301;2 ;3 and l "m; anda2 l /C27 b2 l /C27 g2 l /C301 (37) for l /C301;2 ;3:: These two identities may be combined into the single identity al am /C27 bl bm /C27 gl gm /C30 dlm ; (38) where dlm is the KRONECKER DELTA . Direction Vector UNIT VECTOR # 1999 /C1/001 Wolfram Research, Inc. Directional Derivative 9uf /C139f /C215u ujj8lim h00f(x /C27 hu) /C28 f(x) h: (1) /9ufx0 ;y0 ;z0 ðÞ is the rate at which the function w /C30 f(x; y;z) changes at x0 ;y0 ;z0 ðÞ in the direction u : Let u be a UNIT VECTOR in CARTESIAN COORDINATES ,so ujj/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 x /C27u2y /C27u2zq /C301; (2) then 9uf/C30@f @xux/C27@f @yuy/C27@f @zuz: (3) The directional derivative is often written in the notation d ds/C13ˆs/C2159/C30sx@ @x/C27sy@ @y/C27sz@ @z: (4) Directly Proportional Two quantities yand xare said to be directly proportional, proportional, or "in direct proportion"ifyis given by a constant multiple of x, i.e., y/C30cxfor ca constant. This relationship is commonly written y8x:: / See also INVERSELY PROPORTIONAL ,PROPORTIONAL #1999/C1/001 Wolfram Research, Inc. Directly Similar Two figures are said to be SIMILAR when all corre- sponding ANGLES are equal, and are directly similar when all corresponding ANGLES are equal and de- scribed in the same rotational sense. Any two directly similar figures are related either by a TRANSLATION or by a SPIRAL SIMILARITY (Coxeter and Greitzer 1967, p. 97). See also DOUGLAS- NEUMANN THEOREM ,FUNDAMEN- TAL THEOREM OF DIRECTLY SIMILAR FIGURES ,HOMO- THETIC ,INVERSELY SIMILAR ,S IMILAR ,S PIRAL SIMILARITY References Casey, J. "Two Figures Directly Similar." Supp. Ch. §2in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 173 /C1/79, 1888. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 95, 1967. Lachlan, R. "Properties of Two Figures Directly Similar" and "Properties of Three Figures Directly Similar." §213 /C1/19 and 223 /C1/43 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 135 /C1/38 and 140 /C1/43, 1893. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 12, 1991. Director A PLANE parallel to two (or more) SKEW LINES , also called a director plane. The orientation of a director is fixed, but it is specified uniquely only if a point lying on it is also specified. A director of two SKEW LINES is perpendicular to the line of shortest distance of these two lines (Altshiller- Court 1979, p. 1). See also SKEW LINES References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 1, 1979. # 1999 /C1/001 Wolfram Research, Inc. Director Curve The curve d(u) in the RULED SURFACE parameteriza- tion x(u;v) /C30b(u) /C27vd(u): See also DIRECTOR ,D IRECTRIX (RULED SURFACE ), RULED SURFACE ,RULING References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 431, 1997. Director Plane DIRECTOR # 1999 /C1/001 Wolfram Research, Inc.Directrix DIRECTRIX (CONIC SECTION ), DIRECTRIX (GRAPH ), DIRECTRIX (RULED SURFACE ) Directrix (Conic Section) The LINE which, together with the point known as the FOCUS , serves to define a CONIC SECTION as the LOCUS of points whose distance from the FOCUS is propor- tional to the horizontal distance from the directrix. If the ratio r /C301, the conic is a PARABOLA ,ifr B1, it is an ELLIPSE , and if r /C211, it is a HYPERBOLA (Hilbert and Cohn-Vossen 1999, p. 27). HYPERBOLAS and noncircular ELLIPSES have two distinct FOCI and two associated DIRECTRICES , each DIRECTRIX being PERPENDICULAR to the line joining the two foci (Eves 1965, p. 275). See also CONIC SECTION ,ELLIPSE ,FOCUS ,HYPERBO- LA,PARABOLA References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115 /C1/16, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 141 /C1/44, 1967. Eves, H. "The Focus-Directrix Property." §6.8 in A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, pp. 272 /C1/ 75, 1965. Hilbert, D. and Cohn-Vossen, S. "The Directrices of the Conics." Ch. 1, Appendix 2 in Geometry and the Imagina- tion. New York: Chelsea, pp. 27 /C1/9, 1999. Directrix (Graph) A GRAPH CYCLE . See also GRAPH CYCLE Directrix (Ruled Surface) The curve b(u) in the RULED SURFACE parameteriza- tion x(u; v) /C30b(u) /C27vd(u) is called the directrix (or BASE CURVE ). See also DIRECTOR CURVE ,RULED SURFACE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 431, 1997. Dirichlet Beta Function b(x) /C13X/C12 n/C300(/C281)n(2n /C271)/C28x (1) b(x) /C302/C28x F/C281;x;1 2 ! ; (2) where F(z ;s ;a) is the LERCH TRANSCENDENT . The beta function can be written in terms of the HURWITZ ZETA FUNCTION z(x;a)by b(x) /C301 4xz x;14 ! /C28 z x;34 !"# : (3) The beta function can be evaluated directly for POSITIVE ODD x as b(2k /C271) /C30( /C281)kE2k 2(2k)!12 p ! 2k /C271 ; (4) where En is an EULER NUMBER . The beta function can be defined over the whole COMPLEX PLANE usingANALYTIC CONTINUATION , b(1 /C28z) /C302 p !z sin12 pz ! G(z) b(z) ; (5) where G(z) is the GAMMA FUNCTION . Particular values for b are b(1) /C3014 p (6) b(2) /C13K (7) b(3) /C301 32 p3 ; (8) where K is CATALAN’S CONSTANT . See also CATALAN’S CONSTANT ,DIRICHLET ETA FUNC- TION ,DIRICHLET LAMBDA FUNCTION ,HURWITZ ZETA FUNCTION ,LEGENDRE’S CHI-FUNCTION ,LERCH TRANS- CENDENT ,RIEMANN ZETA FUNCTION ,ZETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807 /C1/08, 1972. Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25 /C1/3, 1987. Dirichlet Boundary Conditions PARTIAL DIFFERENTIAL EQUATION BOUNDARY CONDI- TIONS which give the value of the function on a surface, e.g., T/C30f(r;t):/ See also BOUNDARY CONDITIONS ,CAUCHY BOUNDARY CONDITIONS References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 679, 1953. Dirichlet Conditions DIRICHLET BOUNDARY CONDITIONS ,DIRICHLET FOUR- IERSERIES CONDITIONS Dirichlet Divisor Problem Let the DIVISOR FUNCTION d(n)/C30n(n)/C30s0(n) be the number of DIVISORS ofn(including nitself). For a PRIME p,n(p)/C302:In general, Xn k/C301n(k) /C30n lnn /C27(2g/C281)n /C27O nu0CB0C@ ; where g is the EULER- MASCHERONI CONSTANT . Dirich- let originally gave u :1=2 (Hardy 1999, pp. 67 /C1/8), and Landau (1916) showed than u ]1 =4 (Hardy 1999, p. 81). The following table summarizes incremental progress on the upper limit (Hardy 1999, p. 81). /u/ approx. citation 7/22 0.31818 1988 27/82 0.32927 van der Corput 1928 33/100 0.33000 van der Corput 1922 1/3 0.33333 Voronoi 1903 1/2 0.50000 Dirichlet See also DIVISOR FUNCTION ,GAUSS’S CIRCLE PROBLEM References Bohr, H. and Crame ´r. Enzykl. d. Math. Wiss. II C 8, 815 /C1/22, 1922. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 262 /C1/63, 1979. van der Corput. Math. Ann. 98, 697 /C1/17, 1928. Dirichlet Energy Let h be a real-valued HARMONIC FUNCTION on a bounded DOMAIN V; then the Dirichlet energy is defined as fV9hjj2dx;where 9is the GRADIENT . See also ENERGYDirichlet Eta Function The function defined by h(x)/C13X/C12 n/C301(/C281)n/C281n/C28x/C301/C2821/C28x0CB0C@ z(x); (1) where n/C301, 2, ..., and z(x) is the R IEMANN ZETA FUNCTION . Note that Borwein and Borwein (1986, p. 289) use the notation a(s) instead of h(s)::Parti- cular values are given in Abramowitz and Stegun (1972, p. 811).The eta function is related to the R IEMANN ZETA FUNCTION and D IRICHLET LAMBDA FUNCTION by z(n) 2n/C30l(n) 2n/C281/C30h(n) 2n/C282(2) and z(n)/C27h(n)/C302l(n) (3) (Spanier and Oldham 1987). The eta function is also aspecial case of the POLYLOGARITHM function, h(x)/C30/C28Lix(/C281): (4) The value h(1) may be computed by noting that the MACLAURIN SERIES for ln(1 /C27x) for/C2815x51i s ln(1/C27x)/C30x/C281 2x2/C2713x 3/C2814x 4/C27/C1/C1/C1 (5) Therefore, ln2/C30ln(1/C271)/C301/C281 2/C2713/C2814/C27/C1/C1/C1 /C30X/C12 n/C301( /C281)n/C281 n/C30 h(1): (6) The derivative of the eta function is given by h?ðxÞ¼/C2821 /C28x ln 2 zðxÞþð1 /C2821/C28x Þz?ðxÞ; ð7Þ or in the special case x /C300, by limx 00d dx h(x)"# /C30/C28ln2 /C28 z?(0) /C30/C28ln2 /C271 2ln(2 p) /C30/C28lnffiffiffi 2 ps ! /C3012 ln12 p ! : (8) This latter fact provides a remarkable proof of the W ALLIS FORMULA . Values for EVEN INTEGERS are related to the analy- tical values of the RIEMANN ZETA FUNCTION . h(0) is defined to be1 2 : h(0) /C301 2 h(1) /C30ln2 h(2) /C30p2 12 h(3) /C300:90154... h(4) /C307p4 720 : See also DEDEKIND ETA FUNCTION ,DIRICHLET BETA FUNCTION ,DIRICHLET L-SERIES ,DIRICHLET LAMBDA FUNCTION ,RIEMANN ZETA FUNCTION ,ZETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807 /C1/08, 1972. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25 /C1/3, 1987. Dirichlet Fourier Series Conditions A piecewise regular function which 1. Has a finite number of finite discontinuities and 2. Has a finite number of extrema can be expanded in a FOURIER SERIES which con- verges to the function at continuous points and themean of the POSITIVE and NEGATIVE limits at points of discontinuity. See also FOURIER SERIES Dirichlet Function Let c and d "c be REAL NUMBERS (usually taken as c /C301 and d /C300). The Dirichlet function is defined by D(x) /C30c for x rational d for x irrational0C1n (1) and is discontinuous everywhere. The Dirichlet func- tion can be written analytically as D(x) /C30 lim m0/C12lim n 0/C12cos2n(m!px) : (2) Because the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version can be defined as DM(x) /C300 for x irrational 1 =b for x /C30a=b a reduced fraction0C1n (3) (Dixon 1991), illustrated above. This function is continuous at irrational x and discontinuous at rational x (although a small interval around an irrational point x contains infinitely many ration points, these rationals will have very large denomi- nators). When viewed from a corner along the line y/C30xin normal perspective, a QUADRANT of E UCLID’S ORCHARD turns into the modified Dirichlet function (Gosper). See also CONTINUOUS FUNCTION ,EUCLID’S ORCHARD , IRRATIONAL NUMBER ,RATIONAL NUMBER References Dixon, R. Mathographics. New York: Dover, pp. 177 and 184/C1/86, 1991. Tall, D. "The Gradient of a Graph." Math. Teaching 111, 48/C1/2, 1985. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Dirichlet Integrals There are several types of integrals which go under the name of a "Dirichlet integral." The integral D[u]/C30gV½9u½2dV (1) appears in D IRICHLET’S PRINCIPLE . The integral 1 2pgp /C28pf(x)sin n/C271 2 ! x"# sin12x ! dx; (2) where the kernel is the D IRICHLET KERNEL , gives the nth partial sum of the F OURIER SERIES . Another integral is denoted dk/C131 pg/C12 /C28/C12sinakrk rkeirkgkdrk/C300 for½gk½>ak 1 for½gk½Bak0C1n (3) fork/C301, ..., n. There are two types of Dirichlet integrals which are denoted using the letters C,D,I, and J. The type 1 Dirichlet integrals are denoted I,J, and IJ, and the type 2 Dirichlet integrals are denoted C,D, and CD. The type 1 integrals are given by I/C13gg...gft1/C27t2/C27:::/C27tn ðÞ ta1/C281 1ta2/C281 2...tan/C281 ndt1dt2dtn /C30Ga1ðÞGa2ðÞ :::GanðÞ GP nan0CB0C@ g1 0frðÞrX na !/C281 dr; (4) where G(z) is the GAMMA FUNCTION . In the case n/C302, I/C30ggTxpyqdxdy/C30p!q! (p/C27q/C272)!/C30B(p/C271;q/C271) p/C27q/C272;(5) where the integration is over the TRIANGLE T bounded by the X-AXIS ,Y-AXIS , and line x/C27y/C301 and B(x;y) is the BETA FUNCTION . The type 2 integrals are given for b-D vectors aandr, and 05c5b; C(b) a(r;m)/C30G(m/C27R) G(m)Qb i/C301GriðÞga1 0/C1/C1/C1gab 0 /C2Qbi/C301xri/C281 idxi 1/C27Pbi/C301xi0C@n0C@om/C27R (6)D(b) a(r;m)/C30G(m/C27R) G(m)Qbi/C301GriðÞg/C12 a1/C1/C1/C1g/C12 ak /C2Qbi/C301xri/C281 idxi 1/C27Pbi/C301xi0C@n0C@om/C27R (7) CD(c;d/C28c) a (r;m) /C30G(m/C27R) G(m)Qb i/C301GriðÞgac 0g/C12 ac/C271g/C12 abQbi/C301xri/C281 idxi 1/C27Pbi/C301xi0C@n0C@om/C27R;(8) where R/C13Xk i/C301ri (9) ai/C13pi 1/C28Pk i/C301pi; (10) and piare the cell probabilities. For equal probabil- ities, ai/C301:The Dirichlet Dintegral can be expanded as a MULTINOMIAL SERIES as D(b) a(r;m)/C301 1/C27Pbi/C3010C@n0C@om /C2X x1Br1/C1/C1/C1X xbBrbm/C281/C27Pba/C301xi m/C281;x1...;xb0C@80C@9 Y i/C301bai 1/C27Pb k¼1ak !xi : (11) For small b,Cand Dcan be expressed analytically either partially or fully for general arguments and ai/C301: C(1) 1r2;r1 ðÞ /C30Gr1/C27r2 ðÞ2Fir2;r1/C27r2;1/C27r2;/C281 ðÞ r2Gr1ðÞGr2ðÞ (12) C(2)1r2;r3;r1 ðÞ /C30Gr1/C27r2/C27r3 ðÞ r2Gr1ðÞGr2ðÞGr3ðÞ /C2g1 02F1yra/C281(1/C27y)/C28r1/C27r2/C27r3 ðÞdy; (13) where 2F1/C132F1r2;r1/C27r2/C27r3;1/C27r2;/C28(1/C27y)/C2810C@n0C@o (14) is a HYPERGEOMETRIC FUNCTION . D(1) 1r2;r1 ðÞ /C30Gr1/C27r2 ðÞ2F1r1;r1/C27/C27 r2;1/C27r1;/C281 ðÞ r1Gr1ðÞGr2ðÞ (15) D 2ðÞ 1r2 ;r3;r1 ðÞ /C30G r1 /C27 r2 /C27 r3 ðÞ r1 /C27 r3 ðÞ G r1ðÞG r2ðÞG r3ðÞg/C12 12F1yr3/C281dy ; (16) where 2F1 /C132 F1r1 /C27r3 ;r1 /C27r2 /C27r3;1/C27r1 /C27r3; /C281 /C28y ðÞ : (17) References Jeffreys, H. and Jeffreys, B. S. "Dirichlet Integrals." §15.08 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 468 /C1/70, 1988. Sobel, M.; Uppuluri, R. R.; and Frankowski, K. Selected Tables in Mathematical Statistics, Vol. 4: Dirichlet Dis- tribution--Type 1. Providence, RI: Amer. Math. Soc., 1977. Sobel, M.; Uppuluri, R. R.; and Frankowski, K. Selected Tables in Mathematical Statistics, Vol. 9: Dirichlet Inte- grals of Type 2 and Their Applications. Providence, RI: Amer. Math. Soc., 1985. Weisstein, E. W. "Dirichlet Integrals." MATHEMATICA NOTE- BOOK DIRICHLET INTEGRALS.M . Dirichlet Kernel The Dirichlet kernel DM nis obtained by integrating the CHARACTER ei(j ;x) over the BALL ½ j½5M ; DM n /C30/C281 2prd drDMn/C282 : The Dirichlet kernel of a DELTA SEQUENCE is given by dn(x) /C131 2psin n /C271 2 ! x"# sin12 x ! : The integral of this kernel is called the D IRICHLET INTEGRAL Du½/C138:/ See also DELTA SEQUENCE ,D IRICHLET INTEGRALS , DIRICHLET’S LEMMADirichlet Lambda Function l(x) /C13X/C12 n/C3002n /C301 ðÞ/C28x/C30 1 /C282 /C28xðÞ z xðÞ (1) for x /C302, 3, ..., where z(x) is the RIEMANN ZETA FUNCTION . The function is undefined at x /C301. It can be computed in closed form where z(x) can, that is for EVEN POSITIVE n. It is related to the RIEMANN ZETA FUNCTION and DIRICHLET ETA FUNCTION by z( n) 2n/C30l( n) 2n /C28 1 /C30h( n) 2 n /C28 2 (2) and z(n) /C27 h(n) /C302l(n) (3) (Spanier and Oldham 1987). Special values of l(n) include l(2) /C30p2 8(4) l(4)/C30p4 96: (5) See also DIRICHLET BETA FUNCTION ,DIRICHLET ETA FUNCTION ,L EGENDRE’S CHI-FUNCTION ,R IEMANN ZETA FUNCTION ,ZETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807 /C1/08, 1972. Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25 /C1/3, 1987. Dirichlet L-Series Series OF THE FORM Lk(s;x)/C13X/C12 n/C301xk(n)n/C28s; (1) where the CHARACTER xk(n)i sa n INTEGER FUNCTION with period m, are called Dirichlet L-series. These series are very important in ADDITIVE NUMBER THE- ORY (they were used, for instance, to prove D IRI- CHLET’S THEOREM ), and have a close connection with MODULAR FORMS . Dirichlet L-series can be written as sums of L ERCH TRANSCENDENTS with zaPOWER of e2pi=m:/ The D IRICHLET ETA FUNCTION h(s)/C13X/C12 n/C301(/C281)n/C271 ns/C301/C2821/C28s0CB0C@ z(s) (2) (fors"1);DIRICHLET BETA FUNCTION L/C284(s)/C30b(s)/C13X/C12 n/C300(/C281)n (2n/C271)s; (3) and R IEMANN ZETA FUNCTION L/C271(s)/C30z(s)/C13X/C12 n/C3001 ns(4) are all Dirichlet L-series (Borwein and Borwein 1987, p. 289). Hecke found a remarkable connection between each MODULAR FORM with F OURIER SERIES f(r)/C30c(0)/C27X/C12 n/C301c(n)e2pint(5) and the Dirichlet L-series f(s)/C30X/C12 m/C301c(n) ns(6) This Dirichlet series converges absolutely for s/C30 Rs½/C138>k/C271 (iffis a CUSP FORM ) and s>2kiffis not aCUSP FORM . In particular, if the coefficients /cðnÞ/ satisfy the multiplicative property cmðÞcnðÞ/C30X d½m;nðÞd2k/C281cmn d2 ! ; (7) then the Dirichlet L-series will have a representation OF THE FORM fsðÞ/C30Y p1 1/C28cpðÞp/C28s/C27p2k/C281p/C282s; (8)which is absolutely convergent with the Dirichlet series (Apostol 1997, pp. 136 /C1/37). In addition, let k] 4b ea n EVEN integer, then f(s) can be ANALYTICALLY CONTINUED beyond the line s/C30ksuch that 1. If c(0)/C300;then f(s)i sa n ENTIRE FUNCTION ofs, 2. If c(0)"0;f(s) is analytic for all sexcept a single SIMPLE POLE ats/C30kwith RESIDUE (/C281)k=2c(0)(2p)k G(k); (9) where G(k) is the GAMMA FUNCTION , and 3.f(s) satisfies (2p)/C28sG(s)f(s)/C30(/C281)k/C2152(2p)s/C28kG(k/C28s)f(k/C28s) (10) (Apostol 1997, p. 137). The CHARACTER xkis called primitive if the CONDUC- TORf(x)/C30k:Otherwise, xkis imprimitive. A primitive L-series modulo kis then defined as one for which xk(n) is primitive. All imprimitive L-series can be expressed in terms of primitive L-series. LetP/C301o r P/C30Qt i/C301pi;where piare distinct ODD PRIMES . Then there are three possible types of primitive L-series with REAL COEFFICIENTS . The requirement of REAL COEFFICIENTS restricts the CHARACTER toxk(n)/C3091 for all kand n. The three type are then 1. Ifk/C30P(e.g., k/C301, 3, 5, ...) or k/C304P(e.g., k/C304, 12, 20, ...), there is exactly one primitive L-series. 2. If k/C308P(e.g., k/C308, 24, ...), there are two primitive L-series. 3. If k/C302P;Ppi;or 2aPwhere a>3 (e.g., k/C302, 6, 9, ...), there are no primitive L-series (Zucker and Robertson 1976). All primitive L-series are ALGEBRAICALLY INDEPENDENT and divide into two types according to xkk/C281 ðÞ /C3091: (11) Primitive L-series of these types are denoted L9:For a primitive L-series with REAL CHARACTER (NUMBER THEORY ), ifk/C30P, then Lk/C30L/C28kif P/C133 mod4ðÞ Lkif P/C131 mod4ðÞ:0C1n (12) Ifk/C304P;then Lk/C30L/C28kif P/C131 mod4ðÞ Lkif P/C133 mod4ðÞ;0C1n (13) and if k/C308P;then there is a primitive function of each type (Zucker and Robertson 1976). The first few primitive NEGATIVE L-series are L/C283; L/C284;L/C287;L/C288;L11;L/C2815;L/C2819;L/C2820;L/C2823;L/C2824;L/C2831; L/C2835 ; L/C2839 ; L/C2840 ; L/C2843 ; L/C2847 ; L/C2851 ; L/C2852 ; L/C2855 ; L/C2856 ; L/C2859 ; L/C2867 ; L/C2868 ; L/C2871 ; L/C2879 ; L/C2883 ; L/C2884 ; L/C2887 ; L/C2888 ; L/C2891 ; L/C2895 ; ... (Sloane’s A003657), corresponding to the negated discriminants of IMAGINARY QUADRATIC FIELDS . The first few primitive POSITIVE L-series are L/C271 ; L /C275 ; L /C278 ; L /C2712 ; L /C2713 ; L /C2717 ; L /C2721 ; L /C2724 ; L/C2728 ; L/C2729 ; L/C2733 ; L/C2737 ; L/C2740 ; L/C2741 ; L/C2744 ; L/C2753 ; L/C2756 ; L/C2757 ; L/C2760 ; L/C2761 ; L/C2765 ; L/C2769 ; L/C2773 ; L/C2776 ; L/C2777 ; L/C2785 ; L/C2788 ; L/C2789 ; L/C2792 ; L/C2793 ; L/C2797 ; ... (Sloane’s A046113). The KRONECKER SYMBOL is a REAL CHARACTER mod- ulo k, and is in fact essentially the only type of REAL primitive CHARACTER (Ayoub 1963). Therefore, L/C27d(s) /C30X/C12 n /C301d½nðÞ n/C28s (14) L/C28d(s) /C30X/C12 n /C301/C28d½n ðÞ n/C28s ; (15) where d½nðÞ is the KRONECKER SYMBOL (Borwein and Borwein 1986, p. 293). The functional equations for L9 are L/C28k(s) /C302s ps/C281k/C28s/C271 =2 G(1 /C28s) cos1 2 s p ! L /C28k(1 /C28s) ð16Þ L/C27k(s) /C302s ps/C281k/C28s/C271 =2 G(1 /C28s) sin12 sp ! L /C27k(1 /C28s) :ð17Þ For m a POSITIVE INTEGER L/C27k(/C282m) /C300 (18) L/C28k(1 /C282m) /C300 (19) L/C27k(2m) /C30Rk/C281=2 p2m (20) L/C28k(2m /C281) /C30R?k /C281 =2 p2m/C281 (21) L/C27k(1 /C282m) /C30(/C281)m(2m /C28 1)!R (2k)2m/C281 (22) L/C28k(/C282k) /C30( /C281)mR?(2m)! (2k)2m (23) where R and R? are RATIONAL NUMBERS . Nothing general appears to be known about L/C28k(2m)or L/C27k ð2m /C281Þ; although it is possible to express all L9(1) in terms of known transcendentals (Zucker and Robertson 1976). /L/C27k(1) can be expressed in terms of transcendentals by Ld(1) /C30h(d) k(d) ; (24) where h(d) is the CLASS NUMBER and k(d) is the DIRICHLET STRUCTURE CONSTANT . Some specific va-lues of primitive L-series are L/C2815(1) /C302pffiffiffiffiffiffi 15p L/C2811(1) /C30pffiffiffiffiffiffi 11p L/C288(1) /C30p 2ffiffiffi2p L /C287(1) /C30pffiffiffi 7p L/C284(1) /C301 4 p L/C283(1) /C30p 3ffiffiffi 3p L/C275(1) /C302ffiffiffi5p ln1 /C27ffiffiffi5p 2 ! L /C27s(1) /C30ln 1 /C27ffiffiffi2p0CB0C@ ffiffiffi2p L /C2712(1) /C30ln(2 /C27ffiffiffi3p )ffiffiffi 3p L /C2713(1) /C302ffiffiffiffiffiffi 13p ln3 /C27ffiffiffiffiffiffi13p 2 ! L /C2717(1) /C302ffiffiffiffiffiffi17p ln(4 /C27ffiffiffiffiffiffi 17p ) L /C2721(1) /C302ffiffiffiffiffiffi 21p ln5 /C27ffiffiffiffiffiffi21p 2 ! L /C2724(1)/C30ln(5/C272ffiffiffi 6p )ffiffiffi6p : In particular, L /C283(1)/C30L(1;x)/C30X/C12 n/C3001 (3n/C271)(3n/C272)(25) forxa nontrivial Dirichlet character modulo 3 (Ire- land and Rosen 1990, p. 266). No general forms are known for L/C28k(2m) and L/C27kð2m/C281Þin terms of known transcendentals. For example, L/C2842ðÞ/C30b2ðÞ/C13K; (26) where Kis defined as C ATALAN’S CONSTANT . See also DIRICHLET BETA FUNCTION ,DIRICHLET ETA FUNCTION ,DIRICHLET SERIES ,DOUBLE SUM,HECKE L-SERIES ,MODULAR FORM,PETERSSON CONJECTURE References Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976. Apostol, T. M. "Modular Forms and Dirichlet Series" and "Equivalence of Ordinary Dirichlet Series." §6.16 and §8.8 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 /C1/37 and 174 /C1/76, 1997. Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Buell, D. A. "Small Class Numbers and Extreme Values of L-Functions of Quadratic Fields." Math. Comput. 139, 786 /C1/96, 1977. Hecke, E. "U¨ ber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung." Math. Ann. 112, 664 /C1/ 99, 1936. Ireland, K. and Rosen, M. "Dirichlet L-Functions." Ch. 16 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 249 /C1/68, 1990. Koch, H. "L-Series." Ch. 7 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 203 /C1/58, 2000. Sloane, N. J. A. Sequences A003657/M2332 and A046113 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Weisstein, E. W. "Class Numbers." MATHEMATICA NOTE- BOOK CLASS NUMBERS.M . Zucker, I. J. and Robertson, M. M. "Some Properties of Dirichlet L-Series." J. Phys. A: Math. Gen. 9, 1207 /C1/214, 1976. Dirichlet Problem The problem of finding the connection between a continuous function f on the boundary @R of a region R with a HARMONIC FUNCTION taking on the value f on @R: In general, the problem asks if such a solution exists and, if so, if it is unique. The Dirichlet problem is extremely important in mathematical physics (Courant and Hilbert 1989, pp. 179 /C1/80 and 240; Logan 1997; Krantz 1999b). If f is a CONTINUOUS FUNCTION on the boundary of the open unit disk @D 0; 1ðÞ ; then define uzðÞ/C301 2p g2p 0feic0CB0C@ 1 /C28 zjj2 z /C28 eic jj2 d c f ðzÞif z /C23 D ð0; 1Þ if z /C23@Dð0; 1Þ;8 >< >: where @D 0;1ðÞ ; is the boundary of D(0;1): Then u is continuous on the closed unit disk D(0;1) and har- monic on D(0;1) (Krantz 1999a, p. 93). See also POISSON INTEGRAL ,POISSON KERNEL References Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, pp. 179 /C1/80 and 240, 1989. Krantz, S. G. "The Dirichlet Problem" and "Application of Conformal Mapping to the Dirichlet Problem." §7.3.3, 7.7.1, and 14.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 93, 97 /C1/8, and 164 /C1/68, 1999a.Krantz, S. G. A Panorama of Harmonic Analysis. Washing- ton, DC: Math. Assoc. Amer., 1999b. Logan, J. D. Applied Mathematics, 2nd ed. New York: Wiley, 1997. Dirichlet Region VORONOI POLYGON Dirichlet Series A series X anðÞe/C28 l nðÞz ; where a(n) and z are COMPLEX and l(n) fg is a MONOTONIC increasing sequence of REAL NUMBERS is called a general Dirichlet series. The numbers l(n) are called the exponents, and a(n) are called the coefficients. When l(n) /C30lnn; then e /C28 l nðÞz /C30n /C28z ; the series is a normal DIRICHLET L-SERIES . The Dirichlet series is a special case of the LAPLACE- STIELTJES TRANSFORM . See also DIRICHLET L-SERIES ,L APLACE- STIELTJES TRANSFORM ,MODULAR FORM,MODULAR FUNCTION References Apostol, T. M. "General Dirichlet Series and Bohr’s Equiva- lence Theorem." Ch. 8 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 161 /C1/89, 1997. Bohr, H. "Zur Theorie der allgemeinen Dirichletschen Reihen." Math. Ann. 79, 136 /C1/56, 1919. Dirichlet Structure Constant k dðÞ/C302ln h dðÞffiffiffi dp for d > 0 2p wdðÞffiffiffiffiffiffi djjp for d > 08 >>>< >>>: where h dðÞis the FUNDAMENTAL UNIT and wdðÞis the number of substitutions which leave the BINARY QUADRATIC FORM unchanged wdðÞ/C306 for d/C30/C283 4 for d/C30/C284 2 otherwise :8 < : See also CLASS NUMBER ,DIRICHLET L-SERIES References Weisstein, E. W. "Class Numbers." M ATHEMATICA NOTE- BOOK CLASS NUMBERS.M . Dirichlet Tessellation VORONOI DIAGRAM Dirichlet’s Approximation Theorem Given any REAL NUMBER u and any POSITIVE INTEGER N, there exist integers h and k with 0 5k 5N such that ku /C28h jjB1 N: A slightly weaker form of the theorem states that for every real u; there exist integers h and k with k /C210 and h;kðÞ 1 /C301 such that u/C28h k0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1B 1 k2 : See also HURWITZ’S IRRATIONAL NUMBER THEOREM , IRRATIONALITY MEASURE ,L IOUVILLE’S APPROXIMA- TION THEOREM ,R ATIONAL APPROXIMATION ,R OTH’S THEOREM ,THUE- SIEGEL- ROTH THEOREM References Apostol, T. M. "Dirichlet’s Approximation Theorem." §7.2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 143 /C1/45, 1997. Dirichlet’s Box Principle A.k.a. the PIGEONHOLE PRINCIPLE . Given n boxes and m /C21n objects, at least one box must contain more than one object. This statement has important appli- cations in NUMBER THEORY and was first stated by Dirichlet in 1834. See also FUBINI PRINCIPLE References Chartrand, G. Introductory Graph Theory. New York: Dover, p. 38, 1985. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 38, 1951. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 161, 1993. Dirichlet’s Boxing-In Principle DIRICHLET’S BOX PRINCIPLE Dirichlet’s Formula If g is continuous and m; n > 0 ; then gt 0t /C28 j ðÞm/C281djg j 0j /C28x ðÞn/C281g j;xðÞ dx /C30gt 0dxgt xt /C28 j ðÞm/C281j /C28x ðÞn/C281g j;xðÞ dj:Dirichlet’s Lemma g p 0sin n /C271 2 ! x"# 2 sin12 x ! dx /C3012 p ; where the KERNEL is the DIRICHLET KERNEL . See also DIRICHLET KERNEL References Cohn, H. Advanced Number Theory. New York: Dover, p. 37, 1980. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1101, 2000. Dirichlet’s Principle Dirichlet’s principle, also known as Thomson’s prin- ciple, states that here exists a function u that minimizes the functional D[u] /C30gV½9u½2dV (called the DIRICHLET INTEGRAL ) for VƒR2 or R3 among all the functions /u /C23 Cð1 ÞðVÞS C ð0 ÞðVÞ/ which take on given values f on the boundary @V of V; and that function u satisfies 92 /C300in V; u½@V/C30f ; u /C23 C 2ðÞVðÞS C 0ðÞ ¯V0CB0C@ : Weierstrass showed that Dirichlet’s argument contained a subtle fallacy. As a result, it can be claimed only that there exists a lower bound to which Du½/C138comes arbitrarily close without being forced to actually reach it. Kneser, however, obtained a valid proof of Dirichlet’s principle. See also DIRICHLET’S BOX PRINCIPLE ,D IRICHLET INTEGRALS References Monna, A. F. Dirichlet’s Principle: A Mathematical Comedy of Errors and Its Influence on the Development of Analysis. Utrecht, Netherlands: Osothoek, Scheltema, and Holk-ema, 1975. Dirichlet’s Test Let Xp n/C301an0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1BK; where Kis independent of p. Then if f n]fn/C271>0 and lim n0/C12fn/C300; it follows that X/C12 n /C301anfn CONVERGES . See also CONVERGENCE TESTS Dirichlet’s Theorem Given an ARITHMETIC SERIES of terms an /C27b; for n /C301, 2, ..., the series contains an infinite number of PRIMES if a and b are RELATIVELY PRIME , i.e., (a;b) /C301 : Dirichlet proved this theorem using DIRICH- LET L-SERIES , but the proof is challenging enough that, in their classic text on NUMBER THEORY , the usually explicit Hardy and Wright (1979) report "this theorem is too difficult for insertion in this book." See also PRIME ARITHMETIC PROGRESSION ,P RIME PATTERNS CONJECTURE ,R ELATIVELY PRIME ,S IER- PINSKI’S PRIME SEQUENCE THEOREM References Courant, R. and Robbins, H. "Primes in Arithmetical Progressions." §1.2b in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 26 /C1/7, 1996. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 13 /C1/4, 1979. Landau, E. Vorlesungen u¨ber Zahlentheorie, Vol. 1. New York: Chelsea, pp. 79 /C1/6, 1970. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, pp. 422 /C1/46, 1974. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 22 /C1/3, 1993. Dirichlet-Hardy Test If, in an interval of x, Sn r/C301 is uniformly bounded with respect to n and x, and fvr g is a sequence of positive non-increasing quantities tending to zero, then aar(x)vr is uniformly convergent in the interval. References Jeffreys, H. and Jeffreys, B. S. "Dirichlet-Hardy Test." §1.1155 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 42 /C1/3, 1988. Disc DISK Disconnected Form A FORM which is the sum of two FORMS involving separate sets of variables.Disconnected Graph A graph is said to be disconnected if it is not CONNECTED , i.e., if there exist two nodes is G such that no edge in G having those nodes as endpoints. The numbers of disconnected simple unlabeled graphs on n /C301, 2, ... nodes are 0, 1, 2, 5, 13, 44, 191, ... (Sloane’s A000719). If G is disconnected, then its complement ¯G is connected (Skiena 1990, p. 171; Bolloba ´s 1998). How- ever, the converse is not true, as can be seen using the example of the CYCLE GRAPH C5which is connected and isomorphic to its complement. See also CONNECTED GRAPH ,CUT SET, K-CONNECTED GRAPH References Bolloba ´s, B. Modern Graph Theory. New York: Springer- Verlag, 1998. Harary, F. "The Number of Linear, Directed, Rooted, and Connected Graphs." Trans. Amer. Math. Soc. 78, 445 /C1/63, 1955. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A000719/M1452 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stein, M. L. and Stein, P. R. "Enumeration of Linear Graphs and Connected Linear Graphs Up to p /C3018 Points." Report LA-3775. Los Alamos, NM: Los Alamos National Laboratory, Oct. 1967. Disconnectivity Disconnectivities are mathematical entities which stand in the way of a SPACE being contractible (i.e., shrunk to a point, where the shrinking takes place inside the SPACE itself). When dealing with TOPOLO- GICAL SPACES , a disconnectivity is interpreted as a "HOLE " in the space. Disconnectivities in SPACE are studied through the EXTENSION PROBLEM or the LIFTING PROBLEM . See also EXTENSION PROBLEM ,H OLE,LIFTING PRO- BLEM Discontinuity A point at which a mathematical object is DISCONTIN- UOUS . Discontinuous Not CONTINUOUS . A point at which a function is discontinuous is called a DISCONTINUITY , or some- times a JUMP . See also CONTINUOUS ,DISCONTINUITY References Yates, R. C. "Functions with Discontinuous Properties." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 100 /C1/07, 1952. Discordant Permutation MARRIED COUPLES PROBLEM Discrepancy Theorem Let s1 ; s2 ; ... be an infinite series of real numbers lying between 0 and 1. Then corresponding to any arbi- trarily large K, there exists a positive integer n and two subintervals of equal length such that the number of svwith n /C301; 2, ..., n which lie in one of the subintervals differs from the number of such sn that lie in the other subinterval by more than K (van der Corput 1935ab, van Aardenne-Ehrenfest 1945, 1949, Roth 1954). This statement can be refined as follows. Let N be a large integer and s1 ; s2 ; ..., sN be a sequence of N real numbers lying between 0 and 1. Then for any integer 1 5n 5N and any real number a satisfying 0 B a B1; let DnaðÞdenote the number of snwith v /C301; 2, ..., n that satisfy /0 5sn B a/. Then there exist n and a such that DnaðÞ/C28na jj > c1ln ln N ln ln ln N where c1 is a positive constant. This result can be further strengthened, which is most easily done by reformulating the problem. Let N /C211 be an integer and P1 ; P2 ; ..., PNbe N (not necessarily distinct) points in the square 0 5x 51; 0 5y 51: Then g1 0 g1 0Sx;yðÞ/C28Nxy ½/C1382dxdy > c2 lnN ; where c2is a positive constant and Su;vðÞ is the number of points in the rectangle 0 5x Bu; 0 5y Bv (Roth 1954). Therefore, Sx;yðÞ/C28Nxy jj > c3ffiffiffiffiffiffiffiffiffi lnNp ; and the original result can be stated as the fact that there exist n and a such that DnaðÞ/C28na jj >c4ffiffiffiffiffiffiffiffiffi lnNp : The randomly distributed points shown in the above squares have Sx;yðÞ/C28Nxy jj2/C306:40 and 9.11, respec- tively. Similarly, the discrepancy of a set of Npoints in a unit d-HYPERCUBE satisfies Sx;yðÞ/C28Nxy jj >clnNðÞd/C281 ðÞ =2 (Roth 1954, 1976, 1979, 1980). See also 18-POINT PROBLEM ,CUBE POINT PICKING References Berlekamp, E. R. and Graham, R. L. "Irregularities in the Distributions of Finite Sequences." J. Number Th. 2, 152/C1/ 61, 1970. Roth, K. F. "On Irregularities of Distribution." Mathematika 1,7 3/C1/9, 1954. Roth, K. F. "On Irregularities of Distribution. II." Comm. Pure Appl. Math. 29, 739/C1/44, 1976. Roth, K. F. "On Irregularities of Distribution. III." Acta Arith. 35, 373/C1/84, 1979. Roth, K. F. "On Irregularities of Distribution. IV." Acta Arith. 37,6 7/C1/5, 1980. van Aardenne-Ehrenfest, T. "Proof of the Impossibility of a Just Distribution of an Infinite Sequence Over an Inter- val." Proc. Kon. Ned. Akad. Wetensch. 48,3/C1/, 1945. van Aardenne-Ehrenfest, T. Proc. Kon. Ned. Akad. We- tensch. 52, 734/C1/39, 1949. van der Corput, J. G. Proc. Kon. Ned. Akad. Wetensch. 38, 813/C1/21, 1935a. van der Corput, J. G. Proc. Kon. Ned. Akad. Wetensch. 38, 1058/C1/066, 1935b. Discrete Distribution A STATISTICAL DISTRIBUTION whose variables can take on only discrete values. Abramowitz and Stegun (1972, p. 929) give a table of the parameters of most common discrete distributions. See also BERNOULLI DISTRIBUTION ,B INOMIAL DIS- TRIBUTION ,CONTINUOUS DISTRIBUTION ,G EOMETRIC DISTRIBUTION ,H YPERGEOMETRIC DISTRIBUTION ,NE- GATIVE BINOMIAL DISTRIBUTION ,POISSON DISTRIBU- TION ,P ROBABILITY ,S TATISTICAL DISTRIBUTION , STATISTICS ,UNIFORM DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 927 and 929, 1972. Evans, M.; Hastings, N.; and Peacock, B. Statistical Dis- tributions, 3rd ed. New York: Wiley, 2000. McLaughlin, M. "Common Probability Distributions." http:// www.geocities.com/~mikemclaughlin/math_stat/Dists/ Compendium.html. Wilmmer, G. and Altmann, G. Thesaurus of Univariate Discrete Probability Distributions. Essen, Germany: STAMM, 1999. Discrete Fourier Transform The FOURIER TRANSFORM is defined as f nðÞ/C30F ftðÞ½/C138/C30g/C12 /C28/C12ftðÞe/C282 pintdt: (1) Now consider generalization to the case of a discrete function, ftðÞ0 ftkðÞby letting fk /C13ftkðÞ; where tk /C13 kD; with k /C300, ..., N /C281 : Choose the frequency step such that nn /C30n N D; (2) with n /C30/C28N =2; ..., 0, ..., N =2: There are N /C271 values of n, so there is one relationship between the frequency components. Writing this out as per Press et al. (1989) F f(t)½/C138/C30XN /C281 k /C300fke /C282 pin=N D ðÞ k DD/C30DXN /C281 k /C300fke /C282 pink=N ; (3) and Fn /C13XN /C281 k /C300fke /C282 pink=N : (4) The inverse transform is fk /C301 NXN /C281 n/C300Fne2 pink =N : (5) Note that F/C28n /C30FN /C28n ; n /C301, 2, ..., so an alternate formulation isnn /C30n N D; (6) where the NEGATIVE frequencies /C28nc B n B0 have N =2 /C271 5n 5N /C281; POSITIVE frequencies 0 B n B nc have 1 5n 5N =2 /C281 ; with zero frequency n /C300. n /C30 N =2 corresponds to both n /C30 ncand n /C30/C28nc : The discrete Fourier transform can be computed using a FAST FOURIER TRANSFORM . The discrete Fourier transform is a special case of the Z-TRANSFORM . It can be computed for a list l of COMPLEX NUMBERS using the Mathematica command Fourier [l]. The above plot shows the 2-D discrete Fourier trans- form of the reciprocals of the greatest common divisor GCD (i ;j) for i ;j /C23 1 ;512½/C138 (Trott 2000). See also FAST FOURIER TRANSFORM ,FOURIER TRANS- FORM ,HARTLEY TRANSFORM ,WINOGRAD TRANSFORM , Z-TRANSFORM References Arfken, G. "Discrete Orthogonality--Discrete Fourier Trans- form." §14.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 787 /C1/92, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Fourier Transform of Discretely Sampled Data." §12.1 in Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, pp. 494 /C1/98, 1989. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Discrete Geometry See also COMPUTATIONAL GEOMETRY References Goodman, J. E. and O’Rourke, J. Handbook of Discrete and Computational Geometry. Boca Raton, FL: CRC Press, 1997. Discrete Group See also CONTINUOUS GROUP ,FINITE GROUP Discrete Logarithm MULTIPLICATIVE ORDER Discrete Mathematics The branch of mathematics dealing with objects which can assume only certain "discrete" values. Discrete objects can be characterized by INTEGERS , whereas continuous objects require REAL NUMBERS . The study of how discrete objects combine with one another and the probabilities of various outcomes is known as COMBINATORICS . See also COMBINATORICS ,D ISCRETE DISTRIBUTION , DISCRETE FOURIER TRANSFORM ,D ISCRETE GEOME- TRY,DISCRETE LOGARITHM References Balakrishnan, V. K. Introductory Discrete Mathematics. New York: Dover, 1997. Bobrow, L. S. and Arbib, M. A. Discrete Mathematics: Applied Algebra for Computer and Information Science. Philadelphia, PA: Saunders, 1974. Dossey, J. A.; Otto, A. D.; Spence, L.; and Eynden, C. V. Discrete Mathematics, 3rd ed. Reading, MA: Addison- Wesley, 1997. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Hall, C. and O’Donnell, J. Discrete Mathematics Using a Computer. London: Springer-Verlag, 2000. Lipschutz, S. and Lipson, M. L. 2000 Solved Problems in Discrete Mathematics. New York: McGraw-Hill, 1991. Lipschutz, S. and Lipson, M. L. Schaum’s Outline of Discrete Mathematics, 2nd ed. New York: McGraw-Hill, 1997. Rosenstein, J. G.; Franzblau, D. S.; and Roberts, F. S. Discrete Mathematics in the Schools. Providence, RI: Amer. Math. Soc., 1997. Skiena, S. Implementing Discrete Mathematics. Reading, MA: Addison-Wesley, 1990. Weisstein, E. W. "Books about Discrete Mathematics." http://www.treasure-troves.com/books/DiscreteMathema- tics.html. Discrete Set A set S is discrete in a larger TOPOLOGICAL SPACE X if every point x /C23 S has a NEIGHBORHOOD U such that S S U /C30 xfg:: The points of S are then said to be ISOLATED (Krantz 1999, p. 63). Typically, a discrete set is either finite or COUNTABLY INFINITE . For example, the set of integers is discrete on the REAL LINE. Another example of an infinite discrete set is the set 1=n for all integers n > 1 fg : On any reason- able space, a finite set is discrete. A set is discrete if it has the DISCRETE TOPOLOGY , that is, if every subset is open. In the case of a subset S, as in the examples above, one uses the RELATIVE TOPOLOGY on S. Sometimes a discrete set is also closed. Then there cannot be any ACCUMULATION POINTS of a discrete set. On a COM- PACT SET such as the SPHERE , a closed discrete set must be finite because of this. See also ACCUMULATION POINT ,C OMPACT SPACE , DISCRETE TOPOLOGY ,ISOLATED POINT ,N EIGHBOR- HOOD ,TOPOLOGICAL SPACEReferences Krantz, S. G. "Discrete Sets and Isolated Points." §4.6.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 63 /C1/4, 1999. Discrete Topology A topology is given by a collection of subsets of a TOPOLOGICAL SPACE X. The smallest topology has two OPEN SETS , f and X. The largest topology contains all subsets as open sets, and is called the discrete topology. In particular, every point in X is an OPEN SET in the discrete topology. See also DISCRETE MATHEMATICS ,D ISCRETE SET, TOPOLOGICAL SPACE Discrete Uniform Distribution EQUALLY LIKELY OUTCOMES DISTRIBUTION DiscreteDelta KRONECKER DELTA Discriminant A discriminant is a quantity (usually invariant under certain classes of transformations) which charac- terizes certain properties of a quantity’s ROOTS . The concept of the discriminant is used for BINARY QUAD- RATIC FORMS , ELLIPTIC CURVES , METRICS , MODULES , POLYNOMIALS , QUADRATIC CURVES , QUADRATIC FIELDS , QUADRATIC FORMS , and in the SECOND DERI- VATIVE TEST . See also DISCRIMINANT (BINARY QUADRATIC FORM), DISCRIMINANT (CIRCLE ), DISCRIMINANT (CONIC SEC- TION ), DISCRIMINANT (ELLIPTIC CURVE ), DISCRIMI- NANT (METRIC ), MODULAR DISCRIMINANT , DISCRIMINANT (MODULE ), DISCRIMINANT (POLYNO- MIAL ), DISCRIMINANT (QUADRATIC CURVE ), DISCRIMI- NANT (SECOND DERIVATIVE TEST) Discriminant (Binary Quadratic Form) The discriminant of a BINARY QUADRATIC FORM au2 /C27buv /C27cv2 is defined by d /C13b2 /C284ac: It is equal to four times the corresponding DETERMI- NANT . See also CLASS NUMBER Discriminant (Circle) In H OMOGENEOUS COORDINATES (x1;x2;x3);the equa- tion of a CIRCLE Cis a(x2 1/C27x22)/C272fx2x3/C272gx1x3/C27cx23/C300: The discriminant of this circle is defined as D/C30a 0 g 0 af gf c0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30a(ac /C28f 2 /C28g2); and the quadratic form q(C) /C30ac /C28f2 /C28g2 is the basic invariant. See also DISCRIMINANT (CONIC SECTION ) References Barth, W. and Bauer, T. "Poncelet Theorems." Expos. Math. 14, 125 /C1/44, 1996. Discriminant (Conic Section) The discriminant of the general CONIC SECTION ax2 1 /C27bx22 /C27cx23 /C272fx2x3 /C272gx1x3 /C272hx1x2 /C300 is defined as D/C30ahg hbf gfc0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1/C30abc /C272fgh /C28af 2 /C28bg2 /C28ch2 : If b /C30a and g /C30h /C300; then simplifies to the DISCRI- MINANT of a CIRCLE . See also DISCRIMINANT (CIRCLE ) References Salmon, G. Conic Sections, 6th ed. New York: Chelsea, p. 266, 1960. Discriminant (Elliptic Curve) An ELLIPTIC CURVE is the set of solutions to an equation of the form y2 /C27a1xy /C27a3y /C30x3 /C27a2x2 /C27a4x /C27a6 : (1) By changing variables, y 0 2y /C27a1x /C27a3 ; assuming the CHARACTERISTIC is not 2, the equation becomes y2 /C304x3 /C27b2x2 /C272b4x /C27b6 (2) where b2 /C30a2 1 /C274a2 (3) b4 /C302a4 /C27a1a3 (4) b6 /C30a23 /C274a6 : (5) Define also the quantity b8 /C30a21a6 /C274a2a6 /C28a1a3a4 /C27a2a23 /C28a24 ; (6) then the discriminant is given by D/C30/C28b22b8 /C288b34 /C2827b26 /C279b2b4b6 : (7) The discriminant depends on the choice of equations, and can change after a change of variables, unlike the J-INVARIANT . If the CHARACTERISTIC of the FIELD is neither 2 or 3, then its equation can be written asy2 /C30x3 /C27Ax /C27B ; (8) in which case, the discriminant is given by D/C30/C2816(4A3 /C2727B2) : (9) Algebraically, the discriminant is nonzero when the right-hand side has three distinct roots. In the classical case of an ELLIPTIC CURVE over the COMPLEX NUMBERS , the discriminant has a geometric interpre- tation. If D"0 ; then the elliptic curve is nonsingular and has GENUS 1, i.e., it is a TORUS .If D/C300 and A /C300, then it has a CUSP singularity, in which case there is one tangent direction at the singularity. If D/C300 and A "0 ; then its singularity is called an ORDINARY DOUBLE POINT (or node), in which case the singularity has two distinct tangent directions. Note that the discriminant of an ELLIPTIC CURVE is not the same as the DISCRIMINANT of the correspond- ing polynomial, but the two kinds of discriminants vanish for the same values of AandB. See also ALGEBRAIC GEOMETRY ,E LLIPTIC CURVE , FREY CURVE ,ISOGENY , J -INVARIANT ,L EGENDRE FORM,MINIMAL DISCRIMINANT ,W EIERSTRASS FORM References Silverman, J. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1986. Discriminant (Elliptic Function) If /y2¼4x3þb2x2þ2b4xþb6/and b2are the INVAR- IANTS of a W EIERSTRASS ELLIPTIC FUNCTION a2 1/C274a2 with periods b4and /2a4þa1a3/, then the discriminant is defined by b6 (1) Letting a23/C274a6:;then b8/C30a2 1a6/C274a2a6/C28a1a3a4/C27a2a23/C28a24;r>1D /C30/C28b22b8/C288b34/C2827b26/C279b2b4b6: /C30y2/C30x3/C27Ax/C27B; (2) ¼D¼/C2816ð4A3þ27B2ð3Þ The F OURIER SERIES of for D"0;where His the UPPER HALF-PLANE ,i s A ¼ 0 ð4Þ where A "0; is the TAU FUNCTION , and A "0; are integers (Apostol 1997, p. 20). The discriminant can also be expressed in terms of the DEDEKIND ETA FUNCTION ga b by g /C13det(ga; b) /C30 g11g12 g21g220C@10C@10C@10C@10C@10C@10C@10C@1/C30g11g22 /C28 g12ðÞ2: (5) (Apostol 1997, p. 51). See also DEDEKIND ETA FUNCTION ,INVARIANT (EL- LIPTIC FUNCTION ), KLEIN’S ABSOLUTE INVARIANT ,TAU FUNCTION ,W EIERSTRASS ELLIPTIC FUNCTION References Apostol, T. M. "The Discriminant ¯g/" and "The Fourier Expansions of and ¯g/C30D2g::/"§1.11 and 1.15 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 14 and 20 /C1/2, 1997. Discriminant (Metric) Given a METRIC gab;the discriminant is defined by g/C13det(ga;b)/C30g11g12 g21g220C@10C@10C@10C@10C@10C@10C@10C@1/C30g 11g22/C28g12ðÞ2: (1) Let gbe the discriminant and ¯gthe transformed discriminant, then ¯g/C30D2g (2) g/C30¯D2¯g; (3) where D/C13@u1;u2ðÞ @¯u1;¯u2 ðÞ/C30@u1 @u1@u1 @u2 @u2 @u1@u2 @u20C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (4) ¯D/C13 @¯u1;¯u2ðÞ @u1;u2 ðÞ/C30@¯u1 @u1@¯u1 @u2 @¯u2 @u1@¯u2 @u20C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1: (5) Discriminant (Module) Let a MODULE Min an INTEGRAL DOMAIN D1for R(ffiffiffiffiffiffi D)p be expressed using a two-element basis as M/C30[j1;j2]; where j1andj2are in D1:Then the DIFFERENT of the MODULE is defined as D/C30D(M)/C30j1j2 j?2j?20C@10C@10C@10C@10C@10C@10C@10C@1/C30j 1j?2/C28j?1j2and the discriminant is defined as the square of the DIFFERENT (Cohn 1980). For IMAGINARY QUADRATIC FIELDS QffiffiffinpðÞ (with nB0), the discriminants are given in the following table. /C281 //C2822/ /C2833 //C2822/C2153/C21511//C2867/C2867 /C282 //C2823/ /C2834 //C2823/C21517//C2869 //C2822/C2153/C21523/ /C283/C283 /C2835 //C285/C2157/ /C2870 //C2823/C2155/C2157/ /C285 //C2822/C2155//C2837 //C2822/C21537//C2871/C2871 /C286 //C2823/C2153//C2839 //C283/C21513//C2873 //C2822/C21573/ /C287/C287 /C2841 //C2822/C21541//C2874 //C2823/C21537/ /C2810 //C2823/C2155//C2842 //C2823/C2153/C2157//C2877 //C2822/C2157/C21511/ /C2811/C2811 /C2843/C2843 /C2878 //C2823/C2153/C21513/ /C2813 //C2822/C21513//C2846 //C2823/C21523//C2879/C2879 /C2814 //C2823/C2157//C2847/C2847 /C2882 //C2823/C21541/ /C2815 //C283/C2155//C2851 //C283/C21517//C2883/C2883 /C2817 //C2822/C21517//C2853 //C2822/C21553//C2885 //C2822/C2155/C21517/ /C2819/C2819 /C2855 //C285/C21511//C2886 //C2823/C21543/ /C2821 //C2822/C2153/C2157//C2857 //C2822/C2153/C21519//C2887 //C283/C21529/ /C2822 //C2823/C21511//C2858 //C2823/C21529//C2889 //C2822/C21589/ /C2823/C2823 /C2859/C2859 /C2891 //C287/C21513/ /C2826 //C2823/C21513//C2861 //C2822/C21561//C2893 //C2822/C2153/C21531/ /C2829 //C2822/C21529//C2862 //C2823/C21531//C2894 //C2823/C21547/ /C2830 //C2823/C2153/C2155//C2865 //C2822/C2155/C21513//C2895 //C285/C21519/ /C2831/C2831 /C2866 //C2823/C2153/C21511//C2897 //C2822/C21597/ The discriminants of REAL QUADRATIC FIELDS QffiffiffinpðÞ (n/C210) are given in the following table. 22 334 /23/C21517/67 /67 /C21522/ 3 /3/C21522 /35 /7/C21522/C2155/69 /3/C21523/ 5 5 37 37 70 /7/C21523/C2155/ 6 /3/C21523/38 /19 /C21523/71 /71 /C21522/ 7 /7/C21522/39 /3/C21522/C21513/73 73 10 /23/C2155/41 41 74 /23/C21537/ 11 /11 /C21522 /42 /3/C21523/C2157/77 /7/C21511/ 13 13 43 /43 /C21522/78 /3/C21523/C21513/ 14 /7/C21523/46 /23 /C21523/79 /79 /C21522/ 15 /3/C21522/C2155/47 /47 /C21522 /82 /23/C21541/ 17 17 51 /3/C21522/C21517/83 /83 /C21522 / 19 /19 /C21522/53 53 85 /5/C21517/ 21 /3 /C2157/ 55 /11 /C21522 /C2155/ 86 /43 /C21523/ 22 /11 /C21523/ 57 /3 /C21519/ 87 /3 /C21522 /C21513/ 23 23 /C2152258 /23 /C21529/ 89 89 26 /23 /C21513/ 59 /59 /C21522 / 91 /7 /C21522 /C21513/ 29 29 61 61 93 /3 /C21531/ 30 /3 /C21523 /C2155/ 62 /31 /C21523/ 94 /47 /C21523/ 31 /31 /C21522 / 65 /5 /C21513/ 95 /19 /C21522 /C2155/ 33 /3 /C21511/ 66 /3 /C21523 /C21511/ 97 97 See also DIFFERENT ,FUNDAMENTAL DISCRIMINANT , MODULE References Cohn, H. Advanced Number Theory. New York: Dover, pp. 72 /C1/3 and 261 /C1/74, 1980. Discriminant (Polynomial) The PRODUCT of the SQUARES of the differences of the POLYNOMIAL ROOTS ri : The discriminant of a poly- nomial is only defined up to sign. For a POLYNOMIAL anzn /C27an/C281zn/C281 /C27/C1/C1/C1/C27a1z /C27a0 /C300 (1) of degree n, Dn /C30Yn i;j iBj(ri /C28rj)2 : (2) It is also common to consider discriminants D?nfor an /C131 or discriminants D??nobtained from Dnby multiplying by a2(n/C281) n : If desired, powers ancan be inserted mentally so that each term is of degree 2(n /C28 1) and the whole expression is divided by a2(n/C281) n ::/ The discriminant is closely related to RESULTANTS and can be implemented in Mathematica as Discriminant[p_?PolynomialQ,x_] : /C30 With[{n /C30 Exponent[p,x]}, Cancel[ ((-1)^(n(n-1)/2)Resultant[p,D[p,x],x])/ Coefficient[p,x,n]^(2n-1) ] ] The discriminant of the QUADRATIC EQUATION a2z2 /C27a1z /C27a0 /C300 (3) is given by D2 /C30a2 1 /C28 4a0a2 a2 2: (4) The discriminant of the CUBIC EQUATION a3z3 /C27a2z2 /C27a1z /C27a0 /C300 (5)is given by D3 /C30a2 1a22 /C28 4a0a32 /C28 4a31a3 /C27 18a0a1a2a3 /C28 27a20a23 a4 3 (6) The discriminant of a QUARTIC EQUATION z4 /C27a3z3 /C27a2z2 /C27a1z /C27a0 /C300 (7) is D4 /C301 a6 4(a2 1a22a23 /C284a31a32 /C284a21a32a40C1 þ18a31a2a3a4/C2827a41a24þ256a30a34Þ /C27a0(/C284a32a33/C2718a1a2a33/C2716a42a4 /C2880a1a22a3a4/C286a21a23a4/C27144a21a2a24) /C27a20(/C2827a43/C27144a2a23a4/C28128a22a24/C28192a1a3a24)] (Beeler et al. 1972, Item 4). See also CUBIC EQUATION ,N EWTON’S RELATIONS , POLYNOMIAL ,QUADRATIC EQUATION ,QUARTIC EQUA- TION ,RESULTANT ,SUBRESULTANT References Schroeppel, R. Item 4 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 4, Feb. 1972. Discriminant (Quadratic Curve) Given a general QUADRATIC CURVE Ax2/C27Bxy/C27Cy2/C27Dx/C27Ey/C27F/C300; (1) the quantity Xis known as the discriminant, where X/C13B2/C284AC; (2) and is invariant under ROTATION . Using the COEFFI- CIENTS from QUADRATIC EQUATIONS for a rotation by an angle u; A?/C301 2A1/C27cos(2 u) ½/C138 /C2712B sin(2 u)/C2712C1/C28cos(2 u) ½/C138 A/C27C 2/C27B 2sin(2 u)/C27A/C28C 2cos(2 u) (3) B?/C30Gcos 2 u/C27d/C28p 2 ! /C30Gsin(2u/C27d) (4) C?/C3012A1/C28cos(2 u) ½/C138 /C2812Bsin 2 u/C2712 ! C1/C27cos(2 u) ½/C138 /C30A/C27C 2/C28B 2sin 2 uðÞ/C27C/C28A 2cos 2 uðÞ : (5) Now let G /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 /C27 A /C28C ðÞ2q (6) d /C13tan /C281 B C /C28 A ! (7) d2 /C13tan/C281A /C28 C B ! /C30/C28cot /C281 B C /C28 A ! ; (8) and use cot/C281(x) /C131 2 p /C28tan/C281(x) (9) d2 /C30 d /C2812 p (10) to rewrite the primed variables A?/C30A /C27 C 2/C2712G cos(2 u /C27 d) (11) B ?/C30B cos(2 u) /C27(C /C28A) sin(2u) /C30G(2u /C27 d 2) ð12Þ C?/C30A /C27 C 2/C2812 G cos(2 u /C27 d) : (13) From (11) and (13), it follows that 4A?C ?/C30(A /C27C) 2 /C28G2 cos(2 u /C27 d) : (14) Combining with (12) yields, for an arbitrary u X /C13B ?2 /C284A?C? /C30G2 sin2(2u /C27 d) /C27G2 cos2(2u /C27 d) /C28(A /C27C)2 /C30G2 /C28(A /C27C)2 /C30B2 /C27(A /C28C)2 /C28(A /C27C)2 /C30B2 /C284AC ; (15) which is therefore invariant under rotation. This invariant therefore provides a useful shortcut to determining the shape represented by a QUADRATIC CURVE . Choosing u to make B?/C300 (see QUADRATIC EQUATION ), the curve takes on the form A?x2 /C27C ?y2 /C27D ?x /C27E ?y /C27F /C300 : (16) COMPLETING THE SQUARE and defining new variables gives A?x?2 /C27C ?y ?2 /C30H : (17) Without loss of generality, take the sign of H to be positive. The discriminant is X /C30B?2 /C274A?C ?/C30/C28 4A?C ?: (18) Now, if /C284A?C ?B0 ; then A? and C ? both have the same sign, and the equation has the general form of an ELLIPSE (if A? and B? are positive). If /C284A?C?> 0; then A? and C ? have opposite signs, and the equation has the general form of a HYPERBOLA .If/C284A?C ?/C300 ; theneither A? or C? is zero, and the equation has the general form of a PARABOLA (if the NONZERO A? or C? is positive). Since the discriminant is invariant, these conclusions will also hold for an arbitrary choice of u; so they also hold when /C284A?C? is replaced by the original B2 /C284AC: The general result is 1. If B2 /C284AC B0 ; the equation represents an ELLIPSE ,a CIRCLE (degenerate ELLIPSE ), a POINT (degenerate CIRCLE ), or has no graph. 2. If B2 /C284AC > 0; the equation represents a HYPERBOLA or pair of intersecting lines (degener- ate HYPERBOLA ). 3. If B2/C284AC/C300;the equation represents a PARABOLA ,a LINE (degenerate PARABOLA ), a pair ofPARALLEL lines (degenerate PARABOLA ), or has no graph. Discriminant (Quadratic Form) DISCRIMINANT (BINARY QUADRATIC FORM) Discriminant (Second Derivative Test) D/C13fxxfyy/C28fxyfyx/C30fxxfyy/C28f2 xy; where fijare PARTIAL DERIVATIVES . See also SECOND DERIVATIVE TEST Disdyakis Dodecahedron The DUAL POLYHEDRON of the Archimedean GREAT RHOMBICUBOCTAHEDRON A3and Wenninger dual W15; also called the HEXAKIS OCTAHEDRON . If the original GREAT RHOMBICUBOCTAHEDRON has unit side lengths, then the resulting dual has edge lengths s1 /C302 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 /C283ffiffiffi 2pq (1) s2 /C303 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi62/C27ffiffiffi 2p0C@n0C@or (2) s 3 /C302 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi610/C27ffiffiffi 2p0C@n0C@or : (3) The INRADIUS is r /C303ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 9715 /C278ffiffiffi2p0C@n0C@os : (4) Scaling the disdyakis dodecahedron so that s 1 /C301 gives a solid with SURFACE AREA and VOLUME S /C306 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 783 /C27436ffiffiffi 2pq (5) V /C301 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2194 /C271513ffiffiffi 2p 0C@n0C@or : (6) See also A RCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , GREAT DISDYAKIS DODECAHEDRON ,O CTATETRAHE- DRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 25 /C1/6, 1983. Disdyakis Triacontahedron The DUAL POLYHEDRON of the Archimedean GREAT RHOMBICOSIDODECAHEDRON A2and Wenninger dual W16 : It is also called the HEXAKIS ICOSAHEDRON . See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLIDReferences Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 25 and 27, 1983. Disjoint Sets Two SETS A1 and A2 are disjoint if their INTERSECTION A1 S A2 /C13Ø ; where Ø is the EMPTY SET. n sets A1 ; A2 ; ..., An are disjoint if Ai S Aj /C13Ø for i "j : For example, A;B ;C fg and D ;Efg are disjoint, but A;B ;C fg and C ;D; E fg are not. Disjoint sets are also said to be mutually exclusive or independent. See also EMPTY SET,INDEPENDENT SET,INTERSEC- TION ,SET Disjoint Union The disjoint union of two SETS A and B is a BINARY OPERATOR that combines all distinct elements of a pair of given sets, while retaining the original set membership as a distinguishing characteristic of the union set. The disjoint union is denoted A @+ B /C30 A /C29 0fg ðÞ @ B /C29 1fg ðÞ /C13A+@ B+; where /A /C29S/ is a SET DIRECT PRODUCT . For example, the disjoint union of sets /A ¼f1; 2;3; 4;5g/ and /B ¼f1 ;2;3 ;4;5 g/ can be computed by finding A+/C30 1;0ðÞ ; 2;0ðÞ ; 3;0ðÞ ; 4 ;0ðÞ ; 5 ;0ðÞ fg B +/C30 1;1ðÞ ; 2 ;1ðÞ ; 3 ;1ðÞ ; 4; 1ðÞ fg ; so A @+ B /C30A+@ B + ¼fð1;0 Þ;ð2;0Þ;ð3;0Þ;ð4 ;0Þ;ð5 ;0Þ; ð1;1 Þ;ð2;1Þ;ð3 ;1Þ;ð4 ;1Þg See also UNION References Armstrong, M. A. Basic Topology, rev. ed. New York: Springer-Verlag, 1997. Disjunction The term in logic used to describe the operation commonly known as OR. See also CONJUNCTION ,DISJUNCTIVE NORMAL FORM, DISJUNCTIVE SYLLOGISM ,OR Disjunctive Game NIM-HEAP Disjunctive Normal Form A statement is in disjunctive normal form if it is a DISJUNCTION (sequence of ORs) consisting of one or more disjuncts, each of which is a CONJUNCTION (AND) of one or more statement letters and negations of statement letters. Examples of disjunctive normal forms include A (1) A fflB ðÞ/C150 !A fflC ðÞ (2) A fflB ffl!A ðÞ /C150 C ffl!B ðÞ /C150 A ffl!C ðÞ (3) A fflB ðÞ (4) A /C150 B fflC ðÞ ; (5) where /C150 denotes OR, ffl denotes AND, and ! denotes NOT. Every statement in logic consisting of a combi- nation of multiple ffl;/C150; and !/s can be written in conjunctive normal form. See also CONJUNCTIVE NORMAL FORM References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 27, 1997. Disk An n-D disk (or DISC)of RADIUS r is the collection of points of distance 5r (CLOSED DISK)orBr (OPEN DISK) from a fixed point in EUCLIDEAN n-space. A disk is the SHADOW of a BALL on a PLANE PERPENDICULAR to the BALL -RADIANT POINT line. The n-disk for n ]3 is called a BALL , and the boundary of the n-disk is a (n /C281)/-HYPERSPHERE . The standard n-disk, denoted Dn (or Bn) ; has its center at the ORIGIN and has RADIUS r /C301. See also BALL,CLOSED DISK,DISK COVERING PRO- BLEM ,FIVE DISKS PROBLEM ,H YPERSPHERE ,LOWER HALF-DISK,M ERGELYAN- WESLER THEOREM ,O PEN DISK,POLYDISK ,SPHERE ,U NIT DISK,U PPER HALF- DISK Disk Algebra This entry contributed by RONALD M. AARTS A disk algebra is an ALGEBRA of functions which are analytic on the OPEN UNIT DISK in C and continuous up to the boundary. A representative measure for a point x in the CLOSED DISK is a nonnegative MEASURE m such that Int(fdm ) /C30f(x) for all f in A. These measures form a COMPACT ,CONVEX SET Mxin the linear space of all measures. See also ALGEBRADisk Covering Problem N.B. A detailed online essay by S. Finch was the starting point for this entry. Given a UNIT DISK , find the smallest RADIUS rnðÞ required for nequal disks to completely cover the UNIT DISK . For a symmetrical arrangement with n/C305 (the FIVE DISKS PROBLEM ),r5ðÞ/C30f/C281/C301=f/C30 0:6180340 . . . ;where fis the GOLDEN RATIO . How- ever, the radius can be reduced in the general disk covering problem where symmetry is not required.The first few such values are r(1)/C301 r(2)/C301 r(3)/C301 2ffiffiffi 3p r(4)/C301 2ffiffiffi 2p r(5)/C300:609382864 . . . r(6)/C300:555 r(7)/C301 2 r(8)/C300:437 r(9)/C300:422 r(10)/C300:398: Here, values for n/C306, 8, 9, 10 were obtained using computer experimentation by Zahn (1962). The value r(5) is equal to cos( u/C27f=2);where uand fare solutions to 2 sin u/C28sinuþ1 2fþc ! /C28sinc/C28u/C2812f ! ¼0 (1) 2 sinf/C28sinu/C2712f/C27x ! /C28sinx/C28u/C2812f ! /C300 (2) 2 sinu/C27sin(x/C27u)/C28sin(x/C28u)/C28sin(c/C27f) /C28sin(c/C28f)/C282 sin( c/C282u)/C300 (3) cos(2 c/C28x/C27f)/C28cos(2 c/C27x/C28f)/C282 cosx /C27cos(2 c/C27x/C282u)/C27cos(2 c/C28x/C282u)/C300 (4) (Neville 1915). It is also given by 1 =x;where xis the largest real root of a(y)x 6/C28b(y)x5/C27c(y)x4/C28d(y)x3/C27e(y)x2/C28f(y)x/C27g(y) /C300 (5) maximized over all y, subject to the constraints ffiffiffi 2p BxB2y/C271 (6) /C281 By B1; (7) and with a(y) /C3080y2 /C2764y (8) b(y) /C30416y3 /C27384y2 /C2764y (9) c(y) /C30848y4 /C27928y3 /C27352y2 /C2732y (10) d(y) /C30768y5 /C27992y4 /C27736y3 /C27288y2 /C2796y e(y) /C30256y6 /C27384y5 /C27592y4 /C27480y3 /C27336y2 /C2796y /C2716 (11) f(y) /C30128y5 /C27192y4 /C27256y3 /C27160y2 /C2796y /C2732 ð12Þ g(y) /C3064y2 /C2764y /C2716 (13) (Bezdek 1983, 1984). Letting N(o) be the smallest number of DISKS of RADIUS o needed to cover a disk D, the limit of the ratio of the AREA of D to the AREA of the disks is given by lim o 0 0/C271 o2N( o) /C303ffiffiffi 3p 2p (14) (Kershner 1939, Verblunsky 1949). See also CIRCLE COVERING ,FIVE DISKS PROBLEM References Ball, W. W. R. and Coxeter, H. S. M. "The Five-Disc Pro- blem." In Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 97 /C1/9, 1987. Bezdek, K. "Uuml;ber einige Kreisu ¨berdeckungen." Beitra ¨ge Algebra Geom. 14,7/C1/3, 1983. Bezdek, K. "U¨ ber einige optimale Konfigurationen von Kreisen." Ann. Univ. Sci. Budapest Eotvos Sect. Math. 27, 141 /C1/51, 1984. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/circle/circle.html. Kershner, R. "The Number of Circles Covering a Set." Amer. J. Math. 61, 665 /C1/71, 1939. Neville, E. H. "On the Solution of Numerical Functional Equations, Illustrated by an Account of a Popular Puzzle and of its Solution." Proc. London Math. Soc. 14, 308 /C1/26, 1915. Verblunsky, S. "On the Least Number of Unit Circles which Can Cover a Square." J. London Math. Soc. 24, 164 /C1/70, 1949. Zahn, C. T. "Black Box Maximization of Circular Coverage." J. Res. Nat. Bur. Stand. B 66, 181 /C1/16, 1962. Disk Lattice Points GAUSS’S CIRCLE PROBLEMDisk Line Picking Using DISK POINT PICKING , x /C30ffiffiffirpcosu (1) y /C30ffiffiffirpsinu (2) for r /C23 0 ;1½/C138 ; u /C23 0 ;2p ½Þ ; choose two points at random in a UNIT DISK and find the distribution of distances s between the two points. Without loss of generality, take the first point as (r; u) /C30(r1 ; 0) and the second point as (r2 ; u) : Then> ¯s /C30ng1 0 g1 0 g2 p 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 þ r2 /C28 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1r2cos upq dr1dr2du g1 0 g1 0 g2p 0dr1dr2du(3) /C30128 45p(4) (Uspensky 1937, p. 258). This is a special case of BALL LINE PICKING with n/C302, so the full probability function for a disk of radius Ris P2(s)/C304s pR2cos/C281s 2R ! /C282s2 pR3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28s2 4R2s (5) (Solomon 1978, p. 129). See also BALL LINE PICKING ,CIRCLE LINE PICKING References Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978. Uspensky, J. V. Ch. 12, Problem 5 in Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 257 /C1/58, 1937. Disk Packing CIRCLE PACKING Disk Point Picking To generate random points over the UNIT DISK,itis incorrect to use two uniformly distributed variables r /C23 0;1½/C138 ; and u /C23 0 ;2p ½Þ ; and then take x /C30r cosu (1) y /C30r sinu : (2) Because the area element is given by dA /C302prdr ; (3) this gives a concentration of points in the center (left figure above). The correct transformation is instead given by x/C30ffiffiffirpcosu (4) y/C30ffiffiffirpsinu (5) (right figure above). See also CIRCLE POINT PICKING ,DISK LINE PICKING , POINT PICKING ,SPHERE POINT PICKING Disk Triangle Picking Pick three points P/C30(x1;y1);Q/C30(x2;y2);and R/C30 (x3;y3) distributed independently and uniformly in a UNIT DISK K. Then the average area of the TRIANGLE determined by these points is¯A/C30ggP/C23KggQ/C23KggR/C23K1 2x1y11 x2y21 x3y310C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1dy 3dy3dy1dx3dx2dx1 ggP/C23KggQ/C23KggR/C23Kdy3dy3dy1dx3dx2dx1 (1) which can be evaluated using C ROFTON’S FORMULA and polar coordinates to yield ¯A/C3035=(48p2) (Wool- house 1967; Solomon 1987; Pfiefer 1989). This pro- blem is very closely related to S YLVESTER’S FOUR- POINT PROBLEM , and can be derived as the limit as n0/C12of the general POLYGON TRIANGLE PICKING problem. The probability P2that three random points in a disk form an ACUTE TRIANGLE is P2/C304 p2/C281 8(2) (Woolhouse 1886). The problem was generalized by Hall (1982) to n-D BALL TRIANGLE PICKING , and Buchta (1986) gave closed form evaluations for Hall’sintegrals. Let the VERTICES of a triangle in n-D be NORMAL (GAUSSIAN ) variates. The probability that a Gaussian triangle in n-D is OBTUSE is Pn/C303G(n) G21 2n !g1=3 0x(n/C282)=2 (1/C27x)ndx /C303G(n) G212n ! 2n/C281gp=3 0sinn/C281udu /C306G(n)2F112n;n;1/C2712n;/C2813 ! 3 n=2nG212n ! ; (3) where G(n) is the GAMMA FUNCTION and2F1(a;b;c;x) is the HYPERGEOMETRIC FUNCTION . For EVEN n/C132k; P2k/C303X2k/C281 j/C30k2k/C281 j0C@80C@91 4 !j34 ! 2k/C281/C28j (4) (Eisenberg and Sullivan 1996). The first few cases are explicitly P2/C303 4/C300:75 (5) P3/C301/C283ffiffiffi 3p 4p/C300:586503 . . . (6) P4 /C3015 32 /C300:46875 (7) P5 /C301 /C289ffiffiffi 3p 8p/C300:37975499... (8) See also BALL TRIANGLE PICKING ,H EXAGON TRIAN- GLE PICKING ,OBTUSE TRIANGLE ,SQUARE TRIANGLE PICKING ,SYLVESTER’S FOUR- POINT PROBLEM ,TRIAN- GLE TRIANGLE PICKING References Buchta, C. "Zufallspolygone in konvexen Vielecken." J. reine angew. Math. 347, 212 /C1/20, 1984. Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653 /C1/59, 1986. Eisenberg, B. and Sullivan, R. "Random Triangles n Dimensions." Amer. Math. Monthly 103, 308 /C1/18, 1996. Guy, R. K. "There are Three Times as Many Obtuse-Angled Triangles as There are Acute-Angled Ones." Math. Mag. 66, 175 /C1/78, 1993. Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob. 19, 712 /C1/15, 1982. Pfiefer, R. E. "The Historical Development of J. J. Sylves- ter’s Four Point Problem." Math. Mag. 62, 309 /C1/17, 1989. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978. Woolhouse, W. S. B. Solution to Problem 1350. Mathemati- cal Questions, with Their Solutions, from the Educational Times, Vol. 1. London: F. Hodgson and Son, pp. 49 /C1/1, 1886. Woolhouse, W. S. B. "Some Additional Observations on the Four-Point Problem." Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 7. London: F. Hodgson and Son, p. 81, 1867. Disk-Cyclide Coordinates A coordinate system defined by the transformationequations x /C30a Lcn m cn n cos c (1) y /C30a Lcn m cn n sinc (2) z /C30a Lsn m dn m sn n dn n ; (3) where L/C131 /C28dn2 m sn2v (4) and for m /C23 [0; K]; n /C23 [0;K ?] ; and c /C23 0; 2pi ½Þ :: Surfaces of constant m are given by the cyclides of rotation x2 /C27 y2 a2cn2 m /C27k2sn2 m a2dn2mz2 !2 /C282x2/C27y2ðÞ a2cn2m/C282k2sn2m a2dn2mz2/C271/C3000(5) surfaces of constant nby the disk cyclides cn2n a2x2/C27y20CB0C@ /C27k?2sn2n a2dn2nz2"#2 /C282cn2n a2x2/C27y20CB0C@ /C282k?2sn2n a2dn2nz2/C271/C300; (6) and surfaces of constant cby the half-planes tanc/C30y x: (7) See also CAP-CYCLIDE COORDINATES ,CYCLIDIC CO- ORDINATES ,FLAT-RING CYCLIDE COORDINATES References Moon, P. and Spencer, D. E. "Disk-Cyclide Coordinates (m;n;c):/" Fig. 4.10 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 129 /C1/ 32, 1988. Dispersion (Sequence) An array B/C30bij;i;j]1o f POSITIVE INTEGERS is called a dispersion if 1. The first column of B is a strictly increasing sequence, and there exists a strictly increasing sequence fsk g such that 2. b12 /C30s1 ]2;/ 3. The complement of the SET fbi1 : i ]1 g is the SET fsk g;/ 4. bij /C30sbi;j/C281for all j ]3 for i /C301 and for all g ]2 for all i ]2 ::/ If an array B /C30bij ; is a dispersion, then it is an INTERSPERSION . See also INTERSPERSION References Kimberling, C. "Interspersions and Dispersions." Proc. Amer. Math. Soc. 117, 313 /C1/21, 1993. Dispersion (Statistics) ( Du)2 i /C13 ui /C28 ¯u ðÞ2; where ¯u is the average of fui g::/ See also ABSOLUTE DEVIATION ,SIGNED DEVIATION , VARIANCE Dispersion Numbers MAGIC GEOMETRIC CONSTANTS Dispersion Relation Any pair of equations giving the REAL PART of a function as an integral of its IMAGINARY PART and the IMAGINARY PART as an integral of its REAL PART . Dispersion relationships imply causality in physics. Let fx0ðÞ/C13ux0ðÞ/C27iv x0ðÞ ; (1) then ux0ðÞ/C301 pPV g/C12 /C28/C12v(x)dx x /C28 x0(2) vx0ðÞ/C30/C281 pPV g/C12 /C28/C12u(x)dx x /C28 x0; (3) where PV denotes the CAUCHY PRINCIPAL VALUE and u(x0) and v(x0) are HILBERT TRANSFORMS of each other. If the COMPLEX function is symmetric such that f(/C28x) /C30f +(x); then ux0ðÞ/C302 pPV g/C12 0xv(x)dx x2 /C28 x2 0(4) vx0ðÞ/C30/C282 pPV g/C12 0xu(x)dx x2 /C28 x2 0: (5)See also HILBERT TRANSFORM Dispersive Long-Wave Equation The system of PARTIAL DIFFERENTIAL EQUATIONS ut /C30 u2 /C28 nx /C272v0CB0C@ x vt /C30 2uv /C27vx ðÞx : References Boiti, M.; Leon, J. J.-P.; and Pempinelli, F. "Integrable Two- Dimensional Generalisation of the Sine- and Sinh-Gordon Equations." Inverse Prob. 3,37/C1/9, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Disphenocingulum JOHNSON SOLID J90 ::/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Disphenoid ATETRAHEDRON with identical ISOSCELES orSCALENE faces. See also SNUB DISPHENOID Dissection Any two rectilinear figures with equal AREA can be dissected into a finite number of pieces to form each other. This is the W ALLACE- BOLYAI-GERWEIN THEO- REM. For minimal dissections of a TRIANGLE ,PENTA- GON, and OCTAGON into a SQUARE , see Stewart (1987, pp. 169 /C1/70) and Ball and Coxeter (1987, pp. 89 /C1/1). The TRIANGLE toSQUARE dissection ( HABERDASHER’S PROBLEM ) is particularly interesting because it can be built from hinged pieces which can be folded andunfolded to yield the two shapes (Gardner 1961;Stewart 1987, p. 169; Pappas 1989; Steinhaus 1983, pp. 3/C1 /; Wells 1991, pp. 61 /C1/2). Laczkovich (1988) proved that the CIRCLE can be squared in a finite number of dissections ( /(/C21050):): Furthermore, any shape whose boundary is composed of smoothly curving pieces can be dissected into a SQUARE . The situation becomes considerably more difficultmoving from 2-D to 3-D. In general, a POLYHEDRON cannot be dissected into other POLYHEDRA of a specified type. A CUBE canbe dissected into n3CUBES , where nis any INTEGER . In 1900, Dehn proved that not every PRISM can be dissected into a TETRAHEDRON (Lenhard 1962, Ball and Coxeter 1987) The third ofH ILBERT’S PROBLEMS asks for the determination of two TETRAHEDRA which cannot be decomposed into congruent TETRAHEDRA directly or by adjoining con- gruent TETRAHEDRA . Max Dehn showed this could not be done in 1902, and W. F. Kagon obtained the sameresult independently in 1903. A quantity growing outof Dehn’s work which can be used to analyze thepossibility of performing a given solid dissection is theD EHN INVARIANT . The table below is an updated version of the one givenin Gardner (1991, p. 50). Many of the improvementsare due to G. Theobald (Frederickson 1997). Theminimum number of pieces known to dissect aregular n-gon (where nis a number in the first column) into a k-gon (where kis a number is the bottom row) is read off by the intersection of thecorresponding row and column. In the table, fng denotes a regular n-gon, GR a GOLDEN RECTANGLE , GC a G REEK CROSS ,L CaL ATIN CROSS ,M Ca MALTESE CROSS ,S Wa SWASTIKA ,f5=2ga five-point star (solid PENTAGRAM ),f6=2ga six-point star (i.e., HEXAGRAM or solid STAR OF DAVID ), and f8=3gthe solid OCTAGRAM . /f4g/ 4 /f5g/ 66 /f6g/ 55 7 /f7g/ 87 98 /f8g/ 75 98 1 1 /f9g/ 8 9 12 11 14 13 /f10g/ 7 7 10 9 11 10 13 /f12g/ 8 6 10 6 11 10 14 12 G R 43 65 76 9 6 7 G C 54 77 99 1 2 1 0 65 L C 55 86 88 1 1 1 0 75 7 MC 7 14 8 SW 6 12 8 9 /f5=2g/ 7 7 9 9 11 10 14 6 12 7 10 10/f6=2g/ 55 86 98 1 1 9 95 88 1 1 /f8=3g/ 8 8 9 9 12 6 13 12 12 7 10 11 13 10 /f3g//f4g//f5g//f6g//f7g//f8g// f9g//f10g//f12g/GR GC LC MC SW /f5=2g//f6=2g/ Wells (1991) gives several attractive dissections of the regular DODECAGON . The best-known dissections of one regular convex n-gon into another are shown for n/C303, 4, 5, 6, 7, 8, 9, 10, and 12 in the following illustrations due to Theobald. The best-known dissections of regular concave poly- gons are illustrated below for f5=2g;f6=2g; and f8=3g (Theobald). The best-known dissections of various crosses are illustrated below (Theobald). The best-known dissections of the GOLDEN RECTAN- GLE are illustrated below (Theobald). See also BANACH- TARSKI PARADOX ,BLANCHE’S DIS- SECTION ,C UNDY AND ROLLETT’S EGG,D ECAGON , DEHN INVARIANT ,D IABOLICAL CUBE,D ISSECTION PUZZLES ,DODECAGON ,EHRHART POLYNOMIAL ,EQUI- DECOMPOSABLE ,E QUILATERAL TRIANGLE ,G OLDEN RECTANGLE ,HEPTAGON ,HEXAGON ,HEXAGRAM ,HIL- BERT’S PROBLEMS ,L ATIN CROSS ,M ALTESE CROSS , NONAGON ,OCTAGON ,OCTAGRAM ,PENTAGON ,PENTA- GRAM ,P OLYHEDRON DISSECTION ,P YTHAGOREAN SQUARE PUZZLE ,PYTHAGOREAN THEOREM ,REP-TILE, SOMA CUBE,SQUARE ,STAR OF LAKSHMI ,SWASTIKA ,T- PUZZLE ,TANGRAM ,W ALLACE- BOLYAI- GERWEIN THEO- REM References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 87 /C1/4, 1987. Coffin, S. T. The Puzzling World of Polyhedral Dissections. New York: Oxford University Press, 1990. Coffin, S. T. and Rausch, J. R. The Puzzling World of Polyhedral Dissections CD-ROM. Puzzle World Produc- tions, 1998. Cundy, H. and Rollett, A. Ch. 2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Eppstein, D. "Dissection." http://www.ics.uci.edu/~eppstein/ junkyard/dissect.html. Eppstein, D. "Dissection Tiling." http://www.ics.uci.edu/ ~eppstein/junkyard/distile/. Eriksson, K. "Splitting a Polygon into Two Congruent Pieces." Amer. Math. Monthly 103, 393 /C1/00, 1996. Frederickson, G. Dissections: Plane and Fancy. New York: Cambridge University Press, 1997. Gardner, M. "Mathematical Games: About Henry Ernest Dudeney, A Brilliant Creator of Puzzles." Sci. Amer. 198, 108 /C1/12, Jun. 1958. Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, 1961. Gardner, M. "Paper Cutting." Ch. 5 in Martin Gardner’s New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 58 /C1/9, 1966. Gardner, M. The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: Chicago University Press, 1991. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 65 /C1/7, 1975. Keil, J. M. "Polygon Decomposition." Ch. 11 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 491 /C1/18, 2000. Kraitchik, M. "Dissection of Plane Figures." §8.1 in Mathe- matical Recreations. New York: W. W. Norton, pp. 193 /C1/ 98, 1942. Laczkovich, M. "Von Neumann’s Paradox with Translation." Fund. Math. 131,1/C1/2, 1988. Lenhard, H.-C. "U¨ ber fu¨nf neue Tetraeder, die einem Wu¨rfel a¨quivalent sind." Elemente Math. 17, 108 /C1/09, 1962. Lindgren, H. "Geometric Dissections." Austral. Math. Tea- cher 7,7/C1/0, 1951. Lindgren, H. "Geometric Dissections." Austral. Math. Tea- cher 9,17/C1/1, 1953. Lindgren, H. "Going One Better in Geometric Dissections." Math. Gaz. 45,94/C1/7, 1961. Lindgren, H. Recreational Problems in Geometric Dissection and How to Solve Them. New York: Dover, 1972. Madachy, J. S. "Geometric Dissection." Ch. 1 in Madachy’s Mathematical Recreations. New York: Dover, pp. 15 /C1/3, 1979. Pappas, T. "A Triangle to a Square." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 9 and 230, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, 1987. Weisstein, E. W. "Books about Dissections." http:// www.treasure-troves.com/books/Dissections.html.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 56 /C1/7 and 243 /C1/44, 1991. Dissection Puzzles A puzzle in which one object is to be converted to another by making a finite number of cuts and reassembling it. The cuts are often, but not always, restricted to straight lines. Sometimes, a given puzzle is precut and is to be re-assembled into two or more given shapes. See also CUNDY AND ROLLETT’S EGG,PYTHAGOREAN SQUARE PUZZLE ,T-PUZZLE ,TANGRAM Dissipative System A DYNAMICAL SYSTEM in which the PHASE SPACE volume contracts along a trajectory. This means that the generalized DIVERGENCE is less than zero, @fi @xiB0; where EINSTEIN SUMMATION has been used. See also DYNAMICAL SYSTEM ,PHASE SPACE Dissymmetric An object that is not superimposable on its MIRROR IMAGE is said to be disymmetric. All asymmetric objects are dissymmetric, and an object with no IMPROPER ROTATION (rotoinversion) axis must also be disymmetric. The opposite of dissymmetric is ENANTIOMORPHOUS . See also AMPHICHIRAL KNOT,CHIRAL ,DISSYMMETRIC , ENANTIOMER ,E NANTIOMORPHOUS ,H ANDEDNESS , MIRROR IMAGE ,REFLEXIBLE Distance The distance between two points is the length of the path connecting them. In the plane, the distancebetween points x 1;y1 ðÞ and x2;y2 ðÞ is given by the PYTHAGOREAN THEOREM , d/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C28x1 ðÞ2/C27y2/C28y1 ðÞ2:q (1) In Euclidean 3-space, the distance between points x1;y1;z1 ðÞ and x2;y2;z2 ðÞ is d/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C28x1 ðÞ2/C27y2/C28y1 ðÞ2/C27z2/C28z1 ðÞ2q : (2) In general, the distance between points xandyin a EUCLIDEAN SPACE Rnis given by d/C30x/C28y jj/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn i/C301xi/C28yi jj2:vuut(3) For curved or more complicated surfaces, the so- calledMETRIC can be used to compute the distance between two points by integration. When unqualified, "the" distance generally means the shortest distance between two points. For example, there are an infinite number of paths between two points on a SPHERE but, in general, only a single shortest path. The shortest distance between two points is the length of a so-called GEODESIC between the points. In the case of the sphere, the geodesic is a segment of a GREAT CIRCLE containing the two points. Let g tðÞbe a smooth curve in a MANIFOLD M from x to y with g 0ðÞ/C30x: and g 1ðÞ/C30y:: Then g? tðÞ/C23 Tg tðÞ; where Tx is the TANGENT SPACE of M at x. The LENGTH of g with respect to the Riemannian structure is given by g1 0g? tðÞkkg tðÞdt; (4) and the distance dx;yðÞ between x and y is the shortest distance between x and y given by dx;yðÞ/C30inf gix to y g g ?(t) kkg tðÞdt : (5) In order to specify the relative distances of n /C211 points in the plane, 1 /C272 n /C282 ðÞ /C302n /C283 coordinates are needed, since the first can always be taken as (0, 0) and the second as x;0ðÞ ; which defines the X-AXIS . The remaining n /C282 points need two coordinates each. However, the total number of distances is n 20C@80C@9 /C30n! 2! n /C28 2 ðÞ ! /C301 2nn/C281 ðÞ ; (6) wheren k0CB0C@ is a BINOMIAL COEFFICIENT . The distances between n /C211 points are therefore subject to m relationships, where m /C1312nn/C281 ðÞ /C28 2n /C283 ðÞ /C3012n /C282 ðÞ n /C283 ðÞ : (7) For n /C301, 2, ..., this gives 0, 0, 0, 1, 3, 6, 10, 15, 21, 28, ... (Sloane’s A000217) relationships, and the number of relationships between n points is the TRIANGULAR NUMBER /Tn/C283/. Although there are no relationships for n /C302 and n /C303 points, for n /C304(a QUADRILATERAL ), there is one (Weinberg 1972): 0 /C30d4 12d234 /C27d413d224 /C27d414d223 /C27d423d214 /C27d424d213 /C27d434d212 /C27d212d223d231 /C27d212d224d241 /C27d213d234d241 /C27d223d234d242 /C28d212d223d234 /C28d213d232d224 /C28d212d224d243 /C28d214d242d223 /C28d213d234d242 /C28d214d243d232 /C28d223d231d214 /C28d221d213d234 /C28d224d241d213 /C28d221d214d243 /C28d231d212d224 /C28d232d221d214 : (8)This equation can be derived by writing dij /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xi /C28xj0CB0C@2/C27 yi /C28yj0CB0C@2q (9) and eliminating xiand yjfrom the equations for d12 ; d13 ; d14 ; d23 ; d24 ; and d34 :: This results in a CAYLEY- MENGER DETERMINANT 0 /C3001 1 1 1 10 d2 12d213d214 1 d221 0 d223d224 1 d231d232 0 d234 1 d241d242d243 00C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1; (10) as observed by Uspensky (1948, p. 256). See also A RC LENGTH ,CUBE POINT PICKING ,EXPAN- SIVE,GEODESIC ,LENGTH (CURVE ), METRIC ,PLANAR DISTANCE ,POINT DISTANCES ,POINT- LINE DISTANCE– 2-D, POINT- LINE DISTANCE–3- D, POINT- PLANE DIS- TANCE ,POINT- POINT DISTANCE–1- D, POINT- POINT DIS- TANCE–2- D, POINT- POINT DISTANCE–3- D, SPHERE References Gray, A. "The Intuitive Idea of Distance on a Surface." §15.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 341 /C1/45, 1997. Sloane, N. J. A. Sequences A000217/M2535 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Uspensky, J. V. Theory of Equations. New York: McGraw- Hill, p. 256, 1948. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972. Distance Graph Let D be a set of positive numbers containing 1, then the D-distance graph XDðÞon a nonempty subset X of Euclidean space is the GRAPH with vertex set X and edge set x;yðÞ : dx; yðÞ/C23 D fg ; where dx;yðÞ is the Euclidean distance between vertices x and y. See also PRIME- DISTANCE GRAPH ,U NIT-DISTANCE GRAPH ,UNIT NEIGHBORHOOD GRAPH References Maehara, H. "Distance Graphs in Euclidean Space." Ryukyu Math. J. 5,33/C1/1, 1992. Distance-Regular Graph A CONNECTED GRAPH G is called distance-regular if there are integers dx;yðÞ such that for any two vertices x;y /C23 G ar distance i /C30dx;yðÞ ; there are exactly cineighbors of y/C23Gi/C281xðÞand bineighbors ofy/C23Gi/C271xðÞ::/ See also INTERSECTION ARRAY ,M OORE GRAPH ,REG- ULAR GRAPH References Bendito, E.; Carmona, A.; and Encinas, A. M. "Shortest Paths in Distance-Regular Graphs." Europ. J. Combin. 21, 153 /C1/66, 2000. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, 1989. Distinct Prime Factors The number of distinct prime factors of a number n is denoted (n): The first few values for n /C301, 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, ... (Sloane’s A001221; Abramowitz and Stegun 1972, Kac 1959). This sequence is given by the inverse MO¨ BIUS TRANSFORM of bn /C301 for n prime and bn /C300 for n (Sloane and Plouffe 1995, p. 22). The first few values of the SUMMATORY FUNCTION Xn k /C302v kðÞ are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, ... (Sloane’s A013939), and the asymptotic value is Xn k /C302v kðÞ/C30n ln lnn /C27B1n /C27onðÞ; where B1is M ERTENS CONSTANT . In addition, Xn k/C302vkðÞ½/C1382/C30nln ln n ðÞ2/C27Ol nl n n ðÞ : The numbers consisting only of distinct prime factors are precisely the SQUAREFREE numbers. See also DIVISOR FUNCTION ,ERDOS- KAC THEOREM , GREATEST PRIME FACTOR ,HARDY- RAMANUJAN THEO- REM,HETEROGENEOUS NUMBERS ,LEAST PRIME FAC- TOR,M ERTENS CONSTANT ,P RIME FACTORS , SQUAREFREE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 844, 1972. Hardy, G. H. and Wright, E. M. "The Number of Prime Factors of n" and "The Normal Order of s(n) and VnðÞ::/" §22.10 and 22.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 354 /C1/58, 1979. Kac, M. Statistical Independence in Probability, Analysis and Number Theory. Washington, DC: Math. Assoc. Amer., p. 64, 1959.Sloane, N. J. A. Sequences A001221/M0056 and A013939 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Distribution (Generalized Function) The class of all regular sequences of PARTICULARLY WELL-BEHAVED FUNCTIONS equivalent to a given regular sequence. A distribution is sometimes also called a "generalized function" or "ideal function." As its name implies, a generalized function is a general-ization of the concept of a FUNCTION . For example, in physics, a baseball being hit by a bat encounters aforce from the bat, as a function of time. Since thetransfer of momentum from the bat is modeled astaking place at an instant, the force is not actually a function. Instead, it is a multiple of the DELTA FUNCTION . The set of distributions contains functions (LOCALLY INTEGRABLE ) and R ADON MEASURES . Note that the term "distribution" is closely related to STATISTICAL DISTRIBUTIONS . Generalized functions are defined as continuous linear FUNCTIONALS over a SPACE of infinitely differ- entiable functions such that all continuous functionshave derivatives which are themselves generalizedfunctions. The most commonly encountered general- ized function is the DELTA FUNCTION . Vladimirov (1984) contains a nice treatment of distributions from a physicist’s point of view, while the multi-volume work by Gel’fand and Shilov (1977) is a classic and rigorous treatment of the field. While it is possible to add distributions, it is not possible to multiply distributions when they havecoinciding singular support. Despite this, it is possi- ble to take the DERIVATIVE of a distribution, to get another distribution. Consequently, they may satisfy a linear PARTIAL DIFFERENTIAL EQUATION , in which case the distribution is called a weak solution. Forexample, given any locally integrable function fit makes sense to ask for solutions uof P OISSON’S EQUATION 92u/C30f (1) by only requiring the equation to hold in the sense ofdistributions, that is, both sides are the same dis-tribution. The definitions of the derivatives of a distribution pxðÞare given by g/C12 /C28/C12p?xðÞfxðÞdx/C30/C28g/C12 /C12pxðÞf?xðÞdx (2) g/C12 /C28/C12pnðÞxðÞfxðÞdx/C30/C28 1ðÞng/C12 /C28/C12pxðÞfnðÞxðÞdx: (3) Distributions also differ from functions because they are COVARIANT , that is, they push forward. Given a SMOOTH FUNCTION a:V10V2;a distribution TonV1 pushes forward to a distribution on V2 : In contrast, a REAL FUNCTION f on V2 : pulls back to a function on V1 ; namely f a xðÞðÞ :/ Distributions are, by definition, the dual to the SMOOTH FUNCTIONS of COMPACT SUPPORT , with a particular TOPOLOGY . For example, the DELTA FUNC- TION d is the LINEAR FUNCTIONAL d fðÞ/C30f 0ðÞ: The distribution corresponding to a function g is TgfðÞ/C30gVfg; (4) and the distribution corresponding to a MEASURE m is TmfðÞ/C30gVfdm : (5) The PUSHFORWARD MAP of a distribution T along a is defined by a/C31TfðÞ/C30Tf( aðÞ ; (6) and the derivative of T is defined by DT fðÞ/C30TD/C31fðÞ where D/C31 is the FORMAL ADJOINT of D. For example, the first derivative of the DELTA FUNCTION is given by d dxd fðÞ½/C138/C30/C28df dx j x /C300: (7) As is the case for any function space, the topology determines which LINEAR FUNCTIONALS are continu- ous, that is, are in the DUAL SPACE . The topology is defined by the family of SEMINORMS , NK ;afðÞ/C30sup kD af0C@C0C@C0C@C0C@C; (8) where sup denotes the SUPREMUM . It agrees with the C-INFINITY TOPOLOGY on compact subsets. In this topology, a sequence converges, fn 0 f ; IFF there is a compact set K such that all fnare supported in Kand every derivative Dafnconverges uniformly to Dafin K. Therefore, the constant function 1 is a distribu- tion, because if fn0f;then T1fnðÞ/C30gKfn0gKf/C30T1fðÞ: (9) See also CONVOLUTION ,D ELTA FUNCTION ,D ELTA SEQUENCE ,F OURIER SERIES ,F UNCTIONAL ,L INEAR FUNCTIONAL ,M ICROLOCAL ANALYSIS ,S TATISTICAL ANALYSIS ,TEMPERED DISTRIBUTION ,ULTRADISTRIBU- TION References Friedlander, F. G. Introduction to the Theory of Distribu- tions, 2nd ed. Cambridge, England: Cambridge University Press, 1999. Gel’fand, I. M.; Graev, M. I.; and Vilenkin, N. Ya. General- ized Functions, Vol. 5: Integral Geometry and Representa- tion Theory. New York: Harcourt Brace, 1977.Gel’fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 1: Properties and Operations. New York: Harcourt Brace, 1977. Gel’fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 2: Spaces of Fundamental and Generalized Func-tions. New York: Harcourt Brace, 1977. Gel’fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 3: Theory of Differential Equations. New York: Harcourt Brace, 1977. Gel’fand, I. M. and Vilenkin, N. Ya. Generalized Functions, Vol. 4: Applications of Harmonic Analysis. New York: Harcourt Brace, 1977. Griffel, D. H. Applied Functional Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1984. Halperin, I. and Schwartz, L. Introduction to the Theory of Distributions, Based on the Lectures Given by Laurent Schwarz. Toronto, Canada: University of Toronto Press, 1952. Lighthill, M. J. Introduction to Fourier Analysis and Gen- eralised Functions. Cambridge, England: Cambridge Uni- versity Press, 1958. Richards, I. and Young, H. The Theory of Distributions: A Nontechnical Introduction. Cambridge, England: Cam- bridge University Press, 1990. Rudin, W. Functional Analysis, 2nd ed. New York: McGraw- Hill, 1991. Strichartz, R. Fourier Transforms and Distribution Theory. Boca Raton, FL: CRC Press, 1993. Vladimirov, V. S. Equations of Mathematical Physics. Mos- cow: Mir, 1984. Weisstein, E. W. "Books about Generalized Functions." http://www.treasure-troves.com/books/GeneralizedFunc- tions.html. Yoshida, K. Functional Analysis. Berlin: Springer-Verlag, pp. 28 /C1 /9 and 46 /C1/2, 1974. Zemanian, A. H. Distribution Theory and Transform Ana- lysis: An Introduction to Generalized Functions, withApplications. New York: Dover, 1987. Distribution (Statistical) STATISTICAL DISTRIBUTION Distribution Function The distribution function DxðÞ;sometimes also called the PROBABILITY DISTRIBUTION FUNCTION , describes the probability that a trial Xtakes on a value less than or equal to a number x. The distribution function is therefore related to a continuous PROB- ABILITY DENSITY FUNCTION PxðÞby DxðÞ/C30PX5x ðÞ/C13gx /C28/C12Px?ðÞdx?; (1) soPxðÞ(when it exists) is simply the derivative of the distribution function PxðÞ/C30D?xðÞ/C30Px?ðÞ½/C138x /C28/C12/C30PxðÞ/C28P/C28/C12ðÞ : (2) Similarly, the distribution function is related to a discrete probability PxðÞby DxðÞ/C30PX5x ðÞ /C30X X5xPxðÞ: (3) In general, there exist distributions which are neither continuous nor discrete. A JOINT DISTRIBUTION FUNCTION can be defined if outcomes are dependent on two parameters: Dx ;yðÞ/C13PX5x;Y 5y ðÞ (4) DxxðÞ/C13Dx ;/C12ðÞ (5) DyyðÞ/C13D /C12;y ðÞ : (6) Similarly, a multiple distribution function can be defined if outcomes depend on n parameters: Da1 ;:::; an ðÞ /C13Px1 5a1 ;:::; xn 5an ðÞ : (7) Given a continuous PxðÞ; assume you wish to gen- erate numbers distributed as PxðÞusing a random number generator. If the random number generator yields a uniformly distributed value yiin 0;1½/C138 for each trial i, then compute DxðÞ/C13gx Px?ðÞdx?: (8) The FORMULA connecting yiwith a variable distrib- uted as PxðÞis then xi /C30D/C281 yiðÞ; (9) where D /C28i ðxÞ is the inverse function of DxðÞ;: For example, if PxðÞwere a GAUSSIAN DISTRIBUTION so that DxðÞ/C301 21 /C27erfx- m sffiffiffi 2p !"# ; (10) then xi /C30 sffiffiffi 2p erf /C281 2yi /C281 ðÞ /C27 m: (11) A distribution with constant VARIANCE of y for all values of x is known as a HOMOSCEDASTIC distribu- tion. The method of finding the value at which the distribution is a maximum is known as the MAXIMUM LIKELIHOOD method. See also BERNOULLI DISTRIBUTION ,BETA DISTRIBU- TION ,BINOMIAL DISTRIBUTION ,BIVARIATE DISTRIBU- TION ,CAUCHY DISTRIBUTION ,CHI DISTRIBUTION ,CHI- SQUARED DISTRIBUTION ,CORNISH- FISHER ASYMPTO- TIC EXPANSION ,CORRELATION COEFFICIENT ,DOUBLE EXPONENTIAL DISTRIBUTION ,EQUALLY LIKELY OUT- COMES DISTRIBUTION ,E XPONENTIAL DISTRIBUTION , EXTREME VALUE DISTRIBUTION , F-DISTRIBUTION , FERMI- DIRAC DISTRIBUTION ,F ISHER’S Z-DISTRIBU- TION ,FISHER- TIPPETT DISTRIBUTION ,GAMMA DISTRI- BUTION ,G AUSSIAN DISTRIBUTION ,G EOMETRIC DISTRIBUTION ,HALF-NORMAL DISTRIBUTION ,HYPER- GEOMETRIC DISTRIBUTION ,J OINT DISTRIBUTION FUNCTION ,LAPLACE DISTRIBUTION ,LATTICE DISTRI- BUTION ,LE´ VY DISTRIBUTION ,LOGARITHMIC DISTRIBU- TION ,L OG-SERIES DISTRIBUTION ,L OGISTIC DISTRIBUTION ,LORENTZIAN DISTRIBUTION ,M AXWELL DISTRIBUTION ,N EGATIVE BINOMIAL DISTRIBUTION ,NORMAL DISTRIBUTION ,PARETO DISTRIBUTION ,PAS- CAL DISTRIBUTION ,PEARSON TYPE III DISTRIBUTION , POISSON DISTRIBUTION ,PO´ LYA DISTRIBUTION ,RAN- DOM NUMBER ,RATIO DISTRIBUTION ,RAYLEIGH DIS- TRIBUTION ,R ICE DISTRIBUTION ,S NEDECOR’S F- DISTRIBUTION ,STATISTICAL DISTRIBUTION ,STUDENT’S T-DISTRIBUTION ,S TUDENT’S Z-DISTRIBUTION ,U NI- FORM DISTRIBUTION ,W EIBULL DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Probability Functions." Ch. 26 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 925 /C1/64, 1972. Iyanaga, S. and Kawada, Y. (Eds.). "Distribution of Typical Random Variables." Appendix A, Table 22 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1483 /C1/486, 1980. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 92 /C1/4, 1984. Distribution Parameter The distribution parameter of a NONCYLINDRICAL RULED SURFACE parameterized by x u;vðÞ/C30 s uðÞ/C27v d uðÞ; (1) where s is the STRICTION CURVE and d the DIRECTOR CURVE , is the function p defined by p /C30det s? dd? ðÞ d?: d?: (2) The GAUSSIAN CURVATURE of a RULED SURFACE is given in terms of its distribution parameter by K/C30/C28puðÞ½/C1382 puðÞ½/C1382/C27v2no2: (3) See also NONCYLINDRICAL RULED SURFACE ,RULED SURFACE ,STRICTION CURVE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 447, 1997. Distributive Elements of an ALGEBRA which obey the identity AB/C27C ðÞ /C30AB/C30AC are said to be distributive over the operation +. See also ASSOCIATIVE ,COMMUTATIVE ,TRANSITIVE Distributive Lattice ALATTICE which satisfies the identities (xffly)/C150(xffly)/C30xffl(y/C150z) (x /C150y) ffl(x /C150z) /C30x /C150(y fflz) is said to be distributive. See also LATTICE ,MODULAR LATTICE References Gra¨tzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, pp. 35 /C1/6, 1971. Ditrigonal Dodecadodecahedron The UNIFORM POLYHEDRON U41 ; also called the DITRI- GONAL DODECAHEDRON , whose DUAL POLYHEDRON is the MEDIAL TRIAMBIC ICOSAHEDRON . It has WYTHOFF SYMBOL 3½5 35: Its faces are 1252no /C2712 5fg: It is a FACETED version of the SMALL DITRIGONAL ICOSIDO- DECAHEDRON . The CIRCUMRADIUS for unit edge length is R /C301 2ffiffiffi 3p : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 123 /C1/24, 1989. Ditrigonal Dodecahedron DITRIGONAL DODECADODECAHEDRON Divergence The divergence of a VECTOR FIELD F is given by div(F) /C139/C215F /C13lim V 00GsF /C215 da V: (1) Define F /C13F1 ˆu1 /C27F2 ˆu2 /C27F3 ˆu3 : (2) Then in arbitrary orthogonal CURVILINEAR COORDI- NATES , div(F) /C139/C215F /C131 h1h2h3@ @u1h2h3F1 ðÞ /C27@ @u2h3h1F2 ðÞ" /C27@ @u3h1h2F3 ðÞ0C1@ : (3)If 9/C215F /C300; then the field is said to be a DIVERGENCE- LESS FIELD . For divergence in individual coordinate systems, see CURVILINEAR COORDINATES . 9/C215Ax xjj/C30Tr(A) xjj/C28xT(Ax) xjj3: (4) The divergence of a TENSOR A is 9/C215A /C13Aa ia (5) /C30Ak;k /C27Gk jkAj ; (6) /C301 g1 =2g1 =2Ak0CB0C@ ;k (7) where Aa iais the COVARIANT DERIVATIVE , Ak ;kis the COMMA DERIVATIVE , gij is the METRIC TENSOR , and g /C30 det gij0CB0C@ ; (Arfken 1985, p. 165). Expanding the terms gives Aa ; a /C30Aa; a /C27G a aaAa /C27G abaAb /C27GagaAg0C@n0C@o /C27Ab ; b /C27GbabAa /C27G bbbAb /C27G bgbAg0C@n0C@o /C27Ag ; g /C27Gg agAa /C27G g bgAb /C27G g ggAg0C@n0C@o : (8) See also COMMA DERIVATIVE ,COVARIANT DERIVATIVE , CURL,CURL THEOREM ,DIVERGENCE THEOREM ,GRA- DIENT ,GREEN’S THEOREM ,VECTOR DERIVATIVE References Arfken, G. "Divergence, ." §1.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 37 /C1/2, 1985. Divergence Tests If lim k 0/C12uk "0; then the series unfg diverges. See also CONVERGENCE TESTS ,CONVERGENT SERIES , DINI’S TEST,SERIES Divergence Theorem A.k.a. G AUSS’S THEOREM . Let Vbe a region in space with boundary @V:Then gV9/C215FðÞ dV/C30g@VF/C215da: (1) LetSbe a region in the plane with boundary @S: gS9:FdA/C30g@SF:nds: (2) If the VECTOR FIELD Fsatisfies certain constraints, simplified forms can be used. If F(x;y;z) /C30v(x;y ;z)c where c is a constant vector "0; then gSF :da /C30c /C215gSvda: (3) But 9/C215(fv) /C30(9f) /C215v /C27f( 9/C215v) ; (4) so gV9/C215(cv)dV /C30c /C215gV( 9v /C27v 9/C215c)dV /C30c :gV9vdV (5) c /C215gSvda /C28gV9vdV0C@80C@9 /C300: (6) But c "0; and c :f(v) must vary with v so that c :f(v) cannot always equal zero. Therefore, gSvda /C30gV9vdV : (7) If F(x;y; z) /C30c /C29P(x;y; z) ; where c is a constant vector "0; then gSda /C29PgV9/C29PdV : (8) See also CURL THEOREM ,GRADIENT ,GREEN’S THEO- REM References Arfken, G. "Gauss’s Theorem." §1.11 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57 /C1/1, 1985. Divergenceless Field A divergenceless field, also called a SOLENOIDAL FIELD ,isa FIELD for which 9/C215F /C130: Therefore, there exists a G such that F /C309/C29G : Furthermore, F can be written as F /C309/C29(Tr) /C2792(Sr) /C13T /C27S; (1) where T /C139/C29(Tr) /C30/C28r /C29( 9T) (2) S /C1392(Sr) /C309@ @r (rS)"# /C28r92S: (3) Following Lamb, T and S are called TOROIDAL FIELD and POLOIDAL FIELD . See also BELTRAMI FIELD ,IRROTATIONAL FIELD , POLOIDAL FIELD,SOLENOIDAL FIELD,TOROIDAL FIELD Divergent Sequence A divergent sequence is a SEQUENCE for which the LIMIT exists but is not CONVERGENT .See also CONVERGENT SEQUENCE ,DIVERGENT SERIES Divergent Series A SERIES which is not CONVERGENT . Series may diverge by marching off to infinity or by oscillating. Divergent series have some curious properties. For example, rearranging the terms of 1 /C281 /C271 /C281 /C271 /C28 /C1/C1/C1 gives both (1 /C281) /C27(1 /C281) /C27(1 /C281) /C27/C1/C1/C1/C300 and 1 /C28(1 /C281) /C28(1 /C281) /C27/C1/C1/C1/C301 ::/ The RIEMANN SERIES THEOREM states that, by a suitable rearrangement of terms, a CONDITIONALLY CONVERGENT SERIES may be made to converge to any desired value, or to diverge. No less an authority than N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever" (Gardner 1984, p. 171; Hoffman 1998, p. 218). However, divergent series can actually be "summed" rigorously by using extensions to the usual summation rules (e.g., so-called Abel and Cesa`ro sums). For example, the divergent series 1 /C281 /C271 /C28 1 /C271 /C28/C1/C1/C1 has both Abel and Cesa`ro sums of 1/2. See also ABSOLUTE CONVERGENCE ,C ONDITIONAL CONVERGENCE ,CONVERGENT SERIES ,DIVERGENT SE- QUENCE References Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 170 /C1/71, 1984. Hardy, G. H. Divergent Series. New York: Oxford University Press, 1949. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Diversity Condition For any group of k men out of N, there must be at least k jobs for which they are collectively qualified. Divide To divide is to perform the operation of DIVISION , i.e., to see how many times a DIVISOR dgoes into another number n.ndivided by dis written n=dorn}d:The result need not be an INTEGER , but if it is, some additional terminology is used. d½nis read " ddivides n" and means that dis a DIVISOR ofn. In this case, n is said to be DIVISIBLE byd. Clearly, 1 ½nandn½n:By convention, n½0 for every nexcept 0 (Hardy and Wright 1979). The "divisibility" relation satisfies b½a for c ½b[c½a b½a[bc½ac c ½a and c ½b [c ½ ma /C27nb ðÞ ; where the symbol [means IMPLIES . /d?¶n is read "/d? does not divide n" and means that d? is not a DIVISOR of n. ak ½½b means ak divides b exactly. If n and d are RELATIVELY PRIME , the notation (n;d) /C301 or sometimes n /C222d is used. See also CONGRUENCE ,DIVISIBLE ,DIVISIBILITY TESTS , DIVISION ,D IVISOR ,G REATEST DIVIDING EXPONENT , RELATIVELY PRIME References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 1, 1979. Divided Difference The divided difference fx1 ;x2 ;:::;xn ½/C138 on n points x1 ; x2 ; ..., xn of a function f(x) is defined by f[x1] /C13fx1ðÞand fx1 ;x2 ;:::; xn ½/C138 /C30fx1 ;:::; xn ½/C138 /C28 fx2 ;:::; xn ½/C138 x1 /C28 xn(1) for n ]2 : The first few differences are x0 ;x1 ½/C138 /C30f0 /C28 f1 x0 /C28 x1(2) x0 ; x1 ;x2 ½/C138 /C30x0 ;x1 ½/C138 /C28 x1 ; x2 ½/C138 x0 /C28 x2(3) x0 ;x1 ;:::; xn ½/C138 /C30x0 ;:::; xn/C281 ½/C138 /C28 x1 ;:::; xn ½/C138 x0 /C28 xn: (4) Defining pn(x) /C13 x /C28x0 ðÞ x /C28x1 ðÞ/C1/C1/C1 x /C28xn ðÞ (5) and taking the DERIVATIVE p?nxkðÞ/C30 xk /C28x0 ðÞ ::: xk /C28xk /C281 ðÞ ::: xk /C28xn ðÞ (6) gives the identity x0 ;x1 ;:::; xn ½/C138 /C30Xn k /C300fk p?nxkðÞ: (7) Consider the following question: does the property fx1 ;x2 ;:::; xn ½/C138 /C30hx1 /C27x2 /C27:::/C27xn ðÞ (8) for n ]2 and h(x) a given function guarantee that f(x) is a POLYNOMIAL of degree 5n/? Acze´l (1985) showed that the answer is "yes" for n /C302, and Bailey (1992) showed it to be true for n /C303 with differentiable f(x): Schwaiger (1994) and Andersen (1996) subsequently showed the answer to be "yes" for all n ]3 with restrictions on f(x)orh(x):/ See also HORNER’S METHOD ,INTERPOLATION ,N EW- TON’S DIVIDED DIFFERENCE INTERPOLATION FORMU- LA,RECIPROCAL DIFFERENCEReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877 /C1/78, 1972. Acze´l, J. "A Mean Value Property of the Derivative of Quadratic Polynomials--Without Mean Values and Deri- vatives." Math. Mag. 58,42/C1/5, 1985. Andersen, K. M. "A Characterization of Polynomials." Math. Mag. 69, 137 /C1/42, 1996. Bailey, D. F. "A Mean-Value Property of Cubic Polynomials-- Without Mean Values." Math. Mag. 65, 123 /C1/24, 1992. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 439 /C1/40, 1987. Jeffreys, H. and Jeffreys, B. S. "Divided Differences." §9.012 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 260 /C1/64, 1988. Schwaiger, J. "On a Characterization of Polynomials by Divided Differences." Aequationes Math. 48, 317 /C1/23, 1994. Whittaker, E. T. and Robinson, G. "Divided Differences" and "Theorems on Divided Differences." §11 /C1/2inThe Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 20 /C1/4, 1967. Dividend A quantity that is divided by another quantity. See also DIVISION ,DIVISOR Divine Proportion GOLDEN RATIO Divisibility Tests Write a positive decimal integer aout digit by digit in the form an/C1/C1/C1a3a2a1a0:The following rules then determine if aisDIVISIBLE by another number by examining the CONGRUENCE properties of its digits. In CONGRUENCE notation, n/C13kmodm ðÞ means that the remainder when nis divided by a modulus misk. (Note that it is always true that 100/C301/C131 for any base.) 1. All integers are DIVISIBLE by 1. 2. 101/C130(mod2) ;so 10n/C130(mod2) for n]1:There- fore, if the last digit a0isDIVISIBLE by 2 (i.e., is EVEN ), then so is a. 3. 100/C131;101/C131;102/C131;..., 10n/C131 (mod 3). Therefore, if an i/C300ajisDIVISIBLE by 3, so is a(Wells 1986, p. 48). 4. 101/C132;102/C130;.../10n/C130 (mod 4). So if the last two digits are DIVISIBLE by 4, more specifically if r/C13a0/C272a1is, then so is a. 5. 101/C130(mod5) ;so 10n/C130(mod5) for n]1:There- fore, if the last digit a0isDIVISIBLE by 5 (i.e., is 5 or 0), then so is a. 6. 101/C13/C282;102/C13/C282;..., 10n/C13/C282 (mod 6). There- fore, if r/C13a0/C282an i/C301aiisDIVISIBLE by 6, so is a.A simpler rule states that if a is DIVISIBLE by 3 and is EVEN , then a is also DIVISIBLE by 6. 7a. 101 /C133; 102 /C132; 103 /C13/C281 ; 104 /C13/C283 ; 105 /C13/C282; 106 /C131 (mod 7), and the sequence then repeats. Therefore, if r /C13 a0 /C273a1 /C272a2 /C28a3 /C283a4 /C282a5 ðÞ /C27 a6 /C273a7 /C27/C1/C1/C1 ðÞ /C27/C1/C1/C1 is DIVISIBLE by 7, so is a. 7b. An alternate test proceeds by multiplying an by 3 and adding to an/C281 ; then repeating the procedure up through a0 : The final number can then, of course, be further reduced using the same proce- dure. If the result is divisible by 7, then so is the original number (Wells 1986, p. 70). 7c. A third test multiplies a0 by 5 and adds it to a1 ; proceeding up through an : The final number can then, of course, be further reduced using the same procedure. If the result is divisible by 7, then so is the original number (Wells 1986, p. 70). 8. 101 /C132; 102 /C134; 103 /C130; ..., 10n /C130 (mod 8). Therefore, if the last three digits are DIVISIBLE by 8, more specifically if r /C13a0 /C272a1 /C274a2 is, then so is a (Wells 1986, p. 72). 9. 100 /C131; 101 /C131; 102 /C131; ..., 10n /C131 (mod 9). Therefore, if an i/C300ai is DIVISIBLE by 9, so is a (Wells 1986, p. 74). 10. 101 /C130 (mod 10), so if the last digit is 0, then a is DIVISIBLE by 10. 11. 101 /C13/C281; 102 /C131; 103 /C13/C281; 104 /C131; ... (mod 11). Therefore, if r /C13a0 /C28a1 /C27a2 /C28a3 /C27/C1/C1/C1 is DIVISIBLE by 11, then so is a. 12. 101 /C13/C282 ; 102 /C134; 103 /C134; ... (mod 12). There- fore, if r /C13a0 /C282a1 /C274 a2 /C27a3 /C27/C1/C1/C1 ðÞ is DIVISIBLE by 12, then so is a. Divisibility by 12 can also be checked by seeing if a is DIVISIBLE by 3 and 4. 13. 101 /C13/C283; 102 /C13/C284 ; 103 /C13/C281 ; 104 /C133; 105 /C134; 106 /C131 (mod 13), and the pattern repeats. There- fore, if r /C13ða0 /C283a1 /C284a2 /C28a3 þ 3a4 þ 4a5 Þþða6 /C28 3a7 þ ...Þþ... isDIVISIBLE by 13, so is a. For additional tests for 13, see Gardner (1991). See also CONGRUENCE ,DIVISIBLE ,DIVISOR ,MODULUS (CONGRUENCE ) References Burton, D. M. "Special Divisibility Tests." §4.3 in Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, pp. 89 /C1/6, 1989. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 337 /C1/ 46, 1952. Gardner, M. "Tests of Divisibility." Ch. 14 in The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 160 /C1/69, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 48, 1986.Divisible A number n is said to be divisible by d if d is a DIVISOR of n. The product of any n consecutive integers is divisible by n! : The sum of any n consecutive integers is divisible by n if n is ODD, and by n=2ifn is EVEN . See also DIVIDE ,DIVISIBILITY TESTS,DIVISOR ,DIVI- SOR FUNCTION References Guy, R. K. "Divisibility." Ch. B in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 44 /C1/04, 1994. Nagell, T. "Divisibility." Ch. 1 in Introduction to Number Theory. New York: Wiley, pp. 11 /C1/6, 1951. Division Taking the RATIO x=y of two numbers x and y, also written x }y: Here, x is called the DIVIDEND , y is called the DIVISOR , and x=y is called a QUOTIENT . The symbol "/" is called a SOLIDUS (or DIAGONAL ), and the symbol "/}/" is called the OBELUS . If left unevaluated, x=y is called a FRACTION , with x known as the NUMERATOR and y known as the DENOMINATOR . Division in which the fractional (remainder) is dis- carded is called INTEGER DIVISION , and is sometimes denoted using a backslash, \. Division is the inverse operation of MULTIPLICATION , so that if a /C29b /C30c; then a can be recovered as a /C30c }b as long as b "0: In general, DIVISION BY ZERO is not defined since the ability to "invert" a /C29b /C30c to recover a breaks down if b /C300 (in which case c is always 0, independent of a). Cutting or separating an object into two or more parts is also called division. See also ADDITION ,C OMPLEX DIVISION ,C UTTING , DENOMINATOR ,D IVIDE ,D IVIDEND ,D IVISION BY ZERO,D IVISOR ,INTEGER DIVISION ,LONG DIVISION , MULTIPLICATION ,NUMERATOR ,OBELUS ,ODDS,PLANE DIVISION BY LINES,Q UOTIENT ,R ATIO,S KELETON DIVISION ,S OLIDUS ,S PACE DIVISION BY SPHERES , SUBTRACTION ,TRIAL DIVISION ,VECTOR DIVISION Division Algebra A division algebra, also called a "division ring" or "skew field," is a RING in which every NONZERO element has a multiplicative inverse, but multiplica-tion is not COMMUTATIVE . In French, the term "corps non commutatif" is used to mean division algebra, while "corps" alone means FIELD . Explicitly, a division algebra is a set together with two BINARY OPERATORS S /C27;+ðÞ satisfying the follow- ing conditions: 1. Additive associativity: For all a ;b ;c /C23 S; (a /C27b) /C27c /C30a /C27(b /C27c) ;/ 2. Additive commutativity: For all a ;b /C23 S; a /C27b /C30b /C27a ;/ 3. Additive identity: There exists an element 0 /C23 S such that for all a /C23 S ; 0 /C27a /C30a /C270 /C30a ;/ 4. Additive inverse: For every a /C23 S there exists an element /C28a /C23 S such that a /C30(/C28a) /C30(/C28a) /C27a /C300;/ 5. Multiplicative associativity: For all a;b ;c /C23 S; a+bðÞ +c /C30a + b+cðÞ ;/ 6. Multiplicative identity: There exists an element 1 /C23 S not equal to 0 such that for all a /C23 S; 1+a /C30a+1 /C30a ;/ 7. Multiplicative inverse: For every a /C23 S not equal to 0, there exists a /C281 /C23 S such that a+a/C281 /C30a /C281 +a /C301 ;/ 8. Left and right distributivity: For all a;b ;c /C23 S; a+(b /C27c) /C30(a+b) /C27(a +c) and (b /C27c) +a /C30(b+a) /C27(c +a):/ Thus a division algebra S;/C27;+ ðÞ is a UNIT RING for which S /C28f0g;+ ðÞ is a GROUP . A division algebra must contain at least two elements. A COMMUTATIVE divi- sion algebra is called a FIELD . In 1878 and 1880, Frobenius and Peirce proved that the only associative REAL division algebras are REAL NUMBERS , COMPLEX NUMBERS , and QUATERNIONS (Mishchenko and Solovyov 2000). The CAYLEY ALGE- BRA is the only NONASSOCIATIVE DIVISION ALGEBRA . Hurwitz (1898) proved that the ALGEBRAS of REAL NUMBERS , COMPLEX NUMBERS , QUATERNIONS , and CAYLEY NUMBERS are the only ones where multi- plication by unit "vectors" is distance-preserving. Adams (1956) proved that n-dimensional vectors form an ALGEBRA in which division (except by 0) is always possible only for n /C301, 2, 4, and 8. Bott and Milnor (1958) proved that the only finite dimensional real division algebras occur for dimensions n /C301, 2, 4, and 8. Each gives rise to an ALGEBRA with particu- larly useful physical applications (which, however, is not itself necessarily nonassociative), and these four cases correspond to REAL NUMBERS , COMPLEX NUM- BERS , QUATERNIONS , and CAYLEY NUMBERS , respec- tively. See also ALTERNATIVE ALGEBRA ,CAYLEY NUMBER , FIELD ,G ROUP ,JORDAN ALGEBRA ,L IE ALGEBRA , NONASSOCIATIVE ALGEBRA ,POWER ASSOCIATIVE AL- GEBRA ,QUATERNION ,SCHUR’S LEMMA ,UNIT RING References Albert, A. A. (Ed.). Studies in Modern Algebra. Washington, DC: Math. Assoc. Amer., 1963. Bott, R. and Milnor, J. "On the Parallelizability of the Spheres." Bull. Amer. Math. Soc. 64,87/C1/9, 1958.Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL: University of Chicago Press, 1923. Dixon, G. M. Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Dordrecht, Netherlands: Kluwer, 1994. Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, pp. 326 /C1/29, 1975. Hurwitz, A. "Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln." Nachr. Ko¨nigl. Gesell. Wiss. Go¨ttingen. Math.-phys. Klasse, 309 /C1/16, 1898. Joye, M. "Introduction e´le´mentaire a` la the´orie des courbes elliptiques." http://www.dice.ucl.ac.be/crypto/introductory/ courbes_elliptiques.html. Kurosh, A. G. General Algebra. New York: Chelsea, pp. 221 /C1/43, 1963. Mishchenko, A. and Solovyov, Y. "Quaternions." Quantum 11,4/C1/ and 18, 2000. Petro, J. "Real Division Algebras of Dimension > 1 contain C:/" Amer. Math. Monthly 94, 445 /C1/49, 1987. Saltman, D. D. Lectures on Division Algebras. Providence, RI: Amer. Math. Soc., 1999. Division by Zero Division by zero is the operation of taking the QUOTIENT of any number x and 0, i.e., x=0: The uniqueness of DIVISION breaks down when dividing by zero, since the product 0 /C215y /C300 is the same for any y,soy cannot be recovered by inverting the process of MULTIPLICATION . 0 is the only number with this property and, as a result, division by zero is UNDE- FINED for REAL NUMBERS and can produce a fatal condition called a "division by zero error" in computer programs. There are, however, contexts in which division by zero can be considered as defined. For example, division by zero z=0 for z /C23C /C31"0 in the EXTENDED COMPLEX PLANE C* is defined to be a quantity known as COMPLEX INFINITY . This definition expresses the fact that, for z "0; limw 00z=w /C30/C12 (i.e., COMPLEX INFINITY ). However, even though the formal state- ment 1=0 /C30/C12 is permitted in C*, note that this does notmean that 1 /C300/C215/C12:Zero does not have a multi- plicative inverse under any circumstances.Although division by zero is not defined for reals, LIMITS involving division by a real quantity xwhich approaches zero may be in fact be WELL DEFINED . For example, lim x00sinx x/C301: Of course, such limits may also approach INFINITY , lim x00/C271 x/C30/C12: See also C*,COMPLEX INFINITY ,COMPLEX NUMBER , DIVISION ,E XTENDED COMPLEX PLANE ,F ALLACY , FIELD,LIMIT, REAL NUMBER ,RING,ZERO Division Lemma When ac is DIVISIBLE by a number b that is RELATIVELY PRIME to a, then c must be DIVISIBLE by b. Division Ring DIVISION ALGEBRA Divisor A divisor of a number N is a number d which DIVIDES N, also called a FACTOR . The total number of divisors for a given number N can be found as follows. Write a number in terms of its PRIME FACTORIZATION N /C30p a1 1 p a2 2/C1/C1/C1p arr : (1) For any divisor d of N, N /C30dd? where d /C30pd1 1 pd2 2/C1/C1/C1pdrr; (2) so d?/C30p a1/C28d1 1p a2/C28d2 2 /C1/C1/C1p ar/C28drr: (3) Now, d1 /C300 ;1;...; a1 ; so there are a1 /C271 possible values. Similarly, for dn ; there are an /C271 possible values, so the total number of divisors v(N)ofN is given by n(N) /C30Yr n /C301an /C271 ðÞ : (4) The function n(N) is also sometimes denoted d(N)or s0(N): The product of divisors can be found by writing the number N in terms of all possible products N /C30d(1)d?(1) n d(n)d?(n);8 < : (5) so N n(N) /C30 d(1) /C1/C1/C1d(n)0C10CC d ?(1)d?(n)0C10CC /C30Yn i /C301diYn i/C301d? i /C30Y d0C@n0C@o2 ; (6) and Y d /C30N n(N) =2 : (7) The GEOMETRIC MEAN of divisors is G /C13Y d0C@n0C@o1= n(N) /C30 N n(n)=20C10CC 1 =n(N)/C30ffiffiffiffiffi Np : (8) The ARITHMETIC MEAN is A(N) /C13s(N) n(N): (9)The HARMONIC MEAN is 1 H /C131 NX1 d ! : (10) But N /C30dd ?; so 1 =d /C30d?=N and X1 d /C301 NX d?/C301 NX d /C30s(N) N; (11) and we have 1 H(N) /C301 n(N)s(N) N/C30A(N) N (12) N /C30A(N)H(N) : (13) Given three INTEGERS chosen at random, the prob- ability that no common factor will divide them all is z(3)½/C138/C281:1 :20206/C281 :0:831907 ; (14) where z(3) is APE´ RY’S CONSTANT . The smallest numbers having exactly 0, 1, 2, ... divisors (other than 1) are 1, 2, 4, 6, 16, 12, 64, 24, 36, ... (Sloane’s A005179). Letf(n) be the number of elements in the greatest subset of [1 ;n] such that none of its elements are divisible by two others. For nsufficiently large, 0:6725 . . . 5f(n) n50:673 . . . (15) (Le Lionnais 1983, Lebensold 1976/1977). See also ALIQUANT DIVISOR ,ALIQUOT DIVISOR ,ALI- QUOT SEQUENCE ,DIRICHLET DIVISOR PROBLEM ,DIVI- DEND ,D IVISION ,D IVISOR (CURVE ), DIVISOR FUNCTION ,D IVISOR THEORY , E-DIVISOR ,E XPONEN- TIAL DIVISOR ,GREATEST COMMON DIVISOR ,IMPROPER DIVISOR ,INFINARY DIVISOR , K-ARY DIVISOR ,PERFECT NUMBER ,PROPER DIVISOR ,UNITARY DIVISOR References Guy, R. K. "Solutions of d(n)/C30d(n/C271):/"§B18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 73 /C1/5, 1994. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 43, 1983. Lebensold, K. "A Divisibility Problem." Studies Appl. Math. 56, 291/C1/94, 1976/1977. Nagell, T. "Divisors." §1i n Introduction to Number Theory. New York: Wiley, pp. 11 /C1/2, 1951. Sloane, N. J. A. Sequences A005179/M1026 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Divisor Function /sk(n) for nan integer is defined as the sum of the kth POWERS of the DIVISORS ofn. As an illustrative example, consider the number 140, which has DIVI- SORS di/C301;2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140 (for a total of N/C3012 of them). Therefore, d(140)/C30s0(140)/C30N/C3012 (1) s(140)/C30s1(140)/C30XN i/C301di/C30336 (2) s2(140)/C30XN i/C301d2 i/C3027;300 (3) s3(140)/C30XN i/C301d3i/C303;164;112: (4) The divisor function can also be generalized to GAUSSIAN INTEGERS . The function s0(n) gives the total number of DIVISORS ofnand is often denoted d(n);n(n);t(n);orVnðÞ: (Hardy and Wright 1979, pp. 354 /C1/55). The first few values of s0(n) are 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ... (Sloane’s A000005). These values can be found as theinverse M O¨BIUS TRANSFORM of 1, 1, 1, ... (Sloane and Plouffe 1995, p. 22). Heath-Brown (1984) proved thats 0(n)/C30s0(n/C271) infinitely often. The function s1(n) is equal to the sum of DIVISORS ofn and is often denoted s(n):The first few values of s(n) are 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, ... (Sloane’s A000203).The first few values of s 2(n) are 1, 5, 10, 21, 26, 50, 50, 85, 91, 130, ... (Sloane’s A001157). The first fewvalues of s 3(n) are 1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, ... (Sloane’s A001158). The sum of the DIVISORS ofnexcluding nitself (i.e., the PROPER DIVISORS ofn) is called the RESTRICTED DIVISOR FUNCTION and is denoted s(n):The first few values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... (Sloane’s A001065). The sum of divisors s(N) can be found as follows. Let N/C13abwith a"band ( a;b)/C301:For any divisor dof N,d/C30aibi;where aiis a divisor of aandbiis a divisorofb. The divisors of aare 1, a1;a2;..., and a. The divisors of bare 1, b1;b2/, ..., b. The sums of the divisors are then s(a)/C301/C27a1/C27a2/C27:::/C27a (5) s(b)/C301/C27b1/C27b2/C27:::/C27b: (6) For a given ai; ai1/C27b1/C27b2/C27:::/C27b ðÞ /C30ais(b): (7) Summing over all ai; 1/C27a1/C27a2/C27:::/C27a ðÞ s(b)/C30s(a)s(b); (8) sos(N)/C30s(ab)/C30s(a)s(b):Splitting aand binto prime factors, s(N)/C30spa1 1ðÞspa2 2ðÞ ...spar rðÞ : (9) For a prime POWER pai i;the divisors are 1, pi;p2 i;...,pai i; so spai i0CB0C@ /C301/C27pi/C27p2 i/C27:::/C27pai i/C30pai/C271 i/C281 pi/C281: (10) ForN, therefore, s(N)/C30Yr i/C301pa/C271 i/C281 pi/C281: (11) For the special case of NaPRIME , (11) simplifies to s(p)/C30p2/C281 p/C281/C30p/C271: (12) ForNaPOWER of two, (11) simplifies to s2aðÞ/C302a/C271/C281 2/C281/C302a/C271/C281: (13) The identity (9) can be generalized to sk(N)/C30skpa1 1ðÞskpa2 2ðÞ :::skpar rðÞ : (14) In general, sk(n)/C13X d½ndk: (15) Thes(n) function has the series expansion s(n)/C301 6p2n1/C27/C281ðÞn 22/C272 cos2 3np ! 322 66664 /C272cos1 2np ! 42/C272cos25np ! /C27cos45np !0C1@ 5 2/C27...2 666643 77775(16) (Hardy 1999). Ramanujan gave the beautiful formula X/C12 n/C301sa(n)sb(n) ns /C30z(s)z(s/C28a)z(s/C28b)z(s/C28a/C28b) z(2s/C28a/C28b); (17) where z(n) is the ZETA FUNCTION and / R½s/C138;R½s/C28a/C138;R½s/C28b/C138;R½s/C28a/C28b/C138/C211/(Wilson 1923), which was used by Ingham in a proof of the PRIME NUMBER THEOREM (Hardy 1999, pp. 59 /C1/0). This gives the special case X/C12 n/C301d(n) ½/C1382 ns/C30z(s)½/C1384 z(2s)(18) (Hardy 1999, p. 59). The divisor function also satisfies the INEQUALITY s(n) nln ln n5eg/C2721/C28ffiffiffi 2p0CB0C@ /C27g/C28ln(4p)ffiffiffiffiffiffiffiffi lnnp ln ln n /C27o1ffiffiffiffiffiffiffiffi lnnp ln ln n ðÞ2 ! ; (19) where gis the E ULER- MASCHERONI CONSTANT (Robin 1984, Erdos 1989). Let a number nhave prime factorization n/C30Yr j/C301paj j; (20) then s(n)/C30Yr j/C301paj/C271 j/C281 pj/C281(21) (Berndt 1985). G RONWALL’S THEOREM states that lim n0/C12s(n) nln ln n/C30eg; (22) where gis the E ULER- MASCHERONI CONSTANT .s(n)i s a power of 2 IFFn/C301o r nis a product of distinct MERSENNE PRIMES (Sierpinski1958/59, Sivaramak- rishnan 1989, Kaplansky 1999). The first few such nare 1, 3, 7, 21, 31, 93, 127, 217, 381, 651, 889, 2667, ... (Sloane’s A046528), and the powers of 2 these correspond to are 0, 2, 3, 5, 5, 7, 7, 8, 9, 10, 10, 12, 12, 13, 14, ... (Sloane’s A048947). Curious identities derived using MODULAR FORM theory are given by s3(n)/C28s7(n)/C27120Xn/C281 k/C301s3(k)s3(n/C28k)/C300 (23) /C2810s3(n)/C2721s5(n)/C2811s9(n)/C275040Xn/C281 k/C301s3(k)s5(n/C28k) /C300 (24)(Apostol 1997, p. 140), together with 21s5(n)/C2820s7nðÞ/C28s13(n)/C2710080Xn/C281 k/C301s5(n/C28k)s7(k) /C300 (25) /C2810s3(n)/C2711s9nðÞ/C28s13(n)/C272640Xn/C281 k/C301s3(n/C28k)s9(k) /C300 (26) /C2821s5(n)/C2722s9nðÞ/C28s13(n)/C282904Xn/C281 k/C301s9(n/C28k)s9(k) /C27504Xn/C281 k/C301s5(n/C28k)s13(k)/C300 (27) (M. Trott).The divisor function is ODD IFF nis a SQUARE NUMBER or twice a SQUARE NUMBER . The divisor function satisfies the CONGRUENCE ns(n)/C132 (mod f(n)); (28) for all PRIMES and no COMPOSITE NUMBERS with the exception of 4, 6, and 22 (Subbarao 1974). r(n)i s PRIME whenever s(n) is (Honsberger 1991). Factor- izations of spaðÞfor PRIME pare given by Sorli. In 1838, Dirichlet showed that the average number of DIVISORS of all numbers from 1 to nis asymptotic to Pn i/C301s0(i) n/C2lnn/C272g/C281 (29) (Conway and Guy 1996; Hardy 1999, p. 55), as illustrated above, where the thin solid curve plots the actual values and the thick dashed curve plots the asymptotic function. This is related to the D IRICHLET DIVISOR PROBLEM , which seeks to find the "best" coefficient uin Xn k/C301n(k)/C30nlnn/C27(2g/C281)n/C27Onu0CB0C@ : (30) A more precise formula is given by Xn k /C302s0(k) /C30n ln lnn /C27B2n /C27o(n) ; (31) where B2 /C30 g /C27X p primeln 1 /C28p /C2810CB0C@ /C271 p /C28 1"# :1 :034653 (32) (Hardy and Wright 1979, p. 355). The SUMMATORY FUNCTIONS for sa with a /C211 are Xn k/C301sa(k) /C30z(a /C27 b) a /C27 1na /C271 /C27O naðÞ : (33) Fora/C301, Xn k/C301s1(k)/C30p2 12n2/C27O(nlnn): (34) See also DIRICHLET DIVISOR PROBLEM ,D IVISOR , DIVISOR PRODUCT ,EVEN DIVISOR FUNCTION ,FACTOR , GREATEST PRIME FACTOR ,G RONWALL’S THEOREM , LEAST PRIME FACTOR ,M ULTIPLY PERFECT NUMBER , ODD DIVISOR FUNCTION ,O RE’S CONJECTURE ,PER- FECT NUMBER ,RESTRICTED DIVISOR FUNCTION ,SIL- VERMAN CONSTANT ,SUM OF SQUARES FUNCTION ,TAU FUNCTION ,T OTIENT FUNCTION ,T OTIENT VALENCE FUNCTION ,TWIN PEAKS ,UNITARY DIVISOR FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Divisor Func- tions." §24.3.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 827, 1972. Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 140, 1997. Berndt, B. C. Ramanujan’s Notebooks: Part I. New York: Springer-Verlag, p. 94, 1985. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 260 /C1/61, 1996. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 279 /C1/ 25, 1952. Dirichlet, G. L. "Sur l’usage des se ´ries infinies dans la the´orie des nombres." J. reine angew. Math. 18, 259/C1/74, 1838. Erdos, P. "Ramanujan and I." In Proceedings of the Inter- national Ramanujan Centenary Conference held at AnnaUniversity, Madras, Dec. 21, 1987. (Ed. K. Alladi). New York: Springer-Verlag, pp. 1 /C1 /0, 1989. Guy, R. K. "Solutions of ms(m)/C30ns(n);/" "Analogs with d(n); sk(n);/" "Solutions of s(n)/C30s(n/C271);/" and "Solutions of s(q)/C27s(r)/C30s(q/C27r):/"§B11, B12, B13 and B15 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 67 /C1/0, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 55 and 141, 1999. Hardy, G. H. and Weight, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 354 /C1/55, 1979.Heath-Brown, D. R. "A Parity Problem from Sieve Theory." Mathematika 29,1/C1/, 1982. Heath-Brown, D. R. "The Divisor Function at Consecutive Integers." Mathematika 31, 141/C1/49, 1984. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 250 /C1/51, 1991. Kaplansky, I. "The First Two Chapters of Dickson’s History." Unpublished manuscript, Apr. 1999. Nagell, T. Introduction to Number Theory. New York: Wiley, pp. 26 /C1/7, 1951. Robin, G. "Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann." J. Math. Pures Appl. 63, 187/C1/13, 1984. Sierpinski, W. "Sur les nombres dont la somme de diviseurs est une puissance du nombre 2." Calcutta Math. Soc. Golden Jubilee Commemoration 1958/59, Part I. Cal- cutta: Calcutta Math. Soc., pp. 7 /C1/, 1963. Sloane, N. J. A. Sequences A000005, A000203, A001065, A001157, A001158, A046528, and A048947 in "An On-Line Version of the Encyclopedia of Integer Sequences."http://www.research.att.com/~njas/sequences/eisonli-ne.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Sivaramakrishnan, R. Classical Theory of Arithmetic Func- tions. New York: Dekker, 1989. Subbarao, M. V. "On Two Congruences for Primality." Pacific J. Math. 52, 261/C1 /68, 1974. Wilson, B. M. "Proofs of Some Formulae Enunciated by Ramanujan." Proc. London Math. Soc. 21, 235/C1/55, 1923. Divisor Product Letp(n) denote the product of the divisors of n including nitself. For n/C301, 2, ..., the first few values are 1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, ... (Sloane’s A007955). The following table gives values ofnfor which p(n)i sa Pth power. Lionnet (1879) considered the case P/C301. PSloane n 1 Sloane’s A0489431, 6, 8, 10, 14, 15, 16, 21, 22,24, 26, ... 2 Sloane’s A0489441, 4, 8, 9, 12, 18, 20, 25, 27,28, 32, ... 3 Sloane’s A0489451, 24, 30, 40, 42, 54, 56, 66,70, 78, ... 4 Sloane’s A0489461, 16, 32, 48, 80, 81, 112, 144,162, ... Write the prime factorization of a number n, n/C30p a1 1pa2 2/C1/C1/C1parr: Then the power of pioccurring in p(n)i s 1 2aia1/C271 ðÞ a2/C271 ðÞ /C1 /C1 /C1 ar/C271 ðÞ (Kaplansky 1999). This allows rules for determining when p(n) is a power of nto be determined, as considered by Halcke (1719) and Lionnet (1879). Let p, q, and r be distinct primes, then the following table gives the conditions and first few n for which p(n)isa given power P of n (Dickson 1952, Ireland and Rosen 1990, Kaplansky 1999). The case of third powers corresponds to numbers having exactly six divisors, the case of forth powers to numbers having eight divisors, and so on. P Forms Sloane n 2 /p3 ; pq A007422 6, 8, 10, 14, 15, 21, 22, ... 3 /p5 ; p2q/ A030515 12, 18, 20, 28, 32, 44, ... 4 /p7 ; p3q; pqrA030626 24, 30, 40, 42, 54, 56, ... 5 /p9 ; p4q/ A030628 48, 80, 112, 162, 176, ... References Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 58, 1952. Halcke, P. Exs. 150 /C1/52 in Deliciae Mathematicae; oder, Mathematisches sinnen-confect. Hamburg, Germany: N. Sauer, p. 197, 1719. Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, p. 19, 1990. Kaplansky, I. "The First Two Chapters of Dickson’s History." Unpublished manuscript, Apr. 1999. Lionnet, E. "Note sur les nombres parfaits." Nouv. Ann. Math. 18, 306 /C1/08, 1879. Lucas, E. Ex. 6 in The´orie des nombres. Paris: Gauthier- Villars, p. 373, 1891. Sloane, N. J. A. Sequences A007422/M4068, A007955, A030515, A030626, A030628, A048943, A048944, A048945, and A048946 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993. Divisor Theory A generalization by Kronecker of Kummer’s theory of PRIME IDEAL factors. A divisor on a full subcategory C of mod( A) is an additive mapping x on C with values in a SEMIGROUP of IDEALS on A. See also IDEAL ,IDEAL NUMBER ,PRIME IDEAL ,SEMI- GROUP References Edwards, H. M. Divisor Theory. Boston, MA: Birkha ¨user, 1989. Vasconcelos, W. V. Divisor Theory in Module Categories. Amsterdam, Netherlands: North-Holland, pp. 63 /C1/4, 1974.Divorce Digraph A binary relation associated with an instance of the STABLE MARRIAGE PROBLEM . Stable marriages corre- spond to vertices with outdegree 0 in the divorce digraph (Skiena 1990, p. 252). See also STABLE MARRIAGE PROBLEM References Gusfield, D. and Irving, R. W. The Stable Marriage Problem: Structure and Algorithms. Cambridge, MA: MIT Press, 1989. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Dixon’s Factorization Method In order to find INTEGERS xandysuch that x2/C13y2(mod n) (1) (a modified form of F ERMAT’S FACTORIZATION METHOD ), in which case there is a 50% chance that GCD( n;x/C28y)i sa FACTOR ofn, choose a RANDOM INTEGER ri;compute griðÞ/C13r2 i(mod n); (2) and try to factor griðÞ:IfgriðÞis not easily factorable (up to some small trial divisor d), try another ri:In practice, the trial rs are usually taken to beffiffiffinpbc/C27k; with k/C301, 2, ..., which allows the QUADRATIC SIEVE factorization method to be used. Continue finding and factoring griðÞ /s until N/C13pdare found, where pis the PRIME COUNTING FUNCTION . Now for each griðÞ;write griðÞ/C30pa1i 1ipa2i 2i:::paNi Ni; (3) and form the EXPONENT VECTOR vriðÞ/C30a1i a2i n aNi2 6643 775: (4) Now, if akiare even for any k, then griðÞis a SQUARE NUMBER and we have found a solution to (1). If not, look for a LINEAR COMBINATION aicivriðÞsuch that the elements are all even, i.e., c1a11 a21 n aN12 6643 775/C27c2a12 a22 n aN22 6643 775/C27/C1/C1/C1/C27cNa1N a2N n aNN2 6643 775/C300 0 n 02 6643 775 mod2ðÞ(5) a11a12 /C1/C1/C1 a1N a21a22 /C1/C1/C1 a2N nn:::n aN1aN2/C1/C1/C1 aNN2 6643 775c 1 c2 n cN2 6643 775/C300 0 n 02 6643 775mod2ðÞ : (6) Since this must be solved only mod 2, the problem can be simplified by replacing the a ij/s with bij /C300 1for aij even for aij odd :0C1n (7) GAUSSIAN ELIMINATION can then be used to solve bc /C30z (8) for c, where z is a VECTOR equal to 0 (mod2) . Once c is known, then we have Y kgrkðÞ/C13Y kr2 k (mod n) ; (9) where the products are taken over all k for which ck /C30 1: Both sides are PERFECT SQUARES , so we have a 50% chance that this yields a nontrivial factor of n.Ifit does not, then we proceed to a different z and repeat the procedure. There is no guarantee that this method will yield a factor, but in practice it produces factors faster than any method using trial divisors. It is especially amenable to parallel processing, since each processor can work on a different value of r. References Bressoud, D. M. Factorization and Prime Testing. New York: Springer-Verlag, pp. 102 /C1/04, 1989. Dixon, J. D. "Asymptotically Fast Factorization of Integers." Math. Comput. 36, 255 /C1/60, 1981. Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673 /C1/15, 1990. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/485, 1996. Dixon’s Identity Xn k/C30/C28n/C281ðÞk n /C27b n /C27k0C@80C@9 n /C27c c /C27k0C@80C@9 b /C27c b /C27k0C@80C@9 /C30G b /C27 c /C27 n /C27 1 ðÞ n! G b /C27 1 ðÞ G c /C27 1 ðÞ; (1) wheren k0CB0C@ is a BINOMIAL COEFFICIENT and G(x)isa GAMMA FUNCTION . See also DIXON’S THEOREM References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 11 and 18 /C1/9, 1998. Dixon’s Random Squares Factorization Method DIXON’S FACTORIZATION METHOD Dixon’s Theorem 3F2n;/C28x;/C28y x /C27n /C271;y /C27n /C2710C1B0C1@ /C30G(x /C27n /C271)G(y /C27n /C271)G/C21 2n /C271 ! G x /C27y /C2712 n /C271 ! /C29G(n /C271)G x /C27y /C27n /C271 ðÞ G /C2 x /C2712 n /C271 ! G y /C2712 n /C271 ! ; (1) where 3F2(a; b;c;d;e;z)isa GENERALIZED HYPERGEO- METRIC FUNCTION and G(z) is the GAMMA FUNCTION .It can be derived from the DOUGALL- RAMANUJAN IDEN- TITY. It can be written more symmetrically as 3F2a;b ;c;d ;e;1 ðÞ /C3012 a ! !(a /C28 b)!(a /C28 c)!12 a /C28 b /C28 c ! ! a!12 a /C28 b ! !12a /C28 c ! ! a /C28 b /C28 c ðÞ !; (2) where 1 /C27a=2 /C28b /C28c has a positive REAL PART , d /C30 a /C28b /C271 ; and e /C30a /C28c /C271 (Bailey 1935, p. 13; Pet- kovsek 1996; Koepf 1998, p. 32). The identity can also be written as the beautiful symmetric sum X k/C281ðÞk a /C27b a /C27k0C@80C@9 a /C27c c /C27k0C@80C@9 b /C27c b /C27k0C@80C@9 /C30a /C27 b /C27 c ðÞ ! a!b!c!(3) (Petkovsek 1996). In this form, it closely resembles DIXON’S IDENTITY . See also DIXON’S IDENTITY ,D OUGALL- RAMANUJAN IDENTITY ,G ENERALIZED HYPERGEOMETRIC FUNC- TION ,ZEILBERGER- BRESSOUD THEOREM References Bailey, W. N. "Dixon’s Theorem." §3.1 in Generalised Hy- pergeometric Series. Cambridge, England: Cambridge University Press, pp. 13 /C1/4, 1935. Cartier, P. and Foata, D. Proble `mes combinatoires de commutation et re ´arrangements. New York: Springer- Verlag, 1969. Dixon, A. C. "On the Sum of the Cubes of the Coefficients in Certain Expansion by the Binomial Theorem." Messenger Math. 20,7 9/C1/0, 1891. Dixon, A. C. "Summation of Certain Series." Proc. London Math. Soc. 35, 285/C1/89, 1903. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 104 and 111, 1999. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 18 /C1/9, 1998. MacMahon P. A. "The Sums of the Powers of the Binomial Coefficients." Quart. J. Math. 33, 274/C1/88, 1902. Morley, F. "On the Series 1 /C27p 10C@n0C@o3 /C27p(p/C271) 1/C2152no2 /C27...:/"Proc. London Math. Soc. 34, 397/C1/02, 1902. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, p. 43, 1996. Richmond, H. W. "The Sum of the Cubes of the Coefficients in 1/C28x ðÞ2n:/"Messenger Math. 21,7 7/C1/8, 1892. Watson, G. N. "Dixon’s Theorem on Generalized Hypergeo- metric Functions." Proc. London Math. Soc. 22, xxxii- xxxiii (Records for 17 May, 1923), 1924. Zeilberger, D. and Bressoud, D. "A Proof of Andrews’ q- Dyson Conjecture." Disc. Math. 54, 201 /C1/24, 1985. Dixon-Ferrar Formula Let Jn(z)beaB ESSEL FUNCTION OF THE FIRST KIND , Yn(z)aB ESSEL FUNCTION OF THE SECOND KIND , and Kn(z)a MODIFIED BESSEL FUNCTION OF THE FIRST KIND . Also let R[z] > 0 and R[z] jjB1 =2: Then J2 n (z) /C27Y2 n (z) /C308 cos(n p) p2 g/C12 0K2n(2z sinhdt): See also NICHOLSON’S FORMULA ,W ATSON’S FORMULA References Gradshteyn, I. S. and Ryzhik, I. M. Eqn. 6.518 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 671, 2000. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1476, 1980. dn JACOBI ELLIPTIC FUNCTIONS # 1999 /C1/001 Wolfram Research, Inc. D-Number A NATURAL NUMBER n /C213 such that njðan/C282 /C28a Þ whenever /ða ;nÞ¼1/ (a and n are RELATIVELY PRIME ) and /a 5n/. (Here, /njm/ means that n DIVIDES m.) There are an infinite number of such numbers, the first few being 9, 15, 21, 33, 39, 51, ... (Sloane’s A033553). See also DIVIDE ,KNO¨ DEL NUMBERS References Makowski, A. "Generalization of Morrow’s D-Numbers." Simon Stevin 36, 71, 1962/1963. Sloane, N. J. A. Sequences A033553 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Dobinski’s Formula The general formula states that fn(x)/C30e/C28xX/C12 k/C300kn k!xk; (1) where fn(x)i sa n EXPONENTIAL POLYNOMIAL (Roman 1984, p. 66). Setting x/C301 gives the special case of the nth B ELL NUMBER ,Bn/C301 eX/C12 k/C300kn k!: (2) It can be derived by dividing the formula for a STIRLING NUMBER OF THE SECOND KIND bym!;yield- ing mn m!/C30Xm k/C301n k0C1n0C1o1 (m/C28k)!: (3) Then X/C12 k/C301mn m!lm/C30Xn k/C301n k0C1n0C1o lk !X/C12 k/C300lj j! ! ; (4) and Xn k/C301n k0C1n0C1o lk/C30e/C28lX/C12 m/C301mn m!lm: (5) Now setting l/C301 gives the identity (Dobinski 1877; Rota 1964; Berge 1971, p. 44; Comtet 1974, p. 211;Roman 1984, p. 66; Lupas 1988; Wilf 1990, p. 106;Chen and Yeh 1994; Pitman 1997). References Berge, C. Principles of Combinatorics. New York: Academic Press, 1971. Chen, B. and Yeh, Y.-N. "Some Explanations of Dobinski’s Formula." Studies Appl. Math. 92, 191/C1/99, 1994. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Dobinski, G. "Summierung der Reihe /amm=n!/fu¨rm/C301, 2, 3, 4, 5, ...." Grunert Archiv (Arch. Math. Phys.) 61, 333/C1/36, 1877. Foata, D. La se´rie ge ´ne´ratrice exponentielle dans les proble `- mes d’e ´nume ´ration. Vol. 54 of Se´minaire de Mathe ´ma- tiques supe ´rieures. Montre ´al, Canada: Presses de l’Universite ´de Montre ´al, 1974. Lupas, A. "Dobinski-Type Formula for Binomial Polyno- mials." Stud. Univ. Babes-Bolyai Math. 33,3 0/C1/4, 1988. Pitman, J. "Some Probabilistic Aspects of Set Partitions." Amer. Math. Monthly 104, 201/C1/09, 1997. Roman, S. The Umbral Calculus. New York: Academic Press, p. 66, 1984. Rota, G.-C. "The Number of Partitions of a Set." Amer. Math. Monthly 71, 498/C1/04, 1964. Wilf, H. Generatingfunctionology, 2nd ed. San Diego, CA: Academic Press, 1990. Dodecadodecahedron The UNIFORM POLYHEDRON U36whose DUAL POLYHE- DRON is the MEDIAL RHOMBIC TRIACONTAHEDRON . The solid is also called the GREAT DODECADODECAHEDRON , and its DUAL POLYHEDRON is also called the SMALL STELLATED TRIACONTAHEDRON . The dodecadodecahe- dron has SCHLA ¨ FLI SYMBOL5 2 ;5no and WYTHOFF SYMBOL 2525:0C@10C@10C@1 Its faces are 125 2no /C2712 5fg; and its CIRCUMRADIUS for unit edge length is R /C301 : It can be obtained by TRUNCATING a GREAT DODECA- HEDRON or FACETING a ICOSIDODECAHEDRON with PENTAGONS and covering remaining open spaces with PENTAGRAMS (Holden 1991, p. 103). A FACETED version is the GREAT DODECAHEMICOSAHE- DRON . The CONVEX HULL of the dodecadodecahedron is an ICOSIDODECAHEDRON and the dual of the ICOSIDODECAHEDRON is the RHOMBIC TRIACONTAHE- DRON , so the dual of the dodecadodecahedron is one of the RHOMBIC TRIACONTAHEDRON STELLATIONS (Wen- ninger 1983, p. 41). References Cundy, H. and Rollett, A. "Great Dodecadodecahedron. /ð5 /C2155 2Þ2 /." §3.9.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 123, 1989. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 41, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 112, 1989. Dodecagon A 12-sided polygon. The regular dodecagon is CON- STRUCTIBLE denoted using the SCHLA ¨ FLI SYMBOL f12 g: The INRADIUS r, CIRCUMRADIUS R, and AREA A can be computed directly from the formulas for a general REGULAR POLYGON with side length s and n /C3012 sides, r /C301 2s cotp 12 ! /C30122 /C27ffiffiffi 3p0C@n0C@o s (1) R /C301 2 s cscp 12 ! /C3012ffiffiffi 2p /C27ffiffiffi 6p0C@n0C@o s (2)A /C301 4 ns2 cotp 12 ! /C3032/C27ffiffiffi 3p0C@n0C@o s2 : (3) KURSCHA ´ K’S THEOREM gives the AREA of the dodeca- gon inscribed in a UNIT CIRCLE with R /C301, A/C301 2nR2sin2p n ! /C303 (4) (Wells 1991, p. 137). APLANE PERPENDICULAR to aC5axis of a DODECAHE- DRON orICOSAHEDRON cuts the solid in a regular DECAGONAL CROSS SECTION (Holden 1991, pp. 24 /C1/5). The G REEK ,LATIN, and M ALTESE CROSSES are all irregular dodecagons. See also DECAGON ,D ODECAGRAM ,D ODECAHEDRON , GREEK CROSS ,K URSCHA ´ K’S THEOREM ,K URSCHA ´ K’S TILE,LATIN CROSS ,M ALTESE CROSS ,TRIGONOMETRY VALUES PI/12,UNDECAGON References Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 56 /C1/7 and 137, 1991. Dodecagram The STAR POLYGON f12 =5g:/ See also POLYGON ,POLYGRAM ,STAR POLYGON ,TRI- GONOMETRY VALUES PI/12 Dodecahedral Conjecture In any unit SPHERE PACKING , the volume of any VORONOI CELL around any sphere is at least as large as a regular DODECAHEDRON of INRADIUS 1. If true, this would provide a bound on the densest possible sphere packing greater than any currently known. It is not, however, sufficient to establish the KEPLER CONJECTURE . See also KEPLER CONJECTURE ,SPHERE PACKING Dodecahedral Graph The PLATONIC GRAPH corresponding to the connectiv- ity of the vertices of a DODECAHEDRON . Finding a HAMILTONIAN CIRCUIT on this graph is known as the ICOSIAN GAME . The dodecahedral graph has 20 nodes, 30 edges, VERTEX CONNECTIVITY 3, EDGE CONNECTIV- ITY 3, GRAPH DIAMETER 5, GRAPH RADIUS 5, and GIRTH 5. See also CUBICAL GRAPH ,ICOSAHEDRAL GRAPH , ICOSIAN GAME,O CTAHEDRAL GRAPH ,P LATONIC GRAPH ,TETRAHEDRAL GRAPH References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, 1987. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976. Chartrand, G. Introductory Graph Theory. New York: Dover, 1985. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 198, 1990.Dodecahedral Space POINCARE ´MANIFOLD Dodecahedron The regular dodecahedron is the P LATONIC SOLID P4 composed of 20 VERTICES ,3 0 EDGES , and 12 PENTA- GONAL FACES ,1 2f5g:It is also UNIFORM POLYHEDRON U23and Wenninger model W5:It is given by the SCHLA ¨FLI SYMBOL f5;3gand the W YTHOFF SYMBOL 3½25:/ Crystals of pyrite /(FeS2) resemble slightly distorted dodecahedra (Steinhaus 1983, pp. 207 /C1/08), and spha- lerite (ZnS) crystals are irregular dodecahedra bounded by congruent deltoids (Steinhaus 1983,pp. 207 and 209). The HEXAGONAL SCALENOHEDRON is another irregular dodecahedron. The D ELTOIDAL HEXECONTAHEDRON and TRIAKIS TETRAHEDRON are irregular dodecahedra composed of a single type of face, and the CUBOCTAHEDRON and TRUNCATED TET- RAHEDRON are dodecahedral A RCHIMEDEAN SOLIDS consisting of multiple types of faces. Dodecahedra were known to the Greeks, and 90 models of dodecahedra with knobbed vertices have been found in a number of archaeological excavations in Europe dating from the Gallo-Roman period inlocations ranging from military camps to public bath houses to treasure chests (Schuur). The dodecahedron has the ICOSAHEDRAL GROUP Ihof symmetries. The connectivity of the vertices is given by the DODECAHEDRAL GRAPH . There are three DODE- CAHEDRON STELLATIONS . The DUAL POLYHEDRON of the dodecahedron is the ICOSAHEDRON , so the centers of the faces of an ICOSAHEDRON form a dodecahedron, and vice versa (Steinhaus 1983, pp. 199 /C1/01). APLANE PERPENDICULAR to a C3axis of a dodecahe- dron cuts the solid in a regular HEXAGONAL CROSS SECTION (Holden 1991, p. 27). A PLANE PERPENDICU- LARto aC5axis of a dodecahedron cuts the solid in a regular DECAGONAL CROSS SECTION (Holden 1991, p. 24). ACUBE can be constructed from the dodecahedron’s vertices taken eight at a time (above left figure;Steinhaus 1983, pp. 198 /C1 /99; Wells 1991). Five such cubes can be constructed, forming the CUBE 5-COM- POUND . In addition, joining the centers of the faces gives three mutually PERPENDICULAR GOLDEN REC- TANGLES (right figure; Wells 1991). The short diagonals of the faces of the RHOMBIC TRIACONTAHEDRON give the edges of a dodecahedron (Steinhaus 1983, pp. 209 /C1/10). The following table gives polyhedra which can beconstructed by CUMULATION of a dodecahedron by pyramids of given heights h. h /(r/C27h)=h/ Result //C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10(5/C28ffiffiffi 5p )s // 2ffiffiffi 5p /C283/ 60-faced dimpled DELTA- HEDRON /1 19ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5(65/C2722ffiffiffi 5p )s //3 19(10/C28ffiffiffi5p ) /PENTAKIS DODE- CAHEDRON /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10(5/C28ffiffiffi 5p )s // 2ffiffiffi 5p /C283/ 60-faced star DELTAHEDRON /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5(5/C272ffiffiffi 5p )s //ffiffiffi 5p / SMALL STEL- LATED DODECA- HEDRON When the dodecahedron with edge lengthffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pp is oriented with two opposite faces parallel to the xy- PLANE , the vertices of the top and bottom faces lie at z/C309(f/C271) and the other VERTICES lie at z/C309(f/C281); where fis the GOLDEN RATIO . The explicit coordi- nates are 92 cos2 5pi ! ;2 sin25pi ! ;f/C271 ! (1) 92fcos25pi ! ;2fsin25pi ! ;f/C281 ! (2) with i/C300, 1, ..., 4, where fis the GOLDEN RATIO . The VERTICES of a dodecahedron can be given in a simple form for a dodecahedron of side length a/C30ffiffiffi 5p /C281 by (0, 9f/C281;9f);(/9f;0,9f/C281);(/9f/C281;9f;0), and (91,91,91). For a dodecahedron of unit edge length a/C301, the CIRCUMRADIUS R?and INRADIUS r?of a PENTAGONAL FACE are R?/C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2710ffiffiffi 5pq (3) r?/C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5p :q (4) The SAGITTA xis then given by x/C13R?/C28r?/C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 125/C2810ffiffiffi 5p :q (5) Now consider the following figure. Using the P YTHAGOREAN THEOREM on the figure then gives z2 1/C27m2/C30R?/C27r ðÞ2(6) z22/C27(m/C28x)2/C301 (7)z1/C27z2 2 !2 /C27R?2/C30z1/C28z2 2 !2 /C27m/C27r? ðÞ2: (8) Equation (3) can be written z1z2/C27r2/C30m/C27r? ðÞ2: (9) Solving (1), (2), and (9) simultaneously gives m/C30r?/C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pq (10) z1/C302r?/C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pq (11) z2/C30R?/C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2710ffiffiffi 5pq : (12) The INRADIUS of the dodecahedron is then given by r/C301 2z1/C27z2 ðÞ ; (13) so r2/C301 4025/C2711ffiffiffi 5p0C@n0C@o ; (14) and solving for rgives r/C301 20ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 250/C27110ffiffiffi 5pq /C301:11351 . . . (15) Now, R2/C30R?2/C27r2/C303 83/C27ffiffiffi 5p0C@n0C@o ; (16) so the CIRCUMRADIUS is R/C301 4ffiffiffiffiffiffi 15p /C27ffiffiffi3p0C@n0C@o /C301:40125 . . . (17) The INTERRADIUS is given by r2/C30r?2/C27r2/C301 87/C273ffiffiffi 5p0C@n0C@o ; (18) so r/C301 43/C27ffiffiffi 5p0C@n0C@o /C301:30901 . . . (19) The DIHEDRAL ANGLE is a/C30cos/C281/C281 5ffiffiffi 5p ! :116:57/C14: (20) The AREA of a single FACE is the AREA of a PENTAGON , A/C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5p :q (21) The VOLUME of the dodecahedron can be computed by summing the volume of the 12 constituent PENTAGO- NAL PYRAMIDS , V /C30121 3Ar ! /C301415 /C277ffiffiffi 5p0C@n0C@o : (22) Apollonius showed that the VOLUME V and SURFACE AREA A of the dodecahedron and its DUAL the ICOSAHEDRON are related by Vicosahedron Vdodecahedron/C30Aicosahedron Adodecahedron(23) See also AUGMENTED DODECAHEDRON ,AUGMENTED TRUNCATED DODECAHEDRON ,C AIRO TESSELLATION , CUBOCTAHEDRON ,D ELTOIDAL HEXECONTAHEDRON , DODECAGON ,DODECAHEDRON 2-COMPOUND ,DODECA- HEDRON 3-COMPOUND ,DODECAHEDRON 5-COMPOUND , DODECAHEDRON- ICOSAHEDRON COMPOUND ,DODECA- HEDRON- SMALL TRIAMBIC ICOSAHEDRON COMPOUND , DODECAHEDRON STELLATIONS ,ELONGATED DODECA- HEDRON ,GREAT DODECAHEDRON ,GREAT STELLATED DODECAHEDRON ,H YPERBOLIC DODECAHEDRON ,ICO- SAHEDRON ,METABIAUGMENTED DODECAHEDRON ,ME- TABIAUGMENTED TRUNCATED DODECAHEDRON , PARABIAUGMENTED DODECAHEDRON ,PARABIAUGMEN- TED TRUNCATED DODECAHEDRON ,P YRITOHEDRON , RHOMBIC DODECAHEDRON ,R HOMBIC TRIACONTAHE- DRON ,SMALL STELLATED DODECAHEDRON ,STELLA- TION ,T RIAKIS TETRAHEDRON ,T RIAUGMENTED DODECAHEDRON ,TRIAUGMENTED TRUNCATED DODE- CAHEDRON ,TRIGONAL DODECAHEDRON ,TRIGONOME- TRY VALUES PI/5,T RUNCATED DODECAHEDRON , TRUNCATED TETRAHEDRON References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 228, 1987. Cundy, H. and Rollett, A. "Dodecahedron. 53." §3.5.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 87, 1989. Davie, T. "The Dodecahedron." http://www.dcs.st-and.ac.uk/ ~ad/mathrecs/polyhedra/dodecahedron.html. Harris, J. W. and Stocker, H. "Dodecahedron." §4.4.5 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 101, 1998. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Schuur, W. A. "Pentagonale Dodecaeder." http:// home.wxs.nl/~wschuur/dcaeder.htm. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 195 /C1/99, 1999. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 57 /C1/8, 1991. Wenninger, M. J. "The Dodecahedron." Model 5 in Polyhe- dron Models. Cambridge, England: Cambridge University Press, p. 19, 1989.Dodecahedron 2-Compound A compound of two dodecahedra having the symme- try of the CUBE arises by combining two dodecahedra rotated 908 with respect to each other about a common C2 axis (Holden 1991, p. 37). See also DODECAHEDRON ,D ODECAHEDRON 3-COM- POUND ,DODECAHEDRON 5-COMPOUND ,POLYHEDRON COMPOUND References Holden, A. Shapes, Space, and Symmetry. New York: Dover, p. 37, 1991. Dodecahedron 3-Compound See also DODECAHEDRON ,D ODECAHEDRON 2-COM- POUND ,DODECAHEDRON 5-COMPOUND Dodecahedron 5-Compound There are at least two attractive 5-dodecahedra compounds. The one illustrated in the left figure above has the symmetry of the ICOSAHEDRON and can be constructed by taking a DODECAHEDRON with top and bottom vertices aligned along the Z-AXIS and one vertex oriented in the direction of the x-axis, rotating about the Y-AXIS by an angle a /C30cos /C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 155 /C27ffiffiffi 5p0C@n0C@os ! ; and then rotating this solid by angles 2pi=5 for i /C300, 1, ..., 4. The compound shown at right can be obtained by combining five dodecahedra, each rotated by 1/10 of a turn about the line joining the centroids of opposite faces. See also DODECAHEDRON ,D ODECAHEDRON 2-COM- POUND ,DODECAHEDRON 3-COMPOUND ,POLYHEDRON COMPOUND References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 145 /C1/47, 1983. Dodecahedron Stellations The dodecahedron has three STELLATIONS : the SMALL STELLATED DODECAHEDRON , GREAT DODECAHEDRON , and GREAT STELLATED DODECAHEDRON (Wenninger 1989, pp. 35 and 38 /C1/0). Bulatov has produced 270 stellations of a deformed dodecahedron. See also DODECAHEDRON ,ICOSAHEDRON STELLA- TIONS ,STELLATED POLYHEDRON ,STELLATION References Bulatov, V. "270 Stellations of Deformed Dodecahedron." http://www.physics.orst.edu/~bulatov/polyhedra/do- deca270/. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, pp. 35 and 38 /C1/0, 1989.Dodecahedron-Icosahedron Compound APOLYHEDRON COMPOUND consisting of a DODECAHE- DRON and its dual the ICOSAHEDRON . It is most easily constructed by adding 20 triangular PYRAMIDS , con- structed as above, to an ICOSAHEDRON . In the com- pound, the DODECAHEDRON and ICOSAHEDRON are rotated p=5 radians with respect to each other, and the ratio of the ICOSAHEDRON toDODECAHEDRON edges lengths are the GOLDEN RATIO f:/ If the DODECAHEDRON is chosen to have unit edge length, the resulting compound has side lengths s1/C301 2(1) s2/C30141/C27ffiffiffi 5p0C@n0C@o : (2) Normalizing so that s 1/C301 gives SURFACE AREA and VOLUME S¼15ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13þ5ffiffiffi 5p þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ð25þ11ffiffiffi 5pqr ð3Þ V/C305 215/C277ffiffiffi 5p0C@n0C@o : (4) The above figure shows compounds composed of a DODECAHEDRON of unit edge length and ICOSAHEDRA having edge lengths varying fromffiffiffi 5p =2 (inscribed in the dodecahedron) to 2 (circumscribed about the dodecahedron). The intersecting edges of the compound form the DIAGONALS of the 30 RHOMBUSES constituting the TRIACONTAHEDRON , which is the DUAL POLYHEDRON of the ICOSIDODECAHEDRON (Ball and Coxeter 1987). The dodecahedron-icosahedron compound is also the first STELLATION of the ICOSIDODECAHEDRON . See also DUAL POLYHEDRON ,DODECAHEDRON ,ICOSA- HEDRON ,ICOSIDODECAHEDRON ,P LATONIC SOLID , POLYHEDRON COMPOUND ,R HOMBIC TRIACONTAHE- DRON References Cundy, H. and Rollett, A. "Dodecahedron Plus Icosahedron." §3.10.3 in Mathematical Models, 2nd ed. Stradbroke, England: Tarquin Pub., p. 131, 1989. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Wenninger, M. J. "First Stellation of the Icosidodecahe- dron." §47 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 76, 1989. Dodecahedron-Small Triambic Icosahedron Compound A stellated form of a truncated icosahedron, but a different truncation than in the TRUNCATED ICOSAHE- DRON ARCHIMEDEAN SOLID . It contains curious but attractive patterns of raised regular pentagrams and irregular hexagrams. For the solid constructed from a DODECAHEDRON with unit edge lengths, the SURFACE AREA is given by the root of a 10 order polynomial with large integer coefficients, and the VOLUME is given by V /C301 2035 /C2715ffiffiffiffiffiffi 15p /C284ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 650 /C28290ffiffiffi 5pq 0C@80C@9 : See also DODECAHEDRON ,SMALL TRIAMBIC ICOSAHE- DRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 51 /C1/2 1983. Dodecic Surface An ALGEBRAIC SURFACE of degree 12.See also ALGEBRAIC SURFACE ,SARTI DODECIC Dolbeault Cohomology See also CALABI- YAU SPACE ,DOLBEAULT OPERATORS Dolbeault Operators See also DEL BAR OPERATOR ,DOLBEAULT COHOMOL- OGY Domain A CONNECTED OPEN SET. The term domain is also used to describe the set of values D for which a FUNCTION is defined. The set of values to which D is sent by the function (MAP) is then called the RANGE . See also CODOMAIN ,CONNECTED SET,M AP,ONE-TO- ONE,ONTO,RANGE (IMAGE ), REINHARDT DOMAIN References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 76, 1999. Domain Invariance Theorem The Invariance of domain theorem states that if f : A 0 Rn is a ONE-TO-ONE continuous MAP from A, then a compact subset of Rn ; then the interior of A is mapped to the interior of f(A):/ See also DIMENSION INVARIANCE THEOREM Dome BOHEMIAN DOME,G EODESIC DOME,H EMISPHERE , SPHERICAL CAP,TORISPHERICAL DOME,VAULT Dominance The dominance RELATION on a SET of points in EUCLIDEAN n-space is the INTERSECTION of the n coordinate-wise orderings. A point p dominates a point q provided that every coordinate of p is at least as large as the corresponding coordinate of q. A PARTITION pa dominates a PARTITION pb if, for all k, the sum of the k largest parts of pa is ]the sum of the k largest parts of pb : For example, for n /C307, f7g dominates all other PARTITIONS , while f1; 1;1;1 ;1;1 ;1g is dominated by all others. In con- trast, f3; 1;1; 1;1g and f2;2; 2;1g do not dominate each other (Skiena 1990, p. 52). The dominance orders in Rnare precisely the POSETS ofDIMENSION at most n. See also DOMINATING SET,D OMINATION NUMBER , PARTIALLY ORDERED SET,REALIZER References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Stanton, D. and White, D. Constructive Combinatorics. New York: Springer-Verlag, 1986. Dominant Set DOMINANCE ,DOMINATING SET Dominating Set This entry contributed by NICOLAS BRAY For a GRAPH G and a subset S of the VERTEX SET V(G); denote by NG[S] the set of vertices in G which are in S or adjacent to a vertex in S.IfNG[S] /C30V(G); then S is said to be a dominating set (of vertices in G). See also DOMINANCE ,DOMINATION NUMBER Domination Number This entry contributed by NICOLAS BRAY The domination number of a graph G, denoted g(G) ; is the minimum size of a DOMINATING SET of vertices in G. See also DOMINANCE ,DOMINATING SET,VIZING CON- JECTURE References Clark, W. E. and Suen, S. "An Inequality Related to Vizing’s Conjecture." Electronic J. Combinatorics 7, No. 1, N4, 1 /C1/, 2000. http://www.combinatorics.org/Volume_7/ v7i1toc.html#N4. Haynes, T. W.; Hedetniemi, S. T.; and Slater, P. J. Domina- tion in Graphs--Advanced Topics. New York: Dekker, 1998. Haynes, T. W.; Hedetniemi, S. T.; and Slater, P. J. Funda- mentals of Domination in Graphs. New York: Dekker, 1998. Domineering A two-player game, also called crosscram, in which player H has horizontal DOMINOES and player V has vertical DOMINOES . The two players alternately place a domino on a BOARD until the other cannot move, in which case the player having made the last move wins (Gardner 1974, Lachmann et al. 2000). Depend- ing on the dimension of the board, the winner will be H, V, 1 (the player making the first move), or 2 (the player making the second move). For example, the 2 /C292 ðÞ board is a win for the first player. Berlekamp (1988) solved the general problem for 2 /C29 n board for odd n. Solutions for the 2 /C29n board are summarized in the following table, with 2 /C29n a win for H for n ]28::/ n win n win n win 02 1 01 2 0 H1V1 112 1H 21 1 2H 2 2 H 31 1 32 2 3 14H1 4 124H 5V1 512 5H 61 1 6H 2 6 H 71 1 7H 2 7 18H1 8 128H 9V1 912 9H Lachmann et al. (2000) have solved the game k /C29n for widths of n /C302, 3, 4, 5, 7, 9, and 11, obtaining the results summarized in the following table for k/C300, 1, .... nwinner 3 2 ,V ,1 ,1 ,H ,H ,. . . 4 H for even k]8 and all k]22 / 5 2 ,V ,H ,V ,H ,2 ,H ,H ,. . . 7 H for n]8/ 9 H for n]22/ 11 H for n]56/ See also DOMINO References Berlekamp, E. R. "Blockbuster and Domineering." J. Com- bin. Th. Ser. A 49,6 7/C1/16, 1988. Berlekamp, E. R.; Conway, J. H.; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982. Breuker, D. M.; Uiterwijk, J. W. H. M.; van den Herik, H. J. "Solving 8 /C298 Domineering." Theor. Comput. Sci. 122, 43/C1/8, 2000. Conway, J. H. On Numbers and Games. New York: Aca- demic Press, 1976. Gardner, M. "Mathematical Games: Cram, Crosscram and Quadraphage: New Games having Elusive Winning Stra-tegies." Sci. Amer. 230, 106/C1 /08, Feb. 1974. Lachmann, M.; Moore, C.; and Rapaport, I. Who Wins Domineering on Rectangular Boards? 8 Jun 2000. http://xxx.lanl.gov/abs/math.CO/0006066/. Uiterwijk, J. W. H. M. and van den Herik, H. J. "The Advantage of the Initiative." Info. Sci. 122,4 3/C1 /8, 2000. Wolfe, D. "The Gamesman’s Toolkit." In Games of No Chance. (Ed. R. J. Nowakowski). Cambridge, England: Cambridge University Press, 1998. Domino The unique 2-POLYOMINO consisting of two equal squares connected along a complete EDGE . The FIBONACCI NUMBER Fn/C271gives the number of ways for 2 /C291 dominoes to cover a 2 /C29n CHECKER- BOARD , as illustrated in the following diagrams (Dickau). See also DOMINEERING ,F IBONACCI NUMBER ,G O- MORY’S THEOREM ,HEXOMINO ,PENTOMINO ,POLYOMI- NO,POLYOMINO TILING ,TETROMINO ,TRIOMINO References Culin, S. "Kol-hpai, Bone Tablets--Dominoes." §81 in Games of the Orient: Korea, China, Japan. Rutland, VT: Charles E. Tuttle, pp. 102 /C1/03, 1965. Cohn, H. "2-adic Behavior of Numbers of Domino Tilings." Electronic J. Combinatorics 6, No. 1, R14, 1 /C1/, 1999. http:// www.combinatorics.org/Volume_6/v6i1toc.html#R14. Dickau, R. M. "Fibonacci Numbers." http://www.prairiene- t.org/~pops/fibboard.html. Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer- ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 124 /C1/40, 1959. Kraitchik, M. "Dominoes." §12.1.22 in Mathematical Recrea- tions. New York: W. W. Norton, pp. 298 /C1/02, 1942. Lei, A. "Domino." http://www.cs.ust.hk/~philipl/omino/dom- ino.html. Madachy, J. S. "Domino Recreations." Madachy’s Mathema- tical Recreations. New York: Dover, pp. 209 /C1/19, 1979. Schroeppel, R. Item 111 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 48, Feb. 1972. Domino Problem WANG’S CONJECTUREDonaldson Invariants Distinguish between smooth MANIFOLDS in 4-D. See also DONALDSON THEORY Donaldson Theory See also DONALDSON INVARIANTS Donkin’s Theorem The product of three translations along the directed sides of a TRIANGLE through twice the lengths of these sides is the IDENTITY MAP. Donut TORUS Doob’s Theorem A theorem proved by Doob (1942) which states that any random process which is both GAUSSIAN and MARKOV has the following forms for its correlation function Cy( t) ; spectral density Gy(f) ; and probability densities p1(y) and p2(y1 ½y2 ; t):: Cy t ¼ s2 ye /C28 t=tr Gy(f) /C304t /C281 ts2 y 2 pfðÞ2/C27t /C282 t p1(y) /C301ffiffiffiffiffiffiffi 2 ps2 yp e /C28(y/C28y)2 =2 s2 y p2(y1 =y2 ; t) /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p 1 /C28 e/C28 t= tt ðÞ s2 yq exp /C2/C28y2 /C28 ¯y ðÞ /C28 e /C28 t=tty1 /C28 ¯y ðÞ0C10CC2 21/C28 e /C282t= tt ðÞ s2 y() ; where ¯y is the MEAN , sy the STANDARD DEVIATION , and tr the relaxation time. References Doob, J. L. "Topics in the Theory of Markov Chains." Trans. Amer. Math. Soc. 52,37/C1/4, 1942. Dorman-Luke Construction DUAL POLYHEDRON Dot The "dot" /C215 has several meanings in mathematics, including MULTIPLICATION /(a:bis pronounced " a times b"), computation of a DOT PRODUCT (a/C215bis pronounced " adotb"). See also DERIVATIVE ,DOT PRODUCT ,OVERDOT ,TIMES Dot Product The dot product can be defined for two VECTORS X and Y by X /C215Y /C30½X ½½Y ½ cos u ; ð1Þ where u is the ANGLE between the VECTORS . It follows immediately that X /C215Y /C300ifX is PERPENDICULAR to Y. The dot product therefore has the geometric inter- pretation as the length of the PROJECTION of X onto the UNIT VECTOR Y when the two vectors are placed so that their tails coincide. By writing Ax /C30A cos uABx /C30B cos uB (2) Ay /C30A sin uABy /C30B sin uB ; (3) it follows that (1) yields A /C215B /C30AB cos uA /C28 uB ðÞ /C30AB cos uA cos uB /C27sin uA sin uB ðÞ /C30A cos uAB cos uB /C27A sin uAB sin uB /C30AxBx /C27AyBy : (4) So, in general, X /C215Y /C30Xn i/C301xiyi /C30x1y1 /C27/C1/C1/C1/C27xnyn : (5) This can be written very succinctly using EINSTEIN SUMMATION notation as X /C215Y /C30xiyi : (6) The dot product is implemented in Mathematica as Dot[a, b], or simply by using a period, a . b. The dot product is COMMUTATIVE X /C215Y /C30Y /C215X ; (7) ASSOCIATIVE (rX) /C215Y /C30r(X /C215Y) ; (8) and DISTRIBUTIVE X /C215(Y /C27Z) /C30X /C215Y /C27X /C215Z : (9) The DERIVATIVE of a dot product of VECTORS is d dtr1(t) /C215r2(t) ½/C138 /C30r1(t) /C215dr2 dt/C27dr1 dt/C215r2(t) : (10)The dot product is invariant under rotations A ?:B ?/C30A?i :B ?i /C30aijAjaikBk /C30 aijaik0CB0C@ AjBk /C30 djkAjBk /C30AjBj /C30A /C215B ; (11) where EINSTEIN SUMMATION has been used. The dot product is also called the scalar product and INNER PRODUCT . In the latter context, it is usually written a ;bhi : The dot product is also defined for TENSORS A and B by A /C215B /C13AaBa : (12) See also CROSS PRODUCT ,E INSTEIN SUMMATION , INNER PRODUCT ,OUTER PRODUCT ,VECTOR ,VECTOR MULTIPLICATION ,W EDGE PRODUCT References Arfken, G. "Scalar or Dot Product." §1.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 13 /C1/8, 1985. Jeffreys, H. and Jeffreys, B. S. "Scalar Product." §2.06 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 65 /C1/7, 1988. Douady’s Rabbit Fractal AJ ULIA SET with c /C30/C280 :123 /C270:745i; also known as the dragon fractal. See also DENDRITE FRACTAL ,JULIA SET,SAN MARCO FRACTAL ,SIEGEL DISK FRACTAL References Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 176, 1991. Double Bar The symbol k used to denote certain kinds of NORMS in mathematics ( /xkkðÞ :):/ See also BAR References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 277, 1997. Double Bubble A double bubble is pair of BUBBLES which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illus- trations courtesy of J. M. Sullivan). In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal 1208 ANGLES ) has the minimum PERIMETER for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995). It had been conjectured that two equal partial SPHERES sharing a boundary of a flat disk separate two volumes of air using a total SURFACE AREA that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995), who reduced the problem to a set of 200,260 integrals which they carried out on an ordinary PC. Frank Morgan, Michael Hutchings, Manuel Ritore ´, and Antonio Ros finally proved the conjecture for arbitrary double bubbles in early 2000. In this case of two unequal partial spheres, Morgan et al. showed that the separating boundary which minimizes total surface area is a portion of a SPHERE which meets the outer spherical surfaces at DIHEDRAL ANGLES of 1208. Furthermore, the CURVATURE of the partition is simply the difference of the CURVATURES of the two bubbles.Amazingly, a group of undergraduates has extended the theorem to 4-dimensional double bubbles, as well as certain cases in 5-space and higher dimensions. The corresponding triple bubble conjecture remains open (Cipra 2000). See also A PPLE ,BUBBLE ,CIRCLE- CIRCLE INTERSEC- TION ,ISOVOLUME PROBLEM ,SPHERE- SPHERE INTER- SECTION References Alfaro, M.; Brock, J.; Foisy, J.; Hodges, N.; and Zimba, J. "The Standard Double Bubble in R2 Uniquely Minimized Perimeter." Pacific J. Math. 159,47/C1/9, 1993. Almgren, F. J. and Taylor, J. "The Geometry of Soap Films and Soap Bubbles." Sci. Amer. 235,82/C1/3, 1976.Campbell, P. J. (Ed.). Reviews. Math. Mag. 68, 321, 1995. Cipra, B. "Rounding Out Solutions to Three Conjectures." Science 287, 1910 /C1/911, 2000. Haas, J.; Hutchings, M.; and Schlafy, R. "The Double Bubble Conjecture." Electron. Res. Announc. Amer. Math. Soc. 1, 98 /C1/02, 1995. Haas, J. "General Double Bubble Conjecture in R3 Solved." Focus: The Newsletter of the Math. Assoc. Amer. , No. 5, pp. 4 /C1/, May/June 2000. Hutchings, M.; Morgan, F.; Ritore ´, M.; and Ros, A. "Proof of the Double Bubble Conjecture." http://www.williams.edu/ Mathematics/fmorgan/ann.html. Morgan, F. "The Double Bubble Conjecture." FOCUS 15,6/C1/, 1995. Morgan, F. "Double Bubble Conjecture Proved." http:// www.maa.org/features/mathchat/math- chat_3_18_00.html. Peterson, I. "Toil and Trouble over Double Bubbles." Sci. News 148, 101, Aug. 12, 1995. Ritore ´, M. "Proof of the Double Bubble Conjecture Preprint." http://www.ugr.es/~ritore/bubble/bubble.htm. Sullivan, J. M. "Double Bubble Images." http:// www.math.uiuc.edu/~jms/Images/dubble.html. Double Bubble Conjecture DOUBLE BUBBLE Double Cone Two CONES placed apex to apex. The double cone is given by algebraic equation x2 c2/C30x2/C27y2 a2: See also BICONE ,CONE,NAPPE Double Contact Theorem If S1 ; S2 ; and S3 are three conics having the property that there is a point X, not on any of the conics, lying on a common chord of each pair of the three conics (with the chords in question being distinct), then there exists a conic S4 that has a double contact with each of S1 ; S2 ; and S3 (Evelyn et al. 1974, p. 18). The converse of the theorem states that if three conics S1 ; S2 ; and S3 all have double contact with another S4 then each two of S1 ; S2 ; and S3 have a "distinguished" pair of opposite common chords, the three such pairs of common chords being the pairs of opposite sides of a COMPLETE QUADRANGLE (Evelyn et al. 1974, p. 19). The dual theorems are stated as follows. If three conics are such that, taken by pairs, they have couples of common tangents intersecting at three distinct points on a line (that is not itself a tangent to any of the conics), then (a) the conics have this property in four different ways, and (b) the conics all have double contact with a fourth. And, conver- sely, if three conics each have double contact with a fourth, then certain of their common tangents inter- sect by pairs at the vertices of a COMPLETE QUAD- RILATERAL (Evelyn et al. 1974, p. 22). A degenerate case of the theorem gives the result that the six SIMILITUDE CENTERS of three circles taken by pairs are the vertices of a COMPLETE QUADRILATERAL (Evelyn et al. 1974, pp. 21 /C1/2). See also CONIC SECTION ,SIMILITUDE CENTER References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Double-Contact Theorem." §2.3 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 18 /C1/2, 1974. Double Contraction Relation A TENSOR t is said to satisfy the double contraction relation when ¯tm ij tnij /C30 dmn : This equation is satisfied by ˆt0 /C302ˆzˆz /C28 ˆxˆx /C28 ˆyˆyffiffiffi 6pˆt91 /C30/C141 2(ˆxˆz /C27ˆzˆx) /C2812i(ˆyˆz-ˆzˆy) ˆt 92 /C30/C141 2 (ˆxˆx /C27ˆyˆy) /C2812 i(ˆxˆy-ˆyˆx); where the hat denotes zero trace, symmetric unit TENSORS . These TENSORS are used to define the SPHERICAL HARMONIC TENSOR . See also SPHERICAL HARMONIC TENSOR ,TENSOR References Arfken, G. "Alternating Series." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 140, 1985. Double Cusp DOUBLE POINT Double Dagger The symbol % which is not used very commonly in mathematics. The double dagger is also known as the double obelisk or diesis (Bringhurst 1997, p. 275). See also DAGGER References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 277, 1997. Double Dot A pair of OVERDOTS placed over a symbol, as in ¨x; most commonly used to denote a second derivative with respect to time, i.e., ¨x /C30d2x=dt2 :/ See also OVERDOT Double Exponential Distribution FISHER- TIPPETT DISTRIBUTION ,L APLACE DISTRIBU- TION Double Exponential Integration An fairly good NUMERICAL INTEGRATION technique used by Maple V R4†(Waterloo Maple Inc.) for numerical computation of integrals. The method is also available in Mathematica using the option Method- /C21DoubleExponential toNIntegrate . See also INTEGRAL ,INTEGRATION ,NUMERICAL INTE- GRATION ,QUADRATURE References Davis, P. J. and Rabinowitz, P. Methods of Numerical Integration, 2nd ed. New York: Academic Press, p. 214, 1984. Di Marco, G.; Favati, P.; Lotti, G.; and Romani, F. "Asymp- totic Behaviour of Automatic Quadrature." J. Complexity 10, 296 /C1/40, 1994. Mori, M. Developments in the Double Exponential Formula for Numerical Integration. Proceedings of the Interna- tional Congress of Mathematicians, Kyoto 1990. New York: Springer-Verlag, pp. 1585 /C1/594, 1991. Mori, M. and Ooura, T. "Double Exponential Formulas for Fourier Type Integrals with a Divergent Integrand." In Contributions in Numerical Mathematics (Ed. R. P. Agar- wal). New York: World Scientific, pp. 301 /C1/08, 1993. Ooura, T. and Mori, M. "The Double Exponential Formula for Oscillatory Functions over the Half Infinite Interval." J. Comput. Appl. Math. 38, 353 /C1/60, 1991. Takahasi, H. and Mori, M. "Double Exponential Formulas for Numerical Integration." Pub. RIMS Kyoto Univ. 9, 721 /C1/41, 1974. Toda, H. and Ono, H. "Some Remarks for Efficient Usage of the Double Exponential Formulas." Kokyuroku RIMS Kyoto Univ. 339,74/C1/09, 1978. Double Factorial The double factorial is a generalization of the usual FACTORIAL n! defined by n!! /C13n /C215(n /C282)...5 :3 :1 n odd n /C215(n /C282)...6 :4 :2 n even 1 n /C30/C281;0:8 < : (1) Note that /C281!! /C300!! /C301; by definition (Arfken 1985, p. 547). For n /C300, 1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (Sloane’s A006882). The double factorial is implemented in Mathematica as n!! or Factorial2 [n]. The double factorial is a special case of the MULTIFACTORIAL . The double factorial can be expressed in terms of the GAMMA FUNCTION by G n /C271 2 ! /C30(2n /C28 1)!! 2nffiffiffipp(2) (Arfken 1985, p. 548). There are many identities relating double factorials to FACTORIALS . Since (2n /C271)!!2nn! /C30[(2n /C271)(2n /C281)...1][2 n][2(n /C281)][2( n /C282)]...2(1) /C30[(2n /C271)(2n /C281) /C1/C1/C11][2n(2n /C282)(2n /C284) /C1/C1/C12] /C30(2n /C271)(2n)(2n /C281)(2n /C282)(2n /C283)(2n /C284) /C1/C1/C12(1) /C30(2n /C271)!; (3) it follows that (2n /C271)!! /C30(2n/C271)! 2nn!: For n /C300, 1, ..., the first few values are 1, 3, 15, 105, 945, 10395, ... (Sloane’s A001147). Also, since (2n /C271)!! /C30(2n)(2n /C282)(2n /C284) /C1/C1/C12 /C30[(2n)][2(n /C281)][2( n /C282)] /C1/C1/C12 /C302nn!; (4) it follows that (2n)!! /C302nn!: For n /C300, 1, ..., the firstfew values are 1, 2, 8, 48, 384, 3840, 46080, ... (Sloane’s A000165). Finally, since (2n /C281)!!2nn! /C30[(2n /C281)(2n /C283) /C1/C1/C11][(2n)][2(n /C281)] /C2[2(n /C282)] /C1/C1/C12(1) /C30(2n /C281)(2n /C283) /C1/C1/C11[2n(2n /C282)(2n /C284) /C1/C1/C12] /C302n(2n /C281)(2n /C282)(2n /C283)(2n /C284) /C1/C1/C12(1) /C30(2n)!; (5) it follows that (2n /C281)!! /C30(2n)! 2nn!: (6) The double factorial can also be extended to negative odd integers using the definition (/C282n /C281)!! /C30( /C281)n (2n /C28 1)!! /C30( /C281)n2nn! (2n)! (7) for n /C300, 1, ... (Arfken 1985, p. 547). Similarly, the double factorial can be extended to complex argu- ments as z!! /C302[1/C272x/C28cos( px)] =4 p[cos(px)/C281]=4 G 1 /C271 2 x ! : (8) For n ODD, n! n!!/C30n(n/C281)(n/C282)/C1/C1/C1(1) n(n/C282)(n/C284)/C1/C1/C1(1) /C30(n/C281)(n/C283)/C1/C1/C1(1)/C30(n/C281)!!: (9) FornEVEN , n! n!!/C30n(n/C281)(n/C282)/C1/C1/C1(2) n(n/C282)(n/C284)/C1/C1/C1(2) (n/C281)(n/C283)/C1/C1/C1(2)/C30(n/C281)!!: (10) Therefore, for any n, n! n!!/C30(n/C281)!! (11) n!/C30n!!(n/C281)!!: (12) A closed-form sum due to Ramanujan is given by X/C12 n/C300(/C281)n(2n/C281)!! (2n)!!"#3 /C30G9 80C@n0C@o G5 40C@n0C@o G780C@n0C@o2 4352 (13) (Hardy 1999, p. 106). Whipple (1926) gives a general- ization of this sum (Hardy 1999, pp. 111 /C1/12). See also FACTORIAL ,G AMMA FUNCTION ,M ULTIFAC- TORIAL References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 544 /C1/45 and 547 /C1/48, 1985. Sloane, N. J. A. Sequences A000165/M1878, A001147/ M3002, and A006882/M0876 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Whipple, F. J. W. "On Well-Poised Series, Generalised Hypergeometric Series Having Parameters in Pairs, Each Pair with the Same Sum." Proc. London Math. Soc. 24, 247 /C1/63, 1926. Double Folium BIFOLIUM Double Gamma Function BARNES G-FUNCTION ,DIGAMMA FUNCTION Double Integral MULTIPLE INTEGRAL Double Mersenne Number A number OF THE FORM MMn/C3022n /C281 ðÞ /C281 ; where Mnis a MERSENNE NUMBER (T. Forbes). The following table gives known factors of these numbers. n factors reference 2 prime 3 prime 5 prime 7 prime 13 338193759479 Wilfrid Keller (1976) 17 231733529 Raphael Robinson (1957) 19 62914441 Raphael Robinson (1957) 31 295257526626031 Guy Haworth (1983) See also MERSENNE NUMBER ,MERSENNE PRIME Double Normal A CHORD which is a normal at each end. A CENTRO- SYMMETRIC SET K ƒRd has d double normals through the center (Croft et al. 1991). For a CURVE OF CONSTANT WIDTH , all normals are double normals. See also CENTROSYMMETRIC SETReferences Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 15, 1991. Kuiper, N. H. "Double Normals of Convex Bodies." Israel J. Math. 2,71/C1/0, 1964. Double Obelisk DOUBLE DAGGER Double Overdot DOUBLE DOT Double Point A point traced out twice as a closed curve is traversed. The maximum number of double points for a non- degenerate QUARTIC CURVE is three. An ORDINARY DOUBLE POINT is called a NODE . Arnold (1994) gives pictures of spherical and PLANE CURVES with up to five double points, as well as other curves. See also BIPLANAR DOUBLE POINT ,CONIC DOUBLE POINT ,C RUNODE ,C USP,E LLIPTIC CONE POINT , GAUSS’S DOUBLE POINT THEOREM ,NODE (ALGEBRAIC CURVE ), ORDINARY DOUBLE POINT ,Q UADRUPLE POINT ,RATIONAL DOUBLE POINT ,SPINODE ,TACNODE , TRIPLE POINT ,UNIPLANAR DOUBLE POINT References Aicardi, F. Appendix to "Plane Curves, Their Invariants, Perestroikas, and Classifications." In Singularities & Bifurcations (Ed. V. I. Arnold). Providence, RI: Amer. Math. Soc., pp. 80 /C1/1, 1994. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 12 /C1/3, 1986. Double Prime A symbol used to distinguish a third quantity xƒ ("x double prime") from two other related quantities x and x? ("x PRIME ƒ). Double primes are most commonly used to denote transformed coordinates, conjugate points, and DERIVATIVES . A double prime is also used to denote the number of arc seconds in an angle measure, or the number of inches in a length. See also PRIME References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 277, 1997. Double Series ASERIES having terms depending on two indices, X i;jaij: Identities involving double sums include the follow- ing: X/C12 p¼0Xp q¼0aq;p/C28q¼X/C12 m¼0X/C12 n¼0an;m¼X/C12 r¼0Xr=2bc s¼0as;r/C282s; (1) where r=2bc ¼1 2rr even 1 2(r/C281)rodd8 >>>< >>>:(2) is theFLOOR FUNCTION , and X/C12 i¼1Xp j¼1xixj¼n2x20C@B0C@@ : (3) Consider the series S(a;b;c;s)¼X (m;n)"(0;0)am2/C27bmn/C27cn20CB0C@/C28s(4) over binary QUADRATIC FORMS .I fScan be decom- posed into a linear sum of products of D IRICHLET L- SERIES , it is said to be solvable. The related sums S1(a;b;c;s)¼X (m;n)"(0;0)/C281ðÞmam2/C27bmn/C27cn20CB0C@/C28sð5Þ S2(a;b;c;s)¼X (m;n)"(0;0)/C281ðÞnam2/C27bmn/C27cn20CB0C@/C28sð6Þ S1;2(a;b;c;s) ¼X (m;n)"(0;0)/C281ðÞm/C27nam2/C27bmn/C27cn20CB0C@/C28sð7Þ can also be defined, which gives rise to such impress- ive FORMULAS as S1(1;0;58; 1)/C30/C28pln 27/C275ffiffiffiffiffiffi 29p0CB0C@ ffiffiffiffiffiffi 58p (8) (Glasser and Zucker 1976b). A complete table of the principal solutions of all solvable S(a;b;c;s) is given in Glasser and Zucker (1980, pp. 126 /C1/31). The LATTICE SUM b2(2s) can be separated into two pieces, b2(2s)/C30X/C12 i;j/C30/C28/C12(/C281)i/C27j i2/C27j2 ðÞs/C30X/C12 i/C301X/C12 j/C301(/C281)i/C27j i2/C27j2 ðÞ2/C27X/C12 i/C301X/C28/C12 j/C30/C281(/C281)i/C27j i2/C27j2 ðÞ2 /C27X/C28/C12 i/C30/C281X/C12 j/C301(/C281)i/C27j i2/C27j2 ðÞ2/C27X/C28/C12 i/C30/C281X/C28/C12 j/C30/C281(/C281)i/C27j i2/C27j2 ðÞ2 /C27X/C281 j/C30/C28/C12(/C281)j j2s/C27X/C12 j/C301(/C281)j j2s/C27X/C281 j/C30/C28/C12(/C281)i i2s/C27X/C12 i/C301(/C281)i i2s /C304X/C12 i;j/C301(/C281)i/C27j i2/C27j2 ðÞs/C27X/C12 i/C301(/C281)i i2s"# /C304X/C12 i;j/C301(/C281)i/C27j i2/C27j2 ðÞs/C27h(2s)"# (9) where h(n) is the D IRICHLET ETA FUNCTION . Using the analytic form of the LATTICE SUM b2(s)/C30/C284b(s)h(s)/C304S1;2(1;0;1;s)/C28h(2s)0C10CC ; (10) where b(s) is the D IRICHLET BETA FUNCTION gives the sum S1;2(1;0;1;s)/C30X/C12 i;j/C301/C281ðÞi/C27j i2/C27j2 ðÞ2/C30h(2s)/C28h(s)b(s):(11) Borwein and Borwein (1986, p. 291) show that for R[s]>1; X/C12 i;j/C30/C28/C121 i2/C27j2 ðÞs/C304b(s)&(s) (12) X/C12 i;j/C30/C28/C12(/C281)j i2/C27j2 ðÞs/C302/C28sb2(2s); (13) where z(s) is the R IEMANN ZETA FUNCTION , and for appropriate s, X/C12 i;j/C30/C281(/C281)i/C27j (i/C27j)s/C30h(s)/C28h(s/C281) (14) X/C12 i;j/C301(/C281)i/C27j (i/C27j)s/C302/C28sz(s) (15) X/C12 i;j/C3011 (i/C27j)s/C30z(s/C281)/C28z(s) (16) X/C12 i;j/C30/C28/C12(/C281)i/C27j/C271 ijj/C27jjj ðÞs/C304h(s/C281) (17) X/C12 i;j/C30/C28/C121 (i/C27j)s/C304z(s/C281) (18) X/C12 i ; j/C30/C28/C12(/C281)i /C27j (2i /C27 j /C27 1)s /C301 2(1 /C282 /C28s) h(s) /C2712 b(s) (19) (Borwein and Borwein 1986, p. 305). Another double series reduction is given by X /C12 m;n/C30/C28/C12F(2m /C27 2n /C27 1 jj ) cosh[(2 n /C27 1)u] cosh(2 nu) /C302X/C12 n/C300(2n /C27 1)F(2n /C27 1) sinh[(2 n /C27 1)u]; (20) where F denotes any function (Glasser 1974). See also EULER SUM,L ATTICE SUM,M ADELUNG CONSTANTS ,SERIES ,W EIERSTRASS’S DOUBLE SERIES THEOREM References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Glasser, M. L. "Reduction Formulas for Multiple Series." Math. Comp. 28, 265 /C1/66, 1974. Glasser, M. L. and Zucker, I. J. "Lattice Sums." In Perspec- tives in Theoretical Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring). New York: Academic Press, pp. 67 /C1/39, 1980. Hardy, G. H. "On the Convergence of Certain Multiple Series." Proc. London Math. Soc. 2,24/C1/8, 1904. Hardy, G. H. "On the Convergence of Certain Multiple Series." Proc. Cambridge Math. Soc. 19,86/C1/5, 1917. Jeffreys, H. and Jeffreys, B. S. "Double Series." §1.053 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 16 /C1/7, 1988. Meyer, B. "On the Convergence of Alternating Double Series." Amer. Math. Monthly 60, 402 /C1/04, 1953. Mo´ricz, F. "Some remarks on the notion of regular conver- gence of multiple series." Acta Math. Hungar. 41, 161 /C1/68, 1983. Wilansky, A. "On the Convergence of Double Series." Bull. Amer. Math. Soc. 53, 793 /C1/99, 1947. Zucker, I. J. and Robertson, M. M. "Some Properties of Dirichlet L-Series." J. Phys. A: Math. Gen. 9, 1207 /C1/214, 1976a. Zucker, I. J. and Robertson, M. M. "A Systematic Approach to the Evaluation of a(m ;n"0;0)am2 /C27bmn /C27cn2ðÞ/C28s:/" J. Phys. A: Math. Gen. 9, 1215 /C1/225, 1976b. Double Sixes Two sextuples of SKEW LINES on the general CUBIC SURFACE such that each line of one is SKEW to one LINE in the other set. In all, there are 30 points, with two lines through each point, and 12 lines with five points on each line. Two lines can be placed in the plane of each of the faces of a cube. The double sixes were discovered by Schla ¨fli. See also BOXCARS ,CONFIGURATION ,CUBIC SURFACE , SKEW LINES,SOLOMON’S SEAL LINESReferences Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 11, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 224, 1991. Double Sum DOUBLE SERIES Double Torus A SPHERE with two HANDLES , i.e., a genus-2 TORUS . See also HANDLE ,TORUS ,TRIPLE TORUS Double-Angle Formulas Formulas expressing trigonometric functions of an angle 2x in terms of functions of an angle x, sin(2 x) /C302 sinx cosx (1) cos(2 x) /C30cos2x /C28sin2x (2) /C302 cos2x /C281 (3) /C301 /C282 sin2x (4) tan(2 x) /C302 tanx 1 /C28 tan2x : (5) The corresponding hyperbolic function double-angle formulas are sinh(2 x) /C302 sinhx coshx (6) cosh(2 x) /C282 cosh2x /C281 (7) tanh(2 x)/C302 tanh x 1/C27tanh2x: (8) See also HALF-ANGLE FORMULAS ,HYPERBOLIC FUNC- TIONS ,M ULTIPLE- ANGLE FORMULAS ,PROSTHAPHAER- ESIS FORMULAS ,T RIGONOMETRIC ADDITION FORMULAS ,TRIGONOMETRIC FUNCTIONS ,TRIGONOME- TRY Double-Free Set A SET of POSITIVE INTEGERS is double-free if, for any integer x, the SET fx;2xg¢S (or equivalently, x /C23 S IMPLIES 2x QS): For example, of the subsets of f1; 2;3g; the sets Ø ;f1g;f2 g;f2; 3g;f1 ;3g; and f3 g are double- free, while f1 ;2g and f1;2; 3g are not. The number a(n) of double-free subsets of f1 ;2;...;ng can be computed using a(1) /C302 and the RECURRENCE RELATION a(n) /C30a(n /C281)Fb(n) /C273 Fb(n) /C272; (1) where Fnis a FIBONACCI NUMBER ,1,1,2,3,5,8,... (Sloane’s A000045), and b(n) is the BINARY CARRY SEQUENCE giving the number of trailing 0s is the BINARY representation of n,0,1,0,2,0,1,3,0,1,... (Sloane’s A007814) (C. Bower). For n /C301, 2, ..., a(n) are given by are 2, 3, 6, 10, 20, 30, 60, 96, 192, ... (Sloane’s A050291). Define r(n) /C30max f sjj: S ƒf1;2 ;...;n g is double -free g; (2) where Sjjis the CARDINAL NUMBER of (number of members in) S. Then for n /C301, 2, ..., rnðÞis given by 1, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, ... (Sloane’s A050292). An explicit formula for rnðÞis given by r(n) /C30Xn i/C301p(i); (3) where p(i) /C301if b(i) is even 0i f b(i) is odd0C1n (4) where b(n) is defined above and the first few values of p(i) are 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ... (Sloane’s A035263; C. Bower). A simple RECURRENCE RELATION for rnðÞis given by f(n) /C301 2 n&’ /C27f14 n$% ! (5) with f(0) /C300 (Wang 1989), where xbcis the FLOOR FUNCTION and xde is the CEILING FUNCTION .An asymptotic formula for rnðÞis given by r(n) /C22 3n /C27O log4n ðÞ (6) (Wang 1989). See also A-SEQUENCE ,K LARNER- RADO SEQUENCE , SUM-FREE SET,TRIPLE- FREE SET References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/triple/triple.html.Sloane, N. J. A. Sequences A000045/M0692, A007814, A035263, A050291 and A050292 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wang, E. T. H. "On Double-Free Sets of Integers." Ars Combin. 28,97/C1/00, 1989. Doublestruck A letter of the alphabet drawn with doubled vertical strokes is called doublestruck, or sometimes black- board bold (because doublestruck characters provide a means of indicating bold font weight when writing on a blackboard). For example, A; B; C ; D ; E ; .... Important SETS in mathematics are commonly de- noted using doublestruck characters, e.g., C for the set of complex numbers and R for the real numbers. Doublestruck characters can be encoded using the AMSFonts extended fonts for LATEX using the syntax \mathbb {C}, and typed in Mathematica using the syntax \[DoubleStruckC] or \[DoundStruckCa- pitalC] , where C denotes any letter. Doublet Function y /C30 d?(x /C28a) ; where d(x) is the DELTA FUNCTION . See also DELTA FUNCTION References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 324, 1993. Doubly Even Number An even number N for which N /C130 (mod4) : The first few POSITIVE doubly even numbers are 4, 8, 12, 16, ... (Sloane’s A008586). See also EVEN FUNCTION ,ODD NUMBER ,SINGLY EVEN NUMBER References Sloane, N. J. A. Sequences A008586 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Doubly Magic Square BIMAGIC SQUARE Doubly Periodic Function A function f(z) is said to be doubly periodic if it has two periods v1 and v2 whose ratio v2 =v1 is not real. See also ELLIPTIC FUNCTION ,PERIODIC FUNCTION References Apostol, T. M. "Doubly Periodic Functions." §1.2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 1 /C1/, 1997. Knopp, K. "Doubly-Periodic Functions; in Particular, Elliptic Functions." §9in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 73 /C1/ 2, 1996. Doubly Ruled Surface A surface that contains two families of rulings. The only three doubly ruled surfaces are the PLANE , HYPERBOLIC PARABOLOID , and single-sheeted HYPER- BOLOID . See also HYPERBOLIC PARABOLOID ,H YPERBOLOID , PLANE ,RULED SURFACE References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 15, 1999. Doubly Stochastic Matrix A doubly stochastic matrix is a matrix A /C30(aij) such that aij ]0 and X iaij /C30X jaij /C301 is some field for all i and j. In other words, both the matrix itself and its transpose are STOCHASTIC . The following tables give the number of distinct doubly stochastic matrices (and distinct nonsingular doubly stochastic matrices) over Zm for small m. m doubly stochastic n /C29n matrices over Zm/ 2 1, 2, 16, 512, ... 3 1, 3, 81, ... 4 1, 4, 256, ... m doubly stochastic nonsingular n /C29n matrices over Zm/ 2 1, 2, 6, 192, ... 3 1, 2, 54, ... 4 1, 4, 192, ... Horn (1954) proved that if y /C30Ax; where x and y are complex n-vectors, A is doubly stochastic, and c1 ; c2 ; ..., Cn are any complex numbers, then an i/C301ciyi lies in the CONVEX HULL of all the points ani/C301cixai ; a /C23 Rn ;where Rn is the set of all permutations of f1;:::; ng: Sherman (1955) also proved the converse. Birkhoff (1946) proved that any doubly stochastic n /C29 n matrix is in the CONVEX HULL of m PERMUTATION MATRICES for m 5(n /C281)2 /C271: There are several proofs and extensions of this result (Dulmage and Halperin 1955, Mendelsohn and Dulmage 1958, Mirsky 1958, Marcus 1960). See also STOCHASTIC MATRIX References Birkhoff, G. "Three Observations on Linear Algebra." Univ. Nac. Tucuma ´n. Rev. Ser. A 5, 147 /C1/51, 1946. Dulmage, L. and Halperin, I. "On a Theorem of Frobenius- Ko¨nig and J. von Neumann’s Game of Hide and Seek." Trans. Roy. Soc. Canada Sect. III 49,23/C1/9, 1955. Horn, A. "Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix." Amer. J. Math. 76, 620 /C1/30, 1954. Marcus, M. "Some Properties and Applications of Doubly Stochastic Matrices." Amer. Math. Monthly 67, 215 /C1/21, 1960. Mendelsohn, N. S. and Dulmage, A. L. "The Convex Hull of Subpermutation Matrices." Proc. Amer. Math. Soc. 9, 253 /C1/54, 1958. Mirsky, L. "Proofs of Two Theorems on Doubly Stochastic Matrices." Proc. Amer. Math. Soc. 9, 371 /C1/74, 1958. Schreiber, S. "On a Result of S. Sherman Concerning Doubly Stochastic Matrices." Proc. Amer. Math. Soc. 9, 350 /C1/53, 1958. Sherman, S. "A Correction to ‘On a Conjecture Concerning Doubly Stochastic Matrices."’ Proc. Amer. Math. Soc. 5, 998 /C1/99, 1954. Sherman, S. "Doubly Stochastic Matrices and Complex Vector Spaces." Amer. J. Math. 77, 245 /C1/46, 1955. Dougall’s Formula ForR[a/C27b/C28c/C28d]B/C281 and aandbnot integers, X/C12 n/C30/C28/C12G(a/C27n)G(b/C27n) G(c/C27n)G(d/C27n) /C30p2csc(pa)csc(pb)G(c/C27d/C28a/C28b/C281) G(c/C28a)G(d/C28a)G(c/C28b)G(d/C28b): See also GAMMA FUNCTION References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 7, 1981. Dougall’s Theorem 5F41 2n/C271;n;/C28x;/C28y;/C28z 12n;x/C27n/C271;y/C27n/C271;z/C27n/C2712 66643 7775 /C30G(x/C27n/C271)G(y/C27n/C271)G(z/C27n/C271)G(x/C27y/C27z/C27n/C271) G(n/C271)G(x/C27y/C27n/C271)G(y/C27z/C27n/C271)G(x/C27z/C27n/C271); where5F4(a ;b;c ;d ;e;f ;g ;h;i;z)isa GENERALIZED HYPERGEOMETRIC FUNCTION and G(z) is the GAMMA FUNCTION . Bailey (1935, pp. 25 /C1/6) called the DOUGALL- RAMANU- JAN IDENTITY "Dougall’s theorem." See also DOUGALL- RAMANUJAN IDENTITY ,GENERAL- IZED HYPERGEOMETRIC FUNCTION References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, pp. 25 /C1/7, 1935. Dougall, J. "On Vandermonde’s Theorem and Some More General Expansions." Proc. Edinburgh Math. Soc. 25, 114 /C1/32, 1907. Hardy, G. H. "A Chapter from Ramanujan’s Note-Book." Proc. Cambridge Philos. Soc. 21, 492 /C1/03, 1923. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 84, 1998. Whipple, F. J. W. "On Well-Poised Series, Generalized Hypergeometric Series Having Parameters in Pairs, Each Pair with the Same Sum." Proc. London Math. Soc. 24, 247 /C1/63, 1926. Dougall-Ramanujan Identity A hypergeometric identity discovered by Ramanujan around 1910. From Hardy (1999, pp. 13 and 102 /C1/03), X/C12 n/C300(/C281)n(s /C272n)s(n)(x /C27 y /C27 z /C27 u /C27 2s /C27 1)(n) (x /C27 y /C27 z /C27 u /C28 s)(n)Y x;y ;x;u /C2x(n) (x /C27 s /C27 1)(n) /C30s G(s /C27 1)G(x /C27 y /C27 z /C27 u /C27 s /C27 1)Y x;y ;z;u /C2G(x /C27 s /C27 1)G(y /C27 z /C27 u /C27 s /C27 1) G(z /C27 u /C27 s /C27 1): (1) where a(n) /C13a(a /C271) /C1/C1/C1(a /C27n /C281) (2) is the RISING FACTORIAL (a.k.a. POCHHAMMER SYM- BOL, a(n) /C13a(a /C281) /C1/C1/C1(a /C28n /C271) (3) is the FALLING FACTORIAL (Hardy 1999, p. 101), G(z)is a GAMMA FUNCTION , and one of x; y;z; u;/C28x /C28y /C28z /C28u /C282s /C281 (4) is a POSITIVE INTEGER . Equation (1) can also be rewritten as7F6s ;1 /C271 2 s ;/C28x;/C28y;/C28z ;/C28u;x /C28y /C27z /C27u /C272s /C271 12s ;x /C27s /C271;y /C27s /C271;z /C27s /C271;u /C27s /C271; /C28x /C28y /C28z /C28u /C28s;12 6666643 777775 /C30 1 G(s /C27 1)G(x /C27 y /C27 z /C27 u /C27 s /C27 1)Y x;y ;z ;u /C2G(x /C27 s /C27 1)G(y /C27 z /C27 u /C27 s /C27 1) G(z /C27 u /C27 s /C27 1): (5) (Hardy 1999, p. 102). In a more symmetric form, if n /C302a1 /C271 /C30a2 /C27a3 /C27a4 /C27a5 ; a6 /C301 /C27a1 =2; a7 /C30/C28n; and bi /C301 /C27a1 /C28ai/C271 for i /C301, 2, ..., 6, then 7F6a1 ;a2 ;a3 ;a4 ;a5 ;a6 ; a7 b1;b2;b3;b4;b5;b6;10C1B0C1@ /C30(a1/C271)n(a1/C28a2/C28a3/C271)n (a1/C28a2/C271)n(a1/C28a3/C271)n /C2(a1/C28a2/C28a4/C271)n(a1/C28a3/C28a4/C271)n (a1/C28a4/C271)n(a1/C28a2/C28a3/C28a4/C271)n;(6) where ( a)nis the P OCHHAMMER SYMBOL (Petkovsek et al.1996). The identity is a special case of J ACKSON’S IDENTITY , and gives D IXON’S THEOREM ,SAALSCHU ¨TZ’S THEOREM , and M ORLEY’S FORMULA as special cases. See also BAILEY’S TRANSFORMATION ,DIXON’S THEO- REM,DOUGALL’S THEOREM ,GENERALIZED HYPERGEO- METRIC FUNCTION ,H YPERGEOMETRIC FUNCTION , JACKSON’S IDENTITY ,M ORLEY’S FORMULA ,R OGERS- RAMANUJAN IDENTITIES ,SAALSCHU ¨ TZ’S THEOREM References Bailey, W. N. "An Elementary Proof of Dougall’s Theorem." §5.1 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 25 /C1/6 and 34, 1935. Dixon, A. C. "Summation of a Certain Series." Proc. London Math. Soc. 35, 285/C1/89, 1903. Dougall, J. "On Vandermonde’s Theorem and Some More General Expansions." Proc. Edinburgh Math. Soc. 25, 114/C1/32, 1907. Hardy, G. H. "A Chapter from Ramanujan’s Note-Book." Proc. Cambridge Philos. Soc. 21, 492/C1/03, 1923. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, pp. 43, 126 /C1/27, and 183 /C1/84, 1996. Doughnut TORUS Douglas-Neumann Theorem If the lines joining corresponding points of two DIRECTLY SIMILAR figures are divided proportionally, then the LOCUS of the points of the division will be a figure DIRECTLY SIMILAR to the given figures. See also DIRECTLY SIMILAR References Eves, H. "Solution to Problem E521." Amer. Math. Monthly 50, 64, 1943. Musselman, J. R. "Problem E521." Amer. Math. Monthly 49, 335, 1942. Dovetailing Problem CUBE DOVETAILING PROBLEM Dowker Notation A simple way to describe a knot projection. The advantage of this notation is that it enables a KNOT DIAGRAM to be drawn quickly. For an oriented ALTERNATING KNOT with n crossings, begin at an arbitrary crossing and label it 1. Now follow the undergoing strand to the next crossing, and denote it 2. Continue around the knot following the same strand until each crossing has been numbered twice. Each crossing will have one even number and one odd number, with the numbers running from 1 to 2n:/ Now write out the ODD NUMBERS 1, 3, ..., 2n /C281ina row, and underneath write the even crossing number corresponding to each number. The Dowker NOTA- TION is this bottom row of numbers. When the sequence of even numbers can be broken into two permutations of consecutive sequences (such as f4; 6;2gf10 ;12 ;8g) ; the knot is composite and is not uniquely determined by the Dowker notation. Other- wise, the knot is prime and the NOTATION uniquely defines a single knot (for amphichiral knots) or corresponds to a single knot or its MIRROR IMAGE (for chiral knots). For general nonalternating knots, the procedure is modified slightly by making the sign of the even numbers POSITIVE if the crossing is on the top strand, and NEGATIVE if it is on the bottom strand. These data are available for knots, but not for links, from Berkeley’s gopher site. References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 35 /C1/0, 1994. Dowker, C. H. and Thistlethwaite, M. B. "Classification of Knot Projections." Topol. Appl. 16,19/C1/1, 1983. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Thistlethwaite, M. B. "Knot Tabulations and Related To- pics." In Aspects of Topology in Memory of Hugh Dowker1912 /C1/982 (Ed. I. M. James and E. H. Kronheimer). Cam- bridge, England: Cambridge University Press, pp. 2 /C1/6, 1985. Down Arrow Notation An inverse of the up ARROW NOTATION defined by e ¡n /C30ln n e ¡¡ n /C30ln /C31n e ¡¡¡ n /C30ln /C31/C31n; where ln /C31n is the number of times the NATURAL LOGARITHM must be iterated to obtain a value 5e :/ See also ARROW NOTATION References Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 12 and 231 /C1/32, 1991. Dozen 12. See also BAKER’S DOZEN ,DUODECIMAL ,GROSS Dragon Curve Nonintersecting curves which can be iterated to yield more and more sinuosity. They can be constructed by taking a path around a set of dots, representing a left turn by 1 and a right turn by 0. The first-order curveis then denoted 1. For higher order curves, add a 1 to the end, then copy the string of digits preceding it to the end but switching its center digit. For example,the second-order curve is generated as follows: (1)1 0 (1)1(0) 0110, and the third as: (110)1 0(110)1(100) 01101100. Continuing gives 110110011100100... (Sloane’s A014577). The OCTAL representation se- quence is 1, 6, 154, 66344, ...(Sloane’s A003460). The dragon curves of orders 1 to 9 are illustrated below. This procedure is equivalent to drawing a RIGHT ANGLE and subsequently replacing each RIGHT ANGLE with another smaller RIGHT ANGLE (Gardner 1978). In fact, the dragon curve can be written as a LINDEN- MAYER SYSTEM with initial string "FX" , STRING REWRITING rules "X" 0 "X/C27YF/C27", "Y" 0 " /C28FX-Y" , and angle 908. See also LINDENMAYER SYSTEM ,PEANO CURVE References Bulaevsky, J. "The Dragon Curve or Jurassic Park Fractal." http://www.best.com/~ejad/java/fractals/jurasic.shtml. Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Dixon, R. Mathographics. New York: Dover, pp. 180 /C1/81, 1991. Dubrovsky, V. "Nesting Puzzles, Part I: Moving Oriental Towers." Quantum 6,53/C1/7 (Jan.) and 49 /C1/1 (Feb.), 1996. Dubrovsky, V. "Nesting Puzzles, Part II: Chinese Rings Produce a Chinese Monster." Quantum 6,61/C1/5 (Mar.) and 58 /C1/9 (Apr.), 1996. Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 207 /C1/09 and 215 /C1/20, 1978. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 48 /C1/3, 1991. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, p. 284, 1988. Sloane, N. J. A. Sequences A003460/M4300 and A014577 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Vasilyev, N. and Gutenmacher, V. "Dragon Curves." Quan- tum 6,5/C1/0, 1995. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 59, 1991. Dragon Fractal DOUADY’S RABBIT FRACTAL Draughts CHECKERS Draw The ending of a GAME in which neither of two players wins, sometimes also called a "tie." A GAME in which no draw is possible is called a CATEGORICAL GAME . See also CATEGORICAL GAME,GAME,UNFAIR GAME Drinfel’d-Sokolov-Wilson Equation The system of PARTIAL DIFFERENTIAL EQUATIONS ut /C303wwx wt ¼ 2wxxx þ 2uwx þ uxw:References Hirota, R.; Grammaticos, B.; and Ramani, A. "Soliton Structure of the Drinfel’d-Sokolov-Wilson Equation." J. Math. Phys. 27, 1499 /C1/505, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Drinfeld Module See also MODULE References Gekeler, E.-U.; van der Put, M.; Reversat, M.; and van Geel, J. (Eds.). Proceedings of the Workshop on Drinfeld Mod- ules, Modular Schemes and Applications: Alden-Biesen, 9/C1/4 September 1996. Singapore: World Scientific, 1997. Drinfeld’s Symmetric Space A set of points which do not lie on any of a certain class of HYPERPLANES . References Teitelbaum, J. "The Geometry of p-adic Symmetric Spaces." Not. Amer. Math. Soc. 42, 1120 /C1/126, 1995. Droz-Farny Circles The following amazing property of a triangle, firstgiven by Steiner and then proved by Droz-Farny(1901), is related to the so-called Droz-Farny circles.Draw a CIRCLE with center at the ORTHOCENTER H which cuts the lines M2M3;M3M1;andM1M2(where Miare the MIDPOINTS of their respective sides) at P1; Q1;P2;Q2; and P3;Q3respectively, then the line segments AiPi/C30AiQiare all equal: A1P1/C30A2P2/C30A3P3/C30A1Q1/C30A2Q2/C30A3Q3: Conversely, if equal CIRCLES are drawn about the VERTICES of a TRIANGLE (dashed circles in the above figure), they cut the lines joining the MIDPOINTS of the corresponding sides in six points P1;Q1;P2;Q2;P3; and Q3;which lie on a CIRCLE whose center is the ORTHOCENTER .I fris the RADIUS of the equal CIRCLES centered on the vertices A1;A2;andA3;andR0is the RADIUS of the CIRCLE about H, then R2 0/C304R2/C27r2/C281 2a2 1/C27a22/C27a230CB0C@ (Johnson 1929, p. 257). In the special case that ris taken as the CIRCUMRA- DIUS of the original triangle, then a circle D1;known as the Droz-Farny circle (in particular, the "vertex- circumcenter Droz-Farny circle"), is obtained, having center Hand RADIUS R2 0/C305R2/C281 2a2 1/C27a22/C27a230CB0C@ (Johnson 1929, pp. 257 /C1/78). The "altitude feet-circumcenter" Droz-Farny circle D?1 is obtained by drawing circles with centers at the feet of the altitudes and passing through the CIRCUMCEN-TER. These circles cut the corresponding sides in six concyclic points, having the same center H and the same radius R0as the vertex-circumcenter Droz- Farny circle. This is the first Droz-Farny circle. The first Droz-Farny circle D1therefore passes through 12 notable points, two on each of the sides and two on each of the lines joining midpoints of the sides, as illustrated in the rather busy figure above. The circles about the midpoints of the sides and passing though Hcut the sides in six points lying on another circle D2:This is the second Droz-Farny circle, which has RADIUS equal to that of D1;but whose center is the CIRCUMCENTER Oinstead of the ORTHOCENTER H. There is a beautiful generalization of the Droz-Farnycircles motivated by the observation that the ORTHO- CENTER and CIRCUMCENTER are ISOGONAL CONJU- GATES . Let P and Q be any pair of ISOGONAL CONJUGATES of a triangle DABC ; and let D, E, and F be the feet of the perpendiculars to the sides from one of the points (say, P), and let circles with centers D, E, and F be drawn to pass through Q. Then the three pairs of points on the sides of DABC which are determined by these circles always lie on a circle with center P, and the two circles constructed in this way are congruent (Honsberger 1995). See also CIRCUMCENTER ,ORTHOCENTER References Droz-Farny. "Notes sur un the´ore`me de Steiner." Mathesis 21,22/C1/4, 1901. Goormaghtigh, R. "Droz-Farny’s Theorem." Scripta Math. 16, 268 /C1/71, 1950. Honsberger, R. "The Droz-Farny Circles." §7.4 (ix) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 69 /C1/ 2, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 256 /C1/58, 1929. Droz-Farny Theorem If two perpendicular lines are drawn through the ORTHOCENTER H of any triangle, these lines intercept each side (or its extension) in two points (labeled P12 ; P?12 ; P13 ; P ?13 ; P23 ; P?23) : Then the MIDPOINTS M12 ; M12 ; and M23 of these three segments are COLLINEAR . See also COLLINEAR ,MIDPOINT References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 73, 1995. Drum ISOSPECTRAL MANIFOLDSds JACOBI ELLIPTIC FUNCTIONS # 1999 /C1/001 Wolfram Research, Inc. D-Statistic KOLMOGOROV- SMIRNOV TEST D-Triangle Let the CIRCLES /c2/ and /c ?3/ used in the construction of the BROCARD POINTS which are tangent to /A2A3/ at /A2/ and /A3/, respectively, meet again at D1 : The points / D1D2D3/ then define the D-triangle. The VERTICES of the D-triangle lie on the respective APOLLONIUS CIRCLES . See also APOLLONIUS CIRCLES ,BROCARD POINTS References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 284 /C1/85, 296 and 307, 1929. Du Bois Reymond Constants The constants Cndefined by Cn/C13g/C12 0d dtsint t !n0C@10C@10C@10C@10C@10C@10C@10C@10C@10C@1dt/C281: (1) These constants can also be written as C n/C302X/C12 k/C3011/C27x2 k0CB0C@/C28n=2; (2) where xkis the kth root of t/C30tant: (3) /C1diverges, and the first few constant are numeri- cally given by C2:0:1945280494 (4) C3:0:028254 (5) C4:0:005240704678 : (6) Rather surprisingly, the even-ordered du Bois Rey- mond constants (and, in particular, C2; Le Lionnais 1983) can be computed analytically as polynomials in e2 ; C2 /C301 2e2 /C2870CB0C@ (7) C4 /C3018e 4 /C284e2 /C28250CB0C@ (8) C6 /C301 32e6 /C286e4 /C273e2 /C28980CB0C@ : (9) These have the explicit formula Cn /C30/C283 /C282Res x/C30ix2 1 /C27 x2 ðÞn(tanx /C28 x) ! ; (10) where n is even and Res denotes a RESIDUE (V. Adam- chik). See also INFINITE SERIES References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 23, 1983. Plouffe, S. "Dubois-Raymond 2nd Constant." http://www.la- cim.uqam.ca/piDATA/dubois.txt. Dual Basis Given a CONTRAVARIANT BASIS f /C8e1 ;...; /C8en g; its dual COVARIANT basis is given by /C0e a /C215 /C0e b /C30g( /C0ea ; /C0e b) /C30 da b ; where g is the METRIC and d abis the mixed KRO- NECKER DELTA .InE UCLIDEAN SPACE with an ORTHO- NORMAL BASIS , /C0ej /C30 /C0ej ; so the BASIS and its dual are the same. See also DUAL SPACE Dual Bivector A dual BIVECTOR is defined by ˜Xab /C131 2 eabcdXcd ; and a self-dual BIVECTOR by X /C31 ab /C13Xab /C27i ˜Xab : See also BIVECTOR Dual Bundle Given a VECTOR BUNDLE p : E 0 M ; its dual bundle is a VECTOR BUNDLE p/C31 : E/C310 M : The FIBER BUNDLE ofE /C31 over a point p /C23 M is the DUAL VECTOR SPACE to the fiber of E. See also DUAL SPACE ,VECTOR BUNDLE Dual Graph Given a PLANAR GRAPH G,aGEOMETRIC DUAL GRAPH and COMBINATORIAL DUAL GRAPH can be defined. Whitney showed that these are equivalent (Harary 1994), so that one make speak of "the" dual graph G/C31: The illustration above shows the process of construct- ing a GEOMETRIC DUAL GRAPH . The dual graph G/C31 of a POLYHEDRAL GRAPH G has VERTICES each of which corresponds to a face of G and each of whose faces corresponds to a VERTEX of G. Two nodes in G /C31 are connected by an EDGE if the corresponding faces in G have a boundary EDGE in common. The dual graph of a WHEEL GRAPH is itself a wheel (Skiena 1990, p. 147). See also COMBINATORIAL DUAL GRAPH ,GEOMETRIC DUAL GRAPH ,PLANAR GRAPH ,SELF-DUAL GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 113 /C1/14, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Wagon, S. "An April Fool’s Hoax." Mathematica in Educ. Res. 7,4 6/C1/2, 1998. Wagon, S. Mathematica in Action, 2nd ed. New York: Springer-Verlag, pp. 536 /C1/37, 1999. Dual Map PULLBACK MAP Dual Number A number x/C27ey;where x;y/C23Randois a UNIT with the property that e2/C300:/ References Brand, L. Vector and Tensor Analysis. New York: Wiley, 1947. Dual Polyhedron By the DUALITY PRINCIPLE , for every POLYHEDRON , there exists another POLYHEDRON in which faces and VERTICES occupy complementary locations. This POLY- HEDRON is known as the dual, or RECIPROCAL . The process of taking the dual is also called RECIPROCA- TION , or polar reciprocation. Bru ¨ckner (1900) was among the first to give a precise definition of duality (Wenninger 1983, p. 1). The dual of a P LATONIC SOLID or A RCHIMEDEAN SOLID can be computed by connecting the midpoints of the sides surrounding each VERTEX (the VERTEX FIGURE ; left figure), and constructing the corresponding TAN- GENTIAL POLYGON (tangent to the CIRCUMCIRCLE of the VERTEX FIGURE ; right figure.) This is sometimes called the Dorman-Luke construction (Wenninger 1983, p. 30). The dual polyhedron of a P LATONIC SOLID or A RCHI- MEDEAN SOLID can be also drawn by constructing EDGES tangent to the MIDSPHERE (sometimes also known as the reciprocating sphere or intersphere) which are PERPENDICULAR to the original EDGES . Furthermore, let rbe the INRADIUS of the dual polyhedron (corresponding to the INSPHERE , which touches the faces of the dual solid), rbe the MIDRADIUS of both the polyhedron and its dual (corresponding to the MIDSPHERE , which touches the edges of both the polyhedron and its duals), and Rthe CIRCUMRADIUS (corresponding to the CIRCUMSPHERE of the solid which touches the vertices of the solid).Since the CIRCUMSPHERE and INSPHERE are dual to each other, r,R, and robey the polar relationship Rr/C30r2 (Cundy and Rollett 1989, Table II following p. 144). The process of forming duals is illustrated above forthe P LATONIC SOLIDS . The top row shows the original solid, the middle row shows the vertex figures of the original solid as lines superposed on the tangentialpolygons forming the dual faces. The POLYHEDRON COMPOUNDS consisting of a POLYHEDRON and its dual are generally very attractive, and are illustrated in the bottom row. For an A RCHIMEDEAN SOLID with vvertices, ffaces, and eedges, the dual polyhedron has fvertices, v faces, and eedges. The dual of an isogonal solid (i.e., all vertices are alike) is isohedral (i.e., all faces are alike) (Wenninger 1983, p. 5). The dual of any non-convex UNIFORM POLYHEDRON is a stellated form of the CONVEX HULL of the given polyhedron (Wenninger 1983, pp. 3 /C1/and 40). The following table gives a list of the duals of the PLATONIC SOLIDS and K EPLER- POINSOT SOLIDS , to- gether with the names of the POLYHEDRON -dual COMPOUNDS . (Note that the duals of the P LATONIC SOLIDS are themselves P LATONIC SOLIDS , so no new solids are formed by taking the duals of the Platonic solids.) Duals can also be taken of other polyhedrons, includ- ing the Archimedean solids and Uniform solids. Thenames of some solids and their duals are given in thetable below. POLYHEDRON Dual POLYHEDRON COMPOUND CSA´SZA´R POLYHE- DRONSZILASSI POLY- HEDRON CUBE OCTAHEDRON CUBE-OCTAHE- DRON COMPOUND CUBOCTAHEDRON RHOMBIC DODE- CAHEDRON DODECAHEDRON ICOSAHEDRON DODECAHEDRON- ICOSAHEDRON COMPOUND GREAT DODECA-HEDRONSMALL STEL-LATED DODECA- HEDRONGREAT DODECA- HEDRON-SMALL STELLATED DODE- CAHEDRON COM-POUND GREAT ICOSAHE-DRONGREAT STEL-LATED DODECA-HEDRONGREAT ICOSAHE-DRON-GREATSTELLATED DODE-CAHEDRON COM-POUND GREAT STEL-LATED DODECA-HEDRONGREAT ICOSAHE-DRONGREAT ICOSAHE-DRON-GREATSTELLATED DODE-CAHEDRON COM-POUND ICOSAHEDRON DODECAHEDRON DODECAHEDRON- ICOSAHEDRONCOMPOUND OCTAHEDRON CUBE CUBE-OCTAHE- DRON COMPOUND SMALL STEL- LATED DODECA- HEDRONGREAT DODECA- HEDRONGREAT DODECA- HEDRON-SMALL STELLATED DODE- CAHEDRON COM- POUND SZILASSI POLYHE- DRONCSA´ SZA´ R POLY- HEDRON TETRAHEDRON TETRAHEDRON STELLA OCTANGU- LA When a POLYCHORON with SCHLA ¨ FLI SYMBOL fp ;q;r g and its dual are in reciprocal positions, the vertices of fp ;q;r g/’s bounding polyhedra can be found by select- ing those vertices of fp ;q;r g closest to each vertex of fr ;q; pg:/ See also ARCHIMEDEAN SOLID,D UALITY PRINCIPLE , PLATONIC SOLID ,POLYHEDRON ,POLYHEDRON COM- POUND ,R ECIPROCATING SPHERE ,R ECIPROCATION , SELF-DUAL POLYHEDRON ,U NIFORM POLYHEDRON , ZONOHEDRON References Bru¨ckner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Hart, G. "Duality." http://www.georgehart.com/virtual-poly- hedra/duality.html. Weisstein, E. W. "Polyhedron Duals." MATHEMATICA NOTE- BOOK DUALS.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 60, 1991. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Dual Scalar Given a third RANK TENSOR , Vijk /C13det[ABC ] ; where det is the DETERMINANT , the dual scalar is defined as V /C131 3! eijkVijk ; where eijk is the LEVI-CIVITA TENSOR . See also DUAL TENSOR ,LEVI-CIVITA TENSOR Dual Solid DUAL POLYHEDRON Dual Space The dual space to a real VECTOR SPACE V is the VECTOR SPACE of LINEAR FUNCTIONS f : V 0 R ; and isdenoted V /C31: In the dual to a COMPLEX VECTOR SPACE , the linear functions take complex values. In either case, the dual space has the same DIMEN- SION as V. Given a BASIS v1 ;...; vn for V there exists a DUAL BASIS for V /C31; written v/C31 1 ;...; v/C31n ; where v/C31ivj0CB0C@ /C30 dij and dij is the KRONECKER DELTA . Another way to realize an isomorphism with V is through an INNER PRODUCT .A REAL VECTOR SPACE can have a symmetric INNER PRODUCT ;hi in which case a vector v corresponds to a dual element by fv(w) /C30 w;vhi : Then a basis corresponds to its dual basis only if it is an ORTHONORMAL BASIS , in which case v/C31i /C30/C28;vi0C@B0C@@ : A COMPLEX VECTOR SPACE can have a HERMITIAN INNER PRODUCT , in which case fv(w) /C30 w; vhi is a conjugate-linear isomorphism of V with V /C31; i.e., fav /C30 ¯afv :/ Dual spaces can describe many objects in linear algebra. When V and W are finite dimensional vector spaces, an element of the tensor product V /C31/C156W ; say aaijv/C31j /C156wi ; corresponds to the linear transformation T(v) /C30aaijv/C31j (w)wi : That is, V /C31/C156W #Hom( V ;W) : For example, the identity transformation is v1 /C156v /C311 /C27.../C27 vn /C156v/C31n : A BILINEAR FORM on V, such as an inner product, is an element of V /C31/C156V /C31:/ When V is infinite dimensional, care has to be taken of the topology. The dual space of V is the VECTOR SPACE of CONTINUOUS LINEAR FUNCTIONALS on V. See also BASIS (VECTOR SPACE ), BILINEAR FORM, DISTRIBUTION (GENERALIZED FUNCTION ), DUAL VEC- TOR SPACE ,LINEAR FUNCTIONAL ,MATRIX ,SELF-DUAL, VECTOR SPACE Dual Tensor Given an antisymmetric second RANK TENSOR Cij ; a dual pseudotensor Ci is defined by Ci /C131 2 eijkCjk ; (1) where Ci /C13C23 C31 C122 435 (2) C jk /C130 C12 /C28C31 /C28C120 C23 C31/C28C2302435: (3) See also D UAL SCALAR References Arfken, G. "Pseudotensors, Dual Tensors." §3.4 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 128 /C1/37, 1985. Dual Tessellation The dual of a regular TESSELLATION is formed by taking the center of each polygon as a vertex and joining the centers of adjacent polygons. The triangular and hexagonal tessellations are duals of each other, while the square tessellation it its own dual. Williams (1979, pp. 37 /C1/1) illustrates the dual tessel- lations of the semiregular tessellations. See also CAIRO TESSELLATION ,TESSELLATION References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 60 /C1/1, 1991.Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, p. 37, 1979. Dual Vector Space Given a VECTOR SPACE X, the dual vector space X + is the set of all bounded LINEAR FUNCTIONALS on X. See also DUAL SPACE ,LINEAR FUNCTIONAL ,VECTOR SPACE Dual Voting A term in SOCIAL CHOICE THEORY meaning each alternative receives equal weight for a single vote. See also ANONYMOUS ,MONOTONIC VOTING Duality Principle All the propositions in PROJECTIVE GEOMETRY occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line." The principle was enunciated by Gergonne (1826; Cremona 1960, p. x). A similar duality exists for RECIPROCATION as first enunciated by Poncelet (1818; Casey 1893; Lachlan 1893; Cremona 1960, p. x). Example of dual geometric objects include BRIAN- CHON’S THEOREM and PASCAL’S THEOREM , the 15 PLU¨ CKER LINES and 15 SALMON POINTS , the 20 CAYLEY LINES and 20 STEINER POINTS , the 60 PASCAL LINES and 60 KIRKMAN POINTS , DUAL POLYHEDRA , and DUAL TESSELLATIONS . Propositions which are equivalent to their duals are said to be SELF-DUAL . See also BRIANCHON’S THEOREM ,CONSERVATION OF NUMBER PRINCIPLE ,D ESARGUES’ THEOREM ,D UAL POLYHEDRON ,P APPUS’S HEXAGON THEOREM ,P AS- CAL’S THEOREM ,P ERMANENCE OF MATHEMATICAL RELATIONS PRINCIPLE ,PROJECTIVE GEOMETRY ,RECI- PROCAL ,RECIPROCATION ,SELF-DUAL References Casey, J. "Theory of Duality and Reciprocal Polars." Ch. 13 inA Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nded., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 382 /C1 /92, 1893. Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 78, 1928. Gergonne, J. D. Ann. Math. 16, 209, 1826. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 26 /C1/7 and 41 /C1/3, 1930. Lachlan, R. "The Principle of Duality." §7 and 284 /C1/99 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 3 /C1/ and 174 /C1/82, 1893. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 107 /C1/10, 1990. Poncelet, J.-V. Ann. Math. 8, 201, 1818. Duality Theorem Dual pairs of LINEAR PROGRAMS are in "strong duality" if both are possible. The theorem was first conceived by John von Neumann. The first written proof was an Air Force report by George Dantzig, but credit is usually given to Tucker, Kuhn, and Gale. See also LINEAR PROGRAMMING Duffing Differential Equation The most general forced form of the Duffing equation is ¨x/C27d˙x/C27bx39v2 0x0CB0C@ /C30Asin(vt/C27f): (1) If there is no forcing, the right side vanishes, leaving ¨x/C27d˙x/C27bx39v20x0CB0C@ /C300: (2) Ifd/C300 and we take the plus sign, ¨x/C27v20x/C27bx3/C300 (3) (Bender and Orszag 1978, p. 547; Zwillinger 1997, p. 122). This equation can display chaotic behavior. For b>0; the equation represents a "hard spring," and for bB0; it represents a "soft spring." If bB0;the phase portrait curves are closed. Returning to (1), take b/C30 1;v0/C301;A/C300, and use the minus sign. Then the equation is ˙x/C27d˙x/C27x3/C28x0CB0C@ /C300 (4) (Ott 1993, p. 3). This can be written as a system of first-order ordinary differential equations by writing ˙x/C30y; (5) ˙y/C30x/C28x3/C28dy: (6) The fixed points of these differential equations ˙x/C30y/C300; (7) soy/C300, and ˙y/C30x/C28x3/C28dy/C30x1/C28x20CB0C@ /C280 (8) giving x/C300;91:Differentiating, ¨x/C30˙y/C30x/C28x3/C28dy (9) ¨y/C301/C283x20CB0C@ ˙x/C28d˙y (10)¨x ¨y0C1B0C1@ /C3001 1/C283x2/C28d0C1B0C1@ ˙x ˙y0C1B0C1@ : (11) Examine the stability of the point (0,0): 0/C28l 1 1/C28d/C28l0C@10C@10C@10C@10C@10C@10C@10C@1/C30l(l/C27d)/C281/C30l 2/C27ld/C281/C300 (12) l(0;0) 9/C301 2/C28d9ffiffiffiffiffiffiffiffiffiffiffiffiffi d2/C274p0C@n0C@o : (13) Butd2]0;sol(0;0) 9is real. Sinceffiffiffiffiffiffiffiffiffiffiffiffiffi d2/C274p >djj;there will always be one POSITIVE ROOT , so this fixed point is unstable. Now look at ( 91, 0). 0/C28l 1 /C282/C28d/C28l0C@10C@10C@10C@10C@10C@10C@10C@1/C30l(l/C27d)/C272/C30l 2/C27ld/C272/C300 (14) lð91;0Þ 9/C301 2/C28d9ffiffiffiffiffiffiffiffiffiffiffiffiffi d2/C288p0C@n0C@o : (15) Ford>0;Rl(91;0) 90C10CC B0;so the point is asymptotically stable. If d/C300;l(91;0) /C27/C309iffiffiffi 2p ;so the point is linearly stable. If d/C23(/C282ffiffiffi2p ;0);the radical gives an IMAGINARY PART and the REAL PART is>0;so the point is unstable. If d/C30/C282ffiffiffi2p ;l(91;0) 9/C30ffiffiffi2p ;which has a POSITIVE REAL ROOT , so the point is unstable. If dB /C282ffiffiffi2p ;then djjBffiffiffiffiffiffiffiffiffiffiffiffiffi d2/C288p ;so both ROOTS are POSITIVE and the point is unstable. The following table sum- marizes these results. /d>0/asymptotically stable /d/C300/linearly stable (superstable) /dB0/unstable Now specialize to the case d/C300;which can be integrated by quadratures. In this case, the equations become ˙x/C30y (16) ˙y/C30x/C28x3: (17) Differentiating (16) and plugging in (17) gives ¨x/C30˙y/C30x/C28x3: (18) Multiplying both sides by ˙xgives ¨x˙x/C28˙xx/C27˙xx3/C300 (19) d dt1 2˙x2/C2812x 2/C2814x 4 ! /C300; (20) so we have an invariant of motion h, h/C1312˙x 2/C2812x/C2714x 4: (21) Solving for ˙x2 gives ˙x2 /C30dx dt !2 /C302h /C27x2 /C281 2x4 ; (22) dx dt /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h /C27x2 /C271 2x2s ; (23) so t /C30gdt /C30gdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h /C27 x2 /C271 2 x2s : (24) Note that the invariant of motion h satisfies ˙x /C30@h @ ˙x /C30@h @y (25) @h @x /C30/C28x /C27x3 /C30/C28˙y; (26) so the equations of the Duffing oscillator are given by the HAMILTONIAN SYSTEM ˙x /C30@h @y ˙y /C30/C28@h @x:8 >>>< >>>:(27) References Bender, C. M. and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill, p. 547, 1978. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Duhamel’s Convolution Principle Can be used to invert a LAPLACE TRANSFORM .Dumbbell Curve y2 /C30a2 x4 /C28x60CB0C@ : See also BUTTERFLY CURVE ,EIGHT CURVE ,PIRIFORM References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Dummy Variable A variable that appears in a calculation only as a placeholder and which disappears completely in the final result. For example, in the integral gx 0f(x?)dx?; /x? is a dummy variable since it is "integrated out" in the final answer. Any variable name other than x could therefore be used in the above expression, e.g. fx 0 f(l)dl ;fx 0 f(q)dq; etc. Dummy variables are also called BOUND VARIABLES or dead variables. Comtet (1974) adopts a notation in which dummy variable appearing as indices in sums are denoted by placing a dot underneath them (i.e., indicating them with an UNDERDOT ), e.g., X ˙c:1 /C27 ˙c:2 /C30nc1c2 /C301 6 nn2 /C2810CB0C@ (Comtet 1974, p. 33). See also BOUND VARIABLE ,UNDERDOT References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 32 /C1/3, 1974. Duodecillion In the American system, 1039. See also LARGE NUMBER Duodecimal The base-12 number system composed of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B. Such a system has been advocated by no less than Herbert Spencer, John Quincy Adams, and George Bernard Shaw (Gardner 1984). Some aspects of a base-12 system are pre- served in the terms DOZEN and GROSS . The following table gives the duodecimal equivalents of the first few decimal numbers. 1 1 11 B 21 19 2 2 12 10 22 1A 3 3 13 11 23 1B 4 4 14 12 24 20 5 5 15 13 25 21 6 6 16 14 26 22 7 7 17 15 27 23 8 8 18 16 28 24 9 9 19 17 29 25 10A20183026 See also BASE (NUMBER ), DOZEN ,GROSS References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 104 /C1/05, 1984. Dupin’s Cyclide CYCLIDE Dupin’s Indicatrix A pair of conics obtained by expanding an equation in MONGE’S FORM z /C30Fx; yðÞ in a MACLAURIN SERIES z /C30z 0;0ðÞ/C27z1x /C27z2y /C271 2z11x2 /C272z12xy /C27z22y20CB0C@ /C27::: /C301 2b11x2 /C272b12xy /C27b22y20CB0C@ : This gives the equation b11x2 /C272b12xy /C27b22y2 /C3091 : Amazingly, the radius of the indicatrix in any direc- tion is equal to the SQUARE ROOT of the RADIUS OF CURVATURE in that direction (Coxeter 1969). References Coxeter, H. S. M. "Dupin’s Indicatrix" §19.8 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 363 /C1/65, 1969.Dupin’s Theorem In three mutually orthogonal systems of surfaces, the LINES OF CURVATURE on any surface in one of the systems are its intersections with the surfaces of the other two systems. Duplication Formula ABEL’S DUPLICATION FORMULA ,DOUBLE- ANGLE FOR- MULAS ,LEGENDRE DUPLICATION FORMULA Duplication of the Cube CUBE DUPLICATION Durand’s Rule Let the values of a function fxðÞbe tabulated at points xiequally spaced by h /C30xi /C271 /C28xi ; so f1 /C30fx1ðÞ ;f2/C30 fx2ðÞ ;...,fn/C30fxnðÞ :Then Durand’s rule approximat- ing the integral of fxðÞis given by the N EWTON- COTES -like formula gx1 xif(x)dx/C30h2 5f1/C271110f 2/C27f3/C27:::/C27fn/C282/C271110f n/C281/C2725f n ! : See also BODE’S RULE,HARDY’S RULE,NEWTON- COTES FORMULAS ,S IMPSON’S 3/8 RULE,S IMPSON’S RULE, TRAPEZOIDAL RULE,W EDDLE’S RULE References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987. Du¨rer’s Conchoid These curves appear in Du ¨rer’s work Instruction in Measurement with Compasses and Straight Edge (1525) and arose in investigations of perspective. Du¨rer constructed the curve by drawing lines QRP and P?QR of length 16 units through Q(q;0) and R(r;0);where q/C27r/C3013:The locus of PandP?is the curve, although Du¨rer found only one of the two branches of the curve. The ENVELOPE of the lines QRP and P ?QR is a PARABOLA , and the curve is therefore a GLISSETTE of a point on a line segment sliding between a PARABOLA and one of its TANGENTS . Du¨rer called the curve "muschellini," which means CONCHOID . However, it is not a true CONCHOID and so is sometimes called DU¨ RER’S SHELL CURVE . The Cartesian equation is 2y2 x2 /C27y20CB0C@ /C282by2(x /C27y) /C27 b2 /C283a20CB0C@ y2 /C28a2x2 /C272a2b(x /C27y) /C27a2 a2 /C28b20CB0C@ /C300: The above curves are for (a;b) /C30(3;1); (3;3); (3;5): There are a number of interesting special cases. If b /C300, the curve becomes two coincident straight lines x /C300. For a /C300, the curve becomes the line pair x /C30 b=2 ; x /C30/C28b=2; together with the CIRCLE x /C27y /C30b : If a /C30b =2; the curve has a CUSP at (/C282a ;a):/ References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 157 /C1/59, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 163, 1967. MacTutor History of Mathematics Archive. "Du¨rer’s Shell Curves." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Durers.html. Du¨ rer’s Magic Square Du¨rer’s magic square is a MAGIC SQUARE with MAGIC CONSTANT 34 used in an engraving entitled Melenco- lia I by Albrecht Du¨rer (The British Museum, Burton 1989, Gellert et al. 1989). The engraving shows a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Du¨rer’s magic square is located in the upper right- hand corner of the engraving. The numbers 15 and 14 appear in the middle of the bottom row, indicating the date of the engraving, 1514. Du¨rer’s magic square has the additional property that the sums in any of the four quadrants, as well asthe sum of the middle four numbers, are all 34 (Hunter and Madachy 1975, p. 24). See also DU¨ RER’S SOLID ,MAGIC SQUARE References Boyer, C. D. and Merzbach, U. C. A History of Mathematics. New York: Wiley, pp. 296 /C1/97, 1991. Burton, D. M. Cover illustration of Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989. Gellert, W.; Gottwald, S.; Hellwich, M.; Ka ¨stner, H.; and Ku¨nstner, H. (Eds.). Appendix, Plate 19. VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, p. 24, 1975. Rivera, C. "Melancholia." http://www.primepuzzles.net/mel- ancholia.htm. Du¨rer’s Shell Curve DU¨RER’S CONCHOID Du¨rer’s Solid The 8-faced solid depicted in an engraving entitled Melencolia I by Albrecht Du ¨rer (The British Museum, Burton 1989, Gellert et al. 1989), the same engraving in which D U¨RER’S MAGIC SQUARE appears, which depicts a disorganized jumble of scientific equipmentlying unused while an intellectual sits absorbed in thought. Although Du ¨rer does not specify how his solid is constructed, Schreiber (1999) has noted that it appears to consist of a distorted CUBE which is first stretched to give rhombic faces with angles of 72 8, and then truncated on top and bottom to yield boundingtriangular faces whose vertices lie on the CIRCUM- SPHERE of the azimuthal cube vertices. Starting with a unit cube oriented parallel to the axes of the coordinate system, rotate it by E ULER ANGLES c/C30p=4 and u/C30sec/C281ffiffiffi 3p to align a threefold symme- try axis along the z-axis. The stretch factor needed to produce rhombic angles of 72 8is then s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ3ffiffiffi 5p:s ð1Þ The azimuthal points are a distance /d ¼ s =2/ away from the origin, and in order for the vertices of the triangles obtained by truncation to lie at this same distance, the TRUNCATION must be done a distance / ð3 /C28ffiffiffi5p Þ=2 / along the edge from one of the azimuthal points, which corresponds to a height h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23ffiffiffi5p/C281 4 :s ð2Þ The resulting solid has six 126 /C1/08 /C1/2 /C1/08 /C1/268 penta- gonal faces and two equilateral triangular faces, and the lengths of the sides are in the ratio 1:1 2 ð3 þffiffiffi 5p Þ :ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ð5 þffiffiffi 5p Þ:q ð3Þ Examination of this solid shows it to be identical to the dimensions of the solid reconstructed from its perspective picture (Schro ¨der 1980, p. 70; Schreiber 1999). See also DU¨ RER’S MAGIC SQUARE References Burton, D. M. Cover illustration of Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989. Gellert, W.; Gottwald, S.; Hellwich, M.; Ka¨stner, H.; and Ku¨nstner, H. (Eds.). Appendix, Plate 19. VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989. Schreiber, P. "A New Hypothesis on Du¨rer’s Enigmatic Polyhedron in His Copper Engraving ‘Melancholia I’." Historia Math. 26, 369 /C1/77, 1999. Schro ¨der, E. Du¨rer--Kunst und Geometrie. Berlin: Akade- mie-Verlag, 1980. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Durfee Polynomial Let FnðÞbe a family of PARTITIONS of n and let Fn ;dðÞ denote the set of PARTITIONS in FnðÞwith DURFEE SQUARE of size d. The Durfee polynomial of FnðÞis then defined as the polynomial PF ;n /C30X Fn ;dðÞjj yd ; where 0 5d 5ffiffiffinp:/ See also DURFEE SQUARE ,PARTITION References Canfield, E. R.; Corteel, S.; and Savage, C. D. "Durfee Polynomials." Electronic J. Combinatorics 5, No. 1, R32, 1 /C1/1, 1998. http://www.combinatorics.org/Volume_5/ v5i1toc.html#R32.Durfee Square The length of the largest-sized SQUARE contained within the FERRERS DIAGRAM of a PARTITION . Its size can be determined using DurfeeSquare [f] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). The size of the Durfee square remains unchanged between a partition and its CONJUGATE PARTITION (Skiena 1990, p. 57). In the plot above, the Durfee square has size 3. See also CONJUGATE PARTITION ,D URFEE POLYNO- MIAL ,FERRERS DIAGRAM ,PARTITION References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Dust CANTOR DUST,FATOU DUST Dvoretzky’s Theorem Each centered convex body of sufficiently high dimen- sion has an "almost spherical" k-dimensional central section. Dyad Dyads extend VECTORS to provide an alternative description to second RANK TENSORS . A dyad D A ;BðÞ of a pair of VECTORS A and B is defined by D A;BðÞ/C13 AB: The DOT PRODUCT is defined by A :BC /C13 A :BðÞ C AB:C/C13AB :CðÞ ; and the COLON PRODUCT by AB : CD /C13C:AB:D/C30A:CðÞ B:DðÞ See also DYADIC ,TENSOR References Morse, P. M. and Feshbach, H. "Dyadics and Other Vector Operators." §1.6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 54 /C1/2, 1953. Dyadic A linear POLYNOMIAL of DYADS AB /C27CD /C27::: consist- ing of nine components Aij which transform as Aij0CB0C@ 0 /C30X m;nhmhn h?ih?j@xm @x0 i@xn @x0 jAmn (1) /C30X m;nh0 ih0 j hmhn@x0 i @xm@xj @xnAmn (2) /C30X m;nh0 ihn hmh?j@x0 i @xm@xm @x0 jAmn : (3) Dyadics are often represented by Gothic capital letters. The use of dyadics is nearly archaic since TENSORS perform the same function but are notation- ally simpler. A unit dyadic is also called the IDEMFACTOR and is defined such that I:A /C13A : (4) In CARTESIAN COORDINATES , I /C30ˆxˆx /C27ˆyˆy /C27ˆzˆz ; (5) and in SPHERICAL COORDINATES I /C309r : (6) See also DYAD,TENSOR ,TETRADIC References Arfken, G. "Dyadics." §3.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 137 /C1/ 40, 1985. Jeffreys, H. and Jeffreys, B. S. "Dyadic Notation." §3.04 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 89, 1988. Morse, P. M. and Feshbach, H. "Dyadics and Other Vector Operators." §1.6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 54 /C1/2, 1953. Dyck Language The simplest ALGEBRAIC LANGUAGE , denoted D: If X is the alphabet fx;xg; then D is the set of words u of X which satisfy 1. ujjx/C30 ujj¯x ; where ujjxis the numbers of letters x in the word u, and 2. if u is factored as vw, where v and w are words of X /C31; then vjjx] vjj¯x :/ See also ALGEBRAIC LANGUAGEReferences Bousquet-Me ´lou, M. "Convex Polyominoes and Heaps of Segments." J. Phys. A: Math. Gen. 25, 1925 /C1/934, 1992. Dyck Path A LATTICE PATH from 0;0ðÞ to (n, n) which never crosses (but may touch) the line y /C30x. There are Cn /C301 n /C27 12n n0C@80C@9 Dyck paths, where Cn is a CATALAN NUMBER . See also LATTICE PATH References Degenhardt, S. L. and Milne, S. C. "Weighted Inversion Statistics and Their Symmetry Groups." Preprint. Dyck’s Surface The surface with three CROSS-CAPS (Francis and Collins 1993, Francis and Weeks 1999). See also CROSS- CAP References Francis, G. and Collins, B. "On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art." Ch. 11 in The Visual Mind: Art and Mathematics (Ed. M. Emmer). Cambridge, MA: MIT Press, 1993. Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/99, 1999. # 1999 /C1/001 Wolfram Research, Inc. Dyck’s Theorem HANDLES and CROSS-HANDLES are equivalent in the presence of a CROSS-CAP . See also CROSS- CAP,CROSS- HANDLE ,H ANDLE , VON DYCK’S THEOREM References Dyck, W. "Beitra ¨ge zur Analysis situs I." Math. Ann. 32, 459 /C1/12, 1888. Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/99, 1999. Dye’s Theorem For any two ergodic measure-preserving transforma- tions on nonatomic PROBABILITY SPACES , there is an ISOMORPHISM between the two PROBABILITY SPACES carrying orbits onto orbits. See also ERGODIC THEORY Dyet INEXACT DIFFERENTIAL Dymaxion Buckminster Fuller’s term for the CUBOCTAHEDRON . See also CUBOCTAHEDRON ,MECON Dynamical System A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the INTEGERS on another object (usually a MANIFOLD ). When the reals are acting, the system is called a continuous dynamical system, and when the INTE- GERS are acting, the system is called a discrete dynamical system. If f is any CONTINUOUS FUNCTION , then the evolution of a variable x can be given by the formula xn /C271 /C30fxnðÞ : (1) This equation can also be viewed as a difference equation xn/C271 /C28xn /C30fxnðÞ/C28xn ; (2) so defining gxðÞ/C13fxðÞ/C28x (3) gives xn/C271 /C28xn /C30gxnðÞ+1; (4) which can be read "as n changes by 1 unit, x changes bygxðÞ:/" This is the discrete analog of the DIFFER- ENTIAL EQUATION x0nðÞ/C30gxnðÞðÞ : (5) See also ANOSOV DIFFEOMORPHISM ,ANOSOV FLOW, AXIOM AD IFFEOMORPHISM ,AXIOM AF LOW,BIFURCA- TION THEORY ,CHAOS ,ERGODIC THEORY ,G EODESIC FLOW References Aoki, N. and Hiraide, K. Topological Theory of Dynamical Systems. Amsterdam, Netherlands: North-Holland, 1994. Golubitsky, M. Introduction to Applied Nonlinear Dynami- cal Systems and Chaos. New York: Springer-Verlag, 1997. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed.New York: Springer-Verlag, 1997. Jordan, D. W. and Smith, P. Nonlinear Ordinary Differen- tial Equations: An Introduction to Dynamical Systems,3rd ed. Oxford, England: Oxford University Press, 1999. Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion, 2nd ed. New York: Springer-Verlag, 1994. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, 1990. Strogatz, S. H. Nonlinear Dynamics and Chaos, with Appli- cations to Physics, Biology, Chemistry, and Engineering.Reading, MA: Addison-Wesley, 1994.Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Dynkin Diagram Every SEMISIMPLE LIE ALGEBRA gis classified by its Dynkin diagram. A Dynkin diagram is a GRAPH with a few different kinds of possible edges. The CON- NECTED COMPONENTS of the graph correspond to the irreducible subalgebras of g:So a SIMPLE LIE ALGE- BRA’s Dynkin diagram has only one component. The rules are restrictive. In fact, there are only certain possibilities for each component, corresponding to theclassification of SEMI-SIMPLE LIE ALGEBRAS . The roots of a complex L IE ALGEBRA form a LATTICE of rank kin a C ARTAN SUBALGEBRA hƒg;where kis the RANK ofg:Hence, the ROOT LATTICE can be considered a lattice in Rk:A vertex, or node, in the Dynkin diagram is drawn for each SIMPLE ROOT , which corresponds to a generator of the ROOT LATTICE . Between two nodes aandb;an edge is drawn if the simple roots are not perpendicular. One line is drawnif the angle between them is 2 p=3;two lines if the angle is 3 p=3;and three lines are drawn if the angle is 5p=6:There are no other possible angles between SIMPLE ROOTS . Alternatively, the number of lines N between the simple roots aandbis given by N/C30AabAba/C302a;bhi ajj22b;ahi bjj2/C304 cos2u; where Aabis an entry in the C ARTAN MATRIX .I na Dynkin diagram, an arrow is drawn from the longerroot to the shorter root (when the angle is 3 p=3o r 5p=6): / The picture above shows the two simple roots for G2; at an angle of 5 p=6;in the ROOT LATTICE . Therefore, the Dynkin diagram for G2has two nodes, with three lines between them. Here are some properties of admissible Dynkin diagrams. 1. A diagram obtained by removing a node from an admissible node is admissible. 2. An admissible diagram has no loops. 3. No node has more than three lines attached to it. 4. A sequence of nodes with only two single lines can be collapsed to give an admissible diagram. 5. The only connected diagram with a triple line has two nodes. AC OXETER- DYNKIN DIAGRAM , also called a Coxeter graph, is the same as a Dynkin diagram, without the arrows, although sometimes these are also called Dynkin diagrams. The Coxeter diagram is sufficient to characterize the algebra, as can be seen by enumerating connected diagrams. The simplest way to recover a SIMPLE LIE ALGEBRA from its Dynkin diagram is to first reconstruct its CARTAN MATRIX Aij0CB0C@ : The ith node and jth node are connected by AijAji lines. Since Aij /C300 IFF Aji /C300 ; and otherwise Aji /C23/C283 ;/C282;/C281 fg ; it is easy to find Aijand Aji ; up to order, from their product. The arrow in the diagram indicates which is larger. For example, if node 1 and node 2 have two lines between them, from node 1 to node 2, then A12 /C30/C281 and A21 /C30/C282:/ However, it is worth pointing out that each SIMPLE LIE ALGEBRA can be constructed concretely. For instance, the infinite families An ; Bn ; Cn ; and Dn correspond to sln/C271C the SPECIAL LINEAR LIE ALGE- BRA, so2n/C271C the odd ORTHOGONAL LIE ALGEBRA , sp2nC the SYMPLECTIC LIE ALGEBRA , and so2nC the even ORTHOGONAL LIE ALGEBRA . The other simple Lie algebras are called EXCEPTIONAL LIE ALGEBRAS , and have constructions related to the OCTONIONS . See also CARTAN MATRIX ,COXETER- DYNKIN DIAGRAM , KILLING FORM,L IE ALGEBRA ,L IE GROUP ,R OOT LATTICE ,ROOT (LIE ALGEBRA ), SIMPLE LIE ALGEBRA , WEYL GROUP References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Hsiang, W. Y. Lectures on Lie Groups. Singapore: World Scientific, pp. 98 /C1/02, 2000.Huang, J.-S. "Dynkin Diagrams." §4.6 in Lectures on Representation Theory. Singapore: World Scientific, pp. 39 /C1/4, 1999. Jacobson, N. "The Determination of the Cartan Matrices." §4.5 in Lie Algebras. New York: Dover, pp. 128 /C1/35, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Dyson’s Conjecture Based on a problem in particle physics, Dyson (1962abc) conjectured that the constant term in the LAURENT SERIES Y 1 5i"j5n1 /C28xi xj !ai is the MULTINOMIAL COEFFICIENT a1 /C27 a2 /C27:::/C27 an ðÞ a1!a2!:::an! The theorem was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970). See also MACDONALD’S CONSTANT- TERM CONJECTURE , ZEILBERGER- BRESSOUD THEOREM References Andrews, G. E. "The Zeilberger-Bressoud Theorem." §4.3 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 36 /C1/8, 1986. Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. I." J. Math. Phys. 3, 140/C1/56, 1962a. Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. II." J. Math. Phys. 3, 157/C1/65, 1962b. Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. III." J. Math. Phys. 3, 166/C1/75, 1962c. Good, I. J. "Short Proof of a Conjecture by Dyson." J. Math. Phys. 11, 1884, 1970. Gunson, J. "Proof of a Conjecture of Dyson in the Statistical Theory of Energy Levels." J. Math. Phys. 3, 752/C1/53, 1962. Wilson, K. G. "Proof of a Conjecture by Dyson." J. Math. Phys. 3, 1040 /C1/043, 1962. #1999/C1/001 Wolfram Research, Inc. E Ear A PRINCIPAL VERTEX xiof a SIMPLE POLYGON P is called an ear if the diagonal [xi/C281 ; xi /C271] that bridges xi lies entirely in P. Two ears xiand xjare said to overlap if int[xi /C281 ; xi ; xi /C271] S int[xj/C281 ; xj ; xj/C271] "¥: The TWO-EARS THEOREM states that, except for TRI- ANGLES , every SIMPLE POLYGON has at least two nonoverlapping ears. See also ANTHROPOMORPHIC POLYGON ,MOUTH ,TWO- EARS THEOREM References Meisters, G. H. "Polygons Have Ears." Amer. Math. Monthly 82, 648 /C1/51, 1975. Meisters, G. H. "Principal Vertices, Exposed Points, and Ears." Amer. Math. Monthly 87, 284 /C1/85, 1980. Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122,31/C1/5, 1991. Early Election Results Let Jones and Smith be the only two contestants in an election that will end in a deadlock when all votes for Jones (J) and Smith (S) are counted. What is the EXPECTATION VALUE of Xk /C13 S /C28J jj after k votes are counted? The solution is /C142Xk /C143/C302NN /C28 1 k=2bc;j1z;j1} N /C28 1 k =2bc/C28 1;j1z;j1} 2N k;j1z;j1} /C30k(2N /C28 k) 2NN k =2;j1z;j1}22N k;j1z;j1}/C281 for k even k(2N /C28 k /C27 1) 2NN (k /C281)=2;j1z;j1}22N k /C281;j1z;j1}/C281 for k odd:8 >>>>>>>< >>>>>>>: References Handelsman, M. B. Solution to Problem 10248. "Early Returns in a Tied Election." Amer. Math. Monthly 102, 554 /C1/56, 1995. Eban Number The sequence of numbers whose names (in English) do not contain the letter "e" (i.e., "e" is "banned"). The first few eban numbers are 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, ... (Sloane’s A006933); i.e., two, four, six, thirty, etc.References Sloane, N. J. A. Sequences A006933/M1030 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Eberhart’s Conjecture If qnis the nth prime such that Mqnis a MERSENNE PRIME , then qn /C2(3=2)n : It was modified by Wagstaff (1983) to yield WAG- STAFF’S CONJECTURE , qn /C2(2e/C28g )n ; where g is the EULER- MASCHERONI CONSTANT . See also WAGSTAFF’S CONJECTURE References Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 412, 1996. Wagstaff, S. S. "Divisors of Mersenne Numbers." Math. Comput. 40, 385 /C1/97, 1983. Eccentric Not CONCENTRIC . See also CONCENTRIC ,CONCYCLIC Eccentric Angle The angle u measured from the CENTER of an ELLIPSE to a point on the ELLIPSE . See also ECCENTRICITY ,ELLIPSE Eccentric Anomaly The ANGLE obtained by drawing the AUXILIARY CIRCLE of an ELLIPSE with center Oand FOCUS F, and drawing a LINE PERPENDICULAR to the SEMIMAJOR AXIS and intersecting it at A. The ANGLE Eis then defined as illustrated above. Then for an ELLIPSE with ECCENTRICITY e, AF/C30OF/C28AO/C30ae/C28acosE (1) But the distance AFis also given in terms of the distance from the FOCUS r/C30FPand the SUPPLEMENT of the ANGLE from the SEMIMAJOR AXIS vby AF/C30rcos(p/C28v)/C30/C28rcosv: (2) Equating these two expressions gives r /C30a(cos E /C28 e) cos v; (3) which can be solved for cos v to obtain cos v /C30a(cos E /C28 e) r: (4) To get E in terms of r, plug (4) into the equation of the ELLIPSE r /C30a(1 /C28 e2) 1 /C27 cos v : (5) Rearranging, r(1 /C27e cos v) /C30a(1 /C28e2) (6) and plugging in (4) then gives r 1 /C27ae cos E r/C28e2 r ! /C30r /C27ae cos E /C28e2a /C30a(1 /C28e2) : (7) Solving for r gives r /C30a(1 /C28e2) /C28ea cos E /C27e2a /C30a(1 /C28e cos E) ; (8) so differentiating yields the result ˙r /C30ae ˙E sin E: (9) The eccentric anomaly is a very useful concept in orbital mechanics, where it is related to the so-called mean anomaly M by KEPLER’S EQUATION M /C30E /C28e sin E : (10) M can also be interpreted as the AREA of the shaded region in the above figure (Finch). See also ECCENTRICITY ,ELLIPSE ,KEPLER’S EQUATION References Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed., rev. ed. Richmond, VA: Willmann-Bell, 1988. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lpc/lpc.html. Montenbruck, O. and Pfleger, T. Astronomy on the Personal Computer, 4th ed. Berlin: Springer-Verlag, p. 62, 2000. Eccentricity A quantity defined for a CONIC SECTION which can be given in terms of SEMIMAJOR a and SEMIMINOR AXES b. interval curve e e /C300 CIRCLE 0/0 Be B1/ ELLIPSE /ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28b2 a2s / e /C301 PARABOLA 1 e /C211 HYPERBOLA /ffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 b2 a2s / The eccentricity can also be interpreted as the fraction of the distance to the semimajor axis at which the FOCUS lies, e /C30c a ; where c is the distance from the center of the CONIC SECTION to the FOCUS . See also CIRCLE ,CONIC SECTION ,ECCENTRIC ANOM- ALY,ELLIPSE ,FLATTENING ,FOCAL PARAMETER ,H Y- PERBOLA ,P ARABOLA ,SEMIMAJOR AXIS,SEMIMINOR AXIS Echidnahedron ICOSAHEDRON STELLATION #4. References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 65, 1971. Eckardt Point On the CLEBSCH DIAGONAL CUBIC , all 27 of the complex lines present on a general smooth CUBIC SURFACE are real. In addition, there are 10 points on the surface where three of the 27 lines meet. These points are called Eckardt points (Fischer 1986). See also CLEBSCH DIAGONAL CUBIC ,CUBIC SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 11, 1986. Eckart Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27ah 1/C27h/C27bh (1/C27h)2/C27g"# y/C300; where h/C30edx:/ References Barut, A. O.; Inomata, A.; and Wilson, R. "Algebraic Treat- ment of Second Po¨schl-Teller, Morse-Rosen, and Eckart Equations." J. Phys. A: Math. Gen. 20, 4083 /C1/096, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Eckert IV Projection The equations are x /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p(4 /C27 p)p (l /C28 l0)(1 /C27cos u) (1) y /C302ffiffiffiffiffiffiffiffiffiffiffiffiffi p 4 /C27 ps sin u; (2) where u is the solution to u /C27sin u cos u /C272 sin u /C30(2 /C271 2 p) sin f : (3) This can be solved iteratively using NEWTON’S METH- OD with u0 /C30 f=2 to obtain Du /C30/C28u /C27 sin u cos u /C27 2 sin u /C28 (2 /C2812 p) sin f 2 cos u(1 /C27 cos u) : (4) The inverse FORMULAS are f /C30sin /C281u /C27 sin u cos u /C27 2 sin u 2 /C271 2 p ! (5) l /C30 l0 /C27pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C27 pp x 1 /C27 cos u; (6) where u /C30sin/C281y 2ffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C27 p ps ! : (7) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 253 /C1/58, 1987.Eckert VI Projection The equations are x /C30( l /C28 l0)(1 /C27 cos u)ffiffiffiffiffiffiffiffiffiffiffiffiffi2 /C27 pp (1) y /C30 2 uffiffiffiffiffiffiffiffiffiffiffiffiffi2 /C27 pp ; (2) where u is the solution to u /C27sin u /C30(1 /C271 2 p) sin f: (3) This can be solved iteratively using NEWTON’S METH- OD with u0 /C30 f to obtain D u /C30/C28u /C27 sin u /C28 (1 /C271 2 p) sin f 1 /C27 cos u: (4) The inverse FORMULAS are f /C30sin/C281u /C27 sin u 1 /C2712 p ! (5) l /C30 l0 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27 pp x 1 /C27 cos u ; (6) where u /C301 2ffiffiffiffiffiffiffiffiffiffiffi 2 /C27 pp y: (7) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 253 /C1/58, 1987. Economical Number A number n is called an economical number if the number of digits in the prime factorization of n (including powers) uses fewer digits than the number of digits in n. The first few economical numbers are 125, 128, 243, 256, 343, 512, 625, 729, ... (Sloane’s A046759). Pinch shows that, under a plausible hypothesis related to the TWIN PRIME CONJECTURE , there are arbitrarily long sequences of consecutive economical numbers, and exhibits such a sequence of length nine starting at 1034429177995381247. See also EQUIDIGITAL NUMBER ,W ASTEFUL NUMBER References Hess, R. I. "Solution to Problem 2204(b)." J. Recr. Math. 28, 67, 1996 /C1/997. Pinch, R. G. E. "Economical Numbers." http://www.chalce- don.demon.co.uk/publish.html#62. Rivera, C. "Problems & Puzzles: Puzzle Sequences of Con- secutive Economical Numbers.-053." http://www.prime- puzzles.net/puzzles/puzz_053.htm. Santos, B. R. "Problem 2204. Equidigital Representation." J. Recr. Math. 27,58/C1/9, 1995. Sloane, N. J. A. Sequences A046759 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Economized Rational Approximation AP ADE´ APPROXIMANT perturbed with a CHEBYSHEV POLYNOMIAL OF THE FIRST KIND to reduce the leading COEFFICIENT in the ERROR . See also PADE´ APPROXIMANT Eddington Number 136 /C215 2256 :1:575 /C291079 : According to Eddington, the exact number of protons in the universe, where 136 was the RECIPROCAL of the fine structure constant as best as it could be mea- sured in his time. See also LARGE NUMBER References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 15 and 49, 1999. Edge (Graph) For an UNDIRECTED GRAPH , an unordered pair of nodes which specify the line connecting them are said to form an edge. For a DIRECTED GRAPH , the edge is an ordered pair of nodes. The terms "line," "arc," "branch," and "1-simplex" are sometimes used instead of edge (Skiena 1990, p. 80; Harary 1994). Harary (1994) calls an edge of a graph a "line." See also EDGE NUMBER ,HYPEREDGE ,NULL GRAPH , TAIT COLORING ,TAIT CYCLE ,VERTEX (GRAPH ) References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Edge (Polygon) A LINE SEGMENT on the boundary of a FACE , also called a SIDE. See also EDGE (POLYHEDRON ), VERTEX (POLYGON ) Edge (Polyhedron) A LINE SEGMENT where two FACES of a POLYHEDRON meet, also called a SIDE. See also EDGE (POLYGON ), VERTEX (POLYHEDRON ) Edge (Polytope) A 1-D LINE SEGMENT where two 2-D FACES of an n-D POLYTOPE meet, also called a SIDE. See also EDGE (POLYGON ), EDGE (POLYHEDRON ) Edge Chromatic Number The fewest number of colors necessary to color each EDGE of a GRAPH so that no two EDGES incident on the same VERTEX have the same color. The edge chro- matic number of a graph must be at least D;the largest VERTEX DEGREE of the graph (Skiena 1990, p. 216). However, Vizing (1964) and Gupta (1966) showed that any graph can be edge-colored with atmostD/C271 colors. The edge chromatic number of a COMPLETE BIPARTITE GRAPH isD:/ Determining the edge chromatic number of a graph isan NP -COMPLETE PROBLEM (Holyer 1981; Skiena 1990, p. 216). The edge chromatic number of a graph can be computed using EdgeChromaticNumber [g]i n theMathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also CHROMATIC NUMBER ,EDGE COLORING References Gupta, R. P. "The Chromatic Index and the Degree of a Graph." Not. Amer. Math. Soc. 13, 719, 1966. Holyer, I. "The NP-Completeness of Edge Colorings." SIAM J. Comput. 10, 718 /C1/20, 1981. Skiena, S. "Edge Colorings." §5.5.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 216, 1990. Vizing, V. G. "On an Estimate of the Chromatic Class of a p- Graph" [Russian]. Diskret. Analiz 3,23/C1/0, 1964. # 1999 /C1/001 Wolfram Research, Inc. Edge Coloring An edge coloring of a GRAPH G is a coloring of the edges of G such that adjacent edges (or the edges bounding different regions) receive different colors. BRELAZ’S HEURISTIC ALGORITHM can be used to find a good, but not necessarily minimal, edge coloring. Finding the minimum vertex coloring is equivalent to finding the minimum VERTEX COLORING of its LINE GRAPH (Skiena 1990, p. 216). The EDGE CHROMATIC NUMBER gives the minimum number of colors with which a graph can be colored. An edge coloring of a graph can be computed using EdgeColoring [g] in the Mathematica add-on pack- ageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also BRELAZ’S HEURISTIC ALGORITHM ,CHROMATIC NUMBER ,EDGE CHROMATIC NUMBER , K-COLORING References Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 13, 1986. Skiena, S. "Edge Colorings." §5.5.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 216, 1990. # 1999 /C1/001 Wolfram Research, Inc. Edge Connectivity The minimum number of edges l(G) whose deletion from a GRAPH G disconnects G, also called the line connectivity. The edge connectivity of a DISCON- NECTED GRAPH is 0, while that of a CONNECTED GRAPH with a BRIDGE is 1.Let k(G) be the VERTEX CONNECTIVITY of a graph G and d(G) its minimum degree, then for any graph, k(G) 5 l(G) 5 d(G) (Whitney 1932, Harary 1994, p. 43). The edge-connectivity of a graph can be determined with the command EdgeConnectivity [g] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). See also DISCONNECTED GRAPH , K -CONNECTED GRAPH ,VERTEX CONNECTIVITY References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 43, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 177 /C1/78, 1990. Whitney, H. "Congruent Graphs and the Connectivity of Graphs." Amer. J. Math. 54, 150 /C1/68, 1932. Edge Cover A subset of edges defined similarly to the VERTEX COVER (Skiena 1990, p. 219). Gallai (1959) showed that the size of the minimum edge cover plus the side of the maximum number of independent edges equals the number of vertices of a graph. See also VERTEX COVER References Gallai, T. "U¨ ber extreme Punkt- und Kantenmengen." Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 2, 133 /C1/38, 1959. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 178, 1990. # 1999 /C1/001 Wolfram Research, Inc. Edge Number The number of EDGES in a GRAPH , denoted Ejj:/ See also EDGE (GRAPH ) Edge Set The edge set of a GRAPH is simply a set of all edges of the graph. See also VERTEX SET # 1999 /C1/001 Wolfram Research, Inc. Edge-Graceful Graph A generalization of the GRACEFUL GRAPH . See also GRACEFUL GRAPH ,S KOLEM- GRACEFUL GRAPH ,SUPER- EDGE-GRACEFUL GRAPH References Sheng-Ping, L. "One Edge-Graceful Labeling of Graphs." Congressus Numer. 50,31/C1/41, 1985. Edge-Transitive Graph A GRAPH such that any two edges are equivalent under some element of its automorphism group. Every nontrivial graph that is edge-transitive but not VERTEX-TRANSITIVE contains at least 20 vertices (Skiena 1990, p. 186). The smallest known CUBIC GRAPH that is edge- but not VERTEX-TRANSITIVE is the GRAY GRAPH . See also GRAY GRAPH ,FOLKMAN GRAPH ,V ERTEX- TRANSITIVE GRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. # 1999 /C1/001 Wolfram Research, Inc. Edgeworth Series Let a distribution to be approximated be the distribu- tion Fn of standardized sums Yn /C30Pn i /C301(Xi /C28 ¯X)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i/C301s2 Xp : (1) In the CHARLIER SERIES , take the component random variables identically distributed with mean m ; var- iance s2 ; and higher cumulants sr lrfor r ]3: Also, take the developing function C(t) as the standard NORMAL DISTRIBUTION FUNCTION F(t) ; so we have k1 /C28 g1 /C300 (2) k2 /C28 g2 /C300 (3) k3 /C28 g3 /C30lr nr=2/C281 : (4) Then the Edgeworth series is obtained by collecting terms to obtain the asymptotic expansion of the CHARACTERISTIC FUNCTION (PROBABILITY ) OF THE FORM fn(t) /C30 1 /C27X/C12 r/C301Pr(it) nr=2"# e /C28t2 =2 ; (5) where Pr is a polynomial of degree 3r with coefficients depending on the cumulants of orders 3 to r /C272: If the powers of C are interpreted as derivatives, then the distribution function expansion is given by Fn(x) /C30C(x) /C27X/C12 r/C301Pr( /C28F(x)) nr=2 (6) (Wallace 1958). The first few terms of this expansion are then given byf(t) /C30C(t) /C28l3 C(3)(t) 6ffiffiffinp/C271 nl4 C(4)(t) 24/C27l2 3 C(6)(t) 72"# /C27... (7) Crame ´r (1928) proved that this series is uniformly valid in t. See also CHARLIER SERIES ,CORNISH- FISHER ASYMP- TOTIC EXPANSION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 935, 1972. Charlier, C. V. L. "U¨ ber dir Darstellung willku ¨rlicher Funk- tionen." Ark. Mat. Astr. och Fys. 2, No. 20, 1 /C1/5, 1906. Crame ´r, H. "On the Composition of Elementary Errors." Skand. Aktuarietidskr. 11,13/C1/4 and 141 /C1/80, 1928. Edgeworth, F. Y. "The Law of Error." Cambridge Philos. Soc. 20,36/C1/6 and 113 /C1/41, 1905. Esseen, C. G. "Fourier Analysis of Distribution Functions." Acta Math. 77,1/C1/25, 1945. Hsu, P. L. "The Approximate Distribution of the Mean and Variance of a Sample of Independent Variables." Ann. Math. Stat. 16,1/C1/9, 1945. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 107 /C1/08, 1951. Wallace, D. L. "Asymptotic Approximations to Distribu- tions." Ann. Math. Stat. 29, 635 /C1/54, 1958. e-Divisor d is called an e-divisor (or exponential divisor) of a number n with PRIME FACTORIZATION n/C30pa1 1pa2 2/C1/C1/C1parr if/djn/and d/C30pb1 1pb2 2/C1/C1/C1pbr r; where bj½ajfor 15j5r:For example, the e-divisors of 36 are 2 /C2153;4/C2153;2/C2159;and 4 /C2159:/ See also E-PERFECT NUMBER References Guy, R. K. "Exponential-Perfect Numbers." §B17 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 73, 1994. Straus, E. G. and Subbarao, M. V. "On Exponential Divi- sors." Duke Math. J. 41, 465/C1/71, 1974. Edmonds’ Map A nonreflexible regular map of GENUS 7 with eight VERTICES ,2 8 EDGES , and eight HEPTAGONAL faces. Effective Action AGROUP ACTION G/C29X0Xis effective if there are no nontrivial actions. In particular, this means that there is no element of the GROUP (besides the IDENTITY ELEMENT ) which does nothing, leaving every point where it is. This can be expressed as Sx /C23X Gx /C30 fe g; where Gx is the ISOTROPY GROUP at x and e is the identity of G. It is possible for a LIE GROUP G to have an effective action on a smaller dimensional space M. However, N(M) /C30max fdim G½G is a compact Lie group ; acting effectively on M g is finite, and is called the degree of symmetry of M. See also FREE ACTION ,G ROUP ,ISOTROPY GROUP , MATRIX GROUP ,O RBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION ,TOPOLOGICAL GROUP , TRANSITIVE References Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 4 /C1/ and 221 /C1/24, 1987. Efron’s Dice A set of four nontransitive DICE such that the probabilities of A winning against B, B against C, C against D, and D against A are all 2:1. A set in which ties may occur, in which case the DICE are rolled again, which gives ODDS of 11:6 is See also DICE,SICHERMAN DICE References Gardner, M. "Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indiffer- ence." Sci. Amer. 223, 110 /C1/14, Dec. 1970. Honsberger, R. "Some Surprises in Probability." Ch. 5 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 94 /C1/7, 1979. E-Function For any a /C23A (where A denotes the set of ALGEBRAIC NUMBERS ), letajjdenote the maximum of moduli of allconjugates of a: Then a function f(z) /C30X/C12 n/C300cnzn n! is said to be an E-function if the following conditions hold (Nesterenko 1999). 1. All coefficients cn belong to the same ALGEBRAIC NUMBER FIELD K of finite degree over Q. 2. If e > 0 is any positive number, then cnjj/C30O(nen) as n 0/C12:/ 3. For any e > 0; there exists a sequence of natural numbers fqn gn]1 such that qnck /C23ZK for k /C300, ..., n and that qn /C30O(nen) :/ Every E-function is an ENTIRE FUNCTION , and the set of E-functions is a RING under the operations of ADDITION and MULTIPLICATION . Furthermore, if f(z) is an E-function, then f ?(z) and fz 0f(t) dt are E- functions, and for any ALGEBRAIC NUMBER a; the function f( az) is also an E-function (Nesterenko 1999). See also SHIDLOVSKII THEOREM References Nesterenko, Yu. V. §1.2 in A Course on Algebraic Indepen- dence: Lectures at IHP 1999. http://www.math.jussieu.fr/ ~nesteren/. Siegel, C. L. Transcendental Numbers. New York: Chelsea, 1965. Egg An OVAL with one end more pointed than the other. See also ELLIPSE ,MOSS’S EGG,OVAL,OVOID ,THOM’S EGGS Egyptian Fraction EGYPTIAN NUMBER ,UNITFRACTION Egyptian Number A number nis called an Egyptian number if it is the sum of the DENOMINATORS in some UNIT FRACTION representation of a positive whole number not con- sisting entirely of 1s. For example, 1/C301 2/C2713/C2716; so 2/C273/C276/C3011 is an Egyptian number. The num- bers which are notEgyptian are 2, 3, 5, 6, 7, 8, 12, 13, 14, 15, 19, 21, and 23 (Sloane’s A028229; Konhauser et al. 1996, p. 147). Ifnis the sum of denominators of a unit fraction representation composed of distinct denominators which are not all 1s, then it is called a strictly Egyptian number. For example, by virtue of 1 /C301 2 /C2712 ; 2 /C27 2 /C304 is Egyptian, but it is not strictly Egyptian. Graham (1963) proved that every number ]78 is strictly Egyptian. Numbers which are strictly Egyp- tian are 11, 24, 30, 31, 32, 37, 38, 43, ... (Sloane’s A052428), and those which are not are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, ... (Sloane’s A051882). See also U NIT FRACTION References Graham, R. L. "A Theorem on Partitions." J. Austral. Math. Soc. 3, 435 /C1/41, 1963. Konhauser, J. D. E.; Vellman, D.; and Wagon, S. Which Way Did the Bicycle Go and Other Intriguing Mathematical Mysteries. Washington, DC: Amer. Math. Soc., 1996. Sloane, N. J. A. Sequences A028229, A051882, and A052428 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Ehrhart Polynomial Let D denote an integral convex POLYTOPE of DIMEN- SION n in a lattice M, and let lD(k) denote the number of LATTICE POINTS in D dilated by a factor of the integer k, lD(k) /C30#(kDS M) (1) for k /C23Z/C27: Then lDis a polynomial function in k of degree n with rational coefficients lD(k) /C30ankn /C27an/C281kn/C281 /C27.../C27a0 (2) called the Ehrhart polynomial (Ehrhart 1967, Pom- mersheim 1993). Specific coefficients have important geometric interpretations. 1. an is the CONTENT of D:/ 2. an/C281is half the sum of the CONTENTS of the (n /C281)/-D faces of D:/ 3. a0 /C301 :/ Let S2(D) denote the sum of the lattice lengths of the edges of D; then the case n /C302 corresponds to PICK’S THEOREM , lD(k) /C30Vol( D)k2 /C271 2 S2( D) /C271 : (3) Let S3(D) denote the sum of the lattice volumes of the 2-D faces of D; then the case n /C303 gives lD(k) /C30Vol( D)k3 /C271 2 S3(D)k2 /C27a1k /C271; (4) where a rather complicated expression is given by Pommersheim (1993), since a1 can unfortunately not be interpreted in terms of the edges of D: The Ehrhart polynomial of the tetrahedron with vertices at (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)islD(k) /C3016abck3 /C2714(ab /C27ac /C27bc /C27d)k2 /C271 12ac b/C27bc a/C27ab c/C27d2 abc ! /C271 4(a /C27b /C27c /C27A /C27B /C27C)" /C28Asbc d;aA d ! /C28Bsac d;bB d ! /C28Csab d;cC d !;j21 k /C271; (5) where s(x;y)isaD EDEKIND SUM, A /C30GCD( b ; c) ; B /C30 GCD( a ; c) ; C /C30GCD( a ; b) (here, GCD is the GREAT- EST COMMON DIVISOR ), and d /C30ABC (Pommersheim 1993). See also DEHN INVARIANT ,PICK’S THEOREM References Ehrhart, E. "Sur une proble `me de ge´ome´trie diophantine line´aire." J. reine angew. Math. 227,1/C1/9, 1967. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 215, 1984. Macdonald, I. G. "The Volume of a Lattice Polyhedron." Proc. Camb. Phil. Soc. 59, 719 /C1/26, 1963. McMullen, P. "Valuations and Euler-Type Relations on Certain Classes of Convex Polytopes." Proc. London Math. Soc. 35, 113 /C1/35, 1977. Pommersheim, J. "Toric Varieties, Lattices Points, and Dedekind Sums." Math. Ann. 295,1/C1/4, 1993. Reeve, J. E. "On the Volume of Lattice Polyhedra." Proc. London Math. Soc. 7, 378 /C1/95, 1957. Reeve, J. E. "A Further Note on the Volume of Lattice Polyhedra." Proc. London Math. Soc. 34,57/C1/2, 1959. Ei EXPONENTIAL INTEGRAL , EN-FUNCTION Eigenform Given a DIFFERENTIAL OPERATOR D on the space of DIFFERENTIAL FORMS , an eigenform is a form a such that D a /C30 la for some constant l : For example, on the TORUS , the DIRAC OPERATOR D/C30/C28i(d/C27d/C31) acts on the form b/C303ei(3x/C274y)/C275ei(3x/C274y)dx/C284ei(3x/C274y)dxffldy; giving Db/C3015ei(3x/C274y)/C2725ei(3x/C274y)dx/C2820ei(3x/C274y)dxffldy; i.e.,Db/C305b:/ See also DIRAC OPERATOR ,L APLACIAN ,SPECTRUM (OPERATOR ) Eigenfunction If ˜L is a linear OPERATOR on a FUNCTION SPACE , then f is an eigenfunction for ˜L and l is the associated EIGENVALUE whenever ˜Lf /C30 lf :/ See also EIGENVALUE ,EIGENVECTOR ,FUNCTIONAL Eigenspace IfAis an n/C29nmatrix, and lis an EIGENVALUE ofA; then the union of the ZERO VECTOR and the set of all EIGENVECTORS corresponding to lis a SUBSPACE ofRn known as the EIGENSPACE ofl:/ Eigenvalue LetAbe a linear transformation represented by a MATRIX A:If there is a VECTOR X/C23Rn"0 such that AX/C30lX (1) for some SCALAR l;then lis called the eigenvalue of A with corresponding (right) EIGENVECTOR X. Eigenva- lues are also known as characteristic roots, proper values, or latent roots (Marcus and Minc 1988,p. 144). Letting Abe a k/C29k MATRIX , a11a12/C1/C1/C1 a1k a21a22/C1/C1/C1 a2k nn:::n ak1ak2/C1/C1/C1 akk2 6643 775(2) with eigenvalue l;then the corresponding EIGENVEC- TORS satisfy a11a12/C1/C1/C1 a1k a21a22/C1/C1/C1 a2k nn:::n ak1ak2/C1/C1/C1 akk2 6643 775x 1 x2 n xk2 6643 775/C30lx 1 x2 n xk2 6643 775; (3) which is equivalent to the homogeneous system a 11/C28l a12 /C1/C1/C1 a1k a21 a22/C28l/C1/C1/C1 a2k nn:::n ak1 ak2 /C1/C1/C1 akk/C28l2 6643 775x 1 x2 n xk2 6643 775/C300 0 n 02 6643 775: (4) Equation (4) can be written compactly as (A/C28lI)X/C300; (5) where Iis the IDENTITY MATRIX . This MATRIX EQUA- TION can then be solved for l:/ Eigenvalues are given by the solutions of the CHAR- ACTERISTIC EQUATION of a given matrix. For example, for a 2 /C292 matrix, the eigenvalues are l9/C301 2(a11/C27a22)9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4a12a21/C27(a11/C28a22)2q ;j2r;j21 ; (6) which arises as the solutions of the CHARACTERISTIC EQUATIONx2/C28x(a11/C27a22)/C27(a11a22/C28a12a21)/C300; (7) which can be written x2/C28xTr(A)/C27det(A)/C300; (8) where Tr( A) is the TRACE ofAand det( A) is its DETERMINANT . The CHARACTERISTIC EQUATION for the 3/C293 case is x3/C28Tr(A)x2/C281 2(aijaji/C28aiiajj)(1/C28dij)x/C28det(A)/C300;(9) where dijis the K RONECKER DELTA and E INSTEIN SUMMATION has been used. The corresponding analy- tic eigenvalue expressions for 4 /C294 and larger ma- trices are very complicated. As shown in C RAMER’S RULE , a system of linear equations has nontrivial solutions only if the DETER- MINANT vanishes, so we obtain the CHARACTERISTIC EQUATION A/C28lI jj /C300: (10) If all kl/s are different, then plugging these back in gives k/C281 independent equations for the kcompo- nents of each corresponding EIGENVECTOR . The EI- GENVECTORS will then be orthogonal and the system is said to be nondegenerate. If the eigenvalues are n- fold DEGENERATE , then the system is said to be degenerate and the EIGENVECTORS are not linearly independent. In such cases, the additional constraint that the EIGENVECTORS beORTHOGONAL , Xi/C215Xj/C30XijjXj;j12;j12;j12;j12d ij; (11) where dijis the K RONECKER DELTA , can be applied to yield nadditional constraints, thus allowing solution for the EIGENVECTORS . Assume A has nondegenerate eigenvalues l1;l2;...;lkand corresponding linearly indepen- dent EIGENVECTORS X1;X2;...;Xkwhich can be denoted x11 x12 n x1k2 6643 775;x21 x22 n x2k2 6643 775;/C1/C1/C1xk1 xk2 n xkk2 6643 775: (12) Define the matrices composed of eigenvectors P/C13[X1X2/C1/C1/C1Xk]/C30x11x21/C1/C1/C1 xk1 x12x22/C1/C1/C1 xk2 nn:::n x1kx2k/C1/C1/C1 xkk2 6643 775(13) and eigenvalues D/C13l10/C1/C1/C1 0 0l2/C1/C1/C1 0 nn:::n 00 /C1/C1/C1lk2 6643 775; (14) where Dis a DIAGONAL MATRIX . Then AP /C30A[X1X2/C1/C1/C1 Xk] /C30[AX1AX2/C1/C1/C1 AXk] /C30[ l1X1l2X2/C1/C1/C1 lkXk] /C30l1x11l2x21/C1/C1/C1 lkxk1 l1x12l2x22/C1/C1/C1 lkxk2 nn::: n l1x1kl2x2k/C1/C1/C1 lkxkk2 6643 775 /C30x 11x21/C1/C1/C1 xk1 x12x22/C1/C1/C1 xk2 nn::: n x1kx2k/C1/C1/C1 xkk2 6643 775l10 /C1/C1/C1 0 0 l2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 lk2 6643 775 /C30PD ; (15) so A /C30PDP /C281 : (16) Furthermore, A2 /C30(PDP /C281)(PDP /C281) /C30PD(P/C281P)DP /C281 /C30PD2P /C281 : (17) By induction, it follows that for n /C210, An /C30PDnP /C281 : (18) The inverse of A is A /C281 /C30(PDP /C281) /C281 /C30PD /C281P/C281 ; (19) where the inverse of the DIAGONAL MATRIX D is trivially given by D /C281 /C30l/C281 1 0 /C1/C1/C1 0 0 l /C281 2 /C1/C1/C1 0 nn::: n 00 /C1/C1/C1 l /C281 k2 6643 775: (20) Equation (18) therefore holds for both POSITIVE and NEGATIVE n. A further remarkable result involving the matrices P and D follows from the definition eA /C13X/C12 n/C300An n!/C30X/C12 n/C300PDnP/C281 n! /C30PP/C12 n/C300Dn n! ! P/C281 /C30PeDP /C281 : (21) Since D is a DIAGONAL MATRIX , eD /C30X/C12 n/C300Dn n!/C30X/C12 n/C3001 n!ln 10 /C1/C1/C1 0 0 ln2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 lnk2 6643 775/C30P/C12 n/C300ln 1 n!0 /C1/C1/C1 0 0P/C12 n/C300ln 2 n!/C1/C1/C1 0 nn::: n 00 /C1/C1/C1P/C12 n /C300ln k n!2 66666666643 7777777775 /C30e l1 0 /C1/C1/C1 0 0 e l2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 e lk2 6643 775; (22) /eD can be found using Dn /C30ln 10 /C1/C1/C1 0 0 ln2/C1/C1/C1 0 nn::: n 00 /C1/C1/C1 lnk2 6643 775: (23) Assume we know the eigenvalue for AX /C30 lX : (24) Adding a constant times the IDENTITY MATRIX to A; (A /C27cI)X /C30(l /C27c)X /C13 l ? pX ; (25) so the new eigenvalues equal the old plus c. Multi- plying A by a constant c (cA)X /C30c(lX) /C13 l ?X ; (26) so the new eigenvalues are the old multiplied by c. Now consider a SIMILARITY TRANSFORMATION ofA:Let Ajjbe the DETERMINANT ofA;then Z/C281AZ/C28lI;j12;j12;j12;j12/C30Z /C281(A/C28lI)Z;j12;j12;j12;j12 /C30ZjjA/C28lI jj Z /C281;j12;j12;j12;j12/C30A/C28lI jj ; (27) so the eigenvalues are the same as for A: / See also BRAUER’S THEOREM ,C OMPLEX MATRIX , CONDITION NUMBER ,EIGENFUNCTION ,EIGENVECTOR , FROBENIUS THEOREM ,GERGORIN CIRCLE THEOREM , LYAPUNOV’S FIRST THEOREM ,L YAPUNOV’S SECOND THEOREM ,O STROWSKI’S THEOREM ,PERRON’S THEO- REM,P ERRON- FROBENIUS THEOREM ,P OINCARE ´SE- PARATION THEOREM ,RANDOM MATRIX ,REAL MATRIX , SCHUR’S INEQUALITIES ,STURMIAN SEPARATION THEO- REM,SYLVESTER’S INERTIA LAW,W IELANDT’S THEO- REM References Arfken, G. "Eigenvectors, Eigenvalues." §4.7 in Mathemati- cal Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 229 /C1/37, 1985. Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 145, 1988. Nash, J. C. "The Algebraic Eigenvalue Problem." Ch. 9 in Compact Numerical Methods for Computers: Linear Alge- bra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 102 /C1/18, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Eigensystems." Ch. 11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 449 /C1/89, 1992. Eigenvector A right eigenvector satisfies AX /C30 lX ; (1) where X is a column VECTOR . The right EIGENVALUES therefore satisfy A /C28 lI jj /C300: (2) A left eigenvector satisfies XA /C30 lX ; (3) where X is a row VECTOR ,so (XA)T /C30 lLXT ; (4) ATXT /C30 lLXT ; (5) where XT is the transpose of X. The left EIGENVALUES satisfy AT /C28 lLI;j12;j12;j12;j12/C30AT /C28 lLIT;j12;j12;j12;j12/C30(A /C28 l LI)T;j12;j12;j12;j12;j12;j12/C30(A /C28 l LI) jj ; (6) (since Ajj/C30AT;j12;j12;j12;j12) where Ajjis the DETERMINANT of A: But this is the same equation satisfied by the right EIGENVALUES , so the left and right EIGENVALUES are the same. Let XR be a MATRIX formed by the columns of the right eigenvectors and XLbe a MATRIX formed by the rows of the left eigenvectors. Let D /C13l1/C1/C1/C1 0 n::: n 0 /C1/C1/C1 ln2 435: (7) Then AX R /C30XRD XLA /C30DXL (8) XLAXR /C30XLXRD XLAXR /C30DXLXR ; (9) so XLXRD /C30DXLXR : (10) But this equation is OF THE FORM CD /C30DC where D is a DIAGONAL MATRIX , so it must be true that C /C13XLXR is also diagonal. In particular, if A is a SYMMETRIC MATRIX , then the left and right eigenvectors are transposes of each other. If A is a SELF-ADJOINT MATRIX , then the left and right eigenvectors are conjugate HERMITIAN MATRICES . Eigenvectors are sometimes known as characteristic vectors, proper vectors, or latent vectors (Marcus and Minc 1988, p. 144). Given a 3 /C293 MATRIX A with eigenvectors x1 ; x2 ; and x3and corresponding EIGENVALUES l1 ; l2 ; and l3 ;then an arbitrary VECTOR y can be written y /C30b1x1 /C27b2x2 /C27b3x3 : (11) Applying the MATRIX A ; Ay /C30b1Ax1 /C27b2Ax2 /C27b3Ax3 /C30l1b1x1/C27l2 l1b2x2/C27l3 l1b3x3 ! ; (12) so Any/C30ln 1b1x1/C27l2 l1 !n b2x2/C27l3 l1 !n b3x3"# : (13) Ifl1>l2;l3;it therefore follows that lim n0/C12Any/C30ln 1b1x1; (14) so repeated application of the matrix to an arbitrary vector results in a vector proportional to the EIGEN- VECTOR having the largest EIGENVALUE . See also EIGENFUNCTION ,EIGENVALUE References Arfken, G. "Eigenvectors, Eigenvalues." §4.7 in Mathemati- cal Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 229 /C1/37, 1985. Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 145, 1988. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Eigensystems." Ch. 11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press,pp. 449 /C1 /89, 1992. Eight Curve A curve also known as the G ERONO LEMNISCATE .I ti s given by C ARTESIAN COORDINATES x4/C30a2(x2/C28y2); (1) POLAR COORDINATES , r2/C30a2sec4ucos(2 u); (2) and PARAMETRIC EQUATIONS x /C30a sin t (3) y /C30a sin t cos t: (4) The CURVATURE and TANGENTIAL ANGLE are k(t) /C30/C283 sin t /C27 sin(3 t) 2[cos2 t /C27 cos2(2t)]3 =2 (5) f(t) /C30/C28tan /C281[cos t sec(2 t)] : (6) See also BUTTERFLY CURVE ,DUMBBELL CURVE ,EIGHT SURFACE ,PIRIFORM References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 124 /C1/26, 1972. MacTutor History of Mathematics Archive. "Eight Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/ Eight.html. Eight Surface The SURFACE OF REVOLUTION given by the PARA- METRIC EQUATIONS x(u; v) /C30cos u sin(2 v) (1) y(u; v) /C30sin u sin(2 v) (2) z(u; v) /C30sin v (3) for u /C23 [0; 2 p) and v /C23 [/C28p=2; p=2]::/ See also EIGHT CURVE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 310, 1997.Eight-Point Circle Theorem Let ABCD be a QUADRILATERAL with PERPENDICULAR DIAGONALS . The MIDPOINTS of the sides (a, b, c, and d) determine a PARALLELOGRAM (the VARIGNON PAR- ALLELOGRAM ) with sides PARALLEL to the DIAGONALS . The eight-point circle passes through the four MID- POINTS and the four feet of the PERPENDICULARS from the opposite sides a?; b ?; c ?; and d?:/ See also FEUERBACH’S THEOREM References Brand, L. "The Eight-Point Circle and the Nine-Point Circle." Amer. Math. Monthly 51,8 4/C1/5, 1944. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 11 /C1/3, 1976. Eikonal Equation Xn i/C301@u @xi !2 /C301: Eilenberg-Mac Lane Space For any A BELIAN GROUP Gand any NATURAL NUMBER n, there is a unique SPACE (up to HOMOTOPY type) such that all HOMOTOPY GROUPS except for the nth are trivial (including the 0th HOMOTOPY GROUPS , meaning the SPACE is path-connected), and the nth HOMOTOPY GROUP isISOMORPHIC to the GROUP G.I n the case where n/C301, the GROUP Gcan be non- ABELIAN as well. Eilenberg-Mac Lane spaces have many important applications. One of them is that every TOPOLOGICAL SPACE has the HOMOTOPY type of an iterated FIBRA- TION of Eilenberg-Mac Lane spaces (called a POST- NIKOV SYSTEM ). In addition, there is a spectral sequence relating the COHOMOLOGY of Eilenberg- Mac Lane spaces to the HOMOTOPY GROUPS of SPHERES . Eilenberg-Mac Lane-Steenrod-Milnor Axioms EILENBERG- STEENROD AXIOMS Eilenberg-Steenrod Axioms A family of FUNCTORS Hn( /C215) from the CATEGORY of pairs of TOPOLOGICAL SPACES and continuous maps, to the CATEGORY of ABELIAN GROUPS and group homo- morphisms satisfies the Eilenberg-Steenrod axioms if the following conditions hold. 1. LONG EXACT SEQUENCE OF A PAIR AXIOM . For every pair (X, A), there is a natural long exact sequence ... 0 Hn(A) 0 Hn(X) 0 Hn(X ; A) 0 Hn/C281(A) 0 ...; (1) where the MAP Hn(A) 0 Hn(X) is induced by the INCLUSION MAP A 0 X and Hn(X) 0 Hn(X ; A)is induced by the INCLUSION MAP (X ; f) 0 (X ; A): The MAP Hn(X ; A) 0 Hn/C281(A) is called the BOUND- ARY MAP. 2. HOMOTOPY AXIOM .Iff :(X ; A) 0 (Y ; B) is homo- topic to g :(X ; A) 0 (Y ; B) ; then their INDUCED MAPS f/C31 : Hn(X ; A) 0 Hn(Y ; B) and g /C31 : Hn(X ; A) 0 Hn(Y ; B) are the same. 3. EXCISION AXIOM .IfX is a SPACE with SUBSPACES A and U such that the CLOSURE of A is contained in the interior of U, then the INCLUSION MAP (XU ; AU) 0 (X ; A) induces an isomorphism Hn(XU ; AU) 0 Hn(X ; A) :/ 4. DIMENSION AXIOM . Let X be a single point space. Hn(X) /C300 unless n /C300, in which case H0(X) /C30G where G are some GROUPS . The H0are called the COEFFICIENTS of the HOMOLOGY theory H(/C215) :/ These are the axioms for a generalized homology theory. For a cohomology theory, instead of requiring that H( /C215)bea FUNCTOR , it is required to be a co- functor (meaning the INDUCED MAP points in the opposite direction). With that modification, the ax- ioms are essentially the same (except that all the induced maps point backwards). See also ALEKSANDROV- CECH COHOMOLOGY Ein Function Ein(z) /C13gz 0(1 /C28 e/C28t) dt t/C30E1(z) /C27ln z /C27 g ; where g is the EULER- MASCHERONI CONSTANT and E/1 is the EN-FUNCTION with n/C301. See also EN-FUNCTIONEinstein Field Equations The 16 coupled hyperbolic-elliptic nonlinear PARTIAL DIFFERENTIAL EQUATIONS that describe the gravita- tional effects produced by a given mass in general relativity. The equations state that Gmn/C308pTmn; where Tmnis the stress-energy tensor, and Gmn/C30Rmn/C281 2gmnR is the E INSTEIN TENSOR , with Rmnthe R ICCI TENSOR andRthe SCALAR CURVATURE . #1999/C1/001 Wolfram Research, Inc. Einstein Functions The functions E1(x)/C30x2ex (ex/C281)2(1) E2(x)/C30x ex/C281(2) E3(x)/C30ln(1/C28e/C28x) (3) E4(x)/C30x ex/C281/C28ln(1/C28e/C28x): (4) /E1(x) has an inflection point at Eƒ1(x)/C3018csch4(12x)[(x2/C272) cosh x /C272(x2/C282xsinh x/C281)]/C300 (5) which can be solved numerically to give x:2:34693 : E1(x) has an inflection point at Eƒ2(x)/C30ex[x/C272/C27ex(x/C282)] (ex/C281)3/C300; (6) which can be solved numerically to give x:17:5221 :/ References Abramowitz, M. and Stegun, C. A. (Eds.). "Debye Func- tions." §27.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 999 /C1/000, 1972. Einstein Summation The convention that repeated indices are implicitly summed over. This can greatly simplify and shorten equations involving TENSORS . For example, using Einstein summation, aiai /C13X iaiai and aikaij /C30X iaikaij : The convention was introduced by Einstein (1916), who later jested to a friend,"I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." (Kollros 1956; Pais 1982, p. 216). References Einstein, A. Ann. der Physik 49, 769, 1916. Kollros, L. "Albert Einstein en Suisse Souvenirs." Helv. Phys. Acta. Supp. 4, 271 /C1/81, 1956. Pais, A. Subtle is the Lord: The Science and the Life of Albert Einstein. New York: Oxford University Press, p. 216, 1982. Einstein Tensor Gab /C30Rab /C281 2 Rgab ; where Rabis the RICCI TENSOR , R is the SCALAR CURVATURE , and gabis the METRIC TENSOR . (Wald 1984, pp. 40 /C1/1). It satisfies G mn ; n /C300 (Misner et al. 1973, p. 222). See also METRIC TENSOR ,R ICCI TENSOR ,SCALAR CURVATURE References Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravita- tion. San Francisco: W. H. Freeman, 1973. Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984. # 1999 /C1/001 Wolfram Research, Inc. Eisenstein Integer The numbers a /C27bv; where v /C131 2(/C281 /C27iffiffiffi 3p ) is one of the ROOTS of z3 /C301 ; the others being 1 and v2 /C131 2(/C281 /C28iffiffiffi 3p ) : Eisenstein integers are members of the IMAGINARYQUADRATIC FIELD Q(ffiffiffiffiffiffi /C283p ); and the COMPLEX NUMBERS Z v½/C138: Every Eisenstein integer has a unique factor- ization. Specifically, any NONZERO Eisenstein integer is uniquely the product of POWERS of -1, v; and the "positive" EISENSTEIN PRIMES (Conway and Guy 1996). Every Eisenstein integer is within a distance njj=ffiffiffi 3p of some multiple of a given Eisenstein integer n. Do¨rrie (1965) uses the alternative notation J /C131 2(1 /C27iffiffiffi 3p ) (1) O /C131 2(1 /C28iffiffiffi 3p ) : (2) for /C28v2 and /C28v; and calls numbers OF THE FORM aJ /C27 bO G-NUMBERS . O and J satisfy J /C27O /C301 (3) JO /C301 (4) J2 /C27O /C300 (5) O2 /C27J /C300 (6) J3 /C30/C281 (7) O3 /C30/C281: (8) The sum, difference, and products of G numbers are also G numbers. The norm of a G number is N(aJ /C27bO) /C30a2 /C27b2 /C28ab : (9) The analog of F ERMAT’S THEOREM for Eisenstein integers is that a PRIME NUMBER pcan be written in the form a2/C28ab/C27b2/C30(a/C27bv)(a/C27bv2) IFF3¶p/C271:These are precisely the PRIMES OF THE FORM 3m2/C27n2(Conway and Guy 1996). See also EISENSTEIN PRIME ,EISENSTEIN UNIT,GAUS- SIAN INTEGER ,INTEGER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220 /C1/23, 1996. Cox, D. A. §4A in Primes of the Form x2/C27ny2:Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1989. Do¨rrie, H. "The Fermat-Gauss Impossibility Theorem." §21 in100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 96 /C1/04, 1965. Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes." §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33 /C1/6, 1994. Riesel, H. Appendix 4 in Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨u- ser, 1994. Wagon, S. "Eisenstein Primes." Mathematica in Action. New York: W. H. Freeman, pp. 278 /C1/79, 1991. Eisenstein Prime Letvbe the CUBE ROOT of unity ( /C281/C27iffiffiffi 3p )=2:Then the Eisenstein primes are 1. Ordinary PRIMES CONGRUENT to 2 (mod 3), 2. 1/C28vis prime in Zv½/C138;/ 3. Any ordinary PRIME CONGRUENT to 1 (mod 3) factors as aa/C31;where each of aanda/C31are primes in Zv½/C138andaanda/C31are not "associates" of each other (where associates are equivalent modulo multi- plication by an E ISENSTEIN UNIT ). References Cox, D. A. §4A in Primes of the Form x2/C27ny2:Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1989. Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes." §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33 /C1/6, 1994. Wagon, S. "Eisenstein Primes." Mathematica in Action. New York: W. H. Freeman, pp. 278 /C1/79, 1991. Eisenstein Series Gr(t)/C30X m;n?1 (m/C27nt)2r; (1) where the sum S?excludes m/C30n/C300;/T½t/C138/C210/, and r is an INTEGER with r/C212. The Eisenstein series satisfies the remarkable property Grat/C27b ct/C27d ! /C30(ct/C27d)2rEr(t): (2) Furthermore, each Eisenstein series is expressible as a polynomial of the INVARIANTS g2and g3of the WEIERSTRASS ELLIPTIC FUNCTION with positive ra- tional coefficients (Apostol 1997).The Eisenstein series of EVEN order satisfy G2k(t)/C302z(2k)/C272(2pi)2k (2k/C281)!X/C12 n/C301s2k/C281(n)e2pint; (3)where z(z) is the R IEMANN ZETA FUNCTION andsk(n)i s the DIVISOR FUNCTION (Apostol 1997, pp. 24 and 69). Writing the NOME qas q/C30epti/C30e/C28pK?(k)=K(k)(4) where K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND ,K?(k)/C13K(ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p );kis the MODULUS , and defining E2k(q)/C13G2k(t) 2z(2k); (5) we have E2n(q)/C301/C27c2nX/C12 k/C301kn/C281q2k 1/C28q2k(6) /C301/C27c2nX/C12 k/C301s2n/C281(k)q2k: (7) where c2n/C30(2pi)2k (2k/C281)!z(2k)/C30(/C281)k(2p)2k G(2k)z(2k): (8) /C30/C284n B2n; (9) where Bnis a B ERNOULLI NUMBER . For n/C301, 2, ..., the first few values of c2nare -24, 240, -504, 480, -264, 65520 =691;... (Sloane’s A006863 and A001067). The first few values of E2n(q) are therefore E2(q)/C301/C2824X/C12 k/C301s1(k)q2k(10) E4(q)/C301/C27240X/C12 k/C301s3(k)q2k(11) E6(q)/C301/C28504X/C12 k/C301s5(k)q2k(12) E8(q)/C301/C27480X/C12 k/C301s7(k)q2k(13) E10(q)/C301/C28264X/C12 k/C301s9(k)q2k(14) E12(q)/C301/C2765520 691X/C12 k/C301s11(k)q2k(15) E14(q)/C301/C2824X/C12 k/C301s13(k)q2k; (16) (Apostol 1997, p. 139). Ramanujan used the notations P(z)/C30E2(ffiffiffizp);Q(z)/C30E4(ffiffiffizp);and R(z)/C30E 6(ffiffiffizp);and these functions satisfy the system of differential equations qP /C301 12(P2 /C28Q) (17) qQ /C301 3(PQ /C28R) (18) qR /C3012(PR /C28Q2) (19) (Nesterenko 1999), where q/C30zd=dz is the DIFFEREN- TIAL OPERATOR . /E2n(q) can also be expressed in terms of complete ELLIPTIC INTEGRALS OF THE FIRST KIND K(k)as E4(q) /C302K(k) p !4 (1 /C28k2k?2) (20) E6(q) /C302K(k) p !6 (1 /C282k2)(1 /C271 2 k2k?2) (21) (Ramanujan 1913 /C1/914), where k is the MODULUS . The following table gives the first few Eisenstein series En(q) for even n. n Sloane lattice /En(q)/ 2 A006352 /1 /C2824q2 /C2872q4 /C2896q6 /C28168q8 /C28/C1/C1/C1/ 4 A004009 /E8// 1 /C27240q2 /C272160 q4 /C276720 q6 /C27/C1/C1/C1/ 6 A013973 /1 /C28504q2 /C2816632 q4 /C28122976 q6 /C28/C1/C1/C1/ 8 A008410 /E8 /C154E8//1 /C27480q2 /C2761920 q4 /C271050240 q6 /C27/C1/C1/C1/ 10 A013974 /y /C30r? sin u?:/ Ramanujan (1913 /C1/914) used the notation L(q)to refer to the closely related function L(q) /C301 /C2724X/C12 k /C301s(0) 1(n)(/C281)kqk (22) /C301 /C2824X/C12 k /C301(2k /C28 1)q2k/C281 1 /C27 q2k/C281 /C302K(k) p !2 (1 /C282k2) (23) /C301 /C2824q /C2724q2 /C2896q3 /C27/C1/C1/C1 (24) (Sloane’s A004011), where s(0)1(n) /C13X d½nd oddd (25) is the ODD DIVISOR FUNCTION . Ramanujan used the notation M(q) and N(q) to refer to E4(q) and E6(q); respectively. See also DIVISOR FUNCTION ,INVARIANT (ELLIPTICFUNCTION ), KLEIN’S ABSOLUTE INVARIANT ,L EECH LATTICE ,PI,THETA SERIES ,W EIERSTRASS ELLIPTIC FUNCTION References Apostol, T. M. "The Eisenstein Series and the Invariants g2 and g3/" and "The Eisenstein Series G2( t):/" §1.9 and 3.10 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 12 /C1/3 and 69 /C1/1, 1997. Borcherds, R. E. "Automorphic Forms on Os /C272;2(R)/C27 and Generalized Kac-Moody Algebras." In Proc. Internat. Congr. Math., Vol. 2. pp. 744 /C1/52, 1994. Borwein, J. M. and Borwein, P. B. "Class Number Three Ramanujan Type Series for 1=p:/" J. Comput. Appl. Math. 46, 281 /C1/90, 1993. Bump, D. Automorphic Forms and Representations. Cam- bridge, England: Cambridge University Press, p. 29, 1997. Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lat- tices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 119 and 123, 1993. Coxeter, H. S. M. "Integral Cayley Numbers." The Beauty of Geometry: Twelve Essays. New York: Dover, pp. 20 /C1/9, 1999. Gunning, R. C. Lectures on Modular Forms. Princeton, NJ: Princeton Univ. Press, p. 53, 1962. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 166, 1999. Milne, S. C. Hankel Determinants of Eisenstein Series. 13 Sep 2000. http://xxx.lanl.gov/abs/math.NT/0009130/. Nesterenko, Yu. V. §8.1 in A Course on Algebraic Indepen- dence: Lectures at IHP 1999. http://www.math.jussieu.fr/ ~nesteren/. Ramanujan, S. "Modular Equations and Approximations to p:/" Quart. J. Pure Appl. Math. 45, 350 /C1/72, 1913 /C1/914. Shimura, G. Euler Products and Eisenstein Series. Provi- dence, RI: Amer. Math. Soc., 1997. Sloane, N. J. A. Sequences A001067, A004009/M5416, A004011/M5140, A006863/M5150, A008410, A013973, and A013974 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Eisenstein Unit The Eisenstein units are the EISENSTEIN INTEGERS 91, 9v;9v2 ; where v¼1 2ð/C281þiffiffiffi 3p Þ v2/C301 2(/C281/C28iffiffiffi 3p ): See also EISENSTEIN INTEGER ,EISENSTEIN PRIME References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220 /C1/23, 1996. Eisenstein-Jacobi Integer EISENSTEIN INTEGER Elastica The elastica formed by bent rods and considered in physics can be generalized to curves in a RIEMANNIAN MANIFOLD which are a CRITICAL POINT for F l( g) /C30gg(k2 /C27 l); where k is the GEODESIC CURVATURE of g ; l is a REAL NUMBER , and g is closed or satisfies some specified boundary condition. The curvature of an elastica must satisfy 0 /C302 k ƒ(s) /C27 k3(s) /C272k(s)G(s) /C28 lk(s) ; where k is the signed curvature of g; G(s) is the GAUSSIAN CURVATURE of the oriented Riemannian surface M along g ; k ƒ is the second derivative of k with respect to s, and l is a constant. References Barros, M. and Garay, O. J. "Free Elastic Parallels in a Surface of Revolution." Amer. Math. Monthly 103, 149 /C1/ 56, 1996. Bryant, R. and Griffiths, P. "Reduction for Constrained Variational Problems and f(k2 =s) ds:/" Amer. J. Math. 108, 525 /C1/70, 1986. Langer, J. and Singer, D. A. "Knotted Elastic Curves in R3 :/" J. London Math. Soc. 30, 512 /C1/20, 1984. Langer, J. and Singer, D. A. "The Total Squared of Closed Curves." J. Diff. Geom. 20,1/C1/2, 1984. Elation A perspective COLLINEATION in which the center and axis are incident. See also HOMOLOGY (GEOMETRY ) References Coxeter, H. S. M. "Collineations and Correlations." §14.6 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 247 /C1/52, 1969. Elder’s Theorem A generalization of STANLEY’S THEOREM . It states that the total number of occurrences of an INTEGER k among all unordered PARTITIONS of n is equal to the number of occasions that a part occurs k or more times in a PARTITION , where a PARTITION which contains r parts that each occur k or more times contributes r to the sum in question. See also STANLEY’S THEOREM References Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer, pp. 8 /C1/, 1985. Election EARLY ELECTION RESULTS ,VOTINGElectric Motor Curve DEVIL’S CURVE Elegant Graph See also GRACEFUL GRAPH ,HARMONIOUS GRAPH Element If x is a member of a set A, then x is said to be an element of A, written x /C23 A: If x is not an element of A, this is written x QA: The term element also refers to a particular member of a GROUP , or entry aijin a MATRIX A or unevaluated DETERMINANT det(A) :/ See also SET THEORY Elementary Cellular Automaton The simplest class of 1-D cellular automata. They have two possible values for each cell, and rules that depend only on nearest neighbor values. They can be indexed with an 8-bit binary number, as shown by Stephen Wolfram (1983). Wolfram further restricted the number from /28 ¼ 256 / to 32 by requiring certain symmetry conditions. The illustrations above show automata numbers 30 and 90 propagated for 256 generations. Rule 30 is chaotic, with central column given by 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, ... (Sloane’s A051023). See also CELLULAR AUTOMATON References Sloane, N. J. A. Sequences A051023 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wolfram Research, Inc. "Cellular Automata." http://librar- y.wolfram.com/demos/v4/CellularAutomata.nb. Wolfram, S. "Statistical Mechanics of Cellular Automata." Rev. Mod. Phys. 55, 601 /C1/44, 1983. Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2001. Elementary Function A function built up of a finite combination of constant functions, field operations ( ADDITION , MULTIPLICA- TION , DIVISION , and ROOT EXTRACTIONS –the ELEMEN- TARY OPERATIONS )–and algebraic, exponential, and logarithmic functions and their inverses under re-peated compositions (Shanks 1993, p. 145; Chow 1999). Among the simplest elementary functions are the LOGARITHM , EXPONENTIAL FUNCTION (including the HYPERBOLIC FUNCTIONS ), POWER function, and TRIGONOMETRIC FUNCTIONS . Following Liouville (1837, 1838, 1839), Watson (1966,p. 111) defines the elementary TRANSCENDENTAL FUNCTIONS as l1(z) /C13l(z) /C13ln(z) e1(z) /C13e(z) /C13ez z1f(z) /C13 zf(z) /C13g f(z) dz ; and lets l2 /C13l(l(z)); etc. Not all functions are elementary. For example, the NORMAL DISTRIBUTION FUNCTION F(x) /C131ffiffiffiffiffiffi 2ppgx 0e/C28t2 =2 dt is a notorious example of a nonelementary function. The ELLIPTIC INTEGRAL gffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x4p dx is another. See also ALGEBRAIC FUNCTION ,ELEMENTARY OPERA- TION ,L IOUVILLE’S PRINCIPLE ,R ISCH ALGORITHM , SPECIAL FUNCTION ,SYMMETRIC POLYNOMIAL ,TRANS- CENDENTAL FUNCTION References Bronstein, M. Symbolic Integration I: Transcendental Func- tions. New York: Springer-Verlag, 1997. Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440 /C1/48, 1999. Geddes, K. O.; Czapor, S. R.; and Labahn, G. "Elementary Functions." §12.2 in Algorithms for Computer Algebra. Amsterdam, Netherlands: Kluwer, pp. 512 /C1/19, 1992. Hardy, G. H. Orders of Infinity, the ‘infinitarcalcul’ of Paul Du Bois-Reymond, 2nd ed. Cambridge, England: Cam- bridge University Press, 1924. Knopp, K. "The Elementary Functions." §23 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 96 /C1/8, 1996. Liouville. J. Math. 2,56/C1/05, 1837. Liouville. J. Math. 3, 523 /C1/47, 1838. Liouville. J. Math. 4, 423 /C1/56, 1839.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, p. 111, 1966. Elementary Matrix The elementary MATRICES are the PERMUTATION MATRIX pij and the SHEAR MATRIX e ƒij :/ See also ELEMENTARY ROW AND COLUMN OPERATIONS References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 41, 1962. Elementary Matrix Operations ELEMENTARY ROW AND COLUMN OPERATIONS Elementary Number A number which can be specified implicitly or explicitly by exponential, logarithmic, and algebraic operations. See also LIOUVILLIAN NUMBER References Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440 /C1/48, 1999. Ritt, J. Integration in Finite Terms: Liouville’s Theory of Elementary Models. New York: Columbia University Press, 1948. Elementary Operation One of the operations of ADDITION , SUBTRACTION , MULTIPLICATION , DIVISION , and integer (or rational) ROOT EXTRACTION . See also ABEL’S IMPOSSIBILITY THEOREM ,ALGEBRAIC FUNCTION ,ELEMENTARY FUNCTION Elementary Proof APROOF which can be accomplished using only REAL NUMBERS (i.e., REAL ANALYSIS instead of COMPLEX ANALYSIS ; Hoffman 1998, pp. 92 /C1/3). See also PROOF References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 22, 1986. Elementary Row and Column Operations The MATRIX operations of 1. Interchanging two rows or columns, 2. Adding a multiple of one row or column to another, 3. Multiplying any row or column by a nonzero element. See also GAUSSIAN ELIMINATION ,MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 39, 1962. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 390, 1998. Elementary Symmetric Function The elementary symmetric functions 1 /C28 24 a/C12 k¼1ð2k/C281 Þq21 /C281 1 þq2k /C281 on p(n) variables2K(k) p;j1ffl;j1{2 (1 /C282k2) are defined by 1 /C2824q /C2724q2 /C2896q3 /C27.../C30 s(0) 1(n) /C13X djnd oddd (1) M(q) /C30N(q) (2) E4(q) /C30E6(q) (3) G2( t) /C30Os/C272 ; 2(R) /C27 (4) 1 =p 91 /C309v (5) Alternatively, 9v2 can be defined as the coefficient of v in the GENERATING FUNCTION 1 2(/C281 /C27iffiffiffi 3p ) (6) For example, on four variables v2 ; ...,1 2(/C281 /C28iffiffiffi 3p ) ; the elementary symmetric functions are 1 /C2824q /C2724q2 /C2896q3 /C27.../C30F l( g) /C30gg( k2 /C27 l) ; (7) M(q) /C30 k (8) E4(q) /C300 /C302k ƒ(s) /C27 k3(s) /C272k(s)G(s) /C28 lk(s); (9) G2( t) /C30G(s) (10) Define k ƒ as the coefficients of the GENERATING FUNCTION sg(k2 =s) ds ð11Þ so the first few values are R3 /C30x /C23 A (12) x QA /C30aij (13) ð14Þ 28 /C30256 /C30l1(z) /C13l(z) /C13ln(z) e1(z) /C13e(z) /C13ez z1f(z) /C13 zf(z) /C13g f(z) dz;(15) In general, l2 /C13l(l(z)) can be computed from the DETERMINANT F(x) /C131ffiffiffiffiffiffi 2 ppgx 0e /C28t2 =2 dt (16) (Littlewood 1958, Cadogan 1971). Then the elemen- tary symmetric functions satisfy the relationship gffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x4p dx (17) In particular, pij/C301/C2824q/C2724q2/C2896q3/C27. . . (18) es ij/C30sa (19) sb/C30sc (20) Y/C30DABC (21) (Schroeppel 1972), as can be verified by plugging in and multiplying through. See also FUNDAMENTAL THEOREM OF SYMMETRIC FUNCTIONS ,NEWTON’S RELATIONS ,SYMMETRIC FUNC- TION References Cadogan, C. C. "The Mo ¨bius Function and Connected Graphs." J. Combin. Th. B 11, 193/C1/00, 1971. Littlewood, J. E. A University Algebra, 2nd ed. London: Heinemann, 1958. Schroeppel, R. Item 6 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 4, Feb. 1972. Elementary Transcendental Function ELEMENTARY FUNCTION #1999/C1/001 Wolfram Research, Inc. Elements The classic treatise in geometry written by Euclid and used as a textbook for more than 1,000 years in western Europe. An Arabic version The Elements appears at the end of the eighth century, and the first printed version was produced in 1482 (Tietze 1965, p. 8). The Elements , which went through more than 2,000 editions and consisted of 465 propositions, are divided into 13 "books" (an archaic word for "chap- ters"rpar;. Book Contents 1 TRIANGLES 2 RECTANGLES 3 CIRCLES 4 POLYGONS 5 proportion 6 SIMILARITY 7 /C1/0 NUMBER THEORY 11 solid geometry 12 PYRAMIDS 13 PLATONIC SOLIDS The elements started with 23 definitions, five POSTU- LATES , and five "common notions," and systematically built the rest of plane and solid geometry upon this foundation. The five EUCLID’S POSTULATES are 1. It is possible to draw a straight LINE from any POINT to another POINT . 2. It is possible to produce a finite straight LINE continuously in a straight LINE. 3. It is possible to describe a CIRCLE with any CENTER and RADIUS . 4. All RIGHT ANGLES are equal to one another. 5. If a straight LINE falling on two straight LINES makes the interior ANGLES on the same side less than two RIGHT ANGLES , the straight LINES (if extended indefinitely) meet on the side on which the ANGLES which are less than two RIGHT ANGLES lie. (Dunham 1990). Euclid’s fifth postulate is known as the PARALLEL POSTULATE . After more than two millennia of study, this POSTULATE was found to be independent of the others. In fact, equally valid NON- EUCLIDEAN GEOMETRIES were found to be possible by changing the assumption of this POSTULATE . Unfortu- nately, Euclid’s postulates were not rigorously com- plete and left a large number of gaps. Hilbert needed a total of 20 postulates to construct a logically complete geometry. See also PARALLEL POSTULATEReferences Casey, J. A Sequel to the First Six Books of the Elements of Euclid, 6th ed. Dublin: Hodges, Figgis, & Co., 1892. Dixon, R. Mathographics. New York: Dover, pp. 26 /C1/7, 1991. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 30 /C1/3, 1990. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 1: Books I and II. New York: Dover, 1956. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 2: Books III-IX. New York: Dover, 1956. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 3: Books X-XIII. New York: Dover, 1956. Joyce, D. E. "Euclid’s Elements." http://aleph0.clarku.edu/ ~djoyce/java/elements/elements.html Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, pp. 8 /C1/, 1965. Elevator Paradox A fact noticed by physicist G. Gamow when he had an office on the second floor and physicist M. Stern had an office on the sixth floor of a seven-story building (Gamow and Stern 1958, Gardner 1986). Gamow noticed that about 5/6 of the time, the first elevator to stop on his floor was going down, whereas about the same fraction of time, the first elevator to stop on the sixth floor was going up. This actually makes perfect sense, since 5 of the 6 floors 1, 3, 4, 5, 6, 7 are above the second, and 5 of the 6 floors 1, 2, 3, 4, 5, 7 are below the sixth. However, the situation takes some unexpected turns if more than one elevator is involved, as discussed by Gardner (1986). References Gamow, G. and Stern, M. Puzzle Math. New York: Viking, 1958. Gardner, M. "Elevators." Ch. 10 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 123 /C1/32, 1986. Elevatum A positive-height (outward-pointing) PYRAMID used in CUMULATION . The term was introduced by B. Gru¨n- baum. See also CUMULATION ,INVAGINATUM # 1999 /C1/001 Wolfram Research, Inc. Elkies Point Given POSITIVE numbers sa ; sb ; and sc ; the Elkies point is the unique point Y in the interior of a TRIANGLE DABC such that the respective INRADII ra ; rb ; rcof the TRIANGLES DBYC ;DCYA ; and DAYB satisfy ra : rb : rc /C30sa : sb : sc :/ See also CONGRUENT INCIRCLES POINT ,INRADIUS References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Kimberling, C. and Elkies, N. "Problem 1238 and Solution." Math. Mag. 60, 116/C1/17, 1987. Ellipse A curve which is the LOCUS of all points in the PLANE the SUM of whose distances r1and r2from two fixed points F1andF2(the FOCI) separated by a distance of 2cis a given POSITIVE constant 2 a(Hilbert and Cohn- Vossen 1999, p. 2). This results in the two-center BIPOLAR COORDINATE equation r1/C27r2/C302a; (1) where ais the SEMIMAJOR AXIS and the ORIGIN of the coordinate system is at one of the FOCI. The ellipse was first studied by Menaechmus, inves- tigated by Euclid, and named by Apollonius. The FOCUS and DIRECTRIX of an ellipse were considered by Pappus. In 1602, Kepler believed that the orbit ofMars was OVAL ; he later discovered that it was an ellipse with the Sun at one FOCUS . In fact, Kepler introduced the word " FOCUS " and published his discovery in 1609. In 1705 Halley showed that the comet which is now named after him moved in anelliptical orbit around the Sun (MacTutor Archive).An ellipse rotated about its minor axis gives an OBLATE SPHEROID , while an ellipse rotated about its major axis gives a PROLATE SPHEROID . A ray of light passing through a FOCUS will pass through the other focus after a single bounce (Hilbert and Cohn-Vossen 1999, p. 3). Reflections not passing through a FOCUS will be tangent to a confocal HYPERBOLA orELLIPSE , depending on whether the ray passes between the FOCI or not. Let an ellipse lie along the X-AXIS and find the equation of the figure (1) where F1and F2are at ( /C28c;0) and ( c;0):In CARTESIAN COORDINATES , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C27c)2/C27y2q /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x/C28c) 2/C27y2q /C302a: (2) Bring the second term to the right side and square both sides, (x/C27c)2/C27y2 /C304a2/C284affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C28c)2/C27y2q /C27(x/C28c)2/C27y2: (3) Now solve for the SQUARE ROOT term and simplify ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C28c)2/C27y2q /C30/C281 4a(x2/C272xc/C27c2/C27y2/C284a2/C28x2/C272xc/C28c2/C28y2)/C30/C281 4a(4xc/C284a2)/C30a/C28c ax: (4) Square one final time to clear the remaining SQUARE ROOT , x2/C282xc/C27c2/C27y2/C30a2/C282cx/C27c2 a2x2: (5) Grouping the xterms then gives x2a2/C28c2 a2/C27y2/C30a2/C28c2; (6) which can be written in the simple form x2 a2/C27y2 a2/C28c2/C301: (7) Defining a new constant b2/C13a2/C28c2(8) puts the equation in the particularly simple form x2 a2/C27y2 b2/C301: (9) The parameter bis called the SEMIMINOR AXIS by analogy with the parameter a, which is called the SEMIMAJOR AXIS . The fact that bas defined above is actually the SEMIMINOR AXIS is easily shown by letting r1andr2be equal. Then two RIGHT TRIANGLES are produced, each with HYPOTENUSE a, base c, and height b/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28c2p :Since the largest distance along the MINOR AXIS will be achieved at this point, bis indeed the SEMIMINOR AXIS . If, instead of being centered at (0, 0), the CENTER of the ellipse is at /(x0;y0);equation (9) becomes (x/C28x0)2 a2/C27(y/C28y0)2 b2/C301: (10) The ellipse can also be defined as the LOCUS of points whose distance from the FOCUS is proportional to the horizontal distance from a vertical line known as the DIRECTRIX , where the ratio is B1:Letting rbe the ratio and dthe distance from the center at which the directrix lies, then in order for this to be true, it must hold at the extremes of the major and minor axes, so r/C30a/C28c d/C28a/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C27c2p d: (11) Solving gives d/C30a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28b2p /C30a2 c(12) r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2/C28b2p a/C30c a: (13) The FOCAL PARAMETER of the ellipse is p/C30b2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28b2p (14) /C30a2/C28c2 c(15) /C30a(1/C28e2) e: (16) Like HYPERBOLAS , noncircular ellipses have two distinct FOCI and two associated DIRECTRICES , each DIRECTRIX being PERPENDICULAR to the line joining the two foci (Eves 1965, p. 275). As can be seen from the C ARTESIAN EQUATION for the ellipse, the curve can also be given by a simple parametric form analogous to that of a CIRCLE , but with the xand ycoordinates having different scal- ings, x/C30acost (17) y/C30bsint: (18) The unit TANGENT VECTOR of the ellipse so parame- terized is xT(t)/C30/C28asintffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2cos2t/C27a2sin2tp (19) yT(t)/C30bcostffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2cos2t/C27a2sin2tp : (20) A sequence of NORMAL and TANGENT VECTORS are plotted below for the ellipse. InPOLAR COORDINATES , the ANGLE u?measured from the center of the ellipse is called the ECCENTRIC ANGLE . Writing r?for the distance of a point from the ellipse center, the equation in POLAR COORDI- NATES is just given by the usual x/C30r?cosu? (21) y/C30r?sinu?: (22) Here, the coordinates u?and r?are written with primes to distinguish them from the more common polar coordinates for an ellipse which are centered on afocus. Plugging the polar equations into the Cartesian equation (9) and solving for r?2gives r?2/C30b2a2 b2cos2u?/C27a2sin2u?: (23) Define a new constant 0 5eB1 called the ECCENTRI- CITY (where e/C300 is the case of a CIRCLE ) to replace b e/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2 a2s ; (24) from which it also follows from (8) that a2e2/C30a2/C28b2/C13c2(25) c/C30ae (26) b2/C30a2(1/C28e2): (27) Therefore (23) can be written as r?2/C30a2(1/C28e2) 1/C28e2cos2u?(28) r?/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2 1/C28e2cos2u?s : (29) Ife/C101;then r?/C30af1/C281 2e2sin2u?/C281 16e4 /C2[5/C273 cos(2 u?)] sin2u?/C27...g;(30) so Dr? a/C13a/C28r? a:1 2e2sin2u?: (31) Summarizing relationships among the parameters a, b,c, and echaracterizing an ellipse, b/C30affiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2/C28c2p (32) c/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28b2p /C30ae (33) e/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2 a2s /C30c a: (34) The ECCENTRICITY can therefore be interpreted as the position of the FOCUS as a fraction of the SEMIMAJOR AXIS. Ifranduare measured from a FOCUS Finstead of from the center C(as they commonly are in orbital mechanics ) then the equations of the ellipse are x/C30c/C27rcosu (35) y/C30rsinu; (36) and (9) becomes (c/C27rcosu)2 a2/C27r2sin2u b2/C301: (37) Clearing the DENOMINATORS gives b2(c2/C272crcosu/C27r2cos2u)/C27a2r2sin2u/C30a2b2(38) b2c2/C272rcb2cosu/C27b2r2cos2u/C27a2r2/C28a2r2cos2u /C30a2b2: (39) Plugging in (26) and (27) to re-express band cin terms of aande, a2(1/C28e2)a2e2/C272aea2(1/C28e2)rcosu/C27a2(1/C28e2)r2 /C2cos2u/C27a2r2/C28a2r2cos2u/C30a2[a2(1/C28e2)]:(40) Simplifying, /C28r2/C27[ercosu/C28a(1/C28e2)]2/C300 (41) r/C309[ercosu/C28a(1/C28e2)]: (42) The sign can be determined by requiring that rmust bePOSITIVE . When e/C300, (42) becomes r/C309(/C28a);but since ais always POSITIVE , we must take the NEGATIVE sign, so (42) becomesr/C30a(1/C28e2)/C28ercosu (43) r(1/C27ecosu)/C30a(1/C28e2) (44) r/C30a(1/C28e2) 1/C27ecosu: (45) The distance from a FOCUS to a point with horizontal coordinate x(where the origin is taken to lie at the center of the ellipse) is found from cosu/C30x/C28c r: (46) Plugging this into (45) yields r/C27e(x/C28c)/C30a(1/C28e2) (47) r/C30a(1/C28e2)/C28e(x/C28c): (48) InPEDAL COORDINATES with the PEDAL POINT at the FOCUS , the equation of the ellipse is b2 p2/C302a r/C281: (49) To find the RADIUS OF CURVATURE , return to the parametric coordinates centered at the center of the ellipse and compute the first and second derivatives, x?/C30/C28 asint (50) y?/C30bcost (51) xƒ/C30/C28acost (52) yƒ/C30/C28bsint: (53) Therefore, R/C30(x?2/C27y?2)3=2 x?yƒ/C28xƒy? /C30(a2sin2t/C27b2cos2t)3=2 /C28asint(/C28bsint)/C28(acost)(bcost) /C30(a2sin2t/C27b2cos2t)3=2 ab(sin2t/C27cos2t) /C30(a2sin2t/C27b2cos2t)3=2 ab: (54) Similarly, the unit TANGENT VECTOR is given by ˆT/C30/C28asint bcost;j2r;j211ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2sin2t/C27b2cos2tp : (55) The ARC LENGTH of the ellipse can be computed using s(t)/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2/C27y?2q dt/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2sin2t/C27b2cos2tp dt /C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2sin2t/C27b2(1/C28sin2t)q dt /C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2/C27(a2/C28b2) sin2tq dt /C30bgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2/C28a2 b2sin2ts /C30bgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sin2tp dt/C30bE(t;k); (56) where E(f;k) is an incomplete ELLIPTIC INTEGRAL OF THE SECOND KIND with MODULUS k/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C28a2 b2s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 e2/C281s : (57) Again, note that tis a parameter which does not have a direct interpretation in terms of an ANGLE . How- ever, the relationship between the polar angle from the ellipse center uand the parameter tfollows from u/C30tan/C281y x ! /C30tan/C281b atant ! : (58) This function is illustrated above with ushown as the solid curve and tas the dashed, with b=a/C300:6:Care must be taken to make sure that the correct branch of the INVERSE TANGENT function is used. As can be seen, uweaves back and forth around t, with cross- ings occurring at multiples of p=2:/ The CURVATURE and TANGENTIAL ANGLE of the ellipseare given by k(t)/C30ab (b2cos2t/C27a2sin2t)3=2(59) f(t)/C30tan/C281a btant ! : (60) The entire PERIMETER pof the ellipse is given by setting t/C302p(corresponding to u/C302p);which is equivalent to four times the length of one of theellipse’s QUADRANTS , p/C30bE2p;1/C28a2 b2 ! /C304bE1 2p;1/C28a2 b2 ! /C304bE1/C28a2 b2 ! ; (61) where E(k) is a complete ELLIPTIC INTEGRAL OF THE SECOND KIND with MODULUS k. The PERIMETER can be computed using the rapidly converging G AUSS- KUM- MER SERIES as p/C30p(a/C27b)X/C12 n/C30012 n;j1z;j1}2 hn(62) /C30p(a/C27b)2F1(/C281 2;/C2812;1 ;h2) (63) /C304E(h)/C272(h2/C281)K(h) p(64) /C30p(a/C27b)(1/C2714h/C271 64h2/C271 256h3/C27. . .) (65) (Sloane’s A056981 and A056982), where h/C13a/C28b a/C27b !2 ; (66) /2F1(a;b;c;z)i sa HYPERGEOMETRIC FUNCTION ,K(k) is a complete ELLIPTIC INTEGRAL of the first kind, and n k;jr;j1 is a BINOMIAL COEFFICIENT . Approximations to the PERIMETER include p:pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(a2/C27b2Þp (67) :p[3(a/C27b)/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C273b)(3a/C27b)p ] (68) :p(a/C27b)1/C273h 10/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C283hp ! ; (69) where the last two are due to Ramanujan (1913 /C1/4), and (69) has a relative error of /C23/C2152/C2817h5for small values of h. The error surfaces are illustrated above for these functions. The maximum and minimum distances from the FOCUS are called the APOAPSIS and PERIAPSIS , and are given by r/C27/C30rapoapsis /C30a(1 /C27e) (70) r/C28/C30rperiapsis /C30a(1 /C28e): (71) The AREA of an ellipse may be found by direct INTEGRATION A /C30ga /C28agbffiffiffiffiffiffiffiffiffiffi a2 /C28x2p =a /C28bffiffiffiffiffiffiffiffiffiffi a2 /C28x2p =ady dx /C30ga /C28a2b affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28x2p dx /C302b a1 2xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28x2p /C27a2 sin/C281x ajj ! "#()a x /C30/C28a /C30ab[sin /C281 1 /C28sin/C281(/C281)] /C30abp 2 /C28/C28p 2 !"# /C30 pab : ð72Þ The AREA can also be computed more simply by making the change of coordinates x?/C13(b=a)x and y?/C13 y from the elliptical region R to the new region R?: Then the equation becomes 1 a2a bx? !2 /C27y?2 b2 /C301; (73) or x?2 /C27y?2 /C30b2 ; so R? is a CIRCLE of RADIUS b. Since @x @x?/C30@x? @x !/C281 /C30b a !/C281 /C30a b ; (74) the JACOBIAN is @(x; y) @(x?; y?);j12;j12;j12;j12;j12;j12;j12;j12;j12;j12/C30@x @x?@y? @x? @x @y?@y @y?;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12/C30 a b0 01;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12/C30 a b : (75) The AREA is therefore ggRdx dy /C30ggR?@(x; y) @(x?; y?);j12;j12;j12;j12;j12;j12;j12;j12;j12;j12 dx ? dy? /C30 a b ggR?dx ? dy?/C30a b (pb2) /C30 pab ; (76) as before. The AREA of an arbitrary ellipse given by the QUADRATIC EQUATION ax2 /C27bxy /C27cy2 /C301 (77) isA /C302pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ac /C28 b2p : (78) The AREA of an ELLIPSE with semiaxes a and b with respect to a PEDAL POINT P is A /C301 2 p(a2 /C27b2 /C27 OPjj2) : (79) The ellipse INSCRIBED in a given TRIANGLE and tangent at its MIDPOINTS is called the MIDPOINT ELLIPSE . The LOCUS of the centers of the ellipses INSCRIBED in a TRIANGLE is the interior of the MEDIAL TRIANGLE . Newton gave the solution to inscribing an ellipse in a convex QUADRILATERAL (Do¨rrie 1965, p. 217). The centers of the ellipses INSCRIBED in a QUADRILATERAL all lie on the straight line segment joining the MIDPOINTS of the DIAGONALS (Chakerian 1979, pp. 136 /C1/39). The AREA of an ellipse with BARYCENTRIC COORDI- NATES ( a; b; g) INSCRIBED in a TRIANGLE of unit AREA is D/C30 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1 /C282a)(1 /C282b)(1 /C282g)p : (80) (Chakerian 1979, pp. 142 /C1/45). The LOCUS of the apex of a variable CONE containing an ellipse fixed in 3-space is a HYPERBOLA through the FOCI of the ellipse. In addition, the LOCUS of the apex of a CONE containing that HYPERBOLA is the original ellipse. Furthermore, the ECCENTRICITIES of the ellipse and HYPERBOLA are reciprocals. The LOCUS of centers of a P APPUS CHAIN ofCIRCLES is an ellipse. Surprisingly, the locus of the end of a garage door mounted on rollers along a vertical track but extend- ing beyond the track is a quadrant of an ellipse (Wells 1991, p. 66). (The ENVELOPE of the ladder’s positions is an ASTROID .) See also CIRCLE ,CONIC SECTION ,ECCENTRIC ANOM- ALY,E CCENTRICITY ,E LLIPTIC CONE,E LLIPSE TAN- GENT ,E LLIPTIC CURVE ,E LLIPTIC CYLINDER , HYPERBOLA ,M IDPOINT ELLIPSE ,PARABOLA ,PARABO- LOID ,Q UADRATIC CURVE ,R EFLECTION PROPERTY , SALMON’S THEOREM ,STEINER’S ELLIPSE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 126, 198 /C1/99, and 217, 1987. Casey, J. "The Ellipse." Ch. 6 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, withNumerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 201 /C1 /49, 1893. Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., 1979. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.Oxford, England: Oxford University Press, p. 75, 1996. Coxeter, H. S. M. "Conics" §8.4 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115 /C1 /19, 1969. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965. Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, 1965. Fukagawa, H. and Pedoe, D. "Ellipses," "Ellipses and One Circle," "Ellipses and Two Circles," "Ellipses and Three Circles," "Ellipses and Many Circles," "Ellipses and Tri-angles," "Ellipses and Quadrilaterals," "Ellipses, Circles,and Rectangles," and "Ellipses, Circles and Rhombuses."§5.1, 6.1 /C1 /.2 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage ResearchFoundation, pp. 50 /C1 /8, 135 /C1/60, 1989. Harris, J. W. and Stocker, H. "Ellipse." §3.8.7 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 93, 1998. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 2 /C1/, 1999. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 4, 1948. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 72 /C1/8, 1972. Lockwood, E. H. "The Ellipse." Ch. 2 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 13 /C1/4, 1967. MacTutor History of Mathematics Archive. "Ellipse." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Ellip- se.html. Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/72, 1913 /C1/914. Sloane, N. J. A. Sequences A056981 and A056982 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 63 /C1 /7, 1991. Yates, R. C. "Conics." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 36 /C1/6, 1952. Ellipse Caustic Curve For an ELLIPSE given by x/C30rcost (1) y/C30sint (2) with light source at ( x;0);the CAUSTIC is x/C30Nx Dx(3) y/C30Ny Dy; (4) where Nx/C302rx(3/C285r2)/C27(/C286r2/C276r4/C283x2/C279r2x2) cos t /C276rx(1/C28r2) cos(2 t) /C27(/C282r2/C272r4/C28x2/C28r2x2) cos(3 t) (5) Dx/C302r(1/C272r2/C274x2)/C273x(1/C285r2) cos t /C27(6r/C276r3) cos(2 t)/C27x(1/C28r2) cos(3 t) (6) Ny/C308r(/C281/C27r2/C28x2) sin3t (7)Dy/C302r(/C281/C28r2/C284x2)/C273(/C28x/C275r2) cos t /C276r(1/C28r2) cos(2 t)/C27x(/C281/C27r2) cos(3 t): (8) At (/C12;0); x/C30cost[/C281/C275r2/C28cos(2 t)(1/C27r2)] 4r(9) y/C30sin3t: (10) Ellipse Envelope Consider the family of ELLIPSES x2 c2/C27y2 (1/C28c)2/C281/C300 (1) for /c/C23½0;1/C138/. The PARTIAL DERIVATIVE with respect to c is /C282x2 c3/C272y2 (1/C28c)3/C300 (2) x2 c3/C28y2 (1/C28c)3/C300: (3) Combining (1) and (3) gives the set of equations 1 c21 (1/C28c)2 1 c3/C281 (1/C28c)32 66643 7775x 2 y2;j2r;j21 /C301 0;j2r;j21 (4) x2 y2;j2r;j21 /C301 D/C281 (1 /C28 c)3/C281 (1 /C28 c)2 /C281 c31 c22 66643 77751 0;j2r;j21 /C30 1 D/C281 (1 /C28 c)3 /C281 c32 66643 7775; (5) where the DISCRIMINANT is D/C30/C281 c2(1 /C28 c)3 /C281 c3(1 /C28 c)2 /C30/C281 c3(1 /C28 c)3 ; (6) so (5) becomes x2 y2;j2r;j21 /C30c3 (1 /C28c)3;j2r;j21 : (7) Eliminating c then gives x2 =3 /C27y2 =3 /C301; (8) which is the equation of the ASTROID . If the curve is instead represented parametrically, then x /C30c cos t (9) y /C30(1 /C28c) sin t: (10) Solving @x @t@y @c /C28@x @c@y @t /C30(/C28c sin t)(/C28sin t) /C28(cos t)[(1 /C28c) cos t] /C30c(sin2 t /C27cos2 t) /C28cos2 t /C30c /C28cos2 t /C300 (11) for c gives c /C30cos2 t; (12) so substituting this back into (9) and (10) gives x /C30(cos2 t) cos t /C30cos3 t (13) y /C30(1 /C28cos2 t) sin t /C30sin3 t; (14) the PARAMETRIC EQUATIONS of the ASTROID . See also ASTROID ,ELLIPSE ,ENVELOPEEllipse Evolute The EVOLUTE of an ELLIPSE is given by the PARA- METRIC EQUATIONS x /C30a2 /C28 b2 acos3 t (1) y /C30b2 /C28 a2 bsin3 t; (2) which can be combined and written (ax)2 =3 /C27(by)2=3 /C30[(a2 /C28b2) cos3 t]2 =3 /C27[(b2 /C28a2)] sin3 t]2=3 /C30(a2 /C28b2)2 =3(sin2 t /C27cos2 t) /C30(a2 /C28b2)2 =3 /C30c4 =3 ; (3) which is a stretched ASTROID sometimes called the LAME´ CURVE . From a point inside the EVOLUTE , four NORMALS can be drawn to the ellipse, but from a point outside, only two NORMALS can be drawn. See also ASTROID ,ELLIPSE ,EVOLUTE ,LAME´ CURVE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 217, 1987. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 99 /C1/01, 1997. Ellipse Involute From ELLIPSE , the TANGENT VECTOR is T/C30/C28asint bcost;j2r;j21 ; (1) and the ARC LENGTH is s /C30agffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28e2 sin2 tp dt /C30aE(t; e) ; (2) where E(t; e) is an incomplete ELLIPTIC INTEGRAL OF THE SECOND KIND . Therefore, ri /C30r /C28s ˆT /C30 a cos t b sin t;j2r;j21 /C28aeE(t; e) /C28a sin t b cos t;j2r;j21 (3) /C30afcos t /C27aeE(t; e) sin tg bfsin t /C28aeE(t; e) cos tg;j2r;j21 : (4) Ellipse Pedal Curve The pedal curve of an ellipse with semimajor axis a, semiminor axis b, and PEDAL POINT (x0 ; y0) is given by f /C30a[ax0 sin2 t /C27 b cos t(b /C28 y0 sin t)] b2 cos2 t /C27 a2 sin2 t g /C30b[a2 sin2 t /C28 ax0 cos t sin t /C27 by0 cos2 t] b2 cos2 t /C27 a2 sin2 t : The pedal curve of an ellipse with PEDAL POINT at the FOCUS is a CIRCLE (Hilbert and Cohn-Vossen, pp. 25 /C1/ 6). For other pedal points, the pedal curves are more complicated. See also ELLIPSE ,PEDAL CURVE References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999.Ellipse Point Picking To inscribe an EQUILATERAL TRIANGLE in an ELLIPSE , place the top VERTEX at (0; b) ; then solve to find the (x, y) coordinate of the other two VERTICES . ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27(b /C28y)2q /C302x (1) x2 /C27(b /C28y)2 /C304x2 (2) 3x2 /C30(b /C28y)2 : (3) Now plugging in the equation of the ELLIPSE x2 a2 /C27y2 b2 /C301; (4) gives 3a21 /C28y2 b2 ! /C30b2 /C282by /C27y2 (5) y21 /C273a2 b2 ! /C282by /C27(b2 /C283a2) /C300 (6) y /C302b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4b2 /C28 4(b2 /C28 3a2)1/C27 3a2 b2 !vuut 21/C27 3a2 b2 ! /C301 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281/C283a2 b2 ! 1/C273a2 b2 !vuut 1/C273a2 b2b; (7) and x/C309affiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28y2 b2s : (8) See also ELLIPSE ,EQUILATERAL TRIANGLE #1999/C1/001 Wolfram Research, Inc. Ellipse Tangent The normal to an ellipse at a point P intersects the ellipse at another point Q. The angle corresponding to Q can be found by solving the equation (P /C28Q) /C215dP dt/C300 (1) for t?; where P(t) /C30(a cos t; b sin t) and Q(t) /C30 (a cos t?; b sin t?) : This gives solutions t?/C309cos/C281 9N(t) a4 sin2 t /C27 b4 cos2 t"# ; (2) where N(t) /C13b2 cos t[a2 /C27b2(b2 /C28a)2 cos(2 t)] /C27a2(a /C28b)(a /C27b) cos t sin2 t; (3) of which (/C27;/C28) gives the valid solution. Plugging this in to obtain Q then gives d(t) /C30½P /C28Q ½ /C30ffiffiffi 2p ab[a2 /C27 b2 /C27 (b2 /C28 a2) cos(2 t)]3 =2 a4 /C27 b4 /C27 (b4 /C28 a4) cos(2 t)ð4Þ /C302ab(b2 cos2 t /C27 a2 sin2 t)3 =2 b4 cos2 t /C27 a4 sin2 t: (5) To find the maximum distance, take the derivative and set equal to zero, d?(t) /C302ab(a /C28 b)(a /C27 b) cos t sin tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 cos2 t /C27 a2 sin2 tp (b4 cos2 t /C27 a4 sin2 t)2 /C29(a4 sin2 t /C27b4 cos2 t /C282a2b2) ¼ 0; (6) which simplifies to a4 sin2 t /C27b4 cos2 t /C282a2b2 /C300: (7)Substituting for sin2 t and solving gives cos2 t /C30a4 /C28 2a2b2 a4 /C28 b4 (8) sin2 t /C302a2b2 /C28 b4 a4 /C28 b4: (9) Plugging these into d(t) then gives dmin /C303ffiffiffi 3p a2b2 (a2 /C27 b2)3 =2 : (10) This problem was given as a SANGAKU PROBLEM on a tablet from Miyagi Prefecture in 1912 (Rothman 1998). There is probably a clever solution to this problem which does not require calculus, but it is unknown if calculus was used in the solution by theoriginal authors (Rothman 1998). See also E LLIPSE References Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85/C1/1, May 1998. #1999/C1/001 Wolfram Research, Inc. Ellipsoid AQUADRATIC SURFACE which is given in C ARTESIAN COORDINATES by x2 a2/C27y2 b2/C27z2 c2/C301; (1) where the semi-axes are of lengths a,b, and c.I n SPHERICAL COORDINATES , this becomes r2cos2usin2f a2/C27r2sin2usin2f b2/C27r2cos2f c2/C301:(2) The PARAMETRIC EQUATIONS are x/C30acosusinf (3) y¼bsinusinf ð4Þ z/C30ccosf: (5) foru/C23[0;2p) and f/C23[0;p]:/ If the lengths of two axes of an ellipsoid are the same, the figure is called a SPHEROID (depending on whether c Ba or c /C21a,an OBLATE SPHEROID or PROLATE SPHEROID , respectively), and if all three are the same, it is a SPHERE . Tietze (1965, p. 28) calls the general ellipsoid a "triaxial ellipsoid." There are two families of parallel CIRCULAR CROSS SECTIONS in every ellipsoid. However, the two coin- cide for SPHEROIDS (Hilbert and Cohn-Vossen 1999, pp. 17 /C1/9). If the two sets of circles are fastened together by suitably chosen slits so that are free to rotate without sliding, the model is movable. Further- more, the disks can always be moved into the shape of a SPHERE (Hilbert and Cohn-Vossen 1999, p. 18). In 1882, Staude discovered a "thread" construction for an ellipsoid analogous to the taught pencil and string construction of the ELLIPSE (Hilbert and Cohn-Vossen 1999, pp. 19 /C1/2). This construction makes use of a fixed framework consisting of an ELLIPSE and a HYPERBOLA . The SURFACE AREA of an ellipsoid (Bowman 1961, pp. 31 /C1/2) is given by S /C302pc2 /C272 pbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28 c2p [(a2 /C28c2)E( u) /C27c2 u] ; (6) where /E ðuÞ/ is a COMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND , e2 1 /C13a2 /C28 c2 a2 (7) e22 /C13b2 /C28 c2 b2 (8) k /C13e2 e1; (9) and u is given by inverting the expression e1 /C30sn(u ; k) ; (10) where sn(u ; k)isaJ ACOBI ELLIPTIC FUNCTION . The VOLUME of an ellipsoid is V /C304 3 pabc : (11) A different parameterization of the ellipsoid is the so- called stereographic ellipsoid, given by the PARA- METRIC EQUATIONS x(u; v) /C30a(1 /C28 u2 /C28 v2) 1 /C27 u2 /C27 v2 (12) y(u; v) /C302bu 1 /C27 u2 /C27 v2 (13) z(u; v) /C302cv 1 /C27 u2 /C27 v2 : (14) A third parameterization is the Mercator parameter- ization x(u;v)/C30asech vcosu (15) y(u;v)/C30bsech vsinu (16) z(u;v)/C30ctanh v (17) (Gray 1997). The SUPPORT FUNCTION of the ellipsoid is h/C30x2 a4/C27y2 b4/C27z2 c4 !/C281=2 ; (18) and the G AUSSIAN CURVATURE is K/C30h4 a2b2c2(19) (Gray 1997, p. 296). See also CONFOCAL ELLIPSOIDAL COORDINATES ,CON- FOCAL QUADRICS ,C ONVEX OPTIMIZATION THEORY , ELLIPSOID PACKING ,G OURSAT’S SURFACE ,O BLATE SPHEROID ,PROLATE SPHEROID ,SPHERE ,SPHEROID , SUPERELLIPSOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 226, 1987. Bowman, F. Introduction to Elliptic Functions, with Appli- cations. New York: Dover, 1961. Fischer, G. (Ed.). Plate 65 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 60, 1986. Gray, A. "The Ellipsoid" and "The Stereographic Ellipsoid." §13.2 and 13.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 301 /C1/03, 1997. Harris, J. W. and Stocker, H. "Ellipsoid." §4.10.1 in Hand- book of Mathematics and Computational Science. New York: Springer-Verlag, p. 111, 1998. Hilbert, D. and Cohn-Vossen, S. "The Thread Construction of the Ellipsoid, and Confocal Quadrics." §4i n Geometry and the Imagination. New York: Chelsea, pp. 19 /C1/5, 1999. JavaView. "Classic Surfaces from Differential Geometry: Ellipsoid." http://www-sfb288.math.tu-berlin.de/vgp/java-view/demo/surface/common/PaSurface_Ellipsoid.html. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, pp. 28 and 40 /C1 /1, 1965. Ellipsoid Geodesic An ELLIPSOID can be specified parametrically by x /C30a cos u sin v (1) y /C30b sin u sin v (2) z /C30c cos v: (3) The GEODESIC parameters are then P /C30sin2 v(b2 cos2 u /C27a2 sin2 u) (4) Q /C301 4(b2 /C28a2) sin(2 u) sin(2 v) (5) R /C30cos2 v(a2 cos2 u /C27b2 sin2 u) /C27c2 sin2 v : (6) When the coordinates of a point are on the QUADRIC x2 a/C27y2 b/C27z2 c/C301 (7) and expressed in terms of the parameters p and q of the confocal quadrics passing through that point (in other words, having a /C27p ; b /C27p; c /C27p; and a /C27q ; b /C27 q; c /C27q for the squares of their semimajor axes), then the equation of a GEODESIC can be expressed in the form qdqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q(a /C27 q)(b /C27 q)(c /C27 q)(u /C27 q)p 9pdpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip(a /C27 p)(b /C27 p)(c /C27 p)(u /C27 p)p /C300 ; (8) with u an arbitrary constant, and the ARC LENGTH element ds is given by /C282ds pq /C30dqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq(a /C27 q)(b /C27 q)(c /C27 q)(u /C27 q)p 9 dpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip(a /C27 p)(b /C27 p)(c /C27 p)( u /C27 p)p ; (9) where upper and lower signs are taken together. See also O BLATE SPHEROID GEODESIC ,SPHERE GEO- DESIC References Eisenhart, L. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, pp. 236 /C1/41, 1960. Forsyth, A. R. Calculus of Variations. New York: Dover, p. 447, 1960. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, pp. 28 /C1/9 and 40 /C1/1, 1965. Ellipsoid of Revolution OBLATE SPHEROID ,PROLATE SPHEROID ,SPHEROIDEllipsoid Packing Bezdek and Kuperberg (1991) have constructed pack- ings of identical ellipsoids of densities , greater than the maximum density possible for identical spheres (Sloane 1998). See also SPHERE PACKING References Bezdek, A. and Kuperberg, W. In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift (Ed. P. Gritzmann and B. Sturmfels). Providence, RI: Amer. Math. Soc., pp. 71 /C1/0, 1991. Sloane, N. J. A. "Kepler’s Conjecture Confirmed." Nature 395, 435/C1/36, 1998. Ellipsoidal Calculus Ellipsoidal calculus is a method for solving problems in control and estimation theory having unknown but bounded errors in terms of sets of approximating ellipsoidal-value functions. Ellipsoidal calculus hasbeen especially useful in the study of LINEAR PRO- GRAMMING . References Kurzhanski, A. B. and Va ´lyi, I. Ellipsoidal Calculus for Estimation and Control. Boston, MA: Birkha ¨user, 1996. Papadimitriou, C. H. and Steiglitz, K. Combinatorial Opti- mization: Algorithms and Complexity. New York: Dover, 1998. Ellipsoidal Coordinates CONFOCAL ELLIPSOIDAL COORDINATES Ellipsoidal Harmonic ELLIPSOIDAL HARMONIC OF THE FIRST KIND,ELLIP- SOIDAL HARMONIC OF THE SECOND KIND Ellipsoidal Harmonic of the First Kind The first solution to L AME´’S DIFFERENTIAL EQUATION , denoted Em n(x) for m/C301, ..., 2 n/C271:They are also called L AME´FUNCTIONS . The product of two ellipsoi- dal harmonics of the first kind is a SPHERICAL HARMONIC . Whittaker and Watson (1990, pp. 536 /C1/ 37) write Up/C30x2 a2/C27up/C27y2 b2/C27up/C27z2 c2/C27up/C281 (1) P(U)/C13U1U2/C1/C1/C1Um; (2) and give various types of ellipsoidal harmonics and their highest degree terms as 1. P(U):2m/ 2. xP(U) ; yP( U); z P(U):2m /C271/ 3. yz P( U) ; zx P( U); xyP( U):2m /C272/ 4. xyz P( U):2m /C273 :/ A Lame ´ function of degree n may be expressed as ( u /C27a2) k1 ( u /C27b2) k2 (u /C27c2) k3Ym p/C301( u /C28 up) ; (3) where ki /C300 or 1/2, uiare REAL and unequal to each other and to /C28a2 ;/C28b2 ; and /C28c2 ; and 1 2 n /C30m /C27 k1 /C27 k2 /C27 k3 : (4) Byerly (1959) uses the RECURRENCE RELATIONS to explicitly compute some ellipsoidal harmonics, which he denotes by K(x) ; L(x) ; M(x); and N(x); K0(x) /C301 L0(x) /C300 M0(x) /C300 N0(x) /C300 K1(x) /C30x L1(x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28b2p M1(x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C28c2p N1(x) /C300 Kp1 2(x) /C30x2 /C281 3[b2 /C27c2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b2 /C27c2)2 /C283b2c2q ] Kp2 2(x) /C30x2 /C281 3[b2 /C27c2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b2 /C27c2)2 /C283b2c2q ] L2(x) /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28b2p M2(x) /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C28c2p N2(x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x2 /C28b2)(x2 /C28c2)p Kp1 3(x) /C30x3 /C281 5 x[2(b2 /C27c2) /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4(b2 /C27c2)2 /C2815b2c2q ] Kp2 3(x) /C30x3 /C281 5 x[2(b2 /C27c2) /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4(b2 /C27c2)2 /C2815b2c2q ] Lq1 3 (x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28b2p [x2 /C281 5(b2 /C272c2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b2 /C272c2)2 /C285b2c2q )]Lq2 3 (x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28b2p [x2 /C281 5(b2 /C272c2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b2 /C272c2)2 /C285b2c2q )] Mq1 3(x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28c2p [x2 /C281 5(2b2 /C27c2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2b2 /C27c2)2 /C285b2c2q )] Mq2 3(x) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28c2p [x2 /C281 5(2b2 /C27c2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2b2 /C27c2)2 /C285b2c2q )] Mq3 3(x) /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x2 /C28b2)(x2 /C28c2)p See also ELLIPSOIDAL HARMONIC OF THE SECOND KIND,STIELTJES’ THEOREM References Byerly, W. E. "Laplace’s Equation in Curvilinear Coo¨rdi- nates. Ellipsoidal Harmonics." Ch. 8 in An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 238 /C1/66, 1959. Humbert, P. Fonctions de Lame ´ et Fonctions de Mathieu. Paris: Gauthier-Villars, 1926. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Ellipsoidal Harmonic of the Second Kind Given by Fp m(x) /C30(2m /C271)Ep m(x)g/C12 xdx (x2 /C28 b2)(x2 /C28 c2)[Ep m(x)]2 : Ellipsoidal Wave Equation The ORDINARY DIFFERENTIAL EQUATION yƒ/C28(a/C27bk2sn2x/C27qk4sn4x)y/C300; where sn x/C30sn(x;k)i saJ ACOBI ELLIPTIC FUNCTION (Arscott 1981). See also LAME´ ’S DIFFERENTIAL EQUATION References Arscott, F. M. "The Land beyond Bessel: A Survey of Higher Special Functions." In Ordinary and Partial Differential Equations: Proceeding of the Sixth Conference held at the University of Dundee, March 31-April 4, 1980 (Ed. W. N. Everitt and B. D. Sleeman). New York: Springer-Verlag, pp. 26 /C1 /5, 1981. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Elliptic Alpha Function Elliptic alpha functions relate the complete ELLIPTIC INTEGRALS OF THE FIRST K(kr) and SECOND KINDS E(kr)a t ELLIPTIC INTEGRAL SINGULAR VALUES kr according to a(r) /C30e ?(kr) k(kr) /C28p 4[k(kr)]2 (1) /C30p 4[k(kr)]2 /C27ffiffiffirp/C28e(kr)ffiffiffirp k(kr) (2) /C30p/C281 /C28 4ffiffiffirpqdq4(q) dq1 q4(q) q4 3(q) ; (3) where q3(q)isaJ ACOBI THETA FUNCTION and kr /C30 l /C31(r) (4) q ¼ e /C28 p ffiffirp;ð5Þ and l /C31(r) is the ELLIPTIC LAMBDA FUNCTION . The elliptic alpha function is related to the ELLIPTIC DELTA FUNCTION by a(r) /C301 2[ffiffiffirp/C28 d(r)]: (6) It satisfies a(4r) /C30 (1 /C27 kr)2 a(r) /C282ffiffiffirpkr ; (7) and has the limit lim r0/C12a(r) /C281 p"# :8ffiffiffirp/C281 p ! e /C28 p ffiffirp (8) (Borwein et al. 1989). A few specific values (Borwein and Borwein 1987, p. 172) are að1Þ¼1 2 að2Þ¼ffiffiffi 2p /C281 að3Þ¼1 2 ðffiffiffi 3p /C281 Þ að4Þ¼2ðffiffiffi 2p /C281Þ2 að5Þ¼1 2 ðffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffi 5p /C282q Þ að6 Þ¼5ffiffiffi6p þ 6ffiffiffi3p /C288ffiffiffi 2p /C2811 að7Þ¼ 1 2 ðffiffiffi 7p /C282 Þ að8 Þ¼2 ð10 þ 7ffiffiffi2p Þð1 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8p /C282q Þ 2 að9Þ¼1 2 ½3 /C2833 =4ffiffiffi 2p ðffiffiffi3p /C281Þ/C138 að10 Þ¼/C28103 þ 72ffiffiffi2p /C2846ffiffiffi5p þ 33ffiffiffiffiffiffi10pað12 Þ¼264 þ 154ffiffiffi3p /C28188ffiffiffi 2p /C28108ffiffiffi 6p að13 Þ¼ 1 2 ðffiffiffiffiffiffi 13p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 74ffiffiffiffiffiffi 13p /C28258q Þ að15 Þ¼1 2ðffiffiffiffiffiffi 15p /C28ffiffiffi5p /C281Þ að16 Þ¼4ðffiffiffi 8p /C28 1 ð21 =4 þ 1Þ4 að18 Þ¼/C283057 þ 2163ffiffiffi 2p þ 1764ffiffiffi 3p /C281248ffiffiffi6p að22Þ¼/C2812479 /C288824ffiffiffi2p þ 3762ffiffiffiffiffiffi11p þ 2661ffiffiffiffiffiffi22p að25Þ¼ 5 2 ½1 /C28251=4 ð7 /C283ffiffiffi 5p Þ/C138 að27Þ¼3½1 2 ðffiffiffi 3p þ 1 Þ/C2821 =3 /C138 að30 Þ¼1 2ffiffiffiffiffiffi 30p /C28ð2 þffiffiffi5p Þ2 ð3 þffiffiffiffiffiffi10p Þ2 /C29ð/C286 /C285ffiffiffi 2p /C283ffiffiffi 5p /C282ffiffiffiffiffiffi10p þffiffiffi6pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 57 þ 40ffiffiffi 2pq /C29½56 þ 38ffiffiffi2p þffiffiffiffiffiffi 30p ð2 þffiffiffi5p Þð3 þffiffiffiffiffiffi10p Þ/C138g að37 Þ¼ 1 2ffiffiffiffiffiffi 37p /C28ð171 /C2825ffiffiffiffiffiffi37p Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi37p /C286q ;j2r;j21 að46 Þ¼ 1 2 ½ffiffiffiffiffiffi 46p þð18 þ 13ffiffiffi 2p þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 661 þ 468ffiffiffi 2pq Þ2 /C29ð18 þ 13ffiffiffi2p /C283ffiffiffi2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 147 þ 104ffiffiffi 2pq þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 661 þ 468ffiffiffi 2pq Þ /C29ð200 þ 14ffiffiffi 2p þ 26ffiffiffiffiffiffi 23p þ 18ffiffiffiffiffiffi46p þffiffiffiffiffiffi46pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 661 þ 468ffiffiffi 2pq Þ/C138 að49Þ/C307 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ½ffiffiffi 2p 73=4 ð33011 þ 12477ffiffiffi 7p Þ/C2821 ð9567 þ 3616ffiffiffi7p Þ/C138q að58Þ¼½1 2ðffiffiffiffiffiffi 29p þ 5 Þ/C1386 ð99ffiffiffiffiffiffi29p /C28444 Þð99ffiffiffi 2p /C2870 /C2813ffiffiffiffiffiffi 29p Þ ¼ 3 ð/C2840768961 þ 2882008ffiffiffi 2p /C287570606ffiffiffiffiffiffi 29p þ 5353227 /C2ffiffiffiffiffiffi 58p Þ a(64)/C308[2(ffiffiffi 8p /C281)/C28(21=4/C281)4] (ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p /C271p /C2725=8)4: J. Borwein has written an ALGORITHM which uses lattice basis reduction to provide algebraic values for a(n):/ See also ELLIPTIC INTEGRAL OF THE FIRST KIND, ELLIPTIC INTEGRAL OF THE SECOND KIND,ELLIPTIC INTEGRAL SINGULAR VALUE ,ELLIPTIC LAMBDA FUNC- TION References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanu- jan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201 /C1/19, 1989. Weisstein, E. W. "Elliptic Singular Values." MATHEMATICA NOTEBOOK ELLIPTIC SINGULAR.M . Elliptic Cone A CONE with ELLIPTICAL CROSS SECTION . The PARA- METRIC EQUATIONS for an elliptic cone of height h, SEMIMAJOR AXIS a, and SEMIMINOR AXIS b are x /C30(h /C28z)a cos u y /C30(h /C28z)b sin u z /C30z; where u/C23[0;2p) and z/C23[0;h]:The elliptic cone is a QUADRATIC RULED SURFACE , and has VOLUME V/C301 3pab: See also CONE,ELLIPTIC CYLINDER ,ELLIPTIC PARA- BOLOID ,H YPERBOLIC PARABOLOID ,QUADRATIC SUR- FACE ,RULED SURFACE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 226, 1987. Fischer, G. (Ed.). Plate 68 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 63, 1986. Elliptic Cone Point ISOLATED SINGULARITY Elliptic Coordinates CONFOCAL ELLIPSOIDAL COORDINATES #1999/C1/001 Wolfram Research, Inc. Elliptic Curve Informally, an elliptic curve is a type of CUBIC CURVE whose solutions are confined to a region of space which is topologically equivalent to a TORUS . TheWEIERSTRASS ELLIPTIC FUNCTION /C212(z;g2;g3) de- scribes how to get from this TORUS to the algebraic form of an elliptic curve. Formally, an elliptic curve over a FIELD Kis a nonsingular CUBIC CURVE in two variables, f(X;Y)/C30 0;with a K-rational point (which may be a POINT AT INFINITY ). The FIELD Kis usually taken to be the COMPLEX NUMBERS C;REALS R;RATIONALS Q;alge- braic extensions of Q;P-ADIC NUMBERS Qp;or a FINITE FIELD . By an appropriate change of variables, a general elliptic curve over a FIELD ofCHARACTERISTIC "2;3 Ax3/C27Bx2y/C27Cxy2/C27Dy3/C27Ex2/C27Fxy/C27Gy2/C27Hx /C27Iy/C27J/C300; (1) where A,B, ..., are elements of K, can be written in the form y2/C30x3/C27ax/C27b; (2) where the right side of (2) has no repeated factors. If Khas CHARACTERISTIC three, then the best that can be done is to transform the curve into y2/C30x3/C27ax2/C27bx/C27c (3) (the x2term cannot be eliminated). If Khas CHAR- ACTERISTIC two, then the situation is even worse. A general form into which an elliptic curve over any K can be transformed is called the W EIERSTRASS FORM , and is given by y2/C27ay/C30x3/C27bx2/C27cxy/C27dx/C27e; (4) where a,b,c,d, and eare elements of K. Luckily, Q; R;andCall have CHARACTERISTIC zero. Whereas CONIC SECTIONS can be parameterized by the rational functions, elliptic curves cannot. The simplest parameterization functions are ELLIPTIC FUNCTIONS .A BELIAN VARIETIES can be viewed as generalizations of elliptic curves. If the underlying FIELD of an elliptic curve is algebraically closed, then a straight line cuts anelliptic curve at three points (counting multiple roots at points of tangency). If two are known, it is possible to compute the third. If two of the intersection points are K-RATIONAL , then so is the third. Mazur and Tate (1973/74) proved that there is no elliptic curve over Q having a RATIONAL POINT of order 13. Let (x1 ; y1) and (x2 ; y2) be two points on an elliptic curve E with DISCRIMINANT DE /C30/C2816(4a3 /C2727b2) (5) satisfying DE "0: (6) A related quantity known as the J-INVARIANT of E is defined as j(E) /C132833a3 4a3 /C27 27b2 : (7) Now define l /C30y1 /C28 y2 x1 /C28 x2for x1 "x2 3x2 1 /C27 a 2y1for x1 /C30x2 :8 >>>< >>>:(8) Then the coordinates of the third point are x 3 /C30 l2 /C28x1 /C28x2 (9) y3 /C30 l(x3 /C28x1) /C27y1 : (10) For elliptic curves over Q; Mordell proved that there are a finite number of integral solutions. The MOR- DELL- WEIL THEOREM says that the GROUP of RATIONAL POINTS of an elliptic curve over Q is finitely gener- ated. Let the ROOTS of y2be r1 ; r2 ; and r3 : The discriminant is then D/C30k(r1 /C28r2)2(r1 /C28r3)2(r2 /C28r3)2 : (11) The amazing TANIYAMA- SHIMURA CONJECTURE states that all rational elliptic curves are also modular. This fact is far from obvious, and despite the fact that the conjecture was proposed in 1955, it was not even partially proved until 1995. Even so, Wiles’ proof for the semistable case surprised most mathematicians, who had believed the conjecture unassailable. As a side benefit, Wiles’ proof of the TANIYAMA- SHIMURA CONJECTURE also laid to rest the famous and thorny problem which had baffled mathematicians for hun- dreds of years, FERMAT’S LAST THEOREM . Curves with small CONDUCTORS are listed in Swin- nerton-Dyer (1975) and Cremona (1997). Methods for computing integral points (points with integral co- ordinates) are given in Gebel et al. and Stroeker and Tzanakis (1994). The SCHOOF- ELKIES-ATKIN ALGO- RITHM can be used to determine the order of an elliptic curve E=Fpover the FINITE FIELD Fp:/ See also CUBIC CURVE ,ELLIPTIC CURVE GROUP LAW, FERMAT’S LAST THEOREM ,FREY CURVE , J-INVARIANT ,MINIMAL DISCRIMINANT ,M ORDELL- WEIL THEOREM , OCHOA CURVE ,R IBET’S THEOREM ,SCHOOF- ELKIES- ATKIN ALGORITHM ,SIEGEL’S THEOREM ,SWINNERTON- DYER CONJECTURE ,T ANIYAMA- SHIMURA CONJEC- TURE ,W EIERSTRASS ELLIPTIC FUNCTION ,W EIER- STRASS FORM References Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Math. Comput. 61,2 9/C1/8, 1993. Cassels, J. W. S. Lectures on Elliptic Curves. New York: Cambridge University Press, 1991. Cremona, J. E. Algorithms for Modular Elliptic Curves, 2nd ed.Cambridge, England: Cambridge University Press, 1997. Du Val, P. Elliptic Functions and Elliptic Curves. Cam- bridge, England: Cambridge University Press, 1973. Gebel, J.; Petho, A.; and Zimmer, H. G. "Computing Integral Points on Elliptic Curves." Acta Arith. 68, 171/C1/92, 1994. Ireland, K. and Rosen, M. "Elliptic Curves." Ch. 18 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 297 /C1/18, 1990. Joye, M. "Some Interesting References on Elliptic Curves." http://www.dice.ucl.ac.be/crypto/joye/biblio_ell.html. Katz, N. M. and Mazur, B. Arithmetic Moduli of Elliptic Curves. Princeton, NJ: Princeton University Press, 1985. Knapp, A. W. Elliptic Curves. Princeton, NJ: Princeton University Press, 1992. Koblitz, N. Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993. Lang, S. Elliptic Curves: Diophantine Analysis. Berlin: Springer-Verlag, 1978. Mazur, B. and Tate, J. "Points of Order 13 on Elliptic Curves." Invent. Math. 22,4 1/C1/9, 1973/74. Riesel, H. "Elliptic Curves." Appendix 7 in Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 317 /C1/26, 1994. Silverman, J. H. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1986. Silverman, J. H. The Arithmetic of Elliptic Curves II. New York: Springer-Verlag, 1994. Silverman, J. H. and Tate, J. T. Rational Points on Elliptic Curves. New York: Springer-Verlag, 1992. Stillwell, J. "Elliptic Curves." Amer. Math. Monthly 102, 831/C1/37, 1995. Stroeker, R. J. and Tzanakis, N. "Solving Elliptic Diophan- tine Equations by Estimating Linear Forms in EllipticLogarithms." Acta Arith. 67, 177/C1 /96, 1994. Swinnerton-Dyer, H. P. F. "Correction to: ‘On 1 /-adic Repre- sentations and Congruences for Coefficients of ModularForms."’ In Modular Functions of One Variable, Vol. 4, Proc. Internat. Summer School for Theoret. Phys., Univ.Antwerp, Antwerp, RUCA, July-Aug. 1972. Berlin: Springer-Verlag, 1975. Weisstein, E. W. "Books about Elliptic Curves." http:// www.treasure-troves.com/books/EllipticCurves.html. Elliptic Curve Factorization Method A factorization method, abbreviated ECM, which computes a large multiple of a point on a random ELLIPTIC CURVE modulo the number to be factored N. It tends to be faster than the P OLLARD RHO FACTOR- IZATION and P OLLARD P-1 FACTORIZATION METHODS . Zimmermann maintains a table of the largest factorsfound using the ECM. The largest factor found using this algorithm is a prime factor of 54 digits of the 127- digit cofactor C of n /C30b4 /C28b2 /C271 /C3013 /C215733 /C2157177 /C215C ; where b /C306343 /C281; found by N. Lygeros and M. Miz- ony in Dec. 1999. See also ATKIN- GOLDWASSER- KILIAN- MORAIN CERTI- FICATE ,ELLIPTIC CURVE PRIMALITY PROVING ,ELLIP- TIC PSEUDOPRIME References Atkin, A. O. L. and Morain, F. "Finding Suitable Curves for the Elliptic Curve Method of Factorization." Math. Com- put. 60, 399 /C1/05, 1993. Brent, R. P. "Some Integer Factorization Algorithms Using Elliptic Curves." Austral. Comp. Sci. Comm. 8, 149 /C1/63, 1986. Brent, R. P. "Parallel Algorithms for Integer Factorisation." In Number Theory and Cryptography (Ed. J. H. Loxton). New York: Cambridge University Press, pp. 26 /C1/7, 1990. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn 91; b /C302,3,5,6,7,10,11,12 Up to High Powers, rev. ed. Provi- dence, RI: Amer. Math. Soc., p. lxxxiii, 1988. Eldershaw, C. and Brent, R. P. "Factorization of Large Integers on Some Vector and Parallel Computers." Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Num- ber Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). Amsterdam: Netherlands, Elsevier, pp. 673 /C1/15, 1990. Lenstra, H. W. Jr. "Factoring Integers with Elliptic Curves." Ann. Math. 126, 649 /C1/73, 1987. Montgomery, P. L. "Speeding the Pollard and Elliptic Curve Methods of Factorization." Math. Comput. 48, 243 /C1/64, 1987. Zimmermann, P. "The ECMNET Project." http://www.lor- ia.fr/~zimmerma/records/ecmnet.html. Zimmermann, P. "ECM Top 100 Table." http://www.loria.fr/ ~zimmerma/records/top100.html. Elliptic Curve Group Law The GROUP of an ELLIPTIC CURVE which has been transformed to the form y2 /C30x3 /C27ax /C27b is the set of K-RATIONAL POINTS , including the single POINT AT INFINITY . The group law (addition) is defined as follows: Take 2 K-RATIONAL POINTS P and Q. Now ‘draw’ a straight line through them and compute the third point of intersection R (also a K-RATIONAL POINT ). Then P /C27Q /C27R /C300 gives the identity POINT AT INFINITY . Now find the inverse of R, which can be done by setting R /C30(a ; b) giving /C28R /C30(a ;/C28b) :/ This remarkable result is only a special case of a more general procedure. Essentially, the reason is that this type of ELLIPTIC CURVE has a single POINT AT INFINITY which is an inflection point (the line at infinity meets the curve at a single POINT AT INFINITY , so it must be an intersection of multiplicity three).Elliptic Curve Primality Proving A class of algorithm, abbreviated ECPP, which provides certificates of primality using sophisticated results from the theory of ELLIPTIC CURVES . A detailed description and list of references are given by Atkin and Morain (1990, 1993). Adleman and Huang (1987) designed an independent algorithm using elliptic curves of genus two. See also ATKIN- GOLDWASSER- KILIAN- MORAIN CERTI- FICATE ,E LLIPTIC CURVE FACTORIZATION METHOD , ELLIPTIC PSEUDOPRIME References Adleman, L. M. and Huang, M. A. "Recognizing Primes in Random Polynomial Time." In Proc. 19th STOC, New York City, May 25 /C1/7, 1986. New York: ACM Press, pp. 462 /C1/ 69, 1987. Atkin, A. O. L. Lecture notes of a conference, Boulder, CO, Aug. 1986. Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Res. Rep. 1256, INRIA, June 1990. Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Math. Comput. 61,2 9/C1/8, 1993. Bosma, W. "Primality Testing Using Elliptic Curves." Techn. Rep. 85 /C1/2, Math. Inst., Univ. Amsterdam, 1985. Chudnovsky, D. V. and Chudnovsky, G. V. "Sequences of Numbers Generated by Addition in Formal Groups and New Primality and Factorization Tests." Res. Rep. RC11262, IBM, Yorktown Heights, NY, 1985. Cohen, H. Cryptographie, factorisation et primalite ´: l’utilisa- tion des courbes elliptiques. Paris: C. R. J. Soc. Math. France, Jan. 1987. Kaltofen, E.; Valente, R.; and Yui, N. "An Improved Las Vegas Primality Test." Res. Rep. 89 /C1 /2, Rensselaer Poly- technic Inst., Troy, NY, May 1989. Elliptic Cylinder ACYLINDER with ELLIPTICAL CROSS SECTION . The PARAMETRIC EQUATIONS for the laterals sides of an elliptic cylinder of height h,SEMIMAJOR AXIS a, and SEMIMINOR AXIS bare x/C30acosu y/C30bsinu z/C30z; where u/C23[0;2p) and z/C23[0;h]:/ The elliptic cylinder is a QUADRATIC RULED SURFACE . See also CONE,CYLINDER ,ELLIPTIC CONE,ELLIPTIC PARABOLOID ,QUADRATIC SURFACE ,RULED SURFACE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 12, 1999. Elliptic Cylindrical Coordinates The v coordinates are the asymptotic angle of confocal HYPERBOLIC CYLINDERS symmetrical about the X-AXIS . The u coordinates are confocal ELLIPTIC CYLINDERS centered on the origin. x /C30a cosh u cos v (1) y /C30a sinh u sin v (2) z /C30z ; (3) where u /C23 [0;/C12) ; v /C23 [0; 2p) ; and z /C23 (/C28/C12;/C12) : They are related to CARTESIAN COORDINATES by x2 a2 cosh2 u /C27y2 a2 sinh2 u /C301 (4) x2 a2 cos2 v /C28y2 a2 sin2 v /C301 : (5)The SCALE FACTORS are h1 /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh2 u sin2 v /C27sinh2 u cos2 vp (6) /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh(2 u) /C28 cos(2 v) 2s (7) /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisinh2 u /C27sin2 vp (8) h2 /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisinh2 u sin2 v /C27sinh2 u cos2 vp (9) /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh(2 u) /C28 cos(2 v) 2s (10) /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh2 u /C27sin2 vp (11) h3 /C301: (12) The LAPLACIAN is 92 /C301 a2(sinh2 u /C27 sin2 v)@2 @u2 /C27@2 @v2 ! /C27@2 @z2 : (13) Let q1 /C30cosh u (14) q2 /C30cos v (15) q3 /C30z : (16) Then the new SCALE FACTORS are hq1/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 1/C28q22 q21/C281s (17) hq2/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q21/C28q22 1/C28q21s (18) hq3/C301: (19) The H ELMHOLTZ DIFFERENTIAL EQUATION isSEPAR- ABLE . See also CYLINDRICAL COORDINATES ,H ELMHOLTZ DIFFERENTIAL EQUATION– ELLIPTIC CYLINDRICAL CO- ORDINATES References Arfken, G. "Elliptic Cylindrical Coordinates ( u,v,z)."§2.7 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 95 /C1/7, 1970. Moon, P. and Spencer, D. E. "Elliptic-Cylinder Coordinates / ðh;f;zÞ/." Table 1.03 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 17 /C1/0, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 657, 1953. Elliptic Delta Function d(r) /C30ffiffiffirp/C282a(r) ; where a(r) is the ELLIPTIC ALPHA FUNCTION . See also ELLIPTIC ALPHA FUNCTION ,ELLIPTIC INTE- GRAL SINGULAR VALUE References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Weisstein, E. W. "Elliptic Singular Values." MATHEMATICA NOTEBOOK ELLIPTIC SINGULAR.M . Elliptic Exponential Function The inverse of the ELLIPTIC LOGARITHM eln(x) /C13g/C12 xdtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t3 /C27 at2 /C27 btp : It is doubly periodic in the COMPLEX PLANE . Elliptic Fixed Point (Differential Equations) A FIXED POINT for which the STABILITY MATRIX is purely IMAGINARY , l9/C309i v (for v > 0):/ See also DIFFERENTIAL EQUATION ,F IXED POINT , HYPERBOLIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), PARABOLIC FIXED POINT ,STABLE IMPROPER NODE,STABLE NODE,STABLE SPIRAL POINT ,STABLE STAR,UNSTABLE IMPROPER NODE,UNSTABLE NODE, UNSTABLE SPIRAL POINT ,UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Elliptic Fixed Point (Map) A FIXED POINT of a LINEAR TRANSFORMATION (MAP) for which the rescaled variables satisfy ( d /C28 a)2 /C274bg B0: See also HYPERBOLIC FIXED POINT (MAP), LINEAR TRANSFORMATION ,PARABOLIC FIXED POINT Elliptic Function A DOUBLY PERIODIC FUNCTION with periods 2v1and 2v2 such that f(z /C272v1) /C30f(z /C272 v2) /C30f(z) ; (1) which is ANALYTIC and has no singularities except for POLES in the finite part of the COMPLEX PLANE . TheHALF-PERIOD RATIO t /C13 v2 = v1 must not be purely real, because if it is, the function reduces to a singly periodic function if t is rational, and a constant if t is irrational (Jacobi 1835). v1 and v2 are labeled such that I[ t] /C13I[ v2 =v1] > 0; where I[z] is the IMAGINARY PART . A "cell" of an elliptic function is defined as a parallelogram region in the COMPLEX PLANE in which the function is not multi-valued. Properties obeyed by elliptic functions include 1. The number of POLES in a cell is finite. 2. The number of ROOTS in a cell is finite. 3. The sum of RESIDUES in any cell is 0. 4. LIOUVILLE’S ELLIPTIC FUNCTION THEOREM :An elliptic function with no POLES in a cell is a constant. 5. The number of zeros of f(z) /C28c (the "order"rpar; equals the number of POLES of f(z) :/ 6. The simplest elliptic function has order two, since a function of order one would have a simple irreducible POLE , which would need to have a NONZERO residue. By property (3), this is impos- sible.7. Elliptic functions with a single POLE of order 2 with RESIDUE 0 are called WEIERSTRASS ELLIPTIC FUNCTIONS . Elliptic functions with two simple POLES having residues a0and /C28a0are called JACOBI ELLIPTIC FUNCTIONS . 8. Any elliptic function is expressible in terms of either WEIERSTRASS ELLIPTIC FUNCTION or JACOBI ELLIPTIC FUNCTIONS . 9. The sum of the AFFIXES of ROOTS equals the sum of the AFFIXES of the POLES . 10. An algebraic relationship exists between any two elliptic functions with the same periods. The elliptic functions are inversions of the ELLIPTIC INTEGRALS . The two standard forms of these functions are known as J ACOBI ELLIPTIC FUNCTIONS and W EIER- STRASS ELLIPTIC FUNCTIONS .JACOBI ELLIPTIC FUNC- TIONS arise as solutions to differential equations OF THE FORM d2x dt2/C30A/C27Bx/C27Cx2/C27Dx3; (2) and W EIERSTRASS ELLIPTIC FUNCTIONS arise as solu- tions to differential equations OF THE FORM d2x dt2/C30A/C27Bx/C27Cx2: (3) See also DOUBLY PERIODIC FUNCTION ,E LLIPTIC CURVE ,E LLIPTIC INTEGRAL ,H ALF-PERIOD RATIO , JACOBI ELLIPTIC FUNCTIONS ,JACOBI THETA FUNC- TIONS ,L IOUVILLE’S ELLIPTIC FUNCTION THEOREM , MODULAR FORM,MODULAR FUNCTION ,NEVILLE THE- TA FUNCTIO NS,T HETA FUNCTIO NS,W EIERSTRASS ELLIPTIC FUNCTIONS References Akhiezer, N. I. Elements of the Theory of Elliptic Functions. Providence, RI: Amer. Math. Soc., 1990. Apostol, T. M. "Elliptic Functions." §1.4 in Modular Func- tions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 4 /C1/, 1997. Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Bowman, F. Introduction to Elliptic Functions, with Appli- cations. New York: Dover, 1961. Byrd, P. F. and Friedman, M. D. Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., rev. Berlin: Springer-Verlag, 1971. Cayley, A. An Elementary Treatise on Elliptic Functions, 2nd ed. London: G. Bell, 1895. Chandrasekharan, K. Elliptic Functions. Berlin: Springer- Verlag, 1985. Du Val, P. Elliptic Functions and Elliptic Curves. Cam- bridge, England: Cambridge University Press, 1973. Dutta, M. and Debnath, L. Elements of the Theory of Elliptic and Associated Functions with Applications. Calcutta, India: World Press, 1965. Eagle, A. The Elliptic Functions as They Should Be: An Account, with Applications, of the Functions in a New Canonical Form. Cambridge, England: Galloway and Porter, 1958. Greenhill, A. G. The Applications of Elliptic Functions. London: Macmillan, 1892. Hancock, H. Lectures on the Theory of Elliptic Functions. New York: Wiley, 1910. Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum Borntrae- ger, 1829. King, L. V. On the Direct Numerical Calculation of Elliptic Functions and Integrals. Cambridge, England: Cambridge University Press, 1924. Knopp, K. "Doubly-Periodic Functions; in Particular, Elliptic Functions." §9in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 73 /C1/ 2, 1996. Lang, S. Elliptic Functions, 2nd ed. New York: Springer- Verlag, 1987. Lawden, D. F. Elliptic Functions and Applications. New York: Springer Verlag, 1989. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 427 and 433 /C1/34, 1953. Murty, M. R. (Ed.). Theta Functions. Providence, RI: Amer. Math. Soc., 1993. Neville, E. H. Jacobian Elliptic Functions, 2nd ed. Oxford, England: Clarendon Press, 1951. Oberhettinger, F. and Magnus, W. Anwendung der Ellip- tischen Funktionen in Physik und Technik. Berlin: Springer-Verlag, 1949. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Elliptic Function Identities." §1.8 in A /C30B. Wellesley, MA: A. K. Peters, pp. 13 /C1/5, 1996. Prasolov, V. and Solovyev, Y. Elliptic Functions and Elliptic Integrals. Providence, RI: Amer. Math. Soc., 1997. Siegel, C. L. Topics in Complex Function Theory, Vol. 1: Elliptic Functions and Uniformization Theory. New York: Wiley, 1988. Walker, P. L. Elliptic Functions: A Constructive Approach. New York: Wiley, 1996.Weisstein, E. W. "Books about Elliptic Functions." http:// www.treasure-troves.com/books/EllipticFunctions.html. Whittaker, E. T. and Watson, G. N. Chs. 20 /C1/2in A Course of Modern Analysis, 4th ed. Cambridge, England: Uni- versity Press, 1943. Elliptic Functional COERCIVE FUNCTIONAL Elliptic Geometry A constant curvature NON- EUCLIDEAN GEOMETRY which replaces the PARALLEL POSTULATE with the statement "through any point in the plane, there exist no lines PARALLEL to a given line." Elliptic geometry is sometimes also called R IEMANNIAN GEOMETRY . It can be visualized as the surface of a SPHERE on which "lines" are taken as GREAT CIRCLES . In elliptic geometry, the sum of angles of a TRIANGLE is>180/C14:/ See also EUCLIDEAN GEOMETRY ,HYPERBOLIC GEOME- TRY,NON-EUCLIDEAN GEOMETRY Elliptic Group Modulo p /E(a;b)=pdenotes the elliptic GROUP modulo pwhose elements are 1 and /C12together with the pairs of INTEGERS (x, y) with 0 5x;yBpsatisfying y2/C13x3/C27ax/C27b(mod p) (1) with aandbINTEGERS such that 4a3/C2727b2f0 (mod p): (2) Given ( x1;y1);define (xi;yi)/C13(x1;y1)i(mod p): (3) The ORDER hofE(a;b)=pis given by h/C301/C27Xp x/C301x3/C27ax/C27b p ! /C271"# ; (4) where x3/C27ax/C27b=pis the L EGENDRE SYMBOL , although this FORMULA quickly becomes impractical. However, it has been proven that p/C271/C282ffiffiffipp5h(E(a;b)=p)5p/C271/C272ffiffiffipp: (5) Furthermore, for pa PRIME >3 and INTEGER nin the above interval, there exists aandbsuch that h(E(a;b)=p)/C30n; (6) and the orders of elliptic GROUPS mod pare nearly uniformly distributed in the interval. Elliptic Helicoid A generalization of the HELICOID to the PARAMETRIC EQUATIONS x(u ; v) /C30av cos u y(u; v) /C30bv sin u z(u ; v) /C30cu : See also HELICOID References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 422, 1997. Elliptic Hyperboloid The elliptic hyperboloid is the generalization of the HYPERBOLOID to three distinct semimajor axes. The elliptic hyperboloid of one sheet is a RULED SURFACE and has Cartesian equation x2 a2 /C27y2 b2 /C28z2 c2 /C301; (1) and PARAMETRIC EQUATIONS x(u; v) /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27u2p cos v (2)y(u; v) /C30bffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27u 2p sin v (3) z(u ; v) /C30cu (4) for v /C23 [0; 2p) ; or x(u; v) /C30a(cos u /C14v sin u) (5) y(u; v) /C30b(sin u 9v cos u) (6) z(u; v) /C309cv ; (7) or x(u; v) /C30a cosh v cos u (8) y(u; v) /C30b cosh v sin u (9) z(u; v) /C30c sinh v: (10) The two-sheeted elliptic hyperboloid oriented along the Z-AXIS has Cartesian equation x2 a2 /C27y2 a2 /C28z2 c2 /C30/C281; (11) and PARAMETRIC EQUATIONS x /C30a sinh u cos v (12) y /C30b sinh u sin v (13) z /C30c 9cosh u: (14) The two-sheeted elliptic hyperboloid oriented along the X-AXIS has Cartesian equation x2 a2/C28y2 a2/C28z2 c2/C301 (15) and PARAMETRIC EQUATIONS x/C30acosh ucosh v (16) y/C30bsinh ucosh v (17) z/C30csinh v: (18) See also HYPERBOLOID ,RULED SURFACE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 404 /C1/06 and 470, 1997. Elliptic Integral An elliptic integral is an INTEGRAL OF THE FORM gA(x)/C27B(x)ffiffiffiffiffiffiffiffiffi S(x)p A(x)/C27D(x)ffiffiffiffiffiffiffiffiffiS(x)p dx; (1) or gA(x)dx B(x)ffiffiffiffiffiffiffiffiffiS(x)p ; (2) where A(x);B(x);C(x);andD(x) are POLYNOMIALS inx, and S(x)i sa POLYNOMIAL of degree 3 or 4. Stated more simply, an elliptic integral is an integral OF THE FORM gR(w;x)dx; (3) where R(w;x)i sa RATIONAL FUNCTION ofxandw,w2 is a function of xthat is CUBIC orQUARTIC inx, R(w;x) contains at least one ODD POWER ofw, and w2 has no repeated factors (Abramowitz and Stegun 1972, p. 589). Elliptic integrals can be viewed as generalizations of the inverse TRIGONOMETRIC FUNCTIONS and provide solutions to a wider class of problems. For instance,while the ARC LENGTH of a CIRCLE is given as a simple function of the parameter, computing the ARC LENGTH of an ELLIPSE requires an elliptic integral. Similarly, the position of a pendulum is given by a TRIGONO- METRIC FUNCTION as a function of time for small angle oscillations, but the full solution for arbitrarily large displacements requires the use of elliptic integrals. Many other problems in electromagnetism and grav-itation are solved by elliptic integrals. A very useful class of functions known as ELLIPTIC FUNCTIONS is obtained by inverting elliptic integrals to obtain generalizations of the trigonometric func- tions. E LLIPTIC FUNCTIONS (among which the J ACOBI ELLIPTIC FUNCTIONS and W EIERSTRASS ELLIPTIC FUNCTION are the two most common forms) provide a powerful tool for analyzing many deep problems in NUMBER THEORY , as well as other areas of mathe- matics. All elliptic integrals can be written in terms of three "standard" types. To see this, write R(w;x)/C13P(w;x) Q(w;x)/C30wP(w;x)Q(/C28w;x) wQ(w;x)Q(/C28w;x): (4) But since w2/C30f(x); Q(w;x)Q(/C28w;x)/C13Q1(w;x)/C30Q1(/C28w;x); (5) then wP(w;x)Q(/C28w;x)/C30A/C27Bx/C27Cw/C27Dx2/C27Ewx /C27Fw2/C27Gw2x/C27Hw3x /C30(A/C27Bx/C27Dx2/C27Fw2/C27Gw2x) /C27w(c/C27Ex/C27Hw2x/C27... ) /C30P1(x)/C27wP2(x); (6) so R(w;x)/C30P1(x)/C27wP2(x) wQ1(w)/C30R1(x) w/C27R2(x): (7) But any function fR2(x)dxcan be evaluated in termsof elementary functions, so the only portion that needbe considered is gR1(x) wdx: (8) Now, any quartic can be expressed as S1S2where S1/C13a1x2/C272b1x/C27c1 (9) S2/C13a2x2/C272b2x/C27c2: (10) The COEFFICIENTS here are real, since pairs of COMPLEX ROOTS are COMPLEX CONJUGATES [x/C28(R/C27Ii)][x/C28(R/C28Ii)] /C30x2/C27x(/C28R/C27Ii/C28R/C28Ii)/C27(R2/C28I2i) /C30x2/C282Rx/C27(R2/C27I2): (11) If all four ROOTS are real, they must be arranged so as not to interleave (Whittaker and Watson 1990,p. 514). Now define a quantity lsuch that S 1/C27lS2 (a1/C28la2)x2/C28(2b1/C282b2l)x/C27(c1/C28lc2) (12) is a SQUARE NUMBER and 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a1/C28la2)(c1/C28l2)p /C302(b1/C28b2l) (13) (a1/C28la2)(c1/C28lc2)/C28(b1/C28lb2)2/C300: (14) Call the ROOTS of this equation l1andl2;then S1/C28l1S2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a1/C28l1a2)xp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1/C28lc2p hi2 /C30(a1/C28l1a2)x/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1/C28l1c2 a1/C28l1a2s ! /C13(a1/C28l1a2)(x/C28a)2(15) S1/C28l2S2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a1/C28l1a2)xp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1/C28lc2p hi2 /C30(a1/C28l1a2)x/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1/C28l2c2 a1/C28l2a2s ! /C13(a1/C28l2a2)(x/C28b)2: (16) Taking (15)-(16) and l2(1)/C28l1(2) gives S2(l2/C28l1)/C30(a1/C28l1a2)(x/C28a)2/C28(a1/C28l2a2) /C2(x/C28b)2(17) S1(l2/C28l1)/C30l2(a1/C28l1a2)(x/C28a)2/C28l1(a1/C28l2a2) /C2(x/C28b2): (18) Solving gives S1/C30a1/C28l1a2 l2/C28l1(x/C28a)2/C28a1/C28l2a2 l2/C28l1(x/C28b)2 /C13A1(x/C28a)2/C27B1(x/C28b)2(19) S2/C30l2(a1/C28l1a2) l2/C28l1(x/C28a)2/C28l1(a1/C28l2a2) l2/C28l1(x/C28b)2 /C13A2(x/C28a)2/C27B2(x/C28b)2; (20) so we have w2/C30S1S2/C30[A1(x/C28a)2/C27B1(x/C28b)2] /C2[A2(x/C28a)2/C27B2(x/C28b)2]: (21) Now let t/C13x/C28a x/C28b(22) dy/C30[(x/C28b)/C281/C28(x/C28a)(x/C28b)/C282]dx /C30(x/C28b)/C28(x/C28a) (x/C28b)2dx /C30a/C28b (x/C28b)2dx; (23) so w2/C30(x/C28b)4A1x/C28a x/C28b !2 /C27B12 435A 2x/C28a x/C28b ! /C27B2"# /C30(x/C28b)4(A1t2/C27B1)(A2t2/C27B2); (24) and w/C30(x/C28b)2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A1t2/C27B1)(A2t2/C27B2)p (25) dx w/C30ðx/C0bÞ2 a/C28bdt"# 1 ðx/C28bÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA1t2þB1ÞðA2t2þB2Þq /C30dt (a/C28b)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A1t2/C27B1)(A2t2/C27B2)p : (26) Now let R3(t)/C13R1(x) a/C28b; (27) so gR1(x)dx w/C30gR3(t)dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(A 1t2/C27B1)(A2t2/C27B2)p : (28) Rewriting the EVEN and ODD parts R3(t)/C27R3(/C28t)/C132R4(t2) (29) R3(t)/C28R3(/C28t)/C132tR5(t2); (30) gives R3(t)/C131 2(Reven/C28Rodd)/C30R4(t2)/C27tR5(t2); (31) so we havegR1(x)dx w/C30gR4(t2)dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A1t2/C27B1)(A2t2/C27B2)p /C27gR5(t2)td tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(A 1t2/C27B1)(A2t2/C27B2)p : (32) Letting u/C13t2(33) du/C302td t (34) reduces the second integral to 1 2gR5(u)duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A1u/C27B1)(A2u/C27B2)p ; (35) which can be evaluated using elementary functions. The first integral can then be reduced by INTEGRA- TION BY PARTS to one of the three Legendre elliptic integrals (also called Legendre-Jacobi ELLIPTIC INTE- GRALS ), known as incomplete elliptic integrals of the first, second, and third kind, denoted F(f;k);E(f;k); andQ(n;f;k);respectively (von Ka ´rma´n and Biot 1940, Whittaker and Watson 1990, p. 515). If f/C30p=2; then the integrals are called complete elliptic inte-grals and are denoted K(k);E(k);Q(n;k): / Incomplete elliptic integrals are denoted using a MODULUS k,PARAMETER m/C13k2;orMODULAR ANGLE a/C13sin/C281k:An elliptic integral is written I(f½m) when the PARAMETER is used, I(f;k) when the MODULUS is used, and I(f_a) when the MODULAR ANGLE is used. Complete elliptic integrals are defined when f/C30p=2 and can be expressed using the expansion (1/C28k2sin2u)/C281=2/C30X/C12 n/C300(2n/C281)!! (2n)!!k2nsin2nu:(36) An elliptic integral in standard form gx adxffiffiffiffiffiffiffiffi f(x)p ; (37) where f(x)/C30a4x4/C27a3x3/C27a2x2/C27a1x/C27a0; (38) can be computed analytically (Whittaker and Watson 1990, p. 453) in terms of the W EIERSTRASS ELLIPTIC FUNCTION with invariants g2/C30a0a4/C284a1a3/C273a2 2 (39) g3/C30a0a2a4/C282a1a2a3/C28a4a21/C28a23a0: (40) Ifa/C13x0is a root of f(x)/C300;then the solution is x/C30x0/C271 4f?(x0)[/C212(z;g2;g3)/C281 24fƒ(x0)]/C281: (41) For an arbitrary lower bound, x/C30a /C27ffiffiffiffiffiffiffiffiffi f(a)p /C212?(z)1 2f?(a)[/C212(z)/C281 24fƒ(a)]/C271 24f(a)f§(a) 2[/C212(z)/C281 24fƒ(a)]2/C281 48f(a)f(iv)(a); (42) where /C212(z)/C13/C212(z;g2;g3)i saW EIERSTRASS ELLIPTIC FUNCTION (Whittaker and Watson 1990, p. 454). A generalized elliptic integral can be defined by the function T(a;b)/C132 pgp=2 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2cos2u/C27b2sin2up (43) /C302 pgp=2 0du cosuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2tan2up (44) (Borwein and Borwein 1987). Now let t/C13btanu (45) dt/C30bsec2udu: (46) But secu/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27tan2up ; (47) so dt/C30b cosusecudu/C30b cosuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27tan2up du /C30b cosuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27t b !2vuutdu /C30du cosuffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C27t2p ; (48) and du cosu/C30dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C27t2p ; (49) and the equation becomes T(a;b)/C302 pg/C12 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a2/C27t2)(b2/C27t2)p /C301 pg/C12 /C28/C12dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(a2/C27t2)(b2/C27t2)p : (50) Now we make the further substitution u/C131 2(t/C28ab=t): The differential becomes du/C301 2(1/C27ab=t2)dt; (51) but 2 u/C30t/C28ab=t;so 2u=t/C301/C28ab=t2(52) ab=t2/C301/C282u=t (53) and1/C27ab=t2/C302/C282u=t/C302(1/C28u=t): (54) However, the left side is always positive, so 1/C27ab=t2/C302/C282u=t/C302½1/C28u=t½ (55) and the differential is dt/C30du 1/C28u t;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12: (56) We need to take some care with the limits of integration. Write (50) as g/C12 /C28/C12f(t)dt/C30g0/C28 /C28/C12f(t)dt/C27g/C12 0/C27f(t)dt: (57) Now change the limits to those appropriate for the u integration g/C12 /C28/C12g(u)du/C27g/C12 /C28/C12g(u)du/C302g/C12 /C28/C12g(u)du;(58) so we have picked up a factor of 2 which must beincluded. Using this fact and plugging (56) in (50)therefore gives T(a;b)/C30 2 pg/C12 /C28/C12du 1/C28u t;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2b2/C27(a2/C27b2)t2/C27t4p: (59) Now note that u2/C30t4/C282abt2/C27a2b2 4t2(60) 4u2t2/C30t4/C282abt2/C27a2b2(61) a2b2/C27t4/C304u2t2/C272abt2: (62) Plug (62) into (59) to obtain T(a;b)/C302 pg/C12 /C28/C12du 1/C28u t;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4u 2t2/C272abt2/C27(a2/C27b2)t2p /C302 pg/C12 /C28/C12du ½t/C28u½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4u2/C27(a/C27b)2p : (63) But 2ut/C30t2/C28ab (64) t2/C282ut/C28ab/C300 (65) t/C301 2(2u9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4u2/C274abp Þ/C30u9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu 2/C27abp ; (66) so t/C28u/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu 2/C27abp ; (67) and (63) becomes T(a;b)/C302 pg/C12 /C28/C12duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [4u2/C27(a/C27b)2]/C27(u2/C27ab)p /C301 pg/C12 /C28/C12duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2/C27a/C27b 2 !22 435(u 2/C27ab)vuuut: (68) We have therefore demonstrated that T(a;b)/C30T( 1 2(a/C27b);ffiffiffiffiffiffi abp ): (69) We can thus iterate ai/C271/C301 2(ai/C27bi) (70) bi/C271/C30ffiffiffiffiffiffiffiffiffi aibip ; (71) as many times as we wish, without changing the value of the integral. But this iteration is the same asand therefore converges to the ARITHMETIC-GEO- METRIC MEAN , so the iteration terminates at ai/C30bi/C30 M(a0;b0);and we have T(a0;b0)/C30T(M(a0;b0);M(a0;b0)) /C301 pg/C12 /C28/C12dt M2(a0;b0)/C27t2 /C301 pM(a0;b0)tan/C281 t M(a0;b0) !"#/C12 /C28/C12 /C301 pM(a0;b0)p 2/C28/C28p 2 !"# /C301 M(a0;b0): (72) Complete elliptic integrals arise in finding the arc length of an ELLIPSE and the period of a pendulum. They also arise in a natural way from the theory of THETA FUNCTIONS . Complete elliptic integrals can be computed using a procedure involving the ARITH- METIC-GEOMETRIC MEAN . Note that T(a;b)/C132 pgp=2 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2cos2u/C27b2sin2up /C302 pgp=2 0du affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2u/C27b a !2 sin2uvuut /C302 apgp=2 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281/C28b2 a2 !2 sin2uvuut: (73)So we have T(a;b)/C302 apK1/C28b2 a2 ! /C301 M(a;b); (74) where K(k) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND . We are free to let a/C13a0/C131 and b/C13b0/C13 k?;so 2 pK(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k?2p )/C302 pK(k)/C301 M(1;k?); (75) since k/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k?2p ;so K(k)/C30p 2M(1;k?): (76) But the ARITHMETIC-GEOMETRIC MEAN is defined by ai/C301 2(ai/C281/C27bi/C281) (77) bi/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ai/C281/C27bi/C281p (78) ci/C301 2(ai/C281/C28bi/C281)i>0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 0/C28b20p i/C300;( (79) where cn/C281/C301 2an/C28bn/C30c2 n 4an/C2715c2n 4M(a0;b0); (80) so we have K(k)/C30p 2aN; (81) where aNis the value to which anconverges. Similarly, taking instead a?0/C301 and b?0/C30kgives K?(k)/C30p 2a?N: (82) Borwein and Borwein (1987) also show that defining U(a;b)/C13p 2gp=2 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2cos2/C27b2sin2up du /C30aE?b a ! (83) leads to 2U(an/C271;bn/C271)/C28U(an;bn)/C30anbnT(an;bn);(84) so K(k)/C28E(k) K(k)/C301 2(c2 0/C272c21/C2722c22/C27.../C272nc2n) (85) fora0/C131 and b0/C13k?;and K?(k)/C28E?(k) K?(k)/C301 2(c?02/C272c?12/C2722c?22/C27.../C272nc?n2):(86) The elliptic integrals satisfy a large number of identities. The complementary functions and moduli are defined by K?(k)/C13K(ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p )/C30K(k?): (87) Use the identity of generalized elliptic integrals T(a;b)/C30T(1 2(a/C27b);ffiffiffiffiffiffi abp ) (88) to write 1 aKffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2 a2s ! /C302 a/C27bKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28 4ab (a/C27b)2s ! /C302 a/C27bKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2/C282ab (a/C27b)2s ! /C302 a/C27bKa/C28b a/C27b ! (89) Kffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28 b2 a2s ! /C302 1/C27b aK1/C28b a 1/C27b a0 BBB@1 CCCA: (90) Define k?/C13 b a; (91) and use k/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k?2p ; (92) so K(k)/C302 1/C27k?K1/C28k? 1/C27k? ! : (93) Now letting l/C13(1/C28k?)=(1/C27k?) gives l(1/C27k?)/C301/C28k?[k?(l/C271)/C301/C28l (94) k?/C301/C28l 1/C27l(95) k/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28k? 2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281/C28l 1/C27l !2vuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27l)2/C28(1/C28l)2 (1/C27l)2s /C302ffiffi lp 1/C27l; (96) and1 2(1/C27k?)/C301 21/C271/C28l 1/C27l ! /C3012(1/C27l)/C27(1/C28l) 1/C27l"# /C301 1/C27l: (97) Writing kinstead of l, k(k)/C301 k/C271K2ffiffiffi kp 1/C27k ! : (98) Similarly, from Borwein and Borwein (1987), E(k)/C301/C27k 2E2ffiffiffi kp 1/C27k ! /C27k?2 2K(k) (99) E(k)/C30(1/C27k?)E1/C28k? 1/C27k? ! /C28k?K(k): (100) Expressions in terms of the complementary function can be derived from interchanging the moduli and their complements in (93), (98), (99), and (100). K?(k)/C30K(k?)/C302 1/C27kK1/C28k 1/C27k ! /C302 1/C27kK?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281/C28k 1/C27k !2vuut0 B@1 CA /C302 1/C27kK?2ffiffiffi kp 1/C27k ! (101) K?(k)/C301 1/C27k?K2ffiffiffiffik?p 1/C27k? ! /C30 1 1/C27k?K?1/C28k? 1/C27k? ! ; (102) and E?(k)/C30(1/C27k)E?2ffiffiffikp 1/C27k ! /C28kK?(k) (103) E?(k)/C301/C27k? 2 ! E?1/C28k? 1/C27k? ! /C27k2 2K?(k): (104) Taking the ratios K?(k) K(k)/C302K?2ffiffiffi kp 1/C27k ! K2ffiffiffi kp 1/C27k ! /C301 2K?1/C28k? 1/C27k? ! K1/C28k? 1/C27k? ! (105) gives the MODULAR EQUATION of degree 2. It is also true that K(x) /C304 (1 /C27ffiffiffiffi x?p )2 K1 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x4p 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28 x4p"#20 @1A: (106) See also A BELIAN INTEGRAL ,AMPLITUDE ,ARGUMENT (ELLIPTIC INTEGRAL ), CHARACTERISTIC (ELLIPTIC IN- TEGRAL ), DELTA AMPLITUDE ,E LLIPTIC FUNCTION , ELLIPTIC INTEGRAL OF THE FIRST KIND,E LLIPTIC INTEGRAL OF THE SECOND KIND,ELLIPTIC INTEGRAL OF THE THIRD KIND,ELLIPTIC INTEGRAL SINGULAR VALUE ,H EUMAN LAMBDA FUNCTION ,JACOBI ZETA FUNCTION ,M ODULAR ANGLE ,M ODULUS (ELLIPTIC INTEGRAL ), NOME,PARAMETER References Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Inte- grals." Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587 /C1/07, 1972. Arfken, G. "Elliptic Integrals." §5.8 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 321 /C1/27, 1985. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.New York: Wiley, 1987. Hancock, H. Elliptic Integrals. New York: Wiley, 1917. Ka´rma´n, T. von and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treat-ment of Engineering Problems. New York: McGraw-Hill, p. 121, 1940. King, L. V. The Direct Numerical Calculation of Elliptic Functions and Integrals. London: Cambridge University Press, 1924. Prasolov, V. and Solovyev, Y. Elliptic Functions and Elliptic Integrals. Providence, RI: Amer. Math. Soc., 1997. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Elliptic Integrals and Jacobi Elliptic Func-tions." §6.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 254 /C1 /63, 1992. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon & Breach, 1986. Timofeev, A. F. Integration of Functions. Moscow and Leningrad: GTTI, 1948. Weisstein, E. W. "Books about Elliptic Integrals." http:// www.treasure-troves.com/books/EllipticIntegrals.html. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Woods, F. S. "Elliptic Integrals." Ch. 16 in Advanced Calcu- lus: A Course Arranged with Special Reference to theNeeds of Students of Applied Mathematics. Boston, MA: Ginn, pp. 365 /C1 /86, 1926. Elliptic Integral of the First Kind Let the MODULUS ksatisfy 0 Bk2B1;and the AMPLI- TUDE be given by f/C30amu:The incomplete elliptic integral of the first kind is then defined as u/C30F(f;k)/C30gf 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sin2up : (1) Lett/C13sinu (2) dt/C30cosudu/C30ffiffiffiffiffiffiffiffiffiffiffiffi 1/C28t2p du; (3) then (1) can be written as F(f;k)/C30gsinf 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2t2pdtffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28t2p /C30gsinf 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2t2pffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28t2p : (4) Let v/C13tanu (5) dv/C13sec2udu/C30(1/C27v2)du; (6) then the integral can also be written as F(f;k)/C30gtanf 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2v2 1/C27u2sdu 1/C27v2 /C30gtanf 0dvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27v2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27v2)/C28k2v2p (7) /C30gtanf 0dvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27v2)(1/C27k?v2)p ; (8) where k?2/C131/C28k2is the complementary MODULUS . The elliptic integral of the first kind is implemented inMathematica asEllipticK [phi,m](note the use of the parameter m /C30k2instead of the modulus k ). The inverse function of F(f;k) is given by the AMPLITUDE F/C281(u;k)/C30f/C30am(u;k)/C30amu: (9) The integral I/C301ffiffiffi 2pgu0 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosu/C28cosu0p ; (10) which arises in computing the period of a pendulum, is also an elliptic integral of the first kind. Use cosu/C301/C282 sin2(1 2u) (11) sin(1 2u)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28cosu 2s (12) to write ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosu/C28cosu0p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C282 sin2(1 2u)/C28cosu0q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28cosu0pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C282 1/C28cosu0sin2(1 2u)s /C30ffiffiffi 2p sin(1 2u0)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28csc2(1 2u0) sin2(12u)q ; ð13Þ so I/C301 2gu0 0du sin(1 2u0)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28csc2(1 2u0) sin2(12u)q :(14) Now let sin(1 2u)/C30sin(12u0) sin f; (15) so the angle uis transformed to f/C30sin/C281sin(1 2u) sin(1 2u0)"# ; (16) which ranges from 0 to p=2a s uvaries from 0 to u0: Taking the differential gives 1 2cos(12u)du/C30sin(12u0) cos fdf; (17) or 12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2(1 2u0) sin2fq du/C30sin(12u0) cos fdf:(18) Plugging this in gives I/C30gp=2 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2(1 2u0) sin2fqsin(1 2u0) cos fdf sin(1 2u0)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2fq /C30gp=2 0dfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2(1 2u0) sin2fq /C30K(sin(12u0));(19) so I/C301ffiffiffi 2pgu0 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosu/C28cosu0p /C30K(sin(1 2u0)): (20) Making the slightly different substitution f/C30u=2;so du/C302dfleads to an equivalent, but more compli- cated expression involving an incomplete elliptic integral of the first kind, I/C3021ffiffiffi 2p1ffiffiffi2pcsc(1 2u0)gu0 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28csc2(1 2u0) sin2fq /C30csc(12u0)F(12u0;csc(12u0)): (21) Therefore, we have proven the identity cscxF(x;cscx)/C30K(sinx): (22) The elliptic integral of the first kind satisfies F(/C28f;k)/C30/C28F(f;k): (23) Special values of F(f;k) include F(0;k)/C300 (24) F(12p;k)/C30K(k); (25) where K(k) is known as the complete elliptic integral of the first kind. The complete elliptic integral of the first kind, illustrated above as a function of m/C30k2;is defined by K(k)/C13F(1 2p;k) (26) /C30X/C12 n/C300(2n/C281)!! (2n)!!k2ng2p 0sin2nudu (27) /C3012pq2 3(q) (28) /C30X/C12 n/C300(2n/C281)!! (2n)!!k2np 2(2n/C281)!! (2n)!! /C30p 2X/C12 n/C300(2n/C281)!! (2n)!!"#2 k2n(29) /C301 2p2F1(12;12;1 ;k2) (30) /C30p 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p P/C281=21/C27k2 1/C28k2 ! ; (31) where q/C30e/C28lK?(k)=K(k)(32) is the NOME (for½q½B1);2F1(a;b;c;x) is the HYPER- GEOMETRIC FUNCTION , and Pn(x)i saL EGENDRE POLYNOMIAL .K(k) satisfies the L EGENDRE RELATION E(k)K?(k)/C27E?(k)K(k)/C28K(k)K?(k)/C301 2p; (33) where K(k) and E(k) are complete elliptic integrals of the first and SECOND KINDS , respectively, and K?(k) and E?(k) are the complementary integrals. The modulus kis often suppressed for conciseness, so that K(k) and E(k) are often simply written KandE, respectively. The DERIVATIVE of K(k)is dK dk/C13g1 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1 /C28 t2)(1 /C28 k?2t2)p /C30E(k) k(1 /C28 k2) /C28K(k) k(34) and K(k) satisfies the differential equation d dkkk ?2dK dk ! /C30kK(k) ; (35) so E /C30k(1 /C28k2)dK dk/C27K(k) k ! (36) /C30(1 /C28k2) kdK dk/C27K(k) ! (37) (Whittaker and Watson 1990, pp. 499 and 521). Be- sides y /C30K(k) ; the other solution to the differential equation d dkk(1 /C28k2)dy dk"# /C28ky /C300 (38) (Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is MEIJER’S G-FUNCTION y /C30G2; 0 2; 2k21 2 ;12 0 ; 0;j12;j12;j12;j12;j1} :;j1z (39) See also AMPLITUDE ,CHARACTERISTIC (ELLIPTIC IN- TEGRAL ), ELLIPTIC INTEGRAL OF THE SECOND KIND, ELLIPTIC INTEGRAL OF THE THIRD KIND,E LLIPTIC INTEGRAL SINGULAR VALUE ,G AUSS’S TRANSFORMA- TION ,LANDEN’S TRANSFORMATION ,LEGENDRE RELA- TION ,M ODULAR ANGLE ,M ODULUS (ELLIPTIC INTEGRAL ), PARAMETER References Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Inte- grals." Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587 /C1/07, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Spanier, J. and Oldham, K. B. "The Complete Elliptic Integrals K(p) and E(p)/" and "The Incomplete Elliptic Integrals F(p;f) and E(p;f):/" Chs. 61 /C1/2i n An Atlas of Functions. Washington, DC: Hemisphere, pp. 609 /C1/33, 1987. To¨lke, F. "Parameterfunktionen." Ch. 3 in Praktische Funk- tionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83 /C1/15, 1966. To¨lke, F. "Umkehrfunktionen der Jacobischen elliptischen Funktionen und elliptische Normalintegrale erster Gat- tung. Elliptische Amplitudenfunktionen sowie Legen-dresche F- und E-Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und HeumanscheLambda-Funktionen," and "Normalintegrale dritter Gat-tung. LegendrescheQ /-Funktion. Zuru ¨ckfu¨hrung des all- gemeinen elliptischen Integrals auf Normalintegraleerster, zweiter, und dritter Gattung." Chs. 6 /C1 /inPrak- tische Funktionenlehre, dritter Band: Jacobische ellip-tische Funktionen, Legendresche elliptischeNormalintegrale und spezielle Weierstraßsche Zeta- undSigma Funktionen. Berlin: Springer-Verlag, pp. 58 /C1 /44, 1967. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Elliptic Integral of the Second Kind Let the MODULUS ksatisfy 0 Bk2B1:(This may also be written in terms of the PARAMETER m/C13k2or MODULAR ANGLE a/C13sin/C281k:/) The incomplete elliptic integral of the second kind is then defined as E(f;k)/C13gf 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sin2up du: (1) The elliptic integral of the second kind is implemen- ted in Mathematica asEllipticE [phi,m](note the use of the parameter m /C30k2instead of the modulus k ). To place the elliptic integral of the second kind in a slightly different form, let t/C13sinu (2) dt/C30cosudu/C30ffiffiffiffiffiffiffiffiffiffiffiffi 1/C28t2p du; (3) so the elliptic integral can also be written as E(f;k)/C30gsinf 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2t2p dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28t2p /C30gsinf 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2t2 1/C28t2s dt: (4) The complete elliptic integral of the second kind, illustrated above as a function of the PARAMETER m,is defined by E(k) /C13E(1 2 p; k) (5) /C30p 21 /C28X/C12 n/C301(2n /C28 1)!! (2n)!!"#2k2n 2n /C28 18 < :9 = ; (6) /C301 2 p 2F1(/C2812;12;1;k2) (7) /C30gK 0dn2 udu ; (8) where2F1(a ; b; c; x) is the HYPERGEOMETRIC FUNC- TION and dn u is a JACOBI ELLIPTIC FUNCTION . The complete elliptic integral of the second kind satisfies the LEGENDRE RELATION E(k)K ?(k) /C27E ?(k)K(k) /C28K(k)K ?(k) /C301 2 p; (9) where K(k) and E(k) are complete ELLIPTIC INTEGRALS OF THE FIRST and second kinds, respectively, and K ?(k) and E?(k) are the complementary integrals. The DERIVATIVE is dE dk /C30E(k) /C28 K(k) k (10) (Whittaker and Watson 1990, p. 521). Besides y /C30 E(k); the other solution to the differential equation k?2d dkkdy dk ! /C27ky /C300 (11) (Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is MEIJER’S G-FUNCTION y /C30G2; 0 2; 2k21 2 ;32 0 ; 0;j12;j12;j12;j12;j1} :;j1z (12) If k r is a singular value (i.e., kr /C30 l /C31(r) ; (13) where l/C31 is the ELLIPTIC LAMBDA FUNCTION ), and K(kr) and the ELLIPTIC ALPHA FUNCTION a(r) are also known, then E(k)/C30K(k)ffiffiffirpp 3[K(k)]2/C28a(r)"# /C27K(k): (14) A generalization replacing sin uwith sinh uin (1) gives /C28iE(if;/C28k)/C30gf 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sinh2up du: (15) See also ELLIPTIC INTEGRAL OF THE FIRST KIND, ELLIPTIC INTEGRAL OF THE THIRD KIND,E LLIPTICINTEGRAL SINGULAR VALUE References Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Inte- grals." Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587 /C1/07, 1972. Spanier, J. and Oldham, K. B. "The Complete Elliptic Integrals K(p) and E(p)/" and "The Incomplete Elliptic Integrals F(p;f) and E(p;f):/" Chs. 61 and 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609 /C1/33, 1987. To¨lke, F. "Parameterfunktionen." Ch. 3 in Praktische Funk- tionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83 /C1/15, 1966. To¨lke, F. "Umkehrfunktionen der Jacobischen elliptischen Funktionen und elliptische Normalintegrale erster Gat- tung. Elliptische Amplitudenfunktionen sowie Legen-dresche F- und E-Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und Heumansche Lambda-Funktionen," and "Normalintegrale dritter Gat- tung. LegendrescheQ /-Funktion. Zuru ¨ckfu¨hrung des all- gemeinen elliptischen Integrals auf Normalintegraleerster, zweiter, und dritter Gattung." Chs. 6 /C1 /inPrak- tische Funktionenlehre, dritter Band: Jacobische ellip-tische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 58 /C1 /44, 1967. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Elliptic Integral of the Third Kind Let 0Bk2B1:The incomplete elliptic integral of the third kind is then defined as P(n;f;k)/C30gf 0du (1/C28nsin2u)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sin2up (1) /C30gsinf 0dt (1/C28nt2)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C28t2)(1/C28k2t2)p ; (2) where nis a constant known as the CHARACTERISTIC . The complete elliptic integral of the third kind P(n½m) /C30P(n;1 2 p½m) (3) is illustrated above. See also ELLIPTIC INTEGRAL OF THE FIRST KIND, ELLIPTIC INTEGRAL OF THE SECOND KIND,ELLIPTIC INTEGRAL SINGULAR VALUE References Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Integrals" and "Elliptic Integrals of the Third Kind." Ch. 17 and §17.7 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587 /C1/07, 1972. To¨lke, F. "Normalintegrale dritter Gattung. LegendrescheQ /-Funktion. Zuru ¨ckfu¨hrung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Ch. 7 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 100 /C1/44, 1967. Elliptic Integral Singular Value When the MODULUS khas a singular value, the complete elliptic integrals may be computed in analytic form in terms of GAMMA FUNCTIONS . Abel (quoted in Whittaker and Watson 1990, p. 525) proved that whenever K?(k) K(k)/C30a/C27bffiffiffinp c/C27dffiffiffinp; (1) where a,b,c,d, and nare INTEGERS ,K(k)i sa complete ELLIPTIC INTEGRAL OF THE FIRST KIND , and K?(k)/C13K(ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p ) is the complementary complete ELLIPTIC INTEGRAL OF THE FIRST KIND , then the MODULUS kis the ROOT of an algebraic equation with INTEGER COEFFICIENTS . AMODULUS krsuch that K?(kr) K(kr)/C30ffiffiffirp; (2) is called a singular value of the elliptic integral. The ELLIPTIC LAMBDA FUNCTION l/C31(r) gives the value of kr: Selberg and Chowla (1967) showed that K(l/C31(r)) and E(l/C31(r)) are expressible in terms of a finite number of GAMMA FUNCTIONS . The complete ELLIPTIC INTEGRALS OF THE SECOND KIND e(kr) and e?(kr) can be expressed in terms of k(kr) and k?(kr) with the aid of the ELLIPTIC ALPHA FUNCTION a(r):/ The following table gives the values of k(kr) for small integral rin terms of GAMMA FUNCTIONS G(z):/ K(k1)/C30G2(1 4) 4ffiffiffipp K(k2)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p /C271p G(1 8)G(38) 213=4ffiffiffippK(k3)/C3031=4G3(1 3) 27=3p K(k4)/C30(ffiffiffi 2p /C271)G2(1 4) 27=2ffiffiffipp K(k5)/C30(ffiffiffi 5p /C272)1=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G(1 20)G(3 20)G(7 20)G(9 20) 160pvuut K(k 6)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ffiffiffi 2p /C281)(ffiffiffi 3p /C27ffiffiffi 2p )(2/C27ffiffiffi 3p )q /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G(1 24)G(5 24)G(7 24)G(11 24) 384pvuut K(k 7)/C30G(1 7)G(27)G(47) 71=44p K(k8)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C275ffiffiffi 2pp 4ffiffiffi2pvuut(ffiffiffi 2p /C271)1=4G(1 8)G(38) 8ffiffiffipp K(k9)/C3031=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 3pp 12ffiffiffippG2(1 4) K(k10)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2/C273ffiffiffi 2p /C27ffiffiffi 5p )q /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G(1 40)G(7 40)G(9 40)G(11 40)G(1340)G(1940)G(2340)G(3740) 256p3vuut K(k 11)/C30[2/C27(17/C273ffiffiffiffiffiffi 33p )1=3/C28(3ffiffiffiffiffiffi33p /C2817)1=3]2 /C2G(1 11)G(3 11)G(4 11)G(5 11)G(9 11) 111=4144p2 K(k12)/C3031=4(ffiffiffi 2p /C271)(ffiffiffi3p /C27ffiffiffi2p )ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 3pp G 3(1 3) 213=3p K(k13)/C30(18/C275ffiffiffiffiffiffi 13p )1=4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6656p5p /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G(1 52)G(7 52)G(9 52)G(11 52)G(1552)G(1752)G(1952)G(2552)G(2952)G(3152)Gq K(k15)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ffiffiffi 5p /C271)G(1 15)G(2 15)G(4 15)G(8 15) 240pvuut K(k 16)/C30(21=4/C271)2G2(1 4) 29=2ffiffiffipp K(k17)/C30C1G(1 68)G(3 68)G(7 68)G(11 68)G(1368) G(5 68)G(15 68)G(1968)G(2968)"#1=4 /C2[G(21 68)G(2568)G(2768)G(3168)G(3368)]1=4 K(k25) /C30ffiffiffi 5p /C27 2 20G2(1 4) ffiffiffipp ; where G(z) is the GAMMA FUNCTION and C1is an algebraic number (Borwein and Borwein 1987, p. 298). Borwein and Zucker (1992) give amazing expressions for singular values of complete elliptic integrals in terms of CENTRAL BETA FUNCTIONS b(p) /C13B(p ; p) : (3) Furthermore, they show that K(kn)isalways expres- sible in terms of these functions for n /C131; 2 (mod 4): In such cases, the G(z) functions appearing in the expression are OF THE FORM G(t=4n) where 1 5t 5 (2n /C281) and (t;4n) /C301: The terms in the numerator depend on the sign of the KRONECKER SYMBOL ft=4ng: Values for the first few n are K(k1) /C302/C282 b(1 4) K(k2) /C302 /C2813 =4 b(18) K(k3) /C302 /C284 =33/C281 =4 b(13) /C302 /C285=33/C283 =4 b(16) K(k5) /C302/C2833=205/C285 =8(11 /C275ffiffiffi 5p )1=4sin(1 20 p)b(1 2) /C302/C2829=205/C283 =8(1 /C27ffiffiffi 5p )1 =4sin(3 20 p) b(3 20) K(k6) /C302 /C2847 =123/C283=4(ffiffiffi2p /C281)(ffiffiffi3p /C271)b( 1 24) /C302/C2843 =123 /C281 =4(ffiffiffi3p /C281)b(5 24) K(k7) /C302 /C215 7/C283 =4 sin(1 7 p) sin(27 p)B(17 ;27) /C302 /C282 =77/C281 =4b(1 7) b(27) b(1 14) K(k10) /C302/C2861=205/C281 =4(ffiffiffi 5p /C282)1 =2(ffiffiffiffiffiffi10p /C273)b(1 8)b(7 40) b(1 340) /C302/C2815=45 /C283 =4(ffiffiffi 5p /C282)1 =2b(1 40) b(1 940) b(3 8) K(k11) /C30R /C215 2 /C287 =11 sin(1 11 p) sin(3 11 p)B(1 22;3 22) K(k13) /C302 /C28313 /C285 =8(5ffiffiffiffiffiffi 13p /C2718)1=4 /C2[tan(1 52 p) tan(3 52 p) tan(9 52 p)]1 =2b1 52;j1ffl;j1{ b9 52;j1ffl;j1{ b23 52;j1ffl;j1{K(k14) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ffiffiffi 2p /C272q /C27ffiffiffi2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffi 2p /C281qr /C215 2/C2813 =47/C283=8tan(5 56 p) tan(13 56 p) tan(11 56 p)"#1 =4 /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b(5 56) b(1356)b(18) b(11 56)vuut K(k 15) /C302/C2813/C283 =45 /C287=12B(1 15 ;4 15) /C302 /C2823/C283 =45/C283 =4(ffiffiffi 5p /C28 1)b(1 15) b(4 15) b(1 3) K(k17) /C30C2b(1 68)b(3 68) b(7 68) b(9 68)b(1168)b(1368) b(5 68) b(15 68)"#1=4 ; where R is the REAL ROOT of x3 /C284x /C304 /C300 (4) and C2is an algebraic number (Borwein and Zucker 1992). Note that K(k11) is the only value in the above list which cannot be expressed in terms of CENTRAL BETA FUNCTIONS . Using the ELLIPTIC ALPHA FUNCTION , the ELLIPTIC INTEGRALS OF THE SECOND KIND can also be found from E/C30p 4ffiffiffirpK/C271/C28a(r)ffiffiffirp"# K (5) E?/C30p 4k/C27a(r)K; (6) and by definition, K?/C30Kffiffiffinp: (7) See also CENTRAL BETA FUNCTION ,ELLIPTIC ALPHA FUNCTION ,ELLIPTIC DELTA FUNCTION ,ELLIPTIC IN- TEGRAL OF THE FIRST KIND,ELLIPTIC INTEGRAL OF THE SECOND KIND,E LLIPTIC LAMBDA FUNCTION , GAMMA FUNCTION ,MODULUS (ELLIPTIC INTEGRAL ) References Abel, N. H. "Recherches sur les fonctions elliptiques." J. reine angew. Math. 3, 160/C1/90, 1828. Reprinted in Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., p. 377, 1988. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987. Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua- tion of the Gamma Function at Rational Values of SmallDenominator." IMA J. Numerical Analysis 12, 519/C1 /26, 1992. Bowman, F. Introduction to Elliptic Functions, with Appli- cations. New York: Dover, pp. 75, 95, and 98, 1961. Glasser, M. L. and Wood, V. E. "A Closed Form Evaluation of the Elliptic Integral." Math. Comput. 22, 535/C1/36, 1971. Selberg, A. and Chowla, S. "On Epstein’s Zeta-Function." J. reine angew. Math. 227,8 6/C1/10, 1967. Weisstein, E. W. "Elliptic Singular Values." M ATHEMATICA NOTEBOOK ELLIPTIC SINGULAR.M . Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 524 /C1/28, 1990. Wrigge, S. "An Elliptic Integral Identity." Math. Comput. 27, 837/C1/40, 1973. Zucker, I. J. "The Evaluation in Terms of G/-Functions of the Periods of Elliptic Curves Admitting Complex Multiplica- tion." Math. Proc. Cambridge Phil. Soc. 82, 111/C1/18, 1977. Elliptic Integral Singular Value k1 The first singular value k1of the ELLIPTIC INTEGRAL OF THE FIRST KIND K(k);corresponding to K?(k1)/C30K(k1); (1) is given by k1/C301ffiffiffi 2p (2) k?1/C301ffiffiffi2p: (3) The value K(k 1) is given by K1ffiffiffi2p ! /C13g1 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C28t2)(1/C281 2t2)q ; (4) which can be transformed to K1ffiffiffi 2p ! /C30ffiffiffi 2pg1 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28t4p : (5) Let u/C13t4(6) du/C304t3dt/C304u3=4dt (7) dt/C301 4u/C283=4du; (8) then k1ffiffiffi 2p ! /C30ffiffiffi2p 4g1 0u/C283=4(1/C28u)/C281=2du /C30ffiffiffi2p 4B(1 4;12)/C30G(14)G(12) G(3 4)ffiffiffi 2p 4: (9) where B(a;b) is the BETA FUNCTION andG(z) is the GAMMA FUNCTION . Now use G(1 2)/C30ffiffiffipp(10) and1 G(1/C28x)/C30sin(px) pG(x); (11) so 1 G(3 4)/C301 G(1/C2814)/C30sinp 4 ! pG(1 4)/C301 pffiffiffi 2pG(1 4): (12) Therefore, K1ffiffiffi 2p ! /C30G2(1 4)ffiffiffippffiffiffi 2p 4pffiffiffi 2p/C30G2(1 4) 4ffiffiffipp: (13) Now consider E1ffiffiffi 2p ! /C13g1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281 2t2 1/C28t2vuutdt: (14) Let t2/C131/C28u2(15) 2td t/C30/C282ud u (16) dt/C30/C281 tud u/C30u(1/C28u2)/C281=2du; (17) so E1ffiffiffi 2p ! /C30g1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281 2(1/C28u2) 1/C28(1/C28u2)vuutu(1/C28u2)/C281=2du /C30g1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2(1/C27u2) uvuutu(1/C28u2)/C281=2du /C301ffiffiffi 2pg1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27u2) (1/C28u2)s du: (18) Now note that 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u4p /C27u2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28u4p !2 /C30(1/C27u2)2 1/C28u4/C30(1/C27u2)2 (1/C27u2)(1/C28u2) /C301/C27u2 1/C28u2; (19) so E1ffiffiffi 2p ! /C301ffiffiffi2pg1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u2 1/C28u2s du /C301ffiffiffi2pg1 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u4p /C27u2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28u4p ! du /C301 2K1ffiffiffi 2p ! /C271ffiffiffi2pg1 0u2duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u4p : (20) Now let t /C13u4 (21) dt /C304u3 du; (22) so g1 0u2 duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 u4p /C301 4 g1 0t1 =2t/C283 =4(1 /C28t) /C281=2 dt /C301 4g1 0t/C281=4(1 /C28t)/C281 =2 dt /C3014 B(34 ;12) /C30G(34) G(12) 4 G(54): (23) But [ G(5 4)]/C281 /C30[14 G(14)] /C281 (24) G(34) /C30 pffiffiffi 2p [G(1 4)] /C281 (25) G(12) /C30ffiffiffipp; (26) so g1 0u2 duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 u4p /C301 4pffiffiffi 2p /C215 4ffiffiffipp G2(1 4)/C30ffiffiffi 2p p3 =2 G2(1 4)(27) E1ffiffiffi 2p ! /C301 2 K /C27p3 =2 G2(1 4) /C30G2(1 4) 8ffiffiffipp/C27p3 =2 G2(1 4) /C301 4ffiffiffi p 2s G(1 4) G(3 4) /C27G(3 4) G(5 4)"# : (28) Summarizing (13) and (28) gives K1ffiffiffi 2p ! /C30G2(1 4) 4ffiffiffipp K ?1ffiffiffi 2p ! /C30G2(1 4) 4ffiffiffipp E1ffiffiffi 2p ! /C30G2(1 4) 8ffiffiffipp/C27p3 =2 G2(1 4) E?1ffiffiffi 2p ! /C30G2(1 4) 8ffiffiffipp/C27p3 =2 G2(1 4) : Elliptic Integral Singular Value k2 The second SINGULAR VALUE k2 ; corresponding to K ?(k2) /C30ffiffiffi 2p K(k2) ; (1) is given byk2 /C30tanp 8 ! /C30ffiffiffi2p /C281; (2) k? 2 /C30ffiffiffi2p (ffiffiffi2p /C281): (3) For this modulus, E(ffiffiffi 2p /C281) /C30 1 4ffiffiffi p 4s G(1 8) G(5 8) /C27G(5 8) G(9 8)"# : (4) Elliptic Integral Singular Value k3 The third SINGULAR VALUE k3 ; corresponding to K ?(k3) /C30ffiffiffi 3p K(k3) ; (1) is given by k3 /C30sinp 12 ! /C301 4(ffiffiffi 6p /C28ffiffiffi 2p ) : (2) As shown by Legendre, K(k3) /C30ffiffiffipp 2 /C215 33 =4G(1 6) G(2 3) (3) (Whittaker and Watson 1990, p. 525). In addition, E(k3) /C30p 4ffiffiffi 3p1 K /C27ffiffiffi 3p /C27 1 2ffiffiffi 3p K /C301 4pffiffiffi 3p !1 =2 1 /C271ffiffiffi3p ! G(1 3) G(56) /C272G(5 6) G(13)"# ; (4) and E ?(k3) /C30pffiffiffi 3p 41 K ?(k3) /C27ffiffiffi3p /C28 1 2ffiffiffi3p K ?(k 3) : (5) Summarizing, K[1 4(ffiffiffi 6p /C28ffiffiffi 2p )] /C30ffiffiffipp 2 /C215 33 =4G(1 6) G(2 3) (6) K ?[1 4(ffiffiffi 6p /C28ffiffiffi2p )] /C30ffiffiffi3p K /C30ffiffiffipp 2 /C215 31 =4G(1 6) G(2 3)(7) E[14(ffiffiffi 6p /C28ffiffiffi 2p )] /C301 4pffiffiffi 3p !1=2 1/C271ffiffiffi3p ! G(1 3) G(5 6)/C272G(5 6) G(1 3)"# (8) E?[14(ffiffiffi 6p /C28ffiffiffi2p )]/C30ffiffiffipp 233=4G(2 3) G(16)/C27ffiffiffi 3p /C281 2 /C21533=4G(1 6) G(23)"# :(9) (Whittaker and Watson 1990). See also JACOBI THETA FUNCTIONS References Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/72, 1913 /C1/914. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 525 /C1/27 and 535, 1990. Elliptic Lambda Function ThelGROUP is the SUBGROUP of the GAMMA GROUP with aanddODD;bandcEVEN . The function l(t)/C13l(q)/C13k2(q)/C30q4 2(0;q) q43(0;q); (1) where the NOME qis given by q/C13eipr(2) is al/-MODULAR FUNCTION defined on the UPPER HALF- PLANE andqi(z;q) are THETA FUNCTIONS . The lambda elliptic function is given by the Mathematica com- mandModularLambda [tau], and satisfies the func- tional equations l(t/C272)/C30l(t) (3) lt 2t/C271 ! /C30l(t): (4) /l/C31(r) gives the value of the MODULUS krfor which the complementary and normal complete ELLIPTIC INTE- GRALS OF THE FIRST KIND are related by K?(kr) K(kr)/C30ffiffiffirp: (5) It can be computed from l/C31(r)/C13k(q)/C30q2 2(q) q23(q); (6) where q/C13e/C28pffiffirp ; (7) andqiis a J ACOBI THETA FUNCTION . From the definition of the lambda function, l/C31(r?)/C30l/C311 r ! /C30l/C31?(r): (8) For all rational r,K(l/C31(r)) and E(l/C31(r)) are expres- sible in terms of a finite number of GAMMA FUNCTIONS (Selberg and Chowla 1967). l/C31(r) is related to the RAMANUJAN G- AND G-FUNCTIONS by l/C31(n)/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27G/C2812 nq /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28G /C2812 nq ;j1ffl;j1{ (9) l/C31(n)/C30g6 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g12 n/C27g/C2812 nq /C28g6 n;j1ffl;j1{ : (10) Special values arel/C31(2 29)/C3013ffiffiffiffiffiffi 58p /C2899;j1ffl;j1{ ffiffiffi 2p /C271;j1ffl;j1{6 l/C31(2 5)/C30ffiffiffiffiffiffi 10p /C283;j1ffl;j1{ ffiffiffi 2p /C271;j1ffl;j1{2 l/C31(2 3)/C302/C28ffiffiffi 3p;j1ffl;j1{ffiffiffi 2p /C27ffiffiffi 3p;j1ffl;j1{ l/C31(3 4)/C30ffiffiffi 3p /C28ffiffiffi 2p;j1ffl;j1{2ffiffiffi2p /C271;j1ffl;j1{ 2 l/C31(1)/C301ffiffiffi 2p l/C31(2)/C30ffiffiffi 2p /C281 l/C31(3)/C301 4ffiffiffi 2pffiffiffi3p /C281;j1ffl;j1{ l/C31(4)/C303/C282ffiffiffi 2p l/C31(5)/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5p /C281q /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3/C28ffiffiffi 5pq ;j1z;j1} l/C31(6)/C302/C28ffiffiffi 3p;j1ffl;j1{ffiffiffi3p /C28ffiffiffi 2p;j1ffl;j1{ l/C31(7)/C301 8ffiffiffi 2p 3/C28ffiffiffi 7p;j1ffl;j1{ l/C31(8)/C30ffiffiffi 2p /C271/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffi 2p /C272q;j1z;j1} 2 l/C31(9)/C301 2ffiffiffi 2p /C2831=4;j1ffl;j1{ ffiffiffi3p /C281;j1ffl;j1{ l/C31(10)/C30ffiffiffiffiffiffi 10p /C283;j1ffl;j1{ ffiffiffi 2p /C281;j1ffl;j1{ 2 l/C31(11)/C301 12ffiffiffi 6p /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C272x11/C284x/C281 11q /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11/C272x 11/C284x/C281 11q ;j1z;j1} l/C31(12)/C30ffiffiffi 3p /C28ffiffiffi2p;j1ffl;j1{ 2ffiffiffi2p /C281;j1ffl;j1{ 2 /C3015/C2810ffiffiffi 2p /C278ffiffiffi 3p /C286ffiffiffi6p l/C31(13)/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5ffiffiffiffiffiffi 13p /C2817q /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19/C285ffiffiffiffiffiffi 13pq ;j1z;j1} l/C31(14)/C30/C2811/C288ffiffiffi 2p /C282ffiffiffi2p /C272;j1ffl;j1{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C274ffiffiffi 2pq /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11/C278ffiffiffi 2pq 2/C272ffiffiffi2p /C27ffiffiffi2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C274ffiffiffi 2pq ;j1z;j1} l/C31(15)/C30 1 16ffiffiffi2p 3/C28ffiffiffi 5p;j1ffl;j1{ffiffiffi5p /C28ffiffiffi3p;j1ffl;j1{ 2/C28ffiffiffi3p;j1ffl;j1{ l/C31(16)/C30 (21=4/C281)2 (21=4/C271)2 l/C31(17)/C301 4ffiffiffi 2p (42/C2710ffiffiffiffiffiffi 17p /C2813ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C283/C27ffiffiffiffiffiffi17pq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C27ffiffiffiffiffiffi17pq /C283ffiffiffiffiffiffi 17pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C283 /C27ffiffiffiffiffiffi17pqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffiffiffiffi17pq /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C2838 /C2810ffiffiffiffiffiffi17p /C2713ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C283 /C27ffiffiffiffiffiffi17pqrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffiffiffiffi 17pq /C273ffiffiffiffiffiffi17pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C283 /C27ffiffiffiffiffiffi17pqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffiffiffiffi17pq ) l /C31(18) /C30ffiffiffi2p /C281;j1ffl;j1{ 3 2 /C28ffiffiffi 3p;j1ffl;j1{2 l/C31(22) /C30 3ffiffiffiffiffiffi 11p /C287ffiffiffi2p;j1ffl;j1{ 10 /C283ffiffiffiffiffiffi11p;j1ffl;j1{ l /C31(30) /C30ffiffiffi3p /C28ffiffiffi 2p;j1ffl;j1{ 2 2 /C28ffiffiffi 3p;j1ffl;j1{ffiffiffi6p /C28ffiffiffi5p;j1ffl;j1{ 4 /C28ffiffiffiffiffiffi15p;j1ffl;j1{ l /C31(34) /C30ffiffiffi 2p /C281;j1ffl;j1{ 2 3ffiffiffi2p /C28ffiffiffiffiffiffi 17p;j1ffl;j1{ /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 297 /C2772ffiffiffiffiffiffi 17pq /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 296 /C2772ffiffiffiffiffiffi 17pq ;j1z;j1} l/C31(42) /C30ffiffiffi 2p /C281;j1ffl;j1{ 2 2 /C28ffiffiffi 3p;j1ffl;j1{2 ffiffiffi7p /C28ffiffiffi6p;j1ffl;j1{ 8 /C283ffiffiffi7p;j1ffl;j1{ l /C31(58) /C30 13ffiffiffiffiffiffi58p /C2899;j1ffl;j1{ ffiffiffi 2p /C281;j1ffl;j1{ 6 l /C31(210) /C30ffiffiffi2p /C281;j1ffl;j1{ 2 2 /C28ffiffiffi 3p;j1ffl;j1{ffiffiffi7p /C28ffiffiffi6p;j1ffl;j1{ 2 8 /C283ffiffiffi7p;j1ffl;j1{ /C29ffiffiffiffiffiffi10p /C283;j1ffl;j1{ 2 4 /C28ffiffiffiffiffiffi15p;j1ffl;j1{ 2 ffiffiffiffiffiffi15p /C28ffiffiffiffiffiffi14p;j1ffl;j1{ 6 /C28ffiffiffiffiffiffi35p;j1ffl;j1{ ; where x 11 /C13 17 /C273ffiffiffiffiffiffi33p;j1ffl;j1{ 1 =3 : In addition, l /C31(1?) /C301ffiffiffi 2p l /C31(2?) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffi 2p /C282p l /C31(3?) /C301 4ffiffiffi 2pffiffiffi 3p /C271;jr;j1 l/C31(4?) /C3021 =4 2ffiffiffi 2p /C282;jr;j1 l/C31(5?) /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5p /C281p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C28ffiffiffi 5pp;j1ffl;j1{ l /C31(7?) /C301 8ffiffiffi 2p 3 /C27ffiffiffi7p;jr;j1 l /C31(9?) /C301 2ffiffiffi 2p /C2731 =4;jr;j1 ffiffiffi 3p /C281;jr;j1 l /C31(12?) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28208 /C27147ffiffiffi 2p /C28120ffiffiffi 3p /C2785ffiffiffi6p p : See also D EDEKIND ETA FUNCTION ,ELLIPTIC ALPHA FUNCTION ,ELLIPTIC INTEGRAL OF THE FIRST KIND, JACOBI THETA FUNCTIONS ,KLEIN’S ABSOLUTE INVAR- IANT,M ODULAR FUNCTION ,M ODULUS (ELLIPTIC IN- TEGRAL ), RAMANUJAN G- AND G-FUNCTIONS References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.Bowman, F. Introduction to Elliptic Functions, with Appli- cations. New York: Dover, pp. 75, 95, and 98, 1961. Selberg, A. and Chowla, S. "On Epstein’s Zeta-Function." J. reine angew. Math. 227,86/C1/10, 1967. Watson, G. N. "Some Singular Moduli (1)." Quart. J. Math. 3,81/C1/8, 1932. Elliptic Logarithm A generalization of integrals OF THE FORM gx /C12dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 /C27 atp ; which can be expressed in terms of logarithmic and inverse trigonometric functions to eln(x) /C13g/C12 xdtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t3 /C27 at2 /C27 btp : The inverse of the elliptic logarithm is the ELLIPTIC EXPONENTIAL FUNCTION . Elliptic Modular Function MODULAR FUNCTION Elliptic Modulus MODULUS (ELLIPTIC INTEGRAL ) Elliptic Nome NOME Elliptic Paraboloid A QUADRATIC SURFACE which has ELLIPTICAL CROSS SECTION . The elliptic paraboloid of height h, SEMIMA- JOR AXIS a, and SEMIMINOR AXIS bcan be specified parametrically by x/C30affiffiffiupcosv y/C30bffiffiffiupsinv z/C30u: forv/C23[0;2p) and u/C23[0;h]: / See also ELLIPTIC CONE,ELLIPTIC CYLINDER ,PARA- BOLOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987. Fischer, G. (Ed.). Plate 66 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 61, 1986. JavaView. "Classic Surfaces from Differential Geometry: Elliptic Paraboloid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Elliptic-Paraboloid.html. Elliptic Partial Differential Equation A second-order PARTIAL DIFFERENTIAL EQUATION , i.e., one OF THE FORM Auxx /C272Buxy /C27Cuyy /C27Dux /C27Euy /C27F /C300; (1) is called elliptic if the MATRIX Z /C13AB BC;j2r;j21 (2) is POSITIVE DEFINITE . Elliptic partial differential equations have applications in almost all areas of mathematics, from harmonic analysis to geometry to Lie theory, as well as numerous applications in physics. As with a general PDE, elliptic PDE mayhave non-constant coefficients and be non-linear. Despite this variety, the elliptic equations have a well-developed theory. The basic example of an elliptic partial differential equation is L APLACE’S EQUATION 92u /C300 (3) in n-dimensional Euclidean space, where the L APLA- CIAN 92 is defined by 92 /C30Xn i/C301@2 @x2 i: Other examples of elliptic equations include the nonhomogeneous P OISSON’S EQUATION 92u /C30f(x) (4) and the non-linear minimal surface equation. For an elliptic partial differential equation, BOUND- ARY CONDITIONS are used to give the constraint u(x; y) /C30g(x ; y)on @V; where uxx /C27uyy /C30f(ux ; uy ; u; x; y) (5) holds in V:/ One property of constant coefficient elliptic equations is that their solutions can be studied using the FOURIER TRANSFORM . Consider P OISSON’S EQUATION with periodic f(x): The F OURIER SERIES expansion is then given by /C28 zjj2 ˆu( z) /C30ˆf( z) ; (6) where zjj2 is called the "principal symbol," and so wecan solve for u. Except for z /C300 ; the multiplier is nonzero. In general, a PDE may have non-constant coefficients or even be non-linear. A linear PDE is elliptic if its principal symbol, as in the theory of PSEUDODIFFER- ENTIAL OPERATORS , is nonzero away from the origin. For instance, (3) has as its principal symbol zjj4 ; which is non-zero for zjj"0; and is an elliptic PDE. A nonlinear PDE is elliptic at a solution u if its linearization is elliptic at u. One simply calls a non- linear equation elliptic if it is elliptic at any solution, such as in the case of harmonic maps between Riemannian manifolds. See also HARMONIC FUNCTION ,H ARMONIC MAP, HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION ,LA- PLACE’S EQUATION ,M INIMAL SURFACE ,P ARABOLIC PARTIAL DIFFERENTIAL EQUATION ,PARTIAL DIFFER- ENTIAL EQUATION ,PSEUDODIFFERENTIAL OPERATOR Elliptic Plane The REAL PROJECTIVE PLANE with elliptic METRIC where the distance between two points P and Q is defined as the RADIAN ANGLE between the projection of the points on the surface of a SPHERE (which is tangent to the plane at a point S) from the ANTIPODE N of the tangent point. References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 94, 1969. Elliptic Point A point p on a REGULAR SURFACE M /C23R3 is said to be elliptic if the G AUSSIAN CURVATURE K(p)>0o r equivalently, the PRINCIPAL CURVATURES k1andk2 have the same sign. See also ANTICLASTIC ,ELLIPTIC FIXED POINT (DIFFER- ENTIAL EQUATIONS ), ELLIPTIC FIXED POINT (MAP), GAUSSIAN CURVATURE ,H YPERBOLIC POINT ,P ARA- BOLIC POINT ,PLANAR POINT ,SYNCLASTIC References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 375, 1997. Elliptic Pseudoprime Let E be an ELLIPTIC CURVE defined over the FIELD of RATIONAL NUMBERS Qffiffiffiffiffiffiffi /C28dp;j1ffl;j1{ having equation y2 /C30x3 /C27ax /C27b with a and b INTEGERS . Let P be a point on E with integer coordinates and having infinite order in the additive group of rational points of E, and let n be a COMPOSITE NATURAL NUMBER such that (/C28d=n) /C30/C281; where (/C28d=n) is the JACOBI SYMBOL . Then if (n /C271)P /C130 (mod n) ; n is called an elliptic pseudoprime for (E, P). See also ATKIN- GOLDWASSER- KILIAN- MORAIN CERTI- FICATE ,ELLIPTIC CURVE PRIMALITY PROVING ,STRONG ELLIPTIC PSEUDOPRIME References Balasubramanian, R. and Murty, M. R. "Elliptic Pseudo- primes. II." In Se´minaire de The´orie des Nombres, Paris 1988 /C1/989 (Ed. C. Goldstein). Boston, MA: Birkha ¨user, pp. 13 /C1/5, 1990. Gordon, D. M. "The Number of Elliptic Pseudoprimes." Math. Comput. 52, 231 /C1/45, 1989. Gordon, D. M. "Pseudoprimes on Elliptic Curves." In Num- ber Theory--The ´orie des nombres: Proceedings of the International Number Theory Conference Held at Univer- site´ Laval in 1987 (Ed. J. M. DeKoninck and C. Lev- esque). Berlin: de Gruyter, pp. 290 /C1/05, 1989. Miyamoto, I. and Murty, M. R. "Elliptic Pseudoprimes." Math. Comput. 53, 415 /C1/30, 1989. Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 132 /C1/34, 1996. Elliptic Rotation The transformation x?/C30x cos u /C28y sin u y?/C30x sin u /C27y sin u which leaves the CIRCLE x2 /C27y2 /C301 invariant. See also EQUIAFFINITY Elliptic Theta Function JACOBI THETA FUNCTIONS ,N EVILLE THETA FUNC- TIONSElliptic Torus A SURFACE OF REVOLUTION which is generalization of the RING TORUS . It is produced by rotating an ELLIPSE in the xz-plane about the z-axis, and is given by the PARAMETRIC EQUATIONS x(u ; v) /C30(a /C27b cos v) cos u y(u; v) /C30(a /C27b cos v) sin u z(u ; v) /C30c sin v: See also RING TORUS ,S URFACE OF REVOLUTION , TORUS References Gray, A. "Tori." §11.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 210 and 304 /C1/05, 1997. Elliptic Umbilic Catastrophe A CATASTROPHE which can occur for three control factors and two behavior axes. The elliptical umbilic is catastrophe of codimension 3 that has the equation F(x;y;u;v;w)/C30x3=3/C28xy2/C27w(x2/C27y2)/C28ux/C28vy:/ See also CATASTROPHE THEORY ,HYPERBOLIC UMBILIC CATASTROPHE References Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. Elliptical Projection MOLLWEIDE PROJECTION Elliptic-Cylinder Coordinates ELLIPTIC CYLINDRICAL COORDINATES EllipticE ELLIPTIC INTEGRAL OF THE SECOND KIND # 1999 /C1/001 Wolfram Research, Inc. EllipticExp ELLIPTIC EXPONENTIAL FUNCTION # 1999 /C1/001 Wolfram Research, Inc. EllipticExpPrime ELLIPTIC EXPONENTIAL FUNCTION # 1999 /C1/001 Wolfram Research, Inc. EllipticF ELLIPTIC INTEGRAL OF THE FIRST KIND # 1999 /C1/001 Wolfram Research, Inc. Ellipticity Given a SPHEROID with equatorial radius a and polar radius c, e /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28 c2 a2s a > c (oblate spheroid) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28 a2 a2s : a Bc (prolate spheroid)8 >>>>< >>>>: See also F LATTENING ,O BLATE SPHEROID ,PROLATE SPHEROID ,SPHEROID EllipticK ELLIPTIC INTEGRAL OF THE FIRST KIND # 1999 /C1/001 Wolfram Research, Inc. EllipticLog ELLIPTIC LOGARITHM EllipticNomeQ NOME # 1999 /C1/001 Wolfram Research, Inc. EllipticPi ELLIPTIC INTEGRAL OF THE THIRD KIND # 1999 /C1/001 Wolfram Research, Inc.EllipticTheta JACOBI THETA FUNCTIONS # 1999 /C1/001 Wolfram Research, Inc. EllipticThetaPrime JACOBI THETA FUNCTIONS # 1999 /C1/001 Wolfram Research, Inc. Ellison-Mende `s-France Constant Qffiffiffiffiffiffiffi /C28dp;j1ffl;j1{ where e :K g/C285=7 pg/C272 =7is the EULER- MASCHERONI CONSTANT , and (/C28d=n) /C30/C281 is the Ellision-Mende `s-France constant (given incor- rectly by Le Lionnais 1983). References Ellison, W. J. and Mende `s-France, M. Les nombres pre- miers. Paris: Hermann, 1975. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 47, 1983. Elongated Cupola A n-gonal CUPOLA adjoined to a 2n/-gonal PRISM . See also ELONGATED PENTAGONAL CUPOLA ,E LON- GATED SQUARE CUPOLA ,E LONGATED TRIANGULAR CUPOLA Elongated Dipyramid ELONGATED PENTAGONAL DIPYRAMID ,E LONGATED SQUARE DIPYRAMID ,ELONGATED TRIANGULAR DIPYR- AMID Elongated Dodecahedron ASPACE-FILLING POLYHEDRON and PARALLELOHE- DRON . References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29 /C1/0 and 257, 1973. Elongated Gyrobicupola ELONGATED PENTAGONAL GYROBICUPOLA ,E LON- GATED SQUARE GYROBICUPOLA ,ELONGATED TRIANGU- LARGYROBICUPOLA Elongated Gyrocupolarotunda ELONGATED PENTAGONAL GYROCUPOLAROTUNDA Elongated Orthobicupola ELONGATED PENTAGONAL ORTHOBICUPOLA ,E LON- GATED TRIANGULAR ORTHOBICUPOLA Elongated Orthobirotunda ELONGATED PENTAGONAL ORTHOBIROTUNDA Elongated Orthocupolarotunda ELONGATED PENTAGONAL ORTHOCUPOLAROTUNDA Elongated Pentagonal Cupola JOHNSON SOLID J20:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Dipyramid JOHNSON SOLID J16:/References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Gyrobicupola JOHNSON SOLID J39:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Gyrobirotunda JOHNSON SOLID J43:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Gyrocupolarotunda JOHNSON SOLID J41:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Orthobicupola JOHNSON SOLID J38:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Orthobirotunda JOHNSON SOLID J42:/References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Orthocupolarotunda JOHNSON SOLID J40:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Pyramid JOHNSON SOLID J9:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Pentagonal Rotunda A PENTAGONAL ROTUNDA adjoined to a decagonal PRISM which is JOHNSON SOLID J21 :/ Elongated Pyramid An n-gonal PYRAMID adjoined to an n-gonal PRISM . See also ELONGATED PENTAGONAL PYRAMID ,ELON- GATED SQUARE PYRAMID ,E LONGATED TRIANGULAR PYRAMID ,GYROELONGATED PYRAMID Elongated Rotunda ELONGATED PENTAGONAL ROTUNDA Elongated Square Cupola JOHNSON SOLID J19 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Elongated Square Dipyramid JOHNSON SOLID J15 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Square Gyrobicupola A nonuniform POLYHEDRON obtained by rotating the bottom third of a SMALL RHOMBICUBOCTAHEDRON (Ball and Coxeter 1987, p. 137). It is also called Miller’s solid, the Miller-askinuze solid, or the pseu- dorhombicuboctahedron, and is JOHNSON SOLID J37 :/ Although some writers have suggested that the elongated square gyrobicupola should be considered a fourteenth ARCHIMEDEAN SOLID , its twist allows vertices "near the equator" and those "in the polar regions" to be distinguished. Therefore, it is not a true Archimedean like the SMALL RHOMBICUBOCTAHE- DRON , whose vertices cannot be distinguished (Crom- well 1997, pp. 91 /C1/2). See also ARCHIMEDEAN SOLID ,JOHNSON SOLID , SMALL RHOMBICUBOCTAHEDRON References Askinuze, V. G. "O cisle polupravil’nyh mnogogrannikov." Math. Prosvesc. 1, 107/C1/18, 1957. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 137 /C1/38, 1987. Coxeter, H. S. M. "The Polytopes with Regular-Prismatic Vertex Figures." Phil. Trans. Roy. Soc. 229, 330/C1/25, 1930. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 91 /C1/2, 1997. Miller, J. C. P. "Polyhedron." Encyclopædia Britannica, 11th ed. Elongated Square Pyramid JOHNSON SOLID J8:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Triangular Cupola JOHNSON SOLID J18:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Elongated Triangular Dipyramid JOHNSON SOLID J14:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Triangular Gyrobicupola JOHNSON SOLID J36:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Triangular Orthobicupola JOHNSON SOLID J35:/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elongated Triangular Pyramid JOHNSON SOLID J7 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Elsasser Function E(y; u) /C13g1=2 /C281 =2exp /C282pyu sinh(2 py) cosh(2 py) /C28 cos(2 px)"# dx: Embeddable Knot A KNOT K is an n-embeddable knot if it can be placed on a GENUS n standard embedded surface without crossings, but K cannot be placed on any standardly embedded surface of lower GENUS without crossings. Any KNOT is an n-embeddable knot for some n. The FIGURE-OF-EIGHT KNOT is a 2-EMBEDDABLE KNOT .A knot with BRIDGE NUMBER b is an n-embeddable knot where n 5b :/ See also EMBEDDABLE SURFACE ,TUNNEL NUMBEREmbeddable Surface EMBEDDED SURFACE Embedded Surface A SURFACE S is n-embeddable if it can be placed in Rn /-space without self-intersections, but cannot be similarly placed in any Rk for k Bn. A surface so embedded is said to be an embedded surface. The COSTA MINIMAL SURFACE is embeddable in R3 ; but the KLEIN BOTTLE is not (the commonly depicted R3 representation requires the surface to pass through itself). There is particular interest in surfaces which are minimal, complete, and embedded. See also EMBEDDABLE KNOT,MINIMAL SURFACE References Collin, P. "Topologie et courbure des surfaces minimales proprement plonge ´es de R3 :/" Ann. Math. 145,1/C1/1, 1997. Hoffman, D. and Karcher, H. "Complete Embedded Minimal Surfaces of Finite Total Curvature." In Minimal Surfaces (Ed. R. Osserman). Berlin: Springer-Verlag, pp. 267 /C1/72, 1997. Nikolaos, K. "Complete Embedded Minimal Surfaces of Finite Total Curvature." J. Diff. Geom. 47,96/C1/69, 1997. Pe´rez, J. and Ros, A. "The Space of Properly Embedded Minimal Surfaces with Finite Total Curvature." Indiana Univ. Math. J. 45, 177 /C1/04, 1996. Ros, A. "Compactness of Spaces of Properly Embedded Minimal Surfaces with Finite Total Curvature." Indiana Univ. Math. J. 44, 139 /C1/52, 1995. Embedding An embedding is a representation of a topological object, MANIFOLD , GRAPH , FIELD , etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. For example, a FIELD embedding preserves the algebraic structure of plus and times, an embedding of a TOPOLOGICAL SPACE preserves OPEN SETS , and a GRAPH EMBEDDING pre- serves connectivity. One space X is embedded in another space Y when the properties of Y restricted to X are the same as the properties of X. For example, the rationals are embedded in the reals, and the integers are embedded in the rationals. In geometry, the sphere is embedded inR3as the unit sphere. See also CAMPBELL’S THEOREM ,EMBEDDABLE KNOT, EMBEDDED SURFACE ,EXTRINSIC CURVATURE ,FIELD, GRAPH EMBEDDING ,H YPERBOLOID EMBEDDING ,IN- JECTION ,M ANIFOLD ,N ASH’S EMBEDDING THEOREM , SPHERE EMBEDDING ,SUBMANIFOLD Emden Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION (x2y?)?/C27x2yn /C300: See also MODIFIED EMDEN DIFFERENTIAL EQUATION References Leach, P. G. L. "First Integrals for the Modified Emden Equation ¨q /C27 a(t)˙q /C27qn /C300:/" J. Math. Phys. 26, 2510 /C1/514, 1985. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Emden-Fowler Differential Equation The ORDINARY DIFFERENTIAL EQUATION (xpy?) ?9xsyn /C300 : References Bellman, R. Ch. 7 in Stability Theory of Differential Equa- tions. New York: McGraw-Hill, 1953. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Emden-Fowler Equation The ORDINARY DIFFERENTIAL EQUATION References Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995. Emirp A PRIME whose REVERSAL is also prime, but which is not a PALINDROMIC PRIME . The first few are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, ... (Sloane’s A006567). See also PALINDROMIC PRIME ,REVERSAL References Gardner, M. The Magic Numbers of Dr Matrix. Buffalo, NY: Prometheus, p. 230, 1985. Rivera, C. "Problems & Puzzles: Puzzle Reversible Primes.- 020." http://www.primepuzzles.net/puzzles/puzz_020.htm. Sloane, N. J. A. Sequences A006567/M4887 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Empty Graph An empty graph on n nodes consists of n isolated nodes with no edges. The empty graph on 0 nodes is called the NULL GRAPH . The empty graph on n vertices is the complement of the COMPLETE GRAPH Kn :/ See also COMPLETE GRAPH ,GRAPH ,NULL GRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 141, 1990. Empty Set The SET containing no elements, denoted ¥: Stran- gely, the empty set is both OPEN and CLOSED for any SET X and TOPOLOGY . A GROUPOID , SEMIGROUP , QUASIGROUP , RINGOID , and SEMIRING can be empty. MONOIDS , GROUPS , and RINGS must have at least one element, while DIVISION RINGS and FIELDS must have at least two elements. See also SET,URELEMENT References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 266, 1996. e-Multiperfect Number A number n is called a ke-perfect number if/ se ðnÞ¼kn /, where se(n) is the SUM of the E-DIVISORS of n. See also E-DIVISOR , E-PERFECT NUMBER References Guy, R. K. "Exponential-Perfect Numbers." §B17 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 73, 1994. Enantiomer Two objects which are MIRROR IMAGES of each other are called enantiomers. The term enantiomer is synonymous with ENANTIOMORPH . See also AMPHICHIRAL KNOT,CHIRAL ,DISSYMMETRIC , HANDEDNESS ,MIRROR IMAGE ,REFLEXIBLE References Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 130 /C1/61, 1987. Enantiomorph ENANTIOMER Enantiomorphous Of opposite symmetry under reflection; MIRROR IMAGES . See also DISSYMMETRIC ,ENANTIOMER ,MIRROR IMAGE Encoding An encoding is a way of representing a number or expression in terms of another (usually simpler) one. However, multiple expressions can also be encoded as a single expression, as in, for example, (a ; b) /C131 2[(a /C27b)2 /C273a /C27b] which encodes a and b uniquely as a single number. ab (a, b) 00 0 01 1 10 2 02 3 11 4 20 5 See also CODE,CODING THEORY ,HUFFMAN CODING , PRU¨ FER CODE,RUN-LENGTH ENCODING Encroaching List Set A structure consisting of an ordered set of sorted lists such that the head and tail entries of later lists nest within earlier ones. For example, an encroaching list set for f6; 7; 1; 8; 2; 5; 9; 3; 4g is given by ff1; 6; 7; 8; 9g;f2; 5g;f3; 4gg: Encroaching list sets can be computed usingEncroachingListSet [l] in the Mathematica add-on package Discrete- Math‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). It is conjectured that the number of encroaching lists associated with a RANDOM PERMUTATION of size n is /C2ffiffiffiffiffiffi 2np for sufficiently large n (Skiena 1988; Skiena 1990, p. 78). References Skiena, S. "Encroaching Lists as a Measure if Presorted- ness." BIT 28, 775 /C1/84, 1988.Skiena, S. "Encroaching List Sets." §2.3.7 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 75 /C1/6, 1990. Endogenous Variable An economic variable which is independent of the relationships determining the equilibrium levels, but nonetheless affects the equilibrium. See also EXOGENOUS VARIABLE References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 458, 1980. Endomorphism A SURJECTIVE MORPHISM from an object to itself. The term derives from the Greek adverb ondon (endon ) "inside" and mor8 vsi& (morphosis ) "to form" or "to shape." In ERGODIC THEORY , let X be a SET, F a SIGMA ALGEBRA on X and m a PROBABILITY MEASURE .A MAP T : X 0 X is called an endomorphism or MEA- SURE-PRESERVING TRANSFORMATION if 1. T is SURJECTIVE , 2. T is MEASURABLE , 3. m(T /C281A) /C30m(A) for all A /C23 F :/ An endomorphism is called ERGODIC if it is true that T/C281A/C30A IMPLIES m(A)/C300 or 1, where T/C281A/C30fx/C23X:T(x)/C23Ag:/ See also MEASURABLE FUNCTION ,M EASURE- PRESER- VING TRANSFORMATION ,M ORPHISM ,SIGMA ALGEBRA , SURJECTIVE Endoscopy References Arthur, J. "Stability and Endoscopy: Informal Motivation." InRepresentation Theory and Automorphic Forms: Papers from the Instructional Conference Held in Edinburgh, March 17 /C1/9, 1996 (Ed. T. N. Bailey and Knapp, A. W.). Providence, RI: Amer. Math. Soc., pp. 433 /C1/42, 1997. Hales, T. "On the Fundamental Lemma for Standard Endoscopy: Reduction to Unit Elements." Canad. J. Math. 47, 974/C1/94, 1995. Endpoint A node of a GRAPH of degree 1 (left figure; Harary 1994, p. 15), or, a POINT at the boundary of LINE SEGMENT or CLOSED INTERVAL (right figure). See also CLOSED INTERVAL ,INTERVAL ,ISOLATED POINT ,LINE SEGMENT ,POINT ,ROOT NODE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Endrass Octic Endraß surfaces are a pair of OCTIC SURFACES which have 168 ORDINARY DOUBLE POINTS . This is the maximum number known to exist for an OCTIC SUR- FACE , although the rigorous upper bound is 174. The equations of the surfaces X 9 8are 64(x2 /C28w2)(y2 /C28w2)[(x /C27y)2 /C282w2] [(x /C28y)2 /C282w2] /C28f/C284(1 9ffiffiffi 2p )(x2 /C27y2)2 /C27[8(2 9ffiffiffi2p )z2 /C272(2 97ffiffiffi2p )w2](x2 /C27y2) /C2816z4 /C278(1 /C142ffiffiffi 2p )z2w2 /C28(1 /C2712ffiffiffi2p )w4 g2 /C300; where w is a parameter taken as w /C301 in the above plots. All ORDINARY DOUBLE POINTS of are real, while 24 of those in are complex. The surfaces were discovered in a 5-D family of octics with 112 nodes, and are invariant under the GROUP D8 /C156Z2 :/ See also ALGEBRAIC SURFACE ,OCTIC SURFACE References Endraß, S. "Octics with 168 Nodes." http://enriques.mathe- matik.uni-mainz.de/kon/docs/Eendrassoctic.shtml. Endraß, S. "Fla¨chen mit vielen Doppelpunkten." DMV- Mitteilungen 4,17/C1/0, 4/1995. Endraß, S. "A Proctive Surface of Degree Eight with 168 Nodes." J. Algebraic Geom. 6, 325 /C1/34, 1997.Energy The term energy has an important physical meaning in physics and is an extremely useful concept. A much more abstract mathematical generalization is defined as follows. Let V be a SPACE with MEASURE m ]0 and letF(P;Q) be a real function on the PRODUCT SPACE V/C29V:When (m;n)/C30ggF(P;Q)dm(Q)dn(P) /C30gF(P;m)dn(P) exists for measures m;n]0;(m;n) is called the MUTUAL ENERGY and ( m;m) is called the ENERGY . See also DIRICHLET ENERGY ,MUTUAL ENERGY References Iyanaga, S. and Kawada, Y. (Eds.). "General Potential." §335.B in Encyclopedic Dictionary of Mathematics. Cam- bridge, MA: MIT Press, p. 1038, 1980. En-Function The En(x) function is defined by the integral En(x)/C13g/C12 1e/C28xtdt tn(1) and is given by the Mathematica functionExpInte- gralE [n,x]. Defining t/C13h/C281so that dt/C30/C28h/C282dh; En(x)/C30g1 0e/C28x=hhh/C282dh (2) En(0)/C301 n/C281: (3) The function satisfies the RECURRENCE RELATIONS E?n(x)/C30/C28En/C281(x) (4) nEn/C271(x)/C30e/C28x/C28xEn(x): (5) Equation (4) can be derived from En(x) /C30g/C12 1e /C28tx tndt (6) E ?n(x) /C30d dx g/C12 1e /C28tx tndt /C30g/C12 1d dxe /C28tx tn ! dt /C30/C28g/C12 1te/C28tx tndt /C30/C28g/C12 1e /C28tx tn/C281dt /C30/C28En/C281(x) ; (7) and (5) using INTEGRATION BY PARTS , letting u /C301 tndv /C30e /C28tx dt (8) du /C30/C28n tn/C271dt v /C30/C28e /C28tx x (9) gives En(x) /C30g/C12 1udv/C30[uv] /C12 1/C28g/C12 1vdu /C30/C28e /C28tx xtn"#/C12 t/C301/C28n x g/C12 1e /C28tx tn/C271dt /C30 0 /C28/C28e /C28x x !"# /C28n x g/C12 1e /C28tx tn /C271dt /C30e /C28x x/C28n xEn/C271(x) : (10) Solving (10) for nEn/C271(x) then gives (5). An ASYMPTOTIC SERIES is given by (n /C281)!En(x) /C30(/C28x)n/C281E1(x) /C27e/C28xXn s/C300/C282(n /C28s /C282)!(/C28x)s ; (11) so En(x) /C30e /C28x x1 /C28n x /C27n(n /C27 1) x2/C27/C1/C1/C1"# : (12) The special case n /C301 gives E1(x) /C13/C28ei(/C28x) /C30g/C12 1e /C28txdt t/C30g/C12 xe /C28udu u; (13) where ei(x) is the EXPONENTIAL INTEGRAL , which is also equal to E1(x) /C30/C28g /C28ln x /C28X/C12 n/C301(/C281)nxn n!n; (14) where g is the EULER- MASCHERONI CONSTANT .E1(0) /C30/C12 (15) E1(ix) /C30/C28ci(x) /C27i si(x); (16) where ci(x) and si(x) are the COSINE INTEGRAL and SINE INTEGRAL . See also COSINE INTEGRAL , ET-FUNCTION ,EXPONEN- TIAL INTEGRAL ,GOMPERTZ CONSTANT ,SINE INTEGRAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Exponential Integral and Related Functions." Ch. 5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 227 /C1/33, 1972. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Exponential Integrals." §6.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 215 /C1/19, 1992. Spanier, J. and Oldham, K. B. "The Exponential Integral Ei(x) and Related Functions." Ch. 37 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 351 /C1/60, 1987. Engel’s Theorem A finite-dimensional LIE ALGEBRA all of whose ele- ments are ad-NILPOTENT is itself a NILPOTENT LIE ALGEBRA . Enlargement See also EXPANSION Enneacontagon A 90-sided POLYGON . The regular enneacontagon is CONSTRUCTIBLE . Enneacontahedron A ZONOHEDRON constructed from the 10 diameters of the DODECAHEDRON which has 90 faces, 30 of which are RHOMBS of one type and the other 60 of which are RHOMBS of another. The enneacontahedron somewhat resembles a figure of Sharp. See also DODECAHEDRON ,RHOMB ,ZONOHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 142 /C1/43, 1987. Sharp, A. Geometry Improv’d: 1. By a Large and Accurate Table of Segments of Circles, with Compendious Tables for Finding a True Proportional Part, Exemplify’d in Makingout Logarithms from them, there Being a Table of them forall Primes to 1100, True to 61 Figures. 2. A ConciseTreatise of Polyhedra, or Solid Bodies, of Many Bases. London: R. Mount, p. 87, 1717. Enneadecagon A 19-sided POLYGON , sometimes also called the ENNEAKAIDECAGON . Enneagon NONAGON Enneagonal Number NONAGONAL NUMBER Enneakaidecagon ENNEADECAGON Enneper’s Minimal Surface A self-intersecting MINIMAL SURFACE which can be generated using the E NNEPER- WEIERSTRASS PARAME- TERIZATION with f(z)/C301 (1) g(z)/C30z: (2) Letting z/C30reifand taking the REAL PART give x/C30Rreif/C281 3r3e3ifhi (3) /C30rcosf/C2813r3cos(3 f) (4) y/C30R[ireif/C271 3ir3e3if] (5) /C30/C2813r[3 sin f/C27r2sin(3f)] (6) z/C30R[r2e2if] (7) /C30r2cos(2 f); (8) where r/C23[0;1] and f/C23[/C28p;p):The coefficients of the FIRST FUNDAMENTAL FORM areE/C30/C282 cos(2 f) (9) F/C304rcosfsinf (10) G/C302r2cos(2 f); (11) the SECOND FUNDAMENTAL FORM coefficients are e/C30(1/C27r2)2(12) f/C300 (13) g/C30r2(1/C27r2)2; (14) and the G AUSSIAN and MEAN CURVATURES are K/C30/C284 (1/C27r2)4(15) H/C300: (16) Letting z/C30u/C27ivgives the figure above, with para- metrization x/C30u/C281 3u3/C27uv2(17) y/C30/C28v/C28u2v/C2713v3(18) z/C30u2/C28v2(19) (do Carmo 1986, Gray 1997, Nordstrand). In this parameterization, the coefficients of the FIRST FUNDA- MENTAL FORM are E/C30(1/C27u2/C27v2)2(20) F/C300 (21) G/C30(1/C27u2/C27v2)2; (22) the SECOND FUNDAMENTAL FORM coefficients are e/C30/C282 (23) f/C300 (24) g/C302; (25) the AREA ELEMENT is dA/C30(1/C27u2/C27v2)duffldv; (26) and the G AUSSIAN and MEAN CURVATURES are K/C30/C284 (1/C27u2/C27v2)4(27) H/C300: (28) Nordstrand gives the implicit form y2/C28x2 2z/C272 9z2/C2723 !3 /C286(y2 /C28 x2) 4z/C281 4(x2 /C27y2 /C2789 z2) /C2729"#2 /C300: (29) See also ENNEPER- WEIERSTRASS PARAMETERIZATION References Dickson, S. "Minimal Surfaces." Mathematica J. 1,38/C1/0, 1990. do Carmo, M. P. "Enneper’s Surface." §3.5C in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986. Enneper, A. "Analytisch-geometrische Untersuchungen." Z. Math. Phys. 9,96/C1/25, 1864. Gray, A. "Examples of Minimal Surfaces," "The Associated Family of Enneper’s Surface," and "Enneper’s Surface of Degree n." §30.2 and 31.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 358, 684 /C1/85, and 726 /C1/32, 1997. JavaView. "Classic Surfaces from Differential Geometry: Enneper." http://www-sfb288.math.tu-berlin.de/vgp/java- view/demo/surface/common/PaSurface_Enneper.html. Maeder, R. The Mathematica Programmer. San Diego, CA: Academic Press, pp. 150 /C1/51, 1994. Nordstrand, T. "Enneper’s Minimal Surface." http:// www.uib.no/people/nfytn/enntxt.htm. Osserman, R. A Survey of Minimal Surfaces. New York: Dover, p. 65, 87, and 143, 1986. Wolfram Research "Mathematica Version 2.0 Graphics Gallery." http://www.mathsource.com/cgi-bin/ msitem22?0207 /C1/55. Enneper’s Negative Curvature Surfaces The Enneper surfaces are a three-parameter family of surfaces with constant negative curvature (and non- constant MEAN CURVATURE ). In general, they are described by ELLIPTIC FUNCTIONS . However, a special case which can be specified parametrically using ELEMENTARY FUNCTIONS is the KUEN SURFACE . See also KUEN SURFACE References Enneper, A. "Analytisch-geometrische Untersuchungen." Nachr. Ko¨nigl. Gesell. Wissensch. Georg-Augustus-Univ. Go¨ttingen 12, 258 /C1/77, 1868. Fischer, G. (Ed.). Plate 92 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 88, 1986. Reckziegel, H. "Enneper’s Surfaces." §3.4.4 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 37 /C1/9, 1986. Enneper-Weierstrass Parameterization A parameterization of a MINIMAL SURFACE in terms of two functions f(z) and g(z)as x(r ; f) y(r ; f) z(r ; f)2 435/C30R gf(1/C28g2) if(1/C27g2) 2fg2435dz;where z/C30re ifandRis the REAL PART . Examples are given in the following table. Surface /f(z)// g(z)/ ENNEPER’S MINIMAL SURFACE 1 z HENNEBERG’S MINIMAL SURFACE /2(1/C28z/C284)/z BOUR’S MINIMAL SURFACE 1 /ffiffiffizp / TRINOID /(z3/C281)/C282 //z2/ See also BOUR’S MINIMAL SURFACE ,ENNEPER’S MINI- MAL SURFACE ,H ENNEBERG’S MINIMAL SURFACE , MINIMAL SURFACE ,TRINOID References Dickson, S. "Minimal Surfaces." Mathematica J. 1,3 8/C1/0, 1990. do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 41, 1986. Gray, A. "Minimal Surfaces via the Weierstrass Representa- tion." Ch. 32 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 735 /C1/60, 1997. Weierstrass, K. "U ¨ber die Fla ¨chen deren mittlere Kru ¨m- mung u ¨berall gleich null ist." Monatsber. Berliner Akad., 612/C1/25, 1866. Wolfram Research, Inc. "Minimal Surfaces a `la Weierstrass." http://library.wolfram.com/demos/WeierstrassSurfa- ces.nb. Enormous Theorem CLASSIFICATION THEOREM Enriques Surfaces An Enriques surface Xis a smooth compact complex surface having irregularity q(X)/C300 and nontrivial canonical sheaf KXsuch that K2 X/C30OX(Endraß). Such surfaces cannot be embedded in projective 3-space, but there nonetheless exist transformations onto singular surfaces in projective 3-space. There exists a family of such transformed surfaces of degree sixwhich passes through each edge of a TETRAHEDRON twice. A subfamily with tetrahedral symmetry isgiven by the two-parameter ( r, c) family of surfaces f rx0x1x2x3/C27c(x2 0x21x22/C27x20x21x23/C27x20x22x23/C27x21x22x23/C300 and the polynomial fris a sphere with radius r, fr/C30(3/C28r)(x20/C27x21/C27x22/C27x23) /C282(1/C27r)(x0x1/C27x0x2/C27x0x3/C27x1x2/C27x1x3/C27x2x3) (Endraß). References Angermu ¨ller, G. and Barth, W. "Elliptic Fibres on Enriques Surfaces." Compos. Math. 47, 317/C1/32, 1982. Barth, W. and Peters, C. "Automorphisms of Enriques Surfaces." Invent. Math. 73, 383 /C1/11, 1983. Barth, W. P.; Peters, C. A.; and van de Ven, A. A. Compact Complex Surfaces. New York: Springer-Verlag, 1984. Barth, W. "Lectures on K3- and Enriques Surfaces." In Algebraic Geometry, Sitges (Barcelona) 1983, Proceedings of a Conference Held in Sitges (Barcelona), Spain, October 5 /C1/2, 1983 (Ed. E. Casas-Alvero, G. E. Welters, and S. Xambo ´-Descamps). New York: Springer-Verlag, pp. 21 /C1/7, 1983. Endraß, S. "Enriques Surfaces." http://enriques.mathemati- k.uni-mainz.de/kon/docs/enriques.shtml. Enriques, F. Le superficie algebriche. Bologna, Italy: Zani- chelli, 1949. Enriques, F. "Sulla classificazione." Atti Accad. Naz. Lincei 5, 1914. Hunt, B. The Geometry of Some Special Arithmetic Quoti- ents. New York: Springer-Verlag, p. 317, 1996. Kim, Y. "Normal Quintic Enriques Surfaces." J. Korean Math. Soc. 36, 545 /C1/66, 1999. Entire Function If a COMPLEX FUNCTION is ANALYTIC at all finite points of the COMPLEX PLANE C ; then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112). See also ANALYTIC FUNCTION ,FINITE ORDER ,HADA- MARD FACTORIZATION THEOREM ,H OLOMORPHIC FUNCTION ,L IOUVILLE’S BOUNDEDNESS THEOREM , MEROMORPHIC FUNCTION ,WEIERSTRASS FACTOR THE- OREM References Knopp, K. "Entire Transcendental Functions." Ch. 9 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 112 /C1/16, 1996. Krantz, S. G. "Entire Functions and Liouville’s Theorem." §3.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 31 /C1/2, 1999. Entire Modular Form A MODULAR FORM which is not allowed to have poles in the UPPER HALF-PLANE H or at i /C12:/ See also MODULAR FORM Entringer Number The Entringer numbers E(n; k) are the number of PERMUTATIONS of f1; 2; ...; n /C271 g; starting with k /C27 1; which, after initially falling, alternately fall then rise. The Entringer numbers are given by E(0; 0) /C301 E(n; 0) /C300 together with the RECURRENCE RELATION E(n; k) /C30E(n; k /C271) /C27E(n /C281; n /C28k) : The numbers E(n) /C30E(n; n) are the SECANT and TANGENT NUMBERS given by the MACLAURIN SERIESsec x /C27tan x /C30A0 /C27A1x /C27A2x2 2! /C27A3x3 3! A4x4 4! /C27A5x5 5! /C27...: See also ALTERNATING PERMUTATION ,BOUSTROPHE- DON TRANSFORM ,EULER ZIGZAG NUMBER ,PERMUTA- TION ,SECANT NUMBER ,SEIDEL- ENTRINGER- ARNOLD TRIANGLE ,T ANGENT NUMBER ,Z AG NUMBER ,Z IG NUMBER References Bauslaugh, B. and Ruskey, F. "Generating Alternating Permutations Lexographically." BIT 80,17/C1/6, 1990. Entringer, R. C. "A Combinatorial Interpretation of the Euler and Bernoulli Numbers." Nieuw. Arch. Wisk. 14, 241 /C1/46, 1966. Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Operation on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76,44/C1/4, 1996. Poupard, C. "De nouvelles significations enumeratives des nombres d’Entringer." Disc. Math. 38, 265 /C1/71, 1982. Ruskey, F. "Information of Alternating Permutations." http://www.theory.csc.uvic.ca/~cos/inf/perm/Alterna- ting.html. Sloane, N. J. A. Sequences A000111/M1492 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Entropy In physics, the word entropy has important physical implications as the amount of "disorder" of a system. In mathematics, a more abstract definition is used. The (Shannon) entropy of a variable X is defined as H(X) /C13/C28X xp(x) ln[p(x)] ; where p(x) is the probability that X is in the state x, andplnpis defined as 0 if p/C300. The joint entropy of variables X1;...,Xnis then defined by H(X1;...;Xn) /C13/C28X x1/C1/C1/C1X xnp(x1;...;xn) ln[p(x1;...;xn)]: See also INFORMATION THEORY ,KOLMOGOROV EN- TROPY ,KOLMOGOROV- SINAI ENTROPY ,M AXIMUM EN- TROPY METHOD ,M ETRIC ENTROPY ,O RNSTEIN’S THEOREM ,REDUNDANCY ,RELATIVE ENTROPY ,SHAN- NON ENTROPY ,TOPOLOGICAL ENTROPY References Ellis, R. S. Entropy, Large Deviations, and Statistical Mechanics. New York: Springer-Verlag, 1985. Khinchin, A. I. Mathematical Foundations of Information Theory. New York: Dover, 1957. Lasota, A. and Mackey, M. C. Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd ed. New York: Springer-Verlag, 1994. Ott, E. "Entropies." §4.5 in Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 138 /C1/44, 1993. Rothstein, J. Science 114, 171, 1951. Schnakenberg, J. "Network Theory of Microscopic and Macroscopic Behavior of Master Equation Systems." Rev. Mod. Phys. 48, 571 /C1/85, 1976. Shannon, C. E. "A Mathematical Theory of Communication." The Bell System Technical J. 27, 379 /C1/23 and 623 /C1/56, July and Oct. 1948. http://cm.bell-labs.com/cm/ms/what/ shannonday/shannon1948.pdf. Shannon, C. E. and Weaver, W. Mathematical Theory of Communication. Urbana, IL: University of Illinois Press, 1963. Entscheidungsproblem DECISION PROBLEM Enumerable DENUMERABLE SET Enumerate A GENERATING FUNCTION F(x) /C30X nanxn is said to enumerate an (Hardy 1999, p. 85). See also GENERATING FUNCTION References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Enumeration Problem The problem of determining (or counting) the set of all solutions to a given problem. See also CLASSIFICATION ,COMBINATORICS ,EXISTENCE PROBLEM References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 22, 1984. Enumerative Geometry Schubert’s application of the CONSERVATION OF NUM- BER PRINCIPLE . See also CONSERVATION OF NUMBER PRINCIPLE , DUALITY PRINCIPLE ,H ILBERT’S PROBLEMS ,P ERMA- NENCE OF MATHEMATICAL RELATIONS PRINCIPLE References Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 340, 1945.Envelope The envelope of a one-parameter family of curves given implicitly by U(x; y; c) /C300; (1) or in parametric form by (f(t; c); g(t; c)); is a curve which touches every member of the family. For a curve represented by (f(t; c) ; g(t; c)) ; the envelope is found by solving 0 /C30@f @t@g @c /C28@f @c@g @t: (2) For a curve represented implicitly, the envelope is given by simultaneously solving @U @c/C300 (3) U(x;y;c)/C300: (4) See also ASTROID ,CARDIOID ,CATACAUSTIC ,CAUSTIC , CAYLEYIAN CURVE ,D U¨ RER’S CONCHOID ,E LLIPSE ENVELOPE ,E NVELOPE THEOREM ,E VOLUTE ,G LIS- SETTE ,HEDGEHOG ,KIEPERT’S PARABOLA ,LINDELOF’S THEOREM ,NEGATIVE PEDAL CURVE References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 33 /C1/4, 1972. Yates, R. C. "Envelopes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 75 /C1/0, 1952. Envelope (Form) Given a DIFFERENTIAL P-FORM qin the EXTERIOR ALGEBRA fflpV/C31;its envelope is the smallest SUBSPACE Wsuch that qis in the subspace fflpW/C31ƒfflpV/C31: Alternatively, Wis spanned by the vectors that can be written as the CONTRACTION ofqwith an element offflp/C281V:/ For example, the envelope of dxinV/C30R2isW/C30 /C142@=@x/C143;and the envelope of dx1ffldx2/C27dx3ffldx4in V/C30R4is all of V. Here is a Mathematica function which will compute the envelope of an ANTISYMMETRIC TENSOR . BBDiscreteMath‘Combinatorica‘; ContractAll[a_List, b_List] : /C30Module[{k /C30 TensorRank[a] - TensorRank[b]}, If[k /C21/C300, Map[Flatten[#1].Flatten[b] &, a, {k}], ContractAll[b, a] ] ] Envelope[a_List?VectorQ] : /C30Select[{a}, #1 ! /C30Table[0, {Length[a]}] &] Envelope[a_List] : /C30Module[ { z, inds, vects, d/C30Dimensions[a][[1]], r /C30TensorRank[a] }, z /C30 Table[0, ##1] & @@ Table[{d}, {r - 1}]; inds /C30 KSubsets[Range[d], r - 1]; vects /C30 Map[ContractAll[a, ReplacePart[z, 1, #1]] &, inds]; Select[RowReduce[vects], #1 ! /C30 Table[0, {d}] &] ] See also DECOMPOSABLE ,DIFFERENTIAL FORM,DIF- FERENTIAL IDEAL ,E XTERIOR ALGEBRA ,V ECTOR SPACE ,W EDGE PRODUCT Envelope Theorem Relates EVOLUTES to single paths in the CALCULUS OF VARIATIONS . Proved in the general case by Darboux and Zermelo (1894) and Kneser (1898). It states: "When a single parameter family of external paths from a fixed point O has an ENVELOPE , the integral from the fixed point to any point A on the ENVELOPE equals the integral from the fixed point to any second point B on the ENVELOPE plus the integral along the envelope to the first point on the ENVELOPE , JOA /C30JOB /C27JBA :/" References Kimball, W. S. Calculus of Variations by Parallel Displace- ment. London: Butterworth, p. 292, 1952. Envyfree An agreement in which all parties feel as if they have received the best deal. See also CAKE CUTTING References Robertson, J. and Webb, W. Cake Cutting Algorithms: Be Fair If You Can. Natick, MA: Peters, 1998. Stewart, I. "Mathematical Recreations." Sci. Amer. , p. 86, Jan. 1999. E-Operator SUMMATION BY PARTS e-Perfect Number A number n is called an e-perfect number if se(n) /C30 2n; where se(n) is the SUM of the E-DIVISORS of n.Ifm is SQUAREFREE , then se(m) /C30m: As a result, if n is e- perfect and m is SQUAREFREE with m /C222b ; then mn is e-perfect. The first few e-perfect numbers are 36, 180, 252, 396, 468, ... (Sloane’s A054979). There are no ODD e- perfect numbers. The first few primitive e-perfect numbers are 36, 1800, 2700, 17424, ... (Sloane’s A054980). See also E-DIVISORReferences Guy, R. K. "Exponential-Perfect Numbers." §B17 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 73, 1994. Sloane, N. J. A. Sequences A054979 and A054980 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Subbarao, M. V. and Suryanarayan, D. "Exponential Perfect and Unitary Perfect Numbers." Not. Amer. Math. Soc. 18, 798, 1971. Epicycloid The path traced out by a point Pon the edge of a CIRCLE ofRADIUS brolling on the outside of a CIRCLE ofRADIUS a. An epicycloid is therefore an EPITRO- CHOID with h/C30b. Epicycloids are given by the PARAMETRIC EQUATIONS x/C30(a/C27b) cos f/C28bcosa/C27b bf ! (1) y/C30(a/C27b) sin f/C28bsina/C27b bf ! : (2) A polar equation can be derived by computing x2/C30(a/C27b)2cos2f/C282b(a/C27b) cos fcosa/C27b bf ! /C27b2cos2a/C27b bf ! (3) y2 /C30(a /C27b)2 sin2 f /C282b(a /C27b) sin f sina /C27 b bf ! /C27b2 sin2a /C27 b bf ! ; (4) so r2 /C30x2 /C27y2 /C30(a /C27b)2 /C27b2 /C282b(a /C27b) /C2 cosa b /C271 ! f"# cos f /C27sina b /C271 ! f"# sin f() : (5) But cos a cos b /C27sin a sin b /C30cos(a /C28 b) ; (6) so r2 /C30(a /C27b)2 /C27b2 /C282b(a /C27b) cosa b /C271 ! f /C28 f"# /C30(a /C27b)2 /C27b2 /C282b(a /C27b) cosa bf ! : (7) Note that f is the parameter here, not the polar angle. The polar angle from the center is tan u /C30y x /C30(a /C27 b) sin f /C28 b sina /C27 b bf ! (a /C27 b)cos f /C28 b cosa /C27 b bf ! : (8) To get n CUSPS in the epicycloid, b /C30a=n; because then n rotations of b bring the point on the edge back to its starting position. r2 /C30a21 /C271 n !2 /C271 n !2 /C2821 n ! 1 /C271 n ! cos(nf)2 435 /C30a 2 1 /C272 n /C271 n2 /C271 n2 /C282 n ! n /C27 1 n ! cos(n f)"# /C30a2n2 /C27 2n /C27 2 n2/C282(n /C27 1) n2cos(nf)"# /C30a2 n2(n2 /C272n /C272) /C282(n /C271) cos(nf);j2;j3 ; (9) so tan u /C30an /C27 1 n ! sin f /C28a nsin[(n /C27 1)f] an /C27 1 n ! cos f /C28a ncos[(n /C27 1)f]/C30(n /C27 1) sin f /C28 sin[(n /C27 1)f] (n /C27 1) cos f /C28 cos[(n /C27 1)f] : (10) An epicycloid with one cusp is called a CARDIOID , one with two cusps is called a NEPHROID , and one with five cusps is called a RANUNCULOID . n-epicycloids can also be constructed by beginning with the DIAMETER of a CIRCLE , offsetting one end by a series of steps while at the same time offsetting the other end by steps n times as large. After traveling around the CIRCLE once, an n-cusped epicycloid is produced, as illustrated above (Madachy 1979). Epicycloids have TORSION t/C300 (11) and satisfy s2 a2/C27r2 b2/C301; (12) where ris the RADIUS OF CURVATURE (/1=k):/ See also CARDIOID ,CYCLIDE ,CYCLOID ,EPICYCLOID–1- CUSPED ,E PICYCLOID EVOLUTE ,E PICYCLOID INVO- LUTE ,EPICYCLOID PEDAL CURVE ,EPITROCHOID ,HY- POCYCLOID ,NEPHROID ,RANUNCULOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 217, 1987. Bogomolny, A. "Cycloids." http://www.cut-the-knot.com/ pythagoras/cycloids.html. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 160 /C1/64 and 169, 1972. Lemaire, J. Hypocycloı ¨des et epicycloı ¨des. Paris: Albert Blanchard, 1967. MacTutor History of Mathematics Archive. "Epicycloid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Epi- cycloid.html. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 219 /C1/25, 1979. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 50 /C1/2, 1991. Yates, R. C. "Epi- and Hypo-Cycloids." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Ed- wards, pp. 81 /C1/5, 1952. Epicycloid Evolute The EVOLUTE of the EPICYCLOID x/C30(a/C27b) cos t/C28bcosa/C27b b ! t"# y/C30(a/C27b) sin t/C28bsina/C27b b ! t"# is another EPICYCLOID given by x/C30a a/C272b(a/C27b) cos t/C27bcosa/C27b b ! t"# () y/C30a a/C272b(a/C27b) sin t/C27bcosa/C27b b ! t"# () : Epicycloid Involute The INVOLUTE of the EPICYCLOID x/C30(a/C27b) cos t/C28bcosa/C27b b ! t"# y/C30(a/C27b) sin t/C28bsina/C27b b ! t"#is another EPICYCLOID given by x/C30a/C272b a(a/C27b) cos t/C27bcosa/C27b b ! t"# () y/C30a/C272b a(a/C27b) sin t/C27bcosa/C27b b ! t"# () : Epicycloid Pedal Curve The PEDAL CURVE of an EPICYCLOID with PEDAL POINT at the center, shown for an epicycloid with four cusps, is not a ROSE as claimed by Lawrence (1972). References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 204, 1972. Epicycloid Radial Curve The RADIAL CURVE of an EPICYCLOID is shown above for an epicycloid with four cusps. It is not a ROSE ,a s claimed by Lawrence (1972). References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 202, 1972. Epicycloid1-Cusped A 1-cusped epicycloid has b /C30a,son /C301. The radius measured from the center of the large circle for a 1- cusped epicycloid is given by EPICYCLOID equation (9) with n /C301so r2 /C30a2 n2 [(n2 /C272n /C272) /C282(n /C271) cos (nf)] /C30a2[(12 /C272 /C215 1 /C272) /C282(1 /C271) cos(1 /C215 f)] /C30a2(5 /C284 cos f) (1) r /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C284 cos fp ; (2) and tan u /C302 sin f /C28 sin (2f) 2 cos f /C28 cos (2f) : (3) The 1-cusped epicycloid is just an offset CARDIOID . Epicycloid–2-Cusped NEPHROID Epimenides Paradox A version of the LIAR’S PARADOX , attributed to the philosopher Epimenides in the sixth century BC. "All Cretans are liars...One of their own poets has said so." This is not a true paradox since the poet may have knowledge that at least one Cretan is, in fact, honest, and so be lying when he says that all Cretans are liars. There therefore need be no self-contradiction in what could simply be a false statement by a person who is himself a liar. A sharper version of the paradox (which has no such loophole) is the EUBULIDES PARADOX , "This statement is false." See also EUBULIDES PARADOX ,L IAR’S PARADOX , SOCRATES’ PARADOX References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 5 /C1/, 1977.Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 58 /C1/0, 1998. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 115, 1998. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989. Prior, A. N. "Epimenides the Cretan." J. Symb. Logic 23, 261 /C1/66, 1958. Epimorphism A MORPHISM f : Y 0 X in a CATEGORY is an epimorph- ism if, for any two morphisms u; v : X 0 Z; uf /C30vf implies u/C30v. See also CATEGORY ,MORPHISM Epispiral A plane curve with polar equation r/C30asec(nu): There are nsections if nisODD and 2 nifnisEVEN . References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 /C1/93, 1972. Epispiral Inverse Curve The INVERSE CURVE of the EPISPIRAL r/C30asec(nu) with INVERSION CENTER at the origin and inversion radius k is the ROSE r /C30k cos(nu) a: See also EPISPIRAL ,INVERSE CURVE ,ROSE Epitrochoid The ROULETTE traced by a point P attached to a CIRCLE of radius b rolling around the outside of a fixed CIRCLE of radius a. These curves were studied by Du¨rer (1525), Desargues (1640), Huygens (1679), Leibniz, Newton (1686), L’Hospital (1690), Jakob Bernoulli (1690), la Hire (1694), Johann Bernoulli (1695), Daniel Bernoulli (1725), Euler (1745, 1781). An epitrochoid appears in Du¨rer’s work Instruction in Measurement with Compasses and Straight Edge (1525). He called epitrochoids SPIDER LINES because the lines he used to construct the curves looked like a spider. The PARAMETRIC EQUATIONS for an epitrochoid are x /C30(a /C27b) cos t /C28h cosa /C27 b bt ! y /C30(a /C27b) sin t /C28h sina /C27 b bt ! ; where h is the distance from P to the center of the rolling CIRCLE . Special cases include the LIMAC ¸ ON with a /C30b, the CIRCLE with a /C300, and the EPICYCLOID with h /C30b. See also EPICYCLOID ,H YPOTROCHOID ,SPIROGRAPH , TROCHOID References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 168 /C1/70, 1972.Epitrochoid Evolute The PARAMETRIC EQUATIONS of the EVOLUTE of an EPITROCHOID specified by circle radii a and b with offset h are x /C30ah(a /C27 b)c1(t) cos t /C27 bc2(t) cos(a /C27 b)t b"# b3 /C27 (a /C27 b)h2 /C28 b(a /C27 2b)h cosat b ! (1) y /C30ah(a /C27 b)c1(t) sin t /C27 bc2(t) sin(a /C27 b)t b"# b3 /C27 (a /C27 b)h2 /C28 b(a /C27 2b)h cosat b ! ; (2) where c1(t) /C13h /C28b cosat b ! (3) c2(t) /C13b /C28h cosat b ! : (4) See also EPITROCHOID ,EVOLUTE Epsilon In mathematics, a small POSITIVE INFINITESIMAL quantity, usually denoted e or o ; whose LIMIT is usually taken as e 0 0:/ The late mathematician P. Erdos also used the term "epsilons" to refer to children (Hoffman 1998, p. 4). See also EPSILON CONJECTURE ,W YNN’S EPSILON METHOD References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Epsilon Conjecture The conjecture that Frey’s ELLIPTIC CURVE was not modular. The conjecture was quickly proved by Ribet (RIBET’S THEOREM ) in 1986, and was an important step in the proof of FERMAT’S LAST THEOREM and the TANIYAMA- SHIMURA CONJECTURE . See also FERMAT’S LAST THEOREM ,RIBET’S THEOREM , TANIYAMA- SHIMURA CONJECTURE Epsilon-Delta Definition CONTINUOUS FUNCTION ,LIMIT Epsilon-Neighborhood NEIGHBORHOOD Epstein Zeta Function Z g h;j12;j12;j12;j12;j12;j12;j12;j12(q;s) /C30X 1e /C282pih /C215 1 [q(1 /C27 g)]s=2 ; where g and h are arbitrary VECTORS , the SUM runs over a d-dimensional LATTICE , and 1 /C30/C28g is omitted if g is a lattice VECTOR . See also ZETA FUNCTION References Glasser, M. L. and Zucker, I. J. "Lattice Sums in Theoretical Chemistry." In Theoretical Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring). New York: Academic Press, pp. 69 /C1/0, 1980. Shanks, D. "Calculation and Applications of Epstein Zeta Functions." Math. Comput. 29, 271 /C1/87, 1975. Equal Two quantities are said to be equal if they are, in some WELL DEFINED sense, equivalent. Equality of quantities a and b is written a /C30b. Equal is im- plemented in Mathematica as Equal [A, B, ...], or A /C30/C30 B /C30/C30 .... A symbol with three horizontal line segments (//C13) resembling the equals sign is used to denote both equality by definition (e.g., A /C13B means A is DEFINED to be equal to B) and CONGRUENCE (e.g., 13 /C13 1 (mod 12) means 13 divided by 12 leaves a REMAIN- DERof 1–a fact known to all readers of analog clocks). See also CONGRUENCE ,DEFINED ,DIFFERENT ,EQUAL BY DEFINITION ,EQUALITY ,EQUIVALENT ,ISOMORPH- ISM,UNEQUAL Equal by Definition DEFINEDEqual Detour Point The center of an outer S ODDY CIRCLE . It has TRIANGLE CENTER FUNCTION a/C301/C272D a(b/C27c/C28a)/C30sec(1 2A) cos(12B) cos(12C)/C271: Given a point Ynot between Aand B, a detour of length ½AY½/C27½YB½/C28½AB½ is made walking from AtoBviaY, the point is of equal detour if the three detours from one side to another via Yare equal. If ABC has no ANGLE />2 sin/C281(4=5);then the point given by the above TRILINEAR COORDINATES is the unique equal detour point. Otherwise, the ISOPERIMETRIC POINT is also equal detour. References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Kimberling, C. "Isoperimetric Point and Equal Detour Point." http://cedar.evansville.edu/~ck6/tcenters/recent/ isoper.html. Veldkamp, G. R. "The Isoperimetric Point and the Point(s) of Equal Detour." Amer. Math. Monthly 92, 546/C1/58, 1985. Equal Incircles Theorem INCIRCLE Equal Parallelians Point The point of intersection of the three LINE SEGMENTS , each parallel to one side of a TRIANGLE and touching the other two, such that all three segments are of the same length. The TRILINEAR COORDINATES are bc(ca/C27ab/C28bc):ca(ab/C27bc/C28ca):ab(bc/C27ca/C28ab): References Kimberling, C. "Equal Parallelians Point." http://cedar.e- vansville.edu/~ck6/tcenters/recent/eqparal.html. Equal-Area Projection AMAP PROJECTION in which areas on a sphere, and the areas of any features contained on it, are mappedto the plane in such a way that two are related by a constant scaling factor. No projection can be both equal-area and CONFORMAL , and projections which are neither equal-area nor CONFORMAL are sometimes called APHYLACTIC (Snyder 1987, p. 4). Equal-area projections are also called EQUIVALENT ,HOMOLO- GRAPHIC ,HOMALOGRAPHIC ,AUTHALIC ,o r EQUIAREAL (Lee 1944; Snyder 1987, p. 4). See also ALBERS EQUAL- AREA CONIC PROJECTION , APHYLACTIC PROJECTION ,B EHRMANN CYLINDRICAL EQUAL- AREA PROJECTION ,CONFORMAL PROJECTION , CYLINDRICAL EQUAL- AREA PROJECTION ,EQUIDISTANT PROJECTION ,H AMMER- AITOFF EQUAL- AREA PROJEC- TION ,LAMBERT AZIMUTHAL EQUAL- AREA PROJECTION , MAP PROJECTION References Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190 /C1/00, 1944. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987. Equality A mathematical statement of the equivalence of two quantities. The equality "A is equal to B" is written A /C30B. See also EQUAL ,FORMULA ,INEQUALITY Equally Likely Outcomes Distribution Let there be a set S with N elements, each of them having the same probability. Then P(S) /C30P @N i/C301Ei;j1z;j1} /C30XN i/C301P(Ei) /C30P(Ei)XN i/C3011 /C30NP(Ei) : Using P(S) /C131 gives P(Ei) /C301 N: See also UNIFORM DISTRIBUTION Equation A mathematical expression stating that two or more quantities are the same as one another, also called an EQUALITY , FORMULA ,or IDENTITY . See also EQUALITY ,FORMULA ,IDENTITY ,INEQUATION Equiaffinity An AREA -preserving AFFINITY . Equiaffinities include the CROSSED HYPERBOLIC ROTATION , ELLIPTIC ROTA- TION , HYPERBOLIC ROTATION , and PARABOLIC ROTA- TION . Equiangular Polygon A POLYGON whose vertex angles are equal (Williams 1979, p. 32).See also EQUILATERAL POLYGON ,POLYGON ,REGULAR POLYGON References Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Equiangular Spiral LOGARITHMIC SPIRAL Equianharmonic Case The case of the WEIERSTRASS ELLIPTIC FUNCTION with invariants g2 /C300 and g3 /C301:/ See also LEMNISCATE CASE,PSEUDOLEMNISCATE CASE References Abramowitz, M. and Stegun, C. A. (Eds.). "Equianharmonic Case ( /g2/C300;g3/C301):/"§18.13 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 652, 1972. Equiareal Projection EQUAL- AREAPROJECTION Equi-Brocard Center The point Yfor which the TRIANGLES BYC ,CYA , and AYB have equal B ROCARD ANGLES . References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Equichordal Point A point pfor which all the CHORDS of a curve C passing through pare of the same length. In other words, pis an equichordal point if, for every chord [x;y] of length pof the curve C,psatisfies ½x/C28p½/C27½y/C28p½/C30p: A function r(u) satisfying r(0)/C30p/C28r(p) corresponds to a curve with equichordal point (0, 0) and chord length pdefined by letting r(u) be the polar equation of the half-curve for 0 5u5pand then superimposing the polar equation r(u)/C28pover the same range. The curves illustrated above correspond to polar equations OF THE FORM r( u) /C30x /C27(1 2 /C28x) cos(2 u) for various values of x. Although it long remained an outstanding problem (the EQUICHORDAL POINT PROBLEM ), it is now known that a plane convex region can have two equichordal points. See also CHORD ,E QUICHORDAL POINT PROBLEM , EQUIPRODUCT POINT ,EQUIRECIPROCAL POINT References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 9, 1991. Dirac, G. A. "Ovals with Equichordal Points." J. London Math. Soc. 27, 429 /C1/37, 1952. Dirac, G. A. J. London Math. Soc. 28, 245, 1953. Hallstrom, A. P. "Equichordal and Equireciprocal Points." Bogasici Univ. J. Sci. 2,83/C1/8, 1974. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 152, 1999. Zindler, K. "Uuml;ber konvexe Gebilde, II." Monatshefte f. Math. u. Phys. 3,25/C1/9, 1921. Equichordal Point Problem Is there a plane CONVEX SET having two distinct EQUICHORDAL POINTS ? The problem was first pro- posed by Fujiwara (1916) and Blaschke et al. (1917), but long defied solution. Rogers went so far as to remark, "If you are interested in studying the problem, my first advice is: ‘Don’t"’ (Croft et al. 1991, p. 9). This advice to the contrary, the problem was recently solved by Rychlik (1997). See also EQUICHORDAL POINT References Blaschke, W.; Rothe, W.; and Weitzenbo ¨ck, R. "Aufgabe 552." Arch. Math. Phys. 27, 82, 1917. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "The Equichor- dal Point Problem." §A1 in Unsolved Problems in Geome- try. New York: Springer-Verlag, pp. 9 /C1/1, 1991. Fujiwara, M. "U¨ ber die Mittelkurve zweier geschlossenen konvexen Kurven in Bezug auf einen Punkt." Toˆhoku Math. J. 10,99/C1/03, 1916. Rychlik, M. "The Equichordal Point Problem." Elec. Res. Announcements Amer. Math. Soc. 2, 108 /C1/23, 1996. Rychlik, M. "A Complete Solution to the Equichordal Problem of Fujiwara, Blaschke, Rothe, and Weitzenbo ¨ck." Invent. Math. 129, 141 /C1/12, 1997. Wirsing, E. "Zur Analytisita ¨t von Doppelspeichkurven." Arch. Math. 9, 300 /C1/07, 1958. Equicross RANGES and PENCILS which have equal CROSS-RATIOS are said to be equicross. See also CROSS- RATIO,PENCIL ,R ANGE (LINE SEG- MENT )References Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 74 /C1/6, 1928. Lachlan, R. §422 /C1/28 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 269 /C1/74, 1893. Equidecomposable The ability of two plane or space regions to be DISSECTED into each other. Equidigital Number A number n is called equidigital if the number of digits in the prime factorization of n (including powers) uses the same number of digits as the number of digits in n. The first few equidigital numbers are 1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, ... (Sloane’s A046758). See also ECONOMICAL NUMBER ,W ASTEFUL NUMBER References Pinch, R. G. E. "Economical Numbers." http://www.chalce- don.demon.co.uk/publish.html#62. Santos, B. R. "Problem 2204. Equidigital Representation." J. Recr. Math. 27,58/C1/9, 1995. Sloane, N. J. A. Sequences A046758 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Equidistance Postulate PARALLEL lines are everywhere equidistant. This POSTULATE is equivalent to the PARALLEL AXIOM . References Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 54, 1990. Equidistant Projection A MAP PROJECTION in which the distances between one or two points and every other point on the map differ from the corresponding distances on the sphere by only a constant scaling factor (Snyder 1987, p. 4). See also AZIMUTHAL EQUIDISTANT PROJECTION ,CON- FORMAL PROJECTION ,C ONIC EQUIDISTANT PROJEC- TION ,CYLINDRICAL EQUIDISTANT PROJECTION ,EQUAL- AREA PROJECTION ,EQUIDISTANT PROJECTION ,M ILL- ER EQUIDISTANT PROJECTION References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987. Equidistributed Sequence A sequence of REAL NUMBERS fxn g is equidistributed if the probability of finding xnin any subinterval is proportional to the subinterval length. Consider the distribution of the FRACTIONAL PARTS of nr in the intervals bounded by 0, 1 =n; 2=n; ..., /(n /C281)=n; 1. In particular, the number of empty intervals for n /C301, 2, ..., are given below for E, the EULER- MASCHERONI CONSTANT g ; the GOLDEN RATIO f; and PI. r Sloane # Empty Intervals for n /C301, 2, ..., e Sloane’s A0364120, 0, 0, 0, 1, 0, 0, 1, 1, 3, 1, 4, 4, 7, 5, ... / g/ Sloane’sA0461570, 0, 0, 1, 0, 0, 0, 1, 2, 2, 3, 0, 3, 5, 3, ... / f/ Sloane’s A0364140, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, ... / p/ Sloane’sA0364160, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 4, 5, 7, 7, ... The values of n for which no bins are left blank are given in the following table. r Sloane n with no empty intervals e Sloane’sA0364131, 2, 3, 4, 6, 7, 32, 35, 39, 71, 465, 536, 1001, ... / g/ Sloane’sA0461581, 2, 3, 5, 6, 7, 12, 19, 26, 97, 123, 149, 272, 395, ... / f/ Sloane’s A0364151, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89, 144, ... /p/Sloane’sA0364171, 6, 7, 106, 112, 113, 33102,33215, ... Steinhaus (1983) remarks that the highly uniformdistribution of frac( nf) has its roots in the form of the CONTINUED FRACTION forf:/See also PISOT- VIJAYARAGHAVAN CONSTANT ,UNIFORM DISTRIBUTION ,W EYL’S CRITERION References Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, 1974. Po´lya, G. and Szego, G. Problems and Theorems in Analysis I.New York: Springer-Verlag, p. 88, 1972. Sloane, N. J. A. Sequences A036412, A036413, A036414, A036415, A036416, A036417, A046157, and A046158 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 155 /C1 /56, 1991. Equilateral Hyperbola RECTANGULAR HYPERBOLA Equilateral Polygon APOLYGON whose side are equal (Williams 1979, pp. 31 /C1/2). See also EQUIANGULAR POLYGON ,EQUILATERAL TRI- ANGLE ,POLYGON ,REGULAR POLYGON References Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Equilateral Triangle An equilateral triangle is a TRIANGLE with all three sides of equal length a. An equilateral triangle also has three equal 60 8ANGLES . The ALTITUDE hof an equilateral triangle is h/C301 2ffiffiffi 3p a; (1) where ais the side length, so the AREA is A/C301 2ah/C3014ffiffiffi 3p a2: (2) The INRADIUS r, CIRCUMRADIUS R, and AREA A can be computed directly from the formulas for a general REGULAR POLYGON with side length a and n /C303 sides, r /C301 2 a cotp 3 ! /C301 2 a tanp 6 ! /C3016ffiffiffi 3p a (3) R /C301 2 a cscp 3 ! /C3012 a secp 6 ! /C3013ffiffiffi 3p a (4) A /C301 4 na2 cotp 3 ! /C301 4ffiffiffi 3p a2 : (5) The AREAS of the INCIRCLE and CIRCUMCIRCLE are Ar /C30 pr2 /C301 12 pa2 (6) AR /C30 pR2 /C301 3 pa2 : (7) GEOMETRIC CONSTRUCTION of an equilateral consists of drawing a diameter of a circle OPOand then constructing its perpendicular bisector P3OB: Bisect OB in point D, and extend the line P1P2 through D. The resulting figure P1P2P3is then an equilateral triangle. An equilateral triangle may also be con- structed (although not using the usual Greek rules, which do not permit angle trisection) by TRISECTING all three ANGLES of any TRIANGLE (MORLEY’S THEO- REM). NAPOLEON’S THEOREM states that if three equilateral triangles are drawn on the LEGS of any TRIANGLE (either all drawn inwards or outwards) and the centers of these triangles are connected, the result is another equilateral triangle. Given the distances of a point from the three corners of an equilateral triangle, a, b, and c, the length of a side s is given by3(a4 /C27b4 /C27c4 /C27s4) /C30(a2 /C27b2 /C27c2 /C27s2)2 (8) (Gardner 1977, pp. 56 /C1/7 and 63). There are infinitely many solutions for which a, b, and c are INTEGERS .In these cases, one of a, b, c, and s is DIVISIBLE by 3, one by 5, one by 7, and one by 8 (Guy 1994, p. 183). Begin with an arbitrary TRIANGLE and find the EXCENTRAL TRIANGLE . Then find the EXCENTRAL TRIANGLE of that triangle, and so on. Then the resulting triangle approaches an equilateral triangle. The only RATIONAL TRIANGLE is the equilateral triangle (Conway and Guy 1996). A POLYHEDRON composed of only equilateral triangles is known as a DELTAHEDRON . Let any RECTANGLE be circumscribed about an EQUI- LATERAL TRIANGLE . Then X /C27Y /C30Z; (9) where X, Y, and Z are the AREAS of the triangles in the figure (Honsberger 1985). The smallest equilateral triangle which can be in- scribed in a UNIT SQUARE (left figure) has side length and area s/C301 (10) A/C301 4ffiffiffi 3p :0:4330 : (11) The largest equilateral triangle which can be in- scribed (right figure) is oriented at an angle of 15 8and has side length and area s/C30sec (15/C14)/C30ffiffiffi 6p /C28ffiffiffi2p (12) A/C302ffiffiffi3p /C283:0:4641 (13) (Madachy 1979). See also A CUTE TRIANGLE ,D ELTAHEDRON ,EQUILIC QUADRILATERAL ,FERMAT POINTS ,G YROELONGATED SQUARE DIPYRAMID ,ICOSAHEDRON ,ISOSCELES TRIAN- GLE,MORLEY’S THEOREM ,OCTAHEDRON ,PENTAGONAL DIPYRAMID ,REULEAUX TRIANGLE ,RIGHT TRIANGLE , SCALENE TRIANGLE ,S NUB DISPHENOID ,T ETRAHE- DRON ,TRIANGLE ,TRIANGLE PACKING ,TRIANGULAR DIPYRAMID ,TRIAUGMENTED TRIANGULAR PRISM ,VI- VIANI’S THEOREM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987. Conway, J. H. and Guy, R. K. "The Only Rational Triangle." In The Book of Numbers. New York: Springer-Verlag, pp. 201 and 228 /C1/39, 1996. Dixon, R. Mathographics. New York: Dover, p. 33, 1991. Fukagawa, H. and Pedoe, D. "Circles and Equilateral Triangles." §2.1 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 23 /C1/5 and 100 /C1/02, 1989. Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, 1977. Guy, R. K. "Rational Distances from the Corners of a Square." §D19 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 181 /C1/85, 1994. Honsberger, R. "Equilateral Triangles." Ch. 3 in Mathema- tical Gems I. Washington, DC: Math. Assoc. Amer., 1973. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 19 /C1/1, 1985. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 115 and 129 /C1/31, 1979. Equilateral Triangle Packing TRIANGLE PACKING Equilibrium Point An equilibrium point in GAME THEORY is a set of strategies fˆx1 ; ...; ˆxn g such that the ith payoff function Ki(x) is larger or equal for any other ith strategy, i.e., Ki(ˆx1 ; ...; ˆxn) ]Ki(ˆx1 ; ...; ˆxi /C281 ; xi ; ˆxi /C271 ; ...; ˆxn) : NASH EQUILIBRIUM Equilic Quadrilateral A QUADRILATERAL in which a pair of opposite sides have the same length and are inclined at 608 to each other (or equivalently, satisfy /C142A/C143/C27/C142B/C143/C30120/C14): Some interesting theorems hold for such quadrilat- erals. Let ABCD be an equilic quadrilateral with AD /C30BC and /C142A/C143/C27/C142B /C143/C30120/C14: Then1. The MIDPOINTS P, Q, and R of the diagonals and the side CD always determine an EQUILATERAL TRIANGLE . 2. If EQUILATERAL TRIANGLE PCD is drawn out- wardly on CD, then DPAB is also an EQUILATERAL TRIANGLE . 3. If EQUILATERAL TRIANGLES are drawn on AC, DC, and DB away from AB, then the three new VERTICES P, Q, and R are COLLINEAR . See Honsberger (1985) for additional theorems. References Garfunkel, J. "The Equilic Quadrilateral." Pi Mu Epsilon J. 7, 317 /C1/29, 1981. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 32 /C1/5, 1985. Equinumerous Let A and B be two classes of POSITIVE INTEGERS . Let A(n) be the number of integers in A which are less than or equal to n, and let B(n) be the number of integers in B which are less than or equal to n. Then if A(n) /C2B(n) ; A and B are said to be equinumerous. The four classes of PRIMES 8k /C271 ; 8k /C273; 8k /C275 ; 8k /C27 7 are equinumerous. Similarly, since 8k /C271 and 8k /C27 5 are both of the form 4 k/C271;and 8 k/C273 and 8 k/C277 are both OF THE FORM 4k/C273;4k/C271 and 4 k/C273 are also equinumerous. See also BERTRAND’S POSTULATE ,CHOQUET THEORY , PRIME COUNTING FUNCTION References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 21 /C1/2 and 31 /C1/2, 1993. Equipollent Two statements in LOGIC are said to be equipollent if they are deducible from each other. Two sets AandBare said to be equipollent IFFthere is a one-to-one function (i.e., a BIJECTION ) from Aonto B(Moore 1982, p. 10; Rubin 1967, p. 67; Suppes 1972, p. 91).The term equipotent is sometimes used instead of equipollent. References Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Equipotent EQUIPOLLENT Equipotential Curve A curve in 2-D on which the value of a function f(x; y) is a constant. Other synonymous terms are ISARITHM and ISOPLETH . A plot of several equipotential curves is called a CONTOUR PLOT . See also CONTOUR PLOT,LEMNISCATE Equiproduct Point A point, such as interior points of a disk, such that (px)(py) /C30[const] ; where p is the CHORD length. See also EQUICHORDAL POINT ,E QUIRECIPROCAL POINT Equireciprocal Point p is an equireciprocal point if, for every chord [x; y]of a curve C, p satisfies ½x /C28p½/C281 /C27½y /C28p ½/C281 /C30c for some constant c. The FOCI of an ELLIPSE are equichordal points. See also EQUICHORDAL POINT ,EQUIPRODUCT POINT References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 10, 1991. Falconer, K. J. "On the Equireciprocal Point Problem." Geom. Dedicata 14, 113 /C1/26, 1983. Hallstrom, A. P. "Equichordal and Equireciprocal Points." Bogasici Univ. J. Sci. 2,83/C1/8, 1974. Klee, V. "Can a Plane Convex Body have Two Equireciprocal Points?" Amer. Math. Monthly 76,54/C1/5, 1969. Klee, V. "Correction to ‘Can a Plane Convex Body have Two Equireciprocal Points?"’ Amer. Math. Monthly 78, 114, 1971. Equirectangular Projection A CYLINDRICAL EQUIDISTANT PROJECTION , also called a RECTANGULAR PROJECTION , PLANE CHART , PLATECARRE ,or UNPROJECTED MAP, in which the horizontal coordinate is the longitude and the vertical coordinate is the latitude, so the standard parallel is taken as f1 /C300:/ See also CYLINDRICAL EQUIDISTANT PROJECTION Equiripple A distribution of ERROR such that the ERROR remain- ing is always given approximately by the last term dropped. Equitangential Curve TRACTRIX Equivalence BICONDITIONAL ,EQUIVALENT Equivalence Class An equivalence class is defined as a SUBSET OF THE FORM fx /C23 X : xRa g; where a is an element of X and the NOTATION "xRy" is used to mean that there is an EQUIVALENCE RELATION between x and y. It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of X. For all a ; b /C23 X ; we have aRb IFF a and b belong to the same equivalence class. A set of CLASS REPRESENTATIVES is a SUBSET of X which contains EXACTLY ONE element from each equivalence class. For n a POSITIVE INTEGER , and a, b INTEGERS , consider the CONGRUENCE a /C13b (mod n); then the equivalence classes are the sets f...;/C282n;/C28n;0;n;2n;...g; f...;1/C282n;1/C28 n;1;1/C27n;1/C272n;...getc. The standard CLASS RE- PRESENTATIVES are taken to be 0, 1, 2, ..., n/C281:/ See also CONGRUENCE ,COSET References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 56 /C1/7, 1993. Equivalence Moves REIDEMEISTER MOVES Equivalence Problem METRIC EQUIVALENCE PROBLEM Equivalence Relation An equivalence relation on a set Xis a SUBSET ofX/C29 X;i.e., a collection Rof ordered pairs of elements of X, satisfying certain properties. Write " xRy" to mean ( x, y) is an element of R, and we say " xis related to y," then the properties are 1. Reflexive: aRa for all a /C23 X ;/ 2. Symmetric: aRb IMPLIES bRa for all a ; b /C23 X/ 3. Transitive: aRb and bRc imply aRc for all a; b; c /C23 X ;/ where these three properties are completely indepen- dent. Other notations are often used to indicate a relation, e.g., a /C13b or a /C2b :/ See also EQUIVALENCE CLASS ,TEICHMU ¨ LLER SPACE References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 18, 1990. Stewart, I. and Tall, D. The Foundations of Mathematics. Oxford, England: Oxford University Press, 1977. Equivalent If A [B and B [A (i.e, A [B fflB [A; where [ denotes IMPLIES ), then A and B are said to be equivalent, a relationship which is written symboli- cally as A /C13B (Carnap 1958, p. 8), A UB ; or A XB: Equivalence is implemented in Mathematica as Equal [A, B, ...]. Binary equivalence has the following TRUTH TABLE (Carnap 1958, p. 10). AB /A /C13B/ TTT TFF FTFFFT Similarly, ternary equivalence has the following TRUTH TABLE . ABC /A /C13B /C13C/ TTTTTTFF TFTF TFFFFTTFFTFF FFTF FFFT The opposite of being equivalent is being NONEQUI- VALENT .Note that the symbol /C13is confusingly used in at least two other different contexts. If A and B are "equiva- lent by definition" (i.e., A is DEFINED to be B), this is written A /C13B ; and "a is CONGRUENT to b modulo m" is written a /C13b (mod m) :/ See also BICONDITIONAL ,CONNECTIVE ,DEFINED ,IFF, IMPLIES ,NONEQUIVALENT References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 8, 1958. Equivalent Matrix Two matrices A and B are equal to each other, written A /C30B ; if they have the same dimensions m /C29n and the same elements aij /C30bij for i /C301, ..., n and j /C301, ..., m. Gradshteyn and Ryzhik (2000) call an m /C29n MATRIX A "equivalent" to another m /C29n MATRIX B IFF B/C30PAQ forPandQany suitable nonsingular m/C29nandn/C29n MATRICES , respectively. See also MATRIX References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1103, 2000. Equivalent Projection EQUAL- AREAPROJECTION Eratosthenes Sieve An ALGORITHM for making tables of PRIMES . Sequen- tially write down the INTEGERS from 2 to the highest number nyou wish to include in the table. Cross out all numbers >2 which are divisible by 2 (every second number). Find the smallest remaining number >2:It is 3. So cross out all numbers >3 which are divisible by 3 (every third number). Find the smallest remaining number >3/. It is 5. So cross out all numbers > 5 which are divisible by 5 (every fifth number). Continue until you have crossed out all numbers divisible byffiffiffinpbc ; where xbcis the FLOOR FUNCTION . The numbers remaining are PRIME . This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out PRIMES up toffiffiffiffiffiffi 50p;j4;j| /C307: If the procedure is then continued up to n, then the number of cross-outs gives the number of distinct PRIME FACTORS of each number. References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 127 /C1/30, 1996. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 20 /C1/1, 1996. Erdos Number The number of "hops" needed to connect the author of a paper with the prolific late mathematician Paul Erdos. An author’s Erdos number is 1 if he has co- authored a paper with Erdos, 2 if he has co-authored a paper with someone who has co-authored a paper with Erdos, etc. (Hoffman 1998, p. 13). References de Castro, R. and Grossman, J. W. "Famous Trails to Paul Erdos." Math. Intell. 21,51/C1/3, 1999. Grossman, J. and Ion, P. "The Erdos Number Project." http:// www.acs.oakland.edu/~grossman/erdoshp.html. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Lewandowski, J.; Nurowski, P.; and Abramowicz, M. A. "Erdos Number Updates." Math. Intell. 22, 3, 2000. Erdos Reciprocal Sum Constants A-SEQUENCE , B2-SEQUENCE ,N ONAVERAGING SE- QUENCE Erdos Squarefree Conjecture The CENTRAL BINOMIAL COEFFICIENT2n n;jr;j1 is never SQUAREFREE for n /C214. This was proved true for all sufficiently large n by SA´ RKOZY’S THEOREM . Goetghe- luck (1988) proved the CONJECTURE true for 4 Bn 5 242205184 and Vardi (1991) for 4 Bn B2774840978 : The conjecture was proved true in its entirety by Gran- ville and Ramare (1996). See also CENTRAL BINOMIAL COEFFICIENT References Erdos, P. and Graham, R. L. Old and New Problems and Results in Combinatorial Number Theory. Geneva, Swit- zerland: L’Enseignement Mathe ´matique Universite ´ de Gene`ve, Vol. 28, p. 71, 1980. Goetgheluck, P. "Prime Divisors of Binomial Coefficients." Math. Comput. 51, 325 /C1/29, 1988. Granville, A. and Ramare, O. "Explicit Bounds on Exponen- tial Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43,73/C1/07, 1996.Sander, J. W. "On Prime Divisors of Binomial Coefficients." Bull. London Math. Soc. 24, 140 /C1/42, 1992. Sander, J. W. "A Story of Binomial Coefficients and Primes." Amer. Math. Monthly 102, 802 /C1/07, 1995. Sa´rkozy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20,70/C1/0, 1985. Vardi, I. "Applications to Binomial Coefficients." Computa- tional Recreations in Mathematica. Reading, MA: Addi- son-Wesley, pp. 25 /C1/8, 1991. Erdos-Anning Theorem If an infinite number of points in the PLANE are all separated by INTEGER distances, then all the points lie on a straight LINE. Erdos-Heilbronn Conjecture Erdos and Heilbronn (Erdos and Graham 1980) posed the problem of estimating from below the number of sums a /C27b where a /C23 A and b /C23 B range over given sets A; B ⁄Z=pZ of residues modulo a prime p, so that a "b : Dias da Silva and Hamidoune (1994) gave a solution, and Alon et al. (1995) developed a poly- nomial method that allows one to handle restrictions of the type f(a ; b) "0; where f is a polynomial in two variables over Z=pZ:/ References Alon, N.; Nathanson, M. B.; and Ruzsa, I. Z. "Adding Distinct Congruence Classes Modulo a Prime." Amer. Math. Monthly 102, 250 /C1/55, 1995. Dias da Silva, J. A. and Hamidoune, Y. O. "Cyclic Spaces for Grassmann Derivatives and Additive Theory." Bull. Lon- don Math. Soc. 26, 140 /C1/46, 1994. Erdos, P. and Graham, R. L. Old and New Problems and Results in Combinatorial Number Theory. Geneva, Swit- zerland: L’Enseignement Mathe ´matique Universite ´ de Gene`ve, Vol. 28, 1980. Lev, V. F. "Restricted Set Addition in Groups, II. A General- ization of the Erdos-Heilbronn Conjecture.." Electronic J. Combinatorics 7, No. 1, R4, 1 /C1/0, 2000. http://www.combi- natorics.org/Volume_7/v7i1toc.html. Erdos-Ivic Conjecture There are infinitely many primes m which divide some value of the PARTITION FUNCTION P. See also NEWMAN’S CONJECTURE ,PARTITION FUNC- TION P References Erdos, P. and Ivic, A. "The Distribution of Certain Arithme- tical Functions at Consecutive Integers." In Proc. Buda- pest Conf. Number Th., Coll. Math. Soc. J. Bolyai 51,4 5/C1/ 1, 1989. Ono, K. "Distribution of the Partition Functions Modulo m." Ann. Math. 151, 293/C1/07, 2000. Erdos-Kac Theorem A deeper result than the H ARDY- RAMANUJAN THEO- REM. Let N(x;a;b) be the number of INTEGERS in [3;x] such that inequality a 5v(n) /C28 ln ln nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ln np 5b holds, where v(n) is the number of DISTINCT PRIME FACTORS of n. Then lim x0/C12N(x; a ; b) /C30(x /C27 o(x))ffiffiffiffiffiffi 2ppgb ae /C28t2 =2 dt: The theorem is discussed in Kac (1959). See also DISTINCT PRIME FACTORS References Kac, M. Statistical Independence in Probability, Analysis and Number Theory. New York: Wiley, 1959. Riesel, H. "The Erdos-Kac Theorem." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 158 /C1/59, 1994. Erdos-Mordell Theorem If O is any point inside a TRIANGLE /DABC /, and P, Q, and R are the feet of the perpendiculars from O upon the respective sides BC, CA, and AB, then OA /C27OB /C27OC ]2(OP /C27OQ /C27OR) : Oppenheim (1961) and Mordell (1962) also showed that OA /C29OB /C29OC ](OQ /C27OR)(OR /C27OP)(OP /C27OQ) : References Bankoff, L. "An Elementary Proof of the Erdos-Mordell Theorem." Amer. Math. Monthly 65, 521, 1958. Brabant, H. "The Erdos-Mordell Inequality Again." Nieuw Tijdschr. Wisk. 46, 87, 1958/1959. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, 6th ed. Dublin: Hodges, Figgis, & Co., p. 253, 1892. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 9, 1969. Erdos, P. "Problem 3740." Amer. Math. Monthly 42, 396, 1935. Fejes-To ´th, L. Lagerungen in der Ebene auf der Kugel und im Raum. Berlin: Springer, 1953. Mordell, L. J. "On Geometric Problems of Erdos and Oppen- heim." Math. Gaz. 46, 213 /C1/15, 1962. Mordell, L. J. and Barrow, D. F. "Solution to Problem 3740." Amer. Math. Monthly 44, 252 /C1/54, 1937. Oppenheim, A. "The Erdos Inequality and Other Inequal- ities for a Triangle." Amer. Math. Monthly 68, 226 /C1/30 and 349, 1961. Veldkamp, G. R. "The Erdos-Mordell Inequality." Nieuw Tijdschr. Wisk. 45, 193 /C1/96, 1957/1958. Erdos-Moser Equation The DIOPHANTINE EQUATION Xm/C281 j /C301jn /C30mn : Erdos conjectured that there is no solution to thisequation other than the trivial solution 11 /C2721 /C3031 ; although this remains unproved (Guy 1994, pp. 153 /C1/ 54). Moser (1953) proved that there is no solution for m B10106 ; and Butske et al. (1999) extended this to m B109 :3 /C29106 ; or more specifically, m B1:485 /C29109321155 :/ References Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation ap ½N 1 =p /C271 =N /C301; Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407 /C1/20, 1999. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Moree, P. "Diophantine Equations of Erdos-Moser Type." Bull. Austral. Math. Soc. 53, 281 /C1/92, 1996. Moser, L. "On the Diophantine Equation 1n /C272n /C273n /C27.../C27(m /C281)n /C30mn :/" Scripta Math. 19,84/C1/ 8, 1953. Erdos-Selfridge Function The Erdos-Selfridge function g(k) is defined as the least integer bigger than k /C271 such that the LEAST PRIME FACTOR ofg(k) k;jr;j1 exceeds k (Ecklund et al. 1974, Erdos et al. 1993). The best lower bound known is g(k) ]exp cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [ln k]3 ln ln ks ! (Granville and Ramare 1996). Scheidler and Williams (1992) tabulated g(k)upto k /C30140, and Lukes et al. (1997) tabulated g(k) for 135 5k 5200: The values for n /C302, 3, ... are 4, 7, 7, 23, 62, 143, 44, 159, 46, 47, 174, 2239, ... (Sloane’s A046105). See also BINOMIAL COEFFICIENT ,G OOD BINOMIAL COEFFICIENT ,LEAST PRIME FACTOR References Ecklund, E. F. Jr.; Erdos, P.; and Selfridge, J. L. "A New Function Associated with the prime factors ofn k;jr;j1 : Math. Comput. 28, 647 /C1/49, 1974. Erdos, P.; Lacampagne, C. B.; and Selfridge, J. L. "Esti- mates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215 /C1/24, 1993. Granville, A. and Ramare, O. "Explicit Bounds on Exponen- tial Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43,73/C1/07, 1996. Lukes, R. F.; Scheidler, R.; and Williams, H. C. "Further Tabulation of the Erdos-Selfridge Function." Math. Com- put. 66, 1709 /C1/717, 1997. Scheidler, R. and Williams, H. C. "A Method of Tabulating the Number-Theoretic Function g(k):/" Math. Comput. 59, 251 /C1/57, 1992. Sloane, N. J. A. Sequences A046105 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Erdos-Stone Theorem A generalization of T URA´N’S THEOREM to non- COM- PLETE GRAPHS . See also CLIQUE ,EXTREMAL GRAPH THEORY ,TURA´ N’S THEOREM References Chva´tal, V. and Szemere ´di, E. "On the Erdos-Stone Theo- rem." J. London Math. Soc. 23, 207 /C1/14, 1981. Pach, J. and Agarwal, P. K. Combinatorial Geometry. New York: Wiley, 1995. Erdos-Szekeres Theorem Suppose a ; b /C23N; n /C30ab /C271 ; and x1 ; ..., xnis a sequence of n REAL NUMBERS . Then this sequence contains a MONOTONIC increasing (decreasing) sub- sequence of a /C271 terms or a MONOTONIC decreasing (increasing) subsequence of b /C271 terms. DILWORTH’S LEMMA is a generalization of this theorem. See also COMBINATORICS ,DILWORTH’S LEMMA References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 54 /C1/5, 1998. Erdos-Tura ´n Theorem For any integers ai with 1 5a1 Ba2 B/C1/C1/C1Bak 5n; the proportion of PERMUTATIONS in the SYMMETRIC GROUP Snwhose cyclic decompositions contain no cycles of lengths a1 ; a2 ; ...,ak is at most Xk i/C3011 ai ! /C281 (Erdos and Tura ´n 1967, Dixon 1969). See also CYCLE (PERMUTATION ), SYMMETRIC GROUP References Dixon, J. D. "The Probability of Generating the Symmetric Group." Math. Z. 110, 199/C1/05, 1969. Erdos, P. and Tura ´n, P. "On Some Problems in Statistical Group Theory. II." Acta Math. Acad. Sci. Hung. 18, 151/C1/ 63, 1867.Erf The "error function" encountered in integrating the GAUSSIAN DISTRIBUTION (which is a normalized form of the G AUSSIAN FUNCTION ), erf(z)/C132ffiffiffippgz 0e/C28t2dt (1) /C301/C28erfc(z) (2) /C30p/C281=2g(1 2;z2); (3) where ERFC is the complementary error function and g(x;a) is the incomplete GAMMA FUNCTION . It can also be defined as a M ACLAURIN SERIES erf(z)/C302ffiffiffippX/C12 n/C300(/C281)nz2n/C271 n!(2n/C271): (4) Erf has the values erf(0)/C300 (5) erf(/C12)/C301: (6) It is an ODD FUNCTION erf(/C28z)/C30/C28erf(z); (7) and satisfies erf(z)/C27erfc(z)/C301: (8) Erf may be expressed in terms of a CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND Mas erf(z)/C302zffiffiffippM(1 2;32;/C28z2)/C302zffiffiffippe/C28z2M(1;3 2;z2):(9) Erf is bounded by 1 x /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 2p Bex2g/C12 xe /C28t2 dt 51 x /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C274 pq : (10) Its DERIVATIVE is dn dznerf(z) /C30(/C281)n/C2812ffiffiffipp Hn /C281(z)e /C28z2 ; (11) where Hnis a HERMITE POLYNOMIAL . The first DERIVATIVE is d dzerf(z) /C302ffiffiffipp e /C28z2 ; (12) and the integral is g erf(z) dz /C30z erf(z) /C27e /C28z2 ffiffiffipp : (13) For x /C101; erf may be computed from erf(x) /C302ffiffiffippgx 0e /C28t2 dt (14) /C302ffiffiffippgx 0X/C12 k/C300( /C28t2)k k!dt /C302ffiffiffippgx 0X/C12 k/C300( /C281)kt2k k!dt /C302ffiffiffippX/C12 k /C300x2k /C271( /C281)k k!(2k /C27 1) (15) /C302ffiffiffipp (x /C281 3 x3 /C271 10 x5 /C281 42 x7 /C271 216 x9 /C281 1320 x11 /C27...) (16) /C302ffiffiffipp e /C28x2 x 1 /C272x2 1 /C215 3 /C27(2x2)2 1 /C215 3 /C215 5 /C27..."# (17) (Acton 1990). For x /C271; erf(x) /C302ffiffiffippg/C12 0e/C28t2 dt /C28g/C12 xe /C28t2 dt;j1z;j1} /C301 /C282ffiffiffippg/C12 xe /C28t2 dt : (18)Using INTEGRATION BY PARTS gives g/C12 xe /C28t2 dt /C30/C281 2 g/C12 x1 td(e /C28t2 ) /C30/C2812e /C28t2 t"#/C12 x/C281 2 g/C12 xe /C28t2 dt t2 /C30e /C28x2 2x/C2714 g/C12 x1 t3d(e/C28t2 ) /C30e/C28x2 2x/C28e/C28x2 4x3 /C28... ; (19) so erf(x) /C301 /C28e /C28x2 ffiffiffippx1 /C281 2x2 /C28... ! (20) and continuing the procedure gives the ASYMPTOTIC SERIES erf(x) /C301 /C28e /C28x2 ffiffiffipp /C2(x /C281 /C281 2 x/C283 /C2734 x/C285 /C2815 8x/C287 /C27105 16x/C289 /C27...): (21) Ramanujan rediscovered the CONTINUED FRACTION formula ga 0e /C28t2 dt /C301 2ffiffiffipperf a /C301 2ffiffiffipp/C28e/C28a2 2a/C271 a /C272 2a /C273 a /C274 2a /C27 ...; (22) first stated by Laplace and proved by Jacobi (Watson 1928; Hardy 1999, pp. 8 /C1/). ACOMPLEX generalization of erf xis defined as wðzÞ¼e/C28z2erfcð/C28izÞð 23Þ /C30e/C28z21þ2iffiffiffippþ2iffiffiffippgz 0et2dt ! ð24Þ /C30i pg/C12 /C28/C12e/C28t2dt z/C28t/C302iz pg/C12 0e/C28t2dt z/C28t: ð25Þ See also DAWSON’S INTEGRAL ,ERFC,ERFI,FRESNEL INTEGRALS ,G AUSSIAN FUNCTION ,GAUSSIAN INTE- GRAL ,NORMAL DISTRIBUTION FUNCTION ,PROBABILITY INTEGRAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Error Function and Fresnel Integrals." Ch. 7 in Handbook of Mathema- tical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 297 /C1/09, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 16, 1990. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 568 /C1/69, 1985. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Spanier, J. and Oldham, K. B. "The Error Function erf( x) and Its Complement erfc( x):/" Ch. 40 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385 /C1/93, 1987. Watson, G. N. "Theorems Stated by Ramanujan (IV): The- orems on Approximate Integration and Summation ofSeries." J. London Math. Soc. 3, 282/C1 /89, 1928. Whittaker, E. T. and Robinson, G. "The Error Function." §92 inThe Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 179 /C1/82, 1967. Erfc The "complementary error function" defined by erfc(x)/C131/C28erf(x) (1) /C302ffiffiffippg/C12 xe/C28t2dt (2) /C30ffiffiffippg(1 2;z2); (3) where gis the incomplete GAMMA FUNCTION . It has the values erfc(0) /C301 (4) lim x0/C12erfc(x)/C300 (5) erfc(/C28x)/C302/C28erfc(x) (6)g/C12 0erfc(x)dx/C301ffiffiffipp (7) g/C12 0erfc2(x)dx/C302/C28ffiffiffi 2p ffiffiffipp : (8) A generalization is obtained from the ERFC DIFFER- ENTIAL EQUATION d2y dz2/C272zdy dz/C282ny/C300 (9) (Abramowitz and Stegun 1972, p. 299; Zwillinger 1997, p. 122). The general solution is then y/C30Aerfcn(z)/C27Berfcn(/C28z); (10) where erfcn(z) is the repeated erfc integral. For integral n]1; erfcn(z)/C30g/C1/C1/C1g|fflfflfflffl{zfflfflfflffl} nerfc(z)dz (11) /C302ffiffiffi 2pg/C12 z(t/C28z)n n!e/C28t2dt (12) /C302/C28ne/C28z21F1(1 2(n/C271);12;z2) G(1/C2712n)/C282z1F1(1/C2712n;32;z2) G(12(n/C271))"# (13) (Abramowitz and Stegun 1972), where1F1(a;b;z)i s aCONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND andG(z)i sa GAMMA FUNCTION . The first few values, extended by the definition for n/C30/C28 1 and 0, are given by erfc/C281(z)/C302ffiffiffippe/C28z2(14) erfc0(z)/C30erfc(z) (15) erfc1(z)/C30e/C28z2 ffiffiffipp/C28zerfc(z) (16) erfc2(z) /C301 4(1 /C272z2) erfc(z) /C282ze/C28z2 ffiffiffipp"# : (17) See also ERF,ERFC DIFFERENTIAL EQUATION ,ERFI References Abramowitz, M. and Stegun, C. A. (Eds.). "Repeated Inte- grals of the Error Function." §7.2 in Handbook of Math- ematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 299 /C1/00, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 568 /C1/69, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209 /C1/14, 1992. Spanier, J. and Oldham, K. B. "The Error Function erf(x) and Its Complement erfc(x)/" and "The exp(x) and erfc(ffiffiffixp) and Related Functions." Chs. 40 and 41 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385 /C1/93 and 395 /C1/03, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Erfc Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C272xy?/C282ny /C300; (1) whose solutions may be written either y /C30A erfcn(x) /C27B erfcn(/C28x); (2) where erfcn(x) is the repeated integral of the ERFC function (Abramowitz and Stegun 1972, p. 299), or y /C30C1e /C28x2 H /C28n/C281(x) /C27C21F1(1 2(n /C271);12; x2); (3) where Hn(x)isaH ERMITE POLYNOMIAL and 1F1(a; b; z)isa CONFLUENT HYPERGEOMETRIC FUNC- TION OF THE FIRST KIND . See also ERFC References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 299, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. # 1999 /C1/001 Wolfram Research, Inc.Erfi erfi(z) /C13/C28i erf(iz) : A ASYMPTOTIC SERIES for the erfi function is given by erfi(x)/C2p/C281=2x/C281ex2: See also DAWSON’S INTEGRAL ,ERF,ERFC Ergodic Measure An ENDOMORPHISM is called ergodic if it is true that T/C281A/C30AIMPLIES m(A)/C300 or 1, where T/C281A/C30fx/C23 X:T(x)/C23Ag:Examples of ergodic endomorphisms include the MAP X02xmod 1 on the unit interval with L EBESGUE MEASURE , certain AUTOMORPHISMS of the TORUS , and "Bernoulli shifts" (and more generally "Markov shifts"). Given a MAP Tand a SIGMA ALGEBRA , there may be many ergodic measures. If there is only one ergodic measure, then Tis called uniquely ergodic. An example of a uniquely ergodic transformation is the MAP x/C2x/C27amod 1 on the unit interval when ais irrational. Here, the unique ergodic measure isL EBESGUE MEASURE . Ergodic Theory Ergodic theory can be described as the statistical and qualitative behavior of measurable group and semi- group actions on MEASURE SPACES . The GROUP is most commonly N, R, R /C27, and Z. Ergodic theory had its origins in the work of Boltz- mann in statistical mechanics problems where time- and space-distribution averages are equal. Steinhaus (1983, pp. 237 /C1/39) gives a practical application to ergodic theory to keeping one’s feet dry ( when walking along a shoreline without having to con- stantly turn one’s head to anticipate incoming waves. The mathematical origins of ergodic theory are due to von Neumann, Birkhoff, and Koopman in the 1930s. It has since grown to be a huge subject and has applications not only to statistical mechanics, but also to NUMBER THEORY , DIFFERENTIAL GEOMETRY , FUNC- TIONAL ANALYSIS , etc. There are also many internal problems (e.g., ergodic theory being applied to ergodic theory) which are interesting. See also AMBROSE- KAKUTANI THEOREM ,BIRKHOFF’S ERGODIC THEOREM ,D YE’S THEOREM ,D YNAMICAL SYSTEM ,HOPF’S THEOREM ,ORNSTEIN’S THEOREM References Billingsley, P. Ergodic Theory and Information. New York: Wiley, 1965. Cornfeld, I.; Fomin, S.; and Sinai, Ya. G. Ergodic Theory. New York: Springer-Verlag, 1982. Katok, A. and Hasselblatt, B. An Introduction to the Modern Theory of Dynamical Systems. Cambridge, England: Cam- bridge University Press, 1996. Nadkarni, M. G. Basic Ergodic Theory. India: Hindustan Book Agency, 1995. Parry, W. Topics in Ergodic Theory. Cambridge, England: Cambridge University Press, 1982. Petersen, K. Ergodic Theory. Cambridge, England: Cam- bridge University Press, 1983. Radin, C. "Ergodic Theory." Ch. 1 in Miles of Tiles. Provi- dence, RI: Amer. Math. Soc., pp. 17 /C1/4, 1999. Sinai, Ya. G. Topics in Ergodic Theory. Princeton, NJ: Princeton University Press, 1993. Smorodinsky, M. Ergodic Theory, Entropy. Berlin: Springer- Verlag, 1971. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 237 /C1/39, 1999. Walters, P. Ergodic Theory: Introductory Lectures. New York: Springer-Verlag, 1975. Walters, P. Introduction to Ergodic Theory. New York: Springer-Verlag, 2000. Ergodic Transformation A transformation which has only trivial invariant SUBSETS is said to be ergodic. Erlang Distribution Given a POISSON DISTRIBUTION with a rate of change l ; the DISTRIBUTION FUNCTION D(x) giving the waiting times until the hth Poisson event is D(x) /C301 /C28G(h; xl) G(h) (1) for x /C23 [0;/C12); where G(x) is a complete GAMMA FUNC- TION , and G(a; x)an INCOMPLETE GAMMA FUNCTION . With h explicitly an integer, this distribution is known as the Erlang distribution, and has probability functionP(x) /C30l(lx)h/C281 (h /C28 1)!e /C28 lx : (2) It is closely related to the GAMMA DISTRIBUTION , which is obtained by letting a /C13h (not necessarily an integer) and defining u /C131 =l: When h /C301, it simplifies to the EXPONENTIAL DISTRIBUTION . See also EXPONENTIAL DISTRIBUTION ,G AMMA DIS- TRIBUTION #1999/C1/001 Wolfram Research, Inc. Erlanger Program A program initiated by F. Klein in an 1872 lecture to describe geometric structures in terms of their AUTO- MORPHISM GROUPS . References Klein, F. "Vergleichende Betrachtungen u ¨ber neuere geome- trische Forschungen." 1872. Yaglom, I. M. Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century. Boston, MA: Birkha ¨user, 1988. Ermakoff’s Test The series af(n) for a monotonic nonincreasing f(x)i s convergent if lim x0/C12exf(ex) f(x)B1 and divergent if lim x0/C12exf(ex) f(x)>1: References Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 43, 1991. Ernst Equation The PARTIAL DIFFERENTIAL EQUATION R[u]urr/C27ur r/C27uzz ! /C30u2 r/C27u2z; where R[u] is the REAL PART ofu(Calogero and Degasperis 1982, p. 62; Zwillinger 1997, p. 131). References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. #1999/C1/001 Wolfram Research, Inc. Errera Graph The 17-node PLANAR GRAPH illustrated above which tangles the Kempe chains in Kempe’s algorithm and thus provides an example of how Kempe’s supposed proof of the FOUR-COLOR THEOREM fails. See also FOUR- COLOR THEOREM ,KITTELL GRAPH References Wagon, S. Mathematica in Action, 2nd ed. New York: Springer-Verlag, pp. 522 /C1/24, 1999. # 1999 /C1/001 Wolfram Research, Inc. Error The difference between a quantity and its estimated or measured quantity. See also ABSOLUTE ERROR ,P ERCENTAGE ERROR , RELATIVE ERROR Error Curve GAUSSIAN FUNCTION Error Function ERF,ERFC Error Function Distribution ANORMAL DISTRIBUTION with MEAN 0, P(x)/C30hffiffiffippe/C28h2x2: (1) The CHARACTERISTIC FUNCTION is f(t)/C30e/C28t2=(4h2): (2) The MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS are m/C300 (3)s2/C301 2h2(4) g1/C300 (5) g2/C300: (6) The CUMULANTS are k1/C300 (7) k2/C301 2h2(8) kn/C300 (9) forn]3:/ Error Propagation Given a FORMULA y/C30f(x) with an ABSOLUTE ERROR in xofdx, the ABSOLUTE ERROR isdy. The RELATIVE ERROR isdy=y:Ifx/C30f(u;v);then xi/C28¯x/C30(ui/C28¯u)@x @u/C27(vi/C28¯v)@x @v/C27...; (1) where ¯xdenotes the MEAN ,s o s2 x/C131 N/C281XN i/C301(xi/C28¯x)2 /C301 N/C281XN i/C301;j2r (ui/C28¯u)2@x @u !2 /C27(vi/C28¯v)2@x @v !2 /C272(ui/C28¯u)(vi/C28¯v)@x @u ! @x @v ! /C27...;j21 : (2) The definitions of VARIANCE and COVARIANCE then give s2u/C131 N/C281XN i/C301(ui/C28¯u)2(3) s2v/C131 N/C281XN i/C301(vi/C28¯v)2(4) suv/C131 N/C281XN i/C301(ui/C28¯u)(vi/C28¯v) (5) (where sii/C13s2 i);so s2 x/C30s2u@x @u !2 /C27s2v@x @v !2 /C272suv@x @u ! @x @v ! /C27...:(6) Ifuandvare uncorrelated, then suv/C300s o s2x/C30s2u@x @u !2 /C27s2v@x @v !2 : (7) Now consider addition of quantities with errors. For x /C30au 9bv;@x =@u /C30a and @x =@v /C309b; so s2 x /C30a2 s2u /C27b2 s2v 92ab suv : (8) For division of quantities with x /C309au =v ;@x=@u /C30 9a=v and @x=@v /C30/C14au =v2 ; so s2x /C30a2 v2s2u /C27a2u2 v4s2v /C282a vau v2suv : (9) sx x !2 /C30a2 v2v2 a2u2s2u /C27a2u2 v4v2 a2u2 /C282a v ! au v2 ! suv /C30su u !2 /C27sv v !2 /C282suv u ! suv v ! : (10) For exponentiation of quantities with x /C30a 9bu /C30(eln a) 9bu /C30e 9b(ln a)u ; (11) @x @u /C309b(ln a)e 9b ln au /C309b(ln a)x ; (12) so sx /C30 sub(ln a)x (13) sx x/C30b ln a su : (14) If a /C30e, then sx x/C30b su : (15) For LOGARITHMS of quantities with x /C30a ln(9bu); @x =@u /C30a(9b)=(9bu) /C30a=u ; so s2x /C30 s2ua2 u2 ! (16) sx /C30asu u: (17) For multiplication with x /C309auv ;@x=@u /C309av and @x =@v /C309au ; so s2x /C30a2v2 s2u /C27a2u2 s2v /C272a2uv suv (18) sx x !2 /C30a2v2 a2u2v2s2u /C27a2u2 a2u2v2s2v /C272a2uv a2u2v2suv /C30su u !2 /C27sv v !2 /C272suv u ! suv v ! : (19) For POWERS , with x/C30au9b;@x=@u/C309abu9b/C281/C30 9bx=u;sos2x/C30s2ub2x2 u2(20) sx x/C30bsu u: (21) See also ABSOLUTE ERROR ,COVARIANCE ,PERCENTAGE ERROR ,RELATIVE ERROR ,VARIANCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, pp. 58 /C1/4, 1969. Error-Correcting Code An error-correcting code is an algorithm for expres- sing a sequence of numbers such that any errorswhich are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and theassociated mathematics is known as CODING THEORY . Error detection is much simpler than error correction, and one or more "check" digits are commonly em- bedded in credit card numbers in order to detectmistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute CHECKSUMS , which are sums of the digits in a given transmissionmodulo some number. The ISBN used to identify books also incorporates a check DIGIT . A powerful check for 13 DIGIT numbers consists of the following. Write the number as a string of DIGITS a1;a2;a3...a13:Take a1þa3þ/C1/C1/C1þ a13and double. Now add the number of DIGITS inODD positions which are >4 to this number. Now add a2/C27a4/C27/C1/C1/C1/C27a12: The check number is then the number required tobring the last DIGIT to 0. This scheme detects all single DIGIT errors and all TRANSPOSITIONS of adja- cent DIGITS except 0 and 9. LetA(n;d) denote the maximal number of n(0,1)- vectors having the property that any two of the set differ in at least dplaces. The corresponding vectors can correct [( d/C281)=2] errors. A(n;d;w) is the num- ber of A(n;d)/s with precisely w1s (Sloane and Plouffe 1995). Since it is not possible for n-vectors to differ in d/C21nplaces and since n-vectors which differ in all nplaces partition into disparate sets of two, A(n; d) /C301 n Bd 2 n /C30d:;j2ffl Values of A(n; d) can be found by labeling the 2n (0,1)- n-vectors, finding all unordered pairs (ai ; aj)of n- vectors which differ from each other in at least d places, forming a GRAPH from these unordered pairs, and then finding the CLIQUE NUMBER of this graph. Unfortunately, finding the size of a clique for a given GRAPH is an NP-COMPLETE PROBLEM . d Sloane /A(n; d)/ 1 A000079 2, 4, 8, 16, 32, 64, 128, ... 2 1,2,4,8,... 3 1,1,2,2,... 4 A005864 1, 1, 1, 2, 4, 8, 16, 20, 40, ... 5 1,1,1,1,2,... 6 A005865 1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, ... 7 1,1,1,1,1,1,2,... 8 A005866 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, ... See also CHECKSUM ,CLIQUE ,CLIQUE NUMBER ,COD- ING THEORY ,F INITE FIELD ,H ADAMARD MATRIX , HAMMING CODE, ISBN, UPC References Baylis, J. Error Correcting Codes: A Mathematical Introduc- tion. Boca Raton, FL: CRC Press, 1998. Berlekamp, E. R. Algebraic Coding Theory, rev. ed. New York: McGraw-Hill, 1968. Brouwer, A. E.; Shearer, J. B.; Sloane, N. J. A.; and Smith, W. D. "A New Table of Constant Weight Codes." IEEE Trans. Inform. Th. 36, 1334 /C1/380, 1990. Calderbank, A. R.; Hammons, A. R. Jr.; Kumar, P. V.; Sloane, N. J. A.; and Sole´, P. "A Linear Construction for Certain Kerdock and Preparata Codes." Bull. Amer. Math. Soc. 29, 218 /C1/22, 1993. Conway, J. H. and Sloane, N. J. A. "Quaternary Construc- tions for the Binary Single-Error-Correcting Codes of Julin, Best and Others." Des. Codes Cryptogr. 4,31/C1/2, 1994. Conway, J. H. and Sloane, N. J. A. "Error-Correcting Codes." §3.2 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 75 /C1/8, 1993. Gallian, J. "How Computers Can Read and Correct ID Numbers." Math Horizons , pp. 14 /C1/5, Winter 1993. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 119 /C1/21, 1994. MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error- Correcting Codes. Amsterdam, Netherlands: North-Hol- land, 1977. Sloane, N. J. A. Sequences A000079/M1129, A005864/ M1111, A005865/M0240, and A005866/M0226 in "An On- Line Version of the Encyclopedia of Integer Sequences."http://www.research.att.com/~njas/sequences/eisonli- ne.html. Sloane, N. J. A. and Plouffe, S. Figure M0240 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Escher’s Map The function f(b;z)/C2z(1/C27cosb/C27isinb)=2; illustrated above for b/C300:4:/ Escher’s Solid The solid illustrated on the right pedestal in M. C. Escher’s Waterfall woodcut. It can be con- structed by CUMULATION of the RHOMBIC DODECAHE- DRON with cumulation height 5/2. See also CUBE 3-COMPOUND ,CUMULATION ,RHOMBIC DODECAHEDRON #1999/C1/001 Wolfram Research, Inc. Escribed Circle EXCIRCLE Essential Singularity ASINGULAR POINT a for which f(z)(z/C28a)nis not DIFFERENTIABLE for any INTEGER n/C210. See also PICARD’S THEOREM ,P OLE,R EMOVABLE SINGULARITY ,SINGULAR POINT (FUNCTION ), WEIER- STRASS- CASORATI THEOREM References Knopp, K. "Essential and Non-Essential Singularities or Poles." §31 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 123 /C1/26, 1996. Krantz, S. G. "Removable Singularities, Poles, and Essential Singularities." §4.1.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 42, 1999. Essential Supremum The essential supremum is the proper generalization to MEASURABLE FUNCTIONS of the MAXIMUM . The technical difference is that the values of a function on a set of MEASURE ZERO don’t affect the essential supremum. Given a MEASURABLE FUNCTION f : X 0 R ; where X is a MEASURE SPACE with measure m; the essential supremum is the smallest number a such that m( fx such that f(x) > ag has MEASURE ZERO . If no such number exists, as in the case of f(x) /C301=x on (0; 1); then the essential supremum is /C12:/ The essential supremum of the absolute value of a function ½f ½ is usually denoted ½½f ½½/C12; and this serves as the norm for L-INFINITY-SPACE . See also L-INFINITY- SPACE , LP-SPACE , L2-SPACE , MEASURE ,MEASURABLE FUNCTION ,MEASURE SPACE # 1999 /C1/001 Wolfram Research, Inc. Estimate An estimate is an educated guess for an unknown quantity or outcome based on known information. The making of estimates is an important part of statistics, since care is needed to provide as accurate an estimate as possible using as little input data as possible. Often, an estimate for the uncertainty DE of an estimate E can also be determined statistically. A rule that tells how to calculate an estimate based on the measurements contained in a sample is called an ESTIMATOR .See also BIAS (ESTIMATOR ), ERROR ,ESTIMATOR References Iyanaga, S. and Kawada, Y. (Eds.). "Statistical Estimation and Statistical Hypothesis Testing." Appendix A, Table 23 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1486 /C1/489, 1980. Estimator An estimator is a rule that tells how to calculate an ESTIMATE based on the measurements contained in a sample. For example, the "sample MEAN " AVERAGE ¯x is an estimator for the population MEAN m:/ The mean square error of an estimator ˜u is defined by MSE /C13 (˜u /C28 u)2DE : Let B be the BIAS, then MSE /C30 [(˜u /C28 ˜u;j1r;j11 ) /C27B(˜u)]2;j1r;j11 /C30 (˜u /C28 ˜u;j1r;j11 )2DE /C27B2(˜u) /C13V(˜u) /C27B2(˜u); where V is the estimator VARIANCE . See also BIAS (ESTIMATOR ), ERROR ,E STIMATE , K- STATISTIC ,UNBIASED ESTIMATOR Eta Function DEDEKIND ETA FUNCTION ,DIRICHLET ETA FUNCTION , JACOBI THETA FUNCTIONS Et-Function A function which arises in FRACTIONAL CALCULUS . Et( n ; a) /C301 G( n)eatgt 0xn/C281e /C28ax dx /C30tneat g( n ; at) ; (1) where gða ; jÞ is the incomplete GAMMA FUNCTION and G(z) the complete GAMMA FUNCTION . The Etfunction satisfies the RECURRENCE RELATION Et( n ; a) /C30aEt( n /C271; a) /C27tn G(n/C271): (2) A special value is Et(0;a)/C30eat: (3) See also EN-FUNCTION ,FRACTIONAL CALCULUS References Abramowitz, M. and Stegun, C. A. (Eds.). "Exponential Integral and Related Functions." Ch. 5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 227 /C1/33, 1972. Ethiopian Multiplication RUSSIAN MULTIPLICATION Etruscan Venus Surface A 3-D shadow of a 4-D KLEIN BOTTLE . See also IDA SURFACE ,KLEIN BOTTLE References Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 42 /C1/4, 1990. Eubulides Paradox The PARADOX "This statement is false," stated in the fourth century BC. It is a sharper version of the EPIMENIDES PARADOX , "All Cretans are liars...One of their own poets has said so." See also EPIMENIDES PARADOX ,SOCRATES’ PARADOX References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 63 /C1/4, 1998. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989. Euclid Number The nth Euclid number is defined by En /C131 /C27Yn i /C301pi /C301 /C27pn#; where piis the ith PRIME and pn# is the PRIMORIAL . The first few Enare 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (Sloa- ne’s A006862; Tietze 1965, p. 19). The largest factors of Enfor n /C301, 2, ... are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (Sloane’s A002585). The n of the first few PRIME Euclid numbers En are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, ... (Sloane’s A014545), and the largest known Euclid number is E4413 : It is not known if there are an INFINITE number of PRIME Euclid numbers (Guy 1994, Ribenboim 1996). See also EUCLID- MULLIN SEQUENCE ,P RIMORIAL , SMARANDACHE SEQUENCES References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Sloane, N. J. A. Sequences A006862/M2698, A002585/ M2697, and A014545 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, 1965.Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 35 /C1/7, 1991. Euclid’s Axioms EUCLID’S POSTULATES Euclid’s Elements ELEMENTS Euclid’s Fifth Postulate EUCLID’S POSTULATES Euclid’s Orchard An array of "trees" of unit height located at integer- coordinate points in a POINT LATTICE . When viewed from a corner along the line y /C30x in normal perspec- tive, a QUADRANT of Euclid’s orchard turns into the modified DIRICHLET FUNCTION (Gosper). See also DIRICHLET FUNCTION ,G REATEST COMMON DIVISOR ,ORCHARD- PLANTING PROBLEM Euclid’s Postulates 1. A straight LINE SEGMENT can be drawn joining any two points. 2. Any straight LINE SEGMENT can be extended indefinitely in a straight LINE. 3. Given any straight LINE SEGMENT ,aCIRCLE can be drawn having the segment as RADIUS and one endpoint as center.4. All RIGHT ANGLES are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two RIGHT ANGLES , then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the PARALLEL POSTULATE . Euclid’s fifth postulate cannot be proven as a theo-rem, although this was attempted by many people.Euclid himself used only the first four postulates ( for the first 28 propositions of the E LEMENTS , but was forced to invoke the PARALLEL POSTULATE on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "NON- EUCLIDEAN GEOMETRIES " could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.) See also ABSOLUTE GEOMETRY ,CIRCLE ,ELEMENTS , LINE SEGMENT ,NON-EUCLIDEAN GEOMETRY ,PARAL- LEL POSTULATE ,PASCH’S THEOREM ,RIGHT ANGLE References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 88 /C1/2, 1989. Euclid’s Principle EUCLID’S THEOREMS Euclid’s Theorems A theorem sometimes called "Euclid’s First Theorem" or EUCLID’S PRINCIPLE states that if p is a PRIME and p ½ab ; then p½a or p ½b (where ½ means DIVIDES ). A COROLLARY is that p ½an [p ½a (Conway and Guy 1996). The FUNDAMENTAL THEOREM OF ARITHMETIC is an- other COROLLARY (Hardy and Wright 1979). Euclid’s Second Theorem states that the number of PRIMES is INFINITE . This theorem, also called the INFINITUDE OF PRIMES theorem, was proved by Euclid in Proposition IX.20 of the ELEMENTS (Tietze 1965, pp. 7 /C1/). Ribenboim (1989) gives nine (and a half) proofs of this theorem. Euclid’s elegant proof proceeds as follows. Given a finite sequence of consecutive PRIMES 2, 3, 5, ..., p, the number N /C302 /C2153 /C2155 /C1/C1/C1p /C271 ; (1) known as the ith EUCLID NUMBER when p /C30pi is the ith PRIME , is either a new PRIME or the product of PRIMES .IfN is a PRIME , then it must be greater than the previous PRIMES , since one plus the product of PRIMES must be greater than each PRIME composing the product. Now, if N is a product of PRIMES , then at least one of the PRIMES must be greater than p. This can be shown as follows. If N is COMPOSITE and has no prime factors greater than p, then one of its factors (say F) must be one of the PRIMES in the sequence, 2, 3, 5, ..., p. It therefore DIVIDES the product 2 /C2153 /C2155 /C1/C1/C1p: However, since it is a factor of N, it also DIVIDES N. But a number which DIVIDES two numbers a and b Ba also DIVIDES their difference a /C28b; so F must also divide N /C28(2 /C2153 /C2155 /C1/C1/C1p) /C30(2 /C2153 /C2155 /C1/C1/C1p /C271) /C28(2 /C2153 /C2155 /C1/C1/C1p) /C301: (2) However, in order to divide 1, F must be 1, which is contrary to the assumption that it is a PRIME in the sequence 2, 3, 5, .... It therefore follows that if N is composite, it has at least one factor greater than p. Since N is either a PRIME greater than p or contains a prime factor greater than p,aPRIME larger than the largest in the finite sequence can always be found, so there are an infinite number of PRIMES . Hardy (1967) remarks that this proof is "as fresh and significant as when it was discovered" so that "two thousand years have not written a wrinkle" on it.A similar argument shows that p! 91 and 1 /C2153 /C2155 /C2157 /C1/C1/C1p /C271 (3) must be either PRIME or be divisible by a PRIME > p: Kummer used a variation of this proof, which is also a proof by contradiction. It assumes that there exist only a finite number of PRIMES N /C30p1 ; p2 ; ..., pr : Now consider N /C281: It must be a product of PRIMES ,soit has a PRIME divisor pi in common with N. Therefore, pi ½N /C28(N /C281) /C301 which is nonsense, so we have proved the initial assumption is wrong by contra- diction. It is also true that there are runs of COMPOSITE NUMBERS which are arbitrarily long. This can be seen by defining n/C13j!/C30Yj i/C301i; (4) where j!i sa FACTORIAL . Then the j/C281 consecutive numbers n/C272;n/C273;...,n/C27jare COMPOSITE , since n/C272/C30(1 /C2152/C1/C1/C1j)/C272/C302(1 /C2153/C2154/C1/C1/C1n/C271) (5) n/C273/C30(1 /C2152/C1/C1/C1j)/C273/C303(1 /C2152/C2154/C2155/C1/C1/C1n/C271) (6) n/C27j/C30(1 /C2152/C1/C1/C1j)/C27j/C30j[1 /C2152/C1/C1/C1(j/C281)/C271]: (7) Guy (1981, 1988) points out that while p1p2/C1/C1/C1pn/C271 is not necessarily PRIME , letting qbe the next PRIME after p1p2/C1/C1/C1pn/C271;the number q/C28p1p2/C1/C1/C1pn/C271i s almost always a PRIME , although it has not been proven that this must always be the case. See also DIVIDE ,EUCLID NUMBER ,PRIME NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 60, 1987. Conway, J. H. and Guy, R. K. "There are Always New Primes!" In The Book of Numbers. New York: Springer- Verlag, pp. 133 /C1/34, 1996. Cosgrave, J. B. "A Remark on Euclid’s Proof of the Infinitude of Primes." Amer. Math. Monthly 96, 339/C1/41, 1989. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 22, 1996. Dunham, W. "Great Theorem: The Infinitude of Primes." Journey through Genius: The Great Theorems of Mathe-matics. New York: Wiley, pp. 73 /C1 /5, 1990. Guy, R. K. §A12 in Unsolved Problems in Number Theory. New York: Springer-Verlag, 1981. Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697/C1/12, 1988. Hardy, G. H. A Mathematician’s Apology. Cambridge, Eng- land: Cambridge University Press, 1992. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 3 /C1/2, 1989. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod-ern Times. New York: Graylock Press, pp. 7 /C1 /, 1965. Euclidean Algorithm An ALGORITHM for finding the GREATEST COMMON DIVISOR of two numbers aandb, also called Euclid’s algorithm. The algorithm can also be defined for more general RINGS than just the integers Z. There are even PRINCIPAL RINGS which are not E UCLIDEAN but where one can define the equivalent of the Euclidean algorithm. The algorithm for rational numbers was given in Book VII of Euclid’s Elements , and the algorithm for reals appeared in Book X, and is the earliest example of an INTEGER RELATION algorithm (Ferguson et al. 1999). The Euclidean algorithm is an example of a P - PROBLEM whose time complexity is bounded by a quadratic function of the length of the input values (Banach and Shallit). Let a/C30bq/C27r;then find a number uwhich DIVIDES both aand b(so that a/C30suandb/C30tu), then ualso DIVIDES rsince r/C30a/C28bq/C30su/C28qtu/C30(s/C28qt)u: (1) Similarly, find a number vwhich DIVIDES bandr(so that b/C30s?vandr/C30t?v);then vDIVIDES asince a/C30bq/C27r/C30s?vq/C27t?v/C30(s?q/C27t?)v: (2) Therefore, every common DIVISOR ofaand bis a common DIVISOR ofbandr, so the procedure can be iterated as follows. q1/C30a b$% a/C30bq1/C27r1r1/C30a/C28bq1 (3) q2/C30b r1$% b/C30q2r1/C27r2r2/C30b/C28q2r1 (4) q3/C30r1 r2$% r1/C30q3r2/C27r3r3/C30r1/C28q3r2 (5) q4/C30r2 r3$% r2/C30q4r3/C27r4r4/C30r2/C28q4r3 (6) qn/C30rn/C282 rn/C281$% rn/C282/C30qnrn/C281/C27rnrn/C30rn/C282/C28qnrn/C281 (7) qn/C271/C30rn/C281 rn$% rn/C281/C30qn/C271rn/C270rn/C30rn/C281=qn/C271:(8) For integers, the algorithm terminates when qn/C271 divides rn/C281exactly, at which point rncorresponds to the GREATEST COMMON DIVISOR ofaand b,/ GCD( a;b)/C30rn:For real numbers, the algorithm yields either an exact relation or an infinite sequenceof approximate relations (Ferguson et al. 1999). Lame ´showed that the number of steps needed to arrive at the GREATEST COMMON DIVISOR for two numbers less than nissteps5log10n log10f/C27log10ffiffiffi 5p log10f(9) where fis the GOLDEN MEAN ,o r55 times the number of digits in the smaller number (Wells 1986, p. 59). Numerically, Lame ´’s expression evaluates to steps54:785 log10n/C271:6723 : (10) As shown by L AME´’S THEOREM , the worst case occurs when the ALGORITHM is applied to two consecutive FIBONACCI NUMBERS . Heilbronn showed that the average number of steps is 12 ln 2 =p2log10n/C30 0:843 log10nfor all pairs ( n, b) with bBn. Kronecker showed that the shortest application of the ALGO- RITHM uses least absolute remainders. The QUOTI- ENTS obtained are distributed as shown in the following table (Wagon 1991). Quotient /%/ 1 41.5 2 17.0 3 9.3 For details, see Uspensky and Heaslet (1939) or Knuth (1973). Let T(m;n) be the number of divisions required to compute GCD( m;n) using the Euclidean algorithm, and define T(m;0)/C300i fm]0:Then the function T(m;n) is given by the RECURRENCE RELA- TION T(m;n)/C301/C27T(n;mmod n) for m]n 1/C27T(n;m) for mBn:;j2ffl (11) Tabulating this function for 0 5mBngives 0 01 012011201232011122 (Sloane’s A051010). The maximum numbers of steps for a given n/C301, 2, 3, ... are 1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 4, 5, ... (Sloane’s A034883). Define the functions T(n)/C30 1 nX 05mBnT(m;n) (12) t(n)/C301 f(n)X 0BmBnGCD( m;n)/C301T(m;n) (13) A(N)/C301 N2X 15mBN15n5NT(m;n); (14) where f(n) is the TOTIENT FUNCTION , T(n) is the average number of divisions when n is fixed and m chosen at random, t(n) is the average number of divisions when n is fixed and m is a random number coprime to n, and A(N) is the average number of divisions when m and n are both chosen at random in [1; N] : The first few values of T(n) are 0, 1/2, 1, 1, 8/5, 7/6, 13/7, 7/4, ... (Sloane’s A051011 and A051012). Norton (1990) showed that T(n) /C3012 ln 2 p2ln n /C28X d½nL(d) d"# /C27C /C271 nX d½nf(d)O(d/C281=6 /C27e) ; (15) where L(d) is the VON MANGOLDT FUNCTION and C is PORTER’S CONSTANT . Porter (1975) showed that t(n) /C3012 ln 2 p2ln n /C27C /C27O(n/C281 =6 /C27 e) ; (16) and Norton (1990) proved that A(N) /C3012 ln 2 p2ln N /C281 2 /C276 p2z?(2)"# /C27C /C2812 /C27O(N /C281 =6 /C27 e); (17) where z?(z) is the derivative of the RIEMANN ZETA FUNCTION . There exist 21 QUADRATIC FIELDS in which there is a Euclidean algorithm (Inkeri 1947, Barnes and Swin- nerton-Dyer 1952). Although various attempts were made to generalize the algorithm to find INTEGER RELATIONS between n ]3 variables, none were successful until the dis- covery of the FERGUSON- FORCADE ALGORITHM (Fergu- son et al. 1999). Several other INTEGER RELATION algorithms have now been discovered. See also BLANKINSHIP ALGORITHM ,EUCLIDEAN RING, FERGUSON- FORCADE ALGORITHM ,INTEGER RELATION , QUADRATIC FIELD References Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, 1996. Barnes, E. S. and Swinnerton-Dyer, H. P. F. "The Inhomo- geneous Minima of Binary Quadratic Forms. I." Acta Math 87, 259 /C1/23, 1952. Chabert, J.-L. (Ed.). "Euclid’s Algorithm." Ch. 4 in A History of Algorithms: From the Pebble to the Microchip. New York: Springer-Verlag, pp. 113 /C1/38, 1999. Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993. Courant, R. and Robbins, H. "The Euclidean Algorithm." §2.4 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 42 /C1/1, 1996.Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 69 /C1/0, 1990. Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351 /C1/69, 1999. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/porter/porter.html. Inkeri, K. "U¨ ber den Euklidischen Algorithmus in quad- ratischen Zahlko ¨rpern." Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 1947 ,1/C1/5, 1947. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Motzkin, T. "The Euclidean Algorithm." Bull. Amer. Math. Soc. 55, 1142 /C1/146, 1949. Nagell, T. "Euclid’s Algorithm." §7in Introduction to Num- ber Theory. New York: Wiley, pp. 21 /C1/3, 1951. Norton, G. H. "On the Asymptotic Analysis of the Euclidean Algorithm." J. Symb. Comput. 10,53/C1/8, 1990. Porter, J. W. "On a Theorem of Heilbronn." Mathematika 22,20/C1/8, 1975. Se´roul, R. "Euclidean Division" and "The Euclidean Algo- rithm." §2.1 and 8.1 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 5 and 169 /C1/61, 2000. Sloane, N. J. A. Sequences A034883, A051010, A051011, and A051012 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, 1939. Wagon, S. "The Ancient and Modern Euclidean Algorithm" and "The Extended Euclidean Algorithm." §8.1 and 8.2 in Mathematica in Action. New York: W. H. Freeman, pp. 247 /C1/52 and 252 /C1/56, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 59, 1986. Euclidean Construction GEOMETRIC CONSTRUCTION Euclidean Domain A more common way to describe a E UCLIDEAN RING . See also ALGEBRAIC NUMBER THEORY ,E UCLIDEAN RING Euclidean Geometry AGEOMETRY in which E UCLID’S FIFTH POSTULATE holds, sometimes also called PARABOLIC GEOMETRY .2 - D Euclidean geometry is called PLANE GEOMETRY , and 3-D Euclidean geometry is called SOLID GEOMETRY . Hilbert proved the CONSISTENCY of Euclidean geome- try. See also ELLIPTIC GEOMETRY ,GEOMETRIC CONSTRUC- TION ,GEOMETRY ,H YPERBOLIC GEOMETRY ,N ON-EU- CLIDEAN GEOMETRY ,PLANE GEOMETRY References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., 1967 Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, 1913. Greenberg, M. J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W. H. Freeman, 1994. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 1: Books I and II. New York: Dover, 1956. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 2: Books III-IX. New York: Dover, 1956. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 3: Books X-XIII. New York: Dover, 1956. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Klee, V. "Some Unsolved Problems in Plane Geometry." Math. Mag. 52, 131 /C1/45, 1979. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory, rev. ed. Washington, DC: Math. Assoc. Amer., 1991. Weisstein, E. W. "Books about Plane Geometry." http:// www.treasure-troves.com/books/PlaneGeometry.html. Euclidean Graph A WEIGHTED GRAPH in which the weights are equal to the Euclidean lengths of the edges in a specified embedding (Skiena 1990, pp. 201 and 252). References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Euclidean Group The GROUP of ROTATIONS and TRANSLATIONS . See also ROTATION ,TRANSLATION References Lomont, J. S. Applications of Finite Groups. New York: Dover, 1987. Euclidean Metric The FUNCTION f : Rn /C29Rn 0 R that assigns to any two VECTORS (/x1 ; ..., xn) and (/y1 ; ..., yn) the number ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x1 /C28y1)2 /C27.../C27(xn /C28yn)2q ; and so gives the "standard" distance between any two VECTORS in Rn :/Euclidean Motion A Euclidean motion of Rn is an AFFINE TRANSFORMA- TION whose linear part is an ORTHOGONAL TRANSFOR- MATION . See also RIGID MOTION References Gray, A. "Euclidean Motions." §6.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 128 /C1/34, 1997. Euclidean Norm L2-NORM Euclidean Number A Euclidean number is a number which can be obtained by repeatedly solving the QUADRATIC EQUA- TION . Euclidean numbers, together with the RA- TIONAL NUMBERS , can be constructed using classical GEOMETRIC CONSTRUCTIONS . However, the cases for which the values of the TRIGONOMETRIC FUNCTIONS SINE, COSINE , TANGENT , etc., can be written in closed form involving square roots of REAL NUMBERS are much more restricted. See also ALGEBRAIC INTEGER ,ALGEBRAIC NUMBER , CONSTRUCTIBLE NUMBER ,RADICAL INTEGER References Conway, J. H. and Guy, R. K. "Three Greek Problems." In The Book of Numbers. New York: Springer-Verlag, pp. 192 /C1/94, 1996. Klein, F. "Algebraic Equations Solvable by Square Roots." Part I, Ch. 1 in "Famous Problems of Elementary Geome- try: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 5 /C1/2, 1980. Euclidean Plane The 2-D EUCLIDEAN SPACE denoted R2 :/ See also COMPLEX PLANE ,EUCLIDEAN SPACE Euclidean Ring A RING without zero divisors in which an integer norm and an associated division algorithm (i.e., a EUCLIDEAN ALGORITHM ) can be defined. For signed integers, the usual norm is the ABSOLUTE VALUE and the division algorithm gives the ordinary QUOTIENT and REMAINDER . For polynomials, the norm is the degree. Important examples of Euclidean rings (besides Z) are the GAUSSIAN INTEGERS and C[x], the RING of polynomials with complex coefficients. All Euclidean rings are also PRINCIPAL RINGS . See also EUCLIDEAN ALGORITHM ,PRINCIPAL RING, RING References Wilson, J. C. "A Principle Ring that is Not a Euclidean Ring." Math. Mag. 34 /C1/8, 1973. Euclidean Space Euclidean n-space is the SPACE of all n-tuples of REAL NUMBERS ,( /x1 ;x2 ; ..., xn) and is denoted Rn : It is sometimes also called Cartesian space. Rn is a VECTOR SPACE and has LEBESGUE COVERING DIMENSION n. Elements of Rn are called n-VECTORS . R1 /C30R is the set of REAL NUMBERS (i.e., the REAL LINE), and R2 is called the EUCLIDEAN PLANE . In Euclidean space, COVAR- IANT and CONTRAVARIANT quantities are equivalent so /C0ej /C30 /C0ej :/ See also EUCLIDEAN PLANE ,P SEUDO- EUCLIDEAN SPACE ,REAL LINE,VECTOR References Gray, A. "Euclidean Spaces." §1.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 2 /C1/, 1997. Euclid-Mullin Sequence The sequence of numbers obtained by letting ai /C302; and defining an /C301pf 1 /C27Yn/C281 k/C301ak ! where lpf(n) is the LEAST PRIME FACTOR . The first few terms are 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ... (Sloane’s A000945). Only 43 terms of the sequence are known; the 44th requires factoring a composite 180-digit number. See also EUCLID NUMBER ,LEAST PRIME FACTOR References Guy, R. K. and Nowakowski, R. "Discovering Primes with Euclid." Delta (Waukesha) 5,4 9/C1/3, 1975. Mullin, A. A. "Recursive Function Theory." Bull. Amer. Math. Soc. 69, 737, 1963. Naur, T. "Mullin’s Sequence of Primes Is Not Monotonic." Proc. Amer. Math. Soc. 90,4 3/C1/4, 1984. Sloane, N. J. A. Sequences A000945/M0863 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wagstaff, S. S. "Computing Euclid’s Primes." Bull. Institute Combin. Applications 8,2 3/C1/2, 1993. Eudoxus’s Kampyle KAMPYLE OF EUDOXUSEuler Angles According to E ULER’S ROTATION THEOREM , any ROTA- TION may be described using three ANGLES . If the ROTATIONS are written in terms of ROTATION MA- TRICES B;C;and D;then a general ROTATION Acan be written as A/C30BCD : (1) The three angles giving the three rotation matrices are called Euler angles. There are several conven- tions for Euler angles, depending on the axes about which the rotations are carried out. Write the MATRIX Aas A/C13a11a12a13 a21a22a23 a31a32a332 435: (2) The so-called " x-convention," illustrated above, is the most common definition. In this convention, the rotation given by Euler angles ( f;u;c));where the first rotation is by an angle fabout the Z-AXIS , the second is by an angle u/C23[0;p] about the X-AXIS , and the third is by an angle cabout the Z-AXIS (again). Note, however, that several notational conventions for the angles are in common use. Goldstein (1960,pp. 145 /C1 /48) and Landau and Lifschitz (1976) use (f;u;c);Tuma (1974) says ( c;u;f) is used in aeronautical engineering in the analysis of spacevehicles (but claims that ( f;u;c) is used in the analysis of gyroscopic motion), while Bate et al. (1971) use ( V;i;v):Goldstein remarks that conti- nental authors usually use ( c;u;f);and warns that left-handed coordinate systems are also in occasionaluse (Osgood 1937, Margenau and Murphy 1956 /C1 /4). Here, the notation ( f;u;c) is used, a convention also followed by Mathematica ’sRotateMatrix3D [phi, theta ,psi] in the Mathematica add-on package Geometry‘Rotations‘ (which can be loaded with the command BBGeometry‘ ) andRotateSha- pe[g,phi,theta ,psi] in the Mathematica add-on packageGraphics‘Shapes‘ (which can be loaded with the command BBGraphics‘ ) commands. In thex-convention, the component rotations are then given by D/C13cosfsinf0 /C28sinfcosf0 00 12 435 (3) C /C1310 0 0 cos u sin u 0 /C28sin u cos u2 435 (4) B /C13cos c sin c 0 /C28sin c cos c 0 00 12435; (5) so a 11 /C30cos c cos f /C28cos u sin f sin c a12 /C30cos c sin f /C27cos u cos f sin c a13 /C30sin c sin u a21 /C30/C28sin c cos f /C28cos u sin f cos c a22 /C30/C28sin c sin f /C27cos u cos f cos c a23 /C30cos c sin u a31 /C30sin u sin f a32 /C30/C28sin u cos f a33 /C30cos u To obtain the components of the ANGULAR VELOCITY v in the body axes, note that for a MATRIX A /C13 A1A2A3 ½/C138 ; (6) it is true that a11a12a13 a21a22a23 a31a32a332435v x vy vz2435/C30a 11 vx /C27a12 vy /C27a13 vz a21 vx /C27a22 vy /C27a23 vz a31 vx /C27a32 vy /C27a33 vz2435ð7Þ /C30A 1 vx /C27A2 vy /C27A3 vz : (8) Now, vzcorresponds to rotation about the f axis, so look at the vz component of Av; vf /C30A1 vz /C30sin c sin u cos c sin u cos u2 435˙f : (9) The line of nodes corresponds to a rotation by u about the j /-axis, so look at the vj component of B v; vu /C30B1 vj /C30B1 ˙u /C30cos c /C28sin c 02435˙u: (10) Similarly, to find rotation by c about the remaining axis, look at the v ccomponent of Bv; vc/C30B3vc/C30B3c/C300 0 12 435˙c: (11) Combining the pieces gives v/C30sincsinu˙f/C27cosc˙u coscsinu˙f/C28sinc˙u cosu˙f/C27˙c:2 435 (12)For more details, see Goldstein (1980, p. 176) and Landau and Lifschitz (1976, p. 111). The x-convention Euler angles are given in terms of the C AYLEY- KLEIN PARAMETERS by f/C30/C282iln9a1=2g1=4 b1=4(1/C27bg)1=4"# ;/C282iln9ia1=2g1=4 b1=4(1/C27bg)1=4"# (13) c/C30/C282iln9a1=2b1=4 g1=4(1/C27bg)1=4"# ;/C282iln9ia1=2b1=4 g1=4(1/C27bg)1=4"# (14) u/C3092 cos/C2819ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27bgp;j1ffl;j1{ : (15) In the " y-convention," fx/C13fy/C271 2p (16) cx/C13cy/C281 2p: (17) Therefore, sinfx/C30cosfy (18) cosfx/C30/C28sinfy (19) sincx/C30/C28coscy (20) coscx/C30sincy; (21) giving rotation matrices D/C13/C28sinfcosf0 /C28cosf/C28sinf0 00 12 435 (22) C/C1310 0 0 cos usinu 0/C28sinucosu2 435 (23) B/C13sinc/C28cosc0 coscsinc0 00 12 435 (24) andAis given by a 11/C30/C28sincsinf/C27cosucosfcosc a12/C30sinccosf/C27cosusinfcosc a13/C30/C28coscsinu a21/C30/C28coscsinf/C28cosucosfsinc a22/C30cosccosf/C28cosusinfsinc a23/C30sincsinu a31/C30sinucosf a32/C30sinusinf a33/C30cosu: In the " xyz" (pitch-roll-yaw) convention, uis pitch, c is roll, and fis yaw. D /C13cos f sin f 0 /C28sin f cos f 0 00 12 435 (25) C /C13cos u 0 /C28sin u 01 0 sin u 0 cos u2 435 (26) B /C1310 0 0 cos c sin c 0 /C28sin c cos c2435 (27) and A is given by a 11 /C30cos u cos f a12 /C30cos u sin f a13 /C30/C28sin u a21 /C30sin c sin u cos f /C28cos c sin f a22 /C30sin c sin u sin f /C27cos c cos f a23 /C30cos u sin c a31 /C30cos c sin u cos f /C27sin c sin f a32 /C30cos c sin u sin f /C28sin c cos f a33 /C30cos u cos c: Varshalovich (1988, pp. 21 /C1/3) use the notation ( a; b; g)or( a?; b?; g ?) to denote the Euler angles, and give three different angle conventions, none of which corresponds to the x-convention. A set of parameters sometimes used instead of angles are the EULER PARAMETERS e0 ; e1 ; e2 and e3 ; defined by e0 /C13cosf 2 ! (28) e /C13e1 e2 e32 435/C30ˆn sin f 2 ! : (29) Using EULER PARAMETERS (which are QUATERNIONS ), an arbitrary ROTATION MATRIX can be described by a11 /C30e2 0 /C27e21 /C28e22 /C28e23 a12 /C302(e1e2 /C27e0e3) a13 /C302(e1e3 /C28e0e2) a21 /C302(e1e2 /C28e0e3) a22 /C30e20 /C28e21 /C27e22 /C28e23 a23 /C302(e2e3 /C27e0e1) a31 /C302(e1e3 /C27e0e2) a32 /C302(e2e3 /C28e0e1) a33 /C30e20 /C28e21 /C28e22 /C27e23 (Goldstein 1960, p. 153). If the coordinates of two pairs of n points xi and x?i are known, one rotated with respect to the other, then the Euler rotation matrix can be obtained in a straight- forward manner using LEAST SQUARES FITTING . Write the points as arrays of vectors, so [x?i /C1/C1/C1x ?n] /C30A[x1 /C1/C1/C1xn] : (30) Writing the arrays of vectors as matrices givesX?/C30AX (31) X ?XT /C30AXXT ; (32) and solving for A gives A /C30X ?XT(XXT)/C281 : (33) However, we want the angles u; f ; and c; not their combinations contained in the MATRIX A : Therefore, write the 3 /C293 MATRIX A /C30f1( u; f ; c) f2(u ; f ; c) f3( u; f; c) f4( u; f ; c) f5(u ; f ; c) f6( u; f; c) f7(u;f;c)f7(u;f;c)f9(u;f;c)2 435 (34) as a 1 /C299 VECTOR f/C30f1(u;f;c) n f9(u;f;c)2 435: (35) Now set up the matrices @f1 @uj ui;fi;ci@f1 @fj ui;fi;ci@f1 @cj ui;fi;cinnn @f9 @uj ui;fi;ci@f9 @fj ui;fi;ci@f9 @cj ui;fi;ci266666643 7777775du df dc2 435/C30df:(36) Using NONLINEAR LEAST SQUARES FITTING then gives solutions which converge to ( u;f;c):/ See also CAYLEY- KLEIN PARAMETERS ,EULER PARA- METERS ,EULER’S ROTATION THEOREM ,INFINITESIMAL ROTATION ,QUATERNION ,ROTATION ,ROTATION FOR- MULA ,ROTATION MATRIX References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 198 /C1/00, 1985. Bate, R. R.; Mueller, D. D.; and White, J. E. Fundamentals of Astrodynamics. New York: Dover, 1971. Goldstein, H. "The Euler Angles" and "Euler Angles in Alternate Conventions." §4/C1/and Appendix B in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 143 /C1/48 and 606 /C1/10, 1980. Kraus, M. "LiveGraphics3D Example: Euler Angles." http:// wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGra- phics3D/examples/Euler.html. Landau, L. D. and Lifschitz, E. M. Mechanics, 3rd ed. Oxford, England: Pergamon Press, 1976. Margenau, H. and Murphy, G. M. The Mathematics of Physics and Chemistry, 2 vols. Princeton, NJ: Van Nostrand, 1956 /C1/4. Osgood, W. F. Mechanics. New York: Macmillan, 1937. Tuma, J. J. Dynamics. New York: Quantum Publishers, 1974. Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Description of Rotation in Terms of the Euler Angles." §1.4.1 in Quantum Theory of Angular Momen- tum. Singapore: World Scientific, pp. 21 /C1/3, 1988. Euler Brick A RECTANGULAR PARALLELEPIPED ("BRICK ") with in- teger edges a > b > c and face diagonals dij given by dab /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2p (1) dac /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2 /C27c2p (2) dbc /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 /C27c2p : (3) The problem is also called the brick problem, diag- onals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem. The smallest solution with integer edges and face diagonals has sides (a ; b; c) /C30(240 ; 117; 44) and face DIAGONALS dab /C30267; dac /C30244; and dbc /C30125; and was discovered by Halcke (1719; Dickson 1952, pp. 497 /C1/00). Interest in this problem was high during the 18th century, and Saunderson (1740) found a parametric solution, while Euler (1770, 1772) found at least two parametric solutions. Kraitchik gave 257 cuboids with the ODD edge less than 1 million (Guy 1994, p. 174). F. Helenius has compiled a list of the 5003 smallest (measured by the longest edge) Euler bricks. The first few are (240, 117, 44), (275, 252, 240), (693, 480, 140), (720, 132, 85), (792, 231, 160), ... (Sloane’s A031173, A031174, and A031175). Para- metric solutions for Euler bricks are also known. No solution is known to the more general problem in which the oblique SPACE DIAGONAL dabc /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2 /C27b2 /C27c2p (4) is also an INTEGER . If such a brick exists, the smallest side must be at least 1,281,000,000 (R. Rathbun 1996). Such a solution is equivalent to solving the DIOPHANTINE EQUATIONS A2 /C27B2 /C30C2 (5) A2 /C27D2 /C30E2 (6) B2 /C27D2 /C30F2 (7) B2 /C27E2 /C30G2 : (8) A solution with integral SPACE DIAGONAL and two out of three face diagonals is a /C30672, b /C30153, and c /C30104, giving dab /C303ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 52777p ;dac/C30680;dbc/C30185; anddabc/C30697;which was known to Euler. A solution giving integral space and face diagonals with only a single nonintegral EDGE is a/C3018720, b/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 211773121p ;and c/C307800, giving dab/C3023711 ;dac/C30 20280 ;dbc/C3016511 ;anddabc/C3024961 :/See also CUBOID ,CYCLIC QUADRILATERAL ,DIAGONAL (POLYHEDRON ), PARALLELEPIPED ,P YTHAGOREAN QUADRUPLE References Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952. Guy, R. K. "Is There a Perfect Cuboid? Four Squares whose Sums in Pairs are Square. Four Squares whose Differ- ences are Square." §D18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 173 /C1/81, 1994. Halcke, P. Deliciae Mathematicae; oder, Mathematisches sinnen-confect. Hamburg, Germany: N. Sauer, p. 265, 1719. Helenius, F. First 1000 Primitive Euler Bricks. NOTEBOOKS/ EULER BRICKS.DAT . Leech, J. "The Rational Cuboid Revisited." Amer. Math. Monthly 84, 518/C1/33, 1977. Erratum in Amer. Math. Monthly 85, 472, 1978. Sloane, N. J. A. Sequences A031173, A031174, and A031175 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Rathbun, R. L. Personal communication, 1996.Saunderson, N. The Elements of Algebra in 10 Books, Vol. 2. Cambridge, England: University Press, pp. 429 /C1 /31, 1740. Spohn, W. G. "On the Integral Cuboid." Amer. Math. Monthly 79,5 7/C1/9, 1972. Spohn, W. G. "On the Derived Cuboid." Canad. Math. Bull. 17, 575/C1/77, 1974. Wells, D. G. The Penguin Dictionary of Curious and Inter- esting Numbers. London: Penguin, p. 127, 1986. Euler Chain ACHAIN whose EDGES consist of all graph EDGES . Euler Characteristic Let a closed surface have GENUS g. Then the POLY- HEDRAL FORMULA generalizes to the P OINCARE ´FOR- MULA x/C13V/C28E/C27F/C30x(g); (1) where x(g)/C302/C282g (2) is the Euler characteristic, sometimes also known as the E ULER- POINCARE ´CHARACTERISTIC . The POLYHE- DRAL FORMULA corresponds to the special case g/C300. The only compact closed surfaces with Euler char-acteristic 0 are the K LEIN BOTTLE and TORUS (Dodson and Parker 1997, p. 125). In terms of the INTEGRAL CURVATURE of the surface K, ggKd a/C302px: (3) The Euler characteristic is sometimes also called the EULER NUMBER . It can also be expressed as x/C30p0/C28p1/C27p2; (4) where piis the ith B ETTI NUMBER of the space. See also CHROMATIC NUMBER ,EULER NUMBER (FI- NITE COMPLEX ), MAP COLORING ,POINCARE ´ FORMULA , POLYHEDRAL FORMULA References Coxeter, H. S. M. "Poincare ´’s Proof of Euler’s Formula." Ch. 9 in Regular Polytopes, 3rd ed. New York: Dover, pp. 165 /C1/72, 1973. Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, 1997. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 635, 1997. Euler Constant E,EULER- MASCHERONI CONSTANT ,M ACLAURIN- CAU- CHY THEOREM Euler Curvature Formula The curvature of a surface satisfies k/C30k1cos2u/C27k2sin2u; where kis the normal CURVATURE in a direction making an ANGLE uwith the first principal direction andk1andk2are the PRINCIPAL CURVATURES . See also PRINCIPAL CURVATURES Euler Differential Equation The general nonhomogeneous differential equation is given by x2d2y dx2/C27axdy dx/C27by/C30S(x); (1) and the homogeneous equation is x2yƒ/C27axy?/C27by/C300 (2) yƒ/C27a xy?/C27b x2y/C300: (3) Now attempt to convert the equation from yƒ/C27p(x)y?/C27q(x)y/C300 (4) to one with constant COEFFICIENTS d2y dz2/C27Ady dz/C27By/C300 (5) by using the standard transformation for linear SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS . Comparing (3) and (5), the functions p(x) and q(x) are p(x)/C13a x/C30ax/C281(6) q(x)/C13b x2/C30bx/C282: (7)LetB/C13band define z/C13B/C281=2gffiffiffiffiffiffiffiffiffi q(x)p dx/C30b/C281=2gffiffiffiffiffiffiffiffiffiffiffibx /C282p dx /C30gx/C281dx/C30lnx: (8) Then Ais given by A/C13q?(x)/C272p(x)q(x) 2[q(x)]3=2B1=2 /C30/C282bx/C283/C272(ax/C281)(bx/C282) 2(bx/C282)3=2b1=2 /C30a/C281; (9) which is a constant. Therefore, the equation becomes a second-order ODE with constant COEFFICIENTS d2y dz2/C27(a/C281)dy dz/C27by/C300: (10) Define r1/C131 2/C28A/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2/C284Bp;j1ffl;j1{ /C301 21/C28a/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C281)2/C284bq;j2r;j21 (11) r2/C131 2/C28A/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2/C284Bp;j1ffl;j1{ /C301 21/C28a/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C281)2/C284bq;j2r;j21 (12) and a/C131 2(1/C28a) (13) b/C1312ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4b/C28(a/C281)2q : (14) The solutions are y/C30c1er1z/C27c2er2z(a/C281)2>4b c1/C27c2z)eaz(a/C281)2/C304 caz[c1cos(bz)/C27c2sin (bz)] (a/C281)2B4b:8 < :ð15Þ In terms of the original variable x, y/C30c1½x½r1/C27c2½x½r2 (a/C281)2>4b (c1/C27c2ln½x½)½x½a(a/C281)2/C304b ½x½a[c1cos(bln½x½)/C27c2sin (bln½x½)] (a/C281)2B4b:8 < : ð16Þ Zwillinger (1997, p. 120) gives two other types of equations known as Euler differential equations, y?/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ay4/C27by3/C27cy2/C27dy/C27e ax3/C27bx3/C27cx2/C27dx/C27es (17) (Valiron 1950, p. 201) and y ?/C27y2 /C30 axm (18) (Valiron 1950, p. 212), the latter of which can be solved in terms of Bessel functions. See also EULER’S EQUATIONS OF INVISCID MOTION References Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Brookline, MA: Math. Sci. Press, 1950. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Euler Equation EULER DIFFERENTIAL EQUATION ,EULER’S EQUATIONS OF INVISCID MOTION ,E ULER FORMULA ,E ULER- LA- GRANGE DIFFERENTIAL EQUATION Euler Formula The Euler formula states eix /C30cos x /C27i sin x; (1) where I is the IMAGINARY NUMBER . Note that Euler’s POLYHEDRAL FORMULA is sometimes also called the Euler formula, as is the EULER CURVATURE FORMULA . The equivalent expression ix /C30ln(cos x /C27i sin x) (2) had previously been published by Cotes (1714). The special case of the formula with x /C30 p gives the beautiful identity eip /C271 /C300; (3) an equation connecting the fundamental numbers I, PI, E, 1, and 0 (ZERO ). The Euler formula can be demonstrated using a series expansion eix /C30X/C12 n/C300ðixÞn n! /C30X/C12 n/C300( /C281)nx2n (2n)!/C27iX/C12 n/C301( /C281)n/C281x2n/C281 (2n /C28 1)! /C30cos x /C27i sin x: (4) It can also be proven using a COMPLEX integral. Let z /C13cos u /C27i sin u (5) dz /C30(/C28sin u /C27i cos u) du /C30i(cos u /C27i sin u) du /C30iz d u (6) gdz z/C30g idu (7) ln z /C30iu ; (8)so z /C30eiu /C13cos u /C27i sin u : (9) See also DE MOIVRE’S IDENTITY ,POLYHEDRAL FOR- MULA References Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67 /C1/8, 1988. Conway, J. H. and Guy, R. K. "Euler’s Wonderful Relation." The Book of Numbers. New York: Springer-Verlag, pp. 254 /C1/56, 1996. Cotes, R. Philosophical Transactions 29, 32, 1714. Euler, L. Miscellanea Berolinensia 7, 179, 1743. Euler, L. Introductio in Analysin Infinitorum, Vol. 1. Lau- sanne, p. 104, 1748. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 212, 1998. Euler Four-Square Identity The amazing polynomial identity (a2 1 /C27a22 /C27a23 /C27a24)(b21 /C27b22 /C27b23 /C27b24) /C30(a1b1 /C28a2b2 /C28a3b3 /C28a4b4)2 /C27(a1b2 /C27a2b1 /C27a3b4 /C28a4b3)2 /C27(a1b3 /C28a2b4 /C27a3b1 /C27a4b2)2 /C27(a1b4 /C27a2b3 /C28a3b2 /C27a4b1)2 ; communicated by Euler in a letter to Goldbach on April 15, 1750 (incorrectly given as April 15, 1705– before Euler was born–in Conway and Guy 1996, p. 232). The identity also follows from the fact thatthe norm of the product of two QUATERNIONS is the product of the norms (Conway and Guy 1996). See also FIBONACCI IDENTITY ,L AGRANGE’S FOUR- SQUARE THEOREM ,LEBESGUE IDENTITY References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 232, 1996. Nagell, T. Introduction to Number Theory. New York: Wiley, pp. 191 /C1/92, 1951. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, p. 8, 1996. Euler Graph EULERIAN GRAPH Euler Identity For½z½B1; Y/C12 k/C301(1/C27zk)/C30Y/C12 k/C301(1/C28z2k/C281)/C281: Expanding and taking a series expansion about zero for either side gives 1 /C27z /C27z2 /C272z3 /C272z4 /C273z5 /C274z6 /C275z7 /C27...; giving 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, ... (Sloane’s A000009), the number of partitions of n into distinct parts. See also JACOBI TRIPLE PRODUCT ,PARTITION FUNC- TION P, Q-SERIES References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, p. 72, 1935. Franklin. Comptes Rendus 92, 448 /C1/50, 1881. Hardy, G. H. §6.2 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 83 /C1/5, 1999. Hardy, G. H. and Wright, E. M. §19.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clar- endon Press, 1979. MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 21 /C1/3, 1960. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 55, 1951. Sloane, N. J. A. Sequences A000009/M0281 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Euler Integral Euler integration was defined by Schanuel and subsequently explored by Rota, Chen, and Klain. The Euler integral of a FUNCTION f : R 0 R (assumed to be piecewise-constant with finitely many disconti- nuities) is the sum of f(x) /C281 2[f(x/C27) /C27f(x/C28)] over the finitely many discontinuities of f. The n-D Euler integral can be defined for classes of functions Rn0R:Euler integration is additive, so the Euler integral of f/C27gequals the sum of the Euler integrals offandg. See also EULER MEASURE Euler Law POLYHEDRAL FORMULA Euler L-Function A special case of the A RTIN L-FUNCTION for the POLYNOMIAL x2/C271:It is given by L(s)/C30Y podd prime1 1/C28x/C28(p)p/C28s; where x/C28(p)/C131 for p/C131 (mod 4) /C281 for p/C133 (mod 4)/C30/C281 p ! ;( where ( /C281=p)i saL EGENDRE SYMBOL .References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537/C1/49, 1996. Euler Line The line on which the ORTHOCENTER H,CENTROID G, CIRCUMCENTER O,DELONGCHAMPS POINT L,NINE- POINT CENTER F, and the TANGENTIAL TRIANGLE CIRCUMCIRCLE OTof a TRIANGLE lie. The INCENTER lies on the Euler line only if the TRIANGLE is an ISOSCELES TRIANGLE . The Euler line consists of all points with TRILINEAR COORDINATES a:b:gwhich satisfy abg cosA cosB cosC cosBcosCcosCcosAcosAcosB;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12/C300; (1) which simplifies to acosA(cos 2B/C28cos2C)/C27bcosB(cos2C/C28cos2A) /C27gcosC(cos2A/C28cos2B)/C300: (2) This can also be written asin(2 A) sin( B/C28C)/C27bsin(2 B) sin( C/C28A) /C27gsin(2 C) sin( A/C28B)/C300: (3) The Euler line may also be given parametrically in EXACT TRILINEAR COORDINATES by P(l)/C30O/C27lH (4) where the following table summarized important TRIANGLES CENTERS corresponding to various values ofl(including the factor of 1/2 omitted by Oldknow 1996). /l/TRIANGLE CENTER -1 POINT AT INFINITY //C281 2/DELONGCHAMPS POINT L 0CIRCUMCENTER O /1 2/CENTROID G 1NINE-POINT CENTER F //C12/ORTHOCENTER H The CIRCUMCENTER O, NINE-POINT CENTER F, CEN- TROID G, and ORTHOCENTER H form a HARMONIC RANGE with GO /C301 2 HG (5) OG /C301 3 HO (6) OF /C301 2 HO (7) FG /C3016 HO (8) (Honsberger 1995, p. 7). The Euler line intersects the SODDY LINE in the DE LONGCHAMPS POINT , and the GERGONNE LINE in the EVANS POINT . The ISOTOMIC CONJUGATE of the Euler line is called JERABEK’S HYPERBOLA (Casey 1893, Vandeghen 1965). See also CENTROID (TRIANGLE ), CIRCUMCENTER , EVANS POINT ,GERGONNE LINE,JERABEK’S HYPERBO- LA, DE LONGCHAMPS POINT ,N INE-POINT CENTER , ORTHOCENTER ,SODDY LINE,TANGENTIAL TRIANGLE References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2ndrev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Coxeter, H. S. M. and Greitzer, S. L. "The Medial Triangle and Euler Line." §1.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 18 /C1 /0, 1967. Do¨rrie, H. "Euler’s Straight Line." §27 in 100 Great Problems of Elementary Mathematics: Their History and Solutions.New York: Dover, pp. 141 /C1 /42, 1965. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 28, 1928. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 7, 1995. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 117 /C1/19, 1990. Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319/C1/29, 1996. Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian-gle." Amer. Math. Monthly 72, 1091 /C1 /094, 1965. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 69, 1991. Euler Measure Define the Euler measure of a polyhedral set as the EULER INTEGRAL of its indicator function. It is easy to show by induction that the Euler measure of a closedbounded convex POLYHEDRON is always 1 (indepen- dent of dimension), while the Euler measure of a d-D relative-open bounded convex POLYHEDRON is (/C281)d:/Euler Number The Euler numbers, also called the SECANT NUMBERS orZIG NUMBERS , are defined for xjjBp=2b y sech x/C281/C13/C28E/C311x2 2!/C27E/C312x4 4!/C28E/C313x6 6!/C27... ( 1 ) secx/C281/C13E/C311x2 2!/C27E/C312x4 4!/C28E/C313x6 6!/C27...; (2) where sech is the HYPERBOLIC SECANT and sec is the SECANT . Euler numbers give the number of ODD ALTERNATING PERMUTATIONS and are related to G EN- OCCHI NUMBERS . The base Eof the NATURAL LOGA- RITHM is sometimes known as Euler’s number. Some values of the Euler numbers are E/C311/C301 E/C312/C305 E/C313/C3061 E/C314/C301;385 E/C315/C3050;521 E/C316/C302;702;765 E/C317/C30199;360;981 E/C318/C3019;391;512;145 E/C319/C302;404;879;675;441 E/C3110/C30370;371;188;237;525 E/C3111/C3069;348;874;393;137;901 E/C3112/C3015;514;534;163;557;086;905 (Sloane’s A000364). The first few PRIME Euler num- bers En/C31occur for n/C302, 3, 19, 227, 255, ... (Sloane’s A014547) up to a search limit of n/C301415. The slightly different convention defined by E2n/C30(/C281)nE/C31n (3) E2n/C271/C300 (4) is frequently used. These are, for example, the Euler numbers computed by the Mathematica function EulerE [n]. This definition has the particularly sim- ple series definition sech x/C13X/C12 k/C300Ekxk k!(5) and is equivalent to En /C302nEn(1 2) ; (6) where En(x)isanE ULER POLYNOMIAL . The Euler numbers have the ASYMPTOTIC SERIES E2n /C2(/C281)n8ffiffiffi n ps 4n pe !2n : (7) To confuse matters further, the EULER CHARACTER- ISTIC is sometimes also called the "Euler number." See also BERNOULLI NUMBER ,EULER NUMBER (FINITE COMPLEX ), EULERIAN NUMBER ,EULER POLYNOMIAL , EULER ZIGZAG NUMBER ,GENOCCHI NUMBER References Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804 /C1/06, 1972. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 110 /C1/11, 1996. Guy, R. K. "Euler Numbers." §B45 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 101, 1994. Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen fu¨r Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995. Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663 /C1/ 88, 1967. Sloane, N. J. A. Sequences A0003644019 and A014547 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Spanier, J. and Oldham, K. B. "The Euler Numbers, En :/" Ch. 5 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 39 /C1/2, 1987. Young, P. T. "Congruences for Bernoulli, Euler, and Stirling Numbers." J. Number Th. 78, 204 /C1/27, 1999. Euler Number (Finite Complex) The Euler number of a finite complex Kis defined by x(K)/C30X (/C281)prank( Cp(K)): The Euler number is a topological invariant. See also EULER CHARACTERISTIC ,LEFSCHETZ NUMBER References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, p. 124, 1993. Euler Parameters The four parameters e0;e1;e2;and e3describing a finite rotation about an arbitrary axis. The Euler parameters are defined bye0/C13cosf 2 ! (1) e/C13e1 e2 e32 435/C30ˆnsin f 2 ! ; (2) and are a QUATERNION in scalar-vector representation (e0;e)/C30e0/C27e1i/C27e2j/C27e3k: (3) Because E ULER’S ROTATION THEOREM states that an arbitrary rotation may be described by only three parameters, a relationship must exist between thesefour quantities e 2 0/C27e /C215e/C30e20/C27e21/C27e22/C27e23/C301 (4) (Goldstein 1980, p. 153). The rotation angle is then related to the Euler parameters by cosf/C302e2 0/C281/C30e20/C28e /C215e/C30e20/C28e21/C28e22/C28e23 (5) ˆnsinf/C302ee0: (6) The Euler parameters may be given in terms of the EULER ANGLES by e0/C30cos[1 2(f/C27c)] cos(12u) (7) e1/C30sin[1 2(f/C28c)] sin(12u) (8) e2/C30cos[1 2(f/C28c)] sin(12u) (9) e3/C30sin[1 2(f/C27c)] cos(12u) (10) (Goldstein 1980, p. 155). Using the Euler parameters, the ROTATION FORMULA becomes r?/C30r(e2 0/C28e21/C28e22/C28e23)/C272e(e /C215r)/C27(r/C29ˆn) sin f;(11) and the ROTATION MATRIX becomes x? y? z?2 435/C30Ax y z2 435; (12) where the elements of the matrix are a ij/C30dij(e2 0/C28ekek)/C272eiej/C272eijke0ek: (13) Here, E INSTEIN SUMMATION has been used, dijis the KRONECKER DELTA , and eijkis the PERMUTATION SYMBOL . Written out explicitly, the matrix elements are a11/C30e20/C27e21/C28e22/C28e23 (14) a12/C302(e1e2/C27e0e3) (15) a13/C302(e1e3/C28e0e2) (16) a21/C302(e1e2/C28e0e3) (17) a22 /C30e2 0 /C28e21 /C27e22 /C28e23 (18) a23 /C302(e2e3 /C27e0e1) (19) a31 /C302(e1e3 /C27e0e2) (20) a32 /C302(e2e3 /C28e0e1) (21) a33 /C30e20 /C28e21 /C28e22 /C27e23 : (22) See also EULER ANGLES ,Q UATERNION ,R OTATION FORMULA ,ROTATION MATRIX References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 198 /C1/00, 1985. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980. Landau, L. D. and Lifschitz, E. M. Mechanics, 3rd ed. Oxford, England: Pergamon Press, 1976. Euler Point The MIDPOINTS MHA ; MHB ; MHC of the segments which join the VERTICES of a triangle and the ORTHOCENTER H are called Euler points. They are three of the nine prominent points of a triangle through which the NINE-POINT CIRCLE passes. See also FEUERBACH’S THEOREM ,NINE-POINT CIRCLE References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 6, 1995. Euler Polyhedral Formula POLYHEDRAL FORMULA Euler Polynomial The Euler polynomial En(x) is given by the APPELL SEQUENCE with g(t) /C301 2(et /C271); (1)giving the GENERATING FUNCTION 2ext et /C27 1 /C13X/C12 n/C300En(x)tn n! : (2) Roman (1984, p. 100) defines a generalization E(a) n(x) for which En(x) /C30E(1) n(x): Euler polynomials are re- lated to the BERNOULLI NUMBERS by En /C281(x) /C302n nBnx /C27 1 2 ! /C28Bnx 2 ! "# (3) /C302 nBn(x) /C282nBnx 2 ! "# (4) En/C282(x) /C302n 2;j1z;j1}/C281Xn /C282 k/C300n 2;j1z;j1} [(2n/C28k /C281)Bn/C28kBk(x)];ð5Þ wheren k;jr;j1 is a BINOMIAL COEFFICIENT . Setting x /C301=2 and normalizing by 2n gives the EULER NUMBER En /C302nEn(1 2) : (6) Call E ?n /C30En(0) ; then the first few terms are /C281 =2; 0, 1/4, /C281=2; 0, 17/8, 0, 31/2, 0, .... The terms are the same but with the SIGNS reversed if x /C301. These values can be computed using the double sum En(0) /C302/C28nXn j/C301(/C281)j/C27n/C271jkXn /C28j k /C300n /C271 k;j1z;j1}"# : (7) The BERNOULLI NUMBERS Bnfor n /C211 can be ex- pressed in terms of the E ?n by Bn /C30/C28nE ?n/C281 2(2n /C28 1) : (8) The Newton expansion of the Euler polynomials is given by En(x) /C30Xn j /C300Xn k /C30j/C281 j;j1z;j1}1 2j(k)jS(n ; k)(x)k /C28j ; (9) wheren k;jr;j1 is a BINOMIAL COEFFICIENT ,(k)jis a FALLING FACTORIAL , and S(n;k)i saS TIRLING NUMBER OF THE SECOND KIND (Roman 1984, p. 101). The Euler polynomials satisfy the identity Xn k/C300n 2;j1z;j1} Ek(z)En/C28k(w) /C302(1/C28w/C28z)En(z/C27w)/C272En/C271(z/C27w) (10) fornaNONNEGATIVE INTEGER . See also APPELL SEQUENCE ,BERNOULLI POLYNOMIAL , EULER NUMBER ,GENOCCHI NUMBER References Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804 /C1/06, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Zeta Function z(s; x) ; Bernoulli Poly- nomials Bn(x); Euler Polynomials En(x) ; and Polyloga- rithms Lin(x) :/" §1.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 23 /C1/4, 1990. Roman, S. "The Euler Polynomials." §4.2.3 in The Umbral Calculus. New York: Academic Press, pp. 100 /C1/06, 1984. Spanier, J. and Oldham, K. B. "The Euler Polynomials /En(x) :/" Ch. 20 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 175 /C1/81, 1987. Euler Polynomial Identity EULER FOUR- SQUARE IDENTITY Euler Power Conjecture EULER’S SUM OF POWERS CONJECTURE Euler Product For s /C211, the RIEMANN ZETA FUNCTION is given by z(s) /C13X/C12 n/C3011 ns /C30Y/C12 n/C3011 1 /C281 ps n; where piis the ith PRIME . This is Euler’s product (Whittaker and Watson 1990). Let s 0 1 ; then the terms in the product for upper limits n /C301, 2, ..., are given by 2, 4, 6, 15/2, 35/4, 77/8, 1001/96, 17017/1536, ... (Sloane’s A050298 and A050299). The limiting case as n 0/C12 gives MERTENS THEOREM , e g /C30 lim n0/C121 ln nYn i/C3011 1 /C281 pi; where g is the EULER- MASCHERONI CONSTANT . See also DEDEKIND FUNCTION ,EULER- MASCHERONI CONSTANT ,M ERTENS THEOREM ,R IEMANN ZETA FUNCTION ,STIELTJES CONSTANTS References Hardy, G. H. and Wright, E. M. "The Zeta Function." §17.2 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 245 /C1/47, 1979. Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, p. 216, 1996. Shimura, G. Euler Products and Eisenstein Series. Provi- dence, RI: Amer. Math. Soc., 1997. Sloane, N. J. A. Sequences A050298 and A050299 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Whittaker, E. T. and Watson, G. N. "Euler’s Product for /z(s) :/" §13.3 in A Course in Modern Analysis, 4th ed.Cambridge, England: Cambridge University Press, pp. 271 /C1/72, 1990. Euler Pseudoprime An Euler pseudoprime is a composite number n which satisfies 2(n/C281)=2 /C1391 (mod n) : The first few base-2 Euler pseudoprimes are 341, 561, 1105, 1729, 1905, 2047, ... (Sloane’s A006970). See also EULER- JACOBI PSEUDOPRIME ,PSEUDOPRIME , STRONG PSEUDOPRIME References Sloane, N. J. A. Sequences A006970/M5442 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Euler Quartic Conjecture Euler conjectured that there are no POSITIVE INTEGER solutions to the quartic DIOPHANTINE EQUATION A4 /C30B4 /C27C4 /C27D4 : This conjecture was disproved by Elkies (1988), who found an infinite class of solutions. See also DIOPHANTINE EQUATION–4TH POWERS ,EU- LER’S SUM OF POWERS CONJECTURE References Berndt, B. C. and Bhargava, S. "Ramanujan--For Low- brows." Amer. Math. Monthly 100, 644/C1/56, 1993. Elkies, N. "On A4/C27B4/C27C4/C30D4:/"Math. Comput. 51, 825/C1/ 35, 1988. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/40, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 201, 1998. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. Ward, M. "Euler’s Problem on Sums of Three Fourth Powers." Duke Math. J. 15, 827/C1/37, 1948. Wiles, A. "The Birch and Swinnerton-Dyer Conjecture." http://www.claymath.org/prize_problems/birchsd.pdf. Euler Square A square ARRAY made by combining nobjects of two types such that the first and second elements form LATIN SQUARES . Euler squares are also known as GRAECO- LATIN SQUARES ,GRAECO- ROMAN SQUARES ,o r LATIN- GRAECO SQUARES . For many years, Euler squares were known to exist for n/C303, 4, and for every ODD nexcept n/C303k:EULER’S GRAECO-ROMAN SQUARES CONJECTURE maintained that there do not exist Euler squares of order n/C304k/C272 for k/C301, 2, .... However, such squares were found to exist in 1959, refuting the CONJECTURE . See also LATIN RECTANGLE ,LATIN SQUARE ,ROOM SQUARE References Beezer, R. "Graeco-Latin Squares." http://buzzard.ups.edu/ squares.html. Fisher, R. A. The Design of Experiments, 8th ed. New York: Hafner, 1971. Kraitchik, M. "Euler (Graeco-Latin) Squares." §7.12 in Mathematical Recreations. New York: W. W. Norton, pp. 179 /C1/82, 1942. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 31 /C1/3, 1999. Euler Sum In response to a letter from Goldbach, Euler consid- ered DOUBLE SUMS OF THE FORM sh(m;n)/C30X/C12 k/C3011/C271 2/C27.../C271k ! m (k/C271)/C28n(1) /C30X/C12 k/C301[g/C27c0(k/C271)]m(k/C271)/C28n(2) with m]1 and n]2 and where gis the E ULER- MASCHERONI CONSTANT andC(x)/C30c0(x) is the DI- GAMMA FUNCTION . Euler found explicit formulas in terms of the R IEMANN ZETA FUNCTION fors(1;n) with n]2;and E. Au-Yeung numerically discovered X/C12 k/C3011/C271 2/C27.../C271k ! 2 k/C282/C3017 4z(4); (3) where z(z) is the R IEMANN ZETA FUNCTION , which was subsequently rigorously proven true (Borwein and Borwein 1995). Sums involving k/C28ncan be re-ex- pressed in terms of sums the form ( k/C271)/C28nvia X/C12 k/C3011/C271 2m/C27.../C271 km ! k/C28n /C30X/C12 k/C3001/C272 2m/C27.../C271 (k/C271)m"# (k/C271)/C28n /C30X/C12 k/C3011/C271 2m/C27.../C271 km ! (k/C271)/C28n/C27X/C12 k/C301k/C28(m/C27n) /C13sh(m;n)/C27z(m/C27n) (4) X/C12 k/C3011/C271 2/C27.../C271k ! 2 k/C28n /C30sh(2;n)/C272sh(1;n/C271)/C27z(n/C272); (5) where shis defined below. Bailey et al. (1994) subsequently considered sums OF THE FORM ssh(m;n)/C30X/C12 k/C3011/C271 2/C27.../C271k ! m (k/C271)/C28n(6) sa(m;n)/C30X/C12 k/C3011/C2812/C27.../C27(/C281)k/C271 k"#m (k/C271)/C28n(7) ah(m;n)/C30X/C12 k/C3011/C2712/C27.../C271k ! m (/C281)k/C271(k/C271)/C28n(8) aa(m;n)/C30X/C12 k/C3011/C2812/C27.../C27(/C281)k/C271 k !m (/C281)k/C271 /C2(k/C271)/C28nð9Þ sh(m;n)/C30X/C12 k/C3011/C271 2m/C27.../C271 km ! (k/C271)/C28n(10) sa(m;n)/C30X/C12 k/C3011/C281 2m/C27.../C27(/C281)k/C271 km ! (k/C271)/C28n(11) ah(m;n)/C30X/C12 k/C3011/C271 2m/C27.../C271 km ! (/C281)k/C271 /C2(k/C271)/C28nð12Þ aa(m;n)/C30X/C12 k/C3011/C281 2m/C27.../C27(/C281)k/C271 km ! (/C281)k/C271 /C2(k/C271)/C28n; ð13Þ where shandsahave the special forms sh/C30X/C12 k/C301[g/C27c0(n/C271)]m(k/C271)/C28n(14) aa/C30X/C12 k/C301fln 2/C271 2(/C281)n[c0(12n/C2712)/C28c0(12n/C271)]gm /C2(k/C271)/C28m: (15) Analytic single or double sums over z(z) can be constructed for sh(2;n)/C301 3n(n/C271)z(n/C272)/C27z(2)z(n) /C281 2nXn/C282 k/C300z(n/C28k)z(k/C272) ð16Þ sh(2;2n/C281)/C3016(2n2/C287n/C283)z(2n/C271)/C27z(2)z(2n/C281) /C2812Xn/C282 k/C301(2k/C281)z(2n/C281/C282k)z(2k/C272) ð17Þ sh(2;2n/C281) /C30/C281 2(2n2/C27n/C271)z(2n/C271)/C27z(2)z(2n/C281) ð18Þ sh(meven ;nodd) /C301 2m/C27n m;j1z;j1} /C281;j2r;j21 z(m/C27n)/C27z(m)z(n) /C28Xm/C27n j/C3012j/C282 m/C281;j1z;j1} /C272j/C282 n/C281;j1z;j1};j2r;j21 ð19Þ sh(modd;neven) /C30/C2812m/C27n m;j1z;j1} /C271;j2r;j21 z(m/C27n) /C27Xm/C27n k/C3012j/C282 m/C281;j1z;j1} /C272j/C282 n/C281;j1z;j1};j2r;j21 ð20Þ wheren m;jr;j1 is a BINOMIAL COEFFICIENT . Explicit for- mulas inferred using the PSLQ ALGORITHM include sh(2;2)/C303 2z(4)/C2712[z(2)]2(21) /C3011 360p4(22) sh(2;4)/C3023z(6)/C2813z(2)z(4)/C2713[z(2)]3/C28[z(3)]2(23) /C3037 22680p6/C28[z(3)]2(24) sh(3;2)/C3015 2z(5)/C27z(2)z(3) (25) sh(3;3)/C30/C283316z(6)/C272[z(3)]2(26) sh(3;4)/C30119 16z(7)/C2833 4z(3)z(4)/C272z(2)z(5) (27) sh(3;6)/C30197 24z(9)/C2833 4z(4)z(5)/C2837 8z(3)z(6)/C27[z(3)]3 /C273z(2)z(7) (28) sh(4;2)/C30859 24z(6)/C273[z(3)]2(29) sh(4;3)/C30/C28109 8z(7)/C2737 2z(3)z(4)/C285z(2)z(5) (30) sh(4;5)/C30/C2829 2z(9)/C2737 2z(4)z(5)/C2733 4z(3)z(6)/C2883[z(3)]3 /C287z(2)z(7) (31) sh(5;2)/C301855 16z(7)/C2733z(3)z(4)/C2757 2z(2)z(5) (32) sh(5;4)/C30890 9z(9)/C2766z(4)z(5)/C284295 24z(3)z(6)/C285[z(3)]3 /C27265 8z(2)z(7) (33) sh(6;3)/C30/C283073 12z(9)/C28243z(4)z(5)/C272097 4z(3)z(6) /C2767 3[z(3)]3/C28651 8z(2)z(7) (34) sh(7;2)/C30134701 36z(9)/C2715697 8z(4)z(5)/C2729555 24z(3)z(6) /C2756[z(3)]3/C273287 4z(2)z(7); (35) ah(2;2)/C30/C282Li4(1 2)/C281 12(ln2)4/C279948z(4)/C2874z(3)ln 2 /C271 2z(2)(ln 2)2(36)ah(2;3)/C30/C284Li5(12)/C284(ln 2)Li4(12)/C282 15(ln 2)5/C27107 32z(5) /C2874z(3)(ln 2)2/C2723z(2)(ln 2)3/C2738z(2)z(3) ð37Þ ah(3;2)/C306Li5(12)/C276(ln 2)Li4(12)/C2715(ln 2)5/C2833 8z(5) /C2721 8z(3)(ln 2)2/C28z(2)(ln 2)3/C281516z(2)z(3); (38) and aa(2;2)/C30/C284Li4(1 2)/C2816(ln 2)4/C273716z(4)/C2774z(3)(ln 2) /C282z(ln 2)2(39) aa(2;3)/C304(ln 2)Li4(12)/C2716(ln 2)5/C287932z(5)/C2711 8z(4)(ln 2) /C28z(2)(ln 2)3(40) aa(3;2)/C3030Li5(12)/C2814(ln 2)5/C281813 64z(5)/C27285 16z(4)(ln 2) /C2721 8z(3)(ln 2)2/C287 2z(2)(ln 2)3/C2734z(2)z(3); (41) where Linis a POLYLOGARITHM , and z(z) is the RIEMANN ZETA FUNCTION (Bailey and Plouffe). Of these, only sh(3;2);sh(3;3) and the identities for sa(m;n);ah(m;n) and aa(m;n) have been rigorously established. References Adamchik, V. "On Stirling Numbers and Euler Sums." J. Comput. Appl. Math. 79, 119/C1/30, 1197. http://members.- wri.com/victor/articles/stirling.html. Bailey, D. and Plouffe, S. "Recognizing Numerical Con- stants." http://www.cecm.sfu.ca/organics/papers/bailey/. Bailey, D. H.; Borwein, J. M.; and Girgensohn, R. "Experi- mental Evaluation of Euler Sums." Exper. Math. 3,1 7/C1/0, 1994. Berndt, B. C. Ramanujan’s Notebooks: Part I. New York: Springer-Verlag, 1985. Borwein, D. and Borwein, J. M. "On an Intriguing Integral and Some Series Related to z(4):/"Proc. Amer. Math. Soc. 123, 1191 /C1/198, 1995. Borwein, D.; Borwein, J. M.; and Girgensohn, R. "Explicit Evaluation of Euler Sums." Proc. Edinburgh Math. Soc. 38, 277/C1/94, 1995. de Doelder, P. J. "On Some Series Containing C(x)/C28C(y) and (C(x)/C28C(y))2for Certain Values of xandy."J. Comp. Appl. Math. 37, 125/C1/41, 1991. Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351/C1/69, 1999. Flajolet, P. and Salvy, B. "Euler Sums and Contour Integral Representation." Experim. Math. 7,1 5/C1/5, 1998. Euler System A mathematical structure first introduced by Koly- vagin (1990) and defined as follows. Let Tbe a finite- dimensional p-adic representation of the G ALOIS GROUP of a NUMBER FIELD K. Then an Euler system forTis a collection of COHOMOLOGY CLASSES cF/C23 H1(F;T) for a family of Abelian extensions FofK, with a relation between cF?andcFwhenever FƒF? (Rubin 2000, p. 4). Wiles’ proof of FERMAT’S LAST THEOREM via the TANIYAMA- SHIMURA CONJECTURE made use of Euler systems. References Kolyvagin, V. A. "Euler Systems." In The Grothendieck Festschrift, Vol. 2 (Ed. P. Cartier et al. ). Boston, MA: Birkha ¨user, pp. 435 /C1/83, 1990. Rubin, K. Euler Systems. Princeton, NJ: Princeton Univer- sity Press, 2000. Euler Totient Function TOTIENT FUNCTION Euler Transform There are (at least) three types of Euler transforms (or transformations). The first is a set of transforma- tions of HYPERGEOMETRIC FUNCTIONS , called EULER’S HYPERGEOMETRIC TRANSFORMATIONS . The second type of Euler transform is a technique for SERIES CONVERGENCE IMPROVEMENT which takes a convergent alternating series X/C12 k /C300(/C281)kak /C30a0 /C28a1 /C27a2 /C28... (1) into a series with more rapid convergence to the same value to s /C30X/C12 k /C300( /C281)k Dka0 2k /C271; (2) where the FORWARD DIFFERENCE is defined by Dka0 /C30Xk m/C300/C13(/C281)m k m;j1z;j1} ak /C28m (3) (Abramowitz and Stegun 1972; Beeler et al. 1972). The third type of Euler transform is a relationship between certain types of INTEGER SEQUENCES (Sloane and Plouffe 1995, pp. 20 /C1/1). If a1 ; a2 ; ... and b1 ; b2 ; ... are related by 1 /C27X/C12 n/C301bnxn /C30Y/C12 i /C3011 (1 /C28 xi)a1(4) or, in terms of GENERATING FUNCTIONS A(x) and B(x); 1 /C27B(x) /C30expX/C12 k /C301A(xk) k"# ; (5) then fbn g is said to be the Euler transform of fan g (Sloane and Plouffe 1995, p. 20). The Euler transform can be effected by introducing the intermediate series c1 ; c2 ; ... given by cn /C30X d½ndad ; (6) thenbn /C301 ncn /C27Xn/C281 k /C301ckbn/C28k"# ; (7) with b1 /C30c1 : Similarly, the inverse transform can be effected by computing the intermediate series as cn /C30nbn /C28Xn/C281 k /C301ckbn/C281 ; (8) then an /C301 nX d½nmn d ! cd ; (9) where m(n) is the MO¨ BIUS FUNCTION . In GRAPH THEORY ,ifanis the number of UNLABELED CONNECTED GRAPHS on n nodes satisfying some property, then bnis the total number of UNLABELED GRAPHS (connected or not) with the same property. This application of the Euler transform is called RIDDELL’S FORMULA for unlabeled graph (Sloane and Plouffe 1995, p. 20). There are also important number theoretic applica- tions of the Euler transform. For example, if there are a1 kinds of parts of size 1, a2 kinds of parts of size 2, etc., in a given type of partition, then the Euler transform bnof anis the number of partitions of n into these integer parts. For example, if an/C301 for all n, then bnis the number of partitions of ninto integer parts. Similarly, if an/C301 for nPRIME andan/C300 for n composite, then bnis the number of partitions of n into prime parts (Sloane and Plouffe 1995, p. 21). Other applications are given by Andrews (1986), Andrews and Baxter (1989), and Cameron (1989). See also BINOMIAL TRANSFORM ,EULER’S HYPERGEO- METRIC TRANSFORMATIONS ,F ORWARD DIFFERENCE , INTEGER SEQUENCE ,M O¨ BIUS TRANSFORM ,RIDDELL’S FORMULA ,STIRLING TRANSFORM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics,and Computer Algebra. Providence, RI: Amer. Math. Soc.,1986. Andrews, G. E. and Baxter, R. J. "A Motivated Proof of the Rogers-Ramanujan Identities." Amer. Math. Monthly 96, 401/C1 /09, 1989. Beeler, M. et al. Item 120 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 55, Feb. 1972. Bernstein, M. and Sloane, N. J. A. "Some Canonical Se- quences of Integers." Linear Algebra Appl. 226//228 ,5 7/C1/ 2, 1995. Cameron, P. J. "Some Sequences of Integers." Disc. Math. 75,8 9/C1/02, 1989. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1163, 1980. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 20 /C1/1, 1995. Euler Triangle Formula Let O and I be the CIRCUMCENTER and INCENTER of a TRIANGLE with CIRCUMRADIUS R and INRADIUS r. Let d be the distance between O and I. Then d2 /C30R2 /C282rR: This is the simplest case of PONCELET’S PORISM . See also PONCELET’S PORISM Euler Walk EULERIAN TRAIL Euler Zigzag Number The number of ALTERNATING PERMUTATIONS for n elements is sometimes called an Euler zigzag num- ber. Denote the number of ALTERNATING PERMUTA- TIONS on n elements for which the first element is k by E(n; k) : Then E(1; 1) /C301 and E(n ; k) /C30 0 for k ]n or k B1 E(n; k /C271) /C27E(n /C281; n /C28k) otherwise :;j2ffl where E(n; k)isanE NTRINGER NUMBER . See also ALTERNATING PERMUTATION ,E NTRINGER NUMBER ,SECANT NUMBER ,TANGENT NUMBER References Ruskey, F. "Information of Alternating Permutations." http://www.theory.csc.uvic.ca/~cos/inf/perm/Alterna- ting.html. Sloane, N. J. A. Sequences A000111/M1492 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Euler’s 6n /C271 Theorem Every PRIME OF THE FORM 6n /C271 can be written in the form x2 /C273y2 :/ Euler’s Addition Theorem Let g(x) /C13(1 /C28x2)(1 /C28k2x2) : Then ga 0dxffiffiffiffiffiffiffiffiffi g(x)p/C27gb 0dxffiffiffiffiffiffiffiffiffig(x)p/C30gc 0dxffiffiffiffiffiffiffiffiffig(x)p ; wherec /C13bffiffiffiffiffiffiffiffiffig(a)p /C27 affiffiffiffiffiffiffiffiffig(b)p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 k 2a2b2p : Euler’s Circle NINE-POINT CIRCLE Euler’s Conjecture Define g(k) as the quantity appearing in WARING’S PROBLEM , then Euler conjectured that g(k) /C302k /C273 2 !k66647775/C282 ; where xbcis the FLOOR FUNCTION . See also WARING’S PROBLEM Euler’s Criterion For p an ODD PRIME and a POSITIVE INTEGER a which is not a multiple of p, a(p /C281)=2 /C13a p ! (mod p) ; where (a ½p) is the LEGENDRE SYMBOL . See also LEGENDRE SYMBOL ,QUADRATIC RESIDUE References Nagell, T. "Euler’s Criterion and Legendre’s Symbol." §38 in Introduction to Number Theory. New York: Wiley, pp. 133 /C1/36, 1951. Rosen, K. H. Ch. 9 in Elementary Number Theory and Its Applications, 3rd ed. Reading, MA: Addison-Wesley, 1993. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 33 /C1/7, 1993. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 293, 1991. Euler’s Dilogarithm DILOGARITHM Euler’s Displacement Theorem The general displacement of a rigid body (or coordi- nate frame) with one point fixed is a ROTATION about some axis. Furthermore, a ROTATION may be de- scribed in any basis using three ANGLES . See also EUCLIDEAN MOTION ,EULER ANGLES ,RIGID MOTION ,ROTATION ,TRANSLATION Euler’s Distribution Theorem For signed distances on a LINE SEGMENT , AB /C215CD/C27AC /C215DB/C27AD /C215BC/C300; since (b /C28a)(d /C28c) /C27(c /C28a)(b /C28d) /C27(d /C28a)(c /C28b) /C300: References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 3, 1929. Euler’s Equations of Inviscid Motion The system of PARTIAL DIFFERENTIAL EQUATIONS describing fluid flow in the absence of viscosity, given by @u @t/C27(u /C2159)u /C30/C289P r; where u is the fluid velocity, P is the pressure, and r is the fluid density. See also EULER DIFFERENTIAL EQUATION References Landau, L. D. and Lifschitz, E. M. Fluid Mechanics, 2nd ed. Oxford, England: Pergamon Press, p. 3, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Euler’s Factorization Method A factorization algorithm which works by expressing N as a QUADRATIC FORM in two different ways. Then N /C30a2 /C27b2 /C30c2 /C27d2 ; (1) so a2 /C28c2 /C30d2 /C28b2 (2) (a /C28c)(a /C27c) /C30(d /C28b)(d /C27b) : (3) Let k be the GREATEST COMMON DIVISOR of a /C28c and d /C28b so a /C28c /C30kl (4) d /C28b /C30km (5) (l ; m) /C301 ; (6) (where (l, m) denotes the GREATEST COMMON DIVISOR of l and m), and l(a /C27c) /C30m(d /C27b) : (7) But since (l ; m) /C301; m½a /C27c and a /C27c /C30mn; (8) which gives b /C27d /C30ln ; (9) so we have [(1 2 k)2 /C27(12 n)2](l2 /C27m2) /C3014(k2 /C27n2)(l2 /C27m2)/C3014[(kn)2 /C27(kl)2 /C27(nm)2 /C27(nl)2] /C3014[(d /C28b)2 /C27(a /C28c)2 /C27(a /C27c)2 /C27(d /C27b)2] /C3014(2a2 /C272b2 /C272c2 /C272d2) /C3014(2N /C272N) /C30N : (10) See also PRIME FACTORIZATION ALGORITHMS Euler’s Graeco-Roman Squares Conjecture Euler conjectured that there do not exist GRAECO- ROMAN SQUARES (now known as EULER SQUARES )of order n /C304k /C272 for k /C301, 2, .... In fact, MacNeish (1921 /C1/922) published a purported proof of this con- jecture (Bruck and Ryser 1949). While it is true that no such square of order six exists, such squares were found to exist for all other orders of the form 4k /C272 by Bose, Shrikhande, and Parker in 1959 (Wells 198, p. 77), refuting the CONJECTURE (and establishing unequivocally the invalidity of MacNeish’s "proof"). See also 36 OFFICER PROBLEM ,EULER SQUARE ,LATIN SQUARE References Bose, R. C. "On the Application of the Properties of Galois Fields to the Problem of Construction of Hyper-Graeco- Latin Squares." Indian J. Statistics 3, 323/C1/38, 1938. Bose, R. C.; Shrikhande, S. S.; and Parker, E. T. "Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler’s Conjecture." Canad. J. Math. 12, 189, 1960. Bruck, R. H. and Ryser, H. J. "The Nonexistence of Certain Finite Projective Planes." Canad. J. Math. 1,8 8/C1/3, 1949. Levi, F. W. Second lecture in Finite Geometrical Systems. Calcutta, India: University of Calcutta, 1942. MacNeish, H. F. "Euler Squares." Ann. Math. 23, 221/C1/27, 1921/C1/922. Mann, H. B. "On Orthogonal Latin Squares." Bull. Amer. Math. Soc. 51, 185/C1/97, 1945. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 77, 1986. Euler’s Homogeneous Function Theorem Letf(x;y)b ea HOMOGENEOUS FUNCTION of order nso that f(tx ; ty) /C30tnf(x; y): (1) Then define x?/C13xt and y ?/C13yt: Then ntn /C281f(x; y) /C30@f @x?@x? @t/C27@f @y?@y? @t /C30x@f @x?/C27y@f @y?/C30x@f @(xt) /C27y@f @(yt) : (2) Let t /C301, then x@f @x /C27y@f @y /C30nf(x; y) : (3) This can be generalized to an arbitrary number of variables xi@f @xi/C30nf(x); (4) where EINSTEIN SUMMATION has been used. Euler’s Hypergeometric Transformations 2F1(a ; b; c; z) /C30g1 0tb/C281(1 /C28 t)c/C28b /C281 (1 /C28 tz)a dt ; (1) where2F1(a ; b; c; z)isa HYPERGEOMETRIC FUNC- TION . The solution can be written using the Euler’s transformations t 0 t (2) t 0 1 /C28t (3) t 0 (1 /C28z /C28tz) /C281 (4) t 01 /C28 t 1 /C28 tz (5) in the equivalent forms 2F1(a; b; c; z) /C30(1 /C28z) /C28a 2F1(a ; c /C28b; c; z =(z /C281)) (6) /C30(1 /C28z) /C28b 2F1(c /C28a ; b; c; z =(z /C281)) (7) /C30(1 /C28z)c/C28a /C28b 2F1(c /C28a ; c /C28b; c; z) : (8) See also HYPERGEOMETRIC FUNCTION References Euler, L. Nova Acta Acad. Petropol. 7, p. 58, 1778. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 585 /C1/91, 1953. Euler’s Idoneal Number IDONEAL NUMBEREuler’s Machin-Like Formula The MACHIN-LIKE FORMULA 1 4 p /C30tan/C281(12) /C27tan/C281(13) : The other 2-term MACHIN-LIKE FORMULAS are HER- MANN’S FORMULA , HUTTON’S FORMULA , and MACHIN’S FORMULA . See also INVERSE TANGENT Euler’s Pentagonal Number Theorem PENTAGONAL NUMBER THEOREM Euler’s Phi Function TOTIENT FUNCTION Euler’s Polygon Division Problem The problem of finding in how many ways En a PLANE convex POLYGON of n sides can be divided into TRIANGLES by diagonals. Euler first proposed it to Christian Goldbach in 1751, and the solution is the CATALAN NUMBER En /C30Cn/C282 :/ See also CATALAN NUMBER ,CATALAN’S PROBLEM References Forder, H. G. "Some Problems in Combinatorics." Math. Gaz. 41, 199 /C1/01, 1961. Guy, R. K. "Dissecting a Polygon Into Triangles." Bull. Malayan Math. Soc. 5,57/C1/0, 1958. Euler’s Quadratic Residue Theorem A number D that possesses no common divisor with a prime number p is either a QUADRATIC RESIDUE or nonresidue of p, depending whether D(p /C281)=2 is con- gruent mod p to 9 1. Euler’s Rotation Theorem An arbitrary ROTATION may be described by only three parameters. See also EULER ANGLES ,EULER PARAMETERS ,ROTA- TION MATRIX Euler’s Rule The numbers 2npqand 2nrare an AMICABLE PAIR if the three INTEGERS p/C132m(2n/C28m/C271)/C281 (1) q/C132n(2n/C28m/C271)/C281 (2) r/C132n/C27m(2n/C28m/C271)2/C281 (3) are all PRIME NUMBERS for some POSITIVE INTEGER m satisfying 1 5m5n/C281 (Dickson 1952, p. 42). How- ever, there are many AMICABLE PAIRS which do not satisfy Euler’s rule, so it is a SUFFICIENT but not NECESSARY condition for amicability. Euler’s rule is a generalization of THAˆ BIT IBN KURRAH RULE . For example, Euler’s rule is satisfied for (n; m) /C30 (2; 1); (4; 4); (6; 7); (8; 1); (40 ; 29) ; ..., corresponding to the triples (p; q; r) /C30(5; 11; 71); (23, 47, 1151), (191, 383, 73727), ..., giving the AMICABLE PAIRS (220, 284), (17296, 18416), (9363584, 9437056), .... See also AMICABLE PAIR,THAˆ BIT IBN KURRAH RULE References Borho, W. "On Thabit ibn Kurrah’s Formula for Amicable Numbers." Math. Comput. 26, 571 /C1/78, 1972. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, 1952. Euler, L. "De Numeris Amicabilibus." In Leonhardi Euleri Opera Omnia, Ser. 1, Vol. 2. Leipzig, Germany: Teubner, pp. 63 /C1/62, 1915. te Riele, H. J. J. "Four Large Amicable Pairs." Math. Comput. 28, 309 /C1/12, 1974. Euler’s Series Transformation Accelerates the rate of CONVERGENCE for an ALTER- NATING SERIES S /C30X/C12 s/C300(/C281)sus /C30u0 /C28u1 /C27u2 /C28.../C28un /C281 /C27X/C12 s/C300( /C281)2 2s/C271[Dsun] (1) for n EVEN and D the FORWARD DIFFERENCE operator Dkun /C13Xk m/C300(/C281)m k m;j1z;j1} un/C27k/C28m ; (2) wherek m;jr;j1 are BINOMIAL COEFFICIENTS . The POSITIVE terms in the series can be converted to an ALTERNAT- ING SERIES using X/C12 r/C301vr /C30X/C12 r/C301(/C281)r/C281wr ; (3) where wr /C13vr /C272v2r /C274v4r /C278v8r /C27...: (4) See also ALTERNATING SERIES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Euler’s Spiral CORNU SPIRALEuler’s Sum of Powers Conjecture Euler conjectured that at least nnth POWERS are required for n /C212 to provide a sum that is itself an nth POWER . The conjecture was disproved by Lander and Parkin (1967) with the counterexample 275 /C27845 /C271105 /C271335 /C301445 : Ekl (1998) defined Euler’s extended conjecture as the assertion that there are no solutions to the k:m:n DIOPHANTINE EQUATION ak 1 /C27ak2 /C27.../C27akm /C30bk1 /C27bk2 /C27.../C27bkn ; with ai and bi not necessarily distinct, such that m /C27 n Bk: There are no known counterexamples to this conjecture (Ekl 1998). Ekl (1998) defines the Euler conjecture number as the minimum known value of D/C13m /C27n /C28k: The following table gives the smallest known values. k Soln. / D/ Reference 4 4.1.3 0 Elkies 1988 5 5.1.4 0 Lander et al. 1967 6 6.3.3 0 Subba Rao 1934 7 7.4.4 1 Ekl 1996 8 8.5.5 2 Letac 1942 9 9.6.6 3 Lander et al. 1967 10 10.7.7 4 Moessner 1939 See also DIOPHANTINE EQUATION–5TH POWERS ,EU- LER QUARTIC CONJECTURE References Ekl, R. L. "Equal Sums of Four Seventh Powers." Math. Comput. 65, 1755 /C1/756, 1996. Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309 /C1/315, 1998. Elkies, N. "On A4/C27B4/C27C4/C30D4:/"Math. Comput. 51, 828/C1/ 38, 1988. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 195, 1998. Lander, L. J. and Parkin, T. R. "A Counterexample to Euler’s Sum of Powers Conjecture." Math. Comput. 21, 101/C1/03, 1967. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446/C1/59, 1967. Letac, A. Gazetta Mathematica 48,6 8/C1/9, 1942. Moessner, A. "Einige Numerische Identitaten." Proc. Indian Acad. Sci. Sect. A 10, 296/C1/06, 1939. Subba Rao, K. "On Sums of Sixth Powers." J. London Math. Soc. 9, 172/C1/73, 1934. Euler’s Theorem A generalization of FERMAT’S LITTLE THEOREM . Euler published a proof of the following more general theorem in 1736. Let f(n) denote the TOTIENT FUNC- TION . Then af(n) /C131 (mod n) for all a RELATIVELY PRIME to n. See also CHINESE HYPOTHESIS ,E ULER’S DISPLACE- MENT THEOREM ,E ULER’S DISTRIBUTION THEOREM , FERMAT’S LITTLE THEOREM ,TOTIENT FUNCTION References Se´roul, R. "The Theorems of Fermat and Euler." §2.8 in Programming for Mathematicians. Berlin: Springer-Ver- lag, p. 15, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 21 and 23 /C1/5, 1993. Euler’s Totient Rule The number of bases in which 1=p is a REPEATING DECIMAL (actually, repeating b-ary) of length l is the same as the number of FRACTIONS 0 =(p /C281); 1=(p /C281); ..., (p /C282)=(p /C281) which have reduced DENOMINATOR l. For example, in bases 2, 3, ..., 6, 1/7 is given by 1 7 /C300:001001001001...2 /C300 :010212010212...3 /C300 :021021021020...4 /C300 :032412032412...5 /C300:050505050505...6 ; which have periods 3, 6, 3, 6, and 2, respectively, corresponding to the DENOMINATORS 6, 3, 2, 3, and 6 of 1 6 ;13 ;12 ;23 ; and56 : See also C YCLIC NUMBER ,R EPEATING DECIMAL , TOTIENT FUNCTION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 167 /C1/68, 1996. Euler’s Triangle The triangle of numbers An; k given by An; 1 /C30An; n /C301 and the RECURRENCE RELATION An/C271; k /C30kAn; k /C27(n /C272 /C28k)An; k /C281 for k /C23 [2; n]; where An ; k are EULERIAN NUMBERS .1 11 141 11 11 11 12 66 62 61 1 57 302 302 57 1 The numbers 1, 1, 1, 1, 4, 1, 1, 11, 11, 1, ... are Sloane’s A008292. Amazingly, the Z-TRANSFORMS of tn (z/C281)n TnzZ[tn]/C30(1/C28z)n Tnzlim x00@n @xnz z/C28e/C28xT ! are generators for Euler’s triangle. ASPHERICAL TRIANGLE is sometimes also called Euler’s triangle. See also CLARK’S TRIANGLE ,E ULERIAN NUMBER , LEIBNIZ HARMONIC TRIANGLE ,LOSSNITSCH’S TRIAN- GLE,NUMBER TRIANGLE ,PASCAL’S TRIANGLE ,SEIDEL- ENTRINGER- ARNOLD TRIANGLE ,SPHERICAL TRIANGLE , Z-TRANSFORM References Sloane, N. J. A. Sequences A008292 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Euler-Bernoulli Triangle SEIDEL- ENTRINGER- ARNOLD TRIANGLE Euler-Darboux Equation The PARTIAL DIFFERENTIAL EQUATION uxy/C27aux/C28buy x/C28y/C300: See also EULER- POISSON- DARBOUX EQUATION References Miller, W. Jr. Symmetry and Separation of Variables. Reading, MA: Addison-Wesley, 1977. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997. EulerE EULER NUMBER ,EULER POLYNOMIAL EulerGamma EULER- MASCHERONI CONSTANT #1999/C1/001 Wolfram Research, Inc. Eulerian Circuit An EULERIAN TRAIL which starts and ends at the same VERTEX . In other words, it is a GRAPH CYCLE which uses each EDGE exactly once. The term EU- LERIAN CYCLE is also used synonymously with Euler- ian circuit. For technical reasons, Eulerian circuits are easier to study mathematically than are HAMIL- TONIAN CIRCUITS . As a generalization of the KO¨ NIGS- BERG BRIDGE PROBLEM , Euler showed (without proof) that a CONNECTED GRAPH has an Eulerian circuit IFF it has no VERTICES of ODD DEGREE . FLEURY’S ALGORITHM is an elegant, but inefficient, method of generating Eulerian circuit. An Eulerian cycle of a graph may be found usingEulerianCyc- le[g] in the Mathematica add-on package Discre- teMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also CHINESE POSTMAN PROBLEM ,EULER GRAPH , HAMILTONIAN CIRCUIT ,UNICURSAL CIRCUIT References Bolloba ´s, B. Graph Theory: An Introductory Course. New York: Springer-Verlag, p. 12, 1979. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 94 /C1/6, 1984. Hierholzer, C. "U¨ ber die Mo¨glichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechnung zu umfahren." Math. Ann. 6,30/C1/2, 1873. Lucas, E. Re´cre´ations Mathe ´matiques. Paris: Gauthier- Villars, 1891. Skiena, S. "Eulerian Cycles." §5.3.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 192 /C1/ 96, 1990. Eulerian Cycle EULERIAN CIRCUIT Eulerian Graph A GRAPH containing an EULERIAN CIRCUIT . Finding the largest SUBGRAPH of graph having an odd number of vertices which is Eulerian is an NP-COMPLETE PROBLEM (Skiena 1990, p. 194). An UNDIRECTED GRAPH is Eulerian IFF every VERTEX has EVEN DEGREE . The numbers of Eulerian graphs with n /C301, 2, ... nodes are 1, 1, 2, 3, 7, 16, 54, 243, ...(Sloane’s A002854; Robinson 1969; Mallows and Sloane 1975; Buekenhout 1995, p. 881; Colbourn and Dinitz 1996, p. 687). There is an explicit formula giving these numbers. Euler showed (without proof) that a CONNECTED GRAPH is Eulerian IFF it has no VERTICES of ODD DEGREE . The numbers of connected Eulerian graphs with n /C301, 2, ... nodes are 1, 0, 1, 1, 4, 8, 37, 184, ... (Sloane’s A003049; Robinson 1969; Liskovec 1972; Harary and Palmer 1973, p. 117). ADIRECTED GRAPH is Eulerian IFFevery VERTEX has equal INDEGREE and OUTDEGREE . A planar BIPARTITE GRAPH isDUAL to a PLANAR Eulerian graph and vice versa. The numbers of Eulerian digraphs on n/C301, 2, ... nodes are 1, 1, 3, 12, .... See also HAMILTONIAN GRAPH ,TWO-GRAPH References Bolloba ´s, B. Graph Theory: An Introductory Course. New York: Springer-Verlag, p. 12, 1979. Buekenhout, F. (Ed.). Handbook of Incidence Geometry: Building and Foundations. Amsterdam, Netherlands: North-Holland, 1995. Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, 1996. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 94, 1984. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, p. 117, 1973. Liskovec, V. A. "Enumeration of Euler Graphs" [Russian]. Review MR#6557 in Math. Rev. 44, 1195, 1972. Mallows, C. L. and Sloane, N. J. A. "Two-Graphs, Switching Classes, and Euler Graphs are Equal in Number." SIAM J. Appl. Math. 28, 876/C1/80, 1975. Robinson, R. W. "Enumeration of Euler Graphs." In Proof Techniques in Graph Theory (Ed. F. Harary). New York: Academic Press, pp. 147 /C1/53, 1969. Skiena, S. "Eulerian Cycles." §5.3.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 192 /C1/ 96, 1990. Sloane, N. J. A. Sequences A002854/M0846 and A003049/ M3344 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Eulerian Integral of the First Kind Legendre and Whittaker and Watson’s (1990) term for the BETA INTEGRAL g1 0xp(1 /C28x)q dx; whose solution is the BETA FUNCTION B(p /C271; q /C271):/ See also BETA FUNCTION ,BETA INTEGRAL ,EULERIAN INTEGRAL OF THE SECOND KIND References Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Eulerian Integral of the Second Kind For R[n] >/C281 and R[z] > 0; Y (z ; n) /C30nzg1 0(1 /C28x)nxz/C281 dx (1) /C30n! (z)n/C271nz (2) /C30B(z; n /C271); (3) where (z)n is the POCHHAMMER SYMBOL and B(p ; q)is the BETA FUNCTION . See also BETA FUNCTION ,BETA INTEGRAL ,EULERIAN INTEGRAL OF THE FIRST KIND Eulerian Number The number of PERMUTATION RUNS of length n with k 5n; denotedn k;j1r;j11 ; An; k ; or A(n ; k): The Eulerian numbers are given explicitly by the sum n k;j2z;j2} /C30Xk j/C300(/C281)j n þ 1 j;j1z;j1} (k /C28j)n : (1) Making the definition bn ; 1 /C301 (2) b1 ; n /C301 (3) together with the RECURRENCE RELATIONbn; k /C30nbn; k /C281 /C27kbn/C281 ; k (4) for n /C21k then gives n k;j2z;j2} /C30bk ; n/C28k /C271 : (5) The arrangement of the numbers into a triangle gives EULER’S TRIANGLE , whose entries are 1, 1, 1, 1, 4, 1, 1, 11, 11, 1, ... (Sloane’s A008292). Therefore, they represent a sort of generalization of the BINOMIAL COEFFICIENTS where the defining RECURRENCE RELA- TION weights the sum of neighbors by their row and column numbers, respectively. The Eulerian numbers satisfy Xn k /C301n k;j2z;j2} /C30n!: (6) Eulerian numbers also arise in the surprising context of integrating the SINC FUNCTION , and also in sums of the form X/C12 k/C301knrk/C30Li/C28n(r)/C30r (1/C28r)n/C271Xn i/C301n k;j2z;j2} rn/C28i; (7) where Lim(z) is the POLYLOGARITHM function. See also COMBINATION LOCK,EULER NUMBER ,EU- LER’S TRIANGLE ,EULER ZIGZAG NUMBER ,PERMUTA- TION RUN,P OLYLOGARITHM ,S IMON NEWCOMB’S PROBLEM ,SINC FUNCTION ,W ORPITZKY’S IDENTITY , Z-TRANSFORM References Abramson, M. and Moser, W. O. J. "Permutations without Rising or Falling v/-Sequences." Ann. Math. Statist. 38, 1245/C1/254, 1967. Andre ´, D. "Me ´moir sur les couples actifs de permutations." Mem. della Pontificia Acad. Romana dei Nuovo Lincei 23, 189/C1/23, 1906. Carlitz, L. "Note on a Paper of Shanks." Amer. Math. Monthly 59, 239/C1/41, 1952. Carlitz, L. "Eulerian Numbers and Polynomials." Math. Mag. 32, 247/C1/60, 1959. Carlitz, L. "Eulerian Numbers and Polynomials of Higher Order." Duke Math. J. 27, 401/C1/23, 1960. Carlitz, L. "A Note on the Eulerian Numbers." Arch. Math. 14, 383/C1/90, 1963. Carlitz, L. and Riordan, J. "Congruences for Eulerian Numbers." Duke Math. J. 20, 339/C1/43, 1953. Carlitz, L.; Roselle, D. P.; and Scoville, R. "Permutations and Sequences with Repetitions by Number of Increase." J. Combin. Th. 1, 350/C1/74, 1966. Cesa`ro, E. "De ´rive´es des fonctions de fonctions." Nouv. Ann. 5, 305/C1/27, 1886. Comtet, L. "Permutations by Number of Rises; Eulerian Numbers." §6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 240 /C1/46, 1974. David, F. N.; Kendall, M. G.; and Barton, D. E. Symmetric Function and Allied Tables. Cambridge, England: Cam- bridge University Press, p. 260, 1966. Dillon, J. F.; Roselle, D. P. "Eulerian Numbers of Higher Order." Duke Math. J. 35, 247/C1/56, 1968. Foata, D. and Schu¨tzenberger, M.-P. The´orie Ge´ome´trique des Polyno ˆmes Eule´riens. Berlin: Springer-Verlag, 1970. Frobenius, F. G. "Ueber die Bernoullischen Zahlen und die Eulerischen Polynome." Sitzungsber. Preuss. Akad. Wiss. , pp. 808 /C1/47, 1910. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Eulerian Numbers." §6.2 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 267 /C1/72, 1994. Kimber, A. C. "Eulerian Numbers." Supplement to Encyclo- pedia of Statistical Sciences. (Eds. S. Kotz, N. L. Johnson, and C. B. Read). New York: Wiley, pp. 59 /C1/0, 1989. Poussin, F. "Sur une proprie ´te´ arithme ´tique de certains polynomes associe ´s aux nombres d’Euler." C. R. Acad. Sci. Paris Se´r. A-B 266, A392-A393, 1968. Salama, I. A. and Kupper, L. L. "A Geometric Interpretation for the Eulerian Numbers." Amer. Math. Monthly 93,51/C1/ 2, 1986. Schrutka, L. "Eine neue Einleitung der Permutationen." Math. Ann. 118, 246 /C1/50, 1941. Shanks, E. B. "Iterated Sums of Powers of the Binomial Coefficients." Amer. Math. Monthly 58, 404 /C1/07, 1951. Sloane, N. J. A. Sequences A008292 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Tomic, M. "Sur une nouvelle classe de polyno ˆmes de la the´orie des fonctions spe´ciales." Publ. Fac. Elect. U. Belgrade, No. 38, 1960. Toscano, L. "Su due sviluppi della potenza di un binomio, q- coefficienti di Eulero." Bull. S. M. Calabrese 16,1/C1/, 1965. Eulerian Tour EULERIAN TRAIL Eulerian Trail A WALK on the EDGES of a GRAPH which uses each EDGE exactly once. A CONNECTED GRAPH has an Eulerian trail IFF it has at most two VERTICES of ODD DEGREE . See also EULERIAN CIRCUIT ,EULERIAN GRAPH KO¨ - NIGSBERG BRIDGE PROBLEM References Edmonds, J. and Johnson, E. L. "Matching, Euler Tours, and the Chinese Postman." Math. Programm. 5,88/C1/24, 1973. Wilson, R. J. "An Eulerian Trail through Ko¨nigsberg." J. Graph Th. 10, 265 /C1/75, 1986. Euler-Jacobi Pseudoprime An Euler-Jacobi pseudoprime to a base a is an ODD COMPOSITE numbers such that (a ; n) /C301 and the JACOBI SYMBOL (a=n) satisfies a n ! /C13a(n/C281)=2 (mod n) : (Guy 1994; but note that Guy calls these simply "Euler pseudoprimes"). No ODD COMPOSITE number is an Euler-Jacobi pseudoprime for all bases a RELA- TIVELY PRIME to it. This class includes some CARMI- CHAEL NUMBERS , all STRONG PSEUDOPRIMES to base a, and all EULER PSEUDOPRIMES to base a. An Eulerpseudoprime is pseudoprime to at most 1/2 of all possible bases less than itself. The first few base-2 Euler-Jacobi pseudoprimes are 561, 1105, 1729, 1905, 2047, 2465, ... (Sloane’s A047713), and the first few base-3 Euler-Jacobi pseudoprimes are 121, 703, 1729, 1891, 2821, 3281, 7381, ... (Sloane’s A048950). The number of base-2 Euler-Jacobi primes less than 102,103, ... are 0, 1, 12, 36, 114, ... (Sloane’s A055551). See also EULER PSEUDOPRIME ,PSEUDOPRIME References Guy, R. K. "Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes." §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 27 /C1/0, 1994. Pinch, R. G. E. "The Pseudoprimes Up to 1013." ftp:// ftp.dpmms.cam.ac.uk/pub/PSP/. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, 1994. Sloane, N. J. A. Sequences A047713/M5461, A048950, and A055551 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Euler-Lagrange Derivative The derivative dL dq/C13@L @q/C28d dt@L @˙q ! appearing in the E ULER- LAGRANGE DIFFERENTIAL EQUATION . Euler-Lagrange Differential Equation A fundamental equation of CALCULUS OF VARIATIONS which states that if Jis defined by an INTEGRAL OF THE FORM J/C30gf(x;y;˙y)dx; (1) where ˙y/C13dy dt; (2) then Jhas a STATIONARY VALUE if the Euler- Lagrange differential equation @f @y/C28d dt@f @˙y ! /C300 (3) is satisfied. If time DERIVATIVE NOTATION is replaced instead by space variable notation, the equation becomes @f @y/C28d dx@f @yx/C300: (4) In many physical problems, fx(the PARTIAL DERIVA- TIVE offwith respect to x) turns out to be 0, in which case a manipulation of the Euler-Lagrange differen- tial equation reduces to the greatly simplified and partially integrated form known as the B ELTRAMI IDENTITY , f/C28yx@f @yx/C30C: (5) For three independent variables (Arfken 1985, pp. 924 /C1/44), the equation generalizes to @f @u/C28@ @x@f @ux/C28@ @y@f @uy/C28@ @z@f @uz/C300: (6) Problems in the CALCULUS OF VARIATIONS often can be solved by solution of the appropriate Euler-Lagrangeequation. To derive the Euler-Lagrange differential equation, examine dJ/C13dgL(q;˙q;t)dt/C30g@L @qdq/C27@L @˙qd˙q ! dt /C30g@L @qdq/C27@L @˙qdðdqÞ dt"# dt; ð7Þ since d˙q/C30d(dq)=dt:Now, integrate the second term byPARTS using u/C30@L @˙qdv/C30d(dq) (8) du/C30d dt@L @˙q ! dt v /C30dq; (9) so g@L @˙qd(dq) dtdt/C30g@L @˙qd(dq) /C30@L @˙qdq"#t2 t1/C28gt2 t1d dt@L @˙qdt ! dq: (10) Combining (7) and (10) then gives dJ/C30@L @˙qdq"#t2 t1/C27gt2 t1@L @q/C28d dt@L @˙q ! dqd t : (11) But we are varying the path only, not the endpoints,sodq(t 1)/C30dq(t2)/C300 and (11) becomes dJ/C30gt2 t1@L @q/C28d dt@L @˙q ! dqd t : (12) We are finding the STATIONARY VALUES such that dJ/C300:These must vanish for any small change dq; which gives from (12),@L @q/C28d dt@L @˙q ! /C300: (13) This is the Euler-Lagrange differential equation. The variation in Jcan also be written in terms of the parameter kas dJ/C30g[f(x;y/C27kv;˙y/C27k˙v)/C28f(x;y;˙y)]dt /C30kI1/C271 2k2I2/C2716k3I3/C271 24k4I4/C27...; (14) where v/C30dy (15) ˙v/C30d˙y (16) and the first, second, etc., variations are I1/C30g(vfy/C27˙vf˙y)dt (17) I2/C30g(v2fyy/C272v˙vfy˙y/C27˙v2f˙y˙y)dt (18) I3/C30g(v3fyyy/C273v2˙vfyy˙y/C273v˙v2fy˙y˙y/C27˙v3f˙y˙y˙y)dt (19) I4/C30g(v4fyyyy/C274v3˙vfyyy˙y/C276v2˙v2fyy˙y˙y/C274v˙v3fy˙y˙y˙y /C27˙v4f˙y˙y˙y˙y)dt: (20) The second variation can be re-expressed using d dt(v2l)/C30v2˙l/C272v˙vl; (21) so I2/C27[v2l]1 2/C30g2 1[v2(fyy/C27˙l)/C272v˙v(fy˙y/C27l)/C27˙v2f˙y˙y]dt: (22) But [v2l]12/C300: (23) Now choose lsuch that f˙y˙y(fyy/C27˙l)/C30(fy˙y/C27l)2(24) andzsuch that fy˙y/C27l/C30/C28f˙y˙y zdz dt(25) so that zsatisfies f˙y˙y¨z/C27˙f˙y˙y˙z/C28(fyy/C28˙fy˙y)z/C300: (26) It then follows that I2 /C30g f˙y˙y˙v /C27fy˙y /C27 l f˙y˙yv !2 dt /C30g f˙y˙y˙v /C28v zdz dt !2 :(27) See also BELTRAMI IDENTITY ,B RACHISTOCHRONE PROBLEM ,C ALCULUS OF VARIATIONS ,E ULER- LA- GRANGE DERIVATIVE References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985. Forsyth, A. R. Calculus of Variations. New York: Dover, pp. 17 /C1/0 and 29, 1960. Morse, P. M. and Feshbach, H. "The Variational Integral and the Euler Equations." §3.1 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 276 /C1/80, 1953. Euler-Lucas Pseudoprime Let U(P;Q) and V(P;Q)b eL UCAS SEQUENCES generated by PandQ, and define D/C13P2/C284Q: Then U(n/C28(D=n))=2/C130 (mod n) when ( Q=n)/C301 V(n/C28(D=n))=2/C13D(mod n) when ( Q=n)/C30/C281;;j2ffl where ( Q=n) is the L EGENDRE SYMBOL .A n ODD COMPOSITE NUMBER nsuch that ( n;QD)/C301 (i.e., n and QD are RELATIVELY PRIME ) is called an Euler- Lucas pseudoprime with parameters ( P, Q ). See also PSEUDOPRIME ,STRONG LUCAS PSEUDOPRIME References Ribenboim, P. "Euler-Lucas Pseudoprimes (elpsp( P, Q )) and Strong Lucas Pseudoprimes (slpsp( P, Q ))." §2.X.C in The New Book of Prime Number Records. New York: Springer- Verlag, pp. 130 /C1/31, 1996. Euler-Maclaurin Integration Formulas The Euler-Maclaurin integration and sums formulas can be derived from D ARBOUX’S FORMULA by substi- tuting the B ERNOULLI POLYNOMIAL Bn(t) in for the function f(t):Differentiating the identity Bn(t/C271)/C28Bn(t)/C30ntn/C281(1) /n/C28ktimes gives B(n/C28k) n(t/C271)/C28f(n/C28k) n(t)/C30n(n/C281)/C1/C1/C1ktk/C281: (2) Plugging in t/C300 gives B(n/C28k) n(1)/C30B(n/C28k) n(0):From the Maclaurin series of Bn(z) with k/C210, we have B(n/C282k/C281) n (0)/C300 (3) B(n/C282k) n (0)/C30n! (2k)!B2k (4)B(n/C281) n(0)/C301 2n! (5) B(n) n(0)/C30n!; (6) where Bnis a B ERNOULLI NUMBER , and substituting these values of B(n/C28k) n(1) and B(n/C28k) n(0) into D ARBOUX’S FORMULA gives (z/C28a)f?(a)/C30f(z)/C28f(a)/C28z/C28a 2[f?(z)/C28f?(a)] /C27Xn/C281 m/C301B2m(z/C28a)2m (2m)![f(2m)(z)/C28f(2m)(a)] /C28(z/C28a)2n/C271 (2n)!g1 0B2n(t)f(2n/C271)[a/C28(z/C28a)t]dt; (7) which is the Euler-Maclaurin integration formula (Whittaker and Watson 1990, p. 128). In certain cases, the last term tends to 0 as n0/C12; and an infinite series can then be obtained for f(z)/C28 f(a):In such cases, SUMS may be converted to INTEGRALS by inverting the formula to obtain the Euler-Maclaurin sum formula Xn/C281 k/C301fk/C30gn 0f(k)dk/C281 2[f(0)/C27f(n)] /C27X/C12 k/C301B2n (2n)![f(2n/C281)(n)/C28f(2n/C281)(0)]; (8) which, when expanded, gives Xn/C281 k/C301fk/C30gn 0f(k)dk/C281 2[f(0)/C27f(n)]/C271 12[f?(n)/C28f?(0)] /C281 720[f§(n)/C28f§(0)]/C271 30240[f(5)(n)/C28f(5)(0)] /C281 1209600[f(7)/C28f(7)(0)]/C27... ( 9 ) (Abramowitz and Stegun 1972, p. 16). The Euler- Maclaurin sum formula is implemented in Mathema- tica as the function NSum with option Method- /C21Integrate . The second Euler-Maclaurin integration formula isused when f(x) is tabulated at nvalues f 3=2;f5=2;..., fn/C281=2:/ gxn x1f(x)dx/C30h[f3=2/C27f5=2/C27f7=2/C27.../C27fn/C283=2/C27fn/C281=2] /C28X/C12 k/C301B2kh2k (2k)!(1/C282/C282k/C271)[f(2k/C281) n/C28f(2k/C281) 1 ]: (10) See also DARBOUX’S FORMULA ,SUM,WYNN’S EPSILON METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 16 and 806, 1972. Apostol, T. M. "An Elementary View of Euler’s Summation Formula." Amer. Math. Monthly 106, 409/C1/18, 1999. Arfken, G. "Bernoulli Numbers, Euler-Maclaurin Formula." §5.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 327 /C1/38, 1985. Borwein, J. M.; Borwein, P. B.; and Dilcher, K. "Pi, Euler Numbers, and Asymptotic Expansions." Amer. Math. Monthly 96, 681/C1/87, 1989. Euler, L. Comm. Acad. Sci. Imp. Petrop. 6, 68, 1738. Knopp, K. Theory and Application of Infinite Series. New York: Hafner, 1951. Maclaurin, C. Treatise of Fluxions. Edinburgh, p. 672, 1742. Vardi, I. "The Euler-Maclaurin Formula." §8.3 in Computa- tional Recreations in Mathematica. Reading, MA: Addi- son-Wesley, pp. 159 /C1/63, 1991. Whittaker, E. T. and Robinson, G. "The Euler-Maclaurin Formula." §67 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 134 /C1/36, 1967. Whittaker, E. T. and Watson, G. N. "The Euler-Maclaurin Expansion." §7.21 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press,pp. 127 /C1 /28, 1990. Euler-Maclaurin Sum Formula EULER- MACLAURIN INTEGRATION FORMULAS Euler-Mascheroni Constant The Euler-Mascheroni constant is denoted g(or sometimes C) and has the numerical value g:0:577215664901532860606512090082402431042 . . . (1) (Sloane’s A001620). The Euler-Mascheroni constant was denoted gand calculated to 16 digits by Euler in 1781. It is therefore sometimes known as Euler’s constant. No quadratically converging algorithm forcomputing gis known (Bailey 1988). X. Gourdon and P. Demichel computed a record 108 million digits of g in October 1999 (Gourdon and Sebah). The Euler-Mascheroni constant is implemented in Mathematica asEulerGamma . It is not known if this constant is IRRATIONAL , let alone TRANSCENDENTAL (Wells 1986, p. 28). If gis a simple fraction a=b;then it is known that b>1010;000(Brent 1977; Wells 1986, p. 28). Conway and Guy (1996) are "prepared to betthat it is transcendental," although they do not expecta proof to be achieved within their lifetimes. The CONTINUED FRACTION of the Euler-Mascheroni constant is [0, 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (Sloane’s A002852). The first few CON- VERGENTS are 1, 1/2, 3/5, 4/7, 11/19, 15/26, 71/123, 228/395, 3035/5258, 15403/26685, ... (Sloane’s A046114 and A046115). The positions at which the digits 1, 2, ... first occur in the CONTINUED FRACTION are 2, 4, 9, 8, 11, 69, 24, 14, 139, 52, 22, ... (Sloane’s A033149). The sequence of largest terms in theCONTINUED FRACTION is 1, 2, 4, 13, 40, 49, 65, 399, 2076, ... (Sloane’s A033091), which occur at positions 2, 4, 8, 10, 20, 31, 34, 40, 529, ... (Sloane’s A033092). The Euler-Mascheroni constant arises in many inte- grals g/C13/C28g/C12 0e/C28xlnxd x (2) /C30g/C12 01 1/C28e/C28x/C281 x ! e/C28xdx (3) /C30g/C12 01 x1 1/C27x/C28e/C28x ! dx (4) (Whittaker and Watson 1990, p. 246), and sums g/C131/C27X/C12 k/C3021 k/C27lnk/C281 k !"# (5) /C30lim n0/C12(Hn/C28lnn) (6) /C30X/C12 n/C302(/C281)nz(n) n(7) /C30ln4 p ! /C28X/C12 n/C301(/C281)nz(n/C271) 2n(n/C271); (8) where /Hn/is a HARMONIC NUMBER (Graham et al. 1994, p. 278) and z(z) is the R IEMANN ZETA FUNCTION . /gis also given by the E ULER PRODUCT eg/C30lim n0/C121 lnnYn i/C3011 1/C281 pi; (9) where the product is over PRIMES p. Another connec- tion with the PRIMES was provided by Dirichlet’s 1838 proof that the average number of DIVISORS of all numbers from 1 to nis asymptotic to Pn i/C301s0(i) n/C2lnn/C272g/C281 (10) (Conway and Guy 1996). de la Valle ´e Poussin (1898) proved that, if a large number nis divided by all PRIMES5n;then the average amount by which the QUOTIENT is less than the next whole number is g:/ INFINITE PRODUCTS involving galso arise from the BARNES’ G-FUNCTION with POSITIVE INTEGER n. The cases G(2) and G(3) give Y/C12 n/C301e/C281/C271=2(n)1/C271 n !n /C30e1/C27g=2 ffiffiffiffiffiffi 2pp (11) Y/C12 n/C301e/C282/C272=n1/C272 n !n /C30e3/C272g ffiffiffiffiffiffi 2pp : (12) The Euler-Mascheroni constant is also given by the limits g /C30/C28G?(1) (13) (Whittaker and Watson 1990, p. 236), g /C30lim s01z(s) /C281 s /C28 1 (14) (Whittaker and Watson 1990, p. 271), and g /C30lim x 0/C12x /C28G1 x !"# (15) (Le Lionnais 1983). The difference between the nth convergent in (6) and g is given by Xn k /C3011 k /C28ln n /C28 g /C30g/C12 nx /C28 xbc x2dx; (16) where xbcis the FLOOR FUNCTION , and satisfies the INEQUALITY 1 2(n /C27 1) BXn k/C3011k /C28ln n /C28 g B1 2n (17) (Young 1991). A series with accelerated convergence is g /C303 2 /C28ln 2 /C28X/C12 m/C302(/C281)mm /C28 1 m[z(m) /C281] (18) (Flajolet and Vardi 1996). Another series is g /C30X/C12 n/C301(/C281)n1gnbc n (19) (Vacca 1910, Gerst 1969), where LG is the LOGARITHM to base 2. The convergence of this series can be greatly improved using Euler’s CONVERGENCE IM- PROVEMENT transformation to g /C30X/C12 k/C3012/C28(k /C271)Xk /C281 j/C3001 2k /C28j /C27 j j;j1z;j1} ; (20) wherea b;jr;j1 is a BINOMIAL COEFFICIENT (Beeler et al. 1972, with k /C28j replacing the undefined i). Bailey (1988) gives g /C302n e2nX/C12 m/C3002mn (m /C27 1)!Xm t/C3001 t /C27 1 /C28n ln 2 /C27O1 2ne2n ! ; (21) which is an improvement over Sweeney (1963).The symbol g is sometimes also used for g ?/C13e g :1 :781072 (22) (Gradshteyn and Ryzhik 2000, p. xxvii). Odena (1982 /C1/983) gave the strange approximation (0:11111111)1 =4 /C300 :577350... ; (23) and Castellanos (1988) gave (7 83)2=9/C300:57721521 . . . (24) 5202/C2722 524 !1=6 /C300:5772156634 . . . (25) 803/C2792 614 !1=6 /C300:57721566457 . . . (26) 9903/C28553/C28792/C2842 705/C300:5772156649015295 . . . : ð27Þ See also EULER PRODUCT ,M ERTENS THEOREM , STIELTJES CONSTANTS References Anastassow, T. Die Mascheroni’sche Konstante: Eine histor- isch-analytisch zusammenfassende Studie. Thesis. Bonn, Germany: Universita ¨t Bonn. Wetzikon: J. Wirz, 1914. Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving p;e, and Euler’s Constant." Math. Comput. 50, 275/C1/81, 1988. Beeler, M. et al. Item 120 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 55, Feb. 1972. Brent, R. P. "Computation of the Regular Continued Frac- tion for Euler’s Constant." Math. Comput. 31, 771/C1/77, 1977. Brent, R. P. and McMillan, E. M. "Some New Algorithms for High-Precision Computation of Euler’s Constant." Math. Comput. 34, 305/C1/12, 1980. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1/8, 1988. Conway, J. H. and Guy, R. K. "The Euler-Mascheroni Number." In The Book of Numbers. New York: Springer- Verlag, pp. 260 /C1/61, 1996. de la Valle ´e Poussin, C.-J. Untitled communication. Annales de la Soc. Sci. Bruxelles 22,8 4/C1/0, 1898. DeTemple, D. W. "A Quicker Convergence to Euler’s Con- stant." Amer. Math. Monthly 100, 468/C1/70, 1993. Dirichlet, G. L. "Sur l’usage des se ´ries infinies dans la the´orie des nombres." J. reine angew. Math. 18, 259/C1/74, 1838. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 1, 1981. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/euler/euler.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript, 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Gerst, I. "Some Series for Euler’s Constant." Amer. Math. Monthly 76, 273/C1/75, 1969. Glaisher, J. W. L. "On the History of Euler’s Constant." Messenger of Math. 1,25/C1/0, 1872. Gourdon, X. and Sebah, P. "The Euler Constant: g :/" http:// xavier.gourdon.free.fr/Constants/Gamma/gamma.html. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Knuth, D. E. "Euler’s Constant to 1271 Places." Math. Comput. 16, 275 /C1/81, 1962. Krantz, S. G. "The Euler-Mascheroni Constant." §13.1.7 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 156 /C1/57, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983. Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u- qam.ca/pi/records.html. Sloane, N. J. A. Sequences A001620/M3755, A002852/ M0097, A033091, A033092, A033149, A046114, and A046115 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sweeney, D. W. "On the Computation of Euler’s Constant." Math. Comput. 17, 170 /C1/78, 1963. Vacca, G. "A New Series for the Eulerian Constant." Quart. J. Pure Appl. Math. 41, 363 /C1/68, 1910. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 28, 1986. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 235 /C1/36 and 271, 1990. Young, R. M. "Euler’s Constant." Math. Gaz. 75, 187 /C1/90, 1991. Euler-Mascheroni Integrals Define In /C13(/C281)ng/C12 0(ln z)ne /C28z dz ; (1) then I0 /C30g/C12 0e/C28z dz /C30[/C28e/C28z] /C12 0/C30(0 /C271) /C301 (2) I1 /C30/C28g/C12 0(ln z)e/C28z dz /C30 g (3) I2 /C30 g2 /C271 6 p2 (4) I3 /C30 g3 /C2712 gp2 /C272z(3) (5) I4 /C30 g4 /C27 g2 p2 /C283 20 p4 /C278gz(3) ; (6) where g is the EULER- MASCHERONI CONSTANT and z(3) is APE´ RY’S CONSTANT . EulerPhi TOTIENT FUNCTION Euler-Poincare ´ Characteristic EULER CHARACTERISTICEuler-Poisson-Darboux Equation The PARTIAL DIFFERENTIAL EQUATION uxy /C27N(ux /C27 uy) x /C27 y/C300 : See also EULER- DARBOUX EQUATION References Ames, W. F. "Ad Hoc Exact Techniques for Nonlinear Partial Differential Equations." §3.3 in Nonlinear Partial Differ- ential Equations in Engineering (Ed. W. F. Ames). New York: Academic Press, 1967. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997. Eutactic Star An orthogonal projection of a CROSS onto a 3-D SUBSPACE . It is said to be normalized if the CROSS vectors are all of unit length. See also HADWIGER’S PRINCIPAL THEOREM Evans Point The intersection of the GERGONNE LINE and the EULER LINE. It does not appear to have a simple parametric representation. See also EULER LINE,GERGONNE LINE References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319/C1/29, 1996. Eve APPLE ,R OOT,S NAKE ,S NAKE EYES,S NAKE OIL METHOD ,SNAKE POLYIAMOND Even Divisor Function The sum of powers of EVEN DIVISORS of a number. It is the analog of the DIVISOR FUNCTION for even divisors only and is written s(e) k(n):It is given simply in terms of the usual DIVISOR FUNCTION by s(e) k(n) /C300 for n odd 2k sk(n=2) for n even :;j2ffl See also DIVISOR FUNCTION ,ODD DIVISOR FUNCTION Even Function A function f(x) such that f(x) /C30f(/C28x) : An even func- tion times an ODD FUNCTION is odd. Even Node A NODE in a GRAPH is said to be an even node if its VERTEX DEGREE is EVEN . See also GRAPH ,NODE (GRAPH ), ODD NODE,VERTEX DEGREE Even Number An INTEGER OF THE FORM N /C302n; where n is an INTEGER . The even numbers are therefore ..., -4, -2, 0, 2, 4, 6, 8, 10, ... (Sloane’s A005843). Since the even numbers are integrally divisible by two, N /C13 0 (mod 2) for even N. An even number N for which N /C132 (mod 4) is called a SINGLY EVEN NUMBER , and an even number N for which N /C130 (mod 4) is called a DOUBLY EVEN NUMBER . An integer which is not even is called an ODD NUMBER . The GENERATING FUNCTION of the even numbers is 2x (x /C28 1)2 /C302x /C274x2 /C276x3 /C278x4 /C27... : See also DOUBLY EVEN NUMBER ,EVEN FUNCTION , ODD NUMBER ,SINGLY EVEN NUMBER References Commission on Mathematics of the College Entrance Ex- amination Board. Informal Deduction in Algebra: Proper- ties of Odd and Even Numbers. Princeton, NJ, 1959. Sloane, N. J. A. Sequences A005843/M0985 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Even Part The even part Ev(n) of a positive integer n is defined by Ev(n) /C302b(n) ; where b(n) is the EXPONENT of the exact power of 2 dividing n. The values for n /C301, 2, ..., are 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, ... (Sloane’s A006519). The even part function can be implemented in Mathematica as EvenPart[0]: /C301 EvenPart[n_Integer]: /C302^IntegerExponent[n,2] See also GREATEST DIVIDING EXPONENT ,ODD PART References Sloane, N. J. A. Sequences A006519/M0162 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Even Prime The unique EVEN PRIME NUMBER 2. All other PRIMES are ODD PRIMES . The sequence 2, 4, 6, 10, 14, 22, 26, 34, 38, ... (Sloane’s A001747) consisting of the number 2 together with the PRIMES multiplied by 2 is sometimes also called the even primes, since these are the even numbers n /C302k that are divisible by just 1, 2, k, and 2k:/ See also EVEN NUMBER ,ODD PRIME ,PRIME NUMBER References Sloane, N. J. A. Sequences A001747 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 44, 1986. Event An event is a certain subset of a PROBABILITY SPACE . Events are therefore collections of OUTCOMES on which probabilities have been assigned. Events are sometimes assumed to form a BOREL FIELD (Papoulis 1984, p. 29). See also EXPERIMENT ,INDEPENDENT EVENTS ,M U- TUALLY EXCLUSIVE EVENTS ,OUTCOME ,TRIAL References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 24 and 29 /C1/0, 1984. Eventually Periodic A PERIODIC SEQUENCE such as f1; 1; 1; 2; 1; 2; 1; 2; 1; 2 ; 1 ; 1 ; 2 ; 1; ...g which is periodic from some point onwards. See also PERIODIC SEQUENCE Everett Interpolation EVERETT’S FORMULA Everett’s Formula fp /C30(1 /C28p)f0 /C27pf1 /C27E2 d2 0 /C27F2 d21 /C27E4 d40 /C27F4 d41 /C27E6 d6 0 /C27F6 d61 /C27...; (1) for p /C23 [0; 1]; where d is the CENTRAL DIFFERENCE and E2n /C13G2n /C28G2n/C271 /C13B2n /C28B2n /C271 (2) F2n /C13G2n/C271 /C13B2n /C27B2n/C271 ; (3) where Gkare the COEFFICIENTS from GAUSS’S BACK- WARD FORMULA and GAUSS’S FORWARD FORMULA and Bkare the COEFFICIENTS from BESSEL’S FINITE DIF- FERENCE FORMULA . The Ek/s and Fk/s also satisfy E2n(p) /C30F2n(q) (4) F2n(p) /C30E2n(q) ; (5) for q /C131 /C28p : (6) See also BESSEL’S FINITE DIFFERENCE FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 880 /C1/81, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 92 /C1/3, 1990. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Whittaker, E. T. and Robinson, G. "The Laplace-Everett Formula." §25 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 40 /C1/1, 1967. Eversion A curve on the unit sphere S2 is an eversion if it has no corners or cusps (but it may be self-intersecting). These properties are guaranteed by requiring that the curve’s velocity never vanishes. A mapping s : S1 0 S2 forms an immersion of the CIRCLE into theSPHERE IFF, for all u /C23R; d d u[ s(eiu)];j12;j12;j12;j12;j12;j12;j12;j12;j12;j12> 0: Smale (1958) showed it is possible to turn a SPHERE inside out (SPHERE EVERSION ) using eversion. See also SPHERE EVERSION References Smale, S. "A Classification of Immersions of the Two- Sphere." Trans. Amer. Math. Soc. 90, 281/C1/90, 1958. Evolute An evolute is the locus of centers of curvature (the envelope) of a plane curve’s normals. The original curve is then said to be the INVOLUTE of its evolute. Given a plane curve represented parametrically by (f(t);g(t));the equation of the evolute is given by x/C30f/C28Rsint (1) y/C30g/C27Rcost; (2) where ( x, y) are the coordinates of the running point, Ris the RADIUS OF CURVATURE R/C30(f?2/C27g?2)3=2 f?gƒ/C28fƒg?; (3) andtis the angle between the unit TANGENT VECTOR ˆT/C30x? ½x?½/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f?2/C27g?2pf? g?;j2r;j21 (4) and the X-AXIS , cost/C30ˆT /C215ˆx (5) sint/C30ˆT /C215ˆy: (6) Combining gives x/C30f/C28(f?2/C27g?2)g? f?gƒ/C28fƒg?(7) y/C30g/C27(f?2/C27g?2)f? f?gƒ/C28fƒg?: (8) The definition of the evolute of a curve is independent of parameterization for any differentiable function (Gray 1997). If Eis the evolute of a curve I, then Iis said to be the INVOLUTE ofE. The centers of the OSCULATING CIRCLES to a curve form the evolute to that curve (Gray 1997, p. 111). The following table lists the evolutes of some common curves, some of which are illustrated above. Curve Evolute ASTROID ASTROID 2 times as large CARDIOID CARDIOID 1/3 as large CAYLEY’S SEXTIC NEPHROID CIRCLE point (0, 0) CYCLOID equal CYCLOID DELTOID DELTOID 3 times as large ELLIPSE ELLIPSE EVOLUTE EPICYCLOID enlarged EPICYCLOID HYPOCYCLOID similar HYPOCYCLOID LIMAC ¸ ON CIRCLE CATACAUSTIC for a point source LOGARITHMIC SPIRALequal LOGARITHMIC SPIRAL NEPHROID NEPHROID 1/2 as large PARABOLA NEILE’S PARABOLA TRACTRIX CATENARY See also ENVELOPE ,INVOLUTE ,OSCULATING CIRCLE , ROULETTE References Cayley, A. "On Evolutes of Parallel Curves." Quart. J. Pure Appl. Math. 11, 183 /C1/99, 1871. Dixon, R. "String Drawings." Ch. 2 in Mathographics. New York: Dover, pp. 75 /C1/8, 1991. Gray, A. "Evolutes." §5.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 98 /C1/03, 1997. Jeffrey, H. M. "On the Evolutes of Cubic Curves." Quart. J. Pure Appl. Math. 11,78/C1/1 and 145 /C1/55, 1871. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40 and 202, 1972. Lockwood, E. H. "Evolutes and Involutes." Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166 /C1/71, 1967.Yates, R. C. "Evolutes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 86 /C1/2, 1952. Evolution Strategies A DIFFERENTIAL EVOLUTION method used to minimize functions of real variables. Evolution strategies are significantly faster at numerical optimization than traditional GENETIC ALGORITHMS and also more likely to find a function’s true GLOBAL EXTREMUM . See also DIFFERENTIAL EVOLUTION ,GENETIC ALGO- RITHM ,OPTIMIZATION THEORY References Price, K. and Storn, R. "Differential Evolution." Dr. Dobb’s J.,18/C1/8, Apr. 1997. Exact Covering System A system of congruences aimod niwith 1 5i 5k is called a COVERING SYSTEM if every INTEGER y satisfies y /C13ai (mod n) for at least one value of i. A covering system in which each integer is covered by just one congruence is called an exact covering system. See also COVERING SYSTEM References Guy, R. K. "Exact Covering Systems." §F14 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 253 /C1/56, 1994. Exact Differential A differential OF THE FORM df/C30P(x;y)dx/C27Q(x;y)dy (1) is exact (also called a TOTAL DIFFERENTIAL )i f fdfis path-independent. This will be true if df/C30@f @xdx/C27@f @ydy; (2) soPandQmust be OF THE FORM P(x;y)/C30@f @xQ(x;y)/C30@f @y: (3) But @P @y/C30@2f @y@x(4) @Q @x/C30@2f @x@y; (5) so @P @y/C30@Q @x: (6) See also PFAFFIAN FORM,INEXACT DIFFERENTIAL Exact Period LEAST PERIOD Exact Sequence An exact sequence is a sequence of maps ai : Ai 0 Ai/C271 (1) between a sequence of spaces Ai ; which satisfies im ai /C30ker ai/C271 ; (2) where "im" denotes the IMAGE and "ker" the KERNEL . That is, for a /C23 Ai ; ai(a) /C300 IFF a /C30 ai /C281(b) for some b /C23 Ai/C281 : It follows that ai/C271(ai /C300: The notion of exact sequence makes sense when the spaces are GROUPS , MODULES , CHAIN COMPLEXES ,or SHEAVES . The nota- tion for the maps may be suppressed and the sequence written on a single line as ... 0 Ai/C281 0 Ai 0 Ai/C271 0 ...: (3) An exact sequence may be of either finite or infinite length. The special case of length five, 0 0 A 0 B 0 C 0 0; (4) beginning and ending with zero, meaning the zero module f0g; is called a SHORT EXACT SEQUENCE .An infinite exact sequence is called a LONG EXACT SEQUENCE . For example, the sequence where Ai /C30 Z=4Z and ai is given by multiplying by 2, ...0/C292Z=4Z 0/C292Z=4Z 0/C292... ; (5) is a long exact sequence because at each stage the kernel and image are equal to the SUBGROUP f0 ; 2 g:/ Special information is conveyed when one of the spaces Aiis the ZERO MODULE . For instance, the sequence 0 0 A 0 B (6) is exact IFF the map A 0 B is INJECTIVE . Similarly, A 0 B 0 0 (7) is exact IFF the map A 0 B is SURJECTIVE . See also CHAIN COMPLEX ,HOMOLOGY ,LONG EXACT SEQUENCE ,SHORT EXACT SEQUENCE References Atiyah, M. F. and MacDonald, I. G. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, pp. 22 /C1/ 4, 1969. Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, p. 144, 1995. Hilton, P. and Stammbach, U. A Course in Homological Algebra. New York: Springer-Verlag, 1997.Munkres, J. Elements of Algebraic Topology. Reading, MA: Addison-Wesley, pp. 130 /C1/33, 1984. Exact Trilinear Coordinates The TRILINEAR COORDINATES a : b : g of a point P relative to a TRIANGLE are PROPORTIONAL to the directed distances a ? : b? : c ? from P to the side lines (i.e, a ?/C30k a; b ?/C30k b; c ?/C30kg): Letting k be the constant of proportionality, k /C132D a a /C27 b b /C27 c g ; where D is the AREA of DABC and a, b, and c are the lengths of its sides. When the trilinears are chosen so that k /C301, the coordinates are known as exact tri- linear coordinates. See also TRILINEAR COORDINATES Exactly One "Exactly one" means "one and only one," sometimes also referred to as "JUST ONE." J. H. Conway has also humorously suggested "onee" (one and only one) by analogy with IFF (if and only if), "twoo" (two and only two), and "threee" (three and only three). This refinement is sometimes needed in formal mathema- tical discourse because, for example, if you have two apples, you also have one apple, but you do not have exactly one apple. In 2-valued LOGIC , exactly one is equivalent to the exclusive or operator XOR, P(E) XOR P(F) /C30P(E) /C27P(F) /C282P(E S F) : See also IFF,PRECISELY UNLESS , XNOR, XOR Exactly When IFF Excenter The center Jiof an EXCIRCLE . There are three excenters for a given TRIANGLE , denoted J1 ; J2 ; J3 : The INCENTER I and excenters Ji of a TRIANGLE are an ORTHOCENTRIC SYSTEM . OI2 /C27OJ12/C27OJ22/C27OJ32/C3012R2 ; where O is the CIRCUMCENTER , Jiare the excenters, and R is the CIRCUMRADIUS (Johnson 1929, p. 190). Denote the MIDPOINTS of the original TRIANGLE M1 ; M2 ; and M3 : Then the lines J1M1 ; J2M2 ; and J3M3 intersect in a point known as the MITTENPUNKT . See also CENTROID (ORTHOCENTRIC SYSTEM ), EXCEN- TER-EXCENTER CIRCLE ,EXCENTRAL TRIANGLE ,EXCIR- CLE,INCENTER ,MITTENPUNKT References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 13, 1967. Dixon, R. Mathographics. New York: Dover, pp. 58 /C1/9, 1991. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 115 /C1/16, 1991. Excenter-Excenter Circle Given a TRIANGLE DA1A2A3 ; the points A1 ; I, and J1 lie on a line, where I is the INCENTER and J1is the EXCENTER corresponding to A1 : Furthermore, the circle with J2J3as the diameter has Q as its center, where P is the intersection of A1J1 with the CIRCUM- CIRCLE of A1A2A3 and Q is the point opposite P on the CIRCUMCIRCLE . The circle with diameter J2J3also passes through A2andA3and has radius r/C301 2a1csc12a1;j1ffl;j1{ /C302Rcos12a1;j1ffl;j1{ : It arises because the points I,J1;J2;andJ3form an ORTHOCENTRIC SYSTEM . See also EXCENTER ,INCENTER- EXCENTER CIRCLE , ORTHOCENTRIC SYSTEM References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 185 /C1/86, 1929.Excentral Triangle The TRIANGLE J/C30DJ1J2J3with VERTICES correspond- ing to the EXCENTERS of a given TRIANGLE A, also called the TRITANGENT TRIANGLE . Beginning with an arbitrary TRIANGLE A, find the excentral triangle J. Then find the excentral triangle J?of that TRIANGLE , and so on. Then the resulting TRIANGLE J(/C12)approaches an EQUILATERAL TRIANGLE . Given a triangle DABC ;draw the excentral triangle DJAJBJCand MEDIAL TRIANGLE DMAMBMC:Then the ORTHOCENTER HofDABC ;INCENTER Imof DMAMBMC;and CIRCUMCENTER OeofDJAJBJCare COLLINEAR with Im the MIDPOINT of HOe (Honsberger 1995). The INCENTER I of DABC coincides with the ORTHO- CENTER Heof DJAJBJC ; and the CIRCUMCENTER O of DABC coincides with the NINE-POINT CENTER Neof DJAJBJC : Furthermore, Ne /C30O is the MIDPOINT of the line segment joining the ORTHOCENTER Heand CIR- CUMCENTER Oe of DJAJBJC (Honsberger 1995). Call T the TRIANGLE tangent externally to the EXCIRCLES of A. Then the INCENTER IT of K coincides with the CIRCUMCENTER CJof TRIANGLE DJ1J2J3 ; where Ji are the EXCENTERS of A. The INRADIUS rT of the INCIRCLE of T is rT /C302R /C27r /C301 2(r /C27r1 /C27r2 /C27r3) ; where R is the CIRCUMRADIUS of A, r is the INRADIUS , and ri are the EXRADII (Johnson 1929, p. 192). See also EXCENTER ,E XCENTER- EXCENTER CIRCLE , EXCIRCLE ,GERGONNE POINT ,M ITTENPUNKT ,SODDY CIRCLESReferences Honsberger, R. "A Trio of Nested Triangles." §3.2 in Episodes in Nineteenth and Twentieth Century Euclidean Geome- try. Washington, DC: Math. Assoc. Amer., pp. 27 /C1/0, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Exceptional Binomial Coefficient A BINOMIAL COEFFICIENTN k;jr;j1 is said to be exceptional if lpf N k;jr;j1 > N =k: The following tables gives the excep- tion binomial coefficients which are also GOOD BINO- MIAL COEFFICIENTS , are not OF THE FORMN N/C281;jr;j1 ; and have specified least prime factors p /C215. p Exceptional Binomial Coefficients 13 /3574 406;jr;j1 / 17 /241 16;jr;j1 ;439 33;jr;j1 ;317 56;jr;j1 ;482 130;jr;j1 ;998256;jr;j1 ;/ /998 260;jr;j1 ;14273 896;jr;j1 ;13277 900;jr;j1 / 19 /62 6;jr;j1 ;959 56;jr;j1 / 23 /474 66;jr;j1 / 29 /284 28;jr;j1 / See also GOOD BINOMIAL COEFFICIENT ,LEAST PRIME FACTOR References Erdos, P.; Lacampagne, C. B.; and Selfridge, J. L. "Esti- mates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215 /C1/24, 1993. Exceptional Jordan Algebra AJ ORDAN ALGEBRA which is not isomorphic to a subalgebra. See also JORDAN ALGEBRA ,SPECIAL JORDAN ALGEBRA References Albert, A. A. "A Construction of Exceptional Jordan Division Algebras." Ann. Math. 67,1/C1/8, 1958. Albert, A. A. and Jacobson, N. "On Reduced Exceptional Simple Jordan Algebra." Ann. Math. 66, 400/C1/17, 1957. Exceptional Set of Goldbach Numbers GOLDBACH NUMBER Excess The KURTOSIS of a distribution is sometimes called the excess, or excess coefficient. The term is also used to refer to the quantity e/C13n/C28f0(n;g) for a GRAPH G with n vertices and GIRTH g, where f0(v; g) /C30v(v /C28 1)r /C28 2 v /C28 2for g /C302r /C271 2(v /C28 1)r /C28 2 v /C28 2for g /C302r8 >>>< >>>: (Biggs and Ito 1980, Wong 1982). A (v, g)- CAGE GRAPH having f(v; g) /C30f0(v; g) vertices (i.e., the minimal number, so that the excess is e /C300) is called a MOORE GRAPH . See also CAGE GRAPH ,KURTOSIS ,MOORE GRAPH References Biggs, N. L. and Ito, T. "Graphs with Even Girth and Small Excess." Math. Proc. Cambridge Philos. Soc. 88,1/C1/0, 1980. Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/2, 1982. Excess Coefficient KURTOSIS Excessive Number ABUNDANT NUMBER Exchange Shuffle A SHUFFLE of a deck of cards obtained by successively exchanging the cards in position 1, 2, ..., n with cards in randomly chosen positions. For 4 5n 517; the most frequent permutation is (n; ...; m /C27 1)(m; ...; 1) ; where m /C30n=2ifn is even and either (n /C281)=2or( n /C271)=2ifn is odd (Goldstine and Moews 2000). Amazingly, for n ]18 cards, the identity permutation (i.e., the original state before the cards were shuffled) is the most likely (Goldstine and Moews 2000). See also SHUFFLE References Goldstein, D. ad Moews, D. The Identity Is the Most Likely Exchange Shuffle for Large n. 6 Oct 2000. http://xxx.lanl.- gov/abs/math.CO/0010066/. Robbins, D. P. and Bolker, E. D. "The Bias of Three Pseudo- Random Shuffles." Aeq. Math 22, 268/C1/92, 1981. Schmidt, F. and Simion, R. "Card Shuffling and a Transfor- mation on Sn:/"Aeq. Math 44,1 1/C1/4, 1992.Excircle Given a TRIANGLE , extend two nonadjacent sides. The CIRCLE tangent to these two lines and to the other side of the TRIANGLE is called an ESCRIBED CIRCLE ,o r excircle. The CENTER Jiof the excircle is called the EXCENTER and lies on the external ANGLE BISECTOR of the opposite ANGLE . Every TRIANGLE has three ex- circles, and the TRILINEAR COORDINATES of the EX- CENTERS are/C281:1:1 ;1:/C281:1 ;a n d 1:1: /C281:The RADIUS riof the excircle iis called its EXRADIUS . Note that the three excircles are not necessarily tangent to the INCIRCLE , and so these four circles are not equivalent to the configuration of the S ODDY CIRCLES . Given a TRIANGLE with INRADIUS r, let hibe the ALTITUDES of the excircles, and ritheir RADII (the EXRADII ). Then 1 h1/C271 h2/C271 h3/C301 r1/C271 r2/C271 r3/C301 r (Johnson 1929, p. 189). There are four CIRCLES that are tangent all three sides (or their extensions) of a given TRIANGLE : the INCIRCLE I and three excircles J1 ; J2 ; and J3 : These four circles are, in turn, all touched by the NINE-POINT CIRCLE N. Given a TRIANGLE DABC ; construct the INCIRCLE with INCENTER I and EXCIRCLE with EXCENTER JA : Let Ti be the tangent point of DABC with its incircle, Tebe the tangent point of DABC with its EXCIRCLE JA ; HA the foot of the ALTITUDE to vertex A, M the MIDPOINT of AHA ; and construct Q such that QTi is a DIAMETER of the INCIRCLE . Then M, I, and Te are COLLINEAR ,as are A, Q, and Te (Honsberger 1995). See also EXCENTER ,E XCENTER- EXCENTER CIRCLE , EXCENTRAL TRIANGLE ,FEUERBACH’S THEOREM ,N A- GEL POINT ,TRIANGLE TRANSFORMATION PRINCIPLE References Coxeter, H. S. M. and Greitzer, S. L. "The Incircle and Excircles." §1.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 10 /C1/3, 1967. Honsberger, R. "An Unlikely Collinearity." §3.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geome- try. Washington, DC: Math. Assoc. Amer., pp. 30 /C1/1, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 176 /C1/77 and 182 /C1/94, 1929. Lachlan, R. "The Inscribed and the Escribed Circles." §126 /C1/ 28 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 72 /C1/4, 1893. Excision Axiom One of the EILENBERG- STEENROD AXIOMS which states that, if X is a SPACE with SUBSPACES A and U such that the CLOSURE of A is contained in the interior of U, then the INCLUSION MAP (XU ; AU) 0 (X ; A) induces an isomorphism Hn(XU ; AU) 0 Hn(X ; A) :/ Excluded Middle Law A law in (2-valued) LOGIC which states there is no third alternative to TRUTH or FALSEHOOD . In other words, for any statement A, either A or not-A must be true and the other must be false. This law no longer holds in THREE-VALUED LOGIC or FUZZY LOGIC .See also BIVALENT ,F UZZY LOGIC ,T HREE- VALUED LOGIC References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 64 /C1/5, 1998. Excludent A method which can be used to solve any QUADRATIC CONGRUENCE EQUATION . This technique relies on the fact that solving x2 /C13b (mod p) is equivalent to finding a value y such that b /C27py /C30x2 : Pick a few small moduli m.Ify mod m does not make b /C27py a quadratic residue of m, then this value of y may be excluded. Furthermore, values of y > p =4 are never necessary. See also QUADRATIC CONGRUENCE EQUATION Excludent Factorization Method Also known as the difference of squares method. It was first used by Fermat and improved by Gauss. Gauss looked for INTEGERS x and y satisfying y2 /C13x2 /C28N (mod E) for various moduli E. This allowed the exclusion of many potential factors. This method works best when factors are of approximately the same size, so it is sometimes better to attempt mN for some suitably chosen value of m. See also PRIME FACTORIZATION ALGORITHMS Exclusion METHOD OF EXCLUSIONS Exclusive Disjunction A DISJUNCTION that is true if only one, but not both, of its arguments are true, and is false if neither or both are true, which is equivalent to the XOR connective. By contrast, the INCLUSIVE DISJUNCTION is true if either or both of its arguments are true. This is equivalent to the OR CONNECTIVE . See also DISJUNCTION ,INCLUSIVE DISJUNCTION , OR, XOR Exclusive Nor XNOR Exclusive Or XOR Excosine Circle If the tangents at B and C to the CIRCUMCIRCLE of a TRIANGLE DABC intersect in a point K1 ; then the CIRCLE with center K1and which passes through B and C is called the excosine circle, and cuts AB and AC in two points which are extremities of a DIA- METER . See also COSINE CIRCLE References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 75, 1893. Exeter Point Define A? to be the point (other than the VERTEX A) where the MEDIAN through A meets the CIRCUMCIR- CLE of ABC , and define B? and C? similarly. Then the Exeter point is the PERSPECTIVE CENTER of the TRIANGLE A?B?C ? and the TANGENTIAL TRIANGLE .It has TRIANGLE CENTER FUNCTION a /C30a(b4 /C27c4 /C28a4): References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Kimberling, C. "Exeter Point." http://cedar.evansville.edu/ ~ck6/tcenters/recent/exeter.html. Kimberling, C. and Lossers, O. P. "Problem 6557 and Solution." Amer. Math. Monthly 97, 535 /C1/37, 1990. Exhaustion Method The method of exhaustion was a INTEGRAL -like limit- ing process used by Archimedes to compute the AREA and VOLUME of 2-D LAMINA and 3-D SOLIDS . See also INTEGRAL ,LIMIT Existence If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Frequently, mathematicians seek to prove the exis- tence of solutions (the EXISTENCE PROBLEM ) and then investigate their UNIQUENESS . See also EXISTENCE PROBLEM ,E XISTS ,P ICARD’S EXISTENCE THEOREM ,UNIQUE Existence Problem The question of whether a solution to a given problem exists. The existence problem can be solved in the affirmative without actually finding a solution to the original problem. Such a demonstration is said to benonconstructive, and is called a NONCONSTRUCTIVE PROOF or an existence proof. See also ENUMERATION PROBLEM ,EXISTENCE ,NON- CONSTRUCTIVE PROOF ,PICARD’S EXISTENCE THEOREM References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 22, 1984. Richman, F. "Existence Proofs." Amer. Math. Monthly 106, 303 /C1/08, 1999. Existence Proof EXISTENCE PROBLEM ,NONCONSTRUCTIVE PROOF Existential Closure A class of processes which attempt to round off a domain and simplify its theory by adjoining elements. See also MODEL COMPLETION References Manders, K. L. "Domain Extension and the Philosophy of Mathematics." J. Philos. 86, 553 /C1/62, 1989. Existential Formula UNIVERSAL FORMULA Existential Quantifier The EXISTS QUANTIFIER /C215:/ See also EXISTS ,F OR ALL,G ENERAL QUANTIFIER , QUANTIFIER Existential Sentence See also UNIVERSAL SENTENCE References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 34, 1958. Exists If there exists an A, this is written /C215A:Similarly, " A does not exist" is written ~A:/C215is one of the two mathematical objects known as QUANTIFIERS . InMathematica 4.0, the command ExistsRealQ [i- neqs ,vars] can be used to determine if there exist real values of the variables vars satisfying the system of real equations and inequalities ineqs . See also EXISTENCE ,FOR ALL,IMPLIES ,QUANTIFIER Exmedian The line through the VERTEX of a TRIANGLE which is PARALLEL to the opposite side. References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 176, 1929. Exmedian Point The point of intersection of two EXMEDIANS . References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 176, 1929. Exogenous Variable An economic variable that is related to other eco- nomic variables and determines their equilibrium levels. See also ENDOGENOUS VARIABLE References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 458, 1980. Exotic R4 Donaldson (1983) showed there exists an exotic smooth DIFFERENTIAL STRUCTURE on R4 : Donaldson’s result has been extended to there being precisely a CONTINUUM of nondiffeomorphic DIFFERENTIAL STRUCTURES on R4 :/ See also EXOTIC SPHERE ,SMOOTH STRUCTURE References Donaldson, S. K. "Self-Dual Connections and the Topology of Smooth 4-Manifold." Bull. Amer. Math. Soc. 8,81/C1/3, 1983. Monastyrsky, M. Modern Mathematics in the Light of the Fields Medals. Wellesley, MA: A. K. Peters, 1997. Exotic Sphere Milnor (1963) found more than one smooth structure on the 7-D HYPERSPHERE . Generalizations have sub- sequently been found in other dimensions. Using SURGERY theory, it is possible to relate the number of DIFFEOMORPHISM classes of exotic spheres to higher homotopy groups of spheres (Kosinski 1992). Kervaire and Milnor (1963) computed a list of the number N(d) of distinct (up to DIFFEOMORPHISM ) DIFFERENTIAL STRUCTURES on spheres indexed by the DIMENSION d of the sphere. For d /C301, 2, ..., assuming the POINCARE ´ CONJECTURE , they are 1, 1, 1, ]2; 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, ... (Sloane’s A001676). The status of d /C304 is still unresolved: at least one exotic structure exists, but it is not known if others do as well. The only exotic Euclidean spaces are a CONTINUUM of EXOTIC R4 structures. See also EXOTIC R4,HYPERSPHERE ,SMOOTH STRUC-TURE References Kervaire, M. A. and Milnor, J. W. "Groups of Homotopy Spheres: I." Ann. Math. 77, 504 /C1/37, 1963. Kosinski, A. A. §X.6 in Differential Manifolds. Boston, MA: Academic Press, 1992. Milnor, J. "Topological Manifolds and Smooth Manifolds." In Proc. Internat. Congr. Mathematicians (Stockholm, 1962). Djursholm: Inst. Mittag-Leffler, pp. 132 /C1/38, 1963. Milnor, J. W. and Stasheff, J. D. Characteristic Classes. Princeton, NJ: Princeton University Press, 1973. Monastyrsky, M. Modern Mathematics in the Light of the Fields Medals. Wellesley, MA: A. K. Peters, 1997. Novikov, S. P. (Ed.). Topology I. New York: Springer-Verlag, 1996. Sloane, N. J. A. Sequences A001676/M5197 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Exp EXPONENTIAL FUNCTION Expansion An AFFINE TRANSFORMATION (sometimes called an enlargement or dilation) in which the scale is in- creased. It is the opposite of a CONTRACTION , and is also sometimes called an enlargement. A CENTRAL DILATION corresponds to an expansion plus a TRANS- LATION . See also AFFINE TRANSFORMATION ,CENTRAL DILA- TION ,CONTRACTION (GEOMETRY ), DILATION ,H OMO- THETIC ,TRANSFORMATION References Coxeter, H. S. M. and Greitzer, S. L. "Dilation." §4.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 94 /C1/5, 1967. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 13, 1999. Expansive Letfbe a MAP. Then fis expansive if the statement that the DISTANCE d(fnx;fny)Bdfor all n/C23Zimplies that x/C30y. Equivalently, fis expansive if the orbits of two points xandyare never very close. Expectation Value The expectation value of a function f(x) in a variable x is denoted /C142f(x)/C143orEff(x)g:For a single discrete variable, it is defined by /C142f(x)/C143/C30X xf(x)P(x): (1) For a single continuous variable it is defined by, /C142f(x)/C143/C30gf(x)P(x)dx: (2) The expectation value satisfies /C142ax /C27by /C143/C30a/C142x/C143/C27b /C142y/C143 (3) /C142a/C143/C30a (4) X xDE /C30X /C142x/C143: (5) For multiple discrete variables /C142f(x1 ; ... ; xn)/C143 /C30X x1 ; ...; xnf(x1 ; ...; xn)P(x1 ; ...; xn) : (6) For multiple continuous variables /C142f(x1 ; ...; xn) /C143 /C30g f(x1 ; ...; xn)P(x1 ; ... ; xn) dx1 /C1/C1/C1dxn : (7) The (multiple) expectation value satisfies /C142(x /C28 mx)(y /C28 my) /C143/C30/C142xy /C28 mxy /C28 myx /C27 mx my /C143 /C30/C142xy/C143/C28 mx my /C28 my mx /C27 mx my /C30/C142xy/C143/C28/C142x/C143/C142y/C143; (8) where mi is the MEAN for the variable i. See also CENTRAL MOMENT ,ESTIMATOR ,M AXIMUM LIKELIHOOD ,MEAN,MOMENT ,RAW MOMENT ,WALD’S EQUATION References Papoulis, A. "Expected Value; Dispersion; Moments." §5 /C1/ in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 139 /C1/52, 1984. Expected Value EXPECTATION VALUE Experiment An experiment E(S ; F ; P) is defined (Papoulis 1984, p. 30) as a mathematical object consisting of the following elements. 1. A set S (the PROBABILITY SPACE ) of elements. 2. A BOREL FIELD F consisting of certain subsets of S called EVENTS . 3. A number P(X) satisfying the PROBABILITY AXIOMS , called the probability, that is assigned to every event A. See also EVENT ,O UTCOME ,P ROBABILITY AXIOMS , PROBABILITY SPACE ,TRIAL References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Experimental Design DESIGNExpIntegralE EN-FUNCTION ExpIntegralEi EXPONENTIAL INTEGRAL Exploration Problem JEEP PROBLEM Exponent The POWER p in an expression ap :/ See also BASE (NUMBER ), POWER ,EXPONENT LAWS, EXPONENT VECTOR ,HAUPT- EXPONENT Exponent Laws The laws governing the combination of EXPONENTS (POWERS ), sometimes called the laws of indices (Hig- gens 1998). The laws are given by xm/C215xn/C30xm/C27n(1) xm xn/C30xm/C28n(2) (xm)n/C30xmn(3) (xy)m/C30xmym(4) x y !n /C30xn yn(5) x/C28n/C301 xn(6) xy ! /C28n /C30yx ! n ; (7) where quantities in the DENOMINATOR are taken to be nonzero. Special cases include x1/C30x (8) and x0/C301 (9) forx"0:The definition 00/C301 is sometimes used to simplify formulas, but it should be kept in mind that this equality is a definition and not a fundamental mathematical truth. See also EXPONENT ,EXPONENTIAL FUNCTION ,POWER References Higgins, P. M. Mathematics for the Curious. Oxford, Eng- land: Oxford University Press, 1998. Krantz, S. G. "Laws of Exponentiation." §1.2.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 8, 1999. Exponent Vector Let pi denote the ith PRIME , and write m /C30Y ipvi i: Then the exponent vector is v(m) /C30(v1;v2;... ):/ See also DIXON’S FACTORIZATION METHOD References Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/485, 1996. Exponential EXPONENTIAL FUNCTION Exponential Digital Invariant NARCISSISTIC NUMBER Exponential Distribution Given a P OISSON DISTRIBUTION with rate of change l; the distribution of waiting times between successive changes (with k/C300) is D(x)/C13P(X5x)/C301/C28P(X>x) /C301/C28(lx)0e/C28lx 0!/C301/C28e/C28lx(1) P(x)/C30D?(x)/C30le/C28lx; (2) which is normalized since g/C12 0P(x)dx/C30lg/C12 0e/C28lxdx /C30/C28[e/C28lx]/C12 0/C30/C28(0/C281)/C301: (3) This is the only MEMORYLESS RANDOM DISTRIBUTION . Define the MEAN waiting time between successive changes as u/C13l/C281:Then P(x)/C301 ue/C28x=ux]0 0 xB0:;j2ffl (4)The MOMENT-GENERATING FUNCTION is M(t)/C30g/C12 0etx1 u ! e/C28x=udx/C301ug/C12 0e/C28(1/C28ut)x=udx /C30e/C28(1/C28ut)x=u 1/C28ut"#/C12 0/C301 1/C28ut(5) M?(t)/C30u (1/C28ut)2(6) Mƒ(t)/C302u2 (1/C28ut)3; (7) so R(t)/C13lnM(t)/C30/C28ln(1/C28ut) (8) R?(t)/C30u 1/C28ut(9) Rƒ(t)/C30u2 (1/C28ut)2(10) m/C30R?(0)/C30u (11) s2/C30Rƒ(0)/C30u2: (12) The CHARACTERISTIC FUNCTION is f(t)/C30Ffle/C28lx[1 2(1/C27sgnx)]g (13) /C30il t/C27il; (14) where F[f] is the F OURIER TRANSFORM with para- meters a/C30b/C301:/ The SKEWNESS and KURTOSIS are given by g1/C302 (15) g2/C306: (16) The MEAN and VARIANCE can also be computed directly xhi/C13g/C12 0P(x)dx/C301 sg/C12 0xe/C28x=sdx: (17) Use the integral gxeaxdx/C30eax a2(ax/C281) (18) to obtain xhi/C301 se/C28x=s /C281 s !2/C281 s ! x/C281()2 666643 77775/C12 0 /C30/C28se/C28x =s1 /C27x s !"#/C12 0 /C30/C28s(0 /C281) /C30s : (19) Now, to find x2;j1r;j11 /C301 s g/C12 0x2e/C28x =s dx ; (20) use the integral g x2e /C28x=s dx /C30eax a3 (2 /C282ax /C27a2x2) (21) x2;j1r;j11 /C301 se /C28x=s /C281 s !32 /C272 sx /C271 s2x2 !2 666643 77775/C12 0 /C30/C28s2(0 /C282) /C302s2 ; (22) giving s2 /C13 x2;j1r;j11 /C28 xhi2 /C302s2 /C28s2 /C30s2 (23) s /C13ffiffiffiffiffiffiffiffiffiffiffiffiffi var(x)p /C30s : (24) If a generalized exponential probability function is defined by P(a ; b)(x) /C301 be /C28(x/C28 a)=b ; (25) for x ] a; then the CHARACTERISTIC FUNCTION is f(t) /C30eiat 1 /C28 ibt ; (26) and the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m/C30a/C27b (27) s2/C30b2(28) g1/C302 (29) g2/C306: (30) See also DOUBLE EXPONENTIAL DISTRIBUTION References Balakrishnan, N. and Basu, A. P. The Exponential Distribu- tion: Theory, Methods, and Applications. New York: Gordon and Breach, 1996. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 534 /C1/35, 1987. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992.Exponential Divisor E-DIVISOR Exponential Function The exponential function is defined by exp(x)/C13ex; (1) where Eis the constant 2.718.... It satisfies the identity exp(x/C27y)/C30exp(x) exp( y): (2) Ifz/C13x/C27iy; ez/C30ex/C27iy/C30exeiy/C30ex(cosy/C27isiny): (3) The exponential function satisfies the identities ex/C30cosh x/C27sinh x (4) /C30sec(gd x)/C27tan(gd x) (5) /C30tan1 4p/C2712gdx;j1ffl;j1{ (6) /C301/C27sin(gd x) cos(gd x); (7) where gd xis the G UDERMANNIAN FUNCTION (Beyer 1987, p. 164; Zwillinger 1995, p. 485). The exponential function has M ACLAURIN SERIES exp(x)/C30X/C12 n/C300xn n!; (8) and satisfies the LIMIT exp(x)/C30lim n0/C121/C27x n !n : (9) If a /C27bi /C30ex/C27iy ; (10) then y /C30tan /C281b a ! (11) x /C30ln b csc tan/C281b a !"#() /C30ln a sec tan/C281b a !"#() : (12) The above plot shows the function e1=z :/ See also CIS, E,EULER FORMULA ,EXPONENT LAWS, EXPONENTIAL RAMP,FOURIER TRANSFORM– EXPONEN- TIAL FUNCTION ,GUDERMANNIAN FUNCTION ,PHASOR , POWER ,SIGMOID FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Exponential Function." §4.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 69 /C1/1, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 217, 1987. Finch, S. "Unsolved Mathematics Problems: Linear Inde- pendence of Exponential Functions." http://www.math- soft.com/asolve/sstein/sstein.html. Fischer, G. (Ed.). Plates 127 /C1/28 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 124 /C1/25, 1986. Krantz, S. G. "The Exponential and Applications." §1.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 7 /C1/2, 1999. Spanier, J. and Oldham, K. B. "The Exponential Function exp(bx /C27c)/" and "Exponentials of Powers exp(/C28ax n) :/"Chs. 26 /C1/7in An Atlas of Functions. Washington, DC: Hemisphere, pp. 233 /C1/61, 1987. Yates, R. C. "Exponential Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 86 /C1/7, 1952. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995. Exponential Generating Function An exponential generating function for the integer sequence a0 ; a1 ; ... is a function E(x) such that E(x) /C30X/C12 k /C300akxk k! /C30a0 /C27a1x 1!/C27a2x2 2!/C27...: See also GENERATING FUNCTION References Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 9, 1995. Exponential Inequality ForcB1, xcB1/C27c(x/C281): Forc/C211, xc>1/C27c(x/C281): Exponential Integral Let E1(x) be the EN-FUNCTION with n/C301, E1(x)/C13g/C12 1e/C28txdt t/C30g/C12 xe/C28udu u: (1) Then define the exponential integral ei(x)by E1(x) /C30/C28ei(/C28x) ; (2) where the retention of the /C28ei(/C28x) NOTATION is a historical artifact. Then ei(x) is given by the integral ei(x) /C30/C28g/C12 /C28xe /C28t dt t: (3) This function is given by the Mathematica function ExpIntegralEi [x]. The exponential integral can also be written ei(ix) /C30ci(x) /C27i si(x) ; (4) where ci(x) and si(x) are COSINE and SINE INTEGRAL . The real ROOT of the exponential integral occurs at 0.37250741078..., which is not known to be expres- sible in terms of other standard constants. The quantity /C28e ei(/C281) /C300 :596347362... is known as the GOMPERTZ CONSTANT . lim x00/C27e2ei(/C28x) x2/C30e2 g ; (5) where g is the EULER- MASCHERONI CONSTANT . The TAYLOR SERIES of ei(/C28x) is given by ei(/C28x) /C30 g /C27ip /C27ln x /C28x /C271 4 x2 /C281 18 x3 /C271 96 x4 /C281 600 x5 /C27...; (6) where the denominators of the coefficients are given by n /C215 n! (Sloane’s A001563; van Heemert 1957, Mundfrom 1994). See also COSINE INTEGRAL , EN-FUNCTION ,GOMPERTZ CONSTANT ,SINE INTEGRAL References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 566 /C1/68, 1985. Jeffreys, H. and Jeffreys, B. S. "The Exponential and Related Integrals." §15.09 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 470 /C1/72, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 434 /C1/35, 1953. Mundfrom, D. J. "A Problem in Permutations: The Game of ‘Mousetrap’." European J. Combin. 15, 555 /C1/60, 1994. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Exponential Integrals." §6.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 215 /C1/19, 1992. Sloane, N. J. A. Sequences A001563/M3545 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Exponential Integral Ei(x) and Related Functions." Ch. 37 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 351 /C1/60, 1987. van Heemert, A. "Cyclic Permutations with Sequences and Related Problems." J. reine angew. Math. 198,56/C1/2, 1957.Exponential Map On a LIE GROUP , exp is a MAP from the LIE ALGEBRA to its LIE GROUP . If you think of the LIE ALGEBRA as the TANGENT SPACE to the identity of the LIE GROUP , exp(v) is defined to be h(1) ; where h is the unique LIE GROUP HOMEOMORPHISM from the REAL NUMBERS to the LIE GROUP such that its velocity at time 0 is v. On a RIEMANNIAN MANIFOLD , exp is a MAP from the TANGENT BUNDLE of the MANIFOLD to the MANIFOLD , and exp(v) is defined to be h(1) ; where h is the unique GEODESIC traveling through the base-point of v such that its velocity at time 0 is v. The three notions of exp (exp from COMPLEX ANALY- SIS, exp from LIE GROUPS , and exp from Riemannian geometry) are all linked together, the strongest link being between the LIE GROUPS and Riemannian geometry definition. If G is a compact LIE GROUP ,it admits a left and right invariant R IEMANNIAN ME- TRIC. With respect to that metric, the two exp maps agree on their common domain. In other words, one- parameter subgroups are geodesics. In the case of the MANIFOLD S1;the CIRCLE , if we think of the tangent space to 1 as being the IMAGINARY axis ( Y-AXIS ) in the COMPLEX PLANE , then expRiemannian geometry (v)/C30expLie Groups (v) /C30expcomplex analysis (v); and so the three concepts of the exponential all agree in this case. See also EXPONENTIAL FUNCTION ,M ATRIX EXPONEN- TIAL References Huang, J.-S. "The Exponential Map." §7.3 in Lectures on Representation Theory. Singapore: World Scientific, pp. v, 1999. Exponential Map Matrix MATRIX EXPONENTIAL Exponential Matrix MATRIX EXPONENTIAL Exponential Polynomial Polynomials fn(x) (sometimes called the B ELL POLY- NOMIALS ) which form the associated SHEFFER SE- QUENCE for f(t) /C30ln(1 /C27t); (1) and therefore have GENERATING FUNCTION Xn k /C300fk(x) k!tk /C30e(et/C281)x : (2) Additional GENERATING FUNCTIONS are given by fn(x) /C13e /C28xX/C12 k /C300knxk k! (3) or fn(x) /C30xXn k /C301n /C281 k /C281;j1z;j1} fk /C281(x) ; (4) with f0(x) /C301; where n k;jr;j1 is a BINOMIAL COEFFICIENT . The exponential polynomials have the explicit for- mula fn(x) /C30Xn k/C300S(n; k)xk ; (5) where S(n; k)isaS TIRLING NUMBER OF THE SECOND KIND . The binomial identity fn(x /C27y) /C30Xn k /C300n k;j1z;j1} fk(x) fn/C28k(y); (6) wheren k;jr;j1 is a BINOMIAL COEFFICIENT , and the recurrence formula is fn/C271(x) /C30x[ fn(x) /C27 f?n(x)]: (7) The Bell polynomials are defined such that fn(1) /C30 Bn ; where Bnis a BELL NUMBER . The first few Bell polynomials are f0(x) /C301 f1(x) /C30x f2(x) /C30x /C27x2 f3(x) /C30x /C273x2 /C27x3 f4(x) /C30x /C277x2 /C276x3 /C27x4 f5(x) /C30x /C2715x2 /C2725x3 /C2710x4 /C27x5 f6(x) /C30x /C2731x2 /C2790x3 /C2765x4 /C2715x5 /C27x6 : See also ACTUARIAL POLYNOMIAL ,B ELL NUMBER , DOBINSKI’S FORMULA ,L AH NUMBER ,SHEFFER SE-QUENCE ,STIRLING NUMBER OF THE SECOND KIND References Bell, E. T. "Exponential Polynomials." Ann. Math. 35, 258 /C1/ 77, 1934. Roman, S. "The Exponential Polynomials." §4.1.3. in The Umbral Calculus. New York: Academic Press, pp. 63 /C1/7, 1984. Exponential Ramp The curve y/C301/C28eax illustrated above. See also EXPONENTIAL FUNCTION ,SIGMOID FUNCTION References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 158, 1993. Exponential Sum Formulas XN/C281 n/C300einx/C301/C28eiNx 1/C28eix/C30/C28eiNx=2(e/C28iNx=2/C28eiNx=2) /C28eix=2(e/C28ix=2/C28eix=2) /C30sin1 2Nx;j1ffl;j1{ sin12x;j1ffl;j1{ eix(N/C281)=2; (1) where XN/C281 n/C300rn/C301/C28rN 1/C28r(2) has been used. Similarly, XN /C281 n/C300pneinx /C301 /C28 pNeiNx 1 /C28 peix (3) X/C12 n/C300pneinx /C301 eipx /C28 1 /C301 /C28 pe /C28ix 1 /C28 2p cos x /C27 p2 : (4) By looking at the REAL and IMAGINARY PARTS of these FORMULAS , sums involving sines and cosines can be obtained. Exponential Sum Function The exponential sum function en(x) ; sometimes also denoted expn(x) ; is defined by en(x) /C13Xn k/C300xk k! /C30ex G(n /C27 1 ; x) G(n /C27 1); where G(a ; x) is the upper INCOMPLETE GAMMA FUNCTION and G(x) is the (complete) GAMMA FUNC- TION . See also GAMMA FUNCTION ,INCOMPLETE GAMMA FUNCTION Exponential Transform The exponential transform is the transformation of a sequence a1 ; a2 ; ... into a sequence b1 ; b2 ; ... according to the equation 1 /C27X/C12 n /C301bnxn n!/C30expX/C12 n/C301anxn n! ! : The inverse ("logarithmic"rpar; transform is then given by X/C12 n/C301anxn n!/C30ln 1 /C27X/C12 n/C301bnxn n! ! : The exponential transform relates the number anof labeled CONNECTED GRAPHS on n nodes satisfying some property with the corresponding total number bn (not necessarily connected) of labeled GRAPHS on nnodes. In this application, the transform is called RIDDELL’S FORMULA for labeled graphs. See also BINOMIAL TRANSFORM ,EULER TRANSFORM , LOGARITHMIC TRANSFORM ,MO¨ BIUS TRANSFORM ,RID- DELL’S FORMULA ,STIRLING TRANSFORM References Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 19 /C1/0, 1995. Expression See also QUANTITY Exradius The RADIUS of an EXCIRCLE . Let a TRIANGLE have exradius r1(sometimes denoted r1);opposite side of length a1and angle a1;AREAD;and SEMIPERIMETER s. Then r2 1/C30D s/C28a1 !2 (1) /C30s(s/C28a2)(s/C28a3) s/C28a1(2) /C304Rsin1 2a1;j1ffl;j1{ cos12a2;j1ffl;j1{ cos12a3;j1ffl;j1{ (3) (Johnson 1929, p. 189), where Ris the CIRCUMRA- DIUS. Let rbe the INRADIUS , then 4R/C30r1/C27r2/C27r3/C28r (4) 1 r1/C271 r2/C271 r3/C301 r(5) rr1r2r3/C30D2: (6) Some fascinating FORMULAS due to Feuerbach are r(r2r3 /C27r3r1 /C27r1r2) /C30s D/C30r1r2r3 (7) r(r1 /C27r2 /C27r3) /C30a2a3 /C27a3a1 /C27a1a2 /C28s2 (8) rr1 /C27rr2 /C27rr3 /C27r1r2 /C27r2r3 /C27r3r1 /C30a2a3 /C27a3a1 /C27a1a2 (9) r2r3 /C27r3r1 /C27r1r2 /C28rr1 /C28rr2 /C28rr3 /C301 2(a2 1 /C27a22 /C27a23) (10) (Johnson 1929, pp. 190 /C1/91). See also CIRCLE ,CIRCUMRADIUS ,EXCIRCLE ,INRADIUS , RADIUS References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles of a Triangle." Proc. Edinburgh Math. Soc. 12,86/C1/05. Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles of a Triangle." Proc. Edinburgh Math. Soc. 13, 103 /C1/04. Exsecant exsec x /C13sec x /C281; where sec x is the SECANT . See also COVERSINE ,HAVERSINE ,SECANT ,VERSINE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 78, 1972. Extended Binary Tree A BINARY TREE in which special nodes are added wherever a null subtree was present in the original tree so that each node in the original tree (except the root node) has degree three (Knuth 1997, p. 399). See also BINARY TREE References Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997.Extended Complex Plane The COMPLEX PLANE with a POINT AT INFINITY attached: C @f/C12g; where /C12 denotes COMPLEX INFI- NITY. The extended complex plane is denoted C*. See also C*,C OMPLEX INFINITY ,C OMPLEX PLANE , RIEMANN SPHERE References Krantz, S. G. "The Topology of the Extended Complex Plane." §6.3.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 83, 1999. Extended Cycloid PROLATE CYCLOID Extended Goldbach Conjecture GOLDBACH CONJECTURE Extended Greatest Common Divisor GREATEST COMMON DIVISOR Extended Mean-Value Theorem Let the functions f and g be DIFFERENTIABLE on the OPEN INTERVAL (a, b) and CONTINUOUS on the CLOSED INTERVAL [a, b]. If g ?(x) "0 for any x /C23 (a ; b); then there is at least one point c /C23 (a ; b) such that f ?(c) g ?(c) /C30f(b) /C28 f(a) g(b) /C28 g(a) : See also MEAN-VALUE THEOREM Extended Real Number (Affine) This entry contributed by D AVID W.CANTRELL The set R@f/C27/C12;/C28/C12gobtained by adjoining two improper elements to the set Rof real numbers is normally called the set of (affinely) extended real numbers. Although the notation for this set is not completely standardized, ¯Ris commonly used. The set may also be written in interval notation as [ /C28/C12;/C27/C12]: With an appropriate topology, ¯Ris the two-point COMPACTIFICATION (or affine closure) of R:The im- proper elements, the affine infinities /C27/C12and/C28/C12; correspond to ideal points of the number line. Note that these improper elements are notreal numbers, and that this system of extended real numbers is nota FIELD . Instead of writing /C27/C12;many authors write simply /C12: However, the compound symbol /C27/C12will be used here to represent the positive improper element of ¯R; allowing the individual symbol /C12 to be used unam- biguously to represent the unsigned improper ele- ment of R /C31; the one-point COMPACTIFICATION (or projective closure) of R :/ A very important property of ¯R ; which R lacks, is that every subset S of ¯R has an INFIMUM (greatest lower bound) and a SUPREMUM (least upper bound). In particular, sup ¥/C30/C28/C12 and, if S is unbounded above, then sup S /C30/C27/C12: Similarly, inf ¥/C30/C27/C12 and, if S is unbounded below, then inf S /C30/C28/C12:/ Order relations can be extended from R to ¯R ; and arithmetic operations can be partially extended. For x /C23 ¯R ; /C28/C12B x B/C27/C12 if x "9/C12 ;/C28/C12B/C27/C12 (1) /C28(/C27/C12) /C30/C28/C12;/C28(/C28/C12) /C30/C27/C12 (2) x /C27(/C27/C12) /C30/C27/C12/C27x /C30/C27/C12 if x "/C28/C12 (3) x /C27(/C28/C12) /C30/C28/C12/C27x /C30/C28/C12 if x "/C27/C12 (4) x /C215(9/C12) /C309/C12 /C215 x /C309/C12 if x > 0 (5) x /C215(9/C12) /C309/C12 /C215 x /C30/C14/C12 if x B0 (6) x 9/C12/C300if x "9/C12 (7) x 0;j12;j12;j12;j12;j12;j12;j12;j12;j12;j12/C30/C27/C12 if x "0 (8) However, the expressions /C27/C12/C27(/C28/C12) ;/C28/C12/C27(/C27/C12); and x=0 are UNDEFINED . The above statements which define results of arith- metic operations on ¯R may be considered as abbrevia- tions of statements about determinate LIMIT forms. For example, /C28(/C27/C12) /C30/C28/C12 may be considered as an abbreviation for "If x increases without bound, then /C28x decreases without bound." Most descriptions of ¯R also make a statement concerning the products of the improper elements and 0, but there is no consensus as to what that statement should be. Some authors (e.g., Kolmogorov 1995, p. 193) state that, like /C27/C12/C27(/C28/C12) and /C28/C12/C27(/C27/C12); 0 /C215 (9/C12) and 9/C12 /C215 0 should be UNDEFINED , presumably because of the INDETERMI- NATE status of the corresponding LIMIT forms. Other authors (such as McShane 1983, p. 2) accept 0 /C215 (9/C12) /C309/C12 /C2150 /C300; at least as a convention which is useful in certain contexts. Many results for other operations and functions can be obtained by considering determinate LIMIT forms. For example, a partial extension of the function f(x; y) /C30xy can be obtained for x; y /C23 ¯R as (/C27/C12)y /C300i f y B0 /C27/C12 if y > 0;j2ffl (9) x/C27/C12/C300i f 0 Bx B1 /C27/C12 if x > 1;j2ffl (10)x/C28/C12/C30/C27/C12 if 0 Bx B1 0i f x > 1:;j2ffl (11) The functions ex and ln xjjcan be fully extended to ¯R; with e /C28/C12/C300 (12) e/C27/C12/C30/C27/C12 (13) ln 0jj/C30/C28/C12 (14) ln9/C12jj/C30/C27/C12: (15) Some other important functions (e.g., tanh( 9/C12)/C3091 and tan/C281(9/C12)/C309p=2) can be extended to ¯R;while others (e.g., sin x;cosx) cannot. Evaluations of expressions involving /C27/C12and/C28/C12;derived by con- sidering determinate LIMIT forms, are routinely used by computer algebra systems such as Mathematica when performing simplifications. See also CLOSURE (SET), COMPACTIFICATION ,E X- TENDED REAL NUMBER (PROJECTIVE ), INDETERMI- NATE ,LIMIT,R,R -,R/C27,REAL NUMBER References Kolmogorov, N. A. "Infinity." Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia," 2nd ed., Vol. 3. (Mana- ging Ed. M. Hazewinkel). Dordrecht, Netherlands: Reidel,1995. McShane, E. J. Unified Integration. Orlando, FL: Academic Press, p. 2, 1983. Extended Real Number (Projective) This entry contributed by D AVID W.CANTRELL The set R@f/C12g;obtained by adjoining one improper element to the set Rof real numbers, is the set of projectively extended real numbers. Although nota- tion is not completely standardized, R/C31is used here to denote this set of extended real numbers. With anappropriate topology, R/C31is the one-point COMPACTI- FICATION (or projective closure) of R:As shown above, the cross section of the R IEMANN SPHERE consisting of its "real axis" and "north pole" can be used tovisualize R/C31:The improper element, projective in- finity ( //C12);then corresponds with the ideal point, the "north pole." In contrast to the signed affine infinities (//C27/C12 and /C28/C12) of the affinely EXTENDED REAL NUMBERS ¯R; projective infinity, /C12; is unsigned, like 0. Regrettably, /C12 is also unordered, i.e., for x /C23R /C31 it can be said neither that x B/C12 nor that x >/C12: For this reason, R /C31 is used much less often in real analysis than is ¯R: Thus, if context is not specified, "the extended real numbers" normally refers to ¯R; not R/C31:/ Arithmetic operations can be partially extended from R to R /C31; /C28( /C12) /C30/C12; x /C27/C12/C30/C12/C27x /C30/C12 if x "/C12 ; x /C215/C12/C30/C12 /C215 x /C30/C12 if x "0; x=/C12/C300i f x "/C12 ; and x=0 /C30/C12 if x "0 (by contrast, x=0is UNDEFINED in ¯R) : The expressions Kn and 0 /C215/C12 are most often left UNDEFINED in R /C31:/ The exponential function ex cannot be extended to R/C31: On the other hand, R /C31 is useful when dealing with rational functions and certain other functions. For example, if R /C31 is used as the range of tan x; then by taking tan((2 n /C271)p=2) /C30/C12 for integer n, the domain of the function can be extended to all of R: Extended real numbers are sometimes used in the implementa- tion of FLOATING-POINT ARITHMETIC (Hauser 1996, pp. 158 /C1/59). See also COMPACTIFICATION ,C LOSURE (SET), EX- TENDED REAL NUMBER (AFFINE ), REAL NUMBER , RIEMANN SPHERE References Hauser, J. R. "Handling Floating-Point Exceptions in Nu- meric Programs." ACM Trans. Program. Lang. Sys. 18, 139 /C1/74, 1996. http://www.cs.berkeley.edu/~jhauser/excep- tions/HandlingFloatingPointExceptions.html. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia," Vol. 3. Dordrecht, Netherlands: Reidel, p. 193, 1988. Extended Riemann Hypothesis The first quadratic nonresidue mod p of a number is always less than 2(ln p)2 :/ See also RIEMANN HYPOTHESIS References Bach, E. Analytic Methods in the Analysis and Design of Number-Theoretic Algorithms. Cambridge, MA: MIT Press, 1985. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 295, 1991. ExtendedGCD GREATEST COMMON DIVISORExtension (Ideal) The extension of a; an IDEAL in COMMUTATIVE RING A, in a RING B, is the IDEAL generated by its image f(a) under a RING HOMOMORPHISM f. Explicitly, it is any finite sum OF THE FORM a yif(xi) where yi is in B and xi is in a: Sometimes the extension of a is denoted ae :/ The image f( a) may not be an ideal if f is not SURJECTIVE . For instance, f : Z 0 Z[x] is a ring homomorphism and the image of the even integers is not an ideal since it does not contain any non- constant polynomials. The extension of the even integers in this case is the set of polynomials with even coefficients. The extension of a PRIME IDEAL may not be prime. For example, consider f : Z 0 Zffiffiffi 2p;j2;j3 : Then the extension of the even integers is not a prime ideal since 2 /C30ffiffiffi 2p /C215ffiffiffi2p : / See also ALGEBRAIC NUMBER THEORY ,CONTRACTION (IDEAL ), IDEAL ,PRIME IDEAL ,RING References Atiyah, M. F. and MacDonald, I. G. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, pp. 9 /C1/0, 1969. Extension (Set) The definition of a SET by enumerating its members. An extensional definition can always be reduced to an INTENTIONAL one. An EXTENSION FIELD is sometimes also called simply an extension. See also EXTENSION FIELD ,INTENSION References Russell, B. "Definition of Number." Introduction to Mathe- matical Philosophy. New York: Simon and Schuster, 1971. Extension Field A FIELD K is said to be an extension field (or field extension, or extension), denoted K =F ; of a field F if F is a SUBFIELD of K. The COMPLEX NUMBERS are an extension field of the REAL NUMBERS , and the REAL NUMBERS are an extension field of the RATIONAL NUMBERS . The DEGREE ) (or relative degree, or index) of an extension field K =F ; denoted [K : F] ; is the dimension ofKas a VECTOR SPACE over F, i.e., [K:F]/C30dimFK: See also DEGREE (EXTENSION FIELD), FIELD,PYTHA- GOREAN EXTENSION ,SPLITTING FIELD,SUBFIELD References Dummit, D. S. and Foote, R. M. "Basic Theory of Field Extensions." §13.1 in Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 422 /C1/32, 1998. Extension Problem Given a SUBSPACE A of a SPACE X and a MAP from A to a SPACE Y, is it possible to extend that MAP to a MAP from X to Y? See also LIFTING PROBLEM Extensions Calculus EXTERIOR ALGEBRA Extent The RADIUS of the smallest CIRCLE centered at one of the points of an N-CLUSTER , which contains all the points in the N-CLUSTER . See also N-CLUSTER Exterior That portion of a region lying "outside" a specified boundary. See also INTERIOR Exterior Algebra The ALGEBRA of the EXTERIOR PRODUCT , also called an alternating algebra or Grassmann algebra. The study of exterior algebra is also called Ausdehnungslehre and extensions calculus. Exterior algebras are GRADED ALGEBRAS . In particular, the exterior algebra of a VECTOR SPACE is the DIRECT SUM over kin the natural numbers of the VECTOR SPACES of alternating k-forms on that VECTOR SPACE . The product on this algebra is then the wedge product of forms. The exterior algebra for a VECTOR SPACE Vis constructed by forming monomials u,vfflw;xfflyfflz;etc., where u,v,w,x,y, and zare vectors in Vandfflis asymmetric multiplication. The sums formed from LINEAR COMBINATIONS of the MONOMIALS are the elements of an exterior algebra. The exterior algebra of a VECTOR SPACE can also be described as a QUOTIENT VECTOR SPACE , LpV/C30/C156pV=Wp; (1) where Wpis the subspace of p-tensors generated by transpositions such as W2/C30x/C156y/C27y/C156x hi and/C156 denotes the TENSOR PRODUCT . The EQUIVALENCE CLASS [x1/C156.../C156xp] is denoted x1ffl...fflxp:For in- stance,xffly/C27yfflx/C300; (2) since the representatives add to an element of W2: Consequently, xffly/C30/C28yfflx:Sometimes LpVis called thepth exterior power of V, and may also be denoted by AltpV:/ The alternating products are a SUBSPACE of the tensor products. Define the linear map Alt :/C156pV0/C156pV (3) by Alt(vi1/C156.../C156vip)/C301 p!X sp(s)vis(1)/C156.../C156vis(p);(4) where sranges over all PERMUTATIONS off1;...;pg; andp(s) is the signature of the PERMUTATION , given by the PERMUTATION SYMBOL . Then LpVis the image of Alt, as Wpis its NULLSPACE . The constant factor 1=p! , which is sometimes not used, makes Alt into a PROJECTION OPERATOR . For example, if Vhas the BASIS fe1;e2;e3;e4g;then L0V/C301hi (5) L1V/C30e1;e2;e3;e4 hi (6) L2V/C30e1ffle2;e1ffle3;e1ffle4;e2ffle3;e2ffle4;e3ffle4 hi (7) L3V/C30e1ffle2ffle3;e1ffle2ffle4;e1ffle3ffle3; h /C2e2ffle3ffle4i (8) L4V/C30e1ffle2ffle3ffle4 hi ; (9) andLkV/C30f0gwhere k>dimV:For a general VECTOR SPACE Vof dimension n, the space LpVhas dimensionn p;j1ffl;j1{ :/ Here is a Mathematica function that implements the Alt operator, whose image is the alternating subspaceof the p-tensors. Alt[x_] : /C30Module[ {p/C30TensorRank[x], perms}, perms /C30Permutations[Range[p]]; Sum[ Signature[perms[[i]]] Transpose[x, perms[[i]]], {i, p!} ]/p! ] Here is a Mathematica function which tests whether ap-tensor is alternating by testing transpositions. Transpositions[n_] : /C30Module[{i}, Table[Range[n] /. {i - /C21i/C271, i/C271-/C21i}, {i, n - 1}] ] AlternatingQ[a_] : /C30(And[##1] &) @@ ((a /C30/C30 -Transpose[a, #1] &) /@ Transpositions[TensorRank[a]]) The space L/C31/C30/C156p LpV becomes an ALGEBRA with the WEDGE PRODUCT , defined using the function Alt. Also, if T : V 0 W is a LINEAR TRANSFORMATION , then the map T /C31;p : LpV 0LpW sends v1 ffl...fflvpto T(v1) ffl ...fflT(vp): If n /C30dim V and T(v) /C30Av where A is a SQUARE MATRIX , then /T /C31;n (e1 ffl...fflen) /C30/ /(det A)e1 ffl...fflen :/ The alternating algebra, also called the exterior algebra, L/C31V is a 2n dimensional ALGEBRA .InMath- ematica , an element of the alternating algebra can be represented by an n-nested binary list. For example, {{{1, 2}, {0, 0}}, {{3, 0}, {4, 5}}} represents e1 ffle2 ffl e3 /C272e1 ffle3 /C273e2 ffle3 /C274e3 /C275 : The WEDGE PRODUCT can defined by the following Mathematica function sgntmp[a_, b_] : /C30 (-1)^(Mod[Sum[b[[i]], {i, Length[b]}], 2]) a sgn[a_] : /C30 Module[{d /C30 TensorRank[a]}, MapIndexed[sgntmp, a, {d}] ] wedge[{a_, b_}, {c_, d_}] : /C30 Module[{rnk /C30 TensorRank[a]}, If[rnk /C30/C30 0, {a d /C27 b c, b d}, {wedge[a, d] /C27 wedge[ sgn[b], c], wedge[b, d]} ] ] The following Mathematica function gives the p powers of an element a in the exterior algebra as a tensor. ExtToTensor[a_, p_] : /C30 Module[{d /C30 TensorRank[a], tmp, ind, indices}, tmp /C30 Table[2, {d}]; If[p /C30/C30 0, (a[[##1]] &) @@ tmp, Array[ (Block[{b}, b /C30 {##1}; ind /C30 ReplacePart[tmp, 1, Transpose[{b}]]; Signature[b]/p! (a[[##1]] &) @@ ind] &), Table[d, {p}]]] ] The rank of an alternating form has a couple different definitions. The rank of a form, used in studying integral manifolds of differential ideals, is the dimen- sion of its ENVELOPE . Another definition is its rank as a TENSOR . The DIFFERENTIAL K-FORMS in modern geometry are an exterior algebra, and play a role in multivariable calculus. In general, it is only necessary for V to have the structure of a MODULE . So exterior algebras come up in REPRESENTATION THEORY . For example, if V is a REPRESENTATION of a group G, then Sym2V /C156L2V is a decomposition of V /C156V into two representations. See also DIFFERENTIAL FORM,E NVELOPE (FORM), REPRESENTATION ,SYMMETRIC GROUP ,TENSOR PRO- DUCT ,VECTOR SPACE ,W EDGE PRODUCTReferences Flanders, H. Differential Forms with Applications to the Physical Sciences. New York: Academic Press, 1963. Forder, H. G. The Calculus of Extension. Cambridge, Eng- land: Cambridge University Press, 1941. Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, pp. 472 /C1/75, 1991. Lounesto, P. "Counterexamples to Theorems Published and Proved in Recent Literature on Clifford Algebras, Spinors, Spin Groups, and the Exterior Algebra." http://www.hit.fi/ ~lounesto/counterexamples.htm. Peano, G. Geometric Calculus According to the Ausdehnung- slehre of H. Grassmann. Boston: Birkha ¨user, 2000. Sternberg, S. Differential Geometry. New York: Chelsea, pp. 14 /C1/0, 1983. Exterior Angle The angle ai formed between a side of a polygon and the extension of an adjacent side. Since there are two directions in which a side can be extended, there are two exterior angles at each vertex. However, since corresponding angles are opposite, they are also equal. The sum of exterior angles in a convex polygon is equal to 2p RADIANS (360 8), since this corresponds to one complete rotation of the polygon. See also ANGLE ,EXTERIOR ANGLE BISECTOR Exterior Angle Bisector The exterior bisector of an ANGLE is the LINE orLINE SEGMENT which cuts it into two equal ANGLES on the opposite "side" as the ANGLE . For a TRIANGLE , the exterior angle bisector bisects the SUPPLEMENTARY ANGLE at a given VERTEX . It also divides the opposite side externally in the ratio of adjacent sides. The points A?; B ?; and C ? determined on opposite sides of a triangle DABC by an ANGLE BISECTOR from each vertex, lie on a straight line if either (1) all or (2) one out of the three bisectors is an external angle bisector (Honsberger 1995). See also ANGLE BISECTOR ,ISODYNAMIC POINTS References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 12, 1967. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 149 /C1/50, 1995. Exterior Angle Theorem In any TRIANGLE , if one of the sides is extended, the exterior angle is greater than both the interior and opposite angles. See also EXTERIOR ANGLE References Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 41, 1990. Exterior Derivative The exterior derivative of a function fis the ONE- FORM df/C30X i@f @xidxi (1) written in a COORDINATE CHART (x1;...;xn):Think- ing of a function as a zero-form, the exterior deriva- tive extends linearly to all DIFFERENTIAL K-FORMS using the formula d(afflb)/C30dafflb/C27(/C281)paffldb; when ais a k-form and where fflis the WEDGE PRODUCT .The exterior derivative of a k-form is a ( k/C271)/-form. For example, for a DIFFERENTIAL K-FORM v1/C30b1dx1/C27b2dx2; (2) the exterior derivative is dv1/C30db1ffldx1/C27db2ffldx2: (3) Similarly, consider v1/C30b1(x1;x2)dx1/C27b2(x1;x2)dx2: (4) Then dv1/C30db1ffldx1/C27db2ffldx2 /C30@b1 @x1dx1/C27@b1 @x2dx2 ! ffldx1 /C27@b2 @x1dx1/C27@b2 @x2dx2 ! ffldx2:(5) Denote the exterior derivative by Dt/C13@ @xfflt: (6) Then for a 0-form t, (Dt)m/C13@t @xm; (7) for a 1-form t, (Dt)mn/C131 2@tn @xm/C28@tm @xn ! ; (8) and for a 2-form t, (Dt)ijk/C131 3eijk@t23 @x1/C27@t31 @x2/C27@t12 @x3 ! ; (9) where eijkis the PERMUTATION TENSOR . It is always the case that d(da)/C300:When da/C300;then ais called a CLOSED FORM .A TOP-DIMENSIONAL FORM is always a CLOSED FORM . When a/C30dhthen ais called an EXACT FORM , so any EXACT FORM is also CLOSED . An example of a CLOSED FORM which is not EXACT isduon the circle. Since uis a function defined up to a constant multiple of 2 p;duis a WELL DEFINED ONE-FORM , but there is no function for which it is the EXTERIOR DERIVATIVE . The exterior derivative is linear and commutes with the PULLBACK v/C31ofDIFFERENTIAL K-FORMS v:That is, df/C31(a)/C30f/C31(da): (10) Hence the PULLBACK of a CLOSED FORM is closed and the PULLBACK of an EXACT FORM is exact. Moreover, a DERHAM COHOMOLOGY class [ a] has a WELL DEFINED PULLBACK MAP [f/C31(a)]:/ In Mathematica ,a k-form can be written as an ANTISYMMETRIC k-tensor. Using this format, the following Mathematica function computes the exter- ior derivative of the form a in the (ordered) variables vars . Alt[x_List] : /C30 Module[ { p /C30 TensorRank[x], perms }, perms /C30 Permutations[Range[p]]; Sum[Signature[perms[[i]]] Transpose[x, perms[[i]]],{i, p!}]/p! ] ExtD1[a_List, vars_?List] : /C30 Alt[Outer[D[#2, #1] &, vars , a]] It is also possible to use an n-nested binary tree to represent the algebra of differential forms. Using this format, the following Mathematica function computes the exterior derivative recursively. ExtD2[{a_List, b_List}, vars_List] : /C30 {D[b, First[vars]] - ExtD2[a, Rest[vars]], ExtD2[b, Rest[vars]]} ExtD2[{a_?(! ListQ[#1] &), b_?(! ListQ[#1] &)}, var_?ListQ] : /C30 {D[b, First[var]], 0} See also DIFFERENTIAL K-FORM,EXTERIOR ALGEBRA , HODGE STAR,JACOBIAN ,MANIFOLD ,POINCARE ´ ’S LEM- MA,STOKES’ THEOREM ,TANGENT BUNDLE ,TENSOR , WEDGE PRODUCT References Berger, M. Differential Geometry. New York: Springer- Verlag, p. 152, 1988. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 2nd ed. Houston, TX: Publish or Perish Press, pp. 286 /C1/05, 1999. Sternberg, S. Differential Geometry. New York: Chelsea, pp. 99 /C1/04, 1983. Exterior Dimension A type of DIMENSION which can be used to character- ize FAT FRACTALS . See also FAT FRACTAL References Grebogi, C.; McDonald, S. W.; Ott, E.; and Yorke, J. A. "Exterior Dimension of Fat Fractals." Phys. Let. A 110, 1 /C1/, 1985. Grebogi, C.; McDonald, S. W.; Ott, E.; and Yorke, J. A. Erratum to "Exterior Dimension of Fat Fractals." Phys. Let. A 113, 495, 1986. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 98, 1993. Exterior Power The kth exterior power of an element a in an EXTERIOR ALGEBRA LV is given by the WEDGE PRO- DUCT of a with itself k times. Note that if a has odd degree, then any higher power of a must be zero. Thesituation for even degree forms is different. For example, if a /C30e1 ffle2 /C27e3 ffle4 /C27e5 ffle6 ; (1) then a2 /C302e1 ffle2 ffle3 ffle4 /C272e1 ffle2 ffle5 ffle6 /C272e3 ffle4 ffle5 ffle6(2) a3 /C306e1 ffle2 ffle3 ffle4 ffle5 ffle6 ; (3) a4 /C300: (4) See also EXTERIOR ALGEBRA ,W EDGE PRODUCT Exterior Product WEDGE PRODUCT Exterior Snowflake The FRACTAL illustrated above. See also FLOWSNAKE FRACTAL ,K OCH ANTISNOW- FLAKE ,KOCH SNOWFLAKE ,PENTAFLAKE References Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 193 /C1/95, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . External Contact TANGENT EXTERNALLY External Direct Product The term external direct product is used to refer to either the EXTERNAL DIRECT SUM of groups under the group operation of multiplication, or over infinitelymany spaces in which the sum is not required to befinite. In the latter case, the operation is also called the C ARTESIAN PRODUCT . See also CARTESIAN PRODUCT ,EXTERNAL DIRECT SUM External Direct Sum The C ARTESIAN PRODUCT of a finite or infinite set of modules over a ring with only finitely many nonzero entries in each sequence. See also CARTESIAN PRODUCT ,E XTERNAL DIRECT PRODUCT External Path Length The sum over all external (square) nodes of the paths from the root of an EXTENDED BINARY TREE to each node. For example, in the tree above, the external path length is 25 (Knuth 1997, p. 399 /C1/00). The INTERNAL and external path lengths are related by E /C30I /C272n; where n is the number of internal nodes. See also EXTENDED BINARY TREE,INTERNAL PATH LENGTH References Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. External Tensor Product Suppose that V is a REPRESENTATION of G, and W is a REPRESENTATION of H. Then the TENSOR PRODUCT V /C156W is a REPRESENTATION of the GROUP DIRECT PRODUCT G /C29H : An element (g, h)ofG /C29H acts on a basis element v /C156w by (g ; h)(v /C156w) /C30gv /C156hw : To distinguish from the TENSOR PRODUCT of repre- sentations, the external tensor product is denoted V/C156W ; although the only possible confusion would occur when G /C30H. When V and W are IRREDUCIBLE REPRESENTATIONS of G and H respectively, then so is the external tensor product. In fact, all IRREDUCIBLE REPRESENTATIONS of G /C29H arise as external direct products of IRREDUCI- BLE REPRESENTATIONS . See also GROUP ,IRREDUCIBLE REPRESENTATION , REPRESENTATION ,T ENSOR PRODUCT (REPRESENTA- TION ), TENSOR PRODUCT (VECTOR SPACE ), VECTOR SPACE Externally Tangent TANGENT EXTERNALLYExtra Strong Lucas Pseudoprime Given the LUCAS SEQUENCE Un(b ;/C281) and Vn(b;/C281); define D/C30b2 /C284: Then an extra strong Lucas pseu- doprime to the base b is a COMPOSITE NUMBER n /C30 2rs /C27( D=n) ; where s is ODD and (n; 2 D) /C301 such that either Us/C130 (mod n) and Vs/C1392 (mod n);orV2ts/C13 0 (mod n) for some twith 05tBr/C281:An extra strong Lucas pseudoprime is a STRONG LUCAS PSEU- DOPRIME with parameters ( b;/C281):COMPOSITE nare extra strong pseudoprimes for at most 1/8 of possible bases (Grantham 1997). See also LUCAS PSEUDOPRIME ,STRONG LUCAS PSEU- DOPRIME References Grantham, J. "Frobenius Pseudoprimes." http://www.clar- k.net/pub/grantham/pseudo/pseudo1.ps Grantham, J. "A Frobenius Probable Prime Test with High Confidence." 1997. http://www.clark.net/pub/grantham/ pseudo/pseudo2.ps Jones, J. P. and Mo, Z. "A New Primality Test Using Lucas Sequences." Preprint. Extrapolation RICHARDSON EXTRAPOLATION Extremal Coloring EXTREMAL GRAPH Extremal Graph In general, an extremal graph is the largest graph of order nwhich does not contain a given graph Gas a SUBGRAPH (Skiena 1990, p. 143). Tura ´n studied ex- tremal graphs that do not contain a COMPLETE GRAPH Kpas a SUBGRAPH . One much-studied type of extremal graph is a two-coloring of a COMPLETE GRAPH Knofnnodes which contains exactly the number N/C13(R/C27B)minofMONO- CHROMATIC FORCED TRIANGLES and no more (i.e., a minimum of R/C27Bwhere RandBare the numbers of red and blue TRIANGLES ). Goodman (1959) showed that for an extremal graph of this type, N(n)/C301 3m(m/C281)(m/C282) for n/C302m 132m(m/C281)(4m/C271) for n/C304m/C271 132m(m/C271)(4m/C281) for n/C304m/C273:8 >< >: This is sometimes known as G OODMAN’S FORMULA . Schwenk (1972) rewrote it in the form N(n)/C30n 3;j1z;j1} /C281 2n14(n/C281)2jkjk ; sometimes known as S CHWENK’S FORMULA , where xbc is the FLOOR FUNCTION . The first few values of N(n) for n /C301, 2, ... are 0, 0, 0, 0, 0, 2, 4, 8, 12, 20, 28, 40, 52, 70, 88, ... (Sloane’s A014557). See also BICHROMATIC GRAPH ,BLUE-EMPTY GRAPH , EXTREMAL GRAPH THEORY ,G OODMAN’S FORMULA , MONOCHROMATIC FORCED TRIANGLE ,S CHWENK’S FORMULA ,TURA´ N GRAPH References Goodman, A. W. "On Sets of Acquaintances and Strangers at Any Party." Amer. Math. Monthly 66, 778 /C1/83, 1959. Schwenk, A. J. "Acquaintance Party Problem." Amer. Math. Monthly 79, 1113 /C1/117, 1972. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 143, 1990. Sloane, N. J. A. Sequences A014557 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Extremal Graph Theory The study of how the intrinsic structure of graphs ensures certain types of properties (e.g., CLIQUE - formation and GRAPH COLORINGS ) under appropriate conditions. See also ERDOS- STONE THEOREM ,EXTREMAL GRAPH , RAMSEY THEORY ,S TRUCTURAL RAMSEY THEORY , SZEMERE ´ DI’S REGULARITY LEMMA ,T URA´ N GRAPH , TURA´ N’S THEOREM References Bolloba ´s, B. Extremal Graph Theory. New York: Academic Press, 1978. Bolloba ´s, B. Extremal Graph Theory with Emphasis on Probabilistic Methods. Providence, RI: Amer. Math. Soc., 1986. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 143, 1990. Extremals A field of extremals is a plane region which is SIMPLY CONNECTED by a one-parameter family of extremals. The concept was invented by Weierstrass. Extreme and Mean Ratio GOLDEN MEAN Extreme Value Distribution N.B. A detailed online essay by S. Finch was the starting point for this entry. Let Mndenote the "extreme" (i.e., largest) ORDER STATISTIC Xnhifor a distribution of nelements Xi taken from a continuous UNIFORM DISTRIBUTION . Then the distribution of the Mnis PðMnBxÞ¼0 xn 1ifxB0 if 05x51 ifx/C218 < :ð1Þ and the MEAN and VARIANCE arem/C30n n/C271(2) s2/C30n (n/C271)2(n/C272): (3) IfXiare taken from a STANDARD NORMAL DISTRIBU- TION , then its cumulative distribution is F(x)/C301ffiffiffiffiffiffi 2xpgx /C28/C12e/C28t2=2dt/C301 2/C27F(x); (4) where F(x) is the NORMAL DISTRIBUTION FUNCTION . The probability distribution of Mnis then P(MnBx)/C30[F(x)]n/C30nffiffiffiffiffiffi 2npgx /C28/C12[F(t)]n/C281e/C28t2=2dt:(5) The MEAN m(n) and VARIANCE s2(n) are expressible in closed form for small n, m(1)/C300 (6) m(2)/C301ffiffiffipp (7) m(3)/C303 2ffiffiffipp (8) m(4)/C303 2ffiffiffipp 1/C272 psin/C2811 3;j1ffl;j1{"# (9) m(5)/C305 4ffiffiffipp 1/C276 psin/C2811 3;j1ffl;j1{"# (10) and s2(1)/C301 (11) s2(2)/C301/C281 p(12) s2(3)/C304p/C289/C272ffiffiffi 3p 4p(13) s2(4)/C301/C27ffiffiffi3p p/C28[m(4)]2(14) s2(5)/C301/C275ffiffiffi3p 4p/C275ffiffiffi3p 2p2sin/C2811 4;j1ffl;j1{ /C28[m(5)]2: (15) No exact expression is known for m(6) or s2(6);but there is an equation connecting them [m(6)]2/C27s2(6)/C301/C275ffiffiffi 3p 4p/C2715ffiffiffi3p 2p2sin/C2811 4;j1ffl;j1{ : (16) An analog to the CENTRAL LIMIT THEOREM states that the asymptotic normalized distribution of Mnsatisfies one of the three distributions P(y) /C30exp(/C28e/C28y) (17) P(y) /C300i f y 50 exp[/C28(/C28y /C28a)] if y > 0;j2ffl (18) P(y) /C30exp[/C28(/C28y)a]i fy50 1i f y > 0;;j2ffl (19) also known as GUMBEL , Fre´chet, and WEIBULL DIS- TRIBUTIONS , respectively. See also FISHER- TIPPETT DISTRIBUTION ,ORDER STA- TISTIC References Balakrishnan, N. and Cohen, A. C. Order Statistics and Inference. New York: Academic Press, 1991. David, H. A. Order Statistics, 2nd ed. New York: Wiley, 1981. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/extval/extval.html. Gibbons, J. D. and Chakraborti, S. Nonparametric Statisti- cal Inference, 3rd rev. ext. ed. New York: Dekker, 1992. Extreme Value Theorem If a function f(x) is continuous on a closed interval [a, b], then f(x) has both a MAXIMUM and a MINIMUM on [a, b]. If f(x) has an extreme value on an open interval (a, b), then the extreme value occurs at a CRITICAL POINT . This theorem is sometimes also called the WEIERSTRASS EXTREME VALUE THEOREM . Extremum A MAXIMUM or MINIMUM . An extremum may be LOCAL (a.k.a. a RELATIVE EXTREMUM ; an extremum in a given region which is not the overall MAXIMUM or MINIMUM )or GLOBAL . Functions with many extrema can be very difficult to GRAPH . Notorious examples include the functions cos(1 =x) and sin(1 =x) near x /C300 and sin(e2x /C279) near 0 and 1. The latter hase11 p/C281 2$% /C28e9 p/C2812$% /C271 /C3019085 /C282579 /C271 /C3016480 extrema in the CLOSED INTERVAL [0,1] (Mulcahy 1996). See also GLOBAL EXTREMUM ,G LOBAL MAXIMUM , GLOBAL MINIMUM ,K UHN- TUCKER THEOREM ,L A- GRANGE MULTIPLIER ,LOCAL EXTREMUM ,LOCAL MAX- IMUM ,LOCAL MINIMUM ,MAXIMUM ,MINIMUM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Mulcahy, C. "Plotting and Scheming with Wavelets." Math. Mag. 69, 323 /C1/43, 1996. Tikhomirov, V. M. Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., 1991. Extremum Test Consider a function f(x) in 1-D. If f(x) has a relative extremum at (x0) ; then either f ?(x0) /C300or f is not DIFFERENTIABLE at (x0) : Either the first or second DERIVATIVE tests may be used to locate relative extrema of the first kind. A NECESSARY condition for f(x) to have a MINIMUM (MAXIMUM )at( x0)is f ?(x0) /C300; and f ƒ(x0) ]0(f ƒ(x0) 50): A SUFFICIENT condition is f ?(x0) /C300 and f ƒ(x0) > 0/ (/f ƒ(x0) B0): Let f ?(x0) /C300; f ƒ(x0) /C300 ; ..., f(n)(x0) /C300; but f(n/C271)(x0) "0: Then f(x) has a RELATIVE MAXIMUM at (x0)ifn is ODD and f(n/C271)(x0) > 0; and f(x) has a RELATIVE MINIMUM at (x0)ifn is ODD and f(n/C271)(x0) > 0: There is a SADDLE POINT at (x0)ifn is EVEN . See also EXTREMUM ,FIRST DERIVATIVE TEST,RELA- TIVE MAXIMUM ,RELATIVE MINIMUM ,SADDLE POINT (FUNCTION ), SECOND DERIVATIVE TEST Extrinsic Curvature A curvature of a SUBMANIFOLD of a MANIFOLD which depends on its particular EMBEDDING . Examples of extrinsic curvature include the CURVATURE and TOR- SION of curves in 3-space, or the mean curvature of surfaces in 3-space. See also CURVATURE ,INTRINSIC CURVATURE ,M EAN CURVATURE Eyeball Theorem Given two circles, draw the tangents from the center of each circle to the sides of the other. Then the line segments AB and CD are of equal length. See also CIRCLE References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 70, 1991. F Faa´ di Bruno’s Formula If f(t) and g(t) are functions for which all necessary derivatives are defined, then Dnf(g(t)) /C30 X n! k1! /C1/C1/C1kn!Dkf/C0/C1 (g(t))Dg(t) 1! !k1 ...Dng(t) n! !kn ; where k /C30k1 /C27.../C27knand the sum of over all k1 ; ..., kn for which k1 /C272k2 /C27.../C27nkn /C30n (Roman 1980). See also LEIBNIZ IDENTITY ,UMBRAL CALCULUS References Bertrand, J. Cours de calcul diffe´rentiel er inte´gral, tome I. Paris: Gauthier-Villars, p. 138, 1864. Cesa`ro. "De´rive´es des fonctions de fonctions." Nouvelles Ann. 4,41/C1/5, 1885. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 137 /C1/39, 1974. Dederick. "Successive Derivatives of a Function of Several Functions." Ann. Math. 27, 385 /C1/94, 1926. Faa´ di Bruno. "Sullo sviluppo delle funzione." Ann. di Scienze Matem. et Fisiche di Tortoloni 6, 479 /C1/80, 1855. Faa´ di Bruno. "Note sur un nouvelle formule de calcul diffe´rentiel." Quart. J. Math. 1, 359 /C1/60, 1857. Franc ¸ais. "Du calcul des de´rivations ramere ´ a` ses ve´ritables principes...." Ann. Gergonne 6,61/C1/11, 1815. Joni, S. A. and Rota, C.-G. "The Faa´ di Bruno Bialgebra." §IX in "Coalgebras and Bialgebras in Combinatorics." Umbral Calculus and Hopf Algebras. Contemp. Math. 6,18/C1/1, 1982. Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, p. 33, 1965. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, p. 50, 1997. Marchand. "Sur le changement de variables." Ann. E´ cole Normale Sup. 3, 137 /C1/88 and 343 /C1/88, 1886. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, pp. 35 /C1/7, 1958. Roman, S. "The Formula of Faa di Bruno." Amer. Math. Monthly 87, 805 /C1/09, 1980. Teixeira. "Sur les de´rive´es d’ordre quelconque." Giornale di Matem. di Battaglini 18, 306 /C1/16, 1880. Wall. "On the n-th Derivative of f(x) :/" Bull. Amer. Math. Soc. 44, 395 /C1/98, 1938. Faber Polynomial Let f(x) /C30z /C27a1 /C27a2z/C281 /C27a3z /C282 /C27.../C30zX/C12 n/C300anz/C28n /C13zg(1=z) (1) be a LAURENT POLYNOMIAL with a0 /C301 : Then theFaber polynomial Pm(f)inf(z) of degree m is defined such that Pm(f) /C30zm /C27cm1z /C281 /C27cm2z/C282 /C27.../C30zm /C27Gm(1=z) ; (2) where Gm(x) /C30X/C12 n/C301cmnxn (3) (Schur 1945). Writing [g(x)]m /C30X/C12 k /C300amkxl (4) for m /C30 1, 2, ... gives the relationship am;m/C27n /C30cmn /C27am1cm/C281 ;n /C27am2cm/C282 ;n /C27.../C27am;m/C281c1n : (5) connecting amn and cmn :/ This polynomial can be used to calculate the number of LATTICE PATHS from a point (r ;0) to a point (a, b) that remain below the line y /C30 cx. See also LATTICE PATH References Gessel, I. M. Ree, S. "Lattice Paths and Faber Polynomials." In Advances in Combinatorial Methods and Applications to Probability and Statistics (Ed. N. Balakrishnan). Bos- ton, MA: Birkha ¨user, 1997. Pommerenke, C. "U¨ ber die Faberschen Polynome schlichter Funktionen." Math. Z. 85, 197 /C1/08, 1964. Schiffer, M. "Faber Polynomials in the Theory of Univalent Functions." Bull. Amer. Math. Soc. 54, 503 /C1/17, 1948. Schur, I. "On Faber Polynomials." Amer. J. Math. 67,33/C1/1, 1945. Fabry Imbedding A representation of a PLANAR GRAPH as a planar straight line graph such that no two EDGES cross. See also PLANAR GRAPH Face The intersection of an n-DPOLYTOPE with a tangent HYPERPLANE . 0-D faces are known as VERTICES (nodes), 1-D faces as EDGES ,(n /C282)/-D faces as RIDGES , and (n /C281)/-D faces as FACETS . See also EDGE (POLYHEDRON ), FACET ,P OLYTOPE , RIDGE ,VERTEX (POLYHEDRON ) Face-Regular Polyhedron JOHNSON SOLID Facet An (n /C281)/-D FACE of an n-D POLYTOPE . A procedure for generating facets is known as FACETING . Faceting Using a set of corners of a SOLID that lie in a plane to form the VERTICES of a new POLYGON is called faceting. Such POLYGONS may outline new FACES that join to enclose a new SOLID , even if the sides of the POLYGONS do not fall along EDGES of the original SOLID . References Holden, A. Shapes, Space, and Symmetry. New York: Columbia University Press, p. 94, 1971. Factor A factor is a portion of a quantity, usually an INTEGER or POLYNOMIAL that, when MULTIPLIED by all other factors, give the entire quantity. The determination of factors is called FACTORIZATION (or sometimes "FAC- TORING "). It is usually desired to break factors down into the smallest possible pieces so that no factor is itself factorable. For INTEGERS , the determination of factors is called PRIME FACTORIZATION . For large quantities, the determination of all factors is usually very difficult except in exceptional circumstances. See also DIVISOR ,FACTORIZATION ,GREATEST PRIME FACTOR ,L EAST PRIME FACTOR ,M ULTIPLICATION , POLYNOMIAL FACTORIZATION ,PRIME FACTORIZATION , PRIME FACTORIZATION ALGORITHMS Factor (Graph) A 1-factor of a GRAPH G with n VERTICES is a set of n=2 separate EDGES which collectively contain all n of the VERTICES of G among their endpoints. See also GRAPH Factor Base The primes with LEGENDRE SYMBOL (n=p) /C301 (less than N /C30p(d) for trial divisor d) which need be considered when using the QUADRATIC SIEVE factor- ization method.See also DIXON’S FACTORIZATION METHOD References Morrison, M. A. and Brillhart, J. "A Method of Factoring and the Factorization of F7:/"Math. Comput. 29, 183/C1/05, 1975. Factor Group QUOTIENT GROUP Factor Level A grouping of statistics. Factor Ring QUOTIENT RING Factor Space QUOTIENT SPACE Factorial The factorial n! is defined for a POSITIVE INTEGER nas n!/C13n/C215(n/C281)/C1/C1/C12/C2151n/C301;2;... 1 n/C300:/C27 (1) The factorial n! gives the number of ways in which n objects can be permuted. For example, 3! /C306;since the six possible permutations of f1;2;3gare f1;2;3g; f1;3;2g;f2;1;3g;f2;3;1g;f3;1;2g;f3;2;1g:Since there is a single permutation of zero elements (the EMPTY SET ¥);0!/C301:The first few factorials for /C28n/C300, 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (Sloane’s A000142). An older NOTATION for the factorial is n (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996). Asngrows large, factorials begin acquiring tails of trailing ZEROS . To calculate the number Zof trailing ZEROS forn!;use Z/C30Xkmax k/C301n 5k$% ; (2) where kmax/C13lnn ln5$% (3) and xbcis the FLOOR FUNCTION (Gardner 1978, p. 63; Ogilvy and Anderson 1988, pp. 112 /C1/14). For n/C301, 2, ..., the number of trailing zeros are 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, ... (Sloane’s A027868). This is aspecial application of the general result that the POWER of a PRIME pdividing n!i s /C23p(n)/C30X k]0n pk$% (4) (Landau 1974, pp. 75 /C1/6; Hardy and Wright 1979, pp. 342; Ingham 1990, p. 20; Graham et al. 1994; Vardi 1991; Hardy 1999, pp. 18 and 21). Stated another way, the exact POWER of a PRIME pwhich divides n!i s n/C28sum of digits of the base /C28prepresentation of n p/C281 (5) Leta(n) be the last nonzero digit in n!;then the first few values are 2, 6, 4, 2, 2, 4, 2, 8, 8, 8, 6, 8, ...(Sloane’s A008904). This sequence was studied by Kakutani (1967), who showed that this sequence is"5-automatic," meaning roughly that there exists afinite automaton which, when given the digits of nin base-5, will wind up in a state for which an outputmapping specifies a(n):The exact distribution of digits follows from this result. By noting that n!/C13G(n/C271); (6) where G(n) is the GAMMA FUNCTION for INTEGERS n, the definition can be generalized to COMPLEX values z!/C13G(z/C271)/C13g/C12 0e/C28ttzdt: (7) This defines z! for all COMPLEX values of z, except when zis a NEGATIVE INTEGER , in which case z!/C30/C12: Using the identities for GAMMA FUNCTIONS , the values of (1 2n)! (half integral values) can be written explicitly /C281 2 ! !/C30ffiffiffipp(8) 1 2 ! !/C3012ffiffiffipp(9) n/C281 2 ! !/C30ffiffiffipp 2n(2n/C281)!! (10) n/C271 2 ! !/C30ffiffiffipp 2n/C271(2n/C281)!!; (11) where n!! is a DOUBLE FACTORIAL . For INTEGERS sandnwith sBn, (s/C28n)! 2s/C282n ðÞ !/C30(/C281)n/C28s(2n/C282s)! (n/C28s)!: (12) The LOGARITHM ofz! is frequently encounteredln(z!)/C301 2lnpz sin(pz)"# /C28g/C28X/C12 n/C301z(2n/C271) 2n/C271z2n/C271(13) /C3012lnpz sin(pz)"# /C2812ln1/C27z 1/C28z ! /C27(1/C28g)z /C28X/C12 n/C301z(2n/C271)/C281 ½/C138z2n/C271 2n/C271(14) /C30ln lim n0/C12n! (z/C271)(z/C272)/C1/C1/C1(z/C27n)nz"# (15) /C30lim n0/C12[ln(n!)/C27zlnn/C28ln(z/C271)/C28ln(z/C272)/C28... /C28ln(z/C27n)] (16) /C30X/C12 n/C301zn n!Fn/C281(0) (17) /C30/C28gz/C27X/C12 n/C302(/C281)nzn nz(n) (18) /C30/C28ln(1/C27z)/C27z(1/C28g)/C27X/C12 n/C302(/C281)n[z(n)/C281]zn n;(19) where gis the E ULER- MASCHERONI CONSTANT ,z(z)i s the R IEMANN ZETA FUNCTION , and Fn(z) is the POLY- GAMMA FUNCTION . The factorial can be expanded in a series z!/C30 ffiffiffiffiffiffi 2pp zz/C271=2e/C28z1/C271 2z/C281/C271 288z/C282/C28139 51840z/C283/C27... ! (20) (Sloane’s A001163 and A001164). S TIRLING’S SERIES gives the series expansion for ln( z!); ln(z!)/C301 2ln(2p)/C27z/C2712 ! lnz/C28z/C27B2 2z/C27... /C27B2n 2n(2n/C281)z2n/C281/C27... /C301 2ln(2p)/C27z/C2712 ! lnz/C28z/C271 12z/C281/C281 360z/C283 /C271 1260z/C285/C28. . . (21) (Sloane’s A046968 and A046969), where Bnis a BERNOULLI NUMBER . Lethbe the exponent of the greatest POWER of a PRIME pdividing n!:Then h /C30X i/C301 pi 5nn pi$% : (22) Let g be the number of 1s in the BINARY representa- tion of n. Then g /C27h /C30n (23) (Honsberger 1976). In general, as discovered by Legendre in 1808, the POWER m of the PRIME p dividing n! is given by m /C30X/C12 k /C300n pk$% /C30n /C28 (n0 /C27 n1 /C27 ... /C27 nN p /C28 1; (24) where the INTEGERS n1 ; ..., nNare the digits of n in base p (Ribenboim 1989). The numbers n! /C271 are prime for n /C30 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, ... (Sloane’s A002981; Wells 1986, p. 70), and the numbers n! /C281 are prime for n /C30 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, ... (Sloane’s A002982). In general, the power-product sequences (Mudge 1997) are given by S9 k (n) /C30(n!)k 91: The first few terms of S/C27 2 (n) are 2, 5, 37, 577, 14401, 518401, ... (Sloane’s A020549), and S/C27 2 (n)is PRIME for n /C30 1, 2, 3, 4, 5, 9, 10, 11, 13, 24, 65, 76, ... (Sloane’s A046029). The first few terms of S /C28 2 (n) are 0, 3, 35, 575, 14399, 518399, ... (Sloane’s A046032), but S/C282 (n)is PRIME for only n /C30 2 since S /C282 (n) /C30(n!)2 /C281 /C30(n! /C271)(n! /C281) for n /C21 2. The first few terms of S /C28 3 (n) are 0, 7, 215, 13823, 1727999, ... (Sloane’s A046033), and the first few terms of S/C27 3 (n) are 2, 9, 217, 13825, 1728001, ... (Sloane’s A019514). The first few numbers n such that the sum of the factorials of their digits is equal to the PRIME COUNT- ING FUNCTION p(n) are 6500, 6501, 6510, 6511, 6521, 12066, 50372, ... (Sloane’s A049529). This sequence is finite, with the largest term being a23 /C3011 ;071;599:/ There are three numbers less than 200,000 for which (n/C281)!/C271/C130(mod n2); (25) namely 5, 13, and 563 (Le Lionnais 1983). B ROWN NUMBERS are pairs ( m, n )o f INTEGERS satisfying the condition of B ROCARD’S PROBLEM , i.e., such that n!/C271/C30m2; (26) Only three such numbers are known: (5, 4), (11, 5), (71, 7). Erdos conjectured that these are the only three such pairs (Guy 1994, p. 193). See also ALLADI- GRINSTEAD CONSTANT ,B ROCARD’S PROBLEM ,B ROWN NUMBERS ,C ENTRAL FACTORIAL , DOUBLE FACTORIAL ,FACTORIAL PRIME ,FACTORIAL PRODUCTS ,FACTORIAL SUMS,FACTORION ,FALLING FACTORIAL ,G AMMA FUNCTION ,H YPERFACTORIAL , MULTIFACTORIAL ,POCHHAMMER SYMBOL ,PRIMORIAL , RISING FACTORIAL ,R OMAN FACTORIAL ,S TIRLING’SSERIES ,S UBFACTORIAL ,S UPERFACTORIAL ,W ILSON PRIME References Caldwell, C. K. "The Top Twenty: Primorial and Factorial Primes." http://www.utm.edu/research/primes/lists/top20/ PrimorialFactorial.html. Conway, J. H. and Guy, R. K. "Factorial Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 65 /C1/6, 1996. Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 96, 1970. Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusionsand Other Mathematical Sleight-of-Mind from ScientificAmerican. New York: Vintage, pp. 50 /C1 /5, 1978. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Factorial Factors." §4.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 111--115, 1994. Guy, R. K. "Equal Products of Factorials," "Alternating Sums of Factorials," and "Equations Involving Factorialn."§B23, B43, and D25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 80, 100, and 193 /C1 /94, 1994. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 2, 1976. Ingham, A. E. The Distribution of Prime Numbers. Cam- bridge, England: Cambridge University Press, 1990. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 462 /C1/63, 1988. Kakutani, S. "Ergodic Theory of Shift Transformations." In Proc. 5th Berkeley Symposium on Mathematical Statisticsand Probability, Vol. 2. Berkeley, CA: University of California Press, pp. 405 /C1 /14, 1967. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, 1958. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/factorial-.Z and ftp://sable.ox.ac.uk/pub/math/factors/factorial /C27.Z. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 174, 1979. Mellin, H. "Abrißeiner einheitlichen Theorie der Gamma- und der hypergeometrischen Funktionen." Math. Ann. 68, 305/C1/37, 1909. Mudge, M. "Not Numerology but Numeralogy!" Personal Computer World, 279/C1/80, 1997. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, 1988. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, p. 86, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gamma Function, Beta Function, Factorials,Binomial Coefficients." §6.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206 /C1 /09, 1992. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 22 /C1/4, 1989. Sloane, N. J. A. Sequences A000142/M1675, A001163/ M5400, A001164/M4878, A002981/M0908, A002982/ M2321, A008904, A019514, A020549, A027868, A046029, A046032, A046033, A046968, A046969, and A049529 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Spanier, J. and Oldham, K. B. "The Factorial Function n! and Its Reciprocal." Ch. 2 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 19 /C1/3, 1987. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 67, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 70, 1986. Factorial Moment v(r) /C13X xx(r)f(x) ; where x(r) /C13x(x /C281) /C1/C1/C1(x /C28r /C271): See also MOMENT Factorial Number FACTORIAL Factorial Prime A PRIME OF THE FORM n! 91: n! /C271is PRIME for 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, ... (Sloane’s A002981). No others are known, but N. Kuosa is coordinating a search in the range 23; 000 Bn B30 ;000:/ /n! /C281is PRIME for 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, ... (Sloane’s A002982). See also FACTORIAL ,PRIME NUMBER ,PRIMORIAL References Borning, A. "Some Results for k! /C271 and 2 /C2153 /C2155 /C215p /C271:/" Math. Comput. 26, 567 /C1/70, 1972. Buhler, J. P.; Crandall, R. E.; and Penk, M. A. "Primes of the Form M! /C271 and 2 /C2153 /C2155 /C1/C1/C1p /C271:/" Math. Comput. 38, 639 /C1/43, 1982. Caldwell, C. K. "Prime Links/C27/C27: Resources in theory: special_forms: near_products: factorial." http://primes.utm.edu/links/theory/special_forms/near_ products/factorial/. Caldwell, C. K. "On the Primality of N! /C271 and 2 /C2153 /C2155 /C1/C1/C1p 91 :/" Math. Comput. 64, 889 /C1/90, 1995. Dubner, H. "Factorial and Primorial Primes." J. Rec. Math. 19, 197 /C1/03, 1987. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 7, 1994. Kuosa, N. "Search of [sic] the Next Prime of the Form n! /C271:/" http://www.hut.fi/~nkuosa/primeform/. Sloane, N. J. A. Sequences A002981/M0908 and A0029822321 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html.Temper, M. "On the Primality of k! /C271 and /C215/3 /C2155 /C1/C1/C1p /C271:/" Math. Comput. 34, 303 /C1/04, 1980. Factorial Products The only known factorials which are products of factorials in an ARITHMETIC SEQUENCE are 0!1! ¼ 1! 1!2! ¼ 2! 0!1!2! ¼ 2! 6!7! ¼ 10! 1!3!5! ¼ 6! 1!3!5!7! ¼ 10! (Madachy 1979). There are no identities OF THE FORM n! /C30a1!a2! /C1/C1/C1ar! (1) for r ]2 with ai ]aj ]2 for i B j for n 518160 except 9! ¼ 7!3!3!2! (2) 10! ¼ 7!6! ¼ 7!5!3! (3) 16!¼14!5!2! (4) (Guy 1994, p. 80). See also FACTORIAL ,FACTORIAL SUMS References Guy, R. K. "Equal Products of Factorials," "Alternating Sums of Factorials," and "Equations Involving Factorial n."§B23, B43, and D25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 80, 100, and 193 /C1/94, 1994. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 174, 1979. Factorial Sums The sum-of-factorials function is defined by X (n)/C13Xn k/C301k! /C30/C28e/C27ei(1)/C27pi/C27E2n/C271(/C281)]G(n/C272) e; (1) /C30/C28e/C27ei(1)/C27R[E2n/C271(/C281)]G(n/C272) e; (2) where ei(1) :1:89512 is the EXPONENTIAL INTEGRAL , Enis the EN-FUNCTION ,R[z] is the REAL PART ofz, and Iis the IMAGINARY NUMBER . The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, ... (Sloane’s A007489). a(n) cannot be written as a hypergeometric term plus a constant (Petkovsek et al.1996). However the sum X ?(n)/C13Xn k/C301kk!/C30(n/C271)!/C281 (3) has a simple form, with the first few values being 1, 5, 23, 119, 719, 5039, ... (Sloane’s A033312). There are only four INTEGERS equal to the sum of the factorials of their digits. Such numbers are called FACTORIONS . While no factorial greater than 1! is a SQUARE NUMBER , D. Hoey listed sums B1012of dis- tinct factorials which give SQUARE NUMBERS , and J. McCranie gave the one additional sum less than 21!/C305:1/C291019: 0!þ1!þ2!¼22 1!þ2!þ3!¼32 1!þ4!¼52 1!þ5!¼112 4!þ5!¼122 1!þ2!þ3!þ6!¼272 1!þ5!þ6!¼292 1!þ7!¼712 4!þ5!þ7!¼722 1!þ2!þ3!þ7!þ8!¼2132 1!þ4!þ5!þ6!þ7!þ8!¼2152 1!þ2!þ3!þ6!þ9!¼6032 1!þ4!þ8!þ9!¼6352 1!þ2!þ3!þ6!þ7!þ8!þ10!¼19172 1!/C272!/C273!/C277!/C278!/C279!/C2710!/C2711!/C2712!/C2713!/C2714!/C2715! /C3011838932 (Sloane’s A014597). The first few values of the alternating SUM a(n)/C13Xn i/C301(/C281)n/C28ii! (4) /C30(/C281)n/C281/C28eei(/C281)/C27(/C281)nEn/C272(1)G(n/C272)/C2/C6 ; (5) where ei( x) is the EXPONENTIAL INTEGRAL ,En(x) is the EN-FUNCTION , and G(x) is the GAMMA FUNCTION , are 1, 1, 5, 19, 101, 619, 4421, 35899, ... (Sloane’s A005165), and the first few values nfor which a(n) are prime are n/C303, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, ...(Sloane’s A001272, Guy 1994, p. 100). Zivkovic (1999)has shown that the number of such primes is finite. Sums with powers of an index in the NUMERATOR and products of FACTORIALS in the DENOMINATOR can often be done analytically. For example, for numera- tor 1, Xn i/C3011 k1/C27i/C0/C1 !k2/C27i/C0/C1 !/C301˜F2(1; 2/C27k1;2/C27k2;1 ) /C281˜F2(1;n/C27k1/C272;n/C27k2/C272; 1) (6) Xn i/C3011 k1/C28i ðÞ !k2/C27i ðÞ !/C302˜F1;1/C28k1;k2/C272;/C281 ðÞ Gk1ðÞ/C282˜F1;n/C28k1/C271;n/C27k2/C272;/C281 ðÞ Gk1/C28n ðÞ(7) wherep˜Fqis a REGULARIZED HYPERGEOMETRIC FUNC- TION . For numerator i, Xn i/C301i k1/C27i ðÞ !k2/C27i ðÞ ! /C30/C28(n/C271)1˜F21;n/C27k1/C272;n/C27k2/C272; 1 ðÞ /C271˜F22;k1/C272;k2/C272; 1 ðÞ /C281˜F2(2;n/C27k1/C273;n/C27k2/C273; 1) (8) Xn i/C301i k1/C28i ðÞ !k2/C27i ðÞ ! /C28(n/C271)2˜F11;n/C28k1/C271;n/C27k2/C272;/C281 ðÞ Gk1/C28n ðÞ /C272˜F12;1/C28k1;k2/C272;/C281 ðÞ Gk1ðÞ /C282˜F12;n/C28k1/C272;n/C27k2/C273;/C281 ðÞ Gk1/C28n/C281 ðÞ: (9) These sums simplify substantially for special values ofk1andk2:For example, with k1/C30k2/C30n; Xn i/C3011 (n/C28i)!(n/C27i)!/C3022n/C281 G(2n/C271)/C281 2[G(n)]2(10) Xn i/C301i (n/C28i)!(n/C27i)!/C301 2G(n)G(n/C271)(11) Xn i/C301i2 (n/C28i)!(n/C27i)! /C301 2G(n)G(n/C271)/C2722˜F1(3;2/C28n;n/C273;/C281) G(n/C281):(12) With k1/C30nandk2/C30n/C281; Xn i/C3011 (n/C28i)!(n/C281/C27i)!/C304n/C281 G(2n)(13) Xn i/C301i (n/C28i)!(n/C281/C27i)!/C301 2[G(n)]2/C2722n/C283 G(2n): (14) With k1/C30nandk2/C30n/C271; Xn i/C3011 (n/C28i)!(n/C271/C27i)! /C304n G(2n/C272)/C281 G(n/C271)G(n/C272)(15) Xn i/C3011 (n /C28 i)!(n /C27 1 /C27 i)! /C30G(n) /C27G(n /C27 1) 2G(n) G(n /C27 1)G(n /C27 2) /C2822n /C281 G(2n /C27 2)(16) Sums of factorial POWERS include X/C12 n /C300(n!)2 (2n)! /C302 2718 /C27ffiffiffi 3p p/C17/C15 (17) X/C12 n/C300(n!)3 (3n)! /C303F2 1; 1;1;1 3 ;23 ;1 27 ! (18) /C30g1 0P(t) /C27Q(t)cos/C281R(t)/C2/C6 dt; (19) where P(t) /C3028/C27 7t2 /C28 7t3ðÞ 4 /C28 t2 /C27 t3 ðÞ2 (20) Q(t) /C304t(1 /C28 t)5/C27 t2 /C28 t3ðÞ 4 /C28 t2 /C27 t3 ðÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1 /C28 t)4/C28 t2 /C27 t3 ðÞp (21) R(t) /C301 /C281 2t2 /C28t3/C0/C1 (22) (Schroeppel and Gosper 1972). In general, X/C12 n/C300(n!)k (kn)! /C30k Fk/C281 1;...;1;|fflfflfflfflffl{zfflfflfflfflffl} k1 k ;2k ;...;k /C28 1 k;1 kk0 @1A: (23) Identities satisfied by sums of factorials include X /C12 k /C3001 k! /C30e /C302 :718281828... (24) X/C12 k /C300( /C281)k k!/C30e/C281 /C300 :3678794411... (25) X/C12 k/C3001 ðk!Þ2 /C30I0 ð2 Þ/C302 :279585302... (26) X/C12 k /C300( /C281)k (k!)2 /C30J0(2) /C300:2238907791... (27) X/C12 k /C3001 (2k)! /C30cosh 1 /C301:543080634... (28) X/C12 k /C300( /C281)k (2k)!/C30cos 1 /C300 :5403023058... (29) X/C12 k /C3001 (2k /C27 1)! /C30sinh 1 /C301:175201193... (30)X/C12 k /C300( /C281)k (2k /C27 1)! /C30sin 1 /C300 :8414709848... (31) (Spanier and Oldham 1987), where I0(x)isa MODIFIED BESSEL FUNCTION OF THE FIRST KIND , J0(x)isa BESSEL FUNCTION OF THE FIRST KIND , cosh x is the HYPERBOLIC COSINE , cos x is the COSINE , sinh x is the HYPERBOLIC SINE, and sin x is the SINE. See also BINOMIAL SUMS,F ACTORIAL ,F ACTORIAL PRODUCTS References Guy, R. K. "Equal Products of Factorials," "Alternating Sums of Factorials," and "Equations Involving Factorial n." §B23, B43, and D25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 80, 100, and 193 /C1/94, 1994. Schroeppel, R. and Gosper, R. W. Item 116 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM- 239, p. 54, Feb. 1972. Sloane, N. J. A. Sequences A001272, A005165/M3892, A007489/M2818, A014597, and A033312 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Factorial Function n! and Its Reciprocal." Ch. 2 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 19 /C1/3, 1987. Zivkovic, M. "The Number of Primes an i /C301(/C281)n/C28ii! is Finite." Math. Comput. 68, 403 /C1/09, 1999. Factorial2 DOUBLE FACTORIAL Factoring FACTORIZATION Factorion A factorion is an INTEGER which is equal to the sum of FACTORIALS of its digits. There are exactly four such numbers: 1 /C301! (1) 2/C302! (2) 145/C301!/C274!/C275! (3) 40;585/C304!/C270!/C275!/C278!/C275! (4) (Sloane’s A014080; Gardner 1978, Madachy 1979, Pickover 1995). Obviously, the factorion of an n-digit number cannot exceed n/C2159!:/ See also FACTORIAL ,FACTORIAL SUMS References Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 61 and 64, 1978. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 167, 1979. Pickover, C. A. "The Loneliness of the Factorions." Ch. 22 in Keys to Infinity. New York: W. H. Freeman, pp. 169 /C1/71 and 319 /C1/20, 1995. Sloane, N. J. A. Sequences A014080 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Factorization The determination of FACTORS (DIVISORS ) of a given INTEGER ("PRIME FACTORIZATION "), POLYNOMIAL ("POLYNOMIAL FACTORIZATION "), etc. In many cases of interest (particularly PRIME FACTORIZATION , factor- ization is unique, and so gives the "simplest" repre- sentation of a given quantity in terms of smaller parts. The terms "factorization" and "factoring" are used synonymously. See also FACTOR ,POLYNOMIAL FACTORIZATION ,PRIME FACTORIZATION ,PRIME FACTORIZATION ALGORITHMS Fagnano’s Point The point of coincidence of P and p? in FAGNANO’S THEOREM . See also FAGNANO’S THEOREM Fagnano’s Problem In a given ACUTE TRIANGLE DABC ; find the INSCRIBED TRIANGLE whose PERIMETER is as small as possible. The answer is the ORTHIC TRIANGLE of DABC : The problem was proposed and solved using calculus by Fagnano in 1775 (Coxeter and Greitzer 1967, p. 88). See also ACUTE TRIANGLE ,ORTHIC TRIANGLE ,PERI- METER References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 21, 1969. Coxeter, H. S. M. and Greitzer, S. L. "Fagnano’s Problem." §4.5 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 88 /C1/9, 1967.Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 347, 1996. Kazarinoff, N. D. Geometric Inequalities. New York: Ran- dom House, pp. 76 /C1/7, 1961. Morley, F. and Morley, F. V. Inversive Geometry. Boston, MA: Ginn, p. 37, 1933. Fagnano’s Theorem If P(x;y) and P(x?;y ?) are two points on an ELLIPSE x2 a2 /C27y2 b2 /C301; (1) with ECCENTRIC ANGLES f and f? such that tan f tan f?/C30b a (2) and A /C30P(a;0) and B /C30P(0;b) : Then arcBP /C27arcBP ?/C30e2xx ? a/C215 (3) This follows from the identity E(u; k) /C27E(v; k) /C28E(k) /C30k2 sn(u;k) sn(v;k); (4) where E(u;k) is an incomplete ELLIPTIC INTEGRAL OF THE SECOND KIND , E(k) is a complete ELLIPTIC INTEGRAL OF THE SECOND KIND , and sn(v; k)isa JACOBI ELLIPTIC FUNCTION .IfP and p? coincide, the point where they coincide is called FAGNANO’S POINT . See also ELLIPSE ,FAGNANO’S POINT Fair Dice DICE,ICOSAHEDRON Fair Division CAKE CUTTING Fair Game A GAME which is not biased toward any player. See also FUTILE GAME,GAME,MARTINGALE Fairy Chess A variation of CHESS involving a change in the form of the board, the rules of play, or the pieces used. For example, the normal rules of chess can be used but with a cylindrical or MO¨ BIUS STRIP connection of the edges. See also CHESS References Kraitchik, M. "Fairy Chess." §12.2 in Mathematical Recrea- tions. New York: W. W. Norton, pp. 276 /C1/79, 1942. Faithful Group Action A GROUP ACTION f : G /C29X 0 X is called faithful if there are no group elements g such that gx /C30 x for all x /C23 X : Equivalently, the map f induces an INJEC- TION of G into the SYMMETRIC GROUP Sx : So G can be identified with a PERMUTATION SUBGROUP . Most actions that arise naturally are faithful. An example of an action which is not faithful is the action ei(x /C27y)of G /C30R2 /C30f(x;y) g on X /C30S1 /C30 eiufg ; i.e., f x;y;eiuðÞ /C30ei(u /C27x/C27y) :/ See also ADO’S THEOREM ,EFFECTIVE ACTION ,FREE ACTION ,G ROUP ,IWASAWA’S THEOREM ,O RBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), TRANSITIVE References Huang, J.-S. "Faithful Irreducible Representations." §9.3 in Lectures on Representation Theory. Singapore: World Scientific, pp. 124 /C1/28, 1999. Rotman, J. Theory of Groups. New York: Allyn and Bacon, p. 180, 1984. Falkner-Skan Differential Equation The third-order ORDINARY DIFFERENTIAL EQUATION y§/C27 ayyƒ/C27 b 1 /C28y?2/C0/C1 /C300: References Cebeci, T. and Keller, H. B. "Shooting and Parallel Shooting Methods for Solving Falkner-Shan Boundary Layer Equa- tion." J. Comput. Phys. 71, 289 /C1/00, 1971. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997. Fallacy A fallacy is an incorrect result arrived at by appar- ently correct, though actually specious reasoning. The great Greek geometer Euclid wrote an entire book on geometric fallacies which, unfortunately, has not survived (Gardner 1984, p. ix). The most common example of a mathematical fallacy is the "proof" that 1 /C30 2 as follows. Let a /C30 b, then ab /C30a2 (1) ab /C28b2 /C30a2 /C28b2 (2) b(a /C28b) /C30(a /C27b)(a /C28b) (3) b /C30a /C27b (4) b /C302b (5) 1 /C302: (6) The incorrect step is (4), in which DIVISION BY ZERO/(a /C28b /C300) is performed, which is not an allowed algebraic operation. Similarly flawed reasoning can be used to show that 0 /C30 1, or any number equals any other number. Ball and Coxeter (1987) give other such examples in the areas of both arithmetic and geometry. See also DIVISION BY ZERO References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 41 /C1/5 and 76/C1/4, 1987. Barbeau, E. J. Mathematical Fallacies, Flaws, and Flim- flam. Washington, DC: Math. Assoc. Amer., 1999. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984. Pappas, T. "Geometric Fallacy & the Fibonacci Sequence." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 191, 1989. Falling Factorial Forn]0;the falling factorial is defined by (x)n/C30x(x/C281)/C1/C1/C1(x/C28n/C271); (1) and is related to the RISING FACTORIAL x(n)(a.k.a. POCHHAMMER SYMBOL )b y (x)n/C30(/C281)n(/C28x)(n): (2) The falling factorial can be implemented in Mathe- matica as FallingFactorial[x_, n_] : /C30(-1)Pochhammer[- x, n] The falling factorial is also called a binomial poly- nomial or lower factorial. Unfortunately, there are two notations used for the falling and rising factorials, ( x)nand x(n);which are unfortunately polar opposites of one another. Incombinatorial usage, the falling factorial is denoted(x) nand the RISING FACTORIAL is denoted ( x)(n) (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999,p. 101), whereas in the calculus of FINITE DIFFER- ENCES and the theory of special functions, the falling factorial is denoted x(n)and the RISING FACTORIAL is denoted ( x)n(Roman 1984, p. 5; Abramowitz and Stegun 1972, p. 256; Spanier 1987). Extreme cautionis therefore needed in interpreting the meanings of the notations ( x) nandx(n):In this work, the notation (x)nis used for the falling factorial , potentially causing confusion with the P OCHHAMMER SYMBOL (another name for the RISING FACTORIAL , which is universally denoted ( x)n):/ The first few falling factorials are (x)0/C301 (x)1 /C30x (x)2 /C30x(x /C281) /C30x2 /C28x (x)3 /C30x(x /C281)(x /C282) /C30x3 /C283x2 /C272x (x)4 /C30x(/C281)(x /C282)(x /C283) /C30x4 /C286x3 /C2711x2 /C286x: A sum formula connecting the falling factorial (x)n and rising factorial x(n) ; (x)n /C30Xn k /C300cnkx(k) ; (3) is given using the Sheffer formalism with g(t) /C301 (4) f(t) /C30et /C281 (5) h(t) /C301 (6) l(t) /C301 /C28e /C28t ; (7) which gives the GENERATING FUNCTION X/C12 n /C300tn(x) n!tn /C30X/C12 n/C3001 n!Xn k /C300cnkxktk /C30etx=(1/C27t) ; (8) /C301 /C27xt /C271 2x2 /C282x/C0/C1 t2 /C2716x 3 /C286x2 /C276x/C0/C1 t3 /C271 24x4 /C2812x3 /C2736x2 /C2824x/C0/C1 t4 /C27...; (9) where tn(x) /C30Xn k /C300cnkxk : (10) Reading the coefficients off gives c00 /C301 c11 /C301 c10 /C300 c22 /C301 c21 /C30/C282 c20 /C300 c33 /C301 c32 /C30/C286 c31 /C306 c30 /C300; so, (x)0 /C30x(0) (11) (x)1 /C30x(1) (12) (x)2 /C30x(2) /C282x(1) (13) (x)3 /C30x(3) /C286x(2) /C276x(1) ; (14) etc. (and the formula given by Roman 1984, p. 133, is incorrect). The falling factorial is an associated SHEFFER SE- QUENCE withf(t) ¼ et /C281 (15) (Roman 1984, p. 29), and has GENERATING FUNCTION X/C12 k /C300(x)k k!tk /C30ex ln(1/C27t) /C30(1 /C27t)x ; (16) which is equivalent to the BINOMIAL THEOREM X/C12 k /C300x k/C1Y/C1Q tk /C30(1 /C27t)x /C215 (17) The binomial identity of the SHEFFER SEQUENCE is (x /C27y)n /C30Xn k /C300n k/C1Y/C1Q (x)k(y)n/C28k ; (18) wheren k/C0/C1 is a BINOMIAL COEFFICIENT , which can be rewritten as x /C27y n/C1Y/C1Q /C30X/C12 k/C300x k/C1Y/C1Q y n /C28k/C1Y/C1Q ; (19) known as the CHU-VANDERMONDE IDENTITY . The falling factorials obey the RECURRENCE RELATION x(x)n /C30(x)n/C271 /C27n(x)n (20) (Roman 1984, p. 61). See also BINOMIAL THEOREM ,CENTRAL FACTORIAL , CHU-VANDERMONDE IDENTITY ,R ISING FACTORIAL , SHEFFER SEQUENCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 101, 1999. Roman, S. "The Lower Factorial Polynomial." §1.2 in The Umbral Calculus. New York: Academic Press, pp. 5, 28 /C1/ 9, and 56 /C1/3, 1984. Spanier, J. and Oldham, K. B. "The Pochhammer Polyno- mials (x)n :/" Ch. 18 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 149 /C1/65, 1987. False A statement which is rigorously not TRUE . Regular two-valued LOGIC allows statements to be only TRUE or false, but FUZZY LOGIC treats "truth" as a con- tinuum which can have a value between 0 and 1. The symbol ]is sometimes used to denote "false," although "F" is more commonly used in TRUTH TABLES . See also ALETHIC ,BOOLEANS ,FUZZY LOGIC ,LOGIC , TRUE,TRUTH TABLE ,UNDECIDABLE False Position Method METHOD OF FALSE POSITION False Spiral References Fraser, J. Brit. J. Psychol. Jan. 1908. Pappas, T. "The False Spiral Optical Illusion." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 114, 1989. Faltung (Form) Let A and B be bilinear forms A /C30A(x;y) /C30XX aijxiyi B /C30B(x; y) /C30XX bijxiyi and suppose that A and B are bounded in [p ;p ?] with bounds M and N. Then F /C30F(A;B) /C30XX fijxiyj ; where the series fij /C30X kaikbkj is absolutely convergent, is called the faltung of A and B. F is bounded in [p ;p ?]; and its bound does not exceed MN. References Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 210 /C1/11, 1988. Faltung (Function) CONVOLUTION Family Number HOME PRIME Fan A SPREAD in which each node has a FINITE number of children. See also SPREAD (TREE) Fano Configuration FANO PLANEFano Plane The 2-D finite PROJECTIVE PLANE over GF(2) ("of order two"), illustrated above. It is a BLOCK DESIGN with n /C307 ; k /C303, l /C301 ; r /C303, and b /C307, the STEINER TRIPLE SYSTEM S(7) ; and the unique 73CONFIGURA- TION . The Fano plane also solves the TRANSYLVANIA LOT- TERY , which picks three numbers from the INTEGERS 1 /C1/4. Using two Fano planes we can guarantee matching two by playing just 14 times as follows. Label the VERTICES of one Fano plane by the INTE- GERS 1 /C1/, the other plane by the INTEGERS 8 /C1/4. The 14 tickets to play are the 14 lines of the two planes. Then if (a;b ;c) is the winning ticket, at least two of a ;b;c are either in the interval [1, 7] or [8, 14]. These two numbers are on exactly one line of the corresponding plane, so one of our tickets matches them. The Lehmers (1974) found an application of the Fano plane for factoring INTEGERS via QUADRATIC FORMS . Here, the triples of forms used form the lines of the PROJECTIVE GEOMETRY on seven points, whose planes are Fano configurations corresponding to pairs of residue classes mod 24 (Lehmer and Lehmer 1974, Guy 1975, Shanks 1985). The group of AUTOMORPH- ISMS (incidence-preserving BIJECTIONS ) of the Fano plane is the SIMPLE GROUP of ORDER 168 (Klein 1870). See also CONFIGURATION ,D ESIGN ,P ROJECTIVE PLANE ,STEINER TRIPLE SYSTEM ,TRANSYLVANIA LOT- TERY References Guy, R. "How to Factor a Number." Proc. Fifth Manitoba Conf. on Numerical Math. ,49/C1/9, 1975. Lehmer, D. H. and Lehmer, E. "A New Factorization Technique Using Quadratic Forms." Math. Comput. 28, 625 /C1/35, 1974. Shanks, D. Solved and Unsolved Problems in Number Theory, 3rd ed. New York: Chelsea, pp. 202 and 238, 1985. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 72, 1991. Fano’s Axiom The three diagonal points of a COMPLETE QUADRILAT- ERAL are never COLLINEAR . Far Out A phrase used by Tukey to describe data points which are outside the outer FENCES . See also FENCE References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 44, 1977. Farey Fraction FAREY SEQUENCE Farey Sequence The Farey sequence Fn for any POSITIVE INTEGER n is the set of irreducible RATIONAL NUMBERS a=b with 0 5 a 5b 5n and (a ;b) /C301 arranged in increasing order. The first few are F1 /C300 1 ;11() (1) F 2 /C300 1 ;12 ;11() (2) F 3 /C300 1 ;13 ;12 ;23 ;11() (3) F 4 /C3001 ;14 ;13 ;12 ;23 ;34 ;11() (4) F 5 /C300 1 ;15 ;14 ;13 ;25 ;12 ;35 ;23 ;34 ;45 ;11() (5) (Sloane’s A006842 and A006843). Except for F 1 ; each Fn has an ODD number of terms and the middle term is always 1/2. Let p=q ; p?=q ?; and p ƒ=qƒ be three successive terms in a Farey series. Then qp ?/C28pq ?/C301 (6) p ? q?/C30p /C27 p ƒ q /C27 q ƒ/C215 (7) These two statements are actually equivalent (Hardy and Wright 1979, p. 24). For a method of computing a successive sequence from an existing one of n terms, insert the MEDIANT fraction (a /C27b) =(c /C27d) between terms a=c and b=d when c /C27d 5n (Hardy and Wright 1979, pp. 25 /C1/6; Conway and Guy 1996; Apostol 1997). Given 0 5a =b Bc =d 51 with bc /C28ad /C301; let h=k be the MEDIANT of a=b and c =d: Then a =b Bh=k Bc =d; and these fractions satisfy the unimodular relations bh /C28ak ¼ 1 (8) ck /C28dh /C301 (9) (Apostol 1997, p. 99). The number of terms N(n) in the Farey sequence for the INTEGER n isN(n) /C301 /C27Xn k /C301f(k) /C301 /C27F(n); (10) where f(k) is the TOTIENT FUNCTION and F(n) is the SUMMATORY FUNCTION of f(k); giving 2, 3, 5, 7, 11, 13, 19, ... (Sloane’s A005728). The asymptotic limit for the function N(n)i s N(n)/C23n2 p2/C300:3039635509 n2(11) (Vardi 1991, p. 155). FORD CIRCLES provide a method of visualizing the Farey sequence. The Farey sequence Fndefines a subtree of the S TERN- BROCOT TREE obtained by pruning unwanted branches (Graham et al. 1994). See also FORD CIRCLE ,MEDIANT ,MINKOWSKI’S QUES- TION MARK FUNCTION ,R ANK (SEQUENCE ), STERN- BROCOT TREE References Apostol, T. M. "Farey Fractions." §5.4 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 97 /C1/9, 1997. Beiler, A. H. "Farey Tails." Ch. 16 in Recreations in the Theory of Numbers: The Queen of Mathematics Enter- tains. New York: Dover, 1966. Bogomolny, A. "Farey Series, A Story." http://www.cut-the- knot.com/blue/FareyHistory.html. Conway, J. H. and Guy, R. K. "Farey Fractions and Ford Circles." The Book of Numbers. New York: Springer- Verlag, pp. 152 /C1/54 and 156, 1996. Devaney, R. "The Mandelbrot Set and the Farey Tree, and the Fibonacci Sequence." Amer. Math. Monthly 106, 289/C1/ 02, 1999. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 155 /C1/ 58, 1952. Farey, J. "On a Curious Property of Vulgar Fractions." London, Edinburgh and Dublin Phil. Mag. 47, 385, 1816. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed.Reading, MA: Addison-Wesley, pp. 118 /C1 /19, 1994. Guy, R. K. "Mahler’s Generalization of Farey Series." §F27 inUnsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 263 /C1/65, 1994. Hardy, G. H. and Wright, E. M. "Farey Series and a Theorem of Minkowski." Ch. 3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 23 /C1/7, 1979. Sloane, N. J. A. Sequences A005728/M0661, A006842/ M0041, and A006843/M0081 in "An On-Line Version ofthe Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Sylvester, J. J. "On the Number of Fractions Contained in Any Farey Series of Which the Limiting Number is Given." London, Edinburgh and Dublin Phil. Mag. (5th Series) 15, 251, 1883. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 155, 1991. Weisstein, E. W. "Plane Geometry." M ATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Farey Series FAREY SEQUENCE Farkas’s Lemma The system Ax ¼ x; x ]0 has no solution IFF the system ATw 50; bT > 0 has a solution (Fang and Puthenpura 1993, p. 60). This LEMMA is used in the proof of the KUHN- TUCKER THEOREM . See also KUHN- TUCKER THEOREM ,LAGRANGE MULTI- PLIER References Fang, S.-C. and Puthenpura, S. Linear Optimization and Extensions: Theory and Algorithms. Englewood Cliffs, NJ: Prentice-Hall, p. 60, 1993. Faro Shuffle RIFFLE SHUFFLE Far-Out Point For a TRIANGLE with side lengths a,b, and c, the far- out point has TRIANGLE CENTER FUNCTION a/C30ab4/C27c4/C28a4/C28b2c2/C0/C1 : Asa:b:capproaches 1 : 1 : 1 ;this point moves out along the E ULER LINE to infinity. References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Kimberling, C.; Lyness, R. C.; and Veldkamp, G. R. "Pro- blem 1195 and Solution." Crux Math. 14, 177/C1/79, 1988. Fast Fibonacci Transform For a general second-order RECURRENCE RELATION fn/C271/C30xfn/C27yfn/C281; (1) define a multiplication rule on ordered pairs by (A;B)(C;D)/C30(AD/C27BC/C27xAC ;BD/C27yAC): (2) The inverse is then given by (A;B)/C281/C30(/C28A;xA/C27B) B2/C27xAB/C28yA2; (3) and we have the identity f1;yf0 ðÞ (1;0)n/C30fn/C271;yfn/C0/C1 (4) (Beeler et al. 1972, Item 12). References Gosper, R. W. and Salamin, G. Item 12 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge,MA: MIT Artificial Intelligence Laboratory, Memo AIM- 239, p. 6, Feb. 1972. Fast Fourier Transform The fast Fourier transform (FFT) is a DISCRETE FOURIER TRANSFORM ALGORITHM which reduces the number of computations needed for Npoints from 2N2to 2NlgN;where LGis the base-2 LOGARITHM .I f the function to be transformed is not harmonically related to the sampling frequency, the response of an FFT looks like a SINC FUNCTION (although the integrated POWER is still correct). A LIASING (LEAKAGE ) can be reduced by APODIZATION using a TAPERING FUNCTION . However, ALIASING reduction is at the expense of broadening the spectral response. FFTs were first discussed by Cooley and Tukey (1965), although Gauss had actually described the critical factorization step as early as 1805 (Gergkand 1969, Strang 1993). A DISCRETE FOURIER TRANSFORM can be computed using an FFT by means of the DANIELSON- LANCZOS LEMMA if the number of points Nis a POWER of two. If the number of points Nis not a POWER of two, a transform can be performed on sets of points corresponding to the prime factors of Nwhich is slightly degraded in speed. An efficient real Fourier transform algorithm or a fast H ARTLEY TRANSFORM (Bracewell 1999) gives a further increase in speed by approximately a factor of two. Base-4 and base-8 fast Fourier transforms use optimized code, and can be20/C1 /0% faster than base-2 fast Fourier transforms. PRIME factorization is slow when the factors are large, but discrete Fourier transforms can be made fast forN/C302, 3, 4, 5, 7, 8, 11, 13, and 16 using the W INOGRAD TRANSFORM ALGORITHM (Press et al. 1992, pp. 412 /C1/ 13, Arndt). Fast Fourier transform algorithms generally fall into two classes: decimation in time, and decimation infrequency. The Cooley-Tukey FFT ALGORITHM first rearranges the input elements in bit-reversed order,then builds the output transform (decimation intime). The basic idea is to break up a transform of length Ninto two transforms of length N=2 using the identity X N/C281 n/C300ane/C282pink=N /C30XN=2/C281 n/C300a2ne/C282pi(2n)k=N/C27XN=2/C281 n/C300a2n/C271e/C282pi(2n/C271)k=N /C30XN=2/C281 n/C300aeven ne/C282pink=(N=2)/C27e/C282pik=N /C2XN=2/C281 n/C300aoddne/C282pink=(N=2); sometimes called the D ANIELSON- LANCZOS LEMMA . The easiest way to visualize this procedure is perhaps via the FOURIER MATRIX . The Sande-Tukey ALGORITHM (Stoer and Bulirsch 1980) first transforms, then rearranges the output values (decimation in frequency). See also DANIELSON- LANCZOS LEMMA ,D ISCRETE FOURIER TRANSFORM ,F OURIER MATRIX ,F OURIER TRANSFORM ,HARTLEY TRANSFORM ,NUMBER THEORE- TIC TRANSFORM ,W INOGRAD TRANSFORM References Arndt, J. "FFT Code and Related Stuff." http://www.jjj.de/ fxt/. Bell Laboratories. "Netlib FFTPack." http://netlib.bell-labs.- com/netlib/fftpack/. Blahut, R. E. Fast Algorithms for Digital Signal Processing. New York: Addison-Wesley, 1984. Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999. Brigham, E. O. The Fast Fourier Transform and Applica- tions. Englewood Cliffs, NJ: Prentice Hall, 1988. Chu, E. and George, A. Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms. Boca Raton, FL: CRC Press, 2000. Cooley, J. W. and Tukey, O. W. "An Algorithm for the Machine Calculation of Complex Fourier Series." Math. Comput. 19, 297 /C1/01, 1965. Duhamel, P. and Vetterli, M. "Fast Fourier Transforms: A Tutorial Review." Signal Processing 19, 259 /C1/99, 1990. Gergkand, G. D. "A Guided Tour of the Fast Fourier Trans- form." IEEE Spectrum 6,41/C1/2, July 1969. Lipson, J. D. Elements of Algebra and Algebraic Computing. Reading, MA: Addison-Wesley, 1981. Nussbaumer, H. J. Fast Fourier Transform and Convolution Algorithms, 2nd ed. New York: Springer-Verlag, 1982. Papoulis, A. The Fourier Integral and its Applications. New York: McGraw-Hill, 1962. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Fast Fourier Transform." Ch. 12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 490 /C1/29, 1992. Ramirez, R. W. The FFT: Fundamentals and Concepts. Englewood Cliffs, NJ: Prentice-Hall, 1985. Stoer, J. and Bulirsch, R. Introduction to Numerical Analy- sis. New York: Springer-Verlag, 1980. Strang, G. "Wavelet Transforms Versus Fourier Trans- forms." Bull. Amer. Math. Soc. 28, 288 /C1/05, 1993. Van Loan, C. Computational Frameworks for the Fast Fourier Transform. Philadelphia, PA: SIAM, 1992. Walker, J. S. Fast Fourier Transform, 2nd ed. Boca Raton, FL: CRC Press, 1996. Fast Gossiping GOSSIPING Fat Fractal AC ANTOR SET with LEBESGUE MEASURE greater than 0. See also CANTOR SET,EXTERIOR DERIVATIVE ,FRAC- TAL,LEBESGUE MEASUREReferences Ott, E. "Fat Fractals." §3.9 in Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 97 /C1/00, 1993. Fatou Dust FATOU SET Fatou Set AJ ULIA SET J consisting of a set of isolated points which is formed by taking a point outside an under- lying set M (e.g., the MANDELBROT SET). If the point is outside but near the boundary of M, the Fatou set resembles the JULIA SET for nearby points within M. As the point moves further away, however, the set becomes thinner and is called FATOU DUST . See also JULIA SET References Schroeder, M. Fractals, Chaos, Power Laws. New York: W. H. Freeman, p. 39, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 72 /C1/3, 1991. Fatou’s Lemma If ffn g is a SEQUENCE of NONNEGATIVE measurable functions, then glim inf n0/C12fndm5lim inf n0/C12gfndm: See also ALMOST EVERYWHERE CONVERGENCE ,M EA- SURE THEORY ,POINTWISE CONVERGENCE References Browder, A. Mathematical Analysis: An Introduction. New York: Springer-Verlag, 1996. Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995. Fatou’s Theorems Letf(u)b eL EBESGUE INTEGRABLE and let f(r;u)/C301 2pgp /C28pf(t)1/C28r2 1/C282rcos(t/C28u)/C27r2dt (1) be the corresponding P OISSON INTEGRAL . Then AL- MOST EVERYWHERE in/C28p5u5p lim r00/C28f(r;u)/C30f(u): (2) Let F(z)/C30c0/C27c1z/C27c2z2/C27.../C27cnzn/C27... ( 3 ) be regular for ½z½B1;and let the integral 1 2pgp /C28p½F(reiu) ½2du (4) be bounded for r B1. This condition is equivalent to the convergence of ½C0 ½2 /C27½C1 ½2 /C27.../C27½Cn ½2 /C27... (5) Then almost everywhere in /C28p5 u 5p; lim r00/C28F(reiu) /C30F(ei u) : (6) Furthermore, F(eiu) is measurable, ½F(eiu) ½2 is LEBES- GUE INTEGRABLE , and the FOURIER SERIES of F(ei u)is given by writing z /C30eiu :/ References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 274, 1975. Faulhaber’s Formula In a 1631 edition of Academiae Algebrae , J. Faulha- ber published the general formula for the POWER SUM of the first n POSITIVE INTEGERS , Xn k/C301kp /C301 p /C27 1Xp/C271 i/C301/C281ðÞdipp /C271 i/C1Y/C1Q Bp/C271 /C28ini ; (1) where dip is the KRONECKER DELTA , n i/C0/C1 is a BINOMIAL COEFFICIENT , and Biis the ith BERNOULLI NUMBER . Computing the sums for p /C30 1, ..., 10 gives Xn k /C301k /C301 2n2 /C27n/C0/C1 (2) Xn k /C301k2 /C30162n 3 /C273n2 /C27n/C0/C1 (3) Xn k /C301k3 /C3014n 4 /C272n3 /C27n2/C0/C1 (4) Xn k/C301k4 /C301 306n5 /C2715n4 /C2710n3 /C28n/C0/C1 (5) Xn k /C301k5 /C301 122n6 /C276n5 /C275n4 /C28n2/C0/C1 (6) Xn k /C301k6 /C301 426n7 /C2721n6 /C2721n5 /C287n3 /C27n/C0/C1 (7) Xn k/C301k7 /C301 243n8 /C2712n7 /C2714n6 /C287n4 /C272n2/C0/C1 (8) Xn k /C301k8 /C301 9010n9 /C2745n8 /C2760n7 /C2842n5 /C2720n3 /C283n/C0/C1 (9)Xn k /C301k9 /C301 202n10 /C2710n9 /C2715n8 /C2814n6 /C2710n4 /C283n2/C0/C1 (10) Xn k /C301k10 /C301 666n11 /C2733n10 /C2755n9 /C2866n5 /C2833n3 /C275n/C0/C1 : (11) See also POWER ,POWER SUM,SUM References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996. Fault-Free Rectangle A DISSECTION of a RECTANGLE into smaller RECTAN- GLES such that the original rectangle is not divided into two subrectangles. Rectangle dissections into 3, 4, or 6 pieces cannot be fault-free but, as illustrated above, a dissection into five or more pieces may be fault-free. See also BLANCHE’S DISSECTION ,M RS. PERKINS’ QUILT,PERFECT SQUARE DISSECTION ,RECTANGLE References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 85, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 73, 1991. Favard Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. LetTn(x) be an arbitrary trigonometric POLYNOMIAL Tn(x)/C301 2a0/C27Xn k/C301akcos(kx)/C27bksin(kx) ½/C138() ; (1) where the COEFFICIENTS are real. Let the rth deriva- tive of Tn(x) be bounded in [ /C281;1];then there exists a POLYNOMIAL Tn(x) for which f(x)/C28Tn(x) jj 5Kr (n/C271)r; (2) for all x, where Kris the rth Favard constant, which is the smallest constant possible, Kr/C304 pX/C12 k/C300(/C281)k 2k/C271"#r/C271 ; (3) which can be written in terms of the LERCH TRANS- CENDENT as Kr /C302 /C28(r/C271) F (/C281)r/C271 ; r /C271;1 2 ! : (4) These can be expressed by Kr /C304 p l(r /C271) for r odd 4 p b(r /C271) for r even ;8 >>>< >>>:(5) where l(x) is the D IRICHLET LAMBDA FUNCTION and b(x) is the DIRICHLET BETA FUNCTION . Explicitly, K0 /C301 K1 /C301 2 p K2 /C3018 p 2 K3 /C301 24 p3 K4 /C305 384 p4 K5 ¼1 240 p5 (Sloane’s A050970 and A050971). See also DIRICHLET BETA FUNCTION ,D IRICHLET LAMBDA FUNCTION References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/favard/favard.html. Kolmogorov, A. N. "Zur Gro¨ssenordnung des Restgliedes Fourierscher reihen differenzierbarer Funktionen." Ann. Math. 36, 521 /C1/26, 1935. Sloane, N. J. A. Sequences A050970 and A050970 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Zygmund, A. G. Trigonometric Series, Vols. 1 /C1/, 2nd ed. New York: Cambridge University Press, 1959. F-Distribution A continuous statistical distribution which arises in the testing of whether two observed samples have the same VARIANCE . Let x2 mand x2nbe independent variates distributed as CHI-SQUARED with m and n DEGREES OF FREEDOM . Define a statistic Fn;mas the ratio of the dispersions of the two distributions Fn;m /C13x2n =n x2 m =m : (1)This statistic then has an F-distribution with prob- ability function fn;m(x) and cumulative distribution function Fn;m(x) given by fn;m(x) /C30Gn /C27 m 2 ! nn=2mm=2 Gn 2 ! Gm 2 !xn=2 /C281 (m /C27 nx)(n/C27m)=2 (2) /C30mm=2nn=2xn=2 /C281 (m /C27 nx)(n/C27m)=2B1 2 n;12 m ! (3) F n ;m(x) /C30I 1;1 2 m;12 n ! /C28Im m /C27 nx;12 m;12n ! ; (4) where G(z) is the GAMMA FUNCTION , B(a; b) is the BETA FUNCTION , and I(x;a; b) is the REGULARIZED BETA FUNCTION . The MEAN , VARIANCE , SKEWNESS and KURTOSIS are m /C30m m /C28 2 (5) s2 /C302m2(m /C27 n /C28 2) n(m /C28 2)2(m /C28 4) (6) g1 /C302(m /C27 2n /C28 2) m /C28 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(m /C28 4) n(m /C27 n /C28 2)s (7) g2 /C3012 /C2816 /C27 20m /C28 8m2 /C27 m3 /C27 44n ðÞ n(m /C28 6)(m /C28 8)(n /C27 m /C28 2) /C2712 /C2832mn /C27 5m2n /C28 22n2 /C27 5mn2ðÞ n(m /C28 6)(m /C28 8)(n /C27 m /C28 2): (8) The probability that F would be as large as it is if the first distribution has a smaller variance than the second is denoted Q(Fn ;m) :/ The noncentral F-distribution is given by P(x)/C30e/C28l=2/C27ln1x ðÞ =2n2/C27n1x ðÞ½/C138nn1=2 1nn2=2 2xn1=2/C281 /C2n2/C27n1x ðÞ/C28n1/C27n2 ðÞ =2 /C29G1 2n1 ! G1/C2712n 2 ! Ln1=2/C281 n2=2/C28ln1x 2n2/C27n1x ðÞ ! B1 2n1;12n 2 ! G12n 1/C27n2 ðÞ"# ; (9) where G(z) is the GAMMA FUNCTION ,B(a;b) is the BETA FUNCTION , and Ln m(z) is an associated L AGUERRE POLYNOMIAL . See also BETA FUNCTION ,G AMMA FUNCTION ,H O- TELLING T-SQUARED DISTRIBUTION ,R EGULARIZED BETA FUNCTION ,SNEDECOR’S F-DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 946 /C1/49, 1972. David, F. N. "The Moments of the z and F Distributions." Biometrika 36, 394 /C1/03, 1949. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Incomplete Beta Function, Student’s Distribu- tion, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 219 /C1/23, 1992. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 117 /C1/18, 1992. Feigenbaum Constant A universal constant for functions approaching CHAOS via period doubling. It was discovered by Feigenbaum in 1975 and demonstrated rigorously by Lanford (1982) and Collet and Eckmann (1979, 1980). The Feigenbaum constant d characterizes the geometric approach of the bifurcation parameter to its limiting value. Let mkbe the point at which a period 2k cycle becomes unstable. Denote the converged value by m/C12: Assuming geometric convergence, the difference be- tween this value and mk is denoted lim k 0/C12m/C12/C28 mk /C30G dk ; (1) where G is a constant and d is a constant > 1: Solving for d gives d /C30 lim n0/C12mn /C271 /C28 mn mn/C272 /C28 mn/C271(2) (Rasband 1990, p. 23). For the LOGISTIC EQUATION , d /C304 :669201609102990... (3) G/C302 :637 ... (4) m/C12/C303:5699456 ... (5) Stoschek gives the approximation d /C3041 /C27122 163/C274 :122 /C27 31 4 :1632/C27 ... 1 /C27102 163/C27102 /C27 30 1632/C27 ...(6) :4:66920160933975 : Amazingly, the Feigenbaum constant d :4:669 is "universal" (i.e., the same) for all 1-D MAPS f(x)if f(x) has a single locally quadratic MAXIMUM . More specifically, the Feigenbaum constant is universal for 1-D MAPS if the SCHWARZIAN DERIVATIVE DSchwarzian /C30f §(x) f ?(x)/C283 2f ƒ(x) f ?(x)"#2 (7)is NEGATIVE in the bounded interval (Tabor 1989, p. 220). Examples of maps which are universal include the HE´ NON MAP, LOGISTIC MAP, LORENZ SYSTEM , Navier-Stokes truncations, and sine map xn/C271 /C30a sin( pxn) : The value of the Feigenbaum con- stant can be computed explicitly using functional group renormalization theory. The universal constant also occurs in phase transitions in physics and, curiously, is very nearly equal to p/C27tan /C281 e pðÞ/C304:669201932... (8) For an AREA-PRESERVING 2-D MAP with xn/C271 /C30fxn ;yn ðÞ (9) yn /C271 /C30gxn ;yn ðÞ ; (10) the Feigenbaum constant is d /C308 :7210978... (Tabor 1989, p. 225). For a function OF THE FORM f(x) /C301 /C28a ½x½n (11) with a and n constant and n an INTEGER , the Feigenbaum constant for various n is given in the following table (Briggs 1991, Briggs et al. 1991, Finch), which updates the values in Tabor (1989, p. 225). n / d// a/ 3 5.9679687038... 1.9276909638... 4 7.2846862171... 1.6903029714... 5 8.3494991320... 1.5557712501... 6 9.2962468327... 1.4677424503... An additional constant a; defined as the separation of adjacent elements of PERIOD DOUBLED ATTRACTORS from one double to the next, has a value lim n0/C12dn dn/C271/C13/C28a/C30/C282:502907875 . . . (12) for "universal" maps (Rasband 1990, p. 37). This value may be approximated from functional group renormalization theory to the zeroth order by 1/C28a/C281/C301/C28a/C282 1/C28a/C2821/C28a/C281 ðÞ ½/C1382; (13) which, when the QUINTIC EQUATION is numerically solved, gives a/C30/C282:48634 . . . ;only 0.7% off from the actual value (Feigenbaum 1988). See also ATTRACTOR ,B IFURCATION ,F EIGENBAUM FUNCTION ,LINEAR STABILITY ,LOGISTIC EQUATION , PERIOD DOUBLING References Briggs, K. "A Precise Calculation of the Feigenbaum Con- stants." Math. Comput. 57, 435 /C1/39, 1991. Briggs, K.; Quispel, G.; and Thompson, C. "Feigenvalues for Mandelsets." J. Phys. A: Math. Gen. 24 3363 /C1/368, 1991. Collet, P. and Eckmann, J.-P. "Properties of Continuous Maps of the Interval to Itself." Mathematical Problems in Theoretical Physics (Ed. K. Osterwalder). New York: Springer-Verlag, 1979. Collet, P. and Eckmann, J.-P. Iterated Maps on the Interval as Dynamical Systems. Boston, MA: Birkha ¨user, 1980. Eckmann, J.-P. and Wittwer, P. Computer Methods and Borel Summability Applied to Feigenbaum’s Equations. New York: Springer-Verlag, 1985. Feigenbaum, M. J. "Presentation Functions, Fixed Points, and a Theory of Scaling Function Dynamics." J. Stat. Phys. 52, 527 /C1/69, 1988. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/fgnbaum/ fgnbaum.html. Finch, S. "Generalized Feigenbaum Constants." http:// www.mathsoft.com/asolve/constant/fgnbaum/gener-al.html. Lanford, O. E. "A Computer-Assisted Proof of the Feigen- baum Conjectures." Bull. Amer. Math. Soc. 6, 427 /C1 /34, 1982. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, 1990. Stephenson, J. W. and Wang, Y. "Numerical Solution of Feigenbaum’s Equation." Appl. Math. Notes 15,68/C1/8, 1990. Stephenson, J. W. and Wang, Y. "Relationships Between the Solutions of Feigenbaum’s Equations." Appl. Math. Let. 4, 37 /C1/9, 1991. Stoschek, E. "Modul 33: Algames with Numbers." http:// marvin.sn.schule.de/~inftreff/modul33/task33.htm. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Feigenbaum Function Consider an arbitrary 1-D MAP xn/C271 /C30FxnðÞ (1) at the onset of CHAOS . After a suitable rescaling, the Feigenbaum function g(x) /C30 lim n0/C121 F 2nðÞ(0)F 2nðÞxF 2nðÞ(0)/C0/C1 (2) is obtained. This function satisfies g(g(x)) /C30/C281 a g( ax) ; (3) with a /C302 :50290... ; a quantity related to the FEI- GENBAUM CONSTANT . See also BIFURCATION ,C HAOS ,F EIGENBAUM CON- STANT References Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/08, 1983.Feit-Thompson Conjecture The conjecture that there are no PRIMES p and q for which (pq /C281)=(p /C281) and (qp /C281)=(q /C281) have a common factor. Parker noticed that if this were true, it would greatly simplify the lengthy proof of the FEIT-THOMPSON THEOREM (Guy 1994, p. 81). However, the counterexample (p /C3017 ;q /C303313) with a common factor 112,643 was subsequently found by Stephens (1971). There are no other such pairs with both values less than 400,000. See also FEIT-THOMPSON THEOREM References Apostol, T. M. "The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx):/" Math. Comput. 29,1/C1/, 1975. Feit, W. and Thompson, J. G. "A Solvability Criterion for Finite Groups and Some Consequences." Proc. Nat. Acad. Sci. USA 48, 968 /C1/70, 1962. Feit, W. and Thompson, J. G. "Solvability of Groups of Odd Order." Pacific J. Math. 13, 775 /C1/029, 1963. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 81, 1994. Stephens, N. M. "On the Feit-Thompson Conjecture." Math. Comput. 25, 625, 1971. Wells, D. G. The Penguin Dictionary of Curious and Inter- esting Numbers. London: Penguin, p. 17, 1986. Feit-Thompson Theorem Every FINITE SIMPLE GROUP (which is not CYCLIC ) has EVEN ORDER , and the ORDER of every FINITE SIMPLE noncommutative group is DOUBLY EVEN , i.e., divisible by 4 (Feit and Thompson 1963). See also BURNSIDE PROBLEM ,FEIT-THOMPSON CON- JECTURE ,F INITE GROUP ,O RDER (GROUP ), SIMPLE GROUP References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 81, 1994. Feit, W. and Thompson, J. G. "A Solvability Criterion for Finite Groups and Some Consequences." Proc. Nat. Acad. Sci. USA 48, 968/C1/70, 1962. Feit, W. and Thompson, J. G. "Solvability of Groups of Odd Order." Pacific J. Math. 13, 775/C1/029, 1963. Fejes To ´th’s Integral 1 2p(n/C271)gp /C28pf(x)sin1 2(n/C271)x"# sin12x !8 >>>>< >>>>:9 >>>>= >>>>;2 dx gives the nth C ESA`RO MEAN of the F OURIER SERIES of f(x):/ References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 12, 1975. Fejes To´th’s Problem SPHERICAL CODE Feldman’s Theorem Any nondegenerate closed SPACE CURVE may be nondegenerately deformed into either of the two curves illustrated above. Neither of these can be nondegenerately transformed into the other. References Feldman, E. A. "Deformations of Closed Space Curves." J. Diff. Geom. 2,67/C1/5, 1968. Pohl, W. F. "The Self-Linking Number of a Closed Space Curve." J. Math. Mech. 17, 975 /C1/85, 1968. Feller’s Coin-Tossing Constants COIN TOSSING Feller-Le ´vy Condition Given a sequence of independent random variates X1 ; X2 ; ..., if s2 k /C30var(Xk) and r2 n /C13max k5ns2 k s2 n ! ; then lim n0/C12r2 n /C300: This means that if the LINDEBERG CONDITION holds for the sequence of variates X1 ; ..., then the VARIANCE of an individual term in the sum Snof Xkis asymptotically negligible. For such sequences, the LINDEBERG CONDITION is NECESSARY as well as SUFFICIENT for the LINDEBERG- FELLER CENTRAL LIMIT THEOREM to hold. See also BERRY- ESSE´ EN THEOREM ,C ENTRAL LIMIT THEOREM ,LINDEBERG CONDITION References Lindeberg, J. W. "Eine neue Herleitung des Exponentialge- setzes in der Wahrschienlichkeitsrechnung." Math. Z. 15, 211 /C1/25, 1922. Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483 /C1/94, 1995.Fence Values one STEP outside the HINGES are called inner fences, and values two steps outside the HINGES are called outer fences. Tukey calls values outside the outer fences FAR OUT. See also ADJACENT VALUE References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 44, 1977. Fence Poset A PARTIAL ORDER defined by /(i /C281); i), /(i /C271); i) for ODD i. See also PARTIAL ORDER References Ruskey, F. "Information on Ideals of Partially Ordered Sets." http://www.theory.csc.uvic.ca/~cos/inf/pose/Ideals.html. Ferguson-Forcade Algorithm The first practical algorithm for determining if there exist integers ai for given real numbers xi such that a1x1 /C27a2x2 /C27.../C27anxn /C300 ; or else establish bounds within which no such INTEGER RELATION can exist (Ferguson and Forcade 1979). The algorithm therefore became the first viable generalization of the EUCLIDEAN ALGORITHM to n ]3 variables. A nonrecursive variant of the original algorithm was subsequently devised by Ferguson (1987). The Fer- guson-Forcade algorithm has been shown to be polynomial-time in the logarithm in the size of a smallest relation, but has not been shown to be polynomial in dimension (Ferguson et al. 1999). See also CONSTANT PROBLEM ,E UCLIDEAN ALGO- RITHM ,INTEGER RELATION , PSLQ ALGORITHM References Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving p;e, and Euler’s Constant." Math. Comput. 50, 275/C1/81, 1988. Bergman, G. "Notes on Ferguson and Forcade’s Generalized Euclidean Algorithm." Unpublished notes. Berkeley, CA: University of California at Berkeley, Nov. 1980. Ferguson, H. R. P. "A Short Proof of the Existence of Vector Euclidean Algorithms." Proc. Amer. Math. Soc. 97,8/C1/0, 1986. Ferguson, H. R. P. "A Non-Inductive GL( n, Z ) Algorithm that Constructs Linear Relations for nZ-Linearly Depen- dent Real Numbers." J. Algorithms 8, 131/C1/45, 1987. Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351/C1/69, 1999. Ferguson, H. R. P. and Forcade, R. W. "Generalization of the Euclidean Algorithm for Real Numbers to All Dimen-sions Higher than Two." Bull. Amer. Math. Soc. 1, 912/C1 / 14, 1979. Ferguson, H. R. P. and Forcade, R. W. "Multidimensional Euclidean Algorithms." J. reine angew. Math. 334, 171 /C1/ 81, 1982. Fermat 4n /C271 Theorem Every PRIME p OF THE FORM p /C304n /C271 is a sum of two SQUARE NUMBERS in one unique way (up to the order of SUMMANDS ). The theorem was stated by Fermat, but the first published proof was by Euler. The first few primes p which are 1 or 2 (mod 4) are 2, 5, 13, 17, 29, 37, 41, 53, 61, ... (Sloane’s A002313) (with the only prime congruent to 2 mod 4 being 2). The numbers (x, y) such that x2 /C27y2equal these primes are (1, 1), (1, 2), (2, 3), (1, 4), (2, 5), (1, 6), ... (Sloane’s A002331 and A002330). See also SIERPINSKI’S PRIME SEQUENCE THEOREM , SQUARE NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 146 /C1/47, 1996. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 13 and 219, 1979. Se´roul, R. "Prime Number and Sum of Two Squares." §2.11 in Programming for Mathematicians. Berlin: Springer- Verlag, pp. 18 /C1/9, 2000. Sloane, N. J. A. Sequences A002313/M1430, A002330/ M000462, and A002331/M0096 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Fermat Compositeness Test The COMPOSITENESS TEST consisting of the applica- tion of FERMAT’S LITTLE THEOREM Fermat Conic A PLANE CURVE OF THE FORM y /C30xn : For n /C210, the curve is a generalized PARABOLA ; for n B0itisa generalized HYPERBOLA . See also CONIC SECTION ,HYPERBOLA ,PARABOLA Fermat Difference Equation PELL EQUATION Fermat Diophantine Equation PELL EQUATION Fermat Elliptic Curve Theorem The only whole number solution to the DIOPHANTINE EQUATION y3 /C30x2 /C272is y /C303, x /C3095: This theorem was offered as a problem by Fermat , who suppressed his own proof. Fermat Equation The DIOPHANTINE EQUATION xn /C27yn /C30zn : The assertion that this equation has no nontrivial solutions for n /C212 has a long and fascinating history and is known as FERMAT’S LAST THEOREM . See also FERMAT’S LAST THEOREM Fermat Number A BINOMIAL NUMBER OF THE FORM Fn /C3022n /C271 : The first few for n /C30 0, 1, 2, ... are 3, 5, 17, 257, 65537, 4294967297, ... (Sloane’s A000215). The number of DIGITS for a Fermat number is D(n) /C30 log 22n /C271/C0/C1/C2/C6 /C271/C4/C3 : log 22n/C0/C1 /C271/C4/C3 /C30 2n log2 /C271 bc : (1) Being a Fermat number is the NECESSARY (but not SUFFICIENT ) form a number Nn/C132n/C271 (2) must have in order to be PRIME . This can be seen by noting that if Nn/C302n/C271i st ob e PRIME , then n cannot have any ODD factors bor else Nnwould be a factorable number OF THE FORM 2n/C271/C302aðÞb/C271/C302a/C271 ðÞ /C22a(b/C281)/C282a(b/C282)/C272a(b/C283)/C28.../C271/C2/C6 :(3) Therefore, for a PRIME Nn;nmust be a POWER of 2. No two Fermat numbers have a common divisor greater than 1 (Hardy and Wright 1979, p. 14). Fermat conjectured in 1650 that every Fermat num- ber is PRIME and Eisenstein (1844) proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88). At present, however, only COMPOSITE Fermat numbers Fnare known for n]5:An anonymous writer proposed that numbers OF THE FORM 22/C271;222/C271;2222/C271 were PRIME . However, this conjecture was refuted when Selfridge (1953) showed that F16/C302216/C271/C3022222 /C271 (4) isCOMPOSITE (Ribenboim 1996, p. 88). Numbers OF THE FORM a2n/C27b2nare called generalized Fermat numbers (Ribenboim 1996, pp. 359 /C1/60). Fermat numbers satisfy the RECURRENCE RELATION Fm/C30F0F1...Fm/C281/C272: (5) /Fncan be shown to be PRIME IFF it satisfies PE´PIN’S TEST 3(Fn/C281)=2/C13/C281(mod Fn): (6) PE´PIN’S THEOREM 322n/C281 /C13/C281(mod Fn) (7) is also NECESSARY and SUFFICIENT . In 1770, Euler showed that any FACTOR ofFnmust have the form 2n/C271K/C271; (8) where Kis a POSITIVE INTEGER . In 1878, Lucas increased the exponent of 2 by one, showing that FACTORS of Fermat numbers must be OF THE FORM 2n/C272L/C271: (9) If F/C30p1p2...pr (10) is the factored part of Fn/C30FC(where Cis the cofactor to be tested for primality), compute A/C133Fn/C281(mod Fn) (11) B/C133F/C281(mod Fn) (12) R/C13A/C28B(mod C): (13) Then if R/C130;the cofactor is a PROBABLE PRIME to the base 3F; ; otherwise CisCOMPOSITE . In order for a POLYGON to be circumscribed about a CIRCLE (i.e., a CONSTRUCTIBLE POLYGON ), it must have a number of sides Ngiven by N/C302kF0...Fn; (14) where the Fnaredistinct Fermat primes (as stated by Gauss and first published by Wantzel 1836). This is equivalent to the statement that the trigonometric functions sin( kp=N);cos(kp=N);etc., can be computed in terms of finite numbers of additions, multiplica- tions, and square root extractions IFFNis of the above form. The only known Fermat PRIMES are F0/C303 F1/C305 F2/C3017 F3/C30257 F4/C3065537 and it seems unlikely that any more exist. Factoring Fermat numbers is extremely difficult as a result of their large size. In fact, only F5toF11have been complete factored, as summarized in the follow- ing table. Written out explicitly, the complete factor-izations are F 5/C30641 /C2156700417F6/C30274177 /C21567280421310721 F7/C3059649589127497217 /C2155704689200685129054721 F8/C301238926361552897 /C21593461639715357977769163 /C1/C1/C1 /C1/C1/C1558199606896584051237541638188580280321 F9/C302424833 /C21574556028256478842083373957362004 /C1/C1/C1 /C1/C1/C154918783366342657 /C215P99 F10/C3045592577 /C2156487031809 /C21546597757852200185 /C1/C1/C1 /C1/C1/C143264560743076778192897 /C215P252 F11/C30319489 /C215974849 /C215167988556341760475137 /C2153560841906445833920513 /C215P564: Here, the final large PRIME is not explicitly given since it can be computed by dividing Fnby the other given factors. The following table summarizes the properties of completely factored Fermat numbers. /Fn/Digits Factors Digits Reference 5 10 2 3, 7 Euler 1732 6 20 2 6, 14 Landry 1880 7 39 2 7, 22 Morrison and Brillhart 1975 8 78 2 16, 62 Brent and Pollard 1981 9 155 3 7, 49, 99 Manasse and Lenstra (In Cipra 1993) 10 309 4 8, 10, 40, 252Brent 1995 11 617 5 6, 6, 21, 22, 564Brent 1988 Tables of known factors of Fermat numbers are given by Keller (1983), Brillhart et al. (1988), Young and Buell (1988), Riesel (1994), and Pomerance (1996). Young and Buell (1988) discovered that F20isCOM- POSITE , and Crandall et al. (1995) that F22is COMPOSITE . In 1999, Crandall et al. showed that F24 isCOMPOSITE . A current list of the known factors of Fermat numbers is maintained by Keller, and repro- duced in the form of a Mathematica notebook by Weisstein. In these tables, since all factors are OF THE FORM k2n/C271;the known factors are expressed in the concise form ( k, n). The number of factors for Fermat numbers Fn for n /C30 0, 1, 2, ... are 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, .... See also CULLEN NUMBER ,P E´ PIN’S TEST,P E´ PIN’S THEOREM ,P OCKLINGTON’S THEOREM ,P OLYGON , PROTH’S THEOREM ,S ELFRIDGE- HURWITZ RESIDUE , WOODALL NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 68 /C1/9 and 94 /C1/5, 1987. Brent, R. P. "Factorization of the Eighth Fermat Number." Amer. Math. Soc. Abstracts 1, 565, 1980. Brent, R. P. "Factorisation of F10." http://cslab.anu.edu.au/ ~rpb/F10.html. Brent, R. P "Factorization of the Tenth Fermat Number." Math. Comput. 68, 429 /C1/51, 1999. Brent, R. P. and Pollard, J. M. "Factorization of the Eighth Fermat Number." Math. Comput. 36, 627 /C1/30, 1981. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn 91 ; b /C30 2, 3; 5; 6;7 ;10;11; 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., pp. 1xxxvii and 2 /C1/ of Update 2.2, 1988. Caldwell, C. K. "The Top Twenty: Fermat Divisors." http:// www.utm.edu/research/primes/lists/top20/FermatDivi- sor.html. Cipra, B. "Big Number Breakdown." Science 248, 1608, 1990. Conway, J. H. and Guy, R. K. "Fermat’s Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 137 /C1/ 41, 1996. Cormack, G. V. and Williams, H. C. "Some Very Large Primes of the Form k /C2152m /C271 :/" Math. Comput. 35, 1419 /C1/ 421, 1980. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 25 /C1/6 and 119, 1996. Crandall, R.; Doenias, J.; Norrie, C.; and Young, J. "The Twenty-Second Fermat Number is Composite." Math. Comput. 64, 863 /C1/68, 1995. Crandall, R. "F24 Resolved--Official Announcement." [email protected] posting, 29 Sep 1999. Dickson, L. E. "Fermat Numbers Fn /C3022n /C271 :/" Ch. 15 in History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 375 /C1/80, 1952. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Euler, L. "Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus." Acad. Sci. Petropol. 6, 103 /C1/07, ad annos 1732 /C1/3 (1738). In Leon- hardi Euleri Opera Omnia, Ser. I, Vol. II. Leipzig: Teubner, pp. 1 /C1/, 1915. Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243,18/C1/8, Dec. 1980. Gostin, G. B. "A Factor of F17 :/" Math. Comput. 35, 975 /C1/76, 1980. Gostin, G. B. "New Factors of Fermat Numbers." Math. Comput. 64, 393 /C1/95, 1995. Gostin, G. B. and McLaughlin, P. B. Jr. "Six New Factors of Fermat Numbers." Math. Comput. 38, 645 /C1/49, 1982. Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape k /C2152n /C272:/" §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8 /C1/3, 1994. Hallyburton, J. C. Jr. and Brillhart, J. "Two New Factors of Fermat Numbers." Math. Comput. 29, 109 /C1/12, 1975.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 14 /C1/5 and 19, 1979. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 200, 1998. Keller, W. "Factor of Fermat Numbers and Large Primes of the Form k /C2152n /C271:/" Math. Comput. 41, 661 /C1/73, 1983. Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k /C2152n /C271; II." In prep. Keller, W. "Prime Factors k /C2152n /C271 of Fermat Numbers Fm and Complete Factoring Status." http://vamri.xray.u- fl.edu/proths/fermat.html. Kraitchik, M. "Fermat Numbers." §3.6 in Mathematical Recreations. New York: W. W. Norton, pp. 73 /C1/5, 1942. Landry, F. "Note sur la de´composition du nombre 264 /C271 (Extrait)." C. R. Acad. Sci. Paris , 91, 138, 1880. Lenstra, A. K.; Lenstra, H. W. Jr.; Manasse, M. S.; and Pollard, J. M. "The Factorization of the Ninth Fermat Number." Math. Comput. 61, 319 /C1/49, 1993. Morrison, M. A. and Brillhart, J. "A Method of Factoring and the Factorization of F7 :/" Math. Comput. 29, 183 /C1/05, 1975. Po´lya, G. and Szego, G. Problem 94, Part 8 in Problems and Theorems in Analysis. Berlin: Springer-Verlag, 1976. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/485, 1996. Ribenboim, P. "Fermat Numbers" and "Numbers k /C292n 91:/" §2.6 and 5.7 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 83 /C1/0 and 355 /C1/60, 1996. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkha ¨user, pp. 384 /C1/88, 1994. Robinson, R. M. "A Report on Primes of the Form k /C2152n /C271 and on Factors of Fermat Numbers." Proc. Amer. Math. Soc. 9, 673 /C1/81, 1958. Selfridge, J. L. "Factors of Fermat Numbers." Math. Com- put. 7, 274 /C1/75, 1953. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 13 and 78 /C1/0, 1993. Shorey, T. N. and Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers, 2." J. London Math. Soc. 23,17/C1/3, 1981. Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers." Proc. London Math. Soc. 35, 425/C1/47, 1977. Sloane, N. J. A. Sequences A000215/M2503 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wantzel, M. L. "Recherches sur les moyens de reconnaı ˆtre si un proble `me de ge ´ome´trie peut se re ´soudre avec la re `gle et le compas." J. Math. pures appliq. 1, 366/C1/72, 1836. Weisstein, E. W. "Fermat Numbers." M ATHEMATICA NOTE- BOOK FERMAT.M . Wrathall, C. P. "New Factors of Fermat Numbers." Math. Comput. 18, 324/C1/25, 1964. Young, J. and Buell, D. A. "The Twentieth Fermat Number is Composite." Math. Comput. 50, 261/C1/63, 1988. Fermat Number (Lucas) A number OF THE FORM 2n/C281 obtained by setting x/C301i naF ERMAT POLYNOMIAL is called a M ERSENNE NUMBER . See also FERMAT- LUCAS NUMBER ,M ERSENNE NUM- BER Fermat Points In a given ACUTE TRIANGLE DABC ; the Fermat point X (or "first Fermat point" F1 ; also called the Torricelli point) is the point which minimizes the sum of distances from A, B, and C, AXjj/C27BXjj/C27CXjj : (1) This problem is called FERMAT’S PROBLEM or STEI- NER’S PROBLEM (Courant and Robbins 1941) and was proposed by Fermat to Torricelli. Torricelli’s solution was published by his pupil Viviani in 1659 (Johnson 1929). If all ANGLES of the TRIANGLE are less than 1208 / 2p=3 ðÞ ; then the Fermat point is the interior point X from which each side subtends an ANGLE of 1208, i.e., /C218BXC /C30/C218CXA /C30/C218AXB /C30120( : (2) The Fermat point can be constructed by drawing EQUILATERAL TRIANGLES on the outside of the given TRIANGLE and connecting opposite VERTICES . The three diagonals in the figure then intersect in the Fermat point. Similarly, the second Fermat point F2 is constructed using equilateral triangles pointing inwards. The Fermat points are also known as the isogonic centers, since they are ISOGONAL CONJU- GATES of the ISODYNAMIC POINTS . The TRIANGLE CENTER FUNCTIONS of the Fermat points are a1 /C30csc A /C271 3 p ! (3) bc c2a2 /C27(c2 /C27a2 /C28b2)2hi a2b2 /C28(a2 /C27b2 /C28c2)2hi /C2 4 D/C28ffiffiffi 3p (b2 /C27c2 /C28d2)hi (4) a2 /C30csc A /C281 3 p ! (5) The ANTIPEDAL TRIANGLE of F1is EQUILATERAL and has AREAD?/C302 D 1 /C27cot v cotp3 ! "# ; (6) where v is the B ROCARD ANGLE . The ANTIPEDAL TRIANGLE of F2 is also an EQUILATERAL and has AREA 2 D/C30/C28 1 /C27cot v cot13p ! "# : (7) Given three POSITIVE REAL NUMBERS l;m;n;the "generalized" Fermat point is the point Pof a given ACUTE TRIANGLE DABC such that l/C215PA/C27m /C215PB/C27n/C215PC (8) is a minimum (Greenberg and Robertello 1965, van de Lindt 1966, Tong and Chua 1995) See also BROCARD ANGLE ,EQUILATERAL TRIANGLE , FERMAT POINTS ,ISODYNAMIC POINTS ,ISOGONAL CON- JUGATE ,LESTER CIRCLE References Courant, R. and Robbins, H. What is Mathematics?, 2nd ed. Oxford, England: Oxford University Press, 1941. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 107, 1913. Greenberg, I. and Robertello, R. A. "The Three Factory Problem." Math. Mag. 38,6 7/C1/2, 1965. Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 24 /C1/4, 1973. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 221 /C1/22, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Kimberling, C. "Fermat Point." http://cedar.evansville.edu/ ~ck6/tcenters/class/fermat.html. Mowaffaq, H. "An Advanced Calculus Approach to Finding the Fermat Point." Math. Mag. 67,2 9/C1/4, 1994. Nelson, D. "Napoleon Revisited." Math. Gaz. No. 404, 1974. Pottage, J. Geometrical Investigations. Reading, MA: Addi- son-Wesley, 1983. Spain, P. G. "The Fermat Point of a Triangle." Math. Mag. 69, 131/C1/33, 1996. Tong, J. and Chua, Y. S. "The Generalized Fermat’s Point." Math. Mag. 68, 214/C1/15, 1995. van de Lindt, W. J. "A Geometrical Solution of the Three Factory Problem." Math. Mag. 39, 162/C1/65, 1966. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 75 /C1/6, 1991. Fermat Polynomial The POLYNOMIALS obtained by setting p(x)/C303xand q(x)/C30/C282 in the L UCAS POLYNOMIAL SEQUENCES . The first few Fermat polynomials are F(x)/C301 F2(x)/C303x F3(x)/C309x2/C282 F4(x)/C3027x3/C2812x F5(x) /C3081x4 /C2854x2 /C274; and the first few Fermat-Lucas polynomials are f1(x) /C303x f2(x) /C309x2 /C284 f3 ¼ 27x3 /C2818x f4(x) /C3081x4 /C2872x2 /C278 f5(x) /C30243x5 /C28270x3 /C2760x: Fermat and Fermat-Lucas POLYNOMIALS satisfy Fn(1) /C30Fn fn(1) /C30fn where Fnare FERMAT NUMBERS and fnare FERMAT- LUCAS NUMBERS . Fermat Prime AF ERMAT NUMBER Fn /C3022n /C271 which is PRIME . See also CONSTRUCTIBLE POLYGON ,FERMAT NUMBER Fermat Pseudoprime A Fermat pseudoprime to a base a, written psp(a), is a COMPOSITE NUMBER n such that an/C281 /C131ðmod nÞ (i.e., it satisfies FERMAT’S LITTLE THEOREM , some- times with the requirement that n must be ODD; Pomerance et al. 1980). psp(2)s are called POULET NUMBERS or, less commonly, SARRUS NUMBERS or FERMATIANS (Shanks 1993). The first few EVEN psp(2)s (including the PRIME 2 as a pseudoprime) are 2, 161038, 215326, ... (Sloane’s A006935). If base 3 is used in addition to base 2 to weed out potential COMPOSITE NUMBERS , only 4709 COMPOSITE NUMBERS remain B25 /C29109 : Adding base 5 leaves 2552, and base 7 leaves only 1770 COMPOSITE NUM- BERS . See also CARMICHAEL NUMBER ,F ERMAT’S LITTLE THEOREM ,POULET NUMBER ,PSEUDOPRIME References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 182, 1998. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. "The Pseudoprimes to 25 /C215109 :/" Math. Comput. 35, 1003 /C1/026, 1980. Available electronically from ftp://sable.ox.ac.uk/ pub/math/primes/ps2.Z. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 115, 1993. Sloane, N. J. A. Sequences A006935/M2190 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Fermat Quotient The Fermat quotient for a number a and a PRIME base p is defined as qp(a) /C13ap /C281 /C28 1 p/C215 (1) If p¶ab ; then qp(ab) /C30qp(a) /C27qp(b) (2) qp(p 91) /C30/C141 (3) qp(2) /C301 p1 /C281 2 /C2713 /C2814 /C27/C1/C1/C1/C281 p /C28 1 ! (4) all (mod p). The quantity qp(2) /C30(2p /C281 /C281)=p is known to be SQUARE for only two PRIMES : the so- called WIEFERICH PRIMES 1093 and 3511 (Lehmer 1981, Crandall 1986). See also WIEFERICH PRIME References Crandall, R. Projects in Scientific Computation. New York: Springer-Verlag, 1986. Lehmer, D. H. "On Fermat’s Quotient, Base Two." Math. Comput. 36, 289/C1/90, 1981. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 70, 1986. Fermat’s Algorithm FERMAT’S FACTORIZATION METHOD Fermat’s Congruence FERMAT’S LITTLE THEOREM Fermat’s Conjecture FERMAT’S LASTTHEOREM Fermat’s Divisor Problem In 1657, Fermat posed the problem of finding solu- tions to s(x3)/C30y2(1) and s(x2)/C30y3; (2) where s(n) is the DIVISOR FUNCTION (Dickson 1952). The first few solutions to s(x3)/C30y2are ( x;y)/C30(1;1); (7, 20), (751530, 1292054400) (Sloane’s A008849 and A048948) .... Lucas stated that there are an infinite number of solutions (Dickson 1952, p. 56), but only solutions up to the fourth are known to be complete. The first few solutions to s(x2) /C30y3 are (x ;y) /C30(1;1); (43098, 1729), ... (Sloane’s A008850 and A048949), with only solutions up to the second known to be complete. See also DIVISOR FUNCTION ,W ALLIS’S PROBLEM References Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, p. 9, 1966. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 54 /C1/8, 1952. Sloane, N. J. A. Sequences A008849, A008850, A048948, and A048949 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Fermat’s Factorization Method Given a number n, look for INTEGERS x and y such that n /C30x2 /C28y2 : Then n /C30(x /C28y)(x /C27y) (1) and n is factored. Any ODD NUMBER can be repre- sented in this form since then n /C30 ab, a and b are ODD, and a /C30x /C27y (2) b /C30x /C28y: (3) Adding and subtracting, a /C27b /C302x (4) a /C28b /C302y; (5) so solving for x and y gives x /C301 2(a /C27b) (6) y /C301 2 (a /C28b): (7) Therefore, x2 /C28y2 /C301 4a /C27b ðÞ2/C28 a /C28b ðÞ2hi /C30ab : (8) As the first trial for x, try x1ffiffiffinpde ; where xdeis the CEILING FUNCTION . Then check if Dx1 /C30x2 1 /C28n (9) is a SQUARE NUMBER . There are only 22 combinations of the last two digits which a SQUARE NUMBER can assume, so most combinations can be eliminated. If Dx1 is not a SQUARE NUMBER , then try x2 /C30x1 /C271 ; (10)so Dx2 /C30x22 /C28n /C30 x1 /C271 ðÞ2/C28n /C30x21 /C272x1 /C271 /C28n /C30Dx1 /C272x1 /C271: (11) Continue with Dx3 /C30x23 /C28n /C30 x2 /C271 ðÞ2/C28n /C30x22 /C272x2 /C271 /C28n /C30Dx2 /C272x2 /C271 /C30Dx2 þ 2x1 þ 3; (12) so subsequent differences are obtained simply by adding two. Maurice Kraitchik sped up the ALGORITHM by looking for x and y satisfying x2 /C13y2(mod n); (13) i.e., n½(x2 /C28y2) : This congruence has uninteresting solutions x /C139y(mod n) and interesting solutions /x f9y(mod n) : It turns out that if n is ODD and DIVISIBLE by at least two different PRIMES , then at least half of the solutions to x2 /C13y2(mod n) with xy COPRIME to n are interesting. For such solutions, (n, x/C28y) is neither n nor 1 and is therefore a nontrivial factor of n (Pomerance 1996). This ALGO- RITHM can be used to prove primality, but is not practical. In 1931, Lehmer and Powers discovered how to search for such pairs using CONTINUED FRACTIONS . This method was improved by Morrison and Brillhart (1975) into the CONTINUED FRACTION FACTORIZATION ALGORITHM , which was the fastest ALGORITHM in use before the QUADRATIC SIEVE factor- ization method was developed. See also PRIME FACTORIZATION ALGORITHMS ,SMOOTH NUMBER References Lehmer, D. H. and Powers, R. E. "On Factoring Large Numbers." Bull. Amer. Math. Soc. 37, 770/C1/76, 1931. McKee, J. "Speeding Fermat’s Factoring Method." Math. Comput. 68, 1729/C1/738, 1999. Morrison, M. A. and Brillhart, J. "A Method of Factoring and the Factorization of F7:/"Math. Comput. 29, 183/C1/05, 1975. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473/C1/485, 1996. Fermat’s Last Theorem A theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original isnow lost. However, a copy was preserved in a book published by Fermat’s son. In the note, Fermat claimed to have discovered a proof that the D IOPHAN- TINE EQUATION xn/C27yn/C30znhas no INTEGER solutions forn/C212. The full text of Fermat’s statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato- quadratum in duos quadrato-quadratos, et generali- ter nullam in infinitum ultra quadratum potestatemin duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc mar- ginis exiguitas non caperet" (Nagell 1951, p. 252). Intranslation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is apower greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin istoo narrow to contain." As a result of Fermat’s marginal note, the proposition that the D IOPHANTINE EQUATION xn/C27yn/C30zn; (1) where x,y,z, and nare INTEGERS , has no NONZERO solutions for n/C212 has come to be known as Fermat’s Last Theorem. It was called a " THEOREM " on the strength of Fermat’s statement, despite the fact thatno other mathematician was able to prove it forhundreds of years. Note that the restriction n/C212 is obviously necessary since there are a number of elementary formulas for generating an infinite number of P YTHAGOREAN TRIPLES (x;y;z) satisfying the equation for n/C302, x2/C27y2/C30z2: (2) A first attempt to solve the equation can be made byattempting to factor the equation, giving z n=2/C27yn=2/C0/C1 zn=2/C28yn=2/C0/C1 /C30xn: (3) Since the product is an exact POWER , zn=2/C27yn=2/C302n/C281pn zn=2/C28yn=2/C302qn orzn=2/C27yn=2/C302pn zn=2/C28yn=2/C302n/C281qn:/C27 /C27 (4) Solving for yandzgives zn=2/C302n/C282pn/C27qn yn=2/C302n/C282pn/C28qnorzn=2/C30pn/C272n/C282qn yn=2/C30pn/C282n/C282qn;/C27 /C27 (5) which give z/C302n/C282pn/C27qnðÞ2=n y/C302n/C282pn/C28qnðÞ2=norz/C30pn/C272n/C282qnðÞ2=n y/C30pn/C282n/C282qnðÞ2=n:( ( (6) However, since solutions to these equations in RA- TIONAL NUMBERS are no easier to find than solutions to the original equation, this approach unfortunately does not provide any additional insight. It is sufficient to prove Fermat’s Last Theorem by considering PRIME POWERS only, since the arguments can otherwise be written xmðÞp/C27ymðÞp/C30zmðÞp; (7)so redefining the arguments gives zp/C27yp/C30zp: (8) The so-called "first case" of the theorem is for exponents which are RELATIVELY PRIME tox,y, and z(p¶x;y;z) and was considered by Wieferich. Sophie Germain proved the first case of Fermat’s Last Theorem for any ODD PRIME pwhen 2 p/C271 is also a PRIME . Legendre subsequently proved that if pis a PRIME such that 4 p/C271;8p/C271;10p/C271;14p/C271;or 16p/C271 is also a PRIME , then the first case of Fermat’s Last Theorem holds for p. This established Fermat’s Last Theorem for pB100. In 1849, Kummer proved it for all REGULAR PRIMES and COMPOSITE NUMBERS of which they are factors (Vandiver 1929, Ball andCoxeter 1987). Kummer’s attack led to the theory of IDEALS , and Vandiver developed V ANDIVER’S CRITERIA for deciding if a given IRREGULAR PRIME satisfies the theorem. Genocchi (1852) proved that the first case is true for p if (p;p/C283) is not an IRREGULAR PAIR . In 1858, Kummer showed that the first case is true if either (p;p/C283) or ( p;p/C285) is an IRREGULAR PAIR , which was subsequently extended to include ( p;p/C287) and ( p;p/C28 9) by Mirimanoff (1905). Vandiver (1920ab) pointedout gaps and errors in Kummer’s memoir which, in his view, invalidate Kummer’s proof of Fermat’s LastTheorem for the irregular primes 37, 59, and 67, although he claims Mirimanoff’s proof of FLT for exponent 37 is still valid. Wieferich (1909) proved that if the equation is solved in integers RELATIVELY PRIME to an ODD PRIME p, then 2p/C281/C131 mod p2/C0/C1 : (9) (Ball and Coxeter 1987). Such numbers are calledW IEFERICH PRIMES . Mirimanoff (1909) subsequently showed that 3p/C281/C131 mod p2/C0/C1 (10) must also hold for solutions RELATIVELY PRIME to an ODD PRIME p, which excludes the first two W IEFERICH PRIMES 1093 and 3511. Vandiver (1914) showed 5p/C281/C131 mod p2/C0/C1 ; (11) and Frobenius extended this to 11p/C281;17p/C281/C131 mod p2/C0/C1 : (12) It has also been shown that if pwere a PRIME OF THE FORM 6x/C281;then 7p/C281;13p/C281;19p/C281/C131 mod p2/C0/C1 ; (13) which raised the smallest possible pin the "first case" to 253,747,889 by 1941 (Rosser 1941). Granville andMonagan (1988) showed if there exists a PRIME p satisfying Fermat’s Last Theorem, then qp /C281 /C131 mod p2/C0/C1 (14) for q /C30 5, 7, 11, ..., 71. This establishes that the first case is true for all PRIME exponents up to 714,591,416,091,398 (Vardi 1991). The "second case" of Fermat’s Last Theorem (for p ½x;y ;z) proved harder than the first case. Euler proved the general case of the theorem for n /C303, Fermat n /C304, Dirichlet and Lagrange n /C305. In 1832, Dirichlet established the case n /C3014. The n /C307 case was proved by Lame ´ (1839; Wells 1986, p. 70), using the identity X /C27Y /C27Z ðÞ7/C28 X7 /C27Y7 /C27Z7/C0/C1 /C307 X /C27Y ðÞ X /C27Z ðÞ Y /C27Z ðÞ /C2 X2 /C27Y2 /C27Z2 /C27XY /C27XZ /C27YZ/C0/C12/C27XYZ X /C27Y /C27Z ðÞhi : (15) Although some errors were present in this proof, these were subsequently fixed by Lebesgue (1840). Much additional progress was made over the next 150 years, but no completely general result had been obtained. Buoyed by false confidence after his proof that PI is TRANSCENDENTAL , the mathematician Lin- demann proceeded to publish several proofs of Fer- mat’s Last Theorem, all of them invalid (Bell 1937, pp. 464 /C1/65). A prize of 100,000 German marks, known as the Wolfskehl Prize, was also offered for the first valid proof (Ball and Coxeter 1987, p. 72; Barner 1997; Hoffman 1998, pp. 193 /C1/94 and 199). A recent false alarm for a general proof was raised by Y. Miyaoka (Cipra 1988) whose proof, however, turned out to be flawed. Other attempted proofs among both professional and amateur mathemati- cians are discussed by vos Savant (1993), although vos Savant erroneously claims that work on the problem by Wiles (discussed below) is invalid. By the time 1993 rolled around, the general case of Fermat’s Last Theorem had been shown to be true for all exponents up to 4 /C29106 (Cipra 1993). However, given that a proof of Fermat’s Last Theorem requires truth for all exponents, proof for any finite number of exponents does not constitute any significant pro- gress towards a proof of the general theorem (although the fact that no counterexamples were found for this many cases is highly suggestive). In 1993, a bombshell was dropped. In that year, the general theorem was partially proven by Andrew Wiles (Cipra 1993, Stewart 1993) by proving the SEMISTABLE case of the TANIYAMA- SHIMURA CONJEC- TURE . Unfortunately, several holes were discovered in the proof shortly thereafter when Wiles’ approach via the TANIYAMA- SHIMURA CONJECTURE became hung up on properties of the SELMER GROUP using a tool called an EULER SYSTEM . However, the difficulty was cir- cumvented by Wiles and R. Taylor in late 1994 (Cipra 1994, 1995ab) and published in Taylor and Wiles (1995) and Wiles (1995). Wiles’ proof succeeds by (1) replacing ELLIPTIC CURVES with Galois representa-tions, (2) reducing the problem to a CLASS NUMBER FORMULA , (3) proving that FORMULA , and (4) tying up loose ends that arise because the formalisms fail in the simplest degenerate cases (Cipra 1995a). The proof of Fermat’s Last Theorem marks the end of a mathematical era. Since virtually all of the tools which were eventually brought to bear on the problem had yet to be invented in the time of Fermat, it is interesting to speculate about whether he actually was in possession of an elementary proof of the theorem. Judging by the temerity with which the problem resisted attack for so long, Fermat’s alleged proof seems likely to have been illusionary. This conclusion is further supported by the fact that Fermat searched for proofs for the cases n /C304 and n/C305, which would have been superfluous had he actually been in possession of a general proof. See also ABC CONJECTURE ,B EAL’S CONJECTURE , BOGOMOLOV- MIYAOKA- YAU INEQUALITY ,EULER SYS- TEM,F ERMAT- CATALAN CONJECTURE ,G ENERALIZED FERMAT EQUATION ,M ORDELL CONJECTURE ,PYTHA- GOREAN TRIPLE ,RIBET’S THEOREM ,SELMER GROUP , SOPHIE GERMAIN PRIME ,SZPIRO’S CONJECTURE ,TA- NIYAMA- SHIMURA CONJECTURE ,VOJTA’S CONJECTURE , WARING FORMULA References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 69 /C1/3, 1987. Barner, K. "Paul Wolfskehl and the Wolfskehl Prize." Not. Amer. Math. Soc. 44, 1294 /C1/303, 1997. Beiler, A. H. "The Stone Wall." Ch. 24 in Recreations in the Theory of Numbers: The Queen of Mathematics Enter- tains. New York: Dover, 1966. Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1937. Bell, E. T. The Last Problem. New York: Simon and Schuster, 1961. Cipra, B. A. "Fermat Theorem Proved." Science 239, 1373, 1988. Cipra, B. A. "Mathematics--Fermat’s Last Theorem Finally Yields." Science 261,3 2/C1/3, 1993. Cipra, B. A. "Is the Fix in on Fermat’s Last Theorem?" Science 266, 725, 1994. Cipra, B. A. "Fermat’s Theorem--At Last." What’s Happen- ing in the Mathematical Sciences, 1995 /C1/996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 2 /C1/4, 1996. Cipra, B. A. "Princeton Mathematician Looks Back on Fermat Proof." Science 268, 1133/C1/134, 1995b. Courant, R. and Robbins, H. "Pythagorean Numbers and Fermat’s Last Theorem." §2.3 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideasand Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 40 /C1 /2, 1996. Cox, D. A. "Introduction to Fermat’s Last Theorem." Amer. Math. Monthly 101,3/C1/4, 1994. Darmon, H. and Merel, L. "Winding Quotients and Some Variants of Fermat’s Last Theorem." J. reine angew. Math. 490,8 1/C1/00, 1997. Dickson, L. E. "Fermat’s Last Theorem, axr/C27bys/C30czt;and the Congruence xn/C27yn/C13zn(mod p)." Ch. 26 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 731 /C1/76, 1952. Edwards, H. M. Fermat’s Last Theorem: A Genetic Introduc- tion to Algebraic Number Theory. New York: Springer- Verlag, 1977. Edwards, H. M. "Fermat’s Last Theorem." Sci. Amer. 239, 104/C1/22, Oct. 1978. Granville, A. "Review of BBC’s Horizon Program, ‘Fermat’s Last Theorem’." Not. Amer. Math. Soc. 44,2 6/C1/8, 1997. Granville, A. and Monagan, M. B. "The First Case of Fermat’s Last Theorem is True for All Prime Exponents up to 714,591,416,091,389." Trans. Amer. Math. Soc. 306, 329/C1/59, 1988. Guy, R. K. "The Fermat Problem." §D2 in Unsolved Pro- blems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 144 /C1/46, 1994. Hanson, A. "Fermat Project." http://www.cica.indiana.edu/ projects/Fermat/. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.New York: Hyperion, pp. 183 /C1 /99, 1998. Kolata, G. "Andrew Wiles: A Math Whiz Battles 350-Year- Old Puzzle." New York Times , June 29, 1993. Lynch, J. "Fermat’s Last Theorem." BBC Horizon television documentary. http://www.bbc.co.uk/horizon/fermat.shtml. Lynch, J. (Producer and Writer). "The Proof." NOVA televi- sion episode. 52 mins. Broadcast by the U. S. PublicBroadcasting System on Oct. 28, 1997. Mirimanoff, D. "Sur le dernier the ´ore`me de Fermat et le crite´rium de Wiefer." Enseignement Math. 11, 455/C1 /59, 1909. Mordell, L. J. Fermat’s Last Theorem. New York: Chelsea, 1956. Murty, V. K. (Ed.). Fermat’s Last Theorem: Proceedings of the Fields Institute for Research in Mathematical Scienceson Fermat’s Last Theorem, Held 1993 /C1 /994 Toronto, Ontario, Canada. Providence, RI: Amer. Math. Soc., 1995. Nagell, T. "Fermat’s Last Theorem." §68 in Introduction to Number Theory. New York: Wiley, pp. 251 /C1/53, 1951. Osserman, R. (Ed.). Fermat’s Last Theorem. The Theorem and Its Proof: An Exploration of Issues and Ideas. 98 min. videotape and 56 pp. book. 1994. Ribenboim, P. 13 Lectures on Fermat’s Last Theorem. New York: Springer-Verlag, 1979. Ribenboim, P. Fermat’s Last Theorem for Amateurs. New York: Springer-Verlag, 1999. Ribet, K. A. and Hayes, B. "Fermat’s Last Theorem and Modern Arithmetic." Amer. Sci. 82, 144/C1/56, March/April 1994. Ribet, K. A. and Hayes, B. Correction to "Fermat’s Last Theorem and Modern Arithmetic." Amer. Sci. 82, 205, May/June 1994. Rosser, B. "On the First Case of Fermat’s Last Theorem." Bull. Amer. Math. Soc. 45, 636/C1/40, 1939. Rosser, B. "A New Lower Bound for the Exponent in the First Case of Fermat’s Last Theorem." Bull. Amer. Math. Soc. 46, 299/C1/04, 1940. Rosser, B. "An Additional Criterion for the First Case of Fermat’s Last Theorem." Bull. Amer. Math. Soc. 47, 109/C1/ 10, 1941. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 144 /C1/49, 1993. Singh, S. Fermat’s Enigma: The Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker & Co., 1997. Stewart, I. "Fermat’s Last Time-Trip." Sci. Amer. 269, 112/C1/ 15, 1993. Swinnerton-Dwyer, P. Nature 364,1 3/C1/4, 1993. Taylor, R. and Wiles, A. "Ring-Theoretic Properties of Certain Hecke Algebras." Ann. Math. 141, 553/C1/72, 1995. van der Poorten, A. Notes on Fermat’s Last Theorem. New York: Wiley, 1996.Vandiver, H. S. "On Kummer’s Memoir of 1857 Concerning Fermat’s Last Theorem." Proc. Nat. Acad. Sci. 6, 266/C1/69, 1920a. Vandiver, H. S. "On the Class Number of the Field Ve2ip=pn/C0/C1 and the Second Case of Fermat’s Last Theorem." Proc. Nat. Acad. Sci. 6, 416/C1/21, 1920b. Vandiver, H. S. "On Fermat’s Last Theorem." Trans. Amer. Math. Soc. 31, 613/C1/42, 1929. Vandiver, H. S. Fermat’s Last Theorem and Related Topics in Number Theory. Ann Arbor, MI: 1935. Vandiver, H. S. "Fermat’s Last Theorem: Its History and the Nature of the Known Results Concerning It." Amer. Math. Monthly, 53, 555/C1/78, 1946. Vandiver, H. S. "A Supplementary Note to a 1946 Article on Fermat’s Last Theorem." Amer. Math. Monthly 60, 164/C1/ 67, 1953. Vandiver, H. S. "Examination of Methods of Attack on the Second Case of Fermat’s Last Theorem." Proc. Nat. Acad. Sci. 40, 732/C1/35, 1954. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 59 /C1/1, 1991. vos Savant, M. The World’s Most Famous Math Problem. New York: St. Martin’s Press, 1993. Weisstein, E. W. "Books about Fermat’s Last Theorem." http://www.treasure-troves.com/books/FermatsLastTheor- em.html. Wieferich, A. "Zum letzten Fermat’schen Theorem." J. reine angew. Math. 136, 293/C1/02, 1909. Wiles, A. "Modular Elliptic-Curves and Fermat’s Last Theorem." Ann. Math. 141, 443/C1/51, 1995. Fermat’s Lesser Theorem FERMAT’S LITTLE THEOREM Fermat’s Little Theorem Ifpis a PRIME NUMBER and aaNATURAL NUMBER , then ap/C13amod p ðÞ : (1) Furthermore, if p¶a(pdoes not divide a), then there exists some smallest exponent dsuch that ad/C281/C130 mod p ðÞ (2) andddivides p/C281:Hence, ap/C281/C281/C130 mod p ðÞ : (3) This is a generalization of the C HINESE HYPOTHESIS and a special case of E ULER’S THEOREM . It is some- times called F ERMAT’S PRIMALITY TEST and is a NECESSARY but not SUFFICIENT test for primality. Although it was presumably proved (but suppressed) by Fermat, the first proof was published by Euler in1749. The theorem is easily proved using mathematical INDUCTION . Suppose p½ap/C28a:Then examine a/C271 ðÞp/C28a/C271 ðÞ : (4) From the BINOMIAL THEOREM , a /C271 ðÞp /C30ap /C27p 1/C1Y/C1Q ap /C281 /C27p 2/C1Y/C1Q ap /C282 /C27/C1/C1/C1/C27p p /C281/C1Y/C1Q a /C271 : (5) Rewriting, a /C271 ðÞp/C28ap /C281 /C30p 1/C1Y/C1Q ap /C281 /C27p 2/C1Y/C1Q ap /C282 /C27:::/C27p p /C281/C1Y/C1Q a : (6) But p divides the right side, so it also divides the left side. Combining with the induction hypothesis gives that p divides the sum a /C271 ðÞp/C28ap /C281 ½/C138 /C27 ap /C28a ðÞ /C30 a /C271 ðÞp/C28 a /C271 ðÞ ; (7) as assumed, so the hypothesis is true for any a. The theorem is sometimes called FERMAT’S SIMPLE THEO- REM.W ILSON’S THEOREM follows as a COROLLARY of Fermat’s little theorem. Fermat’s little theorem shows that, if p is PRIME , there does not exist a base a Bp with (a; p) /C301 such that ap /C281 /C281 possesses a nonzero residue modulo p.If such base a exists, p is therefore guaranteed to be composite. However, the lack of a nonzero residue in Fermat’s little theorem does not guarantee that p is PRIME . The property of unambiguously certifying composite numbers while passing some PRIMES make Fermat’s little theorem a COMPOSITENESS TEST which is sometimes called the FERMAT COMPOSITE- NESS TEST . A number satisfying Fermat’s little theo- rem for some nontrivial base and which is not known to be composite is called a PROBABLE PRIME . COMPOSITE NUMBERS known as FERMAT PSEUDO- PRIMES (or sometimes simply "PSEUDOPRIMES ") have zero residue for some as and so are not identified as composite. Worse still, there exist numbers known as CARMICHAEL NUMBERS (the smallest of which is 561) which give zero residue for any choice of the base a RELATIVELY PRIME to p. However, FERMAT’S LITTLE THEOREM CONVERSE provides a criterion for certifying the primality of a number. A table of the smallest PSEUDOPRIMES P for the first 100 bases a follows (Sloane’s A007535; Beiler 1966, p. 42 with typos corrected). aPaPaPaPaP 2 341 22 69 42 205 62 63 82 91 3 91 23 33 43 77 63 341 83 105 41 52 42 54 44 5646 58 48 5 5 124 25 28 45 76 65 112 85 129 6 35 26 27 46 133 66 91 86 8772 52 76 54 76 5678 58 79 1 8 9 28 45 48 49 68 69 88 91 92 8 2 93 5 4 96 6 6 98 58 99 9 10 33 30 49 50 51 70 169 90 91 11 15 31 49 51 65 71 105 91 115 12 65 32 33 52 85 72 85 92 9313 21 33 85 53 65 73 111 93 30114 15 34 35 54 55 74 75 94 95 15 341 35 51 55 63 75 91 95 141 16 51 36 91 56 57 76 77 96 13317 45 37 45 57 65 77 247 97 10518 25 38 39 58 133 78 341 98 99 19 45 39 95 59 87 79 91 99 145 20 21 40 91 60 341 80 81 100 15321 55 41 105 61 91 81 85 See also B INOMIAL THEOREM ,CARMICHAEL NUMBER , CHINESE HYPOTHESIS ,COMPOSITE NUMBER ,COMPO- SITENESS TEST,EULER’S THEOREM ,FERMAT’S LITTLE THEOREM CONVERSE ,FERMAT PSEUDOPRIME ,M ODU- LO MULTIPLICATION GROUP ,P RATT CERTIFICATE , PRIMALITY TEST,P RIME NUMBER ,P SEUDOPRIME , RELATIVELY PRIME ,TOTIENT FUNCTION ,W IEFERICH PRIME ,W ILSON’S THEOREM ,W ITNESS References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 61, 1987. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 141 /C1/42, 1996. Courant, R. and Robbins, H. "Fermat’s Theorem." §2.2 in Supplement to Ch. 1 in What is Mathematics?: An Ele- mentary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 37 /C1/8, 1996. Nagell, T. "Fermat’s Theorem and Its Generalization by Euler." §21 in Introduction to Number Theory. New York: Wiley, pp. 71 /C1/3, 1951. Se´roul, R. "The Theorems of Fermat and Euler." §2.8 in Programming for Mathematicians. Berlin: Springer-Ver- lag, p. 15, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 20, 1993. Sloane, N. J. A. Sequences A007535/M5440 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Fermat’s Little Theorem Converse The converse of F ERMAT’S LITTLE THEOREM is also known as L EHMER’S THEOREM . It states that, if an INTEGER xisPRIME tomand xm/C281/C131 mod m ðÞ and there is no INTEGER e Bm /C281 for which xe /C13 1 mod m ðÞ ; then m is PRIME . Here, x is called a WITNESS to the primality of m. This theorem is the basis for the PRATT PRIMALITY CERTIFICATE . See also FERMAT’S LITTLE THEOREM ,PRATT CERTIFI- CATE ,PRIMALITY CERTIFICATE ,W ITNESS References Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, p. 96, 1994. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 278 /C1/79, 1991. Fermat’s Polygonal Number Theorem In 1638, Fermat proposed that every POSITIVE IN- TEGER is a sum of at most three TRIANGULAR NUM- BERS , four SQUARE NUMBERS , five PENTAGONAL NUMBERS , and nn -POLYGONAL NUMBERS . Fermat claimed to have a proof of this result, although Fermat’s proof has never been found. Gauss proved the triangular case, and noted the event in his diary on July 10, 1796, with the notation /C31/C31E Y RHKA num ¼DþDþD: This case is equivalent to the statement that every number OF THE FORM 8m /C273 is a sum of three ODD SQUARES (Duke 1997). More specifically, a number is a sum of three SQUARES IFF it is not OF THE FORM 4b 8m /C277 ðÞ for b ]0, as first proved by Legendre in 1798. Euler was unable to prove the square case of Fermat’s theorem, but he left partial results which were subsequently used by Lagrange. The square case was finally proved by Jacobi and independently by Lagrange in 1772. It is therefore sometimes known as LAGRANGE’S FOUR-SQUARE THEOREM . In 1813, Cauchy proved the proposition in its entirety. See also FIFTEEN THEOREM ,L AGRANGE’S FOUR- SQUARE THEOREM ,S UM OF SQUARES FUNCTION , VINOGRADOV’S THEOREM ,W ARING’S PROBLEM References Cassels, J. W. S. Rational Quadratic Forms. New York: Academic Press, 1978. Cauchy, A. "De´monstration du the´ore`me ge´ne´ral de Fermat sur les nombres polygones." In Oeuvres comple `tes d’Au- gustin Cauchy, Vol. VI (II Se´rie). Paris: Gauthier-Villars, pp. 320 /C1/53, 1905. Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane, N. J. A. "The Primary Pretenders." Acta Arith. 78, 307 /C1/ 13, 1997. Duke, W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer. Math. Soc. 44, 190 /C1/96, 1997. Nathanson, M. B. "A Short Proof of Cauchy’s Polygonal Number Theorem." Proc. Amer. Math. Soc. 9,22/C1/4, 1987. Savin, A. "Shape Numbers." Quantum 11,14/C1/8, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 143 /C1/44, 1993.Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 91, 1984. Fermat’s Primality Test FERMAT’S LITTLE THEOREM Fermat’s Principle of Conjunctive Probability The probability that two events will both happen is hk, where h is the probability that the first event will happen, and k is the probability that the second event will happen when the first even is known to have happened. See also CONDITIONAL PROBABILITY References Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 317, 1967. Fermat’s Problem In a given ACUTE TRIANGLE DABC ; locate a point whose distances from A, B, and C have the smallest possible sum. The solution is the point from which each side subtends an angle of 1208, known as the first FERMAT POINT . See also ACUTE TRIANGLE ,FERMAT POINTS Fermat’s Right Triangle Theorem The AREA of a RATIONAL RIGHT TRIANGLE cannot be a SQUARE NUMBER . This statement is equivalent to "a CONGRUUM cannot be a SQUARE NUMBER ." See also CONGRUUM ,R ATIONAL TRIANGLE ,R IGHT TRIANGLE ,SQUARE NUMBER Fermat’s Simple Theorem FERMAT’S LITTLE THEOREM Fermat’s Spiral An A RCHIMEDEAN SPIRAL with m/C302 having polar equation r /C30a u1 =2 ; discussed by Fermat in 1636 (MacTutor Archive). It is also known as the PARABOLIC SPIRAL . For any given POSITIVE value of u; there are two corresponding values of r of opposite signs. The resulting spiral is therefore symmetrical about the origin. The CURVA- TURE is kuðÞ/C303a2 4u/C27 a2 u a2 4u/C27 a2 u !3 =2 : See also ARCHIMEDEAN SPIRAL ,F ERMAT’S SPIRAL INVERSE CURVE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987. Dixon, R. "The Mathematics and Computer Graphics of Spirals in Plants." Leonardo 16,86/C1/0, 1983. Dixon, R. Mathographics. New York: Dover, p. 121, 1991. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 90 and 96, 1997. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967. MacTutor History of Mathematics Archive. "Fermat’s Spiral." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Fermats.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 74 /C1/5, 1991. Fermat’s Spiral Inverse Curve The INVERSE CURVE of FERMAT’S SPIRAL with the origin taken as the INVERSION CENTER is the LITUUS . References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 /C1/87, 1972. Fermat’s Theorem A PRIME p can be represented in an essentially unique manner in the form x2 /C27y2for integral x and y IFF p /C131 mod 4 ðÞ or p /C30 2. It can be restated by letting Qx;yðÞ/C13x2 /C27y2 ; then all RELATIVELY PRIME solutions (x, y) to the problem of representing Qx;yðÞ/C30m for m any IN- TEGER are achieved by means of successive applica- tions of the GENUS THEOREM and COMPOSITION THEOREM . There is an analog of this theorem for EISENSTEIN INTEGERS .See also EISENSTEIN INTEGER ,SQUARE NUMBER References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 142 /C1/43, 1993. Fermat’s Two-Square Theorem FERMAT’S THEOREM Fermat-Catalan Conjecture The conjecture that there are only finitely many triples of RELATIVELY PRIME integer powers xp ; yq ; zr for which xp /C27yq /C30zr with 1 p /C271 q /C271 r B1: Darmon and Merel (1997) have shown that there are no relatively prime solutions (x; x;3) with x ]3 : Ten solutions are known, 1 /C2723 /C3032 25 /C2772 /C3034 73 /C27132 /C3029 27 /C27173 /C30712 35 /C27114 /C301222 177 /C27762713 /C30210639282 14143/C2722134592/C30657 92623/C27153122832/C301137 438/C27962223/C30300429072 338/C2715490342/C30156133 (Mauldin 1997). See also FERMAT’S LAST THEOREM References Darmon, H. and Granville, A. "On the Equations zm/C30F(x;y) andAxp/C27Byq/C30Czr:/"Bull. London Math. Soc. 27, 513/C1/43, 1995. Darmon, H. and Merel, L. "Winding Quotients and Some Variants of Fermat’s Last Theorem." J. reine angew. Math. 490,8 1/C1/00, 1997. Mauldin, R. D. "A Generalization of Fermat’s Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer. Math. Soc. 44, 1436/C1/437, 1997. Fermat-Euler Theorem FERMAT’S LITTLE THEOREM Fermatian POULET NUMBER Fermat-Lucas Number A number OF THE FORM 2n /C271 obtained by setting x /C30 1inaF ERMAT- LUCAS POLYNOMIAL . The first few are 3, 5, 9, 17, 33, ... (Sloane’s A000051). See also FERMAT NUMBER (LUCAS ) References Shorey, T. N. and Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers, 2." J. London Math. Soc. 23,17/C1/3, 1981. Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers." Proc. London Math. Soc. 35, 425 /C1/47, 1977. Sloane, N. J. A. Sequences A000051/M0717 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Fermi-Dirac Distribution A distribution which arises in the study of half- integral spin particles in physics, R kðÞ/C30ka ek /C28 m /C27 1 : Its integral is g/C12 0kadk ek /C28 m /C27 1 /C30e m G s /C271 ðÞ F/C28em ;s /C271 ;1 ðÞ ; where F z;s ;a ðÞ is the LERCH TRANSCENDENT . Fern BARNSLEY’S FERN Ferrari’s Identity a2 /C272ac /C282bc /C28b2/C0/C14/C27 b2 /C282ab /C282ac /C28c2/C0/C14 /C27 c2 /C272ab /C272bc /C28a2/C0/C14 /C302 a2 /C27b2 /C27c2 /C28ab /C27ac /C27bc/C0/C14: See also DIOPHANTINE EQUATION–4TH POWERS References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 96 /C1/7, 1994. Ferrars Diagram FERRERS DIAGRAM # 1999 /C1/001 Wolfram Research, Inc.Ferrers Diagram A Ferrers diagram represents PARTITIONS as patterns of dots, with the nth row having the same number of dots as the nth term in the PARTITION . The spelling "Ferrars" (Skiena 1990, pp. 53 and 78) is sometimes also used, and the diagram is sometimes called a graphical representation or Ferrers graph (Andrews 1998, p. 6). A Ferrers diagram of the PARTITION n /C30a /C27b /C27:::/C27c ; for a list a, b, ..., c of k POSITIVE INTEGERS with a ] b ]...]c is therefore the arrangement of n dots or square boxes in k rows, such that the dots or boxes are left-justified, the first row is of length a, the second row is of length b, and so on, with the kth row of length c. The above diagram corresponds to one of the possible partitions of 100. See also CONJUGATE PARTITION ,D URFEE SQUARE , SELF-CONJUGATE PARTITION ,YOUNG DIAGRAM References Andrews, G. E. The Theory of Partitions. Cambridge, Eng- land: Cambridge University Press, pp. 6 /C1/, 1998. Comtet, L. "Ferrers Diagrams." §2.4 in Advanced Combina- torics: The Art of Finite and Infinite Expansions, rev. enl. ed.Dordrecht, Netherlands: Reidel, pp. 98 /C1/02, 1974. Liu, C. L. Introduction to Combinatorial Mathematics. New York: McGraw-Hill, 1968. MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 3 /C1/, 1960. Propp, J. "Some Variants of Ferrers Diagrams." J. Combin. Th. A 52,9 8/C1/28, 1989. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, pp. 108 /C1/09, 1980. Skiena, S. "Ferrers Diagrams." §2.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 53 /C1/5, 1990. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, 1999. Stanton, D. and White, D. Constructive Combinatorics. New York: Springer-Verlag, 1986. Ferrers Graph FERRERS DIAGRAM #1999/C1/001 Wolfram Research, Inc. Ferrers Graph Polygon A SELF-AVOIDING POLYGON containing three corners of its minimal bounding rectangle. The anisotropic area and perimeter generating function Gx;yðÞ and partial generating functions HmyðÞ; connected by G(x;y;q) /C30X m]1Hmy;qðÞ xm ; satisfy the self-reciprocity and inversion relations Hm(1=y;1=q) /C30(/C281)mym/C282q(m3/C283m)=2Hm(y;q) and G(x;y) /C28y2G(/C28x=y;1 =y) /C300 (Bousquet-Me ´lou et al. 1999). See also LATTICE POLYGON ,SELF-AVOIDING POLYGON References Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/ 9908123/. # 1999 /C1/001 Wolfram Research, Inc. Ferrers’ Function An alternative name for an associated LEGENDRE POLYNOMIAL . See also LEGENDRE POLYNOMIAL References Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 246, 1991. Ferrier’s Prime According to Hardy and Wright (1979), the largest PRIME found before the days of electronic computers is the 44-digit number F /C131 17(2148 /C271) /C3020988936657440586486151264256610222593863921 ;which was found using only a mechanical calculator. Mathematica can verify primality of this number in a (small) fraction of a second, showing how far the art of numerical computation has advanced in the inter- vening years, In[1]: /C30 PrimeQ[(2^148 /C27 1)/17] // Timing Out[1] /C30 {0.0333333 Second, True} See also PRIME NUMBER References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 16 /C1/2, 1979. Feuerbach Circle NINE-POINT CIRCLE Feuerbach Point The point F at which the INCIRCLE and NINE-POINT CIRCLE are tangent. It has TRIANGLE CENTER FUNC- TION a/C301/C28cosB/C28C ðÞ : See also FEUERBACH’S THEOREM References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 200, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Salmon, G. Conic Sections, 6th ed. New York: Chelsea, p. 127, 1960. Feuerbach’s Conic Theorem The LOCUS of the centers of all CONICS through the VERTICES and ORTHOCENTER of a TRIANGLE (which are RECTANGULAR HYPERBOLAS when not degenerate), is a CIRCLE through the MIDPOINTS of the sides, the points half way from the ORTHOCENTER to the VERTICES , and the feet of the ALTITUDE . See also ALTITUDE ,C ONIC SECTION ,F EUERBACH’S THEOREM ,KIEPERT’S HYPERBOLA ,MIDPOINT ,ORTHO- CENTER ,RECTANGULAR HYPERBOLA References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 198, 1959. Feuerbach’s Theorem There are two theorems commonly known as Feuer- bach’s theorem. The first states that CIRCLE which passes through the feet of the PERPENDICULARS dropped from the VERTICES of any TRIANGLE on the sides opposite them passes also through the MID- POINTS of these sides as well as through the MIDPOINT of the segments which join the VERTICES to the point of intersection of the PERPENDICULAR . Such a circle is called a NINE-POINT CIRCLE . The proposition most frequently called Feuerbach’s theorem states that the NINE-POINT CIRCLE of any TRIANGLE is TANGENT internally to the INCIRCLE and TANGENT externally to the three EXCIRCLES . This theorem was first published by Feuerbach (1822). Many proofs have been given (Elder 1960), with the simplest being the one presented by McClelland (1891, p. 225) and Lachlan (1893, p. 74). See also EXCIRCLE ,FEUERBACH POINT ,HART CIRCLE , INCIRCLE ,M IDPOINT ,NINE-POINT CIRCLE ,PERPENDI- CULAR ,TANGENT References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 107, 273, and 290, 1952.Baker, H. F. Appendix to Ch. 12 in An Introduction to Plane Geometry. Cambridge, England: Cambridge University Press, 1943. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 39, 1971. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 117 /C1/19, 1967. Dixon, R. Mathographics. New York: Dover, p. 59, 1991. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 117, 1928. Elder, A. E. "Feuerbach’s Theorem: A New Proof." Amer. Math. Monthly 67, 905 /C1/06, 1960. F. Gabriel-Marie. Exercices de ge´ome´trie. Tours, France: Maison Mame, pp. 595 /C1/97, 1912. Feuerbach, K. Eigenschaften einiger merkwu ¨rdigen Punkte des geradlinigen Dreiecks und weiterer durch sie bestimm- ten Linien und Figuren. Nu¨rnberg, Germany: 1822. Kroll, W. "Elementarer Beweis des Satzes von Feuerbach." Praxis der Math. 40, 251 /C1/54, 1998. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillan, 1893. McClelland, W. J. Geometry of the Circle. London, 1891. Rouche ´, E. and de Comberousse, C. Traite ´ de ge´ome´trie plane. Paris: Gauthier-Villars, pp. 307 /C1/09, 1900. Sawayama, Y. "De´monstration e´le´mentaire du the´ore`me de Feuerbach." L’enseign. math. 7, 479 /C1/82, 1905. Sawayama, Y. "8 nouvelles de´monstrations d’un the´ore`me relatif au cercle des 9 points." L’enseign. math. 13,31/C1/9, 1911. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 76 /C1/7, 1991. Feynman Point The sequence of six 9s which begins at the 762nd decimal place of PI, p /C303:14159...134 999999|fflfflfflffl{zfflfflfflffl} six 9s837... (Wells 1986, p. 51). The positions of the first occur- rences of strings of 1, 2, ... consecutive 9s are 5, 44, 762, 762, 762, 762, 1722776, ... (Sloane’s A048940). There is no string of seven 9s in the first million digits ofPI. See also PI DIGITS References Sloane, N. J. A. Sequences A048940 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 51, 1986. FFT FASTFOURIER TRANSFORM Fiber A fiber of a map f:X0Yis the PREIMAGE of an element y/C23Y:That is, f /C281(y) /C30 x /C23 X such that f(x) /C30y fg : For instance, let X and Y be the COMPLEX NUMBERS C: When f(z) /C30z2 ; every fiber consists of two points z;/C28z fg ; except for the fiber over 0 ; which has one point. Note that a fiber may be the EMPTY SET. In special cases, the fiber may be independent, in some sense, of the choice of y /C23 Y : For instance, if f is a COVERING MAP, then the fibers are all DISCRETE and have the same CARDINALITY . The example f(z) /C30z2 is a covering map away from zero, i.e., f(z) /C30z2 from the punctured plane C /C28 0fgto itself has a fiber consist- ing of two points. When p : E 0 M is a FIBER BUNDLE , then every fiber is ISOMORPHIC , in whatever CATEGORY is being used. For instance, when E is a REAL VECTOR BUNDLE of RANK k, every fiber is isomorphic to Rk :/ See also COMPLEX NUMBER ,COVERING MAP,FIBER BUNDLE ,MAP,RANK (BUNDLE ), WHITNEY SUM Fiber Bundle A fiber bundle (also called simply a BUNDLE ) with FIBER F is a MAP f : E 0 B where E is called the TOTAL SPACE of the fiber bundle and B the BASE SPACE of the fiber bundle. The main condition for the MAP to be a fiber bundle is that every point in the BASE SPACE b /C23 B has a NEIGHBORHOOD U such that f /C281(U)is HOMEOMORPHIC to U /C29F in a special way. Namely, if h : f /C281(U) 0 U /C29F is the HOMEOMORPHISM , then projU(h /C30f f /C281(U) jj ; where the MAP projUmeans projection onto the U component. The homeomorphisms h which "commute with projection" are called local TRIVIALIZATIONS for the fiber bundle f. In other words, E looks like the product B /C29F (at least locally), except that the fibers f /C281(x) for x /C23 B may be a bit "twisted." A fiber bundle is the most general kind of BUNDLE . Special cases are often described by replacing the word "fiber" with a word that describes the fiber being used, e.g., VECTOR BUNDLES and PRINCIPAL BUNDLES .Examples of fiber bundles include any product B /C29 F 0 B (which is a bundle over B with FIBER F), the MO¨ BIUS STRIP (which is a fiber bundle over the CIRCLE with FIBER given by the unit interval [0,1]; i.e, the BASE SPACE is the CIRCLE ), and S3 (which is a bundle over S2 with fiber S1) : A special class of fiber bundle is the VECTOR BUNDLE , in which the FIBER is a VECTOR SPACE . A basic example of a nontrivial bundle is the MO¨ BIUS STRIP , which is a fiber bundle with the circle as its base, B /C30S /C281 ; and the interval F /C30(/C281;1) as its fiber. Some of the properties of graphs of functions f : B 0 F carry over to fiber bundles. A GRAPH of such a function sits in B /C29F as (b; f(b)) : A graph always projects ONTO the base B and is ONE-TO-ONE . A fiber bundle E is a TOTAL SPACE and, like B /C29F ; it has a projection p : E 0 B : The PREIMAGE , p/C281(b) ; of any point b is isomorphic to F. Unlike B /C29F ; there is no canonical projection from E to F. Instead, maps to F only make sense locally on B. Near any point b in the base B, there is a TRIVIALIZATION of E in which there are actual functions from a neighborhood to F. These local functions can sometimes be patched together to give a (GLOBAL ) SECTION s : B 0 E such that the projection of s is the identity. This is analogous to the map from a domain X of a function f : X 0 Y to its graph in X /C29Y by ˜f(x) /C30(x;f(x)):/ A fiber bundle also comes with a GROUP ACTION on the fiber. This group action represents the different ways the fiber can be viewed as equivalent. For instance, in topology, the GROUP might be the group of HOME- OMORPHISMS of the fiber. The group on a vector bundle is the group of INVERTIBLE LINEAR MAPS , which reflects the equivalent descriptions of a VECTOR SPACE using different BASES . Fiber bundles are not always used to generalize functions. Sometimes they are convenient descrip- tions of interesting manifolds. A common example in GEOMETRIC TOPOLOGY is a torus bundle on the circle. See also BUNDLE ,F IBER SPACE ,F IBRATION ,G EO- METRIC TOPOLOGY ,PRINCIPAL BUNDLE ,SHEAF ,TAN- GENT BUNDLE ,VECTOR BUNDLE Fiber Direct Sum See also DIRECT SUM # 1999 /C1/001 Wolfram Research, Inc. Fiber Space A fiber space, depending on context, means either a FIBER BUNDLE or a FIBRATION . See also FIBER BUNDLE ,FIBRATION Fibonacci FIBONACCI NUMBER ,FIBONACCI POLYNOMIAL # 1999 /C1/001 Wolfram Research, Inc. Fibonacci Coefficient The coefficient defined by m k/C20/C21 F/C30FmFm/C281 /C1/C1/C1Fm/C28k/C271 F1F2 /C1/C1/C1Fk; wherem 0/C2/C6 F/C301 and Fn is a FIBONACCI NUMBER . This coefficient satisfies 2n m/C20/C21 F/C30Lnm /C281 n/C20/C21 /C27Lm/C28nm /C281 n /C281/C20/C21 F; where Lnis a L UCAS NUMBER . See also FIBONACCI NUMBER ,LUCAS NUMBER #1999/C1/001 Wolfram Research, Inc. Fibonacci Dual Theorem Let Fnbe the nth F IBONACCI NUMBER . Then the sequence Fnfg/C12 n/C302/C301;2;3;5;8;... fg isCOMPLETE , even if one is restricted to subsequences in which no two consecutive terms are both passed over (until thedesired total is reached; Brown 1965, Honsberger 1985). See also C OMPLETE SEQUENCE ,FIBONACCI NUMBER . References Brown, J. L. Jr. "A New Characterization of the Fibonacci Numbers." Fib. Quart. 3,1/C1/, 1965. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 130, 1985. Fibonacci Hyperbolic Functions Let c/C131/C27f/C301 2(3/C27ffiffiffi 5p ):2:618034 (1) where fis the GOLDEN RATIO , and a¼lnf:0:4812118 : (2) Define the Fibonacci hyperbolic sine bysFh(x)/C13cx/C28c/C28x ffiffiffi 5p (3) /C30f2x/C28f/C282x ffiffiffi5p (4) /C302ffiffiffi5psinh[2 xa]: (5) The function satisfies sFh(/C28x)/C30/C28sFh(x); (6) and for n/C23Z;sFh(n)/C30F 2nwhere Fnis a F IBONACCI NUMBER . Define the Fibonacci hyperbolic cosine by cFh xðÞ/C13cx/C271=2/C27c/C28x/C271=2 ðÞ ffiffiffi5p (7) /C30f2x/C271 ðÞ/C27f/C282x/C271 ðÞ ffiffiffi5p (8) /C302ffiffiffi5pcosh 2 x/C271 ðÞ a ½Þ : (9) This function satisfies cFh(/C28x)/C30cFh(x/C281); (10) and for n/C23Z;cFh(n)/C30F 2n/C271where Fnis a F IBONACCI NUMBER . Similarly, the Fibonacci hyperbolic tangent is defined by sFh(x) /C13cFh(x) cFh(x) ; and for x /C23Z ; cFh(n) /C30F2n =F2n /C271 :/ References Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and Modified Numerical Triangles." Fib. Quart. 34, 129 /C1/38, 1996. # 1999 /C1/001 Wolfram Research, Inc. Fibonacci Identity Since a /C27ib ðÞ c /C27id ðÞ jj /C30a /C27ib jj c /C27di jj (1) j(ac /C28bd) /C27i(bc /C27ad) j/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C27d2p ; (2) it follows that (a2 /C27b2)(c2 /C27d2) /C30 ac /C28bd ðÞ2/C27 bc /C27ad ðÞ2/C13e2 /C27f2 : (3) This identity implies the 2-dimensional CAUCHY’S INEQUALITY . See also CAUCHY’S INEQUALITY ,EULER FOUR- SQUARE IDENTITY ,LEBESGUE IDENTITY References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 9, 1996. Fibonacci Matrix A SQUARE MATRIX related to the FIBONACCI NUMBERS . The simplest is the FIBONACCI Q-MATRIX . Fibonacci n-Step Number An n-step Fibonacci sequence is given by defining Fk /C300 for k 50; F1 /C30F2 /C301; F3 /C302; and Fk /C30Xk i /C301Fn /C28i (1) for k /C213. The case n /C301 corresponds to the degen- erate 1, 1, 2, 2, 2, 2 ..., n /C302 to the usual FIBONACCI NUMBERS 1, 1, 2, 3, 5, 8, ... (Sloane’s A000045), n /C303 to the TRIBONACCI NUMBERS 1, 1, 2, 4, 7, 13, 24, 44, 81, ... (Sloane’s A000073), n /C304 to the TETRANACCI NUMBERS 1, 1, 2, 4, 8, 15, 29, 56, 108, ... (Sloane’s A000078), etc. The limit limk 0/C12Fk =Fk/C281 is given by solving xn(2 /C28x) /C301; (2) or equivalentlyxn /C28xn/C281 /C28xn/C282 /C28/C1/C1/C1/C28x /C281 /C300; (3) for x and then taking the REAL ROOT x /C211. For EVEN n, there are exactly two real roots, one greater than 1 and one less than 1, and for ODD n, there is exactly one real root, which is always ]1:/ If n /C302, equation (2) reduces to x2(2 /C28x) /C301 (4) x3 /C282x2 /C271 /C30(x /C281) x2 /C28x /C281/C0/C1 /C300; (5) giving solutions x /C301 ;1 21 9ffiffiffi 5p/C17/C15 : (6) The ratio is therefore x /C301 21 /C27ffiffiffi 5p/C17/C15 /C30 f /C301 :618:::; (7) which is the GOLDEN RATIO , as expected. The analytic solutions for n/C301, 2, ... are given by x1/C301 x2/C301 21/C27ffiffiffi 5p/C17/C15 x3/C301 31/C2719/C283ffiffiffiffiffiffi 33p/C17/C151=3 /C2719/C273ffiffiffiffiffiffi33p/C17/C15 1=3/C20/C21 and numerically by 1, 1.61803, 1.83929, 1.92756, 1.96595, ..., approaching 2 as n0/C12:/ See also FIBONACCI NUMBER ,TRIBONACCI NUMBER References Sloane, N. J. A. Sequences A000045/M0692, A000073/ M1074, and A000078/M1108 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Fibonacci Number The sequence of numbers Fndefined by the Unin the LUCAS SEQUENCE , which can be viewed as a particu- lar case of the F IBONACCI POLYNOMIALS Fn(x) with Fn/C30Fn(1):They are companions to the L UCAS NUM- BERS and satisfy the same RECURRENCE RELATION , Fn/C13Fn/C282/C27Fn/C281 (1) forn/C303, 4, ..., with F1/C30F2/C301:The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, ... (Sloane’s A000045). The Fibonacci numbers give thenumber of pairs of rabbits nmonths after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old), as first described by Leonardo of Pisa in his bookLiber Abaci. Kepler also described the Fibonacci numbers (Kepler 1966; Wells 1986, pp. 61 /C1 /2 and 65). The ratios of successive Fibonacci numbers Fn=Fn/C281 approaches the GOLDEN RATIO fasnapproaches infinity, as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62). The ratiosof alternate Fibonacci numbers are given by the CONVERGENTS tof/C282;where fis the GOLDEN RATIO , and are said to measure the fraction of a turnbetween successive leaves on the stalk of a plant ( PHYLLOTAXIS ): 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). The Fibonacci numbers are sometimes called PINE CONE NUMBERS (Pappas 1989, p. 224). The role of the Fibonacci numbers in botanyis sometimes called L UDWIG’S LAW (Szymkiewicz 1928; Wells 1986, p. 66; Steinhaus 1983, p. 299). Another RECURRENCE RELATION for the Fibonacci numbers is Fn/C271/C30Fn1/C27ffiffiffi 5p/C0/C1 /C271 2$% /C30fFn/C271 2$% ; (2) where xbcis the FLOOR FUNCTION andfis the GOLDEN RATIO . This expression follows from the more general RECURRENCE RELATION that Fnþ1 Fnþ2 /C1/C1/C1 Fnþk Fnþkþ1 Fnþkþ2 /C1/C1/C1 Fnþ2k nn:::n Fnþkðk/C281Þþ1Fnþkðk/C281Þþ2/C1/C1/C1 Fnþk2/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12¼0: (3) The GENERATING FUNCTION for the Fibonacci num- bers is g(x)/C30X/C12 n/C300Fnxn/C30x 1/C28x/C28x2 /C30x/C27x2/C272x3/C273x4/C275x5/C27:::: (4) By plugging in x/C301=10;this gives the curious addition tree illustrated below, X/C12 n/C300Fn 10n/C3010 89; (5) so X/C12 n/C300Fn 10n/C271/C301 89(6) Yuri Matiyasevich (1970) showed that there is apolynomial Pinn,m, and a number of other variables x,y,z, ... having the property that n/C30 F2mIFFthere exist integers x,y,z, ... such that p(n;m;x;y;z;... )/C300:This led to the proof of the impossibility of the tenth of H ILBERT’S PROBLEMS (does there exist a general method for solving D IO- PHANTINE EQUATIONS ?) by Julia Robinson and Martin Davis in 1970 (Reid 1997, p. 107). The Fibonacci number Fn/C271gives the number of ways for 2/C291DOMINOES to cover a 2 /C29nCHECKERBOARD , as illustrated in the following diagrams (Dickau). The number of ways of picking a SET(including the EMPTY SET ) from the numbers 1, 2, ..., nwithout picking two consecutive numbers is Fn/C272:The number of ways of picking a set (including the EMPTY SET ) from the numbers 1, 2, ..., nwithout picking two consecutive numbers (where 1 and nare now con- secutive) is Ln/C30Fn/C271/C27Fn/C281;where Lnis a L UCAS NUMBER . The probability of not getting two heads in a row in ntosses of a COIN isFn/C272=2n(Honsberger 1985, pp. 120 /C1/22). Fibonacci numbers are also related to the number of ways in which nCOIN TOSSES can be made such that there are not three consecutive heads or tails. The number of ideals of an n-element FENCE POSET is the Fibonacci number Fn:/ Given a RESISTOR NETWORK ofn1-/Vresistors, each incrementally connected in series or parallel to thepreceding resistors, then the net resistance is a RATIONAL NUMBER having maximum possible denomi- nator of Fn/C271:/ The Fibonacci numbers are given in terms of theC HEBYSHEV POLYNOMIAL OF THE SECOND KIND by Fn/C30in/C281Un/C281/C281 2i ! : (7) Sum identities include Xn k/C301Fk/C30Fn/C272/C281: (8) F1/C27F3/C27F5/C27.../C27F2k/C271/C30F2k/C272 (9) 1/C27F2/C27F4/C27F6/C27.../C27F2k/C30F2k/C271 (10) Xn k/C301F2 k/C30FnFn/C271 (11) F2n/C30F2 n/C271/C28F2 n/C281 (12) F3n/C30F3 n/C271/C27F3 n/C27F3 n/C281: (13) There are a number of particular pretty algebraic identities involving the Fibonacci numbers, including F2 n/C271/C304FnFn/C281/C27F2 n/C282 (14) (Brousseau 1972), C ATALAN’S IDENTITY F2 n/C28Fn/C27rFn/C28r/C30/C28 1ðÞn/C28rF2 r; (15) D’OCAGNE’S IDENTITY FmFn/C271/C28FnFm/C271/C30/C28 1ðÞnFm/C28n; (16) and the G ELIN- CESA`RO IDENTITY F4 n/C28Fn/C282Fn/C281Fn/C271Fn/C272/C301: (17) Letting r/C301 in (15) gives C ASSINI’S IDENTITY Fn/C281Fn/C271/C28F2 n/C30/C28 1ðÞn; (18) sometimes also called Simson’s formula since it was also discovered by Simson (Coxeter and Greitzer1967, p. 41; Coxeter 1969, pp. 165 /C1 /68; Petkovsek et al.1996, p. 12). The Fibonacci numbers obey the negation formula F/C28n/C30/C28 1ðÞn/C271Fn; (19) the addition formula Fm/C27n/C301 2FmLn/C27LmFn ðÞ ; (20) where Lnis a L UCAS NUMBER , the subtraction formula Fm/C28n/C3012(/C281)F mLn/C28LmFn ðÞ ; (21) the fundamental identity L2 n/C285F2 n/C304/C281ðÞn(22) conjugation relation Fn/C281 5Ln/C281/C27Ln/C271/C0/C1 ; (23) successor relationFn/C271/C301 2Fn/C27Ln ðÞ ; (24) double-angle formula F2n/C30FnLn; (25) multiple-angle recurrence Fkn/C30LkFk(n/C281)/C28/C28 1ðÞkFk(n/C282); (26) multiple-angle formulas Fkn/C301 2k/C281X(k/C281)=2 bc i/C300k 2i/C271/C1Y/C1Q 5iF2i/C271 nLk/C281/C282i n (27) /C30FnX(k/C281)=2 bc i/C300k/C281/C28i i/C1Y/C1Q /C281ðÞi(n/C271)Lk/C281/C282i n (28) /C30LnP(k/C282)=2 i/C300k/C281/C28i i/C1Y/C1Q /C281ðÞin5k=2/C281/C28iFk/C281/C282i n for k even Pk=2bc i/C300k k/C28ik/C28i i/C1Y/C1Q /C281ðÞin5k=2bc/C28iFk/C282i n for k odd8 >>>< >>>: (29) /C30Xk i/C300k i/C1Y/C1Q FiFi nFk/C28i n/C281; (30) product expansions FmFn/C301 5Lm/C27n/C28/C28 1ðÞnLm/C28n/C2/C6 (31) and FmLn/C30Fm/C27n/C27/C28 1ðÞnFm/C28n; (32) square expansion, F2 n/C301 5L2n/C282/C281ðÞn½/C138 ; (33) and power expansion Fk n/C301 2:5/C28k=2/C29Xk i/C300k i/C1Y/C1Q /C281ðÞi(n/C271) /C29F(k/C282i)nfor k odd L(k/C282i)nfor k even :/C27 (34) Honsberger (1985, p. 107) gives the general relations Fn/C27m/C30Fn/C281Fm/C27FnFm/C271 (35) F(k/C271)n/C30Fn/C281Fkn/C27FnFkn/C271 (36) Fn/C30FlFn/C28l/C271/C27Fl/C281Fn/C28l: (37) In the case l/C30n/C28l/C271;then l/C30(n/C271)=2 and for n ODD, Fn/C30F2 (n/C271)=2/C27F2 ðn/C281Þ=2: (38) Similarly, for nEVEN , Fn/C30F2 n=2/C271/C28F2 n=2/C281: (39) Letting k/C13(n/C281)=2 gives the identities F2k/C271/C30F2 k/C271/C27F2 k (40) F2 n/C272/C28F2 n/C271/C30FnFn/C273 (41) F2 n/C30F2 n/C281/C273F2 n/C282/C272Fn/C282Fn/C283: (42) Sum FORMULAS forFninclude Fn/C301 2n/C281n 1/C1Y/C1Q /C275n 3/C1Y/C1Q /C2752n 5/C1Y/C1Q /C27.../C20/C21 (43) Fn/C271/C30n 0/C1Y/C1Q /C27n/C281 1/C1Y/C1Q /C27n/C282 2/C1Y/C1Q /C27. . . (44) (Wells 1986, p. 63). Additional identities can be found throughout the Fibonacci Quarterly journal. A list of 47 generalized identities are given by Halton (1965). In terms of the L UCAS NUMBER Ln; F2n/C30FnLn (45) F2nL2 2n/C281/C0/C1 /C30F6n (46) Fm/C27p/C27/C28 1ðÞp/C271Fm/C28p/C30FpLm (47) Xa/C274n k/C30a/C271Fk/C30Fa/C274n/C272/C28Fa/C272/C30F2nLa/C272n/C272 (48) (Honsberger 1985, pp. 111 /C1/13). A remarkable iden- tity is exp L1x/C271 2L2x2/C2713L 3x3/C27... ! /C30F1/C27F2x/C27F3x3/C27. . . (49) (Honsberger 1985, pp. 118 /C1/19). It is also true that L2 n/C28/C28 1ðÞaL2n/C27a F2 n/C28/C28 1ðÞaF2 n/C27a/C305 (50) foraODD, and L2 n/C27L2n/C27a/C288/C281ðÞn F2 n/C27F2 n/C27a/C305 (51) foraEVEN (Freitag 1996). The equation (1) is a LINEAR RECURRENCE SEQUENCE xn/C30Axx/C281/C27Bxn/C282n]3; (52) so the closed form for Fnis given by Fn/C30an/C28bn a/C28b; (53) where aandbare the roots of x2/C30Ax/C27B:Here, A/C30 B/C301;so the equation becomes x2/C28x/C281/C300; (54)which has ROOTS x/C301 219ffiffiffi 5p/C17/C15 : (55) The closed form is therefore given by Fn/C301/C27ffiffiffi 5p/C0/C1 n/C281/C28ffiffiffi5p/C0/C1 n 2nffiffiffi5p ; (56) This is known as B INET’S FIBONACCI NUMBER FOR- MULA (Wells 1986, p. 62). Another closed form is Fn/C301ffiffiffi5p 1/C27ffiffiffi5p 2 ! n "# /C30fn ffiffiffi5p"# ; (57) where x½/C138is the NINT function (Wells 1986, p. 62). From (1), the RATIO of consecutive terms is Fn Fn/C281/C301/C27Fn/C282 Fn/C281/C301/C271 Fn/C281 Fn/C282 /C301/C271 1/C271 Fn/C283 Fn/C282/C301;1;...;F2 F1"# /C301;1;...;1 ½/C138 ;|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} n/C281(58) which is just the first few terms of the CONTINUED FRACTION for the GOLDEN RATIO f:Therefore, lim n0/C12Fn Fn/C281/C30f: (59) The " SHALLOW DIAGONALS "o fP ASCAL’S TRIANGLE sum to Fibonacci numbers (Pappas 1989), Xn k/C301k n/C28k/C1Y/C1Q /C30/C281ðÞn 3F21;2;1/C28n;1 23/C28n ðÞ ;2/C2812n;/C2814 ! p2/C283n/C27n 2 ðÞ /C30Fn/C271; (60) where3F2a;b;c;d;e;z ðÞ is a GENERALIZED HYPERGEO- METRIC FUNCTION . Guy (1990) notes the curious fact that en/C281 ðÞ =2/C7/C5 forn /C300, 1, ... gives 1, 1, 2, 5, 8, 13, 21, 34, 55, ..., but then continues 91, 149, ... (Sloane’s A005181). Taking the product of the first nFibonacci numbers and adding 1 forn/C301, 2, ... gives the sequence 2, 2, 3, 77, 31, 241, ... (Sloane’s A052449). If these, 2, 2, 3, 7, 31, 241,3121, ... (Sloane’s A053413) are prime, i.e., the terms 1, 2, 3, 4, 5, 6, 7, 8, 22, 28, ... (Sloane’s A053408). The sequence of final digits in Fibonacci numbers repeats in cycles of 60. The last two digits repeat in300, the last three in 1500, the last four in 15,000, etc. The number of Fibonacci numbers between nand 2 n is either 1 or 2 (Wells 1986, p. 65). Cesa`ro derived the finite sums X n k/C300n k/C1Y/C1Q Fk/C30F2n (61) Xn k/C300n k/C1Y/C1Q 2kFk/C30F3n (62) (Honsberger 1985, pp. 109 /C1/10). The Fibonacci num- bers satisfy the power recurrence Xt/C271 j/C300/C281ðÞjj/C271 ðÞ =2t/C271 j/C20/C21 FFt n/C28j/C300; (63) wherea b/C2/C6 Fis a F IBONACCI COEFFICIENT , the reciprocal sum Xn k/C301/C281ðÞk FkFk/C27a/C30Fn FaXa k/C301/C281ðÞk FkFk/C27n; (64) the convolution Xn k/C300FkFn/C28k/C301 5nLn/C28Fn ðÞ ; (65) the partial fraction decomposition 1 Fn/C27aFn/C27bFn/C27c/C30A Fn/C27a/C27B Fn/C27b/C27C Fn/C27c; (66) where A/C30/C281ðÞn/C28a Fb/C28aFc/C28a(67) B/C30/C281ðÞn/C28b Fc/C28bFa/C28b(68) C/C30/C281ðÞn/C28c Fa/C28cFb/C28c; (69) and the summation formulaXn k/C300xkFak/C27b/C30g(n/C271)/C28g(0) 1/C28Lax/C27/C28 1ðÞax2; (70) where g(n)/C30/C28 1ðÞaFan/C281 ðÞ/C27bxn/C271/C28Fan/C27bxn: (71) Infinite sums include X/C12 n/C301/C281ðÞn FnFn/C272/C302/C28ffiffiffi 5p (72) (Clark 1995) and X/C12 n/C301/C281ðÞn/C271 Fn/C271Fn/C272/C30f/C282(73) X/C12 n/C3011overF2nF2n/C272/C30f/C282(74) where fis the GOLDEN RATIO (Wells 1986, p. 65). Forn]3;FnjFmIFFnjm(Wells 1986, p. 65). LnjLmIFF ndivides into manEVEN number of times. Fm;Fn ðÞ /C30 Fm;nðÞ (Michael 1964; Honsberger 1985, pp. 131 /C1/32). No ODD Fibonacci number is divisible by 17 (Hon- sberger 1985, pp. 132 and 242). No Fibonacci number >8 is ever OF THE FORM p/C281o r p/C271 where pis a PRIME NUMBER (Honsberger 1985, p. 133). Consider the sum sk/C30Xk n/C3021 Fn/C281Fn/C271/C30Xk n/C3021 Fn/C281Fn/C281 FnFn/C271 ! :(75) This is a TELESCOPING SUM ,s o sk/C301/C281 Fk/C271Fk/C272; (76) thus S/C13lim k0/C12sk/C301 (77) (Honsberger 1985, pp. 134 /C1/35). Using B INET’S FIBO- NACCI NUMBER FORMULA , it also follows that Fn/C27r Fn/C30an/C27r/C28bn/C27r an/C28bn/C30an/C27r an1/C28b a !n/C27r 1/C28b a !n; (78) where a/C301 21/C27ffiffiffi 5p/C17/C15 (79) b/C301 21/C28ffiffiffi 5p/C17/C15 (80) so lim n0/C12Fn/C27r Fn/C30 ar : (81) S?/C30X/C12 n/C301Fn Fn/C271Fn/C272/C301 (82) (Honsberger 1985, pp. 138 and 242 /C1/43). The MILLIN SERIES has sum Sƒ/C13X/C12 n/C3001 F2n/C301 27 /C28ffiffiffi 5p/C17/C15 (83) (Honsberger 1985, pp. 135 /C1/37). The Fibonacci numbers are COMPLETE . In fact, drop- ping one number still leaves a COMPLETE SEQUENCE , although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). Dropping two terms from the Fibonacci numbers produces a sequence which is not even WEAKLY COMPLETE (Honsberger 1985, p. 128). However, the sequence F ?n /C13Fn /C28/C28 1ðÞn(84) is WEAKLY COMPLETE , even with any finite subse- quence deleted (Graham 1964). F2 n/CY/CQ is not COM- PLETE , but F2 n/CY/CQ /C27 F2 n/CY/CQ are. 2N /C281 copies of FN n/CY/CQ are COMPLETE . For a discussion of SQUARE Fibonacci numbers, see Cohn (1964), who proved that the only SQUARE NUMBER Fibonacci numbers are 1 and F12 /C30144 (Cohn 1964, Guy 1994). Ming (1989) proved that the only TRIANGULAR Fibonacci numbers are 1, 3, 21, and 55. The Fibonacci and LUCAS NUMBERS have no common terms except 1 and 3. The only CUBIC Fibonacci numbers are 1 and 8. FnFn /C273 ;2Fn/C271Fn/C272 ;F2n /C273 /C30F2 n/C271 /C27F2 n/C272/C0/C1 (85) is a PYTHAGOREAN TRIPLE . F2 4n /C278F2nF2n /C27F6n ðÞ /C30 3F4n ðÞ2(86) is always a SQUARE NUMBER (Honsberger 1985, p. 243). In 1975, James P. Jones showed that the Fibonacci numbers are the POSITIVE INTEGER values of the POLYNOMIAL P(x; y) /C30/C28y5 /C272y4x /C27y3x2 /C282y2x3 /C28yx4 /C282/C0/C1 (87) for GAUSSIAN INTEGERS x and y (Le Lionnais 1983). If n and k are two POSITIVE INTEGERS , then between nk and nk /C271 ; there can never occur more than n Fibonacci numbers (Honsberger 1985, pp. 104 /C1/05). Every Fn that is PRIME has a PRIME index n, with the exception of F4 /C303: However, the converse is not true (i.e., not every prime index p gives a PRIME Fp) : The first few PRIME Fibonacci numbers Fn are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (Sloane’s A005478), which occur for n /C30 3, 4, 5, 7, 11, 13, 17, 23, 29, 43,47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, ... (Sloane’s A001605; Dubner and Keller 1999). Gard- ner’s statement that F531is prime is incorrect, especially since 531 is not even PRIME (Gardner 1979, p. 161). It is not known if there are an INFINITE number of Fibonacci primes. The Fibonacci numbers Fn ; are SQUAREFUL for n /C30 6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (Sloane’s A037917) and SQUAREFREE for n /C30 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (Sloane’s A037918). 4 F6nj and 25 F25nj for all n, and there is at least one n 52m such that mFn:j No SQUAREFUL Fibonacci numbers Fpare known with pPRIME . See also CASSINI’S IDENTITY ,C ATALAN’S IDENTITY , D’OCAGNE’S IDENTITY ,FAST FIBONACCI TRANSFORM , FIBONACCI COEFFICIENT ,FIBONACCI DUAL THEOREM , FIBONACCI N-STEP NUMBER ,FIBONACCI POLYNOMIAL , FIBONACCI Q-MATRIX ,GELIN- CESA` RO IDENTITY ,GEN- ERALIZED FIBONACCI NUMBER ,INVERSE TANGENT , LINEAR RECURRENCE SEQUENCE ,LUCAS SEQUENCE , NEAR NOBLE NUMBER ,P ELL SEQUENCE ,R ABBIT CONSTANT ,RANDOM FIBONACCI SEQUENCE ,STOLARS- KY ARRAY ,TETRANACCI NUMBER ,TRIBONACCI NUM- BER ,W YTHOFF A RRAY ,Z ECKENDORF REPRESENTATION ,ZECKENDORF’S THEOREM References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 56 /C1/7, 1987. Basin, S. L. and Hoggatt, V. E. Jr. "A Primer on the Fibonacci Sequence." Fib. Quart. 1, 1963. Basin, S. L. and Hoggatt, V. E. Jr. "A Primer on the Fibonacci Sequence--Part II." Fib. Quart. 1,6 1/C1/8, 1963. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. 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Quart. 2,5 9/C1/6, 1964. Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer for the Fibonacci Numbers: Part XII." Fib. Quart. 11, 317/C1/31, 1973. Honsberger, R. "A Second Look at the Fibonacci and Lucas Numbers." Ch. 8 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985. Kepler, J. The Six-Cornered Snowflake. Oxford, England: Oxford University Press, 1966. Knott, R. "Fibonacci Numbers and the Golden Section." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 146, 1983. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/fibonacci.Z. Matiyasevich, Yu. V. "Solution to of the Tenth Problem of Hilbert." Mat. Lapok 21,8 3/C1 /7, 1970. Matijasevich, Yu. V. Hilbert’s Tenth Problem. Cambridge, MA: MIT Press, 1993. http://www.informatik.uni-stutt- gart.de/ifi/ti/personen/Matiyasevich/H10Pbook/. Michael, G. "A New Proof for an Old Property." Fib. Quart. 2,5 7/C1/8, 1964. Ming, L. "On Triangular Fibonacci Numbers." Fib. Quart. 27,9 8/C1/08, 1989. Ogilvy, C. S. and Anderson, J. T. "Fibonacci Numbers." Ch. 11 in Excursions in Number Theory. New York: Dover, pp. 133 /C1/44, 1988. Pappas, T. "Fibonacci Sequence," "Pascal’s Triangle, the Fibonacci Sequence & Binomial Formula," "The Fibonacci Trick," and "The Fibonacci Sequence & Nature." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 28 /C1/9, 40/C1/1, 51, 106, and 222 /C1/25, 1989. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, p. 12, 1996.Ram, R. "Fibonacci Formulae." http://users.tellurian.net/ hsejar/maths/fibonacci/. Reid, C. Julia: A Life in Mathematics. Washington, DC: Math. Assoc. Amer., 1997. Reiter, C. "Fast Fibonacci Numbers." Mathematica J. 2,5 8/C1/ 0, 1992. Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 49 /C1/ 7, 1991. Se´roul, R. "The Fibonacci Numbers." §2.13 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 21 /C1/2, 2000. Shorey, T. N. and Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers, 2." J. London Math. Soc. 23,1 7/C1/3, 1981. Sloane, N. J. A. Sequences A000045/M0692, A001605/ M2309, A005181/M0693, A005478/M0741, A037917, A037918, A053408, A052449, and A053413 in "An On-Line Version of the Encyclopedia of Integer Sequences."http://www.research.att.com/~njas/sequences/eisonli-ne.html. Smith, H. J. "Fibonacci Numbers." http://pweb.netcom.com/ ~hjsmith/Fibonacc.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 46 /C1 /7 and 299, 1999. Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers." Proc. London Math. Soc. 35, 425/C1/47, 1977. Szymkiewicz, D. "Sur la porte ´e de la loi de Ludwig." Acta Soc. Botanicorum Poloniae 5, 390/C1/95, 1928. Vogler, P. "Das ,Ludwig’sche Gipfelgesetz‘ und seine Trag- weite." Flora 104, 123/C1/28, 1912. Vorob’ev, N. N. Fibonacci Numbers. New York: Blaisdell, 1961. Weisstein, E. W. "Books about Fibonacci Numbers." http:// www.treasure-troves.com/books/FibonacciNumbers.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 61 /C1/7, 1986. Zylinski, E. "Numbers of Fibonacci in Biological Statistics." Atti del Congr. internaz. matematici 4, 153/C1/56, 1928. Fibonacci Polynomial TheWPOLYNOMIALS obtained by setting p(x)/C30xand q(x)/C301 in the L UCAS POLYNOMIAL SEQUENCE . (The corresponding w POLYNOMIALS are called L UCAS POLYNOMIALS .) The Fibonacci polynomials are defined by the RECURRENCE RELATION Fn/C271(x)/C30xFn(x)/C27Fn/C281(x); (1) with F1(x)/C301 and F2(x)/C30x:They are also given by the explicit sum formula Fn(x) /C30X(n /C281)=2 bc j/C300n /C28j /C281 j/C1Y/C1Q xn /C282j/C281 ; (2) where xbcis the FLOOR FUNCTION andn m/C0/C1 is a BINOMIAL COEFFICIENT . The first few Fibonacci poly- nomials are F1(x) /C301 F2(x) /C30x F3(x) /C30x2 /C271 F4(x) /C30x3 /C272x F5(x) /C30x4 /C273x2 /C271: The Fibonacci polynomials are normalized so that Fn(1) /C30Fn ; (3) where the Fn/s are FIBONACCI NUMBERS . The Fibonacci polynomials are related to the MOR- GAN- VOYCE POLYNOMIALS by F2n/C271(x) /C30bnx2/C0/C1 (4) F2n /C27n2(x) /C30xBnx2/C0/C1 (5) (Swamy 1968). See also BRAHMAGUPTA POLYNOMIAL ,F IBONACCI NUMBER ,MORGAN- VOYCE POLYNOMIAL References Swamy, M. N. S. "Further Properties of Morgan-Voyce Polynomials." Fib. Quart. 6, 167 /C1/75, 1968. Fibonacci Pseudoprime Consider a LUCAS SEQUENCE with P /C21 0 and Q /C3091: A Fibonacci pseudoprime is a COMPOSITE NUMBER n such that Vn /C13P (mod n) : There exist no EVEN Fibonacci pseudoprimes with parameters P /C301 and Q /C30/C28 1 (Di Porto 1993) or P /C30 Q /C301 (Andre ´-Jeannin 1996). Andre ´-Jeannin (1996) also proved that if (P ;Q) "(1;/C281) and (P;Q) "(1;1); then there exists at least one EVEN Fibonacci pseu- doprime with parameters P and Q. See also PSEUDOPRIME References Andre ´-Jeannin, R. "On the Existence of Even Fibonacci Pseudoprimes with Parameters P and Q." Fib. Quart. 34, 75 /C1/8, 1996. Di Porto, A. "Nonexistence of Even Fibonacci Pseudoprimes of the First Kind." Fib. Quart. 31, 173 /C1/77, 1993. Ribenboim, P. "Fibonacci Pseudoprimes." §2.X.A in The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 127 /C1/29, 1996.Fibonacci Q-Matrix AF IBONACCI MATRIX OF THE FORM M /C30m 1 10/C20/C21 /C215 (1) If U and V are defined as BINET FORMS Un /C30mUn/C281 /C27Un/C282U0 /C300;U1 /C301 ðÞ (2) Vn /C30mVn /C281 /C27Vn/C282V0 /C302;V1 /C30m ðÞ ; (3) then M /C30Un/C271 Un Un Un/C281/C20/C21 (4) M /C281 /C30M /C28ml /C3001 1 /C28m/C20/C21 /C215 (5) Defining Q /C13F2F1 F1F0/C20/C21 /C301110/C20/C21 ; (6) then Q n /C30Fn/C271 Fn Fn Fn/C281/C20/C21 (7) (Honsberger 1985, pp. 106 /C1/07). See also BINET FORMS ,FIBONACCI NUMBER References Honsberger, R. "A Second Look at the Fibonacci and Lucas Numbers." Ch. 8 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985. Fibonacci Sequence FIBONACCI NUMBER Fibration If f : E 0 B is a FIBER BUNDLE with B a PARACOMPACT TOPOLOGICAL SPACE , then f satisfies the HOMOTOPY LIFTING PROPERTY with respect to all TOPOLOGICAL SPACES . In other words, if g :[0;1] /C29X 0 B is a HOMOTOPY from g0to g1 ; and if g ?0is a LIFT of the MAP g0 with respect to f, then g has a LIFT to a MAP g? with respect to f. Therefore, if you have a HOMOTOPY of a MAP into B, and if the beginning of it has a LIFT, then that LIFT can be extended to a LIFT of the HOMOTOPY itself. A fibration is a MAP between TOPOLOGICAL SPACES f: E0Bsuch that it satisfies the HOMOTOPY LIFTING PROPERTY . See also FIBER BUNDLE ,FIBER SPACE Fiedler Vector The EIGENVECTOR corresponding to the second smal- lest EIGENVALUE (i.e., the ALGEBRAIC CONNECTIVITY ) of the LAPLACIAN MATRIX of a graph G. The Fiedler vector is used in SPECTRAL GRAPH PARTITIONING . See also ALGEBRAIC CONNECTIVITY ,C ONNECTED GRAPH ,LAPLACIAN MATRIX ,SPECTRAL GRAPH PARTI- TIONING References Chung, F. R. K. Spectral Graph Theory. Providence, RI: Amer. Math. Soc., 1997. Demmel, J. "CS 267: Notes for Lecture 23, April 9, 1999. Graph Partitioning, Part 2." http://www.cs.berkeley.edu/ ~demmel/cs267/lecture20/lecture20.html. # 1999 /C1/001 Wolfram Research, Inc. Field A field is any set of elements which satisfies the FIELD AXIOMS for both addition and multiplication and is a commutative DIVISION ALGEBRA . An archaic name for a field is RATIONAL DOMAIN . The French term for a field is corps and the German word is Ko¨rper, both meaning "body." A field with a finite number of members is known as a FINITE FIELD or Galois field. Because the identity condition must be different for addition and multiplication, every field must have at least two elements. Examples include the COMPLEX NUMBERS (/C) ; RATIONAL NUMBERS /(Q) ; and REAL NUMBERS /(R) ; but not the INTEGERS (F), which form only a RING . It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements are equivalent to the field of COMPLEX NUMBERS . See also ADJUNCTION ,C HARACTERISTIC (FIELD ), COEFFICIENT FIELD,C YCLOTOMIC FIELD,D IVISION ALGEBRA ,EXTENSION FIELD,FIELD AXIOMS ,FINITE FIELD,FUNCTION FIELD,LOCAL FIELD,M AC LANE’S THEOREM ,M ODULE ,N UMBER FIELD,PYTHAGOREAN FIELD,QUADRATIC FIELD,RING,SKEW FIELD,SPLIT- TING FIELD,SUBFIELD ,VECTOR FIELD References Allenby, R. B. Rings, Fields, and Groups: An Introduction to Abstract Algebra, 2nd ed. Oxford, England: Oxford Uni- versity Press, 1991. Dummit, D. S. and Foote, R. M. "Field Theory." Ch. 13 in Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice- Hall, pp. 422 /C1/70, 1998. Ellis, G. Rings and Fields. Oxford, England: Oxford Uni- versity Press, 1993. Ferreiro ´s, J. "A New Fundamental Notion for Algebra: Fields." §3.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 90 /C1/4, 1999. Joye, M. "Introduction e´le´mentaire a` la the´orie des courbes elliptiques." http://www.dice.ucl.ac.be/crypto/introductory/ courbes_elliptiques.html. Nagell, T. "Moduls, Rings, and Fields." §6inIntroduction to Number Theory. New York: Wiley, pp. 19 /C1/1, 1951.Field Axioms The field axioms are generally written in additive and multiplicative pairs. Name Addition Multiplication Commutativity /a /C27b /C30b /C27a/ ab /C30 ba Associativity /(a /C27b) /C27c /C30a /C27(b /C27c)// (ab)c /C30a(bc)/ Distributivity /a(b /C27c) /C30ab /C27ac// (a /C27b)c /C30ac /C27bc/ Identity /a /C270 /C30a /C300 /C27a// a /C2151 /C30a /C301 /C215a/ Inverses /a /C27(/C28a) /C300 /C30(/C28a) /C27a//aa/C281/C301/C30a/C281aifa"0/ See also ALGEBRA ,FIELD References Apostol, T. M. "The Field Axioms." §I 3.2 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 17 /C1/9, 1967. Field Extension EXTENSION FIELD Fields Medal Portions of this entry contributed by M ICHEL BARRAN The mathematical equivalent of the Nobel Prize (there is no Nobel Prize in mathematics) which isawarded by the International Mathematical Union every four years to one or more outstanding research- ers. "Fields Medals" are more properly known bytheir official name, "International medals for out-standing discoveries in mathematics." The Field medals were first proposed at the 1924 International Congress of Mathematicians in Tor- onto, where a resolution was adopted stating that ateach subsequent conference, two gold medals should be awarded to recognize outstanding mathematical achievement. Professor J. C. Fields, a Canadianmathematician who was secretary of the 1924 Con- gress, later donated funds establishing the medals which were named in his honor. Consistent withFields’ wish that the awards recognize both existing work and the promise of future achievement, it was agreed to restrict the medals to mathematicians notover forty at the year of the Congress. In 1966 it wasagreed that, in light of the great expansion of mathematical research, up to four medals could be awarded at each Congress. The Fields Medal is the highest scientific award for mathematicians, and is presented every four years at the International Congress of Mathematicians, to- gether with a prize of 15,000 Canadian dollars. The first Fields Medal was awarded in 1936 at the World Congress in Oslo. The Fields Medal is made of gold, and shows the head of Archimedes (287 /C1/12 BC) together with a quotation attributed to him: "Transire suum pectus mundoque potiri" ("Rise above oneself and grasp the world"). The reverse side bears the inscription: "Congregati ex toto orbe mathematici ob scripta insignia tribuere" ("the mathematicians as- sembled here from all over the world pay tribute for outstanding work"). Nobel prizes were created in the will of the Swedish chemist and inventor of dynamite Alfred Nobel, but Nobel, who was an inventor and industrialist, did not create a prize in mathematics because he was not particularly interested in mathematics or theoretical science. In fact, his will speaks of prizes for those "inventions or discoveries" of greatest practical ben- efit to mankind. While it is commonly stated that Nobel decided against a Nobel prize in math because of anger over the romantic attentions of a famous mathematician (often claimed to be Gosta Mittag- Leffler ) to a women in his life, there is no historical evidence to support the story. Furthermore, Nobel was a lifelong batchelor, although he did has a Viennese woman named Sophie Hess as his mistress (Lopez-Ortiz). The following table summarizes Fields Medals win- ners together with their institutions. year winners 1936 Lars Valerian Ahlfors (Harvard University) Jesse Douglas (Massachusetts Institute of Tech- nology) 1950 Laurent Schwartz (University of Nancy) Alte Selberg (Institute for Advanced Study, Princeton) 1954 Kunihiko Kodaira (Princeton University) Jean-Pierre Serre (University of Paris) 1958 Klaus Friedrich Roth (University of London) Rene´ Thom (University of Strasbourg) 1962 Lars V. Ho¨rmander (University of Stockholm) John Willard Milnor (Princeton University) 1966 Michael Francis Atiyah (Oxford University) Paul Joseph Cohen (Stanford University) Alexander Grothendieck (University of Paris) Stephen Smale (University of California, Berke- ley) 1970 Alan Baker (Cambridge University)Heisuke Hironaka (Harvard University) Serge P. Novikov (Moscow University) John Griggs Thompson (Cambridge University) 1974 Enrico Bombieri (University of Pisa) David Bryant Mumford (Harvard University) 1978 Pierre Rene´ Deligne (Institut des Hautes E´ tudes Scientifiques) Charles Louis Fefferman (Princeton University) Gregori Alexandrovitch Margulis (Moscow Uni- versity)Daniel G. Quillen (Massachusetts Institute of Technology) 1982 Alain Connes (Institut des Hautes E´ tudes Scientifiques)William P. Thurston (Princeton University) Shing-Tung Yau (Institute for Advanced Study, Princeton) 1986 Simon Donaldson (Oxford University) Gerd Faltings (Princeton University) Michael Freedman (University of California, San Diego) 1990 Vladimir Drinfeld (Phys. Inst. Kharkov) Vaughan Jones (University of California, Ber- keley)Shigefumi Mori (University of Kyoto?) Edward Witten (Institute for Advanced Study, Princeton) 1994 Pierre-Louis Lions (Universite ´de Paris-Dau- phine)Jean-Christophe Yoccoz (Universite ´de Paris- Sud)Jean Bourgain (Institute for Advanced Study, Princeton) Efim Zelmanov (University of Wisconsin) 1998 Richard E. Borcherds (Cambridge University) W. Timothy Gowers (Cambridge University) Maxim Kontsevich (IHES Bures-sur-Yvette) Curtis T. McMullen (Harvard University) See also BURNSIDE PROBLEM ,M ATHEMATICS PRIZES , POINCARE ´ CONJECTURE ,ROTH’S THEOREM ,TAU CON- JECTURE References Albers, D. J.; Alexanderson, G. L.; and Reid, C. Interna- tional Mathematical Congresses, An Illustrated History 1893 /C1/986, rev. ed., incl. 1986. New York: Springer Verlag, 1987. Fields Institute. "Fields Medal Winners." http://www.field- s.toronto.edu/medal.html. International Mathematical Union. "Fields Medals and Rolf Nevanlinna Prize." http://elib.zib.de/IMU/medals/. Joyce, D. "History of Mathematics: Fields Medals." http:// aleph0.clarku.edu/~djoyce/mathhist/fieldsmedal.html. Lopez-Ortiz, A. "Fields Medal: Historical Introduction." http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/no- de19.html. Lopez-Ortiz, A. "Why Is There No Nobel In Mathematics?" http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/no-de21.html. MacTutor History of Mathematics Archives. "The Fields Medal." http://www-groups.dcs.st-and.ac.uk/~history/So- cieties/FieldsMedal.html. Monastyrsky, M. Modern Mathematics in the Light of the Fields Medals. Wellesley, MA: A. K. Peters, 1997. Technische Universita ¨t Berlin. "The Four Fields Medallists and the Nevanlinna Prize Winner of The International Congress of Mathematicians, Berlin 1998." http://www.tu- berlin.de/presse/pi/1998/pi182e.htm. Tropp, H. S. "The Origins and History of the Fields Medal." Historia Math. 3, 167 /C1 /81, 1976. Fifteen Theorem A theorem due to Conway et al. (1997) which states that, if a positive definite QUADRATIC FORM with INTEGER MATRIX entries represents all natural num- bers up to 15, then it represents all natural numbers. This theorem contains L AGRANGE’S FOUR-SQUARE THEOREM , since every number up to 15 is the sum of at most four SQUARES . See also INTEGER MATRIX ,INTEGER- MATRIX FORM, LAGRANGE’S FOUR- SQUARE THEOREM ,Q UADRATIC FORM References Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane, N. J. A. "The Primary Pretenders." Acta Arith. 78, 307/C1/ 13, 1997. Duke, W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer. Math. Soc. 44, 190/C1/96, 1997. Figurate Number A number which can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a REGULAR POLYGON , the number is called a POLYGONAL NUMBER . The poly- gonal numbers illustrated above are called triangu-lar, square, pentagonal, and hexagon numbers,respectively. Figurate numbers can also form other shapes such as centered polygons, L-shapes, 3-dimen- sional solids, etc. The nth regular r-polytopic number is given by P r(n)/C30n/C27r/C281 n/C1Y/C1Q /C301 r!n(r); wheren k/C0/C1 is a BINOMIAL COEFFICIENT and n(k)is a RISING FACTORIAL ,s o P2(n)/C301 2n(n/C271) are the TRIANGULAR NUMBERS , P3(n)/C301 6n(n/C271)(n/C272) the TETRAHEDRAL NUMBERS , P4(n)/C301 24n(n/C271)(n/C271)(n/C273) the PENTATOPE NUMBERS , and so on (Dickson 1952, p. 7). The following table lists the most common types of figurate numbers. Name FORMULA BIQUADRATIC NUMBER /n4/ CENTERED CUBE NUMBER /(2n/C281)(n2/C28n/C271)/ CENTERED PENTAGONAL NUM- BER/1 2(5n2/C275n/C272)/ CENTERED SQUARE NUMBER /n2/C27(n/C281)2 / CENTERED TRIANGULAR NUM- BER/1 2(3n2/C283n/C272)/ CUBIC NUMBER /n3/ DECAGONAL NUMBER /4n2/C283n/ GNOMONIC NUMBER /2n/C281/ Hauy OCTAHEDRAL NUMBER /13(2n/C281)(2n 2/C282n/C273)/ Hauy RHOMBIC DODECAHE- DRAL NUMBER/(2n/C281)(8n2/C2814n/C277)/ HEPTAGONAL NUMBER /12n(5n/C283) / HEX NUMBER /3n2/C283n/C271/ HEPTAGONAL PYRAMIDAL NUM- BER/1 6n(n/C271)(5n/C282)/ HEXAGONAL NUMBER /n(2n/C281)/ HEXAGONAL PYRAMIDAL NUM- BER/1 6n(n/C271)(4n/C281)/ OCTAGONAL NUMBER /n(3n /C282)/ OCTAHEDRAL NUMBER /1 3n(2n2 /C271)/ PENTAGONAL NUMBER /12n(3n /C281) / PENTAGONAL PYRAMIDAL NUM- BER/12n 2(n /C271)/ PENTATOPE NUMBER /1 24n(n /C271)(n /C272)(n /C273)/ PRONIC NUMBER /n(n /C271)/ RHOMBIC DODECAHEDRAL NUMBER/(2n /C281)(2n2 /C282n /C271)/ SQUARE NUMBER /n2/ SQUARE PYRAMIDAL NUMBER /16n(n /C271)(2n /C271) / STELLA OCTANGULA NUMBER /n(2n2 /C281)/ TETRAHEDRAL NUMBER /16n(n /C271)(n /C272) / TRIANGULAR NUMBER /12n(n /C271) / TRUNCATED OCTAHEDRAL NUMBER/16n3 /C2833n2 /C2724n /C286/ TRUNCATED TETRAHEDRAL NUMBER/16n 23n 2 /C2827n /C2710/C0/C1 / See also BIQUADRATIC NUMBER ,C ENTERED CUBE NUMBER ,C ENTERED PENTAGONAL NUMBER ,C EN- TERED POLYGONAL NUMBER ,C ENTERED SQUARE NUMBER ,C ENTERED TRIANGULAR NUMBER ,C UBIC NUMBER ,D ECAGONAL NUMBER ,FIGURATE NUMBER TRIANGLE ,G NOMONIC NUMBER ,H EPTAGONAL NUM- BER,H EPTAGONAL PYRAMIDAL NUMBER ,H EX NUM- BER,HEX PYRAMIDAL NUMBER ,HEXAGONAL NUMBER , HEXAGONAL PYRAMIDAL NUMBER ,N EXUS NUMBER , OCTAGONAL NUMBER ,OCTAHEDRAL NUMBER ,PENTA- GONAL NUMBER ,PENTAGONAL PYRAMIDAL NUMBER , PENTATOPE NUMBER ,POLYGONAL NUMBER ,PRONIC NUMBER ,PYRAMIDAL NUMBER ,RHOMBIC DODECAHE- DRAL NUMBER ,SQUARE NUMBER ,SQUARE PYRAMIDAL NUMBER ,S TELLA OCTANGULA NUMBER ,T ETRAHE- DRAL NUMBER ,T RIANGULAR NUMBER ,T RUNCATED OCTAHEDRAL NUMBER ,T RUNCATED TETRAHEDRAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30 /C1/2, 1996. Dickson, L. E. "Polygonal, Pyramidal, and Figurate Num- bers." Ch. 1 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 1 /C1/9, 1952. Goodwin, P. "A Polyhedral Sequence of Two." Math. Gaz. 69, 191 /C1/97, 1985. Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 147 /C1/50, 1994. Kraitchik, M. "Figurate Numbers." §3.4 in Mathematical Recreations. New York: W. W. Norton, pp. 66 /C1/9, 1942.Savin, A. "Shape Numbers." Quantum 11,14/C1/8, 2000. Figurate Number Triangle AP ASCAL’S TRIANGLE written in a square grid and padded with zeroes, as written by Jakob Bernoulli (Smith 1984). The figurate number triangle therefore has entries aij /C30i j/C1Y/C1Q : where i is the row number, j the column number, and i j/C17/C15 a BINOMIAL COEFFICIENT . Written out explicitly (beginning each row with j /C30 0), 1000000 /C1/C1/C1 1100000 /C1/C1/C1 1210000 /C1/C1/C1 1331000 /C1/C1/C1 1464100 /C1/C1/C1 1 5 10 10 5 1 0 /C1/C1/C1 1 6 15 20 15 6 1 /C1/C1/C1 1 7 21 35 35 21 7::: n n nnnn n:::2 66666666666643 7777777777775 Then we have the sum identities X i j/C300aij/C302i Xi j/C301aij/C302i/C281 Xn i/C300aij/C30a(n/C271);(j/C271)/C30n/C271 j/C271anj: See also BINOMIAL COEFFICIENT ,FIGURATE NUMBER , PASCAL’S TRIANGLE References Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 86, 1984. Figure Eight Knot FIGURE-OF- EIGHT KNOT Figure Eight Surface EIGHT SURFACE Figure-of-Eight Knot The knot 04 /C1/01, which is the unique PRIME KNOT of four crossings, and which is a 2-EMBEDDABLE KNOT .It is AMPHICHIRAL . It is also known as the FLEMISH KNOT and SAVOY KNOT , and it has BRAID WORD s1 s/C281 2s1 s/C281 2:/ References Francis, G. K. A Topological Picture Book. New York: Springer-Verlag, 1987. Owen, P. Knots. Philadelphia, PA: Courage, p. 16, 1993. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 78 /C1/9, 1991. Figures A number x is said to have "n figures" if it takes n DIGITS to express it. The number of figures is there- fore equal to one more than the POWER of 10 in the SCIENTIFIC NOTATION representation of the number. The word is most frequently used in reference to monetary amounts, e.g., a "six-figure salary" would fall in the range of $100,000 to $999,999. See also DIGIT,SCIENTIFIC NOTATION ,SIGNIFICANT FIGURES Filon’s Integration Formula A formula for NUMERICAL INTEGRATION , gxn x0f(x) cos(tx)dx /C30h fa(th) f2n sin tx2nðÞ/C28f0 sin tx0ðÞ ½/C138 /C27 b(th)C2n /C27 g(th)C2n/C281 /C272 45th4S?2n/C281 g/C28Rn ; (1) where C2n /C30Xn i/C300f2i cos tx2iðÞ/C281 2f2n cos tx2nðÞ ½ /C27f0 cos tx0ðÞ /C138 (2) C2n/C281 /C30Xn i/C301f2i/C281 cos tx2i/C281 ðÞ (3) S ?2n /C281 /C30Xn i/C301f(3) 2i /C281 sin(tx2i /C281) (4)a( u) /C301u /C27sin(2u) 2u2/C282 sin2 u u3 (5) b( u) /C3021 /C27 cos2 u u2 /C28sin(2u) u3"# (6) g(u) /C304sin u u3 /C28cos u u2 ! ; (7) and the remainder term is Rn /C301 90nh5f(4)(j) /C27O th7/C0/C1 : (8) See also NUMERICAL INTEGRATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 890 /C1/91, 1972. Tukey, J. W. In On Numerical Approximation: Proceedings of a Symposium Conducted by the Mathematics Research Center, United States Army, at the University of Wiscon- sin, Madison, April 21 /C1/3, 1958 (Ed. R. E. Langer). Madison, WI: University of Wisconsin Press, p. 400, 1959. Filter Let S be a nonempty set, then a filter on S is a nonempty collection F of subsets of S having the following properties: 1. fiQF ;/ 2. If A;B /C23 F ; then A S B /C23 F ;/ 3. If A /C23 F and A ⁄B ⁄S then B /C23 F/ If S is an infinite set, then the collection FS /C30fA ⁄ S : S /C28A is finite g is a filter called the COFINITE (or Fre´chet) filter on S. In signal processing, a filter is a function or procedure which removes unwanted parts of a signal. The concept of filtering and filter functions is particularly useful in engineering. One particularly elegant method of filtering F OURIER TRANSFORMS a signal into frequency space, performs the filtering operation there, then transforms back into the original space (Press et al. 1992). See also COFINITE FILTER ,REMEZ ALGORITHM ,SA- VITZKY- GOLAY FILTER ,ULTRAFILTER ,W IENER FILTER References Hamming, R. W. Digital Filters. New York: Dover, 1998. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Digital Filtering in the Time Domain." §13.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 551 /C1/56, 1992. Filtration # 1999 /C1/001 Wolfram Research, Inc. Fine’s Equation The Q-SERIES identity Y n/C3011 /C28 q2nðÞ 1 /C28 q3nðÞ 1 /C28 q8nðÞ 1 /C28 q12nðÞ 1 /C28 qn ðÞ 1 /C28 q24n ðÞ /C301 /C27X N /C301E1;5 ;7;11(N;24)qN ; where E1;5 ;7 ;11(N;24) is the sum of the DIVISORS of N CONGRUENT to 1, 5, 7, and 11 (mod 24) minus the sum of DIVISORS of N CONGRUENT to -1, -5, -7, and -11 (mod 24). See also Q-SERIES Finite A SET which contains a NONNEGATIVE integral num- ber of elements is said to be finite. A SET which is not finite is said to be INFINITE . A finite or COUNTABLY INFINITE set is said to be COUNTABLE . While the meaning of the term "finite" is fairly clear in common usage, precise definitions of FINITE and INFINITE are needed in technical mathematics and especially in SET THEORY . See also COUNTABLE SET,C OUNTABLY INFINITE , INFINITE ,SET THEORY ,UNCOUNTABLY INFINITE Finite Difference The finite difference is the discrete analog of the DERIVATIVE . The finite FORWARD DIFFERENCE of a function fpis defined as Dfp/C13fp/C271/C28fp; (1) and the finite BACKWARD DIFFERENCE as 9fp/C13fp/C28fp/C281: (2) If the values are tabulated at spacings h, then the notation fp/C13fx0/C27ph ðÞ /C13f(x) (3) is used. The kthFORWARD DIFFERENCE would then be written as Dkfp;and similarly, the kthBACKWARD DIFFERENCE as9kfp:/ However, when fpis viewed as a discretization of the continuous function f(x);then the finite difference is sometimes written Df(x)/C13fx/C271 2 ! /C28fx/C2812 ! /C302II(x)+f(x); (4)where +denotes CONVOLUTION and II(x) is the odd IMPULSE PAIR . The finite difference operator can therefore be written ˜D/C302II+: (5) AnnthPOWER has a constant nth finite difference. For example, take n/C303 and make a DIFFERENCE TABLE , x 1 23 4 5x 3 18 2764 125D 7 19 37 61D 2 12 1824D 3 66D 4 0: (6) TheD3column is the constant 6. Finite difference formulas can be very useful for extrapolating a finite amount of data in an attempt to find the general term. Specifically, if a function f(n) is known at only a few discrete values n/C300, 1, 2, ... and it is desired to determine the analytical form of f, the following procedure can be used if fis assumed to be a POLYNOMIAL function. Denote the nth value in the SEQUENCE of interest by an:Then define bnas the FORWARD DIFFERENCE Dn/C13an/C271/C28an;cnas the second FORWARD DIFFERENCE D2 n/C13bn/C271/C28bn;etc., construct- ing a table as follows a0/C13f(0) a1/C13f(1) a2/C13f(2) . . . ap/C13f(p) b0/C13a1/C28a0b1/C13a2/C28a1... bp/C281/C13ap/C28ap/C281 c0/C13b1/C28b0... ... ::: (7) Continue computing d0;e0;etc., until a 0 value is obtained. Then the POLYNOMIAL function giving the values anis given by f(n)/C30Xp k/C300akn k/C1Y/C1Q /C30a0/C27b0n/C27c0n(n/C281) 2/C27d0n(n/C281)(n/C282) 2/C2153 /C27. . . (8) When the notation D0/C13a0;D20/C13b0;etc., is used, this beautiful equation is called N EWTON’S FORWARD DIFFERENCE FORMULA . To see a particular example, consider a SEQUENCE with first few values of 1, 19, 143, 607, 1789, 4211, and 8539. The difference table is then given by 1 19 143 607 1789 4211 8539 18 124 464 1182 2422 4328 106 340 718 1240 1906 234 378 522 666 144 144 144 00 Reading off the first number in each row gives a0 /C301; b0 /C3018; c0 /C30106; d0 /C30234; e0 /C30144: Plugging these in gives the equation f(n) /C301 /C2718n /C2753n(n /C281) /C2739n(n /C281)(n /C282) /C276n(n /C281)(n /C282)(n /C283); (9) which simplifies to f(n) /C306n4 /C273n3 /C272n2 /C277n /C271; and indeed fits the original data exactly! Beyer (1987) gives formulas for the derivatives hndnf(x0 /C27 ph) dxn/C13hndnfp dxn /C13dnfp dpn (10) (Beyer 1987, pp. 449 /C1/51) and integrals gx/C12 x0f(x)dx /C30hgn 0fpdp (11) (Beyer 1987, pp. 455 /C1/56) of finite differences. Finite differences lead to DIFFERENCE EQUATIONS , finite analogs of DIFFERENTIAL EQUATIONS . In fact, UMBRAL CALCULUS displays many elegant analogs of well-known identities for continuous functions. Com- mon finite difference schemes for PARTIAL DIFFEREN- TIAL EQUATIONS include the so-called Crank- Nicholson, Du Fort-Frankel, and Laasonen methods. See also BACKWARD DIFFERENCE ,B ESSEL’S FINITE DIFFERENCE FORMULA ,DIFFERENCE EQUATION ,DIF- FERENCE TABLE ,EVERETT’S FORMULA ,FINITE ELE- MENT METHOD ,F ORWARD DIFFERENCE ,G AUSS’S BACKWARD FORMULA ,G AUSS’S FORWARD FORMULA , INTERPOLATION ,JACKSON’S DIFFERENCE FAN,N EW- TON’S BACKWARD DIFFERENCE FORMULA ,N EWTON- COTES FORMULAS ,N EWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA ,NEWTON’S FORWARD DIF- FERENCE FORMULA ,Q UOTIENT- DIFFERENCE TABLE , STEFFENSON’S FORMULA ,STIRLING’S FINITE DIFFER- ENCE FORMULA ,UMBRAL CALCULUS References Abramowitz, M. and Stegun, C. A. (Eds.). "Differences." §25.1 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877 /C1/78, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 429 /C1/15, 1987. Boole, G. and Moulton, J. F. A Treatise on the Calculus of Finite Differences, 2nd rev. ed. New York: Dover, 1960. Conway, J. H. and Guy, R. K. "Newton’s Useful Little Formula." In The Book of Numbers. New York: Springer- Verlag, pp. 81 /C1/3, 1996. Iyanaga, S. and Kawada, Y. (Eds.). "Interpolation." Appen- dix A, Table 21 in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 1482 /C1/483, 1980. Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965. Levy, H. and Lessman, F. Finite Difference Equations. New York: Dover, 1992.Milne-Thomson, L. M. The Calculus of Finite Differences. London: Macmillan, 1951. Richardson, C. H. An Introduction to the Calculus of Finite Differences. New York: Van Nostrand, 1954. Spiegel, M. Calculus of Finite Differences and Differential Equations. New York: McGraw-Hill, 1971. Stirling, J. Methodus differentialis, sive tractatus de sum- mation et interpolation serierum infinitarium. London, 1730. English translation by Holliday, J. The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series. 1749. Tweedie, C. James Stirling: A Sketch of his Life and Works Along with his Scientific Correspondence. Oxford, Eng- land: Oxford University Press, pp. 30 /C1/5, 1922. Weisstein, E. W. "Books about Finite Difference Equations." http://www.treasure-troves.com/books/FiniteDifferenceE- quations.html. Zwillinger, D. (Ed.). "Difference Equations." §3.9 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 228 /C1/35, 1995. Finite Element Method A method for solving an equation by approximating continuous quantities as a set of quantities at discrete points, often regularly spaced into a so-called GRID or MESH . Because finite element methods can be adapted to problems of great complexity and unusual geome- try, they are an extremely powerful tool in the solution of important problems in heat transfer, fluid mechanics, and mechanical systems. Furthermore, the availability of fast and inexpensive computers allows problems which are intractable using analyticor mechanical methods to be solved in a straightfor- ward manner using finite element methods. See also F INITE DIFFERENCE ,LATTICE POINT References Akin, J. E. Finite Elements for Analysis and Design. San Diego: Academic Press, 1994. Brenner, S. C. and Scott, L. R. The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag, 1994. Gallagher, R. H. Finite Element Analysis: Fundamentals. Englewood Cliffs, NJ: Prentice-Hall, 1975. Kwon, Y. W. and Bang, H. The Finite Element Method Using MATLAB. Boca Raton, FL: CRC Press, 1996. O¨zisik, M. N. Finite Difference Methods in Heat Transfer. Boca Raton, FL: CRC Press, 1994. Reddy, J. N. and Gartling, D. K. The Finite Element Method in Heat Transfer and Fluid Dynamics. Boca Raton, FL: CRC Press, 1994. White, R. E. An Introduction to the Finite Element Method with Applications to Nonlinear Problems. New York: Wiley, 1985. Finite Field A finite field is a FIELD with a finite ORDER (number of elements), also called a Galois field. The order of a finite field is always a PRIME or a POWER of a PRIME (Birkhoff and Mac Lane 1996). For each PRIME POWER , there exists exactly one (with the usual caveat that "exactly one" means "exactly one up toan ISOMORPHISM ") finite field GF( /pn);often written as Fpnin current usage. GF(p) is called the PRIME FIELD of order p, and is the FIELD of RESIDUE CLASSES modulo p, where the p elements are denoted 0, 1, ..., p /C281 : a /C30 b in GF(p) means the same as a /C13b(mod p) : Note, however, that 2 /C292 /C130(mod4) in the RING of residues modulo 4, so 2 has no reciprocal, and the RING of residues modulo 4 is distinct from the finite field with four elements. Finite fields are therefore denoted GF( /pn) ; instead of GF(k), where k /C30pn ; for clarity. The finite field GF(2) consists of elements 0 and 1 which satisfy the following addition and multiplica- tion tables. //C27/ 01 001 110 //C29/ 01 000 101 If a subset S of the elements of a finite field F satisfies the axioms above with the same operators of F, then S is called a SUBFIELD . Finite fields are used extensively in the study of ERROR-CORRECTING CODES . When n /C211, GF( /pn) can be REPRESENTED AS the FIELD of EQUIVALENCE CLASSES of POLYNOMIALS whose COEFFICIENTS belong to GF(p). Any IRREDUCIBLE POLYNOMIAL of degree n yields the same FIELD up to an ISOMORPHISM . For example, for GF(23), the mod- ulus can be taken as x3 /C27x2 /C271; x3 /C27x /C271 ; or any other IRREDUCIBLE POLYNOMIAL of degree 3. Using the modulus x3 /C27x /C271; the elements of GF(23)–writ- ten 0, x0 ; x1 ; ...–can be REPRESENTED AS POLYNOMIALS with degree less than 3. For instance, x3 /C13/C28x /C281 /C13x /C271 x4 /C13x(x3) /C13x(x /C271) /C13x3 /C27x x5 /C13xx2 /C27x/C0/C1 /C13x3 /C27x2 /C13x2 /C28x /C281 /C13x2 /C27x /C271 x6 /C13x(x2 /C27x /C271) /C13x3 /C27x2 /C27x /C13x2 /C281 /C13x2 /C271 x7 /C13x(x2 þ 1) /C13x3 þ x /C13/C281 /C131 /C13x0 : Now consider the following table which contains several different representations of the elements of a finite field. The columns are the power, polynomial representation, triples of polynomial representation COEFFICIENTS (the vector representation), and the binary INTEGER corresponding to the vector represen- tation (the regular representation).Power Polynomial Vector Regular 0 0 (000) 0 /x0 / 1 (001) 1 /x1/ x (010) 2 /x2// x2/ (100) 4 /x3 // x /C271/ (011) 3 /x4// x2 /C27x/ (110) 6 /x5// x2 /C27x /C271/ (111) 7 /x6// x2 /C271/ (101) 5 The set of POLYNOMIALS in the second column is CLOSED under ADDITION and MULTIPLICATION modulo x3 /C27x /C271; and these operations on the set satisfy the AXIOMS of finite field. This particular finite field is said to be an extension field of degree 3 of GF(2), written GF(23), and the field GF(2) is called the base field of GF(23). If an IRREDUCIBLE POLYNOMIAL gen- erates all elements in this way, it is called a PRIMITIVE POLYNOMIAL . For any PRIME or PRIME POWER q and any POSITIVE INTEGER n, there exists a primitive irreducible polynomial of degree n over GF(q). For any element cof GF( q),cq/C30c;and for any NONZERO element dof GF( q),dq/C281/C301:There is a smallest POSITIVE INTEGER nsatisfying the sum condition e/C27e/C27.../C27e/C300|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ntimesfor some element ein GF(q),. This number is called the CHARACTERISTIC of the finite field GF( q). The CHARACTERISTIC is a PRIME NUMBER for every finite field, and it is true that (x/C27y)p/C30xp/C27yp over a finite field with characteristic p. See also CHARACTERISTIC (FIELD), FIELD,HADAMARD MATRIX ,IRREDUCIBLE POLYNOMIAL ,PRIMITIVE POLY- NOMIAL ,RING,SUBFIELD References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 73 /C1/5, 1987. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, p. 413, 1996. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. viii, 1952. Dummit, D. S. and Foote, R. M. "Finite Fields." §14.3 in Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice- Hall, pp. 499 /C1/05, 1998. Lidl, R. and Niederreiter, H. Introduction to Finite Fields and Their Applications, rev. ed. Cambridge, England: Cambridge University Press, 1994. Lidl, R. and Niederreiter, H. (Eds.). Finite Fields, 2nd ed. Cambridge, England: Cambridge University Press, 1997. Finite Game A GAME in which each player has a finite number of moves and a finite number of choices at each move. See also GAME,HYPERGAME ,ZERO-SUM GAME References Dresher, M. The Mathematics of Games of Strategy: Theory and Applications. New York: Dover, p. 2, 1981. Finite Group AGROUP of finite ORDER . Examples of finite groups are the MODULO MULTIPLICATION GROUPS and the POINT GROUPS . The CLASSIFICATION THEOREM of finite SIMPLE GROUPS states that the finite SIMPLE GROUPS can be classified completely into one of five types. The following table gives the numbers and names of the first few groups of ORDER h. In the table, NA denotes the number of non-Abelian groups, Adenotes the number of A BELIAN GROUPS , and Nthe total number of groups. In addition, Zndenotes a CYCLIC GROUP ofORDER n,AnanALTERNATING GROUP ,Dna DIHEDRAL GROUP ,Q8the group of the QUATERNIONS , Tthe cubic group, and /C29denotes GROUP DIRECT PRODUCT . hName AN AN 1F INITE GROUP E 10 1 2F INITE GROUP Z2 10 1 3F INITE GROUP Z3 10 1 4F INITE GROUP Z2Z2,FINITE GROUP Z420 2 5F INITE GROUP Z5 10 1 6F INITE GROUP Z6,FINITE GROUP D311 2 7F INITE GROUP Z7 10 1 8F INITE GROUP Z2Z2Z2,FINITE GROUP Z2Z4,FINITE GROUP Z8, FINITE GROUP Q8,FINITE GROUP D432 5 9 /Z3/C29Z3;Z9/ 20 2 10 /Z10;D5/ 11 2 11 /Z11/ 10 1 12 /Z2/C29Z6;Z12;A4;D6;T/ 23 5 13 /Z13/ 10 1 14 /Z14;D7/ 11 2 15 /Z15/ 10 1The problem of determining the nonisomorphic finitegroups of order hwas first considered by Cayley (1854). There is no known FORMULA to give the number of possible finite groups g(h) as a function of the ORDER h. However, there are simple formulas for special forms of h. g(1)/C301 (1) g(p)/C301 (2) g(pq)/C301i f p ¶(q/C281) 2i f p½(q/C281)/C27 (3) gp2/C0/C1 /C302 (4) gp3/C0/C1 /C305; (5) where pand q/C21pare distinct primes. In addition, there is a beautiful algorithm due to Ho ¨lder (Ho ¨lder 1895, Alonso 1976) for determining g(h) for square- free h, namely g(h)/C30X d½nY p½d p"1pop(n=d)/C281 p/C281; (6) where op(m) is the number of primes psuch that q½m andp½(q/C281) (Dennis). Miller (1930) gave the number of groups for orders 1 /C1/ 00, including an erroneous 297 as the number of groups of ORDER 64. Senior and Lunn (1934, 1935) subsequently completed the list up to 215, but omitted 128 and 192. The number of groups of ORDER 64 was corrected in Hall and Senior (1964). James et al.(1990) found 2328 groups in 115 ISOCLINISM families of ORDER 128, correcting previous work, and O’Brien (1991) found the number of groups of ORDER 256. Currently, the number of groups is known for orders up to 2000, excluding 1024 (Besche andEick 1999a), with the difficult cases of orders 512 (g(512)/C3010;494;213; Eick and O’Brien 1999b) and 768 (Besche and Eick 2000) now put to rest. The numbers of nonisomorphic finite groups Nof each ORDER hfor the first few hundred orders are given in the table below (Sloane’s A000001–the very firstsequence). The number of nonisomorphic groups of orders 2 nforn/C300, 1, ... are 1, 1, 2, 5, 14, 51, 267, 2328, 56092, ... (Sloane’s A000679). The smallest orders hfor which there exist n/C301, 2, ... nonisomorphic groups are 1, 4, 75, 28, 8, 42, ... (Sloane’s A046057). The incrementally largest num- ber of nonisomorphic finite groups are 1, 2, 5, 14, 15, 51, 52, 267, 2328, ... (Sloane’s A046058), which occurfor orders 1, 4, 8, 16, 24, 32, 48, 64, 128, ... (Sloane’s A046059). Dennis has conjectured that the number of groups g(h) of order hassumes every positive integer as a value an infinite number of times. It is simple to determine the number of A BELIAN GROUPS using the K RONECKER DECOMPOSITION THEO- REM, and there is at least one A BELIAN GROUP for every finite order h. The number Aof A BELIAN GROUPS ofORDER h/C301, 2, ... are given by 1, 1, 1, 2, 1, 1, 1, 3, ... (Sloane’s A000688). The following table summarizes the total number of finite groups Nand the number of Abelian finite groups Afor orders h from 1 to 400. A table of orders up to 1000 is given by Royle; the GAP software package includes a table of the number of finite groups up to order 2000,excluding 1024. h N Ah N Ah N Ah N A 11 15 1 11 1 0 1 11 1 5 1 11 2 1 1 52 5 2 102 4 1 152 12 3 31 15 3 11 1 0 3 11 1 5 3 224 2 2 54 15 3 104 14 3 154 4 151 15 5 21 1 0 5 21 1 5 5 21 6 2 1 56 13 3 106 2 1 156 18 2 71 15 7 21 1 0 7 11 1 5 7 118 5 3 58 2 1 108 45 6 158 2 192 25 9 11 1 0 9 11 1 5 9 11 10 2 1 60 13 2 110 6 1 160 238 711 1 1 61 1 1 111 2 1 161 1 112 5 2 62 2 1 112 43 5 162 55 513 1 1 63 4 2 113 1 1 163 1 1 14 2 1 64 267 11 114 6 1 164 5 2 15 1 1 65 1 1 115 1 1 165 2 116 14 5 66 4 1 116 5 2 166 2 117 1 1 67 1 1 117 4 2 167 1 1 18 5 2 68 5 2 118 2 1 168 57 3 19 1 1 69 1 1 119 1 1 169 2 220 5 2 70 4 1 120 47 3 170 4 1 21 2 1 71 1 1 121 2 2 171 5 2 22 2 1 72 50 6 122 2 1 172 4 223 1 1 73 1 1 123 1 1 173 1 124 15 3 74 2 1 124 4 2 174 4 1 25 2 2 75 3 2 125 5 3 175 2 2 26 2 1 76 4 2 126 16 2 176 42 527 5 3 77 1 1 127 1 1 177 1 128 4 2 78 6 1 128 2328 15 178 2 1 29 1 1 79 1 1 129 2 1 179 1 130 4 1 80 52 5 130 4 1 180 37 4 31 1 1 81 15 5 131 1 1 181 1 132 51 7 82 2 1 132 10 2 182 4 133 1 1 83 1 1 133 1 1 183 2 1 34 2 1 84 15 2 134 2 1 184 12 3 35 1 1 85 1 1 135 5 3 185 1 136 14 4 86 2 1 136 15 3 186 6 137 1 1 87 1 1 137 1 1 187 1 1 38 2 1 88 12 3 138 4 1 188 4 2 39 2 1 89 1 1 139 1 1 189 13 340 14 3 90 10 2 140 11 2 190 4 141 1 1 91 1 1 141 1 1 191 1 1 42 6 1 92 4 2 142 2 1 192 1543 11 43 1 1 93 2 1 143 1 1 193 1 144 4 2 94 2 1 144 197 1 194 2 145 2 2 95 1 1 145 1 1 195 2 1 46 2 1 96 230 7 146 2 1 196 17 4 47 1 1 97 1 1 147 6 2 197 1 148 52 5 98 5 2 148 5 2 198 10 249 2 2 99 2 2 149 1 1 199 1 1 50 2 2 100 16 4 150 13 2 200 52 6 h N A h NA h NA h NA 201 2 1 251 1 1 301 2 1 351 14 3 202 2 1 252 46 4 302 2 1 352 195 7 203 2 1 253 2 1 303 1 1 353 1 1204 12 2 254 2 1 304 42 5 354 4 1205 2 1 255 1 1 305 2 1 355 2 1 206 2 1 256 56092 22 306 10 2 356 5 2 207 2 2 257 1 1 307 1 1 357 2 1208 51 5 258 6 1 308 9 2 358 2 1209 1 1 259 1 1 309 2 1 359 1 1 210 12 1 260 15 2 310 6 1 360 162 6 211 1 1 261 2 2 311 1 1 361 2 2212 5 2 262 2 1 312 61 3 362 2 1213 1 1 263 1 1 313 1 1 363 3 2 214 2 1 264 39 3 314 2 1 364 11 2 215 1 1 265 1 1 315 4 2 365 1 1 216 177 9 266 4 1 316 4 2 366 6 1 217 1 1 267 1 1 317 1 1 367 1 1 218 2 1 268 4 2 318 4 1 368 42 5 219 2 1 269 1 1 319 1 1 369 2 2 220 15 2 270 30 3 320 1640 11 370 4 1 221 1 1 271 1 1 321 1 1 371 1 1 222 6 1 272 54 5 322 4 1 372 15 2 223 1 1 273 5 1 323 1 1 373 1 1 224 197 7 274 2 1 324 176 10 374 4 1 225 6 4 275 4 2 325 2 2 375 7 3 226 2 1 276 10 2 326 2 1 376 12 3 227 1 1 277 1 1 327 2 1 377 1 1 228 15 2 278 2 1 328 15 3 378 60 3 229 1 1 279 4 2 329 1 1 379 1 1 230 4 1 280 40 3 330 12 1 380 11 2 231 2 1 281 1 1 331 1 1 381 2 1 232 14 3 282 4 1 332 4 2 382 2 1 233 1 1 283 1 1 333 5 2 383 1 1 234 16 2 284 4 2 334 2 1 384 20169 15 235 1 1 285 2 1 335 1 1 385 2 1 236 4 2 286 4 1 336 228 5 386 2 1 237 2 1 287 1 1 337 1 1 387 4 2 238 4 1 288 1045 14 338 5 2 388 5 2 239 1 1 289 2 2 339 1 1 389 1 1 240 208 5 290 4 1 340 15 2 390 12 1241 1 1 291 2 1 341 1 1 391 1 1 242 5 2 292 5 2 342 18 2 392 44 6 243 67 7 293 1 1 343 5 3 393 1 1244 5 2 294 23 2 344 12 3 394 2 1245 2 2 295 1 1 345 1 1 395 1 1 246 4 1 296 14 3 346 2 1 396 30 4 247 1 1 297 5 3 347 1 1 397 1 1248 12 3 298 2 1 348 12 2 398 2 1249 1 1 299 1 1 349 1 1 399 5 1 250 15 3 300 49 4 350 10 2 400 221 10 See also ABELIAN GROUP ,ABHYANKAR’S CONJECTURE ,ALTERNATING GROUP ,BURNSIDE’S LEMMA ,BURNSIDE PROBLEM ,CHEVALLEY GROUPS ,CLASSIFICATION THE- OREM ,C OMPOSITION SERIES ,C ONTINUOUS GROUP , DIHEDRAL GROUP ,DISCRETE GROUP ,FEIT-THOMPSON THEOREM ,GROUP ,INFINITE GROUP ,JORDAN- HO¨ LDER THEOREM ,K RONECKER DECOMPOSITION THEOREM , LIE GROUP ,LIE-TYPE GROUP ,LINEAR GROUP ,M OD- ULO MULTIPLICATION GROUP ,O RDER (GROUP ), ORTHOGONAL GROUP , P-GROUP ,POINT GROUPS ,SIM- PLE GROUP ,SPORADIC GROUP ,SYMMETRIC GROUP , SYMPLECTIC GROUP ,TWISTED CHEVALLEY GROUPS , UNITARY GROUP References Alonso, J. "Groups of Square-Free Order, an Algorithm." Math. Comput. 30, 632/C1/37, 1976. Arfken, G. "Discrete Groups." §4.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 243 /C1/51, 1985. Artin, E. "The Order of the Classical Simple Groups." Comm. Pure Appl. Math. 8, 455/C1/72, 1955. Aschbacher, M. Finite Group Theory, 2nd ed. Cambridge, England: Cambridge University Press, 2000. Aschbacher, M. The Finite Simple Groups and Their Classification. New Haven, CT: Yale University Press, 1980. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 73 /C1/5, 1987. Besche, H.-U. and Eick, B. "Construction of Finite Groups." J. Symb. Comput. 27, 387/C1/04, 1999. Besche, H.-U. and Eick, B. "The Groups of Order at Most 1000 Except 512 and 768." J. Symb. Comput. 27, 405/C1/13, 1999. Besche, H.-U. and Eick, B. "The Groups of Order qn/C215p:/"I n preparation, 2000. Cayley, A. "On the Theory of Groups as Depending on the Symbolic Equation un/C301:/"Philos. Mag. 7,3 3/C1/9, 1854. Cayley, A. "On the Theory of Groups as Depending on the Symbolic Equation un/C301:/--Part II." Philos. Mag. 7, 408/C1/ 09, 1854. Cayley, A. "On the Theory of Groups as Depending on the Symbolic Equation un/C301:/--Part III." Philos. Mag. 18,3 4/C1/ 7, 1859. Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Sub- groups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, 1985. Dennis, K. "The Number of Groups of Order n." Preprint. Eick, B. and O’Brien, E. A. "Enumerating p-Groups." J. Austral. Math. Soc. Ser. A 67, 191/C1/05, 1999a. Eick, B. and O’Brien, E. A. "The Groups of Order 512." In Algorithmic Algebra and Number Theory: Selected Papersfrom the Conference held at the University of Heidelberg,Heidelberg, October 1997 (Ed. B. H. Matzat, G.-M. Greuel, and G. Hiss). Berlin: Springer-Verlag, pp. 379 /C1 / 80, 1999b. GAP Group. "GAP--Groups, Algorithms, and Programming." http://www-history.mcs.st-and.ac.uk/~gap/. Hall, M. Jr. and Senior, J. K. The Groups of Order 2n(n56):/ New York: Macmillan, 1964. Ho¨lder, O. "Die Gruppen der Ordnung p3;pq2;pqr,p4:/" Math. Ann. 43, 300/C1/12, 1893. Ho¨lder, O. "Die Gruppen mit quadratfreier Ordnungszahl." Nachr. Ko ¨nigl. Gesell. Wissenschaft. Go ¨ttingen, Math.- Phys. Kl. , 211/C1/29, 1895. Huang, J.-S. "Finite Groups." Part I in Lectures on Repre- sentation Theory. Singapore: World Scientific, pp. 1 /C1/5, 1999. James, R. "The Groups of Order p6(pan Odd Prime)." Math. Comput. 34, 613/C1/37, 1980. James, R.; Newman, M. F.; and O’Brien, E. A. "The Groups of Order 128." J. Algebra 129, 136/C1/58, 1990. Laue, R. "Zur Konstruktion und Klassifikation endlicher auflo¨sbarer Gruppen." Bayreuther Mathemat. Schriften 9, 1982. Miller, G. A. "Determination of All the Groups of Order 64." Amer. J. Math. 52, 617/C1/34, 1930. Miller, G. A. "Orders for which a Given Number of Groups Exist." Proc. Nat. Acad. Sci. 18, 472/C1/75, 1932. Miller, G. A. "Orders for which there Exist Exactly Four or Five Groups." Proc. Nat. Acad. Sci. 18, 511/C1/14, 1932. Miller, G. A. "Groups whose Orders Involve a Small Number of Unity Congruences." Amer. J. Math. 55,2 2/C1/8, 1933. Miller, G. A. "Historical Note on the Determination of Abstract Groups of Given Orders." J. Indian Math. Soc. 19, 205/C1/10, 1932. Miller, G. A. "Enumeration of Finite Groups." Math. Stu- dent 8, 109/C1/11, 1940. Murty, M. R. and Murty, V. K. "On the Number of Groups of a Given Order." J. Number Th. 18, 178/C1/91, 1984. Neubu ¨ser, J. Die Untergruppenverba ¨nde der Gruppen der Ordnung 5100mit Ausnahme der Ordnungen 64 und 96. Habilitationsschrift. Kiel, Germany: Universita ¨t Kiel, 1967. O’Brien, E. A. "The Groups of Order 256." J. Algebra 143, 219/C1/35, 1991. O’Brien, E. A. and Short, M. W. "Bibliography on Classifica- tion of Finite Groups." Manuscript, Australian National University, 1988. Royle, G. "Numbers of Small Groups." http://www.cs.uwa.e- du.au/~gordon/remote/group1000.html. Senior, J. K. and Lunn, A. C. "Determination of the Groups of Orders 101 /C1/61, Omitting Order 128." Amer. J. Math. 56, 328/C1/38, 1934. Senior, J. K. and Lunn, A. C. "Determination of the Groups of Orders 162 /C1/15, Omitting Order 192." Amer. J. Math. 57, 254/C1/60, 1935. Simon, B. Representations of Finite and Compact Groups. Providence, RI: Amer. Math. Soc., 1996. Sloane, N. J. A. Sequences A000001/M0098, A000679/ M1470, A000688/M0064, A046057, A046058, andA046059 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Spiro, C. A. "Local Distribution Results for the Group- Counting Function at Positive Integers." Congr. Numer. 50, 107/C1 /10, 1985. University of Sydney Computational Algebra Group. "The Magma Computational Algebra for Algebra, NumberTheory and Geometry." http://www.maths.usyd.e-du.au:8000/u/magma/. Weisstein, E. W. "Groups." M ATHEMATICA NOTEBOOK GROUPS.M . Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/.Finite Group D3 The DIHEDRAL GROUP D3is one of the two groups of ORDER 6. It is the non-Abelian group of smallest ORDER . Examples of D3include the POINT GROUPS known as C3h;C3v;S3;D3;the symmetry group of the EQUILATERAL TRIANGLE , and the group of permuta- tion of three objects. Its elements Aisatisfy A3 i/C301; and four of its elements satisfy A2i/C301;where 1 is the IDENTITY ELEMENT . The CYCLE GRAPH is shown above, and the MULTIPLICATION TABLE is given below (Cotton 1990, p. 12). /D3/1ABCDE 11 ABCDE AA 1DEBC BBE 1DCA CC D E 1AB DDCABE 1 EEBCA 1D The CONJUGACY CLASSES aref1g(which is always in a class by itself), fA;B;Cg; A/C281AA/C30A (1) B/C281AB/C30C (2) C/C281AC/C30B (3) D/C281AD/C30C (4) E/C281AE/C30B; (5) and fD;Eg; A/C281DA/C30E (6) B/C281DB/C30D: (7) A reducible 2-D representation using REAL MATRICES can be found by performing the spatial rotations corresponding to the symmetry elements of C3v:Take the Z-AXIS along the C3axis. I/C30Rz(0)/C3010 01/C20/C21 (8) A/C30Rz2 3P ! /C30cos23P ! sin23P ! /C28sin23P ! cos23P !2 666643 77775 /C30/C28 1 2/C2812ffiffiffi 3p 1 2ffiffiffi 3p /C281 22 66643 7775(9) B/C30R z4 3P ! /C30/C281 212ffiffiffi 3p /C281 2ffiffiffi 3p /C281 22 66643 7775(10) C/C30R c(P)/C30/C2810 01/C20/C21 (11) D/C30RD(P)/C30CB/C301 2/C2812ffiffiffi 3p /C281 2ffiffiffi 3p /C281 22 66643 7775(12) E/C30R E(P)/C30CA/C301 212ffiffiffi 3p 1 2ffiffiffi 3p /C281 22 66643 7775(13) To find the irreducible representation, note that there are three CONJUGACY CLASSES .GROUP rule 5 requires that there be three irreducible representations satis- fying h/C30l2 1/C27l22/C27l23/C306; (14) so it must be true that l1/C30l2/C301;l3/C302: (15) By GROUP rule 6, we can let the first representation have all 1s. /D3/1ABCDE /G1/111 1 1 1 To find a representation orthogonal to the totally symmetric representation, we must have three /C271 and three /C281CHARACTERS . We can also add the constraint that the components of the IDENTITY ELEMENT 1 be positive. The three CONJUGACY CLASSES have 1, 2, and 3 elements. Since we need a total of three/C271/s and we have required that a /C271 occur for the CONJUGACY CLASS ofORDER 1, the remaining /C271s must be used for the elements of the CONJUGACY CLASS ofORDER 2, i.e., DandE./D3/1ABC D E /G1/1 111 1 1 /G2/1/C281/C281/C28111 Using GROUP rule 1, we see that 12/C2712/C27x2 3(1)/C306 (16) so the final representation for 1 has CHARACTER 2. Orthogonality with the first two representations (GROUP rule 3) then yields the following constraints: 1/C2151/C2152/C271/C2152/C215x2/C271/C2153/C215x3/C302/C272x2/C273x3/C300 (17) 1/C2151/C2152/C271/C2152/C215x2/C27(/C281) /C2153/C215x3/C302/C272x2/C283x3/C300:(18) Solving these simultaneous equations by adding andsubtracting (18) from (17), we obtain x 2/C30/C281;x3/C300: The full CHARACTER TABLE is then /D3/1ABCDE /G1/1 11111 /G2/1/C281/C281/C2811 1 /G3/2 000 /C281/C281 Since there are only three CONJUGACY CLASSES , this table is conventionally written simply as /D3/1 /A/C30B/C30C/D/C30E /G1/11 1 /G2/1/C2811 /G3/20 /C281 Writing the irreducible representations in matrix form then yields 1/C301000 0100001000012 6643 775(19) A/C3010 0 0 01 0 0 00 /C28 1 2/C2812ffiffiffi 3p 001 2ffiffiffi 3p /C281 22 666666643 77777775(20) B /C3010 0 0 01 0 0 00 /C281 212ffiffiffi 3p 00 /C281 2ffiffiffi 3p /C281 22 666666643 77777775(21) C /C3010 00 0 /C28100 00 /C2810 00 012 6643 775 (22) D /C3010 0 0 01 0 0 00 /C28 1 212ffiffiffi 3p 00 /C281 2ffiffiffi 3p /C281 22 666666643 77777775(23) E /C3010 0 0 0 /C2810 0 00 1 212ffiffiffi 3p 001 2ffiffiffi 3p /C281 22 666666643 77777775(24) See also D IHEDRAL GROUP ,FINITE GROUP D4,FINITE GROUP Z6 Finite Group D4 The DIHEDRAL GROUP D4 is one of the two non-Abelian groups of the five groups total of ORDER 8. It is sometimes called the octic group. Examples of D4 include the symmetry group of the SQUARE . The CYCLE GRAPH is shown above. See also DIHEDRAL GROUP ,FINITE GROUP D3,FINITE GROUP Z8,FINITE GROUP Z2Z2Z2,FINITE GROUP Z2Z4,FINITE GROUP Z8 References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990. Finite Group e The unique (and trivial) group of ORDER 1 is denoted ehi: It is (trivially) ABELIAN and CYCLIC . Examplesinclude the POINT GROUP C1and the integers modulo 1 under addition. / ehi / 1 11 Its only conjugacy class is f1g:/ Finite Group Q8 One of the two non-Abelian groups of the five groups total of ORDER 8. The group Q8has the MULTIPLICA- TION TABLE of 91;i ;j ;k; where 1, i, j, and k are the QUATERNIONS . The CYCLE GRAPH is shown above. See also FINITE GROUP D4,FINITE GROUP Z2Z2Z2, FINITE GROUP Z2Z4,FINITE GROUP Z8,QUATERNION Finite Group Z2 The unique group of ORDER 2.Z2is both A BELIAN and CYCLIC . Examples include the POINT GROUPS Cs;Ci; andC2;the integers modulo 2 under addition, and the MODULO MULTIPLICATION GROUPS M3;M4;and M6: The elements Aisatisfy A2 i/C301;where 1 is the IDENTITY ELEMENT . The CYCLE GRAPH is shown above, and the MULTIPLICATION TABLE is given below. /Z2/1A 11 A AA 1 The CONJUGACY CLASSES are f1gand fAg:The irreducible representation for the C2group is f1;/C281g:/ Finite Group Z2Z2 One of the two groups of ORDER 4. The name of this group derives from the fact that it is a GROUP DIRECT PRODUCT of two Z2SUBGROUPS . Like the group Z4; Z2/C29Z2is an A BELIAN GROUP . Unlike Z4;however, it is not CYCLIC . In addition to satisfying A4 i/C301 for each element Ai;it also satisfies A2i/C301;where 1 is the IDENTITY ELEMENT . Examples of the Z2/C29Z2group include the VIERGRUPPE ,POINT GROUPS D2;C2h;and C2v;and the MODULO MULTIPLICATION GROUPS M8and M12:That M8;the RESIDUE CLASSES prime to 8 given byf1;3;5;7g;are a group of type Z2/C29Z2can be shown by verifying that 12/C30132/C309/C13152/C3025/C131 72/C3049/C131 (mod 8)(1) and 3/C2155/C3015/C1373 /C2157/C3021/C1355 /C2157/C3035/C133 (mod 8) :(2) /Z2/C29Z2is therefore a MODULO MULTIPLICATION GROUP . The CYCLE GRAPH is shown above, and the multi- plication table for the Z2/C29Z2group is given below (Cotton 1990, p. 11). /Z2/C29Z2/1ABC 11 ABC AA 1CB BB C 1A C CBA 1 The CONJUGACY CLASSES are f1g;fAg; A/C281AA/C30A (3) B/C281AB/C30A (4) C/C281AC/C30A; (5) /fBg; A/C281BA/C30B (6) C/C281BC/C30B; (7) and fCg:/Now explicitly consider the elements of the C2vPOINT GROUP . /C2v/E /C2//sv//sv/ EE /C2//sv//s?v/ /C2//C2/E /s?v//sv/ /sv//sv//s?v/E /C2/ /s?v//s?v//sv//C2/E In terms of the VIERGRUPPE elements VI /V1//V2//V3/ I /V1//V2//V3//V4/ /V1//V1/I /V3//V2/ /V2//V2//V3/I /V1/ /V3//V3//V2//V1/I A reducible representation using 2-D REAL MATRICES is 1/C3010 01/C20/C21 (8) A/C30/C2810 0/C281/C20/C21 (9) B/C3001 10/C20/C21 (10) C/C300/C281 /C2810/C20/C21 : (11) Another reducible representation using 3-D REAL MATRICES can be obtained from the symmetry ele- ments of the D2group (1, C2(z);C2(y);and C2(x)) or C2vgroup (1, C2;sv;ands?v):Place the C2axis along the Z-AXIS ,svin the x-yplane, and s?vin the y-z plane. 1/C30E/C30E/C30100 010 0012 435 (12) A/C30R x(P)/C30sv/C30100 0/C2810 0012 435 (13) C/C30R z(P)/C30C2/C30/C28100 0/C2810 00 12435 (14) B /C30Ry( P) /C30 s ?n /C30/C28100 010 0012 435: (15) In order to find the irreducible representations, note that the traces are given by x(1) /C303 ; x C 2ðÞ/C30/C281 and xsvðÞ/C30 xs? vðÞ/C301 Therefore, there are at least three distinct CONJUGACY CLASSES . However, we see from the MULTIPLICATION TABLE that there are actually four CONJUGACY CLASSES ,so GROUP rule 5 requires that there must be four irreducible representations. By GROUP rule 1, we are looking for POSITIVE INTEGERS which satisfy l2 1 /C27l22 /C27l23 /C27l24 /C304: (16) The only combination which will work is l1 /C30l2 /C30l3 /C30l4 /C301; (17) so there are four one-dimensional representations. GROUP rule 2 requires that the sum of the squares equal the ORDER h /C304, so each 1-D representation must have CHARACTER 91. GROUP rule 6 requires that a totally symmetric representation always exists, so we are free to start off with the first representation having all 1s. We then use orthogonality (GROUP rule 3) to build up the other representations. The simplest solution is then given by /C2v/ 1 /C2//sv//s?v/ / G1/ 11 11 / G2/ 1-1-11 / G3/ 1-1 1-1 / G4/ 1 1 -1 -1 These can be put into a more familiar form by switching G1 and G3 ; giving the CHARACTER TABLE /C2v/ 1 /C2//sv//s?v/ / G3/ 1-1 1-1 / G2/ 1-1-11 / G1/ 11 11 / G4/ 1 1 -1 -1 The matrices corresponding to this representation are now 1 /C301000 0100 0010 00012 6643 775 (18)C2 /C30/C281000 0 /C28100 0010 00012 6643 775 (19) s v /C301000 0 /C2810 0 0010 000 /C2812 6643 775 (20) sv ?/C30/C28100 0 0100 0010 000 /C2812 6643 775 (21) which consist of the previous representation with an additional component. These matrices are now ortho- gonal, and the order equals the matrix dimension. As before, xs vðÞ/C30 xs1 vðÞ :/ See also CYCLIC GROUP ,FINITE GROUP Z4 References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990. Finite Group Z2Z2Z2 One of the three Abelian groups of the five groups total of ORDER 8. Examples include the MODULO MULTIPLICATION GROUP M24 : The elements Aiof this group satisfy A2i/C301;where 1 is the IDENTITY ELE- MENT . The CYCLE GRAPH is shown above. See also FINITE GROUP D4,FINITE GROUP Q8,FINITE GROUP Z2Z4,FINITE GROUP Z8 Finite Group Z2Z4 One of the three Abelian groups of the five groups total of ORDER 8. Examples include the MODULO MULTIPLICATION GROUPS M15;M16;M20;and M30: The elements Aiof this group satisfy A4 i/C301;where 1 is the IDENTITY ELEMENT , and four of the elements satisfy A2 i /C301: The CYCLE GRAPH is shown above. See also FINITE GROUP D4,FINITE GROUP Q8,FINITE GROUP Z2Z2Z2,FINITE GROUP Z8 Finite Group Z3 The unique group of ORDER 3. It is both ABELIAN and CYCLIC . Examples include the POINT GROUPS C3and D3and the integers under addition modulo 3. The elements Ai of the group satisfy A3i /C301 where 1 is the IDENTITY ELEMENT . The CYCLE GRAPH is shown above, and the MULTIPLICATION TABLE is given below (Cotton 1990, p. 10). /Z3/ 1 AB 11 AB AAB 1 BB 1 A The CONJUGACY CLASSES are f1g;fAg; A/C281AA /C30A B /C281AB /C30A; and fB g; A/C281BA /C30B B /C281BB /C30B: The irreducible representation (CHARACTER TABLE )is therefore /G/1AB /G1/111 /G2/11 /C281 /G3/1/C2811 See also CYCLIC GROUP References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, 1990.Finite Group Z4 One of the two groups of ORDER 4. Like Z2/C29Z2;it is ABELIAN , but unlike Z2/C29Z2;it is a CYCLIC . Examples include the POINT GROUPS C4andS4and the MODULO MULTIPLICATION GROUPS M5andM10:Elements Aiof the group satisfy A4i/C301;where 1 is the IDENTITY ELEMENT , and two of the elements satisfy A2i/C301:/ The CYCLE GRAPH is shown above. The MULTIPLICA- TION TABLE for this group may be written in three equivalent ways */denoted here by Z(1) 4;Z(2)4;and Z(3)4/*/by permuting the symbols used for the group elements. (Cotton 1990, p. 11). /Z(1)4/1ABC 11 ABC AA B C 1 BB C 1A CC 1AB The MULTIPLICATION TABLE forZ(2)4is obtained from Z(1)4by interchanging AandB. /Z(2) 4/1ABC 11 ABC AA 1CB BB C A 1 CC B 1A The MULTIPLICATION TABLE forZ(3) 4is obtained from Z(1)4by interchanging AandC. /Z(3)4/1ABC 11 ABC AA C 1B BB 1CA CC B A 1 The CONJUGACY CLASSES of Z4 are f1g;fAg; A/C281AA /C30A (1) B/C281AB /C30A (2) C /C281AC /C30A; (3) /fB g; A/C281BA /C30B (4) B /C281BB /C30B (5) C /C281BC /C30B ; (6) and fC g:/ The group may be given a reducible representation using COMPLEX NUMBERS 1 /C301 (7) A /C30i (8) B /C30/C281 (9) C /C30/C28i; (10) or REAL MATRICES 1 /C3010 01/C20/C21 (11) A /C300 /C281 10/C20/C21 (12) B /C30/C2810 0 /C281/C20/C21 (13) C /C3001 /C2810/C20/C21 : (14) See also CYCLIC GROUP ,FINITE GROUP Z2Z2 References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990. Finite Group Z5 The unique GROUP of ORDER 5, which is ABELIAN . Examples include the POINT GROUP C5and the integers mod 5 under addition. The elements Aisatisfy A5 i /C301; where 1 is the IDENTITY ELEMENT . The CYCLE GRAPH is shown above, and the MULTI- PLICATION TABLE is illustrated below. /Z5/ 1 ABCD 11 ABCD AABCD 1 BBCD 1 A CCD 1 AB DD 1 ABC The CONJUGACY CLASSES are f1g;fAg;fB g;fCg; and fDg:/ See also CYCLIC GROUP Finite Group Z6 One of the two groups of ORDER 6 which, unlike D3;is ABELIAN . It is also a CYCLIC . It is isomorphic to Z2/C29 Z3::Examples include the POINT GROUPS C6and S6; the integers modulo 6 under addition, and the MODULO MULTIPLICATION GROUPS M7;M9;and M14: The elements Aiof the group satisfy A6i/C301;where 1 is the IDENTITY ELEMENT , three elements satisfy A3i/C301; and two elements satisfy A2i/C301:The CYCLE GRAPH is shown above, and the MULTIPLICATION TABLE is given below. /Z6/1ABCDE 11 ABCDE AABCDE /1/ BBCDE 1A CCDE 1AB DDE 1ABC EE 1ABCD The CONJUGACY CLASSES are f1g;fAg;fB g;fCg;fDg; and fE g:/ See also CYCLIC GROUP ,FINITE GROUP D3 Finite Group Z7 The unique GROUP of ORDER 7. It is ABELIAN and CYCLIC . Examples include the POINT GROUP C7and the integers modulo 7 under addition. The elements Ai of the group satisfy A7 i /C301; where 1 is the IDENTITY ELEMENT . The CYCLE GRAPH is shown above. /Z7/ 1 ABCDEF 11 ABCDEF AABCDEF 1 BBCDEF 1 A CCDEF 1 AB DDEF 1 ABC EEF 1 ABCD FF 1 ABCDE The CONJUGACY CLASSES are f1g;fAg;fB g;fCg;fDg; fE g; and fF g:/ See also CYCLIC GROUP Finite Group Z8 One of the three Abelian groups of the five groups total of ORDER 8. An example is the residue classes modulo 17 which QUADRATIC RESIDUES , i.e., f1; 2;4; 8;9;13 ;15 ;16 g under multiplication modulo17. The elements Aisatisfy A8i /C301; four of them satisfy A4i /C301; and two satisfy A2i /C301: The CYCLE GRAPH is shown above. See also CYCLIC GROUP ,FINITE GROUP D4,FINITE GROUP Q8,F INITE GROUP Z2Z4,F INITE GROUP Z2Z2Z2 Finite Mathematics The branch of mathematics which does not involve infinite sets, limits, or continuity. See also COMBINATORICS ,DISCRETE MATHEMATICS References Hildebrand, F. H. and Johnson, C. G. Finite Mathematics. Boston, MA: Prindle, Weber, and Schmidt, 1970. Kemeny, J. G.; Snell, J. L.; and Thompson, G. L. Introduc- tion to Finite Mathematics, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1974. Marcus, M. A Survey of Finite Mathematics. New York: Dover, 1993. Weisstein, E. W. "Books about Finite Mathematics." http:// www.treasure-troves.com/books/FiniteMathematics.html. Finite Order An ENTIRE FUNCTION f is said to be of finite order if there exist numbers a; r > 0 such that ½f(z)½5exp ½z ½aðÞ for all ½z½> r : The INFIMUM of all numbers a for which this inequality holds is called the ORDER of f, denoted l /C30 l(f) :/ See also ENTIRE FUNCTION ,ORDER (FUNCTION ) References Krantz, S. G. "Finite Order." §9.3.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 121, 1999. Finite Projective Plane PROJECTIVE PLANE Finite Simple Group SIMPLE GROUP Finite Simple Group Classification Theorem CLASSIFICATION THEOREM Finitely Generated A GROUP G is said to be finitely generated if there exists a finite set of GENERATORS forG. See also GENERATOR (GROUP ) Finite-to-One Factor AMAPc:M0M;where Mis a MANIFOLD , is a finite- to-one factor of a MAPC:X0Xif there exists a continuous ONTO MAP P : X 0 M such that c( P/C30 P( C and P/C281(x) ƒX is finite for each x /C23 M :/ Finsler Geometry The geometry of F INSLER SPACE . Finsler Manifold FINSLER SPACE Finsler Metric A continuous real function L(x;y) defined on the TANGENT BUNDLE T(M)ofan n-D DIFFERENTIABLE MANIFOLD M is said to be a Finsler metric if 1. L(x;y)is DIFFERENTIABLE at x "y;/ 2. L(x ; ly) /C30½l ½L(x;y) for any element ( x;y) /C23 T(M) and any REAL NUMBER l ;/ 3. Denoting the METRIC gij(x;y) /C301 2@2 L(x;y) ½/C1382 @yi @yj; then /gij/ is a POSITIVE DEFINITE MATRIX . A DIFFERENTIABLE MANIFOLD M with a Finsler metric is called a F INSLER SPACE . See also DIFFERENTIABLE MANIFOLD ,FINSLER SPACE , TANGENT BUNDLE References Iyanaga, S. and Kawada, Y. (Eds.). "Finsler Spaces." §161 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 540 /C1/42, 1980. Finsler Space A general space based on the LINE ELEMENT ds /C30Fx1 ;...;xn;dx1 ;...;dxn/C0/C1 ; with F(x;y) > 0 for y "0 a function on the TANGENT BUNDLE T(M) ; and homogeneous of degree 1 in y. Formally, a Finsler space is a DIFFERENTIABLE MANI- FOLD possessing a FINSLER METRIC . Finsler geometry is RIEMANNIAN GEOMETRY without the restriction that the LINE ELEMENT be quadratic and OF THE FORM F2 /C30gij(x)dxidxj : A compact boundaryless Finsler space is locally Minkowskian IFF it has 0 "flag curvature." See also FINSLER METRIC ,H ODGE’S THEOREM ,RIE- MANNIAN GEOMETRY ,TANGENT BUNDLE References Akbar-Zadeh, H. "Sur les espaces de Finsler a` courbures sectionnelles constantes." Acad. Roy. Belg. Bull. Cl. Sci. 74, 281 /C1/22, 1988.Bao, D.; Chern, S.-S.; and Shen, Z. (Eds.). Finsler Geometry. Providence, RI: Amer. Math. Soc., 1996. Chern, S.-S. "Finsler Geometry is Just Riemannian Geome- try without the Quadratic Restriction." Not. Amer. Math. Soc. 43, 959 /C1/63, 1996. Iyanaga, S. and Kawada, Y. (Eds.). "Finsler Spaces." §161 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 540 /C1/42, 1980. Finsler-Hadwiger Theorem Let the SQUARES IABCD and IAB ?C?D ? share a common VERTEX A. The midpoints Q and S of the segments B?D and BD? together with the centers of the original squares R and T then form another square IQRST : This theorem is a special case of the FUNDAMENTAL THEOREM OF DIRECTLY SIMILAR FIG- URES (Detemple and Harold 1996). See also DIRECTLY SIMILAR ,FUNDAMENTAL THEOREM OF DIRECTLY SIMILAR FIGURES ,SQUARE References Detemple, D. and Harold, S. "A Round-Up of Square Problems." Math. Mag. 69,1 5/C1/7, 1996. Finsler, P. and Hadwiger, H. "Einige Relationen im Dreieck." Comment. Helv. 10, 316/C1/26, 1937. Fisher, J. C.; Ruoff, D.; and Shileto, J. "Polygons and Polynomials." In The Geometric Vein: The Coxeter Fes- tschrift. New York: Springer-Verlag, 321 /C1/33, 1981. First Curvature CURVATURE First Derivative Test Suppose f(x)i s CONTINUOUS at a STATIONARY POINT x0:/ 1. If f?(x)>0o na n OPEN INTERVAL extending left from x0andf?(x)B0o na n OPEN INTERVAL extend- ing right from x0 ; then f(x) has a RELATIVE MAXIMUM (possibly a GLOBAL MAXIMUM )atx0 :/ 2. If f ?(x) B0onan OPEN INTERVAL extending left from x0 and f ?(x) > 0onan OPEN INTERVAL extend- ing right from x0 ; then f(x) has a RELATIVE MINIMUM (possibly a GLOBAL MINIMUM )atx0 :/ 3. If f ?ðxÞ has the same sign on an OPEN INTERVAL extending left from x0and on an OPEN INTERVAL extending right from x0 ; then f(x) does not have a RELATIVE EXTREMUM at x0 :/ See also EXTREMUM ,G LOBAL MAXIMUM ,G LOBAL MINIMUM ,INFLECTION POINT ,M AXIMUM ,M INIMUM , RELATIVE EXTREMUM ,RELATIVE MAXIMUM ,RELATIVE MINIMUM ,S ECOND DERIVATIVE TEST,S TATIONARY POINT References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. First Digit Law BENFORD’S LAW First Digit Phenomenon BENFORD’S LAW First Fundamental Form Let M be a REGULAR SURFACE with vp ;wppoints in the TANGENT SPACE MPof M. Then the first funda- mental form is the INNER PRODUCT of tangent vectors, Ivp ;wp/C0/C1 /C30vp /C215wp : (1) The first fundamental form satisfies I axu /C27bxv ;axu /C27bxv ðÞ /C30Ea2 /C272Fab /C27Gb2 : (2) The first fundamental form (or LINE ELEMENT )is given explicitly by the RIEMANNIAN METRIC ds2 /C30Edu2 /C272Fdudv /C27Gdv2 : (3) It determines the ARC LENGTH of a curve on a surface. The coefficients are given by E /C30xuu /C30@x @u/C12/C12/C12/C12/C12/C12/C12/C12/C12/C122 (4) F /C30xuv /C30@x @u /C215@x @v (5) G /C30xvv /C30@x @v/C12/C12/C12/C12/C12/C12/C12/C12/C12/C122 : (6) The coefficients are also denoted guu /C30E; guv /C30F ; and gvv /C30G : In CURVILINEAR COORDINATES (where F /C30 0), the quantitieshu /C13ffiffiffiffiffiffiffiguup/C30ffiffiffiffi Ep (7) hv /C13ffiffiffiffiffiffiffigvvp/C30ffiffiffiffi Gp (8) are called SCALE FACTORS . See also FUNDAMENTAL FORMS ,SECOND FUNDAMEN- TAL FORM,THIRD FUNDAMENTAL FORM References Gray, A. "The Three Fundamental Forms." §16.6 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 380 /C1/82, 1997. First Kind Special functions which arise as solutions to second order ordinary differential equations are commonly said to be "of the first kind" if they are nonsingular at the origin, while the corresponding linearly indepen- dent solutions which are singular are said to be "of the second kind." Common examples of functions of the first kind defined in this way include the BESSEL FUNCTION OF THE FIRST KIND ,CHEBYSHEV POLYNO- MIAL OF THE FIRST KIND , CONFLUENT HYPERGEO- METRIC FUNCTION OF THE FIRST KIND ,H ANKEL FUNCTION OF THE FIRST KIND , and so on. The term "first kind" is also used in a more general context to distinguish between two or more types of mathematical objects which, however, all satisfy some common overall property. Examples of objects of this kind include the CHRISTOFFEL SYMBOL OF THE FIRST KIND , ELLIPTIC INTEGRAL OF THE FIRST KIND , FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND , STIRLING NUMBER OF THE FIRST KIND ,V OLTERRA INTEGRAL EQUATION OF THE FIRST KIND , and so on. See also BESSEL FUNCTION OF THE FIRST KIND, CHEBYSHEV POLYNOMIAL OF THE FIRST KIND,CON- FLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND,E LLIPTIC INTEGRAL OF THE FIRST KIND, FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND, HANKEL FUNCTION OF THE FIRST KIND,SECOND KIND, SPECIAL FUNCTION ,STIRLING NUMBER OF THE FIRST KIND,THIRD KIND,VOLTERRA INTEGRAL EQUATION OF THE FIRST KIND First Multiplier Theorem LetDbe a planar Abelian DIFFERENCE SET andtbe any DIVISOR ofn. Then tis a numerical multiplier of D, where a multiplier is defined as an automorphism aof a GROUP Gwhich takes Dto a translation g/C27Dof itself for some g/C23G:IfaisOF THE FORM a:x0txfor t/C23Zrelatively prime to the order of G, then ais called a numerical multiplier. References Gordon, D. M. "The Prime Power Conjecture is True for nB2;000;000:/"Electronic J. Combinatorics 1,R 61 /C1/, 1994. http://www.combinatorics.org/Volume_1/volu- me1.html#R6. First-Countable Space A TOPOLOGICAL SPACE in which every point has a countable BASE for its neighborhood system. Fischer Groups The SPORADIC GROUPS Fi22 ; Fi23 ; and Fi?24 : These groups were discovered during the investigation of 3-TRANSPOSITION GROUPS . See also SPORADIC GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#spo. Fischer’s Baby Monster Group BABY MONSTER GROUP Fish Bladder LENS Fisher Index The statistical INDEX PB /C13ffiffiffiffiffiffiffiffiffiffiffiffiffi PLPP ;p where PLis LASPEYRES’ INDEX and PPis PAASCHE’S INDEX . See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962. Fisher Kurtosis g2 /C13b2 /C13m4 m2 2/C283 /C30m4 s4 /C283; where mi is the ith MOMENT about the MEAN and s /C30ffiffiffiffiffim2pis the STANDARD DEVIATION . See also FISHER SKEWNESS ,K URTOSIS ,P EARSON KURTOSIS Fisher Sign Test A robust nonparametric test which is an alternative to the PAIRED T-TEST . This test makes the basic assumption that there is information only in the signs of the differences between paired observations, not in their sizes. Take the paired observations, calculate the differences, and count the number of /C27sn /C27 and /C28/s n/C28; whereN /C13n/C27/C27n/C28 is the sample size. Calculate the BINOMIAL COEFFI- CIENT B /C13N n/C27/C1Y/C1Q : Then B =2N gives the probability of getting exactly this many /C27s and /C28sif POSITIVE and NEGATIVE values are equally likely. Finally, to obtain the P- VALUE for the test, sum all the COEFFICIENTS that are 5B and divide by 2N :/ See also HYPOTHESIS TESTING Fisher Skewness g1 /C30m3 m3 =2 2/C30m3 s3 ; where miis the i MOMENT about the MEAN , and s /C30ffiffiffiffiffim 2pis the STANDARD DEVIATION . See also FISHER KURTOSIS ,M OMENT ,S KEWNESS , STANDARD DEVIATION Fisher’s Block Design Inequality A balanced incomplete BLOCK DESIGN (v, k, l; r, b) exists only for b ]v (or, equivalently, r ]k):/ See also BRUCK- RYSER- CHOWLA THEOREM References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1 /C1/2, 1992. Fisher’s Equation The PARTIAL DIFFERENTIAL EQUATION ut/C30Duxx/C27u/C28u2: References Kaliappan, P. "An Exact Solution for Travelling Waves of ut/C30Duxx/C27u/C28uk:/"Physica D 11, 368/C1/74, 1984. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Fisher’s Estimator Inequality Given TanUNBIASED ESTIMATOR of/u/so that //C142T/C143/C30u/. Then var(T) ]1 Ng/C12 /C28/C12@(ln f) @ u"#2 fdx; where var is the VARIANCE . Fisher’s Exact Test A STATISTICAL TEST used to determine if there are nonrandom associations between two CATEGORICAL VARIABLES . Let there exist two such variables X and Y, with m and n observed states, respectively. Now form an n /C29 m MATRIX in which the entries aijrepresent the number of observations in which x /C30 i and y /C30 j. Calculate the row and column sums Riand Cj ; respectively, and the total sum N /C30X iRi /C30X jCj (1) of the MATRIX . Then calculate the CONDITIONAL PROBABILITY of getting the actual matrix given the particular row and column sums, given by Pcutoff /C30R1!R2!...Rm! ðÞ C1!C2!...Cn! ðÞ N!Q i ;j aij! ; (2) which is a multivariate generalization of the HYPER- GEOMETRIC probability function. Now find all possible MATRICES of NONNEGATIVE INTEGERS consistent with the row and column sums Riand Cj : For each one, calculate the associated CONDITIONAL PROBABILITY using (2), where the sum of these probabilities must be 1. To compute the P-VALUE of the test, the tables must then be ordered by some criterion that measures dependence, and those tables that represent equal or greater deviation from independence than the ob- served table are the ones whose probabilities are added together. There are a variety of criteria that can be used to measure dependence. In the 2 /C292 case, which is the one Fisher looked at when he developed the exact test, either the Pearson chi-square or the difference in proportions (which are equivalent) is typically used. Other measures of association, such as the likelihood-ratio-test, G-squared, or any of the other measures typically used for association in contingency tables, can also be used. The test is most commonly applied to 2 /C292 MATRICES , and is computationally unwieldy for large m or n. For tables larger than 2 /C292 ; the difference in proportion can no longer be used, but the other measures mentioned above remain applicable (and in practice, the Pearson statistic is most often used to order the tables). In the case of the 2 /C292 matrix, the P-VALUE of the test can be simply computed by the sum of all P- values which are 5Pcutoff :/For an example application of the 2 /C292 test, let X be a journal, say either Mathematics Magazine or Science , and let Y be the number of articles on the topics of mathematics and biology appearing in a given issue of one of these journals. If Mathematics Magazine has five articles on math and one on biology, and Science has none on math and four on biology, then the relevant matrix would be Math : Mag : Science math 5 0 R1 /C305 biology 1 4 R2 /C305 C1 /C306 C2 /C304 N /C3010 : Computing Pcutoff gives Pcutoff /C305!26!4! 10! 5!0!1!4!ðÞ/C300:0238 ; and the other possible matrices and their Ps are 41 23/C20/C21 P /C300 :2381 32 32/C20/C21 P /C300 :4762 23 41/C20/C21 P /C300 :2381 14 50/C20/C21 P /C300:0238 ; which indeed sum to 1, as required. The sum of P- values less than or equal to Pcutoff /C300:0238 is then 0.0476 which, because it is less than 0.05, is SIG- NIFICANT . Therefore, in this case, there would be a statistically significant association between the jour- nal and type of article appearing. Fisher’s Theorem LetAbe a sum of squares of nindependent normal standardized variates xi;and suppose A/C30B/C27C where Bis a quadratic form in the xi;distributed as CHI-SQUARED with hDEGREES OF FREEDOM . Then Cis distributed as x2with n/C28hDEGREES OF FREEDOM and is independent of B. The converse of this theorem is known as C OCHRAN’S THEOREM . See also CHI-SQUARED DISTRIBUTION ,C OCHRAN’S THEOREM Fisher’s z’-Transformation Letrbe the CORRELATION COEFFICIENT . Then defin- ing z?/C13tanh/C281r (1) z/C13tanh/C281p; (2) gives sz?/C30(N /C283)/C281=2 (3) var(z ?) /C301 n /C274 /C28 r2 2n2/C27... (4) g1 /C30rr2 /C289 16/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12 n 3 =2 (5) g2 /C3032 /C28 3r4 16N; (6) where n /C13N /C281:/ See also CORRELATION COEFFICIENT References David, F. N. "The Moments of the z and F Distributions." Biometrika 36, 394 /C1/03, 1949. Fisher’s z-Distribution g(z) /C302nn1 =2 1nn2 =2 2 Bn1 2;n2 2 !en1z n1e2z /C27 n2 ðÞn1 /C27n1 ðÞ =2 (1) (Kenney and Keeping 1951). This general distribution includes the CHI-SQUARED DISTRIBUTION and STU- DENT’S T-DISTRIBUTION as special cases. Let u2 and v2be INDEPENDENT UNBIASED ESTIMATORS of the VARIANCE of a NORMALLY DISTRIBUTED variate. Define z /C13lnu v ! /C301 2lnu2 v2 ! : (2) Then let F /C13u2 v2 /C30Ns2 1 n1 Ns22 n2(3) so that n1F =n2 is a ratio of CHI-SQUARED variates n1F n2/C30x2 n1ðÞ x2 n2ðÞ; (4) which makes it a ratio of GAMMA DISTRIBUTION variates, which is itself a BETA PRIME DISTRIBUTION variate, gn1 2 ! gn2 2 !/C30 b?n1 2;n2 2 ! (5)giving f(F) /C30n1F n2 !n1 =2 /C281 1 /C27n1F n2 !/C28 n1 /C27n2 ðÞ =2n1 n2 Bn1 2;n2 2 ! : (6) The MEAN is Fhi/C30n2 n2 /C28 2 ; (7) and the MODE is n2 n2 /C27 2n1 /C28 2 n1: (8) See also BETA DISTRIBUTION ,BETA PRIME DISTRIBU- TION ,CHI-SQUARED DISTRIBUTION ,GAMMA DISTRIBU- TION ,N ORMAL DISTRIBUTION ,S TUDENT’S T - DISTRIBUTION References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 180 /C1/81, 1951. Fisher-Behrens Problem The determination of a test for the equality of MEANS for two NORMAL DISTRIBUTIONS with different VAR- IANCES given samples from each. There exists an exact test which, however, does not give a unique answer because it does not use all the data. Therealso exist approximate tests which do not use all the data. See also N ORMAL DISTRIBUTION References Aspin, A. A. "An Examination and Further Development of a Formula Arising in the Problem of Comparing Two Mean Values." Biometrika 35,8 8/C1/6, 1948. Chernoff, H. "Asymptotic Studentization in Testing of Hypothesis." Ann. Math. Stat. 20, 268/C1/78, 1949. Fisher, R. A. "The Fiducial Argument in Statistical Infer- ence." Ann. Eugenics 6, 391/C1/98, 1935. Kenney, J. F. and Keeping, E. S. "The Behrens-Fisher Test." §9.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 257 /C1/60 and 261 /C1/64, 1951. Sukhatme, P. V. "On Fisher and Behrens’ Test of Signifi- cance of the Difference in Means of Two Normal Samples." Sankhya 4, 39, 1938. Trickett, W. H. and Welch, B. L. "On the Comparison of Two Means: Further Discussion of Iterative Methods for Calculating Tables." Biometrika 41, 361/C1/74, 1954. Trickett, W. H.; Welch, B. L.; and James, G. S. "Further Critical Values for the Two-Means Problems." Biometrika 43, 203/C1/05, 1956. Wallace, D. L. "Asymptotic Approximations to Distribu- tions." Ann. Math. Stat. 29, 635/C1/54, 1958. Wald, A. "Testing the Difference Between the Means of Two Normal Populations with Unknown Standard Deviations." In Selected Papers in Statistics and Probability by Abra- ham Wald. New York: McGraw-Hill, pp. 669 /C1/95, 1955. Welch, B. L. "The Generalization of ‘Student’s’ Problem when Several Different Populations are Involved." Biome- trika 34,28/C1/5, 1947. Fisher-Tippett Distribution Also called the EXTREME VALUE DISTRIBUTION and LOG-WEIBULL DISTRIBUTION . It is the limiting distri- bution for the smallest or largest values in a large sample drawn from a variety of distributions. P(x) /C30e(a/C28x)=b /C28e(a/C28x) =b b (1) D(x) /C30e/C28e(a/C28x)=b : (2) These can be computed directly be defining z /C13expa /C28 x b ! (3) x /C30a /C28b ln z (4) dz /C30/C281 bexpa /C28 x b ! dx: (5) Then the MOMENTS about the origin are m?n /C13g/C12 /C28/C12xnP(x)dx /C301b g/C12 /C28/C12xnexpa /C28 x b ! exp /C28e(a /C28x)=b/C2/C6 dx /C30/C28g0 /C12(a /C28b ln z)ne /C28zdz /C30g/C12 0(a /C28b ln z)ne /C28zdz /C30Xn k /C300n k/C1Y/C1Q (/C281)kan/C28kbkg/C12 0(ln z)ke /C28zdz /C30Xn k /C300n k/C1Y/C1Q an/C28kbkI(k); (6) where I(k) are EULER- MASCHERONI INTEGRALS . Plug-ging in the EULER- MASCHERONI INTEGRALS I(k) gives m?0 /C301 (7) m ?1 /C30a /C27bg (8) m ?2 /C30a2 /C272ab g /C27b2 g2 /C2716 p 2 ! (9) m?3 /C30a3 /C273a2bg /C273ab2 g2 /C2716 p 2 ! /C27b3 g3 /C2712 gp 2 /C272z(3)"# (10) m?4 /C30a4 /C274a3b g /C276a2b2 g2 /C2716 p 2 ! /C274ab3 g3 /C2712 gp 2 /C272z(3)"# /C27b4 g4 /C27 g2 p2 /C273 20 p4 /C278gz(3)"# ; (11) where g is the EULER- MASCHERONI CONSTANT and z(3) is APE´ RY’S CONSTANT . The corresponding moments about the mean m /C30 m?1 are therefore m2 /C301 6b2 p2 (12) m3 /C302z(3)b3 (13) m4 ¼3 20b4 p2 ; (14) giving MEAN , VARIANCE , SKEWNESS , and KURTOSIS of m /C30a /C27b g (15) s2 /C30 m2 /C28 m2 1 /C301 6 p2b2 (16) g1 /C30m3 s3 /C3012ffiffiffi 6p z(3) p3 (17) g2 /C30m4 s4/C283/C3012 5: (18) The CHARACTERISTIC FUNCTION is f(t)/C30G(1/C28ibt)eiat; (19) where G(z) is the GAMMA FUNCTION (Abramowitz and Stegun 1972, p. 930). The special case of the Fisher-Tippett distribution with a/C300,b/C301 is called GUMBEL’S DISTRIBUTION . See also EULER- MASCHERONI INTEGRALS ,G UMBEL’S DISTRIBUTION Fitting Subgroup The unique smallest NORMAL NILPOTENT SUBGROUP of H, denoted F(H) : The generalized fitting subgroup is defined by F /C31 HðÞ/C30FHðÞEHðÞ; where EHðÞ is the commuting product of all components of H, and F is the fitting subgroup of H. Fitzhugh-Nagumo Equations The system of PARTIAL DIFFERENTIAL EQUATIONS ut /C30uxx /C27u(u /C28a)(1 /C28u) /C27w wt /C30eu : References Sherman, A. S. and Peskin, C. S. "A Monte Carlo Method for Scalar Reaction Diffusion Equations." SIAM J. Sci. Stat. Comput. 7, 1360 /C1/372, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Five Circles Theorem MIQUEL FIVE CIRCLES THEOREM Five Cubes CUBE 5-COMPOUND Five Disks Problem Given five equal DISKS placed symmetrically about a given center, what is the smallest RADIUS r for which the RADIUS of the circular AREA covered by the five disks is 1? The answer is r /C30 f /C281 /C301=f /C30 0:6180339 ... ; where f is the GOLDEN RATIO , andthe centers ci of the disks i /C30 1, ..., 5 are located at ci /C301 fcos2pi 5 ! 1 fsin2pi 5 !2 666643 77775: The GOLDEN RATIO enters here through its connection with the regular PENTAGON . If the requirement that the disks be symmetrically placed is dropped (the general DISK COVERING PROBLEM ), then the RADIUS for n /C305 disks can be reduced slightly to 0.609383... (Neville 1915). See also ARC,C IRCLE COVERING ,D ISK COVERING PROBLEM ,FIVE CIRCLES THEOREM ,FLOWER OF LIFE, SEED OF LIFE References Ball, W. W. R. and Coxeter, H. S. M. "The Five-Disc Pro- blem." In Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 97 /C1/9, 1987. Neville, E. H. "On the Solution of Numerical Functional Equations, Illustrated by an Account of a Popular Puzzle and of its Solution." Proc. London Math. Soc. 14, 308 /C1/26, 1915. Five Tetrahedra Compound TETRAHEDRON 5-COMPOUND Fixed When referring to a planar object, "fixed" means that the object is regarded as fixed in the plane so that it may not be picked up and flipped. As a result, MIRROR IMAGES are not necessarily equivalent for fixed objects. See also FREE,MIRROR IMAGE Fixed Element FIXED POINT (MAP) Fixed Point A point which does not change upon application of a MAP, system of DIFFERENTIAL EQUATIONS , etc. See also FIXED POINT (DIFFERENTIAL EQUATIONS ), FIXED POINT (GROUP ), FIXED POINT (MAP), FIXED POINT THEOREM References Shashkin, Yu. A. Fixed Points. Providence, RI: Amer. Math. Soc., 1991. Fixed Point (Differential Equations) Points of an AUTONOMOUS system of ordinary differ- ential equations at which dx1 dt/C30f1x1 ;...;xn ðÞ /C300 n dxn dt/C30fnx1 ;...; xn ðÞ /C3008 >>>>>< >>>>>: If a variable is slightly displaced from a FIXED POINT , it may (1) move back to the fixed point ("asymptoti- cally stable" or "superstable"), (2) move away ("un- stable"), or (3) move in a neighborhood of the fixed point but not approach it ("stable" but not "asympto- tically stable"). Fixed points are also called CRITICAL POINTS or EQUILIBRIUM POINTS . If a variable starts at a point that is not a CRITICAL POINT , it cannot reach a critical point in a finite amount of time. Also, a trajectory passing through at least one point that is not a CRITICAL POINT cannot cross itself unless it is a CLOSED CURVE , in which case it corresponds to a periodic solution. A fixed point can be classified into one of several classes using LINEAR STABILITY analysis and the resulting STABILITY MATRIX . See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), HYPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,S TABLE NODE,S TABLE SPIRAL POINT ,S TABLE STAR,U N- STABLE IMPROPER NODE,UNSTABLE NODE,UNSTABLE SPIRAL POINT ,UNSTABLE STAR Fixed Point (Group) The set of points of X fixed by a GROUP ACTION are called the group’s set of fixed points, defined by x : gx /C30x for all g /C23 G fg : In some cases, there may not be a group action, but a single operator T. Then {x:x /C23 X, Tx=x } still makes sense even when T is not invertible (as is the case in a GROUP ACTION ). See also FIXED POINT ,GROUP ,GROUP ACTION References Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 4 /C1/ and 31 /C1/5, 1987. Fixed Point (Map) A point x+ which is mapped to itself under a MAP G,so that x+/C30G(x+) : Such points are sometimes also called INVARIANT POINTS ,or FIXED ELEMENTS (Woods 1961). Stable fixed points are called elliptical. Unstable fixed points, corresponding to an intersection of a stable and unstable invariant MANIFOLD , are called HYPER- BOLIC (or SADDLE ). Points may also be called asymp- totically stable (a.k.a. superstable). See also CRITICAL POINT ,INVOLUTORYReferences Shashkin, Yu. A. Fixed Points. Providence, RI: Amer. Math. Soc., 1991. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 14, 1961. Fixed Point (Transformation) FIXED POINT (MAP) Fixed Point Theorem If g is a continuous function g(x) /C23 a;b½/C138 FOR ALL x /C23 [a;b]; then g has a FIXED POINT in [a, b]. This can be proven by noting that g(a) ]ag (b) 5b g(a) /C28a ]0 g(b) /C28b 50: Since g is continuous, the INTERMEDIATE VALUE THEOREM guarantees that there exists a c /C23 [a ;b] such that g(c) /C28c /C300; so there must exist a c such that g(c) /C30c ; so there must exist a FIXED POINT /C23 [a ;b]:/ See also BANACH FIXED POINT THEOREM ,BROUWER FIXED POINT THEOREM ,H AIRY BALL THEOREM , KAKUTANI’S FIXED POINT THEOREM ,LEFSHETZ FIXED POINT FORMULA ,LEFSHETZ TRACE FORMULA ,POIN- CARE ´ -BIRKHOFF FIXED POINT THEOREM ,SCHAUDER FIXED POINT THEOREM References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, p. 80, 1991. Flag A collection of FACES of an n-DPOLYTOPE orSIMPLI- CIAL COMPLEX , one of each DIMENSION 0, 1, ..., n/C281; which all have a common nonempty INTERSECTION .I n normal 3-D, the flag consists of a half-plane, its bounding RAY, and the RAY’s endpoint. Flag Manifold For any SEQUENCE ofINTEGERS 0Bn1B...Bnk;there is a flag manifold of type ( /n1;...,nk) which is the collection of ordered pairs of vector SUBSPACES of Rnk(V1;...,Vk) with dim( Vi)/C30niandViaSUBSPACE of Vi/C271:There are also COMPLEX flag manifolds with COMPLEX subspaces of Cnkinstead of REAL SUBSPACES of a REAL nk/-space. These flag manifolds admit the structure of MANI- FOLDS in a natural way and are used in the theory of LIE GROUPS . See also GRASSMANN MANIFOLD References Lu, J.-H. and Weinstein, A. "Poisson Lie Groups, Dressing Transformations, and the Bruhat Decomposition." J. Diff. Geom. 31, 501 /C1/26, 1990. Flat A set in Rd formed by translating an affine subspace or by the intersection of a set of HYPERPLANES . See also FLAT (MANIFOLD ) Flat (Manifold) See also FLAT Flat Norm The flat norm on a CURRENT is defined by F(S) /C30g Area T /C27vol R : S /C28T /C30@Rg; f where @R is the boundary of R. See also COMPACTNESS THEOREM ,CURRENT References Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103, 369 /C1/76, 1996. Flat Space Theorem If it is possible to transform a coordinate system to a form where the metric elements g mn are constants independent of xm ; then the space is flat. Flat Surface A REGULAR SURFACE and special class of MINIMAL SURFACE for which the GAUSSIAN CURVATURE vanishes everywhere. A TANGENT DEVELOPABLE , GEN- ERALIZED CONE , and GENERALIZED CYLINDER are all flat surfaces. See also GAUSSIAN CURVATURE ,M INIMAL SURFACE , PLANE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 374, 1997.Flat-Ring Cyclide Coordinates A coordinate system similar to TOROIDAL COORDI- NATES but with fourth-degree instead of second- degree surfaces for constant mso that the toroids of circular CROSS SECTION are replaced by flattened rings, and the spherical bowls are replaced by cyclides of rotation for constant n:The transformation equations are x/C30a Lsnmdnncosc (1) y/C30a Lsnmdnnsinc (2) z/C30a Lcnmdnmsnncnn; (3) where L/C131/C28dn2msn2n (4) and with m/C23[0;K];n/C23[0;K?];andc/C23[0;2P):Surfaces of constant mare given by the flat-ring cyclides x2/C27y2/C27z2/C0/C12/C27a2 k4 /C21/C28k2ðÞ2/C2821/C28k2ðÞ dn2m/C271/C27k2ðÞ dn4m dn2mcn2mz2 /C28a2sn2m/C271 sn2m ! x2/C27y2/C0/C1 /C27a4 k2 /C300; (5) surfaces of constant n by the cyclides of rotation dn2 n a2x2 /C27y2/C0/C1 /C27cn2 n a2 sn2 nz2"#2 /C282cn2 n a2sn2 n z2 /C282dn2 n a2 /C2 x2 /C27y2/C0/C1 /C271 /C300; (6) and surfaces of constant c by the half-planes tan c /C30x y : (7) See also CYCLIDIC COORDINATES ,TOROIDAL COORDI- NATES References Moon, P. and Spencer, D. E. "Flat-Ring Cyclide Coordinates ( m; n ; c) :/" Fig. 4.09 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 126 /C1/ 29, 1988. Flattening The flattening of a SPHEROID (also called OBLATENESS ) is denoted /C23 or f. It is defined as /C23/C13a /C28 c a/C30 1 /C28c aoblate c /C28 a a/C30c a /C281 prolate ;8 >>>< >>>: where c is the polar RADIUS and a is the equatorial RADIUS . See also ECCENTRICITY ,ELLIPSOID ,O BLATE SPHER- OID,PROLATE SPHEROID ,SPHEROID Flemish Knot FIGURE-OF- EIGHT KNOT Fletcher Point The intersection Fl of the GERGONNE LINE and the SODDY LINE. In the above figure, D?; E ?; and F ? are theNOBBS POINTS , I is the INCENTER , Ge is the GER- GONNE POINT , and S and S? are the SODDY POINTS . See also GERGONNE LINE,SODDY LINE,SODDY POINTS References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/29, 1996. Fleury’s Algorithm An elegant algorithm for constructing an EULERIAN CIRCUIT (Skiena 1990, p. 193). See also EULERIAN CIRCUIT References Lucas, E. Re´cre´ations Mathe ´matiques. Paris: Gauthier- Villars, 1891. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Flexagon An object created by FOLDING a piece of paper along certain lines to form loops. The number of states possible in an n-FLEXAGON is a CATALAN NUMBER .By manipulating the folds, it is possible to hide and reveal different faces. See also FLEXATUBE ,FOLDING ,HEXAFLEXAGON ,TET- RAFLEXAGON References Crampin, J. "On Note 2449." Math. Gazette 41,5 5/C1/6, 1957. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 205 /C1/07, 1989. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 62 /C1/4, 1979. Gardner, M. "Hexaflexagons." Ch. 1 in The Scientific Amer- ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 1 /C1/4, 1959. Gardner, M. "Tetraflexagons." Ch. 2 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 24 /C1/1, 1961. Maunsell, F. G. "The Flexagon and the Hexaflexagon." Math. Gazette 38, 213/C1/14, 1954. Oakley, C. O. and Wisner, R. J. "Flexagons." Amer. Math. Monthly 64, 143/C1/54, 1957. Wheeler, R. F. "The Flexagon Family." Math. Gaz. 42,1/C1/, 1958. Flexatube AFLEXAGON -like structure created by connecting the ends of a strip of four squares after folding along 45 8 diagonals. Using a number of folding movements, it is possible to flip the flexatube inside out so that the faces originally facing inward face outward. Gardner (1961) illustrated one possible solution, and Stein- haus (1983) gives a second. See also FLEXAGON ,HEXAFLEXAGON ,TETRAFLEXAGON References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 205, 1989. Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 29 /C1/1, 1961. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 177 /C1/81 and 190, 1999. Flexible Graph A GRAPH G is said to be flexible if the vertices of G can be moved continuously so that (1) the distances between adjacent vertices are unchanged, and (2) at least two nonadjacent vertices change their mutual distances. A graph which is not flexible is said to be RIGID . See also RIGID GRAPH References Maehara, H. "Distance Graphs in Euclidean Space." Ryukyu Math. J. 5,33/C1/1, 1992. Flexible Polyhedron Although the RIGIDITY THEOREM states that if the faces of a convex POLYHEDRON are made of metal plates and the EDGES are replaced by hinges, the POLYHEDRON would be RIGID , concave polyhedra need not be RIGID . A nonrigid polyhedron may be "SHAKY " (infinitesimally movable) or flexible (continuously movable; Wells 1991). In 1897, Bricard constructed several self-intersecting flexible octahedra (Cromwell 1997, p. 239). Connelly (1978) found the first example of a true flexible polyhedron, consisting of 18 triangular faces (Crom- well 1997, pp. 242 /C1/44). Mason discovered a 34-sided flexible polyhedron constructed by erecting a pyramid on each face of a CUBE adjoined square ANTIPRISM (Cromwell 1997). Kuiper and Deligne modified Con- nelly’s polyhedron to create a flexible polyhedron having 18 faces and 11 vertices (Cromwell 1997,p. 245), and Steffen found a flexible polyhedron with only 14 triangular faces and 9 vertices (shown above; Cromwell 1997, pp. 244 /C1/47; Mackenzie 1998). Mak- simov (1995) proved that Steffen’s is the simplest possible flexible polyhedron composed of only trian- gles (Cromwell 1997, p. 245). Connelly et al. (1997) proved that a flexible polyhe- dron must keep its VOLUME constant, confirming the so-called BELLOWS CONJECTURE (Mackenzie 1998). See also BELLOWS CONJECTURE ,POLYHEDRON ,QUAD- RICORN ,R IGID POLYHEDRON ,R IGIDITY THEOREM , SHAKY POLYHEDRON References Cauchy, A. L. "Sur les polygones et les polye `dres." XVIe Cahier IX,8 7/C1/9, 1813. Connelly, R. "A Flexible Sphere." Math. Intel. 1, 130/C1/31, 1978. Connelly, R.; Sabitov, I.; and Walz, A. "The Bellows Conjecture." Contrib. Algebra Geom. 38,1/C1/0, 1997. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 222, 224, and 239 /C1/47, 1997. Mackenzie, D. "Polyhedra Can Bend But Not Breathe." Science 279, 1637, 1998. Maksimov, I. G. "Polyhedra with Bendings and Riemann Surfaces." Uspekhi Matemat. Nauk 50, 821/C1/23, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 161 /C1/62, 1991. Flip Bifurcation Letf:R/C29R0Rbe a one-parameter family of C3 maps satisfying f(0;0)/C300 @f @x"# m/C300;x/C300/C30/C281 @2f @x2"# m/C30o;x/C300B0 @3f @x3"# m/C300;x/C300B0: Then there are intervals m1;0 ðÞ ;0;m2 ðÞ ;ando>0 such that 1. If m/C23(0;m2);then fm(x) has one unstable fixed point and one stable orbit of period two for x/C23 (/C28e;e);and 2. If m/C23m1;0/C0/C1 ;then fm(x) has a single stable fixed point for x/C23(/C28e;e):/ This type of BIFURCATION is known as a flip bifurca- tion. An example of an equation displaying a flip bifurcation is f ðx Þ¼ m /C28x /C28x2 : See also BIFURCATION References Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 27 /C1/0, 1990. Floating-Point Arithmetic ARITHMETIC performed on real numbers by computers or other automated devices using a fixed number of bits. ARITHMETIC References Hauser, J. R. "Handling Floating-Point Exceptions in Nu- meric Programs." ACM Trans. Program. Lang. Sys. 18, 139 /C1/74, 1996. http://www.cs.berkeley.edu/~jhauser/excep- tions/HandlingFloatingPointExceptions.html. Severance, C. (Ed.). "IEEE 754: An Interview with William Kahan." Computer , 114 /C1/15, Mar. 1998. Stevenson, D. "A Proposed Standard for Binary Floating- Point Arithmetic: Draft 8.0 of IEEE Task P754." IEEE Comput. 14 51 /C1/2, 1981. Floor FLOOR FUNCTION Floor Function The function floor function xbc; also called the great- est integer function, gives the largest INTEGER less than or equal to x. In many computer languages, the floor function is called the INTEGER PART function and is denotedint(x) . The name and symbol for the floor function were coined by K. E. Iverson (Graham et al. 1990). Unfortunately, in many older and current works (e.g., Steinhaus 1983, p. 300; Shanks 1993; Ribenboim1996; Hilbert and Cohn-Vossen 1999, p. 38; Hardy 1999, p. 18), the symbol x½/C138is used instead of xbc (Graham et al. 1990, p. 67). Because of the elegant symmetry of the floor function and CEILING FUNCTION symbols xbcand xde; and because x½/C138is such a useful symbol when interpreted as an IVERSON BRACKET , the use of x½/C138to denote the floor function should be deprecated. In this work, the symbol x½/C138is used to denote the NEAREST INTEGER FUNCTION since it naturally falls between the xbcand xdesymbols. Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used (D. W. Cantrell). Here, S&O indicates Spanier and Oldham (1987). notation name S&O Graham et al.Mathema- tica / xbc/ integer- value/Int(x)/ floor or integer partFloor [ x] /sgn(x) xjjbc / integer-part/Ip(x)/ no name Integer- Part [ x] /x /C28 xbc/ fractional-value/frac( x)/ fractionalpart or xfg /no name /sgn(x) xjj/C28 xjjbc ðÞ / fractional-part/FP(x)/ no name Fractio- nalPart[ x] There are infinitely many integers OF THE FORM (3=2)nbc and (4=3)nbc which are composite, where xbc is the FLOOR FUNCTION (Forman and Shapiro, 1967; Guy 1994, p. 220). The first few composite (3 =2)nbc occur for n/C308, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, ... (Sloane’s A046037), and the few composite (4=3)nbc occur for n/C305, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ... (Sloane’s A046038). Numbers OF THE FORM frac (3 =2)nðÞ ;where frac( x) is the FRACTIONAL PART also appear in W ARING’S PROBLEM . See also CEILING FUNCTION ,FRACTIONAL PART,INT, IVERSON BRACKET ,N EAREST INTEGER FUNCTION , QUOTIENT ,S HIFT TRANSFORMATION ,S TAIRCASE FUNCTION References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Forman, W. and Shapiro, H. N. "An Arithmetic Property of Certain Rational Powers." Comm. Pure Appl. Math. 20, 561/C1/73, 1967. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer Functions." Ch. 3 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 67 /C1/01, 1994. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Iverson, K. E. A Programming Language. New York: Wiley, p. 12, 1962. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 180 /C1/82, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 14, 1993. Sloane, N. J. A. Sequences A046037 and A046038 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Spanier, J. and Oldham, K. B. "The Integer-Value Int(x) and Fractional-Value frac(x) Functions." Ch. 9 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 71 /C1/8, 1987. Floquet Analysis Given a system of periodic ORDINARY DIFFERENTIAL EQUATIONS OF THE FORM d dtx y vx vy2 6643 775/C30/C280 0 F xx Fxy0 0 Fyy Fyy/C281 0000 /C281 002 6643 775x y v x vy2 6643 775; (1) the solution can be written as a LINEAR COMBINATION of functions OF THE FORM x(t) y(t) vx vy2 6643 775/C30x 0 y0 vx0 vy02 6643 775e mtP m(t) ; (2) where Pm(t) is a function periodic with the same period T as the equations themselves. Given an ORDINARY DIFFERENTIAL EQUATION OF THE FORM ¨x /C27g(t)x /C300; (3) where g(t) is periodic with period T, the ODE has a pair of independent solutions given by the REAL and IMAGINARY PARTS of x /C27w(t)eic(t) (4) ˙x /C30( ˙w /C27iw˙c)ei c (5) ¨x /C30 ¨w /C27i ˙w˙c /C27i( ˙w˙c /C27w¨c /C27iw˙c2)/C2/C6 eic /C30 ( ¨w /C28w˙c2) /C27i(2 ˙w˙c /C27w¨c)/C2/C6 eic : (6) Plugging these into (3) gives ¨w /C272i ˙w˙c /C27w(g /C27i¨c /C28˙c2) /C300; (7) so the REAL and IMAGINARY PARTS are ¨w /C27w(g /C28˙c2) /C300 (8) 2 ˙w˙c /C27w¨c /C300: (9)From (9), 2 ˙w w/C27¨c ˙c /C302d dt(ln w) /C27d dt[ln(˙c)] /C30d dtln(˙cw2) /C300: (10) Integrating gives ˙c /C30c w2 ; (11) where C is a constant which must equal 1, so c is given by c /C30gt todt w2 : (12) The REAL solution is then x(t) /C30w(t) cos [ c(t)]; (13) so ˙x /C30 ˙w cos c /C28w c sin c /C30 ˙wx w/C28w˙csinc /C30˙wx w/C28w1 w2sinc/C30˙wx w/C281 wsinc (14) and 1/C30cos2c/C27sin2c/C30x2w/C282/C27w˙wx w/C28˙x !"#2 /C30x2w/C282/C27(˙wx/C28w˙x)2/C13i(x;˙x;t);(15) which is an integral of motion. Therefore, although w(t) is not explicitly known, an integral Ialways exists. Plugging (10) into (8) gives ¨w/C27g(t)w/C281 w3/C300; (16) which, however, is not any easier to solve than (3). See also FLOQUET’S THEOREM ,HILL’S DIFFERENTIAL EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 727, 1972. Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 175, 1987. Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, p. 32, 1983. Margenau, H. and Murphy, G. M. The Mathematics of Physics and Chemistry, 2 vols. Princeton, NJ: Van Nostrand, 1956 /C1/4. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 556 /C1/57, 1953. Floquet’s Theorem Let Q(x) be a real or complex piecewise-continuous function of the real variable x defined for all values of x that is periodic with minimum period p so that Q(x /C27 p) /C30Q(x) : (1) Then the differential equation yn /C27Q(x)y /C300 (2) has two continuously differentiable solutions y1(x) and y2(x) ; and the characteristic equation is r2 /C28[y1( p) /C27y0 2( p)] r /C271 /C300 ; (3) with eigenvalues r1 /C30eiap and r2 /C30e /C28iap . The Floquet’s theorem states that if the roots r1and r2 are different from each other, then (2) has two linearly independent solutions f1(x) /C30eiaxp1(x) (4) f2(x) /C30e /C28iaxp2(x) ; (5) where p1(x) and p2(x) are period with period p (Magnus and Winkler 1979, p. 4). See also FLOQUET ANALYSIS ,H ILL’S DIFFERENTIAL EQUATION References Magnus, W. and Winkler, S. "Floquet’s Theorem." §1.2 in Hill’s Equation. New York: Dover, pp. 3 /C1/, 1979. Flow An ACTION with G /C30R: Flows are generated by VECTOR FIELDS and vice versa. See also ACTION ,A MBROSE- KAKUTANI THEOREM , ANOSOV FLOW,AXIOM AF LOW,CASCADE ,GEODESIC FLOW,SEMIFLOW Flow Line A flow line for a map on a VECTOR FIELD F is a path s(t) such that s?(t) /C30F(s(t)) :/ Flower DAISY,FLOWER OF LIFE,ROSEFlower of Life One of the beautiful arrangements of CIRCLES found at the Temple of Osiris at Abydos, Egypt (Rawles 1997). The CIRCLES are placed with six-fold symme- try, forming a mesmerizing pattern of CIRCLES and LENSES . See also CIRCLE COVERING ,FIVE DISKS PROBLEM , REULEAUX TRIANGLE ,SEED OF LIFE,VENN DIAGRAM References Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 15, 1997. Wein, J. "La Fleur de Vie." http://www2.cruzio.com/~flower/ fleur.htm. Weisstein, E. W. "Flower of Life." MATHEMATICA NOTEBOOK FLOWER OFLIFE.M . Flowsnake PEANO- GOSPER CURVE Flowsnake Fractal GOSPER ISLAND Floyd’s Algorithm An algorithm for finding the shortest path between two VERTICES . See also DIJKSTRA’S ALGORITHM Fluent Newton’s term for a variable in his method of FLUXIONS (differential calculus). See also CALCULUS ,FLUXION References Newton, I. Methodus fluxionum et serierum infinitarum. 1664 /C1/671. Fluxion The term for DERIVATIVE in Newton’s CALCULUS . See also CALCULUS ,DERIVATIVE ,FLUENT References Newton, I. Methodus fluxionum et serierum infinitarum. 1664 /C1/671. Flype A 1808 rotation of a TANGLE . The word "flype" is derived from the old Scottish verb meaning "to turn or fold back." Tait (1898) used this word to indicate a different knot transformation than the one under- stood in the modern definition, illustrated above (Hoste et al. 1998). See also FLYPING CONJECTURE ,TANGLE References Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Tait, P. G. "On Knots I, II, and III." Scientific Papers, Vol. 1. Cambridge, England: University Press, pp. 273 /C1/47, 1898. Flyping Conjecture Also called the TAIT FLYPING CONJECTURE . Given two reduced alternating projections of the same KNOT , they are equivalent on the SPHERE IFF they are related by a series of FLYPES . The conjecture was proved by Menasco and Thistlethwaite (1991, 1993) using properties of the JONES POLYNOMIAL . It allows all possible REDUCED alternating projections of a given ALTERNATING KNOT to be drawn. See also ALTERNATING KNOT,F LYPE ,R EDUCIBLE CROSSING ,TAIT’S KNOT CONJECTURES References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 164 /C1/65, 1994. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Menasco, W. and Thistlethwaite, M. "The Tait Flyping Conjecture." Bull. Amer. Math. Soc. 25, 403 /C1/12, 1991. Menasco, W. and Thistlethwaite, M. "The Classification of Alternating Links." Ann. Math. 138, 113 /C1/71, 1993. Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, pp. 284 /C1/85, 1987. The following table gives properties of different types of conic sections, where k is the Focal Parameter The distance p (sometimes also denoted k) from the FOCUS to the DIRECTRIX of a CONIC SECTION . The following table gives the focal parameter for the different types of conics, where a is the SEMIMAJORAXIS, c is the distances from the origin to the FOCUS , and e is the ECCENTRICITY . conic e /p(a; b)// p(a ;c)//p(a;e)/ ELLIPSE /0 Be B1//b2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28 b2p //a2 /C28 c2 c//a(1 /C28 e2) e/ PARABOLA e /C30 1 /2a// 2a// 2a/ HYPERBOLA e /C21 1 /b2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2p //c2 /C28 a2 c//ae2 /C28 1 ðÞ e/ See also CONIC SECTION ,DIRECTRIX (CONIC SECTION ), ECCENTRICITY ,FOCUS Focus A point related to the construction and properties of CONIC SECTIONS .H YPERBOLAS and noncircular EL- LIPSES have two distinct foci and two associated DIRECTRICES , each DIRECTRIX being PERPENDICULAR to the line joining the two foci (Eves 1965, p. 275). See also DIRECTRIX (CONIC SECTION ), ELLIPSE ,ELLIP- SOID,FOCAL PARAMETER ,HYPERBOLA ,HYPERBOLOID , PARABOLA ,PARABOLOID ,REFLECTION PROPERTY References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 141 /C1/44, 1967. Eves, H. "The Focus-Directrix Property." §6.8 in A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, pp. 272 /C1/ 75, 1965. Foias Constant A problem listed in a fall issue of Gazeta Matematica in the mid-1970s posed the question if x1>0 and xn/C271/C301/C271 xn !n (1) forn/C301, 2, ..., then are there any values for which xn0/C12/? The problem, listed as one given on an entrance exam to prospective freshman in the mathe- matics department at the University of Bucharest, was solved by C. Foias. It turns out that there exists exactly one real number a:1:187452351126501 (2) such that if x1 /C30 a; then xn 0/C12: However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. Moreover, in this case, lim n0/C12xnln n n/C301; (3) which can be rewritten as lim n0/C12xn p(n) /C301 ; (4) where p(n) is the PRIME COUNTING FUNCTION . How- ever, Ewing and Foias (2000) believe that this connection with the PRIME NUMBER THEOREM is fortuitous. Foias also discovered that the problem stated in the journal was a misprint of the actual exam problem, which used the recurrence xn/C271 /C30 1 /C271=xn ðÞxn(Ewing and Foias 2000). In this form, the recurrence con- verges to x/C12:2:2931662874118610315080282912508 (5) for all starting values of x1 ; which is simply the root of x /C30 1 /C271 x !x : (6) See also GROSSMAN’S CONSTANT References Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse and G. Paun). London: Springer-Verlag, pp. 119 /C1/26, 2000. Fold Bifurcation Let f : R /C29R 0 R be a one-parameter family of C2 MAP satisfying f(0;0) /C300 @f @x"# m/C300 ;x/C300/C300 @2f @x2"# m/C300 ;x/C300/C210 @f @ m"# m/C300 ;x/C300/C210 ; then there exist intervals m1 ;0 ðÞ ; 0 ; m2 ðÞ and o > 0 such that1. If m /C23 m1 ;0 ðÞ ; then fm(x) has two fixed points in (/C28e ; e) with the positive one being unstable and the negative one stable, and 2. If m /C23 (0; m2) ; then f m(x) has no fixed points in (/C28e ; e) :/ This type of BIFURCATION is known as a fold bifurca- tion, sometimes also called a SADDLE-NODE BIFURCA- TION or TANGENT BIFURCATION . An example of an equation displaying a fold bifurcation is x:/C30 m /C28x2 (Guckenheimer and Holmes 1997, p. 145). See also BIFURCATION References Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. New York: Springer-Verlag, pp. 145 /C1/49, 1997. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 27 /C1/8, 1990. Fold Catastrophe A catastrophe which can occur for one control factor and one behavior axis. It is the universal unfolding of the singularity f(x) /C30x3and has the equation F(x;u)/C30x3/C27ux:/ See also CATASTROPHE THEORY References Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. Folding The points accessible from cby a single fold which leaves a1;...,anfixed are exactly those points interior to or on the boundary of the intersection of the CIRCLES through cwith centers at ai;fori/C301, ..., n. Given any three points in the plane a,b, and c, there is an EQUILATERAL TRIANGLE with VERTICES x, y, and zfor which a,b, and care the images of x,y, andzunder a single fold. Given any four points in the plane a, b, c, and d, there is some SQUARE with VERTICES x, y, z, and w for which a, b, c, and d are the images of x, y, z, and w under a sequence of at most three folds. In addition, any four collinear points are the images of the VERTICES of a suitable SQUARE under at most two folds. Every five (six) points are the images of the VERTICES of suitable regular PENTAGON (HEXAGON ) under at most five (six) folds. Wells (1991) illustrates a PENTAGON , HEXAGON , HEPTAGON , and OCTAGON constructed using paper folding. The least number of folds required for n ]4 is not known, but some bounds are. In particular, every set of n points is the image of a suitable REGULAR n-gon under at most F(n) folds, where F(n) 51 2 (3n /C282) for n even 12 (3n /C283) for n odd:8 >>>< >>>: The first few values are 0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, ... (Sloane’s A007494). See also F LEXAGON ,M AP FOLDING ,ORIGAMI ,RUDIN- SHAPIRO SEQUENCE ,STAMP FOLDING References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Hilton, P.; Holton, D.; and Pedersen, J. "Paper-Folding and Number Theory." Ch. 4 in Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 87 /C1/42, 1997. Klein, F. "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, p. 42, 1980. Sabinin, P. and Stone, M. G. "Transforming n-gons by Folding the Plane." Amer. Math. Monthly 102, 620 /C1/27, 1995. Sloane, N. J. A. Sequences A007494 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 191 /C1/92, 1991.Foliation Let Mn be an n-MANIFOLD and let F /C30 Fafg denote a PARTITION of Mn into DISJOINT path-connected SUB- SETS . Then F is called a foliation of Mn of codimension c (with 0 Bc Bn) if there exists a COVER of Mn by OPEN SETS U, each equipped with a HOMEOMORPHISM h : U 0 Rn or h : U 0 Rn /C27which throws each none- mpty component of Fa S U onto a parallel translation of the standard HYPERPLANE Rn/C28c in Rn : Each Fais then called a LEAF and is not necessarily closed or compact. See also CONFOLIATION ,C OVER ,H OMEOMORPHISM , LEAF (FOLIATION ), MANIFOLD ,REEB FOLIATION References Candel, A. and Conlon, L. Foliations I. Providence, RI: Amer. Math. Soc., 1999. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 284, 1976. Folium The word "folium" means leaf-shaped. The polar equation is r /C30cos u(4a sin2 u /C28b) : If b ]4a ; it is a single folium. If b /C300, it is a BIFOLIUM . If 0 Bb B4a ; it is a TRIFOLIUM . The simple folium is the PEDAL CURVE of the DELTOID where the PEDAL POINT is one of the CUSPS . See also BIFOLIUM ,FOLIUM OF DESCARTES ,KEPLER’S FOLIUM ,QUADRIFOLIUM ,ROSE,TRIFOLIUM References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152 /C1/53, 1972. MacTutor History of Mathematics Archive. "Folium." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Fo- lium.html. Folium of Descartes A plane curve proposed by Descartes to challenge Fermat’s extremum-finding techniques. In para- metric form, x /C303at 1 /C27 t3 (1) y /C303at2 1 /C27 t3 : (2) The curve has a discontinuity at t /C30/C281. The left wing is generated as t runs from /C281 to 0, the loop as t runs from 0 to /C12; and the right wing as t runs from /C28/C12 to /C281. The CURVATURE and TANGENTIAL ANGLE of the folium of Descartes, illustrated above, are k(t) /C3021/C27 t3ðÞ4 31/C27 4t2 /C28 4t3 /C28 4t5 /C27 4t6 /C27 t8 ðÞ3 =2 (3) f(t) /C301 2p /C27tan/C2811 /C28 2t3 t4 /C28 2t ! /C28tan /C2812t3 /C28 1 t4 /C28 2t ! "# : 17 /C274ffiffiffiffiffiffi 18p (4) Converting the PARAMETRIC EQUATIONS to POLAR COORDINATES gives r2 /C303atðÞ21 /C27 t2ðÞ 1 /C27 t3 ðÞ2 (5) u /C30tan/C281y x ! /C30tan/C281t; (6) so du /C30dt 1 /C27 t2 : (7)The AREA enclosed by the curve is A ¼1 2 gr2 du ¼12 g/C12 0(3at)2(1 /C27 t2) (1 /C27 t3)2dt 1 /C27 t2 /C303 2 a2 g/C12 03t2dt 1 /C27 t3 ðÞ2 : (8) Now let u /C131 /C27t3 so du /C303t2dt A /C3032 a 2 g/C12 1du u2 /C3032 a 2 /C281 u"#/C12 1/C303 2a2(/C280 /C271) /C3032 a 2 (9) In CARTESIAN COORDINATES , x3 /C27y3 /C303atðÞ31 /C27 t3ðÞ 1 /C27 t3 ðÞ3/C303atðÞ3 1 /C27 t3 ðÞ2 /C303axy (10) (MacTutor Archive). The equation of the ASYMPTOTE is y /C30/C28a /C28x: (11) References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, 1987. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 77 /C1/2, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 106 /C1/09, 1972. MacTutor History of Mathematics Archive. "Folium of Descartes." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Foliumd.html. Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes’ Folium." Math. Mag. 61, 172 /C1/87, 1988. Yates, R. C. "Folium of Descartes." In A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 98 /C1/9, 1952. Folkman Graph A graph which is EDGE-TRANSITIVE but not VERTEX- TRANSITIVE , and has the minimum possible number of nodes (20) for a nontrivial graph satisfying these properties (Skiena 1990, p. 186). See also EDGE-TRANSITIVE GRAPH ,V ERTEX- TRANSI- TIVE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 235, 1976. Folkman, J. "Regular Line-Symmetric Graphs." J. Combin. Th.3, 215/C1/32, 1967. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 186 /C1/87, 1990. Follows SUCCEEDS Fontene ´ Theorems 1. If the sides of the PEDAL TRIANGLE of a point P meet the corresponding sides of a TRIANGLE DO1O2O3at X1 ; X2 ; and X3 ; respectively, then P1X1 ; P2X2 ; P3X3 meet at a point L common to the CIRCLES O1O2O3and P1P2P3 : In other words, L is one of the intersections of the NINE-POINT CIRCLE of A1A2A3 and the PEDAL CIRCLE of P. 2. If a point moves on a fixed line through the CIRCUMCENTER , then its PEDAL CIRCLE passes through a fixed point on the NINE-POINT CIRCLE . 3. The PEDAL CIRCLE of a point is tangent to the NINE-POINT CIRCLE IFF the point and its ISOGONAL CONJUGATE lie on a LINE through the ORTHOCEN- TER.FEUERBACH’S THEOREM is a special case of this theorem. See also CIRCUMCENTER ,F EUERBACH’S THEOREM , ISOGONAL CONJUGATE ,N INE-POINT CIRCLE ,ORTHO- CENTER ,PEDAL CIRCLE References Bricard, R. "Note au sujet de l’article pre´ce´dent." Nouv. Ann. Math. 6,59/C1/1, 1906. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 52, 1971. Fontene ´, G. "Extension du the´ore`me de Feuerbach." Nouv. Ann. Math. 5, 504 /C1/06, 1905. Fontene ´, G. "Sur les points de contact du cercle des neuf point d’un triangle avec les cercles tangents aux trois coˆte´s." Nouv. Ann. Math. 5, 529 /C1/38, 1905. Fontene ´, G. "Sur le cercle pe´dal." Nouv. Ann. Math. 65,55/C1/ 8, 1906. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 245 /C1/47, 1929. Foot PERPENDICULAR FOOT Football LEMON For All If a proposition P is true for all B, this is written P /C214B: /C214is one of the two so-called QUANTIFIERS . In Mathematica 4.0, the command ForAllRealQ [i- neqs , vars] can be used to determine if the system of real equations and inequalities ineqs is satisfied for all real values of the variables vars. See also ALMOST ALL,EXISTS ,IMPLIES ,QUANTIFIER , UNIVERSAL QUANTIFIER Forced Polygon HAPPY END PROBLEMForcing A technique in SET THEORY invented by P. Cohen (1963, 1964, 1966) and used to prove that the AXIOM OF CHOICE and CONTINUUM HYPOTHESIS are indepen- dent of one another in ZERMELO- FRAENKEL SET THEORY . See also AXIOM OF CHOICE ,CONTINUUM HYPOTHESIS , SET THEORY ,ZERMELO- FRAENKEL SET THEORY References Cohen, P. J. "The Independence of the Continuum Hypoth- esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143 /C1/148, 1963. Cohen, P. J. "The Independence of the Continuum Hypoth- esis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105 /C1/10, 1964. Cohen, P. J. Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966. Todorchevich, S. and Farah, I. Some Applications of the Method of Forcing. Moscow: Yenisei, 1995. Ford Circle Pick any two INTEGERS h and k, then the CIRCLE C(h ; k)of RADIUS 1= 2k2ðÞ centered at h=k ;91= 2k2ðÞ ðÞ is known as a Ford circle. No matter what and how many hs and ks are picked, none of the Ford circles intersect (and all are tangent to the X-AXIS ). This can be seen by examining the squared distance between the centers of the circles with ( h, k) and h0;k0/C0/C1 ; d2/C30h0 k0/C28h k !2 /C271 2k02/C281 2k2 !2 : (1) Letsbe the sum of the radii s/C30r1/C27r2/C301 2k2/C271 2k02; (2) then d2/C28s2/C30h0k/C28hk0/C0/C12/C281 k2k02: (3) But h0k/C28k0h/C0/C12]1;sod2/C28s2]0 and the distance between circle centers is ]the sum of the CIRCLE RADII , with equality (and therefore tangency) IFF h0k/C28k0h/C12/C12/C12/C12/C301:Ford circles are related to the FAREY SEQUENCE (Conway and Guy 1996). If h1 =k1 ; h2 =k2 ; and h3 =k3 are three consecutive terms in a FAREY SEQUENCE , then the circles c(h1 ; k1) and c(h2 ;k2) are tangent at a1 /C30h2 k2/C28k1 k2k2 2 /C27 k21 ðÞ;1 k22 /C27 k21 ! (4) and the circles c(h2 ; k2) and Ch3 ;k3 ðÞ intersect in a2 /C30h2 k2/C28k3 k2k22 /C27 k23 ðÞ;1 k22 /C27 k23 ! : (5) Moreover, a1lies on the circumference of the SEMI- CIRCLE with diameter h1 =k1 ;0 ðÞ /C28 h2 =k2 ;0 ðÞ and a2 lies on the circumference of the SEMICIRCLE with diameter h2 =k2 ;0 ðÞ /C28 h3 =k3 ;0 ðÞ (Apostol 1997, p. 101). See also ADJACENT FRACTION ,APOLLONIAN GASKET , FAREY SEQUENCE ,STERN- BROCOT TREE References Apostol, T. M. "Ford Circles." §5.5 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 99 /C1/02, 1997. Conway, J. H. and Guy, R. K. "Farey Fractions and Ford Circles." The Book of Numbers. New York: Springer- Verlag, pp. 152 /C1/54, 1996. Ford, L. R. "Fractions." Amer. Math. Monthly 45, 586 /C1/01, 1938. Pickover, C. A. "Fractal Milkshakes and Infinite Archery." Ch. 14 in Keys to Infinity. New York: W. H. Freeman, pp. 117 /C1/25, 1995. Rademacher, H. Higher Mathematics from an Elementary Point of View. Boston, MA: Birkha ¨user, 1983. Ford’s Theorem Let a, b, and k be INTEGERS with k ]1 : For j /C30 0, 1, 2, let Sj /C13X i/C13j ðmod 3 Þð/C281Þj k i/C1Y/C1Q ak/C28ibi : Then 2ða2 þ ab þ b2 Þ2k ¼ðS0 /C0 S1 Þ4 þðS1 /C0 S2 Þ4 /C27ðS2 /C0 S0 Þ4See also BHARGAVA’S THEOREM ,DIOPHANTINE EQUA- TION–4TH POWERS References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 100 /C1/01, 1994. Forest An acyclic graph (i.e., a GRAPH without any CIRCUITS ). Forests therefore consist only of (possibly discon- nected) TREES , hence the name "forest." A forest with k components and n nodes has n /C28k EDGES . The numbers of forests on n /C30 1, 2, ... nodes are 1, 2, 3, 6, 10, 20, 37, ... (Sloane’s A005195). A graph can be tested to determine if it is acyclic using AcylicQ [g] in the Mathematica add-on package Discrete- Math‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). CONNECTED forests are TREES . See also ACYCLIC DIGRAPH ,C ONNECTED GRAPH , GRAPH CYCLE ,TREE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 32, 1994. Palmer, E. M. and Schwenk, A. J. "On the Number of Trees in a Random Forest." J. Combin. Th. B 27, 109 /C1/21, 1979. Skiena, S. "Acyclic Graphs." §5.3.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 188 /C1/ 90, 1990. Sloane, N. J. A. Sequences A005195/M0776 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Fork A fork of a TREE Tis a node of Twhich is the endpoint of two or more BRANCHES . See also BRANCH ,TREE Form CANONICAL FORM,C USP FORM,D IFFERENTIAL K- FORM,F ORM (GEOMETRIC ), F ORM (POLYNOMIAL ), MODULAR FORM,N ORMAL FORM,P FAFFIAN FORM, QUADRATIC FORM Form (Geometric) A 1-D geometric object such as a PENCIL or RANGE . Form (Polynomial) A HOMOGENEOUS POLYNOMIAL in two or more vari- ables. See also DIFFERENTIAL K-FORM,DISCONNECTED FORM Formal Logic SYMBOLIC LOGIC Formal Power Series A formal power series of a FIELD F is an infinite sequence a0 ;a1 ; a2 ;::: fg over F. Equivalently, it is a function from the set of nonnegative integers to F, 0; 1;2;::: fg 0 F: A formal power series is often written a0 /C27a1x /C27a2x2 /C27:::/C27anxn /C27:::; but with the understanding that no value is assigned to the symbol x. See also POWER SERIES References Henrici, P. "Definition and Algebraic Properties of Formal Series." §1.2 in Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, pp. 9 /C1/3, 1988. Formosa Theorem CHINESE REMAINDER THEOREM Formula A mathematical equation or a formal logical expres- sion. The correct Latin plural form of formula is "formulae," although the less pretentious-sounding "formulas" is more commonly used. See also EQUALITY ,EQUATION ,IDENTITY References Carr, G. S. Formulas and Theorems in Pure Mathematics. New York: Chelsea, 1970. Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968. Tallarida, R. J. Pocket Book of Integrals and Mathematical Formulas, 3rd ed. Boca Raton, FL: CRC Press, 1992. Weisstein, E. W. "Books about Handbooks of Mathematics." http://www.treasure-troves.com/books/Handbooksof- Mathematics.html.Fortunate Prime Let Xk /C131 /C27pk#; where pk is the kth PRIME and p is the PRIMORIAL , and let qkbe the NEXT PRIME (i.e., the smallest PRIME greater than Xk) ; qk /C30p1/C27 p(xk) /C30p1 /C27p(1/C27pk#) where p(n) is the PRIME COUNTING FUNCTION . Then R. F. Fortune conjectured that Fk /C13qk /C28Xk /C271is PRIME for all k. The first values of Fkare 3, 5, 7, 13, 23, 17, 19, 23, ... (Sloane’s A005235), and all known values of Fk are indeed PRIME (Guy 1994). The indices of these primes are 2, 3, 4, 6, 9, 7, 8, 9, 12, 18, .... In numerical order with duplicates removed, the For- tunate primes are 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, ... (Sloane’s A046066). See also ANDRICA’S CONJECTURE ,PRIMORIAL References Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243,1 8/C1/8, Dec. 1980. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 7, 1994. Sloane, N. J. A. Sequences A005235/M2418 and A046066 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Forward Difference The forward difference is a FINITE DIFFERENCE de- fined by Dan/C13an/C271/C28an: (1) Higher order differences are obtained by repeated operations of the forward difference operator, Dkan/C30Dk/C281an/C271/C28Dk/C281an; (2) so D2an/C30D2 n/C30D(Dn)/C30D(an/C271/C28an) /C30Dn/C271/C28Dn/C30an/C272/C282an/C271/C27an: (3) In general, Dk n /C13Dkan /C13Xk i/C300(/C281)i k i/C1Y/C1Q an/C27k/C28i ; (4) wherek m/C0/C1 is a BINOMIAL COEFFICIENT (Sloane and Plouffe 1985, p. 10). NEWTON’S FORWARD DIFFERENCE FORMULA expresses an as the sum of the nth forward differences an /C30a0 /C27n D0 /C271 2!n(n /C271)D20 /C271 3! n(n /C271)(n /C272)D30 /C27... (5) where Dn0is the first nth difference computed from the difference table. Furthermore, if the differences am ;Dam ;D2am ; ..., are known for some fixed value of m, then a formula for the nth term is given by an/C27m /C30Xn k/C300n k/C1Y/C1Q Dkam (6) (Sloane and Plouffe 1985, p. 10). See also BACKWARD DIFFERENCE ,CENTRAL DIFFER- ENCE ,DIFFERENCE EQUATION ,DIVIDED DIFFERENCE , RECIPROCAL DIFFERENCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 877, 1972. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 10, 1995. Fountain An (n, k) fountain is an arrangement of n coins in rows such that exactly k coins are in the bottom row and each coin in the (i /C271)/st row touches exactly two in the ith row. A generalized Rogers-Ramanujan-type continued fraction is closely related to the enumera- tion of coins in a fountain (Berndt 1991, 1985). References Berndt, B. C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, p. 79, 1985. Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan’s Lost Notebook." To appears in Trans. Amer. Math. Soc.Four Coins Problem Given three coins of possibly different sizes which are arranged so that each is tangent to the other two, find the coin which is tangent to the other three coins. The solution is the inner SODDY CIRCLE , illustrated above. See also APOLLONIUS CIRCLES ,A POLLONIUS’ PRO- BLEM ,ARBELOS ,BEND (CURVATURE ), CIRCUMCIRCLE , COIN,D ESCARTES CIRCLE THEOREM ,H ART’S THEO- REM,PAPPUS CHAIN ,SODDY CIRCLES ,SPHERE PACK- ING,STEINER CHAIN ,TANGENT CIRCLES References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319/C1/29, 1996. Four Conics Theorem If two intersections of each pair of three conics S1;S2; and S3lie on a conic , then the lines joining the other two intersections of each pair are CONCURRENT (Evelyn et al. 1974, pp. 23 and 25). The dual theorem states that if two common tangents of each pair of three conics touch a fourth conic, then the remaining common tangents of each pair inter- sect in three COLLINEAR points (Evelyn et al. 1974, pp. 24 /C1/5). See also CONIC SECTION ,THREE CONICS THEOREM References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Four-Conics Theorem." §2.4 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 22 /C1/9, 1974. Four Dog Problem MICE PROBLEM Four Exponentials Conjecture Let x1and x2be two linearly independent complex numbers, and let y1and y2be two linearly indepen- dent complex numbers. Then the four exponential conjecture posits that at least one of ex1y1 ;ex1y2 ;ex2y1 ;ex2y2 is TRANSCENDENTAL (Waldschmidt 1979, p. 3.5). The corresponding statement obtained by replacing y1 ; y2 with y1 ;y2 ;y3 has been proven and is known as the SIX EXPONENTIALS THEOREM . See also HERMITE- LINDEMANN THEOREM ,SIX EXPO- NENTIALS THEOREM ,TRANSCENDENTAL NUMBER References Finch, S. "Powers of 3/2 Modulo One." http://www.mathsoft.- com/asolve/pwrs32/pwrs32.html. Waldschmidt, M. Transcendence Methods. Queen’s Papers in Pure and Applied Mathematics, No. 52. Kingston, Ontario, Canada: Queen’s University, 1979. Waldschmidt, M. "On the Transcendence Method of Gelfond and Schneider in Several Variables." In New Advances in Transcendence Theory (Ed. A. Baker). Cambridge, Eng- land: Cambridge University Press, 1988. Four Travelers Problem Let four LINES in a PLANE represent four roads in GENERAL POSITION , and let one traveler Ti be walking along each road at a constant (but not necessarily equal to any other traveler’s) speed. Say that two travelers Tiand Tjhave "met" if they were simulta- neously at the intersection of their two roads. Then if T1has met all other three travelers (/T2 ; T3 ; and T4) and T2 ; in addition to meeting T1 ; has met T3 and T4 ; then T3 and T4 have also met! References Bogomolny, A. "Four Travellers Problem." http://www.cut- the-knot.com/gproblems.html. Four-Bug Problem MICE PROBLEMFour-Color Problem FOUR- COLOR THEOREM Four-Color Theorem The four-color theorem states that any map in a PLANE can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. This problem is sometimes also called GUTHRIE’S PROBLEM after F. Guthrie, who first conjectured the theorem in 1853. The CONJECTURE was then communicated to de Morgan and thence into the general community. In 1878, Cayley wrote the first paper on the conjecture. Fallacious proofs were given independently by Kempe (1879) and Tait (1880). Kempe’s proof was accepted for a decade until Heawood showed an error using a map with 18 faces (although a map with nine faces suffices to show the fallacy). The HEAWOOD CONJEC- TURE provided a very general assertion for map coloring, showing that in a GENUS 0 SPACE (i.e., either the SPHERE or PLANE ), six colors suffice. This number can easily be reduced to five, but reducing the number of colors all the way to four proved very difficult. (The KLEIN BOTTLE is the sole exception to the HEAWOOD CONJECTURE , requiring five colors instead of the six expected for a surface of genus 0.) Finally, Appel and Haken (1977) announced a com- puter-assisted proof that four colors were SUFFICIENT . However, because part of the proof consisted of an exhaustive analysis of many discrete cases by a computer, some mathematicians do not accept it. However, no flaws have yet been found, so the proof appears valid. A potentially independent proof has recently been constructed by N. Robertson, D. P. Sanders, P. D. Seymour, and R. Thomas. Martin Gardner (1975) played an April Fool’s joke by (incorrectly) claiming that the map of 110 regions illustrated above requires five colors and constitutesa counterexample to the four-color theorem. However, the coloring of Wagon (1998; 1999, pp. 535 /C1 /36) clearly shows that this map is, in fact, four-colorable. See also CHROMATIC NUMBER ,ERRERA GRAPH ,GRAPH COLORING ,HEAWOOD CONJECTURE ,KITTELL GRAPH , MAP COLORING ,SIX-COLOR THEOREM ,TORUS COLOR- ING References Appel, K. and Haken, W. "Every Planar Map is Four- Colorable, II: Reducibility." Illinois J. Math. 21, 491 /C1/67, 1977. Appel, K. and Haken, W. "The Solution of the Four-Color Map Problem." Sci. Amer. 237, 108 /C1/21, 1977. Appel, K. and Haken, W. "The Four Color Proof Suffices." Math. Intell. 8,10/C1/0 and 58, 1986. Appel, K. and Haken, W. Every Planar Map is Four-Color- able. Providence, RI: Amer. Math. Soc., 1989. Appel, K.; Haken, W.; and Koch, J. "Every Planar Map is Four Colorable. I: Discharging." Illinois J. Math. 21, 429 /C1/ 90, 1977. Barnette, D. Map Coloring, Polyhedra, and the Four-Color Problem. Providence, RI: Math. Assoc. Amer., 1983. Birkhoff, G. D. "The Reducibility of Maps." Amer. Math. J. 35, 114 /C1/28, 1913. Chartrand, G. "The Four Color Problem." §9.3 in Introduc- tory Graph Theory. New York: Dover, pp. 209 /C1/15, 1985. Coxeter, H. S. M. "The Four-Color Map Problem, 1840 /C1/ 890." Math. Teach. 52, 283 /C1/89, 1959. Franklin, P. "Note on the Four Color Problem." J. Math. Phys. 16, 172 /C1/84, 1937 /C1/938. Franklin, P. The Four-Color Problem. New York: Scripta Mathematica, Yeshiva College, 1941. Gardner, M. "Mathematical Games: The Celebrated Four- Color Map Problem of Topology." Sci. Amer. 203, 218 /C1/22, Sep. 1960. Gardner, M. "The Four-Color Map Theorem." Ch. 10 in Martin Gardner’s New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 113 /C1/23, 1966. Gardner, M. "Mathematical Games: Six Sensational Dis- coveries that Somehow or Another have Escaped Public Attention." Sci. Amer. 232, 127 /C1/31, Apr. 1975. Gardner, M. "Mathematical Games: On Tessellating the Plane with Convex Polygons." Sci. Amer. 232, 112 /C1/17, Jul. 1975. Harary, F. "The Four Color Conjecture." Graph Theory. Reading, MA: Addison-Wesley, p. 5, 1994. Heawood, P. J. "Map Colour Theorems." Quart. J. Math. 24, 332 /C1/38, 1890. Kempe, A. B. "On the Geographical Problem of Four-Colors." Amer. J. Math. 2, 193 /C1/00, 1879. Kraitchik, M. §8.4.2 in Mathematical Recreations. New York: W. W. Norton, p. 211, 1942. May, K. O. "The Origin of the Four-Color Conjecture." Isis 56, 346 /C1/48, 1965. Morgenstern, C. and Shapiro, H. "Heuristics for Rapidly 4- Coloring Large Planar Graphs." Algorithmica 6, 869 /C1/91, 1991. Ore, Ø. The Four-Color Problem. New York: Academic Press, 1967. Ore, Ø. and Stemple, G. J. "Numerical Methods in the Four Color Problem." Recent Progress in Combinatorics (Ed. W. T. Tutte). New York: Academic Press, 1969. Pappas, T. "The Four-Color Map Problem: Topology Turns the Tables on Map Coloring." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 152 /C1/53, 1989. Robertson, N.; Sanders, D. P.; and Thomas, R. "The Four- Color Theorem." http://www.math.gatech.edu/~thomas/ FC/fourcolor.html. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986.Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 210, 1990. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 274 /C1/75, 1999. Tait, P. G. "Note on a Theorem in Geometry of Position." Trans. Roy. Soc. Edinburgh 29, 657 /C1/60, 1880. Wagon, S. "An April Fool’s Hoax." Mathematica in Educ. Res. 7,46/C1/2, 1998. Wagon, S. Mathematica in Action, 2nd ed. New York: Springer-Verlag, pp. 535 /C1/36, 1999. Weisstein, E. W. "Books about Four-Color Problem." http:// www.treasure-troves.com/books/Four-ColorProblem.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 57, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 81 /C1/2, 1991. Four-Dimensional Geometry 4-DIMENSIONAL GEOMETRY Fourier Analysis FOURIER SERIES Fourier Cosine Series Iff(x)i sa n EVEN FUNCTION , then bn/C300 and the FOURIER SERIES collapses to f(x)/C301 2a0/C27X/C12 n/C301ancos(nx); (1) where a0/C301 pgp /C28pf(x)dx/C302 pgp 0f(x)dx (2) an/C301pgp /C28pf(x) cos( nx)dx /C302pgp 0f(x) cos( nx)dx (3) where the last equality is true because f(x) cos( nx)/C30f(/C28x) cos(/C28nx) (4) Letting the range go to L, a0/C302 LgL 0f(x)dx (5) an/C302 LgL 0f(x) cosnpx L ! dx: (6) See also EVEN FUNCTION ,FOURIER COSINE TRANS- FORM ,FOURIER SERIES ,FOURIER SINE SERIES Fourier Cosine Transform The Fourier cosine transform is the REAL PART of the full complex FOURIER TRANSFORM , Fcf(x)½/C138/C30R F f(x)½/C138½/C138 : In Mathematica 4.0, the Fourier cosine transform Fc(k) of a function f(x) is implemented as Four- ierCosTransform [f, x, k], and different choices of a and b can be used by passing the optional Four- ierParameters - /C21{a, b} option. In this work, a /C30 0 and b /C30/C282p:/ In version 4.1, the discrete Fourier cosine transform of a list l of real numbers can be computed using FourierCos [l] in the Mathematica add-on package LinearAlgebra‘FourierTrig‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also FOURIER SINE TRANSFORM ,FOURIER TRANS- FORM References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "FFT of Real Functions, Sine and Cosine Transforms." §12.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 504 /C1/15, 1992. Fourier Integral FOURIER TRANSFORM Fourier Matrix The n /C29n SQUARE MATRIX F/n with entries given by Fjk /C30e2 pijk=n /C13 vjk (1) for j ;k /C300; 1, 2, ..., n /C281; where I is the IMAGINARY NUMBER i /C30ffiffiffiffiffiffi /C281p ; and normalized by 1ffiffiffinpto make it a UNITARY . The Fourier matrix F2 is given by F2 /C301ffiffiffi 2p11 1 i2/C20/C21 ; (2) and the F4 matrix by F4 /C301ffiffiffi 4p1111 1 ii2i3 1 i2i4i6 1 i3i6i92 6643 775 /C301 211 1 i 1 /C281 1 /C28i2 6643 77511 1 i 2 111 i 22 6643 7751 1 1 12 6643 775: (3) In general, F 2n /C30 InDn In/C28Dn/C20/C21 Fn Fn/C20/C21 even -odd shuffle/C20/C21 ; (4)with Fn Fn/C20/C21 /C30In=2Dn=2 In=2/C28Dn=2/C20 In=2Dn=2 In=2/C28Dn=2/C21 /C29Fn=2 Fn=2 Fn=2 Fn=22 6643 775even -odd 0 ;2(mod4) even -odd 1 ;3(mod4)2 6643 775; (5) where I nis the n /C29n IDENTITY MATRIX and Dnis the DIAGONAL MATRIX with entries 1, v; ..., vn/C281 : Note that the factorization (which is the basis of the FAST FOURIER TRANSFORM ) has two copies of F2in the center factor MATRIX . See also FAST FOURIER TRANSFORM ,FOURIER TRANS- FORM References Strang, G. "Wavelet Transforms Versus Fourier Trans- forms." Bull. Amer. Math. Soc. 28, 288/C1/05, 1993. Fourier Series Fourier series are expansions of PERIODIC FUNCTIONS f(x) in terms of an infinite sum of SINES and COSINES OF THE FORM f(x)/C30X/C12 n/C300a?ncos(nx)/C27X/C12 n/C300b?nsin(nx): (1) Fourier series make use of the ORTHOGONALITY relationships of the SINE and COSINE functions, which can be used to calculate the coefficients anandbnin the sum. The computation and study of Fourier series is known as HARMONIC ANALYSIS . To compute a Fourier series, use the integral iden- tities gp /C28psin(mx) sin( nx)dx/C30pdmnforn;m"0 (2) gp /C28pcos(mx) cos( nx)dx/C30pdmnforn;m"0 (3) gp /C28psin(mx) cos( nx)dx/C300 (4) gp /C28psin(mx)dx/C300 (5) gp /C28pcos(mx)dx/C300; (6) where dmnis the K RONECKER DELTA . Now, expand your function f(x) as an infinite series OF THE FORM f(x)/C30X/C12 n/C300a?ncos(nx)/C27X/C12 n/C300b?nsin(nx) /C301 2a0/C27X/C12 n/C301ancos(nx)/C27X/C12 n/C301bnsin(nx) (7) where we have relabeled the a0/C302a?0term for future convenience but set bn/C30b?nand left an/C30a?nforn]1: Assume the function is periodic in the interval /C28p;p ½/C138 : Now use the orthogonality conditions to obtain gp /C28pf(x)dx /C30gp /C28pX/C12 n/C301ancos(nx)/C27X/C12 n/C301bnsin(nx)/C2712a 0"# dx /C30X/C12 n/C301gp /C28pancos(nx)/C27bnsin(nx) ½/C138 dx/C271 2a0gp /C28pdx /C30X/C12 n/C3010/C270 ðÞ /C27pa0/C30pa0 (8) and gp /C28pf(x) sin( mx)dx /C30gp /C28pX/C12 n/C301ancos(nx)/C27X/C12 n/C301bnsin(nx)/C2712a 0"# /C29sin(mx)dx /C30X/C12 n/C301gp /C28pancos(nx) sin( mx)/C27bnsin(nx) sin( mx) ½/C138 dx /C2712a 0gp /C28psin(mx)dx /C30X/C12 n/C3010/C27bnpdmn ðÞ /C270/C30pbn; (9) so gp /C28pf(x) cos( mx)dx/C30gp /C28pX/C12 n/C301ancos(nx)" /C27X/C12 n/C301bnsin(nx)/C271 2a0/C138cos(mx)dx /C30X/C12 n/C301gp /C28pancos(nx) cos( mx) ½ /C27bnsin(nx) cos( mx)/C138dx/C2712a 0gp /C28pcos(mx)dx/C30X/C12 n/C301anpdmn/C270 ðÞ /C270/C30pan: (10) Plugging back into the original series then gives a0/C301pgp /C28pf(x)dx (11) an/C301pgp /C28pf(x) cos( nx)dx (12) bn/C301 pgp /C28pf(x) sin( nx)dx (13) forn/C301, 2, 3, .... The series expansion converges to the function ¯f(equal to the original function at points of continuity or to the average of the two limits at points of discontinuity) ¯f/C131 2limx0x0/C28f(x)/C27limx0x0/C27f(x)hi for/C28pBx0Bp 12lim x0p/C27f(x)/C27limx0p/C28f(x) ½/C138 forx0/C30/C28p;p8 >>>>>>>< >>>>>>>:(14) if the function satisfies the D IRICHLET CONDITIONS . Near points of discontinuity, a "ringing" known as the GIBBS PHENOMENON , illustrated above, occurs. For a function f(x) periodic on an interval [ /C28L;L];use a change of variables to transform the interval ofintegration to [ /C281;1]:Let x/C13px? L(15) dx/C30pdx? L: (16) Solving for x?;x?/C30Lx=p:Plugging this in gives f(x?)/C301 2a0/C27X/C12 n/C301ancosnpx? L ! /C27X/C12 n/C301bnsinnpx? L ! (17) a0/C301 LgL /C28Lf(x?)dx? an/C301 LgL /C28Lf(x?) cosnpx? L ! dx? bn/C301 LgL /C28Lf(x?) sinnpx? L ! dx?8 >>>>>>>>>>< >>>>>>>>>>:(18) If a function is EVEN so that f(x) /C30f(/C28x); then f(x) sin(nx)is ODD. (This follows since sin(nx)is ODD and an EVEN FUNCTION times an ODD FUNCTION is an ODD FUNCTION .) Therefore, bn /C300 for all n. Similarly, if a function is ODD so that f(x) /C30/C28f(/C28x); then f(x) cos(nx)is ODD. (This follows since cos(nx)is EVEN and an EVEN FUNCTION times an ODD FUNCTION is an ODD FUNCTION .) Therefore, an /C300 for all n. Because the SINES and COSINES form a COMPLETE ORTHOGONAL BASIS , the SUPERPOSITION PRINCIPLE holds, and the Fourier series of a LINEAR COMBINA- TION of two functions is the same as the LINEAR COMBINATION of the corresponding two series. The COEFFICIENTS for Fourier series expansions for a few common functions are given in Beyer (1987, pp. 411 /C1/ 12) and Byerly (1959, p. 51). The notion of a Fourier series can also be extended to COMPLEX COEFFICIENTS . Consider a real-valued func- tion f(x): Write f ðxÞ¼X/C12 n/C30/C28/C12Aneinx: (19) Now examine gp /C28pf(x)e /C28imx dx /C30gp /C28pX/C12 n/C30/C28/C12Aneinx ! e /C28imx dx /C30X/C12 n /C30/C28/C12Angp /C28pei(n/C28m)x dx /C30X/C12 n/C30/C28/C12Angp /C28pcos (n /C28m)x ½/C138 /C27i sin (n /C28m)x ½/C138 fg dx /C30X/C12 m/C30/C28/C12An2pdmn /C302 pAm ; (20) so An ¼1 2p gp /C28pf(x)e /C28inx dx: (21) The COEFFICIENTS can be expressed in terms of those in the FOURIER SERIES An /C301 2pgp /C28pf(x) cos(nx) /C28i sin(nx) ½/C138 dx /C301 2pgp /C28pf(x) cos(nx) /C27i sin(nx) ½/C138 dx n B0 1 2pgp /C28pf(x) dx n /C300 1 2pgp /C28pf(x) cos(nx) /C28i sin(nx) ½/C138 dx n > 08 >>>>>>>>< >>>>>>>>:/C30 1 2(an /C27ibn Þ for n B0 12a0 for n /C300 1 2(an /C28ibn Þ for n /C2108 >< >:(22) For a function periodic in [/C28L =2; L =2]; these become f(x) /C30X/C12 n/C30/C28/C12Anei(2pnx=L)(23) An/C301 LgL=2 /C28L=2f(x)e/C28i(2pnx=L)dx: (24) These equations are the basis for the extremely important F OURIER TRANSFORM , which is obtained by transforming Anfrom a discrete variable to a continuous one as the length L0/C12:/ See also DIRICHLET FOURIER SERIES CONDITIONS , FOURIER COSINE SERIES ,F OURIER SINE SERIES , FOURIER TRANSFORM ,G IBBS PHENOMENON ,LEBES- GUE CONSTANTS (FOURIER SERIES ), LEGENDRE SER- IES,RIESZ- FISCHER THEOREM ,SCHLO ¨ MILCH’S SERIES References Arfken, G. "Fourier Series." Ch. 14 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 760 /C1/93, 1985. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value Problems, 5th ed. New York: McGraw- Hill, 1993. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics.New York: Dover, 1959. Carslaw, H. S. Introduction to the Theory of Fourier’s Series and Integrals, 3rd ed., rev. and enl. New York: Dover, 1950. Davis, H. F. Fourier Series and Orthogonal Functions. New York: Dover, 1963. Dym, H. and McKean, H. P. Fourier Series and Integrals. New York: Academic Press, 1972. Folland, G. B. Fourier Analysis and Its Applications. Pacific Grove, CA: Brooks/Cole, 1992. Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York: Cambridge University Press, 1996. Ko¨rner, T. W. Fourier Analysis. Cambridge, England: Cam- bridge University Press, 1988. Ko¨rner, T. W. Exercises for Fourier Analysis. New York: Cambridge University Press, 1993. Krantz, S. G. "Fourier Series." §15.1 in Handbook of Com- plex Analysis. Boston, MA: Birkha ¨user, pp. 195 /C1 /02, 1999. Lighthill, M. J. Introduction to Fourier Analysis and Gen- eralised Functions. Cambridge, England: Cambridge Uni- versity Press, 1958. Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994. Sansone, G. "Expansions in Fourier Series." Ch. 2 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 39 /C1/68, 1991. Weisstein, E. W. "Books about Fourier Transforms." http:// www.treasure-troves.com/books/FourierTransforms.html. Whittaker, E. T. and Robinson, G. "Practical Fourier Ana- lysis." Ch. 10 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 260 /C1/84, 1967. Fourier Series * /Power Series For f(x) /C30xk on the INTERVAL [/C28L;L) and periodic with period 2L ; the FOURIER SERIES is given by an /C301 L gL /C28Lxk cosnpx L ! dx /C302Lk 1 /C27 k 1F21 /C271 2k 1212(3 /C27k); /C281 4 p2n2 ! bn /C301 L gL /C28Lxk sinnpx L ! dx /C302npLk 2 /C27 k 1F21 /C2712k 322/C2712k; /C281 4 p2n2 ! ; where1F2(a; b; c; x) is a generalized HYPERGEO- METRIC FUNCTION . Fourier Series * /Sawtooth Wave Consider a string of length 2L plucked at the right end, then a0 /C301 L g2L 0x 2Ldx /C301 2L212x2hiL 0/C301 4L2 (2L)2 /C301 an /C301 L g2L 0x 2Lcosn px L ! dx /C302n p cos(n p) /C28 sin(np) ½/C138 sin(np) n2 p2 /C300 bn /C301 L g2L 0x 2Lsinnpx L ! dx /C30/C282n p cos(2 np) /C27 sin(2 n p) 2n2 p2 /C30/C281 np: The Fourier series is therefore f(x) /C301 2 /C281 pX/C12 n/C3011 nsinnpx L ! : See also FOURIER SERIES ,FOURIER SERIES– SQUARE WAVE,SAWTOOTH WAVEFourier Series * /Square Wave Consider a square wave of length 2L: Since the function is ODD, a0 /C30an /C300; and bn /C302 L gL 0sinnpx L ! dx /C304 npsin2(1 2np) /C304 np0 n even 1 n odd:/C27 The Fourier series is therefore f(x) /C304 pX/C12 n /C301 ;3 ;5;...1 nsinnpx L ! : See also FOURIER SERIES ,FOURIER SERIES– SAWTOOTH WAVE,SQUARE WAVE Fourier Series * /Triangle Let a string of length 2 Lhave a y-displacement of unity when it is pinned an x-distance which is ( /(1=m))/ th of the way along the string. The displacement as a function of xis then fm(x)/C30mx 2L05x52L m m 1/C28mx 2L/C281 ! 2L m5x52L:8 >>>>< >>>>: The COEFFICIENTS are therefore a0 /C301 L g2L=m 0nx 2Ldx /C27g2L 2L=mn 1 /C28 nx 2L /C281 ! dx"# /C301 an /C30m 1 /C28 m /C28 cos(2 pn) /C27 m cos2np m ! "# 2(m /C28 1)n2 p2 /C30m2 cos2np m ! /C28 1"# 2(m /C28 1)m2 p2 bn /C30mm sin2pn m ! /C28 sin(2pn)"# 2(m /C28 1)n2 p2 /C30m2 sin2pn m ! 2(m /C28 1)n2 p2 : The Fourier series is therefore fm(x) /C301 2 /C27m2 2(m /C28 1)p2 /C29X/C12 n/C3011 n2cos2np m ! /C281"# cosnpx L ! ( /C27sin2pn m ! n2sinn px L !/C25 : If m /C302, then an and bn simplify to an /C30/C284 n2 p2 sin212np/C17/C15 /C30/C284 n2 p20 n /C300;2 ;... 1 n /C301;3 ;.../C27 bn /C300; giving f2(x) /C3012 /C284 p2X/C12 n/C301 ;3;5 ;...1 n2cosnpx L ! : See also FOURIER SERIESFourier Series * /Triangle Wave Consider a triangle wave of length 2L : Since the function is ODD, a0 /C30an /C300; and bn /C302 L/C27gL =2 0x L =2sinnpx L ! dx /C27g0 L =21 /C282 Lx /C281 2L !"# sinnpx L ! dx/C25 dx /C3032 p2n2 cos14np ! sin 314 np ! /C3032 p2n20 n /C300;4; ... 1 4n /C301 ;5;... 0 n /C302;6; ... /C281 4n /C303;7 ;...8 >>< >>: /C308 p2n2(/C281)(n/C281)=2fornodd 0 for neven :/C27 The Fourier series is therefore f(x)/C308 p2X/C12 n/C301;3;5;...(/C281)(n/C281)=2 n2sinnpx L ! : See also FOURIER SERIES Fourier Sine Series Iff(x)i sa n ODD FUNCTION , then an¼0 and the FOURIER SERIES collapses to f(x)/C30X/C12 n/C301bnsin(nx); (1) where bn/C301 pgp /C28pf(x) sin( nx)dx/C302pgp 0f(x) sin( nx)dx (2) forn/C301, 2, 3, .... The last EQUALITY is true because f(x) sin( nx)/C30/C28 f(/C28x) ½/C138 /C28sin(/C28nx) ½/C138 /C30f(/C28x) sin(/C28nx): (3) Letting the range go to L, bn /C302 L gL 0f(x)sinnpx L ! dx: (4) See also FOURIER COSINE SERIES ,FOURIER SERIES , FOURIER SINE TRANSFORM Fourier Sine Transform The Fourier sine transform is the IMAGINARY PART of the full complex FOURIER TRANSFORM , Fs f(x)½/C138/C30I F f(x)½/C138½/C138 : In Mathematica 4.0, the Fourier sine transform Fs(k) of a function f(x) is implemented as FourierSin- Transform [f, x, k], and different choices of a and b can be used by passing the optional FourierPara- meters - /C21{a, b} option. In this work, a /C300 and b /C30/C282p:/ In version 4.1, the discrete Fourier sine transform of a list l of real numbers can be computed using FourierSin [l] in the Mathematica add-on package LinearAlgebra‘FourierTrig‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also FOURIER COSINE TRANSFORM ,F OURIER TRANSFORM References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "FFT of Real Functions, Sine and Cosine Transforms." §12.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 504 /C1/15, 1992. Fourier Transform The Fourier transform is a generalization of the COMPLEX FOURIER SERIES in the limit as L0/C12: Replace the discrete Anwith the continuous F(k)dk while letting n=L0k:Then change the sum to an INTEGRAL , and the equations become f(x)/C30g/C12 /C28/C12F(k)e2pikxdk (1) F(k)/C30g/C12 /C28/C12f(x)e/C282pikxdx: (2) Here, F(k)/C30F[f(x)]/C30g/C12 /C28/C12f(x)e/C282pikxdx (3) is called the forward /(/C28i) Fourier transform, and f(x)/C30F/C281[F(k)]/C30g/C12 /C28/C12F(k)e2pikxdk (4) is called the inverse /(/C27i) Fourier transform. The notation fffl(k) and f/C150(x) are sometimes used for theFourier transform and inverse Fourier transform, respectively (Krantz 1999, p. 202). Note that some authors (especially physicists) prefer to write the transform in terms of angular frequency v/C132pninstead of the oscillation frequency n:How- ever, this destroys the symmetry, resulting in the transform pair HðvÞ¼F½hðtÞ/C138 ¼g/C12 /C28/C12hðtÞe/C28ivtdt (5) h(t)/C30F/C281[H(v)]/C301 2pg/C12 /C28/C12H(v)eivtdv: (6) To restore the symmetry of the transforms, the convention g(y)/C30F[f(t)]/C301ffiffiffiffiffiffi 2ppg/C12 /C28/C12f(t)e/C28iytdt (7) f(t)/C30F/C281[g(y)]/C301ffiffiffiffiffiffi2ppg/C12 /C28/C12g(y)eiytdy (8) is sometimes used (Mathews and Walker 1970, p. 102). In general, the Fourier transform pair may be defined using two arbitrary constants aandbas F(v)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½b½ (2p)1/C28as g/C12 /C28/C12f(t)eibvtdt (9) f(t)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½b½ (2p)1/C27as g/C12 /C28/C12F(v)e/C28ibvtdw: (10) InMathematica 4.0, the Fourier transform F(k)o fa function f(x) is implemented as FourierTrans- form [f,x,k], and different choices of aand bcan be used by passing the optional FourierPara- meters -/C21{a,b} option. By default, Mathematica takesFourierParameters as (0 ;1):Unfortunately, a number of other conventions are in widespread use. For example, (0 ;1) is used in modern physics, (1 ;/C281) is used in pure mathematics and systems engineer- ing, (1 ;1) is used in probability theory for the computation of the CHARACTERISTIC FUNCTION , (/C281;1) is used in classical physics, and (0 ;/C282p)i s used in signal processing. In this work, followingBracewell (1999, pp. 6 /C1 /),it is always assumed that a/C300 and b/C30/C282punless otherwise stated. This choice often results in greatly simplified transformsof common functions such as 1, cos(2 pk 0x);etc. Since any function can be split up into EVEN and ODD portions E(x) and O(x); f(x)/C301 2[f(x)/C27f(/C28x)]/C2712[f(x)/C28f(/C28x)]/C30E(x)/C27O(x); (11) a Fourier transform can always be expressed in terms of the F OURIER COSINE TRANSFORM and F OURIER SINE TRANSFORM as F[f(x)]/C30g/C12 /C28/C12E(x) cos(2 pkx)dx /C28ig/C12 /C28/C12O(x) sin(2 pkx)dx: (12) A function f(x) has a forward and inverse Fourier transform such that f(x)/C30g/C12 /C28/C12e2pikxg/C12 /C28/C12f(x)e/C282pikxdx/C20/C21 dk forf(x) continuous at x 1 2f(x/C27)/C27f(x/C28)/C2/C6 forf(x) discontinous at x;8 >>>>< >>>>:(13) provided that 1.f/C12 /C28/C12½f(x)½dxexists. 2. There are a finite number of discontinuities. 3. The function has bounded variation. A SUFFI- CIENT weaker condition is fulfillment of the LIPSCHITZ CONDITION (Ramirez 1985, p. 29). The smoother a function (i.e.,the larger the number of continuous DERIVATIVES ), the more compact its Fourier transform. The Fourier transform is linear, since if f(x) and g(x) have Fourier transforms F(k) and G(k);then g[af(x)/C27bg(x)]e/C282pikxdx /C30ag/C12 /C28/C12f(x)e/C282pikxdx/C27bg/C12 /C28/C12g(x)e/C282pikxdx aF(k)/C27bG(k): (14) Therefore, F[af(x)/C27bg(x)]/C30aF[f(x)]/C27bF(g(x)] /C30aF(k)/C27bG(k):(15) The Fourier transform is also symmetric since F(k)/C30 F[f(x)] implies F(/C28k)/C30F[f(/C28x)]:/ Letf+gdenote the CONVOLUTION , then the trans- forms of convolutions of functions have particularly nice transforms, F(f+g)/C30F[f]F[g] (16) F[fg]/C30F[f]+F[g] (17) F/C281[F(f)F(g)]/C30f+g (18) F/C281[F(f)+F(g)]/C30fg: (19) The first of these is derived as follows: F[f+g]/C30g/C12 /C28/C12g/C12 /C28/C12e/C282pikxf(x?)g(x /C28x?)dx?dx/C30g/C12 /C28/C12g/C12 /C28/C12e/C282pikx?f(x?)dx?/C2/C6 /C2e/C282pik(x/C28x?)g(x/C28x?)dx/C2/C6 /C30g/C12 /C28/C12e/C282pikx?f(x?)dx?/C20/C21g/C12 /C28/C12e/C282pikxƒg(xƒ)dxƒ/C20/C21 /C30F[f]F[g]; (20) where xƒ/C13x/C28x?:/ There is also a somewhat surprising and extremely important relationship between the AUTOCORRELA- TION and the Fourier transform known as the WIENER- KHINTCHINE THEOREM . Let F[f(x)]/C30F(k); and ¯fdenote the COMPLEX CONJUGATE off, then the Fourier transform of the ABSOLUTE SQUARE ofF(k)i s given by F[jF(k)j2]/C30g/C12 /C28/C12f(t)f(t/C27x)dt: (21) The Fourier transform of a DERIVATIVE f?ðxÞof a function f(x) is simply related to the transform of the function f(x) itself. Consider Ff?(x) ½/C138/C30g/C12 /C28/C12f?(x)e/C282pikxdx: (22) Now use INTEGRATION BY PARTS gvdu/C30[uv]/C28gudv (23) with du/C30f?(x)dx v/C30e/C282pikx(24) u/C30f(x)dv/C30/C282pike/C282pikxdx; (25) then Ff?(x) ½/C138/C30f(x)e/C282pikx/C2/C6 /C12 /C28/C12/C28g/C12 /C28/C12f(x)(/C282pike/C282pikxdx): (26) The first term consists of an oscillating function times f(x):But if the function is bounded so that lim x09/C12f(x)/C300 (27) (as any physically significant signal must be), thenthe term vanishes, leaving Ff?(x) ½/C138/C302pikg/C12 /C28/C12f(x)e/C282pikxdx/C302pikF f (x)½/C138 :(28) This process can be iterated for the nthDERIVATIVE to yield Ff(n)(x)/C2/C6 /C30(2pik)nFf(x)½/C138 : (29) The important MODULATION THEOREM of Fourier transforms allows Fcos(2 pk0x)f(x) ½/C138 to be expressed in terms of F[f(x)]/C30F(k) as follows, F cos(2 pk0x)f(x) ½/C138 /C13g/C12 /C28/C12f(x) cos(2 pk0x)e /C282 pikxdx /C301 2g/C12 /C28/C12f(x)e2 pik0xe /C282pikxdx /C2712g/C12 /C28/C12f(x)e/C282 pik0xe /C282pikxdx /C301 2g/C12 /C28/C12f(x)e /C282 pi(k /C28k0)xdx /C2712g/C12 /C28/C12f(x)e /C282pi(k/C27k0)xdx /C301 2F(k /C28k0) /C27F(k /C27k0) ½/C138 : (30) Since the DERIVATIVE of the Fourier transform is given by F ?(k) /C13d dxF f(x)½/C138/C30g/C12 /C28/C12(/C282pix)f(x)e /C282 pikxdx; (31) it follows that F ?(0) /C30/C282 pig/C12 /C28/C12xf(x)dx: (32) Iterating gives the general FORMULA mn /C13g/C12 /C28/C12xnf(x)dx /C30F(n)(0) ( /C282pi)n : (33) The VARIANCE of a FOURIER TRANSFORM is s2 f /C30/C142(xf /C28/C142xf /C143)2 /C143; (34) and it is true that sf /C27g /C30 sf /C27 sg : (35) If f(x) has the Fourier transform F(k) ; then the Fourier transform has the shift property g/C12 /C28/C12f(x /C28x0)e/C282 pikxdx /C30g/C12 /C28/C12f(x /C28x0)e /C282pi(x/C28x0)ke /C282 pi(kx0)d(x /C28x0) /C30e /C282 pikx0 F(k) ; (36) so f(x /C28x0) has the Fourier transform Ff(x /C28x0) ½/C138 /C30e/C282 pikx0 F(k) : (37) If f(x) has a Fourier transform F(k) ; then the Fourier transform obeys a similarity theorem. g/C12 /C28/C12f(ax)e /C282 pikxdx /C301 ajjg/C12 /C28/C12f(ax)e /C282 pi(ax)(k =a)d(ax) /C301 ajjFk a ! ; (38) so f(ax) has the Fourier transform ajj/C281F ðk=aÞ:/The "equivalent width" of a Fourier transform is wo /C13g/C12 /C28/C12f(x)dx f(0)/C30F(0) g/C12 /C28/C12F(k)dx: (39) The "autocorrelation width" is wa /C13g/C12 /C28/C12f + ¯fdx f + ¯f/C2/C6 0/C30g/C12 /C28/C12fdxg/C12 /C28/C12¯fdx g/C12 /C28/C12f ¯fdx; (40) where /f + g/ denotes the CROSS-CORRELATION of f and g and ¯f is the COMPLEX CONJUGATE . Any operation on f(x) which leaves its AREA un- changed leaves F(0) unchanged, since g/C12 /C28/C12f(x)dx /C30Ff(0)½/C138/C30f(0) : (41) In 2-D, the Fourier transform becomes F(x;y) /C30g/C12 /C28/C12g/C12 /C28/C12f(kx ;ky)e /C282 pi(kxx/C27kyy)dkxdky (42) F(kx ;ky) /C30g/C12 /C28/C12g/C12 /C28/C12f(x;y)e2pi(kxx/C27kyy)dxdy : (43) Similarly, the n-D Fourier transform can be defined fork,x/C23Rnby F(x)/C30g/C12 /C28/C12/C1/C1/C1g/C12 /C28/C12|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} nf(k)e/C282pik/C215xdnk (44) f(k)/C30g/C12 /C28/C12/C1/C1/C1g/C12 /C28/C12|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} nF(x)e/C282pik /C215xdnx: (45) See also AUTOCORRELATION ,CONVOLUTION ,DISCRETE FOURIER TRANSFORM ,F AST FOURIER TRANSFORM , FOURIER SERIES ,F OURIER- STIELTJES TRANSFORM , FOURIER TRANSFORM–1 ,F OURIER TRANSFORM– CO- SINE,FOURIER TRANSFORM– DELTA FUNCTION ,FOUR- IER TRANSFORM– EXPONENTIAL FUNCTION ,F OURIER TRANSFORM– GAUSSIAN ,FOURIER TRANSFORM– HEAVI- SIDE STEP FUNCTION ,FOURIER TRANSFORM– INVERSE FUNCTION ,FOURIER TRANSFORM– LORENTZIAN FUNC- TION ,FOURIER TRANSFORM– RAMP FUNCTION ,FOUR- IER TRANSFORM– RECTANGLE FUNCTION ,H ANKEL TRANSFORM ,HARTLEY TRANSFORM ,INTEGRAL TRANS- FORM ,L APLACE TRANSFORM ,S TRUCTURE FACTOR , WINOGRAD TRANSFORM References Arfken, G. "Development of the Fourier Integral," "Fourier Transforms--Inversion Theorem," and "Fourier Transform of Derivatives." §15.2/C1/5.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 794 /C1/ 10, 1985. Blackman, R. B. and Tukey, J. W. The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, 1959. Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999. Brigham, E. O. The Fast Fourier Transform and Applica- tions. Englewood Cliffs, NJ: Prentice Hall, 1988. Folland, G. B. Real Analysis: Modern Techniques and their Applications, 2nd ed. New York: Wiley, 1999. James, J. F. A Student’s Guide to Fourier Transforms with Applications in Physics and Engineering. New York: Cambridge University Press, 1995. Ko¨rner, T. W. Fourier Analysis. Cambridge, England: Cam- bridge University Press, 1988. Krantz, S. G. "The Fourier Transform." §15.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 202 /C1/12, 1999. Mathews, J. and Walker, R. L. Mathematical Methods of Physics, 2nd ed. Reading, MA: W. A. Benjamin/Addison- Wesley, 1970. Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994. Morse, P. M. and Feshbach, H. "Fourier Transforms." §4.8 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 453 /C1/71, 1953. Oberhettinger, F. Fourier Transforms of Distributions and Their Inverses: A Collection of Tables. New York: Aca- demic Press, 1973. Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, 1989. Ramirez, R. W. The FFT: Fundamentals and Concepts. Englewood Cliffs, NJ: Prentice-Hall, 1985. Sansone, G. "The Fourier Transform." §2.13 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 158 /C1/68, 1991. Sneddon, I. N. Fourier Transforms. New York: Dover, 1995. Sogge, C. D. Fourier Integrals in Classical Analysis. New York: Cambridge University Press, 1993. Spiegel, M. R. Theory and Problems of Fourier Analysis with Applications to Boundary Value Problems. New York: McGraw-Hill, 1974. Stein, E. M. and Weiss, G. L. Introduction to Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton University Press, 1971. Strichartz, R. Fourier Transforms and Distribution Theory. Boca Raton, FL: CRC Press, 1993. Titchmarsh, E. C. Introduction to the Theory of Fourier Integrals, 3rd ed. Oxford, England: Clarendon Press, 1948. Tolstov, G. P. Fourier Series. New York: Dover, 1976. Walker, J. S. Fast Fourier Transforms, 2nd ed. Boca Raton, FL: CRC Press, 1996. Weisstein, E. W. "Books about Fourier Transforms." http:// www.treasure-troves.com/books/FourierTransforms.html. Fourier Transform * /1 The F OURIER TRANSFORM of the CONSTANT FUNCTION f(x)/C301 is given by F[1]/C30g/C12 /C28/C12e/C282pikxdx/C30d(k); according to the definition of the DELTA FUNCTION .See also DELTA FUNCTION ,FOURIER TRANSFORM Fourier Transform * /Cosine Fcos 2pk0x ðÞ½/C138 /C30g/C12 /C28/C12e2pikxe2pik0x/C27e/C282pik0x 2 ! dx /C301 2g/C12 /C28/C12e/C282pik/C28k0 ðÞ x/C27e/C282pik/C27k0 ðÞ x/C2/C6 dx /C3012dk/C28k0 ðÞ /C27dk/C27k0 ðÞ ½/C138 ; where d(x) is the DELTA FUNCTION . See also COSINE ,F OURIER TRANSFORM ,F OURIER TRANSFORM– SINE Fourier Transform * /Delta Function The F OURIER TRANSFORM of the DELTA FUNCTION is given by Fdx/C28x0 ðÞ½/C138 /C30g/C12 /C28/C12dx/C28x0 ðÞ e/C282pikxdx/C30e/C282pikx0: See also DELTA FUNCTION ,FOURIER TRANSFORM Fourier Transform * /Exponential Function The F OURIER TRANSFORM ofe/C28k0½x½is given by Fe/C28k0½x½/C2/C6 /C30g/C12 /C28/C12e/C28k0½x½e/C282pikxdx /C30g0 /C28/C12e/C282pikxe2pxk0dx/C27g/C12 0e/C282pikxe/C282pk0xdx /C30g0 /C28/C12cos(2pkx)/C28isin(2 kx) ½/C138 e2pk0xdx: /C27g/C12 0cos(2pkx)/C28isin(2pkx) ½/C138 e/C282pk0xdx: (1) Now let u/C13/C28xsodu/C30/C28dx;then Fe/C28k0½x½/C2/C6 /C30g/C12 0cos(2pku)/C27isin(2pku) ½/C138 e/C282pk0udu /C27g/C12 0cos(2pku)/C28isin(2pku) ½/C138 e/C282pk0udu ¼2g/C12 0cos(2pku)e/C282pkoudu; (2) which, from the DAMPED EXPONENTIAL COSINE INTE- GRAL , gives F e/C282 pk0 xjj/C2/C6 /C301 pk0 k2 /C27 k2 0; (3) which is a LORENTZIAN FUNCTION . See also DAMPED EXPONENTIAL COSINE INTEGRAL , EXPONENTIAL FUNCTION ,FOURIER TRANSFORM ,LOR- ENTZIAN FUNCTION Fourier Transform * /Gaussian The FOURIER TRANSFORM of a GAUSSIAN FUNCTION f(x) /C13e /C28ax2 is given by F(k) /C30g/C12 /C28/C12e/C28ax2 e /C282 pikxdx /C30g/C12 /C28/C12e /C28ax2 [cos(2 pkx) /C28i sin(2pix)]dx /C30g/C12 /C28/C12e /C28ax2 cos(2pkx)dx /C28ig/C12 /C28/C12e/C28ax2 sin(2pkx)dx: The second integrand is ODD, so integration over a symmetrical range gives 0. The value of the first integral is given by Abramowitz and Stegun (1972, p. 302, equation 7.4.6), so F(k) /C30ffiffiffi p as e /C28p2k2 =a ; and a GAUSSIAN transforms to a GAUSSIAN . See also GAUSSIAN FUNCTION ,FOURIER TRANSFORM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 302, 1972. Fourier Transform * /Heaviside Step Function The FOURIER TRANSFORM of the HEAVISIDE STEP FUNCTION H(x) is given by F[H(x)] /C30g/C12 /C28/C12e /C282 pikxH(x)dx /C301 2d(k) /C28i pk"# ; where d(k) is the DELTA FUNCTION . See also FOURIER TRANSFORM ,H EAVISIDE STEP FUNCTION Fourier Transform * /Inverse Function The FOURIER TRANSFORM of the GENERALIZED FUNC- TION 1 =x is given byF /C28PV1 px ! /C30/C281 pPV g/C12 /C28/C12e /C282 pikx xdx (1) /C30PVg/C12 /C28/C12cos(2 pkx) /C28 i sin(2pkx) x dx (2) /C30/C282i p g/C12 0sin(2pkx) xdx for k B0 2i p g/C12 0sin(2pkx) xdx for k > 08 >>>< >>>:(3) /C30/C28i for k B0 i for k > 0 ;/C27 (4) where PV denotes the C AUCHY PRINCIPAL VALUE . Equation (4) can also be written as the single equation F /C28PVi px ! /C30i 1 /C282H(/C28k) ½/C138 ; (5) where H(x) is the HEAVISIDE STEP FUNCTION . The integrals follow from the identity g/C12 0sin(2 pkx) xdx /C30g/C12 0sin(2pkx) 2pkxd(2pkx) /C30g/C12 0sinczdz/C301 2 p: (6) See also FOURIER TRANSFORM Fourier Transform * /Lorentzian Function F1 p1 2G (x/C28x0)2/C2712G/C17/C1522 643 75/C30e/C282pikx0/C28Gpkjj: This transform arises in the computation of the CHARACTERISTIC FUNCTION of the C AUCHY DISTRIBU- TION . See also FOURIER TRANSFORM ,LORENTZIAN FUNCTION Fourier Transform * /Ramp Function LetR(x) be the RAMP FUNCTION , then the F OURIER TRANSFORM ofR(x) is given by FR(x) ½/C138/C30g/C12 /C28/C12e/C282pikxR(x)dx/C30pid?(2pk)/C281 4p2k2; where d?(x) is the DERIVATIVE of the DELTA FUNCTION . See also RAMP FUNCTION Fourier Transform * /Rectangle Function Let P(x) be the RECTANGLE FUNCTION , then the FOURIER TRANSFORM is F II(x) ½/C138/C30sinc( pk) ; where sinc(x) is the SINC FUNCTION . See also FOURIER TRANSFORM ,RECTANGLE FUNCTION , SINC FUNCTION Fourier Transform * /Sine F sin(2 pk0x) ½/C138 /C30g/C12 /C28/C12e /C282pikxe2pik0x /C28 e/C282pik0x 2i ! dx /C301 2ig/C12 /C28/C12/C28e /C282 pi(k/C28k0)x /C27e /C282pi(k/C27k0)x/C2/C6 dt /C3012i d(k /C27k0) /C28 d(k /C28k0) ½/C138 ; where d(x) is the DELTA FUNCTION . See also FOURIER TRANSFORM ,FOURIER TRANSFORM– COSINE ,SINE Fourier-Bessel Series BESSEL FUNCTION FOURIER EXPANSION ,SCHLO ¨ MIL- CH’S SERIES Fourier-Bessel Transform HANKEL TRANSFORM Fourier-Budan Theorem For any real a and b such that b > a; let p( a) "0 and p(b) "0 be real polynomials of degree n, and v(x) denote the number of sign changes in the sequence p(x) ;p ?(x);:::; p(n)(x)/CY/CQ : Then the number of zeros in the interval a; b½/C138 (each zero counted with proper multiplicity) equals v( a) /C28v( b) minus an even non- negative integer. References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, p. 443, 1988. Fourier-Mellin Integral The inverse of the LAPLACE TRANSFORM F(t) /C30L/C281 f(s)½/C138/C301 2pi g g/C27i/C12 g/C28i/C12estf(s)dsf(s) /C30L F(t)½/C138/C30g/C12 0F(t)e /C28stdt : See also BROMWICH INTEGRAL ,LAPLACE TRANSFORM Fourier-Stieltjes Transform Let f(x) be a positive definite, measurable function on the INTERVAL (/C28/C12;/C12) : Then there exists a monotone increasing, real-valued bounded function a(t) such that f(x) /C30g/C12 /C28/C12eitxda(t) for "ALMOST ALL" x.If a(t) is nondecreasing and bounded and f(x) is defined as above, then f(x)is called the Fourier-Stieltjes transform of a(t) ; and is both continuous and positive definite. See also FOURIER TRANSFORM ,LAPLACE TRANSFORM References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 618, 1980. Four-Knot FIGURE-OF- EIGHT KNOT Four-Square Theorem LAGRANGE’S FOUR- SQUARE THEOREM Four-Vector A four-element vector am/C30a0 a1 a2 a32 6643 775; (1) which transforms under a L ORENTZ TRANSFORMATION like the POSITION FOUR-VECTOR . This means it obeys a?m/C30Lm vav(2) am/C215bm/C13ambm(3) am/C215bm/C30a?mb?m (4) where Lmmis the L ORENTZ TENSOR . Multiplication of two four-vectors with the METRIC gmngives products OF THE FORM gmnxmxv/C30(x0)2/C28(x1)2/C28(x2)2/C28(x3)2: (5) In the case of the POSITION FOUR-VECTOR ,x0/C30ct (where cis the speed of light) and this product is an invariant known as the spacetime interval. See also GRADIENT FOUR- VECTOR ,LORENTZ TRANS- FORMATION ,P OSITION FOUR- VECTOR ,Q UATERNION , TENSOR ,VECTOR References Morse, P. M. and Feshbach, H. "The Lorentz Transforma- tion, Four-Vectors, Spinors." §1.7 in Methods of Theore- tical Physics, Part I. New York: McGraw-Hill, pp. 93 /C1/07, 1953. Four-Vertex Theorem A closed embedded smooth PLANE CURVE has at least four vertices, where a vertex is defined as an extremum of CURVATURE . See also CURVATURE References Tabachnikov, S. "The Four-Vertex Theorem Revisited--Two Variations on the Old Theme." Amer. Math. Monthly 102, 912 /C1/16, 1995. Fox’s H-Function A very general function defined by H(z) /C30Hm;n p;qz(a1 ; a1) ;...;(ap ; ap) (b1 ; b1) ;...;(bp ; bp)/C12/C12/C12/C12/C21 /C20 /C301 2pi gCPm j/C301 G(bj /C28 bis)Pnj/C301 G(1 /C28 aj /C27 ajs) Pq j/C30m/C271 G(1 /C28 bj /C27 bjs)Pqpj/C30n/C271 G(aj /C28 ajs) /C2zsds ; where 0 5m 5q; 0 5n 5p ; aj ; bj > 0; and aj ;bjare COMPLEX NUMBERS such that the pole of G(bj /C28 bjs) for j /C301, 2, ..., m coincides with any POLE of G(1 /C28aj /C27 ajs) for j /C301, 2, ..., n. In addition C,isa CONTOUR in the complex s-plane from v /C28i /C12 to v /C27i /C12 such that (bj/C27k)=bjand ( aj/C281/C28k)=ajlie to the right and left of C, respectively. A. Kilbas has derived a complete description for the asymptotic expansion of the H-function. See also KAMPE DE FERIET FUNCTION ,M ACROBERT’S E-FUNCTION ,MEIJER’S G-FUNCTION References Carter, B. D. and Springer, M. D. "The Distribution of Products, Quotients, and Powers of Independent H-Func- tions." SIAM J. Appl. Math. 33, 542/C1/58, 1977. Fox, C. "The Gand H-Functions as Symmetrical Fourier Kernels." Trans. Amer. Math. Soc. 98, 395/C1/29, 1961. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. "Evaluation of Integrals and the Mellin Transform." Itogi Nauki i Tekhniki, Seriya Matemat. Analiz 27,3/C1/46, 1989. Yakubovich, S. B. and Luchko, Y. F. The Hypergeometric Approach to Integral Transforms and Convolutions. Am- sterdam, Netherlands: Kluwer, 1994. F-Polynomial KAUFFMAN POLYNOMIAL FFrac FRACTIONAL PART Fractal An object or quantity which displays SELF-SIMILARITY , in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whoseslope is said to be the FRACTAL DIMENSION . The prototypical example for a fractal is the length of a coastline measured with different length RULERS . The shorter the RULER , the longer the length measured, a PARADOX known as the COASTLINE PARADOX . Illustrated above are the fractals known as the GOSPER ISLAND ,K OCH SNOWFLAKE ,BOX FRACTAL , SIERPINSKI SIEVE ,BARNSLEY’S FERN , and M ANDEL- BROT SET . See also BACKTRACKING ,B ARNSLEY’S FERN,B OX FRACTAL ,B UTTERFLY FRACTAL ,C ACTUS FRACTAL , CANTOR SET,C ANTOR SQUARE FRACTAL ,C AROTID- KUNDALINI FRACTAL ,CESA` RO FRACTAL ,CHAOS GAME, CIRCLES-AND- SQUARES FRACTAL ,C OASTLINE PARA- DOX,D RAGON CURVE ,F AT FRACTAL ,F ATOU SET, FRACTAL DIMENSION ,G OSPER ISLAND ,H -FRACTAL , HE´ NON MAP,ITERATED FUNCTION SYSTEM ,JULIA FRACTAL ,K APLAN- YORKE MAP,K OCH ANTISNOW- FLAKE ,K OCH SNOWFLAKE ,L E´ VY FRACTAL ,L E´ VY TAPESTRY ,L INDENMAYER SYSTEM ,M ANDELBROT SET,MANDELBROT TREE,MENGER SPONGE ,MINKOWS- KI SAUSAGE ,M IRA FRACTAL ,NESTED SQUARE ,NEW- TON’S METHOD ,P ENTAFLAKE ,P YTHAGORAS TREE, RABINOVICH- FABRIKANT EQUATION ,S AN MARCO FRACTAL ,S IERPINSKI CARPET ,S IERPINSKI CURVE , SIERPINSKI SIEVE,STAR FRACTAL ,ZASLAVSKII MAP References Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993. Bogomolny, A. "Fractal Curves and Dimension." http:// www.cut-the-knot.com/do_you_know/dimension.html. Brandt, C.; Graf, S.; and Za¨hle, M. (Eds.). Fractal Geometry and Stochastics. Boston, MA: Birkha ¨user, 1995. Bunde, A. and Havlin, S. (Eds.). Fractals and Disordered Systems, 2nd ed. New York: Springer-Verlag, 1996. Bunde, A. and Havlin, S. (Eds.). Fractals in Science. New York: Springer-Verlag, 1994. Devaney, R. L. Complex Dynamical Systems: The Mathe- matics Behind the Mandelbrot and Julia Sets. Providence, RI: Amer. Math. Soc., 1994. Devaney, R. L. and Keen, L. Chaos and Fractals: The Mathematics Behind the Computer Graphics. Providence, RI: Amer. Math. Soc., 1989. Edgar, G. A. (Ed.). Classics on Fractals. Reading, MA: Addison-Wesley, 1993. Eppstein, D. "Fractals." http://www.ics.uci.edu/~eppstein/ junkyard/fractal.html. Falconer, K. J. The Geometry of Fractal Sets, 1st pbk. ed., with corr. Cambridge, England Cambridge University Press, 1986. Feder, J. Fractals. New York: Plenum Press, 1988. Giffin, N. "The Spanky Fractal Database." http://spanky.- triumf.ca/www/welcome1.html. Hastings, H. M. and Sugihara, G. Fractals: A User’s Guide for the Natural Sciences. New York: Oxford University Press, 1994. Kaye, B. H. A Random Walk Through Fractal Dimensions, 2nd ed. New York: Wiley, 1994. Lauwerier, H. A. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press, 1991. le Me´haute, A. Fractal Geometries: Theory and Applications. Boca Raton, FL: CRC Press, 1992. Mandelbrot, B. B. Fractals: Form, Chance, & Dimension. San Francisco, CA: W. H. Freeman, 1977. Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983. Massopust, P. R. Fractal Functions, Fractal Surfaces, and Wavelets. San Diego, CA: Academic Press, 1994. Pappas, T. "Fractals--Real or Imaginary." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 78 /C1/9, 1989. Peitgen, H.-O.; Ju¨rgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer- Verlag, 1992. Peitgen, H.-O.; Ju¨rgens, H.; and Saupe, D. Fractals for the Classroom, Part 1: Introduction to Fractals and Chaos. New York: Springer-Verlag, 1992. Peitgen, H.-O. and Richter, D. H. The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag, 1986. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, 1988. Pickover, C. A. (Ed.). The Pattern Book: Fractals, Art, and Nature. World Scientific, 1995. Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York: St. Martin’s Press, 1996. Rietman, E. Exploring the Geometry of Nature: Computer Modeling of Chaos, Fractals, Cellular Automata, and Neural Networks. New York: McGraw-Hill, 1989. Russ, J. C. Fractal Surfaces. New York: Plenum, 1994.Schroeder, M. Fractals, Chaos, Power Law: Minutes from an Infinite Paradise. New York: W. H. Freeman, 1991. Sprott, J. C. "Sprott’s Fractal Gallery." http://sprott.phy- sics.wisc.edu/fractals.htm. Stauffer, D. and Stanley, H. E. From Newton to Mandelbrot, 2nd ed. New York: Springer-Verlag, 1995. Stevens, R. T. Fractal Programming in C. New York: Henry Holt, 1989. Takayasu, H. Fractals in the Physical Sciences. Manchester, England: Manchester University Press, 1990. Tricot, C. Curves and Fractal Dimension. New York: Springer-Verlag, 1995. Triumf Mac Fractal Programs. http://spanky.triumf.ca/pub/ fractals/programs/MAC/. Vicsek, T. Fractal Growth Phenomena, 2nd ed. Singapore: World Scientific, 1992. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Weisstein, E. W. "Books about Fractals." http://www.trea- sure-troves.com/books/Fractals.html. Yamaguti, M.; Hata, M.; and Kigami, J. Mathematics of Fractals. Providence, RI: Amer. Math. Soc., 1997. Fractal Dimension The term "fractal dimension" is sometimes used to refer to what is more commonly called the CAPACITY DIMENSION (which is, roughly speaking, the exponent Din the expression n(e)/C30e/C28D;where n(e) is the minimum number of OPEN SETS of diameter eneeded to cover the set). However, it can more generally refer to any of the dimensions commonly used to charac- terize fractals (e.g., CAPACITY DIMENSION ,CORRELA- TION DIMENSION , INFORMATION DIMENSION , LYAPUNOV DIMENSION ,M INKOWSKI- BOULIGAND DI- MENSION ). See also BOX-COUNTING DIMENSION ,C APACITY DI- MENSION ,CORRELATION DIMENSION ,FRACTAL DIMEN- SION ,H AUSDORFF DIMENSION ,I NFORMATION DIMENSION ,LYAPUNOV DIMENSION ,MINKOWSKI- BOU- LIGAND DIMENSION ,POINTWISE DIMENSION , Q-DIMEN- SION References Rasband, S. N. "Fractal Dimension." Ch. 4 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 71 /C1/3, 1990. Fractal Land CAROTID- KUNDALINI FRACTAL Fractal Process A 1-D MAP whose increments are distributed accord- ing to a NORMAL DISTRIBUTION . Let y(t/C28Dt) and y(t/C27 Dt) be values, then their correlation is given by the BROWN FUNCTION r/C3022H/C281/C281: When H/C301=2;r/C300 and the fractal process corre- sponds to 1-D Brownian motion. If H>1=2;then r/C210 and the process is called a PERSISTENT PROCESS . If H B1=2 ; then r B0 and the process is called an ANTIPERSISTENT PROCESS . See also ANTIPERSISTENT PROCESS ,PERSISTENT PRO- CESS References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, 1993. Fractal Sequence Given an INFINITIVE SEQUENCE fxngwith associated array a(i;j);then fxngis said to be a fractal sequence 1. If i/C271/C30xn;then there exists mBnsuch that i/C30xm;/ 2. If hBi, then, for every j, there is exactly one k such that a(i;j)Ba(h;k)Ba(i;j/C271):/ (Asiand jrange through N, the array A/C30a(i;j); called the associative array of x, ranges through all of N.) An example of a fractal sequence is 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, .... Iffxngis a fractal sequence, then the associated array is an INTERSPERSION .I fxis a fractal sequence, then the UPPER-TRIMMED SUBSEQUENCE is given by l(x)/C30x; and the LOWER-TRIMMED SUBSEQUENCE V(x) is an- other fractal sequence. The SIGNATURE of an IRRA- TIONAL NUMBER is a fractal sequence. See also INFINITIVE SEQUENCE References Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157/C1/68, 1997. Fractal Valley CAROTID- KUNDALINI FUNCTION Fractile QUANTILE Fraction ARATIONAL NUMBER expressed in the form a=b(in- line notation) ora b(traditional "display" notation), where ais called the NUMERATOR andbis called the DENOMINATOR . When written in-line, the slash "/" between NUMERATOR and DENOMINATOR is called a SOLIDUS . APROPER FRACTION is a fraction such that a=bB1; and a LOWEST TERMS FRACTION is a fraction with common terms canceled out of the NUMERATOR and DENOMINATOR . The Egyptians expressed their fractions as sums (and differences) of UNIT FRACTIONS . Conway and Guy (1999) give a table of Roman NOTATION for fractions,in which multiples of 1/12 (the UNCIA ) were given separate names. See also ADJACENT FRACTION ,ANOMALOUS CANCEL- LATION ,C OMMON FRACTION ,C OMPLEX FRACTION , CONTINUED FRACTION ,D ENOMINATOR ,E GYPTIAN FRACTION ,FAREY SEQUENCE ,GOLDEN RULE,H ALF, LOWEST TERMS FRACTION ,M ATRIX FRACTION ,M ED- IANT ,M IXED FRACTION ,N UMERATOR ,P ANDIGITAL FRACTION ,PROPER FRACTION ,PYTHAGOREAN FRAC- TION ,QUARTER ,RATIONAL NUMBER ,SOLIDUS ,U NIT FRACTION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 22 /C1/3, 1996. Courant, R. and Robbins, H. "Decimal Fractions. Infinite Decimals." §2.2.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 61 /C1/3, 1996. Fractional Calculus The study of an extension of derivatives and integrals to noninteger orders. Fractional calculus is based onthe definition of the FRACTIONAL INTEGRAL as D/C28nf(t)/C301 G(n)gt 0(t/C28j)n/C281f(j)dj; where G(v) is the GAMMA FUNCTION . From this equation, FRACTIONAL DERIVATIVES can also be de- fined. See also DERIVATIVE ,FRACTIONAL DERIVATIVE ,FRAC- TIONAL DIFFERENTIAL EQUATION ,FRACTIONAL INTE- GRAL ,INTEGRAL ,MULTIPLE INTEGRAL References Butzer, P. L. and Westphal, U. "An Introduction to Frac- tional Calculus." Ch. 1 in Applications of Fractional Calculus in Physics (Ed. R. Hilfer). Singapore: World Scientific, pp. 1 /C1/5, 2000. McBride, A. C. Fractional Calculus. New York: Halsted Press, 1986. Nishimoto, K. Fractional Calculus. New Haven, CT: Uni- versity of New Haven Press, 1989. Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993. Spanier, J. and Oldham, K. B. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974. Fractional Derivative The fractional derivative of f(t) of order m>0 (if it exists) can be defined in terms of the FRACTIONAL INTEGRAL D/C28nf(t)a s Dmf(t)/C30DmD/C28(m/C28m)f(t)/C2/C6 ; (1) where mis an integer ]mde;where xdeis the CEILING FUNCTION . The SEMIDERIVATIVE corresponds to m/C301=2:/ The fractional derivative of the function tl is given by Dmtl /C30Dm D/C28(m/C28 m)tl/C2/C6 ¼ Dn Gðl þ 1 Þ Gðl þ m /C28 m þ 1Þ tlþm/C28 m"# ¼Gðl þ 1 Þðl /C28 m þ m Þðl /C28 m þ m /C28 1Þ/C1/C1/C1ðl /C28 m þ 1 Þ Gð1 þ m þ l /C28 mÞ t l/C28 m ¼Gðl þ 1Þð1 þ l /C28 m Þm Gð1 þ m þ l /C28 m Þtl/C28 m ¼Gðl þ 1Þ Gðl /C28 m þ 1 Þtl/C28 m (2) for l >/C281; m > 0 : The fractional derivative of the CONSTANT FUNCTION f(t) /C30c is then given by D mc /C30clim l 00G( l /C27 1) G( l /C28 m /C27 1) tl/C28 m /C30ct /C28m G(1 /C28 m) : (3) The fractional derivate of the ET-FUNCTION is given by DrEt( n ;a) /C30Et(n /C28 r ;a) (4) for n > 0; r "0:/ It is always true that, for m ; n > 0 ; D/C28 mD/C28 nf(t) /C30D/C28(m/C27 n) (5) but not always true that DmDn /C30Dm/C27 n (6) A FRACTIONAL INTEGRAL can also be similarly defined. The study of fractional derivatives and integrals is called FRACTIONAL CALCULUS . See also FRACTIONAL CALCULUS ,SEMIDERIVATIVE References Love, E. R. "Fractional Derivatives of Imaginary Order." J. London Math. Soc. 3, 241 /C1/59, 1971. Miller, K. S. "Derivatives of Noninteger Order." Math. Mag. 68, 183 /C1/92, 1995. Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993. Spanier, J. and Oldham, K. B. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974. Fractional Differential Equation The solution to the differential equation D2v /C27aDv /C27bD0/C2/C6 y(t) /C300 isy(t) /C30ea(t) /C28e b(t) for a " b te at ;Pq /C281 k /C30/C28(q /C281) ak q /C28 kjj ðÞ D1 /C28(k/C271)v te aqtðÞ for a /C30 b "0 t2v /C281 G(2v) for a /C30 b /C300;8 >>>>>>>>< >>>>>>>>: where q /C30 1 v eb(t) /C30Xq /C281 k /C300bq /C28k /C281Et /C28kv ; bqðÞ ; /Et(a ;x) is the ET-FUNCTION , and G(n) is the GAMMA FUNCTION . See also FRACTIONAL CALCULUS References Miller, K. S. "Derivatives of Noninteger Order." Math. Mag. 68, 183 /C1/92, 1995. Fractional Fourier Transform The fractional Fourier transform is generally under- stood to correspond to a rotation in time-frequency phase space, where the usual FOURIER TRANSFORM corresponds to a rotation of 908 (/p=2 radians). A fractional Fourier transform can be used to detect frequencies which are not INTEGER multiples of the lowest DISCRETE FOURIER TRANSFORM frequency. See also DISCRETE FOURIER TRANSFORM ,FOURIER TRANSFORM References Namias, V. "The Fractional Fourier Transform and Its Application to Quantum Mechanics." J. Inst. Math. Appl. 25, 241/C1/65, 1980. Ozaktas, H. M. "Fractional Fourier Transform and Its Applications in Optics and Signal Processing--A Biblio- graphy." http://www.ee.bilkent.edu.tr/~haldun/ffbiblio.ps. Ozaktas, H. M. "Publications Related to Fractional Fourier Transforms." http://www.ee.bilkent.edu.tr/~haldun/frac-fourpub.ps. Fractional Integral Denote the nthDERIVATIVE Dnand the n-fold INTE- GRAL D/C28n:Then D/C281f(t)/C30gt 0f(j)dj: (1) Now, if the equation D/C28nf(t)/C301 (n/C281)!gt 0(t/C28j)n/C281f(j)dj (2) for the MULTIPLE INTEGRAL is true for n, then D/C28(n/C271)f(t) /C30D-1 1 (n /C28 1)! gt 0(t /C28 j)n/C281f(j) dj"# /C30gt 01 (n /C28 1)! gx 0(x /C28 j)n/C281f( j)dj"# dx : (3) Interchanging the order of integration gives D/C28(n/C271)f(t) /C301 n! gt 0(t /C28 j)nf( j) dj: (4) But (2) is true for n /C301, so it is also true for all n by INDUCTION . The fractional integral of f(t) of order n > 0 can then be defined by D /C28nf(t) /C301 G(v) gt 0(t /C28 j)v /C281f(j)dj; (5) where G( n) is the GAMMA FUNCTION . The fractional integral of order 1/2 is called a SEMI- INTEGRAL . The fractional integral can only be given in terms of elementary functions for a small number of functions. For example, D/C28 ntl /C30G( l /C27 1) G( l /C27 n /C27 1)tl/C27 n for l >/C281; n > 0 (6) D /C28neat /C301 G( n)eat gt 0xn/C281e /C28axdx /C30a/C28 neat g(n ;at) G( n)/C30/C13Et( n ;a); (7) where g(a ;x) is a lower incomplete GAMMA FUNCTION and Et( n ;a) is the ET-FUNCTION . From (6), the frac- tional integral of the CONSTANT FUNCTION f(t) /C30c is given by D/C28 nc /C30clim l00G(l/C271) G(l/C27n/C271)tl/C27n/C30tm G(n/C271): (8) AFRACTIONAL DERIVATIVE can also be similarly defined. The study of fractional derivatives and integrals is called FRACTIONAL CALCULUS . See also FRACTIONAL CALCULUS ,SEMI-INTEGRAL References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993.Spanier, J. and Oldham, K. B. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974. Fractional Part The function frac xgiving the fractional (noninteger) part of a REAL NUMBER x. The symbol xfgis some- times used instead of frac x(Graham et al. 1994, p. 70), but this notation is not used in this work due topossible confusion with the SETcontaining the ele- ment x. Unfortunately, there is no universal agreement onthe meaning of frac xforxB0 and there are two common definitions. Let xbcbe the FLOOR FUNCTION , then the Mathematica command Fractional- Part [x] is defined as fracx/C13x/C28xbc x/C28xbc/C281x]0 xB0/C27 (1) (left figure). This definition has the benefit thatfracx/C27intx/C30x;where int xis the INTEGER PART of x. Although Spanier and Oldham (1987) use the same definition as Mathematica , they mention the formula only very briefly and then say it will not be usedfurther. Graham et al. (1994, p. 70), and perhaps most other mathematicians, use the different defini-tion fracx/C30x/C28xbc; (2) (right figure). Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used (D. W. Cantrell). Here, S&O indicates Spanier andOldham (1987). notation name S&O Graham et al.Mathematica /xbc/ integer- value/Int(x)/floor or in-teger partFloor [x] /sgn(x) xjjbc / integer- part/Ip(x)/ no name Integer- Part [ x] /x /C28 xbc/ fractional-value/frac( x)/ fractionalpart or xfg /no name /sgn(x) xjj/C28 xjjbc ðÞ / fractional-part/FP(x)/ no name Fractional Part [ x] The (possibly scaled) periodic waveform correspond- ing to the latter definition is known as the SAWTOOTH WAVE . The fractional part of 1=x has the interesting analytic integrals g1 1 =2frac1 x ! dx /C30g1 1=21 x /C281 ! dx /C30ln 2 /C281 2 (3) g1=2 1 =3frac1 x ! dx /C30g1=2 1 =31 x /C282 ! dx /C30ln 3 /C28ln 2 /C2813(4) g1=3 1 =4frac1 x ! dx /C30g1=3 1 =41 x /C283 ! dx /C30ln 4 /C28ln 3 /C281 4 : (5) The integral I /C30g1 0frac1 x ! dx (6) is therefore a TELESCOPING SUM given by I ¼g1 0frac1 x ! dx ¼ lim n0/C12ln n /C28Xn k¼21 k"# /C301 /C28 g /C27lim n0/C12ln n /C28 C0(1 /C27n) ðÞ ; (7) where g is the EULER- MASCHERONI CONSTANT and Ck(x) is the POLYGAMMA FUNCTION . The quantity on the right is 0, so I /C301 /C28 g : (8) A consequence of WEYL’S CRITERION is that the sequence ffrac( nx)g is dense and EQUIDISTRIBUTED in the interval [0;1] for irrational x, where n /C301, 2, ... (finch). Hardy and Littlewood (1914) proved that the se- quence frac xnðÞ fg is EQUIDISTRIBUTED for almost all real numbers x /C211 (i.e., the exceptional set has LEBESGUE MEASURE ZERO ). Exceptional numbers in- clude the positive integers, 1 /C27ffiffiffi 2p (Finch), and the GOLDEN RATIO f : The plots above illustrate the distribution of frac xnðÞ for x /C30e, f; and 1 /C27ffiffiffi 2p : Candidate members of the measure one set are easy to find, but difficult to proven. However, Levin has explicitly constructed such an example (Drmota and Tichy 1997). The properties of frac (3=2)nðÞ fg ; the simplest such sequence for a rational number x /C211 have been extensively studied (Finch). For example, frac (3=2)nðÞ fg has infinitely many ACCUMULATION POINTS in both [0;1=2] and [1=2;1] (Pisot 1938, Vijayaraghavan 1941). Furthermore, Flatto et al. (1995) proved that any subinterval of [0;1] containing all but at most finitely many ACCUMULATION POINTS of frac (3=2)nðÞ must have length at least 1/3. Surpris- ingly, the sequence frac (3=2)nðÞ fg is also connected with the COLLATZ PROBLEM and with WARING’S PROBLEM . In particular, WARING’S PROBLEM can be solved completely if the inequality frac3 2 !n"# 51/C2834 ! n (9) holds. No counterexample to this inequality is known, and it is even believed that can be extended to 3 4 !n Bfrac32 ! n"# B1/C2834 ! n (10) forn/C217 (Finch; Bennett 1993, 1994). Furthermore, the constant 3/4 can be decreased to 0.5769 (Beukers 1981 and Dubitskas 1990). Unfortunately, theseinequalities have not been proved. See also B EATTY SEQUENCE ,C EILING FUNCTION , EQUIDISTRIBUTED SEQUENCE ,FLOOR FUNCTION ,IN- TEGER PART,N EAREST INTEGER FUNCTION ,ROUND , SAWTOOTH WAVE,S HIFT TRANSFORMATION ,T RUN- CATE ,W HOLE NUMBER References Bennett, M. A. "Fractional Parts of Powers of Rational Numbers." Math. Proc. Cambridge Philos. Soc. 114, 191/C1/01, 1993. Bennett, M. A. "An Ideal Waring Problem with Restricted Summands." Acta Arith. 66, 125/C1/32, 1994. Beukers, F. "Fractional Parts of Powers of Rational Num- bers." Math. Proc. Cambridge Philos. Soc. 90,1 3/C1/0, 1981. Drmota, M. and Tichy, R. F. Sequences, Discrepancies and Applications. New York: Springer-Verlag, 1997. Dubitskas, A. K. "A Lower Bound for the Quantity (3=2)nfg :/" Russian Math. Survey 45, 163 /C1/64, 1990. Finch, S. "Powers of 3/2 Modulo One." http://www.mathsoft.- com/asolve/pwrs32/pwrs32.html. Flatto, L.; Lagarias, J. C.; Pollington, A. D. "On the Range of Fractional Parts j(p =q)nfg :/" Acta Arith. 70, 125 /C1/47, 1995. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Miklavc, A. "Elementary Proofs of Two Theorems on the Distribution of Numbers fnx g (mod 1)." Proc. Amer. Math. Soc. 39, 279 /C1/80, 1973. Spanier, J. and Oldham, K. B. "The Integer-Value Int(x) and Fractional-Value frac(x) Functions." Ch. 9 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 71 /C1/8, 1987. Vijayaraghavan, T. "On the Fractional Parts of the Powers of a Number (I)." J. London Math. Soc. 15, 159 /C1/60, 1940. Vijayaraghavan, T. "On the Fractional Parts of the Powers of a Number (II)." Proc. Cambridge Phil. Soc. 37, 349 /C1/57, 1941. Vijayaraghavan, T. "On the Fractional Parts of the Powers of a Number (III)." J. London Math. Soc. 17, 137 /C1/38, 1942. Fractran Fractran is an algorithm applied to a given list f1 ; f2 ; ..., fkof FRACTIONS . Given a starting INTEGER N, the Fractran algorithm proceeds by repeatedly multi- plying the integer at a given stage by the first element ftgiven an integer PRODUCT . The algorithm terminates when there is no such ft :/ The list 17 91 ;7885 ;1951 ;2338 ;2933 ;7729 ;9523 ;7719 ;1 17 ;1113 ;1311 ;15 2;17 ;55 1 with starting integer N /C302 generates a sequence 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ... (Sloane’s A007542). Conway (1987) showed that the only other powers of 2 which occur are those with PRIME exponent: 22,23,25,27, .... References Conway, J. H. "Unpredictable Iterations." In Proceedings of the 1972 Number Theory Conference Held at the Univer- sity of Colorado, Boulder, Colo., Aug. 14 /C1/8, 1972. Boulder, CO: University of Colorado, pp. 49 /C1/2, 1972. Conway, J. H. "Fractran: A Simple Universal Programming Language for Arithmetic." Ch. 2 in Open Problems in Communication and Computation (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, pp. 4 /C1/6, 1987. Sloane, N. J. A. Sequences A007542/M2084 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Frame A closed curve associated with a knot which is displaced along the normal by a small amount. For K is parameterized by xm(s) for 0 5s 5L along the length of the knot by parameter s, the frame Kf associated with K is ym /C30xm(s) /C27 enm(s);where e is a small parameter, n m(s) is a unit VECTOR FIELD normal to the curve at s. See also FRAMEWORK References Kaul, R. K. Topological Quantum Field Theories--A Meeting Ground for Physicists and Mathematicians. 15 Jul 1999. http://xxx.lanl.gov/abs/hep-th/9907119/. Framework Consider a finite collection of points p /C30(p1 ;:::; pn); pi /C23Rd EUCLIDEAN SPACE (known as a CONFIGURA- TION ) and a graph G whose VERTICES correspond to pairs of points that are constrained to stay the same distance apart. Then the graph G together with the configuration p, denoted G(p) ; is called a framework. See also BAR (EDGE), CONFIGURATION ,RIGID GRAPH , TENSEGRITY References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 56, 1967. Franel Number One of the numbers an k/C300n k/C0/C13; wheren k/C0/C1 is a BINOMIAL COEFFICIENT . The first few values for n /C300, 1, ... are 1, 2, 10, 56, 346, ... (Sloane’s A000172). See also BINOMIAL SUMS References Franel, J. "On a Question of Laisant." L’interme ´diaire des mathe ´maticiens 1,45/C1/7, 1894. Franel, J. "On a Question of J. Franel." L’interme ´diaire des mathe ´maticiens 2,33/C1/5, 1895. Sloane, N. J. A. Sequences A000172/M1971 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Franklin Graph The 12-vertex graph illustrated above which provides the minimal coloring of the K LEIN BOTTLE using six colors, providing the sole counterexample to the HEAWOOD CONJECTURE . See also HEAWOOD CONJECTURE ,KLEIN BOTTLE References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 244, 1976. Franklin, P. "A Six Color Problem." J. Math. Phys. 13, 363 /C1/ 79, 1934. Franklin Magic Square Benjamin Franklin constructed the above 8 /C298 PAN- MAGIC SQUARE having MAGIC CONSTANT 260. Any half- row or half-column in this square totals 130, and the four corners plus the middle total 260. In addition, bent diagonals (such as 52 /C1/-5 /C1/4 /C1/0 /C1/7 /C1/3 /C1/6) also total 260 (Madachy 1979, p. 87). See also MAGIC SQUARE ,PANMAGIC SQUARE References Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in Madachy’s Mathematical Recreations. New York: Dover, pp. 103 /C1/13, 1979. Pappas, T. "The Magic Square of Benjamin Franklin." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 97, 1989. Franse ´n-Robinson Constant F /C13g/C12 0dx G(x) /C302 :8077702420 :::; where G(x) is the GAMMA FUNCTION . The above plots show the functions G(x) and 1=G(x): No closed-form expression in terms of other constants in known for F. See also GAMMA FUNCTIONReferences Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/fran/fran.html. Franse ´n, A. "Accurate Determination of the Inverse Gamma Integral." BIT 19, 137 /C1/38, 1979. Franse ´n, A. "Addendum and Corrigendum to ‘High-Preci- sion Values of the Gamma Function and of Some Related Coefficients."’ Math. Comput. 37, 233 /C1/35, 1981. Franse ´n, A. and Wrigge, S. "High-Precision Values of the Gamma Function and of Some Related Coefficients." Math. Comput. 34, 553 /C1/66, 1980. Plouffe, S. "Fransen-Robinson Constant." http://www.laci- m.uqam.ca/piDATA/fransen.txt. F-Ratio The RATIO of two independent estimates of the VARIANCE of a NORMAL DISTRIBUTION . See also F-DISTRIBUTION ,N ORMAL DISTRIBUTION , VARIANCE F-Ratio Distribution F-DISTRIBUTION Frattini Extension If F is a group, then the extensions G of F of order o with G=f(G) $F ; where f(G) is the FRATTINI SUB- GROUP , are called Frattini extensions. See also FRATTINI FACTOR ,FRATTINI SUBGROUP References Besche, H.-U. and Eick, B. "Construction of Finite Groups." J. Symb. Comput. 27, 387 /C1/04, 1999. Gaschu ¨tz, W. "U¨ ber F-Untergruppen endlicher Gruppen." Math. Z. 58, 160 /C1/70, 1953. Frattini Factor A group given by G =f(G); where f(G) is the FRATTINI SUBGROUP of a given group G. See also FRATTINI EXTENSION ,FRATTINI SUBGROUP References Besche, H.-U. and Eick, B. "Construction of Finite Groups." J. Symb. Comput. 27, 387/C1/04, 1999. Gaschu ¨tz, W. "U ¨berF-Untergruppen endlicher Gruppen." Math. Z. 58, 160/C1/70, 1953. Frattini Subgroup The intersection f(G) of all maximal subgroups of a given group G. See also FRATTINI EXTENSION ,FRATTINI FACTOR References Besche, H.-U. and Eick, B. "Construction of Finite Groups." J. Symb. Comput. 27, 387/C1/04, 1999. Gaschu ¨tz, W. "U ¨berF-Untergruppen endlicher Gruppen." Math. Z. 58, 160/C1/70, 1953. Fre´chet Bounds Any bivariate distribution function with marginal distribution functions F and G satisfies max fF(x) /C27G(y) /C281;0 g5H(x; y) 5min fF(x) ;G(y)g: Fre´chet Derivative A function f is Fre´chet differentiable at a if lim x0af(x) /C28 f(a) x /C28 a exists. This is equivalent to the statement that f has a removable DISCONTINUITY at a, where f(x) /C13f(x) /C28 f(a) x /C28 a: Every function which is Fre´chet differentiable is also Carathe ´odory differentiable. See also CARATHE ´ ODORY DERIVATIVE ,DERIVATIVE Fre´chet Filter COFINITE FILTER Fre´chet Space A complete metrizable space, sometimes also with the restriction that the space be locally convex. A Fre´chet space is a TOPOLOGICAL VECTOR SPACE which is COMPLETE . Its topology is also defined by a COUNTA- BLE family of SEMINORMS . For example, the space of SMOOTH FUNCTIONS on [0;1] is a Fre´chet space. Its topology is the C-INFINITY TOPOLOGY , which is given by the countable family of SEMINORMS , fkka/C30sup Dafjj : Because fn 0 f in this topology implies that f is smooth, i.e., D afn 0 Daf ; any CAUCHY SEQUENCE has a limit in the space of SMOOTH FUNCTIONS , i.e., it is COMPLETE . See also BANACH SPACE ,HILBERT SPACE ,TOPOLOGI- CAL VECTOR SPACE Fredholm Alternative See also SPECTRAL THEORYFredholm Integral Equation of the First Kind An INTEGRAL EQUATION OF THE FORM f(x) /C30g/C12 /C28/C12K(x;t) f(t)dt f(x) /C301 2 pg/C12 /C28/C12F(v) K(v)e /C28ivxd v: See also FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND,INTEGRAL EQUATION ,VOLTERRA INTE- GRAL EQUATION OF THE FIRST KIND,V OLTERRA INTEGRAL EQUATION OF THE SECOND KIND References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 865, 1985. Fredholm Integral Equation of the Second Kind An INTEGRAL EQUATION OF THE FORM f(x) /C30f(x) /C27 lg/C12 /C28/C12K(x; t)f(t)dt f(x) /C301ffiffiffiffiffiffi 2ppg/C12 /C28/C12F(t)e/C28ixtdt 1/C28ffiffiffiffiffiffi2pp lK(t): See also F REDHOLM INTEGRAL EQUATION OF THE FIRST KIND,INTEGRAL EQUATION ,NEUMANN SERIES (INTEGRAL EQUATION ), VOLTERRA INTEGRAL EQUA- TION OF THE FIRST KIND,V OLTERRA INTEGRAL EQUATION OF THE SECOND KIND References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 865, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Fredholm Equations of the Second Kind." §18.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 782 /C1/85, 1992. Fredholm’s Theorem This entry contributed by V IKTOR BENGTSSON Fredholm’s theorem states that, if Ais an m/C29n matrix, then the ORTHOGONAL COMPLEMENT of the ROW SPACE ofAis the NULLSPACE ofA;and the ORTHOGONAL COMPLEMENT of the COLUMN SPACE ofA is the NULLSPACE ofA/C222; (Row A)/C222/C30Null A (Col A)/C222/C30Null A/C222: See also COLUMN SPACE ,NULLSPACE ,ORTHOGONAL DECOMPOSITION ,ROW SPACE Free When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, MIRROR IMAGES are equivalent for free objects. The word "free" is also used in technical senses to refer to a FREE GROUP , FREE SEMIGROUP , FREE TREE , FREE VARIABLE , etc. In ALGEBRAIC TOPOLOGY , a free abstract mathemati- cal object is generated by n elements in a "free manner" ("FREELY "), i.e., such that the n elements satisfy no nontrivial relations among themselves. To make this more formal, an algebraic GADGET X is freely generated by a SUBSET G if, for any function f : G 0 Y where Y is any other algebraic GADGET , there exists a unique HOMOMORPHISM (which has different meanings depending on what kind of GAD- GETS you’re dealing with) g : X 0 Y such that g restricted to G is f. If the algebraic GADGETS are VECTOR SPACES , then G freely generates X IFF G is a BASIS for X. If the algebraic GADGETS are ABELIAN GROUPS , then G freely generates X IFF X is a DIRECT SUM of the INTEGERS , with G consisting of the standard BASIS . See also FIXED ,F REE GROUP ,F REE VARIABLE , FREELY ,GADGET ,MIRROR IMAGE ,RANK Free Abelian Group A free Abelian group is a group G with a subset which generates the group G with the only relation being ab /C30ba. That is, it has no TORSION . All such groups are a DIRECT PRODUCT of the INTEGERS Z ; and have rank given by the number of copies of Z : For example, Z /C29Z /C30 (n;m) fg is a free Abelian group of rank 2. A minimal subset b1 ; ..., bn that generates a free Abelian group is called a basis, and gives G as G /C30Zb1 /C27/C1/C1/C1/C27Zbn : A free Abelian group is an ABELIAN GROUP , but is not a FREE GROUP (except when it has rank one, i.e., Z): Free Abelian groups are the FREE MODULES in the case when the RING is the ring of integers Z:/ See also ABELIAN GROUP ,FREE GROUP ,FREE MOD- ULE,GROUP ,TORSION (GROUP ) Free Action A group action G /C29X 0 X is called free when there are no FIXED POINTS . That is, for any point x there is at least one transformation which does not fix x. The group is said to act freely.The basic example of a free group action is the action of a group on itself by left multiplication L : G /C29G 0 G : As long as the group has more than the IDENTITY ELEMENT , there is no element h which satisfies gh /C30h for all g. An example of a free action which is not TRANSITIVE is the action of S1 on S3 ƒC2 by eiu /C215 Z1 ;Z2 ðÞ /C30 eiuZ1 ; ei uZ2 ðÞ ; which defines the HOPF FIBRATION . See also EFFECTIVE ACTION ,FREE ACTION ,GROUP , ISOTROPY GROUP ,M ATRIX GROUP ,O RBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION , TOPOLOGICAL GROUP ,TRANSITIVE GROUP ACTION Free Group The generators of a group G are defined to be the smallest subset of group elements such that all other elements of G can be obtained from them and their inverses. A GROUP is a free group if no relation exists between its generators (other than the relationship between an element and its inverse required as one of the defining properties of a group). For example, the additive group of whole numbers is free with a single generator, 1. See also FREE ABELIAN GROUP ,FREE SEMIGROUP Free Semigroup A SEMIGROUP with a noncommutative product in which no PRODUCT can ever be expressed more simply in terms of other ELEMENTS . See also FREE GROUP ,SEMIGROUP Free Tree A TREE which is not ROOTED , i.e., a normal TREE with no node singled out for special treatment (Skiena 1990, p. 107). See also ROOTED TREE,TREE References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Free Variable An occurrence of a variable in a LOGIC FORMULA which is not inside the scope of a QUANTIFIER . See also BOUND ,QUANTIFIER ,SENTENCE References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 112, 1977. Freely A group acts freely if there are no FIXED POINTS .A point which is fixed by every group element would not be free to move. See also EFFECTIVE ACTION ,FIXED POINT (GROUP ), FREE ACTION ,G ROUP ,G ROUP ACTION ,ISOTROPY GROUP ,M ATRIX GROUP ,O RBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION ,TOPOLOGICAL GROUP ,TRANSITIVE Freemish Crate An IMPOSSIBLE FIGURE box which can be drawn but not built. References Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 120 /C1/22, 1996. Jablan, S. "Are Impossible Figures Possible?" http://mem- bers.tripod.com/~modularity/kulpa.htm. Pappas, T. "The Impossible Tribar." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 13, 1989. Freeth’s Nephroid A STROPHOID of a CIRCLE with the POLE O at the CENTER of the CIRCLE and the fixed point P on the CIRCUMFERENCE of the CIRCLE . In a paper published by the London Mathematical Society in 1879, T. J. Freeth described it and various other STRO- PHOIDS (MacTutor Archive). If the line through P PARALLEL to the Y-AXIS cuts the NEPHROID at A, then ANGLE AOP is 3 p=7; so this curve can be used toconstruct a regular HEPTAGON . The POLAR equation is r /C30a 1 /C272 sin1 2 u/C17/C15hi : See also STROPHOID References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 175 and 177 /C1/78, 1972. MacTutor History of Mathematics Archive. "Freeth’s Ne- phroid." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Freeths.html. Fre´gier’s Theorem Pick any point P on a CONIC SECTION , and draw a series of RIGHT ANGLES having this point as their vertices. Then the line segments connecting the rays of the RIGHT ANGLES where they intersect the conic section concur in a point p ?; as illustrated above. See also CONIC SECTION ,RIGHT ANGLE References Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, p. 83, 1991. Freiman’s Constant The end of the last gap in the LAGRANGE SPECTRUM , given by F /C132221564096 /C27 283748ffiffiffiffiffiffiffiffi 462p 491993569/C304 :5278295661... : REAL NUMBERS greater than Fare members of the MARKOV SPECTRUM . See also LAGRANGE SPECTRUM ,MARKOV SPECTRUM References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 188 /C1/89, 1996. French Curve French curves are plastic (or wooden) templates having an edge composed of several different curves. French curves are used in drafting (or were before computer-aided design) to draw smooth curves of almost any desired curvature in mechanical draw- ings. Several typical French curves are illustrated above. While an undergraduate at MIT, Feynman (1997, p. 23) used a French curve to illustrate the fallacy of learning without understanding. When he pointed out to his colleagues in a mechanical drawing class the "amazing" fact that the TANGENT at each point on the curve was horizontal, none of his classmates realized that this was trivially true, since the DERI- VATIVE (tangent) at an extremum (lowest or highest point) of any curve is zero (horizontal), as they had already learned in CALCULUS class. See also CORNU SPIRAL References Feynman, R. P. and Leighton, R. "Who Stole the Door?" In ‘Surely You’re Joking, Mr. Feynman!’: Adventures of a Curious Character. New York: W. W. Norton, 1997. French Metro Metric The French metro metric is an example for disproving apparently intuitive but false properties of METRIC SPACES . The metric consists of a distance function on the plane such that for all a; b /C23R2 ; d(a;b) /C30a /C28b jj if a /C30cb for some c /C23R ajj/C27bjj otherwise ;/C27 where ajjis the normal distance function on the plane. This metric has the property that for r B ajj; the OPEN BALL of radius r around a is an open line segment along vector a, while for r > ajj; the OPEN BALL is the union of a line segment and an OPEN DISK around the origin. Frenet Formulas Also known as the Serret-Frenet formulas, these vector differential equations relate inherent proper- ties of a parametrized curve. In matrix form, they can be written˙T ˙N ˙B2 435/C300 k 0 /C28k 0 t 0 /C28t 02435T N B2435; where T is the unit TANGENT VECTOR , N is the unit NORMAL VECTOR , B is the unit BINORMAL VECTOR , t is the TORSION , k is the CURVATURE , and ˙x denotes dx=ds :/ See also CENTRODE ,F UNDAMENTAL THEOREM OF SPACE CURVES ,NATURAL EQUATION References Frenet, F. "Sur les courbes a` double courbure." The`se. Toulouse, 1847. Abstract in J. de Math. 17, 1852. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 186, 1997. Kreyszig, E. "Formulae of Frenet." §15 in Differential Geometry. New York: Dover, pp. 40 /C1/3, 1991. Serret, J. A. "Sur quelques formules relatives a` la the´orie des courbes a` double courbure." J. de Math. 16, 1851. Frequency Curve A smooth curve which corresponds to the limiting case of a HISTOGRAM computed for a frequency distribution of a continuous distribution as the number of data points becomes very large. See also FREQUENCY DISTRIBUTION ,F REQUENCY POLYGON ,GAUSSIAN FUNCTION References Kenney, J. F. and Keeping, E. S. "Frequency Curves." §2.5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 26 /C1/8, 1962. Frequency Distribution The tabulation of raw data obtained by dividing it into CLASSES of some size and computing the number of data elements (or their fraction out of the total)falling within each pair of CLASS BOUNDARIES . The following table shows the frequency distribution ofthe data set illustrated by the histogram below. class intervalclass markabsolute frequencyrelative frequencycumulative absolute frequencyrelative cumulative frequency 0.00 /C1/9.99 5 1 0.01 1 0.01 10.00 /C1/9.99 15 3 0.03 4 0.04 20.00 /C1/9.99 25 8 0.08 12 0.12 30.00 /C1/9.99 35 18 0.18 30 0.30 40.00 /C1/9.99 45 24 0.24 54 0.54 50.00 /C1/9.99 55 22 0.22 76 0.76 60.00 /C1/9.99 65 15 0.15 91 0.91 70.00 /C1/9.99 75 8 0.08 99 0.99 80.00 /C1/9.99 85 0 0.00 99 0.99 90.00 /C1/9.99 95 1 0.01 100 1.00 See also ABSOLUTE FREQUENCY ,CLASS ,CUMULATIVE FREQUENCY ,CLASS BOUNDARIES ,HISTOGRAM ,RELA- TIVE FREQUENCY ,RELATIVE CUMULATIVE FREQUENCY References Kenney, J. F. and Keeping, E. S. "Frequency Distributions." §1.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 12 /C1/9, 1962. Frequency Polygon A distribution of values of a discrete variate repre- sented graphically by plotting points (x1 ;f1) ; (x2 ;f2) ; ..., (xk ; fk); and drawing a set of straight line segmentsconnecting adjacent points. It is usually preferable to use a HISTOGRAM for grouped distributions. See also FREQUENCY CURVE ,FREQUENCY DISTRIBU- TION ,HISTOGRAM ,OGIVE References Kenney, J. F. and Keeping, E. S. "Frequency Polygons" and "Cumulative Frequency Polygons." §2.3 and 2.6 in Mathe- matics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 24 /C1/5 and 28 /C1/9, 1962. Fresnel Integrals In physics, the Fresnel integrals are most often defined by C(u)/C27iS(u)/C13gu 0eipx2=2dx /C30gu 0cos1 2px2/C17/C15 dx/C27igu 0sin12px2/C17/C15 dx; (1) so C(u)/C13gu 0cos1 2px2/C17/C15 dx (2) S(u)/C13gu 0sin1 2px2/C17/C15 dx: (3) The Fresnel integrals are implemented in Mathema- tica asFresnelC [z] andFresnelC [z] They satisfy C(9/C12) /C30/C281 2 (4) S(9/C12) /C301 2 : (5) Related functions are defined as C1(z) /C13ffiffiffi 2 ps gx 0cos t2dt (6) S1(z) /C13ffiffiffi 2ps gx 0sin t2dt (7) C2(z) /C131ffiffiffiffiffiffi 2ppgcos tffiffi tp dt (8) S2(z) /C131ffiffiffiffiffiffi 2ppgsin tffiffi tp dt: (9) An asymptotic expansion for x /C271 gives C(u) :1 2 /C271 pusin1 2 pu2/C17/C15 (10) S(u) :1 2 /C281 pucos1 2 pu2/C17/C15 : (11) Therefore, as u 0/C12; C(u) /C301=2 and S(u) /C301=2 : The Fresnel integrals are sometimes alternatively defined as x(t) /C30gt 0cos v2/C0/C1 dv (12) y(t) /C30gt 0sin v2/C0/C1 dv: (13) Letting x /C30v2so dx /C302vdv/C302ffiffiffixpdv ; and dv /C30 x/C281 =2dx=2 x(t) /C301 2gffiffi tp 0x/C281=2 cos xdx (14) y(t) /C3012gffiffi tp 0x/C281=2 sin xdx : (15) In this form, they have a particularly simple expan- sion in terms of SPHERICAL BESSEL FUNCTIONS OF THE FIRST KIND . Using j0(x) /C30sin x x (16) n1(x) /C30/C28j /C281(x) /C30/C28cos x x; (17) where n1(x)isa SPHERICAL BESSEL FUNCTION OF THESECOND KIND xt2/C0/C1 /C30/C2812gt 0n1(x)x1=2dx /C301 2gt 0j/C281(x)x1=2dx/C30x1=2X/C12 n/C300j2n(x) (18) yt2/C0/C1 /C3012gt 0j0(x)x1=2dx /C30x1=2X/C12 n/C300j2n/C271(x): (19) See also CORNU SPIRAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Fresnel Inte- grals." §7.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 300 /C1/02, 1972. Leonard, I. E. "More on Fresnel Integrals." Amer. Math. Monthly 95, 431/C1/33, 1988. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Fresnel Integrals, Cosine and Sine Integrals." §6.79 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 248 /C1/52, 1992. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Fresnel Integrals S(x;n) and C(x;n):/" §1.3 in Integrals and Series, Vol. 3: More Special Func- tions. Newark, NJ: Gordon and Breach, p. 24, 1990. Spanier, J. and Oldham, K. B. "The Fresnel Integrals S(x) and C(x):/" Ch. 39 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 373 /C1/83, 1987. Fresnel’s Elasticity Surface AQUARTIC SURFACE given by r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2x2/C27b2y2/C27c2z2p ; where r2/C13x?2/C27y?2/C27z?2; also known as Fresnel’s wave surface. It was intro- duced by Fresnel in his studies of crystal optics. Theimage above shows one particular case of the Fresnel surface (JavaView). See also QUARTIC SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 16, 1986. Fischer, G. (Ed.). Plates 38 /C1/9in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 38 /C1/9, 1986. JavaView. "Classic Surfaces from Differential Geometry: Fresnel (Single Eigenvalue)." http://www-sfb288.math.tu- berlin.de/vgp/javaview/demo/surface/common/PaSurface_- Fresnel.html. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 304, 1993. Fresnel’s Wave Surface FRESNEL’S ELASTICITY SURFACE FresnelC FRESNEL INTEGRALS FresnelS FRESNEL INTEGRALS Frey Curve Let ap /C27bp /C30cpbe a solution to FERMAT’S LAST THEOREM . Then the corresponding Frey curve is y2 /C30xx/C28apðÞ x /C27bpðÞ : (1) Frey showed that such curves cannot be MODULAR ,so if the TANIYAMA- SHIMURA CONJECTURE were true, Frey curves couldn’t exist and FERMAT’S LAST THEO- REM would follow with b EVEN and a /C13/C281 (mod4) : Frey curves are SEMISTABLE . Invariants include the DISCRIMINANT ap /C280 ðÞ2/C28bp /C280 ðÞ ap /C28(/C28b)p½/C1382/C30a2pb2pc2p : (2) The MINIMAL DISCRIMINANT is D/C302 /C288a2pb2pc2p ; (3) the CONDUCTOR is N /C30Y l½abcl; (4) and the J-INVARIANT is j /C3028 a2p /C27 b2p /C27 apbpðÞ3 a2pb2pc2p /C3028 c2p /C28 bpcpðÞ3 (abc)2p : (5) See also ELLIPTIC CURVE ,FERMAT’S LAST THEOREM , TANIYAMA- SHIMURA CONJECTUREReferences Cox, D. A. "Introduction to Fermat’s Last Theorem." Amer. Math. Monthly 101,3/C1/4, 1994. Gouve ˆa, F. Q. "A Marvelous Proof." Amer. Math. Monthly 101, 203 /C1/22, 1994. Frey Elliptic Curve FREY CURVE Friend A friend of a number n is another number m such that (m, n)isa FRIENDLY PAIR. See also FRIENDLY PAIR,SOLITARY NUMBER References Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly Integers." Amer. Math. Monthly 84,6 5/C1/6, 1977. Friendly Giant Group MONSTER GROUP Friendly Number AMICABLE PAIR,FRIENDLY NUMBER Friendly Pair Define X (n)/C13s(n) n; where s(n) is the DIVISOR FUNCTION . Then a PAIR of distinct numbers ( k, m ) is a friendly pair (and kis said to be a FRIEND ofm)i f X (k)/C30X (m): For example, (4320, 4680) are a friendly pair, since s(4320) /C3015120 ;s(4680) /C3016380 ;and X (4320)/C1315120 4320/C307 2 X (4680)/C1316380 4680/C307 2: The first few friendly pairs, ordered by smallest maximum element are (6, 28), (30, 140), (80, 200),(40, 224), (12, 234), (84, 270), (66, 308), ... (Sloane’sA050972 and A050973). Numbers which do not have FRIENDS are called SOLITARY NUMBERS . A sufficient (but not necessary) condition for nto be a SOLITARY NUMBER is that (s(n);n)/C301;where ( a, b) is the GREATEST COMMON DIVISOR ofaandb. Hoffman (1998, p. 45) uses the term "friendly num- bers" to describe AMICABLE PAIRS . See also ALIQUOT SEQUENCE ,A MICABLE PAIR, FRIEND ,SOLITARY NUMBER References Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly Integers." Amer. Math. Monthly 84,65/C1/6, 1977. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Sloane, N. J. A. Sequences A050972 and A050973 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Frieze Pattern In general, a frieze consists of repeated copies of a single motif. b ad c Conway and Guy (1996) define a frieze pattern as an arrangement of numbers at the intersection of two sets of perpendicular diagonals such that a /C27d /C30 b /C27c /C271 (for an additive frieze pattern) or ad /C30bc /C271 (for a multiplicative frieze pattern) in each diamond. See also TESSELLATION ,TILING References Conway, J. H. and Coxeter, H. S. M. "Triangulated Polygons and Frieze Patterns." Math. Gaz. 57,87/C1/4, 1973. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 74 /C1/6 and 96 /C1/7, 1996. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 83 /C1/4, 1991. Frivolous Theorem of Arithmetic Almost all natural numbers are very, very, very large. See also LARGE NUMBER References Steinbach, P. Field Guide to Simple Graphs. Albuquerque, NM: Design Lab, 1990. Frobenius Map A map x/C2xpwhere pis a PRIME . Frobenius Method Ifx0is an ordinary point of the ORDINARY DIFFER- ENTIAL EQUATION , expand yin a T AYLOR SERIES about x0;letting y/C30X/C12 n/C300anxn: (1)Plug yback into the ODE and group the COEFFI- CIENTS byPOWER . Now, obtain a RECURRENCE RELA- TION for the nth term, and write the T AYLOR SERIES in terms of the an/s. Expansions for the first few derivatives are y/C30X/C12 n/C300anxn(2) y?/C30X/C12 n/C301nanxn/C281/C30X/C12 n/C300(n/C271)an/C271xn(3) yƒ/C30X/C12 n/C302n(n/C281)anxn/C282/C30X/C12 n/C300(n/C272)(n/C271)an/C272xn:(4) Ifx0is a regular singular point of the ORDINARY DIFFERENTIAL EQUATION , P(x)yƒ/C27Q(x)y?/C27R(x)y/C300; (5) solutions may be found by the Frobenius method or by expansion in a L AURENT SERIES . In the Frobenius method, assume a solution OF THE FORM y/C30xkX/C12 n/C300anxn; (6) so that y/C30xkX/C12 n/C300anxn/C30X/C12 n/C300anxn/C27k(7) y?/C30X/C12 n/C300an(n/C27k)xk/C27n/C281(8) yƒ/C30X/C12 n/C300an(n/C27k)(n/C27k/C281)xk/C27n/C282: (9) Now, plug yback into the ODE and group the COEFFICIENTS by POWER to obtain a recursion FOR- MULA for the an/th term, and then write the T AYLOR SERIES in terms of the an/s. Equating the a0term to 0 will produce the so-called INDICIAL EQUATION , which will give the allowed values of kin the T AYLOR SERIES . FUCHS’S THEOREM guarantees that at least one POWER SERIES solution will be obtained when applying the Frobenius method if the expansion point is anordinary, or regular, SINGULAR POINT . For a regular SINGULAR POINT ,aL AURENT SERIES expansion can also be used. Expand yin a L AURENT SERIES , letting y¼c/C28nx/C28n/C27/C1/C1/C1/C27c0/C27c1x/C27/C1/C1/C1/C27cnxn/C27/C1/C1/C1 ð 10Þ Plug yback into the ODE and group the COEFFI- CIENTS byPOWER . Now, obtain a recurrence FORMULA for the cn/th term, and write the T AYLOR EXPANSION in terms of the cn/s. See also FUCHS’S THEOREM ,ORDINARY DIFFERENTIAL EQUATION References Arfken, G. "Series Solutions--Frobenius’ Method." §8.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 454 /C1/67, 1985. Frobenius Pseudoprime Let f(x)bea MONIC POLYNOMIAL of degree d with discriminant D: Then an ODD INTEGER n with (n;f(0)D) /C301 is called a Frobenius pseudoprime with respect to f(x) if it passes a certain algorithm given by Grantham (1996). A Frobenius pseudoprime with respect to a POLYNOMIAL f(x) /C23Z[x] is then a compo- site Frobenius probably prime with respect to the POLYNOMIAL x /C28a :/ While 323 is the first LUCAS PSEUDOPRIME with respect to the Fibonacci polynomial x2 /C28x /C281; the first Frobenius pseudoprime is 5777. If f(x) /C30x3 /C28 rx2 /C27sx /C281; then any Frobenius pseudoprime n with respect to f(x) is also a PERRIN PSEUDOPRIME . Gran- tham (1997) gives a test based on Frobenius pseudo- primes which is passed by COMPOSITE NUMBERS with probability at most 1/7710. See also PERRIN PSEUDOPRIME ,P SEUDOPRIME , STRONG FROBENIUS PSEUDOPRIME References Grantham, J. "Frobenius Pseudoprimes." 1996. http:// www.clark.net/pub/grantham/pseudo/pseudo1.ps Grantham, J. "A Frobenius Probable Prime Test with High Confidence." 1997. http://www.clark.net/pub/grantham/ pseudo/pseudo2.ps Grantham, J. "Pseudoprimes/Probable Primes." http:// www.clark.net/pub/grantham/pseudo/. Frobenius Theorem Let A /C30aij be a MATRIX with POSITIVE COEFFICIENTS so that aij > 0 for all i ;j /C301; 2, ..., n, then A has a POSITIVE EIGENVALUE l0 ; and all its EIGENVALUES lie on the CLOSED DISK ½z ½5 l0 : See also CLOSED DISK,OSTROWSKI’S THEOREM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000. Frobenius Triangle Identities Let CL;Mbe a PADE´ APPROXIMANT . Then C(L /C271)=MS(L /C281)=M /C28CL=(M /C271)SL=(M /C271) /C30CL =MSL=M (1)CL=(M /C271)S(L/C271)=M /C28C(L/C271)=MSL =(M /C271) /C30C(L/C271)=(M /C271)XSL =M (2) C(L/C271)=MSL =M /C28CL =MS(L /C271)=M /C30C(L/C271)=(M /C271)xSL =(M /C281)(3) CL=(M /C271)SL =M /C28CL =MSL=(M /C271) /C30C(L/C271)=(M /C271)xS(L/C281)=M ; (4) where SL =M /C30G(x)PL(x) /C27H(x)QM(x) (5) and C is the C-DETERMINANT . See also C-DETERMINANT ,PADE´ APPROXIMANT References Baker, G. A. Jr. Essentials of Pade´ Approximants in Theo- retical Physics. New York: Academic Press, p. 31, 1975. Frobenius-Ko ¨nig Theorem The PERMANENT of an n /C29n INTEGER MATRIX with all entries either 0 or 1 is 0 IFF the MATRIX contains an r/C29ssubmatrix of 0s with r/C27s/C30n/C271:This result follows from the K O¨NIG-EGEVA ´RY THEOREM . See also INTEGER MATRIX ,KO¨NIG-EGEVA ´RYTHEOREM , PERMANENT Frobenius-Perron Equation rn/C271(x)/C30grn(y)dx/C28M(y) ½/C138 dy; where d(x)i sa DELTA FUNCTION ,M(x) is a map, and r is the NATURAL INVARIANT . See also NATURAL INVARIANT ,P ERRON- FROBENIUS OPERATOR References Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 51, 1993. Frontier BOUNDARY Frucht Graph The smallest CUBIC GRAPH whose automorphism group consists only of the IDENTITY ELEMENT (Skiena 1990, p. 185). See also CUBIC GRAPH ,GRAPH AUTOMORPHISM References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 235, 1976. Frucht, R. "Herstellung von Graphen mit vorgegebener abstrakter Gruppe." Compos. Math. 6, 239 /C1/50, 1939. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Frugal Number WASTEFUL NUMBER Frullani’s Integral If S? is continuous and the integral converges, g/C12 0f(ax) /C28 f(bx) xdx /C30 f(0) /C28f( /C12) ½/C138 lnb a ! : References Jeffreys, H. and Jeffreys, B. S. "Frullani’s Integrals." §12.16 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 406 /C1/07, 1988. Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968. Frustum The portion of a solid which lies between two PARALLEL PLANES cutting the solid. Degenerate cases are obtained for finite solids by cutting with a single PLANE only. See also CONICAL FRUSTUM ,PYRAMIDAL FRUSTUM , SPHERICAL SEGMENT Fubini Principle If the average number of envelopes per pigeonhole is a, then some pigeonhole will have at least a envel- opes. Similarly, there must be a pigeonhole with at most a envelopes. See also PIGEONHOLE PRINCIPLE Fubini Theorem This entry contributed by RONALD M. AARTS A theorem that establishes a connection between a MULTIPLE INTEGRAL and a REPEATED one. Under certain assumptions the following equality holds:ggRm/C27nf(x;y)d(x ;y) /C30gRndygRmf(x;y)dx: See also MULTIPLE INTEGRAL ,REPEATED INTEGRAL References Fubine, G. "Sugli integrali multipli." Opere scelte, Vol. 2. Cremonese, pp. 243 /C1/49, 1958. Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 9, 1993. Fuchs’s Theorem At least one POWER SERIES solution will be obtained when applying the FROBENIUS METHOD if the expan- sion point is an ordinary, or regular, SINGULAR POINT . The number of ROOTS is given by the ROOTS of the INDICIAL EQUATION . References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 462 /C1/63, 1985. Fuchsian System A system of linear differential equations dy dz /C30A(z)y; with A(z)an ANALYTIC n /C29n MATRIX , for which the MATRIX A(z)is ANALYTIC in C_ fa1 ;...;aN g and has a POLE of order 1 at ajfor j /C301, ..., N. A system is Fuchsian IFF there exist n /C29n matrices B1 ; ..., BN with entries in Z such that A(z) /C30XN j/C301Bj z /C28 aj XN j/C301Bj /C30v : Fuglede’s Conjecture Fuglede (1974) conjectured that a domain V admits a SPECTRUM IFF it is possible to tile Rd by a family of translates of V: Fuglede proved the conjecture in the special case that the tiling set or the spectrum are lattice subsets of Rd and Iosevich et al. (1999) proved that no smooth symmetric convex body V with at least one point of nonvanishing G AUSSIAN CURVATURE can admit an orthogonal basis of exponentials. However, the general conjecture is still far from being proved (Iosevich et al. 1999). See also SPECTRUM (OPERATOR ) References Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101 /C1/21, 1974. Iosevich, A.; Katz, N. H.; and Tao, T. Convex Bodies with a Point of Curvature Do Not Have Fourier Bases. 23 Nov 1999. http://xxx.lanl.gov/abs/math.CA/9911167/. Jorgensen, P. E. T. and Pedersen, S. "Orthogonal Harmonic Analysis of Fractal Measures." Elec. Res. Announc. Amer. Math. Soc. 4,35/C1/2, 1998. Lagarias, J. and Wang, Y. "Spectral Sets and Factorizations of Finite Abelian Groups." J. Func. Anal. 145,73/C1/8, 1997. Fuhrmann Center The center of the FUHRMANN CIRCLE , given by the MIDPOINT of the line joining the NAGEL POINT and ORTHOCENTER (which forms a DIAMETER of the FUHR- MANN CIRCLE ). See also FUHRMANN CIRCLE ,NAGEL POINT ,ORTHO- CENTER Fuhrmann Circle The CIRCUMCIRCLE of the FUHRMANN TRIANGLE . The ORTHOCENTER H,NAGEL POINT Na, and at least six other noteworthy points lie on the Fuhrmann circle (Honsberger 1995, p. 49). In particular, HNa is a DIAMETER of the Fuhrmann circle. It also passes through the points T, U, and V which are a distance 2r along the ALTITUDES from the vertices, where r is the INRADIUS of DABC (Honsberger 1995, p. 52). See also ALTITUDE ,FUHRMANN TRIANGLE ,INRADIUS , MID-ARC POINTS ,NAGEL POINT ,ORTHOCENTER References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 58, 1971. Fuhrmann, W. Synthetische Beweise Planimetrischer Sa¨tze. Berlin, p. 107, 1890.Honsberger, R. "The Fuhrmann Circle." Ch. 6 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 49 /C1/2, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 228 /C1/29, 1929. Fuhrmann Triangle The Fuhrmann triangle of a TRIANGLE DABC is the TRIANGLE DFCFBFAformed by reflecting the MID-ARC POINTS MAB ; MAC ; MBCabout the lines AB, AC, and BC. The CIRCUMCIRCLE of the Fuhrmann triangle is called the FUHRMANN CIRCLE , and the lines FAMBC ; FBMAC ; and FCMAB CONCUR at the CIRCUMCENTER O. See also FUHRMANN CENTER ,F UHRMANN CIRCLE , MID-ARC POINTS References Fuhrmann, W. Synthetische Beweise Planimetrischer Sa ¨tze. Berlin, p. 107, 1890. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 228 /C1/29, 1929. Fuhrmann’s Theorem Let the opposite sides of a convex CYCLIC HEXAGON be a,a?;b,b?;c, and c?;and let the DIAGONALS e,f, and g be so chosen that a,a?;andehave no common VERTEX (and likewise for b,b?;andf), then efg/C30aa?e/C27bb?f/C27cc?g/C27abc/C27a?b?c?: This is an extension of PTOLEMY’S THEOREM to the HEXAGON . See also CYCLIC HEXAGON ,H EXAGON ,P TOLEMY’S THEOREM References Fuhrmann, W. Synthetische Beweise Planimetrischer Sa¨tze. Berlin, p. 61, 1890. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 65 /C1/6, 1929. Full Angle An ANGLE equal to 360 8. See also ACUTE ANGLE ,A NGLE ,O BTUSE ANGLE , REFLEX ANGLE ,RIGHT ANGLE ,STRAIGHT ANGLE Full Reptend Prime A PRIME p for which 1=p has a maximal period DECIMAL EXPANSION of p /C281 DIGITS , sometimes called a long prime (Conway and Guy 1996, pp. 157 /C1/63 and 166 /C1/71). A prime is full reptend IFF 10 is a PRIMITIVE ROOT modulo p. No general method is known for finding full reptend primes. The first few numbers with maximal decimal expansions are 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (Sloane’s A001913). See also DECIMAL EXPANSION ,PRIMITIVE ROOT References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996. Sloane, N. J. A. Sequences A001913/M4353 and A006883/ M1745 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 71, 1986. Full Width at Half Maximum The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value. The follow- ing table gives the analytic and numerical full widths for several common curves.Function Formula FWHM Bartlett /1 /C28½x½ a/ a Blackman /0 :810957 a/ Connes / 1 /C28x2 a2/C1Y/C1Q //ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C282ffiffiffiffiffiffi 2app / Cosine /cospx 2a/C1Y/C1Q //4 3 a/ Gaussian /e/C28x2=(2s2)// 2ffiffiffiffiffiffiffiffiffiffiffiffiffi 2ln2p s/ Hamming /1 :05543 a/ Hanning a Lorentzian /1 2 G x2 /C271 2 G/C1Y/C1Q 2//G/ Welch /1 /C28x2 a2//ffiffiffi 2p a/ See also APODIZATION FUNCTION ,MAXIMUM Fuller Dome GEODESIC DOME Function A relation which uniquely associates members of one SETwith members of another SET. More formally, a function from AtoBis an object fsuch that every a/C23Ais uniquely associated with an object f(a)/C23B:A function is therefore a MANY-TO-ONE (or sometimes ONE-TO-ONE ) relation. Examples of functions include sinx(MANY-TO-ONE ),x(ONE-TO-ONE ),x2(two-to-one except for the single point x/C300), etc. The term " MAP" is synonymous with function. Several notations are commonly used to represent functions. The most rigorous notation is f:x0f(x); which specifies that fis function acting upon a single number x(i.e., fis a univariate, or one-variable, function) and returning a value f(x):To be even more precise, a notation like " f:R0R;where f(x)/C30x2/"i s sometimes used to explicitly specify the domain and range of the function. The slightly different "maps to" notation f : x /C2f(x) is sometimes also used when the function is explicitly considered as a "map." Generally speaking, the symbol f refers to the func- tion itself, while f(x) refers to the value taken by the function when evaluated at a point x. However, especially in more introductory texts, the notation f(x) is commonly used to refer to the function f itself (as opposed to the value of the function evaluated at x). In this context, the argument x is considered to be a DUMMY VARIABLE whose presence indicates that the function f takes a single argument (as opposed to f(x; y); etc.). While this notation is deprecated by professional mathematicians, it is the more familiar one for most nonprofessionals. Therefore, unless indicated otherwise by context, the notation f(x)is taken in this work to be a shorthand for the more rigorous f : x 0 f(x):/ Poincare ´ remarked with regard to the proliferation of pathological functions, "Formerly, when one invented a new function, it was to further some practical purpose; today one invents them in order to make incorrect the reasoning of our fathers, and nothing more will ever be accomplished by these inventions." References Abramowitz, M. and Stegun, C. A. (Eds.). "Miscellaneous Functions." Ch. 27 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 997 /C1/010, 1972. Arfken, G. "Special Functions." Ch. 13 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 712 /C1/59, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Special Functions." Ch. 6 in Numerical Re- cipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 205 /C1/65, 1992. Weisstein, E. W. "Books about Special Functions." http:// www.treasure-troves.com/books/SpecialFunctions.html. Function Element A function element is an ORDERED PAIR (f, U) where U is a disk DZ0 ;r ðÞ and f is an ANALYTIC FUNCTION defined on U.IfW is an OPEN SET, then a function element in W is a pair (f, U) such that U ⁄W :/ References Krantz, S. G. "Function Elements." §10.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 128, 1999. Function Field A finite extension K /C30Z(z)(w) of the FIELD C(z)of RATIONAL FUNCTIONS in the indeterminate z, i.e., w is a ROOT of a POLYNOMIAL a0 /C27a1 a /C27a2 a2 /C27:::/C27an an ; where ai /C23C(z): Function fields are sometimes called algebraic function fields. See also LOCAL FIELD,N UMBER FIELD,R IEMANN SURFACEFunction of the First Kind FIRST KIND Function of the Second Kind SECOND KIND Function of the Third Kind THIRD KIND Function Space /f(I) is the collection of all real-valued continuous functions defined on some interval I. f(n)(I) is the collection of all functions /C23 f(I) with continuous nth DERIVATIVES . A function space is a TOPOLOGICAL VECTOR SPACE whose "points" are functions. See also FUNCTIONAL ,FUNCTIONAL ANALYSIS ,OPERA- TOR Functional A functional is a real-valued function on a VECTOR SPACE V, usually of functions. For example, the ENERGY functional on the UNIT DISK D assigns a number to any differentiable function f : D 0 R ; E(f):gD ½½9f ½½2dA: For the functional to be continuous, it is necessary for the VECTOR SPACE V of functions to have an appro- priate TOPOLOGY . The widespread use of functionals in applications, such as the CALCULUS OF VARIATIONS , gave rise to FUNCTIONAL ANALYSIS . The reason the term "functional" is used is because V can be a space of functions, e.g., V /C30ff :[0;1] 0 R such that f is continuous g in which case T(f) /C30f(0) is a LINEAR FUNCTIONAL on V. See also CALCULUS OF VARIATIONS ,COERCIVE FUNC- TIONAL ,C URRENT ,E LLIPTIC FUNCTIONAL ,E ULER- LAGRANGE DIFFERENTIAL EQUATION ,F UNCTIONAL ANALYSIS ,F UNCTIONAL EQUATION ,G ENERALIZED FUNCTION ,LAPLACIAN ,LAX-MILGRAM THEOREM ,LIN- EAR FUNCTIONAL ,OPERATOR ,RIESZ REPRESENTATION THEOREM ,VECTOR SPACE Functional Analysis A branch of mathematics concerned with infinite dimensional spaces (mainly FUNCTION SPACES ) and mappings between them. The SPACES may be of different, and possibly INFINITE ,DIMENSIONS . These mappings are called OPERATORS or, if the range is on the REAL line or in the COMPLEX PLANE , FUNCTIONALS . See also FUNCTIONAL ,FUNCTIONAL EQUATION ,GEN- ERALIZED FUNCTION ,OPERATOR References Balakrishnan, A. V. Applied Functional Analysis, 2nd ed. New York: Springer-Verlag, 1981. Berezansky, Y. M.; Us, G. F.; and Sheftel, Z. G. Functional Analysis, Vol. 1. Boston, MA: Birkha ¨user, 1996. Berezansky, Y. M.; Us, G. F.; and Sheftel, Z. G. Functional Analysis, Vol. 2. Boston, MA: Birkha ¨user, 1996. Birkhoff, G. and Kreyszig, E. "The Establishment of Func- tional Analysis." Historia Math. 11, 258 /C1/21, 1984. Hutson, V. and Pym, J. S. Applications of Functional Analysis and Operator Theory. New York: Academic Press, 1980. Kreyszig, E. Introductory Functional Analysis with Applica- tions. New York: Wiley, 1989. Yoshida, K. Functional Analysis and Its Applications. New York: Springer-Verlag, 1971. Zeidler, E. Nonlinear Functional Analysis and Its Applica- tions. New York: Springer-Verlag, 1989. Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995. Functional Calculus An early name for CALCULUS OF VARIATIONS . The term is also sometimes used in place of PREDICATE CALCULUS . Functional Congruence A CONGRUENCE OF THE FORM f(x) /C13g(x)( mod n) where f(x) and g(x) are both INTEGER POLYNOMIALS . Functional congruences are sometimes also called "identical congruences" (Nagell 1951, p. 74). See also CONGRUENCE References Nagell, T. "Algebraic Congruences and Functional Con- gruences." §22 in Introduction to Number Theory. New York: Wiley, pp. 73 /C1/6, 1951. Functional Derivative A generalization of the concept of the DERIVATIVE to GENERALIZED FUNCTIONS . Functional Distribution GENERALIZED FUNCTION Functional Equation An equation OF THE FORM f(x;y;:::) /C300; where f contains a finite number of independent variables, known functions, and unknown functions which areto be solved for. Many properties of functions can be determined by studying the types of functional equations they satisfy. For example, the GAMMA FUNCTION G(z) satisfies the functional equations G(1 /C27z) /C30z G(z) G(1 /C28z) /C30/C28zG(/C28z) : See also ABEL’S DUPLICATION FORMULA ,A BEL’S FUNCTIONAL EQUATION ,FUNCTIONAL ANALYSIS References Kuczma, M. Functional Equations in a Single Variable. Warsaw, Poland: Polska Akademia Nauk, 1968. Kuczma, M. An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jen- sen’s Inequality. Warsaw, Poland: Uniwersitet Slaski, 1985. Kuczma, M.; Choczewski, B.; and Ger, R. Iterative Func- tional Equations. Cambridge, England: Cambridge Uni- versity Press, 1990. Functional Graph A functional graph is a DIGRAPH in which each vertex has outdegree one, and can therefore be specified by a function mapping f1 ;:::; ng onto itself. Functional graphs are implemented asFunctionalGraph [f, n] in the Mathematica add-on package Discrete- Math‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). References Skiena, S. "Functional Graphs." §4.5.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 164 /C1/65, 1990. Functor A function between CATEGORIES which maps objects to objects and MORPHISMS to MORPHISMS . Functors exist in both covariant and contravariant types. See also CATEGORY ,EILENBERG- STEENROD AXIOMS , MORPHISM ,SCHUR FUNCTOR Fundamental Class The canonical generator of the nonvanishing HOMOL- OGY GROUP on a TOPOLOGICAL MANIFOLD . See also CHERN NUMBER ,P ONTRYAGIN NUMBER , STIEFEL- WHITNEY NUMBER Fundamental Continuity Theorem Given two UNIVARIATE POLYNOMIALS of the same order whose first pCOEFFICIENTS (but notthe first p/C281) are 0 where the COEFFICIENTS of the second approach the corresponding COEFFICIENTS of the first as limits, the second POLYNOMIAL will have exactly p roots that increase indefinitely. Furthermore, exactly k ROOTS of the second will approach each ROOT of multiplicity k of the first as a limit. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 4, 1959. Fundamental Discriminant //C28D is a fundamental discriminant if D is a POSITIVE INTEGER which is not DIVISIBLE by any square of an ODD PRIME and which satisfies D/C133 (mod 4) or D/C134;8 (mod 16) :/ See also DISCRIMINANT References Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Math. Comput. 61,2 9/C1/8, 1993. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 294, 1987. Cohn, H. Advanced Number Theory. New York: Dover, 1980. Dickson, L. E. History of the Theory of Numbers, Vols. 1 /C1/. New York: Chelsea, 1952. Fundamental Forms There are three types of so-called fundamental forms. The most important are the first and second (sincethe third can be expressed in terms of these). The fundamental forms are extremely important and useful in determining the metric properties of asurface, such as LINE ELEMENT ,AREA ELEMENT , NORMAL CURVATURE ,G AUSSIAN CURVATURE , and MEAN CURVATURE . Let Mbe a REGULAR SURFACE with vP;wPpoints in the TANGENT SPACE MPofM. Then the FIRST FUNDAMENTAL FORM is the INNER PRODUCT of tangent vectors, IvP;wP ðÞ /C30vP/C215wP: (1) ForM/C23R3;the SECOND FUNDAMENTAL FORM is the symmetric bilinear form on the TANGENT SPACE MP; II vp;wp/C0/C1 /C30Svp/C0/C1 /C215wp; (2) where Sis the SHAPE OPERATOR . The THIRD FUNDA- MENTAL FORM is given by III vp;wp/C0/C1 /C30Svp/C0/C1 /C215Swp/C0/C1 : (3) The FIRST and SECOND FUNDAMENTAL FORMS satisfy IaXu/C27bXv;aXu/C27bXv ðÞ /C30Ea2/C272Fab/C27Gb2(4) IIaXu/C27bXv;aXu/C27bXv ðÞ /C30ea2/C272fab/C27gb2(5) where x:U0R3is a REGULAR PATCH andxuandxv are the partial derivatives of xwith respect to parameters uand v, respectively. Their ratio is simply the NORMAL CURVATUREkvp/C0/C1 /C30II vp/C0/C1 Ivp/C0/C1 (6) for any nonzero TANGENT VECTOR . The third funda- mental form is given in terms of the first and secondforms by III/C282HII/C27KI/C300; (7) where His the MEAN CURVATURE and Kis the GAUSSIAN CURVATURE . The first fundamental form (or LINE ELEMENT )i s given explicitly by the R IEMANNIAN METRIC ds2/C30Edu2/C272Fdudv /C27Gdv2: (8) It determines the ARC LENGTH of a curve on a surface. The coefficients are given by E/C30xuu/C30@x @u/C12/C12/C12/C12/C12/C12/C12/C12/C12/C122 (9) F/C30xuv/C30@x @u/C215@x @v(10) G/C30xvv/C30@x @v/C12/C12/C12/C12/C12/C12/C12/C12/C12/C122 : (11) The coefficients are also denoted guu/C30E;guv/C30F;and gvv/C30G:InCURVILINEAR COORDINATES (where F/C300), the quantities hu/C13ffiffiffiffiffiffiffiguup/C30ffiffiffiffi Ep (12) hv/C13ffiffiffiffiffiffiffigvvp/C30ffiffiffiffi Gp (13) are called SCALE FACTORS . The second fundamental form is given explicitly by ed u2/C272fd udv/C27gd v2(14) where e/C30X iXi@2xi @u2(15) f/C30X iXi@2xi @u@v(16) g/C30X iXi@2xi @v2; (17) and Xiare the DIRECTION COSINES of the surface normal. The second fundamental form can also be written e/C30/C28Nu/C215xu/C30N/C215xuu (18) f/C30/C28Nv/C215xu/C30N/C215xuv/C30Nvu/C215xvu /C30Nu/C215xv (19) g /C30/C28Nv /C215xv /C30N /C215xvv ; (20) where N is the NORMAL VECTOR ,or e /C30det(xuuxuxv)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG /C28 F2p (21) f /C30det(xuvxuxv)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEG /C28 F2p (22) g /C30det(xvvxuxv)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG /C28 F2p : (23) See also ARC LENGTH ,AREA ELEMENT ,FIRST FUNDA- MENTAL FORM,G AUSSIAN CURVATURE ,G EODESIC , KA¨ HLER MANIFOLD ,LINE OF CURVATURE ,LINE ELE- MENT ,M EAN CURVATURE ,NORMAL CURVATURE ,RIE- MANNIAN METRIC ,S CALE FACTOR ,S ECOND FUNDAMENTAL FORM,SURFACE AREA,THIRD FUNDA- MENTAL FORM,W EINGARTEN EQUATIONS References Gray, A. "The Three Fundamental Forms." §16.6 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 380 /C1/82, 1997. Fundamental Group The fundamental group of an ARCWISE-CONNECTED setXis the GROUP formed by the sets of EQUIVALENCE CLASSES of the set of all LOOPS , i.e., paths with initial and final points at a given BASEPOINT p, under the EQUIVALENCE RELATION ofHOMOTOPY . The IDENTITY ELEMENT of this group is the set of all paths HOMO- TOPIC to the degenerate path consisting of the point p. The fundamental groups of HOMEOMORPHIC spaces are ISOMORPHIC . In fact, the fundamental group only depends on the HOMOTOPY TYPE ofX. The funda- mental group of a TOPOLOGICAL SPACE was introduced by Poincare ´(Munkres 1993, p. 1). The following is a table of the fundamental group for some common spaces, where p1denotes the funda- mental group, H1is the first integral HOMOLOGY ,/C29 denotes the GROUP DIRECT PRODUCT ,Zdenotes the RING of integers, and Znis the CYCLIC GROUP of order n. space symbol /p1// H1/ CIRCLE /S1 // Z// Z/ figure eight /Z‘Z//Z/C29Z/ SPHERE /S2 / 00 TORUS /T// Z/C29Z//Z/C29Z/ TORUS of genus g /ag// Fg// Z2g /REAL PROJECTIVE PLANE/RP2 //Z2// Z2/ KLEIN BOTTLE /Z‘Z aba/C281b ðÞ//Z/C29Z2/ COMPLEX PROJECTIVE SPACE/CPn / 00 n-torus /Tn // Zn // Zn / The group product a+bofLOOP aand LOOP bis given by the path of afollowed by the path of b. The identity element is represented by the constant path, and the inverse of ais given by traversing ain the opposite direction. The fundamental group is inde-pendent of the choice of basepoint because any loopthrough pis HOMOTOPIC to a loop through any other point q. So it makes sense to say the "fundamental group of X." The diagram above shows that a loop followed by the opposite loop is homotopic to the constant loop, i.e., the identity. That is, it starts by traversing the patha, and then turns around and goes the other way, a /C281:The composition is deformed, or homotoped, to the constant path, along the original path a. A space with a trivial fundamental group (i.e., everyloop is homotopic to the constant loop), is called SIMPLY CONNECTED . For instance, any CONTRACTIBLE space, like E UCLIDEAN SPACE , is simply connected. The SPHERE isSIMPLY CONNECTED , but not CONTRAC- TIBLE . By definition, the UNIVERSAL COVER ˜Xis simply connected, and loops in Xlift to paths in ˜X: The lifted paths in the universal cover define the DECK TRANSFORMATIONS , which form a GROUP iso- morphic to the fundamental group. The underlying set of the fundamental group of Xis the set of based HOMOTOPY CLASSES from the circle to X, denoted S1;X/C2/C6 :For general spaces XandY, there is no natural group structure on [ X, Y ], but when there is, Xis called a H-SPACE . Besides the circle, every SPHERE Snis a H-SPACE , defining the HOMO- TOPY GROUPS . In general, the fundamental group is NON- ABELIAN . However, the higher HOMOTOPY GROUPS are Abelian. In some special cases, the fundamental group is Abelian. For example, the animation above shows that a + b /C30b + a in the TORUS . The red path goes before the green path. The animation is a homotopy between the loop that goes around the inside first and the loop that goes around the outside first. Since the first integral HOMOLOGY H1(X ;Z)ofX is also represented by loops, which are the only 1- dimensional objects with no boundary, there is a GROUP HOMOMORPHISM a : p1(X) 0 H1(X ;Z) ; which is SURJECTIVE . In fact, the KERNEL of a is the COMMUTATOR SUBGROUP and a is called ABELIANIZA- TION . The fundamental group of X can be computed using VAN KAMPEN’S THEOREM , when X can be written as a union X /C30@i Xiof spaces whose fundamental groups are known. When f : X 0 Y is a continuous map, then the fundamental group pushes forward. That is, there is a map f+ : p1(X) 0p1(Y) defined by taking the image of loops from X. The pushforward is natural, i.e., (f(g)+/C30f+(g +whenever the composition of two maps is defined. See also ALGEBRAIC FUNDAMENTAL GROUP ,CAYLEY GRAPH ,C ONNECTED SET,D ECK TRANSFORMATION , HOMOLOGY ,H OMOTOPY GROUP ,G ROUP ,M ILNOR’S THEOREM ,UNIVERSAL COVER , VAN KAMPEN’S THEO- REM References Dodson, C. T. J. and Parker, P. E. "The Fundamental Group." §2.5 in A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 45 /C1/7, 1997. Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, pp. 165 /C1/03, 1995. Massey, W. S. A Basic Course in Algebraic Topology. New York: Springer-Verlag, pp. 35 /C1/8, 1991. Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Fundamental Homology Class FUNDAMENTAL CLASS Fundamental Lemma of Calculus of Variations If gb aM(x)h(x)dx /C300 //C214h(x) with CONTINUOUS second PARTIAL DERIVATIVES , then M(x) /C300 on the OPEN INTERVAL (a, b).Fundamental Polytope PRIMITIVE POLYTOPE Fundamental Region Let G be a SUBGROUP of the MODULAR GROUP GAMMA . Then an open subset RG of the UPPER HALF-PLANE H is called a fundamental region of G if 1. No two distinct points of RGare equivalent under G, 2. If t /C23 H ; then there is a point t? in the closure of RG such that t ? is equivalent to t under G. A fundamental region RGof the MODULAR GROUP GAMMA is given by t /C23 H such that tjj> 1 and ½t /C27¯t ½B 1; illustrated above, where t is the COMPLEX CON- JUGATE of t (Apostol 1997, p. 31). Borwein and Borwein (1987, p. 113) define the boundaries of the region slightly differently by including the boundary points with R[ t] 50 :/ See also MODULAR GROUP GAMMA ,M ODULAR GROUP LAMBDA ,UPPER HALF-PLANE ,VALENCE References Apostol, T. M. "Fundamental Region." §2.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 30 /C1/4, 1997. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 112 /C1/13, 1987. Fundamental System A set of ALGEBRAIC INVARIANTS for a QUANTIC such that any invariant of the QUANTIC is expressible as a POLYNOMIAL in members of the set. In 1868, Gordan proved the existence of finite fundamental systems of algebraic invariants and covariants for any binary QUANTIC . In 1890, Hilbert (1890) proved the HILBERT BASIS THEOREM , which is a finiteness theorem for the related concept of SYZYGIES . See also HILBERT BASIS THEOREM ,SYZYGY References Hilbert, D. "U ¨ber die Theorie der algebraischen Formen." Math. Ann. 36, 473/C1/34, 1890. Fundamental Theorem of Algebra Every POLYNOMIAL EQUATION having COMPLEX COEF- FICIENTS and degree ]1 has at least one COMPLEX ROOT . This theorem was first proven by Gauss. It is equivalent to the statement that a POLYNOMIAL P(z)of degree n has n values zi(some of them possibly degenerate) for which PziðÞ/C300 : Such values are called POLYNOMIAL ROOTS . An example of a POLYNO- MIAL with a single ROOT of multiplicity > 1is z2 /C28 2z /C271 /C30(z /C281)(z /C281); which has z /C301asa ROOT of multiplicity 2. For RINGS more general than the complex polyno- mials C[x]; there does not necessarily exist a unique factorization. However, a PRINCIPAL RING is a struc- ture for which the proof of the unique factorization property is sufficiently easy while being quite general and common. See also DEGENERATE ,F RIVOLOUS THEOREM OF ARITHMETIC ,POLYNOMIAL ,POLYNOMIAL FACTORIZA- TION ,POLYNOMIAL ROOTS ,PRINCIPAL RING References Courant, R. and Robbins, H. "The Fundamental Theorem of Algebra." §2.5.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 101 /C1/03, 1996. Krantz, S. G. "The Fundamental Theorem of Algebra." §1.1.7 and 3.1.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 7 and 32 /C1/3, 1999. Fundamental Theorem of Arithmetic Any POSITIVE INTEGER can be represented in exactly one way as a PRODUCT of PRIMES . The theorem is also called the UNIQUE FACTORIZATION THEOREM . The fundamental theorem of arithmetic is a COROLLARY of the first of EUCLID’S THEOREMS (Hardy and Wright 1979). See also ABNORMAL NUMBER ,EUCLID’S THEOREMS , INTEGER ,PRIME NUMBER References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 23, 1996. Davenport, H. The Higher Arithmetic: An Introduction to the Theory of Numbers, 6th ed. Cambridge, England: Cam- bridge University Press, p. 20, 1992. Hardy, G. H. and Wright, E. M. "Statement of the Funda- mental Theorem of Arithmetic," "Proof of the Fundamen- tal Theorem of Arithmetic," and "Another Proof of the Fundamental Theorem of Arithmetic." §1.3, 2.10 and 2.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 3 and 21, 1979. Hasse, H. "U¨ ber eindeutige Zerlegung in Primelemente oder in Primhauptideale in Integrita ¨tsbereichen." J. reine angew. Math. 159,3/C1/2, 1928. Lindemann, F. A. "The Unique Factorization of a Positive Integer." Quart. J. Math. 4, 319 /C1/20, 1933. Nagell, T. "The Fundamental Theorem." §4in Introduction to Number Theory. New York: Wiley, pp. 14 /C1/6, 1951.Zermelo, E. "Elementare Betrachtungen zur Theorie der Primzahlen." Nachr. Gesellsch. Wissensch. Go¨ttingen 1, 43 /C1/6, 1934. Fundamental Theorem of Curves The CURVATURE and TORSION functions along a SPACE CURVE determine it up to an orientation-preserving ISOMETRY . Fundamental Theorem of Directly Similar Figures Let F0 and F1 denote two DIRECTLY SIMILAR figures in the plane, where P1 /C23 F1 corresponds to P1 /C23 F0 under the given similarity. Let r /C23 (0;1); and define Fr /C30 (1 /C28r)P0 /C27rP1 : P0 /C23 F0 ; P1 /C23 F1 fg : Then /Fr/ is also directly similar to F0 :/ See also DIRECTLY SIMILAR ,FINSLER- HADWIGER THE- OREM References Detemple, D. and Harold, S. "A Round-Up of Square Problems." Math. Mag. 69,15/C1/7, 1996. Eves, H. Solution to Problem E521. Amer. Math. Monthly 50, 64, 1943. Fundamental Theorem of Gaussian Quadrature The ABSCISSAS of the N-point GAUSSIAN QUADRATURE FORMULA are precisely the ROOTS of the ORTHOGONAL POLYNOMIAL for the same INTERVAL and WEIGHTING FUNCTION . See also GAUSSIAN QUADRATURE Fundamental Theorem of Genera Consider h/C27(d) proper equivalence classes of forms with discriminant d equal to the field discriminant, then they can be subdivided equally into 2r/C281 genera of h/C27(d) =2r /C281 forms which form a SUBGROUP of the proper equivalence class group under composition (Cohn 1980, p. 224), where r is the number of distinct prime divisors of d. This theorem was proved by Gauss in 1801.See also G ENUS (FORM), GENUS THEOREM References Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." http:// www.math.uiuc.edu/Algebraic-Number-Theory/0009/. Cohn, H. Advanced Number Theory. New York: Dover, 1980. Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT: Yale University Press, 1966. Fundamental Theorem of Number Theory FUNDAMENTAL THEOREM OF ARITHMETIC Fundamental Theorem of Plane Curves Two unit-speed plane curves which have the same CURVATURE differ only by a EUCLIDEAN MOTION . See also FUNDAMENTAL THEOREM OF SPACE CURVES References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 136 /C1/38, 1997. Fundamental Theorem of Projective Geometry A PROJECTIVITY is determined when three points of one RANGE and the corresponding three points of the other are given. See also PROJECTIVE GEOMETRY Fundamental Theorem of Riemannian Geometry On a RIEMANNIAN MANIFOLD , there is a unique CONNECTION which is TORSION -free and compatible with the METRIC . This CONNECTION is called the LEVI- CIVITA CONNECTION . See also COVARIANT DERIVATIVE ,LEVI-CIVITA CON- NECTION ,RIEMANNIAN MANIFOLD ,RIEMANNIAN ME- TRIC Fundamental Theorem of Space Curves If two single-valued continuous functions k(s)(CUR- VATURE ) and t(s)(TORSION ) are given for s /C210, then there exists EXACTLY ONE SPACE CURVE , determined except for orientation and position in space (i.e., up to aE UCLIDEAN MOTION ), where s is the ARC LENGTH , k is the CURVATURE , and t is the TORSION . See also ARC LENGTH ,CURVATURE ,EUCLIDEAN MO- TION ,FUNDAMENTAL THEOREM OF PLANE CURVES , TORSION (DIFFERENTIAL GEOMETRY ) References Gray, A. "The Fundamental Theorem of Space Curves." §7.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 219 /C1/22, 1997. Struik, D. J. Lectures on Classical Differential Geometry. New York: Dover, p. 29, 1988. Fundamental Theorems of Calculus The first fundamental theorem of calculus states that, if f is CONTINUOUS on the CLOSED INTERVAL [a, b] and F is the ANTIDERIVATIVE (INDEFINITE INTE- GRAL )off on [a, b], then gb af(x)dx /C30F(b) /C28F(a) : (1)The second fundamental theorem of calculus lets f be CONTINUOUS on an OPEN INTERVAL I and lets a be any point in I.IfF is defined by F(x) /C30gx af(t)dt; (2) then F ?(x) /C30f(x) (3) at each point in I. The fundamental theorem of calculus along curves states that if f(z) has a CONTINUOUS ANTIDERIVATIVE F(z) in a region R containing a parameterized curve g : z /C30z(t) for a 5t 5 b; then ggf(z)dz/C30Fz(b) ðÞ/C28Fz(a)ðÞ : (4) See also CALCULUS ,DEFINITE INTEGRAL ,INDEFINITE INTEGRAL ,INTEGRAL References Krantz, S. G. "The Fundamental Theorem of Calculus along Curves." §2.1.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 22, 1999. Fundamental Unit In a REAL QUADRATIC FIELD , there exists a special UNIT hknown as the fundamental unit such that all units rare given by r/C309hm;form/C300,91,92, .... The notation o0is sometimes used instead of h (Zucker and Robertson 1976). The fundamental units for REAL QUADRATIC FIELDS Q(ffiffiffiffi Dp ) may be computed from the fundamental solution of the P ELL EQUATION T2/C28DU2/C3094; where the sign is taken such that the solution ( T, U ) has smallest possible positive T(LeVeque 1977; Cohn 1980, p. 101; Hua 1982; Borwein and Borwein 1986, p. 294). If the positive sign is taken, then one solutionis simply given by ( T;U)/C30(2x;2y);where ( x, y) is the solution to the P ELL EQUATION x2/C28Dy2/C301 However, this need not be the minimal solution. For example, the solution to Pell equation x2/C2821y2/C301 is (x;y)/C30(55;12);so (T;U)/C30(2x;2y)/C30(110 ;24);but (T;U)/C30(5;1) is the minimal solution. Given a mini- mal ( T, U ) (Sloane’s A048941 and A048942), the fundamental unit is given by h /C301 2 (T /C27Uffiffiffiffi Dp ) (Cohn 1980, p. 101). The following table gives fundamental units for small D. D / h(D)/ D /h(D)/ 2 /1 /C27ffiffiffi 2p / 54 /485 /C2766ffiffiffiffiffiffi54p / 3 /2 /C27ffiffiffi3p / 55 /89 /C2712ffiffiffiffiffiffi55p / 5 /1 2(1 /C27ffiffiffi 5p )/ 56 /15 /C272ffiffiffiffiffiffi 56p / 6 /5 /C272ffiffiffi6p / 57 /151 /C2720ffiffiffiffiffiffi57p / 7 /8 /C273ffiffiffi7p / 58 /99 /C2713ffiffiffiffiffiffi58p / 8 /1 2(1 /C272ffiffiffi 8p )/ 59 /530 /C2769ffiffiffiffiffiffi 59p / 10 /3 /C27ffiffiffiffiffiffi10p / 60 /1 2 (8 /C27ffiffiffiffiffiffi 60p )/ 11 /10 /C273ffiffiffiffiffiffi 11p / 61 /1 2(39 /C275ffiffiffiffiffiffi 61p )/ 12 /7 /C272ffiffiffiffiffiffi 12p / 62 /63 /C278ffiffiffiffiffiffi62p / 13 /1 2(3 /C27ffiffiffiffiffiffi 13p )/ 63 /8 /C27ffiffiffiffiffiffi 63p / 14 /15 /C274ffiffiffiffiffiffi14p / 65 /8 /C27ffiffiffiffiffiffi 65p / 15 /4 /C27ffiffiffiffiffiffi 15p / 66 /65 /C278ffiffiffiffiffiffi66p / 17 /4 /C27ffiffiffiffiffiffi17p / 67 /48842 /C275967ffiffiffiffiffiffi67p / 18 /17 /C274ffiffiffiffiffiffi18p / 68 /1 2 (8 /C27ffiffiffiffiffiffi 68p )/ 19 /170 /C2739ffiffiffiffiffiffi 19p / 69 /1 2(25 /C273ffiffiffiffiffiffi 69p )/ 20 /1 2(4 /C27ffiffiffiffiffiffi 20p )/ 70 /251 /C2730ffiffiffiffiffiffi 70p / 21 /1 2 /C27(5 /C27ffiffiffiffiffiffi 21p )/ 71 /3480 /C27413ffiffiffiffiffiffi 71p / 22 /197 /C2742ffiffiffiffiffiffi 22p / 72 /17 /C272ffiffiffiffiffiffi 72p / 23 /24 /C275ffiffiffiffiffiffi23p / 73 /1068 /C27125ffiffiffiffiffiffi73p / 24 /5 /C27ffiffiffiffiffiffi 24p / 74 /43 /C275ffiffiffiffiffiffi 74p / 26 /5 /C27ffiffiffiffiffiffi26p / 75 /26 /C273ffiffiffiffiffiffi75p / 27 /26 /C275ffiffiffiffiffiffi27p / 76 /170 /C2739ffiffiffiffiffiffi19p / 28 /1 2(16 /C273ffiffiffiffiffiffi 28p )/ 77 /1 2 (9 /C27ffiffiffiffiffiffi 77p )/ 29 /1 2(5 /C27ffiffiffiffiffiffi 29p )/ 78 /53 /C276ffiffiffiffiffiffi 78p /30 /11 /C272ffiffiffiffiffiffi30p / 79 /80 /C279ffiffiffiffiffiffi79p / 31 /1520 /C27273ffiffiffiffiffiffi31p / 80 /9 /C27ffiffiffiffiffiffi80p / 32 /1 2(6 /C27ffiffiffiffiffiffi 32p )/ 82 /9 /C27ffiffiffiffiffiffi 82p / 33 /23 /C274ffiffiffiffiffiffi 33p / 83 /82 /C279ffiffiffiffiffiffi83p / 34 /35 /C276ffiffiffiffiffiffi34p / 84 /55 /C276ffiffiffiffiffiffi84p / 35 /6 /C27ffiffiffiffiffiffi35p / 85 /1 2 (9 /C27ffiffiffiffiffiffi 85p )/ 37 /6 /C27ffiffiffiffiffiffi 37p / 86 /10405 /C271122ffiffiffiffiffiffi86p / 38 /37 /C276ffiffiffiffiffiffi38p / 87 /28 /C273ffiffiffiffiffiffi87p / 39 /25 /C274ffiffiffiffiffiffi39p / 88 /197 /C2721ffiffiffiffiffiffi88p / 40 /1 2(6 /C27ffiffiffiffiffiffi 40p )/ 89 /500 /C2753ffiffiffiffiffiffi 89p / 41 /32 /C275ffiffiffiffiffiffi41p / 90 /19 /C272ffiffiffiffiffiffi90p / 42 /13 /C272ffiffiffiffiffiffi 42p / 91 /1574 /C27165ffiffiffiffiffiffi 91p / 43 /3482 /C27531ffiffiffiffiffiffi43p / 92 /1 2(48 /C275ffiffiffiffiffiffi 92p )/ 44 /1 2(20 /C273ffiffiffiffiffiffi 44p )/ 93 /1 2(29 /C273ffiffiffiffiffiffi 93p )/ 45 /1 2(7 /C27ffiffiffiffiffiffi 45p )/ 94 /2143295 /C27221064ffiffiffiffiffiffi 94p / 46 /24335 /C273588ffiffiffiffiffiffi46p / 95 /39 /C274ffiffiffiffiffiffi95p / 47 /48/C277ffiffiffiffiffiffi47p / 96 /1 2(10/C27ffiffiffiffiffiffi 96p )/ 48 /7/C27ffiffiffiffiffiffi 48p / 97 /5604/C27569ffiffiffiffiffiffi97p / 50 /7/C27ffiffiffiffiffiffi50p / 98 /99/C2710ffiffiffiffiffiffi98p / 51 /50/C277ffiffiffiffiffiffi51p / 99 /10/C27ffiffiffiffiffiffi99p / 52 /18/C275ffiffiffiffiffiffi 13p / 101 /10/C27ffiffiffiffiffiffiffiffi101p / 53 /1 2(7/C27ffiffiffiffiffiffi 53p )/ 102 /101/C2710ffiffiffiffiffiffiffiffi 102p / See also PELL EQUATION ,REAL QUADRATIC FIELD, UNIT References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Cohn, H. "Fundamental Units" and "Construction of Funda- mental Units." §6.4 and 6.5 in Advanced Number Theory. New York: Dover, pp. 98 /C1/02, and 261 /C1/74, 1980. Hua, L. K. Introduction to Number Theory. Berlin: Springer-Verlag, 1982. Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, p. 192, 1990. LeVeque, W. J. Fundamentals of Number Theory. Reading, MA: Addison-Wesley, 1977. Narkiewicz, W. Elementary and Analytic Number Theory of Algebraic Numbers. Warsaw: Polish Scientific Publishers, 1974. Stark, H. M. An Introduction to Number Theory. Chicago, IL: Markham, 1970. Weisstein, E. W. "Class Numbers." MATHEMATICA NOTE- BOOK CLASS NUMBERS.M . Zucker, I. J. and Robertson, M. M. "Some Properties of Dirichlet L-Series." J. Phys. A: Math. Gen. 9, 1207 /C1/214, 1976. Funnel The funnel surface is a REGULAR SURFACE and SUR- FACE OF REVOLUTION defined by the Cartesian equa- tion Z /C301 2ln x2 /C27y2/C0/C1 (1) and the PARAMETRIC EQUATIONS x(u;v) /C30u cos v (2) y(u;v) /C30u sin v (3) z(u;v) /C30ln u (4) for u /C210 and v /C23 [0;2 p) : The coefficients of the FIRST FUNDAMENTAL FORM are E /C301 /C271 u2 (5) F /C300 (6) G /C30u2 ; (7) the coefficients of the SECOND FUNDAMENTAL FORMare e /C30/C281 uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 u2p (8) f /C300 (9) g /C30uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 u2p ; (10) the AREA ELEMENT is dA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27u2p du ffldv; (11) and the Gaussian and mean curvatures are K /C30/C281 1 /C27 u2 ðÞ2 (12) H /C301 2u 1 /C27 u2 ðÞ3 =2 : (13) Both the surface area and volume of the solid are infinite. See also GABRIEL’S HORN,PSEUDOSPHERE ,SINCLAIR’S SOAP FILM PROBLEM References Gray, A. "The Funnel Surface." Modern Differential Geome- try of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 423 /C1/26, 1997. Fuss’s Problem BICENTRIC POLYGON Futile Game A GAME which permits a draw ("tie") when played properly by both players. See also CATEGORICAL GAME,FAIR GAME,GAME References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 16, 1999. Fuzzy Logic An extension of two-valued LOGIC such that state- ments need not be TRUE or FALSE , but may have a degree of truth between 0 and 1. Such a system can be extremely useful in designing control logic for real- world systems such as elevators. See also ALETHIC ,FALSE ,LOGIC ,TRUE References McNeill, D. Fuzzy Logic: A Practical Approach. New York: Academic Press, 1994. McNeill, D. and Freiberger, P. Fuzzy Logic: The Discovery of a Revolutionary Computer Technology and How It is Changing Our World. New York: Simon and Schuster, 1993. Nguyen, H. T. and Walker, E. A. A First Course in Fuzzy Logic. Boca Raton, FL: CRC Press, 1996. Weisstein, E. W. "Books about Fuzzy Logic." http:// www.treasure-troves.com/books/FuzzyLogic.html. Yager, R. R. and Zadeh, L. A. (Eds.). An Introduction to Fuzzy Logic Applications in Intelligent Systems. Boston, MA: Kluwer, 1992.Zadeh, L. and Kacprzyk, J. (Eds.). Fuzzy Logic for the Management of Uncertainty. New York: Wiley, 1992. FWHM FULLWIDTH AT HALFMAXIMUM G Gabor Function The computer animation format MPEG-7 uses Gabor functions to specify texture descriptors. References Gabor, D. "Theory of Communication." J. Inst. Electr. Engineering, London 93, 429 /C1/57, 1946. Hubbard, B. B. The World According to Wavelets: The Story of a Mathematical Technique in the Making, 2nd rev. upd. ed. New York: A. K. Peters, pp. 26, 28, and 187 /C1/88, 1998. International Organisation for Standardisation. "MPEG-7 Frequently Asked Questions." http://www.cselt.it/mpeg/ faq/faq_mpeg-7.htm. Gabriel’s Horn The SURFACE OF REVOLUTION of the function y /C301=x about the X-AXIS for x ]1: It has FINITE VOLUME V /C30g/C12 1py2 dx /C30 pg/C12 1dx x2 /C30 p /C281 x"#/C12 1/C30 p[0 /C28(/C281)] /C30 p; but INFINITE SURFACE AREA , since S /C30g/C12 12pyffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y ?2q dx /C212pg/C12 1ydx/C302pg/C12 1dx x/C302p[ln x] /C12 1 /C302 p[ln /C12/C280] /C30/C12: This leads to the paradoxical consequence that while Gabriel’s horn can be filled up with p cubic units of paint, an INFINITE number of square units of paint are needed to cover its surface! See also FUNNEL ,PSEUDOSPHEREGabriel’s Staircase The SUM X/C12 k /C301krk /C30r (1 /C28 r)2 ; valid for 0 Br B1:/ Gadget A term of endearment used by ALGEBRAIC TOPOLO- GISTS when talking about their favorite power tools such as ABELIAN GROUPS , BUNDLES , HOMOLOGY GROUPS , HOMOTOPY GROUPS , K-THEORY ,M ORSE THE- ORY, OBSTRUCTIONS , stable homotopy theory, VECTOR SPACES , etc. See also ABELIAN GROUP ,A LGEBRAIC TOPOLOGY , BUNDLE ,F REE,H OMOLOGY GROUP ,H OMOTOPY GROUP , K-THEORY ,O BSTRUCTION ,M ORSE THEORY , VECTOR SPACE References Page, W. Topological Uniform Structures. New York: Dover, 1994. Galerkin Method A method of determining coefficients akin a power series solution y(x) /C30y0(x) /C27Xn k /C301ak yk(x) of the ORDINARY DIFFERENTIAL EQUATION L[y(x)] /C300 so that the DIFFERENTIAL OPERATOR L[y(x)] is ortho- gonal to every yk(x) for k /C301, ..., n. References Itoˆ, K. (Ed.). "Methods Other than Difference Methods." §303I in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 1139, 1980. Gale-Ryser Theorem Let p and q be PARTITIONS of a POSITIVE INTEGER , then there exists a (0,1)-matrix (i.e., a BINARY MATRIX ) such that c() /C30p ; r() /C30q IFF q is dominated by p/C31:/ See also BINARY MATRIX ,PARTITION References Brualdi, R. and Ryser, H. J. §6.2.4 in Combinatorial Matrix Theory. New York: Cambridge University Press, 1991. Krause, M. "A Simple Proof of the Gale-Ryser Theorem." Amer. Math. Monthly 103, 335/C1/37, 1996. Robinson, G. §1.4 in Representation Theory of the Symmetric Group. Toronto, Canada: University of Toronto Press, 1961. Ryser, H. J. "The Class A(R;S):/"Combinatorial Mathe- matics. Buffalo, NY: Math. Assoc. Amer., pp. 61 /C1/5, 1963. Galilean Transformation A transformation from one reference frame to another moving with a constant VELOCITY v with respect to the first for classical motion. However, special rela- tivity shows that the transformation must be mod- ified to the LORENTZ TRANSFORMATION for relativistic motion. The forward Galilean transformation is t? x? y? z ?2 6643 775/C301000 /C28v 100 0010 00012 6643 775t x y z2 6643 775; and the inverse transformation is t x y z2 6643 775/C301000 v 100 0010 00012 6643 775t? x? y? z?2 6643 775: See also L ORENTZ TRANSFORMATION Gall Isographic Projection A CYLINDRICAL EQUIDISTANT PROJECTION with stan- dard parallel f1 /C3045/C14:/ See also CYLINDRICAL EQUIDISTANT PROJECTIONGall Orthographic Projection A CYLINDRICAL EQUAL-AREA PROJECTION with stan- dard parallel of 458. See also BALTHASART PROJECTION ,BEHRMANN CY- LINDRICAL EQUAL- AREA PROJECTION ,C YLINDRICAL EQUAL- AREA PROJECTION ,EQUAL- AREA PROJECTION , GALL ISOGRAPHIC PROJECTION ,LAMBERT AZIMUTHAL EQUAL- AREA PROJECTION ,P ETERS PROJECTIO N, STEREOGRAPHIC PROJECTION ,TRISTAN EDWARDS PRO- JECTION References Dana, P. H. "Map Projections." http://www.colorado.edu/ geography/gcraft/notes/mapproj/mapproj_f.html. Gall, J. "Uses of Cylindrical Projections for Geographical, Astronomical, and Scientific Purposes." Scottish Geogra- phical Mag. 1, 119 /C1/23, 1885. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 76, 1987. Gall Stereographic Projection GALL ORTHOGRAPHIC PROJECTION Gallows Schroeder (1991) calls the CEILING FUNCTION symbols /C26 and /C27 the "gallows" because of their similarity in appearance to the structure used for hangings. See also CEILING FUNCTION References Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 57, 1991. Gallucci’s Theorem If three SKEW LINES all meet three other SKEW LINES , any TRANSVERSAL to the first set of three meets any TRANSVERSAL to the second set of three. See also SKEW LINES,TRANSVERSAL LINE Galois Extension This entry contributed by N ICOLAS BRAY An extension F of a field K is said to be a Galois extension of K, if for every x /C23 F /C28K ; there is an element of the GALOIS GROUP of the extension which does not fix x (i.e., there exits s /C23 AutKF such that s(x) "x)):/ See also GALOIS EXTENSION FIELD Galois Extension Field If K is the SPLITTING FIELD over a FIELD F of a separable POLYNOMIAL f(x) ; then the EXTENSION FIELD K =F is a Galois extension field. See also EXTENSION FIELD ,G ALOIS EXTENSION , SPLITTING FIELD References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 475 /C1/76, 1998. Galois Field FINITE FIELD Galois Group Let L be a FIELD EXTENSION of K, denoted L=K ; and let G be the set of AUTOMORPHISMS of L =K ; that is, the set of AUTOMORPHISMS s of L such that s(x) /C30x for every x /C23 K ; so that K is fixed. Then G is a GROUP of transformations of L, called the Galois group of L=K :/ The Galois group of (C =R) consists of the IDENTITY ELEMENT and COMPLEX CONJUGATION . These func- tions both take a given REAL to the same real. See also ABHYANKAR’S CONJECTURE ,FINITE GROUP , GROUP References Birkhoff, G. and Mac Lane, S. "The Galois Group." §15.2 in A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 397 /C1/01, 1996. Jacobson, N. Basic Algebra I, 2nd ed. New York: W. H. Freeman, p. 234, 1985. Galois Imaginary A mathematical object invented to solve irreducible CONGRUENCES OF THE FORM F(x) /C130 (mod p) ; where p is PRIME . Galois Theory If there exists a ONE-TO-ONE correspondence between two SUBGROUPS and SUBFIELDS such that G(E(G?)) /C30G ? E(G(E ?)) /C30E ?; then E is said to have a Galois theory. See also ABEL’S IMPOSSIBILITY THEOREM ,SUBFIELDReferences Artin, E. Galois Theory, 2nd ed. Notre Dame, IN: Edwards Brothers, 1944. Birkhoff, G. and Mac Lane, S. "Galois Theory." Ch. 15 in A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 395 /C1/21, 1996. Dummit, D. S. and Foote, R. M. "Galois Theory." Ch. 14 in Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice- Hall, pp. 471 /C1/70, 1998. Galois’s Theorem An algebraic equation is algebraically solvable IFFits GROUP isSOLVABLE . In order that an irreducible equation of PRIME degree be solvable by radicals, it isNECESSARY and SUFFICIENT that all its ROOTS be rational functions of two ROOTS . See also ABEL’S IMPOSSIBILITY THEOREM ,SOLVABLE GROUP Galoisian An algebraic extension EofFfor which every IRREDUCIBLE POLYNOMIAL inFwhich has a single ROOT inEhas allits ROOTS inEis said to be Galoisian. Galoisian extensions are also called algeb- raically normal. Gambler’s Ruin Let two players each have a finite number of pennies (say, n1for player one and n2for player two). Now, flip one of the pennies (from either player), with each player having 50% probability of winning, and give the penny to the winner. Now repeat the process untilone player has all the pennies. If the process is repeated indefinitely, the probability that oneof the two player will eventually lose all his pennies must be 100%. In fact, the chances P 1andP2 that players one and two, respectively, will berendered penniless are P 1/C30n2 n1/C27n2 P2/C30n1 n1/C27n2; i.e., your chances of going bankrupt are equal to the ratio of pennies your opponent starts out to the totalnumber of pennies. Therefore, the player starting out with the smallest number of pennies has the greatest chance of going bankrupt. Even with equal odds, the longer yougamble, the greater the chance that the playerstarting out with the most pennies wins. Since casinos have more pennies than their individual patrons, this principle allows casinos to alwayscome out ahead in the long run. And the common practice of playing games with odds skewed in favor of the house makes this outcome just that much quicker. See also COIN TOSSING ,M ARTINGALE ,SAINT PETERS- BURG PARADOX References Cover, T. M. "Gambler’s Ruin: A Random Walk on the Simplex." §5.4 in Open Problems in Communications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, p. 155, 1987. Hajek, B. "Gambler’s Ruin: A Random Walk on the Simplex." §6.3 in Open Problems in Communications and Computa- tion. (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, pp. 204 /C1/07, 1987. Kraitchik, M. "The Gambler’s Ruin." §6.20 in Mathematical Recreations. New York: W. W. Norton, p. 140, 1942. Game A game is defined as a conflict involving gains and losses between two or more opponents who follow formal rules. The study of games belongs to a branch of mathematics known as GAME THEORY . See also BOARD ,CARDS ,CATEGORICAL GAME,DRAW, FAIR GAME,F INITE GAME,F UTILE GAME,G AME THEORY ,HYPERGAME ,UNFAIR GAME References Falkener, E. Games Ancient and Oriental and How to Play Them. New York: Dover, 1961. Sackson, S. A Gamut of Games. New York: Random House, 1969. University of Waterloo. "Museum and Archive of Games." http://www.ahs.uwaterloo.ca/~museum/. Game Expectation Let the elements in a PAYOFF MATRIX be denoted aij ; where the is are player A’s STRATEGIES and the js are player B’s STRATEGIES . Player A can get at least min j5naij (1) for STRATEGY i. Player B can force player A to get no more than maxj5m aijfor a STRATEGY j. The best STRATEGY for player A is therefore max i5mmin j5naij ; (2) and the best STRATEGY for player B is min j5nmax i5maij : (3) In general, max i 5mmin j5naij 5min j5nmax i5maij : (4) Equality holds only if a SADDLE POINT is present, in which case the quantity is called the VALUE of the game. See also GAME,P AYOFF MATRIX ,S ADDLE POINT (GAME), STRATEGY ,VALUEGame Matrix PAYOFF MATRIX Game of Life LIFE Game Theory A branch of MATHEMATICS and LOGIC which deals with the analysis of GAMES (i.e., situations involving parties with conflicting interests). In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real- world problems as diverse as economics, property division, politics, and warfare. See also BOREL DETERMINACY THEOREM ,CATEGORI- CAL GAME,C HECKERS ,C HESS ,D ECISION THEORY , EQUILIBRIUM POINT ,F INITE GAME,F UTILE GAME, GAME EXPECTATION ,G O,H I-Q, IMPARTIAL GAME, MEX,M INIMAX THEOREM ,M IXED STRATEGY ,N ASH EQUILIBRIUM ,N ASH’S THEOREM ,N IM,N IM-VALUE , PARTISAN GAME,PAYOFF MATRIX ,PEG SOLITAIRE , PERFECT INFORMATION ,SADDLE POINT (GAME), SAFE, SPRAGUE- GRUNDY FUNCTION ,STRATEGY ,TACTIX ,TIT- FOR-TAT,U NSAFE ,VALUE ,W YTHOFF’S GAME,ZERO- SUM GAME References Ahrens, W. Mathematische Unterhaltungen und Spiele. Leipzig, Germany: Teubner, 1910. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Games in General. London: Academic Press, 1982. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games inParticular. London: Academic Press, 1982. Conway, J. H. On Numbers and Games. New York: Aca- demic Press, 1976. Dresher, M. The Mathematics of Games of Strategy: Theory and Applications. New York: Dover, 1981. Eppstein, D. "Combinatorial Game Theory." http://www.ic- s.uci.edu/~eppstein/cgt/. Gardner, M. "Game Theory, Guess It, Foxholes." Ch. 3 in Mathematical Magic Show: More Puzzles, Games, Diver-sions, Illusions and Other Mathematical Sleight-of-Mindfrom Scientific American. New York: Vintage, pp. 35 /C1 /9, 1978. Gardner, R. Games for Business and Economics. New York: Wiley, 1994. Isaacs, R. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Opti-mization. New York: Dover, 1999. Karlin, S. Mathematical Methods and Theory in Games, Programming, and Economics, 2 Vols. Vol. 1: MatrixGames, Programming, and Mathematical Economics.Vol. 2: The Theory of Infinite Games. New York: Dover, 1992. Kuhn, H. W. (Ed.). Classics in Game Theory. Princeton, NJ: Princeton University Press, 1997. McKinsey, J. C. C. Introduction to the Theory of Games. New York: McGraw-Hill, 1952. Me´ro¨, L. Moral Calculations: Game Theory, Logic and Human Frailty. New York: Springer-Verlag, 1998. Neumann, J. von and Morgenstern, O. Theory of Games and Economic Behavior, 3rd ed. New York: Wiley, 1964. Packel, E. The Mathematics of Games and Gambling. Washington, DC: Math. Assoc. Amer., 1981. Stahl, S. A Gentle Introduction to Game Theory. Providence, RI: Amer. Math. Soc., 1999. Straffin, P. D. Jr. Game Theory and Strategy. Washington, DC: Math. Assoc. Amer., 1993. Vajda, S. Mathematical Games and How to Play Them. New York: Routledge, 1992. Walker, P. "An Outline of the History of Game Theory." http://william-king.www.drexel.edu/top/class/histf.html. Weisstein, E. W. "Books about Game Theory." http:// www.treasure-troves.com/books/GameTheory.html. Williams, J. D. The Compleat Strategyst, Being a Primer on the Theory of Games of Strategy. New York: Dover, 1986. Gamma GAMMA FUNCTION ,INCOMPLETE GAMMA FUNCTION Gamma Distribution A general type of STATISTICAL DISTRIBUTION which is related to the BETA DISTRIBUTION and arises naturally in processes for which the waiting times between POISSON DISTRIBUTED events are relevant. Gamma distributions have two free parameters, labeled aand u;a few of which are illustrated above. Given a P OISSON DISTRIBUTION with a rate of change l;the DISTRIBUTION FUNCTION D(x) giving the waiting times until the hth Poisson event is D(x)/C30P(X5x)/C301/C28P(x>x)/C301/C28Xh/C281 k/C300(lx)ke/C28lx k! /C301/C28e/C28lxXh/C281 k/C300(lx)k k!/C301/C28G(h;xl) G(h)(1) forx/C23[0;/C12);where G(x) is a complete GAMMA FUNC- TION , and G(a;x)a n INCOMPLETE GAMMA FUNCTION . With han integer, this distribution is a DISCRETE DISTRIBUTION known as the E RLANG DISTRIBUTION . The probability function P(x) is then obtained by differentiating D(x);P(x)/C30D?(x)/C30le/C28lxXh/C281 k/C300(lx)k k!/C28e/C28lxXh/C281 k/C300k(lx)k/C281l k! /C30le/C28lx/C27le/C28lxXh/C281 k/C301(lx)k k!/C28e/C28lxXh/C281 k/C301k(lx)k/C281l k! /C30le/C28lx/C28le/C28lxXh/C281 k/C301k(lx)k/C281 k!/C28(lx)k k!"# /C30le/C28lx1/C28Xh/C281 k/C301(lx)k/C281 (k/C281)!/C28(lx)k k!"#() /C30le/C28lx1/C281/C28(lx)h/C281 (h/C281)!"#() /C30l(lx)h/C281 (h/C281)!e/C28lx:(2) Now let a/C13h(not necessarily an integer) and define u/C131=lto be the time between changes. Then the above equation can be written P(x)xa/C281e/C28x=u G(a)ua(3) forx/C23[0;/C12):The CHARACTERISTIC FUNCTION describ- ing this distribution is f(t)/C30Fx/C28x=uxa/C281 G(a)ua[1 2(1/C27sgnx)]() /C30(1/C28itu)/C28a;(4) where F[f] is the F OURIER TRANSFORM with para- meters a/C30b/C301;and the MOMENT-GENERATING FUNC- TION is M(t)/C30g/C12 0etxxa/C281e/C28x=udx G(a)ua/C30g/C12 0xa/C281e/C28(1/C28ut)x=udx G(a)ua:(5) giving moments about 0 of m?r/C30urG(a/C27r) G(a)(6) (Papoulis 1984, p. 147). In order to explicitly find the MOMENTS of the distribution using the MOMENT-GENERATING FUNC- TION , let y/C13(1/C28ut)x u(7) dy/C301/C28ut udx; (8) so M(t)/C30g/C12 0uy 1/C28ut !a/C281e/C28y G(a)uaudy 1/C28ut /C301 (1/C28ut)aG(a)g/C12 0ya/C281e/C28ydy /C301 (1/C28ut)a; (9) giving the logarithmic MOMENT-GENERATING FUNC- TION as R(t)/C13lnM(t)/C30/C28aln(1/C28ut) (10) R?(t)/C30au 1/C28ut(11) Rƒ(t)/C30au2 (1/C28ut)2: (12) The MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS are then m/C30R?(0)/C30au (13) s2/C30Rƒ(0)au2(14) g1/C302ffiffiffiap (15) g2/C306 a: (16) The gamma distribution is closely related to other statistical distributions. If X1;X2;...,Xnare indepen- dent random variates with a gamma distribution having parameters ( a1;u);(a2;u);..., (an;u);then an i/C301Xiis distributed as gamma with parameters a/C30Xn i/C301ai (17) u/C30u: (18) Also, if X1and X2are independent random variates with a gamma distribution having parameters ( a1;u) and ( a2;u);then X1=(X1/C27X2)i sa BETA DISTRIBUTION variate with parameters ( a1;a2):Both can be derived as follows. P(x;y)/C301 G(a1)G(a2)ex1/C27x2xa1/C281 1xa2/C281 2: (19) Let u/C30x1/C27x2 x1/C30uv (20) v/C30x1 x1/C27x2x2/C30u(1/C28v); (21) then the J ACOBIAN is Jx1;x2 u;v ! /C30vu 1/C28v/C28u=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n/C30/C28u; (22) so g(u;v)du dv/C30f(x;y)dx dy/C30f(x;y)ud ud v :(23)g(u;v)/C30u G(a1)G(a2)e/C28u(uv)a1/C281ua2/C281(1/C28v)a2/C281 /C301 G(a1)G(a2)e/C28uua1/C27a2/C281va1/C281(1/C28v)a2/C281:(24) The sum X1/C27X2therefore has the distribution f(u)/C30f(x1/C27x2)/C30g1 0g(u;v)dv/C30e/C28uua1/C27a2/C281 G(a1/C27a2);(25) which is a gamma distribution, and the ratio X1=(X1/C27X2) has the distribution h(v)/C30hx1 x1/C27x2 ! /C30g/C12 0g(u;v)du /C30va1/C281(1/C28v)a2/C281 B(a1;a2); (26) where Bis the BETA FUNCTION , which is a BETA DISTRIBUTION . IfXand Yare gamma variates with parameters a1 and a2;theX=Yis a variate with a BETA PRIME DISTRIBUTION with parameters a1anda2:Let u/C30x/C27yv /C30x y; (27) then the J ACOBIAN is Ju;v x;y ! /C3011 1 y/C28x y2=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n/C30/C28 x/C27y y2/C30/C28(1/C27v)2 u; (28) so dx dy/C30u (1/C27v)2du dv (29) g(u;v)/C301 G(a1)G(a2)e/C28uuv 1/C27v !a1/C281u 1/C27v !a2/C281 /C2u (1/C27v)2 /C301 G(a1)G(a2)e/C28uua1/C27a2/C281va2/C281(1/C27v)/C28a1/C28a2:(30) The ratio X=Ytherefore has the distribution h(v)/C30g/C12 0(g(u;v)du/C30va1/C281(1/C27v)/C28a1/C28a2 B(a1;a2); (31) which is a BETA PRIME DISTRIBUTION with parameters (a1;a2):/ The "standard form" of the gamma distribution is given by letting y/C13x=u;sody/C30dx=uand P(y)dy/C30xa/C281e/C28x=u G(a)uadx/C30(uy)a/C281e/C28y G(a)ua(udy) /C30ya /C281e /C28y G( a)dy; (32) so the MOMENTS about 0 are vr /C301 G( a) g/C12 0e /C28xxa/C281 /C27r dx /C30G( a /C27 r) G(a)/C30(a)r ; (33) where (a)ris the POCHHAMMER SYMBOL . The MO- MENTS about m /C30 m1 are then m1 /C30 a (34) m2 /C30 a (35) m3 /C302a (36) m4 /C303 a2 /C276 a: (37) The MOMENT-GENERATING FUNCTION is M(t) /C301 (1 /C28 t) a ; (38) and the CUMULANT-GENERATING FUNCTION is K(t) /C30 a ln(1 /C28t) /C30 a(t /C271 2 t2 /C2713 t3 /C27...); (39) so the CUMULANTS are kr /C30 aG(r) : (40) If x is a NORMAL variate with MEAN m and STANDARD DEVIATION s;then y/C13(x/C28m)2 2s2(41) is a standard gamma variate with parameter a/C301=2:/ See also BETA DISTRIBUTION ,CHI-SQUARED DISTRIBU- TION ,ERLANG DISTRIBUTION References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 534, 1987. Jambunathan, M. V. "Some Properties of Beta and Gamma Distributions." Ann. Math. Stat. 25, 401/C1/05, 1954. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 103 /C1/04, 1984.Gamma Function The complete gamma function G(n) is defined to be an extension of the FACTORIAL toCOMPLEX and REAL NUMBER arguments. It is related to the FACTORIAL by G(n)/C30(n/C301)!:It is ANALYTIC everywhere except at z/C300,/C281,/C282, ..., and the residue at z/C30/C28kis Res z/C30/C28kG(z)/C30(/C281)k k!: (1) There are no points zat which G(z)/C300:The gamma function is implemented in Mathematica asGam- ma[z]. The gamma function can be defined as a DEFINITE INTEGRAL forR[z]>0 (Euler’s integral form) G(z)/C13g/C12 0tz/C281e/C28tdt (2) /C302g/C12 0e/C28t2t2z/C281dt; (3) or G(z)/C13g1 0ln1 t !"#z/C281 dt: (4) Plots of the real and imaginary parts of G(z) in the complex plane are illustrated above. INTEGRATING (2) by parts for a REAL argument, it can be seen that G(x)/C30g/C12 0tx/C281e/C28tdt /C30[/C28tx/C281e/C28t]/C12 0/C27g/C12 0(x/C281)tx/C282e/C28tdt /C30(x/C281)g/C12 0tx/C282e/C28tdt/C30(x/C281)G(x/C281): (5) Ifxis an INTEGER n/C301, 2, 3, ... then G(n)/C30(n/C281)G(n/C281)/C30(n/C281)(n/C282)G(n/C282) /C30(n/C281)(n/C282)/C1/C1/C11/C30(n/C281)!; (6) so the gamma function reduces to the FACTORIAL for a POSITIVE INTEGER argument. The second of B INET’S LOG GAMMA FORMULAS is lnG(a)/C30(a/C281 2)lna/C28a/C2712ln(2p) /C272g/C12 0tan(z a) e2pz/C281dz (7) forR[a]>0 (Whittaker and Watson 1990, p. 251). Another formula for ln G(z) is given by M ALMSTE ´N’S FORMULA , and ln G(z) is implemented in Mathematica asLogGamma [z]. The gamma function can also be defined by an INFINITE PRODUCT form (Weierstrass Form) G(z)/C13zegzY/C12 r/C3011/C27z r ! e/C28z=r"#/C281 ; (8) where gis the E ULER- MASCHERONI CONSTANT (Krantz 1999, p. 157). This can be written G(z)/C301 zexpX/C12 k/C301(/C281)ksk kzk"# ; (9) where s1/C13g (10) sk/C13z(k) (11) fork]2;where z(z) is the R IEMANN ZETA FUNCTION (Finch). Taking the logarithm of both sides of (8), /C28ln[G(z)]/C30lnz/C27gz/C27X/C12 n/C301ln 1/C27z n ! /C28z n"# : (12) Differentiating, /C28G?(z) G(z)/C301 z/C27g/C27X/C12 n/C3011 n 1/C27z n/C281 n0 BBB@1 CCCA/C301 z/C27g/C27X/C12 n/C3011 n/C27z/C281 n ! (13) G?(z)/C30/C28G(z)1 z/C27g/C27X/C12 n/C3011 n/C27z/C281 n !"# (14) /C13G(z)C(z)/C30G(z)c0(z) (15) G?(1)/C30/C28G(1) /C281/C27g/C27(1 2/C281)/C27(13/C2812)/C27.../C271 n/C271/C281 n ! /C27..."#() /C30/C28(1/C27g/C281)/C30/C28g (16) G?(n)/C30/C28G(n) /C21 n/C27g/C271 1/C27n/C281 ! /C271 2/C27n/C281 2 ! "( /C271 3/C27n/C281 3 ! /C27...=zn1=zn+ /C30/C28(n/C281)!1 n/C27g/C28Xn k/C3011 k ! ; (17) where C(z) is the DIGAMMA FUNCTION andc0(z) is the POLYGAMMA FUNCTION .nth derivatives are given in terms of the POLYGAMMA FUNCTIONS cn;cn/C281;...,c0:/ The minimum value x0ofG(x) for REAL POSITIVE x/C30x0 is achieved when G?(x0)/C30G(x0)c0(x0)/C300 (18) c0(x0)/C300; (19) This can be solved numerically to give x0/C301:46163 . . . (Sloane’s A030169; Wrench 1968), which has CONTIN- UED FRACTION [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...] (Sloane’s A030170). At x0;G(x0) achieves the value 0.8856031944... (Sloane’s A030171), which has CON- TINUED FRACTION [0, 1, 7, 1, 2, 1, 6, 1, 1, ...] (Sloane’s A030172). The Euler limit form is 1 G(z)/C30zlim m0/C12e(1/C271=2/C27.../C271=m/C28lnm)zhi /C2lim m0/C12Ym n/C3011/C27z n ! e/C28z=n()"# /C301 zY/C12 n/C3011/C271 n !z 1/C27z n !/C2812 435; (20) so G(z)/C13lim n0/C121 /C2152 /C2153/C1/C1/C1n z(z/C271)(z/C272)/C1/C1/C1(z/C27n)nz(21) (Krantz 1999, p. 156). One over the gamma function is also given by 1 G(z)/C30zexpgz/C28X/C12 k/C302(/C281)kz(k)zk k"# ; (22) where gis the E ULER- MASCHERONI CONSTANT andz(z) is the R IEMANN ZETA FUNCTION (Wrench 1968). An ASYMPTOTIC SERIES for /1=G(z)/is given by 1 G(z)/C2z/C27gz2/C271 12(6g2/C28p2)z3/C271 12[2g3/C28gp2/C274z(3)]z4 /C27...: (23) Writing 1 G(z)/C30X/C12 k/C301akzk; (24) theaksatisfy an/C30na1an/C28a2an/C281/C27Xn k/C302(/C281)kz(k)an/C28k (25) (Bourget 1883, Isaacson and Salzer 1942, Wrench 1968). Wrench (1968) numerically computed thecoefficients for the series expansion about 0 of 1 z(1/C27z)G(z) /C301/C27(g/C281)z/C271/C271 2(g/C282)g/C281 12p2hi z2/C27...:(26) The L ANCZOS APPROXIMATION forz/C210i s G(z/C271)/C30(z/C27g/C271 2)z/C271=2ez/C27g/C271=2ffiffiffiffiffiffi 2pp /C29c0/C27c1 z/C271/C27c2 z/C272/C27.../C27cn z/C27n/C27o"# ; (27) where gis the E ULER- MASCHERONI CONSTANT . The gamma function satisfies the FUNCTIONAL EQUA- TIONS G(1/C27z)/C30zG(z) (28) G(1/C28z)/C30/C28zG(/C28z): (29) Additional identities are G(x)G(/C28x)/C30/C28p xsin(px)(30) G(x)G(1/C28x)/C30p sin(px)(31) ln[G(x/C27iy/C271)] /C30ln(x2/C27y2)/C27itan/C281y x ! /C27ln[G(x/C27iy)] (32) ½(ix)!½2/C30px sinh( px)(33)½(n/C27ix)!½/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px sinh( px)s Yn s/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2/C27x2p : (34) For integer n/C301, 2, ..., the first few values of G(n) are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... (Sloane’s A000142). For half integer arguments, / G(n=2)/has the special form G1 2n=z1*=z1+ /C30(n/C282)!!ffiffiffipp 2(n/C281)=2; (35) where n!! is a DOUBLE FACTORIAL . The first few values forn/C301, 3, 5, ..., are therefore G(1 2)/C30ffiffiffipp(36) G(3 2)/C3012ffiffiffipp(37) G(5 2)/C3034ffiffiffipp; (38) /15ffiffiffipp=8;105ffiffiffipp=16 /, ... (Sloane’s A001147 and A000079; Wells 1986, p. 40). In general, for na POSITIVE INTEGER n/C301, 2, ... G1 2/C27n=z1*=z1+ /C301 /C2153 /C2155/C1/C1/C1(2n/C281) 2nffiffiffipp /C30(2n/C281)!! 2nffiffiffipp(39) G(1 2/C28n)/C30(/C281)n2n 1 /C2153 /C2155/C1/C1/C1(2n/C281)ffiffiffipp /C30(/C281)n2n (2n/C281)!!ffiffiffipp: (40) For /R[x]/C30/C281 2/, ½(/C281 2/C27iy)!½2/C30p cosh( py): (41) Gamma functions of argument 2 zcan be expressed using the L EGENDRE DUPLICATION FORMULA G(2z)/C30(2p)/C281=222z/C281=2G(z)G(z/C2712): (42) Gamma functions of argument 3 zcan be expressed using a triplication FORMULA G(3z)/C30(2p)/C28133z/C281=2G(z)G(z/C271 3)G(z/C2723): (43) The general result is the G AUSS MULTIPLICATION FORMULA G(z)G(z/C271 n)/C1/C1/C1G(z/C27n/C281 n)/C30(2p)(n/C281)=2n1=2/C28nzG(nz):(44) The gamma function is also related to the R IEMANN ZETA FUNCTION z(z)b y Gs 2 ! p/C28s=2z(s)/C30G1/C28s 2 ! p/C28(1/C28s)=2z(1/C28s): (45) Borwein and Zucker (1992) give a variety of identities relating gamma functions to square roots and ELLIP- TIC INTEGRAL SINGULAR VALUES /kn/, i.e., MODULI /kn/ such that K?(kn) K(kn)/C30ffiffiffinp; (46) where K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND and /K?(k)/C30K(k)?/C30K(ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p )/is the com- plementary integral. M. Trott has developed an algo- rithm for automatically generating hundreds of suchidentities. G( 1 3)/C3027=93/C281=12p1=3[K(k3)]1=3(47) G(14)/C302p1=4[K(k1)]1=2(48) G(16)/C302/C281=331=2p/C281=2[G(13)]2(49) G(18)G(38)/C30(ffiffiffi 2p /C281)1=2213=4p1=2K(k2) (50) G(1 8) G(3 8)/C302(ffiffiffi 2p /C271)1=2p/C281=4[K(k1)]1=2(51) G(1 12)/C302/C281=433=8(ffiffiffi 3p /C271)1=2p/C281=2G(1 4)G(13) (52) G(5 12)/C3021=43/C281=8(ffiffiffi 3p /C281)1=2p1=2G(1 4) G(1 3)(53) G(1 24)G(11 24) G(5 24)G(7 24)/C30ffiffiffi 3pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 3pq (54) G(1 24)G(5 24) G(7 24)G(11 24)/C304/C21531=4(ffiffiffi 3p /C27ffiffiffi 2p )p/C281=2K(k1) (55) G(1 24)G(7 24) G(5 24)G(11 24)/C30225=1831=3(ffiffiffi 2p /C271)p/C281=3[K(k3)]2=3(56) G(1 24)G(5 24)G(7 24)G(11 24) /C30384(ffiffiffi 2p /C271)(ffiffiffi 3p /C28ffiffiffi 2p )(2/C28ffiffiffi 3p )p[K(k6)]2(57) G(1 10)/C302/C287=1051=4(ffiffiffi 5p /C271)1=2p/C281=2G(1 5)G(25) (58) G(3 10)/C302/C283=5(ffiffiffi 5p /C281)p1=2G(1 5) G(2 5)(59) G(1 15)G(4 15)G(7 15) G(2 15)/C302/C21531=251=6sin(2 15p)[G(1 3)]2(60) G(1 15)G(2 15)G(7 15) G(4 15)/C3022/C21532=5sin(1 5p) sin(4 15p)[G(15)]2(61)G(2 15)G(4 15)G(7 15) G(1 15)/C302/C283=23/C281=551=4(ffiffiffi 5p /C281)1=2[G(2 5)]2 sin(4 15p)(62) G(1 15)G(2 15)G(4 15) G(7 15)/C3060(ffiffiffi 5p /C281) sin(7 15p)[K(k15)]2(63) G(1 20)G(9 20) G(3 20)G(7 20)/C302/C28151=4(ffiffiffi 5p /C271) (64) G(1 20)G(3 20) G(7 20)G(9 20)/C3024=5(10/C282ffiffiffi 5p )1=2p/C281sin(7 20p) sin(9 20p) /C2[G(1 5)]2(65) G(1 20)G(7 20) G(3 20)G(9 20)/C3023=5(10/C272ffiffiffi 5p )1=2p/C281sin(3 20p) sin(9 20p) /C2[G(2 5)]2(66) G(1 20)G(3 20)G(7 20)G(9 20)/C30160(ffiffiffi 5p /C282)1=2p[K(k5)]2:(67) Several of these are also given in Campbell (1966, p. 31). A few curious identities include Y8 n/C301G1 3n=z1*=z1+ /C30640 36pffiffiffi 3p !3 (68) [G1 4=z1*=z1+ ]4 16p2/C3032 32/C28152/C281 5272 72/C281/C1/C1/C1 (69) G?(1) G(1)/C28G?12=z1*=z1+ G1 2=z1*=z1+/C302 ln 2 (70) (Magnus and Oberhettinger 1949, p. 1). Ramanujan also gave a number of fascinating identities: G2(n/C271) G(n/C27xi/C271)G(n/C28xi/C271)/C30Y/C12 k/C3011/C27x2 (n/C27k)2"# (71) f(m;n)f(n;m)/C30G3(m/C271)G3(n/C271) G(2m/C27n/C271)G(2n/C27m/C271) /C29cosh p(m/C27n)ffiffiffi 3p=zn=zo /C28cos[p(m/C28n)] 2p2(m2/C27mn/C27n2); (72) where f(m;n)/C13Y/C12 k/C3011/C27m/C27n k/C27m !32 435; (73) Y /C12 k/C3011/C27n k !32435Y /C12 k/C3011/C273n n/C272k !22435 /C30G1 2 n=z1*=z1+ G1 2(n /C27 1)hicosh pnffiffiffi 3p=z;=z1 /C28 cos(pn) 2n/C272 p3 =2n (74) (Berndt 1994). Ramanujan gave the infinite sums 1 /C2791 4=z1*=z1+4 /C27171 /C215 5 4 /C215 8 !4 /C27251 /C215 5 /C215 9 4 /C215 8 /C215 12 !4 /C27... /C30X/C12 k/C300(8k /C271)G k /C271 4=z1*=z1+ k!G1 4=z1*=z1+2 4354 /C3023=2 ffiffiffippG3 4=z1*=z1+hi2 (75) and 1 /C2851 2=z1*=z1+5 /C2791 /C215 3 2 /C215 4 !5 /C28131 /C215 3 /C215 5 2 /C215 4 /C215 6 !5 /C27... /C30X/C12 k /C300(/C281)k(4k /C271)(2k /C28 1)!! (2k)!!"#5 /C302 G34=z1*=z1+hi4 : (76) (Hardy 1923; Hardy 1924; Whipple 1926; Watson 1931; Bailey 1935; Hardy 1999, p. 7). The following ASYMPTOTIC SERIES is occasionally useful in probability theory (e.g., the 1-D RANDOM WALK ): G J /C2712=z1*=z1+ G(J) /C30ffiffiffiffi Jp 1 /C281 8J /C271 128J2 /C275 1024 J3 /C2821 32768 J4 /C27... ! (77) (Graham et al. 1994). This series also gives a nice asymptotic generalization of STIRLING NUMBERS OF THE FIRST KIND to fractional values. It has long been known that G(1 4) p/C281 =4 is TRANSCEN- DENTAL (Davis 1959), as is G(1 3) (Le Lionnais 1983), and Chudnovsky has apparently recently proved that G(1 4) is itself TRANSCENDENTAL . The complete gamma function G(x) can be generalized to the upper INCOMPLETE GAMMA FUNCTION G(a;x) and lower INCOMPLETE GAMMA FUNCTION g(a;x):/ See also BAILEY’S THEOREM ,BARNES’ G-FUNCTION , BINET’S FIBONACCI NUMBER FORMULA ,B OHR- MOL- LERUP THEOREM ,DIGAMMA FUNCTION ,DOUBLE GAM- MA FUNCTION ,FRANSE ´ N-ROBINSON CONSTANT GAUSS MULTIPLICATION FORMULA ,INCOMPLETE GAMMA FUNCTION ,K NAR’S FORMULA ,L AMBDA FUNCTION , LANCZOS APPROXIMATION ,L EGENDRE DUPLICATION FORMULA ,M ALMSTE ´ N’S FORMULA ,M ELLIN’S FORMU- LA,M U FUNCTION ,NU FUNCTION ,PEARSON’S FUNC- TION ,POLYGAMMA FUNCTION ,REGULARIZED GAMMA FUNCTION ,STIRLING’S SERIES ,SUPERFACTORIALReferences Abramowitz, M. and Stegun, C. A. (Eds.). "Gamma (Factor- ial) Function" and "Incomplete Gamma Function." §6.1 and 6.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 255 /C1/58 and 260 /C1/63, 1972. Arfken, G. "The Gamma Function (Factorial Function)." Ch. 10 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 339 /C1/41 and 539 /C1/72, 1985. Artin, E. The Gamma Function. New York: Holt, Rinehart, and Winston, 1964. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, 1935. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 334 /C1/42, 1994. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, 1987. Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua- tion of the Gamma Function at Rational Values of Small Denominator." IMA J. Numerical Analysis 12, 519/C1/26, 1992. Bourguet, L. "Sur les inte ´grales Euleriennes et quelques autres fonctions uniformes." Acta Math. 2, 261/C1/95, 1883. Campbell, R. Les inte ´grales eule ´riennes et leurs applications. Paris: Dunod, 1966. Davis, H. T. Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, 1933. Davis, P. J. "Leonhard Euler’s Integral: A Historical Profile of the Gamma Function." Amer. Math. Monthly 66, 849/C1/ 69, 1959. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Gamma Function." Ch. 1 in Higher Transcen- dental Functions, Vol. 1. New York: Krieger, pp. 1 /C1/5, 1981. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/fran/fran.html. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Answer to problem 9.60 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Hardy, G. H. "Some Formulae of Ramanujan." Proc. London Math. Soc. (Records of Proceedings at Meetings) 22, xii- xiii, 1924. Hardy, G. H. "A Chapter from Ramanujan’s Note-Book." Proc. Cambridge Philos. Soc. 21, 492/C1/03, 1923. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Isaacson and Salzer. Math. Tab. Aids Comput. 1, 124, 1943. Koepf, W. "The Gamma Function." Ch. 1 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 4 /C1/0, 1998. Krantz, S. G. "The Gamma and Beta Functions." §13.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 155 /C1/58, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983. Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea, 1949. Nielsen, N. "Handbuch der Theorie der Gammafunktion." Part I in Die Gammafunktion. New York: Chelsea, 1965. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gamma Function, Beta Function, Factorials,Binomial Coefficients" and "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumu- lative Poisson Function." §6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206 /C1/09 and 209 /C1/14, 1992. Sloane, N. J. A. Sequences A000079/M1129, A000142/ M1675, A001147/M3002, A030169/M030170, and A030171/M030172 in "An On-Line Version of the Ency- clopedia of Integer Sequences." http://www.research.att.- com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Gamma Function G(x)/" and "The Incomplete Gamma g(n; x) and Related Func- tions." Chs. 43 and 45 in An Atlas of Functions. Washing- ton, DC: Hemisphere, pp. 411 /C1/21 and 435 /C1/43, 1987. Watson, G. N. "Theorems Stated by Ramanujan (XI)." J. London Math. Soc. 6,59/C1/5, 1931. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986. Whipple, F. J. W. "A Fundamental Relation Between Gen- eralised Hypergeometric Series." J. London Math. Soc. 1, 138 /C1/45, 1926. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput. 22, 617 /C1/26, 1968. Gamma Group MODULAR GROUP Gamma Matrices DIRAC MATRICES Gamma Statistic gr /C13kr sr/C272 ; where krare CUMULANTS and s is the STANDARD DEVIATION . See also KURTOSIS ,SKEWNESS Gamma-Modular Function The GAMMA GROUP G is the set of all transformations w OF THE FORM w(t) /C30at /C27 b ct /C27 d ; where a, b, c, and d are INTEGERS and ad /C28bc /C301:G/- modular functions are then defined as in Borwein and Borwein (1987, p. 114). See also JACOBI THETA FUNCTIONS ,K LEIN’S ABSO- LUTE INVARIANT ,LAMBDA GROUP References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 127 /C1/32, 1987. GammaRegularized REGULARIZED GAMMA FUNCTIONGarage Door ASTROID Ga˚rding’s Inequality Gives a lower bound for the inner product (Lu, u), where L is a linear elliptic real differential operator of order m, and u has compact support. References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Garman-Kohlhagen Formula Vt /C30e /C28y tStN(d1) /C28e /C28rtKN(d2) ; where N is the cumulative NORMAL DISTRIBUTION and d1 ; d2 /C30logSt K=z1*=z1+ /C27 r /C28 y 91 2 s2=z1*=z1+ t sffiffiffitp : If y /C300, this is the standard form of the Black-Scholes formula. See also BLACK- SCHOLES THEORY References Garman, M. B. and Kohlhagen, S. W. "Foreign Currency Option Values." J. International Money and Finance 2, 231/C1/37, 1983. Price, J. F. "Optional Mathematics is Not Optional." Not. Amer. Math. Soc. 43, 964/C1/71, 1996. Garsia-Haiman Conjecture N!THEOREM Garsia-Milne Involution Principle LetC/C30C/C27@C/C28(where C/C27SC/C28/C30f) be the DISJOINT UNION of two finite components C/C27andC/C28:Letaand bbe two involutions on C, each of whose fixed points lie in C/C27:LetFa(respectively, Fb) denote the fixed point set of a(respectively, b):Stipulate that a(C/C27/C28 Fa)ƒC/C28anda(C/C28)ƒC/C27;and similarly b(C/C27/C28Fb)ƒ C/C28andb(C/C28)ƒC/C27(i.e., outside the fixed point sets), both aandbmap each component into the other. Then either a cycle of the PERMUTATION D/C30ab contains no fixed points of either aorb;or it contains exactly one element of Faand one of Fb:/ References Andrews, G. E. " q-Series and Schur’s Theorem" and "Bres- soud’s Proof of Schur’s Theorem." §6.2/C1/.3 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra.Providence, RI: Amer. Math. Soc., pp. 53 /C1 /8, 1986. Gasket APOLLONIAN GASKET ,SIERPINSKI GASKET Gasser-Mu ¨ ller Technique References Gasser, T. and Mu¨ller, H. "Kernel Estimation of Regression Functions." In Smoothing Techniques for Curve Estima- tion: Proceedings of a Workshop Held in Heidelberg, April 2 /C1/, 1979 (Ed. T. Gasser and M. Rosenblatt). Berlin: Springer-Verlag, pp. 23 /C1/8, 1979. Gate Function Bracewell’s term for the RECTANGLE FUNCTION . References Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999. Gauche Conic SKEW CONIC Gauge Theory References Friedman, R. and Morgan, J. W. (Eds.). Gauge Theory and the Topology of Four-Manifolds. Providence, RI: Amer. Math. Soc., 1998. Gaullist Cross A CROSS also called the CROSS OF LORRAINE or PATRIARCHAL CROSS . See also CROSS ,DISSECTION Gauss Equations If x is a regular patch on a REGULAR SURFACE in R3 with normal ˆN ; then xuu /C30G1 11xu /C27G211xv /C27e ˆN (1) xuv /C30G112xu /C27G212xv /C27f ˆN (2) xvv /C30G122xu /C27G222xv /C27g ˆN ; (3) where e, f, and g are coefficients of the second FUNDAMENTAL FORM and Gkijare CHRISTOFFEL SYM- BOLS OF THE SECOND KIND . See also CHRISTOFFEL SYMBOL OF THE SECOND KIND, FUNDAMENTAL FORMS ,M AINARDI- CODAZZI EQUA- TIONSReferences Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 511 /C1/12, 1997. Gauss Integral Consider two closed oriented SPACE CURVES f1 : C1 0 R3 and f2 : C2 0 R3 ; where C1and C2are distinct CIRCLES , f1 and f2 are differentiable C1 functions, and f1(C1) and f2(C3) are disjoint loci. Let Lk(f1 ; f2) be the LINKING NUMBER of the two curves, then the Gauss integral is Lk(f1 ; f2) /C301 4 p gC1 /C29 C2dS : See also CALUGAREANU THEOREM ,LINKING NUMBER References Pohl, W. F. "The Self-Linking Number of a Closed Space Curve." J. Math. Mech. 17, 975 /C1/85, 1968. Gauss Map The Gauss map is a function from an ORIENTABLE SURFACE M in EUCLIDEAN SPACE to a SPHERE .It associates to every point on the surface its oriented NORMAL VECTOR . For a COMPACT SURFACE M in 3- space, the Gauss map of M has DEGREE given by half the EULER CHARACTERISTIC of the surface ggMKdA/C302px(M) /C28X ai /C28g@Tkg ds ; where this formula holds only for ORIENTABLE SUR- FACES . See also CURVATURE ,N IRENBERG’S CONJECTURE , PATCH References Gray, A. "The Local Gauss Map" and "The Gauss Map via Mathematica." §12.3 and §17.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 279 /C1/80 and 403 /C1/08, 1997. Gauss Measure The standard Gauss measure of a finite dimensional REAL HILBERT SPACE Hwith norm ½½/C215½½Hhas the B OREL MEASURE mH(dh)/C30(ffiffiffiffiffiffi 2pp )/C28dim(H)exp(1 2½½h½½2 H)lH(dh); where lHis the L EBESGUE MEASURE onH. Gauss Multiplication Formula (2np)(n/C281)=2n1 =2 /C28nz G(nz) /C30G(z) G z /C271 n ! G z /C272 n ! /C1/C1/C1G z /C27n /C28 1 n ! /C30Yn /C281 k/C300G z /C27k n ! ; where G(z) is the GAMMA FUNCTION . See also GAMMA FUNCTION ,LEGENDRE DUPLICATION FORMULA ,POLYGAMMA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 256, 1972. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 4 /C1/, 1981. Gauss Plane COMPLEX PLANE Gauss’s Backward Formula fp /C30f0 /C27p d/C281=2 /C27G /C31 2 d2 0 /C27G3 d3/C281=2 /C27G /C31 4 d4 0 /C27G5 d5/C281=2 /C27...; for p /C23 [0; 1]; where d is the CENTRAL DIFFERENCE and G/C31 2n/C30p/C27n 2n=z1r=z1> G2n/C271/C30p/C27n 2n/C271=z1r=z1> ; wheren k=z;=z1 is a BINOMIAL COEFFICIENT . See also CENTRAL DIFFERENCE ,G AUSS’S FORWARD FORMULA References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Whittaker, E. T. and Robinson, G. "The Newton-Gauss Backward Formula." §22 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 37 /C1/8, 1967.Gauss’s Circle Problem Count the number of LATTICE POINTS N(r) inside the boundary of a CIRCLE ofRADIUS rwith center at the origin. The exact solution is given by the SUM N(r)/C301/C274rbc/C274Xrbc i/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28i2pjk (1) /C301/C274Xr2 i/C301(/C281)i/C281r2 2i/C281$% (2) (Hilbert and Cohn-Vossen 1999, p. 39). The first few values for r/C300, 1, ... are 1, 5, 13, 29, 49, 81, 113, 149, ... (Sloane’s A000328).The series for N(r) is intimately connected with r(n); the number of representations of nby two squares, since N(r)/C30X r2 n/C300r(n) (3) (Hardy 1999, p. 67). N(r) is also closely connected with the L EIBNIZ SERIES since 1 4N(r) r2/C281 r2"# /C301/C2813/C2715/C2817/C27...91 r; (4) so taking the limit r0/C12gives 1 4p/C301/C2813/C2715/C2817/C2719/C27. . . (5) (Hilbert and Cohn-Vossen 19991, p. 39). Gauss showed that N(r)/C30pr2/C27E(r); (6) where ½E(r)½52ffiffiffi 2p pr (7) (Hardy 1999, p. 67). Writing ½E(r) ½5Cru ; the best bounds on u are 1=2 B u 546 =73 :0 :630137 (Huxley 1990). The lower limit 1/2 was obtained independently by Hardy and Landau in 1915. The following table summarizes incremental improve- ments in the upper limit (Hardy 1999, p. 81). /u/ approx. citation 46/73 0.63014 Huxley 1990 7/11 0.63636 24/37 0.64864 Cheng 1963 34/53 0.64150 Vinogradov 37/56 0.66071 Littlewood and Walfisz 1924 2/3 0.66667 Sierpinski1906, van der Corput 1923 The problem has also been extended to CONICS , ellipsoids (Hardy 1915), and higher dimensions. See also CIRCLE LATTICE POINTS ,DIRICHLET DIVISOR PROBLEM ,LEIBNIZ SERIES ,SUM OF SQUARES FUNC- TION References Bohr, H. and Crame ´r.Enzykl. d. Math. Wiss. II C 8 , 823/C1/24, 1922. Cheng, J. R. "The Lattice Points in a Circle." Sci. Sinica 12, 633/C1/49, 1963. Cilleruello, J. "The Distribution of Lattice Points on Circles." J. Number Th. 43, 198/C1/02, 1993. Guy, R. K. "Gauß’s Lattice Point Problem." §F1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 240 /C1/417, 1994. Hardy, G. H. Quart. J. Math. 46, 283, 1915. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 268 /C1/69, 1979. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 33 /C1/5, 1999. Huxley, M. N. "Exponential Sums and Lattice Points." Proc. London Math. Soc. 60, 471/C1/02, 1990. Huxley, M. N. "Corrigenda: ‘Exponential Sums and Lattice Points’." Proc. London Math. Soc. 66, 70, 1993. Landau, E. Vorlesungen u ¨ber Zahlentheorie, Vol. 2. New York: Chelsea, pp. 183 /C1/08 1970. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 24, 1983. Littlewood, J. E. and Walfisz. Proc. Roy. Soc. (A) 106, 478/C1/ 88, 1924. Sloane, N. J. A. Sequences A000328/M3829 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Titchmarsh. Quart. J. Math. (Oxford) 2, 161/C1/73, 1931.Titchmarsh. Proc. London Math. Soc. 38,9 6/C1/15 and 555, 1935. Weisstein, E. W. "Circle Lattice Points." M ATHEMATICA NOTEBOOK CIRCLE LATTICE POINTS.M . Gauss’s Class Number Conjecture In his monumental treatise Disquisitiones Arithme- ticae, Gauss conjectured that the CLASS NUMBER h(/C28d)o fa n IMAGINARY QUADRATIC FIELD with DIS- CRIMINANT /C28dtends to infinity with d. A proof was finally given by Heilbronn (1934), and Siegel (1936) showed that for any e>0;there exists a constant ce> 0 such that h(/C28d)>ced)1=2/C28e asd0/C12:However, these results were not effective in actually determining the values for a given mof a complete list of fundamental discriminants /C28dsuch that h(/C28d)/C30m;a problem known as G AUSS’S CLASS NUMBER PROBLEM . Goldfeld (1976) showed that if there exists a "Weil curve" whose associated D IRICHLET L-SERIES has a zero of at least third order at s/C301, then for any e>0; there exists an effectively computable constant ce such that h(/C28d)>ce(lnd)1/C28e: Gross and Zaiger (1983) showed that certain curves must satisfy the condition of Goldfeld, and Goldfeld’sproof was simplified by Oesterle ´(1985). See also C LASS NUMBER ,G AUSS’S CLASS NUMBER PROBLEM ,HEEGNER NUMBER References Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." http:// www.math.uiuc.edu/Algebraic-Number-Theory/0009/. Bo¨cherer, S. "Das Gauß’sche Klassenzahlproblem." Mitt. Math. Ges. Hamburg 11, 565/C1/89, 1988. Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT: Yale University Press, 1966. Goldfeld, D. M. "The Class Number of Quadratic Fields and the Conjectures of Birch and Swinnerton-Dyer." Ann. Scuola Norm. Sup. Pisa 3, 623/C1/63, 1976. Gross, B. and Zaiger, D. "Points de Heegner et derive ´es de fonctions L."C. R. Acad. Sci. Paris 297,8 5/C1/7, 1983. Heilbronn, H. "On the Class Number in Imaginary Quad- ratic Fields." Quart. J. Math. Oxford Ser. 25, 150/C1/60, 1934. Oesterle ´, J. "Nombres de classes des corps quadratiques imaginaires." Aste´rique 121/C1/22, 309/C1/23, 1985. Siegel, C. L. "Uuml;ber die Klassenzahl quadratischer Zahlko ¨rper." Acta. Arith. 1,8 3/C1/6, 1936. Gauss’s Class Number Problem For a given m, determine a complete list of funda- mental DISCRIMINANTS /C28dsuch that the CLASS NUM- BER is given by h(/C28d)/C30m:Heegner (1952) gave a solution for m/C301, but it was not completely accepted due to a number of apparent gaps. However, subse- quent examination of Heegner’s proof showed it to be "essentially" correct (Conway and Guy 1996). Conway and Guy (1996) therefore call the nine values of n(/C28d) having h(/C28d) /C301 where /C28d is the DISCRIMINANT corresponding to an QUADRATIC FIELD a /C27bffiffiffiffiffiffiffi/C28np (n /C30/C281, /C282, /C283, /C287, /C2811, /C2819, /C2843, /C2867, and /C28163; Sloane’s A003173) the HEEGNER NUMBERS . The HEEGNER NUMBERS have a number of fascinating properties. Stark (1967) and Baker (1966) gave independent proofs of the fact that only nine such numbers exist; both proofs were accepted. Baker (1971) and Stark (1975) subsequently and independently solved the generalized class number problem completely for m /C302. Oesterle ´ (1985) solved the case m /C303, and Arno (1992) solved the case m /C304. Wagner (1996) solve the cases n /C305, 6, and 7. Arno et al. (1993) solved the problem for ODD m satisfying 5 5m 523: In his thesis, M. Watkins has solved the problem for all m 516:/ See also CLASS NUMBER ,G AUSS’S CLASS NUMBER CONJECTURE ,HEEGNER NUMBER References Arno, S. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith. 40, 321 /C1/34, 1992. Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." Dec. 1993. http://www.math.uiuc.edu/Algebraic-Number-The- ory/0009/. Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers. I." Mathematika 13, 204 /C1/16, 1966. Baker, A. "Imaginary Quadratic Fields with Class Number 2." Ann. Math. 94, 139 /C1/52, 1971. Conway, J. H. and Guy, R. K. "The Nine Magic Discrimi- nants." In The Book of Numbers. New York: Springer- Verlag, pp. 224 /C1/26, 1996. Goldfeld, D. M. "Gauss’ Class Number Problem for Imagin- ary Quadratic Fields." Bull. Amer. Math. Soc. 13,23/C1/7, 1985. Heegner, K. "Diophantische Analysis und Modulfunktio- nen." Math. Z. 56, 227 /C1/53, 1952. Heilbronn, H. A. and Linfoot, E. H. "On the Imaginary Quadratic Corpora of Class-Number One." Quart. J. Math. (Oxford) 5, 293 /C1/01, 1934. Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, p. 192, 1990. Lehmer, D. H. "On Imaginary Quadratic Fields whose Class Number is Unity." Bull. Amer. Math. Soc. 39, 360, 1933. Montgomery, H. and Weinberger, P. "Notes on Small Class Numbers." Acta. Arith. 24, 529 /C1/42, 1974. Oesterle ´, J. "Nombres de classes des corps quadratiques imaginaires." Aste´rique 121 /C1/22, 309 /C1/23, 1985. Oesterle ´, J. "Le proble `me de Gauss sur le nombre de classes." Enseign Math. 34,43/C1/7, 1988. Serre, J.-P. D/C30b2 /C284ac:/" Math. Medley 13,1/C1/0, 1985. Shanks, D. "On Gauss’s Class Number Problems." Math. Comput. 23, 151 /C1/63, 1969. Sloane, N. J. A. Sequences A003173/M0827 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stark, H. M. "A Complete Determination of the Complex Quadratic Fields of Class Number One." Michigan Math. J. 14,1/C1/7, 1967.Stark, H. M. "On Complex Quadratic Fields with Class Number Two." Math. Comput. 29, 289 /C1/02, 1975. Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65, 785 /C1/00, 1996. Gauss’s Constant The RECIPROCAL of the ARITHMETIC-GEOMETRIC MEAN of 1 andffiffiffi 2p ; G /C131 M(1;ffiffiffi2p ) (1) /C302 p g1 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x4p dx (2) /C302 p g p =2 0duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 sin2 up (3) /C30ffiffiffi 2p pK1ffiffiffi 2p ! (4) /C301 (2p)3=2 [G(1 4)]2 (5) /C300 :83462684167... (6) (Sloane’s A014549), where K(k) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND andG(z)i s the GAMMA FUNCTION . Gauss’s constant has CONTIN- UED FRACTION [0, 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, ...] (Sloane’s A053002). The inverse of Gauss’s constant is given by 1 G/C301:1981402347355922074399 . . . (7) (Sloane’s A053004), and has [1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, ...] (Sloane’s A053003). See also ARITHMETIC- GEOMETRIC MEAN,GAUSS- KUZ- MIN-WIRSING CONSTANT ,PYTHAGORAS’S CONSTANT References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 5, 1987. Goldman, J. R. The Queen of Mathematics: An Historically Motivated Guide to Number Theory. Natick, MA: A. K. Peters, p. 92, 1997. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/gauss/gauss.html. Sloane, N. J. A. Sequences A014549, A053002, A053003, and A053004 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Gauss’s Criterion Letpbe an ODD PRIME andbaPOSITIVE INTEGER not divisible by p. Then for each POSITIVE ODD INTEGER 2k/C281Bp;letrkbe rk /C13(2k /C281)b (mod p) with 0 Brk Bp ; and let t be the number of EVEN rk/s. Then (b=p) /C30(/C281)t ; where (b=p) is the LEGENDRE SYMBOL . References Shanks, D. "Gauss’s Criterion." §1.17 in Solved and Un- solved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 38 /C1/0, 1993. Gauss’s Cyclotomic Formula Let p /C21 3bea PRIME NUMBER , then 4xp /C28 yp x /C28 y/C30R2(x; y) /C28(/C281)(p /C281)=2pS2(x; y) ; where R(x; y) and S(x; y) are HOMOGENEOUS POLY- NOMIALS in x and y with integer COEFFICIENTS . Gauss (1965, p. 467) gives the coefficients of R and S up to p /C3023. Kraitchik (1924) generalized Gauss’s formula to odd SQUAREFREE integers n /C213. Then Gauss’s formula can be written in the slightly simpler form 4Fn(z) /C30A2 n(z) /C28(/C281)(n/C281)=2nz2B2n(z) ; where An(z) and Bn(z) have integer coefficients and are of degree f(n) =2 and f(n) =2 /C282; respectively, with f(n) the TOTIENT FUNCTION and Fn(z)a CYCLO- TOMIC POLYNOMIAL . In addition, An(z) is symmetric if n is EVEN ; otherwise it is antisymmetric. Bn(z)is symmetric in most cases, but it antisymmetric if n is OF THE FORM 4k /C273 (Riesel 1994, p. 436). The follow- ing table gives the first few An(z) and Bn(z)/s (Riesel 1994, pp. 436 /C1/42). n /An(z)// Bn(z)/ 5 /2z2 /C27z /C272/ 1 7 /2z3 /C27z2 /C28z /C282// z /C271/ 11 /2z5 /C27z4 /C282z3 /C272z2 /C28z /C282//z3 /C271/ See also AURIFEUILLEAN FACTORIZATION ,C YCLO- TOMIC POLYNOMIAL ,LUCAS’S THEOREM References Gauss, C. F. §356 /C1/57 in Untersuchungen u¨ber ho¨here Arithmetik. New York: Chelsea, pp. 425 /C1/28 and 467, 1965. Kraitchik, M. Recherches sue la the´orie des nombres, tome I. Paris: Gauthier-Villars, pp. 93 /C1/29, 1924. Kraitchik, M. Recherches sue la the´orie des nombres, tome II. Paris: Gauthier-Villars, pp. 1 /C1/, 1929. Riesel, H. "Gauss’s Formula for Cyclotomic Polynomials." In tables at end of Prime Numbers and Computer Methodsfor Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 436 /C1/42, 1994. Gauss’s Digamma Theorem At rational arguments p =q; the DIGAMMA FUNCTION c0(p =q) is given by c0p q ! /C30/C28g /C28ln(2q) /C281 2 p cotp qp ! /C272Xq =2de/C281 k /C301cos2ppk q ! ln sinpk q !"# (1) for 0 Bp Bq (Knuth 1997, p. 94). These give the special values co(1 2) /C30/C28g /C282 ln 2 (2) c0(13) /C3016(/C286g /C28 pffiffiffi 3p /C289 ln 3) (3) c0(2 3) /C3016(/C286g /C27 pffiffiffi 3p /C289 ln 3) (4) c0(1 4) /C3012(/C282 g /C28 p /C286 ln 2) (5) c0(34) /C3012(/C282 g /C27 p /C286 ln 2) (6) c0(16) /C30/C28g /C2812ffiffiffi 3p p /C282ln2/C283 2 ln 3) (7) c0(56) /C30/C28g /C2712ffiffiffi 3p p /C282ln2/C283 2 ln 3) (8) c0(1) /C30/C28g ; (9) where gis the E ULER- MASCHERONI CONSTANT . See also DIGAMMA FUNCTION References Bo¨hmer, E. Differenzengleichungen und bestimmte Inte- grale. Leipzig, Germany: Teubner, p. 77, 1939. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The cFunction." §1.7 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 15 /C1/0, 1981. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Gauss’s Double Point Theorem If a sequence of DOUBLE POINTS is passed as a CLOSED CURVE is traversed, each DOUBLE POINT appears once in an EVEN place and once in an ODD place. References Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 61 /C1/6, 1957. Gauss’s Equation (Radius Derivatives) Expresses the second derivatives of the RADIUS VECTOR rin terms of the C HRISTOFFEL SYMBOL OF THE SECOND KIND . rij /C30Gk ijrk /C27(rij/C215 n)n : Gauss’s Formulas Let a SPHERICAL TRIANGLE have sides a, b, and c with A, B, and C the corresponding opposite angles. Then sin[1 2(a /C28 b)] sin(1 2 c)/C30sin[1 2(A /C28 B)] cos(1 2 C) (1) sin[1 2(a /C27 b)] sin(1 2 c)/C30cos[1 2(A /C28 B)] sin(1 2 C) (2) cos[1 2(a /C28 b)] cos(12 c)/C30sin[12(A /C27 B)] cos(12 C) (3) cos[1 2(a /C27 b)] cos(1 2 c)/C30cos[1 2(A /C27 B)] sin(1 2 C): (4) These formulas are also known as Delambre’s analo- gies (Smart 1960, p. 22). See also SPHERICAL TRIGONOMETRY References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 147 /C1/50, 1987. Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, 1960. Zwillinger, D. (Ed.). "Spherical Geometry and Trigonome- try." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 468 /C1/71, 1995. Gauss’s Forward Formula fp /C30f0 /C27pd1=2 /C27G2 d2 0 /C27G3 d31=2 /C27G4 d40 /C27G5 d51=2 /C27...; for p /C23 [0; 1]; where d is the CENTRAL DIFFERENCE and G2n /C30p /C27n /C281 2n=z1r=z1> G2n/C271 /C30p /C27n 2n /C271=z1r=z1> ; wheren k=z;=z1 is a BINOMIAL COEFFICIENT . See also CENTRAL DIFFERENCE ,GAUSS’S BACKWARD FORMULA References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Whittaker, E. T. and Robinson, G. "The Newton-Gauss Formula for Interpolation." §21 in The Calculus of Ob- servations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 36 /C1/7, 1967.Gauss’s Harmonic Function Theorem If a function f is HARMONIC in a SPHERE , then the value of f at the center of the SPHERE is the ARITHMETIC MEAN of its value on the surface. Gauss’s Hypergeometric Theorem 2F1(a ; b; c;1)/C30(c /C28 b)/C28a (c)/C28a/C30G(c) G(c /C28 a /C28 b) G(c /C28 a) G(c /C28 b) for R[c /C28a /C28b] > 0; where2F1(a ; b; c; x)isa (Gauss) HYPERGEOMETRIC FUNCTION .Ifa is a NEGA- TIVE INTEGER /C28n; this becomes 2F1(/C28n; b; c;1)/C30(c /C28 b)n (c)n; which is known as the VANDERMONDE THEOREM . See also DOUGALL’S FORMULA ,GENERALIZED HYPER- GEOMETRIC FUNCTION ,HYPERGEOMETRIC FUNCTION , THOMAE’S THEOREM ,VANDERMONDE THEOREM References Bailey, W. N. "Gauss’s Theorem." §1.3 in Generalised Hy- pergeometric Series. Cambridge, England: Cambridge University Press, pp. 2 /C1/, 1935. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 104, 1999. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 31, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, pp. 42 and 126, 1996. Gauss’s Inequality If a distribution has a single MODE atm0;then P(½x/C28m0½]lt)54 9l2; where t2/C13s2/C27(m/C28m0)2: Gauss’s Interpolation Formula f(x):tn(x)/C30X2n k/C300fkzk(x); where tn(x) is a trigonometric POLYNOMIAL of degree n such that tn(xk) /C30fk for k /C30 0, ..., 2n; and zk(x) /C30sin1 2(x /C28 x0)hi /C1/C1/C1sin12(x /C28 xk/C281)hi sin1 2(xk /C28 x0)hi /C1/C1/C1sin12(xk /C28 xk/C281)hi /C2sin12(x /C28 xk /C271)hi /C1/C1/C1sin12(x /C28 x2n)hi sin1 2(xk /C28 xk /C271)hi /C1/C1/C1sin12(xk /C28 x2n)hi : References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 881, 1972. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 442 /C1/43, 1987. Gauss’s Lemma Let the multiples m,2m; ..., [(p /C281)=2]m of an INTEGER such that p¶m be taken. If there are an EVEN NUMBER r of least POSITIVE RESIDUES mod p of these numbers > p=2 ; then m is a QUADRATIC RESIDUE of p.Ifr is ODD, m is a QUADRATIC NONRESIDUE . Gauss’s lemma can therefore be stated as (m½p) /C30 (/C281)r ; where (m½p) is the LEGENDRE SYMBOL . It was proved by Gauss as a step along the way to the QUADRATIC RECIPROCITY THEOREM (Nagell 1951). Another result known as Gauss’s lemma states that for any two integer a and b, suppose d½ab : Then if d is RELATIVELY PRIME to a, then d divides b (Se´roul 2000, p. 10). See also LEGENDRE SYMBOL ,QUADRATIC RECIPROCITY THEOREM References Nagell, T. "Gauss’s Lemma." §40 in Introduction to Number Theory. New York: Wiley, pp. 139 /C1/41, 1951. Se´roul, R. "Gauss’s Lemma." §2.4.2 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 10 /C1/1, 2000. Gauss’s Machin-Like Formula The MACHIN-LIKE FORMULA 14 p /C3012 cot /C281 18 /C278 cot /C281 57 /C285 cot /C281 239: Gauss’s Mean-Value Theorem Let f(z)bean ANALYTIC FUNCTION in ½z /C28a½BR: Then f(z) /C301 2p g2 p 0f(z /C27rei u) du for 0 Br BR:/Gauss’s Polynomial Identity For even h, 1 /C281 /C28 xh 1 /C28 x/C27(1 /C28 xh)(1 /C28 xh/C281) (1 /C28 x)(1 /C28 x2) /C28(1 /C28 xh)(1 /C28 xh/C281)(1 /C28 xh/C282) (1 /C28 x)(1 /C28 x2)(1 /C28 x3)/C27... /C30(1 /C28x)(1 /C28x3)(1 /C28x5) /C1/C1/C1(1 /C28xh/C281) (1) (Nagell 1951, p. 176). Writing out explicitly, Xh n /C300( /C281)n Pn/C281 k /C300(1 /C28 xh/C28k) Pn k/C301/C30Y(h/C281)=2 k /C3001 /C28x2k /C271 : (2) For example, for h /C30 2, 1 /C281 /C28 x2 1 /C28 x/C27(1 /C28 x)(1 /C28 x2) (1 /C28 x)(1 /C28 x2) /C302 /C281 /C28 x2 1 /C28 x/C301 /C28x; (3) and for h /C304, 1 /C281 /C28 x4 1 /C28 x/C27(1 /C28 x4)(1 /C28 x3) (1 /C28 x)(1 /C28 x2) /C28(1 /C28 x4)(1 /C28 x3)(1 /C28 x2) (1 /C28 x)(1 /C28 x2)(1 /C28 x3) /C27(1 /C28 x)(1 /C28 x2)(1 /C28 x3)(1 /C28 x4) (1 /C28 x)(1 /C28 x2)(1 /C28 x3)(1 /C28 x4) /C302 /C282(1 /C28 x4) 1 /C28 x/C27(1 /C28 x3)(1 /C28 x4) (1 /C28 x)(1 /C28 x2) /C30(1 /C28x)(1 /C28x3) : (4) See also Q-SERIES References Nagell, T. "A Polynomial Identity of Gauss." §52 in Introduc- tion to Number Theory. New York: Wiley, pp. 174 /C1/76, 1951. Gauss’s Polynomial Theorem If an INTEGER POLYNOMIAL f(x) /C30xN /C27C1xN /C281 /C27C2xN /C282 /C27.../C27CN is divisible into a product of two POLYNOMIALS f /C30 cf c/C30xm/C27a1xm/C281/C27.../C27am f/C30xn/C27b1xn/C281/C27.../C27bn; then the COEFFICIENTS of these POLYNOMIALS are INTEGERS . See also ABEL’S IRREDUCIBILITY THEOREM ,A BEL’S LEMMA ,KRONECKER’S POLYNOMIAL THEOREM ,POLY- NOMIAL ,SCHO¨ NEMANN’S THEOREM References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 119, 1965. Gauss’s Reciprocity Theorem QUADRATIC RECIPROCITY THEOREM Gauss’s Test If un > 0 and given B(n) a bounded function of n as n 0/C12; express the ratio of successive terms as un un /C271=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n/C301 /C27 h n /C27B(n) nr for r /C211. The SERIES converges for h /C211 and diverges for h 51 (Courant and John 1999, p. 567). See also CONVERGENCE TESTS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 287 /C1/88, 1985. Courant, R. and John, F. Introduction to Calculus and Analysis, Vol. 1. New York: Springer-Verlag, 1999. Gauss’s Theorem DIVERGENCE THEOREM ,GAUSS’S DIGAMMA THEOREM , GAUSS’S DOUBLE POINT THEOREM ,G AUSS’S HYPER- GEOMETRIC THEOREM ,GAUSS’S THEOREMA EGREGIUM Gauss’s Theorema Egregium Gauss’s theorema egregium states that the GAUSSIAN CURVATURE of a surface embedded in 3-space may be understood intrinsically to that surface. "Residents" of the surface may observe the GAUSSIANCURVATURE of the surface without ever venturing into full 3- dimensional space; they can observe the curvature of the surface they live in without even knowing about the 3-dimensional space in which they are embedded. In particular, GAUSSIAN CURVATURE can be measured by checking how closely the ARC LENGTH of small RADIUS CIRCLES correspond to what they should be in EUCLIDEAN SPACE ,2pr : If the ARC LENGTH of CIRCLES tends to be smaller than what is expected in EU- CLIDEAN SPACE , then the space is positively curved; if larger, negatively; if the same, 0 GAUSSIAN CURVA- TURE . Gauss (effectively) expressed the theorema egregium by saying that the GAUSSIAN CURVATURE at a point is given by /C28R(v; w)v; w where R is the RIEMANN TENSOR , and v and w are an orthonormal basis for the TANGENT SPACE . See also CHRISTOFFEL SYMBOL OF THE SECOND KIND, GAUSS EQUATIONS ,GAUSSIAN CURVATUREReferences Gray, A. "Gauss’s Theorema Egregium." §22.2 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 507 /C1/09, 1997. Reckziegel, H. In Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, pp. 31 /C1/2, 1986. Gauss’s Transformation If (1 /C27x sin2 a)sin b /C30(1 /C27x)sin a; then (1 /C27x)g a 0dfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2 sin2 fq /C30g b 0dfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C284x (1 /C27 x)2 sin2 fs : See also ELLIPTIC INTEGRAL OF THE FIRST KIND, LANDEN’S TRANSFORMATION Gauss-Bodenmiller Theorem The CIRCLES on the DIAGONALS of a COMPLETE QUAD- RILATERAL as DIAMETERS are COAXAL . Furthermore, the ORTHOCENTERS of the four TRIANGLES of a COM- PLETE QUADRILATERAL are COLLINEAR on the RADICAL AXIS of the COAXAL CIRCLES . See also COAXAL CIRCLES ,C OLLINEAR ,C OMPLETE QUADRILATERAL ,D IAGONAL (POLYGON ), ORTHOCEN- TER,RADICAL AXIS References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 172, 1929. Gauss-Bolyai-Lobachevsky Space A non-Euclidean space with constant NEGATIVE GAUSSIAN CURVATURE . See also LOBACHEVSKY- BOLYAI- GAUSS GEOMETRY , NON-EUCLIDEAN GEOMETRY Gauss-Bonnet Formula The Gauss-Bonnet formula has several formulations. The simplest one expresses the total G AUSSIAN CURVATURE of an embedded triangle in terms of the total GEODESIC CURVATURE of the boundary and the JUMP ANGLES at the corners. More specifically, if Mis any 2-D R IEMANNIAN MANIFOLD (like a surface in 3-space) and if Tis an embedded triangle, then the Gauss-Bonnet formulastates that the integral over the whole triangle of theG AUSSIAN CURVATURE with respect to AREA is given by 2pminus the sum of the JUMP ANGLES minus the integral of the GEODESIC CURVATURE over the whole of the boundary of the triangle (with respect to ARC LENGTH ), ggTKd A/C302p/C28X ai/C28g@Tkgds; (1) where Kis the G AUSSIAN CURVATURE ,dAis the AREA measure, the ai/s are the JUMP ANGLES of@T;andkgis the GEODESIC CURVATURE of@T;with dsthe ARC LENGTH measure. The next most common formulation of the Gauss- Bonnet formula is that for any compact, boundaryless2-D R IEMANNIAN MANIFOLD , the integral of the GAUSSIAN CURVATURE over the entire MANIFOLD with respect to AREA is 2ptimes the E ULER CHAR- ACTERISTIC of the MANIFOLD , ggMKd A/C302px(M): (2) This is somewhat surprising because the total G AUS- SIAN CURVATURE is differential-geometric in charac- ter, but the E ULER CHARACTERISTIC is topological in character and does not depend on differential geo-metry at all. So if you distort the surface and change the curvature at any location, regardless of how you do it, the same total curvature is maintained. Another way of looking at the Gauss-Bonnet theorem for surfaces in 3-space is that the G AUSS MAP of the surface has DEGREE given by half the E ULER CHAR- ACTERISTIC of the surface ggMKd A/C302px(M)/C28X ai/C28g@Mkgds; (3) which works only for ORIENTABLE SURFACES where M isCOMPACT . This makes the Gauss-Bonnet theorem a simple consequence of the POINCARE- HOPF INDEX THEOREM , which is a nice way of looking at things if you’re a topologist, but not so nice for a differentialgeometer. This proof can be found in Guillemin and Pollack (1974). Millman and Parker (1977) give a standard differential-geometric proof of the Gauss-Bonnet theorem, and Singer and Thorpe (1996) give a G AUSS’S THEOREMA EGREGIUM -inspired proof which is entirely intrinsic, without any reference to the ambi- ent E UCLIDEAN SPACE . A general Gauss-Bonnet formula that takes into account both formulas can also be given. For any compact 2-D R IEMANNIAN MANIFOLD with corners, the integral of the G AUSSIAN CURVATURE over the 2- MANIFOLD with respect to AREA is 2ptimes the E ULER CHARACTERISTIC of the MANIFOLD minus the sum of the JUMP ANGLES and the total GEODESIC CURVATURE of the boundary. References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.Guillemin, V. and Pollack, A. Differential Topology. Engle- wood Cliffs, NJ: Prentice-Hall, 1974. Millman, R. S. and Parker, G. D. Elements of Differential Geometry. Prentice-Hall, 1977. Reckziegel, H. In Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 31, 1986. Singer, I. M. and Thorpe, J. A. Lecture Notes on Elementary Topology and Geometry. New York: Springer-Verlag, 1996. Gauss-Bonnet Theorem GAUSS- BONNET FORMULA Gaussian Approximation Algorithm ARITHMETIC- GEOMETRIC MEAN Gaussian Bivariate Distribution The Gaussian bivariate distribution is given by P(x1;x2)/C301 2ps1s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p exp/C28z 2(1/C28r2)"# ;(1) where z/C13(x1/C28m1)2 s2 1/C282r(x1/C28m1)(x2/C28m2) s1s2/C27(x2/C28m2)2 s22;(2) and r/C13cor(x1;x2)/C30s12 s1s2(3) is the CORRELATION ofx1andx2(Kenney and Keeping 1951, pp. 92 and 202 /C1/05; Whittaker and Robinson 1967, p. 329). The Gaussian bivariate distribution is implemented in Mathematica asMultinormalDis- tribution [{mu1 ,mu2 }, {{ sigma11 ,sigma12 }, {sigma12 ,sigma22 }}, {x1,x2}] in the Mathematica add-on package Statistics‘MultinormalDis- tribution‘ (which can be loaded with the command BBStatistics‘ ). The MARGINAL PROBABILITIES are then P(x1)/C30g/C12 /C28/C12P(x1;x2)dx 2/C301 s1ffiffiffiffiffiffi 2pp e/C28(x1/C28m1)2=(2s2 1)(4) and P(x2)/C30g/C12 /C28/C12P(x1;x2)dx 1 /C301 s2ffiffiffiffiffiffi 2pp exp/C28(x2/C28m2)2 2s2 2ðÞ"# (5) (Kenney and Keeping 1951, p. 202). Letz1andz2be two independent Gaussian variables with MEANS mi/C300 and s2 i/C301 for i/C301, 2. Then the variables a1and a2defined below are Gaussian bivariates with unit VARIANCE and CROSS-CORRELA- TION COEFFICIENT r: a1/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27r 2s z1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r 2s z2 (6) a2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27r 2s z1/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r 2s z2/C215 (7) To derive the Gaussian bivariate probability function, letX1and X2be normally and independently dis- tributed variates with MEAN 0 and VARIANCE 1, then define Y1/C13m1/C27s11X1/C27s12X2 (8) Y2/C13m2/C27s21X1/C27s22X2 (9) (Kenney and Keeping 1951, p. 92). The variates Y1 andY2are then themselves normally distributed with MEANS m1andm2;VARIANCES s2 1/C13s211/C27s212(10) s22/C13s221/C27s222; (11) and COVARIANCE V12/C13s11s21/C27s12s22: (12) The COVARIANCE matrix is defined by Vij/C30s2 1rs1s2 rs1s2s22=zn;=zn1 ; (13) where r/C13V12 s1s2/C30s11s21/C27s12s22 s1s2/C215 (14) Now, the joint probability density function for x1and x2is f(x1;x2)dx1dx2/C301 2pe/C28(x2 1/C27x22)=2dx1dx2; (15) but from (8) and (9), we have y1/C28m1 y2/C28m2=zn;=zn1 /C30s11s12 s21s22=zn;=zn1 x1 x2=zn;=zn1 /C215 (16) As long as s11s12 s21s22=zn;=zn1 "0; (17) this can be inverted to give x1 x2=zn;=zn1 /C30s11s12 s21s22=zn;=zn1/C281y1/C28m1 y2/C28m2=zn;=zn1 /C301 s11s22/C28s12s21s22/C28s12 /C28s21s11=zn;=zn1 y1/C28m1 y2/C28m2=zn;=zn1 :(18) Therefore,x2 1/C27x22/C30s22(y1/C28m1)/C28s12(y2/C28m2) ½/C1382 (s11s22/C28s12s21)2 /C27/C28s21(y1/C28m1)/C28s11(y2/C28m2) ½/C1382 (s11s22/C28s12s21)2; (19) and expanding the NUMERATOR of (19) gives s222(y1/C28m1)2/C282s12s22(y1/C28m1)(y2/C28m2)/C27s212(y2/C28m2)2 /C27s222(y1/C28m1)2/C282s11s21(y1/C28m1)(y2/C28m2) /C27s211(y2/C28m2)2; (20) so (x21/C27x22)(s11s22/C28s12s21)2 /C30(y1/C28m1)2(s221/C27s222)/C282(y1/C28m1)(y2/C28m2) /C2(s11s21/C27s12s22)/C27(y2/C28m2)2(s221/C27s212) /C30s22(y1/C28m1)2/C282(y1/C28m1)(y2/C28m2)(rs1s2)/C27s21(y2/C28m2)2 /C30s21s22(y1/C28m1)2 s2 1/C282r(y1/C28m1)(y2/C28m2) s1s2/C27(y2/C28m2)2 s22"# /C215 (21) Now, the DENOMINATOR of (19) is s2 11s221/C27s211s222/C27s212s221/C27s212s222/C28s211s221 /C282s11s12s21s22/C28s212s222 /C30(s11s22/C28s12s21)2; (22) so 1 1/C28r2/C301 1/C28V2 12 s2 1s22/C30s2 1s22 s2 1s22/C28V2 12 /C30s2 1s22 (s2 11/C27s212)(s221/C27s222)/C28(s11s21/C27s12s22)2 /C215 (23) can be written simply as 1 1/C28r2/C30s2 1s22 (s11s22/C28s12s21)2; (24) and x2 1/C27x22/C301 1/C28r2 /C2(y1/C28m1)2 s2 1/C282r(y1/C28m1)(y2/C28m2) s1s2/C27(y2/C28m2)2 s22"# : (25) Solving for x1andx2and defining r?/C13s1s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p s11s22/C28s12s21(26) gives x1/C30s22(y1/C28m1)/C28s12(y2/C28m2) r?(27) x2/C30/C28s21(y1/C28m1)/C27s11(y2/C28m2) r?/C215 (28) But the J ACOBIAN is Jx1;x2 y1;y2 ! /C30@x1 @y1@x1 @y2 @x2 @y1@x2 @y2=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n/C30 s22 r?/C28s12 r? /C28s21 r?s11 r?=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n /C30 1 r?2(s11s22/C28s12s21)/C301 r?/C301 s1s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p ; (29) so dx1dx2/C30dy1dy2 s1s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p (30) and 1 2pe/C28(x2 1/C27x22)=2dx1dx2 /C301 2ps1s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p exp/C28z 2(1/C28r2)"# dy1dy2;(31) where z/C13(y1/C28m1)2 s2 1/C282r(y1/C28m1)(y2/C28m2) s1s2/C27(y2/C28m2)2 s22:(32) Q.E.D. In the singular case that s11s12 s21s22=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n/C300 (33) (Kenney and Keeping 1951, p. 94), it follows that s 11s12/C30s12s21 (34) y1/C30mu1/C27s11x1/C27s12x2 (35) y2/C30m1/C27s12s21 s11x2/C30m2/C27s11s21x1/C27s12s21x2 s11 /C30m2/C27s21 s11(s11x1/C27s12x2); (36) so y1/C30m1/C27x3 (37) y2/C30m2/C27s21 s11x3; (38) wherex3/C30y1/C28m1/C30s11 s21(y2/C28m2): (39) The CHARACTERISTIC FUNCTION of the Gaussian bi- variate distribution is given by f(t1;t2)/C13g/C12 /C28/C12g/C12 /C28/C12ei(t1x1/C27t2x2)P(x1;x2)dx1dx2 /C30Ng/C12 /C28/C12g/C12 /C28/C12ei(t1x1/C27t2x2)exp/C28z 2(1/C28r2)"# dx1dx2;(40) where z/C13(x1/C28m1)2 s2 1/C282r(x1/C28m1)(x2/C28m2) s1s2/C27(x2/C28m2)2 s22"# (41) and N/C131 2ps1s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p : (42) Now let u/C13x1/C28m1 (43) w/C13x2/C28m2: (44) Then f(t1;t2) /C30N?g/C12 /C28/C12eit2wexp/C281 2(1/C28r2)w2 s2 2"# ! g/C12 /C28/C12evet1ududw ; (45) where v/C13/C281 2(1/C28r2)1 s21u2/C282rs1w s2u"# N?/C13ei(t1m1/C27t2m2) 2ps1s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28r2p : (46) COMPLETE THE SQUARE in the inner integral g/C12 /C28/C12exp/C281 2(1/C28r2)1 s2 1u2/C282rs1w s2u"# () et1udu /C30g/C12 /C28/C12exp/C281 2s21(1/C28r2)u/C28r1s1w s2"#28 < :9 = ; /C21 2s2 1(1/C28r2)r1s1w s2 !28 < :9 = ;eit1udu: (47) Rearranging to bring the exponential depending on w outside the inner integral, letting v/C13u/C28rs1w s2; (48) and writing eit1u /C30cos(t1u) /C27i sin(t1u) (49) gives f(t1 ; t2) /C30N ?g/C12 /C28/C12eit2w exp /C281 2 s2 2(1 /C28 r2)w2"# /C2expr2 2s22(1 /C28 r2) w2"# g/C12 /C28/C12exp /C281 2s22(1 /C28 r2) v2"# /C2 cos t1v /C27rs1w s2 !"# /C27i sin t1v /C27rs1w s2 !"# () dvdw : (50) Expanding the term in braces gives cos(t1v)cosrs1wt1 s2 ! /C28sin(t1v)sinrs1w s2t1 ! "# /C27i sin(t1v)cosrs1w s2t1 ! /C27cos(t1v)sinrs1wt1 s2 ! "# /C30 cosrs1wt1 s2 ! /C27i sinrs1wt1 s2 ! "# /C2 [cos(t1v) /C27i sin(t1v)] /C30expi rs1w s2t1 ! [cos(t1v) /C27i sin(t1v)] : (51) But e /C28ax2 sin(bx)is ODD, so the integral over the sine term vanishes, and we are left with f(t1 ; t2) /C30N ?g/C12 /C28/C12eit2w exp /C28w2 2s22"# expr2w2 2s22(1 /C28 r2)"# /C29expi rs1wt1 s2"# dwg/C12 /C28/C12exp /C28v2 2s21(1 /C28 r2)"# cos(t1v) dv /C30N ?g/C12 /C28/C12exp iw t2 /C27t1rs1 s2 ! !"# exp /C28w2 2 s22"# dw g/C12 /C28/C12exp /C28v2 2 s21(1 /C28 r2)"# cos(t1v) dv : (52) Now evaluate the GAUSSIAN INTEGRAL g/C12 /C28/C12eikxe /C28ax2 dx /C30g/C12 /C28/C12e /C28ax2 cos(kx) dx /C30ffiffiffi p as e /C28k2 =4a (53) to obtain the explicit form of the CHARACTERISTIC FUNCTION ,f(t1 ; t2) /C30ei(t1 m1 /C27t2 /C27m2) 2ps1 s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 r2p /C2 s2ffiffiffiffiffiffi 2pp exp /C281 4t2 /C27 rs1 s2t1=z1*=z1+2 2s2 2=zn;=zn1=zn*=zn+ /C2 s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p(1 /C28p2)p exp[/C282 12s21(1 /C28 r2)]no /C30ei(t1m1/C27t2m2)exp f/C281 2[t2 2s22/C272rs1s2t1t2/C27r2s21t21 /C27(1/C28r2)s2 1t21]g /C30exp[i(t1m1/C27t2m2)/C281 2(s2 1t21/C272rs1s2t1t2/C27s21t21)]:(54) See also BOX-MULLER TRANSFORMATION ,G AUSSIAN DISTRIBUTION ,G AUSSIAN MULTIVARIATE DISTRIBU- TION ,NORMAL DISTRIBUTION ,PRICE’S THEOREM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 936 /C1/37, 1972. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Kotz, S.; Balakrishnan, N.; and Johnson, N. L. "Bivariate and Trivariate Normal Distributions." Ch. 46 in Contin- uous Multivariate Distributions, Vol. 1: Models and Ap-plications, 2nd ed. New York: Wiley, pp. 251 /C1 /48, 2000. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 118, 1992. Whittaker, E. T. and Robinson, G. "Determination of the Constants in a Normal Frequency Distribution with TwoVariables" and "The Frequencies of the Variables TakenSingly." §161/C1 /62 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 324 /C1/28, 1967. Gaussian Brackets A notation published by Gauss in Disquisitiones Arithmeticae and defined by ½/C138/C301 (1) a1½/C138/C30a1 (2) a1;a2 ½/C138 /C30a1½/C138a2/C27½/C138 (3) [a1;a2;...;an] /C30[a1;a2;...;an/C281]an/C27[a1;a2;...;an/C282]:(4) Gaussian brackets are useful for treating CONTINUED FRACTIONS because 1 a1/C271 a2/C271 a3/C27.../C271 an/C30a2;an ½/C138 a1;an ½/C138: (5) The NOTATION [x] conflicts with that of G AUSSIAN POLYNOMIALS and the NINT function. References Herzberger, M. Modern Geometrical Optics. New York: Interscience Publishers, pp. 457 /C1/62, 1958. Gaussian Coefficient Q-BINOMIAL COEFFICIENT Gaussian Coordinate System A coordinate system which has a METRIC satisfying gii/C30/C281 and @gij=@xj/C300:/ Gaussian Curvature An intrinsic property of a space independent of the coordinate system used to describe it. The Gaussian curvature of a REGULAR SURFACE inR3at a point pis formally defined as K(p)/C30det(S(p)); (1) where Sis the SHAPE OPERATOR and det denotes the DETERMINANT . Ifx:U0R3is a REGULAR PATCH , then the Gaussian curvature is given by K/C30eg/C28f2 EG/C28F2; (2) where E,F, and Gare coefficients of the first FUNDAMENTAL FORM and e,f, and gare coefficients of the second FUNDAMENTAL FORM (Gray 1997, p. 377). The Gaussian curvature can be given entirelyin terms of the first FUNDAMENTAL FORM ds2/C30Ed u2/C272Fd ud v /C27Gd v2(3) and the DISCRIMINANT g/C13EG/C28F2(4) by K/C301 ffiffiffigp@ @vffiffiffigp EG2 11 ! /C28@ @uffiffiffigp EG2 12 ! "# ; (5) where Gkijare the CONNECTION COEFFICIENTS . Equiva- lently, K/C301 g2EF@F @v/C281 2@G @u FG1 2@G @v 1 2@E @uk23 k33=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n/C28 1 g2EF1 2@E @v FG1 2@G @u 1 2@E @v12@G @v0=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n; (6) where k23/C13@F @u/C281 2@E @v(7) k33/C13/C2812@2E @v2/C27@2F @u@v/C2812@2G @u2: (8)Writing this out, K/C301 2g2@2F @u@v/C28@2E @v2/C28@2G @u2"# /C28G 4g2@E @u2@F @v/C28@G @u ! /C28@E @v !22 435/C27 F 4g2 /C2@E @u@G @v/C282@E @v@G @u/C272@F @u/C28@E @v ! 2@F @v/C28@G @u ! "# /C28E 4g2@G @v2@F @u/C28@E @v ! /C28@G @u !22435: (9) The Gaussian curvature is also given by K/C30 det(xuuxuxv)det(xvvxuxv)/C28[det(xuvxuxv)]2 [½xu½2½xv½2/C28(xu /C215xv)2]2(10) (Gray 1997, p. 380), as well as K/C30[ˆNˆN1ˆN2] ffiffiffigp/C30eij[ˆNˆTˆTi]jffiffiffigp ; (11) where eijis the L EVI-CIVITA SYMBOL ,ˆNis the unit NORMAL VECTOR and ˆTis the unit TANGENT VECTOR . The Gaussian curvature is also given by K/C30/C28R 2/C30k1k2/C301 R1R2; (12) where Ris the CURVATURE SCALAR ,k1andk2the PRINCIPAL CURVATURES , and R1andR2the PRINCIPAL RADII OF CURVATURE . For a M ONGE PATCH with z/C30 h(u;v); K/C30huuhvv/C28h2 uv (1/C27h2 u/C27h2v)2: (13) The Gaussian curvature Kand MEAN CURVATURE H satisfy H2]K; (14) with equality only at UMBILIC POINTS , since H2/C28K/C301 4(k1/C28k2)2: (15) Ifpis a point on a REGULAR SURFACE MƒR3andvp and wpare tangent vectors to Matp, then the Gaussian curvature of Matpis related to the SHAPE OPERATOR Sby S(vP)/C29S(wP)/C30K(p)vP/C29wP: (16) LetZbe a nonvanishing VECTOR FIELD onMwhich is everywhere PERPENDICULAR toM, and let VandWbe VECTOR FIELDS tangent to Msuch that V/C29W/C30Z; then K /C30Z /C215 (DVZ /C29 DWZ) 2½Z ½4 (17) (Gray 1997, p. 410). For a SPHERE , the Gaussian curvature is K /C301=a2 : For EUCLIDEAN SPACE , the Gaussian curvature is K /C300. For GAUSS- BOLYAI- LOBACHEVSKY SPACE , the Gaussian curvature is K /C30/C281=a2 : A FLAT SURFACE is a REGULAR SURFACE and special class of MINIMAL SURFACE on which Gaussian curvature vanishes everywhere. A point p on a REGULAR SURFACE M /C23R3 is classified based on the sign of K(p) as given in the following table (Gray 1997, p. 375), where S is the SHAPE OPERATOR . Sign Point /K(p) > 0/ ELLIPTIC POINT /K(p) B0/ HYPERBOLIC POINT /K(p) /C300 but S(p) "0/ PARABOLIC POINT /K(p) /C300 and S(p) /C300/ PLANAR POINT A surface on which the Gaussian curvature K is everywhere POSITIVE is called SYNCLASTIC , while a surface on which K is everywhere NEGATIVE is called ANTICLASTIC . Surfaces with constant Gaussian cur- vature include the CONE , CYLINDER ,KUEN SURFACE , PLANE ,PSEUDOSPHERE , and SPHERE . Of these, the CONE and CYLINDER are the only FLAT SURFACES OF REVOLUTION . See also ANTICLASTIC ,BRIOSCHI FORMULA ,DEVELOP- ABLE SURFACE ,E LLIPTIC POINT ,F LAT SURFACE , HYPERBOLIC POINT ,INTEGRAL CURVATURE ,M EAN CURVATURE ,M ETRIC TENSOR ,M INIMAL SURFACE , PARABOLIC POINT ,PLANAR POINT ,SYNCLASTIC ,UM- BILIC POINT References Gray, A. "The Gaussian and Mean Curvatures" and "Sur- faces of Constant Gaussian Curvature." §16.5 and Ch. 21 inModern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 373 /C1/80 and 481 /C1/00, 1997. Gaussian Curve GAUSSIAN DISTRIBUTION Gaussian Differential Equation HYPERGEOMETRIC DIFFERENTIAL EQUATIONGaussian Distribution The Gaussian probability distribution with MEAN m and STANDARD DEVIATION sis a normalized G AUSSIAN FUNCTION OF THE FORM P(x)/C301 sffiffiffiffiffiffi 2pp e/C28(x/C28m)2=(2s2); (1) where P(x)dxgives the probability that a variate with a Gaussian distribution takes on a value in the range [ x;x/C27dx]:Statisticians commonly call this distribution the NORMAL DISTRIBUTION and, because of its curved flaring shape, social scientists refer to itas the "bell curve." The distribution P(x) is properly normalized for x/C23(/C28/C12/C12 ) since g/C12 /C28/C12P(x)dx/C301: (2) The cumulative DISTRIBUTION FUNCTION , which gives the probability that a variate will assume a value 5x; is then the integral of the G AUSSIAN FUNCTION , D(x)/C30gx /C28/C12P(x)dx/C301 sffiffiffiffiffiffi 2ppgx /C28/C12e/C28(x/C28m)2=(2s2)dx:(3) Gaussian distributions have many convenient proper- ties, so random variates with unknown distributionsare often assumed to be Gaussian, especially in physics and astronomy. Although this can be a dangerous assumption, it is often a good approxima-tion due to a surprising result known as the CENTRAL LIMIT THEOREM . This theorem states that the MEAN of any set of variates with any distribution having afinite MEAN and VARIANCE tends to the Gaussian distribution. Many common attributes such as test scores, height, etc., follow roughly Gaussian distribu- tions, with few members at the high and low ends andmany in the middle. Gaussian distributions are frequently invoked in situations where they may not be applicable. As Lippmann stated, "Everybody be-lieves in the exponential law of errors: the experi- menters, because they think it can be proved by mathematics; and the mathematicians, because theybelieve it has been established by observation" (Whit- taker and Robinson 1967, p. 179). Making the transformation z/C13 x/C28m s; (4) so that dz/C30dx=s;gives a variate with VARIANCE s2/C30 1 and MEAN m/C300;transforming P(x)dxinto P(z)dz/C301ffiffiffiffiffiffi 2pp e/C28z2=2dz: (5) The distribution having this probability function is known as a standard NORMAL DISTRIBUTION , and z defined in this way is known as a Z-SCORE . The NORMAL DISTRIBUTION FUNCTION F(z) gives the probability that a standard normal variate assumes avalue in the interval [0 ;z]; F(z)/C131ffiffiffiffiffiffi 2ppgz 0e/C28x2=2dx/C301 2erfzffiffiffi 2p ! ; (6) where ERF is a function sometimes called the error function. Neither F(z) nor ERF can be expressed in terms of finite additions, subtractions, multiplica- tions, and ROOT EXTRACTIONS , and so both must be either computed numerically or otherwise approxi-mated. The value of afor which P(x) falls within the interval [ /C28a;a] with a given probability Pis called theP CONFIDENCE INTERVAL . The Gaussian distribution is also a special case of the CHI-SQUARED DISTRIBUTION , since making the substi- tution 1 2z/C13(x/C28m)2 2s2(7) gives d(12z)/C30(x/C28m) s2dx/C30ffiffiffizp sdx: (8) Now, the real line x/C23(/C28/C12/C12 ) is mapped onto the half-infinite interval z/C23[0;/C12) by this transforma- tion, so an extra factor of 2 must be added to d(z=2); transforming P(x)dxinto P(z)dz/C301 sffiffiffiffiffiffi 2pp e/C28z=2sffiffiffizp2(1 2dz)/C30e/C28z=2z/C281=2 21=2G1 2=z1*=z1+ dz (9) (Kenney and Keeping 1951, p. 98), where use has been made of the identity G(1=2)/C30ffiffiffipp:As promised, (9) is a CHI-SQUARED DISTRIBUTION inzwith r/C301 (and also a GAMMA DISTRIBUTION with a/C301=2 and (u/C302)):/ The ratio X=Yof independent Gaussian-distributed variates with zero MEAN is distributed with a C AUCHY DISTRIBUTION . This can be seen as follows. Let Xand Yboth have MEAN 0 and standard deviations of sx andsy;respectively, then the joint probability density function is the G AUSSIAN BIVARIATE DISTRIBUTION with r/C300; f(x;y)/C301 2psxsye/C28[x2=(2s2 x)/C27y2=(2s2y)]: (10) From RATIO DISTRIBUTION , the distribution of U/C30 Y=XisP(u)/C30g/C12 /C28/C12xjjf(x;ux)dx /C301 2psxsyg/C12 /C28/C12xjje/C28[x2=(2s2x)/C27u2x2=(2s2y)]dx /C301 psxsyg/C12 0xexp/C28x21 2s2 x/C27u2 2s2y !"# dx:(11) But g/C12 0xe/C28ax2dx /C30/C281 2ae/C28ax2"#/C12 0/C301 2a[0/C28(/C281)]/C301 2a; (12) so P(u)/C301 psxsy1 21 2s2 x/C27u2 2s2y ! /C301 psxsy u2s2 x/C27s2y /C301 psy sx u2/C27sy sx !2; (13) which is a C AUCHY DISTRIBUTION with MEAN m/C300 and full width G/C302sy sx: (14) The CHARACTERISTIC FUNCTION for the Gaussian distribution is f(t)/C30eimt/C28s2t2=2; (15) and the MOMENT-GENERATING FUNCTION is M(t)/C30etxhi/C30g/C12 /C28/C12etx sffiffiffiffiffiffi 2pp e/C28(x/C28m)2=2s2dx: /C301 sffiffiffiffiffiffi 2ppg/C12 /C28/C12exp/C281 2s2[x2/C282(m/C27s2t)x/C27m2]() dx: (16) COMPLETING THE SQUARE in the exponent, 1 2s2[x2/C282(m/C27s2t)x/C27m2] /C301 2s2f[x/C28(m/C27s2t)]2/C27[m2/C28(m/C27s2t)2]g (17) Let y/C13x/C28(m/C27s2t) (18) dy/C30dx (19) a/C131 2s2: (20) The integral then becomes M(t)/C301 sffiffiffiffiffiffi 2ppg/C12 /C28/C12exp/C28ay2/C272ms2t/C27s4t2 2s2"# dy /C301 sffiffiffiffiffiffi 2ppg/C12 /C28/C12exp[/C28ay2/C27mt/C271 2s2t2]dy /C301 sffiffiffiffiffiffi 2pp emt/C27s2t2=2g/C12 /C28/C12e/C28ay2dy /C301 sffiffiffiffiffiffi2ppffiffiffi p as emt/C27s2t2=2/C30ffiffiffiffiffiffiffiffiffiffiffi 2s2pp sffiffiffiffiffiffi 2pp emt/C27s2t2=2 /C30emt/C27s2t2=2; (21) so M?(t)/C30(m/C27s2t)emt/C27s2t2=2(22) M?(t)/C30s2emt/C27s2t2=2/C27emt/C27s2t2=2(m/C27ts2)2; (23) and m/C30M?(0)/C30m (24) s2/C30M??(0)/C28[M?(0)]2/C30(s2/C27m2)/C28m2/C30s2: (25) These can also be computed using R(t)/C30ln[M(t)]/C30mt/C271 2s2t2(26) R?(t)/C30m/C27s2t (27) Rƒ(t)/C30s2; (28) yielding, as before, m/C30R?(0)/C30m (29) s2/C30Rƒ(0)/C30s2: (30) The raw moments can also be computed directly by computing the MOMENTS about the origin m?n/C13xnhi; m?n/C301 sffiffiffiffiffiffi 2ppg/C12 /C28/C12xne/C28(x/C28m)2=2s2dx: (31) (Papoulis 1984, pp. 147 /C1/48). Now let u/C13x/C28mffiffiffiffiffiffi2sp (32) du/C30dxffiffiffiffiffiffi2sp (33) x/C30suffiffiffi 2p /C27m; (34) giving the raw moments in terms of G AUSSIAN INTEGRALS ,m?n/C30ffiffiffiffiffiffi 2sp sffiffiffiffiffiffi2ppg/C12 /C28/C12xne/C28u2du/C301ffiffiffippg/C12 /C28/C12xne/C28u2du:(35) Evaluating these integrals gives m?0/C301 (36) m?1/C30m (37) m?2/C30m2/C27s2(38) m?3/C30m(m2/C273s2) (39) m?4/C30m4/C276m2s2/C273s4: (40) Now find the MOMENTS about the MEAN , m1/C300 (41) m2/C30s2(42) m3/C300 (43) m4/C303s4; (44) so the VARIANCE ,SKEWNESS , and KURTOSIS are given by var(x)/C30s2(45) g1/C30m3 s3/C300 (46) g2/C30m4 s4/C283/C303s4 s4/C283/C300 (47) Cramer showed in 1936 that if Xand Yare INDE- PENDENT variates and X/C27Yhas a Gaussian distribu- tion, then both XandYmust be Gaussian (C RAMER’S THEOREM ). An easier result states that the sum of n variates each with is Gaussian distribution also has a Gaussian distribution. This follows from the result Pn(x)/C30F/C281f[f(t)]ng/C30e/C28(x/C28nm)2=(2ns2) ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pns2p ; (48) where f(t) is the CHARACTERISTIC FUNCTION and F/C281[f] is the inverse F OURIER TRANSFORM , taken with parameters a/C30b/C301:/ The VARIANCE of the SAMPLE VARIANCE s2for a general distribution is given by var(s2)/C30(N/C281)[(N/C281)m?4/C28(N/C283)m?22] N3; (49) which simplifies in the case of a Gaussian distribu- tion to var(s2)/C302(N/C281)(m4/C272Nm2s2/C27Ns4) N3(50) which, if m/C300;further simplifies to var(s2) /C302s4(N /C28 1) N2 (51) (Kenney and Keeping 1951, p. 164). The CUMULANT-GENERATING FUNCTION for a Gaussian distribution is K(h) /C30ln(e n1hes2h2 =2) /C30 n1h /C271 2 s2h2 ; (52) so k1 /C30 n1 (53) k2 /C30 s2 (54) kr /C300 for r > 2: (55) For Gaussian variates, kr /C300 for r /C212, so the var- iance of K-STATISTIC k3 is var(k3) /C30k6 N /C279 k2 k4 N /C28 1 /C279 k2 3 N /C28 1 /C276k32 N(N /C28 1)(N /C28 2) /C306k32 N(N /C28 1)(N /C28 2) : (56) Also, var(k4) /C3024k42N(N /C28 1)2 (N /C28 3)(N /C28 2)(N /C27 3)(N /C27 5)(57) var(g1) /C306N(N /C28 1) (N /C28 2)(N /C27 1)(N /C27 3)(58) var(g2) /C3024N(N /C28 1)2 (N /C28 3)(N /C28 2)(N /C27 3)(N /C27 5) ; (59) where g1 /C13k3 k3=2 2(60) g2 /C13k4 k2 2: (61) If P(x) is a Gaussian distribution, then D(x) /C301 21 /C27erfx /C28 m sffiffiffi 2p !"# ; (62) so variates xiwith a Gaussian distribution can be generated from variates yihaving a UNIFORM DIS- TRIBUTION in (0,1) via xi /C30 sffiffiffi 2p erf /C281(2yi /C281) /C27 m: (63) However, a simpler way to obtain numbers with a Gaussian distribution is to use the BOX-MULLER TRANSFORMATION . The Gaussian distribution is an approximation to the BINOMIAL DISTRIBUTION in the limit of large numbers,P(n1) /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pNpqp exp /C28(n1 /C28 Np)2 2Npq"# ; (64) where n1is the number of steps in the POSITIVE direction, N is the number of trials ( (N /C13n1 /C27n2)); and p and q are the probabilities of a step in the POSITIVE direction and NEGATIVE direction (/ (q /C131 /C28p)):/ The differential equation having a Gaussian distribu- tion as its solution is dy dx/C30y(m/C28x) s2; (65) since dy y/C30m/C28x s2dx (66) lny yo ! /C30/C281 2s2(m/C28x)2(67) y/C30y0e/C28(x/C28m)2=2s2: (68) This equation has been generalized to yield more complicated distributions which are named using the so-called P EARSON SYSTEM . See also BINOMIAL DISTRIBUTION ,B OX-MULLER TRANSFORMATION ,C ENTRAL LIMIT THEOREM ,E RF, GAUSSIAN BIVARIATE DISTRIBUTION ,G AUSSIAN DIS- TRIBUTION– LINEAR COMBINATION OF VARIATES ,GAUS- SIAN FUNCTION ,L OGIT TRANSFORMATION ,N ORMAL DEVIATES ,N ORMAL DISTRIBUTION ,N ORMAL DISTRI- BUTION FUNCTION ,PEARSON SYSTEM ,RATIO DISTRI- BUTION , Z-SCORE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 533 /C1/34, 1987. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Kraitchik, M. "The Error Curve." §6.4 in Mathematical Recreations. New York: W. W. Norton, pp. 121 /C1/23, 1942. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 109 /C1/11, 1992. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 285 /C1/90, 1999. Whittaker, E. T. and Robinson, G. "Normal Frequency Distribution." Ch. 8 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 164 /C1/08, 1967. Gaussian Distribution Linear Combination of Variates IfxisNORMALLY DISTRIBUTED with MEAN mand VARIANCE s2;then a linear function of x, y/C30ax/C27b; (1) is also NORMALLY DISTRIBUTED . The new distribution has MEAN am/C27band VARIANCE a2s2;as can be derived using the MOMENT-GENERATING FUNCTION M(t) /C30 et(ax /C27b)=z1;=z11 /C30etb eatxhi/C30etbe mat /C27s2(at)2 =2 etb /C27mat /C27s2a2t2 =2 /C30e(b/C27a m)t /C27a2 s2t2 =2 ; (2) which is of the standard form with m ?/C30b /C27a (3) s?2 /C30a2 s2 : (4) For a weighted sum of independent variables y /C13Xn i /C301aixi ; (5) the expectation is given by M(t) /C30 eythi/C30 exp tXn i/C301aixi !*+ /C30 ea1tx1 ea2tx2 /C1/C1/C1eantxn hi /C30Yn i/C301eaitxi hi /C30Yn i /C301exp(ai mit /C271 2 a2 i s2i t2) : (6) Setting this equal to exp( mt /C271 2 s2t2) (7) gives m /C13Xn i/C301ai mi (8) s2 /C13Xn i/C301a2 i s2i : (9) Therefore, the MEAN and VARIANCE of the weighted sums of n RANDOM VARIABLES are their weighted sums. If xiare INDEPENDENT and NORMALLY DISTRIBUTED with MEAN 0 and VARIANCE s2 ; define yi /C13X jcijxj ; (10) where c obeys the ORTHOGONALITY CONDITION cikcjk /C30 dij ; (11) with dijthe K RONECKER DELTA . Then yiare also independent and normally distributed with MEAN 0 and VARIANCE s2:/ See also GAUSSIAN DISTRIBUTION Gaussian Elimination A method for solving MATRIX EQUATIONS OF THE FORM Ax/C30b: (1)To perform Gaussian elimination starting with the system of equations a11a12 /C1/C1/C1 a1k a21a22 /C1/C1/C1 a2k nn:::n ak1ak2/C1/C1/C1 akk2 6643 775x 1 x2 n xk2 6643 775/C30b 1 b2 n bk2 6643 775; (2) compose the "augmented matrix equation" a 11a12 /C1/C1/C1 a1k a21a22 /C1/C1/C1 a2k nn:::n ak1ak2/C1/C1/C1 akkb1 b2 n bk=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n3 775x 1 x2 n xk2 6643 775:2 664(3) Here, the COLUMN VECTOR in the variables xis carried along for labeling the matrix rows. Now, perform ELEMENTARY ROW AND COLUMN OPERATIONS to put the augmented matrix into the UPPER TRIAN- GULAR form a?11a?12 /C1/C1/C1 a?1k 0a?22 /C1/C1/C1 a?2k nn:::n 00 /C1/C1/C1 a?kkb?1 b?2 n b?k=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n3 775:2 664(4) Solve the equation of the kth row for x k;then substitute back into the equation of the ( k/C281)/st row to obtain a solution for xk/C281;etc., according to the formula xi/C301 a?iib?i/C28Xk j/C30i/C271a?ijxj ! : (5) For example, consider the MATRIX EQUATION 934 4341112 435x 1 x2 x32435/C307 832 435: (6) In augmented form, this becomes 934 434111783=z1n=z1n=z1n=z1n=z1n=z1n3 5x 1 x2 x32435:24 (7) Switching the first and third rows gives 111 434934387=z1n=z1n=z1n=z1n=z1n=z1n3 5x 1 x2 x32435:24 (8) Subtracting 9 times the first row from the third row gives 111 434 0 /C286 /C2853 8 /C2820=z1n=z1n=z1n=z1n=z1n=z1n3 5x 1 x2 x32435:24 (9) Subtracting 4 times the first row from the second row gives 111 0 /C2810 0 /C286 /C2853 /C284 /C2820=z1n=z1n=z1n=z1n=z1n=z1n3 5x 1 x2 x32435:24 (10) Finally, adding /C286 times the second column to the third one gives 111 0 /C2810 00 /C2853 /C284 4=z1n=z1n=z1n=z1n=z1n=z1n3 5x 1 x2 x32435:24 (11) Restoring the transformed matrix equation gives 111 0 /C2810 00 /C2852 435x 1 x2 x32435/C303 /C284 42 435; (12) which can be solved immediately to give x 3 /C30/C284=5; back-substituting to obtain x2 /C304 (which actually follows trivially in this example), and then again back-substituting to find x1/C30/C281=5/ See also CONDENSATION ,E LEMENTARY ROW AND COLUMN OPERATIONS ,G AUSS- JORDAN ELIMINATION , LU DECOMPOSITION ,M ATRIX EQUATION ,S QUARE ROOT METHOD References Bareiss, E. H. "Multistep Integer-Preserving Gaussian Elimination." Argonne National Laboratory Report ANL- 7213, May 1966. Bareiss, E. H. "Sylvester’s Identity and Multistep Integer- Preserving Gaussian Elimination." Math. Comput. 22, 565/C1/78, 1968. Garbow, B. S. "Integer-Preserving Gaussian Elimination." Program P-158 (3600F), Applied Mathematics Division,Argonne National Laboratory, Nov. 21, 1966. Gentle, J. E. "Gaussian Elimination." §3.1 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer-Verlag, pp. 87 /C1 /1, 1998.Gaussian Function In 1-D, the Gaussian function is the function from the GAUSSIAN DISTRIBUTION , f(x)/C301 sffiffiffiffiffiffi 2ppe/C28(x/C28m)2=2s2; (1) sometimes also called the FREQUENCY CURVE . The FULL WIDTH AT HALF MAXIMUM (FWHM) for a Gaus- sian is found by finding the half-maximum points x0: The constant scaling factor can be ignored, so we must solve e/C28(x0/C28m)2=2s2/C301 2f(xmax) (2) Butf(xmax) occurs at xmax/C30m;so e/C28(x0/C28m)2=2s2/C3012f(m)/C3012: (3) Solving, e/C28(x0/C28m)2=2s2/C302/C281(4) /C28(x0/C28m)2 2s2/C30/C28ln 2 (5) (x0/C28m)2/C302s2ln 2 (6) x09sffiffiffiffiffiffiffiffiffiffiffiffiffi 2l n2p /C27m: (7) The FULL WIDTH AT HALF MAXIMUM is therefore given by FWHM /C13x/C27/C28x /C302ffiffiffiffiffiffiffiffiffiffiffiffiffi 2ln2p s :2:3548s : (8) In 2-D, the circular Gaussian function is the distribu- tion function for uncorrelated variables x and y having a GAUSSIAN BIVARIATE DISTRIBUTION and equal STANDARD DEVIATION s /C30 sx /C30 sy ; f(x; y) /C301 2ps2 e /C28[(x/C28 mz)2/C27(y/C28my)2]=2s2 : (9) The corresponding elliptical Gaussian function corre- sponding to sx " sy is given by f(x; y) /C301 2psx sye /C28[(x /C28 mz)2 =2s2 z/C27(y/C28my)2 =2 s2y ] : (10) The Gaussian function can also be used as an APODIZATION FUNCTION , shown above with the corre- sponding INSTRUMENT FUNCTION . The HYPERGEOMETRIC FUNCTION is also sometimes known as the Gaussian function. See also ERF,ERFC,FOURIER TRANSFORM– GAUSSIAN , GAUSSIAN BIVARIATE DISTRIBUTION ,G AUSSIAN DIS- TRIBUTION ,NORMAL DISTRIBUTION References MacTutor History of Mathematics Archive. "Frequency Curve." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Frequency.html. Gaussian Hypergeometric Series HYPERGEOMETRIC FUNCTION Gaussian Integer A COMPLEX NUMBER a /C27bi where a and b are INTEGERS . The Gaussian integers are members of the IMAGINARY QUADRATIC FIELD Q(ffiffiffiffiffiffi /C281p ) and form aRING often denoted Z[i]: The sum, difference, and product of two Gaussian integers are Gaussian integers, but (a /C27bi) ½(c /C27di) only if there is an e /C27fi such that (a /C27bi)(e /C27fi) /C30(ae /C28bf) /C27(af /C27be)i /C30c /C27di : Gaussian integers can be uniquely factored in terms of other Gaussian integers (known as GAUSSIAN PRIMES )upto POWERS of i and rearrangements. The units of Z[i] are 9 1 and 9i; and the norm of a Gaussian integer is defined by n(x /C27iy) /C30x2 /C27y2 : Every Gaussian integer is within njj=ffiffiffi 2p of a multiple of a Gaussian integer n. See also COMPLEX NUMBER ,E ISENSTEIN INTEGER , GAUSSIAN PRIME ,INTEGER ,OCTONION References Conway, J. H. and Guy, R. K. "Gauss’s Whole Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 217 /C1/23, 1996. Se´roul, R. "The Gaussian Integers." §9.1 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 225 /C1/34, 2000. Shanks, D. "Gaussian Integers and Two Applications." §50 in Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 149 /C1/51, 1993. Gaussian Integral The Gaussian integral, also called the PROBABILITY INTEGRAL and closely related to the ERFfunction, is the integral of the 1-D G AUSSIAN FUNCTION over (/C28/C12;/C12):It can be computed using the trick of combining two 1-D Gaussians g/C12 /C28/C12e/C28x2dx/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g/C12 /C28/C12e/C28y2dy=z1r=z1>g/C12 /C28/C12e/C28x2dx=z1r=z1>s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g/C12 /C28/C12g/C12 /C28/C12e/C28(x2/C27y2)dy dxs (1) and switching to POLAR COORDINATES , g/C12 /C28/C12e/C28x2dx/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2p 0g/C12 0e/C28r2rd rd us /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p/C281 2e/C28r2hi/C12 0r /C30ffiffiffipp: (2) However, a simple proof can also be given which does not require transformation to POLAR COORDINATES (Nicholas and Yates 1950). The integral from 0 to a finite upper limit acan be given by the CONTINUED FRACTION ga 0e/C28t2dt/C301 2ffiffiffipperfa 1 2ffiffiffipp/C28e/C28a2 2a/C271 a/C272 2a3 a/C274 2a/C27...; (3) first stated by Laplace, proved by Jacobi, and redis- covered by Ramanujan (Watson 1928; Hardy 1999, pp. 8/C1/). The general class of integrals OF THE FORM In(a)/C13g/C12 0e/C28ax2xndx (4) can be solved analytically by setting x/C13a/C281=2y (5) dx/C30a/C281=2dy (6) y2/C30ax2: (7) Then In(a)/C30a/C281=2g/C12 0e/C28y2(a/C281=2y)ndy /C30a/C28(n/C271)=2g/C12 0e/C28y2yndy: (8) Forn/C300, this is just the usual Gaussian integral, so I0(a)/C30ffiffiffipp 2a/C281=2/C301 2ffiffiffiffiffi pa:s (9) Forn/C301, the integrand is integrable by quadrature, I 1(a)/C30a/C281g/C12 0e/C28y2yd y/C30a/C281/C281 2e/C28y2hi/C12 0/C301 2a/C281:(10) To compute In(a) for n/C211, use the identity /C28@ @aIn/C282(a)/C30/C28@ @ag/C12 0e/C28ax2xn/C282dx /C30/C28g/C12 0/C28x2e/C28ax2xn/C282dx /C30g/C12 0e/C28ax2xndx/C30In(a): (11) Forn/C302sEVEN , In(a)/C30/C28@ @a ! In/C282(a)/C30/C28@ @a !2 In/C284 /C30.../C30/C28@ @a !n=2 I0(a) /C30@n=2 @an=2I0(a)/C30ffiffiffipp 2@n=2 @an=2a/C281=2; (12) sog/C12 0x2se/C28ax2dx/C30(s/C281 2)! 2as/C271=2/C30(2s/C281)!! 2s/C271asffiffiffi p as : (13) Ifn/C302s/C271i s ODD, then In(a)/C30/C28@ @a ! In/C282(a)/C30/C28@ @a !2 In/C284(a) /C30.../C30/C28@ @a !(n/C281)=2 I1(a) /C30@(n/C281)=2 @a(n/C281)=2I1(a)/C301 2@(n/C281)=2 @a(n/C281)=2a/C281; (14) so g/C12 0x2s/C271e/C28ax2dx/C30s! 2as/C271: (15) The solution is therefore g/C12 0e/C28ax2xndx /C30(n/C281)!! 2n=2/C271an=2ffiffiffi p as forneven 1 2(n/C281)hi ! 2a(n/C271=2)fornodd:8 >>>>>< >>>>>:(16) The first few values are therefore I 0(a)/C301 2ffiffiffi p as (17) I1(a)/C301 2a(18) I2(a)/C301 4affiffiffi p as (19) I3(a)/C301 2a2(20) I4(a)/C303 8a2ffiffiffi p as (21) I5(a)/C301 a3(22) I6(a)/C3015 16a3ffiffiffi p as : (23) A related, often useful integral is Hn(a)/C131ffiffiffippg/C12 /C28/C12e/C28ax2xndx; (24) which is simply given by Hn(a) /C302In(a)ffiffiffipp for n even 0 for n odd:8 < : (25) The more general integral of xne /C28ax2/C27bxhas the following closed forms g/C12 /C28/C12xne /C28ax2/C27bx dx /C30i /C28na/C28(n/C271)=2ffiffiffippeb2 =(4a)U(/C281 2 n;12; /C28b2 =4a) (26) /C30ffiffiffi p as eb2 =(4a)Xn/C281 k /C300n! k!(n /C28 2k)!(2b)n/C282k (4a)n/C28k (27) /C30ffiffiffi p as eb2 =(4a)Xn/C281 k /C300n 2kðÞ(2k /C281)!!(2 a)k /C28nbn/C282k (28) for integer n /C210 (F. Pilolli), where U(a; b; x)isa CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SEC- OND KIND andn k=z;=z1 is a BINOMIAL COEFFICIENT . See also DIFFERENTIATING UNDER THE INTEGRAL SIGN,ERF,GAUSSIAN DISTRIBUTION ,GAUSSIAN FUNC- TION References Guitton, E. "De´monstration de la formule." Nouv. Ann. Math. 65, 237 /C1/39, 1906. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Nicholas, C. B. and Yates, R. C. "The Probability Integral." Amer. Math. Monthly 57, 412 /C1/13, 1950. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 147 /C1/48, 1984. Watson, G. N. "Theorems Stated by Ramanujan (IV): The- orems on Approximate Integration and Summation of Series." J. London Math. Soc. 3, 282 /C1/89, 1928. Gaussian Joint Variable Theorem Also called the MULTIVARIATE THEOREM . Given an EVEN number of variates from a NORMAL DISTRIBU- TION with MEANS all 0, x1x2 hi/C30 x1hix2hi; (1) x1x2x3x4 hi /C30 x1x2 hi x3x4 hi/C27 x1x3 hi x2x4 hi/C27 x1x4 hi x2x3 hi ; (2) etc. Given an ODD number of variates, x1hi/C300; (3) x1x2x3 hi /C300; (4) etc. Gaussian Mountain Range CAROTID- KUNDALINI FUNCTIONGaussian Multinormal Distribution GAUSSIAN MULTIVARIATE DISTRIBUTION Gaussian Multivariate Distribution A Gaussian p-variate multinormal (or multivariate) distribution is a generalization of the GAUSSIAN BIVARIATE DISTRIBUTION . The p-multivariate distri- bution with mean vector m and COVARIANCE MATRIX S is denoted Np( m; Sigma) : The Gaussian multivariate distribution is implemented in Mathematica asMul- tinormalDistribution [{mu1 , mu2 , ...}, {{sigma11 , sigma12 , ...}, {sigma12 , sigma22 , ...}...}, {x1, x2, ...}] in the Mathematica add-on package Statistics‘MultinormalDistribution‘ (which can be loaded with the command BBStatistics‘ ) (where the matrix a is symmetrical since sij/C30sji):/ See also GAUSSIAN BIVARIATE DISTRIBUTION ,GAUS- SIAN DISTRIBUTION ,JOINT THEOREM ,M ULTIVARIATE THEOREM Gaussian Polynomial Q-BINOMIAL COEFFICIENT ,Q-BRACKET Gaussian Prime Gaussian primes are G AUSSIAN INTEGERS z/C30a/C27bi satisfying one of the following properties. 1. If both aand bare nonzero then, a/C27biis a Gaussian prime IFFa2/C27b2is an ordinary PRIME . 2. If a/C300, then biis a Gaussian prime IFFbjjis an ordinary PRIME andb/C133:/ 3. Ifb/C300, then ais a Gaussian prime IFFajjis an ordinary PRIME anda/C133:/ The above plot of the COMPLEX PLANE shows the Gaussian primes as filled squares. The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ... (Sloane’s A002145). The Gaus-sian primes with ajj;bjj55 are given by /C285/C284i;/C285/C28 2i;/C285/C272i;/C285/C274i;/C284/C285i;/C284/C28i;/C284/C27i;/C284/C275i; /C283/C282i;-3,/C283/C272i;/C282/C285i;/C282/C283i;/C282/C28i;/C282/C27i; /C282/C273i;/C282/C275i;/C281/C284i;/C281/C282i;/C281/C28i;/C281/C27i;/C281/C27 2i;/C281/C274i;/C283i;3i;1/C284i;1/C282i;1/C28i;1/C27i;1/C272i;1/C27 4i ; 2 /C285i ; 2 /C283i ; 2 /C28i; 2 /C27i; 2 /C273i ; 2 /C275i; 3 /C282i ; 3; 3 /C27 2i ; 4 /C285i; 4 /C28i; 4 /C27i ; 4 /C275i ; 5 /C284i; 5 /C282i ; 5 /C272i ; 5 /C274i:/ See also EISENSTEIN INTEGER ,G AUSSIAN INTEGER , MOAT-CROSSING PROBLEM References Gethner, E.; Wagon, S.; and Wick, B. "A Stroll Through the Gaussian Primes." Amer. Math. Monthly 105, 327/C1/37, 1998. Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes." §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33 /C1/6, 1994. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Rademacher, H. Topics in Analytic Number Theory. New York: Springer-Verlag, 1973. Sloane, N. J. A. Sequences A002145/M2624 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Smith, H. J. "Gaussian Primes." http://pweb.netcom.com/ ~hjsmith/GPrimes.html. Wagon, S. "Gaussian Primes." §9.4 in Mathematica in Action. New York: W. H. Freeman, pp. 298 /C1/03, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 85, 1991. Zariski, O. and Samuel, P. Commutative Algebra I. New York: Springer-Verlag, 1958. Gaussian Quadrature Seeks to obtain the best numerical estimate of an integral by picking optimal ABSCISSAS xiat which to evaluate the function f(x):The FUNDAMENTAL THEO- REM OF GAUSSIAN QUADRATURE states that the opti- mal ABSCISSAS of the m-point G AUSSIAN QUADRATURE FORMULAS are precisely the roots of the orthogonal POLYNOMIAL for the same interval and WEIGHTING FUNCTION . Gaussian quadrature is optimal because it fits all POLYNOMIALS up to degree 2 mexactly. Slightly less optimal fits are obtained from R ADAU QUADRA- TURE and L AGUERRE QUADRATURE . /W(x)/ interval /xiare roots of 1 /(/C281;1)//Pn(x)/ /e/C28t// (0;/C12)//Ln(x)/ /e/C28t2 // (/C28/C12;/C12)//Hn(x)/ /(1/C28t2)/C281=2 //(/C281;1)//Tn(x)/ /(1/C28t2)1=2 //(/C281;1)//Un(x)/ /x1=2// (0;1)// x/C281=2P2n/C271(ffiffiffixp)/ /x/C281=2 // (0;1)// PnffiffiffixpðÞ / To determine the weights corresponding to the Gaussian ABSCISSAS xi;compute a L AGRANGE INTER- POLATING POLYNOMIAL forf(x) by lettingp(x)/C30Ym j/C301(x/C28xj) (1) (where Chandrasekhar 1967 uses Finstead of p);so p?(xj)/C30dp dx"# x/C30xj/C30Ym i/C301 i"j(xj/C28xi): (2) Then fitting a L AGRANGE INTERPOLATING POLYNOMIAL through the mpoints gives f(x)/C30Xm j/C301p(x) (x/C28xj)p?(xj)f(xj) (3) for arbitrary points x. We are therefore looking for a set of points xjand weights wjsuch that for a WEIGHTING FUNCTION W(x); gb af(x)W(x)dx/C30gb aXm j/C301p(x)W(x) (x/C28xj)p?(xj)dx f(xj) /C13Xm j/C301wjf(xj); (4) with WEIGHT wj/C301 p?(xj)gb ap(x)W(x) x/C28xjdx: (5) The weights wjare sometimes also called the C HRIS- TOFFEL NUMBER (Chandrasekhar 1967). For orthogo- nal POLYNOMIALS fj(x) with j/C301, ..., n, fj(x)/C30Ajp(x) (6) (Hildebrand 1956, p. 322), where Anis the COEFFI- CIENT ofxninfn(x);then wj/C301 f?n(xj)gb aW(x)f(x) x/C28xjdx /C30/C28An/C271gn Anf?n(xj)fn/C271(x); (7) where gm/C30g[fm(x)]2W(x)dx: (8) Using the relationship fn/C271(xi)/C30/C28An/C271An/C281 A2 ngn gn/C281fn/C281(xi) (9) (Hildebrand 1956, p. 323) gives wj/C30An An/C281gn/C281 f?n(xj)fn/C281(xj): (10) (Note that Press et al. 1992 omit the factor An=An/C281:/) In Gaussian quadrature, the weights are all POSITIVE . The error is given by En /C30f(2n)( j) (2n)!gb aW(x)[p(x)]2 dx /C30gn A2 nf(2n)( j) (2n)!; (11) where a B j Bb (Hildebrand 1956, pp. 320 /C1/21). Other curious identities are Xm k /C300[ fk(x)]2 gk /C30Am Am/C271 gm[ f?m/C271(x)fm(x) /C28 f?m(x) fm/C271(x)] (12) and Xm k /C300[fk(x)]2 gk/C30/C28Am f?m(xi)fm/C271(xi) Am/C271 gm/C301 wi(13) (Hildebrand 1956, p. 323). In the NOTATION of Szego (1975), let x1n B...Bxnn be an ordered set of points in [a, b], and let l1n ; ..., lnn be a set of REAL NUMBERS .Iff(x) is an arbitrary function on the CLOSED INTERVAL [a, b], write the MECHANICAL QUADRATURE as Qn(f) /C30Xn n/C301lnnf(xnn) : (14) Here xnnare the ABSCISSAS and lnnare the COTES NUMBERS . See also CHEBYSHEV QUADRATURE ,C HEBYSHEV- GAUSS QUADRATURE ,C HEBYSHEV -RADAU QUADRA- TURE ,FUNDAMENTAL THEOREM OF GAUSSIAN QUAD- RATURE ,H ERMITE- GAUSS QUADRATURE ,J ACOBI- GAUSS QUADRATURE ,LAGUERRE- GAUSS QUADRATURE , LEGENDRE- GAUSS QUADRATURE ,L OBATTO QUADRA- TURE ,MEHLER QUADRATURE ,RADAU QUADRATURE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 887 /C1/88, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 103, 1990. Arfken, G. "Appendix 2: Gaussian Quadrature." Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 968 /C1/74, 1985. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 461, 1987. Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, 1967. Gauss, C. F. "Methodus nova integralium valores per ap- prox. inveniendi." Werke, Vol. 3. p. 163. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319 /C1/23, 1956. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gaussian Quadratures and Orthogonal Poly- nomials." §4.5 in Numerical Recipes in FORTRAN: The Artof Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 140 /C1/55, 1992. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 37 /C1/8 and 340 /C1/49, 1975. Whittaker, E. T. and Robinson, G. "Gauss’s Formula of Numerical Integration." §80 in The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152 /C1/63, 1967. Gaussian Sum A sum OF THE FORM S(p ; q) /C13Xq /C281 r/C300e /C28pir2p=q ; (1) where p and q are RELATIVELY PRIME INTEGERS . The symbol 8 is sometimes used instead of S. Although the restriction to RELATIVELY PRIME INTEGERS is often useful, it is not necessary, and Gaussian sums can be written so as to be valid for all integer q (Borwein and Borwein 1987, pp. 83 and 86). If (n ; n?) /C301; then S(m; nn?) /C30S(mn?; n)S(mn ; n?) (2) (Nagell 1951, p. 178). Gauss showed that S(1;q)/C301/C28iq 1/C28iffiffiffiqp(3) for ODD q. Written explicitly S(1;q)/C30(i/C271)ffiffiffiqpforq/C130 (mod 4)ffiffiffiqpforq/C131 (mod 4) 0 for q/C132 (mod 4) iffiffiffiqpforq/C133 (mod 4)8 >>< >>:(4) (Nagell 1951, p. 177). Forpandqof opposite PARITY (i.e., one is EVEN and the other is ODD), SCHAAR’S IDENTITY states 1 ffiffiffiqpXq/C281 r/C300e/C28pir2=q/C30e/C28pi=4 ffiffiffippXp/C281 r/C300epir2q=p: (5) Such sums are important in the theory of QUADRATIC RESIDUES . See also KLOOSTERMAN’S SUM,QUADRATIC RESIDUE , SCHAAR’S IDENTITY ,SINGULAR SERIES References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Evans, R. and Berndt, B. "The Determination of Gauss Sums." Bull. Amer. Math. Soc. 5, 107/C1/29, 1981. Katz, N. M. Gauss Sums, Kloosterman Sums, and Mono- dromy Groups. Princeton, NJ: Princeton University Press, 1987. Nagell, T. "The Gaussian Sums." §53 in Introduction to Number Theory. New York: Wiley, pp. 177 /C1/80, 1951. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 132 /C1/ 34, 1994. Gauss-Jackson Method A method for numerical solution of a second-order ordinary differential equation yƒ/C30f(x ; y) first expounded by Gauss. It proceeds by introducing a function d /C282f whose second differences are f. The advantage of this method is that summation to get d/C282 can be done exactly and that each rounding-off error in the correction term arises only a single time (Jeffreys and Jeffreys 1988, p. 300). References Cowell. Appendix to Greenwich Observations. 1909. Jackson, J. Monthly Not. Roy. Astron. Soc. 84, 602 /C1/06, 1924. Jeffreys, H. and Jeffreys, B. S. "The Gauss-Jackson Method." §9.14 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 300 /C1/01, 1988. Gauss-Jacobi Mechanical Quadrature If x1 Bx2 B...Bxndenote the zeros of pn(x); there exist REAL NUMBERS l1 ; l2 ; ... ; ln such that gb ar(x) da(x) /C30 l1 r(x1) /C27 l2 r(x2) /C27.../C27 ln r(xn) ; for an arbitrary POLYNOMIAL of order 2n /C281 and the l ?ns are called CHRISTOFFEL NUMBERS . The distribu- tion da(x) and the INTEGER n uniquely determine these numbers ln :/ References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 47, 1975. Gauss-Jordan Elimination A method for finding a MATRIX INVERSE . To apply Gauss-Jordan elimination, operate on a MATRIX [AI] /C13a11 /C1/C1/C1 a1n10 /C1/C1/C1 0 a21 /C1/C1/C1 a2n01 /C1/C1/C1 0 n::: nnn::: n an1/C1/C1/C1 ann00 /C1/C1/C1 12 6643 775; where I is the IDENTITY MATRIX , to obtain a MATRIX OF THE FORM 10 /C1/C1/C1 0 b11 /C1/C1/C1 b1n 01 /C1/C1/C1 0 b21 /C1/C1/C1 b2n nn ::: nn ::: n 00 /C1/C1/C1 1 bn1/C1/C1/C1 bnn2 6643 775: The MATRIX B /C13b11 /C1/C1/C1 b1n b21 /C1/C1/C1 b2n n::: n bn1/C1/C1/C1 bnn2 6643 775 is then the MATRIX INVERSE of A: The procedure isnumerically unstable unless PIVOTING (exchanging rows and columns as appropriate) is used. Picking the largest available element as the pivot is usually a good choice. See also CONDENSATION ,GAUSSIAN ELIMINATION ,LU DECOMPOSITION ,MATRIX EQUATION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gauss-Jordan Elimination" and "Gaussian Elimination with Backsubstitution." §2.1 and 2.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 27 /C1/2 and 33 /C1/4, 1992. Gauss-Kronrod Quadrature An adaptive GAUSSIAN QUADRATURE method for numerical integration in which error is estimation based on evaluation at special points known as "Kronrod points." By suitably picking these points, abscissas from previous iterations can be reused as part of the new set of points, whereas usual GAUSSIAN QUADRATURE would require recomputation of all abscissas at each iteration. This is particularly important when some specified degree of accuracy is needed but the number of points needed to achieve this accuracy is not known ahead of time. Kronrod (1964) showed how to pick Kronrod points optimally from Gauss-Legendre quadrature, and Patterson (1968, 1969) showed how to compute continued extensions of this kind (Press et al. 1992, p. 154). WithMethod- /C21Automatic , the Mathematica NIn- tegrate command uses Gauss-Kronrod quadrature for 1-D integrals. See also GAUSSIAN QUADRATURE ,N UMERICAL INTE- GRATION ,QUADRATURE References Calvetti, D.; Golub, G. H.; Gragg, W. B. and Reichel, L. "Computation of Gauss-Kronrod Quadrature Rules." Math. Comput. 69, 1035/C1/052, 2000. Calvetti, D.; Golub, G. H.; Gragg, W. B. and Reichel, L. "Computation of Gauss-Kronrod Quadrature Rules." Stan-ford University Scientific Computing/Computational Mathematics Report SCCM-98 /C1 /9. http://www-sccm.stan- ford.edu/nflash/nf-publications-tech.html#start-1998. Kronrod, A. S. [Russian]. Doklady Akad. Nauk SSSR 154, 283/C1/86, 1964. Patterson, T. N. L. Math. Comput. 22, 847/C1/56 and C1-C11, 1968. Patterson, T. N. L. Math. Comput. 23, 892, 1969. Pessens, R.; de Doncker, E.; Uberhuber, C. W.; and Kaha- ner, D. K. QUADPACK: A Subroutine Package for Auto- matic Integration. New York: Springer-Verlag, 1983. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 154, 1992. Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, pp. 105 /C1/06, 1997. Gauss-Kummer Series 2F1(/C281 2;/C2812;1; h2) /C30X/C12 n /C30012 n=z1r=z1>2 h2n /C301 /C2714 h2 /C271 64 h4 /C271 256 h6 /C27... (Sloane’s A056981 and A056982), where 2F1(a ; b; c; x)isa HYPERGEOMETRIC FUNCTION . This can be derived using KUMMER’S QUADRATIC TRANS- FORMATION . The Gauss-Kummer series is closely related to the PERIMETER of an ellipse. See also ELLIPSE References Sloane, N. J. A. Sequences A056981 and A056982 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/eisonline.html. Gauss-Kuzmin-Wirsing Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Letx0be a random number from [0 ;1] written as a simple CONTINUED FRACTION x0/C300/C271 a1/C271 a2/C271 a3/C27...: (1) Define the SHIFT TRANSFORMATION by xn/C300/C271 an/C271/C271 an/C272/C271 an/C273/C27...: (2) /C301 xn/C281/C281 xn/C281$% ; (3) where xbcis the FLOOR FUNCTION . In a letter to Laplace dated January 30, 1812, Gauss said that he could prove by a simple argument that if F(n;x) is the probability that xnBx;then lim n0/C12F(n;x)/C30ln(1/C27x) ln 2(4) (Rockett and Szu ¨sz 1992, pp. 151 /C1/52). However, Gauss was unable to describe the behavior of the correction term in F(n;x)/C30ln(1/C27x) ln 2/C27e(n): (5)Kuzmin (1928) published the first analysis of theasymptotic behavior of F(n;x);obtaining F(n;x)/C30ln(1/C27x) ln 2/C27O(qffiffinp ) (6) with 0BqB1:Using a different method, Le ´vy (1929) obtained F(n;x)/C30ln(1/C27x) ln 2/C27O(qn) (7) with q/C300:7:Wirsing (1974) subsequently showed, among other results, that lim n0/C12F(n;x)/C28ln(1/C27x) ln 2 (/C28l)n/C30C(x); (8) where l/C300:3036630029 . . . and C(x) is an analytic function with C(0)/C30C(1)/C300:This constant is con- nected to the efficiency of the E UCLIDEAN ALGORITHM (Knuth 1981). See also CONTINUED FRACTION ,E UCLIDEAN ALGO- RITHM ,SHIFT TRANSFORMATION References Babenko, K. I. "On a Problem of Gauss." Soviet Math. Dokl. 19, 136/C1/40, 1978. Daude ´, H.; Flajolet, P.; and Valle ´e, B. "An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduc- tion." Submitted. Durner, A. "On a Theorem of Gauss-Kuzmin-Le ´vy." Arch. Math. 58, 251/C1/56, 1992. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/kuzmin/kuzmin.html. Flajolet, P. and Valle ´e, B. "On the Gauss-Kuzmin-Wirsing Constant." Unpublished memo. 1995. http://pauillac.in-ria.fr/algo/flajolet/Publications/gauss-kuzmin.ps. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Kuzmin, R. O. "Sur un proble `me de Gauss." Anni Congr. Intern. Bologne 6,8 3/C1 /9, 1928. MacLeod, A. J. "High-Accuracy Numerical Values of the Gauss-Kuzmin Continued Fraction Problem." Computers Math. Appl. 26,3 7/C1/4, 1993. Rockett, A. M. and Szu ¨sz, P. "The Gauss-Kuzmin Theorem." §5.5 in Continued Fractions. New York: World Scientific, pp. 151 /C1/55, 1992. Wirsing, E. "On the Theorem of Gauss-Kuzmin-Le ´vy and a Frobenius-Type Theorem for Function Spaces." Acta Arith. 24, 507/C1/28, 1974. Gauss-Laguerre Quadrature LAGUERRE- GAUSS QUADRATURE Gauss-Manin Connection A connection defined on a smooth ALGEBRAIC VARIETY defined over the COMPLEX NUMBERS . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 81, 1980. Gauss-Salamin Formula BRENT- SALAMIN FORMULA GCD GREATEST COMMON DIVISOR GCD-Closed Set A set S is said to be GCD-closed if GCD( xi ; xj) /C23 S for 1 5i ; j 5n:/ See also BOURQUE- LIGH CONJECTURE References Hong, S. "On the Bourque-Ligh Conjecture of Least Common Multiple Matrices." J. Algebra 218, 216 /C1/28, 1999. Gear Curve A curve resembling a gear with n teeth given by the PARAMETRIC EQUATIONS x /C30r cos t y /C30r sin t; where r /C30a /C271 btanh[ b sin(nt)] : The above curve has n /C3012, a /C301, and b /C3010. Gear Graph A WHEEL GRAPH with a VERTEX added between each pair of adjacent VERTICES . Gegenbauer Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION (1 /C28x2)yƒ/C282(m /C271)xy?/C27( n /C28 m)(n /C27 m /C271)y /C300 (1)sometimes called the hyperspherical differential equation (Iyanaga and Kawada 1980, p. 1480; Zwil- linger 1997, p. 123). The solution to this equation is y /C30(x2 /C281)/C28 m=2[C1Pm n (x) /C27C2Qmn (x)]; (2) where Pm n (x) is an associated LEGENDRE FUNCTION OF THE FIRST KIND and Q mn (x) is an associated LEGENDRE FUNCTION OF THE SECOND KIND . A number of other forms of this equation are some- times also known as the ultraspherical or Gegen- bauer differential equation, including (1 /C28x2)yƒ/C28(2m /C271)xy?/C27n( n /C272m)y /C300: (3) The general solutions to this equation are y /C30(x2 /C281)(1/C282 m)=4 /C2 [C1P1 =2 /C28 m /C281 =2 /C27m/C27 n(x) /C27C2Q1 =2/C28 m /C281 =2 /C27 m/C27 n(x)] : (4) However, if m is an integer, then the second part of this equation no longer provides a solution, and the solutions are known as the GEGENBAUER POLYNO- MIALS C(m) n(x); also known as ultraspherical polyno- mials (possibly depending on normalization). The form (1/C28x2)yƒ/C28(2m/C273)xy?/C27ly/C300 (5) is also given by Infeld and Hull (1951, pp. 21 /C1/8) and Zwillinger (1997, p. 122). It has the solution y/C30(x2/C281)/C28(2m/C271)=4 /C2C1P1=2/C27m /C281=2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27m)2/C27lp (x)/C27C2Q1=2/C27m /C281=2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27m)2/C27lp (x)=zn;=zn1 : (6) See also GEGENBAUER POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 547 /C1/49, 1953. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Gegenbauer Function GEGENBAUER POLYNOMIAL Gegenbauer Polynomial The Gegenbauer polynomials C(l) n(x) are solutions to the G EGENBAUER DIFFERENTIAL EQUATION for INTE- GER nandlB1=2:They are generalizations of the associated L EGENDRE POLYNOMIALS to ( n/C272)/-D space, and are proportional to (or, depending on the normalization, equal to) the ultraspherical polyno- mials P(l) n(x) :/ Following Szego, in this work, Gegenbauer polyno- mials are given in terms of the JACOBI POLYNOMIALS P( a; b) n(x) with a /C30 b /C30 l /C281=2by C( l) n(x) /C30G(l /C271 2) G(2l)G(n /C27 2l) G(n /C27 l /C271 2) P( l/C281 =2 ; l /C281 =2) n (x) (1) (Szego 1975, p. 80), thus making them equivalent to the Gegenbauer polynomials implemented in Mathe- matica as GegenbauerC [n, lambda , x]. These poly- nomials are also given by the GENERATING FUNCTION 1 (1 /C28 2xt /C27 t2) l /C30X/C12 n/C300C(l) n(x)tn : (2) The first few Gegenbauer polynomials are C( l) 0(x) /C301 (3) C(l) 1(x) /C302 lx (4) C(l) 2(x) /C30/C28l /C272l(1 /C27 l)x2 (5) C( l) 3(x) /C30/C282 l(1 /C27 l)x /C274 3 l(1 /C27 l)(2 /C27 l)x3 : (6) In terms of the HYPERGEOMETRIC FUNCTIONS , C(l) n(x) /C30n /C272 l /C281 n=z1r=z1> /C22F1(/C28n ; n /C272l; l /C271 2 ;12(1 /C28x)) (7) /C302n n /C27 l /C281 n=z1r=z1> (x /C281)n 2F1 /C2/C28n ;/C28n /C28 l /C271 2; /C282n /C282l /C271;2 1 /C28 x ! (8) /C30n /C272 l /C271 n=z1r=z1>x /C27 1 2 !n 2F1 /C2/C28n;/C28n /C28 l /C271 2 ; l /C2712 ;x /C28 1 x /C27 1 ! : (9) They are normalized by g1 /C281(1 /C28x2) l/C281 =2[C(l) n]2 dx /C3021 /C282l pG(n /C27 2l) (n /C27 l) G2( l) G(n /C27 1) : (10) Derivative identities include d dxC(l) n(x) /C302lC( l/C271) n /C281(x) (11) (1 /C28x2)d dx[C(l) n ] /C30[2(n /C27 l)]/C281[(n /C272l /C281)/C29(n /C272l)C( l) n/C281(x) /C28n(n /C271)C( l) n /C271(x)] (12) /C30/C28nxC(l) n(x) /C27(n /C272l /C281)C(l) n/C281(x) (13) /C30(n /C272l)xC(l) n(x) /C28(n /C271)C(l) n/C271(x) (14) nC(l) n(x) /C30xd dx [C(l) n (x)] /C28d dx [C(l) n/C281(x)] (15) (n /C272l)C( l) n(x) /C30d dx [C( l) n /C271(x)] /C28xd dx [C(l) n (x)] (16) d dx[C(l) n/C271(x) /C28C(l) n/C281(x)] /C302(n /C27 l)C(l) n C(l) n (x) (17) /C302l[C(l /C271) n(x) /C28C(l/C271) n/C282(x)] (18) (Szego 1975, pp. 80 /C1/3). A RECURRENCE RELATION is nC(l) n(x) /C302(n /C27 l /C281)xC(l) n/C281(x) /C28(n /C272 l /C282)C(l) n/C282(x) (19) for n /C302, 3, .... Special double- /n FORMULAS also exist C( l) 2n(x)/C302n/C272l/C281 2n=z1r=z1> 2F1(/C28n;n/C27l;l/C2712;1/C28x2) (20) /C30(/C281)nn/C27l/C281 n=z1r=z1> 2F1(/C28n;n/C27l;12;x2) (21) C(l) 2n/C271(x)/C302n/C272l 2n/C271=z1r=z1> x2F1(/C28n;n/C27l/C271;l/C2712;1/C28x2) (22) /C30(/C281)n2ln/C27l n=z1r=z1> x2F1(/C28n;n/C27l/C271;32;x2):(23) Koschmieder (1920) gives representations in terms of ELLIPTIC FUNCTIONS forl/C30/C283=4 and l/C30/C282=3:/ See also BIRTHDAY PROBLEM ,CHEBYSHEV POLYNO- MIAL OF THE SECOND KIND,E LLIPTIC FUNCTION , GEGENBAUER DIFFERENTIAL EQUATION ,H YPERGEO- METRIC FUNCTION ,JACOBI POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771 /C1/02, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 643, 1985. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. New York: Krieger, p. 175, 1981. Infeld, L. and Hull, T. E. "The Factorization Method." Rev. Mod. Phys. 23,2 1/C1/8, 1951. Iyanaga, S. and Kawada, Y. (Eds.). "Gegenbauer Polyno- mials (Gegenbauer Functions)." Appendix A, Table 20.I inEncyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1477 /C1 /478, 1980. Koekoek, R. and Swarttouw, R. F. "Gegenbauer / Ultra- spherical." §1.8.1 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Neth- erlands: Technische Universiteit Delft, Faculty of Techni- cal Mathematics and Informatics Report 98 /C1/7, pp. 40 /C1/1, 1998. ftp://www.twi.tudelft.nl/publications/tech-reports/ 1998/DUT-TWI-98 /C1/7.ps.gz. Koschmieder, L. "Uuml;ber besondere Jacobische Poly- nome." Math. Zeitschrift 8, 123 /C1/37, 1920. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 547 /C1/49 and 600 /C1/04, 1953. Roman, S. "A Particular Delta Series and the Gegenbauer Polynomials." §6.3 in The Umbral Calculus. New York: Academic Press, pp. 166 /C1/74, 1984. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 122 /C1/23, 1997. Gegenbauer C ULTRASPHERICAL POLYNOMIAL Gelfand Space References Stengers, I. and Prigogine, I. The End of Certainty: Time, Chaos, and the New Laws of Nature. Free Press, p. 96, 1997. Gelfand Transform The Gelfand transform x /C2 ˆx is defined as follows. If f : B 0 C is linear and multiplicative in the senses f(ax /C27by) /C30a f(x) /C27bf(y) and f(xy) /C30 f(x) f(y); where B is a commutative BANACH ALGEBRA , then write ˆx(f) /C30 f(x) : The Gelfand transform is automa- tically bounded. For example, if B /C30L1(R) with the usual norm, then B is a BANACH ALGEBRA under convolution and the Gelfand transform is the FOURIER TRANSFORM . (In fact, R may be replaced by any locally compact Abelian group, and then B has a unit if and only if the group is discrete.) See also BANACH ALGEBRA References Katznelson, Y. An Introduction to Harmonic Analysis. New York: Dover, 1976. Rudin, W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, 1987. Gelfond’s Theorem Also called the Gelfond-Schneider theorem, Gelfond’s theorem states that ab is TRANSCENDENTAL if1. a is ALGEBRAIC "0; 1 and 2. b is ALGEBRAIC and IRRATIONAL . This provides a partial solution to the seventh of HILBERT’S PROBLEMS . Gelfond’s theorem is implied by SCHANUEL’S CONJECTURE (Chow 1999). See also ALGEBRAIC NUMBER ,HILBERT’S PROBLEMS , IRRATIONAL NUMBER ,S CHANUEL’S CONJECTURE , TRANSCENDENTAL NUMBER References Baker, A. Transcendental Number Theory. London: Cam- bridge University Press, 1990. Chow, T. Y. "What is a Closed-Form Number?" Amer. Math. Monthly 106, 440 /C1/48, 1999. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 107, 1996. Gelfond-Schneider Constant The number 2ffiffi 2p /C302 :66514414... which is known to be TRANSCENDENTAL by GELFOND’S THEOREM . References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 107, 1996. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 45, 1986. Gelfond-Schneider Theorem GELFOND’S THEOREM Gelin-Cesa `ro Identity The identity F4 n/C28Fn/C282Fn/C281Fn/C271Fn/C272/C301; where Fnis a F IBONACCI NUMBER . See also FIBONACCI NUMBER Genaille Rods Numbered rods which can be used to perform multi- plication. See also NAPIER’S BONES References Gardner, M. "Napier’s Bones." Ch. 7 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 85 /C1/3, 1986. Genera FUNDAMENTAL THEOREM OF GENERA General Confluent Hypergeometric Differential Equation yƒ/C272a x/C272f ?/C27bh? h/C28h ?/C28hƒ h ! y? /C27bh ? h/C28h?/C28hƒ h? ! a x /C27f ? ! /C27a(a /C28 1) x2/C272af ? x" /C27f ƒ/C27f ?2 /C28ah ?2 h=zn1 /C300: See also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 505, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997. General Linear Group The general linear group GLn(q) is the set of n /C29n MATRICES with entries in the FIELD Fqwhich have NONZERO DETERMINANT . See also LANGLANDS RECIPROCITY ,PROJECTIVE GEN- ERAL LINEAR GROUP ,PROJECTIVE SPECIAL LINEAR GROUP ,SPECIAL LINEAR GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GLn(q); SLn(q) ; PGLn(q); and PSLn(q) /C30Ln(q) :/" §2.1 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. General Orthogonal Group The general orthogonal group GOn(q; F) is the SUB- GROUP of all elements of the PROJECTIVE GENERAL LINEAR GROUP that fix the particular nonsingular QUADRATIC FORM F. The determinant of such an element is 9 1. See also PROJECTIVE GENERAL LINEAR GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GOn(q) ; SOn(q) ; PGOn(q); and PSOn(q) ; and On(q) :/" §2.4 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xi-xii, 1985.General Position An arrangement of points with no three COLLINEAR , or of lines with no three CONCURRENT . See also CONCURRENT ,ORDINARY LINE,NEAR-PENCIL References Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/09, 1989. General Prismatoid A solid such that the AREA Ay of any section parallel to and a distance y from a fixed PLANE can be expressed as Ay /C30ay3 /C27by2 /C27cy /C27d: The volume of such a solid is the same as for a PRISMATOID , V /C301 6 h(A1 /C274M /C27A2): Examples include the CONE , CONICAL FRUSTUM , CY- LINDER , PRISMATOID , PYRAMIDAL FRUSTUM , SPHERE , SPHERICAL SEGMENT , and SPHEROID . See also PRISMATOID ,PRISMOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987. Kern, W. F. and Bland, J. R. "The General Prismatoid." Ch. 8 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 120 /C1/30, 1948. General Quantifier The FOR ALL QUANTIFIER /C214:/ See also EXISTENTIAL QUANTIFIER ,EXISTS ,FOR ALL, QUANTIFIER General Unitary Group The general unitary group GUn(q) is the SUBGROUP of all elements of the GENERAL LINEAR GROUP GL(q2) that fix a given nonsingular Hermitian form. This is equivalent, in the canonical case, to the definition of GUnas the group of UNITARY MATRICES . References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GUn(q);SUn(q);PGUn(q); and PSUn(q)/C30Un(q):/"§2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. Generalized Completeness Theorem The proposition that every CONSISTENT generalized theory has a MODEL . The theorem is true if the AXIOM OF CHOICE is assumed. See also AXIOM OF CHOICE References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 121, 1997. Generalized Cone A RULED SURFACE is called a generalized cone if it can be parameterized by x(u; v) /C30p /C27vy(u) ; where p is a fixed point which can be regarded as the vertex of the cone. A generalized cone is a REGULAR SURFACE wherever vy /C29y?"0: The above surface is a general- ized cone over a CARDIOID . A generalized cone is a FLAT SURFACE , and is sometimes called "conical sur- face." See also CONE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 439 /C1/41, 1997. Kern, W. F. and Bland, J. R. "Conical Surfaces." §23 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 57, 1948. Generalized Cylinder A RULED SURFACE is called a generalized cylinder if it can be parameterized by x(u; v) /C30vp /C27y(u); where p is a fixed point. A generalized cylinder is a REGULAR SURFACE wherever y?/C29p "0: The above surface is a generalized cylinder over a CARDIOID . A generalizedcylinder is a FLAT SURFACE , and is sometimes called a "cylindrical surface." See also CYLINDER References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 439 /C1/41, 1997. Harris, J. W. and Stocker, H. "General Cylinder." §4.6.1 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 103, 1998. Kern, W. F. and Bland, J. R. "Cylindrical Surface." §14 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 32 /C1/6, 1948. Generalized Diameter The farthest DISTANCE between two points on the boundary of a closed figure. The diameter of a SUBSET E of a EUCLIDEAN SPACE Rn is therefore given by diam E /C30sup f½x /C28y½ : x; y /C23 Eg; where sup denotes the SUPREMUM (Croft et al. 1991). See also BLASCHKE’S THEOREM ,BORSUK’S CONJEC- TURE ,DIAMETER References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Eppstein, D. "Width, Diameter, and Geometric Inequalities." http://www.ics.uci.edu/~eppstein/junkyard/diam.html. Generalized Euclidean Algorithm INTEGER RELATION Generalized Fermat Equation A generalization of the equation whose solution is desired in FERMAT’S LAST THEOREM xn /C27yn /C30zn to xn /C27yn /C30czn for x, y, z, and c positive constants, with trivial solutions having x /C300, y /C300, or z /C300 being excluded. n /C301 is trivial to solve by taking x /C30y /C30c and z /C302. n /C302 is more difficult, but can be solved by noting that solutions exist for values of cwhich can be written as a sum of two SQUARES , the first few of which are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, ... (Sloane’s A001481). See also FERMAT’S LAST THEOREM ,SQUARE NUMBER References Finch, S. "Unsolved Mathematics Problems: On a General- ized Fermat-Wiles Equation." http://www.mathsoft.com/ asolve/fermat/fermat.html. Sloane, N. J. A. Sequences A001481/M0968 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Generalized Fibonacci Number A generalization of the FIBONACCI NUMBERS defined by 1 /C30G1 /C30G2 /C30.../C30Gc /C281and the RECURRENCE RE- LATION Gn /C30Gn/C281 /C27Gn/C28c : (1) These are the sums of elements on successive diag- onals of a left-justified PASCAL’S TRIANGLE beginning in the left-most column and moving in steps of c /C281 up and 1 right. The case c /C302 equals the usual FIBONACCI NUMBER . These numbers satisfy the iden- tities G1 /C27G2 /C27G3 /C27.../C27Gn /C30Gn/C273 /C281 (2) G3 /C27G6 /C27G9 /C27.../C27G3k /C30G3k /C271 /C281 (3) G1 /C27G4 /C27G7 /C27.../C27G3k /C271 /C30G3k /C272 (4) G2 /C27G5 /C27G8 /C27.../C27G3k /C272 /C30G3k /C273 (5) (Bicknell-Johnson and Spears 1996). For the special case c /C303, Gn/C27w /C30Gw /C282Gn /C27Gw/C283Gn/C271 /C27Gw /C281Gn/C272 : (6) Bicknell-Johnson and Spears (1996) give many further identities. Horadam (1965) defined the generalized Fibonacci numbers fwn g as wn /C30wn(a ; b; p; q) ; where a, b, p, and q are INTEGERS , w0 /C30a ; w1 /C30b; and wn /C30pwn/C281 /C28 qwn/C282 for n ]2 : They satisfy the identities wnwn/C272r /C28eqnUr /C30w2 n/C27r (7) 4wnw2n/C271wn/C272 /C27(wqn)2 /C30(wnwn/C272 /C27w2n/C271)2(8) wnwn/C271wn/C273wn/C274 /C30w4n/C272 /C27eqn(p2 /C27q)w2n/C272 /C27e2q2n/C271p2 (9) 4wnwn/C271wn/C272wn/C274wn/C275wn/C276 /C27e2q2n(wnU4U5 /C28wn/C271U2U6 /C28wnU1U8)2 /C30(wn/C271wn/C272wn/C276 /C27wnwn /C274wn/C275)2 ; (10) where e /C13pab /C28qa2 /C28b2 (11) Un /C13wn(0; 1; p; q) (12) (Dujella 1996). The final above result is due to Morgado (1987) and is called the MORGADO IDENTITY . Another generalization of the Fibonacci numbers is denoted xn : Given x1and x2 ; define the generalized Fibonacci number by xn /C13xn/C282 /C27xn/C281 for n ]3 ;Xn i/C301xn /C30xn/C272 /C28x2 (13) X10 i/C301xn /C3011x7 (14) x2n /C28xn/C281xn /C272 /C30(/C281)n(x22 /C28x21 /C28x1x2); (15) where the plus and minus signs alternate. See also FIBONACCI N-STEP NUMBER ,F IBONACCI NUMBER References Bicknell, M. "A Primer for the Fibonacci Numbers, Part VIII: Sequences of Sums from Pascal’s Triangle." Fib. Quart. 9, 74 /C1/1, 1971. Bicknell-Johnson, M. and Spears, C. P. "Classes of Identities for the Generalized Fibonacci Numbers Gn /C30Gn/C281 /C27Gn/C28c for Matrices with Constant Valued Determinants." Fib. Quart. 34, 121 /C1/28, 1996. Dujella, A. "Generalized Fibonacci Numbers and the Pro- blem of Diophantus." Fib. Quart. 34, 164 /C1/75, 1996. Horadam, A. F. "Generating Functions for Powers of a Certain Generalized Sequence of Numbers." Duke Math. J. 32, 437 /C1/46, 1965. Horadam, A. F. "Generalization of a Result of Morgado." Portugaliae Math. 44, 131 /C1/36, 1987. Horadam, A. F. and Shannon, A. G. "Generalization of Identities of Catalan and Others." Portugaliae Math. 44, 137 /C1/48, 1987. Morgado, J. "Note on Some Results of A. F. Horadam and A. G. Shannon Concerning a Catalan’s Identity on Fibonacci Numbers." Portugaliae Math. 44, 243 /C1/52, 1987. Generalized Function DISTRIBUTION (GENERALIZED FUNCTION ) Generalized Helicoid The SURFACE generated by a twisted curve C when rotated about a fixed axis Aand, at the same time, displaced PARALLEL toAso that the velocity of displacement is always proportional to the ANGULAR VELOCITY ofROTATION . See also GENERALIZED HELIX,HELICOID ,HELIX References do Carmo, M. P.; Fischer, G.; Pinkall, U.; and Reckziegel, H. "General Helicoids." §3.4.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 36 /C1/7, 1986. Fischer, G. (Ed.). Plate 89 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 85, 1986. Kreyszig, E. Differential Geometry. New York: Dover, p. 88, 1991. Generalized Helix The GEODESICS on a general cylinder generated by lines PARALLEL to a line l with which the TANGENT makes a constant ANGLE . See also HELIX Generalized Hyperbolic Functions In 1757, V. Riccati first recorded the generalizations of the HYPERBOLIC FUNCTIONS defined by F a n;r(x) /C13X/C12 k /C300ak (nk /C27 r)!xnk /C27r ; (1) for r /C300, ..., n /C281; where a is COMPLEX , with the value at x /C300 defined by F a n ;0(0) /C301 : (2) This is called the a/-hyperbolic function of order n of the rth kind. The functions F a n;rsatisfy f(k)(x) /C30 af(x) ; (3) where f(k)(0) /C300 k "r; 0 5k 5n /C281; 1 k /C30r:=zn* (4) In addition, d dxF a n; r(x) /C30F a n;r/C281(x) for 0 Br 5n /C281 aF a n;n/C281(x) for r /C300:=zn* (5) The functions give a generalized EULER FORMULA effiffiap /C30Xn/C281 r/C300(ffiffiffiap)rF a n; r(x): (6) Since there are nnth roots of a; this gives a system of n linear equations. Solving for F a n; rgives F a n; r(x) /C301 n(ffiffiffiap) /C28r Xn/C281 k /C300v/C28rk nexp( vk nffiffiffiffiffiaxp) ; (7) where vn /C30exp2pi n ! (8) is a PRIMITIVE ROOT OF UNITY . The LAPLACE TRANSFORM is g/C12 0e/C28stF a n;r(at) dt /C30sn /C28r/C281ar sn /C27 aan: (9) The generalized hyperbolic function is also related to the MITTAG- LEFFLER FUNCTION Eg(x)by F1 n ;0(x) /C30En(xn): (10) The values n /C301 and n /C302 give the exponential andcircular/hyperbolic functions (depending on the sign of a) ; respectively. F a 1 ;0(x) /C30eax (11) F a 2 ;0(x) /C30cosh(ffiffiffiapx) (12) F a 2 ;1(x) /C30sinh(ffiffiffiapx)ffiffiffiap : (13) For a /C301; the first few functions are F1 1;0(x) /C30ex F1 2 ;0(x) /C30cosh x F1 2;1(x) /C30sinh x F1 3 ;0(x) /C301 3[ex /C272e/C28x =2 cos(12ffiffiffi 3p x)] F1 3;1(x)/C301 3ex/C272e/C28x=2cos12ffiffiffi 3p x/C271 3p=z1*=z1+ hi F1 3;2(x)/C301 3ex/C272e/C28x=2cos12ffiffiffi 3p x/C281 3p=z1*=z1+ hi F1 4;0(x)/C3012(cosh x/C27cosx) F1 4;1(x)/C3012(sinh x/C27sinx) F1 4;2(x)/C3012(cosh x/C28cosx) F1 4;3(x)/C3012(sinh x/C27sinx): See also HYPERBOLIC FUNCTIONS ,M ITTAG- LEFFLER FUNCTION References Kaufman, H. "A Biographical Note on the Higher Sine Functions." Scripta Math. 28,2 9/C1/6, 1967. Muldoon, M. E. and Ungar, A. A. "Beyond Sin and Cos." Math. Mag. 69,3/C1/4, 1996. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, 1996. Ungar, A. "Generalized Hyperbolic Functions." Amer. Math. Monthly 89, 688/C1/91, 1982. Ungar, A. "Higher Order Alpha-Hyperbolic Functions." Indian J. Pure. Appl. Math. 15, 301/C1/04, 1984. Generalized Hypergeometric Differential Equation The GENERALIZED HYPERGEOMETRIC FUNCTION F(x)/C30pFqa1;a2;...;ap b1;b2;...;bq;x=zn;=zn1 satisfies the equation ˜D(˜D/C27b1/C281)/C1/C1/C1(˜D/C27bq/C281)F(x) /C30x( ˜D /C27 a1)( ˜D /C27 a2) /C1/C1/C1( ˜D /C27 ap)F(x) ; where ˜D is the DIFFERENTIAL OPERATOR . See also GENERALIZED HYPERGEOMETRIC FUNCTION References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 26, 1998. Miller, W. Jr. Symmetry and Separation of Variables. Reading, MA: Addison-Wesley, p. 271, 1977. Rainville, E. D. Special Functions. New York: Chelsea, 1971. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997. Generalized Hypergeometric Function The generalized hypergeometric function is given by aHYPERGEOMETRIC SERIES , i.e., a series for which the ratio of successive terms can be written ak/C271 ak/C30P(k) Q(k) /C30(k/C27a1)(k/C27a2)/C1/C1/C1(k/C27ap) (k/C27b1)(k/C27b2)/C1/C1/C1(k/C27bq)(k/C271)x: (1) (The factor of k/C271 in the DENOMINATOR is present for historical reasons of notation.) The resulting general-ized hypergeometric function is written X k/C270akxk/C30pFqa1;a2;...;ap b1;b2;...;bq;x=zn;=zn1 (2) /C30X/C12 k/C300(a1)k(a2)k/C1/C1/C1(ap)k (b1)kb(b2)k/C1/C1/C1(bq)kxk k!; (3) where ( a)kis the P OCHHAMMER SYMBOL orRISING FACTORIAL (a)k/C13G(a/C27k) G(a)/C30a(a/C271)/C1/C1/C1(a/C27k/C281): (4) This notation was introduced by Barnes (1907)(Hardy 1999, p. 111). If the argument x/C301, then the function is abbreviated pFqa1;a2;...;ap b1;b2;...;bq=zn;=zn1 /C13pFqa1;a2;...;ap b1;b2;...;bq;x=zn;=zn1 :(5) The KAMPE DE FERIET FUNCTION is a generalization of the generalized hypergeometric function to two vari- ables. The generalized hypergeometric function Fn(x)/C30 pFqa1;a2;...;ap b1;b2;...;bq;xhi satisfies qFn(x)/C30n[Fn/C271(x)/C28Fn(x)] (6) for any of its numerator parameters n/C30ak;and qFn(x)/C30(n/C281)[Fn/C281(x)/C28Fn(x)] (7) for any of its denominator parameters n/C30bk;whereq/C30zd dz(8) (Rainville 1971, Koepf 1998, p. 27). /2F1(a;b;c;z) is "the" HYPERGEOMETRIC FUNCTION , and1F1(a;b;z)/C13M(z) is the CONFLUENT HYPERGEO- METRIC FUNCTION . A function OF THE FORM 0F1(;b;z) is called a CONFLUENT HYPERGEOMETRIC LIMIT FUNCTION . The generalized hypergeometric function p/C271Fpa1;a2;...;ap/C271 b1;b2;...;bp;z=zn;=zn1 (9) is a solution to the DIFFERENTIAL EQUATION [q(q/C27b/C281)/C1/C1/C1(q/C27bp/C281)/C28z(q/C27a1) /C2(q/C27a2)/C1/C1/C1(q/C27ap/C271)]y /C300: (10) The other linearly independent solution is z1/C28b1p/C271Fp /C21/C27a1/C28b1;1/C28a2/C28b2;...;1/C27ap/C271/C28b1 2/C28b1;1/C28b2/C28b1;...;1/C28bp/C28b1;z=zn;=zn1 :(11) A generalized hypergeometric functionq/C271Fpcon- verges absolutely on the unit circle if RXq j/C301bj/C28Xq/C271 j/C301aj ! >0 (12) (Rainville 1971, Koepf 1998). Many sums can be written as generalized hypergeo- metric functions by inspection of the ratios of con-secutive terms in the generating HYPERGEOMETRIC SERIES . For example, for f(n)/C13X k(/C281)k2n k=z1r=z1>2 ; (13) the ratio of successive terms is ak/C271 ak/C30(/C281)k/C2712n k/C271=z1r=z1>2 (/C281)k2n k=z1r=z1>2/C30/C28(k/C282n)2 (k/C271)2; (14) yielding f(n)/C302F1/C282n;/C282n 1;/C281=zn;=zn1 /C302F1(/C282n;/C282n;1 ;/C281) (15) (Petkovsek 1996, pp. 44 /C1/5). Gosper (1978) discovered a slew of unusual hypergeo- metric function identities, many of which were sub- sequently proven by Gessel and Stanton (1982). Animportant generalization of Gosper’s technique, called Z EILBERGER’S ALGORITHM , in turn led to the powerful machinery of the WILF-ZEILBERGER PAIR (Zeilberger 1990). Special hypergeometric identities include GAUSS’S HYPERGEOMETRIC THEOREM 2F1(a ; b ; c;1)/C30G(c)G(c /C28 a /C28 b) G(c /C28 a)G(c /C28 b)(16) for R[c /C28a /C28b] > 0; KUMMER’S FORMULA 2F1(a; b; c; /C281) /C30G(1 2 b /C27 1)G(b /C28 a /C27 1) G(b /C27 1)G(1 2 b /C28 a /C27 1) ; (17) where a /C28b /C27c /C301 and b is a positive integer, SAALSCHU ¨ TZ’S THEOREM 3F2(a ; b ; c; d; e;1)/C30(d /C28 a)½c½(d /C28 b) ½c½ (d)½c½(d /C28 a /C28 b) ½c½(18) for d /C27e /C30a /C27b /C27c /C271 with c a negative integer and (a)n the POCHHAMMER SYMBOL ,DIXON’S THEOREM 3F2(a; b; c; d ; e;1) /C30(12 a)!(a /C28 b)!(a /C28 c)!(12 a /C28 b /C28 c)! a!(12 a /C28 b)!(12 a /C28 c)!(a /C28 b /C28 c)! ; (19) where 1 /C27a=2 /C28b /C28c has a positive REAL PART , d /C30 a /C28b /C271 ; and e /C30a /C28c /C271 ; the CLAUSEN FORMULA 4F3a; b; c ; d e ; f ; g;1=zn;=zn1 /C30(2a)½d½(a /C27 b) ½d½(2b) ½d½ (2a /C27 2b)½d½a ½d ½b½d½; (20) for a /C27b /C27c /C28d /C301 =2; e /C30a /C27b /C271=2 ; a /C27f /C30d /C271 /C30 b /C27g; d a nonpositive integer, and the DOUGALL- RAMANUJAN IDENTITY 7F6a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 b1 ; b2 ; b3 ; b4 ; b5 ; b6;1=zn;=zn1 /C30(a1 /C27 1)n(a1 /C28 a2 /C28 a3 /C27 1)n (a1 /C28 a2 /C27 1)n(a1 /C28 a3 /C27 1)n /C2(a1 /C28 a2 /C28 a4 /C27 1)n(a1 /C28 a3 /C28 a4 /C27 1) (a1 /C28 a4 /C27 1)n(a1 /C28 a2 /C28 a3 /C28 a4 /C27 1)n; (21) where n /C302a1 /C271 /C30a2 /C27a3 /C27a4 /C27a5 ; a6 /C301 /C27a1 =2; a7 /C30/C28n; and bi /C301 /C27a1 /C28ai/C271 for i /C30 1, 2, ..., 6. For all these identities, (a)n is the POCHHAMMER SYMBOL . Gessel (1994) found a slew of new identities using WILF-ZEILBERGER PAIRS , including the following: 5F4/C28a /C28b; n /C271 ; n /C27c /C271 ; 2n /C28a /C28b /C271; n /C271 2(3 /C28a /C28b) n /C28a /C28b /C28c /C271; n /C28a /C28b /C271 ; 2n /C272 ; n /C2712(1 /C28a /C28b);1"# /C300 (22) 3F2/C283n;23 /C28c ; 3n /C272 32; 1 /C283c;34"# /C30(c /C2723)n(13)n (1 /C28 c)n(43)n(23)3F2/C283b;/C2832n;12(1 /C283n) /C283n;23 /C28b /C28n;4 3"# /C30(13 /C28 b)n (1 3 /C27 b)n(24) 4F332 /C2715 n;23;/C28n; 2n /C272 n /C2711 6 ;43;15 n /C2712;2 27"# /C30(52)n(11 6 )n (32)n(72)n(25) (Petkovsek et al. 1996, pp. 135 /C1/37). The following table gives various named identities ordered by the orders (p, q) of thepFq/s they involve. Bailey (1935) gives a large number of such identities. /2F1/ GAUSS’S HYPERGEOMETRIC THEOREM ,KUM- MER’S THEOREM , ORR’S THEOREM ,RAMANU- JAN’S HYPERGEOMETRIC IDENTITY /3F2/ DARLING’S PRODUCTS ,DIXON’S THEOREM , RAMANUJAN’S HYPERGEOMETRIC IDENTITY , SAALSCHU ¨ TZ’S THEOREM , THOMAE’S THEO- REM,W ATSON’S THEOREM , WHIPPLE’S IDEN- TITY /4F3/ CLAUSEN FORMULA , WHIPPLE’S TRANSFOR- MATION /5F4/ DOUGALL’S THEOREM /6F5/ WHIPPLE’S IDENTITY /7F6/ DOUGALL- RAMANUJAN IDENTITY , WHIPPLE’S TRANSFORMATION /9F8/BAILEY’S TRANSFORMATION Nørlund (1955) gave the general transformation nFn/C281a1;a2;...;an b1;b2;...;bn/C281;xz=zn;=zn1 /C30(1/C28z)/C28a1X/C12 n/C300(a1)n n!nFn/C28n;a2;a3;...;an b1;b2;...;bn/C281;x=zn;=zn1 /C2z z/C281 !n ; (26) where ( a)nis the P OCHHAMMER SYMBOL . This identity is based on the transformation due to Euler X/C12 n/C300(a)n n!anzn/C30(1/C28z)/C28aX/C12 n/C300(a)n n!Dna0z 1/C28z !n ;(27) where Dis the FORWARD DIFFERENCE and Dka0/C30Xk m/C300(/C281)mk m=z1r=z1> ak/C28m (28) (Nørlund 1955). See also CARLSON’S THEOREM ,CLAUSEN FORMULA , CONFLUENT HYPERGEOMETRIC FUNCTION ,C ONFLU- ENT HYPERGEOMETRIC LIMIT FUNCTION ,D IXON’S THEOREM ,D OUGALL- RAMANUJAN IDENTITY ,D OU- GALL’S THEOREM ,GOSPER’S ALGORITHM ,H EINE HY- PERGEOMETRIC SERIES ,HYPERGEOMETRIC FUNCTION , HYPERGEOMETRIC IDENTITY ,H YPERGEOMETRIC SER- IES,JACKSON’S IDENTITY , K-BALANCED ,K AMPE DE FERIET FUNCTION ,KUMMER’S THEOREM ,LAURICELLA FUNCTIONS ,N EARLY- POISED ,R AMANUJAN’S HYPER- GEOMETRIC IDENTITY ,S AALSCHU ¨ TZ’S THEOREM , SAALSCHU ¨ TZIAN ,SISTER CELINE’S METHOD ,THOMAE’S THEOREM ,W ATSON’S THEOREM ,W ELL-POISED ,W HIP- PLE’S IDENTITY ,W HIPPLE’S TRANSFORMATION ,W ILF- ZEILBERGER PAIR,ZEILBERGER’S ALGORITHM References Bailey, W. N. "Some Identities Involving Generalized Hy- pergeometric Series." Proc. London Math. Soc. Ser. 2 29, 503 /C1/16, 1929. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, 1935. Barnes. Proc. London Math. Soc. 5,59/C1/16 1907. Dwork, B. Generalized Hypergeometric Functions. Oxford, England: Clarendon Press, 1990. Exton, H. Multiple Hypergeometric Functions and Applica- tions. New York: Wiley, 1976. Exton, H. Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, 1978. Gessel, I. "Finding Identities with the WZ Method." Theoret. Comput. Sci. To appear. Gessel, I. M. "Finding Identities with the WZ Method. Symbolic Computation in Combinatorics D1(Ithaca, NY, 1993)." J. Symbolic Comput. 20, 537 /C1/66, 1995. Gessel, I. and Stanton, D. "Strange Evaluations of Hyper- geometric Series." SIAM J. Math. Anal. 13, 295 /C1/08, 1982. Gosper, R. W. "Decision Procedures for Indefinite Hypergeo- metric Summation." Proc. Nat. Acad. Sci. USA 75,40/C1/2, 1978. Hardy, G. H. "Hypergeometric Series." Ch. 7 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 101 /C1/12, 1999. Klein, F. Vorlesungen u¨ber die hypergeometrische Funktion. Berlin: J. Springer, 1933. Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analo- gue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, 1 /C1/68, 1998. ftp://www.twi.tudelft.nl/publications/ tech-reports/1998/DUT-TWI-98 /C1/7.ps.gz. Koepf, W. "Hypergeometric Database." Ch. 3 in Hypergeo- metric Summation: An Algorithmic Approach to Summa- tion and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 12 and 31 /C1/3, 1998. Nørlund, N. E. "Hypergeometric Functions." Acta Math. 94, 289 /C1/49, 1955. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Rainville, E. D. Special Functions. New York: Chelsea, 1971. Saxena, R. K. and Mathai, A. M. Generalized Hypergeo- metric Functions with Applications in Statistics and Physical Sciences. New York: Springer-Verlag, 1973. Slater, L. J. Generalized Hypergeometric Functions. Cam- bridge, England: Cambridge University Press, 1966. Zeilberger, D. "A Fast Algorithm for Proving Terminating Hypergeometric Series Identities." Discrete Math. 80, 207 /C1/11, 1990.Generalized Matrix Inverse MOORE- PENROSE GENERALIZED MATRIX INVERSE Generalized Mean A generalized version of the MEAN m(t) /C131 nXn k /C301at k ! 1=t (1) with parameter t which gives the GEOMETRIC MEAN , ARITHMETIC MEAN , and HARMONIC MEAN as special cases: lim t00m(t) /C30G (2) m(1) /C30A (3) m(/C281) /C30H : (4) See also MEAN Generalized Polygon Let O be an incidence geometry, i.e., a set with a symmetric, reflexive binary relation I. Let e and f be elements of O. Let an incidence plane be an incidence geometry whose object set is the disjoint union of two sets P and L such that for e ; f /C23 P or e ; f /C23 L; (e ; f) /C23 I only if e /C30f. Then a generalized polygon is an incidence plane such that for all e ; f /C23 O; 1. There exists a CHAIN of length at most n from e to f, and. 2. There exists at most one irreducible CHAIN of length less than n from e to f. (Feit and Higman 1964). The only CUBIC generalized polygons are the general- ized 2-gon K3;3(UTILITY GRAPH ), generalized triangle PG2;2(HEAWOOD GRAPH ), generalized quadrangle W2 (the L EVI GRAPH ), and generalized hexagon GH2;2 (Feit and Higman 1964, Royle). See also CAGE GRAPH ,MOORE GRAPH References Feit, W. and Higman, G. "The Non-Existence of Certain Generalized Polygons." J. Algebra 1, 114/C1/31, 1964. Royle, G. "Cubic Cages." http://www.cs.uwa.edu.au/~gordon/ cages/. Tits, J. "Sur la trialite ´et certains groupes qui s’en de ´dui- sent." Publ. Math. I.H.E.S. Paris 2,1 4/C1/0, 1959. Tits, J. "The´ore`me de Bruhat er sous-groupes paraboliques." C. R. Acad. Sci. Paris 254, 2910 /C1/912, 1962. Generalized Remainder Method An algorithm for computing a UNIT FRACTION . See also UNIT FRACTION References Eppstein, D. Egypt.ma Mathematica notebook. http:// www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma. Generating Function A POWER SERIES f(x) /C30X/C12 n/C300anxn (1) whose COEFFICIENTS give the SEQUENCE fa0 ; a1 ; ...g: The Mathematica function PowerSum in the Mathe- matica add-on package DiscreteMath‘RSolve‘ (which can be loaded with the command BBDiscreteMath‘ ) gives the generating function of a given expression, andExponentialPowerSum in the Mathematica add-on packageDiscreteMath‘R- Solve‘ (which can be loaded with the command BBDiscreteMath‘ ) gives the so-called EXPONEN- TIAL GENERATING FUNCTION . The generating function f(x) is sometimes said to "ENUMERATE " an(Hardy 1999, p. 85). Generating functions for the first few powers a(p) nare given in the following table. /np//f(x)/ series 1 /x 1/C28x// x /C27x2 /C27x3 /C27... / n /x (1/C28x)2// x /C272x2 /C273x3 /C274x4 /C27... / /n2 //x(x /C271) (1/C28x)3// x /C274x2 /C279x3 /C2716x4 /C27... / /n3//x(x2 /C274x /C271) (1/C28x)4 // x /C278x2 /C2727x3 /C27... / /n4//x(x /C271)(x2 /C2710x /C271) (1/C28x)5 //x /C2716x2 /C2781x3 /C27... / There are many beautiful generating functions for special functions in number theory. A few particu- larly nice examples are f(x) /C301Q/C12 k/C3011 /C28 xk /C301 /C27x /C272x2 /C273x3 /C27... (2) for the PARTITION FUNCTION P, and f(x) /C30X/C12 n/C300Fnxn /C30x 1 /C28 x /C28 x2 /C30x /C27x2 /C272x3 /C273x4 /C27... (3) for the FIBONACCI NUMBERS Fn :/The generating function of G(t) of a sequence of numbers f(n) given by the Z-TRANSFORM of f(n)in the variable 1=t (Germundsson 2000). See also CUMULANT- GENERATING FUNCTION ,E NU- MERATE ,EXPONENTIAL GENERATING FUNCTION ,M O- MENT- GENERATING FUNCTION ,R ECURRENCE RELATION , Z-TRANSFORM References Bender, E. A. and Goldman, J. R. "Enumerative Uses of Generating Functions." Indiana U. Math. J. 20, 753/C1/65, 1970/1971. Bergeron, F.; Labelle, G.; and Leroux, P. "The ´orie des espe`ces er Combinatoire des Structures Arborescentes." Publications du LACIM. Que ´bec, Montre ´al, Canada: Univ. Que´bec Montre ´al, 1994. Cameron, P. J. "Some Sequences of Integers." Disc. Math. 75,8 9/C1/02, 1989. Doubilet, P.; Rota, G.-C.; and Stanley, R. P. "The Idea of Generating Function." Ch. 3 in Finite Operator Calculus (Ed. G.-C. Rota). New York: Academic Press, pp. 83 /C1/34, 1975. Germundsson, R. " Mathematica Version 4." Mathematica J. 7, 497/C1/24, 2000. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 85, 1999. Leroux, P. and Miloudi, B. "Ge ´ne´ralisations de la formule d’Otter." Ann. Sci. Math. Que ´bec16,5 3/C1/0, 1992. Riordan, J. Combinatorial Identities. New York: Wiley, 1979. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980. Sloane, N. J. A. and Plouffe, S. "Recurrences and Generat- ing Functions." §2.4 in The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 9 /C1/0, 1995. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, p. 63, 1996. Viennot, G. "Une The ´orie Combinatoire des Polyno ˆmes Orthogonaux Ge ´ne´raux." Publications du LACIM. Que ´bec, Montre ´al, Canada: Univ. Que ´bec Montre ´al, 1983. Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1990. Generation In population studies, the direct offspring of a reference population (roughly) constitutes a single generation. For a CELLULAR AUTOMATON , the funda- mental unit of time during which the rules of reproduction are applied once is called a generation. Generator (Digitaddition) An INTEGER used to generate a DIGITADDITION .A number can have more than one generator. If a number has no generator, it is called a SELF NUMBER . Generator (Group) A member of a CYCLIC GROUP , the POWERS of which generate the entire GROUP . See also FINITELY GENERATED References Arfken, G. "Generators." §4.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 261 /C1/ 67, 1985. Generic Character For a form Q, the generic character xi(Q) OF THE FORM is defined as the values of xi(m) where (m; 2d) /C301 and Q represents m: x1(Q); x2(Q); ..., xr(Q) (Cohn 1980, p. 223). The characters apply to the class of properly equivalent forms as they represent the same numbers. See also GENUS (FORM) References Cohn, H. "Compositions, Order, and Genera." Ch. 8 in Advanced Number Theory. New York: Dover, 1980. Generic Cylindrical Algebraic Decomposition A CYLINDRICAL ALGEBRAIC DECOMPOSITION that omits sets of measure zero. Generic cylindrical algebraic decompositions are generally much quicker to com- pute than are normal decompositions. Generic cylind- rical algebraic decomposition is implemented in Mathematica as GenericCyclindricalAlgeb- raicDecomposition [ineqs , vars]. See also CYLINDRICAL ALGEBRAIC DECOMPOSITION References Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525 /C1/41, 2000. Genetic Algorithm An adaptive STOCHASTIC OPTIMIZATION ALGORITHM involving search and optimization that was first used by John Holland. Holland created an electronic organism as a binary string ("chromosome"), and then used genetic and evolutionary principles of fitness-proportionate selection for reproduction (in- cluding random crossover and mutation) to search enormous solution spaces efficiently. So-called ge- netic programming languages apply the same princi- ples, using an expression tree instead of a bit string as the "chromosome." See also CELLULAR AUTOMATON ,DIFFERENTIAL EVO- LUTION ,EVOLUTION STRATEGIES ,OPTIMIZATION THE- ORY,STOCHASTIC OPTIMIZATION References Bengtsson, M. "Genetic Algorithms Notebook." http:// www.mathsource.com/cgi-bin/msitem?0204 /C1/47.Genocchi Number A number given by the GENERATING FUNCTION 2t et /C27 1 /C30X/C12 n/C301Gntn n! : It satisfies G1 /C301 ; G3 /C30G5 /C30G7 /C30.../C300; and even coefficients are given by G2n /C3021/C2822n=z;=z1 B2n /C302nE2n/C281(0) ; where Bnis a BERNOULLI NUMBER and En(x)isan EULER POLYNOMIAL . The first few Genocchi numbers for n EVEN are /C281, 1, /C283, 17, /C28155, 2073, ... (Sloane’s A001469). See also BERNOULLI NUMBER ,EULER POLYNOMIAL References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 49, 1974. Kreweras, G. "An Additive Generation for the Genocchi Numbers and Two of its Enumerative Meanings." Bull. Inst. Combin. Appl. 20,99/C1/03, 1997. Kreweras, G. "Sur les permutations compte ´es par les nombres de Genocchi de 1-ie`re et 2-ie`me espe`ce." Europ. J. Comb. 18,49/C1/8, 1997. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Sloane, N. J. A. Sequences A001469/M3041 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Gentle Diagonal PASCAL’S TRIANGLE Gentle Giant Group MONSTER GROUP Genus (Curve) One of the PLU¨ CKER CHARACTERISTICS , defined by p /C131 2(n /C281)(n /C282) /C28( d /C27 k) /C3012(m /C281)(m /C282) /C28( t /C27 i) ; where m is the class, n the order, d the number of nodes, k the number of CUSPS , i the number of stationary tangents (INFLECTION POINTS ), and t the number of BITANGENTS . See also RIEMANN CURVE THEOREM References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 100, 1959. Genus (Form) Consider the forms Qfor which the GENERIC CHAR- ACTERS xi(Q) are equal to some preassigned array of signs ei/C301o r/C281, e1 ; e2 ; ...; er ; subject toQr i/C301ei /C301: There are 2r/C281 possible arrays, where r is the number of distinct prime divisors of a field discriminant d, and the set of forms correspond- ing to each array is called a genus of forms. The forms for which all ei /C301 are called the principal genus of forms, and each genus is also a collection of proper EQUIVALENCE CLASSES (Cohn 1980, pp. 223 /C1/24). See also EQUIVALENCE CLASS ,FUNDAMENTAL THEO- REM OF GENERA ,GENERIC CHARACTER References Cohn, H. "Compositions, Order, and Genera." Ch. 8 in Advanced Number Theory. New York: Dover, pp. 212 /C1/ 30, 1980. Genus (Knot) The least genus of any SEIFERT SURFACE for a given KNOT . The UNKNOT is the only KNOT with genus 0. Genus (Surface) A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of HOLES in a surface. The genus of a surface, also called the geometric genus, is related to the EULER CHARACTERISTIC x by x /C302 /C282g : See also EULER CHARACTERISTIC References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 635, 1997. Genus Theorem The DIOPHANTINE EQUATION x2 /C27y2 /C30p can be solved for p a PRIME IFF p /C131 (mod4) or p /C302. The representation is unique except for changes of sign or rearrangements of x and y. This theorem is intimately connected with the QUADRATIC RECIPRO- CITY THEOREM , and generalizes to the QUARTIC RE- CIPROCITY THEOREM . See also COMPOSITION THEOREM ,DIOPHANTINE EQUA- TION–4TH POWERS ,FERMAT’S THEOREM ,FUNDAMEN- TAL THEOREM OF GENERA ,GENUS (FORM), QUADRATIC RECIPROCITY THEOREMGeocentric Latitude An AUXILIARY LATITUDE given by fg/C30tan/C2811/C28e2=z;=z1=zn=zo tanf]: The series expansion is fg/C30f/C28e2sin 2 fðÞ/C271 2e2 2sin 4 fðÞ/C271 3e3 2sin 6 fðÞ/C27...; where e2/C13e2 2/C28e2: See also LATITUDE References Adams, O. S. "Latitude Developments Connected with Geo- desy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections." Spec. Pub.No. 67. U. S. Coast and Geodetic Survey, 1921. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington,DC: U. S. Government Printing Office, pp. 17 /C1 /8, 1987. Geodesic Given two points on a surface, the geodesic is defined as the shortest path on the surface connecting them.Geodesics also preserve a direction on a surface (Tietze 1965, pp. 26 /C1 /7) and have many other inter- esting properties. The NORMAL VECTOR to any point of aGEODESIC arc lies along the normal to a surface at that point (Weinstock 1974, p. 65). Furthermore, no matter how badly a SPHERE is distorted, there exist an infinite number of closed geodesics on it. This general result, demonstrated in the early 1990s, extended earlier work by Birkhoff, who proved in 1917 that there exists at least oneclosed geodesic on a distorted sphere, and Lyusternik and Schnirelmann, who proved in 1923 that there exist at least three closed geodesics on such a sphere(Cipra 1993, p. 28). For a surface given parametrically by x/C30x(u;v);y/C30 y(u;v);and z/C30z(u;v);the geodesic can be found by minimizing the ARC LENGTH L/C13gds/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2/C27dy2/C27dz2p : (1) But dx/C30@x @udu/C27@x @vdv (2) dx2/C30@x @u !2 du2/C272@x @u@x @vdu dv/C27@x @v !2 dv2;(3) and similarly for dy2anddz2:Plugging in, L/C30g@x @u !2 /C27@y @u !2 /C27@z @u !22 435du 28 < : /C272@x @u@x @v/C27@y @u@y @v/C27@z @u@z @v"# du dv /C27@x @v !2 /C27@y @v !2 /C27@z @v !22 435dv 2=zn+1=2 :(4) This can be rewritten as L/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C272Qv?/C27Rv?2q du (5) /C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pu?2/C272Qu?/C27Rp dv; (6) where v?/C13dv du(7) u?/C13du dv(8) and P/C13@x @u !2 /C27@y @u !2 /C27@z @u !2 (9) Q/C13@x @u@x @v/C27@y @u@y @v/C27@z @u@z @v(10) R/C13@x @v !2 /C27@y @v !2 /C27@z @v !2 : (11) Taking derivatives, @L @v/C301 2P/C272Qv?/C27Rv?2=z;=z1 /C281=2@P @v/C272@Q @vv?/C27@R @vv?2 ! (12) @L @v?/C3012P/C272Qv?/C27Rv?2=z;=z1 /C281=22Q/C272Rv? ðÞ ; (13) so the E ULER- LAGRANGE DIFFERENTIAL EQUATION then gives @P @v/C272v?@Q @v/C27v?2@R @v 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C272Qv?/C27Rv?2p /C28d duQ/C27Rv?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP/C272Qv?/C27Rv?2p ! /C300: (14) In the special case when P,Q, and Rare explicit functions of uonly,Q/C27Rv?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C272Qv?/C27Rv?2p /C30c1 (15) Q2/C272QRv?/C27R2v?2 P/C272Qv?/C27Rv?2/C30c2 1 (16) v?2RR/C28c21=z;=z1 /C272v?QR/C28c21=z;=z1 /C27Q2/C28Pc21=z;=z1 /C300 (17) v?/C301 2R(R/C28c2 1) /C22Qc2 1/C28R=z;=z1 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Q2R/C28c2 1 ðÞ /C284RR/C28c21 ðÞ Q2/C28Pc21 ðÞq =zn;=zn1 : (18) Now, if PandRare explicit functions of uonly and Q/C300, v?/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4RR/C28c2 1 ðÞ Pc21p 2RR/C28c21 ðÞ/C30c1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P RR/C28c21 ðÞs ; (19) so v/C30c1gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P RR/C28c21 ðÞs du: (20) In the case Q/C300 where Pand Rare explicit functions of vonly, then @P @v/C27v?2@R @v 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C27Rv?2p /C28d duRv?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP/C27Rv?2p ! /C300; (21) so @P @v/C27v?2@R @v /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C27Rv?2p Rvƒffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C27Rv?2p /C27/C281 2=z1*=z1+v?2Rv0vƒ ðÞ P/C27Rv?2=z;=z1 3=2"# /C300 (22) @P @v/C27v?2@R @v/C282Rvƒ/C272R2v?2vƒ P/C27Rv?2/C300 (23) Rv?2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C27Rv?2p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P/C27Rv?2p /C30c1 (24) Rv02/C28P/C27Rv?2=z;=z1 /C30c1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP/C27Rv?2p (25) p c1 !2 /C30P/C27Rv?2(26) P2/C28c2 1P Rc2 1/C30v?2; (27) and u /C30c1gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R P2 /C28 c2 1Ps dv: (28) For a SURFACE OF REVOLUTION in which y /C30g(x)is rotated about the X-AXIS so that the equation of the surface is y2 /C27z2 /C30g2(x) ; (29) the surface can be parameterized by x /C30u (30) y /C30g(u) cos v (31) z /C30g(u) sin v : (32) The equation of the geodesics is then v /C30c1gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 g?(u) ½/C1382q du g(u)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g(u) ½/C1382/C28c2 1q : (33) See also ELLIPSOID GEODESIC ,GEODESIC CURVATURE , GEODESIC DOME,G EODESIC EQUATION ,G EODESIC MAPPING ,G EODESIC TRIANGLE ,G RAPH GEODESIC , GREAT CIRCLE ,H ARMONIC MAP,O BLATE SPHEROID GEODESIC ,PARABOLOID GEODESIC References 2 /C1/ Cipra, B. What’s Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., p. 28, 1993. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, pp. 27 and 40, 1965. Tietze, H. Mathematische Analyse des Raumproblems. Berlin, 1923. Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, pp. 26 /C1/8 and 45 /C1/6, 1974. Weyl, H. §17 in Space--Time--Matter. New York: Dover, 1952. Geodesic Curvature For a unit speed curve on a surface, the length of the surface-tangential component of acceleration is the geodesic curvature kg : Curves with kg /C300 are called GEODESICS . For a curve parameterized as a(t) /C30 x(u(t) ; v(t)) ; the geodesic curvature is given by kg /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG /C28F2p /C28G2 11u ?3 /C27G122v?3 /C28(2G212 /C28G111)u?2v?=zn /C27(2G112 /C28G222)u?v?2 /C27uƒv?/C28v ƒu?/C138; where E,F, and Gare coefficients of the first FUNDAMENTAL FORM andGkijare C HRISTOFFEL SYM- BOLS OF THE SECOND KIND . See also GEODESICReferences Gray, A. "Geodesic Curvature and Torsion." §22.4 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 513 /C1/18, 1997. Geodesic Dome ATRIANGULATION of a P LATONIC SOLID or other POLYHEDRON to produce a close approximation to a SPHERE (or HEMISPHERE ). The nth order geodesation operation replaces each polygon of the polyhedron by the projection onto the CIRCUMSPHERE of the order- n regular tessellation of that polygon. The above figure shows geodesations of orders 1 to 3 (from top to bottom) of the TETRAHEDRON ,CUBE ,OCTAHEDRON , DODECAHEDRON , and ICOSAHEDRON (from left to right), computed using Geodesate [poly,n] in the Mathematica add-on package Graphics‘Polyhe- dra‘ (which can be loaded with the command BBGraphics‘ ). R. Buckminster Fuller designed the first geodesic dome (i.e., geodesation of a HEMISPHERE ). Fuller’s dome was constructed from an ICOSAHEDRON by adding ISOSCELES TRIANGLES about each VERTEX and slightly repositioning the VERTICES . In such domes, neither the VERTICES nor the centers of faces neces- sarily lie at exactly the same distances from the center. However, these conditions are approximately satisfied. In the geodesic domes discussed by Kniffen (1994), the sum of VERTEX angles is chosen to be a constant. Given a P LATONIC SOLID , let e?/C132e=vbe the number ofEDGES meeting at a VERTEX andnbe the number of EDGES of the constituent POLYGON . Call the angle of the old VERTEX point Aand the angle of the new VERTEX point F. Then A/C30B (1) 2e?A/C30nF (2) 2A /C27F /C30180/C14: (3) Solving for A gives 2A /C272e ? nA /C302A 1 /C27e ? n ! /C30180/C14 (4) A /C3090 /C14n e ?/C27n ; (5) and F /C302e ? nA /C30180/C14e ? e ?/C27n : (6) The VERTEX sum is S/C30nF /C30180/C14e?n e ?/C27n : (7) Solid fv /e ?/ nA F /a/ TETRAHEDRON 3345 8 908 2708 CUBE 24 14 3 4 /513 7/C14//8137/C14 //30847/C14 / OCTAHEDRON 43 /3847/C14 //10847/C14 //30847/C14 / DODECAHEDRON 60 32 3 5 /561 4/C14//7114/C14 //33712/C14 / ICOSAHEDRON 53 /3334/C14 //11834/C14 //33712/C14 / Wenninger and Messer (1996) give general formulas for solving any geodesic chord factor and dihedral angle in a geodesic dome. See also SPHERE ,SPHERICAL TRIANGLE ,TRIANGULAR SYMMETRY GROUP References Kenner, H. Geodesic Math and How to Use It. Berkeley, CA: University of California Press, 1976. Kniffen, D. "Geodesic Domes for Amateur Astronomers." Sky & Telescope 88,90/C1/4, Oct. 1994. Messer, P. W. "Mathematical Formulas for Geodesic Do- mes." Appendix to Wenninger, M. Spherical Models. New York: Dover, pp. 145 /C1/49, 1999. Pappas, T. "Geodesic Dome of Leonardo da Vinci." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 81, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 85 /C1/6, 1991. Wenninger, M. J. and Messer, P. W. "Patterns on the Spherical Surface." Internat. J. Space Structures 11, 183 /C1/92, 1996. Wenninger, M. "Geodesic Domes." Ch. 4 in Spherical Mod- els. New York: Dover, pp. 80 /C1/24, 1999. Geodesic Equation d t2 /C30/C28hab dj a dj b ;or d2 ja dr2 /C300 : See also GEODESIC Geodesic Flow A type of FLOW technically defined in terms of the TANGENT BUNDLE of a MANIFOLD . See also DYNAMICAL SYSTEM Geodesic Mapping A geodesic mapping f : M 0 N between two RIEMAN- NIAN MANIFOLDS is a DIFFEOMORPHISM sending GEO- DESICS of M into GEODESICS of N, whose inverse also sends GEODESICS to GEODESICS (Ambartzumian 1982, p. 26). See also BELTRAMI’S THEOREM ,GEODESIC References Ambartzumian, R. V. Combinatorial Integral Geometry. Chichester, England: Wiley, 1982. Kreyszig, E. Differential Geometry. New York: Dover, 1991. Geodesic Triangle A TRIANGLE formed by the arcs of three GEODESICS on a smooth surface. See also INTEGRAL CURVATURE ,SPHERICAL TRIANGLE Geodetic Latitude LATITUDE Geodetic Number Let I(x; y) denote the set of all vertices lying on an (x, y)-GRAPH GEODESIC in G, then a set S with I(S) /C30 V(G) is called a geodetic set in G and is denoted g(G):/ See also HULL NUMBER References Chartrand, G.; Harary, F.; and Zhang, P. "The Forcing Hull Number of a Graph." To appear in J. Comb. Math. Comb. Combin. Chartrand, G. and Zhang, P. "The Geodetic Number of a Graph." To appear in Networks. Chartrand, G. and Zhang, P. "The Forcing Geodetic Number of a Graph." Discuss. Math. Graph Th. 19,4 5/C1/8, 1999. Chartrand, G. and Zhang, P. "Realizable Ratios in Graph Theory: Geodesic Parameters." Bull. Inst. Comb. Appl. 27, 69/C1/0, 1999. Chartrand, G. and Zhang, P. "The Geodetic Number of an Oriented Graph." Europ. J. Combin. 21, 181/C1/89, 2000. Geographic Latitude LATITUDE Geometric Construction In antiquity, geometric constructions of figures and lengths were restricted to the use of only a STRAIGHT- EDGE and COMPASS (or in Plato’s case, a COMPASS only; a so-called M ASCHERONI CONSTRUCTION ). Although the term " RULER " is sometimes used instead of "STRAIGHTEDGE ," no markings which could be used to make measurements were allowed according to the Greek prescription. Furthermore, the " COMPASS " could not even be used to mark off distances by setting it and then "walking" it along, so the COMPASS had to be considered to automatically collapse whennot in the process of drawing a CIRCLE . Because of the prominent place Greek geometricconstructions held in Euclid’s E LEMENTS , these con- structions are sometimes also known as E UCLIDEAN CONSTRUCTIONS . Such constructions lay at the heart of the GEOMETRIC PROBLEMS OF ANTIQUITY ofCIRCLE SQUARING ,CUBE DUPLICATION , and TRISECTION of an ANGLE . The Greeks were unable to solve these problems, but it was not until hundreds of years laterthat the problems were proved to be actually im-possible under the limitations imposed. Simple algebraic operations such as a/C27b;a/C28b;ra (forra RATIONAL NUMBER ),a=b;ab, andffiffiffixpcan be performed using geometric constructions (bold 1982, Courant and Robbins 1996). Other more complicatedconstructions, such as the solution of A POLLONIUS’ PROBLEM and the construction of INVERSE POINTS can also accomplished. One of the simplest geometric constructions is theconstruction of a BISECTOR of a LINE SEGMENT , illustrated above. The Greeks were very adept at constructing POLY- GONS , but it took the genius of Gauss to mathemati- cally determine which constructions were possible and which were not. As a result, Gauss determinedthat a series of POLYGONS (the smallest of which has 17 sides; the HEPTADECAGON ) had constructions un- known to the Greeks. Gauss showed that the CON- STRUCTIBLE POLYGONS (several of which are illustrated above) were closely related to numberscalled the F ERMAT PRIMES . Wernick (1982) gave a list of 139 sets of three locatedpoints from which a TRIANGLE was to be constructed. Of Wernick’s original list of 139 problems, 20 had notyet been solved as of 1996 (Meyers 1996). It is possible to construct RATIONAL NUMBERS and EUCLIDEAN NUMBERS using a STRAIGHTEDGE and COMPASS construction. In general, the term for a number which can be constructed using a COMPASS and STRAIGHTEDGE is a CONSTRUCTIBLE NUMBER . Some IRRATIONAL NUMBERS , but noTRANSCENDENTAL NUMBERS , can be constructed. It turns out that all constructions possible with a COMPASS and STRAIGHTEDGE can be done with a COMPASS alone, as long as a line is considered constructed when its two endpoints are located. The reverse is also true, since Jacob Steiner showed that all constructions possible with STRAIGHTEDGE and COMPASS can be done using only a straightedge, as long as a fixed CIRCLE and its center (or two inter- secting CIRCLES without their centers, or three non- intersecting CIRCLES ) have been drawn beforehand. Such a construction is known as a S TEINER CON- STRUCTION . GEOMETROGRAPHY is a quantitative measure of the simplicity of a geometric construction. It reduces geometric constructions to five types of operations,and seeks to reduce the total number of operations(called the " SIMPLICITY "rpar; needed to effect a geo- metric construction. Dixon (1991, pp. 34 /C1/1) gives approximate construc- tions for some figures (the HEPTAGON and NONAGON ) and lengths ( PI) which cannot be rigorously con- structed. Ramanujan (1913 /C1/4) and Olds (1963) give geometric constructions for 355 =113:p:Gardner (1966, pp. 92 /C1/3) gives a geometric construction for 3 /C2716 113 /C303 :1415929 ... : p: Kochansky’s approximate construction for p yields KOCHANSKY’S APPROXIMATION ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40 3/C282ffiffiffi 3ps /C303 :141533 ... : p Steinhaus (1983, p. 143). Constructions for p are approximate (but inexact) forms of CIRCLE SQUARING . See also CIRCLE SQUARING ,COMPASS ,CONSTRUCTI- BLE NUMBER ,CONSTRUCTIBLE POLYGON ,CUBE DU- PLICATION ,E LEMENTS ,FERMAT PRIME ,G EOMETRIC PROBLEMS OF ANTIQUITY ,G EOMETROGRAPHY ,K O- CHANSKY’S APPROXIMATION ,M ASCHERONI CONSTRUC- TION ,M ATCHSTICK CONSTRUCTION ,N APOLEON’S PROBLEM ,NEUSIS CONSTRUCTION ,PLANE GEOMETRY , POLYGON ,PONCELET- STEINER THEOREM ,RECTIFICA- TION ,SIMPLICITY ,STEINER CONSTRUCTION ,STRAIGHT- EDGE ,TRISECTION References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 96 /C1/7, 1987. Bold, B. "Achievement of the Ancient Greeks" and "An Analytic Criterion for Constructibility." Chs. 1 /C1/inFa- mous Problems of Geometry and How to Solve Them. New York: Dover, pp. 1 /C1/7, 1982. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 191 /C1/02, 1996. Coolidge, J. L. "Famous Problems in Construction." Ch. 3 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 166 /C1/88, 1971. Courant, R. and Robbins, H. "Geometric Constructions. The Algebra of Number Fields." Ch. 3 in What is Mathe- matics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 117 /C1/64, 1996. Dantzig, T. Number, The Language of Science. New York: Macmillan, p. 316, 1954. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352 /C1/86, 1955. Dixon, R. Mathographics. New York: Dover, 1991. Dummit, D. S. and Foote, R. M. "Classical Straightedge and Compass Constructions." §13.3 in Abstract Algebra, 2nd ed.Englewood Cliffs, NJ: Prentice-Hall, pp. 443 /C1/48, 1998. Eppstein, D. "Geometric Models." http://www.ics.uci.edu/ ~eppstein/junkyard/model.html. Gardner, M. "The Transcendental Number Pi." Ch. 8 in Martin Gardner’s New Mathematical Diversions fromScientific American. New York: Simon and Schuster, pp. 91 /C1 /02, 1966. Gardner, M. "Mascheroni Constructions." Ch. 17 in Mathe- matical Circus: More Puzzles, Games, Paradoxes andOther Mathematical Entertainments from Scientific Amer-ican. New York: Knopf, pp. 216 /C1 /31, 1979. Harris, J. W. and Stocker, H. "Basic Constructions." §3.2 in Handbook of Mathematics and Computational Science.New York: Springer-Verlag, pp. 60 /C1 /2, 1998. Herterich, K. Die Konstruktion von Dreiecken. Stuttgart: Ernst Klett Verlag, 1986.Kro¨tenheerdt, O. "Zur Theorie der Dreieckskonstruktionen." Wissenschaftliche Zeitschrift der Martin-Luther-Univ.Halle-Wittenberg, Math. Naturw. Reihe 15, 677/C1 /00, 1966. Meyers, L. F. "Update on William Wernick’s ‘Triangle Constructions with Three Located Points."’ Math. Mag. 69,4 6/C1/9, 1996. Olds, C. D. Continued Fractions. New York: Random House, pp. 59 /C1/0, 1963. Petersen, J. Methods and Theories for the Solution of Problems of Geometrical Constructions Applied to 410 Problems. New York: Stechert, 1923. Reprinted in String Figures and Other Monographs. New York: Chelsea, 1960. Plouffe, S.. "The Computation of Certain Numbers Using a Ruler and Compass." J. Integer Sequences 1, No. 98.1.3, 1998. http://www.research.att.com/~njas/sequences/JIS/ compass.html. Posamentier, A. S. and Wernick, W. Advanced Geometric Constructions. Palo Alto, CA: Dale Seymour, 1988. Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/72, 1913 /C1/914. Smogorzhevskii, A. S. The Ruler in Geometrical Construc- tions. New York: Blaisdell, 1961. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Sykes, M. Source Book of Problems for Geometry. Palo Alto, CA: Dale Seymour, 1997. Weisstein, E. W. "Books about Geometric Construction." http://www.treasure-troves.com/books/GeometricCon-struction.html. Wernick, W. "Triangle Constructions with Three Located Points." Math. Mag. 55, 227/C1 /30, 1982. Geometric Distribution ADISCRETE DISTRIBUTION forn/C301, 2, ... with prob- ability function P(n)/C30qn/C281P (1) /C30p(1/C28p)n/C281; (2) where 0 BpB1 and ( q/C131/C28p):P(n) is normalized, since X/C12 n/C301P(n)/C30X/C12 n/C301qn/C281p/C30pX/C12 n/C300qn/C30p 1/C28q/C30p p/C301 (3) The corresponding DISTRIBUTION FUNCTION is D(n)/C30Xn k/C301P(k)/C301/C28qn: (4) The MOMENT-GENERATING FUNCTION is given by f(t) /C30p 1 /C28(1 /C28p)eit=zn=zo /C281; (5) or M(t) /C30 etnhi/C30X/C12 n/C301etnpqn/C281 /C30pX/C12 n /C300et(n/C271)qn /C30petX/C12 n/C300ettðÞn/C30pet 1 /C28 etq (6) M ?(t) /C30pet 1 /C28 etq ðÞ2 (7) M ƒ(t) /C30pet 1 /C27 qetðÞ 1 /C28 etq ðÞ3 (8) M §(t) /C30pet 1 /C27 4et(1 /C28 p) /C27 e2t(1 /C28 p)2hi 1 /C28 et /C27 etp ðÞ4 : (9) Therefore, the RAW MOMENTS are M ?(0) /C30 m?1 /C30 m /C30p (1 /C28 q)2 /C30p p2 /C301 p(10) M ƒ(0) /C30 m?2 /C30p(1 /C27 q) (1 /C28 q)3 /C30p(2 /C28 p) p3/C302 /C28 p p2 (11) M §(0) /C30 m ?3 /C306 /C28 6p /C27 p2ðÞ p3 (12) M4(0) /C30 m?4 /C30(p /C28 2) /C28p2 /C27 12p /C28 12 ðÞ p4 ; (13) giving CENTRAL MOMENTS m2 /C30q p2 (14) m3 /C30(p /C28 1)(p /C28 2) p3 (15) m4 /C30(p /C28 1) /C28p2 /C27 9p /C28 9 ðÞ p4 ; (16) so the MEAN , VARIANCE , SKEWNESS , and KURTOSIS aregiven by m /C13 m?1 /C301 p (17) s2 /C30 m2 /C30q p2 (18) g1 /C30m3 m3 =2 2/C302 /C28 p ffiffiffiqp (19) g2 /C30m4 m2 2/C283 /C30p2 /C28 6p /C27 6 1 /C28 p: (20) In fact, the moments of the distribution are given analytically in terms of the POLYLOGARITHM function, m ?k /C13X/C12 n/C301p(n)nk /C30X/C12 n /C301p(1 /C28p)n/C281nk /C30pLi/C28k(1 /C28 p) 1 /C28 p: (21) For the case p /C301=2 (corresponding to the distribu- tion of the number of COIN TOSSES needed to win in the SAINT PETERSBURG PARADOX ) the formula (21) gives m ?k jp /C301=2 /C30Li /C28k1 2=z1*=z1+ : (22) The first few raw moments are therefore 2, 6, 26, 150, 1082, ... (Sloane’s A000629), which have EXPONENTIAL GENERATING FUNCTIONS f(x) /C30/C28ln 2 /C28exðÞ and g(x) /C30 ex = 2 /C28exðÞ :From (22), the MEAN ,VARIANCE ,SKEW- NESS , and KURTOSIS are m/C302 (23) s2/C302 (24) g1/C303 2ffiffiffi 2p (25) g2/C3013 2: (26) The first CUMULANT of the geometric distribution is k1/C301/C28p p; (27) and subsequent CUMULANTS are given by the RECUR- RENCE RELATION kr/C271/C30(1/C28p)dkr dp: (28) See also SAINT PETERSBURG PARADOX References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 531 /C1/32, 1987. Sloane, N. J. A. Sequences A000629 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 118, 1992. Geometric Dual Graph Given a PLANAR GRAPH G, its geometric dual G/C31 is constructed by placing a vertex in each region of G (including the exterior region) and, if two regions have an edge x in common, joining the corresponding vertices by an edge X /C31 crossing only x. The result is always a planar PSEUDOGRAPH . However, an abstract graph with more than one embedding on the sphere can give rise to more than one dual. Whitney showed that the geometric dual graph and COMBINATORIAL DUAL GRAPH are equivalent (Harary 1994, p. 115), and so may simply be called "the" DUAL GRAPH . See also COMBINATORIAL DUAL GRAPH ,DUAL GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 113 /C1/15, 1994. Geometric Genus GENUS (SURFACE ) Geometric Invariant Theory INVARIANT Geometric Mean The geometric mean of a sequence aifgn i/C301is defined by Ga1 ; ... ; an ðÞ /C13Yn i/C301ai !1 =n : (1) Thus, Ga1 ; a2 ðÞ /C30ffiffiffiffiffiffiffiffiffiffia1a2p(2)Ga1 ; a2 ; a3 ðÞ /C30 a1a2a3 ðÞ1 =3; (3) and so on. Hoehn and Niven (1985) show that Ga1 /C27c ; a2 /C27c; ...; an /C27c ðÞ /C21c /C27Ga1 ; a2 ; ... ; an ðÞ (4) for any POSITIVE constant c. See also ARITHMETIC MEAN,ARITHMETIC- GEOMETRIC MEAN,C ARLEMAN’S INEQUALITY ,H ARMONIC MEAN, MEAN,ROOT-MEAN-SQUARE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151 /C1/56, 1985. Kenney, J. F. and Keeping, E. S. "Geometric Mean." §4.10 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 54 /C1/5, 1962. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995. Geometric Mean Index The statistical INDEX PG /C13Ypn p0 !vo"# 1 =S vo ; where pnis the price per unit in period n, qnis the quantity produced in period n, and vn /C13pnqnthe value of the n units. See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 69, 1962. Geometric Modeling References Strasser, W.; Klein, R.; and Rau, R. (Eds.). Geometric Modeling: Theory and Practice, the State of the Art. Berlin: Springer-Verlag, 1997. Geometric Probability The study of the probabilities involved in geometric problems, e.g., the distributions of length, area, volume, etc. for geometric objects under stated con- ditions. See also BERTRAND’S PROBLEM ,B UFFON- LAPLACE NEEDLE PROBLEM ,BUFFON’S NEEDLE PROBLEM ,CIR- CLE INSCRIBING ,COMPUTATIONAL GEOMETRY ,INTE- GRAL GEOMETRY ,P OINT PICKING ,S TOCHASTIC GEOMETRY ,SYLVESTER’S FOUR- POINT PROBLEM References Ambartzumian, R. V. (Ed.). Stochastic and Integral Geome- try. Dordrecht, Netherlands: Reidel, 1987. Isaac, R. The Pleasures of Probability. New York: Springer- Verlag, 1995. Kendall, M. G. and Moran, P. A. P. Geometric Probability. New York: Hafner, 1963. Kendall, W. S.; Barndorff-Nielson, O.; and van Lieshout, M. C. Current Trends in Stochastic Geometry: Likelihood and Computation. Boca Raton, FL: CRC Press, 1998. Klain, D. A. and Rota, G.-C. Introduction to Geometric Probability. New York: Cambridge University Press, 1997. Santalo ´,L.A. Introduction to Integral Geometry. Paris: Hermann, 1953. Santalo ´,L.A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978. Stoyan, D.; Kendall, W. S.; and Mecke, J. Stochastic Geo- metry and Its Applications, with a Foreword by D. G. Ken- dall. New York: Wiley, 1987. Weisstein, E. W. "Books about Geometric Probability." http://www.treasure-troves.com/books/GeometricProbabil- ity.html. Geometric Problems of Antiquity The Greek problems of antiquity were a set of geometric problems whose solution was sought using only COMPASS and STRAIGHTEDGE : 1. CIRCLE SQUARING . 2. CUBE DUPLICATION . 3. TRISECTION of an ANGLE . Only in modern times, more than 2,000 years after they were formulated, were all three ancient pro- blems proved insoluble using only COMPASS and STRAIGHTEDGE . Another ancient geometric problem not proved im- possible until 1997 is ALHAZEN’S BILLIARD PROBLEM . As Ogilvy (1990) points out, constructing the general REGULAR POLYHEDRON was really a "fourth" unsolved problem of antiquity. See also ALHAZEN’S BILLIARD PROBLEM ,C IRCLE SQUARING ,COMPASS ,CONSTRUCTIBLE NUMBER ,CON- STRUCTIBLE POLYGON ,C UBE DUPLICATION ,G EO- METRIC CONSTRUCTION ,R EGULAR POLYHEDRON , STRAIGHTEDGE ,TRISECTION References Conway, J. H. and Guy, R. K. "Three Greek Problems." In The Book of Numbers. New York: Springer-Verlag, pp. 190 /C1/91, 1996. Courant, R. and Robbins, H. "The Unsolvability of the Three Greek Problems." §3.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 117 /C1/18 and 134 /C1/40, 1996. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 135 /C1/38, 1990. Pappas, T. "The Impossible Trio." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 130 /C1/32, 1989.Jones, A.; Morris, S.; and Pearson, K. Abstract Algebra and Famous Impossibilities. New York: Springer-Verlag, 1991. Stoschek, E. "Modul 41 Literatur." http://marvin.sn.schu- le.de/~inftreff/modul41/lit41.htm. Stoschek, E. "Modul 41. Three Geometric Problems of Antiquity: Their Approximate Solutions in Automata Representation--Integrated Control Processors for Nano- technology." http://marvin.sn.schule.de/~inftreff/modul41/ task41.htm. Geometric Progression GEOMETRIC SEQUENCE Geometric Realization If the ABSTRACT SIMPLICIAL COMPLEX S is isomorphic with the VERTEX SCHEME of the SIMPLICIAL COMPLEX K, then K is said to be a geometric realization of S, and is uniquely determined up to a linear isomorph- ism. See also ABSTRACT SIMPLICIAL COMPLEX ,V ERTEX SCHEME References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Geometric Sequence A geometric sequence is a SEQUENCE akfg ;k/C301, 2, ..., such that each term is given by a multiple rof the previous one. Another equivalent definition is that a sequence is geometric IFFit has a zero BIAS. If the multiplier is r, then the kth term is given by ak/C30rak/C281/C30r2ak/C282/C30a0rk: Without loss of generality, take a0/C301;giving ak/C30rk: Geometric Series A geometric series akakis a series for which the ratio of each two consecutive terms ak/C271=akis a constant function of the summation index k. The more general case of the ratio a RATIONAL FUNCTION of the summation index kproduces a series called a HYPER- GEOMETRIC SERIES . For the simplest case of the ratio ak/C271=ak/C30requal to a constant r, the terms akare OF THE FORM ak/C30a0rk: Letting a0/C301;the GEOMETRIC SEQUENCE akfgn k/C300 with constant ½r½B1 is given by Sn/C30Xn k/C300ak/C30Xn k/C300rk(1) is given by Sn /C13Xn k/C300rk /C301 /C27r /C27r2 /C27.../C27rn : (2) Multiplying both sides by r gives rSn /C30r /C27r2 /C27r3 /C27.../C27rn/C271 ; (3) and subtracting (3) from (2) then gives (1 /C28r)Sn /C30(1 /C27r /C27r2 /C27...rn) /C28(r /C27r2 /C27r3 /C27.../C27rn/C271) /C301 /C28rn/C271 ; (4) so Sn /C13Xn k/C300rk /C301 /C28 rn/C271 1 /C28 r: (5) For /C281 Br B1; the sum converges as n 0/C12;/ in which case S /C13S/C12/C30X/C12 k/C300rk /C301 1 /C28 r (6) Similarly, if the sums are taken starting at k /C301 instead of k /C300, Xn k /C301rk /C30r 1 /C28 rnðÞ 1 /C28 r (7) X/C12 k /C301rk /C30r 1 /C28 r ; (8) the latter of which is valid for ½r ½B1:/ See also ARITHMETIC SERIES ,GABRIEL’S STAIRCASE , HARMONIC SERIES ,H YPERGEOMETRIC SERIES ,S T. IVES PROBLEM ,W HEAT AND CHESSBOARD PROBLEM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 278 /C1/79, 1985. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987. Courant, R. and Robbins, H. "The Geometric Progression." §1.2.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 13 /C1/4, 1996. Pappas, T. "Perimeter, Area & the Infinite Series." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 134 /C1/35, 1989. Geometrization Conjecture THURSTON’S GEOMETRIZATION CONJECTUREGeometrography A quantitative measure of the simplicity of a GEO- METRIC CONSTRUCTION which reduces geometric con- structions to five steps. It was devised by E` . Lemoine. /S1/ Place a STRAIGHTEDGE ’s EDGE through a given POINT , /S2/ Draw a straight LINE, /C1 Place a POINT of a COMPASS on a given POINT , /C2 Place a POINT of a COMPASS on an indeterminate POINT on a LINE, /C3/ Draw a CIRCLE . Geometrography seeks to reduce the number of operations (called the "SIMPLICITY "rpar; needed to effect a construction. If the number of the above operations are denoted /m1 ; m2/, n1 ; n2 ; and /n3/, respectively, then the SIMPLICITY is/ m1 /C27m2 /C27n1 /C27n2 /C27n3/ and the symbol is/ m1S1 /C27m2S2 /C27n1C1 /C27n2C2 /C27n3C3/. It is apparently an unsolved problem to determine if a given GEO- METRIC CONSTRUCTION is of the smallest possible simplicity. See also SIMPLICITY References De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97/C1/08, 1991. Eves, H. An Introduction to the History of Mathematics, 6th ed.New York: Holt, Rinehart, and Winston, 1990. Geometry Geometry is the study of figures in a SPACE of a given number of dimensions and of a given type. The most common types of geometry are PLANE GEOMETRY (dealing with objects like the LINE,CIRCLE ,TRIANGLE , and POLYGON ),SOLID GEOMETRY (dealing with objects like the LINE,SPHERE , and POLYHEDRON ), and SPHE- RICAL GEOMETRY (dealing with objects like the SPHE- RICAL TRIANGLE and SPHERICAL POLYGON ). Geometry was part of the QUADRIVIUM taught in medieval universities. Historically, the study of geometry proceeds from a small number of accepted truths ( AXIOMS orPOSTU- LATES ), then builds up true statements using a systematic and rigorous step-by-step PROOF . How- ever, there is much more to geometry than thisrelatively dry textbook approach, as evidenced by some of the beautiful and unexpected results of PROJECTIVE GEOMETRY (not to mention Schubert’s powerful but questionable ENUMERATIVE GEOMETRY ). The late mathematician E. T. Bell has described geometry as follows (Coxeter and Greitzer 1967, p. 1): "With a literature much vaster than those of ALGEBRA and ARITHMETIC combined, and at least as extensive as that of ANALYSIS , geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics." While the literature of ALGEBRA , ARITHMETIC , and ANALYSIS has grown extensively since Bell’s day, the remainder of his commentary holds even more so today. Formally, a geometry is defined as a complete locally homogeneous RIEMANNIAN METRIC .InR2 ; the possible geometries are Euclidean planar, hyperbolic planar, and elliptic planar. In R3 ; the possible geometries include Euclidean, hyperbolic, and elliptic, but also include five other types. See also ABSOLUTE GEOMETRY ,AFFINE GEOMETRY , CARTESIAN COORDINATES ,C OMBINATORIAL GEOME- TRY,COMPUTATIONAL GEOMETRY ,COORDINATE GEO- METRY ,D IFFERENTIAL GEOMETRY ,D ISCRETE GEOMETRY ,ENUMERATIVE GEOMETRY ,FINSLER GEO- METRY ,INVERSIVE GEOMETRY ,K AWAGUCHI GEOME- TRY,M INKOWSKI GEOMETRY ,N IL GEOMETRY ,N ON- EUCLIDEAN GEOMETRY ,ORDERED GEOMETRY ,PLANE GEOMETRY ,PROJECTIVE GEOMETRY ,SOL GEOMETRY , SOLID GEOMETRY ,SPHERICAL GEOMETRY ,STOCHAS- TIC GEOMETRY ,THURSTON’S GEOMETRIZATION CON- JECTURE References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952. Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, 1964. Brown, K. S. "Geometry." http://www.seanet.com/~ksbrown/ igeometr.htm. Cinderella, Inc. "Cinderella: The Interactive Geometry Soft- ware." http://www.cinderella.de/. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. The Beauty of Geometry: Twelve Essays. New York: Dover, 1999. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., 1967. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994. Davis, C.; Gru ¨nbaum, B.; and Scherk, F.A. The Geometric Vein: The Coxeter Festschrift. New York: Springer, 1981. Eppstein, D. "Geometry Junkyard." http://www.ics.uci.edu/ ~eppstein/junkyard/. Eppstein, D. "Many-Dimensional Geometry." http://www.ic- s.uci.edu/~eppstein/junkyard/highdim.html. Eppstein, D. "Planar Geometry." http://www.ics.uci.edu/ ~eppstein/junkyard/2d.html. Eppstein, D. "Three-Dimensional Geometry." http://www.ic- s.uci.edu/~eppstein/junkyard/3d.html. Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, 1972. Ghyka, M. C. The Geometry of Art and Life, 2nd ed. New York: Dover, 1977. Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: The Open Court Publishing Co., 1921. Ivins, W. M. Art and Geometry. New York: Dover, 1964. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. King, J. and Schattschneider, D. (Eds.). Geometry Turned On: Dynamic Software in Learning, Teaching and Re-search. Washington, DC: Math. Assoc. Amer., 1997.Klee, V. "Some Unsolved Problems in Plane Geometry." Math. Mag. 52, 131/C1 /45, 1979. Klein, F. Famous Problems of Elementary Geometry and Other Monographs. New York: Dover, 1956. Melzak, Z. A. Invitation to Geometry. New York: Wiley, 1983. Meschkowski, H. Unsolved and Unsolvable Problems in Geometry. London: Oliver & Boyd, 1966. Moise, E. E. Elementary Geometry from an Advanced Standpoint, 3rd ed. Reading, MA: Addison-Wesley, 1990. Ogilvy, C. S. "Some Unsolved Problems of Modern Geome- try." Ch. 11 in Excursions in Geometry. New York: Dover, pp. 143 /C1/53, 1990. Playfair, J. Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Circle and theGeometry of Solids to which are added Elements of Plane and Spherical Trigonometry. New York: W. E. Dean. Simon, M. U¨ber die Entwicklung der Elementargeometrie im XIX Jahrhundert. Berlin, pp. 97 /C1 /05, 1906. Townsend, R. Chapters on the Modern Geometry of the Point, Line, and Circle, 2 vols. Dublin: Hodges, Smith and Co., 1863. Uspenskii, V. A. Some Applications of Mechanics to Mathe- matics. New York: Blaisdell, 1961. Weisstein, E. W. "Books about Geometry." http://www.trea- sure-troves.com/books/Geometry.html. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, 1961. Geometry of Position PROJECTIVE GEOMETRY Gergonne Line The perspective line for the CONTACT TRIANGLE DDEF and its TANGENTIAL TRIANGLE DABC :It is determined by the NOBBS POINTS D?;E?;andF?:/ In addition to the NOBBS POINTS , the F LETCHER POINT and E VANS POINT also lie on the Gergonne line where it intersects the S ODDY LINE and E ULER LINE , respec- tively. The D and D ? coordinates are given by D /C30B /C27f eC D?/C30B /C28f eC ; so BDCD ? form a HARMONIC RANGE . The equation of the Gergonne line is a d /C27b e /C27g f/C300: See also CONTACT TRIANGLE ,E ULER LINE,E VANS POINT ,FLETCHER POINT ,NOBBS POINTS ,SODDY LINE, TANGENTIAL TRIANGLE References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/29, 1996. Gergonne Point The common point Ge of the CONCURRENT lines from the CONTACT TRIANGLE TRIANGLE’S INCIRCLE to the opposite VERTICES . It has TRIANGLE CENTER FUNCTION a /C30[a(b /C27c /C28a)] /C281 /C301 2sec2 A: The Gergonne point Ge is the ISOTOMIC CONJUGATE POINT of the NAGEL POINT Na. The CONTACT TRIAN- GLE and TANGENTIAL TRIANGLE are perspective from the Gergonne point, and the Gergonne point of a triangle is the SYMMEDIAN POINT of its CONTACT TRIANGLE (Honsberger 1995). See also ADAMS’ CIRCLE ,CONTACT TRIANGLE ,G ER- GONNE LINE,NAGEL POINT References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed. New York: Barnes and Noble, pp. 160 /C1/64, 1952. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. New York: Random House, pp. 11 /C1/3, 1967. Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, p. 83, 1972. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 22, 1913. Honsberger, R. "The Gergonne Point." §7.4 (iv) in Episodes in Nineteenth and Twentieth Century Euclidean Geome-try.Washington, DC: Math. Assoc. Amer., pp. 61 /C1 /2, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 184 and 216, 1929. Kimberling, C. "Gergonne Point." http://cedar.evansvil- le.edu/~ck6/tcenters/class/gergonne.html. Gergonne’s Theorem The internal (external) bisecting plane of a DIHEDRAL ANGLE of a TETRAHEDRON divides the opposite edge in the ratio of the areas of the adjacent faces. References Altshiller-Court, N. "Gergonne’s Theorem." §235 in Modern Pure Solid Geometry. New York: Chelsea, p. 71, 1979. Le Grand, Ferriot, Lambert, et al. "Questions Re ´solues: De´monstrations des deux the ´ore`mes de ge ´ome´trie e´nonce ´s a`la page 196 de ce volume." Ann. de math. 3, 317/C1/23, 1812/C1/813. Germain Primes SOPHIE GERMAIN PRIME Gerono Lemniscate EIGHT CURVE Gergorin Circle Theorem Gives a region in the COMPLEX PLANE containing all the EIGENVALUES of a COMPLEX SQUARE MATRIX . Define Ri/C30Xn i/C301 j"i½ai;j½; (1) then each EIGENVALUE of the MATRIX of order nis in at least one of the disks fz:½z/C28aii½5Rig: (2) The theorem can be made stronger as follows. Let r be an INTEGER with /1 5r 5n/, then each EIGENVALUE of is either in one of the disks /G1 fz : ½z /C28ajj ½5S(r/C281) j g; (3) or in one of the regions z :Xr i/C301½z /C28aii ½5Xr i/C301Ri() ; (4) where /S(r/C281) j/ is the sum of magnitudes of the /r /C281/ largest off-diagonal elements in column j. References Brualdi, R. A. and Mellendorf, S. "Regions in the Complex Plane Containing the Eigenvalues of a Matrix." Amer. Math. Monthly 101, 975 /C1/85, 1994. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1120 /C1/121, 2000. Piziak, R. and Turner, D. "Exploring Gerschgorin Circles and Cassini Ovals." Mathematica Educ. 3,13/C1/1, 1994. Taussky-Todd, O. "A Recurring Theorem on Determinants." Amer. Math. Monthly 56, 672 /C1/76, 1949. G-Function As defined by Erde´lyi et al. (1981, p. 20), the G- function is given by G(z) /C13 c0(1 2 /C27hz) /C28 c0(12 z) ; (1) where c0(z) is the DIGAMMA FUNCTION . Integralrepresentations are given by G(z) /C302g1 0tz/C281 1 /C27 tdt (2) /C302g/C12 0e /C28zt 1 /C27 e/C28tdt (3) for R[z] > 0: G(z) is also given by the series G(z) /C302X/C12 n/C300(/C281)n z /C27 n; (4) and in terms of the HYPERGEOMETRIC FUNCTION by G(z) /C302z/C281 2F1(1;z;1/C27z;/C281): (5) It obeys the functional relations G(1/C27z)/C302z/C281/C28G(z) (6) G(1/C28z)/C302pcsc(pz)/C28G(z) (7) G(mz)/C30/C282 mXm/C281 r/C300(/C281)rc0(z/C27r m) for meven 1 mXm/C281 r/C300(/C281)rG(z/C27r m) for modd:8 >>>>< >>>>:(8) See also B ARNES’ G-FUNCTION ,DIGAMMA FUNCTION , MEIJER’S G-FUNCTION ,RAMANUJAN G- AND G-FUNC- TIONS References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Function G(z):/"§1.8 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 20 and 44 /C1/6, 1981. Ghost If the sampling of an interferogram is modulated at a definite frequency instead of being uniformly sampled, spurious spectral features called "ghosts"are produced (Brault 1985). Periodic ruling or sam-pling errors introduce a modulation superposed on top of the expected fringe pattern due to uniform stage translation. Because modulation is a multi-plicative process, spurious features are generated in spectral space at the sum and difference of the true fringe and ghost fringe frequencies, thus throwing power out of its spectral band. Ghosts are copies of the actual spectrum, but appear at reduced strength. The above shows the power spectrum for a pure sinusoidal signal sampled by translating a Fourier transform spectrometer mirror at constant speed. The small blips on either side of the main peaks are ghosts. In order for a ghost to appear, the process producing it must exist for most of the interferogram. However, if the ruling errors are not truly sinusoidal but vary across the length of the screw, a longer travel path can reduce their effect. See also JITTER References Brault, J. W. "Fourier Transform Spectroscopy." In High Resolution in Astronomy: 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics (Ed. A. Benz, M. Huber, and M. Mayor). Geneva Observatory, Sauverny, Switzerland, 1985. Gibbs Constant WILBRAHAM- GIBBS CONSTANT Gibbs Effect GIBBS PHENOMENON Gibbs Phenomenon An overshoot of FOURIER SERIES and other EIGEN- FUNCTION series occurring at simple DISCONTINU- ITIES . it can be removed with the LANCZOS SIGMA FACTOR . See also FOURIER SERIES References Arfken, G. "Gibbs Phenomenon." §14.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 783 /C1/87, 1985. Foster, J. and Richards, F. B. "The Gibbs Phenomenon for Piecewise-Linear Approximation." Amer. Math. Monthly 98,47/C1/9, 1991. Gibbs, J. W. "Fourier Series." Nature 59, 200 and 606, 1899.Hewitt, E. and Hewitt, R. "The Gibbs-Wilbraham Phenom- enon: An Episode in Fourier Analysis." Arch. Hist. Exact Sci. 21, 129 /C1/60, 1980. Jeffreys, H. and Jeffreys, B. S. "The Gibbs Phenomenon." §14.07 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 445 /C1/46, 1988. Sansone, G. "Gibbs’ Phenomenon." §2.10 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 141 /C1/ 48, 1991. Gift Wrap Theorem No subspace of Rn can be homeomorphic to Sn :/ References Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, p. 121, 1997. Gigantic Prime APRIME with 10,000 or more decimal digits. As of Nov. 15, 1995, 127 were known. See also TITANIC PRIME References Caldwell, C. "The Ten Largest Known Primes." http:// www.utm.edu/research/primes/largest.html#largest. Gilbrat’s Distribution ACONTINUOUS DISTRIBUTION in which the LOGARITHM of a variable xhas a NORMAL DISTRIBUTION , P(x)/C301 xffiffiffiffiffiffi 2pp e/C28(lnx)2=2; (1) defined over the interval [0 ;/C12):It is a special case of the LOG NORMAL DISTRIBUTION P(x)/C301 Sxffiffiffiffiffiffi 2pp e/C28(lnx/C28M)2=(2S2)(2) with S/C301 and M/C300, and so has distribution function D(x)/C301 21/C27erflnxffiffiffi 2p !"# : (3) The MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS are then given by m /C30ffiffiffiep(4) s2 /C30e(e /C281) (5) g1 /C30(e /C272)ffiffiffiffiffiffiffiffiffiffiffi e /C281p (6) g2 /C30e4 /C272e3 /C273e2 /C283: (7) See also LOG NORMAL DISTRIBUTION Gilbreath’s Conjecture Let the DIFFERENCE of successive PRIMES be defined by dn /C13pn/C271 /C28pn ; and dk nby dk n /C13dn for k /C301 ½dk /C281 n /C271 /C28dk /C281 n½ for k > 1:=zn* N. L. Gilbreath claimed that dk 1 /C301 for all k (Guy 1994). It has been verified for k B63,419 and all PRIMES up to p(1013) ; where p(x) is the PRIME COUNT- ING FUNCTION . See also PRIME DIFFERENCE FUNCTION References Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243,18/C1/8, Dec. 1980. Guy, R. K. "Gilbreath’s Conjecture." §A10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 25 /C1/6, 1994. Kilgrove, R. B. and Ralston, K. E. "On a Conjecture Con- cerning the Primes." Math. Tables Aids Comput. 13, 121 /C1/ 22, 1959. Gill’s Method A formula for numerical solution of differential equations, yn /C271 /C30yn /C271 6[k1 /C27(2 /C28ffiffiffi 2p )k2 /C27(2 /C27ffiffiffi2p )k 3 /C27k4] /C27O(h5); where k1/C30hf(xn;yn) k2/C30hf(xn/C271 2h;yn/C2712k1) k3/C30hf[xn/C271 2h;yn/C2712(/C281/C27ffiffiffi 2p )k1/C27(1/C281 2ffiffiffi 2p )k2] k4/C30hf[xn/C27h;yn/C281 2ffiffiffi 2p k2/C27(1/C271 2ffiffiffi 2p )k3]: See also ADAMS’ METHOD ,MILNE’S METHOD ,PREDIC- TOR-CORRECTOR METHODS ,RUNGE- KUTTA METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. New York: Dover, p. 896, 1972. Gingerbreadman Map A 2-D piecewise linear MAP defined by xn/C271/C301/C28yn/C27½xn½ yn/C271/C30xn: The map is chaotic in the filled region above and stable in the six hexagonal regions. Each point in theinterior hexagon defined by the vertices (0, 0), (1, 0), (2, 1), (2, 2), (1, 2), and (0, 1) has an orbit with period six (except the point (1, 1), which has period 1). Orbitsin the other five hexagonal regions circulate from one to the other. There is a unique orbit of period five, with all others having period 30. The points havingorbits of period five are (-1, 3), (-1, -1), (3, -1), (5, 3), and (3, 5), indicated in the above figure by the black line. However, there are infinitely many distinctperiodic orbits which have an arbitrarily long period. References Devaney, R. L. "A Piecewise Linear Model for the Zones of Instability of an Area Preserving Map." Physica D 10, 387/C1/93, 1984. Peitgen, H.-O. and Saupe, D. (Eds.). "A Chaotic Ginger- breadman." §3.2.3 in The Science of Fractal Images. New York: Springer-Verlag, pp. 149 /C1/50, 1988. Gini Coefficient This entry contributed by C HRISTIAN DAMGAARD The Gini coefficient (or Gini ratio) Gis a summary statistic of the L ORENZ CURVE and a measure of inequality in a population. The Gini coefficient is most easily calculated from unordered size data as the "relative mean difference," i.e., the mean of thedifference between every possible pair of individuals, divided by the mean size m; G /C30Pn i /C301Pnj/C301 ½xi /C28 xj ½ 2n2 m Alternatively, if the data is ordered by increasing size of individuals, G is given by G /C30Pn i/C301(2i /C28 n /C28 1)x?i n2 m: The Gini coefficient ranges from a minimum value of zero, when all individuals are equal, to a theoretical maximum of one in an infinite population in which every individual except one has a size of zero. It has been shown that the sample Gini coefficients defined above need to be multiplied by n=(n /C281) in order to become UNBIASED ESTIMATORS for the population coefficients. See also LORENZ ASYMMETRY COEFFICIENT ,LORENZ CURVE References Dixon, P. M.; Weiner, J.; Mitchell-Olds, T.; and Woodley, R. "Bootstrapping the Gini Coefficient of Inequality." Ecology 68, 1548 /C1/551, 1987. Gini, C. "Variabilita ´ e mutabilita." 1912. Reprinted in Memorie di metodologia statistica (Ed. E. Pizetti and T. Salvemini.) Rome: Libreria Eredi Virgilio Veschi, 1955. Glasser, G. J. "Variance Formulas for the Mean Difference and Coefficient of Concentration." J. Amer. Stat. Assoc. 57, 648 /C1/54, 1962. Sen, A. On Economic Inequality. Oxford, England: Claren- don Press, 1973. Ginzburg-Landau Equation The PARTIAL DIFFERENTIAL EQUATION ut /C30(1 /C27ia)uxx /C27(1 /C27ic)u /C28(1 /C27id) ½u½2u: References Katou, K. "Asymptotic Spatial Patterns on the Complex Time-Dependent Ginzburg-Landau Equation." J. Phys. A: Math. Gen. 19, L1063-L1066, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 133, 1997. Girard’s Spherical Excess Formula Let a SPHERICAL TRIANGLE D have angles A, B, and C. Then the SPHERICAL EXCESS is given by D/C30A /C27B /C27C /C28 p: See also ANGULAR DEFECT ,L’HUILIER’S THEOREM , SPHERICAL EXCESS ,SPHERICAL TRIANGLE References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 94 /C1/5, 1969. Girard, A. Invention nouvelle en algebra. Amsterdam, Netherlands, 1629.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 469, 1995. Girko’s Circular Law Let l be (possibly complex) EIGENVALUES of a set of random n /C29n REAL MATRICES with entries indepen- dent and taken from a standard normal distribution. Then as n 0/C12; l =ffiffiffinpis uniformly distributed on the UNIT DISK in the COMPLEX PLANE . For small n, the distribution shows a concentration along the REAL LINE accompanied by a slight paucity above and below (with interesting embedded structure). However, as n0/C12;the concentration about the line disappears and the distribution becomes truly uniform. See also EIGENVALUE ,MATRIX References Bai, Z. D. "Circular Law." Ann. Prob. 25, 494/C1/29, 1997. Bai, Z. D. and Yin, Y. Q. "Limiting Behavior of the Norm Products of Random Matrices and Two Problems of Geman-Hwang." Probab. Theory Related Fields 73, 555/C1/ 69, 1986. Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32,1/C1/7, 1995. Edelman, A. "The Probability that a Random Real Gaussian Matrix has kReal Eigenvalues, Related Distributions, and the Circular Law." J. Multivariate Anal. 60, 203/C1/32, 1997. Geman, S. "The Spectral Radius of Large Random Matrices." Ann. Probab. 14, 1318/C1/328, 1986. Girko, V. L. "Circular Law." Theory Probab. Appl. 29, 694/C1/ 06, 1984. Girko, V. L. Theory of Random Determinants. Boston, MA: Kluwer, 1990. Mehta, M. L. Random Matrices, 2nd rev. enl. ed. New York: Academic Press, 1991. Girth The length of the shortest GRAPH CYCLE (if any) in a GRAPH . Acyclic graphs are considered to have infinite girth (Skiena 1990, p. 191). The girth of a graph may be found using Girth [g] in the Mathematica add-on packageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The following table gives examples of graphs with various girths. girth example 3 TETRAHEDRAL GRAPH , COMPLETE GRAPH Kn/ 4 CUBICAL GRAPH , UTILITY GRAPH 5P ETERSEN GRAPH 6H EAWOOD GRAPH 7M CGEE GRAPH 8L EVI GRAPH See also CAGE GRAPH ,G RAPH CIRCUMFERENCE , GRAPH CYCLE ,MOORE GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 13, 1994. Skiena, S. "Girth." §5.3.2 in Implementing Discrete Mathe- matics: Combinatorics and Graph Theory with Mathema- tica. Reading, MA: Addison-Wesley, pp. 190 /C1/92, 1990. Giuga Number Any COMPOSITE NUMBER n with p½(n=p /C281) for all PRIME DIVISORS p of n. n is a Giuga number IFF Xn/C281 k /C301k f(n) /C13/C281 (mod n) where f is the TOTIENT FUNCTION and IFF X p ½n1 p /C28Y p½n1 p/C23N: n is a Giuga number IFF nBf(n) /C13/C281 (mod n) ; where Bkis a BERNOULLI NUMBER and f is the TOTIENT FUNCTION . Every counterexample to Giuga’s conjecture is a contradiction to ARGOH’S CONJECTURE and vice versa. The smallest known Giuga numbers are 30 (3 factors), 858, 1722 (4 factors), 66198 (5 factors), 2214408306, 24423128562 (6 factors), 432749205173838, 14737133470010574, 5508433913 09130318 (7 factors), 244197000982499715087866346, 5540799146170708 01288578559178 (8 factors), ... (Sloane’s A007850). It is not known if there are an infinite number of Giuga numbers. All the above numbers have sum minus product equal to 1, and any Giuga number of higher order must have at least 59 factors. The smallest ODD Giuga number must have at least nine PRIME FACTORS . See also ARGOH’S CONJECTURE ,BERNOULLI NUMBER , PRIMARY PSEUDOPERFECT NUMBER ,TOTIENT FUNC- TIONReferences Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,40/C1/0, 1996. Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation ap ½N 1 =p /C271 =N /C301; Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407 /C1/20, 1999. Sloane, N. J. A. Sequences A007850 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Giuga Sequence A finite, increasing sequence of INTEGERS fn1 ; ...; nm g such that Xm i/C3011 ni/C28Ym i/C3011 ni/C23N: A sequence is a Giuga sequence IFF it satisfies ni ½(n1 /C1/C1/C1ni /C281/C215 ni/C271/C215 nm /C281) for i /C301, ..., m. There are no Giuga sequences of length 2, one of length 3 (/f2; 3; 5g) ; two of length 4 ( f2; 3; 7; 41g and f2 ; 3; 11 ; 13 g) ; 3 of length 5 ( f2; 3; 7; 43; 1805 g; f2; 3; 7; 83; 85g; and f2; 3; 11; 17 ; 59 g) ; 17 of length 6, 27 of length 7, and hundreds of length 8. There are infinitely many Giuga sequences. It is possible to generate longer Giuga sequences from shorter ones satisfying certain properties. See also CARMICHAEL SEQUENCE References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,4 0/C1/0, 1996. Giuga’s Conjecture Ifn/C211 and n½1n/C281/C272n/C281/C27.../C27(n/C281)n/C281/C271; isnnecessarily a PRIME ? In other words, defining sn/C13Xn/C281 k/C301kn/C281; does there exist a COMPOSITE nsuch that sn/C13/C281(mod n)/? It is known that sn/C13/C281(mod n)IFF for each prime divisor pofn,(p/C281)½(n=p/C281) and p½(n=p/C281) (Giuga 1950, Borwein et al. 1996); there- fore, any counterexample must be SQUAREFREE .A composite INTEGER nsatisfies sn/C13/C281(mod n)IFFit is both a C ARMICHAEL NUMBER and a G IUGA NUMBER . Giuga showed that there are no exceptions to the conjecture up to 101000. This was later improved to 101700(Bedocchi 1985) and 1013800(Borwein et al. 1996). See also ARGOH’S CONJECTURE References Bedocchi, E. "The Z(ffiffiffiffiffiffi 14p ) Ring and the Euclidean Algo- rithm." Manuscripta Math. 53, 199 /C1/16, 1985. Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,40/C1/0, 1996. Giuga, G. "Su una presumibile propertieta ` caratteristica dei numeri primi." Ist. Lombardo Sci. Lett. Rend. A 83, 511 /C1/ 28, 1950. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 20 /C1/1, 1989. GL GENERAL LINEAR GROUP Glaisher GLAISHER- KINKELIN CONSTANT Glaisher Constant GLAISHER- KINKELIN CONSTANT Glaisher-Kinkelin Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Define K(n) /C1300112233 /C1/C1/C1(n /C281)n/C281 (1) G(n) /C13[ G(n)]n K(n)/C301i f n /C300 0!1!2! /C1/C1/C1(n /C281)! if n > 0 :=zn* (2) where G(n)isB ARNES’ G-FUNCTION and K(n) is the K- FUNCTION . Then lim n0/C12K(n /C27 1) nn2 =2/C27n=2/C271 =12e/C28n2 =4 /C30A (3) (Voros 1987) and lim n0/C12G(n) nn2 =2 /C281=12(2p)n=2e /C283n2 =4 /C30e1 =12 A; (4) where A /C30exp[1 12 /C28 z?(/C281)] /C301:28242713... (5) is called the Glaisher-Kinkelin constant (Voros 1987) and z?(z) is the derivative of the RIEMANN ZETA FUNCTION (Kinkelin 1860, Glaisher 1877, 1878, 1893, 1894). The constant A is implemented in Mathematica 4.0 asGlaisher . Glaisher (1877) also obtained A /C3027=36 p/C281 =6exp1 3 /C2723 g1 =2 0ln[G(x /C271)] dx() : (6) Glaisher (1894) showed that 11 =121 =231 =941 =1651 =25 .../C30A12 2pe g !p2 =6 (7)11 =131 =951 =2571 =4991 =81 .../C30A12 24 =3 peg !p2 =8 (8) 11=151 =12591 =729 ... 31 =2771 =343111=1331 .../C30A 25=32 p1 =32e3 =32 /C27 g=48/C27s=4 !p3 ; (9) where s /C13z(3) 3 /C215 4 /C215 51 43 /C27z(5) 5 /C215 6 /C215 71 45 /C27z(7) 7 /C215 8 /C215 91 47 /C27... (10) The constant appears in a number of sums and integrals, especially those involving GAMMA FUNC- TIONS and ZETA FUNCTIONS (Wolfram 1999, p. 757). See also BARNES’ G-FUNCTION ,HYPERFACTORIAL , K- FUNCTION References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/glshkn/glshkn.html. Glaisher, J. W. L. "On a Numerical Continued Product." Messenger Math. 6,71/C1/6, 1877. Glaisher, J. W. L. "On the Product 112233 /C1/C1/C1nn :/" Messenger Math. 7,43/C1/7, 1878. Glaisher, J. W. L. "On Certain Numerical Products." Mes- senger Math. 23, 145 /C1/75, 1893. Glaisher, J. W. L. "On the Constant which Occurs in the Formula for 112233 /C1/C1/C1nn :/" Messenger Math. 24,1/C1/6, 1894. Kinkelin. "U¨ ber eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integral- rechnung." J. reine angew. Math. 57, 122 /C1/58, 1860. Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439 /C1/ 65, 1987. Wolfram, S. The Mathematica Book, 4th ed. Cambridge, England: Cambridge University Press, pp. 756 /C1/57, 1999. Glide A product of a REFLECTION in a line and TRANSLATION along the same line. See also REFLECTION ,TRANSLATION References Addington, S. "The Four Types of Symmetry in the Plane." http://forum.swarthmore.edu/sum95/suzanne/symsu- san.html. Glide Reflection GLIDE Glissette The LOCUS of a point P(or the envelope of a line) fixed in relation to a curve Cwhich slides between fixed curves. For example, if Cis a line segment and Pa point on the line segment, then Pdescribes an ELLIPSE when Cslides so as to touch two ORTHOGO- NALstraight LINES . The glissette of the LINE SEGMENT Citself is, in this case, an ASTROID . See also ROULETTE References Besant, W. H. Notes on Roulettes and Glissettes, 2nd enl. ed. Cambridge, England: Deighton, Bell & Co., 1890. Lockwood, E. H. "Glissettes." Ch. 20 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 160 /C1/65, 1967. Yates, R. C. "Glissettes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 108 /C1/12, 1952. Global See also LOCAL Global Analytic Continuation Analytic continuation gives an equivalence relation between function elements, and the equivalence classes induced by this relation are called global analytic functions. See also ANALYTIC CONTINUATION ,DIRECT ANALYTIC CONTINUATION References Krantz, S. G. The Elements of Advanced Mathematics. Boca Raton, FL: CRC Press, 1995. Krantz, S. G. "Global Analytic Continuation." §10.1.6 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 129 /C1/30, 1999. Global Extremum A GLOBAL MINIMUM or GLOBAL MAXIMUM .Itis impossible to construct an algorithm that will find a global extremum for an arbitrary function. See also LOCAL EXTREMUM Global Field A global field is either a NUMBER FIELD ,aFUNCTION FIELD on an ALGEBRAIC CURVE , or an extension of TRANSCENDENCE DEGREE one over a FINITE FIELD . From a modern point of view, a global field may refer to a FUNCTION FIELD on a complex ALGEBRAIC CURVE as well as one over a FINITE FIELD . A global field contains a canonical SUBRING , either the ALGEBRAIC INTEGERS or the POLYNOMIALS . By choosing a PRIME IDEAL in its SUBRING , a global field can be TOPOLOGI- CALLY COMPLETED to give a LOCAL FIELD . For exam- ple, the RATIONAL NUMBERS are a global field. By choosing a PRIME NUMBER p, the RATIONALS can be completed in the P-ADIC NORM to form the P-ADIC NUMBERS Qp :/ A global field is called global because of the special case of a complex ALGEBRAIC CURVE , for which the field consists of global functions, (i.e., functions that are defined everywhere). These functions differ from functions defined near a point, whose completion is called a LOCAL FIELD . Under favorable conditions, thelocal information can be patched together to yield global information (e.g., the HASSE PRINCIPLE ). See also ALGEBRAIC CURVE ,C LASS FIELD,F IELD, FUNCTION FIELD,H ASSE PRINCIPLE ,LOCAL FIELD, NUMBER FIELD,RIEMANN SURFACE References Cohn, H. Advanced Number Theory. New York: Dover, 1980. Weil, A. Ch. 8 in Basic Number Theory. New York: Springer- Verlag, 1974. Global Maximum The largest overall value of a set, function, etc., over its entire range. It is impossible to construct an algorithm that will find a global maximum for an arbitrary function. See also GLOBAL MINIMUM ,LOCAL MAXIMUM ,M AX- IMUM Global Minimum The smallest overall value of a set, function, etc., over its entire range. It is impossible to construct an algorithm that will find a global minimum for an arbitrary function. See also GLOBAL MAXIMUM ,KUHN- TUCKER THEOREM , LOCAL MINIMUM ,MINIMUM Global Optimization References Floudas, C. A.; Pardalos, P. M.; Adjiman, C. S.; Esposito, W. R.; Gu¨mu¨s, Z. H.; Harding, S. T.; Klepeis, J. L.; Meyer, C. A.; and Schweiger, C. A. Handbook of Test Problems in Local and Global Optimization. Dordrecht, Netherlands: Kluwer, 1999. To¨rn, A. and Zilinskas, A. Global Optimization. New York: Springer-Verlag, 1989. Globe A SPHERE which acts as a model of a spherical (or ellipsoidal) celestial body, especially the Earth, and on which the outlines of continents, oceans, etc. are drawn. See also LATITUDE ,LONGITUDE ,SPHERE Glome A 3-sphere x2/C27y2/C27z2/C27w2/C30r2 (as opposed to the usual 2- SPHERE ). The term derives from the Latin ‘glomus’ meaning ‘ball of string.’ See also HYPERSPHERE ,SPHERE Glove Problem Let there be mdoctors and n5mpatients, and let all mnpossible combinations of examinations of patients by doctors take place. Then what is the minimum number of surgical gloves needed G(m; n) so that no doctor must wear a glove contaminated by a patient and no patient is exposed to a glove worn by another doctor? In this problem, the gloves can be turned inside out and even placed on top of one another if necessary, but no "decontamination" of gloves is permitted. The optimal solution is g(m; n) /C302 m /C30n /C302 1 2(m /C271) n /C301 ; m /C302k /C271 12(m) /C2723 nlm otherwise ;8 >< >: where xdeis the CEILING FUNCTION (Vardi 1991). The case m /C30n /C302 is straightforward since two gloves have a total of four surfaces, which is the number needed for mn /C304 examinations. References Gardner, M. Aha! Insight. New York: Scientific American, 1978. Gardner, M. Science Fiction Puzzle Tales. New York: Crown, pp. 5, 67, and 104 /C1/50, 1981. Hajnal, A. and Lova´sz, L. "An Algorithm to Prevent the Propagation of Certain Diseases at Minimum Cost." §10.1 in Interfaces Between Computer Science and Operations Research (Ed. J. K. Lenstra, A. H. G. Rinnooy Kan, and P. van Emde Boas). Amsterdam: Matematisch Centrum, 1978. Orlitzky, A. and Shepp, L. "On Curbing Virus Propagation." Exercise 10.2 in Technical Memo. Bell Labs, 1989. Vardi, I. "The Condom Problem." Ch. 10 in Computational Recreations in Mathematica. Redwood City, CA: Addison- Wesley, pp. 203 /C1/22, 1991. Glue Vector A VECTOR specifying how layers are stacked in a LAMINATED LATTICE . Gnomon A shape which, when added to a figure, yields another figure SIMILAR to the original. References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 123, 1993. Gnomon Magic Square A3/C293 array of numbers in which the elements in each 2 /C292 corner have the same sum. See also MAGIC SQUARE References Stapleton, H. E. "The Gnomon as a Possible Link Between (a) One Type of Mesopotamian Ziggurat and (b) the Magic Square Numbers on which Jaribian Alchemy was Based." Ambix: J. Soc. Study Alchemy and Early Chem. 6,1/C1/, 1957 /C1/958.Gnomonic Number A FIGURATE NUMBER OF THE FORM gn /C302n /C281 which are the areas of square gnomons, obtained by remov- ing a SQUARE of side n /C281 from a SQUARE of side n, gn /C30n2 /C28(n /C281)2 /C302n /C281: The gnomonic numbers are therefore equivalent to the ODD NUMBERS , and the first few are 1, 3, 5, 7, 9, 11, ... (Sloane’s A005408). The GENERATING FUNCTION for the gnomonic numbers is x(1/C27x) (x/C281)2/C30x/C273x2/C275x3/C277x4/C27...: See also FIGURATE NUMBER ,ODD NUMBER References Sloane, N. J. A. Sequences A005408/M2400 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Gnomonic Projection A nonconformal MAP PROJECTION obtained by project- ing points P1(orP2) on the surface of sphere from a sphere’s center Oto point Pin a plane that is tangent to the south pole S(Coxeter 1969, p. 93). Since this projection obviously sends ANTIPODAL POINTS P1and P2to the same point Pin the plane, it can only be used to project one HEMISPHERE as a time. In a gnomonic projection, ORTHODROMES are straight LINES . The transformation equations for a point at LATITUDE f and LONGITUDE l are given by x /C30cos f sin ( l /C28 l0) cos c (1) y /C30cos f1 sin f /C28 sin f1 cos f cos ( l /C28 l0) cos c ; (2) where l0is the central longitude, f1is the central latitude, and c is the angular distance of the point (x, y) from the center of the projection, given by cos c /C30sin f1sin f /C27cos f1 cos f cos(l /C28 l0) : (3) The inverse FORMULAS are f /C30sin/C281cos f sin f1 /C27y sin u cos u cos f1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2p ! ; (4) l /C30 l0 /C27tan/C281 x sin uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2p cos f1 cos u /C28 y sin f1 sin u ! ; (5) where u /C30tan /C281(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p ) : (6) See also STEREOGRAPHIC PROJECTION References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 93 and 289 /C1/90, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 150 /C1/53, 1967. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 164 /C1/68, 1987. G-Number EISENSTEIN INTEGER Go There are estimated to be about 4:63 /C2910170 possible positions on a 19 /C2919 board (Beeler et al. , Flam- menkamp). The number of n-move Go games are 1, 362, 130683, 47046242, ... (Sloane’s A007565). References Beeler, M. et al. Item 96 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 35, Feb. 1972. Bewersdorff, J. "Go und Mathematik." http://home.t-onli- ne.de/home/joerg.bewersdorff/go.htm. Culin, S. "Pa-tok--Pebble Game." §75 in Games of the Orient: Korea, China, Japan. Rutland, VT: Charles E. Tuttle, pp. 91 /C1/01, 1965. Kraitchik, M. "Go." §12.4 in Mathematical Recreations. New York: W. W. Norton, pp. 279 /C1/80, 1942. Lasker, E. Go and Go-Moku. New York: Dover, 1960.Sloane, N. J. A. Sequences A007565/M5447 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Warkentyne, K. "Ken’s Go Page." http://nngs.cosmic.org/ hmkw/. Warkentyne, K. "The Web Go Page Index." http://nngs.cos- mic.org/hmkw/golinks.html. Goat Grazing Problem GOAT PROBLEM Goat Problem Let a circular field of unit radius be fenced in, and tie a goat to a point on the interior of the fence with a chain of length r. What length of chain must be used in order to allow the goat to graze exactly one half the area of the field? The answer is obtained by using the equation for a CIRCLE-CIRCLE INTERSECTION A /C30r2 cos /C281d2 /C27 r2 /C27 R2 2dr ! /C27R2 cos/C281d2 /C27 R2 /C27/C28 r2 2dR ! /C281 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (/C28d/C27r/C27R)(d/C27r/C28R)(d/C28r/C27R)(d/C27r/C27R)p (1) with R/C30d/C301 and A/C30p=2 (i.e., half of pR2):This leads to the equation /C281 2rffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C28r2p /C27r2cos/C281(1 2r)/C27cos/C281(1/C2812r2)/C3012p;(2) which cannot be solved exactly, but which has approximate solution r:1:15872847 : (3) See also CIRCLE- CIRCLE INTERSECTION ,LENS Go¨bel’s Sequence Consider the RECURRENCE RELATION xn/C301/C27x2 0/C27x21/C27.../C27x2n/C281 n; (1) with x0/C301:The first few iterates of xnare 1, 2, 3, 5, 10, 28, 154, ... (Sloane’s A003504). The terms grow extremely rapidly, but are given by the asymptotic formula xn :(n2 /C272n /C281 /C274n/C281 /C2821n /C282 /C27137n/C283 /C28...)C2n ; (2) where C /C301 :04783144757641122955990946274313755459 ::: (3) (Zagier). It is more convenient to work with the transformed sequence sn /C302 /C27x2 1 /C27x22 /C27.../C27x2n/C281 /C30nxn ; (4) which gives the new recurrence sn/C271 /C30sn /C27s2 n n2 (5) with initial condition s1 /C302: Now, sn/C271will be non- integral IFF n¶sn : The smallest p for which sp f0 (mod p) therefore gives the smallest nonintegral sp /C271 : In addition, since p¶sp ; xp /C30sp =p is also the smallest nonintegral xp :/ For example, we have the sequences fsn (mod k)gk n /C301 : 2; 6 /C132 ;5 4 /C130 ; 0; 0 (mod 5) (6) 2 ; 6 ; 15 /C131;5 4 /C130; 0; 0; 0 (mod 7) (7) 2; 6; 15 /C134;52 9 /C137 ;161 16 /C138;264 5 /C130; 0; ...; 0 (mod 11) (8) Testing values of k shows that the first nonintegral xn is x43 : Note that a direct verification of this fact is impossible since x43 :5 :4093 /C2910178485291567 (9) (calculated using the asymptotic formula) is much too large to be computed and stored explicitly. A sequence even more striking for assuming integer values only for many terms is the 3-Go¨bel sequence xn /C301 /C27 x3 0 /C27 x31 /C27 ... /C27 x3n/C281 n : (10) The first few terms of this sequence are 1, 2, 5, 45, 22815, ... (Sloane’s A005166). The Go¨bel sequences can be generalized to k powers by xn /C301 /C27 xk 0 /C27 xk1 /C27 ... /C27 xkn/C281 n : (11) See also SOMOS SEQUENCEReferences Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697 /C1/12, 1988. Guy, R. K. "A Recursion of Go¨bel." §E15 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 214 /C1/15, 1994. Sloane, N. J. A. Sequences A003504/M0728 and A005166/ M1551 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Zaiger, D. "Solution: Day 5, Problem 3." http://www- groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.3.html. Goblet Illusion An ILLUSION in which the eye alternately sees two black faces, or a white goblet. References Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 111 and 115, 1996. Rubin, E. Synoplevede Figurer. Copenhagen, Denmark: Gyldendalske, 1915. Go¨del Number AGo¨del number is a unique number associated with a statement about arithmetic. It is formed as the PRODUCT of successive PRIMES raised to the POWER of the number corresponding to the individual sym- bols that comprise the sentence. For example, the statement (/C215x)(x /C30sy) that reads "there EXISTS an x such that x is the immediate SUCCESSOR of y" is coded (28)(34)(513)(79)(118)(1313)(175)(197)(2316)(299); where the numbers in the set (8, 4, 13, 9, 8, 13, 5, 7, 16, 9) correspond to the symbols that make up(/C215x)(x/C30sy): / See also GO¨ DEL’S INCOMPLETENESS THEOREM References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 18, 1989. Go¨del’s Completeness Theorem IfTis a set of AXIOMS in a first-order language, and a statement pholds for any structure Msatisfying T, then pcan be formally deduced from Tin some appropriately defined fashion. See also GO¨ DEL’S INCOMPLETENESS THEOREM ,LO¨ W- ENHEIM- SKOLEM THEOREM References Beth, E. W. The Foundations of Mathematics. Amsterdam, Netherlands: North-Holland, 1959. Go¨del’s Incompleteness Theorem Informally, Go¨del’s incompleteness theorem states that all CONSISTENT axiomatic formulations of NUM- BER THEORY include undecidable propositions (Hof- stadter 1989). This is sometimes called Go¨del’s first incompleteness theorem, and answers in the negative HILBERT’S PROBLEM asking whether mathematics is "complete" (in the sense that every statement in the language of NUMBER THEORY can be either proved or disproved). Formally, Go¨del’s theorem states, "To every v/-consistent recursive class k of FORMULAS , there correspond recursive class-signs r such that neither (v Gen r) nor Neg( v Gen r) belongs to Flg( /k); where v is the FREE VARIABLE of r" (Go¨del 1931). A statement sometimes known as Go¨del’s second incompleteness theorem states that if NUMBER THE- ORY is consistent, then a proof of this fact does not exist using the methods of first-order PREDICATE CALCULUS . Stated more colloquially, any formal sys- tem that is interesting enough to formulate its own consistency can prove its own consistency IFF it is inconsistent. Gerhard Gentzen showed that the consistency and completeness of arithmetic can be proved if "transfi- nite" induction is used. However, this approach does not allow proof of the consistency of all mathematics. See also CONSISTENCY ,GO¨ DEL’S COMPLETENESS THE- OREM ,H ILBERT’S PROBLEMS ,K REISEL CONJECTURE , NATURAL INDEPENDENCE PHENOMENON ,N UMBER THEORY ,RICHARDSON’S THEOREM ,UNDECIDABLE References Barrow, J. D. Pi in the Sky: Counting, Thinking, and Being. Oxford, England: Clarendon Press, p. 121, 1993. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 74 /C1/5, 1998. Franze ´n, T. "Go¨del on the Net." http://www.sm.luth.se/ ~torkel/eget/godel.html. Go¨del, K. "Uuml;ber Formal Unentscheidbare Sa¨tze der Principia Mathematica und Verwandter Systeme, I." Monatshefte fu¨r Math. u. Physik 38, 173 /C1/98, 1931. Go¨del, K. On Formally Undecidable Propositions of Princi- pia Mathematica and Related Systems. New York: Dover, 1992. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989. Kolata, G. "Does Go¨del’s Theorem Matter to Mathematics?" Science 218, 779 /C1/80, 1982. Smullyan, R. M. Go¨del’s Incompleteness Theorems. New York: Oxford University Press, 1992. Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, 1927. Gog Triangle MONOTONE TRIANGLEGolay-Rudin-Shapiro Sequence RUDIN- SHAPIRO SEQUENCE Goldbach Conjecture Goldbach’s original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states that every INTEGER > 5is the SUM of three PRIMES (Dickson 1957, p. 421). As re- expressed by Euler, an equivalent of this CONJECTURE (called the "strong" or "binary" Goldbach conjecture) asserts that all POSITIVE EVEN INTEGERS ]4 can be expressed as the SUM of two PRIMES . According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like ‘Gold- bach’s Theorem’, which have never been proved andwhich any fool could have guessed." Schnirelman (1939) proved that every EVEN number can be written as the sum of not more than 300,000 PRIMES (Dunham 1990), which seems a rather far cry from a proof for two PRIMES ! Pogorzelski (1977) claimed to have proven the Goldbach conjecture, but his proof is not generally accepted (Shanks 1993). The following table summarizes bounds nsuch that the strong Goldbach conjecture has been shown to be true for numbers Bn:/ bound reference /1/C29104/Desboves 1885 /1/C29105/Pipping 1938 /1/C29108/Stein and Stein 1965ab /2/C291010/Granville et al. 1989 /4/C291011/Sinisalo 1993 /1/C291014/Deshouillers et al. 1998 /4/C291014/Richstein 2000 (quoted in Peterson 2000) The conjecture that all ODD numbers ]9 are the SUM of three ODD PRIMES is called the "weak" Goldbach conjecture. Vinogradov proved that all ODD INTEGERS starting at some sufficiently large value are the SUM of three PRIMES (Guy 1994). The original "sufficiently large" N]3315:ee16:573:3:25/C29106;846;168was subse- quently reduced to ee11:503:3:33/C291043;000by Chen and Wang (1989). Chen (1973, 1978) also showed that all sufficiently large EVEN NUMBERS are the sum of a PRIME and the PRODUCT of at most two PRIMES (Guy 1994, Courant and Robbins 1996). It has been shown that if the weak Goldbach conjecture is false, then there are only a FINITE number of exceptions. A stronger version of the weak conjecture, namely that every odd number >5 can be expressed as the sum of a prime plus twice a prime has been formulated by C. Eaton. This con- jecture has been verified for n5109(Corbit). Other variants of the Goldbach conjecture include the statements that every EVEN number ]6 is the SUM of two ODD PRIMES , and every INTEGER > 17 the sum of exactly three distinct PRIMES . Let R(n) be the number of representations of an EVEN INTEGER n as the sum of two PRIMES . Then the "extended" Goldbach con- jecture states that R(n) /C22Y 2Y k /C302 pk ½npk /C28 1 pk /C28 2 gx 2dx (ln x)2 ; whereQ 2 is the TWIN PRIMES CONSTANT (Halberstam and Richert 1974). If the Goldbach conjecture is true, then for every number m, there are PRIMES p and q such that f(p) /C27 f(q) /C302m; where f(x) is the TOTIENT FUNCTION (Guy 1994, p. 105). Vinogradov (1937ab, 1954) proved that every suffi- ciently large ODD NUMBER is the sum of three PRIMES (Nagell 1951, p. 66), and Estermann (1938) proves that almost all EVEN NUMBERS are the sums of two PRIMES . See also CHEN’S THEOREM , DE POLIGNAC’S CONJEC- TURE ,GOLDBACH NUMBER ,PRIME PARTITION ,SCHNIR- ELMANN’S THEOREM ,W ARING’S PRIME NUMBER CONJECTURE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 64, 1987. Caldwell, C. K. "Prime Links /C27/C27: Resources in theory: conjectures: Goldbach." http://primes.utm.edu/links/the- ory/conjectures/Goldbach/. Chen, J.-R. "On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most TwoPrimes.’ Sci. Sinica 16, 157/C1 /76, 1973. Chen, J.-R. "On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most TwoPrimes, II." Sci. Sinica 21, 421/C1 /30, 1978. Chen, J.-R. and Wang, T.-Z. "On the Goldbach Problem." Acta Math. Sinica 32, 702/C1/18, 1989. Corbit, D. sci.math posting. Nov 19, 1999. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.Oxford, England: Oxford University Press, pp. 30 /C1 /1, 1996. Desboves, A. Nouv. Ann. Math. 14, 293, 1855. Deshouillers, J.-M.; te Riele, H. J. J.; and Saouter, Y. "New Experimental Results Concerning The Goldbach Conjec- ture." In Algorithmic Number Theory: Proceedings of the 3rd International Symposium (ANTS-III) held at Reed College, Portland, OR, June 21 /C1/5, 1998 (Ed. J. P. Buh- ler). Berlin: Springer-Verlag, pp. 204 /C1/15, 1998. Devlin, K. Mathematics: The New Golden Age. London: Penguin Books, 1988. Dickson, L. E. "Goldbach’s Empirical Theorem: Every In- teger is a Sum of Two Primes." In History of the Theory ofNumbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 421 /C1/24, 1952. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 83, 1990. Estermann, T. "On Goldbach’s Problem: Proof that Almost All Even Positive Integers are Sums of Two Primes." Proc. London Math. Soc. Ser. 2 44, 307/C1/14, 1938. Granville, A.; van der Lune, J.; and te Riele, H. J. J. "Checking the Goldbach Conjecture on a Vector Compu-ter." In Number Theory and Applications: Proceedings of the NATO Advanced Study Institute held in Banff,Alberta, April 27-May 5, 1988 (Ed. R. A. Mollin). Dor- drecht, Netherlands: Kluwer, pp. 423 /C1 /33, 1989. Guy, R. K. "Goldbach’s Conjecture." §C1 in Unsolved Pro- blems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 105 /C1/07, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Littlewood, J. E. "Some Problems of ‘Partitio Numerorum.’ III. On the Expression of a Numberas a Sum of Primes." Acta Math. 44,1/C1 /0, 1922. Hardy, G. H. and Littlewood, J. E. "Some Problems of Partitio Numerorum (V): A Further Contribution to theStudy of Goldbach’s Problem." Proc. London Math. Soc. Ser. 2 22,4 6/C1 /6, 1924. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 19, 1979. Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 66, 1951. Peterson, I. "Prime Conjecture Verified to New Heights." Sci. News 158, 103, Aug. 12, 2000. Pipping, N. "Die Goldbachsche Vermutung und der Gold- bach-Vinogradovsche Satz." Acta. Acad. Aboensis, Math. Phys. 11,4/C1/5, 1938. Pogorzelski, H. A. "Goldbach Conjecture." J. reine angew. Math. 292,1/C1/2, 1977. Richstein, J. To appear in Math. Comput. Schnirelman, L. G. Uspekhi Math. Nauk 6,3/C1/, 1939. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 30 /C1/1 and 222, 1985. Sinisalo, M. K. "Checking the Goldbach Conjecture up to 4/C2151011:/"Math. Comput. 61, 931/C1/34, 1993. Stein, M. L. and Stein, P. R. "New Experimental Results on the Goldbach Conjecture." Math. Mag. 38,7 2/C1/0, 1965a. Stein, M. L. and Stein, P. R. "Experimental Results on Additive 2 Bases." BIT 38, 427/C1/34, 1965b. Vinogradov, I. M. "Representation of an Odd Number as a Sum of Three Primes." Comtes rendus (Doklady) de l’Acade ´mie des Sciences de l’U.R.S.S. 15, 169/C1/72, 1937a. Vinogradov, I. "Some Theorems Concerning the Theory of Primes." Recueil Math. 2, 179/C1/95, 1937b. Vinogradov, I. M. The Method of Trigonometrical Sums in the Theory of Numbers. London: Interscience, p. 67, 1954. Wang, Y. (Ed.). |it Goldbach Conjecture. Singapore: World Scientific, 1984. Woon, M. S. C. On Partitions of Goldbach’s Conjecture 4 Oct 2000. http://xxx.lanl.gov/abs/math.GM/0010027/. Yuan, W. Goldbach Conjecture. Singapore: World Scientific, 1984. Goldbach Number A positive integer which is the sum of two ODD PRIMES is called a Goldbach number (Li 1999). Let E(x) (the "exceptional set of Goldbach numbers") denote the number of even numbers not exceeding xwhich cannot be written as a sum of two primes. Then the GOLDBACH CONJECTURE is equivalent to proving that E(x) /C302 for every x ]4 : Li (1999) proved that for sufficiently large x, E(x)/C30O(x0:921): See also GOLDBACH CONJECTURE References Chen, J. "The Exceptional Set of Goldbach Numbers (II)." Sci. Sinica 26, 714/C1/31, 1983. Chen, J. and Liu, J. "The Exceptional Set of Goldbach Numbers (III)." Chinese Quart. J. Math. 4,1/C1/5, 1989. Chen, J. and Pan, C. "The Exceptional Set of Goldbach Numbers." Sci. Sinica 23, 416/C1/30, 1980. Li, H. "The Exceptional Set of Goldbach Numbers." Quart. J. Math. Oxford 50, 471/C1/82, 1999. Montgomery, H. L. and Vaughan, R. C. "The Exceptional Set of Goldbach’s Problem." Acta. Arith. 27, 353/C1/70, 1975. Goldbach’s Theorem GOLDBACH CONJECTURE Golden Mean GOLDEN RATIO Golden Ratio A number often encountered when taking the ratios of distances in simple geometric figures such as the PENTAGRAM ,DECAGON and DODECAGON . It is denoted f;or sometimes t(which is an abbreviation of the Greek "tome," meaning "to cut"). fis also known as the DIVINE PROPORTION ,GOLDEN MEAN , and GOLDEN SECTION and is a P ISOT- VIJAYARAGHAVAN CONSTANT . It has surprising connections with CONTINUED FRAC- TIONS and the E UCLIDEAN ALGORITHM for computing the GREATEST COMMON DIVISOR of two INTEGERS . Given a RECTANGLE having sides in the ratio 1 : f;f is defined such that partitioning the original RECTAN- GLE into a SQUARE and new RECTANGLE results in a new RECTANGLE having sides with a ratio 1 : f:Such aRECTANGLE is called a GOLDEN RECTANGLE , and successive points dividing a GOLDEN RECTANGLE into SQUARES lie on a LOGARITHMIC SPIRAL . This figure is known as a WHIRLING SQUARE . This means that 1 f/C281/C30f (1)f2/C28f/C281/C300: (2) So, by the QUADRATIC EQUATION , f/C301 2(19ffiffiffiffiffiffiffiffiffiffiffi 1/C274p )/C301 2(1/C27ffiffiffi 5p ) (3) /C301:618033988749894848204586834365638117720 . . . (4) (Sloane’s A001622). The golden ratio is given by the INFINITE SERIES f/C3013 8/C27X/C12 n/C300(/C281)n/C271(2n/C271)! (n/C272)!n!42n/C273(5) (B. Roselle). A geometric definition can be given in terms of the above figure. Let the ratio x/C13BC=AB:The NUMERA- TOR and DENOMINATOR can then be taken as AB/C30a andBC/C30xwithout loss of generality. Now define the position of Bby AB BC/C30BC AC: (6) Plugging in gives 1 x/C30x 1/C27x; (7) or x2/C28x/C281/C300; (8) which can be solved using the QUADRATIC EQUATION to obtain f/C13x/C301/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (/C281)2/C284(1)(/C281)p 2/C301 2(1/C27ffiffiffi 5p ); (9) where the plus sign has been taken to give the solution with x/C211. /fis the "most" IRRATIONAL number because it has a CONTINUED FRACTION representation f/C30[1;1;1;. . .] (10) (Sloane’s A000012; Williams 1979, p. 52; Steinhaus1983, p. 45). Another infinite representation in terms of a NESTED RADICAL is f/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27...pqrs : (11) Ramanujan gave the curious CONTINUED FRACTION identities 1 (ffiffiffiffiffiffiffiffiffiffi fffiffiffi 5pq )e2p=5/C301/C27e/C282p 1/C27e/C284p 1/C27e/C286p 1/C27e/C288p 1/C27e/C2810p 1/C27...(12) 1ffiffiffi 5p 1/C27[53=4(f/C281)5=2/C281]/C28f() e2p=ffiffi 5p /C301/C27e/C282pffiffi 5p 1/C27e/C284pffiffi 5p 1/C27e/C286pffiffi 5p 1/C27e/C288pffiffi 5p 1/C27e/C2810pffiffi 5p 1/C27...(13) (Ramanathan 1984). The SINE of certain complex numbers involving f gives particularly simplex answers, sin(ilnf)/C301 2i (14) sin(12p/C28ilnf)/C3012ffiffiffi 5p (15) (Hoey). A curious approximation due to D. Barron is given by f:1 2Kg/C2819=7p2=7/C27g; (16) where Kis C ATALAN’S CONSTANT andgis the E ULER- MASCHERONI CONSTANT , which is good to two digits. Steinhaus (1983, pp. 48 /C1/9) considers the distribution of the FRACTIONAL PARTS ofnfin the intervals bounded by 0, 1 =n;2=n;..., (n/C281)=n;1, and notes that they are much more uniformly distributed than would be expected due to chance (i.e., frac( nf) is close to an EQUIDISTRIBUTED SEQUENCE ). In particular, the number of empty intervals for n/C301, 2, ..., are a mere 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, ... (Sloane’sA036412). The values of nfor which nobins are left blank are then given by 1, 2, 3, 4, 5, 6, 8, 10, 13, 16,21, 34, 55, 89, 144, ... (Sloane’s A036413). Steinhaus(1983) remarks that the highly uniform distribution has its roots in the CONTINUED FRACTION forf:/The legs of a GOLDEN TRIANGLE are in a golden ratio to its base. In fact, this was the method used by Pythagoras to construct f:Euclid used the following construction. Draw the SQUARE IABCD ;callEthe MIDPOINT of AC, so that AE/C30EC/C13x:Now draw the segment BE, which has length xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 22/C2712p /C30xffiffiffi 5p ; (17) and construct EFwith this length. Now construct FG/C30EF, then f/C30FC CD/C30EF/C27CE CD/C30x(ffiffiffi 5p /C271) 2x/C301 2(ffiffiffi 5p /C271):(18) The ratio of the CIRCUMRADIUS to the length of the side of a DECAGON is also f; R s/C301 2cscp 10 ! /C3012(1/C27ffiffiffi 5p )/C30f: (19) Similarly, the legs of a GOLDEN TRIANGLE (an ISO- SCELES TRIANGLE with a VERTEX ANGLE of 368) are in a golden ratio to the base. Bisecting a G AULLIST CROSS also gives a golden ratio (Gardner 1961, p. 102). In the figure above, three TRIANGLES can be IN- SCRIBED in the RECTANGLE /C176ABCD of arbitrary aspect ratio 1 : rsuch that the three RIGHT TRIANGLES have equal areas by dividing AB and BC in the golden ratio. Then KDADE/C301 2/C215r(1/C27f)/C2151/C3012rf2(20) KDBEF/C301 2/C215rf /C215f/C3012rf2(21) KDCDF/C301 2(1/C27f)/C215r/C3012rf2; (22) which are all equal. The golden ratio also satisfies the RECURRENCE RELATION fn /C30 fn/C281 /C27 fn/C282 ; (23) so taking n /C300 gives f /C30 f/C281 /C271 : (24) The powers of the golden ratio also satisfy fn /C30Fn f /C27Fn /C281 ; (25) where Fn is a FIBONACCI NUMBER (Wells 1986, p. 39). For the difference equations x0 /C301 xn /C301 /C271 xn/C281for n /C301; 2 ; 3 ;8 < : (26) /f is also given by f /C30 lim n 0/C12xn : (27) In addition, f /C30 lim n0/C12Fn Fn/C281; (28) where Fnis the nth FIBONACCI NUMBER , as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62). The SUBSTITUTION MAP 0 0 01 (29) 1 0 0 (30) gives 0 0 01 0 010 0 01001 0 ...; (31) giving rise to the sequence 0100101001001010010100100101... (32) (Sloane’s A003849). Here, the zeros occur at positions 1, 3, 4, 6, 8, 9, 11, 12, ... (Sloane’s A000201), and the ones occur at positions 2, 5, 7, 10, 13, 15, 18, ... (Sloane’s A001950). These are complementary BEATTY SEQUENCES generated by nfbc and nf2=z4=z5 : The sequence also has many connections with the FIBONACCI NUMBERS . Salem showed that the set of P ISOT- VIJAYARAGHAVAN CONSTANTS is closed, with fthe smallest accumula- tion point of the set (Le Lionnais 1983). See also BERAHA CONSTANTS ,DECAGON ,FIVE DISKS PROBLEM ,GOLDEN RATIO CONJUGATE ,GOLDEN REC- TANGLE ,G OLDEN TRIANGLE ,ICOSIDODECAHEDRON , NOBLE NUMBER ,PENTAGON ,PENTAGRAM ,PHI NUM- BER SYSTEM ,PHYLLOTAXIS ,PISOT- VIJAYARAGHAVAN CONSTANT ,SECANT METHODReferences Boyer, C. B. History of Mathematics. New York: Wiley, p. 56, 1968. Coxeter, H. S. M. "The Golden Section, Phyllotaxis, and Wythoff’s Game." Scripta Mathematica 19, 135/C1/43, 1953. Dixon, R. Mathographics. New York: Dover, pp. 30 /C1/1 and 50, 1991. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/gold/gold.html. Gardner, M. "Phi: The Golden Ratio." Ch. 8 in The Second Scientific American Book of Mathematical Puzzles & Diversions, A New Selection. New York: Simon and Schuster, pp. 89 /C1/03, 1961. Gardner, M. "Notes on a Fringe-Watcher: The Cult of the Golden Ratio." Skeptical Inquirer 18, 243/C1/47, 1994. Hambridge, J. The Elements of Dynamic Stability. New York: Dover, 1967. Herz-Fischler, R. A Mathematical History of the Golden Number. New York: Dover, 1998. Huntley, H. E. The Divine Proportion. New York: Dover, 1970. Knott, R. "Fibonacci Numbers and the Golden Section." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 40, 1983. Markowsky, G. "Misconceptions About the Golden Ratio." College Math. J. 23,2/C1 /9, 1992. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 122 /C1/34, 1990. Olariu, A. Golden Section and the Art of Painting. 18 Aug 1999. http://xxx.lanl.gov/abs/physics/9908036/. Pappas, T. "Anatomy & the Golden Section." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 32 /C1/3, 1989. Ramanathan, K. G. "On Ramanujan’s Continued Fraction." Acta. Arith. 43, 209/C1/26, 1984. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 148, 1986. Sloane, N. J. A. Sequences A000012/M0003, A000201/ M2322, A001622/M4046, A001950/M1332, and A003849in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 45, 1999. van Zanten, A. J. "The Golden Ratio in the Arts of Painting, Building, and Mathematics." Nieuw Arch. Wisk. 17, 229/C1 / 45, 1999. Weisstein, E. W. "Books about Golden Ratio." http:// www.treasure-troves.com/books/GoldenRatio.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 36 /C1/9, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 87 /C1/8, 1991. Williams, R. "The Golden Proportion." §2/C1/inThe Geome- trical Foundation of Natural Structure: A Source Book ofDesign. New York: Dover, pp. 52 /C1 /3, 1979. Zeising, A. Neue Lehre von den Proportionen des menschli- chen Ko ¨rpers. Golden Ratio Conjugate The quantity fC/C131 f/C30f/C281/C30ffiffiffi 5p /C281 2:0:6180339887 ; (1) where f is the GOLDEN RATIO . The golden ratio conjugate is sometimes also called the SILVER RATIO . A quantity similar to the FEIGENBAUM CONSTANT can be found for the nth CONTINUED FRACTION represen- tation [a0 ; a1 ; a2 ; ...]: (2) Taking the limit of dn /C13sn /C28 sn/C281 sn /C28 sn/C271(3) gives d /C13 lim n0/C12/C301 /C27 f /C302 /C27 fC : (4) See also GOLDEN RATIO,SILVER RATIO Golden Rectangle Given a RECTANGLE having sides in the ratio 1 : f ; the GOLDEN RATIO f is defined such that partitioning the original RECTANGLE into a SQUARE and new RECTAN- GLE results in a new RECTANGLE having sides with a ratio 1 : f: Such a RECTANGLE is called a golden rectangle, and successive points dividing a golden rectangle into SQUARES lie on a LOGARITHMIC SPIRAL (Wells 1986, p. 39). The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated below. If the top left corner of the original square ispositioned at (0, 0), the center of the spiral occurs at the position x0 /C30X/C12 n/C3001 f4n /C271 f4n /C271 /C281 f4n/C272 /C281 f4n/C273 ! /C30(1 /C27 f/C281 /C28 f/C282 /C28 f/C283)X/C12 n/C3001 f4n /C302 f /C27 1 f /C27 2 /C301 10(5 /C273ffiffiffi 5p ) :1:17082 (1) y0 /C30X/C12 n/C300/C281 f4n /C271 f4n/C271 /C271 f4n/C272 /C281 f4n/C273 ! /C30(/C281 /C27 f /C281 /C27 f/C282 /C28 f/C283)X/C12 n/C300/C281 2 /C27 f /C301 10(ffiffiffi 5p /C285) :/C280:276393 ; (2) and the parameters of the spiral aeb u are given by a /C30(4 5)1=4 f(tan/C281 2)= p (3) b /C302lnf p:0 :306349 : (4) See also GOLDEN RATIO,GOLDEN TRIANGLE ,LOGA- RITHMIC SPIRAL ,RECTANGLE References Bicknell, M.; and Hoggatt, V. E. Jr. "Golden Triangles, Rectangles, and Cuboids." Fib. Quart. 7,73/C1/1, 1969. Cook, T. A. The Curves of Life, Being an Account of Spiral Formations and Their Application to Growth in Nature, To Science and to Art. New York: Dover, 1979. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 70, 1989. Pappas, T. "The Golden Rectangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 102 /C1/06, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 45 /C1/7, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 88, 1991. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, p. 53, 1979. Golden Root GOLDEN RATIO Golden Rule The mathematical golden rule states that, for any FRACTION , both NUMERATOR and DENOMINATOR may be multiplied by the same number without changing the fraction’s value. See also DENOMINATOR ,FRACTION ,NUMERATOR References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 151, 1996. Golden Section GOLDEN RATIO Golden Theorem QUADRATIC RECIPROCITY THEOREM Golden Triangle An ISOSCELES TRIANGLE with VERTEX angles 36 8. Such TRIANGLES occur in the PENTAGRAM and DECAGON . The legs are in a GOLDEN RATIO to the base. For such a TRIANGLE , sin(18 /C14) /C30sin(1 10 p) /C301 2 b l (1) b /C302a sin(1 10 p) /C302affiffiffi 5p /C28 1 4/C301 2 a(ffiffiffi 5p /C281) (2) b /C27l /C301 2 a(ffiffiffi 5p /C271) (3) b /C27 a a/C30ffiffiffi 5p /C27 1 2/C30 f: (4) Kimberling (1991) defines a second type of golden triangle in which the ratio of angles is f :1; where f is the GOLDEN RATIO . See also DECAGON ,GOLDEN RATIO,GOLDEN RECTAN- GLE,ISOSCELES TRIANGLE ,PENTAGRAM References Bicknell, M.; and Hoggatt, V. E. Jr. "Golden Triangles, Rectangles, and Cuboids." Fib. Quart. 7,73/C1/1, 1969. Hoggatt, V. E. Jr. The Fibonacci and Lucas Numbers. Boston, MA: Houghton Mifflin, 1969. Kimberling, C. "A New Kind of Golden Triangle." In Applications of Fibonacci Numbers: Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications,’ Wake Forest University (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dor- drecht, Netherlands: Kluwer, pp. 171 /C1/76, 1991. Pappas, T. "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188 /C1/89, 1989. Schoen, R. "The Fibonacci Sequence in Successive Partitions of a Golden Triangle." Fib. Quart. 20, 159 /C1/63, 1982.Goldschmidt Solution The discontinuous solution of the SURFACE OF REVO- LUTION AREA minimization problem for surfaces con- necting two CIRCLES . When the CIRCLES are sufficiently far apart, the usual CATENOID is no longer stable and the surface will break and form two surfaces with the CIRCLES as boundaries. See also CALCULUS OF VARIATIONS ,S URFACE OF REVOLUTION Go¨llnitz’s Theorem LetA(n) denote the number of PARTITIONS ofninto parts/C132;5;11 (mod 12), let B(n) denote the number ofPARTITIONS ofninto distinct parts /C132;4;5 (mod 6), and let C(n) denote the number of PARTITIONS ofn of the form n/C30b1/C27b2/C27.../C27bt; (1) where bi/C28bi/C271]6;with strict inequality if bi/C130;1 or 3 (mod 6), and bt"1;3:Then A(n)/C30B(n)/C30C(n) (2) (Andrews 1986, p. 101). The values of A(n)/C30B(n)/C30C(n) for n/C301, 2, ... are 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, ... (Sloane’s A056970). For example, for n/C3024, there are eight partitions satisfying these conditions, as summarized in the following table. /A(24)/C308// B(24)/C308// C(24)/C308/ /17/C275/C272/ 22/C2722 4 /14/C275/C275/ 20/C2742 2 /C272 /14/C272/C272/C272/C272/C272//17/C275/C272/ 20/C274 /11/C2711/C272/ 16/C2781 9 /C275 /11/C275/C272/C272/C272/C272/14/C2710 18 /C276 /5/C275/C275/C275/C272/C272// 14/C278/C272// 17/C27/ /5/C275/C272/C272/C272/C272/C272/ //C272/C272//11/C278/C275/ 16/C278 /2/C272/C272/C272/C272/C272/C272/ //C272/C272/C272/C272/C272//10/C278/C274/C272//14/C278/C272/ The identity A(n)/C30B(n) can be established using the identity X/C12 n/C300B(n)qn/C30Y/C12 n/C300(1/C27q6n/C272)(1/C27q6n/C274)(1/C27q6n/C275) (3) /C30Y/C12 n/C300(1/C28q12n/C274)(1/C28q12n/C278)(1/C28q12n/C2710) (1/C28q6n/C272)(1/C28q6n/C274)(1/C28q6n/C275)(4) /C30Y/C12 n/C3001 (1 /C28 q12n/C272)(1 /C28 q12n/C275)(1 /C28 q12n/C2711)(5) /C30X/C12 n/C300A(n)qn (6) (Andrews 1986, p. 101). The assertion B(n) /C30C(n)is significantly more difficult, and no simple proof is known. However, it can be established with the aid of computer algebra and the following refinement of the Go¨llnitz theorem. Let B(n; m) denote the number of partitions of n into m distinct parts /C132 ; 4; 5; 4, 5 (mod 6). Let C(n; m) denote the number of partitions of n of the form n /C30b1 /C27b2 /C27.../C27bn ; (7) where bi /C28bi/C271 ]6 ; with strict inequality if bi /C300; 1, 3 (mod 6), where bs "1; 3, and m is the number of bi /C13 2;4;5 plus twice the number of bi/C130;1;3:Then B(n;m)/C30C(n;m) for each nand m(Go¨llnitz 1967; Andrews 1986, p. 102). See also SCHUR’S PARTITION THEOREM References Alladi, K. and Berkovich, A. A Double Bounded Key Identity for Go ¨llnitz’s (BIG) Partition Theorem. 1 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007001/. Andrews, G. E. "Physics, Ramanujan, and Computer Alge- bra." In Proc. Conf. Computer Algebra as a Tool for Researchers in Mathematics and Physics (Ed. D. Chud- novsky and G. Chudnovsky). New York: Springer-Verlag. Andrews, G. E. "Go ¨llnitz’s Theorem." §10.6 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 101 /C1/04, 1986. Go¨llnitz, H. "Partitionen mit Differenzenbedingungen." J. reine angew. Math. 225, 154/C1/90, 1967. Sloane, N. J. A. Sequences A056970 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Go¨llnitz-Gordon Identities X/C12 n/C300qn2(/C28q;q2)n (q2;q2)n/C301 (q;q8)/C12(q4;q8)/C12(q7;q8)/C12 X/C12 n/C300qn(n/C272)(/C28q;q2)n (q2;q2)n/C301 (q3;q8)/C12(q4;q8)/C12(q5;q8)/C12: References Go¨llnitz, H. "Partitionen mit Differenzenbedingungen." J. reine angew. Math. 225, 154/C1/90, 1967. Gordon, B. "Some Continued Fractions of the Rogers- Ramanujan Type." Duke Math. J. 32, 741/C1/48, 1965. Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." To appear in J. London Math. Soc. 2000. Selberg, A. "U ¨ber die Mock-Thetafunktionen siebenter Ordnung." Arch. Math. og Naturvidenskab 41,3/C1/5, 1938.Golomb Constant GOLOMB- DICKMAN CONSTANT Golomb Ruler Ann-mark Golomb ruler is a set of ndistinct nonnegative integers ( a1;a2;...;an);called "marks," such that the positive differences ½ai/C28aj½;computed over all possible pairs of different integers, are distinct. Let anbe the largest integer in an n-mark Golomb ruler. Then an optimal Golomb ruler with n marks is an n-mark Golomb ruler having largest mark ancharacterized by the property that there exist no other n-mark Golomb rulers having smaller an:In such a case, anis the called the "length" of the optimal n-mark ruler. For example, the set (0, 1, 3, 7) is 4-mark Golombruler since its differences are (1 /C301/C1 /,2/C303/C1/,3/C303/C1/, 4/C307/C1/,6/C307/C1/,7/C307/C1/), all of which are distinct. However, the unique optimal Golomb 4-mark ruleris (0, 1, 4, 6), which measures the distances (1, 2, 3, 4, 5, 6) (and is therefore also a PERFECT RULER ). As a further example, it turns out that the length of an optimal 6-mark Golomb ruler is 17. In fact, there are a total of four distinct 6-mark Golomb rulers, all of length 17, one of which is given by (0, 1, 4, 10, 12, 17). In general, the lengths of the optimal n-mark Golomb rulers for n/C302, 3, 4, ... are 1, 3, 6, 11, 17, 25, 34, ... (Sloane’s A003022, Vanderschel and Garry). Although the lengths of the optimal n-mark Golomb rulers are not known for n]23;the known 21, 22, and 23-mark rulers were proved optimal by the Golomb ruler search project in 1998 and 1999. The number of inequivalent optimal n-mark Golomb rulers for n/C302, 3, ... are 1, 1, 1, 2, 4, 5, 1, 1, 1, ... (Sloane’s A036501), and the number of distances inan optimal n-mark ruler is given by the TRIANGULAR NUMBER Tn/C30n(n/C281)=2;so for n/C301, 2, ..., the first few are 0, 1, 3, 6, 10, 15, ... (Sloane’s A000217). The following table gives the optimal Golomb rulers for small n. A more complete table is maintained by J. B. Shearer. noptimal rulers 2 (0, 1) 3 (0, 1, 3) 4 (0, 1, 4, 6) 5 (0, 1, 4, 9, 11), (0, 3, 4, 9, 11) 6 (0, 1, 4, 10, 12, 17), (0, 1, 4, 10, 15, 17), (0, 3, 5, 9, 16, 17), (0, 4, 6, 9, 16, 17) 7 (0, 1, 4, 10, 18, 23, 25), (0, 2, 3, 10, 16, 21, 25), (0, 2, 6, 9, 14, 24, 25), (0, 1, 7, 11, 20, 23, 25), (0, 3, 4, 12, 18, 23, 25) 8 (0, 1, 4, 9, 15, 22, 32, 34) See also PERFECT DIFFERENCE SET,PERFECT RULER , RULER ,TAYLOR’S CONDITION ,W EIGHING References Atkinson, M. D.; Santoro, N.; and Urrutia, J. "Integer Sets with Distinct Sums and Differences and Carrier Fre- quency Assignments for Nonlinear Repeaters." IEEE Trans. Comm. 34, 614/C1/17, 1986. Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 315, 1996. Dewdney, A. K. "Computer Recreations." Sci. Amer. 253, 16, June 1985. Dewdney, A. K. "Computer Recreations." Sci. Amer. 254, 20, Mar. 1986. distributed.net. "Project OGR." http://www.distributed.net/ ogr/. Golomb, S. W. "How to Number a Graph." In Graph Theory and Computing (Ed. R. C. Read). New York: Academic Press, pp. 23 /C1/7, 1972. Guy, R. K. "Modular Difference Sets and Error Correcting Codes." §C10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 118 /C1/21, 1994. Hewgill, G. "distributed.net OGR Project." http://www.hew- gill.com/ogr/. Kotzig, A. and Laufer, P. J. "Sum Triangles of Natural Numbers Having Minimum Top." Ars. Combin. 21,5/C1/3, 1986. Lam, A. W. and D. V. Sarwate, D. V. "On Optimum Time Hopping Patterns." IEEE Trans. Comm. 36, 380/C1/82, 1988. Miller, L. "Golomb Rulers." http://www.cuug.ab.ca/~millerl/ g3-records.html. Robinson, J. P. and Bernstein, A. J. "A Class of Binary Recurrent Codes with Limited Error Propagation." IEEE Trans. Inform. Th. 13, 106/C1/13, 1967. Shearer, J. B. "Golomb Rulers." http://www.research.ibm.- com/people/s/shearer/grule.html. Sloane, N. J. A. Sequences A000217/M2535, A003022/ M2540, A036501, and A039953 in "An On-Line Versionof the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M2540 in The Encyclopedia of Integer Sequences. San Diego, CA: Aca- demic Press, 1995. Vanderschel, D. and Garry, M. "In Search of the Optimal 20, 21, & 22 Mark Golomb Rulers." http://members.aol.com/golomb20/. Golomb-Dickman Constant N.B. A detailed online essay by S. Finch was the starting point for this entry.LetPbe a PERMUTATION ofnelements, and let aibe the number of CYCLES of length iin this PERMUTA- TION . Picking PatRANDOM gives X/C12 j/C301aj*+ /C30Xn i/C3011 i/C30lnn/C27g/C27O1 n ! (1) varX/C12 j/C301aj ! /C30Xn i/C301i/C281 i2/C30lnn/C27g/C281 6p2/C27O1 n ! (2) lim n0/C12P(a1/C300)/C301 e(3) (Shepp and Lloyd 1966, Wilf 1990). Goncharov (1942) showed that lim n0/C12P(aj/C30k)/C301 k!e/C281=jj/C28k; (4) which is a P OISSON DISTRIBUTION , and lim n0/C12PX/C12 j/C301aj/C28lnn ! (lnn)/C281=25x"# /C30F(x); (5) which is a NORMAL DISTRIBUTION ,gis the E ULER- MASCHERONI CONSTANT , and F(x) is the NORMAL DISTRIBUTION FUNCTION . Let M(a)/C13max fj:aj>0g; (6) i.e., the length of the longest cycle in P:Then Golomb (1959) derived l/C13lim n0/C12/C142M(a)/C143 n/C300:6243299885 . . . ; (7) which is known as the G OLOMB CONSTANT or Golomb- Dickman constant. Knuth (1981) asked for the con-stants bandcsuch that lim n0/C12nb/C142M(a)/C143/C28ln/C281 2lhi /C30c; (8) and Gourdon (1996) showed that /C142M(a)/C143/C30l(n/C2712)/C28eg 24n/C271 48eg/C2818(/C281)n n2 /C2717 3840eg/C271 8(/C281)n/C2716j1/C272n/C2716j2/C27n n3;(9) where j/C13e2pi=3: (10) /lcan be expressed in terms of the function f(x) defined by f(x)/C301 for 15x52 and df dx/C30/C28f(x/C281) x/C281(11) for x /C212, by l /C30g/C12 1f(x) x2dx : (12) Shepp and Lloyd (1966) derived l /C30g/C12 0exp /C28x /C28g/C12 xe /C28y ydy ! /C30g1 0expgx 0dy ln y ! dx: (13) Mitchell (1968) computed l to 53 decimal places. Surprisingly enough, there is a connection between l and PRIME FACTORIZATION (Knuth and Pardo 1976, Knuth 1981, pp. 367 /C1/68, 395, and 611). Dickman (1930) investigated the probability P(x; n) that the largest PRIME FACTOR p of a random INTEGER between 1 and n satisfies p Bnx for x /C23 (0; 1): He found that F(x) /C13 lim n0/C12P(x; n) /C301i f x ]1 gx 0Ft 1 /C28 t ! dt tif 0 5x 51:8 < : (14) Dickman then found the average value of x such that p /C30nx ; obtaining m /C13 lim n0/C12/C142x/C143/C30 lim n0/C12ln p ln n*+ /C30g1 0xdF dxdx /C30g1 0F1 1 /C28 t ! dt /C300:62432999 ; (15) which is l :/ References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/golomb/golomb.html. Gourdon, X. 1996. http://www.mathsoft.com/asolve/constant/ golomb/gourdon.html. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Knuth, D. E. and Pardo, L. T. "Analysis of a Simple Factorization Algorithm." Theor. Comput. Sci. 3, 321 /C1/ 48, 1976. Mitchell, W. C. "An Evaluation of Golomb’s Constant." Math. Comput. 22, 411 /C1/15, 1968. Purdom, P. W. and Williams, J. H. "Cycle Length in a Random Function." Trans. Amer. Math. Soc. 133, 547 /C1/ 51, 1968. Shepp, L. A. and Lloyd, S. P. "Ordered Cycle Lengths in Random Permutation." Trans. Amer. Math. Soc. 121, 350 /C1/57, 1966. Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1993.Golygon A PLANE path on a set of equally spaced LATTICE POINTS , starting at the ORIGIN , where the first step is one unit to the north or south, the second step is two units to the east or west, the third is three units to the north or south, etc., and continuing until the ORIGIN is again reached. No crossing or backtracking is allowed. The simplest golygon is (0, 0), (0, 1), (2, 1), (2, /C282), ( /C282, /C282), ( /C282, /C287), ( /C288, /C287), ( /C288, 0), (0, 0). A golygon can be formed if there exists an EVEN INTEGER n such that 91 93 9...9(n /C281) /C300 (1) 92 94 9...9n /C300 (2) (Vardi 1991). Gardner proved that all golygons are OF THE FORM n /C308k: The number of golygons of length n (EVEN ), with each initial direction counted separately, is the PRODUCT of the COEFFICIENT ofxn2=8in (1/C27x)(1/C27x3)/C1/C1/C1(1/C27xn/C281); (3) with the COEFFICIENT ofxn(n=2/C271)=8in (1/C27x)(1/C27x2)/C1/C1/C1(1/C27xn=2): (4) The number of golygons N(n) of length 8 nfor the first fewnare 4, 112, 8432, 909288, ... (Sloane’s A006718) and is asymptotic to N(n)/C23 /C21528n/C284 pn2(4n/C271)(5) (Sallows et al. 1991, Vardi 1991). See also CANONICAL POLYGON ,LATTICE PATH,LAT- TICE POLYGON References Dudeney, A. K. "An Odd Journey Along Even Roads Leads to Home in Golygon City." Sci. Amer. 263, 118/C1/21, July 1990. Sallows, L. C. F. "New Pathways in Serial Isogons." Math. Intell. 14,5 5/C1/7, 1992. Sallows, L.; Gardner, M.; Guy, R. K.; and Knuth, D. "Serial Isogons of 90 Degrees." Math Mag. 64, 315/C1/24, 1991. Sloane, N. J. A. Sequences A006718/M3707 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Smith, H. J. "Golygons." http://pweb.netcom.com/~hjsmith/ Golygons.html. Vardi, I. "American Science." §5.3 in Computational Recrea- tions in Mathematica. Redwood City, CA: Addison-Wes- ley, pp. 90 /C1/6, 1991. Gomory’s Theorem Regardless of where one white and one black square are deleted from an ordinary 8 /C298 CHESSBOARD , the reduced board can always be covered exactly with 31 DOMINOES (of dimension 2 /C291):/ See also CHESSBOARD Gompertz Constant G /C13g/C12 0e /C28u 1 /C27 udu /C30/C28e ei(/C281) /C300:596347362 .. . ; where ei(x) is the EXPONENTIAL INTEGRAL . Stieltjes showed it has the CONTINUED FRACTION representa- tion G /C301 2/C2812 4 /C2822 6/C2832 8/C28/C1/C1/C1: See also EXPONENTIAL INTEGRAL References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 29, 1983. Gompertz Curve The function defined by y /C30abqx : It is used in actuarial science for specifying a simplified mortality law (Kenney and Keeping 1962, p. 241). Using s(x) as the probability that a newborn will achieve age x, the Gompertz law is s(x) /C30exp[/C28m(cx /C281)] ; for c /C211, x ]0 (Gompertz 1832). See also LAW OF GROWTH ,LIFE EXPECTANCY ,LOGIS- TIC GROWTH CURVE ,M AKEHAM CURVE ,POPULATION GROWTH References Bowers, N. L. Jr.; Gerber, H. U.; Hickman, J. C.; Jones, D. A.; and Nesbitt, C. J. Actuarial Mathematics. Itasca, IL: Society of Actuaries, p. 71, 1997. Gompertz, B. "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies." Phil. Trans. Roy. Soc. London 123, 513 /C1/85, 1832. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.Gon GRADIAN Gonal Number POLYGONAL NUMBER Good Binomial Coefficient A BINOMIAL COEFFICIENTN k=z;=z1 with k ]2 is called good if its LEAST PRIME FACTOR satisfies lpfN k=z1r=z1> > k (Erdos et al. 1993). This is equivalent to the require- ment that GCDN k=z1r=z1> ; k!=z1r=z1> /C301: The first few good binomial coefficients are therefore 3 2=z;=z1 ;54=z;=z1 ;62=z;=z1 ;72=z;=z1 ;73=z;=z1 ;74=z;=z1 ;76=z;=z1 ;10 2=z;=z1 ; .... Good binomial coefficients are closely related to the ERDOS-SELF- RIDGE FUNCTION g(k); which gives the least integer N > k /C271 such thatN k=z;=z1 is good. See also BINOMIAL COEFFICIENT ,DEFICIENCY ,ERDOS- SELFRIDGE FUNCTION ,EXCEPTIONAL BINOMIAL COEF- FICIENT References Erdos, P.; Lacampagne, C. B.; and Selfridge, J. L. "Esti- mates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215 /C1/24, 1993. Good Path P-GOOD PATH Good Prime A PRIME pn is called "good" if p2 n > pn/C28ipn/C27i for all 1 5i 5n /C281 (there is a typo in Guy 1994 in which the is are replaced by 1s). There are infinitely many good primes, and the first few are 5, 11, 17, 29, 37, 41, 53, ... (Sloane’s A028388). See also ANDRICA’S CONJECTURE ,L ANDAU’S PRO- BLEMS ,PO´ LYA CONJECTURE References Guy, R. K. "‘Good’ Primes and the Prime Number Graph." §A14 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 32 /C1/3, 1994. Sloane, N. J. A. Sequences A028388 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Goodman’s Formula A two-coloring of a COMPLETE GRAPH Knofnnodes which contains exactly the number of MONOCHRO- MATIC FORCED TRIANGLES and no more (i.e., a mini- mum of R /C27B where R and B are the number of red and blue TRIANGLES ) is called an EXTREMAL GRAPH . Goodman (1959) showed that for an extremal graph, R /C27B /C301 3 m(m /C281)(m /C282) for n /C302m 23 m(m /C281)(4m /C271) for n /C304m /C271 23 m(m /C271)(4m /C281) for n /C304m /C273:8 >< >: Schwenk (1972) rewrote the equation in the form R /C27B /C30n 3=z1r=z1> /C281 2 n14(n /C281)2jkjk ; wheren k=z;=z1 is a BINOMIAL COEFFICIENT and xbcis the FLOOR FUNCTION . See also BLUE-EMPTY GRAPH ,E XTREMAL GRAPH , MONOCHROMATIC FORCED TRIANGLE References Goodman, A. W. "On Sets of Acquaintances and Strangers at Any Party." Amer. Math. Monthly 66, 778 /C1/83, 1959. Schwenk, A. J. "Acquaintance Party Problem." Amer. Math. Monthly 79, 1113 /C1/117, 1972. Goodstein Sequence Given a HEREDITARY REPRESENTATION of a number n in BASE b, let B[b](n) be the NONNEGATIVE INTEGER which results if we syntactically replace each b by b /C271 (i.e., B[b] is a base change operator that ‘bumps the base’ from b up to b /C271): The HEREDITARY REPRESENTATION of 266 in base 2 is 266 /C3028 /C2723 /C272 /C30222 /C271 /C2722 /C271 /C272 ; so bumping the base from 2 to 3 yields B[2](266) /C30333/C271 /C2733/C271 /C273: Now repeatedly bump the base and subtract 1, G0(266) /C30266 /C30222 /C271 /C2722 /C271 /C272 G1(266) /C30B[2](266) /C281 /C30333 /C271 /C2733 /C271 /C272 G2(266) /C30B[3](G1) /C281 /C30444/C271 /C2744 /C271 /C271 G3(266) /C30B[4](G2) /C281 /C30555/C271 /C2755 /C271 /C271 G4(266) /C30B[5](G3) /C281 /C30666/C271 /C2766 /C271 /C281 /C30666/C271 /C275 /C215 66 /C275 /C215 65 /C27.../C275 /C215 6 /C275 G5(266) /C30B[6](G4) /C281 /C30777/C271 /C275 /C215 77 /C275 /C215 75 /C27.../C275 /C215 7 /C274; etc. Starting this procedure at an INTEGER n gives the Goodstein sequence fGk(n) g: Amazingly, despite the apparent rapid increase in the terms of the sequence,GOODSTEIN’S THEOREM states that Gk(n) is 0 for any n and any sufficiently large k. See also GOODSTEIN’S THEOREM ,HEREDITARY REPRE- SENTATION References Goodstein, R. L. "On the Restricted Ordinal Theorem." J. Symb. Logic 9,33/C1/1, 1944. Henle, J. M. An Outline of Set Theory. New York: Springer- Verlag, 1986. Goodstein’s Theorem For all n, there exists a k such that the kth term of the GOODSTEIN SEQUENCE Gk(n) /C300: In other words, every GOODSTEIN SEQUENCE converges to 0. The secret underlying Goodstein’s theorem is that the HEREDITARY REPRESENTATION of n in base b mimics an ordinal notation for ordinals less than some number. For such ordinals, the base bumping opera- tion leaves the ordinal fixed whereas the subtraction of one decreases the ordinal. But these ordinals are well ordered, and this allows us to conclude that a Goodstein sequence eventually converges to zero. Goodstein’s theorem cannot be proved in PEANO ARITHMETIC (i.e., formal NUMBER THEORY ). See also NATURAL INDEPENDENCE PHENOMENON , PEANO ARITHMETIC References Goodstein, R. L. "On the Restricted Ordinal Theorem." J. Symb. Logic 9,33/C1/1, 1944. Henle, J. M. An Outline of Set Theory. New York: Springer- Verlag, 1986. Googol A LARGE NUMBER equal to 10100 (i.e., a 1 with 100 zeros following it). Written out explicitly, 10000000000000000000000000000000000000000000- 00000000000000000000000000000000000000000000-0000000000000. See also G OOGOLPLEX ,LARGE NUMBER References Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Tempus Books, pp. 20 /C1/7, 1989. Pappas, T. "Googol & Googolplex." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 76, 1989. Googolplex A LARGE NUMBER equal to 1010100 (i.e., 1 with a GOOGOL number of 0s written after it).See also G OOGOL ,LARGE NUMBER References Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Tempus Books, pp. 23 /C1/7, 1989. Pappas, T. "Googol & Googolplex." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 76, 1989. Gordian Distance A metric characterizing the difference between two knots K and K ? in S3 :/ References Murakami, H. "Some Metrics on Classical Knots." Math. Ann. 270,35/C1/5, 1985. Gordon Function Another name for the CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND , defined by G(1 /C28 c) G(1 /C28 a)e /C28 pc /C27sin[ p(a /C28 c)] sin(pa)"# 1F1(a; c; z)( /C282G(c /C28 1) G(c /C28 a)z1 /C28c 1F1(a /C28c /C271; 2 /C28c; z)=zn+ ; where G(x) is the GAMMA FUNCTION and1F1(a; b; z)is the CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND . See also CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 671 /C1/72, 1953. Gordon Matrix PRIME ARRAY Gordon-Luecke Theorem Two distinct knots cannot have the same exterior. Or, equivalently, a knot is completely determined by its KNOT EXTERIOR (Adams 1994, p. 261). The question was first posed by Tietze in 1908, and finally proved by Gordon and Luecke (1989). See also KNOT EXTERIOR References Adams, C. C. "The Poincare ´ Conjecture, Dehn Surgery, and the Gordon-Luecke Theorem." §9.3 in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 257 /C1/63, 1994. Gordon, C. and Luecke, J. "Knots Are Determined by Their Complements." J. Amer. Math. Soc. 2, 371 /C1/15, 1989. Gorenstein Ring An algebraic RING which appears in treatments of duality in ALGEBRAIC GEOMETRY . Let A be a local ARTINIAN RING with m ƒA its maximal IDEAL . Then A is a Gorenstein ring if the ANNIHILATOR of m has DIMENSION 1asa VECTOR SPACE over K /C30A=m:/ See also CAYLEY- BACHARACH THEOREMReferences Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach Theorems and Conjectures." Bull. Amer. Math. Soc. 33, 295 /C1/24, 1996. Gosper Island A modification of the KOCH SNOWFLAKE which has FRACTAL DIMENSION D /C302ln3 ln 7/C301:12915... : The term "Gosper island" was used by Mandelbrot (1977) because this curve bounds the space filled by the PEANO- GOSPER CURVE ; Gosper and Gardner use the term FLOWSNAKE FRACTAL instead. Gosper islands can TILE the PLANE . See also KOCH SNOWFLAKE ,PEANO- GOSPER CURVE References Mandelbrot, B. B. Fractals: Form, Chance, & Dimension. San Francisco, CA: W. H. Freeman, Plate 46, 1977. Gosper’s Algorithm An ALGORITHM for finding closed form HYPERGEO- METRIC IDENTITIES . The algorithm treats sums whose successive terms have ratios which are RATIONAL FUNCTIONS . Not only does it decide conclusively whether there exists a hypergeometric sequence zn such that tn/C30zn/C271/C28zn; (1) but actually produces znif it exists. If not, it produces an/C281 k/C300tk:An outline of the algorithm follows (Petkov- sek 1996): 1. For the ratio r(n) /C30tn/C271 =tn which is a RATIONAL FUNCTION of n. 2. Write r(n) /C30a(n) b(n)c(n /C27 1) c(n); (2) where a(n) ; b(n) ; and c(n) are polynomials satisfy- ing GCD( a(n) ; b(n /C27h)) /C301 (3) for all nonnegative integers h. 3. Find a nonzero polynomial solution x(n)of a(n)x(n /C271) /C28b(n /C281)x(n) /C30c(n) ; (4) if one exists. 4. Return b(n /C281)x(n) =c(n)tn and stop. Petkovsek et al. (1996) describe the algorithm as "one of the landmarks in the history of computerization of the problem of closed form summation." Gosper’s algorithm is vital in the operation of ZEILBERGER’S ALGORITHM and the machinery of W ILF-ZEILBERGER PAIRS . See also HYPERGEOMETRIC IDENTITY ,SISTER CELINE’S METHOD ,W ILF-ZEILBERGER PAIR,ZEILBERGER’S AL- GORITHM References Gessel, I. and Stanton, D. "Strange Evaluations of Hyper- geometric Series." SIAM J. Math. Anal. 13, 295/C1/08, 1982. Gosper, R. W. "Decision Procedure for Indefinite Hypergeo- metric Summation." Proc. Nat. Acad. Sci. USA 75,4 0/C1/2, 1978. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Koepf, W. "Algorithms for m-fold Hypergeometric Summa- tion." J. Symb. Comput. 20, 399/C1/17, 1995. Koepf, W. "Gosper’s Algorithm." Ch. 5 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 61 /C1/9, 1998. Lafron, J. C. "Summation in Finite Terms." In Computer Algebra Symbolic and Algebraic Computation, 2nd ed. (Ed. B. Buchberger, G. E. Collins, and R. Loos). NewYork: Springer-Verlag, 1983. Paule, P. and Schorn, M. "A Mathematica Version of Zeilberger’s Algorithm for Proving Binomial CoefficientIdentities." J. Symb. Comput. 20, 673/C1 /98, 1995. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Gosper’s Algorithm." Ch. 5 in A/C30B.Wellesley, MA: A. K. Peters, pp. 73 /C1/9, 1996. Zeilberger, D. "The Method of Creative Telescoping." J. Symb. Comput. 11, 195/C1/04, 1991. Gosper’s Method GOSPER’S ALGORITHM Gossip Problem GOSSIPINGGossiping This entry contributed by R ONALD M.AARTS Gossiping and broadcasting are two problems of information dissemination described for a group of individuals connected by a communication network. In gossiping, every person in the network knows aunique item of information and needs to communicateit to everyone else. In broadcasting, one individual has an item of information which needs to be com- municated to everyone else (Hedetniemi et al. 1988). A popular formulation assumes there are npeople, each one of whom knows a scandal which is not known to any of the others. They communicate by telephone, and whenever two people place a call, theypass on to each other as many scandals as they know. How many calls are needed before everyone knows about all the scandals? Denoting the scandal-sprea-ders as A,B,C, and D, a solution for n/C304 is given by fA;Bg;fC;Dg;fA;Cg;fB;Dg:The n/C304 solution can then be generalized to n/C214 by adding the pair fA;Xgto the beginning and end of the previous solution, i.e., fA;Eg;fA;Bg;fC;Dg;fA;Cg;fB;Dg; fA;Eg: / Gossiping (which is also called total exchange or all-to-all communication) was originally introduced indiscrete mathematics as a combinatorial problem in GRAPH THEORY , but it also has applications in com- munications and distributed memory multiprocessorsystems (Bermond et al. 1998). Moreover, the gossip problem is implicit in a large class of parallelcomputation problems, such as linear system solving,the DISCRETE FOURIER TRANSFORM , and SORTING . Surveys are given in Hedetniemi et al. (1988) and Hromkovic et al. (1995). Letf(n) be the number of minimum calls necessary to complete gossiping among npeople, where any pair of people may call each other. Then f(1)/C300;f(2)/C301; f(3)/C303;and f(n)/C302n/C284 forn]4:This result was proved by (Tijdeman 1971), as well as many others. In the case of one-way communication ("polarized telephones"), e.g., where communication is done by letters or telegrams, the graph becomes a DIRECTED GRAPH and the minimum number of calls becomes f(n)/C302n/C282 forn]4 (Harary and Schwenk 1974). References Bermond, J.-C.; Gargano, L.; Rescigno, A. A.; and Vaccaro, U. "Fast Gossiping by Short Messages." SIAM J. Comput. 27, 917/C1/41, 1998. Harary, F. and Schwenk, A. J. "The Communication Pro- blem on Graphs and Digraphs." J. Franklin Inst. 297, 491/C1/95, 1974. Hedetniemi, S. M.; Hedetniemi, S. T.; and Liestman, A. L. "A Survey of Gossiping and Broadcasting in Communica- tion Networks." Networks 18, 319 /C1/49, 1988. Hromkovic, J.; Klasing, R.; Monien, B.; and Peine, R. "Dissemination of Information in Interconnection Net- works (Broadcasting and Gossiping)." In Combinatorial Network Theory (Ed. F. Hsu and D.-A. Du). Norwell, MA: Kluwer, pp. 125 /C1/12, 1995. Tijdeman, R. "On a Telephone Problem." Nieuw Archief voor Wiskunde 19, 188 /C1/92, 1971. Gould and Hsu Matrix Inversion Formula Let (ai) be a sequence of complex numbers and let the LOWER TRIANGULAR MATRICES F /C30(F(n; k)) and G /C30 (G(n; k)) be defined as F(n ; k) /C30Qn/C281 j/C30k (aj /C27 k) (n /C28 k)! and G(n; k) /C30(/C281)n/C28kak /C27 k an /C27 nQn j/C30k/C271(aj /C27 n) (n /C28 k)!; where the product over an EMPTY SET is 1. Then F and G are MATRIX INVERSES (Bhatnagar 1995, pp. 15 /C1/6 and 50 /C1/1). The KRATTENTHALER MATRIX INVERSION FORMULA is a generalization of this result. See also KRATTENTHALER MATRIX INVERSION FORMU- LA References Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, 1995. Carlitz, L. "Some Inversion Relations." Duke Math. J. 40, 803 /C1/01, 1972. Chu, W. C. and Hsu, L. C. "Some New Applications of Gould-Hsu Inversions." J. Combin. Inform. System Sci. 14,1/C1/, 1990. Gessel, I. and Stanton, D. "Application of q-Lagrange Inversion to Basic Hypergeometric Series." Trans. Amer. Math. Soc. 277, 173 /C1/01, 1983. Gould, H. W. and Hsu, L. C. "Some New Inverse Series Relations." Duke Math. J. 40, 885 /C1/91, 1973. Riordan, J. Combinatorial Identities. New York: Wiley, 1979. Gould Polynomial The polynomials Gn(x; a ; b) given by the associated SHEFFER SEQUENCE with f(t) /C30eat(ebt /C281); where b "0: The INVERSE FUNCTION (and therefore GENERATING FUNCTION ) cannot be computed algeb- raically, but the GENERATING FUNCTION X/C12 k/C300Gk(x; a; b) k!tk /C30exf /C281(t) (1) can be given in terms of the sumf /C281(t) /C30X/C12 k /C3011 b/C28(b /C27ak) =b k /C281=z1r=z1>tk k: (2) This results in Gn(x; a ; b) /C30x x /C28 anx /C28 an b ! n where (x)n is a FALLING FACTORIAL . The first few are G0(x; a ; b) /C301 G1(x; a ; b) /C30x b G2(x; a; b) /C30/C28(2a /C27 b /C28 x) b2 G3(x; a ; b) /C30(3a /C27 b /C28 x)(3a /C27 2b /C28 x)x b3 G4(x; a;b) /C30/C28(4a /C27 b /C28 x)(4a /C27 2b /C28 x)(4a /C27 3b /C28 x)x b4 : The binomial identity obtained from the SHEFFER SEQUENCE gives the generalized CHU-VANDERMONDE IDENTITY x /C27 y x /C27 y /C28 an(x /C27y /C28an) =b n=z1r=z1> /C30Xn k /C300x x /C28 aky y /C28 a(n /C28 k)x /C28 ak b k0 @1Ay /C28 a(n /C28 k) a n /C28k0@1A(3) (Roman 1984, p. 69). In the special case a /C30/C28b=2; the function f(t) simpli- fies to f(t)/C30ebt=2/C28e/C28bt=2/C302 sinh(1 2bt); (4) which gives the GENERATING FUNCTION X/C12 k/C300Gk(x;/C2812b;b) k!tk/C30exp2xsinh/C281(12t) b"# ; (5) giving the polynomials G0(x;/C28b=2;b)/C301 G1(x;/C28b=2;b)/C30x b G2(x;/C28b=2;b)/C30x2 b2 G3(x;/C28b=2;b)/C30/C28(b/C282x)x(b/C272x) 4b3 G4(x;/C28b=2;b)/C30/C28(b/C28x)x2(b/C27x) b4: See also CENTRAL FACTORIAL ,FALLING FACTORIAL , SHEFFER SEQUENCE References Gould, H. W. "Note on a Paper of Sparre-Anderson." Math. Scand. 6, 226 /C1/30, 1958. Gould, H. W. "Stirling Number Representation Problems." Proc. Amer. Math. Soc. 11, 447 /C1/51, 1960. Gould, H. W. "A Series of Transformation for Finding Convolution Identities." Duke Math. J. 28, 193 /C1/02, 1961. Gould, H. W. "Note on a Paper of Klamkin Concerning Stirling Numbers." Amer. Math. Monthly 68, 477 /C1/79, 1961. Gould, H. W. "A New Convolution Formula and Some New Orthogonal Relations for the Inversion of Series." Duke Math. J. 29, 393 /C1/04, 1962. Gould, H. W. "Congruences Involving Sums of Binomial Coefficients and a Formula of Jensen." Amer. Math. Monthly 69, 400 /C1/02, 1962. Roman, S. "The Gould Polynomials and he Central Factorial Polynomials." §4.1.4 in The Umbral Calculus. New York: Academic Press, pp. 67 /C1/0, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Goursat Problem For the HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION uxy /C30F(x; y; u; p; q) (1) p /C30ux (2) q /C30uy (3) on a domain V; Goursat’s problem asks to find a solution u(x; y) of (3) from the BOUNDARY CONDITIONS u(0; t) /C30 f(t) (4) u(t; 1) /C30 c(t) (5) f(1) /C30 f(0) (6) for 0 5t 51 that is regular in V and continuous in the closure ¯V; where f and c are specified continuously differentiable functions. The linear Goursat problem corresponds to the solu- tion of the equation ˜Lu /C30uxy /C27aux /C27buy /C27cu /C30f ; (7) which can be effected using the so-called RIEMANN FUNCTION R(x; y; j; h) : The use of the RIEMANN FUNCTION to solve the linear Goursat problem is called the RIEMANN METHOD . See also BOUNDARY VALUE PROBLEM ,H YPERBOLIC PARTIAL DIFFERENTIAL EQUATION ,F UNCTION ,R IE- MANN METHOD References Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 2. New York: Wiley, 1989. Goursat, E. Cours d’analyse mathe ´matique, Vol. 3, Part 1. Paris: Gauthier-Villars, 1923. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 289, 1988.Tricomi, F. G. Integral Equations. New York: Interscience, 1957. Goursat’s Surface A general QUARTIC SURFACE defined by x4 /C27y4 /C27z4 /C27a(x2 /C27y2 /C27z2)2 /C27b(x2 /C27y2 /C27z2) /C27c (Gray 1997, p. 314). The above two images correspond to a /C30b /C300 ; c /C30/C28 1, and a /C300, b /C30/C28 2, c /C30/C281, respectively. The related surface xn /C27yn /C27zn /C301 for n ]2 an even integer is considered by Gray (1997, p. 292), and might appropriately be called a SUPER- ELLIPSOID . See also CHMUTOV SURFACE ,CUBE,SUPERELLIPSOID , TOOTH SURFACE References Banchoff, T. F. "Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order." In Geo- metric Analysis and Computer Graphics: Proceedings of a Workshop Held May 23 /C1/5, 1988 (Eds. P. Concus, R. Finn, D. A. Hoffman). New York: Springer-Verlag, pp. 31 /C1/7, 1991. Goursat, E. "Eacute;tude des surfaces qui admettent tous les plans de syme ´trie d’un polye `dre re ´gulier." Ann. Sci. E ´cole Norm. Sup. 4, 159/C1/000, 1897. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 292 and 314, 1997. Graceful Graph ALABELED GRAPH which can be "gracefully num- bered" is called a graceful graph. Label the nodes with distinct NONNEGATIVE INTEGERS . Then label the EDGES with the absolute differences between node values. If the EDGE numbers then run from 1 to e, the graph is gracefully numbered. In order for a graph tobe graceful, it must be without loops or multiple EDGES . Golomb showed that the number of EDGES connecting the EVEN -numbered and ODD-numbered sets of nodes is (e /C271 =)2 bc ; where e is the number of EDGES .In addition, if the nodes of a graph are all of EVEN ORDER , then the graph is graceful only if (e /C271=)2 bc is EVEN . The only ungraceful simple graphs with 55 nodes are shown below. There are exactly e! graceful graphs with e EDGES (Sheppard 1976), where e!=2 of these correspond to different labelings of the same graph. Golomb (1974) showed that all complete bipartite graphs are grace- ful. CATERPILLAR GRAPHS ; COMPLETE GRAPHS K2 ; K3 ; K4 /C30W4 /C30T (and only these; Golomb 1974); CYCLIC GRAPHS Cn when n /C130 or 3(mod 4); when the number of consecutive chords k /C302, 3, or n /C283 (Koh and Punnim 1982), or when they contain a Pkchord (Delorme et al. 1980, Koh and Yap 1985, Punnim and Pabhapote 1987); GEAR GRAPHS ; PATH GRAPHS ; the PETERSEN GRAPH ; POLYHEDRAL GRAPHS T /C30K4 /C30 W4 ; C, O, D, and I (Gardner 1983); STAR GRAPHS ; the THOMSEN GRAPH (Gardner 1983); and WHEEL GRAPHS (Frucht 1988) are all graceful. Some graceful graphs have only one numbering, but others have more than one. It is conjectured that alltrees are graceful (Bondy and Murty 1976), but this has only been proved for trees with 516VERTICES .I t has also been conjectured that all unicyclic graphs aregraceful. See also H ARMONIOUS GRAPH ,LABELED GRAPH References Abraham, J. and Kotzig, A. "All 2-Regular Graphs Consist- ing of 4-Cycles are Graceful." Disc. Math. 135,1/C1/4, 1994. Abraham, J. and Kotzig, A. "Extensions of Graceful Valua- tions of 2-Regular Graphs Consisting of 4-Gons." Ars Combin. 32, 257/C1/62, 1991. Bloom, G. S. and Golomb, S. W. "Applications of Numbered Unidirected Graphs." Proc. IEEE 65, 562/C1/70, 1977. Bolian, L. and Xiankun, Z. "On Harmonious Labellings of Graphs." Ars Combin. 36, 315/C1/26, 1993. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 248, 1976. Brualdi, R. A. and McDougal, K. F. "Semibandwidth of Bipartite Graphs and Matrices." Ars Combin. 30, 275/C1/ 87, 1990. Cahit, I. "Are All Complete Binary Trees Graceful?" Amer. Math. Monthly 83,3 5/C1/7, 1976. Delorme, C.; Maheo, M.; Thuillier, H.; Koh, K. M.; and Teo, H. K. "Cycles with a Chord are Graceful." J. Graph Theory 4, 409/C1/15, 1980. Frucht, R. W. and Gallian, J. A. "Labelling Prisms." Ars Combin. 26,6 9/C1/2, 1988. Gallian, J. A. "A Survey: Recent Results, Conjectures, and Open Problems in Labelling Graphs." J. Graph Th. 13, 491/C1/04, 1989. Gallian, J. A. "Open Problems in Grid Labeling." Amer. Math. Monthly 97, 133/C1/35, 1990. Gallian, J. A. "A Guide to the Graph Labelling Zoo." Disc. Appl. Math. 49, 213/C1/29, 1994. Gallian, J. A.; Prout, J.; and Winters, S. "Graceful and Harmonious Labellings of Prism Related Graphs." Ars Combin. 34, 213/C1/22, 1992. Gardner, M. "Golomb’s Graceful Graphs." Ch. 15 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 152 /C1/65, 1983. Golomb, S. W. "How to Number a Graph." In Graph Theory and Computing (Ed. R. C. Read). New York: Academic Press, pp. 23 /C1/7, 1972. Golomb, S. W. "The Largest Graceful Subgraph of the Complete Graph." Amer. Math. Monthly 81, 499/C1/01, 1974. Guy, R. "Monthly Research Problems, 1969 /C1/5."Amer. Math. Monthly 82, 995/C1/004, 1975. Guy, R. "Monthly Research Problems, 1969 /C1/979." Amer. Math. Monthly 86, 847/C1/52, 1979. Guy, R. K. "The Corresponding Modular Covering Problem. Harmonious Labelling of Graphs." §C13 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 127 /C1/28, 1994. Huang, J. H. and Skiena, S. "Gracefully Labelling Prisms." Ars Combin. 38, 225/C1/42, 1994. Koh, K. M. and Punnim, N. "On Graceful Graphs: Cycles with 3 /-Consecutive Chords." Bull. Malaysian Math. Soc. 5,4 9/C1/4, 1982. Jungreis, D. S. and Reid, M. "Labelling Grids." Ars Combin. 34, 167/C1/82, 1992. Koh, K. M. and Yap, K. Y. "Graceful Numberings of Cycles with a P3/-Chord." Bull. Inst. Math. Acad. Sinica 13,4 1/C1/8, 1985. Moulton, D. "Graceful Labellings of Triangular Snakes." Ars Combin. 28,3/C1/3, 1989. Punnim, N. and Pabhapote, N. "On Graceful Graphs: Cycles with a Pk/-Chord, k]4:/"Ars Combin. A 23, 225/C1/28, 1987. Rosa, A. "On Certain Valuations of the Vertices of a Graph." In Theory of Graphs, International Symposium, Rome, July 1966. New York: Gordon and Breach, pp. 349 /C1/55, 1967. Sheppard, D. A. "The Factorial Representation of Balanced Labelled Graphs." Discr. Math. 15, 379 /C1/88, 1976. Sierksma, G. and Hoogeveen, H. "Seven Criteria for Integer Sequences Being Graphic." J. Graph Th. 15, 223 /C1/31, 1991. Slater, P. J. "Note on k-Graceful, Locally Finite Graphs." J. Combin. Th. Ser. B 35, 319 /C1/22, 1983. Snevily, H. S. "New Families of Graphs That Have a/- Labellings." Preprint. Snevily, H. S. "Remarks on the Graceful Tree Conjecture." Preprint. Xie, L. T. and Liu, G. Z. "A Survey of the Problem of Graceful Trees." Qufu Shiyuan Xuebao 1,8/C1/5, 1984. Graceful Permutation A graceful permutation s on n letters is a PERMUTA- TION such that f½ s(i) /C28 s(i /C271)½ : i /C301;2; ...; n /C281 g /C30f1;2; ...; n /C281g: For example, there are four graceful permutations on f1; 2;3; 4g : f1;4 ;2;3 g;f2;3 ;1;4 g;f3; 2;4;1 g; and f4; 1;3; 2g: The number of graceful permutations on n letters for n /C301, 2, ... are 1, 2, 4, 4, 8, 24, 32, 40, ... (Sloane’s A006967). References Sloane, N. J. A. Sequences A006967/M3229 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos’ 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press, pp. 557 /C1/62, 1997. Wilf, H. S. and Yoshimura, N. "Ranking Rooted Trees and a Graceful Application." In Discrete Algorithms and Com- plexity (Proceedings of the Japan-US Joint Seminar June 4 /C1/, 1986, Kyoto, Japan) (Ed. D. Johnson, T. Nishizeki, A. Nozaki and H. S. Wilf). Boston, MA: Academic Press, pp. 341 /C1/50, 1987. Grade GRADIAN Graded Algebra If A is a GRADED MODULE and there EXISTS a degree- preserving linear map f : A /C156A 0 A; then (A; f)is called a graded algebra. COHOMOLOGY is a graded algebra. In addition, the GRADING SET is MONOID having a compatibility rela- tion such that if A is in the a grading of the algebra M, and B is in the b grading of the algebra M, then AB is in the ab grading of the algebra (where A and B are multiplied in M, and a and b are multiplied in the index monoid). For example, cohomology of a space is a graded algebra over the integers (i.e., aGRADED RING ), since if A is an n-dimensional coho- mology class and B is an m-dimensional cohomology class, then the CUP PRODUCT AB is an m /C27n dimen- sional cohomology class. The GROUP RING of a GROUP G over a RING R is a graded R-algebra with grading G. See also COHOMOLOGY ,G RADED MODULE ,G RADED RING,GROUP RING References Jacobson, N. Lie Algebras. New York: Dover, p. 163, 1979. Graded Module A decomposition of a MODULE into a DIRECT SUM of SUBMODULES . The INDEX SET for the collection of SUBMODULES is then called the GRADING SET. Graded modules arise naturally in HOMOLOGY .In particular, for every integer i, there exists an ith HOMOLOGY GROUP of a space Hi(X) ; and usually the "total homology" of the space is considered to be the direct sum of all the Hi(X)/s. This makes the "total" homology of X a module graded over the integers. See also GRADED ALGEBRA Graded Ring A GRADED ALGEBRA over the integers Z: COHOMOLOGY of a space is a graded ring. See also GRADED ALGEBRA Gradian A unit of angular measure in which the angle of an entire CIRCLE is 400 gradians. A RIGHT ANGLE is therefore 100 gradians. A gradian is sometimes also called a GON or a GRADE . See also DEGREE ,RADIAN References Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 63, 1998. Gradient The gradient is a VECTOR operator denoted 9and sometimes also called D ELorNABLA . It is most often applied to a real function of three variables f(u1;u2;u3);and may be denoted 9f/C13grad(f) : (1) For general CURVILINEAR COORDINATES , the gradient is given by 9f/C301 h1@f @u1ˆu1/C271 h2@f @u2ˆu2/C271 h3@f @u3ˆu3; (2) which simplifies to 9f(x; y; z) /C30@ f @xˆx /C27@ f @yˆy /C27@ f @zˆz (3) in CARTESIAN COORDINATES . The direction of 9f is the orientation in which the DIRECTIONAL DERIVATIVE has the largest value and 9fjj is the value of that DIRECTIONAL DERIVATIVE . Furthermore, if 9f "0; then the gradient is PERPEN- DICULAR to the LEVEL CURVE through (x0 ; y0)ifz /C30 f(x; y) and PERPENDICULAR to the level surface through (x0 ; y0 ; z0)ifF(x; y ; z) /C300:/ In TENSOR notation, let ds2 /C30gm dx2 m (4) be the LINE ELEMENT in principal form. Then 9/C0ea /C0e b /C309 a /C0e b /C301 ffiffiffiffiffiffigap@ @xa/C0e b : (5) For a MATRIX /A/, 9jAxj/C30(Ax)TA jAx j: (6) For expressions giving the gradient in particular coordinate systems, see CURVILINEAR COORDINATES . See also CONVECTIVE DERIVATIVE ,C URL,D IVER- GENCE ,LAPLACIAN ,VECTOR DERIVATIVE References Arfken, G. "Gradient, 9/" and "Successive Applications of 9:/" §1.6 and 1.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 33 /C1/7 and 47 /C1/1, 1985. Gradient Descent Method STEEPEST DESCENT METHOD Gradient Four-Vector The 4-dimensional version of the GRADIENT , encoun- tered frequently in general relativity and special relativity, is 9m /C301 c@ @t @ @x @ @y @ @z2 666666666666643 77777777777775; which can be written ( 9 m)2 /C13I2 ; where I2 is the D’ALEMBERTIAN . See also D’ALEMBERTIAN ,GRADIENT ,TENSOR ,VECTORReferences Morse, P. M. and Feshbach, H. "The Differential Operator 9:/ " §1.4 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 31 /C1/4, 1953. Gradient Theorem ga b( 9f) /C215 ds /C30f(b) /C28f(a); where 9 is the GRADIENT , and the integral is a LINE INTEGRAL . It is this relationship which makes the definition of a scalar potential function f so useful in gravitation and electromagnetism as a concise way to encode information about a VECTOR FIELD . See also DIVERGENCE THEOREM ,GREEN’S THEOREM , LINE INTEGRAL ,POINCARE ´ ’S THEOREM Grading Set The INDEX SET for the collection of SUBMODULES in a GRADED MODULE . See also GRADED MODULE Graeco-Latin Square EULER SQUARE Graeco-Roman Square EULER SQUARE Graeffe Iteration GRAEFFE’S METHOD Graeffe’s Method AROOT -finding method which was among the most popular methods for finding roots of UNIVARIATE POLYNOMIALS in the 19th and 20th centuries. It was invented independently by Graeffe, dandelin, and Lobachevsky (Householder 1959, Malajovich and Zubelli 1999). Graeffe’s method has a number of drawbacks, among which are that its usual formula-tion leads to exponents exceeding the maximumallowed by floating-point arithmetic and also that it can map well-conditioned polynomials into ill-condi- tioned ones. However, these limitations are avoidedin an efficient implementation by Malajovich and Zubelli (1999). The method proceeds by multiplying a POLYNOMIAL f(x)b yf(/C28x) and noting that f(x)/C30(x/C28a1)(x/C28a2)/C1/C1/C1(x/C28an) (1) f(/C28x)/C30(/C281)n(x/C27a1)(x/C27a2)/C1/C1/C1(x/C27an) (2) so the result is f(x)f(/C28x)/C30(/C281)n(x2/C28a2 1)(x2/C28a22)/C1/C1/C1(x2/C28a2n):(3) repeat ntimes, then write this in the form yn /C27b1yn/C281 /C27.../C27bn /C300 (4) where y /C13x2 n : Since the coefficients are given by NEWTON’S RELATIONS b1 /C30/C28(y1 /C27y2 /C27.../C27yn) (5) b2 /C30(y1y2 /C27y1y3 /C27.../C27yn/C281yn) (6) bn /C30(/C281)ny1y2 /C1/C1/C1yn ; (7) and since the squaring procedure has separated the roots, the first term is larger than rest. Therefore, b1 :/C28y1 (8) b2 :y1y2 (9) bn :(/C281)ny1y2 /C1/C1/C1yn ; (10) giving y1 :/C28b1 (11) y2 :/C28b2 b1(12) yn :/C28bn bn/C281: (13) Solving for the original roots gives a1 :ffiffiffiffiffiffiffiffi /C28b1p (14) a2 :ffiffiffiffiffiffiffiffiffi /C28b2 b1s (15) an :ffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28bn bn/C281s : (16) This method works especially well if all roots are real. References Bini, D. and Pan, V. Y. "Graeffe’s, Chebyshev-Like, and Cardinal’s Processes for Splitting a Polynomial Into Factors." J. Complexity 12, 492 /C1/11, 1996. Brodetsky, S. and Smeal, G. "on Graeffe’s Method for Complex Roots of Algebraic Equations." Proc. Cambridge Philos. Soc. 22,83/C1/7, 1924. Dedieu, J.-P. "A` Propos de la me´thode de Dandelin-Graeffe." C. R. Acad. Sci. Paris Se´r. I Math 309, 1019 /C1/022, 1989. Grau, A. A. "On the Reduction of Number Range in the Use of the Graeffe Process." J. Assoc. Comput. Mach. 10, 538 /C1/ 44, 1963. Householder, A. S. "dandelin, Lobacevskii, or Graeffe?" Amer. Math. Monthly 66, 464 /C1/66, 1959. Jana, P. and Sinha, B. "Fast Parallel Algorithms for Graeffe’s Root Squaring." Comput. Math. Appl. 35,71/C1/ 0, 1998. Ka´rma´n, T. Von and Biot, M. a. "Squaring the Roots (Graeffe’s Method)." §5.8.C in Mathematical Methods in Engineering: an Introduction to the Mathematical Treat- ment of Engineering Problems. New York: Mcgraw-Hill, pp. 194 /C1/96, 1940. Malajovich, G. and Zubelli, J. P. "On the Geometry of Graeffe Iteration." Informes de Mathema ´tica, Se´rie B- 118, IMPA.Malajovich, G. and Zubelli, J. P. Tangent Graeffe Iteration. 27 Aug 1999. http://xxx.lanl.gov/abs/math.AG/9908150/. Ostrowski, A. "Recherches sur la me´thode de Graeffe et les ze´ros des polynomes et des se´ries de Laurent." Acta Math. 72,99/C1/55, 1940. Ostrowski, A. "Recherches sur la me´thode de Graeffe et les ze´ros des polynomes et des se´ries de Laurent. Chapitres III et IV." Acta Math. 72, 157 /C1/57, 1940. Pan, V. Y. "Solving a Polynomial Equation: Some History and Recent Progress." SIAM Rev. 39, 187 /C1/20, 1997. Whittaker, E. T. and Robinson, G. "The Root-Squaring Method of Dandelin, Lobachevsky, and Graeffe." §54 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 106 /C1/12, 1967. Graham’s Biggest Little Hexagon The largest possible (not necessarily regular) HEXA- GON for which no two of the corners are more than unit distance apart. In the above figure, the heavy lines are all of unit length. The AREA of the hexagon is A/C300:674981 . . . ;where Ais the second-largest real ROOT of 4096 A10/C278192 A9/C283008 A8/C2830;848A7/C2721;056A6 /C27146;496A5/C28221;360A4/C271232 A3/C27144;464A2 /C2878;488A/C2711;993 /C300: Note that the sign of the A9is positive, not negative as erroneously given in Conway and Guy (1996). See also CALABI’S TRIANGLE References Conway, J. H. and Guy, R. K. "Graham’s Biggest Little Hexagon." In The Book of Numbers. New York: Springer- Verlag, pp. 206 /C1/07, 1996. Graham, R. L. "The Largest Small Hexagon." J. Combin. Th. Ser. A 18, 165/C1/70, 1975. Graham’s Number The smallest dimension nof a HYPERCUBE such that if the lines joining all pairs of corners are two-colored, a PLANAR COMPLETE GRAPH K4of one color will be forced. Stated colloquially, this is equivalent to con- sidering every possible committee from some numberof people nand enumerating every pair of commit- tees. Now assign each pair of committees to one of two groups, and find the smallest nthat will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees (Hoffman 1998, p. 54). An answer was proved to exist by R. L. Graham and B. L. Rothschild. However, although the actual an- swer is believed to be 6, the best bound proved is 643 /C160/C160/C160/C160 3|fflfflffl{zfflfflffl} 3 /C1603|{z} n|fflffl{zfflffl} 3 /C16038 >>>>>>>< >>>>>>>: where /C160 is stacked ARROW NOTATION . It is less than 3 0 3 0 3 0 3; where CHAINED ARROW NOTATION has been used. See also ARROW NOTATION ,CHAINED ARROW NOTA- TION ,EXTREMAL GRAPH THEORY ,RAMSEY THEORY , SKEWES NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 61 /C1/2, 1996. Gardner, M. "Mathematical Games." Sci. Amer. 237,18/C1/8, Nov. 1977. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 18 and 54, 1998. Gram Determinant The DETERMINANT G(f1 ; f2 ; ... ; fn) /C30g f2 1 dtg f1f2 dt ... g f1fn dt g f2f1 dtg f2 2 dt ... g f2fn dt nn::: n g f1fn dt g f1fn dt /C1/C1/C1g f2 n dt=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n: See also G RAM- SCHMIDT ORTHONORMALIZATION , WRONSKIAN References Andrews, G. E.; Askey, R.; and Roy, R. "Jacobi Polynomials and Gram Determinants." §6.3 in Special Functions. Cambridge, England: Cambridge University Press, pp. 293 /C1/97, 1999. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 2, 1991. Gram Matrix Given m points with n-D vector coordinates vi ; let M be the n /C29m matrix whose jth column consists of the coordinates of the vector vj ; with j /C301, ..., m. Then define the m /C29m Gram matrix of dot products aij /C30 vi/C215 vj as A /C30MTM ;where AT denotes the TRANSPOSE . The Gram matrix determines the vectors vi up to ISOMETRY . Gram Series G(x) /C301 /C27X/C12 k/C301(ln x)k kk! z(k /C27 1) ; where z(z) is the RIEMANN ZETA FUNCTION (Hardy 1999, p. 24). This approximation to the PRIME COUNT- ING FUNCTION is 10 times better than Li(x) for x B109 but has been proven to be worse infinitely often by Littlewood (Ingham 1990). An equivalent formulation due to Ramanujan is G(x) /C134 pX/C12 k /C301( /C281)k/C281k B2k(2k /C28 1)ln x 2p !2k/C281 /C2 p(x) (Berndt 1994; Hardy 1999, p. 23), where B2kis a BERNOULLI NUMBER . The integral analog, also found by Ramanujan, is J(x) /C13g/C12 0(ln x)t dt tG(t /C27 1)z(t /C27 1) /C2 p(x) (Berndt 1994; Hardy 1999, p. 23). The Gram series is equivalent to the RIEMANN PRIME NUMBER FORMULA (Hardy 1999, pp. 24 /C1/5). See also RIEMANN PRIME NUMBER FORMULA References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 124 /C1/29, 1994. Gram, J. P. "Undersøgelser angaaende Maengden af Primtal under en given Graeense." K. Videnskab. Selsk. Skr. 2, 183/C1/08, 1884. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Ingham, A. E. Ch. 5 in The Distribution of Prime Numbers. New York: Cambridge, 1990. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 225, 1996. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 74, 1991. Gram’s Inequality Letf1(x);...,fn(x)b e REAL INTEGRABLE FUNCTIONS over the CLOSED INTERVAL [a, b], then the DETERMINANT of their integrals satisfies gb af2 1 (x) dxgb af1(x)f2(x) dx /C1/C1/C1gb af1(x)fn(x) dx gb af2(x)f1(x) dxgb af2 2 (x) dx /C1/C1/C1gb af2(x)fn(x) dx nn::: n gb afn(x)f1(x) dxgb afn(x)f2(x) dx /C1/C1/C1gb afn(x)fn(x) dx=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n=z1n ]0: See also G RAM- SCHMIDT ORTHONORMALIZATION References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1100, 2000. Gram-Charlier Series EDGEWORTH SERIES Gram-Schmidt Orthonormalization A procedure which takes a nonorthogonal set of LINEARLY INDEPENDENT functions and constructs an ORTHOGONAL BASIS over an arbitrary interval with respect to an arbitrary WEIGHTING FUNCTION w(x):/ Given an original set of linearly independent func- tions fung/C12 n/C300;letfcng/C12n/C300denote the orthogonalized (but not normalized) functions, ffng/C12n/C300denote the orthonormalized functions, and define c0(x)/C13u0(x) (1) f0(x)/C13c0(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gc2 0(x)w(x)dxs : (2) Then take c1(x)/C30u1(x)/C27a10f0(x); (3) where we require gc1f0wd x/C30gu1f0wd x/C27a10gf20wd x/C300:(4) By definition, gf20wd x/C301; (5) so a10/C30/C28gu1f0wd x : (6) The first orthogonalized function is therefore c1/C30u1(x)/C28gu1f0wd x=zn;=zn1 f0; (7) and the corresponding normalized function isf1/C30c1(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gc21wd xs : (8) By mathematical induction, it follows that fi(x)/C30ci(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gc2iwd xs ; (9) where ci(x)/C30ui/C27ai0f0/C27ai1f1.../C27ai;i/C281fi/C281 (10) and aij/C13/C28guifjwd x : (11) If the functions are normalized to Njinstead of 1, then gb a[fj(x)]2wd x/C30N2 j (12) fi(x)/C30Nici(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gc2iwd xs (13) aij/C30/C28guifjwd x N2 j: (14) ORTHOGONAL POLYNOMIALS are especially easy to generate using G RAM- SCHMIDT ORTHONORMALIZA- TION . Use the notation xi½xj=z1;=z11 /C13xi½w½xj=z1;=z11 /C13gb axi(x)xj(x)w(x)dx; (15) where w(x)i sa WEIGHTING FUNCTION , and define the first few POLYNOMIALS , p0(x)/C131 (16) p1(x)/C30x/C28xp0½p0 hi p0½p0 hi"# p0: (17) As defined, p0andp1are ORTHOGONAL POLYNOMIALS , as can be seen from p0½p1 hi /C30 x/C28xp0½p0 hi p0½p0 hi"# p0*+ /C30xp0hi/C28xp0½p0 hi p0½p0 hip0hi /C30xp0hi/C28xp0hi/C300: (18) Now use the RECURRENCE RELATION pi/C271(x)/C30x/C28xpi½pi hi pi½pi hi"# pi/C28pi½pi hi pi/C281½pi/C281 hi"# pi/C281 (19) to construct all higher order POLYNOMIALS . To verify that this procedure does indeed produce ORTHOGONAL POLYNOMIALS , examine pi /C271 ½pi=z1;=z11 /C30 x /C28xpi ½pi hi pi ½pi hi"# pi ½pi*+ /C28pi ½pi hi pi/C281 ½pi/C281 hipi/C281 ½pi*+ /C30 xpi ½pi hi /C28xpi ½pi hi pi ½pi hipi ½pi hi/C28pi ½pi hi pi/C281 ½pi/C281 hi /C2 pi/C281 ½pi hi /C30/C28pi ½pi hi pi/C281 ½pi/C281 hipi/C281 ½pi hi /C30/C28pi ½pi hi pi/C281 ½pi/C281 hi/C28pi/C281 ½pj/C281=z1;=z11 pj/C282 ½pj/C282=z1;=z11 pj /C282 ½pj/C281=z1;=z11"# /C30.../C30(/C281)jpj ½pj=z1;=z11 p0 ½p0 hip0 ½p1 hi /C300 ; (20) since p0 ½p1 hi /C300: Therefore, all the POLYNOMIALS pi(x) are orthogonal. Many common ORTHOGONAL POLYNOMIALS of mathe- matical physics can be generated in this manner. Unfortunately, the process turns out to be numeri- cally unstable (Golub and van Loan 1989). See also GRAM DETERMINANT ,G RAM’S INEQUALITY , LATTICE REDUCTION ,ORTHOGONAL POLYNOMIALS References Arfken, G. "Gram-Schmidt Orthogonalization." §9.3 in Math- ematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 516 /C1/20, 1985. Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed.Baltimore, MD: Johns Hopkins, 1989. Pohst, M. and Zassenhaus, H. "Methods from the Geometry of Numbers." Ch. 3 in Algorithmic Algebraic Number Theory. Cambridge, England: Cambridge University Press, 1989. Granny Knot ACOMPOSITE KNOT of seven crossings consisting of a KNOT SUM ofTREFOILS . The granny knot has the sameALEXANDER POLYNOMIAL (x2/C28x/C271)2as the SQUARE KNOT . Graph A mathematical object composed of points known as VERTICES orNODES and lines connecting some (possi- bly empty) SUBSET of them, known as EDGES . For- mally, a graph is a binary relation on a set of vertices. If this relation is symmetric, the graph is said to be UNDIRECTED ; otherwise, the graph is said to be DIRECTED . Graphs in which at most one edge connects any two nodes are said to be SIMPLE GRAPHS . Vertices are usually not allowed to be self-connected, but thisrestriction is sometimes relaxed to allow such "loops." The edges of a graph may be assigned specific values or labels, in which case the graph is called a LABELED GRAPH . The study of graphs is known as GRAPH THEORY , and was first studied systematically by D. Ko ¨nig in the 1930s (Gardner 1984, p. 91). As Gardner (1984, p. 91) notes, "The confusion of this term with the ‘ GRAPHS ’o f analytic geometry is regrettable, but the term has stuck." Graphs are 1-D COMPLEXES , and there are always an EVEN NUMBER ofODD NODES in a graph. GRAPH SUMS , differences, powers, UNIONS , and PRODUCTS can be defined, as can GRAPH EIGENVALUES . The number of nonisomorphic simple undirected graphs with v NODES for v /C301, 2, ..., are 1, 2, 4, 11, 34, 156, 1044, ... (Sloane’s A000088; see above figure). The P O´LYA ENUMERATION THEOREM can be used to determine these numbers. In order to apply the PO´LYA ENUMERATION THEOREM , define the quantity hj/C30p!Qp i/C301ijiji!; (1) where p! is the FACTORIAL ofp, and the related polynomial Zp(S)/C30X ihjiYp k/C301f(ji)k k; (2) where the ji/C30(j1;...;jp)iare all of the p-VECTORS satisfying j1/C272j2/C273j3/C27.../C27pjp/C30p: (3) For example, for p/C303, the three possible values of j are j1/C30(3;0;0);since (1 /C2153)/C27(2 /C2150)/C27(3 /C2150)/C303; giving hj1/C303! (133!)(200!)(300!)/C301 (4) j2/C30(1;1;0);since (1 /C2151)/C27(2 /C2151)/C27(3 /C2150)/C303; giving hj2/C303! (111!)(211!)(300!)/C303; (5) j3/C30(0;0;1);since (1 /C2150)/C27(2 /C2150)/C27(3 /C2151)/C303 giving hj3/C303! (100!)(200!)(311!)/C302: (6) Therefore, Z3(S)/C30f3 1/C273f1f2/C272f3: (7) For small p, the first few values of Zp(S) are given by Z2(S)/C30f2 1/C27f2 (8) Z3(S)/C30f3 1/C273f1f2/C272f3 (9) Z4(S)/C30f4 1/C276f2 1f2/C273f2 2/C278f1f3/C276f4 (10) Z5(S)/C30f5 1/C2710f3 1f2/C2715f1f2 2/C2720f2 1f3/C2720f2f3 /C2730f1f4/C2724f5 (11) Z6(S)/C30f6 1/C2715f4 1f2/C2745f2 1f2 2/C2715f3 2/C2740f3 1f3/C27120f1f2f3 /C2740f2 3/C2790f2 1f4/C2790f2f4/C27144f1f5/C27120f6(12) Z7(S)/C30f7 1/C2721f5 1f2/C27105f3 1f2 2/C27105f1f3 2/C2770f4 1f3 /C27420f2 1f2f3/C27210f2 2f3/C27280f1f2 3/C27210f3 1f4 /C27630f1f2f4/C27420f3f4/C27504f2 1f5/C27504f2f5 /C27840f1f6/C27720f7: (13)Application of the P O´LYA ENUMERATION THEOREM then gives the formula Z(R)/C301 p!X (j)hjY(p/C281)=2 bc n/C300gnj2n/C271/C27(2n/C271)j2n/C271 2ðÞ 2n/C271 /C29Yp=2 bc n/C301[(gng2n)n/C281]j2ng2nj2n 2ðÞ 2n /C29Yp q/C301Yp r/C30q/C271gjqjrGCD( q;r) LCM( q;r); (14) where xbcis the FLOOR FUNCTION ,n m=z;=z1 is a BINOMIAL COEFFICIENT , LCM is the LEAST COMMON MULTIPLE , GCD is the GREATEST COMMON DIVISOR , and the SUM (j) is over all jisatisfying the sum identity described above. The first few generating functions Zp(R) are Z2(R)/C302g1 (15) Z3(R)/C30g3 1/C273g1g2/C272g3 (16) Z4(R)/C30g61/C279g21g22/C278g23/C276g2g4 (17) Z5(R)/C30g101/C2710g41g32/C2715g21g42/C2720g1g33/C2730g2g24 /C2724g25/C2720g1g3g6 (18) Z6(R)/C30g151/C2715g71g42/C2760g31g62/C2740g31g43/C2740g53 /C27180g1g2g34/C27144g35/C27120g1g2g23g6 /C27120g3g26 (19) Z7(R)/C30g211/C2721g111g52/C27105g51g82/C27105g31g92/C2770g61g53 /C27280g73/C27210g31g2g44/C27630g1g22g44/C27504g1g45 /C27420g21g22g33g6/C27210g21g22g3g26/C27840g3g36 /C27720g37/C27504g1g25g10/C27420g2g3g4g12:(20) Letting gi/C301/C27xithen gives a POLYNOMIAL Si(x); which is a GENERATING FUNCTION for (i.e., the terms ofxigive) the number of graphs with iEDGES . The total number of graphs having iedges is Si(1):The first few Si(x) are S2/C301/C27x (21) S3/C301/C27x/C27x2/C27x3(22) S4/C301/C27x/C272x2/C273x3/C272x4/C27x5/C27x6(23) S5/C301/C27x/C272x2/C274x3/C276x4/C276x5/C276x6/C274x7/C272x8 /C27x9/C27x10(24) S6/C301/C27x/C272x2/C275x3/C279x4/C2715x5/C2721x6/C2724x7 /C2724x8/C2721x9/C2715x10/C279x11/C275x12/C272x13 /C27x14/C27x15(25) S7 /C301 /C27x /C272x2 /C275x3 /C2710x4 /C2721x5 /C2721x6 /C2724x7 /C2741x6 /C2765x7 /C2797x8 /C27131x9 /C27148x10 /C27148x11 /C27131x12 /C2797x13 /C2765x14 /C2741x15 /C2721x16 /C2710x17 /C275x18 /C272x19 /C27x20 /C27x21 ; (26) giving the number of graphs with n nodes as 1, 2, 4, 11, 34, 156, 1044, ... (Sloane’s A000088). King and Palmer (cited in Read 1981) have calculated Sn up to n /C3024, for which S24 /C30195; 704; 906; 302;078;447;922;174;862;416;/C1/C1/C1 /C1/C1/C1726;256;004;122;075;267;063;365;754;368:(27) See also BIPARTITE GRAPH ,C ATERPILLAR GRAPH , CAYLEY GRAPH ,CIRCULANT GRAPH ,COCKTAIL PARTY GRAPH ,COMPARABILITY GRAPH ,COMPLEMENT GRAPH , COMPLETE GRAPH ,CONE GRAPH ,CONNECTED GRAPH , COXETER GRAPH ,CUBICAL GRAPH , DE BRUIJN GRAPH , DEGREE SEQUENCE ,D IGRAPH ,D IRECTED GRAPH , DODECAHEDRAL GRAPH ,E ULER GRAPH ,E XTREMAL GRAPH ,G EAR GRAPH ,G RACEFUL GRAPH ,G RAPH DIAMETER ,GRAPH THEORY ,H ANOI GRAPH ,H ARARY GRAPH ,H ARMONIOUS GRAPH ,H OFFMAN- SINGLETON GRAPH ,ICOSAHEDRAL GRAPH ,INTERVAL GRAPH ,ISO- MORPHIC GRAPHS ,LABELED GRAPH ,LADDER GRAPH , LATTICE GRAPH ,MATCHSTICK GRAPH ,MINOR GRAPH , MOORE GRAPH ,MULTIGRAPH ,NULL GRAPH ,OCTAHE- DRAL GRAPH ,PATH GRAPH ,PETERSEN GRAPH ,PLANAR GRAPH ,P SEUDOGRAPH ,R ANDOM GRAPH ,R EGULAR GRAPH ,SEQUENTIAL GRAPH ,SIMPLE GRAPH ,STAR GRAPH ,S UBGRAPH ,S UPERGRAPH ,S UPERREGULAR GRAPH ,S YLVESTER GRAPH ,T ETRAHEDRAL GRAPH , THOMASSEN GRAPH ,T OURNAMENT ,T RIANGULAR GRAPH ,TURAN GRAPH ,TUTTE’S GRAPH ,U NIVERSAL GRAPH ,UTILITY GRAPH ,W EB GRAPH ,W HEEL GRAPH References Bogomolny, A. "Graph Puzzles." http://www.cut-the-knot.- com/do_you_know/graphs2.html. Fujii, J. N. Puzzles and Graphs. Washington, DC: National Council of Teachers, 1966. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 91, 1984. Harary, F. "The Number of Linear, Directed, Rooted, and Connected Graphs." Trans. Amer. Math. Soc. 78, 445/C1/63, 1955. Pappas, T. "Networks." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 126 /C1/27, 1989. Read, R. "The Graph Theorists Who Count--And What They Count." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 326 /C1/45, 1981. Read, R. C. and Wilson, R. J. Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Sloane, N. J. A. Sequences A000088/M1253 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M1253 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.Weisstein, E. W. "Graphs." M ATHEMATICA NOTEBOOK GRAPHS.M . Weisstein, E. W. "Books about Graph Theory." http:// www.treasure-troves.com/books/GraphTheory.html. Wilson, J. C. On the Traversing of Geometrical Figures. Oxford, England: Oxford University Press, 1905. Graph (Function) Given a FUNCTION f(x1;...;xn) defined on a DOMAIN U, the graph of fis defined as the set of points (which often form a CURVE orSURFACE ) showing the values taken by fover U(or some portion of U). Technically, for real functions, graph f(x)/C13f(x;f(x))/C23R2:x/C23Ug graph f(x1;...;xn)/C13 f(x1;...;xn;f(x1;...;xn))/C23Rn/C271:(x1;...;xn)/C23Ug: A graph is sometimes also called a PLOT . Commenting on the unfortunate choice of the word "graph" in the completely different context of so-called GRAPH THE- ORY, Gardner (1984, p. 91) notes, "The confusion of this term with the ‘graphs’ of analytic geometry isregrettable, but the term has stuck." 2-D and 3-D graphs can be produced in Mathematica using the commands Plot [f,{x,xmin ,xmin }] and Plot3D [f,{x,xmin ,xmin }, {y,ymin ,ymax }], respec- tively. Several examples of continuous functions which are notoriously difficult to graph are shown above:sin(1 =x);the FRACTIONAL PART frac(1 =x);and the WEIERSTRASS FUNCTION . Good routines for plotting graphs use adaptive algorithms which plot morepoints in regions where the function varies most rapidly (Wagon 1991, Math Works 1992, Heck 1993, Wickham-Jones 1994). Tupper (1996) has developedan algorithm that rigorously proves the pixels itgenerates are "on" if and only if there exists a mathematical point within the region of space repre- sented by that pixel that is a solution to the relationbeing graphed. Although this method attempts to produce graphs that satisfy strict mathematical relationships, the problem of graphing is ultimately intractable, so no fixed algorithm can produce correct graphs for arbitrary relations. See also CURVE ,D ATA CUBE,E XTREMUM ,G RAPH , HISTOGRAM ,MAXIMUM ,MINIMUM References Cleveland, W. S. The Elements of Graphing Data, rev. ed. Summit, NJ: Hobart, 1994. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 91, 1984. Heck, A. Introduction to Maple, 2nd ed. New York: Springer-Verlag, pp. 303 /C1/04, 1993. Math Works. Matlab Reference Guide. Natick, MA: The Math Works, p. 216, 1992. Tufte, E. R. The Visual Display of Quantitative Information. Cheshire, CN: Graphics Press, 1983. Tufte, E. R. Envisioning Information. Cheshire, CN: Gra- phics Press, 1990. Tupper, J. Graphing Equations with Generalized Interval Arithmetic. M.Sc. Thesis. Department of Computer Science. Toronto: University of Toronto, 1996. http:// www.dgp.toronto.edu/~mooncake/msc.html. Tupper, J. "GrafEq." http://www.peda.com/grafeq/. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 24 /C1/5, 1991. Weisstein, E. W. "Books about Graphing." http://www.trea- sure-troves.com/books/Graphing.html. Wickham-Jones, T. Computer Graphics with Mathematica. Santa Clara, CA: TELOS, pp. 579 /C1/84, 1994. Yates, R. C. "Sketching." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 188 /C1/05, 1952. Graph Automorphism An automorphism of a GRAPH is a GRAPH ISOMORPH- ISM with itself. The sets of automorphisms define a PERMUTATION GROUP . For every GROUP G; there exists a GRAPH whose automorphism group is isomorphic to G (Frucht 1939; Skiena 1990, p. 185). The automorph- ism groups of a graph characterize its symmetries, and are therefore very useful in determining certain of its properties. The automorphism group of a GRAPH COMPLEMENT is the same as that for the original graph. See also FRUCHT GRAPH ,GRAPH ISOMORPHISM ,ISO- MORPHIC GRAPHS References Duijvestijn, A. J. W. "Algorithmic Calculation of the Order of the Automorphism Group of a Graph." Memorandum No. 221. Enschede, Netherlands: Twente Univ. Technol- ogy, 1978. Frucht, R. "Herstellung von Graphen mit vorgegebener abstrakter Gruppe." Compos. Math. 6, 239 /C1/50, 1939. Lipton, R.; North, S.; and Sandberg, J. "A Method for Drawing Graphs." In Proc. First ACM Symposium on Computation Geometry. pp. 153 /C1/60, 1985.Skiena, S. "Automorphism Groups." §5.2.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 184 /C1/87, 1990. Graph Cartesian Product The Cartesian graph product G /C30G1IG2of graphs G1 and G2 with disjoint point sets V1 and V2 and edge sets X1 and X2 is the graph with point set V1 /C29V2 and u /C30(u1 ; u2) adjacent with v /C30(v1 ; v2) whenever [u1 /C30 v1 and u2 adj v2]or[ u2 /C30v2 and u1 adj v1] (Harary 1994, p. 22). Graph Cartesian products can be computed using GraphProduct [G1, G2] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also GRAPH COMPOSITION ,G RAPH PRODUCT , VIZING CONJECTURE References Clark, W. E. and Suen, S. "An Inequality Related to Vizing’s Conjecture." Electronic J. Combinatorics 7, No. 1, N4, 1 /C1/, 2000. http://www.combinatorics.org/Volume_7/ v7i1toc.html#N4. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Hartnell, B. and Rall, D. "Domination in Cartesian Products: Vizing’s Conjecture." In Domination in Graphs--Advanced Topics (Ed. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater). New York: Dekker, pp. 163 /C1/89, 1998. Sabidussi, G. "Graph Multiplication." Math. Z. 72, 446 /C1/57, 1960. Skiena, S. "Products of Graphs." §4.1.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 133 /C1/35, 1990. Vizing, V. G. "The Cartesian Product of Graphs." Vycisl. Sistemy 9,30/C1/3, 1963. Graph Categorical Product This entry contributed by NICOLAS BRAY The GRAPH PRODUCT denoted G/C29Hand defined by the adjacency relations ( gadjg?andhadjh?):/ See also GRAPH PRODUCT Graph Center The center of a GRAPH G is the set of vertices of GRAPH ECCENTRICITY equal to the GRAPH RADIUS (i.e., the set of CENTRAL POINTS ). In the above illustration, center nodes are shown in red. The following table gives the number of n-node simple unlabeled graphs having k center nodes. k Sloane n /C30 1, 2, ... 1 A052437 1, 0, 1, 2, 8, 29, 180, ... 2 A052438 0, 2, 0, 2, 4, 19, 84, ... 3 A052439 0, 0, 3, 0, 4, 18, 119, ... 4 A052340 0, 0, 0, 7, 0, 18, 118, ... 5 A052341 0, 0, 0, 0, 18, 0, 129, ... 6 0, 0, 0, 0, 0, 72, 0, ... 7 0, 0, 0, 0, 0, 0, 414, ... See also BICENTERED TREE,CENTRAL POINT ,CEN- TERED TREE,GRAPH ECCENTRICITY ,GRAPH RADIUS References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 35, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 107, 1990. Sloane, N. J. A. Sequences A052437, A052438, A052439, A052340, and A052341 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Graph Circumference The length of any longest cycle in a GRAPH . See also GIRTH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 13, 1994.Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 192, 1990. Graph Coloring The assignment of labels or colors to the edges or vertices of a graph. The most common types of graph colorings are EDGE COLORING and VERTEX COLORING . See also EDGE COLORING ,FOUR- COLOR THEOREM , K- COLORING ,VERTEX COLORING References Jensen, T. R. and Toft, B. Graph Coloring Problems. New York: Wiley, 1994. Morgenstern, C. and Shapiro, H. "Heuristics for Rapidly 4- Coloring Large Planar Graphs." Algorithmica 6, 869 /C1/91, 1991. Opsut, R. J. and Roberts, F. S. "On the Fleet Maintenance, Mobile Radio Frequency, Task Assignment, and Traffic Phasing Problems." In The Theory and Applications of Graphs (Ed. G. Chartrand, Y. Alavi, D. L. Goldsmith, L. Lesniak-Foster, and D. R. Lick). New York: Wiley, pp. 479 /C1/92, 1981. Skiena, S. "Graph Coloring." §5.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 210 /C1/ 16, 1990. Wagon, S. "An April Fool’s Hoax." Mathematica in Educ. Res. 7,46/C1/2, 1998. Wagon, S. "Coloring Planar Maps and Graphs." Ch. 24 in Mathematica in Action, 2nd ed. New York: Springer- Verlag, pp. 507 /C1/37, 1999. Graph Complement The complement of a graph Gnon n nodes is the graph G?n (sometimes denoted ¯Gn) on the same nodes, but with the vertices in Gnomitted and the omitted vertices in Gnincluded. The GRAPH SUM Gn /C27G?nis therefore the COMPLETE GRAPH Kn : A graph comple- ment can be given by the Mathematica command GraphComplement [graph ] in the Mathematica add- on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also COMPLETE GRAPH ,GRAPH SUM,SELF-COM- PLEMENTARY GRAPH References Skiena, S. "The Complement of a Graph." §3.2.3 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 93, 1990. Graph Composition The composition G /C30G1[G2] of graphs G1 and G2 with disjoint point sets V1 and V2 and edge sets X1 and X2 is the graph with point set V1 /C29V2and u /C30(u1 ; u2) adjacent with v /C30(v1 ; v2) whenever [u1 adj v1]or [u1 /C30v1 and u2 adj v2] (Harary 1994, p. 22). See also GRAPH PRODUCT References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 22, 1994. Graph Contraction The contraction of an edge fvi ; vj g of a GRAPH is the graph obtained by replacing the two nodes v1 and v2 with a single node v such that v is adjacent to the union of the nodes to which v1 and v2 were originally adjacent. The figure above shows a random graph contracted on vertices v7and v9 : Graph contraction can be implemented using Contract [g,{v1, v2}] in the Mathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 91, 1990.Graph Cycle A cycle of a GRAPH is a subset of the EDGE -set of the GRAPH which forms a CHAIN , the first node of which is also the last. This type of cycle is also called a CIRCUIT . Cycle graphs can be constructed using Cycle [n] in the Mathematica add-on package Dis- creteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The minimum number of swaps between vertices in a random circular embedding of a cycle to put in its standard configuration is considered by Bjo¨rner and Wachs (1982) and (Stanley 1986). See also ACYCLIC DIGRAPH ,CHAIN (GRAPH ), CYCLE GRAPH ,EULERIAN CIRCUIT ,EULERIAN GRAPH ,FOR- EST,H AMILTONIAN CIRCUIT ,H AMILTONIAN GRAPH , WALK References Bjo¨rner, A. and Wachs, M. "Bruhat Order of Coxeter Groups and Shellability." Adv. Math. 43,87/C1/00, 1982. Skiena, S. "Cycles in Graphs." §5.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 188 /C1/ 02, 1990. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, 1999. Graph Diameter The length maxu; v d(u; v) of the "longest shortest path" (i.e., the longest GRAPH GEODESIC ) between any two VERTICES (u, v)ofa GRAPH . In other words, a graph’s diameter is the largest number of vertices which must be traversed in order to travel from one vertex to another when paths which backtrack, detour, or loop are excluded from consideration. The above RANDOM GRAPHS on 10 vertices have diameters 3, 4, 5, and 7, respectively. See also DIAMETER ,G RAPH ,G RAPH ECCENTRICITY , GRAPH GEODESIC ,MOORE GRAPH ,PERIPHERAL POINT References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 14, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 107, 1990. Graph Difference The graph difference of graphs GandHis the graph with ADJACENCY MATRIX given by the difference of adjacency matrices of GandH. A graph difference is defined when the orders of Gand Hare the same, and can be computed using GraphDifference [g,h] in the Mathematica add-on package Discrete- Math‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also GRAPH SUM References Skiena, S. "Sum and Difference." §4.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 131, 1990. Graph Eccentricity The eccentricity of a node v in a CONNECTED GRAPH G is length maxu d(u; v) of the longest of all the shortest paths between v and every other point in G. The maximum eccentricity is the GRAPH DIAMETER . The minimum graph eccentricity is called the GRAPH RADIUS . See also CENTRAL POINT ,G RAPH CENTER ,G RAPH DIAMETER ,GRAPH RADIUS ,PERIPHERAL POINT References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 35, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 107, 1990. Graph Eigenvalue The eigenvalues of a GRAPH are defined as the EIGENVALUES of its ADJACENCY MATRIX . The set of eigenvalues of a GRAPH is called a GRAPH SPECTRUM . See also GRAPH SPECTRUM References Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993. Cvetkovic, D.; Doob, M.; and Sachs, H. Spectra of Graphs. New York: Academic Press, 1980. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 85, 1990.Graph Embedding A particular drawing of a GRAPH (with sometimes added constraint that the embedding be planar , i.e., has no crossing edges). The above figure shows the first several circular embeddings of the CUBICAL GRAPH . While the underlying object is independent of theembedding, a clever choice of embedding can lead to particularly illuminating diagrams. For example, the circular embedding of the CUBICAL GRAPH depicted above illustrates this graph’s inherent symmetries. Skiena (1990) considers a number of different types of embeddings, including circular, ranked, radial, rooted, and spring. See also EMBEDDING References Chung, F.; Leighton, T.; and Rosenberg, A. "Embeddings Graphs in Books: A Layout Problem with Applications to VLSI Design." SIAM J. Algebraic Disc. Meth. 8,3 3/C1/8, 1987. Di Battista, G.; Eades, P.; Tamassia, R.; and Tollis, I. G. Graph Drawing: Algorithms for the Visualization of Graphs. Englewood Cliffs, NJ: Prentice-Hall, 1998. Eades, P. "A Heuristic for Graph Drawing." Congr. Numer. 42, 149 /C1/60, 1984. Eades, P.; Fogg, I.; and Kelly, D. SPREMB: A System for Developing Graph Algorithms. Technical Report. Depart- ment of Computer Science. St. Lucia, Queensland, Aus- tralia: University of Queensland, 1988. Eades, P. and Tamassia, R. "Algorithms for Drawing Graphs: An Annotated Bibliography." Technical Report CS-89 /C1/9. Department of Computer Science. Providence, RI: Brown University, Feb. 1989. Kamada, T. and Kawai, S. "An Algorithm for Drawing General Undirected Graphs." Inform. Processing Lett. 31,7/C1/5, 1989. Malitz, S. M. "Genus g Graphs Have Pagenumber O(ffiffiffigp) :/"In Proc. 29th Sympos. Found. Computer Sci. IEEE Press, pp. 458 /C1/68, 1988. Reingold, E. and Tilford, J. "Tidier Drawings of Trees." IEEE Trans. Software Engin. 7, 223 /C1/28, 1981. Skiena, S. "Graph Embeddings." §3.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 81 and 98 /C1/18, 1990. Supowit, K. and Reingold, E. "The Complexity of Drawing Trees Nicely." Acta. Inform. 18, 377 /C1/92, 1983. Tamassia, R. "Graph Drawing." Ch. 21 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 937 /C1/71, 2000. Vaucher, J. "Pretty Printing of Trees." Software Pract. Experience 10, 553 /C1/61, 1980. Wetherell, C. and Shannon, A. "Tidy Drawings of Trees." IEEE Trans. Software Engin. 5, 514 /C1/20, 1979. Graph Genus The genus of a graph is the minimum number of handles that must be added to the plane to embed the graph without any crossings. See also CROSSING NUMBER (GRAPH ), PLANAR GRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Graph Geodesic A shortest path between two VERTICES (u, v)ofa GRAPH (Skiena 1990, p. 225). There may be more than one different shortest paths, all of the same length. Graph geodesics may be found using a BREADTH-FIRST TRAVERSAL (Moore 1959) or using DIJKSTRA’S ALGO- RITHM (Skiena 1990, p. 225). A graph geodesic can be found usingShortestPath [g, s, e] in the Mathema- tica add-on package DiscreteMath‘Combinator-ica‘ (which can be loaded with the command BBDiscreteMath‘ ). The length of the maximum graph geodesic in a given graph is called the GRAPH DIAMETER . See also ALL-PAIRS SHORTEST PATH,G RAPH DIA- METER References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 14, 1994. Moore, E. F. "The Shortest Path through a Maze." In Proc. Internat. Symp. Switching Th., Part II. Cambridge, MA: Harvard University Press, pp. 285 /C1/92, 1959. Skiena, S. "Shortest Paths." §6.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 225 /C1/ 53, 1990. Graph Intersection Let S be a set and F /C30fS1 ; ...; Sp g a nonempty family of distinct nonempty subsets of S whose union is @ p i/C301Si /C30S: The intersection graph of F is denoted V(F) and defined by V(V(F)) /C30F ; with Siand Sj adjacent whenever i "j and Si S Sj "¥: Then a GRAPH G is an intersection graph on S if there exists a family F of subsets for which G and V(F) are ISOMORPHIC GRAPHS (Harary 1994, p. 19). Graph intersections can be computed using GraphInter- section [g, h] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also GRAPH UNION ,INTERSECTION NUMBER References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. "Unions and Intersections." §4.1.1 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 129 /C1/31, 1990. Graph Isomorphism An isomorphism between two graphs is a one-to-one mapping between their two sets of vertices. See also GRAPH AUTOMORPHISM ,ISOMORPHIC GRAPHS References Du, D.-Z. and Ko, K.-I. Theory of Computational Complexity. New York; Wiley, p. 117, 2000. Skiena, S. "Graph Isomorphism." §5.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 181 /C1/87, 1990. Graph Join The join G /C30G1 /C27G2of graphs G1and G2with disjoint point sets V1and V2and edge sets X1and X2is the GRAPH UNION G1 @ G2together with all the edges joining V1and V2(Harary 1994, p. 21). Graph joins can be computed using GraphJoin [G1, G2]in the Mathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). A complete k-partite graph ki; j; ... is the graph join of empty graphs on i, j, ... nodes. A WHEEL GRAPH is the join of a CYCLE GRAPH and the singleton graph. Finally, a STAR GRAPH is the join of an EMPTY GRAPH and the singleton graph (Skiena 1990, p. 132). See also GRAPH SUM,GRAPH UNION References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. "Joins of Graphs." §4.1.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 131 /C1/ 32, 1990. Graph Lexographic Product This entry contributed by NICOLAS BRAY The GRAPH PRODUCT denoted G+H and defined by the adjacency relations ( g adj g ?)or( /g /C30g ? and h adj h?):/ See also GRAPH PRODUCTGraph Power The kth power of a GRAPH G is a graph with the same set of vertices as G and an edge between two vertices IFF there is a path of length at most k between them (Skiena 1990, p. 229). Since a path of length two between vertices u and v exists for every vertex w such that fu; w g and fw; vg are edges in G, the square of the ADJACENCY MATRIX of G counts the number of such paths. Similarly, the (u, v)th element of the kth power of the ADJACENCY MATRIX of G gives the number of paths of length k between vertices u and v. The graph kth power is then defined as the graph whose adjacency matrix given by the sum of the first k powers of the ADJACENCY MATRIX , adj(Gk) /C30Xk i/C301[adj(G)]i ; which counts all paths of length up to k (Skiena 1990, p. 230). Raising any graph to the power of its GRAPH DIAMETER gives a COMPLETE GRAPH . The square of any BICONNECTED GRAPH is HAMILTONIAN (Fleischner 1974, Skiena 1990, p. 231). Mukhopadhyay (1967) has considered "square root graphs," whose square gives a given graph G (Skiena 1990, p. 253). See also ADJACENCY MATRIX ,P O´ SA’S THEOREM , SEYMOUR CONJECTURE References Fleischner, H. "The Square of Every Two-Connected Graph Is Hamiltonian." J. Combin. Th. Ser. B 16,2 9/C1/4, 1974. Mukhopadhyay, A. "The Square Root of a Graph." J. Combin. Th. 2, 290/C1/95, 1967. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Graph Product This entry contributed by NICOLAS BRAY In general, a graph product of two graphs G and H is a new graph whose VERTEX SET is V(G) /C29V(H) and where, for any two vertices (g, h) and (g ?; h?) in the product, the adjacency of those two vertices is determined entirely by the adjacency (or equality, or non-adjacency) of g and g ?; and that of h and h?: There are 3 /C293 /C281 /C308 cases to be decided (three possibili- ties for each, with the case where both are equal eliminated) and thus there are 28 /C30256 different types of graph products that can be defined. The most commonly used graph products, given by conditions sufficient and necessary for adjacency, are summarized in the following table (Hartnell and Rall 1998). Note that the terminology is not quite stan- dardized, so these products may actually be referred to by different names by different sources. Many other graph products can be found in Jensen and Toft (1994). graph product namesymbol definition GRAPH CARTESIAN PRODUCT/GIH/ (/g /C30g ? and h adj h?) or (/g adj g ? and h /C30h ?)/ GRAPH CATEGORI- CAL PRODUCT/G /C29H/ ( g adj g ? and h adj h?)/ GRAPH LEXO- GRAPHIC PRODUCT/G /C215 H/ (/g adj g?)or( /g /C30g ? and h adj h ?)/ GRAPH STRONG PRODUCT/GGH/ (/g /C30g ? and h adj h?) or (/g adj g ? and h /C30 h ?)or( /g adj g? and h adj h?)/ See also GRAPH CARTESIAN PRODUCT References Hartnell, B. and Rall, D. "Domination in Cartesian Products: Vizing’s Conjecture." In Domination in Graphs--Advanced Topics (Ed. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater). New York: Dekker, pp. 163 /C1/89, 1998. Jensen, T. R. and Toft, B. Graph Coloring Problems. New York: Wiley, 1994.Graph Radius The minimum GRAPH ECCENTRICITY of any VERTEX in a GRAPH . See also CENTRAL POINT ,G RAPH CENTER ,G RAPH ECCENTRICITY References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 35, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 107, 1990. Graph Section A section of a GRAPH obtained by finding its intersec- tion with a PLANE . Graph Spectrum The set of GRAPH EIGENVALUES is called the spectrum of the graph. The spectrum of a graph may be computed using Spectrum [g] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). Two nonisomorphic graphs can share the same spectrum, e.g., the GRAPH UNION C4@K1and STAR GRAPH S5(Skiena 1990, p. 85). The maximum degree of a CONNECTED GRAPH Gis an eigenvalue of GIFFG is a REGULAR GRAPH . See also GRAPH EIGENVALUE References Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993. Cvetkovic, D.; Doob, M.; and Sachs, H. Spectra of Graphs. New York: Academic Press, 1980. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 85, 1990. Wilf, H. "Graphs and Their Spectra: Old and New Results." Congr. Numer. 50,3 7/C1/3, 1985. Graph Strong Product This entry contributed by NICOLAS BRAY The GRAPH PRODUCT denoted GGH and defined by the adjacency relations (/g /C30g ? and h adj h?)or (g adj g ? and h /C30h?)or( g adj g ? and h adj h ?):/ See also GRAPH PRODUCT Graph Sum The graph sum of graphs G and H is the graph with ADJACENCY MATRIX given by the sum of adjacency matrices of G and H. A graph sum is defined when the orders of G and H are the same, and can be computed usingGraphSum [g, h] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also GRAPH DIFFERENCE ,G RAPH JOIN,G RAPH UNION References Skiena, S. "Sum and Difference." §4.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 131, 1990. Graph Theory The mathematical study of the properties of the formal mathematical structures called GRAPHS . See also ADJACENCY MATRIX ,ADJACENCY RELATION , ARTICULATION VERTEX ,B LUE- EMPTY COLORING , BRIDGE ,CHROMATIC NUMBER ,CHROMATIC POLYNO- MIAL ,C IRCUIT RANK,C ROSSING NUMBER (GRAPH ), CYCLOMATIC NUMBER ,D EGREE ,D IJKSTRA’S ALGO- RITHM ,ECCENTRICITY ,EDGE COLORING ,EDGE CON- NECTIVITY ,E ULERIAN CIRCUIT ,E ULERIAN TRAIL, FACTOR (GRAPH ), FLOYD’S ALGORITHM ,GIRTH,GRAPH CYCLE ,G RAPH DIAMETER ,G RAPH RADIUS ,G RAPH TWO-COLORING ,GROUP THEORY ,H AMILTONIAN CIR- CUIT,H ASSE DIAGRAM ,H UB,INDEGREE ,INTEGRAL DRAWING ,ISTHMUS ,JOIN (GRAPH ), LOCAL DEGREE , MONOCHROMATIC FORCED TRIANGLE ,O UTDEGR EE, PARTY PROBLEM ,P O´ LYA ENUMERATION THEOREM , PO´ LYA POLYNOMIAL ,RAMSEY NUMBER ,R E-ENTRANT CIRCUIT ,SEPARATING EDGE,TAIT COLORING ,TAIT CYCLE ,TRAVELING SALESMAN PROBLEM ,TREE,TUT- TE’S THEOREM ,UNICURSAL CIRCUIT ,VERTEX COLOR- ING,VERTEX DEGREE ,W ALKReferences Beinecke, L. W. and Wilson, R. J. (Eds.). Graph Connec- tions: Relationships Between Graph Theory and Other Areas of Mathematics. Oxford, England: Oxford Univer- sity Press, 1997. Berge, C. Graphs and Hypergraphs. Amsterdam, Nether- lands: North-Holland, 1976. Berge, C. The Theory of Graphs and Its Applications. New York: Wiley, 1962. Bogomolny, A. "Graphs." http://www.cut-the-knot.com/ do_you_know/graphs.html. Bolloba ´s, B. Graph Theory: An Introductory Course. New York: Springer-Verlag, 1979. Bolloba ´s, B. Modern Graph Theory. New York: Springer- Verlag, 1998. Caldwell, C. K. "Graph Theory Tutorials." http://www.ut- m.edu/departments/math/graph/. Chartrand, G. Introductory Graph Theory. New York: Dover, 1985. Emden-Weinert, T. "Graphs: Theory-Algorithms-Complex- ity." http://people.freenet.de/Emden-Weinert/graphs.html. Foulds, L. R. Graph Theory Applications. New York: Springer-Verlag, 1992. Chung, F. and Graham, R. Erdos on Graphs: His Legacy of Unsolved Problems. New York: A. K. Peters, 1998. Gardner, M. "Graph Theory." Ch. 10 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 91 /C1 /03, 1984. Gould, R. (Ed.). Graph Theory. Menlo Park, CA: Benjamin- Cummings, 1988. Grossman, I. and Magnus, W. Groups and Their Graphs. Washington, DC: Math. Assoc. Amer., 1965. Harary, F. "Graphical Enumeration Problems." In Graph Theory and Theoretical Physics (Ed. F. Harary). London: Academic Press, pp. 1 /C1/1, 1967. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction, 2nd ed. San Diego, CA: Academic Press, 1994. Locke, S. C. "Graph Theory." http://www.math.fau.edu/ locke/graphthe.htm. Locke, S. C. "Graph Theory Books." http://www.math.- fau.edu/locke/graphstx.htm. Mehlhorn, K. and Na ¨her, S. LEDA: A Platform for Combi- natorial and Geometric Computing. Cambridge, England: Cambridge University Press, 1999. Ore, Ø.Graphs and Their Uses. New York: Random House, 1963. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Ruskey, F. "Information on (Unlabelled) Graphs." http:// www.theory.csc.uvic.ca/~cos/inf/grap/GraphInfo.html. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Redwood City, CA: Addison-Wesley, 1988. Trudeau, R. J. Introduction to Graph Theory. New York: Dover, 1994. Tutte, W. T. Graph Theory as I Have Known It. Oxford, England: Oxford University Press, 1998. Weisstein, E. W. "Graphs." M ATHEMATICA NOTEBOOK GRAPHS.M . Weisstein, E. W. "Books about Graph Theory." http:// www.treasure-troves.com/books/GraphTheory.html. Woo, L. "Definitions of Graph Theory." http://www.simmon- s.edu/~woo/graphtheory/definition.html. Graph Thickness The thickness of a GRAPH G is the minimum number of PLANAR SUBGRAPHS of g whose GRAPH UNION is g (skiena 1990, p. 251). References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Graph Two-Coloring Assignment of each EDGE of a GRAPH to one of two color classes ("red" or "green"). See also BLUE- EMPTY GRAPH ,M ONOCHROMATIC FORCED TRIANGLE Graph Union The union G /C30G1 @ G2of graphs G1and G2with disjoint point sets V1 and V2 and edge sets X1 and X2 is the graph with V /C30V1 @ V2and X /C30X1 @ X2(Har- ary 1994, p. 21). Graph unions can be computed using GraphUnion [g, h] in the Mathematica add-on pack- ageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also GRAPH INTERSECTION ,GRAPH JOIN References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. "Unions and Intersections." §4.1.1 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 129 /C1/31, 1990. Graphic Sequence A graphic sequence is a sequence of numbers which can be the DEGREE SEQUENCE of some GRAPH .A sequence can be checked to determine if it is graphic usingGraphicQ [g] in the Mathematica add-on pack- ageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). Erdos and Gallai (1960) proved that a DEGREE SEQUENCE fd1 ; ...; dn g is graphic IFF the sequence obeys the propertyXr i/C301di 5r(r /C281) /C27Xn i/C30r/C271min( r ; di) for each integer r B n (Skiena 1990, p. 157), and this condition also generalizes to DIRECTED GRAPHS .In addition, Hakimi (1962) and Havel (1955) showed that if a DEGREE SEQUENCE is graphic, then there exists a GRAPH G such that the node of highest degree is adjacent to the D(G) next highest degree vertices of G, where D(G) is the maximum degree of G. No degree sequence can be graphic if all the degrees occur with multiplicity 1 (Behzad and Chartrand 1967, p. 158; Skiena 1990, p. 158). Any degree se- quence whose sum is EVEN can be realized by a MULTIGRAPH having loops (Hakimi 1962; Skiena 1990, p. 158). See also DEGREE SEQUENCE ,GRAPHICAL PARTITION , VERTEX DEGREE References Behzad, M. and Chartrand, G. "No Graph is Perfect." Amer. Math. Monthly 74, 962/C1/63, 1967. Eggleton, R. B. "Graphic Sequences and Graphic Polyno- mials." In Infinite and Finite Sets (Ed. A. Hajnal). Am- sterdam, Netherlands: North-Holland, pp. 385 /C1/93, 1975. Erdos, P. and Gallai, T. "Graphs with Prescribed Degrees of Vertices" [Hungarian]. Mat. Lapok. 11, 264/C1/74, 1960. Fulkerson, D. R. "Upsets in Round Robin Tournaments." Canad. J. Math. 17, 957/C1/69, 1965. Fulkerson, D. R.; Hoffman, A. J.; and McAndrew, M. H. "Some Properties of Graphs with Multiple Edges." Canad. J. Math. 17, 166/C1/77, 1965. Hakimi, S. "On the Realizability of a Set of Integers as Degrees of the Vertices of a Graph." SIAM J. Appl. Math. 10, 496/C1/06, 1962. Havel, V. "A Remark on the Existence of Finite Graphs" [Czech]. Casopis Pest. Mat. 80, 477/C1/80, 1955. Ryser, H. J. "Combinatorial Properties of Matrices of Zeros and Ones." Canad. J. Math. 9, 371/C1/77, 1957. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 157, 1990. Graphical Partition A partition fa1;...;angis called graphical if there exists a GRAPH Ghaving DEGREE SEQUENCE fa1;...;ang:The number of graphical partitions on n-node graphs is therefore the same as the number of n-node graphs with no ISOLATED POINTS . A graphical partition of order pis one for which the sum of degrees is p.Ap-graphical partition only exists for EVEN p. It is possible for two topologically distinct graphs to have the same DEGREE SEQUENCE . For n /C302, 4, 6, ..., the numbers of graphical partitions pg(n) are 1, 2, 5, 9, 17, ... (Sloane’s A000569). Erdos and Richmond (1989) showed that lim inf n0/C12ffiffiffiffiffiffi 2np pg(2n) ]pffiffiffi 6p and lim sup npg(2n) 50:4258 : See also CUT,DEGREE SEQUENCE ,SPECTRAL GRAPH PARTITIONING References Barnes, T. M. and Savage, C. D. "A Recurrence for Counting Graphical Partitions." Electronic J. Combinatorics 2, R11 1 /C1/0, 1995. http://www.combinatorics.org/Volume_2/volu- me2.html#R11. Barnes, T. M. and Savage, C. D. "Efficient Generation of Graphical Partitions." Disc. Appl. Math. 78,17/C1/6, 1997. Erdos, P. and Richmond, L. B. "On Graphical Partitions." Combinatorics and Optimization Research Report COPR 89 /C1/2. Waterloo, Ontario: University of Waterloo, pp. 1 /C1/3, 1989. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 57, 1994. Ruskey, F. "Information on Graphical Partitions." http:// www.theory.csc.uvic.ca/~cos/inf/nump/GraphicalParti- tion.html. Sloane, N. J. A. Sequences A000569 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos’ 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press, pp. 557 /C1/62, 1997. Graphical Representation FERRERS DIAGRAM Graphoid A graphoid consists of a set M of elements together with two collections C and D of nonempty subsets of M, called circuits and cocircuits respectively, such that1. For any C /C23C and D /C23D;½C S D ½"1;/ 2. No circuit properly contains another circuit and no cocircuit properly contains another cocircuit, 3. For any painting of M with colors exactly one element green and the rest either red or blue, there exists either (a) a circuit C containing the green element and no red elements, or (b) a cocircuit D containing the green element and no blue elements. See also MATROID References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 41, 1994. Grassmann Algebra EXTERIOR ALGEBRA Grassmann Coordinates An ( m/C271)/-D SUBSPACE Wof an ( n/C271)/-D VECTOR SPACE Vcan be specified by an ( m/C271)/C29(n/C271) MATRIX whose rows are the coordinates of a BASIS of W. The set of alln/C271 m/C271=z1*=z1+ (m/C271)/C29(m/C271)MINORS of this MATRIX are then called the Grassmann (or sometimes Plu¨cker; Stofli 1991) coordinates of w, wherea b=z;=z1 is a BINOMIAL COEFFICIENT . Hodge and Pedoe (1952) give a thorough treatment of Grassmann coordinates. See also CHOW COORDINATES References Hodge, W. V. D. and Pedoe, D. Methods of Algebraic Geo- metry. Cambridge, England: Cambridge University Press, 1952. Stofli, J. Oriented Projective Geometry. New York: Academic Press, 1991. Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 43, 1117/C1/124, 1996. Grassmann Manifold A special case of a FLAG MANIFOLD . A Grassmann manifold is a certain collection of vector SUBSPACES of aVECTOR SPACE . In particular, gn;kis the Grassmann manifold of k-dimensional subspaces of the VECTOR SPACE Rn:It has a natural MANIFOLD structure as an orbit-space of the S TIEFEL MANIFOLD vn;kof orthonor- mal k-frames in Gn:One of the main things about Grassmann manifolds is that they are classifying spaces for VECTOR BUNDLES . Gray Code An encoding of numbers so that adjacent numbershave a single DIGIT differing by 1. A BINARY Gray code with nDIGITS corresponds to a H AMILTONIAN PATH on ann-D HYPERCUBE (including direction reversals). The term Gray code is often used to refer to a"reflected" code, or more specifically still, the binary reflected Gray code. To convert a BINARY number d1d2 /C1/C1/C1dn /C281dnto its corresponding binary reflected Gray code, start at the right with the digit dn (the nth, or last, DIGIT ). If the dn/C281is 1, replace dnby 1 /C28dn; otherwise, leave it unchanged. Then proceed to dn /C281 : Continue up to the first DIGIT d1 ; which is kept the same since d0is assumed to be a 0. The resulting number g1g2 /C1/C1/C1gn/C281gn is the reflected binary Gray code. To convert a binary reflected Gray code g1g2 /C1/C1/C1gn/C281gn to a BINARY number, start again with the nth digit, and compute X n/C13Xn/C281 i /C301gi (mod 2): If an is 1, replace gn by 1 /C28gn; otherwise, leave it the unchanged. Next compute X n/C281/C13Xn/C282 i /C301gi (mod 2); and so on. The resulting number d1d2 /C1/C1/C1dn/C281dn is the BINARY number corresponding to the initial binary reflected Gray code. The code is called reflected because it can be gener- ated in the following manner. Take the Gray code 0, 1. Write it forwards, then backwards: 0, 1, 1, 0. Then append 0s to the first half and 1s to the second half: 00, 01, 11, 10. Continuing, write 00, 01, 11, 10, 10, 11, 01, 00 to obtain: 000, 001, 011, 010, 110, 111, 101, 100, ... (Sloane’s A014550). Each iteration therefore dou- bles the number of codes. The Gray codes correspond- ing to the first few nonnegative integers are given in the following table. 0 0 20 11110 40 111100 1 1 21 11111 41 111101 2 11 22 11101 42 111111 3 10 23 11100 43 111110 4 110 24 10100 44 111010 5 111 25 10101 45 111011 6 101 26 10111 46 111001 7 100 27 10110 47 111000 8 1100 28 10010 48 101000 9 1101 29 10011 49 101001 10 1111 30 10001 50 101011 11 1110 31 10000 51 101010 12 1010 32 110000 52 101110 13 1011 33 110001 53 10111114 1001 34 110011 54 101101 15 1000 35 110010 55 101100 16 11000 36 110110 56 10010017 11001 37 110111 57 100101 18 11011 38 110101 58 100111 19 11010 39 110100 59 100110 The binary reflected Gray code is closely related to the solutions of the TOWERS OF HANOI and BAGUE- NAUDIER , as well as to Hamiltonian circuits of hypercube graphs (Skiena 1990, p. 149). See also BAGUENAUDIER ,BINARY ,H ILBERT CURVE , RYSER FORMULA ,THUE- MORSE SEQUENCE ,TOWERS OF HANOI References Gardner, M. "The Binary Gray Code." Ch. 2 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986. Gilbert, E. N. "Gray Codes and Paths on the n-Cube." Bell System Tech. J. 37, 815/C1/26, 1958. Gray, F. "Pulse Code Communication." United States Patent Number 2,632,058. March 17, 1953. Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gray Codes." §20.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.Cambridge, England: Cambridge University Press,pp. 886 /C1 /88, 1992. Skiena, S. "Gray Code." §1.5.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 42 /C1/3 and 149, 1990. Sloane, N. J. A. Sequences A014550 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 111 /C1/12 and 246, 1991. Wilf, H. S. Combinatorial Algorithms: An Update. Philadel- phia, PA: SIAM, 1989. Gray Graph ACUBIC GRAPH on 54 vertices that is EDGE- but not VERTEX-TRANSITIVE ; the smallest known such exam- ple. It was discovered by Marion C. Gray in 1932, and was first published by Bouwer (1968). It has GIRTH 8, GRAPH DIAMETER 6, has Aut G jj /C301296 ; and is the Levi graph of two dual, triangle-free, point-, line-, and flag- transitive, non-self-dual 273configurations (Maruvic and Pisanski 2000). The symmetric embedding illu- strated above is due to (Maruvic and Pisanski 2000). It can be constructed by taking three copies of the COMPLETE BIPARTITE GRAPH K3;3 and, for a particular edge e, subdividing e in each of the three copies, joining the resulting three vertices to a new vertex, and repeating with each edge. See also COMPLETE BIPARTITE GRAPH ,CUBIC GRAPH , EDGE-TRANSITIVE GRAPH ,VERTEX- TRANSITIVE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 235, 1976. Bouwer, I. Z. "An Edge But Not Vertex Transitive Cubic Graph." Bull. Canad. Math. Soc. 11, 533/C1/35, 1968. Bouwer, I. Z. "On Edge But Not Vertex Transitive Regular Graphs." J. Combin. Th. B 12,3 2/C1/0, 1972. Maruvic, D. and Pisanski, T. "The Gray Graph Revisited." J. Graph Th. 35,1/C1/, 2000. Pisanski, T. and Randic, M. "Bridged Between Geometry and Graph Theory." To appear. Weisstein, E. W. "Graphs." M ATHEMATICA NOTEBOOK GRAPHS.M . Grazing Goat Problem GOATPROBLEM Great Circle A great circle is a SECTION of a SPHERE which contains aDIAMETER of the SPHERE (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called SMALL CIRCLES . The shortest path between two points on a SPHERE ,also known as an ORTHODROME , is a segment of a great circle. To find the great circle ( GEODESIC ) distance between two points located at LATITUDE d and LONGITUDE lof (d1;l1) and ( d2;l2)o na SPHERE ofRADIUS a, convert SPHERICAL COORDINATES to CARTESIAN COORDINATES using ri/C30acoslicosdi sinlicosdi sindi2 435: (1) (Note that the LATITUDE dis related to the COLATI- TUDE fofSPHERICAL COORDINATES byd/C3090/C14/C28f;so the conversion to C ARTESIAN COORDINATES replaces sinfand cos fby cos dand sin d;respectively.) Now find the ANGLE abetween r1and r2using the DOT PRODUCT , cosa/C30ˆr1/C215ˆr2 /C30cosd1cosd2(sinl1sinl2/C27cosl1cosl2) /C27sind1sind2 /C30cosd1cosd2cos(l1/C28l2)/C27sind1sind2: (2) The great circle distance is then d/C30acos/C281[cosd1cosd2cos(l1/C28l2) /C27sind1sind2]: (3) For the Earth, the equatorial RADIUS isa:6378 km, or 3963 (statute) miles. Unfortunately, the FLATTEN- INGof the Earth cannot be taken into account in this simple derivation, since the problem is considerably more complicated for a SPHEROID orELLIPSOID (each of which has a RADIUS which is a function of LATITUDE ). This leads to extremely complicated ex- pressions for OBLATE SPHEROID GEODESICS and GEO- DESICS on other ELLIPSOIDS . A great circle becomes a straight line in a GNOMONIC PROJECTION (Steinhaus 1983, pp. 220 /C1/21). The equation of the great circle can be explicitlycomputed using the GEODESIC formalism. Writing u/C30l (4) v/C30d/C301 2p/C28f (5) gives the P,Q, and Rparameters of the GEODESIC (which are just combinations of the PARTIAL DERIVA- TIVES )a s P/C13@x @u !2 /C27@y @u !2 /C27@z @u !2 /C30a2sin2v (6) Q/C13@x @u@x @v/C27@y @u@y @v/C27@z @u@z @v/C300 (7) R/C13@x @v !2 /C27@y @v !2 /C27@z @v !2 /C30a2: (8) The GEODESIC differential equation then becomes cos v sin4 v /C272 cos v sin2 vv ?2 /C27cos vv ?4 /C28sin vv ƒ /C300: (9) However, because this is a special case of Q /C300 with P and R explicit functions of v only, the GEODESIC solution takes on the special form v /C30c1gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R P2 /C28 c2 1Ps dv /C30c1gdv a2 sin4 v /C28 c21 sin2 v /C30gdv sin vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c1 !2 sin2 v /C28 1vuut /C30/C28tan /C281 cos vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c1 !2 /C281vuut2 66666643 7777775/C27c 2 (10) (Gradshteyn and Ryzhik 2000, p. 174, eqn. 2.599.6), which can be rewritten as v /C30/C28sin/C281 cot vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c1 !2 /C281vuut0 BBBBBB@1 CCCCCCA/C27c 2 : (11) It therefore follows that (sin c2)a sin v cos u /C28(cos c2)a sin v sin u /C28a cos vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c1 !2 /C281vuut /C300 : (12) This equation can be written in terms of the CARTE- SIAN COORDINATES as x sin c2 /C28y cos c2 /C28zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c1 !2 /C281vuut/C300; (13) which is simply aPLANE passing through the center of the SPHERE and the two points on the surface of the SPHERE . See also GEODESIC ,G REAT SPHERE ,L OXODROME , MIKUSINSKI’S PROBLEM ,OBLATE SPHEROID GEODESIC , ORTHODROME ,POINT- POINT DISTANCE–2- D, PSEUDO- CIRCLE ,SMALL CIRCLE ,SPHEREReferences Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 183 and 217, 1999. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, pp. 24 /C1/5, 1965. Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, pp. 26 /C1/8 and 62/C1/3, 1974. Great Cubicuboctahedron The UNIFORM POLYHEDRON U14whose DUAL POLYHE- DRON is the GREAT HEXACRONIC ICOSITETRAHEDRON . It has W YTHOFF SYMBOL 34½4 3and is Wenninger model W77:Its faces are 8 f3g/C276f4g/C276f8 3g:It is a FACETED version of the CUBE . The CIRCUMRADIUS of a great cubicuboctahedron with unit edge length is r/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 2pq : The CONVEX HULL of the great cubicuboctahedron is the Archimedean TRUNCATED CUBE A9;whose dual is the SMALL TRIAKIS OCTAHEDRON , so the dual of the great cubicuboctahedron (i.e., the GREAT HEXACRONIC ICOSITETRAHEDRON ) is one of the stellations of the SMALL TRIAKIS OCTAHEDRON (Wenninger 1983, p. 57). References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 57 /C1/8, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 118 /C1/19, 1989. Great Deltoidal Hexecontahedron The DUAL of the uniform GREAT RHOMBICOSIDODECA- HEDRON U67 and Wenninger dual W105 :/ See also DUAL POLYHEDRON ,GREAT RHOMBICOSIDO- DECAHEDRON (UNIFORM ) References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 88, 1983. Great Deltoidal Icositetrahedron The DUAL of the uniform GREAT RHOMBICUBOCTAHE- DRON and Wenninger dual W85 :/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 59, 1983. Great Dirhombicosidodecacron The DUAL of the GREAT DIRHOMBICOSIDODECAHEDRON U75 and Wenninger dual W119 :/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 139, 1983. Great Dirhombicosidodecahedron The UNIFORM POLYHEDRON U75whose DUAL is the GREAT DIRHOMBICOSIDODECACRON . This POLYHEDRONis exceptional because it cannot be derived from SCHWARZ TRIANGLES and because it is the only UNI- FORM POLYHEDRON with more than six POLYGONS surrounding each VERTEX (four SQUARES alternating with two TRIANGLES and two PENTAGRAMS ). This unique polyhedron has features in common with both snub forms and hemipolyhedra, and its octagrammic faces pass through the origin. It has pseudo-W YTHOFF SYMBOL3 253 352 :=z1n=z1n=z1n Its faces are 40 f3g/C2760 f4g/C2724 f5 2 g; and its CIRCUMRADIUS for unit edge length is R /C3012ffiffiffi 2p : See also UNIFORM POLYHEDRON References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 200 /C1/03, 1989. Great Disdyakis Dodecahedron The DUAL of the GREAT TRUNCATED CUBOCTAHEDRON U20 and Wenninger dual W93 :/ See also DUAL POLYHEDRON ,G REAT TRUNCATED CUBOCTAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 92, 1983. Great Disdyakis Triacontahedron The DUAL of the GREAT TRUNCATED ICOSIDODECAHE- DRON U68 and Wenninger dual W108 :/ See also DUAL POLYHEDRON ,G REAT TRUNCATED ICOSIDODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 96, 1983. Great Ditrigonal Dodecacronic Hexecontahedron The DUAL of the GREAT DITRIGONAL DODECICOSIDODE- CAHEDRON U42 and Wenninger dual W81 :/ See also DUAL POLYHEDRON ,G REAT DITRIGONAL DODECICOSIDODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 62, 1983.Great Ditrigonal Dodecicosidodecahedron The UNIFORM POLYHEDRON U42whose DUAL is the GREAT DITRIGONAL DODECACRONIC HEXECONTAHE- DRON . It has W YTHOFF SYMBOL 35½5 3:Its faces are 20f3g/C2712f5g/C2712f10 3g;and its CIRCUMRADIUS for unit edge length is R/C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34/C286ffiffiffi 5pq : The CONVEX HULL of the great ditrigonal dodecicosi- dodecahedron is a regular DODECAHEDRON , whose dual is the ICOSAHEDRON , so the dual of the great ditrigonal dodecicosidodecahedron (the GREAT TRIAM- BIC ICOSAHEDRON ) is one of the ICOSAHEDRON STELLA- TIONS (Wenninger 1983, p. 42). References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 125, 1989. Great Ditrigonal Icosidodecahedron The UNIFORM POLYHEDRON U47whose DUAL is the GREAT TRIAMBIC ICOSAHEDRON . It has W YTHOFF SYMBOL3 2½35:Its faces are 20 f3g/C2712f5g;and its CIRCUMRADIUS for unit edge length is R/C3012ffiffiffi 3p : The CONVEX HULL of the great triambic icosahedron is a regular DODECAHEDRON , whose dual is the ICOSA- HEDRON , so the dual of the great ditrigonal icosido- decahedron (the GREAT TRIAMBIC ICOSAHEDRON )i s one of the ICOSAHEDRON STELLATIONS . References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 42, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 135 /C1/36, 1989. Great Dodecacronic Hexecontahedron The DUAL of the GREAT DODECICOSIDODECAHEDRON U61 and Wenninger dual W99 :/ See also DUAL POLYHEDRON ,GREAT DODECICOSIDO- DECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 88, 1983. Great Dodecadodecahedron DODECADODECAHEDRON Great Dodecahedron The KEPLER- POINSOT SOLID which is the DUAL of the SMALL STELLATED DODECAHEDRON . It is also UNIFORM POLYHEDRON U35and Wenninger model W20 : Its SCHLA ¨ FLI SYMBOL is f5;5 2 g; and its WYTHOFF SYMBOL is5 2 ½25: Its faces are 12f5 g: Its CIRCUMRADIUS for unit edge length isR /C3012 51 =4 f1 =2a /C3014 51 =4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(1 /C27ffiffiffi 5p )q ; where f is the GOLDEN RATIO . It can be constructed by CUMULATION of a unit edge-length ICOSAHEDRON by a pyramid with height /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 6(7 /C283ffiffiffi 5pq :: This gives side of lengths s1 /C301 2(ffiffiffi 5p /C281) /C30 f /C281 (1) s2 /C301 (2) The result solid has SURFACE AREA and VOLUME S /C3015ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C282ffiffiffi 5pq (3) V /C305 4(ffiffiffi 5p /C281): (4) Schla ¨fli (1901, p. 134) did not recognize the great dodecahedron because it, like the SMALL STELLATED DODECAHEDRON , satisfies N0 /C28N1 /C27N2 /C3012 /C2830 /C2712 /C30/C286; (5) where N0 is the number of vertices, N1 the number of edges, and N2the number of faces (Coxeter 1973, p. 172), thus violating the POLYHEDRAL FORMULA . The CONVEX HULL of the great dodecahedron is a regular ICOSAHEDRON and the dual of the ICOSAHE- DRON is the DODECAHEDRON , so the dual of the great dodecahedron (the SMALL STELLATED DODECAHE- DRON ) is one of the DODECAHEDRON STELLATIONS (Wenninger 1983, pp. 35 and 40) See also DODECAHEDRON ,G REAT ICOSAHEDRON , GREAT STELLATED DODECAHEDRON ,KEPLER- POINSOT SOLID,SMALL STELLATED DODECAHEDRON ,STELLA- TION References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Cundy, H. and Rollett, A. "The Great Dodecahedron. 55=2:/" §3.6.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 92 /C1/3, 1989. Fischer, G. (Ed.). Plate 105 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 104, 1986. Schla¨fli, L. "Theorie der vielfachen Kontinuita ¨t." Denkschrif- ten der Schweizerischen naturforschenden Gessel. 38,1/C1/ 37, 1901. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 39, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 35 and 39, 1989. Great Dodecahedron-Small Stellated Dodecahedron Compound A POLYHEDRON COMPOUND in which the GREAT DODECAHEDRON is interior to the SMALL STELLATED DODECAHEDRON . See also POLYHEDRON COMPOUND Great Dodecahemicosacron The DUAL of the GREAT DODECAHEMICOSAHEDRON U65 and Wenninger dual W102 : When rendered, the SMALL DODECAHEMICOSACRON and great dodecahemicosa- cron appear the same. See also DUAL POLYHEDRON ,GREAT DODECAHEMICO- SAHEDRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 107, 1983. Great Dodecahemicosahedron The UNIFORM POLYHEDRON U65whose DUAL is the GREAT DODECAHEMICOSACRON . It has WYTHOFF SYM- BOL5 4 5½3: Its faces are 10f6 g/C276f5g/C276f54 g: It is a FACETED DODECADODECAHEDRON . The CIRCUMRADIUS for unit edge length is R /C302.References Wenninger, M. J. "Great Dodecahemicosahedron." Model 102 in Polyhedron Models. Cambridge, England: Cam- bridge University Press, p. 158, 1989. Great Dodecahemidodecacron The DUAL of the GREAT DODECAHEMIDODECAHEDRON U70and Wenninger dual W107 : When rendered, the great dodecahemidodecacron and GREAT ICOSIHEMI- DODECACRON look the same, both consisting of a compound of six infinite f10 =3g prisms. See also DUAL POLYHEDRON ,GREAT DODECAHEMIDO- DECAHEDRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 107, 1983. Great Dodecahemidodecahedron The UNIFORM POLYHEDRON U70whose DUAL is the GREAT DODECAHEMIDODECACRON . It has W YTHOFF SYMBOL5352½53:Its faces are 12 f52g/C276f10 3g:Its CIRCUM- RADIUS for unit edge length is R/C30f/C281; where fis the GOLDEN RATIO . References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 165, 1989. Great Dodecicosacron The DUAL of the GREAT DODECICOSAHEDRON and Wenninger dual W101 :/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 67, 1983. Great Dodecicosahedron The UNIFORM POLYHEDRON U63whose DUAL is the GREAT DODECICOSACRON . It has WYTHOFF SYMBOL 35 3 ½32 5 2j: Its faces are 20 f6g/C2712 f10 3 g: Its CIRCUMRADIUS for unit edge length is R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34 /C286ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 156 /C1/57, 1989. Great Dodecicosidodecahedron The UNIFORM POLYHEDRON U61whose DUAL is the GREAT DODECACRONIC HEXECONTAHEDRON . Its WYTH- OFF SYMBOL is 25 2½3: Its faces are 20f6g/C2712f52 g; andits CIRCUMRADIUS for unit edge length is R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 58 /C2818ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 148, 1989. Great Hexacronic Icositetrahedron The DUAL of the GREAT CUBICUBOCTAHEDRON and Wenninger model W77 :/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 58, 1983. Great Hexagonal Hexecontahedron The DUAL of the GREAT SNUB DODECICOSIDODECAHE- DRON and Wenninger dual W115 :/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 1356 1983. Great Icosacronic Hexecontahedron The DUAL of the GREAT ICOSICOSIDODECAHEDRON U48 and Wenninger dual W88:/ See also DUAL POLYHEDRON ,GREAT ICOSICOSIDODE- CAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 65, 1983. Great Icosahedron One of the K EPLER- POINSOT SOLIDS whose DUAL is the GREAT STELLATED DODECAHEDRON . It is also UNIFORM POLYHEDRON U53;Wenninger model W22;and has SCHLA ¨FLI SYMBOL f3;5 2gand W YTHOFF SYMBOL 352½53: Its faces are 20 f3g/C2712f52g/C2712f10 3g:/ The great icosahedron can most easily be constructed by building a "squashed" dodecahedron (top right figure) from the corresponding net (top left). Then,using the net shown in the bottom left figure, build 12 PENTAGRAMMIC PYRAMIDS (bottom middle figure) and affix them into the dimples (bottom right). Thismethod of construction is given in Cundy and Rollett(1989, pp. 98 /C1 /9). If the edge lengths of the dodecahe- dron are unity, then the height of the pentagrammicpyramid (above the dodecahedron faces) is given bysolving the equation for the SLANT HEIGHT of a PENTAGONAL PYRAMID s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C271 10(5/C27ffiffiffi 5p )a2q (1) with a/C301, giving h/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5(5/C272ffiffiffi 5p )q : (2) The distance from the center of the dodecahedron to the apex of a pyramid is then given by H/C30h/C27r/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(25/C2711ffiffiffi 5p )q ; (3) where ris the INRADIUS of the DODECAHEDRON . The dimensions of the pentagrammic pyramid can be by examining a triangular section of the great icosahedron. In this triangle, each side is divided inthe ratios f:1:f;and lines are drawn as shown. Then the light shaded portions on the left and rightcorrespond to sides of two pyramids and the centershaded portion is the "lip" of the pyramid between the first two pyramids. Furthermore, the filled portion of the diagram corresponds to one face of the ICOSAHE- DRON inscribed in the great icosahedron. In the notation of the figure above, ½MP½/C301 10ffiffiffiffiffiffi 15p (4) ½MT2½/C301 2ffiffiffi 3p (5) ½T1T3½/C301 2(ffiffiffi 5p /C281)/C30f/C281 (6) ½CP2½/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5(7/C273ffiffiffi 5p )q (7) ½PA2½/C301 5ffiffiffiffiffiffi 10p : (8) The great icosahedron constructed from the DODECA- HEDRON with unit edge lengths has edge lengths (where edges are interpreted to be broken where facial plane intersect) given by s1/C301 5ffiffiffiffiffiffi 10p (9) s2/C301 (10) s3/C301 2(1/C27ffiffiffi 5p ) (11) s4/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5(7/C273ffiffiffi 5p )q : (12) Its CIRCUMRADIUS is R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(25/C2711ffiffiffi 5p )q ; (13) and the SURFACE AREA and VOLUME are then S/C303ffiffiffi 3p (5/C274ffiffiffi5p ) (14) V /C301 4(25 /C279ffiffiffi 5p ) : (15) The CONVEX HULL of the great icosahedron is a regular ICOSAHEDRON and the dual of the ICOSAHE- DRON is the DODECAHEDRON , so the dual of the great icosahedron is one of the DODECAHEDRON STELLA- TIONS (Wenninger 1983, p. 40) See also GREAT DODECAHEDRON ,GREAT STELLATED DODECAHEDRON ,K EPLER- POINSOT SOLID ,S MALL STELLATED DODECAHEDRON ,TRUNCATED GREAT ICO- SAHEDRON References Cundy, H. and Rollett, A. "The Great Icosahedron. 35 =2 :/" §3.6.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 96 /C1/9, 1989. Fischer, G. (Ed.). Plate 106 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 105, 1986. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 40, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 154, 1989. Great Icosahedron-Great Stellated Dodecahedron Compound A POLYHEDRON COMPOUND of the GREAT ICOSAHE- DRON and GREAT STELLATED DODECAHEDRON most easily constructed by adding the VERTICES OF THE FORM er to the latter. See also GREAT ICOSAHEDRON ,G REAT STELLATED DODECAHEDRON ,POLYHEDRON COMPOUNDReferences Cundy, H. and Rollett, A. "Great Icosahedron Plus Great Stellated Dodecahedron." §3.10.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 132 /C1/33, 1989. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 51 /C1/3 1983. Great Icosicosidodecahedron The UNIFORM POLYHEDRON U48whose DUAL is the GREAT ICOSACRONIC HEXECONTAHEDRON . It has WYTHOFF SYMBOL3 25½3:Its faces are 20 f3g/C2720f6g/C27 12f5g:Its CIRCUMRADIUS for unit edge length is R/C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34/C286ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 137 /C1/39, 1989. Great Icosidodecahedron AUNIFORM POLYHEDRON U54whose DUAL is the GREAT RHOMBIC TRIACONTAHEDRON (also called the GREAT STELLATED TRIACONTAHEDRON ). It is a STELLATED ARCHIMEDEAN SOLID . It has S CHLA ¨FLI SYMBOL3 5 2no / and W YTHOFF SYMBOL 2½35 2:Its faces are 20 f3g/C27 12f5 2g:Its CIRCUMRADIUS for unit edge length is R/C30f/C281; where fis the GOLDEN RATIO . References Cundy, H. and Rollett, A. "Great Icosidodecahedron. (3 /C2155 2)2 /" §3.9.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 124, 1989. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 147, 1989. Great Icosihemidodecacron The DUAL of the GREAT ICOSIHEMIDODECAHEDRON U71 and Wenninger dual W106 : When rendered, the GREAT DODECAHEMIDODECACRON and great icosihemidode- cacron look the same, both consisting of a compound of six infinite f10=3g prisms. See also DUAL POLYHEDRON ,GREAT ICOSIHEMIDODE- CAHEDRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 107, 1983. Great Icosihemidodecahedron The UNIFORM POLYHEDRON U71whose DUAL is the GREAT ICOSIHEMIDODECACRON . It has WYTHOFF SYM- BOL3 2 3½53 : Its faces are 20f3 g/C276f10 3 g: For unit edge length, its CIRCUMRADIUS is R /C30 f/C281 ; where f is the GOLDEN RATIO . References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 164, 1989. Great Inverted Pentagonal Hexecontahedron The DUAL of the GREAT INVERTED SNUB ICOSIDODECA- HEDRON U69 and Wenninger dual W116 :/References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 126, 1983. Great Inverted Retrosnub Icosidodecahedron GREAT RETROSNUB ICOSIDODECAHEDRON Great Inverted Snub Icosidodecahedron The UNIFORM POLYHEDRON U69whose DUAL is the GREAT INVERTED PENTAGONAL HEXECONTAHEDRON .It has WYTHOFF SYMBOL ½2352 : Its faces are 80 f3g/C27 12 f5 2g: For unit edge length, it has CIRCUMRADIUS R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 /C215 22 =3 /C28 16x /C27 21 =3x2 8 /C215 22 =3 /C28 10x /C27 21 =3x2s /C300:816080674799923 ; where x /C13 49 /C2827ffiffiffi 5p /C273ffiffiffi6pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 93 /C2849ffiffiffi 5pq =z1r=z1> 1 =3 : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 179, 1989. Great Pentagonal Hexecontahedron The DUAL of the GREAT SNUB ICOSIDODECAHEDRON U57 and Wenninger dual W113:/ See also DUAL POLYHEDRON ,GREAT SNUB ICOSIDO- DECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 123, 1983. Great Pentagrammic Hexecontahedron The DUAL of the GREAT RETROSNUB ICOSIDODECAHE- DRON and Wenninger dual W117 :/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 128, 1983. Great Pentakis Dodecahedron The DUAL of the SMALL STELLATED TRUNCATED DODE- CAHEDRON U58 and Wenninger dual W97 :/ See also DUAL POLYHEDRON ,S MALL STELLATED TRUNCATED DODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 70, 1983. Great Quasitruncated Icosidodecahedron GREAT TRUNCATED ICOSIDODECAHEDRON Great Retrosnub Icosidodecahedron The UNIFORM POLYHEDRON U74 ; also called the GREAT INVERTED RETROSNUB ICOSIDODECAHEDRON , whoseDUAL is the GREAT PENTAGRAMMIC HEXECONTAHE- DRON . It has WYTHOFF SYMBOL ½23 253: Its faces are 80 f3g/C2712 f5 2 g: For unit edge length, it has CIRCUMRA- DIUS R /C3012ffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28 x 1 /C28 xs :0:5800015 ; where x is the smaller NEGATIVE root of x3 /C272x2 /C28 f /C282 /C300; with f the GOLDEN MEAN . References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 189 /C1/93, 1989. Great Rhombic Triacontahedron A ZONOHEDRON which is the DUAL of the GREAT ICOSIDODECAHEDRON and Wenninger model W94:It is also called the GREAT STELLATED TRIACONTAHE- DRON , and is one of the RHOMBIC DODECAHEDRON STELLATIONS . See also DUAL POLYHEDRON ,GREAT ICOSIDODECAHE- DRON ,RHOMBIC DODECAHEDRON STELLATIONS ,ZONO- HEDRON References Cundy, H. and Rollett, A. "Great Stellated Triacontahedron." V( 3 :5 2)2:/"§3.9.4 in Mathematical Models, 3rd ed. Strad- broke, England: Tarquin Pub., p. 126, 1989. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 54 /C1/5, 1983. Great Rhombicosidodecahedron (Archimedean) The 62-faced ARCHIMEDEAN SOLID A2with faces 30 f4g/C2720 f6g/C2712 f10 g: It is also known as the rhombitruncated icosidodecahedron, and is some- times improperly called the truncated icosidodecahe- dron, a name which is inappropriate since TRUNCATION would yield RECTANGULAR instead of SQUARE . The great rhombicosidodecahedron is also UNIFORM POLYHEDRON U28and Wenninger model W16 : It has SCHLA ¨ FLI SYMBOL t3 5=zr=z> and WYTHOFF SYMBOL 235½:/Its DUAL is the DISDYAKIS TRIACONTAHEDRON , also called the HEXAKIS ICOSAHEDRON . The INRADIUS of the dual, MIDRADIUS of the solid and dual, and CIRCUM- RADIUS of the solid for a /C301 are r /C301 241(105 /C276ffiffiffi 5p )ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31 /C2712ffiffiffi 5pq :3:73665 r /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C2712ffiffiffi 5pq :3:76938 R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31/C2712ffiffiffi 5pq :3:80239 : See also SMALL RHOMBICOSIDODECAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 137, 1987. Cundy, H. and Rollett, A. "Great Rhombicosidodecahedron or Truncated Icosidodecahedron. 4 :6:10:/"§3.7.12 in Math- ematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 112 /C1/13, 1989. Wenninger, M. J. "The Rhombitruncated Icosidodecahe- dron." Model 16 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 30, 1989. Great Rhombicosidodecahedron (Uniform) The UNIFORM POLYHEDRON U67;also called the QUA- SIRHOMBICOSIDODECAHEDRON , whose DUAL is the GREAT DELTOIDAL HEXECONTAHEDRON . It has S CHLA ¨- FLI SYMBOL r’3 5 2no :It has W YTHOFF SYMBOL 35 2½2:Its faces are 20 f3g/C2730f4g/C2712f52g:For unit edge length, itsCIRCUMRADIUS is R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11/C284ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 162 /C1/63, 1989. Great Rhombicuboctahedron (Archimedean) The 26-faced ARCHIMEDEAN SOLID A3consisting of faces 12f4 g/C278f6 g/C276f8 g: It is sometimes (impro- perly) called the truncated cuboctahedron, and is also called the rhombitruncated cuboctahedron. It is UNI- FORM POLYHEDRON U11 and Wenninger model W15 : It has SCHLA ¨ FLI SYMBOL t3 4=zr=z> and WYTHOFF SYMBOL 234½:/ The SMALL CUBICUBOCTAHEDRON is a FACETED ver- sion of the great rhombicuboctahedron. Its DUAL is the DISDYAKIS DODECAHEDRON , also called the HEXAKIS OCTAHEDRON . The INRADIUS r of the dual, MIDRADIUS r of the solid and dual, and CIRCUM- RADIUS R of the solid for a /C301 are r /C303 97(14 /C27ffiffiffi 2p )ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 /C276ffiffiffi 2pq :2 :20974 r /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 /C276ffiffiffi 2pq :2:26303 R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 /C276ffiffiffi 2pq :2 :31761 : Additional quantities aret /C30tan(1 8 p) /C30ffiffiffi 2p /C281 l /C302t /C302(ffiffiffi 2p /C281) h /C301 /C27l sin(1 4 p) /C303 /C28ffiffiffi 2p : The distances between the solid center and centroids of the square and octagonal faces are r4/C301 2(3/C27ffiffiffi 2p ) (1) r8/C301 2(1/C272ffiffiffi 2p ): (2) The SURFACE AREA and VOLUME are S/C3012(2/C27ffiffiffi2p /C27ffiffiffi 3p ) (3) V/C3022/C2714ffiffiffi 2p : (4) See also A RCHIMEDEAN SOLID ,GREAT RHOMBICUBOC- TAHEDRON (UNIFORM ), GREAT TRUNCATED CUBOCTA- HEDRON ,S MALL RHOMBICUBOCTAHEDRON , OCTATETRAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 138, 1987. Cundy, H. and Rollett, A. "Great Rhombicuboctahedron or Truncated Cuboctahedron. 4 :6:8:/"§3.7.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 106, 1989. Wenninger, M. J. "The Rhombitruncated Cuboctahedron." Model 15 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 29, 1989. Great Rhombicuboctahedron (Uniform) The UNIFORM POLYHEDRON U17;also known as the QUASIRHOMBICUBOCTAHEDRON , whose DUAL is the GREAT DELTOIDAL ICOSITETRAHEDRON . It has S CHLA ¨- FLI SYMBOL r’f3 4g;WYTHOFF SYMBOL324½2;and is Wenninger model W85:Its faces are 18 f4g/C278f3=2g: Its CIRCUMRADIUS for unit edge length is R/C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 2pq : The CONVEX HULL of the great cubicuboctahedron is the Archimedean TRUNCATED CUBE A9 ; whose dual is the SMALL TRIAKIS OCTAHEDRON , so the dual of the great rhombicuboctahedron (i.e., the GREAT DELTOI- DAL ICOSITETRAHEDRON ) is one of the stellations of the SMALL TRIAKIS OCTAHEDRON (Wenninger 1983, p. 57). References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 57 and 59, 1983. Wenninger, M. J. Model 85 in Polyhedron Models. Cam- bridge, England: Cambridge University Press, pp. 132 /C1/ 33, 1989. Great Rhombidodecacron The DUAL of the GREAT RHOMBIDODECAHEDRON U73 and Wenninger dual W109:/ See also DUAL POLYHEDRON ,GREAT RHOMBIDODECA- HEDRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 88, 1983.Great Rhombidodecahedron The UNIFORM POLYHEDRON U73whose DUAL is the Great Rhombidodecacron. It has W YTHOFF SYMBOL 25 3½32 5 4j: Its faces are 30 f4g/C2712f10 3g:Its CIRCUMRADIUS for unit edge length is R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11/C284ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 168 /C1/70, 1989. Great Rhombihexacron The DUAL of the GREAT RHOMBIHEXAHEDRON U21and Wenninger dual W103:/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 60, 1983. Great Rhombihexahedron The UNIFORM POLYHEDRON U21whose DUAL is the GREAT RHOMBIHEXACRON . It is Wenninger model W103:Maeder gives its W YTHOFF SYMBOL as4 3322½; and its faces as 6 f4g/C273f8 3g/C273f85g/C276f43g;while Wen- ninger (1989) gives the W YTHOFF SYMBOL as 2433 2 4 2j and its faces as 12 f4g/C276f83g:The CIRCUMRADIUS for a great rhombihexahedron of unit edge length is R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 2pq : The CONVEX HULL of the great rhombihexahedron is the Archimedean TRUNCATED CUBE A9;whose dual is the SMALL TRIAKIS OCTAHEDRON , so the dual of the great rhombihexahedron (i.e., the GREAT RHOMBIHEX- ACRON ) is one of the stellations of the SMALL TRIAKIS OCTAHEDRON (Wenninger 1983, p. 57). References Maeder, R. E. Polyhedra.m andPolyhedraExamples Mathematica notebooks. http://www.inf.ethz.ch/depart- ment/TI/rm/programs.html. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 57 and 160, 1983. Wenninger, M. J. "Great Rhombihexahedron." Model 103 in Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 159 /C1/60, 1989.Great Snub Dodecicosidodecahedron The UNIFORM POLYHEDRON U64whose DUAL is the GREAT HEXAGONAL HEXECONTAHEDRON . It has W YTH- OFF SYMBOL ½35 352:Its faces are 80 f3g/C2724f52g:Its CIRCUMRADIUS for unit edge length is R/C301 2ffiffiffi 2p : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 183 /C1/85, 1989. Great Snub Icosidodecahedron The UNIFORM POLYHEDRON U57whose DUAL is the GREAT PENTAGONAL HEXECONTAHEDRON . It has WYTHOFF SYMBOL ½235 3:Its faces are 80 f3g/C2712f52g: For unit edge length, it has CIRCUMRADIUS R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffi 2/C28x 1/C28xs :0:6450202 ; where xis the most NEGATIVE ROOT of x3/C272x2/C28f/C282/C300; with fthe GOLDEN RATIO . References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 186 /C1/88, 1989. Great Sphere The great sphere on the surface of a HYPERSPHERE is the 3-D analog of the GREAT CIRCLE on the surface of a SPHERE . Let 2 hbe the number of reflecting SPHERES , and let great spheres divide a HYPERSPHERE into g4- DTETRAHEDRA . Then for the POLYTOPE with S CHLA ¨- FLI SYMBOL fp; q; r g; 64h g/C3012 /C28p /C282q /C28r /C274 p /C274 r: See also GREAT CIRCLE Great Stellapentakis Dodecahedron The DUAL of the GREAT TRUNCATED ICOSAHEDRON U55 and Wenninger dual W95 :/ See also DUAL POLYHEDRON ,G REAT TRUNCATED ICOSAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 75, 1983. Great Stellated Dodecahedron One of the KEPLER- POINSOT SOLIDS . It is also UNI- FORM POLYHEDRON U52 ; Wenninger model W41 ; and is the third DODECAHEDRON STELLATION (Wenninger 1989). Its DUAL is the GREAT ICOSAHEDRON . The great stellated dodecahedron has SCHLA ¨ FLI SYMBOL f5 2 ; 3g and WYTHOFF SYMBOL 3½25 2 : Its faces are 12 f52g: Its CIRCUMRADIUS for unit edge length is R /C301 2ffiffiffi 3p f/C281 /C301 4ffiffiffi 3p (ffiffiffi5p /C281): (1) The easiest way to construct a great stellated dode- cahedron is by CUMULATION , i.e., to making 20 TRIANGULAR PYRAMIDS with side length f /C30 (1 /C27ffiffiffi 5p )=2 (the GOLDEN RATIO ) times the base and attaching them to the sides of an ICOSAHEDRON . The height of these pyramids is thenffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 6(7 /C273ffiffiffi 5p )q :/ Cumulating a DODECAHEDRON to construct a great stellated dodecahedron produces a solid with edge lengths s1 /C301 (2) s2 /C30 f /C301 2(1 /C27ffiffiffi 5p ): (3) The SURFACE AREA and VOLUME of such a great stellated dodecahedron are S /C3015ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq (4) V /C305 4(3 /C27ffiffiffi 5p ): (5) The CONVEX HULL of the great stellated dodecahedron is a regular DODECAHEDRON and the dual of the DODECAHEDRON is the ICOSAHEDRON , so the dual of the great stellated dodecahedron (i.e., the GREAT ICOSAHEDRON ) is one of the ICOSAHEDRON STELLA- TIONS (Wenninger 1983, p. 40) See also DODECAHEDRON ,D ODECAHEDRON STELLA- TIONS ,GREAT DODECAHEDRON ,GREAT ICOSAHEDRON , GREAT STELLATED TRUNCATED DODECAHEDRON ,KE- PLER- POINSOT SOLID ,SMALL STELLATED DODECAHE- DRON ,STELLATION References Cundy, H. and Rollett, A. "Great Stellated Dodecahedron. (5 2)3:/"§3.6.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 94 /C1/5, 1989. Fischer, G. (Ed.). Plate 104 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 103, 1986. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 39 /C1/0, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 35 and 40, 1989. Great Stellated Triacontahedron GREAT RHOMBIC TRIACONTAHEDRON Great Stellated Truncated Dodecahedron The UNIFORM POLYHEDRON U66 ; also called the QUASI- TRUNCATED GREAT STELLATED DODECAHEDRON , whose DUAL is the GREAT TRIAKIS ICOSAHEDRON . It has SCHLA ¨ FLI SYMBOL t’ f5 2 ; 3 g and WYTHOFF SYMBOL 23½5 3: Its faces are 20 f3g/C2712f10 3 g: Its CIRCUMRADIUS for unit edge length is R /C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 74 /C2830ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 161, 1989. Great Triakis Icosahedron The DUAL of the GREAT STELLATED TRUNCATED DODE- CAHEDRON U66 and Wenninger dual W104 :/ See also DUAL POLYHEDRON ,G REAT STELLATED TRUNCATED DODECAHEDRONReferences Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 77, 1983. Great Triakis Octahedron The DUAL of the STELLATED TRUNCATED HEXAHEDRON U19 and Wenninger dual W92/ See also DUAL POLYHEDRON ,SMALL TRIAKIS OCTAHE- DRON ,STELLATED TRUNCATED HEXAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 57, 1983. Great Triambic Icosahedron The DUAL of the GREAT DITRIGONAL ICOSIDODECAHE- DRON U47and Wenninger model /W87/whose appear- ance is the same as the MEDIAL TRIAMBIC ICOSAHEDRON (the dual of the DITRIGONAL DODECA- DODECAHEDRON ), since internal vertices are hidden from view (Wenninger 1983, p. 42). The MEDIAL TRIAMBIC ICOSAHEDRON has hidden pentagrammic faces, while the great triambic icosahedron has hidden triangular faces (Wenninger 1983, pp. 45, 47, and 48 /C1/0). The CONVEX HULL of the GREAT DITRIGONAL ICOSIDO- DECAHEDRON is a regular DODECAHEDRON , whose dual is the ICOSAHEDRON , so the dual of the GREAT DITRIGONAL ICOSIDODECAHEDRON (the great triambic icosahedron) is one of the ICOSAHEDRON STELLATIONS (Wenninger 1983, p. 42). See also DUAL POLYHEDRON ,G REAT DITRIGONAL ICOSIDODECAHEDRON ,ICOSAHEDRON STELLATIONS , MEDIAL TRIAMBIC ICOSAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 41 and 46, 1983. Wenninger, M. J. "Ninth Stellation of the Icosahedron." §34 in Polyhedron Models. New York: Cambridge University Press, p. 55, 1989. Great Truncated Cuboctahedron The UNIFORM POLYHEDRON U20 ; also called the quasi- truncated cuboctahedron, whose DUAL is the GREAT DISDYAKIS DODECAHEDRON . Its faces consist of 8f6 g/C27 12 f4g/C276 f8 3g: It has SCHLA ¨ FLI SYMBOL t’ f34 g and WYTHOFF SYMBOL4 3 23½: Its CIRCUMRADIUS for unit edge length is R /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 /C286ffiffiffi 2pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 145 /C1/46, 1989. Great Truncated Icosahedron The UNIFORM POLYHEDRON U55 ; also called the TRUN- CATED GREAT ICOSAHEDRON , whose DUAL is the GREAT STELLAPENTAKIS DODECAHEDRON . It has SCHLA ¨ FLI SYMBOL t f3;5 2g and WYTHOFF SYMBOL 252 ½3: Its faces are 20f6 g/C2712 f5 2g: Its CIRCUMRADIUS for unit edge length isR /C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 58 /C2818ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 148, 1989. Great Truncated Icosidodecahedron The UNIFORM POLYHEDRON U68 ; also called the GREAT QUASITRUNCATED ICOSIDODECAHEDRON , whose DUAL is the GREAT DISDYAKIS TRIACONTAHEDRON . It has SCHLA ¨ FLI SYMBOL t?3 5 2no / and WYTHOFF SYMBOL 235 3 j: Its faces are 20f6 g/C2730f4g/C2712 f10 3 g: Its CIRCUMRA- DIUS for unit edge length is R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31 /C2812ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 166 /C1/67, 1989. Greater A quantity a is said to be greater than b if a is larger than b, written a /C21 b.Ifa is greater than or EQUAL to b, the relationship is written a ]b : If a is MUCH GREATER than b, this is written a /C27b : Statements involving greater than and LESS than symbols are called INEQUALITIES . See also EQUAL ,GREATER THAN/ LESS THAN SYMBOL , INEQUALITY ,LESS,MUCH GREATER Greater Than/Less Than Symbol When applied to a system possessing a length R at which solutions in a variable r change character (such as the gravitational field of a sphere as r runs from the interior to the exterior), the symbols r/C21/C13max( r;R) rB/C13min( r;R) are sometimes used. See also EQUAL ,GREATER ,LESS Greatest Common Denominator GREATEST COMMON DIVISOR Greatest Common Divisor The greatest common divisor GCD( a;b) of two positive integers aand b, sometimes written ( a, b), is the largest DIVISOR common to aand b. For example, GCD(3 ;5)/C301;GCD(12 ;60)/C3012;and GCD(12 ;90)/C306:The greatest common divisor GCD( a;b;c;. . .) can also be defined for three or more positive integers as the largest divisor shared by all of them. The plot above shows GCD(1 ;b) with rational b/C30m=n:/ The greatest common divisor of aand bis imple- mented in Mathematica asGCD[a,b, ...]. Ifdis the greatest common divisor of aandb, then d is the largest possible integer satisfying a/C30dx (1) b/C30dy (2) with xandypositive integers. Therefore, there exists anINTEGER RELATION between aandbOF THE FORM ay/C28bx/C300: (3) The E UCLIDEAN ALGORITHM can be used to find the greatest common divisor of two integers. The notion can also be generalized to more general RINGS than simply the integers Z:However, even for EUCLIDEAN RINGS , the notion of GCD of two elements of a ring is not the same as the GCD of two ideals of a ring. This is sometimes a source of confusion whenstudying rings other than Z;such as polynomial rings in several variables. To compute the GCD, write the PRIME FACTORIZA- TIONS ofaandb, a/C13Y ipai i (4) b/C13Y ipbi i; (5) where the pi/s are all PRIME FACTORS ofaandb, and if pidoes not occur in one factorization, then the corresponding exponent is taken as 0. Then the greatest common divisor GCD( a;b) is given by GCD( a;b)/C30Y ipmin( ai;bi) i ; (6) where min denotes the MINIMUM . For example, con-sider GCD(12 ;30): 12/C3022/C21531/C21550(7) 30/C3021/C21531/C21551; (8) so GCD(12 ;30)/C3021/C21531/C21550/C306: (9) The GCD is DISTRIBUTIVE GCD( ma;mb)/C30mGCD( a;b) (10) GCD( ma;mb;mc)/C30mGCD( a;b;c); (11) and ASSOCIATIVE GCD( a;b;c)/C30GCD(GCD( a;b);c) /C30GCD( a;GCD( b;c)) (12) GCD( ab;cd)/C30GCD( a;c)GCD( b;d) /C29GCDa GCD( a;c);d GCD( b;d) ! /C29GCDc GCD( a;c);b GCD( b;d) ! :(13) Ifa/C30a1GCD( a;b) and b/C30b1GCD( a;b);then GCD( a;b)/C30GCD( a1GCD( a;b);b1GCD( a;b)) /C30GCD( a;b) GCD( a1;b1); (14) so GCD( a1;b1)/C301 and a1and b1are said to be RELATIVELY PRIME . The GCD is also IDEMPOTENT GCD( a;a)/C30a; (15) COMMUTATIVE GCD( a;b)/C30GCD( b;a); (16) and satisfies the ABSORPTION LAW LCM( a;GCD( a;b))/C30a: (17) The probability that two INTEGERS picked at random are RELATIVELY PRIME is [z(2)]/C281/C306=p2;where z(z)i s the R IEMANN ZETA FUNCTION . Polezzi (1997) observed that GCD( m;n)/C30k;where kis the number of LATTICE POINTS in the PLANE on the straight LINE connecting the VECTORS (0, 0) and ( m, n ) (excluding (m, n ) itself). This observation is intimately con- nected with the probability of obtaining RELATIVELY PRIME integers, and also with the geometric inter- pretation of a REDUCED FRACTION y=xas a string through a LATTICE of points with ends at (1,0) and ( x, y). The pegs it presses against ( xi;yi) give alternate CONVERGENTS yi=xiof the CONTINUED FRACTION for y=x;while the other CONVERGENTS are obtained from the pegs it presses against with the initial end at (0, 1). Knuth showed that gcd(2p /C281; 2q /C281) /C302gcd(p ; q) /C281: (18) The extended greatest common divisor of two INTE- GERS m and n can be defined as the greatest common divisor GCD( m; n)ofm and n which also satisfies the constraint GCD( m; n) /C30rm /C27sn for r and s given INTEGERS . It is used in solving LINEAR DIOPHANTINE EQUATIONS . See also BE´ ZOUT NUMBERS ,B E´ ZOUT’S THEOREM , DIRICHLET FUNCTION ,E UCLID’S ORCHARD ,E UCLI- DEAN ALGORITHM ,GAUSS’S LEMMA ,LEAST COMMON MULTIPLE ,LEAST PRIME FACTOR ,ORCHARD- PLANTING PROBLEM ,STAR OF DAVID THEOREM References Nagell, T. "Least Common Multiple and Greatest Common Divisor." §5inIntroduction to Number Theory. New York: Wiley, pp. 16 /C1/9, 1951. Polezzi, M. "A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor." Amer. Math. Monthly 104, 445 /C1/46, 1997. Se´roul, R. "The Greatest Common Divisor." §2.4 in Program- ming for Mathematicians. Berlin: Springer-Verlag, pp. 9 /C1/ 1, 2000. Greatest Common Divisor Theorem Given m and n, it is possible to choose c and d such that cm /C27dn is a common factor of m and n. Greatest Common Factor GREATEST COMMON DIVISOR Greatest Dividing Exponent The greatest dividing exponent gde(n; b) of a base b with respect to a number n is the largest integer value of k such that bk n;j where bk 5n : It is implemented as the Mathematica command Inte- gerExponent [n, b]. See also DIVIDE ,EVEN PART,ODD PART Greatest Integer Function FLOOR FUNCTION Greatest Lower Bound INFIMUMGreatest Prime Factor For an INTEGER n ]2; let gpf(x) denote the greatest prime factor of n, i.e., the number pk in the factoriza- tion n /C30pa1 1...pak k; with pi Bpj for i B j. For n /C302, 3, ..., the first few are 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, ... (Sloane’s A006530). The greatest multiple prime factors for SQUAREFUL integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 3, ... (Sloane’s A046028). The probability that the GREATEST PRIME FACTOR of a RANDOM integer n is greater thanffiffiffinpis ln 2 (Schroeppel 1972). See also DICKMAN FUNCTION ,DISTINCT PRIME FAC- TORS ,F ACTOR ,L EAST COMMON MULTIPLE ,L EAST PRIME FACTOR ,M ANGOLDT FUNCTION ,PRIME FAC- TORS ,TWIN PEAKS References Erdos, P. and Pomerance, C. "On the Largest Prime Factors ofnandn/C271:/"Aequationes Math. 17, 211/C1/21, 1978. Guy, R. K. "The Largest Prime Factor of n."§B46 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 101, 1994. Heath-Brown, D. R. "The Largest Prime Factor of the Integers in an Interval." Sci. China Ser. A 39, 449/C1/76, 1996. Mahler, K. "On the Greatest Prime Factor of axm/C27byn:/" Nieuw Arch. Wiskunde 1, 113/C1/22, 1953. Schroeppel, R. Item 29 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 13, Feb. 1972. Sloane, N. J. A. Sequences A006530/M0428 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Grebe Point SYMMEDIAN POINT Greedy Algorithm An algorithm used to recursively construct a SETof objects from the smallest possible constituent parts. Given a SETofkINTEGERS (/a1;a2;...,ak) with a1B a2B...Bak;a greedy algorithm can be used to find a VECTOR of coefficients (/c1 ; c2 ; ..., ck) such that Xk i /C301ciai /C30c /C215 a /C30n ; (1) where c /C215 a is the DOT PRODUCT , for some given INTEGER n. This can be accomplished by letting ci /C30 0 for i /C301, ..., k /C281 and setting ck /C30n ak$% ; (2) where xbcis the floor function. Now define the difference between the representation and n as D/C13n /C28c /C215 a : (3) If D/C300 at any step, a representation has been found. Otherwise, decrement the NONZERO ai term with least i, set all aj /C300 for j B i, and build up the remaining terms from cj /C30Dj ak"# (4) for j /C30i /C281 ; ..., 1 until D/C300 or all possibilities have been exhausted. For example, MCNUGGET NUMBERS are numbers which are representable using only (a1 ; a2 ; a3) /C30 (6; 9; 20): Taking n /C3062 and applying the algorithm iteratively gives the sequence (0, 0, 3), (0, 2, 2), (2, 1, 2), (3, 0, 2), (1, 4, 1), at which point D/C300: 62 is therefore a MCNUGGET NUMBER with 62 /C30(1 /C215 6) /C27(4 /C215 9) /C27(1 /C215 20): (5) If any INTEGER n can be represented with ci /C300or1 using a sequence (/a1 ; a2 ; ...), then this sequence is called a COMPLETE SEQUENCE . A greedy algorithm can also be used to break down arbitrary fractions into UNIT FRACTIONS in a finite number of steps. For a FRACTION a =b; find the least INTEGER x1 such that 1=x1 5a=b ; i.e., x1 /C30bde a; (6) where xdeis the CEILING FUNCTION . Then find the least INTEGER x2such that 1=x2 5a=b /C281=x1 : Iterate until there is no remainder. The ALGORITHM gives two or fewer terms for 1=n and 2 =n; three or fewer terms for 3 =n; and four or fewer for 4 =n:/ See also COMPLETE SEQUENCE ,INTEGER RELATION , LEVINE- O’SULLIVAN GREEDY ALGORITHM ,MCNUGGET NUMBER ,R EVERSE GREEDY ALGORITHM ,S QUARE NUMBER ,SYLVESTER’S SEQUENCE ,UNIT FRACTIONGreek Cross An irregular DODECAHEDRON CROSS in the shape of a PLUS SIGN. See also CROSS ,DISSECTION ,DODECAHEDRON ,LATIN CROSS ,PLUS SIGN,SAINT ANDREW’S CROSS References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 89, 1991. Greek Problems GEOMETRIC PROBLEMS OF ANTIQUITY Green Space A G-SPACE provides local notions of harmonic, hyper- harmonic, and superharmonic functions. When there exists a nonconstant superharmonic function greater than 0, it is a called a Green space. Examples are Rn (for n ]3) and any bounded domain of Rn :/ See also G-SPACE Green’s Function A Green’s function is an integrating kernal which can be used to solve an inhomogeneous differential equation with boundary conditions. It serves roughlyan analogous role in partial differential equations as does F OURIER ANALYSIS in the solution of ordinary differential equations. As a special case, consider the 1-D DIFFERENTIAL OPERATOR ˜L/C30˜Dn/C27an/C281(t)˜Dn/C281/C27.../C27a1(t)˜D/C27a0(t); (1) with ai(t)CONTINUOUS fori/C300, 1, ..., n/C281 on the interval I, and assume we wish to find the solution y(t) to the equation ˜Ly(t)/C30h(t); (2) where h(t) is a given CONTINUOUS FUNCTION onI.T o solve equation (2), we look for a function g:Cn(I)/C2 C(I) such that ˜L(g(h))/C30h;where y(t)/C30g(h(t)): (3) This is a CONVOLUTION equation OF THE FORM y/C30g+h; (4) so the solution is y(t)/C30gt t0g(t/C28x)h(x)dx; (5) and the function g(t) is called the Green’s function for ˜L on I. Now, note that if we take h(t) /C30 d(t); then y(t) /C30gt t0g(t /C28x)d(x) dx /C30g(t) ; (6) so the Green’s function g(t) can be defined by ˜Lg(t) /C30 d(t): (7) However, the Green’s function is determined un- iquely only if some initial or boundary conditions are given. For an arbitrary linear differential operator ˜L in 3-D, the Green’s function G(r ; r ?) is defined by analogy with the 1-D case by ˜LG(r; r?) /C30 d(r /C28r?) : (8) The solution to ˜Lf /C30f is then f(r) /C30g G(r; r?)f(r?)d3r ?: (9) Explicit expressions for G(r; r?) can often be found in terms of a basis of given eigenfunctions fn(r1)by expanding the Green’s function G(r1 ; r2 ) /C30X/C12 n/C300an(r2)fn(r1) (10) and DELTA FUNCTION , d3(r1 /C28r2) /C30X/C12 n/C300bn fn(r1): (11) Multiplying both sides by fm(r2) and integrating over r1 space, g fm(r2) d3(r1 /C28r2)d3r1 /C30X/C12 n/C300bng fm(r2)fn(r1)d3r1 (12) fm(r2) /C30X/C12 n /C300bn dnm /C30bm ; (13) so d3(r1 /C28r2) /C30X/C12 n/C300fn(r1) fn(r2) : (14) By plugging in the differential operator, solving for the an/s, and substituting into G, the original non- homogeneous equation then can be solved. The coefficient S of ln(1 =r) in all normalized funda- mental Green’s function solutions f(x; y; x0 ; y0) /C30S(x; y; x0 ; y0) ln(1 =r) /C27T(x; y; x0 ; y0) (15) withr /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x /C28x0)2 /C27(y /C28y0)2q (16) of the ELLIPTIC PARTIAL DIFFERENTIAL EQUATION Ku /C30uxx /C27vyy /C27A(x; y)ux /C27B(x; y)uy /C27C(x; y)u /C300 (17) with analytic coefficients is an analytic function of four variables and is equal to the RIEMANN FUNCTION S /C30R/C31( j; h; j0 ; h0) of the conjugate equation K /C31v /C30v( j; h) /C28(av)(j) /C28(bv)( h) /C27cv /C300 (18) which can be produced from Ku/C300 by the change of variables j/C30x/C27iy (19) h/C30x/C28iy (20) j0/C30x0/C27iy0 (21) h0/C30x0/C28iy0 (22) 4a(j;h)/C30A(x;y)/C27iB(x;y) (23) 4b(j;h)/C30A(x;y)/C28iB(x;y) (24) 4c(j;h)/C30C(x;y) (25) (Garabedian 1964, Marichev 1990). See also GREEN’S FUNCTION– HELMHOLTZ DIFFEREN- TIAL EQUATION ,GREEN’S FUNCTION– POISSON’S EQUA- TION ,RIEMANN METHOD References Arfken, G. "Nonhomogeneous Equation--Green’s Function," "Green’s Functions--One Dimension," and "Green’s Func- tions--Two and Three Dimensions." §8.7 and §16.5/C1/6.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 480 /C1/91 and 897 /C1/24, 1985. Garabedian, P. R. Partial Differential Equations. New York: Wiley, 1964. Marichev, O. I. "Funktionen vom hypergeometrischen Typ und einige Anwendungen auf Integral- under Differen-tialgleichungen." Ph.D. dissertation. Jena, Germany: Frie-drich-Schiller-Universita ¨t, p. 266, 1990. Green’s Function * /Helmholtz Differential Equation The inhomogeneous H ELMHOLTZ DIFFERENTIAL EQUA- TION is 92c(r)/C27k2c(r)/C30r(r); (1) where the Helmholtz operator is defined as ˜L/C1392/C27 k2:The Green’s function is then defined by (92/C27k2)G(r1;r2)/C30d3(r1/C28r2): (2) Define the basis functions fnas the solutions to the homogeneous H ELMHOLTZ DIFFERENTIAL EQUATION 92fn(r)/C27k2 nfn(r)/C300: (3) The Green’s function can then be expanded in terms of the fn/s, G(r1;r2)/C30X/C12 n/C300an(r2)fn(r1); (4) and the DELTA FUNCTION as d3(r1/C28r2)/C30X/C12 n/C300fn(r1)fn(r2): (5) Plugging (4) and (5) into (2) gives 92X/C12 n/C300an(r2)fn(r1)"# /C27k2X/C12 n/C300an(r2)fn(r1) /C30X/C12 n/C300fn(r1)fn(r2): (6) Using (3) gives /C28X/C12 n/C300an(r2)k2 nfn(r1)/C27k2X/C12 n/C300an(r2)fn(r1) /C30X/C12 n/C300fn(r1)fn(r2) (7) X/C12 n/C300an(r2)fn(r1)(k2/C28k2n)/C30X/C12 n/C300fn(r1)fn(r2): (8) This equation must hold true for each n,s o an(r2)fn(r1)(k2/C28k2n)/C30fn(r1)fn(r2) (9) an(r2)/C30fn(r2) k2/C28k2 n; (10) and (4) can be written G(r1;r2)/C30X/C12 n/C300fn(r1)fn(r2) k2/C28k2 n: (11) The general solution to (1) is therefore c(r1)/C30gG(r1;r2)r(r2)d3r2 /C30X/C12 n/C300gfn(r1)fn(r2)r(r2) k2/C28k2 nd3r2: (12) References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 529 /C1/30, 1985. Green’s Function * /Poisson’s Equation POISSON’S EQUATION equation is 92f/C304pr; (1) where fis often called a potential function and radensity function, so the differential operator in this case is ˜L/C3092:As usual, we are looking for a Green’s function G(r1;r2) such that 92G(r1;r2)/C30d3(r1;r2): (2) But from L APLACIAN , 92 1 r/C28r? jj ! /C30/C284pd3(r/C28r?); (3) so G(r;r?)/C30/C281 4pr/C28r? jj; (4) and the solution is f(r)/C30gG(r;r?)[4pr(r?)]d3r?/C30/C28gr(r?)d3r? r/C28r? jj:(5) Expanding G(r1;r2) in the SPHERICAL HARMONICS Ym l gives G(r1;r2) /C30X/C12 l/C300Xl m/C30/C28l1 2l/C271rl B rl/C271 >Ym l(u1;f1)˜Ym t(u2;f2);(6) where rBand r>are GREATER THAN/LESS THAN SYMBOLS . this expression simplifies to g(r1;r2)/C301 4pX/C12 l/C300rl B rl/C271 >pl(cosg); (7) where plare L EGENDRE POLYNOMIALS , and cos g/C13 r1/C215r2:Equations (6) and (7) give the addition theorem for L EGENDRE POLYNOMIALS . InCYLINDRICAL COORDINATES , the Green’s function is much more complicated, G(r1;r2)/C301 2p2X/C12 m/C30/C28/C12g/C12 0Im(krB)Km /C2(kr>)eim(f1/C28f2)cos[k(z1/C28z2)]dk: (8) where Im(x) and Km(x) are MODIFIED BESSEL FUNC- TIONS OF THE FIRST and SECOND KINDS (Arfken 1985). References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 485 /C1/86, 905, and 912, 1985. Green’s Identities Green’s identities are a set of three vector derivative/ integral identities which can be derived starting withthe vector derivative identities 9 /C215(c9f)/C30c9 2f/C27(9c)/C215(9f) (1) and 9 /C215 ( f 9c) /C30 f 92 c /C27( 9f) /C215( 9c) ; (2) where 9/C215 is the DIVERGENCE , 9 is the GRADIENT , 92 is the LAPLACIAN , and a /C215 b is the DOT PRODUCT . From the DIVERGENCE THEOREM , gV( 9/C215F) dV /C30gSF /C215 da: (3) Plugging (2) into (3), gSf( 9c) /C215 da /C30gV[f 92 c /C27( 9f) /C215(9 c)] dV : (4) This is Green’s first identity. Subtracting (2) from (1), 9 /C215 ( f9 c /C28 c9 f) /C30 f 92 c /C28 c92 f: (5) Therefore, gV( f92 c /C28 c92 f) dV /C30gS(f 9c /C28 c9 f) /C215 da: (6) This is Green’s second identity. Let u have continuous first PARTIAL DERIVATIVES and be HARMONIC inside the region of integration. Then Green’s third identity is u(x; y) /C301 2p GCln1 r ! @u @n /C28u@ @nln1 r ! "# ds (7) (Kaplan 1991, p. 361). References Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1991. Green’s Theorem Green’s theorem is a vector identity which is equiva- lent to the CURL THEOREM in the PLANE . Over a region D in the plane with boundary @D ; g@Df(x ; y) dx /C27g(x; y) dy /C30ggD@g @x /C28@f @y ! dx dy g@DF /C215 ds /C30ggD( 9/C29F) /C215 k dA: If the region D is on the left when traveling around @D ; then AREA of D can be computed using A /C301 2g@Dxdy/C28ydx : See also CURL THEOREM ,DIVERGENCE THEOREM References Arfken, G. "Gauss’s Theorem." §1.11 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57 /C1/1, 1985.Greene’s Method A method for predicting the onset of widespread CHAOS . It is based on the hypothesis that the dissolu- tion of an invariant torus can be associated with the sudden change from stability to instability of nearly closed orbits (Tabor 1989, p. 163). See also OVERLAPPING RESONANCE METHOD References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Greenwood-Gleason Graph Kalbfleisch and Stanton (1968) showed that in a 3- edge coloring of the COMPLETE GRAPH K16without monochromatic triangles, the subgraph induced bythe edges of any one color is isomorphic to the graphillustrated above, known as the Greenwood-Gleasongraph. References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 242, 1976. Kalbfleisch, J. and Stanton, R. "On the Maximal Triangle- Free Edge-Chromatic Graph in Three Colors." J. Combin. Th.5,9/C1/0, 1968. Gregory Number A number tx/C30tan/C2811 x ! /C30cot/C281x; where xis an INTEGER orRATIONAL NUMBER , tan/C281x is the INVERSE TANGENT , and cot/C281xis the INVERSE COTANGENT . Gregory numbers arise in the determi- nation of M ACHIN-LIKE FORMULAS . Every Gregory number txcan be expressed uniquely as a sum of tn/s where the ns are STøRMER NUMBERS . References Conway, J. H. and Guy, R. K. "Gregory’s Numbers" In The Book of Numbers. New York: Springer-Verlag, pp. 241 /C1/ 42, 1996. Gregory’s Formula There are at least two formulas associated with Gregory. The first is a series PI FORMULA found by Gregory and Leibniz and obtained by plugging x /C301 into the LEIBNIZ SERIES , p 4 /C301 /C281 3 /C2715 /C27/C1/C1/C1 (Wells 1986, p. 50). The formula, also called the L EIBNIZ SERIES , converges very slowly, but its con- vergence can be accelerated using certain transfor- mations, in particular p /C30X/C12 k /C3013k /C28 1 4kz(k /C271); where z(z) is the RIEMANN ZETA FUNCTION (Vardi 1991). The second is the formula gy 0p(u)du /C30X k]0(eyt /C28 1)k p(x) ji k!(et /C281)kp(x) ;* discovered by Gregory in 1670 and reported to be the earliest formula in NUMERICAL INTEGRATION (Jordan 1950, Roman 1984). See also LEIBNIZ SERIES ,M ACHIN’S FORMULA ,M A- CHIN- LIKE FORMULAS ,N UMERICAL INTEGRATION ,PI FORMULAS References Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, p. 284, 1965. Roman, S. The Umbral Calculus. New York: Academic Press, p. 59, 1984. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 157 /C1/58, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 50, 1986. Gregory-Newton Formula NEWTON’S FORWARD DIFFERENCE FORMULA Grelling’s Paradox A semantic PARADOX , also called the HETEROLOGICAL PARADOX , which arises by defining "heterological" tomean "a word which does not describe itself." The word "heterological" is therefore heterological IFF it is not. See also RUSSELL’S PARADOX References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 6, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 83 /C1/4, 1998. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 20 /C1/1, 1989. Grenz-Formel An equation derived by Kronecker: X ?/C12 x; y; z/C30/C28/C12(x2 /C27y2 /C27dz2) /C28s /C304z(s)h(s) /C272p s /C28 1z(2s /C28 2) ds/C281/C272ps G(s)d(1/C28s)=2 /C2X/C12 n /C301n(s /C281)=2X u2 ½nrn u2 ! u2a/C282 g/C12 0e pffiffiffiffi ndp (y/C27y/C281)ys/C282 dy; where r(n) is the SUM OF SQUARES FUNCTION , z(z)is the RIEMANN ZETA FUNCTION , h(z) is the DIRICHLET ETA FUNCTION , G(z) is the GAMMA FUNCTION , and the primed sum omits terms with zero DENOMINATOR (Selberg and Chowla 1967). See also DIRICHLET ETA FUNCTION ,EPSTEIN ZETA FUNCTION ,SUM OF SQUARES FUNCTION References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 296 /C1/97, 1987. Selberg, A. and Chowla, S. "On Epstein’s Zeta-Function." J. reine angew. Math. 227,8 6/C1/10, 1967. Grid This entry contributed by D ANIEL SCOTT UZNANSKI A grid usually refers to two or more infinite sets of evenly-spaced parallel lines at particular angles toeach other in a plane, or the intersections of such lines. The two most common types of grid are orthogonal grids, with two sets of lines perpendicularto each other, and isometric grids, with three sets of lines at 60-degree angles to each other. It should be noted that in most grids with three or more sets of lines, every intersection includes one element of each set. There are other types of planar grids, like hexagonal grids, which are formed by tessellating regular hexagons in the plane. These are often found in strategy and role-playing games because of the lack of single points of contact characteristic of isometric and orthogonal grids. The collection of cells created by a grid is often called a "BOARD " when these cells are used as resting places for pieces in a game. Grids can be generalized into n-D space by using the centers of packed n-spheres or n-cubes as the points. See also BOARD ,FINITE ELEMENT METHOD ,LATTICE POINT References Bern, M. W.; Flaherty, J. E.; and Luskin, M. (Eds.). Grid Generation and Adaptive Algorithms. New York: Springer-Verlag, 1999. Liseikin, V. D. Grid Generation Methods. Berlin: Springer- Verlag, 1999. Grid Graph An m /C29n grid graph Gm;nis the product of PATH GRAPHS on m and n vertices. A grid graph Gn;1is called a PATH GRAPH . The grid graph G2 ;2 is the CYCLE GRAPH C4 :/ A grid graph is HAMILTONIAN if either the number of rows or columns is even (Skiena 1990, p. 148). Grid graphs are also bipartite (Skiena 1990, p. 148). See also PATH GRAPH References Reddy, V. and Skiena, S. "Frequencies of Large Distances in Integer Lattices." Technical Report, Department of Com- puter Science. Stony Brook, NY: State University of New York, Stony Brook, 1989. Skiena, S. "Grid Graphs." §4.2.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 147 /C1/ 48, 1990.Griffiths Points "The" Griffiths point Gr is the fixed point in GRIF- FITHS’ THEOREM . Given four points on a CIRCLE and a line through the center of the CIRCLE , the four corresponding Griffiths points are COLLINEAR (Tabov 1995).The points Gr /C30I /C274Ge Gr ?/C30I /C284Ge; are known as the first and second Griffiths points, where I is the INCENTER and Ge is the GERGONNE POINT (Oldknow 1996). The Griffiths points lie on the SODDY LINE. See also GERGONNE POINT ,G RIFFITHS’ THEOREM , INCENTER ,OLDKNOW POINTS ,RIGBY POINTS ,SODDY LINE References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/29, 1996. Tabov, J. "Four Collinear Griffiths Points." Math. Mag. 68, 61 /C1/4, 1995. Griffiths’ Theorem When a point P moves along a line through the CIRCUMCENTER of a given TRIANGLE D; the PEDAL CIRCLE of P with respect to D passes through a fixed point (the GRIFFITHS POINT ) on the NINE-POINT CIRCLE of D:/ See also CIRCUMCENTER ,G RIFFITHS POINTS ,N INE- POINT CIRCLE ,PEDAL CIRCLE Grimm’s Conjecture Grimm conjectured that if n/C271;n/C272;...,n/C27kare all COMPOSITE NUMBERS , then there are distinct PRIMES pijsuch that pij½(n/C27j) for 15j5k:/ References Guy, R. K. "Grimm’s Conjecture." §B32 in Unsolved Pro- blems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 86, 1994. Grinberg Formula A formula satisfied by all H AMILTONIAN CIRCUITS with nnodes. Let fjbe the number of regions inside the circuit with jsides, and let gjbe the number of regions outside the circuit with j sides. If there are d interior diagonals, then there must be d /C271 regions [# regions in interior] /C30d /C271 /C30f2 /C27f3 /C27.../C27fn : (1) Any region with j sides is bounded by j EDGES , so such regions contribute jfjto the total. However, this counts each diagonal twice (and each EDGE only once). Therefore, 2f2 /C273f3 /C27...nfn /C302d /C27n: (2) Take (2) minus 2/C29/(1), f3 /C272f4 /C273f5 /C27.../C27(n /C282)fn /C30n /C282 : (3) Similarly, g3 /C272g4 /C27.../C27(n /C282)gn /C30n /C282; (4) so (f3 /C28g3) /C272(f4 /C28g4) /C273(f5 /C28g5) /C27.../C27(n /C282)(fn /C28gn) /C300: (5) Gro¨bner Basis A Gro¨bner basis for a system of POLYNOMIALS is an equivalence system that possesses useful properties, for example, that another polynomial f is a combina- tion of those in the system IFF the remainder of f with respect to the system is 0. (Here, the division algorithm requires an ORDER of a certain type on the MONOMIALS .) Furthermore, the set of polynomials in a Gro¨bner basis have the same collection of roots as the original polynomials. For linear functions in any number of variables, a Gro¨bner basis is equivalent to GAUSSIAN ELIMINATION . Gro¨bner bases are pervasive in the construction of symbolic algebra algorithms, and Gro¨bner bases with respect to LEXICOGRAPHIC ORDER are very useful for solving equations and for elimination of variables. The algorithm for computing Gro¨bner bases is known as BUCHBERGER’S ALGORITHM . The determination of a Gro¨bner basis is very roughly analogous to computing an ORTHONORMAL BASIS from a set of BASIS VECTORS and can be described roughly as a combination of GAUSSIAN ELIMINATION (for linear systems) and the EUCLIDEAN ALGORITHM (for UNIVARIATE POLYNOMIALS over a FIELD ). The time and memory required to calculate a Gro¨bner basis depend very much on the variable ordering, MONOMIAL ordering, and on which variables are regarded as constants. Gro¨bner bases are used implicitly in many routines in Mathematica , and can be called explicitly with the command Groeb- nerBasis [{poly1 , poly2 , ...}, {x1, x2, ...}]. See also BUCHBERGER’S ALGORITHM ,COMMUTATIVE ALGEBRA ,EUCLIDEAN ALGORITHM ,G AUSSIAN ELIM- INATION ,MONOMIAL ,ORTHONORMAL BASISReferences Adams, W. W. and Loustaunau, P. An Introduction to Gro¨bner Bases. Providence, RI: Amer. Math. Soc., 1994. Becker, T. and Weispfenning, V. Gro¨bner Bases: A Computa- tional Approach to Commutative Algebra. New York: Springer-Verlag, 1993. Boege, W.; Gebauer, R.; and Kredel, H. "Some Examples for Solving Systems of Algebraic Equations by Calculating Gro¨bner Bases." J. Symb. Comput. 1,83/C1/8, 1986. Buchberger, B. "Gro¨bner Bases: An Algorithmic Method in Polynomial Ideal Theory." Ch. 6 in Multidimensional Systems Theory (Ed. N. K. Bose). New York: van Nos- trand Reinhold, 1982. Cox, D.; Little, J.; and O’Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer- Verlag, 1996. Eisenbud, D. Commutative Algebra with a View toward Algebraic Geometry. New York: Springer-Verlag, 1995. Faugere, J. C.; Gianni, P.; Lazard, D.; and Mora, T. "Efficient Computation of Zero-Dimensional Groebner Bases by Change of Ordering." J. Symb. Comput. 16, 329 /C1/44, 1993. Harris, J. "Rearranging Expressions by Patterns." Mathe- matica J. 4,82/C1/5, 1994. Heck, A. "A Bird’s-Eye View of Gro¨bner Bases." http:// www.can.nl/CA_Library/Groebner/Tutorials/Heck/AI- HENP96.html. Helzer, G. "Gro¨bner Bases." Mathematica J. 5,67/C1/3, 1995. Nakos, G. and Glinos, M. "Computing Gro¨bner Bases over the Integers." Mathematica J. 4,70/C1/5, 1994. Lichtblau, D. "Gro¨bner Bases in Mathematica 3.0." Mathe- matica J. 6,81/C1/8, 1996. Mishra, B. Algorithmic Algebra. New York: Springer-Ver- lag, 1993. Robbiano, L. "Term Ordering on the Polynomial Ring." In EUROCAL ’85: European Conference on Computer Alge- bra, 1985 Linz, Austria, Vol. 2: Research Contribu- tions 0387159843 New York: Springer-Verlag, 1986. Stoutemyer, D. "Which Polynomial Representation is Best? Surprises Abound!" In Proceedings of the Third MAC- SYMA Users’ Conference, Schenectady, NY. pp. 221 /C1/43, 1984. Trott, M. "Applying GroebnerBasis to Three Problems in Geometry." Mathematica Educ. Res. 6,15/C1/8, 1997. Wang, D. Elimination Methods. Berlin: Springer-Verlag, 1999. Groemer Packing A honeycomb-like packing that forms HEXAGONS . See also GROEMER THEOREM References Stewart, I. "A Bundling Fool Beats the Wrap." Sci. Amer. 268, 142 /C1/44, 1993. Groemer Theorem Given n CIRCLES and a PERIMETER p, the total AREA of the CONVEX HULL is AConvex Hull /C302ffiffiffi 3p (n /C281) /C27p(1 /C281 2ffiffiffi 3p ) /C27 p(ffiffiffi3p /C281) Furthermore, the actual AREA equals this value IFF the packing is a GROEMER PACKING . The theorem was proved in 1960 by Helmut Groemer. See also CONVEX HULL Gronwall’s Theorem Let s(n) be the DIVISOR FUNCTION . Then lim n0/C12s(n) n ln ln n /C30e g ; where g is the EULER- MASCHERONI CONSTANT . Rama- nujan independently discovered a less precise version of this theorem (Berndt 1994). Robin (1984) showed that the validity of the inequality s(n) Begn ln ln n for n ]5041 is equivalent to the RIEMANN HYPOTH- ESIS. References Berndt, B. C. Ramanujan’s Notebooks: Part I. New York: Springer-Verlag, p. 94, 1985. Gronwall, T. H. "Some Asymptotic Expressions in the Theory of Numbers." Trans. Amer. Math. Soc. 37, 113 /C1/ 22, 1913. Nicolas, J.-L. "On Highly Composite Numbers." In Rama- nujan Revisited: Proceedings of the Centenary Conference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). Boston, MA: Academic Press, pp. 215 /C1/44, 1988. Robin, G. "Grandes Valeurs de la fonction somme des diviseurs et hypothe `se de Riemann." J. Math. Pures Appl. 63, 187 /C1/13, 1984. Gross A DOZEN DOZEN , or the SQUARE NUMBER 144. See also 12,DOZEN ,DUODECIMAL Gro¨ssencharakter In the original formulation, a quantity associated with ideal class groups. According to Chevalley’s formulation, a Gro¨ssencharakter is a MULTIPLICATIVE CHARACTER of the group of ADE´ LES that is trivial on the diagonally embedded k /C29; where k is a NUMBER FIELD . See also ADE´ LE,MULTIPLICATIVE CHARACTER References Hecke, E. Math. Z. 1, 1918. Hecke, E. Math. Z. 5, 1920. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 24, 1980. Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Tate, J. "Fourier Analysis in Number Fields and Hecke’s Zeta Functions." Ch. 15 in Algebraic Number Theory (Ed. J. W. S. Cassels and A. Fro¨hlich). New York: Academic Press, 1950. Grossman’s Constant Define the sequence a0 /C301 ; a1 /C30x; and an/C272 /C30an 1 /C27 an /C271 for n ]0: Janssen and Tjaden (1987) showed that thissequence converges for exactly one value of x, x /C30 0:73733830336929 ... ; confirming Grossman’s conjec- ture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. See also FOIAS CONSTANT References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/grssmn/grssmn.html. Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Pro- blem 86 /C1/. Math. Intel. 9,40/C1/3, 1987. Grothendieck’s Constant Let A be an n /C29n REAL SQUARE MATRIX and let xi and yjbe real numbers with xijj; yijjB0: Then Grothen- dieck showed that there exists a constant K indepen- dent of both A and n satisfying jX 1 5i; j5naij /C142xi ; yj /C143j5K (1) in which the vectors xi and yj have a norm B1 in any HILBERT SPACE . The Grothendieck constant is the smallest REAL NUMBER for which this inequality has been proven. Krivine (1977) showed that 1:676... 5KG 51:782... ; (2) and has postulated that KG /C13p 2ln(1 /C27ffiffiffi 2p ) /C301:7822139 ... ; (3) which is related to KHINTCHINE’S CONSTANT . References Krivine, J. L. "Sur la constante de Grothendieck." C. R. A. S. 284, 8, 1977. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 42, 1983. Grothendieck’s Theorem Let E and F be paired spaces with S a family of absolutely convex bounded sets of F such that the sets of S generate F and, if B1 ;B2 /C23 S; then there exists a B3 /C23 S such that B3 ‡B1 and B3 ‡B2 : Then ES is complete IFF algebraic linear functional f(y)ofF that is weakly continuous on every B/C23Sis expressed asf(y)/C30/C142x;y/C143for some x/C23E:When ESis not complete, the space of all linear functionals satisfying this condition gives the completion ˆESofES:/ See also MACKEY’S THEOREM References Iyanaga, S. and Kawada, Y. (Eds.). "Grothendieck’s Theo- rem." §407L in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1274, 1980. Ground Set A PARTIALLY ORDERED SET is defined as an ordered pair P /C30(X ;5) : Here, X is called the GROUND SET of P and 5is the PARTIAL ORDER of P. See also PARTIAL ORDER ,PARTIALLY ORDERED SET Group A group Gis a finite or infinite set of elements together with a BINARY OPERATION which together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverseproperty. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation.Elements A,B,C, ... with binary operation between AandBdenoted ABform a group if 1. Closure: If AandBare two elements in G, then the product ABis also in G. 2. Associativity: The defined multiplication is associative, i.e., for all /A;B;C/C23G/,/(AB)C/C30A(BC)/. 3. Identity: There is an IDENTITY ELEMENT I(a.k.a. 1;E,o re) such that IA/C30AI/C30Afor every element A/C23G:/ 4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element B/C30A/C281such that AA/C281/C30A/C281A/C30Ifor each element of G. A group is therefore a MONOID for which every element is invertible, and a group must contain at least one element. The study of groups is known as GROUP THEORY .I f there are a finite number of elements, the group is called a FINITE GROUP and the number of elements is called the ORDER of the group. A subset of a group that is CLOSED under the group operation and the inverse operation is called a SUBGROUP .SUBGROUPS are also groups, and many commonly encounteredgroups are in fact special subgroups of some moregeneral larger group. A basic example of a FINITE GROUP is the SYMMETRIC GROUP an;which is the group of PERMUTATIONS (or "under permutation") of nobjects. The simplest infinite group is the set of INTEGERS under usual ADDITION . For continuous groups, one can consider the real numbers or the set of n/C29ninvertible MATRICES . These last two are examples of L IE GROUPS . One very common type of group is the CYCLIC GROUPS . This group is isomorphic to the group of integers (modulo n), is denoted Zn;Zn;orZ=nZ;and is defined for every integer n/C211. It is CLOSED under addition, associative, and has unique inverses. The numbersfrom 0 to n/C281 represent its elements, with the IDENTITY ELEMENT represented by 0 ;and the inverse ofiis represented by n/C28i:/ A map between two groups which preserves the identity and the group operation is called a HOMO- MORPHISM . If a homomorphism has an inverse which is also a homomorphism, then it is called an ISO- MORPHISM and the two groups are called isomorphic. Two groups which are isomorphic to each other areconsidered to be "the same" when viewed as abstractgroups. For example, the group of rotations of a square, illustrated below, is the CYCLIC GROUP Z4:/ In general, a GROUP ACTION is when a group acts on a set, permuting its elements, so that the map from the group to the PERMUTATION GROUP of the set is a homomorphism. For example, the rotations of asquare are a SUBGROUP of the PERMUTATIONS of its corners. One important GROUP ACTION for any group Gis its action on itself by CONJUGATION . These are just some of the possible GROUP AUTOMORPHISMS . Another important kind of GROUP ACTION is a REPRE- SENTATION of a group, where the group acts on a VECTOR SPACE byINVERTIBLE LINEAR MAPS . When the FIELD of the VECTOR SPACE is the complex numbers, sometimes a representation is called a C GMODULE . GROUP ACTIONS , and in particular representations, are very important in applications, not only to group theory, but also to physics and chemistry. Since agroup can be thought of as an abstract mathematical object, the same group may arise in different con- texts. It is therefore useful to think of a representa-tion of the group as one particular incarnation of the group, which may also have other representations. An IRREDUCIBLE REPRESENTATION of a group is a representation for which there exists no UNITARY TRANSFORMATION which will transform the represen- tation MATRIX into block diagonal form. The irreduci- ble representations have a number of remarkable properties, as formalized in the GROUP ORTHOGONAL- ITY THEOREM . See also GROUP THEORY ,SEMIGROUP Group Action A GROUP G is said to act on a space X when there is a map f : G /C29X 0 X such that the following conditions hold for all elements x /C23 X : 1. f(e ; x) /C30x where e is the identity element of G. 2. f(g ; f(h; x)) /C30 f(gh ; x) for all g ; h /C23 G:/ In this case, G is called a TRANSFORMATION GROUP , X is a called a G-set, and f is called the group action. (5793468201) In a group action, a GROUP permutes the elements of X. The identity does nothing, while a composition of actions corresponds to the action of the composition. For example, as illustrated above, the SYMMETRIC GROUP S10 acts on the digits 0 to 9 by permutations. For a given x, the set fgxg; where the group action moves x, is called the ORBIT of x. The SUBGROUP which fixes x is the ISOTROPY GROUP of x. For example, the group Z2 /C30f[0] ; [1]g acts on the real numbers by multiplication by (/C281)n : The identity leaves everything fixed, while [1] sends x to (/C28x): Note that [1] /C215 [1] /C30[0]; which corresponds to /C28(/C28x) /C30 x: For x "0; the orbit of x is fx;/C28xg; and the isotropy subgroup is trivial, f[0]g: The only FIXED POINT of this action is x /C30 0. In a REPRESENTATION , a group acts by invertible LINEAR TRANSFORMATIONS of a VECTOR SPACE V.In fact, a representation is a GROUP HOMOMORPHISM from G to GL(V) ; the GENERAL LINEAR GROUP of V. Some groups are described in a representation, such as the SPECIAL LINEAR GROUP , although they may have different representations. Historically, the first group action studied was the action of the G ALOIS GROUP on the roots of a POLY- NOMIAL . However, there are numerous examples and applications of group actions in many branches of mathematics, including ALGEBRA ,TOPOLOGY ,GEOME- TRY,NUMBER THEORY , and ANALYSIS , as well as the sciences, including chemistry and physics. See also BLOCK (GROUP ACTION ), EFFECTIVE ACTION , FREE ACTION ,G ALOIS GROUP ,G ROUP ,ISOTROPY GROUP ,M ATRIX GROUP ,O RBIT (GROUP ), PRIMITIVE (GROUP ACTION ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION ,TOPOLOGICAL GROUP ,TRANSITIVE References Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 1 /C1/, 1987.Group Convolution The convolution of two COMPLEX -valued functions on aGROUP Gis defined as (a+b)(g)/C30X k/C23Ga(k)b(k/C281g) where the SUPPORT (set which is not zero) of each function is finite. References Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744/C1/52, 1996. Group Direct Product Given two GROUPS GandH, there are several ways to form a new group. The simplest is the direct product, denoted G/C29H:As a set, the group direct product is the C ARTESIAN PRODUCT of ordered pairs ( g, h), and the group operation is componentwise, so (g1;h1)/C29(g2;h2)/C30(g1g2;h1h2): For example, R/C29Ris isomorphic to R2under VECTOR ADDITION . In a similar fashion, one can take the direct product of any number of groups by taking theCartesian product and operating componentwise.Note that Gis ISOMORPHIC to the SUBGROUP of elements g;eHwhere eHis the IDENTITY ELEMENT in H. Similarly, Hcan be realized as a SUBGROUP . The intersection of these two subgroups is the identity(e G;eH);and the two subgroups are NORMAL . Like the RING DIRECT PRODUCT , the group direct product has the UNIVERSAL PROPERTY that if any group Xhas a HOMOMORPHISM toGand a homo- morphism to H, then these homomorphisms factor through G/C29Hin a unique way. If one has REPRESENTATIONS RGofGand RHofH, then there is a representation RG/C156RHsometimes called the EXTERNAL TENSOR PRODUCT , given by the TENSOR PRODUCT /C156:In this case, the group CHAR- ACTER satisfies x(g /C156h) /C30 xRG(g) xRH(h) : See also CARTESIAN PRODUCT ,E XTERNAL TENSOR PRODUCT ,H OMOMORPHISM ,R EPRESENTATION ,SUB- GROUP ,UNIVERSAL PROPERTY References Riesel, H. "The Direct Product of Two Given Groups." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 251 /C1/52, 1994. Group Homomorphism A group homomorphism is a map f : G 0 H between two groups such that 1. The group operation is preserved: f(g1g2) /C30f(g1)f(g2)/ 2. The identity is mapped to the identity: f(eG) /C30eH ;/ where the product on the left-hand side is in G and on the right-hand side in H. Note that a homomorphism must preserve the inverse map because f(g)f(g /C281) /C30 f(gg /C281) /C30f(eG) /C30eH ; so f(g) /C281 /C30f(g /C281) :/ In particular, the image of G is a SUBGROUP of H and the kernel, i.e., f /C281(eH)isa SUBGROUP of G. The kernel is actually a NORMAL SUBGROUP , as is the PREIMAGE of any NORMAL SUBGROUP of H. Hence, any homomorphism from a SIMPLE GROUP must be IN- JECTIVE . See also HOMOMORPHISM ,G ROUP ,N ORMAL SUB- GROUP ,REPRESENTATION Group Orthogonality Theorem Let G be a representation for a GROUP of ORDER h, then X RGi(R)mn Gj(R)m?n?/C31/C30hffiffiffiffiffiffi liljq dij dmm? dnn?: The proof is nontrivial and may be found in Eyring et al. (1944). See also CHARACTER (GROUP ), GROUP ,IRREDUCIBLE REPRESENTATION References Eyring, H.; Walker, J.; and Kimball, G. E. Quantum Chem- istry. New York: Wiley, p. 371, 1944. Group Representation GROUP ,IRREDUCIBLE REPRESENTATION ,REPRESENTA- TION Group Residue Theorem If two groups are residual to a third, every group residual to one is residual to the other. The Gambierextension of this theorem states that if two groups are pseudoresidual to a third, then every group pseudor- esidual to the first with an excess greater than or equal to the excess of the first minus the excess of the second is pseudoresidual to the second, with an excess ]0:/ References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 30 /C1/1, 1959. Group Ring The set of sums ax axx ranging over a multiplicative GROUP and aiare elements of a FIELD with all but a finite number of ai /C300 : Group rings are GRADED ALGEBRAS . See also GRADED ALGEBRA Group Theory The study of GROUPS . Gauss developed but did not publish parts of the mathematics of group theory, but Galois is generally considered to have been the first to develop the theory. Group theory is a powerful formal method for analyzing abstract and physical systems in which SYMMETRY is present and has surprising importance in physics, especially quantum me- chanics. See also FINITE GROUP ,G ROUP ,H IGHER DIMEN- SIONAL GROUP THEORY ,PLETHYSM ,SYMMETRY References Alperin, J. L. and Bell, R. B. Groups and Representations. New York: Springer-Verlag, 1995. Arfken, G. "Introduction to Group Theory." §4.8 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 237 /C1/76, 1985. Burnside, W. Theory of Groups of Finite Order, 2nd ed. New York: Dover, 1955. Burrow, M. Representation Theory of Finite Groups. New York: Dover, 1993. Carmichael, R. D. Introduction to the Theory of Groups of Finite Order. New York: Dover, 1956. Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Sub- groups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, 1985. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, 1990. Dixon, J. D. Problems in Group Theory. New York: Dover, 1973. Farmer, D. Groups and Symmetry. Providence, RI: Amer. Math. Soc., 1995. Grossman, I. and Magnus, W. Groups and Their Graphs. Washington, DC: Math. Assoc. Amer., 1965. Hamermesh, M. Group Theory and Its Application to Physical Problems. New York: Dover, 1989. Lomont, J. S. Applications of Finite Groups. New York: Dover, 1987. Magnus, W.; Karrass, A.; and Solitar, D. Combinatorial Group Theory: Presentations of Groups in Terms ofGenerators and Relations. New York: Dover, 1976. Mirman, R. Group Theory: An Intuitive Approach. River Edge, NJ: World Scientific, 1995. Robinson, D. J. S. A Course in the Theory of Groups, 2nd ed. New York: Springer-Verlag, 1995. Rose, J. S. A Course on Group Theory. New York: Dover, 1994. Rotman, J. J. An Introduction to the Theory of Groups, 4th ed. New York: Springer-Verlag, 1995. Scott, W. R. Group Theory. New York: Dover, 1987. Weisstein, E. W. "Groups." MATHEMATICA NOTEBOOK GROUPS.M . Weisstein, E. W. "Books about Group Theory." http:// www.treasure-troves.com/books/GroupTheory.html. Weyl, H. The Classical Groups: Their Invariants and Representations. Princeton, NJ: Princeton University Press, 1997. Wybourne, B. G. Classical Groups for Physicists. New York: Wiley, 1974. Groupoid There are at least two definitions of "groupoid" currently in use. The first type of groupoid is an algebraic structure on a SET with a BINARY OPERATOR . The only restriction on the operator is closure (i.e., applying the BINARY OPERATOR to two elements of a given set S returns a value which is itself a member of S). Associativity, commutativity, etc., are not required (Rosenfeld 1968, pp. 88 /C1/03). A groupoid can be empty. The numbers of nonisomorphic groupoids of this type having n ele- ments are 1, 1, 10, 3330, 178981952, ... (Sloane’s A001329), and the numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, ... (Sloane’s A001424). An associative groupoid is called a SEMIGROUP . The second type of groupoid is an algebraic structure first defined by Brandt (1926) and also known as a VIRTUAL GROUP . A groupoid with base B is a set G with mappings a and b from G onto B and a partially defined binary operation (g ; h) /C2gh ; satisfying the following four conditions: 1. gh is defined only when b(g) /C30 a(h) for certain maps a and b from G onto R2 with a :(x ; g ; y) /C2x and b :(x; g ; y) /C2y/ 2. ASSOCIATIVITY : If either (gh)k or g(hk) is defined, then so is the other and (gh)k /C30g(hk) :/ 3. For each g in G, there are left and right IDENTITY ELEMENTS lgand rgsuch that lgg /C30g /C30grg :/ 4. Each g in G has an inverse g /C281 for which gg /C281 /C30 lg and g/C281g /C30 rg/ (Weinstein 1996). A groupoid is a small CATEGORY with every morphism invertible. See also BINARY OPERATOR ,INVERSE SEMIGROUP ,LIE ALGEBROID ,LIE GROUPOID ,M ONOID ,Q UASIGROUP , SEMIGROUP ,TOPOLOGICAL GROUPOID References Brandt, W. "U¨ ber eine Verallgemeinerung des Gruppen- griffes." Math. Ann. 96, 360 /C1/66, 1926.Brown, R. "From Groups to Groupoids: A Brief Survey." Bull. London Math. Soc. 19, 113 /C1/34, 1987. Brown, R. Topology: A Geometric Account of General Topology, Homotopy Types, and the Fundamental Group- oid. New York: Halsted Press, 1988. Higgins, P. J. Notes on Categories and Groupoids. London: Van Nostrand Reinhold, 1971. Ramazan, B. "Groupoids Home Page." http://www.labo- math.univ-orleans.fr/descriptions/ramazan/groupoi- des.html. Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968. Sloane, N. J. A. Sequences A001329/M4760 and A001424 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744 /C1/52, 1996. Growth A general term which refers to an increase (or decrease in the case of the oxymoron "negative growth") in a given quantity. See also LAW OF GROWTH ,LIFE EXPECTANCY ,POPU- LATION GROWTH Growth Function BLOCK GROWTH Growth Spiral LOGARITHMIC SPIRAL Gru¨nbaum Graph Gru¨nbaum conjectured that for every m /C211, n /C212, there exists an m-regular, m-chromatic graph of GIRTH at least n. This result is trivial for n /C302 and m /C302;3; but only two other such graphs are known: the Gru¨nbaum graph illustrated above, and the CHVA´TAL GRAPH . See also CHVA´ TAL GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 241 /C1/42, 1976. Gru¨nbaum, B. "A Problem in Graph Coloring." Amer. Math. Monthly 77, 1088 /C1/092, 1970. Grundy’s Game A special case of NIM played by the following rules. Given a heap of size n, two players alternately select a heap and divide it into two unequal heaps. A player loses when he cannot make a legal move because all heaps have size 1 or 2. Flammenkamp gives a table of the extremal SPRAGUE- GRUNDY VALUES for this game. The first few values of Grundy’s game are 0, 0, 0, 1, 0, 2, 1, 0, 2, ... (Sloane’s A002188). References Sloane, N. J. A. Sequences A002188/M0044 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Grundy-Sprague Number NIM-VALUE G-Space A G-space is a special type of HAUSDORFF SPACE . Consider a point x and a HOMEOMORPHISM of an open NEIGHBORHOOD V of x onto an OPEN SET of Rn : Then a space is a G-space if, for any two such NEIGHBOR- HOODS v? and v ƒ; the images of v?@ v ƒ under the different HOMEOMORPHISMS are ISOMETRIC .Ifn /C302, the HOMEOMORPHISMS need only be conformal (but not necessarily orientation-preserving). Hsiang (2000, p. 1) terms a space X with a topological (resp. differentiable, linear) transformation of a given GROUP G a topological (resp. differentiable, linear) G- space. See also GREEN SPACE References Hsiang, W. Y. Lectures on Lie Groups. Singapore: World Scientific, p. 1, 2000. G-Transform The G-transform of a function f(x) is defined by the integral (Gf)(x) /C30 Gmn pqj ap=z;=z1 bq=z;=z1jf(t) ! (x) (1) /C301 2pi g sGbmðÞ/C27s ; 1 /C28(a)n /C28s an/C271 p=z1*=z1+ /C27s ; 1 /C28 bm/C271 q=z1*=z1+ /C28s"# f /C31(s)x/C28sds ; (2) where Gmn pqis MEIJER’S G-FUNCTION ,GbmðÞ/C27s ; 1 /C28 anðÞ/C28s an/C271 p=z1*=z1+ /C27s ; 1 /C28 bm/C271 q=z1*=z1+ /C28s"# /C30Gb1 /C27s; ...; bm /C27s; 1 /C28a1 /C28s; ...; 1 /C28an /C28s an/C271 /C27s; ...; ap /C27s; 1 /C28bm /C271 /C28s ; ...; 1 /C28bq /C28s=zn;=zn1 (3) /C30Qm j /C301G(bj /C27 s)Qnj /C301G 1 /C28 aj /C28 s=z;=z1 Qp j /C30n/C271G(aj /C27 s)Qqj /C30m /C271G 1 /C28 bj /C28 s ðÞ; (4) /f /C31(s) is the MELLIN TRANSFORM of a function f(x); s is the CONTOUR s /C30f1=2 /C28i /C12; 1=2 /C27i/C12g; anðÞ/C30 a1 ; a2 ; ...; an ; (an/C271 p) /C30an/C271 ; an /C272 ; ... ; ap ; bmðÞ/C30 b1 ; ...bm ; (bm/C271 q) /C30bm/C271 ; ...; bq ; and the components of the vectors (ap) and (bq) are complex numbers satisfying the conditions R ap=zn=zo "1=2;3=2;5=2; ::: / andRbq=zn=zo "/C281=2;/C283=2;/C285=2; ::: /. See also MEIJER’S G-FUNCTION , W-TRANSFORM References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. "Definition of the G-Transform. The Spaces M/C281 c;gandL(c;g) 2and Their Characterization." §36.1 in Fractional Integrals and Deri- vatives. Yverdon, Switzerland: Gordon and Breach, pp. 704 /C1/09, 1993. Gudermannian Function The ODD FUNCTION denoted either g(x) or gd( x) which arises in the inverse equations for the M ERCATOR PROJECTION .f(y)/C30gd(y) expresses the LATITUDE fin terms of the vertical position yin this projection, so the Gudermannian function is defined by gd(x) /C13gx 0dt cosh t (1) /C30tan /C281(sinh x) (2) 2 tan/C281(ex) /C281 2 p (3) The INVERSE FUNCTION of the Gudermannian func- tion y /C30gd/C281 f gives the vertical position y in the MERCATOR PROJECTION in terms of the LATITUDE f; so gd/C281(x) /C13gx 0dt cos t (4) /C30ln[tan(14 p /C2712 x)] (5) /C30ln(sec x /C27tan x) : (6) The derivatives of the function and its inverse are given by d dxgd(x) /C30sech x (7) d dxgd/C281(x) /C30sec x: (8) The Gudermannian connects the TRIGONOMETRIC and HYPERBOLIC FUNCTIONS via sin(gd x) /C30tanh x (9) cos(gd x) /C30sech x (10) tan(gd x) /C30sinh x (11) cot(gd x) /C30csch x (12) sec(gd x) /C30cosh x (13) csc(gd x) /C30coth x: (14) The Gudermannian is related to the EXPONENTIAL FUNCTION by ex /C30sec(gd x) /C30tan(gd x) (15) /C30tan(14 p /C2712 gd x) (16) /C301 /C27 sin(gd x) cos(gd x) (17) (Beyer 1987, p. 164; Zwillinger 1995, p. 485). Other fundamental identities are tanh(12 x) /C30tan(12 gd x) (18) i gd/C281 x /C30gd /C281(ix) : If gd(x /C27iy) /C30a /C27ib; thentan a /C30sinh x cos y (19) tanh b /C30sin y cosh x (20) tanh x /C30sin a cosh b (21) tan y /C30sin b cosh a (22) (Beyer 1987, p. 164; Zwillinger 1995, p. 485). See also EXPONENTIAL FUNCTION ,HYPERBOLIC FUNC- TIONS ,HYPERBOLIC SECANT ,MERCATOR PROJECTION , SECANT ,TRACTRIX ,TRIGONOMETRIC FUNCTIONS References Beyer, W. H. "Gudermannian Function." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 164, 1987. Zwillinger, D. (Ed.). "Gudermannian Function." §6.9 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 484 /C1/86, 1995. Guldinus Theorem PAPPUS’S CENTROID THEOREM Gumbel’s Distribution A special case of the FISHER- TIPPETT DISTRIBUTION with a /C300, b /C301. The MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C30 g s2 /C3016 p2 g1 /C3012ffiffiffi 6p z(3) p3 g2 /C3012 5 : where g is the EULER- MASCHERONI CONSTANT , and z(3) is A PE´RY’S CONSTANT . See also FISHER- TIPPETT DISTRIBUTION Guthrie’s Problem The problem of deciding if four colors are sufficient to color any map on a PLANE orSPHERE . See also COLORING ,FOUR- COLOR THEOREM Gutschoven’s Curve KAPPA CURVE Guy’s Conjecture Guy’s conjecture, which has not yet been proven or disproven, states that the CROSSING NUMBER for a COMPLETE GRAPH of order n is 1 4n 2$% n /C28 1 2$% n /C28 2 2$% n /C28 3 2$% ; where xbcis the FLOOR FUNCTION , which can be rewritten 1 64 n(n /C282)2(n /C284) for n even 1 64(n /C281)2(n /C283)2for n odd:( The first few values are 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, ... (Sloane’s A000241). See also CROSSING NUMBER (GRAPH ) References Sloane, N. J. A. Sequences A000241/M2772 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Gyrate Bidiminished Rhombicosidodecahedron JOHNSON SOLID J82 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyrate Rhombicosidodecahedron JOHNSON SOLID J72 :/References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyrobicupola A BICUPOLA in which the bases are in opposite orientations. See also BICUPOLA ,P ENTAGONAL GYROBICUPOLA , SQUARE GYROBICUPOLA Gyrobifastigium JOHNSON SOLID J26 ; consisting of two joined triangu- lar PRISMS . Gyrobirotunda A BIROTUNDA in which the bases are in opposite orientations. Gyrocupolarotunda A CUPOLAROTUNDA in which the bases are in opposite orientations. See also ORTHOCUPOLAROTUNDA Gyroelongated Cupola A n-gonal CUPOLA adjoined to a 2n/-gonal ANTIPRISM . See also GYROELONGATED PENTAGONAL CUPOLA , GYROELONGATED SQUARE CUPOLA ,GYROELONGATED TRIANGULAR CUPOLA Gyroelongated Dipyramid GYROELONGATED PYRAMID ,GYROELONGATED SQUARE DIPYRAMID Gyroelongated Pentagonal Bicupola JOHNSON SOLID J46;which consists of a PENTAGONAL ROTUNDA adjoined to a decagonal ANTIPRISM . Gyroelongated Pentagonal Birotunda JOHNSON SOLID J48:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Pentagonal Cupola JOHNSON SOLID J24:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M .Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Pentagonal Cupolarotunda JOHNSON SOLID J47:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Pentagonal Pyramid JOHNSON SOLID J11:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Pentagonal Rotunda JOHNSON SOLID J25:/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Pyramid An n-gonal pyramid adjoined to the top of an n-gonal ANTIPRISM . In the 3-gonal gyroelongated pyramid, the pyramid and lateral antiprism are coplanar. How- ever, the 4-gonal and 5-gonal gyroelongated pyramids correspond to JOHNSON SOLIDS J10and J11 ; respec- tively. See also ANTIPRISM ,ELONGATED PYRAMID ,G YROE- LONGATED DIPYRAMID ,G YROELONGATED PENTAGO- NAL PYRAMID ,GYROELONGATED SQUARE DIPYRAMID , GYROELONGATED SQUARE PYRAMID Gyroelongated Rotunda GYROELONGATED PENTAGONAL ROTUNDA Gyroelongated Square Bicupola JOHNSON SOLID J45 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Gyroelongated Square Cupola JOHNSON SOLID J23 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Square Dipyramid One of the eight convex DELTAHEDRA built up from 16 equilateral triangles. It consists of two oppositely faced SQUARE PYRAMIDS rotated 458 to each other and separated by a 4-ANTIPRISM .ItisJ OHNSON SOLID /J17/. If the centroid is at the origin and the sides are of unit length, the equations of the 4-ANTIPRISM give height of the middle points as 92/C285 =4 : Adding the height of the SQUARE PYRAMIDS gives apex heights of 9(2/C285 =4 /C27 2/C281=2):The SURFACE AREA and VOLUME of the solid are S/C304ffiffiffi 3p V/C3021=4 3(1/C27ffiffiffi 2p /C2721=4): See also ANTIPRISM ,D ELTAHEDRON ,SNUB DISPHE- NOID ,SQUARE PYRAMID Gyroelongated Square Pyramid JOHNSON SOLID J10 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Triangular Bicupola JOHNSON SOLID J44 :/References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroelongated Triangular Cupola JOHNSON SOLID J22 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Gyroid An infinitely connected periodic MINIMAL SURFACE containing no straight lines. See also MINIMAL SURFACE References Osserman, R. Frontispiece to A Survey of Minimal Surfaces. New York: Dover, 1986. H HA Measurement INNER QUERMASS Haar Condition This entry contributed by RONALD M. AARTS A set of VECTORS in n-space is said to satisfy the Haar condition if every set of n vectors is LINEARLY INDEPENDENT (Cheney 1999). Expressed otherwise, each selection of n vectors from such a set is a basis for n-space. A system of functions satisfying the Haar condition is sometimes termed a Tchebycheff system (Cheney 1999). References Cheney, E. W. Introduction to Approximation Theory, 2nd ed. Providence, RI: Amer. Math. Soc., 1999. Haar Function Define c(x) /C13105x 51 2 /C28112 5x 51 0 otherwise8 >< >:(1) and cjk(x) /C13 c 2jx /C28kP+$P+’ ; (2)where the FUNCTIONS plotted above are c00 /C30 c(x) c10 /C30 c(2x) c11 /C30 c(2x /C281) c20 /C30 c(4x) c21 /C30 c(4x /C281) c22 /C30 c(4x /C282) c23 /C30 c(4x /C283): Then a FUNCTION f(x) can be written as a series expansion by f(x) /C30c0 /C27X/C12 j /C300X2j /C281 k /C300cjk cjk(x): (3) The FUNCTIONS cjkand c are all ORTHOGONAL in [0; 1]; with g1 0f(x) fjk(x) dx /C300 (4) g1 0fjk(x)flm(x) dx /C300: (5) These functions can be used to define WAVELETS . Let a FUNCTION be defined on n intervals, with n a POWER of 2. Then an arbitrary function can be considered as an n-VECTOR f, and the COEFFICIENTS in the expan- sion b can be determined by solving the MATRIX EQUATION f /C30Wnb (6) for b, where W is the MATRIX of c basis functions. For example, the fourth-order Haar function WAVELET MATRIX is given by W4/C301110 11 /C2810 1/C28101 1/C2810 /C2812 6643 775 /C3011 00 1/C2810 0 00 1100 1 /C2812 6643 7751000 0010010000012 6643 77511 0 0 1/C28100 00 1 000 0 12 6643 775: See also W AVELET ,W AVELET MATRIX ,W AVELET TRANSFORM References Haar, A. "Zur Theorie der orthogonalen Funktionensys- teme." Math. Ann. 69, 331/C1/71, 1910. Strang, G. "Wavelet Transforms Versus Fourier Trans- forms." Bull. Amer. Math. Soc. 28, 288/C1/05, 1993. Haar Integral The INTEGRAL associated with the HAAR MEASURE . See also HAAR MEASURE Haar Measure Any locally compact Hausdorff topological group has a unique (up to scalars) NONZERO left invariant measure which is finite on compact sets. If the group is Abelian or compact, then this measure is also right invariant and is known as the Haar measure. Haar Transform A 1-D transform which makes use of the HAAR FUNCTIONS . See also H-TRANSFORM ,HAAR FUNCTION References Haar, A. "Zur Theorie der orthogonalen Funktionensys- teme." Math. Ann. 69, 331 /C1/71, 1910. Haberdasher’s Problem With four cuts, DISSECT an EQUILATERAL TRIANGLE into a SQUARE . First proposed by Dudeney (1907) and discussed in Gardner (1961, p. 34), Stewart (1987, p. 169), and Wells (1991, pp. 61 /C1/2). The solution can be hinged so that the three pieces collapse into either the TRIANGLE or the SQUARE . Two of the hinges bisect sides of the triangle, while the third hinge and the corner of the large piece on the base cut the base in the approximate ratio 0:982 : 2 : 1:018:/ See also DISSECTION References Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 27, 1958. Gardner, M. "Mathematical Games: About Henry Ernest Dudeney, A Brilliant Creator of Puzzles." Sci. Amer. 198, 108 /C1/12, Jun. 1958. Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, 1961. Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, 1987. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 61 /C1/2, 1991. Habiro Move A KNOT MOVE illustrated above. Two knots cannot be distinguished using VASSILIEV INVARIANTS of order 5n IFF they are related by a sequence of such moves (Habiro 2000). There is a correspondence between the Habiro move and solution of the BAGUENAUDIER puzzle (Przytycki and Sikora 2000). See also BAGUENAUDIER ,KNOT MOVE References Habiro, K. "Claspers and Finite Type Invariants of Links." Geom. Topol. 4,1/C1/3, 2000. Przytycki, J. H. and Sikora, A. S. Topological Insights from the Chinese Rings. 21 Jul 2000. http://xxx.lanl.gov/abs/ math.GT/0007134/. Hadamard Design A SYMMETRIC BLOCK DESIGN (/4n /C273 ; 2n /C271 ; n) which is equivalent to a HADAMARD MATRIX of order 4n /C274: It is conjectured that Hadamard designs exist for all integers n /C210, but this has not yet been proven. This elusive proof (or disproof) remains one of the most important unsolved problems in COMBINATORICS . See also HADAMARD MATRIX ,S YMMETRIC BLOCK DESIGN References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1 /C1/2, 1992. Hadamard Factorization Theorem Letfbe an ENTIRE FUNCTION ofFINITE ORDER land ajP+vP+u the zeros of f, listed with MULTIPLICITY , then the rank poffis defined as the least positive integer such that X an"0anjj/C28(p/C271)B/C12 : (1) Then the canonical Weierstrass product is given by f(z)/C30eg(z)P(z); (2) and ghas degree q5l:The genus moffis then defined as max( p;q);and the Hadamard factorization theory states that an ENTIRE FUNCTION ofFINITE ORDER lis also of finite genus m;and m5l: (3) References Krantz, S. G. "The Hadamard Factorization Theorem." §9.3.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 121 /C1/22, 1999. Hadamard Gap Theorem OSTROWSKI- HADAMARD GAPTHEOREM Hadamard Matrix A class of SQUARE MATRIX invented by Sylvester (1867) under the name of ANALLAGMATIC PAVEMENT . A Hadamard matrix is a SQUARE MATRIX containing only 1s and /C281s such that when any two columns or rows are placed side by side, HALF the adjacent cells are the same SIGN and half the other (excepting from the count an L-shaped "half-frame" bordering the matrix on two sides which is composed entirely of 1s). When viewed as pavements, cells with 1s are colored black and those with /C281s are colored white. There- fore, the n /C29n Hadamard matrix Hn must have n(n /C28 1)=2 white squares ( /C281s) and n(n /C271)=2 black squares (1s). A Hadamard matrix of order n is a solution to HADAMARD’S MAXIMUM DETERMINANT PROBLEM , i.e., has the maximum possible DETERMINANT (in absolute value) of any n /C29n COMPLEX MATRIX with elements aijP+’2P+’2P+’2P+’251 (Brenner 1972), namely nn=2 : An equivalent definition of the Hadamard matrices is given by HnHT n/C30nI n; (1) where Inis the n /C29n IDENTITY MATRIX . A Hadamard matrix of order 4n /C274 corresponds to a HADAMARD DESIGN (/4n /C273 ; 2n /C271; n). Hadamard (1893) remarked that a NECESSARY condi- tion for a Hadamard matrix to exist is that n /C301, 2, or a positive multiple of 4 (Brenner 1972). PALEY’S THEOREM guarantees that there always exists a Hadamard matrix Hnwhen n is divisible by 4 and OF THE FORM 2 e pm /C271 ðÞ ; where p is an ODD PRIME .In such cases, the MATRICES can be constructed using a PALEY CONSTRUCTION . The PALEY CLASS k is unde- fined for the following values of m B1000: 92, 116, 156, 172, 184, 188, 232, 236, 260, 268, 292, 324, 356, 372, 376, 404, 412, 428, 436, 452, 472, 476, 508, 520, 532, 536, 584, 596, 604, 612, 652, 668, 712, 716, 732, 756, 764, 772, 808, 836, 852, 856, 872, 876, 892, 904, 932, 940, 944, 952, 956, 964, 980, 988, 996. Sawade (1985) constructed H268: It is conjectured (and verified up to n B428) that Hnexists for all n DIVISIBLE by 4 (van Lint and Wilson 1993). However, the proof of this CONJECTURE remains an important problem in CODING THEORY . The number of Hada- mard matrices of order 4n are 1, 1, 1, 5, 3, 60, 487, ... (Sloane’s A007299).If Hn and H m are known, then H nm can be obtained by replacing all 1s in Hmby Hn and all /C281s by /C28H n: For n 5100; Hadamard matrices with n /C3012, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, and 100 cannot be built up from lower order Hadamard matrices. H2/C3011 /C2811P+2$P+2’ (2) H4/C30H2H2 /C28H2H2P+2$P+2’ /C3011 /C2811P+2$P+2’ 11 /C2811P+2$P+2’ /C2811 /C2811P+2$P+2’ 11 /C2811P+2$P+2’2 6643 775 /C30111 1 /C2811 /C2811 /C281/C2811 1 1/C281/C28112 6643 775: (3) /H8can be similarly generated from H4:Hadamard matrices can also be expressed in terms of the W ALSH FUNCTIONS Cal and Sal H8/C30Cal(0 ;t) Sal(4 ;t) Sal(2 ;t) Cal(2 ;t) Sal(1 ;t) Cal(3 ;t) Cal(1 ;t) Sal(3 ;t)2 666666666643 77777777775: (4) Hadamard matrices can be used to make ERROR- CORRECTING CODES . See also HADAMARD DESIGN ,HADAMARD’S MAXIMUM DETERMINANT PROBLEM ,INTEGER MATRIX ,P ALEY CONSTRUCTION ,PALEY’S THEOREM ,WALSH FUNCTION References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 107 /C1/09 and 274, 1987. Beth, T.; Jungnickel, D.; and Lenz, H. Design Theory. New York: Cambridge University Press, 1986. Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626/C1/30, 1972. Colbourn, C. J. and Dinitz, J. H. (Eds.). "Hadamard Ma- trices and Designs." Ch. 24 in CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, pp. 370 /C1/77, 1996. Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102/C1/07, May 1975. Geramita, A. V. Orthogonal Designs: Quadratic Forms and Hadamard Matrices. New York: Dekker, 1979. Golomb, S. W. and Baumert, L. D. "The Search for Hada- mard Matrices." Amer. Math. Monthly 70,1 2/C1/7, 1963. Hadamard, J. "Re ´solution d’une question relative aux de´terminants." Bull. Sci. Math. 17,3 0/C1/1, 1893. Hall, M. Combinatorial Theory, 2nd ed. New York: Wiley, 1998. Hedayat, A. and Wallis, W. D. "Hadamard Matrices and Their Applications." Ann. Stat. 6, 1184 /C1/238, 1978. Kimura, H. "Classification of Hadamard Matrices of Order 28." Disc. Math. 133, 171 /C1/80, 1994. Kimura, H. "Classification of Hadamard Matrices of Order 28 with Hall Sets." Disc. Math. 128, 257 /C1/69, 1994. Kitis, L. "Paley’s Construction of Hadamard Matrices." http://www.mathsource.com/cgi-bin/msitem?0205 /C1/60. Ogilvie, G. A. "Solution to Problem 2511." Math. Questions and Solutions 10,74/C1/6, 1868. Paley, R. E. A. C. "On Orthogonal Matrices." J. Math. Phys. 12, 311 /C1/20, 1933. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 104 /C1/22, 1963. Sawade, K. "A Hadamard Matrix of Order-268." Graphs Combinatorics 1, 185 /C1/87, 1985. Seberry, J. and Yamada, M. "Hadamard Matrices, Se- quences, and Block Designs." Ch. 11 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 431 /C1/60, 1992. Sloane, N. J. A. Sequences A007299/M3736 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Spence, E. "Classification of Hadamard Matrices of Order 24 and 28." Disc. Math 140, 185 /C1/43, 1995. Sylvester, J. J. "Thoughts on Orthogonal Matrices, Simulta- neous Sign-Successions, and Tessellated Pavements in Two or More Colours, with Applications to Newton’s Rule, Ornamental Tile-Work, and the Theory of Numbers." Phil. Mag. 34, 461 /C1/75, 1867. Sylvester, J. J. "Problem 2511." Math. Questions and Solu- tions 10, 74, 1868. van Lint, J. H. and Wilson, R. M. A Course in Combinato- rics. New York: Cambridge University Press, 1993. Wallis, W. D.; Street, A. P.; and Wallis, J. S. Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices. New York: Springer-Verlag, 1972. Williamson, J. "Hadamard’s Determinant Theorem and the Sum of Four Squares." Duke. Math. J. 11,65/C1/1, 1944. Williamson, J. "Note on Hadamard’s Determinant Theo- rem." Bull. Amer. Math. Soc. 53, 608 /C1/13, 1947. Hadamard Transform A FAST FOURIER TRANSFORM -like ALGORITHM which produces a hologram of an image. Hadamard’s Determinant Problem HADAMARD’S MAXIMUM DETERMINANT PROBLEM Hadamard’s Inequality LetA/C30aikbe an arbitrary n/C29nnonsingular MATRIX with REAL elements and DETERMINANT Ajj;then Ajj25Yn i/C301Xn k/C301a2 ik ! : See also HADAMARD’S THEOREM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1110, 2000.Hadamard’s Maximum Determinant Problem Find the largest possible DETERMINANT (in absolute value) for any n/C29nmatrix whose elements are taken from some set. Hadamard (1893) proved that the DETERMINANT of any COMPLEX n/C29nmatrix Awith entries in the closed UNIT DISK aijP+’2P+’2P+’2P+’251 satisfies detA jj5nn=2; (1) with equality attained by the V ANDERMONDE MATRIX of the nROOTS OF UNITY (Faddeev and Sominskii 1965, p. 331; Brenner 1972). The first few values for max(det An) for n/C301, 2, ... are 1, 2, 3ffiffiffi 3p ;16, 25ffiffiffi5p ; 216, ..., and the squares of these are 1, 4, 27, 256, 3125, ... (Sloane’s A000312). A matrix having such amaximal determinant is known as a H ADAMARD MATRIX (Brenner 1972). For real entries, Hadamard’s bound can be improvedfor real matrices to detA jj5(n/C271)(n/C271)=2 2n(2) (Faddeev and Sominskii 1965, problem 523; Brenner1972). For an n/C29n BINARY MATRIX , i.e., a (0,1)-matrix, the largest possible determinants bnforn/C301, 2, ... are 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, ... (Sloane’s A003432). The numbers of distinct n/C29n binary matrices having the largest possible determi-nant are 1, 3, 3, 60, ... (Sloane’s A051752). n matrices 1 /[1] / 2 10 01P+2$P+2’ 1011P+2$P+2’ 1101P+2$P+2’ 30111011102 435;101 1100112 435;110 0111012 435 For an n/C29n(/C281;1) /-matrix, the largest possible determinants anforn/C301, 2, ... are 1, 2, 4, 16, 48, 160, ... (Sloane’s A003433; Ehrlich and Zeller 1962, Ehrlich 1964). The numbers of distinct n/C29n(/C281;1)/- matrices having the largest possible determinant are 1, 4, 96, 384, .... anis related to the largest possible (0;1)/-matrix determinant bn/C281by an /C302n/C281 bn/C281 (3) (Williamson 1946, Brenner 1972). n matrices 1 [1] 2 /C281 /C281 1 /C281P+2$P+2’ ;/C2811 /C281 /C281P+2$P+2’ ;1 /C281 1 /C281P+2$P+2’ ;11 /C2811P+2$P+2’ For an n /C29n (/C281 ; 0; 1)/-matrix, the largest possible determinants gnare the same as an(Ehrlich 1964, Brenner 1972). The numbers of n /C29n (/C281; 0; 1)/- matrices having maximum determinants are 1, 4, 240, ... (Sloane’s A051753). See also DETERMINANT ,HADAMARD MATRIX ,INTEGER MATRIX References Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626 /C1/30, 1972. Cohn, J. H. E. "Determinants with Elements 91." J. Lon- don Math. Soc. 14, 581 /C1/88, 1963. Ehrlich, H. "Determinantenabscha ¨tzungen fu¨r bina¨re Ma- trizen." Math. Z. 83, 123 /C1/32, 1964. Ehrlich, H. and Zeller, K. "Bina ¨re Matrizen." Z. angew. Math. Mechanik 42, T20 /C1/1, 1962. Faddeev, D. K. and Sominskii, I. S. Problems in Higher Algebra. San Francisco: W. H. Freeman, 1965. Hadamard, J. "Re´solution d’une question relative aux de´terminants." Bull. Sci. Math. 17,30/C1/1, 1893. Hall, M. Combinatorial Theory, 2nd ed. New York: Wiley, 1998. Kaplansky, I. "Never Too Late." Amer. Math. Monthly 102, 259, 1995. MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error- Correcting Codes. Amsterdam, Netherlands: North-Hol- land, p. 54, 1978. Sloane, N. J. A. Sequences A003432/M0720, A003433/ M1291, A051752, and A051753 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Williamson, J. "Determinants Whose Elements are 0 and 1." Amer. Math. Monthly 53, 427 /C1/34, 1946. Yang, C. H. "Some Designs for Maximal (/C271 ;/C281)/-Determi- nant of Order n /C132 (mod 4):/" Math. Comput. 20, 147 /C1/48, 1966. Yang, C. H. "A Construction for Maximal (/C271 ;/C281)/-Matrix of Order 54." Bull. Amer. Math. Soc. 72, 293, 1966. Yang, C. H. "On Designs of Maximal (/C271 ;/C281)/-Matrices of Order n /C132 (mod 4):/" Math. Comput. 22, 174 /C1/80, 1968. Yang, C. H. "On Designs of Maximal (/C271 ;/C281)/-Matrices of Order n /C132 (mod 4) II." Math. Comput. 23, 201 /C1/05, 1969. Hadamard’s Theorem Let Ajjbe an n /C29n DETERMINANT with COMPLEX (or REAL ) elements aij ; then Ajj"0ifaiijj >Xn j/C301 j"iaijP+’2P+’2P+’2P+’2: See also HADAMARD’S INEQUALITY References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1110, 2000. Hadamard-Valle ´e Poussin Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. The sum of RECIPROCALS ofPRIMES diverges, but lim n0/C12Xp(n) k/C3011 pk/C28ln(ln n)"# /C30g/C27X/C12 k/C301ln 1/C281 pk ! /C271 pk"# /C13C1/C300:2614972128 :::; (1) where p(n) is the PRIME COUNTING FUNCTION andgis the E ULER- MASCHERONI CONSTANT (Le Lionnais 1983). Hardy and Wright (1985) show that, if /v(n)/is the number of distinct PRIME FACTORS ofn, then lim n0/C121 nXn k/C301v(k)/C28ln(ln n)"# /C30C1: (2) Furthermore, if V(n) is the total number of PRIME FACTORS ofn, then lim n0/C121 nXn k/C301V(k)/C28ln(ln n)"# /C30C1/C27X/C12 k/C3011 pk(pk/C281) /C301:0346538819 ::: : (3) Similarly, lim n0/C12Xp(n) k/C301lnpk pk/C28lnn ! /C30/C28g/C28X/C12 j/C302X/C12 k/C301lnpk pj k/C13/C28C2 /C30/C281:3325822757 ::: : (4) References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hdmrd/hdmrd.html. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1985. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 24, 1983. Rosser, J. B. and Schoenfeld, L. "Approximate Formulas for Some Functions of Prime Numbers." Ill. J. Math. 6,6 4/C1/4, 1962. Hadwiger Number References Kostochka, A. V. "On Hadwiger Numbers of a Graph and Its Complement." In Finite and Infinite Sets, Colloq. Math. Soc. Ja´nos Bolyai, Vol. 37 (Ed. A. Hajnal, L. Lova´sz, and V. T. So´s). pp. 537 /C1/45, 1981. Zelinka, B. "Hadwiger Number of Finite Graphs." Math. Slov. 26,23/C1/0, 1976. Hadwiger Problem What is the largest number of subcubes (not necessa- rily different) into which a CUBE cannot be divided by plane cuts? The answer is 47. See also CUBE DISSECTION ,CUTTING Hadwiger’s Principal Theorem The VECTORS 9a1 ; ..., 9anin a 3-space form a normalized EUTACTIC STAR IFF Tx /C30x for all x in the 3-space. Hafner-Sarnak-McCurley Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Given two randomly chosen n /C29n INTEGER MATRICES , what is the probability D(n) that the corresponding DETERMINANTS are RELATIVELY PRIME ? Hafner et al. (1993) showed that D(n) /C30Y/C12 k /C3011 /C28 1 /C28Yn j /C3011 /C28p /C28j kP+$P+’"#28 < :9 = ;; (1) where pn is the nth PRIME . The case /D1/ is just the probability that two random INTEGERS are RELATIVELY PRIME , D(1) /C306 p2 /C300:6079271019... (2) No analytic results are known for n ]2: Approximate values for the first few n are given by D(2) :0 :453103 (3) D(3) :0 :397276 (4) D(4) :0 :373913 (5)D(5) :0:363321 : (6) Vardi (1991) computed the limit s /C13 lim n0/C12D(n) /C300 :3532363719... : (7) The speed of convergence is roughly /C20 :57n (Flajolet and Vardi 1996). See also INTEGER MATRIX ,RELATIVELY PRIME References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hafner/hafner.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Hafner, J. L.; Sarnak, P.; and McCurley, K. "Relatively Prime Values of Polynomials." In Contemporary Mathe- matics Vol. 143 (Ed. M. Knopp and M. Seingorn). Provi- dence, RI: Amer. Math. Soc., 1993. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, 1991. Hahn Polynomial The orthogonal polynomials defined by h(a;b) n(x;N)/C30(/C281)n(N/C28x/C28n)n(b/C27x/C271)n n! /C23F2/C28n;/C28x;a/C27N/C28x N/C28x/C28n;/C28b/C28x/C28n;1P+’vP+’u (1) /C30(/C281)n(N/C28n)n(b/C271)n n! /C23F2/C28n;/C28x;a/C27b/C27n/C271 b/C271;1/C28N;1P+’vP+’u ; (2) where ( x)nis the P OCHHAMMER SYMBOL and 3F2(a;b;c;d;e;z)i sa GENERALIZED HYPERGEO- METRIC FUNCTION (Koepf 1998). The first few are given by h(a;b) 0(x;N)/C301 h(a;b) 1(x;N)/C30x(a/C27b/C272)/C28(N/C281)(b/C271): Koekoek and Swarttouw (1998) define another Hahn polynomial Qn(x;a;b;N)/C303F2/C28n;n/C27a/C27b/C271;/C28x a/C271;/C28N;1P+’vP+’u ;(3) the dual Hahn polynomial Rn(l(x);g;d;N) /C303F2/C28n;/C28x;x/C27g/C27d/C271 g/C271;/C28N;1P+’vP+’u ; (4) the continuous Hahn polynomial pn(x;a;b;c;d)/C30in(a/C27c)n(a/C27d)n n! /C293F2/C28n ; n /C27a /C27b /C27c /C27d /C281 ; a /C27ix a /C27c ; a /C27d ;1P+’vP+’u ; (5) and the continuous dual Hahn polynomial Sn(x2; a; b; c) (a /C27 b)n(a /C27 c)n/C303 F2/C28n; a /C27ix; a /C28ix a /C27b; a /C27c;1P+’vP+’u ; (6) for n /C300, 1, ..., N, and where l(x) /C30x(x /C27 g /C27 d /C271): (7) References Koekoek, R. and Swarttouw, R. F. "Continuous Dual Hahn," "Continuous Hahn," "Hahn," and "Dual Hahn." §1.3 /C1/.6 in The Askey-Scheme of Hypergeometric Orthogonal Polyno- mials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, pp. 29 /C1/6, 1998. ftp://www.twi.- tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1/ 7.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 115, 1998. Hahn-Banach Theorem A linear FUNCTIONAL defined on a SUBSPACE of a VECTOR SPACE V and which is dominated by a sublinear function defined on V has a linear exten- sion which is also dominated by the sublinear func- tion. References Casti, J. L. "The Hahn-Banach Theorem." Ch. 4 in Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics. New York: Wiley, pp. 155 /C1/05, 2000. Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995. Hailstone Number Sequences of INTEGERS generated in the COLLATZ PROBLEM . For example, for a starting number of 7, the sequence is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, .... Such sequences are called hailstone sequences because the values typically rise and fall, somewhat analogously to a hailstone inside a cloud. While a hailstone eventually becomes so heavy that it falls to ground, every starting INTEGER ever tested has produced a hailstone sequence that eventually drops down to the number 1 and then "bounces" into the small loop 4, 2, 1, .... See also COLLATZ PROBLEM References Schwartzman, S. The Words of Mathematics: An Etymologi- cal Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994.Hairy Ball Theorem There does not exist an everywhere NONZERO tangent VECTOR FIELD on the 2-SPHERE S2 : This implies that somewhere on the surface of the Earth, there is a point with zero horizontal wind velocity. The theorem can be generalized to the statement that the n-sphere Sn has a nonzero tangent vector field IFF n is ODD. See also FIXED POINT THEOREM References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 279 /C1/81, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, p. 90, 1991. Hajnal-Szemere ´di Theorem Every GRAPH with n vertices and maximum VERTEX DEGREE D(G) 5k is (k /C271)/-colorable with all color classes of size n=(k /C271) bc or n =(k /C271) de ; where xbcis the FLOOR FUNCTION and xdeis the CEILING FUNCTION . See also SEYMOUR CONJECTURE References Hajnal, A. and Szemere ´di, E. "Proof of a Conjecture of Erdos." In Combinatorial Theory and Its Applications, Vol. 2 (Ed. P. Erdos, A. Re´nyi, and V. T. So´s). Amster- dam, Netherlands: North-Holland, pp. 601 /C1/23, 1970. Komlo ´s, J.; Sa´rkozy, G. N.; and Szemere ´di, E. "Proof of the Seymour Conjecture for Large Graphs." Ann. Comb. 2, 43 /C1/0, 1998. Hajo´s Number The Hajo´s number h(G)ofa GRAPH G is the maximum k such that G contains a subdivision of the COMPLETE GRAPH Kk :/ References Erdos, P. and Fajtlowicz, S. "On the Conjecture of Hajo´s." Combinatorica 1, 141 /C1/43, 1981. Gutin, G.; Kostochka, A. V.; and Toft, B. "On the Hajo´s Number of Graphs." Discr. Math. 213, 153/C1/61, 2000. Half The UNIT FRACTION /1=2:/ See also QUARTER ,SQUARE ROOT,UNIT FRACTION Half-Angle Formulas Formulas expressing trigonometric functions of an angle x=2 in terms of functions of an angle x, sin1 2xP+’kP+’7 /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28cosx 2s (1) cos1 2 xP+’kP+’7 /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 cos x 2s (2) tan1 2 xP+’kP+’7 /C30sin x 1 /C27 cos x (3) /C301 /C28 cos x sin x (4) /C301 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 tan2 xp tan x (5) /C30tan x sin x tan x /C27 sin x : (6) The corresponding hyperbolic function double-angle formulas are sinh1 2 xP+’kP+’7 /C30sgn xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh x /C28 1 2s (7) cosh12 xP+’kP+’7 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh x /C27 1 2s (8) tanh1 2 xP+’kP+’7 /C30sinh x cosh x /C27 1 (9) /C30cosh x /C28 1 sinh x: (10) See also DOUBLE- ANGLE FORMULAS ,H YPERBOLIC FUNCTIONS ,M ULTIPLE- ANGLE FORMULAS ,PROSTHA- PHAERESIS FORMULAS ,T RIGONOMETRIC ADDITION FORMULAS ,TRIGONOMETRIC FUNCTIONS ,TRIGONOME- TRY Half-Closed Interval An INTERVAL in which one endpoint is included but not the other. A half-closed interval is denoted [a, b) or (a, b] and is also called a HALF-OPEN INTERVAL . The non-standard notation [a; b[ and ]a ; b] is sometimes also used. See also CLOSED INTERVAL ,INTERVAL ,OPEN INTER- VALHalf-Normal Distribution A NORMAL DISTRIBUTION with MEAN 0 and STANDARD DEVIATION 1=u limited to the domain x /C23 [0;/C12): P(x) /C302u pe /C28x2 u2 = p (1) D(x) /C30erfuxffiffiffipp ! : (2) The MOMENTS are m1 /C301 u (3) m2 /C30p 2u2 (4) m3 /C30p u3 (5) m4 /C303 p2 4u4 ; (6) so the MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS are m/C301 u(7) s2/C30p/C282 2u2(8) g1/C302ffiffiffi 2 ps (9) g2/C300: (10) See also NORMAL DISTRIBUTION Half-Open Interval HALF-CLOSED INTERVAL Half-Period Ratio The ratio t/C30v1=v2of the two half-periods v1andv2 of an ELLIPTIC FUNCTION (Whittaker and Watson 1990, p. 475). The notation tis sometimes used instead of t:The half-period ratio is most commonly encountered in the definition of the NOME qas1284 Half-Closed Interval Half-Period Ratio q(k) /C13e pi t /C30e /C28 pK ?(k)=K(k) /C30e /C28pKffiffiffiffiffiffiffiffiffi 1 /C28k2pðÞ =K(k)(1) (Borwein and Borwein 1987, pp. 41, 109, and 114; Whittaker and Watson 1990, p. 463) where K(k)is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND , m /C30k2 is the PARAMETER , k is the MODULUS , K ?(k) /C30 K(k?) ; and k? is the complementary MODULUS . /t is defined such that the IMAGINARY PART I[t] > 0:/ See also JACOBI THETA FUNCTIONS ,MODULAR ANGLE , MODULUS (ELLIPTIC INTEGRAL ), INVERSE NOME, NOME,PARAMETER References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Half-Plane This entry contributed by DANIEL SCOTT UZNANSKI A half-plane is a planar region consisting of all points on one side of an infinite straight line, and no points on the other side. See also HALF-SPACE ,LOWER HALF-PLANE ,PLANE , UPPER HALF-PLANE Half-Space A half-space is that portion of an n-dimensional SPACE obtained by removing that part lying on one side of an (n /C281)/-dimensional hyperplane. For exam- ple, half a Euclidean space is given by the 3-dimen- sional region satisfying x /C210, /C28/C12B y B/C12 ; /C28/C12B z B/C12 ; while a HALF-PLANE is given by the 2- dimensional region satisfying x /C210, //C28/C12B y B/C12 :/ See also HALF-PLANE ,SIEGEL’S UPPER HALF-SPACE Half-Turn A ROTATION through 1808 (/p radians). See also ROTATIONReferences Coxeter, H. S. M. and Greitzer, S. L. "Half-Turn." §4.3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 85 /C1/6, 1967. Hall’s Theorem There exists a system of distinct representatives for a family of sets S1 ; S2 ; ..., Sm IFF the union of any k of these sets contains at least k elements for all k from 1 to m (Harary 1994, p. 53). References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Halley’s Irrational Formula A ROOT -finding ALGORITHM which makes use of a third-order TAYLOR SERIES f(x) /C30fxnðÞ/C27f ? xnðÞ x /C28xn ðÞ /C271 2 f ƒ xnðÞ x /C28xn ðÞ2/C27...: (1) A ROOT of f(x) satisfies f(x) /C300 ; so 0 :fxnðÞ/C27f ? xnðÞ xn/C271 /C28xnP+$P+’ /C2712 f ƒ xnðÞ xn /C271 /C28xnP+$P+’2: (2) Using the QUADRATIC EQUATION then gives xn/C271 /C30xn /C27/C28f ? xnðÞ9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ? xnðÞ½/C1382/C282fxnðÞf ƒ xnðÞq f ƒ xnðÞ: (3) Picking the plus sign gives the iteration function Cf (x) /C30x /C281 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C282f(x)f ƒ(x) [f ?(x)]2s f ƒ(x) f ?(x): (4) This equation can be used as a starting point for deriving HALLEY’S METHOD . If the alternate form of the QUADRATIC EQUATION is used instead in solving (2), the iteration function becomes instead Cf (x) /C30x /C282f(x) f ?(x) 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [f ?(x)]2 /C28 2f(x)f ƒ(x)q : (5) This form can also be derived by setting n /C302in LAGUERRE’S METHOD . Numerically, the SIGN in the DENOMINATOR is chosen to maximize its ABSOLUTE VALUE . Note that in the above equation, if f ƒ(x) /C300; then NEWTON’S METHOD is recovered. This form of Halley’s irrational formula has cubic convergence, and is usually found to be substantially more stable than N EWTON’S METHOD . However, it does run into difficulty when both f(x) and f?(x)o rf?(x) and fƒ(x) are simultaneously near zero. See also HALLEY’S METHOD ,HOUSEHOLDER’S METH- OD,LAGUERRE’S METHOD ,NEWTON’S METHOD References Gourdon, X. and Sebah, P. "Newton’s Iteration." http:// xavier.gourdon.free.fr/Constants/Algorithms/new- ton.html. Ortega, J. M. and Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, PA: SIAM, 2000. Qiu, H. "A Robust Examination of the Newton-Raphson Method with Strong Global Convergence Properties." Master’s Thesis. University of Central Florida, 1993. Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley’s Method." Amer. Math. Monthly 102, 417 /C1/26, 1995. Halley’s Method Also known as the TANGENT HYPERBOLAS METHOD or HALLEY’S RATIONAL FORMULA .AsinH ALLEY’S IRRA- TIONAL FORMULA , take the second-order TAYLOR POLYNOMIAL f(x) /C30fxnðÞ/C27f ? xnðÞ x /C28xn ðÞ /C271 2 f ƒ xnðÞ x /C28xn ðÞ2/C27...: (1) A ROOT of f(x) satisfies f(x) /C300; so 0 :fxnðÞ/C27f ? xnðÞ xn/C271 /C28xnP+$P+’ /C2712 f ƒ xnðÞ xn/C271 /C28xnP+$P+’2: (2) Now write 0 /C30fxnðÞ/C27 xn/C271 /C28xnP+$P+’ /C2 f ? xnðÞ/C271 2 f ƒ xnðÞ xn/C271 /C28xnP+$P+’ hi ; (3) giving xn/C271 /C30xn /C28fxnðÞ f ? xnðÞ/C271 2 f ƒ xnðÞ xn/C271 /C28 xnP+$P+’ : (4) Using the result from NEWTON’S METHOD , xn/C271 /C28xn /C30/C28fxnðÞ f ? xnðÞ: (5) gives xn/C271 /C30xn /C282f(xn)f ?(xn) 2[f ?(xn)]2 /C28 f(xn)f ƒ(xn) ; (6) so the iteration function is Hf (x) /C30x /C282f(x)f ?(x) 2[f ?(x)]2 /C28 f(x)f ƒ(x) : (7) This satisfies H ?f ( a) /C30H ƒf (a) /C300 where a is a ROOT ,soit is third order for simple zeros. Curiously, the third derivative H §f ( a) /C30/C28f §( a) f ?( a)/C283 2f ƒ( a) f ?(a)"#28 < :9 = ; (8) is the SCHWARZIAN DERIVATIVE . Halley’s method may also be derived by applying NEWTON’S METHOD to ff ?/C281 =2 : It may also be derived by using an OSCULAT- ING CURVE OF THE FORMy(x) /C30x /C28 xn ðÞ /C27 c ax/C28 xn ðÞ /C27 b : (9) Taking derivatives, fxnðÞ/C30c b (10) f ? xnðÞ/C30b /C28 ac b2 (11) f ƒ xnðÞ/C302a(ac /C28 b) b3; (12) which has solutions a /C30/C28f ƒ xnðÞ 2 f ? xnðÞ½/C1382/C28fxnðÞf ƒ xnðÞ (13) b /C302f ? xnðÞ 2 f ? xnðÞ½/C1382/C28fxnðÞf ƒ xnðÞ (14) c /C302fxnðÞf ? xnðÞ 2 f ? xnðÞ½/C1382/C28fxnðÞf ƒ xnðÞ; (15) so at a ROOT , yxn/C271P+$P+’ /C300 and xn /C271 /C30xn /C28c ; (16) which is Halley’s method. See also HALLEY’S IRRATIONAL FORMULA ,H OUSE- HOLDER’S METHOD ,LAGUERRE’S METHOD ,NEWTON’S METHOD References Ortega, J. M. and Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, PA: SIAM, 2000. Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley’s Method." Amer. Math. Monthly 102, 417 /C1/26, 1995. Halley’s Rational Formula HALLEY’S METHOD Hall-Janko Group The SPORADIC GROUP HJ, also denoted J2:/ See also JANKO GROUPS Hall-Littlewood Polynomial Let nbe an integer such that n]l1;where l/C30 l1;l2;... ðÞ is a PARTITION ofn/C30ljjifl1]l2]...]0; where liare a sequence of positive integers stabiliz- ing 0 such that aili/C30n:Also let mi(l) be the number of parts of lof size i. Then the PERMUTATION w/C23Sn; where Snis the symmetric group, acts on the vari- ables x1;...,xnby sending xitoxw(i):Letting tbe a COMPLEX NUMBER , the Hall-Littlewood polynomials are defined by Pl(x1 ; ... ; xn; t) /C301 Q i]0Qmi( l) r/C3011 /C28 tr 1 /C28 tX w /C23Snwxl1 1/C1/C1/C1xln nY iBjxi /C28 txj xi /C28 xj ! : These polynomials interpolate between the Schur functions (with t /C300) and the monomial symmetric functions (with t /C301; Fulman 1999). References Fulman, J. "The Rogers-Ramanujan Identities, the Finite General Linear Groups, and the Hall-Littlewood Polyno- mials." Proc. Amer. Math. Soc. 128,17/C1/5, 1999. Macdonald, I. G. Symmetric Functions and Hall Polyno- mials, 2nd ed. Oxford, England: Oxford University Press, p. 208, 1995. Halm’s Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION (1 /C27x2)2 /C27yƒ/C27 ly /C300 (Hille 1969, p. 357; Zwillinger 1997, p. 122). References Hille, E. Lectures on Ordinary Differential Equations. Reading, MA: Addison-Wesley, 1969. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Halphen Constant ONE-NINTH CONSTANT Halphen’s Transformation A curve and its polar reciprocal with regard to the fixed CONIC have the same Halphen transformation. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 346 /C1/47, 1959. Halting Problem The determination of whether a TURING MACHINE will come to a halt given a particular input program. This problem is UNDECIDABLE , as first proved by Turing. See also BUSY BEAVER ,CHAITIN’S CONSTANT ,TURING MACHINE ,UNDECIDABLE References Chaitin, G. J. "Computing the Busy Beaver Function." §4.4 in Open Problems in Communication and Computation (Ed. T. M. Cover and B. Gopinath). New York: Springer- Verlag, pp. 108 /C1/12, 1987. Davis, M. "What It a Computation." In Mathematics Today: Twelve Informal Essays (Ed. L. A. Steen). New York: Springer-Verlag, pp. 241 /C1/67, 1978. Penrose, R. The Emperor’s New Mind: Concerning Compu- ters, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 63 /C1/6, 1989.Ham Sandwich Theorem The volumes of any nn-D solids can always be simultaneously bisected by a (n /C281)/-D HYPERPLANE . Proving the theorem for n /C302 (where it is known as the PANCAKE THEOREM ) is simple and can be found in Courant and Robbins (1978). The theorem was proved for n /C213 by Stone and Tukey (1942). See also CUTTING ,PANCAKE THEOREM References Chinn, W. G. and Steenrod, N. E. First Concepts of Topol- ogy. Washington, DC: Math. Assoc. Amer., 1966. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods. Oxford, England: Oxford University Press, 1978. Davis, P. J. and Hersh, R. The Mathematical Experience. Boston, MA: Houghton Mifflin, pp. 274 /C1/84, 1981. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 67 /C1/9, 1975. Steinhaus, H. "Sur la division des ensembles de l’espace par les plans et des ensembles plans par les cercles." Funda- menta Math. 33, 245 /C1/63, 1945. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 145, 1999. Stone, A. H. and Tukey, J. W. "Generalized ‘Sandwich’ Theorems." Duke Math. J. 9, 356 /C1/59, 1942. Hamburger Moment Problem This entry contributed by RONALD M. AARTS A NECESSARY and SUFFICIENT condition that there should exist at least one nondecreasing function a(t) such that mn /C30g/C12 /C28/C12tn da(t) for n /C300, 1, 2, ..., with all the integrals converging, is that sequence mnfg/C12 0is positive (Widder 1941, p. 129). References Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Hamel Basis This entry contributed by KEVIN O’BRYANT A basis for the real numbers R ; considered as a VECTOR SPACE over the rationals Q; i.e., a set of real numbers Uafg such that every real number b has a unique representation of the form b/C30Xn i/C301riUai; where riis rational and ndepends on b:/ The AXIOM OF CHOICE is equivalent to the statement: "Every VECTOR SPACE has a BASIS ," and this is the only justification for the existence of a Hamel basis. See also AXIOM OF CHOICE ,BASIS,BASIS (VECTOR SPACE ) Hamilton’s Equations The equations defined by ˙q /C30@H @p (1) ˙p /C30/C28@H @q (2) where ˙x /C13dx=dt and H is the so-called Hamiltonian, are called Hamilton’s equations. These equations frequently arise in problems of celestial mechanics. The vector form of these equations is ˙xi /C30Hpi(t; x; p) (3) ˙pi /C30Hxi(t; x; p) (4) (Zwillinger 1997, p. 136; Iyanaga and Kawada 1980, p. 1005). Another formulation related to Hamilton’s equation is p /C30@L @ ˙q; (5) where L is the so-called Lagrangian. References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1005, 1980. Morse, P. M. and Feshbach, H. "Hamilton’s Principle and Classical Dynamics." §3.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 280 /C1/01, 1953. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997. Hamilton’s Rules The rules for the MULTIPLICATION of QUATERNIONS . See also QUATERNION Hamilton-Connected Graph A graph G is Hamilton-connected if every two vertices of G are connected by a HAMILTONIAN PATH(Bondy and Murty 1976, p. 61). All COMPLETE GRAPHS are Hamilton-connected. The numbers of Hamilton- connected simple graphs on n /C301, 2, ... nodes are 1, 1, 1, 1, 3, 13, 116, ... (Sloane’s A057865). See also HAMILTONIAN GRAPH ,H AMILTONIAN PATH, HYPOTRACEABLE GRAPH ,TRACEABLE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 61, 1976. Sloane, N. J. A. Sequences A057865 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Hamiltonian Circuit AGRAPH CYCLE (i.e., closed loop) through a GRAPH that visits each node exactly once (Skiena 1990, p. 196). A graph possessing a Hamiltonian circuit is said to be a H AMILTONIAN GRAPH . The Hamiltonian circuit is named after Sir William Rowan Hamilton, who devised a puzzle in which such a path along the EDGES of an ICOSAHEDRON was sought (the ICOSIAN GAME ). All P LATONIC SOLIDS have a Hamiltonian circuit, as illustrated above. Although not explicitly stated by Gardner (1957), all ARCHIMEDEAN SOLIDS have Hamiltonian circuits as well, several of which are illustrated above. The Archimedean dual RHOMBIC DODECAHEDRON is Ha- miltonian (Gardner 1984, p. 98). All PLANAR 4-con- nected graphs also have Hamiltonian circuits. The number of Hamiltonian circuits on an n-HYPER- CUBE is 2, 8, 96, 43008, ... (Sloane’s A006069; Gardner 1986, pp. 23 /C1/4). In general, the problem of finding a Hamiltonian circuit is NP-COMPLETE (Garey and Johnson 1983), so the only known way to determine whether a given general GRAPH has a Hamiltonian circuit is to under- take an exhaustive search. See also CHVA´ TAL’S THEOREM ,D IRAC’S THEOREM , EULERIAN CIRCUIT ,EULER GRAPH ,G RINBERG FOR- MULA ,H AMILTONIAN GRAPH ,H AMILTONIAN PATH, ICOSIAN GAME,K OZYREV- GRINBERG THEORY ,O RE’S THEOREM ,PO´ SA’S THEOREM ,SMITH’S NETWORK THE- OREM ,TOUR,UNICURSAL CIRCUIT References Bolloba ´s, B. Graph Theory: An Introductory Course. New York: Springer-Verlag, p. 12, 1979. Chartrand, G. Introductory Graph Theory. New York: Dover, p. 68, 1985. Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150 /C1/56, May 1957. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 96 /C1/7, 1984. Gardner, M. "The Binary Gray Code." In Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 23 /C1/4, 1986. Garey, M. R. and Johnson, D. S. Computers and Intract- ability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman, 1983. Lederberg, J. "Hamilton Circuits of Convex Trivalent Poly- hedra (up to 18 Vertices)." Amer. Math. Monthly 74, 522 /C1/ 27, 1967. Ore, O. "A Note on Hamiltonian Circuits." Amer. Math. Monthly 67, 55, 1960. Skiena, S. "Hamiltonian Cycles." §5.3.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 196 /C1/98, 1990. Sloane, N. J. A. Sequences A006069/M1903 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hamiltonian Cycle HAMILTONIAN CIRCUITHamiltonian Graph A GRAPH possessing a HAMILTONIAN CIRCUIT .By convention, the trivial graph on a single node is considered Hamiltonian, but the connected graph on two nodes is not. The numbers of simple Hamiltonian graphs on n nodes for n /C301, 2, ... are then 1, 0, 1, 3, 8, 48, 383, ... (Sloane’s A003216). Testing whether a graph is Hamiltonian is an NP- COMPLETE PROBLEM (Skiena 1990, p. 196). An algo- rithm to test graphs is implemented as Hamilto- nianQ [g] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). All Hamiltonian graphs are BICONNECTED , although the converse is not true (Skiena 1990, p. 197). If the sums of the degrees of nonadjacent vertices in a graph Gis greater than the number of nodes nfor all subsets of nonadjacent vertices, then Gis Hamilto- nian (Ore 1960; Skiena 1990, p. 197). See also BARNETTE’S CONJECTURE ,BICUBIC GRAPH , CHVA´ TAL’S THEOREM ,E ULERIAN GRAPH ,H AMILTO- NIAN CIRCUIT ,HAMILTON- CONNECTED GRAPH ,HAMIL- TONIAN PATH ,H YPOHAMILTONIAN GRAPH , HYPOTRACEABLE GRAPH ,ORE GRAPH ,TAIT’S HAMIL- TONIAN GRAPH CONJECTURE ,TUTTE CONJECTURE References Bolloba ´s, B. Graph Theory: An Introductory Course. New York: Springer-Verlag, p. 12, 1979. Chartrand, G. Introductory Graph Theory. New York: Dover, p. 68, 1985. Chartrand, G.; Kapoor, S. F.; and Kronk, H. V. "The Many Facets of Hamiltonian Graphs." Math. Student 41, 327/C1/ 36, 1973. Dolch, J. P. "Names of Hamiltonian Graphs." In 4th S-E Conf. Combin., Graph Theory, Computing. Congress. Numer. 8, 259/C1/71, 1973. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, p. 219, 1973. Ore, O. "A Note on Hamiltonian Circuits." Amer. Math. Monthly 67, 55, 1960. Skiena, S. "Hamiltonian Cycles." §5.3.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 196 /C1/98, 1990. Sloane, N. J. A. Sequences A003216/M2764 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hamiltonian Group A non-Abelian GROUP all of whose SUBGROUPS are self-conjugate. References Carmichael, R. D. "Hamiltonian Groups." §31 in Introduc- tion to the Theory of Groups of Finite Order. New York: Dover, pp. 113 /C1/16, 1956. Hamiltonian Integer A LINEAR COMBINATION of basis QUATERNIONS with integer coefficients. See also QUATERNION References Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351 /C1/69, 1999. Hamiltonian Map Consider a 1-D Hamiltonian MAP OF THE FORM H(p; q) /C301 2 p2 /C27V(q); (1) which satisfies HAMILTON’S EQUATIONS ˙q /C30@H @p (2) ˙p /C30/C28@H @q: (3) Now, write ˙qi /C30qi/C271 /C28 qiP+$P+’ Dt; (4) where qi /C30q(t) (5) qi/C271 /C30q(t /C27Dt): (6) Then the equations of motion become qi /C271 /C30qi /C27pi Dt (7) pi/C271 /C30pi /C28Dt@V @qi ! q /C30qi(8) Note that equations (7) and (8) are not AREA-PRESER- VING , since@(qi /C271 ; pi/C271) @(qi ; pi)/C301 /C28Dt@2V @q2 i Dt 1P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C301 /C27(Dt) 2@2V @q2 i"1 : (9) However, if we take instead of (7) and (8), qi/C271 /C30qi /C27pi Dt (10) pi/C271 /C30pi /C28Dt@V @qi ! q /C30qi /C271(11) @ qi/C271 ; pi/C271P+$P+’ @ qi ; pi ðÞ/C301 /C28Dt@ @qi@V @q ! q/C30qi /C271 Dt 1P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2 /C301 /C27( Dt) 2@2V @q2 i/C301; (12) which is AREA-PRESERVING . See also AREA-PRESERVING MAP Hamiltonian Path A path between two vertices of a GRAPH that visits each vertex exactly once. A Hamiltonian path that is also a GRAPH CYCLE is called a HAMILTONIAN CIRCUIT (or Hamiltonian cycle). Every TOURNAMENT has an ODD NUMBER of Hamiltonian paths (Re´dei 1934; Szele 1943; Skiena 1990, p. 175). The number of Hamiltonian paths on an n-HYPER- CUBE is 0, 0, 48, 48384, ... (Sloane’s A006070; Gardner 1986, pp. 23 /C1/4). See also HAMILTONIAN CIRCUIT ,H AMILTONIAN GRAPH ,TOURNAMENT References Gardner, M. "The Binary Gray Code." In Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 23 /C1/4, 1986. Re´dei, L. "Ein Kombinatorischer Satz." Acta Litt. Szeged. 7, 39/C1/3, 1934. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 175, 1990. Sloane, N. J. A. Sequences A006070/M5295 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Szele, T. "Kombinatorische Untersuchungen u ¨ber den ger- ichteten vollsta ¨ndigen Graphen." Mat. Fiz. Lapok 50, 223/C1/56, 1943. Hamiltonian System A system of variables which can be written in the form of H AMILTON’S EQUATIONS . Hammer’s X-Ray Problems Let a homogeneous solid contain a convex hole Kand take x-rays so that the "darkness" at each point on aphotographic plate determines the length of the chord ofKalong the line of propagation of an x-ray. Then how many x-ray pictures must be taken to exactly reconstruct K if 1. The x-rays originate from a point source, 2. The x-rays originate from a source at infinity and so are parallel? See also RADON TRANSFORM References Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Hammer’s X- Ray Problems." §A2 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 11 /C1/4, 1991. Hammer-Aitoff Equal-Area Projection A MAP PROJECTION whose inverse is defined using the intermediate variable z /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C281 4 xP+’kP+’72 /C2812 yP+’kP+’72r : Then the longitude and latitude are given by l /C302 tan/C281 zx 22z2 /C28 1 ðÞ"# f /C30sin/C281 (yz) : See also EQUAL- AREA PROJECTION Hamming Code A binary Hamming code Hr of length n /C302r /C281 (with r ]2) is a linear code with parity-check matrix H whose columns consist of all nonzero binary vectors of length r, each used once. Hris an (n /C302r /C281; k /C30 2r /C281 /C28r ; d /C303) code. Hamming codes are PERFECT single ERROR-CORRECTING CODES . See also ERROR- CORRECTING CODE,PERFECT CODE References MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error- Correcting Codes. Amsterdam, Netherlands: North-Hol- land, 1977. Hamming Function An APODIZATION FUNCTION chosen to minimize the height of the highest sidelobe (Hamming and Tukey, Blackman and Tukey 1959). The Hamming function is given by A(x) /C300:54 /C270:46 cospx a ! ; (1)and its FULL WIDTH AT HALF MAXIMUM is 1:05543 a: The corresponding INSTRUMENT FUNCTION is I(k) /C30a(1:08 /C28 0:64a2k2) sinc(2 pak) 1 /C28 4a2k2 : (2) This APODIZATION FUNCTION is close to the one produced by the requirement that the APPARATUS FUNCTION goes to 0 at ka /C305 =4: From APODIZATION FUNCTION , a general symmetric apodization function A(x) can be written as a FOURIER SERIES A(x) /C30a0 /C272X/C12 n /C301an cosnpx b ! ; (3) where the COEFFICIENTS satisfy a0 /C272X/C12 n/C301an /C301: (4) The corresponding apparatus function is I(t) /C302ba0 sinc(2 pkb) f /C27X/C12 n/C301[sinc(2 pkb /C27n p) /C27sinc(2 pkb /C28np)] g: (5) To obtain an APODIZATION FUNCTION with zero at ka /C303=4; use a0 /C272a1 /C301 ; (6) so a0 sinc5 2 pP+’kP+’7 /C27a1sinc72 pP+’kP+’7 /C27sinc32 pP+’kP+’7 /C300h (7) 1 /C282a1 ðÞ2 5p /C28a12 7p /C272 3 p ! /C30 1 /C282a1 ðÞ15 /C28a117 /C2713P+’kP+’7 /C300 (8) a11 7 /C2713 /C2725P+’kP+’7 /C3015 (9) a1 /C3015 2 5 /C2717 /C2713/C307 /C215 3 2 /C215 3 /C215 7 /C27 3 /C215 5 /C27 5 /C215 7 /C302192 :0:2283 (10) a0 /C301 /C282a1 /C3092/C282 /C21521 92/C3092/C2842 92 /C305092/C302546:0:5435 : (11) The FWHM is 1.81522, the peak is 1.08, the peak NEGATIVE and POSITIVE sidelobes (in units of the peak) are /C280:00689132 and 0.00734934, respectively. See also APODIZATION FUNCTION ,H ANNING FUNC- TION ,INSTRUMENT FUNCTION References Blackman, R. B. and Tukey, J. W. "Particular Pairs of Windows." In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 98 /C1/9, 1959. Hamming, R. W. and Tukey, J. W. "Measuring Noise Color." Unpublished memorandum. Handedness Objects which are identical except for a mirror reflection are said to display handedness and to be CHIRAL . See also AMPHICHIRAL ,CHIRAL ,ENANTIOMER ,M IR- ROR IMAGE Handkerchief Surface A surface given by the PARAMETRIC EQUATIONS x(u; v) /C30u y(u; v) /C30v z(u; v) /C301 3 u3 /C27uv2 /C272 u2 /C28v2P+$P+’ : References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 948 /C1/49, 1997. Handle A handle is a topological structure which can be thought of as the object produced by puncturing asurface twice, attaching a ZIP around each puncture travelling in opposite directions, pulling the edges of the zips together, and then zipping up. Handles are to MANIFOLDS as CELLS are to CW - COMPLEXES .If M is a MANIFOLD together with a (k /C281)/-SPHERE Sk /C281 embedded in its boundary with a trivial TUBULAR NEIGHBORHOOD , we attach a k- handle to M by gluing the tubular NEIGHBORHOOD of the (k /C281)/-SPHERE Sk /C281 to the TUBULAR NEIGHBOR- HOOD of the standard (k /C281)/-SPHERE Sk/C281in the dim(M)-dimensional DISK. In this way, attaching a k-handle is essentially just the process of attaching a fattened-up k-DISK to M along the (k /C281)/-SPHERE Sk/C281 : The embedded DISK in this new MANIFOLD is called the k-handle in the UNION of M and the handle. DYCK’S THEOREM states that HANDLES and cross- handles are equivalent in the presence of a CROSS- CAP. See also CAP,C LASSIFICATION THEOREM OF SUR- FACES ,C ROSS- CAP,C ROSS- HANDLE ,H ANDLEBODY , SURGERY ,TUBULAR NEIGHBORHOOD References Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/99, 1999. Handlebody A handlebody of type (n, k)isan n-D MANIFOLD that is attained from the standard n-DISK by attaching only k-D HANDLES . See also HANDLE ,HEEGAARD SPLITTING ,SURGERY References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 46, 1976. Handsome Number POWERFUL NUMBER Hankel Contour The CONTOUR Ce illustrated above. See also HANKEL FUNCTION References Krantz, S. G. "The Hankel Contour and Hankel Functions." §13.2.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 159, 1999. Hankel Function There are two types of functions known as Hankel functions. The more common one is a COMPLEX FUNCTION (also called a Bessel function of the third kind, or Weber Function) which is a LINEAR COMBINA- TION of BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS . These are called the HANKEL FUNCTIONS OF THE FIRST and SECOND KINDS . Another type of Hankel function is defined by the CONTOUR INTEGRAL He(z) /C30gCo(/C28w)z/C281e /C28w 1 /C28 e/C28wdw for I[w] B0; arg(/C28w) jj Bp; e "2pk > 0 ; where Ceis a HANKEL CONTOUR . The RIEMANN ZETA FUNCTION can be expressed in terms of He(z)as z(z) /C30/C28He(z) 2i sin (pz)G(z) for 0 B e B2p and R[z] > 1; where G(z) is the GAMMA FUNCTION (Krantz 1999, p. 160). See also HANKEL CONTOUR ,H ANKEL FUNCTION OF THE FIRST KIND,HANKEL FUNCTION OF THE SECOND KIND,SPHERICAL HANKEL FUNCTION OF THE FIRST KIND,SPHERICAL HANKEL FUNCTION OF THE SECOND KIND,THIRD KIND References Arfken, G. "Hankel Functions." §11.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 604 /C1/10, 1985. Hankel, H. "Die Cylinderfunctionen erster und zweiter Art." Math. Ann. 1, 467 /C1/01, 1869. Hankel, H. "Bestimmte Integrale mit Cylinderfunctionen." Math. Ann. 8, 453 /C1/70, 1875. Krantz, S. G. "The Hankel Contour and Hankel Functions." §13.2.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 159, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 623 /C1/24, 1953. Hankel Function of the First Kind H(1) n(z) /C13Jn(z) /C28iYn(z) ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND and Yn(z)isaB ESSEL FUNCTION OF THE SECOND KIND . Hankel functions of the first kind can be REPRE- SENTED AS a CONTOUR INTEGRAL over the UPPER HALF-PLANE using H(1) n(z) /C301 i p g/C12 0[upper half plane]e(z=2)(t/C281=t) tn/C271dt: The plots above show the structure of H(1) 0(z) in the COMPLEX PLANE . See also BESSEL FUNCTION OF THE FIRST KIND, BESSEL FUNCTION OF THE SECOND KIND,D EBYE’S ASYMPTOTIC REPRESENTATION ,HANKEL FUNCTION OF THE SECOND KIND,MACDONALD FUNCTION ,WATSON- NICHOLSON FORMULA ,W EYRICH’S FORMULA References Arfken, G. "Hankel Functions." §11.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 604 /C1/10, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 623 /C1/24, 1953. Hankel Function of the Second Kind H(2) n(z)/C13Jn(z)/C28iYn(z); where Jn(z)i saB ESSEL FUNCTION OF THE FIRST KIND andYn(z)i saB ESSEL FUNCTION OF THE SECOND KIND . Hankel functions of the second kind can be REPRE- SENTED AS aCONTOUR INTEGRAL using H(2) n(z)/C301 ipg0 /C28/C12[lower half plane]e(z=2)(t/C281=t) tn/C271dt: The plots above show the structure of H(2) 0(z) in the COMPLEX PLANE . See also BESSEL FUNCTION OF THE FIRST KIND, BESSEL FUNCTION OF THE SECOND KIND,H ANKEL FUNCTION OF THE FIRST KIND,W ATSON- NICHOLSON FORMULA References Arfken, G. "Hankel Functions." §11.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 604 /C1/10, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 623 /C1/24, 1953. Hankel Matrix A MATRIX Hnwhere the first row (and column) consists of the integers 1, 2, ..., n, the second row (and column) is given by 2, 3, ..., n, 0, and so on, with the nth row (and column) given by n, 0 ; ...; 0|fflfflfflfflfflffl{zfflfflfflfflfflffl} n/C281: A Hankel matrix can be given byHankelMatrix [m, n] in the Mathematica add-on package LinearAl- gebra‘MatrixManipulation‘ (which can be loaded with the command BBLinearAlgebra‘ ). The first few such matrices are H2/C30 12 20P+2$P+2’ H3/C301232303002 435 H 4/C301234 23403400 40002 6643 775: The elements of the Hankel matrix are given expli- citly by h ij /C300i f i /C27j /C281 > n i /C27j /C281 otherwise :P+2k The DETERMINANT of Hnis given by det(Hn) /C30 (/C281) n=2bcnn;where nbcis the FLOOR FUNCTION , so the first few values are 1, /C284,/C2827, 256, 3125, /C2846656, /C28823543, 16777216, ... (Sloane’s A000312). See also TRIANGULAR MATRIX References Sloane, N. J. A. Sequences A000312/M3619 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Hankel Transform Equivalent to a 2-D F OURIER TRANSFORM with a radially symmetric KERNEL , and also called the F OUR- IER-BESSEL TRANSFORM . g(u;v)/C30F[f(r)]/C30g/C12 /C28/C12g/C12 /C28/C12f(r)e/C282pi(ux/C27vy)dx dy :(1) Let x/C27iy/C30reiu(2) u/C27iv/C30qeif(3) so that x/C30rcosu (4) y/C30rsinu (5) r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p (6) u/C30qcosf (7) v/C30qsinf (8) q/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2/C27v2p : (9) Then g(q)/C30g/C12 0g2p 0f(r)e/C282pirq(cosfcosu/C27sinfsinu)rd rd u /C30g/C12 0g2p 0f(r)e/C282pirqcos(u/C28f)rd rd u /C30g/C12 0g2p/C28f /C28ff(r)e/C282pirqcosurd rd u /C30g/C12 0g2p 0f(r)e/C282pirqcosurd rd u /C30g/C12 0f(r)g2p 0e/C282pirqcosudu"# rd r /C302pg/C12 0f(r)J0(2pqr)rd r ; (10) where J0(z) is a zeroth order B ESSEL FUNCTION OF THE FIRST KIND . Therefore, the Hankel transform pairs are g(q)/C302pg/C12 0f(x)J0(2pqr)rd r (11) f(r)/C302pg/C12 0g(q)J0(2pqr)qd q : (12) The following table gives Hankel transforms for a number of common functions (Bracewell 1999, p. 249). Here, Jn(x)i saB ESSEL FUNCTION andQ a(r) is a RECTANGLE FUNCTION equal to 1 for 0 5r5aand 0 otherwise, and M(x) /C302 p x/C283gx 0J0(x) dx /C28x/C282J0(x)P+2$P+2’ (13) /C30p2 x2J1(x)H0(x) /C28J0(x)H1(x) ½/C138 ; (14) where Jn(x)isaB ESSEL FUNCTION OF THE FIRST KIND , Hn(x)isaS TRUVE FUNCTION and Ln(x)isa MODIFIED STRUVE FUNCTION . /f(r)// g(q)/ /Q a(r)//aJ1(2paq) q/ /sin(2par) r//Q(q=(2a))ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28 q2p / /1 2 d(r /C28a)// paJ0(2paq)/ /M(ar)// aLq 2aP+’vP+’u / /e /C28 pr2 // e /C28pq2 / / a2 /C27r2ðÞ/C281=2 //e /C282 paq q/ / a2 /C27r2ðÞ/C281=3 //2pe /C282paq a/ /1 a2 /C27 r2// 2pK0(2paq)/ /2a2 a2 /C27 r2 ðÞ2// 4p2aqK1(2paq)/ /4a4 a2 /C27 r2 ðÞ3// 4p3a2q2K2(2paq)/ / a2 /C28r2ðÞQ a(r)//a2 pq2 J2(2paq)/ /1 r//1 q/ /e /C28ar //2pa a2 /C27 4 p2q2 ðÞ3 =2/ /e /C28ar r//2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 4 p2q2p / /d(r) 2pr// 1/ /r2e/C28 pr2//e /C28pq2 1 /C28 pq2ðÞ p/ //C28r2f(r)//d2f dq2 /C271 qdF dqP+’vP+’u /C3092f/ See also BESSEL FUNCTION OF THE FIRST KIND, FOURIER TRANSFORM ,LAPLACE TRANSFORM References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 795, 1985.Bracewell, R. "The Hankel Transform." The Fourier Trans- form and Its Applications, 3rd ed. New York: McGraw- Hill, pp. 244 /C1/50, 1999. Oberhettinger, F. Tables of Bessel Transforms. New York: Springer-Verlag, 1972. Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993. Hankel’s Integral Jm(x) /C30xm 2m/C281ffiffiffippG m /C271 2P+’kP+’7g1 0cos (xt) /C2 1 /C28t2P+$P+’m /C281=2dt ; where Jm(x)isaB ESSEL FUNCTION OF THE FIRST KIND and G(z) is the GAMMA FUNCTION . Hankel’s integral can be derived from SONINE’S INTEGRAL . See also POISSON INTEGRAL ,SONINE’S INTEGRAL Hankel’s Symbol The symbol defined by (v; n) /C132/C282n4v2 /C28 1 ðÞ 4v2 /C28 32ðÞ/C1/C1/C1 4v2 /C28 2n /C28 1 ðÞ2hi no n! (1) /C30( /C281)n cos(pv) G1 2 /C27 n /C28 vP+’kP+’7 G12 /C27 n /C27 vP+’kP+’7 xn! ; (2) where G(z) is the GAMMA FUNCTION .Ifv is an integer, then this simplifies to (v; n) /C30( /C281)n/C27v G1 2 /C27 n /C28 vP+’kP+’7 G12 /C27 n /C27 vP+’kP+’7 pn! ; (3) given incorrectly by Erde´lyi et al. (1981, p. 52). See also KRAMP’S SYMBOL ,POCHHAMMER SYMBOL References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 52, 1981. Hann Function HANNING FUNCTION Hanning Function An APODIZATION FUNCTION , also called the H ANN FUNCTION , frequently used to reduce ALIASING in FOURIER TRANSFORMS . The illustrations above show the Hanning function, its INSTRUMENT FUNCTION , and a blowup of the INSTRUMENT FUNCTION sidelobes. It is named after the Austrian meteorologist Julius von Hann (Blackman and Tukey 1959, pp. 98 /C1/9). The Hanning function is given by f(x) /C30cos2px 2a ! /C301 2 /C2812cospx a ! : (1) The INSTRUMENT FUNCTION for Hanning apodization can also be written a sinc(2 pka) /C271 2sinc(2 pka /C28 p) /C2712sinc(2 pka /C27 p)hi : (2) Its FULL WIDTH AT HALF MAXIMUM is a. It has APPARATUS FUNCTION A(x) /C30ga /C28a12 /C2812cospx a !"# e/C282 pikx dx /C3012ga /C28ae /C282 pikx dx /C2812ga /C28ae /C282 pikx dx /C1312A1 /C27A2 ðÞ : (3) The first integral is I1 /C30ga /C28ae /C282pikx dx /C30sin(2pka) pk/C302a sinc(2 pka) : (4) The second integral can be rewritten I2 /C30g0 /C28acospx a ! e /C282 pikx dx /C27g0 /C28acospx a ! e /C282pikx dx /C30ga 0cospx a ! e2pikx /C27e /C282 pikxP+$P+’ dx /C302ga 0cospx a ! cos(2 pkx) dx /C302sinpa /C28 2 pkP+’kP+’7 x 2p a /C28 2pkP+’kP+’7 /C27sinp a /C27 2pkP+’kP+’7 x 2p a /C27 2pkP+’kP+’78 < :9 = ;a 0 /C30asin( p /C28 2 pka) p /C28 2pka/C27sin( p /C27 2pka) p /C27 2pka"# /C30a psin(2pka) 1 /C28 2ka/C28sin(2pka) 1 /C27 2ka"# /C30a[sinc( p /C282pka) /C27sinc(p /C272 pka)]: (5)Combining (4) and (5) gives A(x) /C30a sinc(2 pka) /C271 2sinc( p /C282pka) /C2712sinc( p /C272pka)hi : (6) To find the extrema, define x /C132pka and rewrite (6) as A(x) /C30a sin x /C271 2sinc( x /C28 p) /C2712sinc( x /C27 p)hi : (7) Then solve dA dx /C30p2 /C28x3 cos x /C27 3x2 sin x /C27 p2x cos x /C28 p2 sin x ðÞ x2 p2 /C28 x2 ðÞ2 /C300 (8) to find the extrema. The roots are x /C307 :42023 and 10.7061, giving a peak NEGATIVE sidelobe of /C280:026708 and a peak POSITIVE sidelobe (in units of a) of 0.00843441. The peak in units of a is 1, and the full-width at half maximum is given by setting (7) equal to /1=2/ and solving for x, yielding x1 =2 /C302 pk1 =2a /C30 p: (9) Therefore, with L /C132a; the FULL WIDTH AT HALF MAXIMUM is FWHM /C302k1=2 /C301 a /C302 L : (10) See also APODIZATION FUNCTION ,H AMMING FUNC- TION References Blackman, R. B. and Tukey, J. W. "Particular Pairs of Windows." In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, 1959. Hanoi Graph A GRAPH Hnarising in conjunction with the TOWERS OFHANOI problem. The above figure is the Hanoi graph H3:/ See also TOWERS OF HANOI Hanoi Towers TOWERS OF HANOI Hansen Chain An ADDITION CHAIN for which there is a SUBSET H of members such that each member of the chain uses the largest element of H which is less than the member. See also ADDITION CHAIN ,BRAUER CHAIN ,H ANSEN NUMBER References Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111 /C1/13, 1994. Hansen Number A number n for which a shortest chain exists (which is also a HANSEN CHAIN ) is called a Hansen number. References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111 /C1/12, 1994. Hansen’s Problem A SURVEYING PROBLEM : from the position of two known but inaccessible points A and B, determine the position of two unknown accessible points P and P? by bearings from A, B, P ? to P and A, B, P to P?:/ See also SURVEYING PROBLEMS References Do¨rrie, H. "Annex to a Survey." §40 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 193 /C1/97, 1965. Hansen-Bessel Formula Jn(z)1 2p g p /C28peiz cos tein(t/C28 p=2) dt /C30i/C28n pg p 0eiz cos t cos(nt) dt /C301 p g p 0cos(z sin t /C28nt) dt for n /C300, 1, 2, ..., where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1472, 1980.Happy End Problem The problem of determining the smallest number of points g(n)in GENERAL POSITION in the plane (i.e., no three of which are COLLINEAR ), which always deter- mine a CONVEX POLYGON of n sides. The problem was so-named by Erdos when two investigators who first worked on the problem, E. Klein and G. Szekeres, became engaged and subsequently married (Hoffman 1998, p. 76). E. Klein proved that g(4) /C305 by showing that any arrangement of five points must fall into one of the three cases (left figure), and E. Makai proved g(5) /C309 after demonstrating that a counterexample could be found for eight points (right figure; Hoffman 1998, pp. 75 /C1/6). Erdos and Szekeres (1935) showed that g(n) exists and derived the bound 2n/C282 /C271 5g(n) 52n /C284 n /C282P+’vP+’u /C271 ; (1) wheren kP+$P+’ is a BINOMIAL COEFFICIENT . For n ]4; this has since been reduced to g(n) 52n /C284 n /C282P+’vP+’u (2) by Chung and Graham (1998), g(n) 52n /C284 n /C282P+’vP+’u /C277 /C282n (3) by Kleitman and Pachter (1998), and g(n)52n/C285 n/C282P+’vP+’u /C272 (4) by To ´th and Valtr (1998). For g(6);these bounds give 71, 70, 65, and 37, respectively (Hoffman 1998, p. 78). The values of (4) for n/C306, 7, ... are 37, 128, 464, 1718, ... (Sloane’s A052473). See also CONVEX HULL,CONVEX POLYGON References Chung, F. R. K. and Graham, R. L. "Forced Convex n-gons in the Plane." Discr. Comput. Geom. 19, 367/C1/71, 1998. Erdos, P. and Szekeres, G. "A Combinatorial Problem in Geometry." Compositio Math. 2, 463/C1/70, 1935. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 75 /C1/8, 1998. Kleitman, D. and Pachter, L. "Finding Convex Sets among Points in the Plane." Discr. Comput. Geom. 19, 405/C1/10, 1998. Sloane, N. J. A. SequencesA052473 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. To´th, G. and Valtr, P. "Note on the Erdos-Szekeres Theo- rem." Discr. Comput. Geom. 19, 457 /C1/59, 1998. Happy Number Let the sum of the SQUARES of the DIGITS of a POSITIVE INTEGER s0 be represented by s1 : In a similar way, let the sum of the SQUARES of the DIGITS of s1be represented by s2 ; and so on. If si /C301 for some i ]1; then the original INTEGER s0 is said to be happy. Once it is known whether a number is happy (or not), then any number in the sequence s1 ; s2 ; s3 ; ... will also be happy (or not). A number which is not happy is called UNHAPPY . Unhappy numbers have EVENTUALLY PERIODIC sequences of si which do not reach 1 (e.g., 4, 16, 37, 58, 89, 145, 42, 20, 4, ...). Any PERMUTATION of the DIGITS of an UNHAPPY or happy number must also be unhappy or happy. This follows from the fact that ADDITION is COMMUTATIVE . The first few happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, ... (Sloane’s A007770). These are also the numbers whose 2-RECURRING DIGITAL INVARIANT sequences have period 1. The first few happy primes are 7, 13, 19, 23, 31, 79, 97, 103, 109, 139, ... (Sloane’s A035497). See also KAPREKAR NUMBER ,R ECURRING DIGITAL INVARIANT ,UNHAPPY NUMBER References Dudeney, H. E. Problem 143 in 536 Puzzles & Curious Problems. New York: Scribner, pp. 43 and 258 /C1/59, 1967. Guy, R. K. "Happy Numbers." §E34 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 234 /C1/35, 1994. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 163 /C1/65, 1979. Rivera, C. "Problems & Puzzles: Puzzle Happy Primes.-021." http://www.primepuzzles.net/puzzles/puzz_021.htm. Schwartzman, S. The Words of Mathematics: An Etymologi- cal Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994. Sloane, N. J. A. Sequences A007770 and A035497 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Harada-Norton Group The SPORADIC GROUP HN. References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/HN.html.Harary Graph The smallest K-CONNECTED GRAPH Hk ; nwith n VERTICES , having kn =2 de edges, where xdeis the CEILING FUNCTION (Skiena 1990, p. 179). When n or k is even, Hk ; nis a CIRCULANT GRAPH . Hn/C281; nis the COMPLETE GRAPH Kn(Skiena 1990, p. 180). See also K-CONNECTED GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, 1976. Harary, F. "The Maximum Connectivity of a Graph." Proc. Nat. Acad. Sci. USA 48, 1142 /C1/146, 1962. Skiena, S. "Harary Graphs." §5.1.6 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 179 /C1/ 80, 1990. Harary-Read Number POLYHEX Harborth’s Tiling ATILING consisting of a RHOMBUS such that 17 rhombuses fit around a point and a second tile in the shape of six rhombuses stuck together. These two tiles can fill the plane in exactly four different ways. Two tiles which tile the plane in n ways can be constructed using a rhombus of a shape such that 6n /C287 pack around a point together with a complex piece made by sticking 2n /C282 rhombuses together (Wells 1991). References Harborth, H. "Prescribed Numbers of Tiles and Tilings." Math. Gaz. 61, 296 /C1/99, 1977. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 90 /C1/1, 1991. Hard Hexagon Entropy Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. A constant related to the HARD SQUARE ENTROPY CONSTANT . This constant is given by kh /C13 lim N 0/C12[G(N)]1 =N /C301:395485972... ; (1) where G(N) is the number of configurations of nonattacking KINGS on an n /C29n CHESSBOARD with regular hexagonal cells, where N /C13n2 : Amazingly, kh is algebraic and given by kh /C13 k1 k2 k3 k4 ; (2) where k1 /C134/C28135 =411/C285 =12c /C282 (3) k2 /C13 1 /C28ffiffiffiffiffiffiffiffiffiffiffi 1 /C28cp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27c /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27c /C27c2pq P+2$P+2’ 2 (4) k3 /C13/C281 /C28ffiffiffiffiffiffiffiffiffiffiffi 1 /C28cp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27c /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27c /C27c2pq P+2$P+2’ 2 (5) k4 /C13ffiffiffiffiffiffiffiffiffiffiffi 1 /C28ap /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27a /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27a /C27a2pqP+2$P+2’ /C281 =2 (6) a /C13/C28124 363 111 =3 (7) b /C132501 11979 331 =2 (8) c /C131 4 /C2738 a (b /C271)1 =3 /C28(b /C281)1 =3hino1 =3 : (9) (Baxter 1980, Joyce 1988). References Baxter, R. J. "Partition Function of the Eight-Vertex Lattice Model." Ann. Phys. 70, 193 /C1/28, 1972. Baxter, R. J. "Hard Hexagons: Exact Solution." J. Physics A 13, 1023 /C1/030, 1980. Baxter, R. J. Exactly Solved Models in Statistical Me- chanics. New York: Academic Press, 1982. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/square/square.html. Gaunt, D. S. "Hard-Sphere Lattice Gases. II. Plane-Trian- gular and Three-Dimensional Lattices." J. Chem. Phys. 46, 3237 /C1/259, 1967.Gaunt, D. S. and Fisher, M. E. "Hard-Sphere Lattice Gases. I. Plane-Square Lattice." J. Chem. Phys. 43, 2840 /C1/863, 1965. Joyce, G. S. "On the Hard Hexagon Model and the Theory of Modular Functions." Phil. Trans. Royal Soc. London A 325, 643 /C1/02, 1988. Joyce, G. S. "Exact Results for the Activity and Isothermal Compressibility of the Hard-Hexagon Model." J. Phys. A: Math. Gen. 21, L983-L988, 1988. Plouffe, S. "Hard Hexagons Constant." http://www.lacim.u- qam.ca/piDATA/hardhex.html. Hard Lefschetz Theorem See also LEFSCHETZ THEOREMS Hard Square Entropy Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let F(m; n) be the number of m /C29n BINARY MATRICES with no adjacent 1s (in either columns or rows). For n /C301, 2, ..., F(n ; n) is given by 2, 7, 63, 1234, ... (Sloane’s A006506). The hard square entropy constant is defined by k /C13 lim n0/C12[F(n; n)]1 =n2 /C301:503048082... : The quantity ln k arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the HARD HEXAGON ENTROPY CONSTANT can also be defined. See also BINARY MATRIX References Baxter, R. J.; Enting, I. G.; and Tsang, S. K. "Hard-Square Lattice Gas." J. Statist. Phys. 22, 465 /C1/89, 1980. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/square/square.html. Pearce, P. A. and Seaton, K. A. "A Classical Theory of Hard Squares." J. Statist. Phys. 53, 1061 /C1/072, 1988. Sloane, N. J. A. Sequences A006506/M1816 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hardy Function RIEMANN- SIEGEL FUNCTIONS Hardy Space If 0 Bp B/C12 ; then the Hardy space Hp(D) is the class of functions holomorphic on the disk D and satisfying the growth condition fkkHpsup 0BrB11 2pg2p 0fr eiuP+$P+’P+’2P+’2P+’2P+’2pdu"#1=p B/C12 ; where fkkHpis the Hardy norm. See also BERGMAN SPACE References Duren, P. L. Theory of Hp Spaces. New York: Academic Press, 1970. Garnett, J. Bounded Analytic Functions. New York: Aca- demic Press, 1981. Koosis, P. Introduction to Hp Spaces, 2nd ed. Cambridge, England: Cambridge University Press, 1998. Krantz, S. G. "Hardy Spaces." §12.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 152 /C1/54, 1999. Hardy Z-Function RIEMANN- SIEGEL FUNCTIONS Hardy’s Inequality Let anfg be a NONNEGATIVE SEQUENCE and f(x)a NONNEGATIVE integrable FUNCTION . Define An /C30Xn k /C301ak (1) and F(x) /C30gx 0f(t) dt (2) and take p /C211. For sums, X/C12 n /C301An n !p Bp p /C28 1 !pX/C12 n/C301anðÞp(3) (unless all an /C300); and for integrals, g/C12 0F(x) x"#p dx Bp p /C28 1 !p g/C12 0[f(x)]p dx (4) (unless f is identically 0). See also CARLEMAN’S INEQUALITY References Broadbent, T. A. A. "A Proof of Hardy’s Convergence Theo- rem." J. London Math. Soc. 3, 232 /C1/43, 1928. Elliot, E. B. "A Simple Exposition of Some Recently Proved Facts as to Convergency." J. London Math. Soc. 1,93/C1/6, 1926. Grandjot, K. "On Some Identities Relating to Hardy’s Convergence Theorem." J. London Math. Soc. 3, 114 /C1/ 17, 1928. Hardy, G. H. "Note on a Theorem of Hilbert." Math. Z. 6, 314 /C1/17, 1920. Hardy, G. H. "Notes on Some Points in the Integral Calculus. LX." Messenger Math. 54, 150 /C1/56, 1925. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "Hardy’s Inequality." §9.8 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 239 /C1/43, 1988. Kaluza, T. and Szego, G. "Uuml;ber Reihen mit lauter positiven Gliedern." J. London Math. Soc. 2, 266 /C1/72, 1927. Knopp, K. "U¨ ber Reihen mit positiven Gliedern." J. London Math. Soc. 3, 205 /C1/11, 1928. Landau, E. "A Note on a Theorem Concerning Series of Positive Terms." J. London Math. Soc. 1,38/C1/9, 1926. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. New York: Kluwer, 1991.Opic, B. and Kufner, A. Hardy-Type Inequalities. Essex, England: Longman, 1990. Hardy’s Rule Let the values of a function f(x) be tabulated at points xiequally spaced by h /C30xi /C271 /C28xi ; so f1 /C30fx1ðÞ ; f2 /C30 fx2ðÞ ; ..., f7 /C30fx7ðÞ : Then Hardy’s rule approximating the integral of f(x) is given by the NEWTON- COTES -like formula gx7 x1f(x) dx /C301 100 h 28f1 /C27162f2 /C27220f4 /C27162f6 /C2728f7 ðÞ : See also BODE’S RULE,D URAND’S RULE,N EWTON- COTES FORMULAS ,SHOVELTON’S RULE,SIMPSON’S 3/8 RULE,SIMPSON’S RULE,T RAPEZOIDAL RULE,W ED- DLE’S RULE References King, A. E. "Approximate Integration. Note on Quadrature Formulae: Their Construction and Application to Actuar- ial Functions." Trans. Faculty of Actuaries 9, 218 /C1/31, 1923. Sheppard, W. F. "Some Quadrature-Formulæ." Proc. Lon- don Math. Soc. 32, 258 /C1/77, 1900. Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 151, 1967. Hardy-Littlewood Conjectures The first Hardy-Littlewood conjecture is called the K- TUPLE CONJECTURE . It states that the asymptotic number of PRIME CONSTELLATIONS can be computed explicitly. The second Hardy-Littlewood conjecture states that p(x /C27y) /C28 p(x) 5 p(y) for all x and y, where p(x) is the PRIME COUNTING FUNCTION . Although it is not obvious, Richards (1974) proved that this conjecture is incompatible with the first Hardy-Littlewood conjecture. See also PRIME CONSTELLATION ,PRIME COUNTING FUNCTION References Richards, I. "On the Incompatibility of Two Conjectures Concerning Primes." Bull. Amer. Math. Soc. 80, 419/C1/38, 1974. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 61 /C1/2 and 68 /C1/9, 1994. Hardy-Littlewood Constants PRIME CONSTELLATION Hardy-Littlewood k-Tuple Conjecture PRIME PATTERNS CONJECTURE Hardy-Littlewood Tauberian Theorem Let an ]0 and suppose X/C12 n/C301ane /C28an /C21 a as a 0 0/C27: Then X n5xan /C2x as x 0/C12: This theorem is a step in the proof of the PRIME NUMBER THEOREM , but has subsequently been superseded by an approach due to Wiener (Hardy 1999, p. 34). See also TAUBERIAN THEOREM References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 118 /C1/19, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 34 /C1/5, 1999. Hardy, G. H. and Littlewood, J. E. Quart. J. Math. 46, 215 /C1/ 19, 1915. Hardy, G. H. and Littlewood, J. E. Acta Math. 41, 119 /C1/96, 1918. Karamata. Math. Z. 32, 319 /C1/20, 1930. Hardy-Ramanujan Number The smallest nontrivial TAXICAB NUMBER , i.e., the smallest number representable in two ways as a sum of two CUBES . It is given by 1729 /C3013 /C27123 /C3093 /C27103 : The number derives its name from the following story G. H. Hardy told about Ramanujan. "Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, ‘rather a dull number,’ adding that he hoped that wasn’t a bad omen. ‘No, Hardy,’ said Ramanujan, ‘it is a very interesting number. It is the smallest number expressible as the sum of two [POSITIVE ] cubes in two different ways"’ (Hofstadter 1989, Kanigel 1991, Snow 1993; Hardy 1999, pp. 13 and 68). See also DIOPHANTINE EQUATION–3RD POWERS ,TAXI- CAB NUMBER References Guy, R. K. "Sums of Like Powers. Euler’s Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/44, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 564, 1989.Kanigel, R. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Washington Square Press, p. 312, 1991. Snow, C. P. Foreword to Hardy, G. H. A Mathematician’s Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 37, 1993. Hardy-Ramanujan Theorem Let v(n) be the number of DISTINCT PRIME FACTORS of n.If C(x) tends steadily to infinity with x, then ln ln x /C28C(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ln xp B v(n) Bln ln x /C27C(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln ln xp for ALMOST ALL numbers n Bx."ALMOST ALL" means here the frequency of those INTEGERS n in the interval 1 5n 5x for which v(n) /C28ln ln x jj >C(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln ln xp approaches 0 as x0/C12: / See also DISTINCT PRIME FACTORS ,ERDOS- KAC THE- OREM Harmonic The word "harmonic" has several distinct meanings in mathematics, none of which is obviously related tothe others. SIMPLE HARMONIC MOTION or "harmonic oscillation" refers to oscillations with a sinusoidal waveform. Such functions satisfy the differential equation d2x dt2/C27v2x/C300; (1) which has solution x/C30Acos(vt/C27f1)/C27Bsin(vt/C27f2): (2) The word HARMONIC ANALYSIS is therefore used to describe F OURIER ANALYSIS , which breaks an arbi- trary function into a superposition of sinusoids. In complex analysis, a HARMONIC FUNCTION refers to a real-valued function f(x;y) which satisfies L APLA- CE’S EQUATION 92f(x;y)/C300; (3) where 92is the L APLACIAN . Although this definition is similar to that of harmonic oscillation, it omits the second term in the differential equation. The H ELM- HOLTZ DIFFERENTIAL EQUATION is obtained if it is added back in, 92f(x;y)/C27k2f(x;y)/C300: (4) For distances along a line segment, a HARMONIC RANGE is a set of four COLLINEAR points A,B,C, andDarranged such that AB:BC/C302 : 1 (5) AD:DC/C306:3 : (6) This use of the term probably arises from the use of "harmonics" to refer to ratios of notes in small integers producing an attractive sound, known in music theory as "harmony." For a set of data points xi ; the HARMONIC MEAN is defined by 1 H /C131 nXn i /C3011 xi: (7) The connection of this use of "harmonic" with the preceding ones is not obvious. See also HARMONIC FORM,H ARMONIC FUNCTION , HARMONIC RANGE ,SIMPLE HARMONIC MOTION Harmonic Addition Theorem To convert an equation OF THE FORM f( u) /C30a cos u /C27b sin u (1) to the form f( u) /C30c cos(u /C27 d) ; (2) expand (2) using the trigonometric addition formulas to obtain f( u) /C30c cos u cos d /C28c sin u sin d: (3) Now equate the COEFFICIENTS of (1) and (3) a /C30c cos d (4) b /C30/C28c sin d ; (5) so tan d /C30/C28b a (6) a2 /C27b2 /C30c2 ; (7) and we have d /C30tan /C281/C28b a ! (8) c /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2p : (9) Given two general sinusoidal functions with fre- quency v : c1 /C30A1 sin vt /C27 d1 ðÞ (10) c2 /C30A2 sin vt /C27 d2 ðÞ ; (11) their sum c can be expressed as a sinusoidal function with frequency vc /C13 c1 /C27 c2 /C30A1 sin( vt) cos d1 /C27sin d1 cos(vt) ½/C138 /C27A2 sin( vt) cos d2 /C27sin d2 cos(vt) ½/C138 /C30 A1 cos d1 /C27A2 cos d2 ½/C138 sin( vt) /C27 A1 sin d1 /C27A2 sin d2 ½/C138 cos(vt) : (12) Now, define A cos d /C13A1 cos d1 /C27A2 cos d2 (13) A sin d /C13A1 sin d1 /C27A2 sin d2 : (14) Then (12) becomes A cos d sin(vt) /C27A sin d cos(vt) /C30A sin( vt /C27 d) : (15) Square and add (13) and (14) A2 /C30A2 1 /C27A22 /C272A1A2 cos d2 /C28 d1 ðÞ : (16) Also, divide (14) by (13) tan d /C30A1 sin d1 /C27 A2 sin d2 A1 cos d1 /C27 A2 cos d2; (17) so c /C30A sin( vt /C27 d) ; (18) where A and d are defined by (16) and (17). This procedure can be generalized to a sum of n harmonic waves, giving c /C30Xn i /C301Ai cos vt /C27 di ðÞ /C30A cos(vt /C27 d) ; (19) where A2/C13Xn i/C301Xn j/C301AiAjcosdi/C28djP+$P+’ (20) /C30Xn i/C301A2i/C272Xn i/C301Xn j>1AiAjcosdi/C28djP+$P+’ (21) and tand/C30Pn i/C301AisindiPn i/C301Aicosdi: (22) Harmonic Analysis FOURIER SERIES Harmonic Brick A right-angled PARALLELEPIPED with dimensions a/C29 ab/C29abc;where a,b, and care INTEGERS . See also BRICK, DE BRUIJN’S THEOREM ,EULER BRICK Harmonic Conjugate Function The harmonic conjugate to a given function u(x; y)is a function v(x; y) such that f(x; y) /C30u(x; y) /C27iv(x; y) is COMPLEX DIFFERENTIABLE (i.e., satisfies the CAU- CHY-RIEMANN EQUATIONS ). It is given by v(z) /C30gz z0ux dy /C28uy dx /C27C ; where ux /C13@u =@x; uy /C13@u=@y; and C is a CONSTANT OF INTEGRATION . Note that ux dy /C28uy dx is a CLOSED FORM since u is HARMONIC , uxx /C27vyy /C300: The LINE INTEGRAL is WELL DEFINED on a SIMPLY CONNECTED domain because it is closed. However, on a domain which is not simply connected (such as the punctured disk), the harmonic conjugate may not exist. See also CAUCHY- RIEMANN EQUATIONS ,C OMPLEX DIFFERENTIABLE ,H ARDY SPACE ,H ARMONIC FUNC- TION ,HILBERT TRANSFORM ,SIMPLY CONNECTED References Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 350 /C1/52, 1987. Harmonic Conjugate Points Given COLLINEAR points W, X, Y, and Z, Y and Z are harmonic conjugates with respect to W and X if WYjj YXjj/C30WZjj XZjj: The distances between such points are said to be in HARMONIC RATIO , and the LINE SEGMENT depicted above is called a HARMONIC SEGMENT . Harmonic points divide a LINE SEGMENT internally and exter- nally in the same ratio. If WZjj/C301 ; then WYjj/C30a(1 /C28 a) 1 /C27 a WXjj/C302a a /C27 1 : Harmonic conjugate points are also defined for a TRIANGLE .IfW and X have TRILINEAR COORDINATES a : b : g and a? : b? : g ?; then the TRILINEAR COORDI- NATES of the harmonic conjugates are Y /C30 a /C27 a? : b /C27 b? : g /C27 g ? Z /C30 a /C28 a? : b /C28 b? : g /C28 g? (Kimberling 1994).See also HARMONIC RANGE ,HARMONIC RATIO,POLAR , POLE (INVERSION ) References Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 65, 1928. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Lachlan, R. "Harmonic Ranges and Pencils." Ch. 4 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 24 /C1/6, 1893. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 13 /C1/4, 1990. Phillips, A. W. and Fisher, I. Elements of Geometry. New York: American Book Co., 1896. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Viking Penguin, p. 92, 1992. Harmonic Coordinates Harmonic coordinates satisfy the condition Gl /C13g mv Gl mv /C300 ; (1) or equivalently, @ @xkffiffiffigpg lkP+$P+’ /C300: (2) It is always possible to choose such a system. Using the D’ALEMBERTIAN , I2 f /C13 g lk f; lP+$P+’ ; k/C30g lk @2f @xl@xk/C28Gl@f @xl: (3) But since Gl/C130 for harmonic coordinates, the result is a generalization of the harmonic equation 92x/C300 (4) to I2xm/C300: (5) See also D’ALEMBERTIAN References Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972. Harmonic Decomposition A polynomial function of the elements of a VECTOR x can be uniquely decomposed into a sum of HARMONIC POLYNOMIALS times POWERS ofxjj:/ See also HARMONIC FUNCTION Harmonic Divisor Number A number nfor which the HARMONIC MEAN of the DIVISORS ofn, i.e., nd(n)=s(n);is an INTEGER , where d(n) is the number of POSITIVE integral DIVISORS ofn ands(n) is the DIVISOR FUNCTION . For example, the divisors of n /C30140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140, giving d(140) /C3012 s(140) /C30336 140d(140) s(140)/C30140 /C215 12 336/C305 ; so 140 is a harmonic divisor number. Harmonic divisor numbers are also called ORE NUMBERS . Garcia (1954) gives the 45 harmonic divisor numbers less than 107. The first few are 1, 6, 140, 270, 672, 1638, ... (Sloane’s A007340). For distinct PRIMES p and q, harmonic divisor numbers are equivalent to EVEN PERFECT NUMBERS for numbers OF THE FORM prq : Mills (1972) proved that if there exists an ODD POSITIVE harmonic divisor number n, then n has a prime- POWER factor greater than 107. Another type of number called "harmonic" is the HARMONIC NUMBER . See also DIVISOR FUNCTION ,HARMONIC NUMBER References Edgar, H. M. W. "Harmonic Numbers." Amer. Math. Monthly 99, 783/C1/89, 1992. Garcia, M. "On Numbers with Integral Harmonic Mean." Amer. Math. Monthly 61,8 9/C1/6, 1954. Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed.New York: Springer-Verlag, pp. 45 /C1/3, 1994. Mills, W. H. "On a Conjecture of Ore." Proceedings of the 1972 Number Theory Conference. University of Colorado, Boulder, pp. 142 /C1/46, 1972. Ore, Ø. "On the Averages of the Divisors of a Number." Amer. Math. Monthly 55, 615/C1/19, 1948. Pomerance, C. "On a Problem of Ore: Harmonic Numbers." Unpublished manuscript, 1973. Sloane, N. J. A. Sequences A007340/M4299 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M4299 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and Ore Numbers." Bull. Soc. Math. Gre ´ce (New Ser.) 13,1 2/C1/ 2, 1972. Harmonic Equation LAPLACE’S EQUATION Harmonic Function Any REAL FUNCTION u(x;y) with continuous second PARTIAL DERIVATIVES which satisfies L APLACE’S EQUA- TION , 92u(x;y)/C300; (1) is called a harmonic function. Harmonic functions are called POTENTIAL FUNCTIONS in physics and engineer-ing. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a 3-component VECTOR FIELD to a 1- component SCALAR FUNCTION . A scalar harmonic function is called a SCALAR POTENTIAL , and a vector harmonic function is called a VECTOR POTENTIAL . To find a class of such functions in the PLANE , write the L APLACE’S EQUATION inPOLAR COORDINATES urr/C271 rur/C271 r2uuu/C300; (2) and consider only radial solutions urr/C271 rur/C300: (3) This is integrable by quadrature, so define v/C13du=dr; dv dr/C271 rv/C300 (4) dv v/C30/C28dr r(5) lnv A ! /C30/C28lnr (6) v A/C301 r(7) v/C30du dr/C30A r(8) du/C30Adr r; (9) so the solution is u/C30Alnr: (10) Ignoring the trivial additive and multiplicative con- stants, the general pure radial solution then becomes u/C30ln (x/C28a)2/C27(y/C28b)2hi1=2 /C301 2ln (x/C28a)2/C27(y/C28b)2hi : (11) Other solutions may be obtained by differentiation, such as u/C30x/C28a (x/C28a)2/C27(y/C28b)2(12) v/C30y/C28b (x/C28a)2/C27(y/C28b)2; (13) u/C30exsiny (14) v/C30excosy; (15) and tan /C281y /C28 b x /C28 a ! : (16) Harmonic functions containing azimuthal depen- dence include u /C30rn cos(nu) (17) v /C30rn sin(nu) : (18) The POISSON KERNEL u(r ; R; u; f) /C30R2 /C28 r2 R2 /C28 2rR cos(u /C28 f) /C27 r2(19) is another harmonic function. See also CONFORMAL MAPPING ,DIRICHLET PROBLEM , HARMONIC ANALYSIS ,H ARMONIC DECOMPOSITION , HARNACK’S INEQUALITY ,HARNACK’S PRINCIPLE ,KEL- VIN TRANSFORMATION ,LAPLACE’S EQUATION ,POISSON INTEGRAL ,P OISSON KERNEL ,S CALAR POTENTIAL , SCHWARZ REFLECTION PRINCIPLE ,S UBHARMONIC FUNCTION ,VECTOR POTENTIAL References Ash, J. M. (Ed.). Studies in Harmonic Analysis. Washing- ton, DC: Math. Assoc. Amer., 1976. Axler, S.; Bourdon, P.; and Ramey, W. Harmonic Function Theory. Springer-Verlag, 1992. Benedetto, J. J. Harmonic Analysis and Applications. Boca Raton, FL: CRC Press, 1996. Cohn, H. Conformal Mapping on Riemann Surfaces. New York: Dover, 1980. Krantz, S. G. "Harmonic Functions." §1.4.1 and Ch. 7 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 16 and 89 /C1/01, 1999. Weisstein, E. W. "Books about Potential Theory." http:// www.treasure-troves.com/books/PotentialTheory.html. Harmonic Homology A PERSPECTIVE COLLINEATION with center O and axis o not incident is called a HOMOLOGY .A HOMOLOGY is said to be harmonic if the points A and A? on a line through O are harmonic conjugates with respect to O and o /C215 a: Every PERSPECTIVE COLLINEATION of period two is a harmonic homology. See also HOMOLOGY (GEOMETRY ), PERSPECTIVE COL- LINEATION References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 248, 1969. Harmonic Logarithm For all INTEGERS n and NONNEGATIVE INTEGERS t, the harmonic logarithms l(t) n (x) of order t and degree n are defined as the unique functions satisfying1. l(t) n (x) /C30(ln x)t ;/ 2. l(t) n (x) has no constant term except l(0) 0(x) /C301;/ 3.d dxl(t) n (x) /C30 nbel(t) n/C281(x) ;/ where the "ROMAN SYMBOL " / nbe / is defined by nbe/C13n for n "0 1 for n /C300P+2k (1) (Roman 1992). This gives the special cases l(0)n(x) /C30xnfor n ]0 0 for n B0P+2k (2) l(1)n(x) /C30xn(ln x /C28Hn) for n ]0 xn for n B0;P+2k (3) where Hn is a HARMONIC NUMBER Hn /C13Xn k/C3011 k : (4) The harmonic logarithm has the INTEGRAL g l(1) n (x) dx /C301 n /C27 1 bel(1)n/C271(x) : (5) The harmonic logarithm can be written l(t) n (x) /C30 nbe! ˜D/C28n(ln x)t ; (6) where ˜D is the DIFFERENTIAL OPERATOR , (so ˜D/C28n is the nth INTEGRAL ). Rearranging gives ˜Dk l(t) n (x) /C30nbe! n /C28 k be$’ !l(t) n/C28k(x) : (7) This formulation gives an analog of the BINOMIAL THEOREM called the LOGARITHMIC BINOMIAL FORMU- LA. Another expression for the harmonic logarithm is l(t) n(x)/C30xnXt j/C300(/C281)j(t)jc(j) n(lnx)t/C28j; (8) where ( t)j/C30t(t/C281)/C1/C1/C1(t/C28j/C271) is a P OCHHAMMER SYMBOL and c(j) nis a two-index HARMONIC NUMBER (Roman 1992). See also LOGARITHM ,ROMAN FACTORIAL References Loeb, D. and Rota, G.-C. "Formal Power Series of Logarith- mic Type." Advances Math. 75,1/C1/18, 1989. Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641/C1/48, 1992. Harmonic Map A map u:M0N;between two COMPACT RIEMAN- NIAN MANIFOLDS , is a harmonic map if it is a critical point for the energy functional gMdujj2d mM : The norm of the differential dujj is given by the metric on M and N and dmMis the measure on M. Typically, the class of allowable maps lie in a fixed HOMOTOPY CLASS of maps. The EULER- LAGRANGE DIFFERENTIAL EQUATION for the energy functional is a non-linear ELLIPTIC PAR- TIAL DIFFERENTIAL EQUATION . For example, when M is the circle, then the Euler-Lagrange equation is the same as the geodesic equation. Hence, u is a closed geodesic iff u is harmonic. The map from the circle to the equator of the standard 2-sphere is a harmonic map, and so are the maps that take the circle and map it around the equator n times, for any integer n. Note that these all lie in the same HOMOTOPY CLASS . A higher dimensional example is a MEROMORPHIC FUNCTION on a compact RIEMANN SURFACE , which is a harmonic map to the RIEMANN SPHERE . A harmonic map may not always exist in a HOMOTOPY CLASS , and if it does it may not be unique. When N is negatively curved, a harmonic representative exists for each HOMOTOPY CLASS , and is also unique. For surfaces, the harmonic maps have been classified, and are precisely the holomorphic maps and the anti- holomorphic maps. Thus by HODGE’S THEOREM for surfaces, there are no non-trivial harmonic maps from the SPHERE to the TORUS . A harmonic map between RIEMANNIAN MANIFOLDS can be viewed as a generalization of a GEODESIC when the domain DIMENSION is one, or of a HARMONIC FUNCTION when the range is a EUCLIDEAN SPACE . See also BOCHNER IDENTITY ,CALCULUS OF VARIA- TIONS ,C URVATURE ,E UCLIDEAN SPACE ,E ULER- LA- GRANGE DIFFERENTIAL EQUATION ,G EODESIC , HARMONIC FUNCTION ,HODGE’S THEOREM ,HOMOTOPY CLASS,RIEMANNIAN MANIFOLD ,RIEMANN SURFACE References Burstal, F.; Lemaire, L.; and Rawnsley, J. "Harmonic Maps Bibliography." http://www.bath.ac.uk/~masfeb/harmo- nic.html. Eels, J. and Lemaire, L. "A Report on Harmonic Maps." Bull. London Math. Soc. 10,1/C1/8, 1978. Eels, J. and Lemaire, L. "Another Report on Harmonic Maps." Bull. London Math. Soc. 20, 385 /C1/24, 1988. Harmonic Mean The harmonic mean Hx1 ; ...; xn ðÞ of n points xi (where i /C301, ..., n)is 1 H /C131 nXn i /C3011 xi: (1) The special cases of n /C302 and n /C303 are therefore given byHx1 ; x2 ðÞ /C302x1x2 x1 /C27 x2(2) Hx1 ; x2 ; x3 ðÞ /C303x1x2x3 x1x2 /C27 x1x3 /C27 x2x3; (3) and so on. The VOLUME -to-SURFACE AREA ratio for a cylindrical container with height h and radius r and the MEAN CURVATURE of a general surface are related to the harmonic mean. Hoehn and Niven (1985) show that Ha1 /C27c; a2 /C27c ; ...; an /C27c ðÞ > c /C27Ha1 ; a2 ; ...; an ðÞ (4) for any POSITIVE constant c. See also ARITHMETIC MEAN,ARITHMETIC- GEOMETRIC MEAN,G EOMETRIC MEAN,H ARMONIC- GEOMETRIC MEAN,HARMONIC RANGE ,ROOT-MEAN-SQUARE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151 /C1/56, 1985. Kenney, J. F. and Keeping, E. S. "Harmonic Mean." §4.13 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 57 /C1/8, 1962. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995. Harmonic Mean Index The statistical INDEX PH /C13Pv0 Pv0p0 pn/C30Pp0q0 Pp2 0q0 pn; where pnis the price per unit in period n, qnis the quantity produced in period n, and vn/C13pnqnthe value of the nunits, and subscripts 0 indicate the reference year. See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 69, 1962. Harmonic Number A number OF THE FORM Hn/C30Xn k/C3011 k: (1) This can be expressed analytically as Hn/C30g/C27c0(n/C271); (2) where gis the E ULER- MASCHERONI CONSTANT and C(x)/C30c0(x) is the DIGAMMA FUNCTION . The number formed by taking alternate signs in the sum also has an analytic solution H?n/C30Xn k/C301(/C281)k/C271 k(3) /C30ln 2/C271 2(/C281)nc012n/C2712P+’kP+’7 /C28c012n/C271P+’kP+’7 hi : (4) The first few harmonic numbers Hnare 1, 3 =2;11=6; 25=12;137=60;... (Sloane’s A001008 and A002805). The harmonic numbers are implemented in Mathe- matica 4.0 asHarmonicNumber [n]. The harmonic number Hnis never an INTEGER except forH1;which can be proved by using the strong triangle inequality to show that the 2-ADIC VALUE of Hnis greater than 1 for n/C211. This result was proved in 1915 by Taeisinger, and the more general results that any number of consecutive terms not necessarily starting with 1 never sum to an integer was proved byKu¨rscha´k in 1918 (Hoffman 1998, p. 157). The harmonic numbers have ODD NUMERATORS and EVEN DENOMINATORS . The nth harmonic number is given asymptotically by Hn/C2lnn/C27g/C271 2n; (5) where gis the E ULER- MASCHERONI CONSTANT (Con- way and Guy 1996). Gosper gave the interestingidentity X /C12 i/C300ziHi i!/C30/C28ezX/C12 k/C301(/C28z)k kk!/C30ez[lnz/C27G(0;z)/C27g];(6) where G(0;z) is the incomplete GAMMA FUNCTION and gis the E ULER- MASCHERONI CONSTANT . Borwein and Borwein (1995) show that X/C12 n/C301H2 n (n/C271)2/C3011 4z(4)/C3011 360p4(7) X/C12 n/C301H2 n n2/C3017 4z(4)/C3017 360p4(8) X/C12 n/C301Hn n3/C305 4z(4)/C301 72p4; (9) where z(z) is the R IEMANN ZETA FUNCTION . The first of these had been previously derived by de Doelder (1991), and the last by Euler (1775). These identities are corollaries of the identity 1 pgp 0x2ln 2 cos1 2xP+’kP+’7hino2 dx/C3011 2z(4)/C3011 180p4(10) (Borwein and Borwein 1995). Additional identitiesdue to Euler are X/C12 n/C301Hn n2/C302z(3) (11) 2X/C12 n/C301Hn nm/C30(m/C272)z(m/C271) /C28Xm/C282 n/C301z(m/C28n)z(n/C271) (12) form/C302, 3, ... (Borwein and Borwein 1995), where z(3) is A PE´RY’S CONSTANT . These sums are related to so-called E ULER SUMS . There is an unexpected connection between the harmonic numbers and the R IEMANN HYPOTHESIS . Harmonic numbers of order rcan be defined by the relationship H(r) n/C30Xn k/C3011 kr: (13) These number are built into Mathematica 4.0 as HarmonicNumber [n,r]. These numbers obey the unexpected identity 9H(n) 8/C2819H(n) 9/C2710H(n) 10/C27Xn/C281 k/C301H(n/C28k) 8H(k) 9/C28H(n/C28k) 9H(k) 9P+2 /C28H(n/C28k) 8H(k) 10/C27H(n/C28k) 9H(k) 10/C138/C300 (14) (M. Trott). Conway and Guy (1996) define the second harmonic number by H2 n/C13Xn i/C301Hi/C30(n/C271)Hn/C271/C281P+$P+’ /C30(n/C271)Hn/C271/C28H1P+$P+’ ; (15) the third harmonic number by H3 n/C13Xn i/C301H(2) i/C30n/C272 2P+’vP+’u Hn/C272/C28H2P+$P+’ ; (16) and the nth harmonic number by Hk n/C30n/C27k/C281 k/C281P+’vP+’u (Hn/C27k/C281/C28Hk/C281): (17) A slightly different definition of a two-index harmonicnumber c (j) nis given by Roman (1992) in connection with the HARMONIC LOGARITHM . Roman (1992) de- fines this by c(0) n/C301 for n]0 0 for nB0P+2k (18) c(j) 0/C301 for j/C300 0 for j"0P+2k (19) plus the RECURRENCE RELATION cn(j) n /C30c(j/C281) n/C27nc(j) n/C281 : (20) For general n /C210 and j /C210, this is equivalent to c(j) n /C30Xn i/C3011 ic(j/C281) i ; (21) and for n /C210, it simplifies to c(j) n /C30Xn i/C301n iP+’vP+’u (/C281)i/C281i /C28j : (22) For n B0, the harmonic number can be written c(j) n /C30(/C281)j /C28n /C27!s(/C28n; j) ; (23) where nbe! is the R OMAN FACTORIAL and s is a STIRLING NUMBER OF THE FIRST KIND . A separate type of number sometimes also called a "harmonic number" is a HARMONIC DIVISOR NUMBER (or O RE NUMBER ). See also APE´ RY’S CONSTANT ,EULER SUM,HARMONIC LOGARITHM ,HARMONIC SERIES ,ORE NUMBER ,RAMA- NUJAN FUNCTION ,UNIT FRACTION References Borwein, D. and Borwein, J. M. "On an Intriguing Integral and Some Series Related to z(4):/" Proc. Amer. Math. Soc. 123, 1191 /C1/198, 1995. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 143 and 258 /C1/59, 1996. de Doelder, P. J. "On Some Series Containing C(x) /C28C(y) and (C(x) /C28C(y))2 for Certain Values of x and y." J. Comp. Appl. Math. 37, 125 /C1/41, 1991. Flajolet, P. and Salvy, B. "Euler Sums and Contour Integral Representation." Experim. Math. 7,15/C1/5, 1998. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Harmonic Numbers" and "Harmonic Summation." §6.3 and 6.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 272 /C1/ 82, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641 /C1/48, 1992. Roman, S. The Umbral Calculus. New York: Academic Press, p. 99, 1984. Sloane, N. J. A. Sequences A001008/M2885 and A002805/ M1589 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Harmonic Progression HARMONIC SERIES Harmonic Range A set of four COLLINEAR points A, B, C, and Darranged such that AB : BC /C302:1 AD : DC /C306:3 : Hardy (1967) uses the term HARMONIC SYSTEM OF POINTS to refer to a harmonic range. See also BIVALENT RANGE ,EULER LINE,GERGONNE LINE,HARMONIC CONJUGATE POINTS ,SODDY LINE References Casey, J. "Theory of Harmonic Section." §6.3 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 87 /C1/4, 1888. Durell, C. V. "Harmonic Ranges and Pencils." Ch. 6 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 65 /C1/7, 1928. Graustein, W. C. "Harmonic Division." Ch. 4 in Introduction to Higher Geometry. New York: Macmillan, pp. 50 /C1/4, 1930. Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press,pp. 99 and 106, 1967. Lachlan, R. "Harmonic Properties." §288/C1 /90 in An Elemen- tary Treatise on Modern Pure Geometry. London: Macmil- lian, pp. 177 and 267 /C1/68, 1893. Harmonic Ratio HARMONIC RANGE Harmonic Segment HARMONIC CONJUGATE POINTS Harmonic Series The SUM X/C12 k/C3011 k(1) is called the harmonic series. It can be shown to DIVERGE using the INTEGRAL TEST by comparison with the function 1 =x:The divergence, however, is very slow. The generalization of the harmonic series z(n)/C13X/C12 k/C3011 kn(2) is known as the R IEMANN ZETA FUNCTION . The sum X/C12 k/C3011 pk(3) taken over all PRIMES pkalso diverges (Wells 1986, p. 41) with asymptotic behavior Xx k /C3011 pk/C2ln ln x /C27O(1) (4) (Hardy 1999, p. 50). Rather surprisingly, the ALTERNATING SERIES X/C12 k/C301( /C281)k /C281 k/C30ln 2 (5) converges to the natural logarithm of 2. An explicit formula for the partial sum of the alternating series is given by Xn k /C301(/C281)k/C281 k /C30ln 2 /C271 2(/C281)n c012 /C2712 nP+’kP+’7 /C28 c01 /C2712 nP+’kP+’7 hi : (6) Gardner (1984) notes that this series never reaches an integral sum. The sum of the first few terms of the harmonic series is given analytically by the nth HARMONIC NUMBER Hn /C30Xn j/C3011 j/C30 g /C27 c0(n /C271); (7) where g is the EULER- MASCHERONI CONSTANT and C(x) /C30 c0(x) is the DIGAMMA FUNCTION . The number of terms needed to exceed 1, 2, 3, ... are 1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, ... (Sloane’s A004080). Using the analytic form shows that after 2:5 /C29108 terms, the sum is still less than 20. Furthermore, to achieve a sum greater than 100, more than 1:509 /C291043 terms are needed! Written explicitly, the number of terms is 15,092,688,622,113,788,323,693,563,264,538,101,449, 859,497 (Gardner 1984, p. 167). Progressions OF THE FORM 1 a1;1 a1 /C27 d ;1 a1 /C27 2d ; ... (8) are also sometimes called harmonic series (Beyer 1987). The partial sums of the harmonic series are plotted in the left figure above, together with two related series. See also ARITHMETIC SERIES ,BERNOULLI’S PARADOX , BOOK STACKING PROBLEM ,E ULER SUM,M ERTENS CONSTANT , Q-HARMONIC SERIES ,ZIPF’S LAW References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 279 /C1/80, 1985.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987. Boas, R. P. and Wrench, J. W. "Partial Sums of the Harmo- nic Series." Amer. Math. Monthly 78, 864 /C1/70, 1971. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 165 /C1/72, 1984. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 217, 1998. Honsberger, R. "An Intriguing Series." Ch. 10 in Mathema- tical Gems II. Washington, DC: Math. Assoc. Amer., pp. 98 /C1/03, 1976. Rosenbaum, B. "Solution to Problem E46." Amer. Math. Monthly 41, 48, 1934. Sloane, N. J. A. Sequences A004080 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 41, 1986. Harmonic System of Points HARMONIC RANGE Harmonic-Geometric Mean Let an/C271 /C302an bn an /C27 bn bn /C271 /C30ffiffiffiffiffiffiffiffiffiffi an bnp ; then H(a0 ; b0) /C13 lim n 0/C12an /C301 M a/C281 0; b/C281 0P+$P+’ ; where M is the ARITHMETIC-GEOMETRIC MEAN . See also ARITHMETIC MEAN,ARITHMETIC- GEOMETRIC MEAN,GEOMETRIC MEAN,HARMONIC MEAN Harmonious Graph A connected LABELED GRAPH with n EDGES in which all VERTICES can be labeled with distinct INTEGERS (mod n) so that the sums of the PAIRS of numbers at the ends of each EDGE are also distinct (mod n). The LADDER GRAPH , FAN, WHEEL GRAPH ,PETERSEN GRAPH , TETRAHEDRAL GRAPH , DODECAHEDRAL GRAPH , and ICOSAHEDRAL GRAPH are all harmonious (Graham and Sloane 1980). See also GRACEFUL GRAPH ,LABELED GRAPH ,POST- AGE STAMP PROBLEM ,SEQUENTIAL GRAPH References Gallian, J. A. "Open Problems in Grid Labeling." Amer. Math. Monthly 97, 133/C1/35, 1990. Gardner, M. Wheels, Life, and other Mathematical Amuse- ments. New York: W. H. Freeman, p. 164, 1983. Graham, R. L. and Sloane, N. "On Additive Bases and Harmonious Graphs." SIAM J. Algebraic Discrete Math. 1, 382 /C1/04, 1980. Guy, R. K. "The Corresponding Modular Covering Problem. Harmonious Labelling of Graphs." §C13 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 127 /C1/28, 1994. Harmonograph A device consisting of two coupled pendula, usually oscillating at right angles to each other, which are attached to a pen. The resulting damped SIMPLE HARMONIC MOTION can produce beautiful, complicated curves which eventually terminate in a point as the motion of the pendula is damped by friction. In the absence of friction, the figures produced by a harmo- nograph would be LISSAJOUS CURVES . See also LISSAJOUS CURVE ,SIMPLE HARMONIC MO- TION ,SPIROGRAPH References Cundy, H. and Rollett, A. "The Harmonograph." §5.5.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 244 /C1/48, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 92 /C1/3, 1991. Harnack’s Inequality Let D /C30D(z0 ; R)bean OPEN DISK, and let u be a HARMONIC FUNCTION on D such that u(z) ]0 for all z /C23 D: Then for all z /C23 D ; we have 0 5u(z) 5R R /C28 z /C28 z0 jj !2 u(z0) : See also HARMONIC FUNCTION ,HARNACK’S PRINCIPLE , LIOUVILLE’S CONFORMALITY THEOREM References Flanigan, F. J. "Harnack’s Inequality." §2.5.1 in Complex Variables: Harmonic and Analytic Functions. New York: Dover, pp. 88 /C1/0, 1983. Krantz, S. G. "The Harnack Inequality." §7.6.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 97, 1999. Harnack’s Principle Let u1 5u2 5... be HARMONIC FUNCTIONS on a con- nected open set U ⁄C : Then either uj 0/C12 uniformly on compact sets or there is a finite-values HARMONIC FUNCTION u on U such that uj0uuniformly on compact sets.See also HARMONIC FUNCTION ,HARNACK’S INEQUAL- ITY References Krantz, S. G. "Harnack’s Principle." §7.6.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 97, 1999. Harnack’s Theorems Letsibe the orders of singular points on a curve (Coolidge 1959, p. 56). Harnack’s first theorem states that a real irreducible curve of order ncannot have more than 1 2(n/C281)(n/C282)/C28X si(si/C281)/C271 circuits (Coolidge 1959, p. 57). Harnack’s second theorem states that there exists a curve of every order with the maximum number of circuits compatible with that order and with a certain number of double points, provided that number is notpermissible for a curve of lower order (Coolidge 1959,p. 61). References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, 1959. Harry Dym Equation The PARTIAL DIFFERENTIAL EQUATION ut/C30uxxxu3: References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 53, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 133, 1997. Harshad Number APOSITIVE INTEGER which is DIVISIBLE by the sum of itsDIGITS , also called a Niven number (Kennedy et al. 1980) or a multidigital number (Kaprekar 1955). The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (Sloane’s A005349). Grundman (1994) proved that there is no sequence of more than 20 consecutiveHarshad numbers, and found the smallest sequenceof 20 consecutive Harshad numbers, each member of which has 44,363,342,786 digits. Grundman (1994) defined an n-Harshad (or n-Niven) number to be a POSITIVE INTEGER which is DIVISIBLE by the sum of its digits in base n]2:Cai (1996) showed that for n/C302 or 3, there exists an infinite family of sequences of consecutive n-Harshad num- bers of length 2 n:/ Define an all-Harshad (or all-Niven) number as a positive integer which is divisible by the sum of its digits in all bases n ]2: Then only 1, 2, 4, and 6 are all-Harshad numbers (A. Kertesz). References Cai, T. "On 2-Niven Numbers and 3-Niven Numbers." Fib. Quart. 34, 118 /C1/20, 1996. Cooper, C. N. and Kennedy, R. E. "Chebyshev’s Inequality and Natural Density." Amer. Math. Monthly 96, 118 /C1/24, 1989. Cooper, C. N. and Kennedy, R. "On Consecutive Niven Numbers." Fib. Quart. 21, 146 /C1/51, 1993. Grundman, H. G. "Sequences of Consecutive n-Niven Num- bers." Fib. Quart. 32, 174 /C1/75, 1994. Kaprekar, D. R. "Multidigital Numbers." Scripta Math. 21, 27, 1955. Kennedy, R. E. and Cooper, C. N. "On the Natural Density of the Niven Numbers." Abstract 816 /C1/1 /C1/19, Abstracts Amer. Math. Soc. 6, 17, 1985. Kennedy, R.; Goodman, R.; and Best, C. "Mathematical Discovery and Niven Numbers." MATYC J. 14,21/C1/5, 1980. Sloane, N. J. A. Sequences A005349/M0481 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Vardi, I. "Niven Numbers." §2.3 in Computational Recrea- tions in Mathematica. Redwood City, CA: Addison-Wes- ley, pp. 19 and 28 /C1/1, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 171, 1986. Hart Circle The CIRCLE H which touches the INCIRCLES I, IA ; IB ; and ICof a CIRCULAR TRIANGLE ABC and its ASSO- CIATED TRIANGLES . It is either externally tangent to I and internally tangent to incircles of the ASSOCIATED TRIANGLES IA ; IB ; and IC(as in the above figure), or vice versa. The Hart circle has several properties which are analogous to the properties on the NINE- POINT CIRCLE of a linear triangle. There are eight Hart circles associated with a given CIRCULAR TRIAN- GLE. The Hart circle of any CIRCULAR TRIANGLE and the Hart circles of the three ASSOCIATED TRIANGLES havea common tangent circle which touches the former in the opposite sense to that which it touches the latter (Lachlan 1893, p. 254). In addition, the CIRCUMCIR- CLE of any CIRCULAR TRIANGLE is the Hart circle of the CIRCULAR TRIANGLE formed by the circumcircles of the inverse associated triangles (Lachlan 1893, p. 254). See also ASSOCIATED TRIANGLES ,CIRCLE ,CIRCULAR TRIANGLE References Casey, J. "On the Equations and Properties--(1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane." Proc. Roy. Irish Acad. 9, 396 /C1/23, 1864 /C1/866. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 43, 1971. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 127 /C1/28, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 254 /C1/57, 1893. Larmor, A. "Contacts of Systems of Circles." Proc. London Math. Soc. 23, 136 /C1/57, 1891. Hart’s Inversor A LINKAGE which draws the inverse of a given curve. It can also convert circular to linear motion. The rods satisfy AB/C30CD and BC/C30DA, and O,P, and P? remain COLLINEAR . Coxeter (1969, p. 428) shows that ifAO/C30mAB;then OP/C29OP?/C30m(1/C28m)(AD2/C28AB2): See also LINKAGE ,PEAUCELLIER INVERSOR References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods. Oxford, England: Oxford University Press, p. 157, 1978. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 82 /C1/3, 1969. Mannheim, A. "Sur l’inverseur de Hart." Messenger Math. , p. 151, Nov. 1896. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 124 /C1/29, 1957. Hart’s Theorem Any one of the eight A POLLONIUS CIRCLES of three given CIRCLES isTANGENT to a CIRCLE Hknown as a HART CIRCLE , as are the other three APOLLONIUS CIRCLES having (1) like contact with two of the given CIRCLES and (2) unlike contact with the third. See also APOLLONIUS CIRCLES ,HART CIRCLE References Casey, J. "On the Equations and Properties--(1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane." Proc. Roy. Irish Acad. 9, 396 /C1/23, 1864 /C1/866. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 106 /C1/07, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 43, 1971. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 127 /C1/28, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 254 /C1/57, 1893. Larmor, A. "Contacts of Systems of Circles." Proc. London Math. Soc. 23, 136 /C1/57, 1891. Hartley Transform An INTEGRAL TRANSFORM which shares some features with the FOURIER TRANSFORM , but which (in the discrete case), multiplies the KERNEL by cos2pkn N ! /C28sin2pkn N ! (1) instead of e /C282pikn=N /C30cos2pkn N ! /C28i sin2 pkn N ! : (2) The Hartley transform produces REAL output for a REAL input, and is its own inverse. It therefore can have computational advantages over the DISCRETE FOURIER TRANSFORM , although analytic expressions are usually more complicated for the Hartley trans- form. The discrete version of the Hartley transform can be written explicitly as H[a] /C131ffiffiffiffiffi NpXN /C281 n/C300ancos2pkn N ! /C28sin2pkn N ! "# (3) /C30RF[a] /C28IF[a] ; (4) where F denotes the FOURIER TRANSFORM . The Hartley transform obeys the CONVOLUTION property H[a +b]k /C301 2AkBk /C28 ¯Ak¯Bk /C27Ak¯Bk /C27 ¯AkBkP+$P+’ ; (5) where ¯a0 /C13a0 (6)¯an=2 /C13an=2 (7) ¯ak /C13an/C28k (8) (Arndt). Like the FAST FOURIER TRANSFORM , there is a "fast" version of the Hartley transform. A decimation in time algorithm makes use of Hleft n[a] /C13Hn=2 aeven½/C138/C27XHn=2aoddP+2P+3 (9) Hright n[a] /C13Hn=2 aeven½/C138/C28XHn=2aoddP+2P+3 ; (10) where X denotes the sequence with elements an cospn N ! /C28 ¯an sinpn N ! : (11) A decimation in frequency algorithm makes use of Heven n[a] /C30Hn=2aleft /C27arightP+2P+3 ; (12) Hoddn[a] /C30Hn=2X aleft /C28arightP+$P+’P+2P+3 : (13) The DISCRETE FOURIER TRANSFORM Ak /C13F[a] /C30XN /C281 n/C300e /C282 pikn=Nan (14) can be written Ak A/C28kP+2$P+2’ /C30XN /C281 n/C300e /C282pikn=N 0 0 e/C282pikn=NP+2$P+2’ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Fan anP+2$P+2’ (15) /C30XN/C281 n/C3001 21/C28i1/C27i 1/C27i1/C28iP+2$P+2’ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} T/C281cos2pkn N ! sin2pkn N ! /C28sin2pkn N ! cos2pkn N !2 666643 77775 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} H /C21 21/C27i1/C28i 1/C28i1/C27iP+2$P+2’ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Tan anP+2$P+2’ ; (16) so F/C30T/C281HT: (17) See also DISCRETE FOURIER TRANSFORM ,FAST FOUR- IER TRANSFORM ,FOURIER TRANSFORM References Arndt, J. "The Hartley Transform (HT)." Ch. 2 in "Remarks on FFT Algorithms." http://www.jjj.de/fxt/. Bracewell, R. N. The Fourier Transform and Its Applica- tions, 3rd ed. New York: McGraw-Hill, 1999. Bracewell, R. N. The Hartley Transform. New York: Oxford University Press, 1986. Haruki’s Theorem Given three circles, each intersecting the other two in two points, the line segments connecting their points of intersection satisfy ace bdf/C301 (Honsberger 1995). See also CIRCULAR TRIANGLE ,T RIQUETRA ,V ENN DIAGRAM References Honsberger, R. "Haruki’s Cevian Theorem for Circles." §12.4 inEpisodes in Nineteenth and Twentieth Century Eucli- dean Geometry. Washington, DC: Math. Assoc. Amer., pp. 144 /C1/46, 1995. Hash Function A hash function Hprojects a value from a set with many (or even an infinite number of) members to a value from a set with a fixed number of (fewer)members. Hash functions are not reversible. A hash function Hmight, for instance, be defined as / y/C30H(x)/C3010x(mod 1) bc /, where x/C23R;y/C23[0;9];and xbcis the FLOOR FUNCTION . Hash functions can be used to determine if two objects are equal (possibly with a fixed averagenumber of mistakes). Other common uses of hash functions are CHECKSUMS over a large amount of data (e.g., the CYCLIC REDUNDANCY CHECK [CRC]) and finding an entry in a database by a key value. TheUNIX c-shell (csh) uses a hash table to store the location of executable programs. As a result addingnew executables in a user’s search path requiresregeneration of the hash table using the rehash command before these programs can be executedwithout specifying the complete path. To illustrate the use of hash functions in database lookups, consider a database consisting of an arraycontaining an index n, a name, and a telephone number, with names listed in arbitrary order. n Name Number 0 Parker 12345 1 (empty)2 Davis 43534 3 Harris 32452 4 Corea 465325 Hancock 965626 Brecker 37811 7 (empty) /N/C281/Marsalis 54323 To look up Hancock from this array, you would start at the beginning of the array, compare the names, then try the next until the names match. This very simple algorithm finds any entry in 1 to Nsteps, giving an average seek time of N=2:The seek time is therefore proportional to N. A much faster result can generally be achieved, if the database is sorted. n Name 0 Brecker 1 Corea2 Davis3 Hancock 4 Harris 5 Marsalis6 Parker7 (empty) /N/C281/(empty) An efficient algorithm on this sorted array first checks entry N=2;and then recursively uses bisection to check entries in intervals [0 ;N=2/C281] or [ N=2/C27 1;N/C281];depending wether the most recently looked-up name precedes or succeeds the name sought. The average seek time of this procedure this is proportional to ln N:/ The idea behind using a hash function here is thatalthough the possible number of combinations of characters in a name is quite large, only a subset of them is usually found in practice (i.e., names such as "Kwqrst" are much less common than names like "Jones.") Therefore, when you insert an entry into the database at an index that can somehow be calculated using a key (which is also available at the time you search for it), you might be able to find it later at the first location you check. Consider the following simple example in which the hash function H is simply the sum of ASCII codes of characters in a name (considered to be all in lower- case) computed mod N /C3013. Name H Brecker 6 Corea 2 Davis 2 Hancock 12 Harris 12 Marsalis 2 Parker 8 The above example illustrates that the hash function can give the same results for different keys. This difficulty is typically circumvented by introducing a second hash function H2whose results are designed to be completely different from that of H. For illustrative purposes, let H2be one plus the bitwise exclusive or of all codes in a name (again taken as all lower-case) mod N /C281 : This gives the following table. Name /H2/ Brecker 11 Corea 3 Davis 10 Hancock 4 Harris 8 Marsalis 3 Parker 8 A new index can then be calculated as the sum of the first index and H2(mod N) until an empty slot is found where new data can be stored. Note that when using H2 as an offset to walk through the database, it is not, in general, guaranteed that any key will eventually reach any slot. However, for certain values of N, namely N a PRIME NUMBER , such behavior isguaranteed, so N is always chosen to be PRIME . After computing H2 with N /C3013 (a PRIME ), the above phone list would look like this for names added in alphabetic order. Index Key Compares To Find 0 (empty) 1 (empty) 2 Corea 1 3 Hancock 2 4 (empty) 5 Marsalis 2 6 Brecker 1 7 Harris 2 8 Parker 1 9 (empty) 10 (empty) 11 (empty) 12 Davis 2 The average seek time for locating a name in this table depends on the kind of data, N, and the quality of the hash functions used. However, for reasonable choices of hash functions, it will be much smaller than ln N :/ See also COLLISION- FREE HASH FUNCTION ,CRYPTO- GRAPHIC HASH FUNCTION ,C YCLIC REDUNDANCY CHECK ,O NE-WAY HASH FUNCTION ,H ASH TABLE , UNIVERSAL HASH FUNCTION Hash Table A database accessed by one or more HASH FUNCTIONS . See also HASH FUNCTION HashLife A LIFE ALGORITHM that achieves remarkable speed by storing subpatterns in a HASH FUNCTION table, and using them to skip forward, sometimes thousands of generations at a time. HashLife takes tremendous amounts of memory and can’t show patterns at every step, but can quickly calculate the outcome of a pattern that takes millions of generations to com- plete. See also HASH FUNCTION ,LIFE Hasse Diagram A graphical rendering of a PARTIALLY ORDERED SET displayed via the COVER relation of the PARTIALLY ORDERED SET with an implied upward orientation. A point is drawn for each element of the POSET , and line segments are drawn between these points according to the following two rules: 1. If x By in the poset, then the point correspond- ing to x appears lower in the drawing than the point corresponding to y. 2. The line segment between the points corre- sponding to any two elements x and y of the poset is included in the drawing IFF x covers y or y covers x. Hasse diagrams are also called UPWARD DRAWINGS . A Hasse diagram of a GRAPH may be generated using HasseDiagram [g] in the Mathematica add-on pack- ageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). References Skiena, S. "Hasse Diagrams." §5.4.2 in Implementing Dis- crete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 163 and 206 /C1/08, 1990. Hasse Principle A collection of equations satisfies the Hasse principle if, whenever one of the equations has solutions in R and all the Qp ; then the equations have solutions in the RATIONALS Q: Examples include the set of equations ax2 /C27bxy /C27cy2 /C300 with a, b, and c INTEGERS , and the set of equations x2 /C27y2 /C30a for a rational. The trivial solution x /C30y /C300 is usually not taken into account when deciding if a collection of homogeneous equations satisfies the Hasse principle. The Hasse principle is sometimes called the local- global principle. See also GLOBAL FIELD,LOCAL FIELD Hasse’s Algorithm COLLATZ PROBLEM Hasse’s Conjecture Define the ZETA FUNCTION of a VARIETY over a NUMBER FIELD by taking the product over all PRIME IDEALS of the ZETA FUNCTIONS of this VARIETY reduced modulo the PRIMES . Hasse conjectured that this product has a MEROMORPHIC continuation over the whole plane and a functional equation.See also MEROMORPHIC FUNCTION ,PRIME IDEAL References Lang, S. "Some History of the Shimura-Taniyama Conjec- ture." Not. Amer. Math. Soc. 42, 1301 /C1/307, 1995. Hasse’s Resolution Modulus Theorem The JACOBI SYMBOL (a =y) /C30 x(y)asa CHARACTER can be extended to the KRONECKER SYMBOL (f(a) =y) /C30 x/C31(y) so that x/C31(y) /C30 x(y) whenever x(y) "0 : When y is RELATIVELY PRIME to f(a) ; then x /C31(y) "0; and for NONZERO values x/C31(y1) /C30 x /C31(y2) IFF y1 /C13y2mod /C27f(a): In addition, f(a) jj is the minimum value for which the latter congruence property holds in any extension symbol for x(y) :/ See also CHARACTER (NUMBER THEORY ), JACOBI SYMBOL ,KRONECKER SYMBOL References Cohn, H. Advanced Number Theory. New York: Dover, pp. 35 /C1/6, 1980. Hasse-Davenport Relation Let F be a FINITE FIELD with q elements, and let Fs be a FIELD containing F such that Fs : F ½/C138 /C30s : Let x be a nontrivial MULTIPLICATIVE CHARACTER of F and x?/C30 x(NFs =F a character of Fs : Then /C28g(x) ðÞs/C30/C28g x ?ðÞ; where g(x)isaG AUSSIAN SUM. See also GAUSSIAN SUM,MULTIPLICATIVE CHARACTER References Ireland, K. and Rosen, M. "A Proof of the Hasse-Davenport Relation." §11.4 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 162 /C1/65, 1990. Hasse-Minkowski Theorem Two nonsingular forms are equivalent over the rationals IFF they have the same DETERMINANT and the same P-SIGNATURES for all p. Hat The hat is a CARET -shaped symbol most commonly used to denote a UNIT VECTOR (e.g., ˆv)oran ESTIMA- TOR (e.g., ˆx): The symbol ˆx is voiced "x-hat." The hat symbol is more commonly known as the circumflex (Bringhurst 1997, p. 274). See also BAR,C ARET ,E STIMATOR ,M ACRON ,U NIT VECTOR References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, 1997. Hat-Box Theorem ARCHIMEDES’ HAT-BOX THEOREM Haupt-Exponent The smallest exponent e for which be /C131 (mod 1); where b and n are given numbers, is called the haupt-exponent (or sometimes "ORDER ") of b (mod n). The number of bases having a haupt-exponent e is f(e) ; where f(e) is the TOTIENT FUNCTION . Cunning- ham (1922) published the haupt-exponents for primes to 25409 and bases 2, 3, 5, 6, 7, 10, 11, and 12. Haupt-exponents exists for n which are not factors of b. For example, the haupt-exponent of 10 (mod 7) is 6, since 106 /C131 (mod 7): The haupt-exponent of 10 mod an integer n relatively prime to 10 gives the period of the DECIMAL EXPAN- SION of the reciprocal of n (Glaisher 1878, Lehmer 1941). For example, the haupt-exponent of 10 (mod 13) is 6, and 1 13 /C300:0769230 ; which has period 6. The haupt-exponent of 2 mod an integer n relatively prime to 2 gives the multiplica- tive order of 2 (mod 2n /C271) (Golomb 1961). The following table gives the first few haupt-expo- nents for bases b (mod p) with p /C301, 2, .... b Sloane haupt-exponents 2 A002326 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, 20, 18, ... 3 A050975 1, 2, 4, 6, 2, 4, 5, 3, 6, 4, 16, 18, 4, 5, ... 4 A050976 1, 2, 3, 3, 5, 6, 2, 4, 9, 3, 11, 10, 9, 14, ... 5 A050977 1, 2, 1, 2, 6, 2, 6, 5, 2, 4, 6, 4, 16, 6, 9, ... 6 A050978 1, 2, 10, 12, 16, 9, 11, 5, 14, ... 7 A050979 1, 1, 2, 4, 1, 2, 3, 4, 10, 2, 12, 4, 2, 16, ... 8 A050980 2, 4, 1, 2, 10, 4, 4, 8, 6, 2, 11, 20, 6, 28, ... 9 A050981 1, 1, 2, 3, 1, 2, 5, 3, 3, 2, 8, 9, 2, 5, 11, ... 10 A002329 1, 6, 1, 2, 6, 16, 18, 6, 22, 3, 28, ... See also COMPLETE RESIDUE SYSTEM ,M ULTIPLICA- TIVE ORDER ,ORDER (MODULO ), ORDER (POLYNOMIAL ), PRIMITIVE ROOTReferences Cunningham, A. Haupt-Exponents, Residue Indices, Primi- tive Roots. London: F. Hodgson, 1922. Glaisher, J. W. L. "Periods of Reciprocals of Integers Prime to 10." Proc. Cambridge Philos. Soc. 3, 185 /C1/06, 1878. Golomb, S. W. "Permutations by Cutting and Shuffling." SIAM Rev. 3, 293 /C1/97, 1961. Lehmer, D. H. "Guide to Tables in the Theory of Numbers." Bulletin No. 105. Washington, DC: National Research Council, pp. 7 /C1/2, 1941. Nagell, T. "Exponent of an Integer Modulo n." §31 in Introduction to Number Theory. New York: Wiley, pp. 102 /C1/06, 1951. Sloane, N. J. A. Sequences A0023260936, A0023294045, A050975, A050976, A050977, A050978, A050979, A050980, and A050981 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Hausdorff HAUSDORFF SPACE Hausdorff Axioms The axioms formulated by Hausdorff (1914) for his concept of a TOPOLOGICAL SPACE . These axioms describe the properties satisfied by subsets of ele- ments x in a NEIGHBORHOOD SET E of x. 1. There corresponds to each point x at least one NEIGHBORHOOD U(x) ; and each NEIGHBORHOOD U(x) contains the point x. 2. If U(x) and V(x) are two NEIGHBORHOODS of the same point x, there must exist a NEIGHBORHOOD W(x) that is a subset of both. 3. If the point y lies in U(x) ; there must exist a NEIGHBORHOOD U(y) that is a SUBSET of U(x) :/ 4. For two different points x and y, there are two corresponding NEIGHBORHOODS U(x) and U(y) with no points in common. See also HAUSDORFF SPACE ,TOPOLOGICAL SPACE References Hausdorff, F. Grundzu ¨ge der Mengenlehre. Leipzig, Ger- many: von Veit, 1914. Republished as Set Theory, 2nd ed. New York: Chelsea, 1962. Hausdorff Dimension Informally, SELF-SIMILAR objects with parameters N andsare described by a power law such as N/C30sd; where d/C30lnN lns is the " DIMENSION " of the scaling law, known as the Hausdorff dimension. Formally, let Abe a SUBSET of a METRIC SPACE X. Then the Hausdorff dimension D(A)o f Ais the INFIMUM of d ]0 such that the d-dimensional HAUS- DORFF MEASURE of A is 0 (which need not be an INTEGER ). In many cases, the Hausdorff dimension correctly describes the correction term for a resonator with FRACTAL PERIMETER in Lorentz’s conjecture. How- ever, in general, the proper dimension to use turns out to be the MINKOWSKI- BOULIGAND DIMENSION (Schroeder 1991). See also CAPACITY DIMENSION ,FRACTAL ,FRACTAL DIMENSION ,M INKOWSKI- BOULIGAND DIMENSION , SELF-SIMILARITY References Duvall, P.; Keesling, J.; and Vince, A. "The Hausdorff Dimension of the Boundary of a Self-Similar Tile." J. London Math. Soc. 61, 649 /C1/60, 2000. Federer, H. Geometric Measure Theory. New York: Springer-Verlag, 1969. Harris, J. W. and Stocker, H. "Hausdorff Dimension." §4.11.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 113 /C1/14, 1998. Hausdorff, F. "Dimension und a¨ußeres Maß." Math. Ann. 79, 157 /C1/79, 1919. Ott, E. "Appendix: Hausdorff Dimension." Chaos in Dyna- mical Systems. New York: Cambridge University Press, pp. 100 /C1/03, 1993. Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 41 /C1/ 5, 1991. Hausdorff Measure Let X be a METRIC SPACE , A be a SUBSET of X, and d a number ]0 : The d-dimensional Hausdorff measure of A, Hd(A) ; is the INFIMUM of POSITIVE numbers y such that for every r /C210, A can be covered by a countable family of closed sets, each of diameter less than r, such that the sum of the dth POWERS of their diameters is less than y. Note that Hd(A) may be infinite, and d need not be an INTEGER . References Federer, H. Geometric Measure Theory. New York: Springer-Verlag, 1969. Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, p. 103, 1993. Rogers, C. A. Hausdorff Measures, 2nd ed. Cambridge, England: Cambridge University Press, 1999. Hausdorff Moment Problem MOMENT PROBLEM Hausdorff Paradox For n ]3; there exist no additive finite and invariant measures for the group of displacements in Rn :/ References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 49, 1983.Hausdorff Space A TOPOLOGICAL SPACE in which any two points have disjoint NEIGHBORHOODS . A space that is Hausdorff is sometimes said to "have Hausdorff topology" or "be Hausdorff." See also HAUSDORFF MEASURE ,TOPOLOGICAL SPACE References Porter, J. R. Extensions and Absolutes of Hausdorff Spaces. New York: Springer-Verlag, 1987. Hausdorff Topology HAUSDORFF SPACE Hausdorff-Besicovitch Dimension CAPACITY DIMENSION Hauy Construction The construction of polyhedra using identical build- ing blocks. The illustrations above show such con- structions for the OCTAHEDRON and RHOMBIC DODECAHEDRON . In Book XIII of the ELEMENTS , Euclid used a Hauy construction to build the DODE- CAHEDRON (Wells 1991). See also OCTAHEDRAL NUMBER ,OCTAHEDRON ,RHOM- BIC DODECAHEDRAL NUMBER ,RHOMBIC DODECAHE- DRON References Hauy, R.-J. "Essai d’une the ´orie sur la structure des crystals applique ´ea`plusieurs genres de substances crystallise ´es." 1784. Weisstein, E. W. "Ha ¨uy Construction." M ATHEMATICA NOTE- BOOK HAUY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 93, 1991. Haversine hav(x)/C131 2vers( x)/C3012(1/C28cosx); where vers( x) is the VERSINE and cos xis the COSINE . Using a trigonometric identity, the haversine is equal to hav(x) /C30sin2(1 2 x) : See also COSINE ,COVERSINE ,EXSECANT ,SPHERICAL TRIGONOMETRY ,VERSINE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 78, 1972. Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, p. 18, 1960. h-Cobordism An h-cobordism is a COBORDISM W between two MANIFOLDS M1and M2such that W is SIMPLY CONNECTED and the inclusion maps M1 0 W and M2 0 W are HOMOTOPY equivalences. h-Cobordism Theorem If W is a SIMPLY CONNECTED , COMPACT MANIFOLD with a boundary that has two components, M1and M2 ; such that inclusion of each is a HOMOTOPY equivalence, then W is DIFFEOMORPHIC to the product M1 /C29[0; 1] for dim M1ðÞ]5: In other words, if M and M ? are two simply connected MANIFOLDS of DIMEN- SION ]5 and there exists an H-COBORDISM W between them, then W is a product M /C29I and M is DIFFEO- MORPHIC to M ?:/ The proof of the h-cobordism theorem can be accom- plished using SURGERY . A particular case of the h- cobordism theorem is the POINCARE ´ CONJECTURE in dimension n ]5: Smale proved this theorem in 1961. See also DIFFEOMORPHISM ,POINCARE ´ CONJECTURE , SURGERY References Smale, S. "Generalized Poincare ´’s Conjecture in Dimensions Greater than Four." Ann. Math. 74, 391 /C1/06, 1961.Heads-Minus-Tails Distribution A fair COIN is tossed an even 2n number of times. Let D /C13 H /C28T jj be the absolute difference in the number of heads and tails obtained. Then the probability distribution is given by P(D /C302k) /C301 2P+’kP+’72n2n nP+’vP+’u k /C300 21 2P+’kP+’72n2n n /C27kP+’vP+’u k /C301 ; 2; ...;8 >>< >>: where P(D /C302k /C281) /C300: The most probable value of D is D /C302, and the expectation value is D nhi/C30n2n nP+’vP+’u 22n /C281: The generating function for Dhiis given by X Dnhixn/C281 /C30(1 /C28x) /C283 =2 /C301 /C273 2 x /C2715 8x2 /C273516 x3 /C27... (Sloane’s A001803 and A046161; Abramowitz and Stegun 1972, Pre´vost 1933; Hughes 1995). These numbers also arise in 1-D RANDOM WALKS . See also BERNOULLI DISTRIBUTION ,C OIN,C OIN TOSSING ,RANDOM WALK–1- D References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 798, 1972. Handelsman, M. B. Solution to Problem 436, "Distributing ‘Heads’ Minus ‘Tails."’ College Math. J. 22, 444/C1/46, 1991. Pre´vost, G. Tables de Fonctions Sphe ´riques. Paris: Gau- thier-Villars, pp. 156 /C1/57, 1933. Hughes, B. D. Eq. (7.282) in Random Walks and Random Environments, Vol. 1: Random Walks. New York: Oxford University Press, p. 513, 1995. Sloane, N. J. A. Sequences A001803/M2986 and A046161 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Heap ASEQUENCE anfgN n/C301forms a (binary) heap if it satisfies aj=2bc5ajfor 25j5N;where xbcis the FLOOR FUNCTION , which is equivalent to /aiBa2i/and ai Ba2i/C271for 1 5i 5(i /C281)=2: The first member must therefore be the smallest. A heap can be viewed as a labeled BINARY TREE in which the label of the ith node is smallest than the labels of any of its descendents (Skiena 1990, p. 35). Heaps support arbitrary inser- tion and seeking/deletion of the minimum value in O(ln n) times per update (Skiena 1990, p. 38). A list can be converted to a heap in O(n) times using an algorithm due to Floyd (1964). A binary heap can be generated from a PERMUTATION p using Heapi- fy[p] in the Mathematica add-on package Discre- teMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). For ex- ample, given the RANDOM PERMUTATION f6; 2; 7; 9; 5; 3; 4; 8; 10; 1g; Floyd’s algorithm gives the heap f1; 2; 3; 8 ; 5 ; 7 ; 4 ; 9 ; 10 ; 6 g (left figure). The right figure shows a heap containing 30 elements. A PERMUTATION can be tested to see if it is a heap using the following Mathematica functions. BBDiscreteMath‘Combinatorica‘; HeapQ[a_List?PermutationQ] : /C30 Module[{i, n /C30 Length[a]}, And @@ Table[a[[Floor[i/2]]] B a[[i]], {i, 2, n}] ] n heaps 1 {1} 2 {1, 2} 3 {1, 2, 3}, {1, 3, 2} 4 {1, 2, 3, 4}, {1, 2, 4, 3}, {1, 3, 2, 4} The numbers of heaps on n /C301, 2, ... elements are 1, 1, 2, 3, 8, 20, 80, 896, 3360, ... (Sloane’s A056971), the first few of which are summarized in the above table. The number of heaps of l levels (or equivalently, the number of heaps of 2l /C281 elements) is given by the RECURRENCE RELATION Sl /C302l /C282 2l /C281 /C281P+’vP+’u S2 l/C281 with S1 /C301 (Skiena 1990, p. 36), the values of which for l /C301, 2, ... are 1, 2, 80, 21964800, 74836825861835980800000, ... (Sloane’s A056972).See also BINARY TREE,C OMPLETE BINARY TREE, HEAPSORT ,PRIORITY QUEUE References Floyd, R. W. "Algorithm 245: Treesort 3." Comm. ACM 7, 701, 1964. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. Skiena, S. "Heaps." §1.4.4 in Implementing Discrete Mathe- matics: Combinatorics and Graph Theory with Mathema- tica. Reading, MA: Addison-Wesley, pp. 35 /C1/9, 1990. Skiena, S. S. "Heaps." §1.4.4 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 35 /C1/9, 1997. Sloane, N. J. A. Sequences A056971 and A056972 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Heapsort An O(n lg n) SORTING ALGORITHM which is not quite as fast as QUICKSORT . It is a "sort-in-place" algorithm and requires no auxiliary storage, which makes it particularly concise and elegant to implement. See also HEAP,QUICKSORT ,SORTING References Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, pp. 144 /C1/48, 1998. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Heapsort." §8.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 327 /C1/29, 1992. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 38 /C1/9, 1990. Heart Surface A heart-shaped surface given by the SEXTIC EQUATION 2x2/C272y2/C27z2/C281P+$P+’3/C281 10x2z3/C28y2z3/C300: See also ARCHIMEDEAN SPIRAL ,BONNE PROJECTION , CARDIOID ,PIRIFORM References Nordstrand, T. "Heart." http://www.uib.no/people/nfytn/ hearttxt.htm. Heat Conduction Equation APARTIAL DIFFERENTIAL diffusion equation OF THE FORM @T @t/C30k92T: (1) Physically, the equation commonly arises in situa- tions where kis the thermal diffusivity and Tthe temperature. The 1-D heat conduction equation is @T @t/C30k@2T @x2: (2) This can be solved by SEPARATION OF VARIABLES using T(x;t)/C30X(x)T(t): (3) Then XdT dt/C30kTd2X dx2: (4) Dividing both sides by kXTgives 1 kTdT dt/C301 Xd2X dx2/C30/C281 l2; (5) where each side must be equal to a constant. Anticipating the exponential solution in T, we have picked a negative separation constant so that thesolution remains finite at all times and lhas units of length. The Tsolution is T(t)/C30Ae /C28kt=l2; (6) and the Xsolution is X(x)/C30Ccosx l ! /C27Dsinxl ! : (7) The general solution is then T(x;t)/C30T(t)X(x) /C30Ae /C28kt=l2Ccosx l ! /C27Dsinx l ! "# /C30e/C28kt=l2Dcosx l ! /C27Esinx l ! "# : (8) If we are given the boundary conditions T(0;t)/C300 (9) and T(L;t)/C300; (10)then applying (9) to (8) gives Dcosxl ! /C300[D/C300; (11) and applying (10) to (8) gives EsinL l ! /C300[L l/C30np[l/C30L np; (12) so (8) becomes Tn(x;t)/C30Ene/C28k(np=L)2tsinnpx L ! : (13) Since the general solution can have any n, T(x;t)/C30X/C12 n/C301cnsinnpx L ! e/C28k(np=L)2t: (14) Now, if we are given an initial condition T(x;0);we have T(x;0)/C30X/C12 n/C301cnsinnpx L ! : (15) Multiplying both sides by sin( mpx=L) and integrating from 0 to Lgives gL 0sinmpx L ! T(x;0)dx /C30gL 0X/C12 n/C301cnsinmpx L ! sinnpx L ! dx: (16) Using the ORTHOGONALITY of sin( nx) and sin( mx); X/C12 n/C301cngL 0sinnpx L ! sinmpx L ! dx/C30X/C12 n/C3011 2pdmncn /C3012pcm/C30gL 0sinmpx L ! T(x;0)dx; (17) so cn/C302 pgL 0sinmpx L ! T(x;0)dx: (18) If the boundary conditions are replaced by the requirement that the derivative of the temperaturebe zero at the edges, then (9) and (10) are replaced by @T @xj (0;t)/C300 (19) @T @xj (L;t)/C300: (20) Following the same procedure as before, a similar answer is found, but with sine replaced by cosine: T(x; t) /C30X/C12 n/C301cn cosn px L ! e /C28 k(np =L)2t ; (21) where cn /C302 p gL 0cosmpx L ! @T(x; 0) @xj t/C300dx: (22) Heat Conduction EquationDisk To solve the HEAT CONDUCTION EQUATION on a 2-D disk of radius R /C301, try to separate the equation using T(r; u; t) /C30R(r) U(u)T(t) : (1) Writing the u and r terms of the LAPLACIAN in SPHERICAL COORDINATES gives 92 /C30d2R dr2 /C272 rdR dr/C271 r2d2 U du2 ; (2) so the HEAT CONDUCTION EQUATION becomes RU kd2T dt2 /C30d2R dr2UT /C272 rdR drUT /C271 r2d2 U du2 RT : (3) Multiplying through by r2 =RUT gives r2 kTd2T dt2 /C30r2 Rd2R dr2 /C272r RdR dr/C27d2 U du21 U: (4) The u term can be separated. d2 U du21 U/C30/C28n(n /C271); (5) which has a solution U( u) /C30A cosffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n(n /C271)p uhi /C27B sinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin(n /C271)p uhi : (6) The remaining portion becomes r2 kTd2T dt2 /C30r2 Rd2R dr2 /C272r RdR dr/C28n(n /C271): (7) Dividing by r2 gives 1 kTd2T dt2 /C301 Rd2R dr2 /C272 rRdR dr/C28n(n /C27 1) r2/C30/C281 l2 ; (8) where a NEGATIVE separation constant has been chosen so that the t portion remains finite T(t) /C30Ce /C28 kt=l2 : (9) The radial portion then becomes1 Rd2R dr2 /C272 rRdR dr/C28n(n /C27 1) r2/C271 l2 /C300 (10) r2d2R dr2 /C272rdR dr/C27r2 l2 /C28n(n /C271)"# R /C300 ; (11) which is the SPHERICAL BESSEL DIFFERENTIAL EQUA- TION . If the initial temperature is T(r ; 0) /C300 and the boundary condition is T(1; t) /C301; the solution is T(r ; t) /C301 /C282X/C12 n/C301J0( anr) anJ1( an)e a2 nt ; (12) where anis the nth POSITIVE zero of the BESSEL FUNCTION OF THE FIRST KIND J0(x) :/ Heaviside Calculus The study, first developed by Boole, of SHIFT-INVAR- IANT OPERATORS which are polynomials in the DIF- FERENTIAL OPERATOR ˜D:Heaviside calculus can be used to solve any ORDINARY DIFFERENTIAL EQUATION OF THE FORM p(˜D)f(x)/C30g(x) with p(0)"0;and is frequently implemented using LAPLACE TRANSFORMS . See also DIFFERENTIAL OPERATOR ,LAPLACE TRANS- FORM ,SHIFT- INVARIANT OPERATOR References Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684/C1/60, 1973. Heaviside Step Function A discontinuous "step" function also called the unit step, and defined by H(x)/C300xB0 1 2x/C300 1x>0:8 < :(1) It is related to the BOXCAR FUNCTION by Y (x)/C30Hx/C271 2P+’kP+’7 /C28Hx/C2812P+’kP+’7 (2) and can be defined in terms of the SGN function by H(x)/C301 2[1/C27sgn(x)]: (3) The shorthand notation Hc(x)/C13H(x/C28c) (4) is sometimes also used. The Heaviside step function is given by the Mathematica command UnitStep [x]. The DERIVATIVE is given by d dxH(x)/C30d(x); (5) where d(x) is the DELTA FUNCTION , and the step function is related to the RAMP FUNCTION /R(x)/by d dxR(x)/C30/C28H(x) (6) R(x)/C30xH(x) (7) R(x)/C30H(x)+H(x); (8) where +denotes CONVOLUTION . Bracewell (1999) gives many identities, some of which include the following. Letting +denote the CONVOLU- TION , H(x)+f(x)/C30gx /C28/C12f(x?)dx? (9) H(t)+H(t)/C30g/C12 /C28/C12H(u)H(t/C28u)du (10) /C30H(0)g/C12 0H(t/C28u)du /C30H(0)H(t)gt 0du/C30tH(t): (11) In addition, H(ax/C27b)/C30Hx/C27b a ! H(a)/C27H/C28x/C28b a ! H(/C28a) /C30Hx/C27b a ! a>0 H/C28x/C28b a ! aB0:8 >>>>< >>>>:(12) The Heaviside step function can be defined by the following limits, H(x)/C30lim t001 2/C271 ptan/C281x t !"# (13) /C301ffiffiffipplim t00g/C12 /C28xt/C281e/C28u2=t2du /C301 2lim t00erfc/C28x t ! (14) /C301 plim t00gx /C28/C12t/C281sincu t ! du /C301 plim t00gx /C28/C121 usinu t ! (15) /C3012/C271plim t00sipx t ! (16) /C30lim t001 2ex=tforx50 1/C281 2e/C28x=tforx]0( (17) /C30lim t001 1/C27e/C28x=t(18) /C30lim t00ee/C28x=t(19) /C301 2lim t001/C27tanhx t !"# (20) /C30lim t00gx /C28/C12t/C281Lx/C281 2t t ! dx; (21) where erfc( x) is the ERFC function, si( x) is the SINE INTEGRAL , sinc xis the SINC FUNCTION , andL(x) is the one-argument TRIANGLE FUNCTION . The first four of these are illustrated above for t/C300:2;0.1, and 0.01. Of course, any monotonic function with constant unequal horizontal asymptotes is a Heaviside step function under appropriate scaling and possiblereflection. The F OURIER TRANSFORM of the Heaviside step function is given by F[H(x)] /C30g/C12 /C28/C12e /C282 pikxH(x) dx /C301 2d(k) /C28i pk"# ; (22) where d(k) is the DELTA FUNCTION . See also ABSOLUTE VALUE ,BOXCAR FUNCTION ,DELTA FUNCTION ,F OURIER TRANSFORM– HEAVISIDE STEP FUNCTION ,RAMP FUNCTION ,RAMP FUNCTION ,REC- TANGLE FUNCTION ,SGN,SQUARE WAVE,TRIANGLE FUNCTION References Bracewell, R. "Heaviside’s Unit Step Function, H(x) :/" The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 57 /C1/1, 1999. Spanier, J. and Oldham, K. B. "The Unit-Step u(x /C28a) and Related Functions." Ch. 8 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 63 /C1/9, 1987. Heawood Conjecture The bound for the number of colors which are SUFFICIENT for MAP COLORING on a surface of GENUS g, g(g) /C301 2(7 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 48g /C271p )jk is the best possible, where xbcis the FLOOR FUNCTION . g(g) is called the CHROMATIC NUMBER , and the first few values for g /C300, 1, ... are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, ... (Sloane’s A000934). The fact that g(g) is also NECESSARY was proved by Ringel and Youngs (1968) with two exceptions: the SPHERE (PLANE ), and the KLEIN BOTTLE . When the FOUR-COLOR THEOREM was proved in 1976, the KLEIN BOTTLE was left as the only exception, in that the Heawood formula gives seven, but the correct bound is six (as demonstrated by the FRANKLIN GRAPH ). The four most difficult cases to prove in the FOUR-COLOR THEOREM were g /C3059, 83, 158, and 257. See also CHROMATIC NUMBER ,FOUR- COLOR THEO- REM,FRANKLIN GRAPH ,M AP COLORING ,SIX-COLOR THEOREM ,TORUS COLORING References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 244, 1976. Franklin, P. "A Six Color Problem." J. Math. Phys. 13, 363 /C1/ 79, 1934. Heawood, P. J. "Map Colour Theorem." Quart. J. Math. 24, 332 /C1/38, 1890. Ringel, G. Map Color Theorem. New York: Springer-Verlag, 1974. Ringel, G. and Youngs, J. W. T. "Solution of the Heawood Map-Coloring Problem." Proc. Nat. Acad. Sci. USA 60, 438 /C1/45, 1968. Sloane, N. J. A. Sequences A000934/M3292 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wagon, S. "Map Coloring on a Torus." §7.5 in Mathematica in Action. New York: W. H. Freeman, pp. 232 /C1/37, 1991.Heawood Graph The seven-color torus map on 14 nodes illustrated above. The Heawood graph is a CAGE GRAPH and is 4- transitive, but not 5-transitive (Harary 1994, p. 173). The Heawood graph is the point/line INCIDENCE GRAPH on the F ANO PLANE (Royle). See also CAGE GRAPH ,FANO PLANE ,SZILASSI POLY- HEDRON ,TORUS COLORING References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 236 and 244, 1976. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 173, 1994. Royle, G. "Cubic Cages." http://www.cs.uwa.edu.au/~gordon/ cages/. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 192, 1990. Weisstein, E. W. "Graphs." M ATHEMATICA NOTEBOOK GRAPHS.M . Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/2, 1982. Hebesphenomegacorona JOHNSON SOLID J89:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Hecke Algebra An associative RING , also called a HECKE RING , which has a technical definition in terms of commensurable SUBGROUPS . Hecke L-Function A generalization of the EULER L-FUNCTION associated with a GRO¨ SSENCHARAKTER . See also EULER L-FUNCTION ,G RO¨ SSENCHARAKTER , HECKE L-SERIES References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Hecke L-Series See also HECKE L-FUNCTION References Koch, H. "Applications of Hecke L-Series." Ch. 8 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 259 /C1/73, 2000. Hecke Operator A family of operators mapping each SPACE Mkof MODULAR FORMS onto itself. For a fixed integer k and any POSITIVE INTEGER n, the Hecke operator Tnis defined on the set Mkof entire modular forms of weight k by (Tnf)(t) /C30nk/C281X djnd/C28kXd/C281 b /C300fnt /C27 bd d2 ! : (1) For n a PRIME p, the operator collapses to (Tpf)(t) /C30pk /C281f(p t) /C271 pXp /C281 b/C300t /C27 b p ! : (2) If f /C23 Mk has the FOURIER SERIES f(t) /C30X/C12 m/C300c(m)e2 pim t ; (3) then Tnf has FOURIER SERIES TnfðÞ (t) /C30X/C12 m/C300gn(m)e2 pim t ; (4) where gn(m) /C30X d½(n; m)dk /C281cmn d2 ! (5) (Apostol 1997, p. 121). If (m; n) /C301; the Hecke operators obey the composi- tion propertyTmTn /C30Tmn : (6) Any two Hecke operators T(n) and T(m)on Mk COMMUTE with each other, and moreover T(m)T(n) /C30X d½(m; n)dk /C281Tmn d2 ! (7) (Apostol 1997, pp. 126 /C1/27). Each Hecke operator Tnhas eigenforms when the dimension of Mk is 1, so for k /C304, 6, 8, 10, and 14, the eigenforms are the EISENSTEIN SERIES G4 ; G6 ; G8 ; G10 ; and G14 ; respectively. Similarly, each Tnhas eigen- forms when the dimension of the set of CUSP FORMS Mk;0is 1, so for k/C3012, 16, 18, 20, 22, and 26, the eigenforms are D;DG4;DG6;DG8;DG10;andDG14; respectively, where Dis the MODULAR DISCRIMINANT of the W EIERSTRASS ELLIPTIC FUNCTION (Apostol 1997, p. 130). See also HECKE ALGEBRA ,MODULAR FORM References Apostol, T. M. "The Hecke Operators." §6.7 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed.New York: Springer-Verlag, pp. 120 /C1/22, 1997. Hecke Ring HECKE ALGEBRA Hectogon A 100-sided POLYGON , virtually indistinguishable in appearance from a CIRCLE except at very high magnification. Hedgehog An envelope parameterized by its G AUSS MAP . The PARAMETRIC EQUATIONS for a hedgehog are x/C30p(u) cos u/C27p?(u) sin u y/C30p(u) sin u/C27p?(u) cos u: A plane convex hedgehog has at least four VERTICES where the CURVATURE has a stationary value. A plane convex hedgehog of constant width has at least six VERTICES (Martinez-Maure 1996). References Langevin, R.; Levitt, G.; and Rosenberg, H. "He ´rissons et Multihe ´rissons (Enveloppes parame ´tre´es par leur applica- tion de Gauss." Warsaw: Singularities, 245 /C1/53, 1985. Banach Center Pub. 20, PWN Warsaw, 1988. Martinez-Maure, Y. "A Note on the Tennis Ball Theorem." Amer. Math. Monthly 103, 338 /C1/40, 1996. Heegaard Diagram A diagram expressing how the gluing operation that connects the HANDLEBODIES involved in a HEEGAARD SPLITTING proceeds, usually by showing how the meridians of the HANDLEBODY are mapped. See also HANDLEBODY ,HEEGAARD SPLITTING References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 239, 1976. Heegaard Splitting A Heegaard splitting of a connected orientable 3- MANIFOLD M is any way of expressing M as the UNION of two (3,1)- HANDLEBODIES along their boundaries. The boundary of such a (3,1)- HANDLEBODY is an orientable SURFACE of some GENUS , which determines the number of HANDLES in the (3,1)- HANDLEBODIES . Therefore, the HANDLEBODIES involved in a Heegaard splitting are the same, but they may be glued together in a strange way along their boundary. A diagram showing how the gluing is done is known as aH EEGAARD DIAGRAM . References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 255, 1994. Heegner Number The values of /C28d for which IMAGINARY QUADRATIC FIELDS Q(ffiffiffiffiffiffiffi /C28dp ) are uniquely factorable into factors OF THE FORM a /C27bffiffiffiffiffiffiffi /C28dp ): Here, a and b are half-integers, except for d /C301 and 2, in which case they are INTEGERS . The Heegner numbers therefore corre- spond to DISCRIMINANTS /C28d which have CLASS NUM- BER h(/C28d) equal to 1, except for Heegner numbers /C281 and /C282, which correspond to d /C30/C28 4 and /C288, respectively. The determination of these numbers is called GAUSS’S CLASS NUMBER PROBLEM , and it is now known that there are only nine Heegner numbers: /C281, /C282, /C283, /C287, /C2811, /C2819, /C2843, /C2867, and /C28163 (Sloane’s A003173), corresponding to discriminants /C284, /C288, /C283, /C287, /C2811, /C2819, /C2843, /C2867, and /C28163, respec- tively. Heilbronn and Linfoot (1934) showed that if a larger d existed, it must be 109 : Heegner (1952) published a proof that only nine such numbers exist, but his proof was not accepted as complete at the time. Subsequent examination of Heegner’s proof show it to be "essen- tially" correct (Conway and Guy 1996).The Heegner numbers have a number of fascinating connections with amazing results in PRIME NUMBER theory. In particular, the J-FUNCTION provides stun- ning connections between e, p; and the ALGEBRAIC INTEGERS . They also explain why Euler’s PRIME- GENERATING POLYNOMIAL n2 /C28n /C2741 is so surpris- ingly good at producing PRIMES . See also CLASS NUMBER ,D ISCRIMINANT (BINARY QUADRATIC FORM), GAUSS’S CLASS NUMBER PROBLEM , J-FUNCTION ,PRIME- GENERATING POLYNOMIAL ,QUAD- RATIC FIELD,RAMANUJAN CONSTANT References Conway, J. H. and Guy, R. K. "The Nine Magic Discrimi- nants." In The Book of Numbers. New York: Springer- Verlag, pp. 224 /C1/26, 1996. Heegner, K. "Diophantische Analysis und Modulfunktio- nen." Math. Z. 56, 227 /C1/53, 1952. Heilbronn, H. A. and Linfoot, E. H. "On the Imaginary Quadratic Corpora of Class-Number One." Quart. J. Math. (Oxford) 5, 293 /C1/01, 1934. Sloane, N. J. A. Sequences A003173/M0827 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Heesch Number The Heesch number of a closed plane figure is the maximum number of times that figure can be com- pletely surrounded by copies of itself. The determina- tion of the maximum possible (finite) Heesch number is known as HEESCH’S PROBLEM . The Heesch number of a TRIANGLE , QUADRILATERAL , regular HEXAGON ,or any other shape that can TILE or TESSELLATE the plane, is infinity. Conversely, any shape with infinite Heesch number must tile the plane (Eppstein). A tile invented by R. Ammann has Heesch number is three (Senechal 1995), and Mann has found an infinite family of tiles with Heesch number five(illustrated above), the largest (finite) number known. See also H EESCH’S PROBLEM ,TILING References Eppstein, D. "Heesch’s Problem." http://www.ics.uci.edu/ ~eppstein/junkyard/heesch/. Fontaine, A. "An Infinite Number of Plane Figures with Heesch Number Two." J. Comb. Th. A 57, 151 /C1/56, 1991. Friedman, E. "Heesch Tiles with Surround Numbers 3 and 4." http://www.stetson.edu/~efriedma/papers/heesch/ heesch.html. Gru¨nbaum, B. and Sheppard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1986. Mann, C. "Heesch’s Problem." http://www.math.unl.edu/ ~cmann/math/heesch/heesch.htm. Raedschelders, P. "Heesch Tiles Based on Regular Poly- gons." Combinatorics 7, 101 /C1/06, 1998. Raedschelders, P. "Heesch-Tiles Based on n-gons." http:// home.planetinternet.be/~praedsch/heersch.htm. Senechal, M. Quasicrystals and Geometry. New York: Cam- bridge University Press, 1995. Thompson, M. "Self-Surrounding Tiles." http://home.flash.- net/~markthom/html/self-surrounding_tiles.html. Heesch’s Problem How many times can a shape be completely sur- rounded by copies of itself without being able to TILE the entire plane, i.e., what is the maximum (finite) HEESCH NUMBER ? References Eppstein, D. "Heesch’s Problem." http://www.ics.uci.edu/ ~eppstein/junkyard/heesch/. Height The vertical length of an object from top to bottom. See also LENGTH (SIZE), POLYNOMIAL HEIGHT ,WIDTH (SIZE) Heilbronn Triangle Problem N.B. A detailed online essay by S. Finch was the starting point for this entry. Given any arrangement of npoints within a UNIT SQUARE , let Hnbe the smallest value for which there is at least one TRIANGLE formed from three of the points with AREA5Hn:The first few values are H3/C301 2 H4/C301 2 H5/C3019ffiffiffi 3p H6/C301 8 H7]1 12 H8]1 4(2/C28ffiffiffi 3p ) H9]1 21 H10]1 32(3ffiffiffiffiffiffi 17p /C2811) H11]1 27 H12]1 33H13]0:030 H14]0:022 H15]0:020 H16]0:0175 : Komlo ´set al. (1981, 1982) have shown that there are constants csuch that clnn n25Hn5C n8=7/C28e; for any e>0 and all sufficiently large n. Using an EQUILATERAL TRIANGLE of unit AREA instead gives the constants h3/C301 h4/C301 3 h5/C303/C282ffiffiffi 2p h6/C301 8: References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hlb/hlb.html. Friedman, E. "The Heilbronn Problem." http://www.stetso- n.edu/~efriedma/heilbronn/. Goldberg, M. "Maximizing the Smallest Triangle Made by N Points in a Square." Math. Mag. 45, 135/C1/44, 1972. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 242 /C1/44, 1994. Komlos, J.; Pintz, J.; and Szemere ´di, E. "On Heilbronn’s Triangle Problem." J. London Math. Soc. 24, 385/C1/96, 1981. Komlos, J.; Pintz, J.; and Szemere ´di, E. "A Lower Bound for Heilbronn’s Triangle Problem." J. London Math. Soc. 25, 13/C1/4, 1982. Roth, K. F. "Developments in Heilbronn’s Triangle Pro- blem." Adv. Math. 22, 364/C1/85, 1976. Heine Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION y??/C271 21 x/C28a1/C272 x/C28a3 ! y?/C2714 /C2A0/C27A1x/C27A2x2/C27A3x3 (x/C28a1)(x/C28a2)2(x/C28a3)2"# y /C300 (Moon and Spencer 1961, p. 157; Zwillinger 1997, p. 123). References Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997. Heine Hypergeometric Series Q-HYPERGEOMETRIC FUNCTION Heine-Borel Theorem If a CLOSED SET of points on a line can be covered by a set of intervals so that every point of the set is an interior point of at least one of the intervals, then there exist a finite number of intervals with the covering property. The Heine-Borel theorem gives the BOLZANO- WEIER- STRASS THEOREM as a special case. See also BOLZANO- WEIERSTRASS THEOREM References Baker, H. F. Cited in Lamb, H. Proc. London Math. Soc. 35, 459 /C1/60, 1903. Heine, E. "Die Elemente der Functionenlehre." J. reine angew. Math. 74, 172 /C1/88, 1871. Jeffreys, H. and Jeffreys, B. S. "The Heine-Borel Theorem" and "The Modified Heine-Borel Theorem." §1.0621 /C1/.0622 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 20 /C1/1, 1988. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 9, 1996. Young, W. H. "Overlapping Intervals." Proc. London Math. Soc. 35, 384 /C1/88, 1903. Heisenberg Ferromagnet Equation The system of PARTIAL DIFFERENTIAL EQUATIONS St /C30S /C29Sxx : References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 56, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Heisenberg Group The Heisenberg group Hn in n COMPLEX variables is the GROUP of all (z, t) with z /C23Cn and t /C23R having multiplication (w ; t)(z; t?) /C30(w /C27z; t /C27t?/C27I[w /C31z]) where w /C31 is the adjoint. The Heisenberg group is ISOMORPHIC to the group of MATRICES 1 zT 1 2zjj2/C27it 01 z 00 12 435; and satisfies (z; t) /C281 /C30(/C28z ;/C28t): Every finite-dimensional unitary representation is trivial on Z and therefore factors to a REPRESENTA- TION of the quotient Cn :/See also NIL GEOMETRY References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Heisenberg Space The boundary of COMPLEX HYPERBOLIC 2-SPACE . See also HYPERBOLIC SPACE Held Group The SPORADIC GROUP He. References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/He.html. Helen of Geometers CYCLOID Helicoid The MINIMAL SURFACE having a HELIX as its bound- ary. It is the only RULED MINIMAL SURFACE other than the PLANE (Catalan 1842, do Carmo 1986). For many years, the helicoid remained the only known example of a complete embedded MINIMAL SURFACE of finite topology with infinite CURVATURE . However, in 1992 a second example, known as H OFFMAN’S MINIMAL SUR- FACE and consisting of a helicoid with a HOLE , was discovered ( Sci. News 1992). The helicoid is the only non-rotary surface which can glide along itself(Steinhaus 1983, p. 231). The equation of a helicoid in CYLINDRICAL COORDI- NATES is z/C30cu: (1) In C ARTESIAN COORDINATES ,i ti s y x/C30tanz c ! : (2) It can be given in parametric form by x/C30ucosv (3) y/C30usinv (4) z/C30cv; (5) which has an obvious generalization to the ELLIPTIC HELICOID . Writing z /C30/C28cu instead of z /C30cv gives a CONE instead of a helicoid. The FIRST FUNDAMENTAL FORM coefficients of the helicoid are given by E /C301 (6) F /C300 (7) G2 /C30c2 /C27u2 ; (8) and the SECOND FUNDAMENTAL FORM coefficients are e /C300 (9) f /C30/C28cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C27 u2p (10) g /C300; (11) giving AREA ELEMENT dS /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C27u2p du L dv : (12) Integrating over v /C23 [0; u] and u /C23 [0; r] then gives S /C30g u 0gr 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic 2 /C27u2p du dv /C301 2 u rffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C27r2p /C27c2 lnr /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C27 r2p c ! "# : (13) The GAUSSIAN CURVATURE is given by K /C30/C28c2 (c2 /C27 u2)2 ; (14) and the MEAN CURVATURE is H /C300 (15) making the helicoid a MINIMAL SURFACE . The helicoid can be continuously deformed into a CATENOID by the transformation x(u ; v) /C30cos a sinh v sin u /C27sin a cosh v cos u (16) y(u; v) /C30/C28cos a sinh v cos u /C27sin a cosh v sin u (17) z(u ; v) /C30u cos a /C27v sin a; (18)where a /C300 corresponds to a helicoid and a /C30 p=2toa CATENOID . If a twisted curve C (i.e., one with TORSION t "0) rotates about a fixed axis A and, at the same time, is displaced parallel to Asuch that the speed of displacement is always proportional to the angular velocity of rotation, then Cgenerates a GENERALIZED HELICOID . See also CALCULUS OF VARIATIONS ,CATENOID ,CONE, ELLIPTIC HELICOID ,GENERALIZED HELICOID ,HELIX, HOFFMAN’S MINIMAL SURFACE ,H YPERBOLIC HELI- COID ,MINIMAL SURFACE References Catalan E. "Sur les surfaces re ´gle´es dont l’aire est un minimum." J. Math. Pure Appl. 7, 203/C1/11, 1842. do Carmo, M. P. "The Helicoid." §3.5B in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,pp. 44 /C1 /5, 1986. Fischer, G. (Ed.). Plate 91 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume.Braunschweig, Germany: Vieweg, p. 87, 1986. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 449 and 644, 1997. Kreyszig, E. Differential Geometry. New York: Dover, p. 88, 1991. Meusnier, J. B. "Me ´moire sur la courbure des surfaces." Me´m. des savans e ´trangers 10(lu 1776), 477 /C1 /10, 1785. Ogawa, A. "Helicatenoid." Mathematica J. 2, 21, 1992. Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 17 /C1/8, 1986. Peterson, I. "Three Bites in a Doughnut." Sci. News 127, 168, Mar. 16, 1985. "Putting a Handle on a Minimal Helicoid." Sci. News 142, 276, Oct. 24, 1992. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 231 /C1/32, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 94, 1991. Wolfram, S. The Mathematica Book, 3rd ed. Champaign, IL: Wolfram Media, p. 164, 1996. Helix A helix is also called a CURVE OF CONSTANT SLOPE .I t can be defined as a curve for which the TANGENT makes a constant ANGLE with a fixed line. The shortest path between two points on a cylinder (one not directly above the other) is a fractional turn of a helix, as can be seen by cutting the cylinder along one of its sides, flattening it out, and noting that a straight line connecting the points becomes helical upon re-wrapping (Steinhaus 1983, p. 229). It is for this reason that squirrels chasing one another up and around tree trunks follow helical paths. Helices come in enantiomorphous left- (coils counter- clockwise as it "goes away") and right-handed forms (coils clockwise). Standard screws, nuts, and bolts are all right-handed, as are both the helices in a double- stranded molecule of DNA (Gardner 1984, pp. 2 /C1/). Large helical structures in animals (such as horns) usually appear in both mirror-image forms, although the teeth of a male narwhal, usually only one which grows into a tusk, are both left-handed (Bonner 1951; Gardner 1984, p. 3; Thompson 1992). Gardner (1984) contains a fascinating discussion of helices in plants and animals, including an allusion to Shakespeare’s A Midsummer Night’s Dream. The helix is a SPACE CURVE with PARAMETRIC EQUA- TIONS x /C30r cos t (1) y /C30r sin t (2) z /C30ct ; (3) where r is the radius of the helix and c is a constant giving the vertical separation of the helix’s loops. The CURVATURE of the helix is given by k /C30r r2 /C27 c2 ; (4) and the LOCUS of the centers of CURVATURE of a helix is another helix. The ARC LENGTH is given by s /C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2 /C27y?2 /C27z ?2q dt /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27c2p t: (5) The TORSION of a helix is given by t /C301 r2(r2 /C27 c2)/C28r sin t /C28r cos tr sin t r cos t /C28r sin t /C28r cos t c 00P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2 /C30 c r2 /C27 c2 ; (6) so k t /C30r r2 /C27 c2 c r2 /C27 c2/C30r c ; (7) which is a constant. In fact, LANCRET’S THEOREM states that a NECESSARY and SUFFICIENT condition for a curve to be a helix is that the ratio of CURVATUREto TORSION be constant. The OSCULATING PLANE of the helix is given by z1 /C28r cos tz2 /C28r sin tz3 /C28ct /C28r sin tr cos tc /C28r cos t /C28r sin t 0P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C300 (8) z 1c sin t /C28z2c cos t /C27(z3 /C28ct)r /C300: (9) The MINIMAL SURFACE of a helix is a HELICOID . See also GENERALIZED HELIX,HELICOID ,SPHERICAL HELIX,SPIRAL References Bonner, J. T. "The Horn of the Unicorn." Sci. Amer. , Mar. 1951. Gardner, M. "The Helix." Ch. 1 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 1 /C1/, 1984. Gray, A. "The Helix and Its Generalizations." §8.5 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 198 /C1/00, 1997. Isenberg, C. Plate 4.11 in The Science of Soap Films and Soap Bubbles. New York: Dover, 1992. Pappas, T. "The Helix--Mathematics & Genetics." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 166 /C1/68, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 229, 1999. Thompson, D’A. W. On Growth and Form, 2nd ed., compl. rev. ed. New York: Cambridge University Press, 1992. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 95, 1991. Wolfram, S. The Mathematica Book, 3rd ed. Champaign, IL: Wolfram Media, p. 163, 1996. Helly Number Given a Euclidean n-space, Hn /C13n /C271: See also EUCLIDEAN SPACE ,HELLY’S THEOREM Helly’s Theorem If F is a family of more than n bounded closed convex sets in Euclidean n-space Rn ; and if every Hn (where Hn is the HELLY NUMBER ) members of F have at least one point in common, then all the members of F have at least one point in common. See also CARATHE ´ ODORY’S FUNDAMENTAL THEOREM , HELLY NUMBER References Eckhoff, J. "Helly, Radon, and Carathe ´odory Type Theo- rems." Ch. 2.1 in Handbook of Convex Geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 389 /C1/48, 1993. Helmholtz Differential Equation An ELLIPTIC PARTIAL DIFFERENTIAL EQUATION given by 92 c /C27k2 c /C300 ; (1) where c is a SCALAR FUNCTION and 92 is the scalar LAPLACIAN ,or 92A /C27k2A /C300; (2) where A is a VECTOR FUNCTION and 92 is the vector Laplacian (Moon and Spencer 1988, pp. 136 /C1/43). When k /C300, the Helmholtz differential equation reduces to LAPLACE’S EQUATION . When k2 B0 (i.e., for imaginary k), the equation becomes the space part of the diffusion equation. The Helmholtz differential equation can be solved by SEPARATION OF VARIABLES in only 11 coordinate systems, 10 of which (with the exception of CONFOCAL PARABOLOIDAL COORDINATES ) are particular cases of the CONFOCAL ELLIPSOIDAL system: CARTESIAN , CON- FOCAL ELLIPSOIDAL , CONFOCAL PARABOLOIDAL , CON- ICAL, CYLINDRICAL , ELLIPTIC CYLINDRICAL , OBLATE SPHEROIDAL , PARABOLOIDAL , PARABOLIC CYLINDRICAL , PROLATE SPHEROIDAL , and SPHERICAL COORDINATES (Eisenhart 1934). LAPLACE’S EQUATION (the Helm- holtz differential equation with k /C300) is separable in the two additional BISPHERICAL COORDINATES and TOROIDAL COORDINATES . If Helmholtz’s equation is separable in a 3-D coordi- nate system, then Morse and Feshbach (1953, pp. 509 /C1/10) show that h1h2h3 h2 n/C30fn(un)gn(ui ; uj) ; (3) where i "j "n: The LAPLACIAN is therefore OF THE FORM 92 /C301 h1h2h3g1(u2 ; u3)@ @u1f1(u1)@ @u1"# ( /C27g2(u1 ; u3)@ @u2f2(u2)@ @u2"# /C27g3(u1 ; u3)@ @u3f3(u3)@ @u3"#P+27 ; (4) which simplifies to 92 /C301 h2 1f1@ @u1f1(u1)@ @u1"# /C271 h22f2@ @u2f2(u2)@ @u2"# /C271 h23f3@ @u3f3(u3)@ @u3"# : (5) Such a coordinate system obeys the ROBERTSON CONDITION , which means that the STA¨ CKEL DETERMI-NANT is OF THE FORM S /C30h1h2h3 f1(u1)f2(u2)f3(u3) : (6) See also LAPLACE’S EQUATION ,POISSON’S EQUATION , SEPARATION OF VARIABLES ,SPHERICAL BESSEL DIF- FERENTIAL EQUATION ,STA¨ CKEL DETERMINANT References Eisenhart, L. P. "Separable Systems in Euclidean 3-Space." Physical Review 45, 427 /C1/28, 1934. Eisenhart, L. P. "Separable Systems of Sta¨ckel." Ann. Math. 35, 284 /C1/05, 1934. Eisenhart, L. P. "Potentials for Which Schroedinger Equa- tions Are Separable." Phys. Rev. 74,87/C1/9, 1948. Moon, P. and Spencer, D. E. "Eleven Coordinate Systems" and "The Vector Helmholtz Equation." §1 and 5 in Field Theory Handbook, Including Coordinate Systems, Differ- ential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1 /C1/8 and 136 /C1/43, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 125 /C1/26, 271, and 509 /C1/10, 1953. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997. Helmholtz Differential Equation * /Bipolar Coordinates In BIPOLAR COORDINATES , the HELMHOLTZ DIFFEREN- TIAL EQUATION is not separable, but LAPLACE’S EQUA- TION is. See also LAPLACE’S EQUATION– BIPOLAR COORDINATES Helmholtz Differential Equation * / Bispherical Coordinates The HELMHOLTZ DIFFERENTIAL EQUATION is not se- parable in BISPHERICAL COORDINATES . See also BISPHERICAL COORDINATES ,H ELMHOLTZ DIFFERENTIAL EQUATION ,L APLACE’S EQUATION– BI- SPHERICAL COORDINATES Helmholtz Differential Equation * / Cartesian Coordinates In 2-D C ARTESIAN COORDINATES , attempt SEPARATION OF VARIABLES by writing F(x;y)/C30X(x)Y(y); (1) then the H ELMHOLTZ DIFFERENTIAL EQUATION be- comes d2X dx2Y/C27d2Y dy2X/C27k2XY/C300: (2) Dividing both sides by XY gives 1 Xd2X dx2 /C271 Yd2Y dy2 /C27k2 /C300 : (3) This leads to the two coupled ordinary differential equations with a separation constant m2 ; 1 Xd2X dx2 /C30m2 (4) 1 Yd2Y dy2 /C30/C28(m2 /C27k2); (5) where X and Y could be interchanged depending on the boundary conditions. These have solutions X /C30Amemx /C27Bme/C28mx (6) Y /C30Cmeiffiffiffiffiffiffiffiffiffiffi m2/C27k2p y/C27Dme /C28iffiffiffiffiffiffiffiffiffiffi m2/C27k2p y /C30Em sin(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 /C27k2p y) /C27Fm cos(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim 2 /C27k2p y) : (7) The general solution is then F(x; y) /C30X/C12 m/C301(Amemx /C27Bme /C28mx) /C29[Em sin(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim 2 /C27k2p y) /C27Fm cos(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim 2 /C27k2p y)] : (8) In 3-D CARTESIAN COORDINATES , attempt SEPARATION OF VARIABLES by writing F(x; y; z) /C30X(x)Y(y)Z(z) ; (9) then the HELMHOLTZ DIFFERENTIAL EQUATION be- comes d2X dx2YZ /C27d2Y dy2XZ /C27d2Z dz2XY /C27k2XY /C300: (10) Dividing both sides by XYZ gives 1 Xd2X dx2 /C271 Yd2Y dy2 /C271 Zd2Z dz2 /C27k2 /C300: (11) This leads to the three coupled differential equations 1 Xd2X dx2 /C30t2 (12) 1 Yd2Y dy2 /C30m2 (13) 1 Zd2Z dz2 /C30(k2 /C27l2 /C27m2) ; (14) where X, Y, and Z could be permuted depending on boundary conditions. The general solution is there- foreF(x;y;z)/C30X/C12 l/C301X/C12 m/C301(Alelx/C27Ble/C28lx)(Cmemy/C27Dme/C28my) /C29(Elme/C28iffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2/C27l2/C27m2p z/C27Flmeiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2/C27l2/C27m2p z): (15) See also CARTESIAN COORDINATES ,HELMHOLTZ DIF- FERENTIAL EQUATION References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 501 /C1/02, 513/C1/14 and 656, 1953. Helmholtz Differential Equation * / Circular Cylindrical Coordinates InCYLINDRICAL COORDINATES , the SCALE FACTORS are hr/C301;hu/C30r;hz/C301;so the L APLACIAN is given by 92F/C301 r@ @rr@F @r ! /C271 r2@2F @u2/C27@2F @z2: (1) Attempt SEPARATION OF VARIABLES in the H ELMHOLTZ DIFFERENTIAL EQUATION 92F/C27k2F/C300 (2) by writing F(r;u;z)/C30R(r)U(u)Z(z); (3) then combining (1) and (2) gives d2R dr2UZ/C271 rdR drUZ/C271 r2d2U du2RZ/C27d2Z dz2RU/C27k2RUZ /C300: (4) Now multiply by r2=(RUZ); r2 Rd2R dr2/C27r RdR dr ! /C271 Ud2U du2/C27r2 Zd2Z dz2/C27k2r2/C300;(5) so the equation has been separated. Since the solution must be periodic in ufrom the definition of the circular cylindrical coordinate system, the solutionto the second part of (5) must have a NEGATIVE separation constant 1 Ud2u du2/C30/C28m2; (6) which has a solution U(u)/C30Cmcos(mu)/C27Dmsin(mu): (7) Plugging (7) back into (5) gives r2 Rd2R dr2/C27r RdR dr/C28m2/C27r2 Zd2Z dz2/C27k2r2/C300; (8) and dividing through by r2results in 1 Rd2R dr2 /C271 rRdR dr/C28m2 r2 /C271 Zd2Z dz2 /C27k2 /C300 : (9) The solution to the second part of (9) must not be sinusoidal at /9/C12 / for a physical solution, so the differential equation has a POSITIVE separation con- stant 1 Zd2Z dz2 /C30n2 ; (10) and the solution is Z(z) /C30Ene /C28nx /C27Fnenx : (11) Plugging (11) back into (9) and multiplying through by R yields d2R dr2 /C271 rdR dr/C27 n2 /C27k2 /C28m2 r2 ! R /C300 (12) But this is just a modified form of the BESSEL DIFFERENTIAL EQUATION , which has a solution R(r) /C30AmnJm(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 /C27k2p r) /C27BmnYm(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin 2 /C27k2p r) ; (13) where Jn(x) and Yn(x) are BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS , respectively. The general solution is therefore F(r ; u ; z) /C30X/C12 m/C30oX/C12 n/C300[AmnJm(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 /C27n2p r) /C27BmnYm(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 /C27n2p r)] /C29[Cm cos(mu) /C27Dm sin(mu)](Ene /C28nz /C27Fnenz) : (14) In the notation of Morse and Feshbach (1953), the separation functions are /f1(r) /C30r/, f2( u) /C301; f3(z) /C301/,so the STA¨ CKEL DETERMINANT is 1. The H ELMHOLTZ DIFFERENTIAL EQUATION is also separable in the more general case of k2OF THE FORM k2(r;u;z)/C30f(r)/C27g(u) r2/C27h(z)/C27k?2: (15) See also CYLINDRICAL COORDINATES ,H ELMHOLTZ DIFFERENTIAL EQUATION References Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 15 /C1/7, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 656/C1/57, 1953.Helmholtz Differential Equation * / Confocal Ellipsoidal Coordinates Using the NOTATION of Byerly (1959, pp. 252 /C1/53), LAPLACE’S EQUATION can be reduced to 92F/C30(m2/C28n2)@2F @a2/C27(l2/C28n2)@2F @b2/C27(l2/C28m2)@2F @g2 /C300; (1) where a/C30cgl cdlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (l2/C28b2)(l2/C28c2)p /C30Fb c;p 2 ! /C28Fb c;sin/C281c l ! ! (2) b/C30cgm bdmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (c2/C28m2)(m2/C28b2)p /C30Fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2/C28c2p ;sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2 m2 1/C28b2 c2vuuuuuut0 BBBB@1 CCCCA0 BBBB@1 CCCCA(3) g/C30c gn 0dnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b2/C28n2)(c2/C28n2)p /C30Fb c;sin/C281n b ! ! : (4) In terms of a;b;andg; l/C30cdca;b c ! (5) m/C30bndb;ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2 c2s ! (6) n/C30bsng;b c ! : (7) Equation (1) is not separable using a function OF THE FORM F/C30L(a)M(b)N(g); (8) but it is if we let 1 Ld2L da2/C30X aklk(9) 1 Md2M db2/C30X bkmk(10) 1 Nd2N dg2/C30X cknk: (11) These give a0 /C30/C28b0 /C30c0 (12) a2 /C30/C28b2 /C30c2 ; (13) and all others terms vanish. Therefore (1) can be broken up into the equations d2L da2 /C30(a0 /C27a2 l2)L (14) d2M db2 /C30/C28(a0 /C27a2 m2)M (15) d2N dg2 /C30(a0 /C27a2 n2)N : (16) For future convenience, now write a0 /C30/C28(b2 /C27c2)p (17) a2 /C30m(m /C271); (18) then d2L da2 /C28[m(m /C271)l2 /C28(b2 /C27c2)p]L /C300 (19) d2M db2 /C27[m(m /C271)m2 /C28(b2 /C27c2)p]M /C300 (20) d2N dg2 /C28[m(m /C271)n2 /C28(b2 /C27c2)p]N /C300 : (21) Now replace a; b; and g to obtain ( l2 /C28b2)( l2 /C28c2)d2L dl2 /C27 l( l2 /C28b2 /C27 l2 /C28c2)dL dl /C28[m(m /C271)l2 /C28(b2 /C27c2)p]L /C300 (22) ( m2 /C28b2)(m2 /C28c2)d2M dm2 /C27 m( m2 /C28b2 /C27 m2 /C28c2)dM dm /C28[m(m /C271)m2 /C28(b2 /C27c2)p]M /C300 (23) ( n2 /C28b2)( n2 /C28c2)d2N dn2 /C27 n(n2 /C28b2 /C27 n2 /C28c2)dN dn /C28[m(m /C271)n2 /C28(b2 /C27c2)p]N /C300: (24) Each of these is a LAME´ ’S DIFFERENTIAL EQUATION , whose solution is called an ELLIPSOIDAL HARMONIC . Writing L(l) /C30Ep m( l) (25) M( l) /C30Epm(m) (26) N( l) /C30Epm( n) (27) gives the solution to (1) as a product of ELLIPSOIDAL HARMONICS Ep m(x):F /C30Ep m( l)Epm(m)Epm( n) : (28) See also CONFOCAL ELLIPSOIDAL COORDINATES , HELMHOLTZ DIFFERENTIAL EQUATION References Arfken, G. "Confocal Ellipsoidal Coordinates ( j1 ; j2 ; j3):/" §2.15 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 117 /C1/18, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 251 /C1/58, 1959. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 43 /C1/4, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953. Helmholtz Differential Equation * / Confocal Paraboloidal Coordinates As shown by Morse and Feshbach (1953), the H ELM- HOLTZ DIFFERENTIAL EQUATION is separable in CON- FOCAL PARABOLOIDAL COORDINATES . See also CONFOCAL PARABOLOIDAL COORDINATES , HELMHOLTZ DIFFERENTIAL EQUATION References Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 47 /C1/8, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 664, 1953. Helmholtz Differential Equation * /Conical Coordinates InCONICAL COORDINATES ,LAPLACE’S EQUATION can be written @2V @a2/C27@2V @b2/C27(m2/C28n2)@ @ll2@V @l ! /C300; (1) where a/C30gm admffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (m2/C28a2)(b2/C28m2)p (2) b/C30gn 0dnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a2/C28n2)(b2/C28n2)p (3) (Byerly 1959). Letting V/C30U(u)R(r) (4) breaks (1) into the two equations, d drr2dR dr ! /C30m(m/C271)R (5) @2U @ a2 /C27@2U @ b2 /C27m(m /C271)(m2 /C28 n2)U /C300 (6) Solving these gives R(r) /C30Arm /C27Br /C28m/C281 (7) U(u) /C30Ep m( m)Epm( n); (8) where Ep mare ELLIPSOIDAL HARMONICS . The regular solution is therefore V /C30ArmEp m(m)Epm(n) ; (9) However, because of the cylindrical symmetry, the solution Ep m( m)Epm( n)isan mth degree SPHERICAL HARMONIC . See also CONICAL COORDINATES ,HELMHOLTZ DIFFER- ENTIAL EQUATION References Arfken, G. "Conical Coordinates ( j1;j2;j3):/"§2.16 in Math- ematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 118 /C1/19, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics.New York: Dover, p. 263, 1959. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations,and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 39 /C1 /0, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 659, 1953. Helmholtz Differential Equation * /Elliptic Cylindrical Coordinates In ELLIPTIC CYLINDRICAL COORDINATES , the SCALE FACTORS arehu/C30hv/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh2u/C27sin2vp ;hz/C301;and the separation functions are f1(u)/C30f2(v)/C30f3(z)/C301; giving a STA¨CKEL DETERMINANT ofS/C30(sin2v/C27 sinh2u):The Helmholtz differential equation is 1 sinh2u/C27sin2v@2F @u2/C27@2F @v2 ! /C27@2F @z2/C27k2F/C300:(1) Attempt SEPARATION OF VARIABLES by writing F(u;v;z)/C30U(u)V(v)Z(z); (2) then the H ELMHOLTZ DIFFERENTIAL EQUATION be- comes Z sinh2u/C27sin2vVd2U du2/C27Ud2V dv2 ! /C27UVd2Z dz2 /C27k2UVZ /C300: (3) Now divide by UVZ to give1 sinh2u/C27sin2v1 U@2U @u2/C271 V@2V @v2 ! /C271 Z@2Z @z2/C27k2 /C300: (4) Separating the Zpart, 1 Zd2Z dz2/C30/C28(k2/C27m2) (5) 1 sinh2u/C27sin2v1 U@2U @u2/C271 V@2V @v2 ! /C30m2(6) so @2Z dz2/C30/C28(k2/C27m2)Z; (7) which has the solution Z(z)/C30Akmcos(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2/C27m2p z)/C27Bkmsin(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2/C27m2p z):(8) Rewriting (6) gives 1 Ud2U du2/C28m2sinh2u ! /C271 Vd2V dv2/C28m2sin2v ! /C300; (9) which can be separated into 1 Ud2U du2/C28m2sinh2u/C30c (10) c/C271 Vd2V dv2/C28m2sin2v/C300; (11) so d2U du2/C28(c/C27m2sinh2u)U/C300 (12) d2V dv2/C27(c/C28m2sin2v)V/C300: (13) Now use sinh2u/C301 2[cosh (2 u)/C281] (14) sin2v/C3012[1/C28cos (2 v)] (15) to obtain d2U du2/C28fc/C271 2m2[cosh (2 u)/C281]gU/C300 (16) d2V dv2/C27fc/C2812m2[1/C28cos (2 v)]gV/C300: (17) Regrouping gives d2U du2 /C28[(c /C281 2m2) /C2712m2cosh (2u)]U /C300 (18) d2V dv2 /C27[(c /C2812m2) /C2712m2cos (2v)]V /C300: (19) Let a /C13c /C28m2 =2 and q /C13/C28m2 =4 ; then these become d2V dv2 /C27[a /C282q cos (2v)]V /C300 (20) d2U du2 /C28[a /C282q cosh (2u)]U /C300: (21) Here, (20) is the MATHIEU DIFFERENTIAL EQUATION and (21) is the modified MATHIEU DIFFERENTIAL EQUATION . These solutions are known as MATHIEU FUNCTIONS . See also ELLIPTIC CYLINDRICAL COORDINATES ,HELM- HOLTZ DIFFERENTIAL EQUATION ,MATHIEU DIFFEREN- TIAL EQUATION ,MATHIEU FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Mathieu Func- tions." Ch. 20 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 721 /C1/46, 1972. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 17 /C1/9, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 657, 1953. Helmholtz Differential Equation * /Oblate Spheroidal Coordinates As shown by Morse and Feshbach (1953) and Arfken (1970), the HELMHOLTZ DIFFERENTIAL EQUATION is separable in OBLATE SPHEROIDAL COORDINATES . See also HELMHOLTZ DIFFERENTIAL EQUATION ,OB- LATE SPHEROIDAL COORDINATES References Arfken, G. "Oblate Spheroidal Coordinates ( u;v;8):/"§2.11 inMathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 107 /C1/09, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 242 and 245 /C1/47, 1959. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations,and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 33 /C1 /4, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 662, 1953.Helmholtz Differential Equation * / Parabolic Coordinates The SCALE FACTORS arehu/C30hv/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2/C27v2p ;hu/C30uv and the separation functions are fu(u)/C30u;f2(v)/C30v; f3(u)/C301;given a STA¨CKEL DETERMINANT ofS/C30u2/C27 v2:The L APLACIAN is 1 u2/C27v21 u@F @u/C27@2F @u2/C271 v@F @v/C27@2F @v2 ! /C271 u2v2@2F @u2/C27k2F /C300: (1) Attempt SEPARATION OF VARIABLES by writing F(u;v;u)/C13U(u)V(v)U(u); (2) then the H ELMHOLTZ DIFFERENTIAL EQUATION be- comes 1 u2/C27v2VU1 udU du/C27d2U du2 ! /C27UU1 vdV dv/C27d2V dv2 ! "# /C27UV u2v2d2U du2/C27k2UVU/C300: (3) Now multiply through by u2v2=(UVU); u2v2 u2/C27v21 U1 udU du/C27d2U du2 ! /C271 V1 vdV dv/C27d2V dv2 ! "# /C271 Ud2U du2/C27k2u2v2/C300: (4) Separating the Upart gives 1 Ud2u du2/C30/C28m2; (5) which has solution U(u)/C30Amcos(mu)/C27Bmsin(mu): (6) Plugging (5) back into (4) and multiplying by ( u2/C27 v2)=(u2v2) gives 1 U1 udU du/C27d2U du2 ! /C271 V1 vdV dv/C27d2V dv2 ! "# /C28m2u2/C27v2 u2v2/C27k2(u2/C27v2) (7) Rewriting, 1 U1 udU du/C27d2U du2 ! /C271 V1 vdV dv/C27d2V dv2 ! "# /C28m21 v2/C271 u2 ! /C27k2(u2/C27v2): (8) This can be rearranged into two terms, each contain- ing only uorv, 1 U1 udU du /C27d2U du2 ! /C27k2u2 /C28m2 u2"# /C271 V1 vdV dv /C27d2V dv2 ! /C27k2v2 /C28m2 v2"# (9) and so can be separated by letting the first part equal c and the second equal /C28c; giving d2U du2 /C271 udU du /C27 k2u2 /C28m2 u2 /C28c ! U /C300 (10) d2V dv2 /C271 vdV dv /C27 k2v2 /C28m2 v2 /C27c ! V /C300 : (11) See also HELMHOLTZ DIFFERENTIAL EQUATION ,PARA- BOLIC COORDINATES References Arfken, G. "Parabolic Coordinates ( j; h; f) :/" §2.12 in Math- ematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109 /C1/11, 1970. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 36, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York McGraw-Hill, pp. 514 /C1/15 and 660, 1953. Helmholtz Differential Equation * / Parabolic Cylindrical Coordinates In PARABOLIC CYLINDRICAL COORDINATES , the SCALE FACTORS are hu /C30hv /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 /C27v2p ; hz /C301 and the se- paration functions are f1(u) /C30f2(v) /C30f3(z) /C301; giving STA¨ CKEL DETERMINANT of s /C30u2 /C27v2 : the HELMHOLTZ DIFFERENTIAL EQUATION is 1 u2 /C27 v2@2f @u2 /C27@2f @v2 ! /C27@2f @z2 /C27k2f /C300: (1) attempt SEPARATION OF VARIABLES by writing f(u; v; z) /C13u(u)v(v)z(z) ; (2) then the HELMHOLTZ DIFFERENTIAL EQUATION be- comes 1 u2 /C27 v2VZd2U du2 /C27UZd2V dv2 ! /C27UVd2Z dz2 /C27k2UVZ /C300: (3) Divide by UVZ , 1 u2 /C27 v21 Ud2U du2 /C271 Vd2V dv2 ! /C271 Zd2Z dz2 /C27k2 /C300 : (4) Separating the Z part,1 Zd2Z dz2 /C30/C28(k2 /C27m2) (5) 1 u2 /C27 v21 Ud2U du2 /C271 Vd2V dv2 ! /C28k2 /C300: (6) 1 Ud2U du2 /C271 Vd2V dv2 /C28k2(u2 /C27v2) /C300; (7) so @2Z dz2 /C30/C28(k2 /C27m2)Z ; (8) which has solution Z(z) /C30A cos(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 /C27m2p z) /C27B sin(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 /C27m2p z); (9) and 1 Ud2U du2 /C28k2u2 ! /C271 Vd2V dv2 /C28k2v2 ! /C300 : (10) This can be separated 1 Ud2U du2/C28k2u2/C30c (11) 1 Vd2V dv2/C28k2v2/C30/C28c; (12) so d2U du2/C28(c/C27k2u2)U/C300 (13) d2V dv2/C28(c/C27k2v2)V/C300: (14) These are the W EBER DIFFERENTIAL EQUATIONS , and the solutions are known as P ARABOLIC CYLINDER FUNCTIONS . See also HELMHOLTZ DIFFERENTIAL EQUATION ,PARA- BOLIC CYLINDER FUNCTION ,PARABOLIC CYLINDRICAL COORDINATES ,W EBER DIFFERENTIAL EQUATIONS References Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 36, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 515 and 658, 1953. Helmholtz Differential Equation * /Polar Coordinates In 2-D POLAR COORDINATES , attempt SEPARATION OF VARIABLES by writing F(r;u)/C30R(r)U(u); (1) then the HELMHOLTZ DIFFERENTIAL EQUATION be- comes d2R dr2 U/C271 rdR drU/C271 r2d2 U du2 R /C27k2RU/C300: (2) Divide both sides by RU r2 Rd2R dr2 /C27r RdR dr ! /C271 Ud2 U d u2 /C27k2 ! /C300: (3) The solution to the second part of (3) must be periodic, so the differential equation is d2 U du21 U/C30/C28(k2 /C27m2) ; (4) which has solutions U(u) /C30c1eiffiffiffiffiffiffiffiffiffiffi k2/C27m2p u/C27c2e /C28iffiffiffiffiffiffiffiffiffiffi k2/C27m2p u /C30c3 sin(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 /C27m2p u) /C27c4 cos(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 /C27m2p u): (5) Plug (4) back into (3) r2Rƒ/C27rR ?/C28m2R /C300: (6) This is an EULER DIFFERENTIAL EQUATION with a /C131 and b /C13/C28m2 : The roots are r /C309m: So for m /C300, r /C300 and the solution is R(r) /C30c1 /C27c2 In r : (7) But since In r blows up at r /C300, the only possible physical solution is R(r) /C30c1 : When m /C210, r /C309m; so R(r) /C30c1rm /C27c2r /C28m : (8) But since r /C28m blows up at r /C300, the only possible physical solution is Rm(r) /C30c1rm : The solution for R is then Rm(r) /C30cmrm (9) for m /C300, 1, ...and the general solution is F(r ; u) /C30X/C12 m/C300[amrm sin(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 /C27m2p u) /C27bmrm cos(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 /C27m2p u)] : (10) References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York McGraw-Hill, pp. 502 /C1/04, 1953. Helmholtz Differential Equation * /Prolate Spheroidal Coordinates As shown by Morse and Feshbach (1953) and Arfken (1970), the H ELMHOLTZ DIFFERENTIAL EQUATION is separable in PROLATE SPHEROIDAL COORDINATES .See also HELMHOLTZ DIFFERENTIAL EQUATION ,PRO- LATE SPHEROIDAL COORDINATES References Arfken, G. "Prolate Spheroidal Coordinates ( u;v;8):/"§2.10 inMathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103 /C1/07, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics.New York: Dover, pp. 243 /C1 /44, 1959. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations,and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 30, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953. Helmholtz Differential Equation * / Spherical Coordinates InSPHERICAL COORDINATES , the SCALE FACTORS are hr/C301;hu/C30rsinf;hf/C30r;and the separation func- tions are f1(r)/C30r2;f2(u)/C301;f3(f)/C30sinf;giving a STA¨CKEL DETERMINANT ofS/C301. The L APLACIAN is 92/C131 r2@ @rr2@ @r ! /C271 r2sin2f@2 @u2/C271 r2sinf@ @f /C2sinf@ @f ! : (1) To solve the H ELMHOLTZ DIFFERENTIAL EQUATION in SPHERICAL COORDINATES , attempt SEPARATION OF VARIABLES by writing F(r;u;f)/C30R(r)U(u)F(f): (2) Then the H ELMHOLTZ DIFFERENTIAL EQUATION be- comes d2R dr2FU/C272 rdR drFU/C271 r2sin2fd2U du2FR /C27cosf r2sinfdF dfUR/C271 r2d2F df2UR /C300: (3) Now divide by RUF; r2sin2f FRUFUd2R dr2/C272 rr2sin2f FRUFUdR dr /C271 r2sin2fr2sin2f FRUFRd2U du2 /C27cosf r2sinfr2sin2f FURdF dfUR /C271 r2r2sin2f FRUd2F df2UR/C300 (4) r2sin2f Rd2R dr2/C272rsin2f RdR dr ! /C271 Ud2U du2 ! /C27cosfsinf FdF df/C27sin2f Fd2F df2 ! /C300: (5) The solution to the second part of (5) must be sinusoidal, so the differential equation is d2U du21 U/C30/C28m2; (6) which has solutions which may be defined either as a COMPLEX function with m/C30/C28/C12;...,/C12 U(u)/C30Ameimu; (7) or as a sum of REAL sine and cosine functions with m/C30/C28/C12;...,/C12 U(u)/C30Smsin(mu)/C27Cmcos(mu): (8) Plugging (6) back into (7), r2 Rd2R dr2/C272r RdR dr/C281 sin2fm2/C27cosfsinf F ! dF df /C27sin2f Fd2F df2 /C300: (9) The radial part must be equal to a constant r2 Rd2R dr2/C272r RdR dr/C30l(l/C271) (10) r2d2R dr2/C272rdR dr/C30l(l/C271)R: (11) But this is the E ULER DIFFERENTIAL EQUATION ,s ow e try a series solution OF THE FORM R/C30X/C12 n/C300anrn/C27c(12) Then r2X/C12 n/C300(n/C27c)(n/C27c/C281)anrn/C27c/C282 /C272rX/C12 n/C300(n/C27c)anrn/C27c/C281 /C28l(l/C271)X/C12 n/C300anrn/C27c/C300 (13) X/C12 n/C300(n/C27c)(n/C27c/C281)anrn/C27c/C272X/C12 n/C300(n/C27c)anrn/C27c /C28l(l/C271)X/C12 n/C300anrn/C27c/C300 (14)X/C12 n/C300[(n/C27c)(n/C27c/C281)/C28l(l/C271)]anrn/C27c/C300: (15) This must hold true for all POWERS ofr. For the rc term (with n/C300), c(c/C271)/C30l(l/C271); (16) which is true only if c/C30l;/C28l/C281 and all other terms vanish. So an/C300 for n"l;/C28l/C281:Therefore, the solution of the Rcomponent is given by Rl(r)/C30Alrl/C27Blr/C28l/C281: (17) Plugging (17) back into (9), l(l/C271)/C28m2 sin2f/C27cosf sinf1 FdF df/C271 Fd2F df2/C300 (18) Fƒcosf sinfF?/C27l(l/C271)/C28m2 sin2f"# F/C300; (19) which is the associated L EGENDRE DIFFERENTIAL EQUATION forx/C30cosfand m/C300, ..., l. The general COMPLEX solution is therefore X/C12 t/C300Xl m/C30/C28l(Alrl/C27Blr/C28l/C281)Pm l(cosf)e/C28imu /C13X/C12 t/C300Xl m/C30/C281(Alrl/C27Blr/C28l/C281)Ym l(u;f) (20) where Ym l(u;f)/C13Pml(cosf)e/C28imu(21) are the ( COMPLEX )SPHERICAL HARMONICS . The gen- eral REAL solution is X/C12 t/C300Xl m/C300(Alrl/C27Blr/C28l/C281)Pml(cosf) /C2[Smsin(mu)/C27Cmcos(mu)]: (22) Some of the normalization constants of Pm lcan be absorbed by SmandCm;so this equation may appear in the form X/C12 t/C300Xl m/C300(Alrl/C27Blr/C28l/C281)Pm l(cosf) /C2[Smlsin(mu)/C27Cmlcos(mu)] /C13X/C12 l/C300Xl m/C300(Alrl/C27Blr/C28l/C281) /C29[SmlYm(o) l(u;f)/C27CmlYm(e) l(u;f)]; (23) where Ym(0) l(u;f)/C13Pml(cosu)sin(mu) (24) Ym(e) l(u;f)/C13Pml(cosu)cos(mu) (25) are the EVEN and ODD (real) SPHERICAL HARMONICS .If azimuthal symmetry is present, then U( u) is constant and the solution of the F component is a LEGENDRE POLYNOMIAL Pl(cos f) : The general solution is then F(r; f) /C30X/C12 l /C300(AlrlBlr/C28l/C281)Pl(cos f): (26) Actually, the equation is separable under the more general condition that k2 is OF THE FORM k2(r ; u; f) /C30f(r) /C27g( u) r2/C27h(f) r2 sin u /C27k?2 : (27) See also HELMHOLTZ DIFFERENTIAL EQUATION ,SPHE- RICAL COORDINATES ,SPHERICAL HARMONIC References Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 244, 1959. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 27, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 514 and 658, 1953. Helmholtz Differential Equation * / Spherical Surface On the surface of a SPHERE , attempt SEPARATION OF VARIABLES in SPHERICAL COORDINATES by writing F(u ; f) /C30U( u) F(f); (1) then the HELMHOLTZ DIFFERENTIAL EQUATION be- comes 1 sin2 fd2 U du2 F/C27cos f sin fdF dfU/C27d2 F df2 U/C27k2 UF/C300: (2) Dividing both sides by FU; cos f sin f FdF d f /C27sin2 f Fd2 F df2 ! /C271 Ud2 U du2 /C27k2 ! /C300; (3) which can now be separated by writing d2 U du21 U/C30/C28(k2 /C27m2) : (4) The solution to this equation must be periodic, so m must be an INTEGER . The solution may then be defined either as a COMPLEX function U( u) /C30Ameiffiffiffiffiffiffiffiffiffiffi k2/C27m2p u/C27Bme /C28iffiffiffiffiffiffiffiffiffiffi k2/C27m2p u(5) for m /C30/C28/C12; ..., /C12; or as a sum of REAL sine and cosinefunctions U( u) /C30Sm sinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 /C27m2p uP+’kP+’7 /C27Cm cosffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 /C27m2p uP+’kP+’7 (6) for m /C30 0, ..., /C12: Plugging (4) into (3) gives cos f sin f FdF df /C27sin2 f Fd2 F df2 /C27m2 /C300 (7) Fƒ/C27cos f sin fF?/C27m2 sin2 fF/C300; (8) which is the LEGENDRE DIFFERENTIAL EQUATION for x /C30cos f with m2 /C13l(l /C271); (9) giving l2 /C27l /C28m2 /C300 (10) l /C301 2(/C281 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C274m2p ) : (11) Solutions are therefore LEGENDRE POLYNOMIALS with a COMPLEX index. The general COMPLEX solution is then F(u ; f) /C30X/C12 m/C30/C28/C12Pl(cos f)(Ameimu /C27Bme /C28imu) ; (12) and the general REAL solution is F( u; f) /C30X/C12 m/C300Pl(cos f) /C2[Sm sin(mu) /C27Cm cos(mu)]: (13) Note that these solutions depend on only a single variable m. However, on the surface of a sphere, it is usual to express solutions in terms of the SPHERICAL HARMONICS derived for the 3-D spherical case, which depend on the two variables landm. Helmholtz Differential Equation * / Toroidal Coordinates The H ELMHOLTZ DIFFERENTIAL EQUATION is not se- parable in TOROIDAL COORDINATES See also HELMHOLTZ DIFFERENTIAL EQUATION ,LA- PLACE’S EQUATION– TOROIDAL COORDINATES ,TOROI- DAL COORDINATES Helmholtz’s Theorem Any VECTOR FIELD vsatisfying [9 /C215v]/C12/C300 (1) [9/C29v]/C12/C300 (2) may be written as the sum of an IRROTATIONAL part and a SOLENOIDAL part, v /C30/C289 f /C279/C29A ; (3) where for a VECTOR FIELD F, f /C30/C28gV9 /C215 F 4 p r?/C28r jjd3r ? (4) A /C30gV9/C29 F 4p r ?/C28r jjd3r?: (5) See also IRROTATIONAL FIELD,SOLENOIDAL FIELD, VECTOR FIELD References Arfken, G. "Helmholtz’s Theorem." §1.15 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 78 /C1/4, 1985. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1084, 2000. Helson-Szego Measure An absolutely continuous measure on @D whose density has the form exp(x /C27 ¯y) ; where x and y are real-valued functions in L /C12; ykk/C12B p=2; exp is the EXPONENTIAL FUNCTION , and ykkis the NORM . Hemicylindrical Function A function Sn(z) which satisfies the RECURRENCE RELATION Sn/C281(z) /C28Sn/C271(z) /C302S ?n(z) together with S1(z) /C30/C28S ?0(z) is called a hemicylindrical function. References Sonine, N. "Recherches sur les fonctions cylindriques et le de´veloppement des fonctions continues en se´ries." Math. Ann. 16,1/C1/ and 71 /C1/0, 1880. Watson, G. N. "Hemi-Cylindrical Functions." §10.8 in A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, p. 353, 1966.Hemisphere Half of a SPHERE cut by a PLANE passing through its CENTER . A hemisphere of RADIUS r can be given by the usual SPHERICAL COORDINATES x /C30r cos u sin f (1) y /C30r sin u sin f (2) z /C30r cos f; (3) where u /C23 [0; 2p) and f /C23 [0; p=2]: All CROSS SECTIONS passing through the Z-AXIS are SEMICIRCLES . The VOLUME of the hemisphere is V/C30pgr 0(r2/C28z2)dz/C302 3pr3: (4) The weighted mean of zover the hemisphere is /C142z/C143/C30pgr 0z(r2/C28z2)dz/C3014pr2: (5) The CENTROID is then given by ¯z/C30/C142z/C143 V/C3038r (6) (Beyer 1987). See also SEMICIRCLE ,SPHERE References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 133, 1987. Hemispherical Function The hemisphere function is defined as H(x; y) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a /C28x2 /C28y2p forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 /C27y2p 5a 0 forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p > a:P+2k Watson (1966) defines a hemispherical function as a function S which satisfies the RECURRENCE RELA- TIONS Sn/C281(z) /C28Sn/C271(z) /C302S ?n(z) with S1(z) /C30/C28S ?0(z) See also CYLINDER FUNCTION ,C YLINDRICAL FUNC- TION References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, p. 353, 1966. Hempel’s Paradox A purple cow is a confirming instance of the hypoth- esis that all crows are black. References Carnap, R. Logical Foundations of Probability. Chicago, IL: University of Chicago Press, pp. 224 and 469, 1950. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 79 /C1/1, 1998. Gardner, M. The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 52 /C1/4, 1959. Goodman, N. Ch. 3 in Fact, Fiction, and Forecast. Cam- bridge, MA: Harvard University Press, 1955. Hempel, C. G. "A Purely Syntactical Definition of Confirma- tion." J. Symb. Logic 8, 122 /C1/43, 1943. Hempel, C. G. "Studies in Logic and Confirmation." Mind 54,1/C1/6, 1945. Hempel, C. G. "Studies in Logic and Confirmation. II." Mind 54,97/C1/21, 1945. Hempel, C. G. "A Note on the Paradoxes of Confirmation." Mind 55, 1946.Hosiasson-Lindenbaum, J. "On Confirmation." J. Symb. Logic 5, 133 /C1/48, 1940. Whiteley, C. H. "Hempel’s Paradoxes of Confirmation." Mind 55, 156 /C1/58, 1945. Hendecagon An 11-sided polygon, also variously known as the undecagon or unidecagon. The term "hendecagon" is preferable to the other two since it uses the Greek prefix and suffix instead of mixing a Roman prefix and Greek suffix. The regular 11-sided POLYGON has SCHLA ¨ FLI SYMBOL f11g:/ The hendecagon cannot be constructed using the classical Greek rules of GEOMETRIC CONSTRUCTION , but Conway and Guy (1996) give a NEUSIS CONSTRUC- TION based on TRISECTION . See also DECAGON ,D ODECAGON ,T RIGONOMETRY VALUES PI/11 References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 194 /C1/00, 1996. Henneberg’s Minimal Surface AMINIMAL SURFACE and double ALGEBRAIC SURFACE of 15th order and fifth class which can be given by PARAMETRIC EQUATIONS x(u;v)/C302 sinh ucosv/C282 3sinh(3 u) cos(3 v) (1) y(u;v)/C302 sinh usinv/C272 3sinh(3 u) sin(3 v) (2) z(u;v)/C302 cosh(2 u) cos(2 v): (3) The coefficients of the FIRST FUNDAMENTAL FORM of this parameterization are given by E/C308 cosh2u[cosh(4 u)/C28cos(4 v)] (4) F/C300 (5) G /C308 cosh2 u[cosh(4 u) /C28cos(4 v)]; (6) and the coefficients of the SECOND FUNDAMENTAL FORM are e /C30/C284 cos(2 v) sinh(2 u) (7) f /C304 cosh 2uðÞ sin 2vðÞ (8) g /C304 sinh(2 u) cos(2 v); (9) giving AREA ELEMENT dS /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2[cos(4 v) /C28cosh(4 u)]p (10) and GAUSSIAN and MEAN CURVATURES are K /C30sech4 u 8[cos(4 v) /C28 cosh(4 u)] (11) H /C300: (12) The surface can also be obtained from the ENNEPER- WEIERSTRASS PARAMETERIZATION with f /C302 /C282z/C284 (13) g /C30z ; (14) which gives a parameterization OF THE FORM x /C302(r2 /C28 1)cos f r/C282(r6 /C28 1)cos(3 f) 3r3 (15) y /C30/C286r2(r2 /C28 1)sin f /C27 2(r6 /C28 1)sin(3 f) 3r3 (16) z /C302(r4 /C27 1)cos(2 f) r2 (17) Henneberg’s minimal surface is a NONORIENTABLE SURFACE defined over the UNIT DISK. It is an immer- sion of the REAL PROJECTIVE PLANE that has been multiply PUNCTURED (once at the origin and four times at each of the roots of the metric). Conse- quently, it is not a COMPLETE SURFACE . The total curvature is /C282p:/ See also ENNEPER- WEIERSTRASS PARAMETERIZATION , MINIMAL SURFACE References Darboux, G. §226 in Lecons sur la the´orie ge´ne´rale des surfaces. Paris: Gauthier-Villars, 1941. Eisenhart, L. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, p. 267, 1960. Gray, A. "Henneberg’s Minimal Surface." Modern Differen- tial Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 691 /C1/92, 1997. JavaView. "Classic Surfaces from Differential Geometry: Henneberg." http://www-sfb288.math.tu-berlin.de/vgp/ja- vaview/demo/surface/common/PaSurface_Henne- berg.html. Nitsche, J. C. C. Introduction to Minimal Surfaces. Cam- bridge, England: Cambridge University Press, p. 144, 1989.He´non Attractor HE´ NON MAP He´non Map A quadratic 2-D MAP given by the equations xn/C271 /C301 /C28 ax2 n /C27yn (1) yn/C271 /C30 bxn (2) or xn/C271 /C30xn cos a /C28(yn /C28x2n)sin a (3) yn /C271 /C30xn sin a /C27(yn /C28x2n)cos a: (4) The above map is for a /C301 :4 and b /C300:3: The He´non map has CORRELATION EXPONENT 1.25 9 0.02 (Grass- berger and Procaccia 1983) and CAPACITY DIMENSION 1.261 9 0.003 (Russell et al. 1980). Hitzl and Zele (1985) give conditions for the existence of periods 1 to 6. See also BOGDANOV MAP,LOZI MAP,QUADRATIC MAP References Dickau, R. M. "The He ´non Attractor." http://forum.swarth- more.edu/advanced/robertd/henon.html. Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 144 /C1/53, 1988. Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189/C1/08, 1983. Hitzl, D. H. and Zele, F. "An Exploration of the He ´non Quadratic Map." Physica D 14, 305/C1/26, 1985. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 128 /C1/ 33, 1991. Morosawa, S.; Nishimura, Y.; Taniguchi, M.; and Ueda, T. "Dynamics of Generalized He ´non Maps." Ch. 7 in Holo- morphic Dynamics. Cambridge, England: Cambridge Uni- versity Press, pp. 225 /C1/62, 2000. Peitgen, H.-O. and Saupe, D. (Eds.). "A Chaotic Set in the Plane." §3.2.2 in The Science of Fractal Images. New York: Springer-Verlag, pp. 146 /C1/48, 1988. Russell, D. A.; Hanson, J. D.; and Ott, E. "Dimension of Strange Attractors." Phys. Rev. Let. 45, 1175 /C1/178, 1980. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 95 /C1/7, 1991. He´non-Heiles Equation A nonlinear nonintegrable HAMILTONIAN SYSTEM with ¨x/C30/C28@V @x(1) ¨y /C30/C28@V @y; (2) where the potential energy function is defined by the polar equation V(r ; u) /C301 2 r2 /C2713 r3 sin(3u) ; (3) giving Cartesian potential V(x; y) /C301 2x2 /C27y2 /C272x2y /C2823 y3P+’kP+’7 : (4) The total energy of the system is then given by E /C30V(x; y) /C271 2(˙x2 /C27 ˙y2) ; (5) which is conserved during motion. Integrating the above coupled ordinary differential equations from an arbitrary starting point with x(t /C30 0) /C300 and E /C301=8 gives the motion illustrated above. Computing the values of t at which x /C300 and plotting y(t) vs. ˙y(t) at these values gives a so-called SURFACE OF SECTION . The surfaces of section shown below correspond to E /C301 =12 and E /C301=8:/ The Hamiltonian for a generalized He´non-Heiles potential is H /C3012(p2 x /C27p2y /C27Ax2 /C27By2) /C27Dx2y /C281 3Cy3: (6) The equations of motion are integrable only for 1.D=C/C300;/ 2.D=C/C30/C281;A=B/C301;/ 3.D=C/C30/C281=6;and 4.D=C/C30/C281=16;A=B/C301=6:/ See also STANDARD MAP,SURFACE OF SECTIONReferences Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 144 /C1/53, 1988. He´non, M. and Heiles, C. "The Applicability of the Third Integral of Motion: Some Numerical Experiments." As- tron. J. 69,7 3/C1/9, 1964. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 171 /C1/72, 1990. Tabor, M. "The He ´non-Heiles Hamiltonian." §4.1.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 121 /C1/22, 1989. Henry VIII Prime TRUNCATABLE PRIME Hensel’s Lemma An important result in VALUATION THEORY which gives information on finding roots of POLYNOMIALS . Hensel’s lemma is formally stated as follow. Let (K; /C215jj) be a complete NON- ARCHIMEDEAN FIELD , and letRbe the corresponding VALUATION RING . Let f(x) be a POLYNOMIAL whose COEFFICIENTS are in Rand suppose a0satisfies f(a0) jjBf?(a0) jj2; (1) where f?is the (formal) DERIVATIVE off. Then there exists a unique element a/C23Rsuch that f(a)/C300 and a/C28a0 jj5f(a0) f?(a0)P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2: (2) Less formally, if f(x)i sa POLYNOMIAL with " INTEGER " COEFFICIENTS andf(a0) is "small" compared to f?(a0); then the equation f(x)/C300 has a solution "near" a0:In addition, there are no other solutions near a0; although there may be other solutions. The proof of the LEMMA is based around the Newton-Raphson method and relies on the non-Archimedean nature of the valuation. Consider the following example in which Hensel’s lemma is used to determine that the equation x2/C30/C281 is solvable in the 5-adic numbers Q5(and so we can embed the G AUSSIAN INTEGERS inside Q5in a nice way). Let Kbe the 5-adic numbers Q5;letf(x)/C30 x2/C271;and let a0/C302:Then we have f(2)/C305 and f?(2)/C304;so f(2)jj5/C301 5Bf?(2)jj2 5/C301; (3) and the condition is satisfied. Hensel’s lemma then tells us that there is a 5-adic number asuch that a2/C27 1/C300 and a/C282 jj5B/C305 4P+’2P+’2P+’2P+’2P+’2P+’2 5/C301 5: (4) Similarly, there is a 5-adic number bsuch that b2/C27 1/C300 and b/C283 jj5B/C3010 7P+’2P+’2P+’2P+’2P+’2P+’2 5/C301 5: (5) Therefore, we have found both the square roots of /C281 in Q5 : It is possible to find the roots of any POLY- NOMIAL using this technique. See also P-ADIC NUMBER ,VALUATION THEORY References Chevalley, C. C. "Hensel’s Lemma." §3.2 in Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc., pp. 43 /C1/4, 1951. Getz, J. "On Congruence Properties of the Partition Func- tion." Internat. J. Math. Math. Sci. 23, 493/C1/96, 2000. Koch, H. Number Theory: Algebraic Numbers and Func- tions. Providence, RI: Amer. Math. Soc., pp. 115 /C1/17, 2000. Niven, I. M.; Zuckerman, H. S.; and Montgomery, H. L. An Introduction to the Theory of Numbers, 5th ed. New York: Wiley, 1991. Henstock-Kurzweil Integral HK I NTEGRAL Heptacontagon A 70-sided POLYGON . Heptadecagon The REGULAR POLYGON of 17 sides is called the HEPTADECAGON , or sometimes the HEPTAKAIDECAGON . Gauss proved in 1796 (when he was 19 years old) that the heptadecagon is CONSTRUCTIBLE with a COMPASS and STRAIGHTEDGE . Gauss’s proof appears in his monumental work Disquisitiones Arithmeticae. The proof relies on the property of irreducible POLYNO- MIAL equations that ROOTS composed of a finite number of SQUARE ROOT extractions only exist when the order of the equation is a product OF THE FORM 2a3bFc/C215Fd/C1/C1/C1Fe;where the Fnare distinct PRIMES OF THE FORM Fn/C3022n/C271; known as F ERMAT PRIMES . Constructions for the regular TRIANGLE (31),SQUARE (22),PENTAGON (/221/C27 1);HEXAGON (/2131);etc., had been given by Euclid, but constructions based on the F ERMAT PRIMES ]17 were unknown to the ancients. The first explicit construc- tion of a heptadecagon was given by Erchinger inabout 1800. The following elegant construction for the heptade- cagon (Yates 1949, Coxeter 1969, Stewart 1977, Wells1992) was first given by Richmond (1893). 1. Given an arbitrary point O, draw a CIRCLE centered on Oand a DIAMETER drawn through O. 2. Call the right end of the DIAMETER dividing the CIRCLE into a SEMICIRCLE P1:/ 3. Construct the DIAMETER PERPENDICULAR to the original DIAMETER by finding the PERPENDICULAR BISECTOR OB. 4. Construct JaQUARTER the way up OB. 5. Join JP1and find Eso that /C218OJE is a QUARTER of/C218OJP1:/ 6. Find Fso that /C218EJF is 458. 7. Construct the SEMICIRCLE with DIAMETER FP1:/ 8. This SEMICIRCLE cuts OBatK. 9. Draw a SEMICIRCLE with center Eand RADIUS EK. 10. This cuts the extension of OP1atN4:/ 11. Construct a line PERPENDICULAR toOP1 through N4:/ 12. This line meets the original SEMICIRCLE atP4:/ 13. You now have points P1and P4of a heptade- cagon.14. Use P 1andP4to get the remaining 15 points of the heptadecagon around the original CIRCLE by constructing P1;P4;P7;P10;P13;P16[filled circles], P2;P5;P8;P11;P14;P17[single-ringed filled circles], P3;P6;P9;P12;and P15[double-ringed filled circles].15. Connect the adjacent points P ifori/C301 to 17, forming the heptadecagon. This construction, when suitably streamlined, has SIMPLICITY 53. The construction of Smith (1920) has a greater SIMPLICITY of 58. Another construction due to Tietze (1965) and reproduced in Hall (1970) has a SIMPLICITY of 50. However, neither Tietze (1965) nor Hall (1970) provides a proof that this construction is correct. Both Richmond’s and Tietze’s constructions require extensive calculations to prove their validity. De Temple (1991) gives an elegant construction involving the CARLYLE CIRCLES which has GEOMETRO- GRAPHY symbol 8S1 /C274S2 /C2722C1 /C2711C3and SIMPLI- CITY 45. The construction problem has now been automated to some extent (Bishop 1978). See also 257-GON , 65537-GON ,COMPASS ,CONSTRUCTI- BLE POLYGON ,F ERMAT NUMBER ,F ERMAT PRIME , REGULAR POLYGON ,STRAIGHTEDGE ,TRIGONOMETRY VALUES PI/17 References Archibald, R. C. "The History of the Construction of the Regular Polygon of Seventeen Sides." Bull. Amer. Math. Soc. 22, 239/C1/46, 1916. Archibald, R. C. "Gauss and the Regular Polygon of Seven- teen Sides." Amer. Math. Monthly 27, 323/C1/26, 1920. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 95 /C1/6, 1987. Bishop, W. "How to Construct a Regular Polygon." Amer. Math. Monthly 85, 186/C1/88, 1978. Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 63 /C1/9, 1982. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 201 and 229 /C1/30, 1996. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 26 /C1/8, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97/C1/08, 1991. Dickson, L. E. "Construction of the Regular Polygon of 17 Sides." §8.20 in Monographs on Topics of Modern Mathe- matics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 372 /C1/73, 1955. Dixon, R. "Gauss Extends Euclid." §1.4 in Mathographics. New York: Dover, pp. 52 /C1/4, 1991. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. New Haven, CT: Yale University Press, 1965. Hall, T. Carl Friedrich Gauss: A Biography. Cambridge, MA: MIT Press, 1970. Hardy, G. H. and Wright, E. M. "Construction of the Regular Polygon of 17 Sides." §5.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clar- endon Press, pp. 57 /C1/2, 1979. Klein, F. Famous Problems of Elementary Geometry and Other Monographs. New York: Chelsea, 1956. Ore, Ø.Number Theory and Its History. New York: Dover, 1988. Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964. Richmond, H. W. "A Construction for a Regular Polygon of Seventeen Sides." Quart. J. Pure Appl. Math. 26, 206/C1/07, 1893. Smith, L. L. "A Construction of the Regular Polygon of Seventeen Sides." Amer. Math. Monthly 27, 322/C1/23, 1920. Stewart, I. "Gauss." Sci. Amer. 237, 122/C1/31, 1977. Tietze, H. Famous Problems of Mathematics. New York: Graylock Press, 1965.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin, pp. 212 /C1/13, 1991. Yates, R. C. Geometrical Tools. St. Louis, MO: Educational Publishers, 1949. Heptagon The regular seven-sided POLYGON , illustrated above, which has S CHLA ¨FLI SYMBOL 7fg:According to Bank- off and Garfunkel (1973), "since the earliest days of recorded mathematics, the regular heptagon has beenvirtually relegated to limbo." Nevertheless, The ´bault (1913) discovered many beautiful properties of theheptagon, some of which are discussed by Bankoffand Garfunkel (1973). Although the regular heptagon is not a CONSTRUCTI- BLE POLYGON using the classical rules of Greek GEOMETRIC CONSTRUCTION ,i tisconstructible using aN EUSIS CONSTRUCTION (Johnson 1975; left figure above). To implement the construction, place a mark Xon a ruler AZ, and then build a SQUARE of side length AX. Then construct the perpendicular bisector atMtoBC, and draw an arc centered at Cof radius CE. Now place the marked ruler so that it passes through B,Xlies on the arc, and Afalls on the perpendicular bisector. Then 2 u/C30/C218BAC/C30p=7;and two such triangles give the vertex angle 2 p=7o fa regular heptagon. Conway and Guy (1996) give a NEUSIS CONSTRUCTION for the heptagon. In addition, the regular heptagon can be constructed using sevenidentical toothpicks to form 1:3:3 triangles (Finlay1959, Johnson 1975, Wells 1991; right figure above).Bankoff and Garfunkel (1973) discuss the heptagon, including a purported discovery of the N EUSIS CON- STRUCTION by Archimedes (Heath 1931). Madachy (1979) illustrates how to construct a heptagon byfolding and knotting a strip of paper, and the regular heptagon can also be constructed using a CONCHOID OF NICOMEDES . Although the regular heptagon not constructible using classical techniques, Dixon (1991) gives con- structions for several angles very close to 360( =7: While the ANGLE subtended by a side is 360( =7 : 51 :428571( ; Dixon gives constructions containing angles of 2 sin /C281(ffiffiffi 3p =4) :51:3178813( ; tan /C281(5=4) : 51 :340192( ; and 30( /C27sin /C281ffiffiffi3p /C281P+$P+’ =2Þ:51:470701( :/ In the regular heptagon with unit CIRCUMRADIUS and center O, construct the MIDPOINT MAB of AB and the MID-ARC POINT XCB of the arc CB, and let MOXbe the MIDPOINT of OXCB : Then /MOX /C30MAB /C301=ffiffiffi 2p / (Bankoff and Garfunkel 1973). In the regular heptagon, construct the points XCB ; MAB ; and MOXas above. Also construct the midpoint MOXand construct J along the extension of MABB such that MABJ /C30MABXCB : Note that the APOTHEM OMAB of the heptagon has length r /C30cos(p=7): Then 1. The length x /C30MABMOFis equal toffiffiffi2p r /C30ffiffiffi2p cos(p=7); and also to the largest root of 8x 6 /C2820x4 /C2712x2 /C281 /C300; 2. /MOJ /C30ffiffiffi 6p =2/, and 3. MABMOXis tangent to the CIRCUMCIRCLE of DMOFOMAB/ (Bankoff and Garfunkel 1973). Construct a HEPTAGONAL TRIANGLE DABC in a reg- ular heptagon with center O, and let BN and AM bisect /C218ABC and /C218BAC ; respectively, with M and N both lying on the circumcircle. Also define the mid- points MMO;MNO;MMC;andMNC:Then MN/C301 2MMOMNO/C3012MMCMNC (1) /C30ffiffiffi 2p MNOMMC (2) MMOMMC/C30MNOMNC/C301 2(3) MMOMNC/C3012ffiffiffi 2p (4) (Bankoff and Garfunkel 1973). See also CONCHOID OF NICOMEDES ,EDMONDS’ MAP, HEPTAGON THEOREM ,HEPTAGONAL TRIANGLE ,NEU- SIS CONSTRUCTION ,TRIGONOMETRY VALUES PI/7 References Aaboe, A. Episodes from the Early History of Mathematics. Washington, DC: Math. Assoc. Amer., 1964. Bankoff, L. and Garfunkel, J. "The Heptagonal Triangle." Math. Mag. 46,7/C1/9, 1973. Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 59 /C1/0, 1982. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 194 /C1/00, 1996. Courant, R. and Robbins, H. "The Regular Heptagon." §3.3.4 inWhat is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 138 /C1/39, 1996. Dixon, R. Mathographics. New York: Dover, pp. 35 /C1/0, 1991. Finlay, A. H. "Zig-Zag Paths." Math. Gaz. 43, 199, 1959. Heath, T. L. A Manual of Greek Mathematics. Oxford, England: Clarendon Press, pp. 340 /C1/42, 1931. Johnson, C. "A Construction for a Regular Heptagon." Math. Gaz. 59,1 7/C1/1, 1975. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 59 /C1/1, 1979. Bankoff, L. and Demir, H. "Solution to Problem E 1154." Amer. Math. Monthly 62, 584/C1/85, 1955. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. Heptagon Theorem Let H be a heptagon with seven vertices given in cyclic order inscribed in a CONIC . Then the PASCAL LINES of the seven HEXAGONS obtained by omitting each vertex of H in turn and keeping the remaining vertices in the same cyclic order are the sides of a HEPTAGON I which circumscribes a CONIC . Moreover, the BRIANCHON POINTS of the seven HEXA- GONS obtained by omitting the sides of I one at a time and keeping the remaining sides in the natural cyclic order are the vertices of the original HEPTAGON . See also BRIANCHON POINT ,CONIC SECTION ,HEPTA- GON,HEXAGON ,PASCAL LINES References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Heptagon Theorem." §2.1 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8 /C1/1, 1974. Heptagonal Hexagonal Number A number which is simultaneously a HEPTAGONAL NUMBER Hepnand HEXAGONAL NUMBER Hexm : Such numbers exist when 1 2 n(5n /C283) /C30m(2m /C281): (1) COMPLETING THE SQUARE and rearranging gives (10n /C283)2 /C285(4m /C281)2 /C304 : (2) Substituting x /C3010n /C283 and y /C304m /C281 gives the Pell-like quadratic Diophantine equation x2 /C285y2 /C304; (3) which has solutions (x; y) /C30(3; 1); (7, 3), (18, 8), (47, 21), (123, 55), .... The integer solutions in m and n are then given by (n; m) /C30(1; 1); (221, 247), (71065, 79453), (22882613, 25583539), ... (Sloane’s A048902 and A048901), corresponding to the heptagonal hex-agonal numbers 1, 121771, 12625478965, 1309034909945503, ... (Sloane’s A048903). See also HEPTAGONAL NUMBER ,HEXAGONAL NUMBER References Sloane, N. J. A. Sequences A048901, A048902, and A048903 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Heptagonal Number A FIGURATE NUMBER OF THE FORM n(5n /C283)=2 : The first few are 1, 7, 18, 34, 55, 81, 112, ... (Sloane’s A000566). The GENERATING FUNCTION for the hepta- gonal numbers is x(4x/C271) (1/C28x)3/C30x/C277x2/C2718x3/C2734x4/C27...: See also HEPTAGONAL HEXAGONAL NUMBER ,HEPTA- GONAL PENTAGONAL NUMBER ,HEPTAGONAL SQUARE NUMBER ,HEPTAGONAL TRIANGULAR NUMBER ,OCTA- GONAL HEPTAGONAL NUMBER References Sloane, N. J. A. Sequences A000566/M4358 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Heptagonal Pentagonal Number A number which is simultaneously a HEPTAGONAL NUMBER Hnand PENTAGONAL NUMBER Pm:Such numbers exist when 1 2n(5n/C283)/C3012m(3m/C281): (1) COMPLETING THE SQUARE and rearranging gives 3(10n/C283)2/C285(6m/C281)2/C3022: (2) Substituting x/C3010n/C283 and y/C306m/C281 gives the Pell-like quadratic Diophantine equation 3x2/C285y2/C3022; (3) which has solutions ( x;y)/C30(3;1);(7, 5), (17, 13), (53, 41), (133, 103), .... The integer solutions in mandn are then given by ( n;m)/C30(1;1);(42, 54), (2585, 3337), (160210, 206830), (9930417, 12820113) ... (Sloane’s A046198 and A046199), corresponding to the heptagonal pentagonal numbers 1, 4347, 16701685, 64167869935, 246532939589097, ... (Sloa- ne’s A048900). See also HEPTAGONAL NUMBER ,PENTAGONAL NUM- BER References Sloane, N. J. A. Sequences A046198, A046199, and A048900 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Heptagonal Pyramidal Number A PYRAMIDAL NUMBER OF THE FORM n(n /C271)(5n /C28 2)=6; The first few are 1, 8, 26, 60, 115, ... (Sloane’s A002413). The GENERATING FUNCTION for the hepta- gonal pyramidal numbers is x(4x /C27 1) (x /C28 1)4 /C30x /C278x2 /C2726x3 /C2760x4 /C27... References Sloane, N. J. A. Sequences A002413/M4498 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Heptagonal Square Number A number which is simultaneously a HEPTAGONAL NUMBER Hnand SQUARE NUMBER Sm : Such numbers exist when 1 2 n(5n /C283) /C30m2 : (1) COMPLETING THE SQUARE and rearranging gives (10n /C283)2 /C2840m2 /C309 : (2) Substituting x /C3010n /C283 and y /C302m gives the Pell- like quadratic Diophantine equation x2 /C2810y2 /C309 ; (3) which has basic solutions (x; y) /C30(7; 2); (13, 4), and (57, 18). Additional solutions can be obtained from the unit PELL EQUATION , and correspond to integer solutions when (n; m) /C30(1; 1); (6, 9), (49, 77), (961, 1519), ... (Sloane’s A046195 and A046196), corre- sponding to the heptagonal square numbers 1, 81, 5929, 2307361, 168662169, 12328771225, ... (Sloane’s A036354). See also HEPTAGONAL NUMBER ,SQUARE NUMBER References Sloane, N. J. A. Sequences A036354, A046195, and A046196 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html.Heptagonal Triangle The unique (modulo rotations) SCALENE TRIANGLE formed from three vertices of a regular HEPTAGON , having vertex angles p=7;2p=7;and 4 p=7:There are a number of amazing formulas connecting the sides and angles of the heptagonal triangle (Bankoff andGarfunkel 1973). The AREA of the TRIANGLE is A/C301 4ffiffiffi 7p R2; (1) where Ris the triangle’s CIRCUMRADIUS . The sum of squares of sides of the heptagonal triangle is equal to 7R2(Bankoff and Garfunkel 1973). The ratio x/C30r=R ofINRADIUS rtoCIRCUMRADIUS Ris given by the positive root of 8x3/C2728x2/C2714x/C287/C300: (2) Also, 1 a2/C271 b2/C271 c2/C302 R2: (3) The B ROCARD ANGLE Vsatisfies cotV/C30ffiffiffi 7p ; (4) and the EXRADIUS rais equal to the radius of the NINE- POINT CIRCLE ofDABC :/ ais half the HARMONIC MEAN of the other two sides, a/C30bc b/C27c(5) b2/C28a2/C30ac; (6) and so on for all permutations of variables (Bankoff and Garfunkel 1973). Also, b2 a2/C27c2 b2/C27a2 c2/C305: (7) Ifha;hb;andhcare the altitudes, then ha/C30hb/C27hc (8) h2 a/C27h2b/C27h2c/C301 2(a2/C27b2/C27c2): (9) IfA?;B?;andC?are the feet of the altitudes, then BA?/C215 A?C /C301 4 ac (10) and so on (Bankoff and Garfunkel 1973). The internal angle bisectors of C and B are equal to the difference of the adjacent sides and the external angle bisector of A is equal to the sum of adjacent sides. The triangle DDEF joining the feet of the angle bisectors of the heptagonal triangle is an ISOSCELES TRIANGLE with DF /C30EF. The ORTHIC TRIANGLE DHAHBHCand MEDIAN TRIAN- GLE MAMBMCare congruent and perspective. In addition both are similar to DABC ; to the PEDAL TRIANGLE DPAPBPC of DABC with respect to the NINE- POINT CENTER N, and to the triangle DIIBIC formed by the INCENTER I and the exterior angle bisectors IB and IC (Bankoff and Garfunkel 1973). There are also a slew of curious trigonometric identities involving the angles of the heptagonal triangle: sin A sin B sin C /C3018ffiffiffi 7p (11) sin2 A /C27sin2 B /C27 sin2 C /C307 4 (12) sin(2 A) /C27sin(2 B) /C27sin(2 C) /C301 2ffiffiffi 7p (13) sin2 A sin2 B sin2 C /C307 64 (14) sin2 A sin2 B /C27sin2 A sin2 C /C27sin2 B sin2 C /C307 8(15) cos A cos B cos C /C30/C2818 (16) cos2 A /C27cos2 B /C27cos2 C /C3054 (17) cos2 A cos2 B /C27cos2 A cos2 C /C27cos2 B cos2 C /C303 8(18) cos(2 A) /C27cos(2 B) /C27cos(2 C) /C30/C281 2 (19) sin A /C27sin B /C27sin C /C301 2ffiffiffiffiffiffi 14p (20)tan A tan B tan C /C30/C28ffiffiffi 7p (21) cot A /C27cot B /C27cot C /C30ffiffiffi 7p (22) csc2 A /C27csc2 B /C27csc2 C /C308 (23) sec2 A /C27sec2 B /C27sec2 C /C3024 (24) cot2 A /C27cot2 B /C27cot2 C /C305 (25) tan2 A /C27tan2 B /C27tan2 C /C3021 (26) sec4 A /C27sec4 B /C27sec4 C /C30416 (27) cos4 A /C27cos4 B /C27cos4 C /C3013 16 (28) sin4 A /C27sin4 B /C27sin4 C /C3021 16 (29) csc4 A /C27csc4 B /C27csc4 C /C3032 (30) sec(2 A) /C27sec(2 B) /C27sec(2 C) /C30/C284 (31) (Bankoff and Garfunkel 1973). Finally, the heptagonal triangle satisfies the miscel- laneous properties: 1. The first BROCARD POINT corresponds to the NINE-POINT CENTER and the second BROCARD POINT lies on the NINE-POINT CIRCLE . 2. OH /C30Rffiffiffi 2p ; where O is the CIRCUMCENTER , H is the ORTHOCENTER , and R is the CIRCUMRADIUS . 3. IH /C30(R2 /C274r2) =2; where I is the INCENTER and r is the INRADIUS . 4. The two tangents from the ORTHOCENTER Hto the CIRCUMCIRCLE of the heptagonal triangle are mutually perpendicular. 5. The center of the CIRCUMCIRCLE of the TANGEN- TIAL TRIANGLE corresponds with the symmetric point of Owith respect to H. 6. The ALTITUDE from Bis half the length of the internal bisector of the angle A. See also HEPTAGON References Bankoff, L. and Garfunkel, J. "The Heptagonal Triangle." Math. Mag. 46,7/C1/9, 1973. Heptagonal Triangular Number A number which is simultaneously a HEPTAGONAL NUMBER Hnand TRIANGULAR NUMBER Tm:Such numbers exist when 1 2n(5n/C283)/C3012m(m/C271): (1) COMPLETING THE SQUARE and rearranging gives 10n/C283 ðÞ2/C2852m/C271 ðÞ2/C304: (2) Substituting x/C3010n/C283 and y/C302m/C271 gives the Pell-like quadratic Diophantine equation x2/C285y2/C304; (3) which has basic solutions (x; y) /C30(3; 1); (7, 3), and (18, 8). Additional solutions can be obtained from the unit PELL EQUATION , and correspond to integer solutions when (n; m) /C30(1; 1); (5, 10), (221, 493), (1513, 3382), ... (Sloane’s A046193 and A039835), corresponding to the heptagonal triangular numbers 1, 55, 121771, 5720653, 12625478965, ... (Sloane’s A046194). See also HEPTAGONAL NUMBER ,TRIANGULAR NUMBER References Sloane, N. J. A. Sequences A039835, A046193, and A046194 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Heptagram One of the two 7-sided STAR POLYGONS 7=2fg and 7=3fg ; illustrated above. See also HEPTAGON ,STAR POLYGON References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 211, 1999. Heptahedral Graph A POLYHEDRAL GRAPH on seven nodes. There are 34 nonisomorphic heptahedral graphs, as first enumer- ated by Kirkman (1862) and Hermes (1899ab, 1900, 1901; Federico 1969; Duijvestijn and Federico 1981). See also HEPTAHEDRON ,POLYHEDRAL GRAPHReferences Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (3-Connected Planar) Graphs." Math. Comput. 37, 523 /C1/32, 1981. Federico, P. J. "Enumeration of Polyhedra: The Number of 9-Hedra." J. Combin. Th. 7, 155 /C1/61, 1969. Gru¨nbaum, B. Convex Polytopes. New York: Wiley, pp. 288 and 424, 1967. Hermes, O. "Die Formen der Vielflache. I." J. reine angew. Math. 120,27/C1/9, 1899a. Hermes, O. "Die Formen der Vielflache. II." J. reine angew. Math. 120, 305 /C1/53, 1899b. Hermes, O. "Die Formen der Vielflache. III." J. reine angew. Math. 122, 124 /C1/54, 1900. Hermes, O. "Die Formen der Vielflache. IV." J. reine angew. Math. 123, 312 /C1/42, 1901. Kirkman, T. P. "Application of the Theory of the Polyhedra to the Enumeration and Registration of Results." Proc. Roy. Soc. London 12, 341 /C1/80, 1862 /C1/863. Pegg, E. Jr. "The 34 Convex Heptahedra and Their Char- acteristic Polynomials." http://www.mathpuzzle.com/char- poly.htm. Heptahedron A heptahedron is a POLYHEDRON with seven faces. There are 34 topologically distinct convex heptahe- dra, corresponding to the HEPTAHEDRAL GRAPHS . The "regular" heptahedron is a one-sided surface made from four TRIANGLES and three QUADRILAT- ERALS . It is topologically equivalent to the ROMAN SURFACE (Wells 1991). While all of the faces are regular and vertices equivalent, the heptahedron is self-intersecting and is therefore not considered an ARCHIMEDEAN SOLID . There are three semiregular heptahedra: the PENTA- GONAL PRISM and PENTAGRAMMIC PRISM (illustrated above), and a FACETED version of the OCTAHEDRON (Holden 1991). See also ARCHIMEDEAN SOLID ,HEPTAHEDRAL GRAPH , OCTAHEDRON ,POLYHEDRON ,QUADRILATERAL ,ROMAN SURFACE ,SZILASSI POLYHEDRON References Holden, A. Shapes, Space, and Symmetry. New York: Dover, p. 95, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Viking Penguin, p. 98, 1992. Heptakaidecagon HEPTADECAGON Heptaparallelohedron CUBOCTAHEDRON Heptiamond One of the 24 7-polyiamonds. See also HEPTIAMOND TILING ,POLYIAMOND References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 246, 248, and 250 /C1/51, 1984. Heptiamond Tiling See also HEPTIAMOND ,H EXIAMOND TILING ,O CTIA- MOND TILING ,PENTIAMOND TILING References Vichera, M. "Polyiamonds." http://alpha.ujep.cz/~vicher/puz- zle/polyform/iamond/iamonds.htm. Heptic Surface An ALGEBRAIC SURFACE of degree 7. See also ALGEBRAIC SURFACE Heptomino The heptominoes are the 7-POLYOMINOES . There are108 FREE , 760 FIXED , and 196 one-sided heptominoes. There is a single heptomino containing a hole (illu- strated above), making heptominoes the smallest polyominoes for which the existence of a hole is possible. See also DOMINO ,HERSCHEL ,HEXOMINO ,OCTOMINO , PENTOMINO ,PI HEPTOMINO ,POLYOMINO ,TETROMINO , TRIOMINO Herbrand Function References Koch, H. Number Theory: Algebraic Numbers and Func- tions. Providence, RI: Amer. Math. Soc., p. 190, 2000. Herbrand’s Theorem Let an ideal class be in A if it contains an IDEAL whose lth power is PRINCIPAL . Let i be an ODD INTEGER 1 5i 5l and define j by i /C27j /C301: Then A1 /C30 /C142e /C143: If i ]3 and l¶Bj ; then Ai /C30/C142e /C143:/ See also IDEAL References Ireland, K. and Rosen, M. "Herbrand’s Theorem." §15.3 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 241 /C1/48, 1990. Hereditary Representation The representation of a number as a sum of powers of a BASE b, followed by expression of each of the exponents as a sum of powers of b, etc., until the process stops. For example, the hereditary represen- tation of 266 in base 2 is 266/C3028/C2723/C272 /C30222/C271/C2722/C271/C272: See also GOODSTEIN SEQUENCE ,GOODSTEIN’S THEO- REM References Henle, J. M. An Outline of Set Theory. New York: Springer- Verlag, 1986. Heredity A property of a SPACE which is also true of each of its SUBSPACES . Being "COUNTABLE " is hereditary, but having a given GENUS is not. Hermann Grid Illusion A regular 2-D arrangement of squares separated by vertical and horizontal "canals." Looking at the grid produces the illusion of gray spots in the white AREA between square VERTICES . The illusion was noted by Hermann (1870) while reading a book on sound by J. Tyndall. References Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 139 /C1/40, 1996. Hermann’s Formula The MACHIN-LIKE FORMULA 1 4 p /C302 tan/C281(12) /C28tan /C281(17) : The other 2-term MACHIN-LIKE FORMULAS are EU- LER’S MACHIN-LIKE FORMULA ,H UTTON’S FORMULA , and MACHIN’S FORMULA . Hermann-Hering Illusion The illusion in view by staring at the small black dot for a half minute or so, then switching to the white dot. The black squares appear stationary when staring at the white dot, but a fainter grid of moving squares also appears to be present. Hermann-Mauguin Symbol A symbol used to represent the POINT and SPACE GROUPS (e.g., 2=m¯3): Some symbols have abbreviated form. The equivalence between Hermann-Mauguin symbols (a.k.a. "crystallographic symbols"rpar; and SCHO¨ NFLIES SYMBOLS for the POINT GROUPS is given by Cotton (1990). See also POINT GROUPS ,SCHO¨ NFLIES SYMBOL ,SPACE GROUPSReferences Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 379, 1990. Hermit Point ISOLATED POINT Hermite Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. The Hermite constant is defined for DIMENSION n as the value gn /C30supfminxif(x1 ; x2 ; ... ; xn) [discriminant( f)]1 =n (Le Lionnais 1983). In other words, they are given by gn /C304dn Vn !2 =n ; where dn is the maximum lattice PACKING DENSITY for HYPERSPHERE PACKING and Vn is the CONTENT of the n-HYPERSPHERE . The first few values of (gn)n are 1, / 4=3/, 2, 4, 8, 64/3, 64, 256, ... (Sloane’s A007361 and A007362). Values for larger n are not known. For sufficiently large n, 1 2pe5gn n51:744 . . . 2pe: See also DISCRIMINANT ,H YPERSPHERE PACKING , KISSING NUMBER ,SPHERE PACKING References Cassels, J. W. S. An Introduction to the Geometry of Num- bers, 2nd ed. New York: Springer-Verlag, p. 332, 1997. Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lat- tices, and Groups, 2nd ed. New York: Springer-Verlag, p. 20, 1993. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hermit/hermit.html. Gruber, P. M. and Lekkerkerker, C. G. Geometry of Num- bers, 2nd ed. Amsterdam, Netherlands: North-Holland, p. 410, 1987. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 38, 1983. Sloane, N. J. A. Sequences A007361/M3201 and A007362/ M2209 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Hermite Differential Equation The second-order ordinary linear differential equa- tion d2y dx2/C282xdy dx/C27ly/C300: (1) This differential equation has an irregular singular- ity at/C12:It can be solved using the series method X/C12 n/C300(n/C272)(n/C271)an/C272xn/C28X/C12 n/C3012nanxn/C27X/C12 n/C300lanxn /C300 (2) (2a2/C27la0)/C27X/C12 n/C301[(n/C272)(n/C271)]an/C272/C282nan/C27lan]xn /C300: (3) Therefore, a2/C30/C28la0 2(4) and an/C272/C302n/C28l (n/C272)(n/C271)an (5) forn/C301, 2, .... Since (4) is just a special case of (5), an/C272/C302n/C28l (n/C272)(n/C271)an (6) forn/C300, 1, .... The linearly independent solutions are then y1 /C30a01/C28l 2!x2/C28(4/C28l)l 4!x4/C28(8/C28l)(4/C28l)l 6!x6/C28..."# (7) y2/C30a1x/C27(2/C28l) 3!x3/C27(6/C28l)(2/C28l) 5!x5/C27..."# :(8) These can be done in closed form as y/C30a01F1(/C281 4l;12;x2)/C27a1x1F1(/C2814(l/C282);32;x2) (9) /C30a01F1(/C2814l;12;x2)/C27a2Hl=2(x); (10) where1F1(a;b;x)i sa CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND andHn(x)i saH ERMITE POLYNOMIAL . In particular, for l/C300;2, 4, ..., the solutions can be written yl/C300/C30a0/C2712ffiffiffippa1erfi(x) (11) yl/C302/C30a0ex2/C28ffiffiffippxerfi(x)hi /C27xa1 (12) yl/C304/C301 4f2ex2xa1/C28(2x2/C281)[4a0/C27ffiffiffippa1erfi(x)]g;(13) where erfi( x) is the ERFI function. Ifl/C300;then Hermite’s differential equation becomes y??/C282xy?/C300; (14) which is OF THE FORM P2(x)y??/C27P1(x)y?/C300 and so has solutiony/C30c1gdx expgP1 P2dxP+’vP+’u /C27c2 /C30c1gdx expg(/C282x)dx/C27c2 /C30c1gdx e/C28x2/C27c2/C30c1erfi(x)/C27c2: (15) Hermite Interpolation HERMITE’S INTERPOLATING POLYNOMIAL Hermite Polynomial A set of ORTHOGONAL POLYNOMIALS Hn(x);illustrated above for x/C23[0;1] and n/C301, 2, ..., 5. Roman (1984, pp. 87 /C1/3) defines a generalized Hermite polynomial H(n) n(x) of variance n:/ The Hermite polynomials are a S HEFFER SEQUENCE with g(t)/C30et2=4(1) f(t)/C301 2t (2) (Roman 1984, p. 30), giving the GENERATING FUNC- TION exp(2 xt/C28t2)/C13X/C12 n/C300Hn(x)tn n!: (3) Using a T AYLOR SERIES shows that Hn(x)/C30@ @t !n exp(2 xt/C28t2)"# t/C300 /C30ex2@ @t !n e/C28(x/C28t)2"# t/C300: (4) Since @f(x/C28t)=@t/C30/C28 @f(x/C28t)=@x; Hn(x)/C30(/C281)nex2 @ @x !n e/C28(x/C28t)2"# t/C300 /C30(/C281)nex2dn dxne/C28x2: (5) Now define operators ˜O1/C13/C28ex2d dxe/C28x2(6) ˜O2/C13ex2=2x/C28d dx ! e/C28x2=2: (7) It follows that ˜O1f/C30/C28ex2d dx[fe/C28x2]/C302xf/C28df dx(8) ˜O2f/C30ex2=2x/C28d dx ! [fe/C28x2=2] /C30xf/C27xf/C28df dx/C302xf/C28df dx; (9) so ˜O1/C30˜O2; (10) and /C28ex2d dxe/C28x2/C30ex2=2x/C28d dx ! e/C28x2=2(11) (Arfken 1985, p. 720), which means the following definitions are equivalent: exp(2 xt/C28t2)/C13X/C12 n/C300Hn(x)tn n!(12) Hn(x)/C13(/C281)nex2dn dxne/C28x2(13) Hn(x)/C13ex2=2x/C28d dx !n e/C28x2=2(14) (Arfken 1985, pp. 712 /C1/13 and 720). The Hermite polynomials may be written as Hn(x)/C30(2x)n 2F0(/C28n=2;/C28(n/C281)=2; ;/C281=x2) (15) (Koekoek and Swarttouw 1998), or Hn(x)/C302nU(/C281 2n;12;x2); (16) where U(a;b;x)i sa CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND . The Hermite poly- nomials are related to the derivative of the ERROR FUNCTION by Hn(z)/C30(/C281)2ffiffiffipp 2ez2dn/C271 dzn/C271erf(z): (17) They have a CONTOUR INTEGRAL representationHn(x)/C30n! 2pige/C28t2/C272txt/C28n/C281dt: (18) They are orthogonal in the range ( /C28/C12;/C12) with respect to the WEIGHTING FUNCTION /e/C28x2 g/C12 /C28/C12Hm(x)Hn(x)e/C28x2dx/C30dmn2nn!ffiffiffipp: (19) The first few POLYNOMIALS are H0(x)/C301 H1(x)/C302x H2(x)/C304x2/C282 H3(x)/C308x3/C2812x H4(x)/C3016x4/C2848x2/C2712 H5(x)/C3032x5/C28160x3/C27120x H6(x)/C3064x6/C28480x4/C27720x2/C28120 H7(x)/C30128x7/C281344 x5/C273360 x3/C281680 x H8(x)/C30256x8/C283584 x6/C2713440 x4/C2813440 x2/C271680 H9(x)/C30512x9/C289216 x7/C2748348 x5/C2880640 x3/C2730240 x H10(x)/C301024 x10/C2823040 x8/C27161280 x6/C28403200 x4 /C27302400 x2/C2830240 : The Hermite polynomials obey the orthogonality conditions g/C12 /C28/C12un(x)dum dxdx/C30affiffiffiffiffiffiffi n/C271 2q m/C30n/C271 /C28affiffi n 2q m/C30n/C281 0 otherwise8 >>< >>:(20) g/C12 /C28/C12um(x)un(x)dx/C30dmn (21) g/C12 /C28/C12um(x)xun(x)dx/C301 affiffiffiffiffiffiffi n/C271 2q m/C30n/C271 1 affiffi n 2q m/C30n/C281 0 otherwise8 >>< >>:(22) g/C12 /C28/C12um(x)x2un(x)dx /C302n/C271 2a2 m/C30nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n/C271)(n/C272)p 2a2 m/C30n/C272 0 m"n"n928 >< >:(23) g/C12 /C28/C12e /C28x2 HaHbH g dx /C30ffiffiffipp 2s a!b!g! (s /C28 a)!(s /C28 b)!(s /C28 g)! ; (24) if a /C27 b /C27 g /C302s is EVEN and s ] a; s ] b; and s ] g: Otherwise, the last integral is 0 (Szego 1975, p. 390). They also satisfy the RECURRENCE RELATIONS Hn/C271(x) /C302xHn(x) /C282nHn /C281(x) (25) H ?n(x) /C302nHn/C281(x): (26) By solving the HERMITE DIFFERENTIAL EQUATION , the series H2k(x) /C30(/C281)k2k(2k /C281)!! /C2 1 /C27Xk j/C301( /C284k)( /C284k /C27 4) /C1/C1/C1(/C284k /C27 4j /C28 4) (2j)! x2j"# (27) H2k/C271(x) /C30(/C281)k2k /C271(2k /C271)!! /C2 x /C27Xk j/C301( /C284k)(/C284k /C27 4) /C1/C1/C1( /C284k /C27 4j /C28 4) (2j /C27 1)! x2j/C271"# (28) are obtained, where the products in the numerators are equal to (/C284k)(/C284k /C274) /C1/C1/C1(/C284k /C274j /C284) /C304j(/C28k)j ; (29) with (x)n the POCHHAMMER SYMBOL . The DISCRIMINANT is Dn /C3023n(n/C281)=2Yn k /C301kk (30) (Szego 1975, p. 143), a normalized form of the HYPERFACTORIAL , the first few values of which are 1, 32, 55296, 7247757312, 92771293593600000, ... (Sloane’s A054374). The table of RESULTANTS is given by {0}, { /C288, 0}, {0, /C282048, 0}, {192, 16384, 28311552, 0}, ... (Sloane’s A054373). Two interesting identities involving Hn(x /C27y) are given by Xn k/C300n kP+’vP+’u Hk(x)Hn/C28k(y) /C302n=2Hn(2/C281=2(x /C27y)) (31) and Xn k /C300n kP+’vP+’u Hk(x)(2y)n/C28k /C30Hn(x /C27y) (32) (G. Colomer).A set of associated functions is defined by un(x) /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a p1=2n!2ns Hn(ax)e /C28a2x2 =2 : (33) A class of generalized Hermite POLYNOMIALS gm n (x) satisfying emxt /C28tm /C30X/C12 n /C300gm n (x)tn (34) was studied by Subramanyan (1990). A class of related POLYNOMIALS defined by hn;m /C30 gmn2x m ! (35) and with GENERATING FUNCTION e2xt/C28tm/C30X/C12 n/C300hn;m(x)tn(36) was studied by Djordjevic (1996). They satisfy Hn(x)/C30n!hn;2(x): (37) A modified version of the H ERMITE POLYNOMIAL is sometimes defined by Hen(x)/C13Hnxffiffiffi 2p ! : (38) See also MEHLER’S HERMITE POLYNOMIAL FORMULA , WEBER FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771 /C1/02, 1972. Andrews, G. E.; Askey, R.; and Roy, R. "Hermite Polyno- mials." §6.1 in Special Functions. Cambridge, England: Cambridge University Press, pp. 278 /C1/82, 1999. Arfken, G. "Hermite Functions." §13.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 712 /C1/21, 1985. Chebyshev, P. L. "Sur le de ´veloppement des fonctions a `une seule variable." Bull. ph.-math., Acad. Imp. Sc. St. Pe´tersbourg 1, 193/C1/00, 1859. Chebyshev, P. L. Oeuvres, Vol. 1. New York: Chelsea, pp. 49 /C1/08, 1987. Djordjevic, G. "On Some Properties of Generalized Hermite Polynomials." Fib. Quart. 34,2/C1/, 1996. Hermite, C. "Sur un nouveau de ´veloppement en se ´rie de fonctions." Compt. Rend. Acad. Sci. Paris 58,9 3/C1/00 and 266/C1/73, 1864. Reprinted in Hermite, C. Oeuvres com- ple`tes, Vol. 2. Paris, pp. 293 /C1/08, 1908. Hermite, C. Oeuvres comple `tes, Tome III. Paris: Hermann, p. 432, 1912. Iyanaga, S. and Kawada, Y. (Eds.). "Hermite Polynomials." Appendix A, Table 20.IV in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1479 /C1/480, 1980. Jeffreys, H. and Jeffreys, B. S. "The Parabolic Cylinder, Hermite, and Hh Functions" §23.08 in Methods of Math- ematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 620 /C1/22, 1988. Koekoek, R. and Swarttouw, R. F. "Hermite." §1.13 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Uni- versiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, pp. 50 /C1/1, 1998. ftp://www.twi.- tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1/ 7.ps.gz. Roman, S. "The Hermite Polynomials." §4.2.1 in The Umbral Calculus. New York: Academic Press, pp. 87 /C1/3, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "Hermite Polyno- mials." §10 in "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Sansone, G. "Expansions in Laguerre and Hermite Series." Ch. 4 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 295 /C1/85, 1991. Sloane, N. J. A. Sequences A054373 and A054374 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Spanier, J. and Oldham, K. B. "The Hermite Polynomials Hn(x) :/" Ch. 24 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 217 /C1/23, 1987. Subramanyan, P. R. "Springs of the Hermite Polynomials." Fib. Quart. 28, 156 /C1/61, 1990. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. Hermite Quadrature HERMITE- GAUSS QUADRATURE Hermite’s Interpolating Polynomial Let l(x)bean nth degree POLYNOMIAL with zeros at x1 ; ..., xn : Then the fundamental Hermite interpolat- ing polynomials of the first and second kinds are defined by h(1) n(x) /C30 1 /C28l ??(xn) l ?(xn)"# [ln(x)]2 (1) and h(2)n(x) /C30(x /C28xn)[l n(x)]2 (2) for n /C301, 2, .., .n. These polynomials have the proper- ties h(1)n(xm) /C30 d nm (3) h(1)? n(xm) /C300 (4) h(2)n(xm) /C300 (5) h(2)? n(xm) /C30 dnm : (6) for m; n /C301; 2, ..., n. Now let f1 ; ..., fn and f ? 1 ; ..., f ? nbe values. Then the expansion Wn(x) /C30Xn n/C301fnh(1)n(x) /C27Xn n/C301f ? nh(2)n(x) (7) gives the unique Hermite interpolating fundamentalpolynomial for which Wn(xn) /C30f n (8) W ?n(xn) /C30f ?n : (9) If f ?n /C300; these are called STEP POLYNOMIALS . The fundamental polynomials satisfy h1(x) /C27.../C27hn(x) /C301 (10) and Xn n/C301xnh(1)n(x) /C27Xn n/C301h(2)n(x) /C30x: (11) Also, if da(x) is an arbitrary distribution on the interval [a, b], then gb ah(1)n(x) da(x) /C30 ln (12) gb ah(1)? n(x) da(x) /C300 (13) gb axh(1) ? n(x) da(x) /C300 (14) gb ah(2)n(x) da(x) /C300 (15) gb ah(2)? n(x) da(x) /C30 ln (16) gb axh(2)? n(x) da(x) /C30 lnxn ; (17) where ln are CHRISTOFFEL NUMBERS . See also CHRISTOFFEL NUMBER ,LAGRANGE INTERPO- LATING POLYNOMIAL References Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 314 /C1/19, 1956. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 330 /C1/32, 1975. Hermite’s Theorem EisTRANSCENDENTAL . See also E,TRANSCENDENTAL NUMBER . Hermite-Gauss Quadrature Also called H ERMITE QUADRATURE .AG AUSSIAN QUAD- RATURE over the interval ( /C28/C12;/C12) with WEIGHTING FUNCTION W(x)/C30e/C28x2(Abramowitz and Stegun 1972, p. 890). The ABSCISSAS for quadrature order nare given by the roots of the H ERMITE POLYNOMIALS Hn(x);which occur symmetrically about 0. The WEIGHTS are wi /C30/C28An/C271 gn AnH ?n(xi)Hn/C271(xi) /C30An An/C281gn/C281 Hn/C281(x1)H ?n(xi) ; (1) where Anis the COEFFICIENT of xnin Hn(x): For HERMITE POLYNOMIALS , An /C302n ; (2) so An/C271 An/C302: (3) Additionally, gn /C30ffiffiffipp2nn!; (4) so wi /C30/C282n/C271n!ffiffiffipp Hn/C271(xi)H ?n(xi) /C302n(n /C28 1)!ffiffiffipp H n/C281(xi)H ?n(xi) : (5) Using the RECURRENCE RELATION H ?n(x) /C302nHn /C281(x) /C302xHn(x) /C28Hn/C271(x) (6) yields H ?n(xi) /C302nHn /C281(xi) /C30/C28Hn/C271(xi) (7) and gives wi /C302n/C271n!ffiffiffipp [H ? n(xi)]2 /C302n /C271n!ffiffiffipp [H n/C271(xi)]2 : (8) The error term is E /C30n!ffiffiffipp 2n(2n)!f(2n)( j) : (9) Beyer (1987) gives a table of ABSCISSAS and weights up to n /C3012. n /xi// wi/ 2 9 0.707107 0.886227 3 0 1.18164 9 1.22474 0.295409 4 9 0.524648 0.804914 9 1.65068 0.0813128 5 0 0.945309 9 0.958572 0.393619 9 2.02018 0.0199532The ABSCISSAS and weights can be computed analy- tically for small n. n /xi// wi/ 2 /91 2ffiffiffi 2p //1 2ffiffiffipp / 30 /2 3ffiffiffipp / /91 2ffiffiffi 6p //1 6ffiffiffipp / 4 /9ffiffiffiffiffiffiffiffiffiffi 3 /C28ffiffi 6p 2q //ffiffipp 4(3 /C28ffiffi 6p )/ /9ffiffiffiffiffiffiffiffiffiffi 3 /C27ffiffi 6p 2q //ffiffipp 4(3 /C27ffiffi 6p )/ References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 890, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 464, 1987. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 327 /C1/30, 1956. HermiteH HERMITE POLYNOMIAL Hermite-Lindemann Theorem Let ai and A1 be ALGEBRAIC NUMBERS such that the Ai/ s differ from zero and the ai/s differ from each other. Then the expression A1ea1 /C27A2ea2 /C27A3ea3 /C27... cannot equal zero. The theorem was proved by Hermite (1873) in the special case of the Ai/s and ai/s RATIONAL INTEGERS , and subsequently proved for algebraic numbers by Lindemann (1882). The proof was subsequently simplified by Weierstrass (1885)and Gordan (1893). See also A LGEBRAIC NUMBER ,CONSTANT PROBLEM , FOUR EXPONENTIALS CONJECTURE ,INTEGER RELA- TION ,LINDEMANN- WEIERSTRASS THEOREM ,SIX EXPO- NENTIALS THEOREM References Do¨rrie, H. "The Hermite-Lindemann Transcendence Theo- rem." §26 in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 128 /C1/37, 1965. Hermite, C. "Sur la fonction exponentielle." Comptes rendus 77,1 8/C1/4, 1873. Gordan, P. "Transcendenz von eund p:/"Math. Ann. 43, 222/C1/24, 1893. Lindemann, F. "U ¨ber die Ludolph’sche Zahl." Sitzungber. Ko¨nigl. Preuss. Akad. Wissensch. zu Berlin No. 2, pp. 679 /C1/82, 1888. Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1902. Weierstrass, K. "Zu Hrn. Lindemann’s Abhandlung: ‘U¨ ber die Ludolph’sche Zahl’." Sitzungber. Ko¨nigl. Preuss. Akad. Wissensch. zu Berlin No. 2, pp. 1067 /C1/086, 1885. Hermitian Conjugate ADJOINT Hermitian Form A combination of variables x and y given by ax¯x /C27bx¯y /C27 ¯b¯xy /C27cy¯y; where ¯b ; ¯x and ¯y are COMPLEX CONJUGATES . Hermitian Inner Product A Hermitian inner product on a COMPLEX VECTOR SPACE V is a complex-valued BILINEAR FORM on V which is ANTILINEAR in the second slot, and is positive definite. That is, it satisfies the following properties, where ¯z denotes the COMPLEX CONJUGATE of z. 1. /C142u /C27v; w /C143/C30/C142u; w/C143/C27/C142v; w /C143/ 2. /C142u; v /C27w /C143/C30/C142u; v/C143/C27/C142u; w/C143/ 3. /C142 au; v/C143/C30 a/C142u; v/C143/ 4. /C142u; av/C143/C30 ¯a/C142u; v/C143/ 5. /C142u; v/C143/C30/C142v ; u/C143/ 6. /C142u; u/C143]0; with equality only if u /C300 The basic example is the form h(z ; w) /C30X zi ¯wi (1) on Cn ; where z /C30(z1 ; ...; zn) and w /C30(w1 ; ...; wn): Note that by writing zk /C30xk /C27iyk ; it is possible to consider Cn /C2R2n ; in which case R[h] is the Euclidean INNER PRODUCT and I[h] is a nondegenerate alter- nating BILINEAR FORM , i.e., a SYMPLECTIC FORM . Explicitly, in C2 ; the standard Hermitian form is expressed below. h((z11 ; z12) ; (z21 ; z22)) /C30x11 ;x21 /C27x12x22 /C27y11y21 /C27y12y22 /C27i(x21y11 /C28x11y21 /C27x22y12 /C28x12y22) : (2) A generic Hermitian inner product has its REAL PART symmetric positive definite, and its IMAGINARY PART symplectic by properties 5 and 6. A matrix H /C30( hij) defines an antilinear form, satisfying 1 /C1/,by/C142ei ; ej /C143/C30 hij IFF H is a HERMITIAN MATRIX . It is positive definite (satisfying 6) when R[H]isa POSITIVE DEFINITE MATRIX . In matrix form, /C142v; w/C143/C30vTH ¯w (3) and the canonical Hermitian inner product is when H is the IDENTITY MATRIX . See also COMPLEX NUMBER ,H ERMITIAN METRIC , INNER PRODUCT ,P OSITIVE DEFINITE QUADRATIC FORM,SYMPLECTIC FORM,UNITARY BASIS,UNITARY GROUP ,UNITARY MATRIX ,VECTOR SPACEHermitian Matrix A SQUARE MATRIX is called Hermitian if it is SELF- ADJOINT . Therefore, a Hermitian matrix is defined as one for which A /C30A/C31 (1) where A /C31 denotes the ADJOINT MATRIX . For example, A /C3011 /C27i 2i 1 /C28i 5 /C283 /C282i /C28302 435 (2) is a Hermitian matrix. An INTEGER or REAL MATRIX is Hermitian iff it is SYMMETRIC . A matrix m can be tested to see if it is Hermitian using the Mathematica function HermitianQ[m_List?MatrixQ] : /C30 (m /C30/C30/C30 Conjugate@Transpose@m) Hermitian matrices have REAL EIGENVALUES whose EIGENVECTORS form a UNITARY BASIS . For REAL MATRICES , Hermitian is the same as SYMMETRIC . Any MATRIX C which is not Hermitian can be expressed as the sum a Hermitian matrix and a SKEW HERMITIAN MATRIX using C /C301 2(C /C27C/C31) /C2712(C /C28C/C31): (3) Let U be a UNITARY MATRIX and A be a Hermitian matrix. Then the ADJOINT MATRIX of a SIMILARITY TRANSFORMATION is (UAU/C281) /C30[(UA)(U /C281)]/C31/C30(U /C281) /C31(UA) /C31 /C30(U /C31) /C31(A/C31U /C31) /C30UAU /C31/C30UAU/C281 : (4) The specific matrix H(x; y; z) /C30zx /C27iy x /C28iy /C28zP+2$P+2’ /C30xP1 /C27yP2 /C27zP3 ; (5) where Piare PAULI SPIN MATRICES , is sometimes called "the" Hermitian matrix. See also ADJOINT MATRIX ,H ERMITIAN OPERATOR , NORMAL MATRIX ,PAULI SPIN MATRICES ,SKEW HER- MITIAN MATRIX ,SYMMETRIC MATRIX References Arfken, G. "Hermitian Matrices, Unitary Matrices." §4.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 209 /C1/17, 1985. Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 13 and 117 /C1/18, 1962. Hermitian Metric A Hermitian metric on a COMPLEX VECTOR BUNDLE assigns a H ERMITIAN INNER PRODUCT to every FIBER . The basic example is the TRIVIAL BUNDLE p:U/C29 Ck0U;where Uis an OPEN SET inRn:Then a positive definite H ERMITIAN MATRIX Hdefines a Hermitian metric by /C142v ; w /C143/C30vTH ¯w ; where ¯w is the COMPLEX CONJUGATE of w.Bya PARTITION OF UNITY , any COMPLEX VECTOR BUNDLE has a Hermitian metric. In the special case of a COMPLEX MANIFOLD , the complexified TANGENT BUNDLE TM /C156C may have a Hermitian metric, in which case its REAL PART is a RIEMANNIAN METRIC and its IMAGINARY PART is a nondegenerate ALTERNATING MULTILINEAR FORM v: When v is CLOSED , i.e., in this case a SYMPLECTIC FORM , then v is a KA¨ HLER FORM . On a HOLOMORPHIC VECTOR BUNDLE with a Hermitian metric h, there is a unique connection compatible with hand the complex structure. Namely, it must be 9/C30@/C27¯@;where @s/C30h/C281@hsin a TRIVIALIZATION . See also COMPLEX GEOMETRY ,COMPLEX MANIFOLD , COMPLEX VECTOR BUNDLE ,H OLOMORPHIC VECTOR BUNDLE ,KA¨ HLER FORM,KA¨ HLER MANIFOLD ,RIEMAN- NIAN METRIC ,SYMPLECTIC FORM,UNITARY GROUP Hermitian Operator A Hermitian OPERATOR ¯Lis one which satisfies gb a¯v¯Lu dx/C30gb au¯L¯vd x : (1) where ¯zdenotes a COMPLEX CONJUGATE . As shown in STURM- LIOUVILLE THEORY ,i f ¯LisSELF-ADJOINT and satisfies the boundary conditions ¯vpu?½x/C30a/C30¯vpu?½x/C30b; (2) then it is automatically Hermitian. Hermitian opera- tors have REAL EIGENVALUES ,ORTHOGONAL EIGEN- FUNCTIONS , and the corresponding EIGENFUNCTIONS form a COMPLETE set when ¯Lis second-order and linear. In order to prove that EIGENVALUES must be REAL and EIGENFUNCTIONS ORTHOGONAL , consider ¯Lui/C27liwui/C300: (3) Assume there is a second EIGENVALUE ljsuch that ¯Luj/C27ljwuj/C300 (4) ¯L¯uj/C27¯ljw¯uj/C300: (5) Now multiply (3) by ¯ujand (5) by ui ¯uj˜Lu i/C27¯ujlwui/C300 (6) ui˜L¯uj/C27ui¯ljw¯uj/C300 (7) ¯ui˜Lui/C28ui˜L¯uj/C30(¯lj/C28li)wui¯uj: (8) Now integrategb a¯uj˜Lui/C28gb aui˜L¯uj/C30(¯lj/C28li)gb awui¯uj: (9) But because ¯Lis Hermitian, the left side vanishes. (¯lj/C28li)gb awui¯uj/C300: (10) IfEIGENVALUES liandljare not degenerate, then fb awui¯uj/C300;so the EIGENFUNCTIONS are ORTHOGO- NAL. If the EIGENVALUES are degenerate, the EIGEN- FUNCTIONS are not necessarily orthogonal. Now take i /C30j. (¯li/C28li)gb awui¯ui/C300: (11) The integral cannot vanish unless ui/C300;so we have ¯li/C30liand the EIGENVALUES are real. For a Hermitian operator ˜O; /C142f½˜Oc/C143/C30/C142f½˜Oc/C143/C30/C142˜Of½c/C143: (12) In integral notation, g˜Afcdx/C30g¯f˜Acdx: (13) Given Hermitian operators ˜Aand ˜B; /C142f½˜A˜Bc/C143/C30/C142˜Af½˜Bc/C143/C30/C142˜B˜Af½c/C143/C30/C142f½˜B˜Ac/C143:(14) Because, for a Hermitian operator ˜Awith EIGENVA- LUEa, /C142c½˜Ac/C143/C30/C142˜Ac½c/C143 (15) a/C142c½c/C143/C30¯a/C142c½c/C143: (16) Therefore, either /C142c½c/C143/C300o ra/C30¯a:But/C142c½c/C143/C300IFF c/C300;so /C142c½c/C143"0; (17) for a nontrivial EIGENFUNCTION . This means that a/C30 a/C31;namely that Hermitian operators produce REAL expectation values. Every observable must therefore have a corresponding Hermitian operator. Further-more, /C142c n½˜Acm/C143/C30/C142˜Acn½cm/C143 (18) am/C142cn½cm/C143/C30¯an/C142cn½cm/C143/C30an/C142cn½cm/C143; (19) since an/C30¯an:Then (am/C28an)/C142cn½cm/C143/C300 (20) Foram"an(i.e., cn"cm); /C142cn½cm/C143/C300: (21) Foram/C30an(i.e., cn/C30cm); /C142cn½cm/C143/C30/C142cn½cn/C143/C131: (22) Therefore, /C142cn ½cm /C143/C30 dnm ; (23) so the basis of EIGENFUNCTIONS corresponding to a Hermitian operator are ORTHONORMAL . Define the Hermitian conjugate operator ˜A/C31 by /C142 ˜Ac½c /C143/C13/C142c½ ˜A/C31 c/C143: (24) For a Hermitian operator, ˜A /C30 ˜A/C31: Furthermore, given two Hermitian operators ˜A and ˜B ; /C142c2 ½( ˜A ˜B)/C31 c1 /C143/C30/C142( ˜A ˜B)c2 ½ c1 /C143/C30/C142 ˜Bc2 ½ ˜A/C31c1 /C143 /C30/C142c2 ½ ˜B /C31 ˜A/C31c1 /C143; (25) so ( ˜A ˜B)/C31/C30 ˜B /C31 ˜A/C31: (26) By further iterations, this can be generalized to ( ˜A ˜B /C1/C1/C1 ˜Z) /C31/C30 ˜Z/C31/C1/C1/C1 ˜B /C31 ˜A/C31: (27) Given two Hermitian operators ˜A and ˜B ; ( ˜A ˜B)/C31/C30 ˜B /C31 ˜A/C31/C30 ˜B ˜A /C30 ˜A ˜B /C27[ ˜B; ˜A] ; (28) the operator ˜A ˜B equals ( ˜A ˜B) /C31; and is therefore Hermitian, only if [ ˜B; ˜A] /C300 : (29) Given an arbitrary operator ˜A; /C142c1 ½( ˜A /C27 ˜A/C31) c2 /C143/C30/C142( ˜A /C31/C27 ˜A)c1 ½ c2 /C143 /C30/C142( ˜A /C27 ˜A/C31) c1 ½ c2 /C143; (30) so ˜A /C27 ˜A/C31 is Hermitian. /C142c1½i(˜A/C28˜A/C31)c2/C143/C30/C142/C28i(˜A/C31/C28 ˜A)c1½c2/C143 /C30/C142i(˜A/C28˜A/C31)c1½c2/C143; (31) so is Hermitian. Similarly, /C142c1½(˜A˜A/C31)c2/C143/C30/C142˜A/C31c1½˜A/C31c2/C143/C30/C142(˜A˜A/C31)c1½c2/C143;(32) so˜A˜A/C31is Hermitian. See also ADJOINT ,HERMITIAN MATRIX ,SELF-ADJOINT , STURM- LIOUVILLE THEORY References Arfken, G. "Hermitian (Self-Adjoint) Operators." §9.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 504 /C1/06 and 510 /C1/16, 1985. Heron Triangle HERONIAN TRIANGLE Heron’s Formula Gives the AREA of a TRIANGLE in terms of the lengths of the sides a,b, and cand the SEMIPERIMETER s/C301 2(a/C27b/C27c): (1)Heron’s formula then states D/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s/C28a)(s/C28b)(s/C28c)p : (2) Heron’s formula may be stated beautifully using a CAYLEY- MENGER DETERMINANT as /C2816D2/C300abc a0cb bc 0a cba 0P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C3001 1 1 10 c 2b2 1c20a2 1b2a20P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2: (3) Expressing the side lengths a,b, and cin terms of the radii a?;b?;and c’ of the mutually tangent circles centered on the TRIANGLE vertices (which define the SODDY CIRCLES ), a/C30b?/C27c? (4) b/C30a?/C27c? (5) c/C30a?/C27b?; (6) gives the particularly pretty form D/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a?b?c?(a?/C27b?/C27c?)p : (7) Heron’s proof (Dunham 1990) is ingenious but ex- tremely convoluted, bringing together a sequence of apparently unrelated geometric identities and relying on the properties of CYCLIC QUADRILATERALS and RIGHT TRIANGLES . Heron’s proof can be found in Proposition 1.8 of his work Metrica (ca. 100 BC-100 AD). This manuscript had been lost for centuriesuntil a fragment was discovered in 1894 and acomplete copy in 1896 (Dunham 1990, p. 118). More recently, writings of the Arab scholar Abu’l Raihan Muhammed al-Biruni have credited the formula toHeron’s predecessor Archimedes prior to 212 BC (van der Waerden 1961, pp. 228 and 277; Coxeter and Greitzer 1967, p. 59; Kline 1972; Bell 1986, p. 58;Dunham 1990, p. 127). A much more accessible algebraic proof proceeds from the LAW OF COSINES , cos A /C30b2 /C27 c2 /C28 a2 2bc: (8) Then sin A /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28a4 /C28 b4 /C28 c4 /C27 2b2c2 /C27 2c2a2 /C27 2a2b2p 2bc ; (9) giving D/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s /C28a)(s /C28b)(s /C28c)p (10) /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2ab)2 /C28(a2 /C27b2 /C28c2)2q (11) /C301 2 bc sin A (12) /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a /C27b /C27c)(/C28a /C27b /C27c)(a /C28b /C27c)(a /C27b /C28c)p /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(b2c2 /C27c2a2 /C27a2b2) /C28(a4 /C27b4 /C27c4)p (13) (Coxeter 1969). Heron’s formula contains the PYTHA- GOREAN THEOREM as a degenerate case. See also BRAHMAGUPTA’S FORMULA ,BRETSCHNEIDER’S FORMULA ,C AYLEY- MENGER DETERMINANT ,H ERO- NIAN TETRAHEDRON ,H ERONIAN TRIANGLE ,S ODDY CIRCLES , SSS THEOREM ,TRIANGLE References Bell, E. T. Men of Mathematics. New York: Simon and Schuster, p. 58, 1986. Brown, K. S. "Heron’s FOrmula and Brahmagupta’s Gen- eralization." http://www.seanet.com/~ksbrown/ kmath196.htm. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 12, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 59, 1967. Dunham, W. "Heron’s Formula for Triangular Area." Ch. 5 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 113 /C1/32, 1990. Kline, M. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Pappas, T. "Heron’s Theorem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 62, 1989. van der Waerden, B. L. Science Awakening. Oxford, Eng- land: Oxford University Press, pp. 228 and 277, 1961. Heronian Mean The Heronian mean of two numbers m and n is defined as HM(a; b) /C301 3(a /C27ffiffiffiffiffiffi abp /C27b) ; which arises in the determination of the volume of a PYRAMIDAL FRUSTUM . See also PYRAMIDAL FRUSTUMReferences Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, p. 7, 1965. Heronian Tetrahedron A TETRAHEDRON with RATIONAL sides, FACE AREAS , and VOLUME . The smallest examples have pairs of opposite sides (148, 195, 203), (533, 875, 888), (1183, 1479, 1804), (2175, 2296, 2431), (1825, 2748, 2873), (2180, 2639, 3111), (1887, 5215, 5512), (6409, 6625, 8484), and (8619, 10136, 11275). See also HERON’S FORMULA ,HERONIAN TRIANGLE References Guy, R. K. "Simplexes with Rational Contents." §D22 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 190 /C1/92, 1994. Heronian Triangle ATRIANGLE with RATIONAL side lengths and RATIONAL AREA . Brahmagupta gave a parametric solution for integer Heronian triangles (the three side lengths and area can be multiplied by their LEAST COMMON MULTIPLE to make them all INTEGERS ): side lengths c(a2/C27b2);b(a2/C27c2);and ( b/C27c)(a2/C28bc);giving SEMI- PERIMETER s/C30a2(b/C27c) (1) and AREA D/C30abc(a/C27b)(a2/C28bc): (2) The first few integer Heronian triangles sorted by increasing maximal side lengths, are ((3, 4, 5), (5, 5, 6), (5, 5, 8), (6, 8, 10), (10, 10, 12), (5, 12, 13), (10, 13, 13), (9, 12, 15), (4, 13, 15), (13, 14, 15), (10, 10, 16), ...(Sloane’s A055594, A055593, and A055592), having areas 6, 12, 12, 24, 48, 30, 60, 54, ... (Sloane’s A055595). The first few integer Heronian SCALENE TRIANGLES , sorted by increasing maximal side lengths, are (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12,15), (4, 13, 15), (13, 14, 15), (9, 10, 17), ... (Sloane’sA046128, A046129, and A046130), having areas 6, 24,30, 54, 24, 84, 36, ... (Sloane’s A046131). Schubert (1905) claimed that Heronian triangles with two rational MEDIANS do not exist (Dickson 1952). This was shown to be incorrect by Buchholz and Rathbun (1997), who discovered the triangles given in the following table, where miare MEDIAN lengths and Ais the area. ab c /m1// m2/ A 73 51 26 /35 2//97 2/ 420 626 875 291 572 /433 2/ 55440 4368 1241 3673 1657 /7975 2/ 2042040 14791 14384 11257 /21177 2/ 11001 75698280 28779 13816 15155 /3589 2/ 21937 23931600 1823675 185629 1930456 /2048523 2//3751059 2/ 142334216640 See also HERON’S FORMULA ,M EDIAN (TRIANGLE ), PYTHAGOREAN TRIPLE ,TRIANGLE References Buchholz, R. H. On Triangles with Rational Altitudes, Angle Bisectors or Medians. Doctoral Dissertation. Newcastle, England: Newcastle University, 1989. Buchholz, R. H. and Rathbun, R. L. "An Infinite Set of Heron Triangles with Two Rational Medians." Amer. Math. Monthly 104, 107 /C1/15, 1997. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 199 and 208, 1952. Fleenor, C. R. "Heronian Triangles with Consecutive Integer Sides." J. Recr. Math. 28, 113 /C1/15, 1996 /C1/6. Guy, R. K. "Simplexes with Rational Contents." §D22 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 190 /C1/92, 1994. Kraitchik, M. "Heronian Triangles." §4.13 in Mathematical Recreations. New York: W. W. Norton, pp. 104 /C1/08, 1942. Rabinowitz, S. "Problem 2006: Heronian Properties." J. Recr. Math. 24, 309, 1992. Schubert, H. "Die Ganzzahligkeit in der algebraischen Geometrie." In Festgabe 48 Versammlung d. Philologen und Schulma ¨nner zu Hamburg. Leipzig, Germany, pp. 1 /C1/ 6, 1905. Sloane, N. J. A. Sequences A046128, A046129, A046130, A046131, A055592, A055593, A055594, and A055595 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Wells, D. G. The Penguin Dictionary of Curious and Inter- esting Puzzles. London: Penguin Books, p. 34, 1992. Yiu, P. "Construction of Indecomposable Heronian Trian- gles." Rocky Mountain J. Math. 28, 1189 /C1/202, 1998. Herschel A HEPTOMINO shaped like the astronomical symbol for Uranus (which was discovered by William Herschel ). See also HEPTOMINO Herschfeld’s Convergence Theorem For real, NONNEGATIVE terms xn and REAL p with 0 B p B1 ; the expression lim k0/C12x0 /C27(x1 /C27(x2 /C27(. .. /C27(xk)p)p)p)p converges IFF (xn)pn is bounded.See also NESTED RADICAL References Herschfeld, A. "On Infinite Radicals." Amer. Math. Monthly 42, 419 /C1/29, 1935. Jones, D. J. "Continued Powers and a Sufficient Condition for Their Convergence." Math. Mag. 68, 387 /C1/92, 1995. Hesse’s Theorem If two pairs of opposite VERTICES of a COMPLETE QUADRILATERAL are pairs of CONJUGATE POINTS , then the third pair of opposite VERTICES is likewise a pair of CONJUGATE POINTS . See also COMPLETE QUADRILATERAL Hessenberg Matrix A matrix OF THE FORM a11a12a13 /C1/C1/C1 a1(n /C281) a1n a21a22a23 /C1/C1/C1 a2(n /C281) a2n 0 a32a33 /C1/C1/C1 a3(n /C281) a3n 00 a43 /C1/C1/C1 a4(n /C281) a4n 000 /C1/C1/C1 a5(n /C281) a5n nnn::: nn 0000 a(n /C281)(n/C281)a(n /C281)n 0000 an(n/C281) ann2 666666666643 77777777775: See also T OEPLITZ MATRIX ,TRIANGULAR MATRIX References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Reduction of a General Matrix to Hessenberg Form." §11.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 476 /C1/80, 1992. Hessian Covariant H /C13½aa ?a ƒ½axn/C282 a ?xn /C282 a ƒxn /C282 /C300: The nonsingular inflections of a curve are its non- singular intersections with the Hessian. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 79, 95 /C1/8, and 151 /C1/61, 1959. Hessian Determinant The DETERMINANT Hf(x; y) /C30@2f @x2@2f @x@y @2f @y@x@2f @y2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2 appearing in the SECOND DERIVATIVE TEST as D/C13Hf(x;y):/ See also SECOND DERIVATIVE TEST References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1112 /C1/113, 2000. Heteroclinic Point If intersecting stable and unstable MANIFOLDS (SE- PARATRICES ) emanate from FIXED POINTS of different families, they are called heteroclinic points. See also HOMOCLINIC POINT ,MANIFOLD ,SEPARATRIX Heterogeneous Numbers Two numbers are heterogeneous if their PRIME FACTORS are distinct. For example, 6 /C302 /C2153 and 24 /C30 23 /C2153 are not heterogeneous since their factors are each (2, 3). See also DISTINCT PRIME FACTORS ,H OMOGENEOUS NUMBERS References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 146, 1983. Heterological Paradox GRELLING’S PARADOX Heteromecic Number PRONIC NUMBER Heteroscedastic A set of STATISTICAL DISTRIBUTIONS having different VARIANCES . See also HOMOSCEDASTIC ,VARIANCE Heterosquare A heterosquare is an n /C29n ARRAY of the integers from 1to n2 such that the rows, columns, and diagonals have different sums. (By contrast, in a MAGIC SQUARE , they have the same sum.) There are no heterosquares of order two, but heterosquares of every ODD order exist. They can be constructed by placing consecutive INTEGERS in a SPIRAL pattern (Fults 1974, Madachy 1979). An ANTIMAGIC SQUARE is a special case of a hetero- square for which the sums of rows, columns, andmain diagonals form a SEQUENCE of consecutive integers. See also ANTIMAGIC SQUARE ,M AGIC SQUARE ,TALIS- MAN SQUARE References Duncan, D. "Problem 86." Math. Mag. 24, 166, 1951. Fults, J. L. Magic Squares. Chicago, IL: Open Court, 1974. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 101 /C1/03, 1979. Rivera, C. "Problems & Puzzles: Puzzle Primeful Hetero- squares.-069." http://www.primepuzzles.net/puzzles/ puzz_069.htm. Weisstein, E. W. "Magic Squares." MATHEMATICA NOTEBOOK MAGICSQUARES.M . Heuman Lambda Function L0(f ½m) /C13F( f½1 /C28 m) K(1 /C28 m)/C272 pK(m)Z( f½1 /C28m) ; where f is the AMPLITUDE , m is the PARAMETER , Z is the JACOBI ZETA FUNCTION , and F(f ½m?) and K(m) are incomplete and complete ELLIPTIC INTEGRALS OF THE FIRST KIND . See also ELLIPTIC INTEGRAL OF THE FIRST KIND, JACOBI ZETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 595, 1972. To¨lke, F. "Jacobische Zeta- und Heumansche Lambda- Funktionen." §132 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 94 /C1/9, 1967. Heun’s Differential Equation A natural extension of the RIEMANN P-DIFFERENTIAL EQUATION given by d2w dx2 /C27g x/C27d x/C281/C27o x/C28a ! dw dx/C27abx/C28q x(x/C281)(x/C28a)w /C300 where a/C27b/C28g/C28d/C28o/C271/C300: See also RIEMANN P-DIFFERENTIAL EQUATION References Decarreau, A.; Dumont-Lepage, M.-C.; Maroni, P.; Robert, A.; and Ronveaux, A. "Formes canoniques des e ´quations confluentes de l’e ´quation de Heun." Ann. Soc. Sci. de Bruxelles 92,5 3/C1/8, 1978. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, pp. 57 /C1/2, 1981. Heun, K. "Zur Theorie der Riemann’schen Functionen Zweiter Ordnung mit Verzweigungspunkten." Math. Ann. 33, 161 /C1/79. Ronveaux, A. (Ed.). Heun’s Differential Equations. Oxford, England: Oxford University Press, 1995. Valent, G. "An Integral Transform Involving Heun Func- tions and a Related Eigenvalue Problem." SIAM J. Math. Anal. 17, 688 /C1/03, 1986. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 576, 1990. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997. Heuristic (1) Based on or involving trial and error. (2) Convin- cing without being rigorous. See also PARADOX ,PROOF Hex (Polyhex) POLYHEX Hex Game A two-player GAME . There is a winning strategy for the first player if there is an even number of cells on each side; otherwise, there is a winning strategy for the second player. References Gardner, M. "The Game of Hex." Ch. 8 in The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 73 /C1/3, 1959. Hex Number The CENTERED HEXAGONAL NUMBER given by Hn /C301 /C276Tn /C302Hn/C281 /C28Hn /C282 /C276 /C303n2 /C283n /C271; where Tnis the nth TRIANGULAR NUMBER . The first few hex numbers are 1, 7, 19, 37, 61, 91, 127, 169, ... (Sloane’s A003215). The GENERATING FUNCTION of the hex numbers is x(x2 /C27 4x /C27 1) (1 /C28 x)3/C30x /C277x2 /C2719x3 /C2737x4 /C27...: The first TRIANGULAR hex numbers are 1 and 91, and the first few SQUARE ones are 1, 169, 32761, 6355441,... (Sloane’s A006051). SQUARE hex numbers are obtained by solving the DIOPHANTINE EQUATION 3x2 /C271 /C30y2 : The only hex number which is SQUARE and TRIANGU- LAR is 1. There are no CUBIC hex numbers. See also MAGIC HEXAGON ,CENTERED PENTAGONAL NUMBER ,CENTERED SQUARE NUMBER ,STAR NUMBER , TALISMAN HEXAGON References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 41, 1996. Gardner, M. "Hexes and Stars." Ch. 2 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 15 /C1/5, 1988. Hindin, H. "Stars, Hexes, Triangular Numbers, and Pytha- gorean Triples." J. Recr. Math. 16, 191 /C1/93, 1983 /C1/984. Sloane, N. J. A. Sequences A003215/M4362 and A006051/ M5409 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Hex Pyramidal Number A FIGURATE NUMBER which is equal to the CUBIC NUMBER n3 : The first few are 1, 8, 27, 64, ... (Sloane’s A000578). References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 42 /C1/4, 1996. Sloane, N. J. A. Sequences A000578/M4499 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hexa POLYHEX Hexabolo A6- POLYABOLO . Hexacontagon A 60-sided POLYGON . Hexacronic Icositetrahedron GREAT HEXACRONIC ICOSITETRAHEDRON ,SMALL HEX- ACRONIC ICOSITETRAHEDRON Hexad ASETof six. See also MONAD ,QUARTET ,QUINTET ,TETRAD ,TRIAD Hexadecagon A 16-sided POLYGON , sometimes also called a HEX- AKAIDECAGON . The regular hexadecagon is a CON- STRUCTIBLE POLYGON , and the INRADIUS r, CIRCUMRADIUS R, and area A of the regular hexade- cagon of side length 1 are r /C301 2(1 /C27ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(2 /C27ffiffiffi 2p )q ) R /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2(4 /C272ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20 /C2714ffiffiffi 2pq )r A /C304(1 /C27ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(2 /C27ffiffiffi 2p )q ): See also POLYGON ,REGULAR POLYGON ,TRIGONOME- TRY VALUES PI/16 Hexadecimal The base 16 notational system for representing REAL NUMBERS . The digits used to represent numbers using hexadecimal NOTATION are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The following table gives the hexadecimal equivalents of the first few decimal numbers. 1 1 11 B 21 15 2 2 12 C 22 16 3 3 13 D 23 17 4 4 14 E 24 18 5 5 15 F 25 19 6 6 16 10 26 1A 7 7 17 11 27 1B 8 8 18 12 28 1C 9 9 19 13 29 1D 10A2014301E The hexadecimal system is particularly important in computer programming, since four bits (each consist- ing of a one or zero) can be succinctly expressed usinga single hexadecimal digit. Two hexadecimal digits represent numbers from 0 to 255, a common range used, for example, to specify colors. Thus, in theHTML language of the web, colors are specified using three pairs of hexadecimal digitsRRGGBB , where RR is the amount of red, GG the amount of green, and BB the amount of blue. In HEXADECIMAL , numbers with increasing digits are called METADROMES , those with nondecreasing digits are called PLAINDRONES , those with nonincreasing digits are called NIALPDROMES , and those with de- creasing digits are called KATADROMES . See also BASE (NUMBER ), BINARY ,DECIMAL ,DIGIT, KATADROME ,M ETADROME ,N IALPDROME ,O CTAL , PLAINDROME ,QUATERNARY ,TERNARY ,VIGESIMAL References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 105, 1984. Weisstein, E. W. "Bases." MATHEMATICA NOTEBOOK BASES.M . Hexaflexagon A FLEXAGON made by folding a strip into adjacent EQUILATERAL TRIANGLES . The number of states pos- sible in a hexaflexagon is the CATALAN NUMBER C4/C3042:/ See also FLEXAGON ,FLEXATUBE ,TETRAFLEXAGON References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 205 /C1/07, 1989. Gardner, M. "Hexaflexagons." Ch. 1 in The Scientific Amer- ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 1 /C1/4, 1959. Gardner, M. "Tetraflexagons." Ch. 2 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 24 /C1/1, 1961. Maunsell, F. G. "The Flexagon and the Hexaflexagon." Math. Gazette 38, 213/C1/14, 1954. Wheeler, R. F. "The Flexagon Family." Math. Gaz. 42,1/C1/, 1958. Hexafrob POLYHEX Hexagon A six-sided POLYGON . In proposition IV.15, Euclid showed how to inscribe a regular hexagon in a CIRCLE . The INRADIUS r,CIRCUMRADIUS R, and AREA A can be computed directly from the formulas for a general REGULAR POLYGON with side length s and n /C306 sides, r /C301 2 s cotp 6 ! /C3012ffiffiffi 3p s (1) R /C301 2 s cscp 6 ! /C30s (2) A /C301 4 ns2 cotp 6 ! /C3032ffiffiffi 3p s2 : (3) Therefore, for a regular hexagon, R r/C30secp 6 ! /C302ffiffiffi 3p; (4) so AR Ar/C30R r !2 /C304 3 : (5) A PLANE PERPENDICULAR to a C3axis of a CUBE (Gardner 1960), DODECAHEDRON ,or ICOSAHEDRONcuts the solid in a regular HEXAGONAL CROSS SECTION (Holden 1991, pp. 22 /C1/3 and 27). For the CUBE , the PLANE passes through the MIDPOINTS of opposite sides (Steinhaus 1983, p. 170; Cundy and Rollett 1989, p. 157; Holden 1991, pp. 22 /C1/3). Since there are four such axes for the CUBE and OCTAHEDRON , there are four possible HEXAGONAL CROSS SECTIONS .A HEXA- GON is also obtained when the cube is viewed from above a corner along the extension of a space diagonal (Steinhaus 1983, p. 170). Take seven CIRCLES and close-pack them together in a hexagonal arrangement. The PERIMETER obtained by wrapping a band around the CIRCLE then consists of six straight segments of length d (where d is the DIAMETER ) and 6 arcs with total length 1 =6ofa CIRCLE . The PERIMETER is therefore p /C30(12 /C272 p)r /C302(6 /C27 p)r : (6) Given an arbitrary hexagon, take each three con- secutive vertices, and mark the fourth point of the PARALLELOGRAM sharing these three vertices. Taking alternate points then gives two congruent triangles, as illustrated above (Wells 1991). Given an arbitrary hexagon, connecting the centroids of each consecutive three sides gives a hexagon with equal and parallel sides known as the CENTROID HEXAGON (Wells 1991). See also CENTROID HEXAGON ,C OSINE HEXAGON , CUBE,C YCLIC HEXAGON ,D ISSECTION ,D ODECAHE- DRON ,GRAHAM’S BIGGEST LITTLE HEXAGON ,HEPTA- GON THEOREM ,H EXAGON POLYIAMOND ,H EXAGRAM , LEMOINE HEXAGON ,M AGIC HEXAGON ,OCTAHEDRON , PAPPUS’S HEXAGON THEOREM ,P ASCAL’S THEOREM , TALISMAN HEXAGON ,TUCKER HEXAGON References Cadwell, J. H. Topics in Recreational Mathematics. Cam- bridge, England: Cambridge University Press, 1966. Coxeter, H. S. M. and Greitzer, S. L. "Hexagons." §3.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 73 /C1/4, 1967. Cundy, H. and Rollett, A. "Hexagonal Section of a Cube." §3.15.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 157, 1989. Dixon, R. Mathographics. New York: Dover, p. 16, 1991. Gardner, M. "Mathematical Games: More About the Shapes that Can Be Made with Complex Dominoes." Sci. Amer. 203, 186 /C1/98, Nov. 1960. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Pappas, T. "Hexagons in Nature." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 74 /C1/5, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 53 /C1/4, 1991. Hexagon Polyiamond A6- POLYIAMOND . See also HEXAGON References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Hexagon Tiling There are at least three aperiodic tilings of HEXA- GONS , given by the following types: A /C27B /C27C /C30360/C14 a /C30d A /C27B /C27D /C30360/C14 a /C30d; c /C30e A /C30C /C30Ea /C30b ; c /C30d; e /C30f(1) (Gardner 1988). Note that the periodic hexagonal TESSELLATION is a degenerate case of all three tilings with A /C30B /C30C /C30D /C30E /C30Fa/C30b /C30c /C30d /C30e /C30f (2) Amazingly, the number of PLANE PARTITIONS PL(a; b; c) contained in an a /C29b /C29c box also gives the number of hexagon tilings by RHOMBI for a hexagon of side lengths a, b, c, a, b, c (David and Tomei 1989, Fulmek and Krattenthaler 2000). The asymptotic distribution of rhombi in a random hexa-gon tiling by rhombi was given by Cohn et al. (1998). A variety of enumerations for various explicit posi- tions of rhombi are given by Fulmek and Krattentha- ler (1998, 2000). See also PLANE PARTITION ,TILING References Cohn, H.; Larsen, M.; and Propp, J. "The Shape of a Typical Boxed Plane Partition." New York J. Math. 4, 137 /C1/66, 1998. David, G. and Tomei, C. "The Problem of the Calissons." Amer. Math. Monthly 96, 429 /C1/31, 1989. Gardner, M. "Tilings with Convex Polygons." Ch. 13 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 162 /C1/76, 1988. Fulmek, M. and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry Axis, I." Ann. Combin. 2,19/C1/ 0, 1998. Fulmek, M. and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry Axes, II." Europ. J. Combin. 21, 601 /C1/40, 2000. Hexagon Triangle Picking The mean area of a TRIANGLE picked inside a regular HEXAGON with unit area is ¯A /C30289=3888 (Woolhouse 1867, Pfiefer 1989). This is a special case of a general POLYGON TRIANGLE PICKING result due to Alikoski (1939). See also DISK TRIANGLE PICKING ,POLYGON TRIANGLE PICKING ,SQUARE TRIANGLE PICKING ,SYLVESTER’S FOUR- POINT PROBLEM ,TRIANGLE TRIANGLE PICKING References Alikoski, H. A. "U ¨ber das Sylvestersche Vierpunktproblem." Ann. Acad. Sci. Fenn. 51, No. 7, 1 /C1/0, 1939. Pfiefer, R. E. "The Historical Development of J. J. Sylves- ter’s Four Point Problem." Math. Mag. 62, 309/C1/17, 1989. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, p. 114, 1978. Woolhouse, W. S. B. "Question 2471" Mathematical Ques- tions, with Their Solutions, from the Educational Times, Vol. 8. London: F. Hodgson and Son, pp. 100 /C1/05, 1867. Hexagonal Close Packing SPHERE PACKING Hexagonal Number AFIGURATE NUMBER and 6- POLYGONAL NUMBER OF THE FORM n(2n/C281):The first few are 1, 6, 15, 28, 45, ... (Sloane’s A000384). The GENERATING FUNCTION of the hexagonal numbers x(3x /C27 1) (1 /C28 x)3 /C30x /C276x2 /C2715x3 /C2728x4 /C27...: Every hexagonal number is a TRIANGULAR NUMBER since r(2r /C281) /C301 2(2r /C281)[(2r /C281) /C271]: In 1830, Legendre (1979) proved that every number larger than 1791 is a sum of four hexagonal numbers, and Duke and Schulze-Pillot (1990) improved this to three hexagonal numbers for every sufficiently large integer. The numbers 11 and 26 can only be REPRE- SENTED AS a sum using the maximum possible of six hexagonal numbers: 11 /C301 /C271 /C271 /C271 /C271 /C276 26 /C301 /C271 /C276 /C276 /C276 /C276: See also FIGURATE NUMBER ,H EX NUMBER ,H EPTA- GONAL HEXAGONAL NUMBER ,HEXAGONAL PENTAGO- NAL NUMBER ,O CTAGONAL HEXAGONAL NUMBER , TRIANGULAR NUMBER References Duke, W. and Schulze-Pillot, R. "Representations of Integers by Positive Ternary Quadratic Forms and Equidistribu- tion of Lattice Points on Ellipsoids." Invent. Math. 99,49/C1/ 7, 1990. Guy, R. K. "Sums of Squares." §C20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 /C1/38, 1994. Legendre, A.-M. The´orie des nombres, 4th ed., 2 vols. Paris: A. Blanchard, 1979. Sloane, N. J. A. Sequences A000384/M4108 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hexagonal Pentagonal Number A number which is simultaneously PENTAGONAL and HEXAGONAL . Let Pndenote the nth PENTAGONAL NUMBER and Hmthe mth SQUARE NUMBER , then a number which is both pentagonal and hexagonal satisfies the equation Pn /C30Hm ; or 1 2 n(3n /C281) /C30m(2m /C281): (1) COMPLETING THE SQUARE and rearranging gives (6n /C281)2 /C283(4m /C281)2 /C30/C282: (2) Therefore, defining x /C132n /C271 (3) y /C132m (4) gives the Pell-like equationx2 /C283y2 /C30/C282 (5) The first few solutions are (x; y) /C30(1; 1); (5, 3), (19, 11), (71, 74), (265, 153), (989, 571), .... These give the solutions (n; m); (1, 1), (/10=3/, 3), (12, /21 =2/), (/133=3/, / 77 =2/), (165, 143), ..., of which the integer solutions are (1, 1), (165, 143), (31977, 27693), (6203341, 5372251), ... (Sloane’s A046178 and A046179), corresponding to the pentagonal hexagonal numbers 1, 40755, 1533776805, 57722156241751, ... (Sloane’s A046180). See also HEXAGONAL NUMBER ,PENTAGONAL NUMBER References Sloane, N. J. A. Sequences A046178, A046179, and A046180 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Hexagonal Prism A PRISM composed of hexagonal faces. The regular right hexagonal prism has SURFACE AREA and VO- LUME S /C303(2 /C27ffiffiffi 3p ) V /C303 2ffiffiffi 3p : See also HEXAGON ,PRISM Hexagonal Pyramid A PYRAMID with a hexagonal base. The SLANT HEIGHT of a hexagonal pyramid is a special case of the formula for a regular n-gonal PYRAMID with n /C306, given by s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C27a2p ; (1) where h is the height and a is the length of a side of the base. See also HEXAGON ,PYRAMID Hexagonal Pyramidal Number A PYRAMIDAL NUMBER OF THE FORM n(n /C271)(4n /C28 1)=6; The first few are 1, 7, 22, 50, 95, ... (Sloane’s A002412). The GENERATING FUNCTION of the hexago- nal pyramidal numbers is x(3x /C27 1) (x /C28 1)4 /C30x /C277x2 /C2722x3 /C2750x4 /C27...: References Sloane, N. J. A. Sequences A002412/M4374 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hexagonal Scalenohedron An irregular DODECAHEDRON which is also a TRAPE- ZOHEDRON . See also DODECAHEDRON ,TRAPEZOHEDRON References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 63, 1990. Hexagonal Square Number Let Hndenote the nth HEXAGONAL NUMBER and Sm the mth SQUARE NUMBER , then a number which is both hexagonal and square satisfies the equation Hn /C30Sm ; or n(2n /C281) /C30m2 : (1) COMPLETING THE SQUARE and rearranging gives (4n /C281)2 /C288m2 /C301: (2) Therefore, defining x /C134n /C281 (3) y /C132m (4) gives the PELL EQUATION x2 /C282y2 /C301: (5) The first few solutions are (x; y) /C30(3; 2); (17, 12), (99, 70), (577, 408), .... These give the solutions (n; m) /C30 (1; 1); (/9=2/, 6), (25, 35), (/289=2/, 204), ..., giving the integer solutions (1, 1), (25, 35), (841, 1189), (28561,40391), ... (Sloane’s A008844 and A046176). The corresponding hexagonal square numbers are 1, 1225, 1413721, 1631432881, 1882672131025, ... (Sloa- ne’s A046177). See also HEXAGONAL NUMBER ,SQUARE NUMBER References Sloane, N. J. A. Sequences A008844, A046176, and A046177 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Hexagram The STAR POLYGON f6 =2g; also known as the STAR OF DAVID. See also DISSECTION ,PENTAGRAM ,SOLOMON’S SEAL KNOT,STAR FIGURE ,STAR OF LAKSHMI Hexagrammum Mysticum Theorem PASCAL’S THEOREM Hexahedral Graph A POLYHEDRAL GRAPH on six vertices. There are seven topologically distinct hexahedral graphs (Gardner 1966, p. 233), of which three are the PENTAGONAL PYRAMID (first figure), TRIANGULAR PRISM (second figure), and OCTAHEDRON /square dipyramid/ TRIANGU- LAR ANTIPRISM (last figure). The hexahedral graphs were first enumerated by Steiner (1828; Duijvestijn and Federico 1981). See also HEXAHEDRON ,POLYHEDRAL GRAPH References Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral ( /3/-Connected Planar) Graphs." Math. Comput. 37, 523/C1/32, 1981. Gardner, M. Martin Gardner’s New Mathematical Diver- sions from Scientific American. New York: Simon and Schuster, 1966. Steiner, J. "Proble `me de situation." Ann. de Math 19, 36, 1828. Reprinted in Jacob Steiner’s gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 227, 1971. Hexahedron A hexahedron is a POLYHEDRON with six faces. The regular hexahedron is the CUBE , although there are seven topologically different CONVEX hexahedra (Guy 1994, p. 189). Steiner (1828) was the first to enumer- ate the hexahedra (Duijvestijn and Federico 1981). There are exactly two hexahedra composed of iden- tical REGULAR POLYGONS : the regular TRIANGULAR DIPYRAMID (six EQUILATERAL TRIANGLES ; left figure) and the CUBE (six SQUARES ; right figure). See also CUBE,H EXAHEDRAL GRAPH ,POLYHEDRON , TRIANGULAR DIPYRAMID References Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (3-Connected Planar) Graphs." Math. Comput. 37, 523 /C1/32, 1981. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Steiner, J. "Proble `me de situation." Ann. de Math. 19, 36, 1828. Reprinted in Jacob Steiner’s gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 227, 1971. Hexahemioctacron The DUAL POLYHEDRON of the CUBOHEMIOCTAHEDRON U15and Wenninger dual W78 : When rendered, the OCTAHEMIOCTACRON and hexahemioctacron appear the same. See also DUAL POLYHEDRON ,CUBOHEMIOCTAHEDRON , OCTAHEMIOCTACRON ,UNIFORM POLYHEDRONReferences Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 104, 1983. Hexahemioctahedron (6n /C281)2 /C283(4m /C281)2 /C30/C282: The DUAL POLYHEDRON of the CUBOHEMIOCTAHEDRON 2n /C271: When rendered, the OCTAHEMIOCTACRON and hexahemioctahedron appear the same. See also DUAL POLYHEDRON ,CUBOHEMIOCTAHEDRON , OCTAHEMIOCTACRON ,UNIFORM POLYHEDRON Hexakaidecagon HEXADECAGON Hexakis Icosahedron DISDYAKIS TRIACONTAHEDRON Hexakis Octahedron DISDYAKIS DODECAHEDRON Hexecontahedron A 60-faced POLYHEDRON . Taking the RHOMBIC TRIA- CONTAHEDRON , placing a plane along each edge which is perpendicular to the plane of symmetry in which the edge lies, and taking the solid bounded by these planes gives a hexecontahedron (Steinhaus 1999). See also DELTOIDAL HEXECONTAHEDRON ,PENTAGO- NAL HEXECONTAHEDRON ,PENTAKIS DODECAHEDRON , SMALL RHOMBICOSIDODECAHEDRON ,SNUB DODECA- HEDRON ,TRIAKIS ICOSAHEDRON ,TRUNCATED DODE- CAHEDRON ,TRUNCATED ICOSAHEDRON References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 210, 1999. Hexiamond APOLYIAMOND composed of six equilateral triangles. The 12 hexiamonds are illustrated above. They are given the names BAR, CROOK , CROWN , SPHINX , SNAKE , YACHT , CHEVRON , SIGNPOST , LOBSTER , HOOK , HEXA- GON, and BUTTERFLY . See also POLYIAMOND ,HEXIAMOND TILING References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 174 /C1/75, 1984. O’Beirne, T. H. "Pentominoes and Hexiamonds." New Scien- tist 12, 379 /C1/80, 1961. O’Beirne, T. H. "Some Hexiamond Solutions and an Intro- duction to a Set of 25 Remarkable Points." New Scientists 12, 379 /C1/80, 1961. O’Beirne, T. H. "Thirty-Six Triangles Make Six Hexiamonds Make One Triangle." New Scientist 12, 706 /C1/07, 1961. Zimpfer, H. Die 12 Verhext. Baden, Germany: privately printed, 1967. Hexiamond Tiling There are a number of tilings of various shapes by all the 12 order n /C306 polyiamonds, summarized in the following table. Several of these (starred in the table below) are also illustrated above (Beeler 1972). Beeler’s numbers for the side 6 parallelogram of base 6 and side 4 trapezoid (156 and 76, respectively), differ from those quoted in Gardner (1984, p. 182) of 155 and 74, respectively. Size Solutions side 9 D with inverted side 3 D hole 0 side 6 trapezoid with bases 3 and 9 0 two side 6 triangles 0 /3 /C2912 rhomboid 0 /4 /C299 rhomboid* 37 side 4 trapezoid with bases 7 and 11* 76 side 6 parallelogram of base 6* 156 triangle of side 9 with 1, 2, 2 corners removed*5885 trefoil* several The following table gives the number of solutions to various hexiamond tilings using fewer than 12 pieces. Those indicated with asterisks (*) have a solution illustrated above. Size Pieces Solutions 2-hexagon /]1/ 3-hexagon* 9 /]15/ equilateral D/ 0 hexagonal ring 0 6-point star* 8 1 triangular ring 0 /2/C293 rhomboid 0 /2/C296 rhomboid* 4 1 /3/C293 rhomboid 3 0 /3/C294 rhomboid 4 many /3/C295 rhomboid 5 many /3/C296 rhomboid 6 many /3/C297 rhomboid 7 many /3/C298 rhomboid 8 many /3/C299 rhomboid 9 many /3/C2910 rhomboid 10 many /3/C2911 rhomboid* 11 24 /4/C296 rhomboid 8 /]1/ /5/C296 rhomboid 10 many See also HEPTIAMOND TILING ,H EXIAMOND ,O CTIA- MOND TILING ,PENTIAMOND TILING ,POLYHEX TILING , POLYIAMOND ,POLYOMINO TILING References Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 176 /C1/81, 1984. Vichera, M. "Polyiamonds." http://alpha.ujep.cz/~vicher/puz- zle/polyform/iamond/iamonds.htm. Hexlet Consider two mutually tangent (externally) SPHERES A and B together with a larger sphere C inside which A and B are internally tangent. Then construct a chain of spheres each tangent externally to A, B and internally to C (so that C encloses the chain as well as the two original spheres). Surprisingly, every such chain closes into a "necklace" after six SPHERES , regardless of where the first SPHERE is placed. This beautiful and amazing result due to Soddy (1937) is a special case of KOLLROS’ THEOREM . It can be demonstrated using INVERSION of six identical spheres around an equal center sphere, all of which are sandwiched between two planes (Wells 1991, pp. 120 and 232). This result was given in a SANGAKU PROBLEM from Kanagawa Prefecture in 1822, more than a century before it was published by Soddy (Rothman 1998). Moreover, the centers of the six spheres in the necklace and their six points of contact all lie in a plane. Furthermore, there are two planes which touch each of the six spheres, one on either side of the necklace. Finally, the radii riof the spheres are related by 1 r1/C271 r4/C301 r2/C271 r3/C301 r3/C271 r6 (Rothman 1998). Soddy’s BOWL OF INTEGERS contains an infinite number of nested hexlets. The centers of a Soddy hexlet always lie on an ELLIPSE (Ogilvy 1990, p. 63). See also BOWL OF INTEGERS ,COXETER’S LOXODROMIC SEQUENCE OF TANGENT CIRCLES ,D AISY,K OLLROS’ THEOREM ,SEVEN CIRCLES THEOREM ,STEINER CHAIN , TANGENT SPHERES References Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta Math. 18, 113 /C1/21, 1952. Gosset, T. "The Hexlet." Nature 139, 251 /C1/52, 1937. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 49 /C1/0, 1976. Morley, F. "The Hexlet." Nature 139,72/C1/3, 1937. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 60 /C1/2, 1990.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/1, May 1998. Soddy, F. "The Bowl of Integers and the Hexlet." Nature 139,77/C1/9, 1937. Soddy, F. "The Hexlet." Nature 139, 154 and 252, 1937. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 120 and 231 /C1/32, 1991. HexLife An alternative LIFE game similar to Conway’s, which is played on a hexagonal grid. No set of rules has yet emerged as uniquely interesting. See also HIGHLIFE Hexomino One of the 35 6-POLYOMINOES . See also DOMINO ,HEPTOMINO ,OCTOMINO ,PENTOMI- NO,POLYOMINO ,TETROMINO ,TRIOMINO References Pappas, T. "Triangular, Square & Pentagonal Numbers." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989. Heyting Algebra An ALGEBRA which is a special case of a LOGOS . See also LOGOS ,TOPOS H-Fractal The FRACTAL illustrated above. References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 1 /C1/, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . H-Function FOX’S H-FUNCTION Hh Function Let Z(x) /C131ffiffiffiffiffiffi 2pp e /C28x2 =2 (1) Q(x) /C131ffiffiffiffiffiffi2 ppg/C12 xe /C28t2 =2 dt (2) /C301 21 /C28erfxffiffiffi 2p !"# ; (3) where /Z(x)/ and /Q(x)/ are closely related to the NORMAL DISTRIBUTION FUNCTION , then Hh/C28n(x) /C30(/C281)n/C281ffiffiffiffiffiffi 2pp Z(n/C281)(x) (4) Hhn(x) /C30( /C281)n n!Hh/C281(x)dn dxnQ(x) Z(x)"# : (5)The first few values are Hh/C283(x) /C30e /C28x2 =2(x2 /C281) (6) Hh/C282(x) /C30e /C28x2 =2x (7) Hh/C281(x) /C30e /C28x2 =2 (8) Hh0(x) /C300 (9) Hh1(x) /C30e /C28x2 =2 /C28ffiffiffi p 2s x erfcxffiffiffi 2p ! (10) Hh2(x) /C301 4/C282xe /C28x2 =2 /C27ffiffiffiffiffiffi 2pp (x2 /C271)erfcxffiffiffi 2p ! "# (11) Hh3(x) /C301 122e/C28x2 =2(x2 /C272) /C28ffiffiffiffiffiffi 2pp x(x2 /C273)erfcxffiffiffi 2p ! "# : (12) See also NORMAL DISTRIBUTION FUNCTION ,TETRA- CHORIC FUNCTION References Jeffreys, H. and Jeffreys, B. S. "The Parabolic Cylinder, Hermite, and Hh Functions" et seq. §23.08 /C1/3.081 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 620 /C1/27, 1988. Higher Arithmetic An archaic term for NUMBER THEORY . Higher Dimensional Group Theory The term "higher dimensional group theory" was introduced by Brown (1982), and refers to a method for obtaining new homotopical information by gen- eralizing to higher dimensions the fundamental group of a space with a base point. See also GROUP THEORY ,LOW-DIMENSIONAL TOPOL- OGY References Brown, R. "Higher Dimensional Group Theory." In Low- Dimensional Topology: Proceedings of a Conference on Topology in Low Dimension, Bangor, 1979 (Ed. R. Brown and T. L. Thickstun). Cambridge, England: CambridgeUniversity Press, pp. 215 /C1 /38, 1982. Brown, R. "Higher Dimensional Group Theory." http:// www.bangor.ac.uk/~mas010/hdaweb2.htm. Higher Geometry PROJECTIVE GEOMETRY Highest Common Divisor GREATEST COMMON DIVISOR Highest Weight Theorem A theorem proved by E´ . Cartan in 1913 which classifies the irreducible representations of COMPLEX semisimple LIE ALGEBRAS . References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. HighLife An alternate set of LIFE rules similar to Conway’s, but with the additional rule that six neighbors generate a birth. Most of the interest in this variant is due to the presence of a so-called replicator. See also HEXLIFE,LIFE Highly Abundant Number HIGHLY COMPOSITE NUMBER Highly Composite Number A COMPOSITE NUMBER (also called a SUPERABUNDANT NUMBER ) is a number n which has more FACTORS than any other number less than n. In other words, / s(n) =n/ exceeds /s(k) =k/ for all k Bn, where s(n) is the DIVISOR FUNCTION . They were called highly composite numbers by Ramanujan, who found the first 100 or so, and superabundant numbers by Alaoglu and Erdos (1944). There are an infinite number of highly composite numbers, and the first few are 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, ... (Sloane’s A002182). Ramanujan (1915) listed 102 up to 6746328388800 (but omitted 293, 318, 625, 600, and 29331862500). Robin (1983) gives the first 5000 highly composite numbers, and a comprehensive survey is given by Nicholas (1988). If N /C302a2 3a3 /C1/C1/C1pap (1) is the PRIME FACTORIZATION of a highly composite number, then 1. The PRIMES 2, 3, ..., p form a string of consecutive PRIMES , 2. The exponents are nonincreasing, so / a2 ]a3 ]...]ap/, and 3. The final exponent /ap/ is always 1, except for the two cases /N /C304 /C3022 / and /N /C3036 /C3022 /C215 32 /, where it is 2. Let /Q(x)/ be the number of highly composite numbers / 5x/. Ramanujan (1915) showed that lim x0/C12Q(x) ln x /C30/C12: (2) Erdos (1944) showed that there exists a constant /c1 /C210/ such that Q(x) ](ln x)1 /C27c1 (3) Nicholas proved that there exists a constant /c2/C210/ such that Q(x)/C10(lnx)c2: (4) See also ABUNDANT NUMBER ,R OUND NUMBER , ROUNDNESS ,SMOOTH NUMBER References Alaoglu, L. and Erdos, P. "On Highly Composite and Similar Numbers." Trans. Amer. Math. Soc. 56, 448/C1/69, 1944. Andree, R. V. "Ramanujan’s Highly Composite Numbers." Abacus 3,6 1/C1/2, 1986. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, p. 53, 1994. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 323, 1952. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 88 /C1/1, 1998. Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., p. 112, 1973. Honsberger, R. "An Introduction to Ramanujan’s Highly Composite Numbers." Ch. 14 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 193 /C1/07, 1985. Kanigel, R. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Washington Square Press, p. 232, 1991. Nicholas, J.-L. "On Highly Composite Numbers." In Rama- nujan Revisited: Proceedings of the Centenary Conference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin).Boston, MA: Academic Press, pp. 215 /C1 /44, 1988. Ramanujan, S. "Highly Composite Numbers." Proc. London Math. Soc. 14, 347/C1/09, 1915. Ramanujan, S. Collected Papers. New York: Chelsea, 1962. Robin, G. "Me ´thodes d’optimalisation pour un proble `me de the´ories des nombres." RAIRO Inform. The ´or.17, 239/C1/47, 1983. Se´roul, R. "Highly Composite Numbers." §8.14 in Program- ming for Mathematicians. Berlin: Springer-Verlag, pp. 208 /C1/13, 2000. Sloane, N. J. A. Sequences A002182/M1025 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. New York: Penguin Books, p. 128, 1986. Higman-Sims Group The SPORADIC GROUP HS. References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/HS.html. Hilbert Basis A Hilbert basis for the VECTOR SPACE of square summable sequences /(an)/C30a1/,a2;... is given by the standard basis /ei/, where /ei/C30din/, with /din/the K RO- NECKER DELTA . Then (an) /C30X aiei ; with /ajai j2 B/C12 /. Although strictly speaking, the /ei/ are not a BASIS because there exist elements which are not a finite LINEAR COMBINATION , they are given the special term "Hilbert basis." In general, a HILBERT SPACE V has a Hilbert basis /ei/ if the /ei/ are an ORTHONORMAL BASIS and every element v /C23 V can be written v /C30X/C12 i/C301aiei for some /ai/ with / ajai j2 B/C12 /. See also BASIS (VECTOR SPACE ), FOURIER SERIES , HILBERT SPACE , L2-SPACE ,ORTHONORMAL SET Hilbert Basis Theorem If R is a NOETHERIAN RING , then S /C30R[X] is also a NOETHERIAN RING . See also ALGEBRAIC VARIETY ,FUNDAMENTAL SYSTEM , NOETHERIAN RING,SYZYGY References Hilbert, D. "U¨ ber die Theorie der algebraischen Formen." Math. Ann. 36, 473 /C1/34, 1890. Hilbert Curve AL INDENMAYER SYSTEM invented by Hilbert (1891) whose limit is a PLANE-FILLING CURVE which fills a square. Traversing the VERTICES of an n-D HYPER- CUBE in GRAY CODE order produces a generator for the n-D Hilbert curve (Goetz). The Hilbert curve can be simply encoded with initial string "L", STRING REWRITING rules"L"- /C21 " /C27RF-LFL-FR /C27RFR /C27FL- ", and angle 908 (Peitgen and Saupe 1988, p. 278). A related curve is the Hilbert II curve, shown above (Peitgen and Saupe 1988, p. 284). It is also a LINDENMAYER SYSTEM and the curve can be encoded with initial string"X", STRING REWRITING rules"X"- /C21 "XFYFX /C27F/C27YFXFY-F-XFYFX", "Y" - /C21 "YFX- FY-F-XFYFX /C27F/C27YFXFY" , and angle 908. See also LINDENMAYER SYSTEM ,P EANO CURVE , PLANE- FILLING CURVE ,S IERPINSKI CURVE ,S PACE- FILLING CURVEReferences Bogomolny, A. "Plane Filling Curves." http://www.cut-the- knot.com/do_you_know/hilbert.html. Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Dickau, R. M. "Three-Dimensional L-Systems." http://for- um.swarthmore.edu/advanced/robertd/lsys3d.html. Goetz, P. "Phil’s Good Enough Complexity Dictionary." http://www.cs.buffalo.edu/~goetz/dict.html. Hilbert, D. "Uuml;ber die stetige Abbildung einer Linie auf ein Flachenstu ¨ck." Math. Ann. 38, 459 /C1/60, 1891. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, pp. 278 and 284, 1988. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 198 /C1/06, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 100 /C1/01, 1991. Hilbert Function Let /G/C30fp1 ; ...; pm gƒP2 / be a collection of m distinct points. Then the number of conditions imposed by G on forms of degree d is called the Hilbert function /hG/ of G. If curves X1 and X2 of degrees d and e meet in a collection G of/d /C215 e/ points, then for any k, the number /hG(k)/ of conditions imposed by on forms of degree k is independent of X1 and X2 and is given by hG(k) /C30k /C272 2P+’vP+’u /C28k /C28d /C272 2P+’vP+’u /C28k /C28e /C272 2P+’vP+’u /C27k /C28d /C28e /C272 2P+’vP+’u ; where the BINOMIAL COEFFICIENT /(a 2)/ is taken as 0 if a B2 (Cayley 1843). References Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach Theorems and Conjectures." Bull. Amer. Math. Soc. 33, 295 /C1/24, 1996. Hilbert Hotel Let a hotel have a DENUMERABLE set of rooms numbered 1, 2, 3, .... Then any finite number n of guests can be accommodated without evicting the current guests by moving the current guests from room i to room /i /C27n/. Furthermore, a DENUMERABLE number of guests can be similarly accommodated by moving the existing guests from i to /2i/, freeing up a DENUMERABLE number of rooms /2i/C281/. See also CARDINAL NUMBER ,DENUMERABLE SET References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 84 /C1/5, 1998. Fadiman, C. Fantasia Mathematica, Being a Set of Stories, Together with a Group of Oddments and Diversions, All Drawn from the Universe of Mathematics. New York: Simon and Schuster, p. 286, 1958. Gamow, G. One, Two, Three, ... Infinity. New York: Dover, 1988. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 222, 1998. Lauwerier, H. "Hilbert Hotel." In Fractals: Endlessly Re- peated Geometric Figures. Princeton, NJ: Princeton Uni- versity Press, p. 22, 1991. Pappas, T. "Hotel Infinity." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 37, 1989. Hilbert Matrix A MATRIX H with elements Hij /C13(i /C27j /C281)/C281 for /i ; j /C301/, 2, ..., n. Hilbert matrices are given by HilbertMatrix [m, n] in the Mathematica add-on package LinearAlgebra‘MatrixManipulation‘ (which can be loaded with the command BBLinearAlgebra‘ ). Although the MATRIX IN- VERSE is given analytically by (H /C281)ij /C30(/C281)i/C27j i /C27 j /C28 1(n /C27 i /C28 1)!(n /C27 j /C28 1)! [(i /C28 1)!(j /C28 1)!]2(n /C28 i)!(n /C28 j)! ; Hilbert matrices are difficult to invert numerically. The DETERMINANTS for the first few values of Hnare given in the following table, and the numerical values for n /C301, 2, ... are given by one divided by 1, 12, 2160, 6048000, 266716800000, ... (Sloane’s A005249). n det( /H)/ 11 2 8.33333 /C2910 /C282 3 4.62963 /C2910 /C284 4 1.65344 /C2910 /C287 5 3.74930 /C2910 /C2812 6 5.36730 /C2910 /C2818 References Choi, M.-D. "Tricks or Treats with the Hilbert Matrix." Amer. Math. Monthly 90, 301 /C1/12, 1983. Richardson, T. M. 1999. http://xxx.lanl.gov/abs/math.LA/ 9905079/. Sloane, N. J. A. Sequences A005249/M4882 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hilbert Number GELFOND- SCHNEIDER CONSTANTHilbert Polynomial Let G be an ALGEBRAIC CURVE in a projective space of DIMENSION n, and let p be the PRIME IDEAL defining G, and let / x(p; m)/ be the number of linearly indepen- dent forms of degree m modulo p. For large m, / x(p ; m)/ is a POLYNOMIAL known as the Hilbert polynomial. References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 36, 1980. Hilbert Space A Hilbert space is a VECTOR SPACE H with an INNER PRODUCT //C142f ; g/C143/ such that the NORM defined by ½f ½/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C142f ; f /C143p turns H into a COMPLETE METRIC SPACE . If the INNER PRODUCT does not so define a NORM , it is instead known as an INNER PRODUCT SPACE . Examples of FINITE -dimensional Hilbert spaces in- clude 1. The REAL NUMBERS Rn with //C142v; u/C143/ the vector DOT PRODUCT of v and u. 2. The COMPLEX NUMBERS Cnwith //C142v;u/C143/the vector DOT PRODUCT ofvand the COMPLEX CON- JUGATE ofu. An example of an INFINITE -dimensional Hilbert space is/L2 /, the SETof all FUNCTIONS /f:R0R/such that the INTEGRAL of /f2/over the whole REAL LINE isFINITE .I n this case, the INNER PRODUCT is /C142f;g/C143/C30g/C12 /C28/C12f(x)g(x)dx: A Hilbert space is always a B ANACH SPACE , but the converse need not hold. See also BANACH SPACE ,COMPLETE SET OF FUNC- TIONS ,H ILBERT BASIS, L2-NORM, L2-SPACE ,L IOU- VILLE SPACE ,PARALLELOGRAM LAW,VECTOR SPACE References Sansone, G. "Elementary Notions of Hilbert Space." §1.3 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 5/C1/0, 1991. Stone, M. H. Linear Transformations in Hilbert Space and Their Applications Analysis. Providence, RI: Amer. Math. Soc., 1932. Hilbert Symbol For any two nonzero P-ADIC NUMBERS aand b, the Hilbert symbol is defined as (a;b)/C301i f z2/C30ax2/C27by2has a nonzero solution /C281 otherwise :P+2k If the p-adic field is not clear, it is said to be the Hilbert symbol of a and b relative to k. The field can also be the reals (/p /C30/C12/). The Hilbert symbol satisfies the following formulas: 1. /(a ; b) /C30(b; a)/. 2. /(a ; c2) /C301/ for any c. 3. /(a ;/C28a) /C301/. 4. /(a ; 1 /C28a) /C301/. 5. /(a ; b) /C301 [(aa?; b) /C30(a?; b)/. 6. /(a ; b) /C30(a;/C28ab) /C30(a ; (1 /C28a)b)/. The Hilbert symbol depends only the values of a and b modulo squares. So the symbol is a map / k/C31=k /C312 /C29k/C31=k/C312 0f1;/C281g/. Hilbert showed that for any two nonzero rational numbers a and b, 1. /(a ;b)v /C301/ for almost every prime v. 2. /Q(a ;b)v /C301/ where v ranges over every prime, including /v /C30/C12/ corresponding to the reals. See also DIOPHANTINE EQUATION–2ND POWERS , FIELD, P-ADIC NUMBER ,SYMMETRIC BILINEAR FORM (GENERAL FIELDS ), VECTOR SPACE References Serre, J. P. A Course in Arithmetic. New York: Springer- Verlag, pp. 27 /C1/5, 1973. Hilbert Transform The INTEGRAL TRANSFORM g(y) /C30H[f(x)] /C301 p g/C12 /C28/C12f(x) dx x /C28 y f(x) /C30H/C281[g(y)] /C301 p g/C12 /C28/C12g(y) dy y /C28 x; where the CAUCHY PRINCIPAL VALUE is taken in each of the integrals. In the following table, / P(x)/ is the RECTANGLE FUNC- TION , sinc x is the SINC FUNCTION , d(x) is the DELTA FUNCTION , /P(x)/ and /II(x)/ are IMPULSE SYMBOLS , and 1F1(a; b; x)isa CONFLUENT HYPERGEOMETRIC FUNC- TION OF THE FIRST KIND . /f(x)//g(y)/ /sin x//cos y/ /cos x///C28sin y/ /sin x x//cos y /C28 1 y/ /P(x)/ 1 plny /C281 2 y /C271 2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’21 1 /C27 x2 /C28y 1 /C27 y2 sinc ? x/C28p sinc y /C281 2p sinc2(12 py) /d(x)//C281 py /P(x)/ y p(14 /C28 y2) /II(x)//C281 2 p(14 /C28 y2) /e /C28x2 //C282yffiffiffipp 1 F1(a; b; x) See also ABEL TRANSFORM ,F OURIER TRANSFORM , INTEGRAL TRANSFORM ,TITCHMARSH THEOREM ,W I- ENER- LEE TRANSFORM References Bracewell, R. "The Hilbert Transform." The Fourier Trans- form and Its Applications, 3rd ed. New York: McGraw- Hill, pp. 267 /C1/72, 1999. Papoulis, A. "Hilbert Transforms." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 198 /C1/01, 1962. Hilbert’s Axioms The 21 assumptions which underlie the GEOMETRY published in Hilbert’s classic text Grundlagen der Geometrie. The eight INCIDENCE AXIOMS concern collinearity and intersection and include the first of EUCLID’S POSTULATES . The four ORDERING AXIOMS concern the arrangement of points, the five CONGRU- ENCE AXIOMS concern geometric equivalence, and the three CONTINUITY AXIOMS concern continuity. There is also a single parallel axiom equivalent to Euclid’s PARALLEL POSTULATE . See also CONGRUENCE AXIOMS ,CONTINUITY AXIOMS , INCIDENCE AXIOMS ,O RDERING AXIOMS ,P ARALLEL POSTULATE References Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: Open Court, 1980. Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert’s System of Axioms." §163B in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 544 /C1/45, 1980. Hilbert’s Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. Extend HILBERT’S INEQUALITY by letting /p ; q /C211/ and 1 p /C271 q ]1 ; (1) so that 0 B l /C302 /C281 p /C281q 51: (2) Levin (1937) and Steckin (1949) showed that X /C12 m/C301X/C12 n/C301ambn (m /C27 n)l 5 p cscp(q /C28 1) lq"#()lX/C12 m/C301(am)p"# 1 =pX/C12 n /C301(an)q"# 1 =q (3) and g/C12 0g/C12 0f(x)g(y) (x /C27 y) l dx dy B p cscp(q /C28 1) p"#l /C29g/C12 0[f(x)]p dxP+’vP+’u 1 =pg/C12 0[g(x)]q dxP+’vP+’u 1 =q : (4) Mitrinovic et al. (1991) indicate that this constant is the best possible. See also HILBERT’S INEQUALITY References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hilbert/hilbert.html. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, 1991. Steckin, S. B. "On the Degree of Best Approximation to Continuous Functions." Dokl. Akad. Nauk SSSR 65, 135 /C1/ 37, 1949. Hilbert’s Inequality Given a POSITIVE SEQUENCE anfg ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X/C12 j/C30/C28/C12X/C12 n/C30/C28/C12 n"jan j /C28 nP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’22vuuuuuuuuut5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X /C12 n/C30/C28/C12½an ½2vuut; where the a n/s are REAL and "square summable." Another INEQUALITY known as Hilbert’s applies to NONNEGATIVE sequences anfg and bnfg /, X/C12 m/C301X/C12 n/C301ambn m /C27 n B p cscp p !X/C12 m/C301ap m ! 1 =pX/C12 n/C301bqn ! 1 =q unless all anor all /bn/ are 0. If f(x) and g(x) are NONNEGATIVE integrable functions, then the integralform is g/C12 0g/C12 0f(x)g(y) x /C27 ydx dy B p cscp p ! /C29g/C12 0[f(x)]p dxP+’vP+’u 1 =pg/C12 0[g(x)]q dxP+’vP+’u 1 =q : The constant /p csc( p=P)/ is the best possible, in the sense that counterexamples can be constructed for any smaller value. References Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "Hilbert’s Double Series Theorem" and "On Hilbert’s Inequality." §9.1 and Appendix III in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 226 /C1/27 and 308 /C1/09, 1988. Hilbert’s Nullstellensatz Let K be an algebraically closed field and let I be an IDEAL in /K(x)/, where /x(x1 ; x2 ; ...; xn/ is a finite set of indeterminates. Let /p /C23 K(x)/ be such that for any / (c1 ; ...; cn/ in /Kn/, if every element of vanishes when evaluated if we set each (/xi /C30ci/), then p also vanishes. Then /pi/ lies in I for some j. Colloquially, the theory of algebraically closed fields is a complete model. See also ALGEBRAIC SET,IDEAL References Becker, T. and Weispfenning, V. "The Hilbert Nullstellen- satz." §7.4 in Gro¨bner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, pp. 312 /C1/23, 1993. Hartshorne, R. Algebraic Geometry. New York: Springer- Verlag, 1977. Hilbert’s Problems A set of (originally) unsolved problems in mathe- matics proposed by Hilbert. Of the 23 total, ten were presented at the Second International Congress inParis in 1900. These problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathe-matics. 1a. Is there a transfinite number between that of a DENUMERABLE SET and the numbers of the CON- TINUUM ? This question was answered by Go ¨del and Cohen to the effect that the answer dependson the particular version of SET THEORY assumed. 1b. Can the CONTINUUM of numbers be considered aWELL ORDERED SET ? This question is related to Zermelo’s AXIOM OF CHOICE . In 1963, the AXIOM OF CHOICE was demonstrated to be independent of all other AXIOMS inSET THEORY , so there appears to be no universally valid solution to this question either. 2. Can it be proven that the AXIOMS of logic are consistent? GO¨ DEL’S INCOMPLETENESS THEOREM indicated that the answer is "no," in the sense that any formal system interesting enough to formulate its own consistency can prove its own consistency IFF it is inconsistent. 3. Give two TETRAHEDRA which cannot be decom- posed into congruent TETRAHEDRA directly or by adjoining congruent TETRAHEDRA . Max Dehn showed this could not be done in 1902 by inventing the theory of DEHN INVARIANTS , and W. F. Kagon obtained the same result independently in 1903. 4. Find GEOMETRIES whose AXIOMS are closest to those of EUCLIDEAN GEOMETRY if the ORDERING and INCIDENCE AXIOMS are retained, the CONGRU- ENCE AXIOMS weakened, and the equivalent of the PARALLEL POSTULATE omitted. This problem was solved by G. Hamel. 5. Can the assumption of differentiability for functions defining a continuous transformation GROUP be avoided? (This is a generalization of the CAUCHY FUNCTIONAL EQUATION .) Solved by John von Neumann in 1930 for bicompact groups. Also solved for the ABELIAN case, and for the solvable case in 1952 with complementary results by Montgomery and Zipin (subsequently combined by Yamabe in 1953). Andrew Glean showed in 1952 that the answer is also "yes" for all locally bicompact groups. 6. Can physics be axiomized? 7. Let /a "1 "0/ be ALGEBRAIC and b IRRATIONAL .Is / ab/ then TRANSCENDENTAL (Wells 1986, p. 45)? /ab/ is known to be transcendental for the special case of b an ALGEBRAIC NUMBER , as proved in 1934 by Aleksander Gelfond in a result now known as GELFOND’S THEOREM (Courant and Robins 1996). However, the case of general irrational b has not been resolved. 8. Prove the RIEMANN HYPOTHESIS . The CONJEC- TURE has still been neither proved nor disproved. 9. Construct generalizations of the RECIPROCITY THEOREM of NUMBER THEORY . 10. Does there exist a universal algorithm for solving DIOPHANTINE EQUATIONS ? The impossibil- ity of obtaining a general solution was proven by Julia Robinson and Martin Davis in 1970, follow- ing proof of the result that the relation /n /C30F2m/ (where /F2m/ is a FIBONACCI NUMBER ) is Diophantine by Yuri Matijasevich (Matiyasevich 1970; Davis 1973; Davis and Hersh 1973; Davis 1982; Matiya- sevich 1993; Reid 1997, p. 107). More specifically, Matiyasevich showed that there is a polynomial P in n, m, and a number of other variables x, y, z, ...having the property that /n /C30F2m/ IFF there exist integers x, y, z, ... such that /P(n ; m; x;y;z ;...)/C300/. 11. Extend the results obtained for quadratic fields to arbitrary INTEGER algebraic fields. 12. Extend a theorem of Kronecker to arbitrary algebraic fields by explicitly constructing Hilbert class fields using special values. This calls for the construction of HOLOMORPHIC FUNCTIONS in sev- eral variables which have properties analogous to the exponential function and elliptic modular functions (Holzapfel 1995). 13. Show the impossibility of solving the general seventh degree equation by functions of two vari- ables. 14. Show the finiteness of systems of relatively integral functions. 15. Justify Schubert’s ENUMERATIVE GEOMETRY (Bell 1945). 16. Develop a topology of real algebraic curves and surfaces. The TANIYAMA- SHIMURA CONJECTURE postulates just this connection. See Gudkov and Utkin (1978), Ilyashenko and Yakovenko (1995), and Smale (2000). 17. Find a representation of definite form by SQUARES . 18. Build spaces with congruent POLYHEDRA . 19. Analyze the analytic character of solutions tovariational problems. 20. Solve general BOUNDARY VALUE PROBLEMS . 21. Solve differential equations given a MONO- DROMY GROUP . More technically, prove that there always exists a F UCHSIAN SYSTEM with given singularities and a given MONODROMY GROUP . Several special cases had been solved, but a NEGATIVE solution was found in 1989 by B. Boli- bruch (Anasov and Bolibruch 1994). 22. Uniformization.23. Extend the methods of CALCULUS OF VARIA- TIONS . See also GELFOND’S THEOREM ,RIEMANN HYPOTHESIS , TANIYAMA- SHIMURA CONJECTURE ,U NSOLVED PRO- BLEMS References Anasov, D. V. and Bolibruch, A. A. The Riemann-Hilbert Problem. Braunschweig, Germany: Vieweg, 1994. Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 340, 1945. Borowski, E. J. and Borwein, J. M. (Eds.). "Hilbert Pro- blems." Appendix 3 in The Harper Collins Dictionary of Mathematics. New York: Harper-Collins, p. 659, 1991. Boyer, C. and Merzbach, U. "The Hilbert Problems." History of Mathematics, 2nd ed. New York: Wiley, pp. 610 /C1/14, 1991. Browder, Felix E. (Ed.). Mathematical Developments Arising from Hilbert Problems. Providence, RI: Amer. Math. Soc., 1976. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 107, 1996. Davis, M. "Hilbert’s Tenth Problem is Unsolvable." Amer. Math. Monthly 80, 233/C1/69, 1973. Davis, M. and Hersh, R. "Hilbert’s 10th Problem." Sci. Amer. 229,8 4/C1/1, Nov. 1973. Davis, M. "Hilbert’s Tenth Problem is Unsolvable." Appen- dix 2 in Computability and Unsolvability. New York: Dover, 1999 /C1/35, 1982. Gudkov, D. and Utkin, G. A. Nine Papers on Hilbert’s 16th Problem. Providence, RI: Amer. Math. Soc., 1978. Hilbert, D. "Mathematical Problems." Bull. Amer. Math. Soc. 8, 437/C1/79, 1901 /C1/902. Holzapfel, R.-P. The Ball and Some Hilbert Problems. Boston, MA: Birkha ¨user, 1995. Ilyashenko, Yu. and Yakovenko, S. (Eds.). Concerning the Hilbert 16th Problem. Providence, RI: Amer. Math. Soc., 1995. Itoˆ, K. (Ed.). "Hilbert, David." §196 in Encyclopedic Dic- tionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, pp. 736 /C1/37, 1987. Joyce, D. E. "The Mathematical Problems of David Hilbert." http://aleph0.clarku.edu/~djoyce/hilbert/. Matiyasevich, Yu. V. "Solution to of the Tenth Problem of Hilbert." Mat. Lapok 21,8 3/C1/7, 1970. Matijasevich, Yu. V. Hilbert’s Tenth Problem. Cambridge, MA: MIT Press, 1993. http://www.informatik.uni-stutt- gart.de/ifi/ti/personen/Matiyasevich/H10Pbook/. Reid, C. Julia: A Life in Mathematics. Washington, DC: Math. Assoc. Amer., 1997. Schroeppel, R. C. Transcription of Hilbert’s Problems Lec- ture. http://www.cs.arizona.edu/~rcs/hilbert-speech. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 (Ed. V. Ar- nold, M. Atiyah, P. Lax, and B. Mazur). Providence, RI:Amer. Math. Soc., 2000. Vsemirnov, M. "Welcome to Hilbert’s Tenth Problem Page!" http://logic.pdmi.ras.ru/Hilbert10/. Waldschmidt, M. "Schneider’s Solution of Hilbert’s Seventh Problem." §3.1 in Transcendence Methods. Queen’s Papers in Pure and Applied Mathematics, No. 52. Kingston,Ontario, Canada: Queen’s University, pp. 3.1 /C1 /.4, 1979. Weisstein, E. W. "Books about Hilbert’s Problems." http:// www.treasure-troves.com/books/HilbertsProblems.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 45, 1986. Hilbert’s Theorem Every MODULAR SYSTEM has a MODULAR SYSTEM BASIS consisting of a finite number of POLYNOMIALS . Stated another way, for every order nthere exists a nonsingular curve with the maximum number of circuits and the maximum number for any one nest. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 61, 1959. Hilbert-Schmidt Norm The Hilbert-Schmidt norm of a MATRIX Ais defined as ½A½2/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX ija2 ij:sHilbert-Schmidt Theory The study of linear integral equations of the Fred- holm type with symmetric kernels K(x;t)/C30K(t;x): References Arfken, G. "Hilbert-Schmidt Theory." §16.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 890 /C1/97, 1985. Hill Determinant ADETERMINANT which arises in the solution of the second-order ORDINARY DIFFERENTIAL EQUATION x2d2c dx2/C27xdc dx/C271 4h2x2/C2712h2/C28b/C27h2 4x2 ! c/C300:(1) Writing the solution as a POWER SERIES c/C30X/C12 n/C30/C28/C12anxs/C272n(2) gives a RECURRENCE RELATION h2an/C271/C27[2h2/C284b/C2716(n/C271 2s)2]an/C27h2an/C281/C300:(3) The value of scan be computed using the Hill determinant D(s)/C30:::nnn n U /C1/C1/C1(s/C272)/C28a2 4/C28a2b2 4/C28a2 00 /C1/C1/C1 /C1/C1/C1 0/C28b2 a2/C28s2/C28a2 a2/C28b2 a2 /C1/C1/C1 /C1/C1/C1 00 /C28b2 1/C28a2(s/C281)2/C28a2 1/C28a2 /C1/C1/C1 Unnn n:::P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2(4) where s/C30 1 2s (5) a2/C3014b/C2818h2(6) b/C301 4h; (7) and /s/is the variable to solve for. The determinant can be given explicitly by the amazing formula D(s)/C30D(0)/C28sin2(ps=2) sin21 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b/C281 2h2qP+’vP+’u ; (8) where D(0) /C30::: nnnn U /C1/C1/C1 1h2 144/C272h2 /C284b 00 /C1/C1/C1 /C1/C1/C1h2 64 /C272h2 /C284b 1h2 64 /C272h2 /C284b 0 /C1/C1/C1 /C1/C1/C1 0h2 16 /C272h2 /C284b 1h2 16 /C272h2 /C284b/C1/C1/C1 /C1/C1/C1 00h2 2h2 /C284b 1 /C1/C1/C1 /C1/C1/C1 000h2 16 /C272h2 /C284b::: U nnnP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2 (9) leading to the implicit equation for s, sin21 2 psP+’kP+’7 /C30D(0)sin212 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b /C281 2 h2qP+’kP+’7 : (10) See also HILL’S DIFFERENTIAL EQUATION References Hill, G. W. "On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sum and Moon." Acta Math. 8,1/C1/6, 1886. Magnus, W. and Winkler, S. Hill’s Equation. New York: Dover, 1979. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 555 /C1/62, 1953. Hill’s Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION d2y dx2 /C27 u0 /C272X/C12 n /C301un cos(2 nx)"# y /C300 ; (1) where / un/ are fixed constants. A general solution can be given by taking the "DETERMINANT " of an infinite MATRIX . If only the n /C300 term is present, the equation have solution y /C30C1 sin(xffiffiffiffiffi u0p ) /C27C2 cos(xffiffiffiffiffiu 0p ) : (2) If terms /n 51/ are included, the equation becomes the MATHIEU DIFFERENTIAL EQUATION , which has solu- tion y /C30C1C(a;/C281 2 b; x) /C27C2Sa ;/C2812 b; xP+’kP+’7 : (3) If terms /n 52/ are included, it becomes the WHIT- TAKER- HILL DIFFERENTIAL EQUATION . See also HILL DETERMINANT ,W HITTAKER- HILL DIF- FERENTIAL EQUATION References Hill, G. W. "On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon." Acta Math. 8,1/C1/6, 1886. Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 384, 1956. Magnus, W. and Winkler, S. Hill’s Equation. New York: Dover, 1979. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997.Hillam’s Theorem If/f :[a ; b] 0 [a ; b]/ (where [a, b] denotes the CLOSED INTERVAL from a to b on the REAL LINE) satisfies a LIPSCHITZ CONDITION with constant K, i.e., if ½f(x) /C28f(y) ½5K ½x /C28y½ for all /x; y /C23 [a; b]/, then the iteration scheme xn/C271 /C30(1 /C28 l)xn /C27 lf(xn) ; where /l /C301=(K /C271)/, converges to a FIXED POINT of f. References Falkowski, B.-J. "On the Convergence of Hillam’s Iteration Scheme." Math. Mag. 69, 299 /C1/03, 1996. Geist, R.; Reynolds, R.; and Suggs, D. "A Markovian Frame- work for Digital Halftoning." ACM Trans. Graphics 12, 136 /C1/59, 1993. Hillam, B. P. "A Generalization of Krasnoselski’s Theorem on the Real Line." Math. Mag. 48, 167 /C1/68, 1975. Krasnoselski, M. A. "Two Remarks on the Method of Successive Approximations." Uspehi Math. Nauk (N. S.) 10, 123 /C1/27, 1955. Hindu Check CASTING OUT NINES Hinge The upper and lower hinges are descriptive statistics of a set of N data values, where N is OF THE FORM / N /C304n /C275/ with n /C300, 1, 2, .... The hinges are obtained by ordering the data in increasing order a1 ;:::; aN ; and writing them out in the shape of a "w" as illustrated above. The values at the bottom legs are called the hinges H1andH2(and the central peak is the MEDIAN ). In this ordering, H1/C30an/C272/C30a(N/C273)=4 M/C30a2n/C273/C30a(N/C271)=2 H2/C30a3n/C274/C30a(3N/C271)=4: ForNOF THE FORM /4n/C275/, the hinges are identical to the QUARTILES . The difference H2/C28H1is called the H - SPREAD . See also H-SPREAD ,HABERDASHER’S PROBLEM ,M ED- IAN (STATISTICS ), ORDER STATISTIC ,QUARTILE ,TRI- MEAN References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, pp. 32 /C1/4, 1977. Hinged Tessellation A TESSELLATION which can be thought of consisting of a number of pieces which are hinged at their vertices and therefore can be opened or closed to yield a series of tessellations. Examples above are given by Wells (1991). See also BRACED SQUARE ,TESSELLATION References Wells, D. Hidden Connections, Double Meanings. Cam- bridge, England: Cambridge University Press, 1988. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 101 /C1/03, 1991. Hippias’ Quadratrix QUADRATRIX OF HIPPIAS Hippopede A curve also known as a HORSE FETTER and given bythe polar equation r2/C304b(a/C28bsin2u): References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 144 /C1/46, 1972. Hi-Q A triangular version of PEG SOLITAIRE with 15 holes and 14 pegs. Numbering hole 1 at the apex of the triangle and thereafter from left to right on the next lower row, etc., the following table gives possible ending holes for a single peg removed (Beeler 1972).Because of symmetry, only the first five pegs need be considered. Also because of symmetry, removing peg 2 is equivalent to removing peg 3 and flipping theboard horizontally. remove possible ending pegs 11 , 7 /C3010, 13 2 2, 6, 11, 14 43 /C3012, 4, 9, 15 51 3 References Beeler, M. Item 76 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 29, Feb. 1972. Hirota Equation The PARTIAL DIFFERENTIAL EQUATION ut/C27iau/C27ib(uxx/C282h½u2½u)/C27cux/C27d(uxxx/C286h½u½2)/C300: References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 56, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 133, 1997. Hirota-Satsuma Equation The system of PARTIAL DIFFERENTIAL EQUATIONS ut/C301 2uxxx/C273uux/C286wwx wt/C30/C28wxxx/C283uwx: References Weiss, J. "Periodic Fixed Points of Ba¨cklund Transformation and the Korteweg-de Vries Equation." J. Math. Phys. 27, 2647 /C1/656, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Histogram The grouping of data into BINS (spaced apart by the so-called CLASS INTERVAL ) plotting the number of members in each bin versus the bin number. The above histogram shows the number of variates in bins with CLASS INTERVAL 1 for a sample of 100 real variates with a UNIFORM DISTRIBUTION from 0 and 10. Therefore, bin 1 gives the number of variates in the range 0 /C1/, bin 2 gives the number of variates in the range 1 /C1/, etc. See also BAR CHART ,B IN,C LASS INTERVAL ,F RE- QUENCY DISTRIBUTION ,FREQUENCY POLYGON ,OGIVE, PIE CHART ,SHEPPARD’S CORRECTION Kenney, J. F. and Keeping, E. S. "Histograms." §2.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 25 /C1/6, 1962. Hitch A KNOT that secures a rope to a post, ring, another rope, etc., but does not keep its shape by itself. See also CLOVE HITCH,KNOT,LINK,LOOP (KNOT) References Owen, P. Knots. Philadelphia, PA: Courage, p. 17, 1993. Hitting Set VERTEX COVER Hjelmslev’s Theorem When all the points P on one line are related by an ISOMETRY to all points P? on another, the MIDPOINTS of the segments /PP ?/ are either distinct and COLLINEAR or COINCIDENT . HJLS Algorithm An algorithm for finding INTEGER RELATIONS whose running time is bounded by a polynomial in thenumber of real variables (Ferguson and Bailey 1992). Unfortunately, it is numerically unstable and therefore requires extremely high numeric precision. The cause of this instability is not known, but is believed to derive from its reliance on GRAM- SCHMIDT ORTHONORMALIZATION (Ferguson and Bailey 1992), which is known to be numerically unstable (Golub and van Loan 1989). Ro¨ssner, C. and Schnorr (1994) have developed a stable variation of HJLS (Ferguson et al. 1999). See also FERGUSON- FORCADE ALGORITHM ,INTEGER RELATION ,LLL ALGORITHM ,PSLQA LGORITHM , PSOS ALGORITHM References Ferguson, H. R. P. and Bailey, D. H. "A Polynomial Time, Numerically Stable Integer Relation Algorithm." RNR Techn. Rept. RNR-91 /C1/32, Jul. 14, 1992. Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351 /C1/69, 1999. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, 1996. Hastad, J.; Just, B.; Lagarias, J. C.; and Schnorr, C. P. "Polynomial Time Algorithms for Finding Integer Rela- tions Among Real Numbers." SIAM J. Comput. 18, 859 /C1/ 81, 1988. Ro¨ssner, C. and Schnorr, C. P. "A Stable Integer Relation Algorithm." Tech. Rep. TR-94 /C1/16. FB Mathematik/Infor- matik, Universita ¨t Frankfurt, 1 /C1/1, 1994. HK Integral A type of integral named after Henstock and Kurz- weil. Every LEBESGUE INTEGRABLE function is HK integrable with the same value. References Shenitzer, A. and Steprans, J. "The Evolution of Integra- tion." Amer. Math. Monthly 101,66/C1/2, 1994. H-Matrix HADAMARD MATRIX Hoax Number A COMPOSITE NUMBER defined analogously to a SMITH NUMBER except that the SUM of the number’s DIGITS equals the sum of the DIGITS of its distinct PRIME FACTORS (excluding 1). The first few hoax numbers are 22, 58, 84, 85, 94, 136, 160, 166, 202, 234, ... (Sloane’s A019506), and the corresponding sums ofdigits are 4, 13, 12, 13, 13, 10, 7, 13, 4, 9, 7, ... (Sloane’s A050223). See also S MITH NUMBER References Sloane, N. J. A. Sequences A019506 and A050223 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Hodge Conjecture The Hodge conjecture asserts that, for particularly nice types of spaces called PROJECTIVE ALGEBRAIC VARIETIES , the pieces called HODGE CYCLES are actually rational linear combinations of geometric pieces called algebraic cycles. See also HODGE CYCLE ,P ROJECTIVE ALGEBRAIC VARIETY References Clay Mathematics Institute. "The Hodge Conjecture." http:// www.claymath.org/prize_problems/hodge.htm. Deligne, P. "The Hodge Conjecture." http://www.clay- math.org/prize_problems/hodge.pdf. Grothendieck, A. "Hodge’s General Conjecture Is False for Trivial Reasons." Topology 8, 299 /C1/03, 1969. Hodge, W. V. D. "The Topological Invariants of Algebraic Varieties." Proc. Internat. Congress Math., Cambridge, Mass., 1950, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 182 /C1/92, 1952. Hodge Cycle See also HODGE CONJECTURE Hodge Diamond See also HODGE STAR Hodge Identities KA¨ HLER IDENTITIES Hodge Star On an oriented n-D RIEMANNIAN MANIFOLD , the Hodge star is a linear FUNCTION which converts alternating DIFFERENTIAL K-FORMS to alternating (n /C28k)/-forms. If w is an alternating K-FORM , its Hodge star is given by w(v1 ; ...; vk) /C30( /C31w)(vk /C271 ; ... ; vn) when v1 ; ..., vn is an oriented orthonormal basis. See also HODGE DIAMOND ,STOKES’ THEOREM Hodge’s Theorem On a COMPACT oriented FINSLER MANIFOLD without boundary, every COHOMOLOGY class has a UNIQUE harmonic representation. The DIMENSION of the SPACE of all harmonic forms of degree p is the pth BETTI NUMBER of the MANIFOLD . See also BETTI NUMBER ,COHOMOLOGY ,DIMENSION , FINSLER MANIFOLDReferences Chern, S.-S. "Finsler Geometry is Just Riemannian Geome- try without the Quadratic Restriction." Not. Amer. Math. Soc. 43, 959 /C1/63, 1996. Hoehn’s Theorem A geometric theorem related to the PENTAGRAM and also called the PRATT-KASAPI THEOREM . ½V1W1 ½ ½W2V3 ½½V2W2 ½ ½W3V4 ½½V3W3 ½ ½W4V5 ½½V4W4 ½ ½W5V1 ½½V5W5 ½ ½W1V2 ½/C301 ½V1W2 ½ ½W1V3 ½½V2W3 ½ ½W2V4 ½½V3W4 ½ ½W3V5 ½½V4W5 ½ ½W4V1 ½½V5W1 ½ ½W5V2 ½/C301: In general, it is also true that ViWi jj Wi/C271Vi/C272P+’2P+’2P+’2P+’2/C30 ViVi/C271Vi /C274P+’2P+’2P+’2P+’2 V iVi/C271Vi/C272Vi/C274P+’2P+’2P+’2P+’2ViVi /C271Vi/C272Vi /C273P+’2P+’2P+’2P+’2 V i/C272Vi /C273Vi/C271P+’2P+’2P+’2P+’2: This type of identity was generalized to other figures in the plane and their duals by Pinkernell (1996). See also CEVA’S THEOREM ,MENELAUS’ THEOREM References Chou, S. C. Mechanical Geometry Theorem Proving. Dor- drecht, Netherlands: Reidel, 1987. Gru¨nbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254 /C1/68, 1995. Hoehn, L. "A Menelaus-Type Theorem for the Pentagram." Math. Mag. 68, 254 /C1/68, 1995. Pinkernell, G. M. "Identities on Point-Line Figures in the Euclidean Plane." Math. Mag. 69, 377 /C1/83, 1996. Hoffman’s Minimal Surface A MINIMAL EMBEDDED SURFACE discovered in 1992 consisting of a HELICOID with a HOLE and HANDLE (Science News 1992). It has the same topology as a PUNCTURED sphere with a handle, and is only the second complete embedded minimal surface of finite topology and infinite total curvature discovered (the HELICOID being the first). A three-ended MINIMAL SURFACE ofGENUS 1 is some- times also called Hoffman’s minimal surface (Peter- son 1988). See also HELICOID ,MINIMAL SURFACE References Karcher, H.; Wei, F. S.; and Hoffman, D. "The Genus One Helicoid and the Minimal Surfaces that Led to Its Discovery." In Global Analysis in Modern Mathematics. Proceedings of the Symposium in Honor of Richard Palais’ Sixtieth Birthday held at the University of Maine, Orono, Maine, August 8 /C1/0, 1991, and at Brandeis University, Waltham, Massachusetts, August 12, 1992 (Ed. K. Uhlen- beck). Houston, TX: Publish or Perish Press, pp. 119 /C1/70, 1993. Peterson, I. Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 57 /C1/9, 1988. "Putting a Handle on a Minimal Helicoid." Sci. News 142, 276, Oct. 24, 1992. Hoffman-Singleton Graph The only REGULAR GRAPH of VERTEX DEGREE 7, DIAMETER 2, and GIRTH 5. It is the unique (7; 5)/- MOORE GRAPH (and is therefore also a (7,5)- CAGE GRAPH ), and contains many copies of the PETERSEN GRAPH . It can be constructed from the 10 5-cycles illustrated above, with vertex i of Pj joined to vertex i /C27jk (mod 5) of Qk(Robertson 1969; Bondy and Murty 1976, p. 239; Wong 1982). (Note the correction of Wong’s j /C27jk to i /C27jk :/) Other constructions are given by (Benson and Losey 1971; Biggs 1993, p. 163), and a RADIAL EMBEDDING is illustrated above. See also CAGE GRAPH ,H OFFMAN- SINGLETON THEO- REM,MOORE GRAPH ,PETERSEN GRAPH References Benson, C. T.; and Losey, N. E. "On a Graph of Hoffman and Singleton." J. Combin. Th. Ser. B 11,67/C1/9, 1971. Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 235, 1976.Hoffman, A. J. and Singleton, R. R. "On Moore Graphs of Diameter Two and Three." IBM J. Res. Develop. 4, 497 /C1/ 04, 1960. Robertson, N. Graphs Minimal Under Girth, Valency, and Connectivity Constraints. Dissertation. Waterloo, Ontario: University of Waterloo, 1969. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/2, 1982. Hoffman-Singleton Theorem Let G be a k-regular graph with GIRTH 5 and GRAPH DIAMETER 2. (Such a graph is a MOORE GRAPH ). Then, k /C30 2, 3, 7, or 57. A proof of this theorem is difficult (Hoffman and Singleton 1960, Feit and Higman 1964, Damerell 1973, Bannai and Ito 1973), but can be found in Biggs (1993). See also HOFFMAN- SINGLETON GRAPH ,MOORE GRAPH References Bannai, E. and Ito, T. "On Moore Graphs." J. Fac. Sci. Univ. Tokyo Ser. A 20, 191/C1/08, 1973. Biggs, N. L. Ch. 23 in Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993. Damerell, R. M. "On Moore Graphs." Proc. Cambridge Philos. Soc. 74, 227/C1/36, 1973. Feit, W. and Higman, G. "The Non-Existence of Certain Generalized Polygons." J. Algebra 1, 114/C1/31, 1964. Hoffman, A. J. and Singleton, R. R. "On Moore Graphs of Diameter Two and Three." IBM J. Res. Develop. 4, 497/C1/ 04, 1960. Hofstadter Figure-Figure Sequence Define F(1)/C301 and S(1)/C302 and write F(n)/C30F(n/C281)/C27S(n/C281); where the sequence S(n) fg consists of those integers not already contained in F(n) fg :For example, F(2)/C30 F(1)/C27S(1)/C303;so the next term of S(n)i sS(2)/C304; giving F(3)/C30F(2)/C27S(2)/C307:The next integer is 5, so S(3)/C305 and F(4)/C30F(3)/C27S(3)/C3012:Continuing in this manner gives the "figure" sequence F(n)a s1 ,3 , 7, 12, 18, 26, 35, 45, 56, ... (Sloane’s A005228) and the "space" sequence as 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, ... (Sloane’s A030124). References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 73, 1989. Sloane, N. J. A. Sequences A005228/M2629 and A030124 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Hofstadter G-Sequence The sequence defined by G(0)/C300 and G(n)/C30n/C28G(G(n/C281)): The first few terms are 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, ... (Sloane’s A005206). References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 137, 1989. Sloane, N. J. A. Sequences A005206/M0436 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hofstadter H-Sequence The sequence defined by H(0) /C300 and H(n) /C30n /C28H(H(H(n /C281))): The first few terms are 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, ... (Sloane’s A005374). References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 137, 1989. Sloane, N. J. A. Sequences A005374/M0449 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hofstadter Male-Female Sequences The pair of sequences defined by F(0) /C301 ; M(0) /C300; and F(n) /C30n /C28M(F(n /C281)) M(n) /C30n /C28F(M(n /C281)): The first few terms of the "male" sequence M(n) are 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, ... (Sloane’s A005379), and the first few terms of the "female" sequence F(n) are 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, ... (Sloane’s A005378). References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 137, 1989. Sloane, N. J. A. Sequences A005378/M0263 and A005379/ M0278 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Hofstadter Point The r-HOFSTADTER TRIANGLE of a given TRIANGLE DABC is perspective to DABC ; and the PERSPECTIVE CENTER is called the Hofstadter point. The TRIANGLE CENTER FUNCTION is a /C30sin(rA) sin(r /C28 rA) : As r 0 0; the TRIANGLE CENTER FUNCTION approaches a /C30A a; and as r 0 1 ; the TRIANGLE CENTER FUNCTION ap- proachesa /C30a A : See also HOFSTADTER TRIANGLE References Kimberling, C. "Hofstadter Points." Nieuw Arch. Wiskunder 12, 109 /C1/14, 1994. Kimberling, C. "Major Centers of Triangles." Amer. Math. Monthly 104, 431 /C1/38, 1997. Kimberling, C. "Hofstadter Points." http://cedar.evansvil- le.edu/~ck6/tcenters/recent/hofstad.html. Hofstadter Sequences Let b1 /C301 and b2 /C302 and for n ]3; let /bn/ be the least INTEGER > bn/C281 which can be expressed as the SUM of two or more consecutive terms. The resulting se- quence is 1, 2, 3, 5, 6, 8, 10, 11, 14, 16, ... (Sloane’s A005243). Let c1 /C302 and c2 /C303; form all possible expressions OF THE FORM cicj /C281 for 1 5i Bj 5n; and append them. The resulting sequence is 2, 3, 5, 9, 14, 17, 26, 27, ... (Sloane’s A005244). See also HOFSTADTER- CONWAY $10,000 SEQUENCE , HOFSTADTER’S Q-SEQUENCE ,SUM-FREE SET References Guy, R. K. "Three Sequences of Hofstadter." §E31 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 231 /C1/32, 1994. Sloane, N. J. A. Sequences A005243/M0623 and A005244/ M0705 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Hofstadter Triangle For a NONZERO REAL NUMBER r and a TRIANGLE DABC ; swing LINE SEGMENT BC about the vertex B towards vertex A through an ANGLE rB. Call the line along the rotated segment L. Construct a second line L? by rotating LINE SEGMENT BC about vertex C through an ANGLE rC. Now denote the point of intersection of L and L ? by A(r): Similarly, construct B(r) and /C(r)/. The TRIANGLE having these points as vertices is called the Hofstadter r-triangle. Kimber- ling (1994) showed that the Hofstadter triangle is perspective to DABC ;and calls PERSPECTIVE CENTER the H OFSTADTER POINT . See also HOFSTADTER POINT References Kimberling, C. "Hofstadter Points." Nieuw Arch. Wiskunde 12, 109/C1/14, 1994. Kimberling, C. "Hofstadter Points." http://cedar.evansvil- le.edu/~ck6/tcenters/recent/hofstad.html. Hofstadter’s Q-Sequence The INTEGER SEQUENCE given by Q(n) /C30Q(n /C28Q(n /C281)) /C27Q(n /C28Q(n /C282)); with Q(1) /C30Q(2) /C301: The first few values are 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, ... (Sloane’s A005185; illustrated above). These numbers are sometimes called Q- NUMBER . There are currently no rigorous analyses or detailed predictions of the rather erratic behavior of Q(n) (Guy 1994). It has, however, been demonstrated that the chaotic behavior of the Q-numbers shows some signs of order, namely that they exhibit approximate PERIOD DOUBLING , SELF-SIMILARITY and SCALING (Pinn 1998). These properties are shared with the related sequence D(n) /C30D(D(n /C281)) /C27D(n /C281 /C28D(n /C282)) with D(1) /C30D(2) /C301; which exhibits exact PERIOD DOUBLING (Pinn 1998). The chaotic regions of D(n) are separated by predictable smooth behavior. See also HOFSTADTER- CONWAY $10,000 SEQUENCE , MALLOWS’ SEQUENCE ,PERIOD DOUBLING References Conolly, B. W. "Fibonacci and Meta-Fibonacci Sequences." In Fibonacci and Lucas Numbers, and the Golden Section (Ed. S. Vajda). New York: Halstead Press, pp. 127 /C1/38, 1989. Dawson, R.; Gabor, G.; Nowakowski, R.; and Weins, D. "Random Fibonacci-Type Sequences." Fib. Quart. 23, 169 /C1/76, 1985. Guy, R. "Some Suspiciously Simple Sequences." Amer. Math. Monthly 93, 186 /C1/91, 1986. Guy, R. K. "Three Sequences of Hofstadter." §E31 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 231 /C1/32, 1994. Hofstadter, D. R. Go¨del, Escher Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 137 /C1/38, 1980. Kubo, T. and Vakil, R. "On Conway’s Recursive Sequence." Disc. Math. 152, 225 /C1/52, 1996. Mallows, C. L. "Conway’s Challenge Sequence." Amer. Math. Monthly 98,5/C1/0, 1991. Pickover, C. A. "The Crying of Fractal Batrachion 1,489." Ch. 25 in Keys to Infinity. New York: W. H. Freeman, pp. 183 /C1/91, 1995.Pinn, K. Order and Chaos is Hofstadter’s Q(n) Sequence. 1 Jul 1998. http://xxx.lanl.gov/abs/chao-dyn/9803012/. To appear in Complexity. Pinn, K. A Chaotic Cousin of Conway’s Recursive Sequence. 4 Aug 1998. http://xxx.lanl.gov/abs/cond-mat/9808031/.. Submitted to J. Exper. Math. Sloane, N. J. A. Sequences A005185/M0438 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Tanny, S. M. "A Well-Behaved Cousin of the Hofstadter Sequence." Disc. Math. 105, 227 /C1/39, 1992. Hofstadter-Conway $10,000 Sequence The INTEGER SEQUENCE defined by the RECURRENCE RELATION a(n) /C30a(a(n /C281)) /C27a(n /C28a(n /C281)) with a(1) /C30a(2) /C301: The first few values are 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, ... (Sloane’s A004001). Plotting a(n) =n against n gives the BATRACHION plotted below. Con- way (1988) showed that limn 0/C12a(n) =n /C301=2 and offered a prize of $10,000 to the discoverer of a value of n for which a(i) =i /C281=2 jj B1 =20 for i /C21 n. The prize was subsequently claimed by Mallows, after adjust- ment to Conway’s "intended" prize of $1,000 (Schroe-der 1991), who found n/C301489. /a(n)=ntakes a value of 1/2 for nOF THE FORM 2kwith k/C301, 2, .... Pickover (1996) gives a table of analogous values of ncorresponding to different values of a(n)=n/C281=2 jj Be:/ See also BLANCMANGE FUNCTION ,H OFSTADTER’S Q- SEQUENCE ,MALLOWS’ SEQUENCE References Conolly, B. W. "Meta-Fibonacci Sequences." In Fibonacci and Lucas Numbers, and the Golden Section (Ed. S. Vajda). New York: Halstead Press, pp. 127 /C1/38, 1989. Conway, J. "Some Crazy Sequences." Lecture at AT&T Bell Labs, July 15, 1988. Guy, R. K. "Three Sequences of Hofstadter." §E31 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 231 /C1/32, 1994. Kubo, T. and Vakil, R. "On Conway’s Recursive Sequence." Disc. Math. 152, 225/C1/52, 1996. Mallows, C. L. "Conway’s Challenge Sequence." Amer. Math. Monthly 98,5/C1/0, 1991. Pickover, C. A. "The Drums of Ulupu." In Mazes for the Mind: Computers and the Unexpected. New York: St. Martin’s Press, 1993. Pickover, C. A. "The Crying of Fractal Batrachion 1,489." Ch. 25 in Keys to Infinity. New York: W. H. Freeman, pp. 183 /C1/91, 1995. Pinn, K. "A Chaotic Cousin of Conway’s Recursive Se- quence." Exp. Math. 9,55/C1/6, 2000. Schroeder, M. "John Horton Conway’s ‘Death Bet."’ Fractals, Chaos, Power Laws. New York: W. H. Freeman, pp. 57 /C1/9, 1991. Sloane, N. J. A. Sequences A004001/M0276 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Ho¨lder Condition A function f(t) satisfies the Ho¨lder condition on two points t1 and t2 on an arc L when f(t2) /C28 f(t1) jj 5At2 /C28t1 jjm; with A and m POSITIVE REAL constants. See also LIPSCHITZ CONDITION Ho¨lder Integral Inequality If C(r) with p, q /C21 1, then t1 with equality when t2 If f(t2) /C28 f(t1) jj 5At2 /C28t1 jjm; this inequality becomes SCHWARZ’S INEQUALITY . References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000. Ho¨lder, O. "U¨ ber einen Mittelwertsatz." Go¨ttingen Nachr., 44, 1889. Riesz, F. "Untersuchungen u¨ber Systeme integrierbarer Funktionen." Math. Ann. 69, 456, 1910. Riesz, F. "Su alcune disuguaglianze." Boll. Un. Mat. It. 7, 77 /C1/9, 1928. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 32 /C1/3, 1991. Ho¨lder Sum Inequality If C(r) with p, q /C21 1, then 1 p /C271 q /C301with equality when q > 1: If f(t2) /C28 f(t1) jj 5 At2 /C28t1 jjm; this becomes CAUCHY’S INEQUALITY . References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1092, 2000. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 10 /C1/5, 1988. Ho¨lder’s Inequalities Let 1 p /C271 q /C301 (1) with p, q /C21 1. Then Ho¨lder’s inequality for integrals states that gb af(x)g(x) jj dx 5gb af(x)jjpdx"#1 =p gb ag(x) jjqdx"#1 =q ; (2) with equality when g(x) jj/C30cf(x)jjp/C281: Ifp/C30q/C302;this inequality becomes S CHWARZ’S IN- EQUALITY . Similarly, Ho ¨lder’s inequality for sums states that Xn k/C301akbk jj5Xn k/C301akjjp ! 1=pXn k/C301bkjjq ! 1=q ; (3) with equality when bkjj/C30cakjjp/C281:Ifp/C30q/C302;this becomes C AUCHY’S INEQUALITY . See also CAUCHY’S INEQUALITY ,SCHWARZ’S INEQUAL- ITY References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1092 and 1099, 2000. Hardy, G. H.; Littlewood, J. E.; and Po ´lya, G. "Ho ¨lder’s Inequality and Its Extensions." §2.7 and 2.8 in Inequal- ities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 21 /C1/6, 1988. Ho¨lder, O. "U ¨ber einen Mittelwertsatz." Go¨ttingen Nachr. , 38/C1/7, 1889. Riesz, F. "Untersuchungen u ¨ber Systeme integrierbarer Funktionen." Math. Ann. 69, 456, 1910. Riesz, F. "Su alcune disuguaglianze." Boll. Un. Mat. It. 7, 77 /C1/9, 1928. Rogers, L. J. "An Extension of a Certain Theorem in Inequalities." Messenger Math. 17, 145 /C1/50, 1888. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 32 /C1/3, 1991. Holditch’s Theorem Let a CHORD of constant length be slid around a smooth, closed, convex curve C, and choose a point on the CHORD which divides it into segments of lengths p and q. This point will trace out a new closed curve C ?; as illustrated above. Provided certain conditions are met, the area between C and C ? is given by ppq; as first shown by Holditch in 1858. The Holditch curve for a CIRCLE of RADIUS R is another CIRCLE which, from the theorem, has RADIUS r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28pq :p References Bender, W. "The Holditch Curve Tracer." Math. Mag. 54, 128 /C1/29, 1981. Broman, A. "Holditch’s Theorem." Math. Mag. 54,99/C1/08, 1981. Kilic¸, E. and Keles, S. "On Holditch’s Theorem and Polar Inertia Momentum." Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist. 43,41/C1/7, 1996. Weisstein, E. W. "Holditch’s Theorem." MATHEMATICA NOTE- BOOK HOLDITCH.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 103, 1991. Hole A hole in a mathematical object is a TOPOLOGICAL structure which prevents the object from being con- tinuously shrunk to a point. When dealing with TOPOLOGICAL SPACES ,a DISCONNECTIVITY is inter- preted as a hole in the space. Examples of holes are things like the "donut hole" in the center of the TORUS , a domain removed from a plane, and theportion missing from EUCLIDEAN SPACE after cutting a KNOT out from it. Singular HOMOLOGY GROUPS form a MEASURE of the hole structure of a SPACE , but they are one particular measure and they don’t always detect all holes. HOMOTOPY GROUPS of a SPACE are another measure of holes in a SPACE , as well as BORDISM GROUPS , K- THEORY , COHOMOTOPY GROUPS , and so on. There are many ways to measure holes in a space. Some holes are picked up by HOMOTOPY GROUPS that are not detected by HOMOLOGY GROUPS , and some holes are detected by HOMOLOGY GROUPS that are not picked up by HOMOTOPY GROUPS . (For example, in the TORUS , HOMOTOPY GROUPS "miss" the two-dimen- sional hole that is given by the TORUS itself, but the second HOMOLOGY GROUP picks that hole up.) In addition, HOMOLOGY GROUPS don’t detect the varying hole structures of the complement of KNOTS in 3- space, but the first HOMOTOPY GROUP (the funda- mental group) does. See also BRANCH CUT,BRANCH POINT ,CORK PLUG, CROSS- CAP,GENUS (SURFACE ), PEG,PRINCE RUPERT’S CUBE,SINGULAR POINT (FUNCTION ), SPHERICAL RING, TORUS Holographic Projection EQUAL- AREA PROJECTION Holography The mathematical study of a nonlinear equation f(8) /C30y; where f maps from a HILBERT SPACE X to a HILBERT SPACE Y and y /C23 Y which abstracts the construction of optical holograms. References Lannes, A. "Abstract Holography." J. Math. Anal. Appl. 74, 530 /C1/59, 1980. Holomorphic Function A synonym for ANALYTIC FUNCTION , regular function, differentiable function, complex differentiable func- tion, and holomorphic map (Krantz 1999, p. 16). The word derives from the Greek olo& (holos ), meaning "whole," and mor8 h (morphe ), meaning "form" or "appearance." Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic func- tion" (Krantz 1999, p. 16), while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts (Morse and Feshbach 1953, pp. 356 /C1/74; Knopp 1996, pp. 83 /C1/11; Whittaker and Watson 1990, p. 83). See also ANALYTIC FUNCTION ,COMPLEX DIFFERENTI- ABLE ,HOLONOMIC FUNCTION ,HOMEOMORPHIC ,M ER- OMORPHIC FUNCTION References Knopp, K. "Analytic Continuation and Complete Definition of Analytic Functions." Ch. 8 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 83 /C1/11, 1996. Krantz, S. G. "Holomorphic Functions." §1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 12 /C1/6, 1999. Morse, P. M. and Feshbach, H. "Analytic Functions." §4.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 356 /C1/74, 1953. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Holomorphic Line Bundle A COMPLEX LINE BUNDLE is a VECTOR BUNDLE p : E 0 M whose FIBERS p/C281(m) are a copy of C : p is a holomorphic line bundle if it is a HOLOMORPHIC MAP between COMPLEX MANIFOLDS and its TRANSITION FUNCTIONS are HOLOMORPHIC . On a compact RIEMANN SURFACE ,a DIVISOR anipi determines a LINE BUNDLE . For example, consider 2p /C28q on X. Around p there is a COORDINATE CHART U given by the HOLOMORPHIC FUNCTION zpwith zp(p) /C300: Similarly, zqis a HOLOMORPHIC FUNCTION defining a disjoint chart V around q with zq(q) /C300: Then letting W /C30X /C28fp ; q g; the RIEMANN SURFACE is covered by X /C30U @ V @ W : The LINE BUNDLE corre- sponding to 2p /C28q is then defined by the following TRANSITION FUNCTIONS , gUW(x) /C30zp(x)2 defined for x /C23 U S W gVW(x) /C30zq(x) /C281 defined for x /C23 V S W : See also CHERN CLASS ,H ERMITIAN METRIC ,H OLO- MORPHIC FUNCTION ,H OLOMORPHIC TANGENT BUN- DLE,HOLOMORPHIC VECTOR BUNDLE ,LINE BUNDLE , RIEMANN- ROCH THEOREM ,RIEMANN SURFACE ,VEC- TOR BUNDLE Holomorphic Map HOLOMORPHIC FUNCTIONHolomorphic Tangent Bundle The holomorphic tangent bundle to a COMPLEX MANI- FOLD is given by its complexified tangent vectors which are of type (1; 0): In a CHART z /C30(z1 ; ...; zn); the bundle is spanned by the local SECTIONS @=@zk : The antiholomorphic sections are spanned by @=@¯zk ; of type (0; 1); where ¯z denotes the COMPLEX CON- JUGATE . See also COMPLEX STRUCTURE ,CR -STRUCTURE ,HER- MITIAN METRIC ,HOLOMORPHIC LINE BUNDLE ,HOLO- MORPHIC VECTOR BUNDLE ,TANGENT BUNDLE Holomorphic Vector Bundle A COMPLEX VECTOR BUNDLE is a VECTOR BUNDLE p : E 0 M whose FIBERS p/C281(m) are a copy of Ck : p is a holomorphic vector bundle if it is a HOLOMORPHIC MAP between COMPLEX MANIFOLDS and its TRANSITION FUNCTIONS are HOLOMORPHIC . The simplest example is a HOLOMORPHIC LINE BUNDLE , where the fiber is simply a copy of C:/ See also COMPLEX MANIFOLD ,H ERMITIAN METRIC , HOLOMORPHIC FUNCTION ,HOLOMORPHIC LINE BUN- DLE,HOLOMORPHIC TANGENT BUNDLE ,VECTOR BUN- DLE Holonomic Constant A limiting value of a HOLONOMIC FUNCTION near a SINGULAR POINT . Holonomic constants include APE´ RY’S CONSTANT ,C ATALAN’S CONSTANT ,P O´ LYA’S RANDOM WALK CONSTANTS for d /C212, and PI. Holonomic Function A solution of a linear homogeneous ORDINARY DIFFER- ENTIAL EQUATION with POLYNOMIAL COEFFICIENTS . See also HOLOMORPHIC FUNCTION ,HOLONOMIC CON- STANT References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 2, 1998. Zeilberger, D. "A Holonomic Systems Approach to Special Function Identities." J. Comput. Appl. Math. 32, 321 /C1/48, 1990. Holonomy A general concept in CATEGORY THEORY involving the globalization of topological or differential structures. The term derives from the Greek olo& (holos ) "whole" andnomo&(nomos ) "law, rule." See also HOLONOMY GROUP ,MONODROMY Holonomy Group On a RIEMANNIAN MANIFOLD M, tangent vectors can be moved along a path by PARALLEL TRANSPORT , which preserves VECTOR ADDITION and SCALAR MULTI- PLICATION . So a closed loop at a base point p, gives rise to a INVERTIBLE LINEAR MAP of TMp ; the tangent vectors at p. It is possible to compose closed loops by following one after the other, and to invert them by going backwards. Hence, the set of linear transforma- tions arising from PARALLEL TRANSPORT along closed loops is a GROUP , called the holonomy group. Since PARALLEL TRANSPORT preserves the RIEMAN- NIAN METRIC , the holonomy group is contained in the ORTHOGONAL GROUP O(n) : Moreover, if the manifold is ORIENTABLE , then it is contained in the SPECIAL ORTHOGONAL GROUP . A generic RIEMANNIAN METRIC on an ORIENTABLE MANIFOLD has holonomy group SO(n) ; but for some special metrics it can be a subgroup, in which case the manifold is said to have special holonomy. AK A¨ HLER MANIFOLD is a 2n/-dimensional MANIFOLD whose holonomy lies in the UNITARY GROUP U(n) ƒ O(2n) : AC ALABI- YAU MANIFOLD is a SIMPLY CON- NECTED 2n/-dimensional manifold with holonomy in the SPECIAL UNITARY GROUP .A4n/-dimensional mani- fold with holonomy group Sp(n) ; the QUATERNIONIC UNITARY GROUP , is called a HYPER- KA¨ HLER MANIFOLD , and one with holonomy Sp(n)Sp(1) is called a QUA- TERNION KA¨ HLER MANIFOLD . The possible groups that can arise as a holonomy group of the metric compa- tible LEVI-CIVITA CONNECTION were classified by Berger. The other possibilities for a nonproduct, nonsymmetric MANIFOLD are the LIE GROUPS G2 ; Spin(7) ; and Spin(9) :/ On a FLAT MANIFOLD , two homotopic loops give the same linear transformation. Consequently, the hol- onomy group is a REPRESENTATION of the FUNDAMEN- TAL GROUP of M. In general though, the CURVATURE of M changes the PARALLEL TRANSPORT between homo- topic loops. In fact, there is a formula for the difference as an integral of the curvature. See also CALABI- YAU MANIFOLD ,CONNECTION (PRIN- CIPAL BUNDLE ), CONNECTION (VECTOR BUNDLE ), CURVATURE FORM,H OMOGENEOUS SPACE ,K A¨ HLER MANIFOLD ,PARALLEL TRANSPORT ,QUATERNION, REPRESENTATION ,TANGENT BUNDLEReferences Salamon, S. Riemannian Geometry and Holonomy Groups. Essex, England: Longman Group, 1989. Holor Moon, P. and Spencer, D. E. Theory of Holors: A Generalization of Tensors. Cambridge, England: Cambridge University Press, 1986. Holyhedron A polyhedron whose faces and holes are all finite- sided polygons and which contains at least one hole whose boundary shares no point with a face bound- ary. D. Wilson coined the term in 1997, although no actual holyhedron was known until 1999, when a holyhedron of GENUS approximately 54,000,000 was (apparently) constructed (Vinson 2000). J. H. Con- way believes the construction to be correct, although he believes that the minimal GENUS should be closer to 100. See also POLYHEDRON References Vinson, J. "On Holyhedra." Disc. Comput. Geom. 24,85/C1/04, 2000. Homalographic Projection EQUAL- AREA PROJECTION Home Plate Home plate in the game of BASEBALL is an irregular PENTAGON . However, the Little League rulebook’s specification of the shape of home plate (Kreutzer and Kerley 1990), illustrated above, is not physically realizable, since it requires the existence of a (12, 12, 17) RIGHT TRIANGLE , whereas 122 /C27122 /C30288 "289 /C30172 (Bradley 1996). See also BASEBALL ,BASEBALL COVER References Bradley, M. J. "Building Home Plate: Field of Dreams or Reality?" Math. Mag. 69,4 4/C1/5, 1996. Kreutzer, P. and Kerley, T. Little League’s Official How-to- Play Baseball Book. New York: Doubleday, 1990. Home Prime The prime HP(n) reached starting from a number n, concatenating its prime factors, and repeating until a prime is reached. For example, for n /C30 9, 9 /C303 /C2153 0 33 /C303 /C21511 0 311; so 311 is the home prime of 9. For n /C302, 3, ..., the first few are 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ... (Sloane’s A037274). Probabilistic arguments give exactly zero for the chance that the sequence of integers starting at a given number n contains no prime (J. H. Conway, Sloane), so a home prime should exist for every positive integer. Since prime numbers have trivial home primes (themselves), we can restrict attention to composite numbers. The number of steps to arrive at a home prime for composite numbers 4, 6, 8, 9, ... are 1, 13, 2, 4, 1, 5, 4, 4, 1, 15, 1, ... (Sloane’s A037271), and the primes they reach are 211, 23, 3331113965338635107, 311, 773, 223, ... (Sloane’s A037272). The largest home prime for n B100 is HP(49) /C30HP(77) ; although its value is not known. After 55 steps, the sequence reaches 3 /C21573 /C215C105; where C105 is the 105-digit composite number. This number was factored by P. Leyland in November 1999, and subsequently reached a number C137 in December 1999. In June 2000, Leyland factored this number as well, and proceeded a few steps to obtain a C131; which has not yet been factored. The next largest HP(n) for n B100 is HP(80) /C30313; 169; 138; 727; 147; 145; 210; 044; 974; 146; 858; 220; 729; 781; 791; 489: There are about 50 unknown HP(n) with 100 Bn B 1000 (Hoey). References De Geest, P. "Repeated Factorisation of Concatenated Primefactors of the Composite Numbers Up to 100 and Beyond..." http://www.ping.be/~ping6758/topic1.htm. Heleen, J. "Family Numbers: Constructing Primes by Prime Factor Splitting." J. Recr. Math. 28, 116 /C1/19, 1996 /C1/7. Sloane, N. J. A. Sequences A037271, A037272, A037273 and A037274 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Homeoid A shell bounded by two similar ELLIPSOIDS having a constant ratio of axes. Given a CHORD passing through a homeoid, the distance between inner and outer intersections is equal on both sides. Since a spherical shell is a symmetric case of a homeoid, this theorem is also true for spherical shells (CONCENTRIC CIRCLES in the PLANE ), for which it is easily proved by symmetry arguments. See also CHORD ,ELLIPSOIDHomeomorphic There are two possible definitions: 1. Possessing similarity of form, 2. Continuous, ONE-TO-ONE , ONTO , and having a continuous inverse. The most common meaning is possessing intrinsic topological equivalence. Two objects are homeo- morphic if they can be deformed into each other by a continuous, invertible mapping. Such a HOMEO- MORPHISM ignores the space in which surfaces are embedded, so the deformation can be completed in a higher dimensional space than the surface was originally embedded. MIRROR IMAGES are homeo- morphic, as are MO¨ BIUS STRIP with an EVEN number of half-twists, and MO¨ BIUS STRIP with an ODD number of half-twists. In CATEGORY THEORY terms, homeomorphisms are ISOMORPHISMS in the CATEGORY of TOPOLOGICAL SPACES and CONTINUOUS MAPS . See also HOMEOMORPHIC ,H OMOMORPHIC ,ISOGENY , POLISH SPACE References Krantz, S. G. "The Concept of Homeomorphism." §6.4.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 86, 1999. Homeomorphic Type The following three pieces of information completely determine the homeomorphic type of a surface (Mas- sey 1967): 1. Orientability, 2. Number of boundary components, 3. EULER CHARACTERISTIC . See also ALGEBRAIC TOPOLOGY ,EULER CHARACTER- ISTIC References Massey, W. S. Algebraic Topology: An Introduction. New York: Springer-Verlag, 1996. Homeomorphically Irreducible Tree SERIES- REDUCED TREE Homeomorphism An EQUIVALENCE RELATION and one-to-one correspon- dence between points in two geometric figures or topological spaces which is continuous in both direc- tions, also called a continuous transformation. A homeomorphism which also preserves distances iscalled an ISOMETRY .AFFINE TRANSFORMATIONS are another type of common geometric homeomorphism. The similarity in meaning and form of the words "HOMOMORPHISM " and "homeomorphism" is unfortu- nate and a common source of confusion. See also AFFINE TRANSFORMATION ,HOMEOMORPHIC , HOMEOMORPHIC TYPE,H OMOMORPHISM ,ISOMETRY , TOPOLOGICALLY CONJUGATE References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 101, 1967. Krantz, S. G. "The Concept of Homeomorphism." §6.4.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 86, 1999. Ore, Ø. Graphs and Their Uses. New York: Random House, 1963. Homeomorphism Group The homeomorphism group of a TOPOLOGICAL SPACE X is the set of all HOMEOMORPHISMS f : X 0 X ; which forms a GROUP by composition. See also GROUP ,INFINITE GROUP ,T OPOLOGICAL SPACE HOMFLY Polynomial A 2-variable oriented KNOT POLYNOMIAL PL(a; z) motivated by the JONES POLYNOMIAL (Freyd et al. 1985). Its name is an acronym for the last names of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lick- orish, and Yetter (Freyd et al. 1985). Independent work related to the HOMFLY polynomial was also carried out by Prztycki and Traczyk (1987). HOMFLY polynomial is defined by the SKEIN RELATIONSHIP a /C281PL /C27(a ; z) /C28aPL/C28(a; z) /C30zPL0(a;z) (1) (Doll and Hoste 1991), where v is sometimes written instead of a (Kanenobu and Sumi 1993) or, with a slightly different relationship, as aPL/C27(a; z) /C28 a/C281PL/C28( a; z) /C30zPL0( a; z) (2) (Kauffman 1991). It is also defined as PL(l; m)in terms of SKEIN RELATIONSHIP lPL/C27/C27l/C281PL /C28/C27mPL0/C300 (3) (Lickorish and Millett 1988). It can be regarded as a nonhomogeneous POLYNOMIAL in two variables or a homogeneous POLYNOMIAL in three variables. In three variables the SKEIN RELATIONSHIP is written xPL/C27(x; y; z) /C27yPL/C28(x; y; z) /C27zPL0(x; y; z) /C300 : (4) It is normalized so that Punknot /C301: Also, for n unlinked unknotted components, PL(x; y; z) /C30/C28x /C27 y z !n /C281 : (5) This POLYNOMIAL usually detects CHIRALITY but doesnot detect the distinct ENANTIOMERS of the KNOTS 09 /C1/ 42, 10 /C1/48, 10 /C1/71, 10 /C1/91, 10 /C1/04, and 10 /C1/25 (Jones 1987). The HOMFLY polynomial of an oriented KNOT is the same if the orientation is reversed. It is a general- ization of the JONES POLYNOMIAL V(t) ; satisfying V(t) /C30P(a /C30t; z /C30t1 =2 /C28t/C281=2) (6) V(t) /C30P(l /C30it/C281 ; m /C30i(t/C281 =2 /C28t1 =2)): (7) It is also a generalization of the ALEXANDER POLY- NOMIAL 9(z) ; satisfying 9(z) /C30P(a /C301 ; z /C30t1=2 /C28t/C281 =2) : (8) The HOMFLY POLYNOMIAL of the MIRROR IMAGE K /C31 of a KNOT K is given by PK /C31(l; m) /C30PK (l/C281 ; m) ; (9) so P usually but not always detects CHIRALITY . A split union of two links (i.e., bringing two links together without intertwining them) has HOMFLY polynomial P(L1 @ L2) /C30/C28(l /C27l/C281)m/C281P(L1)P(L2) : (10) Also, the composition of two links P(L1#L2) /C30P(L1)P(L2) ; (11) so the POLYNOMIAL of a COMPOSITE KNOT factors into POLYNOMIALS of its constituent knots (Adams 1994). MUTANTS have the same HOMFLY polynomials. In fact, there are infinitely many distinct KNOTS with the same HOMFLY POLYNOMIAL (Kanenobu 1986). Ex- amples include ( 05/C1/01,10/C1/32), (08/C1/08,10/C1/29)(08/C1/16,10/C1/ 56), and ( 10/C1/25,10/C1/56) (Jones 1987). Incidentally, these also have the same J ONES POLYNOMIAL . M. B. Thistlethwaite has tabulated the HOMFLY polynomial for KNOTS up to 13 crossings. See also ALEXANDER POLYNOMIAL ,JONES POLYNO- MIAL ,KNOT POLYNOMIAL References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 171 /C1/72, 1994. Doll, H. and Hoste, J. "A Tabulation of Oriented Links." Math. Comput. 57, 747/C1/61, 1991. Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett, K.; and Oceanu, A. "A New Polynomial Invariant of Knots and Links." Bull. Amer. Math. Soc. 12, 239/C1/46, 1985. Jones, V. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335/C1/88, 1987. Kanenobu, T. "Infinitely Many Knots with the Same Poly- nomial." Proc. Amer. Math. Soc. 97, 158/C1/61, 1986. Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- Bridge Knots through 22 Crossings." Math. Comput. 60, 771/C1/78 and S17-S28, 1993. Kauffman, L. H. Knots and Physics. Singapore: World Scientific, p. 52, 1991. Lickorish, W. B. R. and Millett, B. R. "The New Polynomial Invariants of Knots and Links." Math. Mag. 61,1/C1/3, 1988. Morton, H. R. and Short, H. B. "Calculating the -Variable Polynomial for Knots Presented as Closed Braids." J. Algorithms 11, 117 /C1/31, 1990. Przytycki, J. and Traczyk, P. "Conway Algebras and Skein Equivalence of Links." Proc. Amer. Math. Soc. 100, 744 /C1/ 48, 1987. Stoimenow, A. "Jones Polynomials." http://guests.mpim- bonn.mpg.de/alex/ptab/j10.html. Weisstein, E. W. "Knots and Links." MATHEMATICA NOTE- BOOK KNOTS.M . Homoclinic Point A point where a stable and an unstable SEPARATRIX (invariant MANIFOLD ) from the same fixed point or same family intersect. Therefore, the limits lim k0/C12fk(X) and lim k 0/C28/C12fk(X) exist and are equal. Refer to the above figure. Let X be the point of intersection, with X ? ahead of X on one MANIFOLD and X ƒ ahead of X of the other. The mapping of each of these points TX ? and TX ƒ must be ahead of the mapping of X, TX. The only way this can happen is if the MANIFOLD loops back and crosses itself at a new homoclinic point. Another loop must be formed, with T2X another homoclinic point. Since T2X is closer to the hyperbolic point than TX, the distance between T2X and TX is less than that between X and TX. Area preservation requires the AREA to remain the same, so each new curve (which is closer than the previous one) must extend further. In effect, the loops become longer and thinner. The network of curves leading to a dense AREA of homoclinic points is known as a homoclinic tangle or tendril. Homoclinic points appear where CHAOTIC regions touch in a hyperbolic FIXED POINT . A small DISK centered near a homoclinic point includes infinitely many periodic points of different periods. Poincare ´showed that if there is a single homoclinic point, there are an infinite number. More specifically, there are infinitely many homoclinic points in each small disk (Nusse and Yorke 1996). See also HETEROCLINIC POINT ,M ANIFOLD ,SEPARA- TRIXReferences Nusse, H. E. and Yorke, J. A. "Basins of Attraction." Science 271, 1376 /C1/380, 1996. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 145, 1989. Homogeneous Barycentric Coordinates AREAL COORDINATES Homogeneous Cartesian Coordinates HOMOGENEOUS COORDINATES Homogeneous Coordinates Homogeneous coordinates ( x1;x2;x3) of a finite point (x, y) in the plane are any three numbers for which x1 x3/C30x (1) x2 x3/C30y: (2) Coordinates /(x1;x2;0) for which x2 x3/C30l (3) describe the POINT AT INFINITY in the direction of slope l:/ In homogeneous coordinates, the equation of a LINE a1x/C27a2y/C27a3/C300 (4) is given by a1x1/C27a2x2/C27a3x3/C300: (5) Two points expressed using homogeneous coordinates (a1;a2;a3) and ( b1;b2;b3) are identical IFF a2a3 b2b3P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C30a 3a1 b3b1P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C30a 1a2 b1b2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C300: (6) Two lines expressed using homogeneous coordinates a 1x1/C27a2x2/C27a3x3/C300 (7) b1x1/C27b2x2/C27b3x3/C300 (8) are identical IFF a2a3 b2b3P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C30a 3a1 b3b1P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C30a 1a2 b1b2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2/C300: (9) The intersection of the two lines above is given by x 1/C30a2a3 b2b3P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2(10) x 2/C30a3a1 b3b1P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2(11) x3 /C30 a1a2 b1b2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2: (12) See also T RILINEAR COORDINATES References Graustein, W. C. "Homogeneous Cartesian Coordinates. Linear Dependence of Points and Lines." Ch. 3 in Intro- duction to Higher Geometry. New York: Macmillan, pp. 29 /C1/9, 1930. Homogeneous Function A function which satisfies f(tx; ty) /C30tnf(x; y) for a fixed n.M EANS , the WEIERSTRASS ELLIPTIC FUNCTION , and TRIANGLE CENTER FUNCTIONS are homogeneous functions. A transformation of the variables of a TENSOR changes the TENSOR into another whose components are linear homogeneous functions of the components of the original TENSOR . See also EULER’S HOMOGENEOUS FUNCTION THEOREM Homogeneous Ideal A homogeneous ideal I in a GRADED RING R /C30/C154Aiis an IDEAL generated by a set of homogeneous ele- ments, i.e., each one is contained in only one of the Ai : For example, the POLYNOMIAL RING C[x] /C30/C154Aiis a GRADED RING , where Ai /C30faxi g: The IDEAL I /C30/C142x2 /C143; i.e., all polynomials with no constant or linear terms, is a homogeneous ideal in C[x]: Another homogeneous ideal is I /C30/C142x2 /C27y2 /C27z2 ; xy /C27yz /C27zx ; z5 /C143 in C[x; y; z]:/ Given any finite set of polynomials in n variables, the process of homogenization converts them to homo- geneous polynomials in n /C271 variables. If f /C30 f(x1 ; ...; xn) is a polynomial of degree d then fh(x0 ; x1 ; ...; xn) /C30xd 0f(x1 =x0 ; ...; xn =x0) is the homogenization of f. Similarly, if I is an IDEAL in C x1 ; ...; xn ½/C138 ; then Ih /C30 fhP+vP+’2P+’2f /C23 I g is its homogeni- zation and is a homogeneous ideal. For example, if f /C30x3 1 /C272x1x2 /C283 then fh /C30x31 /C272x0x1x2 /C283x30 : Note that in general, if I /C30/C142f1 ; ... ; fk /C143 then Ih may have more elements than /C142fh 1 ; ...; fh k /C143: However, if f1 ; ..., fk form a GRO¨ BNER BASIS using a graded monomial order, then Ih /C30/C142fh 1 ; ... ; fh k /C143: A polynomial is easily dehomogenized by setting the extra variable x0 /C301 :/ Here is a Mathematica function which takes a polynomial, in variables vars, and homogenizes it with the variable x0. (*dg finds the degree of the polynomial f*) dg[f_?PolynomialQ, {vars_?AtomQ}] : /C30 Exponent[f, vars]; dg[f_?PolynomialQ, vars_?ListQ] : /C30Max[MapIndexed[(dg[#1, Rest[vars]] /C27 #2 - 1 &), CoefficientList[f, First[vars]]]]; (*uses dg /C30 degree of polynomial above*) Homogenize[f_?PolynomialQ, vars_?ListQ, x0_?AtomQ] : /C30 Expand[x0 ^ dg[f, vars] f /. Map[(#1 - /C21 #1/ x0 &), vars]] Here is a Mathematica function which dehomo- genizes a polynomial in the variable x0. Dehomogenize[f_?PolynomialQ, x0_?AtomQ] : /C30 f /. x0 - /C21 1 The AFFINE VARIETY V corresponding to a homoge- neous ideal has the property that x /C23 V IFF cx /C23 V for all COMPLEX c. Therefore, a homogeneous ideal defines an ALGEBRAIC VARIETY in COMPLEX PROJEC- TIVE SPACE . See also ALGEBRAIC VARIETY ,C ATEGORY THEORY , COMMUTATIVE ALGEBRA ,C ONIC SECTION ,IDEAL , PRIME IDEAL ,PROJECTIVE VARIETY ,SCHEME ,ZARISKI TOPOLOGY References Hartshorne, R. Algebraic Geometry. New York: Springer- Verlag, 1977. Homogeneous Numbers Two numbers are homogeneous if they have identical PRIME FACTORS . An example of a homogeneous pair is (6, 72), both of which share PRIME FACTORS 2 and 3: 6 /C302 /C215 3 72 /C3023 /C215 32 : See also HETEROGENEOUS NUMBERS ,PRIME FACTORS , PRIME NUMBER References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 146, 1983. Homogeneous Polynomial A multivariate polynomial (i.e., a POLYNOMIAL in more than one variable) with all terms having the same degree. For example, x3 /C27xyz /C27y2z /C27z3is a homogeneous polynomial of degree three. SYMMETRIC POLYNOMIALS are always homogeneous. See also FORM (POLYNOMIAL ), POLYNOMIAL ,S YM- METRIC POLYNOMIAL Homogeneous Space A homogeneous space Mis a SPACE with a TRANSITIVE GROUP ACTION by a L IE GROUP . Because a TRANSITIVE GROUP ACTION implies that there is only one ORBIT ,M isISOMORPHIC to the QUOTIENT SPACE G=Hwhere H is the ISOTROPY GROUP Gx : The choice of x /C23 M does not affect the isomorphism type of G=Gx because all of the ISOTROPY GROUPS are CONJUGATE . Many common spaces are homogeneous spaces, such as the HYPERSPHERE , Sn /C2O(n /C271)=O(n) ; (1) and the COMPLEX PROJECTIVE SPACE C’n /C2U(n /C271)=U(n) /C29U(1) : (2) The real GRASSMANNIAN of k-dimensional SUBSPACES in Rn/C27k is O(n /C27k)=O(n) /C29O(k) : (3) The projection p : G 0 G=H makes G a PRINCIPAL BUNDLE on G =H with FIBER H. For example, p : SO(3) 0 SO(3)=SO(2) /C2S2 is a SO(2) BUNDLE , i.e., a CIRCLE BUNDLE , on the sphere. The SUBGROUP SO(2) /C3010 0 0 cos t /C28sin t 0 sin t cos t2 435 (4) acts on the right, and does not affect the first column so p(v 1v2v3) /C30v1 /C23S2 is WELL DEFINED . See also EFFECTIVE ACTION ,FREE ACTION ,GROUP , ISOTROPY GROUP ,M ATRIX GROUP ,O RBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION , TOPOLOGICAL GROUP ,TRANSITIVE References Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 41 /C1/9 and 89 /C1/4, 1987. Homographic Any two ranges ABC ... fg and fA?B?C ?...g which are situated on the same or different lines are said to be homographic when the CROSS-RATIO of any four points on one range is equal to the CROSS-RATIO of the corresponding points of the other range. See also CROSS- RATIO,MO¨ BIUS TRANSFORMATION References Lachlan, R. "Homographic Ranges and Pencils." §433 /C1/39 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 279 /C1/82, 1893. Homography A CIRCLE -preserving transformation composed of an EVEN number of inversions. See also ANTIHOMOGRAPHY Homological Algebra An abstract ALGEBRA concerned with results valid for many different kinds of SPACES .M ODULES are the basic tools used in homological algebra.See also MODULE References Enochs, E. E. and Jenda, O. M. G. Relative Homological Algebra. Berlin: de Gruyter, 2000. Hilton, P. and Stammbach, U. A Course in Homological Algebra, 2nd ed. New York: Springer-Verlag, 1997. Weibel, C. A. An Introduction to Homological Algebra. New York: Cambridge University Press, 1994. Homological Projection EQUAL- AREA PROJECTION Homologous Points The extremities of PARALLEL RADII of two CIRCLES are called homologous with respect to the SIMILITUDE CENTER collinear with them. See also ANTIHOMOLOGOUS POINTS ,INVARIABLE POINT ,SIMILITUDE CENTER References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 19, 1929. Homologous Triangles PERSPECTIVE TRIANGLES Homolographic Equal-Area Projection MOLLWEIDE PROJECTION Homology Homology is a concept which is used in many branches of algebra and topology. The basic example is degree one integral homology for a domain in R2 : In this case, a HOMOLOGY CLASS is represented by a finite sum or difference of closed loops. For example, consider the loops in the twice PUNCTURED plane R2 /C28 f(0; 0); (1; 0)g; illustrated above. The equality a /C27 b /C30 g holds in homology because the difference is the BOUNDARY of a COMPACTLY SUP- PORTED region. The homology of a space is an algebraic object which reflects the topology. The algebraic tools used are called HOMOLOGICAL ALGE- BRA, and in that language, the homology is a DERIVED FUNCTOR , the homology of a LONG EXACT SEQUENCE . See also BOUNDARY (HOMOLOGY ), COHOMOLOGY , DERIVED FUNCTOR ,H OMOLOGY CLASS,H OMOLOGY (GEOMETRY ), HOMOLOGY GROUP ,INTERSECTION (HOMOLOGY ), POINCARE DUALITY Homology (Chain) For every p, the kernel of @P : CP 0 CP/C281 is called the group of cycles, ZP /C30fc /C23 CP : @(c) /C300g: (1) The letter Z is short for the German word for cycle, "Zyklus." The image @(CP/C271) is contained in the group of cycles because @( @/C300; and is called the group of boundaries, BP /C30fc /C23 CP : there exists b /C23 CP/C271such that @(b) /C30c g: (2) The quotients HP /C30ZP =BP are the HOMOLOGY GROUPS of the chain. Given a SHORT EXACT SEQUENCE of CHAIN COMPLEXES 0 0 A/C310 B /C310 C /C310 0; (3) there is a LONG EXACT SEQUENCE in homology. ... 0 HP(A) 0 HP(B) 0 HP(C) 0dHP/C281(A) 0 ...: (4) In particular, a cycle a in AP with @a /C300; is mapped to a cycle b in BP : Similarly, a boundary @a ? in APgets mapped to a boundary @b? in BP : Consequently, the map between homologies HP(A) 0 HP(B) is well- defined. The only map which is not that obvious is d; called the CONNECTING HOMOMORPHISM , which is well-defined by the SNAKE LEMMA . Proofs of this nature are (with a modicum of humor) referred to as DIAGRAM CHASING . See also CHAIN COMPLEX ,C HAIN EQUIVALENCE , CHAIN HOMOMORPHISM ,CHAIN HOMOTOPY ,COCHAIN COMPLEX ,HOMOLOGY ,SNAKE LEMMA References Hilton, P. and Stammbach, U. A Course in Homological Algebra. New York: Springer-Verlag, pp. 117 /C1/18, 1997. Munkres, J. Elements of Algebraic Topology. Reading, MA: Addison-Wesley, pp. 58 and 71 /C1/6, 1984. Homology (Geometry) A PERSPECTIVE COLLINEATION in which the center and axis are not incident. The term was first used by Poncelet (Cremona 1960, p. ix). See also ELATION ,H ARMONIC HOMOLOGY ,PERSPEC- TIVE COLLINEATION ,PERSPECTIVE TRIANGLES References Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960. Desargues, G. /(E/uvres de Desargues, re´unies et analyse ´es par M. Pudra, tome 1. Paris, pp. 413 /C1/16, 1864. Lambert, J. H. Freie Perspective, 2nd ed. Zu¨rich, 1774. Homology (Topology) Historically, the term "homology" was first used in a topological sense by Poincare ´. To him, it meant prettymuch what is now called a COBORDISM , meaning that a homology was thought of as a relation between MANIFOLDS mapped into a MANIFOLD . Such MANI- FOLDS form a homology when they form the boundary of a higher-dimensional MANIFOLD inside the MANI- FOLD in question. To simplify the definition of homology, Poincare ´ simplified the spaces he dealt with. He assumed that all the spaces he dealt with had a triangulation (i.e., they were "SIMPLICIAL COMPLEXES "). Then in- stead of talking about general "objects" in these spaces, he restricted himself to subcomplexes, i.e., objects in the space made up only on the simplices in the TRIANGULATION of the space. Eventually, Poin- care´’s version of homology was dispensed with and replaced by the more general SINGULAR HOMOLOGY . SINGULAR HOMOLOGY is the concept mathematicians mean when they say "homology." In modern usage, however, the word homology is used to mean HOMOLOGY GROUP . For example, if someone says "X did Y by computing the homology of Z," they mean "X did Y by computing the HOMOLOGY GROUPS of Z." But sometimes homology is used more loosely in the context of a "homology in a SPACE ," which corresponds to singular homology groups. Singular homology groups of a SPACE measure the extent to which there are finite (compact) boundary- less GADGETS in that SPACE , such that these GADGETS are not the boundary of other finite (compact) GADGETS in that SPACE . A generalized homology or cohomology theory must satisfy all of the EILENBERG- STEENROD AXIOMS with the exception of the DIMENSION AXIOM . See also COHOMOLOGY ,D IMENSION AXIOM ,E ILEN- BERG- STEENROD AXIOMS ,GADGET ,GRADED MODULE , HOMOLOGICAL ALGEBRA ,HOMOLOGY GROUP ,SIMPLI- CIAL COMPLEX ,S IMPLICIAL HOMOLOGY ,S INGULAR HOMOLOGY References Goldberg, S. I. Curvature and Homology, enl. ed. New York: Dover, 1998. Homology Axis PERSPECTIVE AXIS Homology Center PERSPECTIVE CENTER Homology Class A homology class in a singular homology theory is represented by a finite LINEAR COMBINATION of geo- metric subobjects with zero boundary. Such a linear combination is considered to be HOMOLOGOUS to zero if it is the boundary of something having dimension one greater. For instance, two points that can be connected by a path comprise the boundary for that path, so any two points in a component are homo- logous and represent the same homology class. See also COHOMOLOGY ,COHOMOLOGY CLASS ,HOMOL- OGY,HOMOLOGY GROUP ,INTERSECTION (HOMOLOGY ) Homology Group The term "homology group" usually means a singular homology group, which is an ABELIAN GROUP which partially counts the number of HOLES in a TOPOLOGI- CAL SPACE . In particular, singular homology groups form a MEASURE of the HOLE structure of a SPACE , but they are one particular measure and they don’t always pick up everything. In addition, there are "generalized homology groups" which are not singular homology groups. See also HOMOLOGY (TOPOLOGY ) References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Homomorphic Related to one another by a HOMOMORPHISM . Homomorphism A term used in CATEGORY THEORY to mean a general MORPHISM . The term derives from the Greek omo (omo) "alike" and mor 8 vsi& (morphosis ), "to form" or "to shape." The similarity in meaning and form of the words "homomorphism" and "HOMEOMORPHISM "is unfortunate and a common source of confusion. If G and H are GROUPS , then a group homomorphism of G into H is a function f : G 0 H which preserves the group operation, i.e., for all g1 ; g2 /C23 G; (g1g2) f /C30(g1) f(g2) f (Yale 1988, p. 18). See also GROUP HOMOMORPHISM ,HOMEOMORPHISM , MORPHISM ,RING HOMOMORPHISM References Yale, P. B. Geometry and Symmetry. New York: Dover, 1988. Homomorphism (Ring) See also RINGHomoscedastic A set of STATISTICAL DISTRIBUTIONS having the same VARIANCE . See also HETEROSCEDASTIC Homothecy A SIMILARITY TRANSFORMATION which preserves or- ientation, also called a homothety. See also HOMOTHETIC ,SIMILARITY TRANSFORMATION References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 68, 1969. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991. Homothetic Two figures are homothetic if they are related by an EXPANSION or CONTRACTION . This means that they lie in the same plane and corresponding sides are PARALLEL ; such figures have connectors of corre- sponding points which are CONCURRENT at a point known as the HOMOTHETIC CENTER . The HOMOTHETIC CENTER divides each connector in the same ratio k, known as the SIMILITUDE RATIO . For figures which are similar but do not have PARALLEL sides, a SIMILITUDE CENTER exists. See also CONTRACTION (GEOMETRY ), DIRECTLY SIMI- LAR,EXPANSION ,HOMOTHECY ,HOMOTHETIC CENTER , INVERSELY SIMILAR ,PANTOGRAPH ,PERSPECTIVE ,SI- MILAR ,SIMILITUDE RATIO References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 173, 1888. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 1 /C1/, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 129, 1893. Homothetic Center The meeting point of lines that connect corresponding points from HOMOTHETIC figures. In the above figure, O is the homothetic center of the HOMOTHETIC figures ABCDE and A?B?C ?D?E ?: For figures which are similar but do not have PARALLEL sides, a SIMILITUDE CENTER exists (Johnson 1929, pp. 16 /C1/0). Given two nonconcentric CIRCLES , draw RADII PARAL- LEL and in the same direction. Then the line joining the extremities of the RADII passes through a fixed point on the line of centers which divides that line externally in the ratio of RADII . This point is called the external homothetic center, or external center of similitude (Johnson 1929, pp. 19 /C1/0 and 41). If RADII are drawn PARALLEL but instead in opposite directions, the extremities of the RADII pass through a fixed point on the line of centers which divides that line internally in the ratio of RADII (Johnson 1929, pp. 19 /C1/0 and 41). This point is called the internal homothetic center, or internal center of similitude (Johnson 1929, pp. 19 /C1/0 and 41). The position of the homothetic centers for two circles of radii ri ; centers (xi ; yi); and segment angle u are given by solving the simultaneous equations y /C28y2 /C30y2 /C28 y1 x2 /C28 x1(x /C28x2) y /C28y9 2/C30y92/C28 y91 x92/C28 x91(x /C28x92 ) for (x, y), where x9 i/C13xi /C27(/C281)iri cos uy9 i/C13yi /C27(/C281)iri sin u ; and the plus signs give the external homothetic center, while the minus signs give the internal homothetic center. As the above diagrams show, as the angles of the parallel segments are varied, the positions of the homothetic centers remain the same. This fact pro- vides a (slotted) LINKAGE for converting circular motion with one radius to circular motion with another. The six homothetic centers of three circles lie three by three on four lines (Johnson 1929, p. 120), which"enclose" the smallest circle. The homothetic center of triangles is the PERSPECTIVE CENTER ofHOMOTHETIC TRIANGLES . It is also called the SIMILITUDE CENTER (Johnson 1929, pp. 16 /C1/7). See also APOLLONIUS’ PROBLEM ,H OMOTHETIC ,PER- SPECTIVE ,SIMILITUDE CENTER References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 129, 1893. Weisstein, E. W. "Plane Geometry." M ATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Homothetic Position Two similar figures with PARALLEL homologous LINES and connectors of HOMOLOGOUS POINTS CONCURRENT at the HOMOTHETIC CENTER are said to be in homo- thetic position. If two SIMILAR figures are in the same plane but the corresponding sides are not PARALLEL , there exists a self- HOMOLOGOUS POINT which occupies the same homologous position with respect to the two figures. Homothetic Triangles Nonconcurrent TRIANGLES with PARALLEL sides are always HOMOTHETIC . Homothetic triangles are al- ways PERSPECTIVE TRIANGLES . Their PERSPECTIVE CENTER is called their HOMOTHETIC CENTER . Homothety HOMOTHECY Homotopic Two mathematical objects are said to be homotopic when they are the "same" in a certain abstract sense. For instance, the real line is homotopic to a single point, as is any TREE . However, the circle is not CONTRACTIBLE , but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps f0 : X 0 Y and f1 : X 0 Y are homotopic if there is a CONTINUOUS MAP F : X /C29[0; 1] 0 Y such that F(x; 0) /C30f0(x) and F(x; 1) /C30f1(x):/ Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the PUNCTU- RED plane R2 /C280: The puncture can be thought of as an obstacle. However, there is a way to compare two spaces via homotopy without ambient spaces. Two spaces X and Y are homotopy equivalent if there are maps f : X 0 Y and g : X 0 Y such that the composition f(g is homotopic to the IDENTITY MAP of Y and g(f is homotopic to the IDENTITY MAP of X. For example, the circle is not homotopic to a point, for then the constant map would be homotopic to the identity map of a circle, which is impossible because they have different DEGREES . See also HOMEOMORPHISM ,H OMOTOPY ,H OMOTOPY CLASS,H OMOTOPY GROUP ,H OMOTOPY TYPE,TOPO- LOGICAL SPACEHomotopy A continuous transformation from one FUNCTION to another. A homotopy between two functions f and g from a SPACE X to a SPACE Y is a continuous MAP G from X /C29[0; 1] /C2Y such that G(x; 0) /C30f(x) and G(x; 1) /C30g(x) ; where /C29 denotes set pairing. Another way of saying this is that a homotopy is a path in the mapping SPACE Map( X ; Y) from the first FUNCTION to the second. See also H-COBORDISM References Krantz, S. G. "The Concept of Homotopy" §10.3.2 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, pp. 132 /C1/33, 1999. Homotopy Axiom One of the EILENBERG- STEENROD AXIOMS which states that, if f :(X ; A) 0 (Y ; B)is HOMOTOPIC to g : (X ; A) 0 (Y ; B) ; then their INDUCED MAPS f/C31 : Hn(X ; A) 0 Hn(Y ; B) and g/C31 : Hn(X ; A) 0 Hn(Y ; B) are the same. Homotopy Class Given two TOPOLOGICAL SPACES M and N, place an equivalence relationship on the CONTINUOUS MAPS f : M 0 N using homotopies, and write f1 /C2f2if f1is HOMOTOPIC to f2 : Roughly speaking, two maps are HOMOTOPIC if one can be deformed into the other. This equivalence relation is transitive because these homotopy deformations can be composed (i.e., one can follow the other). A simple example is the case of CONTINUOUS MAPS from one CIRCLE to another circle. Consider the number of ways an infinitely stretchable string can be tied around a tree trunk. The string forms the first circle, and the tree trunk’s surface forms the second circle. For any integer n, the string can be wrapped around the tree n times, for positive n clockwise, and negative n counterclockwise. Each integer n corre- sponds to a homotopy class of maps from S1 to S1 :/ After the string is wrapped around the tree n times, it could be deformed a little bit to get another CONTIN- UOUS MAP, but it would still be in the same homotopy class, since it is HOMOTOPIC to the original map. Conversely, any map wrapped around n times can be deformed to any other. See also HOMOTOPY ,H OMOTOPY GROUP ,TOPOLOGI- CAL SPACE Homotopy Group The homotopy groups generalize the FUNDAMENTAL GROUP to maps from higher dimensional spheres, instead of from the circle. The nth homotopy group of aTOPOLOGICAL SPACE Xis the set of HOMOTOPY CLASSES of maps from the H YPERSPHERE toX, with a GROUP structure, and is denoted pn(X) : The FUNDA- MENTAL GROUP is p1(X) ; and, as in the case of p1 ; the maps Sn 0 X must pass through a BASEPOINT p /C23 X : For n /C211, the homotopy group pn(X)isanA BELIAN GROUP . The group operations are not as simple as those for the FUNDAMENTAL GROUP . Consider two maps a : Sn 0 X and b : Sn 0 X ; which pass through p /C23 X : The product a+b : Sn 0 X is given by mapping the equator to the BASEPOINT p. Then the northern hemisphere is mapped to the sphere by collapsing the equator to a point, and then it is mapped to X by a. The southern hemisphere is similarly mapped to X by b. The diagram above shows the product of two spheres. The identity element is represented by the constant map e(x) /C30p : The choice of direction of a loop in the fundamental group corresponds to a ORIENTATION of Sn in a homotopy group. Hence the inverse of a map a is given by switching orientation for the sphere. By describing the sphere in n /C271 coordinates, switching the first and second coordinate changes the orienta- tion of the sphere. Or as a HYPERSURFACE , Sn ƒRn/C271 ; switching orientation reverses the roles of inside and outside. The above diagram shows that a +/C28a is homotopic to the constant map, i.e., the identity. It begins by expanding the equator in a +/C28a ; and then the resulting map is contracted to the BASEPOINT . As with the FUNDAMENTAL GROUP , the homotopy groups do not depend on the choice of BASEPOINT . But the higher homotopy groups are always ABELIAN . The above diagram shows an example of a+b /C30b +a: The BASEPOINT is fixed, and because n /C211 the map can be rotated. When n /C301, i.e., the FUNDAMENTALGROUP , it is impossible to rotate the map while keeping the BASEPOINT fixed. A space with pi /C300 for all i 5n is called n-connected. If X is n /C281/-connected, n /C211, then the HUREWICZ HOMOMORPHISM pn(X) 0 Hn(X) from the nth-homo- topy group to the nth-homology group is an ISO- MORPHISM . When f : X 0 Y is a CONTINUOUS MAP, then f/C31 : pn(X) 0 pn(Y) is defined by taking the images under f of the spheres in X. The pushforward is natural, i.e., (f(g)/C31/C30f /C31(g /C31 whenever the composition of two maps is defined. In fact, given a FIBRATION , F 0 E 0 B where B is PATH-CONNECTED , there is a LONG EXACT SEQUENCE of homotopy groups ... 0 pn(F) 0 pn(E) 0 pn(B) 0 pn/C281(F) 0 ... 0 p0(B) /C300: See also ABELIAN GROUP ,C OHOMOTOPY GROUP , FREUDENTHAL SUSPENSION THEOREM ,FUNDAMENTAL GROUP ,H OMOTOPY EXCISIO N,H UREWICZ HOMO- MORPHISM ,H YPERSPHERE ,GROUP ,RELATIVE HOMO- TOPY GROUP ,W EAK EQUIVALENCE References Dodson, C. T. J. and Parker, P. E. "Homotopy Groups" and "Tables of Homotopy Groups." §2.4 and Appendix D in A User’s Guide to Algebraic Topology. Dordrecht, Nether- lands: Kluwer, pp. 44 /C1/5 and 365 /C1/80, 1997. Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, pp. 324 /C1/25, 1995. Homotopy Theory The branch of ALGEBRAIC TOPOLOGY which deals with HOMOTOPY GROUPS . Homotopy methods can be used to solve systems of polynomials by embedding the polynomials in a family of systems that define the deformation of the original problem into a simpler one whose solutions are known. See also ALGEBRAIC TOPOLOGY ,HOMOTOPY GROUP References Aubry, M. Homotopy Theory and Models. Boston, MA: Birkha ¨user, 1995. Honaker’s Constant PALINDROMIC PRIME Honeycomb ATESSELLATION inn-D, for n]3:The only regular honeycomb in 3-D is f4;3;4g;which consists of eight cubes meeting at each VERTEX . The only quasiregular honeycomb (with regular cells and semiregular VER- TEX FIGURES ) has each VERTEX surrounded by eight TETRAHEDRA and six OCTAHEDRA and is denoted 3 3 ; 4no :/ Ball and Coxeter (1987) use the term "sponge" for a solid which can be parameterized by INTEGERS p, q, and n which satisfy the equation 2 sinp p ! sinp q ! /C30cosp n ! : The possible sponges are fp ; qng/C30f6; 63g;j j f6; 44g;j f4; 64g;j f3; 66g;j and f4; 4 /C12g: j / There are many semiregular honeycombs, such as 3; 3 4P+vP+u ; in which each VERTEX consists of two OCTAHE- DRA f3; 4g and four CUBOCTAHEDRA3 4P+vP+u :/ See also HONEYCOMB CONJECTURE ,MENGER SPONGE , SIERPINSKI SPONGE ,TESSELLATION ,TETRIX ,TILING References Ball, W. W. R. and Coxeter, H. S. M. "Regular Sponges." In Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 152 /C1/53, 1987. Bulatov, V. "Infinite Regular Polyhedra." http://www.phy- sics.orst.edu/~bulatov/polyhedra/infinite/. Coxeter, H. S. M. "Regular Honeycombs in Hyperbolic Space." Proc. International Congress of Math., Vol. 3. Amsterdam, Netherlands: pp. 155 /C1/69, 1954. Coxeter, H. S. M. "Space Filled with Cubes," "Other Honey- combs," and "Polytopes and Honeycombs." §4.6, 4.7, and 7.4 in Regular Polytopes, 3rd ed. New York: Dover, pp. 68 /C1/2 and 126 /C1/28, 1973. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, p. 79, 1997. Gott, J. R. III "Pseudopolyhedrons." Amer. Math. Monthly 73, 497 /C1/04, 1967. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 104 /C1/06, 1991. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Honeycomb Conjecture Any partition of the plane into regions of equal area has PERIMETER as least that of the regular hexagonal honeycomb TILING . Pappus refers to the problem in his fifth book, but the conjecture was finally proven by Hales (1999). See also PERIMETER ,TESSELLATION ,TILING References Hales, T. C. The Honeycomb Conjecture. 8 Jun 1999. http:// xxx.lanl.gov/abs/math.MG/9906042/. Kepler, J. "L’e´trenne ou la neige sexangulaire." C.N.R.S., 1975. Mackenzie, D. "Proving the Perfection of the Honeycomb." Science 285, 1338 /C1/339, 1999.Thompson, D’A. W. On Growth and Form, 2nd ed., compl. rev. ed. New York: Cambridge University Press, 1992. Weyl, H. Symmetry. Princeton, NJ: Princeton University Press, 1952. Hoof CYLINDRICAL WEDGE Hook One of the 12 6-POLYIAMONDS . See also POLYIAMOND References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Hook Length Formula A FORMULA for the number of YOUNG TABLEAUX associated with a given YOUNG DIAGRAM . In each box, write the sum of one plus the number of boxes horizontally to the right and vertically below the box (the "hook length"). The number of tableaux is then n! divided by the product of all "hook lengths". The NumberOfTableaux in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ) function in Mathematica im- plements the hook length formula. See also YOUNG DIAGRAM ,YOUNG TABLEAU References Jones, V. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335/C1/88, 1987. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Hopf Algebra Let a graded module Ahave a multiplication fand a co-multiplication c:Then if fandchave the unity of kas unity and c:(A;f)0(A;f)/C156(A;f)i sa n algebra homomorphism, then ( A;f;c) is called a Hopf algebra. Hopf Bifurcation The BIFURCATION of a FIXED POINT to a LIMIT CYCLE (Tabor 1989). References Casti, J. L. "The Hopf Bifurcation Theorem." Ch. 2 in Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics. New York: Wiley, pp. 35 /C1/9, 2000. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. New York: Springer-Verlag, pp. 150 /C1/54, 1997. Marsden, J. and McCracken, M. Hopf Bifurcation and Its Applications. New York: Springer-Verlag, 1976. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 197, 1989. Hopf Circle HOPF MAP Hopf Fibration HOPF MAP Hopf Link The LINK 02 /C1/2 /C1/1 which has JONES POLYNOMIAL V(t) /C30/C28t /C28t/C281 and HOMFLY POLYNOMIAL P(z ; a) /C30z/C281( a/C281 /C28 a/C283) /C27za/C281 : It has BRAID WORD s2 1 :/ Hopf Map The first example discovered of a MAP from a higher- dimensional SPHERE to a lower-dimensional SPHERE which is not null- HOMOTOPIC . Its discovery was a shock to the mathematical community, since it was believed at the time that all such maps were null- HOMOTOPIC , by analogy with HOMOLOGY GROUPS . The Hopf map f : S3 0 S2 arises in many contexts, and can be generalized to a map S7 0 S4 : For any point p in the sphere, its PREIMAGE f /C281(p) is a circle S1 in S3 : There are several descriptions of the Hopf map, also called the Hopf fibration. As a SUBMANIFOLD of R4 ; the 3-SPHERE is S3 /C30f(X1 ; X2 ; X3 ; X4):X2 1 /C27X2 2 /C27X2 3 /C27X2 4 /C301g (1) and the 2-SPHERE is a SUBMANIFOLD of R3 ; S2 /C30f(x1 ; x2 ; x3):x2 1 /C27x22 /C27x23 /C301g: (2) The Hopf map takes points (/X1 ; X2 ; X3 ; X4)ona3- sphere to points on a 2-sphere (/x1 ; x2 ; x3) x1 /C302(X1X2 /C27X3X4) (3) x2 /C302(X1X4 /C28X2X3) (4) x3 /C30(X2 1 /C27X2 3 ) /C28(X2 2 /C27X2 4 ) : (5) Every point on the 2-SPHERE corresponds to a CIRCLE called the HOPF CIRCLE on the 3-SPHERE . By STEREOGRAPHIC PROJECTION , the 3-sphere can be mapped to R3 ; where the point at infinity corresponds to the north pole. As a map, from R3 ; the Hopf map can be pretty complicated. The diagram above shows some of the preimages f /C281(p); called HOPF CIRCLES . The straight red line is the circle through infinity. By associating R4with C2 ; the map is given by f(z ; w) /C30z =w ; which gives the map to the RIEMANN SPHERE . The Hopf fibration is a FIBRATION S1 0 S3 0 S2 ; (6) and is in fact a PRINCIPAL BUNDLE . The ASSOCIATED VECTOR BUNDLE L /C30S3 /C29C=U(1) ; (7) where ((z; w); v) /C2((eitz ; eitw) ; eitv) (8) is a complex LINE BUNDLE onS2:In fact, the set of line bundles on the sphere forms a group under TENSOR PRODUCT , and the bundle Lgenerates all of them. That is, every line bundle on the sphere is L/C156kfor some k. The sphere S3is the L IE GROUP of unit QUATERNIONS , and can be identified with the SPECIAL UNITARY GROUP SU(2);which is the SIMPLY CONNECTED double cover of SO(3):The Hopf bundle is the quotient map S2$SU(2)=U(1):/ See also FIBRATION ,FIBER BUNDLE ,H OMOGENEOUS SPACE ,PRINCIPAL BUNDLE ,STEREOGRAPHIC PROJEC- TION ,VECTOR BUNDLE References Berger, M. Chs. 4 and 18 in Geometry I. New York: Springer-Verlag, 1987. Kreminski, R. "Visualizing the Hopf Fibration." Mathema- tica Educ. Res. 6,9/C1/4, 1997. Penrose, R. and Rindler, W. Spinors and Space-Time, Vol. 1: Two-Spinor Calculus and Relativistic Fields. Cambridge, England: Cambridge University Press, 1987. Ryder, L. H. Quantum Field Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1996. Whitehead, G. W. Elements of Homotopy Theory. New York: Springer Verlag, 1979. Hopf Trace Theorem Let K be a finite complex, and let f : CP(K) 0 CP(K) be a chain map, then X P(/C281)PTr( f; CP(K)) /C30X P(/C281)PTr( f/C31; HP(K) =TP(K)) : References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, p. 122, 1993. Hopf’s Theorem A NECESSARY and SUFFICIENT condition for a MEA- SURE which is quasi-invariant under a transforma- tion to be equivalent to an invariant PROBABILITY MEASURE is that the transformation cannot (in a measure theoretic sense) compress the SPACE . Horizontal Oriented in position PERPENDICULAR to up-down, and therefore PARALLEL to a flat surface. See also VERTICAL Horizontal Cusp SPINODE Horizontal Cylinder CYLINDRICAL SEGMENT Horizontal Tank CYLINDRICAL SEGMENT Horizontally Convex Polyomino ROW-CONVEX POLYOMINO Horizontal-Vertical Illusion VERTICAL- HORIZONTAL ILLUSION Horn Angle The configuration formed by two curves starting at a point, called the vertex V, in a common direction. Horn angles are concrete illustrations of NON- ARCHI- MEDEAN GEOMETRIES . See also NON-ARCHIMEDEAN GEOMETRY References Kasner, E. "The Recent Theory of the Horn Angle." Scripta Math 11, 263 /C1/67, 1945.Horn Cyclide The INVERSION of a HORN TORUS . If the INVERSION CENTER lies on the TORUS , then the horn cyclide degenerates to a PARABOLIC HORN CYCLIDE . See also CYCLIDE ,H ORN TORUS ,INVERSION ,PARA- BOLIC CYCLIDE ,R ING CYCLIDE ,SPINDLE CYCLIDE , TORUS Horn Function The 34 distinct convergent hypergeometric series of order two enumerated by Horn (1931) and correctedby Bornga ¨sser (1933). There are 14 complete series for which p/C30p?/C30q/C30q?/C302; F 1(a;b;b?;g;x;y)/C30X m;n(a)m/C27n(b)m(b?)n (g)m/C27nm!n!xmyn(1) F2(a;b;b?;g;g?;x;y)/C30X m;n(a)m/C27n(b)m(b?)n (g)m(g?)nm!n!xmyn(2) F3(a;a?;b;b?;g;x;y) /C30X m;n(a)m(a?)n(b)m(b?)n (g)m/C27nm!n!xmyn(3) F4(a;b;g;g?;x;y)/C30X m;n(a)m/C27n(b)m/C27n (g)m(g?)nm!n!xmyn(4) G1(a;b;b?;x;y)/C30X m;n(a)m/C27n(b)n/C28m(b?)m/C28n m!n!xmyn(5) G2(a;a?;b;b?;x;y) /C30X m;n(a)m(a?)n(b)n/C28m(b?)m/C28n m!n!xmyn(6) G3(a;a?;x;y)/C30X m;n(a)2n/C28m(a?)2m/C28n m!n!xmyn(7) H1(a;b;g;d;x;y)/C30X m;n(a)m/C28n(b)m/C27n(g)n (d)mm!n!xmyn(8) H2(a;b;g;d;e;x;y) /C30X m;n(a)m/C28n(b)m(g)n(d)n (e)mm!n!xmyn(9) H3( a; b; g ; x; y) /C30X m; n(a)2m/C27n(b)n ( g)m/C27nm!n!xmyn (10) H4( a; b; g ; d ; x; y) /C30X m; n(a)2m/C27n(b)n (g)m(d)nm!n!xmyn(11) H5( a; b; g ; x; y) /C30X m; n( a)2m/C27n(b)n/C28m ( g)nm!n!xmyn(12) H6( a; b; g ; x; y) /C30X m; n(a)2m/C28n(b)n/C28m(g)n m!n!xmyn(13) H7( a; b; g ; d ; x; y) /C30X m; n(a)2m/C28n(b)n(g)n ( g)mm!n!xmyn(14) (of which F1 ; F2 ; F3 ; and F4are precisely APPELL HYPERGEOMETRIC FUNCTIONS ), and 20 confluent ser- ies with p 5p?/C302; q 5q?/C302; and p, q not both 2, F1( a; b; g ; x; y) /C30X m; n( a)m/C27n( b)n ( g)m/C27nm!n!xmyn (15) F2(b; b?; g ; x; y) /C30X m; n( b)m( b?)m ( g)m/C27nm!n!xmyn (16) F3(b; g ; x; y) /C30X m; n( b)m (g)m/C27nm!n!xmyn (17) C1( a; b; g ; g ?; x; y) /C30X m; n(a)m/C27n(b)m ( g)m( g?)nm!n!xmyn(18) C2(a; g ; g ?; x; y) /C30X m; n( a)m/C27n ( g)m( g ?)nm!n!xmyn (19) J1( a; a?; b; g ; x; y) /C30X m; n(a)m( a?)n( b)m (g)m/C27nm!n!xmyn(20) J2( a; b; g ; x ; y) /C30X m; n( a)m( b)n ( g)m/C27nm!n!xmyn (21) G1( a; b; b?; x; y) /C30X m; n( a)m( b)n /C28m( b?)m/C28n m!n!xmyn(22) G2( b; b?; x; y) /C30X m; n( b)n/C28m( b?)m/C28n m!n!xmyn (23) H1( a; b; d; x; y) /C30X m; n( a)m/C28n(b)m/C27n ( d)mm!n!xmyn(24) H2( a; b; g ; d; x; y) /C30X m; n( a)m/C28n( b)m( g)n (d)mm!n!xmyn(25) H3( a; b; d; x; y) /C30X m; n( a)m/C28n(b)m ( d)mm!n!xmyn (26)H4( a; g ; d; x; y) /C30X m; n( a)m/C28n( g)n ( d)mm!n!xmyn (27) H5(a; d; x; y) /C30X m; n(a)m/C28n ( d)mm!n!xmyn (28) H6( a; g ; x; y) /C30X m; n( a)2m/C27n ( g)m/C27nm!n!xmyn (29) H7(a; g ; d; x ; y) /C30X m;n(a)2m/C27n (g)m(d)nm!n!xmyn(30) H8(a;b;x;y)/C30X m;n(a)2m/C28n(b)n/C28m m!n!xmyn(31) H9(a;b;d;x;y)/C30X m;n(a)2m/C28n(b)n (d)mm!n!xmyn(32) H10(a;d;x;y)/C30X m;n(a)2m/C28n (d)mm!n!xmyn(33) H11(a;b;g;d;x;y)/C30X m;n(a)m/C28n(b)n(g)n (d)mm!n!xmyn(34) (Erde ´lyiet al. 1981, pp. 224 /C1/26). See also APPELL HYPERGEOMETRIC FUNCTION ,KAMPE ´ DE FE´ RIET FUNCTION ,LAURICELLA FUNCTIONS References Bornga ¨sser, L. U¨ber hypergeometrische Funktionen zweier Vera¨nderlichen. Dissertation. Darmstadt, Germany: Uni- versity of Darmstadt, 1933. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Horn’s List" and "Convergence of the Series." §5.7.1 and 5.7.2 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 224 /C1/29, 1981. Horn, J. "Hypergeometrische Funktionen zweier Vera ¨nder- lichen." Math. Ann. 105, 381/C1/07, 1931. Horn Torus One of the three STANDARD TORI given by the PARA- METRIC EQUATIONS x /C30(c /C27a cos v)cos u (1) y /C30(c /C27a cos v)sin u (2) z /C30a sin v (3) with a /C30c. The INVERSION of a horn torus is a HORN CYCLIDE (or PARABOLIC HORN CYCLIDE ). The above figures show a horn torus (left), a cutaway (middle), and a CROSS SECTION of the horn torus through the xz-plane (right). See also CYCLIDE ,H ORN CYCLIDE ,R ING TORUS , SPINDLE TORUS ,STANDARD TORI,TORUS References Gray, A. "Tori." §13.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 304 /C1/06, 1997. Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28 /C1/0, 1986. Horn’s Theorem This entry contributed by FRED MANBY Let X /C30fx1 ]x2 ]/C1/C1/C1]xn ½xi /C23Rg (1) and Y /C30fy1 ]y2 ]/C1/C1/C1]yn ½yi /C23Rg: (2) Then there exists an n /C29n HERMITIAN MATRIX with eigenvalues X and diagonal elements Y IFF Xt i /C301(xi /C28yi) ]0 /C2141 5t 5n (3) and with equality for t/C30n. The theorem is sometimes also known as Schur’s theorem. See also HERMITIAN MATRIX ,M AJORIZATION ,STO- CHASTIC MATRIX References Horn, A. "Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix." Amer. J. Math. 76, 620/C1/30, 1954. Lieb, E. H "Variational Principle for Many-Fermion Sys- tems." Phys. Rev. Lett. 46, 457/C1/59, 1981. Horned Sphere ALEXANDER’S HORNED SPHERE ,ANTOINE’S HORNED SPHERE Horner’s Method A method for finding roots of a polynomial equation f(x)/C300:Now find an equation whose roots are the roots of this equation diminished by r,s o0/C30f(x/C27r) /C30f(r)/C27xf?(r)/C271 2x2fƒ(r)/C2713x3f§(r)/C27...: (1) The expressions for f(r);f?(r);... are then found as in the following example, where f(x)/C13Ax5/C27Bx4/C27Cx3/C27Dx2/C27Ex/C27F: (2) Write the coefficients A,B, ...,Fin a horizontal row, and let a new letter shown as a denominator stand for the sum immediately above it so, in the followingexample, P/C30Ar/C27B:The result is the following table. AB C D E F /Ar P//Pr Q//Qr R//Rr S//Sr v/ /Ar T//Tr U//Ur R//Vr x/ /Ar W//Wr X//Xr c/ /Ar Y//Yr f/ /Ar u/ Solving for the quantities u;f;c;x;andvgives u/C305Ar/C27B/C301 4!f(iv)(r) (3) f/C3010Ar2/C274Br/C27C/C301 3!f§(r) (4) c/C3010Ar3/C276Br2/C273Cr/C27D/C301 2!fƒ(r) (5) x/C305Ar4/C274Br3/C273Cr2/C272Dr/C27E/C30f?(r) (6) v/C30Ar5/C27Br4/C27Cr3/C27Dr2/C27Er/C27F/C30f(r); (7) so the equation whose roots are the roots of f(x)/C300; each diminished by r,i s 0/C30Ax5/C27ux4/C27fx3/C27cx2/C27xx/C27v (8) (Whittaker and Robinson 1967). To apply the procedure, first determine the integer part of the root through whatever means are needed, then reduce the equation by this amount. This gives the second digit, by which the equation is once againreduced (after suitable multiplication by 10) to find the third digit, and so on. 1 /C284 0 5(1 /C2810 /C28500 2000(3 1 /C283/C283 /C283/C283 23 /C287/C2821 /C28521/C281563 437 1 /C282/C282 /C2853 /C284/C2812 /C28523 1 /C2813 /C281 To see the method applied, consider the problem of finding the smallest positive root of x3 /C284x2 /C275 /C300: (9) This root lies between 1 and 2, so diminish the equation by 1, resulting in the left table shown above. The resulting diminished equation is x3 /C28x2 /C285x /C272 /C300 ; (10) and roots which are ten times the roots of this equation satisfy the equation x3 /C2810x2 /C28500x /C272000 /C300 : (11) The root of this equation between 1 and 10 lies between 3 and 4, so reducing the equation by 3 produces the right table shown above, giving the transformed equation x3 /C28x2 /C28533 /C27437 /C300: (12) This procedure can be continued to yield the root as approximately 1.3819659. Horner’s process really boils down to the construction of a DIVIDED DIFFERENCE table (Whittaker and Ro- binson 1967). See also DIVIDED DIFFERENCE ,NEWTON’S METHOD References Boyer, C. B. and Merzbacher, U. C. A History of Mathe- matics, 2nd ed. New York: Wiley, pp. 202 /C1/04, 256, and 307, 1991. Horner, W. G. Philos. Trans. 1, 308, 1819. Pena, J. M. and Sauer, T. SIAM J. Numer. Anal. 37, 1186, 2000. Ruffini, P. Sopra la determinazione della radici. Modena, Italy, 1804. Ruffini, P. Memorie di Mat. e di Fis. della Soc. Italiana delle Scienze. Verona, Italy, 1813. Se´roul, R. "Evaluation of Polynomials: Horner’s Method." §10.6 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 216 /C1/62, 2000. Whittaker, E. T. and Robinson, G. "The Ruffini-Horner Method." §53 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 100 /C1/06, 1967. Horner’s Rule A rule for POLYNOMIAL computation which both reduces the number of necessary multiplications and results in less numerical instability due to potential subtraction of one large number from another. The rule simply factors out POWERS of x,giving anxn /C27an/C281xn /C281 /C27.../C27a0 /C30((anx /C27an/C281)x /C27...)x /C27a0 : Horner’s rule can be implemented to form a POLY- NOMIAL from a list of coefficients in Mathematica as follows. PolynomialFromCoefs[l_List, x_] : /C30 Fold[x#1 /C27 #2 &, 0, l] See also POLYNOMIAL References Borwein, P. and Erde ´lyi, T. "Horner’s Rule." §1.1.E.5 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 8, 1995. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, pp. 467 /C1/69, 1998. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 9, 1991. Horocycle The LOCUS of a point which is derived from a fixed point Qby continuous parallel displacement. References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 300, 1969. Horse Fetter HIPPOPEDE Horseshoe Map SMALE HORSESHOE MAP Horton Graph A graph on 93 nodes providing a counterexample to Tutte’s conjecture that every 3-regular 3-connected bipartite graph is HAMILTONIAN . Two smaller coun- terexamples, each on 78 nodes, are now known (Ellingham 1981, 1982; Ellingham and Horton 1983; Owens 1983). See also HAMILTONIAN GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 61 and 242, 1976. Ellingham, M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs." Research Report No. 28, Dept. of Math., Univ. Melbourne, Melbourne, 1981. Ellingham, M. N. "Constructing Certain Cubic Graphs." In Combinatorial Mathematics, IX: Proceedings of the Ninth Australian Conference held at the University of Queens- land, Brisbane, August 24 /C1/8, 1981) (Ed. E. J. Billington, S. Oates-Williams, and A. P. Street). Berlin: Springer- Verlag, pp. 252 /C1/74, 1982. Ellingham, M. N. and Horton, J. D. "Non-Hamiltonian 3- Connected Cubic Bipartite Graphs." J. Combin. Th. Ser. B 34, 350 /C1/53, 1983. Owens, P. J. "Bipartite Cubic Graphs and a Shortness Exponent." Disc. Math. 44, 327 /C1/30, 1983. Hotelling T2 Distribution A univariate distribution proportional to the F- DISTRIBUTION . If the vector d is Gaussian multi- variate-distributed with zero mean and unit covar- iance matrix X m; n( a)2m/C28n( b)n ( d)mm!n!xmyn and H11( a; b; g ; d; x; y)isan X m; n(a)m/C28n( b)n( g)n ( d)mm!n!xmyn matrix with a WISHART DISTRIBUTION with unit scale matrix and m degrees of freedom X /C30fx1 > x2 ]/C1/C1/C1] xn ½xi /C23R g; then /Y /C30fy1 ]y2 ]/C1/C1/C1]yn jyi /C23R g/ has the Hotelling at i/C281(xi /C28yi) ]0 /C2141 5t 5n distribution with parameters p and m, denoted t /C30n: This distribution is commonly used to describe the sample Mahalanobis distance between two populations, and is implemented as HotellingTSquareDistribu- tion [p, m] in the Mathematica add-on package Statistics‘MultinomialDistribution‘ (which can be loaded with the command BBStatistics‘ ), where p is the dimensionality parameter and m is the number of degrees of free- dom. See also F-DISTRIBUTION ,W ISHART DISTRIBUTION References NIST/SEMATECH. "Hotelling T Squared." §6.5.4.3 in NIST/Sematech Engineering Statistics Internet Hand- book. http://www.itl.nist.gov/div898/handbook/pmc/sec- tion5/pmc543.htm.Hotelling T-Squared Distribution A univariate distribution proportional to the F- DISTRIBUTION . If the vector d is Gaussian multi- variate-distributed with zero mean and unit covar- iance matrix Np(0; I) and M is an m /C29p matrix with a WISHART DISTRIBUTION with unit scale matrix and m degrees of freedom Wp(I;m); then mdTM /C281d has the Hotelling T2 distribution with parameters p and m, denoted T2(p; m) : This distribution is commonly used to describe the sample Mahalanobis distance between two populations, and is implemented as Hotel- lingTSquareDistribution [p, m] in the Mathema- tica add-on package Statistics‘MultinomialDistribution‘ (which can be loaded with the command BBStatistics‘ ), where p is the dimensionality parameter and m is the number of degrees of free- dom. See also F-DISTRIBUTION ,H OTELLING’S T-SQUARED TEST,W ISHART DISTRIBUTION References NIST/SEMATECH. "Hotelling T Squared." §6.5.4.3 in NIST/Sematech Engineering Statistics Internet Hand- book. http://www.itl.nist.gov/div898/handbook/pmc/sec- tion5/pmc543.htm. Hotelling’s T-Squared Test See also HOTELLING T-SQUARED DISTRIBUTION References Winer, B. J. Statistical Principles in Experimental Design. New York: McGraw-Hill, 1962. Hough Transform A technique used to detect boundaries in digital images. Householder’s Method A ROOT -finding algorithm based on the iteration formula xn/C271 /C30xn /C28f(xn) f ?(xn)1 /C27f(xn)f ƒ(xn) 2[f ?(xn)]2() : This method, like NEWTON’S METHOD , has poor con- vergence properties near any point where the DERI- VATIVE f?(x)/C300:/ See also HALLEY’S IRRATIONAL FORMULA ,H ALLEY’S METHOD ,NEWTON’S METHOD References Gourdon, X. and Sebah, P. "Newton’s Iteration." http:// xavier.gourdon.free.fr/Constants/Algorithms/new- ton.html. Householder, A. S. The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, 1970. Ortega, J. M. and Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, PA: SIAM, 2000. Howe’s Theorem Let P be a PRIMITIVE POLYTOPE with eight vertices. Then there is a unimodular map that maps P to the polyhedron whose vertices are (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1), (1, a, b), (1, c, d), and (1, a /C27c; b /C27d) with a; b; c ; d /C23Z; a ; b ; c ; d ]0 ; and ad /C28bc /C30 1: Furthermore, any primitive polyhedron with fewer than eight vertices can be embedded in one with eight vertices. See also PRIMITIVE POLYTOPE References Khan, M. R. "A Counting Formula for Primitive Tetrahedra in Z3 :/" Amer. Math. Monthly 106, 525 /C1/33, 1999. Scarf, H. E. "Integral Polyhedra in Three Space." Math. Oper. Res. 10, 403 /C1/38, 1985. Howell Design Let S be a set of n /C271 symbols, then a Howell design H(s ; 2n) on symbol set S is an s /C29s array H such that 1. Every cell of H is either empty or contains an unordered pair of symbols from S, 2. Every symbol of S occurs once in each row and column of H, and 3. Every unordered pair of symbols occurs in at most one cell of H. References Colbourn, C. J. and Dinitz, J. H. (Eds.). "Howell Designs." Ch. 26 in CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, pp. 381 /C1/85, 1996. H-Spread The difference H2 /C28H1 ; where H1 and H2 are HINGES . It is the same as the INTERQUARTILE RANGE for N /C30 5, 9, 13, ... points. See also HINGE ,INTERQUARTILE RANGE ,STEP References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 44, 1977. h-Statistic An unbiased estimator for a MOMENT of a distribu- tion. See also K-STATISTICH-Transform A 2-D generalization of the HAAR TRANSFORM which is used for the compression of astronomical images. The algorithm consists of dividing the 2N /C292N image into blocks of 2 /C292 pixels, calling the pixels in the block a00 ; a10 ; a01 ; and a11 : For each block, compute the four coefficients h0 /C131 2(a11 /C27a10 /C27a01 /C27a00) hx /C1312(a11 /C27a10 /C28a01 /C28a00) hy /C131 2(a11 /C28a10 /C27a01 /C28a00) hc /C1312(a11 /C28a10 /C28a01 /C27a00) : Construct a 2N /C281 /C292N /C281 image from the h0values, and repeat until only one h0value remains. The H- transform can be performed in place and requires about 16N2 =3 additions for an N /C29N image. See also HAAR TRANSFORM References Capaccioli, M.; Held, E. V.; Lorenz, H.; Richter, G. M.; and Ziener, R. "Application of an Adaptive Filtering Technique to Surface Photometry of Galaxies. I. The Method Tested on NGC 3379." Astron. Nachr. 309,69/C1/0, 1988. Fritze, K.; Lange, M.; Mo¨stle, G.; Oleak, H.; and Richter, G. M. "A Scanning Microphotometer with an On-Line Data Reduction for Large Field Schmidt Plates." Astron. Nachr. 298, 189 /C1/96, 1977. Richter, G. M. "The Evaluation of Astronomical Photo- graphs with the Automatic Area Photometer." Astron. Nachr. 299, 283 /C1/03, 1978. White, R. L.; Postman, M.; and Lattanzi, M. G. "Compres- sion of the Guide Star Digitised Schmidt Plates." In Digitised Optical Sky Surveys: Proceedings of the Con- ference on "Digitised Optical Sky Surveys" held in Edin- burgh, Scotland, 18 /C1/1 June 1991 (Ed. H. T. MacGillivray and E. B. Thompson). Dordrecht, Netherlands: Kluwer, pp. 167 /C1/75, 1992. Hub The central point in a WHEEL GRAPH Wn : The hub has DEGREE n/C281:/ See also WHEEL GRAPH References Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 148, 1986. Huffman Coding A lossless data compression algorithm which uses a small number of bits to encode common characters.Huffman coding approximates the probability for each character as a POWER of 1/2 to avoid complica- tions associated with using a nonintegral number of bits to encode characters using their actual probabil- ities. Huffman coding works on a list of weights fwi g by building an EXTENDED BINARY TREE with minimum weighted PATH LENGTH and proceeds by finding the two smallest ws, w1 and w2 ; viewed as external nodes, and replacing them with an internal node of weight w1 /C27w2 : The procedure is them repeated stepwise until the root node is reached. An individual external node can then encoded by a binary string of 0s (for left branches) and 1s (for right branches). The procedure is summarized below for the weights 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and 41 given by the first 13 primes, and the resulting tree is shown above (Knuth 1997, pp. 402 /C1/03). As is clear from the diagram, the paths to the larger weights are shorter than those to the smaller weights. In this example, the number 13 would be encoded as 1010. 2 3 5 7 11 13 17 19 23 29 31 37 41 5 5 71113171923293137 41 10 71113171923293137 41 17 11 13 17 19 23 29 31 37 41 17 24 17 19 23 29 31 37 41 24 34 19 23 29 31 37 41 24 34 42 29 31 37 41 34 42 53 31 37 41 42 53 65 37 41 42 53 65 78 95 65 78 95 143 238 The following Mathematica code can be used to construct the list of internal nodes and table of iterations. HuffmanStep[l0_List] : /C30 Module[ {l /C30 l0, s2 /C30 Take[Select[Sort[l0], Positive], 2] }, l[[Take[Flatten[Position[l, #] & /@ s2], 2]]] /C30 0; l[[Last[Position[l, 0]]]] /C30 Plus @@ s2; {l, s2} ] HuffmanList[l_List] : /C30 Module[{}, Plus @@@ Last /@ NestWhileList[HuffmanStep[First[#]] &, HuffmanStep[l], Length[Union[First[#]]] /C21 2&] ] HuffmanTable[l_List] : /C30 NestWhileList[First[HuffmanStep[#]] &, l, Length[Union[#]] /C21 2&] References Huffman, D. A. "A Method for the Construction of Mini- mum-Redundancy Codes." Proc. Inst. Radio Eng. 40, 1098 /C1/101, 1952. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, pp. 402 /C1/06, 1997. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Huffman Coding and Compression of Data." Ch. 20.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 896 /C1/01, 1992. Schwarz, E. S. "An Optimum Encoding with Minimum Longest Code and Total Number of Digits." Information and Control 7,37/C1/4, 1964. Hull AFFINE HULL,CONVEX HULL Hull Number Let a set of vertices Ain a CONNECTED GRAPH Gbe called convex if for every two vertices x;y/C23A;the vertex set of every ( x, y )GRAPH GEODESIC lies completely in A. Also define the convex hull A⁄ V(G)o fa GRAPH Gwith vertex set V(G) as the smallest CONVEX SET inGcontaining A. Then the smallest cardinality of a set Awhose convex hull is V(G) is called the hull number of G, denoted h(G):/ See also GEODETIC NUMBER References Chartrand, G. and Zhang, P. "On the Hull Number of a Graph." To appear in Ars. Combin. Chartrand, G. and Zhang, P. "The Forcing Hull Number of a Graph." To appear in J. Combin. Math. Comb. Comput. Chartrand, G. and Zhang, P. "The Geodetic Number of an Oriented Graph." Europ. J. Combin. 21, 181/C1/89, 2000. Everett, M. G. and Seidman, S. B. "The Hull Number of a Graph." Discr. Math. 57, 217/C1/23, 1985. Mulder, H. M. "The Expansion Procedure for Graphs." In Contemporary Methods in Graph Theory (Ed. R. Boden- diek). Mannheim, Germany: Wissenschaftsverlag, pp. 459 /C1/77, 1990. Humbert’s Theorem The NECESSARY and SUFFICIENT condition that an ALGEBRAIC CURVE has an algebraic INVOLUTE is that the ARC LENGTH is a two-valued algebraic function of the coordinates of the extremities. Furthermore, this function is a ROOT of a QUADRATIC EQUATION whose COEFFICIENTS are rational functions of x and y. See also ALGEBRAIC CURVE ,INVOLUTE References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 195, 1959. Hundkurve TRACTRIX Hundred /100 /C30102 : Madachy (1979) gives a number of alge- braic equations using the digits 1 to 9 which evaluate to 100, such as (7 /C285)2 /C2796 /C278 /C284 /C283 /C281 /C30100 32 /C2791 /C277 /C278 /C286 /C285 /C284 /C30100ffiffiffi 9p /C286 /C2772 /C28(1)(3!) /C288 /C2745 /C30100 123 /C2845 /C2867 /C2789 /C30100; and so on. See also 10,B ILLION ,H UNDRED ,L ARGE NUMBER , MILLION ,THOUSAND References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 156 /C1/59, 1979. Hunt’s Surface A SEXTIC SURFACE given by the implicit equation 4(x2 /C27y2 /C27z2 /C2813)3 /C2727(3x2 /C27y2 /C284z2 /C2812)2 /C300: References Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html.Nordstrand, T. "Hunt’s Surface." http://www.uib.no/people/ nfytn/hunttxt.htm. Huntington Axiom An axiom proposed by Huntington (1933) as part of his definition of a BOOLEAN ALGEBRA , H(x; y) /C13!(!x /C150y) /C150!(!x /C150!y) /C30x; (1) where !x denotes NOT and x /C150y denotes OR. Taken together, the three axioms consisting of (1), commu- tativity x /C150y /C30y /C150x (2) and associativity (x /C150y) /C150z /C30x /C150(y /C150z) ; (3) are equivalent to the axioms of BOOLEAN ALGEBRA . The Huntington operator can be defined in Mathe- matica by Huntington : /C30 Function[{x, y}, ! (! x \[Or] y) \[Or] ! (! x \[Or] ! y)] That the Huntington axiom is a true statement in BOOLEAN ALGEBRA can be verified by examining its TRUTH TABLE . xy /H(x; y)/ TTT TFTFTFFFF See also B OOLEAN ALGEBRA ,R OBBINS ALGEBRA , ROBBINS AXIOM ,W INKLER CONDITIONS ,W OLFRAM AXIOM References Huntington, E. V. "New Sets of Independent Postulates for the Algebra of Logic, with Special Reference to Whitehead and Russell’s Principia Mathematica. " Trans. Amer. Math. Soc. 35, 274 /C1/04, 1933. Huntington, E. V. "Boolean Algebra. A Correction." Trans. Amer. Math. Soc. 35, 557 /C1/58, 1933. Huntington Equation An equation proposed by Huntington (1933) as part of his definition of a B OOLEAN ALGEBRA , f:(X;A)0(Y;B) See also ROBBINS ALGEBRA ,ROBBINS EQUATIONH References Huntington, E. V. "New Sets of Independent Postulates for the Algebra of Logic, with Special Reference to Whitehead and Russell’s Principia Mathematica. " Trans. Amer. Math. Soc. 35, 274 /C1/04, 1933. Huntington, E. V. "Boolean Algebra. A Correction." Trans. Amer. Math. Soc. 35, 557 /C1/58, 1933. Hurwitz Equation The DIOPHANTINE EQUATION x2 1 /C27x22 /C27.../C27x2n /C30ax1x2 /C1/C1/C1xn which has no INTEGER solutions for a/C21n. See also LAGRANGE NUMBER (DIOPHANTINE EQUA- TION ) References Guy, R. K. "Markoff Numbers." §D12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 166 /C1/68, 1994. Hurwitz Number A number with a CONTINUED FRACTION whose terms are the values of one or more POLYNOMIALS evaluated on consecutive INTEGERS and then interleaved. This property is preserved by M O¨BIUS TRANSFORMATIONS (Gosper 1972, p. 44). References Gosper, R. W. Item 101b in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 39 /C1/4, Feb. 1972. Hurwitz Polynomial APOLYNOMIAL with REAL POSITIVE COEFFICIENTS and ROOTS which are either NEGATIVE or pairwise con- jugate with NEGATIVE REAL PARTS . Hurwitz Zeta Function A generalization of the R IEMANN ZETA FUNCTION with aFORMULA z(s;a)/C13X/C12 k/C3001 (k/C27a)s; (1) where any term with k/C27a/C300 is excluded. The Hurwitz zeta function can also be given by the functional equation zs;p q ! /C302G(1/C28s) /C2(2pq)s/C281Xq n/C301sinps 2/C272pnp q ! z1/C28s;n q ! (2) (Apostol 1976, Miller and Adamchik), or the integralz(s;a)/C301 2a/C28s/C27a1/C28s s/C281 /C272g/C12 0(a2/C27y2)/C28s=2sinstan/C281y a !"#() dy e2xy/C281: (3) IfRzjjB0 and 0 Ba51;then z(z;a)/C302G(1/C28z) (2p)1/C28z /C2sinpz 2 !X/C12 n/C301cos(2 pan) n1/C28z/C27cospz 2 !X/C12 n/C301sin(2pan) n1/C28z"# (4) (Hurwitz 1882; Whittaker and Watson 1990, pp. 268 /C1/ 69). The Hurwitz zeta function satisfies z(0;a)/C3012/C28a (5) d dsz(0;a)/C30ln[G(a)]/C281 2ln(2p) (6) d dsz(0;0)/C301 2ln(2p); (7) where G(z) is the GAMMA FUNCTION . In the limit, lim s01z(s;a)/C281 s/C281/C30G?(a) G(a)(8) (Whittaker and Watson 1990, p. 271; Allouche 1992). The POLYGAMMA FUNCTION cm(z) can be expressed in terms of the Hurwitz zeta function by cm(z)/C30(/C281)m/C271m!z(1/C27m;z): (9) For POSITIVE INTEGERS k,p, and q/C21p, z?/C282k/C271;p qP+’kP+’7 /C30[c(2k)/C28ln(2pq)]B2k(p=q) 2k/C28[c(2k)/C28ln(2p)]B2k q2k2k /C27(/C281)k/C271p (2pq)2kXq/C281 n/C301sin2ppn q ! c(2k/C281)n q ! /C27(/C281)k/C2712(2k/C281)! (2pq)2kXq/C281 n/C301cos2ppn q ! z?2k;n q ! /C27z?(/C282k/C271) q2k; (10) where Bnis a B ERNOULLI NUMBER ,Bn(x)aB ERNOULLI POLYNOMIAL ,cn(z)i sa POLYGAMMA FUNCTION , and z(z)i saR IEMANN ZETA FUNCTION (Miller and Adam- chik). Miller and Adamchik also give the closed-form expressions z ?(/C282k /C271;1 2) /C30/C28B2k ln 2 4kk/C28(22k /C281) z?(/C282k /C27 1) 22k /C281 (11) z?/C282k /C271;1=3 2=3P+’vP+’u /C30/C14(9k /C28 1)B2k pffiffiffi 3p (32k/C281 /C28 1)8k /C28B2k ln 3 (32k /C281)4k(12) z?/C282k /C271;1=4 3=4P+’vP+’u /C30/C14(4k /C27 1)B2k p 4k/C271k/C27(4k /C281 /C28 1)B2k ln 2 23k/C281k (13) z?/C282k /C271;1=6 5=6P+’vP+’u /C30/C14(9k /C28 1)(22k /C281 /C27 1)B2k pffiffiffi 3p (62k /C281)8k /C27B2k(32k /C281 /C28 1)ln 2 (62k /C281)4k/C27B2k(22k/C281 /C28 1)ln 3 (62k /C281)4k /C14(/C281)k(22k /C281 /C27 1)c2k /C281(1 3) 2ffiffiffi 3p (12p)2k /C281 (14) In these equations, z?(z0 ; a) means dz(z; a)=dz ½z/C30z0; z?(z0) means dz(z) =dz½z/C30z0; and the upper and lower fractions on the left side of the equations correspond to the plus and minus signs, respectively, on the right side. Gauss gave G?(p=q) G(p =q)/C30/C28g /C28ln(2q) /C281 2 p cotpp q ! /C272X 0 BnBq =2cos2 ppn q ! ln sinpn q !"# (15) (Allouche 1992, Knuth 1997, p. 94). See also HURWITZ’S FORMULA ,K HINTCHINE’S CON- STANT ,POLYGAMMA FUNCTION ,PSI FUNCTION ,RIE- MANN ZETA FUNCTION ,ZETA FUNCTION References Adamchik, V. "A Class of Logarithmic Integrals." In Proc. ISSAC’97, Maui, Hawaii (Ed. W. W. Kuechlin). New York: ACM, 1997. Adamchik, V. S. and Srivastava, H. M. "Some Series of the Zeta and Related Functions." Analysis 18, 131 /C1/44, 1998. Allouche, J.-P. "Series and Infinite Products related to Binary Expansions of Integers." 1992. http://algo.inria.fr/ seminars/sem92 /C1/3/allouche.ps. Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995. Berndt, B. C. "On the Hurwitz Zeta-Function." Rocky Mountain J. Math. 2, 151 /C1/57, 1972. Cvijovic, D. and Klinowski, J. "Values of the Legendre Chi and Hurwitz Zeta Functions at Rational Arguments." Math. Comput. 68, 1623 /C1/630, 1999.Elizalde, E.; Odintsov, A. D.; and Romeo, A. Zeta Regular- ization Techniques with Applications. River Edge, NJ: World Scientific, 1994. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Generalized Zeta Function." §1.10 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 24 /C1/7, 1981. Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen fu¨r Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995. Hurwitz. Z. Math. Phys. 27, 95, 1882. Knopfmacher, J. "Generalised Euler Constants." Proc. Edinburgh Math. Soc. 21,25/C1/2, 1978. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. New York: Springer-Verlag, 1966. Miller, J. and Adamchik, V. "Derivatives of the Hurwitz Zeta Function for Rational Arguments." J. Comput. Appl. Math. 100, 201 /C1/06, 1999. http://members.wri.com/victor/ articles/hurwitz.html. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Zeta Function z(s; x) ; Bernoulli Poly- nomials Bn(x); Euler Polynomials En(x) ; and Polyloga- rithms Lin(x) :/" §1.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 23 /C1/4, 1990. Spanier, J. and Oldham, K. B. "The Hurwitz Function z(n; u) :/" Ch. 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 653 /C1/64, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 268 /C1/69, 1950. Hurwitz’s Formula z(1 /C28s ; a) /C30G(s) (2 p)s [e /C28pis=2F(a ; s) /C27e pis=2F(/C28a ; s)]; where z(z; a)isaH URWITZ ZETA FUNCTION , G(z) is the GAMMA FUNCTION , and F(a;s) is the PERIODIC ZETA FUNCTION . See also GAMMA FUNCTION ,H URWITZ ZETA FUNC- TION ,PERIODIC ZETA FUNCTION References Apostol, T. M. Theorem 12.6 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995. Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 71, 1997. Hurwitz’s Irrational Number Theorem As Lagrange showed, any IRRATIONAL NUMBER ahas an infinity of rational approximations p=qwhich satisfy a/C28p qP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2B 1ffiffiffi 5p q2: (1) Furthermore, if there are no integers a;b;c;dwith ad/C28bc jj /C301 and a/C30aa/C27b da/C27c(corresponding to values of a associated with the GOLDEN RATIO f through their CONTINUED FRACTIONS ), then a/C28p qP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2B 1ffiffiffi 8p q2 ; (2) and if values of a associated with the SILVER RATIO 1 /C27ffiffiffi 2p are also excluded, then a/C28p qP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2B 5ffiffiffiffiffiffiffiffi 221p1 q2 : (3) In general, even tighter bounds OF THE FORM a/C28p qP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2B 1 Lnq2(4) can be obtained for the best rational approximation possible for an arbitrary irrational number a; where the Ln are called LAGRANGE NUMBERS and get steadily larger for each "bad" set of irrational numbers which is excluded. See also CONTINUED FRACTION ,IRRATIONALITY MEA- SURE , Hurwitz’s Root Theorem Let ff(x)g be a SEQUENCE of ANALYTIC FUNCTIONS REGULAR in a region G, and let this sequence be UNIFORMLY CONVERGENT in every CLOSED SUBSET of G. If the ANALYTIC FUNCTION lim n 0/C12fn(x) /C30f(x) does not vanish identically, then if x /C30 a is a zero of f(x) of order k,aNEIGHBORHOOD x /C28a jjB d of x /C30 a and a number N exist such that if n /C21 N, fn(x) has exactly k zeros in x /C28a jjB d:/ See also ARGUMENT PRINCIPLE ,ROOT References Krantz, S. G. "Hurwitz’s Theorem." §5.3.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 76, 1999. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 22, 1975. Hurwitz-Radon Theorem Determined the possible values of r and n for which there is an IDENTITY OF THE FORM (x2 1 /C27.../C27x2r )(y21 /C27.../C27y2r ) /C30z21 /C27.../C27z2n : Hutton’s Formula The MACHIN-LIKE FORMULA 1 4 p /C302 tan/C28113P+’kP+’7 /C27tan /C28117P+’kP+’7 : The other two-term MACHIN-LIKE FORMULAS areEULER’S MACHIN-LIKE FORMULA ,H ERMANN’S FOR- MULA , and MACHIN’S FORMULA . Hutton’s Method LAMBERT’S METHOD Hyperasymptotic Series See also ASYMPTOTIC SERIES ,S UPERASYMPTOTIC SERIES References Boyd, J. P. "The Devil’s Invention: Asymptotic, Superasymp- totic and Hyperasymptotic Series." Acta Appl. Math. 56, 1/C1/8, 1999. Hyperbola A hyperbola is a CONIC SECTION defined as the LOCUS of all points Pin the PLANE the difference of whose distances r1/C30F1Pandr2/C30F2Pfrom two fixed points (the FOCI F1and F2) separated by a distance 2 cis a given POSITIVE constant k, r2/C28r1/C30k (1) (Hilbert and Cohn-Vossen 1999, p. 3). Letting Pfall on the left x-intercept requires that k/C30(c/C27a)/C28(c/C28a)/C302a; (2) so the constant is given by k/C302a;i.e., twice the distance between the x-intercepts (left figure above). The hyperbola has the important property that a ray originating at a FOCUS F1reflects in such a way that the outgoing path lies along the line from the other FOCUS through the point of intersection (right figure above). The special case of the RECTANGULAR HYPERBOLA , corresponding to a hyperbola with eccentricity e/C30ffiffiffi 2p ; was first studied by Menaechmus. Euclid and Aris- taeus wrote about the general hyperbola, but onlystudied one branch of it. The hyperbola was given itspresent name by Apollonius, who was the first to study both branches. The FOCUS and DIRECTRIX were considered by Pappus (MacTutor Archive). The hy- perbola is the shape of an orbit of a body on an escape trajectory (i.e., a body with positive energy), such as some comets, about a fixed mass, such as the sun. The hyperbola can be constructed by connecting the free end Xof a rigid bar F1X;where F1is a FOCUS , and the other FOCUS F2with a string F2PX:As the bar AXis rotated about F1andPis kept taut against the bar (i.e., lies on the bar), the LOCUS ofPis one branch of a hyperbola (left figure above; Wells 1991). A theorem of Apollonius states that for a line segmenttangent to the hyperbola at a point Tand intersecting the asymptotes at points Pand Q, then OP/C29OQis constant, and PT/C30QT(right figure above; Wells 1991). Let the point Pon the hyperbola have Cartesian coordinates ( x, y), then the definition of the hyperbola r2/C28r1/C302agives ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C28c)2/C27y2q /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x/C27c) 2/C27y2q /C302a: (3) Rearranging and completing the square gives x2(c2/C28a2)/C28a2y2/C30a2(c2/C28a2); (4) and dividing both sides by a2(c2/C28a2) results in x2 a2/C28y2 c2/C28a2/C301: (5) By analogy with the definition of the ELLIPSE , define b2/C13c2/C28a2; (6) so the equation for a hyperbola with SEMIMAJOR AXIS aparallel to the X-AXIS and SEMIMINOR AXIS b parallel to the Y-AXIS is given byx2 a2/C28y2 b2/C301: (7) or, for a center at the point ( x0;y0) instead of (0 ;0); (x/C28x0)2 a2/C28(y/C28y0)2 b2/C301: (8) Unlike the ELLIPSE , no points of the hyperbola actually lie on the SEMIMINOR AXIS , but rather the ratio b=adetermines the vertical scaling of the hyperbola. The ECCENTRICITY eof the hyperbola (which always satisfies e/C211) is then defined as e/C13c a/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2 a2s : (9) In the standard equation of the hyperbola, the center is located at ( x0;y0);the FOCI are at ( x09c;y0);and the vertices are at ( x09a;y0):The so-called ASYMP- TOTES (shown as the dashed lines in the above figures) can be found by substituting 0 for the 1 onthe right side of the general equation (8), y/C309b a(x/C28x0)/C27y0; (10) and therefore have SLOPES9b=a:/ The special case a/C30b(the left diagram above) is known as a RIGHT HYPERBOLA because the ASYMP- TOTES are PERPENDICULAR . The hyperbola can also be defined as the LOCUS of points whose distance from the FOCUS Fis propor- tional to the horizontal distance from a vertical line L known as the DIRECTRIX , where the ratio is /C211. Letting rbe the ratio and dthe distance from the center at which the directrix lies, then d/C30a2 c(11) r /C30a c: (12) Like noncircular ELLIPSES , hyperbolas have two distinct FOCI and two associated DIRECTRICES , each DIRECTRIX being PERPENDICULAR to the line joining the two foci (Eves 1965, p. 275). The FOCAL PARAMETER of the hyperbola is p /C30b2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2p (13) /C30c2 /C28 a2 c (14) /C30a(e2 /C28 1) e: (15) In POLAR COORDINATES , the equation of a hyperbola centered at the ORIGIN (i.e., with x0 /C30y0 /C300) is r2 /C30a2b2 b2 cos2 u /C28 a2 sin2 u : (16) In POLAR COORDINATES centered at a FOCUS , r /C30a(e2 /C28 1) 1 /C28 e cos u : (17) The two-center BIPOLAR COORDINATES equation with origin at a FOCUS is r1 /C28r2 /C3092a: (18) The PARAMETRIC EQUATIONS for the hyperbola are x /C309a cosh t (19) y /C30b sinh t: (20) The CURVATURE and TANGENTIAL ANGLE are k(t) /C30/C28[cosh(2 t)]/C283 =2 (21) f(t) /C30/C28tan/C281(tanh t) : (22) The LOCUS of the apex of a variable CONE containing an ELLIPSE fixed in 3-space is a hyperbola through the FOCI of the ELLIPSE . In addition, the LOCUS of the apex of a CONE containing that hyperbola is the original ELLIPSE . Furthermore, the ECCENTRICITIES of the ELLIPSE and hyperbola are reciprocals.See also CONIC SECTION ,ELLIPSE ,HYPERBOLA EVO- LUTE ,HYPERBOLA INVERSE CURVE ,HYPERBOLA PED- AL CURVE ,H YPERBOLOID ,JERABEK’S HYPERBOLA , KIEPERT’S HYPERBOLA ,P ARABOLA ,Q UADRATIC CURVE ,R ECTANGULAR HYPERBOLA ,R EFLECTION PROPERTY ,RIGHT HYPERBOLA References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 199 /C1/00 and 218, 1987. Casey, J. "The Hyperbola." Ch. 7 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most RecentExtensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 250 /C1 /84, 1893. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 75 /C1/6, 1996. Coxeter, H. S. M. "Conics" §8.4 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115 /C1/19, 1969. Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, 1965. Fukagawa, H. and Pedoe, D. "The One Hyperbola." §5.2 in Japanese Temple Geometry Problems. Winnipeg, Mani- toba, Canada: Charles Babbage Research Foundation,pp. 51 and 136 /C1 /38, 1989. Gardner, M. "Hyperbolas." Ch. 15 in Penrose Tiles and Trapdoor Ciphers...and the Return of Dr. Matrix, reissueed.New York: W. H. Freeman, pp. 205 /C1 /18, 1989. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 3 /C1/, 1999. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 79 /C1/2, 1972. Lockwood, E. H. "The Hyperbola." Ch. 3 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 24 /C1/3, 1967. MacTutor History of Mathematics Archive. "Hyperbola." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hy-perbola.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 106 /C1 /09, 1991. Yates, R. C. "Conics." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 36 /C1/6, 1952. Hyperbola Evolute The EVOLUTE of a RECTANGULAR HYPERBOLA is the LAME´CURVE (ax)2=3/C28(by)2=3/C30(a/C27b)2=3: From a point between the two branches of the EVOLUTE , two NORMALS can be drawn to the HYPER- BOLA . However, from a point beyond the EVOLUTE , four NORMALS can be drawn. Hyperbola Inverse Curve For a HYPERBOLA with a/C30bwith INVERSION CENTER at the center, the INVERSE CURVE x /C302k cos t a[3 /C28 cos(2 t)] (1) y /C30k sin(2 t) a[3 /C28 cos(2 t)] (2) is a LEMNISCATE . For an INVERSION CENTER at the VERTEX , the INVERSE CURVE x /C30a /C274k cos t sin21 2 tP+’kP+’7 a[5 /C28 4 cos t /C27 cos(2 t) /C28 2 sin(2 t)](3) y /C30a /C27k(tan t /C28 1) a[(sec t /C28 1)2 /C27 (tan t /C28 1)2](4) is a RIGHT STROPHOID . For an INVERSION CENTER at the FOCUS , the INVERSE CURVE x /C30ae /C30k cos t(1 /C28 e cos t) a(cos t /C28 e)2 (5) y /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 /C28 1p k sin(2 t) 2a(cos t /C28 e)2 (6) is a LIMAC ¸ ON, where e is the ECCENTRICITY . For a HYPERBOLA with a /C30ffiffiffi 3p b and INVERSION CEN- TER at the VERTEX , the INVERSE CURVE x /C30b /C272k cos t(ffiffiffi3p /C28 cos t) b[9 /C28 4ffiffiffi3p cos t /C27 cos(2 t) /C28 2 sin(2 t)](7) y /C30b /C27 k(tan t /C28 1) bffiffiffi 3p sec t /C28 1P+$P+’ 2/C27(tan t /C28 1)2hi (8) is a MACLAURIN TRISECTRIX . References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 203, 1972.Hyperbola Pedal Curve The PEDAL CURVE of a HYPERBOLA with the PEDAL POINT at the FOCUS is a CIRCLE (left figure; Hilbert and Cohn-Vossen 1999, p. 26). The PEDAL CURVE of a RECTANGULAR HYPERBOLA with PEDAL POINT at the center is a LEMNISCATE (right figure). See also HYPERBOLA ,PEDAL CURVE References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Hyperbolic Automorphism ANOSOV AUTOMORPHISM Hyperbolic Cosecant The hyperbolic cosecant is defined as csch x/C131 sinh x/C302 e2/C28e/C28x: See also BERNOULLI NUMBER ,BIPOLAR COORDINATES , BIPOLAR CYLINDRICAL COORDINATES ,C OSECANT , HELMHOLTZ DIFFERENTIAL EQUATION– TOROIDAL CO- ORDINATES ,H YPERBOLIC SINE,P OINSOT’S SPIRALS , SURFACE OF REVOLUTION ,TOROIDAL FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83 /C1/6, 1972. Spanier, J. and Oldham, K. B. "The Hyperbolic Secant sech( x) and Cosecant csch( x) Functions." Ch. 29 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 273 /C1/78, 1987. Hyperbolic Cosine The hyperbolic cosine is defined as cosh x /C131 2(ex /C27e /C28x) : The notation ch x is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). This function describes the shape of a hanging cable, known as the CATENARY . See also BIPOLAR COORDINATES ,BIPOLAR CYLINDRI- CAL COORDINATES ,BISPHERICAL COORDINATES ,CA- TENARY ,C ATENOID ,C HI,C ONICAL FUNCTION , CORRELATION COEFFICIENT– GAUSSIAN BIVARIATE DIS- TRIBUTION ,COSINE ,CUBIC EQUATION , DE MOIVRE’S IDENTITY ,ELLIPTIC CYLINDRICAL COORDINATES ,EL- SASSER FUNCTION ,H YPERBOLIC GEOMETRY ,H YPER- BOLIC LEMNISCATE FUNCTION ,H YPERBOLIC SINE, HYPERBOLIC SECANT ,HYPERBOLIC TANGENT ,INVER- SIVE DISTANCE ,LAPLACE’S EQUATION– BIPOLAR COOR- DINATES ,L APLACE’S EQUATION– BISPHERICAL COORDINATES ,LAPLACE’S EQUATION– TOROIDAL COOR- DINATES ,LEMNISCATE FUNCTION ,LORENTZ GROUP , MATHIEU DIFFERENTIAL EQUATION ,MEHLER’S BESSEL FUNCTION FORMULA ,M ERCATOR PROJECTION ,M OD- IFIED BESSEL FUNCTION OF THE FIRST KIND,OBLATE SPHEROIDAL COORDINATES ,P ROLATE SPHEROIDAL COORDINATES ,P SEUDOSPHERE ,R AMANUJAN COS/ COSH IDENTITY ,SINE-GORDON EQUATION ,SURFACE OF REVOLUTION ,TOROIDAL COORDINATESReferences Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83 /C1/6, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Spanier, J. and Oldham, K. B. "The Hyperbolic Sine sinh( x) and Cosine cosh( x) Functions." Ch. 28 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 263 /C1/71, 1987. Hyperbolic Cosine Integral CHI Hyperbolic Cotangent The hyperbolic cotangent is defined as coth x /C13ex /C27 e /C28x ex /C28 e /C28x /C30e2x /C27 1 e2x /C28 1 : The notation cth x is sometimes also used (Gradsh- teyn and Ryzhik 2000, p. xxix). The L AURENT SERIES of coth xis given by coth x/C301 x/C271 3x/C281 45x3/C27...: See also BERNOULLI NUMBER ,BIPOLAR COORDINATES , BIPOLAR CYLINDRICAL COORDINATES ,C OTANGENT , HYPERBOLIC TANGENT ,LAPLACE’S EQUATION– TOROI- DAL COORDINATES ,LEBESGUE CONSTANTS (FOURIER SERIES ), PROLATE SPHEROIDAL COORDINATES ,SUR- FACE OF REVOLUTION ,TOROIDAL COORDINATES ,TOR- OIDAL FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83 /C1/6, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Spanier, J. and Oldham, K. B. "The Hyperbolic Tangent tanh( x) and Cotangent coth( x) Functions." Ch. 30 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 279 /C1/84, 1987. Hyperbolic Cube A hyperbolic version of the Euclidean CUBE . See also HYPERBOLIC DODECAHEDRON ,H YPERBOLIC ICOSAHEDRON ,H YPERBOLIC OCTAHEDRON ,H YPER- BOLIC TETRAHEDRON References Rivin, I. "Hyperbolic Polyhedron Graphics." http:// www.mathsource.com/cgi-bin/msitem22?0201 /C1/88. Trott, M. "The Cover Image: Hyperbolic Platonic Bodies." §8.3.10 in The Mathematica Guidebook, Vol. 2: Graphics in Mathematica. New York: Springer-Verlag, 2000. Hyperbolic Cylinder A QUADRATIC SURFACE given by the equation x2 a2 /C28y2 b2 /C30/C281 :See also ELLIPTIC PARABOLOID ,PARABOLOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 210 /C1/11, 1987. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 12, 1999. Hyperbolic Disk POINCARE ´ HYPERBOLIC DISK Hyperbolic Dodecahedron A hyperbolic version of the Euclidean DODECAHE- DRON . See also HYPERBOLIC CUBE,H YPERBOLIC ICOSAHE- DRON ,H YPERBOLIC OCTAHEDRON ,H YPERBOLIC TET- RAHEDRON References Rivin, I. "Hyperbolic Polyhedron Graphics." http:// www.mathsource.com/cgi-bin/msitem22?0201 /C1/88. Trott, M. "The Cover Image: Hyperbolic Platonic Bodies." §8.3.10 in The Mathematica Guidebook, Vol. 2: Graphics in Mathematica. New York: Springer-Verlag, 2000. Hyperbolic Fixed Point (Differential Equations) A FIXED POINT for which the STABILITY MATRIX has EIGENVALUES l1B0Bl2;also called a SADDLE POINT . See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,S TABLE IMPROPER NODE, STABLE SPIRAL POINT ,S TABLE STAR,U NSTABLE IMPROPER NODE,UNSTABLE NODE,UNSTABLE SPIRAL POINT ,UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Hyperbolic Fixed Point (Map) AFIXED POINT of a LINEAR TRANSFORMATION (MAP) for which the rescaled variables satisfy ( d /C28 a)2 /C274bg > 0 : See also ELLIPTIC FIXED POINT (MAP), LINEAR TRANSFORMATION ,PARABOLIC FIXED POINT Hyperbolic Functions The hyperbolic functions sinh, cosh, tanh, csch, sech, coth (HYPERBOLIC SINE, HYPERBOLIC COSINE , etc.) share many properties with the corresponding CIR- CULAR FUNCTIONS . The hyperbolic functions arise in many problems of mathematics and mathematical physics in which integrals involvingffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27x2p arise (whereas the CIRCULAR FUNCTIONS involveffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p ) :/ For instance, the HYPERBOLIC SINE arises in the gravitational potential of a cylinder and the calcula- tion of the Roche limit. The HYPERBOLIC COSINE function is the shape of a hanging cable (the so-called CATENARY ). The HYPERBOLIC TANGENT arises in the calculation of magnetic moment and rapidity of special relativity. All three appear in the Schwarzs- child metric using external isotropic Kruskal coordi- nates in general relativity. The HYPERBOLIC SECANT arises in the profile of a laminar jet. The HYPERBOLIC COTANGENT arises in the Langevin function for magnetic polarization. The hyperbolic functions are defined by sinh z /C13ez /C28 e /C28z 2/C30/C28sinh(/C28z) (1) cosh z /C13ez /C27 e /C28z 2/C30cosh(/C28z) (2) tanh z /C13ez /C28 e /C28z ez /C27 e /C28z /C30e2z /C28 1 e2z /C27 1 (3) csch z /C132 ez /C28 e /C28z (4) sech z /C132 ez /C27 e/C28z (5) coth z /C13ez /C27 e /C28z ez /C28 e /C28z /C30e2z /C27 1 e2z /C28 1 : (6) For purely IMAGINARY arguments, sinh( iz) /C30i sin z (7) cosh( iz) /C30cos z : (8) The hyperbolic functions satisfy many identities analogous to the trigonometric identities (which can be inferred using OSBORNE’S RULE ) such as cosh2 x /C28sinh2 x /C301 (9) cosh x /C27sinh x /C30ex (10)cosh x/C28sinh x/C30e/C28x: (11) See also Beyer (1987, p. 168). Some HALF-ANGLE FORMULAS are tanhz 2 ! /C30sinh x/C27isiny cosh x/C27cosy(12) cothz 2 ! /C30sinh x/C28isiny cosh x/C28cosy: (13) Some DOUBLE-ANGLE FORMULAS are sinh(2 x)/C302 sinh xcosh x (14) cosh(2 x)/C302 cosh2x/C281/C301/C272 sinh2x (15) Identities for COMPLEX arguments include sinh( x/C27iy)/C30sinh xcosh y/C27icosh xsiny (16) cosh( x/C27iy)/C30cosh xcosy/C27isinh xsiny: (17) The ABSOLUTE SQUARES for COMPLEX arguments are sinh( z) jj2/C30sinh2x/C27sin2y (18) cosh( z) jj2/C30sinh2x/C27cos2y: (19) Integrals involving hyperbolic functions include gdx xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bxp /C30lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bxp /C28ffiffiffiap ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bxp /C27ffiffiffiapP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2(20) /C30ln (ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bxp /C28ffiffiffiap)2 (a/C27bx)/C28aP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2 /C30ln (a/C27bx)/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a(a/C27bx)p /C27a bxP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2: (21) Ifb/C210, then gdx xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bxp /C30ln2a/C27bx/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a(a/C27bx)p bxP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2(22) /C30ln 2a bx/C271 ! /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a bxa bx/C271 !vuutP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2: (23) Letz/C132a=bx/C271;anda=bx/C30(z/C281)=2 and gdx xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bxp /C30lnz/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2(z/C281)12(z/C271)qhi /C30lnz/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (z/C281)(z/C271)phi (24) /C30lnz/C28ffiffiffiffiffiffiffiffiffiffiffiffiffi z2/C281pP+’kP+’7 /C30cosh/C281(z) (25) /C30cosh/C2811/C272a bx ! (26) /C302 tanh /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a /C27 bxs ! : (27) See also DOUBLE- ANGLE FORMULAS ,FIBONACCI HY- PERBOLIC FUNCTIONS ,H ALF-ANGLE FORMULAS ,H Y- PERBOLIC COSECANT ,H YPERBOLIC COSINE , HYPERBOLIC COTANGENT ,GENERALIZED HYPERBOLIC FUNCTIONS ,HYPERBOLIC SECANT ,HYPERBOLIC SINE, HYPERBOLIC TANGENT ,INVERSE HYPERBOLIC FUNC- TIONS ,OSBORNE’S RULE References Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83 /C1/6, 1972. Anderson, J. W. "Trigonometry in the Hyperbolic Plane." §5.7 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 146 /C1/51, 1999. Beyer, W. H. "Hyperbolic Function." CRC Standard Math- ematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 168 /C1/86 and 219, 1987. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 126 /C1/31, 1967. Yates, R. C. "Hyperbolic Functions." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 113 /C1/18, 1952. Hyperbolic Geometry A NON- EUCLIDEAN GEOMETRY , also called LOBA- CHEVSKY- BOLYAI- GAUSS GEOMETRY , having constant SECTIONAL CURVATURE -1. This GEOMETRY satisfies all of EUCLID’S POSTULATES except the PARALLEL POSTU- LATE , which is modified to read: For any infinite straight LINE L and any POINT P not on it, there are many other infinitely extending straight LINES that pass through P and which do not intersect L. In hyperbolic geometry, the sum of ANGLES of a TRIANGLE is less than 1808, and TRIANGLES with the same angles have the same areas. Furthermore, not all TRIANGLES have the same ANGLE sum (cf. the AAA THEOREM for TRIANGLES in Euclidean 2-space). There are no similar triangles in hyperbolic geometry. The best-known example of a hyperbolic space are SPHERES in Lorentzian 4-space. The POINCARE ´ HYPER- BOLIC DISK is a hyperbolic 2-space. Hyperbolic geo- metry is well understood in 2-D, but not in 3-D. Geometric models of hyperbolic geometry include the KLEIN- BELTRAMI MODEL , which consists of an OPEN DISK in the Euclidean plane whose open chords correspond to hyperbolic lines. A 2-D model is the POINCARE ´ HYPERBOLIC DISK. Felix Klein constructed an analytic hyperbolic geometry in 1870 in which a POINT is represented by a pair of REAL NUMBERS (x1 ; x2) with x2 1 /C27x22 B1 (i.e., points of an OPEN DISK in the COMPLEX PLANE )and the distance between two points is given by d(x; X) /C30a cosh /C281 1 /C28 x1X1 /C28 x2X2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2 1 /C28 x22pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28X2 1/C28X2 2p"# : The geometry generated by this formula satisfies all of E UCLID’S POSTULATES except the fifth. The METRIC of this geometry is given by the C AYLEY- KLEIN- HILBERT METRIC , g11/C30a2(1/C28x2 2) (1/C28x2 1/C28x22)2 g12/C30a2x1x2 (1/C28x21/C28x22)2 g22/C30a2(1/C28x2 1) (1/C28x2 1/C28x22)2: Hilbert extended the definition to general bounded sets in a E UCLIDEAN SPACE . See also ELLIPTIC GEOMETRY ,EUCLIDEAN GEOMETRY , HYPERBOLIC METRIC ,KLEIN- BELTRAMI MODEL ,NON- EUCLIDEAN GEOMETRY ,P SEUDOSPHERE ,S CHWARZ- PICK LEMMA References Anderson, J. W. Hyperbolic Geometry. New York: Springer- Verlag, 1999. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 57 /C1/0, 1990. Eppstein, D. "Hyperbolic Geometry." http://www.ics.uci.edu/ ~eppstein/junkyard/hyper.html. Stillwell, J. Sources of Hyperbolic Geometry. Providence, RI: Amer. Math. Soc., 1996. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 109 /C1/10, 1991. Hyperbolic Helicoid The surface with parametric equations x/C30sinh vcos(tu) 1/C27cosh ucosh v(1) y/C30sinh vsin(tu) 1/C27cosh ucosh v(1) z /C30cosh v sinh( u) 1 /C27 cosh u cosh v : (3) where t is a constant (the torsion). See also HELICOID References JavaView. "Classic Surfaces from Differential Geometry: Hyperbolic Helicoid." http://www-sfb288.math.tu-ber- lin.de/vgp/javaview/demo/surface/common/PaSurface_Hy- perbolicHelicoid.html. Hyperbolic Icosahedron A hyperbolic version of the Euclidean ICOSAHEDRON . See also HYPERBOLIC CUBE,HYPERBOLIC DODECAHE- DRON ,HYPERBOLIC OCTAHEDRON ,HYPERBOLIC POLY- HEDRON ,HYPERBOLIC TETRAHEDRON References Trott, M. "The Cover Image: Hyperbolic Platonic Bodies." §8.3.10 in The Mathematica Guidebook, Vol. 2: Graphics in Mathematica. New York: Springer-Verlag, 2000. Hyperbolic Inverse Functions INVERSE HYPERBOLIC FUNCTIONS Hyperbolic Knot A hyperbolic knot is a KNOT that has a complement that can be given a metric of constant curvature -1. All hyperbolic knots are PRIME KNOTS (Hoste et al. 1998). KNOTS which are not hyperbolic are either TORUS KNOTS or SATELLITE KNOTS , as proved by Thurston in 1978. Of the prime knots with 16 or fewer crossings, all but 32 are hyperbolic. Of these 32, 12 are torus knots and the remaining 20 are satellites of the TREFOIL KNOT (Hoste et al. 1998). The nonhyperbolic knots with nine or fewer crossings are all torus knots,including 03 /C1/01 (the (3; 2)/-TORUS KNOT ), 05 /C1/01, 07 /C1/01, 08 /C1/19 (the (4; 3)/-TORUS KNOT ), and 09 /C1/01. The following table gives the number of nonhyper- bolic and hyperbolic knots of n crossing starting with n /C303. type Sloane counts torus A051764 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1 satellite A051765 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10 nonhyperbolic A052407 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 3, 3, 8, 11 hyperbolic A052408 0, 1, 1, 3, 6, 20, 48, 164, 551, 2176, 9985, 46969, 253285, 1388694 Almost all hyperbolic knots can be distinguished by their hyperbolic volumes (exceptions being 05 /C1/02 and a certain 12-crossing knot; see Adams 1994, p. 124). It has been conjectured that the smallest hyperbolic volume is 2.0298..., that of the FIGURE-OF-EIGHT KNOT . MUTANT KNOTS have the same hyperbolic knot volume. The KNOT SYMMETRY group of a hyperbolic knot must be either a finite CYCLIC GROUP or a finite DIHEDRAL GROUP (Riley 1979, Kodama and Sakuma 1992, Hoste et al. 1998). See also MUTANT KNOT,SATELLITE KNOT,TORUS KNOT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 119 /C1/27, 1994. Adams, C.; Hildebrand, M.; and Weeks, J. "Hyperbolic Invariants of Knots and Links." Trans. Amer. Math. Soc. 326,1/C1/6, 1991. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,3 3/C1/8, Fall 1998. Kodama K. and Sakuma, M. "Symmetry Groups of Prime Knots Up to 10 Crossings." In Knot 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka, Japan, 1990 (Ed. A. Kawauchi.) Berlin: de Gruyter, pp. 323 /C1/40, 1992. Riley, R. "An Elliptic Path from Parabolic Representations to Hyperbolic Structures." In Topology of Low-Dimensional Manifolds, Proceedings, Sussex 1977 (Ed. R. Fenn). New York: Springer-Verlag, pp. 99 /C1/33, 1979. Sloane, N. J. A. Sequences A051764, A051765, A052407, A052408 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Knots and Links." M ATHEMATICA NOTE- BOOK KNOTS.M . Hyperbolic Lemniscate Function By analogy with the LEMNISCATE FUNCTIONS , hyper- bolic lemniscate functions can also be defined arcsinhlemn x /C13gx 0(1 /C27t4)1 =2 dt (1) arccoshlemn x /C13g1 0(1 /C27t4)1 =2 dt: (2) Let 0 5 u 5 p=2 and 0 5v 51; and write um 2/C30gv 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 t2p ; (3) where m is the constant obtained by setting u /C30 p=2 and v /C301. Then m /C302 pK1ffiffiffi 2p ! ; (4) where K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND , and Ramanujan showed 2 tan/C281 v /C30 u /C27X/C12 n/C301sin(2 nu) n cosh( np) ; (5) 1 8 p /C2812tan /C281(v2) /C30X/C12 n/C300( /C281)ncos[(2 n /C27 1)u] (2n /C27 1)cosh12(2n /C27 1)phi (6) and ln1 /C27 v 1 /C28 v ! /C30ln tan14 p /C2712 uP+’kP+’7hi /C274X/C12 n/C300( /C281)n sin[(2 n /C27 1)u] (2n /C27 1)[e(2n/C271)p /C28 1](7) (Berndt 1994). See also LEMNISCATE FUNCTION References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 255 /C1/58, 1994. Hyperbolic Map A linear MAP Rn is hyperbolic if none of its EIGENVA- LUES has modulus 1. This means that Rn can be written as a DIRECT SUM of two A-invariant SUB- SPACES Es and Eu (where s stands for stable and u for unstable). This means that there exist constants C /C21 0 and 0 B l B1 such that Anv kk5C ln vkk if v /C23 Es A/C28nv kk5C ln vkk if v /C23 Eu for n /C300, 1, .... See also PESIN THEORYHyperbolic Metric The METRIC for the POINCARE ´ HYPERBOLIC DISK,a model for HYPERBOLIC GEOMETRY . The hyperbolic metric is invariant under conformal maps of the disk onto itself. See also HYPERBOLIC GEOMETRY ,POINCARE ´ HYPER- BOLIC DISK References Bear, H. S. "Part Metric and Hyperbolic Metric." Amer. Math. Monthly 98, 109/C1/23, 1991. Hyperbolic Octahedron A hyperbolic version of the Euclidean OCTAHEDRON , which is a special case of the ASTROIDAL ELLIPSOID with a/C30b/C30c/C301:It is given by the PARAMETRIC EQUATIONS x/C30(cosucosv)3 y/C30(sinucosv)3 z/C30sin3v foru/C23[/C28p=2;p=2] and v/C23[/C28p;p]:/ The FIRST FUNDAMENTAL FORM coefficients are E/C309a6cos2usin2ucos6v (1) F/C309 4a6cos5vsinvsin(4 u) (2) G/C309a6cos2vsin2v[cos2v(cos6u/C27sin6u) /C27sin2v]; (3) the SECOND FUNDAMENTAL FORM coefficients are e/C3024a3cos2usin2ucsc(2 u)cos3vsinvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9/C28cos(4 u)/C28[7/C27cos(4 u)]cos(2 v)p (4) f/C300 (5) g/C3024a3cos2usin2ucsc(2 u)cos3vsinvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9/C28cos(4 u)/C28[7/C27cos(4 u)]cos(2 v)p ; (6) the AREA ELEMENT is dA /C309 8 a6 cos4 v sin v sin(2 u) /C29ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 /C28cos(4 u) /C28[7 /C27cos(4 u)cos(2 v)p ; (7) and the GAUSSIAN CURVATURE is K /C30256 sec4 v 9a6 f[7 /C27 cos(4 u)]cos(2 v) /C27 cos(4 u) /C28 9g2 : (8) The MEAN CURVATURE is given by a complicated expression. See also ASTROIDAL ELLIPSOID ,H YPERBOLIC CUBE, HYPERBOLIC DODECAHEDRON ,HYPERBOLIC ICOSAHE- DRON ,HYPERBOLIC TETRAHEDRON References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 396 /C1/98, 1997. Nordstrand, T. "Astroidal Ellipsoid." http://www.uib.no/peo- ple/nfytn/asttxt.htm. Rivin, I. "Hyperbolic Polyhedron Graphics." http:// www.mathsource.com/cgi-bin/msitem22?0201 /C1/88. Trott, M. "The Cover Image: Hyperbolic Platonic Bodies." §8.3.10 in The Mathematica Guidebook, Vol. 2: Graphics in Mathematica. New York: Springer-Verlag, 2000. Hyperbolic Paraboloid The QUADRATIC and DOUBLY RULED SURFACE given by the Cartesian equation z /C30y2 b2 /C28x2 a2 (1) (left figure). An alternative form is z /C30xy (2) (right figure; Fischer 1986), which has PARAMETRIC EQUATIONS x(u; v) /C30u (3) y(u; v) /C30v (4) z(u; v) /C30uv (5) (Gray 1997, pp. 297 /C1/98). The coefficients of the FIRST FUNDAMENTAL FORM are E /C301 /C27v2 (6)F /C30uv (7) G /C301 /C27u2 ; (8) and the SECOND FUNDAMENTAL FORM coefficients are e /C300 (9) f /C30(1 /C27u2 /C27v2) /C281=2 (10) g /C300; (11) giving SURFACE AREA element dS /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27u2 /C27v2p : (12) The GAUSSIAN CURVATURE is K /C30/C28(1 /C27u2 /C27v2) /C282 (13) and the MEAN CURVATURE is H /C30uv (1 /C27 u2 /C27 v2)3 =2 : (14) Three skew lines always define a one-sheeted HYPER- BOLOID , except in the case where they are all parallel to a single PLANE but not to each other. In this case, they determine a hyperbolic paraboloid (Hilbert and Cohn-Vossen 1999, p. 15). See also DOUBLY RULED SURFACE ,ELLIPTIC PARA- BOLOID ,P ARABOLOID ,R ULED SURFACE ,S ADDLE , SKEW QUADRILATERAL References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 3 /C1/, 1986. Fischer, G. (Ed.). Plates 7 /C1/inMathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 8 /C1/0, 1986. Gray, A. "The Hyperbolic Paraboloid." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nded.Boca Raton, FL: CRC Press, pp. 297 /C1 /98 and 449, 1997. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. JavaView. "Classic Surfaces from Differential Geometry: Hyperbolic Paraboloid." http://www-sfb288.math.tu-ber-lin.de/vgp/javaview/demo/surface/common/PaSurface_Hy-perbolicParaboloid.html. McCrea, W. H. Analytical Geometry of Three Dimensions. Edinburgh: Oliver and Boyd, 1947. Meyer, W. "Spezielle algebraische Fla ¨chen." Encylopa ¨die der Math. Wiss. III ,22B, 1439 /C1 /779. Salmon, G. Analytic Geometry of Three Dimensions. New York: Chelsea, 1979. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 245, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 110 /C1/12, 1991. Hyperbolic Partial Differential Equation APARTIAL DIFFERENTIAL EQUATION of second-order, i.e., one OF THE FORM Auxx /C272Buxy /C27Cuyy /C27Dux /C27Euy /C27F /C300; (1) is called hyperbolic if the MATRIX Z /C13AB BCP+2$P+2’ (2) satisfies det /(Z) B0: The WAVE EQUATION is an exam- ple of a hyperbolic partial differential equation. Initial-boundary conditions are used to give u(x; y; t) /C30g(x; y; t) for x /C23@V; t > 0 (3) u(x; y; 0) /C30v0(x; y)inV (4) ut(x; y; 0) /C30v1(x; y)inV; (5) where uxy /C30f(ux ; ut ; x ; y) (6) holds in V:/ See also ELLIPTIC PARTIAL DIFFERENTIAL EQUATION , PARABOLIC PARTIAL DIFFERENTIAL EQUATION ,PAR- TIAL DIFFERENTIAL EQUATION Hyperbolic Plane In the hyperbolic plane H2 ; a pair of LINES can be PARALLEL (diverging from one another in one direc- tion and intersecting at an IDEAL POINT at infinity in the other), can intersect, or can be HYPERPARALLEL (diverge from each other in both directions). See also EUCLIDEAN PLANE ,RIEMANN SPHERE ,RIGID MOTION References Anderson, J. W. "A Model for the Hyperbolic Plane." §1.1 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 1 /C1/, 1999. Hyperbolic Point A point p on a REGULAR SURFACE M /C23R3 is said to be hyperbolic if the GAUSSIAN CURVATURE K(p) B0or equivalently, the PRINCIPAL CURVATURES k1and k2 ; have opposite signs. See also ANTICLASTIC ,E LLIPTIC POINT ,G AUSSIAN CURVATURE ,H YPERBOLIC FIXED POINT (DIFFEREN- TIAL EQUATIONS ), HYPERBOLIC FIXED POINT (MAP), PARABOLIC POINT ,PLANAR POINT ,SYNCLASTIC References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 375, 1997. Hyperbolic Polyhedron A POLYHEDRON in a HYPERBOLIC GEOMETRY . See also HYPERBOLIC CUBE,HYPERBOLIC DODECAHE- DRON ,HYPERBOLIC ICOSAHEDRON ,HYPERBOLIC OCTA- HEDRON ,HYPERBOLIC TETRAHEDRONReferences Hodgson, C. D. and Riven, I. "A Characterization of Compact Convex Polyhedra in Hyperbolic 3-Space." Invent. Math. 111,77/C1/11, 1993. Kellerhals, R. " Shape and Size Through Hyperbolic Eyes." Math. Intell. 17,21/C1/0, 1995. Kellerhals, R. "Nichteuklidische Geometrie und Volumina hyperbolischer Polyeder." Math. Semesterber. 43, 155 /C1/68, 1996. Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer-Verlag, 1994. Rivin, I. " A Characterization of Ideal Polyhedra in Hyper- bolic 3-Space." Ann. Math. 143,51/C1/0, 1996. Thurston, W. P. and Levy, S. (Eds.). Three-Dimensional Geometry and Topology, Vol. 1. Princeton, NJ: Princeton University Press, 1997. Trott, M. "The Cover Image: Hyperbolic Platonic Bodies." §8.3.10 in The Mathematica Guidebook, Vol. 2: Graphics in Mathematica. New York: Springer-Verlag, 2000. Hyperbolic Rotation Also known as the a Lorentz transformation or Procrustian stretch, a hyperbolic transformation leaves each branch of the HYPERBOLA x?y?/C30xy invar- iant and transforms CIRCLES into ELLIPSES with the same AREA . x?/C30m/C281x y?/C30my: See also CROSSED HYPERBOLIC ROTATION References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 101, 1967. Hyperbolic Secant The hyperbolic secant is defined as sech x /C131 cosh x /C302 ex /C27 e/C28x ; (1) where cosh x is the HYPERBOLIC COSINE . It has a MAXIMUM at x /C300 and inflection points at x /C309sech/C281 1ffiffiffi 2pP+$P+’ :0:881374 :/ Equating coefficients of u0 ; u4 ; and u8 in the RAMA- NUJAN COS/COSH IDENTITY 1 /C272X/C12 n /C301cos(nu) cosh( np)"# /C282 /C27 1 /C272X/C12 n/C301cosh( nu) cosh( np)"# /C282 /C302G43 4P+’kP+’7 p (2) gives the amazing identities X/C12 n/C301sech( pn) /C301 2ffiffiffipp G3 4P+’kP+’7hi2 /C2818 >< >:9 >= >;(3) X/C12 n/C301n4 sech( pn) /C3018 G3 4P+’kP+’7hi2 ffiffiffippX/C12 n/C301n2 sech( pn)"# 2 (4) X/C12 n/C301n8 sech( pn) /C30168[ G(3 4)]2 ffiffiffippX/C12 n/C301n2 sech( pn)"#X/C12 n/C301n6 sech( pn) /C2863000[ G(3 4)]6 p3 =2X/C12 n/C301n2 sech( pn)"# 4 : (5) See also BENSON’S FORMULA ,CATENARY ,CATENOID , EULER NUMBER ,HYPERBOLIC COSINE ,OBLATE SPHER- OIDAL COORDINATES ,PSEUDOSPHERE ,SECANT ,SUR- FACE OF REVOLUTION ,TRACTRIX ,TRACTROID References Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83 /C1/6, 1972. Spanier, J. and Oldham, K. B. "The Hyperbolic Secant sech( x) and Cosecant csch( x) Functions." Ch. 29 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 273 /C1/78, 1987.Hyperbolic Sine The hyperbolic sine is defined as sinh x /C1312(ex /C28e/C28x) : The notation sh x is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). See also BETA EXPONENTIAL FUNCTION ,B IPOLAR COORDINATES ,BIPOLAR CYLINDRICAL COORDINATES , BISPHERICAL COORDINATES ,C ATENARY ,C ATENOID , CONICAL FUNCTION ,CUBIC EQUATION , DE MOIVRE’S IDENTITY ,DIXON- FERRAR FORMULA ,ELLIPTIC CYLIND- RICAL COORDINATES ,E LSASSER FUNCTION ,G UDER- MANNIAN FUNCTION ,H ELICOID ,H ELMHOLTZ DIFFERENTIAL EQUATION– ELLIPTIC CYLINDRICAL CO- ORDINATES ,HYPERBOLIC COSECANT ,LAPLACE’S EQUA- TION– BISPHERICAL COORDINATES ,L APLACE’S EQUATION– TOROIDAL COORDINATES ,LEBESGUE CON- STANTS (FOURIER SERIES ), LORENTZ GROUP ,M ERCA- TOR PROJECTION ,M ILLER CYLINDRICAL PROJECTION , MODIFIED BESSEL FUNCTION OF THE SECOND KIND, MODIFIED SPHERICAL BESSEL FUNCTION ,M ODIFIED STRUVE FUNCTION ,N ICHOLSON’S FORMULA ,OBLATE SPHEROIDAL COORDINATES ,P ARABOLA INVOLUTE , PARTITION FUNCTION P,POINSOT’S SPIRALS ,PROLATE SPHEROIDAL COORDINATES ,RAMANUJAN’S TAU FUNC- TION ,SCHLA ¨ FLI’S FORMULA ,SHI,SINE,SINE-GORDON EQUATION ,SURFACE OF REVOLUTION ,TOROIDAL CO- ORDINATES ,T OROIDAL FUNCTION ,T RACTRIX ,W AT- SON’S FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83 /C1/6, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Spanier, J. and Oldham, K. B. "The Hyperbolic Sine sinh( x) and Cosine cosh( x) Functions." Ch. 28 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 263 /C1/71, 1987. Hyperbolic Sine Integral SHI Hyperbolic Space HYPERBOLIC GEOMETRY Hyperbolic Spiral An ARCHIMEDEAN SPIRAL with POLAR equation r /C30a u : The hyperbolic spiral originated with Pierre Varignon in 1704 and was studied by Johann Bernoulli between 1710 and 1713, as well as by Cotes in 1722 (MacTutor Archive). See also ARCHIMEDEAN SPIRAL ,SPIRAL References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 91, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 and 188, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967. MacTutor History of Mathematics Archive. "Hyperbolic Spiral." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Hyperbolic.html. Hyperbolic Spiral Inverse Curve Taking the pole as the INVERSION CENTER , the HYPERBOLIC SPIRAL inverts to ARCHIMEDES’ SPIRAL r /C30a u:Hyperbolic Spiral Roulette The ROULETTE of the pole of a HYPERBOLIC SPIRAL rolling on a straight line is a TRACTRIX . Hyperbolic Substitution A substitution which can be used to transform integrals involving square roots into a more tractable form. Form Substitution /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27a2p //x /C30a sinh u/ /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28a2p //x/C30acosh u/ See also INTEGRAL ,TRIGONOMETRIC SUBSTITUTION Hyperbolic Tangent By way of analogy with the usual TANGENT tanx/C13sinx cosx; the hyperbolic tangent is defined as tanh x/C13sinh x cosh x/C30ex/C28e/C28x ex/C27e/C28x/C30e2x/C281 e2x/C271; where sinh xis the HYPERBOLIC SINE and cosh xis the HYPERBOLIC COSINE . The notation th xis sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). The hyperbolic tangent can be written using a CONTINUED FRACTION as tanh x /C30x 1 /C27x2 3 /C27x3 5 /C27/C1/C1/C1: See also BERNOULLI NUMBER ,CATENARY ,CORRELA- TION COEFFICIENT– GAUSSIAN BIVARIATE DISTRIBU- TION ,F ISHER’S Z ’-TRANSFORMATION ,H YPERBOLIC COTANGENT ,L ORENTZ GROUP ,M ERCATOR PROJEC- TION ,O BLATE SPHEROIDAL COORDINATES ,PSEUDO- SPHERE ,S URFACE OF REVOLUTION ,T ANGENT , TRACTRIX ,TRACTROID References Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83 /C1/6, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Spanier, J. and Oldham, K. B. "The Hyperbolic Tangent tanh( x) and Cotangent coth( x) Functions." Ch. 30 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 279 /C1/84, 1987. Hyperbolic Tetrahedron A hyperbolic version of the Euclidean TETRAHEDRON . See also HYPERBOLIC CUBE,HYPERBOLIC DODECAHE- DRON ,HYPERBOLIC ICOSAHEDRON ,HYPERBOLIC OCTA- HEDRON ,REULEAUX TETRAHEDRON References Rivin, I. "Hyperbolic Polyhedron Graphics." http:// www.mathsource.com/cgi-bin/msitem22?0201 /C1/88. Trott, M. "The Cover Image: Hyperbolic Platonic Bodies." §8.3.10 in The Mathematica Guidebook, Vol. 2: Graphics in Mathematica. New York: Springer-Verlag, 2000.Hyperbolic Umbilic Catastrophe A CATASTROPHE which can occur for three control factors and two behavior axes. The hyperbolic umbilic is the universal unfolding of the function germ f(x; y) /C30x3 /C27y3 : The CODIMENSION of f is 3, and therefore the universal unfolding F of f has three unfolding parameters. See also CATASTROPHE THEORY ,ELLIPTIC UMBILIC CATASTROPHE References Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. Hyperboloid AQUADRATIC SURFACE which may be one- or two- sheeted. The one-sheeted hyperboloid is a SURFACE OF REVOLUTION obtained by rotating a HYPERBOLA about the perpendicular bisector to the line between the FOCI, while the two-sheeted hyperboloid is a SURFACE OF REVOLUTION obtained by rotating a HYPERBOLA about the line joining the FOCI (Hilbert and Cohn- Vossen 1991, p. 11). The one-sheeted circular hyperboloid is a DOUBLY RULED SURFACE . When oriented along the Z-AXIS , the one-sheeted circular hyperboloid has C ARTESIAN CO- ORDINATES equation x2 a2/C27y2 a2/C28z2 c2/C301; (1) and parametric equation x/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u2p cosv (2) y/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27u 2p sinv (3) z/C30cu (4) forv/C23[0;2p) (left figure). Other parameterizations include x(u;v)/C30a(cosu/C14vsinu) (5) y(u;v)/C30a(sinu9vsinu) (6) z(u;v)/C309cv; (7) (middle figure), or x(u;v)/C30acosh vcosu (8) y(u;v)/C30acosh vsinu (9) z(u;v)/C30csinh v (10) (right figure). A hyperboloid of one sheet is also obtained as the envelope of a CUBE rotated about a space diagonal (Steinhaus 1983, pp. 171 /C1/72). Three skew lines al- ways define a one-sheeted hyperboloid, except in the case where they are all parallel to a single PLANE but not to each other (Hilbert and Cohn-Vossen 1999, p. 15). The VOLUME of a one-sheeted hyperboloid of height h, waist radius a, and top and bottom radii Ris V/C30pha21/C27h2 12b2 ! (11) /C301 3ph(2a2/C27R2); (12) where R2/C30a21/C27h2 4b2 ! (13) (Harris and Stocker 1998). An obvious generalization gives the one-sheeted ELLIPTIC HYPERBOLOID . The hyperboloid of one sheet can be constructed by connecting two concentric vertically offset rings wire tilted wires, as illustrated above (Steinhaus 1983, pp. 242 /C1/43; Hilbert and Cohn-Vossen 1999, p. 11). Surprisingly, when the wires are fastened together so that rotation but not sliding is permitted, the frame- work can be expanded and collapsed as one ring isrotated relative to the other (Hilbert and Cohn-Vossen 1999, pp. 16 /C1 /7 and 29 /C1/1). A two-sheeted circular hyperboloid oriented along the Z-AXIS has C ARTESIAN COORDINATES equation x2 a2/C27y2 a2/C28z2 c2/C30/C281: (14) The PARAMETRIC EQUATIONS are x/C30asinh ucosv (15) y/C30asinh usinv (16) z/C309ccosh u (17) forv/C23[0;2p):Note that the plus and minus signs in z correspond to the upper and lower sheets. The two-sheeted circular hyperboloid oriented along the X- AXIS has Cartesian equation x2 a2 /C28y2 a2 /C28z2 c2 /C301 (18) and PARAMETRIC EQUATIONS x /C309a cosh u cosh v (19) y /C30a sinh u cosh v (20) z /C30c sinh v (21) (Gray 1997, p. 406). The VOLUME of a two-sheeted hyperboloid of half-separation a, height h, and radius R is V /C302ph2b2 a2(a /C271 3 h) (22) /C30 phR2/C28h2b2 3a2 ! ; (23) where R2 /C30hb2 a2(2a /C27h) (24) (Harris and Stocket 1998). Again, an obvious general- ization gives the two-sheeted ELLIPTIC HYPERBOLOID . The SUPPORT FUNCTION of the hyperboloid of one sheet x2 a2 /C27y2 b2 /C28z2 c2 /C301 (25) is h /C30x2 a4 /C27y2 b4 /C27z2 c4 !/C281 =2 ; (26) and the GAUSSIAN CURVATURE is K /C30/C28h4 a2b2c2 : (27) The SUPPORT FUNCTION of the hyperboloid of two sheets x2 a2 /C28y2 b2 /C28z2 c2 /C301 (28) is h /C30x2 a4 /C28y2 b4 /C27z2 c4 !/C281 =2 ; (29) and the GAUSSIAN CURVATURE is K /C30h4 a2b2c2 (30) (Gray 1997, p. 414). See also CATENOID ,CONFOCAL QUADRICS ,D OUBLYRULED SURFACE ,E LLIPSOID ,E LLIPSOIDAL COORDI- NATES ,ELLIPTIC HYPERBOLOID ,HYPERBOLA ,HYPER- BOLOID EMBEDDING ,PARABOLOID ,RULED SURFACE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987. Fischer, G. (Ed.). Plates 67 and 69 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Vo- lume. Braunschweig, Germany: Vieweg, pp. 62 and 64, 1986. Gray, A. "The Hyperboloid of Revolution." §20.5 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, p. 470, 1997. Harris, J. W. and Stocker, H. "Hyperboloid of Revolution." §4.10.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 10 /C1/1, 1999. JavaView. "Classic Surfaces from Differential Geometry: Hyperboloid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Hyperbo- loid.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 112 /C1 /13, 1991. Hyperboloid Embedding A4 - HYPERBOLOID has NEGATIVE CURVATURE , with R2/C30x2/C27y2/C27z2/C28w2(1) 2xdx dw/C272ydy dw/C272zdz dw/C282w/C300: (2) Since r/C13xˆx/C27yˆy/C27zˆz; (3) dw/C30xd x/C27yd y/C27zd z w/C30r /C215drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28R2p : (4) To stay on the surface of the HYPERBOLOID , the LINE ELEMENT is given by ds2/C30dx2/C27dy2/C27dz2/C28dw2 /C30dx2/C27dy2/C27dz2/C28r2dr2 r2/C28R2 /C30dr2/C27r2dV2/C27dr2 1/C28R2 r2: (5) Hypercomplex Number There are at least two definitions of hypercomplex numbers. C LIFFORD ALGEBRAISTS call their higher dimensional numbers hypercomplex, even thoughthey do not share all the properties of complexnumbers and no classical function theory can be constructed over them. According to van der Waerden (1985), a hypercomplex number is a number having properties departing from those of the REAL and COMPLEX NUMBERS . The most common examples are BIQUATERNIONS , EXTER- IOR ALGEBRAS , GROUP algebras, MATRICES , OCTO- NIONS , and QUATERNIONS . One type of hypercomplex number due to Davenport (1996) and sometimes called "the" hypercomplex numbers are defined ac- cording to the multiplication table ij /C30ji /C30k (1) jk /C30kj /C30/C28i (2) ki /C30ik /C30/C28j; (3) and therefore satisfy i2 /C30j2 /C30/C281 (4) k2 /C301 : (5) Unlike QUATERNIONS , multiplication of these hyper- complex numbers is commutative, and unlike real and complex numbers, not all nonzero hypercomplex numbers have a multiplicative inverse. An applica- tion of this sort of hypercomplex number can be found in thejulia_fractal command in POVRay . See also BIQUATERNION ,CAYLEY NUMBER ,CLIFFORD ALGEBRA ,C OMPLEX NUMBER ,E XTERIOR ALGEBRA , GROUP ,M ATRIX ,O CTONION ,Q UATERNION ,R EAL NUMBER ,W EIERSTRASS’S THEOREM References Davenport, C. M. "A Commutative Hypercomplex Algebra with Associated Function Theory." In Clifford Algebras with Numeric and Symbolic Computations (Ed. R. Ab / ½/ amowicz, P. Lounesto, and J. M. Parra). Boston, MA: Birkha ¨user, pp. 213 /C1/27, 1996. Kantor, I. L. and Solodovnikov, A. S. Hypercomplex Num- bers : An Elementary Introduction to Algebras. New York: Springer-Verlag, 1989. van der Waerden, B. L. A History of Algebra from al- Khwarizmi to Emmy Noether. New York: Springer-Verlag, pp. 177 /C1/17, 1985. Hypercube The generalization of a 3-CUBE to n-D, also called a MEASURE POLYTOPE . It is a regular POLYTOPE with mutually PERPENDICULAR sides, and is therefore an ORTHOTOPE . It is denoted gnand has SCHLA ¨ FLI SYMBOL f4 ; 3; 3|ffl{zffl} n /C282g:The number of k-cubes contained in an n-cube can be found from the COEFFICIENTS of (2k /C271)n :/ The 1-hypercube is a LINE SEGMENT , the 2-hypercube is the SQUARE , and the 3-hypercube is the CUBE . The hypercube in R4 ; called a TESSERACT , has the SCHLA ¨ - FLI SYMBOL f4; 3; 3g and VERTICES (91;91;91;91): The above figures show two visualizations of the TESSERACT . The figure on the left is a projection of the TESSERACT in 3-space (Gardner 1977; Williams 1979, p. 26), which also appears on the cover of Born (1926), and the figure on the right is the GRAPH of the TESSERACT symmetrically projected into the PLANE (Coxeter 1973). A TESSERACT has 16 VERTICES ,32 EDGES ,24 SQUARES , and eight CUBES . The dual of the 4-hypercube is the 16-CELL . The above figures show the graphs for the n-hyper- cubes with n/C302 to 7. All hypercubes are H AMILTO- NIAN , and any H AMILTONIAN CIRCUIT of a labeled hypercube defines a G RAY CODE (Skiena 1990, p. 149). See also CROSS POLYTOPE ,CUBE,GLOME ,HAMILTO- NIAN GRAPH ,H YPERCUBE LINE PICKING ,H YPER- SPHERE ,O RTHOTOPE ,P ARALLELEPIPED ,P OLYTOPE , SIMPLEX ,TESSERACT References Born, M. Problems of Atomic Dynamics. Cambridge, MA: MIT Press, 1926. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 123, 1973. Dewdney, A. K. "Computer Recreations: A Program for Rotating Hypercubes Induces Four-Dimensional Demen- tia." Sci. Amer. 254,1 4/C1/3, Mar. 1986. Gardner, M. "Hypercubes." Ch. 4 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scien-tific American. New York: Vintage Books, pp. 41 /C1 /4, 1977. Pappas, T. "How Many Dimensions are There?" The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 204 /C1/05, 1989. Skiena, S. "Hypercubes." §4.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 148 /C1/50, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 113 /C1/14 and 210, 1991. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Hypercube Line Picking N.B. A detailed online essay by S. Finch was the starting point for this entry. Let two points x and y be picked randomly from a unit n-dimensional HYPERCUBE . The expected dis- tance between the points D(N) is then D(N) /C30g1 0/C1/C1/C1g1 0|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 2n[(x1 /C28y1)2 /C27(x2 /C28y2)2 /C27.../C27(xn /C27yn)]1 =2 dx1 /C1/C1/C1dxndy1 ...dyn : (1) This MULTIPLE INTEGRAL has been evaluated analyti- cally only for small values of n. The case D(1) corresponds to the POINT-POINT DISTANCE between two random points in the interval [0; 1]:/ The function D(n) satisfies 1 3 n1 =2 5D(n) 516 nP+’kP+’71=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 31 /C2721/C283 5n !1 =22 435vuuut (2) (Anderssen et al. 1976). The first few numerical and analytic results for D(n) are D(1) /C30 1 3 D(2) /C301 15[ffiffiffi 2p /C272 /C275 ln(1 /C27ffiffiffi2p )] /C300:521405433... D(3) /C301 105[4 /C2717ffiffiffi2p /C286ffiffiffi 3p /C2721 ln(1 /C27ffiffiffi 2p ) /C2742 ln(2 /C27ffiffiffi 3p ) /C287 p] /C300:661707182... D(4) /C300:77766... D(5) /C300:87852... D(6) /C300:96895... D(7) /C301:05159... D(8) /C301:12817... See also C UBE LINE PICKING ,S QUARE TRIANGLE PICKING References Anderssen, R. S.; Brent, R. P.; Daley, D. J.; and Moran, A. P. "Concerning f1 0/C1/C1/C1 f1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 1 /C27.../C27x2 kp dx1/C1/C1/C1 dxkanda Taylor Series Method." SIAM J. Appl. Math. 30,22/C1/0, 1976. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983. Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978. Trott, M. "The Area of a Random Triangle." Mathematica J. 7, 189 /C1/98, 1998. Hypercube Triangulation References Finch, S. "Unsolved Mathematics Problems: Triangulating an n-Dimensional Cube." http://www.mathsoft.com/ asolve/simplex/simplex.html. Hyperdeterminant A technically defined extension of the ordinary DETERMINANT to "higher dimensional" HYPERMA- TRICES . Cayley (1845) originally coined the term, but subsequently used it to refer to an ALGEBRAIC INVARIANT of a multilinear form. The hyperdetermi- nant of the 2 /C292 /C292 HYPERMATRIX A /C30aijk(for i ; j ; k /C300 ; 1) is given by det(A) /C30(a2 000a2111 /C27a2001a2110 /C27a2010a2101 /C27a2011a2100) /C282(a000a001a110a111 /C27a000a010a101a111 /C27a000a011a100a111 /C27a001a010a101a110 /C27a001a011a110a100 /C27a010a011a101a100) /C274(a000a011a101a110 /C27a001a010a100a111) : The above hyperdeterminant vanishes IFF the follow- ing system of equations in six unknowns has a nontrivial solution, a000x0y0 /C27a010x0y1 /C27a100x1y0 /C27a110x1y1 /C300 a001x0y0 /C27a011x0y1 /C27a101x1y0 /C27a111x1y1 /C300 a000x0z0 /C27a001x0z1 /C27a100x1z0 /C27a101x1z1 /C300 a010x0z0 /C27a011x0z1 /C27a110x1z0 /C27a111x1z1 /C300 a000y0z0 /C27a001y0z1 /C27a010y1z0 /C27a011y1z1 /C300 a100y0z0 /C27a101y0z1 /C27a110y1z0 /C27a111y1z1 /C300: Glynn (1998) has found the only known multiplicative hyperdeterminant in dimension larger than two. See also DETERMINANT ,HYPERMATRIX References Cayley, A. "On the Theory of Linear Transformations." Cambridge Math. J. 4, 193/C1/09, 1845. Gel’fand, I. M.; Kapranov, M. M.; and Zelevinsky, A. V. "Hyperdeterminants." Adv. Math. 96, 226/C1/63, 1992. Glynn, D. G. "The Modular Counterparts of Cayley’s Hyper- determinant." Bull. Austral. Math. Soc. 57, 479/C1/97, 1998. Schla¨fli, L. "U ¨ber die Resultante eine Systemes mehrerer algebraischer Gleichungen." Denkschr. Kaiserl. Akad. Wiss., Math.-Naturwiss. Klasse 4, 1852. Hyperedge A connection between two or more vertices of a HYPERGRAPH . A hyperedge connecting just two ver- tices is simply a usual EDGE . See also EDGE (GRAPH ), HYPERGRAPH Hyperellipse yn =m /C27cx aP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2n =m /C28c /C300; with n=m > 2 : If n=m B2; the curve is a HYPOELLIPSE . See also ELLIPSE ,HYPOELLIPSE ,SUPERELLIPSE References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 82, 1993. Hyperelliptic Function ABELIAN FUNCTION Hyperelliptic Integral ABELIAN INTEGRAL Hyperfactorial The function defined by H(n) /C13K(n /C271) /C13112233 /C1/C1/C1nn ; where K(n) is the K-FUNCTION and the first few values for n /C30 1, 2, ... are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (Sloane’s A002109), and these numbers are called hyperfactorials by Sloane and Plouffe (1995). See also BARNES’ G-FUNCTION ,G LAISHER- KINKELIN CONSTANT , K-FUNCTION References Fletcher, A.; Miller, J. C. P.; Rosenhead, L.; and Comrie, L. J. An Index of Mathematical Tables, Vol. 1, 2nd ed. Reading, MA: Addison-Wesley, p. 50, 1962. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 477, 1994. Sloane, N. J. A. Sequences A002109/M3706 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Hyperfinite Set One of the most useful tools in NONSTANDARD ANALY- SIS is the concept of a hyperfinite set. To understand a hyperfinite set, begin with an arbitrary infinite set X whose members are not sets, and form the SUPER- STRUCTURE S(X) over X. Assume that X includes the natural numbers as elements, let N denote the set of natural numbers as elements of X, and let /C31S(X)be an ENLARGEMENT of S(X) : By the TRANSFER PRINCI- PLE, the ordering B on N extends to a strict linearordering on /C31N; which can be denoted with the symbol "/B:/" Since /C31S(X) is an enlargement of S(X); it satisfies the CONCURRENCY PRINCIPLE , so that there is an element n of /C31N such that if n /C23N; then n B n: This follows because the relation Bis a CONCURRENT RELATION on the set of natural numbers. Any member n of /C31N is called an infinite nonstandard natural number, and for any set A /C23/C31S(X) ; if A is in one-to-one correspondence with any element of /C31N; then A is called a hyperfinite set in /C31S(X) : Because there are infinite nonstandard natural numbers in any enlargement /C31S(X)ofS(X); there are hyperfinite sets that are not finite, in any such enlargement. Such hyperfinite sets can be used to study infinite structures satisfying various finiteness conditions. References Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindst- røom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, 1986. Anderson, R. M. "Nonstandard Analysis with Applications to Economics." Ch. 39 in Handbook of Mathematical Economics, Vol. 4 (Ed. W. Hildenbrand and H. Son- nenschein). New York: Elsevier, pp. 2145 /C1/208, 1991. Dauben, J. W. Abraham Robinson: The Creation of Non- standard Analysis, A Personal and Mathematical Odys- sey. Princeton, NJ: Princeton University Press, 1998. Davis, P. J. and Hersch, R. The Mathematical Experience. Boston, MA: Birkha ¨user, 1981. Insall, M. "Nonstandard Methods and Finiteness Conditions in Algebra" Zeitschr. f. Math., Logik, und Grundlagen d. Math. 37, 525 /C1/32, 1991. Keisler, H. J. Elementary Calculus: An Infinitesimal Ap- proach. Boston, MA: PWS, 1986. Lindstrøom, T. "An Invitation to Nonstandard Analysis." In Nonstandard Analysis and Its Applications (Ed. N. Cut- land). New York: Cambridge University Press, 1988. Robinson, A. Non-Standard Analysis. Princeton, NJ: Prin- ceton University Press, 1996. Stewart, I. "Non-Standard Analysis." In From Here to Infinity: A Guide to Today’s Mathematics. Oxford, Eng- land: Oxford University Press, pp. 80 /C1/1, 1996. Hypergame A two-player game in which player 1 chooses any FINITE GAME and player 2 moves first. A PSEUDOPAR- ADOX then arises as to whether the hypergame is itself a FINITE GAME . See also FINITE GAME,GAME Hypergeometric Differential Equation x(x /C281)d2y dx2 /C27[(1 /C27 a /C27 b)x /C28 g]dy dx /C27 aby /C300: It has REGULAR SINGULAR POINTS at 0, 1, and /C12: Every ORDINARY DIFFERENTIAL EQUATION of second- order with at most three REGULAR SINGULAR POINTS can be transformed into the hypergeometric differ- ential equation. See also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATION ,CONFLUENT HYPERGEOMETRIC FUNCTION , GENERALIZED HYPERGEOMETRIC FUNCTION ,H YPER- GEOMETRIC FUNCTION References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: University Press, pp. 1 /C1/, 1935. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 542 /C1/43, 1953. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997. Hypergeometric Distribution Let there be nways for a successful and mways for an unsuccessful trial out of a total of n/C27mpossibi- lities. Take Nsamples and let xiequal 1 if selection i is successful and 0 if it is not. Let xbe the total number of successful selections, x/C13XN i/C301xi: (1) The probability of isuccessful selections is then P(x/C30i)/C30 [# ways for isuccesses][# ways for N/C28iunsuccesses] [total number of ways to select] /C30n iP+’vP+’u m N/C28iP+’vP+’u n/C27m NP+’vP+’u /C30n! i!(n/C28i!)m! (m/C27i/C28N)!(N/C28i)! (n/C27m)! N!(N/C28n/C28m)! /C30n!m!N!(N/C28m/C28n)! i!(n/C28i)!(m/C27i/C28N)!(N/C28i)!(n/C27m)!: (2) The ith selection has an equal likelihood of being in any trial, so the fraction of acceptable selections pis p/C13n n/C27m(3) P(xi/C301)/C30n n/C27m/C13p: (4) The expectation value of xis m/C13/C142x/C143/C30XN i/C301xi*+ /C30XN i/C301/C142xi/C143 /C30XN i/C301n n/C27m/C30nN n/C27m/C30Np: (5) The VARIANCE is var(x)/C13XN i/C301var(xi)/C27XN i/C301XN j/C301 j"1cov(xi;xj): (6)Since xiis a B ERNOULLI variable, var(xi)/C30p(1/C28p)/C30n n/C27m1/C28n n/C27m ! /C30n n/C27m1/C28n n/C27m ! /C30n n/C27mn/C27m/C28n n/C27m ! /C30nm (n/C27m)2; (7) so XN i/C301var(xi)/C30Nnm (n/C27m)2: (8) ForiBj, the COVARIANCE is cov(xi;xj)/C30/C142xixj/C143/C28/C142xi/C143/C142xj/C143: (9) The probability that both iand jare successful for i"jis P(xi/C301;xj/C301)/C30P(xi/C301)P(xj/C301½xi/C301) /C30n n/C27mn/C281 n/C27m/C281 /C30n(n/C281) (n/C27m)(n/C27m/C281): (10) But since xiandxjare random B ERNOULLI variables (each 0 or 1), their product is also a B ERNOULLI variable. In order for xixjto be 1, both xiandxjmust be 1, /C142xixj/C143/C30P(xixj/C301)/C30P(xi/C301;xj/C301) /C30n n/C27mn/C281 n/C27m/C281 /C30n(n/C281) (n/C27m)(n/C27m/C281): (11) Combining (11) with /C142xi/C143/C142xj/C143/C30n n/C27mn n/C27m/C30n2 (n/C27m)2; (12) gives cov(xi;xj)/C30(n/C27m)(n2/C28n)/C28n2(n/C27m/C281) (n/C27m)2(n/C27m/C281) /C30n3/C27mn2/C28n2/C28mn/C28n3/C28n2m/C27n2 (n/C27m)2(n/C27m/C281) /C30/C28mn (n/C27m)2(n/C27m/C281): (13) There are a total of N2terms in a double summation over N. However, i/C30jforNof these, so there are a total of N2/C28N/C30N(N/C281) terms in the COVARIANCE summation XN i/C301Xn j/C301 j"icov(xi;xj)/C30/C28N(N/C281)mn (n/C27m)2(n/C27m/C281):(14) Combining equations (6), (8), (11), and (14) gives the VARIANCE var(x)/C30Nmn (n/C27m)2/C28N(N/C281)mn (n/C27m)2(n/C27m/C281) /C30Nmn (n/C27m)21/C28N/C281 n/C27m/C281 ! /C30Nmn (n/C27m)2N/C27m/C281/C28N/C271 n/C27m/C281 ! /C30Nmn (n/C27m/C28N) (n/C27m)2(n/C27m/C281); (15) so the final result is /C142x/C143/C30Np (16) and, since 1/C28p/C30m n/C27m(17) and np(1/C28p)/C30mn (n/C27m)2; (18) we have s2/C30var(x)/C30Np(1/C28p)1/C28N/C281 n/C27m/C281 ! /C30mnN (m/C27n/C28N) (m/C27n)2(m/C27n/C281): (19) The SKEWNESS is g1/C30q/C28p ffiffiffiffiffiffiffiffiffinpqpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N/C281 N/C28ms N/C282n N/C282 ! /C30(m/C28n)(m/C27n/C282N) m/C27n/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m/C27n/C281 mnN (m/C27n/C28N)s ;(20) and the KURTOSIS is given by the complicated expres- sion g2/C30F(m;n;N) mnN (/C283/C27m/C27n)(/C282/C27m/C27n)(/C28m/C28n/C27N); (21)where F(m;n;N)/C30m3/C28m5/C273m2n/C286m3n/C27m4n/C273mn2 /C2812m2n2/C278m3n2/C27n3/C286mn3/C278m2n3 /C27mn4/C28n5/C286m3N/C276m4N/C2718m2nN /C286m3nN/C2718mn2N/C2824m2n2N/C286n3N /C286mn3N/C276n4N/C276m2N2/C286m3N2 /C2824mnN2/C2712m2nN2/C276n2N2 /C2712mn2N2/C286n3N2: (22) The GENERATING FUNCTION is f(t)/C30m NP+’vP+’u n/C27m NP+’vP+’u2F1(/C28N;/C28n;m/C28N/C271;eit);(23) where2F1(a;b;c;z) is the HYPERGEOMETRIC FUNC- TION . If the hypergeometric distribution is written hn(x;s)/C30np xP+’vP+’u nq s/C28xP+’vP+’u n sP+’vP+’u ; (24) then Xs x/C300hn(x;s)ux/C30A2F1(/C28s;/C28np;nq/C28s/C271;u):(25) References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 532 /C1/33, 1987. Feller, W. "The Hypergeometric Series." §2.6 in An Intro- duction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 41 /C1/5, 1968. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 113 /C1/14, 1992. Hypergeometric Function A GENERALIZED HYPERGEOMETRIC FUNCTION pFq(a1;...;ap;b1;...;bq;x) is a function which can be defined in the form of a HYPERGEOMETRIC SERIES , i.e., a series for which the ratio of successive terms can be written ck/C271 ck/C30P(k) Q(k) /C30(k/C27a1)(k/C27a2)/C1/C1/C1(k/C27ap) (k/C27b1)(k/C27b2)/C1/C1/C1(k/C27bq)(k/C271)x: (1) (The factor of k/C271 in the DENOMINATOR is present for historical reasons of notation.) The function2F1(a;b;c;x) corresponding to p/C302, q/C301 is the first hypergeometric function to be studied (and, in general, arises the most frequently in physical problems), and so is frequently known as"the" hypergeometric equation or, more explicitly, Gauss’s hypergeometric function (Gauss 1812; Barnes 1908). To confuse matters even more, theterm "hypergeometric function" is less commonly used to mean CLOSED FORM , and "hypergeometric series" is sometimes used to mean hypergeometric function. The hypergeometric functions are solutions to the HYPERGEOMETRIC DIFFERENTIAL EQUATION , which has aREGULAR SINGULAR POINT at the ORIGIN . To derive the hypergeometric function based on the HYPERGEO- METRIC DIFFERENTIAL EQUATION , plug y/C30X/C12 n/C300Anzn(2) y?/C30X/C12 n/C300nAnzn/C281(3) yƒ/C30X/C12 n/C300n(n/C281)Anzn/C282(4) into z(1/C28z)yƒ/C27[c/C28(a/C27b/C271)z]y?/C28aby/C300 (5) to obtain X/C12 n/C300n(n/C281)Anzn/C281/C28X/C12 n/C300n(n/C281)Anzn /C27cX/C12 n/C300nAnzn/C281/C27(a/C27b/C271)X/C12 n/C300nAnzn /C28abX/C12 n/C300Anzn/C300 (6) X/C12 n/C302n(n/C281)Anzn/C281/C28X/C12 n/C300n(n/C281)Anzn /C27cX/C12 n/C301nAnzn/C281/C28(a/C27b/C271)X/C12 n/C301nAnzn /C28abX/C12 n/C300Anzn/C300 (7) X/C12 n/C300(n/C271)nAn/C271zn/C28X/C12 n/C300n(n/C281)Anzn /C27cX/C12 n/C300(n/C271)An/C271zn/C28(a/C27b/C271)X/C12 n/C300nAnzn/C28abX/C12 n/C300Anzn/C300 (8) X/C12 n/C300[n(n/C271)An/C271/C28n(n/C281)An/C27c(n/C271)An/C281 /C29X/C12 n/C300f(n/C271)(n/C27c)An/C271 /C28[n(n/C281/C27a/C27b/C271)/C27ab]Angzn/C300 (9) X/C12 n/C300f(n/C271)(n/C27c)An/C271 /C28[n2/C27(a/C27b)n/C27ab]Angzn/C300; (10) so An/C271/C30(n/C27a)(n/C27b) (n/C271)(n/C27c)An (11) and y/C30A01/C27ab 1!cz/C27a(a/C271)b(b/C271) 2!c(c/C271)z2/C27..."# :(12) This is the regular solution and is denoted 2F1(a;b;c;z)/C301/C27ab 1!cz/C27a(a/C271)b(b/C271) 2!c(c/C271)z2/C27... /C30X/C12 n/C300(a)n(b)n (c)nzn n!; (13) where ( a)nare P OCHHAMMER SYMBOLS . The hypergeo- metric series is convergent for REAL/C281BzB1;and forz/C3091i fc>a/C27b:The complete solution to the HYPERGEOMETRIC DIFFERENTIAL EQUATION is y/C30A2F1(a;b;c;z) /C27Bz1/C28c 2F1(a/C271/C28c;b/C271/C28c;2/C28c;z):(14) Derivatives are given by d2F1(a;b;c;z) dz/C30ab c2F1(a/C271;b/C271;c/C271;z) (15) d2 2F1(a;b;c;z) dz2 /C30a(a/C271)b(b/C271) c(c/C271)2F1(a/C272;b/C272;c/C272;z) (16) (Magnus and Oberhettinger 1949, p. 8).An integral giving the hypergeometric function is 2F1(a;b;c;z) /C30G(c) G(b)G(c/C28b)g1 0tb/C281(1/C28t)c/C28b/C281 (1/C28tz)adt (17) as shown by Euler in 1748 (Bailey 1935, pp. 4 /C1/). Barnes (1908) gave the CONTOUR INTEGRAL 2F1(a;b;c;z) /C301 2pigi/C12 /C28i/C12G(a/C27s)G(b/C27s)G(/C28s) G(c/C28s)(/C28z)sds; (18) where arg( /C28z) jj Bpand the path is curved (if neces- sary) to separate the poles s/C30/C28a/C28n;s/C30/C28b/C28n;... (n/C300, 1, ...) from the poles s/C300, 1 ... (Bailey 1935, pp. 4/C1/; Whittaker and Watson 1990). A hypergeometric function can be written using EULER’S HYPERGEOMETRIC TRANSFORMATIONS t0t (19) t01/C28t (20) t0(1/C28z/C28tz)/C281(21) t01/C28t 1/C28tz(22) in any one of four equivalent forms 2F1(a;b;c;z)/C30(1/C28z)/C28a 2F1(a;c/C28b;c;z=(z/C281)) nbsp; (23rpar /C30(1/C28z)/C28b 2F1(c/C28a;b;c;z=(z/C281)) nbsp; (24rpar /C30(1/C28z)c/C28a/C28b 2F1(c/C28a;c/C28b;c;z) nbsp; (25rpar It can also be written as a linear combination 2F1(a;b;c;z) /C30G(c)G(c/C28a/C28b) G(c/C28a)G(c/C28b)2F1(a;b;a/C27b/C271/C28c;1/C28z) /C27G(c)G(a/C27b/C28c) G(a)G(b) /C2(1/C28z)c/C28a/C28b 2F1(c/C28a;c/C28b;1/C27c/C28a/C28b;1 /C28z) (26) (Barnes 1908; Bailey 1935, pp. 3 /C1/; Whittaker and Watson 1990, p. 291). Kummer found all six solutions (not necessarily regular at the origin) to the HYPERGEOMETRIC DIFFER- ENTIAL EQUATION , u1(x)/C302F1(a;b;c;z) (27) u2(x)/C302F1(a;b;a/C27b/C271/C28c;1/C28z) (28)u3(x)/C30z/C28a 2F1(a;a/C271/C28c;a/C271/C28b;z/C281) (29) u4(x)/C30z/C28b 2F1(b/C271/C28c;b;b/C271/C28a;z/C281) (30) u5(x)/C30z1/C28c 2F1(b/C271/C28c;a/C271/C28c;2/C28c;z) (31) u6(x)/C30(1/C28z)c/C28a/C28b 2F1(c/C28a;c/C28b;c/C271/C28a/C28b;1 /C28z) (32) (Abramowitz and Stegun 1972, p. 563). Applying E ULER’S HYPERGEOMETRIC TRANSFORMA- TIONS to the Kummer solutions then gives all 24 possible forms which are solutions to the HYPERGEO- METRIC DIFFERENTIAL EQUATION u(1) 1(x)/C302F1(a;b;c;z) (33) u(2)1(x)/C30(1/C28z)c/C28a/C28b 2F1(c/C28a;c/C28b;c;z) (34) u(3)1(x)/C30(1/C28z)/C28a 2F1(a;c/C28b;c;z=(z/C281)) (35) u(4)1(x)/C30(1/C28z)/C28b 2F1(c/C28a;b;c;z=(z/C281)) (36) u(1)2(x)/C302F1(a;b;a/C27b/C271/C28c;1/C28z) (37) u(2)2(x)/C30z1/C28c 2F1(a/C271/C28c;b/C271/C28c;a/C27b/C271/C28c;1 /C28z) (38) u(3)2(x)/C30z/C28a 2F1(a;a/C271/C28c;a/C27b/C271/C28c;1 /C28z/C281) (39) u(4)2(x)/C30z/C28b 2F1(b/C271/C28c;b;a/C27b/C271/C28c;1 /C28z/C281) (40) u(1) 3(x)/C30(/C28z)/C28a 2F1(a;a/C271/C28c;a/C271/C28b;z/C281) (41) u(2)3(x)/C30(/C28z)b/C28c /C2(1/C28z)c/C28a/C28b 2F1(1/C28b;c/C28b;a/C271 /C28b;z/C281) (42) u(3) 3(x)/C30(1/C28z)/C28a 2F1(a;c/C28b;a/C271/C28b;( 1/C28z)/C281) (43) u(4)3(x)/C30(/C28z)1/C28c /C2(1/C28z)c/C28a/C281 2F1(a/C271/C28c;1/C28b;a/C271 /C28b;( 1/C28z)/C281) (44) u(1)4(x)/C30(/C28z)/C28b 2F1(b/C271/C28c;b;b/C271/C28a;z/C281) (45) u(2)4/C30(/C28z)a/C28c /C2(1/C28z)c/C28a/C28b 2F1(1/C28a;c/C28a;b/C271 /C28a;z/C281) (46) u(3)4(x)/C30(1/C28z)/C28b 2F1(b;c/C28a;b/C271/C28a;( 1/C28z)/C281) (47) u(4)4(x)/C30(/C28z)1/C28c /C2(1/C28z)c/C28b/C281 2F1(b/C271/C28c;1/C28a;b/C271 /C28a;( 1/C28z)/C281) (48) u(1) 5(x) /C30z1/C28c 2F1(a /C271 /C28c ; b /C271 /C28c;2/C28c; z) (49) u(2)5/C30z1 /C28c(1 /C28z)c/C28a /C28b 2F1(1 /C28a ; 1 /C28b;2/C28c; z) (50) u(3) 5(x) /C30z1 /C28c(1 /C28z)c/C28a/C281 2F1(a /C271 /C28c ; 1 /C28b;2 /C28c; z=(z /C281)) (51) u(4) 5(x) /C30z1 /C28c(1 /C28z)c/C28b/C281 2F1(b /C271 /C28c; 1 /C28a;2 /C28c; z =(z /C281)) (52) u(1)6(x) /C30(1 /C28z)c/C28a /C28b 2F1(c /C28a; c /C28b; c /C271 /C28a /C28b;1 /C28z) (53) u(2)6(x) /C30z1 /C28c(1 /C28z)c/C28a /C28b 2F1(1 /C28a ; 1 /C28b; c /C271 /C28a /C28b;1/C28z) (54) u(3) 6(x) /C30za/C28c(1 /C28z)c /C28a /C28b 2F1(c /C28a; 1 /C28a; c /C271 /C28a /C28b;1/C28z/C281) (55) u(4)6(x) /C30zb /C28c(1 /C28z)c/C28a/C28b 2F1(c /C28b; 1 /C28b; c /C271 /C28a /C28b;1/C28z/C281) (56) (Kummer 1836; Erde´lyi et al. 1981, pp. 105 /C1/06). Goursat (1881) and Erde´lyi et al. (1981) give many hypergeometric transformation formulas, including several cubic transformations. Many functions of mathematical physics can be expressed as special cases of the hypergeometric functions. For example, 2F1(/C28l ; l /C271; 1; (1 /C28z) =2) /C30Pl(z); (57) where Pl(z)isaL EGENDRE POLYNOMIAL . (1 /C27z)n /C302 F1(/C28n; b; b; /C28z) (58) ln(1 /C27z) /C30z2F1(1; 1; 2; /C28z) (59) Complete ELLIPTIC INTEGRALS and the RIEMANN P- SERIES can also be expressed in terms of 2F1(a ; b; c; z): Special values include 2F1(a ; b; a /C28b /C271; /C281) /C302 /C28affiffiffipp G(1 /C27 a /C27 b) G 1 /C271 2 a /C28 bP+’kP+’7 G12 /C2712 aP+’kP+’7 (60) 2F1(1;/C28a; a; /C281) /C30ffiffiffipp 2G(a) G a /C271 2P+’kP+’7 /C271 (61) 2F1a ; b; c;12P+’kP+’7 /C302a 2F1(a; c /C28b; c; /C281) (62) 2F1a; b;1 2(a /C27b /C271);12P+’kP+’7 /C30G12P+’kP+’7 G12(1 /C27 a /C27 b)hi G1 2(1 /C27 a)hi G12(1 /C27 b)hi (63)2F1a ; 1 /C28a; c;12P+’kP+’7 /C30G12 cP+’kP+’7 G12(c /C27 1)hi G1 2(a /C27 c)hi G12(1 /C27 c /C28 a)hi (64) 2F1(a; b; c;1)/C30G(c) G(c /C28 a /C28 b) G(c /C28 a) G(c /C28 b) : (65) KUMMER’S FIRST FORMULA gives 2F11 2 /C27m /C28k;/C28n;2m /C271; 1P+’kP+’7 /C30G(2m /C27 1)G m /C2712 /C27 k /C27 nP+’kP+’7 G m /C271 2 /C27 kP+’kP+’7 G(2m /C27 1 /C27 n); (66) where m "/C281=2;/C281, /C283=2 ; .... Many additional identities are given by Abramowitz and Stegun (1972, p. 557). Hypergeometric functions can be generalized to GEN- ERALIZED HYPERGEOMETRIC FUNCTIONS nFm(a1;...;an;b1;...;bm;z): (67) A function OF THE FORM1F1(a;b;z) is called a CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND , and a function OF THE FORM0F1(a;b;z)i s called a CONFLUENT HYPERGEOMETRIC LIMIT FUNC- TION . See also APPELL HYPERGEOMETRIC FUNCTION , BARNES’ LEMMA ,B RADLEY’S THEOREM ,C AYLEY’S HYPERGEOMETRIC FUNCTION THEOREM ,C LAUSEN FORMULA ,C LOSED FORM,C ONFLUENT HYPERGEO- METRIC FUNCTION OF THE FIRST KIND,CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND, CONFLUENT HYPERGEOMETRIC LIMIT FUNCTION ,CON- TIGUOUS FUNCTION ,DARLING’S PRODUCTS ,GENERAL- IZED HYPERGEOMETRIC FUNCTION ,G OSPER’S ALGORITHM ,HYPERGEOMETRIC IDENTITY ,HYPERGEO- METRIC SERIES ,JACOBI POLYNOMIAL ,KUMMER’S FOR- MULAS ,K UMMER’S QUADRATIC TRANSFORMATION , KUMMER’S RELATION ,ORR’S THEOREM ,PFAFF TRANS- FORMATION , Q-HYPERGEOMETRIC FUNCTION ,RAMANU- JAN’S HYPERGEOMETRIC IDENTITY ,SAALSCHU ¨ TZIAN , SISTER CELINE’S METHOD ,ZEILBERGER’S ALGORITHM References Abramowitz, M. and Stegun, C. A. (Eds.). "Hypergeometric Functions." Ch. 15 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 555 /C1/66, 1972. Appell, P. and Kampe ´de Fe ´riet, J. Fonctions hyperge ´o- me´triques et hypersphe ´riques: polynomes d’Hermite. Paris: Gauthier-Villars, 1926. Arfken, G. "Hypergeometric Functions." §13.5 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 748 /C1/52, 1985. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: University Press, 1935. Barnes, E. W. "A New Development in the Theory of the Hypergeometric Functions." Proc. London Math. Soc. 6, 141/C1/77, 1908. Emmanuel, J. "Eacute;valuation rapide de fonctions hyper- ge´ome´triques." Report RT-0242. INRIA, Jul 2000. http:// www.inria.fr.RRRT/RT-0242.html. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, 1981. Exton, H. Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, 1978. Fine, N. J. Basic Hypergeometric Series and Applications. Providence, RI: Amer. Math. Soc., 1988. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Gauss, C. F. "Disquisitiones Generales Circa Seriem Infini- tamab 1 /C215 ghi x /C27a(a /C271) b(b/C271) 1 /C215 2 /C215 g(g/C271)hi x2 /C27a(a /C271)(a /C272) b(b/C271)(b/C272) 1 /C215 2 /C215 3 /C215 g( g/C271)(g/C272)hi x3/C27etc. Pars Prior." Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, Vol. II. 1812. Reprinted in Ge- sammelte Werke, Bd. 3, pp. 123 /C1/63 and 207 /C1/29, 1866. Gessel, I. and Stanton, D. "Strange Evaluations of Hyper- geometric Series." SIAM J. Math. Anal. 13, 295 /C1/08, 1982. Gosper, R. W. "Decision Procedures for Indefinite Hypergeo- metric Summation." Proc. Nat. Acad. Sci. USA 75,40/C1/2, 1978. Goursat, M. E. "Sur l’e´quation diffe´rentielle line´aire qui admet pour inte´grale la se´rie hyperge ´ome´trique." Ann. Sci. E´ cole Norm. Super. Sup. 10, S3-S142, 1881. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Hardy, G. H. "A Chapter from Ramanujan’s Note-Book." Proc. Cambridge Philos. Soc. 21, 492 /C1/03, 1923. Hardy, G. H. "Hypergeometric Series." Ch. 7 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 101 /C1/12, 1999. Iyanaga, S. and Kawada, Y. (Eds.). "Hypergeometric Func- tions and Spherical Functions." Appendix A, Table 18 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1460 /C1/468, 1980. Kampe ´ de Fe´riet, J. La fonction hyperge ´ome´trique. Paris: Gauthier-Villars, 1937. Kohno, M. Global Analysis in Linear Differential Equations. Dordrecht, Netherlands: Kluwer, 1999. Krattenthaler, C. "HYP and HYPQ." J. Symb. Comput. 20, 737 /C1/44, 1995. Kummer, E. E. "U¨ ber die Hypergeometrische Reihe." J. reine angew. Math. 15,39/C1/3 and 127 /C1/72, 1836. Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea, 1949. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 541 /C1/47, 1953. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Hypergeometric Functions." §6.12 in Numer- ical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 263 /C1/65, 1992. Seaborn, J. B. Hypergeometric Functions and Their Applica- tions. New York: Springer-Verlag, 1991. Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. Washington, DC: U. S. Government Printing Office, 1952. Spanier, J. and Oldham, K. B. "The Gauss Function F(a; b; c; x) :/" Ch. 60 in An Atlas of Functions. Washing- ton, DC: Hemisphere, pp. 599 /C1/07, 1987. Thomae. J. reine angew. Math. 87, 222 /C1/49, 1879. Watson, G. N. "Ramanujan’s Note Books." J. London Math. Soc. 6, 137 /C1/53, 1931.Weisstein, E. W. "Books about Hypergeometric Functions." http://www.treasure-troves.com/books/Hypergeometric- Functions.html. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Hypergeometric Identity A relation expressing a sum potentially involving BINOMIAL COEFFICIENTS ,FACTORIALS ,RATIONAL FUNC- TIONS , and power functions in terms of a simple result. Thanks to results by Fasenmyer, Gosper, Zeilberger, Wilf, and Petkovsek, the problem of determining whether a given hypergeometric sum isexpressible in simple closed form and, if so, finding the form, is now (subject to a mild restriction) completely solved. The algorithm which does so hasbeen implemented in several computer algebrapackages and is called Z EILBERGER’S ALGORITHM . See also BINOMIAL SUMS,GENERALIZED HYPERGEO- METRIC FUNCTION ,GOSPER’S ALGORITHM ,HYPERGEO- METRIC SERIES ,S ISTER CELINE’S METHOD ,W ILF- ZEILBERGER PAIR,ZEILBERGER’S ALGORITHM References Koepf, W. "Hypergeometric Identities." Ch. 2 in Hypergeo- metric Summation: An Algorithmic Approach to Summa- tion and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 11 /C1/0, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, p. 18, 1996. Hypergeometric Polynomial JACOBI POLYNOMIAL Hypergeometric Series A hypergeometric series akckis a series for which c0/C301 and the ratio of consecutive terms is a RATIONAL FUNCTION of the summation index k, i.e., one for which ck/C271 ck/C30P(k) Q(k); (1) with P(k) and Q(k)POLYNOMIALS . In this case, ckis called a HYPERGEOMETRIC TERM (Koepf 1998, p. 12). The functions generated by hypergeometric series are called HYPERGEOMETRIC FUNCTIONS or, more gener- ally, GENERALIZED HYPERGEOMETRIC FUNCTIONS .I f the polynomials are completely factored, the ratio ofsuccessive terms can be written ck/C271 ck/C30P(k) Q(k) /C30(k/C27a1)(k/C27a2)/C1/C1/C1(k/C27ap) (k/C27b1)(k/C27b2)/C1/C1/C1(k/C27bq)(k/C271)x; (2) where the factor of k/C271 in the DENOMINATOR is present for historical reasons of notation, and the resulting GENERALIZED HYPERGEOMETRIC FUNCTION is written pFqa1a2/C1/C1/C1 ap b1b2/C1/C1/C1 bq; xP+2$P+2’ /C30X k/C300ckxk : (3) If p /C302 and q /C301, the function becomes a traditional HYPERGEOMETRIC FUNCTION2F1(a ; b; c; x) :/ Many sums can be written as GENERALIZED HYPER- GEOMETRIC FUNCTIONS by inspections of the ratios of consecutive terms in the generating hypergeometric series. See also BINOMIAL SUMS,GENERALIZED HYPERGEO- METRIC FUNCTION ,G EOMETRIC SERIES ,H YPERGEO- METRIC FUNCTION ,H YPERGEOMETRIC IDENTITY , HYPERGEOMETRIC TERM References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Hypergeo- metric Series," "How to Identify a Series as Hypergeo- metric," and "Software That Identifies Hypergeometric Series." §3.2 /C1/.4 in A /C30B. Wellesley, MA: A. K. Peters, pp. 34 /C1/2, 1996. Hypergeometric Summation The analytic summation of a HYPERGEOMETRIC SER- IES. Powerful general techniques of hypergeometric summation include GOSPER’S ALGORITHM ,S ISTER CELINE’S METHOD ,WILF-ZEILBERGER PAIRS , and ZEIL- BERGER’S ALGORITHM . See also BINOMIAL SUMS,G OSPER’S ALGORITHM , SISTER CELINE’S METHOD ,W ILF-ZEILBERGER PAIR, ZEILBERGER’S ALGORITHM References Koepf, W. "Algorithms for m-fold Hypergeometric Summa- tion." J. Symb. Comput. 20, 399 /C1/17, 1995. Hypergeometric Term Given a HYPERGEOMETRIC SERIES ak ck ; ckis called a hypergeometric term (Koepf 1998, p. 12). See also HYPERGEOMETRIC SERIES References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Hypergeometric0F1 CONFLUENT HYPERGEOMETRIC LIMIT FUNCTION Hypergeometric0F1Regularized CONFLUENT HYPERGEOMETRIC LIMIT FUNCTIONHypergeometric1F1 CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND Hypergeometric2F1 HYPERGEOMETRIC FUNCTION HypergeometricU CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND Hypergraph A hypergraph is a GRAPH in which generalized edges (called HYPEREDGES ) may connect more than two nodes. See also GRAPH ,HYPEREDGE ,M ULTIGRAPH ,PSEUDO- GRAPH References Berge, C. Graphs and Hypergraphs. New York: Elsevier, 1973. Berge, C. Hypergraphs: The Theory of Finite Sets. Amster- dam, Netherlands: North-Holland, 1989. Hypergroup A MEASURE ALGEBRA which has many properties associated with the convolution MEASURE ALGEBRA of a GROUP , but no algebraic structure is assumed for the underlying SPACE . References Bloom, W. R.; and Heyer, H. The Harmonic Analysis of Probability Measures on Hypergroups. Berlin: de Gruyter, 1995. Jewett, R. I. "Spaces with an Abstract Convolution of Measures." Adv. Math. 18,1/C1/01, 1975. Hyper-Ka ¨hler Manifold See also KA¨ HLER MANIFOLD Hypermatrix A generalization of the MATRIX to an n1 /C29n2 /C29/C1/C1/C1 array of numbers. See also HYPERDETERMINANT References Gel’fand, I. M.; Kapranov, M. M.; and Zelevinsky, A. V. "Hyperdeterminants." Adv. Math. 96, 226 /C1/63, 1992. Hyperparallel Two lines in HYPERBOLIC GEOMETRY which diverge from each other in both directions. See also ANTIPARALLEL ,IDEAL POINT ,PARALLEL Hyperperfect Number A number n is called k-hyperperfect if n /C301 /C27kX idi /C301 /C27k[ s(n) /C28n /C281]; where s(n) is the DIVISOR FUNCTION and the summa- tion is over the PROPER DIVISORS with 1 Bdi Bn: Rearranging gives ks(n) /C30(k /C271)n /C27k /C281: Taking k /C301 gives the usual PERFECT NUMBERS . If k /C211 is an odd integer, and p /C30(3k /C271)=2 and q /C30 3k /C274 /C302p /C273 are prime, then p2q is k-hyperperfect. McCranie (2000) conjectures that all k-hyperperfect numbers for odd k /C211 are in fact of this form. Similarly, if p and q are distinct odd primes such that k(p /C27q) /C30pq /C281 for some integer k, then n /C30 pq is k-hyperperfect. Finally, if k /C210 and p /C30k /C271is prime, then if q /C30pi /C28p /C271 is prime for some i /C211 B then n /C30pi/C281q is k-hyperperfect (McCranie 2000). The first few hyperperfect numbers (excluding PER- FECT NUMBERS ) are 21, 301, 325, 697, 1333, ... (Sloane’s A007592). If PERFECT NUMBERS are in- cluded, the first few are 6, 21, 28, 301, 325, 496, ... (Sloane’s A034897), whose corresponding values of k are 1, 2, 1, 6, 3, 1, 12, ... (Sloane’s A034898). The following table gives the first few k-hyperperfect numbers for small values of k. McCranie (2000) has tabulated all hyperperfect numbers less than 1011. k Sloane k-hyperperfect number 1 A000396 6 ,28, 496, 8128, ... 2 A007593 21, 2133, 19521, 176661, ... 3 325, ... 4 1950625, 1220640625, ... 6 A028499 301, 16513, 60110701, ... 10 159841, ... 11 10693, ... 12 A028500 697, 2041, 1570153, 62722153, ... See also PERFECT NUMBER References Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45 /C1/3, 1994. McCranie, J. S.. "A Study of Hyperperfect Numbers." J. Integer Sequences 3, No. 00.1.3, 2000. http://www.re- search.att.com/~njas/sequences/JIS/VOL3/mccranie.html. Minoli, D. "Issues in Nonlinear Hyperperfect Numbers." Math. Comput. 34, 639 /C1/45, 1980.Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 177, 1992. Sloane, N. J. A. Sequences A000396/M4186, A007592/ M5113, A007593/M5121, A028499, A028500, A034897, and A034898 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. te Riele, H. J. J. "Hyperperfect Numbers with Three Differ- ent Prime Factors." Math. Comput. 36, 297 /C1/98, 1981. Hyperplane Let a1 ; a2 ; ..., anbe SCALARS not all equal to 0. Then the SET S consisting of all VECTORS X /C30x1 x2 n xn2 6643 775 in R n such that a1x1 /C27a2x2 /C27.../C27anxn /C300 is a SUBSPACE of Rn called a hyperplane. More generally, a hyperplane is any CODIMENSION -1 vector SUBSPACE of a VECTOR SPACE . Equivalently, a hyperplane V in a VECTOR SPACE W is any SUBSPACE such that W =V is 1-dimensional. Equivalently, a hyperplane is the KERNEL of any NONZERO linear MAP from the VECTOR SPACE to the underlying FIELD . Hyperreal Number Hyperreal numbers are an extension of the REAL NUMBERS to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite IFF xjjB n for some INTEGER n. x is said to be infinitesimal IFFxjjB1=nfor all INTEGERS n. See also AX-KOCHEN ISOMORPHISM THEOREM ,N ON- STANDARD ANALYSIS References Keisler, H. J. "The Hyperreal Line." In Real Numbers, Generalizations of the Reals, and Theories of Continua (Ed. P. Ehrlich). Norwell, MA: Kluwer, 1994. Hyperspace ASPACE having DIMENSION n/C213. Hypersphere Then-hypersphere (often simply called the n-sphere) is a generalization of the CIRCLE (n/C302) and SPHERE (n/C303) to dimensions n]4:It is therefore defined as the set of n-tuples of points ( /x1;x2;...,xn) such that x2 1/C27x22/C27.../C27x2n/C30R2; (1) where Ris the RADIUS of the hypersphere. The CONTENT Vn(i.e., n-DVOLUME )o fa n n-hypersphere ofRADIUS Ris given by Vn/C30gR 0Snrn/C281dr/C30SnRn n; (2) where Snis the hyper- SURFACE AREA of an n-sphere of unit radius. But, for a unit hypersphere, it must be true that Sng/C12 0e/C28r2rn/C281dr /C30g/C12 /C28/C12/C1/C1/C1g/C12 /C28/C12|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ne/C28(x2 1/C27/C1/C1/C1/C27x2 n)dx1/C1/C1/C1dxm /C30g/C12 /C28/C12e/C28x2dxP+’vP+’u n : (3) But the GAMMA FUNCTION can be defined by G(m)/C302g/C12 0e/C28r2r2m/C281dr; (4) so 1 2SnG12nP+’kP+’7 /C30G12P+’kP+’7hin /C30(p1=2)n(5) Sn/C302pn=2 G12nP+’kP+’7 : (6) Special forms of G12nP+’kP+’7 fornan integer allow the above expression to be written as Sn/C302(n/C271)=2p(n/C281)=2 (n/C282)!!fornodd 2pn=2 1 2n/C281P+’kP+’7 !forneven ;8 >>>>< >>>>:(7) where n!i sa FACTORIAL and n!! is a DOUBLE FACTORIAL . Equation (6) gives the RECURRENCE RELATION Sn/C272/C302pSn n: (8) Using G(n/C271)/C30nG(n) then gives Vn/C30SnRn n/C30pn=2Rn 1 2nP+’kP+’7 G12nP+’kP+’7/C30pn=2Rn G1/C2712nP+’kP+’7 (9) (Sommerville 1958, p. 136; Conway and Sloane 1993). Strangely enough, the hyper- SURFACE AREA and CONTENT reach MAXIMA and then decrease towards0a s nincreases. The point of MAXIMAL hyper- SUR- FACE AREA satisfies dSn dn/C30pn=2lnp/C28c012nP+’kP+’7hi G1 2nP+’kP+’7 /C300; (10) where c0(x)/C13C(x) is the DIGAMMA FUNCTION . The point of MAXIMAL CONTENT satisfies dVn dn/C30pn=2lnp/C28c01/C271 2nP+’kP+’7hi 2G1/C271 2nP+’kP+’7 /C300: (11) Neither can be solved analytically for n, but the numerical solutions are n/C307:25695 . . . for hyper- SURFACE AREA andn/C305:25695 . . . for CONTENT (Wells 1986, p. 67). As a result, the 7-D and 5-D hyper- spheres have MAXIMAL hyper- SURFACE AREA and CONTENT , respectively (Le Lionnais 1983; Wells 1986, p. 60). n /Vn//Vsphere =Vcube//Sn/ 01 1 0 12 1 22 /p//1 4p// 2p/ 3 /43p//16p// 4p/ 4 /1 2p2//1 32p2// 2p2/ 5 /8 15p2 //1 60p2 //83p2 / 6 /1 6p3//1 384p3// p3/ 7 /16 105p3//1 840p3//1615p3/ 8 /1 24p4//1 6144p4//1 3p4/ 9 /32 945p4 //1 15120p4 //32 105p4 / 10 /1 120p5//1 122880p5//1 12p5/ In 4-D, the generalization of SPHERICAL COORDINATES is defined by x1/C30Rsincsinfcosu (12) x2/C30Rsincsinfsinu (13) x3/C30Rsinccosf (14) x4/C30Rcosc: (15) The equation for a 4-sphere is x2 1/C27x22/C27x23/C27x24/C30R2; (16) and the LINE ELEMENT is ds2/C30R2[dc2/C27sin2c(df2/C27sin2fdu2)]: (17) By defining r/C13Rsinc;the LINE ELEMENT can be rewritten ds2 /C30dr2 1 /C28r2 R2P+’kP+’7 /C27r2(d f2 /C27sin2 f d u2) : (18) The hyper- SURFACE AREA is therefore given by S4 /C30g p 0Rdcg p 0R sin c dfg2 p 0R sin c sin f df /C302p2R3 : (19) See also CIRCLE ,GLOME ,HYPERCUBE ,HYPERSPHERE PACKING ,H YPERSPHERE POINT PICKING ,M AZUR’S THEOREM ,PEG,SPHERE ,TESSERACT References Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958. Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lat- tices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 58, 1983. Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 96 /C1/01, 1988. Hypersphere Packing The analog of face-centered cubic packing is the densest lattice packing in 4- and 5-D. In 8-D, the densest lattice packing is made up of two copies of face-centered cubic. In 6- and 7-D, the densest lattice packings are CROSS SECTIONS of the 8-D case. In 24-D, the densest packing appears to be the LEECH LATTICE . For high dimensions ( /C21000-D), the densest known packings are nonlattice. The densest lattice packings in n-D have been rigorously proved to have PACKING DENSITY 1, p= 2ffiffiffi 3pP+$P+’ ; p= 3ffiffiffi2pP+$P+’ ; p2 =16 ; p2 = 15ffiffiffi2pP+$P+’ ; p3 = 48ffiffiffi3pP+$P+’ ; p3 =105; and p4 =384 (Hilbert and Cohn- Vossen 1999, p. 47; Finch). The densest known non-lattice packings of hyper- spheres in dimensions up to 10 are given by Conway and Sloane (1995). However, there are no proofs that any packing in dimensions greater than 3 is optimal (Sloane 1998). The largest number of UNIT CIRCLES which can touch a given UNIT CIRCLE is six. For SPHERES , the max- imum number is 12. Newton considered this question long before a proof was published in 1874. The maximum number of hyperspheres that can touch another in n-D is the so-called KISSING NUMBER .The following example illustrates the sometimes counterintuitive properties of hypersphere packings. Draw unit n-spheres in an n-D space centered at all 91 coordinates. Now place an additional HYPER- SPHERE at the origin tangent to the other HYPER- SPHERES . For values of n between 2 and 8, the central HYPERSPHERE is contained inside the HYPERCUBE with VERTICES at the centers of the other spheres. However, for n /C309, the central HYPERSPHERE just touches the HYPERCUBE of centers, and for n /C219, the central HYPERSPHERE is partially outside the HYPER- CUBE . This fact can be demonstrated by finding the distance from the origin to the center of one of the n HYPER- SPHERES , which is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (91)2/C27.../C27(91)2q |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n/C30ffiffiffinp: The radius of the central sphere is thereforeffiffiffinp/C281: Now, the distance from the origin to the center of a FACET bounding the HYPERCUBE is always 2 (two hypersphere radii), so the center HYPERSPHERE is tangent to the hypercube whenffiffiffinp/C281/C302;orn/C309, and partially outside it for n/C219. See also C IRCLE PACKING ,ELLIPSOID PACKING ,KE- PLER CONJECTURE ,KISSING NUMBER ,LEECH LATTICE , PEG,SPHERE PACKING References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hermit/hermit.html. Conway, J. H. and Sloane, N. J. A. Disc. Comput. Geom. 13, 383/C1/03, 1995. Gardner, M. Martin Gardner’s New Mathematical Diver- sions from Scientific American. New York: Simon and Schuster, pp. 89 /C1/0, 1966. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 47, 1999. Schnell, U. and Wills, J. M. "Densest Packings of More than Three d-Spheres are Nonplanar." Disc. Comput. Geom. 24, 539/C1/49, 2000. Sloane, N. J. A. "Kepler’s Conjecture Confirmed." Nature 395, 435/C1/36, 1998. Hypersphere Point Picking Marsaglia (1972) has given a simple method for selecting points with a uniform distribution on thesurface of a 4-sphere. This is accomplished by picking two pairs of points ( x 1;x2) and ( x3;x4);rejecting any points for which x2 1/C27x22]1 and x23/C27x24]1:Then the points x/C30x1 (1) y/C30x2 (2) z/C30x3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 1/C28x22 x3 2/C27x2 4s (3) w /C30x4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2 1 /C28 x22 x3 2 /C27 x2 4s (4) have a uniform distribution on the surface of the hypersphere. This extends the method of Marsaglia (1972) for SPHERE POINT PICKING . See also SPHERE POINT PICKING References Hicks, J. S. ad Wheeling, R. F. "An Efficient Method for Generating Uniformly Distributed Points on the Surface of an n-Dimensional Sphere." Comm. Assoc. Comput. Mach. 2,13/C1/5, 1959. Marsaglia, G. "Choosing a Point from the Surface of a Sphere." Ann. Math. Stat. 43, 645 /C1/46, 1972. Hyperspherical Differential Equation ULTRASPHERICAL DIFFERENTIAL EQUATION Hypersurface A generalization of an ordinary two-dimensional sur- face embedded in three-dimensional space to an (n /C281)/-dimensional surface embedded in n-dimen- sional space. A hypersurface is therefore the set of solutions to a single equation f(x1 ; ...; xn) /C300 and so it has CODIMENSION one. For instance, the n- dimension HYPERSPHEREncorresponds to the equa- tion x2 1/C27.../C27x2n/C301:/ See also HYPERSPHERE ,SURFACE Hypervolume CONTENT Hypocycloid The curve produced by fixed point Pon the CIRCUM- FERENCE of a small CIRCLE ofRADIUS brolling around the inside of a large CIRCLE ofRADIUS a/C21b.A hypocycloid is a HYPOTROCHOID with h/C30b. To derive the equations of the hypocycloid, call the ANGLE by which a point on the small CIRCLE rotates about its center q;and the ANGLE from the center of the large CIRCLE to that of the small CIRCLE f:Then (a/C28b)f/C30bq; (1) so q/C30a/C28b bf: (2) Call r/C13a/C282b:Ifx(0)/C30r;then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are x/C30(a/C28b)cosf/C28bcosq /C30(a/C28b)cosf/C28bcosa/C28b bf ! (3) y/C30(a/C28b)sinf/C28bsinq /C30(a/C28b)sinf/C27bsina/C28b bf ! : (4) Ifx(0)/C30ainstead so the first point is at maximum radius (on the CIRCLE ), then the equations of the hypocycloid are x/C30(a/C28b)cosf/C27bcosa/C28b bf ! (5) y/C30(a/C28b)sinf/C28bsina/C28b bf ! : (6) Ann-cusped non-self-intersecting hypocycloid has a=b/C30n:A 2-cusped hypocycloid is a LINE SEGMENT (Steinhaus 1983, p. 145), as can be seen by settinga/C30bin equations (3) and (4) and noting that the equations simplify to x/C30asinf (7) y/C300: (8) A 3-cusped hypocycloid is called a DELTOID orTRICUS- POID , and a 4-cusped hypocycloid is called an ASTROID . Ifa=bis rational, the curve closes on itself and has b cusps. If a=bisIRRATIONAL , the curve never closes and fills the entire interior of the CIRCLE . n-hypocycloids can also be constructed by beginning with the DIAMETER of a CIRCLE , offsetting one end by a series of steps while at the same time offsetting the other end by steps ntimes as large in the opposite direction and extending beyond the edge of the CIRCLE . After traveling around the CIRCLE once, an n-cusped hypocycloid is produced, as illustrated above (Madachy 1979). Letrbe the radial distance from a fixed point. For RADIUS OF TORSION rand ARC LENGTH s, a hypocy- cloid can given by the equation s2/C27r2/C3016r2(9) (Kreyszig 1991, pp. 63 /C1/4). A hypocycloid also satis- fies sin2c/C30r2 a2/C28r2a2/C28r2 r2; (10) where rdr du/C30tanc (11) andcis the ANGLE between the RADIUS VECTOR and the TANGENT to the curve. The ARC LENGTH of the hypocycloid can be computed as follows x?/C30/C28 (a/C28b)sinf/C28(a/C28b)sina/C28b bf ! /C30(a/C28b) sin f/C27sina/C28b bf !"# (12) y?/C30(a/C28b)cosf/C28(a/C28b)cosa/C28b af ! /C30(a/C28b) cos f/C28cosa/C28b bf !"# (13)x?2/C27y?2/C30(a/C28b)2sin2f/C272 sin fsina/C28b bf ! " /C27sin2a/C28b bf ! /C27cos2f/C282 cos fcosa/C28b bf ! /C27cos2a/C28b bfP+’u # /C30(a/C28b)22/C272 sin fsina/C28b af !"( /C28cosfcosa/C28b bfP+’uP+2’ ) /C302(a/C28b)21/C28cosf/C27a/C28b bf !"# /C304(a/C28b)21 21/C28cosa bf !"# /C304(a/C28b)2sin2af 2b ! ; (14) so ds/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2/C27y?2q df/C302(a/C28b)sinaf 2b ! df (15) forf5(b=2a)p:Integrating, s(f)/C30gf 0ds/C302(a/C28b)/C282b acosaf 2b !"#f 0 /C304b(a/C28b) a/C28cosa 2bf ! /C271"# /C308b(a/C28b) asin2a 4bf ! : (16) The length of a single cusp is then s2pb a ! /C308b(a/C28b) asin2p 2 ! /C308b(a/C28b) a:(17) Ifn/C13a=bis rational, then the curve closes on itself without intersecting after ncusps. For n/C13a=band with x(0)/C30a;the equations of the hypocycloid become x/C301 n[(n/C281)cos f/C28cos[(n/C281)f]a; (18) y/C301 n[(n/C281)sin f/C27sin[(n/C281)f]a; (19) and sn/C30n8b(bn/C28b) nb/C308b(n/C281)/C308a(n/C281) n: (20) Compute xy?/C28yx?/C30(a/C28b)cosf/C27bcosa/C28b af ! "# (b/C28a) /C2sinf/C27sina/C28b bf !"# /C28(a/C28b)sinf/C28bsina/C28b bf ! "# (a/C28b) /C2cosf/C28cosa/C28b bf !"# /C302(a2/C283ab/C272b2)sin2af 2b ! : (21) The AREA of one cusp is then A/C301 2g2pb=a 0(xy?/C28yx?)df /C30(a2/C283ab/C272b2)at/C28bsinat bP+’kP+’7 2a2 4352pb=a a /C30(a2/C283ab/C272b2)a2pb aP+’kP+’7 2a2435 /C30 b(a2/C283ab/C272b2) ap: (22) Ifn/C30a=bis rational, then after ncusps, An/C30npb(a2/C283ab/C272b2) a /C30npa na2/C283aa n/C272a2 n2 ! a /C30n2/C283n/C272 n2pa2/C30(n/C281)(n/C282) n2pa2: (23) The equation of the hypocycloid can be put in a form which is useful in the solution of CALCULUS OF VARIATIONS problems with radial symmetry. Consider the case x(0)/C30r;then r2/C30x2/C27y2 /C30(a/C28b)2cos2f/C282(a/C28b)bcosfcosa/C28b bf ! "/C27b2cos2a/C28b bf ! /C27(a/C28b)2sin2f /C272(a/C28b)bsinfsina/C28b bf ! /C27b2sin2a/C28b bf !P+2’ /C30(a/C28b)2/C27b2/C282(a/C28b)bcosfcosa/C28b bf !" ( /C28sinfsina/C28b bf !P+2’P+27 /C30(a/C28b)2/C27b2/C282(a/C28b)bcosa bf ! : (24) Butr/C30a/C282b;sob/C30(a/C28r)=2;which gives (a/C28b)2/C27b2/C30a/C281 2(a/C28r)hi2 /C2712(a/C28r)hi2 /C301 2(a/C27r)hi2 /C2712(a/C28r)hi2 /C301 4(a2/C272ar/C27r2/C27a2/C282ar/C27r2) /C3012(a2/C27r2) (25) 2(a/C28b)b/C302a/C281 2(a/C28r)hi 12(a/C28r) /C301 2(a/C27r)(a/C28r)/C3012(a2/C28r2): (26) Now let 2Vt/C13a bf; (27) so f/C30a/C28r aVt (28) f a/C28r/C30Vt a; (29) then r2/C3012(a2/C27r2)/C2812(a2/C28r2)cosa bf ! /C301 2(a2/C27r2)/C2812(a2/C28r2)cos(2Vt): (30) The POLAR ANGLE is tanu/C13y x/C30(a/C28b)sinf/C27bsina/C28b afP+’kP+’7 (a/C28b)cosf/C27bcosa/C28b afP+’kP+’7 : (31) But b /C301 2(a /C28 r) (32) a /C28b /C3012(a /C27 r) (33) a /C28 b b/C30a /C27 r a /C28 r ; (34) so tan u /C3012(a /C27 r)sin f /C2712(a /C28 r)sina/C27 r z/C28 rfP+’kP+’7 1 2(a /C27 r)cos f /C2812(a /C28 r)cosa /C27r a /C28r fP+’kP+’7 /C30(a /C27 r)sina /C28 r aVtP+’kP+’7 /C27 (a /C28 r)sina /C27 r aVtP+’kP+’7 (a /C27 r)cosa /C28 r aVtP+’kP+’7 /C28 (a /C28 r)cosa /C27r aVtP+’kP+’7 /C30a sina /C28r aVtP+’kP+’7 /C27 sina/C27r aVtP+’kP+’7 hi /C27 r sina /C28r aVtP+’kP+’7 /C28 sina /C27r aVtP+’kP+’7 hi a cosa /C28r aVtP+’kP+’7 /C28 cosa/C27r aVtP+’kP+’7 hi /C27 r cosa /C28r aVtP+’kP+’7 /C27 cosa /C27r aVtP+’kP+’7 hi /C302a sin( Vt)cosr qVtP+’kP+’7 /C28 2r cos( Vt)sinr a VtP+’kP+’7 2a sin( Vt)sinr qVtP+’kP+’7 /C27 2 r cos( Vt)sinr aVtP+’kP+’7 /C30a tan( Vt) /C28 r tanr aVtP+’kP+’7 a tan(Vt)tanr aVtP+’kP+’7 /C27 r: (35) Computing tan u /C27r aVt ! /C30a tan( Vt) /C28 r tanr aVtP+’kP+’7 /C27 tanr aVtP+’kP+’7 hi a tan( Vt)tanr aVtP+’kP+’7 /C27 rhi a tan( Vt)tanr aVtP+’kP+’7 /C27 rhi /C28 a tan( Vt) /C28 r tanr aVtP+’kP+’7 hi tanr aVtP+’kP+’7 /C30a tan( Vt)1/C27 tan2r aVtP+’kP+’7hi r 1 /C27 tan2r aVtP+’kP+’7hi /C30a rtan( Vt) ; (36) then gives u /C30tan /C281a rtan(Vt)"# /C28r aVt: (37) Finally, plugging back in gives u /C30tan/C281a rtana a /C28 rf !"# /C28r aa a /C28 rf /C30tan/C281a rtana a /C28 rf !"# /C28r a /C28 rf (38)This form is useful in the solution of the SPHERE WITH TUNNEL problem, which is the generalization of the BRACHISTOCHRONE PROBLEM , to find the shape of a tunnel drilled through a SPHERE (with gravity vary- ing according to Gauss’s law in a gravitational field such that the travel time between two points on the surface of the SPHERE under the force of gravity is minimized. See also ASTROID ,CYCLOID ,DELTOID ,EPICYCLOID References Bogomolny, A. "Cycloids." http://www.cut-the-knot.com/ pythagoras/cycloids.html. Kreyszig, E. Differential Geometry. New York: Dover, 1991. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 171 /C1/73, 1972. Lemaire, J. Hypocycloı ¨des et epicycloı ¨des. Paris: Albert Blanchard, 1967. MacTutor History of Mathematics Archive. "Hypocycloid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hy- pocycloid.html. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 225 /C1/31, 1979. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 50 /C1/2, 1991. Yates, R. C. "Epi- and Hypo-Cycloids." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Ed- wards, pp. 81 /C1/5, 1952. Hypocycloid Evolute Forx(0)/C30a; x/C30a a/C282b(a/C28b)cosf/C28bcosa/C28b bf ! "# y/C30a a/C282b(a/C28b)sinf/C27bsina/C28b bf ! "# : Ifa=b/C30n;then x/C301 n/C282[(n/C281)cos f/C28cos[(n/C281)f]a y/C301 n/C282[(n/C281)sin f/C28sin[(n/C281)f]a: This is just the original HYPOCYCLOID scaled by the factor (n /C282)=n and rotated by 1 =(2n) of a turn. Hypocycloid Involute The HYPOCYCLOID x /C30a a /C28 2b(a /C28b)cos f /C28b cosa /C28 b bf ! "# y /C30a a /C28 2b(a /C28b)sin f /C27b sina /C28 b bf ! "# has INVOLUTE x /C30a /C28 2b a(a /C28b)cos f /C27b cosa /C28 b bf ! "# y /C30a /C28 2b a(a /C28b)sin f /C28b sina /C28 b bf ! "# ; which is another HYPOCYCLOID . Hypocycloid Pedal Curve The PEDAL CURVE for a PEDAL POINT at the center is a ROSE . Hypocycloid–3-Cusped DELTOID Hypocycloid–4-Cusped ASTROIDHypoellipse yn =m /C27cx aP+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2P+’2n =m /C28c /C300; with n=m B2: If n=m > 2; the curve is a HYPEREL- LIPSE . See also ELLIPSE ,HYPERELLIPSE ,SUPERELLIPSE References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 82, 1993. Hypohamiltonian Graph A graph G is hypohamiltonian if G is not HAMILTO- NIAN , but G /C28v is HAMILTONIAN for every v /C23 V (Bondy and Murty 1976, p. 61). The PETERSEN GRAPH , which has ten nodes and is illustrated above, is the smallest hypohamiltonian graph (Herz et al. 1967; Bondy and Murty 1976, p. 61). There are no hypohamiltonian graphs with 11 or 12 vertices. However, there exists a hypohamiltonian graph on p vertices for every p ]13 with the possible excep- tions of p /C3014, 17, 19. Thomassen (1973) found hypohamiltonian graphs on p /C3020 and 25 vertices, which had previously been open. A graph can be tested to see if it is hypohamiltonian using the following Mathematica function. BBDiscreteMath‘Combinatorica‘; HypohamiltonianQ[g_Graph] : /C30 ! HamiltonianQ[g] && HamiltonianQ /@ And @@ (DeleteVertex[g, #] & /@ Range[V[g]]) See also HAMILTONIAN GRAPH ,H YPOTRACEABLE GRAPH ,TRACEABLE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 61, 1976. Chva´tal, V. "Flip-Flops in Hypohamiltonian Graphs." Ca- nad. Math. Bull. 16,3 3/C1/1, 1973. Gaudin, T.; Herz, J.-C.; and Rossi, P. "Solution de proble `me no. 29." Franc ¸aise Informat. Recherche Ope ´rationnelle 8, 214/C1/18, 1964. Herz, J. C.; Duby, J. J.; and Vigue ´, F. "Recherche syste ´ma- tique des graphes hypohamiltoniens." In Theory of Graphs: Internat. Sympos., Rome 1966 (Ed. P. Rosen- stiehl). Paris: Gordon and Breach, pp. 153 /C1/59, 1967. Lindgren, W. F. "An Infinite Class of Hypohamiltonian Graphs." Amer. Math. Monthly 74, 1087 /C1/089, 1967. Thomassen, C. "Hypohamiltonian and Hypotraceable Graphs." Disc. Math. 9,91/C1/6, 1974. Hypotenuse The longest LEG of a RIGHT TRIANGLE (which is the side opposite the RIGHT ANGLE ). The word derives from the Greek hypo- ("under") and teinein ("to stretch"). Hypothesis A proposition that is consistent with known data, but has been neither verified nor shown to be false. It is synonymous with CONJECTURE . See also BOURGET’S HYPOTHESIS ,CHINESE HYPOTH- ESIS,CONTINUUM HYPOTHESIS ,HYPOTHESIS TESTING , NESTED HYPOTHESIS ,NULL HYPOTHESIS ,POSTULATE , RAMANUJAN’S HYPOTHESIS ,R IEMANN HYPOTHESIS , SCHINZEL’S HYPOTHESIS ,SOUSLIN’S HYPOTHESIS Hypothesis Testing The use of statistics to determine the probability that a given hypothesis is true. See also BONFERRONI CORRECTION ,ESTIMATE ,FISHER SIGN TEST,P AIRED T-TEST,P ERMUTATION TESTS , STATISTICAL TEST,TYPE IE RROR ,TYPE II ERROR , WILCOXON SIGNED RANK TEST References Good, P. Permutation Tests: A Practical Guide to Resam- pling Methods for Testing Hypotheses, 2nd ed. New York: Springer-Verlag, 2000. Hoel, P. G.; Port, S. C.; and Stone, C. J. "Testing Hypoth- eses." Ch. 3 in Introduction to Statistical Theory. New York: Houghton Mifflin, pp. 52 /C1/10, 1971. Iyanaga, S. and Kawada, Y. (Eds.). "Statistical Estimation and Statistical Hypothesis Testing." Appendix A, Table 23 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1486 /C1/489, 1980. Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev. Psych. 46, 561 /C1/84, 1995. Hypotraceable Graph G is a hypotraceable graph if G has no HAMILTONIAN PATH (i.e., it is not a TRACEABLE GRAPH ), but G /C28v has aH AMILTONIAN PATH (i.e., is a TRACEABLE GRAPH ) for every v /C23 V (Bondy and Murty 1976, p. 61). T. Gallai conjectured that there exist no hypotrace- able graphs (there are none on seven or fewer nodes), but the THOMASSEN GRAPH , illustrated above, pro- vides a counterexample (Bondy and Murty 1973, pp. 239 /C1/40). However, a hypotraceable graph with 40 vertices was found by Horton (Gru¨nbaum 1973, Thomassen 1974). Thomassen (1974) showed that for p /C3034, 37, 39, 40, and all p ]42; there exists a hypotraceable graph with p vertices. The smallest of these, the so-called THOMASSEN GRAPH , is illu- strated above. Walter (1969) gave an example of a connected graph in which the longest paths do not have a vertex in common, a property shared by hypotraceable graphs. See also HAMILTON- CONNECTED GRAPH ,THOMASSEN GRAPH ,TRACEABLE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 61 and 239/C1/40, 1976. Gru¨nbaum, B. "Vertices Missed by Longest Paths or Cir- cuits." Preprint, University of Washington, Seattle, May 1973. Kapoor, S. F.; Kronk, H. V.; and Lick, D. R. "On Detours in Graphs." Canad. Math. Bull. 11, 195/C1/01, 1968. Thomassen, C. "Hypohamiltonian and Hypotraceable Graphs." Disc. Math. 9,9 1/C1/6, 1974. Walter, H. "U ¨ber die Nichtexistenz eines Knotenpunktes, durch den alle la ¨ngsten Wege eines Graphen gehen." J. Combin. Th. 6,1/C1/, 1969. Hypotrochoid The ROULETTE traced by a point P attached to a CIRCLE of radius b rolling around the inside of a fixed CIRCLE of radius a, where P is a distance h 5b from the center of the interior circle. The PARAMETRIC EQUATIONS for a hypotrochoid are x /C30(a /C27b) cos t /C28h cosa /C27 b bt ! ; (1) y /C30(a /C27b) sin t /C28h sina /C27 b bt ! ; (2) Special cases include the HYPOCYCLOID with h /C30b, the ELLIPSE with a/C302b;and the ROSE with a/C302nh n/C271(3) b/C30(n/C281)h n/C271: (4) See also EPITROCHOID ,H YPOCYCLOID ,SPIROGRAPH , TROCHOIDReferences Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 165 /C1/68, 1972. MacTutor History of Mathematics Archive. "Hypotrochoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hy- potrochoid.html. Hypotrochoid Evolute The EVOLUTE of the HYPOTROCHOID is illustrated above. Hyzer’s Illusion FREEMISH CRATE I i "The" IMAGINARY NUMBER i (also called the IMAGIN- ARY UNIT ) is defined as the SQUARE ROOT of /C281, i.e., i /C13ffiffiffiffiffiffi /C281p : Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point /C27i and /C28i can then be distinguished. Since either choice is possible, there is no ambiguity in defining i as "the" square root of /C281. In Mathematica , the imaginary number is implemen- ted as I. For some reason engineers and physicists prefer the symbol J to i, probably because the symbol i (or I) is commonly used to denote current. Numbers OF THE FORM iy, where y is a REAL NUMBER , are called IMAGINARY NUMBERS . Numbers OF THE FORM z /C30x /C27iy where x and y are REAL NUMBERS are called COMPLEX NUMBERS , and when z is used to denote a COMPLEX NUMBER , it is sometimes (in older texts) called an "AFFIX ." The SQUARE ROOT of i is ffiffi ip /C309i /C27 1ffiffiffi 2p ; (1) since 1ffiffiffi 2p (i /C271)"#2 /C301 2(i2 /C272i /C271) /C30i : (2) This can be immediately derived from the EULER FORMULA with x /C30 p=2; i /C30eip=2 (3) ffiffi ip /C30ffiffiffiffiffiffiffiffiffi eip=2p /C30eip=4 /C30cos1 4 p1CA}1CA$ /C27i sin14 p1CA}1CA$ /C301 /C27 iffiffiffi 2p : (4) The PRINCIPAL VALUE of ii is ii /C30 ei p=21CC1CA i/C30ei2 p=2 /C30e /C28p=2 /C300 :207879... (5) (Wells 1986, p. 26). See also COMPLEX NUMBER ,I,I MAGINARY IDENTITY , IMAGINARY NUMBER ,REAL NUMBER ,SURREAL NUM- BER References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 89, 1996. Nahin, P. J. An Imaginary Tale: The Story offfiffiffiffiffiffi /C281p :/ Prince- ton, NJ: Princeton University Press, 1998. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 26, 1986.I The double-struck capital letter I, l ; is a symbol sometimes used instead of Z for the RING of INTEGERS . See also I,Z Iamond POLYIAMOND Ice Fractal Z A FRACTAL (square, triangle, etc.) based on a simple generating motif. The above plots show the ice triangle, antitriangle, square, and antisquare. The base curves and motifs for the fractals illustrated above are shown below. See also FRACTAL References Birch, M. W. "The Cross-Stitch Curve." Eureka 21,1 2/C1/3, 1958. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, p. 44, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Icosagon A 20-sided POLYGON . The regular icosagon is a CONSTRUCTIBLE POLYGON , and the regular icosagon of unit side length has INRADIUS r, CIRCUMRADIUS R, and area A given by r /C301 21 /C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq1CA%1CAP R /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C27ffiffiffi 5p /C271 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 /C2722ffiffiffi 5pqr A /C3051/C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq1CA%1CAP :s The SWASTIKA is an irregular icosagon. See also SWASTIKA ,TRIGONOMETRY VALUES PI/20 Icosahedral Equation Hunt (1996) gives the "dehomogenized" icosahedral equation as z20 /C2711CC1CA /C28228 z15 /C28z51CC1CA /C27494z101C|1CA3 /C271728 uz5 z10 /C2711z5 /C2811CC1CA5/C300 : Other forms include I(u;v ;Z) /C30u5v5 u10 /C2711u5v5 /C28v101CC1CA5 /C28 u30 /C27v30 /C2810005 u20v10 /C27u10v201CC1CA 1C| /C27522 u25v5 /C28u5v251CC1CA /C1382Z /C300 and I(z ; 1; z) /C30z5 /C281 /C2711z5 /C27z101CC1CA5 /C28 1 /C27z30 /C2810005 z10 /C27z201CC1CA /C27522 /C28z5 /C27z251CC1CA 1C|1Cffl2z /C300: References Hunt, B. The Geometry of Some Special Arithmetic Quoti- ents. New York: Springer-Verlag, p. 146, 1996. Klein, F. "Sull’ equazione dell’ Icosaedro nella risoluzione delle equazioni del quinto grado [per funzioni ellittiche]." Reale Istituto Lombardo, Rendiconto, Ser. 2 10, 1877.Icosahedral Graph The PLATONIC GRAPH whose nodes have the connec- tivity of the ICOSAHEDRON . The icosahedral graph has 12 vertices, 30 edges, vertex connectivity 5, edge connectivity 5, GRAPH DIAMETER 3, GRAPH RADIUS 3, and GIRTH 3. See also CUBICAL GRAPH ,D ODECAHEDRAL GRAPH , OCTAHEDRAL GRAPH ,P LATONIC GRAPH ,T ETRAHE- DRAL GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976. Icosahedral Group The POINT GROUP Ihof symmetries of the ICOSAHE- DRON and DODECAHEDRON having order 60. The icosahedral group consists of the symmetry opera- tions E,12C5 ; 12C2 5 ; 20C3 ; 15C2 ; i,12S10 ; 12S310 ; 20S6 ; and 15s (Cotton 1990). The icosahedron group is a SUBGROUP of the SPECIAL ORTHOGONAL GROUP SO(3):/ See also BIPOLYHEDRAL GROUP ,D ODECAHEDRON , ICOSAHEDRON ,OCTAHEDRAL GROUP ,POINT GROUPS , POLYHEDRAL GROUP ,SPECIAL ORTHOGONAL GROUP , TETRAHEDRAL GROUP References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, pp. 48 /C1/0, 1990. Coxeter, H. S. M. "The Polyhedral Groups." §3.5 in Regular Polytopes, 3rd ed. New York: Dover, pp. 46 /C1/7, 1973. Lomont, J. S. "Icosahedral Group." §3.10.E in Applications of Finite Groups. New York: Dover, p. 82, 1987. Icosahedron AP LATONIC SOLID P5having 12 VERTICES ,3 0 EDGES , and 20 equivalent EQUILATERAL TRIANGLE faces, 20f3g:It is also UNIFORM POLYHEDRON U22and Wenninger model W4:It is described by the S CHLA ¨FLI SYMBOL f3;5gand W YTHOFF SYMBOL 5½23:/ The icosahedron has the ICOSAHEDRAL GROUP Ihof symmetries. The connectivity of the vertices is given by the ICOSAHEDRAL GRAPH . The DUAL POLYHEDRON of the icosahedron is the DODECAHEDRON , so the centers of the faces of an icosahedron form a DODECAHEDRON , and vice versa (Steinhaus 1983, pp. 199 /C1/01). There are 59 distinct icosahedra when each TRIANGLE is colored differently (Coxeter 1969). Taken eight at a time, the centers of the faces of an icosahedron comprise the vertices of a CUBE . Thisleads to the beautiful CUBE 5-COMPOUND and is the basis for JESSEN’S ORTHOGONAL ICOSAHEDRON . A plane PERPENDICULAR to aC5axis of an icosahedron cuts the solid in a regular DECAGONAL CROSS SECTION (Holden 1991, pp. 24 /C1/5). The long diagonals of the faces of the RHOMBIC TRIACONTAHEDRON give the edges of an icosahedron (Steinhaus 1983, pp. 209 /C1/10). The following table gives polyhedra which can beconstructed by CUMULATION of an icosahedron by pyramids of given heights h. h /(r/C27h)=h/ Result /1 6ffiffiffi 3pffiffiffi5p /C2831CC1CA //3ffiffiffi5p /C2821CC1CA / GREAT DODECAHE- DRON /1 15ffiffiffiffiffiffi 15p //1 5(10/C283ffiffiffi 5p )/ SMALL TRIAMBIC ICOSAHEDRON /1 3ffiffiffi 6p // 1/C283ffiffiffi 2p /C27ffiffiffiffiffiffi 10p /60-faced star DEL- TAHEDRON /1 6ffiffiffi 3p (3/C27ffiffiffi5p ) /3 GREAT STELLATED DODECAHEDRON A construction for an icosahedron with side length a/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2810ffiffiffi 5pp =5 places the end vertices at (0 ;0;91) and the central vertices around two staggered CIR- CLES ofRADII2 5ffiffiffi 5p and heights 91 5ffiffiffi 5p :By a suitable rotation, the VERTICES of an icosahedron of side length 2 can also be placed at (0 ;9f;91);(91;0;9f); and (9f;91;0);where fis the GOLDEN RATIO . These points divide the EDGES of an OCTAHEDRON into segments with lengths in the ratio f:1:Another orientation of the icosahedron places two opposite triangular faces in an orientation parallel to the xy- plane. In this orientation, the distance h0from the top plane to the triangle T of vertices below it is h0/C30ffiffiffi 3p =3; equal to the circumradius of a face. The circumradius RTof T is given by RT /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 6(3 /C27ffiffiffi 5p ) :q (1) To derive the VOLUME of an icosahedron having edge length a, consider the orientation so that two VER- TICES are oriented on top and bottom. The vertical distance between the top and bottom PENTAGONAL DIPYRAMIDS is then given by z /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 /C28x2 ;q (2) where l /C301 2ffiffiffi 3p a (3) is the height of an ISOSCELES TRIANGLE , and the SAGITTA x /C30R?/C28r ? of the pentagon is x /C301 2a1 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 /C2810ffiffiffi 5p a ;q (4) giving x2 /C301 20ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C282ffiffiffi 5pq a2 : (5) Plugging (3) and (5) into (2) gives z /C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 /C2710ffiffiffi 5pq a ; (6) which is identical to the radius of a PENTAGON of side a. The CIRCUMRADIUS is then R /C30h /C271 2z ; (7) where h /C301 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 /C2810ffiffiffi 5pq a (8) is the height of a PENTAGONAL DIPYRAMID . Therefore, R2 /C30(h /C271 2z)2 /C3018(5 /C27ffiffiffi 5p )a2 : (9) Taking the square root gives the CIRCUMRADIUS R /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 8(5 /C27ffiffiffi 5p )q a /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C272ffiffiffi 5pq a :0:95105 a: (10) The INRADIUS isr /C301 12(3ffiffiffi3p /C27ffiffiffiffiffiffi15p )a :0 :75576 a: (11) The square of the MIDRADIUS is r2 /C301 2z1CA}1CA$2 /C27x2 1 /C301 8(3 /C27ffiffiffi 5p )a2 ; (12) so r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 8(3 /C27ffiffiffi 5p )q a /C301 4(1 /C27ffiffiffi 5p )a :0:80901 a : (13) The DIHEDRAL ANGLE is a /C30cos/C281(/C281 3ffiffiffi 5p ) :138:19( : (14) The AREA of one face is the AREA of an EQUILATERAL TRIANGLE A /C301 4a2ffiffiffi 3p : (15) The volume can be computed by taking 20 pyramids of height r V/C30201 3A1CA}1CA$ rhi /C305 12(3/C27ffiffiffi 5p )a3: (16) Apollonius showed that Vicosahedron Vdodecahedron/C30Aicosahedron Adodecahedron; (17) where Vis the volume and Athe SURFACE AREA . See also AUGMENTED TRIDIMINISHED ICOSAHEDRON , CUBE 5-COMPOUND ,D ECAGON ,D ODECAHEDRON , GREAT ICOSAHEDRON ,ICOSAHEDRON STELLATIONS , JESSEN’S ORTHOGONAL ICOSAHEDRON ,M ETABIDIMIN- ISHED ICOSAHEDRON ,R HOMBIC TRIACONTAHEDRON , TRIDIMINISHED ICOSAHEDRON ,T RIGONOMETRY VA- LUES PI/5 References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 228, 1987. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Cundy, H. and Rollett, A. "Icosahedron 35."§3.5.5 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 88, 1989. Davie, T. "The Icosahedron." http://www.dcs.st-and.ac.uk/ ~ad/mathrecs/polyhedra/icosahedron.html. Harris, J. W. and Stocker, H. "Icosahedron." §4.4.6 in Hand- book of Mathematics and Computational Science. New York: Springer-Verlag, p. 101, 1998. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Klein, F. Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. New York: Dover, 1956. Pappas, T. "The Icosahedron & the Golden Rectangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 115, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 199 /C1/01, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 163, 1991. Wenninger, M. J. "The Icosahedron." Model 4 in Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 17 /C1/8, 1989. Icosahedron Stellations Applying the STELLATION process to the ICOSAHEDRON gives 20/C2730/C2760/C2720/C2760/C27120/C2712/C2730/C2760/C2760 cells of ten different shapes and sizes in addition to the ICOSAHEDRON itself. After application of five restrictions due to J. C. P. Miller to define which forms should be considered distinct, 59 stellations are found to be possible. Miller’s restrictions are 1. The faces must lie in the twenty bounding planes of the icosahedron. 2. The parts of the faces in the twenty planes must be congruent, but those parts lying in one placemay be disconnected. 3. The parts lying in one plane must have threefold rotational symmetry with or without reflections.4. All parts must be accessible, i.e., lie on theoutside of the solid. 5. Compounds are excluded that can be divided into two sets, each of which has the full symmetryof the whole. Of these, 32 have full icosahedral symmetry and 27are ENANTIOMERIC forms. Four are POLYHEDRON COMPOUNDS , one is a K EPLER- POINSOT SOLID , and one is the DUAL POLYHEDRON of an A RCHIMEDEAN SOLID . nname 1ICOSAHEDRON 2SMALL TRIAMBIC ICOSAHEDRON 3OCTAHEDRON 5-COMPOUND 4ECHIDNAHEDRON 11 GREAT ICOSAHEDRON 13 MEDIAL TRIAMBIC ICOSAHEDRON 13 GREAT TRIAMBIC ICOSAHEDRON 18 TETRAHEDRON 10-COMPOUND 36 TETRAHEDRON 5-COMPOUND See also ARCHIMEDEAN SOLID STELLATION ,DODECA- HEDRON STELLATIONS ,STELLATION References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 146 /C1/47, 1987. Bulatov, V. "Stellations of Icosahedron." http://www.physic- s.orst.edu/~bulatov/polyhedra/icosahedron/. Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra. Stradbroke, England: Tarquin Publications, 1999. Hart, G. "59 Stellations of the Icosahedron." http:// www.georgehart.com/virtual-polyhedra/stellations-icosa- hedron-index.html. Maeder, R. E. "Icosahedra." http://www.mathsource.com/cgi- bin/msitem?0206 /C1/42. http://www.inf.ethz.ch/department/TI/rm/programs.html. Maeder, R. E. "The Stellated Icosahedra." Mathematica in Education 3, 1994. ftp://ftp.inf.ethz.ch/doc/papers/ti/scs/ icosahedra94.ps.gz. Maeder, R. E. "Stellated Icosahedra." http://www.mathcon- sult.ch/showroom/icosahedra/. Weisstein, E. W. "Corrected version of Maeder’s Icosahedra package." MATHEMATICA NOTEBOOK ICOSAHEDRA.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 77 /C1/8, 1991. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, pp. 41 /C1/5, 1989. Wheeler, A. H. "Certain Forms of the Icosahedron and a Method for Deriving and Designating Higher Polyhedra." Proc. Internat. Math. Congress 1, 701 /C1/08, 1924. Icosian Game The problem of finding a HAMILTONIAN CIRCUIT along the edges of an DODECAHEDRON , i.e., a path such that every vertex is visited a single time, no edge is visited twice, and the ending point is the same as the starting point (left figure). The puzzle was distributed commercially as a pegboard with holes at the nodes of the DODECAHEDRAL GRAPH , illustrated above (right figure). The Icosian Game was invented in 1857 by William Rowan Hamilton. Hamilton sold it to a London game dealer in 1859 for 25 pounds, and the game was subsequently marketed in Europe in a number of forms (Gardner 1957). See also HAMILTONIAN CIRCUIT ,D ODECAHEDR AL GRAPH ,DODECAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, 1987. Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150/C1/56, May 1957. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 4, 1994. Herschel, A. S. "Sir Wm. Hamilton’s Icosian Game." Quart. J. Pure Applied Math. 5, 305, 1862. MacTutor Archive. "Mathematical Games and Recreations." http://www-groups.dcs.st-and.ac.uk/~history/HistTo-Mathematical_games.html#49. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 198, 1990. Icosidodecadodecahedron The UNIFORM POLYHEDRON U44whose DUAL POLYHE- DRON is the MEDIAL ICOSACRONIC HEXECONTAHEDRON . It has W YTHOFF SYMBOL5 35½3:Its faces are 20 f6g/C27 12f5 2g/C2712f5g:ItsCIRCUMRADIUS for unit edge length is R/C301 2ffiffiffi 7p : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 128 /C1/29, 1989. Icosidodecagon A 32-sided polygon. The regular icosidodecagon is a CONSTRUCTIBLE POLYGON , and the regular icosidode- cahedron of side length 1 has INRADIUS r, CIRCUMRA- DIUS R, and AREA A r /C301 21 /C27ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(2 /C27ffiffiffi 2p )q /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(2 /C27ffiffiffiffiffi 2)p 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq1CA%1CAPs "# R /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2(2 /C27ffiffiffiffiffi 2)p 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq1CA%1CAP 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqr !vuut A /C3081/C27ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(2 /C27ffiffiffi 2p )q /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(2 /C27ffiffiffi 2p )2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq1CA%1CAPs "# : See also TRIGONOMETRY VALUES PI/32 Icosidodecahedron An icosidodecahedron is a 32-faced POLYHEDRON . "The" icosidodecahedron is the 32-faced ARCHIME- DEAN SOLID A4with faces 20 f3g/C2712 f5g: It is one of the two convex QUASIREGULAR POLYHEDRA . It also UNIFORM POLYHEDRON U24and Wenninger modelW12 : It has SCHLA ¨ FLI SYMBOL3 51C%1CP and WYTHOFF SYMBOL 2½35:/ The DUAL POLYHEDRON is the RHOMBIC TRIACONTAHE- DRON . The VERTICES of an icosidodecahedron of EDGE length 2f/C281are (92;0 ;0); (0;92; 0); (0;0;92); (91;9f/C281 ;91); (91;9f;9f/C281) ; (9f/C281 ;91;9f) : The 30 VERTICES of an OCTAHEDRON 5-COMPOUND form an icosidodecahedron (Ball and Coxeter 1987). FA- CETED versions include the SMALL ICOSIHEMIDODECA- HEDRON and SMALL DODECAHEMIDODECAHEDRON . The faces of the icosidodecahedron consist of 20 triangles and 12 pentagons. Furthermore, its 60 edges are bisected perpendicularly by those of the reciprocal RHOMBIC TRIACONTAHEDRON (Ball and Coxeter 1987). The INRADIUS r of the dual, MIDRADIUS r of the solid and dual, and CIRCUMRADIUS R of the solid for a /C301 are r /C301 8(5 /C273ffiffiffiffiffi 5)p :1:46353 r /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq :1:53884 R /C301 2(1 /C27ffiffiffi 5p ) /C30 f :1:61803 : The SURFACE AREA and VOLUME for an icosidodecahe- dron are given by S /C305ffiffiffi 3p /C273ffiffiffi5pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq (1) V /C301 645 /C2717ffiffiffi 5p (2) The distance to the centers of the triangular and pentagonal faces are r3/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 67/C273ffiffiffi 5p1CA}1CA$r (3) r5/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 55/C272ffiffiffi 5p1CA}1CA$ :r (4) See also ARCHIMEDEAN SOLID ,GREAT ICOSIDODECA- HEDRON ,ICOSIDODECAHEDRON ,QUASIREGULAR POLY- HEDRON ,S MALL ICOSIHEMIDODECAHEDRON ,S MALL DODECAHEMIDODECAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 137, 1987. Cundy, H. and Rollett, A. "Icosidodecahedron. 3 :5ðÞ2:/"§3.7.8 inMathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 108, 1989. Wenninger, M. J. "The Icosidodecahedron." Model 12 in Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 26 and 73, 1989. Icosidodecahedron Stellation The first stellation is a DODECAHEDRON-ICOSAHEDRON COMPOUND . References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 73 /C1/6, 1989. Icosidodecahedron-Rhombic Triacontahedron Compound The POLYHEDRON COMPOUND of the ICOSIDODECAHE- DRON and its dual, the RHOMBIC TRIACONTAHEDRON . The compound can be constructed from an ICOSIDO- DECAHEDRON of unit edge length by midpoint CUMU- LATION with heights h3 /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C283ffiffiffi 5pq1CA%1CAPs (1) h5 /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15(5 /C272ffiffiffi 5p1CA}1CA$r : (2) The resulting solid has edge lengths s1 /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 125 /C28ffiffiffi 5p1CA}1CA$r (3) s2 /C301 2 (4) s3 /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq (5) s4 /C301 41 /C27ffiffiffi 5p1CA}1CA$ ; (6) CIRCUMRADIUS R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 55/C272ffiffiffi 5p1CA}1CA$r ; (7) SURFACE AREA S given by the largest positive root of /C28612530859375 /C27147622500000 x /C2736267750000 x2 /C288164800000 x3 /C28450360000 x4 /C2782944000 x5 /C28230400 x6 /C28184320 x7 /C274096 x8 /C300 (8) and VOLUMEV /C305 16(27 /C2710ffiffiffi5p ) : (9) See also C UMULATION ,ICOSIDODECAHEDRON ,POLY- HEDRON COMPOUND ,RHOMBIC TRIACONTAHEDRON Icosidodecatruncated Icosidodecahedron ICOSITRUNCATED DODECADODECAHEDRON Icositetragon A 24-sided POLYGON . The regular icositetragon is constructible. For side length 1, the INRADIUS r, CIRCUMRADIUS R, and AREA A are given by r /C301 2(2 /C27ffiffiffi 2p /C27ffiffiffi 3p /C27ffiffiffi6p ) R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2710ffiffiffi 2p /C278ffiffiffi 3p /C276ffiffiffi6pq A /C306(2 /C27ffiffiffi 2p /C27ffiffiffi3p /C27ffiffiffi6p ) : See also T RIGONOMETRY VALUES PI/24 Icositetrahedron A 24-faced POLYHEDRON . See also DELTOIDAL ICOSITETRAHEDRON ,PENTAGO- NAL ICOSITETRAHEDRON ,SMALL RHOMBICUBOCTAHE- DRON ,S MALL TRIAKIS OCTAHEDRON ,S NUB CUBE, TETRAKIS HEXAHEDRON ,TRUNCATED OCTAHEDRON Icositruncated Dodecadodecahedron The UNIFORM POLYHEDRON U45also called the ICOSI- DODECATRUNCATED ICOSIDODECAHEDRON whose DUAL POLYHEDRON is the TRIDYAKIS ICOSAHEDRON . It has WYTHOFF SYMBOL 35 35½: Its faces are 20 f6g/C2712 f10 g/C27 12 f10 3 g: Its CIRCUMRADIUS for unit edge length is R /C302 : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 130 /C1/31, 1989. Ida Surface A 3-D shadow of a 4-D K LEIN BOTTLE . See also KLEIN BOTTLE References Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 44 /C1/5, 1990. Ideal A subset Iof elements in a RING Rwhich forms an additive GROUP and has the property that, whenever xbelongs to Rand ybelongs to I;then xyand yx belong to I:For example, the set of EVEN INTEGERS is an ideal in the RING ofINTEGERS Z:Given an ideal I; it is possible to define a FACTOR RING R=I:Ideals are commonly denoted using a Gothic typeface. An ideal may be viewed as a lattice and specified as the finite list of algebraic integers that form a basis for the lattice. Any two bases for the same lattice are equivalent. Ideals have multiplication, and this isbasically the K RONECKER PRODUCT of the two bases. From the perspective of ALGEBRAIC GEOMETRY , ideals correspond to VARIETIES . For any ideal I;there is an ideal Iisuch that IIi/C30z; (1) where zis a PRINCIPAL IDEAL , (i.e., an ideal of rank 1). Moreover there is a finite list of ideals Iisuch that this equation may be satisfied for every I:The size of this list is known as the CLASS NUMBER . In effect, the above relation imposes an EQUIVALENCE RELATION on ideals, and the number of ideals modulo this relationis the CLASS NUMBER . When the CLASS NUMBER is 1, the corresponding number RING has unique factoriza- tion and, in a sense, the class number is a measure of the failure of unique factorization in the original number ring. Dedekind (1871) showed that every NONZERO ideal in the domain of INTEGERS of a FIELD is a unique product ofPRIME IDEALS , and in fact all ideals of Zare of this form and therefore PRINCIPAL IDEALS . Ideals can be added, multiplied and intersected. The union of ideals usually is not an ideal since it may not be closed under addition. From the perspective ofALGEBRAIC GEOMETRY , the addition of ideals corre- sponds to the intersection of VARIETIES and the intersection of ideals corresponds to the union of varieties. Also, the multiplication of ideals corre-sponds to the union of varieties. Intersection and multiplication are different, for instance consider the ideal a/C30(x)i nZ[x;y]:Then a 2/C30a/C215a/C30x21CAC1CAA : (2) Sometimes they are the same. If b/C30yhi;then ab/C30aSb/C30xyhi : (3) There is also an analog of division, the IDEAL QUOTIENT (a:b);and there is an analog of the RADICAL , also called the RADICAL r(a):Given a ring homomorphism f:A0B;ideals in AEXTEND to ideals in B, while ideals in BCONTRACT to ideals in A. The following formulas summarize operations on ideals, where rcdenotes CONTRACT ,redenotes EXTEN- SION, and ( a:b) denotes an IDEAL QUOTIENT . a(b/C27c)/C30ab/C27ac (4) (a:b)bƒa (5) (Sai:b)/C30S(ai:b) (6) (a:X bi)/C30S(a:bi) (7) aƒr(a) (8) r(r(a))/C30r(a) (9) rðabÞ¼rðaSbÞ¼rðaÞSrðbÞð 10Þ r(a/C27b)/C30r(r(a)/C27r(b)) (11) aƒaec(12) bceƒb (13) bc/C30bcec(14) ae/C30aece(15) a1/C27a2 ðÞe/C30ae 1/C27ae2 (16) bc 1/C27bc2ƒb1/C27b2 ðÞc(17) a1Sa2 ðÞeƒae 1Sae2 (18) bc 1Sbc2/C30b1Sb2 ðÞc(19) ae 1ae2/C30a1a2 ðÞe(20) bc 1bc2ƒb1b2 ðÞc(21) a1:a2 ðÞeƒ(ae 1:ae2) (22) b1:b2 ðÞcƒ(bc 1:bc2) (23) raðÞeƒr(ae) (24) r bðÞc/C30r(bc) (25) See also ALGEBRAIC GEOMETRY ,C LASS NUMBER , CONTRACTION (IDEAL ), DIVISOR THEORY ,EXTENSION (IDEAL ), HERBRAND’S THEOREM ,HILBERT’S NULLSTEL- LENSATZ ,HOMOGENEOUS IDEAL ,IDEAL NUMBER ,IN- TEGRAL DOMAIN ,IDEAL QUOTIENT ,JOSEPH IDEAL , MAXIMAL IDEAL ,P RIME IDEAL ,P RINCIPAL IDEAL , RADICAL ,VARIETY References Atiyah, M. F. and MacDonald, I. G. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, pp. 6 /C1/0, 1969. Dedekind, R. "U¨ ber die Theorie der ganzen algebraischen Zahlen." X. Supplement to Vorlesungen u¨ber Zahlenthe- orie, 2nd ed. Braunschweig, Germany: Vieweg, 1871. Ferreiro ´s, J. "Ideal Factors." §3.3.1 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathe- matics. Basel, Switzerland: Birkha ¨user, pp. 95 /C1/7, 1999. Halter-Koch, F. Ideal Systems: An Introduction to Multi- plicative Ideal Theory. New York: Dekker, 1998. Koch, H. "Dedekind’s Theory of Ideals." Ch. 3 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 65 /C1/02, 2000. Malgrange, B. Ideals of Differentiable Functions. London: Oxford University Press, 1966. Ideal (Partial Order) An ideal I of a PARTIAL ORDER P is a subset of the elements of P which satisfy the property that if y /C23 1 and x By, then x /C23 I : For k disjoint chains in which the ith chain contains nielements, there are (1 /C27 n1)(1 /C27n2) /C1/C1/C1(1 /C27nk) ideals. The number of ideals of a n-element FENCE POSET is the FIBONACCI NUMBER Fn :/ References Ruskey, F. "Information on Ideals of Partially Ordered Sets." http://www.theory.csc.uvic.ca/~cos/inf/pose/Ideals.html. Steiner, G. "An Algorithm to Generate the Ideals of a Partial Order." Operat. Res. Let. 5, 317 /C1/20, 1986. Ideal Function DISTRIBUTION (GENERALIZED FUNCTION ) Ideal Number A type of number involving the ROOTS OF UNITY which was developed by Kummer while trying to solve FERMAT’S LAST THEOREM . Although factorization over the INTEGERS is unique (the FUNDAMENTAL THEOREM OF ALGEBRA ), factorization is not unique over the COMPLEX NUMBERS . Over the ideal numbers, however, factorization in terms of the COMPLEX NUMBERS becomes unique. Ideal numbers were so powerful that they were generalized by Dedekind into the more abstract IDEALS in general RINGS which are a key part of modern abstract ALGEBRA . See also DIVISOR THEORY ,FERMAT’S LAST THEOREM , IDEALReferences Ferreiro ´s, J. "Ideal Factors." §3.3.1 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathe- matics. Basel, Switzerland: Birkha ¨user, pp. 95 /C1/7, 1999. Ideal Point A type of POINT AT INFINITY in which parallel lines in the HYPERBOLIC PLANE intersect at infinity in one direction, while diverging from one another in the other. See also HYPERPARALLEL Ideal Quotient The ideal quotient ( a : b) is an analog of division for IDEALS in a COMMUTATIVE RING R, ( a : b) /C30fx /C23 R : xbƒag: The ideal quotient is always another ideal. However, this operation is not exactly like division. For example, when R is the ring of integers, then 12hi :2hi ðÞ /C30 6hi; which is nice, while 12hi :5hi ðÞ /C30 12hi Þ ; which is not as nice. See also ALGEBRAIC GEOMETRY ,ALGEBRAIC NUMBER THEORY ,IDEAL Idele The multiplicative subgroup of all elements in the product of the multiplicative groups k/C29 nwhose abso- lute value is 1 at all but finitely many n ; where k is a number FIELD and n a PLACE . See also ADE´ LE References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Idemfactor DYADIC Idempotent An OPERATOR ¯A such that ¯A2 /C30 ¯A or an element of an ALGEBRA x such that x2 /C30x:/ See also AUTOMORPHIC NUMBER ,BOOLEAN ALGEBRA , GROUP ,IDEMPOTENT MATRIX ,SEMIGROUP Idempotent Matrix A PERIODIC MATRIX with period 1, so that A2 /C30A :/ See also IDEMPOTENT ,NILPOTENT MATRIX ,PERIODIC MATRIX Idempotent Number The idempotent numbers are given by Bn;k(1;2 ;3;...)/C30n k1CA%1CAP kn/C28k ; where Bn;kis a BELL POLYNOMIAL andn k1CC1CA is a BINOMIAL COEFFICIENT . A table of the first few is given below. n /C301 n /C302 n /C303 n /C304 n /C305 n /C306 n /C307 k A000027 A001788 A036216 A040075 A050982 A050988 A050989 11 221 33614 4 24 12 1 5 5 80 90 20 1 6 6 240 540 240 30 1 7 7 672 2835 2240 525 42 1 8 8 1792 13608 17920 7000 1008 56 9 9 4608 61236 129024 78750 18144 1764 10 10 11520 262440 860160 787500 272160 41160 See also BELL POLYNOMIAL ,LAH NUMBER References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 91, 1974. Roman, S. The Umbral Calculus. New York: Academic Press, p. 85, 1984. Sloane, N. J. A. Sequences A000027/M0472, A001788/ M4161, A036216, A040075, A050982, A050988, and A050989 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Identical Congruence FUNCTIONAL CONGRUENCE Identity An identity is a mathematical relationship equating one quantity to another (which may initially appear to be different). See also ABEL’S DIFFERENTIAL EQUATION IDENTITY , ANDREWS- SCHUR IDENTITY ,BAC -CAB IDENTITY , BEAUZAMY AND DE´ GOT’S IDENTITY ,BELTRAMI IDEN- TITY,BIANCHI IDENTITIES ,BOCHNER IDENTITY ,BRAH- MAGUPTA IDENTITY ,C ASSINI’S IDENTITY ,C AUCHY- LAGRANGE IDENTITY ,C HRISTOFFEL- DARBOUX IDEN- TITY,C HU-VANDERMONDE IDENTITY , DE MOIVRE’SIDENTITY ,D OUGALL- RAMANUJAN IDENTITY ,E ULER FOUR- SQUARE IDENTITY ,E ULER IDENTITY ,E ULER POLYNOMIAL IDENTITY ,FERRARI’S IDENTITY ,FIBONAC- CI IDENTITY ,F ROBENIUS TRIANGLE IDENTITIES , GREEN’S IDENTITIES ,H YPERGEOMETRIC IDENTITY , IMAGINARY IDENTITY ,JACKSON’S IDENTITY ,JACOBI IDENTITIES ,JACOBI’S DETERMINANT IDENTITY ,JOR- DAN IDENTITY ,L AGRANGE’S IDENTITY ,L E CAM’S IDENTITY ,LEIBNIZ IDENTITY ,LIOUVILLE POLYNOMIAL IDENTITY ,M ATRIX POLYNOMIAL IDENTITY ,M ORGADO IDENTITY ,N EWTON’S IDENTITIES ,Q UINTUPLE PRO- DUCT IDENTITY ,RAMANUJAN 6 /C110-8 IDENTITY ,RAMANU- JAN COS/COSH IDENTITY ,R AMANUJAN’S IDENTITY , RAMANUJAN’S SUM IDENTITY ,R EZNIK’S IDENTITY , ROGERS- RAMANUJAN IDENTITIES ,SCHAAR’S IDENTITY , STREHL IDENTITIES ,SYLVESTER’S DETERMINANT IDEN- TITY,TRINOMIAL IDENTITY ,V ISIBLE POINT VECTOR IDENTITY ,W ATSON QUINTUPLE PRODUCT IDENTITY , WORPITZKY’S IDENTITY References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Identities." §2.2 in A /C30B. Wellesley, MA: A. K. Peters, pp. 21 /C1/2, 1996. Identity Element The identity element I (also denoted E, e,orI)ofa GROUP or related mathematical structure S is the unique element such that IA /C30AI /C30A for every element A /C23 S : The symbol "E" derives from the German word for unity, "Einheit." An identity ele- ment is also called a unit element. See also BINARY OPERATOR ,G ROUP ,INVOLUTION (GROUP ), MONOID Identity Function The function f(x)/C30xwhich assigns every REAL NUMBER x to the same REAL NUMBER x. It is identical to the IDENTITY MAP. Identity Map The MAP which assigns every member of a set A to the same element idA : It is identical to the IDENTITY FUNCTION . See also DONKIN’S THEOREM ,IDENTITY FUNCTION , ZERO MAP Identity Matrix The identity matrix is a very special BINARY MATRIX denoted I (or I) and defined such that I(X) /C13X (1) for all VECTORS X. The identity matrix is Iij /C30 dij (2) for i ;j /C301 ;2; ..., n, where dij is the KRONECKER DELTA . Written explicitly, I /C3010 /C1/C1/C1 0 01 /C1/C1/C1 0 nn ::: n 00 /C1/C1/C1 12 6643 775: (3) The notation E (an abbreviation for the German term, "Einheitsmatrix") is sometimes also used (Courant and Hilbert 1989, p. 7). "Square root of identity" matrices can be defined for I n by solving a11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n n /C1/C1/C1::: n an1an2/C1/C1/C1 ann2 6643 775a 11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n n /C1/C1/C1::: n an1an2/C1/C1/C1 ann2 6643 775 /C3010 /C1/C1/C1 0 01 /C1/C1/C1 0 nn ::: 0 00 /C1/C1/C1 12 6643 775: (4) For n /C302, the resulting matrices are I 1 =2 2/C30910 0 911C|C1C|A ;910 c /C1411C|C1C|A ; 91 b 0 /C1411C|C1C|A ;/C28d1 /C28 d2 c2 cd2 435: (5) "Cube root of identity" matrices can take on even more complicated forms. However, one simple class of such matrices is called K-MATRICES . See also BINARY MATRIX ,IDENTITY MATRIX , K-MA- TRIX,ZERO MATRIXReferences Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 10, 1962. Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989. Identity Operator The OPERATOR ¯I which takes a REAL NUMBER to the same REAL NUMBER ¯Ir /C30r:/ See also IDENTITY FUNCTION ,IDENTITY MAP Identity Transformation IDENTITY MAP Identric Mean This entry contributed by RONALD M. AARTS The identric mean is defined by I(a; b) /C301 ebb aa !1 =(b/C28a) for a /C210, b /C210, and a "b: The identric mean has been investigated intensively and many remarkable inequalities for I(a ;b) have been published (Bullen et al. 1988, Alzer 1993). References Alzer, H. "Some Gamma Function Inequalities." Math. Comput. 60, 337 /C1/46, 1993. Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means and Their Inequalities. Dordrecht, Netherlands: Reidel, 1988. Idoneal Number A POSITIVE value of D for which the fact that a number is a MONOMORPH (i.e., the number is expres- sible in only one way as x2 /C27Dy2 or x2 /C28Dy2 where x2 is RELATIVELY PRIME to Dy2) guarantees it to be a PRIME , POWER of a PRIME , or twice one of these. The numbers are also called EULER’S IDONEAL NUMBERS , or SUITABLE NUMBERS . The 65 idoneal numbers found by Gauss and Euler and conjectured to be the only such numbers (Shanks 1969) are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 (Sloane’s A000926). See also MONOMORPH References Shanks, D. "On Gauss’s Class Number Problems." Math. Comput. 23, 151/C1/63, 1969. Sloane, N. J. A. Sequences A000926/M0476 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Iff If and only if (i.e., NECESSARY and SUFFICIENT ). The terms "JUST IF"or" EXACTLY WHEN " are sometimes used instead. A iff B is written symbolically as A l B : A iff B is also equivalent to A [B; together with B [A; where the symbol [denotes "IMPLIES ." J. H. Conway believes that the word originated with P. Halmos and was transmitted through Kelley (1975). Halmos has stated, "To the best of my knowl- edge, I did invent the silly thing, but I wouldn’t swear to it in a court of law. So there–give me credit for it anyway" (D. Asimov 1997). See also EQUIVALENT ,EXACTLY ONE,IMPLIES ,N E- CESSARY ,SUFFICIENT References Asimov, D. "Iff." [email protected] posting, Sept. 19, 1997. Kelley, J. L. General Topology. New York: Springer-Verlag, 1975. Ill-Conditioned Matrix A MATRIX is ill-conditioned if the CONDITION NUMBER is too large (and SINGULAR if it is INFINITE ). See also CONDITION NUMBER ,S INGULAR MATRIX , SINGULAR VALUE DECOMPOSITION References Arfken, G. "Ill-Conditioned Systems." Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 233 /C1/34, 1985. Ill Defined A solution to a PARTIAL DIFFERENTIAL EQUATION that is not a continuous function of its values on the boundary is said to be ill defined. Otherwise, a solution is called WELL DEFINED . The term "ill defined" is also used informally to mean AMBIGUOUS . See also AMBIGUOUS ,W ELL DEFINED Illumination Problem In the early 1950s, Ernst Straus asked 1. Is every POLYGONAL region illuminable from every point in the region? 2. Is every POLYGONAL region illuminable from at least one point in the region? Here, illuminable means that there is a path from every point to every other by repeated reflections. Tokarsky (1995) showed that unilluminable rooms exist in the plane and 3-D, but question (2) remains open. The smallest known counterexample to (1) in the PLANE has 26 sides. See also ART GALLERY THEOREMReferences Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Illumination Problems." §A5 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 18 /C1/9, 1991. Klee, V. "Is Every Polygonal Region Illuminable from Some Point?" Math. Mag. 52, 180, 1969. Tokarsky, G. W. "Polygonal Rooms Not Illuminable from Every Point." Amer. Math. Monthly 102, 867 /C1/79, 1995. Illusion An object or drawing which appears to have proper- ties which are physically impossible, deceptive, or counterintuitive. See also BENHAM’S WHEEL ,B LACK DOT ILLUSION , BULLSEYE ILLUSION ,FREEMISH CRATE ,GOBLET ILLU- SION,H ERMANN GRID ILLUSION ,H ERMANN- HERING ILLUSION ,H YZER’S ILLUSION ,IMPOSSIBLE FIGURE , IRRADIATION ILLUSION ,KANIZSA TRIANGLE ,M U¨ LLER- LYER ILLUSION ,NECKER CUBE,ORBISON’S ILLUSION , PARALLELOGRAM ILLUSION ,PENROSE STAIRWAY ,POG- GENDORFF ILLUSION ,PONZO’S ILLUSION ,RABBIT- DUCK ILLUSION ,T RIBAR ,T RIBOX ,V ERTICAL- HORIZONTAL ILLUSION ,YOUNG GIRL-OLD WOMAN ILLUSION ,ZO¨ LL- NER’S ILLUSION References Ausbourne, B. "A Sensory Adventure." http://www.lainet.- com/illusions/. Ausbourne, B. "Optical Illusions: A Collection." http:// www.lainet.com/~ausbourn/. Ernst, B. Optical Illusions. New York: Taschen, 1996. Fineman, M. The Nature of Visual Illusion. New York: Dover, 1996. Gardner, M. "Optical Illusions." Ch. 1 in Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American.New York: Knopf, pp. 3 /C1 /5, 1979. Gregory, R. L. Eye and Brain, 5th ed. Princeton, NJ: Princeton University Press, 1997. Illusion Works. "Interactive Optical Illusions." http:// www.illusionworks.com/. Jablan, S. "Modularity in Art." http://www.mi.sanu.ac.yu/ ~jablans/d3.htm. Landrigad, D. "Gallery of Illusions." http://dragon.uml.edu/ psych/illusion/.html. Luckiesh, M. Visual Illusions: Their Causes, Characteristics, and Applications. New York: Dover, 1965. Pappas, T. "History of Optical Illusions." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 172 /C1/ 73, 1989. Robinson, J. O. The Psychology of Visual Illusion. New York: Dover, 1998. Tolansky, S. Optical Illusions. New York: Pergamon Press, 1964. Im IMAGINARY PART Image RANGE (IMAGE ) Imaginary Axis The axis in the COMPLEX PLANE corresponding to zero REAL PART , R z½/C138/C300:/ See also COMPLEX PLANE ,IMAGINARY LINE,R EAL AXIS Imaginary Identity I Imaginary Line A "line" having imaginary coefficients in its equations which can arise in algebraic geometry. See also IMAGINARY AXIS,LINE,REAL LINE Imaginary Number A COMPLEX NUMBER which has zero REAL PART ,so that it can be written as a REAL NUMBER multiplied by the "IMAGINARY UNIT " I (equal to the SQUARE ROOTffiffiffiffiffiffi /C281p ) :/ See also COMPLEX NUMBER ,G ALOIS IMAGINARY , GAUSSIAN INTEGER , I,IMAGINARY PART,IMAGINARY UNIT,REAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 211 /C1/16, 1996. Imaginary Part The imaginary part I[z]ofa COMPLEX NUMBER z /C30x /C27iy is the REAL NUMBER multiplying I,soI x /C27iy ½/C138 /C30 y: In terms of z itself, I z½/C138/C30z /C28 ¯z 2i; where ¯z is the COMPLEX CONJUGATE of z. The imaginary part is implemented in Mathematica as Im[z]. See also ABSOLUTE SQUARE ,A RGUMENT (COMPLEX NUMBER ), COMPLEX CONJUGATE ,C OMPLEX PLANE , MODULUS (COMPLEX NUMBER ), REAL PART References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 2, 1999. Imaginary Point A pair of values x and y one or both of which is COMPLEX . References Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 2, 1961. Imaginary Quadratic Field A QUADRATIC FIELD Q(ffiffiffiffi Dp ) with D B0. See also JUGENDTRAUM ,QUADRATIC FIELD Imaginary Unit The IMAGINARY NUMBER i /C30ffiffiffiffiffiffi /C281p ; i.e., the SQUARE ROOT of /C281. The imaginary unit is denoted and commonly referred to as "I." Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point /C27i and /C28i can then be distinguished. Since either choice is possible, there is no ambiguity in defining i as "the" square root of /C281. In Mathe- matica , the imaginary unit is implemented as I. See also COMPLEX NUMBER , I,IMAGINARY NUMBER , UNIT Immanant For an n/C29nmatrix, let Sdenote any permutation e1; e2;...,enof the set of numbers 1, 2, . . ., n, and let x(l)(S) be the character of the symmetric group corresponding to the partition ( l):Then the imma- nant amnjj(l)is defined as amnjj(l)/C30X x(l)(S)PS where the summation is over the n! permutations of the SYMMETRIC GROUP and PS /C30a1e1a2e2/C1/C1/C1anen: See also DETERMINANT ,PERMANENT References Littlewood, D. E. and Richardson, A. R. "Group Characters and Algebra." Philos. Trans. Roy. Soc. London A 233,99/C1/ 41, 1934. Littlewood, D. E. and Richardson, A. R. "Immanants of Some Special Matrices." Quart. J. Math. (Oxford) 5, 269 /C1/82, 1934. Wybourne, B. G. "Immanants of Matrices." §2.19 in Symme- try Principles and Atomic Spectroscopy. New York: Wiley, pp. 12 /C1/3, 1970. Immersed Minimal Surface ENNEPER’S MINIMAL SURFACE Immersion A special nonsingular MAP from one MANIFOLD to another such that at every point in the domain of the map, the DERIVATIVE is an injective linear map. This is equivalent to saying that every point in the DOMAIN has a NEIGHBORHOOD such that, up to DIFFEOMORPH- ISMS of the TANGENT SPACE , the map looks like the inclusion map from a lower-dimensional EUCLIDEAN SPACE to a higher-dimensional EUCLIDEAN SPACE . See also BOY SURFACE ,E VERSION ,SMALE- HIRSCH THEOREM ,SUBMERSION References Boy, W. "U¨ ber die Curvatura integra und die Topologie geschlossener Fla¨chen." Math. Ann 57, 151 /C1/84, 1903. Pinkall, U. "Models of the Real Projective Plane." Ch. 6 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 63 /C1/7, 1986. Immersion Theorem SMALE- HIRSCH THEOREM Impartial Game A GAME in which the possible moves are the same for each player in any position. All positions in all impartial GAMES form an additive ABELIAN GROUP . For impartial games in which the last player wins (normal form games), the nim-value of the sum of two GAMES is the nim-sum of their nim-values. If the last player loses, the GAME is said to be in mise`re form and the analysis is much more difficult. See also FAIR GAME,GAME,PARTISAN GAME Implicit Function A function which is not defined explicitly, but rather is defined in terms of an algebraic relationship (whichcan not, in general, be "solved" for the function in question). For example, the ECCENTRIC ANOMALY E of a body orbiting on an ELLIPSE with ECCENTRICITY e is defined implicitly in terms of the mean anomaly M by KEPLER’S EQUATION M /C30E /C28e sin E : Implicit Function Theorem Given F1(x; y;z;u ;v;w) /C300 (1) F2(x; y;z;u ;v;w) /C300 (2) F3(x; y;z;u ;v;w) /C300 (3) if the JACOBIAN JF(u;v ;w) /C30@(F1 ;F2 ;F3) @(u;v ;w)"0; (4) then u, v, and w can be solved for in terms of x, y, and z and PARTIAL DERIVATIVES of u, v, w with respect to x, y, and z can be found by differentiating implicitly. More generally, let A be an OPEN SET in Rn/C27k and let f : A 0 Rn be a C t FUNCTION . Write f in the form f(x; y); where x and y are elements of Rk and Rn : Suppose that (a, b) is a point in A such that f(a ;b) /C300 and the DETERMINANT of the n /C29n MATRIX whose elements are the DERIVATIVES of the n component FUNCTIONS of f with respect to the n variables, written as y, evaluated at (a, b), is not equal to zero. The latter may be rewritten as rank( Df(a ;b)) /C30n: (5) Then there exists a NEIGHBORHOOD B of a in Rk and a unique C t FUNCTION g : B 0 Rn such that g(a) /C30b and f(x;g(x))/C300 for all x/C23B:/ See also CHANGE OF VARIABLES THEOREM ,JACOBIAN References Munkres, J. R. Analysis on Manifolds. Reading, MA: Ad- dison-Wesley, 1991. Implies The CONNECTIVE inPROPOSITIONAL CALCULUS which has the meaning "‘if Ais true, then Bis also true." In formal terminology, the term CONDITIONAL is often used to refer to this connective (Mendelson 1997, p. 13). The symbol used to denote "implies" is A[B; A‡B(Carnap 1958, p. 8; Mendelson 1997, p. 13), or A0B:InMathematica 4.0, the command Implies- RealQ [ineqs1 ,ineqs2 ] can be used to determine if the system of real algebraic equations and inequalitiesineqs1 implies the system of real algebraic equations and inequalities ineqs2 . /A [B is an abbreviation for !A /C150B ; where !A denotes NOT and /C150denoted OR. [is a binary operator that is implement in Mathematica as Implies [A, B], and can not be extended to more than two arguments. /A [B has the following TRUTH TABLE (Carnap 1958, p. 10; Mendelson 1997, p. 13). AB /A [B/ TTT TFFFTT FFT If A [B and B [A (i.e, A [B fflB [A) ; then A and B are said to be EQUIVALENT , a relationship which is written symbolically as A UB; A XB ; or A /C13B (Car- nap 1958, p. 8). See also CONNECTIVE ,EQUIVALENT ,EXISTS ,FOR ALL, QUANTIFIER References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 8, 1958. Impossible Figure A class of ILLUSION in which an object which is physically unrealizable is apparently depicted. See also FREEMISH CRATE ,H OME PLATE ,ILLUSION , NECKER CUBE,PENROSE STAIRWAY ,TRIBAR References Cowan, T. M. "The Theory of Braids and the Analysis of Impossible Figures." J. Math. Psych. 11, 190 /C1/12, 1974. Cowan, T. M. "Supplementary Report: Braids, Side Seg- ments, and Impossible Figures." J. Math. Psych. 16, 254 /C1/ 60, 1977. Cowan, T. M. "Organizing the Properties of Impossible Figures." Perception 6,41/C1/6, 1977. Cowan, T. M. and Pringle, R. "An Investigation of the Cues Responsible for Figure Impossibility." J. Exper. Psy- ch. Human Perception Performance 4, 112 /C1/20, 1978. Ernst, B. Adventures with Impossible Figures. Stradbroke, England: Tarquin, 1987. Harris, W. F. "Perceptual Singularities in Impossible Pic- tures Represent Screw Dislocations." South African J. Sci. 69,10/C1/3, 1973. Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 119 /C1/22, 1996. Jablan, S. "Impossible Figures." http://members.tripod.com/ ~modularity/impos.htm and "Are Impossible Figures Pos- sible?" http://members.tripod.com/~modularity/kulpa.htm. Kulpa, Z. "Are Impossible Figures Possible?" Signal Proces- sing 5, 201 /C1/20, 1983. Kulpa, Z. "Putting Order in the Impossible." Perception 16, 201 /C1/14, 1987. Sugihara, K. "Classification of Impossible Objects." Percep- tion 11,65/C1/4, 1982.Terouanne, E. "Impossible Figures and Interpretations of Polyhedral Figures." J. Math. Psych. 27, 370 /C1/05, 1983. Terouanne, E. "On a Class of ‘Impossible’ Figures: A New Language for a New Analysis." J. Math. Psych. 22,24/C1/7, 1983. Thro, E. B. "Distinguishing Two Classes of Impossible Objects." Perception 12, 733 /C1/51, 1983. Wilson, R. "Stamp Corner: Impossible Figures." Math. Intell. 13, 80, 1991. Impredicative Definitions about a SET which depend on the entire SET. Improper Divisor A DIVISOR which is not a PROPER DIVISOR . See also DIVISOR ,PROPER DIVISOR Improper Fraction A FRACTION p=q > 1 : A FRACTION with p =q B1is called a PROPER FRACTION . Therefore, the special cases 1/1, 2/2, 3/3, etc. are generally considered to be improper. See also FRACTION ,M IXED FRACTION ,PROPER FRAC- TION Improper Integral An INTEGRAL which has either or both limits INFINITE or which has an INTEGRAND which approaches IN- FINITY at one or more points in the range of integra- tion. See also DEFINITE INTEGRAL ,INDEFINITE INTEGRAL , INTEGRAL ,PROPER INTEGRAL References Jeffreys, H. and Jeffreys, B. S. "Infinite and Improper Integrals." §1.104 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 33 /C1/4, 1988. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Improper Integrals." §4.4 in Numerical Re- cipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 135 /C1/40, 1992. Improper Node A FIXED POINT for which the STABILITY MATRIX has equal nonzero EIGENVECTORS . See also STABLE IMPROPER NODE,UNSTABLE IMPRO- PER NODE Improper Rotation The SYMMETRY OPERATION corresponding to a ROTA- TION followed by an INVERSION OPERATION , also called aROTOINVERSION . This operation is denoted ¯nfor an improper rotation by 360 8/nso the CRYSTALLOGRAPHY RESTRICTION gives only ¯1;¯2;¯3;¯4;¯6 for crystals. The MIRROR PLANE symmetry operation is (x; y;z) 0 (x;y;/C28z); etc., which is equivalent to ¯2:/ See also INVERSION OPERATION ,ROTATION ,SYMME- TRY OPERATION Impulse Pair The even impulse pair is the FOURIER TRANSFORM of cos(pk) ; P(x) /C131 2 d x /C27121CA}1CA$ /C2712d x /C28121CA}1CA$ : (1) It satisfies P(x) + f(x) /C3012 fx/C27121CA}1CA$ /C2712 fx/C28121CA}1CA$ ; (2) where + denotes CONVOLUTION , and g/C12 /C28/C12P(x)dx /C301: (3) The odd impulse pair is the FOURIER TRANSFORM of i sin( ps); II(x) /C1312 d x /C27121CA}1CA$ /C2812d x /C28121CA}1CA$ : (4) Impulse Symbol Bracewell’s term for the DELTA FUNCTION . See also DELTA FUNCTION ,IMPULSE PAIR References Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999. Inaccessible Cardinal An inaccessible cardinal is a CARDINAL NUMBER which cannot be expressed in terms of a smaller number of smaller cardinals. See also CARDINAL NUMBERInaccessible Cardinals Axiom INACCESSIBLE CARDINAL ,LEBESGUE MEASURABILITY PROBLEM Inadmissible A word or string which is not ADMISSIBLE . In-and-Out Curve A curve created by starting with a circle, dividing it into six arcs, and flipping three alternating arcs. Theprocess is then repeated an infinite number of times. Incenter The center Iof a TRIANGLE’S INCIRCLE . It can be found as the intersection of ANGLE BISECTORS , and it is the interior point for which distances to the sides of the triangle are equal. It has TRILINEAR COORDINATES 1:1:1 and homogeneous BARYCENTRIC COORDINATES (a;b;c):The distance between the incenter and CIRCUMCENTER isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R(R/C282r)p :/ The incenter lies on the N AGEL LINE and S ODDY LINE . The incenter lies on the E ULER LINE only for an ISOSCELES TRIANGLE . For an EQUILATERAL TRIANGLE , the CIRCUMCENTER O,CENTROID G,NINE-POINT CEN- TERF,ORTHOCENTER H, and DELONGCHAMPS POINT Zall coincide with I. The incenter and EXCENTERS of a TRIANGLE are an ORTHOCENTRIC SYSTEM . The POWER of the incenter with respect to the CIRCUMCIRCLE is p/C30a1a2a3 a1/C27a2/C27a3 (johnson 1929, p. 190). if the incenters of the TRIAN- GLES DA1H2H3;DA2H3A1;andDA3H1H2areX1;X2; andX3;then X2X3is equal and parallel to I2I3;where Hiare the FEET of the ALTITUDES and Iiare the incenters of the TRIANGLES . Furthermore, X1;X2;X3; are the reflections of Iwith respect to the sides of the TRIANGLE DI1I2I3(Johnson 1929, p. 193). See also CENTROID (ORTHOCENTRIC SYSTEM ), CIRCUM- CENTER ,C YCLIC QUADRILATERAL ,E XCENTER ,G ER- GONNE POINT ,INCIRCLE ,INRADIUS ,O RTHOCENTER , NAGEL LINE References Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 622, 1970. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 10, 1967. Dixon, R. Mathographics. New York: Dover, p. 58, 1991. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 182 /C1/94, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Kimberling, C. "Incenter." http://cedar.evansville.edu/~ck6/ tcenters/class/incenter.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 115 /C1/16, 1991. Incenter-Excenter Circle Given a triangle DA1A2A3 ; the points A1 ; I, and J1 lie on a line, where I is the INCENTER and J1is the EXCENTER corresponding to A1 : Furthermore, the CIRCLE with IJ1 as the DIAMETER has P as its center, where P is the intersection of A1J1 with the CIRCUM- CIRCLE of DA1A2A3 ; and passes through A2and A3 : This CIRCLE has RADIUS r /C301 2a1 sec12 a11CA}1CA$ /C302R sin12 a11CA}1CA$ : It arises because IJ1J2J3forms an ORTHOCENTRIC SYSTEM . See also CIRCUMCIRCLE ,E XCENTER ,E XCENTER- EX- CENTER CIRCLE ,INCENTER ,ORTHOCENTRIC SYSTEM References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 185, 1929.Incidence Axioms The eight of HILBERT’S AXIOMS which concern colli- nearity and intersection; they include the first four of EUCLID’S POSTULATES . See also ABSOLUTE GEOMETRY ,CONGRUENCE AXIOMS , CONTINUITY AXIOMS ,E UCLID’S POSTULATES ,H IL- BERT’S AXIOMS ,ORDERING AXIOMS ,PARALLEL POSTU- LATE References Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: Open Court, 1980. Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert’s System of Axioms." §163B in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 544 /C1/45, 1980. Incidence Matrix The incidence matrix of a GRAPH gives the ( 0,1)- MATRIX which has a row for each vertex and column for each edge, and ( v;e)/C301IFFvertex vis incident upon edge e(Skiena 1990, p. 135). The physicist Kirchhoff (1847) was the first to define the incidence matrix. The incidence matrix of a graph can becomputed using IncidenceMatrix [g] in the Math- ematica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ ). The incidence matrix Cof a graph and ADJACENCY MATRIX Lof its LINE GRAPH are related by L/C30CTC/C282I; where Iis the IDENTITY MATRIX (Skiena 1990, p. 136). For a k-D POLYTOPE Pk;the incidence matrix is defined by hk ij/C301i f Pi k/C281belongs to Pik 0i f Pik/C281does not belong Pik1C|} Theith row shows which Pk/s surround Pi k/C281;and the jth column shows which Pk/C281/s bound Pj k:Incidence matrices are also used to specify PROJECTIVE PLANES . The incidence matrices for a TETRAHEDRON ABCD are /h0 / 1 ABC 11111 / h1/ AD BD CD BC AC AB A 100011 B 010101 C 001110 D 111000 / h2/ BCD ACD ABD ABC AD 0110 BD 1010 CD 1100 BC 1001 AC 0101 AB 0011 /h3/ ABCD BCD 1 ACD 1 ABD 1 ABC 1 See also ADJACENCY MATRIX , K-CHAIN , K-CIRCUIT , INTEGER MATRIX References Bruck, R. H. and Ryser, H. J. "The Nonexistence of Certain Finite Projective Planes." Canad. J. Math. 1,88/C1/3, 1949. Kirchhoff, G. "U¨ ber die Auflo¨sung der Gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer Stro¨me gefu¨hrt wird." Ann. Phys. Chem. 72, 497 /C1/08, 1847. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 135 /C1/36, 1990.Incident Two objects which touch each other are said to be incident. See also CONCUR ,TANGENT CURVES Incircle The INSCRIBED CIRCLE of a TRIANGLE DABC :The center Iof the incircle is called the INCENTER and the RADIUS rthe INRADIUS . The points of intersection of the incircle with Tare the VERTICES of the PEDAL TRIANGLE ofTwith the INCENTER as the PEDAL POINT (cf. TANGENTIAL TRIANGLE ). This TRIANGLE is called the CONTACT TRIANGLE . There are four CIRCLES that are tangent all three sides (or their extensions) of a given TRIANGLE : the incircle Iand three EXCIRCLES J1;J2;and J3:These four circles are, in turn, all touched by the NINE-POINT CIRCLE N. The TRILINEAR COORDINATES of the INCENTER are 1 : 1:1 :The INRADIUS rand horizontal position of the INCENTER xIfor a given triangle with two angles A and Cand adjacent side of length bis given by simultaneously solving the equations tan1 2A1CA}1CA$ /C30r xI(1) tan12C1CA}1CA$ /C30r b/C28xI; (2) giving r/C30tan1 2A1CA}1CA$ tan12C1CA}1CA$ tan1 2A1CA}1CA$ /C27tan12C1CA}1CA$ b (3) xI/C30tan12C1CA}1CA$ tan1 2A1CA}1CA$ /C27tan12C1CA}1CA$ b; (4) whereas the ALTITUDE height hand horizontal posi- tion xhof the ALTITUDE , are given by h/C30tanC tanA/C27tanCb (5) xh/C30tanAtanC tanA/C27tanCb: (6) The AREA Dof the TRIANGLE DABC is given by D/C30DBIC/C27DAIC/C27DAIB /C301 2ar/C2712br/C2712cr/C3012(a/C27b/C27c)r/C30sr; (7) where sis the SEMIPERIMETER , so the INRADIUS is r/C30D s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s/C28a)(s/C28b)(s/C28c) ss (8) Using the incircle of a TRIANGLE as the INVERSION CENTER , the sides of the TRIANGLE and its CIRCUM- CIRCLE are carried into four equal CIRCLES (Honsber- ger 1976, p. 21). Pedoe (1995, p. xiv) gives a GEOMETRIC CONSTRUCTION for the incircle. Let a triangle DABC have INCIRCLE with INCENTER I and let the incircle be tangent to DABC atTA;TC; (and TB; not shown). Then the lines CI,TATC;and the perpendicular to CIthrough ACONCUR in a point P (Honsberger 1995). Given a triangle, draw a C EVIAN to one of the bases which divides it into two triangles having congruent incircles. The positions and sizes of these two cir- cumcircles can then be determined by simultaneouslysolving the eight equations x 1/C30tan1 2u121CA}1CA$ tan12u11/C27tan12u121CA}1CA$1CA}1CA$ d1 (9) x2/C30tan1 2u221CA}1CA$ tan1 2u211CA}1CA$ /C27tan12u221CA}1CA$ d2 (10) a/C30tan12u111CA}1CA$ tan12u121CA}1CA$ tan1 2u111CA}1CA$ /C27tan12u121CA}1CA$ d1 (11) a/C30tan12u211CA}1CA$ tan12u221CA}1CA$ tan1 2u211CA}1CA$ /C27tan12u221CA}1CA$ d2 (12) h/C30tanu11tanu12 tanu11/C27tanu12d1 (13) h/C30tanu21tanu22 tanu21/C27tanu22d2 (14) d/C30d1/C27d2 (15) p/C30u12/C27u21 (16) for the eight variables d1;d2;u12;u21;a,x1;x2;andh, with u11;u22;and dgiven. Generalizing to ncon- gruent circles gives the 4 nequations xi/C30tan12ui21CA}1CA$ tan1 2ui11CA}1CA$ /C27tan12ui21CA}1CA$ di (17) a/C30tan12ui11CA}1CA$ tan12ui21CA}1CA$ tan1 2ui11CA}1CA$ /C27tan12ui21CA}1CA$ di (18) h/C30tanui1tanui2 tanui1/C27tanui2di (19) fori/C301 , ... , n, ui2/C27ui/C271;1/C30p (20) fori/C301 , ... , n/C281;and d/C30Xn i/C301di (21) to be solved for the unknowns diandxi(nof them), ui1 andui2(/n/C282 of each for i/C302 ,... , n/C281);andu12;un1; a, and h, a total of n/C27n/C272(n/C282)/C274/C304nun- knowns. Given an arbitrary TRIANGLE , let n/C281 Cevians be drawn from one of its vertices so all of the ntriangles so determined have equal incircles. Then the incircles determined by spanning 2, 3, ..., n /C281 adjacent triangles are also equal (Wells 1991, p. 67). See also CIRCUMCIRCLE ,C ONGRUENT INCIRCLES POINT ,CONTACT TRIANGLE ,EQUAL INCIRCLES THEO- REM,EXCIRCLE ,INCENTER ,INRADIUS ,JAPANESE THE- OREM ,SEVEN CIRCLES THEOREM ,TANGENT CIRCLES , TANGENTIAL TRIANGLE ,TRIANGLE TRANSFORMATION PRINCIPLE References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 53 /C1/5, 1888. Coxeter, H. S. M. and Greitzer, S. L. "The Incircle and Excircles." §1.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 10 /C1/3, 1967. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., 1976. Honsberger, R. "An Unlikely Concurrence." §3.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geome- try. Washington, DC: Math. Assoc. Amer., pp. 31 /C1/2, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 182 /C1/94, 1929. Lachlan, R. "The Inscribed and the Escribed Circles." §126 /C1/ 28 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 72 /C1/4, 1893. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Inclusion Map Given a SUBSET B of a SET A, the INJECTION f : B 0 A defined by f(b) /C30b for all b /C23 B is called the inclusion map. See also LONG EXACT SEQUENCE OF A PAIR AXIOM Inclusion-Exclusion Principle Let Ajjdenote the CARDINALITY of set A, then it follows immediately that A @ B jj /C30Ajj/C27Bjj/C28 A S B jj ; where @ denotes UNION , and S denotes INTERSECTION . This formula can be generalized in the following beautiful manner. Let A /C30fAi gp i/C301be a P-SYSTEM of S consisting of sets A1 ; ...,Ap ; then A1 @ A2 @ ...@ Ap1CA|1CA|1CA|1CA|/C30X 15i5pAijj/C28X 1 5i1Bi2 5pAi1 S Ai2 jj /C27X 15i1 Bi2 Bi3 5pAi1 S Ai2 S Ai3 jj /C28... /C27(/C281)p /C281 Ai1 S Ai2 S ...S Ap1CA|1CA|1CA|1CA|; where the sums are taken over K-SUBSETS of A : This formula holds for infinite sets S as well as finite sets (Comtet 1972, p. 177).The principle of inclusion-exclusion was used by Nicholas Bernoulli to solve the recontres problem of finding the number of DERANGEMENTS (Bhatnagar 1995, p. 8). The following Mathematica programs give a list of the subsets appearing under each sum and the contribution each sum makes to the total. BB DiscreteMath‘Combinatorica‘; InclusionExclusionSubets[a_List] : /C30 Module[{n, p /C30 Length[a]}, Table[Intersection @@ a[[#]] & /@ KSubsets[Range[p], n], {n, p}] ] InclusionExclusionTerms[a_List] : /C30 Module[{n, p /C30 Length[a]}, Table[(-1)^(n - 1)Plus @@ Length /@ (Intersection @@ a[[#]] & /@ KSubsets[Range[p], n]), {n, p}] ] For example, for the three subsets A1 /C30f2;3 ;7;9 ;10g; A2 /C30f1 ;2;3 ;9g; and A3 /C30f2 ;4;9 ;10g of S /C30 f1; 2;...; 10g; the following table summarizes the terms appearing the sum. # term set length 1 /A1/ {2, 3, 7, 9, 10} 5 /A2/ {1, 2, 3, 9} 4 /A3/ {2, 4, 9, 10} 4 2 /A1SA2/ {2, 3, 9} 3 /A1SA3/ {2, 9, 10} 3 /A2SA3/ {2, 9} 2 3 /A1SA2SA3/{2, 9} 2 /A1@A2@A3 jj is therefore equal to (5 /C274/C274)/C28(3/C27 3/C272)/C272/C307;corresponding to the seven elements A1@A2@A3/C30f1;2;3;4;7;9;10g:/ See also BAYES’ THEOREM References Andrews, G. E. Number Theory. Philadelphia, PA: Saun- ders, pp. 139 /C1/40, 1971. Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 60, 1986. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and Their U (n) Extensions. Ph.D. thesis. Ohio State University, 1995. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 176 /C1/77, 1974. da Silva. "Proprietades geraes." J. de l’Ecole Polytechnique , cah. 30. de Quesada, C. A. "Daniel Augusto da Silva e la theoria delle congruenze binomie." Ann. Sci. Acad. Polytech. Porto, Co/¯1/ mbra 4, 166 /C1/92, 1909. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, pp. 178 /C1/79, 1997. Sylvester, J. "Note sur la the´ore`me de Legendre." C. R. Acad. Sci. Paris 96, 463 /C1/65, 1883. Inclusive Disjunction A DISJUNCTION that remains true if either or both of its arguments are true. This is equivalent to the OR CONNECTIVE . By contrast, the EXCLUSIVE DISJUNCTION is true if only one, but not both, of its arguments are true, and is false if neither or both are true, which is equivalent to the XOR connective. See also DISJUNCTION ,EXCLUSIVE DISJUNCTION , OR, XOR Incommensurate Two lengths are called incommensurate or incom- mensurable if their ratio cannot be expressed as a ratio of whole numbers. IRRATIONAL NUMBERS and TRANSCENDENTAL NUMBERS are incommensurate with the integers. See also FRACTION ,IRRATIONAL NUMBER ,PYTHAGOR- AS’S CONSTANT ,TRANSCENDENTAL NUMBER Incomparable Rectangles Two RECTANGLES , neither of which will fit inside the other, are said to be incomparable. This is equivalent to one rectangle being both longer and narrower. At least seven and at most eight mutually incomparable rectangles are needed to tile a given rectangle (Wells 1991). See also RECTANGLE References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 116 /C1/17, 1991. Incomplete Beta Function A generalization of the complete BETA FUNCTION defined by B(z;a ;b) /C13gz 0ua /C281(1 /C28u)b /C281du /C30za1 a /C271 /C28 b a /C27 1z /C27.../C27(1 /C28 b) /C1/C1/C1(n /C28 b) n!(a /C27 n)zn /C27..."# : The symbol Bz(a ;b) is sometimes also used. The incomplete beta function B(z;a ;b) reduces to the use BETA FUNCTION B(a; b) when z /C301,B(1;a;b) /C30B(a ;b) The incomplete beta function is implemented in Mathematica asBeta [z, a, b]. See also BETA FUNCTION ,REGULARIZED BETA FUNC- TION Incomplete Gamma Function The "complete" GAMMA FUNCTION G(x) can be general- ized to the incomplete gamma function G(a ;x) such that G(a) /C30G(a;0): This "upper" incomplete gamma function is given by G(a ;x) /C13g/C12 xta /C281e /C28tdt: (1) For a an INTEGER n G(n;x) /C30(n /C281)!e /C28xXn /C281 s/C300xs s! /C30(n /C281)!e /C28xen /C281(x); (2) where es is the EXPONENTIAL SUM FUNCTION . The lower incomplete gamma function is given by g(a ;x) /C13gx 0ta /C281e /C28tdt a /C281xae /C28x 1F1(1;1 /C27a;x) a /C281xa 1F1(a;1/C27a; /C28x) ; (3) where1F1(a;b;x) is the CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND . For a an INTEGER n, g(n;x) /C30(n /C281)! 1 /C28e /C28xXn /C281 k/C300xk k! ! /C30(n /C281)! 1 /C28e /C28xen/C281(x) ½/C138 : (4) The function G(a ;z) is denoted Gamma [a, z] and the function g(a ;z) is denoted Gamma [a,0,z]in Mathe- matica . By definition, the two incomplete functions satisfy G(a ;x) /C27g(a; x) /C30G(a): (5) See also GAMMA FUNCTION ,R EGULARIZED GAMMA FUNCTION Incompleteness A formal theory is said to be incomplete if it contains fewer theorems than would be possible while still retaining CONSISTENCY . See also CONSISTENCY ,G O¨ DEL’S INCOMPLETENESS THEOREM References Chaitin, G. J. "G. J. Chaitin’s Home Page." http:// www.cs.auckland.ac.nz/CDMTCS/chaitin/. Increasing Function A function f(x) increases on an INTERVAL I if f(b) > f(a) for all b /C21a, where a;b /C23 I : Conversely, a function f(x) decreases on an INTERVAL I if f(b) Bf(a) for all b /C21a with a ;b /C23 I :/ If the DERIVATIVE f ?(x)ofa CONTINUOUS FUNCTION f(x) satisfies f ?(x) > 0onan OPEN INTERVAL (a, b), then f(x) is increasing on (a, b). However, a function may increase on an interval without having a derivative defined at all points. For example, the function x1 =3 is increasing everywhere, including the origin x /C300, despite the fact that the DERIVATIVE is not defined at that point. See also DECREASING FUNCTION ,DERIVATIVE ,N ON- DECREASING FUNCTION ,NONINCREASING FUNCTION References Jeffreys, H. and Jeffreys, B. S. "Increasing and Decreasing Functions." §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988. Increasing Sequence For a SEQUENCE anfg ; if an/C271 /C28an > 0 for n ]x; then an is increasing for n ]x: Conversely, if an/C271 /C28an B0 for n ]x; then an is DECREASING for n ]x:/ If an > 0 and an/C271 =an > 1 for all n ]x; then anis increasing for n ]x: Conversely, if an > 0 and an/C271 =an B1 for all n ]x; then anis decreasing for n ]x:/ See also DECREASING SEQUENCE ,SEQUENCE Indecomposable A P-FORM a is indecomposable if it cannot be written as the WEDGE PRODUCT of ONE-FORMS a /C30 b1 ffl...ffl bp : A p-form that can be written as such a product is called DECOMPOSABLE . See also DECOMPOSABLE ,DIFFERENTIAL K-FORM Indefinite Integral An INTEGRAL gf(x)dx without upper and lower limits, also called an ANTI- DERIVATIVE . The first FUNDAMENTAL THEOREM OF CALCULUS allows DEFINITE INTEGRALS to be computed in terms of indefinite integrals. If F is the indefinite integral for f(x) ; then gb af(x)dx /C30F(b) /C28F(a) : The question of which definite integrals can beexpressed in terms of elementary function is not susceptible to any established theory. In fact, the problem belongs to transcendence theory, which appears to be "infinitely hard." For example, there are definite integrals that are equal to the EULER- MASCHERONI CONSTANT g : However, the problem of deciding whether g can be expressed in terms of the values at rational values of elementary functions involves the decision as to whether g is rational or algebraic, which is not known. See also ANTIDERIVATIVE ,CALCULUS ,DEFINITE INTE- GRAL ,FUNDAMENTAL THEOREMS OF CALCULUS ,INTE- GRAL Indefinite Quadratic Form A QUADRATIC FORM Q(x) is indefinite if it is less than 0 for some values and greater than 0 for others. The QUADRATIC FORM , written in the form (x;Ax) ; is indefinite if EIGENVALUES of the MATRIX A are of both signs. See also POSITIVE DEFINITE QUADRATIC FORM,POSI- TIVE SEMIDEFINITE QUADRATIC FORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1106, 2000. Indefinite Summation Operator The indefinite summation operator D/C281 for discrete variables, is the equivalent of integration for contin- uous variables. If DY(x) /C30y(x) then D/C281y(x) /C30Y(x) :/ Indegree The number of inward directed EDGES from a given VERTEX in a DIRECTED GRAPH . See also LOCAL DEGREE ,OUTDEGREE Independence Axiom A rational choice between two alternatives should depend only on how they differ. Independence Complement Theorem If sets EandFare INDEPENDENT , then so are Eand F?;where F?is the complement of F(i.e., the set of all possible outcomes not contained in F). Let@denote "or" and Sdenote "and." Then P(E)/C30PE F@EF? ðÞ (1) /C30P(EF)/C27PE F? ðÞ/C28PE FSEF? ðÞ ; (2) where ABis an abbreviation for ASB:ButEandF are independent, so P(EF)/C30P(E)P(F): (3) Also, since FandF?are complements, they contain no common elements, which means that PEFS EF ? ðÞ /C300 (4) for any E. Plugging (4) and (3) into (2) then gives P(E) /C30P(E)P(F) /C27PEF? ðÞ : (5) Rearranging, PEF? ðÞ/C30P(E)[1 /C28P(F)] /C30P(E)PF?ðÞ ; (6) Q.E.D. See also INDEPENDENT SET Independence Number The independence number a(G) of a graph is the cardinality of the largest INDEPENDENT SET. For- mally, a(G) /C30max Ujj: U ƒV independent ðÞ for a GRAPH G, where Ujjdenotes the CARDINALITY of the set U. The independence number of the DE BRUIJN GRAPH of order n is given by 1, 2, 3, 7, 13, 28, ... (Sloane’s A006946). By definition, the independence number of a graph G plus the number of elements in a minimal VERTEX COVER of G equals the number of vertices in the graph. See also INDEPENDENT SET,VERTEX COVER References Skiena, S. "Maximum Independent Set" §5.6.3 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 218 /C1/19, 1990. Sloane, N. J. A. Sequences A006946/M0834 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Independent Equations LINEARLY INDEPENDENT Independent Events Two events A and B are called independent if their probabilities satisfy P(AB) /C30P(A)P(B) (Papoulis 1984, p. 40). See also EVENT ,INDEPENDENT STATISTICS References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Independent Sequence STRONGLY INDEPENDENT ,W EAKLY INDEPENDENT Independent Set Two sets A and B are said to be independent if their INTERSECTION A S B /C30¥; where ¥ is the EMPTY SET.For example, fA;B ;Cg and fD ;E g are independent, but fA;B ;C g and fC ;D; Eg are not. Independent sets are also called DISJOINT or mutually exclusive. An independent set of a GRAPH G is a subset of the vertices such that no two vertices in the subset represent an edge of G. Given a VERTEX COVER of a GRAPH , all vertices not in the cover define an independent set (Skiena 1990, p. 218). The INDEPEN- DENCE NUMBER of a graph is the cardinality of the largest independent set. A maximum independent set of a graph can be computed using MaximumInde- pendentSet [g] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). An independent set of edges can be defined similarly (Skiena 1990, p. 219). Gallai (1959) showed that the size of the minimum EDGE COVER plus the side of the maximum number of independent edges equals the number of vertices of a graph. See also CLIQUE ,DISJOINT SETS,EDGE COVER ,EMPTY SET,INDEPENDENCE NUMBER ,INTERSECTION ,VENN DIAGRAM ,VERTEX COVER References Gallai, T. "U¨ ber extreme Punkt- und Kantenmengen." Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 2, 133 /C1/38, 1959. Skiena, S. "Maximum Independent Set" §5.6.3 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 218 /C1/19, 1990. Independent Statistics Two variates A and B are statistically independent IFF the CONDITIONAL PROBABILITY P(A½B)ofA given B satisfies P(A½B) /C30P(A); (1) in which case the probability of A and B is just P(A;B) /C30P(A S B) /C30P(A)P(B) : (2) Similarly, n events A1 ; A2 ; ...,An are independent IFF p Sn i/C301Ai1CA%1CAP /C30Yn i/C301P(Ai): (3) Statistically independent variables are always UN- CORRELATED , but the converse is not necessarily true. See also BAYES’ FORMULA ,CONDITIONAL PROBABIL- ITY,INDEPENDENT EVENTS ,INDEPENDENCE COMPLE- MENT THEOREM ,UNCORRELATED Independent Vertices A set of VERTICES A of a GRAPH with EDGES V is independent if it contains no EDGES . See also INDEPENDENCE NUMBER Indeterminate Not definitively or precisely determined. Certain forms of LIMITS are said to be indeterminate when merely knowing the limiting behavior of individual parts of the expression is not sufficient to actually determine the overall limit. For example, a LIMIT OF THE FORM 0/0, i.e., limx00 f(x)=g(x) where limx00 f(x) /C30limx00 g(x) /C300; is indeterminate since the value of the overall limit actually depends on the limiting behavior of the combination of the two functions (e.g. limx00 x=x /C301; while limx00 x2 =x /C300):/ See also AMBIGUOUS ,L IMIT,TRIVIAL ,U NDEFINED , WELL DEFINED Indeterminate Problems DIOPHANTINE EQUATION Index The word "index" has a very large number of completely different meanings in mathematics. Most commonly, it is used in the context of an INDEX SET, where it means a quantity which can take on a set of values and is used to designate one out of a number of possible values associated with this value. For exam- ple, the subscript i in the symbol ai could be called the index of a. In a RADICALffiffiffixp; the quantity n is called the index. The word index has a special meaning in economics, where it refers to a single quantity used to quantify the "average" value of a possibly complicated set of quantities. In this context, it is sometimes called an INDEX NUMBER . In TOPOLOGY , INDEX THEORY refers to the study of topological invariants of MANIFOLDS . See also INDEX LOWERING ,INDEX RAISING ,INDEX SET, MANIFOLD ,M ULTIPLICATIVE ORDER ,STATISTICAL IN- DEX Index (Extension Field) DEGREE (EXTENSION FIELD Index (Modulo) MULTIPLICATIVE ORDER Index (Residue) MULTIPLICATIVE ORDER Index (Subgroup) This entry contributed by NICOLAS BRAYFor a SUBGROUP H of a GROUP G, the index of H, denoted (G : H) ; is the CARDINALITY of the set of LEFT COSETS of H in G (which is equal to the CARDINALITY of the set of RIGHT COSETS of H in G). See also COSET ,LAGRANGE’S GROUP THEOREM ,LEFT COSET ,RIGHT COSET Index (Tensor) See also INDEX LOWERING ,INDEX RAISING Index Law EXPONENT LAWS Index Lowering The indices of a CONTRAVARIANT TENSOR Aj can be lowered, turning it into a COVARIANT TENSOR Ai ; by multiplication by a so-called METRIC TENSOR , e.g., gijAj /C30Ai : See also CONTRAVARIANT TENSOR ,COVARIANT TEN- SOR,INDEX RAISING ,INDEX (TENSOR ), TENSOR Index Number ASTATISTIC which assigns a single number to several individual statistics in order to quantify trends. The best-known index in the United States is the con-sumer price index, which gives a sort of "average" value for inflation based on price changes for a group of selected products. The Dow Jones and NASDAQindexes for the New York and American Stock Exchanges, respectively, are also index numbers. Letp nbe the price per unit in period n,qnbe the quantity produced in period n, and vn/C13pnqnbe the value of the nunits. Let qabe the estimated relative importance of a product. There are several types of indices defined, among them those listed in thefollowing table. Index Abbr. Formula B OWLEY INDEX /PB//1 2PL/C27PP ðÞ / FISHER INDEX /PF//ffiffiffiffiffiffiffiffiffiffiffiffi PLPPp / GEOMETRIC MEAN INDEX /PG//Qpn p01CA%1CAP v0"#1=Sv0 / HARMONIC MEAN INDEX /PH//ap0q0 ap2 0q0 nm/ LASPEYRES’ INDEX /PL//apnq0 ap0q0/ MARSHALL- EDGEWORTH INDEX/PME//apn(q0 /C27 qn) a(v0 /C27 vn) / MITCHELL INDEX /PM//apnqn ap0qn/ PAASCHE’S INDEX /PP//apnqn ap0qn/ WALSH INDEX /PW//affiffiffiffiffiffiffiffiffiffiq0qnppn affiffiffiffiffiffiffiffiffiffiq 0qappn/ See also BOWLEY INDEX ,FISHER INDEX ,GEOMETRIC MEAN INDEX ,H ARMONIC MEAN INDEX ,LASPEYRES’ INDEX ,M ARSHALL- EDGEWORTH INDEX ,M ITCHELL IN- DEX,PAASCHE’S INDEX ,W ALSH INDEX References Fisher, I. The Making of Index Numbers: A Study of Their Varieties, Tests and Reliability, 3rd ed. New York: Augustus M. Kelly, 1967. Kenney, J. F. and Keeping, E. S. "Index Numbers." Ch. 5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 64 /C1/4, 1962. Mudgett, B. D. Index Numbers. New York: Wiley, 1951. Index Raising The indices of a COVARIANT TENSOR Aj can be raised, forming a CONTRAVARIANT TENSOR Ai ; by multiplica- tion by a so-called METRIC TENSOR , e.g., gijAj /C30Ai (1) See also CONTRAVARIANT TENSOR ,COVARIANT TEN- SOR,INDEX LOWERING ,INDEX (TENSOR ), TENSOR Index Set A SET whose members index (label) members of another set. For example, in the set A /C30@k /C23K Ak ; the set K is an index set of the set A. See also SET Index Theory A branch of TOPOLOGY dealing with topological invariants of MANIFOLDS . References Roe, J. Index Theory, Coarse Geometry, and Topology of Manifolds. Providence, RI: Amer. Math. Soc., 1996.Upmeier, H. Toeplitz Operators and Index Theory in Several Complex Variables. Boston, MA: Birkha ¨user, 1996. Indicator References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 104, 1971. Indicatrix A spherical image of a curve. The most common indicatrix is DUPIN’S INDICATRIX . See also DUPIN’S INDICATRIX Indicial Equation The RECURRENCE RELATION obtained during applica- tion of the FROBENIUS METHOD of solving a second- order ordinary differential equation. The indicial equation (also called the CHARACTERISTIC EQUATION ) is obtained by noting that, by definition, the lowest order term xk (that corresponding to n /C300) must have a COEFFICIENT of zero. For an example of the construction of an indicial equation, see BESSEL DIFFERENTIAL EQUATION . 1. If the two ROOTS are equal, only one solution can be obtained. 2. If the two ROOTS differ by a noninteger, two solutions can be obtained. 3. If the two ROOTS differ by an INTEGER , the larger will yield a solution. The smaller may or may not. References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 532 /C1/34, 1953. Indifference Principle INSUFFICIENT REASON PRINCIPLE Individual One of the basic objects treated in a given formal language system. The term is sometimes also used as a synonym for URELEMENT . See also URELEMENT References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 4, 1958. Induced Map If f :(X ; A) 0 (Y ;B) is homotopic to g :(X ;A) 0 (Y ; B) ; then f+ : Hn(X ;A) 0 Hn(Y ;B) and g + : Hn(X ;A) 0 Hn(Y ;B) are said to be the induced maps. See also EILENBERG- STEENROD AXIOMS Induced Norm NATURAL NORM Induced Representation If a SUBGROUP H of G has a REPRESENTATION f : H /C29W 0 W ; then there is a unique induced repre- sentation of G on a VECTOR SPACE V. The original space W is contained in V, and in fact, V /C30/C154s /C23G =H sW ; where sW is a copy of W. The induced representation on V is denoted IndG H :/ Alternatively, the induced representation is the /CG/- MODULE IndGH #CG /C156CH W : (1) Also, it can be viewed as W-valued functions on G which commute with the H action. IndG H #ff : G 0 W : hf(g) /C30f(hg)g: (2) The induced representation is also determined by its UNIVERSAL PROPERTY : HomH(W ; Res U) /C30HomG(Ind W ;U) ; (3) where U is any representation of G. Also, the induced representation satisfies the following formulas. 1. Ind /C154Wi /C30/C156Ind Wi :/ 2. U /C156Ind W /C30Ind(Res( U) /C156W) for any REPRESEN- TATION U. 3. IndG H(W) /C30IndGK(IndKHW) when H 5K 5G :/ Some of the CHARACTERS of G can be calculated from the CHARACTERS of H, as induced representations, using FROBENIUS RECIPROCITY .ARTIN’S RECIPROCITY THEOREM says that the induced representations of CYCLIC SUBGROUPS of a FINITE GROUP G generates a LATTICE of finite index in the lattice of VIRTUAL CHARACTERS .BRAUER’S THEOREM says that the vir- tual characters are generated by the induced repre- sentations from P-ELEMENTARY SUBGROUPS . See also ARTIN’S RECIPROCITY THEOREM ,FROBENIUS RECIPROCITY ,GROUP ,IRREDUCIBLE REPRESENTATION , REPRESENTATION ,R ESTRICTION (REPRESENTATION ), TENSOR PRODUCT (VECTOR SPACE ), VECTOR SPACE References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991.Induced Subgraph An induced subgraph is a subset of the edges of a GRAPH G together with any edges whose endpoints are both in this subset. The figure above illustrates the subgraph induced on the COMPLETE GRAPH K5 by the vertex subset f1;2 ;3;5 ;7;10 g: An induced sub- graph that is a COMPLETE GRAPH is called a CLIQUE . Any induced subgraph of a COMPLETE GRAPH forms a CLIQUE . An induced subgraph can be computed using InduceSubgraph [g] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also CLIQUE ,SUBGRAPH References Skiena, S. "Induced Subgraphs." §3.2.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 90 /C1/2, 1990. Induction The use of the INDUCTION PRINCIPLE in a PROOF . Induction used in mathematics is often called MATH- EMATICAL INDUCTION . See also PRINCIPLE OF STRONG INDUCTION ,PRINCIPLE OF TRANSFINITE INDUCTION ,P RINCIPLE OF WEAK INDUCTION References Buck, R. C. "Mathematical Induction and Recursive Defini- tions." Amer. Math. Monthly 70, 128 /C1/35, 1963. Se´roul, R. "Reasoning by Induction." §2.14 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 22 /C1/5, 2000. Induction Axiom The fifth of PEANO’S AXIOMS , which states: If a SET S of numbers contains zero and also the successor of every number in S, then every number is in S. See also PEANO’S AXIOMS Induction Principle The truth of an INFINITE sequence of propositions Pi fori/C301 ,... ,/C12is established if (1) P1is true, and (2) PkIMPLIES Pk/C271for all k. References Courant, R. and Robbins, H. "The Principle of Mathematical Induction" and "Further Remarks on Mathematical In- duction." §1.2.1 and 1.7 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 9 /C1/1 and 18 /C1/0, 1996. Apostol, T. M. "The Principle of Mathematical Induction." §I 4.2 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, p. 34, 1967. Inequality A mathematical statement that one quantity is greater than or less than another. "a is less than b" is denoted a Bb, and "a is greater than b" is denoted a /C21b."a is less than or equal to b" is denoted a 5b; and "a is greater than or equal to b" is denoted a ]b: The symbols a /C10b and a /C27b are used to denote "a is much less than b" and "a is much greater than b," respectively. Solutions to the inequality x /C28a jjBb consist of the set fx : a /C28b Bx /C28a /C27b g; or equivalently fx : a /C28b B x Ba /C27bg: Solutions to the inequality x /C28a jj > b consist of the set fx : x /C28a > b g@fx : x /C28a B/C28b g: If a and b are both POSITIVE or both NEGATIVE and a Bb, then 1=a > 1=b: The portions of the xy-plane satisfying a number of specific inequalities are illu- strated above. In Mathematica 4.0, the command InequalityIn- stance [ineqs , vars] can be used to find a real solution of the system of real equations and inequal- ities ineqs in the variables vars or return the EMPTY SET if no such solution exists. Solution of inequalities can be performed using [ineqs , vars], in the Mathe- matica add-on package Algebra‘Inequality- Solve‘ (which can be loaded with the command BBAlgebra‘ ) or directly using CylindricalAl- gebraicDecomposition [ineqs , vars]. See also CYLINDRICAL ALGEBRAIC DECOMPOSITION , EQUALITY ,EXISTS ,FOR ALL,INEQUATION ,Q UANTI- FIER,STRICT INEQUALITYReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Beckenbach, E. F. and Bellman, Richard E. An Introduction to Inequalities. New York: Random House, 1961. Beckenbach, E. F. and Bellman, Richard E. Inequalities, 2nd rev. print. Berlin: Springer-Verlag, 1965. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, 1952. Kazarinoff, N. D. Geometric Inequalities. New York: Ran- dom House, 1961. Mitrinovic, D. S. Analytic Inequalities. New York: Springer- Verlag, 1970. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical & New Inequalities in Analysis. Dordrecht, Netherlands: Kluwer, 1993. Mitrinovic, D. S.; Pecaric, J. E.; Fink, A. M. Inequalities Involving Functions & Their Integrals & Derivatives. Dordrecht, Netherlands: Kluwer, 1991. Mitrinovic, D. S.; Pecaric, J. E.; and Volenec, V. Recent Advances in Geometric Inequalities. Dordrecht, Nether- lands: Kluwer, 1989. Weisstein, E. W. "Books about Inequalities." http:// www.treasure-troves.com/books/Inequalities.html. Inequation While an equality A /C30B states that two mathematical expressions are equal, an inequation A "B states that two expressions are not equal. See also EQUATION ,INEQUALITY ,STRICT INEQUALITY Inexact Differential An infinitesimal which is not the differential of an actual function and which cannot be expressed as dz /C30@z @x ! ydx /C27@z @y ! zdy; the way an EXACT DIFFERENTIAL can. Inexact differ- entials are denoted with a bar through the d. The most common example of an inexact differential is the change in heat dQ encountered in thermodynamics. See also EXACT DIFFERENTIAL ,PFAFFIAN FORM References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 277, 1997. Zemansky, M. W. Heat and Thermodynamics, 5th ed. New York: McGraw-Hill, p. 38, 1968. Inf INFIMUM ,INFIMUM LIMIT Infimum Portions of this entry contributed by JEROME R. BREITENBACH The infimum is the greatest lower bound of a SET S, defined as a quantity m such that no member of the SET is less than m, but if e is any POSITIVE quantity, however small, there is always one member that is less than m /C27 e (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., R does not exist), the infimum is denoted inf S or infx /C23S x: The infimum can be computed using the Mathematica 4.0 command Infimum [f, constr , vars]. More formally, the infimum inf S for S a (nonempty) SUBSET of the extended reals R /C30R @f9/C12 g is the largest value y /C23R such that for all x /C23 S we have x ]y: Using this definition, infS always exists and, in particular, R /C30/C28/C12:/ Whenever an infimum exists, its value is unique. See also INFIMUM LIMIT,LOWER BOUND ,SUPREMUM References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Jeffreys, H. and Jeffreys, B. S. "Upper and Lower Bounds." §1.044 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 13, 1988. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 6, 1996. Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, p. 31, 1988. Rudin, W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, p. 7, 1987. Infimum Limit Given a sequence of real numbers an ; the infimum limit, also called the lower limit but more often simply pronounced ‘lim-inf’ and written liminf is the limit of An /C30inf k >nak as n 0/C12: Note that by definition, Anis nondecreas- ing, and so either has a limit or tends to /C12: For example, suppose an /C30(/C281)n =n; then for n odd, An /C30 /C281=n; and for n even, An /C30/C281=(n /C271): Another example is an /C30sin n; in which case Anis a constant sequence An /C30/C281 :/ When lim sup an /C30lim inf an ; the sequence converges to the real number lim an /C30lim sup an /C30lim inf an : Otherwise, the sequence does not converge. See also INFIMUM ,LIMIT,LOWER LIMIT,SUPREMUM Infinary Divisor /px is an infinary divisor of py (with y /C210) if px ½y/C281py : This generalizes the concept of the K-ARY DIVISOR .See also INFINARY PERFECT NUMBER , K-ARY DIVISOR References Cohen, G. L. "On an Integer’s Infinary Divisors." Math. Comput. 54, 395 /C1/11, 1990. Cohen, G. and Hagis, P. "Arithmetic Functions Associated with the Infinary Divisors of an Integer." Internat. J. Math. Math. Sci. 16, 373 /C1/83, 1993. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 54, 1994. Infinary Multiperfect Number Let s/C12(n) be the SUM of the INFINARY DIVISORS of a number n. An infinary k-multiperfect number is a number n such that s/C12(n) /C30kn: Cohen (1990) found 13 infinary 3-multiperfects, seven 4-multiperfects, and two 5-multiperfects. See also INFINARY PERFECT NUMBER References Cohen, G. L. "On an Integer’s Infinary Divisors." Math. Comput. 54, 395 /C1/11, 1990. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 54, 1994. Infinary Perfect Number Let s/C12(n) be the SUM of the INFINARY DIVISORS of a number n. An infinary perfect number is a number n such that s/C12(n) /C302n: Cohen (1990) found 14 such numbers. The first few are 6, 60, 90, 36720, ... (Sloane’s A007357). See also INFINARY MULTIPERFECT NUMBER References Cohen, G. L. "On an Integer’s Infinary Divisors." Math. Comput. 54, 395 /C1/11, 1990. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 54, 1994. Sloane, N. J. A. Sequences A007357/M4267 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Infinite Greater than any assignable quantity of the sort in question. In mathematics, the concept of the infinite is made more precise through the notion of an INFINITE SET. See also COUNTABLE SET,C OUNTABLY INFINITE , FINITE ,INFINITE SET,INFINITESIMAL ,INFINITY Infinite Group A group having an infinite number of elements. Some infinite groups, such as the integers or rationals, are not CONTINUOUS GROUPS . See also CONTINUOUS GROUP ,FINITE GROUP Infinite Product N.B. A detailed online essay by S. Finch was the starting point for this entry. APRODUCT involving an INFINITE number of terms. Such products can converge. In fact, for POSITIVE an; the PRODUCTQ/C12 n/C301anconverges to a NONZERO number IFF a/C12 n/C301lnanconverges. Infinite products can be used to define the COSINE cosx/C30Y/C12 n/C3011/C284x2 p2(2n/C281)2"# ; (1) GAMMA FUNCTION G(z)/C30zegzY/C12 r/C3011/C27z r ! e/C28z=r"#/C281 ; (2) SINE, and SINC FUNCTION . They also appear in the POLYGON CIRCUMSCRIBING CONSTANT k/C30Y/C12 n/C3031 cosp n ! : (3) An interesting infinite product formula due to Euler which relates pand the nthPRIME pnis p/C302 P/C12 i/C30n1/C27sin1 2ppn1CA}1CA$ pn2 435(4) /C30 2 P/C12 i/C30n1/C27(/C281)(pn/C281)=2 pn"# (5) (Blatner 1997). K NAR’S FORMULA gives a functional equation for the GAMMA FUNCTION G(x) in terms of the infinite product G(1/C27v)/C3022vY/C12 m/C301p/C281=2G1 2/C272/C28mv1CA}1CA$hi : (6) The class of products Y/C12 n/C302n2/C281 n2/C271/C30pcschp (7) Y/C12 n/C302n3/C281 n3/C271/C302 3(8) Y/C12 n/C302n4/C281 n4/C271 /C30/C281 2psinhpcsc/C281ðÞ1=4phi csc/C281ðÞ3=4phi ; (9) the first of which is given in Borwein and Corless (1999), can be done analytically.The first few products Y/C12 k/C301(1/C27k/C281)2 1/C272k/C281/C302 (10) Y/C12 k/C3011/C27k/C281/C27k/C282ðÞ2 1/C272k/C281/C273k/C282 /C303ffiffiffi 2p cosh21 2pffiffiffi 3p1CA}1CA$ cschpffiffiffi2p1CC1CA p(11) Y/C12 k/C3011/C27k/C281/C27k/C282/C27k/C283ðÞ 2 1/C272k/C281/C273k/C282/C274k/C283 /C30sinh2pP3 i/C301GxiðÞ p2; (12) Y/C12 k/C3011/C27k/C281/C27k/C282/C27k/C283/C27k/C284ðÞ2 1/C272k/C281/C273k/C282/C274k/C283/C275k/C284/C30Y4 i/C301GyiðÞ GziðÞ(13) where xi;yi;andziare the roots of x3/C285x2/C2710x/C2810/C300 (14) y4/C286y3/C2715y2/C2820y/C2715/C300; (15) and z4/C285z3/C2710z2/C2810z/C275/C300; (16) respectively, can also be done analytically. Note that (15) and (16) were unknown to Borwein and Corless (1999). The product Y/C12 n/C3011/C271 np ! (17) has closed form expressions for small POSITIVE inte- gral p]2; Y/C12 n/C3011/C271 n2 ! /C30sinhp p(18) Y/C12 n/C3011/C271 n3 ! /C301 pcosh1 2pffiffiffi 3p1CA}1CA$ (19) Y/C12 n/C3011/C271 n4 ! /C30cosh pffiffiffi 2p1CC1CA /C28cospffiffiffi2p1CC1CA 2p2(20) Y/C12 n/C3011/C271 n5 ! /C30Gexp2 5pi1CA}1CA$hi Gexp65pi1CA}1CA$hi 1CA|1CA|1CA|1CA|1CA|1CA| /C282 (21) The D-ANALOG expression /C12!½/C138d/C30Y/C12 n/C3031/C282d nd ! (22) also has closed form expressions, Y/C12 n/C3031/C284 n2 ! /C301 6(23) Y/C12 n/C3031/C288 n3 ! /C30sinhpffiffiffi 3p1CC1CA 42pffiffiffi3p (24) Y/C12 n/C3031/C2816 n4 ! /C30sinh 2 pðÞ 120p(25) Y/C12 n/C3031/C2832 n5 ! /C30Gexp1 5pi1CA}1CA$hi G2 exp75pi1CA}1CA$hi 1CA|1CA|1CA|1CA|1CA|1CA|/C282 (26) General expressions for infinite products of this type include Y/C12 n/C3011/C28z n !2N2 435/C30 sinpzðÞ pz2N/C281YN/C281 k/C301Gze2pik/C28N ðÞ =(2N)1CC1CA1CA|1CA|1CA|1CA|/C282 (27) Y/C12 n/C3011/C27z n !2N2 435/C30 1 z2NYN k/C301Gzepi2k/C28N ðÞ /C281 ½/C138 =2NðÞ1CC1CA1CA|1CA|1CA|1CA|/C282(28) Y/C12 n/C3011/C27z n !2N/C2712 435 /C30 1 G1/C28z ðÞ z2NYN k/C301Gzepi2(k/C28N ðÞ /C281=2N/C271 ðÞ1CC1CA1CA|1CA|1CA|1CA|/C282(29) Y/C12 n/C3011/C27z n !2N/C2712 435 /C30 1 G1/C27z ðÞ z2NYN k/C301Gze2pik/C28N/C281 ðÞ =2N/C2711CC1CA1CA|1CA|1CA|1CA|/C282(30) where GzðÞis the GAMMA FUNCTION and zjjdenotes the MODULUS (Kahovec). (27) and (28) can also be rewritten as Y/C12 n/C3011/C28z n !2N2 435/C30 sinpzðÞ p3z2sinhpzðÞ pz !mod N/C271;2 ðÞ /C29Y/C26N=2/C27/C281 k/C301cosh2pzsinkp N !"# /C28cos2pzcoskp N !"# (31) Y/C12 n/C3011/C27z n !2N2435/C30 1 p2z2sinhpzðÞ pz !mod N;2ðÞ/C29Y/C28N=2/C29 k/C301cosh2pzsin2k/C281 ðÞ p 2N !"# /C28cos2pzcos2k/C281 ðÞ p 2N !"# ; (32) where xbcis the FLOOR FUNCTION ,xdeis the CEILING FUNCTION , and mod a;mðÞ is the modulus of a(mod m) (Kahovec). Infinite products OF THE FORM Y/C12 k/C3011/C281 nk ! (33) converge for n]2:I am not aware of any analytic expressions, but the first few such products are numerically given by Y/C12 k/C3011/C281 2k ! :0:28878809508660242128 (34) Y/C12 k/C3011/C281 3k ! :0:56012607792794894497 (35) Y/C12 k/C3011/C281 4k ! :0:68853753712033971546 (36) Y/C12 k/C3011/C281 5k ! :0:76033279587123242010 : (37) A class of infinite products derived from the B ARNES’ G-FUNCTION is given by Y/C12 n/C3011/C27z n !n e/C28z/C27z2=2nðÞ/C30GzðÞ 2pðÞp=2ezz/C271 ðÞ/C27gz2½/C138 =2;(38) where gis the E ULER- MASCHERONI CONSTANT . The first few cases are Y/C12 n/C3011/C271 n !n e1=(2n)/C281/C30e1/C27g=2 ffiffiffiffiffiffi 2pp (39) Y/C12 n/C3011/C272 n !n e4=(2n)/C282/C30e3/C272g 2p(40) Y/C12 n/C3011/C273 n !n e9=(2n)/C283/C30e6/C279g=2 2pðÞ3=2(41) Y/C12 n/C3011 /C274 n !n e16 =(2n)/C283 /C30e10 /C278g 2p2: (42) The interesting identities xY/C12 n/C301(1 /C28 x2n)8 (1 /C28 x2n/C281)8 /C30X/C12 n/C30123b(n) s3(Od(n))xn (43) (Ewell 1995, 1999), where b(n) is the exponent of the exact power of 2 dividing n, Od(n) is the ODD PART of n, sk(n) is the DIVISOR FUNCTION of n, and rk(n) is the SUM OF SQUARES FUNCTION , and Y/C12 n/C301(1 /C27x2n/C281)8 /C30Y/C12 n/C301(1 /C28x2n/C281)8 /C2716xY/C12 n/C301(1 /C27x2n)8 (44) (Ewell 1998, 1999) arise is connection with the TAU FUNCTION . See also ARTIN’S CONSTANT ,BARNES’ G-FUNCTION , COSINE , D-ANALOG ,D EDEKIND ETA FUNCTION ,D I- RICHLET ETA FUNCTION ,E ULER IDENTITY ,E ULER- MASCHERONI CONSTANT ,EULER’S PENTAGONAL NUM- BER THEOREM ,EULER PRODUCT ,GAMMA FUNCTION , INFINITE SERIES ,JACOBI TRIPLE PRODUCT ,K NAR’S FORMULA ,P OLYGON CIRCUMSCRIBING CONSTANT , POLYGON INSCRIBING CONSTANT ,POWER TOWER , Q- FUNCTION , Q-SERIES ,RIEMANN ZETA FUNCTION ,SINE, STEPHENS’ CONSTANT References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 75, 1972. Arfken, G. "Infinite Products." §5.11 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 346 /C1/51, 1985. Blatner, D. The Joy of Pi. New York: Walker, p. 119, 1997. Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899/C1/09, 1999. Ewell, J. A. "Arithmetical Consequences of a Sextuple Product Identity." Rocky Mtn. J. Math. 25, 1287 /C1/293, 1995. Ewell, J. A. "A Note on a Jacobian Identity." Proc. Amer. Math. Soc. 126, 421/C1/23, 1998. Ewell, J. A. "New Representations of Ramanujan’s Tau Function." Proc. Amer. Math. Soc. 128, 723/C1/26, 1999. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/infprd/infprd.html. Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975. Jeffreys, H. and Jeffreys, B. S. "Infinite Products." §1.14 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 52 /C1/3, 1988. Kahovec, H. "Basic Infinite Products." http://www.mathsoft.- com/asolve/constant/infprd/kahovec/ip.html. Kahovec, H. "Proof of the Infinite Product Formulas." http:// www.mathsoft.com/asolve/constant/infprd/kahovec/proof01.html. Krantz, S. G. "The Concept of an Infinite Product." §8.1.6 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 104 /C1 /05, 1999.Ritt, J. F. "Representation of Analytic Functions as Infinite Products." Math. Z. 32,1/C1/, 1930. Whittaker, E. T. and Watson, G. N. §7.5/C1/.6 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cam- bridge University Press, 1990. Infinite Series ASERIES with an INFINITE number of terms is called an infinite series. A (possibly infinite) series for which the ratio of each two consecutive terms ak/C271=akis a constant function of the summation index k. The more general case of the ratio a RATIONAL FUNCTION of the summation index kproduces a series called a HYPERGEOMETRIC SERIES . A particular infinite series identity is given by X/C12 k/C301;3;5;...e/C28kxsin(ky) k/C301 2tan/C281siny sinh x ! (1) forx/C210. Apostol (1997, p. 25) gives the analytic sum X/C12 n/C301;3;5;...n4k/C271 1/C27enp/C3024k/C271/C281 8k/C274B4k/C272; (2) where Bkis a B ERNOULLI NUMBER . Infinite series of the following type can also be computed analytically, X/C12 k/C300xk ! p /C30(1/C28x)/C28p(3) /C301 (p/C281)!X/C12 n/C300(n/C27p/C281)! n!xn: (4) /C301 (p/C281)!X/C12 n/C300(n/C271)p/C281xn; (5) where ( n)pis a P OCHHAMMER SYMBOL . An infinite series of the following form can be done inclosed form. X /C12 k/C3011 [1/C27k2p2]n/C30pn(e) 2n/C271n!(e2/C281)n; (6) where Pn(e2)i sa n nth order polynomial in e2:The first few polynomials are P1/C301 P2/C30/C28e4/C278e2/C283 P3/C30/C285e6/C2741e4/C2831e2/C2711 P4/C30/C2833e8/C27286e6/C28344e4/C27250e2/C2863: The related infinite series can also be done in closed form. X/C12 k/C3011 1 /C27 k /C271 21CA}1CA$2 p21C|C1C|An /C30Qn(e) 2n/C271n!(e2 /C27 1)n /C284n (4 /C27p2)n ; (7) where Qn(e2)isan nth order polynomial in e2 : The first few polynomials are Q1 /C30e2 /C281 Q2 /C30e4 /C284e2 /C281 Q3 /C303e6 /C2817e4 /C287e2 /C283 Q4 /C3015e8 /C2894e6 /C2856e4 /C2858e2 /C2815 Q5 /C30105e10 /C28657e8 /C28578e6 /C28982e4 /C28503 /C28105: See also ABSOLUTE CONVERGENCE ,C ONDITIONAL CONVERGENCE ,C ONVERGENT SERIES ,D IVERGENT SERIES ,G EOMETRIC SERIES ,H YPERGEOMETRIC SER- IES,INFINITE PRODUCT ,SERIES References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 25, 1997. Bromwich, T. J. I’a. and MacRobert, T. M. "Alternating Series." §19 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 55 /C1/7, 1991. Gardner, M. "Limits of Infinite Series." Ch. 17 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 163 /C1/72, 1984. Natanson, I. P. Summation of Infinitely Small Quantities. Boston, MA: Heath, 1963. Rainville, E. D. Infinite Series. New York: Macmillan, 1967. Infinite Set A SET of S elements is said to be infinite if the elements of a PROPER SUBSET S0 can be put into ONE- TO-ONE correspondence with the elements of S.An infinite set whose elements can be put into a ONE-TO- ONE correspondence with the set of INTEGERS is said to be COUNTABLY INFINITE ; otherwise, it is called UNCOUNTABLY INFINITE . See also ALEPH-0 ,A LEPH-1 ,C ARDINAL NUMBER , COUNTABLY INFINITE ,CONTINUUM ,FINITE ,INFINITE , INFINITY ,ORDINAL NUMBER ,TRANSFINITE NUMBER , UNCOUNTABLY INFINITE References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 77, 1996.Infinite Sum An infinite sum identity is given by z4 /C285z3 /C2710z2 /C2810z /C275 /C300 ; forY/C12 n/C3011 /C271 np ! : See also INFINITE PRODUCT Infinitesimal A quantity which yields 0 after the application of some LIMITING process. The understanding of infini- tesimals was a major roadblock to the acceptance of CALCULUS and its placement on a firm mathematical foundation. See also INFINITE ,INFINITY ,NONSTANDARD ANALYSIS References Bell, J. L. A Primer of Infinitesimal Analysis. Cambridge, England: Cambridge University Press, 1998. Infinitesimal Analysis An archaic term for CALCULUS . Infinitesimal Matrix Change Let B ; A; and e be square matrices with e small, and define B /C13A(I /C27e) ; (1) where I is the IDENTITY MATRIX . Then the inverse of B is approximately BB/C281/C30(I/C28e)A/C281: (2) This can be seen by multiplying BB/C281/C30(A/C27Ae)(A/C281/C28eA/C281) /C30AA/C281/C28AeA/C281/C27AeA/C281/C28Ae2A/C281 /C30I/C28Ae2A/C281:1: (3) Note that if we instead let B?/C13A/C27e;and look for an inverse OF THE FORM B?/C281/C30A/C281/C27C;we obtain BB0/C281/C30(A/C27e)(A/C281/C27C)/C30AA/C281/C27AC/C27eA/C281/C27eC /C30I/C27AC/C27e(C/C27A/C281)/C13I: (4) In order to eliminate the eterm, we require C/C30/C28A/C281: However, then AC/C30/C28I;soBB/C281/C300so there can be no inverse of this form. The exact inverse of B0can be found as follows. B0/C30A(I/C27e)/C30A(I/C27A/C281e); (5) so B?/C281 /C30[A(I /C27A/C281e)] /C281 : (6) Using a general MATRIX INVERSE identity then gives B ?/C281 /C30 I /C27A /C281e1CC1CA /C281A /C281 : (7) Infinitesimal Rotation An infinitesimal transformation of a VECTOR r is given by r ?/C30(I /C27e)r ; (1) where the MATRIX e is infinitesimal and I is the IDENTITY MATRIX . (Note that the infinitesimal trans- formation may not correspond to an inversion, since inversion is a discontinuous process.) The COMMU- TATIVITY of infinitesimal transformations e1 and e2 is established by the equivalence of I /C27e1 ðÞ (I /C27e2) /C30I2 /C27e1I /C27Ie2 /C27e1e2 :I /C27e1 /C27e2(2) (I /C27e2)(I /C27e1) /C30I2 /C27e2I /C27Ie1 /C27e2e1 :I /C27e2 /C27e1 : (3) Now let A /C13I /C27e; (4) The inverse A/C281 is then I /C28e; since AA /C281 /C30(I /C27e)(I /C28e) /C30I2 /C28e2 :I: (5) Since we are defining our infinitesimal transforma- tion to be a rotation, ORTHOGONALITY of ROTATION MATRICES requires that AT /C30A /C281 ; (6) but A /C281 /C30I /C28e (7) (I /C27e)T /C30IT /C27eT /C30I /C27eT ; (8) so e /C30/C28eT and the infinitesimal rotation is ANTISYM- METRIC . It must therefore have a MATRIX OF THE FORM e /C300 dV3 /C28dV2 /C28dV30 dV1 dV2 /C28dV102 435: (9) The differential change in a vector r upon application of the ROTATION MATRIX is then dr /C13r?/C28r /C30(I /C27e)r /C28r /C30er : (10) Writing in MATRIX form, dr /C30x y z2 4350 dV 3 /C28dV2 /C28d V30 dV1 d V2 /C28dV102435 /C30ydV 3 /C28zdV2 zdV1 /C28xdV3 xdV2 /C28ydV12 435 (11)/C30 ydV 3 /C28zdV2 ðÞ ˆx /C27 zdV1 /C28xdV3 ðÞ ˆy /C27 xdV2 /C28ydV1 ðÞ ˆz /C30r /C29d V: (12) Therefore, dr dt ! rotation ; body/C30r /C29dV dt/C30r /C29 v; (13) where v /C13dV dt/C30ˆndf dt: (14) The total rotation observed in the stationary frame will be a sum of the rotational velocity and the velocity in the rotating frame. However, note that an observer in the stationary frame will see a velocity opposite in direction to that of the observer in the frame of the rotating body, so dr dt ! space/C30dr dt ! body/C27v /C29r : (15) This can be written as an operator equation, known as the ROTATION OPERATOR , defined as d dt ! space/C30d dt ! body/C27v /C29: (16) See also ACCELERATION ,EULER ANGLES ,ROTATION , ROTATION MATRIX ,ROTATION OPERATOR Infinitive Sequence A sequence xnfg is called an infinitive sequence if, for every i, xn /C30i for infinitely many n. Write a(i ;j) for the jth index n for which xn /C30i : Then as i and j range through N, the array A /C30a(i ;j) ; called the associative array of x, ranges through all of N. See also FRACTAL SEQUENCE References Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157/C1/68, 1997. Infinitude of Primes EUCLID’S THEOREMS Infinity An unbounded number greater than every REAL NUMBER , most often denoted as /C12:The symbol /C12 had been used as an alternative to M (1,000) in ROMAN NUMERALS until 1655, when John Wallis suggested it be used instead for infinity. Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results which follow from Georg Cantor’s treatment of INFINITE SETS . Informally, 1 =/C12/C300; a statement which can be made rigorous using the LIMIT concept, lim x0/C121 x /C300 : Similarly, lim x 00/C271 x /C30/C12; where the notation 0/C27 indicates that the LIMIT is taken from the POSITIVE side of the REAL LINE. See also ALEPH ,ALEPH-0 ,ALEPH-1 ,CARDINAL NUM- BER,C OMPLEX INFINITY ,C ONTINUUM ,C ONTINUUM HYPOTHESIS ,H ILBERT HOTEL ,INFINITE ,INFINITE SET,INFINITESIMAL ,LINE AT INFINITY ,L’HOSPITAL’S RULE,P OINT AT INFINITY ,T RANSFINITE NUMBER , UNCOUNTABLY INFINITE ,ZERO References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 19, 1996. Courant, R. and Robbins, H. "The Mathematical Analysis of Infinity." §2.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 77 /C1/8, 1996. Hardy, G. H. Orders of Infinity, the ‘infinitarcalcul’ of Paul Du Bois-Reymond, 2nd ed. Cambridge, England: Cam- bridge University Press, 1924. Lavine, S. Understanding the Infinite. Cambridge, MA: Harvard University Press, 1994. Maor, E. To Infinity and Beyond: A Cultural History of the Infinite. Boston, MA: Birkha ¨user, 1987. Moore, A. W. The Infinite. New York: Routledge, 1991. Morris, R. Achilles in the Quantum Universe: The Definitive History of Infinity. New York: Henry Holt, 1997. Owen, H. P. "Infinity in Theology and Metaphysics." In The Encyclopedia of Philosophy, Vol. 4. New York: Crowell Collier, pp. 190 /C1/93, 1967. Pe´ter, R. Playing with Infinity. New York: Dover, 1976. Rucker, R. Infinity and the Mind: The Science and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1995. Smail, L. L. Elements of the Theory of Infinite Processes. New York: McGraw-Hill, 1923. Thomson, J. "Infinity in Mathematics and Logic." In The Encyclopedia of Philosophy, Vol. 4. New York: Crowell Collier, pp. 183 /C1/90, 1967. Vilenskin, N. Ya. In Search of Infinity. Boston, MA: Bir- kha¨user, 1995. Weisstein, E. W. "Books about Infinity." http://www.trea- sure-troves.com/books/Infinity.html. Wilson, A. M. The Infinite in the Finite. New York: Oxford University Press, 1996. Zippin, L. Uses of Infinity. New York: Random House, 1962. Inflection Point A point on a curve at which the SIGN of the CURVATURE (i.e., the concavity) changes. The FIRST DERIVATIVE TEST can sometimes distinguish inflection points from EXTREMA for DIFFERENTIABLE functions f(x) :/See also CURVATURE ,D IFFERENTIABLE ,EXTREMUM , FIRST DERIVATIVE TEST,STATIONARY POINT Information Dimension Define the "information function" to be I /C30/C28XN i/C301Pi( e)lnPi(e) ½/C138 ; (1) where Pi(e) is the NATURAL MEASURE , or probability that element i is populated, normalized such that XN i/C301Pi( e) /C301: (2) The information dimension is then defined by dinf /C13/C28 lim e00 /C27I ln(e) /C30 lim e00/C27XN i/C301Pi( e)lnPi( e) ½/C138 ln( e): (3) If every element is equally likely to be visited, then Pi( e) is independent of i, and XN i /C301Pi(e) /C30NPi( e) /C301; (4) so Pi( e) /C301 N; (5) and dinf /C30 lim e00/C27XN i/C3011 Nln1 N ! lne /C30lim e00/C27lnN/C281ðÞ lne/C30/C28lim e00/C27lnN lne/C30dcap; (6) where dcapis the CAPACITY DIMENSION . See also CORRELATION EXPONENT References Balatoni, J. and Renyi, A. Pub. Math. Inst. Hungarian Acad. Sci. 1, 9, 1956. Farmer, J. D. "Chaotic Attractors of an Infinite-dimensional Dynamical System." Physica D 4, 366/C1/93, 1982. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 79, 1993. Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 545 /C1/47, 1995. Information Entropy ENTROPY Information Theory The branch of mathematics dealing with the efficient and accurate storage, transmission, and representa- tion of information. See also CODING THEORY ,COMPRESSION ,ENTROPY References Goldman, S. Information Theory. New York: Dover, 1953. Hankerson, D.; Harris, G. A.; and Johnson, P. D. Jr. Intro- duction to Information Theory and Data Compression. Boca Raton, FL: CRC Press, 1998. Lee, Y. W. Statistical Theory of Communication. New York: Wiley, 1960. Pierce, J. R. An Introduction to Information Theory. New York: Dover, 1980. Reza, F. M. An Introduction to Information Theory. New York: Dover, 1994. Singh, J. Great Ideas in Information Theory, Language and Cybernetics. New York: Dover, 1966. Weisstein, E. W. "Books about Information Theory." http:// www.treasure-troves.com/books/InformationTheory.html. Zayed, A. I. Advances in Shannon’s Sampling Theory. Boca Raton, FL: CRC Press, 1993. Initial Ordinal An ORDINAL NUMBER is called an initial ordinal if every smaller ordinal has a smaller CARDINALITY (Moore 1982, p. 248; Rubin 1967, p. 271). The va/s ordinal numbers are just the transfinite initial ordinals (Rubin 1967, p. 272). This PROPER CLASS can be well ordered and put into one-to-one correspondence with the ORDINAL NUM- BERS . For any two WELL ORDERED SETS that are ORDER ISOMORPHIC , there is only one order isomorph- ism between them. Let f be that isomorphism from the ordinals to the transfinite initial ordinals, then va /C30f( a); where v0 /C30 v:/ See also ORDINAL NUMBER References Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Initial Segment Let (A;5)bea WELL ORDERED SET. Then the set fa /C23 A : a Bkg for some k /C23 A is called an initial segment of A (Rubin 1967, p. 161; Dauben 1990, pp. 196 /C1/97; Moore 1982, pp. 90 /C1/1). This term was first used by Cantor, who also proved that if (A;5) and (B ;5) are WELL ORDERED SETS that are not ORDER ISOMORPHIC , then exactly one of the following statements is true:1. A is ORDER ISOMORPHIC to an initial segment of B,or 2. B is ORDER ISOMORPHIC to an initial segment of A (Dauben 1990, p. 198). See also WELL ORDERED SET References Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Initial Value Problem An initial value problem is a problem that has its conditions specified at some time t /C30t0 : Usually, the problem is an ORDINARY DIFFERENTIAL EQUATION or a PARTIAL DIFFERENTIAL EQUATION . For example, @2u @t2 /C2892u /C30f in V u /C30u0 t /C30t0 u /C30u1 on @V;8 >>< >>: where @V denotes the boundary of V; is an initial value problem. See also BOUNDARY CONDITIONS ,BOUNDARY VALUE PROBLEM ,PARTIAL DIFFERENTIAL EQUATION References Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Computational Differential Equations. Lund, Sweden: Studentlitteratur, 1996. Injection ONE-TO- ONE Injective A MAP is injective when it is ONE-TO-ONE , i.e., f is injective when x "y IMPLIES f(x) "f(y) :/ See also ONE-TO- ONE,SURJECTIVE Injective Patch An injective patch is a PATCH such that x(u1 ;v1) /C30 x(u2 ;v2) implies that u1 /C30u2 and v1 /C30v2 : An example of a PATCH which is injective but not REGULAR is the function defined by (u3 ;v3 ;uv) for u ;v /C23 (/C281;1): How- ever, if x : U 0 Rn is an injective regular patch, then xmaps Udiffeomorphically onto x(U):/ See also PATCH ,REGULAR PATCH References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 273, 1997. Inner Automorphism Group A particular type of AUTOMORPHISM GROUP which exists only for GROUPS . For a GROUP G, the inner automorphism group is defined by Inn(G) /C30fsa : a /C23 G gƒAut(G) where sa is an AUTOMORPHISM of G defined by sa(x) /C30axa /C281 : See also AUTOMORPHISM ,AUTOMORPHISM GROUP Inner Product DOT PRODUCT ,HERMITIAN INNER PRODUCT ,INTERIOR PRODUCT , L2-INNER PRODUCT Inner Product Space An inner product space is a VECTOR SPACE which has an INNER PRODUCT . If the INNER PRODUCT defines a NORM , then the inner product space is called a HILBERT SPACE . See also HILBERT SPACE ,INNER PRODUCT ,NORM Inner Quermass The largest area of intersection of a solid body by a plane parallel to a given plane, also called the "HA measurement." See also BRIGHTNESS ,C ROSS SECTION ,S HADOW , STEREOLOGY References Bonnesen, T. "Om Minkowski’s uligheder fur konvexer legemer." Mat. Tidsskr. B, 80, 1926. Bonnesen, R. and Fenchel, W. Theorie der Konvexer Ko ¨rper. New York: Chelsea, p. 140, 1971. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "What Can You Tell About a Convex Body from its Section." §A11 in Unsolved Problems in Geometry. New York: Springer- Verlag, pp. 24 /C1/5, 1991. Klee, V. "Is a Body Spherical if All its HA Measurements are Constant?" Amer. Math. Monthly 76, 539/C1/42, 1969. Zaks, J. "Nonspherical Bodies with Constant HA Measure- ments Exist." Amer. Math. Monthly 78, 513/C1/16, 1971. Inradius The radius of a TRIANGLE’S INCIRCLE or of a POLY- HEDRON ’sINSPHERE , denoted r(or sometimes r):For aTRIANGLE , r/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b/C27c/C28a)(c/C27a/C28b)(a/C27b/C28c) a/C27b/C27cs (1)/C30D s(2) 4Rsin1 2A1CA}1CA$ sin12B1CA}1CA$ sin12C1CA}1CA$ ; (3) where Dis the AREA of the TRIANGLE ,a,b, and care the side lengths, sis the SEMIPERIMETER ,Ris the CIRCUMRADIUS , and A,B, and Care the angles opposite sides a,b, and c(Johnson 1929, p. 189). If two triangle side lengths aandbare known, together with the inradius r, then the length of the third side c can be found by solving (1) for c, resulting in a CUBIC EQUATION . Equation (2) can be derived easily using TRILINEAR COORDINATES . Since the INCENTER is equally spaced from all three sides, its trilinear coordinates are 1:1:1, and its exact trilinear coordinates are r:r:r:The ratio kof the exact trilinears to the homogeneous coordinates is given by k/C302D a/C27b/C27c/C30D s: (4) But since k/C30rin this case, r/C30k/C30D s; (5) Q.E.D. Other equations involving the inradius include Rr/C30abc 4s(6) D2/C30rr1r2r3 (7) cosA/C27cosB/C27cosC/C301/C27r R(8) a2/C27b2/C27c2/C304rR/C278R2; (9) where riare the EXRADII (Johnson 1929, pp. 189 /C1/91). As shown in RIGHT TRIANGLE , the inradius of a RIGHT TRIANGLE side lengths a,b, and cis given by r/C30ab a/C27b/C27c(10) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2(c/C28a)(c/C28b)q (11) /C3012(a/C27b/C28c); (12) where cis the HYPOTENUSE . Let dbe the distance between inradius rand CIRCUMRADIUS R,d/C30rR:Then R2/C28d2/C302Rr (13) 1 R /C28 d /C271 R /C27 d /C301 r (14) (Mackay 1886 /C1/7; Casey 1888, pp. 74 /C1/5). These and many other identities are given in Johnson (1929, pp. 186 /C1/90). For a PLATONIC SOLID or ARCHIMEDEAN SOLID , the inradius of the solid is also the inradius of the DUAL POLYHEDRON . Expressing the MIDRADIUS r and CIR- CUMRADIUS R in terms of the midradius gives r /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C271 4a2q (15) r /C30R2 /C2814a2 R (16) for an ARCHIMEDEAN SOLID . See also CARNOT’S THEOREM ,CIRCUMRADIUS ,JAPA- NESE THEOREM ,MIDRADIUS References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 10, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Mackay, J. S. "Historical Notes on a Geometrical Theorem and its Developments [18th Century]." Proc. Edinburgh Math. Soc. 5,62/C1/8, 1886 /C1/887. Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles of a Triangle." Proc. Edinburgh Math. Soc. 12,86/C1/05. Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles of a Triangle." Proc. Edinburgh Math. Soc. 13, 103 /C1/04. Inscribed A geometric figure which touches only the sides (or interior) of another figure. See also CIRCUMSCRIBED ,INCENTER ,INCIRCLE ,IN- RADIUS Inscribed Angle The ANGLE with VERTEX on a CIRCLE ’s CIRCUMFER- ENCE formed by two points on a CIRCLE ’s CIRCUMFER-ENCE . For ANGLES with the same endpoints, uc /C302ui ; where ucis the CENTRAL ANGLE . See also CENTRAL ANGLE References Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., pp. xxi-xxii, 1995. Inside-Outside Theorem LetP(z) and Q(z)b e UNIVARIATE POLYNOMIALS in a complex variable z, and let the DEGREES ofPandQ satisfy deg( Q)]deg(P/C272):Then ggP(z) Q(z)dz/C302piX ai/C23ARes z/C30aiP(z) Q(z)(1) /C30/C282piX bi/C23BRes z/C30biP(z) Q(z); (2) where gis a simple closed clockwise-oriented CON- TOUR ,Ais the set of ROOTS ofQinside of g;andBis the set of ROOTS ofQoutside of g:/ The first equality is an instance of the RESIDUE THEOREM . On the R IEMANN SPHERE , the simple closed CONTOUR gsplits the sphere into two regions. After the change of variables w/C301=z;the point zero is mapped to infinity and vice versa. What was the "inside" of gbecomes the outside of gin the new coordinate. The second equality is the RESIDUE THEOREM applied to the MEROMORPHIC ONE-FORM a/C30 P=Qd z in the coordinate w, with a minus sign because gtravels clockwise after the coordinate change. The hypothesis on the degrees of Pand Q ensure that adoes not have a POLE atz/C30/C12:/ The above diagram shows two different points of viewof the contour gand the poles of the MEROMORPHIC ONE-FORM P=Qd z on the R IEMANN SPHERE . The usual point of view is centered at z/C300, but the role of inside and outside is switched from the point of view of z/C30 /C12:The poles inside are labeled blue and outside are green. The theorem also follows from taking the CONTOUR INTEGRAL at infinity, i.e., a circle of large radius R. The hypothesis on the degree says that this integral tends to zero. Hence it must actually be zero, because at some point the circle contains all of the poles of / P=Q/. This is a special case of the fact that on a COMPACT RIEMANN SURFACE , in this case the RIE- MANN SPHERE , the sum of the RESIDUES of a MER- OMORPHIC ONE-FORM is zero. See also CONTOUR ,CONTOUR INTEGRAL ,JACOBIAN , RESIDUE (COMPLEX ANALYSIS ), RESIDUE THEOREM , RIEMANN SPHERE ,ROOT Insphere A SPHERE INSCRIBED in a given solid. The figures above depict the inspheres of the Platonic solids. See also CIRCUMSPHERE ,MIDSPHERE Instrument Function The finite FOURIER COSINE TRANSFORM of an APODIZA- TION FUNCTION , also known as an APPARATUS FUNC- TION . The instrument function IxðÞcorresponding to a given APODIZATION FUNCTION AxðÞis then given by I(k) /C30ga /C28acos(2 pkx)A(x)dx: See also APODIZATION FUNCTION ,FOURIER COSINE TRANSFORM Insufficient Reason Principle A principle, also called the indifference principle, that was first enunciated by Johann Bernoulli. The in- sufficient reason principle states that, if we are ignorant of the ways an event can occur and therefore have no reason to believe that one way will occur preferentially to another, it will occur equally likely in any way. Int INTEGER PART Integer One of the numbers ..., -2, -1, 0, 1, 2, .... The SET of INTEGERS forms a RING which is denoted Z: A given INTEGER n may be NEGATIVE ( a /C13Z/C28) ; NONNEGATIVE n /C23Z /C31 ðÞ ; ZERO (n /C300), or POSITIVE n /C23Z /C27/C30N ðÞ : The set of integers is denotedIntegers in Mathematica , and a number x can be tested to see if it is an integer using the command Element[ x, Integers]. Numbers that are integers are sometimes described as "inte- gral" (instead of integer-valued), but this practice may lead to unnecessary confusions with the INTE- GRALS of INTEGRAL CALCULUS .The RING Z of integers has CARDINALITY of ALEPH-0 . The GENERATING FUNCTION for the NONNEGATIVE INTEGERS is f(x) /C30x (1 /C28 x)2 /C30x /C272x2 /C273x3 /C274x4 /C27...: There are several symbols used to perform operations having to do with conversion between REAL NUMBERS and integers. The symbol xbc("FLOOR x") means "the largest integer not greater than x," i.e., int(x) in computer parlance. The symbol x½/C138means "the near- est integer to x"(NINT ), i.e., nint(x) in computer parlance. The symbol xde("CEILING x") means the smallest integer not smaller x," or-int(-x) , where int(x) is the INTEGER PART of x. The German mathematician and logician Kronecker vociferously opposed the work of Georg Cantor on infinite sets and summarized his view that ARITH- METIC and ANALYSIS should be based on whole numbers only by saying, "God made the natural numbers; all else is the work of man" (Bell 1986, p. 477). See also ALGEBRAIC INTEGER ,A LMOST INTEGER , COMPLEX NUMBER ,COUNTING NUMBER ,CYCLOTOMIC INTEGER ,E ISENSTEIN INTEGER ,F RACTIONAL PART, GAUSSIAN INTEGER ,INTEGER PART,N,N ATURAL NUMBER ,N EGATIVE ,P OSITIVE ,R ADICAL INTEGER , REAL NUMBER ,W HOLE NUMBER ,Z,Z -,Z/C27,Z*,ZERO References Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1986. Integer Array See also INTEGER SEQUENCE References Kimberling, C. "Integer Sequences and Arrays." http:// cedar.evansville.edu/~ck6/integer/. Integer Bowl BOWL OF INTEGERS Integer Cuboid EULER BRICK Integer Division DIVISION in which the fractional part (remainder) is discarded is called integer division and is sometimes denoted \. Integer division can be defined as / a_b/C13/C28a=b/C29/, where "/" denotes normal division and xbcis the FLOOR FUNCTION . For example, 10=3 ¼ 3 þ 1=3 10_3 ¼ 3: Integer Exponent GREATEST DIVIDING EXPONENT Integer Factorization PRIME FACTORIZATION Integer Function A FUNCTION defined for all positive integers, some- times also called an "arithmetical function" (Nagell 1951, p. 26). See also COMPLEX MATRIX ,REAL MATRIX References Nagell, T. "Arithmetical Functions." §9in Introduction to Number Theory. New York: Wiley, pp. 26 /C1/9, 1951. Integer Matrix A MATRIX whose entries are all integers. Special cases which arise frequently are those having only (1;/C281) as entries (e.g., HADAMARD MATRIX ), BINARY MATRICES having only (0;1) as entries (e.g., ADJACENCY MATRIX , FROBENIUS- KO¨ NIG THEOREM ,GALE-RYSER THEOREM , HADAMARD’S MAXIMUM DETERMINANT PROBLEM , HARD SQUARE ENTROPY CONSTANT , IDENTITY MATRIX , INCI- DENCE MATRIX ,LAM’S PROBLEM ), and those having (/C281;0; 1) as entries (e.g., ALTERNATING SIGN MATRIX , C-MATRIX ). The ZERO MATRIX could be considered a degenerate case of an integer matrix. See also ALTERNATING SIGN MATRIX ,(-1,0,1)-MATRIX , (-1,1)-MATRIX ,(0,1)-MATRIX ,COMPLEX MATRIX ,FROBE- NIUS- KO¨ NIG THEOREM ,G ALE-RYSER THEOREM , C- MATRIX ,FIFTEEN THEOREM ,GALE-RYSER THEOREM , HADAMARD’S MAXIMUM DETERMINANT PROBLEM ,HA- DAMARD MATRIX ,H AFNER- SARNAK- MCCURLEY CON- STANT ,HARD SQUARE ENTROPY CONSTANT ,IDENTITY MATRIX ,INCIDENCE MATRIX ,INTEGER- MATRIX FORM, INTERSPERSION ,LAM’S PROBLEM ,M ORTAL ,M ORTAL- ITY PROBLEM ,REAL MATRIX ,SMITH NORMAL FORM, SPECIAL MATRIX ,UNIT MATRIX ,ZERO MATRIX Integer-Matrix Form Let QxðÞ/C13QxðÞ/C30Qx1 ;x2 ;...; xn ðÞ be an integer-va- lued n-ary QUADRATIC FORM , i.e., a POLYNOMIAL with integer COEFFICIENTS which satisfies QxðÞ> 0 for REAL x "0: Then QxðÞcan be represented by Q(x) /C30xTAx; whereA /C301 2@2Q(x) @xi @xj is a POSITIVE SYMMETRIC MATRIX (Duke 1997). If A has POSITIVE entries, then QxðÞis called an integer- matrix form. Conway et al. (1997) have proven that, if a POSITIVE integer-matrix quadratic form repre- sents each of 1, 2, 3, 5, 6, 7, 10, 14, and 15, then it represents all POSITIVE INTEGERS . See also FIFTEEN THEOREM References Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane, N. J. A. "The Primary Pretenders." Acta Arith. 78, 307/C1/ 13, 1997. Duke, W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer. Math. Soc. 44, 190/C1/96, 1997. Integer Module ABELIAN GROUP Integer Part The function int xgives the integer part of x. In many computer languages, the function is denoted int(x) . It is related to the FLOOR and CEILING FUNCTIONS xbc and xdeby intx/C30xbc forx]/C12 xde forxB01C|} The integer part function satisfies int(/C28x)/C30/C28int(x) and is implemented in Mathematica asInteger- Part [x]. This definition is chosen so that int x/C27 fracx/C30x;where frac xis the FRACTIONAL PART . Although Spanier and Oldham (1987) use the same definition as Mathematica , they mention the formula only very briefly and then say it will not be used further. Graham et al. (1994), and perhaps most other mathematicians, use the term "integer" part interchangeably with the FLOOR FUNCTION xbc:/ Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used (D. W. Cantrell). Here, S&O indicates Spanier and Oldham (1987). notation name S&O Graham et al.Mathematica / xbc/ integer- value/Int(x)/ floor or integer partFloor [ x] /sgn xðÞ xjjbc / integer- part/Ip xðÞ/ no name Integer-Part [ x] /x /C28 xbc/ fractional- value/frac xðÞ/ fractionalpart or {x}no name /sgn xðÞxjj/C28 xjjbc ðÞ / fractional- part/Fp(x)/ no name Fractional-Part [ x] See also CEILING FUNCTION ,FLOOR FUNCTION ,FRAC- TIONAL PART,INTEGER ,NEAREST INTEGER FUNCTION References Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 67, 1994. Spanier, J. and Oldham, K. B. "The Integer-Value Int(x) and Fractional-Value frac(x) Functions." Ch. 9 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 71 /C1/8, 1987. Integer Polynomial A POLYNOMIAL OF THE FORM f(x) /C30anxn /C27an/C281xn/C281 /C27.../C27a1x /C27a0 having coefficients ai that are all integers. An integer polynomial gives integer values for all integer argu- ments of x (Nagell 1951, p. 73). The set of integer polynomials is denoted Z x½/C138:/ An integer polynomial is called primitive if the GREATEST COMMON DIVISOR a0 /C27a1 ;...;an /C301: ðÞ : In- teger polynomials are sometimes called "integral polynomials," which is an unfortunately confusing choice of nomenclature. See also INTEGER- REPRESENTING POLYNOMIAL ,POLY- NOMIAL ,PRIME DIVISOR References Nagell, T. "Prime Divisors of Integral Polynomials" and "Divisibility of Integral Polynomials with Regard to a Prime Modulus." §25 and 29 in Introduction to Number Theory. New York: Wiley, pp. 73, 81 /C1/3, and 93 /C1/8, 1951. Integer Relation A set of REAL NUMBERS x1;... ,xnis said to possess an integer relation if there exist integers aisuch thata1x1/C27a2x2/C27/C1/C1/C1/C27anxn/C300; with not all ai/C300:For historical reasons, integer relation algorithms are sometimes called generalized Euclidean algorithms or multidimensional continued fraction algorithms. An interesting example of such a relation is the 17- VECTOR (1,x,x2;... ,x16) with x/C3031=4/C2822=4;which has an integer relation (1, 0, 0, 0, -3860, 0, 0, 0, -666, 0, 0, 0, -20, 0, 0, 0, 1), i.e., 1/C283860 x4/C28666x8/C2820x12/C27x16/C300: This is a special case of finding the polynomial ofdegree n/C30rssatisfied by x/C303 1=r/C2821=s:/ Integer relation algorithms can be used to solve SUBSET SUM PROBLEMS , as well as to determine if a given numerical constant is equal to a root of aunivariate polynomial of degree nor less (Bailey and Ferguson 1989, Ferguson and Bailey 1992). One of the simplest cases of an integer relation between two numbers is the one inherent in thedefinition of the GREATEST COMMON DIVISOR . The well-known E UCLIDEAN ALGORITHM solves this pro- blem, as well as the more general problem of aninteger relation between two real numbers, yieldingeither an exact relation or an infinite sequence of approximate relations (Ferguson et al. 1999). Although attempts were made to generalize the algorithm to n]3 by Hermite (1850), Jacobi (1868), Poincare ´(1884), Perron (1907), Brun (1919, 1920, 1957), and Szekeres (1970), all such routines wereknown to fail in certain cases (Ferguson and Forcade1979, Forcade 1981, Hastad et al. 1989). The first successful integer relation algorithm was developedby Ferguson and Forcade (1979) (Ferguson andBailey 1992, Ferguson et al. 1999). Algorithms for finding integer relations include the F ERGUSON- FORCADE ALGORITHM , HJLS ALGORITHM , LLL ALGORITHM , PSLQ ALGORITHM , PSOS ALGO- RITHM , and the algorithm of Lagarias and Odlyzko (1985). Perhaps the simplest (and unfortunately most inefficient) such algorithm is the GREEDY ALGORITHM . Plouffe’s "Inverse Symbolic Calculator" site includes ahuge database of 54 million REAL NUMBERS which are algebraically related to fundamental mathematicalconstants. The F ERGUSON- FORCADE ALGORITHM has shown that there are no algebraic equations of degree 58 with integer coefficients having Euclidean norms below certain bounds for e=p;e/C27p;lnp;g;eg;g=e;g=p; and ln g;where Eis the base for the NATURAL LOGARITHM ,pisPI, and gis the E ULER- MASCHERONI CONSTANT (Bailey 1988). Constant Bound /e=p;// 6:1030/C291014/ /e /C27p;// 2:2753 /C291014/ /ln p;// 8:7697 /C29109/ / g// 3:5739 /C29109 / /e g ;// 1:6176 /C291017 / / g =e ;// 1:8440 /C291011/ / g =p// 6:5403 /C29109/ /ln g// 2:6881 /C291010 / See also CONSTANT PROBLEM ,FERGUSON- FORCADE ALGORITHM ,G REEDY ALGORITHM ,H ERMITE- LINDE- MANN THEOREM , HJLS ALGORITHM ,KNAPSACK PRO- BLEM ,L ATTICE REDUCTION ,L INDEMANN- WEIERSTRASS THEOREM , LLL ALGORITHM ,PSLQ ALGORITHM , PSOS ALGORITHM ,RICHARDSON’S THEO- REM,REAL NUMBER ,SUBSET SUM PROBLEM References Bailey, D. H. and Ferguson, H. R. P. "Numerical Results on Relations Between Numerical Constants Using a New Algorithm." Math. Comput. 53, 649 /C1/56, 1989. Bailey, D. and Plouffe, S. "Recognizing Numerical Con- stants." http://www.cecm.sfu.ca/organics/papers/bailey/. Bernstein, L. The Jacobi-Perron Algorithm: Its Theory and Applications. Berlin: Springer-Verlag, 1971. Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899 /C1/09, 1999. Borwein, J. M. and Lisonek, P. "Applications of Integer Relation Algorithms." To appear in Disc. Math. http:// www.cecm.sfu.ca/preprints/1997pp.html. Brentjes, A. J. "Multi-Dimensional Continued Fraction Al- gorithms." Mathemat. Centre Tracts, No. 145. Amster- dam, Netherlands: Mathemat. Centrum, 1981. Brun, V. "En generalisatiken av kjedeboøken, I." Norske Vidensk. Skrifter I. Matemat. Naturvid. Klasse 6,1/C1/9, 1919. Brun, V. "En generalisatiken av kjedeboøken, II." Norske Vidensk. Skrifter I. Matemat. Naturvid. Klasse 7,1/C1/4, 1920. Brun, V. "Algorithmes euclidiens pour trois et quatre nombres." In Treizie `me Congre `s des mathe ´maticiens Scandinaves, tenu a Helsinki 18 /C1/3 aouˆt 1957. Helsinki: Mercators Trycheri, pp. 46 /C1/4, 1958. Centre for Experimental & Constructive Mathematics. "In- teger Relations." http://www.cecm.sfu/projects/IntegerRe- lations/. Ferguson, H. R. P. and Bailey, D. H. "A Polynomial Time, Numerically Stable Integer Relation Algorithm." RNR Techn. Rept. RNR-91 /C1/32, Jul. 14, 1992. Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351 /C1/69, 1999. Ferguson, H. R. P. and Forcade, R. W. "Generalization of the Euclidean Algorithm for Real Numbers to All Dimen- sions Higher than Two." Bull. Amer. Math. Soc. 1, 912 /C1/ 14, 1979. Forcade, R. W. "Brun’s Algorithm." Unpublished manu- script, 1 /C1/7, Nov. 1981. Hastad, J.; Just, B.; Lagarias, J. C.; and Schnorr, C. P. "Polynomial Time Algorithms for Finding Integer Rela- tions Among Real Numbers." SIAM J. Comput. 18, 859 /C1/ 81, 1988.Hermite, C. "Extraits de lettres de M. Ch. Hermite a` M. Jacobi sur differe ´nts objets de la the´orie de nombres." J. reine angew. Math. 3/4, 261 /C1/15, 1850. Jacobi, C. G. "Allgemeine Theorie der Kettenbruchahnli- chen Algorithmen, in welche jede Zahl aus Drei vorherge- henden gebildet wird (Aus den hinterlassenen Papieren von C. G. Jacobi mitgetheilt durch Herrn E. Heine." J. reine angew. Math. 69,29/C1/4, 1868. Lagarias, J. C. and Odlyzko, A. M. "Solving Low-Density Subset Sum Problems." J. ACM 32, 229 /C1/46, 1985. Lenstra A. K.; Lenstra, H. W. Jr.; and Lova´sz, L. "Factoring Polynomials with Rational Coefficients." Math. Ann. 261, 515 /C1/34, 1982. Perron, O. "Grundlagen fu¨r eine Theorie des Jacobischen Kettenbruchalgorithmus." Math. Ann. 64,1/C1/6, 1907. Plouffe, S. "Inverse Symbolic Calculator." http:// www.cecm.sfu.ca/projects/ISC/. Poincare ´, H. "Sur une ge´ne´ralisation des fractions con- tinues." Comptes Rendus Acad. Sci. Paris 99, 1014 /C1/016, 1884. Szekeres, G. "Multidimensional Continued Fractions." Ann. Univ. Sci. Budapest Eotvos Sect. Math. 13, 113 /C1/40, 1970. Integer-Representing Polynomial A polynomial that represents integers for all integer values of the variables. An INTEGER POLYNOMIAL is a special case of such a polynomial. In general, every integer representing polynomial f(x) of degree nin the variable xcan be written in the form f(x)/C30A0/C27A1x 11CA%1CAP /C27A2x21CA%1CAP /C27.../C27A nxn1CA%1CAP ; where n k1CC1CA is a BINOMIAL COEFFICIENT andA0;A1;... , Anare integers (Nagell 1951, p. 121). See also INTEGER POLYNOMIAL References Nagell, T. "Polynomials Representing Integers." §35 in Introduction to Number Theory. New York: Wiley, pp. 115 /C1/20 and 121, 1951. Integer Sequence ASEQUENCE whose terms are INTEGERS . The most complete printed references for such sequences are Sloane (1973) and its update, Sloane and Plouffe (1995). Sloane also maintains the sequences from both works together with many additional sequencesin an on-line listing. In this listing, sequences are identified by a unique 6- DIGIT A-number. Sequences appearing in Sloane and Plouffe (1995) are ordered lexicographically and identified with a 4- DIGIT M- number, and those appearing in Sloane (1973) areidentified with a 4- DIGIT N-number. Sloane’s huge (and enjoyable) database is accessibleby either e-mail or web browser. To look up sequencesby e-mail, send a message to either mailto:sequen- [email protected] or mailto:superseeker@re- search.att.com containing lines OF THE FORM lookup 5 14 42 132 ...(note that spaces must be used instead of commas). To use the browser version, point to http://www.research.att.com/~njas/se- quences/eisonline.html. Integer sequences can be analyzed by a variety techniques (Sloane and Plouffe 1995, p. 26), including the application a data compression algorithm (Bell et al. 1990) and computation of the DISCRETE FOURIER TRANSFORM (Loxton 1989). There are also a large number of transformations which relate integer sequences to one another, including the EULER TRANSFORM , EXPONENTIAL TRANSFORM ,M O¨ BIUS TRANSFORM , and others (Bower, Sloane). See also ARONSON’S SEQUENCE ,C OMBINATORICS , CONSECUTIVE NUMBER SEQUENCES ,C ONWAY SE- QUENCE ,E BAN NUMBER ,E ULER TRANSFORM ,H OF- STADTER- CONWAY $10,000 SEQUENCE ,H OFSTADTER’S Q-SEQUENCE ,INTEGER ARRAY ,L EVINE- O’SULLIVAN SEQUENCE ,L OOK AND SAY SEQUENCE ,M ALLOW’S SEQUENCE ,M IAN-CHOWLA SEQUENCE ,M O¨ BIUS TRANSFORMATION ,M ORSE- THUE SEQUENCE ,N EW- MAN- CONWAY SEQUENCE ,N UMBER ,P ADOVAN SE- QUENCE ,P ERRIN SEQUENCE , RATS SEQUENCE , SEQUENCE ,SMARANDACHE SEQUENCES References Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential Sequences." Fib. Quart. 11, 429 /C1/37, 1973. Bell, T. C.; Cleary, J. G.; and Witten, I. H. Text Compres- sion. Englewood Cliffs, NJ: 1990. Bernstein, M. and Sloane, N. J. A. "Some Canonical Se- quences of Integers." Linear Algebra Appl. 226//228 ,57/C1/ 2, 1995. Bower, C. G. "Further Transformations of Integer Se- quences." http://www.research.att.com/~njas/sequences/ transforms2.html. Cameron, P. J. "Some Sequences of Integers." Disc. Math. 75,89/C1/02, 1989. Ding, C.; Helleseth, T.; and Niederreiter, H. (Eds.). Se- quences and Their Applications: Proceedings of SETA’ 98. New York: Springer-Verlag, 1999. Erdos, P.; Sa´rko¨zy, E.; and Szemere ´di, E. "On Divisibility Properties of Sequences of Integers." In Number Theory, Colloq. Math. Soc. Ja´nos Bolyai, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 35 /C1/9, 1970. Guy, R. K. "Sequences of Integers." Ch. E in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 199 /C1/39, 1994. Kimberling, C. "Integer Sequences and Arrays." http:// cedar.evansville.edu/~ck6/integer/. Krattenthaler, C. "RATE: A Mathematica Guessing Ma- chine." http://radon.mat.univie.ac.at/People/kratt/rate/ rate.html. Loxton, J. H. "Spectral Studies of Automata." In Irregula- rities of Partitions (Ed. G. Hala´sz and V. T. So´s). New York: Springer-Verlag, pp. 115 /C1/28, 1989. Ostman, H. Additive Zahlentheorie I, II. Heidelberg, Ger- many: Springer-Verlag, 1956. Petit, S. "Encyclopedia of Combinatorial Structures." http:// algo.inria.fr/encyclopedia/. Pomerance, C. and Sa´rko¨zy, A. "Combinatorial Number Theory." In Handbook of Combinatorics (Ed. R. Graham, M. Gro¨tschel, and L. Lova´sz). Amsterdam, Netherlands: North-Holland, 1994. Ruskey, F. "The (Combinatorial) Object Server." http:// www.theory.csc.uvic.ca/~cos/.Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, 1973. Sloane, N. J. A. "Find the Next Term." J. Recr. Math. 7, 146, 1974. Sloane, N. J. A. "An On-Line Version of the Encyclopedia of Integer Sequences." Elec. J. Combin. 1,F11 /C1/, 1994. http://www.combinatorics.org/Volume_1/volu- me1.html#F1. Sloane, N. J. A. "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sloane, N. J. A. "Some Important Integer Sequences." In CRC Standard Mathematical Tables and Formulae. (Ed. D. Zwillinger). Boca Raton, FL: CRC Press, 1995. Sloane, N. J. A. "Transformation of Integer Sequences." http://www.research.att.com/~njas/sequences/trans- forms.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Sto¨hr, A. "Gelo¨ste und ungelo ¨ste Fragen u¨ber Basen der natu¨rlichen Zahlenreihe I, II." J. reine angew. Math. 194, 40/C1/5 and 111 /C1/40, 1955. Tura´n, P. (Ed.). Number Theory and Analysis: A Collection of Papers in Honor of Edmund Landau (1877 /C1/938). New York: Plenum Press, 1969. Weisstein, E. W. "Integer Sequences." M ATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Integers INTEGER Integrable A function for which the INTEGRAL can be computed is said to be integrable. See also DIFFERENTIABLE ,INTEGRABLE (DIFFEREN- TIAL IDEAL ), INTEGRAL ,INTEGRATION ,LOCALLY IN- TEGRABLE Integrable (Differential Ideal) ADIFFERENTIAL IDEAL is an IDEAL Iin the RING of smooth FORMS on a MANIFOLD M. That is, it is closed under addition, scalar multiplication, and WEDGE PRODUCT with an arbitrary form. The IDEAL Iis called integrable if, whenever a/C23I;then also da/C23I; where dis the EXTERIOR DERIVATIVE . For example, in R3;the IDEAL I/C30a1ydx/C27a2dxffldy/C27a3ydxffldz/C27a4dxffldyffldz fg ; (1) where the aiare arbitrary smooth functions, is an integrable differential ideal. However, if the second term were of the form a2ydxffldy;then the ideal would not be integrable because it would not containd ydxðÞ/C30/C28dxffldy: / Given an integral differential ideal IonM,aSMOOTH MAP f:X0Mis called integral if the PULLBACK of every form avanishes on X, i.e., f/C31a/C300:In coordi- nates, an integral manifold solves a system of PARTIAL DIFFERENTIAL EQUATIONS . For example, using I above, a map f/C30f1;f2;f3 ðÞ from an OPEN SET inR2is integral if f2@f1 @x/C300 (2) f2@f1 @y/C300 (3) @f1 @x@f2 @y /C28@f1 @y@f2 @x/C300 (4) f2@f1 @x@f3 @y /C28@f1 @y@f3 @x ! /C300 (5) Conversely, any system of PARTIAL DIFFERENTIAL EQUATIONS can be expressed as an integrable differ- ential ideal on a JET BUNDLE . For instance, @f =@x /C30g on R corresponds to I /C30 df /C28gdx hi onR2/C30x;fðÞfg :/ See also DIFFERENTIAL K-FORM,INTEGRABLE ,JET BUNDLE ,PARTIAL DIFFERENTIAL EQUATION ,W EDGE PRODUCT Integral An integral is a mathematical object which can be interpreted as an AREA or a generalization of AREA . Integrals, together with DERIVATIVES , are the funda- mental objects of CALCULUS . Other words for integral include ANTIDERIVATIVE and PRIMITIVE . The R IEMANN INTEGRAL is the simplest integral definition and the only one usually encountered in physics and elemen-tary CALCULUS . In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases wherethese methods [i.e., generalizations of the Riemannintegral] are applicable and Riemann’s [definition ofthe integral] is not are too rare in physics to repay the extra difficulty." The R IEMANN INTEGRAL of the function f(x) over xfrom atobis written gb af(x)dx: (1) Every definition of an integral is based on a parti- cular MEASURE . For instance, the R IEMANN INTEGRAL is based on J ORDAN MEASURE , and the L EBESGUE INTEGRAL is based on L EBESGUE MEASURE . The process of computing an integral is called INTEGRA- TION (a more archaic term for INTEGRATION isQUAD- RATURE ), and the approximate computation of an integral is termed NUMERICAL INTEGRATION . There are two classes of (Riemann) integrals: DEFI- NITE INTEGRALS such as (1), which have upper and lower limits, and INDEFINITE INTEGRALS , such as gf(x)dx (2) which are written without limits. The first FUNDA- MENTAL THEOREM OF CALCULUS allows DEFINITE INTEGRALS to be computed in terms of INDEFINITE INTEGRALS , since if Fis the INDEFINITE INTEGRAL forf(x);then gb af(x)dx/C30F(b)/C28F(a): (3) WOLFRAM RESEARCH maintains a web site which will integrate many common (and not so common) func- tions. However, Mathematica 4.0 cannot solve some simple indefinite integrals such as gd dxxffiffiffiffiffiffiffiffiffiffiffi sinxp1CA}1CA$"# dx/C30gxcosx 2ffiffiffiffiffiffiffiffiffiffiffi sinxp /C27ffiffiffiffiffiffiffiffiffiffiffi sinxp ! dx (4) gd dxLi2(xlnx)"# dx /C30/C28g(lnx/C271) ln(1 /C28xlnx) xlnx"# dx; (5) where Li2(x) is the DILOGARITHM . Consider integrals of this form I(a)/C30gp=2 0dx 1/C27(tan x)a;(6) can be done trivially by taking advantage of the trigonometric identity tan1 2p/C28x1CA}1CA$ /C30cotx (7) Letting z/C13(tan x)a; I(a)/C30gp=4 0dx 1/C27z/C27gp=2 p=4dx 1/C27z /C30gp=4 0dx 1/C27z/C27gp=4 0dx 1/C271 z /C30gp=4 01 1/C27z/C271 1/C271 z0 BBB@1 CCCAdx/C30gp=4 0dx /C301 4p (8) However, Mathematica 3.0 gives an incorrect answer ofp1/C282ffiffi 3p =ffiffiffi 3p /C2154ffiffi 3p1CA}1CA$ to Iðffiffiffi 3p Þ¼gp=2 0dx 1þðtanxÞffiffi 3p¼1 4p; ð9Þ although integrals of this type remain unevaluated in Mathematica 4.0. Some care is therefore needed in the use of symbolic computer algebra packages for integration. This caveat is further illustrated by the example of the integral fðaÞ¼gp 0lnð1/C282acosxþa2Þdx¼2plnjajð 10Þ that has a simple analytic from for ajj>1 (Woods 1926) using the L EIBNIZ INTEGRAL RULE . However, Mathematica 4.0 gives a very complicated solution because it does not recognize the simple form above. There are a wide range of methods available for NUMERICAL INTEGRATION . Good sources for such techniques include Press et al. (1992) and Hildebrand (1956). The most straightforward numerical integra- tion technique uses the N EWTON- COTES FORMULAS (also called QUADRATURE FORMULAS ), which approx- imate a function tabulated at a sequence of regularlyspaced INTERVALS by various degree POLYNOMIALS .I f the endpoints are tabulated, then the 2- and 3-point formulas are called the TRAPEZOIDAL RULE and SIMPSON’S RULE , respectively. The 5-point formula is called B ODE’S RULE . A generalization of the TRAPE- ZOIDAL RULE isROMBERG INTEGRATION , which can yield accurate results for many fewer function eva- luations. If the analytic form of a function is known (instead of its values merely being tabulated at a fixed number ofpoints), the best numerical method of integration is called G AUSSIAN QUADRATURE . By picking the optimal ABSCISSAS at which to compute the function, Gaus- sian quadrature produces the most accurate approx-imations possible. However, given the speed of modern computers, the additional complication ofthe G AUSSIAN QUADRATURE formalism often makes it less desirable than the brute-force method of simplyrepeatedly calculating twice as many points on aregular grid until convergence is obtained. An ex-cellent reference for G AUSSIAN QUADRATURE is Hil- debrand (1956). Here is a list of common INDEFINITE INTEGRALS : gxrdx/C30xr/C271 r/C271/C27C (11) gdx x/C30lnxjj/C27C (12) gaxdx/C30ax lna/C27C (13) gsinxdx/C30/C28cosx/C27C (14) gcosxdx/C30sinx/C27C (15) gtanxdx/C30ln sec x jj/C27C (16) gcscxdx/C30ln csc x/C28cotx jj /C27C (17)/C30ln tan1 2x1CA}1CA$hi /C27C (18) 12ln1/C28cosx 1/C27cosx ! /C27C (19) gsecxdx ¼lnjsecxþtanxjþC ð20Þ /C30gd/C281(x)/C27C (21) gcotxdx/C30ln sin x jj/C27C (22) gsec2xdx/C30tanx/C27C (23) gcsc2xdx/C30/C28cotx/C27C (24) gsecxtanxdx/C30secx/C27C (25) gcos/C281xdx/C30xcos/C281x/C28ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C27C (26) gsin/C281xdx/C30xsin/C281x/C27ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C27C (27) gtan/C281xdx/C30xtan/C281x/C281 2ln 1/C27x21CC1CA /C27C (28) gdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28x2p /C30sin/C281x a ! /C27C (29) gdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28x2p /C30/C28cos/C281x a ! /C27C (30) gdx a2/C28x2/C301 a ! tan/C281x a ! /C27C (31) gdx a2/C27x2/C30/C281 acot/C281x a ! /C27C (32) gdx xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2/C28a2p /C30/C281 asec/C281x a ! /C27C (33) gdx xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C28a2p /C30/C281 acsc/C281x a ! /C27C (34) gsin2(ax)dx/C30x 2/C281 4asin(2 ax)/C27C (35) gsnud u/C30k/C281ln(dn u/C28kcnu)/C27C (36) gsn2ud u/C30u/C28E(u) k2/C27C (37) gcnud u/C30k/C281sin/C281(ksnu)/C27C (38) gdnud u/C30sin/C281(snu)/C27C; (39) where sin xis the SINE; cos xis the COSINE ; tanxis the TANGENT ; csc xis the COSECANT ; sec xis the SECANT ; cotxis the COTANGENT ; cos/C281xis the INVERSE COSINE ; sin/C281xis the INVERSE SINE ; tan/C281xis the INVERSE TANGENT ;s n u;cnu;and dn uare J ACOBI ELLIPTIC FUNCTIONS ;E(u) is a complete ELLIPTIC INTEGRAL OF THE SECOND KIND ; and gd( x) is the GUDERMANNIAN FUNCTION . To derive (16), let u/C13cosx;sodu/C30/C28sinxdxand gtanx/C30gsinu cosxdx/C30/C28gdu u /C30/C28lnujj/C27C/C30/C28ln cos x jj/C27C /C30ln cos x jj/C281/C27C/C30ln sec x jj/C27C: (40) To derive (17), let u/C13cscx/C28cotx;sodu/C30 (/C28cscxcotx/C27csc2x)dxand gcscxdx/C30gcscxcscx/C28cotx cscx/C28cotxdx /C30gcsc2x/C27cotxcscx cscx/C27cotxdx /C30gdu u/C30lnujj/C27C /C30ln csc x/C28cotx jj /C27C: (41) To derive (20), let u/C13secx/C27tanx; (42) so du/C30secxtanx/C27sec2x1CC1CA dx (43) and gsecxdx/C30gsecxsecx/C27tanx secx/C27tanxdx ¼gsec2xþsecxtanx secxþtanxdx /C30gdu u/C30lnujj/C27C /C30ln sec x/C27tanx jj /C27C: (44) To derive (22), let u/C13sinx;sodu/C30cosxdxand gcotxdx/C30gcosx sinxdx/C30gdu u/C30lnujj/C27C/C30ln sin x jj/C27C: (45) Integral identities include dx dy/C301 dy dx(46) d2x dy2/C30/C28d2y dx2dydx ! /C283 (47) d3x dy3/C303d2y dx2 !2 /C28d3y dx3dydx2 435 dy dx !/C285 (48) Differentiating integrals leads to some useful and powerful identities, for instance d dxgx afx?ðÞdx?/C30f(x); (49) which is the first FUNDAMENTAL THEOREM OF CALCU- LUS. Other derivative-integral identities include d dxgb xfx?ðÞdx?/C30/C28 f(x); (50) the L EIBNIZ INTEGRAL RULE d dxgb af(x;t)dt/C30gb a@ @xf(x;t)dt (51) (Kaplan 1992, p. 275), and its generalization d dxgv(x) u(x)f(x;t)dt /C30v?(x)f(x;v(x))/C28u?f(x;u(x))/C27gv(x) u(x)@ @xf(x;t)dt(52) (Leibniz 1992, p. 258). If f(x;t) is singular or INFINITE , then d dxgx af(x;t)dx /C301 x/C28agx a(x/C28a)@f @x/C27(t/C28a)@f @x/C27f"# dt (53) Other integral identities include gx 0dtngtn 0dtn/C281/C1/C1/C1gt3 0dt2gt2 0ft1ðÞdt1 /C301 (n/C281)!gx 0(x/C28t)n/C281f(t)dt (54) @ @xkxjJk1CC1CA /C30djkJk/C27xj@ @xkJk/C30J/C27r9 /C215J (55) gVJd3r/C30gV@ @xkxiJk ðÞ/C28gVr9 /C215Jd3r /C28gVr9 /C215Jd3r (56) and the amusing integral identity g/C12 /C28/C12F(f(x))dx/C30g/C12 /C28/C12F(x)dx; (57) where Fis any function and f(x)/C30x/C28X/C12 n/C300an x/C27bn(58) as long as an]0 and bnis real (Glasser 1983). Integrals OF THE FORM gb af(x)dx (59) with one INFINITE LIMIT and the other NONZERO may be expressed as finite integrals over transformed functions. If f(x) decreases at least as fast as 1 =x2; then let t/C131 x(60) dt/C30/C28dx x2(61) dx/C30/C28x2dt/C30/C28dt t2; (62) and gb af(x)dx/C30/C28g1=b 1=a1 t2f1 t ! dt/C30g1=a 1=b1 t2f1 t ! dt:(63) Iff(x) diverges as ( x/C28a)gforg/C23[0;1];let x/C13t1=(1/C28g)/C27a (64) dx/C301 1/C28gt1=(1/C28g)/C281dt/C301 1/C28gt[1/C28(1/C28g)]=(1/C28g)dt /C301 g/C281tg=(1/C28g)dt (65) t/C30(x/C28a)1/C28g; (66) and gb af(x)dx/C301 1/C28g/C30g(b/C28a)1/C28g 0tg(1/C28g)ft1=(1/C28g)/C27a1CC1CA dt:(67) Iff(x) diverges as ( x/C27b)gforg/C23[0;1];let x/C13b/C28t1=(1/C28g)(68) dx/C30/C281 g/C281tg=(1/C28g)dt (69)t/C30(b/C28x)1/C28g; (70) and gb af(x)dx/C301 1/C28g/C30g(b/C28a)1/C28g 0tg=(1/C28g)f(b/C28t1=(1/C28g)dt:(71) If the integral diverges exponentially, then let t/C13e/C28x(72) dt/C30/C28e/C28xdx (73) x/C30/C28lnt; (74) and g/C12 af(x)dx/C30ge/C28a 0f(/C28lnt)dt t: (75) Integrals with rational exponents can often be solvedby making the substitution u/C30x 1=n;where nis the LEAST COMMON MULTIPLE of the DENOMINATOR of the exponents. Integration rules include ga af(x)dx/C300 (76) gb af(x)dx/C30/C28ga bf(x)dx: (77) Forc/C23(a;b); gb af(x)dx/C30gc af(x)dx/C27gb cf(x)dx: (78) Ifg?is continuous on [ a, b] and fis continuous and has an antiderivative on an INTERVAL containing the values of g(x) for a5x5b;then gb afg(x) ðÞ g?(x)dx/C30gg(b) g(a)f(u)du: (79) Liouville showed that the integrals ge/C28x2dxgex xdxgsinx xdxgdx lnx(80) cannot be expressed as terms of a finite number of elementary functions. Other irreducibles include gxxdxgx/C28xdxgffiffiffiffiffiffiffiffiffiffiffi sinxp dx: (81) Chebyshev proved that if U,V, and Ware RATIONAL NUMBERS , then gxUA/C27BxV1CC1CA Wdx (82) is integrable in terms of elementary functions IFF (U/C271)=V;W,o rW/C27(U/C271)=Vis an INTEGER (Ritt 1948, Shanks 1993). See also A-INTEGRABLE ,ABELIAN INTEGRAL ,CALCU- LUS,C HEBYSHEV- GAUSS QUADRATURE ,C HEBYSHEV QUADRATURE ,D ARBOUX INTEGRAL ,D EFINITE INTE- GRAL ,D ENJOY INTEGRAL ,D ERIVATIVE ,D OUBLE EX- PONENTIAL INTEGRATION ,E ULER INTEGRAL , FUNDAMENTAL THEOREM OF GAUSSIAN QUADRATURE , GAUSS- JACOBI MECHANICAL QUADRATURE ,GAUSSIAN QUADRATURE ,H AAR INTEGRAL ,H ERMITE- GAUSS QUADRATURE ,HERMITE QUADRATURE ,HKI NTEGRAL , INDEFINITE INTEGRAL ,INTEGRATION ,JACOBI- GAUSS QUADRATURE ,J ACOBI QUADRATURE ,L AGUERRE- GAUSS QUADRATURE ,L AGUERRE QUADRATURE ,L E- BESGUE INTEGRAL ,L EBESGUE- STIELTJES INTEGRAL , LEGENDRE- GAUSS QUADRATURE ,LEGENDRE QUADRA- TURE ,LEIBNIZ INTEGRAL RULE,LOBATTO QUADRA- TURE ,M ECHANICAL QUADRATURE ,M EHLER QUADRATURE ,N EWTON- COTES FORMULAS ,N UMERI- CAL INTEGRATION ,PERRON INTEGRAL ,QUADRATURE , RADAU QUADRATURE ,RECURSIVE MONOTONE STABLE QUADRATURE ,R IEMANN- STIELTJES INTEGRAL ,R OM- BERG INTEGRATION ,RIEMANN INTEGRAL ,STIELTJES INTEGRAL References Beyer, W. H. "Integrals." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 233 /C1/ 96, 1987. Bronstein, M. Symbolic Integration I: Transcendental Func- tions. New York: Springer-Verlag, 1996. Glasser, M. L. "A Remarkable Property of Definite Inte- grals." Math. Comput. 40, 561 /C1/63, 1983. Gordon, R. A. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Providence, RI: Amer. Math. Soc., 1994. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319 /C1/23, 1956. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, p. 29, 1988. Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1992. Piessens, R.; de Doncker, E.; Uberhuber, C. W.; and Kaha- ner, D. K. QUADPACK: A Subroutine Package for Auto- matic Integration. New York: Springer-Verlag, 1983. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Integration of Functions." Ch. 4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 123 /C1/58, 1992. Ritt, J. F. Integration in Finite Terms. New York: Columbia University Press, p. 37, 1948. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 145, 1993. Woods, F. S. Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 143 /C1/44, 1926. Wolfram Research. "The Integrator." http://integrals.wol- fram.com/. Integral Brick EULER BRICKIntegral Calculus That portion of "the" CALCULUS dealing with INTE- GRALS . See also CALCULUS ,DIFFERENTIAL CALCULUS ,INTE- GRAL Integral Cohomology Class See also COHOMOLOGY CLASS Integral Cuboid EULER BRICK Integral Current A RECTIFIABLE CURRENT whose boundary is also a RECTIFIABLE CURRENT . Integral Curvature Given a GEODESIC TRIANGLE (a triangle formed by the arcs of three GEODESICS on a smooth surface), gABCKda/C30A /C27B /C27C /C28p: Given the EULER CHARACTERISTIC x; ggKda/C302px so the integral curvature of a closed surface is not altered by a topological transformation. See also GAUSS- BONNET FORMULA ,GEODESIC TRIAN- GLE Integral Domain A RING that is COMMUTATIVE under multiplication, has an IDENTITY ELEMENT , and has no divisors of 0. The INTEGERS form an integral domain. See also FIELD,IDEAL ,RING References Anderson, D. D. (Ed.). Factorization in Integral Domains. New York: Dekker, 1997. Integral Drawing AGRAPH drawn such that the EDGES have only INTEGER lengths. It is conjectured that every PLANAR GRAPH has an integral drawing. References Harborth, H. and Mo ¨ller, M. "Minimum Integral Drawings of the Platonic Graphs." Math. Mag. 67, 355/C1/58, 1994. Integral Equation If the limits are fixed, an integral equation is called a Fredholm integral equation. If one limit is variable, it is called a Volterra integral equation. If the unknown function is only under the integral sign, the equation is said to be of the "first kind." If the function is both inside and outside, the equation is called of the "second kind." A Fredholm equation of the first kind is OF THE FORM f ðx Þ¼gb aK ðx;tÞfðtÞdt: ð1Þ A Fredholm equation of the second kind is OF THE FORM f(x) /C30f(x) /C27gb aK(x; t)f(t)dt: (2) A Volterra equation of the first kind is OF THE FORM f(x) /C30gx aK(x;t) f(t)dt : (3) A Volterra equation of the second kind is OF THE FORM f(x) /C30f(x) /C27gx aK(x;t) f(t)dt; (4) where the functions K(x ;t) are known as KERNELS . Integral equations may be solved directly if they are SEPARABLE . Otherwise, a NEUMANN SERIES must be used. A KERNEL is separable if K(x;t) /C30 lXn j/C301Mj(x)Nj(t) : (5) This condition is satisfied by all POLYNOMIALS and many TRANSCENDENTAL FUNCTIONS .aF REDHOLM INTEGRAL EQUATION OF THE SECOND KIND with separ- able KERNEL may be solved as follows: f(x) /C30f(x) /C27gb aK(x;t) f(t)dt /C30f(x) /C27 lXn j/C301Mj(x)gb aNj(t)f(t)dt /C30f(x) /C27 lXn j /C301cjMj(x) ; (6) where cj /C13gb aNj(t) f(t)dt: (7) Now multiply both sides of (7) by Ni(x) and integrate over dx.gb af(x)Ni(x)dx /C30gb af(x)Ni(x)dx /C27 lXn j/C301cjgb aMj(x)Ni(x)dx: (8) By (7), the first term is just ci:Now define bi/C13gb aNi(x)f(x)dx (9) aij/C30gb aNi(x)Mj(x)dx; (10) so (8) becomes ci/C30bi/C27lXn j/C301aijcj (11) Writing this in matrix form, C/C30B/C27lAC; (12) so (I/C28lA)C/C30B (13) C/C30(I/C28lA)/C281B (14) See also FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND,FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND,VOLTERRA INTEGRAL EQUATION OF THE FIRST KIND,VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND References Corduneanu, C. Integral Equations and Applications. Cam- bridge, England: Cambridge University Press, 1991. Davis, H. T. Introduction to Nonlinear Differential and Integral Equations. New York: Dover, 1962. Kondo, J. Integral Equations. Oxford, England: Clarendon Press, 1992. Lovitt, W. V. Linear Integral Equations. New York: Dover, 1950. Mikhlin, S. G. Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2nd rev. ed. New York: Macmillan, 1964. Mikhlin, S. G. Linear Integral Equations. New York: Gor- don & Breach, 1961. Pipkin, A. C. A Course on Integral Equations. New York: Springer-Verlag, 1991. Porter, D. and Stirling, D. S. G. Integral Equations: A Practical Treatment, from Spectral Theory to Applica-tions. Cambridge, England: Cambridge University Press, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Integral Equations and Inverse Theory." Ch. 18 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 779 /C1 /17, 1992. Tricomi, F. G. Integral Equations. New York: Dover, 1957. Weisstein, E. W. "Books about Integral Equations." http:// www.treasure-troves.com/books/IntegralEquations.html. Whittaker, E. T. and Robinson, G. "The Numerical Solution of Integral Equations." §183 in The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 376 /C1/81, 1967. Integral Function ENTIRE FUNCTION Integral Geometry See also GEOMETRIC PROBABILITY ,STOCHASTIC GEO- METRY Integral of Motion A function of the coordinates which is constant along a trajectory in PHASE SPACE . The number of DEGREES OF FREEDOM of a DYNAMICAL SYSTEM such as the DUFFING DIFFERENTIAL EQUATION can be decreased by one if an integral of motion can be found. In general, it is very difficult to discover integrals of motion. Integral Polyhedron PRIMITIVE POLYTOPE Integral Polynomial INTEGER POLYNOMIAL Integral Sign The symbol f used to denote an INTEGRAL ff(x)dx : The symbol was invented by Leibniz and chosen to be a stylized script "S" to stand for "summation." See also INTEGRAL ,INTEGRATION UNDER THE INTE- GRAL SIGN Integral Test Let auk be a series with POSITIVE terms and let f(x)be the function that results when k is replaced by x in the FORMULA for uk : If f is decreasing and continuous for x ]1 and lim x 0/C12f(x) /C300; then X/C12 k /C301uk and g/C12 tf(x)dx both converge or diverge, where 1 5t 5/C12 : The test is also called the CAUCHY INTEGRAL TEST or MACLAURIN INTEGRAL TEST . See also CONVERGENCE TESTSReferences Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 283 /C1/284, 1985. Integral Transform A general integral transform is defined by g( a) /C30gb af(t)K( a;t)dt; where K( a;t) is called the KERNEL of the transform. See also BUSCHMAN TRANSFORM ,F OURIER TRANS- FORM ,F OURIER- STIELTJES TRANSFORM , G-TRANS- FORM ,H -TRANSFORM ,H ADAMARD TRANSFORM , HANKEL TRANSFORM ,HARTLEY TRANSFORM ,HOUGH TRANSFORM ,K ONTOROVICH- LEBEDEV TRANSFORM , MEHLER- FOCK TRANSFORM ,M EIJER TRANSFORM , NARAIN G-TRANSFORM ,OPERATIONAL MATHEMATICS , RADON TRANSFORM ,S TIELTJES TRANSFORM , W- TRANSFORM ,W AVELET TRANSFORM , Z-TRANSFORM References Arfken, G. "Integral Transforms." Ch. 16 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 794 /C1/864, 1985. Brychkov, Yu. A. and Prudnikov, A. P. Integral Transforms of Generalized Functions. New York: Gordon and Breach, 1989. Carslaw, H. S. and Jaeger, J. C. Operational Methods in Applied Mathematics. New York: Dover, 1963. Davies, B. Integral Transforms and Their Applications, 2nd ed. New York: Springer-Verlag, 1985. Erde´lyi, A.; Oberhettinger, M. F.; and Tricomi, F. G. Tables of Integral Transforms. Based, in Part, on Notes Left by Harry Bateman and Compiled by the Staff of the Bateman Manuscript Project, 2 vols. McGraw-Hill, 1954. Krantz, S. G. "Transform Theory." Ch. 15 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 195 /C1/217, 1999. Marichev, O. I. Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Ta- bles. Chichester, England: Ellis Horwood, 1982. Poularikas, A. D. (Ed.). The Transforms and Applications Handbook. Boca Raton, FL: CRC Press, 1995. Weisstein, E. W. "Books about Integral Transforms." http:// www.treasure-troves.com/books/IntegralTrans- forms.html. Zayed, A. I. Handbook of Function and Generalized Func- tion Transformations. Boca Raton, FL: CRC Press, 1996. Integrand The quantity being INTEGRATED , also called the KERNEL . For example, in ff(x)dx; f(x) is the integrand. See also INTEGRAL ,INTEGRATION Integrating Factor AFUNCTION by which an ORDINARY DIFFERENTIAL EQUATION is multiplied in order to make it integrable. See also ORDINARY DIFFERENTIAL EQUATION References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 526 /C1/529, 1953. Integration The process of computing or obtaining an INTEGRAL .A more archaic term for integration is QUADRATURE . See also CONTOUR INTEGRATION ,INTEGRAL ,INTEGRA- TION BY PARTS ,M EASURE THEORY ,N UMERICAL IN- TEGRATION References Shenitzer, A. and Steprans, S. J. "The Evolution of Integra- tion." Amer. Math. Monthly 101,66/C1/72, 1994. Integration (Form) A DIFFERENTIAL K-FORM can be integrated on an n- dimensional MANIFOLD . The basic example is an n- form a in the open unit ball in Rn : Since a is a TOP- DIMENSIONAL FORM , it can be written a /C30fdx1 ffl...ffl dxn and so gBa /C30gBfdm; (1) where the integral is the LEBESGUE INTEGRAL . On a MANIFOLD M covered by COORDINATE CHARTS Ui ; there is a PARTITION OF UNITY ri such that 1. ri is SUPPORTED in Ui and 2. a ri /C301:/ Then gMa /C30XgUiri a; (2) where the right-hand side is WELL DEFINED because each integration takes place in a COORDINATE CHART . The integral of the n-form a is WELL DEFINED because, under a change of coordinates g : X 0 Y ; the integral transforms according to the determinant of the JACOBIAN , while an n-form pulls back by the deter- minant of the JACOBIAN . Hence, gXg /C31(a) /C30gXJjjjjf(g(x)) /C30gYf(y) (3) is the same integral in either COORDINATE CHART . For example, it is possible to integrate the 2-form a /C30zdxffldy /C28ydxffldz /C27xdyffldz (4) on the SPHERE S2 : Because a point has MEASURE ZERO , it is enough to integrate a on S2 /C28(0 ;0;1); which can be covered by STEREOGRAPHIC PROJECTION f : R2 0 S2 /C28(0; 0;1): Sincef(x;y)/C302x 1/C27r2;2y 1/C27r2;1/C28r2 1/C27r2 ! (5) the PULLBACK MAP ofais f/C31(a)/C304 1/C27r2 ðÞ2dxffldy; (6) the integral of aonS2is gg4 (1/C27r2)22prdu/C304p: (7) Note that this computation is done more easily by STOKES’ THEOREM , because da/C303dxffldyffldz:/ See also DE RHAM COHOMOLOGY ,STOKES’ THEOREM , SUBMANIFOLD ,T OP-DIMENSIONAL FORM,V OLUME FORM Integration by Parts Integration by parts is a technique for performing definite integration fud v by expanding the differen- tial of a product of functions d(uv) and expressing the original integral in terms of a known integral fvd u : A single integration by parts starts with d(uv)/C30ud v/C27vd u ; (1) and integrates both sides, gd(uv)/C30uv/C30gud v/C27gvd u : (2) Rearranging gives gud v/C30uv/C28gvd u ; (3) so gb aud v/C30[uv]b a/C28gf(b) f(a)vd u ; (4) where [ f]ba/C30f(b)/C28f(a):/ This procedure can also be applied ntimes to ff(n)(x)g(x)dx: u/C30g(x)dv/C30f(n)(x)dx (5) du/C30g?(x)dx v/C30f(n/C281)(x): (6) Therefore, gf(n)g(x)dx/C30g(x)f(n/C281)(x)/C28gf(n/C281)(x)g?(x)dx: (7) But gf(n/C281)(x)g?(x)dx/C30g?(x)f(n/C282)(x)/C28gf(n/C282)(x)gƒ(x)dx(8) gf(n/C282)(x)g ƒ(x)dx /C30g ƒ(x)f(n /C283)(x) /C28gf(n/C283)(x)g(3)(x)dx; (9) so gf(n)(x)g(x)dx /C30g(x)f(n/C281)(x) /C28g ?(x)f(n/C282)(x) /C27g(x)f(n/C283)(x) /C28.../C27(/C281)ngf(x)g(n)(x)dx: (10) Now consider this in the slightly different form f f(x)g(x)dx: Integrate by parts a first time u /C30f(x) dv /C30g(x)dx (11) du /C30f ?(x)dx v /C30gg(x)dx; (12) so gf(x)g(x)dx /C30f(x) gg(x)dx /C28ggg(x)dx1C|C1C|A f ?(x)dx: (13) Now integrate by parts a second time, u /C30f ?(x) dv /C30gg(x)dx (14) du /C30f ƒ(x)dx v /C30ggg(x)(dx)2 ; (15) so gf(x)g(x)dx /C30f(x) gg(x)dx /C28f ?(x)ggg(x)(dx)2 /C27gggg(x)(dx)21C|C1C|A f ƒ(x)dx: (16) Repeating a third time, gf(x)g(x)dx /C30f(x)gg(x)dx /C28f ?(x) ggg(x)(dx)2 /C27f ƒ(x)gggg(x)(dx)3 /C28ggggg(x)(dx)31C|C1C|A f ???(x)dx : (17) Therefore, after n applications, gf(x)g(x)dx /C30f(x)gg(x)dx /C28f ?(x) ggg(x)(dx)2 /C27f ƒ(x)gggg(x)(dx)3 /C28... þ(/C281)n/C271f(n)(x)g/C1/C1/C1g|fflfflffl{zfflfflffl} n/C271g(x)(dx)n /C271/C27(/C281)ngg/C1/C1/C1g|fflfflffl{zfflfflffl} n/C271g(x)(dx)n /C2712 6643 775f (n/C271)(x)dx: (18) If fn/C271(x) /C300 (e.g., for an nth degree POLYNOMIAL ), the last term is 0, so the sum terminates after n terms and gf(x)g(x)dx /C30f(x) gg(x)dx /C28f ?(x)ggg(x)(dx)2 /C27f ƒ(x)gggg(x)(dx)3 /C28... /C27(/C281)n /C271f(n)(x)g/C1/C1/C1g|fflfflffl{zfflfflffl} n /C271g(x)(dx)n/C271 : (19) See also INTEGRAL ,INTEGRATION ,S UMMATION BY PARTS References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 12, 1972. Integration Constant CONSTANT OF INTEGRATION Integration Lattice A discrete subset of Rswhich is CLOSED under addition and subtraction and which contains Zs as a SUBSET . See also LATTICE ,POINT LATTICE References Sloan, I. H. and Joe, S. Lattice Methods for Multiple Integration. New York: Oxford University Press, 1994. Integration Theory MEASURE THEORY Integration Under the Integral Sign The use of the identity gb adxga a0f(x;a)da/C30ga a0dagb af(x;a)dx (1) to compute an INTEGRAL . For example, consider g1 0xadx/C301 a/C271(2) fora>/C281:Multiplying by daand integrating be- tween aandbgives gb cdag1 0xadx /C30gb ada a /C27 1 /C30lnb /C27 1 a /C27 1 ! : (3) But the left-hand side is equal to g1 0dagb ax ada /C30g1 0xb /C28 xa ln xdx; (4) so it follows that g1 0xb /C28 xa ln xdx /C30lnb /C27 1 a /C27 1 ! (5) (Woods 1926, pp. 145 /C1/146). See also INTEGRAL ,INTEGRAL SIGN,INTEGRATION , LEIBNIZ INTEGRAL RULE References Woods, F. S. "Integration Under the Integral Sign." §61 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathe- matics. Boston, MA: Ginn, pp. 145 /C1/146, 1926. Intension A definition of a SET by mentioning a defining property. See also EXTENSION (SET) References Russell, B. "Definition of Number." Introduction to Mathe- matical Philosophy. New York: Simon and Schuster, 1971. Interchange Graph LINE GRAPH Interest Interest is a fee (or payment) made for the borrowing (or lending) of money. The two most common types of interest are SIMPLE INTEREST , for which interest is paid only on the initial PRINCIPAL , and COMPOUND INTEREST , for which interest earned can be re-in- vested to generate further interest. See also COMPOUND INTEREST ,CONVERSION PERIOD , PRESENT VALUE ,RULE OF 72,SIMPLE INTEREST References Kellison, S. G. Theory of Interest, 2nd ed. Burr Ridge, IL: Richard D. Irwin, 1991. Interior That portion of a region lying "inside" a specified boundary. For example, the interior of the SPHERE is a BALL . See also EXTERIORInterior Angle Bisector ANGLE BISECTOR Interior Product The interior product is a dual notion of the EXTERIOR PRODUCT in an EXTERIOR ALGEBRA LV ; where V is a VECTOR SPACE . Given an ORTHONORMAL BASIS fei g of V, the forms fei1ffl...ffleipgi1 B...Bip (1) are an ORTHONORMAL BASIS for LpV : They define a metric on the EXTERIOR ALGEBRA , /C142a; b/C143: The interior product with a form g is the ADJOINT of the EXTERIOR PRODUCT with g : That is, a /C21 g; b hi /C30 a; b ffl g hi (2) for all b: For example, e1 ffle2 /C21 e3 /C300 (3) and e1 ffle2 ffle3 ffle4 /C21 e1 ffle4 /C30e2 ffle3 ; (4) where the eiare ORTHONORMAL , are two interior products. An inner product on V gives an isomorphism e : V # V /C31 with the DUAL SPACE V /C31: The interior product is the composition of this isomorphism with CONTRAC- TION . See also CONTRACTION (TENSOR ), EXTERIOR ALGEBRA , EXTERIOR PRODUCT ,INNER PRODUCT ,W EDGE PRO- DUCT Intermediate Value Theorem If f is continuous on a CLOSED INTERVAL [a, b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the CLOSED INTERVAL such that f(x) /C30c :/ See also WEIERSTRASS INTERMEDIATE VALUE THEO- REM Internal Bisectors Problem STEINER- LEHMUS THEOREM Internal Contact TANGENT INTERNALLY Internal Knot One of the "knots" tp/C271;... ,tm/C28p/C281of a B -SPLINE with control points P0;... ,Pnand KNOT VECTOR T/C30ft0;t1;...;tmg; where p /C13m /C28n /C281: See also B-SPLINE ,KNOT VECTOR Internal Path Length The sum I over all internal (circular) nodes of the paths from the root of an EXTENDED BINARY TREE to each node. For example, in the tree above, the external path length is 11 (Knuth 1997, p. 399 /C1/ 400). The internal and EXTERNAL PATH LENGTHS are related by E /C30I /C272n; where n is the number of internal nodes. See also EXTENDED BINARY TREE,EXTERNAL PATH LENGTH References Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Internally Tangent TANGENT INTERNALLY Interpolation The computation of points or values between ones that are known or tabulated using the surrounding points or values. See also AITKEN INTERPOLATION ,BESSEL’S INTERPO- LATION FORMULA ,EVERETT INTERPOLATION ,EXTRA- POLATION ,F INITE DIFFERENCE ,G AUSS’S INTERPOLATION FORMULA ,HERMITE INTERPOLATION , LAGRANGE INTERPOLATING POLYNOMIAL ,N EWTON- COTES FORMULAS ,N EWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA ,O SCULATING INTERPOLA- TION ,THIELE’S INTERPOLATION FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). "Interpolation." §25.2 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 878 /C1/882, 1972. Iyanaga, S. and Kawada, Y. (Eds.). "Interpolation." Appen- dix A, Table 21 in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 1482 /C1/1483, 1980. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Interpolation and Extrapolation." Ch. 3 in Numerical Recipes in FORTRAN: The Art of ScientificComputing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 99 /C1/122, 1992. Whittaker, E. T. and Robinson, G. "Interpolation with Equal Intervals of the Argument." Ch. 1 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 1 /C1/34, 1967. Interquartile Range Divide a set of data into two groups (high and low) of equal size at the MEDIAN if there is an EVEN number of data points, or two groups consisting of points on either side of the MEDIAN itself plus the MEDIAN if there is an ODD number of data points. Find the MEDIANS of the low and high groups, denoting these first and third quartiles by Q1and Q3 : The inter- quartile range is then defined by IQR /C13Q3 /C28Q1 : See also H-SPREAD ,H INGE ,M EDIAN (STATISTICS ), QUARTILE Interradius MIDRADIUS Intersecting Circles CIRCLE- CIRCLE INTERSECTION Intersecting Cylinders STEINMETZ SOLID Intersecting Lines LINE-LINE INTERSECTION Intersecting Spheres SPHERE- SPHERE INTERSECTION Intersection The intersection of two SETS A and B is the SET of elements common to A and B. This is written A S B; and is pronounced "A intersection B"or" A cap B." The intersection of sets A1through Anis written S n i/C301 Ai :/ The intersection of two LINES AB and CD is written AB S CD: The intersection of two or more geometric objects is the point (points, lines, etc.) at which they CONCUR . See also AND, CIRCLE- CIRCLE INTERSECTION ,CIRCLE- LINE INTERSECTION ,C ONCUR ,C ONCURRENT ,C ONE- SPHERE INTERSECTION ,CONIC SECTION ,CYLINDRICAL SECTION ,LINE-LINE INTERSECTION ,SPHERE- SPHERE INTERSECTION ,SPIRIC SECTION ,STEINMETZ SOLID , TORIC SECTION ,T OTAL INTERSECTION THEOREM , UNION ,VENN DIAGRAM ,VIVIANI’S CURVE Intersection (Homology) When two cycles intersect TRANSVERSALLY X1 S X2 /C30 Y on a SMOOTH MANIFOLD M, then Y is a cycle. Moreover, the homology class that Y represents depends only on the HOMOLOGY CLASS of X1and X2 : The sign of Y is determined by the orientations on M, X1 ; and X2 :/ For example, two curves can intersect in one point on a surface transversally, since dim X1 /C27dim X2 /C301 /C271 /C302 /C30dim M /C280 : The curves can be deformed so that they intersect three times, but two of those intersections sum to zero since two intersect positively and one intersects negatively, i.e., with the ORIENTATION of the curves being the reverse orientation of the ambient space. On the torus illustrated above, the cycles intersect in one point. The binary operation of intersection makes homology on a MANIFOLD into a RING . That is, it plays the role of multiplication, which respects the grading. When a /C23 Hn/C28pand a /C23 Hn/C28q ; then a S b /C23 Hn/C28(p /C27q) : In fact, intersection is the dual to the CUP PRODUCT in POINCARE ´ DUALITY . That is, if a /C23 Hp is the POINCARE ´ DUAL to A /C23 Hn/C28pand b /C23 Hq is the dual to B /C23 Hn/C28q then a ffl b /C23 Hp /C27q is the dual to A S B /C23 Hn/C28(p /C27q) :/ Without the notion of TRANSVERSALITY , intersections are not well-defined in HOMOLOGY . On a more general space, even a manifold with singularities, the homol- ogy does not have a natural ring structure. See also CODIMENSION ,CUP PRODUCT ,H OMOLOGY , MANIFOLD ,ORIENTATION (MANIFOLD ), ORIENTATION (VECTOR SPACE ), POINCARE DUALITY ,TRANSVERSAL INTERSECTION Intersection Array Given a DISTANCE-REGULAR GRAPH G with integers bi ; ci ; i /C300;...;d such that for any two vertices x;y /C23 G at distance i /C30d(x;y) ; there are exactly ci neighbors of y /C23 Gi/C281(x) and bineighbors of y /C23 Gi/C271(x) ; the se-quence i(g) /C30fb0 ;b1 ;...;bd/C281;c1 ; ... ;cd g is called the intersection array of G. References Bendito, E.; Carmona, A.; and Encinas, A. M. "Shortest Paths in Distance-Regular Graphs." Europ. J. Combin. 21, 153 /C1/166, 2000. Intersection Detection See also TESSELLATION References Skiena, S. S. "Intersection Detection" §8.6.8 in The Algo- rithm Design Manual. New York: Springer-Verlag, pp. 370 /C1/373, 1997. Intersection Graph GRAPH INTERSECTION Intersection Number The intersection number v(G) of a given GRAPH G is the minimum number of elements in a set S such that G is an intersection graph on S. See also GRAPH INTERSECTION References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Interspersion An ARRAY A /C30aij ; i ;j ]1of POSITIVE INTEGERS is called an interspersion if 1. The rows of A comprise a PARTITION of the POSITIVE INTEGERS , 2. Every row of A is an INCREASING SEQUENCE , 3. Every column of A is a (possibly FINITE ) INCREASING SEQUENCE , 4. If (uj) and (vj) are distinct rows of A and if p and q are any indices for which up Bvq Bup /C271 ; then up /C271 Bvq /C271 Bup /C272 :/ If an array A /C30aijis an interspersion, then it is a DISPERSION . If an array A /C30a(i ;j) is an interspersion, then the sequence xnfg given by fxn /C30i : n /C30(i ;j) g for some jis a FRACTAL SEQUENCE . Examples of inter- spersion are the S TOLARSKY ARRAY and W YTHOFF ARRAY . See also DISPERSION (SEQUENCE ), FRACTAL SE- QUENCE ,STOLARSKY ARRAY References Kimberling, C. "Interspersions and Dispersions." Proc. Amer. Math. Soc. 117, 313/C1/321, 1993. Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157/C1/168, 1997. Intersphere MIDSPHERE Interval A collection of points on a LINE SEGMENT . If the endpoints a and b are FINITE and are included, the interval is called CLOSED and is denoted [a, b]. If one of the endpoints is 9/C12 ; then the interval still contains all of its LIMIT POINTS ,so[ a;/C12) and (/C28/C12;b] are also closed intervals. If the endpoints are not included, the interval is called OPEN and denoted (a, b). If one endpoint is included but not the other, the interval is denoted [a, b)or(a, b] and is called a HALF-CLOSED (or HALF-OPEN ) interval. The non-standard notation ]a; b[ for an OPEN INTER- VAL and [a ;b[or] a ;b] for a HALF-CLOSED INTERVAL is sometimes also used. See also CLOSED INTERVAL ,HALF-CLOSED INTERVAL , LIMIT POINT ,OPEN INTERVAL ,PENCIL Interval Graph A GRAPH G /C30(V ; E) is an interval graph if it captures the INTERSECTION RELATION for some set of INTERVALS on the REAL LINE. Formally, P is an interval graph provided that one can assign to each v /C23 V an interval Iv such that Iu S Iv is nonempty precisely when uv /C23 E: An interval graph on a list l can be generated using IntervalGraph [l] in the Mathematica add-on pack- ageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). STAR GRAPHS are interval graphs, but CYCLE GRAPHS are not (Skiena 1990, p. 164). Determining if a graph is an interval graph and realizing it can be done in O(n) time (Booth and Lueker 1976; Skiena 1990, p. 164). See also COMPARABILITY GRAPH References Booth, K. S. and Lueker, G. S. "Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity using PQ-Tree Algorithms." J. Comput. System Sci. 13, 335 /C1/379, 1976. Fishburn, P. C. Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. New York: Wiley, 1985. Gilmore, P. C. and Hoffman, A. J. "A Characterization of Comparability Graphs and of Interval Graphs." Canad. J. Math. 16, 539 /C1/548, 1964.Lekkerkerker, C. G. and Boland, J. C. "Representation of a Finite Graph by a Set of Intervals on the Real Line." Fund. Math. 51,45/C1/64, 1962. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 163 /C1/164, 1990. Interval Order A POSET P /C30(X ;5) is an interval order if it is ISOMORPHIC to some set of INTERVALS on the REAL LINE ordered by left-to-right precedence. Formally, P is an interval order provided that one can assign to each x /C23 X an INTERVAL [xL ;xR] such that xR ByLin the REAL NUMBERS IFF x By in P. See also PARTIALLY ORDERED SET References Fishburn, P. C. Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. New York: Wiley, 1985. Wiener, N. "A Contribution to the Theory of Relative Position." Proc. Cambridge Philos. Soc. 17, 441 /C1/449, 1914. Intrinsic Curvature A CURVATURE such as GAUSSIAN CURVATURE which is detectable to the "inhabitants" of a surface and not just outside observers. An EXTRINSIC CURVATURE ,on the other hand, is not detectable to someone who can’t study the 3-dimensional space surrounding the sur- face on which he resides. See also CURVATURE ,EXTRINSIC CURVATURE ,GAUS- SIAN CURVATURE Intrinsic Equation An equation which specifies a CURVE in terms of intrinsic properties such as ARC LENGTH , RADIUS OF CURVATURE , and TANGENTIAL ANGLE instead of with reference to artificial coordinate axes. Intrinsic equa- tions are also called NATURAL EQUATIONS . See also CESA` RO EQUATION ,N ATURAL EQUATION , WHEWELL EQUATION References Yates, R. C. "Intrinsic Equations." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 123 /C1/126, 1952. Intrinsic Variety See also VARIETY Intrinsically Linked A GRAPH is intrinsically linked if any embedding of it in 3-D contains a nontrivial link. A GRAPH is intrinsi- cally linked IFF it contains one of the seven PETERSEN GRAPHS (Robertson et al. 1993). The COMPLETE GRAPH K6(left) is intrinsically linked because it contains at least two linked TRIANGLES . The COMPLETE K-PARTITE GRAPH K3;3 ;1(right) is also intrinsically linked. See also COMPLETE GRAPH ,C OMPLETE K-PARTITE GRAPH ,PETERSEN GRAPH References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 217 /C1/221, 1994. Robertson, N.; Seymour, P. D.; and Thomas, R. "Linkless Embeddings of Graphs in 3-Space." Bull. Amer. Math. Soc. 28,84/C1/89, 1993. Invaginatum A negative-height (inward-pointing) PYRAMID used in CUMULATION . The term was introduced by B. Gru¨n- baum. See also CUMULATION ,ELEVATUM Invariable Point Three concurrent homologous lines pass respectively through three fixed points on the SIMILITUDE CIRCLE which are known as the invariable points. See also HOMOLOGOUS POINTS ,SIMILITUDE CIRCLE References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Invariant A quantity which remains unchanged under certain classes of transformations. Invariants are extremely useful for classifying mathematical objects because they usually reflect intrinsic properties of the object of study. See also ADIABATIC INVARIANT ,ALEXANDER INVAR- IANT,ALGEBRAIC INVARIANT ,ARF INVARIANT ,G EO- METRIC INVARIANT THEORY ,INTEGRAL OF MOTION , INVARIANT (ELLIPTIC FUNCTION ), KNOT POLYNOMIALReferences Hunt, B. "Invariants." Appendix B.1 in The Geometry of Some Special Arithmetic Quotients. New York: Springer- Verlag, pp. 282 /C1/290, 1996. Olver, P. J. Classical Invariant Theory. Cambridge, Eng- land: Cambridge University Press, 1999. Invariant (Elliptic Function) The invariants of a WEIERSTRASS ELLIPTIC FUNCTION /C212(z½ v1 ; v2) are defined by the EISENSTEIN SERIES g2(v1 ; v2) /C1360X ? m;nV/C284 m;n g3( v1 ; v2) /C13140X ? m;nV/C285 m;n : Here, Vmn( v1; v2) /C132mv1 /C282nv2 ; where v1and v2are the periods of the ELLIPTIC FUNCTION . Writing gi(t) /C13gi(1; t) ; g2( t) /C13g2(1; t) /C30 v4 1(v1 ; v2) (1) g3( t) /C13g3(1; t) /C30 v61( v1 ; v2); (2) and the invariants have the FOURIER SERIES g2(t) /C304p4 41 /C27240X/C12 k/C301s3(k)e2 pikt"# (3) g3(t) /C308p6 271 /C28504X/C12 k/C301s5(k)e2 pikt"# (4) where t /C13 v2 =v2and sk(n) is the DIVISOR FUNCTION (Apostol 1997). See also DEDEKIND ETA FUNCTION ,E ISENSTEIN SERIES ,M ODULAR DISCRIMINANT ,T AU FUNCTION , WEIERSTRASS ELLIPTIC FUNCTION References Apostol, T. M. "The Fourier Expansions of g2( t) and g3( t) :/" §1.9 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 12 /C1/13, 1997. Invariant Density NATURAL INVARIANT Invariant Factor The polynomials in the DIAGONAL of the SMITH NORMAL FORM or RATIONAL CANONICAL FORM of a MATRIX are called its invariant factors. See also RATIONAL CANONICAL FORM,SMITH NORMAL FORM References Ayres, F. Jr. "Smith Normal Form." Ch. 24 in Theory and Problems of Matrices. New York: Schaum, pp. 188 /C1/195, 1962. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. Invariant Factors The polynomials in the DIAGONAL of the SMITH NORMAL FORM of a MATRIX . References Ayres, F. Jr. "Smith Normal Form." Ch. 24 in Theory and Problems of Matrices. New York: Schaum, pp. 188 /C1/195, 1962. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. Invariant Manifold When stable and unstable invariant MANIFOLDS intersect, they do so in a HYPERBOLIC FIXED POINT (SADDLE POINT ). The invariant MANIFOLDS are then called SEPARATRICES .A HYPERBOLIC FIXED POINT is characterized by two ingoing stable MANIFOLDS and two outgoing unstable MANIFOLDS . In integrable systems, incoming Ws and outgoing Wu MANIFOLDS all join up smoothly. A stable invariant MANIFOLD Ws of a FIXED POINT Y /C31 is the set of all points Y0such that the trajectory passing through Y0 tends to Y /C31 as j 0/C12:/ An unstable invariant MANIFOLD Wu of a FIXED POINT Y /C31 is the set of all points Y0 such that the trajectory passing through Y0 tends to Y /C31 as j 0/C28/C12:/ See also HOMOCLINIC POINT Invariant Point FIXED POINT (TRANSFORMATION ) Invariant Series An invariant series of a GROUP G is a NORMAL SERIES I /C30A01 A11 ...1 Ar /C30G such that each Ai1G ; where H1G means that H is a NORMAL SUBGROUP of G. See also COMPOSITION SERIES ,NORMAL SERIES References Scott, W. R. Group Theory. New York: Dover, p. 36, 1987. Invariant Subgroup NORMAL SUBGROUPInverse Cosecant The function csc/C281 x; also denoted arccsc( x), where csc x is the COSECANT and the SUPERSCRIPT -1 denotes an INVERSE FUNCTION , not the multiplicative inverse. The inverse cosecant is implemented asArcCsc [x]in Mathematica . The inverse cosecant satisfies csc /C281 x /C30sec /C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 1p ! (1) for POSITIVE or NEGATIVE x, and csc /C281 x /C30p/C27csc /C281(/C28x) (2) for x ]0 : The inverse cosecant has TAYLOR SERIES about infinity of csc /C281 x /C30x/C281 /C271 6x/C283 /C273 40x/C285 /C275 112x/C287 þ ...: (3) The inverse cosecant is given in terms of other inverse trigonometric functions by csc /C281 x /C30cos /C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C281p x ! (4) /C30cot/C281ffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C281p1CA}1CA$ (5) /C301 2p/C28sec/C281x/C30/C2812p/C28sec/C281(/C28x) (6) /C30sin/C2811 x ! (7) forx]0:/ See also COSECANT ,INVERSE SINE,SINE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142 /C1/143, 1987. Inverse Cosine The function cos /C281 x; where cos x is the COSINE and the superscript -1 denotes the INVERSE FUNCTION , not the multiplicative inverse. The notation arccos x or Arccos x is sometimes also used. The inverse cosine is implemented as ArcCos [x]in Mathematica . The inverse cosine satisfies cos /C281 x /C30p/C28cos/C281(/C28x) (1) for POSITIVE and NEGATIVE x, and cos/C281 x /C301 2p/C28cos/C281ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p1CA}1CA$ for 0 5x 51 1 2p/C27cos /C281ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p1CA}1CA$ for /C281 5x 50:8 < : (2) The MACLAURIN SERIES for the inverse cosine with /C281 5x 51is cos/C281 x /C301 2 p/C28x /C2816x3 /C283 40x5 /C285 112x7 /C2835 1152x9 /C28...: (3) The inverse cosine is given in terms of other inverse trigonometric functions by cos /C281 x /C30cot /C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2p ! (4) /C301 2 p/C27sin/C281(/C28x) /C3012 p/C28sin/C281 x (5) ¼1 2 p/C28tan/C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2p ! (6)for POSITIVE or NEGATIVE x, and cos/C281x/C30csc/C281 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x2p ! (7) /C30sec/C2811 x ! (8) /C30sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p1CA}1CA$ (9) /C30tan/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p x ! (10) forx]0:/ See also COSINE ,INVERSE SECANT References Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79 /C1/83, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142 /C1/143 and 219, 1987. Inverse Cotangent The function cot/C281x;also denoted arccot( x), where cotxis the COTANGENT and the superscript -1 denotes an INVERSE FUNCTION and not the multiplicative inverse. The inverse cotangent is implemented as ArcCot [x]i nMathematica . The M ACLAURIN SERIES of the inverse cotangent is given by cot/C281x/C301 2p/C28x/C2713x3/C2815x5/C2717x7/C2819x9/C27...; (1) and L AURENT SERIES by cot /C281 x /C30x/C281 /C281 3x/C283 /C2715x/C285 /C2817x/C287 /C2719x/C289 /C27...: (2) Euler derived the INFINITE series cot /C281 x /C30x1 x2 /C27 1 /C272 3(x2 /C27 1)2 /C272 /C215 4 3 /C215 5(x2 /C27 1)3 /C27..."# (3) (Wetherfield 1996). The inverse cotangent satisfies cot /C281 x /C30tan/C2811 x ! (4) /C30/C28cot /C281(/C28x) (5) for POSITIVE and NEGATIVE x, and cot /C281 x /C30cos/C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 1p ! (6) /C301 2p/C28cot /C2811 x ! (7) ¼ csc /C281(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C271)p (8) /C30sec/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 1p x ! (9) ¼ sin/C281 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 /C27 1p ! (10) /C301 2 p/C28sin/C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 1p ! (11) ¼1 2 p/C27tan/C281(/C28x) (12) ¼1 2 p/C28tan /C281 x (13) for x ]0:/ A number tx /C30cot /C281 x; (14) where x is an INTEGER or RATIONAL NUMBER ,is sometimes called a GREGORY NUMBER . Lehmer (1938a) showed that cot /C281(a=b) can be expressed as a finite sum of inverse cotangents of INTEGER argu- ments cot /C281a b ! /C30Xk i/C301(/C281)i/C281cot /C281 ni ; (15) where ni /C30ai bi$% ; (16) with xbcthe FLOOR FUNCTION , andai/C271 /C30ain /C27i /C27bi (17) bi/C271 /C30ai /C28nibi ; (18) with a0 /C30a and b0 /C30b; and where the recurrence is continued until bk /C271 /C300: If an INVERSE TANGENT sum is written as tan /C281 n /C30X k/C301fk tan/C281 nk /C27f tan /C281 1 ; (19) then equation (15) becomes cot /C281 n /C30X k /C301fk cot /C281 nk /C27c cot/C281 1 ; (20) where c /C302 /C28f /C282X k/C301fk : (21) Inverse cotangent sums can be used to generate MACHIN-LIKE FORMULAS . An interesting inverse cotangent identity attributed to Charles Dodgson (Lewis Carroll) by Lehmer (1938b; Bromwich 1965, Castellanos 1988ab) is cot /C281(p /C27r) /C27tan /C281(p /C27q) /C30tan /C281 p ; (22) where 1/C27p2/C30qr: (23) Other inverse cotangent identities include 2 cot/C281(2x)/C28cot/C281x/C30cot/C281(4x3/C273x) (24) 3 cot/C281(3x)/C28cot/C281x/C30cot/C28127x4/C2718x2/C281 8x ! ;(25) as well as many others (Bennett 1926, Lehmer 1938b). See also COTANGENT ,INVERSE TANGENT ,M ACHIN’S FORMULA ,MACHIN- LIKE FORMULAS ,TANGENT References Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79 /C1/83, 1972. Bennett, A. A. "The Four Term Diophantine Arccotangent Relation." Ann. Math. 27,2 1/C1/24, 1926. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142 /C1/143, 1987. Bromwich, T. J. I. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1/98, 1988a. Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148/C1/163, 1988b. Lehmer, D. H. "A Cotangent Analogue of Continued Frac- tions." Duke Math. J. 4, 323/C1/340, 1938a. Lehmer, D. H. "On Arccotangent Relations for p:/"Amer. Math. Monthly 45, 657/C1/664, 1938b. Weisstein, E. W. "Arccotangent Series." M ATHEMATICA NO- TEBOOK COTSERIES.M . Wetherfield, M. "The Enhancement of Machin’s Formula by Todd’s Process." Math. Gaz., 333 /C1/344, July 1996. Inverse Curve Given a CIRCLE C with CENTER O and RADIUS k, then two points P and Q are inverse with respect to C if OP /C215 OQ /C30k2 : If P describes a curve C1 ; then Q describes a curve C2called the inverse of C1with respect to the circle C (with INVERSION CENTER O). The PEAUCELLIER INVERSOR can be used to construct an inverse curve from a given curve. If the POLAR equation of C is r( u) ; then the inverse curve has polar equation r /C30k2 r( u) : If O /C30 x0 ;y0 ðÞ and P /C30 f(t) ;g(t) ðÞ ; then the inverse has equations x /C30x0 /C27k2 f /C28 x0 ðÞ f /C28 x0 ðÞ2/C27 g /C28 y0 ðÞ2 y /C30y0 /C27k2 g /C28 y0 ðÞ f /C28 x0 ðÞ2/C27 g /C28 y0 ðÞ2 : Curve INVERSION CENTERInverse Curve ARCHIMEDEAN SPIRALORIGIN ARCHIMEDEAN SPIRAL CARDIOID CUSP PARABOLA CIRCLE any point another CIRCLE CISSOID OF DIOCLESCUSP PARABOLA COCHLEOID ORIGIN QUADRATRIX OF HIPPIAS EPISPIRAL ORIGIN ROSE FERMAT’S SPIRAL ORIGIN LITUUS HYPERBOLA center LEMNISCATE HYPERBOLA VERTEX RIGHT STROPHOID HYPERBOLA with a /C30ffiffiffi 3p /VERTEX MACLAURIN TRI- SECTRIX LEMNISCATE center HYPERBOLA LITUUS ORIGIN FERMAT’S SPIRAL LOGARITHMIC SPIRALORIGIN LOGARITHMIC SPIRAL MACLAURIN TRI- SECTRIXFOCUS TSCHIRNHAUSEN’S CUBIC PARABOLA FOCUS CARDIOIDPARABOLA VERTEX CISSOID OF DIOCLES QUADRATRIX OF HIPPIASCOCHLEOID RIGHT STRO- PHOIDORIGIN the same RIGHT STROPHOID IN- VERSE CURVE SINUSOIDAL SPIRALORIGIN SINUSOIDAL SPIRAL TSCHIRNHAUSEN CUBICSINUSOIDAL SPIRAL See also INVERSION ,INVERSION CENTER ,INVERSION CIRCLE ,PEAUCELLIER INVERSOR ,RECIPROCAL ,RECI- PROCATION References Welke, S. "Inversion of Elementary Algebraic Curves with Respect to a Circle." Mathematica Educ. Res. 4,16/C1/22, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 120, 1991. Yates, R. C. "Inversion." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 127 /C1/134, 1952. Inverse Elliptic Nome INVERSE NOME Inverse Filter A linear DECONVOLUTION ALGORITHM . Inverse Fourier Transform FOURIER TRANSFORM Inverse Function Given a FUNCTION f(x); its inverse f /C281(x) is defined by f(f /C281(x)) /C30f /C281(f(x)) /C13x: Therefore, f(x) and f/C281(x) are reflections about the liney/C30x. See also COMPOSITION ,INVERSE FUNCTION THEOREM , SERIES REVERSION References Jeffreys, H. and Jeffreys, B. S. "Inverse Functions." §1.066 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 22 /C1/23, 1988. Inverse Function Theorem Given a SMOOTH FUNCTION f:Rn0Rn;if the J ACO- BIAN is invertible at 0 ;then there is a NEIGHBORHOOD Ucontaining 0 such that f:U0f(U)i sa DIFFEO- MORPHISM . That is, there is a smooth inverse f /C281 : f(U) 0 U :/ See also DIFFEOMORPHISM ,IMPLICIT FUNCTION THE- OREM ,JACOBIAN References Rudin, W. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, 1976. Inverse Hyperbolic Cosecant The INVERSE FUNCTION of the HYPERBOLIC COSECANT , denoted csch /C281 z : It can be defined for complex z by csch/C281 z /C30lnffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C271 z2s /C271 z ! ; (1) or for real x by csch/C281 x /C30ln1 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 x2p x ! : (2) The inverse hyperbolic cosecant is implemented as ArcCsch [x]inMathematica . The inverse hyperbolic cosecant has TAYLOR SERIES csch /C281 x /C30(ln 2 /C28ln x) /C271 4x2 /C283 32x4 /C275 96x6 /C27... (3) csch/C2811 x ! /C30x /C2816x3 /C273 40x5 /C285 112x7 /C27...: (4) See also HYPERBOLIC COSECANT ,INVERSE HYPER- BOLIC FUNCTIONSInverse Hyperbolic Cosine The INVERSE FUNCTION of the HYPERBOLIC COSINE , denoted cosh /C281 z : It can be defined for complex z by cosh /C281 z /C30ln z /C27ffiffiffiffiffiffiffiffiffiffiffi z /C271pffiffiffiffiffiffiffiffiffiffiffi z /C281p1CA}1CA$ ; (1) and for real x by cosh/C281 x /C30ln x 9ffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C281p1CA}1CA$ : (2) The inverse cosine is implemented asArcCosh [x]in Mathematica . The inverse hyperbolic cosine has the TAYLOR SERIES cosh /C281 x /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(x /C281)p /C2 1 /C281 12(x /C281) /C273 160(x /C281)2 /C285 896(x/C281)3/C27...hi (3) cosh/C2811 x ! /C30(ln 2/C28lnx)/C281 4x2/C283 32x4/C285 96x6/C27...:(4) See also HYPERBOLIC COSINE ,INVERSE HYPERBOLIC FUNCTIONS Inverse Hyperbolic Cotangent The INVERSE FUNCTION of the HYPERBOLIC COTAN- GENT , denoted coth /C281 x: It can be defined for complex z as coth /C281 z /C301 2ln 1 /C271 z ! /C28ln 1 /C281 z ! "# ; (1) and for real x as coth /C281 x /C3012 lnx /C27 1 x /C28 1 ! : (2) The inverse hyperbolic cotangent is implemented as ArcCoth [x]inMathematica . It has the special values coth /C281 0 /C30/C281 2i p (3) coth /C281 1 /C30/C12 (4) coth /C281 /C12/C300: (5) coth /C281 i /C30/C2814 pi (6) and the MACLAURIN SERIES coth /C2811 x ! /C30x /C271 3x3 /C2715x5 /C2717x7 /C27...: (7) See also HYPERBOLIC COTANGENT ,INVERSE HYPER- BOLIC FUNCTIONS ,INVERSE HYPERBOLIC TANGENT Inverse Hyperbolic Functions The INVERSE of the HYPERBOLIC FUNCTIONS , denoted cosh /C281 x; coth /C281 x; csch/C281 x; sech/C281 x; sinh/C281 x; and tanh /C281 x: They are defined bysinh /C281 z /C30ln z /C27ffiffiffiffiffiffiffiffiffiffiffiffiffi z2 /C271p1CA}1CA$ (1) cosh/C281 z /C30ln z 9ffiffiffiffiffiffiffiffiffiffiffiffiffi z2 /C281p1CA}1CA$ (2) tanh/C281 z /C301 2 ln1 /C27 z 1 /C28 z ! (3) csch/C281 z /C30ln1 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 z2p z ! (4) sech/C281 z /C30ln1 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 z2p z ! (5) coth /C281 z /C301 2lnz/C271 z/C281 ! : (6) See also HYPERBOLIC FUNCTIONS ,INVERSE HYPER- BOLIC COSECANT ,INVERSE HYPERBOLIC COSINE ,IN- VERSE HYPERBOLIC COTANGENT ,I NVERSE HYPERBOLIC SECANT ,INVERSE HYPERBOLIC SINE, INVERSE HYPERBOLIC TANGENT References Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic Functions." §4.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 86 /C1/89, 1972. Spanier, J. and Oldham, K. B. "The Inverse Hyperbolic Functions." Ch. 31 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 285 /C1/293, 1987. Inverse Hyperbolic Secant The INVERSE FUNCTION of the HYPERBOLIC SECANT , denoted sech /C281 x: It can be defined for complex z as sec /C281 z /C30lnffiffiffiffiffiffiffiffiffiffiffi 1 z /C281sffiffiffiffiffiffiffiffiffiffiffi 1 z /C271s /C271 z ! ; (1) and for real x as sech/C281 x /C30ln1 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2p x ! : (2) The inverse hyperbolic secant is implemented as ArcSech [x]inMathematica . It has MACLAURIN SERIES sech/C281 x /C30(ln 2 /C28ln x) /C281 4x2 /C283 32x4 /C285 96x6 /C2835 1024x8 /C27... (Sloane’s A052468 and A052469) and sech/C2811 x ! /C30i1 2p/C28x /C2816x3 /C283 40x5 /C27...1CA}1CA$ : (3) See also HYPERBOLIC SECANT ,INVERSE HYPERBOLIC FUNCTIONS References Sloane, N. J. A. Sequences A052468 and A052469 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Inverse Hyperbolic Sine The INVERSE FUNCTION of the HYPERBOLIC SINE, denoted sinh/C281 x: It can be defined for complex z as sinh /C281 z /C30ln(z /C27ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27z2p ) :The inverse hyperbolic sine is implemented as Arc- Sinh [x]inMathematica . It has a MACLAURIN SERIES sinh/C281 x /C30x /C281 6x3 /C273 40x5 /C285 112x7 /C2735 1152x9 /C27... (1) sinh /C2811 x ! /C30(ln 2 /C28ln x) /C271 4x2 /C283 32x4 /C275 96x6/C28...:(2) See also HYPERBOLIC SINE,INVERSE HYPERBOLIC FUNCTIONS Inverse Hyperbolic Tangent The INVERSE FUNCTION of the HYPERBOLIC TANGENT , denoted tanh/C281x:It can be defined for complex zas tanh/C281z/C3012[ln(1/C27z)/C28ln(1/C28z)]; (1) and for real xas tanh/C281x/C301 2ln1/C27x 1/C28x ! : (2) The inverse hyperbolic tangent is implemented as ArcTanh [x]i nMathematica . It has special values tanh/C2810/C300 (3) tanh/C2811/C30/C12 (4) tanh/C281/C12/C30/C281 2pi (5) tanh/C281i/C3014pi (6) and M ACLAURIN SERIES tanh/C281 x /C30x /C271 3x3 /C2715x5 þ17x7 þ19x9 /C27...: (7) See also HYPERBOLIC TANGENT ,INVERSE HYPERBOLIC COTANGENT ,INVERSE HYPERBOLIC FUNCTIONS Inverse Laplace Transform BROMWICH INTEGRAL ,LAPLACE TRANSFORM Inverse Matrix MATRIX INVERSE Inverse Nome Solving the NOME q for the PARAMETER m gives m(q) /C30q4 2(0; q) q4 3(0; q) ; where qi(z ;q)isaJ ACOBI THETA FUNCTION . The inverse nome is implemented asInverseElliptic- NomeQ [q]inMathematica . It satisfies lim q00/C27dm dq/C3016: See also JACOBI THETA FUNCTIONS ,NOMEInverse Oblate Spheroidal Coordinates A system of coordinates obtained by INVERSION of the oblate spheroids and one-sheeted hyperboloids in OBLATE SPHEROIDAL COORDINATES . The inverse oblate spheroidal coordinates ( h;u;c) are given by the transformation equations x/C30acosh hsinucosc cosh2h/C28cos2u(1) y/C30acosh hsinusinc cosh2h/C28cos2u(2) z/C30asinh hcosu cos2h/C28cos2u; (3) where h]0;u/C23[0;p];and c/C23[0;2p):Surfaces of constant hare given by the cyclides of rotation x2/C27y2/C27z2/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2 cosh2h/C27z2 sinh2hs ; (4) surfaces of constant uby the cyclides of rotation x2/C27y2/C27z2/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2 sin2u/C28z2 cos2us ; (5) and surfaces of constant cby the half-planes tanc/C30y x: (6) The metric coefficients are given by ghh /C30a2 cosh2 h /C28 sin2 u1CC1CA cosh2 h /C28 cos2 u1CC1CA (7) guu /C30a2 cosh2 h /C28 sin2 u1CC1CA cosh2 h /C28 cos2 u1CC1CA (8) gcc /C30a2 cosh2 h sin2 u cosh2 h /C28 cos2 u1CC1CA 2 : (9) See also INVERSE PROLATE SPHEROIDAL COORDI- NATES ,PROLATE SPHEROIDAL COORDINATES References Moon, P. and Spencer, D. E. "Inverse Oblate Spheroidal Coordinate (h; u; c):/" Fig. 4.06 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 119 /C1/121, 1988. Inverse Permutation An inverse permutation is a permutation in which each number and the number of the place which it occupies are exchanged. For example, p1 /C30f3; 8;5;10 ;9 ;4;6 ;1;7; 2g p2 /C30f8; 10;1 ;6;3 ;7;9 ;2;5; 4g are inverse permutations, since the positions of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 in p1 are p2 ; and the positions of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 in p2 are likewise p1 (Muir 1960, p. 5). The inverse permutation of a given PERMUTATION can be computed using InversePer- mutation [p] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). Inverse permutations are sometimes also called con- jugate or reciprocal permutations (Muir 1960, p. 4). See also PERMUTATION ,P ERMUTATION INVERSION , SELF-CONJUGATE PARTITION References Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960.Inverse Points Points, also called polar reciprocals, which are trans- formed into each other through INVERSION about a given INVERSION CIRCLE C (or INVERSION SPHERE ). The points P and P? are inverse points with respect to the INVERSION CIRCLE if OP /C215 OP0/C30OQ2 /C30k2 (Wenninger 1983, p. 2). In this case, P ? is called the POLE and the line L through P and perpendicular to OP is called the POLAR . In the above figure, the quantity k2 is called the POWER of the point P relative to the circle C. The point P? which is the inverse point of a given point P with respect to an INVERSION CIRCLE C may be constructed geometrically using a COMPASS only (Coxeter 1969, p. 78; Courant and Robbins 1996, pp. 144 /C1/145). Inverse points can also be taken with respect to an INVERSION SPHERE , which is a natural extension of geometric INVERSION from the plane to 3-dimensional space. See also GEOMETRIC CONSTRUCTION ,INVERSION , INVERSION CIRCLE ,INVERSION SPHERE ,L IMITING POINT ,POLAR ,POLE (INVERSION ), POWER (CIRCLE ) References Courant, R. and Robbins, H. "Geometrical Construction of Inverse Points." §3.4.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 144 /C1/145, 1996. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Inverse Problem References Kozhanov, A. I. Composite Type Equations and Inverse Problems. Utrecht, Netherlands: VSP, 1999. Prilepko, A. I.; Orlovsky, D. G.; and Vasin, I. A. Methods for Solving Inverse Problems in Mathematical Physics. New York: Dekker, 1999. Inverse Prolate Spheroidal Coordinates A system of coordinates obtained by INVERSION of the prolate spheroids and two-sheeted hyperboloids in PROLATE SPHEROIDAL COORDINATES . The inverse pro- late spheroidal coordinates ( h; u ; c) are given by the transformation equations x /C30a sinh h sin u cos c cosh2 h /C28 sin2 u (1) y /C30a sinh h sin u sin c cosh2 h /C28 sin2 u (2) z /C30a cosh h cosh u cosh2 h /C28 sin2 u ; (3) with h ]0; u /C23 [0;p] ; and c /C23 [0;2 p) : Surfaces of con- stant h are given by the cyclides of rotation x2 /C27y2 /C27z2 /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2 sinh2 h /C27z2 cosh2 hs ; (4) surfaces of constant u by the cyclides of rotation x2 /C27y2 /C27z2 /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28x2 /C27 y2 sin2 u/C27z2 cosh2 u ;s (5) and surfaces of constant c by the half-planes tan c /C30y x : (6)The metric coefficients are given by ghh /C30a2(sinh2 h /C27 sin2 u) (cosh2 h /C28 sin2 u)2 (7) guu /C30a2(sinh2 h /C27 sin2 u) (cosh2 h /C28 sin2 u)2 (8) gcc /C30a2 sinh2 h sinh2 u (cosh2 h /C28 sin2 u)2 : (9) See also INVERSE OBLATE SPHEROIDAL COORDINATES , OBLATE SPHEROIDAL COORDINATES References Moon, P. and Spencer, D. E. "Inverse Prolate Spheroidal Coordinate (h; u; c):/" Fig. 4.05 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 115 /C1/118, 1988. Inverse Proportion INVERSELY PROPORTIONAL Inverse Quadratic Interpolation The use of three prior points in a ROOT -finding ALGORITHM to estimate the zero crossing. Inverse Scattering Method A method which can be used to solve the initial value problem for certain classes of nonlinear PARTIAL DIFFERENTIAL EQUATIONS . The method reduces the initial value problem to a linear INTEGRAL EQUATION in which time appears only implicitly. However, the solutions u(x;t) and various of their derivatives must approach zero as x09/C12 (Infeld and Rowlands 2000). See also ABLOWITZ- RAMANI- SEGUR CONJECTURE , BA¨ CKLUND TRANSFORMATION References Infeld, E. and Rowlands, G. "Inverse Scattering Method." §7.4 in Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge University Press, pp. 173 /C1/175, 2000. Miura, R. M. (Ed.). Ba¨cklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications. New York: Springer-Verlag, 1974. Inverse Secant The function sec/C281 x; where sec x is the SECANT and the superscript -1 denotes the INVERSE FUNCTION , not the multiplicative inverse. The inverse secant is implemented as ArcSec [x]in Mathematica . The inverse secant satisfies sec/C281 x /C30csc /C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 1p ! (1) for POSITIVE or NEGATIVE x, and sec /C281 x /C30p/C28sec/C281(/C28x) (2) for x ]0: The inverse secant has a TAYLOR SERIES about infinity of sec/C281 x /C301 2 p/C28x/C281 /C2816x/C283 /C283 40x/C285 /C285 112x /C287 /C28...: (3) The inverse secant is given in terms of other inverse trigonometric functions by sec /C281 x /C30cos /C2811 x ! (4) /C30cot /C281 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 1p ! (5) /C301 2p /C28csc /C281 x /C30/C2812 p/C30csc /C281 /C28xðÞ (6) /C30sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 1p x ! (7) /C30tan/C281(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C281)p (8) for x ]0:/ See also INVERSE COSECANT ,SECANTReferences Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 141 /C1/143, 1987. Inverse Semigroup This entry contributed by NICOLAS BRAY A SEMIGROUP S is said to be an inverse semigroup if, for every a in S, there is a unique b (called the inverse of a) such that a /C30aba and b /C30bab. This is equivalent to the condition that every element has at least one inverse and that the IDEMPOTENTS of S COMMUTE (Lawson 1999). Note that if b is an inverse of a, then ba is an IDEMPOTENT . See also SEMIGROUP References Clifford, A. H. and Preston, G. B. The Algebraic Theory of Semigroups, Vol. 1. Providence, RI: Amer. Math. Soc., 1961. Clifford, A. H. and Preston, G. B. The Algebraic Theory of Semigroups, Vol. 2. Providence, RI: Amer. Math. Soc., 1967. Lawson, M. V. Inverse Semigroups: The Theory of Partial Symmetries. Singapore: World Scientific, 1999. Lyapin, E. S. Semigroups. Providence, RI: Amer. Math. Soc., 1974. Shevrin, L. N. "Inversion Semi-Group." In Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia," Vol. 5 (Mana- ging Ed. M. Hazewinkel). Dordrecht, Netherlands: Reidel,pp. 184 /C1 /185, 1988. Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744/C1/752, 1996. Inverse Sine The function sin/C281x;where sin xis the SINE and the superscript -1 denotes the INVERSE FUNCTION ,no t the multiplicative inverse. The notation arcsin xor Arcsin xis sometimes also used. The inverse sine is implemented as ArcSin [x]in Mathematica . The inverse sine satisfies sin/C281 x /C30/C28sin/C281(/C28x) (1) for POSITIVE and NEGATIVE x, and sin/C281 x /C301 2 p/C28sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p1CA}1CA$ for 0 5x 51 /C281 2p/C27sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p1CA}1CA$ for /C281 5x 50:8 < : (2) The MACLAURIN SERIES for the inverse sine with /C281 5 x 51 is given by sin/C281 x /C30x /C271 6x3 /C273 40x5 /C275 112x7 /C2735 1152x9 /C27...: (3) The inverse sine is given in terms of other inverse trigonometric functions by sin/C281 x /C30cos /C281 /C28xðÞ/C281 2p/C3012 p/C28cos/C281 x (4) /C3012 p/C28cot /C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2p ! (5) /C30tan /C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28 x2p ! (6) for POSITIVE or NEGATIVE x, and sin/C281 x /C30cos /C281ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p1CA}1CA$ (7) /C30cot /C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p x ! (8) /C30csc/C2811 x(9) /C30sec/C281 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p ! (10) forx]0:/ See also INVERSE COSINE ,SINE References Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79 /C1/83, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142 /C1/143 and 220, 1987.Inverse Tangent The inverse tangent is also called the arctangent and is denoted either tan/C281xor arctan x, and is the INVERSE FUNCTION of the TANGENT tanx:The inverse tangent is implemented as ArcTan [x]i nMathemati- ca. The ARGUMENT of a COMPLEX NUMBER z/C30x/C27iyis often written as u/C30tan/C281y x ! ; (1) where u;sometimes also denoted f;corresponds to the counterclockwise ANGLE from the POSITIVE REAL AXIS, i.e., the value of usuch that x/C30cosuand y/C30 sinu:This special kind of INVERSE TANGENT takes into account the quadrant in which zlies and is returned by the FORTRAN command ATAN2(X,Y) and the Mathematica command ArcTan [x,y], and is often restricted to the range /C28pBu5p:In the degenerate case when x/C300, f/C30/C281 2p ifyB0 undefined if y/C300 1 2p ify>0:8 >< >:(2) /tan/C281xhas the M ACLAURIN SERIES for/C2815x51o f tan/C281x/C30X/C12 n/C300/C281ðÞnx2n/C271 2n/C271 /C30x/C281 3x3/C2715x5/C2817x7/C27...: (3) A more rapidly converging form due to Euler is given by tan/C281x/C30X/C12 n/C30022nn!ðÞ2 (2n/C271)!x2n/C271 1/C27x2 ðÞn/C271(4) (Castellanos 1988). The inverse tangent satisfies tan/C281x/C30/C28tan/C281(/C28x) (5) for POSITIVE and NEGATIVE x, and tan/C281x/C301 2p/C28tan/C2811 x ! (6) forx]0:The inverse tangent is given in terms of other inverse trigonometric functions by tan/C281x/C3012p/C28cos/C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C271p ! (7) /C30cot/C281(/C28x)/C281 2p/C3012p/C28cot/C281x (8) /C30sin/C281 xffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C271p ! (9) for POSITIVE orNEGATIVE x, and tan/C281x/C30cos/C281 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C271p ! (10) /C30cot/C2811 x ! (11) /C30csc/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2/C271p x ! (12) /C30sec/C281ffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C271p1CA}1CA$ (13) forx]0:/ In terms of the HYPERGEOMETRIC FUNCTION , tan/C281x/C30x2F11;1 2;32;/C28x21CA}1CA$ (14) /C30x 1/C27x22F11;1;32;x2 1/C27x2 ! (15) (Castellanos 1988). Castellanos (1986, 1988) also gives some curious formulas in terms of the F IBO- NACCI NUMBERS , tan/C281x/C30X/C12 n/C300/C281nðÞ f2n/C271t2n/C271 5n(2n/C271)(16) /C305X/C12 n/C300/C281ðÞnf2 2n/C271 (2n/C271)u/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2/C271p1CC1CA 2n/C271 (17) /C30X/C12 n/C300/C281ðÞn5n/C272F3 2n/C271 (2n/C271)v/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2/C275p1CC1CA 2n/C271; (18)where t/C132x 1/C27ffiffiffiffiffiffiffiffi 4x2 5s (19) u/C135 4x1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C2724 25x2s ! ; (20) andvis the largest POSITIVE ROOT of 8xv4/C28100v3/C28450xv2/C27875v/C27625x/C300: (21) The inverse tangent satisfies the addition FORMULA tan/C281x/C27tan/C281y/C30tan/C281x/C27y 1/C28xy ! (22) as well as the more complicated FORMULAS tan/C281 1 a/C28b ! /C30tan/C2811 a ! /C27tan/C281 b a2/C28ab/C271 ! (23) tan/C2811 a ! /C302 tan/C2811 2a ! /C28tan/C281 1 4a3/C273a ! (24) tan/C2811 p ! /C30tan/C2811 p/C27q/C27tan/C281 q p2/C27pq/C271 ! ;(25) the latter of which was known to Euler. The inverse tangent FORMULAS are connected with many inter- esting approximations to PI tan/C281(1/C27x) /C301 4p/C2712x/C2814x2/C271 12x3/C271 40x5/C271 48x6/C271 112x7/C27...:(26) Euler gave tan/C281x/C30y x23y/C272 /C2154 3 /C2155y2/C272 /C2154 /C2156 3 /C2155 /C2157y3/C27... ! ;(27) where y/C13x2 1/C27x2: (28) The inverse tangent has CONTINUED FRACTION repre- sentations tan/C281x/C30x 1/C27x2 3/C274x2 5/C279x2 7/C2716x2 9/C27...(29) /C30x 1 /C27x2 3 /C28 x2 /C279x2 5 /C28 3x2 /C2725x2 7 /C28 5x2 /C27 ...(30) To find tan/C281 x numerically, the following ARITH- METIC-GEOMETRIC MEAN -like ALGORITHM can be used. Let a0 /C30 1 /C27x21CC1CA/C281 =2(31) b0 /C301: (32) Then compute ai/C271 /C301 2ai /C27bi ðÞ (33) bi/C271 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi ai/C271biq ; (34) and the inverse tangent is given by tan /C281 x /C30 lim n0/C12xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 x2p an(35) (Acton 1990). An inverse tangent tan/C281 n with integral n is called reducible if it is expressible as a finite sum OF THE FORM tan/C281 n /C30X k /C301fk tan /C281 nk ; (36) where fkare POSITIVE or NEGATIVE INTEGERS and ni are INTEGERS Bn : tan /C281 m is reducible IFF all the PRIME FACTORS of 1 /C27m2occur among the PRIME FACTORS of 1 /C27n2for n /C301, ..., m /C281: A second NECESSARY and SUFFICIENT condition is that the largest PRIME factor of 1 /C27m2is less than 2m: Equivalent to the second condition is the statement that every GREGORY NUMBER tx /C30cot /C281 x can be uniquely expressed as a sum in terms of tm/s for which m is a STØRMER NUMBER (Conway and Guy 1996). To find this decomposition, write arg(1 /C27in) /C30argY k /C3011 /C27nki ðÞfk; (37) so the ratio r /C30Q k /C3011 /C27 nki ðÞfk 1 /C27 in (38) is a RATIONAL NUMBER . Equation (38) can also be written r2 1 /C27n21CC1CA /C30Y k/C3011 /C27n2 k1CC1CAfk: (39) Writing (36) in the form tan/C281 n /C30X k /C301fk tan/C281 nk /C27f tan/C281 1 (40)allows a direct conversion to a corresponding INVERSE COTANGENT FORMULA cot /C281 n /C30X k /C301fk cot /C281 nk /C27ccot/C281 1 ; (41) where c /C302 /C28f /C282X k /C301fr : (42) Todd (1949) gives a table of decompositions of tan/C281 n for n 5342: Conway and Guy (1996) give a similar table in terms of STøRMER NUMBERS . Arndt and Gosper give the remarkable inverse tangent identity sinX2n/C271 k/C301tan/C281ak ! /C30/C281ðÞn 2n/C271P2n/C271 k/C301Q2n/C271 j/C301aj/C28tanp(j/C28k) 2n/C271 !"# ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2n/C271 j/C301a2 j/C2711CA}1CA$r :(43) See also INVERSE COTANGENT ,TANGENT References Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79 /C1/83, 1972. Acton, F. S. "The Arctangent." In Numerical Methods that Work, upd. and rev. Washington, DC: Math. Assoc. Amer., pp. 6/C1/10, 1990. Arndt, J. "Completely Useless Formulas." http://www.jjj.de/ hfloat/hfloatpage.html#formulas. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142 /C1/143 and 220, 1987. Castellanos, D. "Rapidly Converging Expansions with Fibo- nacci Coefficients." Fib. Quart. 24,7 0/C1/82, 1986. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1/98, 1988. Conway, J. H. and Guy, R. K. "Størmer’s Numbers." The Book of Numbers. New York: Springer-Verlag, pp. 245 /C1/ 248, 1996. Hildebrand, J. D. "Arctan() Appreciation Home Page!" http:// www.undergrad.math.uwaterloo.ca/~jdhildeb/arc-tan.html. Salamin, G. Item 137 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 67 /C1 /68, Feb. 1972. Todd, J. "A Problem on Arc Tangent Relations." Amer. Math. Monthly 56, 517/C1/528, 1949. Inverse Tangent Integral The inverse tangent integral Ti2(x) is defined in terms of the DILOGARITHM Li2(x)b y Li2(ix)/C301 4Li2/C28x21CC1CA /C27iTi2(x) (1) (Lewin 1958, p. 33). It has the series Ti2(x)/C30X/C12 k/C301/C281ðÞk/C281x2k/C281 2k/C281 ðÞ2(2) and gives in closed form the sum X/C12 n/C301sin (4 n/C282)x ½/C138 2n/C281 ðÞ2/C30Ti2(tan x)/C28xln(tan x) (3) that was considered by Ramanujan (Lewin 1958, p. 39). The inverse tangent integral can be expressedin terms of the DILOGARITHM as Ti2(x)/C301 2iLi2(ix)/C28Li2(/C28ix) ½/C138 ; (4) in terms of L EGENDRE’S CHI-FUNCTION as Ti2(x)/C30/C28ix2(ix); (5) in terms of the L ERCH TRANSCENDENT by Ti2(x)/C301 4xF/C28x2;2;121CA}1CA$ ; (6) and as the integral Ti2(x)/C30gx 0tan/C281x?ðÞ x?dx?: (7)/Ti2(x) has derivative dTi2(x) dx/C30tan/C281x x: (8) It satisfies the identities Ti2(x)/C28Ti21 x ! /C301 2psgn(x)l nxjj (9) 12Ti22x 1/C28x2 ! /C30Ti2(x)/C27Ti2(/C28x;1)/C28Ti2(x;1);(10) where Ti2(x;a)/C13gx 0tan/C281x? a/C27x?dx? (11) is the generalized inverse tangent function. /Ti2(x) has the special value Ti2(1)/C30K; (12) where Kis C ATALAN’S CONSTANT , and the functional relationships 3T i2(1)/C282T i2121CA}1CA$ /C28Ti2131CA}1CA$ /C2812Ti2341CA}1CA$ /C3012pln 2 ;(13) the two equivalent identities 3T i 2 /C28ffiffiffi 3p1CA}1CA$ /C302T i2(1)/C281 4pln 2/C28ffiffiffi 3p1CA}1CA$ (14) Ti2tan1 12p1CA}1CA$1CA}1CA$ /C302 3Ti2tan14p1CA}1CA$1CA}1CA$ /C271 12pln tan1 12p1CA}1CA$1CA}1CA$ ; (15) and 3T i 2 /C27ffiffiffi 3p1CA}1CA$ /C302T i2(1)/C275 4pln 2/C27ffiffiffi 3p1CA}1CA$ (16) (Lewin 1958, p. 39). The triplication formula is given by 1 3Ti23x/C28x3 1/C283x2 ! /C30Ti2(x)/C27Ti21/C28xffiffiffi 3p ffiffiffiffiffiffiffiffiffiffiffiffi 3/C27xp ! /C28Ti21/C27xffiffiffi 3p ffiffiffi 3p /C28x ! /C271 6plnffiffiffi 3p /C27x1CC1CA 1/C27xffiffiffi3p1CC1CA 1/C28xffiffiffi 3p1CC1CA ffiffiffi3p /C28x1CC1CA ! ;ð17Þ which leads to Ti 2tan1 24p1CA}1CA$1CA}1CA$ /C28Ti2tan5 24p1CA}1CA$1CA}1CA$ /C272 3Ti2tan18p1CA}1CA$1CA}1CA$ /C271 6plntan5 24p1CA}1CA$ tan18p1CA}1CA$0 @1A/C300 (18) and the algebraic form Ti2ffiffiffi 3p /C28ffiffiffi 2p ffiffiffi2p /C27 1 ! /C28Ti 2ffiffiffi 3p /C28ffiffiffi 2p ffiffiffi2p /C28 1 ! /C272 3Ti2ffiffiffi 2p /C2811CA}1CA$ /C301 6 p lnffiffiffi 2p /C28 1ffiffiffi 3p /C28ffiffiffi2p1CC1CA ffiffiffi2p /C27 11CC1CA ! (19) (Lewin 1958, p. 41). See also D ILOGARITHM ,LEGENDRE’S CHI-FUNCTION , LERCH TRANSCENDENT References Lewin, L. "The Inverse Tangent Integral" and "The General- ized Inverse Tangent Integral." Chs. 2 /C1/3in Dilogarithms and Associated Functions. London: Macdonald, pp. 33 /C1/ 90, 1958. Lewin, L. Polylogarithms and Associated Functions. Am- sterdam, Netherlands: North-Holland, p. 45, 1981. Nielsen, N. "Der Eulersche Dilogarithmus und seine Ver- allgemeinerungen." Nova Acta (Leopold) 90, 121 /C1/212, 1909. Inverse Trigonometric Functions INVERSE FUNCTIONS of the TRIGONOMETRIC FUNC- TIONS written cos/C281 x; cot /C281 x; csc /C281 x; sec /C281 x; sin/C281 x; and tan /C281 x: As noted by Feynman (1997), the notation f /C281x is unfortunate because it conflicts with the common interpretation of a superscripted quantity as indicating a power, i.e., f /C281x /C30 1=fðÞ x /C30x =f :/ The inverse trigonometric functions are generally defined on the following domains. Function Domain /sin/C281 x///C281 2 p5y 512 p/ /cos/C281 x//0 5y 5p/ /tan/C281 x///C281 2 pBy B12 p/ /csc /C281 x//0 5y 51 2 p or p5y 53 p 2/ /sec/C281 x//0 5y 5p/ /cot /C281 x//0 5y 51 2 p or /C28p5y 5/C2812 p/ Inverse-forward identities are tan/C281(cot x) /C3012 p/C28x (1) sin/C281(cos x) /C301 2 p/C28x (2) sec /C281(csc x) /C3012p/C28x; (3) and forward-inverse identities are cos sin/C281 x1CC1CA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p (4)cos tan /C281 x1CC1CA /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 x2p (5) sin cos/C281 x1CC1CA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p (6) sin tan/C281 x1CC1CA /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 x2p (7) tan cos /C281 x1CC1CA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2p x (8) tan sin /C281 x1CC1CA /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2p : (9) Inverse sum identities include sin/C281 x /C27cos/C281 x /C301 2p (10) tan /C281 x /C27cot/C281 x /C301 2 p (11) sec /C281 x /C27csc /C281 x /C3012 p; (12) where (10) follows from x /C30sin sin/C281 x1CC1CA /C30cos12p/C28sin/C281x1CA}1CA$ : (13) Complex inverse identities in terms of LOGARITHMS include sin/C281(z)/C30/C28ilniz9ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28z2p1CA}1CA$ (14) cos/C281(z)/C30/C28ilnz9iffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28z2p1CA}1CA$ (15) tan/C281(z)/C30/C28iln1/C27izffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27z2p ! (16) /C301 2iln1/C28iz 1/C27iz ! : (17) See also INVERSE COSECANT ,INVERSE COSINE ,IN- VERSE COTANGENT ,INVERSE SECANT ,INVERSE SINE, INVERSE TANGENT References Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79 /C1/83, 1972. Feynman, R. P. and Leighton, R. "He Fixes Radios by Thinking!" In ‘Surely You’re Joking, Mr. Feynman!’: Adventures of a Curious Character. New York: W. W. Norton, p. 12, 1997. Spanier, J. and Oldham, K. B. "Inverse Trigonometric Functions." Ch. 35 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 331 /C1/341, 1987. InverseEllipticNomeQ INVERSE NOME InverseJacobiCD JACOBI ELLIPTIC FUNCTIONS InverseJacobiCN JACOBI ELLIPTIC FUNCTIONS InverseJacobiCS JACOBI ELLIPTIC FUNCTIONS InverseJacobiDC JACOBI ELLIPTIC FUNCTIONS InverseJacobiDN JACOBI ELLIPTIC FUNCTIONS InverseJacobiDS JACOBI ELLIPTIC FUNCTIONS InverseJacobiNC JACOBI ELLIPTIC FUNCTIONS InverseJacobiND JACOBI ELLIPTIC FUNCTIONS InverseJacobiNS JACOBI ELLIPTIC FUNCTIONS InverseJacobiSC JACOBI ELLIPTIC FUNCTIONS InverseJacobiSD JACOBI ELLIPTIC FUNCTIONS InverseJacobiSN JACOBI ELLIPTIC FUNCTIONS Inversely Proportional Two quantities y and x are said to be inversely proportional (or "in inverse proportion") if y is given by a constant multiple of 1=x; i.e., y /C30c=x for c a constant. This relationship is commonly written y 8 x/C281 :/ See also DIRECTLY PROPORTIONAL ,PROPORTIONAL Inversely Similar Two figures are said to be SIMILAR when all corre-sponding ANGLES are equal, and are inversely similar when all corresponding ANGLES are equal and de- scribed in the opposite rotational sense. See also DIRECTLY SIMILAR ,HOMOTHETIC ,SIMILAR References Lachlan, R. "Properties of Two Figures Inversely Similar." §220/C1/222 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 138 /C1/139, 1893. InverseWeierstrassP WEIERSTRASS ELLIPTIC FUNCTION Inversion Inversion is the process of transforming points Pto a corresponding set of points P?known as their INVERSE POINTS . Two points Pand P?are said to be inverses with respect to an INVERSION CIRCLE having INVER- SION CENTER O/C30x0;y0 ðÞ and INVERSION RADIUS kifP? is the foot of the altitude of DOQP ;where Qis a point on the circle such that OQ/C222PQ:The analogous notation of inversion can be carried to in 3-dimen- sional space with respect to an INVERSION SPHERE . IfPandP?are inverse points, then the line Lthrough Pand perpendicular to OPis sometimes called a "POLAR " with respect to point P, known as the " POLE ". In addition, the curve to which a given curve istransformed under inversion is called its INVERSE CURVE (or more simply, its "inverse"). This sort of inversion was first systematically investigated byJakob Steiner. From similar triangles, it immediately follows that the inverse points PandP?obey OP k/C30k OP?; (1) or k2/C30OP/C29OP? (2) (Coxeter 1969, p. 78), where the quantity k2is known as the POWER (Coxeter 1969, p. 81). The general equation for the inverse of the point ( x, y) relative to the INVERSION CIRCLE with INVERSION CENTER x0 ; y0 ðÞ and INVERSION RADIUS k is given by x?/C30x0 /C27k2 x /C28 x0 ðÞ x /C28 x0 ðÞ2/C27 y /C28 y0 ðÞ2 (3) y?/C30y0 /C27k2 y /C28 y0 ðÞ x /C28 x0 ðÞ2/C27 y /C28 y0 ðÞ2 : (4) In vector form, x?/C30x0 /C27k2 x /C28 x0 ðÞ x /C28 x0 jj2: (5) Note that a point on the CIRCUMFERENCE of the INVERSION CIRCLE is its own inverse point. In addi- tion, any ANGLE inverts to an opposite ANGLE . Treating LINES as CIRCLES of INFINITE RADIUS , all CIRCLES invert to CIRCLES (Lachlan 1893, p. 221). Furthermore, any two nonintersecting circles can be inverted into concentric circles by taking the INVER- SION CENTER at one of the two so-called LIMITING POINTS of the two circles (Coxeter 1969), and any two circles can be inverted into themselves or into two equal circles (Casey 1888, pp. 97 /C1/98). ORTHOGONAL CIRCLES invert to ORTHOGONAL CIRCLES (Coxeter 1969). The INVERSION CIRCLE itself, circles orthogonal to it, and lines through the INVERSION CENTER are invariant under inversion. Furthermore, inversion is a CONFORMAL MAP, so angles are preserved. The property that inversion transforms circles and lines to circles or lines (and that inversion is con- formal) makes it an extremely important tool of plane analytic geometry. By picking a suitable inversion circle, it is often possible to transform one geometric configuration into another simpler one in which a proof is more easily effected. The illustration above shows examples of the results of geometric inversion. The inverse of a CIRCLE of RADIUS a with CENTER (x, y) with respect to an inversion circle with INVERSION CENTER x0 ; y0 ðÞ and INVERSION RADIUS k is another CIRCLE with CENTER x?/C30x0 /C27sx/C28x0 ðÞ (6) y?/C30y0 /C27sy/C28y0 ðÞ (7) and RADIUS r ?/C30 sjja ; (8) where s /C13k2 x/C28x0 ðÞ2/C27y/C28y0 ðÞ2/C28a2: (9) These equations can also be naturally extended to inversion with respect to a sphere in 3-dimensional space. The above plot shows a CHESSBOARD centered at (0, 0) and its inverse about a small circle also centered at (0, 0) (Gardner 1984, pp. 244 /C1/245; Dixon 1991). See also ARBELOS ,CONFORMAL MAP,CYCLIDE ,HEX- LET,INVERSE CURVE ,INVERSE POINTS ,INVERSION CIRCLE ,INVERSION OPERATION ,INVERSION RADIUS , INVERSION SPHERE ,INVERSIVE DISTANCE ,INVERSIVE GEOMETRY ,L IMITING POINT ,M IDCIRCLE ,P APPUS CHAIN ,PEAUCELLIER INVERSOR ,PERMUTATION INVER- SION,P OLAR ,P OLE (INVERSION ), POWER (CIRCLE ), RADICAL LINE,STEINER CHAIN ,STEINER’S PORISM References Casey, J. "Theory of Inversion." §6.4 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Exam- ples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 95 /C1/112, 1888. Coolidge, J. L. "Inversion." §1.2 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 21 /C1/30, 1971. Courant, R. and Robbins, H. "Geometrical Transformations. Inversion." §3.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 140 /C1/146, 1996. Coxeter, H. S. M. "Inversion in a Circle" and "Inversion of Lines and Circles." §6.1 and 6.3 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 77 /C1/83, 1969. Coxeter, H. S. M. and Greitzer, S. L. "An Introduction to Inversive Geometry." Ch. 5 in Geometry Revisited. Wa- shington, DC: Math. Assoc. Amer., pp. 103 /C1/131, 1967. Darboux, G. Lec¸ons sur les systemes orthogonaux et les coordonne ´es curvilignes. Paris: Gauthier-Villars, 1910. Dixon, R. "Inverse Points and Mid-Circles." §1.6 in Matho- graphics. New York: Dover, pp. 62 /C1/73, 1991. Durell, C. V. "Inversion." Ch. 10 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 105 /C1/ 120, 1928. Fukagawa, H. and Pedoe, D. "Problems Soluble by Inver- sion." §1.8 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 17 /C1/22 and 93 /C1/99, 1989. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984. Jeans, J. H. The Mathematical Theory of Electricity and Magnetism, 5th ed. Cambridge, England: The University Press, 1925. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 43 /C1/57, 1929. Kelvin, W. T. and Tait, P. G. Principles of Mechanics and Dynamics, Vol. 2. New York: Dover, p. 62, 1962. Lachlan, R. "The Theory of Inversion." Ch. 14 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 218 /C1/236, 1893. Liouville, J. "Note au sujet de l’article pre´ce´dent." J. math. pures appl. 12, 265 /C1/290, 1847. Lockwood, E. H. "Inversion." Ch. 23 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 176 /C1/181, 1967. Maxwell, J. C. A Treatise on Electricity and Magnetism, Vol. 1, unabridged 3rd ed. New York: Dover, 1954. Maxwell, J. C. A Treatise on Electricity and Magnetism, Vol. 2, unabridged 3rd ed. New York: Dover, 1954. Morley, F. and Morley, F. V. Inversive Geometry. Boston, MA: Ginn, 1933. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 25 /C1/31, 1990. Schmidt, H. Die Inversion und ihre Anwendung. Munich, Germany: Oldenbourg, 1950. Thomson, W. "Extrait d’un lettre de M. William Thomson a M. Liouville." J. math. pures appl. 10, 364 /C1/367, 1845. Thomson, W. "Extrait de deux lettres adresse ´es a` M. Liou- ville." J. math. pures appl. 12, 256, 1847. Wangerin, A. S. 147 in Theorie des Potentials und der Kugelfunktionen, Bd. II. Berlin: de Gruyter, 1921.Weber, E. Electromagnetic Fields. New York: Wiley, p. 244, 1950. Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 119 /C1/121, 1991. Inversion Center The point that INVERSION OF A CURVE is performed with respect to. See also INVERSE POINTS ,INVERSION CIRCLE ,INVER- SION RADIUS ,INVERSIVE DISTANCE ,LIMITING POINT , POLAR ,POLE (INVERSION ), POWER (CIRCLE ) Inversion Circle The CIRCLE with respect to which an INVERSE CURVE is computed or relative to which INVERSE POINTS are computed. In 3-D, INVERSE POINTS can be computed relative to an INVERSION SPHERE . See also INVERSE POINTS ,INVERSION CENTER ,INVER- SION RADIUS ,INVERSION SPHERE ,INVERSIVE DIS- TANCE ,M IDCIRCLE ,P OLAR ,P OLE (INVERSION ), POWER (CIRCLE ) Inversion Number In DETERMINANT EXPANSION BY MINORS , the minimal number of TRANSPOSITIONS of adjacent columns in a SQUARE MATRIX needed to turn the matrix represent- ing a permutation of /f1;2 ;...; ng/ into the IDENTITY MATRIX . See also DETERMINANT EXPANSION BY MINORS , TRANSPOSITION References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/646. Inversion Operation The SYMMETRY OPERATION ðx;y;z Þ0ð/C28x;/C28y;/C28zÞ: When used in conjunction with a ROTATION ,it becomes an IMPROPER ROTATION . Inversion Poset A relation between permutations p and q that exists if there is a sequence of TRANSPOSITIONS such that each transposition increases the number of inversions (Stanton and White 1986; Skiena 1990, p. 162). See also PERMUTATION References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Stanton, D. W. and White, D. E. Constructive Combinato- rics. New York: Springer-Verlag, 1986. Inversion Radius The RADIUS used in performing an INVERSION with respect to an INVERSION CIRCLE . See also INVERSE POINTS ,INVERSION CENTER ,INVER- SION CIRCLE ,INVERSIVE DISTANCE ,P OLAR ,P OLE (INVERSION ), POWER (CIRCLE ) Inversion Semigroup INVERSE SEMIGROUP Inversion Sphere The SPHERE with respect to which INVERSE POINTS are computed (i.e., with respect to which geometrical INVERSION is performed). For example, the CYCLIDES are inversions in a sphere of TORI. The center of the inversion sphere is called the INVERSION CENTER , and its radius is called the INVERSION RADIUS . When DUAL POLYHEDRA are being considered, the inversion sphere is commonly called the MIDSPHERE (or inter- sphere, or reciprocating sphere). In 2-D, the inversion sphere collapses to an INVER- SION CIRCLE . See also CYCLIDE ,INVERSE POINTS ,INVERSION , INVERSION CENTER ,INVERSION CIRCLE ,INVERSION RADIUS ,INVERSIVE DISTANCE ,M IDCIRCLE ,M ID- SPHERE ,POLAR ,POLE (INVERSION ), POWER (CIRCLE ) Inversion Statistic See also WEIGHTED INVERSION STATISTIC References Milne, S. and Degenhardt, S. "Weighted Inversion Statistics and Their Symmetry Group." To appear in J. Combin. Th. Ser. A. http://www.math.ohio-state.edu/~milne/pre- prints.html. Inversion Vector The number of elements greater than i to the left of i in a PERMUTATION gives the ith element of the inversion vector (Skiena 1990, p. 27). A PERMUTATION p can be converted to an inversion vector using ToInversionVector [p] in the Mathematica add- on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ), and an inversion vector v can be converted to a PERMUTATION usingToInver- sionVector [v]. See also PERMUTATION INVERSION References Skiena, S. "Inversion Vectors." §1.3.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 27 /C1/28, 1990. Thompkins, C. B. Machine Attacks on Problems Whose Variables are Permutations. Providence, RI: Amer. Math. Soc., p. 203, 1956.Inversive Distance The inversive distance is the NATURAL LOGARITHM of the ratio of two concentric circles into which the given circles can be inverted. Let c be the distance between the centers of two nonintersecting CIRCLES of RADII a and b Ba. Then the inversive distance is d /C30cosh/C281a2 /C27 b2 /C28 c2 2ab1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA| (Coxeter and Greitzer 1967). The inversive distance between the S ODDY CIRCLES is given by d /C302 cosh /C281 2; and the CIRCUMCIRCLE and INCIRCLE of a TRIANGLE with CIRCUMRADIUS R and INRADIUS r are at inver- sive distance d /C302 sinh/C2811 2ffiffiffiffi r Rs ! (Coxeter and Greitzer 1967, pp. 130 /C1/131). References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 123 /C1/124 and 127 /C1/131, 1967. Inversive Geometry The GEOMETRY resulting from the application of the INVERSION operation. It can be especially powerful for solving apparently difficult problems such as STEI- NER’S PORISM and APOLLONIUS’ PROBLEM . See also HEXLET ,INVERSE CURVE ,INVERSION ,PEAU- CELLIER INVERSOR ,POLAR ,POLE (INVERSION ), POWER (CIRCLE ), RADICAL LINE References Coxeter, H. S. M. and Greitzer, S. L. "An Introduction to Inversive Geometry." Ch. 5 in Geometry Revisited. Wa- shington, DC: Math. Assoc. Amer., pp. 103 /C1/131, 1967. Ogilvy, C. S. "Inversive Geometry" and "Applications of Inversive Geometry." Chs. 3--4 in Excursions in Geome- try.New York: Dover, pp. 24 /C1/55, 1990. Morley, F. and Morley, F. V. Inversive Geometry. Boston, MA: Ginn, 1933. Inverted Funnel FUNNEL ,SINCLAIR’S SOAPFILMPROBLEM Inverted Snub Dodecadodecahedron The UNIFORM POLYHEDRON U60whose DUAL POLYHE- DRON is the MEDIAL INVERTED PENTAGONAL HEXECON- TAHEDRON . It has WYTHOFF SYMBOL j25 35: Its faces are 12 f5 3g/C2760f3 g/C2712 f5g: It has CIRCUMRADIUS for unit edge length of R :0 :8516302 : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 180 /C1/182, 1989. Invertible Knot A knot which can be deformed via an AMBIENT ISOTOPY into itself but with the orientation reversed. No noninvertible knots were known until Trotter (1964) discovered an infinite family, the smallest of which had nine crossings. The simplest noninvertible knot is 08 /C1/017, illustrated above. The following table gives the numbers of noninvertible and invertible knots of n crossings. type Sloane counts noninvertible A052403 0, 0, 0, 0, 0, 0, 0, 1, 2, 33, 187, 1144, 6919, 38118, 226581, 1309875, ... invertible A052402 0, 0, 1, 1, 2, 3, 7, 20, 47, 132, 365, 1032, 3069, 8854, 26712, 78830, ... No general technique is known for determining if a KNOT is invertible. Burde and Zieschang (1985) give a tabulation from which it is possible to extract the noninvertible knots up to 10 crossings.See also AMPHICHIRAL KNOT References Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, 1985. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Sloane, N. J. A. Sequences A052402 and A052403 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Trotter, H. F. "Noninvertible Knots Exist." Topology 2, 275 /C1/ 280, 1964. Invertible Linear Map An invertible linear transformation T : V 0 W is a map between VECTOR SPACES V and W with an inverse map which is also a LINEAR TRANSFORMATION . When T is given by MATRIX MULTIPLICATION , i.e., T(v) /C30Av ; then T is invertible IFF A is a INVERTIBLE MATRIX . Note that the dimensions of V and W must be the same. See also INVERTIBLE MATRIX ,LINEAR TRANSFORMA- TION ,MATRIX ,VECTOR SPACE Invertible Linear Transformation INVERTIBLE LINEAR MAP Invertible Matrix NONSINGULAR MATRIX Invertible Polynomial Map A POLYNOMIAL MAP ff ; with f /C30 f1 ;...; fn ðÞ /C23 KX1 ;...; Xn ½/C138ðÞmin a FIELD K is called invertible if there exist g1 ;...;gm /C23 KX1 ;...;xn ½/C138 such that gif1 ;...;fn ðÞ /C30Xifor 1 5n 5n so that fg(ff /C30idkn (Becker and Weispfenning 1993, p. 330). GRO¨ BNER BASES provide a means to decide for given fwhether or not ffis invertible. See also JACOBIAN CONJECTURE ,POLYNOMIAL MAP References Becker, T. and Weispfenning, V. Gro¨bner Bases: A Computa- tional Approach to Commutative Algebra. New York: Springer-Verlag, p. 330, 1993. Involuntary ALINEAR TRANSFORMATION of period two. Since a LINEAR TRANSFORMATION has the form, l?¼alþb glþd; (1) applying the transformation a second time gives lƒþal?þb gl?þd¼ða2þbgÞlþbðaþdÞ ðaþdÞglþbgþd2; (2) For an involuntary, /lƒ¼l/,s o g ða þ dÞl2 þðd2 /C28 a2 Þl /C28ða þ dÞb ¼ 0: (3) Since each COEFFICIENT must vanish separately, ag þ gd ¼ 0 (4) d2 /C28 a2 ¼ 0 (5) ab þ bd ¼ 0: ð6Þ The first equation gives /d ¼9a/. Taking / d ¼ a/ would require /g ¼ b ¼ 0/, giving /l ¼ l ?/, the identity transfor- mation. Taking / d ¼/C28a/ gives /d ¼/C28a/,so l ?¼al þ b gl /C28 a (7) the general form of an INVOLUTION . See also CROSS- RATIO,INVOLUTION (LINE) References Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 14 /C1/ 15, 1961. Involute Attach a string to a point on a curve. Extend the string so that it is tangent to the curve at the point of attachment. Then wind the string up, keeping it always taut. The LOCUS of points traced out by the end of the string is the involute of the original curve, and the original curve is called the EVOLUTE of its involute. Although a curve has a unique EVOLUTE ,it has infinitely many involutes corresponding to differ- ent choices of initial point. An involute can also be thought of as any curve ORTHOGONAL to all the TANGENTS to a given curve. The equation of the involute is ri /C30r /C28s ˆT ; (1) where ˆT is the TANGENT VECTOR ˆT /C30dr dt dr dt1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|(2) and s is the ARC LENGTHs /C30gds /C30gds dtdt /C30gffiffiffiffiffiffiffiffi ds2p dtdt /C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ?2 /C27g ?2q dt: (3) This can be written for a parametrically represented function f(t) ;g(t) ðÞ as x(t) /C30f /C28sf ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ?2 /C27 g?2p (4) y(t) /C30g /C28sg ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ?2 /C27 g ?2p : (5) The following table lists the involutes of some common curves, some of which are illustrated above. Curve Involute ASTROID ASTROID 1/2 as large CARDIOID CARDIOID 3 times as large CATENARY TRACTRIX CIRCLE CATACAUSTIC for a point sourceLIMAC ¸ ON CIRCLE CIRCLE INVOLUTE (a SPIRAL ) CYCLOID equal CYCLOID DELTOID DELTOID 1/3 as large ELLIPSE ELLIPSE INVOLUTE EPICYCLOID reduced EPICYCLOID HYPOCYCLOID similar HYPOCY- CLOID LOGARITHMIC SPIRAL equal LOGARITHMIC SPIRAL NEILE’S PARABOLA PARABOLANEPHROID CAYLEY’S SEXTIC NEPHROID NEPHROID 2 times as large See also ENVELOPE ,EVOLUTE ,HUMBERT’S THEOREM , ROULETTE References Cundy, H. and Rollett, A. "Roulettes and Involutes." §2.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 46 /C1/55, 1989. Dixon, R. "String Drawings." Ch. 2 in Mathographics. New York: Dover, pp. 75 /C1/78, 1991. Gray, A. "Involutes." §5.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 103 /C1/107, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40 /C1/42 and 202, 1972. Lockwood, E. H. "Evolutes and Involutes." Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166 /C1/171, 1967. Pappas, T. "The Involute." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 187, 1989. Yates, R. C. "Involutes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 135 /C1/137, 1952. Involution An OPERATOR of period 2, i.e., an OPERATOR + which satisfies aðÞ/C31ðÞ/C31/C30a :/ Involution (Group) An element of order 2 in a GROUP (i.e., an element A of a GROUP such that A2 /C30I ; where I is the IDENTITY ELEMENT ). See also GROUP ,IDENTITY ELEMENT Involution (Line) Pairs of points of a line, the product of whose distances from a FIXED POINT is a given constant. This is more concisely defined as a PROJECTIVITY of period two. If AA?;BB?;CC ? fg is a range in involution, then the ranges AA?; BC fg and A?A;B ?C ? fg are EQUICROSS , and conversely. See also EQUICROSS ,INVOLUTORY References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 133, 1888. Lachlan, R. "Theory of Involutions" and "Involution." Ch. 5 and §426 /C1/427 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 272 /C1/274, 1893. Involution (Operator) An OPERATOR of period 2, i.e., an OPERATOR ¯3 which satisfies sin/C281 x /C30x /C271 6x3 /C273 40x5 /C275 112x7 /C2735 1152x9 /C27...: Involution (Permutation) An involution of a SET S is a PERMUTATION of S which does not contain any CYCLES of length > 2 (i.e., itconsists exclusively of fixed points and TRANSPOSI- TIONS ). Involutions are in one-to-one correspondence with self-conjugate permutations (i.e, permutations that are their own INVERSE PERMUTATION ). For example, the unique permutation involution on 1 element is f1g; the two involution permutations on 2 elements are f1;2g and f2;1g; and the four involution permutations on 3 elements are f1; 2;3g;f1;3 ;2g; f2; 1;3g; and f3; 2;1g: A PERMUTATION p can be tested to determine if it is a permutation using Involu- tionQ [p] in the Mathematica add-on package Dis- creteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The PERMUTATION MATRICES of an involution are SYMMETRIC . The number of involutions on n elements is the same as the number of distinct YOUNG TABLEAUX on n elements (Skiena 1990, p. 32). In general, the number of involution permutations on n letters is given by the formula I(n) /C301 /C27X(n/C282)=2 bc k /C3001 (k /C27 1)!n /C282i 21CA%1CAP ; (1) wheren k1CC1CA is a BINOMIAL COEFFICIENT (Muir 1960, p. 5), or alternatively by I(n) /C30n!Xnbc k /C3001 2kk!(n /C28 2k)! (2) (Skiena 1990, p. 32). Although the number of involu- tions on n symbols cannot be expressed as a fixed number of hypergeometric terms (Petkovsek et al. 1996, p. 160), it can be written in terms of the CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SEC- OND KIND U(a ;b;z)as I(n) /C30/C28 iðÞn2n=2U /C281 2n;12 ;/C28121CA}1CA$ : (3) Breaking this up into n even and odd gives I(n)/C30/C282ðÞkU/C28k;12;/C28121CA}1CA$ forn/C302k /C282ðÞkU/C28k;32;/C28121CA}1CA$ forn/C302k/C2718 < :(4) The number of involutions I(n)o fa SETcontaining the first nintegers is given by the RECURRENCE RELATION IðnÞ¼Iðn/C281Þþðn/C281ÞIðn/C282Þð 5Þ (Muir 1960, pp. 3 /C1/7; Skiena 1990, p. 32). For n/C301, 2, . . ., the first few values of I(n) are 1, 2, 4, 10, 26, 76, . . . (Sloane’s A000085). See also CYCLE (PERMUTATION ), INVERSE PERMUTA- TION ,PERMUTATION ,PERMUTATION MATRIX ,YOUNG TABLEAU References Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. Muir, T. "On Self-Conjugate Permutations." Proc. Royal Soc. Edinburgh 17,7/C1/22, 1889. Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Ruskey, F. "Information on Involutions." http://www.theor- y.csc.uvic.ca/~cos/inf/perm/Involutions.html. Skiena, S. "Involutions." §1.4.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 32 /C1/33, 1990. Sloane, N. J. A. Sequences A000085/M1221 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Involution (Transformation) A TRANSFORMATION of period 2. Involution Principle GARSIA- MILNE INVOLUTION PRINCIPLE Involutory A LINEAR TRANSFORMATION of period two. Since a LINEAR TRANSFORMATION has the form, l ?/C30al /C27 b gl /C27 d; (1) applying the transformation a second time gives lƒ/C30al ?/C27b gl ?/C27d /C30a2 /C27 bg ðÞ l /C27 ba /C27 d ðÞ a /C27 d ðÞ gl /C27 bg /C27 d2 : (2) For an involutory, l ƒ/C30 l ; so ga/C27 d ðÞ l2 /C27 d2 /C28 a21CC1CA l /C28 a /C27 d ðÞ b /C300: (3) Since each COEFFICIENT must vanish separately, ga/C27 d ðÞ /C300 (4) d2 /C28 a2 /C300 (5) ba/C27 d ðÞ /C300: (6) Equation (5) requires d /C309a: Taking d /C30 a in turn requires that g /C30 b /C300; giving l /C30 l ?; i.e., the IDENTITY MAP, while taking d /C30/C28a gives d /C30/C28a; so l ?/C30al /C27 b gl /C28 a; (7) which is the general form of an INVOLUTION . See also CROSS- RATIO,INVOLUTION (LINE) References Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 14 /C1/ 15, 1961.Involutory Matrix A SQUARE MATRIX A such that A2 /C30 l ; where I is the IDENTITY MATRIX . An involutory matrix is its own MATRIX INVERSE . References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 11, 1962. Irradiation Illusion The ILLUSION shown above which was discovered by Helmholtz in the 19th century. Despite the fact that the two above figures are identical in size, the white hole looks bigger than the black one in this ILLUSION . See also ILLUSION References Pappas, T. "Irradiation Optical Illusion." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 199, 1989. Irrational Number A number which cannot be expressed as a FRACTION p=qfor any INTEGERS pand q. The most famous irrational number isffiffiffi 2p ;sometimes called P YTHAGOR- AS’S CONSTANT . Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality offfiffiffi 2p while at sea and, upon notifying his comrades of his great dis- covery, was immediately thrown overboard by the fanatic Pythagoreans . Other examples includeffiffiffi 3p ;e, p;etc. Every TRANSCENDENTAL NUMBER is irrational. Num- bers OF THE FORM n1=mare irrational unless nis the mthPOWER of an INTEGER . Numbers OF THE FORM lognm;where log is the LOGARITHM , are irrational if m andnare INTEGERS , one of which has a PRIME factor which the other lacks. eris irrational for rational r" 0:cosris irrational for every nonnegative rational number r(Niven 1956, Stevens 1999), and cos( u) (for umeasured in degrees) is irrational for every rational 0/C14BuB90/C14with the exception of u/C3060/C14(Niven 1956). tan ris irrational for every rational r"0 (Stevens 1999). The irrationality of Ewas proven by Lambert in 1761; for the general case, see Hardy and Wright (1979, p. 46). pn is irrational for POSITIVE integral n. The irrationality of PI itself was proven by Lambert in 1760; for the general case, see Hardy and Wright (1979, p. 47). APE´ RY’S CONSTANT z(3) (where z(z) is the RIEMANN ZETA FUNCTION ) was proved irrational by Ape´ry (Ape´ry 1979, van der Poorten 1979). In addi- tion, T. Rivoal (2000) recently proved that there are infinitely many integers n such that z(2n /C271) is irrational. From GELFOND’S THEOREM , a number OF THE FORM ab is TRANSCENDENTAL (and therefore irrational) if a is ALGEBRAIC "0; 1 and b is irrational and ALGEBRAIC . This establishes the irrationality of e p (since /C281ðÞ/C28i/C30 eipðÞ/C28i/C30e p)) ; 2ffiffi 2p ; and e p: Nesterenko (1996) proved that p/C27e p is irrational. In fact, he proved that p; e p and G 1 =4ðÞ are ALGEBRAICALLY INDEPENDENT , but it was not previously known that p/C27e p was irrational. Given a POLYNOMIAL equation xm /C27cm/C281xm/C281 /C27.../C27c0 ; (1) where ci are INTEGERS , the roots xi are either integral or irrational. If cos 2 uðÞ is irrational, then so are cos u; sin u; and tan u:/ Irrationality has not yet been established for 2e ;pe ; pffiffi 2p ; or g (where g is the EULER- MASCHERONI CON- STANT ). QUADRATIC SURDS are irrational numbers which have periodic CONTINUED FRACTIONS . HURWITZ’S IRRATIONAL NUMBER THEOREM gives bounds OF THE FORM a/C28p q B1 lnq21CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA| (2) for the best rational approximation possible for an arbitrary irrational number a; where the l n are called LAGRANGE NUMBERS and get steadily larger for each "bad" set of irrational numbers which is excluded. The SERIES X/C12 n/C301sk(n) n!; (3) where sk(n) is the DIVISOR FUNCTION , is irrational for k/C301 and 2, and the series X/C12 n/C3011 2n/C281/C30X/C12 n/C301d(n) 2n; (4) where d(n) is the number of divisors of n, is also irrational (Guy 1994). See also ALGEBRAIC INTEGER ,ALGEBRAIC NUMBER , ALMOST INTEGER ,D IRICHLET FUNCTION , E,FERGU- SON- FORCADE ALGORITHM ,G ELFOND’S THEOREM , HURWITZ’S IRRATIONAL NUMBER THEOREM ,N EAR NOBLE NUMBER ,NOBLE NUMBER ,PI,PYTHAGORAS’SCONSTANT ,P YTHAGORAS’S THEOREM , Q-HARMONIC SERIES ,QUADRATIC IRRATIONAL NUMBER ,RATIONAL NUMBER ,SEGRE’S THEOREM ,TRANSCENDENTAL NUM- BER References Ape´ry, R. "Irrationalite ´dez(2) et z(3):/"Aste´risque 61,1 1/C1/13, 1979. Courant, R. and Robbins, H. "Incommensurable Segments, Irrational Numbers, and the Concept of Limit." §2.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 58 /C1/61, 1996. Guy, R. K. "Some Irrational Series." §B14 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 69, 1994. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Manning, H. P. Irrational Numbers and Their Representa- tion by Sequences and Series. New York: Wiley, 1906. Nagell, T. "Irrational Numbers" and "Irrationality of the numbers eandp:/"§12/C1/13 in Introduction to Number Theory. New York: Wiley, pp. 38 /C1/40, 1951. Nesterenko, Yu. "Modular Functions and Transcendence Problems." C. R. Acad. Sci. Paris Se ´r. I Math. 322, 909/C1/ 914, 1996. Nesterenko, Yu. V. "Modular Functions and Transcendence Questions." Mat. Sb. 187,6 5/C1/96, 1996. Niven, I. M. Irrational Numbers. New York: Wiley, 1956. Niven, I. M. Numbers: Rational and Irrational. New York: Random House, 1961. Pappas, T. "Irrational Numbers & the Pythagoras Theorem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 98 /C1/99, 1989. Rivoal, T. "Irrationalite ´d’une infinite ´de valeurs de la fonction Zeta aux entiers impairs." Preprint 2000 /C1/9. http://www.math.unicaen.fr/~leclerc/publi_labo/2000/in- dex2000.html. Stevens, J. "Zur Irrationalita ¨t von p:/"Mitt. Math. Ges. Hamburg 18, 151/C1/158, 1999. van der Poorten, A. "A Proof that Euler Missed ...Ape´ry’s Proof of the Irrationality of z(3):/"Math. Intel. 1, 196/C1/203, 1979. Weisstein, E. W. "Books about Irrational Numbers." http:// www.treasure-troves.com/books/IrrationalNumbers.html. Irrationality Measure N.B. A detailed online essay by S. Finch was the starting point for this entry. Letxbe a REAL NUMBER , and let Rbe the SETof POSITIVE REAL NUMBERS for which x/C28p q1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|Bq /C28r(1) has (at most) finitely many solutions p=qforpandq INTEGERS . Then the irrationality measure, sometimes called the Liouville-Roth constant, is defined as the threshold at which L IOUVILLE’S APPROXIMATION THE- OREM kicks in and xis no longer approximable by RATIONAL NUMBERS , r(x)/C13inf r/C23Rr: (2) There are three regimes: r(x) /C301 x is rational r(x) /C302 x is algebraic r(x) ]3 x is transcendental8 < : (3) Exact values include r(L) /C30/C12 r(e) /C302; where L is LIOUVILLE’S CONSTANT . The best known upper bounds for other common constants are sum- marized in the following table, where z(3) is APE´ RY’S CONSTANT , Lnq(2) and hq(1) are Q-HARMONIC SERIES , and the lower bounds are 2. constant xupperboundreference / p/ 8.0161 Hata (1992) / p2/ 6.3489 Hata (1992) /ln 2/ 4.13 / z(3) / 7.377956 Hata (2000) /Lnq(2) / 4.80 Amdeberhan and Zeil- berger (1998) /hq(1) / 4.80 Amdeberhan and Zeil- berger (1998) See also LIOUVILLE’S APPROXIMATION THEOREM , ROTH’S THEOREM ,THUE- SIEGEL- ROTH THEOREM References Amdeberhan, T. and Zeilberger, D. "q-Ape´ry Irrationality Proofs by q-WZ Pairs." Adv. Appl. Math. 20, 275 /C1/283, 1998. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lvlrth/lvlrth.html. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford: Clarendon Press, 1979. Hata, M. "Legendre Type Polynomials and Irrationality Measures." J. reine angew. Math. 407,99/C1/125, 1990. Hata, M. "Improvement in the Irrationality Measures of p and p2 :/" Proc. Japan. Acad. Ser. A Math. Sci. 68, 283 /C1/286, 1992. Hata, M. "Rational Approximations to p and Some Other Numbers." Acta Arith. 63 335 /C1/349, 1993. Hata, M. "A Note on Beuker’s Integral." J. Austral. Math. Soc. 58, 143 /C1/153, 1995. Hata, M. "A New Irrationality Measure for z(3):/" Acta Arith. 92,47/C1/57, 2000. Stark, H. M. An Introduction to Number Theory. Cam- bridge, MA: MIT Press, 1978.Irrationality Sequence A sequence of POSITIVE INTEGERS anfg such that / a 1=ðanbn Þ/ is IRRATIONAL for all integer sequences / fbn g/. Erdos showed that /f22n g¼f 1; 2;4; 16;256;...;g/ (Sloane’s A001146) is an irrationality sequence. References Guy, R. K. "Irrationality Sequence." §E24 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 225, 1994. Sloane, N. J. A. Sequences A001146/M1297 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Irreducible Matrix A SQUARE MATRIX which is not REDUCIBLE is said to be irreducible. See also REDUCIBLE MATRIX Irreducible Polynomial APOLYNOMIAL is said to be irreducible if it cannot be factored into nontrivial polynomials over the same FIELD . For example, in the FIELD of rational polynomials Qx½/C138 (i.e., polynomials f(x) with rational coefficients), a f(x) is said to be irreducibility if there do not exist two nonconstant polynomials g(x) and h(x)i n xwith rational coefficients such that f(x)/C30g(x)h(x) (Nagell 1951, p. 160). Similarly, in the FINITE FIELD GF(2), x2/C27x/C271 is irreducible, but x2/C271 is not, since (x/C271)(x/C271)/C30x2/C272x/C271/C13x2/C271 (mod 2). A polyno- mial can be tested to see if it is primitive using the Mathematica function IrreducibleQ[p_,n_] : /C30SameQ[Factor[p, Modu- lus-/C21n], p] In general, the number of irreducible polynomials ofdegree nover the FINITE FIELD GF(q) is given by Lq(n)/C301 nX d½nmn d ! qd; where m(n) is the M O¨BIUS FUNCTION . The number of irreducible polynomials of degree n over GF(2) is equal to the number of n-bead fixed aperiodic NECKLACES of two colors and the number of binary L YNDON WORDS of length n. The first few values for n/C301, 2, ...are 2, 1, 2, 3, 6, 9, 18, ... (Sloane’s A001037). The following table lists the irreducible polynomials (mod 2) of degrees 1 through 5. n irreducible polynomials 11, x 2 /1 /C27x /C27x2 / 3 /1 /C27x /C27x3 ; 1 /C27x2 /C27x3/ 4 /1 /C27x /C27x4 ; 1 /C27x /C27x2 /C27x3 /C27x4 ; 1 /C27x3 /C27x4/ 5 /1 /C27x2 /C27x5 ; 1 /C27x /C27x2 /C27x3 /C27x5 ; 1 /C27x3 /C27x5 ;/ /1 /C27x /C27x3 /C27x4 /C27x5 ; 1 /C27x2 /C27x3 /C27x4 /C27x5 ; 1 /C27x /C27x2 /C27x4 /C27x5/ See also FIELD ,F INITE FIELD ,L YNDON WORD, NECKLACE ,POLYNOMIAL ,PRIMITIVE POLYNOMIAL References Marsh, R. Tables of Irreducible Polynomials of GF(2) through Degree 19. Washington, DC: U. S. Dept. Com- merce., 1957. Nagell, T. "Irreducibility of the Cyclotomic Polynomial." §47 in Introduction to Number Theory. New York: Wiley, pp. 160 /C1/164, 1951. Ruskey, F. "Information on Primitive and Irreducible Poly- nomials." http://www.theory.csc.uvic.ca/~cos/inf/neck/ PolyInfo.html. Sloane, N. J. A. Sequences A001037/M0116 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0564 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Irreducible Representation An irreducible representation of a GROUP is a REPRE- SENTATION that has no nontrivial invariant sub- spaces. For example, the ORTHOGONAL GROUP O(n) has an irreducible representation on Rn :/ Any representation of a finite or SEMISIMPLE LIE GROUP breaks up into a DIRECT SUM of irreducible representations. But in general, this is not the case, e.g., (R;/C27) has a representation on R2 by f(a) /C301 a 011C|C1C|A ; i.e., f(a)(x;y) /C30(x /C27ay ;y) : But the subspace y /C300is fixed, hence f is not irreducible, but there is no complementary invariant subspace. The irreducible representation has a number of remarkable properties, as formalized in the GROUP ORTHOGONALITY THEOREM . Let the ORDER of a GROUP be h, and the dimension of the ith representation (the order of each constituent matrix) be li(a POSITIVE INTEGER ). Let any operation be denoted R, and let the mth row and nth column of the matrix corresponding to a matrix R in the ith IRREDUCIBLE REPRESENTA- TION be Gi(R)mn : The following properties can be derived from the GROUP ORTHOGONALITY THEOREM ,X RGi(R)mn Gj(R)/C31m?n?/C30hffiffiffiffiffiffi liljq dij dmm? dnn?: (1) 1. The DIMENSIONALITY THEOREM : h /C30X il2 i /C30l21 /C27l22 /C27l23 /C27.../C30X ix2i (I) ; (2) where each li must be a POSITIVE INTEGER and x is the CHARACTER (trace) of the representation. 2. The sum of the squares of the CHARACTERS in any IRREDUCIBLE REPRESENTATION i equals h, h /C30X Rx2i (R) : (3) 3. ORTHOGONALITY of different representations X Rxi(R) xi(R) /C300 for i "j : (4) 4. In a given representation, reducible or irredu- cible, the CHARACTERS of all MATRICES belonging to operations in the same class are identical (but differ from those in other representations). 5. The number of IRREDUCIBLE REPRESENTATIONS of a GROUP is equal to the number of CONJUGACY CLASSES in the GROUP . This number is the dimen- sion of the G MATRIX (although some may have zero elements). 6. A one-dimensional representation with all 1s (totally symmetric) will always exist for any GROUP . 7. A 1-D representation for a GROUP with elements expressed as MATRICES can be found by taking the CHARACTERS of the MATRICES . 8. The number aiof IRREDUCIBLE REPRESENTA- TIONS xipresent in a reducible representation cis given by ai/C301 hX Rx(R)xi(R); (5) where his the ORDER of the GROUP and the sum must be taken over all elements in each class. Written explicitly, ai/C301 hX Rx(R)x?i(R)nR; (6) where x?iis the CHARACTER of a single entry in the CHARACTER TABLE and nRis the number of ele- ments in the corresponding CONJUGACY CLASS . Irreducible representations can be indicated using MULLIKEN SYMBOLS . See also CHARACTER (GROUP ), CHARACTER TABLE , FINITE GROUP ,GROUP ,GROUP ORTHOGONALITY THE- OREM ,ITOˆ ’S THEOREM ,M ULLIKEN SYMBOLS ,REPRE- SENTATION ,R EPRESENTATION (LIE ALGEBRA ), SEMISIMPLE LIE GROUP UNITARY TRANSFORMATION , VECTOR SPACE ,W EDDERBURN’S THEOREM References Fulton, W. and Harris, J. Representation Theory. New York:Springer-Verlag, 1991. Jacobson, N. Lie Algebras. New York: Dover, 1979. Huang, J.-S. "Irreducible Representations." §2.3 in Lectures on Representation Theory. Singapore: World Scientific, pp. 11 /C1/14, 1999. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Irreducible Semiperfect Number PRIMITIVE PSEUDOPERFECT NUMBER Irreducible Tensor Given a general second RANK TENSOR Aijand a METRIC gij ; define u /C13Aijgij /C30Ai i (1) vi /C13 eijkAjk (2) sij /C131 2Aij /C27Aji1CC1CA /C2813gijAk k ; (3) where dij is the KRONECKER DELTA and eijk is the LEVI- CIVITA SYMBOL . Then sij /C271 3 ugij /C2712 eijk vk /C3012Aij /C27Aji1CC1CA /C2813gijAk khi /C271 3Ak kgij /C271 2eijk elmkAlm1C|1Cffl /C301 2Aij /C27Aji1CC1CA /C2712dl i dm j /C28 d mi d l j1CA}1CA$ Alm /C301 2Aij /C27Aji1CC1CA /C2712Aij /C28Aji1CC1CA /C30Aij ; (4) where u; vi ; and sij are TENSORS of RANK 0, 1, and 2. See also TENSOR References Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Irreducible Tensors." Ch. 3 in Quantum Theory of Angular Momentum. Singapore: World Scientific, pp. 61 /C1/ 71, 1988. Irreducible Variety An ALGEBRAIC VARIETY is called irreducible if it cannot be written as the union of nonempty algebraic varieties. For example, the set of solutions to xy /C300is reducible because it is the union of the solutions to x /C300 and the solutions to y /C300. See also ALGEBRAIC SET,ALGEBRAIC VARIETY ,PRO- JECTIVE VARIETY Irredundant Ramsey Number Let G1 ; G2 ; ..., Gtbe a t-EDGE coloring of the COMPLETE GRAPH Kn ; where for each i /C301, 2, ..., t, / Gi/ is the spanning SUBGRAPH of Knconsisting of allEDGES colored with the ith color. The irredundant Ramsey number sq1 ;...;qt ðÞ is the smallest INTEGER n such that for any t-EDGE coloring of Kn ; the COMPLEMENT GRAPH Gihas an irredundant set of size qifor at least one i /C301, ..., t. Irredundant Ramsey numbers were introduced by Brewster et al. (1989) and satisfy sq1 ;/C1/C1/C1qt ðÞ 5Rq1 ;...qt ðÞ : For a summary, see Mynhardt (1992). s Bounds Reference /s(3; 3)/ 6 Brewster et al. 1989 /s(3; 4)/ 8 Brewster et al. 1989 /s(3; 5)/ 12 Brewster et al. 1989 /s(3; 6)/ 15 Brewster et al. 1990 /s(3; 7)/ 18 Chen and Rousseau 1995, Cockayne et al. 1991 /s(4; 4)/ 13 Cockayne et al. 1992 /s(3; 3;3)/ 13 Cockayne and Mynhardt 1994 References Brewster, R. C.; Cockayne, E. J.; and Mynhardt, C. M. "Irredundant Ramsey Numbers for Graphs." J. Graph Theory 13, 283 /C1/290, 1989. Brewster, R. C.; Cockayne, E. J.; and Mynhardt, C. M. "The Irredundant Ramsey Number s(3;6) :/" Quaest. Math. 13, 141 /C1/157, 1990. Chen, G. and Rousseau, C. C. "The Irredundant Ramsey Number s(3; 7):/" J. Graph. Th. 19, 263 /C1/270, 1995. Cockayne, E. J.; Exoo, G.; Hattingh, J. H.; and Mynhardt, C. M. "The Irredundant Ramsey Number s(4;4) :/" Util. Math. 41, 119 /C1/128, 1992. Cockayne, E. J.; Hattingh, J. H.; and Mynhardt, C. M. "The Irredundant Ramsey Number s(3;7) :/" Util. Math. 39, 145 /C1/160, 1991. Cockayne, E. J. and Mynhardt, C. M. "The Irredundant Ramsey Number s(3; 3;3) /C3013:/" J. Graph. Th. 18, 595 /C1/ 604, 1994. Hattingh, J. H. "On Irredundant Ramsey Numbers for Graphs." J. Graph Th. 14, 437 /C1/441, 1990. Mynhardt, C. M. "Irredundant Ramsey Numbers for Graphs: A Survey." Congres. Numer. 86,6 5/C1/79, 1992. Irreflexive ARELATION Ron a SETSis irreflexive provided that no element is related to itself; in other words, xRxfor noxinS. See also RELATION Irregular Pair If p divides the NUMERATOR of the BERNOULLI NUMBER B2kfor 0 B2k Bp /C281; then (p; 2k) is called an irregular pair. For p B30000, the irregular pairs of various forms are p /C3016843 for (p ;p /C283); p /C3037 for (p;p /C285); none for (p;p /C287); and p /C3067 ;877 for (p;p /C289):/ See also BERNOULLI NUMBER ,IRREGULAR PRIME References Johnson, W. "Irregular Primes and Cyclotomic Invariants." Math. Comput. 29, 113 /C1/120, 1975. Irregular Prime PRIMES for which Kummer’s theorem on the unsolva- bility of FERMAT’S LAST THEOREM does not apply. An irregular prime p divides the NUMERATOR of one of the BERNOULLI NUMBERS B0 ; B2 ; ..., Bp /C283 ; as shown by Kummer in 1850. The FERMAT EQUATION has no solutions for REGULAR PRIMES . An INFINITE number of irregular primes exist, as proven in 1915 by Jensen. The first few irregular primes are 37, 59, 67, 101, 103, 131, 149, 157, ... (Sloane’s A000928). Of the 283,145 PRIMES less than 4 /C29106 ; 111,597 (or 39.41%) are irregular. The con- jectured FRACTION is 1 /C28e /C281 =2 :39:35% (Ribenboim 1996, p. 415). See also BERNOULLI NUMBER ,FERMAT’S LAST THEO- REM,IRREGULAR PAIR,REGULAR PRIME References Buhler, J.; Crandall, R.; Ernvall, R.; and Metsa ¨nkyla ¨, T. "Irregular Primes and Cyclotomic Invariants to Four Million." Math. Comput. 60, 151 /C1/153, 1993. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 202, 1979. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 192, 1998. Johnson, W. "Irregular Primes and Cyclotomic Invariants." Math. Comput. 29, 113 /C1/120, 1975. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 325 /C1/329 and 414 /C1/425, 1996.Sloane, N. J. A. Sequences A000928/M5260 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stewart, C. L. "A Note on the Fermat Equation." Mathema- tika 24, 130 /C1/132, 1977. Irregular Singularity Consider a second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27P(x)y?/C27Q(x)y /C300: If P(x) and QxðÞremain FINITE at x /C30x0 ; then x0is called an ORDINARY POINT . If either P(x)or QxðÞ diverges as x 0 x0 ; then x0 is called a singular point. If P(x) diverges more quickly than 1= x /C28x0 ðÞ ; so x /C28x0 ðÞ P(x) approaches INFINITY as x 0 x0 ; or QxðÞ diverges more quickly than 1= x /C28x0 ðÞ2Q so that x /C28x0 ðÞ2Q(x) goes to INFINITY as x 0 x0 ; then x0is called an IRREGULAR SINGULARITY (or ESSENTIAL SINGULARITY ). See also ORDINARY POINT ,REGULAR SINGULAR POINT , SINGULAR POINT (DIFFERENTIAL EQUATION ) References Arfken, G. "Singular Points." §8.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 451 /C1/453 and 461 /C1/463, 1985. Irrotational Field A VECTOR FIELD v for which the CURL vanishes, 9/C29v/C300: See also BELTRAMI FIELD,C ONSERVATIVE FIELD , POINCARE ´ ’S THEOREM ,SOLENOIDAL FIELD,V ECTOR FIELD Isarithm EQUIPOTENTIAL CURVE ISBN Publisher Digits Addison-Wesley 0 /C1/201 Amer. Math. Soc. 0 /C1/821 Birkha ¨user Basel 3 /C1/7643 Birkha ¨user Boston 0 /C1/8176 Cambridge University Press 0 /C1/521 CRC Press 0 /C1/8493 Dover 0 /C1/486 McGraw-Hill 0 /C1/070 Oxford University Press 0 /C1/198 Springer-Verlag Berlin 3 /C1/540 Springer-Verlag New York 0 /C1/387 Tarquin Publications 0 /C1/906212 Wiley 0 /C1/471 The International Standard Book Number (ISBN) is a 10-digit CODE which is used to uniquely identify a book. The digits di are arranged in four groups, which are sometimes (but not always) separated by hy- phens. The first group is a single digit which codes country or language in which a publisher is incorpo- rated: 0 for English, 2 for French, 3 for German, 4 for Japanese, 8 for Indian publishers, etc. The next group of digits specifies the publisher, and may range in length from two to seven digits, with fewer digits used for larger publishers. Some publishers with offices in more than one country (at least when different languages are spoken in those countries) have multi- ple publisher codes and initial digits. The third group of digits specifies an individual book, and may be from one to six digits in length. The actual number is eight minus the number of digits in the publisher group, so that small publishers may have only 10 books, while large ones can have up to a millions books. The last digit d10is a check digit which may be in the range 0 /C1/9 or X (where X is the ROMAN NUMERAL for 10). The check digit is computed from the equation 10d1 /C279d2 /C278d3 /C27.../C272d9 /C27d10 /C130 (mod 11) : For example, the number for this book is 0 /C1/8493 /C1/ 9640 /C1/9, and 10 /C2150 /C279 /C2158 /C278 /C2154 /C277 /C2159 /C276 /C2153 /C275 /C2159 /C274 /C2156 /C273 /C2154 /C272 /C2150 /C271 /C2159 /C30275 /C3025 /C21511 /C130 (mod 11) : as required. The ISBN is error-detecting, but not error-correcting (unless it is known that only a single digit is erroneous). The ISBN detects any single-digit error, as well as any two-digit error resulting from trans- posing two digits. See also CODE,CODING THEORY , UPC References Hill, R. First Course in Coding Theory. Oxford, England: Oxford University Press, 1986. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 894, 1992.Iseki’s Formula Let R z½/C138> 0; 0 5 a; b 51 ; and L(a; b; z) /C13X/C12 r/C300l((r /C27 a)z /C28i b) /C27 l((r /C271 /C28 a)z /C27i b) ½/C138 ; (1) where l(x) /C13/C28ln 1 /C28e /C282 px1CC1CA /C30X/C12 m/C301e /C282 pmx m: (2) Then if either 0 5 a 51 and 0 B b B1 ; or 0 B a B1 and 0 5 b 51; L( a; b;z) /C30L 1 /C28 b; a;z/C2811CC1CA /C28pzX2 n/C3002 n1CA%1CAP (iz)/C28nB2/C28n( a)Bn( b); (3) where Bk(x)isaB ERNOULLI POLYNOMIAL , and the second term on the right side can be written explicitly as /C28pza2a/C271 61CA}1CA$ /C27p zb2/C28b/C27161CA}1CA$ /C272pia/C28121CA}1CA$ (b/C28h):(4) See also DEDEKIND ETA FUNCTION References Apostol, T. M. "Iseki’s Transformation Formula" and "De- duction of Dedekind’s Functional Equation from Iseki’s Formula." §3.5/C1/3.6 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 53 /C1/61, 1997. Iseki, S. "The Transformation Formula for the Dedekind Modular Function and Related Functional Equations."Duke Math. J. 24, 653/C1 /662, 1957. I-Signature SIGNATURE (RECURRENCE RELATION ) Island If an integrable QUASIPERIODIC system is slightly perturbed so that it becomes nonintegrable, only a finite number of n-CYCLES remain as a result of MODE LOCKING . One will be elliptical and one will be hyperbolic. Surrounding the ELLIPTIC FIXED POINT is a region of stable ORBITS which circle it, as illustrated above in the STANDARD MAP with K /C301:5: As the map is iteratively applied, the island is mapped to a similar structure surrounding the next point of the elliptic cycle. The map thus has a chain of islands, with the FIXED POINT alternating between ELLIPTIC (at the center of the islands) and HYPERBOLIC (between islands). Because the unperturbed system goes through an INFINITY of rational values, the perturbed system must have an INFINITE number of island chains. See also MODE LOCKING ,ORBIT (MAP), QUASIPERIO- DIC FUNCTION Isobaric Polynomial A POLYNOMIAL in which the sum of SUBSCRIPTS is the same in each term. See also HOMOGENEOUS POLYNOMIAL Isochronous Curve SEMICUBICAL PARABOLA ,TAUTOCHRONE PROBLEM Isoclinal ISOCLINAL LINE,ISOCLINAL PLANE ,ISOCLINE Isoclinal Line A line making equal angles with the edges of a TRIHEDRON is called an isoclinal line of the TRIHE- DRON . See also ISOCLINAL PLANE ,TRIHEDRONReferences Altshiller-Court, N. "Isoclinal Lines and Planes." §2.3 in Modern Pure Solid Geometry. New York: Chelsea, pp. 32 /C1/ 37, 1979. Isoclinal Plane A PLANE making equal angles with the three edges of a TRIHEDRON . See also ISOCLINAL LINE,TETRAHEDRON References Altshiller-Court, N. "Isoclinal Lines and Planes." §2.3 in Modern Pure Solid Geometry. New York: Chelsea, pp. 32 /C1/ 37, 1979. Isocline A graphical method of solving an ORDINARY DIFFER- ENTIAL EQUATION OF THE FORM dy dx /C30f(x;y) by plotting a series of curves f(x;y) /C30[const] ; then drawing a curve PERPENDICULAR to each curve such that it satisfies the initial condition. This curve is the solution to the ORDINARY DIFFERENTIAL EQUATION . See also ISOCLINAL LINE,ISOCLINAL PLANE References Ka´rma´n, T. von and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treat- ment of Engineering Problems. New York: McGraw-Hill, pp. 3 and 7, 1940. Isoclinic Groups Two GROUPS GandHare said to be isoclinic if there are isomorphisms G=Z(G)0H=Z(H) and G?0H?; where Z(G) is the CENTER of the group, which identify the two commutator maps. References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "Isoclinism." §6.7 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xxiii-xxiv, 1985. Isodynamic Points The first and second isodynamic points of a TRIANGLE DABC can be constructed by drawing the triangle’s ANGLE BISECTORS and EXTERIOR ANGLE BISECTORS . Each pair of bisectors intersects a side of the triangle (or its extension) in two points Di1 and Di2 ; for i /C301, 2, 3. The three CIRCLES having D11D12 ; D21D22 ; and D31D32 as DIAMETERS are the APOLLONIUS CIRCLES C1 ; C2 ; and C3 : The points S and S0 in which the three APOLLONIUS CIRCLES intersect are the first and second isodynamic points, respectively. S and S 0 have TRIANGLE CENTER FUNCTIONS a /C30sin A 91 3 p1CA}1CA$ ; respectively. The ANTIPEDAL TRIANGLES of both points are EQUILATERAL and have AREAS D?/C302D cot v cot1 3 p1CA}1CA$hi ; where v is the BROCARD ANGLE . The isodynamic points are ISOGONAL CONJUGATES of the FERMAT POINTS . They lie on the BROCARD AXIS. The distances from either isodynamic point to the VERTICES are inversely proportional to the sides. The PEDAL TRIANGLE of either isodynamic point is an EQUILATERAL TRIANGLE .An INVERSION with either isodynamic point as the INVERSION CENTER trans- forms the triangle into an EQUILATERAL TRIANGLE . The CIRCLE which passes through both the isody- namic points and the CENTROID of a TRIANGLE is known as the PARRY CIRCLE . See also APOLLONIUS CIRCLES ,BROCARD AXIS,CEN- TROID (TRIANGLE ), FERMAT POINTS ,PARRY CIRCLE References Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 106, 1913. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 295 /C1/297, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994.Isoenergetic Nondegeneracy The condition for isoenergetic nondegeneracy for a Hamiltonian H /C30H0(I)/C27/C23 H1(I ; u) is @2H0 @Ii @Ij@H0 @Ii @H0 @Ij01CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|"0; which guarantees the EXISTENCE on every energy level surface of a set of invariant tori whose comple- ment has a small MEASURE . References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 113 /C1/114, 1989. Isogeny A rational homomorphism 8G 0 G ? defined over a FIELD is called an isogeny when dim G /C30dim G ?: Two GROUPS G and G ? are then called isogenous if there exists a third group G ƒ and isogenies Gƒ0G and G ƒ0G ?:/ See also HOMEOMORPHIC References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 47, 1980. Isogonal Conjugate The isogonal conjugate X/C281of a point Xin the plane of the TRIANGLE DABC is constructed by reflecting the lines AX,BX, and CXabout the ANGLE BISECTORS at A,B, and C. The three reflected lines then CONCUR at the isogonal conjugate (Honsberger 1995, pp. 55 /C1/56). The TRILINEAR COORDINATES of the isogonal conjugate of the point with coordinates a:b:g are a/C281 : b/C281 : g /C281 : In the above figure with P and Q isogonal conjugates, x y /C30sr (1) (Honsberger 1995, pp. 54 /C1 /55). Isogonal conjugation maps the interior of a TRIANGLE onto itself. This mapping transforms lines onto CONIC SECTIONS that CIRCUMSCRIBE the TRIANGLE . The type of CONIC SECTION is determined by whether the line d meets the CIRCUMCIRCLE C ?; 1. If d does not intersect C?; the isogonal transform is an ELLIPSE ; 2. If d is tangent to C ?; the transform is a PARABOLA ; 3. If d cuts C ?; the transform is a HYPERBOLA , which is a RECTANGULAR HYPERBOLA if the line passes through the CIRCUMCENTER (Casey 1893, Vandeghen 1965). The isogonal conjugate of a point on the CIRCUMCIR- CLE is a POINT AT INFINITY (and conversely). The sides of the PEDAL TRIANGLE of a point are PERPENDICULAR to the connectors of the corresponding VERTICES with the isogonal conjugate. The isogonal conjugate of a set of points is the LOCUS of their isogonal conjugate points. The product of ISOTOMIC and isogonal conjugation is a COLLINEATION which transforms the sides of a TRIAN- GLE to themselves (Vandeghen 1965). See also ANTIPEDAL TRIANGLE ,COLLINEATION ,ISO- GONAL LINE,ISOTOMIC CONJUGATE POINT ,LINE AT INFINITY ,SYMMEDIAN References Barrow, D. F. "A Theorem about Isogonal Conjugates." Amer. Math. Monthly 20, 251 /C1/253, 1913. Casey, J. "Theory of Isogonal and Isotomic Points, and of Antiparallel and Symmedian Lines." Supp. Ch. §1in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometrywith Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 165 /C1/173, 1888. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 49, 1971. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 93, 1967. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 53 /C1/57, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 153 /C1/158, 1929. Lachlan, R. §10 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 55 /C1/57, 1893. Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091 /C1/1094, 1965. Isogonal Line The line L? through a TRIANGLE VERTEX obtained by reflecting an initial line L (also through a VERTEX ) about the ANGLE BISECTOR . If three lines from the VERTICES of a TRIANGLE DABC are CONCURRENT at X /C30L1L2L3 ; then their isogonal lines are also CON- CURRENT , and the point of concurrence X ?/C30L ?1L?2L?3is called the ISOGONAL CONJUGATE point. See also ISOGONAL CONJUGATE References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 153 /C1/157, 1929. Isogonic Centers FERMAT POINTS Isograph The substitution of reiuforzin a POLYNOMIAL p(z): p(z) is then plotted as a function of ufor a given rin the COMPLEX PLANE . By varying rso that the curve passes through the ORIGIN , it is possible to determine a value for one ROOT of the POLYNOMIAL . Isohedral Tiling LetS(T) be the group of symmetries which map a MONOHEDRAL TILING Tonto itself. The TRANSITIVITY CLASS of a given tile T is then the collection of all tiles to which T can be mapped by one of the symmetries of S(T) : If T has k TRANSITIVITY CLASSES , then T is said to be k-isohedral. Berglund (1993) gives examples of k-isohedral tilings for k /C301, 2, and 4. See also ANISOHEDRAL TILING References Berglund, J. "Is There a k-Anisohedral Tile for k ]5/?" Amer. Math. Monthly 100, 585 /C1/588, 1993. Gru¨nbaum, B. and Shephard, G. C. "The 81 Types of Isohedral Tilings of the Plane." Math. Proc. Cambridge Philos. Soc. 82, 177 /C1/196, 1977. Isohedron S(T) A convex POLYHEDRON with symmetries acting tran- sitively on its faces. Every isohedron has an EVEN number of faces (Gru¨nbaum 1960). The isohedra make fair DICE, and there are 30 of them, many of which are PLATONIC SOLIDS ,ARCHIMEDEAN SOLIDS ,or duals of ARCHIMEDEAN SOLIDS . The 30 isohedra are the CUBE , DISDYAKIS DODECAHE- DRON , DELTOIDAL HEXECONTAHEDRON , DELTOIDAL ICOSITETRAHEDRON , DISDYAKIS TRIACONTAHEDRON , DODECAHEDRON , dyakis dodecahedron, hexakis tetra-hedron, ICOSAHEDRON , isosceles tetrahedron, octahe- dral pentagonal dodecahedron, OCTAHEDRON , PENTAGONAL HEXECONTAHEDRON , PENTAGONAL ICOSI- TETRAHEDRON , PENTAKIS DODECAHEDRON , RHOMBIC DODECAHEDRON , RHOMBIC TRIACONTAHEDRON , sca- lene tetrahedron, tetragonal pentagonal dodecahe- dron, TETRAHEDRON , TETRAKIS HEXAHEDRON , trapezoidal dihedron, trapezoidal dihedron (skewed), trapezoidal dodecahedron, TRIAKIS ICOSAHEDRON , TRIAKIS OCTAHEDRON , TRIAKIS TETRAHEDRON , trian- gular dihedron, triangular dihedron (skewed in-out), triangular dihedron (skewed up-down). A 2-D LAMINA such as a COIN can also be viewed as a degenerate case of a fair 2-sided solid. See also COIN,DICE,POLYHEDRON References Gru¨nbaum, B. "On Polyhedra in E3Having All Faces Congruent." Bull. Research Council Israel 8F, 215/C1/218, 1960. Gru¨nbaum, B. and Shepard, G. C. "Spherical Tilings with Transitivity Properties." In The Geometric Vein: The Coxeter Festschrift (Ed. C. Davis, B. Gru ¨nbaum, and F. Shenk). New York: Springer-Verlag, 1982. Pegg, E. Jr. "Fair Dice." http://www.mathpuzzle.com/Fair- dice.htm. Weisstein, E. W. "Fair Dice." M ATHEMATICA NOTEBOOK FAIRDICE.M . Isolated Point An isolated point on a curve, also known as an ACNODE orHERMIT POINT , is a point which has no other points in its NEIGHBORHOOD . An isolated point of a GRAPH is a node of degree 1 (Harary 1994, p. 15). The number of n-node graphs with no isolated points are 0, 1, 2, 7, 23, 122, 888, ... (Sloane’s A002494), the first few of which are illu- strated below. An isolated point of a DISCRETE SET S is a member of S (Krantz 1999, p. 63). See also ENDPOINT ,NEIGHBORHOOD References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Krantz, S. G. "Discrete Sets and Isolated Points." §4.6.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 63 /C1/64, 1999. Sloane, N. J. A. Sequences A002494/M1762 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Isolated Singular Point ISOLATED SINGULARITY Isolated Singularity An isolated singularity is a SINGULARITY for which there exists a (small) REAL NUMBER e such that there are no other SINGULARITIES within a NEIGHBORHOOD of radius e centered about the SINGULARITY . Isolated singularities are also known as conic double points. The types of isolated singularities possible for CUBIC SURFACES have been classified (Schla ¨fli 1864, Cayley 1869, Bruce and Wall 1979) and are summarized in the following table from Fischer (1986). Name Symbol Normal Form COXETER DIAGRAM conic dou- ble point/C2// x2 /C27y2 /C27z2// A1/ biplanar double point/B3// x2 /C27y2 /C27z3// A2/ biplanardoublepoint/B4// x2 /C27y2 /C27z4// A3/biplanar double point/B5// x2 /C27y2 /C27z5// A4/ biplanardoublepoint/B6// x2 /C27y2 /C27z6// A5/ uniplanardouble point/U6// x2 /C27zy2 /C27z2ðÞ // D4/ uniplanar double point/U7// x2/C27zy2/C27z3ðÞ // D5/ uniplanardoublepoint/U8// x2/C27y3/C27z4// E6/ ellipticcone point– /xy2/C284z3/C28g2x2y/C27g3x3//˜E6/ See also CUBIC SURFACE ,RATIONAL DOUBLE POINT , SINGULAR POINT (FUNCTION ) References Bruce, J. and Wall, C. T. C. "On the Classification of Cubic Surfaces." J. London Math. Soc. 19, 245/C1/256, 1979. Cayley, A. "A Memoir on Cubic Surfaces." Phil. Trans. Roy. Soc. 159, 231/C1/326, 1869. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 12 /C1/13, 1986. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 41, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 380 /C1/381, 1953. Schla¨fli, L. "On the Distribution of Surfaces of Third Order into Species." Phil. Trans. Roy. Soc. 153, 193/C1/247, 1864. Isolating Integral An integral of motion which restricts the PHASE SPACE available to a DYNAMICAL SYSTEM . Isometric AMETRIC SPACE Xis isometric to a METRIC SPACE Yif there is a BIJECTION fbetween Xand Ythat preserves distances. That is, d(a;b)/C30d(f(a);f(b)):In the context of R IEMANNIAN GEOMETRY , two manifolds MandNare isometric if there is a DIFFEOMORPHISM such that the R IEMANNIAN METRIC from one pulls back to the metric on the other. Since the GEODESICS define a distance, a R IEMANNIAN METRIC makes the MANIFOLD MaMETRIC SPACE . An isometry between Riemannian manifolds is also an isometry between the two manifolds, considered as metric spaces. Isometric spaces are considered isomorphic. For instance, the circle of radius one around the origin is isometric to the circle of radius one around (0 ;3):/ See also ISOMETRIC LATITUDE ,ISOMETRY ,M ETRIC SPACE ,RIEMANNIAN METRIC ,TOPOLOGICAL SPACE Isometric Latitude An AUXILIARY LATITUDE which is directly proportional to the spacing of parallels of LATITUDE from the equator on an ellipsoidal MERCATOR PROJECTION .It is defined by c /C30ln tan1 4 p/C2712f1CA}1CA$1 /C28 e sin f 1 /C27 e sin f !e=21CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|1CA|; (1) where the symbol t is sometimes used instead of c: The isometric latitude is related to the CONFORMAL LATITUDE by c /C30ln tan1 4 p/C2712x1CA}1CA$ : (2) The inverse is found by iterating f /C302 tan /C281exp( c)1 /C27 e sin f 1 /C28 e sin f !e=22 435/C28 1 2p; (3) with the first trial as f0 /C302 tan/C281 ec1CC1CA /C2812 p: (4) See also LATITUDE References Adams, O. S. "Latitude Developments Connected with Geo- desy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, 1921. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 15, 1987. Isometry A BIJECTIVE MAP between two METRIC SPACES that preserves distances, i.e., d(f(x) ;f(y)) /C30d(x;y) ; where f is the MAP and d(a ;b) is the DISTANCE function. Isometries are sometimes also called con- gruence transformations. Two figures that can be transformed into each other by an isometry are said to be CONGRUENT (Coxeter and Greitzer 1967, p. 80). An isometry of the PLANE is a linear transformation which preserves length. Isometries include ROTATION , TRANSLATION , REFLECTION , GLIDES , and the IDENTITY MAP. If an isometry has more than one FIXED POINT ,it must be either the identity transformation or a reflection. Every isometry of period two (two applica- tions of the transformation preserving lengths in the original configuration) is either a reflection or a half- turn rotation. Every isometry in the plane is theproduct of at most three reflections (at most two if there is a FIXED POINT ). Every finite group of isometries has at least one FIXED POINT . See also CONGRUENT ,DISTANCE ,EUCLIDEAN MOTION , GLIDE,H JELMSLEV’S THEOREM ,IDENTITY MAP,ISO- METRIC ,LENGTH (CURVE ), REFLECTION ,R OTATION , TRANSLATION References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 80, 1967. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991. Gray, A. "Isometries and Conformal Maps of Surfaces." §15.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 346 /C1/351, 1997. Isomorphic The term "isomorphic" means "having the same form," and is used in many branches of mathematics to identify mathematical objects which have the same structural properties. Objects which may be repre- sented (or "embedded") differently but which have the same essential structure are often said to be "iden- tical up to an isomorphism." The statement "A is isomorphic to B" is denoted A $B (Harary 1994, p. 161). See also ISOMORPHIC GRAPHS ,ISOMORPHIC GROUPS , ORDER ISOMORPHIC ,ISOMORPHIC POSETS ,ISOMORPH- ISM References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Isomorphic Graphs Two GRAPHS which contain the same number of VERTICES connected in the same way are said to be isomorphic. Formally, two graphs G and H with VERTICES Vn /C30 1;2; ... ;n fg are said to be isomorphic if there is a PERMUTATION p of Vn such that fu;vg is in the set of EDGES E(G) IFF fp(u) ;p(v) g is in the set of EDGES E(H) :/ Determining if two GRAPHS are isomorphic is thought to be an NP-HARD PROBLEM (Skiena 1990, p. 181), although this has not been proved. However, a polynomial-time algorithm is known when the max- imum VERTEX DEGREE is bounded by a constant (Luks 1980; Skiena 1990, p. 181). The equivalence or none- quivalence of two graphs can be ascertained using IsomorphicQ [g1,g2] in the Mathematica add-on packageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also GRAPH ,G RAPH AUTOMORPHISM ,G RAPH ISOMORPHISM ,GRAPH THEORY ,ULAM’S CONJECTURE References Chartrand, G. "Isomorphic Graphs." §2.2 in Introductory Graph Theory. New York: Dover, pp. 32 /C1/40, 1985. Corneil, D. G. and Gottlieb, C. C. "An Efficient Algorithm for Graph Isomorphism." J. ACM 17,51/C1/64, 1970. Cvetkovic, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 10 /C1/11, 1994. Luks, E. M. "Isomorphism of Bounded Valence can be Tested in Polynomial Time." In Proc. 21st Annual Symposium on Foundations of Computing. IEEE Press, pp. 42 /C1/49, 1980. Schmidt, D. C. and Druffel, L. E. "A Fast Backtracking Algorithm to Test Directed Graphs for Isomorphism Using Distance Matrices." J. ACM 23, 433 /C1/445, 1976. Skiena, S. "Graph Isomorphism." §5.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 181 /C1/187, 1990. Isomorphic Groups Two GROUPS are isomorphic if the correspondence between them is ONE-TO-ONE and the "multiplication" table is preserved. For example, the POINT GROUPS C2 and D1are isomorphic GROUPS , written C2 $D1or C2 XD1 (Shanks 1993). See also JORDAN- HO¨ LDER THEOREM References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993. Isomorphic Posets Two POSETS are said to be isomorphic if their "structures" are entirely analogous. Formally, POSETS P /C30(X ;5) and Q /C30(X ;5?) are isomorphic if there is a BIJECTION f from X to X ? such that x 5x? precisely when f(x) 5? f(x?) :/ Isomorphism Isomorphism is a very general concept which appears in several areas of mathematics. The word derives from the Greek iso (iso), meaning "equal," and mor 8 vsi& (morphosis ), meaning "to form" or "to shape." Formally, an isomorphism is BIJECTIVE MORPHISM . Informally, an isomorphism is a map which preserves sets and relations among elements. "A is isomorphic to B" is written A $B: Unfortu- nately, this symbol is also used to denote geometric CONGRUENCE . A space isomorphism is a VECTOR SPACE in which addition and scalar multiplication are preserved. An isomorphism of a TOPOLOGICAL SPACE is called a HOMEOMORPHISM . Two groups G1 and G2 with binary operators /C27and /C29 are isomorphic if there exists a map f : G1 0 G2 which satisfies f(x /C27y) /C30f(x) /C29f(y) :An isomorphism preserves the identities and inverses of a GROUP . An isomorphism of a GROUP onto itself is called an AUTOMORPHISM . See also AUTOMORPHISM ,AX-KOCHEN ISOMORPHISM THEOREM ,H OMEOMORPHISM ,ISOMORPHIC GRAPHS , ISOMORPHIC GROUPS ,MORPHISM Isoperimetric Inequality Let a PLANE figure have AREA A and PERIMETER p. Then Q /C134pA p251; where Q is known as the ISOPERIMETRIC QUOTIENT . The equation becomes an EQUALITY only for a CIRCLE . See also ISOPERIMETRIC QUOTIENT References Osserman, R. "Isoperimetric Inequalities." Appendix 3, §3i n A Survey of Minimal Surfaces. New York: Dover, pp. 147 /C1/ 148, 1986. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, p. 35, 1978. Isoperimetric Point The point S?which makes the PERIMETERS of the TRIANGLES DBS?C;DCS?A;andDAS?Bequal. The isoperimetric point exists IFFthe largest ANGLE of the triangle satisfies max( A;B;C)B2 sin/C2814 51CA}1CA$ :1:85459 rad :106:26/C14; or equivalently a/C27b/C27c>4R/C27r; where a,b, and care the side lengths of DABC ;ris the INRADIUS , and Ris the CIRCUMRADIUS . The isoperimetric point is also the center of the outer SODDY CIRCLE ofDABC and has TRIANGLE CENTER FUNCTION a /C301 /C282D a(b /C27 c /C28 a) /C30sec1 2A1CA}1CA$ cos12B1CA}1CA$ cos12C1CA}1CA$ /C281: See also EQUAL DETOUR POINT ,PERIMETER ,SODDY CIRCLES References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Kimberling, C. "Isoperimetric Point and Equal Detour Point." http://cedar.evansville.edu/~ck6/tcenters/recent/ isoper.html. Kimberling, C. and Wagner, R. W. "Problem E 3020 and Solution." Amer. Math. Monthly 93, 650 /C1/652, 1986. Veldkamp, G. R. "The Isoperimetric Point and the Point(s) of Equal Detour." Amer. Math. Monthly 92, 546 /C1/558, 1985. Isoperimetric Problem Find a closed plane curve of a given PERIMETER which encloses the greatest AREA . The solution is a CIRCLE . If the class of curves to be considered is limited to smooth curves, the isoperimetric problem can be stated symbolically as follows: find an arc with PARAMETRIC EQUATIONS x /C30x(t) ; y /C30y(t) for t /C23 t1 ;t2 jj such that x(t1) /C30x(t2) ; y(t1) /C30y(t2) (where no further intersections occur) constrained by l /C30gt2 t1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2 /C27y?2q1CA%1CAP dt such that A /C301 2gt2 t1xy?/C28x?y ðÞ dt is a MAXIMUM . Zenodorus proved that the AREA of the CIRCLE is larger than that of any POLYGON having the same PERIMETER , but the problem was not rigorously solved until Steiner published several proofs in 1841 (Wells 1991). See also CIRCLE ,DIDO’S PROBLEM ,DOUBLE BUBBLE , ISOPERIMETRIC QUOTIENT ,ISOPERIMETRIC THEOREM , ISOVOLUME PROBLEM ,PERIMETER References Bogomolny, A. "Isoperimetric Theorem and Inequality." http://www.cut-the-knot.com/do_you_know/isoperime- tric.html. Isenberg, C. "The Maximum Area Contained by a Given Circumference." Appendix V in The Science of Soap Films and Soap Bubbles. New York: Dover, pp. 171 /C1/173, 1992.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 149 /C1/150, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 122 /C1/124, 1991. Isoperimetric Quotient Portions of this entry contributed by HERMANN KRE- MER The isoperimetric quotient of a closed curve is defined as the ratio of the curve area to the area of a circle with same perimeter as the curve, Q /C134pA p2; (1) where A is the area of the plane figure and p is its PERIMETER . The ISOPERIMETRIC INEQUALITY gives Q 5 1; with equality only in the case of the CIRCLE . For a regular n-gon with INRADIUS r, the area is given by A/C30nr2tanp n ! ; (2) edge length by a/C302rtanp n ! ; (3) and the perimeter is given by p/C30na/C302nrtanp n ! : (4) Thus, Qn/C30p ntanp n ! ; (5) which converges to 1 for n0/C12:/ See also ISOPERIMETRIC INEQUALITY References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 23, 1991. Isoperimetric Theorem Of all convex n-gons of a given PERIMETER , the one which maximizes AREA is the regular n-gon. See also ISOPERIMETRIC INEQUALITY ,ISOPERIMETRIC PROBLEM Isopleth EQUIPOTENTIAL CURVE Isoptic Curve For a given curve C, consider the locus of the point P from where the TANGENTS from P to C meet at a fixed given ANGLE . This is called an isoptic curve of the given curve. Curve Isoptic CYCLOID curtate or prolate CYCLOID EPICYCLOID EPITROCHOID HYPOCYCLOID HYPOTROCHOID PARABOLA HYPERBOLA SINUSOIDAL SPIRAL SINUSOIDAL SPIRAL See also ORTHOPTIC CURVE References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 58 /C1/59 and 206, 1972. Yates, R. C. "Isoptic Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 138 /C1/ 140, 1952. Isosceles Tetrahedron A nonregular TETRAHEDRON in which each pair of opposite EDGES are equal such that all triangular faces are congruent. A TETRAHEDRON is isosceles IFF the sum of the face angles at each VERTEX is 1808, and IFF its INSPHERE and CIRCUMSPHERE are concentric. The only way for all the faces of a general TETRA- HEDRON to have the same PERIMETER or to have the same AREA is for them to be fully congruent, in which case the tetrahedron is isosceles. If the CIRCUMCEN- TER and the INCENTER of a general TETRAHEDRON coincide, then the TETRAHEDRON is isosceles (Altshil- ler-Court 1930, p. 97). See also CIRCUMSPHERE ,INSPHERE ,ISOSCELES TRI- ANGLE ,TETRAHEDRON References Altshiller-Court, N. "The Isosceles Tetrahedron." §4.6b in Modern Pure Solid Geometry. New York: Chelsea, pp. 94 /C1/ 101 and 300, 1979.Biddle, D. Problem 14684. Math. Questions and Solutions from the Educational Times 75, 133 /C1/136, 1901. Biddle, D. Mathesis , p. 91, 1931. Brown, B. H. "Theorem of Bang. Isosceles Tetrahedra." Amer. Math. Monthly 33, 224 /C1/226, 1926. Gentry, E. "Exercices sur le te´trae`dre." Nouvelles ann. de math. 37, 223 /C1/225, 1878. Honsberger, R. "A Theorem of Bang and the Isosceles Tetrahedron." Ch. 9 in Mathematical Gems II. Washing- ton, DC: Math. Assoc. Amer., pp. 90 /C1/97, 1976. Jacobi, C. F. A. In Swinden, J. H. Elemente. p. 457, 1834. Lemoine, E. "Quelques the´ore`mes sur les te´trae`dres dont les areˆtes oppose ´es sont e´gales deux a deux, et solution de la question 1272." Nouvelle ann. de math. 39, 133 /C1/138, 1880. Lemoine, E. Z. Math. u. Physik 29, 321, 1884. Monge, G. Corresp. sur l’E´ cole Polytech. , pp. 1 /C1/6, 1809. Monge, G. Arts. 7 and 8. Ann. de math. 1, 355, 1810 /C1/1811. Morley, F. "Problem 12032." Math. Questions and Solutions from the Educational Times 61,26/C1/27, 1894. Isosceles Trapezoid A TRAPEZOID in which the base angles are equal. See also TRAPEZOID References Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 83, 1998. Isosceles Triangle ATRIANGLE with two equal sides (and two equal ANGLES ). The name derives from the Greek iso(same) and skelos (LEG). The height of the above isosceles triangle can be found from the P YTHAGOREAN THEO- REM as h/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C281 4a2q : (1) The AREA is therefore given by A /C301 2ah /C3012affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C281 4a2q : (2) There is a surprisingly simple relationship between the AREA and VERTEX ANGLE u: As shown in the above diagram, simple TRIGONOMETRY gives h /C30R cos12 u1CA}1CA$ (3) a /C30R sin12 u1CA}1CA$ ; (4) so the AREA is A /C301 2(2a)h /C30ah /C30R2 cos12 u1CA}1CA$ sin12u1CA}1CA$ /C3012R2 sin u : (5) Erecting similar isosceles triangles on the edges of an initial triangle DABC gives another triangle DA?B ?C? such that AA?; BB?; and CC ? concur. The triangles are therefore PERSPECTIVE TRIANGLES . No set of n /C216 points in the PLANE can determine only ISOSCELES TRIANGLES . See also ACUTE TRIANGLE ,EQUILATERAL TRIANGLE , INTERNAL BISECTORS PROBLEM ,ISOSCELES TETRAHE- DRON ,ISOSCELIZER ,K IEPERT’S PARABOLA ,O BTUSE TRIANGLE ,POINT PICKING ,PONS ASINORUM ,RIGHT TRIANGLE ,S CALENE TRIANGLE ,S TEINER- LEHMUS THEOREMIsoscelizer An isoscelizer of an (interior) ANGLE A in a TRIANGLE DABC is a LINE through points IABIACwhere IABlies on AB and IACon AC such that DAIABIACis an ISOSCELES TRIANGLE . An isoscelizer is therefore a line perpendicular to an ANGLE BISECTOR , and if the angle is A, the line is known as an A-isoscelizer. There are obviously an infinite number of isoscelizers for any given angle. Isoscelizers were invented by P. Yff in 1963. Through any point P draw the line parallel to BC as well as the corresponding ANTIPARALLEL . Then the A- isoscelizer through P bisects the angle formed by the parallel and the antiparallel. Another way of saying this is that an isoscelizer is a line which is both parallel and antiparallel to itself (P. Yff). Let u1 /C30 u1x ;u1y1CC1CA and u2 /C30 u2x ;u2y1CC1CA be the unit vectors from a given vertex v /C30 vx ;vy1CC1CA ; let X /C30(x;y) be a point in the interior of a triangle through which an isoscelizer passes, and the side lengths of the isosceles triangle be l. Then setting the POINT-LINE DISTANCE from the vector u1;u2 ðÞ to the point xequal to 0 gives y2/C28y1 ðÞ x0/C28x1 ðÞ /C28x2/C28x1 ðÞ y0/C28y1 ðÞ /C300 (1) lu2y/C28u1y1CC1CA x/C28vx ðÞ /C28lu1x ½/C138 /C28lu2x/C28u1x ðÞ y/C28vy1CC1CA /C28lu1y1C|1Cffl /C300 (2) l/C30x/C28vx ðÞ u2y/C28u1y1CC1CA /C28y/C28vy1CC1CA u2x/C28u1x ðÞ u1xu2y/C28u2xu1y:(3) See also ANGLE BISECTOR ,ANTIPARALLEL ,CONGRU- ENT ISOSCELIZERS POINT ,ISOSCELES TRIANGLE ,YFF CENTER OF CONGRUENCE ,YFF CENTRAL TRIANGLE Isospectral Manifolds DRUMS that sound the same, i.e., have the same eigenfrequency spectrum. Two drums with differing AREA , PERIMETER ,or GENUS can always be distin- guished. However, Kac (1966) asked if it was possible to construct differently shaped drums which have the same eigenfrequency spectrum. This question was answered in the affirmative by Gordon et al. (1992). Two such isospectral manifolds are shown in the right figure above (Cipra 1992). Furthermore, pairs of separate drums (having the same total area) can be constructed which have the same eigenfrequency spectrum when played together (illustrated above). Therefore, you cannot hear the shape of a two-piece band (Zwillinger 1995, p. 426). References Chapman, S. J. "Drums That Sound the Same." Amer. Math. Monthly 102, 124 /C1/138, 1995. Cipra, B. "You Can’t Hear the Shape of a Drum." Science 255, 1642 /C1/1643, 1992. Gordon, C.; Webb, D.; and Wolpert, S. "Isospectral Plane Domains and Surfaces via Riemannian Orbifolds." Invent. Math. 110,1/C1/22, 1992. Gordon, C.; Webb, D.; and Wolpert, S. "You Cannot Hear the Shape of a Drum." Bull. Amer. Math. Soc. 27, 134 /C1/138, 1992. Kac, M. "Can One Hear the Shape of a Drum?" Amer. Math. Monthly 73,1/C1/23, 1966. Zwillinger, D. (Ed.). "Eigenvalues." §5.8 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 425 /C1/426, 1995. Isothermal Parameterization A parameterization is isothermal if, for z /C13u /C27iv and fk(z) /C30@xk @u/C28i@xk @v; the identity f2 1( z) /C27 f22(z) /C27 f23( z) /C300 holds.See also MINIMAL SURFACE ,TEMPERATURE References Osserman, R. "Isothermal Parameters." §4in A Survey of Minimal Surfaces. New York: Dover, pp. 27 /C1/33, 1986. Isotomic Conjugate Point The point of concurrence Q of the ISOTOMIC LINES relative to a point P. The isotomic conjugate a? : b? : g? of a point with TRILINEAR COORDINATES a : b : g is a2 a1CC1CA/C281: b2 b1CC1CA/C281: c2 g1CC1CA/C281: (1) The isotomic conjugate of a LINE d having trilinear equation la /C27mb /C27ng (2) is a CONIC SECTION circumscribed on the TRIANGLE DABC (Casey 1893, Vandeghen 1965). The isotomic conjugate of the LINE AT INFINITY having trilinear equation a a /C27bb /C27c g /C300 (3) is STEINER’S ELLIPSE b? g ? a/C27g ?a? b/C27a?b? c/C300 (4) (Vandeghen 1965). The type of CONIC SECTION to which d is transformed is determined by whether the line d meets STEINER’S ELLIPSE E. 1. If d does not intersect E, the isotomic transform is an ELLIPSE . 2. If d is tangent to E, the transform is a PARABOLA . 3. If d cuts E, the transform is a HYPERBOLA , which is a RECTANGULAR HYPERBOLA if the line passes through the isotomic conjugate of the ORTHOCENTER (Casey 1893, Vandeghen 1965). There are four points which are isotomically self- conjugate: the CENTROID M and each of the points of intersection of lines through the VERTICES PARALLEL to the opposite sides. The isotomic conjugate of the EULER LINE is called JERABEK’S HYPERBOLA (Casey 1893, Vandeghen 1965). Vandeghen (1965) calls the transformation taking points to their isotomic conjugate points the C EVIAN TRANSFORM . The product of isotomic and ISOGONAL is aCOLLINEATION which transforms the sides of a TRIANGLE to themselves (Vandeghen 1965). See also CEVIAN TRANSFORM ,G ERGONNE POINT , ISOGONAL CONJUGATE ,JERABEK’S HYPERBOLA ,N A- GEL POINT ,STEINER’S ELLIPSE References Casey, J. "Theory of Isogonal and Isotomic Points, and of Antiparallel and Symmedian Lines." Supp. Ch. §1in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 165 /C1/173, 1888. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Eddy, R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Trian- gle." Math. Mag. 67, 188 /C1/205, 1994. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 157 /C1/159, 1929. Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091 /C1/1094, 1965. Isotomic Lines Given a point P in the interior of a TRIANGLE DA1A2A3 ; draw the CEVIANS through P from each VERTEX which meet the opposite sides at P1 ; P2 ; and P3 : Now, mark off point Q1 along side A2A3 such that A3P1 /C30A2Q1 ; etc., i.e., so that Qiand Piare equi- distance from the MIDPOINT of AjAk : The lines A1Q1 ; A2Q2 ; and A3Q3 then coincide in a point Q known as the ISOTOMIC CONJUGATE POINT . See also CEVIAN ,ISOTOMIC CONJUGATE POINT ,M ID- POINT Isotone Map A MAP which is monotone increasing and therefore order-preserving. Isotope To rearrange without cutting or pasting. Isotopy A HOMOTOPY from one embedding of a MANIFOLD M in N to another such that at every time, it is an embedding. The notion of isotopy is category inde- pendent, so notions of topological, piecewise-linear,smooth, isotopy (and so on) exist. When no explicit mention is made, "isotopy" usually means "smooth isotopy." See also AMBIENT ISOTOPY ,REGULAR ISOTOPY Isotropic Line A LINE in the COMPLEX PLANE with SLOPE 9i:/ References Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 121, 1930. Isotropic Tensor A TENSOR which has the same components in all rotated coordinate systems. All rank-0 TENSORS (SCA- LARS ) are isotropic, but no rank-1 TENSORS (VECTORS ) are. The unique rank-2 isotropic tensor is the KRO- NECKER DELTA . The number of isotropic tensors of rank 0, 1, 2, ...are 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (Sloane’s A005043). These numbers are called the Motzkin sum numbers and are given by the RECUR- RENCE RELATION a(n) /C30(n /C28 1)[2a(n /C28 1) /C27 3a(n /C28 2)] n /C27 1 with a(1) /C300 and a(2) /C301:/ Starting at rank 5, SYZYGIES play a role in restricting the number of isotropic tensors. In particular, SYZY- GIES occur at rank 5, 7, 8, and all higher ranks. See also KRONECKER DELTA ,SCALAR ,SYZYGY ,TEN- SOR,VECTOR References Jeffreys, H. and Jeffreys, B. S. "Isotropic Tensors." §3.03 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 87 /C1/89, 1988. Kearsley, E. A. and Fong, J. T. ""Linearly Independent Sets of Isotropic Cartesian Tensors of Ranks up to Eight." J. Res. Nat. Bureau Standards 79B,4 9/C1/58, 1975. Sloane, N. J. A. Sequences A005043/M2587 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Smith G. F. "On Isotropic Tensors and Rotation Tensors of Dimension mand Order n."Tensor, N. S. 19,7 9/C1/88, 1968. Isotropy Group Some elements of a GROUP GACTING on a space X may fix a point x. These group elements form a SUBGROUP called the isotropy group, defined by Gx/C30fg/C23Gsuch that gx/C30xg: For example, consider the group SO(3) of all rotations of a sphere S2:Letxbe the north pole (0 ;0;1):Then a rotation which does not change xmust turn about the usual axis, leaving the north pole and the south pole fixed. These rotations correspond to the action of the circle group S1on the equator. When two points x and y are on the same ORBIT , say y /C30gx, then the isotropy groups are CONJUGATE SUBGROUPS . More precisely, Gy /C30gGxg/C281 : In fact, any subgroup conjugate to Gxoccurs as an isotropy group Gy to some point y on the same orbit as x. See also EFFECTIVE ACTION ,FREE ACTION ,G ROUP ACTION ,M ATRIX GROUP ,ORBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION ,TOPOLOGICAL GROUP ,TRANSITIVE References Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 4 and 49 /C1/ 52, 1987. Isovolume Problem Find the surface enclosing the maximum VOLUME per unit SURFACE AREA , I /C13V =S: The solution is a SPHERE , which has Isphere /C304 3pr3 4 pr2 /C301 3r : The fact that a sphere solves the isovolume problem was only proved as recently as 1882 by Schwarz (Haas 2000). See also DIDO’S PROBLEM ,DOUBLE BUBBLE ,ISOPERI- METRIC PROBLEM ,SPHERE ,SURFACE AREA,VOLUME References Bogomolny, A. "Isoperimetric Theorem and Inequality." http://www.cut-the-knot.com/do_you_know/isoperime- tric.html. Haas, J. "General Double Bubble Conjecture in R3 Solved." Focus: The Newsletter of the Math. Assoc. Amer. , No. 5, pp. 4 /C1/5, May/June 2000. Isenberg, C. "The Maximum Volume Contained by a Closed Surface of Fixed Area." Appendix VI in The Science of Soap Films and Soap Bubbles. New York: Dover, pp. 174 /C1/177, 1992. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 214, 1999. Isthmus BRIDGE Iterated Exponential POWER TOWER Iterated Function System A finite set of contraction maps vi for i /C301, 2, ..., N, each with a contractivity factor s B1, which map a compact METRIC SPACE onto itself. It is the basis for FRACTAL image compression techniques. See also BARNSLEY’S FERN,SELF-SIMILARITYReferences Barnsley, M. F. "Fractal Image Compression." Not. Amer. Math. Soc. 43, 657 /C1/662, 1996. Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993. Barnsley, M. F. and Demko, S. G. "Iterated Function Sys- tems and the Global Construction of Fractals." Proc. Roy. Soc. London, Ser. A 399, 243 /C1/275, 1985. Barnsley, M. F. and Hurd, L. P. Fractal Image Compres- sion. Wellesley, MA: A. K. Peters, 1993. Diaconis, P. M. and Shashahani, M. "Products of Random Matrices and Computer Image Generation." Contemp. Math. 50, 173 /C1/182, 1986. Fisher, Y. Fractal Image Compression. New York: Springer- Verlag, 1995. Hutchinson, J. "Fractals and Self-Similarity." Indiana Univ. J. Math. 30, 713 /C1/747, 1981. Wagon, S. "Iterated Function Systems." §5.2 in Mathematica in Action. New York: W. H. Freeman, pp. 149 /C1/156, 1991. Iterated Radical NESTED RADICAL Iteration The repeated application of a transformation. See also ITERATED FUNCTION SYSTEM ,ITERATION SEQUENCE ,POWER TOWER References Chang, G. and Sederberg, T. W. Over and Over Again. Washington, DC: Math. Assoc. Amer., 1997. Iteration Sequence A SEQUENCE aj1C%1CP of POSITIVE INTEGERS is called an iteration sequence if there EXISTS a strictly INCREAS- ING SEQUENCE skfg of POSITIVE INTEGERS such that a1 /C30s1 ]2 and aj /C30saj/C281for j /C302, 3, .... A NECESSARY and SUFFICIENT condition for aj1C%1CP to be an iteration sequence is aj ]2aj/C281 /C28aj/C282 for all j ]3 :/ References Kimberling, C. "Interspersions and Dispersions." Proc. Amer. Math. Soc. 117, 313 /C1/321, 1993. Itoˆ’s Lemma Let W(u)beaW IENER PROCESS . Then Vt /C28V0 /C30gt 0fx(W(u) ;u)dW(u) /C28gt 0ft(W(u) ;u)du /C271 2gt 0fxx(W(u);u)du; where Vt/C30f(W(t);t) for 0 5t/C13T/C28t5T;and f/C23C2;1((0;/C12)/C29[0;T]):/ See also WIENER PROCESS References Karatsas, I. and Shreve, S. Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer-Verlag, 1997. Price, J. F. "Optional Mathematics is Not Optional." Not. Amer. Math. Soc. 43, 964 /C1/971, 1996. Itoˆ’s Theorem The dimension d of any IRREDUCIBLE REPRESENTA- TION of a GROUP G must be a DIVISOR of the index of each maximal normal ABELIAN SUBGROUP of G. See also ABELIAN GROUP ,IRREDUCIBLE REPRESENTA- TION ,SUBGROUP References Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 55, 1993. Iverson Bracket Let S be a mathematical statement, then the Iverson bracket is defined by [S] /C130i f S is false 1if S is true :1C|} This notation conflicts with the brackets sometimesused to denote the FLOOR FUNCTION . (However, because of the elegant symmetry of the FLOOR FUNC- TION and CEILING FUNCTION symbols xbcand xde; the use of x½/C138to denote the FLOOR FUNCTION should be deprecated.) The Iverson bracket is implemented in Mathematica 4.1 asBoole [S]. See also CEILING FUNCTION ,FLOOR FUNCTION References Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Read- ing, MA: Addison-Wesley, p. 24, 1990. Iverson, K. E. A Programming Language. New York: Wiley, p. 11, 1962. Iwasawa’s Theorem Every finite-dimensional LIE ALGEBRA of character- istic p "0 has a FAITHFUL finite-dimensional repre- sentation. See also ADO’S THEOREM ,LIE ALGEBRA References Jacobson, N. Lie Algebras. New York: Dover, pp. 204 /C1/205, 1979. J j The symbol used by engineers and some physicists to denote I, the IMAGINARY NUMBERffiffiffiffiffiffi /C281p : j is probably preferred over i because the symbol i (or I)is commonly used to denote current. Jack Polynomial References Lasalle, M. "Some Combinatorial Conjectures for Jack Polynomials." Ann. Combin. 2,61/C1/3, 1998. Jackknife See also BOOTSTRAP METHODS ,PERMUTATION TESTS , RESAMPLING STATISTICS Jackson’s Difference Fan If, after constructing a DIFFERENCE TABLE , no clear pattern emerges, turn the paper through an ANGLE of 60 8 and compute a new table. If necessary, repeat the process. Each ROTATION reduces POWERS by 1, so the sequence fkn g multiplied by any POLYNOMIAL in n is reduced to 0s by a k-fold difference fan. References Conway, J. H. and Guy, R. K. "Jackson’s Difference Fans." In The Book of Numbers. New York: Springer-Verlag, pp. 84 /C1/5, 1996. Jackson’s Identity The Q-HYPERGEOMETRIC FUNCTION identity r f ?sa; qffiffiffiap;/C28qffiffiffiap;1=b ;1=c ;1=d;1=e ;1 =fffiffiffiap;/C28ffiffiffiap;abq ;acq ;adq;aeq ;afq/C20/C21 /C30 aqðÞm qaqdeðÞmqadecðÞmqaqcdðÞmq aqcðÞmqaqdðÞmqaqeðÞmqaqcdeðÞmq; where a2bcdefq /C301; /r f?s is a Q-HYPERGEOMETRIC FUNCTION , and one of b, c, d, e,or f is equal to qm (Hardy 1999, pp. 108 /C1/09). This identity includes the DOUGALL- RAMANUJAN IDENTITY as a special case. See also DOUGALL- RAMANUJAN IDENTITY , Q-HYPER- GEOMETRIC FUNCTION References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, pp. 66 /C1/2, 1935.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 109 /C1/10, 1959. Jackson, F. H. "Summation of q-Hypergeometric Series." Messenger Math. 50, 101 /C1/12, 1921. Jackson’s Theorem Jackson’s theorem is a statement about the error En(f) of the best uniform approximation to a REAL FUNCTION fxðÞon [/C281;1] by REAL POLYNOMIALS of degree at most n. Let fxðÞbe of bounded variation in [/C281;1] and let M ? and V ? denote the least upper bound of fxðÞjj and the total variation of fxðÞin [/C281;1]; respectively. Given the function FxðÞ/C30F /C281ðÞ/C27gx /C281fxðÞdx; (1) then the coefficients an /C301 22n /C271 ðÞ g1 /C281FxðÞPnxðÞdx (2) of its LEGENDRE SERIES , where Pn(x)isaL EGENDRE POLYNOMIAL , satisfy the inequalities anjjB6ffiffiffipp M ?/C27V ? ðÞ n/C283 =2for n ]1 4ffiffiffipp M ?/C27V ? ðÞ n/C283 =2for n ]18 >>>< >>>:(3) Moreover, the LEGENDRE SERIES of FxðÞconverges uniformly and absolutely to FxðÞin [/C281; 1]:/ Bernstein strengthened Jackson’s theorem to 2nE2n( a) 54n p 2n /C27 1 ðÞB2 p/C300:6366 : (4) A specific application of Jackson’s theorem shows that if a(x) /C30 xjj; (5) then En(a)56 n: (6) See also LEGENDRE SERIES ,PICONE’S THEOREM References Cheney, E. W. Introduction to Approximation Theory, 2nd ed.Providence, RI: Amer. Math. Soc., 1999. Jackson, D. The Theory of Approximation. New York: Amer. Math. Soc., p. 76, 1930. Rivlin, T. J. An Introduction to the Approximation of Func- tions. New York: Dover, 1981. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 205 /C1/08, 1991. Jacobi Algorithm A method which can be used to solve a TRIDIAGONAL MATRIX equation with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approx- imate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the JACOBI METHOD of matrix diagonaliza- tion. See also JACOBI METHOD ,TRIDIAGONAL MATRIX References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 161 /C1/63, 1990. Jacobi Differential Equation 1/C28x2/C0/C1 yƒ/C27b/C28a/C28(a/C27b/C272)x ½/C138 y? /C27n(n/C27a/C27b/C271)y/C300 (1) or d dx1/C28x ðÞa/C2711/C27x ðÞb/C271y?hi /C27n(n/C27a/C27b/C271) /C21/C28x ðÞa1/C27x ðÞby/C300: (2) The solutions are J ACOBI POLYNOMIALS Pa;bðÞ n(x) or, in terms of hypergeometric functions, as y(x)/C30C12F1/C28n;n/C271/C27a/C27b;1/C27a;1 2(x/C281)/C16/C17 /C272ax/C281 ðÞ/C28aC22F1/C28n/C28a;n/C271/C27b;1/C28a;1 2(1/C28x)/C16/C17 : (3) The equation (2) can be transformed to d2y dx2/C27/C201 41/C28a2 1/C28x ðÞ2/C27141/C28b2 1/C27x ðÞ2 /C27nn/C27a/C27b/C271 ðÞ /C271 2a/C271 ðÞ b/C271 ðÞ 1/C28x2/C21 u/C300;(4) where u(x)/C301/C28x ðÞa/C271 ðÞ =21/C27x ðÞ(b/C271)=2Pa;bðÞ n(x); (5) and d2u du2/C271 4/C28a2 4 sin212u/C16/C17/C2714/C28b2 4 cos212u/C16/C17/C27n/C27a/C27b/C271 2 !22 435u /C300; (6) where u(u)/C30sina/C271=21 2u/C16/C17 cosb/C271=212u/C16/C17 Pa;bðÞ ncosu ðÞ : (7) Zwillinger (1997, p. 123) gives a related differentialequation he terms Jacobi’s equation x(1/C28x)yƒ/C27g/C28(a/C271)x ½/C138 y?/C27na/C27n ðÞ y/C300 (8) (Iyanaga and Kawada 1980, p. 1480), which has solution y/C30C12F1(/C28n;n/C27a;g;x) /C28/C281ðÞ/C28gx1/C28gC22F1(1/C28n/C28g;1/C27n/C27a/C28g;2/C28g;x):(9) Zwillinger (1997, p. 120; duplicated twice) also givesanother types of ordinary differential equation called a Jacobi equation, a 1/C27b1x/C27c1y ðÞ xy?/C28y ðÞ /C28a2/C27b2x/C27c2y ðÞ y? /C27a3/C27b3x/C27c3y ðÞ /C300 (10) (Ince 1956, p. 22). In the CALCULUS OF VARIATIONS , the PARTIAL DIFFER- ENTIAL EQUATION d dxVh?/C28Vh/C30d dxfy?yh/C27fy?yh?/C0/C1 /C28fyyh/C27fyy?h?/C0/C1 /C300;(11) where Vx;h;h? ðÞ /C131 2fyyh2/C272fyy?hh?/C27fy?yh?2/C0/C1 (12) is called the Jacobi differential equation. References Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 162 /C1/63, 1925. Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 22, 1956. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1480, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Jacobi Differential Equation (Calculus of Variations) u(x)/C301/C28x ðÞa/C271 ðÞ =21/C27x ðÞ(b/C271)=2Pa;bðÞ n(x); where d2u du2/C271 4/C28a2 4 sin212u/C16/C17/C2714/C28b2 4 cos212u/C16/C17/C27n/C27a/C27b/C271 2 !22 435u /C300; This equations arises in the CALCULUS OF VARIATIONS . References Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 162 /C1/63, 1925. Jacobi Elliptic Functions The Jacobi elliptic functions are standard forms of ELLIPTIC FUNCTIONS . The three basic functions are denoted cn( u;k);dn(u;k);and sn( u;k);where kis known as the MODULUS . The arise from the inversion of the ELLIPTIC INTEGRAL OF THE FIRST KIND , u/C30F(f;k)/C30gf odtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sin2tp ; (1) where 0 Bk2B1;k/C30mod uis the MODULUS , and f/C30 am(u;k)/C30am(u) is the AMPLITUDE , giving f/C30F/C281(u;k)/C30am(u;k)/C30am(u): (2) From this, it follows that sinf/C30sin(am( u;k))/C30sin(am u)/C30sn(u;k)/C30sn(u) (3) cosf/C30cos(am( u;k))/C30cos(am u)/C30cn(u;k)/C30cn(u) (4) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sin2fq /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sin2(am(u;k))q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2sn2up /C30dn(u;k)/C30dn(u): (5) These functions are doubly periodic generalizations of the trigonometric functions satisfying sn(u;0)/C30sinu (6) cn(u;0)/C30cosu (7) dn(u;0)/C301: (8) In terms of J ACOBI THETA FUNCTIONS , sn(u;k)/C30q3 q4q1uq/C282 3/C0/C1 q4uq/C282 3/C0/C1 (9) cn(u;k)/C30q4 q2q2uq/C282 3/C0/C1 q4uq/C282 3/C0/C1 (10) dn(u;k)/C30q4 q3q3uq/C282 3/C0/C1 q4uq/C282 3/C0/C1 (11) (Whittaker and Watson 1990, p. 492), where qi/C13 qi(0) (Whittaker and Watson 1990, p. 464). Ratios of Jacobi elliptic functions are denoted by combining thefirst letter of the NUMERATOR elliptic function with the first of the DENOMINATOR elliptic function. The multiplicative inverses of the elliptic functions aredenoted by reversing the order of the two letters. These combinations give a total of 12 functions: cd,cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn. The AMPLITUDE fis defined in terms of sn uby y/C30sinf/C30sn(u;k): (12) Thekargument is often suppressed for brevity so, for example, sn( u;k) can be written as sn u:/ The Jacobi elliptic functions are periodic in K(k) and K?(k)a s snu/C272mK/C272niK?;k ðÞ /C30/C28 1ðÞmsn(u;k) (13) cnu/C272mK/C272niK?;k ðÞ /C30/C28 1ðÞm/C27ncn(u;k) (14)dnu/C272mK/C272niK?;k ðÞ /C30/C28 1ðÞndn(u;k); (15) where K(k) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND ,K?(k)/C13Kk?ðÞ;andk?/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p (Whittaker and Watson 1990, p. 503). The cn x;dnx;and sn xfunctions may also be defined as solutions to the differential equations d2y dx2/C30/C28 1/C27k2/C0/C1 y/C272k2y3(16) d2y dx2/C30/C28 1/C282k2/C0/C1 y/C282k2y3(17) d2y dx2/C302/C28k2/C0/C1 y/C282y3: (18) The standard Jacobi elliptic functions satisfy the identities sn2u/C27cn2u/C301 (19) k2sn2u/C27dn2u/C301 (20) k2cn2u/C27k?2/C30dn2u (21) cn2u/C27k?2sn2u/C30dn2u: (22) Special values include cn(0 ;k)/C30cn(0)/C301 (23) cn(K(k);k)/C30cn(K(k))/C300 (24) dn(0 ;k)/C30dn(0)/C301 (25) dn(K(k);k)/C30dn(K(k))/C30k?/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p ; (26) sn(0 ;k)/C30sn(0)/C300 (27) sn(K(k);k)/C30sn(K(k))/C301; (28) where K/C30K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND and k?/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p is the complemen- tary MODULUS (Whittaker and Watson 1990, pp. 498 /C1/ 99), and cn(u;1)/C30sech u (29) dn(u;1)/C30sech u (30) sn(u;1)/C30tanh u: (31) In terms of integrals, u/C30gsnu 01/C28t2/C0/C1/C281=21/C28k2t2/C0/C1/C281=2dt (32) /C30g/C12 nsut2/C281/C0/C1/C281=2t2/C28l2/C0/C1/C281=2dt (33) /C30g1 cnu1/C28t2/C0/C1/C281=2k?2/C27k2t2/C0/C1 /C281=2dt (34) /C30gncu 1t2/C281/C0/C1/C281=2k?2t2/C27k2/C0/C1 /C281=2dt (35) /C30g1 dnu1/C28t2/C0/C1/C281=2t2/C28k?2/C0/C1 /C281=2dt (36) /C30gndu 1t2/C281/C0/C1/C281=21/C28k?2t2/C0/C1 /C281=2dt (37) /C30gscu 01/C27t2/C0/C1/C281=21/C27k?2t2/C0/C1 /C281=2dt (38) /C30g/C12 csut2/C271/C0/C1/C281=2t2/C27k?2/C0/C1 /C281=2dt (39) /C30gsdu 01/C28k?2t2/C0/C1 /C281=21/C27k2t2/C0/C1/C281=2dt (40) /C30g/C12 dsut2/C28k?2/C0/C1 /C281=2t2/C27k2/C0/C1/C281=2dt (41) /C30gcdu 11/C28t2/C0/C1/C281=21/C28k2t2/C0/C1/C281=2dt (42) /C30g1 dcut2/C281/C0/C1/C281=2t2/C28k2/C0/C1/C281=2dt (43) (Whittaker and Watson 1990, p. 494). Jacobi elliptic functions addition formulas include sn(u/C27v)/C30snucnvdnv/C27snvcnudnu 1/C28k2sn2usn2v(44) cn(u/C27v)/C30cnucnv/C28snusnvdnudnv 1/C28k2sn2usn2v(45) dn(u/C27v)/C30dnudnv/C28k2snusnvcnucnv 1/C28k2sn2usn2v:(46) Extended to integral periods, sn(u/C27K)/C30cnu dnu(47) cn(u/C27K)/C30k?snu dnu(48) dn(u/C27K)/C30k? dnu(49) sn(u/C272K)/C30/C28snu (50) cnðuþ2KÞ¼/C28cnu ð51Þ dn(u/C272K)/C30dnu (52) For COMPLEX arguments,sn(u/C27iv)/C30sn(u;k)d n v;k? ðÞ 1/C28dn2(u;k)s n2v;k? ðÞ /C27icn(u;k) dn( u;k)s nv;k? ðÞ cnv;k? ðÞ 1/C28dn2(u;k)s n2v;k? ðÞ(53) cn(u/C27iv)/C30cn(u;k)c nv;k? ðÞ 1/C28dn2(u;k)s n2v;k? ðÞ /C27isn(u;k) dn( u;k)s nv;k? ðÞ dnv;k? ðÞ 1/C28dn2(u;k)s n2v;k? ðÞ(54) dn(u/C27iv)/C30dn(u;k)c nv;k? ðÞ dnv;k? ðÞ 1/C28dn2(u;k)s n2v;k? ðÞ /C27ik2sn(u;k) cn(u;k)s nv;k? ðÞ 1/C28dn2(u;k)s n2v;k? ðÞ(55) DERIVATIVES of the Jacobi elliptic functions include dsnu du/C30cnudnu (56) dcnu du/C30/C28snudnu (57) ddnu du/C30/C28k2snucnu (58) (Hille 1969, p. 66; Zwillinger 1997, p. 136).Double-period formulas involving the Jacobi elliptic functions include sn(2u)/C302s n ucnudnu 1/C28k2sn4u(59) cn(2u)/C301/C282s n2u/C27k2sn4u 1/C28k2sn4u(60) dn(2u)/C301/C282k2sn2u/C27k2sn4u 1/C28k2sn4u: (61) Half-period formulas involving the Jacobi ellipticfunctions include sn 1 2K/C16/C17 /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27k?p (62) cn1 2K/C16/C17 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k? 1/C27k?s (63) dn12K/C16/C17 /C30ffiffiffiffi k?p : (64) Squared formulas include sn2u/C301/C28cn(2u) 1/C27dn(2u)(65) cn2u/C30dn(2u)/C27cn(2u) 1/C27dn(2u)(66) dn2 u /C30dn(2u) /C27 cn(2u) 1 /C27 cn(2u): (67) See also AMPLITUDE ,E LLIPTIC FUNCTION ,JACOBI DIFFERENTIAL EQUATION ,JACOBI’S IMAGINARY TRANS- FORMATION ,JACOBI FUNCTION OF THE SECOND KIND, JACOBI THETA FUNCTIONS ,W EIERSTRASS ELLIPTIC FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Jacobian Elliptic Functions and Theta Functions." Ch. 16 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 567 /C1/81, 1972. Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961. Hille, E. Lectures on Ordinary Differential Equations. Reading, MA: Addison-Wesley, 1969. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 433, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Elliptic Integrals and Jacobi Elliptic Func- tions." §6.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 254 /C1/63, 1992. Spanier, J. and Oldham, K. B. "The Jacobian Elliptic Functions." Ch. 63 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 635 /C1/52, 1987. To¨lke, F. "Jacobische elliptische Funktionen und zugeho ¨rige logarithmische Ableitungen," "Umkehrfunktionen der Ja- cobischen elliptischen Funktionen und elliptische Normal- integrale erster Gattung. Elliptische Amplitudenfunktionen sowie Legendresche F- und E- Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und Heumansche Lambda-Funktionen," and "Normalintegrale dritter Gattung. Legendresche P/- Funktion. Zuru¨ckfu¨hrung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Chs. 5 /C1/ in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 1 /C1/44, 1967. To¨lke, F. Praktische Funktionenlehre, vierter Band: Ellip- tische Integralgruppen und Jacobische elliptische Funk- tionen im Komplexen. Berlin: Springer-Verlag, 1967. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Jacobi Function of the First Kind JACOBI POLYNOMIAL Jacobi Function of the Second Kind Q a; bðÞ n(x) /C302/C28n/C281 x /C281 ðÞ/C28ax /C271 ðÞ/C28b /C29g1 /C2811 /C28t ðÞn/C27a1 /C27t ðÞn/C27 bx /C28t ðÞ/C28n/C281dt: In the exceptional case n /C300, a/C27 b /C271 /C300 ; a non- constant solution is given by Q aðÞ(x) /C30ln(x /C271) /C27p/C281 sin paðÞ x /C281 ðÞ/C28ax /C271 ðÞ/C28 b/C29g1 /C2811 /C28 t ðÞa1 /C27 t ðÞb x /C28 tln(1 /C27t)dt: See also JACOBI DIFFERENTIAL EQUATION ,JACOBI POLYNOMIAL References Szego, G. "Jacobi Polynomials." Ch. 4 in Orthogonal Poly- nomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 73 /C1/9, 1975. Jacobi Identities "The" Jacobi identity is a relationship [A;[B; C]] /C27[B;[C ;A]] /C27[C ;[A;B]] /C300; (1) between three elements A, B, and C, where [A, B]is the COMMUTATOR . The elements of a LIE ALGEBRA satisfy this identity. Relationships between the Q-FUNCTIONS Qiare also known as Jacobi identities: Q1Q2Q3 /C301 ; (2) equivalent to the JACOBI TRIPLE PRODUCT (Borwein and Borwein 1987, p. 65) and Q82 /C3016qQ81 /C27Q83 ; (3) where q /C13e /C28pK ? kðÞ=KkðÞ; (4) /K /C30K(k) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND , and K ?(k) /C30Kk?ðÞ/C30Kffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2p/C16/C17 :Using WEBER FUNCTIONS f1/C30q/C281=24Q3 (5) f2/C3021=2q1=12Q1 (6) f/C30q/C281=24Q2; (7) (5) and (6) become f1f2f/C30ffiffiffi 2p (8) f8/C30f8 1/C27f8 2 (9) (Borwein and Borwein 1987, p. 69). See also COMMUTATOR ,JACOBI TRIPLE PRODUCT , PARTITION FUNCTION Q, Q-FUNCTION ,W EBER FUNC- TIONS References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 3, 1996. Jacobi Matrix JACOBI ROTATION MATRIX ,JACOBIAN Jacobi Method A method of diagonalizing a MATRIX A using JACOBI ROTATION MATRICES Ppq : It consists of a sequence of ORTHOGONAL SIMILARITY TRANSFORMATIONS OF THE FORM A ?/C30PT pqAPpq ; each of which eliminates one off-diagonal element. Each application of Ppq affects only rows and columns of A ; and the sequence of such matrices is chosen so as to eliminate the off-diagonal elements. See also JACOBI ALGORITHM ,JACOBI ROTATION MA- TRIX References Gentle, J. E. "Givens Transformations (Rotations)." §3.2.5 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer-Verlag, pp. 99 /C1/02, 1998. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Jacobi Transformation of a Symmetric Ma- trix." §11.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 456 /C1/62, 1992. Jacobi Polynomial Also known as the HYPERGEOMETRIC POLYNOMIALS , they occur in the study of ROTATION GROUPS and in the solution to the equations of motion of the symmetric top. They are solutions to the J ACOBI DIFFERENTIAL EQUATION . Plugging y/C30X/C12 n/C300anx/C281 ðÞv(1) into the differential equation gives the RECURRENCE RELATION g/C28n(n/C27a/C27b/C271) ½/C138 an/C282(n/C271)(n/C27a/C271)an/C271/C300 (2) forn/C300;1, ..., where g/C13n(n/C27a/C27b/C271): (3) Solving the RECURRENCE RELATION gives Pa/C27b ðÞ n(x)/C30/C281ðÞn 2nn!1/C28x ðÞ/C28a1/C27x ðÞ/C28bdn dxn /C21/C28x ðÞa/C27n1/C27x ðÞb/C27nhi (4) fora;b>/C281:They form a complete orthogonal system in the interval [ /C281;1] with respect to the weighting function wn(x)/C301/C28x ðÞa1/C27x ðÞb; (5) and are normalized according toPa;bðÞn(1)/C30n/C27a n/C18/C19 ; (6) wheren k/C0/C1 is a BINOMIAL COEFFICIENT . Jacobi poly- nomials can also be written Pa;b n/C30G(2n/C27a/C27b/C271) n!G(n/C27a/C27b/C271)Gna/C27b/C271;b/C271;1 2(x/C271)/C16/C17 ; (7) where G(z) is the GAMMA FUNCTION and Gn(p;q;x)/C13n!G(n/C27p) G(2n/C27p)Pp/C28q;q/C281 ðÞ n (2x/C281): (8) Jacobi polynomials are ORTHOGONAL satisfying g1 /C281Pa;bðÞ mPa;bðÞn1/C28x ðÞa1/C27x ðÞbdx /C302a/C27b/C271 2n/C27a/C27b/C271G(n/C27a/C271)G(n/C27b/C271) n!G(n/C27a/C27b/C271)dmn:(9) The COEFFICIENT of the term xninPa;bðÞ n(x) is given by An/C30G(2n/C27a/C27b/C271) 2nn!G(n/C27a/C27b/C271): (10) They satisfy the RECURRENCE RELATION 2(n/C271)(n/C27a/C27b/C271)(2n/C27a/C27b)Pa;bðÞ n/C271xðÞ /C30(2n/C27a/C27b/C271)a2/C28b2/C0/C1 /C272n/C27a/C27b ðÞ3x/C2/C3 Pa;bðÞ nxðÞ /C282(n/C27a)(n/C27b)(2n/C27a/C27b/C272)Pa;bðÞ n/C281xðÞ; (11) where mðÞnis the RISING FACTORIAL mðÞn/C13m(m/C271)/C1/C1/C1(m/C27n/C281)/C30(m/C27n/C281)! (m/C281)!:(12) The DERIVATIVE is given by d dxPa;bðÞ nxðÞ/C2/C3 /C301 2n/C27a/C27b/C271 ðÞ Pa/C271;b/C271 ðÞ n/C281 xðÞ: (13) The ORTHOGONAL POLYNOMIALS with WEIGHTING FUNCTION b/C28x ðÞax/C28a ðÞbon the CLOSED INTERVAL [a, b] can be expressed in the form const : ½/C138 Pa;bðÞ n 2x/C28a b/C28a/C281 ! (14) (Szego 1975, p. 58). Special cases with a/C30bare Pa;aðÞ 2nxðÞ/C30G(2n/C27a/C271)G(n/C271) G(n/C27a/C271)G(2n/C271)Pa;/C281=2 ðÞ n 2x2/C281/C0/C1 (15) /C30/C28 1ðÞnG(2n/C27a/C271)G(n/C271) G(n/C27a/C271)G(2n/C271)P/C281=2;a ðÞn1/C282x2/C0/C1 (16) Pa;aðÞ 2n/C271xðÞ/C30G(2n/C27a/C272)G(n/C271) G(n/C27a/C271)G(2n/C272)xPa;1=2 ðÞ n 2x2/C281/C0/C1 (17) /C30/C28 1ðÞnG(2n/C27a/C272)G(n/C271) G(n/C27a/C271)G(2n/C272)xP1=2;a ðÞn 1/C282x2/C0/C1 :(18) Further identities are Pa/C271;b ðÞ n xðÞ/C302 2n/C27a/C27b/C272 /C2n/C27a/C271 ðÞ Pa;bðÞ n/C28n/C271 ðÞ Pa;bðÞ n/C271xðÞ 1/C28x (19) Pa/C27b/C271 ðÞ n xðÞ/C302 2n/C27a/C27b/C272 /C2n/C27b/C271 ðÞ Pa;bðÞ nxðÞ/C27n/C271 ðÞ Pa;bðÞ n/C271xðÞ 1/C27x (20) Xn n/C3002n/C27a/C27b/C271 2a/C27b/C271 /C2G(n/C271)G(n/C27a/C27b/C271) G(n/C27a/C271)G(n/C27b/C271)Pa;bðÞ nxðÞQa;bðÞnyðÞ /C301 2y/C281 ðÞ/C2ay/C271 ðÞ/C2b y/C28x/C272/C2a/C2b 2n/C27a/C27b/C272 /C2G(n/C272)G(n/C27a/C27b/C272) G(n/C27a/C271)G(n/C27b/C271) /C29Pa;bðÞ n/C271xðÞQa;bðÞ nyðÞ/C28Pa;bðÞnxðÞQa;b n/C271yðÞ x/C28y(21) (Szego 1975, p. 79). The KERNEL POLYNOMIAL is Ka;bðÞ n(x;y)/C302/C2a/C2b 2n/C27a/C27b/C272 /C2G(n/C272)G(n/C27a/C27b/C272) G(n/C27a/C271)G(n/C27b/C271) /C29Pa;bðÞ n/C271xðÞPa;bðÞ nyðÞ/C28Pa;bðÞnxðÞPa;bðÞ n/C271yðÞ x/C28y(22) (Szego 1975, p. 71). The DISCRIMINANT is Da;bðÞ n/C302/C28nn/C281 ðÞYn n/C301nn/C282n/C272n/C27a ðÞn/C281n/C27b ðÞn/C281 /C2n/C27n/C27a/C27b ðÞn/C28n(23) (Szego 1975, p. 143). Fora/C30b/C300;P0;0ðÞ nxðÞreduces to a L EGENDRE POLY- NOMIAL . The G EGENBAUER POLYNOMIALGn(p;q;x)/C30n!G(n/C27p) G(2n/C27p)Pp/C28q;q/C281 ðÞ n 2x/C281 ðÞ (24) and C HEBYSHEV POLYNOMIAL OF THE FIRST KIND can also be viewed as special cases of the Jacobi poly- nomials. In terms of the HYPERGEOMETRIC FUNCTION , Pða;bÞ nðxÞ¼nþa n/C18/C19 2F1ð/C28n;nþaþb;aþ1;1 2ð1/C28xÞÞ (25) /C30a/C271 ðÞn n!2F1/C28n;n/C27a/C27b;a/C271;1 21/C28x ðÞ/C16/C17 (26) /C30n/C27a n/C18/C19x/C271 2 !2 /C22F1/C28n;/C28n/C28b;a/C271;x/C281 x/C271 ! ; (27) where aðÞnis the P OCHHAMMER SYMBOL (Koekoek 1998). LetN1be the number of zeros in x/C23(/C281;1);N2the number of zeros in x/C23(/C28/C12;/C281);andN3the number of zeros in x/C23(1;/C12):Define Klein’s symbol E(u)/C300i f u50 ubc ifupositive and nonintegral u/C281i f u/C301;2... ;8 < :(28) where xbcis the FLOOR FUNCTION , and X(a;b)/C30E1 22n/C27a/C27b/C271 jj /C28ajj/C28bjj/C271 ðÞhi (29) Y(a;b)/C30E12/C282n/C27a/C27b/C271 jj /C27ajj/C28bjj/C271 ðÞhi (30) Z(a;b)/C30E12/C282n/C27a/C27b/C271 jj /C28ajj/C27bjj/C271 ðÞhi : (31) If the cases a/C30/C281;/C282, ...,/C28n;b/C30/C281;/C282, ...,/C28n;and n/C27a/C27b/C30/C281;/C282, ..., /C28nare excluded, then the number of zeros of Pa;bðÞ nin the respective intervals are N1a;bðÞ /C3021 2X/C271 ðÞjk for/C281ðÞnn/C27a n/C18/C19 n/C27b n/C18/C19 >0 21 2Xjk /C271 for /C281ðÞnn/C27a n/C18/C19 n/C27b n/C18/C19 B08 >>< >>:(32) N 2a;bðÞ /C3021 2Y/C271 ðÞjk for2n/C27a/C27b n/C18/C19 n/C27b n/C18/C19 >0 21 2Yjk /C271 for2n/C27a/C27b n/C18/C19 n/C27b n/C18/C19 B08 >>< >>:(33) N 3a;bðÞ /C3021 2Z /C271 ðÞjk for2n /C27 a /C27 b n/C18/C19 n /C27 a n/C18/C19 > 0 21 2Zjk /C271 for2n /C27 a /C27 b n/C18/C19 n /C27 a n/C18/C19 B08 >>< >>:(34) (Szego 1975, pp. 144 /C1 /46). The first few POLYNOMIALS are P a ; bðÞ 0 xðÞ/C301 (35) P a ;bðÞ1 xðÞ/C301 2 2 a /C271 ðÞ /C27 a /C27 b /C272 ðÞ x /C281 ðÞ ½/C138 (36) P a ; bðÞ 2 xðÞ/C301 8½4 a /C271 ðÞ2ðÞ/C274 a /C27 b /C273 ðÞ a /C272 ðÞ x /C281 ðÞ /C27 a /C27 b /C273 ðÞ 2ðÞðx /C281)2 /C138; (37) where mðÞnis a RISING FACTORIAL (Abramowitz and Stegun 1972, p. 793). See Abramowitz and Stegun (1972, pp. 782 /C1/93) and Szego (1975, Ch. 4) for additional identities. See also CHEBYSHEV POLYNOMIAL OF THE FIRST KIND, GEGENBAUER POLYNOMIAL ,JACOBI FUNCTION OF THE SECOND KIND,RISING FACTORIAL ,ZERNIKE POLYNO- MIAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771 /C1/02, 1972. Andrews, G. E.; Askey, R.; and Roy, R. "Jacobi Polynomials and Gram Determinants" and "Generating Functions for Jacobi Polynomials." §6.3 and 6.4 in Special Functions. Cambridge, England: Cambridge University Press, pp. 293 /C1/06, 1999. Iyanaga, S. and Kawada, Y. (Eds.). "Jacobi Polynomials." Appendix A, Table 20.V in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1480, 1980. Koekoek, R. and Swarttouw, R. F. "Jacobi." §1.8 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Uni- versiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, pp. 38 /C1/4, 1998. ftp://www.twi.- tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1/ 7.ps.gz. Roman, S. "The Theory of the Umbral Calculus I." J. Math. Anal. Appl. 87,58/C1/15, 1982. Szego, G. "Jacobi Polynomials." Ch. 4 in Orthogonal Poly- nomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. Jacobi Quadrature JACOBI- GAUSS QUADRATURE Jacobi Rotation Matrix A MATRIX used in the JACOBI TRANSFORMATION method of diagonalizing MATRICES . The Jacobi rota- tion matrix Ppqcontains 1s along the DIAGONAL , except for the two elements cos f in rows and columns p and q. In addition, all off-diagonal ele- ments are zero except the elements sin f and /C28sin f: The rotation angle f for an initial matrix A is chosen such thatcot(2 f) /C30aqq /C28 app 2apq: Then the corresponding Jacobi rotation matrix which annihilates the off-diagonal element apq is Ppq /C1310::: n U cos f /C1/C1/C1 0 /C1/C1/C1 sin f /C1/C1/C1 0 /C1/C1/C1 1 /C1/C1/C1 0 /C1/C1/C1 /C28sin f /C1/C1/C1 0 /C1/C1/C1 cos f U n::: 012 6666666643 777777775 See also J ACOBI TRANSFORMATION References Gentle, J. E. "Givens Transformations (Rotations)." §3.2.5 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer-Verlag, pp. 99 /C1/02, 1998. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Jacobi Transformation of a Symmetric Ma-trix." §11.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 456 /C1 /62, 1992. Jacobi Symbol The product of L EGENDRE SYMBOLS n=pi ðÞ for each of the PRIME FACTORS pisuch that m/C30Q ipi;denoted n=mðÞ orn m/C16/C17 :When mis a PRIME , the Jacobi symbol reduces to the L EGENDRE SYMBOL . (The Legendre symbol is equal to 91 depending on whether mis a QUADRATIC RESIDUE modulo m.) Analogously to the Legendre symbol, the Jacobi symbol is commonly generalized to have value n m ! /C300i fmn ;j (1) giving n n ! /C300 (2) as a special case. Note that the Jacobi symbol is not defined form50o r mEVEN . The Jacobi symbol is implemented in Mathematica asJacobiSymbol [n, m]. Use of the Jacobi symbol provides the generalization of the QUADRATIC RECIPROCITY THEOREM m n ! n m ! /C30/C28 1ðÞm/C281 ðÞ n/C281 ðÞ =4(3) formand nRELATIVELY PRIME ODD INTEGERS with n]3 (Nagell 1951, pp. 147 /C1/48). Written another way, m n ! /C30/C28 1ðÞm/C281 ðÞ n/C281 ðÞ =4n m ! (4) or n m ! /C30m n ! for m or n /C131 mod 4 ðÞ /C28m n ! for m; n /C133 mod 4 ðÞ:8 >>>>< >>>>:(5) The Jacobi symbol satisfies the same rules as the L EGENDRE SYMBOL n m ! n m? ! /C30n mm? ðÞ ! (6) n m ! n? m ! /C30nn?ðÞ m ! (7) n2 m ! /C30n m2 ! /C301i f( m;n) /C301 (8) n m ! /C30n? m ! if n /C13n? mod m ðÞ (9) /C281 m ! /C30/C28 1ðÞm/C281 ðÞ =2/C301 for m /C131 mod 4 ðÞ /C281 for m /C13/C281 mod 4 ðÞ/C26 (10) 2 m ! /C30/C28 1ðÞm2/C281ðÞ =8/C301 for m /C1391 mod 8 ðÞ /C281 for m /C1393 mod 8 ðÞ/C26 (11) Bach and Shallit (1996) show how to compute the Jacobi symbol in terms of the SIMPLE CONTINUED FRACTION of a RATIONAL NUMBER n=m:/ See also KRONECKER SYMBOL ,LEGENDRE SYMBOL , QUADRATIC RESIDUE References Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, pp. 343 /C1/44, 1996. Guy, R. K. "Quadratic Residues. Schur’s Conjecture." §F5 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 244 /C1/45, 1994. Nagell, T. "Jacobi’s Symbol and the Generalization of the Reciprocity Law." §42 in Introduction to Number Theory. New York: Wiley, pp. 145 /C1/49, 1951. Riesel, H. "Jacobi’s Symbol." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨u- ser, pp. 281 /C1/84, 1994. Jacobi Tensor Jm nab/C30Jm nba/C131 2Rm anb/C27Rmbna/C16/C17 ; where Ris the R IEMANN TENSOR . See also RIEMANN TENSORJacobi Theta Function THETA FUNCTIONS Jacobi Theta Functions The Jacobi theta functions are the elliptic analogs of the EXPONENTIAL FUNCTION , and may be used to express the J ACOBI ELLIPTIC FUNCTIONS . The theta functions are quasi-doubly periodic, and are most commonly denoted qnz;qðÞ in modern texts, although the notations Unz;qðÞ and unz;qðÞ (Borwein and Borwein 1987) are sometimes also used. Whittakerand Watson (1990, p. 487) gives a table summarizing notations used by various earlier writers. The theta functions are given in Mathematica byEllip- ticTheta [n,z,q]. The theta functions may be expressed in terms of the NOME q, denoted qnz;qðÞ ;or the HALF-PERIOD RATIO t; denoted qnztjÞ; ð where qjjB1 and qandtare related by q/C13eipt: ð1Þ Let the many-valued function qlbe interpreted to stand for elpit:Then for a complex number z, the Jacobi theta functions are defined as q1z;qðÞ/C13X/C12 n/C30/C28/C12/C281ðÞn/C281=2qn/C271=2 ðÞ2e2n/C271 ðÞ iz(2) q2z;qðÞ/C13X/C12 n/C30/C28/C12qn/C271=2 ðÞ2e2n/C271 ðÞ iz(3) q3z;qðÞ/C13X/C12 n/C30/C28/C12qn2e2niz(4) q4z;qðÞ/C13X/C12 n/C30/C28/C12/C281ðÞnqn2e2niz: (5) Writing the doubly infinite sums as singly infinite sums gives the slightly less symmetrical forms q1z;qðÞ/C302X/C12 n/C300/C281ðÞnqn/C271=2 ðÞ2sin[(2 n/C271)z] (6) /C302q1=4X/C12 n/C300/C281ðÞnqnn/C271 ðÞsin[(2 n/C271)z] (7) q2z;qðÞ/C302X/C12 n/C300qn/C271=2 ðÞ2cos[(2 n/C271)z] (8) /C302q1=4X/C12 n/C300qnn/C271 ðÞcos[(2 n/C271)z] (9) q3z;qðÞ/C301/C272X/C12 n/C300qn2cos 2 nzðÞ (10) q4z;qðÞ/C301/C272X/C12 n/C300/C281ðÞnqn2cos(2 nz) (11) (Whittaker and Watson 1990, p. 463 /C1/64). Explicitly writing out the series gives q1z;qðÞ/C302q1=4sinz/C282q9=4sin(3 z)/C272q25=4sin(5 z) /C27. . . (12) q2z;qðÞ/C302q1=4cosz/C272q9=4cos(3 z)/C272q25=4cos(5 z) /C27. . . (13) q3z;qðÞ/C301/C272qcos(2 z)/C272q4cos(4 z)/C272q9cos(6 z) /C27. . . (14) q4z;qðÞ/C301/C282qcos 2 zðÞ/C272q4cos 4 zðÞ/C282q9cos 6 zðÞ /C27. . . (15) (Borwein and Borwein 1987, p. 52; Whittaker and Watson 1990, p. 464). q1(z;q)i sa n ODD FUNCTION of z, while the other three are even functions of z. The following table illustrates the quasi-double per-iodicity of the Jacobi theta functions. /qi//qiz/C27p ðÞ =qizðÞ //qiz/C27tp ðÞ =qizðÞ / /q1/ /C281 //C28N/ /q2/ /C281 N /q3/ 1 N /q4/ 1 //C28N/ Here, N/C13q/C281e/C282iz: (16) The quasi-periodicity can be established as follows for the specific case of q4; q4z/C27p;q ðÞ /C30X/C12 n/C30/C28/C12/C281ðÞnqn2e2nize2nip /C30X/C12 n/C30/C28/C12/C281ðÞnqn2e2niz/C30q4z;qðÞ (17) q4z/C27pt;q ðÞ /C30X/C12 n/C30/C28/C12/C281ðÞnqn2e2nipte2niz /C30X/C12 n/C30/C28/C12/C281ðÞnqn2q2ne2niz /C30/C28q/C281e/C282izX/C12 n/C30/C28/C12/C281ðÞn/C271qn/C271 ðÞ2q2n/C271 ðÞ iz /C30/C28q/C281e/C282izX/C12 n/C30/C28/C12/C281ðÞnqn2q2niz/C30/C28q/C281e/C282izq4z;qðÞ : (18) The Jacobi theta functions can be written in terms of each other: q1z;qðÞ/C30/C28ieiz/C27pit=4q4z/C271 4pt;q/C16/C17 (19) q2z;qðÞ/C30q1z/C271 2p;q/C16/C17 (20) q3z;qðÞ/C30q4z/C271 2p;q/C16/C17 (21) Any Jacobi theta function of given arguments can be expressed in terms of any other two Jacobi theta functions with the same arguments. Define qiqðÞ/C13qiz/C300;q ðÞ (22) to be the Jacobi theta functions with argument z/C300, plotted above. Then the doubly infinite sums (2) to (5)take on the particularly simple forms q1qðÞ/C300 (23) q2qðÞ/C30X/C12 n/C30/C28/C12qn/C271=2 ðÞ2(24) q3qðÞ/C30X/C12 n/C30/C28/C12qn2(25) q4qðÞ/C30X/C12 n/C30/C28/C12/C281ðÞnqn2(26) (Borwein and Borwein 1987, p. 33). The plots above show the Jacobi theta functionsplotted as a function of argument zand NOME q restricted to real values. Particularly beautiful plots are obtained by examin- ing the REAL and IMAGINARY PARTS ofqiz;qðÞ for fixed zin the complex plane for qjjB1;illustrated above. The Jacobi theta functions satisfy an almost bewil-deringly large number of identities involving the fourfunctions, their derivatives, multiples of their argu- ments, and sums of their arguments. Among the unusual identities given by Whittaker and Watson(1990) are q3z;qðÞ/C30q32z;q4/C0/C1 /C27q22z;q4/C0/C1 (27) q3z;qðÞ/C30q32z;q4/C0/C1 /C28q22z;q4/C0/C1 (28) (Whittaker and Watson 1990, p. 464) and q?kz/C27p ðÞ qkz/C27p ðÞ/C30q?kzðÞ qkzðÞ(29) q?kz/C27pg ðÞ qkz/C27pg ðÞ/C30/C282i/C27q?kzðÞ qkzðÞ(30) (Whittaker and Watson 1990, p. 465), for k/C301, ..., 4, where qkzðÞ/C13qkz;qðÞ andqi/C13qi0;qðÞ :A class of identities involving the squares of Jacobi theta func- tions are q2 1zðÞq24/C30q23zðÞq22/C28q22zðÞq23(31) q22zðÞq24/C30q24zðÞq22/C28q21zðÞq23(32) q2 3zðÞq24/C30q24zðÞq23/C28q21zðÞq22(33) q24zðÞq24/C30q23zðÞq23/C28q22zðÞq22(34) (Whittaker and Watson 1990, p. 466). Taking z/C300i n (34) gives the special case q44/C30q43/C28q42; (35) which is the only identity of this type. In addition, q3xðÞ/C30X/C12 n/C30/C28/C12xn2/C301/C272x/C272x4/C272x9/C27. . . (36)q2 3xðÞ/C301 /C274x 1/C28x/C28x3 1/C28x3/C27x5 1/C28x5/C28x7 1/C28x7/C27... ! (37) q4 3xðÞ/C301 /C278x 1/C28x/C272x2 1/C28x2/C273x3 1/C28x3/C274x4 1/C28x4/C27... ! (38) The Jacobi theta functions obey addition rules such as q1y/C27z ðÞ q1y/C28z ðÞ q2 4/C30q23yðÞq22zðÞ/C28q22yðÞq23zðÞ /C30q21yðÞq24zðÞ/C28q24yðÞq21zðÞ (39) q2y/C27z ðÞ q2y/C28z ðÞ q2 4/C30q24yðÞq22zðÞ/C28q21yðÞq23zðÞ /C30q2 2yðÞq24yðÞ/C28q23yðÞq21zðÞ (40) q3y/C27z ðÞ q3y/C28z ðÞ q2 4/C30q24yðÞq23zðÞ/C28q21yðÞq22zðÞ /C30q2 3yðÞq24zðÞ/C28q22yðÞq21zðÞ (41) q4y/C27z ðÞ q4y/C28z ðÞ q24/C30q23yðÞq23zðÞ/C28q22yðÞq22zðÞ /C30q2 4yðÞq24zðÞ/C28q21yðÞq21zðÞ (42) (Whittaker and Watson 1990, p. 487), and q3y/C27z ðÞ q3y/C28z ðÞ q2 2/C30q23yðÞq22zðÞ/C27q24yðÞq21zðÞ /C30q2 2yðÞq23zðÞ/C27q21yðÞq24zðÞ q3y/C27z ðÞ q3y/C28z ðÞ q2 3/C30q21yðÞq21zðÞ/C27q23yðÞq23zðÞ /C30q2 2yðÞq22zðÞ/C28q4yðÞq24zðÞ q4y/C27z ðÞ q4y/C28z ðÞ q2 2/C30q24yðÞq22zðÞ/C27q23yðÞq21zðÞ /C30q22yðÞq24zðÞ/C27q21yðÞq23zðÞ (43) q4y/C27z ðÞ q4y/C28z ðÞ q2 3/C30q24yðÞq23zðÞ/C27q22yðÞq21zðÞ /C30q2 3yðÞq24zðÞ/C27q21yðÞq22zðÞ (44) (Whittaker and Watson 1990, p. 488). q1y9z ðÞ q2y/C14z ðÞ q3q4 /C30q1yðÞq2yðÞq3zðÞq4zðÞ9q3yðÞq4yðÞq1zðÞq2zðÞ(45) q1y9z ðÞ q3y/C14z ðÞ q2q4 /C30q1yðÞq3yðÞq2zðÞq4zðÞ9q2yðÞq4yðÞq1zðÞq3zðÞ(46) q1y9z ðÞ q4y/C14z ðÞ q2q3 /C30q1yðÞq4yðÞq2zðÞq3zðÞ9q2yðÞq3yðÞq1zðÞq4zðÞ(47) q2y9z ðÞ q3y/C14z ðÞ q2q3 /C30q2yðÞq3yðÞq2zðÞq3zðÞ/C14q1yðÞq4yðÞq1zðÞq4zðÞ(48) q2y9z ðÞ q4y/C14z ðÞ q2q4 /C30q2yðÞq4yðÞq2zðÞq4zðÞ/C14q1yðÞq3yðÞq1zðÞq3zðÞ(49) q3y9z ðÞ q4y9z ðÞ q3q4 /C30q3yðÞq4yðÞq3zðÞq4zðÞ/C14q1yðÞq2yðÞq1zðÞq2zðÞ(50) (Whittaker and Watson 1990, p. 488). There are also a series of DUPLICATION FORMULAS q32zðÞq3 3/C30q43zðÞ/C27q41zðÞ (51) q22zðÞq2q2 4/C30q22zðÞq24zðÞ/C28q21zðÞq23zðÞ (52) q32zðÞq3q2 4/C30q23zðÞq24zðÞ/C28q21zðÞq22zðÞ (53) q42zðÞq3 4/C30q43zðÞ/C28q42zðÞ (54) ¼q4 4ðzÞ/C28q41ðzÞð 55Þ q12zðÞq2q3q4/C302q1zðÞq2zðÞq3zðÞq4zðÞ (56) (Whittaker and Watson 1990, p. 488). Ratios of Jacobi theta function derivatives to the functions themselves have the simple forms q?1zðÞ q1zðÞ/C30cotz/C274X/C12 n/C301q2n 1/C28q2nsin(2 nz) (57) q?2zðÞ q2zðÞ/C30/C28tanz/C274X/C12 n/C301/C281ðÞnq2n 1/C28q2nsin(2 nz) (58) q?3zðÞ q3zðÞ/C304X/C12 n/C301/C281ðÞnqn 1/C28q2nsin(2 nz) (59) q?4zðÞ q4zðÞ/C30X/C12 n/C301q2n/C281sin(2 z) 1/C282q2n/C281cos(2 z)/C27q4n/C282(60) /C30X/C12 n/C3014qnsin(2 nz) 1/C28q2n(61) (Whittaker and Watson 1990, p. 489). The Jacobi theta functions can be expressed as products instead of sums by q1zðÞ/C302Gq1=4sinzY/C12 n/C3011/C282q2ncos(2 z)/C27q4n/C2/C3 (62) q2zðÞ/C302Gq1=4coszY/C12 n/C3011/C272q2ncos(2 z)/C27q4n/C2/C3 (63) q3zðÞ/C30GY/C12 n/C3011/C272q2n/C281cos(2 z)/C27q4n/C282/C2/C3 (64) q4zðÞ/C30GY/C12 n/C3011/C282q2n/C281cos(2 z)/C27q4n/C282/C2/C3 ; (65) whereG/C13Y/C12 n/C3011/C28q2n/C0/C1 (66) (Whittaker and Watson 1990, pp. 469 /C1/70). The Jacobi theta functions satisfy the PARTIAL DIF- FERENTIAL EQUATION 1 4pi@2y @z2/C27@y @t/C300; (67) where y/C13qiztjÞ: ð Ratios of the Jacobi theta functions withq4in the DENOMINATOR also satisfy differential equations d dzq1zðÞ q4zðÞ"# /C30q2 4q2zðÞq3zðÞ q2 4zðÞ(68) d dzq2zðÞ q4zðÞ"# /C30/C28q2 3q1zðÞq3zðÞ q2 4zðÞ(69) d dzq3zðÞ q4zðÞ"# /C30q2 2q1zðÞq2zðÞ q2 4zðÞ(70) JACOBI’S IMAGINARY TRANSFORMATION expresses qiz=t/C281=t j Þ ð in terms of qiztjÞ: ð There are a large number of beautiful identities involving Jacobi theta functions of arguments w,x,y, and zand w?;x?;y?; andz?;related by 2w?/C30/C28 w/C27x/C27y/C27z (71) 2x?/C30w/C28x/C28y/C27z (72) 2y?/C30w/C27x/C28y/C27z (73) 2z?/C30w/C27x/C27y/C28z (74) (Whittaker and Watson 1990, pp. 467 /C1/69, 488, and 490). Using the notation qiw/C27p=2;q ðÞ qjx/C27p=2;q ðÞ qky;qðÞqlz;qðÞ/C13ijkl½/C138 (75) qiw?;q ðÞ qjx?;qðÞ qky?/C27p=2;q ðÞ qlz?/C27p=2;q ðÞ /C13ijkl; (76) gives a whopping 288 identities of the form 9a1a2a3a4 ½/C138 9b1b2b3b4 ½/C138 /C309a?1a?2a?3a?49b?1b?2b?3b?4:(77) The complete ELLIPTIC INTEGRALS OF THE FIRST and SECOND KINDS can be expressed using Jacobi theta functions. Let j/C13q1zðÞ q4zðÞ; (78) and plug into (68) dj dz !2 /C30q2 2/C28j2q23/C0/C1 q23/C28j2q22/C0/C1 : (79) Now write jq3 q2/C13y (80) and z q2 3 /C13u: (81) Then dy du !2 /C30 1 /C28y2/C0/C1 1 /C28k2y2/C0/C1 ; (82) where the MODULUS is defined by k /C30k(q) /C30q2 2qðÞ q2 3qðÞ: (83) Define also the complementary MODULUS k?/C30k? qðÞ/C30q24/C28qðÞ q23qðÞ: (84) Now, since q42 /C27q44 /C30q43 ; (85) we have shown k2 /C27k?2 /C301 : (86) The solution to the equation is y /C30q3 q2q1(uq/C282 3jr Þ q4u q/C282 3jr/C0/C1 /C13sn(u; k) ; (87) which is a JACOBI ELLIPTIC FUNCTION with periods 4K(k) /C302pq23(q) (88) and 2iK ?(k) /C30pr q2 3(q): (89) Here, K is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND , K(k) /C301 2 pq2 3(q) : (90) The Jacobi theta functions provide analytic solutions to many tricky problems in mathematics and math- ematical physics. For example, the Jacobi theta functions are related to the SUM OF SQUARES FUNC- TION r2(n) giving the number of representations of n by two squares via q23(q) /C30X/C12 n /C300r2(n)qn (91) q2 4(q) /C30X/C12 n/C300/C281ðÞnr2(n)qn (92) (Borwein and Borwein 1987, p. 34). The general QUINTIC EQUATION is solvable in terms of Jacobi theta functions, and these functions also provide a uni-formly convergent form of the GREEN’S FUNCTION for a rectangular region (Oberhettinger and Magnus 1949). Finally, Jacobi theta functions can be used to uniformize all elliptic and hyperelliptic curves, the classical example being y2/C28xx4/C281/C0/C1 /C300; (93) with x/C30/C28q3ð0j1 2tÞ q4ð0j1 2tÞ(94) y/C30iq1u2 3ð0j1 2tÞq2 2ð0j1 2tÞ q5u2 4ð0j1 2tÞ: (95) See also BLECKSMITH- BRILLHART- GERST THEOREM , ELLIPTIC FUNCTION ,ETA FUNCTION ,EULER’S PENTA- GONAL NUMBER THEOREM ,H ALF-PERIOD RATIO,JA- COBI ELLIPTIC FUNCTIONS ,JACOBI TRIPLE PRODUCT , LANDEN’S FORMULA ,M OCK THETA FUNCTION ,M OD- ULAR EQUATION ,M ODULAR TRANSFORMATION ,M OR- DELL INTEGRAL ,NEVILLE THETA FUNCTIONS ,NOME, POINCARE ´ -FUCHS- KLEIN AUTOMORPHIC FUNCTION , QUINTUPLE PRODUCT IDENTITY ,RAMANUJAN THETA FUNCTIONS ,SCHRO ¨ TER’S FORMULA ,SUM OF SQUARES FUNCTION ,THETA FUNCTIONS ,W EBER FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 576 /C1/79, 1972. Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961. Berndt, B. C. "Theta-Functions and Modular Equations." Ch. 25 in Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 138 /C1/44, 1994. Borwein, J. M. and Borwein, P. B. "Theta Functions and the Arithmetic-Geometric Mean Iteration." Ch. 2 in Pi & the AGM: A Study in Analytic Number Theory and Computa-tional Complexity. New York: Wiley, pp. 33 /C1 /1, 1987. Euler, L. Opera Omnia, Vol. 20. Leipzig, Germany, 1912. Hermite, C. Oeuvres Mathe ´matiques. Paris, 1905 /C1/917. Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum Ellipticarum. Ko¨nigsberg, Germany: Regiomonti, Sumti- bus fratrum Borntraeger, 1829. Reprinted in Gesammelte Mathematische Werke, Vol. 1 , pp. 497 /C1/38. Klein, F. Vorlesungen u ¨ber die Theorie der elliptischen Modulfunctionen, 2 vols. Leipzig, Germany: Teubner, 1890/C1/2. Kronecker, L. J. reine angew. Math. 102, 260/C1/72, 1887. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 430 /C1/32, 1953. Oberhettinger, F. and Magnus, W. Anwendung der Ellip- tischen Funktionen in Physik und Technik. Berlin: Springer-Verlag, 1949. Tannery, J. and Molk, J. Elements de la Theorie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars, 1893/C1/902. To¨lke, F. "Theta-Funktionen" and "Logarithmen der Theta- Funktionen." Chs. 1 /C1/inPraktische Funktionenlehre, zweiter Band: Theta-Funktionen und spezielle Weier- straßsche Funktionen. Berlin: Springer-Verlag, pp. 1 /C1/3, 1966. To¨lke, F. Praktische Funktionenlehre, fu ¨nfter Band: Allge- meine Weierstraßsche Funktionen und Ableitungen nachdem Parameter. Integrale der Theta-Funktionen und Bi-linear-Entwicklungen. Berlin: Springer-Verlag, 1968. Weber, H. Elliptische Funktionen und algebraische Zahlen. Brunswick, Germany, 1891. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Jacobi Transformation JACOBI METHOD Jacobi Triple Product The Jacobi triple product is the beautiful identity Y/C12 n/C3011/C28x2n/C0/C1 1/C27x2n/C281z2/C0/C1 1/C27x2n/C281 z2 ! /C30X/C12 m/C30/C28/C12xm2z2m: (1) In terms of the Q-FUNCTION , (1) is written Q1Q2Q3/C301; (2) which is one of the two J ACOBI IDENTITIES .I n Q- SERIES notation, the Jacobi triple product identity is written q;/C28xq;/C281=x;q ðÞ/C12/C30X/C12 k/C30/C28/C12xkqk2/C27kðÞ =2(3) for 0BqjjB1 and x"0 (Gasper and Rahman 1990, p. 12; Leininger and Milne 1997). Another form of the identity is X/C12 n/C30/C28/C12/C281ðÞnanqn2/C28nðÞ =2 /C30Y/C12 n/C3011/C28aqn/C281/C0/C1 1/C28a/C281qn/C0/C1 1/C28qnðÞ (4) (Hirschhorn 1999). Dividing (4) by 1 /C28aand letting a01 gives the limiting case q;qðÞ3 /C12/C30X/C12 n/C300/C281ðÞn(2n/C271)qnn/C271 ðÞ =2(5) /C301 2X/C12 n/C30/C28/C12/C281ðÞn(2n/C271)qnn/C271 ðÞ =2(6) (Jacobi 1829; Hardy and Wright 1979; Leininger and Milne 1997; Hardy 1999, p. 87; Hirschhorn 1999).For the special case of z/C301, (1) becomes 8(x)/C13G(1)/C30Y/C12 n/C3011/C27x2n/C281/C0/C121/C28x2n/C0/C1 /C30X/C12 m/C30/C28/C12xm2/C301/C272X/C12 m/C301xm2; (7) where 8xðÞis the one-variable R AMANUJAN THETA FUNCTION . In terms of the two-variable R AMANUJAN THETA FUNCTION f(a;b);the Jacobi triple product is equivalent to f(a;b)/C30/C28 a;ab ðÞ/C12/C28b;ab ðÞ/C12ab;ab ðÞ/C12 (8) (Berndt et al. ). One method of proof for the Jacobi identity proceeds by defining the function F(z)/C13Y/C12 n/C3011/C27x2n/C281z2/C0/C1 1/C27x2n/C281 z2 ! /C301/C27xz2/C0/C1 1/C27x z2 ! 1/C27x3z2/C0/C1 1/C27x3 z2 ! 1/C27x5z2/C0/C1 /C21/C27x5 z2 ! /C1/C1/C1; (9) Then F(xz)/C301/C27x3z2/C0/C1 1/C271 xz2 ! 1/C27x5z2/C0/C1 1/C27x z2 ! /C291/C27x7z2/C0/C1 1/C27x3 z2 ! /C1/C1/C1: (10) Taking (10) }(9), F(xz) F(z)/C301/C271 xz2 ! 1 1/C27xz2 ! /C30xz2/C271 xz21 1/C27xz2/C301 xz2; (11) which yields the fundamental relation xz2F(xz)/C30F(z): (12) Now define G(z)/C13F(z)Y/C12 n/C3011/C28x2n/C0/C1 (13) G(xz)/C30F(xz)Y/C12 n/C3011/C28x2n/C0/C1 : (14) Using (12), (14) becomes G(xz)/C30F(z) xz2Y/C12 n/C3011/C28x2n/C0/C1 /C30G(z) xz2; (15) so G(z) /C30xz2G(xz) : (16) Expand G in a LAURENT SERIES . Since G is an EVEN FUNCTION , the LAURENT SERIES contains only even terms. G(z) /C30X/C12 m/C30/C28/C12amz2m : (17) Equation (16) then requires that X/C12 m/C30/C28/C12amz2m /C30xz2X/C12 m/C30/C28/C12amxzðÞ2m /C30X/C12 m/C30/C28/C12amx2m/C271z2m/C272 : (18) This can be re-indexed with m?/C13m /C281 on the left side of (18) X/C12 m/C30/C28/C12amz2m /C30X/C12 m/C30/C28/C12amx2m/C281z2m ; (19) which provides a RECURRENCE RELATION am /C30am/C281x2m/C281 ; (20) so a1 /C30a0x (21) a2 /C30a1x3 /C30a0x3 /C271 /C30a0x4 /C30a0x22 (22) a3 /C30a2x5 /C30a0x5 /C274 /C30a0x9 /C30a0x32 : (23) The exponent grows greater by (2m /C281) for each increase in m of 1. It is given by Xm n/C301(2m /C281) /C302mm/C27 1 ðÞ 2/C28m /C30m2 : (24) Therefore, am /C30a0xm2 : (25) This means that G(z) /C30a0X/C12 m/C30/C28/C12xm2 z2m : (26) The COEFFICIENT a0must be determined by going back to (9) and (13) and letting z /C301. Then F(1) /C30Y/C12 n /C3011 /C27x2n/C281/C0/C1 1 /C27x2n/C281/C0/C1 /C30Y/C12 n/C3011 /C27x2n /C281/C0/C12(27)G(1) /C30F(1)Y/C12 n/C3011 /C28x2n/C0/C1 /C30Y/C12 n/C3011 /C27x2n/C281/C0/C12 Y/C12 n/C3011 /C28x2n/C0/C1 /C30Y/C12 n/C3011 /C27x2n /C281/C0/C121 /C28x2n/C0/C1 ; (28) since multiplication is ASSOCIATIVE . It is clear from this expression that the a0term must be 1, because all other terms will contain higher POWERS of x. Therefore, a0/C301; (29) so we have the Jacobi triple product, G(z)/C30Y/C12 n/C3011/C28x2n/C0/C1 1/C27x2n/C281z2/C0/C1 1/C27x2n/C281 z2 ! /C30X/C12 m/C30/C28/C12xm2z2m: (30) See also EULER IDENTITY ,JACOBI IDENTITIES ,PARTI- TION FUNCTION Q, Q-FUNCTION ,Q UINTUPLE PRO- DUCT IDENTITY ,RAMANUJAN PSI SUM,RAMANUJAN THETA FUNCTIONS ,S CHRO ¨ TER’S FORMULA ,T HETA FUNCTIONS References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 63 /C1/4, 1986. Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan’s Lost Notebook." To appears in Trans. Amer. Math. Soc. Borwein, J. M. and Borwein, P. B. "Jacobi’s Triple Product and Some Number Theoretic Applications." Ch. 3 in Pi & the AGM: A Study in Analytic Number Theory andComputational Complexity. New York: Wiley, pp. 62 /C1 /01, 1987. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hirschhorn, M. D. "Another Short Proof of Ramanujan’s Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580/C1/83, 1999. Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum Borntrae- ger, p. 90, 1829. Leininger, V. E. and Milne, S. C. "Expansions for qðÞn2/C27n /C12and Basic Hypergeometric Series in U(n):/" Preprint. http:// www.math.ohio-state.edu/~milne/preprints.html. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 470, 1990. Jacobi Zeta Function Denoted zn(u;k)orZ(u) : Z ðfjmÞ/C13Eðf jmÞ/C28E(m)F ðf jmÞ K(m); where f is the AMPLITUDE , m is the PARAMETER , and F f mjÞ ð and K(m) are ELLIPTIC INTEGRALS OF THE FIRST KIND , and e(m)isan ELLIPTIC INTEGRAL OF THE SECOND KIND . See Gradshteyn and Ryzhik (2000, p. xxxi) for expressions in terms of THETA FUNCTIONS . The Jacobi zeta functions is implemented in Mathe- matica asJacobiZeta [phi, m]. See also ELLIPTIC INTEGRAL OF THE FIRST KIND, ELLIPTIC INTEGRAL OF THE SECOND KIND,H EUMAN LAMBDA FUNCTION ,ZETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 595, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. To¨lke, F. "Jacobische Zeta- und Heumansche Lambda- Funktionen." §132 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 94 /C1/9, 1967. Jacobi’s Curvature Theorem The principal normal indicatrix of a closed SPACE CURVE with nonvanishing curvature bisects the AREA of the unit sphere if it is embedded. Jacobi’s Determinant Identity Let A /C30BD EC/C20/C21 (1) A /C281 /C30WX YZ/C20/C21 ; (2) where B and W are k /C29k MATRICES . Then det Z ðÞ det A ðÞ /C30det B : (3) The proof follows from equating determinants on the two sides of the block matrices BD EC/C20/C21 IX OZ/C20/C21 /C30BO EI/C20/C21 ; (4) where I is the IDENTITY MATRIX and O is the ZERO MATRIX . References Gantmacher, F. R. The Theory of Matrices, Vol. 1. New York: Chelsea, p. 21, 1960.Horn, R. A. and Johnson, C. R. Matrix Analysis. Cambridge, England: Cambridge University Press, p. 21, 1985. Jacobi’s Imaginary Transformation Transformations which relate elliptic functions to other elliptic functions of the same type but having different arguments. In the case of the JACOBI ELLIPTIC FUNCTIONS sn u; cn u; and dn u; the trans- formations are sn(iu ;k) /C30isn u;k? ðÞ cn u;k ? ðÞ (1) cn(iu ;k) /C301 cn u;k? ðÞ (2) dn(iu ;k) /C30dn u;k? ðÞ cn u;k? ðÞ; (3) where k is the MODULUS , and k?/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2p is the COMPLEMENTARY MODULUS (Abramowitz and Stegun 1972; Whittaker and Watson 1990, p. 505). In the case of the JACOBI THETA FUNCTIONS , Jacobi’s imaginary transformation gives q1z j tðÞ/C30/C28i /C28itðÞ/C281 =2ei t?z2 =pq1z t ? t ?jÞ ð (4) q2z jtðÞ/C30/C28 itðÞ/C281 =2ei t?z2 =pq4zt ? t?jÞ ð (5) q3z jtðÞ/C30/C28 itðÞ/C281 =2ei t?z2 =pq3zt ? t?jÞ ð (6) q4zj tðÞ/C30/C28 itðÞ/C281 =2ei t?z2 =pq2z t ? tjÞ; ð (7) where t ?/C13/C281 t ? (8) and /C28itðÞ/C281 =2is interpreted as satisfying arg /C28i tðÞ jj B p=2 (Whittaker and Watson 1990, p. 475). These transformations were first obtained by Jacobi (1828), but Poisson (1827) had previously obtained a formula equivalent to one of the four, and from whichthe other three follow from elementary algebra (Whittaker and Watson 1990, p. 475). See also J ACOBI ELLIPTIC FUNCTIONS ,JACOBI THETA FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 592 and 595, 1972. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.New York: Wiley, p. 73, 1987. Jacobi, C. G. J. "Suite des notices sur les fonctions ellip- tiques." J. reine angew. Math. 3, 403/C1 /04, 1828. Reprinted inGesammelte Werke, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 264 /C1/65, 1969. Landsberg, G. "Zur Theorie der Gaussschen Summen und der linearen Transformation der Thetafunctionen." J. reine angew. Math. 111, 234/C1/53, 1893. Poisson, S. Me´m. de l’Acad. des Sci. 6, 592, 1827. Whittaker, E. T. and Watson, G. N. "Jacobi’s Imaginary Transformation." §21.51 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 474 /C1/76 and 505, 1990. Jacobi’s Theorem Let Mrbe an r-rowed MINOR of the nth order DETERMINANT Ajjassociated with an n/C29nMATRIX A/C30aijin which the rows i1;i2;...,irare represented with columns k1;k2;...,kr:Define the complementary minor to Mras the ( n/C28k)/-rowed MINOR obtained from Ajjby deleting all the rows and columns associated with Mrand the signed complementary minor MrðÞto Mrto be MrðÞ/C30/C28 1ðÞi1/C27i2/C27.../C27ir/C27k1/C27k2/C27.../C27kr /C29complementary minor to Mr ½/C138 : Let the MATRIX of cofactors be given by D/C30A11A12 /C1/C1/C1 A1n A21A22 /C1/C1/C1 A2n nn:::n An1An2/C1/C1/C1 Ann/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12; with M randM?rthe corresponding r-rowed minors of AjjandD;then it is true that M?r/C30Ajjr/C281MrðÞ: References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1109 /C1/100, 2000. JacobiAmplitude AMPLITUDE Jacobian Given a set y/C30f(x)o fnequations in nvariables x1; ...,xn;written explicitly as y/C13f1(x) f2(x) n fn(x)2 6643 775; (1) or more explicitly as y 1/C30f1x1;...;xn ðÞ n yn/C30fnx1;...;xn ðÞ ;8 < :(2) the Jacobian matrix, sometimes simply called "the Jacobian" (Simon and Blume 1994) is defined byJx1;...;xn ðÞ /C30@y1 @x1/C1/C1/C1@y1 @xn n:::n @yn @x1/C1/C1/C1@yn @xn2 6666643 777775: (3) The Jacobian matrix can be computed using the Mathematica command JacobianMatrix[fns_List, vars_List] : /C30 Outer[D, fns, vars] The DETERMINANT ofJis the Jacobian determinant (confusingly, often called "the Jacobian" as well) andis denoted J/C30@y1;...;yn ðÞ @x1;...;xn ðÞ/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12: (4) It can be computed using the Mathematica command JacobianDeterminant[fns_List, vars_List] : /C30 Module[ { nf/C30Length[fns], nv/C30Length[vars], j/C30JacobianMatrix[fns, vars] }, Which[ nf/C21nv, Sqrt[Det[Transpose[j].j]], nf/C30/C30nv, Det[j], nfBnv, Sqrt[Det[j.Transpose[j]]] ] ] Taking the differential dy/C30yxdx (5) shows that Jis the DETERMINANT of the MATRIX yx; and therefore gives the ratios of n-D volumes ( CON- TENTS )i nyandx, dy1/C1/C1/C1dyn/C30j@y1;...;yn ðÞ @x1;...;xn ðÞ jdx1/C1/C1/C1dxn: (6) The concept of the Jacobian can also be applied to n functions in more than nvariables. For example, considering f(u;v;w) and g(u;v;w);the Jacobians @(f;g) @(u;v)/C30jfufv gugvj(7) @(f;g) @(u;w)/C30jfufw gugwj(8) can be defined (Kaplan 1984, p. 99). For the case of n/C303 variables, the Jacobian takes the special form Jf(x1;x2;x3)/C13j@y @x1/C215@y @x2/C29@y @x3j; (9) where a /C215b is the DOT PRODUCT and b /C29c is the CROSS PRODUCT , which can be expanded to give j@ y1 ;y2 ;y3 ðÞ @ x1 ;x2 ;x3 ðÞ j/C30@y1 @x1@y1 @x2@y1 @x3 @y2 @x1@y2 @x2@y2 @x3 @y3 @x1@y3 @x2@y3 @x3/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12: (10) See also C HANGE OF VARIABLES THEOREM ,C URVI- LINEAR COORDINATES ,IMPLICIT FUNCTION THEOREM References Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 98 /C1/9, 123, and 238 /C1/45, 1984. Simon, C. P. and Blume, L. E. Mathematics for Economists. New York: W. W. Norton, 1994. Jacobian Conjecture If det F ?(x) ½/C138/C301 for a POLYNOMIAL MAP F (where det is the DETERMINANT ), then F is BIJECTIVE with poly- nomial inverse (i.e., F is an INVERTIBLE POLYNOMIAL MAP). See also INVERTIBLE POLYNOMIAL MAP,POLYNOMIAL MAP References Becker, T. and Weispfenning, V. Gro¨bner Bases: A Computa- tional Approach to Commutative Algebra. New York: Springer-Verlag, p. 330, 1993. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 0821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Provi- dence, RI: Amer. Math. Soc., 2000. Jacobian Curve The Jacobian of a linear net of curves of order n is a curve of order 3(n /C281): It passes through all points common to all curves of the net. It is the LOCUS of points where the curves of the net touch one another and of singular points of the curve. See also CAYLEYIAN CURVE ,H ESSIAN COVARIANT , STEINERIAN CURVE References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 149, 1959. Jacobian Determinant JACOBIAN Jacobian Group The Jacobian group of a 1-D linear series is given by intersections of the base curve with the JACOBIAN CURVE of itself and two curves cutting the series.References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 283, 1959. Jacobian Matrix JACOBIAN Jacobi-Anger Expansion eiz cos u /C30X/C12 n/C30/C28/C12inJn(z)ein u ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . The identity can also be written eiz cos u /C30J0(z) /C272X/C12 n/C301inJn(z) cos(nu) : This expansion represents an expansion of plane waves into a series of cylindrical waves. See also BESSEL FUNCTION OF THE FIRST KIND JacobiCD JACOBI ELLIPTIC FUNCTIONS JacobiCN JACOBI ELLIPTIC FUNCTIONS JacobiCS JACOBI ELLIPTIC FUNCTIONS JacobiDC JACOBI ELLIPTIC FUNCTIONS JacobiDN JACOBI ELLIPTIC FUNCTIONS JacobiDS JACOBI ELLIPTIC FUNCTIONS Jacobi-Gauss Quadrature Also called J ACOBI QUADRATURE or M EHLER QUAD- RATURE .AG AUSSIAN QUADRATURE over the interval [/C281;1] with WEIGHTING FUNCTION W(x)/C301/C28x ðÞa1/C27x ðÞb: (1) The ABSCISSAS for quadrature order nare given by the roots of the J ACOBI POLYNOMIALS Pa;bðÞ n(x):The weights are wi/C30/C28An/C271gn AnPa;bðÞ? nxiðÞPa;bðÞ n/C271xiðÞ /C30An An/C281gn/C281 Pa;bðÞn/C281xiðÞPa;bðÞ? nxiðÞ; (2) where Anis the COEFFICIENT of xn in P a; bðÞ n(x): For JACOBI POLYNOMIALS , AnG(2n /C27a/C27 b /C27 1) 2nn!G(n /C27a/C27 b /C27 1) ; (3) where G(z)isa GAMMA FUNCTION . Additionally, gn /C301 22n n!ðÞ222n/C27a/C27 b/C271n! 2n /C27a/C27 b /C27 1 /C2G(n /C27a/C27 1)G(n /C27 b /C27 1) G(n /C27a/C27 b /C27 1); (4) so wi /C302n /C27a/C27 b /C27 2 n /C27a/C27 b /C27 1G(n /C27a/C27 1)G(n /C27 b /C27 1) G(n /C27a/C27 b /C27 1) /C222n/C27a/C27 b/C271n! V ?nxiðÞVn/C271xiðÞ (5) /C30G(n /C27a/C27 1)G(n /C27 b /C27 1) G(n /C27a/C27 b /C27 1)22n/C27a/C27 b/C271n! 1 /C28 x2 i/C0/C1 V ?nxiðÞ½/C1382 ; (6) where Vm /C13P a; bðÞ n(x)2nn! /C281ðÞn : (7) The error term is En /C30G(n /C27a/C27 1)G(n /C27 b /C27 1)G(n /C27a/C27 b /C27 1) 2n /C27a/C27 b /C27 1 ðÞ G 2n /C27a/C27 b /C27 1 ðÞ½/C1382 /C222n /C27a/C27 b/C271n! 2nðÞ!f 2nðÞjðÞ (8) (Hildebrand 1959). References Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 331 /C1/34, 1956. JacobiNC JACOBI ELLIPTIC FUNCTIONS JacobiND JACOBI ELLIPTIC FUNCTIONS JacobiNS JACOBI ELLIPTIC FUNCTIONS JacobiP JACOBI POLYNOMIAL JacobiSC JACOBI ELLIPTIC FUNCTIONSJacobiSD JACOBI ELLIPTIC FUNCTIONS JacobiSN JACOBI ELLIPTIC FUNCTIONS JacobiZeta JACOBI ZETA FUNCTION Jacobson Canonical Form Let A be a matrix with the elementary divisors of its characteristic matrix expressed as powers of its irreducible polynomials in the field F[l] ; and consider an elementary divisor p lðÞ½/C138q: If q /C211, then Cq(p) /C30C(p) M 0 /C1/C1/C1 00 0 C(p) M /C1/C1/C1 00 n:::::::::::: n 000 /C1/C1/C1 C(p) M 000 /C1/C1/C1 0 C(p)2 666643 77775; where M is a matrix of the same order as C(p) having the element 1 in the lower left-hand corner and zeros everywhere else. Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 205 /C1 /06, 1962. Jacobson Radical A special ideal in a COMMUTATIVE RING R. The Jacobson radical is the intersection of the maximal ideals in R. It could be the zero ideal, as in the case of the integers. See also ALGEBRAIC GEOMETRY ,ALGEBRAIC NUMBER THEORY ,IDEAL ,NILRADICAL ,RADICAL (IDEAL ) Jacobsthal Number The Jacobsthal numbers are the numbers obtained by the Un/s in the L UCAS SEQUENCE with P/C301 and Q/C30/C28 2, corresponding to a/C302 and b/C30/C28 1. They and the Jacobsthal-Lucas numbers (the Vn/s) satisfy the RECURRENCE RELATION Jn/C30Jn/C281/C272Jn/C282: (1) The Jacobsthal numbers satisfy J0/C300 and J1/C301 and are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (Sloane’sA001045). The Jacobsthal-Lucas numbers satisfy j 0/C30 2 and j1/C301 and are 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, ... (Sloane’s A014551). The properties of thesenumbers are summarized in Horadam (1996). Theyare given by the closed form expressions J n/C30Xn/C281 ðÞ =2 ½/C138 r/C300n/C281/C28r r/C18/C19 2r(2) jn/C30Xn=2½/C138 r/C300n n/C28rn/C28r r/C18/C19 2r; (3) where xbc is the FLOOR FUNCTION andn k/C0/C1 is a BINOMIAL COEFFICIENT . The Binet forms are Jn/C301 3an/C28bnðÞ /C30132n/C28/C28 1ðÞn½/C138 (4) jn/C30an/C27bn/C302n/C27/C28 1ðÞn: (5) The GENERATING FUNCTIONS are X/C12 i/C301Jixi/C281/C301/C28x/C282x2/C0/C1/C281(6) X/C12 i/C301jixi/C281/C30(1/C274x)1/C28x/C282x2/C0/C1/C281: (7) The Simson FORMULAS are Jn/C271Jn/C281/C28J2 n/C30/C28 1ðÞn2n/C281(8) jn/C271jn/C281/C28j2 n/C309/C281ðÞn/C2812n/C281/C30/C289Jn/C271Jn/C281/C28J2 n/C0/C1 :(9) Summation FORMULAS include Xn i/C302Ji/C301 2Jn/C272/C283/C0/C1 : (10) Xn i/C301ji/C3012jn/C272/C285/C0/C1 : (11) Interrelationships are jnJn/C30J2n (12) jn/C30Jn/C271/C272Jn/C281 (13) 9Jn/C30jn/C271/C272jn/C281 (14) jn/C271/C27jn/C303Jn/C271/C27Jn/C0/C1 /C303/C2152n(15) jn/C271/C28jn/C303Jn/C271/C28Jn/C0/C1 /C274/C281ðÞn/C271 /C302n/C272/C281ðÞn/C271(16) jn/C271/C282jn/C3032Jn/C28Jn/C271/C0/C1 /C303/C281ðÞn/C271(17) 2jn/C271/C27jn/C281/C3032Jn/C271/C27Jn/C281/C0/C1 /C276/C281ðÞn/C271(18) jn/C27r/C27jn/C28r/C303Jn/C27r/C27Jn/C28r/C0/C1 /C274/C281ðÞn/C28r(19) /C302n/C28r22r/C271/C0/C1 /C272/C281ðÞn/C28r(20) jn/C27r/C28jn/C28r/C303Jn/C27r/C28Jn/C28r/C0/C1 /C302n/C28r22r/C281/C0/C1 (21) jn/C303Jn/C272/C281ðÞn(22) 3Jn/C27jn/C302n/C271(23) Jn/C27jn/C302Jn/C271 (24) jn/C272jn/C282/C28j2 n/C30/C289Jn/C272Jn/C282/C28Jn/C0/C12/C309/C281ðÞn2n/C282(25)Jmjn/C27Jnjm/C302Jm/C27n (26) jmjn/C279JmJn/C302jm/C27n (27) j2n/C279J2 n/C302j2n (28) Jmjn/C28Jnjm/C30/C28 1ðÞn2n/C271Jm/C28n (29) jmjn/C289JmJn/C30/C28 1ðÞn2n/C271jm/C28n (30) j2n/C289J2 n/C30/C28 1ðÞn2n/C272(31) (Horadam 1996). References Horadam, A. F. "Jacobsthal and Pell Curves." Fib. Quart. 26,7 9/C1/3, 1988. Horadam, A. F. "Jacobsthal Representation Numbers." Fib. Quart. 34,4 0/C1/4, 1996. Sloane, N. J. A. Sequences A001045/M2482 and A014551 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Jacobsthal Polynomial The Jacobsthal polynomials are the POLYNOMIALS obtained by setting p(x)/C301 and q(x)/C302xin the L UCAS POLYNOMIAL SEQUENCE . The first few Jacobsthal polynomials are J1xðÞ/C301 J2xðÞ/C301 J3xðÞ/C301/C272x J4xðÞ/C301/C274x J5xðÞ/C304x2/C276x/C271; and the first few Jacobsthal-Lucas polynomials are j1xðÞ/C301 j2xðÞ/C304x/C271 j3xðÞ/C306x/C271 j4xðÞ/C308x2/C278x/C271 j5xðÞ/C3020x2/C2710x/C271: Jacobsthal and Jacobsthal-Lucas polynomials satisfy Jn1ðÞ/C30Jn jn1ðÞ/C30jn where Jnis a J ACOBSTHAL NUMBER and jnis a JACOBSTHAL- LUCAS NUMBER . Jacobsthal-Lucas Number JACOBSTHAL NUMBER Jacobsthal-Lucas Polynomial JACOBSTHAL POLYNOMIAL Jaco-Shalen-Johannson Torus Decomposition Irreducible orientable COMPACT 3-MANIFOLDS have a canonical (up to ISOTOPY ) minimal collection of disjointly EMBEDDED incompressible TORI such that each component of the 3-MANIFOLD removed by the TORI is either "atoroidal" or "Seifert-fibered." Janko Groups The SPORADIC GROUPS J1 ; J2 ; J3and J4 : The Janko group J2 is also known as the HALL-JANKO GROUP . See also SPORADIC GROUP References Ivanov, A. A. and Meierfrankenfeld, U. "A Computer-Free Construction of J4 :/" J. Algebra 219, 113 /C1/72, 1999. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#spo. Japanese Temple Problem SANGAKU PROBLEM Japanese Theorem Let a convex CYCLIC POLYGON be TRIANGULATED in any manner, and draw the INCIRCLE to each TRIANGLE so constructed. Then the sum of the INRADII is a constant independent of the TRIANGULATION chosen. This theorem can be proved using CARNOT’S THEO- REM. In the above figures, for example, the INRADII of the left triangulation are 0.142479, 0.156972, 0.232307, 0.498525, and the INRADII of the right triangulation are 0.157243, 0.206644, 0.312037, 0.354359, giving a sum of 1.03028 in each case. According to an ancient custom of Japanese mathe- maticians, this theorem was a SANGAKU PROBLEM inscribed on tablets hung in a Japanese temple to honor the gods and the author in 1800 (Johnson 1929). The converse is also true: if the sum of INRADII does not depend on the TRIANGULATION of a POLYGON , then the POLYGON is CYCLIC . See also CARNOT’S THEOREM ,C YCLIC POLYGON , INCIRCLE ,INRADIUS ,SANGAKU PROBLEM ,TRIANGULA- TIONReferences Hayashi, T. "Sur un soi-disant the´ore`me chinois." Mathesis 6, 257 /C1/60, 1906. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 24 /C1/6, 1985. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 193, 1929. Lambert, T. "The Delaunay Triangulation Maximizes the Mean Inradius." Proc. Sixth Canadian Conf. Comput. Geometry. Saskatoon, Saskatchewan, Canada, pp. 201 /C1/ 06, Aug. 1994. Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 125, 1991. Japanese Triangulation Theorem JAPANESE THEOREM Jarnick’s Inequality Given a CONVEX plane region with AREA A and PERIMETER p, then N /C28A jjB p ; where N is the number of enclosed LATTICE POINTS . See also LATTICE POINT ,NOSARZEWSKA’S INEQUALITY j-Conductor FREYCURVE Jeep Problem Maximize the distance a jeep can penetrate into the desert using a given quantity of fuel. The jeep isallowed to go forward, unload some fuel, and then return to its base using the fuel remaining in its tank. At its base, it may refuel and set out again. When itreaches fuel it has previously stored, it may then useit to partially fill its tank. This problem is also called the EXPLORATION PROBLEM (Ball and Coxeter 1987). Given n/C27f(with 05fB1) drums of fuel at the edge of the desert and a jeep capable of holding one drum (and storing fuel in containers along the way), the maximum one-way distance which can be traveled (assuming the jeep travels one unit of distance perdrum of fuel expended) is d/C30f 2n/C271/C27Xn i/C3011 2i/C281 /C30f 2n/C271/C271 2g/C272l n2/C27c012/C27n/C16/C17 hi ; where gis the E ULER- MASCHERONI CONSTANT and cnzðÞthe POLYGAMMA FUNCTION . For example, the farthest a jeep with n/C301 drum can travel is obviously 1 unit. However, with n/C302 drums of gas, the maximum distance is achieved by filling up the jeep’s tank with the first drum, traveling 1/3 of a unit, storing 1/3 of a drum of fuel there, and then returning to base with the remaining 1/3 of a tank. At the base, the tank is filled with the second drum. The jeep then travels 1/3 of a unit (expending 1/3 of a drum of fuel), refills the tank using the 1/3 of a drum of fuel stored there, and continues an additional 1 unit of distance on a full tank, giving a total distance of 4/3. The solutions for n /C301, 2, ... drums are 1, 4/3, 23/15, 176/105, 563/315, ..., which can also be written as a(n) =b(n) ; where a(n) /C301 1 /C2713 /C27.../C271 2n /C28 1 ! LCM 1;3 ;5;...;2n /C281 ðÞ b(n) /C30LCM 1;3 ;5;...;2n /C281 ðÞ (Sloane’s A025550 and A025547). See also HARMONIC NUMBER References Alway, G. C. "Crossing the Desert." Math. Gaz. 41, 209, 1957. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 32, 1987. Bellman, R. Exercises 54 /C1/5 Dynamic Programming. Prin- ceton, NJ: Princeton University Press, p. 103, 1955. Fine, N. J. "The Jeep Problem." Amer. Math. Monthly 54, 24 /C1/1, 1947. Gale, D. "The Jeep Once More or Jeeper by the Dozen." Amer. Math. Monthly 77, 493 /C1/01, 1970. Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 152 and 157 /C1/59, 1961. Haurath, A.; Jackson, B.; Mitchem, J.; and Schmeichel, E. "Gale’s Round-Trip Jeep Problem." Amer. Math. Monthly 102, 299 /C1/09, 1995. Helmer, O. "A Problem in Logistics: The Jeep Problem." Project Rand Report No. Ra 15015, Dec. 1947. Phipps, C. G. "The Jeep Problem, A More General Solution." Amer. Math. Monthly 54, 458 /C1/62, 1947. Sloane, N. J. A. Sequences A025550 and A025547 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Jenkins’ Theorem This entry contributed by RONALD M. AARTS A theorem in the theory of univalent CONFORMAL MAPPINGS of families of domains on a RIEMANN SUR- FACE , containing an inequality for the coefficients of the mapping functions, as well as conditions to be satisfied by the function so that the inequality becomes an equality. Jenkins’ theorem is an exactexpression and generalization of T EICHMU ¨LLER’S PRINCIPLE (Jenkins 1958, Jenkins 1964). See also CONFORMAL MAPPING ,TEICHMU ¨ LLER’S PRIN- CIPLEReferences Jenkins, J. A. Univalent Functions and Conformal Map- ping. New York: Springer-Verlag, 1958. Jenkins, J. A. "Some Area Theorems and a Special Coeffi- cient Theorem." Illinois J. Math. 8,8 0/C1/9, 1964. Jenkins-Traub Method A complicated POLYNOMIAL ROOT -finding algorithm which is used in the IMSL†(IMSL, Houston, TX) library and which Press et al. (1992) describe as "practically a standard in black-box POLYNOMIAL ROOT -finders." References IMSL, Inc. IMSL Math/Library User’s Manual. Houston, TX: IMSL, Inc. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 369, 1992. Ralston, A. and Rabinowitz, P. §8.9/C1/.13 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978. Jensen Polynomial LetfxðÞbe a real ENTIRE FUNCTION OF THE FORM f(x)/C30X/C12 k/C300gkxk k!; where the gk/s are POSITIVE and satisfy T URA´N’S INEQUALITIES g2 k/C28gk/C281gk/C271]0 fork/C301, 2, .... The Jensen polynomial g(t) associated with fxðÞis then given by gntðÞ/C30Xn k/C300n k/C18/C19 gktk; wherea b/C0/C1 is a BINOMIAL COEFFICIENT . References Csordas, G.; Varga, R. S.; and Vincze, I. "Jensen Polyno- mials with Applications to the Riemann z/-Function." J. Math. Anal. Appl. 153, 112/C1/35, 1990. Jensen’s Formula Portions of this entry contributed by R ONALD M. AARTS A relation connecting the values of a MEROMORPHIC FUNCTION inside a disk with its boundary values on the circumference and with its zeros and poles (Jensen 1899, Levin 1980). Let fbe holomorphic on aNEIGHBORHOOD of the CLOSED DISK ¯D(0;r) and f(0)"0;a1;...,akbe the zeros of fin the OPEN DISK D(0;r) counted according to their multiplicities, and assume that f"0o n @D(0;r):Then ln f(0)jj/C27Xk j/C301lnr aj/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C30 1 2pg2 p 0ln frei u/C0/C1/C12/C12/C12/C12du (Krantz 1999, p. 118). See also CONTOUR INTEGRAL ,JENSEN’S INEQUALITY , MAHLER MEASURE References Borwein, P. and Erde´lyi, T. "Jensen’s Formula." §4.2.E.10c in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 187, 1995. Jensen, J. L. "Sur un nouvel et important the´ore`me de la the´orie des fonctions." Acta Math. 22, 359 /C1/64, 1899. Krantz, S. G. "Jensen’s Formula." §9.1.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 117 /C1/18, 1999. Levin, B. Ya. Distribution of Zeros of Entire Functions. Providence, RI: Amer. Math. Soc., 1980. Jensen’s Inequality For a REAL CONTINUOUS CONCAVE FUNCTION PfxiðÞ n5fPxi n ! (1) if f is concave down, PfxiðÞ n]fPxi n ! (2) if f is concave up, and PfxiðÞ n/C30fPxi n ! (3) IFF x1 /C30x2 /C30.../C30xn : A special case is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix1x2 /C1/C1/C1xnp5x1 /C27 x2 /C27 ... /C27 xn n; (4) with equality IFF x1 /C30x2 /C30.../C30xn :/ See also CONCAVE FUNCTION ,JENSEN’S FORMULA References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1101, 2000. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "Some Theorems Concerning Monotonic Functions." §3.14 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 83 /C1/4, 1988. Jensen, J. L. W. V. "Sur les fonctions convexes et les ine´galite ´s entre les valeurs moyennes." Acta Math. 30, 175 /C1/93, 1906. Krantz, S. G. "Jensen’s Inequality." §9.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 118, 1999. Jensen’s Theorem This entry contributed by RONALD M. AARTSFor fixed v /C30 v1 ;...;vm ðÞ ; the function vkkp/C30Xm i/C301vijjp"# 1=p is a DECREASING FUNCTION of p (Cheney 1999). References Cheney, E. W. Introduction to Approximation Theory, 2nd ed. Providence, RI: Amer. Math. Soc., 1999. Jerabek’s Hyperbola The ISOGONAL CONJUGATE of the EULER LINE.It passes through the vertices of a TRIANGLE , the ORTHOCENTER , CIRCUMCENTER , the SYMMEDIAN POINT , and the ISOGONAL CONJUGATE points of the NINE-POINT CENTER and DE LONGCHAMPS POINT . See also CIRCUMCENTER , DE LONGCHAMPS POINT , EULER LINE,ISOGONAL CONJUGATE ,S YMMEDIAN POINT ,NINE-POINT CENTER ,ORTHOCENTER References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Pinkernell, G. M. "Cubic Curves in the Triangle Plane." J. Geom. 55, 141 /C1/61, 1996. Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091 /C1/094, 1965. Jerk The jerk j is defined as the time DERIVATIVE of the VECTOR ACCELERATION a, j/C13da dt: See also ACCELERATION ,VELOCITY Jessen’s Orthogonal Icosahedron A SHAKY POLYHEDRON constructed by replacing six pairs of adjacent triangles in an ICOSAHEDRON (whose edges form a SKEW QUADRILATERAL ) with pairs of ISOSCELES TRIANGLES sharing a common base. The polyhedron can be constructed by dividing the sides of the ICOSAHEDRON in the GOLDEN RATIO (as used in the construction of the ICOSAHEDRON along the edges of the OCTAHEDRON ), but reversing the long and short segments. The centers of the eight EQUILATERAL TRIANGLES which remain are then the vertices of a CUBE . The polyhedron can be deformed infinitesimally by pinch- ing the angles between the isosceles triangles whose bases act as hinges. If the polyhedron is constructed using paper and tape instead of entirely rigid faces, itis possible to collapse the isosceles triangles onto one another, resulting in an OCTAHEDRON . See also FLEXIBLE POLYHEDRON ,RIGID POLYHEDRON , RIGIDITY THEOREM ,SHAKY POLYHEDRON References Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165/C1/70, 1978. Jessen, B. "Orthogonal Icosahedron." Nordisk Mat. Tidskr. 15,9 0/C1/6, 1967. Weisstein, E. W. "Polyhedra." M ATHEMATICA NOTEBOOK POLYHEDRA.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 161, 1991. j-Function The j-function is defined as j(q)/C131728 JffiffiffiqpðÞ ; (1) where J(q)/C134 271/C28l(q)/C27l2(q)/C2/C3 l2(q)1/C28l(q) ½/C13823 (2) is K LEIN’S ABSOLUTE INVARIANT ,l(q) the ELLIPTIC LAMBDA FUNCTION l(q)/C13k2(q)/C30q2(q) q3(q)"#4 ; (3) /qiaJ ACOBI THETA FUNCTION , and 1728 /C30123:This function can also be specified in terms of the W EBER FUNCTIONS f,f1;f2;g2;andg3as j(z)/C30f24(z)/C2816 ½/C1383 f24(z)(4) /C30f24 1(z)/C2716 ½/C1383 f24 1(z)(5) /C30f24 2(z)/C2716 ½/C1383 f24 2(z)(6) /C30g3 2(z) (7) /C30g23(z)/C271728 (8) (Weber 1902, p. 179; Atkin and Morain 1993). The j-function is a MEROMORPHIC FUNCTION on the UPPER HALF-PLANE which is invariant with respect to the SPECIAL LINEAR GROUP /SLð2;ZÞ/. It has a F OURIER SERIES j(q)/C30X/C12 n/C30/C28/C12cnqn; (9) for the NOME q/C13e2pit(10) withI[t]>0:The coefficients in the expansion of the j-function satisfy: 1.cn/C300 for nB/C281 and c/C281/C301;/ 2. all cn/s are INTEGERS with fairly limited growth with respect to n, and 3.j(q)i sa n ALGEBRAIC NUMBER , sometimes a RATIONAL NUMBER , and sometimes even an INTE- GERat certain very special values of q(ort):/ The latter result is the end result of the massive and beautiful theory of COMPLEX multiplication and the first step of Kronecker’s so-called "J UGENDTRAUM ." Then all of the COEFFICIENTS in the L AURENT SERIES j(q)/C301 q/C27744/C27196884 q/C2721493760 q2/C27864299970 q3 /C2720245856256 q4/C27333202640600 q5/C27... ð11Þ (Sloane’s A000521) are POSITIVE INTEGERS (Rankin 1977, Apostol 1997). Berwick calculated the first seven c(n) in 1916, Zuckerman found the first 24 in 1939, and van Wijngaarden gave the first 100 in 1963. Some remarkable sum formulas involving j(q) for t/C23 H;where His the UPPER HALF-PLANE , and c(n) include 504X/C12 n/C300s5(n)qn"# 2 /C30j(q)/C28123/C2/C3X/C12 n/C301t(n)xn; (12) where sk(n) is the DIVISOR FUNCTION and s5(0)/C30 /C281=504:In addition, 504ðÞ2Xn k/C300s5(k)s5(n/C28k) /C30t(n/C271)/C28984t(n)/C27Xn/C281 k/C301c(k)t(n/C28k) (13) 65520 691s11(n)/C28t(n) ½/C138 /C30t(n/C271)/C2724t(n)/C27Xn/C281 k/C301c(k)t(n/C28k); (14) where t(n) is the TAU FUNCTION (Lehmer 1942; Apostol 1997, p. 92). The latter leads immediately to the remarkable congruencet(n)/C13s11(n) (mod 691) : (15) Lehmer (1942) showed that (n/C271)c(n)/C130 mod 24ðÞ (16) for all n]1;and Lehner (1949) and Apostol (1997, pp. 22, 74, and 90 /C1/1) demonstrated that c(2n)/C130 mod 211/C0/C1 (17) c(3n)/C130 mod 35/C0/C1 (18) c(5n)/C130 mod 52/C0/C1 (19) c(7n)/C130 (mod 7) (20) c(11n)/C130 (mod 11) : (21) More generally, c2anðÞ/C130 mod 23a/C278/C0/C1 (22) c3anðÞ/C130 mod 32a/C273/C0/C1 (23) c5anðÞ/C130 mod 5a/C271/C0/C1 (24) c7anðÞ/C130 mod 7aðÞ (25) (Lehner 1949; Apostol 1997, p. 91). Congruences of this type cannot exist for 13, but Newman (1958)showed c(13np)/C27c(13n)c(13p)/C27p /C281c13n p ! /C130 (mod 13) ; (26) where p/C281p/C131 (mod 13) and c(x)/C300i fxis not an integer (Apostol 1997, p. 91). Congruences for c(kn) have been generalized by Atkin and O’Brien (1967). An asymptotic formula for c(n) was discovered by Petersson (1932), and subsequently independently rediscovered by Rademacher (1938): c(n)/C2e4pffiffinp ffiffiffi 2p n3=4: (27) Letdbe a POSITIVE SQUAREFREE INTEGER , and define t/C13iffiffiffi dp ford/C131 or 2 (mod 4) 1 21/C27iffiffiffi dp/C16/C17 ford/C133 (mod 4) :( (28) Then the NOME is q/C13eipr/C30e2piiffiffi dpðÞford/C131 or 2 mod 4 ðÞ e2pi1/C27iffiffi dpðÞ =2ford/C133 mod 4ðÞ( /C30e/C282pffiffi dp ford/C131 or 2 mod 4 ðÞ /C28e/C28pffiffi dp ford/C133 mod 4ðÞ :/C26 (29) It then turns out that j(q)i sa n ALGEBRAIC INTEGER of degree h(/C28d);where h(/C28d) is the CLASS NUMBER of the DISCRIMINANT /C28dof the QUADRATIC FIELD QffiffiffinpðÞ (Silverman 1986). The first term in the L AURENT SERIES is then q /C281 /C30e/C282 pffiffinp or /C28e /C28pffiffinp ; and all the later terms are POWERS of q /C281 ; which are small numbers. The larger n, the faster the series converges. If h(/C28d) /C301 ; then j(q)isa ALGEBRAIC INTEGER of degree 1, i.e., just a plain INTEGER . Furthermore, the INTEGER is a perfect CUBE . The numbers whose LAURENT SERIES give INTEGERS are those with CLASS NUMBER 1. But these are precisely the HEEGNER NUMBERS -1, -2, -3, -7, -11, -19, -43, -67, -163. The greater (in ABSOLUTE VALUE ) the HEEGNER NUMBER d, the closer to an INTEGER is the expression e pffiffiffiffiffiffi/C28np ; since the initial term in j(q)is the largest and subsequent terms are the smallest. The best approximations with h(/C28d) /C301 are therefore e pffiffiffiffi 43p :9603 /C27744 /C282:2 /C2910 /C284 (30) e pffiffiffiffi 67p :52803 /C27744 /C281 :3 /C2910 /C286 (31) e pffiffiffiffiffiffi 163p :6403203 /C27744 /C287 :5 /C2910 /C2813 : (32) The exact values of j(q) corresponding to the HEEGNER NUMBERS are j /C28e /C28pðÞ /C30123 (33) je/C282 pffiffi 2p/C16/C17 /C30203 (34) j /C28e /C28pffiffi 3p/C16/C17 /C3003 (35) j /C28e /C28pffiffi 7p/C16/C17 /C30/C28153 (36) j /C28e /C28 pffiffiffiffi 11p/C16/C17 /C30/C28323 (37) j /C28e /C28pffiffiffiffi 19p/C16/C17 /C30/C28963 (38) j /C28e /C28pffiffiffiffi 43p/C16/C17 /C30/C289603 (39) j /C28e /C28pffiffiffiffi 67p/C16/C17 /C30/C2852803 (40) j /C28e /C28pffiffiffiffiffiffi 163p/C16/C17 /C30/C286403203 : (41) (The number 5280 is particularly interesting since it is also the number of feet in a mile.) The ALMOST INTEGER generated by the last of these, e pffiffiffiffiffiffi 163p (corre- sponding to the field Qffiffiffiffiffiffiffiffiffiffiffiffi /C28163p/C0/C1 and the IMAGINARY QUADRATIC FIELD of maximal discriminant), is some- times known as the RAMANUJAN CONSTANT . However, this attribution is historically fallacious since this amazing property of e pffiffiffiffiffiffi 163p was first noted by Hermite (1859) and does not seem to appear in any of the works of Ramanujan. /e pffiffiffiffi 22p ; e pffiffiffiffi 37p ; and epffiffiffiffi 58p are also ALMOST INTEGERS . These correspond to binary quadratic forms with discriminants -88, -148, and -232, all of which have CLASS NUMBER two and were noted by Ramanujan (Berndt 1994).It turns out that the j-function also is important in the CLASSIFICATION THEOREM for finite simple groups, and that the factors of the orders of the SPORADIC GROUPS , including the celebrated MONSTER GROUP , are also related. See also ALMOST INTEGER ,HEEGNER NUMBER ,IMA- GINARY QUADRATIC FIELD,KLEIN’S ABSOLUTE INVAR- IANT,RAMANUJAN CONSTANT ,W EBER FUNCTIONS References Apostol, T. M. "The Fourier Expansions of D(t) and J(t)/" and "Congruences for the Coefficients of the Modular Function j."§1.15 and Ch. 4 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 20 /C1/2 and 74 /C1/3, 1997. Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Math. Comput. 61,2 9/C1/8, 1993. Atkin, A. O. L. and O’Brien, J. N. "Some Properties of p(n) andc(n) Modulo Powers of 13." Trans. Amer. Math. Soc. 126, 442/C1/59, 1967. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 90 /C1/1, 1994. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.New York: Wiley, pp. 117 /C1 /18, 1987. Cohn, H. Introduction to the Construction of Class Fields. New York: Dover, p. 73, 1994. Conway, J. H. and Guy, R. K. "The Nine Magic Discrimi- nants." In The Book of Numbers. New York: Springer- Verlag, pp. 224 /C1/26, 1996. Hermite, C. "Sur la the ´orie des e ´quations modulaires." C. R. Acad. Sci. (Paris) 49,1 6/C1/4, 110 /C1/18, and 141 /C1/44, 1859 Oeuvres comple `tes, Tome II. Paris: Hermann, p. 61, 1912. Lehmer, D. H. "Properties of the Coefficients of the Modular Invariant J(t):/"Amer. J. Math. 64, 488/C1/02, 1942. Lehner, J. "Divisibility Properties of the Fourier Coefficients of the Modular Invariant j(t):/"Amer. J. Math. 71, 136/C1/48, 1949. Lehner, J. "Further Congruence Properties of the Fourier Coefficients of the Modular Invariant j(t):/"Amer. J. Math. 71, 373/C1/86, 1949. Morain, F. "Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm." Rapport de Recherche 911, INRIA, Oct. 1988. Newman, M. "Congruences for the Coefficients of Modular Forms and for the Coefficients of j(t):/"Proc. Amer. Math. Soc. 9, 609/C1/12, 1958. Petersson, H. "U ¨ber die Entwicklungskoeffizienten der automorphen formen." Acta Math. 58, 169/C1/15, 1932. Rademacher, H. "The Fourier Coefficients of the Modular Invariant j(t):/"Amer. J. Math. 60, 501/C1/12, 1938. Rankin, R. A. Modular Forms. New York: Wiley, 1985. Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, p. 199, 1977. Serre, J. P. Cours d’arithme ´tique. Paris: Presses Universi- taires de France, 1970. Silverman, J. H. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, p. 339, 1986. Sloane, N. J. A. Sequences A000521/M5477 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1979. Weisstein, E. W. " j-Function." M ATHEMATICA NOTEBOOK JFUNCTION.M . Jinc Function The jinc function is defined as jinc(x) /C13J1(x) x; where J1(x)isaB ESSEL FUNCTION OF THE FIRST KIND , and satisfies limx00 jinc(x) /C301=2: The DERIVATIVE of the jinc function is given by jinc ?(x) /C30/C28J2(x) x: The function is sometimes normalized by multiplying by a factor of 2 so that jinc(0) /C301 (Siegman 1986, p. 729). See also BESSEL FUNCTION OF THE FIRST KIND,SINC FUNCTION References Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, p. 64, 1999. Siegman, A. E. Lasers. Sausalito, CA: University Science Books, 1986. j-Invariant An invariant of an ELLIPTIC CURVE given in the form y2 /C30x3 /C27ax /C27b which is closely related to the DISCRIMINANT and defined by j(E) /C132833a3 4a3 /C27 27b2 : The determination of j as an ALGEBRAIC INTEGER in the QUADRATIC FIELD Q(j) is discussed by Greenhill (1891), Weber (1902), Berwick (1928), Watson (1938),Gross and Zaiger (1985), and Dorman (1988). The norm of j in Q(j) is the CUBE of an INTEGER in Z :/ See also DISCRIMINANT (ELLIPTIC CURVE ), ELLIPTIC CURVE ,FREY CURVE References Berwick, W. E. H. "Modular Invariants Expressible in Terms of Quadratic and Cubic Irrationalities." Proc. London Math. Soc. 28,53/C1/9, 1928. Dorman, D. R. "Special Values of the Elliptic Modular Function and Factorization Formulae." J. reine angew. Math. 383, 207 /C1/20, 1988. Greenhill, A. G. "Table of Complex Multiplication Moduli." Proc. London Math. Soc. 21, 403 /C1/22, 1891. Gross, B. H. and Zaiger, D. B. "On Singular Moduli." J. reine angew. Math. 355, 191 /C1/20, 1985. Stepanov, S. A. "The j-Invariant." §7.2 in Codes on Algebraic Curves. New York: Kluwer, pp. 178 /C1/80, 1999. Watson, G. N. "Ramanujans Vermutung u¨ber Zerfa¨llung- sanzahlen." J. reine angew. Math. 179,97/C1/28, 1938. Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1979. Jitter A SAMPLING phenomenon produced when a waveform is not sampled uniformly at an interval t each time, but rather at a series of slightly shifted intervals t /C27 Dti such that the average Dtihi/C300:/ See also GHOST ,SAMPLING Joachimsthal’s Equation Using CLEBSCH- ARONHOLD NOTATION , an algebraic curve satisfies jn 1an y /C27 jn/C281 1j2an/C281 yax /C271 2n(n /C281)jn/C282 1j2 2an/C282 ya2 x /C27... /C27nj1 jn /C281 2ayan/C281 x/C27 jn 2an x /C300: References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 89, 1959. Johnson Bound A bound on error-correcting codes. Johnson Circle The CIRCUMCIRCLE in JOHNSON’S THEOREM . See also JOHNSON’S THEOREM Johnson Solid The Johnson solids are the CONVEX POLYHEDRA having regular faces and equal edge lengths (with the exception of the completely regular P LATONIC SOLIDS , the " SEMIREGULAR "A RCHIMEDEAN SOLIDS , and the two infinite families of PRISMS and ANTI- PRISMS ). There are 28 simple (i.e., cannot be dissected into two other regular-faced polyhedra by a plane) regular-faced polyhedra in addition to the PRISMS and ANTIPRISMS (Zalgaller 1969), and Johnson (1966) proposed and Zalgaller (1969) proved that there existexactly 92 Johnson solids in all. There is a near-Johnson solid which can be con- structed by inscribing regular nonagons inside theeight triangular faces of a regular octahedron, thenjoining the free edges to the 24 triangles and finally the remaining edges of the triangles to six squares, with one square for each octahedral vertex. It turnsout that the triangles are not quite equilateral,making the edges that bound the squares a slightly different length from that of the enneagonal edge. However, because the differences in edge lengths areso small, the flexing of an average model allows thesolid to be constructed with all edges equal (Ol-shevsky). A database of solids and VERTEX NETS of these solids is maintained on the Bell Laboratories Netlib server, but a few errors exist in several entries. A concate-nated and corrected version of the files is given byWeisstein, together with Mathematica code to display the solids and nets. The following table summarizesthe names of the Johnson solids and gives theirimages and nets. 1. S QUARE PYRAMID 2. P ENTAGONAL PYRAMID 3. T RIANGULAR CUPOLA 4. S QUARE CUPOLA 5. P ENTAGONAL CUPOLA 6. P ENTAGONAL ROTUNDA 7. E LONGATED TRIANGULAR PYRAMID 8. E LONGATED SQUARE PYRAMID 9. E LONGATED PENTAGONAL PYRAMID 10. G YROELONGATED SQUARE PYRAMID 11. G YROELONGATED PENTAGONAL PYRAMID 12. T RIANGULAR DIPYRAMID 13. P ENTAGONAL DIPYRAMID 14. E LONGATED TRIANGULAR DIPYRAMID 15. E LONGATED SQUARE DIPYRAMID 16. E LONGATED PENTAGONAL DIPYRAMID 17. G YROELONGATED SQUARE DIPYRAMID 18. E LONGATED TRIANGULAR CUPOLA 19. E LONGATED SQUARE CUPOLA 20. E LONGATED PENTAGONAL CUPOLA 21. E LONGATED PENTAGONAL ROTUNDA 22. G YROELONGATED TRIANGULAR CUPOLA 23. G YROELONGATED SQUARE CUPOLA 24. G YROELONGATED PENTAGONAL CUPOLA 25. G YROELONGATED PENTAGONAL ROTUNDA 26. G YROBIFASTIGIUM 27. T RIANGULAR ORTHOBICUPOLA 28. S QUARE ORTHOBICUPOLA 29. S QUARE GYROBICUPOLA 30. P ENTAGONAL ORTHOBICUPOLA 31. P ENTAGONAL GYROBICUPOLA 32. P ENTAGONAL ORTHOCUPOLARONTUNDA 33. P ENTAGONAL GYROCUPOLAROTUNDA 34. P ENTAGONAL ORTHOBIROTUNDA 35. E LONGATED TRIANGULAR ORTHOBICUPOLA 36. E LONGATED TRIANGULAR GYROBICUPOLA 37. E LONGATED SQUARE GYROBICUPOLA 38. E LONGATED PENTAGONAL ORTHOBICUPOLA 39. E LONGATED PENTAGONAL GYROBICUPOLA 40. E LONGATED PENTAGONAL ORTHOCUPOLAROTUNDA 41. E LONGATED PENTAGONAL GYROCUPOLAROTUNDA 42. E LONGATED PENTAGONAL ORTHOBIROTUNDA 43. E LONGATED PENTAGONAL GYROBIROTUNDA 44. G YROELONGATED TRIANGULAR BICUPOLA 45. G YROELONGATED SQUARE BICUPOLA 46. G YROELONGATED PENTAGONAL BICUPOLA 47. G YROELONGATED PENTAGONAL CUPOLAROTUNDA 48. G YROELONGATED PENTAGONAL BIROTUNDA 49. A UGMENTED TRIANGULAR PRISM 50. B IAUGMENTED TRIANGULAR PRISM 51. T RIAUGMENTED TRIANGULAR PRISM 52. A UGMENTED PENTAGONAL PRISM 53. B IAUGMENTED PENTAGONAL PRISM 54. A UGMENTED HEXAGONAL PRISM 55. P ARABIAUGMENTED HEXAGONAL PRISM 56. M ETABIAUGMENTED HEXAGONAL PRISM 57. T RIAUGMENTED HEXAGONAL PRISM 58. A UGMENTED DODECAHEDRON 59. P ARABIAUGMENTED DODECAHEDRON 60. M ETABIAUGMENTED DODECAHEDRON 61. T RIAUGMENTED DODECAHEDRON 62. M ETABIDIMINISHED ICOSAHEDRON 63. T RIDIMINISHED ICOSAHEDRON 64. A UGMENTED TRIDIMINISHED ICOSAHEDRON 65. A UGMENTED TRUNCATED TETRAHEDRON 66. A UGMENTED TRUNCATED CUBE 67. B IAUGMENTED TRUNCATED CUBE 68. A UGMENTED TRUNCATED DODECAHEDRON 69. P ARABIAUGMENTED TRUNCATED DODECAHEDRON 70. M ETABIAUGMENTED TRUNCATED DODECAHEDRON 71. T RIAUGMENTED TRUNCATED DODECAHEDRON 72. G YRATE RHOMBICOSIDODECAHEDRON 73. P ARABIGYRATE RHOMBICOSIDODECAHEDRON 74. M ETABIGYRATE RHOMBICOSIDODECAHEDRON 75. T RIGYRATE RHOMBICOSIDODECAHEDRON 76. D IMINISHED RHOMBICOSIDODECAHEDRON 77. P ARAGYRATE DIMINISHED RHOMBICOSIDODECAHE- DRON 78. M ETAGYRATE DIMINISHED RHOMBICOSIDODECAHE- DRON 79. B IGYRATE DIMINISHED RHOMBICOSIDODECAHE- DRON 80. P ARABIDIMINISHED RHOMBICOSIDODECAHEDRON 81. M ETABIDIMINISHED RHOMBICOSIDODECAHEDRON 82. G YRATE BIDIMINISHED RHOMBICOSIDODECAHE- DRON 83. T RIDIMINISHED RHOMBICOSIDODECAHEDRON 84. S NUB DISPHENOID 85. S NUB SQUARE ANTIPRISM 86. S PHENOCORONA 87. A UGMENTED SPHENOCORONA 88. S PHENOMEGACORONA 89. H EBESPHENOMEGACORONA 90. D ISPHENOCINGULUM 91. B ILUNABIROTUNDA 92. T RIANGULAR HEBESPHENOROTUNDA The number of constituent n-gons ({ n}) for each Johnson solid are given in the following table. /Jn/{3} {4} {5} {6} {8} {10} /Jn/{3} {4} {5} {6} {8} {10} 141 4 7 3 557 25 1 4 8 4 0 1 2343 1 4 9624 4 5 1 50 10 15551 1 5 1 1 461 0 6 15 2 4 4 2743 5 3832845 5 445 29551 5 584 2 10 12 1 56 8 4 211 15 1 57 12 3 212 6 58 5 1113 10 59 10 1014 6 3 60 10 1015 8 4 61 15 916 10 5 62 10 217 16 63 5 318 4 9 1 64 7 319 4 13 1 65 8 3 3 20 5 15 1 1 66 12 5 5 21 10 10 6 1 67 16 10 4 22 16 3 1 68 25 5 1 11 23 20 5 1 69 30 10 2 10 24 25 5 1 1 70 30 10 2 10 25 30 6 1 71 35 15 3 9 26 4 4 72 20 30 12 27 8 6 73 20 30 12 28 810 74203012 29 810 75203012 30 10 10 2 76 15 25 11 1 31 10 10 2 77 15 25 11 1 32 15 5 7 78 15 25 11 1 33 15 5 7 79 15 25 11 1 34 20 12 80 10 20 10 2 35 812 81102010 2 36 812 82102010 2 37 8 18 83 5 15 9 3 38 10 20 2 84 12 39 10 20 2 85 24 2 40 15 15 7 86 12 2 41 15 15 7 87 16 1 42 20 10 12 88 16 2 43 20 10 12 89 18 3 44 20 6 90 20 4 45 24 10 91 8 2 4 46 30 10 2 92 13 3 3 1 See also ANTIPRISM ,ARCHIMEDEAN SOLID,CONVEX POLYHEDRON ,KEPLER- POINSOT SOLID ,POLYHEDRON , PLATONIC SOLID ,PRISM ,UNIFORM POLYHEDRON References Bell Laboratories. http://netlib.bell-labs.com/netlib/polyhe- dra/. Bulatov, V. "Johnson Solids." http://www.physics.orst.edu/ ~bulatov/polyhedra/johnson/. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 86 /C1/2, 1997. Hart, G. "NetLib Polyhedra DataBase." http://www.george- hart.com/virtual-polyhedra/netlib-info.html. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Hume, A. Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals. Computer Science Technical Report #130. Murray Hill, NJ: AT&T Bell Laboratories, 1986. Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169 /C1/00, 1966. Pedagoguery Software. Poly . http://www.peda.com/poly/. Pugh, A. "Further Convex Polyhedra with Regular Faces." Ch. 3 in Polyhedra: A Visual Approach. Berkeley, CA: University of California Press, pp. 28 /C1/5, 1976.Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 70 /C1/1, 1991. Zalgaller, V. Convex Polyhedra with Regular Faces. New York: Consultants Bureau, 1969. Johnson’s Equation The PARTIAL DIFFERENTIAL EQUATION @ @xu1 /C27uux /C271 2uxxx /C27u 2t ! /C273a2 2t2 uyy /C300 which arises in the study of water waves. References Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge Univer- sity Press, p. 284, 1990. Johnson’s Theorem Let three equal CIRCLES with centers C1;C2;and C3 intersect in a single point Oand intersect pairwise in the points P,Q, and R. Then the CIRCUMCIRCLE Jof DPQR (the so-called J OHNSON CIRCLE ) is congruent to the original three. See also CIRCUMCIRCLE ,JOHNSON CIRCLE References Emch, A. "Remarks on the Foregoing Circle Theorem." Amer. Math. Monthly 23, 162/C1/64, 1916. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 18 /C1/1, 1976. Johnson, R. "A Circle Theorem." Amer. Math. Monthly 23, 161/C1/62, 1916. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 125 /C1/26, 1991. Join (Graph) Let x and y be distinct nodes of G which are not joined by an EDGE . Then the graph /Guxy/ which is formed by adding the EDGE (x, y)toG is called a join of G. Join (Spaces) Let X and Y be TOPOLOGICAL SPACES . Then their join is the factor space X + Y /C30(X /C29Y /C29I) =/C2; where /C2is the EQUIVALENCE RELATION (x;y;t) /C2(x?;y?;t?) Ut /C30t?/C300 and x /C30x? or t /C30t?/C301 and y /C30y:8 < : See also CONE (SPACE ), SUSPENSION References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 6, 1976. Joint Denial The term used in PROPOSITIONAL CALCULUS for the NOR CONNECTIVE . The notation A ¡B is used for this connective. See also ALTERNATIVE DENIAL , NAND References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 26, 1997. Joint Distribution Function A joint distribution function is a DISTRIBUTION FUNC- TION D(x;y) in two variables defined by D(x;y) /C13P(X 5x;Y 5y) (1) Dx(x) /C13lim y 0/C12D(x;y) (2) Dy(y) /C13lim x 0/C12D(x;y) (3) so that the joint probability function satisfies D (x ;y) /C23 C ½/C138 /C30gg (X ;Y) /C23 CP(X ;Y)dXdY (4)D(x /C23 A;y /C23 B) /C30gY /C23 BgX /C23 AP(X ; Y)dXdY (5) D(x; y) /C30PX/C23 (/C28/C12;x] ;Y /C23 (/C28/C12;y] fg /C30gz /C28/C12gy /C28/C12P(X ;Y)dXdY (6) Da5x 5a /C27da ;b 5y 5b /C27db ðÞ /C30gb/C27db bga/C27da aPX ;YðÞ dXdY :Pa;bðÞ da db : (7) Two random variables X and Y are independent IFF D(x;y) /C30Dx(x)Dy(y) (8) for all x and y and P(x;y) /C30@2D(x;y) @x @y: (9) A multiple distribution function is OF THE FORM Dx1 ;...;xn ðÞ /C13PX1 5x1 ;...; Xn 5xn ðÞ : (10) See also DISTRIBUTION FUNCTION References Grimmett, G. and Stirzaker, D. Probability and Random Processes, 2nd ed. New York: Oxford University Press, 1992. Joint Probability Density Function JOINT DISTRIBUTION FUNCTION Joint Theorem GAUSSIAN JOINT VARIABLE THEOREM Joke Number HOAX NUMBER ,SMITH NUMBER Jonah Formula A formula for the generalized CATALAN NUMBERpdqi : The general formula is n /C28q k /C281/C18/C19 /C30Xk i/C301p dqin /C28pi k /C28i/C18/C19 ; wheren k/C0/C1 is a BINOMIAL COEFFICIENT , although Jonah’s original formula corresponded to p /C302, q/C300 (Hilton and Pederson 1991). See also BINOMIAL COEFFICIENT ,CATALAN NUMBER References Hilton, P. and Pederson, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13,6 4/C1/5, 1991. Jones Polynomial The second KNOT POLYNOMIAL discovered. Unlike the first-discovered A LEXANDER POLYNOMIAL , the Jones polynomial can sometimes distinguish handedness (as can its more powerful generalization, the HOM-FLY POLYNOMIAL ). Jones polynomials are L AURENT POLYNOMIALS intassigned to an R3KNOT . The Jones polynomials are denoted VL(t) for LINKS ,VK(t) for KNOTS , and normalized so that Vunknot (t)/C301: (1) For example, the Jones polynomial of the TREFOIL KNOT is given by Vtrefoil tðÞ/C30t/C27t3/C28t4: (2) If a LINK has an ODD number of components, then VL is a L AURENT POLYNOMIAL over the INTEGERS ; if the number of components is EVEN ,VL(t)i st1=2times a LAURENT POLYNOMIAL . The Jones polynomial of a KNOT SUM L1#L2satisfies VL1#L2/C30VL1/C16/C17 VL2/C16/C17 : (3) The SKEIN RELATIONSHIP for under- and overcrossings is t/C281VL/C27/C28tVL/C28/C30t1=2/C28t/C281=2/C0/C1 VL0: (4) Combined with the link sum relationship, this allowsJones polynomials to be built up from simple knotsand links to more complicated ones. Some interesting identities from Jones (1985) follow. For any LINK L, VL(/C281)/C30DL(/C281); (5) where DLis the A LEXANDER POLYNOMIAL , and VL(1)/C30/C28 2ðÞp/C281; (6) where pis the number of components of L. For any KNOT K, VKe2pi=3/C0/C1 /C301 (7) and d dtVK(1)/C300 (8) LetK/C31denote the MIRROR IMAGE of a KNOT K. Then VK+(t)/C30VKt/C281/C0/C1 : (9) For example, the right-hand and left-hand TREFOILKNOTS have polynomials Vtrefoil(t)/C30t/C27t3/C28t4(10) Vtrefoil+(t)/C30t/C281/C27t/C283/C28t/C284: (11) Jones defined a simplified trace invariant for knots by WK(t)/C301/C28VK(t) 1/C28t3 ðÞ (1/C28t): (12) The A RF INVARIANT ofWKis given by Arf(K)/C30WK(i) (13) (Jones 1985), where Iisffiffiffiffiffiffi /C281p :A table of the W polynomials is given by Jones (1985) for knots of up to eight crossings, and by Jones (1987) for knots of up to 10 crossings. (Note that in these papers, an additional polynomial which Jones calls Vis also tabulated, but it is not the conventionally defined Jones polynomial.) Jones polynomials were subsequently generalized to the two-variable HOMFLY POLYNOMIALS , the rela- tionship being V(t)/C30Pa/C30t;x/C30t1=2/C28t/C281=2/C0/C1 (14) V(t)/C30Pl/C30it;m/C30it/C281=2/C28t1=2/C0/C1/C0/C1 : (15) They are related to the K AUFFMAN POLYNOMIAL Fby V(t)/C30F/C28t/C283=4;t/C281=4/C27t1=4/C0/C1 : (16) Jones (1987) gives a table of BRAID WORDS and W polynomials for knots up to 10 crossings. Jones polynomials for KNOTS up to nine crossings are given in Adams (1994) and for oriented links up to nine crossings by Doll and Hoste (1991). All PRIME KNOTS with 10 or fewer crossings have distinct Jones polynomials. It is not known if there is a nontrivialknot with Jones polynomial 1. The Jones polynomial of an ( m, n )- TORUS KNOT is t(m/C281)(n/C281)=21/C28tm/C271/C28tn/C271/C27tm/C27nðÞ 1/C28t2(17) Letkbe one component of an oriented LINK L. Now form a new oriented LINK L/C31by reversing the orientation of k. Then VL/C31/C30t/C283lVLðÞ; (18) where Vis the Jones polynomial and lis the LINKING NUMBER ofkand L/C28k:No such result is known for HOMFLY POLYNOMIALS (Lickorish and Millett 1988). Birman and Lin (1993) showed that substituting the POWER SERIES forexas the variable in the Jones polynomial yields a POWER SERIES whose COEFFI- CIENTS are V ASSILIEV INVARIANTS . LetLbe an oriented connected LINK projection of n crossings, then n ]span V(L) ; (19) with equality if L is ALTERNATING and has no REMOVABLE CROSSING (Lickorish and Millett 1988). There exist distinct KNOTS with the same Jones polynomial. Examples include (05 /C1/01, 10 /C1/32), (08 /C1/08, 10 /C1/29), (08 /C1/16, 10 /C1/56), (10 /C1/25, 10 /C1/56), (10 /C1/22, 10 /C1/35), (10 /C1/ 41, 10 /C1/94), (10 /C1/43, 10 /C1/91), (10 /C1/59, 10 /C1/06), (10 /C1/60, 10 /C1/83), (10 /C1/71, 10 /C1/04), (10 /C1/73, 10 /C1/86), (10 /C1/81, 10 /C1/09), and (10 /C1/37, 10 /C1/55) (Jones 1987). Incidentally, the first four of these also have the same HOMFLY POLYNOMIAL . Witten (1989) gave a heuristic definition in terms of a topological quantum field theory, and Sawin (1996) showed that the "quantum group" Uqsl2ðÞ gives rise to the Jones polynomial. See also ALEXANDER POLYNOMIAL , HOMFLY POLY- NOMIAL ,K AUFFMAN POLYNOMIAL F,K NOT,L INK, VASSILIEV INVARIANT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. Birman, J. S. and Lin, X.-S. "Knot Polynomials and Vassi- liev’s Invariants." Invent. Math. 111, 225 /C1/70, 1993. Doll, H. and Hoste, J. "A Tabulation of Oriented Links." Math. Comput. 57, 747 /C1/61, 1991. El-Misiery, A. "An Algorithm for Calculating Jones Poly- nomials." Appl. Math. Comput. 74, 249 /C1/59, 1996. Jones, V. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Am. Math. Soc. 12, 103 /C1/11, 1985. Jones, V. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335 /C1/88, 1987. Khovanov, M. A Categorification of the Jones Polynomial. 30 Aug 1999. http://xxx.lanl.gov/abs/math.QA/9908171/. Khovanov, M. "A Categorification of the Jones Polynomial." Duke Math. J. 101, 359 /C1/26, 2000. Lickorish, W. B. R. and Millett, B. R. "The New Polynomial Invariants of Knots and Links." Math. Mag. 61,1/C1/3, 1988. Murasugi, K. "Jones Polynomials and Classical Conjectures in Knot Theory." Topology 26, 297 /C1/07, 1987. Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Praslov, V. V. and Sossinsky, A. B. Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology. Providence, RI: Amer. Math. Soc., 1996. Sawin, S. "Links, Quantum Groups, and TQFTS." Bull. Amer. Math. Soc. 33, 413 /C1/45, 1996. Stoimenow, A. "Jones Polynomials." http://guests.mpim- bonn.mpg.de/alex/ptab/j10.html. Thistlethwaite, M. "A Spanning Tree Expansion for the Jones Polynomial." Topology 26, 297 /C1/09, 1987. Weisstein, E. W. "Knots and Links." MATHEMATICA NOTE- BOOK KNOTS.M . Witten, E. "Quantum Field Theory and the Jones Polyno- mial." Comm. Math. Phys. 121, 351 /C1/99, 1989. Jonquie `re’s Function POLYGAMMA FUNCTIONJordan Algebra A NONASSOCIATIVE ALGEBRA named after physicist Pascual Jordan which satisfies xy /C30yx (1) and (xx)(xy) /C30x((xx)y)): (2) The latter is equivalent to the so-called JORDAN IDENTITY (xy)x2 /C30xyx2/C0/C1 (3) (Schafer 1996, p. 4). An ASSOCIATIVE ALGEBRA A with associative product xy can be made into a Jordan algebra A/C27 by the JORDAN PRODUCT x /C215 y /C301 2(xy /C27yx): (4) Division by 2 gives the nice identity x /C215 x /C30xx; but it must be omitted in characteristic p /C302. Unlike the case of a LIE ALGEBRA , not every Jordan algebra is isomorphic to a SUBALGEBRA of some A/C27: Jordan algebras which are isomorphic to a subalgebra are called SPECIAL JORDAN ALGEBRAS , while those that are not are called EXCEPTIONAL JORDAN ALGE- BRAS . See also ANTICOMMUTATOR ,N ONASSOCIATIVE ALGE- BRA References Jacobson, N. Structure and Representations of Jordan Algebras. Providence, RI: Amer. Math. Soc., 1968. Jordan, P. "U¨ ber eine Klasse nichtassoziativer hyperkom- plexer Algebren." Nachr. Ges. Wiss. Go¨ttingen , 569 /C1/75, 1932. Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, pp. 4 /C1/, 1996. Jordan Basis Given a matrix A ; a Jordan basis satisfies Abi;1 /C30 libi;1 and Abi;j/C30libi;j/C27bi;j/C281; and provides the means by which any COMPLEX MATRIX Acan be written in J ORDAN CANONICAL FORM . See also JORDAN BLOCK ,JORDAN CANONICAL FORM Jordan Block A matrix, also called a canonical box matrix, having zeros everywhere except along the DIAGONAL and SUPERDIAGONAL , with each element of the DIAGONAL consisting of a single number l;and each element of the SUPERDIAGONAL consisting of a 1. For example, l 10 /C1/C1/C1 00 0 l 1::: 00 00 l::: 00 000 ::: 00 n:::::::::::: 1 000 /C1/C1/C1 0 l2 66666643 7777775 (Ayres 1962, p. 206). A J ORDAN CANONICAL FORM consists of one or more Jordan blocks. The convention that 1s be along the SUBDIAGONAL instead of the SUPERDIAGONAL is sometimes adopted instead (Faddeeva 1958, p. 50). See also DIAGONAL MATRIX ,JORDAN CANONICAL FORM,SUBDIAGONAL References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 206, 1962. Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 50, 1958. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 317, 1996. Jordan Canonical Form A BLOCK MATRIX in which the blocks consist of CANONICAL BOX MATRICES with possibly differing constants li ; also called classical canonical form. For example, l110 /C1/C1/C1 0 0 l11::: 0 00 l1::: 0 n::::::::: 1 000 /C1/C1/C1 l1::: lk10 /C1/C1/C1 0 0 lk1::: 0 00 lk::: 0 n::::::::: 1 000 /C1/C1/C1 lk2 666666666666666643 77777777777777775 (Ayres 1962, p. 206). A specific example is given by 510 0 0 0 050 0 0 0 005 0 0 0 0001 /C282i 10 0 0 001 /C282i 1 0 0 00 01 /C282i2 66666643 7777775; which has three J ORDAN BLOCKS . Any COMPLEX MATRIX A can be written in Jordan canonical form by finding a JORDAN BASIS bi;j for each JORDAN BLOCK . In fact, any matrix with coefficients in an algebraically closed FIELD can be put into Jordan canonical form. The dimensions of the blocks corresponding to the EIGENVALUE l can be recovered by the sequence ai /C30dim Null A /C28 l ðÞi:The convention that the submatrices have 1s on the SUBDIAGONAL instead of the SUPERDIAGONAL is also used sometimes (Faddeeva 1958, p. 50). See also JORDAN BASIS,JORDAN BLOCK ,JORDAN MATRIX DECOMPOSITION References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 206, 1962. Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 50, 1958. Jordan Curve A Jordan curve is a plane curve which is topologically equivalent to (a HOMEOMORPHIC image of) the UNIT CIRCLE , i.e., it is SIMPLE and CLOSED . It is not known if every Jordan curve contains all four VERTICES of some SQUARE , but it has been proven true for "sufficiently smooth" curves and closed convex curves (Schnirelman 1944; Steinhaus 1990, p. 104). For every TRIANGLE T and Jordan curve J, J has an INSCRIBED TRIANGLE similar to T. See also CARATHE ´ ODORY’S THEOREM ,CLOSED CURVE , JORDAN CURVE THEOREM ,SQUARE INSCRIBING ,SIM- PLE CURVE ,UNIT CIRCLE References Krantz, S. G. "Closed Curves." §2.1.2 in Handbook of Com- plex Analysis. Boston, MA: Birkha ¨user, pp. 19 /C1/0, 1999. Schnirelman, L. G. "On Certain Geometrical Properties of Closed Curves." Uspehi Matem. Nauk 10,34/C1/4, 1944. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Jordan Curve Theorem If J is a simple closed curve in R2 ; then R2 /C28J has two components (an "inside" and "outside"), with Jthe BOUNDARY of each. See also JORDAN CURVE ,SCHO¨ NFLIES THEOREM References Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 14, 1996. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 9, 1976. Jordan Decomposition Theorem LetV"(0) be a finite dimensional VECTOR SPACE over the COMPLEX NUMBERS , and let Abe a linear operator on V. Then V can be expressed as a DIRECT SUM of cyclic subspaces. References Gohberg, I. and Goldberg, S. "A Simple Proof of the Jordan Decomposition Theorem for Matrices." Amer. Math. Monthly 103, 157 /C1/59, 1996. Jordan Identity The identity (xy)x2 /C30xyx2/C0/C1 satisfied by elements x and y in a JORDAN ALGEBRA . See also JORDAN ALGEBRA References Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 4, 1996. Jordan Matrix Decomposition The Jordan matrix decomposition is the decomposi- tion of a square matrix M into the form M /C30SJS/C281 ; (1) where M and J are SIMILAR MATRICES , J is a matrix of JORDAN CANONICAL FORM , and S /C281 is the MATRIX INVERSE of S : In other words, M is a SIMILARITY TRANSFORMATION of a matrix J in JORDAN CANONICAL FORM . The proof that any square matrix can be brought into JORDAN CANONICAL FORM is rather complicated (Turnbull and Aitken 1932; Faddeeva 1958, p. 49; Halmos 1958, p. 112). Jordan decomposition is also associated with the MATRIX EQUATION AX /C30XB and the special case A /C30B:/ The Jordan matrix decomposition is implemented in Mathematica as JordanDecomposition [m], and returns a list {s, j}. Note that Mathematica takes the CANONICAL BOX MATRICES in the JORDAN CANONI- CAL FORM to have 1s along the SUPERDIAGONAL instead of the SUBDIAGONAL . For example, a Jordan decomposition of M /C3024 /C2860 46 /C283 /C284 0040 04 /C28622 6643 775 (2) is given by S /C301 /C28 1 401 01431 0 020 1 0012 6643 775 (3)J /C302100 0200004000062 6643 775; (4) See also J ORDAN CANONICAL FORM,M ATRIX DECOM- POSITION ,SIMILAR MATRICES References Faddeeva, V. N. "The Jordan Canonical Form." §4i n Computational Methods of Linear Algebra. New York: Dover, pp. 49 /C1/4 and 235, 1958. Frazer, R. A.; Duncan, W. J.; and Collar, A. R. "Collinearity Transformation of a Numerical Matrix to a Canonical Form." §3.16 in Elementary Matrices and Some Applica- tions to Dynamics and Differential Equations. Cambridge, England: Cambridge University Press, pp. 93 /C1/5, 1955. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 317, 1996. Halmos, P. R. Finite-Dimensional Vector Spaces, 2nd ed. Princeton, NJ: Van Nostrand, p. 112, 1958. Turnbull, H. W. and Aitken, A. C. Chs. 5 /C1/inAn Introduc- tion to the Theory of Canonical Matrices. London: Blackie and Sons, 1932. Jordan Measure Let the set Mcorrespond to a bounded, NONNEGATIVE function fon an interval 0 5fxðÞ5cforx/C23[a;b]:The Jordan measure, when it exists, is the common value of the outer and inner Jordan measures of M. The outer Jordan measure is the greatest lowerbound of the areas of the covering of M, consisting of finite unions of RECTANGLES . The inner Jordan measure of Mis the difference between the AREA c(a/C28b) of the RECTANGLE Swith base [ a, b] and height c, and the outer measure of the complement of MinS. References Shenitzer, A. and Steprans, J. "The Evolution of Integra- tion." Amer. Math. Monthly 101,6 6/C1/2, 1994. Jordan Measure Decomposition Ifmis a REAL MEASURE (i.e., a MEASURE that takes on real values), then one can decompose it according towhere it is positive and negative. The positive varia- tion is defined by m /C27/C301 2mjj/C27m ðÞ ; (1) where /jmj/is the TOTAL VARIATION MEASURE . Similarly, the negative variation is m/C28/C301 2mjj/C28m ðÞ : (2) Then the Jordan decomposition of mis defined as m/C30m/C27/C28m/C28: (3) When malready is a positive measure then m/C30m/C27: More generally, if m is ABSOLUTELY CONTINUOUS , i.e., m(E) /C30gEfdx; (4) then so are m/C27 and m/C28: The positive and negative variations can also be written as m/C27(E) /C30gEf /C27dx (5) and m/C28(E) /C30gEf /C28dx; (6) where f /C30f /C27/C28f /C28 is the decomposition of f into its positive and negative parts. The Jordan decomposition has a so-called minimum property. In particular, given any positive measure l; the measure m has another decomposition m /C30 m/C27/C27 l ðÞ /C28 m /C28/C27 l ðÞ : (7) The Jordan decomposition is minimal with respect to these changes. One way to say this is that any decomposition m /C30 l1 /C28 l2must have l1 ] m /C27 and l2 ] m/C28:/ See also MEASURE ,POLAR REPRESENTATION (MEA- SURE ), TOTAL VARIATION MEASURE References Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, p. 119, 1987. Jordan Polygon SIMPLE POLYGON Jordan Product The Jordan product of quantities x and y is defined by x/C215y/C301 2(xy/C27yx): See also ANTICOMMUTATOR ,JORDAN ALGEBRA Jordan’s Inequality For 05x5p=2 2 px5sinx5x: References Yuefeng, F. "Jordan’s Inequality." Math. Mag. 69, 126, 1996. Jordan’s Lemma Jordan’s lemma shows the value of the INTEGRAL I/C13g/C12 /C28/C12f(x)eiaxdx (1) along the REAL AXIS is 0 for "nice" functions which satisfy lim R0/C12fR eiuðÞjj /C300:This is established using a CONTOUR INTEGRAL IRwhich satisfies lim R0/C12IRjj5p alim R0/C12e/C300: (2) To derive the lemma, write x/C13Reiu/C30Rcosu/C27isinu ðÞ (3) dx/C30iReiudu (4) and define the CONTOUR INTEGRAL IR/C30gp 0fR eiu/C0/C1 eiaR cosu/C28aRsinuiReiudu (5) Then IRjj/C30Rgp 0fR eiu/C0/C1/C12/C12/C12/C12eiaRcosu/C12/C12/C12/C12e /C28aRsinu/C12/C12/C12/C12ijje iu/C12/C12/C12/C12du /C30Rgp 0fR eiu/C0/C1/C12/C12/C12/C12e/C28aRsinudu: /C302Rgp=2 0fR eiu/C0/C1/C12/C12/C12/C12e/C28aRsinudu: (6) Now, if lim R0/C12fR eiuðÞjj /C300;choose an esuch that fR eiuðÞjj 5e;so IRjj52Regp=2 0e/C28aRsinudu: (7) But, for u/C230;p=2 ½/C138 ; 2 pu5sinu; (8) so IRjj52Regp=2 0e/C282aRu=pdu /C302eR1 /C28 e /C28aR 2aR p/C30p e a1 /C28e /C28aR/C0/C1 : (9) As long as limR0/C12 f(z)jj/C30 0; Jordan’s lemma lim R0/C12IRjj5p alim R0/C12e /C300 (10) then follows. See also CONTOUR INTEGRATION References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 406 /C1/08, 1985. Jordan’s Symmetric Group Theorem A primitive subgroup of the SYMMETRIC GROUP Snis equal to either the ALTERNATING GROUP Anor Sn whenever it contains at least one PERMUTATION which is a q-cycle for some prime q 5n /C283 :/ References Dixon, J. D. "The Probability of Generating the Symmetric Group." Math. Z. 110, 199 /C1/05, 1969. Wielandt, H. Finite Permutation Groups. New York: Aca- demic Press, 1964. Jordan-Ho ¨lder Theorem The composition QUOTIENT GROUPS belonging to two COMPOSITION SERIES of a FINITE GROUP G are, apart from their sequence, ISOMORPHIC in pairs. In other words, if I ƒHs ƒ...ƒH2 ƒH1 ƒG is one COMPOSITION SERIES and I ƒKt ƒ...ƒK2 ƒK1 ƒG is another, then t /C30s, and corresponding to any composition quotient group Kj =Kj/C271 ; there is a com- position QUOTIENT GROUP Hi =Hi/C271 such that Kj Kj/C271$Hi Hi/C271: This theorem was proven in 1869 /C1/889. See also COMPOSITION SERIES ,FINITE GROUP ,ISO- MORPHIC GROUPS References Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993. Scott, W. R. §2.5.8 in Group Theory. New York: Dover, p. 37, 1987. Joseph Ideal See also IDEALReferences Huang, J.-S. "Joseph Ideals and Minimal Representations." §12.3 in Lectures on Representation Theory. Singapore: World Scientific, pp. 169 /C1/71, 1999. Josephus Problem Given a group of nmen arranged in a CIRCLE under the edict that every mth man will be executed going around the CIRCLE until only one remains, find the position L(n;m) in which you should stand in order to be the last survivor (Ball and Coxeter 1987). The list giving the place in the execution sequence of the first, second, etc. man can be given by Josephus [m,n]i n theMathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). To obtain the or- dered list of men who are consecutively slaughtered, InversePermutation in the Mathematica add- on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ) can be applied to the output ofJosephus . The following array gives the original position of the last survivor out of a group of n/C301, 2, ..., if every mth man is killed: 1 2133 241 1 2 53 4 1 2 6 515147 7426358 1763144 9 31187238 1 0545339178 (Sloane’s A032434). The survivor for m/C302 can be given analytically by L(n;2)/C301/C272n/C282 1/C27/C28lgn/C29; where nbcis the FLOOR FUNCTION and LGis the LOGARITHM to base 2. The first few solutions are therefore 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 1, ... (Sloane’s A006257). The original position of the second-to-last survivor is given in the following table for n/C302, 3, ...:> 11 211311243212511514 6312134 714631348311271379545338164 (Sloane’s A032435). The original position of the second-to-last survivor is given in the following table for n /C302, 3, ...:> 111 2111 31212411312531211261433112 731124112 8141335114 (Sloane’s A032436). The original Josephus problem consisted of a CIRCLE of 41 men with every third man killed (n /C3041, m /C303). In order for the lives of the last two men to be spared, they must be placed at positions 31 (last) and 16 (second-to-last). The complete list in order of execu- tion is 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 1, 5, 10, 14, 19, 23, 28, 32, 37, 41, 7, 13, 20, 26, 34, 40, 8, 17, 29, 38, 11, 25, 2, 22, 4, 35, 16, 31. Another version of the problem considers a CIRCLE of two groups (say, "A" and "B") of 15 men each (giving a total of 30 men), with every ninth man cast over- board. To save all the members of the "A" group, the men must be placed at positions 1, 2, 3, 4, 10, 11, 13, 14, 15, 17, 20, 21, 25, 28, 29. Written out explicitly, the order isAAAABBBBBAABAAABABBAABBBABBAAB : This sequence of letters can be remembered with the aid of the MNEMONIC "From numbers’ aid and art, never will fame depart." Consider the vowels only, assign a /C301, e /C302, i /C303, o /C304, u /C305, and alternately add a number of letters corresponding to a vowel value, so 4A (o), 5B (u), 2A (e), etc. (Mott-Smith 1954, §149, pp. 94 and 209 /C1/10; Ball and Coxeter 1987). If instead every tenth man is thrown overboard, the men from the "A" group must be placed in positions 1, 2, 4, 5, 6, 12, 13, 16, 17, 18, 19, 21, 25, 28, 29. Written out explicitly, AABAAABBBBBAABBAAAABABBBABBAAB which can be constructed using the Latin MNEMONIC "Rex paphi cum gente bona dat signa serena" (Ball and Coxeter 1987). Mott-Smith (1954, §153, pp. 96 and 212) discusses a card game called "Out and Under" in which cards at the top of a deck are alternately discarded and placed at the bottom. This is a Josephus problem withparameter m/C302, and Mott-Smith hints at the above closed-form solution. See also K IRKMAN’S SCHOOLGIRL PROBLEM ,N ECK- LACE References Bachet, C. G. Problem 23 in Proble `mes plaisans et de ´lect- ables, 2nd ed. p. 174, 1624. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 32 /C1/6, 1987. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. Kraitchik, M. "Josephus’ Problem." §3.13 in Mathematical Recreations. New York: W. W. Norton, pp. 93 /C1/4, 1942. Mott-Smith, G. "Decimation Puzzles." Ch. 9, §149/C1/54 in Mathematical Puzzles for Beginners and Enthusiasts, 2nd rev. ed. New York: Dover, pp. 94 /C1/7 and 209 /C1/14, 1954. Odlyzko, A. M. and Wilf, H. S. "Functional Iteration and the Josephus Problem." Glasgow Math. J. 33, 235 /C1/40, 1991. Skiena, S. "Josephus’ Problem." §1.4.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 34 /C1/5, 1990. Sloane, N. J. A. Sequences A0062572216, A032434, A032435, and A032436 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Smith, H. J. "Josephus Permutation Problems." http:// pweb.netcom.com/~hjsmith/Josephus.html. Joyce Sequence The sequence of numbers giving the number of digits in /nnn /. The sequence /nnn / for n /C301, 2, ... is 1, 16, 7625597484987, ... (Sloane’s A002488; Rossier 1948), so the Joyce sequence is 1, 2, 13, 155, 2185, 36306, ... (Sloane’s A054382). Laisant (1906) found the term j(9) ; and Uhler (1947) published the logarithm of this number to 250 decimal places (Wells 1986, p. 208). The sequence is named in honor of the following excerpt from the "Ithaca" chapter of James Joyce’s Ulysses : "Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers." References Joyce, J. "Ithaca" Chapter in Ulysses. New York: Random House, 1986. Rossier, P. "Grands nombres." Elemente der Math. 3, 20, 1948. Sloane, N. J. A. Sequences A002488/M5031 and A054382 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 208, 1986. Jug THREE JUG PROBLEM Jugendtraum The German mathematician Kronecker proved that all the Galois extensions of the RATIONALS Q withABELIAN Galois groups are SUBFIELDS of cyclotomic fields Q(mn); where mnis the group of nth ROOTS OF UNITY . He then sought to find a similar function whose division values would generate the Abelian extensions of an arbitrary NUMBER FIELD . He dis- covered that the J-FUNCTION works for IMAGINARY QUADRATIC FIELDS K, but the completion of this problem, known as Kronecker’s Jugendtraum ("dream of youth"), for more general FIELDS remains one of the great unsolved problems in NUMBER THEORY . See also IMAGINARY QUADRATIC FIELD, J-FUNCTION References Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, NJ: Princeton Uni- versity Press, 1981. Juggling The throwing and catching of multiple objects such that at least one is always in the air. Some aspects of juggling turn out to be quite mathematical. The best examples are the two-handed asynchronous juggling sequences known as " SITESWAPS ." See also SITESWAP References Buhler, J.; Eisenbud, D.; Graham, R.; and Wright, C. "Juggling Drops and Descents." Amer. Math. Monthly 101, 507/C1/19, 1994. Donahue, B. "Jugglers Now Juggle Numbers to Compute New Tricks for Ancient Art." New York Times, pp. B5 and B10, Apr. 16, 1996. Juggling Information Service. "Siteswaps." http://www.jug- gling.org/help/siteswap/. Julia Fractal JULIA SET Julia Set LetR(z)b ea RATIONAL FUNCTION R(z)/C13P(z) Q(z); (1) where /z/C23C/C31/,z/C23C/C31is the R IEMANN SPHERE C@/C12fg ; and Pand Qare POLYNOMIALS without common divisors. The "filled-in" Julia set JRis the set of points zwhich do not approach infinity after R(z)i s repeatedly applied (corresponding to a STRANGE ATTRACTOR ). The true Julia set Jis the boundary of the filled-in set (the set of "exceptional points"). Thereare two types of Julia sets: connected sets (F ATOU SET) and C ANTOR SETS (FATOU DUST ). Quadratic Julia sets are generated by the quadratic mapping zn/C271 /C30z2 n /C27c (2) for fixed c. For almost every c, this transformation generates a FRACTAL . Examples are shown above for various values of c. The resulting object is not a fractal for c /C30-2 (Dufner et al. 1998, pp. 224 /C1/26) and c /C300 (Dufner et al. 1998, pp. 125 /C1/26), although it does not seem to be known if these two are the only such exceptional values. The special case of c on the boundary of the MANDELBROT SET is called a DENDRITE FRACTAL (top left figure, computed using c /C30i),/c ¼/C280 :123 þ 0:745i/ is called DOUADY’S RABBIT FRACTAL (left figure), / c ¼/C280:75/ is called the SAN MARCO FRACTAL (middle figure), and /c ¼/C280 :391 /C280:587i/ is the SIEGEL DISK FRACTAL (right figure). Julia sets can be rendered in Mathematica using the following code. JuliaSet[n_:50,c_,rmax_:3.,{{x1_,x2_},{y1_,- y2_}},opts___]: /C30 DensityPlot[-Length[ FixedPointList[#^2 /C27c&,x/C27I y,n,SameTest- /C21(Abs[#2] /C21rmax&)]],{x,x1,x2},{y,y1,y2},opts,PlotPoints- /C21200,Mesh- /C21False, Frame- /C21False,AspectRatio- /C21Automatic ] The equation for the quadratic Julia set is a CON- FORMAL MAPPING , so angles are preserved. Let J be the JULIA SET, then x?/C2x leaves J invariant. If a point P is on J, then all its iterations are on J. The transformation has a two-valued inverse. If b /C300 and y is started at 0, then the map is equivalent to the LOGISTIC MAP. The set of all points for which J is connected is known as the MANDELBROT SET. For a Julia set Jc with /c /C261/, the CAPACITY DIMENSION is dcapacity ¼ 1 þjcj2 4ln2þ Oðjc j3 Þ: ð3Þ For small c, Jcis also a JORDAN CURVE , although its points are not COMPUTABLE . See also DENDRITE FRACTAL ,D OUADY’S RABBIT FRACTAL ,F ATOU DUST,F ATOU SET,M ANDELBROT SET,N EWTON’S METHOD ,S AN MARCO FRACTAL , SIEGEL DISK FRACTAL ,STRANGE ATTRACTOR References Dickau, R. M. "Julia Sets." http://forum.swarthmore.edu/ advanced/robertd/julias.html. Dickau, R. M. "Another Method for Calculating Julia Sets." http://forum.swarthmore.edu/advanced/robertd/inverseju- lia.html. Douady, A. "Julia Sets and the Mandelbrot Set." In The Beauty of Fractals: Images of Complex Dynamical Systems (Ed. H.-O. Peitgen and D. H. Richter). Berlin: Springer- Verlag, p. 161, 1986. Dufner, J.; Roser, A.; and Unseld, F. Fraktale und Julia- Mengen. Harri Deutsch, 1998. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 124 /C1/ 26, 138 /C1/48, and 177 /C1/79, 1991. Mendes-France, M. "Nevertheless." Math. Intell. 10, 35, 1988. Peitgen, H.-O. and Saupe, D. (Eds.). "The Julia Set," "Julia Sets as Basin Boundaries," "Other Julia Sets," and "Exploring Julia Sets." §3.3.2 to 3.3.5 in The Science of Fractal Images. New York: Springer-Verlag, pp. 152 /C1/63, 1988. Schroeder, M. Fractals, Chaos, Power Laws. New York: W. H. Freeman, p. 39, 1991. Wagon, S. "Julia Sets." §5.4 in Mathematica in Action. New York: W. H. Freeman, pp. 163 /C1/78, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 126 /C1/27, 1991. Jump A point of DISCONTINUITY , also called a LEAP . See also DISCONTINUITY ,JUMP ANGLE ,JUMPING CHAMPION References Jeffreys, H. and Jeffreys, B. S. "Leap at a Discontinuity." §1.094 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 26, 1988. Jump Angle A GEODESIC TRIANGLE with oriented boundary yields a curve which is piecewise DIFFERENTIABLE . Further- more, the TANGENT VECTOR varies continuously at all but the three corner points, where it changes sud- denly. The angular difference of the tangent vectors at these corner points are called the jump angles. See also ANGULAR DEFECT ,GAUSS- BONNET FORMULA Jumping Champion An integer /jðn Þ/ is called a JUMPING CHAMPION if/jðnÞ/ is the most frequently occurring difference between consecutive PRIMES /5n/ (Odlyzko et al. ). This term was coined by J. H. Conway in 1993. There are occasionally several jumping champions in a range. The scatter plots above show the jumping champions for small n, and the ranges of number having given jumping champion sets are summarized in the follow- ing table. j(n) n 13 1, 2 5 27 /C1/00, 103 /C1/06, 109 /C1/12, ... 2, 4 101 /C1/02, 107 /C1/08, 113 /C1/30, ... 4 131 /C1/38, ... 2, 4, 6 179 /C1/80, 467 /C1/90, ... 2, 6 379 /C1/88, 421 /C1/32, ... 6 389 /C1/20, ... Odlyzko et al. give a table of jumping champions for n 51000 ; consisting mainly of 2, 4, and 6. 6 is the jumping champion up to about n :1:74 /C291035 ; at which point 30 dominates. At n :10425 ; 210 becomes champion, and subsequent PRIMORIALS are conjec- tured to take over at larger and larger n. Erdos and Straus (1980) proved that the jumping champions tend to infinity under the assumption of a quantita- tive form of the k-tuples conjecture. Wolf gives a table of approximate values ˜n at which the PRIMORIAL pnðÞ will become a champion. Anestimate for ˜n is given by ˜n /C30nn/C27o(n) : See also PRIME DIFFERENCE FUNCTION ,PRIME GAPS, PRIME NUMBER ,PRIMORIAL References Erdos, P.; and Straus, E. G. "Remarks on the Differences Between Consecutive Primes." Elem. Math. 35, 115 /C1/18, 1980. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Nelson, H. "Problem 654." J. Recr. Math. 11, 231, 1978 /C1/979. Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping Cham- pions." http://www.research.att.com/~amo/doc/re- cent.html. Jumping Octahedron A bistable eight-sided polyhedron discovered by Wunderlich and Schwabe (1986). See also FLEXIBLE POLYHEDRON ,MULTISTABLE ,RIGID POLYHEDRON References Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 222 /C1/23, 1997. Wunderlich, W. and Schwabe, C. "Eine Familie von ges- chlossen gleichflachigen Polyedern, die fast beweglich sind." Elem. Math. 41,88/C1/8, 1986. Jung’s Theorem Every finite set of points with SPAN d has an enclosing CIRCLE with RADIUS no greater thanffiffiffi 3p d=3:/ In 3-D, a generalization of the theorem states that every set of points with SPAN d has an enclosing SPHERE with RADIUS no greater thanffiffiffi6p d=4 (Smar- andache 1992, 1996). See also S PAN (GEOMETRY ) References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 103 /C1/10, 1957. Smarandache, F. "A Generalization in Space of Jung’s Theorem." Gazeta Matematica (Bucharest) , No. 9--12, 352, 1992. Smarandache, F. "A Generalization in Space of Jung’s Theorem." In Collected Papers, Vol. 1. Bucharest, Roma- nia: Tempus, pp. 223 /C1/24, 1996. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 128, 1991. Just If IFF Just One EXACTLY ONE Just Rigid A FRAMEWORK is called "just rigid" if it is RIGID , but ceases to be so when any single bar is removed. Lamb (1928, pp. 93 /C1/4) proved that a NECESSARY (but not SUFFICIENT ) condition that a graph be just rigid is that E /C302V /C283;where E is the number of edges (bars) and V is the node of vertices (i.e., pivots; Coxeter and Greitzer 1967, p. 56). See also RIGID GRAPH References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 56, 1967. Lamb, H. Statics, Including Hydrostatics and the Elements of the Theory of Elasticity, 3rd ed. London: Cambridge University Press, 1928. K Kabon Triangles The largest number N(n) of nonoverlapping TRIAN- GLES which can be produced by n straight LINE SEGMENTS . The first few terms are 1, 2, 5, 7, 11, 15, 21, ... (Sloane’s A006066). References Sloane, N. J. A. Sequences A006066/M1334 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Kac Formula The expected number of REAL zeros Enof a RANDOM POLYNOMIAL of degree n if the coefficients are independent and distributed normally is given by En /C301 p g/C12 /C28/C12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 (t2 /C28 1)2 /C28(n /C27 1)2t2n (t2n/C272 /C28 1)2s dt (1) /C304 p g1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 (1 /C28 t2)2 /C28(n /C27 1)2t2n (1 /C28 t2n/C272)2s dt: (2) (Kac 1943, Edelman and Kostlan 1995). Another form of the equation is given by En /C301 p g/C12 /C28/C12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @2 @x @yln1 /C28 (xy)n/C271 1 /C28 xy"# x/C30y /C30tdtvuut (3) (Kostlan 1993, Edelman and Kostlan 1995). As n 0 /C12; E n /C302 pln n /C27C1 /C272 pn /C27O(n/C282) ; (4) where C1 /C302 pln 2 /C27g/C12 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 /C284e /C282x (1 /C28 e/C282x)2s /C281 x /C27 1"# dx() /C300 :6257358072 ... : (5) The initial term was derived by Kac (1943). See also RANDOM POLYNOMIAL References Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32,1/C1/37, 1995. Kac, M. "On the Average Number of Real Roots of a Random Algebraic Equation." Bull. Amer. Math. Soc. 49, 314 /C1/320, 1943. Kac, M. "A Correction to ‘On the Average Number of Real Roots of a Random Algebraic Equation’." Bull. Amer. Math. Soc. 49, 938, 1943. Kostan, E. "On the Distribution of Roots in a Random Polynomial." Ch. 38 in From Topology to Computation: Proceedings of the Smalefest (Ed. M. W. Hirsch,J. E. Marsden, and M. Shub). New York: Springer-Verlag, pp. 419 /C1/431, 1993. Kac Matrix The (n /C271) /C29(n /C271) TRIDIAGONAL MATRIX (also called the CLEMENT MATRIX ) defined by Sn /C300 n 00 /C1/C1/C1 0 10 n /C2810 /C1/C1/C1 0 02 0 n /C282 /C1/C1/C1 0 nn::::::::: n 00 0 n /C28101 00 0 0 n 02 66666643 7777775: The EIGENVALUES are 2k /C28n for k /C300, 1, ..., n. Kadomtsev-Petviashvili Equation The PARTIAL DIFFERENTIAL EQUATION 3 4 Uy /C27Wx /C300; (1) where Wy /C27Ut /C2814 Uxxx /C2732 UUx /C300 (2) (Krichever and Novikov 1980; Novikov 1999). Zwil- linger (1997, p. 131) and Calogero and Degasperis (1982, p. 54) give the equation as @ @x(ut /C27uxxx /C286uux) 9uyy /C300: (3) The modified Kadomtsev-Petviashvili equation is given by uxt /C30uxxx /C273uyy /C286u2 xuxx /C286uyuxx (4) (Clarkson 1986; Zwillinger 1997, p. 133). See also KADOMTSEV- PETVIASHVILI- BURGERS EQUA- TION ,K ORTEWEG-DE VRIES EQUATION ,K RICHEVER- NOVIKOV EQUATION References Baker, H. F. Abelian Functions: Abel’s Theorem and the Allied Theory, Including the Theory of the Theta Func- tions. New York: Cambridge University Press, p. xix, 1995. Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu-tion Equations. New York: North-Holland, 1982. Clarkson, P. A. "The Painleve ´Property, a Modified Boussi- nesq Equation and a Modified Kadomtsev-PetviashviliEquation." Physica D 19, 447/C1 /450, 1986. Krichever, I. M. and Novikov, S. P. "Holomorphic Bundles over Algebraic Curves, and Nonlinear Equations." Russ. Math. Surv. 35,5 3/C1/80, 1980. English translation of Uspekhi Mat. Nauk 35,4 7/C1/68, 1980. Novikov, D. P. "Algebraic-Geometric Solutions of the Krich- ever-Novikov Equation." Theoret. Math. Phys. 121, 1567 /C1/ 15773, 1999. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Kadomtsev-Petviashvili-Burgers Equation The so-called generalized Kadomtsev-Petviashvili- Burgers equation is the PARTIAL DIFFERENTIAL EQUA- TION @ @xut /C27Ju 2t/C27J1uux /C27J2uxx /C27J3uxxx ! /C27J4(t)uyy /C300 (Brugarino 1986; Zwillinger 1997, p. 131). See also KADOMTSEV- PETVIASHVILI EQUATION References Brugarino, T. "Similarity Solutions of the Generalized Kadomtsev-Petviashvili-Burgers Equation." Nuovo Ci- mento B 92, 142 /C1/156, 1986. Infeld, E. and Rowlands, G. "An Example: The Kadomtsev- Petviashvili Equation." §7.10.4 in Nonlinear Waves, Soli- tons, and Chaos, 2nd ed. Cambridge, England: Cambridge University Press, pp. 196 /C1/199, 2000. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Ka¨hler Form A CLOSED TWO-FORM v on a COMPLEX MANIFOLD M which is also the negative IMAGINARY PART of a HERMITIAN METRIC h /C30g /C28iv is called a Ka¨hler form. In this case, M is called a KA¨ HLER MANIFOLD and g, the REAL PART of the HERMITIAN METRIC ,is called a KA¨ HLER METRIC . The Ka¨hler form combines the metric and the COMPLEX STRUCTURE , indeed g(X ; Y) /C30 v(X ; JY) ; (1) where J is the ALMOST COMPLEX STRUCTURE induced by multiplication by i. Since the Ka¨hler form comes from a HERMITIAN METRIC , it is preserved by J, i.e., since h(X ; Y) /C30h(JX ; JY): The equation dv /C300 im- plies that the metric and the complex structure are related. It gives M aK A¨ HLER STRUCTURE , and has many implications. On C2 ; the Ka¨hler form can be written as v /C30/C281 2 i(dz1 ffldz1 /C27dz2 ffldz2) /C30dx1 ffldy1 /C27dx2 ffldy2 ; (2) where zn /C30xn /C27iyn : In general, the Ka¨hler form can be written in coordinates v /C30X gi ¯k dzi ffld¯zk ; (3) where gi ¯kis a HERMITIAN METRIC , the REAL PART of which is the KA¨ HLER METRIC . Locally, a Ka¨hler form can be written as @ ¯@f ; where f is a function called a KA¨ HLER POTENTIAL . The Ka¨hler form is a real (1 ; 1)/- COMPLEX FORM . Since the Ka¨hler form v is closed, it represents a COHOMOLOGY CLASS in DE RHAM COHOMOLOGY .Ona COMPACT MANIFOLD , it cannot be EXACT because vn =n! "0 is the volume form determined by the metric. In the special case of a PROJECTIVE VARIETY ,the Ka¨hler form represents an INTEGRAL COHOMOL- OGY CLASS . That is, it integrates to an integer on any one-dimensional submanifold, i.e., an ALGEBRAIC CURVE . The KODAIRA EMBEDDING THEOREM says that if the Ka¨hler form represents an INTEGRAL COHOMOLOGY CLASS on a compact manifold, then it must be a PROJECTIVE VARIETY . There exist Ka¨hler forms which are not projective algebraic, but it is an open question whether or not any KA¨ HLER MANIFOLD can be deformed to a PROJECTIVE VARIETY (in the compact case). AKa¨hler form satisfies WIRTINGER’S INEQUALITY , ½v(X ; Y)½5½X fflY ½; (4) where the right-hand side is the volume of the parallelogram formed by the tangent vectors X and Y. Corresponding inequalities hold for the EXTERIOR POWERS ofv:Equality holds IFFXand Yform a complex subspace. Therefore, vis a CALIBRATION FORM , and the complex submanifolds of a Ka ¨hler manifold are CALIBRATED SUBMANIFOLDS . In particu- lar, the complex submanifolds are locally volume minimizing in a Ka ¨hler manifold. For example, the graph of a holomorphic function is a locally area- minimizing surface in C2#R4:/ See also CALABI- YAU SPACE ,C ALIBRATION FORM, COMPLEX MANIFOLD ,C OMPLEX PROJECTIVE SPACE , DOLBEAULT COHOMOLOGY ,KA¨ HLER IDENTITIES ,KA¨ H- LER MANIFOLD ,KA¨ HLER METRIC ,KA¨ HLER POTENTIAL , KA¨ HLER STRUCTURE ,KODAIRA EMBEDDING THEOREM , PROJECTIVE VARIETY ,S YMPLECTIC FORM,W IRTIN- GER’S INEQUALITY References Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, pp. 106 /C1/126, 1994. Weil, A. Introduction a `l’e´tude des varie ´te`sK a¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer- site´de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, 1980. Ka¨hler Identities A collection of identities which hold on a K A¨HLER MANIFOLD , also called the Hodge identities. Let vbe a KA¨HLER FORM ,d/C30@/C27¯@be the EXTERIOR DERIVATIVE , where /¯@/is the DEL BAR OPERATOR ,[A;B]/C30AB/C28BA be the COMMUTATOR of two differential operators, and A/C31denote the FORMAL ADJOINT ofA. The following operators also act on DIFFERENTIAL FORMS on a KA¨HLER MANIFOLD : L(a)/C30afflv (1) L(a)/C30L/C31(a)/C30a/C21v (2) dc/C30/C28JdJ ; (3) where Jis the ALMOST COMPLEX STRUCTURE ,J2/C30/C28I; and //C21/ denotes the INTERIOR PRODUCT . Then [L ; ¯@] /C30[L;@] /C300 (4) [ L; ¯@/C31] /C30[ L;@/C31] /C300 (5) [L; ¯@/C31] /C30/C28i @ (6) [L;@/C31] /C30i ¯@ (7) [L; ¯@] /C30/C28i @/C31 (8) [ L;@] /C30i ¯@: (9) In addition, d/C31dc /C30/C28dcd/C31/C30d/C31Ld /C31/C30/C28 dc Ldc (10) ddc /C31/C30/C28 dc /C31d /C30dc /C31Ldc /C31/C30/C28 dLd (11) @ ¯@/C31/C30/C28 ¯@/C31@/C30/C28i ¯@/C31L ¯@/C31/C30/C28 i @L@ (12) ¯@@/C31/C30/C28 @/C31 ¯@/C30i @/C31L @/C31/C30i ¯@L ¯@: (13) These identities have many implications. For in- stance, the two operators Dd /C30dd/C31/C27d/C31d (14) and D¯@/C30 ¯@ ¯@/C31/C27 ¯@/C31 ¯@ (15) (called Laplacians because they are elliptic operators) satisfy Dd /C302D¯@: At this point, assume that M is also a COMPACT MANIFOLD . Along with HODGE’S THEOREM , this equality of Laplacians proves the HODGE DECOM- POSITION . The operators L and L commute with these Laplacians. By HODGE’S THEOREM , they act on coho- mology, which is represented by HARMONIC FORMS . Moreover, defining H /C30[L;L] /C30X (p /C27q /C28n) Pp ; q ; (16) where Pp ; q is projection onto the (p, q)-DOLBEAULT COHOMOLOGY , they satisfy [L;L] /C30H (17) [H ; L] /C30/C282L (18) [H ;L] /C302 L: (19) In other words, these operators provide a REPRESEN- TATION of the SPECIAL LINEAR LIE ALGEBRA sl2(C)on the complex cohomology of a compact Ka¨hler mani- fold. In effect, this is the content of the HARD LEFSCHETZ THEOREM . See also CALIBRATED MANIFOLD ,C OMPLEX MANI- FOLD ,COMPLEX PROJECTIVE SPACE ,HARD LEFSCHETZ THEOREM ,H ODGE’S THEOREM ,KA¨ HLER FORM,KA¨ H- LER MANIFOLD ,KA¨ HLER POTENTIAL ,KA¨ HLER STRUC- TURE ,P ROJECTIVE VARIETY ,R IEMANNIAN METRIC , SYMPLECTIC MANIFOLDReferences Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, p. 111 /C1/122, 1994. Weil, A. Introduction a` l’e´tude des varie´te`sKa¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer- site´ de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, p. 44, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, pp. 191 /C1/195, 1980. Ka¨hler Manifold A COMPLEX MANIFOLD for which the EXTERIOR DERI- VATIVE of the fundamental form V associated with the given HERMITIAN METRIC vanishes, so dV/C300: In other words, it is a complex manifold with a KA¨ HLER STRUCTURE . It has a KA¨ HLER FORM , so it is also a SYMPLECTIC MANIFOLD . It has a KA¨ HLER METRIC ,soit is also a RIEMANNIAN MANIFOLD . The simplest example of a Ka¨hler manifold is a RIEMANN SURFACE , which is a COMPLEX MANIFOLD of dimension 1. In this case, the IMAGINARY PART of any HERMITIAN METRIC must be a CLOSED FORM since all 2-forms are CLOSED on a two real dimensional MANI- FOLD . See also CALIBRATED MANIFOLD ,C OMPLEX MANI- FOLD ,COMPLEX PROJECTIVE SPACE ,H YPER- KA¨ HLER MANIFOLD ,KA¨ HLER FORM,KA¨ HLER IDENTITIES ,KA¨ H- LER METRIC ,K A¨ HLER POTENTIAL ,K A¨ HLER STRUC- TURE ,P ROJECTIVE VARIETY ,Q UATERNION KA¨ HLER MANIFOLD RIEMANNIAN METRIC ,SYMPLECTIC MANI- FOLD References Amoro ´s, J. Fundamental Groups of Compact Ka ¨hler Mani- folds. Providence, RI: Amer. Math. Soc., 1996. Goldberg, S. I. Curvature and Homology, enl. ed. New York: Dover, 1998. Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley pp. 106 /C1/126, 1994. Iyanaga, S. and Kawada, Y. (Eds.). "Ka ¨hler Manifolds." §232 inEncyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 732 /C1/734, 1980. Weil, A. Introduction a `l’e´tude des varie ´te`sK a¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer- site´de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, 1980. Ka¨hler Metric AK a ¨hler metric is a R IEMANNIAN METRIC gon a COMPLEX MANIFOLD which gives MaK A¨HLER STRUC- TURE , i.e., it is a K A¨HLER MANIFOLD with a K A¨HLER FORM . However, the term "Ka ¨hler metric" can also refer to the corresponding H ERMITIAN METRIC h/C30 g/C28iv;where vis the K A¨HLER FORM , defined by v(X;Y)/C30g(JX;Y):Here, the operator Jis the ALMOST COMPLEX STRUCTURE , a linear map on tan- gent vectors satisfying J2/C30/C28I;induced by multi- plication by i. In coordinates zk/C30xk/C27iyk;the operator J satisfies J(@=@xk) /C30@=@yk and J(@=@yk) /C30/C28 @=@xk :/ The operator J depends on the COMPLEX STRUCTURE , and on a KA¨ HLER MANIFOLD , it must preserve the Ka¨hler metric. For a metric to be Ka¨hler, one additional condition must also be satisfied, namely that it can be expressed in terms of the metric and the complex structure. Near any point p, there exists holomorphic coordinates zk /C30xk /C27iyksuch that the metric has the form g /C30X dxk /C156dxk /C27dyk /C156dyk /C27O(½z½2); where /C156 denotes the TENSOR PRODUCT ; that is, it vanishes up to order two at p. Hence, any geometric equation in Cn involving only the first derivatives can be defined on a Ka¨hler manifold. Note that a generic metric can be written to vanish up to order two, but not necessarily in holomorphic coordinates, using a GAUSSIAN COORDINATE SYSTEM . See also CALIBRATED MANIFOLD ,C OMPLEX MANI- FOLD ,COMPLEX PROJECTIVE SPACE ,KA¨ HLER FORM, KA¨ HLER IDENTITIES ,K A¨ HLER MANIFOLD ,K A¨ HLER POTENTIAL ,K A¨ HLER STRUCTURE ,PROJECTIVE VARI- ETY,RIEMANNIAN METRIC ,SYMPLECTIC MANIFOLD References Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, pp. 106 /C1/126, 1994. Weil, A. Introduction a` l’e´tude des varie´te`sKa¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer- site´ de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, 1980. Ka¨hler Potential The Ka¨hler potential is a real-valued function f on a KA¨ HLER MANIFOLD for which the KA¨ HLER FORM v can be written as v /C30i @ ¯@f : Here, the operators @/C30X @ @zkdzk (1) and ¯@/C30X @ @ ¯zkd¯zk (2) are called the del and DEL BAR OPERATOR , respec- tively. For example, in Cn ; the function f /C30½z ½2 =2isaKa ¨hler potential for the standard Ka¨hler form, because i @ ¯@(1 2 ½z½2) /C3012 i @ ¯@X zk ¯zk /C3012i@X zkd¯zk/C3012iX dzkffld¯zk/C30v: See also CALIBRATED MANIFOLD ,C OMPLEX MANI- FOLD ,COMPLEX PROJECTIVE SPACE ,KA¨ HLER FORM, KA¨ HLER IDENTITIES ,K A¨ HLER MANIFOLD ,K A¨ HLER METRIC ,KA¨ HLER STRUCTURE ,PROJECTIVE VARIETY , RIEMANNIAN METRIC ,SYMPLECTIC MANIFOLD References Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, pp. 106 /C1/126, 1994. Weil, A. Introduction a `l’e´tude des varie ´te`sK a¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer- site´de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, 1980. Ka¨hler Structure AK a ¨hler structure on a COMPLEX MANIFOLD M combines a R IEMANNIAN METRIC on the underlying REAL MANIFOLD with the COMPLEX STRUCTURE . Such a structure brings together geometry and complex analysis, and the main examples come from ALGE- BRAIC GEOMETRY . When Mhas ncomplex dimen- sions, then it has 2 nreal dimensions. A Ka ¨hler structure is related to the UNITARY GROUP U(n); which embeds in SO(2n) as the orthogonal matrices that preserve the ALMOST COMPLEX STRUCTURE (mul- tiplication by ‘ i’). In a COORDINATE CHART , the COM- PLEX STRUCTURE ofMdefines a multiplication by i and the metric defines orthogonality for tangent vectors. On a Ka ¨hler manifold, these two notions (and their derivatives) are related. The following are elements of a Ka ¨hler structure, with each condition SUFFICIENT for a Ka ¨hler structure to exist. 1. A K A¨HLER METRIC . Near any point p, there exists holomorphic coordinates zk/C30xk/C27iyksuch that the metric has the form g/C30X dxk/C156dxk/C27dyk/C156dyk/C27O(½z½2); (1) where /C156denotes the TENSOR PRODUCT ; that is, it vanishes up to order two at p. Hence any geo- metric equation in Cninvolving only the first derivatives can be defined on a K A¨HLER MANIFOLD . Note that a generic metric can be written to vanish up to order two, but not necessarily in holomorphic coordinates, using a G AUSSIAN COORDINATE SYS- TEM. 2. A K A¨HLER FORM vis a real CLOSED nondegene- rate TWO-FORM , i.e., a SYMPLECTIC FORM , for which v(X;JX)>0 for nonzero tangent vectors X. More- over, it must also satisfy v(JX;JY)/C30v(X;Y); where Jis the ALMOST COMPLEX STRUCTURE induced by multiplication by i. That is, J@ @xk ! /C30@ @yk(2) and J@ @yk ! /C30/C28@ @xk: (3) Locally, a Ka¨hler form can be written as @ ¯@f ; where f is a function called a KA¨ HLER POTENTIAL . The Ka¨hler form is a real (1; 1)/-FORM . 3. A HERMITIAN METRIC h /C30g /C28iv where the REAL PART is a KA¨ HLER METRIC , as in item (1) above, and where the IMAGINARY PART is a KA¨ HLER FORM ,as in item (2). 4. A metric for which the ALMOST COMPLEX STRUCTURE J is PARALLEL . Since PARALLEL TRANS- PORT is always an isometry, a HERMITIAN METRIC is well-defined by PARALLEL TRANSPORT along paths from a base point. The HOLONOMY GROUP is contained in the UNITARY GROUP . It is easy to see that a complex SUBMANIFOLD of a KA¨ HLER MANIFOLD inherits its Ka¨hler structure, and so must also be Ka¨hler. The main source of examples are PROJECTIVE VARIETIES , complex submanifolds of COMPLEX PROJECTIVE SPACE which are solutions to algebraic equations. There are several deep consequences of the Ka¨hler condition. For example, the KA¨ HLER IDENTITIES , the HODGE DECOMPOSITION of COHOMOLOGY , and the LEFSCHETZ THEOREMS depend on the Ka¨hler condi- tion for compact manifolds. See also CALIBRATED MANIFOLD ,C OMPLEX MANI- FOLD ,COMPLEX PROJECTIVE SPACE ,COMPLEX STRUC- TURE ,K A¨ HLER FORM,K A¨ HLER IDENTITIES ,K A¨ HLER MANIFOLD ,K A¨ HLER METRIC ,K A¨ HLER POTENTIAL , PROJECTIVE VARIETY ,RIEMANN SURFACE ,SYMPLEC- TIC MANIFOLD References Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, pp. 106 /C1/126, 1994. Weil, A. Introduction a` l’e´tude des varie´te`sKa¨hleriennes. Publications de l’Institut de Mathe ´matiques de l’Univer- site´ de Nancago, VI, Actualites Scientifiques et Indus- trielles, no. 1267. Paris: Hermann, 1958. Wells, R. O. Differential Analysis on Complex Manifolds. New York: Springer-Verlag, 1980. Kakeya Needle Problem What is the plane figure of least AREA in which a line segment of width 1 can be freely rotated (where translation of the segment is also allowed)? When the figure is restricted to be convex, Cunningham and Schoenberg (1965) found there is still no minimum AREA , although Wells (1991) states that Kakeyadiscovered that the smallest convex region is an EQUILATERAL TRIANGLE of unit height. The smallest simple convex domain in which one can put a segment of length 1 which will coincide with itself when rotated by 1808 is 1 24(5 /C282ffiffiffi 2p )p /C300:284258... (Le Lionnais 1983). For a general convex shape, Besicovitch (1928) proved that there is no MINIMUM AREA . This can be seen by rotating a line segment inside a DELTOID , star-shaped 5-oid, star-shaped 7-oid, etc. Another iterative con- struction which tends to as small an area as desired is called a PERRON TREE (Falconer 1990, Wells 1991). See also CURVE OF CONSTANT WIDTH ,L EBESGUE MINIMAL PROBLEM ,PERRON TREE,REULEAUX POLY- GON,REULEAUX TRIANGLE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 99 /C1/101, 1987. Besicovitch, A. S. "On Kakeya’s Problem and a Similar One." Math. Z. 27, 312 /C1/320, 1928. Besicovitch, A. S. "The Kakeya Problem." Amer. Math. Monthly 70, 697 /C1/706, 1963. Cunningham, F. Jr. and Schoenberg, I. J. "On the Kakeya Constant." Canad. J. Math. 17, 946 /C1/956, 1965. Falconer, K. J. The Geometry of Fractal Sets, 1st pbk. ed., with corrections. Cambridge, England: Cambridge Uni- versity Press, 1990. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 24, 1983. Ogilvy, C. S. A Calculus Notebook. Boston, MA: Prindle, Weber, & Schmidt, 1968. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 147 /C1/153, 1990. Pa´l, J. "Ein Minimumproblem fu¨r Ovale." Math. Ann. 88, 311 /C1/319, 1921. Plouffe, S. "Kakeya Constant." http://www.lacim.uqam.ca/ piDATA/kakeya.txt. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 151 /C1/152, 1999. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 50 /C1/52, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 128 /C1/129, 1991. Kakeya Set KAKEYA NEEDLE PROBLEM Kakutani’s Fixed Point Theorem Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point. See also FIXED POINT THEOREM Kakutani’s Problem COLLATZ PROBLEM Kalman Filter An ALGORITHM in CONTROL THEORY introduced by R. Kalman in 1960 and refined by Kalman and R. Bucy. It is an ALGORITHM which makes optimal use of imprecise data on a linear (or nearly linear) system with Gaussian errors to continuously update the best estimate of the system’s current state. See also WIENER FILTER References Casti, J. L. "The Kalman Filter." Ch. 1 in Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics. New York: Wiley, pp. 101 /C1/ 154, 2000. Chui, C. K. and Chen, G. Kalman Filtering: With Real-Time Applications, 2nd ed. Berlin: Springer-Verlag, 1991. Grewal, M. S. Kalman Filtering: Theory & Practice. Engle- wood Cliffs, NJ: Prentice-Hall, 1993. Kalman, H. E. "Transversal Filters." Proc. I.R.E. 28, 302 /C1/ 310, 1940. KAM Theorem KOLMOGOROV- ARNOLD- MOSER THEOREM Kampe ´deFe ´riet Function A SPECIAL FUNCTION generalizes the GENERALIZED HYPERGEOMETRIC FUNCTION to two variables and includes the APPELL HYPERGEOMETRIC FUNCTION F1( a; b; b?; g; x ; y) as a special case. The Kampe de Feriet function can represent derivatives of GENERAL- IZED HYPERGEOMETRIC FUNCTIONS with respect to their parameters, as well as indefinite integrals of two and three MEIJER’S G-FUNCTIONS . Exton and Krupnikov (1998) have derived a large collection of formulas involving this function. Kampe ´ de Fe´riet functions are written in the notation Fp ; r; t q ; s; ucp dqj ar bsj at bu j x; y ! : (1) Special cases include F1; 1; 1 1; 0; 01 =2 3 =2 j 1=2 /C28j/C281=2 /C28j x ; y ! /C301ffiffiffixp E sin/C281(ffiffiffixp);ffiffiffiffiffiffiffiffi y=xp>C16>C17 (2) F1; 1; 1 1; 0; 01 =2 3 =2 j 1=2 /C28j/C281=2 /C28j x; y ! /C301ffiffiffixp F sin /C281(ffiffiffixp);ffiffiffiffiffiffiffiffi y=xp>C16>C17 (3) for x "0 and ½x½;½y½51 ; where E(x; k) is the incom- plete ELLIPTIC INTEGRAL OF THE SECOND KIND and F(x; k) is the incomplete ELLIPTIC INTEGRAL OF THE FIRST KIND , as well asF1 ; 1 ; 1 1 ; 0 ; 01=2 1 j 1 /C28j 1=2 /C28j x; y ! /C302 pP(1; x;ffiffiffiyp) (4) for ½x½;½y½B1; where P(n; x; k) is the incomplete ELLIPTIC INTEGRAL OF THE THIRD KIND (Exton and Krupnikov 1998, p. 1). Additional identities are given by F1/C27p;r;t q;s;u0;cp dqjar bsjat bujx;y ! /C301 (5) Fp;r;t q;s;ucp dqjar bsjat bujx;0 ! /C30Fp/C27r q/C27scp;ar dq;dsjx ! (6) Fp;r;1/C27t q;s;ucp dqjar bsj0;at bujx;y ! /C30Fp/C27r q/C27scp;ar dq;dsjx ! (7) (Exton and Krupnikov 1998, p. 3). See also APPELL HYPERGEOMETRIC FUNCTION ,FOX’S H-FUNCTION ,GENERALIZED HYPERGEOMETRIC FUNC- TION ,H ORN FUNCTION ,L AURICELLA FUNCTIONS , MACROBERT’S E-FUNCTION ,MEIJER’S G-FUNCTION References Appell, P. Sur le fonctions hyperge ´ome´triques de plusieurs variables. Paris: Gauthier-Villars, 1925. Appell, P. and Kampe ´de Fe ´riet, J. Fonctions hyperge ´o- me´triques et hypersphe ´riques: polynomes d’Hermite. Paris: Gauthier-Villars, 1926. Exton, H. "The Kampe ´de Fe ´riet Function." §1.3.2 in Hand- book of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, pp. 24 /C1/25, 1978. Exton, H. Multiple Hypergeometric Functions and Applica- tions. Chichester, England: Ellis Horwood, 1976. Exton, H. and Krupnikov, E. D. A Register of Computer- Oriented Reduction Identities for the Kampe ´de Fe ´riet Function. Draft manuscript. Novosibirsk, 1998. Kampe ´de Fe ´riet, J. La fonction hyperge ´ome´trique. Paris: Gauthier-Villars, 1937. Srivastava, H. M., Karlsson, P. W. Multiple Gaussian Hy- pergeometric Series. Chichester, England: Ellis Horwood, 1985. Kampyle of Eudoxus A curve studied by Eudoxus in relation to the classical problem of CUBE DUPLICATION . It is given by the polar equation r cos2 u /C30a; and the PARAMETRIC EQUATIONS x /C30a sec t y /C30a tan t sec t with t /C23 [/C28p=2; p=2]:/ References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 141 /C1/143, 1972. MacTutor History of Mathematics Archive. "Kampyle of Eudoxus." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Kampyle.html. Kanizsa Triangle An optical ILLUSION , illustrated above, in which the eye perceives a white upright EQUILATERAL TRIANGLE where none is actually drawn. See also ILLUSION References Bradley, D. R. and Petry, H. M. "Organizational Determi- nants of Subjective Contour." Amer. J. Psychology 90, 253 /C1/262, 1977. Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 26, 137, and 156, 1996. Kantorovich Inequality Suppose x1 Bx2 B...Bxnare given POSITIVE num- bers. Let l1 ; ..., ln ]0 and an j/C301 lj /C301: Then Xn j/C301ljxj !Xn j/C301ljx/C281 j ! 5A2G /C282 ; (1) where A /C301 2(x1 /C27xn) (2) G /C30ffiffiffiffiffiffiffiffiffix1xnp(3) are the ARITHMETIC and GEOMETRIC MEAN , respec- tively, of the first and last numbers. The Kantorovich inequality is central to the study of convergence properties of descent methods in optimization (Luen- berger 1984). See also ARITHMETIC MEAN,GEOMETRIC MEANReferences Bauer, F. L. "A Further Generalization of the Kantorovich Inequality." Numer. Math. 3, 117 /C1/119, 1961. Greub, W. and Rheinboldt, W. "On a Generalization of an Inequality of L. V. Kantorovich." Proc. Amer. Math. Soc. 10, 407 /C1/413, 1959. Henrici, P. "Two Remarks of the Kantorovich Inequality." Amer. Math. Monthly 68, 904 /C1/906, 1961. Kantorovic, L. V. "Functional Analysis and Applied Mathe- matics" [Russian]. Uspekhi Mat. Nauk 3,89/C1/185, 1948. Luenberger, D. G. Linear and Nonlinear Programming, 2nd ed. Reading, MA: Addison-Wesley, pp. 217 /C1/219, 1984. Newman, M. "Kantorovich’s Inequality." J. Res. National Bur. Standards 64B,33/C1/34, 1960. Po´lya, G. and Szego, G. Aufgaben und Lehrsa ¨tze der Analysis. Berlin, 1925. Pta´k, V. "The Kantorovich Inequality." Amer. Math. Monthly 102, 820 /C1/821, 1995. Schopf, A H. "On the Kantorovich Inequality." Numer. Math. 2, 344 /C1/346, 1960. Strang, W. G. "On the Kantorovich Inequality." Proc. Amer. Math. Soc. 11, 468, 1960. Kaplan-Yorke Conjecture There are several versions of the Kaplan-Yorke conjecture, with many of the higher dimensional ones remaining unsettled. The original Kaplan-Yorke conjecture (Kaplan and Yorke 1979) proposed that, for a two-dimensional mapping, the CAPACITY DIMEN- SION D equals the KAPLAN- YORKE DIMENSION DKY ; D /C30DKY /C30dLya /C301 /C27s1 s2; where s1and s2are the LYAPUNOV CHARACTERISTIC EXPONENTS . This was subsequently proven to be true in 1982. A later conjecture held that the KAPLAN- YORKE DIMENSION is generically equal to a probabil- istic dimension which appears to be identical to the INFORMATION DIMENSION (Frederickson et al. 1983). This conjecture is partially verified by Ledrappier (1981). For invertible 2-D maps, n /C30 s /C30D ; where n is the CORRELATION EXPONENT , s is the INFORMATION DIMENSION , and D is the CAPACITY DIMENSION (Young 1984). See also CAPACITY DIMENSION ,K APLAN- YORKE DI- MENSION ,L YAPUNOV CHARACTERISTIC EXPONENT , LYAPUNOV DIMENSION References Chen, Z. M. "A Note on Kaplan-Yorke-Type Estimates on the Fractal Dimension of Chaotic Attractors." Chaos, Solitons, and Fractals 3, 575/C1/582, 1994. Frederickson, P.; Kaplan, J. L.; Yorke, E. D.; and Yorke, J. A. "The Liapunov Dimension of Strange Attractors." J. Diff. Eq. 49, 185/C1/207, 1983. Kaplan, J. L. and Yorke, J. A. In Functional Differential Equations and Approximations of Fixed Points (Ed. H.- O. Peitgen and H.-O. Walther). Berlin: Springer-Verlag, p. 204, 1979. Ledrappier, F. "Some Relations Between Dimension and Lyapunov Exponents." Commun. Math. Phys. 81, 229/C1/ 238, 1981. Worzbusekros, A. "Remark on a Conjecture of Kaplan and Yorke." Proc. Amer. Math. Soc. 85, 381 /C1/382, 1982. Young, L. S. "Dimension, Entropy, and Lyapunov Exponents in Differentiable Dynamical Systems." Phys. A 124, 639 /C1/ 645, 1984 Kaplan-Yorke Dimension DKY /C13j /C27s1 /C27 ... /C27 sj ½sj /C271 ½; where s1 5 snare LYAPUNOV CHARACTERISTIC EXPO- NENTS and j is the largest INTEGER for which l1 /C27.../C27 lj ]0: If n /C30 s /C30D ; where n is the CORRELATION EXPONENT , s the INFORMATION DIMENSION , and D the HAUSDORFF DIMENSION , then D 5DKY (Grassberger and Procaccia 1983). References Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/208, 1983. Kaplan-Yorke Map xn/C271 /C302xn yn/C271 /C30ayn /C27cos(4 pxn); where xn ; yn are computed mod 1. (Kaplan and Yorke 1979). The Kaplan-Yorke map with a /C300:2 has CORRELATION EXPONENT 1.4290.02 (Grassberger Pro- caccia 1983) and CAPACITY DIMENSION 1.43 (Russell et al. 1980). References Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/208, 1983. Kaplan, J. L. and Yorke, J. A. In Functional Differential Equations and Approximations of Fixed Points (Ed. H.- O. Peitgen and H.-O. Walther). Berlin: Springer-Verlag, p. 204, 1979. Russell, D. A.; Hanson, J. D.; and Ott, E. "Dimension of Strange Attractors." Phys. Rev. Let. 45, 1175 /C1/1178, 1980. Kappa Curve A curve also known as GUTSCHOVEN’S CURVE which was first studied by G. van Gutschoven around 1662(MacTutor Archive). It was also studied by Newton and, some years later, by Johann Bernoulli. It is given by the Cartesian equation (x2 /C27y2)y2 /C30a2x2 ; (1) by the polar equation r /C30a cot u ; (2) and the PARAMETRIC EQUATIONS x /C30a cos t cot t (3) y /C30a cos t: (4) References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 136 and 139 /C1/141, 1972. MacTutor History of Mathematics Archive. "Kappa Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Kap- pa.html. Kaprekar Number Consider an n-digit number k. Square it and add the right ndigits to the left norn/C281 digits. If the resultant sum is k, then kis called a Kaprekar number. The first few are 1, 9, 45, 55, 99, 297, 703, ... (Sloane’s A006886). 92/C3081 8/C271/C309 2972/C3088;209 88 /C27209/C30297: See also DIGITAL ROOT,DIGITADDITION ,HAPPY NUM- BER,K APREKAR ROUTINE ,N ARCISSISTIC NUMBER , RECURRING DIGITAL INVARIANT References Iannucci, D. E.. "The Kaprekar Numbers." J. Integer Se- quences 3, No. 00.1.2, 2000. http://www.research.att.com/ ~njas/sequences/JIS/VOL3/iann2a.html. Sloane, N. J. A. Sequences A006886/M4625 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 73, 1986. Kaprekar Routine A routine discovered in 1949 by D. R. Kaprekar for 4- digit numbers, but which can be generalized to k- digit numbers. To apply the Kaprekar routine to a number n, arrange the digits in descending /(n?) and ascending /(nƒ) order. Now compute K(n)/C13n?/C28nƒand iterate. The algorithm reaches 0 (a degenerate case),a constant, or a cycle, depending on the number ofdigits in kand the value of n. For a 3-digit number nin base 10, the Kaprekar routine reaches the number 495 in at most six iterations. In base r, there is a unique number ((r /C28 2)=2; r /C281; r =2)r to which n converges in at most (r /C27 2)=2 iterations IFF r is EVEN . For any 4-digit number n in base-10, the routine terminates on the number 6174 after seven or fewer steps (where it enters the 1- cycle K(6174) /C306174) :/ 2. 0, 0, 9, 21, f(45) ; (49) g; ..., 3. 0, 0, (32, 52), 184, (320, 580, 484), ..., 4. 0, 30, f201; (126 ; 138) g; (570, 765), {(2550), (3369), (3873)}, ..., 5. 8, (48, 72), 392, (1992, 2616, 2856, 2232), (7488, 10712, 9992, 13736, 11432), ..., 6. 0, 105, (430, 890, 920, 675, 860, 705), {5600, (4305, 5180)}, {(27195), (33860), (42925), (16840, 42745, 35510)}, ..., 7. 0, (144, 192), (1068, 1752, 1836), (9936, 15072, 13680, 13008, 10608), (55500, 89112, 91800, 72012, 91212, 77388), ..., 8. 21, 252, {(1589, 3178, 2723), (1022, 3122, 3290, 2044, 2212)}, {(17892, 20475), (21483, 25578, 26586, 21987)}..., 9. (16, 48), (320, 400), {(2256, 5312, 3856),(3712, 5168, 5456)}, {41520,(34960, 40080, 55360, 49520, 42240)}, ..., 10. 0, 495, 6174, {(53955, 59994), (61974, 82962, 75933, 63954), (62964, 71973, 83952, 74943)}, ..., See also 196-ALGORITHM ,KAPREKAR NUMBER , RATS SEQUENCE References Eldridge, K. E. and Sagong, S. "The Determination of Kaprekar Convergence and Loop Convergence of All 3- Digit Numbers." Amer. Math. Monthly 95, 105 /C1/112, 1988. Kaprekar, D. R. "An Interesting Property of the Number 6174." Scripta Math. 15, 244 /C1/245, 1955. Trigg, C. W. "All Three-Digit Integers Lead to..." The Math. Teacher , 67,41/C1/45, 1974. Young, A. L. "A Variation on the 2-digit Kaprekar Routine." Fibonacci Quart. 31, 138 /C1/145, 1993. Kaps-Rentrop Methods A generalization of the RUNGE- KUTTA METHOD for solution of ORDINARY DIFFERENTIAL EQUATIONS , also called ROSENBROCK METHODS . See also RUNGE- KUTTA METHOD References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 730 /C1/735, 1992.Kapteyn Series A series OF THE FORM X/C12 n/C300anJn /C27n[( n /C27n)z] ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . Examples include Kapteyn’s original series 1 1 /C28 z /C301 /C272X/C12 n /C300Jn(nz) and z2 2(1/C28z2)/C30X/C12 n/C300J2n(2nz): See also BESSEL FUNCTION OF THE FIRST KIND, LEMON ,NEUMANN SERIES (BESSEL FUNCTION ) References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1473, 1980. Karamata’s Tauberian Theorem References Widder, D. V. Ch. 5 in The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Karatsuba Multiplication It is possible to perform MULTIPLICATION ofLARGE NUMBERS in (many) fewer operations than the usual brute-force technique of "long multiplication." As discovered by Karatsuba and Ofman (1962), MULTI- PLICATION of two n-DIGIT numbers can be done with a BIT COMPLEXITY of less than n2using identities OF THE FORM (a/C27b/C21510n)(c/C27d /C21510n) /C30ac/C27[(a/C27b)(c/C27d)/C28ac/C28bd]10n/C27bd /C215102n:(1) Proceeding recursively then gives BIT COMPLEXITY O(nlg 3);where lg 3 /C301:5 8...B2 (Borwein et al. 1989). The best known bound is O(nlgnlgn) steps forn/C271 (Scho ¨nhage and Strassen 1971, Knuth 1981). However, this ALGORITHM is difficult to imple- ment, but a procedure based on the FAST FOURIER TRANSFORM is straightforward to implement and gives BIT COMPLEXITY O((lgn)2/C27en) (Brigham 1974, Borodin and Munro 1975, Knuth 1981, Borwein et al. 1989). As a concrete example, consider MULTIPLICATION of two numbers each just two "digits" long in base w, N1/C30a0/C27a1w (2) N2 /C30b0 /C27b1w ; (3) then their PRODUCT is P /C13N1N2 /C30a0b0 /C27(a0b1 /C27a1b0)w /C27a1b1w2 /C30p0 /C27p1w /C27p2w2 : (4) Instead of evaluating products of individual digits, now write q0 /C30a0b0 (5) q1 /C30(a0 /C27a1)(b0 /C27b1) (6) q2 /C30a1b1 : (7) The key term is q1 ; which can be expanded, re- grouped, and written in terms of the pj as q1 /C30p1 /C27p0 /C27p2 : (8) However, since p0 /C30q0 ; and p2 /C30q2 ;/ it immediately follows that p0 /C30q0 (9) p1 /C30q1 /C28q0 /C28q2 (10) p2 /C30q2 ; (11) so the three "digits" of p have been evaluated using three multiplications rather than four. The technique can be generalized to multidigit numbers, with the trade-off being that more additions and subtractions are required. Now consider four-"digit" numbers N1 /C30a0 /C27a1w /C27a2w2 /C27a3w3 ; (12) which can be written as a two-"digit" number repre- sented in the base w2 ; N1 /C30(a0 /C27a1w) /C27(a2 /C27a3w) + w2 : (13) The "digits" in the new base are now a ?0 /C30a0 /C27a1w (14) a?1 /C30a2 /C27a3w; (15) and the Karatsuba algorithm can be applied to N1 and N2in this form. Therefore, the Karatsuba algorithm is not restricted to multiplying two-digit numbers, but more generally expresses the multi- plication of two numbers in terms of multiplications of numbers of half the size. The asymptotic speed the algorithm obtains by recursive application to the smaller required subproducts is O(nlg 3) (Knuth 1981). When this technique is recursively applied to multi- digit numbers, a point is reached in the recursion when the overhead of additions and subtractions makes it more efficient to use the usual O(n2) MULTI- PLICATION algorithm to evaluate the partial products. The most efficient overall method therefore relies on acombination of Karatsuba and conventional multi- plication. See also COMPLEX MULTIPLICATION ,MULTIPLICATION , STRASSEN FORMULAS References Borodin, A. and Munro, I. The Computational Complexity of Algebraic and Numeric Problems. New York: American Elsevier, 1975. Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanu- jan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201 /C1/219, 1989. Brigham, E. O. The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice-Hall, 1974. Brigham, E. O. Fast Fourier Transform and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1988. Cook, S. A. On the Minimum Computation Time of Func- tions. Ph.D. Thesis. Cambridge, MA: Harvard University, pp. 51 /C1/77, 1966. Hollerbach, U. "Fast Multiplication & Division of Very Large Numbers." sci.math.research posting, Jan. 23, 1996. Karatsuba, A. and Ofman, Yu. "Multiplication of Many- Digital Numbers by Automatic Computers." Doklady Akad. Nauk SSSR 145, 293 /C1/294, 1962. Translation in Physics-Doklady 7, 595 /C1/596, 1963. Knuth, D. E. The Art of Computing, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 278 /C1/286, 1998. Scho¨nhage, A. and Strassen, V. "Schnelle Multiplikation Grosser Zahlen." Computing 7, 281 /C1/292, 1971. Toom, A. L. "The Complexity of a Scheme of Functional Elements Simulating the Multiplication of Integers." Dokl. Akad. Nauk SSSR 150, 496 /C1/498, 1963. English transla- tion in Soviet Mathematics 3, 714 /C1/716, 1963. Zuras, D. "More on Squaring and Multiplying Large In- tegers." IEEE Trans. Comput. 43, 899 /C1/908, 1994. Karnaugh Map In combinatorial logic minimization, a device known as a Karnaugh map is frequently used. It is similar to a TRUTH TABLE , but the various variables are repre- sented along two axes, and are arranged in such a way that only one input bit changes in going from one square to an adjacent square. See also TRUTH TABLE k-ary Divisor Let a DIVISOR d of n be called a 1-ary divisor if d /C222nd (i.e., d is RELATIVELY PRIME to n=d) : Then d is called a k-ary divisor of n, written d ½kn; if the GREATEST COMMON (k /C281)/-ary divisor of d and (n=d)is1. In this notation, d½½n is written d½0n;and d½½nis written d½1n:pxis an INFINARY DIVISOR ofpy(with y/C210) if px½y/C281py:/ See also BIUNITARY DIVISOR ,D IVISOR ,G REATEST COMMON DIVISOR ,INFINARY DIVISOR ,UNITARY DIVI- SOR References Cohen, G. L. "On an Integer’s Infinary Divisors." Math. Comput. 54, 395/C1/411, 1990. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 54, 1994. Suryanarayana, D. "The Number of k-ary Divisors of an Integer." Monatschr. Math. 72, 445 /C1/450, 1968. Katadrome A katadrome is a number whose HEXADECIMAL digits are in strict descending order. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 32, 33, 48, 49, ... (Sloane’s A023797), corresponding to 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 20, 21, 30, 31, .... See also DIGIT,H EXADECIMAL ,M ETADROME ,N IALP- DROME ,PLAINDROME References Sloane, N. J. A. Sequences A023797 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Katona’s Problem Find the minimum number f(n)of SUBSETS in a SEPARATING FAMILY for a SET of n elements, where a SEPARATING FAMILY is a SET of SUBSETS in which each pair of adjacent elements is found separated, each in one of two DISJOINT SUBSETS . For example, the 26 letters of the alphabet can be separated by a family of nine: (abcdefghi )( jklmnopqr )(stuvwxyz ) (abcjklstu )(defmnovwx )(ghipqryz ) (adgjmpsvy )(behknqtwz )( cfilorux ): The problem was posed by Katona (1973) and solved by C. Mao-Cheng in 1982, f(n) /C30min 2p /C273 log3n 2p !&’ : p /C300; 1; 2() ; where xdeis the CEILING FUNCTION . f(n) is nonde- creasing, and the values for n /C301, 2, ... are 0, 2, 3, 4, 5, 5, 6, 6, 6, 7, ... (Sloane’s A007600). The values at which f(n) increases are 1, 2, 3, 4, 5, 7, 10, 13, 19, 28, 37, ... (Sloane’s A007601), so f(26) /C309 ; as illustrated in the preceding example. See also SEPARATING FAMILY References Honsberger, R. "Cai Mao-Cheng’s Solution to Katona’s Problem on Families of Separating Subsets." Ch. 18 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 224 /C1/239, 1985. Katona, G. O. H. "Combinatorial Search Problem." In A Survey of Combinatorial Theory (Ed. J. N. Srivasta, F. Harary, C. R. Rao, G.-C. Rota, and S. S. Shrikhande). Amsterdam, Netherlands: North-Holland, pp. 285 /C1/308, 1973. Sloane, N. J. A. Sequences A007600/M0456 and A007601/ M0525 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Kauffman Polynomial F A semi-oriented 2-variable KNOT POLYNOMIAL defined by FL(a ; z) /C30a /C28w(L) ½L½hi ; (1) where L is an oriented LINK DIAGRAM , w(L) is the WRITHE of L, ½L ½ is the unoriented diagram corre- sponding to L, and /C142L /C143 is the BRACKET POLYNOMIAL . It was developed by Kauffman by extending the BLM / HO POLYNOMIAL Q to two variables, and satisfies F(1; x) /C30Q(x) : (2) The Kauffman POLYNOMIAL is a generalization of the JONES POLYNOMIAL V(t) since it satisfies V(t) /C30F(/C28t/C283 =4 ; t/C281=4 /C27t1 =4) ; (3) but its relationship to the HOMFLY POLYNOMIAL is not well understood. In general, it has more terms than the HOMFLY POLYNOMIAL , and is therefore more powerful for discriminating KNOTS . It is a semi- oriented POLYNOMIAL because changing the orienta- tion only changes F by a POWER of a. In particular, suppose L /C31 is obtained from L by reversing the orientation of component k, then FL /C31/C30a4 lFL ; (4) where l is the LINKING NUMBER of k with L /C28k (Lickorish and Millett 1988). F is unchanged by MUTATION . FL1/C27FL2/C30F(L1)F(L2) (5) FL1@L2/C30[(a/C281 /C27a)x/C281 /C281]FL1FL2: (6) M. B. Thistlethwaite has tabulated the Kauffman 2- variable POLYNOMIAL for KNOTS up to 13 crossings. See also KAUFFMAN POLYNOMIAL X References Lickorish, W. B. R. and Millett, B. R. "The New Polynomial Invariants of Knots and Links." Math. Mag. 61,1/C1/23, 1988. Stoimenow, A. "Kauffman Polynomials." http://guests.mpim- bonn.mpg.de/alex/ptab/k10.html. Weisstein, E. W. "Knots and Links." M ATHEMATICA NOTE- BOOK KNOTS.M . Kauffman Polynomial X A 1-variable KNOT POLYNOMIAL denoted XorL: LL(A)/C13(/C28A3)/C28w(L)/C142L/C143; (1) where /C142L/C143is the BRACKET POLYNOMIAL and w(L)i s the WRITHE ofL. This POLYNOMIAL is invariant under AMBIENT ISOTOPY , and relates MIRROR IMAGES by LL /C31/C30LL(A/C281) : (2) It is identical to the JONES POLYNOMIAL with the change of variable L(t/C281 =4) /C30V(t) : (3) The X POLYNOMIAL of the MIRROR IMAGE K /C31 is the same as for K but with A replaced by A/C281 :/ See also KAUFFMAN POLYNOMIAL F References Kauffman, L. H. Knots and Physics. Singapore: World Scientific, p. 33, 1991. Kaup’s Equation The system of PARTIAL DIFFERENTIAL EQUATIONS fx /C302fgc(x /C28t) gt /C302fgc(x /C28t) : References Dodd, R. and Fordy, A. "The Prolongation Structures of Quasi-Polynomial Flows." Proc. Roy. Soc. A 385, 389 /C1/429, 1983. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. k-Automatic Set AUTOMATIC SET k-Balanced A GENERALIZED HYPERGEOMETRIC FUNCTION pFqa1 ; a2 ; ...; ap b1 ; b2 ; ...; bq; z>C20>C21 ; is said to be k-balanced if Xq i/C301bi /C30k /C27Xp i/C301ai : See also GENERALIZED HYPERGEOMETRIC FUNCTION , NEARLY- POISED ,SAALSCHU ¨ TZIAN ,W ELL-POISED References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 43, 1998. k-Chain Any sum of a selection of Pk/s, where Pk denotes a k-D POLYTOPE . See also K-CIRCUIT ,POLYTOPEk-Circuit A K-CHAIN whose bounding (K-1)-CHAIN vanishes. See also K-CHAIN k-Coloring A k-coloring of a GRAPH G is an assignment of one of k possible colors to each vertex of G (i.e, a VERTEX COLORING ) such that no two adjacent vertices receive the same color. See also CHROMATIC NUMBER ,CHROMATIC POLYNO- MIAL ,COLORING ,EDGE COLORING ,VERTEX COLORING References Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 13, 1986. k-Connected Graph A graph G is said to be k-connected if there does not exist a set of k /C281 vertices whose removal disconnects the graph, i.e., the VERTEX CONNECTIVITY of G is ]k (Skiena 1990, p. 177). Therefore, a CONNECTED GRAPH is 1-connected, and a BICONNECTED GRAPH is 2- connected (Skiena 1990, p. 177). The following table gives the numbers of k-connected graphs for n-node graphs. Note that there is a unique n-connected n-node graph, namely, the COMPLETE GRAPH Kn : The WHEEL GRAPH is the basic 3-connected graph (Tutte 1961; Skiena 1990, p. 179). kk-connected graphs on 1, 2, ... nodes 1 1, 1, 2, 6, 21, 112, 853, ... 2 0, 1, 1, 3, 10, 56, 468, ... 3 0, 0, 1, 1, 3, 17, 136, ... 4 0,0,0,1,1,4,25,... 5 0,0,0,0,1,1,4,... 6 0,0,0,0,0,1,1,... 7 0,0,0,0,0,0,1,... 8 0,0,0,0,0,0,0,... See also BARNETTE’S CONJECTURE ,B ICONNECTED GRAPH ,CONNECTED GRAPH ,DISCONNECTED GRAPH , HARARY GRAPH , K-EDGE-CONNECTED GRAPH ,M EN- GER’S N-ARC THEOREM ,POLYHEDRAL GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 45, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A000719/M1452, A052442, A052443, A052444, and A052445 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Tutte, W. T. "A Theory of 3-Connected Graphs." Indag. Math. 23, 441 /C1/455, 1961. k-Edge-Connected Graph A graph is k-edge-connected if there does not exist a set of k edges whose removal disconnects the graph (Skiena 1990, p. 177). The maximum edge connectiv- ity of a given graph is the smallest degree of any node, since deleting these edges disconnects the graph. Complete bipartite graphs have maximum edge con- nectivity. The following table gives the numbers of k- edge-connected graphs for n-node graphs. k Sloane n /C301, 2, ... 0 A000719 0, 1, 2, 5, 13, 44, 191, ... 1 A052446 0, 1, 1, 3, 10, 52, 351, ... 2 A052447 0, 0, 1, 2, 8, 41, 352, ... 3 A052448 0, 0, 0, 1, 2, 15, 121, ... 4 0,0,0,0,1,3,25,... 5 0,0,0,0,0,1,3,... 6 0,0,0,0,0,0,1,... See also K-CONNECTED GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 45, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A000719/M1452, A052446, A052447, and A052448 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Kei The IMAGINARY PART of e /C28 npi=2Kn(xepi =4) /C30ker n(x) /C27i kei n(x) ; where Kn(z)isa MODIFIED BESSEL FUNCTION OF THESECOND KIND . The special case n /C300 gives the plots shown above. See also BEI,BER,KER,KELVIN FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 379 /C1/381, 1972. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions bern(x);bein(x);kern(x) and kein(x):/"§1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29 /C1/30, 1990. Keith Number A Keith number is an n-digit INTEGER Nsuch that if a Fibonacci-like sequence (in which each term in the sequence is the sum of the nprevious terms) is formed with the first nterms taken as the decimal digits of the number N, then Nitself occurs as a term in the sequence. For example, 197 is a Keith number since it generates the sequence 1, 9, 7, 17, 33, 57, 107,197, ... (Keith). Keith numbers are also called REPFI- GIT NUMBERS . There is no known general technique for findingKeith numbers except by exhaustive search. Keithnumbers are much rarer than the PRIMES , with only 52 Keith numbers with B15 digits: 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788,7385, 7647, 7909, ... (Sloane’s A007629). The numberof Keith numbers having n/C301, 2, ... digits are 0, 6, 2, 9, 7, 10, 2, 3, 2, 0, 2, 4, 2, 3, 3, 3, 5, 3, 5, ... (Sloane’sA050235; Keith), so there are only 71 less than 10 19. It is not known if there are an INFINITE number of Keith numbers. The known prime Keith numbers are 19, 47, 61, 197, 1084051, 74596893730427, ... (Sloane’s A048970). References --. "Table: Repfigit Numbers (Base 10/C31) Less than 1015." J. Recr. Math. 26, 195, 1994. Esche, H. A. "Non-Decimal Replicating Fibonacci Digits." J. Recr. Math. 26, 193 /C1/194, 1994. Heleen, B. "Finding Repfigits--A New Approach." J. Recr. Math. 26, 184 /C1/187, 1994. Keith, M. "Repfigit Numbers." J. Recr. Math. 19,41/C1/42, 1987. Keith, M. "All Repfigit Numbers Less than 100 Billion (1011)." J. Recr. Math. 26, 181 /C1/184, 1994. Keith, M. "Keith Numbers." http://member.aol.com/s6sj7gt/ mikekeit.htm. Keith, M. "Determination of All Keith Numbers Up to 1019." http://member.aol.com/s6sj7gt/keithnum.htm. Pickover, C. "All Known Replicating Fibonacci Digits Less then One Billion." J. Recr. Math. 22, 176, 1990. Piele, D. "Mathematica Pearls: Keith Numbers." Mathema- tica Res. Educ. 6, No. 3, 50 /C1/52, 1997. Piele, D. "Mathematica Pearls: Keith Numbers." Mathema- tica Res. Educ. 7, No. 1, 44 /C1/45, 1998. Robinson, N. M. "All Known Replicating Fibonacci Digits Less than One Thousand Billion (1012)." J. Recr. Math. 26, 188 /C1/191, 1994. Sherriff, K. "Computing Replicating Fibonacci Digits." J. Recr. Math. 26, 191 /C1/193, 1994. Sloane, N. J. A. Sequences A007629, A048970, and A050235 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Keller’s Conjecture Keller conjectured that tiling an n-D space with n-D HYPERCUBES of equal size yields an arrangement in which at least two hypercubes have an entire (n /C281)/- D "side" in common. The CONJECTURE has been proven true for n /C301 to 6, but disproven for n ]10:/ References Cipra, B. "If You Can’t See It, Don’t Believe It." Science 259, 26 /C1/27, 1993. Cipra, B. What’s Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., p. 24, 1993. Kelvin Differential Equation The second-order complex ORDINARY DIFFERENTIAL EQUATION x2yƒ/C27xy?/C28(ix2 /C27 n2)y /C300 (1) (Abramowitz and Stegun 1972, p. 379; Zwillinger 1997, p. 123), whose solutions can be given in terms of the KELVIN FUNCTIONS y /C30bern x /C27i bei n (2) /C30ber/C28 n x /C27i bei /C28n (3) /C30kern x /C27i kein (4)/C30ker/C28 n x /C27i kei/C28 n (5) (Abramowitz and Stegun 1972, p. 379). See also KELVIN FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 379 /C1/381, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997. Kelvin Functions Kelvin defined the Kelvin functions BEI and BER according to bern(x) /C27i bei n(x) /C30Jn(xe3pi=4) (1) /C30e npiJn(xe/C28 pi=4) ; (2) /C30e npi=2In(xe pi=4) (3) /C30e3npi=2In(xe /C283 pi =4); (4) where Jn(x)isaB ESSEL FUNCTION OF THE FIRST KIND and In(x)isa MODIFIED BESSEL FUNCTION OF THE FIRST KIND . These functions satisfy the KELVIN DIFFERENTIAL EQUATION . Similarly, the functions KEI and KER by kern(x) /C27i kein(x) /C30e /C28 npi=2Kn(xe pi=4) ; (5) where Kn(x)i sa MODIFIED BESSEL FUNCTION OF THE SECOND KIND . For the special case n/C300; J0iffiffi ip x>C16>C17 /C30J01 2ffiffiffi 2p (i/C281)x>C16>C17 /C13ber(x)/C27ibei(x):(6) See also BEI,B ER,K EI,K ELVIN DIFFERENTIAL EQUATION ,KER References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 379 /C1/381, 1972. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions bern(x);bein(x);kern(x) and kein(x):/"§1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29 /C1/30, 1990. Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 543 /C1/554, 1987. Kelvin Problem KELVIN’S CONJECTURE Kelvin Transformation Let Dbe a DOMAIN inRnforn]3:Then the transformation v(x?1 ; ...; x?n) /C30a r ? !n/C282 ua2x?1 r ?2 ; ...;a2x?n r ?2 ! onto a domain D ?; where r ?2 /C30x?12/C27.../C27x?n2 is called a Kelvin transformation. If u(x1 ; ...; xn)isa HARMONIC FUNCTION on D, then v(x?1 ; ...; x?n) is also HARMONIC on D ?:/ See also HARMONIC FUNCTION References Itoˆ, K. (Ed.). "Harmonic Functions and Subharmonic Func- tions: Invariance of Harmonicity." §193B in Encyclopedic Dictionary of Mathematics, 2nd ed. Cambridge, MA: MIT Press, p. 725, 1980. Kelvin’s Conjecture What space-filling arrangement of similar polyhedral cells of equal volume has minimal SURFACE AREA ? Kelvin (Thomson 1887) proposed that the solution was the 14-sided TRUNCATED OCTAHEDRON . The iso- perimetric quotient for the TRUNCATED OCTAHEDRON is given by Q /C3036 pV3 S2/C3036p 8ffiffiffi 2p>C0>C1 2 6 /C27 12ffiffiffi 3p>C0>C1 3 /C3064p 31/C27 2ffiffiffi3p>C0>C1 3 :0:753367 : Despite one hundred years of failed attempts and Weyl’s (1952) opinion that the TRUNCATED OCTAHE- DRON could not be improved upon, Weaire and Phelan (1994) discovered a space-filling unit cell consisting of six 14-sided polyhedra and two 12-sided polyhedra that has 0.3% less SURFACE AREA . See also SPACE- FILLING POLYHEDRON ,T RUNCATED OCTAHEDRON References Gray, J. "Parsimonious Polyhedra." Nature 367, 598 /C1/599, 1994. Matzke, E. Amer. J. Botany 32, 130, 1946. Princen, H. M. and Levinson, P. J. Colloid Interface Sci. 120, 172, 1987. Ross, S. Amer. J. Phys. 46, 513, 1978. Thomson, W. Philos. Mag. 25, 503, 1887. Weaire, D. Philos. Mag. Let. 69, 99, 1994. Weaire, D. and Phelan, R. "A Counter-Example to Kelvin’s Conjecture on Minimal Surfaces." Philos. Mag. Let. 69, 107 /C1/110, 1994. Weaire, D. The Kelvin Problem: Foam Structures of Minimal Surface Area. London: Taylor and Francis, 1996. Weyl, H. Symmetry. Princeton, NJ: Princeton University Press, 1952. Williams, R. Science 161, 276, 1968.Kempe Linkage A double rhomboid LINKAGE which gives rectilinear motion from circular without an inversion. See also PEAUCELLIER INVERSOR References Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 126 /C1/127, 1957. Kepler Conjecture In 1611, Kepler proposed that close packing (cubic or hexagonal) is the densest possible SPHERE PACKING (has the greatest h) ; and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic) packing of spheres is known as the KEPLER PROBLEM . Buckminster Fuller (1975) claimed to have a proof, but it was really a description of face-centered cubic packing, not a proof of its optimality (Sloane 1998). A second putative proof of the Kepler conjecture was put forward by W.-Y. Hsiang (Cipra 1991, Hsiang 1992, Hsiang 1993, Cipra 1993), but was subse- quently determined to be flawed (Conway et al. 1994, Hales 1994, Sloane 1998). According to J. H. Conway, nobody who has read Hsiang’s proof has any doubts about its validity: it is nonsense. Soon thereafter, Hales (1997a) published a detailed plan describing how the Kepler conjecture might be proved using a significantly different approach from earlier attempts and making extensive use of compu- ter calculations. Hales subsequently completed a full proof, which appears in a series of papers totaling more than 250 pages (Cipra 1998) The proof relies extensively on methods from the theory of global optimization, linear programming, and interval ar- ithmetic. The computer files containing the computer code and data files for combinatorics, interval arith- metic, and linear programs require over 3 gigabytes of space for storage. See also DODECAHEDRAL CONJECTURE ,KEPLER PRO- BLEM ,KISSING NUMBER ,SPHERE PACKING References Buckminster Fuller, R. Synergetics. London: Macmillan, 1975. Cipra, B. "Gaps in a Sphere Packing Proof?" Science 259, 895, 1993. Cipra, B. "Packing Challenge Mastered at Last." Science 281, 1267, 1998. Cipra, B. "Music of the Spheres." Science 251, 1028, 1991. Conway, J. H.; Hales, T. C.; Muder, D. J.; and Sloane, N. J. A. "On the Kepler Conjecture." Math. Intel. 16,5 , Spring 1994. Eppstein, D. "Sphere Packing and Kissing Numbers." http:// www.ics.uci.edu/~eppstein/junkyard/spherepack.html. Ferguson, S. P. "Sphere Packings. V." http://www.math.l- sa.umich.edu/~samf/MyStuff/Research/draft.ps.gz. Ferguson, S. P. and Hales, T. C. "A Formulation of the Kepler Conjecture." http://www.math.lsa.umich.edu/ ~hales/countdown/form.ps. Hales, T. C. "The Kepler Conjecture." http://www.math.l- sa.umich.edu/~hales/countdown/. Hales, T. C. "An Overview of the Kepler Conjecture." http:// www.math.lsa.umich.edu/~hales/countdown/sphere0.ps. Hales, T. C. "Recent Progress on the Kepler Conjecture." http://www.math.lsa.umich.edu/~hales/countdown/re- cent.ps. Hales, T. C. "The Sphere Packing Problem." J. Comput. Appl. Math. 44,41/C1/76, 1992. Hales, T. C. "Remarks on the Density of Sphere Packings in 3 Dimensions." Combinatori 13, 181 /C1/197, 1993. Hales, T. C. "The Status of the Kepler Conjecture." Math. Intel. 16,47/C1/58, Summer 1994. Hales, T. C. "Sphere Packings. I." Disc. Comput. Geom. 17, 1 /C1/51, 1997a. http://www.math.lsa.umich.edu/~hales/ countdown/sphere1.ps. Hales, T. C. "Sphere Packings. II." Disc. Comput. Geom. 18, 135 /C1/149, 1997b. http://www.math.lsa.umich.edu/~hales/ countdown/sphere2.ps. Hales, T. C. "Sphere Packings. III." http://www.math.lsa.u- mich.edu/~hales/countdown/sphere3.ps. Hales, T. C. "Sphere Packings. IV." http://www.math.lsa.u- mich.edu/~hales/countdown/sphere4.ps. Hales, T. C. "Sphere Packings. VI." http://www.math.lsa.u- mich.edu/~hales/countdown/sphere6.ps. Hsiang, W.-Y. "On Soap Bubbles and Isoperimetric Regions in Noncompact Symmetrical Spaces. 1." Toˆhoku Math. J. 44, 151 /C1/175, 1992. Hsiang, W.-Y. "On the Sphere Packing Problem and the Proof of Kepler’s Conjecture." Int. J. Math. 4, 739 /C1/831, 1993. Hsiang, W.-Y. "A Rejoinder to Hales’s Article." Math. Intel. 17,35/C1/42, Winter 1995. Sloane, N. J. A. "Kepler’s Conjecture Confirmed." Nature 395, 435/C1/436, 1998. Zong, C. and Talbot, J. Sphere Packings. New York: Springer-Verlag, 1999. Kepler Problem Finding the densest not necessarily periodic SPHERE PACKING . See also KEPLER CONJECTURE ,SPHERE PACKING Kepler Solid KEPLER- POINSOT SOLID Kepler’s Equation Kepler’s equation gives the relation between the polar coordinates of a celestial body (like a planet) and the time elapsed from a given initial point. Kepler’sequation is of fundamental importance in celestial mechanics, but cannot be directly inverted in terms of simple functions in order to determine where the planet will be at a given time.LetMbe the mean anomaly (a parameterization of time) and Ethe ECCENTRIC ANOMALY (a parameter- ization of polar angle) of a body orbiting on an ELLIPSE with ECCENTRICITY e, then M/C30E/C28esinE: (1) ForMnot a multiple of p;Kepler’s equation has a unique solution, but is a TRANSCENDENTAL EQUATION and so cannot be inverted and solved directly for E given an arbitrary M. However, many algorithms have been derived for solving the equation as a resultof its importance in celestial mechanics. Writing a Eas a POWER SERIES inegives E/C30M/C27X/C12 n/C301anen; (2) where the coefficients are given by the L AGRANGE INVERSION THEOREM as an/C301 2n/C281n!Xn=2bc k/C300(/C281)kn k>C18>C19 /C2(n/C282k)n/C281sin[(n/C282k)M] (3) (Wintner 1941, Moulton 1970, Henrici 1974, Finch). Surprisingly, this series diverges for e>0:6627434193 . . . ; (4) a value known as the L APLACE LIMIT . In fact, E converges as a GEOMETRIC SERIES with ratio r/C30e 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27e2p expffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27e2p>C16>C17 (5) (Finch). There is also a series solution in B ESSEL FUNCTIONS OF THE FIRST KIND , E/C30M/C27X/C12 n/C3012 nJn(ne) sin( nM): (6) This series converges for all eB1 like a GEOMETRIC SERIES with ratio r/C30e 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2p expffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2p>C16>C17 : (7) The equation can also be solved by letting cbe the ANGLE between the planet’s motion and the direction PERPENDICULAR to the RADIUS VECTOR . Then tanc/C30esinEffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2p : (8) Alternatively, we can define ein terms of an inter- mediate variable f e /C13sin f; (9) then sin1 2(v /C28E)hi /C30ffiffiffi r ps sin12 f>C16>C17 sin v (10) sin1 2(v /C27E)hi /C30ffiffiffi r ps cos12 f>C16>C17 sin v: (11) Iterative methods such as the simple Ei/C271 /C30M /C27e sin Ei (12) with E0 /C300 work well, as does NEWTON’S METHOD , Ei/C271 /C30Ei /C27M /C27 e sin Ei /C28 Ei 1 /C28 e cos Ei: (13) In solving Kepler’s equation, Stieltjes required the solution to ex(x /C281) /C30e /C28x(x /C271); (14) which is 1.1996678640257734... (Goursat 1959, Le Lionnais 1983). See also ECCENTRIC ANOMALY References Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed., rev. ed. Richmond, VA: Willmann-Bell, 1988. Do¨rrie, H. "The Kepler Equation." §81 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 330 /C1/334, 1965. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lpc/lpc.html. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 101 /C1/102 and 123 /C1/124, 1980. Goursat, E. A Course in Mathematical Analysis, Vol. 2. New York: Dover, p. 120, 1959. Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo-cation of Zeros. New York: Wiley, 1974. Ioakimids, N. I. and Papadakis, K. E. "A New Simple Method for the Analytical Solution of Kepler’s Equation."Celest. Mech. 35, 305/C1 /316, 1985. Ioakimids, N. I. and Papadakis, K. E. "A New Class of Quite Elementary Closed-Form Integrals Formulae for Roots ofNonlinear Systems." Appl. Math. Comput. 29, 185/C1 /196, 1989. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 36, 1983. Marion, J. B. and Thornton, S. T. "Kepler’s Equations." §7.8 inClassical Dynamics of Particles & Systems, 3rd ed. San Diego, CA: Harcourt Brace Jovanovich, pp. 261 /C1/266, 1988. Moulton, F. R. An Introduction to Celestial Mechanics, 2nd rev. ed. New York: Dover, pp. 159 /C1/169, 1970. Montenbruck, O. and Pfleger, T. "Mathematical Treatment of Kepler’s Equation." §4.3 in Astronomy on the Personal Computer, 4th ed. Berlin: Springer-Verlag, pp. 62 /C1/63 and 65/C1/68, 2000.Siewert, C. E. and Burniston, E. E. "An Exact Analytical Solution of Kepler’s Equation." Celest. Mech. 6, 294/C1/304, 1972. Wintner, A. The Analytic Foundations of Celestial Me- chanics. Princeton, NJ: Princeton University Press, 1941. Kepler’s Folium The plane curve with implicit equation [(x/C28b)2/C27y2][x(x/C28b)/C27y2]/C304a(x/C28b)y2: References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 93, 1997. Kepler-Poinsot Solid The Kepler-Poinsot solids are the four regular CON- CAVE POLYHEDRA with intersecting facial planes. They are composed of regular CONCAVE POLYGONS and were unknown to the ancients. Kepler discovered two and described them in his work Harmonice Mundi in 1619. These two were subsequently redis- covered by Poinsot, who also discovered the other two,in 1809. As shown by Cauchy, they are stellated forms of the DODECAHEDRON and ICOSAHEDRON . The Kepler-Poinsot solids, illustrated above, are known as the GREAT DODECAHEDRON ,GREAT ICOSA- HEDRON ,GREAT STELLATED DODECAHEDRON , and SMALL STELLATED DODECAHEDRON . These names probably originated with Arthur Cayley, who first used them in 1859. Cauchy (1813) proved that these four exhaust all possibilities for regular star polyhe-dra (Ball and Coxeter 1987). A table listing these solids, their DUALS , and COM- POUNDS is given below. Like the five Platonic solids, duals of the Kepler-Poinsot solids are themselves Kepler-Poinsot solids (Wenninger 1983, pp. 39 and 43/C1/45). n solid UNIFORM POLYHEDRONSCHLA ¨ FLI SYMBOLWYTHOFF SYMBOLPOINTGROUP 1 GREAT DODECA- HEDRON/U35// 5;5 2no //52 ½ 25// Ih/ 2 GREAT ICOSAHE- DRON/U53// 3;52no // 352 ½53// Ih/ 3 GREAT STEL- LATED DODECA- HEDRON/U52//52; 3no // 3 ½ 252// Ih/ 4 SMALL STEL- LATED DODECA- HEDRON/U34//5 2; 5no // 5 ½ 252// Ih/ The polyhedra f5 2 ; 5g and f5 ;52 g fail to satisfy the POLYHEDRAL FORMULA V /C28E /C27F /C302; where V is the number of vertices, E the number of edges, and F the number of faces, despite the fact that the formula holds for all ordinary polyhedra (Ball and Coxeter 1987). This unexpected result led none less than Schla ¨fli (1860) to erroneously conclude that they could not exist. In 4-D, there are 10 Kepler-Poinsot solids, and in n-D with n ]5 ; there are none. In 4-D, nine of the solids have the same VERTICES as f3; 3; 5g; and the tenth has the same as f5; 3; 3g: Their SCHLA ¨ FLI SYMBOLS are f52 5; 3 g;f3; 5;52g;f5 ;52 ; 5g;f52 ; 3 ; 5 g;f5; 3;52g; f5 2 ; 5 ;52 g;f5;52; 3g;f3;52 ; 5 g;f52 ; 3; 3g; and f3 ; 3 ;52 g:/ Coxeter et al. (1954) have investigated star "Archi- medean" polyhedra. See also ARCHIMEDEAN SOLID ,DELTAHEDRON ,JOHN- SON SOLID ,P LATONIC SOLID ,P OLYHEDRON COM- POUND ,UNIFORM POLYHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 144 /C1/146, 1987. Cauchy, A. L. "Recherches sur les polye`dres." J. de l’E´ cole Polytechnique 9,68/C1/86, 1813. Cayley, A. "On Poinsot’s Four New Regular Solids." Philos. Mag. 17, 123 /C1/127 and 209, 1859. Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401 /C1/450, 1954. Pappas, T. "The Kepler-Poinsot Solids." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989. Quaisser, E. "Regular Star-Polyhedra." Ch. 5 in Mathema- tical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 56 /C1/62, 1986. Schla¨fli. Quart. J. Math. 3,66/C1/67, 1860. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 130 /C1/131, 1991.Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 39 /C1/41, 1983. Ker The REAL PART of e/C28npi=2Kn(xepi=4)/C30kern(x)/C27ikein(x); where Kn(x)i sa MODIFIED BESSEL FUNCTION OF THE SECOND KIND . The special case n/C300 gives the plots shown above. See also BEI,BER,KEI,KELVIN FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 379 /C1/381, 1972. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions bern(x);bein(x);kern(x) and kein(x):/"§1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29 /C1/30, 1990. Keratoid Cusp The PLANE CURVE given by the Cartesian equation y2 /C30x2y /C27x5 : References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Kernel (Integral) The function K( a; t)inan INTEGRAL or INTEGRAL TRANSFORM g( a) /C30gb af(t)K( a; t) dt: Whittaker and Robinson (1967, p. 376) use the term nucleus for kernel. See also BERGMAN KERNEL ,INTEGRAL ,P OISSON KERNEL References Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 376, 1967. Kernel (Linear Algebra) NULLSPACE Kernel Polynomial The function Kn(x0 ; x) /C30Kn(x; x0) /C30Kn(¯x; ¯x0) which is useful in the study of many POLYNOMIALS . References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. Kervaire’s Characterization Theorem Let G be a GROUP , then there exists a piecewise linear KNOT Kn/C282 in Sn for n ]5 with G /C30 p1(Sn /C28K) IFF G satisfies1. G is finitely presentable, 2. The Abelianization of G is infinite cyclic, 3. The normal closure of some single element is all of G, 4. H2(G) /C300; the second homology of the group is trivial. References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 350 /C1/351, 1976. Ket A CONTRAVARIANT VECTOR , denoted cji: The ket is DUAL to the COVARIANT BRA one-forms chj: Taken together, the BRA and ket form an ANGLE BRACKET (bra/C27ket /C30bracket) c½ chi : The ket is commonly encountered in quantum mechanics. See also ANGLE BRACKET ,BRA,BRACKET PRODUCT , CONTRAVARIANT VECTOR ,C OVARIANT VECTOR ,D IF- FERENTIAL K-FORM,ONE-FORM References Dirac, P. A. M. "Bra and Ket Vectors." §6in Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, pp. 16 and 18 /C1/22, 1982. k-Factor A k-factor of a GRAPH is a k-regular SUBGRAPH of order n. k-factors are a generalization of complete matchings. A PERFECT MATCHING is a 1-factor (Skiena 1990, p. 244). See also MATCHING References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. k-Factorable Graph A GRAPH G is k-factorable if it is the union of disjoint K-FACTORS (Skiena 1990, p. 244). See also K-FACTOR References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. k-Form DIFFERENTIAL K-FORM K-Function For positive integer n, the K-function is defined by K(n) /C1300112233 /C1/C1/C1(n /C281)n/C281 (1) and is related to the BARNES’ G-FUNCTION by K(n) /C30[ G(n)]n/C281 G(n); (2) where G(n) is defined by G(n) /C301i f n /C300 0!1!2! /C1/C1/C1(n /C282)! if n > 0 :>C26 (3) The K-function is given by the integral K(z) /C30(2p)/C28(z/C281)=2expz 2>C18>C19 /C27gz/C281 0ln(t!) dt"# (4) and the closed-form expression K(z) /C30exp[ z?(/C281; z) /C28 z?(/C281)] ; (5) where z(z) is the RIEMANN ZETA FUNCTION , z?(z) its DERIVATIVE , z(a ; z) is the HURWITZ ZETA FUNCTION , and z?(a ; z) /C13dz(s ; z) ds"# s/C30a: (6) /K(z) also has a STIRLING -like series K(z /C271) /C30(21 =3 p1z)1=12zz /C271 2>C18>C19 /C29exp1 4 z2 /C271 12 /C28B4 2 /C215 3 /C215 4z2 /C28B6 4 /C215 5 /C215 6z4 /C28... ! ; (7)where p1 /C13 K1 2>C16>C17hi8 (8) /C30e/C28(ln 2)=3/C2812 z?(/C281) (9) /C3022 =3 pe g/C281 /C28 z?(2)= z(2) ; (10) and g is the EULER- MASCHERONI CONSTANT (Gosper). The first few values of K(n) for n /C301, 2, ... are 1, 1, 1, 4, 108, 27648, 86400000, 4031078400000, ... (Sloane’s A002109). These numbers are called HYPERFACTOR- IALS by Sloane and Plouffe (1995). See also BARNES’ G-FUNCTION ,G LAISHER- KINKELIN CONSTANT ,HYPERFACTORIAL ,STIRLING’S SERIES References Sloane, N. J. A. Sequences A002109/M3706 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 264, 1990. K-Graph The GRAPH obtained by dividing a set of VERTICES f1; ...; ng into k /C281 pairwise disjoint subsets with VERTICES of degree n1 ; ..., nk /C281 ; satisfying n /C30n1 /C27.../C27nk /C281 ; and with two VERTICES joined IFF they lie in distinct VERTEX sets. Such GRAPHS are denoted Kn1;...;nk:/ See also BIPARTITE GRAPH ,COMPLETE GRAPH ,COM- PLETE K-PARTITE GRAPH , K-PARTITE GRAPH Khinchin KHINTCHINE’S CONSTANT Khinchin Constant KHINTCHINE’S CONSTANT Khintchine’s Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let x/C30[a0;a1;... ]/C30a0/C271 a1/C271 a2/C271 a3/C27...(1) be the SIMPLE CONTINUED FRACTION of a REAL NUM- BER x, where the numbers aiare the PARTIAL QUOTIENTS . Khintchine (1934) considered the limit of the GEOMETRIC MEAN Gn(x)/C30(a1a2/C1/C1/C1an)1=n(2) asn0/C12:Amazingly enough, this limit is a constant independent ofx–except if xbelongs to a set of MEASURE 0-given by K/C302:685452001 . . . (3) (Sloane’s A002210), as proved in Kac (1959). The constant is built into Mathematica 4.0 asKhinchin . The values Gn(x) are plotted above for n/C301 to 500 and x/C30p;1=p;sin 1 ;the E ULER- MASCHERONI CON- STANT g;and the C OPELAND- ERDOS CONSTANT .REAL NUMBERS xfor which limn0/C12Gn(x)"Kinclude x/C30e,ffiffiffi 2p ;ffiffiffi 3p ;and the GOLDEN RATIO f;plotted below. The CONTINUED FRACTION forKis [2, 1, 2, 5, 1, 1, 2, 1, 1, ...] (Sloane’s A002211; Havermann). It is not known ifKisIRRATIONAL , let alone TRANSCENDENTAL . Bailey et al. (1995) have computed Kto 7350 DIGITS . Explicit expressions for Kinclude K/C30Y/C12 n/C3011/C271 n(n/C272)"#lnn=ln 2 (4) ln 2 ln K/C301 12p2/C271 2(ln 2)2/C27gp 0ln(u½cotu½)du u(5) lnK/C301 ln 2X/C12 m/C301hm/C281 m[z(2m)/C281]; (6) where z(z) is the R IEMANN ZETA FUNCTION andhm/C30Xm j/C301(/C281)j/C281 j(7) (Shanks and Wrench 1959). Gosper gave lnK/C301 ln 2X/C12 j/C302(/C281)j(2/C282j)z?(j) j; (8) where z?(z) is the DERIVATIVE of the R IEMANN ZETA FUNCTION . An extremely rapidly converging sum also due to Gosper is lnK/C301 ln 2X/C12 k/C300>C26 /C28ln(k/C271)[ln( k/C273) /C282 ln( k/C272)/C27ln(k/C271)] /C28(/C281)k(2/C282k/C272) k/C272 /C2ln(k/C271) (k/C271)k/C272/C28z?(k/C272;k/C272)"# /C27ln(k/C271)Xk/C272 s/C301(/C281)s(2/C282s) (k/C271)ss"#>C27 ; (9) where z(s;a) is the H URWITZ ZETA FUNCTION . Khintchine’s constant is also given by the integral ln 2 ln1 2K>C16>C17 /C30g1 01 x(1/C27x)lnpx(1/C28x2) sin(px)"# dx:(10) IfPn=Qnis the nthCONVERGENT of the CONTINUED FRACTION ofx, then lim n0/C12(Qn)1=n/C30lim n0/C12Pn x !1=n /C30ep2=(12 ln 2):3:27582 (11) for almost all REAL x(Le´vy 1936, Finch). This number is sometimes called the L E´VY CONSTANT , and the argument of the exponential is sometimes called the KHINTCHINE- LE´VY CONSTANT . Define the following quantity in terms of the kth partial quotient qk; M(s;n;x)/C301 nXn k/C301qs k ! 1=s : (12) Then lim n0/C12M(1;n;x)/C30/C12 (13) for almost all real x(Khintchine, Knuth 1981, Finch), and M(1;n;x)/C2O(lnn): (14) Furthermore, for sB1, the limiting value lim n0/C12M(s ; n ; x) /C30K(s) (15) exists and is a constant K(s) with probability 1 (Rockett and Szu¨sz 1992, Khintchine 1997). See also CONTINUED FRACTION ,C ONVERGENT , KHINTCHINE- LE´ VY CONSTANT ,LE´ VY CONSTANT ,PAR- TIAL QUOTIENT ,SIMPLE CONTINUED FRACTION References Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khintchine Constant." Math. Comput. 66, 417 /C1/431, 1997. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/khntchn/ khntchn.html. Havermann, H. "Simple Continued Fraction Expansion of Khinchin’s Constant." http://members.home.net/hahaj/ cfk.html. Kac, M. Statistical Independence and Probability, Analysts and Number Theory. Providence, RI: Math. Assoc. Amer., 1959. Khinchin, A. Ya. Continued Fractions. New York: Dover, 1997. Knuth, D. E. Exercise 24 in The Art of Computer Program- ming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 604, 1998. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983. Lehmer, D. H. "Note on an Absolute Constant of Khintch- ine." Amer. Math. Monthly 46, 148 /C1/152, 1939. Phillipp, W. "Some Metrical Theorems in Number Theory." Pacific J. Math. 20, 109 /C1/127, 1967. Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u- qam.ca/pi/records.html. Rockett, A. M. and Szu¨sz, P. Continued Fractions. Singa- pore: World Scientific, 1992. Shanks, D. and Wrench, J. W. "Khintchine’s Constant." Amer. Math. Monthly 66, 148 /C1/152, 1959. Sloane, N. J. A. Sequences A002210/M1564 and A002211/ M0118 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Vardi, I. "Khinchin’s Constant." §8.4 in Computational Recreations in Mathematica. Reading, MA: Addison-Wes- ley, pp. 163 /C1/171, 1991. Wolfram, S. The Mathematica Book, 4th ed. Cambridge, England: Cambridge University Press, pp. 756 /C1/757, 1999. Wrench, J. W. "Further Evaluation of Khintchine’s Con- stant." Math. Comput. 14, 370/C1/371, 1960. Khintchine-Le ´vy Constant A constant related to K HINTCHINE’S CONSTANT and defined by KL/C13p2 12 ln 2/C301:1865691104 . . . : See also KHINTCHINE’S CONSTANT ,LE´ VY CONSTANT References Plouffe, S. "Khintchine-Levy Constant." http://www.lacim.u- qam.ca/piDATA/klevy.txt.Khovanski’s Theorem Iff1;...;fm:Rn0Rare exponential polynomials, then fx/C23Rn:f1(x)/C30/C1/C1/C1fn(x)/C300ghas finitely many connected components. References Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753/C1/759, 1996. Kiepert’s Conics KIEPERT’S HYPERBOLA ,KIEPERT’S PARABOLA Kiepert’s Hyperbola A curve which is related to the solution of L EMOINE’S PROBLEM and its generalization to ISOSCELES TRIAN- GLES constructed on the sides of a given TRIANGLE . The VERTICES of the constructed TRIANGLES are A?/C30/C28 sinf: sin( C/C27f) : sin( B/C27f) (1) B?/C30sin(C/C27f):/C28sinf: sin( A/C27f) (2) C?/C30sin(B/C27f) : sin( A/C27f):/C28sinf; (3) where fis the base ANGLE of the ISOSCELES TRIANGLE . Kiepert showed that the lines connecting the VER- TICES of the given TRIANGLE and the corresponding peaks of the ISOSCELES TRIANGLES CONCUR . The TRILINEAR COORDINATES of the point of concurrence are sin(B/C27f) sin( C/C27f) : sin( C/C27f) sin( A/C27f): sin(A/C27f) sin( B/C27f): (4) The LOCUS of this point as the base ANGLE varies is given by the curve sin(B/C28C) a/C27sin(C/C28A) b/C27sin(A/C28B) g /C30bc(b2/C28c2) a/C27ca(c2/C28a2) b/C27ab(a2/C28b2) g/C300: (5) Writing the TRILINEAR COORDINATES as ai/C30disi; (6) where diis the distance to the side opposite aiof length siand using the POINT-LINE DISTANCE FOR- MULA with ( x0;y0) written as ( x, y), di/C30j(yi/C272/C28yi/C271)(x/C28xi/C271) si /C28(xi/C272/C28xi/C271)(y/C28yi/C271)j si; (7) where y4/C13y1andy5/C13y2gives the FORMULA X3 i/C301si /C271si/C272(s2 i/C271 /C28s2i/C272) /C2si (yi/C272 /C28 yi /C271)(x /C28 xi/C271) /C28 (xi/C272 /C28 xi /C271)(y /C28 yi/C271) /C300 (8) X3 i/C301(s2 i/C271 /C28 s2i /C272) (yi/C272 /C28 yi /C271)(x /C28 xi/C271) /C28 (xi/C272 /C28 xi /C271)(y /C28 yi/C271) /C300: (9) Bringing this equation over a common DENOMINATOR then gives a quadratic in x and y, which is a CONIC SECTION (in fact, a HYPERBOLA ). The curve can also be written as csc(A /C27t) : csc(B /C27t) : csc(C /C27t) ; as t varies over [/C28p=4; p=4]:/ Kiepert’s hyperbola passes through the triangle’s CENTROID M (/f /C300); ORTHOCENTER H (/f /C30 p=2); VERTICES A (/f /C30/C28a if a 5 p=2 and f /C30 p /C28 a if a > p=2); B (/ f /C30/C28b) ; C (/ f /C30/C28g); FERMAT POINTS F1(/f /C30 p=3) and F2(/f /C30/C28p=3); ISOGONAL CONJUGATE of the BROCARD MIDPOINT (/f /C30 v) ; and BROCARD’S THIRD POINT Z3(/f /C30 v) ; where v is the BROCARD ANGLE (Eddy and Fritsch 1994, p. 193). The ASYMPTOTES of Kiepert’s hyperbola are the SIMSON LINES of the intersections of the BROCARD AXIS with the CIRCUMCIRCLE . Kiepert’s hyperbola is a RECTANGULAR HYPERBOLA . In fact, all nondegenerate conics through the VERTICES and ORTHOCENTER of a TRIANGLE are RECTANGULAR HYPERBOLAS the centers of which lie halfway between the FERMAT POINTS and on the NINE-POINT CIRCLE . The LOCUS of centers of these HYPERBOLAS is the NINE-POINT CIRCLE . The ISOGONAL CONJUGATE curve of Kiepert’s hyper- bola is the B ROCARD AXIS . The center of the INCIRCLE of the TRIANGLE constructed from the MIDPOINTS of the sides of a given TRIANGLE lies on Kiepert’s hyperbola of the original TRIANGLE . See also BROCARD ANGLE ,BROCARD AXIS,BROCARD POINTS ,CENTROID (TRIANGLE ), CIRCUMCIRCLE ,FER- MAT POINTS ,ISOGONAL CONJUGATE ,ISOSCELES TRI- ANGLE ,K IEPERT’S PARABOLA ,LEMOINE’S PROBLEM , NINE-POINT CIRCLE ,ORTHOCENTER ,SIMSON LINE References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account ofIts Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Eddy, R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Trian-gle." Math. Mag. 67, 188/C1 /205, 1994. Kelly, P. J. and Merriell, D. "Concentric Polygons." Amer. Math. Monthly 71,3 7/C1/41, 1964. Mineuer, A. "Sur les asymptotes de l’hyperbole de Kiepert." Mathesis 49,3 0/C1/33, 1935. Rigby, J. F. "A Concentrated Dose of Old-Fashioned Geo- metry." Math. Gaz. 57, 296/C1/298, 1953. Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian-gle." Amer. Math. Monthly 72, 1091 /C1 /1094, 1965. Kiepert’s Parabola Let three similar ISOSCELES TRIANGLES DA?BC; DAB?C;andDABC?be constructed on the sides of a TRIANGLE DABC :Then DABC andDA?B?Cƒare PER- SPECTIVE TRIANGLES , and the ENVELOPE of their PERSPECTIVE AXIS as the vertex angle of the erected triangles is varied is a PARABOLA known as Kiepert’s parabola. It has equation sinA(sin2B/C28sin2C) u/C27sinB(sin2C/C28sin2A) v /C27sinC(sin2A/C28sin2B) w/C300 (1) a(b2/C28c2) u/C27b(c2/C28a2) v/C27c(a2/C28b2) w/C300; (2) where [ u;v;w] are the TRILINEAR COORDINATES for a line tangent to the parabola. Kiepert’s parabola is tangent to the sides of the TRIANGLE (or their extensions), the line at infinity, and the L EMOINE LINE . The FOCUS has TRIANGLE CENTER FUNCTION a /C30csc(B /C28C) : (3) The EULER LINE of a triangle is the DIRECTRIX of Kiepert’s parabola. In fact, the DIRECTRICES of all parabolas inscribed in a TRIANGLE pass through the ORTHOCENTER . The BRIANCHON POINT for Kiepert’s parabola is the STEINER POINT of DABC :/ See also BRIANCHON POINT ,ENVELOPE ,EULER LINE, ISOSCELES TRIANGLE ,L EMOINE LINE,P ARABOLA , STEINER POINTS Kieroid Let the center B of a CIRCLE of RADIUS a move along a line BA. Let O be a fixed point located a distance c away from AB. Draw a SECANT LINE through O and D, the MIDPOINT of the chord cut from the line DE (which is parallel to AB) and a distance b away. Then the LOCUS of the points of intersection of OD and the CIRCLE P1 and P2 is called a kieroid. Special Case Curve b /C300 CONCHOID OF NICOMEDES b /C30a CISSOID plus asymptote /b /C30a /C30/C28c/ STROPHOID plus ASYMPTOTE References Yates, R. C. "Kieroid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 141 /C1/142, 1952. Killing Form The Killing form is an INNER PRODUCT on a finite dimensional LIE ALGEBRA g defined by B(X ; Y) /C30Tr(ad)( X) ad(Y)) (1) in the ADJOINT REPRESENTATION , where ad(X) is the adjoint representation of X. (1) is adjoint-invariant in the sense that B(ad(X)Y ; Z) /C30/C28B(Y ; ad(X)Z) : (2) When g is a SEMISIMPLE LIE ALGEBRA , the Killing form is NONDEGENERATE . For example, the SPECIAL LINEAR LIE ALGEBRA sl2(C) has three basis vectors fX ; Y ; H g; where [X ; Y] /C30/ 2H: X /C3001 10>C20>C21 (3) Y /C300 /C281 10>C20>C21 (4)H /C30100 /C281>C20>C21 : (5) The other brackets are given by [X ; H] /C302Y and [Y ; H] /C302X : In the adjoint representation, with the ordered basis fX;Y;Hg;these elements are repre- sented by ad(X)/C300000020202 435 (6) ad(Y)/C30002 000 /C282002 435 (7) ad(H)/C300/C2820 /C2820 0 00 02 435; (8) and so B(u;v)/C30u TBvwhere B/C3080 0 0/C2880 00 82 435: (9) See also C ARTAN MATRIX ,INNER PRODUCT ,L IE ALGEBRA ,S EMISIMPLE LIE ALGEBRA ,S IGNATURE (MATRIX ), SPECIAL LINEAR LIE ALGEBRA ,W EYL GROUP References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Huang, J.-S. "The Killing Form." §4.4 in Lectures on Representation Theory. Singapore: World Scientific, pp. 33 /C1/36, 1999. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Killing Vectors If any set of points is displaced by Xidxiwhere all distance relationships are unchanged (i.e., there is an ISOMETRY ), then the VECTOR FIELD is called a Killing vector. gab/C30@x?c @xa@x?d @xbgcd(x?); (1) so let x?a/C30xa/C27exa(2) @x?a @xb/C30da b/C27exa ;b (3) gab(x)/C30(dca/C27exc ;a)(ddb/C27exd ;b)gcd(xe/C27eXe) /C30(dca/C27exc ;a)(ddb/C27exd ;b)[gcd(x)/C27eXegcd(x);e/C27... ] /C30gab(x) /C27 e[gadXd ;b /C27gbdXd ;a /C27Xegab ;e] /C27O( e2) /C30gab /C27LXgab /C30g?ab ; (4) where L is the LIE DERIVATIVE . An ordinary derivative can be replaced with a COVARIANT DERIVATIVE in a LIE DERIVATIVE ,sowe can take as the definition gab; c/C300 (5) gabgbc /C30 dc a ; (6) which gives KILLING’S EQUATION LXgab /C30Xa; b /C27Xb; a /C302X(a; b) /C300; (7) where X(a; b) denotes the SYMMETRIC TENSOR part and Xa; b is a COVARIANT DERIVATIVE . A Killing vector Xb satisfies gbcXc; ab /C28RabXb /C300 (8) Xa; bc /C30RabcdXd (9) Xa; b ;b /C27Ra c Xc /C300 ; (10) where Rabis the RICCI TENSOR and Rabcd is the RIEMANN TENSOR . A 2-sphere with METRIC ds2 /C30du2 /C27sin2 u df2 (11) has three Killing vectors, given by the angular momentum operators ˜Lx /C30/C28cos f@ @ u /C27cot u sin f@ @ f (12) ˜Ly /C30sin f@ @ u /C27cot u cos f@ @ f (13) ˜Lz /C30@ @ f : (14) The Killing vectors in Euclidean 3-space are x1 /C30@ @x (15) x2 /C30@ @y (16) x3 /C30@ @z (17) x4 /C30y@ @z /C28z@ @y (18)x5 /C30z@ @x /C28x@ @z (19) x6 /C30x@ @y /C28y@ @x : (20) In MINKOWSKI SPACE , there are 10 Killing vectors X m i /C30am ifor i /C301; 2; 3; 4 (21) X0 k /C300 (22) Xl k /C30 elkmxmfor k /C301 ; 2 ; 3 (23) Xk m /C30 d[0zk] m for k /C301; 2 ; 3 : (24) The first group is TRANSLATION , the second ROTATION , and the final corresponds to a "boost. " See also KILLING’S EQUATION ,LIE DERIVATIVE Killing’s Equation The equation defining KILLING VECTORS . LXgab /C30Xa; b /C27Xb; a /C302X(a; b) /C300; where L is the LIE DERIVATIVE and Xb; ais a COVARIANT DERIVATIVE . See also KILLING VECTORS ,LIE DERIVATIVE References Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, pp. 23 /C1/26, 1996. Kilroy Curve The curve defined by the Cartesian equation f(x) /C30lnsin x x>C12>C12>C12>C12>C12>C12>C12>C12>C12>C12/C30 ln sinc x jj : The Kilroy curve arises in the study of spread spectra plotted on a logarithmic (decibel) scale, and is so named because it resembles Kilroy looking over a wall. See also S INC FUNCTION Kimberling Sequence A sequence generated by beginning with the POSITIVE INTEGERS , then iteratively applying the following algorithm: 1. In iteration i, discard the ith element, 2. Alternately write the i /C27k and i /C28k/th elements until k /C30i, 3. Write the remaining elements in order. The first few iterations are therefore The diagonal elements form the sequence 1, 3, 5, 4, 10, 7, 15, ... (Sloane’s A007063). See also PERFECT SHUFFLE ,SHUFFLE References Guy, R. K. "The Kimberling Shuffle." §E35 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 235 /C1/236, 1994. Kimberling, C. "Problem 1615." Crux Math. 17, 44, 1991. Sloane, N. J. A. Sequences A007063/M2387 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Kimberling Shuffle KIMBERLING SEQUENCE King Walk DELANNOY NUMBER Kings Problem The problem of determining how many nonattacking kings can be placed on an n /C29n CHESSBOARD . For n /C308, the solution is 16, as illustrated above (Mada- chy 1979). In general, the solutions are K(n) /C301 4 n2 n even 1 4(n /C271)2n odd( (1) (Madachy 1979), giving the sequence of doubled squares 1, 1, 4, 4, 9, 9, 16, 16, ... (Sloane’s A008794).This sequence has GENERATING FUNCTION 1 /C27 x2 (1 /C28 x2)2(1 /C28 x) /C301 /C27x /C274x2 /C274x3 /C279x4 /C279x5 /C27...: (2) The minimum number of kings needed to attack or occupy all squares on an 8 /C298 CHESSBOARD is nine, illustrated above (Madachy 1979). See also BISHOPS PROBLEM ,CHESS ,HARD HEXAGON ENTROPY CONSTANT ,K NIGHTS PROBLEM ,Q UEENS PROBLEM ,ROOKS PROBLEM References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 39, 1979. Sloane, N. J. A. Sequences A008794 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Kinney’s Set A set of plane MEASURE 0 that contains a CIRCLE of every RADIUS . References Falconer, K. J. The Geometry of Fractal Sets. New York: Cambridge University Press, 1985. Fejzic, H. "On Thin Sets of Circles." Amer. Math. Monthly 103, 582 /C1/585, 1996. Kinney, J. R. "A Thin Set of Circles." Amer. Math. Monthly 75, 1077 /C1/1081, 1968. Kinoshita-Terasaka Knot The KNOT with BRAID WORD s3 1 s23 s2 s/C281 3s/C282 1s2 s /C281 1s/C281 3s/C281 2: Its JONES POLYNOMIAL is t/C284(/C281 /C272t /C282t2 /C272t3 /C27t6 /C282t7 /C272t8 /C282t9 /C27t10) ; thesame as for C ONWAY’S KNOT . It has the same ALEXANDER POLYNOMIAL as the UNKNOT . See also CONWAY’S KNOT,KNOT,UNKNOT References Kinoshita, S. and Terasaka, H. "On Unions of Knots." Osaka Math. J. 9, 131/C1/153, 1959. Kinoshita-Terasaka Mutants References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 49 /C1/50, 1994. Kirby Calculus The manipulation of DEHN SURGERY descriptions by a certain set of operations. See also DEHN SURGERY References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 263, 1994. Kirby’s List A list of problems in low-dimensional TOPOLOGY maintained by R. C. Kirby. The list currently runs about 380 pages. References Kirby, R. "Problems in Low-Dimensional Topology." http:// www.math.berkeley.edu/~kirby/. Kirkman Points The 60 PASCAL LINES of a HEXAGON inscribed in a conic intersect three at a time through 20 STEINER POINTS , and also three at a time in 60 points known as Kirkman points. Each STEINER POINT lines together with three Kirkman points on a total of 20 lines known as CAYLEY LINES . There is a dual relationship between the 60 Kirkman points and the 60 PASCAL LINES . See also CAYLEY LINES,P ASCAL LINES,P ASCAL’S THEOREM ,PLU¨ CKER LINES,SALMON POINTS ,STEINER POINTS References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 236 /C1/237, 1929. Kirkman, T. P. Cambridge Dublin Math. J. 5, 185. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 116, 1893. Salmon, G. "Notes: Pascal’s Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 379 /C1/ 382, 1960. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 172, 1991.Kirkman Triple System A Kirkman triple system of order v /C306n /C273isa STEINER TRIPLE SYSTEM with parallelism (Ball and Coxeter 1987), i.e., one with the following additional stipulation: the set of b /C30(2n /C271)(3n /C271) triples is partitioned into (3n /C271) components such that each component is a (2n /C271)/-subset of triples and each of the v elements appears exactly once in each compo- nent. The STEINER TRIPLE SYSTEMS of order 3 and 9 are Kirkman triple systems with n /C300 and 1. Solu- tion to KIRKMAN’S SCHOOLGIRL PROBLEM requires construction of a Kirkman triple system of order n /C302. Ray-Chaudhuri and Wilson (1971) showed that there exists at least one Kirkman triple system for every NONNEGATIVE order n. Earlier editions of Ball and Coxeter (1987) gave constructions of Kirkman triple systems with 9 5v B99: For n /C301, there is a single unique (up to an isomorphism) solution, while there are 7 different systems for n /C302 (Mulder 1917, Cole 1922, Ball and Coxeter 1987). See also STEINER TRIPLE SYSTEM References Abel, R. J. R. and Furino, S. C. "Kirkman Triple Systems." §I.6.3 in The CRC Handbook of Combinatorial Designs (Ed. C. J. Colbourn and J. H. Dinitz). Boca Raton, FL: CRC Press, pp. 88 /C1/89, 1996. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 287 /C1/289, 1987. Kirkman, T. P. "On a Problem in Combinations." Cambridge and Dublin Math. J. 2, 191/C1/204, 1847. Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. Mulder, P. Kirkman-Systemen. Groningen Dissertation. Leiden, Netherlands, 1917. Ray-Chaudhuri, D. K. and Wilson, R. M. "Solution of Kirk- man’s Schoolgirl Problem." Combinatorics, Proc. Sympos. Pure Math., Univ. California, Los Angeles, Calif., 1968 19, 187/C1/203, 1971. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 101 /C1/102, 1963. Kirkman’s Schoolgirl Problem In a boarding school there are fifteen schoolgirls who always take their daily walks in rows of threes. Howcan it be arranged so that each schoolgirl walks in thesame row with every other schoolgirl exactly once a week? Solution of this problem is equivalent to constructing a K IRKMAN TRIPLE SYSTEM of order n/C302. The following table gives one of the 7 distinct (up to permutations of letters) solutions to theproblem. Sun Mon Tue Wed Thu Fri Sat ABC ADE AFG AHI AJK ALM ANO DHL BIK BHJ BEG CDF BEF BDG EJN CMO CLN CMN BLO CIJ CHK FIO FHN DIM DJO EHM DKN EIL GKM GJL EKO FKL GIN GHO FJM (The table of Do¨rrie 1965 contains four omissions in which the a1 /C30B and a2 /C30C entries for Wednesday and Thursday are written simply as a.) See also JOSEPHUS PROBLEM ,KIRKMAN TRIPLE SYS- TEM,STEINER TRIPLE SYSTEM References Abel, R. J. R. and Furino, S. C. "Kirkman Triple Systems." §I.6.3 in The CRC Handbook of Combinatorial Designs (Ed. C. J. Colbourn and J. H. Dinitz). Boca Raton, FL: CRC Press, pp. 88 /C1/89, 1996. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 287 /C1/289, 1987. Carpmael. Proc. London Math. Soc. 12, 148 /C1/156, 1881. Cole, F. N. "Kirkman Parades." Bull. Amer. Math. Soc. 28, 435 /C1/437, 1922. Do¨rrie, H. §5in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 14 /C1/18, 1965. Frost, A. "General Solution and Extension of the Problem of the 15 School Girls." Quart. J. Pure Appl. Math. 11,26/C1/ 37, 1871. Kirkman, T. P. "On a Problem in Combinatorics." Cam- bridge and Dublin Math. J. 2, 191 /C1/204, 1847. Kirkman, T. P. Lady’s and Gentleman’s Diary . 1850. Kraitchik, M. §9.3.1 in Mathematical Recreations. New York: W. W. Norton, pp. 226 /C1/227, 1942. Peirce, B. "Cyclic Solutions of the School-Girl Puzzle." Astron. J. 6, 169 /C1/174, 1859 /C1/1861. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 101 /C1/102, 1963. Woolhouse. Lady’s and Gentleman’s Diary . 1862 /C1/1863. Kiss Surface The QUINTIC SURFACE given by the equation 1 2x5/C2712x4/C28(y2/C27z2)/C300: See also QUINTIC SURFACEReferences Nordstrand, T. "Surfaces." http://www.uib.no/people/nfytn/ surfaces.htm. Kissing Circles Problem DESCARTES CIRCLE THEOREM ,SODDY CIRCLES Kissing Number The number of equivalent HYPERSPHERES inn-D which can touch an equivalent HYPERSPHERE without any intersections, also sometimes called the N EWTON NUMBER ,CONTACT NUMBER ,COORDINATION NUMBER , or LIGANCY . Newton correctly believed that the kissing number in 3-D was 12, but the first proofs were not produced until the 19th century (Conway and Sloane 1993, p. 21) by Bender (1874), Hoppe (1874), and Gu ¨nther (1875). More concise proofs were published by Schu ¨tte and van der Waerden (1953) and Leech (1956). After packing 12 spheres aroundthe central one (which can be done, for example, byarranging the spheres so that their points of tangencywith the central sphere correspond to the vertices of an ICOSAHEDRON ), there is a significant amount of free space left (above figure), although not enough to fit a 13th sphere. Exact values for lattice packings are known for n/C301 to 9 and n/C3024 (Conway and Sloane 1992, Sloane and Nebe). Odlyzko and Sloane (1979) found the exactvalue for 24-D. The arrangement of npoints on the surface of a sphere, corresponding to the placement of nidentical spheres around a central sphere (not necessarily of the same radius) is called a SPHERICAL PACKING . The following table gives the largest known kissing numbers in DIMENSION Dfor lattice ( L) and non- lattice ( NL) packings (if a nonlattice packing with higher number exists). In nonlattice packings, the kissing number may vary from sphere to sphere, so the largest value is given below (Conway and Sloane1993, p. 15). A more extensive and up-to-date tabula- tion is maintained by Sloane and Nebe. D L NL D L NL 12 1 3 /]918 /]1,130 26 1 4]1,422 ]1,582 31 2 15]2,340 42 4 16]4,320 54 0 17]5,346 67 2 18]7,398 7 126 19 ]10,668 8 240 20 ]17,400 9 272 /]306 / 21 ]27,720 10 /]336 //]500 / 22 /]49 ;896 / 11 /]438 //]582 / 23 ]93,150 12 /]756 //]840 / 24 196,560 The lattices having maximal packing numbers in 12- and 24-D have special names: the COXETER- TODD LATTICE and LEECH LATTICE , respectively. The gen- eral form of the lower bound of n-D lattice densities given by h ]z(n) 2n/C281 ; where z(n) is the RIEMANN ZETA FUNCTION , is known as the MINKOWSKI-HLAWKA THEOREM . See also COXETER -TODD LATTICE ,H ERMITE CON- STANTS ,H YPERSPHERE PACKING ,K EPLER CONJEC- TURE ,L EECH LATTICE ,M INKOWSKI- HLAWKA THEOREM ,SPHERE PACKING References Bender, C. "Bestimmung der gro¨ssten Anzahl gleich Kugeln, welche sich auf eine Kugel von demselben Radius, wie die u¨brigen, auflegen lassen." Archiv Math. Physik (Grunert) 56, 302 /C1/306, 1874. Conway, J. H. and Sloane, N. J. A. "The Kissing Number Problem" and "Bounds on Kissing Numbers." §1.2 and Ch. 13 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 21 /C1/24 and 337 /C1/339, 1993. Edel, Y.; Rains, E. M.; Sloane, N. J. A. "On Kissing Numbers in Dimensions 32 to 128." Electronic J. Combinatorics 5, No. 1, R22, 1 /C1/5, 1998. http://www.combinatorics.org/Vo- lume_5/v5i1toc.html. Gu¨nther, S. "Ein stereometrisches Problem." Archiv Math. Physik 57, 209 /C1/215, 1875. Hoppe, R. "Bemerkung der Redaction." Archiv Math. Physik. (Grunert) 56, 307 /C1/312, 1874. Kuperberg, G. "Average Kissing Numbers for Sphere Pack- ings." Preprint. Kuperberg, G. and Schramm, O. "Average Kissing Numbers for Non-Congruent Sphere Packings." Math. Res. Let. 1, 339 /C1/344, 1994. Leech, J. "The Problem of Thirteen Spheres." Math. Gaz. 40, 22 /C1/23, 1956. Odlyzko, A. M. and Sloane, N. J. A. "New Bounds on the Number of Unit Spheres that Can Touch a Unit Sphere in n Dimensions." J. Combin. Th. A 26, 210 /C1/214, 1979.Schu¨tte, K. and van der Waerden, B. L. "Das Problem der dreizehn Kugeln." Math. Ann. 125, 325 /C1/334, 1953. Sloane, N. J. A. Sequences A001116/M1585 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Nebe, G. "Table of Highest Kissing Numbers Presently Known." http://www.research.att.com/ ~njas/lattices/kiss.html. Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, pp. 82 /C1/84, 1987. Zong, C. and Talbot, J. Sphere Packings. New York: Springer-Verlag, 1999. Kite A planar convex QUADRILATERAL consisting of two adjacent sides of length a and the other two sides of length b. The RHOMBUS is a special case of the kite, and the LOZENGE is a special case of the RHOMBUS . The AREA of a kite is given by A /C301 2 pq ; where p and q are the lengths of the DIAGONALS , which are PERPENDICULAR . See also LOZENGE ,PARALLELOGRAM ,PENROSE TILES, QUADRILATERAL ,RHOMBUS References Harris, J. W. and Stocker, H. "Kite." §3.6.9 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 86, 1998. Kittell Graph A planar 23-node graph which tangles the Kempe chains in Kempe’s algorithm and thus provides an example of how Kempe’s supposed proof of the FOUR- COLOR THEOREM fails. See also ERRERA GRAPH ,FOUR- COLOR THEOREM References Kittell, I. "A Group of Operations on a Partially Colored Map." Bull. Amer. Math. Soc. 41, 407 /C1/413, 1935. Wagon, S. Mathematica in Action, 2nd ed. New York: Springer-Verlag, pp. 533 /C1/534, 1999. Klarner’s Theorem An a /C29b RECTANGLE can be packed with 1 /C29n strips IFF n½a or n½b :/ See also BOX-PACKING THEOREM ,CONWAY PUZZLE , DE BRUIJN’S THEOREM ,R ECTANGLE ,S LOTHOUBER- GRAATSMA PUZZLE References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 88, 1976. Klarner-Rado Sequence The thinnest sequence which contains 1, and when- ever it contains x, also contains 2x; 3x /C272; and 6x /C273: 1, 2, 4, 5, 8, 9, 10, 14, 15, 16, 17, ... (Sloane’s A005658). See also DOUBLE- FREE SET References Guy, R. K. "Klarner-Rado Sequences." §E36 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 237, 1994. Klarner, D. A. and Rado, R. "LINEAR COMBINATIONS of Sets of Consecutive Integers." Amer. Math. Monthly 80, 985 /C1/989, 1973. Sloane, N. J. A. Sequences A005658/M0969 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Klee’s Identity X k]0(/C281)k n k>C18>C19 n /C27k m>C18>C19 /C30(/C281)n n m/C28n>C18>C19 ; wheren k>C0>C1 is a BINOMIAL COEFFICIENT . See also BINOMIAL SUMS References Riordan, J. Combinatorial Identities. New York: Wiley, p. 13, 1979. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684/C1/760, 1973.Klein Bottle A closed NONORIENTABLE SURFACE of E ULER CHAR- ACTERISTIC 0 (Dodson and Parker 1997, p. 125) that has no inside or outside. It can be constructed by gluing both pairs of opposite edges of a RECTANGLE together giving one pair a half-twist, but can be physically realized only in 4-D, since it must pass through itself without the presence of a HOLE . Its TOPOLOGY is equivalent to a pair of CROSS-CAPS with coinciding boundaries (Francis and Weeks 1999). It can be cut in half along its length to make two MO¨BIUS STRIPS (Dodson and Parker 1997, p. 88), but can also be cut into a single MO¨BIUS STRIP (Gardner 1984, pp. 14 and 17). The above picture is an IMMERSION of the Klein bottle inR3(3-space). There is also another possible IMMER- SION called the "figure-8" IMMERSION (Geometry Center).The equation for the usual IMMERSION is given by the implicit equation (x2/C27y2/C27z2/C272y/C281)[(x2/C27y2/C27z2/C272y/C281)2/C288z2] /C2716xz(x2/C27y2/C27z2/C282y/C281)/C300 (1) (Stewart 1991). Nordstrand gives the parametric form x/C30cosucos1 2u>C16>C17ffiffiffi 2p /C27cosv>C16>C17 /C27sin1 2u>C16>C17 sinvcosvhi (2) y/C30sinucos1 2u>C16>C17ffiffiffi 2p /C27cosv>C16>C17 /C27sin1 2u>C16>C17 sinvcosvhi (3) z/C30/C28sin12u>C16>C17ffiffiffi 2p /C27cosv>C16>C17 /C27cos1 2u>C16>C17 sinvcosv:(4) The "figure-8" form of the Klein bottle is obtained by rotating a figure eight about an axis while placing a twist in it, and is given by PARAMETRIC EQUATIONS x(u; v) /C30 a /C27cos1 2 u>C16>C17 sin(v) /C28sin12 u>C16>C17 sin(2 v)hi cos(u) (5) y(u; v) /C30 a /C27cos1 2 u>C16>C17 sin(v) /C28sin12 u>C16>C17 sin(2 v)hi sin(u) (6) z(u; v) /C30sin1 2 u>C16>C17 sin(v) /C27cos12 u>C16>C17 sin(2 v) (7) for u /C23 [0; 2 p); v /C23 [0; 2 p) ; and a /C212 (Gray 1997). The image of the CROSS-CAP map of a TORUS centered at the ORIGIN is a Klein bottle (Gray 1997, p. 339). The MO¨ BIUS SHORTS are topologically equivalent to a Klein bottle with a hole (Gramain 1984, Stewart 2000). Any set of regions on the Klein bottle can be colored using six colors only (Franklin 1934, Saaty and Kainen 1986), providing the sole exception to the HEAWOOD CONJECTURE (Bondy and Murty 1976, p. 244). See also CROSS- CAP,E TRUSCAN VENUS SURFACE , FRANKLIN GRAPH ,HEAWOOD CONJECTURE ,IDA SUR- FACE ,MAP COLORING ,MO¨ BIUS SHORTS ,MO¨ BIUS STRIP References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 244, 1976. Dickson, S. "Klein Bottle Graphic." http://www.mathsource.- com/cgi-bin/msitem22?0201 /C1/801. Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, 1997. Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/399, 1999.Franklin, P. "A Six Colour Problem." J. Math. Phys. 13, 363 /C1/369, 1934. Gardner, M. "Klein Bottles and Other Surfaces." Ch. 2 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 9 /C1/18, 1984. Gramain, A. Topology of Surfaces. Moscow, ID: BCS Associates, 1984. Gray, A. "The Klein Bottle" and "A Different Klein Bottle." §14.4 and 14.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 327 /C1/330, 1997. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 308 /C1/311, 1999. JavaView. "Classic Surfaces from Differential Geometry: Klein Bottle." http://www-sfb288.math.tu-berlin.de/vgp/ja- vaview/demo/surface/common/PaSurface_KleinBot- tle.html. Nordstrand, T. "The Famed Klein Bottle." http:// www.uib.no/people/nfytn/kleintxt.htm. Pappas, T. "The Moebius Strip & the Klein Bottle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 44 /C1/46, 1989. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 45, 1986. Stewart, I. Game, Set and Math. New York: Viking Penguin, 1991. Stewart, I. "Mathematical Recreations: Reader Feedback." Sci. Amer. 283, 101, Sep. 2000. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 131 /C1/132, 1991. Wolfram Research, Inc. "Algebraic Construction of a Klein Bottle." http://library.wolfram.com/demos/v4/KleinBottle- Formula.nb. Klein Bottle Dissection Every MO¨ BIUS STRIP DISSECTION of unequal squares can be glued along its edge to produce a dissection of the Klein bottle. There are no other ways to tile a Klein bottle with six or fewer squares, the situation is unknown for seven or eight squares, but it is known that other types of dissections do exists for nine squares (Stewart 1997). See also CYLINDER DISSECTION ,M O¨ BIUS STRIP DIS- SECTION ,PERFECT SQUARE DISSECTION ,TORUS DIS- SECTION References Stewart, I. "Squaring the Square." Sci. Amer. 277,94/C1/96, July 1997. Klein Four-Group VIERGRUPPE Klein Quartic A 3-holed TORUS . In 1879, Felix Klein discovered that the surface has a 366-fold symmetry, the maximum possible for a surface of its type. See also QUARTIC SURFACE References Levy, S. (Ed.). The Eightfold Way: The Beauty of the Klein Quartic. New York: Cambridge University Press, 1999. Klein’s Absolute Invariant Let v1and v2be periods of a DOUBLY PERIODIC FUNCTION , with t /C30 v2 =v1the HALF-PERIOD RATIO a number with I[ t] "0 : Then Klein’s absolute invariant (also called Klein’s modular function) is defined as J( v1 ; v2) /C13g3 2(v1 ; v2) D( v1 ; v2) ; (1) where g2and g3are the invariants of the WEIER- STRASS ELLIPTIC FUNCTION with MODULAR DISCRIMI- NANT D/C13g3 2 /C2827g23 (2) (Klein 1877). If t /C23 H ; where H is the UPPER HALF- PLANE , then J( t) /C13J(1; t) /C30J(v1 ; v2) (3) is a function of the ratio t only, as are g2 ; g3 ; and D: Furthermore, g2( t) ; g3( t);D(t) ; and J( t) are analytic in H (Apostol 1997, p. 15). /J(t) is invariant under a UNIMODULAR TRANSFORMA- TION ,so Ja t /C27 b c t /C27 d ! /C30J( t); (4) and J( t)isa MODULAR FUNCTION . J( t) takes on the special values J( r /C30e2 pi=3) /C300 (5) J(i) /C301 (6) J(i /C12) /C30/C12: (7) Every RATIONAL FUNCTION of J is a MODULAR FUNC- TION , and every MODULAR FUNCTION can be expressedas a RATIONAL FUNCTION of J (Apostol 1997, p. 40). The FOURIER SERIES of J(t) ; modulo a constant multiplicative factor, is called the J-FUNCTION . Klein’s invariant can be given explicitly by J(q) /C134 27[1 /C28 l(q) /C27 l2(q)]3 l2(q)[1 /C28 l(q)]2/C30[E4(q)]3 [E4(q)]3 /C28 [E6(q)]2 (8) (Klein 1878/79, Cohn 1994), where q /C13eiptis the NOME , l(q) is the ELLIPTIC LAMBDA FUNCTION l(q) /C13k2(q) /C30q2(q) q3(q)"#4 ; (9) /qi(q)isaJ ACOBI THETA FUNCTION , and the Ei(q) are RAMANUJAN- EISENSTEIN SERIES . See also ELLIPTIC LAMBDA FUNCTION , J-FUNCTION , JACOBI THETA FUNCTIONS ,LAMBDA ELLIPTIC FUNC- TION ,PI,RAMANUJAN- EISENSTEIN SERIES References Apostol, T. M. "Klein’s Modular Function J( t);/" "Invariance of J Under Unimodular Transformation," "The Fourier Expansions of D( t) and J(t) ;/" "Special Values of J," and "Modular Functions as Rational Functions of J." §1.12 /C1/ 1.13, 1.15, and 2.5 /C1/2.6 in Modular Functions and Dirich- let Series in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 15 /C1/18, 20 /C1/22, and 39 /C1/40, 1997. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 115 and 179, 1987. Cohn, H. Introduction to the Construction of Class Fields. New York: Dover, p. 73, 1994. Klein, F. "Sull’ equazioni dell’ Icosaedro nella risoluzione delle equazioni del quinto grado [per funzioni ellittiche]." Reale Istituto Lombardo, Rendiconto, Ser. 2 10, 1877. Klein, F. "U¨ ber die Transformation der elliptischen Funk- tionen und die Auflo¨sung der Gleichungen fu¨nften Grades." Math. Ann. 14, 1878/79. Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. http://www.math.jussieu.fr/~neste- ren/. Weisstein, E. W. "j-Function." MATHEMATICA NOTEBOOK JFUNCTION.M . Klein’s Equation If a real ALGEBRAIC CURVE has no singularities except nodes and CUSPS , BITANGENTS , and INFLECTION POINTS , then n /C272t ?2 /C27 i?/C30m /C272d ?2 /C27 k ?; where nis the order, t?is the number of conjugate tangents, i?is the number of REAL inflections, mis the class, d?is the number of REAL conjugate points, and k?is the number of REAL CUSPS . This is also called KLEIN’S THEOREM . See also PLU¨ CKER’S EQUATION References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 114, 1959. Klein’s Modular Function KLEIN’S ABSOLUTE INVARIANT Klein’s Theorem KLEIN’S EQUATION Klein-Beltrami Model The Klein-Beltrami model of HYPERBOLIC GEOMETRY consists of an OPEN DISK in the Euclidean plane whose open chords correspond to hyperbolic lines. Two lines l and m are then considered parallel if their chords fail to intersect and are PERPENDICULAR under the following conditions, 1. If at least one of l and m is a diameter of the DISK, they are hyperbolically perpendicular IFF they are perpendicular in the Euclidean sense. 2. If neither is a diameter, l is perpendicular to m IFF the Euclidean line extending l passes through the pole of m (defined as the point of intersection of the tangents to the disk at the "endpoints" of m). There is an isomorphism between the POINCARE ´ HYPERBOLIC DISK model and the Klein-Beltrami model. Consider a Klein disk in Euclidean 3-space with a SPHERE of the same radius seated atop it, tangent at the ORIGIN . If we now project chords on the disk orthogonally upward onto the SPHERE ’s lower HEMISPHERE , they become arcs of CIRCLES orthogonal to the equator. If we then stereographically project the SPHERE ’s lower HEMISPHERE back onto the plane of the Klein disk from the north pole, the equator will map onto a disk somewhat larger than the Klein disk, and the chords of the original Klein disk will now be arcs of CIRCLES orthogonal to this larger disk. That is, they will be Poincare ´ lines. Now we can say that two Klein lines or angles are congruent IFF their corre- sponding Poincare ´ lines and angles under this iso- morphism are congruent in the sense of the Poincare ´ model. See also HYPERBOLIC GEOMETRY ,POINCARE ´ HYPER- BOLIC DISK Klein-Erdos-Szekeres Problem HAPPY END PROBLEM Klein-Gordon Equation The PARTIAL DIFFERENTIAL EQUATION 1 c2@2 c @t2 /C30@2 c @x2 /C28 m2 c (1) that arises in mathematical physics. The quasilinear Klein-Gordon equation is given by utt /C28 a2uxx /C27 g2u /C30 bu3 (2) (Nayfeh 1972, p. 76; Zwillinger 1997, p. 133), and thenonlinear Klein-Gordon equation by Xn i/C301uxixi/C27 lup /C300 (3) (Matsumo 1987; Zwillinger 1997, p. 133). See also LIOUVILLE’S EQUATION ,SINE-GORDON EQUA- TION ,W AVE EQUATION References Matsumo, Y. "Exact Solution for the Nonlinear Klein- Gordon and Liouville Equations in Four-Dimensional Euclidean Space." J. Math. Phys. 28, 2317 /C1/2322, 1987. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 272, 1953. Nayfeh, A. H. Perturbation Methods. New York: Wiley, 1973. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 129 and 133, 1997. Klein-Gordon-Maxwell Equation The system of PARTIAL DIFFERENTIAL EQUATIONS 92s/C28(½a½2/C271)s/C300 92a/C289(9 /C215a)/C28s2a/C30a: References Deumens, E. "The Klein-Gordon-Maxwell Nonlinear System of Equations." Physica D 18, 371/C1/373, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Kleinian Group A finitely generated discontinuous group of linear fractional transformation acting on a domain in the COMPLEX PLANE . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 425, 1980. Kra, I. Automorphic Forms and Kleinian Groups. Reading, MA: W. A. Benjamin, 1972. KleinInvariantJ KLEIN’S ABSOLUTE INVARIANT Kloosterman’s Sum S(u;v;n)/C13X nexp2pi(uh/C27v¯h) n"# ; (1) where hruns through a complete set of residues RELATIVELY PRIME ton, and ¯his defined by h¯h/C131 (mod n): (2) If (n; n) /C301 (if n and (n?) are RELATIVELY PRIME ), then S(u; v; n)S(u; v ?; n?) /C30S(u; vn?2 /C27v?n2 ; nn?) : (3) Kloosterman’s sum essentially solves the problem introduced by Ramanujan of representing sufficiently large numbers by QUADRATIC FORMS ax2 1 /C27bx22 /C27cx23 /C27 dx2 4 : Weil improved on Kloosterman’s estimate for Ramanujan’s problem with the best possible estimate ½S(u; v; n)½52ffiffiffinp(4) (Duke 1997). See also GAUSSIAN SUM References Duke, W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer. Math. Soc. 44, 190 /C1/196, 1997. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 56, 1979. Katz, N. M. Gauss Sums, Kloosterman Sums, and Mono- dromy Groups. Princeton, NJ: Princeton University Press, 1987. Kloosterman, H. D. "On the Representation of Numbers in the Form ax2 /C27by2 /C27cz2 /C27dt2 :/" Acta Math. 49, 407 /C1/464, 1926. Ramanujan, S. "On the Expression of a Number in the Form ax2 /C27by2 /C27cz2 /C27du2 :/" Collected Papers. New York: Chel- sea, 1962. k-Matrix A k-matrix is a kind of CUBE ROOT of the IDENTITY MATRIX (distinct from the IDENTITY MATRIX ) which is defined by the COMPLEX MATRIX k /C3000 /C28i i 00 01 02 435: It satisfies k 3 /C30I where I is the IDENTITY MATRIX . See also COMPLEX MATRIX ,C UBE ROOT,IDENTITY MATRIX ,QUATERNION K-Means Clustering Algorithm An algorithm for partitioning (or clustering) N data points into K disjoint subsets Sjcontaining Njdata points so as to minimize the sum-of-squares criterion J /C30XK j/C301X n /C23Sj½½xn /C28 mj ½½2 ; where xnis a vector representing the nth data point and mjis the CENTROID of the data points in Sj : In general, the algorithm does not achieve a GLOBAL MINIMUM of J over the assignments. In fact, since the algorithm uses discrete assignment rather than a setof continuous parameters, the "minimum" it reaches cannot even be properly called a LOCAL MINIMUM . Despite these limitations, the algorithm is used fairly frequently as a result of its ease of implementation. The algorithm consists of a simple re-estimation procedure as follows. First, the data points are assigned at random to the K sets. Then the centroid is computed for each set. These two steps are alternated until a stopping criterion is met, i.e., when there is no further change in the assignment of the data points. See also GLOBAL MINIMUM ,LOCAL MINIMUM ,M INI- MUM References Bishop, C. M. Neural Networks for Pattern Recognition. Oxford, England: Oxford University Press, 1995. Knapsack Problem Given a SUM and a set of WEIGHTS , find the WEIGHTS which were used to generate the SUM. The values of the weights are then encrypted in the sum. This system relies on the existence of a class of knapsack problems which can be solved trivially (those in which the weights are separated such that they can be "peeled off" one at a time using a GREEDY -like algorithm), and transformations which convert the trivial problem to a difficult one and vice versa. Modular multiplication is used as the TRAPDOOR ONE-WAY FUNCTION . The simple knapsack system was broken by Shamir in 1982, the Graham-Shamir system by Adleman, and the iterated knapsack by Ernie Brickell in 1984. See also SUBSET SUM PROBLEM ,TRAPDOOR ONE-WAY FUNCTION References Coppersmith, D. "Knapsack Used in Factoring." §4.6 in Open Problems in Communication and Computation (Ed. T. M. Cover and B. Gopinath). New York: Springer-Ver- lag, pp. 117 /C1/119, 1987. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 163 /C1/166, 1985. Knar’s Formula The INFINITE PRODUCT identity G(1 /C27v) /C3022vY/C12 m/C301p/C281 =2 G1 2/C272/C28mv>C16>C17hi ; where G(x) is the GAMMA FUNCTION . See also GAMMA FUNCTION ,INFINITE PRODUCT References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 6, 1981. Kneser-Sommerfeld Formula Let Jn(z)beaB ESSEL FUNCTION OF THE FIRST KIND , Nn(z)aN EUMANN FUNCTION , and j n; n(z)/ the zeros of z/C28 nJn(z) in order of ascending REAL PART . Then for 0 B x BX B1 and R[z] > 0 ; pJn(xz) 4Jn(z) [Jn(z)Nn(Xz) /C28N n(z)J n(Xz)] /C30X/C12 n/C301Jn(j n; nx)Jn(j n; nX) (z2 /C28 j2 n; n)J ?2 n; n(jn; n) : References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1474, 1980. Knight’s Tour A knight’s tour of a CHESSBOARD (or any other grid) is a sequence of moves by a knight CHESS piece (which may only make moves which simultaneously shift one square along one axis and two along the other) such that each square of the board is visited exactly once (i.e., a HAMILTONIAN CIRCUIT ). If the final position is a knight’s move away from the first position, the tour is called re-entrant. The above figures shows six knight’s tours on an 8 /C298 CHESSBOARD , all but the first of which are re-entrant. The final tour has the additional property that it is a SEMIMAGIC SQUARE with row and column sums of 260 and main diagonal sums of 348 and 168 (Steinhaus 1983, p. 30). BACKTRACKING algorithms (in which the knight is allowed to move as far as possible until it comes to a blind alley, at which point it backs up some number of steps and then tries a different path) can be used to find knight’s tours, but such methods can be very slow. Warnsdorff (1823) proposed an algorithm that finds a path without any backtracking by computing ratings for "successor" steps at each position. Here, successors of a position are those squares that have not yet been visited and can be reached by a single move from the given position. The rating is highestfor the successor whose number of successors is least. In this way, squares tending to be isolated are visited first and therefore prevented from being isolated (Roth). The time needed for this algorithm grows roughly linearly with the number of squares of the chessboard, but unfortunately computer implementa- tion show that this algorithm runs into blind alleys for chessboards bigger than 76 /C2976 ; despite the fact that it works well on smaller boards (Roth). Recently, Conrad et al. (1994) discovered another linear time algorithm and proved that it solves the problem for all n ]5: The Conrad et al. algorithm works by decomposition of the chessboard into smal- ler chessboards (not necessarily square) for which explicit solutions are known. This algorithm is rather complicated because it has to deal with many special cases, but has been implemented in Mathematica by A. Roth. Example tours are illustrated above for n /C29n boards with n /C305to8. Lo¨bbing and Wegener (1996) computed the number of cycles covering the directed knight’s graph for an 8 /C29 8 CHESSBOARD . They obtained a2 ; where a /C302,849,759,680, i.e., 8,121,130,233,753,702,400. They also computed the number of undirected tours, obtaining an incorrect answer 33,439,123,484,294 (which is not divisible by 4 as it must be), and so are currently redoing the calculation. The following results are given by Kraitchik (1942). The number of possible tours on a 4k /C294k board for k /C303, 4, ... are 8, 0, 82, 744, 6378, 31088, 189688, 1213112, ... (Kraitchik 1942, p. 263). There are 14 tours on the 3 /C297 rectangle, two of which are symmetrical. There are 376 tours on the 3 /C298 rectangle, none of which is closed. There are 16 symmetric tours on the 3 /C299 rectangle and 8 closed tours on the 3 /C2910 rectangle. There are 58 symmetric tours on the 3 /C2911 rectangle and 28 closed tours on the 3/C2912 rectangle. There are five doubly symmetric tours on the 6 /C296 square. There are 1728 tours on the 5/C295 square, 8 of which are symmetric. The longest "uncrossed" knight’s tours on an n/C29nboard for n/C303, 4, ... are 2, 5, 10, 17, 24, 35, ... (Sloane’s A003192). See also CHESS ,HAMILTONIAN CIRCUIT ,KINGS PRO- BLEM ,K NIGHTS PROBLEM ,M AGIC TOUR,Q UEENS PROBLEM ,TOUR References Ahrens, W. Mathematische Unterhaltungen und Spiele. Leipzig, Germany: Teubner, p. 381, 1910. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 175 /C1/186, 1987. Chartrand, G. "The Knight’s Tour." §6.2 in Introductory Graph Theory. New York: Dover, pp. 133 /C1/135, 1985. Conrad, A.; Hindrichs, T.; Morsy, H.; and Wegener, I. "Solution of the Knight’s Hamiltonian Path Problem on Chessboards." Discr. Appl. Math. 50, 125 /C1/134, 1994. Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 102 /C1/103, 1970. Euler, L. "Solution d’une question curieuse qui ne paroit soumise a aucune analyse." Me´moires de l’Acade ´mie Royale des Sciences et Belles Lettres de Berlin, Anne´e 1759 15, 310 /C1/337, 1766. Gardner, M. "Knights of the Square Table." Ch. 14 in Mathematical Magic Show: More Puzzles, Games, Diver- sions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 188 /C1/ 202, 1978. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 98 /C1/100, 1984. Guy, R. K. "The n Queens Problem." §C18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 133 /C1/135, 1994. Jelliss, G. "Knight’s Tour Notes." http://homepages.stayfree.- co.uk/gpj/ktn.htm. Jelliss, G. "Magic Knight’s Tours." http://homepages.stay- free.co.uk/gpj/mkt.htm. Kraitchik, M. "The Problem of the Knights." Ch. 11 in Mathematical Recreations. New York: W. W. Norton, pp. 257 /C1/266, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 87 /C1/89, 1979. Roget, P. M. Philos. Mag. 16, 305 /C1/309, 1840. Roth, A. "The Problem of the Knight: A Fast and Simple Algorithm." http://www.mathsource.com/cgi-bin/ msitem?0202 /C1/127. Ruskey, F. "Information on the n Knight’s Tour Problem." http://www.theory.csc.uvic.ca/~cos/inf/misc/Knight.html. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 166, 1990. Sloane, N. J. A. Sequences A003192/M1369 and A006075/ M3224 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 30, 1999. van der Linde, A. Geschichte und Literatur des Schachspiels, Vol. 2. Berlin: Springer-Verlag, pp. 101 /C1/111, 1874. Vandermonde, A.-T. "Remarques sur les Proble `mes de Situation." L’Histoire de l’Acade ´mie des Sciences avec les Me´moires, Anne´e 1771. Paris: Me´moirs, pp. 566 /C1/574 and Plate I, 1774. Volpicelli, P. "Soluzione completa e generale, mediante la geometria di situazione, del problema relativo alle corse del cavallo sopra qualunque scacchiere." Atti della Reale Accad. dei Lincei 25,87/C1/162, 1872. Warnsdorff, H. C. von Des Ro¨sselsprungs einfachste und allgemeinste Lo¨sung. Schmalkalden, 1823. Wegener, I. and Lo¨bbing, M. "The Number of Knight’s Tours Equals 33,439,123,484,294--Counting with Binary Deci- sion Diagrams." Electronic J. Combinatorics 3,R51 /C1/4, 1996. http://www.combinatorics.org/Volume_3/volu- me3.html#R5. Knights of the Round Table NECKLACEKnights Problem The problem of determining how many nonattacking knights K(n) can be placed on an n /C29n CHESSBOARD . For n /C308, the solution is 32 (illustrated above). In general, the solutions are K(n)/C301 2n2n>2 even 12(n2/C271)n>1 odd ;( giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (Sloane’s A030978, Dudeney 1970, p. 96; Madachy 1979). The minimal number of knights needed to occupy or attack every square on an n/C29nCHESSBOARD is given by 1, 4, 4, 4, 5, 8, 10, ... (Sloane’s A006075). The number of such solutions are given by 1, 1, 2, 3, 8, 22,3, ... (Sloane’s A006076). See also B ISHOPS PROBLEM ,CHESS ,KINGS PROBLEM , KNIGHT’S TOUR,QUEENS PROBLEM ,ROOKS PROBLEM References Dudeney, H. E. "The Knight-Guards." §319 in Amusements in Mathematics. New York: Dover, p. 95, 1970. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 38 /C1/39, 1979. Moser, L. "King Paths on a Chessboard." Math. Gaz. 39, 54, 1955. Sloane, N. J. A. Sequences A006075/M3224, A006076/ M0884, and A030978 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M3224 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 196 /C1 /197, 1991. Wilf, H. S. "The Problem of Kings." Electronic J. Combina- torics 2,31/C1/7, 1995. http://www.combinatorics.org/Vo- lume_2/volume2.html#3. Kno¨del Numbers For every k ]1 ; let Ckbe the set of COMPOSITE NUMBERS n /C21k such that if 1 Ba Bn; GCD( a ; n) /C301 (where GCD is the GREATEST COMMON DIVISOR ), then an/C28k /C131 (mod n): C1is the set of CARMICHAEL NUM- BERS . Makowski (1962/1963) proved that there are infinitely many members of Ck for k ]2:/ k Sloane /Ck/ 1 A002997 561, 1105, 1729, 2465, 2821, 6601, 8911, ... 2 A050990 4, 6, 8, 10, 12, 14, 22, 24, 26, 30, ... 3 A050991 9, 15, 21, 33, 39, 51, 57, 63, 69, 87, ... 4 A050992 6, 8, 12, 16, 20, 24, 28, 40, 44, 48, ... 5 A050993 25, 65, 85, 145, 165, 185, 205, ... See also CARMICHAEL NUMBER , D-NUMBER ,GREAT- EST COMMON DIVISOR References Makowski, A. "Generalization of Morrow’s D-Numbers." Simon Stevin 36, 71, 1962/1963. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 101, 1989. Sloane, N. J. A. Sequences A002997/M5462, A050990, A050991, A050992, and A050993 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Knot A knot is defined as a closed, non-self-intersecting curve embedded in 3-D. A knot is a single component LINK . Knot theory was given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations (Thompson 1867). P. G. Tait then cataloged possible knots by trial anderror. Much progress has been made in the interven- ing years. Klein proved that knots cannot exist in an EVEN - numbered dimensional space ]4:It has since been shown that a knot cannot exist in anydimension ]4: Two distinct knots cannot have the same KNOT COMPLEMENT (Gordon and Luecke 1989), but two LINKS can! (Adams 1994, p. 261). Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition isperformed) as a KNOT SUM of a class of knots known asPRIME KNOTS , which cannot themselves be further decomposed. Combining PRIME KNOTS gives no new knot types for knots of three to five crossing, but oneadditional COMPOSITE KNOT each for knots of six and seven crossings.Knots are most commonly cataloged based on theminimum number of crossings present (the so-called CROSSING NUMBER . Thistlethwaite has used D OWKER NOTATION to enumerate the number of PRIME KNOTS of up to 13 crossings, and ALTERNATING KNOTS up to 14 crossings. In this compilation, MIRROR IMAGES are counted as a single knot type. Hoste et al. (1998) subsequently tabulated all prime knots up to 16 crossings. Hoste and Weeks are currently begun compiling a list of 17-crossing knots (Hoste et al. 1998). The following table gives the number of distinct PRIME ,ALTERNATING ,NONALTERNATING ,TORUS , and SATELLITE KNOTS , in addition to the number of chiral noninvertible c,/C27amphichiral noninvertible, /C28am- phichiral noninvertible, chiral invertible i, and fully amphichiral and invertible knots aforn/C303t o1 6 (Hoste et al. 1998). n prime alt. nonalt. torus sat. Sloane A002863 A002864 A051763 A051764 A051765 311010 411000 522010 6330007770108 2 1 1 83109 4 9 4 1810 10 165 123 42 1 011 552 367 185 1 012 2176 1288 888 0 013 9988 4878 5110 1 214 46972 19536 27436 1 215 253293 85263 168030 2 616 1388705 379799 1008906 1 10 nc //C27// /C28/ ia Sloane A051766 A051767 A051768 A051769 A052400 3000104000015000206000217000708001 1 649200 4 70 10 27 0 6 125 711 187 0 0 365 0 12 1103 1 40 1015 17 13 6919 0 0 3069 0 14 37885 6 227 8813 41 15 226580 0 1 26712 0 16 1308449 65 1361 78717 113 A pictorial enumeration of PRIME KNOTS of up to 10 crossings appears in Rolfsen (1976, Appendix C). Note, however, that in this table, the PERKO PAIR 10 /C1/161 and 10 /C1/162 are actually identical, and the uppermost crossing in 10 /C1/144 should be changed (Jones 1987). The kth knot having n crossings in this (arbitrary) ordering of knots is given the symbol nk : Another possible representation for knots uses the BRAID GROUP . A knot with n /C271 crossings is a member of the BRAID GROUP n. There is no general ALGORITHM to determine if a tangled curve is a knot or if two given knots are interlocked. Haken (1961) and Hemion (1979) have given ALGORITHMS for rigorously determining if two knots are equivalent, but they are too complex to apply even in simple cases (Hoste et al. 1998). If a knot is AMPHICHIRAL , the "amphichirality" is A /C301, otherwise A /C300 (Jones 1987). ARF INVARIANTS are designated a.BRAID WORDS are denoted b (Jones 1987). CONWAY’S KNOT NOTATION C for knots up to 10 crossings is given by Rolfsen (1976). Hyperbolic volumes are given (Adams, Hildebrand, and Weeks 1991; Adams 1994). The BRAID INDEX i is given by Jones (1987). ALEXANDER POLYNOMIALS D are given in Rolfsen (1976), but with the POLYNOMIALS for 10 /C1/083 and 10 /C1/086 reversed (Jones 1987). The ALEXANDER POLYNOMIALS are normalized according to Conway, and given in abbreviated form [a1 ; a2 ; ... for a1 /C27a2(x/C281 /C27x) /C27...:/ The JONES POLYNOMIALS W for knots of up to 10 crossings are given by Jones (1987), and the JONES POLYNOMIALS V can be either computed from these, or taken from Adams (1994) for knots of up to 9 crossings (although most POLYNOMIALS are associated with the wrong knot in the first printing). The JONES POLYNOMIALS are listed in the abbreviated form fnga0a1 ... for t/C28n(a0 /C27a1t /C27...); and correspond either to the knot depicted by Rolfsen or its MIRROR IMAGE , whichever has the lower POWER of t/C281 : The HOMFLY POLYNOMIAL P(l; m) and KAUFFMAN POLY- NOMIAL F(A, X) are given in Lickorish and Millett (1988) for knots of up to 7 crossings. M. B. Thistlethwaite has tabulated the HOMFLY POLYNOMIAL and KAUFFMAN POLYNOMIAL F for KNOTS of up to 13 crossings. 03 /C1/001 04 /C1/001 05 /C1/001 05 /C1/002 06 /C1/001 06 /C1/002 06 /C1/003 07 /C1/001 07 /C1/002 07 /C1/003 07 /C1/004 07 /C1/005 07 /C1/006 07 /C1/007 08 /C1/001 08 /C1/002 08 /C1/003 08 /C1/004 08 /C1/005 08 /C1/006 08 /C1/007 08 /C1/008 08 /C1/009 08 /C1/010 08 /C1/011 08 /C1/012 08 /C1/013 08 /C1/014 08 /C1/015 08 /C1/016 08 /C1/017 08 /C1/01808 /C1/019 08 /C1/020 08 /C1/021 09 /C1/001 09 /C1/002 09 /C1/003 09 /C1/004 09 /C1/005 09 /C1/006 09 /C1/007 09 /C1/008 09 /C1/009 09 /C1/010 09 /C1/011 09 /C1/012 09 /C1/013 09 /C1/014 09 /C1/015 09 /C1/016 09 /C1/017 09 /C1/018 09 /C1/019 09 /C1/020 09 /C1/021 09 /C1/022 09 /C1/023 09 /C1/024 09 /C1/025 09 /C1/026 09 /C1/027 09 /C1/028 09 /C1/029 09 /C1/030 09 /C1/031 09 /C1/032 09 /C1/033 09 /C1/034 09 /C1/035 09 /C1/036 09 /C1/037 09 /C1/038 09 /C1/039 09 /C1/040 09 /C1/041 09 /C1/042 09 /C1/043 09 /C1/044 09 /C1/045 09 /C1/046 09 /C1/047 09 /C1/048 09 /C1/049 10 /C1/001 10 /C1/002 10 /C1/003 10 /C1/004 10 /C1/005 10 /C1/006 10 /C1/007 10 /C1/008 10 /C1/009 10 /C1/010 10 /C1/011 10 /C1/012 10 /C1/013 10 /C1/014 10 /C1/015 10 /C1/016 10 /C1/017 10 /C1/018 10 /C1/019 10 /C1/020 10 /C1/021 10 /C1/022 10 /C1/023 10 /C1/024 10 /C1/025 10 /C1/026 10 /C1/027 10 /C1/028 10 /C1/029 10 /C1/030 10 /C1/031 10 /C1/032 10 /C1/033 10 /C1/034 10 /C1/035 10 /C1/036 10 /C1/037 10 /C1/038 10 /C1/039 10 /C1/040 10 /C1/041 10 /C1/042 10 /C1/043 10 /C1/044 10 /C1/045 10 /C1/046 10 /C1/047 10 /C1/048 10 /C1/049 10 /C1/050 10 /C1/051 10 /C1/052 10 /C1/053 10 /C1/054 10 /C1/055 10 /C1/056 10 /C1/057 10 /C1/058 10 /C1/059 10 /C1/060 10 /C1/061 10 /C1/062 10 /C1/063 10 /C1/064 10 /C1/065 10 /C1/066 10 /C1/067 10 /C1/068 10 /C1/069 10 /C1/070 10 /C1/071 10 /C1/072 10 /C1/073 10 /C1/074 10 /C1/075 10 /C1/076 10/C1/077 10 /C1/078 10 /C1/079 10 /C1/080 10 /C1/081 10 /C1/082 10 /C1/083 10 /C1/084 10/C1/085 10 /C1/086 10 /C1/087 10 /C1/088 10 /C1/089 10 /C1/090 10 /C1/091 10 /C1/092 10/C1/093 10 /C1/094 10 /C1/095 10 /C1/096 10 /C1/097 10 /C1/098 10 /C1/099 10 /C1/100 10/C1/101 10 /C1/102 10 /C1/103 10 /C1/104 10 /C1/105 10 /C1/106 10 /C1/107 10 /C1/108 10/C1/109 10 /C1/110 10 /C1/111 10 /C1/112 10 /C1/113 10 /C1/114 10 /C1/115 10 /C1/116 10/C1/117 10 /C1/118 10 /C1/119 10 /C1/120 10 /C1/121 10 /C1/122 10 /C1/123 10 /C1/124 10/C1/125 10 /C1/126 10 /C1/127 10 /C1/128 10 /C1/129 10 /C1/130 10 /C1/131 10 /C1/132 10/C1/133 10 /C1/134 10 /C1/135 10 /C1/136 10 /C1/137 10 /C1/138 10 /C1/139 10 /C1/140 10/C1/141 10 /C1/142 10 /C1/143 10 /C1/144 10 /C1/145 10 /C1/146 10 /C1/147 10 /C1/148 10/C1/149 10 /C1/150 10 /C1/151 10 /C1/152 10 /C1/153 10 /C1/154 10 /C1/155 10 /C1/156 10/C1/157 10 /C1/158 10 /C1/159 10 /C1/160 10 /C1/161 10 /C1/162 10 /C1/163 10 /C1/164 10/C1/165 10 /C1/166 See also ALEXANDER POLYNOMIAL ,A LEXANDER’S HORNED SPHERE ,A MBIENT ISOTOPY ,A MPHICHIRAL KNOT,ANTOINE’S NECKLACE ,BEND (KNOT), BENNE- QUIN’S CONJECTURE ,B ORROMEAN RINGS ,B RAID GROUP ,B RUNNIAN LINK,B URAU REPRESENTATION , CHEFALO KNOT,C LOVE HITCH ,C OLORABLE ,C ON- WAY’S KNOT,CROOKEDNESS ,DEHN’S LEMMA ,DOWKER NOTATION ,FIGURE-OF- EIGHT KNOT,G RANNY KNOT, HITCH,INVERTIBLE KNOT,JONES POLYNOMIAL ,K I- NOSHITA- TERASAKA KNOT,KNOT POLYNOMIAL ,KNOT SUM,L INKING NUMBER ,L OOP (KNOT), MARKOV’S THEOREM ,M ENASCO’S THEOREM ,M ILNOR’S CONJEC- TURE ,NASTY KNOT,ORIENTED KNOT,PRETZEL KNOT, PRIME KNOT,REIDEMEISTER MOVES ,RIBBON KNOT, RUNNING KNOT,SATELLITE KNOT,SCHO¨ NFLIES THE- OREM ,SHORTENING ,SIGNATURE (KNOT), SKEIN RELA- TIONSHIP ,S LICE KNOT,S LIP KNOT,S MITH CONJECTURE ,SOLOMON’S SEAL KNOT,SPAN (LINK), SPLITTING ,SQUARE KNOT,STEVEDORE’S KNOT,STICK NUMBER ,STOPPER KNOT,TAIT’S KNOT CONJECTURES , TAME KNOT,T ANGLE ,T ORSION NUMBER ,T ORUS KNOT,TREFOIL KNOT,UNKNOT ,UNKNOTTING NUM- BER,VASSILIEV INVARIANT ,W HITEHEAD LINK References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 280 /C1/286, 1994. Adams, C.; Hildebrand, M.; and Weeks, J. "Hyperbolic Invariants of Knots and Links." Trans. Amer. Math. Soc. 1,1/C1/56, 1991. Alexander, J. W. and Briggs, G. B. "On Types of Knotted Curves." Ann. Math. 28, 562 /C1/586, 1927. Aneziris, C. N. The Mystery of Knots: Computer Program- ming for Knot Tabulation. Singapore: World Scientific, 1999. Ashley, C. W. The Ashley Book of Knots. New York: McGraw-Hill, 1996. Bogomolny, A. "Knots...." http://www.cut-the-knot.com/ do_you_know/knots.html. Bruzelius, L. "Knots and Splices." http://pc-78 /C1/ 120.udac.se:8001/WWW/Nautica/Bibliography/Knots&S- plices.html. Caudron, A. "Classification des noeuds et des enlacements." Prepublication Math. d’Orsay. Orsay, France: Universite ´ Paris-Sud, 1980. Cerf, C. "Atlas of Oriented Knots and Links." Topology Atlas Invited Contributions 3, No. 2, 1 /C1/32, 1998. http://at.yor- ku.ca/t/a/i/c/31.htm. Conway, J. H. "An Enumeration of Knots and Links." In Computational Problems in Abstract Algebra (Ed. J. Leech). Oxford, England: Pergamon Press, pp. 329 /C1/ 358, 1970. Eppstein, D. "Knot Theory." http://www.ics.uci.edu/~epp- stein/junkyard/knot.html. Eppstein, D. "Knot Theory." http://www.ics.uci.edu/~epp- stein/junkyard/knot/. Erdener, K.; Candy, C.; and Wu, D. "Verification and Extension of Topological Knot Tables." ftp://chs.cusd.clar- emont.edu/pub/knot/FinalReport.sit.hqx. Gordon, C. and Luecke, J. "Knots are Determined by their Complements." J. Amer. Math. Soc. 2, 371 /C1/415, 1989. Haken, W. "Theorie der Normalflachen." Acta Math. 105, 245 /C1/375, 1961. Hemion, G. "On the Classification of Homeomorphisms of 2- Manifolds and the Classification of 3-Manifolds." Acta Math. 142, 123 /C1/155, 1979. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Kauffman, L. Knots and Applications. River Edge, NJ: World Scientific, 1995. Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, 1991. Kirkman, T. P. "The Enumeration, Description, and Con- struction of Knots Fewer than Ten Crossings." Trans. Roy. Soc. Edinburgh 32, 1885, 281 /C1/309. Kirkman, T. P. "The 634 Unifilar Knots of Ten Crossings Enumerated and Defined." Trans. Roy. Soc. Edinburgh 32, 483 /C1/506, 1885. Korpega ˚rd, J. "The Knotting Dictionary of Ka¨nnet." http:// www.korpegard.nu/jan/knots.html. Lickorish, W. B. R. and Millett, B. R. "The New Polynomial Invariants of Knots and Links." Math. Mag. 61,1/C1/23, 1988. Listing, J. B. "Vorstudien zur Topologie." Go¨ttingen Studien, University of Go¨ttingen, Germany, 1848. Little, C. N. "On Knots, with a Census of Order Ten." Trans. Connecticut Acad. Sci. 18, 374 /C1/378, 1885. Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993. Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Neuwirth, L. "The Theory of Knots." Sci. Amer. 140,84/C1/96, Jun. 1979. Perko, K. "Invariants of 11-Crossing Knots." Prepublications Math. d’Orsay. Orsay, France: Universite ´ Paris-Sub, 1980. Perko, K. "Primality of Certain Knots." Topology Proc. 7, 109 /C1/118, 1982. Praslov, V. V. and Sossinsky, A. B. Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology. Providence, RI: Amer. Math. Soc., 1996.Przytycki, J. "A History of Knot Theory from Vandermonde to Jones." Proc. Mexican Nat. Congress Math. , Nov. 1991. Reidemeister, K. Knotentheorie. Berlin: Springer-Verlag, 1932. Rolfsen, D. "Table of Knots and Links." Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 280 /C1/287, 1976. Schubert, H. Sitzungsber. Heidelberger Akad. Wiss., Math.- Naturwiss. Klasse, 3rd Abhandlung. 1949. Sloane, N. J. A. Sequences A002863/M0851 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0851 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Suber, O. "Knots on the Web." http://www.earlham.edu/ ~peters/knotlink.htm. Tait, P. G. "On Knots I, II, and III." Scientific Papers, Vol. 1. Cambridge, England: University Press, pp. 273 /C1/347, 1898. Thistlethwaite, M. B. "Knot Tabulations and Related To- pics." In Aspects of Topology in Memory of Hugh Dowker 1912 /C1/1982 (Ed. I. M. James and E. H. Kronheimer). Cambridge, England: Cambridge University Press, pp. 2 /C1/76, 1985. Thistlethwaite, M. B. ftp://chs.cusd.claremont.edu/pub/knot/ Thistlethwaite_Tables/. Thistlethwaite, M. B. "Morwen’s Home Page." http:// www.math.utk.edu/~morwen/. Thompson, W. T. "On Vortex Atoms." Philos. Mag. 34,15/C1/ 24, 1867. Weisstein, E. W. "Knots." MATHEMATICA NOTEBOOK KNOTS.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 132 /C1/135, 1991. Weisstein, E. W. "Books about Knot Theory." http:// www.treasure-troves.com/books/KnotTheory.html. Knot Complement LetR3be the space in which a KNOT Ksits. Then the space "around" the knot, i.e., everything but the knot itself, is denoted R3/C28Kand is called the knot complement of K(Adams 1994, p. 84). If a knot complement is hyperbolic (in the sense thatit admits a complete Riemannian metric of constantG AUSSIAN CURVATURE -1), then this metric is unique (Prasad 1973, Hoste et al. 1998). See also COMPLEMENT ,C OMPRESSIBLE SURFACE , KNOT,KNOT EXTERIOR References Adams, C. C. "Knot Complements and Three-Manifolds." §9.1 in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Free- man, pp. 243 /C1/246, 1994. Cipra, B. "To Have and Have Knot: When are Two Knots Alike?" Science 241, 1291 /C1/1292, 1988. Gordon, C. and Luecke, J. "Knots are Determined by their Complements." J. Amer. Math. Soc. 2, 371/C1/415, 1989. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,3 3/C1/48, Fall 1998. Prasad, G. "Stong Rigidity of Q-Rank 1 Lattices." Invent. Math. 21, 255/C1/286, 1973. Knot Curve (x2 /C281)2 /C30y2(3 /C272y) : References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Knot Determinant The determinant of a knot is ½D(/C281)½; where D(z) is the ALEXANDER POLYNOMIAL . Knot Diagram A picture of a projection of a KNOT onto a PLANE . Usually, only double points are allowed (no more than two points are allowed to be superposed), and the double or crossing points must be "genuine crossings" which transverse in the plane. This means that double points must look like the above left diagram, and not the above right one. Also, it is usually demanded that a knot diagram contain the informa- tion if the crossings are overcrossings or undercross- ings so that the original knot can be reconstructed. The knot diagram of the TREFOIL KNOT is illustrated below. KNOT POLYNOMIALS can be computed from knot diagrams. Such POLYNOMIALS often (but not always) allow the knots corresponding to given diagrams to be uniquely identified. See also NUGATORY CROSSING ,REDUCED KNOT DIA- GRAM ,REIDEMEISTER MOVESReferences Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Knot Exterior The exterior of a knot K is the complement of an open solid TORUS knotted like K. The removed open solid TORUS is called a TUBULAR NEIGHBORHOOD (Adams 1994, p. 258). See also KNOT COMPLEMENT ,GORDON- LUECKE THE- OREM ,TUBULAR NEIGHBORHOOD References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. Knot Invariant A knot invariant is a function from the set of all KNOTS to any other set such that the function does not change as the knot is changed (up to isotopy). In other words, a knot invariant always assigns the same value to equivalent knots (although different knots may have the same knot invariant). Standard knot invariants include the FUNDAMENTAL GROUP of the KNOT COMPLEMENT , numerical knot invariants (such as VASSILIEV INVARIANTS ), polynomial invariants (KNOT POLYNOMIALS such as the ALEXANDER POLY- NOMIAL ,JONES POLYNOMIAL ,KAUFFMAN POLYNOMIAL F, and KAUFFMAN POLYNOMIAL X), and torsion invariants (such as the TORSION NUMBER ). See also ARF INVARIANT ,KNOT,KNOT POLYNOMIAL , LINK INVARIANT ,TORSION NUMBER ,VASSILIEV INVAR- IANT References Aneziris, C. N. "The Knot INvariants." Ch. 5 in The Mystery of Knots: Computer Programming for Knot Tabulation. Singapore: World Scientific, pp. 35 /C1/42, 1999. Knot Linking In general, it is possible to link two n-D HYPER- SPHERES in (n/C272)/-D space in an infinite number of inequivalent ways. In dimensions greater than n/C272 in the piecewise linear category, it is true that these spheres are themselves unknotted. However, they may still form nontrivial links. In this way, they aresomething like higher dimensional analogs of two 1- spheres in 3-D. The following table gives the number of nontrivial ways that two n-D HYPERSPHERES can be linked in k-D. D of spheres D of space Distinct Linkings 23 40 239 31 48 959 102 181 3 102 182 10438319 102 183 3 Two 10-D HYPERSPHERES link up in 12, 13, 14, 15, and 16-D, then unlink in 17-D, link up again in 18, 19, 20, and 21-D. The proof of these results consists of an "easy part" (Zeeman 1962) and a "hard part" (Ravenel 1986). The hard part is related to the calculation of the (stable and unstable) HOMOTOPY GROUPS of SPHERES . References Bing, R. H. The Geometric Topology of 3-Manifolds. Provi- dence, RI: Amer. Math. Soc., 1983. Ravenel, D. Complex Cobordism and Stable Homotopy Groups of Spheres. New York: Academic Press, 1986. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 7, 1976. Zeeman. "Isotopies and Knots in Manifolds." In Topology of 3-Manifolds and Related Topics (Ed. M. K. Fort). Engle- wood Cliffs, NJ: Prentice-Hall, 1962. Knot Move An operation on a knot or link diagram which preserves its crossing number. Thistlethwaite used 13 different moves in generating a list of 16-crossing alternating knots (Hoste et al. 1998). While these moves eliminate all duplicate knots up to 13 crossings with only a single exception, there are 9,868 dupli- cates in his list of 1,018,774 16-crossing knots (Hoste et al. 1998). See also FLYPE ,HABIRO MOVE,MARKOV MOVES ,PASS MOVE,P ERKO MOVE,P OKE MOVE,R EIDEMEISTER MOVES ,SLIDE MOVE,TWIST MOVE References Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Knot Polynomial A knot invariant in the form of a POLYNOMIAL such as the ALEXANDER POLYNOMIAL , BLM /HO POLYNOMIAL , BRACKET POLYNOMIAL ,CONWAY POLYNOMIAL , HOM- FLY POLYNOMIAL ,JONES POLYNOMIAL ,K AUFFMAN POLYNOMIAL F,KAUFFMAN POLYNOMIAL X, and VAS- SILIEV INVARIANT . See also KNOT,LINK References Lickorish, W. B. R. and Millett, K. C. "The New Polynomial Invariants of Knots and Links." Math. Mag. 61,3/C1/23, 1988. Knot Problem The problem of deciding if two KNOTS in 3-space are equivalent such that one can be continuously de- formed into another.Knot Shadow A KNOT DIAGRAM which does not specify whether crossings are under- or overcrossings. Knot Sum Two oriented knots (or links) can be summed by placing them side by side and joining them by straight bars so that orientation is preserved in the sum. This operation is denoted #, so the knot sum of knots K1 and K2 is written K1 # K2 /C30K2 # K1 : The KNOT SUM of any number of knots cannot be the UNKNOT unless each knot in the sum is the UNKNOT (Schubert 1949; Steinhaus 1983, p. 265). See also CONNECTED SUM References Schubert, H. Sitzungsber. Heidelberger Akad. Wiss., Math.- Naturwiss. Klasse, 3rd Abhandlung. 1949. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Knot Symmetry A symmetry of a knot Kis a homeomorphism of R3 which maps Konto itself. More succinctly, a knot symmetry is a homeomorphism of the pair of spaces (R3;K):Hoste et al. (1998) consider four types of symmetry based on whether the symmetry preservesor reverses orienting of R 3andK, 1. preserves R3;preserves K(identity operation), 2. preserves R3;reverses K, 3. reverses R3;preserves K, 4. reverses R3;reverses K. This then gives the five possible classes of symmetrysummarized in the table below. class symmetries knot symmetries c 1 chiral, noninvertible //C27/ 1, 3 //C27amphichiral, noninvertible //C28/ 1, 4 //C28amphichiral, noninvertible i 1, 2 chiral, invertible a 1, 2, 3, 4 //C27and/C28amphichiral, inver- tible In the case of HYPERBOLIC KNOTS , the symmetry group must be finite and either CYCLIC orDIHEDRAL (Riley 1979, Kodama and Sakuma 1992, Hoste et al. 1998). The classification is slightly more complicated for nonhyperbolic knots. Furthermore, all knots with 58 crossings are either amphichiral or invertible (Hoste et al. 1998). Any symmetry of a prime alternating link must be visible up to flypes in any alternating diagram of the link (Bonahon and Sie- bermann, Menasco and Thistlethwaite 1993, Hoste et al. 1998). The following tables (Hoste et al. 1998) give the numbers of n-crossing knots belonging to cyclic symmetry groups Zk(Sloane’s A052411 for Z1and A052412 for Z2) and dihedral symmetry groups Dk (Sloane’s A052415 through A052422). Of knots with 16 or fewer crossings, there are only one each having symmetry groups Z3 ; D14 ; and D16 (above left). There are only two knots with symmetry group D9 ; one hyperbolic (above right), and one a satellite knot. In addition, there are 2, 4, and 10 satellite knots having 14-, 15-, and 16-crossings, respectively, which belong to the dihedral group D/C12:/ n /Z1//Z2//Z3//Z4/ 10 0 0 0 20 0 0 0 30 0 0 0 40 0 0 050 0 0 060 0 0 0 70 0 0 0 80 0 0 092 0 0 0 10 24 3 0 0 11 173 14 0 0 12 1047 57 0 0 13 6709 210 0 0 14 37177 712 0 2 15 224311 2268 1 0 16 1301492 7011 0 11n /D1//D2//D3//D4//D5//D6//D7//D8//D9//D10//D14//D16/ 1 0 00000000000 2 0 00000000000 3 0 00000000000 4 0 01000000000 5 0 10000000000 6 0 20100000000 7 0 40200000000 8 41 20300010000 9 1 32 3 3 4 0 3 0 0 0000 1 0 6 66 2 1 5 0 1 0 0 0100 1 12 1 7 1 3 4 2 1 1 0 0 0 0 00001 27 2 8 3 0 9 6 1 8 0 8 1 2 000013 2391 647 1 21 2 3 1 2 0000 14 7575 1463 4 31 2 2 0 0 0010 15 23517 3065 50 53 3 12 0 2 1400 16 73263 6791 15 89 0 10 1 8 1101 See also AMPHICHIRAL KNOT,CHIRAL KNOT,KNOT References Bonahon, F. and Siebermann, L. "The Classification of Algebraic Links." Unpublished manuscript. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,3 3/C1/48, Fall 1998. Kodama K. and Sakuma, M. "Symmetry Groups of Prime Knots Up to 10 Crossings." In Knot 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka, Japan, 1990 (Ed. A. Kawauchi.) Berlin: de Gruyter, pp. 323 /C1/340, 1992. Menasco, W. and Thistlethwaite, M. "The Classification of Alternating Links." Ann. Math. 138, 113/C1/171, 1993. Riley, R. "An Elliptic Path from Parabolic Representations to Hyperbolic Structures." In Topology of Low-Dimensional Manifolds, Proceedings, Sussex 1977 (Ed. R. Fenn). New York: Springer-Verlag, pp. 99 /C1/133, 1979. Sloane, N. J. A. Sequences A052411, A052412, A052415, A052416, A052417, A052418, A052420, and A052422 in"An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Knot Theory The mathematical study of KNOTS . Knot theory considers questions such as the following: 1. Given a tangled loop of string, is it really knotted or can it, with enough ingenuity and/or luck, be untangled without having to cut it? 2. More generally, given two tangled loops ofstring, when are they deformable into each other? 3. Is there an effective algorithm (or any algorithm to speak of) to make these determinations? Although there has been almost explosive growth in the number of important results proved since the discovery of the JONES POLYNOMIAL , there are still many "knotty" problems and conjectures whose an- swers remain unknown. See also KNOT,LINK Knot Vector B-SPLINE Knuth Number The numbers defined by the RECURRENCE RELATION Kn/C271 /C301 /C27min(2 K n=2bc; 3Kn=3bc) ; with K0 /C301: The first few values for n /C300, 1, 2, ... are 1, 3, 3, 4, 7, 7, 7, 9, 9, 10, 13, ... (Sloane’s A007448). References Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Sloane, N. J. A. Sequences A0074482276 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Ko¨be Function The function fu(z) /C13z (1 /C27 ei uz)2 (1) defined on the UNIT DISK ½z½B1: For u /C23 [0; 2p) ; the Ko¨be function is a SCHLICHT FUNCTION f(z) /C30z /C27X/C12 j/C302ajzj (2)with ½aj ½/C30j for all j (Krantz 1999, p. 149). For u /C300; f0(z) /C30z (z /C28 1)2 ; (3) illustrated above. See also KO¨ BE’S ONE-FOURTH THEOREM ,SCHLICHT FUNCTION References Bombieri, E. "On the Local Maximum of the Koebe Func- tion." Invent. Math. 4,26/C1/67, 1967. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 149, 1999. Pederson, R. and Schiffer, M. "A Proof of the Bieberbach Conjecture for the Fifth Coefficient." Arch. Rational Mech. Anal. 45, 161 /C1/193, 1972. Stewart, I. From Here to Infinity: A Guide to Today’s Mathematics. Oxford, England: Oxford University Press, pp. 164 /C1/165, 1996. Ko¨be’s One-Fourth Theorem If f is a SCHLICHT FUNCTION and D(z0 ; r) is the OPEN DISK of radius r centered at z0;then f(D(0;1))–D(0;1=4); where–denotes a (not necessarily proper) SUPERSET (Krantz 1999, p. 150). See also KO¨ BE FUNCTION ,SCHLICHT FUNCTION References Krantz, S. G. "The Ko ¨be 1/4 Theorem." §12.1.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 150 /C1/ 151, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 24, 1983. Koch Antisnowflake AFRACTAL derived from the K OCH SNOWFLAKE . The base curve and motif for the fractal are illustrated below. The AREA after the nth iteration is An/C30An/C281/C281 3ln/C281 aD 3n; where Dis the area of the original EQUILATERAL TRIANGLE , so from the derivation for the KOCH SNOWFLAKE , A /C13 lim n 0/C12An /C30(1 /C283 5)D/C3025D: See also EXTERIOR SNOWFLAKE ,FLOWSNAKE FRAC- TAL,K OCH SNOWFLAKE ,P ENTAFLAKE ,S IERPINSKI CURVE References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 66 /C1/67, 1989. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 36 /C1/ 37, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 136, 1991. Koch Island KOCH SNOWFLAKE Koch Snowflake AFRACTAL , also known as the K OCH ISLAND , which was first described by Helge von Koch in 1904. It is built by starting with an EQUILATERAL TRIANGLE , removing the inner third of each side, building another EQUILATERAL TRIANGLE at the location where the side was removed, and then repeating the process indefinitely. The Koch snowflake can be simplyencoded as a L INDENMAYER SYSTEM with initial string "F-F-F" ,STRING REWRITING rule"F" -/C21"F/C27F- F/C27F", and angle 60 8. The zeroth through third iterations of the construction are shown above. The fractal can also be constructed using a base curve and motif, illustrated below. LetNnbe the number of sides, Lnbe the length of a single side, lnbe the length of the PERIMETER , and An the snowflake’s AREA after the nth iteration. Further, denote the AREA of the initial n/C300TRIANGLE D;andthe length of an initial n/C300 side 1. Then Nn/C303/C2154n(1) Ln/C301 3>C16>C17n /C303/C28n(2) ln/C13NnLn/C3034 3>C16>C17n (3) An/C30An/C281/C2714NnL2 nD/C30An/C281/C273 /C2154n 41 3 !2n D /C30An/C281/C273 /C2154n/C281 9nD/C30An/C281/C273 /C21544/C281 9 /C2159n/C281D /C30An/C281/C271 349>C16>C17n/C281 D: (4) The CAPACITY DIMENSION is then dcap/C30/C28lim n0/C12lnNn lnLn/C30/C28lim n0/C12ln(3 /C2154)n ln(3/C28n) /C30lim n0/C12ln 3/C27nln 4 nln 3/C30ln 4 ln 3/C302l n2 ln 3 /C301:261859507 . . . : (5) Now compute the AREA explicitly, A0/C30D (6) A1/C30A0/C271349 ! 0 D/C30D1/C271349 ! 08 < :9 = ;(7) A2/C30A1/C271 349 ! 1 D/C30D1/C2713 49 ! 0 /C2749 ! 12 4358 < :9 = ;(8) A n/C301/C271 3Xn k/C30049 ! k2 435D; (9) so as n0/C12; A/C13A /C12/C301/C271 3X/C12 k/C30149 ! k2 435/C301/C27 1 31 1/C284 9 ! D /C3085D: (10) Some beautiful TILINGS , a few examples of which are illustrated above, can be made with iterations toward Koch snowflakes. In addition, two sizes of Koch snowflakes in AREA ratio 1:3 TILE the PLANE , as shown above (Mandel- brot). Another beautiful modification of the Koch snowflake involves inscribing the constituent triangles with filled-in triangles, possibly rotated at some angle. Some sample results are illustrated above for 3 and 4 iterations. See also CESA` RO FRACTAL ,E XTERIOR SNOWFLAKE , GOSPER ISLAND ,KOCH ANTISNOWFLAKE ,PEANO- GOS- PER CURVE ,PENTAFLAKE ,SIERPINSKI SIEVE References Bulaevsky, J. "The Koch Curve Fractal." http://www.best.- com/~ejad/java/fractals/koch.shtml. Cesa`ro, E. "Remarques sur la courbe de von Koch." Atti della R. Accad. della Scienze fisiche e matem. Napoli 12, No. 15, 1905. Reprinted as §228 in Opere scelte, a cura dell’Unione matematica italiana e col contributo del Consiglio nazio- nale delle ricerche, Vol. 2: Geometria, analisi, fisicamatematica. Rome: Edizioni Cremonese, pp. 464 /C1/479, 1964. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 65 /C1/66, 1989. Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Dixon, R. Mathographics. New York: Dover, pp. 175 /C1/177 and 179, 1991. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 227, 1984. Harris, J. W. and Stocker, H. "Koch’s Curve" and "Koch’s Snowflake." §4.11.5 /C1/4.11.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 114 /C1/115, 1998. King, B. W. "Snowflake Curves." Math. Teacher 57, 219/C1/ 222, 1964. Koch, von. Acta Math. 30, 145, 1906. Koch, von. Archiv fo ¨r Matemat., Astron. och Fysik. , pp. 681 /C1/ 702, 1914. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 28 /C1/ 29 and 32 /C1/36, 1991. Pappas, T. "The Snowflake Curve." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 78 and 160 /C1/ 161, 1989. Peitgen, H.-O.; Ju ¨rgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer- Verlag, 1992. Peitgen, H.-O. and Saupe, D. (Eds.). "The von Koch Snow- flake Curve Revisited." §C.2 in The Science of Fractal Images. New York: Springer-Verlag, pp. 275 /C1/279, 1988. Schneider, J. E. "A Generalization of the Von Koch Curves." Math. Mag. 38, 144/C1/147, 1965. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 185 /C1/195, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 135 /C1/136, 1991. Kochansky’s Approximation The approximation for PIgiven by p:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40 3/C282ffiffiffi 3ps /C301 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 120/C2818ffiffiffi 3pq /C303:141533 . . . : In the above figure, let OA/C30AF/C301;and construct the circle centered at A/C30(0;0) of radius 1. This intersects Oat point B/C30(/C28ffiffiffi 3p =2;1=2):Now con- struct the circle about Bwith radius 1. The circles AandBintersect in C/C30(/C28ffiffiffi3p =2;/C281=2);and the line CO intersects the perpendicular to OA through A in the point D /C30(/C28ffiffiffi 3p =3; 0): Now construct the point E /C30(3 /C28ffiffiffi 3p =3; 0) to be a distance 3 along DA. The line segment EF is then of length ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 22 /C27 3 /C281 2ffiffiffi 3p>C16>C172r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40 3/C282ffiffiffi3ps : This construction was given by the Polish Jesuit priest Kochansky (Steinhaus 1983). See also G EOMETRIC CONSTRUCTION ,PI References Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 44, 1982. Kochansky. Acta Eruditorum. 1685. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 143, 1999. Kodaira Embedding Theorem A theorem which states that if a KA¨ HLER FORM represents an INTEGRAL COHOMOLOGY CLASS on a COMPACT MANIFOLD , then it must be a PROJECTIVE VARIETY . See also KA¨ HLER FORM Koenigs-Poincare ´ Theorem Let G denote the group of GERMS of holomorphic diffeomorphisms of (C; 0): Then if ½ l ½"1; then Gl is a conjugacy class, i.e., all f /C23 Gl are linearizable. References Marmi, S. An Introduction to Small Divisors Problems 27 Sep 2000. http://xxx.lanl.gov/abs/math.DS/0009232/. Kolakoski Sequence The self-describing sequence consisting of "blocks" of single and double 1s and 2s, where a "block" is a single or double digit that is different from the digit in the preceding block. To construct the sequence, start with the single digit 1 (the first "block"). Here, the single 1 means that block of length one follows the first block. Therefore, require that the next block is 2, giving the sequence 12. Now, the 2 means that the next (third) block will have length two, so append 11 and obtain the sequence 1211. We have added two 1s, so the fourth and fifth blocks have length one each, giving 12112 and then 121121. As a result of adding 21, we obtain 121121221. As a result of adding 221, we obtain 12112122122112, and so on, giving the sequence 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ... (Sloane’s A006928). The sequence after successive iterations is given by 1, 12, 1211, 121121, 121121221, ..., and the lengths of this sequence after steps n /C301, 2, ... are given by 1, 2, 4, 6, 9, 14, 22, ... (Sloane’s A042942).If the sequence is started with 1, 2, 2 and the above procedure is undertaken beginning with the last 2, then the virtually identical sequence 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ... (Sloane’s A000002) is obtained. (It is the same as Sloane’s A006928, except that the second 2 is doubled.) When presented in this form, the term a(n) gives the length of the nth RUN in the sequence. The lengths after steps n /C301, 2, ... are then 1, 2, 3, 5, 7, 10, 15, ... (Sloane’s A001083), essentially one less than Sloane’s A042942. The question of whether the number of 1s is "asymp- totically" equal to the number of 2s is unsettled, although the above plot (which shows the fraction of1s as a function of number of digits) is certainly consistent with 1 and 2 being equidistributed. See also R UN References Dekking, F. M. "What Is the Long Range Order in the Kolakoski Sequence?" Reports of the Faculty of Technical Mathematics and Informatics, No. 95 /C1/100. Delft, Nether- lands: Delft University of Technology, 1995. Kimberling, C. "Integer Sequences and Arrays." http:// cedar.evansville.edu/~ck6/integer/. Kimberling, C. "Unsolved Problems and Rewards." http:// cedar.evansville.edu/~ck6/integer/unsolved.html. Kolakoski, W. "Problem 5304: Self Generating Runs." Amer. Math. Monthly 72, 674, 1965. Kolakoski, W. "Problem 5304." Amer. Math. Monthly 73, 681/C1/682, 1966. Lagarias, J. C. "Number Theory and Dynamical Systems." InThe Unreasonable Effectiveness of Number Theory (Ed. S. A. Burr). Providence, RI: Amer. Math. Soc., pp. 35 /C1/72, 1992. Paun, G. and Salomaa, A. "Self-Reading Sequences." Amer. Math. Monthly 103, 166/C1/168, 1996. Sellke. Problem 324 in Statistica Neerlandica 50, 222/C1/223, 1996. Sloane, N. J. A. Sequences A000002/M0190, A001083, and A006298/M0070, A042942 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, p. 233, 1991. Kollros’ Theorem For every ring containing pSPHERES , there exists a ring of qSPHERES , each touching each of the p SPHERES , where 1 p /C271 q /C3013 : The HEXLET is a special case with p /C303. See also HEXLET ,SPHERE References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 50, 1976. Kolmogorov Complexity The complexity of a pattern parameterized as the shortest ALGORITHM required to reproduce it. Also known as ALGORITHMIC COMPLEXITY . References Goetz, P. "Phil’s Good Enough Complexity Dictionary." http://www.cs.buffalo.edu/~goetz/dict.html. Kolmogorov Constant The exponent 5/3 in the spectrum of homogeneous turbulence, k/C285 =3 :/ References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 41, 1983. Kolmogorov Criterion STRONG LAW OF LARGE NUMBERS Kolmogorov Entropy Also known as METRIC ENTROPY . Divide PHASE SPACE into D-dimensional HYPERCUBES of CONTENT eD : Let Pi0 ; ... ; inbe the probability that a trajectory is in HYPERCUBE i0at t /C300, i1at t /C30T, i2at t /C302T ; etc. Then define Kn /C30hK /C30/C28X i0 ; ... ; inPi0 ; ... ; inln Pi0 ; ... ; in; (1) where KN /C271 /C28KN is the information needed to predict which HYPERCUBE the trajectory will be in at (n /C271)T given trajectories up to nT. The Kolmogorov entropy is then defined by K/C13lim T00lim e00/C27lim N0/C121 NTXN/C281 n/C300(Kn/C271/C28Kn): (2) The Kolmogorov entropy is related to L YAPUNOV CHARACTERISTIC EXPONENTS by hK/C30gpX si>0sidm: (3) See also HYPERCUBE ,L YAPUNOV CHARACTERISTIC EXPONENTReferences Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 138, 1993. Schuster, H. G. Deterministic Chaos: An Introduction, 3rd ed.New York: Wiley, p. 112, 1995. Kolmogorov-Arnold-Moser Theorem A theorem outlined in 1954 by Kolmogorov which was subsequently proved in the 1960s by Arnold andMoser (Tabor 1989, p. 105). It gives conditions under which CHAOS is restricted in extent. Moser’s 1962 proof was valid for TWIST MAPS u?/C30u/C272pf(I)/C27g(u;I) (1) I?/C30I/C27f(u;I): (2) In 1963, Arnold produced a proof for Hamiltoniansystems H/C30H 0(I)/C27eH1(I): (3) The original theorem required perturbations e/C2 10/C2848;although this has since been significantly increased. Arnold’s proof required C/C12;and Moser’s original proof required C333:Subsequently, Moser’s version has been reduced to C6;then C2/C27e;although counterexamples are known for C2:Conditions for applicability of the KAM theorem are: 1. small perturbations, 2. smooth perturbations, and 3. sufficiently irrational WINDING NUMBER . Moser considered an integrable Hamiltonian functionH 0with a TORUS T0and set of frequencies vhaving an incommensurate frequency vector v/C31(i.e.,v /C215k" 0 for all INTEGERS ki):LetH0be perturbed by some periodic function H1:The KAM theorem states that, if H1is small enough, then for almost every v/C31there exists an invariant TORUS T(v/C31) of the perturbed system such that T(v/C31) is "close to" T0(v/C31):Moreover, the TORI T(v/C31) form a set of POSITIVE measures whose complement has a measure which tends to zero as ½H1½00:A useful paraphrase of the KAM theorem is, "For sufficiently small perturbation, almost all TORI (excluding those with rational frequency vectors) arepreserved." The theorem thus explicitly excludes TORI with rationally related frequencies, that is, n/C281 conditions of the form v /C215k/C300: (4) These TORI are destroyed by the perturbation. For a system with two DEGREES OF FREEDOM , the condition of closed orbits is s/C30v1 v2/C30r s: (5) For a QUASIPERIODIC ORBIT ,sisIRRATIONAL . KAM shows that the preserved TORI satisfy the irration- ality condition v1 v2/C28r s>C12>C12>C12>C12>C12>C12>C12>C12>C12>C12> K( e) s2 :5 (6) for all r and s, although not much is known about K( e) :/ The KAM theorem broke the deadlock of the small divisor problem in classical perturbation theory, and provides the starting point for an understanding of the appearance of CHAOS . For a HAMILTONIAN SYS- TEM, the ISOENERGETIC NONDEGENERACY condition @2H0 @Ij @Ij>C12>C12>C12>C12>C12>C12>C12>C12>C12>C12"0 (7) guarantees preservation of most invariant TORI under small perturbations e /C101: The Arnold version states that Xn k/C301mk vk>C12>C12>C12>C12>C12>C12>C12>C12>C12>C12> K( e)X n k /C301½mk ½ ! /C28n/C281 (8) for all mk /C23Z: This condition is less restrictive than Moser’s, so fewer points are excluded. See also CHAOS ,HAMILTONIAN SYSTEM ,QUASIPERIO- DIC FUNCTION ,TORUS References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Kolmogorov-Sinai Entropy KOLMOGOROV ENTROPY ,METRIC ENTROPY Kolmogorov-Smirnov Test A goodness-of-fit test for any STATISTICAL DISTRIBU- TION . The test relies on the fact that the value of the sample cumulative density function is asymptotically normally distributed. To apply the Kolmogorov-Smirnov test, calculate the cumulative frequency (normalized by the sample size) of the observations as a function of class. Then calculate the cumulative frequency for a true dis- tribution (most commonly, the NORMAL DISTRIBU- TION ). Find the greatest discrepancy between the observed and expected cumulative frequencies, which is called the "D-STATISTIC ." Compare this against the critical D-STATISTIC for that sample size. If the calculated D-STATISTIC is greater than the critical one, then reject the NULL HYPOTHESIS that the distribution is of the expected form. The test is an R-ESTIMATE . See also ANDERSON- DARLING STATISTIC , D-STATISTIC , KUIPER STATISTIC ,N ORMAL DISTRIBUTION , R-ESTI- MATEReferences Boes, D. C.; Graybill, F. A.; and Mood, A. M. Introduction to the Theory of Statistics, 3rd ed. New York: McGraw-Hill, 1974. DeGroot, M. H. Ch. 9 in Probability and Statistics, 3rd ed. Reading, MA: Addison-Wesley, 1991. Knuth, D. E. §3.3.1B in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 45 /C1/52, 1998. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Kolmogorov-Smirnov Test." In Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 617 /C1/620, 1992. Ko¨nig’s Theorem If an ANALYTIC FUNCTION has a single simple POLE at the RADIUS OF CONVERGENCE of its POWER SERIES , then the ratio of the coefficients of its POWER SERIES converges to that POLE . See also POLE References Ko¨nig, J. "U¨ ber eine Eigenschaft der Potenzreihen." Math. Ann. 23, 447 /C1/449, 1884. Ko¨nig-Egeva ´ry Theorem A theorem on BIPARTITE GRAPHS . See also BIPARTITE GRAPH ,FROBENIUS- KO¨ NIG THEO- REM Ko¨nigsberg Bridge Problem The Ko¨nigsberg bridges cannot all be traversed in a single trip without doubling back. This problem was solved by Euler (1736), and represented the begin- ning of GRAPH THEORY . See also CIRCUIT ,EULERIAN CIRCUIT ,GRAPH THEORY , UNICURSAL CIRCUIT References Biggs, N. L.; Lloyd, E. K.; and Wilson, R. J. Graph Theory 1736/C1/1936. Oxford, England: Oxford University Press, 1976. Bogomolny, A. "Graphs." http://www.cut-the-knot.com/ do_you_know/graphs.html. Chartrand, G. "The Ko ¨nigsberg Bridge Problem: An Intro- duction to Eulerian Graphs." §3.1 in Introductory Graph Theory. New York: Dover, pp. 51 /C1/66, 1985. Euler, L. "Solutio problematis ad geometriam situs perti- nentis." Comment. Acad. Sci. U. Petrop. 8, 128/C1/140, 1736. Reprinted in Opera Omnia Ser. I-7 , pp. 1 /C1/10, 1766. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 1/C1/2, 1994. Kraitchik, M. §8.4.1 in Mathematical Recreations. New York: W. W. Norton, pp. 209 /C1/211, 1942. Newman, J. "Leonhard Euler and the Ko ¨nigsberg Bridges." Sci. Amer. 189,6 6/C1/70, 1953. Pappas, T. "Ko ¨nigsberg Bridge Problem & Topology." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, pp. 124 /C1/125, 1989. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 192, 1990. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 256 /C1/259, 1999. Wilson, R. J. "An Eulerian Trail through Ko ¨nigsberg." J. Graph Th. 10, 265/C1/275, 1986. Kontorovich-Lebedev Transform The forward and inverse Kontorovich-Lebedev trans- forms are defined by Kix[f(t)]/C30g/C12 0Kix(t)f(t)dt K/C281 ix[g(t)]/C302 p2xg/C12 0tsinh( pt)Kit(x)g(t)dt; respectively, where Kn(z)i sa MODIFIED BESSEL FUNCTION OF THE SECOND KIND with imaginary index /n/C30ix/. References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 753, 1993. Kontsevich Integral This entry contributed by S ERGEI DUZHIN AND S.CHMUTOV Kontsevich’s integral is a far-reaching generalizationof the G AUSS INTEGRAL for the LINKING NUMBER , and provides a tool to construct the UNIVERSAL VASSILIEV INVARIANT of a KNOT . In fact, any V ASSILIEV KNOT INVARIANT can be derived from it. To construct the Kontsevich integral, represent the three-dimensional space R3as a DIRECT PRODUCT of a complex line Cwith coordinate zand a real line R with coordinate t. The integral is defined for M ORSE KNOTS , i.e., knots Kembedded in R3/C30Cz/C29Rtin such a way that the coordinate tis a M ORSE FUNCTION on K, and its values belong to the GRADED COMPLETION ¯Aof the ALGEBRA OF CHORD DIAGRAMS A:/ The Kontsevich integral Z(K) of the knot Kis defined asZ(K)/C30X/C12 m/C3001 (2pi)m g tminBt1B...BtmBtmax tjare noncriticalX P/C30f(zj;z?j)g(/C281)¡Dp /C2fflm j/C301dzj/C28dz?j zj/C28z?j; (1) where the ingredients of this formula have the following meanings. The real numbers tminand tmax are the minimum and the maximum of the function t onK. The integration domain is the m-dimensional simplex tminBt1B...BtmBtmaxdivided by the critical values into a certain number of connected components. Forexample, for the embedding of the unknot and m/C302 (left figure), the corresponding integration domainhas six connected components, illustrated in the rightfigure above. The number of summands in the integrand is con- stant in each connected component of the integration domain, but can be different for different components.In each plane ft/C30t jgƒR3;choose an unordered pair of distinct points ( zj;tj) and ( z?j;tj)o n Kso that zj(tj) and z?t(tj) are continuous functions. Denote by P/C30 f(zj;z?j)gthe set of such pairs for j/C301, ..., m, then the integrand is the sum over all choices of P. In the example above, for the component ftminBt1B tc1;tc2Bt2Btmaxg;we have only one possible pair of points on the levels ft/C30t1gand ft/C30t2g:Therefore, the sum over Pfor this component consists of only one summand. In contrast, in the component ftminB t1Btc1;tc1Bt2Btc2g;we still have only one possibility for the level ft/C30t1g;but the plane ft/C30t2gintersects our knot Kin four points. So we have4 2>C0>C1 /C306 possible pairs ( z2;z?2);and the total number of summands is six (see the picture below). For a pairing Pthe symbol " /¡/" denotes the number of points ( zj;tj)o r( z?j;tj)i n Pwhere the coordinate t decreases along the ORIENTATION ofK. Fix a pairing P, consider the knot Kas an oriented circle, and connect the points ( zj;tj) and ( z?j;tj)b ya chord to obtain a chord diagram with mchords. The corresponding element of the algebra Ais denoted DP:In the picture above, one of the possible pairings, the corresponding CHORD DIAGRAM with the sign (/C281)¡;and the number of summands of the integrand (some of which are equal to zero in Adue to a ONE- TERM RELATION ) are shown for each connected com- ponent. Over each connected component, /zjandz?jare SMOOTH FUNCTIONS intj:/By fflm j/C301dzj/C28dz?j zj/C28z?j we mean the PULLBACK of this form to the integration domain of variables t1;...,tm:The integration domain is considered with the ORIENTATION of the space Rm defined by the natural order of the coordinates t1;..., tm:/ By convention, the term in the Kontsevich integral corresponding to m/C300 is the (only) CHORD DIAGRAM of order 0 with coefficient one. It represents the unitof the algebra A: / The Kontsevich integral is convergent thanks to ONE- TERM RELATIONS . It is invariant under DEFORMATIONS of the knot in the class of M ORSE KNOTS . Unfortu- nately, the Kontsevich integral is not invariant underdeformations that change the number of critical points of the function t. However, the formula shows how the integral changes under such deformations: In the above equation, the graphical arguments of Z represent two embeddings of an arbitrary knot,differing only in the illustrated fragment, His the hump (i.e, the UNKNOT embedded in R3in the specified way; illustrated above), and the product is the product in the completed algebra ¯AofCHORD DIAGRAMS . The last equality allows the definition of the UNIVERSAL VASSILIEV INVARIANT by the formula I(K)/C30Z(K) Z(H)c=2; (2) where cdenotes the number of critical points of K and quotient means division in the algebra ¯A according to the rule (1 /C27a)/C281/C301/C28a/C27a2/C28a3/C27...: The UNIVERSAL VASSILIEV INVARIANT I(K) is invariant under an arbitrary DEFORMATION ofK. Consider a function won the set of CHORD DIAGRAMS with mchords satisfying ONE- AND FOUR-TERM RELA- TIONS (aWEIGHT SYSTEM ). Applying this function to the UNIVERSAL VASSILIEV INVARIANT w(I(K));we get a numerical knot invariant. This invariant will be aV ASSILIEV INVARIANT of order m, and any V ASSILIEV INVARIANT can be obtained in this way. The Kontsevich integral behaves in a nice way with respect to the natural operations on knots, such as mirror reflection, changing the orientation of the knot, and mutation of knots. In a proper normal-ization it is multiplicative under the CONNECTED SUM of knots: I?(K1#K2)/C30I?(K1)I?(K2); (3) where I?(K)/C30Z(H)I(K):For any knot Kthe coeffi- cients in the expansion of Z(K) over an arbitrary basis consisting of CHORD DIAGRAMS are rational (Kontse- vich 1993, Le and Murakami 1996). The task of computing the Kontsevich integral is very difficult. The explicit expression of the universal Vassiliev invariant I(K) is currently known only for the UNKNOT , I(O)/C30expX/C12 n/C300b2nw2n ! (4) /C301/C27X/C12 n/C300b2nw2n ! /C271 2X/C12 n/C300b2nw2n ! 2 /C27...: (5) (Bar-Natan et al. 1997). Here, b2nare MODIFIED BERNOULLI NUMBERS , i.e., the coefficients of the TAYLOR SERIES X/C12 n/C300b2nx2n/C301 2lnex=2/C28e/C28x=2 1 2x ! (6) (/b2 /C301 =48; b4 /C30/C281=5760 ; ...; Sloane’s A057868), and w2n are the wheels , i.e., diagrams of the form The linear combination is understood as an element of the ALGEBRA OF CHINESE CHARACTERS B; which is isomorphic to the ALGEBRA OF CHORD DIAGRAMS A: Expressed through CHORD DIAGRAMS , the beginning of this series looks as follows: The Kontsevich integral was invented by Kontsevich (1993), and detailed expositions can be found in Arnol’d (1994), Bar-Natan (1995), and Chmutov and Duzhin (2000). See also CHORD DIAGRAM ,GAUSS INTEGRAL ,M ORSE KNOT,VASSILIEV INVARIANT References Arnol’d, V. I. "Vassiliev’s Theory of Discriminants and Knots." In First European Congress of Mathematics, Vol. 1 (Paris, 1992) 3764327987 (Ed. A. Joseph, F. Mignot, F. Murat, B. Prum, and R. Rentschler). Basel, Switzer- land: Birkha ¨user, pp. 3 /C1/29, 1994. Bar-Natan, D.; Garoufalidis, S.; Rozansky, L.; and Thurston, D. "Wheels, Wheeling, and the Kontsevich Integral of the Unknot." Preprint, 1997. Bar-Natan, D. "On the Vassiliev Knot Invariants." Topology 34 423 /C1/472, 1995. Chmutov, S. V. and Duzhin, S. V. "The Kontsevich Inte- gral." To appear in Acta Appl. Math. , 2000. ftp://ftp.bo- tik.ru/pub/local/zmr/ki.ps.gz. Kontsevich, M. "Vassiliev’s Knot Invariants." Adv. Soviet Math. 16, Part 2, 137 /C1/150, 1993. Le, T. Q. T. and Murakami, J. "The Universal Vassiliev- Kontsevich Invariant for Framed Oriented Links." Com- pos. Math. 102,42/C1/64, 1996. Sloane, N. J. A. Sequences A057868 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Vassiliev, V. A. "Cohomology of Knot Spaces." In Theory of Singularities and Its Applications (Ed. V. I. Arnold). Adv. Soviet Math. 1,23/C1/69, 1990. Kontsevich’s Integral See also VASSILIEV INVARIANT Korselt’s Criterion n DIVIDES an /C28a for all INTEGERS a IFF n is SQUARE- FREE and (p /C281)½n=p /C281 for all PRIME DIVISORS p of n. CARMICHAEL NUMBERS satisfy this CRITERION . See also CARMICHAEL NUMBERReferences Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,40/C1/50, 1996. Korteweg de Vries Equation The PARTIAL DIFFERENTIAL EQUATION K0/C301 See also KADOMTSEV- PETVIASHVILI EQUATION ,KRICH- EVER- NOVIKOV EQUATION References Baker, H. F. Abelian Functions: Abel’s Theorem and the Allied Theory, Including the Theory of the Theta Func- tions. New York: Cambridge University Press, p. xix, 1995. Segal, G. "The Geometry of the KdV Equation." Int. J. Math. Phys. A 6, 2859 /C1/2869, 1991. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Korteweg-de Vries Equation The PARTIAL DIFFERENTIAL EQUATION ut/C27uxxx/C286uux/C300 (1) (Lamb 1980; Zwillinger 1997, p. 131), often abbre- viated "KdV." The so-called generalized KdV equation is given by ut/C27uux/C28uxxxxx/C300 (2) (Boyd 1986; Zwillinger 1997, p. 131). The so-called deformed KdV equation is given by ut/C27@ @xuxx/C282hu3/C283 2uu2 x h/C27u2 ! /C300 (3) (Dodd and Fordy 1983; Zwillinger 1997, p. 133), and the modified KdV equation is given by ut/C27uxxx96u2ux/C300 (4) (Calogero and Degasperis 1982, p. 51; Tabor 1990,p. 304; Zwillinger 1997, p. 133), or u t/C27uxxx/C281 8u3 x/C27ux(Aeu/C27B/C27Ce/C28u)/C300 (5) (Dodd and Fordy 1983; Zwillinger 1997, p. 133). The cylindrical KdV equation is given by ut/C27uxxx/C286uux/C27u 2t/C300 (6) (Calogero and Degasperis 1982, p. 50; Zwillinger 1997, p. 131), and the spherical KdV by ut/C27uxxx/C286uux/C27u t/C300 (7) (Calogero and Degasperis 1982, p. 51; Zwillinger 1997, p. 132). See also KADOMTSEV- PETVIASHVILI EQUATION ,KOR- TEWEG-DE VRIES- BURGER EQUATION ,KRICHEVER- NO- VIKOV EQUATION ,R EGULARIZED LONG- WAVE EQUATION ,SOLITON References Baker, H. F. Abelian Functions: Abel’s Theorem and the Allied Theory, Including the Theory of the Theta Func- tions. New York: Cambridge University Press, p. xix, 1995. Boyd, J. P. "Solitons from Sine Waves: Analytical and Numerical Methods of Non-Integrable Solitary and Cnoi- dal Waves." Physica D 21, 227 /C1/246, 1986. Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, 1982. Dodd, R. and Fordy, A. "The Prolongation Structures of Quasi-Polynomial Flows." Proc. Roy. Soc. A 385, 389 /C1/429, 1983. Gardner, C. S. "The Korteweg-de Vries Equation and Gen- eralizations, IV. The Korteweg-de Vries Equation as a Hamiltonian System." J. Math. Phys. 12, 1548 /C1/1551, 1971. Gardner, C. S.; Greene, C. S.; Kruskal, M. D.; and Miura, R. M. "Method for Solving the Korteweg-de Vries Equa- tion." Phys. Rev. Lett. 19, 1095 /C1/1097, 1967. Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge Univer- sity Press, 2000. Korteweg, D. J. and de Vries, F. "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves." Philos. Mag. 39, 422 /C1/443, 1895. Lamb, G. L. Jr. Ch. 4 in Elements of Soliton Theory. New York: Wiley, 1980. Miles, J. W. "The Korteweg-de Vries Equation, A Historical Essay." J. Fluid Mech. 106, 131 /C1/147, 1981. Russell, J. S. "Report on Waves." Report of the 14th Meeting of the British Association for the Advancement of Science. London: Jon Murray, pp. 311 /C1/390, 1844. Segal, G. "The Geometry of the KdV Equation." Int. J. Math. Phys. A 6, 2859 /C1/2869, 1991. Tabor, M. "Nonlinear Evolution Equations and Solitons." Ch. 7 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 278 /C1/321, 1989. Zakharov, V. E. and Faddeev, L. D. "Korteweg-de Vries Equation, A Completely Integrable System." Funct. Anal. Appl. 5, 280 /C1/287, 1971. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Korteweg-de Vries-Burger Equation The PARTIAL DIFFERENTIAL EQUATION ut /C272uux /C28 nuxx /C27 muxxx /C300: See also KORTEWEG-DE VRIES EQUATION References Canosa, J. and Gazdag, J. "The Korteweg-de Vries-Burgers Equation." J. Comput. Phys. 23, 393 /C1/403, 1977.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Kovalevskaya Exponent LEADING ORDER ANALYSIS Kovalevskaya Top Equations The system of ORDINARY DIFFERENTIAL EQUATIONS dm dt/C30 lm /C29m /C27 g /C291 dg dt /C30 lg /C29m: References Haine, L. and Horozov, E. "A Lax Pair for Kowalevski’s Top." Physica D 29, 173 /C1/180, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 136, 1997. Kozyrev-Grinberg Theory A theory of HAMILTONIAN CIRCUITS . See also GRINBERG FORMULA ,HAMILTONIAN CIRCUIT k-Partite Graph A k-partite graph is a GRAPH whose VERTICES can be partitioned into k DISJOINT SETS so that no two vertices within the same set are adjacent. See also COMPLETE K-PARTITE GRAPH , K-GRAPH References Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 12, 1986. Kramers Equation The PARTIAL DIFFERENTIAL EQUATION Pt /C30Pxx /C28uPx /C27@ @x f[u /C28F(x)]P g: References Duck, P. W.; Marshall, T. W.; and Watson, E. J. "First- Passage Times for the Uhlenbeck-Ornstein Process." J. Phys. A: Math. Gen. 19, 3545 /C1/3558, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, 1997. Kramers Rate The characteristic escape rate from a stable state of a potential in the absence of signal. See also STOCHASTIC RESONANCE References Bulsara, A. R. and Gammaitoni, L. "Tuning in to Noise." Phys. Today 49,39/C1/45, March 1996. Kramp’s Symbol The symbol defined by ca =b /C13c(c /C27b)(c /C272b) /C1/C1/C1[c /C27(a /C281)b] (1) /C30bac b ! a(2) /C30ba G a /C27c b ! Gc b ! ; (3) where (a)nis the POCHHAMMER SYMBOL and G(z)is the GAMMA FUNCTION . Note that the definition by Erde´lyi et al. (1981, p. 52) incorrectly gives the PREFACTOR of (3) as ba /C281 :/ See also HANKEL’S SYMBOL ,POCHHAMMER SYMBOL References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 52, 1981. Krattenthaler Matrix Inversion Formula Let (ai) and (bi) be sequences of complex numbers such that bj "bkfor j "k; and let the LOWER TRIAN- GULAR MATRICES F /C30(F(n; k)) and G /C30(G(n; k)) be defined as F(n; k) /C30Qn/C281 j/C30k (aj /C27 k)Qn j/C30k /C271(bj /C28 bk) and G(n ; k) /C30ak /C27 bk an /C27 bnQnj/C30k /C271(aj /C27 bn)Qn/C281 j/C30k (bj /C28 bn); where the product over an EMPTY SET is 1. Then F and G are MATRIX INVERSES (Bhatnagar 1995, pp. 16 /C1/17). This result simplifies to the GOULD AND HSU MATRIX INVERSION FORMULA when bk /C30k; to Carlitz’s q-analog for bk /C30qk (Carlitz 1972), and to Bressoud’s matrix theorem for bk /C30q/C28k /C27aqk and ak /C30/C28(aq /C28j =b) /C28bqj (Bressoud 1983). The formula can be extended to a summation theorem which generalizes Gosper’s bibasic sum (Gasper and Rahman 1990, p. 240; Bhatnagar 1995, p. 19). See also GOULD AND HSU MATRIX INVERSION FOR- MULAReferences Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U (n) Extensions. Ph.D. thesis. Ohio State University, 1995. Bressoud, D. M. "A Matrix Inverse." Proc. Amer. Math. Soc. 88, 446/C1/448, 1983. Carlitz, L. "Some Inversion Relations." Duke Math. J. 40, 803/C1/901, 1972. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Krattenthaler, C. "Operator Methods and Lagrange Inver- sions: A Unified Approach to Lagrange Formulas." Trans. Amer. Math. Soc. 305, 431/C1/465, 1988. Riordan, J. Combinatorial Identities. New York: Wiley, 1979. Krawtchouk Polynomial Leta(x)b ea STEP FUNCTION with the JUMP j(x)/C30N x>C18>C19 pxqN/C28x(1) atx/C300, 1, ..., N, where p>0;q>0;and p/C27q/C301: Then the Krawtchouk polynomial is defined by k(p) n(x;N)/C30Xn n/C300(/C281)n/C28nN/C28x n/C28n>C18>C19 x n>C18>C19 pn/C28nqn; (2) /C30(/C281)nN n>C18>C19 pn 2F1(/C28n;/C28x;/C28N;1=p) (3) /C30(/C281)npn n!G(N/C28x/C271) G(N/C28x/C28n/C271) /C292F1(/C28n;/C28x;N/C28x/C28n/C271; (p/C281)=p): (4) forn/C300, 1, ..., N. The first few Krawtchouk poly- nomials are k(p) 0(x;N)/C301 k(p) 1(x;N)/C30/C28Np/C27x k(p) 2(x;N)/C301 2[N2p2/C27x(2p/C27x/C281)/C28Np(p/C272x)]: Koekoek and Swarttouw (1998) define the Krawtch- ouk polynomial without the leading coefficient as Kn(x;p;N)/C302F1(/C28n;/C28x;/C28N;1=p): (5) The Krawtchouk polynomials have WEIGHT FUNCTION w/C30N!pxqN/C28x G(1/C27x)G(N/C271/C28x); (6) where G(x) is the GAMMA FUNCTION ,RECURRENCE RELATION (n/C271)k(p) n/C271(x;N)/C27pq(N/C28n/C271)k(p) n/C281(x;N) /C30[x/C28n/C28(N/C282)]k(p) n(x;N); (7) and squared norm N! n!(N /C28 n)!(pq)n : (8) It has the limit lim n0/C122 Npq !n=2 n!k(p) n(Np /C27ffiffiffiffiffiffiffiffiffiffiffiffi 2Npqp s ; N) /C30Hn(s); (9) where Hn(x)isaH ERMITE POLYNOMIAL . The Krawtchouk polynomials are a special case of the MEIXNER POLYNOMIALS OF THE FIRST KIND . See also MEIXNER POLYNOMIAL OF THE FIRST KIND, ORTHOGONAL POLYNOMIALS References Koekoek, R. and Swarttouw, R. F. "Krawtchouk." §1.10 in The Askey-Scheme of Hypergeometric Orthogonal Polyno- mials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/17, pp. 46 /C1/47, 1998. ftp:// www.twi.tudelft.nl/publications/tech-reports/1998/DUT- TWI-98 /C1/17.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 115, 1998. Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 35 /C1/37, 1975. Zelenkov, V. "Krawtchouk Polynomial Home Page." http:// www.isir.minsk.by/~zelenkov/physmath/kr_polyn/. Kreisel Conjecture A CONJECTURE in DECIDABILITY theory which postu- lates that, if there is a uniform bound to the lengths of shortest proofs of instances of S(n) ; then the universal generalization is necessarily provable in PEANO AR- ITHMETIC . The CONJECTURE was proven true by M. Baaz in 1988 (Baaz and Pudla ´k 1993). See also DECIDABLE References Baaz, M. and Pudla ´k P. "Kreisel’s Conjecture for /L /C2151/.In Arithmetic, Proof Theory, and Computational Complexity, Papers from the Conference Held in Prague, July 2 /C1/5, 1991 (Ed. P. Clote and J. Krajicek). New York: Oxford University Press, pp. 30 /C1/60, 1993. Dawson, J. "The Go¨del Incompleteness Theorem from a Length of Proof Perspective." Amer. Math. Monthly 86, 740 /C1/747, 1979. Kreisel, G. "On the Interpretation of Nonfinitistic Proofs, II." J. Symbolic Logic 17,43/C1/58, 1952. Krichever-Novikov Equation The PARTIAL DIFFERENTIAL EQUATION ut ux/C301 4uxxx ux/C2838u2 xx u2 x/C273 2p(u) u2 x; wherep(u) /C301 4(4u3 /C28g2u /C28g3) : The special cases p(u) /C30(u /C28e1)2(u /C28e2) and p(u) /C30u3 can be reduced to the KORTEWEG-DE VRIES EQUATION by a change of variables. See also KADOMTSEV- PETVIASHVILI EQUATION ,KOR- TEWEG-DE VRIES EQUATION References Krichever, I. M. and Novikov, S. P. "Holomorphic Bundles over Algebraic Curves, and Nonlinear Equations." Russ. Math. Surv. 35,53/C1/80, 1980. English translation of Uspekhi Mat. Nauk 35,47/C1/68, 1980. Mokhov, O. I. "Canonical Hamiltonian Representation of the Krichever-Novikov Equation." Math. Notes 50, 939 /C1/945, 1991. English translation of Mat. Zametki 50,87/C1/96, 1991. Novikov, D. P. "Algebraic-Geometric Solutions of the Krich- ever-Novikov Equation." Theoret. Math. Phys. 121, 1567 /C1/ 15773, 1999. Sokolov, V. V. "Hamiltonian Property of the Krichever- Novikov Equation." Dokl. Akad. Nauk SSSR 277,48/C1/ 50, 1984. Svinolupov, S. I.; Sokolov, V. V.; and Yamilov, R. I. "Ba¨ck- lund Transformations for Integrable Evolution Equa- tions." Dokl. Akad. Nauk SSSR 271, 802 /C1/805, 1983. English translation of Sov. Math. Dokl. 28, 165 /C1/168, 1983. Kronecker Decomposition Theorem Every FINITE ABELIAN GROUP can be written as a GROUP DIRECT PRODUCT of CYCLIC GROUPS of PRIME POWER ORDERS . In fact, the number of nonisomorphic ABELIAN FINITE GROUPS a(n) of any given ORDER n is given by writing n as n/C30Y ipai i; where the piare distinct PRIME FACTORS , then a(n)/C30Y iP(ai); where P(n) is the PARTITION FUNCTION . This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (Sloane’s A000688). See also ABELIAN GROUP ,F INITE GROUP ,O RDER (GROUP ), PARTITION FUNCTION P References Sloane, N. J. A. Sequences A000688/M0064 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Kronecker Delta The simplest interpretation of the Kronecker delta is as the discrete version of the DELTA FUNCTION defined by dij/C130 for i"j 1 for i/C30j:>C26 (1) It has the COMPLEX GENERATING FUNCTION dmn /C301 2pi g zm/C28n/C281 dz ; (2) where m and n are INTEGERS . In 3-space, the Kronecker delta satisfies the identities dii /C303 (3) dij eijk /C300 (4) eipq ejpg /C302 dij (5) eijk epqk /C30 dip djq /C28 diq djp ; (6) where EINSTEIN SUMMATION is implicitly assumed, i ; j /C301 ; 2 ; 3 ; and eijk is the PERMUTATION SYMBOL . Technically, the Kronecker delta is a TENSOR defined by the relationship dk l@x?i @xk@xl @x?j/C30@x?i @xk@xk @x?j/C30@x?j @x?j: (7) Since, by definition, the coordinates xiand xjare independent for i "j ; @x?i @x?j/C30 d?ji; (8) so d?ji/C30@x?i @xk@xl @x?j@k l ; (9) and dijis really a mixed second- RANK TENSOR .It satisfies djk ab /C30 eabi ejki /C30 dj a dkb /C28 dka dj b (10) dabjk /C30gajgbk /C28gakgbj (11) eaij ebij /C30 dbi ai /C302dba : (12) The generalization of the Kronecker delta viewed as a tensor is called the PERMUTATION TENSOR . See also DELTA FUNCTION ,PERMUTATION SYMBOL , PERMUTATION TENSOR Kronecker Product MATRIX DIRECT PRODUCT Kronecker Symbol An extension of the JACOBI SYMBOL (n=m) to all INTEGERS . It is variously written as (n=m)or(n m) (Cohn 1980) or (n½m) (Dickson 1957). The Kronecker symbol can be computed using the normal rules for the JACOBI SYMBOLab cd ! /C30a cd ! b cd ! /C30ab c ! ab d ! /C30a c ! b c ! a d ! b d ! (1) plus additional rules for m /C30/C28 1, (n=/C281) /C30/C281 for n B0 1 for n > 0 ;>C26 (2) and m /C302. The definition for (n=2) is variously written as (n=2) /C300 for n even 1 for n odd ; n /C1391 (mod 8) /C281 for n odd ; n /C1393 (mod 8)8 < : (3) or (n=2) /C130 for 4½n 1 for n /C131 (mod 8) /C281 for n /C135 (mod 8) undefined otherwise8 >>< >>:(4) (Cohn 1980). Cohn’s form "undefines" (n=2) for SINGLY EVEN NUMBERS n /C132 (mod 4) and n /C13/C281; 3 (mod 8); probably because no other values are needed in applications of the symbol involving the DISCRIMI- NANTS d of QUADRATIC FIELDS , where m /C210 and d always satisfies d /C130 ; 1 (mod 4):/ The KRONECKER SYMBOL is a REAL CHARACTER mod- ulo n, and is, in fact, essentially the only type of REAL PRIMITIVE CHARACTER (Ayoub 1963). See also CHARACTER (NUMBER THEORY ), CLASS NUMBER ,DIRICHLET L-SERIES ,JACOBI SYMBOL ,LE- GENDRE SYMBOL ,PRIMITIVE CHARACTER ,QUADRATIC RESIDUE References Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963. Cohn, H. Advanced Number Theory. New York: Dover, p. 35, 1980. Dickson, L. E. "Kronecker’s Symbol." §48 in Introduction to the Theory of Numbers. New York: Dover, p. 77, 1957. Kronecker’s Algorithm A POLYNOMIAL FACTORIZATION algorithm that pro- ceeds by considering the vector of coefficients of a polynomial P, calculating bi /C30P(i) =ai ; constructing the LAGRANGE INTERPOLATING POLYNOMIALS from the conditions A(i)/C30aiandB(i)/C30bi;and checking to see which are factorizations. See also POLYNOMIAL FACTORIZATION References Hausmann, B. A. "A New Simplification of Kronecker’s Method of Factorization of Polynomials." Amer. Math. Monthly 47, 574/C1/576, 1937. Se´roul, R. "Kronecker’s Factorization Algorithm." §10.14.2 in Programming for Mathematicians. Berlin: Springer-Ver- lag, pp. 288 /C1/289, 2000. Kronecker’s Approximation Theorem If u is a given IRRATIONAL NUMBER , then the sequence of numbers fnu g; where fxg/C13x /C28 xbc; is DENSE in the unit interval. Explicitly, given any a; 0 5 a 51 ; and given any e > 0; there exists a POSITIVE INTEGER k such that ½fkug/C28 a½B e: Therefore, if h /C30 k ubc ; it follows that /jku /C28h /C28 ajB e/. The restriction on a can be removed as follows. Given any real a; any irrational u ; and any e > 0; there exist integers h and k with k /C210 such that ½ku /C28h /C28 a½B e: See also RATIONAL APPROXIMATION References Apostol, T. M. "Kronecker’s Approximation Theorem: The One-Dimensional Case" and "Extension of Kronecker’s Theorem to Simultaneous Approximation." §7.4 and 7.5 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 148 /C1/ 155, 1997. Kronecker’s Constant MERTENS CONSTANT Kronecker’s Polynomial Theorem An algebraically soluble equation of ODD PRIME degree which is irreducible in the natural FIELD possesses either 1. Only a single REAL ROOT ,or 2. All REAL ROOTS . See also ABEL’S IRREDUCIBILITY THEOREM ,A BEL’S LEMMA ,SCHO¨ NEMANN’S THEOREM References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover p. 127, 1965. Krull Dimension If R is a RING (commutative with 1), the height of a PRIME IDEAL p is defined as the SUPREMUM of all n so that there is a chain p0 ƒ/C1/C1/C1pn/C281 ƒpn /C30p where all pi are distinct PRIME IDEALS . Then, the Krull dimension of R is defined as the SUPREMUM of all the heights of all its PRIME IDEALS . See also PRIME IDEALReferences Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry. New York: Springer-Verlag, 1995. Macdonald, I. G. and Atiyah, M. F. Introduction to Commu- tative Algebra. Reading, MA: Addison-Wesley, 1969. Kruskal’s Algorithm An ALGORITHM for finding a GRAPH ’s spanning TREE of minimum length. See also KRUSKAL’S TREE THEOREM References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 248 /C1/249, 1978. Kruskal’s Tree Theorem A theorem which plays a fundamental role in compu- ter science because it is one of the main tools for showing that certain orderings on TREES are well- founded. These orderings play a crucial role in proving the termination of rewriting rules and the correctness of the Knuth-Bendix equational comple- tion procedures. See also KRUSKAL’S ALGORITHM ,NATURAL INDEPEN- DENCE PHENOMENON ,TREE References Gallier, J. "What’s so Special about Kruskal’s Theorem and the Ordinal Gamma[0]? A Survey of Some Results in Proof Theory." Ann. Pure and Appl. Logic 53, 199/C1/260, 1991. KS Entropy METRIC ENTROPY k-Statistic Theithk-statistic kiis an UNBIASED ESTIMATOR of the CUMULANT kiof a given DISTRIBUTION , i.e., kiis defined so that /C142ki/C143/C30ki;where /C142x/C143denotes the EXPECTATION VALUE ofx(Kenney and Keeping 1951, p. 189). For a SAMPLE SIZE n, the first few k- statistics are given by k1/C30m (1) k2/C30n n/C281m2 (2) k3/C30n2 (n/C281)(n/C282)m3 (3) k4/C30n2[(n/C271)m4/C283(n/C281)m2 2] (n/C281)(n/C282)(n/C283); (4) where mis the sample MEAN ,m2is the SAMPLE VARIANCE , and miis the sample ithCENTRAL MOMENT (Kenney and Keeping 1951, pp. 109 /C1/110, 163 /C1/165, and 189; Kenney and Keeping 1962). Thek-statistics can be obtained by defining the sums of the rth powers of the data points as sr/C13Xn i/C301Xr i; (5) then the CENTRAL MOMENTS miare given in terms of thesrby m2/C30/C28s2 1 n2/C27s2 n(6) m3/C302s31 n3/C283s1s2 n2/C27s3 n(7) m4/C30/C283s41 n4/C276s21s2 n3/C284s1s3 n2/C27s4 n: (8) Taking the raw expectations of these equations and expressing the answers in terms of moments miusing mi/C30si n(9) then gives the expectation values of the observed central moments miin terms of the population central moments as /C142m2/C143/C30n/C281 nm2(10) /C142m3/C143/C30(n/C281)(n/C282) n2m3 (11) /C142m4/C143/C30(n/C281)[(n2/C283n/C273)m4/C273(2n/C283)m2 2] n3;(12) together with /C142m2 2/C143/C30(n/C281)[(n/C281)m4/C27(n2/C282n/C273)m2 2] n3(13) (Kenney and Keeping 1951, p. 189). Solving for the population central moments miin terms of the expectation values of the observed central moments then gives the formulas for the k-statistics, e.g., (10) becomes m2/C30n n/C281/C142m2/C143; (14) so k2/C30n n/C281m2 (15) is an UNBIASED ESTIMATOR fork2/C30m2:/ In terms of the power sums, the k-statistics can then be written ask2/C30ns2/C28s2 1 n(n/C281)(16) k3/C302s3 1/C283ns1s2/C27n2s3 n(n/C281)(n/C282)(17) k4/C30/C286s4 1/C2712ns21s2/C283n(n/C281)s22/C284n(n/C271)s1s3/C27n2(n/C271)s4 n(n/C281)(n/C282)(n/C283): (18) The VARIANCE var(k2)o f k2is given by the second central expectation of k2which, when expressed in terms of CUMULANTS , becomes var(k2)/C30k4 n/C272k2 2 n/C281: (19) The UNBIASED ESTIMATOR of var( k2)i s ˆvar(k2)/C302k22n/C27(n/C281)k4 n(n/C271)(20) (Kenney and Keeping 1951, p. 189). The VARIANCE ofk3can be expressed in terms of CUMULANTS by var(k3)/C30k6 n/C279k2k4 n/C281/C279k2 3 n/C281/C276nk32 (n/C281)(n/C282);(21) and the UNBIASED ESTIMATOR for var( k3)i s ˆvar(k3)/C306k22n(n/C281) (n/C282)(n/C271)(n/C273)(22) (Kenney and Keeping 1951, p. 190). For a finite population, let a SAMPLE SIZE nbe taken from a population size N. Then UNBIASED ESTIMATORS M1for the population MEAN m;M2for the population VARIANCE m2;G1for the population SKEWNESS g1;and G2for the population KURTOSIS g2are M1/C30m (23) M2/C30N/C28n n(N/C281)m2 (24) G1/C30N/C282n N/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N/C281 n(N/C28n)s g1 (25) G2/C30(N/C281)(N2/C286Nn/C27N/C276n2)g2 n(N/C282)(N/C283)(N/C28n) /C286N(Nn/C27N/C28n2/C281) n(N/C282)(N/C283)(N/C28n)(26) (Church 1926, p. 357; Carver 1930; Irwin and Ken- dall 1944; Kenney and Keeping 1951, p. 143), whereg 1is the sample SKEWNESS and g2is the sample KURTOSIS . See also CUMULANT ,G AUSSIAN DISTRIBUTION , H- STATISTIC ,K URTOSIS ,M EAN,M OMENT ,SKEWNESS , STATISTIC ,UNBIASED ESTIMATOR ,VARIANCE References Carver, H. C. (Ed.). "Fundamentals of the Theory of Sam- pling." Ann. Math. Stat. 1, 101 /C1/121, 1930. Church, A. E. R. "On the Means and Squared Standard- Deviations of Small Samples from Any Population." Biometrika 18, 321 /C1/394, 1926. Irwin, J. O. and Kendall, M. G. "Sampling Moments of Moments for a Finite Population." Ann. Eugenics 12, 138 /C1/142, 1944. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Kenney, J. F. and Keeping, E. S. "The k-Statistics." §7.9 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 99 /C1/100, 1962. k-Subset A k-subset is a SUBSET of a set on n elements containing exactly k elements. The number of k- subsets on n elements is therefore given by the BINOMIAL COEFFICIENTn k>C0>C1 : For example, there are 3 2>C0>C1 /C303 2-subsets of f1; 2; 3g; namely f1 ; 2g;f1; 3g; and f2 ; 3 g: The k-subsets on a list can be enumerated using KSubsets [list, k] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The total number of distinct k-subsets on a set of n elements (i.e., the number of SUBSETS ) is given by Xn k /C300n k>C18>C19 /C302n : See also BINOMIAL COEFFICIENT ,C OMBINATION , P- SYSTEM ,PERMUTATION ,SUBSET References Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Skiena, S. "Generating k-Subsets." §1.5.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 44 /C1/46, 1990. K-Theory A branch of mathematics which brings together ideas from ALGEBRAIC GEOMETRY , LINEAR ALGEBRA , and NUMBER THEORY . In general, there are two main types of K-theory: topological and algebraic. Topological K-theory is the "true" K-theory in the sense that it came first. Topological K-theory has to do with VECTOR BUNDLES over TOPOLOGICAL SPACES . Elements of a K-theory are STABLE EQUIVALENCE classes of VECTOR BUNDLES over a TOPOLOGICAL SPACE . You can put a RING structure on the collectionof STABLY EQUIVALENT bundles by defining ADDITION through the WHITNEY SUM, and MULTIPLICATION through the TENSOR PRODUCT of VECTOR BUNDLES . This defines "the reduced real topological K-theory of a space." "The reduced K-theory of a space" refers to the same construction, but instead of REAL VECTOR BUNDLES , COMPLEX VECTOR BUNDLES are used. Topological K- theory is significant because it forms a generalized COHOMOLOGY theory, and it leads to a solution to the vector fields on spheres problem, as well as to an understanding of the J-homeomorphism of HOMO- TOPY THEORY . Algebraic K-theory is somewhat more involved. Swan (1962) noticed that there is a correspondence between the CATEGORY of suitably nice TOPOLOGICAL SPACES (something like regular HAUSDORFF SPACES ) and C*- ALGEBRAS . The idea is to associate to every SPACE the C*-ALGEBRA of CONTINUOUS MAPS from that SPACE to the REALS . A VECTOR BUNDLE over a SPACE has sections, and these sections can be multiplied by CONTINUOUS FUNCTIONS to the REALS . Under Swan’s correspon- dence, VECTOR BUNDLES correspond to modules over the C*-ALGEBRA of CONTINUOUS FUNCTIONS , the MOD- ULES being the modules of sections of the VECTOR BUNDLE . This study of MODULES over C*-ALGEBRA is the starting point of algebraic K-theory. The QUILLEN-LICHTENBAUM CONJECTURE connects algebraic K-theory to E ´tale cohomology. See also C*-ALGEBRA References Atiyah, M. F. K-Theory. New York: Benjamin, 1967. Bass, H.; Kuku, A. O.; and Pedrini, C. Proceedings of the Workshop and Symposium: Algebraic K -Theory and Its Applications, ICTP, Trieste, Italy, 1 /C1/19 Sept. 1997. Singapore: World Scientific, 1999. Raskind, W. and Weibel, C. (Eds.). Algebraic K -Theory: AMS-IMS-SIAM Joint Summer Research Conference on Algebraic K-Theory, July 13 /C1/24, 1997, University of Washington, Seattle. Providence, RI: Amer. Math. Soc.,1997. Srinivas, V. Algebraic K -Theory, 2nd ed. Boston, MA: Birkha ¨user, 1995. Swan, R. G. "Vector Bundles and Projective Modules." Trans. Amer. Math. Soc. 105, 264/C1 /277, 1962. k-Tuple Conjecture The first of the H ARDY- LITTLEWOOD CONJECTURES . The k-tuple conjecture states that the asymptotic number of PRIME CONSTELLATIONS can be computed explicitly. In particular, unless there is a trivial divisibility condition that stops p,/p/C27a1; :::; p/C27ak/ from consisting of PRIMES infinitely often, then such PRIME CONSTELLATIONS will occur with an asymptotic density which is computable in terms of a1;...,ak:Let 0Bm1Bm2B...Bmk;then the k-tuple conjecture predicts that the number of PRIMES p5xsuch that p /C272m1 ; p /C272m2 ; ..., p /C272mk are all PRIME is P(x; m1 ; m2 ; ...; mk) /C2C(m1 ; m2 ; ...; mk)gx 2dt lnk /C271 t ; (1) where C(m1 ; m2 ; ...; mk) /C302kY q1 /C28w(q; m1 ; m2 ;...; mk) q 1 /C281 q !k /C271 ; (2) the product is over ODD PRIMES q, and w(q; m1 ; m2 ; ...; mk) (3) denotes the number of distinct residues of 0, m1 ; ..., mk (mod q) (Halberstam and Richert 1974, Odlyzko). If k /C301, then this becomes C(m) /C302Y qq(q /C28 2) (q /C28 1)2Y q jmq /C28 1 q /C28 2 : (4) This conjecture is generally believed to be true, but has not been proven (Odlyzko et al. ). The following special case of the conjecture is sometimes known as the PRIME PATTERNS CONJECTURE . Let S be a FINITE set of INTEGERS . Then it is conjectured that there exist infinitely many k for which fk /C27s : s /C23 S g are all PRIME IFF S does not include all the RESIDUES of any PRIME . The TWIN PRIME CONJECTURE is a special case of the prime patterns conjecture with S/C30f0;2g:This conjecture also implies that there are arbitrarily long ARITHMETIC PROGRESSIONS ofPRIMES . See also ARITHMETIC PROGRESSION ,D IRICHLET’S THEOREM ,H ARDY- LITTLEWOOD CONJECTURES , K-TU- PLE CONJECTURE ,PRIME ARITHMETIC PROGRESSION , PRIME CONSTELLATION ,PRIME QUADRUPLET ,PRIME PATTERNS CONJECTURE ,TWIN PRIME CONJECTURE , TWIN PRIMES References Brent, R. P. "The Distribution of Small Gaps Between Successive Primes." Math. Comput. 28, 315/C1/324, 1974. Brent, R. P. "Irregularities in the Distribution of Primes and Twin Primes." Math. Comput. 29,4 3/C1/56, 1975. Halberstam, E. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974. Hardy, G. H. and Littlewood, J. E. "Some Problems of ‘Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes." Acta Math. 44,1/C1/70, 1922. Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping Cham- pions." Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 66 /C1/ 68, 1994.Kuen Surface A special case of E NNEPER’S NEGATIVE CURVATURE SURFACES which can be given parametrically by x/C302(cos u/C27usinu) sin v 1/C27u2sin2v(1) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u2p cos(u/C28tan/C281u) sin v 1/C27u2sin2v(2) y/C302(sin u/C27ucosu) sin v 1/C27u2sin2v(3) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u2p sin(u/C28tan/C281u) sin v 1/C27u2sin2v(4) z/C30ln tan1 2v>C16>C17hi /C272 cos v 1/C27u2sin2v(5) forv/C230;p½Þ ;u/C23[0;2p) (Reckziegel et al. 1986; Gray 1997, p. 496). The coefficients of the FIRST FUNDAMENTAL FORM are E/C3016u2sin2v [2/C27u2/C28u2cos2(2v)]2(6) F/C300 (7) G/C30csc2v/C2816u2sin2v [2/C27u2/C28u2cos2(2v)]2; (8) the SECOND FUNDAMENTAL FORM coefficients are e/C304u[2/C28u2/C27u2cos2(2v)] sin v [2/C27u2/C28u2cos2(2v)]2; (9) f/C300 (10) g/C304u[2/C28u2/C27u2cos2(2v)] csc v [2/C27u2/C28u2cos2(2v)]2; (11) and the surface area element is dS/C304u[2/C28u2/C27u2cos2(2v)] [2/C27u2/C28u2cos2(2v)]2: (12) The G AUSSIAN and MEAN CURVATURES are K /C30/C281 (13) H /C30/C28csc v 4u /C271 4u sin v 1 /C278 2 /C28 u2 /C27 u2 cos(2 v)"# ; (14) so the Kuen surface has constant NEGATIVE GAUSSIAN CURVATURE , and the PRINCIPAL CURVATURES are k1 /C304u sin v 2 /C28 u2 /C27 u2 cos(2 v) (15) k2 /C30/C28[2 /C28 u2 /C27 u2 cos(2 v)] csc v 4u (16) (Gray 1997, p. 496). See also ENNEPER’S NEGATIVE CURVATURE SURFACES References --. Cover of La Gaceta de la Real Sociedad Matema ´tica Espan ˜ola 2, 1999. Fischer, G. (Ed.). Plate 86 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 82, 1986. Gray, A. "Kuen’s Surface." §21.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 496 /C1/497, 1997. JavaView. "Classic Surfaces from Differential Geometry: Kuen." http://www-sfb288.math.tu-berlin.de/vgp/java- view/demo/surface/common/PaSurface_Kuen.html. Kuen, T. "Ueber Fla¨chen von constantem Kru¨mmungs- maass." Sitzungsber. d. ko¨nigl. Bayer. Akad. Wiss. Math.-phys. Classe, Heft II, 193 /C1/206, 1884. Nordstrand, T. "Kuen’s Surface." http://www.uib.no/people/ nfytn/kuentxt.htm. Reckziegel, H. "Kuen’s Surface." §3.4.4.2 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 38, 1986. Kuhn-Tucker Theorem A theorem in nonlinear programming which states that if a regularity condition holds and f and the functions hjare convex, then a solution x0 which satisfies the conditions hj for a VECTOR of multipliers l is a GLOBAL MINIMUM . The Kuhn-Tucker theorem is a generalization of LAGRANGE MULTIPLIERS .FARKAS’S LEMMA is key in proving this theorem. See also FARKAS’S LEMMA ,LAGRANGE MULTIPLIER Kuiper Statistic A statistic defined to improve the KOLMOGOROV- SMIRNOV TEST in the TAILS . See also ANDERSON- DARLING STATISTIC References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art ofScientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 621, 1992. Kulikowski’s Theorem For every POSITIVE INTEGER n, there exists a SPHERE which has exactly n LATTICE POINTS on its surface. The SPHERE is given by the equation (x /C28a)2 /C27(y /C28b)2 /C27(z /C28ffiffiffi 2p )2 /C30c2 /C272; where a and b are the coordinates of the center of the so-called SCHINZEL CIRCLE x /C281 2>C16>C172 /C27y2 /C3014 5k /C281for n /C302k even x /C2813>C16>C172 /C27y2 /C3019 52kforn/C302k/C271 odd8 >< >: andcis its RADIUS . See also CIRCLE LATTICE POINTS ,LATTICE POINT , SCHINZEL’S THEOREM References Honsberger, R. "Circles, Squares, and Lattice Points." Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117 /C1/127, 1973. Kulikowski, T. "Sur l’existence d’une sphe `re passant par un nombre donne ´aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 5,8 9/C1/90, 1959. Schinzel, A. "Sur l’existence d’un cercle passant par un nombre donne ´de points aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 4,7 1/C1/72, 1958. Sierpinski, W. "Sur quelques proble `mes concernant les points aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 4,2 5/C1/31, 1958. Sierpinski, W. "Sur un proble `me de H. Steinhaus concernant les ensembles de points sur le plan." Fund. Math. 46, 191/C1/194, 1959. Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964. Kullback-Leibler Distance RELATIVE ENTROPY Kummer Extension References Koch, H. "Kummer Extensions." §6.8 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 195 /C1/199, 2000. Kummer Group AGROUP ofLINEAR FRACTIONAL TRANSFORMATIONS which transform the arguments of Kummer solutions to the HYPERGEOMETRIC DIFFERENTIAL EQUATION into each other. Define A(z)/C301/C28z B(z)/C301=z; then the elements of the group are fI;A;B;AB;BA;ABA/C30BAB g::/ Kummer Surface The Kummer surfaces are a family of QUARTIC SURFACES given by the algebraic equation (x2 /C27y2 /C27z2 /C28 m2w2)2 /C28 lpqrs /C300; (1) where l /C133 m2 /C28 1 3 /C28 m2; (2) p, q, r, and s are the TETRAHEDRAL COORDINATES p /C30w /C28z /C28ffiffiffi 2p x (3) q /C30w /C28z /C27ffiffiffi2p x (4) r /C30w /C27z /C27ffiffiffi 2p y (5) s /C30w /C27z /C28ffiffiffi2p y; (6) and w is a parameter which, in the above plots, is set to w /C301. The above plots correspond to m 2 /C301 =3 (3x2 /C273y2 /C273z2 /C271)2 /C300 ; (7) (double sphere), 2/3, 1 x4 /C282x2y2 /C27y4 /C274x2z /C274y2z /C274x2z2 /C274y2z2 /C300 (8) (ROMAN SURFACE ),ffiffiffi 2p ;ffiffiffi 3p [(z /C281)2 /C282x2][y2 /C28(z /C271)2] /C300 (9) (four planes), 2, and 5. The case 0 5 m2 51 =3 corre- sponds to four real points. The following table gives the number of ORDINARY DOUBLE POINTS for various ranges of m2 ; correspond- ing to the preceding illustrations. /0 5 m2 51 3/ 412 / m2 /C301 3/ /1 3 5 m2 B1/ 412 / m2 /C301/ /1 B m2 B3/ 16 0 / m2 /C303/ / m2 > 3/ 16 0The Kummer surfaces can be represented parame- trically by hyperelliptic THETA FUNCTIONS . Most of the Kummer surfaces admit 16 ORDINARY DOUBLE POINTS , the maximum possible for a QUARTIC SUR- FACE . A special case of a Kummer surface is the TETRAHEDROID . Nordstrand gives the implicit equations as x4/C27y4/C27z4/C28x2/C28y2/C28z2/C28x2y2/C28x2z2/C28y2z2/C271/C300 (10) or x4/C27y4/C27z4/C27a(x2/C27y2/C27z2)/C27b(x2y2/C27x2z2/C27y2z2) /C27cxyz/C281/C300: (11) See also QUARTIC SURFACE ,ROMAN SURFACE ,TETRA- HEDROID References Endraß, S. "Fla ¨chen mit vielen Doppelpunkten." DMV- Mitteilungen 4,1 7/C1/20, Apr. 1995. Endraß, S. "Kummer Surfaces." http://enriques.mathemati- k.uni-mainz.de/kon/docs/Ekummer.shtml. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 14 /C1/19, 1986. Fischer, G. (Ed.). Plates 34 /C1/37 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 33 /C1/37, 1986. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 313, 1997. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 183, 1994. Hudson, R. Kummer’s Quartic Surface. Cambridge, Eng- land: Cambridge University Press, 1990. Kummer, E. "U ¨ber die Fla ¨chen vierten Grades mit sechs- zehn singula ¨ren Punkten." Ges. Werke 2, 418/C1/432. Kummer, E. "U ¨ber Strahlensysteme, deren Brennfla ¨chen Fla¨chen vierten Grades mit sechszehn singula ¨ren Punk- ten sind." Ges. Werke 2, 418/C1/432. Nordstrand, T. "Kummer’s Surface." http://www.uib.no/peo- ple/nfytn/kummtxt.htm. Kummer’s Conjecture A conjecture concerning PRIMES . Kummer’s Differential Equation CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUA- TION Kummer’s Formulas Kummer’s first formula is 2F11 2/C27m/C28k;/C28n;2m/C271; 1>C16>C17 /C30G(2m/C271)Gm/C2712/C27k/C27n>C16>C17 G(m/C2712/C27k)G2m/C271/C27n ðÞ; (1) where2F1(a; b; c; z) is the HYPERGEOMETRIC FUNC- TION with m "/C281=2 ;/C281, /C283=2 ; ..., and G(z) is the GAMMA FUNCTION . The identity can be written in the more symmetrical form as 2F1(a ; b; c; /C281) /C30G1 2 b /C27 1>C16>C17 G(b /C28 a /C27 1) G(b /C27 1)G1 2 b /C28 a /C27 1>C16>C17 ; (2) where a /C28b /C27c /C301 and b is a positive integer (Bailey 1935, p. 35; Petkovsek et al. 1996; Koepf 1998, p. 32; Hardy 1999, p. 106). If b is a negative integer, the identity takes the form 2F1(a ; b; c; /C281) /C302 cos12 pb>C16>C17G bjjðÞG(b /C28 a /C27 1) G1 2 b /C28 a /C27 1>C16>C17 (3) (Petkovsek et al. 1996). Kummer’s second formula is 1F112 /C27m;2m /C271; z>C16>C17 /C30M0 ;m(z) /C30zm/C271 =2 1 /C27X/C12 p /C301z2p 24pp!(m /C27 1)(m /C27 2) /C1/C1/C1(m /C27 p)"# ; (4) where1F1(a; b; z) is the CONFLUENT HYPERGEO- METRIC FUNCTION and m "/C281 =2;/C281, /C283=2 ; .... See also CONFLUENT HYPERGEOMETRIC FUNCTION , HYPERGEOMETRIC FUNCTION References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, 1935. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, pp. 42 /C1/43 and 126, 1996. Kummer’s Function CONFLUENT HYPERGEOMETRIC FUNCTION Kummer’s Quadratic Transformation A transformation of a HYPERGEOMETRIC FUNCTION , 2F1a; b;2b;4z (1 /C27 z)2 ! /C30(1 /C27z)2a 2F1a; a /C2712 /C28 b; b /C2712; z2>C16>C17 : Kummer’s Relation An identity which relates HYPERGEOMETRIC FUNC- TIONS ,2F12a ; 2b; a /C27b /C271 2; x>C16>C17 /C302 F1(a ; b; a /C27b /C2712; 4x(1 /C28x)): Kummer’s Series HYPERGEOMETRIC FUNCTION Kummer’s Series Transformation Let a/C12 k /C300 ak /C30a and a/C12k /C300 ck /C30c be convergent series such that lim k 0/C12ak ck/C30 l "0: Then a /C30 lc /C27X/C12 k /C3001 /C28 lck ak ! ak : References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Kummer’s Test Given a SERIES of POSITIVE terms ui and a sequence of finite POSITIVE constants ai ; let r /C13 lim n0/C12anun un/C271/C28an/C271 ! : 1. If r > 0 ; the series converges. 2. IfrB0;the series diverges. 3. Ifr/C300;the series may converge or diverge. The test is a general case of B ERTRAND’S TEST , the ROOT TEST ,G AUSS’S TEST , and R AABE’S TEST . With an/C30nand an/C271/C30n/C271;the test becomes R AABE’S TEST . See also CONVERGENCE TESTS ,RAABE’S TEST References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 285 /C1/286, 1985. Jingcheng, T. "Kummer’s Test Gives Characterizations for Convergence or Divergence of All Series." Amer. Math. Monthly 101, 450/C1/452, 1994. Samelson, H. "More on Kummer’s Test." Amer. Math. Monthly 102, 817/C1/818, 1995. Kummer’s Theorem The identity 2F1(x;/C28x; x /C27n /C271; /C281) /C30G(x /C27 n /C27 1)G1 2 n /C27 1>C16>C17 G x /C271 2 n /C27 1>C16>C17 G(n /C27 1); or equivalently 2F1( a; b;1/C27 a /C28 b; /C281) /C30G(1 /C27 a /C28 b) G 1 /C2712 a>C16>C17 G 1 /C27 a ðÞ G 1 /C271 2 a /C28 b>C16>C17 ; where2F1(a; b; c; z)isa HYPERGEOMETRIC FUNCTION and G(z) is the GAMMA FUNCTION . This formula was first stated by Kummer (1836, p. 53). See also SAALSCHU ¨ TZ’S THEOREM References Bailey, W. N. "Kummer’s Theorem." §2.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 9 /C1/10, 1935. Kummer, E. E. "Ueber die hypergeometrische Reihe." J. fu¨r Math. 15,39/C1/83, 1836. Kupershmidt Equation The PARTIAL DIFFERENTIAL EQUATION ut /C30uxxxxx /C2752 uxxxu /C2725 4uxxux /C2754 u2ux : References Fuchssteiner, B.; Oevel, W.; and Wiwianka, W. "Computer- Algebra Methods for Investigation of Hereditary Opera- tors of High Order Soliton Equations." Comput. Phys. Commun. 44,47/C1/55, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 133, 1997. Kuramoto-Sivashinsky Equation The PARTIAL DIFFERENTIAL EQUATION u1 /C2794u /C2792u /C271292u>C12>C12>C12>C122/C300 ; where 92 is the LAPLACIAN and 94 is the BIHARMONIC OPERATOR . References Michelson, D. "Steady Solutions of the Kuramoto-Siva- shinsky Equation." Physica D 19,89/C1/111, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Kuratowski Reduction Theorem Every nonplanar graph is a SUPERGRAPH of an expansion of the UTILITY GRAPH UG /C30K3 ;3(i.e., the COMPLETE BIPARTITE GRAPH on two sets of three vertices) or the COMPLETE GRAPH K5 : This theorem was also proven earlier by Pontryagin (1927 /C1/1928), and later by Frink and Smith (1930). Kennedy et al.(1985) give a detailed history of the theorem, and there exists a generalization known as the ROBERT- SON-SEYMOUR THEOREM . See also COMPLETE BIPARTITE GRAPH ,C OMPLETE GRAPH ,PLANAR GRAPH ,ROBERTSON- SEYMOUR THEO- REM,UTILITY GRAPH References Harary, F. "Kuratowski’s Theorem." In Graph Theory. Reading, MA: Addison-Wesley, pp. 108 /C1/113, 1994. Kennedy, J. W.; Quintas, L. V.; and Syslo, M. M. "The Theorem on Planar Graphs." Historia Math. 12, 356 /C1/ 368, 1985. Kuratowski, C. "Sur l’operation A de l’analysis situs." Fund. Math. 3, 182 /C1/199, 1922. Kuratowski, C. "Sur le proble `me des courbes gauches en topologie." Fund. Math. 15, 217 /C1/283, 1930. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 247, 1990. Thomassen, C. "Kuratowski’s Theorem." J. Graph Th. 5, 225 /C1/241, 1981. Thomassen, C. "A Link Between the Jordan Curve Theorem and the Kuratowski Planarity Criterion." Amer. Math. Monthly 97, 216 /C1/218, 1990. Kuratowski’s Closure-Component Problem Let X be an arbitrary TOPOLOGICAL SPACE . Denote the CLOSURE of a SUBSET A of X by A/C28 and the COMPLE- MENT of A by A?: Then at most 14 different SETS can be derived from A by repeated application of closure and complementation (Berman and Jordan 1975, Fife 1991). The problem was first proved by Kuratowski (1922) and popularized by Kelley (1955). See also KURATOWSKI REDUCTION THEOREM References Anusiak, J. and Shum, K. P. "Remarks on Finite Topological Spaces." Colloq. Math. 23, 217/C1/223, 1971. Aull, C. E. "Classification of Topological Spaces." Bull. de l’Acad. Pol. Sci. Math. Astron. Phys. 15, 773/C1/778, 1967. Baron, S. Advanced Problem 5569. Amer. Math. Monthly 75, 199, 1968. Beeler et al. Item 105 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 45, Feb. 1972. Berman, J. and Jordan, S. L. "The Kuratowski Closure- Complement Problem." Amer. Math. Monthly 82, 841/C1/ 842, 1975. Buchman, E. "Problem E 3144." Amer. Math. Monthly 93, 299, 1986. Chagrov, A. V. "Kuratowski Numbers, Application of Func- tional Analysis in Approximation Theory." Kalinin: Kali- nin Gos. Univ., pp. 186 /C1/190, 1982. Chapman, T. A. "A Further Note on Closure and Interior Operators." Amer. Math. Monthly 69, 524/C1/529, 1962. Fife, J. H. "The Kuratowski Closure-Complement Problem." Math. Mag. 64, 180/C1/182, 1991. Fishburn, P. C. "Operations on Binary Relations." Discrete Math. 21,7/C1/22, 1978. Graham, R. L.; Knuth, D. E.; and Motzkin, T. S. "Comple- ments and Transitive Closures." Discrete Math. 2,1 7/C1/29, 1972. Hammer, P. C. "Kuratowski’s Closure Theorem." Nieuw Arch. Wisk. 8,7 4/C1/80, 1960. Herda, H. H. and Metzler, R. C. "Closure and Interior in Finite Topological Spaces." Colloq. Math. 15, 211 /C1/216, 1966. Kelley, J. L. General Topology. Princeton: Van Nostrand, p. 57, 1955. Koenen, W. "The Kuratowski Closure Problem in the Topology of Convexity." Amer. Math. Monthly 73, 704 /C1/ 708, 1966. Kuratowski, C. "Sur l’operation A de l’analysis situs." Fund. Math. 3, 182 /C1/199, 1922. Langford, E. "Characterization of Kuratowski 14-Sets." Amer. Math. Monthly 78, 362 /C1/367, 1971. Levine, N. "On the Commutativity of the Closure and Interior Operators in Topological Spaces." Amer. Math. Monthly 68, 474 /C1/477, 1961. Moser, L. E. "Closure, Interior, and Union in Finite Topolo- gical Spaces." Colloq. Math. 38,41/C1/51, 1977. Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975. Peleg, D. "A Generalized Closure and Complement Phenom- enon." Discrete Math. 50, 285 /C1/293, 1984. Shum, K. P. "On the Boundary of Kuratowski 14-Sets in Connected Spaces." Glas. Mat. Ser. III 19, 293 /C1/296, 1984. Shum, K. P. "The Amalgamation of Closure and Boundary Functions on Semigroups and Partially Ordered Sets." In Proceedings of the Conference on Ordered Structures and Algebra of Computer Languages. Singapore: World Scien- tific, pp. 232 /C1/243, 1993. Smith, A. Advanced Problem 5996. Amer. Math. Monthly 81, 1034, 1974. Soltan, V. P. "On Kuratowski’s Problem." Bull. Acad. Polon. Sci. Ser. Sci. Math. 28, 369 /C1/375, 1981. Soltan, V. P. "Problems of Kuratowski Type." Mat. Issled. 65, 121 /C1/131 and 155, 1982. Steen, L. A. and Seebach, J. A. Jr. Counterexamples in Topology. New York: Dover, 1996. Kuratowski’s Theorem KURATOWSKI REDUCTION THEOREM Kurscha ´k’s Theorem The AREA of the DODECAGON (n /C3012) inscribed in a UNIT CIRCLE with R /C301is A /C301 2 nR2 sin2p n ! /C303 : (1) See also DODECAHEDRON References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 137, 1991.Kurscha ´k’s Tile An attractive tiling of the SQUARE composed of two types of triangular tiles. It consists of 16 EQUILATERAL TRIANGLES and 32 15 8-158-150 8 ISOSCELES TRIANGLES arranged in the shape of a DODECAGON . The composition of Ku¨rscha´k’s tile is motivated by drawing inward-pointing EQUILATERAL TRIANGLES on each side of a UNIT SQUARE and then connecting adjacent vertices to form a smaller SQUARE rotated 45 8 with respect to the original SQUARE . Joining the midpoints of the square together with the intersec- tions of the EQUILATERAL TRIANGLES then gives a DODECAGON (Wells 1991) with CIRCUMRADIUS R/C30sinp 12 ! /C3014(ffiffiffi 6p /C28ffiffiffi 2p ): See also DODECAGON ,EQUILATERAL TRIANGLE ,ISO- SCELES TRIANGLE References Alexanderson, G. L. and Seydel, K. "Ku ¨rscha´k’s Tile." Math. Gaz. 62, 192/C1/196, 1978. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 30 /C1/32, 1985. Schoenberg, I. Mathematical Time Exposures. Washington, DC: Math. Assoc. Amer., p. 7, 1982. Weisstein, E. W. "Ku ¨rscha´k’s Tile." M ATHEMATICA NOTE- BOOK KURSCHAKS TILE.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 136 /C1/137, 1991. Kurtosis The degree of peakedness of a distribution, also called the "excess" or "excess coefficient." Kurtosis is anormalized form of the fourth CENTRAL MOMENT of a distribution. There are several flavors of kurtosis commonly encountered, including FISHER KURTOSIS (denoted g2 or b2) and PEARSON KURTOSIS (denoted b2 or a4): If not specifically qualified, then term "kurto- sis" is generally taken to refer to FISHER KURTOSIS .A distribution with a high peak (g2 > 0) is called LEPTOKURTIC , a flat-topped curve ( g2 B0) is called PLATYKURTIC , and the normal distribution ( g2 /C300) is called MESOKURTIC . Let midenote the ith CENTRAL MOMENT . Then the FISHER KURTOSIS is defined by g2 /C13m4 m2 2/C283 /C30m4 s4 /C283; (1) where s2 is the VARIANCE . Similarly, the PEARSON KURTOSIS is defined by b2 /C13m4 m2 2/C30m4 s4 : (2) An ESTIMATOR for the FISHER KURTOSIS g2 is given by ˆg2 /C30k4 k22; (3) where the ks are K-STATISTIC . For a normal distribu- tion, the variance of this estimator is var(g2) :24 N: (4) The following table lists the FISHER KURTOSIS for a number of common distributions. distribution FISHER KURTOSIS BERNOULLI DISTRIBUTION1 1 /C28 p /C271 p /C286 BETA DISTRIBUTION6[a3 /C27 a2(1 /C28 2b) /C27 b2(1 /C27 b) /C28 2ab(2 /C27 b)] ab(2 /C27 a /C27 b)(3 /C27 a /C27 b) BINOMIALDISTRIBUTION6p2 /C28 6p /C27 1 np(1 /C28 p) CHI-SQUARED DISTRIBUTION12 r EXPONENTIAL DISTRIBUTION6 FISHER- TIPPETT DISTRIBUTION/12 5/ GAMMA DISTRIBUTION6 a GEOMETRICDISTRIBUTION 5 /C28p /C271 1 /C28 pHALF-NORMAL DISTRIBUTION8( p /C28 3) (p /C28 2)2 LAPLACE DISTRIBUTION3 LOG NORMAL DISTRIBUTIONe4S2 /C272e3S2 /C273e2S2 /C286 MAXWELL DISTRIBUTION/C284 3 NEGATIVE BINOMIAL DISTRIBUTION6 /C28 p(6 /C28 p) r(1 /C28 p) NORMALDISTRIBUTION0 POISSON DISTRIBUTION1 n RAYLEIGH DISTRIBUTION6p(4/C28p)/C2816 (p/C284)2 STUDENT’S T- DISTRIBUTION6 n/C284 continuous UNIFORM DISTRIBUTION/C286 5 discrete UNIFORM DISTRIBUTION6(n2/C271) 5(n2/C281) See also FISHER KURTOSIS ,M EAN,PEARSON KURTO- SIS,SKEWNESS ,STANDARD DEVIATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972. Darlington, R. B. "Is Kurtosis Really Peakedness?" Amer. Statist. 24,1 9/C1/22, 1970. Dodge, Y. and Rousson, V. "The Complications of the Fourth Central Moment." Amer. Statist. 53, 267/C1/269, 1999. Kenney, J. F. and Keeping, E. S. "Kurtosis." §7.12 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 102 /C1/103, 1962. Moors, J. J. A. "The Meaning of Kurtosis: Darlington Reex- amined." Amer. Statist. 40, 283/C1/284, 1986. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Moments of a Distribution: Mean, Variance,Skewness, and So Forth." §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.Cambridge, England: Cambridge University Press,pp. 604 /C1 /609, 1992. Rupert, D. "What is Kurtosis? An Influence Function Approach." Amer. Statist. 41,1/C1/5, 1987. L L1-Norm A VECTOR NORM defined for a VECTOR x /C30x1 x2 n xn2 6643 775; with COMPLEX entries by xkk1/C30Xn r/C301½xr ½: The vector norm xkk1is implemented as Vector- Norm [m, 1] in the Mathematica add-on package LinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also L1-SPACE , L2-NORM, L-INFINITY- NORM,VEC- TOR NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1114 /C1/125, 2000. L1-Space See also L1-NORM L2-Function Informally, an L2/-function is a function f : X 0 R that is SQUARE INTEGRABLE , i.e., ½½f ½½2 /C30gX½f ½2 dm with respect to the MEASURE m; exists (and is finite), in which case ½½f ½½ is its L2-NORM . Here X is a MEASURE SPACE and the integral is the LEBESGUE INTEGRAL . The collection of L2 functions on X is called L2(X) (ell- two) of L2-SPACE , which is a HILBERT SPACE . On the unit interval (0; 1); the functions f(x) /C301=xp are in L2 for p B1=2: However, the function f(x) /C30 x/C281 =2 is not in L2 since g1 0(x/C281=2)2 dx /C30g1 0dx x does not exist.More generally, there are L2/-COMPLEX FUNCTIONS obtained by replacing the ABSOLUTE VALUE of a REAL NUMBER in the definition with the NORM of the COMPLEX NUMBER . In fact, this generalizes to func- tions from a MEASURE SPACE X to any NORMED SPACE . /L2/-functions play an important role in many areas of ANALYSIS . They also arise in physics, and especially quantum mechanics, where probabilities are given as the integral of the absolute square of a wavefunction c: In this and in the context of energy density, L2 /- functions arise due to the requirement that these quantities remain finite. See also HILBERT SPACE ,LEBESGUE INTEGRAL , LP- SPACE , L2-SPACE ,M EASURE ,M EASURE SPACE , SQUARE INTEGRABLE L2-Inner Product The L2/-inner product of two REAL FUNCTIONS f and g on a MEASURE SPACE X with respect to the MEASURE m is given by /C142f ; g /C143L2 /C30gXfg dm ; sometimes also called the bracket product, where the symbol /C142f ; g/C143 are called ANGLE BRACKETS . If the functions are COMPLEX , the generalization of the HERMITIAN INNER PRODUCT gXf¯gdm is used. See also ANGLE BRACKET ,BRA,HILBERT SPACE ,KET, LEBESGUE INTEGRAL , L2-FUNCTION , L2-SPACE L2-Norm AVECTOR NORM defined for a VECTOR x/C30x1 x2 n xn2 6643 775; (1) with COMPLEX entries by xkk2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn r/C301½xr½2vuut: (2) This discrete norm for a vector is sometimes called the l 2/-norm, while the L2/-norm (denoted with an upper-case L) is reserved for application with a function f(x);where it is defined by fkk2/C13f /C215f/C13/C142f½f/C143/C13g½f(x)½2dx; (3) with/C142f½g/C143denoting an ANGLE BRACKET . The L2/-norm xkk2is also called the Euclidean norm, and is implemented as VectorNorm [m, 2] in the Mathematica add-on package LinearAlgebra‘Ma- trixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also ANGLE BRACKET ,COMPLETE SET OF FUNC- TIONS , L1-NORM, L2-SPACE , L-INFINITY- NORM,PAR- ALLELOGRAM LAW,VECTOR NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1114 /C1/125, 2000. L2-Space On a MEASURE SPACE X, the set of SQUARE INTEGR- ABLE L2-FUNCTIONS is an L2 /-space. Taken together with the L2-INNER PRODUCT (a.k.a. BRACKET PRO- DUCT ) with respect to a MEASURE m; /C142f ; g/C143/C13gXfg dm (1) the L2/-space forms a HILBERT SPACE . The functions in an L2 /-space satisfy /C142f j c/C143/C13g ¯cf dx (2) and /C142fj c/C143/C30/C142cf/C143j (3) /C142fl1 c1 /C27 l2 c2 /C143/C30 l1 /C142 fc1 /C143/C27 l2 /C142 fc2 /C143j j j (4) /C142l1 f1 /C27 l2 f2 c/C143/C30 ¯l1 /C142f1 c/C143/C27 ¯l2 /C142f2 c/C143j/C()/C()/C()/C() (5) /C142 cc/C143/C23R ]0 j (6) ½/C142c 1 c2 /C143½2 5/C142c1 c1 /C143/C142 c2 c2 /C143: j j/C()/C() (7) The inequality (7) is called SCHWARZ’S INEQUALITY . The basic example is when X /C30R with LEBESGUE MEASURE . Another important example is when X is the positive integers, in which case it is denoted as l2 ; or "little ell-two." These are the square summable SERIES . Strictly speaking, L2/-space really consists of EQUIVA- LENCE CLASSES of functions. Two functions represent the same L2 /-function if the set where they differ has measure zero. It is not hard to see that this makes /C142f ; g/C143 an inner product, because /C142f ; f /C143/C300 if and only if f /C300 ALMOST EVERYWHERE . A good way to think of an L2/-function is as a density function, so only its integral on sets with positive measure matter. In practice, this does not cause much trouble, except that some care has to be taken with boundary conditions in DIFFERENTIAL EQUATIONS . The problem is that for any particular point p, the value /f(p)/ isn’t WELL DEFINED for an L2 /-function f.If an L2/-function in EUCLIDEAN SPACE can be repre- sented by a continuous function f, then f is the only continuous representative. In such a case, it is not harmful to consider the L2/-function as the continuous function f. Also, it is often convenient to think of L2(Rn) as the COMPLETION of the CONTINUOUS func- tions with respect to the L2-NORM . See also BRACKET PRODUCT ,COMPLETION ,H ILBERT SPACE , L2-NORM, LP-SPACE , L-FUNCTION ,LEBESGUE INTEGRAL ,LEBESGUE MEASURE ,M EASURE ,M EASURE SPACE ,R IESZ- FISCHER THEOREM ,S CHWARZ’S IN- EQUALITY Labeled Graph A labeled graph G /C30(V ; E) is a finite series of VERTICES V with a set of EDGES E of 2-SUBSETS of V. Given a VERTEX set Vn /C30f1;2; ...; ng; the number of vertex-labeled graphs is given by 2n(n/C281)=2 : Two graphs G and H with VERTICES Vn /C30f1;2 ; ...; n g are said to be ISOMORPHIC if there is a PERMUTATION p of Vnsuch that fu; vg is in the set of EDGES E(G) IFF fp(u); p(v)g is in the set of EDGES E(H):/ The term "labeled graph" when used without qualifi- cation means a graph with each node labeled differ- ently (but arbitrarily), so that all nodes are considered distinct for purposes of enumeration. The total number of (not necessarily connected) labeled n- node graphs is given 1, 2, 8, 64, 1024, 32768, ... (Sloane’s A006125; illustrated above), and the num- bers of connected labeled graphs on n-nodes are given by the LOGARITHMIC TRANSFORM of the preceding sequence, 1, 1, 4, 38, 728, 26704, ... (Sloane’s A001187; Sloane and Plouffe 1995, p. 19). See also 15 PUZZLE , A-CORDIAL GRAPH ,CONNECTED GRAPH ,C ORDIAL GRAPH ,E DGE-GRACEFUL GRAPH , ELEGANT GRAPH ,E QUITABLE GRAPH ,G RACEFUL GRAPH ,G RAPH , H-CORDIAL GRAPH ,H ARMONIOUS GRAPH ,L ABELED TREE,M AGIC GRAPH ,O RIENTED GRAPH ,S UPER- EDGE- GRACEFUL GRAPH ,T AYLOR’S CONDITION ,UNLABELED GRAPH ,W EIGHTED TREE References Cahit, I. "Homepage for the Graph Labelling Problems and New Results." http://www.emu.edu.tr/~cahit/COR- DIAL.htm. Gallian, J. A. "Graph Labeling." Elec. J. Combin. DS6, 1 /C1/2, Apr. 15, 1999. http://www.combinatorics.org/Surveys/. Gilbert, E. N. "Enumeration of Labeled Graphs." Canad. J. Math. 8, 405 /C1/11, 1956. Harary, F. "Labeled Graphs." Graph Theory. Reading, MA: Addison-Wesley, pp. 10 and 178 /C1/80, 1994. Sloane, N. J. A. Sequences A001187/M3671 and A006125/ M1897 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Labeled Tree A TREE with its nodes labeled. The number of labeled trees on n nodes is nn /C282 ; the first few values of which are 1, 1, 3, 16, 125, 1296, ... (Sloane’s A000272). Cayley (1889) provided the first proof of the number of labeled trees (Skiena 1990, p. 151), and a con- structive proof was subsequently provided by Pru¨fer (1918). Pru¨fer’s result gives an encoding for labeled trees known as PRU¨ FER CODE (indicated underneath the trees above, where the trees are depicted using an embedding with root at the node labeled 1). The probability that a random labeled tree is CEN- TERED is asymptotically equal to 1/2 (Szekeres 1983; Skiena 1990, p. 167). See also LABELED GRAPH ,PRU¨ FER CODE,TREE References Biggs, N. L.; Lloyd, E. K.; and Wilson, R. J. Graph Theory 1736 /C1/936. Oxford, England: Oxford University Press, p. 51, 1976. Cayley, A. "A Theorem on Trees." Quart. J. Math. 23, 376 /C1/ 78, 1889. Pru¨fer, H. "Neuer Beweis eines Satzes u¨ber Permutationen." Arch. Math. Phys. 27, 742 /C1/44, 1918.Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 128, 1980. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A000272/M3027 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Szekeres, G. Distribution of Labeled Trees by Diameter. New York: Springer-Verlag, pp. 392 /C1/97, 1983. van Lint, J. H. and Wilson, R. M. A Course in Combinato- rics. New York: Cambridge University Press, 1992. Lacunarity Quantifies deviation from translational invariance by describing the distribution of gaps within a set at multiple scales. The more lacunar a set, the more heterogeneous the spatial arrangement of gaps. Lacunary Function This entry contributed by JONATHAN DEANE A function that has a NATURAL BOUNDARY . See also NATURAL BOUNDARY References Ash, R. B. Ch. 3 in Complex Variables. New York: Academic Press, 1971. Ladder ASTROID ,CROSSED LADDERS PROBLEM ,CROSSED LAD- DERS THEOREM ,LADDER GRAPH Ladder Graph A GRAPH consisting of two rows of paired nodes each connected by an EDGE . Its complement is the COCK- TAIL PARTY GRAPH . See also COCKTAIL PARTY GRAPH Lagerstrom Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27k xy?/C27ey?y/C300: References Rosenblat, S. and Shepherd, J. "On the Asymptotic Solution of the Lagerstrom Model Equation." SIAM J. Appl. Math. 29, 110/C1/20, 1975. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 124, 1997. Lagrange Bracket Let F and G be infinitely differentiable functions of x, u, and p. Then the Lagrange bracket is defined by [F ; G] /C30Xn n/C301@F @pn@G @xp/C27pn@G @u ! /C28@G @pn@F @xn/C27pn@F @u ! "# : (1) The Lagrange bracket satisfies [F ; G] /C30/C28[G ; F] (2) [[F ; G]; H] /C27[[G ; H]; F] /C27[[H ; F] ; G] /C30@F @u[G ; H] /C27@G @u[H ; F] /C27@H @u[F ; G]: (3) If F and G are functions of x and p only, then the Lagrange bracket [F, G] collapses the POISSON BRACKET (F, G). See also LIE BRACKET ,POISSON BRACKET References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1004, 1980. Lagrange-Bu ¨rmann Expansion LAGRANGE INVERSION THEOREM Lagrange-Bu ¨rmann Theorem LAGRANGE INVERSION THEOREM Lagrange Expansion Let y /C30f(x) and y0 /C30f(x0) where f ?(x0) "0; then x /C30x0 /C27X/C12 k/C301(y /C28 y0)k k!dk /C281 dxk /C281x /C28 x0 f(x) /C28 y0"#k8 < :9 = ; x/C30x0 g(x) /C30g(x0) /C27X/C12 k /C301(y /C28 y0)k k! /C2dk/C281 dxk /C281g ?(x)x /C28 x0 f(x) /C28 y0 !k2 4358 < :9 = ; x/C30x0: Expansions of this form were first considered by Lagrange (1770; Lagrange 1868, pp. 680 /C1/93). See also BU¨ RMANN’S THEOREM ,M ACLAURIN SERIES , TAYLOR SERIES ,TEIXEIRA’S THEOREM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Goursat, E. A Course in Mathematical Analysis, Vol. 2, Pt. 1. New York: Dover, p. 106, 1959.Lagrange, J. L. "Nouvelle me ´thode pour re ´soudre les pro- ble`mes inde ´termine ´s en nombres entiers." Me´m. de l’Acad. Roy. des Sci. et Belles-Lettres de Berlin 24, 1770. Rep- rinted in Oeuvres de Lagrange, tome 2, section deuxie `me: Me´moires extraits des recueils de l’Academie royale des sciences et Belles-Lettres de Berlin. Paris: Gauthier-Vil- lars, pp. 655 /C1/26, 1868. Whittaker, E. T. and Watson, G. N. "Lagrange’s Theorem." §7.32 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 132, 1990. Lagrange Interpolating Polynomial The Lagrange interpolating polynomial is the POLY- NOMIAL of degree n/C281 which passes through the n points y1/C30f(x1);y2/C30f(x2);...,yn/C30f(xn):It is given by P(x)/C30Xn j/C301Pj(x); (1) where Pj(x)/C30Yn k/C301 k"jx/C28xk xj/C28xkyj: (2) Written explicitly, P(x)/C30(x/C28x2)(x/C28x3)/C1/C1/C1(x/C28xn) (x1/C28x2)(x1/C28x3)/C1/C1/C1(x1/C28xn)y1 /C27(x/C28x1)(x/C28x3)/C1/C1/C1(x/C28xn) (x2/C28x1)(x2/C28x3)/C1/C1/C1(x2/C28xn)y2/C27/C1/C1/C1 /C27(x/C28x1)(x/C28x2)/C1/C1/C1(x/C28xn/C281) (xn/C28x1)(xn/C28x2)/C1/C1/C1(xn/C28xn/C281)yn: (3) The formula was first published by Waring (1779), rediscovered by Euler in 1783, and published by Lagrange in 1795 (Jeffreys and Jeffreys 1988).Forn/C303 points, P(x)/C30(x/C28x2)(x/C28x3) (x1/C28x2)(x1/C28x3)y1/C27(x/C28x1)(x/C28x3) (x2/C28x1)(x2/C28x3)y2 /C27(x/C28x1)(x/C28x2) (x3/C28x1)(x3/C28x2)y3 (4) P ?(x) /C302x /C28 x2 /C28 x3 (x1 /C28 x2)(x1 /C28 x3)y1 /C272x /C28 x1 /C28 x3 (x2 /C28 x1)(x2 /C28 x3)y2 /C272x /C28 x1 /C28 x2 (x3 /C28 x1)(x3 /C28 x2)y3 (5) Note that the function P(x) passes through the points (xi ; yi); as can be seen for the case n /C303, P(x1) /C30(x1 /C28 x2)(x1 /C28 x3) (x1 /C28 x2)(x1 /C28 x3)y1 /C27(x1 /C28 x1)(x1 /C28 x3) (x2 /C28 x1)(x2 /C28 x3)y2 /C27(x1 /C28 x1)(x1 /C28 x2) (x3 /C28 x1)(x3 /C28 x2)y3 /C30y1 (6) P(x2) /C30(x2 /C28 x2)(x2 /C28 x3) (x1 /C28 x2)(x1 /C28 x3)y1 /C27(x2 /C28 x1)(x2 /C28 x3) (x2 /C28 x1)(x2 /C28 x3)y2 /C27(x2 /C28 x1)(x2 /C28 x2) (x3 /C28 x1)(x3 /C28 x2)y3 /C30y2 (7) P(x3) /C30(x3 /C28 x2)(x3 /C28 x3) (x1 /C28 x2)(x1 /C28 x3)y1 /C27(x3 /C28 x1)(x3 /C28 x3) (x2 /C28 x1)(x2 /C28 x3)y2 /C27(x3 /C28 x1)(x3 /C28 x2) (x3 /C28 x1)(x3 /C28 x2)y3 /C30y3 : (8) Generalizing to arbitrary n, P(xj) /C30Xn k /C301Pk(xj) /C30Xn k /C301djkyk /C30yj : (9) The Lagrange interpolating polynomials can also be written using what Szego (1975) called Lagrange’s fundamental interpolating polynomials. Let p(x) /C13Yn k /C301(x /C28xk) ; (10) p(xj) /C13Yn k /C301(xj /C28xk) ; (11) p?(xj) /C30dp dx"# x /C30xj/C30Yn k/C301 k "j(xj /C28xk) (12) so that p(x)isan nth degree POLYNOMIAL with zeros at x1 ; ..., xn : Then define the fundamental polynomials by pn(x) /C30p(x) p?(xn)(x /C28 xn) ; (13) which satisfy pn(xm) /C30 d nm ; (14) where dnmis the KRONECKER DELTA . Now let y1 /C30 P(x1); ..., yn /C30P(xn) ; then the expansionP(x) /C30Xn k /C301pk(x)yk /C30Xn k /C301p(x) (x /C28 xk) p?(xk)yk (15) gives the unique Lagrange interpolating polynomial assuming the values yk at xk : More generally, let da(x) be an arbitrary distribution on the interval [a, b], fpn(x) g the associated ORTHOGONAL POLYNOMIALS , and l1(x) ; ..., ln(x) the fundamental POLYNOMIALS corresponding to the set of zeros of a polynomial Pn(x):Then gb aln(x)lm(x)da(x)/C30lmdnm (16) forn;m/C301;2, ..., n, where lnare C HRISTOFFEL NUMBERS . Lagrange interpolating polynomials give no errorestimate. A more conceptually straightforward method for calculating them is N EVILLE’S ALGORITHM . See also AITKEN INTERPOLATION ,H ERMITE’S INTER- POLATING POLYNOMIAL ,LEBESGUE CONSTANTS (LA- GRANGE INTERPOLATION ), NEVILLE’S ALGORITHM , NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FOR- MULA References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 878 /C1/79 and 883, 1972. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 439, 1987. Jeffreys, H. and Jeffreys, B. S. "Lagrange’s Interpolation Formula." §9.011 in Methods of Mathematical Physics, 3rd ed.Cambridge, England: Cambridge University Press, p. 260, 1988. Pearson, K. Tracts for Computers 2, 1920. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Polynomial Interpolation and Extrapolation" and "Coefficients of the Interpolating Polynomial." §3.1 and 3.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 102 /C1/04 and 113 /C1/16, 1992. Se´roul, R. "Lagrange Interpolation." §10.9 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 269 /C1/73, 2000. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 329 and 332, 1975. Waring, E. Philos. Trans. 69,5 9/C1/7, 1779. Whittaker, E. T. and Robinson, G. "Lagrange’s Formula of Interpolation." §17 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 28 /C1/0, 1967. Lagrange Interpolation LAGRANGE INTERPOLATING POLYNOMIAL Lagrange Inversion Theorem Letzbe defined as a function of win terms of a parameter aby z/C30w/C27af(z): Then any function of z can be expressed as a POWER SERIES in a which converges for sufficiently small a and has the form F(z) /C30F(w) /C27a 1f(w)F ?(w) /C27a2 1 /C215 2@ @w f[ f(w)]2F ?(w) g /C27.../C27an/C271 (n /C27 1)!@n @wn f[ f(w)]n/C271F ?(w) g/C27...: See also BU¨ RMANN’S THEOREM ,SCHUR- JABOTINSKY THEOREM References Goursat, E. Functions of a Complex Variable, Vol. 2, Pt. 1. New York: Dover, 1959. Henrici, P. "An Algebraic Proof of the Lagrange-Burmann Formula." J. Math. Anal. Appl. 8, 218 /C1/24, 1964. Henrici, P. "The Lagrange-Bu ¨rmann Theorem." §1.9 in Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, pp. 55 /C1/5, 1988. Joni, S. A. "Lagrange Inversion in Higher Dimensions and Umbral Operators." J. Linear Multi-Linear Algebra 6, 111 /C1/21, 1978. Moulton, F. R. An Introduction to Celestial Mechanics, 2nd rev. ed. New York: Dover, p. 161, 1970. Popoff, M. "Sur le reste de la se´rie de Lagrange." Comptes Rendus Herbdom. Se´ances de l’Acad. Sci. 53, 795 /C1/98, 1861. Roman, S. "The Lagrange Inversion Formula." §5.2. in The Umbral Calculus. New York: Academic Press, pp. 138 /C1/ 40, 1984. Whittaker, E. T. and Watson, G. N. "Lagrange’s Theorem." §7.32 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 132 /C1/33, 1990. Williamson, B. "Remainder in Lagrange’s Series." §119 in An Elementary Treatise on the Differential Calculus, 9th ed. London: Longmans, pp. 158 /C1/59, 1895. Lagrange Multiplier Used to find the EXTREMUM of f(x1 ; x2 ; ... ; xn) sub- ject to the constraint g(x1 ; x2 ; ...; xn) /C30C; where f and g are functions with continuous first PARTIAL DERIVATIVES on the OPEN SET containing the curve g(x1 ; x2 ; ...; xn) /C300 ; and 9g "0 at any point on the curve (where 9 is the GRADIENT ). For an EXTREMUM to exist, df /C30@f @x1dx1 /C27@f @x2dx2 /C27.../C27@f @xndxn /C300: (1) But we also have dg /C30@g @x1dx1 /C27@g @x2dx2 /C27.../C27@g @xndxn /C300: (2) Now multiply (2) by the as yet undetermined para- meter l and add to (1), @f @x1/C27 l@q @x1 ! dx1 /C27@f @x2/C27 l@q @x2 ! dx2/C27.../C27@f @xn/C27 l@q @xn ! dxn /C300: (3) Note that the differentials are all independent, so we can set any combination equal to 0, and the remain- der must still give zero. This requires that @f @xk/C27 l@g @xk/C300 (4) for all k /C301, ..., n. The constant l is called the Lagrange multiplier. For multiple constraints, g1 /C30 0; g2 /C300; ..., 9f /C30 l1 9g1 /C27 l2 9g2 /C27...: (5) See also KUHN- TUCKER THEOREM References Arfken, G. "Lagrange Multipliers." §17.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 945 /C1/50, 1985. Lagrange Number (Diophantine Equation) Given a FERMAT DIFFERENCE EQUATION (a quadratic DIOPHANTINE EQUATION ) x2 /C28r2y2 /C304 with r a QUADRATIC SURD , assign to each solution x ½y the Lagrange number z /C131 2(x /C27yr) : The product and quotient of two Lagrange numbers are also Lagrange numbers. Furthermore, every Lagrange number is a POWER of the smallest La- grange number with an integral exponent. See also PELL EQUATION References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 94 /C1/5, 1965. Lagrange Number (Rational Approximation) HURWITZ’S IRRATIONAL NUMBER THEOREM gives the best rational approximation possible for an arbitrary irrational number /C28bas f Theffiffiffi 8p are called Lagrange numbers and get steadily larger for each "bad" set of irrational numbers which is excluded. nExclude /ffiffiffi 8p / 1 none /ffiffiffi 2p / 2 /ffiffiffiffiffiffiffiffi 221p 5//ffiffiffiffiffiffiffiffiffiffiffi 9 /C284 3s ;/ 3 /m// f(x) /C30f(x0) /C27(x /C28x0)f ?(x0) /C27(x /C28 x0)2 2!f ƒ(x0) /C27... / Lagrange numbers are OF THE FORM /C27(x /C28 x0)n n!f(n)(x0) /C27Rn ; where m is a MARKOV NUMBER . The Lagrange numbers form a SPECTRUM called the LAGRANGE SPECTRUM . See also HURWITZ’S IRRATIONAL NUMBER THEOREM , IRRATIONALITY MEASURE ,L IOUVILLE’S APPROXIMA- TION THEOREM ,MARKOV NUMBER ,ROTH’S THEOREM , SPECTRUM SEQUENCE ,THUE- SIEGEL- ROTH THEOREM References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187 /C1/89, 1996. Lagrange Polynomial LAGRANGE INTERPOLATING POLYNOMIAL Lagrange Remainder Given a TAYLOR SERIES f(x) /C30f(x0) /C27(x /C28x0)f ?(x0) /C27(x /C28 x0)2 2!f ƒ(x0) /C27... /C27(x /C28 x0)n n!f(n)(x0) /C27Rn ; (1) the error Rn after n terms is given by Rn /C30gx x0f(n/C271)(t)(x /C28 t)n n!dt: (2) Using the MEAN-VALUE THEOREM , this can be bounded by Rn /C30f(n/C271)(x/C31) (n /C27 1)!(x /C28x0)n/C271 (3) for some x/C31/C23 (x0 ; x) (Abramowitz and Stegun 1972, p. 880). Note that the Lagrange remainder Rnis also some- times taken to refer to the remainder when terms up to the (n /C281)/st power are taken in the TAYLOR SERIES ,and that a notation in which h 0 x /C28x0 ; x/C310 a /C27 uh; and x /C28x/C310 1 /C28 u is sometimes used (Blumenthal 1926; Whittaker and Watson 1990, pp. 95 /C1/6). See also CAUCHY REMAINDER ,SCHLO ¨ MILCH REMAIN- DER,TAYLOR SERIES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Beesack, P. R. "A General Form of the Remainder in Taylor’s Theorem." Amer. Math. Monthly 73,64/C1/7, 1966. Blumenthal, L. M. "Concerning the Remainder Term in Taylor’s Formula." Amer. Math. Monthly 33, 424 /C1/26, 1926. Firey, W. J. "Remainder Formulae in Taylor’s Theorem." Amer. Math. Monthly 67, 903 /C1/05, 1960. Fulks, W. Advanced Calculus. New York: Wiley, p. 137, 1961. Nicholas, C. P. "Taylor’s Theorem in a First Course." Amer. Math. Monthly 58, 559 /C1/62, 1951. Poffald, E. I. "The Remainder in Taylor’s Formula." Amer. Math. Monthly 97, 205 /C1/13, 1990. Whittaker, E. T. and Watson, G. N. "Forms of the Remain- der in Taylor’s Series." §5.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 95 /C1/6, 1990. Lagrange Resolvent A quantity involving primitive cube ROOTS OF UNITY which can be used to solve the CUBIC EQUATION . References Faucette, W. M. "A Geometric Interpretation of the Solution of the General Quartic Polynomial." Amer. Math. Monthly 103,51/C1/7, 1996. Lagrange’s Continued Fraction Theorem The REAL ROOTS of quadratic expressions with inte- gral COEFFICIENTS have periodic CONTINUED FRAC- TIONS , as first proved by Lagrange. See also CONTINUED FRACTION Lagrange’s Equation The PARTIAL DIFFERENTIAL EQUATION (1 /C27f2 y )fxx /C272fxfyfxy /C27(1 /C27f2 x )fyy /C300; whose solutions are called MINIMAL SURFACES . This corresponds to the MEAN CURVATURE H equalling 0 over the surface. D’ALEMBERT’S EQUATION y /C30xf(y?) /C27g(y?) is sometimes also known as Lagrange’s equation (Zwillinger 1997, pp. 120 and 265 /C1/68). See also D’ALEMBERT’S EQUATION ,MEAN CURVATURE , MINIMAL SURFACE References do Carmo, M. P. "Minimal Surfaces." §3.5 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 41 /C1/3, 1986. Zwillinger, D. "Lagrange’s Equation." §II.A.69 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 120 and 265 /C1/68, 1997. Lagrange’s Four-Square Theorem A theorem also known as BACHET’S CONJECTURE which was stated but not proven by Diophantus. It states that every POSITIVE INTEGER can be written as the SUM of at most four SQUARES . Although the theorem was proved by Fermat using infinite descent, the proof was suppressed. Euler was unable to prove the theorem. The first published proof was given by Lagrange in 1770 and made use of the EULER FOUR- SQUARE IDENTITY . Lagrange proved that g(2) /C304; where 4 may be reduced to 3 except for numbers OF THE FORM 4n(8k /C27 7);as proved by Legendre in 1798 (Nagell 1951, p. 194; Wells 1986, pp. 48 and 56; Hardy 1999, p. 12; Savin 2000). See also DIOPHANTINE EQUATION–2ND POWERS ,EU- LER FOUR- SQUARE IDENTITY ,FERMAT’S POLYGONAL NUMBER THEOREM ,F IFTEEN THEOREM ,L EBESGUE IDENTITY ,SUM OF SQUARES FUNCTION ,VINOGRADOV’S THEOREM ,W ARING’S PROBLEM References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. "The Four-Square Theo- rem." §20.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 302 /C1/03, 1979. Landau, E. Vorlesungen u ¨ber Zahlentheorie, Vol. 1. New York: Chelsea, pp. 114 /C1/22, 1970. Nagell, T. "Bachet’s Theorem." §55 in Introduction to Number Theory. New York: Wiley, pp. 191 /C1/95, 1951. Niven, I. M.; Zuckerman, H. S.; and Montgomery, H. L. An Introduction to the Theory of Numbers, 5th ed. New York: Wiley, 1991. Savin, A. "Shape Numbers." Quantum 11,1 4/C1/8, 2000. Se´roul, R. "Sums of Four Squares." §8.13 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 207 /C1/08, 2000. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 48, 1986. Lagrange’s Group Theorem This entry contributed by N ICOLAS BRAY Also known as Lagrange’s lemma. The most general form of Lagrange’s theorem states that for a GROUP G,aSUBGROUP HofG, and a subgroup KofH,(G: K)/C30(G:H)(H:K);where the products are taken as cardinalities (thus the theorem holds even for INFI- NITE GROUPS ) and ( GH) denotes the INDEX . A fre-quently stated corollary (which follows from takingK/C30feg;where eis the IDENTITY ELEMENT ) is that the order of Gis equal to the product of the order of H and the INDEX ofH. The corollary is easily proven in the case of Gbeing a FINITE GROUP , as the LEFT COSETS ofHform a partition of G, and so the number of blocks in the partition (which is ( G:H)) multiplied by the number of elements in each partition (which is just the order ofH). For a FINITE GROUP G, this corollary gives that the order of Hmust divide the order of G. Then, because the order of an element xofGis the order of the cyclic subgroup generated by x, we must have that the order of any element of Gdivides the order of G. The converse of Lagrange’s theorem is not, in general, true (Gallian 1993, 1994). References Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, p. 111, 1996. Gallian, J. A. "On the Converse of Lagrange’s Theorem." Math. Mag. 63, 23, 1993. Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lexington, MA: D. C. Heath, 1994. Herstein, I. N. Abstract Algebra, 3rd ed. New York: Mac- millan, p. 66, 1996. Hogan, G. T. "More on the Converse of Lagrange’s Theo- rem." Math. Mag. 69, 375/C1/76, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 86, 1993. Lagrange’s Identity The algebraic identity Xn k/C301akbk ! 2 /C30Xn k/C301a2 k !Xn k/C301b2k ! /C28X 15kBj5n(akbj/C28ajbk)2(1) (Mitrinovic 1970, p. 41). In determinant form, (a1/C29/C1/C1/C1/C29an/C281)/C215(b1/C29/C1/C1/C1/C29bn/C281) /C30a1/C215b1 /C1/C1/C1 a1/C215bn/C281 n::: n an/C281/C215b1/C1/C1/C1an/C281/C215bn/C281/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C(); (2) where Ajjis the DETERMINANT ofA:Lagrange’s identity is a special case of the B INET- CAUCHY IDENTITY , and C AUCHY’S INEQUALITY inn-D follows from it. It can be coded in Mathematica as follow. BBDiscreteMath‘Combinatorica‘; CauchyLagrangeId[n_] : /C30Module[ {aa/C30Array[a, n], bb /C30Array[b, n]}, Plus @@ (aa^2)Plus @@ (bb^2) /C30/C30 Plus @@ ((a[#1]b[#2] - a[#2]b[#1])^2 & @@@ KSubsets[Range[n], 2]) /C27 (aa.bb)^2 ] Plugging in gives the n /C302 and n /C303 identities (a2 1 /C27a22)(b21 /C27b22) /C30(a1b1 /C27a2b2)2 /C27(a1b2 /C28a2b1)2(3) (a21 /C27a22 /C27a23)(b21 /C27b22 /C27b23) /C30(a1b1 /C27a2b2 /C27a3b3)2 /C27[(a1b2 /C28a2b1)2 /C27(a1b3 /C28a3b1)2 /C27(a2b3 /C28a3b2)2] : (4) See also BINET- CAUCHY IDENTITY ,CAUCHY’S INEQUAL- ITY,VECTOR TRIPLE PRODUCT ,VECTOR QUADRUPLE PRODUCT References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1093, 2000. Mitrinovic, D. S. Analytic Inequalities. New York: Springer- Verlag, 1970. Lagrange’s Inequality CAUCHY’S INEQUALITY Lagrange’s Lemma LAGRANGE’S FOUR- SQUARE THEOREM Lagrange Spectrum A SPECTRUM formed by the LAGRANGE NUMBERS . The only ones less than three are the LAGRANGE NUM- BERS , but the last gaps end at FREIMAN’S CONSTANT . REAL NUMBERS larger than FREIMAN’S CONSTANT are in the M ARKOV SPECTRUM . See also FREIMAN’S CONSTANT ,LAGRANGE NUMBER (RATIONAL APPROXIMATION ), MARKOV SPECTRUM , SPECTRUM SEQUENCE References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187 /C1/89, 1996. Lagrangian Coefficient COEFFICIENTS which appear in L AGRANGE INTERPO- LATING POLYNOMIALS where the points are equally spaced along the ABSCISSA . Lagrangian Derivative CONVECTIVE DERIVATIVE Laguerre Differential Equation xyƒ/C27(1/C28x)y?/C27ly/C300: (1) The Laguerre differential equation is a special case of the more general "associated Laguerre differentialequation" xyƒ/C27(n/C271/C28x)y?/C27ly/C300 (2) (Iyanaga and Kawada 1980, p. 1481; Zwillinger 1997, p. 124) with n/C300:The general solution is t/C30C1U(/C28l;1/C27n;x)/C27C2Ln l(x); (3) where U(a;b;x)i sa CONFLUENT HYPERGEOMETRIC FUNCTION OF THE FIRST KIND and Ln l(x) is an asso- ciated L AGUERRE POLYNOMIAL . Note that in the special case l/C300;the associated Laguerre differential equation is OF THE FORM yƒ(x)/C27P(x)y?(x)/C300; (4) so the solution can be found using an INTEGRATING FACTOR m/C30expgP(x)dx/C(*/C(+ /C30expgn/C271/C28x xdx ! /C30exp[( n/C271) ln x/C28x]/C30xn/C271e/C28x; (5) as y/C30C1gdx m/C27C2/C30C1gex xn/C271dx/C27C2 (6) /C30C2/C28C1x/C28nE1/C27n(/C28x); (7) where En(x) is the EN-FUNCTION . The associated Laguerre differential equation has a REGULAR SINGULAR POINT at 0 and an IRREGULAR SINGULARITY at/C12:It can be solved using a series expansion, xX/C12 n/C302n(n/C281)anxn/C282/C27(n/C271)X/C12 n/C301nanxn/C281 /C28xX/C12 n/C301nanxn/C281/C27lX/C12 n/C300anxn/C300 (8) X/C12 n/C302n(n/C281)anxn/C281/C27(n/C271)X/C12 n/C301nanxn/C281 /C28X/C12 n/C301nanxn/C27lX/C12 n/C300anxn/C300 (9) X/C12 n/C301(n/C271)nan/C271xn/C27(n/C271)X/C12 n/C300(n/C271)an/C271xn /C28X/C12 n/C301nanxn/C27lX/C12 n/C300anxn/C300 (10) [(n /C271)a1 /C27 la0] /C27X/C12 n/C301f[(n /C271)n /C27( n /C271)(n /C271)]an/C271 /C28nan /C27 lan gxn /C300 (11) [(n /C271)a1 /C27 la0] /C27X/C12 n /C301[(n /C271)(n /C27 n /C271)an/C271 /C27(l /C28n)an]xn /C300: (12) This requires a1 /C30/C28l n /C27 1a0 (13) an/C271 /C30n /C28 l (n /C27 1)(n /C27 n /C27 1)an (14) for n /C211. Therefore, an/C271 /C30n /C28 l (n /C27 1)(n /C27 n /C27 1)an (15) for n /C301, 2, ..., so y /C30a01 /C28l n /C27 1x /C28l(1 /C28 l) 2(n /C27 1)( n /C27 2)x2" /C28l(1 /C28 l)(2 /C28 l) 2 /C215 3(n /C27 1)(n /C27 2)(n /C27 3) /C27/C1/C1/C1/C)( : (16) If l is a POSITIVE INTEGER , then the series terminates and the solution is a POLYNOMIAL , known as an associated L AGUERRE POLYNOMIAL (or, if n/C300;simply aLAGUERRE POLYNOMIAL ). See also LAGUERRE POLYNOMIAL References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Laguerre-Gauss Quadrature Also called G AUSS- LAGUERRE QUADRATURE or L A- GUERRE QUADRATURE .AG AUSSIAN QUADRATURE over the interval [0 ;/C12) with WEIGHTING FUNCTION W(x)/C30e/C28x(Abramowitz and Stegun 1972, p. 890). The ABSCISSAS for quadrature order nare given by the ROOTS of the L AGUERRE POLYNOMIALS Ln(x):The weights are wi/C30/C28An/C271gn AnL?n(xi)Ln/C271(xi)/C30An An/C281gn/C281 Ln/C281(xi)L?n(xi);(1) where Anis the COEFFICIENT ofxninLn(x):For LAGUERRE POLYNOMIALS ,An/C30(/C281)n n!; (2) where n!i sa FACTORIAL ,s o An/C271 An/C30/C281 n/C271(3) An An/C281/C30/C281 n: (4) Additionally, gn/C30g/C12 0W(x)[Ln(x)]2dx/C301; (5) so wi/C301 (n/C271)L?n(xi)Ln/C271(xi)/C30/C281 nLn/C281(xi)L?n(xi):(6) Using the RECURRENCE RELATION xL?n(x)/C30nLn(x)/C28nLn/C281(x) /C30(x/C28n/C281)Ln(x)/C27(n/C271)Ln/C271(x) (7) which, since xiis a root of Ln(x);gives nLn(x)/C30(x/C28n/C281)Ln(x)/C300; (8) so (7) becomes xiL?n(xi)/C30/C28nLn/C281(xi)/C30(n/C271)Ln/C271(xi) (9) gives wi/C301 xi[L?n(xi)]2/C30xi (n/C271)2[Ln/C271(xi)]2: (10) The error term is E/C30(n!)2 (2n)!f(2n)(j) (11) (Abramowitz and Stegun 1972, p. 890). Beyer (1987) gives a table of ABSCISSAS and weights up to n/C306. n /xi// wi/ 2 0.585786 0.853553 3.41421 0.146447 3 0.415775 0.711093 2.29428 0.278518 6.28995 0.0103893 4 0.322548 0.603154 1.74576 0.357419 4.53662 0.0388879 9.39507 0.000539295 5 0.26356 0.521756 1.4134 0.398667 3.59643 0.0759424 7.08581 0.00361176 12.6408 0.00002337 The ABSCISSAS and weights can be computed analy- tically for small n. n /xi// wi/ 2 /2 /C28ffiffiffi 2p //1 42 /C27ffiffiffi 2p/CP/C( / /2 /C27ffiffiffi2p //1 42 /C28ffiffiffi 2p/CP/C( / For the associated Laguerre polynomial Lb n(x) with WEIGHTING FUNCTION w(x) /C30xbe /C28x ; An /C30( /C281)n n! (12) is the coefficient of xn in Lbn(x) and gn /C30g/C12 0xbe/C28x[Lb n(x)]2 dx /C30G(n /C27 b /C27 1) n!; (13) where G(z) is the GAMMA FUNCTION . The weights are then wi /C30G(n /C27 b)xi n!(n /C27 b)[Lb n /C281(xi)]2 /C30G(n /C27 b /C27 1)xi n!(n /C27 1)2[L bn/C271(xi)]2;(14) and the error term is En/C30n!G(n/C27b/C271) (2n)!f(2n)(j): (15) See also GAUSSIAN QUADRATURE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 890 and 923, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 463, 1987. Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 64 /C1/5, 1960. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 325 /C1/27, 1956. LaguerreL LAGUERRE POLYNOMIALLaguerre Polynomial Solutions Ln(x) to the L AGUERRE DIFFERENTIAL EQUA- TION with n/C300 are called Laguerre polynomials, illustrated above for x/C23[0;1] and n/C301, 2, ..., 5. The Rodrigues formula for the Laguerre polynomials is Ln(x)/C30ex n!dn dxn(xne/C28x) (1) and the GENERATING FUNCTION for Laguerre polyno- mials is g(x;z)/C30exp/C28zz 1/C28z ! 1/C28z/C301/C27(/C28x/C271)z /C271 2x2/C282x/C271/C(%/C(r z2/C27/C2816x3/C2732x2/C283x/C271/C(%/C(r z3/C27...: (2) ACONTOUR INTEGRAL is given by Ln(x)/C301 2pige/C28xz=(1/C28z) (1/C28z)zn/C271dz: (3) The Laguerre polynomials satisfy the RECURRENCE RELATIONS (n/C271)Ln/C271(x)/C30(2n/C271/C28x)Ln(x)/C28nLn/C281(x) (4) (Petkovsek et al. 1996) and xL?n(x)/C30nLn(x)/C28nLn/C281(x): (5) The first few Laguerre polynomials are L0(x)/C301 L1(x)/C30/C28x/C271 L2(x)/C301 2(x2/C284x/C272) L3(x)/C301 6(/C28x3/C279x2/C2818x/C276): Solutions to the associated L AGUERRE DIFFERENTIAL EQUATION with n"0 are called associated Laguerre polynomials Lk n(x) or, in older literature, Sonine polynomials (Sonine 1880, p. 41; Whittaker and Watson 1990, p. 352). In terms of the unassociated Laguerre polynomials, Ln(x)/C30L0 n(x): (6) The Rodrigues formula for the associated Laguerre polynomials is Lk n(x)/C30exx/C28k n!dn dxn(e/C28xxn/C27k) (7) /C30(/C281)kdk dxk[Ln/C27k(x)] (8) /C30(/C281)nx/C28(k/C271)=2 n!ex=2Wk=2/C27n/C271=2;k=2(x) (9) /C30Xn m/C300(/C281)m (n/C27k)! (n/C28m)!(k/C27m)!m!xm; (10) where Wk;m(x)i saW HITTAKER FUNCTION . The asso- ciated Laguerre polynomials are a S HEFFER SE- QUENCE with g(t)/C30(1/C28t)/C28k/C281(11) f(t)/C30t t/C281; (12) giving the GENERATING FUNCTION g(x;z)/C30exp/C28zz 1/C28z ! (1/C28z)k/C271 /C301/C27(k/C271/C28x)z/C271 2[x2/C282(k/C272)x/C27(k/C271)(k/C272)]z2 /C27...: (13) where the usual factor of n! in the denominator has been suppressed (Roman 1984, p. 31). Many interest- ing properties of the associated Laguerre polynomials follow from the fact that f/C281(t)/C30f(t) (Roman 1984, p. 31). The associated Laguerre polynomials are given ex- plicitly by the formula L(k) n(x)/C301 n!Xn i/C300n! i!k/C27n n/C28i;/C(*/C(+ (/C28x)i; (14) wheren k/CP/C( is a BINOMIAL COEFFICIENT , and have Sheffer identity 1 n!L(k) n(x/C27y)/C30Xn i/C300n i/C(*/C(+1 i!L(k) i(x)1 (n/C28i)!L(/C281) n/C28i(y) (15) (Roman 1984, p. 31). The associated Laguerre poly- nomial can also be written asL(k) n(x)/C30(k/C271)n n!1F1(/C28n;k/C271;x); (16) where ( a)nis the P OCHHAMMER SYMBOL and 1F1(a;b;x)i sa CONFLUENT HYPERGEOMETRIC FUNC- TION (Koekoek and Swarttouw 1998). The associated Laguerre polynomials are orthogonal over [0 ;/C12) with respect to the WEIGHTING FUNCTION xne/C28x: g/C12 0e/C28xxkLk n(x)Lkm(x)dx/C30(n/C27k)! n!dmn; (17) where dmnis the K RONECKER DELTA . They also satisfy g/C12 0e/C28xxk/C271[Lkn(x)]2dx/C30(n/C27k)! n!(2n/C27k/C271):(18) RECURRENCE RELATIONS include Xn n/C300L(k) n(x)/C30L(k/C271) n(x) (19) and L(k) n(x)/C30L(k/C271) n(x)/C28L(k/C271) n/C281(x): (20) The DERIVATIVE is given by d dxL(k) n(x)/C30/C28L(k/C271) n/C281(x) /C30x/C281nL(k) n(x)/C28(n/C27k)L(k) n/C281(x):/C) (21) An interesting identity is X/C12 n/C300L(k) n(x) G(n/C27k/C271)wn/C30ew(xw)/C28k=2Jk2ffiffiffiffiffiffiffixwp/CP/C( ;(22) where G(z) is the GAMMA FUNCTION and Jk(z) is the BESSEL FUNCTION OF THE FIRST KIND (Szego 1975, p. 102). An integral representation is e/C28xxk=2L(k) n(x)/C301 n!g/C12 0e/C28ttn/C27k=2Jk2ffiffiffiffiffi txp/C(%/C(r dt (23) forn/C300, 1, ...and k/C21/C281. The DISCRIMINANT is D(k) n/C30Yn n/C301nn/C282n/C272(n/C27k)n/C281(24) (Szego 1975, p. 143). The KERNEL POLYNOMIAL is K(k) n(x; y) /C30n /C27 1 G(k /C27 1) /C2n /C27k n/C(*/C(+/C281 /C2L(k) n(x)L(k) n /C271(y) /C28 L(k) n/C271(x)Ln(k)(y) x /C28 y ; (25) wheren k/CP/C( is a BINOMIAL COEFFICIENT (Szego 1975, p. 101). The first few associated Laguerre polynomials are Lk 0(x) /C301 Lk1(x) /C30/C28x /C27k /C271 Lk2(x) /C301 2[x2 /C282(k /C272)x /C27(k /C271)(k /C272)] Lk 3(x) /C301 6[/C28x3 /C273(k /C273)x2 /C283(k /C272)(k /C273)x /C27(k /C271)(k /C272)(k /C273)] : See also LAGUERRE DIFFERENTIAL EQUATION ,SONINE POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771 /C1/02, 1972. Andrews, G. E.; Askey, R.; and Roy, R. "Laguerre Polyno- mials." §6.2 in Special Functions. Cambridge, England: Cambridge University Press, pp. 282 /C1/93, 1999. Arfken, G. "Laguerre Functions." §13.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 721 /C1/31, 1985. Chebyshev, P. L. "Sur le de´veloppement des fonctions a` une seule variable." Bull. Ph.-Math., Acad. Imp. Sc. St. Pe´tersbourg 1, 193 /C1/00, 1859. Chebyshev, P. L. Oeuvres, Vol. 1. New York: Chelsea, pp. 499 /C1/08, 1987. Iyanaga, S. and Kawada, Y. (Eds.). "Laguerre Functions." Appendix A, Table 20.VI in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980. Koekoek, R. and Swarttouw, R. F. "Laguerre." §1.11 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Uni- versiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, pp. 47 /C1/9, 1998. ftp://www.twi.- tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1/ 7.ps.gz. Laguerre, E. de. "Sur l’inte´grale f/C27/C12 xx/C281e /C28x dx :/" Bull. Soc. math. France 7,72/C1/1, 1879. Reprinted in Oeuvres, Vol. 1. New York: Chelsea, pp. 428 /C1/37, 1971. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, pp. 61 /C1/2, 1996. Roman, S. "The Laguerre Polynomials." §3.1 i The Umbral Calculus. New York: Academic Press, pp. 108 /C1/13, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "Laguerre Polyno- mials." §11 in "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Sansone, G. "Expansions in Laguerre and Hermite Series." Ch. 4 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 295 /C1/85, 1991. Sonine, N. J. "Sur les fonctions cylindriques et le de´veloppe- ment des fonctions continues en se´ries." Math. Ann. 16, 1 /C1/0, 1880.Spanier, J. and Oldham, K. B. "The Laguerre Polynomials Ln(x) :/" Ch. 23 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 209 /C1/16, 1987. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. Whittaker, E. T. and Watson, G. N. Ch. 16, Ex. 8 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cam- bridge University Press, p. 352, 1990. Laguerre Quadrature AG AUSSIAN QUADRATURE -like FORMULA for numer- ical estimation of integrals. It fits exactly all POLY- NOMIALS of degree 2m /C281 :/ References Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 61, 1960. Laguerre’s Method A ROOT -finding algorithm which converges to a COMPLEX ROOT from any starting position. Pn(x) /C30(x /C28x1)(x /C28x2) /C1/C1/C1(x /C28xn) (1) ln Pn(x) jj/C30ln x /C28x1 jj /C27ln x /C28x2 jj /C27.../C27ln x /C28xn jj (2) P?n(x)/C30(x/C28x2)/C1/C1/C1(x/C28xn)/C27(x/C28x1)/C1/C1/C1(x/C28xn)/C27... /C30Pn(x)1 x/C28x1/C27.../C271 x/C28xn ! (3) dlnPn(x) jj dx/C301 x/C28x1/C271 x/C28x2/C27.../C271 x/C28xn /C30P?n(x) Pn(x)/C13G(x) (4) /C28d2lnPn(x) jj dx2/C301 (x/C28x1)2/C271 (x/C28x2)2/C27.../C271 (x/C28xn)2 /C30P?n(x) Pn(x)"#2 /C28Pƒn(x) Pn(x)/C13H(x): (5) Now let a/C13x/C28x1andb/C13x/C28x1:Then G/C131 a/C27n/C281 b(6) H/C131 a2/C27n/C281 b2; (7) so a/C30n max G9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n/C281)(nH/C28G2)p/C)/Cn : (8) Setting n/C302 gives H ALLEY’S IRRATIONAL FORMULA . See also HALLEY’S IRRATIONAL FORMULA ,H ALLEY’S METHOD ,NEWTON’S METHOD ,ROOT References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 365 /C1/66, 1992. Ralston, A. and Rabinowitz, P. §8.9 /C1/.13 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978. Laguerre’s Repeated Fraction The CONTINUED FRACTION (x /C27 1)n /C28 (x /C28 1)n (x /C27 1)n /C27 (x /C28 1)n /C30n x/C27n2 /C28 1 3x/C27n2 /C28 22 5x /C27 ...: References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 13 and 21, 1959. Watson, G. N. "Ramanujan’s Note Books." J. London Math. Soc. 6, 137 /C1/53, 1931. Watson, G. N. "The Mock Theta Functions (II)." Proc. London Math. Soc. 42, 274 /C1/04, 1937. Lah Number The numbers Bn; k(1!; 2! ; 3! ; ...)/C30n /C281 k /C281/C(*/C(+n! k! ; where Bn; k is a BELL POLYNOMIAL . See also BELL POLYNOMIAL ,IDEMPOTENT NUMBER References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 156, 1974. Roman, S. The Umbral Calculus. New York: Academic Press, p. 86, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Laisant’s Recurrence Formula The RECURRENCE RELATION (n /C281)An/C271 /C30(n2 /C281)An /C27(n /C271)An/C281 /C274(/C281)n with A(1)/C30A(2)/C301 which solves the MARRIED COU- PLES PROBLEM . See also MARRIED COUPLES PROBLEM Lakshmi Star STAR OF LAKSHMI L-Algebraic Number AnL-algebraic number is a number u/C23(0;1) which satisfiesXn k/C300ckL(uk)/C300; (1) where L(x) is the R OGERS L-FUNCTION and ckare integers not all equal to 0 (Gordon and Mcintosh 1997). Loxton (1991, p. 289) gives a slew of similar identities having rational coefficients Xn k/C300ek kL(uk)/C300 (2) instead of integers. The only known L-algebraic numbers of order 1 are L(0)/C300 (3) L(1/C28r)/C302 5(4) L12/C(%/C(r /C3012 (5) L(r)/C3035 (6) L(1)/C301 (7) (Loxton 1991, pp. 287 and 289; Bytsko 1999), where r/C30ffiffiffi 5p /C281/CP/C( =2:/ The only known rational L-algebraic numbers are /1=2/ and /1=3/: L1 64/C(%/C(r /C282L1 8/C(%/C(r /C286L14/C(%/C(r /C272L(1)/C300 (8) L1 9/C(%/C(r /C286L13/C(%/C(r /C272L(1)/C300 (9) (Lewin 1982, pp. 317 /C1/18; Gordon and McIntosh 1997). There are a number of known quadratic L-algebraic numbers. Watson (1937) found L(a)/C28L(a2)/C301 42p2(10) 2L(b)/C27L(b2)/C305 21p2(11) 2L(g)/C27L(g2)/C304 21p2; (12) where a;/C28b;and/C281=gare the roots of x3/C272x2/C281/C300; (13) so that a/C301 2sec27p/C(%/C(r (14) b/C3012sec17p/C(%/C(r (15) g/C302 cos37p/C(%/C(r (16) (Loxton 1991, pp. 287 /C1/88). Higher order algebraic identities include 5L(d3) /C285L( d) /C27L(1) /C300; (17) L( d12) /C282L( d6) /C286L(d4) /C274L(d3) /C273L( d2) /C274L( d) /C284L(1) /C300 (18) 3L( k3) /C289L( k2) /C289K( k) /C277L(1) /C300 (19) 3L( l6) /C286L(l3) /C2827L( l2) /C2718L( l)2L(1) /C300 (20) 3L( m6) /C286L( m3) /C2827L( m2) /C2718L( m) /C282L(1) /C300 (21) 2L(a3) /C282L(a2) /C2811L(a) /C273L(1) /C300 (22) 2L(b6) /C284L(b3) /C2815L(b2) /C2722L(b) /C286L(1) /C300 (23) 2L(c6) /C284L(c3) /C2815L(c2) /C2722L(c) /C284L(1) /C300; where d /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C272ffiffiffi 5pq /C281/C(*/C(+ (24) k /C301 2sec19 p/C(%/C(r (25) l /C301 2sec29 p/C(%/C(r (26) m /C302 cos49 p/C(%/C(r (27) a /C302ffiffiffi 3p cos5p 18 ! /C282 (28) b /C302ffiffiffi 3p cos11 p 18 ! /C272 (29) c /C302ffiffiffi3p cos 7p 18 ! /C281 (30) (Gordon and McIntosh 1997). See also DILOGARITHM ,ROGERS L-FUNCTION References Bytsko, A. G. Two-Term Dilogarithm Identities Related to Conformal Field Theory. 9 Nov 1999. http://xxx.lanl.gov/ abs/math-ph/9911012/. Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431 /C1/48, 1997. Lewin, L. "The Dilogarithm in Algebraic Fields." J. Austral. Soc. Ser. A 33, 302 /C1/30, 1982. Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991. Loxton, J. H. "Special Values of the Dilogarithm Function." Acta Arith. 43, 155 /C1/66, 1984. Loxton, J. H. "Partition Identities and the Dilogarithm." Ch. 13 in Structural Properties of Polylogarithms (Ed. L. Lewin). Providence, RI: Amer. Math. Soc., pp. 287 /C1/99, 1991. Watson, G. N. Quart. J. Math. Oxford Ser. 8, 39, 1937.Lal’s Constant Let P(N) denote the number of PRIMES OF THE FORM n2 /C271 for 1 5n 5N ; then P(N) /C20:68641 li(N) ; (1) where li(N) is the LOGARITHMIC INTEGRAL (Shanks 1960, pp. 321 /C1/32). Let Q(N) denote the number of PRIMES OF THE FORM n4 /C271 for 1 5n 5N ; then Q(N) /C21 4 s1 li(N) /C300:66974 li(N) (2) (Shanks 1961, 1962). Let R(N) denote the number of pairs of PRIMES (n /C281)2 /C271 and (n /C271)2 /C271 for n 5 N /C281 ; then R(N) /C20:487621 li2(N); (3) where li2(N) /C13gN 2dn (ln n)2 (4) (Shanks 1960, pp. 201 /C1/03). Finally, let S(N) denote the number of pairs of PRIMES (n /C281)4 /C271 and (n /C27 1)4 /C271 for n 5N /C281 ; then S(N) /C2 l li2(N) (5) (Lal 1967), where l is called Lal’s constant. Shanks (1967) showed that l :0 :79220 :/ References Lal, M. "Primes of the Form n4 /C271 :/" Math. Comput. 21, 245 /C1/ 47, 1967. Shanks, D. "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes of the Form n2 /C27a:/" Math. Comput. 14, 321 /C1/32, 1960. Shanks, D. "On Numbers of the Form n4 /C271:/" Math. Comput. 15, 186 /C1/89, 1961. Shanks, D. Corrigendum to "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes of the Form n2/C27a:/"Math. Comput. 16, 513, 1962. Shanks, D. "Lal’s Constant and Generalization." Math. Comput. 21, 705/C1/07, 1967. Laman’s Theorem Let a GRAPH Ghave exactly 2 n/C283EDGES , where nis the number of VERTICES inG. Then Gis "generically" RIGID inR2IFFe?52n?/C283 for every SUBGRAPH ofG having n?VERTICES ande?EDGES . See also RIGID GRAPH References Laman, G. "On Graphs and Rigidity of Plane Skeletal Structures." J. Engineering Math. 4, 331/C1/40, 1970. Lambda Calculus Developed by Alonzo Church and Stephen Kleene to address the COMPUTABLE NUMBER problem. In the lambda calculus, lis defined as the ABSTRACTION OPERATOR . Three theorems of lambda calculus are l/- conversion, a/-conversion, and h/-conversion. See also ABSTRACTION OPERATOR ,COMPUTABLE NUM- BER References Hankin, C. Lambda Calculi: A Guide for Computer Scien- tists. Oxford, England: Oxford University Press, 1995. Penrose, R. The Emperor’s New Mind: Concerning Compu- ters, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 66 /C1/0, 1989. Lambda Elliptic Function ELLIPTIC LAMBDA FUNCTION Lambda Function The lambda function defined by Jahnke and Emden (1945) is Ln(z) /C13G(n /C271)Jn(z) 1 2 z/C(%/C(rn (1) where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND and G(x) is the GAMMA FUNCTION . L0(z) /C30J0(z) ; and taking n /C301 gives the special case L1(z) /C13J1(z) 1 2 z/C302 jinc(z) ; (2) where jinc(z) is the JINC FUNCTION . A two-variable lambda function is defined as l(x ; y) /C13gy 0G(t /C27 1) dt xt; (3) where G(z) is the GAMMA FUNCTION (McLachlan et al. 1950, p. 9; Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109). The MANGOLDT FUNCTION is sometimes called the lambda function. See also AIRY FUNCTIONS ,DIRICHLET LAMBDA FUNC- TION ,ELLIPTIC LAMBDA FUNCTION ,JINC FUNCTION , MANGOLDT FUNCTION ,MU FUNCTION ,NU FUNCTION References Gradshteyn, I. S. and Ryzhik, I. M. "The Functions n(x); n(x; a); m(x; b) ; m(x; b; a) ; l(x; y) :/" §9.64 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1109, 2000. Jahnke, E. and Emde, F. Tables of Functions with Formulae and Curves, 4th ed. New York: Dover, 1945.McLachlan, N. W. et al. Supple ´ment au formulaire pour le calcul symbolique. Paris: L’Acad. des Sciences de Paris, Fasc. 113, p. 9, 1950. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, 1990. Lambda Group MODULAR GROUP LAMBDA Lambda Modular Function ELLIPTIC LAMBDA FUNCTION Lambert Azimuthal Equal-Area Projection A special case of a CYLINDRICAL EQUAL-AREA PROJEC- TION with standard parallel of fs /C300( : x /C30k? cos f sin( l /C28 l0) (1) y /C30k?[cos f1 sin f /C28sin f1 cos f cos( l /C28 l0)]; (2) where k ?/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 /C27 sin f1 sin f /C27 cos f1 cos f cos(l /C28 l0)s :(3) The inverse FORMULAS are f/C30sin/C281coscsinf1/C27ysinccosf1 r ! (4) l/C30l0/C27tan/C281 xsinc rcosf1cosc/C28ysinf1sinc ! ;(5) where r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p (6) c/C302 sin/C2811 2r/C(%/C(r : (7) See also AZIMUTHAL PROJECTION ,BALTHASART PRO- JECTION ,BEHRMANN CYLINDRICAL EQUAL- AREA PRO- JECTION ,C YLINDRICAL EQUAL- AREA PROJECTION , EQUAL- AREA PROJECTION ,GALL ORTHOGRAPHIC PRO- JECTION ,P ETERS PROJECTION ,T RISTAN EDWARDS PROJECTION References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 182 /C1/90, 1987. Lambert Conformal Conic Projection Let l be the longitude, l0the reference longitude, f the latitude, f0 the reference latitude, and f1 and f2 the standard parallels. Then the transformation of SPHERICAL COORDINATES to the plane via the Lambert conformal conic projection is given by x /C30 r sin[n( l /C28 l0)] (1) y /C30 r0 /C28 r cos[n(l /C28 l0)] ; (2) where r /C30F cotn1 4 p /C2712 f/C(%/C(r (3) r0 /C30F cotn14 p /C2712 f0/C(%/C(r (4) F /C30cos f1 tann1 4 p /C2712 f1/C(%/C(r n (5) n /C30ln(cos f1 secf2) ln tan14 p /C2712 f2/C(%/C(r cot14 p /C2712 f1/C(%/C(r hi : (6) The inverse formulas are f /C302 tan/C281F r0 !1 =n2 435/C28 1 2 p (7) l /C30 l0 /C27u n ; (8) where r /C30sgn(n)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27(r0 /C28y)2q (9) u /C30tan/C281 x r0 /C28 y ! ; (10) with F, r0 ; and n as defined above. See also CONFORMAL PROJECTION ,CONIC PROJECTIONReferences Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 104 /C1/10, 1987. Lambert Cylindrical Equal-Area Projection A CYLINDRICAL EQUAL-AREA PROJECTION with stan- dard parallel fs/C300/C14:/ See also CYLINDRICAL EQUAL- AREA PROJECTION Lambert Series A series OF THE FORM F(x)/C13X/C12 n/C301anxn 1/C28xn(1) forjxjB1:Then F(x)/C30X/C12 n/C301anX/C12 m/C301xmn/C30X/C12 N/C301bNxN; (2) where bN/C13X n½Nan: (3) Some beautiful series of this type include X/C12 n/C301m(n)xn 1/C28xn/C30x (4) X/C12 n/C301f(n)xn 1/C28xn/C30x (1/C28x)2(5) X/C12 n/C301xn 1/C28xn/C30X/C12 n/C301d(n)xn(6) X/C12 n/C301nkxn 1/C28xn/C30X/C12 n/C301sk(n)xn(7) X/C12 n/C3014(/C281)n/C271xn 1/C28xn/C30X/C12 n/C301r(n)xn(8) X/C12 n/C301l(n)xn 1/C28xn/C30X/C12 n/C301xn2; (9) where m(n) is the M O¨BIUS FUNCTION ,f(n) is the TOTIENT FUNCTION ,d(n)/C30s0(n) is the number of divisors of n,sk(n) is the DIVISOR FUNCTION ,r(n)i s the number of representations of n in the form n /C30 A2 /C27B2 where A and B are rational integers (Hardy and Wright 1979), and l(n) is the LAMBDA FUNCTION . See also DIVISOR FUNCTION ,L AMBDA FUNCTION , MO¨ BIUS FUNCTION ,M O¨ BIUS TRANSFORM ,T OTIENT FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Number Theore- tic Functions." §24.3.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 826 /C1/27, 1972. Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 24 /C1/5, 1997. Erdos, P. "On Arithmetical Properties of Lambert Series." J. Indian Math. Soc. 12,63/C1/6, 1948. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 257 /C1/58, 1979. Lambert’s Method A ROOT -finding method also called BAILEY’S METHOD and HUTTON’S METHOD If g(x) /C30xd /C28r ; then Hg(x) /C30(d /C28 1)xd /C27 (d /C27 1)r (d /C27 1)xd /C27 (d /C28 1)rx: References Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley’s Method." Amer. Math. Monthly 102, 417 /C1/26, 1995. Lambert’s Transcendental Equation An equation proposed by Lambert (1758) and studied by Euler in 1779 (Euler 1921). xa/C28xb/C30(a/C28b)vxa/C27b: When a0b;the equation becomes lnx/C30vxb; which has the solution x/C30exp/C28W(/C28bv) b"# ; where W(x)i sL AMBERT’S W-FUNCTION . See also LAMBERT’S W-FUNCTION References Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; and Knuth, D. E. "On the Lambert WFunction." Adv. Comput. Math. 5, 329/C1/59, 1996. de Bruijn, N. G. Asymptotic Methods in Analysis. Amster- dam, Netherlands: North-Holland, pp. 27 /C1/8, 1961. Euler, L. "De Serie Lambertina Plurismique Eius Insignibus Proprietatibus." Leonhardi Euleri Opera Omnia, Ser. 1. Opera Mathematica, Bd. 6, 1921.Lambert, J. H. "Observations variae in Mathesin Puram." Acta Helvitica, physico-mathematico-anatomico-botanico- medica 3, 128/C1/68, 1758. Lambert’s W-Function The inverse of the function f(W)/C30WeW; (1) also called the omega function. The plots above show the function along the REAL AXIS (left figure) and its RIEMANN SURFACE (right figure). The principal value of the Lambert W-function is implemented in Math- ematica asProductLog [z]. Different branches of the function are available as ProductLog [k,z], where k is any integer and k/C300 corresponds to the principal value. Lambert’s W-function can be used to analytically express the value of the POWER TOWER h(x)/C30x/C160/C160/C12/C30 xxU;where xxxis an abbreviation for x(xx);as h(x)/C30/C28W(/C28lnx) lnx: (2) /W(1) is called the OMEGA CONSTANT and can be considered a sort of " GOLDEN RATIO " of exponentials since exp[/C28W(1)]/C30W(1); (3) giving ln1 W(1)"# /C30W(1): (4) Lambert’s W-Function has the series expansion W(x)/C30X/C12 n/C301(/C281)n/C281nn/C282 (n/C281)!xn/C30x/C28x2/C273 2x3/C2883x4 /C27125 24x5/C2854 5x6/C2716807 720x7/C27... ( 5 ) The L AGRANGE INVERSION THEOREM gives the equiva- lent series expansion W0(z)/C30X/C12 n/C301(/C28n)n/C281 n!zn; (6) where n!i sa FACTORIAL . However, this series oscil- lates between ever larger POSITIVE and NEGATIVE values for REAL z H0:4; and so cannot be used for practical numerical computation. An asymptotic FOR- MULA which yields reasonably accurate results for z H 3is W(z) /C30Ln z /C28ln Ln z /C27X/C12 k /C300X/C12 m/C300ckm(ln Ln z)m/C271 /C2 (Ln z) /C28k /C28m/C281 /C30L1 /C28L2 /C27L2 L1/C27L2(/C282 /C27 L2) 2L2 1/C27L26 /C28 9L2 /C27 2L2 2 ðÞ 6L3 1 /C27L2/C2812 /C27 36L2 /C28 22L2 2 /C27 3L32 ðÞ 12L4 1 /C27L260 /C28 300L2 /C27 350L2 2 /C28 125L32 /C27 12L42 ðÞ 60L5 1 /C27OL2 L1 !62 435; (7) where L 1 /C30Ln z (8) L2 /C30ln Ln z (9) (Corless et al. 1996), correcting a typographical error in de Bruijn (1961). Another expansion due to Gosper is the DOUBLE SUM W(x) /C30a /C27X/C12 n/C300Xn k /C300S1(n; k) lnx a/C(%/C(r /C28 ahik /C281 (n /C28 k /C27 1)!8 >< >:9 >= >; /C2 1 /C28lnx a/C(%/C(r a2 435n ; (10) where S1is a nonnegative STIRLING NUMBER OF THE FIRST KIND and a is a first approximation which can be used to select between branches. Lambert’s W- function is two-valued for /C281=e 5x B0 : For W(x) ]/C281; the function is denoted W0(x) or simply W(x) ; and this is called the principal branch. For W(x) 5/C281; the function is denoted W/C281(x): The DERIVATIVE of W is W ?(x) /C301 [1 /C27 W(x)] exp[W(x)] /C30W(x) x[1 /C27 W(x)](11) for x "0: For the principal branch when z /C210, ln W(z) /C30ln z /C28W(z) (12) See also ABEL POLYNOMIAL ,D IGIT-SHIFTING CON- STANTS ,L AMBERT’S TRANSCENDENTAL EQUATION , OMEGA CONSTANT ,POWER TOWERReferences --. "Time for a New Elementary Function?" FOCUS: News- letter Math. Assoc. Amer. 20, 2, Feb. 2000. Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899 /C1/09, 1999. Briggs, K. "W-ology, or, Some Exactly Solvable Growth Models." http://epidem13.plantsci.cam.ac.uk/~kbriggs/W- ology.html. Corless, R. M.; Jeffrey, D. J.; and Knuth, D. E. "A Sequence of Series for the Lambert W Function." In Proc. ISSAC ’97, Maui, Hawaii (Ed. W. W. Ku¨chlin). New York: ACM, pp. 197 /C1/04, 1997. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; and Knuth, D. E. "On the Lambert W Function." Adv. Comput. Math. 5, 329 /C1/59, 1996. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; and Jeffrey, D. J. "Lambert’s W Function in Maple." Maple Technical Newsletter 9,12/C1/2, Spring 1993. de Bruijn, N. G. Asymptotic Methods in Analysis. Amster- dam, Netherlands: North-Holland, pp. 27 /C1/8, 1961. Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2,29/C1/1, 1783. Reprinted in Euler, L. Opera Omnia I6: Commenta- tiones Algebraicae. pp. 350 /C1/69. Fritsch, F. N.; Shafer, R. E.; and Crowley, W. P. "Algorithm 443: Solution of the Transcendental Equation /wew /C30x/." Comm. ACM 16, 123 /C1/24, 1973. Jeffrey, D. J.; Hare, D. E. G.; and Corless, R. M. "Unwind- ing the Branches of the Lambert W Function." Math. Scientist 21,1/C1/, 1996. Jeffrey, D. J.; Corless, R. M.; Hare, D. E. G.; and Knuth, D. E. "Sur l’inversion de yaˆ ey au moyen des nombres de Stirling associes. " Comptes Rendus Acad. Sci. Paris 320, 1449 /C1/452, 1995. Po´lya, G. and Szego, G. Problems and Theorems in Analysis I. Berlin: Springer-Verlag, 1998. Lame ´ Curve There are two curves commonly known as the Lame ´ curve: the ELLIPSE EVOLUTE and the SUPERELLIPSE . See also ELLIPSE EVOLUTE ,SUPERELLIPSE Lame ´ Function ELLIPSOIDAL HARMONIC Lame ´’s Differential Equation The ORDINARY DIFFERENTIAL EQUATION (x2/C28b2)(x2/C28c2)d2z dx2/C27x(x2/C28b2/C27x2/C28c2)dz dx /C28[m(m/C271)x2/C28(b2/C27c2)p]z/C300: (1) (Byerly 1959, p. 255). The solution is denoted Ep m(x) and is known as a L AME´FUNCTION or an ELLIPSOIDAL HARMONIC . Whittaker and Watson (1990, pp. 554 /C1/55) give the alternative forms 4Dld dlDldL dl"# /C30[n(n/C271)l/C27C]L (2) d2 L dl2 /C271 2 a2 /C27 l /C2712 b2 /C27 l /C2712 c2"# dL dl /C30[n(n /C27 1)l /C27 C] L 4Dl(3) d2 L du2 /C30 n(n /C271)/C212(u) /C27C /C2813 n(n /C271)(a2 /C27b2 /C27c2)hi L (4) d2 L dz2 /C30n(n /C271)k2 sn2(z; k) /C27AL (5) (Whittaker and Watson 1990, pp. 554 /C1/55; Ward 1997; Zwillinger 1997, p. 124). Here, /C212 is a WEIER- STRASS ELLIPTIC FUNCTION , sn(z; k)isaJ ACOBI ELLIPTIC FUNCTION , and L( u) /C13Ym q/C301( u /C28 uq) (6) Dl /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a2 /C27 l)(b2 /C27 l)(c2 /C27 l)p (7) A /C13C /C281 3 n(n /C27 1)(a2 /C27 b2 /C27 c2) /C27 e3n(n /C27 1) e1 /C28 e3: (8) Two other equations named after Lame ´ are given by y ƒ/C27121 x /C28 a1/C271 x /C28 a2/C271 x /C28 a3"# y? /C271 4A0 /C27 A1x (x /C28 a1)(x /C28 a2)(x /C28 a3)"# y /C300 (9) and yƒ/C271 21 x /C271 x /C28 a2/C271 x /C28 a3"# y? /C271 4a2 2 /C27 a23 ðÞ q /C28 p(p /C27 1)x /C27 kx2 x(x /C28 a2)(x /C28 a3)"# y /C300 (10) (Moon and Spencer 1961, p. 157; Zwillinger 1997, p. 124). See also ELLIPSOIDAL WAVE EQUATION ,L AME´ ’S DIFFERENTIAL EQUATION TYPES ,W ANGERIN DIFFER- ENTIAL EQUATION References Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959. Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961. Ward, R. S. "The Nahn Equations, Finite-Gap Potentials and Lame ´ Functions." J. Phys. A: Math. Gen. 20, 2679 /C1/ 683, 1987.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 124, 1997. Lame ´’s Differential Equation Types Whittaker and Watson (1990, pp. 539 /C1/40) write Lame ´’s differential equation for ELLIPSOIDAL HARMO- NICS of the four types as 4 d( u)d d uf( u)d l( u) du"# /C30[2m(2m /C271)u /C27c] l( u) (1) 4d(u)d duf( u)d l( u) d u"# /C30[(2m /C271)(2m /C272)u /C27c] l(u) (2) 4d(u)d duf( u)d l( u) d u"# /C30[(2m /C272)(2m /C273)u /C27c] l(u) (3) 4 d( u)d duf( u)dl(u) du"# /C30[(2m /C273)(2m /C274)u /C27c]l( u) ; (4) where d( u) /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a2 /C27 u)(b2 /C27 u)(c2 /C27 u)p (5) l( u) /C13Ym q /C301( u /C28 uq) : (6) See also LAME´ ’S DIFFERENTIAL EQUATION References Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Lame ´’s Theorem If a is the smallest INTEGER for which there is a smaller INTEGER b such that a and b generate a EUCLIDEAN ALGORITHM remainder sequence with n steps, then a is the FIBONACCI NUMBER /Fn/C272/. Further- more, the number of steps in the EUCLIDEAN ALGO- RITHM never exceeds 5 times the number of digits in the smaller number. See also EUCLIDEAN ALGORITHM References Honsberger, R. "A Theorem of Gabriel Lame ´." Ch. 7 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 54 /C1/7, 1976. Lamina A 2-D planar closed surface L which has a mass M and a surface density s(x; y) (in units of mass per areas squared) such that M /C30gLs(x; y) dx dy: The CENTER OF MASS of a lamina is called its CENTROID . See also CENTROID (GEOMETRIC ), CROSS SECTION , SOLID Laminated Lattice A LATTICE which is built up of layers of n-D lattices in (n /C271)/-D space. The VECTORS specifying how layers are stacked are called GLUE VECTORS . See also GLUE VECTOR ,LATTICE References Conway, J. H. and Sloane, N. J. A. "Laminated Lattices." Ch. 6 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 157 /C1/80, 1993. Lamp Paradox THOMPSON LAMP PARADOX Lam’s Problem Given a 111 /C29111 BINARY MATRIX , fill 11 spaces in each row in such a way that all columns also have 11 spaces filled. Furthermore, each pair of rows must have exactly one filled space in the same column. This problem is equivalent to finding a PROJECTIVE PLANE of order 10. Using a computer program, Lam et al. (1989) showed that no such arrangement exists. Lam’s problem is equivalent to finding nine orthogo- nal L ATIN SQUARES of order 10. See also BINARY MATRIX ,LATIN SQUARE ,PROJECTIVE PLANE References --.Science. 1507/C1/508, Dec. 20, 1988. Beezer, R. "Graeco-Latin Squares." http://buzzard.ups.edu/ squares.html. Browne, M. W. "Is a Math Proof a Proof If No One Can Check It?" New York Times , Sec. 3, p. 1, col. 1, Dec. 20, 1988. Lam, C. W. H.; Thiel, L.; and Swiercz, S. "The Nonexistence of Finite Projective Planes of Order 10." Canad. J. Math. 41, 1117 /C1/123, 1989. Petersen, I. "Search Yields Math Proof No One Can Check." Science News 134, 406, Dec. 24 & 31, 1988.Lancret Equation ds2 N/C30ds2T/C27ds2B; where Nis the NORMAL VECTOR ,Tis the TANGENT , andBis the BINORMAL VECTOR . Lancret’s Theorem ANECESSARY and SUFFICIENT condition for a curve to be a HELIX is that the ratio of CURVATURE toTORSION be constant. Lanczos Algorithm An algorithm for computing the eigenvalues and eigenvectors for large symmetric sparse matrices. References Chung, F. R. K. Spectral Graph Theory. Providence, RI: Amer. Math. Soc., 1997. Demmel, J. "CS 267: Notes for Lecture 23, April 9, 1999. Graph Partitioning, Part 2." http://www.cs.berkeley.edu/ ~demmel/cs267/lecture20/lecture20.html. Lanczos Approximation An approximation for the GAMMA FUNCTION G(z/C271) with z/C210 is given by G(z/C271)/C30ffiffiffiffiffiffi 2pp /C2z/C27s/C271 2/C(%/C(rz/C271=2 e/C28(z/C27s/C271=2)X/C12 k/C300gkHk(z); (1) where sis an arbitrary constant such that R[z/C27s/C27 1=2]>0; gk/C30esok(/C281)k ffiffiffiffiffiffi 2ppXk r/C300(/C281)rk r/C(*/C(+ (k)re r/C27s/C271 2 !r/C271=2 (2) where ( k)ris a P OCHHAMMER SYMBOL and ok/C301 for k/C300 2 otherwise ;/C)% (3) and Hk(z)/C301 (z/C271)k(z/C271)/C28k(4) /C30(/C281)k(/C28z)k (z/C271)k; (5) with H0(z)/C301 (Lanczos 1964; Luke 1969, p. 30). gk satisfies X/C12 k/C300gk/C301; (6) and if zis a POSITIVE INTEGER , then gksatisfies the identity Xn k/C300( /C281)k(/C28n)k (n /C27 1)kgk /C30en/C27 s/C271 =2n!ffiffiffiffiffiffi 2pp (n /C27 s /C27 1=2)n/C271 =2 (7) (Luke 1969, p. 30). A similar result is given by ln[ G(z)] /C30 z /C281 2/C(%/C(r ln z /C28z /C2712ln(2p) /C2712c1 z /C27 1 /C27c2 2(z /C27 1)(z /C27 2) /C27..."# (8) where cn /C30g1 0(x)n(2x /C281) dx; (9) with (x)naP OCHHAMMER SYMBOL . The first few values of cn are c1 /C301 6 c2 /C301 3 c3 /C3059 60 c4 /C3058 15 c5 /C30533 28 (Sloane’s A054379 and A054380; Whittaker and Watson 1990, p. 253). Note that Whittaker and Watson incorrectly give c4 as 227/60. Yet another related result gives ln[ G(z)] /C30 z /C281 2/C(%/C(r ln z /C28z /C2712ln(2p) /C27121 2 /C215 3X/C12 r/C3011 (z /C27 r)2 /C272 3 /C215 4X/C12 r/C3011 (z /C27 r)3" /C273 4 /C215 5X/C12 r/C3011 (z /C27 r)4 /C27.../C)( (10) (Whittaker wand Watson 1990, p. 261). See also GAMMA FUNCTION References Lanczos, C. J. Soc. Indust. Appl. Math. Ser. B: Numer. Anal. 1,86/C1/6, 1964. Luke, Y. L. "An Expansion for G(z /C271):/" §2.10.3 in The Special Functions and their Approximations, Vol. 1. New York: Academic Press, pp. 29 /C1/1, 1969. Sloane, N. J. A. Sequences A054379 and A054379 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990.Lanczos Sigma Factor Writing a FOURIER SERIES as f( u) /C301 2 a0 /C27Xm n/C301sin cnp 2m ! [an cos(nu) /C27bn sin(nu)]; where m is the last term and the sinc x terms are the Lanczos s factor, removes the GIBBS PHENOMENON (Acton 1990). See also FOURIER SERIES ,GIBBS PHENOMENON ,SINC FUNCTION References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 228, 1990. Landau Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let F be the set of COMPLEX analytic functions f defined on an open region containing the closure of the unit disk D /C30fz : ½z½B1 g satisfying f(0) /C300 and df =dz(0) /C301: For each f in F, let (f) be the SUPREMUM of all numbers r such that f(D) contains a disk of radius r. Then L /C13inf fl(f):f /C23 F g: This constant is called the Landau constant, or the BLOCH- LANDAU CONSTANT . Robinson (1938, unpub- lished) and Rademacher (1943) derived the bounds 12 BL 5G1 3/C(%/C(r G56/C(%/C(r G1 6/C(%/C(r/C300 :5432588 ... ; where G(z) is the GAMMA FUNCTION , and conjectured that the second inequality is actually an equality, L /C30G1 3/C(%/C(r G56/C(%/C(r G1 6/C(%/C(r/C300:5432588 . . . : See also BLOCH CONSTANT References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/bloch/bloch.html. Rademacher, H. "On the Bloch-Landau Constant." Amer. J. Math. 65, 387/C1/90, 1943. Landau-Kolmogorov Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. Let½½f½½be the SUPREMUM of½f(x)½;a real-valued function fdefined on (0 ;/C12):Iffis twice differenti- able and both fand fƒare bounded, Landau (1913) showed that ½½f?½½52½½f½½1=2½½fƒ½½1=2; (1) where the constant 2 is the best possible. Schoenberg (1973) extended the result to the nth derivative of f defined on (0 ;/C12) if both fandf(n)are bounded, ½½f(k)½½5C(n;k)½½f½½1/C28k=n½½f(n)½½k=n: (2) An explicit FORMULA forC(n;k) is not known, but particular cases are C(3;1)/C30243 8 !1=3 (3) C(3;2)/C30241=3(4) C(4;1)/C304:288 . . . (5) C(4;2)/C305:750 . . . (6) C(4;3)/C303:708 . . . : (7) Let½½f½½be the SUPREMUM of½f(x)½;a real-valued function fdefined on ( /C28/C12;/C12):Iffis twice differenti- able and both fandfƒare bounded, Hadamard (1914) showed that ½½f?½½5ffiffiffi 2p ½½f½½1=2½½fƒ½½1=2; (8) where the constantffiffiffi 2p is the best possible. Kolmo- gorov (1962) determined the best constants C(n;k) for ½½f(k)½½5C(n;k)½½f½½1/C28k=n½½f(n)½½k=n(9) in terms of the F AVARD CONSTANTS an/C304 pX/C12 j/C300(/C281)j 2j/C271"#n/C271 (10) by C(n;k)/C30an/C28ka/C281/C27k=n n /C215 (11) Special cases derived by Shilov (1937) are C(3;1)/C309 8 !1=3 (12) C(3;2)/C3031=3(13) C(4;1)/C30512 375 !1=4 (14) C(4;2)/C30ffiffiffi 6 5s (15) C(4;3)/C3024 5 !1=4 (16)C(5;1)/C3019531251572864 ! 1=5 (17) C(5;2)/C30125 72 !1=5 : (18) For a real-valued function fdefined on ( /C28/C12;/C12); define ½½f½½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g/C12 /C28/C12[f(x)]2dxs : (19) Iffisndifferentiable and both fandf(n)are bounded, Hardy et al. (1934) showed that ½½f(k)½½5½½f½½1/C28k=n½½f(n)½½k=n; (20) where the constant 1 is the best possible for all nand 0BkBn:/ For a real-valued function fdefined on (0 ;/C12);define ½½f½½/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g/C12 0[f(x)]2dxs : (21) Iffis twice differentiable and both fand fƒare bounded, Hardy et al. (1934) showed that ½½f?½½5ffiffiffi 2p ½½f½½1=2½½f(n)½½1=2; (22) where the constantffiffiffi 2p is the best possible. This inequality was extended by Ljubic (1964) and Kupcov (1975) to ½½f(k)½½5C(n;k)½½f½½1/C28k=n½½f(n)½½k=n(23) where C(n;k) are given in terms of zeros of POLY- NOMIALS . Special cases are C(3;1)/C30C(3;2)/C3031=2[2(21=2/C281)]/C281=3 /C301:84420 . . . (24) C(4;1)/C30C(4;3)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=4/C273/C283=4 as /C302:27432 . . . (25) C(4;2)/C30ffiffiffi 2 bs /C302:97963 . . . (26) C(4;3)/C3024 5 !1=4 (27) C(5;1)/C30C(5;4)/C302:70247 . . . (28) C(5;2)/C30C(5;3)/C304:37800 . . . ; (29) where ais the least POSITIVE ROOT of x8/C286x4/C288x2/C271/C300 (30) andbis the least POSITIVE ROOT of x4/C282x2/C284x/C271/C300 (31) (Franco et al. 1985, Neta 1980). The constants C(n;1) are given by C(n;1)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n/C281)1=n/C27(n/C271)/C281/C271=n cvuut; (32) where cis the least POSITIVE ROOT of gc 0g/C12 0dx dy (x2n/C28yx2/C271)ffiffiffiyp/C30p2 2n: (33) An explicit FORMULA of this type is not known for k/C211. The cases p/C301, 2,/C12are the only ones for which the best constants have exact expressions (Kwong and Zettl 1992, Franco et al. 1983). References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lk/lk.html. Franco, Z. M.; Kaper, H. G.; Kwong, M. N.; and Zettl, A. "Bounds for the Best Constants in Landau’s Inequality on the Line." Proc. Roy. Soc. Edinburgh 95A, 257/C1/62, 1983. Franco, Z. M.; Kaper, H. G.; Kwong, M. N.; and Zettl, A. "Best Constants in Norm Inequalities for Derivatives on a Half Line." Proc. Roy. Soc. Edinburgh 100A ,6 7/C1/4, 1985. Hardy, G. H.; Littlewood, J. E.; and Po ´lya, G. Inequalities. Cambridge, England: Cambridge University Press, 1934. Kolmogorov, A. "On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function onan Infinite Integral." Amer. Math. Soc. Translations, Ser. 12, 233/C1 /43, 1962. Kupcov, N. P. "Kolmogorov Estimates for Derivatives in / L2(0;/C12)/."Proc. Steklov Inst. Math. 138, 101/C1/25, 1975. Kwong, M. K. and Zettl, A. Norm Inequalities for Derivatives and Differences. New York: Springer-Verlag, 1992. Landau, E. "Einige Ungleichungen fu ¨r zweimal different- zierbare Funktionen." Proc. London Math. Soc. Ser. 2 13, 43/C1/9, 1913. Landau, E. "Die Ungleichungen fu ¨r zweimal differentzier- bare Funktionen." Danske Vid. Selsk. Math. Fys. Medd. 6, 1/C1/9, 1925. Ljubic, J. I. "On Inequalities Between the Powers of a Linear Operator." Amer. Math. Soc. Trans. Ser. 2 40,3 9/C1/4, 1964. Neta, B. "On Determinations of Best Possible Constants in Integral Inequalities Involving Derivatives." Math. Com- put. 35, 1191 /C1/193, 1980. Schoenberg, I. J. "The Elementary Case of Landau’s Pro- blem of Inequalities Between Derivatives." Amer. Math. Monthly 80, 121/C1/58, 1973. Landau-Lifshitz Equation The system of PARTIAL DIFFERENTIAL EQUATIONS Ut/C30U /C215Uxx/C27U /C215AU: References Fuchssteiner, B. "On the Hierarchy of the Landau-Lifshitz Equation." Physica D 13, 387/C1/94, 1984. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997.Landau-Ramanujan Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. LetS(x) denote the number of POSITIVE INTEGERS not exceeding xwhich can be expressed as a sum of two squares, then lim x0/C12ffiffiffiffiffiffiffiffiffi lnxp xS(x)/C30K; (1) as proved by Landau (1908). Ramanujan indepen- dently stated the theorem in the slightly differentform that the number of numbers between Aand x which are either squares of sums of two squares is S(x)/C30Kgx Adtffiffiffiffiffiffiffiffi lntp/C27u(x); (2) where K:0:764 and u(x) is very small compared with the previous integral (Hardy 1999, p. 8; Moree and Cazaran 1999). However, the convergence to the constant Kis very slow. The exact value for K/C300:764223653 . . . (3) (sometimes denoted l) is given by K/C301ffiffiffi 2pY pprime /C133(mod 4)1/C281 p2 !/C281=2 (4) (Landau 1908; Le Lionnais 1983, p. 31; Berndt 1994; Hardy 1999; Moree and Cazaran 1999). An equivalentformula is given by K/C30p 4Y pprime /C131(mod 4)1/C281 p2 !/C281=2 : (5) Flajolet and Vardi (1996) give a beautiful FORMULA with fast convergence K/C301ffiffiffi 2pY/C12 n/C3011/C281 22n ! z(2n) b(2n)"#1=(2n/C271) ; (6) where b(s) /C131 4sz s ;1 4/C(%/C(r /C28& s ;34/C(%/C(rhi (7) is the DIRICHLET BETA FUNCTION , and z(z ; a) is the HURWITZ ZETA FUNCTION . Landau proved the even stronger fact lim x0/C12(ln x)3=2 KxS(x)Kxffiffiffiffiffiffiffiffiffi ln xp"# /C30C ; (8) where C /C131 21 /C28lnpeg L !"# /C2814d dslnY p prime p /C304k /C2731 p/C282s0 BBBBBB@1 CCCCCCA2 66666643 7777775 s/C301 /C300 :581948659 ... : (9) Here, L /C305:2441151086 ... (10) is the ARC LENGTH of a LEMNISCATE with a /C301 (the LEMNISCATE CONSTANT to within a factor of 2 or 4), and g is the EULER- MASCHERONI CONSTANT . Landau’s method of proof can be extended to show that B(x) /C2Kxffiffiffiffiffiffiffiffiffi ln xp (11) has an ASYMPTOTIC SERIES B(x) /C30Kxffiffiffiffiffiffiffiffiffiln xp /C2 1 /C27C1 ln x /C27C2 (ln x)2 /C27.../C27Cn (ln x)n /C27O1 (ln x)n/C271 ! "# ; (12) where n can be arbitrarily large and the Cjare constants (Moree and Cazaran 1999). See also SQUARE NUMBER References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 60 /C1/6, 1994. Berndt, B. C. and Rankin, R. A. Ch. 2 in Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc, 1995. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lr/lr.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 9 /C1/0, 55, and 60 /C1/4, 1999. Landau, E. "U¨ ber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihreradditiven Zusammensetzung erforderlichen Quadrate." Arch. Math. Phys. 13, 305 /C1/12, 1908. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, Bd. II, 2nd ed. New York: Chelsea, pp. 641 /C1/ 69, 1953. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Moree, P. and Cazaran, J. "On a Claim of Ramanujan in His First Letter to Hardy." Expos. Math. 17, 289 /C1/12, 1999. Selberg, A. Collected Papers, Vol. II. Berlin: Springer- Verlag, pp. 183 /C1/85, 1991. Shanks, D. "The Second-Order Term in the Asymptotic Expansion of B(x) :/" Math. Comput. 18,75/C1/6, 1964. Shanks, D. "Non-Hypotenuse Numbers." Fibonacci Quart. 13, 319 /C1/21, 1975. Shanks, D. and Schmid, L. P. "Variations on a Theorem of Landau. I." Math. Comput. 20, 551 /C1/69, 1966. Shiu, P. "Counting Sums of Two Squares: The Meissel- Lehmer Method." Math. Comput. 47, 351 /C1/60, 1986. Stanley, G. K. "Two Assertions Made by Ramanujan." J. London Math. Soc. 3, 232 /C1/37, 1928. Stanley, G. K. Corrigendum to "Two Assertions Made by Ramanujan." J. London Math. Soc. 4, 32, 1929. Wolfram Research, Inc. "Computing the Landau-Ramanujan Constant." http://library.wolfram.com/demos/v4/LandauR- amanujan.nb. Landau’s Problems The four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathemati- cians in Cambridge. The four were 1. The GOLDBACH CONJECTURE , 2. TWIN PRIME CONJECTURE , 3. The conjecture that there exists a PRIME p such that n2 Bp B(n /C271)2for every n (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397 /C1/ 98), and 4. The conjecture that there are infinitely many PRIMES p OF THE FORM p /C30n2 /C271 (Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206 /C1/08). The first few PRIMES p which are OF THE FORM p /C30 n2 /C271 are given by 2, 5, 17, 37, 101, 197, 257, 401, ... (Sloane’s A002496). These correspond to n /C301, 2, 4, 6, 10, 14, 16, 20, ... (Sloane’s A005574; Hardy and Wright 1979, p. 19). Although it is not know if there always exists a PRIME psuch that n2BpB(n/C271)2;Chen (1975) has shown that a number Pwhich is either a PRIME orSEMI- PRIME does always satisfy this inequality. Moreover, there is always a prime between n/C28nuandnwhere u/C3023=42 (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest PRIMES between n2and (n/C271)2forn/C301, 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83, ... (Sloane’s A007491). See also GOLDBACH CONJECTURE ,G OOD PRIME , PRIME NUMBER ,TWIN PRIME CONJECTURE References Chen, J. R. "On the Distribution of Almost Primes in an Interval." Sci. Sinica 18, 611/C1/27, 1975. Hardy, G. H. and Wright, W. M. "Unsolved Problems Con- cerning Primes." §2.8 and Appendix §3inAn Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 19 and 415 /C1/16, 1979. Iwaniec, H. and Pintz, J. "Primes in Short Intervals." Monatsh. f. Math. 98, 115 /C1/43, 1984. Ogilvy, C. S. Tomorrow’s Math: Unsolved Problems for the Amateur, 2nd ed. Oxford, England: Oxford University Press, p. 116, 1972. Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 132 /C1/34 and 206 /C1/08, 1996. Sloane, N. J. A. Sequences A002496/M1506, A005574/ M1010, and A007491/Min "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Landau Symbol Let f(z) be a function "0 in an interval containing z /C300. Let g(z) be another function also defined in this interval such that g(z) =f(z) 0 0as z 0 0: Then g(z)is said to be o(f(z)):/ See also ASYMPTOTIC NOTATION Landen’s Formula q3(z; t) q4(z; t) q4(2z ; 2t)/C30q3(0; t) q4(0; t) q4(0; 2t)/C30q2(z; t)q4(z; t) q1(2z ; 2t); where qiare JACOBI THETA FUNCTIONS . This trans- formation was used by Gauss to show that ELLIPTIC INTEGRALS could be computed using the ARITHMETIC- GEOMETRIC MEAN . See also JACOBI THETA FUNCTIONS Landen’s Identity The DILOGARITHM identity Li2(/C28x) /C30/C28Li2x 1 /C27 x ! /C281 2[ln(1 /C27x)]2 : See also DILOGARITHM References Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431 /C1/48, 1997. Landen, J. Mathematical Memoirs Respecting a Variety of Subjects, with an Appendix Containing Tables of Theo- rems, Vol. 1. London: printed for the author, p. 112, 1780 /C1/789.Landen’s Transformation If x sin a /C30sin(2b /C28 a); then (1 /C27x)g a 0dfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2 sin2 fq /C302g b 0dfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C284x (1/C27x)2sin2s f: See also ELLIPTIC INTEGRAL OF THE FIRST KIND, GAUSS’S TRANSFORMATION References Abramowitz, M. and Stegun, C. A. (Eds.). "Ascending Landen Transformation" and "Landen’s Transformation." §16.14 and 17.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9thprinting. New York: Dover, pp. 573 /C1 /74 and 597 /C1/98, 1972. Lane-Emden Differential Equation A second-order ORDINARY DIFFERENTIAL EQUATION arising in the study of stellar interiors, also called the polytropic differential equations. It is given by 1 j2d djj2du dj ! /C27un/C300 (1) 1 j22jdu dj/C27j2d2u dj2 ! /C27un/C30d2u dj2/C272du jdj/C27un/C300 (2) (Zwillinger 1997, pp. 124 and 126). It has the BOUND- ARY CONDITIONS u(0)/C301 (3) du dj"# j/C300/C300: (4) Solutions u(j) for n/C300, 1, 2, 3, and 4 are shown above. The cases n/C300, 1, and 5 can be solved analytically (Chandrasekhar 1967, p. 91); the othersmust be obtained numerically.Forn/C300((g/C30/C12));the L ANE-EMDEN DIFFERENTIAL EQUATION is 1 j2d djj2du dj ! /C271/C300 (5) (Chandrasekhar 1967, pp. 91 /C1/2). Directly solving gives d djj2du dj ! /C271/C30/C28j2(6) gdj2du dj2 ! /C30/C28gj2dj (7) j2du dj/C30c1/C281 3j3(8) du dj/C30c1/C2813j3 j2(9) u(j)/C30gdu/C30gc1/C281 3j3 j2dj (10) u(j)/C30u0/C28c1j/C281/C2816j2: (11) The BOUNDARY CONDITION u(0)/C301 then gives u0/C301 andc1/C300;so u1(j)/C301/C2816j2; (12) andu1(j)i s PARABOLIC . Forn/C301/(g/C302);the differential equation becomes 1 j2d djj2du dj ! /C27u/C300 (13) d djj2du dj ! /C27uj2/C300; (14) which is the SPHERICAL BESSEL DIFFERENTIAL EQUA- TION d drr2dR dr ! /C27[k2r2/C28n(n/C271)]R/C300 (15) with k/C301 and n/C300, so the solution is u(j)/C30Aj0(j)/C27Bn0(j): (16) Applying the BOUNDARY CONDITION u(0)/C301 gives u2(j)/C30j0(j)/C30sinj j; (17) where j0(x)i sa SPHERICAL BESSEL FUNCTION OF THE FIRST KIND (Chandrasekhar 1967, pp. 92). Forn/C305, make Emden’s transformation u/C30Axvz (18)v/C302 n/C281; (19) which reduces the Lane-Emden equation to d2z dt2/C27(2v/C281)dz dt/C27v(v/C281)z/C27An/C281zn/C300 (20) (Chandrasekhar 1967, p. 90). After further manipu- lation (not reproduced here), the equation becomes d2z dt2/C301 4z(1/C28z4) (21) and then, finally, u5(j)/C301/C2713j2/C(%/C(r/C281=2 : (22) References Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, pp. 84 /C1/82, 1967. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 908, 1980. Seshadi, R. and Na, T. Y. Group Invariance in Engineering Boundary Value Problems. New York: Springer-Verlag, p. 193, 1985. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 124 and 126, 1997. Langford’s Problem Arrange copies of the ndigits 1, ..., nsuch that there is one digit between the 1s, two digits between the 2s, etc. For example, the unique (modulo reversal) n/C303 solution is 231213, and the unique (again moduloreversal) n/C304 solution is 23421314. Solutions to Langford’s problem exist only if n/C130;3(mod 4) ;so the next solutions occur for n/C307. There are 26 of these, as exhibited by Lloyd (1971). In lexicographi- cally smallest order (i.e., small digits come first), the first few Langford sequences are 231213, 23421314,14156742352637, 14167345236275, 15146735423627, ... (Sloane’s A050998). The number of solutions for n/C303, 4, 5, ... (modulo reversal of the digits) are 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, ... (Sloane’s A014552). No formula is known for the number of solutions of a given order nf0;3 (mod 4) :: / References Davies, R. O. "On Langford’s Problem. II." Math. Gaz. 43, 253/C1/55, 1959. Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 70 and 77 /C1/8, 1978. Langford, C. D. "Problem." Math. Gaz. 42, 228, 1958. Lloyd, P. R. Correspondence to the Editor. Math. Gaz. 55, 73, 1971. Lorimer, P. "A Method of Constructing Skolem and Langford Sequences." Southeast Asian Bull. Math. 6, 115/C1/19, 1982. Miller, J. "Langford’s Problem." http://www.lclark.edu/ ~miller/langford.html. Miller, J. "Langford’s Problem Bibliography." http:// www.lclark.edu/~miller/langford/langford-biblio.html. Simpson, J. E. "Langford Sequences: Perfect and Hooked." Disc /C21 Math. 44,97/C1/04, 1983. Priday, C. J. "On Langford’s Problem. I." Math. Gaz. 43, 250 /C1/53, 1959. Sloane, N. J. A. Sequences A014552 and A050998 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Langlands Conjectures LANGLANDS PROGRAM Langlands Program A grand unified theory of mathematics which in- cludes the search for a generalization of ARTIN RECIPROCITY (known as LANGLANDS RECIPROCITY )to non-Abelian Galois extensions of NUMBER FIELDS .Ina January 1967 letter to Andre ´ Weil, Langlands pro- posed that the mathematics of algebra (Galois repre- sentations) and analysis (AUTOMORPHIC FORMS ) are intimately related, and that congruences over FINITE FIELDS are related to infinite-dimensional representa- tion theory. In particular, Langlands conjectured that the transformations behind general reciprocity laws could be represented by means of MATRICES (Mack- enzie 2000). In 1998, three mathematicians proved Langlands’ conjectures for LOCAL FIELDS , and in a November 1999 lecture at the Institute for Advanced Study at Princeton University, L. Lafforgue presented a proof of the conjectures for FUNCTION FIELDS . This leaves only the case of NUMBER FIELDS as unresolved (Mackenzie 2000). Langlands was a co-recipient of the 1996 Wolf Prize for the web of conjectures underlying this program. See also ARTIN RECIPROCITY ,AUTOMORPHIC FORM, ENDOSCOPY ,LANGLANDS RECIPROCITY ,RECIPROCITY THEOREM ,TANIYAMA- SHIMURA CONJECTURE References American Mathematical Society. "Langlands and Wiles Share Wolf Prize." Not. Amer. Math. Soc. 43, 221 /C1/22, 1996. Knapp, A. W. "Group Representations and Harmonic Ana- lysis from Euler to Langlands." Not. Amer. Math. Soc. 43, 410 /C1/15, 1996. Mackenzie, D. "Fermat’s Last Theorem’s Cousin." Science 287, 792 /C1/93, 2000. Langlands Reciprocity The conjecture that the ARTIN L-FUNCTION of any n-D GALOIS GROUP representation is an L-FUNCTION obtained from the GENERAL LINEAR GROUP GL1(A) :/ See also ARTIN L-FUNCTIONReferences Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Langton’s Ant A CELLULAR AUTOMATON for which the COHEN- KUNG THEOREM guarantees that the ant’s trajectory is unbounded. See also CELLULAR AUTOMATON ,COHEN- KUNG THEO- REM References Stewart, I. "The Ultimate in Anty-Particles." Sci. Amer. 271, 104/C1/07, 1994. Laplace-Beltrami Operator A self-adjoint elliptic differential operator defined somewhat technically as D/C30dd/C27dd; where dis the EXTERIOR DERIVATIVE anddanddare adjoint to each other with respect to the INNER PRODUCT . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 628, 1980. Laplace Distribution Also called the DOUBLE EXPONENTIAL DISTRIBUTION .I t is the distribution of differences between two inde-pendent variates with identical EXPONENTIAL DISTRI- BUTIONS (Abramowitz and Stegun 1972, p. 930). P(x)/C301 2be/C28½x/C28m½=b(1) D(x)/C301 2[1/C27sgn(x/C28m)(1/C28e/C28½x/C28m½=b)]: (2) The MOMENTS about the MEAN mnare related to the MOMENTS about 0 by mn/C30Xn j/C300n j/C(%/C(r (/C281)n/C28jm?jmn/C28j; (3) wheren k/CP/C( is a BINOMIAL COEFFICIENT ,so mn /C30Xn j/C300Xj=2bc k/C300(/C281)n/C28j n j/C(%/C(r j 2k/CP/C( b2k mn /C282k G(2k /C271) /C30n!bn 0for n even for n odd;/C)% (4) where xbcis the FLOOR FUNCTION and G(2k /C271) is the GAMMA FUNCTION . The MOMENTS can also be com- puted using the CHARACTERISTIC FUNCTION , f(t) /C13g/C12 /C28/C12eitxP(x)dx /C301 2b g/C12 /C28/C12eitxe/C28½x/C28 m½=b dx: (5) Using the FOURIER TRANSFORM OF THE EXPONENTIAL FUNCTION F[e /C282 pk0 ½x ½] /C301 pk0 k2 /C27 k2 0(6) gives f(t) /C30eimt 2b2 b t2 /C271 b/C(%/C(r2 /C30eimt 1 /C27 b2t2 (7) (Abramowitz and Stegun 1972, p. 930). The MOMENTS are therefore mn /C30(/C28i)n f(0) /C30(/C28i)ndn f dtn"# t/C300: (8) The MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C30 m (9) s2 /C302b2 (10) g1 /C300 (11) g2 /C303: (12) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 104, 1984. Laplace-Everett Formula EVERETT’S FORMULA Laplace Limit The value e /C300:6627434193... (Sloane’s A033259) for which Laplace’s formula for solving KEPLER’S EQUA- TION begins diverging. The constant is defined as the value e at which the functionf(x) /C30x expffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 x2p/CP/C( 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 x2p equals f(l) /C301 : The CONTINUED FRACTION of e is given by [0, 1, 1, 1, 27, 1, 1, 1, 8, 2, 154, ...] (Sloane’s A033260). The positions of the first occurrences of n in the CONTINUED FRACTION of e are 2, 10, 35, 13, 15, 32, 101, 9, ... (Sloane’s A033261). The incrementally largest terms in the CONTINUED FRACTION are 1, 27, 154, 1601, 2135, ... (Sloane’s A033262), which occur atpositions 2, 5, 11, 19, 1801, ... (Sloane’s A033263). See also E CCENTRIC ANOMALY ,KEPLER’S EQUATION References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lpc/lpc.html. Plouffe, S. "Laplace Limit Constant." http://www.lacim.u- qam.ca/piDATA/laplace.txt. Sloane, N. J. A. Sequences A033259, A033260, A033261, A033262, and A033263 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Laplace-Mehler Integral pn(cosu)/C301 pg2p 0(cosu/C27isinucosf)ndf /C30ffiffiffi 2p pgu 0cos n/C271 2/C(%/C(r fhi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosf/C28cosup df /C30ffiffiffi 2p pgp usin n/C271 2/C(%/C(r fhi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosu/C28cosfp df: References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1463, 1980. Laplace’s Equation The scalar form of Laplace’s equation is the PARTIAL DIFFERENTIAL EQUATION 92c/C300: (1) Note that the operator 92is commonly written as Dby mathematicians (Krantz 1999, p. 16). Laplace’s equa- tion is a special case of the H ELMHOLTZ DIFFERENTIAL EQUATION 92c/C27k2c/C300 (2) with k/C300, or P OISSON’S EQUATION 92c/C30/C284pr (3) with r/C300:The vector Laplace’s equation is given by 92F/C300: (4) A FUNCTION c which satisfies Laplace’s equation is said to be HARMONIC . A solution to Laplace’s equation has the property that the average value over a spherical surface is equal to the value at the center of the SPHERE (GAUSS’S HARMONIC FUNCTION THEO- REM). Solutions have no local maxima or minima. Because Laplace’s equation is linear, the superposi- tion of any two solutions is also a solution. A solution to Laplace’s equation is uniquely deter- mined if (1) the value of the function is specified on all boundaries (DIRICHLET BOUNDARY CONDITIONS ) or (2) the normal derivative of the function is specified on all boundaries (NEUMANN BOUNDARY CONDITIONS ). Coordinate SystemVariables Solution Func- tions CARTESIAN /X(x)Y(y)Z(z)/ EXPONENTIAL FUNCTIONS , CIR- CULAR FUNC- TIONS , HYPER- BOLIC FUNCTIONS CIRCULAR CY- LINDRICAL/R(r)U( u)Z(z)/ BESSEL FUNC- TIONS , EXPONEN- TIAL FUNCTIONS , CIRCULAR FUNC- TIONS CONICAL ELLIPSOIDAL HARMONICS , POWER ELLIPSOIDAL /L(l)M( m)N( n)/ ELLIPSOIDAL HARMONICS ELLIPTIC CY- LINDRICAL/U(u)V(v)Z(z)/ MATHIEU FUNC- TION , CIRCULAR FUNCTIONS OBLATE SPHER- OIDAL/L(l)M( m)N( n)/ LEGENDRE POLY- NOMIAL , CIRCU- LAR FUNCTIONS PARABOLIC BESSEL FUNC- TIONS , CIRCULAR FUNCTIONS PARABOLIC CY- LINDRICALPARABOLIC CY- LINDER FUNC- TIONS ,BESSEL FUNCTIONS , CIR- CULAR FUNC- TIONS PARABOLOIDAL /U(u)V(v) U( u)/ CIRCULAR FUNC- TIONS PROLATE SPHEROIDAL/L(l)M( m)N( n)/ LEGENDRE POLY- NOMIAL , CIRCU- LAR FUNCTIONSSPHERICAL /R(r)U( u) F(f)/ LEGENDRE POLY- NOMIAL , POWER , CIRCULAR FUNC- TIONS Laplace’s equation can be solved by SEPARATION OF VARIABLES in all 11 coordinate systems that the HELMHOLTZ DIFFERENTIAL EQUATION can. The form these solutions take is summarized in the table above. In addition to these 11 coordinate systems, separation can be achieved in two additional coordinate systems by introducing a multiplicative factor. In these coordinate systems, the separated form is c /C30X1(u1)X2(u2)X3(u3) R(u1 ; u2 ; u3); (5) and setting h1h2h3 h2 i/C30gi(ui /C271 ; ui /C272)fi(ui)R2 ; (6) where hiare SCALE FACTORS , gives the Laplace’s equation X3 i /C3011 h2i Xi1 fid duifidXi dui !"# /C30X3 i/C3011 h2i R1 fi@ @uifi@R @ui !"# : (7) If the right side is equal to /C28k2 1 =F(u1 ; u2 ; u3) ; where k1 is a constant and F is any function, and if h1h2h3 /C30Sf1f2f3R2F ; (8) where S is the STA¨ CKEL DETERMINANT , then the equation can be solved using the methods of the HELMHOLTZ DIFFERENTIAL EQUATION . The two sys- tems where this is the case are BISPHERICAL and TOROIDAL , bringing the total number of separable systems for Laplace’s equation to 13 (Morse and Feshbach 1953, pp. 665 /C1/66). In 2-D BIPOLAR COORDINATES , Laplace’s equation is separable, although the H ELMHOLTZ DIFFERENTIAL EQUATION is not. Zwillinger (1997, p. 128) calls (a0x/C27b0)y(n)/C27(a1x/C27b1)y(n/C281)/C27.../C27(anx/C27bn)y /C300 (9) the Laplace equations. See also BOUNDARY CONDITIONS ,H ARMONIC EQUA- TION ,H ARMONIC FUNCTION ,H ELMHOLTZ DIFFEREN- TIAL EQUATION ,PARTIAL DIFFERENTIAL EQUATION , POISSON’S EQUATION ,S EPARATION OF VARIABLES , STA¨ CKEL DETERMINANT References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959. Eisenhart, L. P. "Separable Systems in Euclidean 3-Space." Physical Review 45, 427 /C1/28, 1934. Eisenhart, L. P. "Separable Systems of Sta¨ckel." Ann. Math. 35, 284 /C1/05, 1934. Eisenhart, L. P. "Potentials for Which Schroedinger Equa- tions Are Separable." Phys. Rev. 74,87/C1/9, 1948. Krantz, S. G. "The Laplace Equation." §7.1.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 16 and 89, 1999. Moon, P. and Spencer, D. E. "Recent Investigations of the Separation of Laplace’s Equation." Proc. Amer. Math. Soc. 4, 302, 1953. Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1 /C1/8, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 125 /C1/26 and 271, 1953. Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Brookline, MA: Math. Sci. Press, pp. 306 /C1/15, 1950. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997. Laplace’s Equation * /Bipolar Coordinates In 2-D BIPOLAR COORDINATES ,LAPLACE’S EQUATION is (cosh v /C28 cos u)2 a2@F2 @u2 /C27@F2 @v2 ! /C300; (1) which simplifies to @F2 @u2 /C27@F2 @v2 /C300 ; (2) so LAPLACE’S EQUATION is separable, although the HELMHOLTZ DIFFERENTIAL EQUATION is not. See also BIPOLAR COORDINATES ,LAPLACE’S EQUATION Laplace’s Equation * /Bispherical Coordinates In BISPHERICAL COORDINATES ,LAPLACE’S EQUATION becomes92f /C30sin u (cosh v /C28 cos u)3@ @usin u cosh v /C28 cos u@f @u !" /C27@ @vsin u cosh v /C28 cos u@f @v ! /C27@ @ f /C2csc u cosh v /C28 cos u@f @ f !/C)( : (1) Attempt SEPARATION OF VARIABLES by plugging in the trial solution fu; v; f ðÞ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh v /C28cos up U(u)V(v)C( c) ; (2) then divide the result by csc2 u(cosh v /C28cos u)5 =2 U(u)V(v) F(f) to obtain /C281 4sinh2 u /C27cos u sin uU ?(u) U(u)/C27sin2 uU ƒ(u) U(u) /C27sin2 uV ƒ(v) V(v)/C27Fƒ(f) F( f)/C300: (3) The function F( f) then separates with Fƒ( f) F( f)/C30/C28m2 ; (4) giving solution C(c) /C30sin cos (mf) /C30X/C12 k/C301[Ak sin(mc) /C27Bk cos(mc)] : (5) Plugging C( c) back in and dividing by sin2 u gives cot uU ?(u) U(u)/C27U ƒ(u) U(u)/C28m2 sin2 u /C281 4 /C27V ƒ(v) V(v)/C300 : (6) The function V(v) then separates with V ƒ(v) V(v)/C30/C28n2 ; (7) giving solution V(v) /C30sin cos (nv) /C30X/C12 k /C301[Ck sin(nv) /C27Dk cos(nv)]: (8) Plugging V(v) back in and multiplying by V(v) gives U ƒ(u) /C27cot uU ?(u) /C28m2 sin2u/C27n2/C271 4/C(%/C(r"# U(u)/C300;(9) so L APLACE’S EQUATION is partially separable in BISPHERICAL COORDINATES . However, the H ELMHOLTZ DIFFERENTIAL EQUATION cannot be separated in this manner. See also BISPHERICAL COORDINATES ,LAPLACE’S EQUA- TION References Arfken, G. "Bispherical Coordinates (j; h; f) :/" §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115 /C1/17, 1970. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665 /C1/66, 1953. Laplace’s Equation * /Spherical Coordinates Laplace’s Equation–Spherical HELMHOLTZ DIFFERENTIAL EQUATION– SPHERICAL CO- ORDINATES Laplace’s Equation * /Toroidal Coordinates In TOROIDAL COORDINATES ,LAPLACE’S EQUATION be- comes 92f /C30sinh u cosh u /C28 cos v ðÞ3@ @usinh u cosh u /C28 cos v@f @u !" /C27@ @vsinh u cosh u /C28 cos v@f @v ! /C27@ @ f /C2csch u cosh u /C28 cos v@f @ f !/C)( (1) Attempt SEPARATION OF VARIABLES by plugging in the trial solution fu; v ; f ðÞ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh u /C28cos up U(u)V(v) C( c); (2) then divide the result by csch2 u(cosh u /C28cos v)5 =2 U(u)V(v) F(f) to obtain 1 4sinh2 u /C27cosh u sinh uU ?(u) U(u)/C27sin2 uU ƒ(u) U(u) /C27sinh2 uV ƒ(v) V(v)/C27Fƒ( f) F(f)/C300: (3) The function F( f) then separates with Fƒ( f) F( f)/C30/C28m2 ; (4) giving solution C(c) /C30sin cos (mf) /C30X/C12 k/C301[Ak sin(mc) /C27Bk cos(mc)]: (5) Plugging C( c) back in and dividing by sinh2 u gives coth uU ?(u) U(u)/C27U ƒ(u) U(u)/C28m2 sinh2 u /C271 4 /C27V ƒ(v) V(v)/C300 : (6) The function V(v) then separates with V ƒ(v) V(v)/C30/C28n2 ; (7)giving solution V(v) /C30sin cos (nv) /C30X/C12 k /C301[Ck sin(nv) /C27Dk cos(nv)]: (8) Plugging V(v) back in and multiplying by V(v) gives U ƒ(u) /C27coth uU ?(u) /C28m2 sinh2 u /C27 n2 /C281 4/C(%/C(r"# U(u) /C300; (9) which can also be written 1 sinh ud dusinh udU du ! /C28m2 sinh2u/C27n2/C281 4/C(%/C(r"# U /C300 (10) (Arfken 1970, pp. 114 /C1/15). L APLACE’S EQUATION is partially separable, although the H ELMHOLTZ DIFFER- ENTIAL EQUATION is not. See also LAPLACE’S EQUATION ,LAPLACIAN ,TOROIDAL COORDINATES References Arfken, G. "Toroidal Coordinates ( j;h;f):/"§2.13 in Math- ematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112 /C1/15, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics.New York: Dover, pp. 264 /C1 /66, 1959. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953. Laplace Series A function f(u;f) expressed as a double sum of SPHERICAL HARMONICS is called a Laplace series. Taking fas a COMPLEX FUNCTION , f(u;f)/C30X/C12 l/C300Xl m/C30/C281almYm l(u;f): (1) Now multiply both sides by ¯Ym? l?sinuand integrate over duanddf: g2p 0gp 0f(u;f)¯Ym? l?sinududf /C30X/C12 l/C300Xl m/C30/C281almg2p 0gp 0¯Ym? l?(u;f)Ym l(u;f) sin ududf: (2) Now use the ORTHOGONALITY of the SPHERICAL HARMONICS g2p 0gp 0Ym l(u;f)¯Ym? l?sinududf/C30dmm?dll?; (3) so (2) becomes g2p 0g p 0f( u; f) ¯Ym? l?sin u du df /C30X/C12 l/C300Xl m/C30/C281alm dmm? dll ? /C30alm ; (4) where dmn is the KRONECKER DELTA . For a REAL series, consider f(u ; f) /C30X/C12 l/C300Xl m/C30/C281[Cm lcos(mf) /C27Smlsin(mf)]Pml(cos u) : (5) Proceed as before, using the orthogonality relation- ships g2 p 0g p 0Pml(cos u) cos(mf)Pm? l?(cos u) cos(m? f) /C2sin( u) du d f /C30/C282 p(l /C27 m)! (2l /C27 1)(l /C28 m)!dmm? dll? (6) g2 p 0g p 0Pm l(cos u) sin(mf)Pm? l ?(cos u) sin(m? f) /C2sin u du df /C30/C282p(l /C27 m)! (2l /C27 1)(l /C28 m)!dmm? dll ?: (7) So Cm land Smlare given by Cm l/C30/C28(2l /C27 1)(l /C28 m)! 2p(l /C27 m)! g2 p 0g p 0f(u ; f) /C2Pmlcos u cos(mf) sin u du df (8) Sml/C30/C28(2l /C27 1)(l /C28 m)! 2p(l /C27 m)! g2 p 0g p 0f( u; f) /C2Pmlcos u sin(mf) sin u du df: (9) Laplace’s Integral Pn(x) /C301 p g p 0du x /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 1p cos u/C(%/C(rn /C271du /C301 p g p 0x /C27ffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C281p cos u/C(%/C(rn du: It can be evaluated in terms of the HYPERGEOMETRIC FUNCTION . Laplace’s Problem BUFFON- LAPLACE NEEDLE PROBLEM Laplace-Stieltjes Transform An integral transform which is often written as an ordinary LAPLACE TRANSFORM involving the DELTA FUNCTION . The L APLACE TRANSFORM and D IRICHLET SERIES are special cases of the Laplace-Stieltjes transform (Apostol 1997, p. 162).See also DIRICHLET SERIES ,LAPLACE TRANSFORM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 1029, 1972. Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 162, 1997. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, 1953. Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Laplace Transform The Laplace transform is an INTEGRAL TRANSFORM perhaps second only to the F OURIER TRANSFORM in its utility in solving physical problems. Due to its useful properties, the Laplace transform is particularly useful in solving linear ORDINARY DIFFERENTIAL EQUATIONS such as those arising in the analysis of electronic circuits. The (one-sided) Laplace transform L(not to be confused with the L IE DERIVATIVE ) is defined by L(s)/C30Lf(t)½/C138/C13g/C12 0f(t)e/C28stdt; (1) where f(t) is defined for t]0:The one-sided Laplace transform is implemented in Mathematica asLa- placeTransform [expr,t,s]. A two-sided Laplace transform is sometimes also defined by L(s)/C30Lf(t)jj/C30g/C12 /C28/C12f(t)e/C28stdt: (2) The Laplace transform existence theorem states that,iff(t)i s PIECEWISE CONTINUOUS function on every finite interval in [0 ;/C12) satisfying f(t)jj5Meat(3) for all t/C23[0;/C12);thenLf(t)½/C138 exists for all s/C21a. The Laplace transform is also UNIQUE , in the sense that, given two functions F1(t) and F2(t) with the same transform so that LF1(t) ½/C138/C30LF2(t) ½/C138/C13f(s); (4) then L ERCH’S THEOREM guarantees that the integral ga 0N(t)dt/C300 (5) vanishes for all a/C210 for a NULL FUNCTION defined by N(t)/C13F1(t)/C28F2(t): (6) The Laplace transform is LINEAR since L[af(t)/C27bg(t)]/C30g/C12 0[af(t)/C27bg(t)]e/C28stdt /C30ag/C12 0f(t)e/C28st dt /C27bg/C12 0g(t)e /C28st dt /C30aL[f(t)] /C27bL[g(t)] : (7) The inverse Laplace transform is given by the BROMWICH INTEGRAL (see also DUHAMEL’S CONVOLU- TION PRINCIPLE ). A table of several important Laplace transforms follows. /f(t)// Lf(t)½/C138 / Range 1 /1 s/ s/C210 t /1 s2/ s/C210 /tn//n! sn/C271// n/C23Z>0/ /ta//G(a/C271) sa/C271/ a/C210 /eat//1 s/C28a/ s/C21a /cos(vt)//s s2/C27v2/ s/C210 /sin(vt)//v s2/C27v2{\it s} \hskip -1.80\ma- threl{{\tf="DM5"\char21}}\hskip - 1.80 0\cr /cosh( vt)//s s2/C28v2/ /s>ajj/ /sinh( vt)//v s2/C28v2// s>ajj/ /eatsin(bt)//b (s/C28a)2/C27b2/ s/C21a /eatcos(bt)//s/C28a (s/C28a)2/C27b2/ s/C21a /d(t/C28c)// e/C28cs/ c/C210 /Hc(t)//e/C28cs s/ s/C210 /J0(t)//1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2/C271p / /Jn(t)//2F11 2(n/C271);12(n/C272);n/C271;/C28s/C282/C(%/C(r 2nsn/C271/ In the above table, J0(t) is the zeroth order B ESSEL FUNCTION OF THE FIRST KIND ,d(t) is the DELTA FUNCTION , and Hc(t) is the H EAVISIDE STEP FUNCTION . The Laplace transform has many important proper- ties.The Laplace transform of a CONVOLUTION is given by L[f(t)+g(t)]/C30L(f(t))L(g(t)) (8) L/C281[F(s)G(s)]/C30L/C281(F(s))+L/C281(G(s)): (9) Now consider DIFFERENTIATION . Let f(t) be continu- ously differentiable n/C281 times in [0 ;/C12):Iff(t)jj5 Meat;then L[f(n)(t)]/C30snL(f(t))/C28sn/C281f(0)/C28sn/C282f?(0)/C28... /C28f(n/C281)(0): (10) This can be proved by INTEGRATION BY PARTS , L[f?(t)]/C30lim a0/C12ga 0e/C28stf?(t)dt /C30lim a0/C12[e/C28stf(t)]a 0/C27sga 0e/C28stf(t)dt/C)%/C)r /C30lim a0/C12[e/C28saf(a)/C28f(0)/C27sga 0e/C28stf(t)dt/C)P/C)( /C30sL[f(t)]/C28f(0): (11) Continuing for higher order derivatives then gives L[fƒ(t)]/C30s2L[f(t)]/C28sf(0)/C28f?(0): (12) This property can be used to transform differential equations into algebraic equations, a procedureknown as the H EAVISIDE CALCULUS , which can then be inverse transformed to obtain the solution. Forexample, applying the Laplace transform to theequation fƒ(t)/C27a 1f?(t)/C27a0f(t)/C300 (13) gives fs2L[f(t)]/C28sf(0)/C28f?(0)g/C27a1fsL[f(t)]/C28f(0)g /C27a0L[f(t)]/C300 (14) L[f(t)](s2/C27a1s/C27a0)/C28sf(0)/C28f?(0)/C28a1f(0)/C300;(15) which can be rearranged to L[f(t)]/C30sf(0)/C27[f?(0)/C27a1f(0)] s2/C27a1s/C27a0: (16) If this equation can be inverse Laplace transformed, then the original differential equation is solved. Consider EXPONENTIATION .I fL[f(t)]/C30F(s) for s>a; thenL(eatf(t))/C30F(s/C28a) for s>a/C27a: F(s/C28a)/C30g/C12 0f(t)e/C28(s/C28a)tdt/C30g/C12 0[f(t)eat]e/C28stdt /C30L[eatf(t)]: (17) Consider INTEGRATION .I ff(t)i s PIECEWISE CONTIN- UOUS and f(t)jj5Meat;then Lgt 0f(t) dt"# /C301 sL[f(t)]: (18) The inverse transform is known as the BROMWICH INTEGRAL , or sometimes the FOURIER- MELLIN INTE- GRAL . See also BROMWICH INTEGRAL ,FOURIER- MELLIN IN- TEGRAL ,FOURIER TRANSFORM ,INTEGRAL TRANSFORM , LAPLACE- STIELTJES TRANSFORM ,O PERATIONAL MATHEMATICS References Abramowitz, M. and Stegun, C. A. (Eds.). "Laplace Trans- forms." Ch. 29 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1019 /C1/030, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 824 /C1/63, 1985. Churchill, R. V. Operational Mathematics. New York: McGraw-Hill, 1958. Doetsch, G. Introduction to the Theory and Application of the Laplace Transformation. Berlin: Springer-Verlag, 1974. Franklin, P. An Introduction to Fourier Methods and the Laplace Transformation. New York: Dover, 1958. Jaeger, J. C. and Newstead, G. H. An Introduction to the Laplace Transformation with Engineering Applications.London: Methuen, 1949. Henrici, P. Applied and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms, Asympto-tics, Continued Fractions. New York: Wiley, pp. 322 /C1 /50, 1991. Krantz, S. G. "The Laplace Transform." §15.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 212 /C1/ 14, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 467 /C1/69, 1953. Oberhettinger, F. Tables of Laplace Transforms. New York: Springer-Verlag, 1973. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 4: Direct Laplace Transforms.New York: Gordon and Breach, 1992. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 5: Inverse Laplace Transforms.New York: Gordon and Breach, 1992. Spiegel, M. R. Theory and Problems of Laplace Transforms. New York: McGraw-Hill, 1965. Weisstein, E. W. "Books about Laplace Transforms." http:// www.treasure-troves.com/books/LaplaceTransforms.html. Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 231 and 543, 1995. Laplacian The Laplacian operator for a SCALAR function /f/is defined by 92f/C301 h1h2h3@ @u1h2h3 h1@ @u1 !"/C27@ @u2h1h3 h2@ @u2 ! /C27@ @u3h1h2 h3@ @u3 !/C)( f (1) inVECTOR notation, where the hiare the SCALE FACTORS of the coordinate system. In TENSOR nota- tion, the Laplacian is written 92f/C30(glkf;l);k/C30glk @2f @xl@xk/C28Gl@f @xl /C301 ffiffiffigp@ @xjffiffiffigpgij@f @xi ! ; (2) where g;kis a COVARIANT DERIVATIVE and Gl/C131 2gmnglk@gkm @xn/C27@gkn @xm/C28@gmn @xk ! : (3) Note that the operator 92is commonly written as Dby mathematicians (Krantz 1999, p. 16). The following table gives the form of the Laplacian in several common coordinate systems. coordinate system /92/ CARTESIAN COORDI- NATES/@2 @x2/C27@2 @y2/C27@2 @z2/ CYLINDRICAL COOR- DINATES/1 r@ @rr@f @r/C(*/C(+ /C271 r2@2f @u2/C27@2f @z2/ PARABOLIC COORDI- NATES/1 uv(u2/C27v2)@ @uuv@f @u/C(*/C(+ /C27@ @vuv@f @v/C(*/C(+ /C)P/C)( / //C271 u2v2@2f @u2/ PARABOLIC CYLINDRI- CAL COORDINATES/1 u2/C27v2@2f @u2/C27@2f @v2/C(*/C(+ /C27@2f @z2/ SPHERICAL COORDI-NATES/1 r2@ @rr2@ @r/C(*/C(+ /C271 r2sin2f@2 @u2/ //C271 r2sinf@ @fsinf@ @f/C(*/C(+ / The finite difference form is 92c(x;y;z)/C301 h2c(x/C27h;y;z)/C27c(x/C28h;y;z) ½ /C27c(x;y/C27h;z)/C27c(x;y/C28h;z)/C27c(x;y;z/C27h) /C27c(x;y;z/C28h)/C286c(x;y;z)/C138: (4) For a pure radial function g(r); 92g(r)/C139 /C215[9g(r)] /C309 /C215@g(r) @rˆr /C271 r@g(r) @ uˆu /C271 r sin u@g(r) @ fˆf"# /C309 /C215 ˆrdg dr ! : (5) Using the VECTOR DERIVATIVE identity 9 /C215(fA) /C30f( 9 /C215 A) /C27( 9f) /C215(A) ; (6) so 92g(r) /C139 /C215[ 9g(r)] /C30dg dr9 /C215 ˆr /C279dg dr ! /C215 ˆr /C302 rdg dr /C27d2g dr2 : (7) Therefore, for a radial power law, 92rn /C302 rnrn/C281 /C27n(n /C281)rn/C282 /C30[2n /C27n(n /C281)]rn/C282 /C30n(n /C271)rn/C282 : (8) A vector Laplacian can also be defined for a VECTOR A by 92A /C309( 9 /C215 A) /C289/C29( 9/C29A) (9) in vector notation. The notation /C19 is sometimes also used for a vector Laplacian (Moon and Spencer 1988, p. 3). In tensor notation, A is written Am ; and the identity becomes 92Am /C30A; l m; l /C30(g lkAm; l); k /C30g l k; kAm; l /C27g lkAm; lk : (10) Similarly, a TENSOR Laplacian can be given by 92Aab /C30A; l ab; l : (11) An identity satisfied by the Laplacian is 92 xAjj/C30Ajj2 2 /C28 (xA)AT/C()/C()/C()/C()2 xAjj3 ; (12) where Ajj2is the HILBERT- SCHMIDT NORM , x is a row VECTOR , and AT is the MATRIX TRANSPOSE of A :/ To compute the LAPLACIAN of the inverse distance function 1=r ; where r /C13 r /C28r? jj ; and integrate the LAPLACIAN over a volume, gV92 1 r /C28 r ? jj ! d3r: (13) This is equal to gV921 rd3r /C30gV9 /C21591 r ! d3r /C30gS91 r ! /C215 da/C30gS@ @r1 r ! ˆr /C215 da /C30gS/C281 r2ˆr /C215 da /C30/C284pR2 r2 ; (14) where the integration is over a small SPHERE of RADIUS R. Now, for r /C210 and R 0 0; the integral becomes 0. Similarly, for r /C30R and R 0 0; the integral becomes /C284p: Therefore, 92 1 r/C28r? jj ! /C30/C284pd3(r/C28r?); (15) where d(x) is the DELTA FUNCTION . The tensor Laplacian is given by 9 /C215(9c)/C301 g1=2(g1=2gikc;k);i; (16) where gijis the METRIC TENSOR ,g/C30det(gij);andA;kis the COMMA DERIVATIVE (Arfken 1985, p. 185). See also ANTILAPLACIAN , D’ALEMBERTIAN ,H ELM- HOLTZ DIFFERENTIAL EQUATION ,L APLACE’S EQUA- TION ,VECTOR LAPLACIAN References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 16, 1999. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, 1988. Laplacian Determinant Expansion by Minors DETERMINANT EXPANSION BY MINORS Laplacian Expansion DETERMINANT EXPANSION BY MINORS Laplacian Matrix The Laplacian matrix L(G) of a graph G, where G/C30 (N;E) is an undirected, unweighted graph without self edges ( i, i) or multiple edges from one node to another, is an Njj/C29Njj SYMMETRIC MATRIX with one row and column for each node. It is defined as follows, Lij(G)/C30degree of node iifi/C30j /C281i f i"jand/C215edge( i;j) 0 otherwise :8 < : A normalized version of the Laplacian matrix, de- noted L;is similar defined by Lij(G) /C301i f i /C30j and dj "0 /C281ffiffiffiffiffiffiffiffiffi didjq if i and j are adjacent 0 otherwise :8 >>>< >>>: See also A LGEBRAIC CONNECTIVITY ,FIEDLER VECTOR , SPECTRAL GRAPH PARTITIONING References Bendito, E.; Carmona, A.; and Encinas, A. M. "Shortest Paths in Distance-Regular Graphs." Europ. J. Combin. 21, 153 /C1/66, 2000. Chung, F. R. K. Spectral Graph Theory. Providence, RI: Amer. Math. Soc., 1997. Demmel, J. "CS 267: Notes for Lecture 23, April 9, 1999. Graph Partitioning, Part 2." http://www.cs.berkeley.edu/ ~demmel/cs267/lecture20/lecture20.html. Large Number There are a wide variety of large numbers which crop up in mathematics. Some are contrived, but some actually arise in proofs. Often, it is possible to prove existence theorems by deriving some potentially huge upper limit which is frequently greatly reduced in subsequent versions (e.g., GRAHAM’S NUMBER ,KOL- MOGOROV- ARNOLD-MOSER THEOREM ,M ERTENS CON- JECTURE , SKEWES NUMBER ,W ANG’S CONJECTURE ). Large decimal numbers beginning with 109are named according to two mutually conflicting nomen- clatures: the American system (in which the prefix stands for n in 103/C273n) and the British system (in which the prefix stands for n in 106n) : However, it should be noted that in more recent years, the "American" system is now widely used in England as well as in the United States. The following table gives the names assigned to various POWERS of 10 (Woolf 1982). American British power of 10 MILLION million 106 BILLION milliard 109 TRILLION billion 1012 QUADRILLION 1015 QUINTILLION trillion 1018 SEXTILLION 1021 SEPTILLION quadrillion 1024 OCTILLION 1027 NONILLION quintillion 1030 DECILLION 1033 UNDECILLION sexillion 1036DUODECILLION 1039 TREDECILLION septillion 1042 QUATTUORDECILLION 1045 QUINDECILLION octillion 1048 SEXDECILLION 1051 SEPTENDECILLION nonillion 1054 OCTODECILLION 1057 NOVEMDECILLION decillion 1060 VIGINTILLION 1063 undecillion 1066 duodecillion 1072 tredecillion 1078 quattuordecillion 1084 quindecillion 1090 sexdecillion 1096 septendecillion 10102 octodecillion 10108 novemdecillion 10114 vigintillion 10120 centillion 10303 centillion 10600 See also 10,ACKERMANN NUMBER ,ARROW NOTATION , BARNES’ G-FUNCTION ,B ILLION ,C IRCLE NOTATION , EDDINGTON NUMBER ,ERDOS- MOSER EQUATION ,FRI- VOLOUS THEOREM OF ARITHMETIC ,G O¨ BEL’S SE- QUENCE ,GOOGOL ,GOOGOLPLEX ,GRAHAM’S NUMBER , HUNDRED ,H YPERFACTORIAL ,JUMPING CHAMPION , LAW OF TRULY LARGE NUMBERS ,M EGA,M EGISTRON , MILLION ,M ONSTER GROUP ,M OSER , N-PLEX,POWER TOWER ,SKEWES NUMBER ,SMALL NUMBER ,STEIN- HAUS- MOSER NOTATION ,S TRONG LAW OF LARGE NUMBERS ,SUPERFACTORIAL ,THOUSAND ,W EAK LAW OF LARGE NUMBERS ,ZILLION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 59 /C1/2, 1996. Crandall, R. E. "The Challenge of Large Numbers." Sci. Amer. 276,7 4/C1/9, Feb. 1997. Davis, P. J. The Lore of Large Numbers. New York: Random House, 1961. Knuth, D. E. "Mathematics and Computer Science: Coping with Finiteness. Advances in Our Ability to Compute Are Bringing Us Substantially Closer to Ultimate Limita-tions." Science 194, 1235 /C1 /242, 1976. Munafo, R. "Large Numbers." http://www.mrob.com/largen- um.html. Spencer, J. "Large Numbers and Unprovable Theorems." Amer. Math. Monthly 90, 669 /C1/75, 1983. Woolf, H. B. (Ed. in Chief). Webster’s New Collegiate Dic- tionary. Springfield, MA: Merriam, p. 782, 1980. Large Prime GIGANTIC PRIME ,LARGE NUMBER ,TITANIC PRIME Largest Prime Factor GREATEST PRIME FACTOR Laspeyres’ Index The statistical INDEX PL /C13PPnq0Pp0q0; where pn is the price per unit in period n and qn is the quantity produced in period n. See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 65 /C1/7, 1962. Latent Root EIGENVALUE Latent Vector EIGENVECTOR Latin Cross An irregular DODECAHEDRON CROSS in the shape of a dagger $: The six faces of a CUBE can be cut along seven EDGES and unfolded into a Latin cross (i.e., the Latin cross is the NET of the CUBE ). Similarly, eight hypersurfaces of a HYPERCUBE can be cut along 17 SQUARES and unfolded to form a 3-D Latin cross. Another cross also called the Latin cross is illustrated above. It is a G REEK CROSS with flared ends, and is also known as the crux immissa or cross pate ´e. See also CROSS ,DISSECTION ,DODECAHEDRON ,GREEK CROSS ,MALTESE CROSSLatin-Graeco Square EULER SQUARE Latin Rectangle Ak/C29nLatin rectangle is a k/C29nMATRIX with elements aij/C23f1;2;...;ngsuch that entries in each row and column are distinct. If k/C30n, the special case of a L ATIN SQUARE results. A normalized Latin rectangle has first row f1;2;...;ngand first column f1;2;...;kg:LetL(k;n) be the number of normal- ized k/C29nLatin rectangles, then the total number of k/C29nLatin rectangles is N(k;n)/C30n!(n/C281)!L(k;n) (n/C28k)! (McKay and Rogoyski 1995), where n!i sa FACTORIAL . Kerewala (1941) found a RECURRENCE RELATION for L(3;n);and Athreya, Pranesachar, and Singhi (1980) found a summation FORMULA forL(4;n):/ The asymptotic value of L(o(n6=7);n) was found by Godsil and McKay (1990). The numbers of k/C29nLatin rectangles are given in the following table from McKay and Rogoyski (1995). The entries L(1;n) and L(n;n) are omitted, since L(1;n)/C301 L(n;n)/C30L(n/C281;n); butL(1;1) and L(2;1) are included for clarity. The values of L(k;n) are given as a "wrap-around" series by Sloane’s A001009. nk /L(k;n)/ 11 1 21 1 32 1 42 343 452 1 1 53 4 6 54 5 662 5 36 3 1064 6 4 6552 6 5 94087 2 3097 3 35792 7 4 1293216 7 5 11270400 7 6 169420808 2 21198 3 1673792 8 4 420909504 8 5 272066580488 6 3353901895688 7 535281401856 9 2 16687 9 3 1034438089 4 2076245602569 5 112681643083776 9 6 12952605404381184 9 7 2243829679166914569 8 377597570964258816 10 2 148329 10 3 8154999232 10 4 14717452105958410 5 74698838307628646410 6 870735405591003709440 10 7 177144296983054185922560 10 8 429203942159185427300352010 9 7580721483160132811489280 References Athreya, K. B.; Pranesachar, C. R.; and Singhi, N. M. "On the Number of Latin Rectangles and Chromatic Polyno- mial of /L(Kr;s)/."Europ. J. Combin. 1,9/C1/7, 1980. Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, 1996. Godsil, C. D. and McKay, B. D. "Asymptotic Enumeration of Latin Rectangles." J. Combin. Th. Ser. B 48,1 9/C1/4, 1990. Kerawla, S. M. "The Enumeration of Latin Rectangle of Depth Three by Means of Difference Equation" [sic]. Bull. Calcutta Math. Soc. 33, 119/C1/27, 1941. McKay, B. D. and Rogoyski, E. "Latin Squares of Order 10." Electronic J. Combinatorics 2,N 31 /C1/, 1995. http:// www.combinatorics.org/Volume_2/volume2.html#N3. Ryser, H. J. "Latin Rectangles." §3.3 in Combinatorial Mathematics. Buffalo, NY: Math. Assoc. of Amer., pp. 35 /C1/7, 1963. Sloane, N. J. A. Sequences A001009 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html.Latin Square Ann/C29nLatin square is a L ATIN RECTANGLE with k/C30n. Specifically, a Latin square consists of nsets of the numbers 1 to narranged in such a way that no orthogonal (row or column) contains the same two numbers. The numbers of Latin squares of ordern/C301, 2, ... are 1, 2, 12, 576, 161280, ... (Sloane’s A002860). For example, the two Latin squares oforder two are given by 12 21/C)P/C)( ;2112/C)P/C)( ; (1) the 12 Latin squares of order three are given by 123 2313122 435;123 3122312 435;132 2133212 435;132 3212132 435; 213 1323212 435;213 3211322 435;231 1233122 435;231 3121232 435; 321 132 2132 435;321 213 1322 435;312 123 2312 435;312 231 1232 435;(2) and two of the whopping 576 Latin squares of order 4 are given by 1234 2143341243212 6643 775and1234 3412432121432 6643 775: (3) A pair of Latin squares is said to be orthogonal if the n 2pairs formed by juxtaposing the two arrays are all distinct. For example, the two Latin squares 321 2131322 435231 1233122 435 (4) are orthogonal. A normalized, or reduced, Latin square is a Latin square with the first row and column given by f1;2;...;ng:General FORMULAS for the number of normalized n /C29nLatin squares L(n;n) are given by Nechvatal (1981), Gessel (1987), and Shao and Wei (1992). The total number of Latin squares N(n;n)o f order ncan then be computed from N(n;n)/C30n!(n/C281)!L(n;n): (5) The numbers of normalized Latin squares of order n/C301, 2, ..., are 1, 1, 1, 4, 56, 9408, ... (Sloane’s A000315). McKay and Rogoyski (1995) give thenumber of normalized L ATIN RECTANGLES L(k;n) for n/C301, ..., 10, as well as estimates for L(n;n) with n/C3011, 12, ..., 15. n /L(n ; n)/ 11 /5 :36 /C291033 / 12 /1 :62 /C291044/ 13 /2 :51 /C291056/ 14 /2 :33 /C291070 / 15 /1:5 /C291086 / See also 36 OFFICER PROBLEM ,EULER SQUARE ,KIRK- MAN TRIPLE SYSTEM ,LAM’S PROBLEM ,PARTIAL LATIN SQUARE ,QUASIGROUP , SOMA References Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, 1996. Gessel, I. "Counting Latin Rectangles." Bull. Amer. Math. Soc. 16,79/C1/3, 1987. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 33 /C1/4, 1975. Kraitchik, M. "Latin Squares." §7.11 in Mathematical Recreations. New York: W. W. Norton, p. 178, 1942. Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. McKay, B. D. and Rogoyski, E. "Latin Squares of Order 10." Electronic J. Combinatorics 2,N31 /C1/, 1995. http:// www.combinatorics.org/Volume_2/volume2.html#N3. Nechvatal, J. R. "Asymptotic Enumeration of Generalised Latin Rectangles." Util. Math. 20, 273 /C1/92, 1981. Rohl, J. S. Recursion via Pascal. Cambridge, England: Cambridge University Press, pp. 162 /C1/65, 1984. Ryser, H. J. "Latin Rectangles." §3.3 in Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 35 /C1/ 7, 1963. Shao, J.-Y. and Wei, W.-D. "A Formula for the Number of Latin Squares." Disc. Math. 110, 293 /C1/96, 1992. Sloane, N. J. A. Sequences A002860/M2051 and A000315/ M3690 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Latitude The latitude of a point on a SPHERE is the elevation of the point from the PLANE of the equator. The latitude d is related to the COLATITUDE (the polar angle in SPHERICAL COORDINATES )by d /C30 f /C2890 /C14: More gener- ally, the latitude of a point on an ELLIPSOID is the ANGLE between a LINE PERPENDICULAR to the surface of the ELLIPSOID at the given point and the PLANE of the equator (Snyder 1987). The equator therefore has latitude 0 8, and the north and south poles have latitude 990/C14; respectively. Latitude is also called GEOGRAPHIC LATITUDE or GEODETIC LATITUDE in order to distinguish it from several subtly different varieties of AUXILIARY LATI- TUDES . The shortest distance between any two points on a SPHERE is the so-called GREAT CIRCLE distance, whichcan be directly computed from the latitudes and LONGITUDES of the two points. See also AUXILIARY LATITUDE ,COLATITUDE ,CONFOR- MAL LATITUDE ,GREAT CIRCLE ,ISOMETRIC LATITUDE , LATITUDE ,LONGITUDE ,SPHERICAL COORDINATES References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 13, 1987. Lattice A lattice is a system K such that //C214A /C23 K ; A ƒA; and if A ƒB and B ƒA; then A /C30B, where ƒ means "is included in." Lattices offer a natural way to formalize and study the ordering of objects using a general concept known as the POSET (partially ordered set). The study of lattices is called LATTICE THEORY . Note that this type of lattice is distinct from the regular array of points known as a POINT LATTICE (or informally as a mesh or grid). The following inequalities hold for any lattice: (x ffly) /C150(x fflz) 5x ffl(y /C150z) x /C150(y fflz) 5(x /C150y) ffl(x /C150z) (x ffly) /C150(y fflz) /C150(z fflx) 5(x /C150y) ffl(y /C150z) ffl(z /C150x) (x ffly) /C150(x fflz) 5x ffl(y /C150(x fflz)) (Gra¨tzer 1971, p. 35). The first three are the distri- butive inequalities, and the last is the modular identity. See also DISTRIBUTIVE LATTICE ,INTEGRATION LAT- TICE,L ATTICE THEORY ,M ODULAR LATTICE ,P OINT LATTICE ,TORIC VARIETY Lattice Algebraic System A generalization of the concept of SET UNIONS and INTERSECTIONS . Lattice Animal A distinct (including reflections and rotations) ar- rangement of adjacent squares on a grid, also called a FIXED POLYOMINO . See also ANIMAL ,PERCOLATION THEORY ,POLYOMINO References Delest, M.-P. and Viennot, G. "Algebraic Languages and Polyominoes [sic] Enumeration." Theoret. Comput. Sci. 34, 169/C1/06, 1984. Read, R. C. "Contributions to the Cell Growth Problem." Canad. J. Math. 14,1/C1/0, 1962. Lattice Basis Reduction LATTICE REDUCTION Lattice Distribution A DISCRETE DISTRIBUTION of a random variable such that every possible value can be represented in the form a /C27bn ; where a; b "0 and n is an INTEGER . References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 927, 1972. Lattice Graph The lattice graph with n nodes on a side is denoted L(n) :/ See also TRIANGULAR GRAPH Lattice Groups In the plane, there are 17 lattice groups, eight of which are pure translation. In R3 ; there are 32 POINT GROUPS and 230 SPACE GROUPS .InR4 ; there are 4783 space lattice groups. See also POINT GROUPS ,SPACE GROUPS ,W ALLPAPER GROUPS Lattice Invariant INVARIANT (ELLIPTIC FUNCTION ) Lattice Path A path composed of connected horizontal and vertical line segments, each passing between adjacent LAT- TICE POINTS . A lattice path is therefore a SEQUENCE of points P0 ; P1 ; ..., Pn with n ]0 such that each Pi is a LATTICE POINT and Pi/C271is obtained by offsetting one unit east (or west) or one unit north (or south). The number of paths of length a /C27b from the ORIGIN (0,0) to a point (a, b) which are restricted to east and north steps is given by the BINOMIAL COEFFICIENT a/C27b a/CP/C( :/ See also BALLOT PROBLEM ,D YCK PATH,F ABER POLYNOMIAL ,G OLYGON ,K INGS PROBLEM ,L ATTICE POINT , P-GOOD PATH,R ANDOM WALK,S TAIRCASE WALK References Dickau, R. M. "Shortest-Path Diagrams." http://forum.s- warthmore.edu/advanced/robertd/manhattan.html. Hilton, P. and Pederson, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13,64/C1/5, 1991.Mohanty, S. G. Lattice Path Counting and Applications. New York: Academic Press, 1979. Moser, L. and Zayachkowski, H. S. "Lattice Paths with Diagonal Steps." Scripta Math. 26, 223 /C1/29, 1963. Narayana, T. V. Lattice Path Combinatorics with Statistical Applications. Toronto, Ontario, Canada: University of Toronto Press, 1979. Lattice Point A POINT at the intersection of two or more grid lines in a POINT LATTICE . See also POINT LATTICE Lattice Polygon A POLYGON whose vertices are points of a POINT LATTICE . Regular lattice n-gons exists only for n /C303, 4, and 6 (Schoenberg 1937, Klamkin and Chrestenson 1963, Maehara 1993). A lattice n-gon in the plane can be equiangular to a regular polygon only for n /C304 and 8 (Scott 1987, Maehara 1993). Maehara (1993) presented a NECESSARY and SUFFI- CIENT condition for a polygon to be angle-equivalent to a lattice polygon in Rn : In addition, Maehara (1993) proved that cos2( au /C23S u)isa RATIONAL NUMBER for any collection S of interior angles of a lattice polygon. See also BAR GRAPH POLYGON ,CANONICAL POLYGON , CONVEX POLYGON ,C ONVEX POLYOMINO ,F ERRERS GRAPH POLYGON ,G OLYGON ,POINT LATTICE ,POLY- OMINO ,SELF-AVOIDING POLYGON ,STACK POLYGON , STAIRCASE POLYGON ,THREE- CHOICE POLYGON References Beeson, M. J. "Triangles with Vertices on Lattice Points." Amer. Math. Monthly 99, 243/C1/52, 1992. Jensen, I. Size and Area of Square Lattice Polygons. 28 Mar 2000. http://xxx.lanl.gov/abs/cond-mat/0003442/. Klamkin, M. and Chrestenson, H. E. "Polygon Imbedded in a Lattice." Amer. Math. Monthly 70,5 1/C1/1, 1963. Maehara, H. "Angles in Lattice Polygons." Ryukyu Math. J. 6,9/C1/9, 1993. Schoenberg, I. J. "Regular Simplices and Quadratic Forms." J. London Math. Soc. 12,4 8/C1/5, 1937. Scott, P. R. "Equiangular Lattice Polygons and Semiregular Lattice Polyhedra." College Math. J. 18, 300/C1/06, 1987. LatticeReduce LLL ALGORITHM Lattice Reduction The process of finding a reduced set of basis vectors for a given LATTICE having certain special properties. Lattice reduction algorithms are used in a number of modern number theoretical applications, including in the discovery of a SPIGOT ALGORITHM for PI. Although determining the shortest basis is possibly an NP- COMPLETE PROBLEM , algorithms such as the LLL ALGORITHM can find a short basis in polynomial time with guaranteed worst-case performance. The LLL ALGORITHM of lattice reduction is implemen- ted in Mathematica using the function LatticeR- educe .Recognize [x, n, t] in the Mathematica add- on packageNumberTheory‘Recognize‘ (which can be loaded with the command BBNumberTheory‘ ) also calls this routine in order to find a polynomial of degree at most n in a variable t such that x is an approximate zero of the polynomial. When used to find integer relations, a typical input to the algorithm consists of an augmented n /C29n IDEN- TITY MATRIX with the entries in the last column consisting of the n elements (multiplied by a large positive constant w to penalize vectors that do not sum to zero) between which the relation is sought. For example, if an equality OF THE FORM a1x /C27a2y /C27a3z /C300 is known to exist, then doing a lattice reduction on the matrix m /C30100 wx 010 wy 001 wz2 435 will produce a new matrix in which one or more entries in the last column being close to zero. This row then gives the coefficients fa 1;a2;a3;0gof the identity. An example lattice reduction calculation is illustrated in both Borwein and Corless (1999) andBorwein and Lisonek. See also G RAM- SCHMIDT ORTHONORMALIZATION ,IN- TEGER RELATION , LLL ALGORITHM , PSLQ ALGORITHM References Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899/C1/09, 1999. Borwein, J. M. and Lisonek, P. "Applications of Integer Relation Algorithms." To appear in Disc. Math. http:// www.cecm.sfu.ca/preprints/1997pp.html. Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993. Coster, M. J.; Joux, A.; LaMacchia, B. A.; Odlyzko, A. M.; Schnorr, C. P.; and Stern, J. "Improved Low-Density Subset Sum Algorithms." Comput. Complex. 2, 111/C1/28, 1992.Hastad, J.; Just, B.; Lagarias, J. C.; and Schnorr, C. P. "Polynomial Time Algorithms for Finding Integer Rela- tions Among Real Numbers." SIAM J. Comput. 18, 859/C1/ 81, 1988. Lagarias, J. C.; Lenstra, H. W. Jr.; and Schnorr, C. P. "Korkin-Zolotarev Bases and Successive Minima of aLattice and Its Reciprocal Lattice." Combinatorica 10, 333/C1 /48, 1990. Schnorr, C. P. "A More Efficient Algorithm for Lattice Basis Reduction." J. Algorithms 9,4 7/C1/2, 1988. Schnorr, C. P. and Euchner, M. "Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset SumProblems." In Fundamentals of Computation Theory (Gosen 1991). Berlin: Springer-Verlag, pp. 68 /C1 /5, 1991. Lattice Sum Cubic lattice sums include the following: b2(2s)/C13X ?/C12 i;j/C30/C28/C12(/C281)i/C27j (i2/C27j2)s(1) b3(2s)/C13X ?/C12 i;j;k/C30/C28/C12(/C281)i/C27j/C27k (i2/C27j2/C27k2)s(2) bn(2s)/C13X ?/C12 k1;...;kn/C30/C28/C12(/C281)k1/C27.../C27kn (k2 1/C27.../C27k2 n)s; (3) where the prime indicates that summation over the original (0 ;0);(0;0;0);... is excluded (Borwein and Borwein 1986, p. 288). As shown in Borwein and Borwein (1987, pp. 288 /C1/ 01), these have closed forms for even n b2(2s)/C30/C284b(s)h(s) (4) b4(2s)/C30/C288h(s)h(s/C281) (5) b8(2s)/C30/C2816z(s)h(s/C283);forR[s]>1 (6) where b(z) is the D IRICHLET BETA FUNCTION ,h(z)i s the D IRICHLET ETA FUNCTION , and z(z) is the R IEMANN ZETA FUNCTION . The lattice sums evaluated at s/C301 are called the M ADELUNG CONSTANTS . An additional form for b2(2s) is given by b2(2s)/C30X/C12 n/C301(/C281)nr2(n) ns(7) forR[s]>1=3;where r2(n) is the SUM OF SQUARES FUNCTION , i.e., the number of representations of nby two squares (Borwein and Borwein 1986, p. 291). Borwein and Borwein (1986) prove that b8(2) con- verges (the closed form for b8(2s) above does not apply fors/C301), but its value has not been computed. A number of other related DOUBLE SERIES can be evaluated analytically. For hexagonal sums, Borwein and Borwein (1987, p. 292) give h2(2s) /C134 3X/C12 m; n /C30/C28/C12 /C2sin[(n /C27 1)u]sin[( m /C27 1)u] /C28 sin(n u)sin[( m /C28 1)u] n /C271 2 m/C(%/C(r2 /C27312 m/C(%/C(r2/C)P/C)(s ; (8) where u /C302p=3: This MADELUNG CONSTANT is expres- sible in closed form for s /C301as h2(2) /C30 p ln 3ffiffiffi 3p : (9) Other interesting analytic lattice sums are given by X/C12 k ; m; n/C30/C28/C12( /C281)k /C27m/C27n k /C271 6/C(%/C(r2 m /C2716/C(%/C(r2 n /C2716/C(%/C(r2/C)P/C)(s /C3012s b(2s /C281); (10) giving the special case X/C12 k ; m; n /C30/C28/C12(/C281)k/C27m/C27n k /C271 6/C(%/C(r2 m /C2716/C(%/C(r2 n /C2716/C(%/C(r2/C)P/C)(1 =2 /C30ffiffiffi 3p (11) (Borwein and Borwein 1986, p. 303), and X/C12 k ; m; n/C30/C28/C12(/C281)k/C27m/C27n/C271 ( ½k ½/C27½m½/C27½n½)s /C302h(s) /C274h(s /C282) (12) (Borwein and Borwein 1986, p. 305). See also BENSON’S FORMULA ,DOUBLE SERIES ,MADE- LUNG CONSTANTS References Borwein, D. and Borwein, J. M. "On Some Trigonometric and Exponential Lattice Sums." J. Math. Anal. 188, 209 /C1/ 18, 1994. Borwein, D.; Borwein, J. M.; and Shail, R. "Analysis of Certain Lattice Sums." J. Math. Anal. 143, 126 /C1/37, 1989. Borwein, D.; Borwein, J. M.; and Taylor, K. F. "Convergence of Lattice Sums and Madelung’s Constant." J. Math. Phys. 26, 2999 /C1/009, 1985. Borwein, D. and Borwein, J. M. "A Note on Alternating Series in Several Dimensions." Amer. Math. Monthly 93, 531 /C1/39, 1986. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/mdlung/mdlung.html. Glasser, M. L. and Zucker, I. J. "Lattice Sums." In Perspec- tives in Theoretical Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring). Lattice Theory Lattice theory is the study of sets of objects known as LATTICES . It is an outgrowth of the study of BOOLEAN ALGEBRAS , and provides a framework for unifying the study of classes or ordered sets in mathematics. The study of lattice theory was given a great boost by aseries of papers and subsequent textbook written by Birkhoff (1967). See also BOOLEAN ALGEBRA ,LATTICE References Birkhoff, G. Lattice Theory, 3rd ed. Providence, RI: Amer. Math. Soc., 1967. Gra¨tzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971. Gra¨tzer, G. General Lattice Theory, 2nd ed. Boston, MA: Birkha ¨user, 1998. Priestly, H. A. and Davey, B. A. Introduction to Lattices and Order. Cambridge, England: Cambridge University Press, 1990. Weisstein, E. W. "Books about Lattice Theory." http:// www.treasure-troves.com/books/LatticeTheory.html. Latus Rectum Twice the SEMILATUS RECTUM of a CONIC SECTION . See also PARABOLA ,SEMILATUS RECTUM References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 116 /C1/18, 1969. Laurent Polynomial A Laurent polynomial with COEFFICIENTS in the FIELD F is an algebraic object that is typically expressed in the form .../C27a/C28nt/C28n /C27a /C28(n/C281)t /C28(n/C281) /C27... /C27a/C281t/C281 /C27a0 /C27a1t /C27.../C27antn /C27...; where the ai are elements of F; and only finitely many of the aiare NONZERO . A Laurent polynomial is an algebraic object in the sense that it is treated as a POLYNOMIAL except that the indeterminant "t" can also have NEGATIVE POWERS . Expressed more precisely, the collection of Laurent polynomials with COEFFICIENTS in a FIELD F form a RING , denoted F[t; t/C281]; with RING operations given by componentwise addition and multiplication according to the relation atn /C215 btm /C30abtn/C27m for all n and m in the INTEGERS . Formally, this is equivalent to saying that F[t; t/C281] is the GROUP RING of the INTEGERS and the FIELD F: This corresponds to F[t] (the POLYNOMIAL ring in one variable for F) being the GROUP RING orMONOID ring for the MONOID of natural numbers and the FIELD F:/ See also POLYNOMIAL ,PRINCIPAL PART References Lang, S. Undergraduate Algebra, 2nd ed. New York: Springer-Verlag, 1990. Laurent Series Let there be two circular contours C2andC1;with the radius of C1larger than that of C2:Letz0be interior toC1andC2;andzbe between C1andC2:Now create a cut line Ccbetween C1andC2;and integrate around the path C/C13C1/C27Cc/C28C2/C28Cc;so that the plus and minus contributions of Cccancel one another, as illustrated above. From the C AUCHY INTEGRAL FOR- MULA , f(z)/C301 2pigCf(z?) z?/C28zdz? /C301 2pigC1f(z?) z?/C28zdz?/C271 2pigCcf(z?) z?/C28zdz? /C281 2pigC1f(z?) z?/C28z/C281 2pigCcf(z?) z?/C28zdz? /C301 2pigC1f(z?) z?/C28zdz?/C281 2pigC2f(z?) z?/C28zdz?: (1) Now, since contributions from the cut line in opposite directions cancel out, f(z)/C301 2pigC1f(z?) (z?/C28z0)/C28(z/C28z0)dz? /C281 2pigC2f(z?) (z?/C28z0)/C28(z/C28z0)dz? /C301 2pigC1f(z?) (z?/C28z0)1/C28z/C28z0 z?/C28z0/C(%/C(r dz? /C281 2pigC2f(z?) (z/C28z0)z?/C28z0 z/C28z0/C281/C(%/C(r dz? /C301 2pigC1f(z?) (z?/C28z0)1/C28z/C28z0 z?/C28z0/C(%/C(r dz? /C281 2pigC2f(z?) (z/C28z0)1/C28z?/C28z0 z/C28z0/C(%/C(r dz? (2) For the first integral, ½z?/C28z0½>½z/C28z0½:For the sec- ond,½z?/C28z0½B½z/C28z0½:Now use the T AYLOR EXPANSION(valid for ½t½B1) 1 1/C28t/C30X/C12 n/C300tn(3) to obtain f(z)/C301 2pigC1f(z?) z?/C28z0X/C12 n/C300z/C28z0 z/C28z0 !n dz?" /C27gC2f(z?) z/C28z0X/C12 n/C300z?/C28z0 z/C28z0 !n dz?/C138 /C301 2piX/C12 n/C300(z/C28z0)ngC1f(z?) (z?/C28z0)n/C271dz? /C271 2piX/C12 n/C300(z/C28z0)/C28n/C281gC2(z?/C28z0)nf(z?)dz? /C301 2piX/C12 n/C300(z/C28z0)ngC1f(z?) (z?/C28z0)n/C271dz? /C271 2piX/C12 n/C301(z/C28z0)/C28ngC2(z?/C28z0)n/C271f(z?)dz?; (4) where the second term has been re-indexed. Re- indexing again, f(z)/C301 2piX/C12 n/C300(z/C28z0)ngC1f(z?) (z?/C28z0)n/C271dz? /C271 2piX/C281 n/C30/C28/C12(z/C28z0)gC2f(z?) (z?/C28z0)n/C271dz?: (5) Now, use the C AUCHY INTEGRAL THEOREM , which requires that any CONTOUR INTEGRAL of a function which encloses no POLES has value 0. But 1 =(z?/C28 z0)n/C271is never singular inside C2forn]0;and 1=(z?/C28z0)n/C271is never singular inside C1forn5/C281: Similarly, there are no POLES in the closed cut Cc/C28 Cc:We can therefore replace C1andC2in the above integrals by Cwithout altering their values, so f(z)/C301 2piX/C12 n/C300(z/C28z0)ngCf(z?) (z?/C28z0)n/C271dz? /C271 2piX/C281 n/C30/C28/C12(z/C28z0)ngCf(z?) (z?/C28z0)n/C271dz? /C301 2piX/C12 n/C30/C28/C12(z/C28z0)ngCf(z?) (z?/C28z0)n/C271dz? /C13X/C12 n/C30/C28/C12an(z/C28z0)n: (6) The only requirement on Cis that it encloses z,s ow e are free to choose any contour gthat does so. The RESIDUES an are therefore defined by an /C131 2 pi g gf(z?) (z ?/C28z0)n/C271 dz ?: (7) See also MACLAURIN SERIES ,PRINCIPAL PART,RESI- DUE (COMPLEX ANALYSIS ), TAYLOR SERIES References Arfken, G. "Laurent Expansion." §6.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 376 /C1/84, 1985. Knopp, K. "The Laurent Expansion." Ch. 10 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 117 /C1/22, 1996. Krantz, S. G. "Laurent Series." §4.2.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 43, 1999. Morse, P. M. and Feshbach, H. "Derivatives of Analytic Functions, Taylor and Laurent Series." §4.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 374 /C1/98, 1953. Lauricella Functions This entry contributed by RONALD M. AARTS Lauricella functions are generalizations of the Gauss hypergeometric functions to multiple variables. Four such generalizations were investigated by Lauricella (1893), and more fully by Appell and Kampe ´ de Fe´riet (1926, p. 117). Let n be the number of variables, then the Lauricella functions are defined by F(n) A(a; b1 ; ...; bn; c1 ; ...; cn; x1 ; ...xn) /C30X(a ; m1 /C27 ... /C27 mn)(b1 ; m1) /C1/C1/C1(bn ; mn)xm1 1/C1/C1/C1xmnn (c1 ; m1) /C1/C1/C1(cn ; mn)m1! /C1/C1/C1mn! (1) F(n) B(a1 ; ...; an ; b1 ; ... ; bn; c; x1 ; ...; xn) /C30X(a1 ; m1) /C1/C1/C1(an ; mn)(b1 ; m1) /C1/C1/C1(bn ; mn)xm1 1/C1/C1/C1xmnn (c ; m1 /C27 ...mn)m1! /C1/C1/C1mn! (2) F(n) C(a ; b; c1 ; ...; cn; x1 ; ...; xn) /C30X(a1 ; m1 /C27 ...mn)(b ; m1 /C27 ...mn)xm1 1/C1/C1/C1xmnn (c1 ; m1) /C1/C1/C1(cn ; mn)m1! /C1/C1/C1mn! (3) F(n) D(a ; b1 ; ... ; bn; c; x1 ;...; xn) /C30X(a ; m1 /C27 ... /C27 mn)(b1 ; m1) /C1/C1/C1(bn ; mn)xm1 1/C1/C1/C1xmnn (c ; m1 /C27 ...mn)m1! /C1/C1/C1mn! : (4) If n /C302, then these functions reduce to the APPELLHYPERGEOMETRIC FUNCTIONS F2 ; F3 ; F4 ; and F1 ; respectively. If n /C301, all four become the Gauss hypergeometric function2F1 (Exton 1978, p. 29). See also APPELL HYPERGEOMETRIC FUNCTION ,GEN- ERALIZED HYPERGEOMETRIC FUNCTION ,HORN FUNC- TION ,KAMPE ´ DE FE´ RIET FUNCTION References Appell, P. and Kampe ´ de Fe´riet, J. Fonctions hyperge ´o- me´triques et hypersphe ´riques: polynomes d’Hermite. Paris: Gauthier-Villars, 1926. Erde´lyi, A. "Hypergeometric Functions of Two Variables." Acta Math. 83, 131 /C1/64, 1950. Exton, H. Ch. 5 in Multiple Hypergeometric Functions and Applications. New York: Wiley, 1976. Exton, H. "The Lauricella Functions and Their Confluent Forms," "Convergence," and "Systems of Partial Differen- tial Equations." §1.4.1 /C1/.4.3 in Handbook of Hypergeo- metric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, pp. 29 /C1/1, 1978. Lauricella, G. "Sulla funzioni ipergeometriche a piu` varia- bili." Rend. Circ. Math. Palermo 7, 111 /C1/58, 1893. Law A law is a mathematical statement which always holds true. Whereas "laws" in physics are generally experimental observations backed up by theoretical underpinning, laws in mathematics are generally THEOREMS which can formally be proven true under the stated conditions. However, the term is also sometimes used in the sense of an empirical observa- tion, e.g., B ENFORD’S LAW . See also ABSORPTION LAW,BENFORD’S LAW,CONTRA- DICTION LAW, DE MORGAN’S DUALITY LAW, DE MOR- GAN’S LAWS,ELLIPTIC CURVE GROUP LAW,EXCLUDED MIDDLE LAW,E XPONENT LAWS,G IRKO’S CIRCULAR LAW,L AW OF COSINES ,L AW OF SINES,L AW OF TANGENTS ,LAW OF TRULY LARGE NUMBERS ,M OR- RIE’S LAW,PARALLELOGRAM LAW,PLATEAU’S LAWS, QUADRATIC RECIPROCITY LAW,S TRONG LAW OF LARGE NUMBERS ,STRONG LAW OF SMALL NUMBERS , SYLVESTER’S INERTIA LAW,TRICHOTOMY LAW,VEC- TOR TRANSFORMATION LAW,W EAK LAW OF LARGE NUMBERS ,ZIPF’S LAW Law of Anomalous Numbers BENFORD’S LAW Law of Cancellation CANCELLATION LAW Law of Cosines Let a, b, and c be the lengths of the legs of a TRIANGLE opposite ANGLES A, B, and C. Then the law of cosines states c2 /C30a2 /C27b2 /C282ab cos C : (1) This law can be derived in a number of ways. The definition of the DOT PRODUCT incorporates the law of cosines, so that the length of the VECTOR from X to Y is given by ½X /C28Y ½2 /C30(X /C28Y) /C215 (X /C28Y) (2) /C30X /C215 X /C282X /C215 Y /C27Y /C215 Y (3) /C30½X ½2 /C27½Y ½2 /C282½X ½½Y ½cos u; (4) where u is the ANGLE between X and Y. The formula can also be derived using a little geometry and simple algebra. From the above dia- gram, c2 /C30(a sin C)2 /C27(b /C28a cos C)2 /C30a2 sin2 C /C27b2 /C282ab cos C /C27a2 cos2 C /C30a2 /C27b2 /C282ab cos C: (5) The law of cosines for the sides of a SPHERICAL TRIANGLE states that cos a /C30cos b cos c /C27sin b sin c cos A (6) cos b /C30cos c cos a /C27sin c sin a cos B (7) cos c /C30cos a cos b /C27sin a sin b cos C (8) (Beyer 1987). The law of cosines for the angles of aSPHERICAL TRIANGLE states that cos A /C30/C28cos B cos C /C27sin B sin C cos a (9) cos B /C30/C28cos C cos A /C27sin C sin A cos b (10) cos C /C30/C28cos A cos B /C27sin A sin B cos c (11) (Beyer 1987). For similar triangles, a generalized law of cosines is given by aa ?/C30bb ?/C27cc?/C28(bc?/C27b ?c)cos A (12) (Lee 1997). Furthermore, consider an arbitrary TET- RAHEDRON A1A2A3A4with triangles T1 /C30DA2A3A4 ; T2 /C30DA1A3A4 ; T3 /C30DA1A2A4 ; and T4 /C30A1A2A3 : Let the areas of these triangles be s1 ; s2 ; s3 ; and s4 ; respectively, and denote the DIHEDRAL ANGLE with respect to Ti and Tj for i "j /C301; 2; 3 ; 4by uij : Then sk /C30X j"k 1 5i54si cos uki ; (13) which gives the law of cosines in a tetrahedron, s2 k /C30X i "k 1 5j54s2j /C282X i ; j"k 15i ;j54sisj cos uij (14) (Lee 1997). A corollary gives the nice identity s1s?1 /C30s2s ?2 /C27s3s ?3 /C27s4s ?4 /C28(s2s?3 /C27s ?2s3)cos u23 /C28(s3s?4 /C27s ?3s4)cos u34 /C28(s2s ?4 /C27s ?2s4)cos u24 (15) See also LAW OF SINES,LAW OF TANGENTS References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 79, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 148 /C1/49, 1987. Lee, J. R. "The Law of Cosines in a Tetrahedron." J. Korea Soc. Math. Ed. Ser. B: Pure Appl. Math. 4,1/C1/, 1997. Law of Exponents EXPONENT LAWS Law of Growth An exponential growth law OF THE FORM y/C30arx characterizing a quantity which increases at a fixed rate proportionally to itself. See also GROWTH ,LOGISTIC GROWTH CURVE ,POPULA- TION GROWTH References Kenney, J. F. and Keeping, E. S. "The Law of Growth." §4.12 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 56 /C1/7, 1962. Law of Indices EXPONENT LAWS Law of Large Numbers STRONG LAW OF LARGE NUMBERS ,W EAK LAW OF LARGE NUMBERS Law of Sines Let a, b, and c be the lengths of the LEGS of a TRIANGLE opposite ANGLES A, B, and C. Then the law of sines states that a sin A /C30b sin B /C30c sin C /C302R; (1) where R is the radius of the CIRCUMCIRCLE . Other related results include the identities a(sin B /C28sin C) /C27b(sin C /C28sin A) /C27c(sin A /C28sin B) /C300 (2) a /C30b cos C /C27c cos B; (3) the LAW OF COSINES cos A /C30c2 /C27 b2 /C28 a2 2bc; (4) and the LAW OF TANGENTS a /C27 b a /C28 b /C30tan1 2(A /C27 B)hi tan1 2(A /C28 B)hi : (5) The law of sines for oblique SPHERICAL TRIANGLESstates that sin a sin A /C30sin b sin B /C30sin c sinC : (6) See also LAW OF COSINES ,LAW OF TANGENTS References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 79, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 148, 1987. Coxeter, H. S. M. and Greitzer, S. L. "The Extended Law of Sines." §1.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1 /C1/, 1967. Law of Small Numbers STRONG LAW OF SMALL NUMBERS Law of Tangents Let a TRIANGLE have sides of lengths a, b, and c and let the ANGLES opposite these sides by A, B, and C. The law of tangents states a /C28 b a /C27 b /C30tan12(A /C28 B)hi tan1 2(A/C27B)hi : An analogous result for oblique SPHERICAL TRIANGLES states that tan1 2(a/C28b)hi tan1 2(a/C27b)hi /C30tan1 2(A/C28B)hi tan1 2(A/C27B)hi : See also LAW OF COSINES ,LAW OF SINES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 79, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 145 and 149, 1987. Law of Truly Large Numbers With a large enough sample, any outrageous thing is likely to happen (Diaconis and Mosteller 1989). Littlewood (1953) considered an event which occursone in a million times to be "surprising." Taking thisdefinition, close to 100,000 surprising events are "expected" each year in the United States alone and, in the world at large, "we can be absolutely sure thatwe will see incredibly remarkable events" (Diaconis and Mosteller 1989). See also COINCIDENCE ,F RIVOLOUS THEOREM OF ARITHMETIC ,S TRONG LAW OF LARGE NUMBERS , STRONG LAW OF SMALL NUMBERS References Diaconis, P. and Mosteller, F. "Methods of Studying Coin- cidences." J. Amer. Statist. Assoc. 84, 853 /C1/61, 1989. Littlewood, J. E. Littlewood’s Miscellany. Cambridge, Eng- land: Cambridge University Press, 1986. Lax-Milgram Theorem Let f be a bounded COERCIVE bilinear FUNCTIONAL on aH ILBERT SPACE H. Then for every bounded linear FUNCTIONAL f on H, there exists a unique xf /C23 H such that f(x) /C30 f(x; xf ) for all x /C23 H :/ References Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990. Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995. Lax Pair A pair of linear OPERATORS L and A associated with a given PARTIAL DIFFERENTIAL EQUATION which can be used to solve the equation. However, it turns out to be very difficult to find the L and A corresponding to a given equation, so it is actually simpler to postulate a given L and A and determine to which PARTIAL DIFFERENTIAL EQUATION they correspond (Infeld and Rowlands 2000). See also PARTIAL DIFFERENTIAL EQUATION References Infeld, E. and Rowlands, G. "Integrable Equations in Two Space Dimensions as Treated by the Zakharov-Shabat Method." §7.10 in Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge University Press, pp. 192 /C1/99, 2000. Layer P-LAYER LCM LEAST COMMON MULTIPLE Leading Digit Phenomenon BENFORD’S LAW Leading Order Analysis A procedure for determining the behavior of an nth order ORDINARY DIFFERENTIAL EQUATION at a REMO- VABLE SINGULARITY without actually solving the equation. Considerdny dzn /C30Fdn/C281y dzn/C281 ;...;dy dx ; y; z ! ; (1) where F is ANALYTIC in z and rational in its other arguments. Proceed by making the substitution y(z) /C13a(z /C28z0) a (2) with a B1 : For example, in the equation d2y dz2 /C306y2 /C27Ay; (3) making the substitution gives aa( a /C281)(z /C28z0)a /C282 /C306a2(z /C28z0)2a /C27Aa(az /C28z0)a : (4) The most singular terms (those with the most NEGATIVE exponents) are called the "dominant bal- ance terms," and must balance exponents and COEF- FICIENTS at the SINGULARITY . Here, the first two terms are dominant, so a /C282 /C302a [ a /C30/C282 (5) 6a /C306a2 [a /C301 ; (6) and the solution behaves as y(z) /C30(z /C28z0) /C282 : The behavior in the NEIGHBORHOOD of the SINGULARITY is given by expansion in a LAURENT SERIES , in this case, y(z) /C30X/C12 j/C300aj(z /C28z0)j/C282 : (7) Plugging this series in yields X/C12 j/C300aj(j /C282)(j /C283)(z /C28z0)j /C284 /C306X/C12 j/C300X/C12 k /C300ajak(z /C28z0)j/C27k /C284 /C27AX/C12 j/C300aj(z /C28z0)j/C282 : (8) This gives RECURRENCE RELATIONS , in this case with a6arbitrary, so the (z /C28z0)6term is called the resonance or K OVALEVSKAYA EXPONENT . At the reso- nances, the COEFFICIENT will always be arbitrary. If no resonance term is present, the POLE present is not ordinary, and the solution must be investigated using aPSI FUNCTION . See also PSI FUNCTION References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 330, 1989. Leaf (Foliation) LetMnbe an n-MANIFOLD and let F/C30fFagdenote a PARTITION ofMinto DISJOINT path-connected SUB- SETS . Then if F is a FOLIATION of M, each Fa is called a leaf and is not necessarily closed or compact. See also FOLIATION References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 284, 1976. Leaf (Tree) An unconnected end of a TREE (i.e., a node of VERTEX DEGREE 1). The following tables gives the total numbers of leaves for various classes of graphs on n /C301, 2, ... nodes. For ROOTED TREES , the ROOT NODE is not counted as a leaf. graph type Sloane leaf count for n /C301, 2, ...nodes GRAPH A055540 0, 2, 4, 14, 38, 153, 766, ... TREE A003228 0, 2, 2, 5, 9, 21, 43, 101, ... LABELED TREEA055541 0, 2, 6, 36, 320, 3750, ... ROOTEDTREE A003227 1, 1, 3, 8, 22, 58, 160, 434, 1204, ... See also BRANCH ,CHILD ,FORK,ROOT NODE,TREE References Robinson, R. W. and Schwenk, A. J. "The Distribution of Degrees in a Large Random Tree." Discr. Math. 12, 359/C1/ 72, 1975. Sloane, N. J. A. Sequences A003227/M2744, A003228/ M0351, A055540, and A055541 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Leakage ALIASINGLeap JUMP Least Bound SUPREMUM Least Common Multiple The least common multiple of two numbers aandb, denoted LCM( a;b)o r[ a, b], is the smallest number mfor which there exist positive integers naand nb such that naa/C30nbb/C30m: (1) The least common multiple LCM( a;b;c;. . .) of more than two numbers is similarly defined. The plot above shows LCM(1 ;r) for rational r/C30m=n;which is equivalent to the NUMERATOR of the reduced form of m=n:/ The least common multiple of a,b,c, ..., is denoted LCM[a,b,c, ...] in Mathematica . The least common multiple of two numbers aandb can be obtained by finding the PRIME FACTORIZATION of each a/C30pa1 1/C1/C1/C1pann (2) b/C30pb1 1/C1/C1/C1pbnn; (3) where the p i/s are all PRIME FACTORS ofaandb, and if p idoes not occur in one factorization, then the corresponding exponent is taken as 0. The least common multiple is then given by LCM( a;b)/C30Yn i/C301pmax( ai;bi) i : (4) For example, consider LCM(12 ;30): 12/C3022/C21531/C21550(5) 30/C3021/C21531/C21551; (6) so LCM (12;30)/C3022/C21531/C21551/C3060: (7) Letmbe a common multiple of aandbso that m/C30ha/C30kb: (8) Write a /C30a1 GCD( a ; b) and b /C30b1 GCD( a ; b) ; where a1and b1are RELATIVELY PRIME by definition of the GREATEST COMMON DIVISOR GCD (a1 ; b1) /C301: Then ha1 /C30kb1 ; and from the DIVISION LEMMA (given that ha1 is DIVISIBLE by b1 and GCD( b1 ; a1) /C301); we have h is DIVISIBLE by b1 ; so h /C30nb1 (9) m /C30ha /C30nb1a /C30nab GCD( a ; b) : (10) The smallest m is given by n /C30 1, LCM( a; b) /C30ab GCD( a ; b) ; (11) so GCD( a ; b)LCM( a ; b) /C30ab (12) The LCM is IDEMPOTENT LCM( a ; a) /C30a (13) COMMUTATIVE LCM( a; b) /C30LCM( b; a) ; (14) ASSOCIATIVE LCM( a ; b ; c) /C30LCM(LCM( a ; b) ; c) /C30LCM( a; LCM( b; c)) ; (15) DISTRIBUTIVE LCM( ma; mb; mc) /C30m LCM( a ; b ; c) ; (16) and satisfies the ABSORPTION LAW GCD( a ; LCM( a ; b)) /C30a: (17) It is also true that LCM( ma; mb) /C30GCD( ma)GCD( mb) GCD( ma; mb)/C30mab GCD( a; b) /C30m LCM( a; b) : (18) See also GREATEST COMMON DIVISOR ,M ANGOLDT FUNCTION ,RELATIVELY PRIME References Guy, R. K. "Density of a Sequence with L.C.M. of Each Pair Less than x." §E2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 200 /C1/01, 1994. Nagell, T. "Least Common Multiple and Greatest Common Divisor." §5inIntroduction to Number Theory. New York: Wiley, pp. 16 /C1/9, 1951. Least Common Multiple Matrix Let S /C30fx1 ; ...; xn g be a set of n distinct POSITIVE INTEGERS . Then the matrix [S]nhaving the LEAST COMMON MULTIPLE LCM( xi ; xj)ofxi and xj as its i, jthentry is called the least common multiple matrix on S. See also BOURQUE- LIGH CONJECTURE References Hong, S. "On the Bourque-Ligh Conjecture of Least Common Multiple Matrices." J. Algebra 218, 216 /C1/28, 1999. Least Deficient Number A number for which s(n) /C302n /C281: A number is least deficient IFF it is a POWERS of 2: 1, 2, 4, 8, 16, 32, 64, ... (Sloane’s A000079). See also DEFICIENT NUMBER ,QUASIPERFECT NUMBER References Sloane, N. J. A. Sequences A000079/M1129 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Least Divisor LEAST PRIME FACTOR Least Period The smallest nfor which a point x0is a PERIODIC POINT of a function fso that fn(x0)/C30x0:For example, for the FUNCTION f(x)/C30/C28x;all points xhave period 2 (including x/C300). However, x/C300 has a least period of 1. The analogous concept exists for a PERIODIC SEQUENCE , but not for a PERIODIC FUNCTION . The least period is also called the exact period. Least Prime Factor Letn/C211 be any integer and let LD( n) be the least integer greatest than 1 that divides n. Then LD( n)i s a prime number, and if nis not prime, then [LD( n)]25n(Se´roul 2000, p. 7). For an INTEGER n]2;let lpf( x) denote the LEAST PRIME FACTOR ofn, i.e., the number p1in the factorization n /C30pa1 1/C1/C1/C1pak k; with pi Bpj for i B j. For n /C302, 3, ..., the first few are 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, ... (Sloane’s A020639). The above plot of the least prime factor function can be seen to resemble a jagged terrain of mountains, which leads to the appellation of "TWIN PEAKS "toa PAIR of INTEGERS (x, y) such that 1. x By, 2. lpf(x) /C30lpf(y) ;/ 3. For all z, x Bz By IMPLIES lpf(z) Blpf(x):/ The least multiple prime factors for SQUAREFUL integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, ... (Sloane’s A046027). Erdos et al. (1993) consider the least prime factor of the BINOMIAL COEFFICIENTS , and define what they term GOOD BINOMIAL COEFFICIENTS and EXCEPTIONAL BINOMIAL COEFFICIENTS . They also conjecture that lpfN k/C(*/C(+ 5max( N=k;29): (1) See also ALLADI- GRINSTEAD CONSTANT ,D ISTINCT PRIME FACTORS ,E RDOS- SELFRIDGE FUNCTION ,E U- CLID- MULLIN SEQUENCE ,E XCEPTIONAL BINOMIAL COEFFICIENT ,FACTOR ,GOOD BINOMIAL COEFFICIENT , GREATEST PRIME FACTOR ,LEAST COMMON MULTIPLE , MANGOLDT FUNCTION ,PRIME FACTORS ,TWIN PEAKS References Erdos, P.; Lacampagne, C. B.; and Selfridge, J. L. "Esti- mates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215/C1/24, 1993. Se´roul, R. "The Lowest Divisor Function." §8.4 in Program- ming for Mathematicians. Berlin: Springer-Verlag, pp. 9 /C1/ 1 and 165 /C1/67, 2000. Sloane, N. J. A. Sequences A020639 and A046027 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Least Squares Fitting A mathematical procedure for finding the best fitting curve to a given set of points by minimizing the sumof the squares of the offsets ("the residuals"rpar; ofthe points from the curve. The sum of the squares of the offsets is used instead of the offset absolute valuesbecause this allows the residuals to be treated as acontinuous differentiable quantity. However, because squares of the offsets are used, outlying points canhave a disproportionate effect on the fit, a property which may or may not be desirable depending on theproblem at hand. In practice, the vertical offsets from a line are almost always minimized instead of the perpendicular off- sets. This allows uncertainties of the data pointsalong the x- and y-axes to be incorporated simply, and also provides a much simpler analytic form forthe fitting parameters than would be obtained using afit based on perpendicular distances. In addition, thefitting technique can be easily generalized from abest-fit line to a best-fit polynomial when sums of vertical distances are used (which is not the caseusing perpendicular distances). For a reasonablenumber of noisy data points, the difference betweenvertical and perpendicular fits is quite small. The linear least squares fitting technique is the simplest and most commonly applied form of LINEAR REGRESSION and provides a solution to the problem of finding the best fitting straight line through a set of points. In fact, if the functional relationship between the two quantities being graphed is known to withinadditive or multiplicative constants, it is commonpractice to transform the data in such a way that the resulting line isa straight line, say by plotting Tvs.ffiffiffi lp instead of Tvs.lin the case of analyzing the period Tof a pendulum as a function of its length l. For this reason, standard forms for EXPONENTIAL , LOGARITHMIC , and POWER laws are often explicitly computed. The formulas for linear least squares fitting were independently derived by Gauss andLegendre. For NONLINEAR LEAST SQUARES FITTING to a number of unknown parameters, linear least squares fitting may be applied iteratively to a linearized form of thefunction until convergence is achieved. Depending on the type of fit and initial parameters chosen, the nonlinear fit may have good or poor convergenceproperties. If uncertainties (in the most general case,error ellipses) are given for the points, points can beweighted differently in order to give the high-qualitypoints more weight. The residuals of the best-fit line for a set of npoints using unsquared perpendicular distances d iof points (xi;yi) are given by R/C222/C13Xn i/C301di: (1) Since the perpendicular distance from a line y/C30a/C27 bxto point iis given by di/C30½yi/C28(a/C27bxi)½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2p ; (2) the function to be minimized is R/C222/C13Xn i/C301½yi/C28(a/C27bxi)½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2p : (3) Unfortunately, because the absolute value function does not have continuous derivatives, minimizing R/C222 is not amenable to analytic solution. However, if thesquare of the perpendicular distances R 2 /C222/C13Xn i/C301[yi/C28(a/C27bxi)]2 1/C27b2(4) is minimized instead, the problem can be solved in closed form. R2 /C222is a minimum when (suppressing the indices) @R2/C222 @a/C302 1/C27b2X [y/C28(a/C27bx)](/C281)/C300 (5) and @R2 /C222 @b/C302 1/C27b2X [y/C28(a/C27bx)](/C28x) /C27X[y/C28(a/C27bx)]2(/C281)(2b) (1/C27b2)2 /C300: (6) The former gives a/C30Py/C28bPx n/C30¯y/C28b¯x; (7) and the latter (1/C27b2)X [y/C28(a/C27bx)]x/C27bX [y/C28(a/C27bx)]2/C300:(8) But [y/C28(a/C27bx)]2/C30y2/C282(a/C27bx)y/C27(a/C27bx)2 /C30y2/C282ay/C282bxy/C27a2/C272abx/C27b2x2; (9) so (8) becomes (1/C27b2)X xy/C28aX x/C28bX x2/C(%/C(r /C27bX y2/C282aX y/C282b/C(% /C2X xy/C27a2X 1/C272abX x/C27b2X x2Þ/C300 (10) [(1/C27b2)(/C28b)/C27b(b2)]X x2/C27[(1/C27b2)/C282b2]X xy/C27bX y2/C27[/C28a(1/C27b2)/C272ab2]X x/C282abX y /C27ba2X 1/C300 (11) /C28bX x2/C27(1/C28b2)X xy/C27bX y2/C27a(b2/C281)X x /C282abX y/C27ba2n/C300: (12) Plugging (7) into (12) then gives /C28bX x2/C27(1/C28b2)X xy/C27bX y2/C271 n(b2/C281) /C2X y/C28bX x/C(%/C(r X x /C282 nX y/C28bX x/C(%/C(r bX y /C271 nbX y/C28bX x/C(%/C(r2 /C300 (13) After a fair bit of algebra, the result is b2/C27Py2/C28Px2/C271 nPx ðÞ2/C28Py ðÞ2hi 1 nPxPy/C28Pxyb/C281 /C300: (14) So define B/C131 2Py2/C281 nPy ðÞ2hi /C28Px2/C281 nPx ðÞ2hi 1 nPxPy/C28Pxy /C3012Py2/C28n¯y2ðÞ /C28Px2/C28n¯x2ðÞ n¯x¯y/C28Pxy; (15) and the QUADRATIC FORMULA gives b/C30/C28B9ffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2/C271p ; (16) with afound using (7). Note the rather unwieldy form of the best-fit parameters in the formulation. In addition, minimizing R2 /C222for a second- or higher-order POLYNOMIAL leads to polynomial equations having higher order, so this formulation cannot be extended. Vertical least squares fitting proceeds by finding the sum of the squares of the vertical deviations R2of a set of ndata points R2/C13X [yi/C28f(xi;a1;a2;...;an)]2(17) from a function f. Note that this procedure does not minimize the actual deviations from the line (whichwould be measured perpendicular to the given func- tion). In addition, although the unsquared sum of distances might seem a more appropriate quantity tominimize, use of the absolute value results in dis- continuous derivatives which cannot be treated ana- lytically. The square deviations from each point aretherefore summed, and the resulting residual is then minimized to find the best fit line. This procedure results in outlying points being given disproportio- nately large weighting. The condition for R2to be a minimum is that @(R2) @ai/C300 (18) fori/C301, ..., n. For a linear fit, f(a;b)/C30a/C27bx; (19) so R2(a;b)/C13Xn i/C301[yi/C28(a/C27bxi)]2(20) @(R2) @a/C30/C282Xn i/C301[yi/C28(a/C27bxi)]/C300 (21) @(R2) @b/C30/C282Xn i/C301[yi/C28(a/C27bxi)]xi/C300: (22) These lead to the equations na/C27bX x/C30X y (23) aX x/C27bX x2/C30X xy; (24) where the subscripts have been dropped for concise- ness. In MATRIX form, nPxPxPx2/C)P/C)( a b/C)P/C)( /C30PyPxy/C)P/C)( ; (25) so a b/C)P/C)( /C30nPxPxPx2/C)P/C)(/C281PyPxy/C)P/C)( : (26) The 2/C292MATRIX INVERSE is a b/C)P/C)( /C301 nPx2/C28Px ðÞ2 /C2PyPx2/C28PxPxy nPxy/C28PxPy/C)P/C)( ; (27) so a/C30PyPx2/C28PxPxy nPx2/C28Px ðÞ2(28) /C30¯yPx2/C28¯xPxyPx2/C28n¯x2(29) b/C30nPxy/C28PxPy nPx2/C28Px ðÞ2(30) /C30Pxy/C28n¯x¯yPx2/C28n¯x2(31) (Kenney and Keeping 1962). These can be rewrittenin a simpler form by defining the sums of squares ssxx/C30Xn i/C301(xi/C28¯x)2/C30X x2/C(%/C(r /C28n¯x2(32) ssyy/C30Xn i/C301(yi/C28¯y)2/C30X y2/C(%/C(r /C28n¯y2(33) ssxy/C30Xn i/C301(xi/C28¯x)(yi/C28¯y)/C30X xy/C(%/C(r /C28n¯x¯y; (34) which are also written as s2 x/C30ssxx (35) s2y/C30ssyy (36) cov(x;y)/C30ssxy: (37) Here, cov( x;y) is the COVARIANCE ands2 xands2yare variances. Note that the quantities axyand ax2can also be interpreted as the DOT PRODUCTS X x2/C30x /C215x (38) X xy/C30x /C215y: (39) In terms of the sums of squares, the REGRESSION COEFFICIENT bis given by b/C30cov(x;y) s2 x/C30ssxy ssxx; (40) andais given in terms of busing (24) as a/C30¯y/C28b¯x: (41) The overall quality of the fit is then parameterized in terms of a quantity known as the CORRELATION COEFFICIENT , defined by r2/C30ss2 xy ssxxssyy; (42) which gives the proportion of ssyywhich is accounted for by the regression. The STANDARD ERRORS foraandbare SE(a)/C30sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n/C27¯x2 ssxxs (43) SE(b)/C30s ffiffiffiffiffiffiffiffissxxp : (44) Let ˆyibe the vertical coordinate of the best-fit line with x-coordinate xi;so ˆyi/C13a/C27bxi; (45) then the error between the actual vertical point yiand the fitted point is given by ei /C13yi /C28 ˆyi : (46) Now define s2 as an estimator for the variance in ei ; s2 /C30Xn i/C301e2 i n /C28 2 : (47) Then s can be given by s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ssyy /C28 bssxy n /C28 2s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ssyy /C28ss2 xy ssxx n /C28 2vuut(48) (Acton 1966, pp. 32 /C1/5; Gonick and Smith 1993, pp. 202 /C1/04). Generalizing from a straight line (i.e., first degree polynomial) to a kth degree POLYNOMIAL y /C30a0 /C27a1x /C27.../C27akxk ; (49) the residual is given by R2 /C13Xn i/C301[yi /C28(a0 /C27a1xi /C27.../C27akxk i )]2 : (50) The PARTIAL DERIVATIVES (again dropping super- scripts) are @(R2) @a0/C30/C282X [y /C28(a0 /C27a1x /C27.../C27akxk)] /C300 (51) @(R2) @a1/C30/C282X [y /C28(a0 /C27a1x /C27.../C27akxk)]x /C300 (52) @(R2) @ak/C30/C282X [y /C28(a0 /C27a1x /C27.../C27akxk)]xk /C300: (53) These lead to the equations a0n /C27a1X x /C27.../C27akX xk /C30X y (54) a0X x /C27a1X x2 /C27.../C27akX xk /C271 /C30X xy (55) a0X xk /C27a1X xk /C271 /C27.../C27akX x2k /C30X xky (56) or, in MATRIX form nPx /C1/C1/C1Pxk PxPx2/C1/C1/C1Pxk/C271 nn::: nPxkPxk /C271/C1/C1/C1Px2k2 6643 775a 0 a1 n ak2 6643 775 /C30PyPxy nPx ky2 6643 775: (57) This is a V ANDERMONDE MATRIX . We can also obtain the MATRIX for a least squares fit by writing1 x1/C1/C1/C1 xk 1 1 x2/C1/C1/C1 xk2 nn ::: n 1 xn/C1/C1/C1 xkn2 6643 775a 0 a1 n ak2 6643 775/C30y 1 y2 n yn2 6643 775: (58) Premultiplying both sides by the TRANSPOSE of the first MATRIX then gives 11 /C1/C1/C1 1 x1x2/C1/C1/C1 xn nn::: n xk 1xk2/C1/C1/C1 xkn2 6643 7751 x1/C1/C1/C1 xk 1 1 x2/C1/C1/C1 xk 2 nn ::: n 1 xn/C1/C1/C1 xkn2 6643 775a0 a1 n ak2 6643 775 /C3011 /C1/C1/C1 1 x1x2/C1/C1/C1 xn nn::: n xk 1xk2/C1/C1/C1 xkn2 6643 775y1 y2 n yn2 6643 775; (59) so nPx /C1/C1/C1Pxn PxPx2/C1/C1/C1Pxn/C271 nn::: nPxnPxn/C271/C1/C1/C1Px2n2 6643 775a0 a1 n ak2 6643 775 /C30PyPxy nPxky2 6643 775: (60) As before, given m points (x i ; yi) and fitting with POLYNOMIAL COEFFICIENTS a0 ; ..., an gives y1 y2 n ym2 6643 775/C301 x 1x2 1/C1/C1/C1 xn1 1 x2x22/C1/C1/C1 xn2 nn ::: n 1 xmx2m/C1/C1/C1 xnm2 6643 775a 0 a1 n an2 6643 775; (61) In MATRIX notation, the equation for a polynomial fit is given by y /C30Xa: (62) This can be solved by premultiplying by the MATRIX TRANSPOSE XT; XTy/C30XTXa: (63) This MATRIX EQUATION can be solved numerically, or can be inverted directly if it is well formed, to yield the solution vector a/C30(XTX)/C281XTy: (64) Setting m/C301 in the above equations reproduces the linear solution. See also CORRELATION COEFFICIENT ,INTERPOLATION , LEAST SQUARES FITTING– EXPONENTIAL ,L EAST SQUARES FITTING– LOGARITHMIC ,LEAST SQUARES FIT- TING– POWER LAW,M OORE- PENROSE GENERALIZED MATRIX INVERSE ,N ONLINEAR LEAST SQUARES FIT- TING ,REGRESSION COEFFICIENT ,SPLINE References Acton, F. S. Analysis of Straight-Line Data. New York: Dover, 1966. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1969. Chatterjee, S.; Hadi, A.; and Price, B. "Simple Linear Regression." Ch. 2 in Regression Analysis by Example, 3rd ed. New York: Wiley, pp. 21 /C1/0, 2000. Gauss, C. F. "Theoria combinationis obsevationum erroribus minimis obnoxiae." Werke, Bd. 4,p.1. Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, 1993. Kenney, J. F. and Keeping, E. S. "Linear Regression, Simple Correlation, and Contingency." Ch. 8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 199 /C1/37, 1951. Kenney, J. F. and Keeping, E. S. "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252 /C1/85, 1962. Lancaster, P. and Salkauskas, K. Curve and Surface Fitting: An Introduction. London: Academic Press, 1986. Laplace, P. S. Ch. 4 in The´orie anal. des prob., Livre 2. 1812. Lawson, C. and Hanson, R. Solving Least Squares Problems. Englewood Cliffs, NJ: Prentice-Hall, 1974. Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 21 /C1/4, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Fitting Data to a Straight Line" "Straight-Line Data with Errors in Both Coordinates," and "General Linear Least Squares." §15.2, 15.3, and 15.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 655 /C1/75, 1992. Whittaker, E. T. and Robinson, G. "The Method of Least Squares." Ch. 9 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 209-, 1967. York, D. "Least-Square Fitting of a Straight Line." Canad. J. Phys. 44, 1079 /C1/086, 1966. Least Squares Fitting * /Exponential To fit a functional form y /C30AeBx ; (1) take the LOGARITHM of both sides ln y /C30ln A /C27Bx : (2)The best-fit values are then a /C30Pln yPx2 /C28P xPx ln y nPx2 /C28Px ðÞ2 (3) b /C30nPx ln y /C28P xPln y nPx2 /C28PxðÞ2 ; (4) where B /C13b and A /C13exp(a) :/ This fit gives greater weights to small y values so, in order to weight the points equally, it is often better to minimize the function X y(ln y /C28a /C28bx)2 : (5) Applying LEAST SQUARES FITTING gives aX y/C27bX xy/C30X ylny (6) aX xy/C27bX x2y/C30X xylny (7) PyPxyPxyPx2y/C)P/C)( a b/C)P/C)( /C30PylnyPxylny/C)P/C)( : (8) Solving for aandb, a/C30P(x2y)P(ylny)/C28P(xy)P(xylny)PyP(x2y)/C28Pxy ðÞ2(9) b/C30PyP(xylny)/C28P(xy)P(ylny)PyP(x2y)/C28Pxy ðÞ2: (10) In the plot above, the short-dashed curve is the fit computed from (3) and (4) and the long-dashed curveis the fit computed from (9) and (10). See also L EAST SQUARES FITTING ,LEAST SQUARES FITTING– LOGARITHMIC ,L EAST SQUARES FITTING– POWER LAW Least Squares Fitting * /Logarithmic Given a function OF THE FORM y/C30a/C27blnx; (1) the COEFFICIENTS can be found from LEAST SQUARES FITTING as b /C30nP(y ln x) /C28P yP(ln x) nP(ln x)2hi /C28P(ln x) ½Þ2 (2) a /C30Py /C28 bP(ln x) n: (3) See also LEAST SQUARES FITTING ,LEAST SQUARES FITTING– EXPONENTIAL ,L EAST SQUARES FITTING– POWER LAW Least Squares Fitting * /Power Law Given a function OF THE FORM y /C30AxB ; (1) LEAST SQUARES FITTING gives the COEFFICIENTS as b /C30nP(ln x ln y) /C28P(ln x)P(ln y) nP[(ln x)2] /C28Pln x ðÞ2 (2) a /C30P(ln y) /C28 bP(lnx) n; (3) where B/C13bandA/C13exp(a):/ See also LEAST SQUARES FITTING ,LEAST SQUARES FITTING– EXPONENTIAL ,L EAST SQUARES FITTING– LOGARITHMIC Least Universal Exponent CARMICHAEL FUNCTION Least Upper Bound SUPREMUM Lebesgue Constants (Fourier Series) N.B. A detailed online essay by S. Finch was the starting point for this entry. Assume a function fis integrable over the interval [/C28p;p] and Sn(f;x) is the nth partial sum of the FOURIER SERIES off, so thatak/C301 pgp /C28pf(t)cos(kt)dt (1) bk/C301pgp /C28pf(t)sin(kt)dt (2) and Sn(f;x)/C301 2a0/C27Xn k/C301[akcos(kx)/C27bksin(kx)]() :(3) If ½f(x)½51 (4) for all x, then Sn(f;x)51 pgp 0sin12(2n/C271)uhi/C()/C()/C()/C()/C()/C() sin 1 2u/C(%/C(r du/C30Ln; (5) andLnis the smallest possible constant for which this holds for all continuous f. The first few values of Ln are L0/C301 (6) L1/C301 3/C272ffiffiffi 3p p/C301:435991124 . . . (7) L2/C301 5/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C282ffiffiffi 5pp p/C301:642188435 . . . (8) L3/C301 7/C271 p4 sin2 7p/C(%/C(r /C282 sin47p/C(%/C(r /C2716 3sin67p/C(%/C(r h /C282 sin8 7p/C(%/C(r /C2723sin12 7p/C(%/C(r /C2743sin18 7p/C(%/C(r /C138 /C301:778322861 . . . : (9) L4/C3039ffiffiffi 3p 18p/C271 9/C271 pi/C274 sin2 9p/C(%/C(r /C272 sin49p/C(%/C(r h /C275 sin8 9p/C(%/C(r /C273 sin16 9p/C(%/C(r /C27sin32 9p/C(%/C(r/C)( (10) /C301:880080599 . . . : Some sum FORMULAS forLninclude Ln/C301 2n/C271/C272 pXn k/C3011 ktanpk 2n/C271 ! /C3016 p2X/C12 k/C301X(2n/C271)k j/C3011 4k2/C2811 2j/C281(11) (Zygmund 1959) and integral FORMULAS include Ln/C304g/C12 0tanh[(2 n/C271)x] tanh xdx p2/C274x2 /C304 p2g/C12 0sinh[(2 n/C271)x] sinh xln coth1 2(2n/C271)xhino dx (12) (Hardy 1942). For large n, 4 p2lnnBLnB3/C274 p2lnn: (13) This result can be generalized for an r-differentiable function satisfying drf dxr/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()51 (14) for all x. In this case, f(x)/C28S n(f;x) jj 5Ln;r/C304 p2lnn nr/C27O1 nr ! ; (15) where Ln;r/C301 pgp /C28pX/C12 k/C30n/C271sin(kx) kr/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()dx forr]1 odd 1 pgp /C28pX/C12 k/C30n/C271cos(kx) kr/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()dx forr]1 even8 >>>>< >>>>:(16) (Kolmogorov 1935, Zygmund 1959). Watson (1930) showed that lim n0/C12Ln/C284 p2ln(2n/C271)"# /C30c; (17) where c/C308 p2X/C12 k/C301lnk 4k2/C281 ! /C284 p2G?1 2/C(%/C(r G1 2/C(%/C(r (18) /C308 p2X/C12 j/C300l(2j/C272)/C281 2j/C271"# /C274 p2(2 ln 2 /C27g) (19) /C300:9894312738 :::; (20) where G(z) is the GAMMA FUNCTION ,l(z) is the DIRICHLET LAMBDA FUNCTION , and gis the E ULER- MASCHERONI CONSTANT . References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lbsg/lbsg.html. Hardy, G. H. "Note on Lebesgue’s Constants in the Theory of Fourier Series." J. London Math. Soc. 17,4/C1/3, 1942. Kolmogorov, A. N. "Zur Gro ¨ssenordnung des Restgliedes Fourierscher reihen differenzierbarer Funktionen." Ann. Math. 36, 521/C1/26, 1935. Watson, G. N. "The Constants of Landau and Lebesgue." Quart. J. Math. Oxford 1, 310/C1/18, 1930. Zygmund, A. G. Trigonometric Series, 2nd ed., Vols. 1 /C1/. Cambridge, England: Cambridge University Press, 1959.Lebesgue Constants (Lagrange Interpolation) N.B. A detailed online essay by S. Finch was the starting point for this entry. Define the nth Lebesgue constant for the L AGRANGE INTERPOLATING POLYNOMIAL by Ln(X)/C13max /C2815x51Xn k/C301Y j"kx/C28xj xk/C28xj/C()/C()/C()/C()/C()/C()/C()/C()/C()/C(): (1) It is true that L n>4 p2lnn/C281: (2) The efficiency of a Lagrange interpolation is related to the rate at which Lnincreases. Erdos (1961) proved that there exists a POSITIVE constant such that Ln>2 plnn/C28C (3) for all n. Erdos (1961) further showed that LnB2 plnn/C274; (4) so (3) cannot be improved upon. References Erdos, P. "Problems and Results on the Theory of Interpola- tion, II." Acta Math. Acad. Sci. Hungary 12, 235/C1/44, 1961. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lbsg/lbsg.html. Lebesgue Covering Dimension An important DIMENSION and one of the first dimen- sions investigated. It is defined in terms of covering sets, and is therefore also called the COVERING DIMENSION . Another name for the Lebesgue covering dimension is the TOPOLOGICAL DIMENSION . ASPACE has Lebesgue covering dimension mif for every open COVER of that space, there is an open COVER that refines it such that the refinement has order at most m/C271:Consider how many elements of the cover contain a given point in a base space. If thishas a maximum over all the points in the base space, then this maximum is called the order of the cover. Ifa SPACE does not have Lebesgue covering dimension mfor any m, it is said to be infinite dimensional. Results of this definition are: 1. Two homeomorphic spaces have the samedimension, 2.R nhas dimension n, 3. A TOPOLOGICAL SPACE can be embedded as a closed subspace of a E UCLIDEAN SPACE IFF it is LOCALLY COMPACT ,HAUSDORFF ,SECOND COUNTA- BLE, and is finite-dimensional (in the sense of the LEBESGUE DIMENSION ), and 4. Every compact metrizable m-dimensional TOPO- LOGICAL SPACE can be embedded in R2m/C271 :/ See also LEBESGUE MINIMAL PROBLEM References Dieudonne, J. A. A History of Algebraic and Differential Topology. Boston, MA: Birkha ¨user, 1994. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 414, 1980. Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975. Lebesgue Decomposition (Measure) Any COMPLEX MEASURE l decomposes into an ABSO- LUTELY CONTINUOUS measure laand a SINGULAR MEASURE lc ; with respect to some positive measure m: This is the LEBESGUE DECOMPOSITION l /C30 la /C27 lc : See also ABSOLUTELY CONTINUOUS ,COMPLEX MEA- SURE ,FUNDAMENTAL THEOREMS OF CALCULUS ,LE- BESGUE MEASURE ,P OLAR REPRESENTATION (MEASURE ), RADON- NIKODYM THEOREM ,S INGULAR MEASURE References Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, p. 121, 1987. Lebesgue Dimension LEBESGUE COVERING DIMENSION Lebesgue Identity (a2 /C27b2 /C27c2 /C27d2)2 /C30(a2 /C27b2 /C28c2 /C28d2)2 /C27(2ac /C272bd)2 /C27(2ad /C282bc)2 (Nagell 1951, pp. 194 /C1/95). See also DIOPHANTINE EQUATION–2ND POWERS ,EU- LER FOUR- SQUARE IDENTITY References Nagell, T. Introduction to Number Theory. New York: Wiley, 1951. Lebesgue Integrable A real-valued function f defined on the reals R is called Lebesgue integrable if there exists a SEQUENCE of STEP FUNCTIONS ffn g such that the following two conditions are satisfied:1. a/C12 n/C301 f fnjjB/C12 ;/ 2. f(x) /C30a/C12n/C301 fn(x) for every x /C23R such that a/C12 n/C301 f fnjjB/C12 :/ Here, the above integral denotes the ordinary RIE- MANN INTEGRAL . Note that this definition avoids explicit use of the LEBESGUE MEASURE . See also INTEGRAL ,LEBESGUE INTEGRAL ,RIEMANN INTEGRAL ,STEP FUNCTION Lebesgue Integral The LEBESGUE INTEGRAL is defined in terms of upper and lower bounds using the LEBESGUE MEASURE of a SET. It uses a LEBESGUE SUM Sn /C30 hi m(Ei) where hi is the value of the function in subinterval i, and m(Ei)is the LEBESGUE MEASURE of the SET Eiof points for which values are approximately hi : This type of integral covers a wider class of functions than does the RIEMANN INTEGRAL . The Lebesgue integral of a function f over a MEASURE SPACE X is written gXf ; or sometimes gXfdm to emphasize that the integral is taken with respect to the MEASURE m:/ See also A-INTEGRABLE ,COMPLETE FUNCTIONS ,IN- TEGRAL ,MEASURE ,MEASURE SPACE References Kestelman, H. "Lebesgue Integral of a Non-Negative Func- tion" and "Lebesgue Integrals of Functions Which Are Sometimes Negative." Chs. 5 /C1/ in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 113 /C1/60, 1960. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 141, 1984. Lebesgue Measurability Problem A problem related to the CONTINUUM HYPOTHESIS which was solved by Solovay (1970) using the IN- ACCESSIBLE CARDINALS AXIOM . It has been proven by Shelah and Woodin (1990) that use of this AXIOM is essential to the proof. See also CONTINUUM HYPOTHESIS ,INACCESSIBLE CARDINALS AXIOM ,LEBESGUE MEASURE References Shelah, S. and Woodin, H. "Large Cardinals Imply that Every Reasonable Definable Set of Reals is Lebesgue Measurable." Israel J. Math. 70, 381/C1/94, 1990. Solovay, R. M. "A Model of Set-Theory in which Every Set of Reals is Lebesgue Measurable." Ann. Math. 92,1/C1/6, 1970. Lebesgue Measure An extension of the classical notions of length and AREA to more complicated sets. Given an open set S /C13 ak(ak ; bk) containing DISJOINT intervals, mL(S) /C13X k(bk /C28ak) : Given a CLOSED SET S?/C13[a ; b] /C28ak(ak ; bk) ; mL(S ?) /C13(b /C28a) /C28X k(bk /C28ak): A unit LINE SEGMENT has Lebesgue measure 1; the CANTOR SET has Lebesgue measure 0. The MIN- KOWSKI MEASURE of a bounded, CLOSED SET is the same as its Lebesgue measure (Ko 1995). See also CANTOR SET,M EASURE ,R IESZ- FISCHER THEOREM References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 4, 1991. Kestelman, H. "Lebesgue Measure." Ch. 3 in Modern The- ories of Integration, 2nd rev. ed. New York: Dover, pp. 67 /C1/ 1, 1960. Ko, K.-I. "A Polynomial-Time Computable Curve whose Interior has a Nonrecursive Measure." Theoret. Comput. Sci. 145, 241/C1/70, 1995. Lebesgue Minimal Problem Find the plane LAMINA of least AREA Awhich is capable of covering any plane figure of unit GENERAL- IZED DIAMETER .A UNIT CIRCLE is too small, but a HEXAGON circumscribed on the UNIT CIRCLE is larger than necessary. Pa ´l (1920) showed that the hexagon can be reduced by cutting off two EQUILATERAL TRIANGLES on the corners of the hexagon which are tangent to the hexagon’s INCIRCLE (Wells 1991; left figure above). Sprague subsequently demonstrated that an additional small curvilinear region could be removed (Wells 1991; right figure above). Theseconstructions give upper bounds. The HEXAGON having INRADIUS r/C301=2 (giving a DIAMETER of 1) has side length a/C302rtanp n ! /C301 3ffiffiffi 3p ; (1) and the area of this HEXAGON is A1/C30nr2tanp n ! /C301 2ffiffiffi 3p :0:866025 : (2) In the above figure, the SAGITTA is given by s/C30rtanp n ! tanp 2n ! /C301 62ffiffiffi 3p /C283/C(%/C(r ; (3) and the other distances by b/C30stanp 3 ! /C30ffiffiffi 3p s (4) h/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2/C27b2p /C302s; (5) so the area of one of the equilateral triangles removed in Pa´l’s reduction is AD/C30bs/C30ffiffiffi 3p s2/C301 127ffiffiffi3p /C3012/C(%/C(r :0:0773505 ; (6) so the area left after removing two of these triangles is A 2/C13A1/C282AD/C302 33/C28ffiffiffi 3p/C(%/C(r :0:845299 : (7) Computing the area of the region removed in Spra- gue’s construction is more involved. First, use similar triangles a/C28h h/C30r2 r1(8) together with r1 /C27r2 /C30r to obtain r2 /C302r(a /C28 h) a/C30ffiffiffi 3p /C281: (9) Then x /C30r2 cosp 3 ! /C301 2ffiffiffi 3p /C281/C(%/C(r ; (10) and the angle u is given by u /C30cos/C281x 2r ! /C30cos/C2811 2ffiffiffi 3p /C281/C(%/C(rhi ; (11) and the angle f is just f /C30 u /C281 3 p: (12) The distance h? is h?/C302r tan f (13) l /C302r sec f; (14) and the area between the triangle and sector is dA(1) 3/C30rh /C281 2(2r)2 f /C302r2(tan f /C28 f) /C3012(tan f /C28 f) :0:000554738 : (15) The area of the small triangle is dA(2) 3/C301 2(l /C282r)(h /C28h?) /C3016(sec f /C281)(2ffiffiffi 3p /C283 /C283 tan f) :0 :0000264307 ; (16) so the total area remaining is A3 /C30A2 /C282(dA(1) 3/C28dA(2)3) /C300:844137 : (17) It is also known that a lower bound for the AREA is given by A >1 8 p /C2714ffiffiffi 3p :0:825712 (18) (Ogilvy 1990). See also AREA,BORSUK’S CONJECTURE ,GENERALIZED DIAMETER ,KAKEYA NEEDLE PROBLEM References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 99, 1987. Coxeter, H. S. M. "Lebesgue’s Minimal Problem." Eureka 21, 13, 1958. Gru¨nbaum, B. "Borsuk’s Problem and Related Questions." Proc. Sympos. Pure Math, Vol. 7. Providence, RI: Amer. Math. Soc., pp. 271 /C1/84, 1963. Kakeya, S. "Some Problems on Maxima and Minima Re- garding Ovals." Sci. Reports Toˆhoku Imperial Univ., Ser. 1 (Math., Phys., Chem.) 6,71/C1/8, 1917. Ogilvy, C. S. Tomorrow’s Math: Unsolved Problems for the Amateur, 2nd ed. New York: Oxford University Press, 1972.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 142 /C1/44, 1990. Pa´l, J. "Ueber ein elementares Variationsproblem." Det Kgl. Danske videnkabernes selskab, Math.-fys. meddelelser 3, Nr. 2, 1 /C1/5, 1920. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 138, 1991. Yaglom, I. M. and Boltyanskii, V. G. Convex Figures. New York: Holt, Rinehart, & Winston, pp. 18 and 100, 1961. Lebesgue-Radon Integral LEBESGUE- STIELTJES INTEGRAL Lebesgue’s Dominated Convergence Theorem Suppose that ffn g is a sequence of MEASURABLE FUNCTIONS , that fn 0 f ; as n 0/C12; and that ½fn ½5g for all n, where g is integrable. Then f is integrable, and gfdm/C30lim n0/C12gfndm: See also ALMOST EVERYWHERE CONVERGENCE ,M EA- SURE THEORY ,POINTWISE CONVERGENCE References Browder, A. Mathematical Analysis: An Introduction. New York: Springer-Verlag, 1996. Lebesgue Singular Integrals Un(f)/C30gb af(x)Kn(x)dx; where fKn(x)gis a SEQUENCE ofCONTINUOUS FUNC- TIONS . Lebesgue-Stieltjes Integral Leta(x) be a monotone increasing function and define an INTERVAL I/C30(x1;x2):Then define the NONNEGA- TIVE function U(I)/C30a(x2/C270)/C28a(x1/C270): The L EBESGUE INTEGRAL with respect to a MEASURE constructed using U(I) is called the Lebesgue- Stieltjes integral, or sometimes the L EBESGUE- RADON INTEGRAL . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 326, 1980. Lebesgue Sum Sn/C30X ihim(Ei); where m(Ei) is the MEASURE of the SET Ei of points on the X-AXIS for which f(x) : hi :/ Le Cam’s Identity Let Snbe the sum of n random variates Xiwith a BERNOULLI DISTRIBUTION with P(Xi /C301) /C30pi : Then X/C12 k /C300P(Sn /C30k) /C28e /C28l lk k!/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()B 2X n i /C301p2 i ; where l /C13Xn i /C301pi : See also BERNOULLI DISTRIBUTION References Cox, D. A. "Introduction to Fermat’s Last Theorem." Amer. Math. Monthly 101,3/C1/4, 1994. Leech Lattice A 24-D Euclidean lattice. An AUTOMORPHISM of the Leech lattice modulo a center of two leads to the CONWAY GROUP Co1 : Stabilization of the 1- and 2-D sublattices leads to the CONWAY GROUPS Co2 and Co3 ; the HIGMAN- SIMS GROUP HS and the MCLAUGHLIN GROUP McL . The Leech lattice appears to be the densest HYPER- SPHERE PACKING in 24-D, and results in each HYPER- SPHERE touching 195,560 others. The number of vectors with norm n in the Leech lattice (i.e., its "theta series"rpar; is given by u(n) /C3065520 691[s11(n) /C28 t(n)] ; (1) where s11 is the DIVISOR FUNCTION giving the sum of the 11th powers of the DIVISORS of n and t(n) is the TAU FUNCTION (Conway and Sloane 1993, p. 135). The first few values for n /C30 1, 2, ... are 0, 196560, 16773120, 398034000, ... (Sloane’s A008408). This is an immediate consequence of the theta function for Leech’s lattice being a weight 12 MODULAR FORM and having no vectors of norm two. u(n) has the generat- ing function f(q) /C30[E2(q)]3 /C28720q2Y/C12 m/C301(1 /C28q2m)24 (2) /C30 1 /C27240X/C12 m/C301s3(m)q2m ! 3 /C28720q2Y/C12 m/C301(1 /C28q2m)24 (3) 1 /C27196560 q4 /C2716773120 q6 /C273980034000 q8 /C27... ; (4) where E2(q) is the RAMANUJAN- EISENSTEIN SERIESwhich is the theta series of the E8lattice (Sloane’s A004009). See also BARNES- WALL LATTICE ,CONWAY GROUPS , COXETER- TODD LATTICE ,E ISENSTEIN SERIES ,H IG- MAN- SIMS GROUP ,H YPERSPHERE ,H YPERSPHERE PACKING ,K ISSING NUMBER ,M CLAUGHLIN GROUP , TAU FUNCTION References Conway, J. H. and Sloane, N. J. A. "The 24-Dimensional Leech Lattice L24 ;/" "A Characterization of the Leech Lattice," "The Covering Radius of the Leech Lattice," "Twenty-Three Constructions for the Leech Lattice," "The Cellular of the Leech Lattice," "Lorentzian Forms for the Leech Lattice." §4.11, Ch. 12, and Chs. 23 /C1/6in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 131 /C1/35, 331 /C1/36, and 478 /C1/26, 1993. Leech, J. "Notes on Sphere Packings." Canad. J. Math. 19, 251 /C1/67, 1967. Sloane, N. J. A. Sequences A004009/M5416 and A008408 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Wilson, R. A. "Vector Stabilizers and Subgroups of Leech Lattice Groups." J. Algebra 127, 387 /C1/08, 1989. Lefschetz Number If K is a finite complex and h : Kjj0 Kjj is a continuous map, then L(h) /C30X (/C281)pTr(h/C31; Hp(K)=Tp(K)) is the Lefschetz number of the map h. See also EULER NUMBER (FINITE COMPLEX ) References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, p. 125, 1993. Lefschetz Theorems Each DOUBLE POINT assigned to an irreducible ALGE- BRAIC CURVE whose GENUS is NONNEGATIVE imposes exactly one condition. See also HARD LEFSCHETZ THEOREM References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 104, 1959. Lefshetz Fixed Point Formula Let K be a finite complex, let h : Kjj0 Kjjbe a continuous map. If L(h) "0; then h has a fixed point. See also LEFSHETZ TRACE FORMULA References Munkres, J. R. "Application: The Lefschetz Fixed-Point Theorem." §22 in Elements of Algebraic Topology. Perseus Press, pp. 121 /C1/28, 1993. Lefshetz Trace Formula A formula which counts the number of FIXED POINTS for a topological transformation. Left Coset Consider a countable SUBGROUP H with ELEMENTS hi and an element x not in H, then xhi for i /C301, 2, ... are the left cosets of the SUBGROUP H with respect to x. See also COSET ,RIGHT COSET Left Half-Plane The portion of the COMPLEX PLANE z /C30x /C27iy with REAL PART R[z] B0 :/ See also COMPLEX PLANE ,LOWER HALF-PLANE ,RIGHT HALF-PLANE ,UPPER HALF-PLANE Left-Handed Coordinate System A three-dimensional COORDINATE SYSTEM in which the axes do not satisfy the RIGHT-HAND RULE . See also CROSS PRODUCT ,RIGHT- HAND RULE,RIGHT- HANDED COORDINATE SYSTEM Leg A leg of a TRIANGLE is one of its sides. For a RIGHT TRIANGLE , the term "leg" generally refers to a side other than the one opposite the RIGHT ANGLE , which is termed the HYPOTENUSE . See also HYPOTENUSE ,TRIANGLE Legendre Addition Theorem SPHERICAL HARMONIC ADDITION THEOREMLegendre Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION (1/C28x2)d2y dx2/C282xdy dx/C27l(l/C271)y/C300; (1) which can be rewritten d dx(1/C28x2)dydx"# /C27l(l/C271)y/C300: (2) The above form is a special case of the associated Legendre differential equation with m/C300. The Le- gendre differential equation has REGULAR SINGULAR POINTS at/C281, 1, and /C12:/ If the variable xis replaced by cos u;then the Legendre differential equation becomes d2y du2/C27cosu sinudy du/C27l(l/C271)y/C300; (3) as is derived below for the associated Legendre differential equation with m/C300. Since the Legendre differential equation is a second- order ORDINARY DIFFERENTIAL EQUATION , it has two linearly independent solutions. A solution Pl(x) which is regular at the origin is called a L EGENDRE FUNC- TION OF THE FIRST KIND , while a solution Ql(x) which is singular at the origin is called a L EGENDRE FUNCTION OF THE SECOND KIND .I flis an integer, the function of the first kind reduces to a polynomialknown as the L EGENDRE POLYNOMIAL . The Legendre differential equation can be solved using the standard method of making a seriesexpansion, y/C30X /C12 n/C300anxn(4) y?/C30X/C12 n/C300nanxn/C281(5) yƒ/C30X/C12 n/C300n(n/C281)anxn/C282: (6) Plugging in, (1/C28x2)X/C12 n/C300n(n/C281)anxn/C282/C282xX/C12 n/C300nanxn/C281 /C27l(l/C271)X/C12 n/C300anxn/C300 (7) X/C12 n/C300n(n/C281)anxn/C282/C28X/C12 n/C300n(n/C281)anxn /C282xX/C12 n/C300nanxn/C281/C27l(l/C271)X/C12 n/C300anxn/C300 (8) X/C12 n/C300n(n/C281)anxn/C282/C28X/C12 n/C300n(n/C281)anxn /C282X/C12 n/C300nanxn/C27l(l/C271)X/C12 n/C300anxn/C300 (9) X/C12 n/C300(n/C272)(n/C271)an/C272xn/C28X/C12 n/C300n(n/C281)anxn /C282X/C12 n/C300nanxn/C27l(l/C271)X/C12 n/C300anxn/C300 (10) X/C12 n/C300f(n/C271)(n/C272)an/C272/C27[/C28n(n/C281) /C282n/C27l(l/C271)]ang/C300; (11) so each term must vanish and (n/C271)(n/C272)an/C272/C27[/C28n(n/C271)/C27l(l/C271)]an/C300 (12) an/C272/C30n(n/C271)/C28l(l/C271) (n/C271)(n/C272)an /C30/C28[l/C27(n/C271)](l/C28n) (n/C271)(n/C272)an: (13) Therefore, a2/C30/C28l(l/C271) 1 /C2152a0 (14) a4/C30/C28(l/C282)(l/C273) 3 /C2154a2 /C30(/C281)2[(l/C282)l][(l/C271)(l/C273)] 1 /C2152 /C2153 /C2154a0 (15) a6/C30/C28(l/C284)(l/C275) 5 /C2156a4 /C30(/C281)3[(l/C284)(l/C282)l][(l/C271)(l/C273)(l/C275)] 1 /C2152 /C2153 /C2154 /C2155 /C2156a0;(16) so the EVEN solution is y1(x)/C301/C27X/C12 n/C301(/C281)n /C2[(l/C282n/C272 )...( l/C282)l][(l/C271)(l/C273 )...( l/C272n/C281)] (2n)!x2n: (17) Similarly, the ODD solution isy2(x)/C30x/C27X/C12 n/C301(/C281)n /C2[(l/C282n/C271)/C1/C1/C1(l/C283)(l/C281)][(l/C272)(l/C274)/C1/C1/C1(l/C272n) (2n/C271)!x2m/C271: (18) Iflis an EVEN INTEGER , the series y1(x) reduces to a POLYNOMIAL of degree lwith only EVEN POWERS ofx and the series y2(x) diverges. If lis an ODD INTEGER , the series y2(x) reduces to a POLYNOMIAL of degree l with only ODD POWERS ofxand the series y1(x) diverges. The general solution for an INTEGER lis then given by the L EGENDRE POLYNOMIALS Pn(x)/C30cny1(x) for leven y2(x) for lodd;/C)% (19) where cnis chosen so as to yield the normalization Pn(1)/C301:/ The associated Legendre differential equation is d dx(1/C28x2)dy dx"# /C27l(l/C271)/C28m2 1/C28x2"# y/C300; (20) which can be written (1/C28x2)d2y dx/C282xdydx/C27l(l/C271)/C28m2 1/C28x2"# y/C300 (21) (Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions Pm l(x) to this equation are called the associated Legendre polynomials (if lis an integer), or associated Legendre functions of the first kind (if lis not an integer). The complete solution is y/C30C1Pm l(x)/C27C2Qml(x); (22) where Qm l(x)i saL EGENDRE FUNCTION OF THE SECOND KIND . The associated Legendre differential equation is often written in a form obtained by setting x/C13cosu:Using the identities dy dx/C30dy d(cosu)/C30/C281 sinudy du(23) xdydx/C30/C28cosu sinudy du; (24) d2y dx2/C301 sinud du1 sinudy du ! /C301 sinu/C28cosu sin2u ! dy du/C271 sin2ud2y du2; (25) and 1/C28x2/C301/C28cos2u/C30sin2u; (26) therefore gives (1 /C28x2)d2y dx2 /C30sin2 u1 sin u/C28cos u sin2 u ! dy d u /C271 sin2 ud2y du2 /C30d2y du2 /C28cos u sin udy du : (27) Plugging (23) into (27) and the result back into (21) gives d2y d u2 /C28cos u sin udy d u ! /C272cos u sin udy du /C27 l(l /C271) /C28m2 sin2 u"# y /C300 (28) d2y du2 /C27cos u sin udy du /C27 l(l /C271) /C28m2 sin2 u"# y /C300 : (29) Moon and Spencer (1961, p. 155) call (1 /C28x2)yƒ/C282xy?/C28 k2a2(x2 /C281) /C28p(p /C271) /C28q2 x2 /C28 1"# y /C300 (30) The Legendre wave function (Zwillinger 1997, p.124). See also LEGENDRE FUNCTION OF THE FIRST KIND, LEGENDRE FUNCTION OF THE SECOND KIND,L E- GENDRE POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 332, 1972. Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995. Legendre Duplication Formula GAMMA FUNCTIONS of argument 2z can be expressed in terms of GAMMA FUNCTIONS of smaller arguments. From the definition of the BETA FUNCTION , B(m; n) /C30G(m) G(n) G(m /C27 n) /C30g1 0um/C281(1 /C28u)n/C281 du: (1) Now, let m /C30n /C13z ; then G(z) G(z) G(2z)/C30g1 0uz/C281(1 /C28u)z/C281 du (2) and u /C13(1 /C27x) =2; so du /C30dx=2 and G(z) G(z) G(2z)/C30g1 01 /C27 x 2 !z /C281 1 /C281 /C27 x 2 !z/C281 (1 2 dx)/C301 2 g1 01 /C27 x 2 !z/C2811 /C27 x 2 !z/C281 dx /C301 21 /C272(z/C281) g1 0(1 /C28x2)z/C281 dx /C3021 /C282xg1 0(1 /C28x2)z/C281 dx: (3) Now, use the BETA FUNCTION identity B(m; n) /C302g1 0x2z/C281(1 /C28x2)z/C281 dx (4) to write the above as G(z) G(z) G(2z)/C3021 /C282zB(1 2 ; z) /C3021 /C282zG(12)G(z) G(z /C271 2) : (5) Solving for G(2x); G(2z) /C30G(z)G(z /C2712)22z/C281 G(12)/C30G(z) G(z /C2712)22z/C281 ffiffiffipp /C30(2p) /C281 =222z /C281 =2 G(z)G(z /C271 2) ; (6) since G(1 2) /C30ffiffiffipp:/ See also GAMMA FUNCTION ,GAUSS MULTIPLICATION FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 256, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 561 /C1/62, 1985. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 5, 1981. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 424 /C1/25, 1953. Legendre Function of the First Kind The (associated) Legendre function of the first kind Pm n (z) is the solution to the LEGENDRE DIFFERENTIAL EQUATION which is regular at the origin. For m, n integers and z real, the Legendre function of the first kind simplifies to a polynomial, called the LEGENDRE POLYNOMIAL . The associated Legendre function of first kind is given by the Mathematica command LegendreP [n,m,z], and the unassociated function byLegendreP [n,z]. See also LEGENDRE DIFFERENTIAL EQUATION ,L E- GENDRE FUNCTION OF THE SECOND KIND,LEGENDRE POLYNOMIAL Legendre Function of the Second Kind The second solution Q1(x) to the LEGENDRE DIFFER- ENTIAL EQUATION . The Legendre functions of the second kind satisfy the same RECURRENCE RELATION as the LEGENDRE POLYNOMIALS . The Legendre func- tions of the second kind are implemented in Mathe- matica asLegendreQ [l, x]. The first few are Q0(x) /C301 2ln1 /C27 x 1 /C28 x ! Q1(x) /C30x 2ln1 /C27 x 1 /C28 x ! /C281 Q2(x) /C303x2 /C28 1 4ln1 /C27 x 1 /C28 x ! /C283x 2 Q3(x) /C305x3 /C28 3x 4ln1 /C27 x 1 /C28 x ! /C285x2 2/C2723 : The associated Legendre functions of the second kind Q m l(x) are the second solution to the associated Legendre differential equation, and are implemented in Mathematica as LegendreQ [l, m, x] Qm v(x) has DERIVATIVE about 0 of dQ mn (x) dx"# x/C300/C302mffiffiffippcos[1 2 p(n /C27 m)]G(12 n /C2712 m /C27 1) G(1 2 n /C2812 m /C2712) (Abramowitz and Stegun 1972, p. 334). The LOGA- RITHMIC DERIVATIVE is dlnQm l(z) dz"# z/C300 /C302exp f1 2pisgn(I[z])g[12(l/C27m)]![12(l/C28m)]! [1 2(l/C27m/C281)]![12(l/C28m/C281)]! (Binney and Tremaine 1987, p. 654). See also LEGENDRE DIFFERENTIAL EQUATION ,L E- GENDRE FUNCTION OF THE FIRST KIND,LEGENDRE POLYNOMIALReferences Abramowitz, M. and Stegun, C. A. (Eds.). "Legendre Func- tions." Ch. 8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 331 /C1/39, 1972. Arfken, G. "Legendre Functions of the Second Kind, Qn(x):/" Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 701 /C1/07, 1985. Binney, J. and Tremaine, S. "Associated Legendre Func- tions." Appendix 5 in Galactic Dynamics. Princeton, NJ: Princeton University Press, pp. 654 /C1/55, 1987. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 597 /C1/00, 1953. Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory.Washington, DC: U. S. Government Printing Office, 1952. Spanier, J. and Oldham, K. B. "The Legendre Functions P n(x) and Qn(x):/" Ch. 59 in An Atlas of Functions. Wa- shington, DC: Hemisphere, pp. 581 /C1/97, 1987. Legendre-Gauss Quadrature Also called "the" G AUSSIAN QUADRATURE or L E- GENDRE QUADRATURE .AG AUSSIAN QUADRATURE over the interval [ /C281;1] with WEIGHTING FUNCTION W(x)/C301:The ABSCISSAS for quadrature order nare given by the roots of the L EGENDRE POLYNOMIALS Pn(x);which occur symmetrically about 0. The weights are wi/C30/C28An/C271gn AnP?n(xi)Pn/C271(xi)/C30An An/C281gn/C281 Pn/C281(xi)P?n(xi);(1) where Anis the COEFFICIENT ofxninPn(x):For LEGENDRE POLYNOMIALS , An/C30(2n)! 2n(n!)2; (2) so An/C271 An/C30[2(n/C271)]! 2n/C271[(n/C271)!]22n(n!)2 (2n)! /C30(2n/C271)(2n/C272) 2(n/C271)2/C302n/C271 n/C271: (3) Additionally, gn/C302 2n/C271; (4) so wi/C30/C282 (n/C271)Pn/C271(xi)P?n(xi)/C302 nPn/C281(xi)P?n(xi):(5) Using the RECURRENCE RELATION (1/C28x2)P?n(x)/C30nxPn(x)/C27nPn/C281(x) /C30(n/C271)xPn(x)/C28(n/C271)Pn/C271(x) (6) gives wi /C30/C282 (1 /C28 x2)[P ?n(xi)]2 /C302(1 /C28 x2 i ) (n /C27 1)2[Pn/C271(xi)]2 : (7) The error term is E /C3022n/C271(n!)4 (2n /C27 1)[(2n)!]3 f(2n)( j): (8) Beyer (1987) gives a table of ABSCISSAS and weights up to n /C3016, and Chandrasekhar (1960) up to n /C308 for n EVEN . n /xi// wi/ 2 9 0.57735 1.000000 3 0 0.888889 9 0.774597 0.555556 4 9 0.339981 0.652145 9 0.861136 0.347855 5 0 0.568889 9 0.538469 0.478629 9 0.90618 0.236927 The ABSCISSAS and weights can be computed analy- tically for small n. n /xi// wi/ 2 /91 3ffiffiffi 3p / 1 30 /8 9/ /91 5ffiffiffiffiffiffi 15p //5 9/ 4 /91 35ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 525 /C2870ffiffiffiffiffiffi 30pp //1 36(18 /C27ffiffiffiffiffiffi30p ) / /91 35ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 525 /C2770ffiffiffiffiffiffi 30pp //1 36(18 /C28ffiffiffiffiffiffi30p ) / 50 /128 225/ /91 21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 245 /C2814ffiffiffiffiffiffi 70pp //1 900(322 /C2713ffiffiffiffiffiffi70p ) / /91 21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 245 /C2714ffiffiffiffiffiffi 70pp //1 900(322 /C2813ffiffiffiffiffiffi70p ) / References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 462 /C1/63, 1987. Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 56 /C1/2, 1960. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 323 /C1/25, 1956.Legendre-Jacobi Elliptic Integral Any of the three standard forms in which an ELLIPTIC INTEGRAL can be expressed. See also ELLIPTIC INTEGRAL OF THE FIRST KIND, ELLIPTIC INTEGRAL OF THE SECOND KIND,ELLIPTIC INTEGRAL OF THE THIRD KIND LegendreP LEGENDRE FUNCTION OF THE FIRST KIND,LEGENDRE POLYNOMIAL Legendre Polynomial The Legendre polynomials, sometimes called Le- gendre functions of the first kind, Legendre coeffi- cients, or ZONAL HARMONICS (Whittaker and Watson 1990, p. 302), are solutions to the L EGENDRE DIFFER- ENTIAL EQUATION .I f lis an INTEGER , they are POLYNOMIALS . The Legendre polynomials Pn(x) are illustrated above for x/C23[0;1] and n/C301, 2, ..., 5. The Legendre polynomials are a special case of the ULTRASPHERICAL FUNCTIONS with a/C301=2;a special case of the J ACOBI POLYNOMIALS P(a;b) nwith a/C30b/C300; and can be written as a HYPERGEOMETRIC FUNCTION using Murphy’s formula Pn(x)/C30P(0;0) n(x)/C302F1(/C28n;n/C271; 1;1 2(1/C28x)) (1) (Bailey 1933; Bailey 1935, p. 101; Koekoek and Swarttouw 1998). The Rodrigues formula provides the GENERATING FUNCTION Pl(x)/C30l 2ll!dl dxl(x2/C281)l; (2) which yields upon expansion Pl(x)/C301 2lXl=2bc k/C300(/C281)k(2l/C282k)! k!(l/C28k)!(l/C282k)!xl/C282k(3) /C301 2lXl=2bc k/C300(/C281)kl k/C(*/C(+ 2l/C282k l/C(*/C(+ xl/C282k(4) where rbcis the FLOOR FUNCTION . Additional sum formulas include Pl(x)/C301 2lXl k/C300l k/C(*/C(+2 (x/C281)l/C28k(x/C271)k(5) /C30Xl k/C300l k/C(*/C(+ /C28l/C281 k/C(*/C(+1/C28x 2 !k (6) (Koepf 1998, p. 1). In terms of HYPERGEOMETRIC FUNCTIONS , these can be written Pn(x)/C30x/C281 2 !n 2F1(/C28n;/C28n;1 ; (x/C271)=(x/C281)) (7) Pn(x)/C302n n/C(*/C(+xn 2n2F1(/C28n=2;(1/C28n)=2; 1 =2/C28n;x/C282) (8) Pn(x)/C302F1(/C28n;n/C271; 1; (1 /C28x)=2) (9) (Koepf 1998, p. 3). AGENERATING FUNCTION forPn(x) is given by g(t;x)/C30(1/C282xt/C27t2)/C281=2/C30X/C12 n/C300Pn(x)tn: (10) Take @g=@t; /C281 2(1/C282xt/C27t2)/C283=2(/C282x/C272t)/C30X/C12 n/C300nPn(x)tn/C281:(11) Multiply (11) by 2 t; /C28t(1/C282xt/C27t2)/C283=2(/C282x/C272t)/C30X/C12 n/C3002nPn(x)tn(12) and add (10) and (12), (1/C282xt/C27t2)/C283=2[(2xt/C282t2)/C27(1/C282xt/C27t2)] /C30X/C12 n/C300(2n/C271)Pn(x)tn(13) This expansion is useful in some physical problems, including expanding the Heyney-Greenstein phasefunction and computing the charge distribution on a SPHERE . Another GENERATING FUNCTION is given by X/C12 n/C300Pn(x) n!zn/C30exzJ0(zffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p ); (14) where J0(x) is a zeroth order B ESSEL FUNCTION OF THE FIRST KIND (Koepf 1998, p. 2). The Legendre polynomials satisfy the RECURRENCE RELATION (l/C271)Pl/C271(x)/C28(2l/C271)xPl(x)/C27lPl/C281(x)/C300 (15) (Koepf 1998, p. 2). The Legendre polynomials are orthogonal over (/C281;1) with WEIGHTING FUNCTION 1 and satisfyg1 /C281Pn(x)Pm(x)dx/C302 2n/C271dmn; (16) where dmnis the K RONECKER DELTA . ACOMPLEX GENERATING FUNCTION is Pl(x)/C301 2pig(1/C282zx/C27z2)/C281=2z/C28l/C281dz; (17) and the Schla ¨fli integral is Pl(x)/C30(/C281)l 2l1 2pig(1/C28z2)l (z/C28x)l/C271dz: (18) Additional integrals (Byerly 1959, p. 172) include g1 0Pm(x)dx /C300 meven"0 (/C281)(m/C281)=2 m!! m(m/C271)(m/C281)!!modd8 < :(19) g1 0Pm(x)Pn(x)dx /C300 m;nboth even or odd m"n (/C281)(m/C27n/C271)=2 /C2m!n! 2m/C27n/C271(m/C28n)(m/C27n/C271)(1 2m)!f[12(n/C281)]!g2 meven ;nodd 1 2n/C271 m/C30n:8 >>>>>>>>>>>>< >>>>>>>>>>>>: (20) Integrals with weighting functions xandx 2are given by g1 /C281xPL(x)PN(x)dx/C302(L/C271) (2L/C271)(2L/C273)N/C30L/C271 2L (2L/C281)(2L/C271)N/C30L/C281( (21) g1 /C281x2PL(x)PN(x)dx /C302(L/C271)(L/C272) (2L/C271)(2L/C273)(2L/C275)N/C30L/C272 2(L2/C272L/C281) (2L/C281)(2L/C271)(2L/C273)N/C30L 2L(L/C281) (2L/C283)(2L/C281)(2L/C271)N/C30L/C2828 >< >:(22) (Arfken 1985, p. 700). An additional identity is 1/C28[Pn(x)]2/C30Xn n/C3011/C28x2 1/C28x2 nPn(x) P?n(xn)(x/C28xn)"#2 ; (23) where xnis the n/th root of Pn(x) (Szego 1975, p. 348). The first few Legendre polynomials are P0(x)/C301 P1(x)/C30x P2(x)/C301 2(3x2/C281) P3(x)/C3012(5x3/C283x) P4(x)/C3018(35x4/C2830x2/C273) P5(x)/C3018(63x5/C2870x3/C2715x) P6(x)/C301 16(231x6/C28315x4/C27105x2/C285): The first few POWERS in terms of Legendre polyno- mials are x/C30P1 x2/C301 3[P0(x)/C272P2(x)] x3/C3015[3P1(x)/C272P3(x)] x4/C301 35[7P0(x)/C2720P2(x)/C278P4(x)] x5/C301 63[27P1(x)/C2728P3(x)/C278P5(x)] x6/C301 231[33P0(x)/C27110P2(x)/C2772P4(x)/C2716P6(x)]: For Legendre polynomials and POWERS up to expo- nent 12, see Abramowitz and Stegun (1972, p. 798). The Legendre POLYNOMIALS can also be generated using G RAM- SCHMIDT ORTHONORMALIZATION in the OPEN INTERVAL (/C281;1) with the WEIGHTING FUNCTION 1. P0(x)/C301 (24) P1(x)/C30x/C28g1 /C281xd x g1 /C281dx2 66643 7775/C2151 /C30x/C281 2[x2]1 /C281 [x]1 /C281/C30x/C281 2(1/C281) 1/C28(/C281)/C30x (25) P2(x)/C30x/C28g1 /C281x3dx g1 /C281x2dx2 66643 7775/C28g1 /C281x2dx g1 /C281dx2 66643 7775/C2151 /C30x/C28 1 4[x4]1 /C281 1 3[x3]1 /C281"# x/C281 3[x3]1 /C281 [x]1/C281/C30x2/C281 3(26) P3(x)/C30x/C28g1 /C281x(x2/C281 3)2dx g1 /C281(x2/C2813)2dx2 66643 7775(x2/C281 3) /C28g1 /C281(x2/C2813)2dx g1 /C281x2dx2 66643 7775x/C30xx2/C281 3/C28(1 5/C2829/C2719)x 13"# /C30x3/C281 3x/C283(15/C2819) /C30x3/C28x(13/C2735/C2813)/C30x3/C2835x: (27) Normalizing so that Pn(1)/C301 gives the expected Legendre polynomials. The "shifted" Legendre polynomials are a set of functions analogous to the Legendre polynomials,but defined on the interval (0, 1). They obey the ORTHOGONALITY relationship g1 0¯Pm(x)¯Pn(x)dx/C301 2n/C271dmn: (28) The first few are ¯P0(x)/C301 ¯P1(x)/C302x/C281 ¯P2(x)/C306x2/C286x/C271 ¯P3(x)/C3020x3/C2830x2/C2712x/C281: The associated Legendre polynomials Pm l(x) are solu- tions to the associated L EGENDRE DIFFERENTIAL EQUATION , where lis a POSITIVE INTEGER andm/C300, ...,l. They can be given in terms of the unassociated polynomials by Pm l(x)/C30(/C281)m(1/C28x2)m=2dm dxmPl(x) /C30(/C281)m 2ll!(1/C28x2)m=2dl/C27m dxl/C27m(x2/C281)l; (29) where Pl(x) are the unassociated L EGENDRE POLYNO- MIALS . Note that some authors (e.g., Arfken 1985, p. 668) omit the C ONDON- SHORTLEY PHASE (/C281)m; while others include it (e.g., Abramowitz and Stegun 1972, Press et al. 1992, and the LegendreP [l,m,z] command of Mathematica ). Abramowitz and Stegun (1972, p. 332) use the notation Plm(X)/C13(/C281)mPl m(x) (30) to distinguish these two cases. Associated polynomials are sometimes called F ER- RERS’ FUNCTIONS (Sansone 1991, p. 246). If m/C300, they reduce to the unassociated POLYNOMIALS . The associated Legendre functions are part of the SPHE- RICAL HARMONICS , which are the solution of L APLA- CE’S EQUATION inSPHERICAL COORDINATES . They are ORTHOGONAL over [/C281;1] with the WEIGHTING FUNC- TION 1 g1 /C281Pm l(x)Pml?(x)dx/C302 2l/C271(l/C27m)! (l/C28m)!dll?; (31) and ORTHOGONAL over [/C281;1] with respect to mwith the WEIGHTING FUNCTION (1/C28x2)/C282 g1 /C281Pm l(x)Pm? l(x)dx 1 /C28 x2 /C30(l /C27 m)! m(l /C28 m)!dmm?: (32) The associated Legendre polynomials also obey the following RECURRENCE RELATIONS (l /C28m)Pml(x) /C30x(2l /C281)Pml/C281(x) /C28(l /C27m /C281)Pml /C282(x): (33) Letting x /C13cos u (commonly denoted m in this con- text), dPm l(m) du/C30l mPml( m) /C28 (l /C27 m)Pml /C281( m)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 m2p (34) (2l /C271)mPm l( m) /C30(l /C27m)Pml/C281( m) /C27(l /C28m /C271)Pml/C271( m) : (35) An identity relating associated POLYNOMIALS with NEGATIVE m to the corresponding functions with POSITIVE m is P /C28m l(x) /C30(/C281)m(l /C28 m)! (l /C27 m)!Pml(x): (36) Additional identities are Pll(x) /C30(/C281)l(2l /C281)!!(1 /C28x2)1 =2 (37) Pll/C271(x) /C30x(2l /C271)Pll(x): (38) Written in terms of x and using the convention without a leading factor of (/C281)m(Arfken 1985, p. 669), the first few associated Legendre polynomials are P0 0(x) /C301 P0 1(x) /C30x P11(x) /C30/C28(1 /C28x2)1=2 P02(x) /C301 2(3x2 /C281) P1 2(x) /C30/C283x(1 /C28x2)1 =2 P22(x) /C303(1 /C28x2) P03(x) /C301 2x(5x2 /C283) P1 3(x) /C303 2(1 /C285x2)(1 /C28x2)1 =2 P2 3(x) /C3015x(1 /C28x2) P33(x) /C30/C2815(1 /C28x2)3 =2 P04(x) /C301 8(35x4 /C2830x2 /C273) P1 4(x) /C305 2x(3 /C287x2)(1 /C28x2)1 =2 P2 4(x) /C3015 2 (7x2 /C281)(1 /C28x2)P34(x) /C30/C28105x(1 /C28x2)3 =2 P44(x) /C30105(1 /C28x2)2 P05(x) /C301 8x(63x4 /C2870x2 /C2715) : Written in terms x /C30cos u (commonly written m /C30 cos u) ; the first few become P0 0(cos u) /C301 P01(cos u) /C30cos u P11(cos u) /C30/C28sin u P02(cos u) /C301 2(3 cos2 u /C281) P1 2(cos u) /C30/C283 sin u cos u P22(cos u) /C303 sin2 u P03(cos u) /C301 2cos u(5 cos2 u /C283) P1 3(cos u) /C30/C283 2(5 cos2 u /C281)sin u P2 3(cos u) /C3015 cos u sin2 u P33(cos u) /C30/C2815 sin3 u: The derivative about the origin is dPm n (x) dx"# x /C300/C302m/C271 sin[1 2p( n /C27 m)] G(12n/C2712m/C271) p/C281=2G(1 2n/C2812m/C2712)(39) (Abramowitz and Stegun 1972, p. 334), and the logarithmic derivative is dlnPm l(z) dz"# z/C300 /C302 tan[1 2p(l/C27m)] /C2[1 2(l/C27m)]![12(l/C28m)]! [1 2(l/C27m/C281)]![12(l/C28m/C281)]!: (40) (Binney and Tremaine 1987, p. 654). See also CONDON- SHORTLEY PHASE ,CONICAL FUNC- TION ,K INGS PROBLEM ,L APLACE’S INTEGRAL ,L A- PLACE- MEHLER INTEGRAL ,LEGENDRE FUNCTION OF THE FIRST KIND,L EGENDRE FUNCTION OF THE SECOND KIND,SUPER CATALAN NUMBER ,TOROIDAL FUNCTION ,TURA´ N’S INEQUALITIES ,U LTRASPHERICAL POLYNOMIAL ,ZONAL HARMONIC References Abramowitz, M. and Stegun, C. A. (Eds.). "Legendre Func- tions" and "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 331 /C1/39 and 771 /C1/02, 1972. Arfken, G. "Legendre Functions." Ch. 12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 637 /C1/11, 1985. Bailey, W. N. "On the Product of Two Legendre Polyno- mials." Proc. Cambridge Philos. Soc. 29, 173 /C1/77, 1933. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, 1935. Binney, J. and Tremaine, S. "Associated Legendre Func- tions." Appendix 5 in Galactic Dynamics. Princeton, NJ: Princeton University Press, pp. 654 /C1/55, 1987. Byerly, W. E. "Zonal Harmonics." Ch. 5 in An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 144 /C1/94, 1959. Iyanaga, S. and Kawada, Y. (Eds.). "Legendre Function" and "Associated Legendre Function." Appendix A, Tables 18.II and 18.III in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1462 /C1/468, 1980. Koekoek, R. and Swarttouw, R. F. "Legendre / Spherical." §1.8.3 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, p. 44, 1998. ftp://www.twi.tudelft.nl/publications/tech-reports/1998/ DUT-TWI-98 /C1/7.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Lagrange, R. Polynomes et fonctions de Legendre. Paris: Gauthier-Villars, 1939. Legendre, A. M. "Sur l’attraction des Sphe´roides." Me´m. Math. et Phys. pre´sente´sa` l’Ac. r. des. sc. par divers savants 10, 1785. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 593 /C1/97, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 252, 1992. Sansone, G. "Expansions in Series of Legendre Polynomials and Spherical Harmonics." Ch. 3 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 169 /C1/94, 1991. Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. Washington, DC: U. S. Government Printing Office, 1952. Spanier, J. and Oldham, K. B. "The Legendre Polynomials Pn(x)/" and "The Legendre Functions P n(x) and Qn(x):/" Chs. 21 and 59 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 183 /C1/92 and 581 /C1/97, 1987. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. Legendre Polynomial of the Second Kind LEGENDRE FUNCTION OF THE SECOND KIND LegendreQ LEGENDRE FUNCTION OF THE SECOND KIND Legendre Quadrature LEGENDRE- GAUSS QUADRATURE Legendre Relation Let E(k) and K(k) be complete ELLIPTIC INTEGRALS OF THE FIRST and SECOND KINDS , with E ?(k) and K ?(k) the complementary integrals. ThenE(k)K ?(k) /C27E ?(k)K(k) /C28K(k)K ?(k) /C301 2 p: References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 591, 1972. Legendre’s Chi-Function Portions of this entry contributed by Joe Keane . The function defined by xn(z) /C30X/C12 k/C300z2k /C271 (2k /C27 1)n (1) for integral n /C302; 3, .... It is related to the POLYLOGA- RITHM by xn(z) /C301 2[Lin(z) /C28Lin(/C28z)] (2) /C30Lin(z) /C282 /C28 nLi n(z2) (3) and to the LERCH TRANSCENDENT by xn(z)/C302/C28nzF(z2;n;1 2): (4) It takes the special values x2(i)/C30iK (5) x2(ffiffiffi 2p /C281)/C301 16p2/C281 4[ln(ffiffiffi 2p /C271)]2(6) x2(1 2(ffiffiffi 5p /C281))/C301 12p2/C283 4[ln(12(ffiffiffi 5p /C271))]2(7) x2(ffiffiffi5p /C282)/C301 24p2/C283 4[ln(12(ffiffiffi 5p /C271))]2(8) x2(/C281)/C30/C281 8p2(9) x2(1)/C3018p2; (10) where Iis the imaginary unit and Kis C ATALAN’S CONSTANT (Lewin, p. 19). Other special values in- clude xn(1)/C30l(n) (11) xn(1)/C30ib(n); (12) where l(n) is the D IRICHLET LAMBDA FUNCTION and b(n) is the D IRICHLET BETA FUNCTION . See also LERCH TRANSCENDENT ,POLYLOGARITHM References Cvijovic, D. and Klinowski, J. "Closed-Form Summation of Some Trigonometric Series." Math. Comput. 64, 205/C1/10, 1995. Edwards, J. A Treatise on the Integral Calculus, Vol. 2. New York: Chelsea, p. 290, 1955. Legendre, A. M. Exercices de calcul inte ´gral, tome 1. p. 247, 1811. Lewin, L. "Legendre’s Chi-Function." §1.8 in Dilogarithms and Associated Functions. London: Macdonald, pp. 17 /C1/9, 1958. Lewin, L. Polylogarithms and Associated Functions. Am- sterdam, Netherlands: North-Holland, pp. 282 /C1/83, 1981. Nielsen, N. "Der Eulersche Dilogarithmus und seine Ver- allgemeinerungen." Nova Acta (Leopold) 90, 121 /C1/12, 1909. Legendre’s Constant The number 1.08366 in Legendre’s guess at the PRIME NUMBER THEOREM p(n) /C30n ln n /C28 A(n) with limn0/C12 A(n) :1:08366 : This expression is cor- rect to leading term only, since it is actually true that this limit approaches 1 (Rosser and Schoenfeld 1962, Panaitopol 1999). See also PRIME COUNTING FUNCTION References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 147, 1983. Panaitopol, L. "Several Approximations of p(x):/" Math. Ineq. Appl. 2, 317 /C1/24, 1999. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Rosser, J. B. and Schoenfeld, L. "Approximate Formulas for Some Functions of Prime Numbers." Ill. J. Math. 6,64/C1/4, 1962. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 28 /C1/9, 1991. Legendre Series Because the LEGENDRE FUNCTIONS OF THE FIRST KIND form a COMPLETE ORTHOGONAL BASIS , any FUNCTION may be expanded in terms of them f(x) /C30X/C12 n/C300anPn(x): (1) Now, multiply both sides by Pm(x) and integrate g1 /C281Pm(x)f(x) dx /C30X/C12 n/C300ang1 /C281Pn(x)Pm(x) dx: (2) Butg1 /C281Pn(x)Pm(x) dx /C302 2m /C27 1dmn ; (3) where dmn is the KRONECKER DELTA ,so g1 /C281Pm(x)f(x) dx /C30X/C12 n/C300an2 2m /C27 1dmn /C302 2m /C27 1am (4) and am /C302m /C27 1 2 g1 /C281Pm(x)f(x) dx: (5) See also FOURIER SERIES ,JACKSON’S THEOREM , LEGENDRE POLYNOMIAL ,M ACLAURIN SERIES ,PICO- NE’S THEOREM ,TAYLOR SERIES Legendre’s Factorization Method A PRIME FACTORIZATION ALGORITHM in which a sequence of TRIAL DIVISORS is chosen using a QUAD- RATIC SIEVE . By using QUADRATIC RESIDUES of N, the QUADRATIC RESIDUES of the factors can also be found. See also PRIME FACTORIZATION ALGORITHMS ,QUAD- RATIC RESIDUE ,QUADRATIC SIEVE,TRIAL DIVISOR Legendre’s Formula Counts the number of POSITIVE INTEGERS less than or equal to a number xwhich are not divisible by any of the first aPRIMES , f(x;a)/C30xbc/C28X x pi$% /C27X x pipj$% /C28X x pipjpk$% /C27...; (1) where xbcis the FLOOR FUNCTION . Taking a/C30xgives f(x;x)/C30p(x)/C28p(ffiffiffixp)/C271 /C30xbc/C28X pi5ffiffixpx pi$% /C27X piBpj5ffiffixpx pipj$% /C28X piBpjBpk5ffiffixpx p ipjpk$% /C27...;(2) where p(n) is the PRIME COUNTING FUNCTION . Legen- dre’s formula holds since one more than the number ofPRIMES in a range equals the number of INTEGERS minus the number of composites in the interval. Legendre’s formula satisfies the RECURRENCE RELA- TION f(x;a)/C30f(x;a/C281)/C28fx pa;a/C281 ! : (3) Let mk /C13p1 p2 /C1/C1/C1pk ; then f(mk ; k) /C30 mkbc/C28Xmk pi$% /C27X mk pipj$% /C28... /C30mk /C28Xmk pi/C27Xmk pipj/C28... /C30mk1 /C281 p /C28 1 ! 1 /C281 p2 ! /C1/C1/C1 1 /C281 pk ! /C30Yk i/C301(pi /C281) /C30 f(mk) ; (4) where f(n) is the TOTIENT FUNCTION , and f(smk /C27t; k) /C30sf(mk) /C27 f(t; k) ; (5) where 0 5t 5mk : If t > mk =2 ; then f(t; k) /C30 f(mk) /C28 f(mk /C28t /C281 ; k): (6) Note that f(n; n) is not practical for computing p(n) for large arguments. A more efficient modification is MEISSEL’S FORMULA . See also LEHMER’S FORMULA ,MAPES’ METHOD ,MEIS- SEL’S FORMULA ,PRIME COUNTING FUNCTION References Se´roul, R. "Legendre’s Formula" and "Implementation of Legendre’s Formula." §8.7.1 and 8.7.2 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 175 /C1/79, 2000. Legendre’s Quadratic Reciprocity Law QUADRATIC RECIPROCITY LAW Legendre Sum LEGENDRE’S FORMULA Legendre Symbol The Legendre symbol is a number theoretic function (m n) which is defined to be equal to 9 1 depending on whether m is a QUADRATIC RESIDUE modulo n. The definition is sometimes generalized to have value 0 if m½n; m n ! /C30(m½n) /C130i f m½n 1i f m is a quadratic residue modulo n /C281if m is a quadratic nonresidue modulo n:8 < : (1) If n is an ODD PRIME , then the JACOBI SYMBOL reduces to the Legendre symbol. The Legendre symbol is implemented in Mathematica via the JACOBI SYMBOL , JacobiSymbol [n, m].The Legendre symbol obeys the identity ab p ! /C30a p ! b p ! : (2) Particular identities include /C281 p ! /C30(/C281)(p /C281)=2 (3) 2 p ! /C30(/C281)(p2/C281)=8 (4) 3 p ! /C301i f p /C131(mod 6) /C281if p /C135(mod 6)/C)% (5) 5 p ! /C301i f p/C1391(mod 10) /C281i f p/C1397(mod 10)/C)% (6) (Nagell 1951, p. 144), as well as the general q p ! /C30p q ! (/C281)[(p/C281)=2][(q/C281)=2]: (7) See also JACOBI SYMBOL ,KRONECKER SYMBOL ,QUAD- RATIC RECIPROCITY THEOREM ,QUADRATIC RESIDUE References Guy, R. K. "Quadratic Residues. Schur’s Conjecture." §F5 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 244 /C1/45, 1994. Hardy, G. H. and Wright, E. M. "Quadratic Residues." §6.5 inAn Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 67 /C1/8, 1979. Nagell, T. "Euler’s Criterion and Legendre’s Symbol." §38 in Introduction to Number Theory. New York: Wiley, pp. 133 /C1/36, 1951. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 33 /C1/4 and 40 /C1/2, 1993. Legendre Transform The Legendre transform of a sequence ckfg is the sequence akfg with terms given by an/C30Xn k/C300ckn k/C(*/C(+ n/C27k k/C(*/C(+ ; wheren k/CP/C( is a BINOMIAL COEFFICIENT (Jin and Dick- inson 2000). Strehl (1994) and Schmidt (1995) showed that Xn k/C300n k/C(*/C(+2n/C27k k/C(*/C(+2 /C30Xn k/C300n k/C(*/C(+ n/C27k k/C(*/C(+Xk j/C300k j/C(*/C(+3 : References Jin, Y. and Dickinson, H. "Ape´ry Sequences and Legendre Transforms." J. Austral. Math. Soc. Ser. A 68, 349 /C1/56, 2000. Schmidt, A. L. "Legendre Transforms and Ape´ry’s Se- quences." J. Austral. Math. Soc. Ser. A 58, 358 /C1/75, 1995. Strehl, V. "Binomial Identities--Combinatorial and Algorith- mic Aspects. Trends in Discrete Mathematics." Disc. Math. 136, 309 /C1/46, 1994. Legendre Transformation Given a function of two variables df /C30@f @xdx /C27@f @ydy /C13udx/C27vdy ; (1) change the differentials from dx and dy to du and dy with the transformation g /C13f /C28ux (2) dg /C30df /C28udx/C28xdu/C30udx/C27vdy/C28udx/C28xdu /C30vdy/C28xdu : (3) Then x /C13/C28@g @u : (4) v /C13@g @y : (5) Lehmer Continued Fraction A CONTINUED FRACTION OF THE FORM b0 /C27e1 b1 /C27e2 b2 /C27e3 b3 /C27::: where (bi;ei/C271)/C30(1;1) or (2, /C281) for x/C231;2½Þ an IRRATIONAL NUMBER (Lehmer 1994, Dajani and Kraaikamp 1999). See also CONTINUED FRACTION References Dajani, K. and Kraaikamp, C. "The Mother of All Continued Fractions." http://www.math.uu.nl/publications/preprints/ 1106.ps.gz. Lehmer, J. "Semiregular Continued Fractions whose Partial Denominators are 1 or 2." In The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry, and Special Func-tions. Conference on the Legacy of Wilhelm Magnus May1/C1 /, 1992 (Brooklyn, NY) (Ed. W. Abikoff, J. S. Birman, and K. Kuiken). Providence, RI: Amer. Math. Soc., 1994. Lehmer Method LEHMER- SCHUR METHODLehmer Number A number generated by a generalization of a L UCAS SEQUENCE . Let aandbbeCOMPLEX NUMBERS with a/C27b/C30ffiffiffiffi Rp (1) ab/C30Q; (2) where Qand Rare RELATIVELY PRIME NONZERO INTEGERS and a=bis a ROOT OF UNITY . Then the Lehmer numbers are Un(ffiffiffiffi Rp ;Q)/C30an/C28bn a/C28b; (3) and the companion numbers VnffiffiffiffiRp ;Q/C(%/C(r /C30an/C27bn a/C27bfornodd an/C27bnforneven8 < :(4) References Lehmer, D. H. "An Extended Theory of Lucas’ Functions." Ann. Math. 31, 419/C1/48, 1930. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 61 and 70, 1989. Shorey, T. N. and Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers, 2." J. London Math. Soc. 23,1 7/C1/3, 1981. Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers." Proc. London Math. Soc. 35, 425/C1/47, 1977. Williams, H. C. "The Primality of N/C302A3n/C281:/"Canad. Math. Bull. 15, 585/C1/89, 1972. Lehmer-Schur Method An ALGORITHM which isolates ROOTS in the COMPLEX PLANE by generalizing 1-D bracketing. References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 196 /C1/98, 1990. Lehmer’s Conjecture LEHMER’S MAHLER MEASURE PROBLEM Lehmer’s Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Lehmer (1938) showed that every POSITIVE IRRA- TIONAL NUMBER xhas a unique infinite continued cotangent representation OF THE FORM x/C30cotX/C12 k/C300/C281ðÞkcot/C281bk"# ; where the bk/s are NONNEGATIVE and bk](bk/C281)2/C27bk/C281/C271: The case for which the convergence is slowest occurs when the inequality is replaced by equality, giving c0 /C300 and ck /C30(ck /C281)2 /C27ck /C281 /C271 for k ]1: The first few values are ckare 0, 1, 3, 13, 183, 33673, ... (Sloane’s A024556), resulting in the constant j /C30cot(cot/C281 0 /C28cot/C281 1 /C27cot/C281 3 /C28cot /C281 13 /C27cot /C281 183 /C28cot /C281 33673 /C27cot /C281 1133904603 /C28cot /C281 1285739649838492213 /C27.../C27(/C281)kck ...) /C30cot1 4 p /C27cot /C281 3 /C28cot /C281 13/C(% /C27cot /C281 183 /C28cot /C281 33673 /C27cot /C281 1133904603 /C28cot /C281 1285739649838492213 /C27.../C27(/C281)kck ...) /C300:59263271... (Sloane’s A030125). j is not an ALGEBRAIC NUMBER of degree less than 4 ;but Lehmer’s approach cannot show whether or not jisTRANSCENDENTAL . See also ALGEBRAIC NUMBER ,T RANSCENDENTAL NUMBER References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lehmer/lehmer.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 29, 1983. Lehmer, D. H. "A Cotangent Analogue of Continued Frac- tions." Duke Math. J. 4, 323/C1/40, 1938. Plouffe, S. "The Lehmer Constant." http://www.lacim.u- qam.ca/piDATA/lehmer.txt. Sloane, N. J. A. Sequences A024556 and A030125 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/eisonline.html. Lehmer’s Formula AFORMULA related to M EISSEL’S FORMULA . p(x)/C30xbc/C28Xa i/C301x pi$% /C27X 15i5j5ax pipj$% /C28... /C271 2(b/C27a/C282)(b/C28a/C271)/C28X a5i5bpx pi ! /C28Xc i/C30a/C271Xbi j/C30ipx pipj ! /C28(j/C281)"# ; where xbcis the FLOOR FUNCTION , a/C13p(x1=4) b/C13p(x1=2) bi/C13pffiffiffiffiffiffiffiffiffi x=pip/C(%/C(rc/C13p(x1=3); andp(n) is the PRIME COUNTING FUNCTION . References Riesel, H. "Lehmer’s Formula." Prime Numbers and Com- puter Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 13 /C1/4, 1994. Lehmer’s Mahler Measure Problem Portions of this entry contributed by K EVIN O’BRYANT An UNSOLVED PROBLEM in mathematics attributed to Lehmer that concerns the minimum M AHLER MEA- SURE M1(P) for a UNIVARIATE POLYNOMIAL P(x) that is not a product of CYCLOTOMIC POLYNOMIALS . Lehmer conjectured that if P(x) is such a polynomial with integer coefficients, then M1(P)]M1(1/C28x/C27x3/C28x4/C27x5/C28x6/C27x7/C28x9/C27x10) /C30m/C31; (1) where m/C31:1:1762 is the largest positive root of this polynomial. The roots of this polynomial, plotted in the left figure above, are very special, since 8 of the 10 lie on the UNIT CIRCLE in the COMPLEX PLANE . The roots of the polynomials (represented by half their coefficients) giving the two next smallest known Mahler measures are also illustrated above (Mos-singhoff, p. S11). The best current bound is that of Smyth (1971), who showed that M(F)>u 1;where Fis a nonzero non- reciprocal polynomial that is not a product of CYCLO- TOMIC POLYNOMIALS (Everest 1999), and u1:1:324 is the real root of x3/C28x/C281/C300:Generalizations of Smyth’s result have been constructed by Lloyd-Smith(1985) and Dubickas (1997). In general, the smallest M AHLER MEASURES occur for polynomials with integers coefficients that are smallin absolute value. The histogram above shows the distribution of measures for random ( /C281, 0, 1)-poly- nomials of random orders 1 to 10. Mossinghoff (1998) gives a table of the smallest known Mahler measures for polynomial degrees up to d /C3024. See also MAHLER MEASURE References Boyd, D. W. "Reciprocal Polynomials Having Small Mea- sure." Math. Comput. 35, 1361 /C1/377, 1980. Boyd, D. W. "Reciprocal Polynomials Having Small Mea- sure. II." Math. Comput. 53, 355 /C1/57 and S1-S5, 1989. Dubickas, A. "Algebraic Conjugates Outside the Unit Cir- cle." In New Trends in Probability and Statistics, Vol. 4: Analytic and Probabilistic Methods in Number Theory. Proceedings of the 2nd International Conference held in Honor of J. Kubilius on His 75th Birthday in Palanga, September 23 /C1/7, 1996 (Ed. A. Laurincikas, E. Manstavi- cius, and V. Stakenas). Utrecht, Netherlands: VSP, pp. 11 /C1/1, 1997. Everest, G. Ch. 1 in Heights of Polynomials and Entropy in Algebraic Dynamics. London: Springer-Verlag, 1999. Lloyd-Smith, C. W. "Algebraic Numbers Near the Unit Circle." Acta Arith. 45,43/C1/7, 1985. Mossinghoff, M. J. "Polynomials with Small Mahler Mea- sure." Math. Comput. 67, 1697 /C1/705 and S11-S14, 1998. Smyth, C. J. "On the Product of the Conjugates Outside the Unit Circle of an Algebraic Integer." Bull. London Math. Soc. 3, 169 /C1/75, 1971. Lehmer’s Phenomenon The appearance of nontrivial zeros (i.e., those along the CRITICAL STRIP with R[z] /C301 =2) of the RIEMANN ZETA FUNCTION z(z) very close together. An example is the pair of zeros z1 2 /C27(7005 /C27t)i/C(%/C(r given by t1 : 0:0606918 and t2 :0:100055 ; illustrated above in the plot of j z(1 2 /C27(7005 /C27t)i) j2 :/ See also CRITICAL STRIP,RIEMANN ZETA FUNCTION References Csordas, G.; Odlyzko, A. M.; Smith, W.; and Varga, R. S. "A New Lehmer Pair of Zeros and a New Lower Bound for the de Bruijn-Newman Constant." Elec. Trans. Numer. Ana- lysis 1, 104 /C1/11, 1993. Csordas, G.; Smith, W.; and Varga, R. S. "Lehmer Pairs of Zeros, the de Bruijn-Newman Constant and the Riemann Hypothesis." Constr. Approx. 10, 107 /C1/29, 1994. Csordas, G.; Smith, W.; and Varga, R. S. "Lehmer Pairs of Zeros and the Riemann z/-Function." In Mathematics of Computation 1943 /C1/993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993). Proc. Sympos. Appl. Math. 48, 553 /C1/56, 1994. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 357 /C1/58, 1991. Lehmer’s Problem LEHMER’S MAHLER MEASURE PROBLEM ,L EHMER’S TOTIENT PROBLEMLehmer’s Theorem FERMAT’S LITTLE THEOREM CONVERSE Lehmer’s Totient Problem Do there exist any COMPOSITE NUMBERS n such that f(n) ½(n /C281); where f(n) is the TOTIENT FUNCTION ?No such numbers are known. In 1932, Lehmer showed that such an n must be ODD and SQUAREFREE , and that the number of distinct PRIME FACTORS d(7) ]7: This was subsequently extended to d(n) ]11 : The best current results are n > 1020 and d(n) ]14 (Cohen and Hagis 1980), if 30¶n ; then d(n) ]26 (Wall 1980), and if 3½n then d(n) ]213 and n ]5:5 /C2910570 (Lieu- wens 1970). See also LEHMER’S MAHLER MEASURE PROBLEM , TOTIENT FUNCTION References Cohen, G. L. and Hagis, P. Jr. "On the Number of Prime Factors of nisf(n)½(n/C281):/"Nieuw Arch. Wisk. 28, 177/C1/ 85, 1980. Lieuwens, E. "Do There Exist Composite Numbers for Which kf(M)/C30M/C281 Holds?" Nieuw. Arch. Wisk. 18, 165/C1/69, 1970. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 27 /C1/8, 1989. Wall, D. W. "Conditions for f(N) to Properly Divide N/C281:/" InA Collection of Manuscripts Related to the Fibonacci Sequence (Ed. V. E. Hoggatt and M. V. E. Bicknell-John- son). San Jose, CA: Fibonacci Assoc., pp. 205 /C1/08, 1980. Lehmus’ Theorem STEINER- LEHMUS THEOREM Leibniz Criterion Also known as the ALTERNATING SERIES TEST . Given a SERIES X/C12 n/C301(/C281)n/C271an with an>0;ifanis monotonic decreasing as n0/C12 and lim n0/C12an/C300 then the series CONVERGES . Leibniz Harmonic Triangle 1 1 1212 131613 14 1 121 1214 15 1 201 301 2015 (Sloane’s A003506). In the Leibniz harmonic triangle, each FRACTION is the sum of numbers below it, with the initial and final entry on each row one over the corresponding entry in PASCAL’S TRIANGLE . The DE- NOMINATORS in the second diagonals are 6, 12, 20, 30, 42, 56, ... (Sloane’s A007622). See also CATALAN’S TRIANGLE ,C LARK’S TRIANGLE , EULER’S TRIANGLE ,LOSSNITSCH’S TRIANGLE ,NUMBER TRIANGLE ,P ASCAL’S TRIANGLE ,SEIDEL- ENTRINGER- ARNOLD TRIANGLE References Sloane, N. J. A. Sequences A003506 and A007622/M4096 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Leibniz Identity dn dxn (uv) /C30dnu dxnv /C27n 1/C(*/C(+dn/C281u dxn/C281dv dx /C27.../C27 n r/C(*/C(+ /C2dn/C28ru dxn/C28rdnv dxr /C27.../C27udnv dxn wheren k/CP/C( is a BINOMIAL COEFFICIENT . This can also be written explicitly as Dnf(t)g(t) /C30Xn k/C300n k/C(*/C(+ Dkf(t)Dn/C28kg(t) (Roman 1980). See also FAA´ DI BRUNO’S FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 12, 1972. Roman, S. "The Formula of Faa di Bruno." Amer. Math. Monthly 87, 805 /C1/09, 1980. Leibniz Integral Rule @ @z gb(z) a(z)f(x; z) dx /C30gb(z) a(z)@f @zdx /C27f(b(z) ; z)@b @z /C28f(a(z) ; z)@a @z: The differentiation of a definite integral whose limits are functions of the differential variable. The rule can be used to evaluate certain unusual definite integrals such as f( a) /C30g p 0ln(1 /C282a cos x /C27 a2) dx /C302 p ln ½a½ for ½a½> 1 (Woods 1926). Although the symbolic mathematics program Mathematica gives an analy-tic solution to this integral, it gives the solution in a much more complicated form. Feynman (1997) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again." See also DERIVATIVE ,INTEGRAL ,INTEGRATION UNDER THE INTEGRAL SIGN References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 232, 1987. Feynman, R. P. and Leighton, R. "A Different Set of Tools." In ‘Surely You’re Joking, Mr. Feynman!’: Adventures of a Curious Character. New York: W. W. Norton, pp. 69 /C1/2, 1997. Kaplan, W. "Integrals Depending on a Parameter--Leibnitz’s Rule.’ §4.9 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 256 /C1/58, 1992. Woods, F. S. "Differentiation of a Definite Integral." §60 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathe- matics. Boston, MA: Ginn, pp. 141 /C1/44, 1926. Leibniz Series The SERIES for the INVERSE TANGENT , tan /C281 x /C30x /C281 3 x3 /C2715 x5 /C27... : Plugging in x /C301 gives GREGORY’S FORMULA 1 4 p /C301 /C2813 /C2715 /C2817 /C2719 /C28...: This series is intimately connected with the number of representations of nbyksquares rk(n);and also with G AUSS’S CIRCLE PROBLEM (Hilbert and Cohn- Vossen 1999, pp. 27 /C1/9). See also GAUSS’S CIRCLE PROBLEM ,GREGORY’S FOR- MULA ,SUM OF SQUARES FUNCTION References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 37, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 50, 1986. Lelong’s Theorem References Morosawa, S.; Nishimura, Y.; Taniguchi, M.; and Ueda, T. "Lelong’s Theorem." §8.2 in Holomorphic Dynamics. Cam- bridge, England: Cambridge University Press, pp. 270 /C1/ 76, 2000. Lemarie ´’s Wavelet A wavelet used in multiresolution representation to analyze the information content of images. The WAVELET is defined by H(v) /C30 2(1 /C28u)4315 /C28 420u /C27 126u2 /C28 4u3 315 /C28 420v /C27 126v2 /C28 4v3"#1 =2 ; where u /C13sin21 2 v/C(%/C(r v /C13sin2 v (Mallat 1989). See also WAVELET References Mallat, S. G. "A Theory for Multiresolution Signal Decom- position: The Wavelet Representation." IEEE Trans. Pattern Analysis Machine Intel. 11, 674 /C1/93, 1989. Mallat, S. G. "Multiresolution Approximation and Wavelet Orthonormal Bases of L2(R) :/" Trans. Amer. Math. Soc. 315,69/C1/7, 1989. Lemma A short THEOREM used in proving a larger THEOREM . Related concepts are the AXIOM ,PORISM ,POSTULATE , PRINCIPLE , and THEOREM . See also ABEL’S LEMMA ,A RCHIMEDES’ LEMMA , BARNES’ LEMMA ,BLICHFELDT’S LEMMA ,BOREL- CAN- TELLI LEMMA ,BURNSIDE’S LEMMA ,DANIELSON- LANC- ZOS LEMMA ,D EHN’S LEMMA ,D ILWORTH’S LEMMA , DIRICHLET’S LEMMA ,D IVISION LEMMA ,F ARKAS’S LEMMA ,FATOU’S LEMMA ,FUNDAMENTAL LEMMA OF CALCULUS OF VARIATIONS ,GAUSS’S LEMMA ,HENSEL’S LEMMA ,ITOˆ ’S LEMMA ,JORDAN’S LEMMA ,LAGRANGE’S LEMMA ,N EYMAN- PEARSON LEMMA ,POINCARE ´ ’S HO- LOMORPHIC LEMMA ,P OINCARE ´ ’S LEMMA ,P O´ LYA- BURNSIDE LEMMA ,R IEMANN- LEBESGUE LEMMA , SCHUR’S LEMMA ,SCHUR’S REPRESENTATION LEMMA , SCHWARZ- PICK LEMMA ,S PIJKER’S LEMMA ,Z ORN’S LEMMA Lemma That Is Not Burnside’s CAUCHY- FROBENIUS LEMMA ,P O´LYA ENUMERATION THEOREM Lemniscate A polar curve also called LEMNISCATE OF BERNOULLI which is the LOCUS of points the product of whosedistances from two fixed points (called the FOCI)a distance 2 aaway is the constant a2:Letting the FOCI be located at ( 9a;0);the Cartesian equation is [(x/C28a)2/C27y2][(x/C27a)2/C27y2]/C30a4; (1) which can be rewritten x4/C27y4/C272x2y2/C302a2(x2/C28y2): (2) Letting a?/C13ffiffiffi 2p a;the POLAR COORDINATES are given by r2/C30a2cos(2 u): (3) An alternate form is r2/C30a2sin(2 u) (4) The PARAMETRIC EQUATIONS for the lemniscate are x/C30acost 1/C27sin2t: (5) y/C30asintcost 1/C27sin2t: (6) The bipolar equation of the lemniscate is rr0/C301 2a2; (7) and in PEDAL COORDINATES with the PEDAL POINT at the center, the equation is pa2/C30r3: (8) The two-center BIPOLAR COORDINATES equation with origin at a FOCUS is r1r2/C30c2: (9) The lemniscate can also be generated as the ENVEL- OPEof circles centered on a RECTANGULAR HYPERBOLA and passing through the center of the HYPERBOLA (Wells 1991). Jakob Bernoulli published an article in Acta Erudi- torum in 1694 in which he called this curve the lemniscus (Latin for "a pendant ribbon"). Jakob Bernoulli was not aware that the curve he was describing was a special case of C ASSINI OVALS which had been described by Cassini in 1680. The general properties of the lemniscate were discovered by G. Fagnano in 1750 (MacTutor Archive). Gauss’s and Euler’s investigations of the ARC LENGTH of the curve led to later work on ELLIPTIC FUNCTIONS . The lemniscate is the INVERSE CURVE of the HYPER- BOLA with respect to its center. The CURVATURE of the lemniscate is k /C303ffiffiffi 2p costffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C28 cos(2 t)p : (10) The ARC LENGTH is more problematic. Using the polar form, ds2 /C30dr2 /C27r2 d u2 (11) so ds /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 rdu dr !2vuutdr : (12) But we have 2rdr/C302a2 sin(2u) du (13) rdr du /C30r2 a2 sin(2u) (14) rdu dr !2 /C30r4 a4 sin2(2u) /C30r4 a4[1 /C28 cos2(2u)] /C30r4 a4 /C28 r4 ; (15) so ds /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27r4 a4 /C28 r4s dr /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 a4 /C28 r4s dr /C30a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 /C28 r4p dr /C30drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28r a/C(%/C(r4r ; (16) and L /C30ga 0ds /C302ga 0ds drdr /C302ga 0drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28r a/C(%/C(r4r : (17) Let t /C13r =a; so dt /C30dr =a ; and L /C302ag1 0(1 /C28t4)/C281 =2 dt (18) which, as shown in LEMNISCATE FUNCTION , is given analytically by L /C30ffiffiffi 2p aK1ffiffiffi 2p ! /C30G21 4/C(%/C(r 23 =2ffiffiffipp a : (19)If a /C301, then L /C305 :2441151086 ::: (20) which is related to GAUSS’S CONSTANT M by L /C302 p M: (21) The quantity L=2or L =4 is called the LEMNISCATE CONSTANT and plays a role for the lemniscate analo- gous to that of pfor the CIRCLE . The AREA of one loop of the lemniscate is A/C301 2gr2du/C3012a2gp=4 /C28p=4cos(2 u)du /C301 4a2sin(2u) ½/C138/C27p=4 /C28p=4 /C301 2a2[sin(2 u)]p=4 0/C3012a2sinp2/C(%/C(r /C28sin 0hi /C3012a2:(22) See also LEMNISCATE FUNCTION ,LICHTENFELS MINI- MAL SURFACE References Ayoub, R. "The Lemniscate and Fagnano’s Contributions to Elliptic Integrals." Arch. Hist. Exact Sci. 29, 131/C1/49, 1984. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 220, 1987. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Gray, A. "Lemniscates of Bernoulli." §3.2 in Modern Differ- ential Geometry of Curves and Surfaces with Mathema-tica, 2nd ed. Boca Raton, FL: CRC Press, pp. 52 /C1 /3, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 120 /C1/24, 1972. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 37, 1983. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, 1967. MacTutor History of Mathematics Archive. "Lemniscate of Bernoulli." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Lemniscate.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 139 /C1/40, 1991. Yates, R. C. "Lemniscate." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 143 /C1/47, 1952. Lemniscate (Mandelbrot Set) A curve on which points of a MAP zn(such as the MANDELBROT SET) diverge to a given value rmax at the same rate. A common method of obtaining lemnis- cates is to define an INTEGER called the COUNT which is the largest n such that ½zn ½Br where r is usually taken as r /C302. Successive COUNTS then define a series of lemniscates, which are called EQUIPOTENTIAL CURVES by Peitgen and Saupe (1988). See also COUNT ,MANDELBROT SET References Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, pp. 178 /C1/79, 1988. Lemniscate Case The case of the WEIERSTRASS ELLIPTIC FUNCTION with invariants g2 /C301 and g3 /C300:/ See also EQUIANHARMONIC CASE,W EIERSTRASS EL- LIPTIC FUNCTION ,PSEUDOLEMNISCATE CASE References Abramowitz, M. and Stegun, C. A. (Eds.). "Lemniscate Case (/g2 /C301; g3 /C300):/" §18.14 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 658 /C1/62, 1972. Lemniscate Constant Let L /C301ffiffiffiffiffiffi 2 pp G1 4/C(%/C(rhi2 /C305 :2441151086... be the ARC LENGTH of a LEMNISCATE with a /C30 1. Then the lemniscate constant is the quantity L=2 (Abra- mowitz and Stegun 1972), or L =4 /C301:311028777... (Todd 1975, Le Lionnais 1983). Todd (1975) cites T. Schneider (1937) as proving Lto be a TRANSCEN- DENTAL NUMBER . See also LEMNISCATEReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/gauss/gauss.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 37, 1983. Todd, J. "The Lemniscate Constant." Comm. ACM 18,1 4/C1/9 and 462, 1975. Lemniscate Function The lemniscate functions arise in rectifying the ARC LENGTH of the LEMNISCATE . The lemniscate functions were first studied by Jakob Bernoulli and Giulio Fagnano. A historical account is given by Ayoub(1984), and an extensive discussion by Siegel (1969). The lemniscate functions were the first functions defined by inversion of an integral, which was firstdone by Gauss. L/C302a g1 0(1/C28t4)/C281=2dt: (1) Define the functions f(x)/C13arcsinlemn xgx 0(1/C28t4)/C281=2dt (2) f?(x)/C13arccoslemn x/C30g1 x(1/C28t4)/C281=2dt; (3) where 6/C13L a; (4) and write x/C30sinlemn f (5) x/C30coslemn f?: (6) There is an identity connecting fandf?since f(x)/C27f?(x)/C30L 2a/C301 26; (7) so sinlemn f/C30coslemn126/C28f/C(%/C(r : (8) These functions can be written in terms of J ACOBI ELLIPTIC FUNCTIONS , u/C30gsd(u;k) 0[(1/C28k?2y2)(1/C27k2y2)]/C281=2dy: (9) Now, if k/C30k?/C301=ffiffiffi 2p ;then u/C30gsd(u;1=ffiffi 2p ) 01/C281 2y2/C(%/C(r 1/C2712y2/C(%/C(rhi/C281=2 dy /C30gsd(u;1=ffiffi 2p ) 01/C2814y4/C(%/C(r/C281=2 dy: (10) Lett/C13y=ffiffiffi 2p sody/C30ffiffiffi2p dt; u/C30ffiffiffi 2p gsd(u;1=ffiffi 2p )=ffiffi 2p 0(1/C28t4)/C281=2dt (11) uffiffiffi 2p/C30gsd(u;1=ffiffi 2p )=ffiffi 2p 0(1/C28t4)/C281=2dt (12) u/C30gsd(uffiffi 2p ;1=ffiffi 2p )=ffiffi 2p 0(1/C28t4)/C281=2dt (13) and sinlemn f/C301ffiffiffi 2psdfffiffiffi 2p ;1ffiffiffi 2p ! : (14) Similarly, u/C30g1 cn(u;k)(1/C28t2)/C281=2(k?2/C27k2t2)/C281=2dt /C30g1 cn(u;1=ffiffi 2p )(1/C28t2)/C281=21 2/C2712t2/C(%/C(r/C281=2 dt /C30ffiffiffi 2pg1 cn(u;1=ffiffi 2p )(1/C28t4)/C281=2dt (15) uffiffiffi 2p/C30g1 cn(u;1=ffiffi 2p )(1/C28t4)/C281=2dt (16) u/C30g1 cn(uffiffi 2p ;1=ffiffi 2p )(1/C28t4)/C281=2dt; (17) and coslemn f/C30cnfffiffiffi 2p ;1ffiffiffi 2p ! : (18) We know coslemn1 26/C(%/C(r /C30cn126ffiffiffi 2p ;1ffiffiffi 2p ! /C300: (19) But it is true that cn(K;k)/C300; (20) so K1ffiffiffi2p ! /C301 2ffiffiffi 2p 6/C301ffiffiffi 2p6 (21)G21 4/C(%/C(r 4ffiffiffipp/C301ffiffiffi 2p6 (22) L/C30a6/C30affiffiffi 2pG21 4/C(%/C(r 4ffiffiffipp/C30G21 4/C(%/C(r 23=2ffiffiffippa: (23) By expanding (1 /C28t4)/C281=2in a BINOMIAL SERIES and integrating term by term, the arcsinlemn function can be written f(x)/C30gv 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28t4p /C30X/C12 n/C3001 2/C(%/C(r nx4n/C271 n!(4n/C271); (24) where ( a)nis the RISING FACTORIAL (Berndt 1994). Ramanujan gave the following inversion FORMULA for f(x):If umffiffiffi 2p/C30X/C12 n/C3001 2/C(%/C(r nx4n/C271 n!(4n/C271); (25) where m/C30G214/C(%/C(r 2p3=2(26) is the constant obtained by letting x/C301 and u/C30p=2; and v/C302/C281=2sd(mu); (27) then m2 2x2/C30csc2u/C281 p/C288X/C12 n/C301ncos(2 nu) e2pn/C281(28) (Berndt 1994). Ramanujan also showed that if 0 B uBp=2;then /C28mffiffiffi 2pX/C12 n/C3001 2/C(%/C(r nv4n/C281 n!(4n/C281) /C30cotu/C27u p/C274X/C12 n/C301sin(2 nu) 22pn/C281; (29) lnv/C271 6p/C2812ln 2/C27X/C12 n/C3001 4/C(%/C(r nv4n 34/C(%/C(r n4n /C30ln(sin u)/C27u2 2p/C282X/C12 n/C301cos(2 nu) n(e2pn/C281); (30) 1 2tan/C281v/C30X/C12 n/C300sin[(2 n/C271)u] (2n/C271)cosh1 2(2n/C271)phi ; (31) 1 4cos/C281(v2)/C30X/C12 n/C300(/C281)ncos[(2 n/C271)u] (2n/C271)cosh1 2(2n/C271)phi ; (32) and ffiffiffi 2p 4 mX/C12 n/C30022n(n!)2 (2n /C27 1)!(4n /C27 3)v4n/C273 /C30pu 8/C28X/C12 n/C300( /C281)nsin[(2 n /C27 1)u] (2n /C27 1)2cosh1 2(2n /C27 1)phi (33) (Berndt 1994). A generalized version of the lemniscate function can be defined by letting 0 5 u 5 p=2 and 0 5v 51 : Write 2 3 um /C30gv 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 t6p ; (34) where m is the constant obtained by setting u /C30 p=2 and v /C301. Then m /C30ffiffiffipp G2 3/C(%/C(r G56/C(%/C(r ; (35) and Ramanujan showed 4m2 9v2 /C30csc2 u /C282 pffiffiffi 3p/C278X/C12 n /C301( /C281)n/C281n cos(2 n u) e pnffiffi 3p /C28 ( /C281)n (36) (Berndt 1994). See also ELLIPTIC FUNCTION ,E LLIPTIC INTEGRAL , HYPERBOLIC LEMNISCATE FUNCTION References Ayoub, R. "The Lemniscate and Fagnano’s Contributions to Elliptic Integrals." Arch. Hist. Exact Sci. 29, 131 /C1/49, 1984. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 245, and 247 /C1/55, 258 /C1/60, 1994. Siegel, C. L. Topics in Complex Function Theory, Vol. 1. New York: Wiley, 1969. Lemniscate Inverse Curve The INVERSE CURVE of a LEMNISCATE in a CIRCLE centered at the origin and touching the LEMNISCATE where it crosses the X-AXIS produces a RECTANGULAR HYPERBOLA (Wells 1991). See also RECTANGULAR HYPERBOLA References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 209, 1991. Lemniscate of Bernoulli LEMNISCATE Lemniscate of Gerono EIGHT CURVE Lemoine Axis LEMOINE LINELemoine Circle Draw lines P1Q1 ; P2Q2 ; and P3Q3through the SYMMEDIAN POINT K and parallel to the sides of the triangle DA1A2A3 : The points where the parallel lines intersect the sides of DA1A2A3then lie on a CIRCLE known as the Lemoine circle, or sometimes the TRIPLICATE-RATIO CIRCLE (Tucker 1883). This circle has center at the MIDPOINT Z of OK, where O is the CIRCUMCENTER , and RADIUS 1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C27r2 cq /C301 2 R sec v; where R is the CIRCUMRADIUS , rcis RADIUS of the COSINE CIRCLE , and v is the BROCARD ANGLE of the original triangle (Johnson 1929, p. 274). The Lemoine circle and BROCARD CIRCLE are concentric, and the triangles DQ1P3K ;DKQ3P2 ; and DP1KQ2are similar to DA1A3A2 (Tucker 1883). The Lemoine circle divides any side into segments proportional to the squares of the sides A2P2 :P2Q3 :Q3A3 /C30a2 3 : a21 : a22 Furthermore, the chords cut from the sides by the Lemoine circle are proportional to the squares of the sides. The COSINE CIRCLE is sometimes called the second Lemoine circle. The Lemoine circle is a special case of aTUCKER CIRCLE . See also COSINE CIRCLE ,L EMOINE HEXAGON ,L E- MOINE LINE,S YMMEDIAN POINT ,T AYLOR CIRCLE , TUCKER CIRCLES References Casey, J. "On the Equations and Properties--(1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on aSphere; (4) of the System of Conics Inscribed to a Conic,and Touching Three Inscribed Conics in a Plane." Proc. Roy. Irish Acad. 9, 396/C1 /23, 1864 /C1/866. Casey, J. "Lemoine’s, Tucker’s, and Taylor’s Circle." Supp. Ch. §3i nA Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 179 /C1/89, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 70, 1971. Honsberger, R. "The Lemoine Circles." §9.2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 88 /C1/9, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 273 /C1/75, 1929. Lachlan, R. "The Lemoine Circle." §131 /C1/32 in An Elemen- tary Treatise on Modern Pure Geometry. London: Macmil- lian, pp. 76 /C1/7, 1893. Lemoine. Assoc. Franc ¸ais pour l’avancement des Sci. 1873. Tucker, R. "The ‘Triplicate Ratio’ Circle." Quart. J. Pure Appl. Math. 19, 342 /C1/48, 1883. Lemoine Hexagon The closed self-intersecting cyclic hexagon formed by joining the adjacent PARALLELS in the construction of the LEMOINE CIRCLE . The sides of this hexagon have the property that, in addition to Q1P2A1A2 ; k Q2P3A2A3 ; k and Q3P2A1A3 ; k the remaining sides Q1P1 ; Q2P2 ; and Q3P3are ANTIPARALLEL to A2A3 ; A1A3 ; and A1A2 ; respectively. The Lemoine hexagon is a special case of a TUCKER HEXAGON . See also COSINE HEXAGON ,LEMOINE CIRCLE ,TUCKER HEXAGON Lemoine Line The Lemoine line, also called the LEMOINE AXIS, is the perspectivity axis of a TRIANGLE and its TANGENTIAL TRIANGLE , and also the TRILINEAR POLAR of the CENTROID of the triangle vertices. It is also the POLAR of K with regard to its CIRCUMCIRCLE , and is PERPEN- DICULAR to the BROCARD AXIS. The centers of the APOLLONIUS CIRCLES L1 ; L2 ; and L3 are COLLINEAR on the LEMOINE LINE. This line is PERPENDICULAR to the BROCARD AXIS OK and is the RADICAL AXIS of the CIRCUMCIRCLE and the BROCARD CIRCLE . It has equation a a /C27b b /C27g c in terms of TRILINEAR COORDINATES (Oldknow 1996).See also APOLLONIUS CIRCLES ,BROCARD AXIS,CEN- TROID (TRIANGLE ), CIRCUMCIRCLE ,C OLLINEAR ,L E- MOINE CIRCLE ,SYMMEDIAN POINT ,POLAR ,RADICAL AXIS,SYMMEDIAN ,TANGENTIAL TRIANGLE ,TRILINEAR POLAR References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 295, 1929. Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/29, 1996. Lemoine Point SYMMEDIAN POINT Lemoine’s Problem Given the vertices of the three EQUILATERAL TRIAN- GLES placed on the sides of a TRIANGLE T, construct T. The solution can be given using KIEPERT’S HYPER- BOLA . See also KIEPERT’S HYPERBOLA Lemon A SURFACE OF REVOLUTION defined by Kepler. It consists of less than half of a circular ARC rotated about an axis passing through the endpoints of the ARC. The equations of the upper and lower boundaries in the xz plane are z9/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28(x /C27r)2q for R /C21 r and x /C23 [/C28(R /C28r) ; R /C28r] : The CROSS SECTION of a lemon is a LENS . The lemon is the inside surface of a SPINDLE TORUS . The American football is shaped like a lemon. See also APPLE ,LENS,O VAL,PROLATE SPHEROID , SPINDLE TORUS References JavaView. "Classic Surfaces from Differential Geometry: Football/Barrel." http://www-sfb288.math.tu-berlin.de/ vgp/javaview/demo/surface/common/PaSurface_Football- Barrel.html. Length (Curve) Let g(t) be a smooth curve in a MANIFOLD M from x to y with g(0) /C30x and g(1) /C30y: Then g?(t) /C23 Tg(t) where Tx is the TANGENT SPACE of M at x. The length of g with respect to the Riemannian structure is given by g1 0½½g ?(t) ½½g(t) dt: See also ARC LENGTH ,DISTANCE Length (Number) The length of a number n in base b is the number of DIGITS in the base- b numeral for n, given by the formula L(n ; b) /C30 logb(n) bc /C271; where xbcis the FLOOR FUNCTION . The MULTIPLICATIVE PERSISTENCE of an n-DIGIT is sometimes also called its length. See also CONCATENATION ,D IGIT,FIGURES ,M ULTI- PLICATIVE PERSISTENCE Length (Partial Order) For a PARTIAL ORDER , the size of the longest CHAIN is called the length. See also WIDTH (PARTIAL ORDER ) Length (Size) The longest dimension of a 3-D object. See also HEIGHT ,W IDTH (SIZE) Length Distribution Function A function giving the distribution of the interpoint distances of a curve. It is defined by p(r) /C301 NX ijdrij/C30r: See also RADIUS OF GYRATION References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 204 /C1/ 06, 1995. Length-Preserving Transformation ISOMETRYLengyel’s Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let Ldenote the partition lattice of the SET f1;2;...;ng:The MAXIMUM element of Lis M/C30ff1;2;...;ngg (1) and the MINIMUM element is m/C30ff1g;f2g;...;fngg: (2) LetZndenote the number of chains of any length in L containing both Mand m. Then Znsatisfies the RECURRENCE RELATION Zn/C30Xn/C281 k/C301s(n;k)Zk; (3) where s(n;k)i saS TIRLING NUMBER OF THE SECOND KIND . Lengyel (1984) proved that the QUOTIENT r(n)/C30Zn (n!)2(2 ln 2)/C28nn1/C28(ln 2) =3(4) is bounded between two constants as n0/C12;and Flajolet and Salvy (1990) improved the result of Babai and Lengyel (1992) to show that L/C13lim n0/C12r(n)/C301:0986858055 . . . : (5) References Babai, L. and Lengyel, T. "A Convergence Criterion for Recurrent Sequences with Application to the Partition Lattice." Analysis 12, 109/C1/19, 1992. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/lngy/lngy.html. Flajolet, P. and Salvy, B. "Hierarchal Set Partitions and Analytic Iterates of the Exponential Function." Unpub-lished manuscript, 1990. Lengyel, T. "On a Recurrence Involving Stirling Numbers." Europ. J. Comb. 5, 313/C1 /21, 1984. Plouffe, S. "The Lengyel Constant." http://www.lacim.u- qam.ca/piDATA/lengyel.txt. Lens A figure composed of two equal and symmetrically placed circular ARCS . It is also known as the FISH BLADDER (Pedoe 1995, p. xii) or VESICA PISCIS . The latter term is often used for the particular lensformed by the intersection of two unit CIRCLES whose centers are offset by a unit distance (Rawles 1997). In this case, the height of the lens is given by letting d /C30r /C30R /C301 in the equation for a CIRCLE-CIRCLE INTERSECTION a /C301 dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4d2R2 /C28(d2 /C28r2 /C27R2)2q ; (1) giving a /C30ffiffiffi 3p : The AREA of the VESICA PISCIS is given by plugging d /C30R into the CIRCLE-CIRCLE INTERSEC- TION area equation with r /C30R, A /C302R2 cos/C281d 2R ! /C281 2 dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4R2 /C28d2p ; (2) giving A /C301 64 p /C283ffiffiffi 3p/C(%/C(r :1:22837 : (3) Renaissance artists frequently surrounded images of Jesus with the vesica piscis (Rawles 1997). An asymmetrical lens is produced by a CIRCLE-CIRCLE INTERSECTION for unequal CIRCLES . A lens-shaped region also arises in the study of BESSEL FUNCTIONS . Letting z /C30ei u ; the inequality z exp(1 /C28 z2) 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 z2p/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()51 holds in the region illustrated above. This region can be parameterized in terms of a variable u as r 2 /C302u sinh(2 u) (4) sin2 u /C30sinh u(u cosh u /C28sinh u) : (5) As u increases from u to its maximum value of 1.19967874... (the root of sinh u(u cosh u /C28sinh u) /C30 0); r decreases from 1 to 0.6627434... (Plummer 1960, p. 47; Watson 1966, p. 270). This curve is very important in the theory of KAPTEYN SERIES . See also CIRCLE ,CIRCLE- CIRCLE INTERSECTION ,DOU- BLE BUBBLE ,F LOWER OF LIFE,G OAT PROBLEM , KAPTEYN SERIES ,LEMON ,LUNE,REULEAUX TRIAN- GLE,SECTOR ,SEED OF LIFE,SEGMENT ,VENN DIA- GRAMReferences Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Plummer, H. An Introductory Treatise of Dynamical Astron- omy. New York: Dover, 1960. Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 11, 1997. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Lens Space A lens space L(p ; q) is the 3-MANIFOLD obtained by gluing the boundaries of two solid TORI together such that the meridian of the first goes to a (p, q)-curve on the second, where a (p, q)-curve has p meridians and q longitudes. References Adams, C. C. "The Three-Sphere and Lens Spaces." §9.2 in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Free- man, pp. 246 /C1/56, 1994. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976. Lenstra Elliptic Curve Method A method of factoring INTEGERS using ELLIPTIC CURVES . References Montgomery, P. L. "Speeding up the Pollard and Elliptic Curve Methods of Factorization." Math. Comput. 48, 243 /C1/ 64, 1987. Le´on Anne’s Theorem Pick a point O in the interior of a QUADRILATERAL which is not a PARALLELOGRAM . Join this point to each of the four VERTICES , then the LOCUS of points O for which the sum of opposite TRIANGLE areas is half the QUADRILATERAL AREA is the line joining the MIDPOINTS M1andM2of the DIAGONALS . See also DIAGONAL (POLYGON ), MIDPOINT ,Q UADRI- LATERAL References Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 174 /C1/75, 1991. Leonardo’s Paradox In the depiction of a row of identical columns parallel to the plane of a PERSPECTIVE drawing, the outer columns should appear wider even though they are farther away. See also PERSPECTIVE ,VANISHING POINT ,ZEEMAN’S PARADOX References Dixon, R. Mathographics. New York: Dover, p. 82, 1991. Leptokurtic A distribution with a high peak so that the KURTOSIS satisfies g2 > 0:/ See also KURTOSIS LerchPhi LERCH TRANSCENDENT Lerch’s Theorem If there are two functions F1(t) and F2(t) with the same integral transform T[F1(t)] /C30T[F2(t)] /C13f(s) ; (1) then a NULL FUNCTION can be defined by d0(t) /C13F1(t) /C28F2(t) (2) so that the integral ga 0d0(t) dt /C300 (3) vanishes for all a /C210. See also NULL FUNCTION Lerch Transcendent A generalization of the HURWITZ ZETA FUNCTION and POLYLOGARITHM function. Many sums of reciprocal POWERS can be expressed in terms of it. It is defined by F(z ; s ; a) /C13X/C12 k /C300zk (a /C27 k)s ; (1) where any term with a /C27k /C300 is excluded. The Lerch transcendent is given by the Mathematica command LerchPhi [z, s, a]. The Lerch transcendent can be used to express the DIRICHLET BETA FUNCTIONb(s) /C13X/C12 k/C300(/C281)k(2k /C271)/C28s2 /C28s F/C281; s ;1 2/C(%/C(r ; (2) the integral of the FERMI- DIRAC DISTRIBUTION g/C12 0ks ek /C28 m /C27 1dk /C30e m G(s /C271)F(/C28e m ; s /C271;1); (3) where G(z) is the GAMMA FUNCTION , and to evaluate the DIRICHLET L-SERIES . See also DIRICHLET BETA FUNCTION ,DIRICHLET L- SERIES ,FERMI- DIRAC DISTRIBUTION ,H URWITZ ZETA FUNCTION ,L EGENDRE’S CHI-FUNCTION ,P OLYLOGA- RITHM References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Function C(z; s ; v) /C30a/C12 n/C300(v /C27n) /C28szn :/" §1.11 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 27 /C1/1, 1981. Less A quantity a is said to be less than b if a is smaller than b, written a B b.Ifa is less than or EQUAL to b, the relationship is written a 5b: If a is MUCH LESS than b, this is written a /C10b : Statements involving GREATER than and less than symbols are called INEQUALITIES . See also EQUAL ,GREATER ,INEQUALITY ,MUCH GREAT- ER,MUCH LESS Lester Circle The CIRCUMCENTER C, NINE-POINT CENTER N, and the first and second FERMAT POINTS F1and F2of a triangle lie on a circle known as the Lester circle. See also CIRCUMCENTER ,FERMAT POINTS ,NINE-POINT CENTER References Kimberling, C. "Lester Circle." Math. Teacher 89, 26, 1996. Lester, J. "Triangles III: Complex Triangle Functions." Aequationes Math. 53,4/C1/5, 1997. Trott, M. "Applying GroebnerBasis to Three Problems in Geometry." Mathematica Educ. Res. 6,15/C1/8, 1997. Trott, M. "A Proof of Lester’s Circle Theorem." http:// library.wolfram.com/demos/v3/GeometryProof.nb. L-Estimate A ROBUST ESTIMATION based on LINEAR COMBINA- TIONS of ORDER STATISTICS . Examples include the MEDIAN and TUKEY’S TRIMEAN . See also M-ESTIMATE , R-ESTIMATE References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Robust Estimation." §15.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 694 /C1/00, 1992. Letter-Value Display A method of displaying simple statistical parameters including HINGES , MEDIAN , and upper and lower values. References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 33, 1977. Leudesdorf Theorem Let t(m) denote the set of the f(m) numbers less than and RELATIVELY PRIME to m, where f(n) is the TOTIENT FUNCTION . Then if Sm /C13X t(m)1 t; then Sm /C130(mod m2)i f 2 ¶m; 3¶m Sm /C130 mod1 3 m2/C(%/C(r if 2¶m; 3¶m Sm /C130 mod12 m2/C(%/C(r 2¶m; 3¶m; m not a power of 2 Sm /C130 mod16 m2/C(%/C(r if 2¶m; 3¶m Sm /C130 mod14 m2/C(%/C(r if m /C302a :8 >>>>>>>>< >>>>>>>>: See also B AUER’S IDENTICAL CONGRUENCE ,TOTIENT FUNCTION References Hardy, G. H. and Wright, E. M. "A Theorem of Leudesdorf." §8.7 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 100 /C1/02, 1979. Level Curve A LEVEL SET in 2-D.See also CONTOUR PLOT,E QUIPOTENTIAL CURVE , LEVEL SURFACE Level Set The level set of c is the SET of points f(x1 ; ...; xn) /C23 U : f(x1 ; ... ; xn) /C30c g/C23Rn ; and is in the DOMAIN of the function. If n /C302, the level set is a plane curve (a LEVEL CURVE ). If n /C303, the level set is a surface (a level surface). See also CONTOUR PLOT,E QUIPOTENTIAL CURVE , LEVEL CURVE ,LEVEL SURFACE References Gray, A. "Level Surfaces in R3 :/" §12.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 291 /C1/93, 1997. Level Surface A LEVEL SET in 3-D. Levenberg-Marquardt Method Levenberg-Marquardt is a popular alternative to the Gauss-Newton method of finding the minimum of a function F(x) that is a sum of squares of nonlinear functions, F(x) /C301 2Xm i/C301[fi(x)]2 : Let the JACOBIAN of fi(x) be denoted Ji(x); then the Levenberg-Marquardt method searches in the direc- tion given by the solution p to the equations (JT k J) /C27 l kI) pk/C30/C28JTkfk; where lkare nonnegative scalars and I is the IDENTITY MATRIX . The method has the nice property that, for some scalar D related to lk ; the vector pkis the solution of the constrained subproblem of mini- mizing ½½Jkp /C27fk ½½2 2 =2 subject to ½½p ½½2 5D (Gill et al. 1981, p. 136). The method is used by the Mathematica 4.0 com- mand FindMinimum [f,{x, x0}] when given the Method- /C21LevenbergMarquardt option. See also MINIMUM ,OPTIMIZATION References Gill, P. R.; Murray, W.; and Wright, M. H. "The Levenberg- Marquardt Method." §4.7.3 in Practical Optimization. London: Academic Press, pp. 136 /C1/37, 1981. Levenberg, K. "A Method for the Solution of Certain Problems in Least Squares." Quart. Appl. Math. 2, 164/C1/ 68, 1944. Marquardt, D. "An Algorithm for Least-Squares Estimation of Nonlinear Parameters." SIAM J. Appl. Math. 11, 431/C1/ 41, 1963. Leviathan Number The number (10666)! ; where 666 is the BEAST NUMBER and n! denotes a FACTORIAL . The number of trailing zeros in the Leviathan number is 25 /C2910664 /C28143 (Pickover 1995). See also 666,A POCALYPSE NUMBER ,A POCALYPTIC NUMBER ,BEAST NUMBER References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97 /C1/ 02, 1995. Levi-Civita Connection On a RIEMANNIAN MANIFOLD M, there is a canonical CONNECTION called the Levi-Civita connection (pro- nounced le-ve shi-vit-), sometimes also known as the Riemannian connection or COVARIANT DERIVATIVE .As a CONNECTION on the TANGENT BUNDLE , it provides a well-defined method for differentiating VECTOR FIELDS , forms, or any other kind of TENSOR . The theorem asserting the existence of the Levi-Civita connection, which is the unique TORSION -free CON- NECTION 9 on the TANGENT BUNDLE TM compatible with the metric, is called the FUNDAMENTAL THEOREM OF RIEMANNIAN GEOMETRY . These properties can be described as follows. Let X, Y, and Z be any VECTOR FIELDS , and /C142;/C143 denote the METRIC . Recall that vector fields act as DERIVATIONS on the ring of smooth functions by the DIRECTIONAL DERIVATIVE , and that this action extends to an action on vector fields. The notation [X, Y] is the COMMU- TATOR of vector fields, XY /C28YX : The Levi-Civita connection is torsion-free, meaning 9X 9Y Z /C289Y 9XZ /C309[X ; Y]Z ; (1) and is compatible with the metric X(Y ; Z) /C30/C1429XY ; Z/C143/C27/C142Y ;9XZ/C143: (2) In coordinates, the Levi-Civita connection can be described using the CHRISTOFFEL SYMBOLS OF THE SECOND KIND Gk i; j : In particular, if ei /C30@=@xi ; then Gk i; j /C30/C1429eiej ; ek /C143; (3) or in other words, 9eiej /C30X kGk i; jek : (4) As a CONNECTION on the TANGENT BUNDLE TM ; it induces a connection on the DUAL BUNDLE T /C31M and on all their TENSOR PRODUCTS TMk /C156TM /C31l : Also, given a SUBMANIFOLD N it restricts to TN to give the Levi-Civita connection from the restriction of the metric to N. The Levi-Civita connection can be used to describe many intrinsic geometric objects. For instance, a path c : R 0 M is a geodesic IFF 9˙c(t) ˙c(t) /C300 where ˙c is thepath’s TANGENT VECTOR . On a more general path c, the equation 9˙c(t)v(t)/C300 defines PARALLEL TRANSPORT for a VECTOR FIELD valong c. The SECOND FUNDA- MENTAL FORM IIof a submanifold Nis given by pQ(9TNwhere TNis the TANGENT BUNDLE ofNand pQis projection onto the NORMAL BUNDLE Q. The CURVATURE ofMis given by 9(9:/ See also CHRISTOFFEL SYMBOL ,CONNECTION ,COVAR- IANT DERIVATIVE ,CURVATURE ,FUNDAMENTAL THEO- REM OF RIEMANNIAN GEOMETRY ,G EODESIC , PRINCIPAL BUNDLE ,RIEMANNIAN MANIFOLD ,RIEMAN- NIAN METRIC References Carmo, M. Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall, pp. 441 /C1/42, 1976. Gallot, S.; Hulin, D.; and Lafontaine, J. §II.B in Riemannian Geometry. New York: Springer-Verlag, 1980. Lee, J. M. Riemannian Manifolds: An Introduction to Curvature. New York: Springer-Verlag, pp. 65 /C1/1, 1997. Sternberg, S. Differential Geometry. New York: Chelsea, 1983. Levi-Civita Density PERMUTATION SYMBOL Levi-Civita Symbol PERMUTATION SYMBOL Levi-Civita Tensor PERMUTATION TENSOR Levi Graph The unique 8- CAGE GRAPH (right figure) consisting of the union of the two leftmost subgraphs illustrated above. It has 45 nodes, 15 edges, and all nodes havedegree 3. The Levi graph is a GENERALIZED POLYGON which is the point/line INCIDENCE GRAPH of the generalized quadrangle W2:The graph is a 4-arc transitive cubic graph, was first discovered by Tutte (1947), and is also called the Tutte-Coxeter graph (Bondy and Murty 1976, p. 237). An alternative embedding is illustrated above. See also CAGE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 276, 1976. Coxeter, H. S. M. "The Chords of the Non-Ruled Quadratic in PG(3,3)." Canad. J. Math. 10, 484 /C1/88, 1958. Coxeter, H. S. M. "Twelve Points in PG(5,3) with 95040 Self- Transformations." Proc. Roy. Soc. London Ser. A 247, 279 /C1/93, 1958. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 174 /C1/75, 1994. Royle, G. "Cubic Cages." http://www.cs.uwa.edu.au/~gordon/ cages/. Tutte, W. T. "A Family of Cubical Graphs." Proc. Cambridge Philos. Soc., 459 /C1/74, 1947. Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966. Tutte, W. T. "The Chords of the Non-Ruled Quadratic in PG(3,3)." Canad. J. Math. 10, 481 /C1/83, 1958. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/2, 1982. Levine-O’Sullivan Greedy Algorithm For a sequence fxi g; the Levine-O’Sullivan greedy algorithm is given by x1 /C301 xi /C30 max 1 5j5i /C281(j /C271)(i /C28 xj) for i /C211. See also GREEDY ALGORITHM ,L EVINE- O’SULLIVAN SEQUENCE References Levine, E. and O’Sullivan, J. "An Upper Estimate for the Reciprocal Sum of a Sum-Free Sequence." Acta Arith. 34, 9 /C1/4, 1977. Levine-O’Sullivan Sequence The sequence generated by the LEVINE- O’SULLIVAN GREEDY ALGORITHM : 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28,32, 36, 40, 45, 50, 55, 60, 65, ... (Sloane’s A014011). The reciprocal sum of this sequence is conjectured to bound the reciprocal sum of all A-SEQUENCE . References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/erdos/erdos.html. Levine, E. and O’Sullivan, J. "An Upper Estimate for the Reciprocal Sum of a Sum-Free Sequence." Acta Arith. 34, 9 /C1/4, 1977. Sloane, N. J. A. Sequences A014011 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Le´vy Constant Let pn =qn be the nth CONVERGENT of a REAL NUMBER x. Then almost all REAL NUMBERS satisfy L /C13 lim n0/C12(qn)1 =n /C30e p2 =(12 ln 2) /C303:27582291872... See also CONTINUED FRACTION ,KHINTCHINE’S CON- STANT ,KHINTCHINE- LE´ VY CONSTANT References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 51, 1983. Le´vy Distribution F[PN(k)] /C30F[exp(/C28N ½k½b)]; where F is the FOURIER TRANSFORM of the probabil- ity PN(k) for N-step addition of random variables. Le´vy showed that b /C23 (0 ; 2) for P(x)tobe NONNEGA- TIVE. The Le´vy distribution has infinite variance and sometimes infinite mean. The case b /C301 gives a CAUCHY DISTRIBUTION , while b /C302 gives a GAUSSIAN DISTRIBUTION . See also CAUCHY DISTRIBUTION ,GAUSSIAN DISTRIBU- TION ,LE´ VY FLIGHT Le´vy Dragon LE´ VY FRACTAL Le´vy Flight RANDOM WALK trajectories which are composed of self-similar jumps. They are described by the LE´ VY DISTRIBUTION . See also LE´ VY DISTRIBUTION References Shlesinger, M.; Zaslavsky, G. M.; and Frisch, U. (Eds.). Le´vy Flights and Related Topics in Physics. New York: Springer-Verlag, 1995. Le´vy Fractal A FRACTAL curve, also called the C-CURVE (Gosper 1972). The base curve and motif are illustrated below. Duvall and Keesling (1999) proved that the HAUS- DORFF DIMENSION of the boundary of the Le´vy fractal is rigorously greater than one, obtaining an estimate of 1.934007183. See also LE´ VY TAPESTRY References Dixon, R. Mathographics. New York: Dover, pp. 182 /C1/83, 1991. Duvall, P. and Keesling, J. The Hausdorff Dimension of the Boundary of the Le´vy Dragon. 22 Jul 1999. http:// xxx.lanl.gov/abs/math.DS/9907145/. Gosper, R. W. Item 135 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 65 /C1/6, Feb. 1972. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 45 /C1/8, 1991. Le´vy, P. "Les courbes planes ou gauches et les surfaces compose ´es de parties semblales au tout." J. l’E´ cole Poly- tech. , 227 /C1/47 and 249 /C1/91, 1938. Le´vy, P. "Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole." In Classics on Fractals (Ed. G. A. Edgar). Reading, MA: Addison-Wesley, pp. 181 /C1/39, 1993. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Le´vy Function BROWN FUNCTION Le´vy Process References Sato, K.-I. Le´vy Processes and Infinitely Divisible Distribu- tions. Cambridge, England: Cambridge University Press, 1999.Le´vy Tapestry The FRACTAL curve illustrated above, with base curve and motif illustrated below. See also LE´ VY FRACTAL References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 45 /C1/8, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Lewis Regulator The ORDINARY DIFFERENTIAL EQUATION y??/C27(1/C28½y½)y?/C27y/C300: References Hagerdorn, P. Non-Linear Oscillations. Oxford, England: Clarendon Press, p. 152, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 124, 1997. Lew k-Gram Diagrams invented by Lewis Carroll which can be used to determine the number of minimal MINIMAL COVERS ofnnumbers with kmembers. References Macula, A. J. "Lewis Carroll and the Enumeration of Minimal Covers." Math. Mag. 68, 269/C1/74, 1995. Lexicographic Order An ordering for the Cartesian product /C29of any two sets Aand Bwith order relations BAandBB; respectively, such that if ( a1;b1) and ( a2;b2) both belong to A/C29B;then ( a1;b1)B(a2;b2)IFFeither 1. a1 BAa2 ; or 2. a1 /C30a2 and b1 BBb2 :/ The lexicographic order can be readily extended to cartesian products of arbitrary length by recursively applying this definition, i.e., by observing that A /C29B /C29C /C30A /C29(B /C29C) :/ When applied to PERMUTATIONS , lexicographic order is increasing numerical order (or equivalently, alpha- betic order for lists of symbols; Skiena 1990, p. 4). For example, the PERMUTATIONS of f1; 2; 3g in lexico- graphic order are 123, 132, 213, 231, 312, and 321. When applied to subsets, two subsets are ordered by their smallest elements (Skiena 1990, p. 44). For example, the subsets of f1 ; 2 ; 3g in lexicographic order are fg;f1g;f1; 2g;f1 ; 2; 3g;f1; 3g;f2 g;f2; 3g; f3g:/ Lexicographic order is sometimes called dictionary order. See also ORDER (ORDERING ), MONOMIAL ORDER , TRANSPOSITION ORDER References Ruskey, F. "Information on Combinations of a Set." http:// www.theory.csc.uvic.ca/~cos/inf/comb/CombinationsIn- fo.html. Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 23, 2000. Skiena, S. "Lexicographically Ordered Permutations" and "Lexicographically Ordered Subsets." §1.1.1 and 1.5.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 3 /C1/ and 43 /C1/4, 1990. Lexis Ratio L /C13s sB; where s is the VARIANCE in a set of s LEXIS TRIALS and sBis the VARIANCE assuming BERNOULLI TRIALS .If L B1, the trials are said to be SUBNORMAL , and if L /C211, the trials are said to be SUPERNORMAL . See also BERNOULLI TRIAL,LEXIS TRIALS ,SUBNOR- MAL,SUPERNORMAL Lexis Trials n sets of s trials each, with the probability of success p constant in each set. varx n ! /C30spq /C27s(s /C281)s2 p ; where s2 pis the VARIANCE of p i:/ See also BERNOULLI TRIAL,LEXIS RATIOL-Function ARTIN L-FUNCTION ,DIRICHLET L-SERIES ,EULER L- FUNCTION ,HECKE L-FUNCTION Lg The LOGARITHM to BASE 2 is denoted lg ; i.e., lg x /C13log2 x: Care is needed in interpreting this symbol, however, since Russian literature uses lg x to denote the base- 10 logarithm denoted in this work by log x:/ See also BASE (LOGARITHM ), E,L N,L OGARITHM , NAPIERIAN LOGARITHM ,NATURAL LOGARITHM L’Hospital’s Cubic TSCHIRNHAUSEN CUBIC L’Hospital’s Rule Let lim stand for the LIMIT limx0c;limx0c/C28;limx0c/C27; limx0/C12;or limx0/C28/C12;and suppose that lim f(x) and lim g(x) are both ZERO or are both 9/C12 :If limf?(x) g?(x) has a finite value or if the LIMIT is9/C12 ;then limf(x) g(x)/C30limf?(x) g?(x): L’Hospital’s rule occasionally fails to yield useful results, as in the case of the function limu0/C12u(u2/C27 1)/C281=2:Repeatedly applying the rule in this case gives expressions which oscillate and never converge, lim u0/C12u (u2/C271)1=2/C30lim u0/C121 u(u2/C271)/C281=2 /C30lim u0/C12(u2/C271)1=2 u/C30lim u0/C12u(u2/C271)/C281=2 1 /C30lim u0/C12u (u2/C271)1=2: (The actual LIMIT is 1.) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 13, 1972. L’Hospital, G. de L’analyse des infiniment petits pour l’intelligence des lignes courbes. 1696. L’Huilier’s Theorem Let a SPHERICAL TRIANGLE have sides of length a,b, and c, and SEMIPERIMETER s. Then the SPHERICAL EXCESS Eis given by tan1 4 E/C(%/C(r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 2 s/C(%/C(r tan12(s /C28a)hi tan12(s /C28b)hi tan12(s /C28c)hir : See also GIRARD’S SPHERICAL EXCESS FORMULA , SPHERICAL EXCESS ,SPHERICAL TRIANGLE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 148, 1987. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 469, 1995. Liar’s Paradox The paradox of a man who states "I am lying." If he is lying, then he is telling the truth, and vice versa. Another version of this paradox is the EPIMENIDES PARADOX . Such paradoxes are often analyzed by creating so-called "metalanguages" to separate state- ments into different levels on which truth and falsity can be assessed independently. For example, Ber- trand Russell noted that, "The man who says, ‘I am telling a lie of order n’ is telling a lie, but a lie of order n /C271/" (Gardner 1984, p. 222). See also EPIMENIDES PARADOX ,EUBULIDES PARADOX References Beth, E. W. The Foundations of Mathematics. Amsterdam, Netherlands: North-Holland, p. 485, 1959. Bochenski, I. M. §23 and 25 in Formale Logik. Munich, Germany, 1956. Church, A. "Paradoxes, Logical." In The Dictionary of Philosophy, rev. enl. ed. (Ed. D. D. Runes). New York: Rowman and Littlefield, p. 224, 1984. Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 5 /C1/, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 108 /C1/11, 1998. Fraenkel, A. A. and Bar-Hillel, Y. Foundations of Set Theory. Amsterdam, Netherlands, p. 11, 1958. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 222, 1984. Kleene, S. C. Introduction to Metamathematics. Princeton, NJ: Van Nostrand, p. 39, 1964. Prior, A. N. "Epimenides the Cretan." J. Symb. Logic 23, 261 /C1/66, 1958. Tarski, A. "The Semantic Conception of Truth and the Foundations of Semantics." Philos. Phenomenol. Res. 4, 341 /C1/76, 1944. Tarski, A. "Der Wahrheitsbegriff in den formalisierten Sprachen." Studia Philos. 1, 261 /C1/05, 1936. Weyl, H. Philosophy of Mathematics and Natural Science. Princeton, NJ, p. 228, 1949. Lichnerowicz Conditions Second and higher derivatives of the METRIC TENSOR gabneed not be continuous across a surface of discontinuity, but gaband gab ; cmust be continuous across it.Lichnerowicz Formula D /C31Dc /C309/C319c /C271 4 Rc /C2812 F /C27 L ( c) ; where D is the Dirac operator D : G(W /C27) 0G(W /C28) ;9 is the COVARIANT DERIVATIVE on SPINORS , R is the CURVATURE SCALAR , and F /C27 Lis the self-dual part of the curvature of L. See also LICHNEROWICZ- WEITZENBOCK FORMULA References Donaldson, S. K. "The Seiberg-Witten Equations and 4- Manifold Topology." Bull. Amer. Math. Soc. 33,45/C1/0, 1996. Lichnerowicz-Weitzenbock Formula D /C31Dc /C309/C319 c /C271 4 Rc; where D is the Dirac operator D : G(S/C27) 0G(S /C28) ;9 is the COVARIANT DERIVATIVE onSPINORS , and Ris the CURVATURE SCALAR . See also LICHNEROWICZ FORMULA References Donaldson, S. K. "The Seiberg-Witten Equations and 4- Manifold Topology." Bull. Amer. Math. Soc. 33,4 5/C1/0, 1996. Lichtenfels Minimal Surface AMINIMAL SURFACE that contains LEMNISCATES as geodesics which is given by the parametric equations x/C30Rffiffiffi 2p cos1 2z/C(%/C(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 3z/C(%/C(rr /C)P/C)( (1) y/C30R/C28ffiffiffi 2p sin1 3z/C(%/C(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 3z/C(%/C(rr /C)P/C)( (2) z/C30R/C2813ffiffiffi 2p igz 0dzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 3z/C(%/C(rr2 6643 775(3) /C30R/C28iffiffiffi 2p Fffiffi 1 3q z;2/C(%/C(rhi ; (4) where F(x; x) is an incomplete ELLIPTIC INTEGRAL OF THE FIRST KIND and z/C30u /C27iv is a COMPLEX NUMBER . A given LEMNISCATE is the intersection of the surface with the xy-plane. The surface is periodic in the direction of the axis with period v /C302g1 0dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 t2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C281 2 t2q /C302K12/C(%/C(r ; (5) where K(x) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND . See also LEMNISCATE ,MINIMAL SURFACE References do Carmo, M. P. "Minimal Surfaces with a Lemniscate as a Geodesic." §3.5F in Mathematical Models from the Collec- tions of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 47, 1986. Lichtenfels, O. von. "Notiz u¨ber eine transcendente Mini- malfla ¨che." Sitzungsber. Kaiserl. Akad. Wiss. Wien 94, 41 /C1/4, 1889. Lie Algebra A NONASSOCIATIVE ALGEBRA obeyed by objects such as the LIE BRACKET and POISSON BRACKET . Elements f, g, and h of a Lie algebra satisfy [f ; f] /C300 (1) [f /C27g ; h] /C30[f ; h] /C27[g; h] ; (2) and [f ;[g ; h]] /C27[g; [h ; f]] /C27[h; [f ; g]] /C300 (3) (the JACOBI IDENTITY ). The relation [f ; f] /C300 implies [f ; g] /C30/C28[g; f]: (4) For characteristic not equal to two, these two rela- tions are equivalent. The binary operation of a Lie algebra is the bracket [fg; h] /C30f[g; h] /C27g[f ; h] : (5) An ASSOCIATIVE ALGEBRA A with associative product xy can be made into a Lie algebra A/C28 by the Lie product [x; y] /C30xy /C28yx: (6) Every Lie algebra L is isomorphic to a SUBALGEBRA of some A/C28 where the associative algebra A may be taken to be the linear operators over a VECTOR SPACE V (the POINCARE ´ -BIRKHOFF- WITT THEOREM ; Jacobson 1979, pp. 159 /C1/60). If L is finite dimensional, then V can be taken to be finite dimensional (ADO’S THEOREM for characteristic p /C300; IWASAWA’S THEOREM for characteristic p "0):/ The classification of finite dimensional simple Lie algebras over an algebraically closed field of char- acteristic 0 can be accomplished by (1) determining matrices called CARTAN MATRICES corresponding toindecomposable simple systems of roots and (2) determining the simple algebras associated with these matrices (Jacobson 1979, p. 128). This is one of the major results in Lie algebra theory, and is frequently accomplished with the aid of diagrams called DYNKIN DIAGRAMS . See also ADO’S THEOREM ,D ERIVATION ALGEBRA , DYNKIN DIAGRAM ,JACOBI IDENTITIES ,LIE ALGEB- ROID ,LIE BRACKET ,IWASAWA’S THEOREM ,POINCARE ´ - BIRKHOFF- WITT THEOREM ,P OISSON BRACKET ,R E- DUCED ROOT SYSTEM ,ROOT SYSTEM ,W EYL GROUP References Humphrey, J. E. Introduction to Lie Algebras and Repre- sentation Theory. New York: Springer-Verlag, 1972. Jacobson, N. Lie Algebras. New York: Dover, 1979. Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 3, 1996. Weisstein, E. W. "Books about Lie Algebra." http:// www.treasure-troves.com/books/LieAlgebra.html. Lie Algebroid The infinitesimal algebraic object associated with a LIE GROUPOID . A Lie algebroid over a MANIFOLD B is a VECTOR BUNDLE A over B with a LIE ALGEBRA structure [ ;](L IE BRACKET ) on its SPACE of smooth sections together with its ANCHOR r:/ See also LIE ALGEBRA References Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744 /C1/52, 1996. Liebmann’s Theorem A SPHERE is rigid. See also SPHERE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 483 and 653 /C1/54, 1997. O’Neill, B. Elementary Differential Geometry, 2nd ed. New York: Academic Press, p. 262, 1997. Lie Bracket The commutation operation [a ; b] /C30ab /C28ba corresponding to the L IE PRODUCT . See also LAGRANGE BRACKET ,POISSON BRACKET Lie Commutator LIEPRODUCT Lie Derivative The Lie derivative of TENSOR Tabwith respect to the VECTOR FIELD Xis defined by LXTab /C13 lim dx 00T ?ab(x?) /C28 Tab(x) dx: (1) Explicitly, it is given by LXTab /C30TabXd ;b /C27TbdXd ;a /C27Tab ; eXe ; (2) where X;a is a COMMA DERIVATIVE . The Lie derivative of a METRIC TENSOR gabwith respect to the VECTOR FIELD X is given by LXgab /C30Xa; b /C27Xb; a /C302X(a; b) ; (3) where X(a; b) denotes the SYMMETRIC TENSOR part and Xa; b is a COVARIANT DERIVATIVE . See also COVARIANT DERIVATIVE ,K ILLING’S EQUA- TION ,KILLING VECTORS ,LIE DERIVATIVE (SPINOR ) Lie Derivative (Spinor) The Lie derivative of a SPINOR c is defined by LX c(x) /C30lim t 00˜ct(x) /C28 c(x) t; where ˜ct is the image of c by a one-parameter group of isometries with X its generator. For a VECTOR FIELD Xa and a COVARIANT DERIVATIVE 9a ; the Lie derivative of c is given explicitly by LX c /C30Xa 9a c /C281 8( 9aXb /C289bXa) ga gb c; where ga and gb are DIRAC MATRICES (Choquet-Bruhat and DeWitt-Morette 2000). See also COVARIANT DERIVATIVE ,D IRAC MATRICES , LIE DERIVATIVE ,SPINOR References Choquet-Bruhat, Y. and DeWitt-Morette, C. Analysis, Mani- folds and Physics, Part II: 92 Applications, rev. ed. Amsterdam, Netherlands: North-Holland, 2000. Lie Group A Lie group is a DIFFERENTIABLE MANIFOLD obeying the group properties and that satisfies the additional condition that the group operations are continuous. The simplest examples of Lie groups are one-dimen- sional. Under addition, the REAL LINE is a Lie group. After picking a specific point to be the IDENTITY ELEMENT , the CIRCLE is also a Lie group. Another point on the circle at angle u from the identity then acts by rotating the circle by the angle u: In general, a Lie group may have a more complicated group structure, such as the ORTHOGONAL GROUP O(n) (i.e., the n /C29n orthogonal matrices), or the GENERAL LINEAR GROUP GL(n) (i.e., the n /C29n invertible ma- trices). The LORENTZ GROUP is also a Lie group. The TANGENT SPACE at the identity of a Lie group always has the structure of a LIE ALGEBRA , and this LIE ALGEBRA determines the local structure of the Liegroup via the EXPONENTIAL MAP. For example, the function eitgives the EXPONENTIAL MAP from the circle’s tangent space (i.e., the reals), to the circle, thought of as a the UNIT CIRCLE in C: A more difficult example is the exponential map eA from SKEW SYM- METRIC n /C29n matrices to the SPECIAL ORTHOGONAL GROUP SO(n) ; the subset of O(n) with determinant 1:/ The topology of a Lie group is fairly restricted. For example, there always exists a nonvanishing VECTOR FIELD . This structure has allowed complete classifica- tion of the finite dimensional SEMISIMPLE LIE GROUPS and their representations. See also COMPACT GROUP ,C ONTINUOUS GROUP , GROUP ,DIFFERENTIABLE MANIFOLD ,L IE ALGEBRA , LIE GROUPOID ,LIE-TYPE GROUP ,LORENTZ GROUP , NIL GEOMETRY ,O RTHOGONAL GROUP ,SEMISIMPLE LIE GROUP ,SOL GEOMETRY ,TANGENT SPACE ,VECTOR FIELD References Arfken, G. "Infinite Groups, Lie Groups." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 251 /C1/52, 1985. Chevalley, C. Theory of Lie Groups. Princeton, NJ: Prince- ton University Press, 1946. Hsiang, W. Y. Lectures on Lie Groups. Singapore: World Scientific, 2000. Knapp, A. W. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Lipkin, H. J. Lie Groups for Pedestrians, 2nd ed. Amster- dam, Netherlands: North-Holland, 1966. Lie Groupoid A GROUPOID G over B for which G and B are differentiable manifolds and a; b; and multiplication are differentiable maps. Furthermore, the derivatives of a and b are required to have maximal RANK everywhere. Here, aandbare maps from GontoR2 with a:(x;g;y)/C2zandb:(x;g;y)/C2y/ See also LIE ALGEBROID ,N ILPOTENT LIE GROUP , SEMISIMPLE LIE GROUP ,SOLVABLE LIE GROUP References Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744/C1/52, 1996. Lie´nard’s Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION y??/C27f(x)y?/C27y/C300: References Villari, G. "Periodic Solutions of Lie ´nard’s Equation." J. Math. Anal. Appl. 86, 379/C1/86, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 124, 1997. Lie Product The multiplication operation corresponding to the LIE BRACKET . Lie-Type Group A finite analog of LIE GROUPS . The Lie-type groups include the CHEVALLEY GROUPS [/PSL(n; q); PSU (n; q) ; PSp(2n; q) ; P Ve(n; q)] ; TWISTED CHEVAL- LEY GROUPS , and the TITS GROUP . See also CHEVALLEY GROUPS ,F INITE GROUP ,L IE GROUP ,LINEAR GROUP ,ORTHOGONAL GROUP ,SIMPLE GROUP ,SYMPLECTIC GROUP ,TITS GROUP ,TWISTED CHEVALLEY GROUPS ,UNITARY GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#lie. Life The most well-known CELLULAR AUTOMATON , in- vented by John Conway and popularized in Martin Gardner’s Scientific American column starting in October 1970. The game was originally played (i.e., successive generations were produced) by hand with counters, but implementation on a computer greatly increased the ease of exploring patterns. The Life CELLULAR AUTOMATON is run by placing a number of filled cells on a 2-D grid. Each generation then switches cells on or off depending on the state of the cells that surround it. The rules are defined as follows. All eight of the cells surrounding the current one are checked to see if they are on or not. Any cells that are on are counted, and this count is then used to determine what will happen to the current cell. 1. Death: if the count is less than 2 or greater than 3, the current cell is switched off. 2. Survival: if (a) the count is exactly 2, or (b) the count is exactly 3 and the current cell is on, the current cell is left unchanged. 3. Birth: if the current cell is off and the count is exactly 3, the current cell is switched on. Hensel gives a JAVA APPLET implementing the Game of Life on his web page. Weisstein gives an extensive alphabetical tabulation of life forms and terms. A pattern which does not change from one generation to the next is known as a still life , and is said to have period 1. Conway originally believed that no pattern could produce an infinite number of cells, and offered a $50 prize to anyone who could find a counter- example before the end of 1970 (Gardner 1983, p. 216). Many counterexamples were subsequently found, including guns and puffer trains. A Life pattern which has no father pattern is known as a Garden of Eden (for obvious biblical reasons). The first such pattern was not found until 1971, andat least 3 are now known. It is not, however, known if a pattern exists which has a father pattern , but no grandfather pattern (Gardner 1983, p. 249). Rather surprisingly, Gosper and J. H. Conway inde- pendently showed that Life can be used to generate a UNIVERSAL TURING MACHINE (Berlekamp et al. 1982, Gardner 1983, pp. 250 /C1/53). Similar CELLULAR AUTOMATON games with different rules are H EXLIFEand H IGHLIFE.HASHLIFEis a life ALGORITHM that achieves remarkable speed by stor- ing subpatterns in a hash table, and using them to skip forward, sometimes thousands of generations ata time. See also C ELLULAR AUTOMATON ,HASHLIFE,HEXLIFE, HIGHLIFE References Berlekamp, E. R.; Conway, J. H.; and Guy, R. K. "What Is Life." Ch. 25 in Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982. Flammenkamp, A. "Game of Life." http://www.uni-biele- feld.de/~achim/gol.html. "The Game of Life." Math Horizons. p. 9, Spring 1994. Gardner, M. "The Game of Life, Parts I-III." Chs. 20 /C1/2i n Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, 1983. Hensel, A. "PC Life Distribution." http://www.mindspring.- com/~alanh/lifep.zip. Hensel, A. "Conway’s Game of Life." Includes a Java applet for the Game of Life. http://www.mindspring.com/~alanh/ life/. Koenig, H. "Game of Life Information." http://www.halcyon.- com/hkoenig/LifeInfo/LifeInfo.html. Poundstone, W. The Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge. New York: Morrow, 1985. Resnick, M. and Silverman, B. "A Zoo of Life Forms." http:// lcs.www.media.mit.edu/groups/el/projects/emergence/life- zoo.html. Toffoli, T. and Margolus, N. Cellular Automata Machines: A New Environment for Modeling. Cambridge, MA: MIT Press, 1987. Wainwright, R. T. "LifeLine." http://members.aol.com/life1- ine/life/lifepage.htm. Wainwright, R. T. LifeLine: A Quarterly Newsletter for Enthusiasts of John Conway’s Game of Life. Nos. 1 /C1/1, 1971/C1/973. Weisstein, E. W. "Eric’s Treasure Trove of Life." http:// www.treasure-troves.com/life/. Life Expectancy Anlxtable is a tabulation of numbers which is used to calculate life expectancies. x /nx//dx//lx//qx//Lx//Tx//ex/ 0 1000 200 1.00 0.20 0.90 2.70 2.70 1 800 100 0.80 0.12 0.75 1.80 2.25 2 700 200 0.70 0.29 0.60 1.05 1.50 3 500 300 0.50 0.60 0.35 0.45 0.90 4 200 200 0.20 1.00 0.10 0.10 0.505 0 0 0.00 – 0.00 0.00 – /S/ 1000 2.70 x: Age category ( x/C300, 1, ..., k). These values can be in any convenient units, but must be chosen so that no observed lifespan extends past category k/C281:/ /nx: Census size, defined as the number of indivi- duals in the study population who survive to thebeginning of age category x. Therefore, n 0/C30N(the total population size) and nk/C300:/ /dx:nx/C28nx/C271;ak i/C300di/C30n0:Crude death rate, which measures the number of individuals who die within age category x. /lx:/C30nx=n0:Survivorship, which measures the proportion of individuals who survive to the beginning of age category x. /qx:/C30dx=nx;qk/C281/C301:Proportional death rate, or "risk," which measures the proportion of indivi-duals surviving to the beginning of age category x who die within that category. /Lx:/C30(lx/C27lx/C271)=2:Midpoint survivorship, which measures the proportion of individuals survivingto the midpoint of age category x. Note that the simple averaging formula must be replaced by a more complicated expression if survivorship is nonlinear within age categories. The sum a k i/C300Lx gives the total number of age categories lived by the entire study population. /Tx:Tx/C281/C28Lx/C281;T0/C30ak i/C300Lx:Measures the total number of age categories left to be lived by all individuals who survive to the beginning of age category x. /ex:/C30Tx=lx;ek/C281/C301=2:Life expectancy, which is the mean number of age categories remaininguntil death for individuals surviving to the begin-ning of age category x. For all x,e x/C271/C271>ex:This means that the total expected lifespan increases monotonically. For in-stance, in the table above, the one-year-olds have anaverage age at death of 2.25 /C271/C303.25, compared to 2.70 for newborns. In effect, the age of death of olderindividuals is a distribution conditioned on the factthat they have survived to their present age. It is common to study survivorship as a semilog plot ofl xvs.x, known as a SURVIVORSHIP CURVE . A so- called lxmxtable can be used to calculate the mean generation time of a population. Two lxmxtables are illustrated below.Population 1 x /lx//mx// lxmx//xlxmx/ 0 1.00 0.00 0.00 0.00 1 0.70 0.50 0.35 0.35 2 0.50 1.50 0.75 1.50 3 0.20 0.00 0.00 0.004 0.00 0.00 0.00 0.00 /R0/C301:10//S/C301:85/ T/C30PxlxmxPlxmx/C301:85 1:10/C301:68 r/C30lnR0 T/C30ln 1 :10 1:68/C300:057: Population 2 x /lx//mx// lxmx//xlxmx/ 0 1.00 0.00 0.00 0.001 0.70 0.00 0.00 0.002 0.50 2.00 1.00 2.00 3 0.20 0.50 0.10 0.30 4 0.00 0.00 0.00 0.00 /R0/C301:10//S/C302:30/ T/C30PxlxmxPlxmx/C302:30 1:10/C302:09 r/C30lnR0 T/C30ln 1 :10 2:09/C300:046: x: Age category ( x/C300, 1, ..., k). These values can be in any convenient units, but must be chosen so that no observed lifespan extends past category k/C281 (as in an lxtable). /lx:/C30nx=n0:Survivorship, which measures the proportion of individuals who survive to the beginning of age category x(as in an lxtable). /mx: The average number of offspring produced by an individual in age category x while in that age category .ak i/C300mxtherefore represents the average lifetime number of offspring produced by an individual of maximum lifespan. /lxmx: The average number of offspring produced by an individual within age category xweighted by the probability of surviving to the beginning of that age category. ak i/C300lxmxtherefore represents the average lifetime number of offspring produced by a member of the study population. It is called the net reproductive rate per generation and is often denoted R0 :/ /xlxmx : A column weighting the offspring counted in the previous column by their parents’ age when they were born. Therefore, the ratio T /C30 a(xlxmx) =a(lxmx) is the mean generation time of the population. The MALTHUSIAN PARAMETER r measures the repro- ductive rate per unit time and can be calculated as r /C30(ln R0)=T : For an exponentially increasing popu- lation, the population size N(t) at time t is then given by N(t) /C30N0ert : In the above two tables, the populations have iden- tical reproductive rates of R0 /C301:10 : However, the shift toward later reproduction in population 2 increases the generation time, thus slowing the rate of POPULATION GROWTH . Often, a slight delay of reproduction decreases POPULATION GROWTH more strongly than does even a fairly large reduction in reproductive rate. See also GOMPERTZ CURVE ,LOGISTIC GROWTH CURVE , MAKEHAM CURVE ,MALTHUSIAN PARAMETER ,POPULA- TION GROWTH ,SURVIVORSHIP CURVE References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 294 /C1/95, 1999. Lift Given a MAP f from a SPACE X to a SPACE Y and another MAP g from a SPACE Z to a SPACE Y, a lift is a MAP h from X to Z such that gh /C30f. In other words, a lift of f is a MAP h such that the diagram (shown below) commutes. If f is the identity from Y to Y,aMANIFOLD , and if g is the BUNDLE PROJECTION from the TANGENT BUNDLE to Y, the lifts are precisely VECTOR FIELDS .Ifg is a bundle projection from any FIBER BUNDLE to Y, then lifts are precisely sections. If f is the identity from Y to Y,a MANIFOLD , and g a projection from the orientation double cover of Y, then lifts exist IFF Y is an orientable MANIFOLD . If f is a MAP from a CIRCLE to Y,ann-MANIFOLD , and g the bundle projection from the FIBER BUNDLE of alternating K-FORMS on Y, then lifts always exist IFF Y is orientable. If f is a MAP from a region in theCOMPLEX PLANE to the COMPLEX PLANE (complex analytic), and if g is the exponential MAP, lifts of f are precisely LOGARITHMS of f. See also LIFTING PROBLEM Lifting Problem Given a MAP f from a SPACE X to a SPACE Y and another MAP g from a SPACE Z to a SPACE Y, does there exist a MAP h from X to Z such that gh /C30 f?If such a map h exists, then h is called a LIFT of f. See also EXTENSION PROBLEM ,LIFT Ligancy KISSING NUMBER Likelihood The hypothetical PROBABILITY that an event which has already occurred would yield a specific outcome. The concept differs from that of a probability in that a probability refers to the occurrence of future events, while a likelihood refers to past events with known outcomes. See also LIKELIHOOD RATIO,M AXIMUM LIKELIHOOD , NEGATIVE LIKELIHOOD RATIO,PROBABILITY Likelihood Ratio A quantity used to test NESTED HYPOTHESES . Let H ? be a NESTED HYPOTHESIS with n? DEGREES OF FREE- DOM within H (which has n DEGREES OF FREEDOM ), then calculate the MAXIMUM LIKELIHOOD of a given outcome, first given H ?; then given H. Then LR /C30[likelihood H ?] [likelyhood H] : Comparison of this ratio to the critical value of the CHI-SQUARED DISTRIBUTION with n /C28n ? DEGREES OF FREEDOM then gives the SIGNIFICANCE of the increase in LIKELIHOOD . The term likelihood ratio is also used (especially in medicine) to test nonnested complementary hypoth- eses as follows, LR/C30[true positive rate] [false positive rate]/C30[sensitivity] 1/C28[specificity]: See also NEGATIVE LIKELIHOOD RATIO,SENSITIVITY , SPECIFICITY Limac ¸on of Pascal LIMAC ¸ON Limac ¸on The limac ¸on is a polar curve OF THE FORM r /C30b /C27a cos u also called the LIMAC ¸ ON OF PASCAL . It was first investigated by Du¨rer, who gave a method for draw- ing it in Underweysung der Messung (1525). It was rediscovered by E´ tienne Pascal, father of Blaise Pascal, and named by Gilles-Personne Roberval in 1650 (MacTutor Archive). The word "limac ¸on" comes from the Latin limax , meaning "snail." If b ]2a ; we have a convex limac ¸on. If 2a > b > a; we have a dimpled limac ¸on. If b /C30 a, the limac ¸on degenerates to a CARDIOID .Ifb B a, we have limac ¸on with an inner loop. If b /C30a =2; it is a TRISECTRIX (but not the MACLAURIN TRISECTRIX ) with inner loop of AREA Ainner loop /C301 4 a2 p /C283ffiffiffi 3 2s ! ; and AREA between the loops of Abetween loops /C301 4 a2 p /C273ffiffiffi 3p/C(%/C(r (MacTutor Archive). The limac ¸on can be generated by specifying a fixed point P, then drawing a sequences of circles with centers on a given circle which all pass through P. The ENVELOPE of these curves is a limac ¸on. If the fixed point is on the CIRCUMFERENCE of the circle, then the ENVELOPE is a CARDIOID . The limac ¸on is an ANALLAGMATIC CURVE , and is also the CATACAUSTIC of a CIRCLE when the RADIANT POINT is a finite (NONZERO ) distance from the CIRCUMFER- ENCE , as shown by Thomas de St. Laurent in 1826 (MacTutor Archive). The limac ¸on is the CONCHOID of a CIRCLE with respect to a point on its CIRCUMFER- ENCE (Wells 1991). See also CARDIOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 220 /C1/21, 1987.Baudoin, P. Les ovales de Descartes et le limac ¸on de Pascal. Paris: Vuibert, 1938. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 113 /C1/17, 1972. Lockwood, E. H. "The Limac ¸on." Ch. 5 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 44 /C1/1, 1967. MacTutor History of Mathematics Archive. "Limacon of Pascal." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Limacon.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 154 /C1/55, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 140 /C1/41, 1991. Yates, R. C. "Limacon of Pascal." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 148 /C1/ 51, 1952. Limac ¸on Evolute The CATACAUSTIC of a CIRCLE for a RADIANT POINT is the limac ¸on evolute. It has PARAMETRIC EQUATIONS x /C30a[4a2 /C27 4b2 /C27 9ab cos t /C28 ab cos(3 t)] 4(2a2 /C27 b2 /C27 3ab cos t) y /C30a2b sin3 t 2a2 /C27 b2 /C27 3ab cos t : Limb A limb of a TREE at a vertex v is the union of one or more BRANCHES atvin the tree. vis then called the base of the limb. See also BRANCH ,TREE References Lu, T. "The Enumeration of Trees with and without Given Limbs." Disc. Math. 154, 153/C1/65, 1996. Schwenk, A. "Almost All Trees are Cospectral." In New Directions in the Theory of Graphs (Ed. F. Harary). New York: Academic Press, pp. 275 /C1/07, 1973. Lim Inf INFIMUM LIMIT Limit A function f(z) is said to have a limit limz0af(z)/C30cif, for all e>0;there exists a d>0 such that ½f(z)/C28c½Be whenever 0 B½z/C28a½Bd:This form of definition is sometimes called an EPSILON-DELTA DEFINITION . Lim- its may be taken from below lim z0a/C28/C30lim x/C160a(1) or from above lim z0a /C27/C30lim z¡a: (2) if the two are equal, then "the" limit is said to exist lim z0a/C30 lim z0a /C28/C30 lim z0a /C27: (3) A LOWER LIMIT h lower lim n0/C12Sn /C30lim n0/C12Sn /C30h (4) is said to exist if, for every e> 0 ;½Sn /C28h½Be for infinitely many values of n and if no number less than h has this property. An UPPER LIMIT k upper lim n0/C12Sn /C30lim n0/C12Sn /C30k (5) is said to exist if, for every e> 0 ;½Sn /C28h½Be for infinitely many values of n and if no number larger than k has this property. INDETERMINATE limit forms of types /C12=/C12 and 0=0 can often be computed with L’HOSPITAL’S RULE . Types 0 /C215/C12 can be converted to the form 0=0 by writing f(x)g(x) /C30f(x) 1 =g(x) : (6) Types 00, /C120 ; and 1/C12 are treated by introducing a dependent variable y /C30f(x)g(x) (7) so that ln y /C30g(x)ln[f(x)]; (8) then calculating lim ln y: The original limit then equals elim ln y ; L /C30lim f(x)g(x) /C30elim ln y (9) The INDETERMINATE form /C12/C28/C12 is also frequently encountered. See also CENTRAL LIMIT THEOREM ,C ONTINUOUS , DERIVATIVE ,D ISCONTINUITY ,INDETERMINATE ,INFI- MUM LIMIT,L’HOSPITAL’S RULE,LIMIT COMPARISON TEST,LIMIT TEST,LOWER LIMIT,PINCHING THEOREM , SQUEEZING THEOREM ,SUPREMUM LIMIT,UPPER LIM- IT References Courant, R. and Robbins, H. "Limits. Infinite Geometrical Series." §2.2.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 63 /C1/6, 1996. Gruntz, D. On Computing Limits in a Symbolic Manipula- tion System. Doctoral thesis. Zu¨rich: Swiss Federal Institute of Technology, 1996. Hight, D. W. A Concept of Limits. New York: Prentice-Hall, 1966.Kaplan, W. "Limits and Continuity." §2.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 82 /C1/ 6, 1992. Miller, N. Limits. Waltham, MA: Blaisdell, 1964. Prevost, S. "Exploring the e/-/ d Definition of Limit with Mathematica." Mathematica Educ. 3,17/C1/1, 1994. Smith, W. K. Limits and Continuity. New York: Macmillan, 1964. Limit Comparison Test Let aakand abkbe two SERIES with POSITIVE terms and suppose lim k 0/C12ak bk/C30 r: If r is finite and r > 0; then the two SERIES both CONVERGE or DIVERGE . See also CONVERGENCE TESTS ,LIMIT,LIMIT TEST Limit Cycle An attracting set to which orbits or trajectories converge and upon which trajectories are periodic. See also HOPF BIFURCATION Limiting Point A point about which INVERSION of two circles pro- duced CONCENTRIC CIRCLES . Every pair of distinct circles has two limiting points. The limiting points correspond to the POINT CIRCLES of a COAXAL SYSTEM , and the limiting points of a COAXAL SYSTEM are INVERSE POINTS with respect to any circle of the system. To find the limiting point of two circles of radii rand Rwith centers separated by a distance d, set up a coordinate system centered on the circle of radius R and with the other circle centered at ( d;0):Then the equation for the position of the center of the inverted circles with inversion center ( x0;0); x?/C30x0 /C27k2(x /C28 x0) (x /C28 x0)2 /C27 (y /C28 y0)2 /C28 a2; (1) becomes x?1 /C30x0 /C27k2(d /C28 x0) (d /C28 x0)2 /C28 r2(2) x?2 /C30x0 /C27k2(0 /C28 x0) (0 /C28 x0)2 /C28 R2(3) for the first and second circles, respectively. Setting x?1 /C30x?2 gives d /C28 x0 (d /C28 x0)2 /C28 r2/C30/C28x0 x2 0 /C28 R2; (4) and solving using the quadratic equation gives the positions of the limiting points as x?/C30d2 /C28 r2 /C27 R2 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (d2 /C28 r2 /C27 R2)2 /C28 4d2R2p 2d : (5) See also COAXAL SYSTEM ,C ONCENTRIC CIRCLES , INVERSE POINTS ,INVERSION CENTER ,POINT CIRCLE References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 43, 1888. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 123 and 130, 1928. Limit Ordinal An ORDINAL NUMBER a > 0 is called a limit ordinal IFF it has no immediate PREDECESSOR , i.e., if there is no ORDINAL NUMBER b such that b /C271 /C30 a (Ciesielski 1997, p. 46; Moore 1982, p. 60; Rubin 1967, p. 182; Suppes 1972, p. 196). The first limit ordinal is v:/ See also ORDINAL NUMBER ,SUCCESSOR References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Limit Point A number x such that for all e> 0; there exists a member of the SET y different from x such that ½y /C28 x½Be: The topological definition of limit point P of A is that P is a point such that every OPEN SET around it intersects A. See also ACCUMULATION POINT ,CLOSED SET,OPEN SETReferences Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 9 /C1/0, 1988. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 25 /C1/6, 1991. Limit Test If lim an "0 or this LIMIT does not exist as n tends to infinity, then the INFINITE SERIES a andoes not CONVERGE . For example, a/C12 n /C301(/C281)n does not converge by the limit test. The limit test is inconclusive when the limit is zero. See also CONVERGENT SERIES ,CONVERGENCE TESTS , LIMIT,LIMIT COMPARISON TEST,SEQUENCE ,SERIES Limit Theorem CENTRAL LIMIT THEOREM ,LEBESGUE’S DOMINATED CONVERGENCE THEOREM LINDEBERG- FELLER CEN- TRAL LIMIT THEOREM ,M ONOTONE CONVERGENCE THEOREM ,POINTWISE CONVERGENCE Lim Sup SUPREMUM LIMIT Lindeberg Condition ASUFFICIENT condition on the L INDEBERG- FELLER CENTRAL LIMIT THEOREM . Given random variates X1; X2;..., let /C142Xi/C143/C300;the VARIANCE s2 iofXibe finite, and VARIANCE of the distribution consisting of a sum of Xi/s Sn/C13X1/C27X2/C27.../C27Xn (1) be s2 n/C13Xn i/C301s2i: (2) In the terminology of Zabell (1995), let Ln(e)/C13Xn k/C301Xk sn !2 :½Xk½ sn]e*+ ; (3) where //C142f:g/C143/denotes the EXPECTATION VALUE off restricted to outcomes g, then the Lindeberg condi- tion is lim n0/C12Ln(e)/C300 (4) for all e>0 (Zabell 1995). In the terminology of Feller (1971), the Lindeberg condition assumed that for each t/C210, 1 s2 nXn k/C301g½y½]tsny2Fkfdyg00; (5) or equivalently 1 s2 nXn k /C301g½y ½B tsny2Fk fdy g0 1: (6) Then the distribution Sn /C31/C30X1 /C27 ... /C27 Xn sn(7) tends to the NORMAL DISTRIBUTION with zero expecta- tion and unit variance (Feller 1971, p. 256). The Lindeberg condition (5) guarantees that the indivi- dual variances s2 k are small compared to their sum s2n in the sense that for given e> 0 for for all SUFFI- CIENTLY LARGE n, sk =sn Be for k /C301, ..., n (Feller 1971, p. 256). See also CENTRAL LIMIT THEOREM ,F ELLER- LE´ VY CONDITION References Feller, W. "Uuml;ber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung." Math. Zeit. 40, 521 /C1/59, 1935. Feller, W. "U¨ ber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung, II." Math. Zeit. 42, 301 /C1/ 12, 1935. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 257 /C1/ 58, 1971. Lindeberg, J. W. "Eine neue Herleitung des Exponential- gesetzes in der Wahrscheinlichkeitsrechnung." Math. Zeit. 15, 211 /C1/35, 1922. Trotter, H. F. "An Elementary Proof of the Central Limit Theorem." Arch. Math. 10, 226 /C1/34, 1959. Wallace, D. L. "Asymptotic Approximations to Distribu- tions." Ann. Math. Stat. 29, 635 /C1/54, 1958. Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483 /C1/94, 1995. Lindeberg-Feller Central Limit Theorem If the random variates X1 ; X2 ; ... satisfy the LINDE- BERG CONDITION , then for all a Bb, lim n 0/C12PaBSn snBb ! /C30F(b) /C28F(a) ; where F is the NORMAL DISTRIBUTION FUNCTION . See also BERRY- ESSE´ EN THEOREM ,C ENTRAL LIMIT THEOREM ,FELLER- LE´ VY CONDITION ,NORMAL DISTRI- BUTION FUNCTION References Feller, W. "U¨ ber den zentralen Genzwertsatz der Wahrscheinlichkeitsrechnung." Math. Z. 40, 521 /C1/59, 1935. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, p. 229, 1968. Lindeberg, J. W. "Eine neue Herleitung des Exponentialge- setzes in der Wahrschienlichkeitsrechnung." Math. Z. 15, 211 /C1/25, 1922. Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483 /C1/94, 1995.Lindelof’s Theorem The SURFACE OF REVOLUTION generated by the ex- ternal CATENARY between a fixed point a and its conjugate on the ENVELOPE of the CATENARY through the fixed point is equal in AREA to the surface of revolution generated by its two Lindelof TANGENTS , which cross the axis of rotation at the point a and are calculable from the position of the points and CATEN- ARY. See also CATENARY ,ENVELOPE ,SURFACE OF REVOLU- TION Lindemann-Weierstrass Theorem If a1 ; ..., an are linearly independent over Q; then e a1 ; ..., e anare ALGEBRAICALLY INDEPENDENT over Q: The Lindemann-Weierstrass theorem is implied by SCHA- NUEL’S CONJECTURE (Chow 1999). See also ALGEBRAICALLY INDEPENDENT ,H ERMITE- LINDEMANN THEOREM ,SCHANUEL’S CONJECTURE References Baker, A. Theorem 2.1 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1990. Chow, T. Y. "What is a Closed-Form Number?" Amer. Math. Monthly 106, 440 /C1/48, 1999. Lindenmayer System A STRING REWRITING system which can be used to generate FRACTALS with DIMENSION between 1 and 2. The term L-system is often used as an abbreviation. See also ARROWHEAD CURVE ,DRAGON CURVE EXTER- IOR SNOWFLAKE ,FRACTAL ,H ILBERT CURVE ,K OCH SNOWFLAKE ,PEANO CURVE ,PEANO- GOSPER CURVE , SIERPINSKI CURVE ,STRING REWRITING References Bulaevsky, J. " L-System Based Fractals." http://www.best.- com/~ejad/java/fractals/lsystems.shtml. Bulaevsky, J. "A Process to Generate Fractals." http:// www.best.com/~ejad/java/fractals/process.shtml. Dickau, R. M. "Two-dimensional L-systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Prusinkiewicz, P. and Hanan, J. Lindenmayer Systems, Fractal, and Plants. New York: Springer-Verlag, 1989. Prusinkiewicz, P. and Lindenmayer, A. The Algorithmic Beauty of Plants. New York: Springer-Verlag, 1990. Stevens, R. T. Fractal Programming in C. New York: Holt, 1989. Wagon, S. "Recursion via String Rewriting." §6.2 in Math- ematica in Action. New York: W. H. Freeman, pp. 190 /C1/ 96, 1991. Line Euclid defined a line as a "breadthless length," and a straight line as a line which "lies evenly with the points on itself" (Kline 1956, Dunham 1990). Lines are intrinsically 1-dimensional objects, but may beembedded in higher dimensional SPACES . An infinite line passing through points AandBis denoted AB:A LINE SEGMENT terminating at these points is denoted AB:A line is sometimes called a STRAIGHT LINE or, more archaically, a RIGHT LINE (Casey 1893), to emphasize that it has no curves anywhere along its length. Harary (1994) called an edge of a graph a "line."Consider first lines in a 2-D PLANE . The line with X- INTERCEPT aand Y-INTERCEPT bis given by the intercept form x a/C27y b/C301: (1) The line through ( x1;y1) with SLOPE mis given by the point-slope form y/C28y1/C30m(x/C28x1): (2) The line with y-intercept band slope mis given by theslope-intercept form y/C30mx/C27b: (3) The line through ( x1;y1) and ( x2;y2) is given by the two point form y/C28y1/C30y2/C28y1 x2/C28x1(x/C28x1): (4) Other forms are a(x/C28x1)/C27b(y/C28y1)/C300 (5) ax/C27by/C27c/C300 (6) xy 1 x1y11 x2y21/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300: (7) A line in 2-D can also be REPRESENTED AS aVECTOR . The VECTOR along the line ax/C27by/C300 (8) is given by t/C28b a/C)P/C)( ; (9) where t/C23R:Similarly, VECTORS OF THE FORM ta b/C)P/C)( (10) are PERPENDICULAR to the line. Three points lie on a line if x1y11 x2y21 x3y31/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300: (11) The ANGLE between lines A1x/C27B1y/C27C1/C300 (12)A2x/C27B2y/C27C2/C300 (13) is tanu/C30A1B2/C28A2B1 A1A2/C27B1B2: (14) The line joining points with TRILINEAR COORDINATES a1:b1:g1anda2:b2:g2is the set of point a:b:g satisfying abg a1b1g1 a2b2g2/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300 (15) (b 1g2/C28g1b2)a/C27(g1a2/C28a1g2)b/C27(a1b2/C28b1a2)g /C300: (16) Three lines CONCUR if their TRILINEAR COORDINATES satisfy l1a/C27m1b/C27n1g/C300 (17) l2a/C27m2b/C27n2g/C300 (18) l3a/C27m3b/C27n3g/C300; (19) in which case the point is m2n3/C28n2m3:n2l3/C28l2n3:l2m3/C28m2l3; (20) or if the COEFFICIENTS of the lines A1x/C27B1y/C27C1/C300 (21) A2x/C27B2y/C27C2/C300 (22) A3x/C27B3y/C27C3/C300 (23) satisfy A1B1C1 A2B2C2 A3B3C3/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300: (24) Two lines CONCUR if their TRILINEAR COORDINATES satisfy l1m1n1 l2m2n2 l3m3n3/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300: (25) The line through P 1is the direction ( a1;b1;c1) and the line through P2in direction ( a2;b2;c2) intersect IFF x2/C28x1y2/C28y1z2/C28z1 a1 b1 c1 a2 b2 c2/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300: (26) The line through a point a?:b?:g? PARALLEL to la/C27mb/C27ng/C300 (27) is abg a? b? g ? bn /C28cm cl /C28an am /C28bl/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300 : (28) The lines l a /C27mb /C27ng /C300 (29) l ?a /C27m?b /C27n ?g /C300 (30) are PARALLEL if a(mn?/C28nm?) /C27b(nl ?/C28ln ?) /C27c(lm ?/C28ml?) /C300 (31) for all (a; b; c) ; and PERPENDICULAR if 2abc(ll?/C27mm ?/C27nn?) /C28(mn ?/C27m?m)cos A /C28(nl ?/C27n?l)cos B /C28(lm ?/C27l?m)cos C /C300 (32) for all (a ; b; c) (Sommerville 1924). The line through a point a? : b? : g ? PERPENDICULAR to (32) is given by abg a? b? g ? l /C28m cos Cm/C28n cos An/C28l cos B /C28n cos B /C28l cos C /C28m cos A/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300 : (33) In 3-D SPACE , the line passing through the point (x0 ; y0 ; z0) and PARALLEL to the NONZERO VECTOR v /C30a b c2 435 (34) has PARAMETRIC EQUATIONS x /C30x0 /C27at (35) y /C30y0 /C27bt (36) z /C30z0 /C27ct ; (37) written concisely as x /C30x0 /C27vt: (38) Similarly, the line in 3-D passing through (x1 ; y1) and (x2 ; y2) has parametric vector equation x /C30x1 /C27(x2 /C28x1)t; (39) where this parametrization corresponds to x(t /C300) /C30 x1 and x(t /C301) /C30x2 :/ See also ASYMPTOTE ,BRANCH LINE,BROCARD LINE, CAYLEY LINES,COLLINEAR ,CONCUR ,CRITICAL LINE, DESARGUES’ THEOREM ,E RDOS- ANNING THEOREM , EULER LINE,FLOW LINE,GERGONNE LINE,IMAGIN- ARY LINE,ISOGONAL LINE,ISOTROPIC LINE,LEMOINE LINE,LINE-LINE INTERSECTION ,LINE-PLANE INTER- SECTION ,LINE SEGMENT ,O RDINARY LINE,P ASCAL LINES,P EDAL LINE,P ENCIL ,P HILO LINE,P OINT , POINT- LINE DISTANCE–2- D, POINT- LINE DISTANCE–3- D, PLANE ,PLU¨ CKER LINES,POLAR LINE,POWER LINE, RADICAL LINE,RANGE (LINE SEGMENT ), RAY,REAL LINE,RHUMB LINE,SECANT LINE,SIMSON LINE,SKEWLINES,SODDY LINE,SOLOMON’S SEAL LINES,STEINER SET,STEINER’S THEOREM ,SYLVESTER’S LINE PRO- BLEM ,S YMMEDIAN ,T ANGENT LINE,T RANSVERSAL LINE,TRILINEAR LINE,W ORLD LINE References Casey, J. "The Right Line." Ch. 2 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 30 /C1/5, 1893. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 32, 1990. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Kern, W. F. and Bland, J. R. "Lines and Planes in Space." §4 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 9 /C1/2, 1948. Kline, M. "The Straight Line." Sci. Amer. 156, 105 /C1/14, Mar. 1956. MacTutor History of Mathematics Archive. "Straight Line." http://www-groups.dcs.st-and.ac.uk/~history/Curves/ Straight.html. Sommerville, D. M. Y. Analytical Conics. London: G. Bell, p. 186, 1924. Spanier, J. and Oldham, K. B. "The Linear Function /bx /C27c/ and Its Reciprocal." Ch. 7 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 53 /C1/2, 1987. Linear Algebra The study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of ROTATIONS in space, LEAST SQUARES FITTING , solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathe- matics, physics, and engineering. The MATRIX and DETERMINANT are extremely useful tools of linear algebra. One central problem of linear algebra is the solution of the matrix equation Ax/C30b forx. While this can, in theory, be solved using a MATRIX INVERSE x/C30A/C281b; other techniques such as G AUSSIAN ELIMINATION are numerically more robust. See also CONTROL THEORY ,CRAMER’S RULE,DETER- MINANT ,GAUSSIAN ELIMINATION ,LINEAR TRANSFOR- MATION ,MATRIX ,VECTOR References Axler, S. Linear Algebra Done Right, 2nd ed. New York: Springer-Verlag, 1997. Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, 1962. Banchoff, T. and Wermer, J. Linear Algebra Through Geometry, 2nd ed. New York: Springer-Verlag, 1992. Bellman, R. E. Introduction to Matrix Analysis, 2nd ed. New York: McGraw-Hill, 1970. BLAS. "BLAS (Basic Linear Algebra Subprograms)." http:// www.netlib.org/blas/. Carlson, D.; Johnson, C. R.; Lay, D. C.; Porter, A. D.; Watkins, A. E.; and Watkins, W. (Eds.). Resources for Teaching Linear Algebra. Washington, DC: Math. Assoc. Amer., 1997. Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, 1958. Golub, G. and van Loan, C. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996. Halmos, P. R. Linear Algebra Problem Book. Providence, RI: Math. Assoc. Amer., 1995. Lang, S. Introduction to Linear Algebra, 2nd ed. New York: Springer-Verlag, 1997. LAPACK. "LAPACK--Linear Algebra PACKage." http:// www.netlib.org/lapack/. Lipschutz, S. Schaum’s Outline of Theory and Problems of Linear Algebra, 2nd ed. New York: McGraw-Hill, 1991. Lumsdaine, J. and Siek, J. "The Matrix Template Library: Generic Components for High Performance Scientific Computing." http://www.lsc.nd.edu/research/mtl/. Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, 1988. Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, 1992. Marcus, M. Matrices and Matlab: A Tutorial. Englewood Cliffs, NJ: Prentice-Hall, 1993. Mirsky, L. An Introduction to Linear Algebra. New York: Dover, 1990. Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, 1990. Petard, H. Problems in Linear Algebra, preliminary ed. New York: W.A. Benjamin, 1967. Strang, G. Linear Algebra and its Applications, 3rd ed. Philadelphia, PA: Saunders, 1988. Strang, G. Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press, 1993. Strang, G. and Borre, K. Linear Algebra, Geodesy, & GPS. Wellesley, MA: Wellesley-Cambridge Press, 1997. Weisstein, E. W. "Books about Linear Algebra." http:// www.treasure-troves.com/books/LinearAlgebra.html. Zhang, F. Matrix Theory: Basic Results and Techniques. New York: Springer-Verlag, 1999. Linear Algebraic Group A linear algebraic group is a GROUP which is also an AFFINE VARIETY . In particular, its elements satisfy polynomial equations. For example, GL(n) ; the GEN- ERAL LINEAR GROUP , is a linear algebraic group because an INVERTIBLE MATRIX is given by n2 entries that satisfy the polynomial det an /C301 : The group operations are required to be given by REGULAR RATIONAL FUNCTIONS . The linear algebraic groups are similar to the LIE GROUPS , except that linear algebraic groups may be defined over any FIELD , including those of positive CHARACTERISTIC . See also AFFINE VARIETY ,ALGEBRAIC GROUP ,FORMAL GROUP ,GROUP ,GROUP SCHEME ,LIE ALGEBRA ,LIE GROUP ,VARIETY Linear Approximation A linear approximation to a function f(x) at a point x0 can be computed by taking the first term in theTAYLOR SERIES f(x0 /C27Dx) /C30f(x0) /C27f ?(x0) Dx /C27... : See also MACLAURIN SERIES ,TAYLOR SERIES Linear Code A linear code over a FINITE FIELD with q elements Fq is a linear SUBSPACE C ƒFn q : The vectors forming the SUBSPACE are called code words. When code words are chosen such that the distance between them is maximized, the code is called error-correcting since slightly garbled vectors can be recovered by choosing the nearest code word. See also CODE,CODING THEORY ,ERROR- CORRECTING CODE,GRAY CODE,HUFFMAN CODING , ISBN, UPC Linear Combination A sum of the elements from some set with constant coefficients placed in front of each. For example, a linear combination of the VECTORS x, y, and z is given by ax /C27by /C27cz ; where a, b, and c are constants. See also BASIS,BASIS (VECTOR SPACE ), SPAN (VECTOR SPACE ) Linear Congruence Equation A linear congruence equation ax/C13b(mod m) (1) is solvable IFFthe CONGRUENCE b/C130 (mod d) (2) is solvable, where d/C13GCD( a;m) is the GREATEST COMMON DIVISOR . Let one solution to the original equation be x0Bm=d:Then the solutions are x/C30x0; x0/C27m=d;x0/C272m=d;...,x0/C27(d/C281)m=d:Ifd/C301, then there is only one solution Bm:The solution of a linear congruence can be found in Mathematica using Solve [ax/C30/C30 b&&Modulus /C30/C30 m,x]. Solution to a linear congruence equation is equivalent to finding the value of a fractional CONGRUENCE , for which a greedy-type algorithm exists. In particular,(1) can be rewritten as x/C13b a(mod m) (3) which can also be written x b/C131 a(mod m): (4) In this form, the solution xcan be found as Mod[by, m] of the solution y returned by the Mathematica command y /C30PowerMod [a, -1, m]. See also CHINESE REMAINDER THEOREM ,C ONGRU- ENCE ,CONGRUENCE EQUATION ,QUADRATIC CONGRU- ENCE EQUATION References Nagell, T. "Linear Congruences." §23 in Introduction to Number Theory. New York: Wiley, pp. 76 /C1/8, 1951. Linear Congruence Method A METHOD for generating RANDOM (PSEUDORANDOM ) numbers using the linear RECURRENCE RELATION Xn/C271 /C30aXn /C27c (mod m); where a and c must assume certain fixed values and X0 is an initial number known as the SEED . See also PSEUDORANDOM NUMBER ,RANDOM NUMBER , SEED References Brunner, D. and Uhl, A. "Optimal Multipliers for Linear Congruential Pseudo Random Number Generators with Prime Moduli: Parallel Computation and Properties." BIT. Numer. Math. 39, 193 /C1/09, 1999. Pickover, C. A. "Computers, Randomness, Mind, and In- finity." Ch. 31 in Keys to Infinity. New York: W. H. Freeman, pp. 233 /C1/47, 1995. Linear Diophantine Equation DIOPHANTINE EQUATION Linear Equation An algebraic equation OF THE FORM y/C30ax/C27b involving only a constant and a first-order (linear) term. See also LINE,POLYNOMIAL ,QUADRATIC EQUATION Linear Equation System When solving a system of nlinear equations with k/C21nunknowns, use MATRIX operations to solve the system as far as possible. Then solve for the first ( k/C28 n) components in terms of the last ncomponents to find the solution space. Linear Extension A linear extension of a PARTIALLY ORDERED SET Pis a PERMUTATION of the elements p1;p2;... of Psuch that iBjIMPLIES piBpj:For example, the linear exten- sions of the PARTIALLY ORDERED SET ((1;2);(3;4)) are 1234, 1324, 1342, 3124, 3142, and 3412, all of which have 1 before 2 and 3 before 4.References Brightwell, G. and Winkler, P. "Counting Linear Exten- sions." Order 8, 225/C1/42, 1991. Bubley, R. and Dyer, M. "Faster Random Generation of Linear Extensions." In Proc. Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, Calif., pp. 350 /C1/54, 1998. Preusse, G. and Ruskey, F. "Generating Linear Extensions Fast." SIAM J. Comput. 23, 373/C1/86, 1994. Ruskey, F. "Information on Linear Extension." http:// www.theory.csc.uvic.ca/~cos/inf/pose/LinearExt.html. Varol, Y. and Rotem, D. "An Algorithm to Generate All Topological Sorting Arrangements." Comput. J. 24,8 3/C1/4, 1981. Linear Fractional Transformation A transformation OF THE FORM w/C30f(z)/C30az/C27b cz/C27d; (1) where a,b,c,d/C23Cand ad/C28bc"0; (2) is a CONFORMAL MAPPING called a linear fractional transformation. The transformation can be extended to the entire extended COMPLEX PLANE C+/C30C@f/C12g by defining f/C28d c ! /C30/C12 (3) f(/C12)/C30a c(4) (Apostol 1997, p. 26). The linear fractional transfor-mation is linear in both wand z, and analytic everywhere except for a simple POLE atz/C30/C28d=c:/ Every linear fractional transformation except f(z)/C30z has one or two FIXED POINTS . The linear fractional transformation sends CIRCLES and lines to CIRCLES or lines. Linear fractional transformations preserve symmetry. The CROSS-RATIO is invariant under a linear fractional transformation. A linear fractional transformation is a composition of translations, rota- tions, magnifications, and inversions. To determine a particular linear fractional transfor- mation, specify the map of three points whichpreserve orientation. A particular linear fractional transformation is then uniquely determined. To determine a general linear fractional transformation,pick two symmetric points aanda S:Define b/C13f(a); restricting bas required. Compute bS:f(aS) then equals bSsince the linear fractional transformation preserves symmetry (the SYMMETRY PRINCIPLE ). Plug inaandaSinto the general linear fractional trans- formation and set equal to bandbS:Without loss of generality, let c/C301 and solve for aandbin terms of b:Plug back into the general expression to obtain a linear fractional transformation. See also CAYLEY TRANSFORM ,M O¨ BIUS TRANSFORM , MODULAR GROUP GAMMA ,SCHWARZ’S LEMMA ,SYM- METRY PRINCIPLE ,UNIMODULAR TRANSFORMATION References Anderson, J. W. "The Group of Mo¨bius Transformations." §2.1 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 19 /C1/5, 1999. Apostol, T. M. "Mo¨bius Transformations." Ch. 2.1 in Mod- ular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 26 /C1/8, 1997. Krantz, S. G. "Linear Fractional Transformations." §6.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 81 /C1/6, 1999. Mathews, J. "The Moebius Transformation." http:// www.ecs.fullerton.edu/~mathews/fofz/mobius/. Linear Function A linear function is a function f which satisfies f(x /C27y) /C30f(x) /C27f(y) and f(ax) /C30 af(x) for all x and y in the DOMAIN , and all SCALARS a:/ See also BILINEAR FUNCTION ,F UNCTION ,V ECTOR SPACE Linear Functional A linear functional on a REAL VECTOR SPACE V is a function T : V 0 R; which satisfies the following properties. 1. /T(v /C27w) /C30T(v) /C27T(w)/, and 2. /T( av) /C30 aT(v)/. When V is a COMPLEX VECTOR SPACE , then T is a linear map into the COMPLEX NUMBERS . DISTRIBUTIONS are a special case of linear func- tionals, and have a rich theory surrounding them. See also DISTRIBUTION (GENERALIZED FUNCTION ), DUAL SPACE ,FUNCTIONAL ,VECTOR SPACE Linear Group See also GENERAL LINEAR GROUP ,LIE-TYPE GROUP , PROJECTIVE GENERAL LINEAR GROUP ,P ROJECTIVE SPECIAL LINEAR GROUP ,SPECIAL LINEAR GROUP References Hsiang, W. Y. "Linear Groups and Linear Representations." Lec. 1 in Lectures on Lie Groups. Singapore: World Scientific, pp. 1 /C1/9, 2000. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#lin.Linear Group Theorem Any linear system of point-groups on a curve with only ordinary singularities may be cut by ADJOINT CURVES . References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 122 and 251, 1959. Linear Map LINEAR TRANSFORMATION Linear Operator An operator ˜L is said to be linear if, for every pair of functions f and g and SCALAR t, ˜L(f /C27g) /C30 ˜Lf /C27 ˜Lg and ˜L(tf) /C30t ˜Lf : See also LINEAR TRANSFORMATION ,OPERATOR Linear Ordinary Differential Equation ORDINARY DIFFERENTIAL EQUATION– FIRST- ORDER , ORDINARY DIFFERENTIAL EQUATION– SECOND- ORDER Linear Programming The problem of maximizing a linear function over a convex polyhedron, also known as OPERATIONS RE- SEARCH , OPTIMIZATION THEORY ,or CONVEX OPTIMIZA- TION THEORY . Linear programming is extensively used in economics and engineering. Examples from economics include Leontief’s input-output model, the determination of shadow prices, etc., while an exam- ple of an engineering application would be maximiz- ing profit in a factory that manufactures a number of different products from the same raw material using the same resources. Linear programming can be solved using the SIMPLEX METHOD (Wood and Dantzig 1949, Dantzig 1949) which runs along EDGES of the visualization solid to find the best answer. In 1979, L. G. Khachian found a O(x5) POLYNOMIAL -time ALGORITHM . A much more efficient POLYNOMIAL -time ALGORITHM was found by Karmarkar (1984). This method goes through the middle of the solid and then transforms and warps, and offers many advantages over the simplex method. Karmarkar’s method is patented, so it has not received much detailed discussion. See also CRISS- CROSS METHOD ,ELLIPSOIDAL CALCU- LUS,K UHN- TUCKER THEOREM ,L AGRANGE MULTI- PLIER ,O PTIMIZATION ,O PTIMIZATION THEORY , STOCHASTIC OPTIMIZATION ,VERTEX ENUMERATION References Bellman, R. and Kalaba, R. Dynamic Programming and Modern Control Theory. New York: Academic Press, 1965. Dantzig, G. B. "Programming of Interdependent Activities. II. Mathematical Model." Econometrica 17, 200 /C1/11, 1949. Dantzig, G. B. Linear Programming and Extensions. Prin- ceton, NJ: Princeton University Press, 1963. Karloff, H. Linear Programming. Boston, MA: Birkha ¨user, 1991. Karmarkar, N. "A New Polynomial-Time Algorithm for Linear Programming." Combinatorica 4, 373 /C1/95, 1984. Pappas, T. "Projective Geometry & Linear Programming." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 216 /C1/17, 1989. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Linear Programming and the Simplex Method." §10.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 423 /C1/36, 1992. Sultan, A. Linear Programming: An Introduction with Applications. San Diego, CA: Academic Press, 1993. Tokhomirov, V. M. "The Evolution of Methods of Convex Optimization." Amer. Math. Monthly 103,65/C1/1, 1996. Weisstein, E. W. "Books about Linear Programming." http:// www.treasure-troves.com/books/LinearProgram- ming.html. Wood, M. K. and Dantzig, G. B. "Programming of Interde- pendent Activities. I. General Discussion." Econometrica 17, 193 /C1/99, 1949. Yudin, D. B. and Nemirovsky, A. S. Problem Complexity and Method Efficiency in Optimization. New York: Wiley, 1983. Linear Recurrence Sequence RECURRENCE SEQUENCE Linear Regression The fitting of a straight LINE through a given set of points according to some specified goodness-of-fit criterion. The most common form of linear regressionis LEAST SQUARES FITTING . See also LEAST SQUARES FITTING ,MULTIPLE REGRES- SION,NONLINEAR LEAST SQUARES FITTING References Edwards, A. L. An Introduction to Linear Regression and Correlation. San Francisco, CA: W. H. Freeman, 1976. Edwards, A. L. Multiple Regression and the Analysis of Variance and Covariance. San Francisco, CA: W. H. Freeman, 1979. Linear Space VECTOR SPACE Linear Stability Consider the general system of two first-order ORDIN- ARY DIFFERENTIAL EQUATIONS ˙x/C30f(x;y) (1) ˙y/C30g(x;y): (2) Letx0andy0denote FIXED POINTS with ˙x/C30˙y/C300;sof(x0;y0)/C300 (3) g(x0;y0)/C300: (4) Then expand about ( x0;y0)s o d˙x/C30fx(x0;y0)dx/C27fy(x0;y0)dy/C27fxy(x0;y0)dxdy /C27/C1/C1/C1 (5) d˙y/C30gx(x0;y0)dx/C27gy(x0;y0)dy/C27gxy(x0;y0)dxdy /C27/C1/C1/C1 (6) To first-order, this gives d dtdx dy/C)P/C)( /C30fx(x0;y0)fy(x0;y0) gx(x0;y0)gy(x0;y0)/C)P/C)( dx dy/C)P/C)( ; (7) where the 2 /C292MATRIX is called the STABILITY MATRIX . In general, given an n-DMAP x?/C30T(x);letx0be a FIXED POINT , so that T(x0)/C30x0: (8) Expand about the fixed point, T(x0/C27dx)/C30T(x0)/C27@T @xdx/C27O(dx)2 /C13T(x0)/C27dT; (9) so dT/C30@T @xdx/C13Adx: (10) The map can be transformed into the principal axisframe by finding the EIGENVECTORS and EIGENVALUES of the MATRIX A (A/C28lI)dx/C300; (11) so the DETERMINANT A/C28lI jj /C300: (12) The mapping is dx? princ/C30l1/C1/C1/C1 0 n:::n 0 /C1/C1/C1ln2 435: (13) When iterated a large number of times, dT? princ00 (14) only if R(li) jjB1 for i/C301, ..., nbut0/C12if any lijj> 1:Analysis of the EIGENVALUES (and EIGENVECTORS ) ofAtherefore characterizes the type of FIXED POINT . The condition for stability is R(li) jjB1 for i/C301, ..., n. See also FIXED POINT ,LYAPUNOV FUNCTION ,N ON- LINEAR STABILITY ,STABILITY MATRIX References Tabor, M. "Linear Stability Analysis." §1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20 /C1/1, 1989. Linear Transformation A linear transformation between two VECTOR SPACES V and W is a MAP T : V 0 W such that the following hold: 1. T(v1 /C27v2) /C30T(v1)T(v2) for any VECTORS v1and v2 in V, and 2. T(av) /C30 aT(v) for any SCALAR a:/ A linear transformation may not be INJECTIVE or ONTO . When V and W have the same DIMENSION ,itis possible for T to be invertible, meaning there exists a T /C281 such that TT /C281 /C30I : It is always the case that T(0) /C300: Also, a linear transformation always maps LINES to LINES (or to zero). nbsp The main example of a linear transformation is given by MATRIX MULTIPLICATION . Given an n /C29m MATRIX A; define /T(v) /C30Av/, where v is written as a COLUMN VECTOR (with m coordinates). For example, consider A /C3001 /C2813 402 435; (1) then T is a linear transformation from R 2 to R3 ; defined by, T(x; y) /C30(y;/C282x /C272y; x): (2) Another example is /T(x; y) /C30(1:4x /C28y; 0:8x)/. The homotopy from the identity transformation to T is illustrated above. When V and W are FINITE dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a BASIS for V andW. When V and W have an INNER PRODUCT , and their BASES , fv1 ;/C1/C1/C1; vm g and fw;/C1/C1/C1; wn g; are ORTHONOR- MAL, it is easy to write the corresponding matrix A /C30 (aij) : In particular, aij /C30 wi ; T(vj)/C(P/C(( : Note that when using the standard basis for Rnand Rm ; the jth column corresponds to the image of the jth standard basis vector. When V and W are INFINITE dimensional, then it is possible for a linear transformation to not be CON- TINUOUS . For example, let V be the space of poly- nomials in one variable, and T be the DERIVATIVE . Then Tx3ðÞ/C30nxn /C281 ; which is not CONTINUOUS because xn =n 0 0 while T(xn =n) does not converge. Linear 2-D transformations have a simple classifica- tion. Consider the 2-D linear transformation rx?1 /C30a11x1 /C27a12x2 (3) rx?2 /C30a21x1 /C27a22x2 : (4) Now rescale by defining l /C13x1 =x2 and l ?/C13x?1 =x?2 : Then the above equations become l ?/C30al /C27 b gl /C27 d (5) where ad /C28 bg "0 and a; b; g and d are defined in terms of the old constants. Solving for l gives l/C30dl?/C28b /C28gl?/C27a; (6) so the transformation is ONE-TO-ONE . To find the FIXED POINTS of the transformation, set l/C30l?to obtain gl2/C27(d/C28a)l/C28b/C300: (7) This gives two fixed points which may be distinct or coincident. The fixed points are classified as follows. variables type /(d/C28a)2/C274bg>0/HYPERBOLIC FIXED POINT /(d/C28a)2/C274bgB0/ELLIPTIC FIXED POINT /(d/C28a)2/C274bg/C300/PARABOLIC FIXED POINT See also BASIS (VECTOR SPACE ), ELLIPTIC FIXED POINT (MAP), GENERAL LINEAR GROUP ,HYPERBOLIC FIXED POINT (MAP), INVERTIBLE LINEAR MAP,INVOL- UTORY ,LINEAR OPERATOR ,M ATRIX ,M ATRIX MULTI- PLICATION ,PARABOLIC FIXED POINT ,VECTOR SPACE References Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 13 /C1/ 5, 1961. Linear Weighted Moment L-MOMENT Linearly Dependent Curves Two curves f and c satisfying f /C27 c /C300 are said to be linearly dependent. Similarly, n curves fi ; i /C30 1, ..., n are said to be linearly dependent if Xn i/C301fi /C300: See also BERTINI’S THEOREM ,STUDY’S THEOREM References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 32 /C1/4, 1959. Linearly Dependent Functions The n functions f1(x) ; f2(x) ; ..., fn(x) are linearly dependent if, for some c1 ; c2 ; ..., cn /C23R not all zero, cifi(x) /C300 (1) (where EINSTEIN SUMMATION is used) for all x in some interval I. If the functions are not linearly dependent, they are said to be linearly independent. Now, if the functions /C23Rn/C281 ; we can differentiate (1) up to n /C281 times. Therefore, linear dependence also requires cif ?i /C300 (2) cif ƒi /C300 (3) cif(n/C281) i /C300; (4) where the sums are over i /C301, ..., n. These equations have a nontrivial solution IFF the DETERMINANT f1 f2 /C1/C1/C1 fn f ?1 f ?2 /C1/C1/C1 f ?2 nn::: n f(n/C281) 1 f(n /C281) 2 /C1/C1/C1 f(n/C281) n/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300 ; (5) where the DETERMINANT is conventionally called the WRONSKIAN and is denoted W(f1 ; f2 ; ...; fn): If the WRONSKIAN "0 for any value c in the interval I, then the only solution possible for (2) is ci /C300(i /C30 1, ..., n), and the functions are linearly independent. If, on the other hand, W /C300 for a range, the functions are linearly dependent in the range. This is equivalent to stating that if the vectors V[f1(c)]; ..., V[fn(c)] defined byV[fi(x)] /C30fi(x) f ?i(x) f ƒi(x) n fn /C281 i(x)2 666643 77775(6) are linearly independent for at least one c /C23 I ; then the functions f i are linearly independent in I. References Sansone, G. "Linearly Independent Functions." §1.2 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 2/C1/, 1991. Linearly Dependent Sequences Sequences x(1) n;x(2)n;...,x(k) nare linearly independent if constants c1;c2;...,ck(not all zero) exist such that Xk i/C301cix(i) n/C300 forn/C300, 1, .... See also CASORATIAN References Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 229, 1995. Linearly Dependent Vectors nVECTORS X1;X2;...,Xnare linearly dependent IFF there exist SCALARS c1;c2;...,cn;not all zero, such that ciXi/C300; (1) where E INSTEIN SUMMATION is used and i/C301, ..., n.I f no such SCALARS exist, then the vectors are said to be linearly independent. In order to satisfy the CRITER- IONfor linear dependence, c1x11 x12 n xn12 6643 775/C27c 2x12 x22 n xn22 6643 775/C27/C1/C1/C1/C27c nx1n x2n n xnn2 6643 775/C300 0 n 02 6643 775(2) x 11x12 /C1/C1/C1 x1n x21x22 /C1/C1/C1 x2n nn:::n xn1xn2/C1/C1/C1 xnn2 6643 775c 1 c2 n cn2 6643 775/C300 0 n 02 6643 775: (3) In order for this MATRIX equation to have a nontrivial solution, the DETERMINANT must be 0, so the VECTORS are linearly dependent if x11x12 /C1/C1/C1 x1n x21x22 /C1/C1/C1 x2n nn:::n xn1xn2/C1/C1/C1 xnn2 6643 775/C300; (4) and linearly independent otherwise. Letpandqben-D VECTORS . Then the following three conditions are equivalent (Gray 1997). 1. p and q are linearly dependent. 2.p /C215 pp /C215 q q /C215 pq /C215 q/C()/C()/C()/C()/C()/C()/C()/C()/C300: / 3. The 2 /C29n MATRIXp qhi has rank less than two. References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 272 /C1/73, 1997. Linearly Independent Two or more functions, equations, or vectors f1 ; f2 ; ..., which are not linearly dependent, i.e., cannot be expressed in the form a1f1 /C27a2f2 /C27/C1/C1/C1/C27anfn /C300 with a1 ; a2 ; ... constants which are not all zero are said to be linearly independent. See also LINEARLY DEPENDENT CURVES ,LINEARLY DEPENDENT FUNCTIONS ,LINEARLY DEPENDENT VEC- TORS ,MAXIMALLY LINEARLY INDEPENDENT Linearly Ordered Set TOTAL ORDER Line at Infinity The straight line on which all POINTS AT INFINITY lie. The line at infinity is given in terms of TRILINEAR COORDINATES by aa /C27bb /C27c g /C300 ; which follows from the fact that a REAL TRIANGLE will have POSITIVE AREA , and therefore that 2D/C30a a /C27bb /C27c g > 0: Instead of the three reflected segments concurring for the ISOGONAL CONJUGATE of a point X on the CIRCUMCIRCLE of a TRIANGLE , they become parallel (and can be considered to meet at infinity). As X varies around the CIRCUMCIRCLE , X /C281 varies through a line called the line at infinity. Every line is PERPENDICULAR to the line at infinity. Poncelet was the first to systematically employ the line at infinity (Graustein 1930). See also POINT AT INFINITY References Lachlan, R. §10 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 6, 1893. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 30, 1930. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 141 /C1/42, 1991.Line Bisector The line bisecting a given LINE SEGMENT P1P2 can be constructed geometrically, as illustrated above. References Courant, R. and Robbins, H. "How to Bisect a Segment and Find the Center of a Circle with the Compass Alone." §3.4.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 145 /C1/46, 1996. Dixon, R. Mathographics. New York: Dover, p. 22, 1991. Line Bundle A line bundle is a special case of a VECTOR BUNDLE in which the fiber is either R; in the case of a real line bundle, or C ; in the case of a complex line bundle. See also MANIFOLD ,P RINCIPAL BUNDLE ,T RIVIAL BUNDLE ,VECTOR BUNDLE Line-Circle Intersection CIRCLE- LINE INTERSECTION Line Connectivity EDGE CONNECTIVITY Line Element Also known as the first FUNDAMENTAL FORM ds2 /C30gabdxadxb : In the principal axis frame for 3-D, ds2 /C30gaa(dxa)2 /C27gbb(dxb)2 /C27gcc(dxc)2 : At ORDINARY POINTS on a surface, the line element is positive definite. See also AREA ELEMENT ,F UNDAMENTAL FORMS , VOLUME ELEMENT Line Graph A LINE GRAPH L(G) (also called an interchange graph) of a graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge IFF the corresponding edges of G meet at one or both endpoints. In the three examples above, the original graphs are the COMPLETE GRAPHS K3 ; K4 ; and K5 :/ The line graph of a GRAPH with n nodes, e edges, and vertex degrees di contains n ?/C30e nodes and e ?/C301 2Xn i/C301d2 i /C28e edges (Skiena 1990, p. 137). The INCIDENCE MATRIX C of a graph and ADJACENCY MATRIX L of its line graph are related by L /C30CTC /C282I ; where I is the IDENTITY MATRIX (Skiena 1990, p. 136). A graph is a line graph IFF if does not contain any of the above graphs as SUBGRAPHS (van Rooij and Wilf 1965; Beineke 1968; Skiena 1990, p. 138). Of the nine, one has four nodes (the STAR GRAPH S4 /C30K1 ; 3) ; two have five nodes, and six have six nodes (including the WHEEL GRAPH W6) :/ The only CONNECTED GRAPH that is isomorphic to its line graph is a CYCLE GRAPH Cn (Skiena 1990, p. 137). Whitney (1932) showed that, with the exception of K3 and K1 ; 3 ; any two CONNECTED GRAPHS with iso- morphic line graphs are isomorphic (Skiena 1990, p. 138).The line graph of an E ULERIAN GRAPH is both Eulerian and H AMILTONIAN (Skiena 1990, p. 138). More information about cycles of line graphs is given by Harary and Nash-Williams (1965) and Chartrand(1968). See also T OTAL GRAPH References Beineke, L. W. "Derived Graphs and Digraphs." In Beitra ¨ge zur Graphentheorie (Ed. H. Sachs, H. Voss, and H. Walther). Leipzig, Germany: Teubner, pp. 17 /C1/3, 1968. Chartrand, G. "On Hamiltonian Line Graphs." Trans. Amer. Math. Soc. 134, 559/C1/66, 1968. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Harary, F. and Nash-Williams, C. J. A. "On Eulerian and Hamiltonian Graphs and Line Graphs." Canad. Math. Bull. 8, 701/C1/09, 1965. Saaty, T. L. and Kainen, P. C. "Line Graphs." §4/C1/inThe Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 108 /C1/12, 1986. Skiena, S. "Line Graph." §4.1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 128 and 135 /C1/39, 1990. van Rooij, A. and Wilf, H. "The Interchange Graph of a Finite Graph." Acta Math. Acad. Sci. Hungar. 16, 263/C1/69, 1965. Whitney, H. "Congruent Graphs and the Connectivity of Graphs." Amer. J. Math. 54, 150/C1/68, 1932. Line Integral The line integral of a VECTOR FIELD F(x) on a curve s is defined by gsF /C215ds/C30gb aF(s(t))/C215s?(t)dt; (1) where a /C215bdenotes a DOT PRODUCT . In Cartesian coordinates, the line integral can be written gsF /C215ds/C30gCF1dx/C27F2dy/C27F3dz; (2) where F/C13F1(x) F2(x) F3(x)2 435: (3) Forz COMPLEX andg:z/C30z(t) a path in the COMPLEX PLANE parameterized by t/C23[a;b]; ggfd z/C30gb af(z(t))z?(t)dt: (4) POINCARE ´’S THEOREM states that if 9/C29F/C300i na simply connected neighborhood U(x) of a point x, then in this neighborhood, Fis the GRADIENT of a SCALAR FIELD f(x); F(x)/C30/C289f(x) (5) forx/C23U(x);where 9is the gradient operator. Conse- quently, the GRADIENT THEOREM gives gsF /C215 ds /C30 f(x1) /C28 f(x2) (6) for any path s located completely within U(x); starting at x1 and ending at x2 :/ This means that if 9/C29F /C300 (i.e., F(x)isan IRROTA- TIONAL FIELD in some region), then the line integral is path-independent in this region. If desired, a Carte- sian path can therefore be chosen between starting and ending point to give g(x; y; z) (a; b; c)F1 dx /C27F2 dy /C27F3 dz /C30g(x ; b ; c) (a ; b ; c)F1 dx /C27g(x; y ; c) (x ; b ; c)F2 dy /C27g(x; y ; z) (x ; y; c)F3 dz : (7) If 9 /C215 F /C300 (i.e., F(x)isa DIVERGENCELESS FIELD , a.k.a. SOLENOIDAL FIELD ), then there exists a VECTOR FIELD A such that F /C309/C29A ; (8) where A is uniquely determined up to a gradient field (and which can be chosen so that /9 /C215 A /C300/). See also CONSERVATIVE FIELD,CONTOUR INTEGRAL , GRADIENT THEOREM ,IRROTATIONAL FIELD ,P ATH INTEGRAL ,POINCARE ´ ’S THEOREM References Krantz, S. G. "The Complex Line Integral." §2.1.6 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, p. 22, 1999. Line-Line Intersection The INTERSECTION of two LINES L1 and L2 in 2-D with, L1containing the points (x1 ; y1) and (x2 ; y2) ; and L2containing the points (x3 ; y3) and (x4 ; y4) ; is given by x /C30x1y1 x2y2/C()/C()/C()/C()/C()/C()/C()/C()x 11 x21/C()/C()/C()/C()/C()/C()/C()/C() x 3y3 x4y4/C()/C()/C()/C()/C()/C()/C()/C()x 31 x41/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C() x 11 x21/C()/C()/C()/C()/C()/C()/C()/C()y 11 y21/C()/C()/C()/C()/C()/C()/C()/C() x 31 x41/C()/C()/C()/C()/C()/C()/C()/C()y 31 y41/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C30x1y1 x2y2/C()/C()/C()/C()/C()/C()/C()/C()x 1 /C28 x2 x3y3 x4y4/C()/C()/C()/C()/C()/C()/C()/C()x 3 /C28 x4/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C() x 1 /C28 x2y1 /C28 y2 x3 /C28 x4y3 /C28 y4/C()/C()/C()/C()/C()/C()/C()/C()(1) y /C30x1y1 x2y2/C()/C()/C()/C()/C()/C()/C()/C()y 11 y21/C()/C()/C()/C()/C()/C()/C()/C() x3y3 x4y4/C()/C()/C()/C()/C()/C()/C()/C()y 31 y41/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C() x 11 x21/C()/C()/C()/C()/C()/C()/C()/C()y 11 y21/C()/C()/C()/C()/C()/C()/C()/C() x 31 x41/C()/C()/C()/C()/C()/C()/C()/C()y 31 y41/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C30 x1y1 x2y2/C()/C()/C()/C()/C()/C()/C()/C()y 1 /C28 y2 x3y3 x4y4/C()/C()/C()/C()/C()/C()/C()/C()y 3 /C28 y4/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C() x 1 /C28 x2y1 /C28 y2 x3 /C28 x4y3 /C28 y4/C()/C()/C()/C()/C()/C()/C()/C(): (2) In 3-D, let the two lines pass through points given by the vectors ( /p1 ; q1) and (/p2 ; q2) and define v1 /C30q1 /C28 p1 q1 /C28 p1 jj (3) v2 /C30q2 /C28 p2 q2 /C28 p2 jj (4) v12 /C30v1 /C29v2 (5) s1 /C30det(p2 /C28p1v2v12) (6) s2/C30det(p2/C28p1v1v12): (7) Then the point of intersection pof the two lines is given by p/C301 2(p1/C27v1s1/C27p2/C27v2s2) (8) (Glassner). See also CONCUR ,CONCURRENT ,INTERSECTION ,LINE, LINE-PLANE INTERSECTION References Glassner, A. S. (Ed.). Graphics Gems. Line Line Picking POINT- POINT DISTANCE–1- D Line of Curvature A curve on a surface whose tangents are always in the direction of PRINCIPAL CURVATURE . The equation of the lines of curvature can be written g11 g12 g22 b11 b12 b22 du2/C28du dv dv2/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300; where g and b are the COEFFICIENTS of the first and second FUNDAMENTAL FORMS . See also DUPIN’S THEOREM ,FUNDAMENTAL FORMS , PRINCIPAL CURVATURES Line-Plane Intersection The PLANE determined by the points x1 ; x2 ; and x3 and the LINE passing through the points x4and x5 intersect in a point which can be determined by solving the four simultaneous equations xyz 1 x1y1z11 x2y2z21 x3y3z31/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C300 (1) x /C30x 4 /C27(x4 /C28x5)t (2) y /C30y4 /C27(y4 /C28y5)t (3) z /C30z4 /C27(z4 /C28z5)t (4) for x, y, z, and t, giving t /C301111 x1x2x3x4 y1y2y3y4 z1z2z3z4/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C() 111 0 x 1x2x3x5 /C28 x4 y1y2y3y5 /C28 y4 z1z2z3z5 /C28 z4/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C(): (5) This value can then be plugged back in to (2), (3), and (4) to give the point of intersection /(x; y; z)/. See also LINE,LINE-LINE INTERSECTION ,PLANE Line Segment A closed interval corresponding to a FINITE portion of an infinite LINE. Line segments are generally labeledwith two letters corresponding to their endpoints, say A and B, and then written AB. The length of the line segment is indicated with an overbar, so the length of the line segment AB would be written AB :/ Curiously, the number of points in a line segment (ALEPH-1 ) is equal to that in an entire 1-D SPACE (a LINE), and also to the number of points in an n-D SPACE , as first recognized by Georg Cantor. See also ALEPH-1 ,C OLLINEAR ,C ONTINUUM ,L INE, RANGE (LINE SEGMENT ), RAY References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 14 /C1/6, 1893. Line Space LIOUVILLE SPACE L-Infinity-Norm A VECTOR NORM defined for a VECTOR x /C30x1 x2 n xn2 6643 775; with COMPLEX entries by xkk/C12/C30max i½xi ½: The vector norm ½x½/C12is implemented as Vector- Norm [m, Infinity] in the Mathematica add-on pack- age LinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also L1-NORM, L2-NORM,VECTOR NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1114 /C1/125, 2000. L-Infinity-Space The SPACE called L/C12 (ell-infinity) generalizes the LP- SPACES to p /C30/C12: No integration is used to define them, and instead, the norm on L/C12 is given by the ESSENTIAL SUPREMUM . More precisely, fkk/C12/C30ess sup½f ½ is the norm which makes L /C12 aB ANACH SPACE .Itis the space of all essentially bounded functions. The space of bounded continuous functions is not DENSE inL/C12:/ See also BANACH SPACE ,COMPLETION ,DENSE ,ESSEN- TIAL SUPREMUM , LP-SPACE , L2-SPACE ,M EASURE , MEASURABLE FUNCTION ,MEASURE SPACE Link Formally, a link is one or more disjointly embedded CIRCLES in 3-space. More informally, a link is an assembly of KNOTS with mutual entanglements. Kuperberg (1994) has shown that a nontrivial KNOT or link in R3 has four COLLINEAR points (Eppstein). Doll and Hoste (1991) list POLYNOMIALS for oriented links of nine or fewer crossings. A listing of the first few simple links follows, ar- ranged by CROSSING NUMBER . The numbers of non- trivial 2-component links of 0, 1, 2, ... crossings are 1, 0, 1, 0, 1, 1, 3, 8, 16, 61, ... (Sloane’s A048952). The numbers of nontrivial 3-component links of 6, 7, ... crossings are 3, 1, 10, 21, ... (Sloane’s A048953). The number of nontrivial 4-component links of 8, 9, ... crossings are 3, 1, .... 00 /C1/2 /C1/102/C1/2 /C1/104/C1/2 /C1/105/C1/2 /C1/106/C1/2 /C1/106/C1/2 /C1/206/C1/2 /C1/307/C1/2 /C1/ 107/C1/2 /C1/207/C1/2 /C1/307/C1/2 /C1/407/C1/2 /C1/507/C1/2 /C1/607/C1/2 /C1/707/C1/2 /C1/808/C1/ 2 /C1/108/C1/2 /C1/208/C1/2 /C1/308/C1/2 /C1/408/C1/2 /C1/508/C1/2 /C1/608/C1/2 /C1/708/C1/2 /C1/8 08 /C1/2 /C1/908/C1/2 /C1/008/C1/2 /C1/108/C1/2 /C1/208/C1/2 /C1/308/C1/2 /C1/408/C1/2 /C1/508/C1/2 /C1/ 609/C1/2 /C1/109/C1/2 /C1/209/C1/2 /C1/309/C1/2 /C1/409/C1/2 /C1/509/C1/2 /C1/609/C1/2 /C1/709/C1/ 2 /C1/809/C1/2 /C1/909/C1/2 /C1/009/C1/2 /C1/109/C1/2 /C1/209/C1/2 /C1/309/C1/2 /C1/409/C1/2 /C1/5 09 /C1/2 /C1/609/C1/2 /C1/709/C1/2 /C1/809/C1/2 /C1/909/C1/2 /C1/009/C1/2 /C1/109/C1/2 /C1/209/C1/2 /C1/ 309/C1/2 /C1/409/C1/2 /C1/509/C1/2 /C1/609/C1/2 /C1/709/C1/2 /C1/809/C1/2 /C1/909/C1/2 /C1/009/C1/ 2 /C1/109/C1/2 /C1/209/C1/2 /C1/309/C1/2 /C1/409/C1/2 /C1/509/C1/2 /C1/609/C1/2 /C1/709/C1/2 /C1/8 09 /C1/2 /C1/909/C1/2 /C1/009/C1/2 /C1/109/C1/2 /C1/209/C1/2 /C1/309/C1/2 /C1/409/C1/2 /C1/509/C1/2 /C1/ 609/C1/2 /C1/709/C1/2 /C1/809/C1/2 /C1/909/C1/2 /C1/009/C1/2 /C1/109/C1/2 /C1/209/C1/2 /C1/309/C1/ 2 /C1/409/C1/2 /C1/509/C1/2 /C1/609/C1/2 /C1/709/C1/2 /C1/809/C1/2 /C1/909/C1/2 /C1/009/C1/2 /C1/1 06 /C1/3 /C1/106/C1/3 /C1/206/C1/3 /C1/307/C1/3 /C1/108/C1/3 /C1/108/C1/3 /C1/208/C1/3 /C1/308/C1/3 /C1/ 408/C1/3 /C1/508/C1/3 /C1/608/C1/3 /C1/708/C1/3 /C1/808/C1/3 /C1/908/C1/3 /C1/009/C1/3 /C1/109/C1/ 3 /C1/209/C1/3 /C1/309/C1/3 /C1/409/C1/3 /C1/509/C1/3 /C1/609/C1/3 /C1/709/C1/3 /C1/809/C1/3 /C1/9 09 /C1/3 /C1/009/C1/3 /C1/109/C1/3 /C1/209/C1/3 /C1/309/C1/3 /C1/409/C1/3 /C1/509/C1/3 /C1/609/C1/3 /C1/ 709/C1/3 /C1/809/C1/3 /C1/909/C1/3 /C1/009/C1/3 /C1/108/C1/4 /C1/108/C1/4 /C1/208/C1/4 /C1/309/C1/ 4 /C1/1 See also ANDREWS- CURTIS LINK,BORROMEAN RINGS, BRUNNIAN LINK,HOPF LINK,KNOT,ORIENTED LINK, WHITEHEAD LINK References Cerf, C. "Atlas of Oriented Knots and Links." Topology Atlas Invited Contributions 3, No. 2, 1 /C1/2, 1998. http://at.yor- ku.ca/t/a/i/c/31.htm. Doll, H. and Hoste, J. "A Tabulation of Oriented Links." Math. Comput. 57, 747 /C1/61, 1991. Eppstein, D. "Colinear Points on Knots." http://www.ics.u- ci.edu/~eppstein/junkyard/knot-colinear.html. Kuperberg, G. "Quadrisecants of Knots and Links." J. Knot Theory Ramifications 3,41/C1/0, 1994. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976. Sloane, N. J. A. Sequences A048952 and A048953 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Weisstein, E. W. "Knots." MATHEMATICA NOTEBOOK KNOTS.M . Link (Simplicial Complex) The setSt v /C28St v; where St v is a CLOSED STAR and St v is a STAR , is called the link of v in a SIMPLICIALCOMPLEX K and is denoted Lkv (Munkres 1993, p. 11). See also CLOSED STAR,SIMPLICIAL COMPLEX ,STAR References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Link Complement KNOT COMPLEMENT Link Diagram A planar diagram depicting a LINK (or KNOT )asa sequence of segments with gaps representing under- crossings and solid lines overcrossings. In such a diagram, only two segments should ever cross at a single point. Link diagrams for the TREFOIL KNOT and FIGURE-OF-EIGHT KNOT are illustrated above. Link Invariant A link invariant is a function from the set of all LINKS to any other set such that the function does not change as the link is changed (up to isotopy). In other words, a link invariant always assigns the same value to equivalent links (although different knots may have the same link invariant). When the link has a single component and therefore generates to a KNOT , the invariant is called a KNOT INVARIANT . See also KNOT,KNOT INVARIANT ,LINK Linkage Sylvester, Kempe and Cayley developed the geometry associated with the theory of linkages in the 1870s. Kempe proved that every finite segment of an algebraic curve can be generated by a linkage in the manner of W ATT’S CURVE . See also HART’S INVERSOR ,KEMPE LINKAGE ,PANTO- GRAPH ,P EAUCELLIER INVERSOR ,SARRUS LINKAGE , WATT’S PARALLELOGRAM References Chuan, J. C. "Machine." http://www.math.ntnu.edu.tw/ ~jcchuan/demo/gear/machine.html. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Kempe, A. B. How to Draw a Straight Line: A Lecture on Linkages. 1977. King, H. C. Configuration Spaces of Linkages in Rn 23 Nov 1998. http://xxx.lanl.gov/abs/math.GT/9811138/. King, H. C. Semiconfiguration Spaces of Planar Linkages. 20 Oct 1998. http://xxx.lanl.gov/abs/math.GT/9810130/. McCarthy, J. M. "Geometric Design of Linkages." http:// www.eng.uci.edu/~mccarthy/. Rademacher, H. and Toeplitz, O. "Producing Rectilinear Motion by Means of Linkages." §18 in The Enjoyment of Mathematics: Selections from Mathematics for the Ama- teur. Princeton, NJ: Princeton University Press, pp. 119 /C1/ 29, 1957. Linking Number A LINK INVARIANT defined for a two-component oriented LINK as the sum of /C271 crossings and /C281 crossing over all crossings between the two links divided by 2. For components a and b; Lk( a; b) /C131 2X p /C23 a /C17be(p); where a/C17b is the set of crossings of a with b; and e(p) is the sign of the crossing. The linking number of a splittable two-component link is always 0. See also CALUGAREANU THEOREM ,GAUSS INTEGRAL , JONES POLYNOMIAL ,LINK,TWIST ,W RITHE References Pohl, W. F. "The Self-Linking Number of a Closed Space Curve." J. Math. Mech. 17, 975/C1/85, 1968. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 132 /C1/33, 1976. Links Curve The curve given by the Cartesian equation (x2/C27y2/C283x)2/C304x2(2/C28x): The origin of the curve is a TACNODE . References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Linnik’s Constant The constant Lin L INNIK’S THEOREM . Heath-Brown (1992) has shown that L55:5;and Schinzel, Sier- pinski, and Kanold (Ribenboim 1989) have conjec- tured that L/C302.References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/linnik/linnik.html. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 13, 1994. Heath-Brown, D. R. "Zero-Free Regions for Dirichlet L- Functions and the Least Prime in an Arithmetic Progres- sion." Proc. London Math. Soc. 64, 265/C1/38, 1992. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, 1989. Linnik’s Theorem Letp(d;a) be the smallest PRIME in the arithmetic progression fa/C27kdgforkanINTEGER >0:Let p(d)/C13max p(d;a) such that 1 5aBdand ( a;d)/C301:Then there exists a d0]2 and an L/C211 such that p(d)BdLfor all d>d0: Lis known as L INNIK’S CONSTANT . References Linnik, U. V. "On the Least Prime in an Arithmetic Progression. I. The Basic Theorem." Mat. Sbornik N. S. 15 (57) , 139/C1/78, 1944. Linnik, U. V. "On the Least Prime in an Arithmetic Progression. II. The Deuring-Heilbronn Phenomenon" Mat. Sbornik N. S. 15 (57) , 347/C1/68, 1944. Lin’s Method An ALGORITHM for finding ROOTS for QUARTIC EQUA- TIONS with COMPLEX ROOTS . References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 198 /C1/99, 1990. Lin-Tsien Equation The PARTIAL DIFFERENTIAL EQUATION 2utx/C27uxuxx/C28uyy/C300: References Ames, W. F. and Nucci, W. N. "Analysis of Fluid Equations by Group Methods." J. Eng. Mech. 20, 181/C1/87, 1985. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Linus Sequence The sequence composed of 1s and 2s obtained by starting with the number 1, and picking subsequent elements to avoid repeating the longest possible substring. The first few terms are 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, ... (Sloane’s A006345). The SALLY SEQUENCE gives the length of the run that was avoided. See also SALLY SEQUENCE References Sloane, N. J. A. Sequences A006345/M0126 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0126 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Liouville Function The function l(n)/C30(/C281)r(n); (1) where r(n) is the number of not necessarily distinct PRIME FACTORS ofn, with r(1)/C300:The first few values of l(n) are 1, /C281,/C281, 1,/C281, 1,/C281,/C281, 1, 1,/C281,/C281, .... The Liouville function is connectedwith the R IEMANN ZETA FUNCTION by the equation z(2s) z(s)/C30X/C12 n/C301l(n) ns(2) (Lehman 1960). The CONJECTURE that the SUMMATORY FUNCTION L(n)/C13Xn k/C301l(n) (3) satisfies L(n)50 for n]2 is called the P O´LYA CON- JECTURE and has been proved to be false. The first n for which L(n)/C300 are for n/C302, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Sloane’s A028488), and n/C30906150257 is, in fact, the first counterexample to the P O´LYA CONJECTURE (Tanaka 1980). However, it is unknown if L(x) changes sign infinitely often (Tanaka 1980). The first few values of L(n) are 1, 0, /C281, 0,/C281, 0,/C281,/C282,/C281, 0,/C281,/C282,/C283,/C282,/C281, 0,/C281,/C282,/C283,/C284, ... (Sloane’s A002819). L(n) also satisfies Xx n/C301Lx n ! /C30ffiffiffixp/Co/C$ ; (4) where xbcis the FLOOR FUNCTION (Lehman 1960). Lehman (1960) also gives the formulas L(x)/C30Xx=w m/C301m(m) /C2ffiffiffiffiffi x ms$% /C28Xv/C281 k/C301l(k)x km$% /C28x mv$% ! () /C28Xx=v l/C30x=w/C281Lx l !Xx=w m½l m/C301m(m) (5) and L(x)/C30Xg k/C301Mx k2 ! /C27Xx=g2 l/C301m(l)ffiffiffi x ls$% /C28Mx g2 ! /C2ffiffiffiffiffi x g2s$% ; (6) where k, l, and m are variables ranging over the POSITIVE INTEGERS , m(n) is the M O¨ BIUS FUNCTION , M(x)isM ERTENS FUNCTION , and v, w, and x are POSITIVE real numbers with v Bw Bx:/ See also PO´ LYA CONJECTURE ,PRIME FACTORS ,RIE- MANN ZETA FUNCTION References Fawaz, A. Y. "The Explicit Formula for L0(x) :/" Proc. London Math. Soc. 1,86/C1/03, 1951. Lehman, R. S. "On Liouville’s Function." Math. Comput. 14, 311 /C1/20, 1960. Sloane, N. J. A. Sequences A002819/M0042 and A028488 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Tanaka, M. "A Numerical Investigation on Cumulative Sum of the Liouville Function." Tokyo J. Math. 3, 187 /C1/89, 1980. Liouville Measure Y idpi dqi ; where piand qiare momenta and positions of particles. See also LIOUVILLE’S PHASE SPACE THEOREM ,PHASE SPACE Liouville Number A Liouville number is a TRANSCENDENTAL NUMBER which has very close RATIONAL NUMBER approxima- tions. An IRRATIONAL NUMBER b is a Liouville number if, for any n, there exist an infinite number of pairs of INTEGERS p and q such that 0 B b/C28p q/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()B 1 qn : LIOUVILLE’S CONSTANT is an example of a Liouville number. Mahler (1953) proved that p is not a Liouville number. See also LIOUVILLE’S CONSTANT ,LIOUVILLE’S APPROX- IMATION THEOREM ,ROTH’S THEOREM ,TRANSCENDEN- TAL NUMBER References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 147, 1997. Mahler, K. "On the Approximation of p:/" Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes Math. 15,30/C1/2, 1953. Liouville Polynomial Identity 6(x2 1 /C27x22 /C27x23 /C27x24) /C30(x1 /C27x2)4 /C27(x1 /C27x3)4 /C27(x2 /C27x3)4 /C27(x1 /C27x4)4 /C27(x2 /C27x4)4 /C27(x3 /C27x4)4 /C27(x1 /C28x2)4/C27(x1 /C28x3)4 /C27(x2 /C28x3)4 /C27(x1 /C28x4)4 /C27(x2 /C28x4)4 /C27(x3 /C28x4)4 : This is proven in Rademacher and Toeplitz (1957). See also WARING’S PROBLEM References Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 55 /C1/6, 1957. Liouville-Roth Constant IRRATIONALITY MEASURE Liouville’s Approximation Theorem For any ALGEBRAIC NUMBER x of degree n ]2 ; a RATIONAL approximation x /C30p =q must satisfy x /C28p q/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()> 1 qn/C271 for sufficiently large q. Writing r /C13n /C271 leads to the definition of the IRRATIONALITY MEASURE of a given number. Apostol (1997) states the theorem in the slightly modified form that for all integers p and q with q /C210, there exists a positive constant C(x) depending only on x such that x /C28p q/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()> C(x) qn: See also DIRICHLET’S APPROXIMATION THEOREM , IRRATIONALITY MEASURE ,LAGRANGE NUMBER (RA- TIONAL APPROXIMATION ), LIOUVILLE’S CONSTANT , LIOUVILLE NUMBER ,M ARKOV NUMBER ,ROTH’S THE- OREM ,THUE- SIEGEL- ROTH THEOREM References Apostol, T. M. "Liouville’s Approximation Theorem." §7.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 146 /C1/48, 1997. Courant, R. and Robbins, H. "Liouville’s Theorem and the Construction of Transcendental Numbers." §2.6.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 104 /C1/07, 1996. Liouville’s Boundedness Theorem A bounded ENTIRE FUNCTION in the COMPLEX PLANE C is constant. The FUNDAMENTAL THEOREM OF ALGEBRA follows as a simple corollary. See also COMPLEX PLANE ,ENTIRE FUNCTION ,FUNDA- MENTAL THEOREM OF ALGEBRA References Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, p. 74, 1996. Krantz, S. G. "Entire Functions and Liouville’s Theorem." §3.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 31 /C1/2, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 381 /C1/82, 1953. Liouville’s Conformality Theorem In SPACE , the only CONFORMAL MAPPINGS are inver- sions, SIMILARITY TRANSFORMATIONS , and CONGRU- ENCE TRANSFORMATIONS . Or, restated, every ANGLE - preserving transformation is a SPHERE -preserving transformation. See also CONFORMAL MAP Liouville’s Conic Theorem The lengths of the TANGENTS from a point P to a CONIC C are proportional to the CUBE ROOTS of the RADII OF CURVATURE of C at the corresponding points of contact. See also CONIC SECTION Liouville’s Constant L /C13X/C12 n/C30110/C28n! /C300 :110001000000000000000001 .. . (Sloane’s A012245). Liouville’s constant is a decimal fraction with a 1 in each decimal place corresponding to a FACTORIAL n!; and ZEROS everywhere else. Liouville (1844) constructed an infinite class of TRANSCENDENTAL NUMBERS using CONTINUED FRAC- TIONS , but the above number was the first decimal constant to be proven TRANSCENDENTAL (Liouville 1850). However, Cantor subsequently proved that "almost all" real numbers are in fact transcendental. Liouville’s constant nearly satisfies 10x6 /C2875x3 /C28190x /C2721 /C300; but plugging x /C30L into this equation gives /C280:0000000059... instead of 0. See also LIOUVILLE NUMBER References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 147, 1997. Conway, J. H. and Guy, R. K. "Liouville’s Number." In The Book of Numbers. New York: Springer-Verlag, pp. 239 /C1/ 41, 1996. Courant, R. and Robbins, H. "Liouville’s Theorem and the Construction of Transcendental Numbers." §2.6.2 in What is Mathematics?: An Elementary Approach to Ideas andMethods, 2nd ed. Oxford, England: Oxford University Press, pp. 104 /C1/07, 1996. Liouville, J. "Sur des classes tre`se´tendues de quantite ´s dont la valeur n’est ni alge´brique, ni meˆme reductible a` des irrationelles alge´briques." C. R. Acad. Sci. Paris 18, 883 /C1/ 85 and 993 /C1/95, 1844. Liouville, J. "Sur des classes tre`s-e´tendues de quantite ´s dont la valeur n’est ni alge´brique, ni meˆme re´ductible a` des irrationelles alge´briques." J. Math. pures appl. 15, 133 /C1/ 42, 1850. Sloane, N. J. A. Sequences A012245 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 26, 1986. Liouville’s Elliptic Function Theorem An ELLIPTIC FUNCTION with no POLES in a FUNDA- MENTAL CELL is a constant. See also ELLIPTIC FUNCTION ,FUNDAMENTAL CELL, POLE References Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 431, 1990. Liouville’s Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27g(y)y?2 /C27f(x)y?/C300 (1) is called Liouville’s equation (Goldstein and Braun 1973; Zwillinger 1997, p. 124), as are the PARTIAL DIFFERENTIAL EQUATIONS Xn i/C301uxixi/C27e lu /C300 (2) (Matsumo 1987; Zwillinger 1997, p. 133) and uxt /C30e hu (3) (Calogero and Degasperis 1982, p. 60; Zwillinger 1997, p. 133). See also KLEIN- GORDON EQUATION References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 60, 1982. Goldstein, M. E. and Braun, W. H. Advanced Methods for the Solution of Differential Equations. NASA SP-316. Washington, DC: U.S. Government Printing Office,p. 98, 1973. Matsumo, Y. "Exact Solution for the Nonlinear Klein- Gordon and Liouville Equations in Four-DimensionalEuclidean Space." J. Math. Phys. 28, 2317 /C1 /322, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 124 and 133, 1997. Liouville Space Also known as LINE SPACE or "extended" HILBERT SPACE , it is the SET DIRECT PRODUCT of two HILBERT SPACES . See also HILBERT SPACE ,SET DIRECT PRODUCT Liouville’s Phase Space Theorem States that for a nondissipative HAMILTONIAN SYS- TEM, phase space density (the AREA between phase space contours) is constant. This requires that, given a small time increment dt, q1 /C30q(t0 /C27dt) /C30q0 /C27@H(q0 ; p0 ; t) @p0dt /C27O(dt2) (1) p1 /C13p(t0 /C27dt) /C30p0 /C28@H(q0 ; p0 ; t) @q0dt /C27O(dt2) ; (2) the JACOBIAN be equal to one: @(q1 ; p1) @(q0 ; p0) /C30@q1 @q0@p1 @q0 @q1 @p0@p1 @p0/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C() /C301 /C27 @2H @q0 @p0dt /C28@2H @q2 0dt @2H @p20dt 1 /C28@2H @q0 @p0dt/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C27O(dt 2) /C301 /C27O(dt2) : (3) Expressed in another form, the integral of the LIOU- VILLE MEASURE , YN i/C301g dpi dqi ; (4) is a constant of motion. SYMPLECTIC MAPS of HAMIL- TONIAN SYSTEMS must therefore be AREA preserving (and have DETERMINANTS equal to 1). See also LIOUVILLE MEASURE ,PHASE SPACE References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Liouville’s Principle Let F be a differential field with constant field K. For f /C23 F ; suppose that the equation g?/C30f (i.e., g /C30f f) has a solution g /C23 G ; where G is an elementary extension of F having the same constant FIELD K. Then there exist v0 ; v1 ; ..., vm /C23 F and constants c1 ; ..., cm /C23 K such that f /C30v?0 /C27Xm i /C301civ ?i vi;In other words, such that g f /C30v0 /C27Xm i /C301ciln vi : See also ELEMENTARY FUNCTION References Geddes, K. O.; Czapor, S. R.; and Labahn, G. "Liouville’s Principle." §12.4 in Algorithms for Computer Algebra. Amsterdam, Netherlands: Kluwer, pp. 523 /C1/29, 1992. Liouville’s Sphere-Preserving Theorem LIOUVILLE’S CONFORMALITY THEOREM Liouvillian Number A member of the smallest algebraically closed SUB- FIELD L of C which is CLOSED under the exponentia- tion and logarithm operations. See also ELEMENTARY NUMBER References Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440 /C1/48, 1999. Richardson, D. "The Elementary Constant Problem." In Proc. Internat. Symp. on Symbolic and Algebraic Compu- tation, Berkeley, July 27 /C1/9, 1992 (Ed. P. S. Wang). ACM Press, 1992. Ritt, J. Integration in Finite Terms: Liouville’s Theory of Elementary Models. New York: Columbia University Press, 1948. Lipschitz Condition A function f(x) satisfies the Lipschitz condition of order a at x /C300if ½f(h) /C28f(0) ½5B ½h½ b for all ½h½B e; where B and b are independent of h, b>0;andais an UPPER BOUND for all bfor which a finite Bexists. See also HILLAM’S THEOREM ,H O¨ LDER CONDITION , LIPSCHITZ FUNCTION References Jeffreys, H. and Jeffreys, B. S. "The Lipschitz Condition." §1.15 in Methods of Mathematical Physics, 3rd ed. Cam- bridge, England: Cambridge University Press, p. 53, 1988. Lipschitz Function A function fsuch that ½f(x)/C28f(y)½5C½x/C28y½ for all xandy, where Cis a constant independent of x andy, is called a Lipschitz function. For example, any function with a bounded first derivative must be Lipschitz. See also LIPSCHITZ CONDITION References Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103, 369 /C1/76, 1996. Lipschitz’s Integral g/C12 0e/C28axJ0(bx) dx /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2p ; where J0(z) is the zeroth order BESSEL FUNCTION OF THE FIRST KIND . References Bowman, F. Introduction to Bessel Functions. New York: Dover, p. 58, 1958. Lissajous Curve Lissajous curves are the family of curves described by the PARAMETRIC EQUATIONS x(t) /C30A cos(vxt /C28 dx) (1) y(t) /C30B cos(vyt /C28 dy) ;: (2) sometimes also written in the form x(t) /C30a sin(nt /C27c) (3) y(t) /C30b sin t: (4) They are sometimes known as BOWDITCH CURVES after Nathaniel Bowditch, who studied them in 1815. They were studied in more detail (indepen- dently) by Jules-Antoine Lissajous in 1857 (MacTutor Archive). Lissajous curves have applications in phy- sics, astronomy, and other sciences. The curves close IFF vx =vy is RATIONAL . Lissajous curves are a special case of the HARMONO- GRAPH with damping constants b1 /C30 b2 /C300:/ See also HARMONOGRAPH References Cundy, H. and Rollett, A. "Lissajous’s Figures." §5.5.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 242 /C1/44, 1989.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 70 /C1/1, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178 /C1/79 and 181 /C1/83, 1972. MacTutor History of Mathematics Archive. "Lissajous Curves." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Lissajous.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 142, 1991. Lissajous Figure LISSAJOUS CURVE List An DATA STRUCTURE consisting of an ordered SET of elements, each of which may be a number, another list, etc. A list is usually denoted (/a1 ; a2 ; ..., an)or fa1 ; a2 ; ... ; an g; and may also be interpreted as a VECTOR . Multiplicity matters in a list, so (1, 1, 2) and (1, 2) are not equivalent. See also MULTISET ,Q UEUE ,SET,STACK ,STRING , VECTOR Little Moment Problem MOMENT PROBLEM Lituus An A RCHIMEDEAN SPIRAL with m/C30/C28 2, having polar equation r2u/C30a2: Lituus means a "crook," in the sense of a bishop’s crosier. The lituus curve originated with Cotes in1722. Maclaurin used the term lituus in his bookHarmonia Mensurarum in 1722 (MacTutor Archive). The lituus is the locus of the point Pmoving such that the AREA of a circular SECTOR remains constant. References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 221, 1987. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 91, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 and 188, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967. MacTutor History of Mathematics Archive. "Lituus." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Li- tuus.html. Lituus Inverse Curve The INVERSE CURVE of the LITUUS is an ARCHIMEDEAN SPIRAL with m /C302, which is FERMAT’S SPIRAL . See also ARCHIMEDEAN SPIRAL ,F ERMAT’S SPIRAL , LITUUS LLL Algorithm A LATTICE REDUCTION algorithm, named after dis- coverers Lenstra, Lenstra, and Lovasz (1982), that produces a lattice basis of "short" vectors. It was noticed by Lenstra et al. (1928) that the algorithm could be used to obtain factors of univariate poly- nomials, which amounts to the determination of INTEGER RELATIONS . However, this application of the algorithm, which later came to be one of its primary applications, was not stressed in the original paper. The Mathematica command LatticeReduce [ma- trix] implements the LLL algorithm to perform LATTICE REDUCTION . Mathematica ’s implementation requires the input to consist of rational numbers, so Rationalize may need to be called first. More recently, other algorithms such as PSLQ, which can be significant faster than LLL, have been devel- oped for finding INTEGER RELATIONS . PSLQ achieves its performance because of clever techniques that allow machine arithmetic to be used at many inter- mediate steps, whereas LLL must use moderate precision (although generally not as much as the HJLS ALGORITHM ). See also FERGUSON- FORCADE ALGORITHM ,HJLS ALGORITHM ,INTEGER RELATION ,L ATTICE REDUC- TION , PSLQ ALGORITHM , PSOS ALGORITHM References Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899 /C1/09, 1999. Borwein, J. M. and Lisonek, P. "Applications of Integer Relation Algorithms." To appear in Disc. Math. http:// www.cecm.sfu.ca/preprints/1997pp.html. Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993. Lenstra, A. K.; Lenstra, H. W.; and Lovasz, L. "Factoring Polynomials with Rational Coefficients." Math. Ann. 261, 515 /C1/34, 1982. Matthews, K. "Keith Matthews’ LLL Page." http:// www.maths.uq.edu.au/~krm/lll.html. Mignotte, M. Mathematics for Computer Algebra. New York: Springer-Verlag, 1991. L-Moment A type of statistic which can be useful for determining asymmetry and tailedness of a population. See also MOMENT ,ORDER STATISTICReferences Hosking, J. R. M. "L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statis- tics." J. Roy. Stat. Soc. B 52, 105 /C1/24, 1990. Ln The LOGARITHM to BASE E, also called the NATURAL LOGARITHM , is denoted ln ; i.e., ln x /C13loge x: See also BASE (LOGARITHM ), E,L G,L OGARITHM , NAPIERIAN LOGARITHM ,NATURAL LOGARITHM Lobachevsky-Bolyai-Gauss Geometry HYPERBOLIC GEOMETRY Lobachevsky’s Formula Given a point P and a LINE AB, draw the PERPENDI- CULAR through P and call it PC. Let PD be any other line from P which meets CB in D.Ina HYPERBOLIC GEOMETRY ,asD moves off to infinity along CB, then the line PD approaches the limiting line PE, which is said to be parallel to CB at P. The angle /C218CPE which PE makes with PC is then called the ANGLE OF PARALLELISM for perpendicular distance x, and is given by Y (x) /C302 tan/C281(e /C28x) ; which is called Lobachevsky’s formula. See also ANGLE OF PARALLELISM ,HYPERBOLIC GEO- METRY References Manning, H. P. Introductory Non-Euclidean Geometry. New York: Dover, p. 58, 1963. Lobatto Quadrature Also called R ADAU QUADRATURE (Chandrasekhar 1960). A G AUSSIAN QUADRATURE with WEIGHTING FUNCTION W(x)/C301 in which the endpoints of the interval [ /C281;1] are included in a total of nABSCISSAS , giving r/C30n/C282 free abscissas. A BSCISSAS are symme- trical about the origin, and the general FORMULA is g1 /C281f(x)dx/C30w1f(/C281)/C27wnf(1)/C27Xn/C281 i/C302wif(xi): (1) The free ABSCISSAS xifori/C302, ..., n/C281 are the roots of the POLYNOMIAL P?n/C281(x) ; where P(x)isaL EGENDRE POLYNOMIAL . The weights of the free abscissas are wi /C30/C282n (1 /C28 x2 i )P ƒn/C281(xi)P?m(xi) (2) /C302 n(n /C28 1)[Pn/C281(xi)]2 ; (3) and of the endpoints are w1 ; n /C302 n(n /C28 1) : (4) The error term is given by E /C30/C28n(n /C28 1)322n/C281[(n /C28 2)!]4 (2n /C28 1)[(2n /C28 1)!]3f(2n/C282)(j) ; (5) for j /C23 (/C281; 1): Beyer (1987) gives a table of para- meters up to n /C3011 and Chandrasekhar (1960) up to n /C309 (although Chandrasekhar’s m3; 4for m /C305is incorrect). n /xi// wi/ 3 0 1.33333 9 1 0.333333 4 9 0.447214 0.833333 9 1 0.166667 5 0 0.711111 9 0.654654 0.544444 9 1 0.100000 6 9 0.285232 0.554858 9 0.765055 0.378475 9 1 0.0666667 The ABSCISSAS and weights can be computed analy- tically for small n. n /xi// wi/ 30 /4 3/ 9 1 /1 3/ 4 /915ffiffiffi 5p //1 6/ 9 1 /5 6/ 50 /32 45/ /91 7ffiffiffiffiffiffi 21p //49 90/ 9 1 /1 10/ See also CHEBYSHEV QUADRATURE ,RADAU QUADRA- TUREReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 888 /C1/90, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 465, 1987. Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 63 /C1/4, 1960. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 343 /C1/45, 1956. Hunter, D. and Nikolov, G. "On the Error Term of Sym- metric Gauss-Lobatto Quadrature Formulae for Analytic Functions." Math. Comput. 69, 269 /C1/82, 2000. Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, p. 105, 1997. Lobster One of the 12 6-POLYIAMONDS . See also POLYIAMOND References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Local A mathematical property Pholds locally if Pis true near every point. In many different areas of mathe- matics, this notion is very useful. For instance, the sphere, and more generally a MANIFOLD , is locally Euclidean. For every point on the sphere, there is a NEIGHBORHOOD which is the same as a piece of EUCLIDEAN SPACE . The description of local as "near every point" has adifferent interpretation in algebra. For instance,given a RING Rand a PRIME IDEAL p, there is the LOCAL RING Rp;which often is simpler to study. It is possible to understand the original ring better bypatching together the information from the local rings. What ties all the notions of local together is the concept of a topology, a collection of open sets. For a SUBMANIFOLD of Euclidean space, or for the set of ideals of a ring, the topology is chosen as is appro- priate. A property P holds locally on a TOPOLOGICAL SPACE if every point has a NEIGHBORHOOD on which P holds. This concept is useful on any topological space. See also GLOBAL ,LOCAL FIELD,LOCAL RING,M ANI- FOLD ,TOPOLOGICAL SPACE Local Cell The POLYHEDRON resulting from letting each SPHERE in a SPHERE PACKING expand uniformly until it touches its neighbors on flat faces. See also LOCAL DENSITY ,SPHERE PACKING Local Class Field Theory The study of NUMBER FIELDS by embedding them in a LOCAL FIELD is called local class field theory. Informa- tion about an equation in a LOCAL FIELD may give information about the equation in a GLOBAL FIELD , such as the rational numbers or a NUMBER FIELD (e.g., the HASSE PRINCIPLE ). Local class field theory is termed "local" because the local fields are LOCALIZED at a PRIME IDEAL in the RING of ALGEBRAIC INTEGERS . The methods of using CLASS FIELDS have developed over the years, from the LEGENDRE SYMBOL , to the CHARACTERS of ABELIAN EXTENSIONS of a number field, and is applied to LOCAL FIELDS . See also ABELIAN EXTENSION ,CLASS FIELD,FIELD, GLOBAL FIELD ,H ASSE PRINCIPLE ,L OCAL FIELD , NUMBER FIELD,UNIQUE FACTORIZATION References Koch, H. "Local Class Field Theory." §10.3 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 321 /C1/22, 2000. Weil, A. Basic Number Theory. New York:Springer-Verlag, Chapter VII, 1974. Local Degree The degree of a VERTEX of a GRAPH is the number of EDGES which touch the VERTEX , also called the LOCAL DEGREE . The VERTEX degree of a point A in a GRAPH , denoted r(A) ; satisfies Xn i/C301r(Ai) /C302E; where E is the total number of EDGES . Directed graphs have two types of degrees, known as the INDEGREE and OUTDEGREE . See also INDEGREE ,OUTDEGREELocal Density Let each SPHERE in a SPHERE PACKING expand uniformly until it touches its neighbors on flat faces. Call the resulting POLYHEDRON the LOCAL CELL . Then the local density is given by r /C13Vsphere Vlocal cell: When the LOCAL CELL is a regular DODECAHEDRON , then rdodecahedron /C30pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffi 5pp 15ffiffiffiffiffiffi10pffiffiffi5p /C28 2/CP/C( /C300:7547... : See also L OCAL CELL,LOCAL DENSITY CONJECTURE , SPHERE PACKING Local Density Conjecture The CONJECTURE that the maximum LOCAL DENSITY is given by rdodecahedron :/ See also DODECAHEDRAL CONJECTURE ,LOCAL DEN- SITY Local Extremum A LOCAL MINIMUM or LOCAL MAXIMUM . See also EXTREMUM ,GLOBAL EXTREMUM Local Field A FIELD which is complete with respect to a discrete VALUATION is called a local field if its FIELD of RESIDUE CLASSES is FINITE . The HASSE PRINCIPLE is one of the chief applications of local field theory. See also FUNCTION FIELD,HASSE PRINCIPLE ,NUMBER FIELD,VALUATION References Iyanaga, S. and Kawada, Y. (Eds.). "Local Fields." §257 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 811 /C1/15, 1980. Local-Global Principle HASSE PRINCIPLE Local Group Theory The study of a FINITE GROUP G using the LOCAL SUBGROUPS of G. Local group theory plays a critical role in the CLASSIFICATION THEOREM . See also SYLOW THEOREMS Local Maximum The largest value of a set, function, etc., within some local neighborhood. See also GLOBAL MAXIMUM ,LOCAL MINIMUM ,M AX- IMUM ,PEANO SURFACE Local Minimum The smallest value of a set, function, etc., within some local neighborhood. See also GLOBAL MINIMUM ,LOCAL MAXIMUM ,M INI- MUM Local Ring AN OETHERIAN RING R with a JACOBSON RADICAL which has only a single MAXIMAL IDEAL . One property of a local ring R is that the SUBSET R /C28m is precisely the set of UNITS , where m is the MAXIMAL IDEAL . This follows because, in a ring, any nonunit belongs to at least one MAXIMAL IDEAL . See also JACOBSON RADICAL ,M AXIMAL IDEAL , NOETHERIAN RING,RESIDUE FIELD,UNIT (RING) References Iyanaga, S. and Kawada, Y. (Eds.). "Local Rings." §281D in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 890 /C1/91, 1980. Local Subgroup A normalizer of a nontrivial SYLOW P-SUBGROUP of a GROUP G. See also LOCAL GROUP THEORY Local Surface PATCH Locally Compact A TOPOLOGICAL SPACE X is locally compact if every point has a NEIGHBORHOOD which is itself contained in a COMPACT SET. Many familiar topological spaces are locally compact, including the EUCLIDEAN SPACE . Of course, any COMPACT SET is locally compact. Some common spaces are not locally compact, such as infinite dimensional BANACH SPACES . For instance, the L2-SPACE of SQUARE INTEGRABLE functions is not locally compact. See also COMPACT SET,LOCALLY COMPACT GROUP , NEIGHBORHOOD ,TOPOLOGICAL SPACE Locally Convex Space LOCALLY PATHWISE- CONNECTED Locally Finite Complex A SIMPLICIAL COMPLEX K is said to be locally finite if each vertex of K belongs only to finitely many SIMPLICES of K.References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Locally Finite Space A locally finite SPACE is one for which every point of a given space has a NEIGHBORHOOD that meets only finitely many elements of the COVER . Locally Integrable A function is called locally integrable if, around every point in the domain, there is a NEIGHBORHOOD on which the function is INTEGRABLE . The space of locally integrable functions is denoted L1 loc : Any integrable function is also locally integrable. One possibility for a nonintegrable function which is locally integrable is if it does not decay at infinity. For instance, f(x) /C301 is locally integrable on R ; as is any CONTINUOUS FUNCTION . See also FRECHET SPACE ,INTEGRABLE ,L EBESGUE INTEGRABLE , L1-SPACE Locally Pathwise-Connected A SPACE X is locally pathwise-connected if for every NEIGHBORHOOD around every point in X, there is a smaller, PATHWISE-CONNECTED NEIGHBORHOOD . See also ARCWISE- CONNECTED ,P ATHWISE- CON- NECTED Locally Pathwise-Connected Space A SPACE X is locally pathwise-connected if for every NEIGHBORHOOD around every point in X, there is a smaller, PATHWISE-CONNECTED NEIGHBORHOOD . Lochs’ Theorem For a real number x /C23 (0; 1); let m be the number of terms in the CONVERGENT to a CONTINUED FRACTION that are required to represent n decimal places of x. Then for almost all x, lim n0/C12m n/C306 ln 2 ln 10 p2/C300:97027014... (Lochs 1964). Therefore, the CONTINUED FRACTION is only slightly more efficient at representing real numbers than is the decimal expansion. The set of x for which this statement does not hold is of measure 0. See also CONTINUED FRACTION References Kintchine, A. "Zur metrischen Kettenbruchtheorie." Com- pos. Math. 3, 276/C1/85, 1936. Le´vy, P. "Sur le developpement en fraction continue d’un nombre choisi au hasard." Compos. Math. 3, 286/C1/03, 1936. Lochs, G. Abh. Hamburg Univ. Math. Sem. 27, 142 /C1/44, 1964. Perron, O. Die Lehre von Kettenbru ¨chen, 3. verb. und erweiterte Aufl. Stuttgart, Germany: Teubner, 1954 /C1/7. Loculus of Archimedes STOMACHION Locus The set of all points (usually forming a curve or surface) satisfying some condition. For example, the locus of points in the plane equidistant from a given point is a CIRCLE , and the set of points in 3-space equidistant from a given point is a SPHERE . References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 5 /C1/, 1888. Log COMMON LOGARITHM ,LOGARITHM ,N ATURAL LOGA- RITHM Log Likelihood Procedure A method for testing NESTED HYPOTHESES . To apply the procedure, given a specific model, calculate the LIKELIHOOD of observing the actual data. Then compare this likelihood to a nested model (i.e., one in which fewer parameters are allowed to vary independently). Log Normal Distribution A CONTINUOUS DISTRIBUTION in which the LOGARITHM of a variable has a NORMAL DISTRIBUTION .Itisa general case of GILBRAT’S DISTRIBUTION , to which the log normal distribution reduces with S /C301 and M /C300. The probability density and cumulative distribution functions for the log normal distribution are P(x) /C301 Sxffiffiffiffiffiffi 2 pp e/C28(ln x /C28M)2 =(2S2) (1) D(x) /C301 21 /C27erfln x /C28 M Sffiffiffi 2p !"# ; (2) where erf(x) is the ERF function. This distribution is normalized, since letting y /C13ln x gives dy /C30dx=x andx /C30ey ; so g/C12 0P(x) dx /C301 Sffiffiffiffiffiffi2ppg/C12 /C28/C12e/C28(y/C28M)2 =2s2 dy /C301: (3) The RAW MOMENTS are m?1 /C30eM /C27S2 =2 (4) m?2 /C30e2(M /C27S)2 (5) m?3 /C30e3M /C279S2 =2 (6) m ?4 /C30e4M /C278S2 ; (7) and the CENTRAL MOMENTS are m2 /C30e2M /C27S2 (eS2 /C281) (8) m3 /C30e3M /C273S2 =2(eS2 /C281)2(eS2 /C272) (9) m4 /C30e4M /C272S2 (eS2 /C281)2(e4S2 /C272e3S2 /C273e2S2 /C283): (10) Therefore, the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are given by m /C30eM /C27S2 =2 (11) s2 /C30eS2/C272M(eS2 /C281) (12) g1 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eS2 /C281p (2 /C27eS2 ) (13) g2 /C30e4S2 /C272e3S2 /C273e2S2 /C286: (14) These can be found by direct integration m /C301 Sffiffiffiffiffiffi 2ppg/C12 0e/C28(ln x /C28M)2 =(2S2) dx /C301 Sffiffiffiffiffiffi2ppg/C12 /C28/C12e/C28(/C28y /C28M)2 =2S2 ey dy /C30eM/C27S2=2; (15) and similarly for s2:/ Examples of variates which have approximately log normal distributions include the size of silver parti- cles in a photographic emulsion, the survival time ofbacteria in disinfectants, the weight and blood pres-sure of humans, and the number of words written in sentences by George Bernard Shaw. See also G ILBRAT’S DISTRIBUTION ,WEIBULL DISTRIBU- TION References Aitchison, J. and Brown, J. A. C. The Lognormal Distribu- tion, with Special Reference to Its Use in Economics. New York: Cambridge University Press, 1957. Balakrishnan, N. and Chen, W. W. S. Handbook of Tables for Order Statistics from Lognormal Distributions with Applications. Amsterdam, Netherlands: Kluwer, 1999. Crow, E. L. and Shimizu, K. (Ed.). Lognormal Distribu- tions:Theory and Applications. New York: Dekker, 1988. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 123, 1951. Logarithm The logarithm logb x for a BASE b and a number x is defined to be the INVERSE FUNCTION of taking x to the POWER b. Therefore, for any x and b, x /C30blogb x ; (1) or equivalently, x /C30logb(bx): (2) Whereas power of trigonometric functions are de- noted using notations like sink x ; lnk x is less com- monly used in favor of the notation (ln x)k :/ For any BASE , the logarithm function has a SINGU- LARITY at x /C300. In the above plot, the solid curve is the logarithm to BASE e (the NATURAL LOGARITHM ), and the dotted curve is the logarithm to BASE 10 (LOG). Logarithms are used in many areas of science and engineering in which quantities vary over a large range. For example, the decibel scale for the loudness of sound, the Richter scale of earthquake magnitudes, and the astronomical scale of stellar brightnesses are all logarithmic scales. The logarithm can also be defined for COMPLEX arguments, as shown above. If the logarithm is taken as the forward function, the function taking the BASE to a given POWER is then called the ANTILOGARITHM . For x /C30log N ; xbcis called the CHARACTERISTIC and x /C28 xbcis called the MANTISSA . Division and multi- plication identities follow from these xy /C30blogb xblogb y /C30blogb x/C27logb y ; (3) from which it follows thatlogb(xy) /C30logb x /C27logb y (4) logbx y ! /C30logb x /C28logb y (5) logb xn /C30n logb x: (6) There are a number of properties which can be used to change from one logarithm BASE to another a /C30aloga b=loga b /C30(aloga b)1 =loga b /C30b1 =loga b (7) logb a /C301 loga b (8) logb x /C30logbylogy x/CP/C( /C30logy x logb y (9) logb x /C30logn x logn b (10) ax /C30bx =loga b /C30bx logb a : (11) The logarithm BASE E is called the NATURAL LOGA- RITHM and is denoted ln x (LN). The logarithm BASE 10 is denoted log x (LOG), (although mathematics texts often use log x to mean ln x) : The logarithm BASE 2is denoted lg x (LG). An interesting property of logarithms follows from looking for a number y such that logb(x /C27y) /C30/C28logb(x /C28y) (12) x /C27y /C301 x/C28y(13) x2/C28y2/C301 (14) y/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C281p ; (15) so logbx/C27ffiffiffiffiffiffiffiffiffiffiffiffiffix 2/C281p/C(%/C(r /C30/C28logbx/C28ffiffiffiffiffiffiffiffiffiffiffiffiffix 2/C281p/C(%/C(r : (16) Numbers OF THE FORM logabare IRRATIONAL ifaand bare INTEGERS , one of which has a PRIME factor which the other lacks. A. Baker made a major step forward in TRANSCENDENTAL NUMBER theory by prov- ing the transcendence of sums of numbers OF THE FORM alnbforaandbALGEBRAIC NUMBERS . See also ANTILOGARITHM ,BASE (LOGARITHM ), COLO- GARITHM , E,E XPONENTIAL FUNCTION ,H ARMONIC LOGARITHM ,L G,L N,L OG,L OGARITHMIC SERIES , LOGARITHMIC NUMBER ,NAPIERIAN LOGARITHM ,NAT- URAL LOGARITHM ,POWER References Abramowitz, M. and Stegun, C. A. (Eds.). "Logarithmic Function." §4.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 67 /C1/9, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 221, 1987. Conway, J. H. and Guy, R. K. "Logarithms." The Book of Numbers. New York: Springer-Verlag, pp. 248 /C1/52, 1996. Beyer, W. H. "Logarithms." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 159 /C1/60, 1987. Pappas, T. "Earthquakes and Logarithms." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 20 /C1/1, 1989. Spanier, J. and Oldham, K. B. "The Logarithmic Function ln(x) :/" Ch. 25 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 225 /C1/32, 1987. Logarithmic Binomial Formula LOGARITHMIC BINOMIAL THEOREM Logarithmic Binomial Theorem For all integers n and ½x½Ba ; l(t) n (x /C27a) /C30X/C12 k /C300n k/C))/C)$ lt n/C28k(a)xk ; where l(t) nis the HARMONIC LOGARITHM andn k/Co/Cr is a ROMAN COEFFICIENT . For t /C300, the logarithmic bino- mial theorem reduces to the classical BINOMIAL THEOREM for POSITIVE n, since l(0) 1(a) /C28cn/C28k for n ] k; l(0)n/C28k(a) /C300 for n Bk, and n k/Co/Cr /C30 n k/CP/C( when n ]k ]0:/ Similarly, taking t /C301 and n B0 gives the NEGATIVE BINOMIAL SERIES . Roman (1992) gives expressions obtained for the case t /C301 and n ]0 which are not obtainable from the BINOMIAL THEOREM . See also HARMONIC LOGARITHM ,ROMAN COEFFICIENT References Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641 /C1/48, 1992. Logarithmic Derivative The logarithmic derivative of a function f is defined as the DERIVATIVE of the LOGARITHM of a function. For example, the DIGAMMA FUNCTION is defined as the logarithmic derivative of the GAMMA FUNCTION , C(z)/C30d dzlnG(z): See also DERIVATIVE ,D IGAMMA FUNCTION ,L OGA- RITHM ,POLYGAMMA FUNCTION References Zwillinger, D. (Ed.). "Logarithmic Derivative." §6.11.8 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 496, 1995.Logarithmic Distribution ACONTINUOUS DISTRIBUTION for a variate x/C23[a;b] with probability function P(x)/C30lnx b(lnb/C281)/C28a(lna/C281)(1) and distribution function D(x)/C30a(1/C28lna)/C28x(1/C28lnx) a(1/C28lna)/C28b(1/C28lnb): (2) The moments about zero are given by m?n/C30an/C271[1/C28(n/C271)ln a]/C28bn/C271[1/C28(n/C271)ln b] (n/C271)2[a(1/C28lna)/C28b(1/C28lnb)]; (3) giving MEAN m/C30a2(1/C282l na)/C28b2(1/C282l nb) 4[a(1/C28lna)/C28b(1/C28lnb)]: (4) The VARIANCE ,SKEWNESS , and KURTOSIS are compli- cated expressions involving the m?n:/ Logarithmic Integral The logarithmic integral is defined by li(x)/C13gx 0du lnu: (1) This function is implemented in Mathematica as LogIntegral [x]. The logarithmic integral obeys the identity li(xm) /C30 g /C27ln ln x /C28lnm /C27X/C12 n /C301(ln x)n n /C215 n!mn (2) (Bromwich and MacRobert 1991, p. 334; Hardy 1999, p. 25). The form of this function appearing in the PRIME NUMBER THEOREM is defined so that Li(2) /C300: Li(x) /C13gx 2du ln u (3) /C30li(x) /C28li(2) :li(x) /C281 :04516 (4) /C30ei(ln x) ; (5) where ei(x) is the EXPONENTIAL INTEGRAL . (Note that the NOTATION Lin(z) is also used for the POLYLOGA- RITHM .) Nielsen (1965, pp. 3 and 11) showed and Ramanujan independently discovered (Berndt 1994) that gx mdt ln t /C30 g /C27ln ln x /C27X/C12 k /C301(ln x)k k!k; (6) where g is the EULER- MASCHERONI CONSTANT and m is SOLDNER’S CONSTANT . Another FORMULA due to Ra- manujan which converges more rapidly is gx mdt ln t /C30 g /C27ln ln x /C27ffiffiffixpX/C12 n/C300( /C281)n/C281(ln x)n n!2n/C281X[(n /C281)=2] k/C3001 2k /C27 1(7) (Berndt 1994). See also POLYLOGARITHM ,P RIME CONSTELLATION , PRIME NUMBER THEOREM ,SKEWES NUMBER References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 126 /C1/31, 1994. Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 334, 1991. de Morgan, A. The Differential and Integral Calculus, Containing Differentiation, Integration, Development, Ser- ies, Differential Equations, Differences, Summation, Equations of Differences, Calculus of Variations, Definite Integrals,--With Applications to Algebra, Plane Geometry, Solid Geometry, and Mechanics. London: Robert Baldwin, p. 662, 1839. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Koosis, P. The Logarithmic Integral I. Cambridge, England: Cambridge University Press, 1998. Nielsen, N. "Theorie des Integrallograrithmus und Ver- wandter Transzendenten." Part II in Die Gammafunktion. New York: Chelsea, 1965.Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 151, 1991. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 45, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 39, 1983. Soldner. Abhandlungen 2, 333, 1812. Logarithmic Number A COEFFICIENT of the MACLAURIN SERIES of 1 ln (1 /C27 x) /C301 x /C271 2 /C281 12 /C271 24x2 /C2819 720x3 /C273 160x4 /C27... (Sloane’s A002206 and A002207), the multiplicative inverse of the MERCATOR SERIES function ln (1 /C27x) :/ See also MERCATOR SERIES References Sloane, N. J. A. Sequences A002206/M5066 and A002207/ M2017 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Logarithmic Series X/C12 k /C301(/C281)kln k /C301 2 ln12 p/C(%/C(r X/C12 k /C301ln k /C3012ln(2 p): See also LOGARITHM References Bromwich, T. J. I’a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 351, 1991. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 37, 1999. Logarithmic Spiral A curve whose equation in POLAR COORDINATES is given by r /C30aeb u ; (1) where r is the distance from the ORIGIN , u is the angle from the X-AXIS , and a and b are arbitrary constants. The logarithmic spiral is also known as the GROWTH SPIRAL , EQUIANGULAR SPIRAL , and SPIRA MIRABILIS .It can be expressed parametrically using cos u /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 tan2 up /C301ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y2 x2q /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2p /C30x r ; (2) which gives x /C30r cos u /C30a cos uebu (3) y /C30x tan u /C30r sin u /C30a sin uebu : (4) The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approached infinity, the sequence of segments approaches the smooth loga- rithmic spiral (Hilton et al. 1997, pp. 2 /C1/). The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw it true to form) together with the words "eadem mutata resurgo" ("I shall arise the same though changed"rpar;. Torricelli worked on it independently and found the length of the curve (MacTutor Ar- chive). The rate of change of RADIUS is dr du /C30abebu /C30br; (5) and the ANGLE between the tangent and radial line at the point (r; u)is c /C30tan/C281r dr du ! /C30tan /C2811 b ! /C30cot /C281b : (6) So, as b 0 0; c 0 p=2 and the spiral approaches a CIRCLE . If P is any point on the spiral, then the length of the spiral from P to the origin is finite. In fact, from the point P which is at distance r from the origin measured along a RADIUS vector, the distance from P to the POLE along the spiral is just the ARC LENGTH . In addition, any RADIUS from the origin meets the spiral at distances which are in GEOMETRIC PROGRES- SION (MacTutor Archive). The ARC LENGTH , CURVATURE , and TANGENTIAL ANGLE of the logarithmic spiral are s/C30gds/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2/C27y?2q dt/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2p bebu /C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2p b(7) k/C30x?yƒ/C28y?xƒ (x?2/C27y?2)3=2/C30affiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2p ebu/C(%/C(r/C281 (8) f/C30gk(s)ds/C30u: (9) The C ESA`RO EQUATION is k/C301 bs: (10) On the surface of a SPHERE , the analog is a LOXO- DROME . This SPIRAL is related to F IBONACCI NUMBERS and the GOLDEN RATIO . See also GOLDEN RECTANGLE ,LOGARITHMIC SPIRAL CAUSTIC CURVE ,L OGARITHMIC SPIRAL EVOLUTE , LOGARITHMIC SPIRAL INVERSE CURVE ,LOGARITHMIC SPIRAL PEDAL CURVE ,LOGARITHMIC SPIRAL RADIAL CURVE ,MICE PROBLEM ,SPIRAL ,W HIRL References Boyadzhiev, K. N. "Spirals and Conchospirals in the Flight of Insects." Coll. Math. J. 30,2 3/C1/1, 1999. Cook, T. A. The Curves of Life, Being an Account of Spiral Formations and Their Application to Growth in Nature, To Science and to Art. New York: Dover, 1979. Gray, A. "Logarithmic Spirals." Modern Differential Geome- try of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 40 /C1/2, 1997. Hilton, P.; Holton, D.; and Pedersen, J. Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 184 /C1/86, 1972. Lockwood, E. H. "The Equiangular Spiral." Ch. 11 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 98 /C1/09, 1967. MacTutor History of Mathematics Archive. "Equiangular Spiral." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Equiangular.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 132 /C1 /36, 1999. Thompson, D’Arcy W. Science and the Classics. Oxford, England: Oxford University Press, pp. 114 /C1/47, 1940. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 67 /C1/8, 1991. Logarithmic Spiral Caustic Curve The CAUSTIC of a LOGARITHMIC SPIRAL , where the pole is taken as the RADIANT POINT , is an equal LOGARITH- MIC SPIRAL . Logarithmic Spiral Evolute InPOLAR COORDINATES r/C30r(u);the RADIUS OF CUR- VATURE is given by R/C30(r2/C27r2 u)3=2 r2/C272r2r2 u/C28rruu; (1) so plugging in the equation of the LOGARITHMIC SPIRAL and its derivatives r/C30aebu(2) ru/C30abebu(3) ruu/C30ab2ebu(4) gives R/C30affiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2p ebu: (5) To find the VELOCITY VECTOR , compute x y/C)P/C)( /C30aebucosu aebusinu/C)P/C)( x? y?/C)P/C)( /C30abebucosu/C28aebusinu abebusinu/C27aebucosu/C)P/C)( /C30aebubcosu/C28sinu bsinu/C27cosu/C)P/C)( ; (6) so ½r?½/C30aebuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (bcosu/C28sinu)2/C27(bsinu/C27cosu)2q /C30aebuffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2p ; (7) and the TANGENT VECTOR is given by ˆT/C30r? ½r?½/C301 aebuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2paebucosu aebusinu/C)P/C)( /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27b2pcosu sinu/C)P/C)( : (8) The coordinates of the EVOLUTE are therefore j/C30/C28abebusinu (9) h/C30/C28abebucosu: (10) Therefore, the EVOLUTE is another logarithmic spiral with a?/C13ab;as first shown by Johann Bernoulli. In some cases, the EVOLUTE is identical to the original, as can be demonstrated by making the substitution to the new variableu/C13f/C281 2p92np: (11) Then the above equations become j/C30/C28abeb(f/C28p=292np)sin(f/C28p=292np) /C30abebfeb(/C28p=292np)cosf (12) h/C30abeb(f/C28p=292np)cos(f/C28p=292np) /C30abebfeb(/C28p=292np)sinf; (13) which are equivalent to the form of the original equation if beb/C281 2p92np/C(%/C(r /C301 (14) lnb/C27b/C2812p92np/C(%/C(r /C300 (15) lnb b/C301 2p/C142np/C30/C28 2n/C2812/C(%/C(r p; (16) where only solutions with the minus sign in /C14exist. Solving gives the values summarized in the following table. n /bn// c/C30cot/C281bn/ 1 0.2744106319... /74/C1439?18:53ƒ/ 2 0.1642700512... /80/C1440?16:80ƒ/ 3 0.1218322508... /83/C1403?13:53ƒ/ 4 0.0984064967... /84/C1422?47:53ƒ/ 5 0.0832810611... /85/C1414?21:60ƒ/ 6 0.0725974881... /85/C1450?51:92ƒ/ 7 0.0645958183... /86/C1418?14:64ƒ/ 8 0.0583494073... /86/C1439?38:20ƒ/ 9 0.0533203211... /86/C1456?52:30ƒ/ 10 0.0491732529... /87/C1411?05:45ƒ/ References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 60 /C1/4, 1991. Logarithmic Spiral Inverse Curve The INVERSE CURVE of the LOGARITHMIC SPIRAL r/C30eau with INVERSION CENTER at the origin and inversion radius kis the LOGARITHMIC SPIRAL r /C30ke/C28a u : Logarithmic Spiral Pedal Curve The PEDAL CURVE of a LOGARITHMIC SPIRAL with parametric equation f /C30eat cos t (1) g /C30eat sin t (2) for a PEDAL POINT at the pole is an identical LOGARITHMIC SPIRAL x /C30(a sin t /C27 cos t)eat 1 /C27 a2 (3) y /C30(sin t /C28 a cos t)eat 1 /C27 a2 (4) so r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p /C30eat ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 a2p : (5) Logarithmic Spiral Radial Curve The RADIAL CURVE of the LOGARITHMIC SPIRAL is another LOGARITHMIC SPIRAL . Logarithmic Transform The inverse transform X/C12 n /C301anxn n!/C30ln 1 /C27X/C12 n/C301bnxn n! ! of the EXPONENTIAL TRANSFORM1 /C27X/C12 n/C301bnxn n!/C30expX/C12 n /C301anxn n! ! which relate sequences a1 ; a2 ; ... and b1 ; b2 ; .... See also EXPONENTIAL TRANSFORM References Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 19 /C1/0, 1995. Logarithmically Concave Function A function f(x) is logarithmically concave on the interval [a, b]iff /C210 and ln f(x)is CONCAVE on [a, b]. The definition can also be extended to Rk 0 (0;/C12) functions (Dharmadhikari and Joag-Dev 1988, p. 18). See also CONCAVE FUNCTION ,L OGARITHMICALLY CONVEX FUNCTION References Dharmadhikari, S. and Joag-Dev, K. Unimodality, Convex- ity, and Applications. Boston, MA: Academic Press, 1988. Logarithmically Convex Function A function f(x) is logarithmically convex on the interval [a, b]iff /C210 and ln f(x)is CONVEX on [a, b]. If f(x) and g(x) are logarithmically convex on the interval [a, b], then the functions f(x) /C27g(x) and f(x)g(x) are also logarithmically convex on [a, b]. The definition can also be extended to Rk 0 (0;/C12) functions (Dharmadhikari and Joag-Dev 1988, p. 18). See also CONVEX FUNCTION ,LOGARITHMICALLY CON- CAVE FUNCTION References Dharmadhikari, S. and Joag-Dev, K. Unimodality, Convex- ity, and Applications. Boston, MA: Academic Press, 1988. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1100, 2000. Logconcave Function LOGARITHMICALLY CONCAVE FUNCTION Logconvex Function LOGARITHMICALLY CONVEX FUNCTION LogGamma GAMMA FUNCTION Logic The formal mathematical study of the methods, structure, and validity of mathematical deductionand proof. In Hilbert’s day, formal logic sought to devise a complete, consistent formulation of mathematics such that propositions could be formally stated and proved using a small number of symbols with WELL DEFINED meanings. The difficulty of formal logic was demonstrated in the monumental Principia Mathe- matica (1925) of Whitehead and Russell’s , in which hundred of pages of symbols were required before the statement 1/C271 /C302 could be deduced. In 1931, Go¨del unexpectedly showed that Hilbert’s goal to be im- possible, and this proved only the first of a number of difficult and counterintuitive results which have since been demonstrated. A very simple form of logic is the study of "TRUTH TABLES " and digital logic circuits in which one or more outputs depend on a combination of circuit elements (AND, OR, NAND, NOR, NOT, XOR, etc.; "gates") and the input values. In such a circuit, values at each point can take on values of only TRUE (1) or FALSE (0). DE MORGAN’S DUALITY LAW is a useful principle for the analysis and simplification of such circuits. A generalization of this simple type of logic in which possible values are TRUE , FALSE , and "undecided" is called THREE-VALUED LOGIC . A further generalization called FUZZY LOGIC treats "truth" as a continuous quantity ranging from 0 to 1. See also ABSORPTION LAW,ALETHIC ,BOOLEAN ALGE- BRA,BOOLEAN CONNECTIVE ,BOUND ,CALIBAN PUZ- ZLE,C ONTRADICTION LAW, DE MORGAN’S DUALITY LAW, DE MORGAN’S LAWS,D EDUCIBLE ,E XCLUDED MIDDLE LAW,FREE,FUZZY LOGIC ,GO¨ DEL’S INCOM- PLETENESS THEOREM ,KHOVANSKI’S THEOREM ,LOGI- CAL PARADOX ,L OGOS ,L O¨ WENHEIM- SKOLEM THEOREM ,M ETAMATHEMATICS ,M ODEL THEORY , QUANTIFIER ,SENTENCE ,TARSKI’S THEOREM ,TAUTOL- OGY,THREE- VALUED LOGIC ,TOPOS ,TRUTH TABLE , TURING MACHINE ,U NIVERSAL TURING MACHINE , VENN DIAGRAM ,W ILKIE’S THEOREM References Adamowicz, Z. and Zbierski, P. Logic of Mathematics: A Modern Course of Classical Logic. New York: Wiley, 1997. Bogomolny, A. "Falsity Implies Anything." http://www.cut- the-knot.com/do_you_know/falsity.html. Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, 1958. Church, A. Introduction to Mathematical Logic, Vol. 1. Princeton, NJ: Princeton University Press, 1996. Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972. Enderton, H. B. Elements of Set Theory. New York: Aca- demic Press, 1977. Heijenoort, J. van. From Frege to Go¨del: A Sourcebook in Mathematical Logic, 1879 /C1/931. Cambridge, MA: Cam- bridge University Press, 1967. Go¨del, K. On Formally Undecidable Propositions of Princi- pia Mathematica and Related Systems. New York: Dover, 1992. Jeffrey, R. C. Formal Logic: Its Scope and Limits. New York: McGraw-Hill, 1967.Kac, M. and Ulam, S. M. Mathematics and Logic: Retrospect and Prospects. New York: Dover, 1992. Kleene, S. C. Introduction to Metamathematics. Princeton, NJ: Van Nostrand, 1971. Smullyan, R. M. First-Order Logic. New York: Dover. Weisstein, E. W. "Books about Logic." http://www.treasure- troves.com/books/Logic.html. Whitehead, A. N. and Russell, B. Principia Mathematica, 2nd ed. Cambridge, England: Cambridge University Press, 1962. Logical And AND Logical Connective CONNECTIVE Logical Not NEGATION SIGN, NOT Logical Or OR Logical Paradox PARADOX LogIntegral Logarithmic Integral Logistic Distribution P(x)/C30e(x/C28m)=b ½b½[1/C27e(x/C28m)=b]2(1) D(x)/C301 1/C27e(m/C28x)=½b½; (2) and the MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS are m/C30m (3) s2/C301 3p2b2(4) g1/C300 (5) g2/C306 5: (6) See also LOGISTIC EQUATION ,L OGISTIC GROWTH CURVE References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 250, 1993. Logistic Equation The logistic equation (sometimes called the V ERHULST MODEL since it was first published in 1845 by the Belgian P.-F. Verhulst) is defined by xn/C271/C30rxn(1/C28xn); (1) where r(sometimes also denoted m)i sa POSITIVE constant (the "biotic potential"). Let an initial point x0 lie in the interval [0 ;1]:Now find appropriate conditions on rwhich keep points in the interval. The maximum value xn/C271can take is found from dxn/C271 dxn/C30r(1/C282xn)/C300; (2) so the largest value of xn/C271occurs for xn/C301=2: Plugging this in, max( xn/C271)/C30r=4:Therefore, to keep the MAP in the desired region, we must have r/C23(0;4]: The J ACOBIAN is J/C30dxn/C271 dxn/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C30½r(1/C282x n)½; (3) and the MAP is stable at a point x0ifJ(x0)B1:/ Now find the FIXED POINTS of the MAP, which occur when xn/C271/C30xn:For convenience, drop the nsubscript onxn f(x)/C30rx(1/C28x)/C30x (4) x[1/C28r(1/C28x)]/C30x(1/C28r/C27rx)/C30rx[x/C28(1/C28r/C281)] /C300; (5) so the FIXED POINTS arex(1) 1/C300 and x(1)2/C301/C28r/C281:/ An interesting thing happens if a value of rgreater than 3 is chosen. The map becomes unstable and we get a PITCHFORK BIFURCATION with two stable orbits of period two corresponding to the two stable FIXED POINTS off2(x):The fixed points of order two must satisfy xn/C272/C30xn;so xn/C272/C30rxn/C271(1/C28xn/C271) /C30r[rxn(1/C28xn)][1/C28rxn(1/C28xn)] /C30r2xn(1/C28xn)(1/C28rxn/C27rx2 n)/C30xn: (6) For convenience, drop the nsubscripts and rewrite xfr2[1/C28x(1/C27r)/C272rx2/C28rx3]/C281g/C300 (7) x[/C28r3x3/C272r3x2/C28r2(1/C27r)x/C27(r2/C281)]/C300 (8) /C28r3x[x/C28(1/C28r/C281)][x2/C28(1/C27r/C281)x/C27r/C281(1/C27r/C281)] /C300: (9) Notice that we have found the first-order FIXED POINTS as well, since two iterations of a first-orderFIXED POINT produce a trivial second-order FIXED POINT . The true 2- CYCLES are given by solutions to the quadratic part x(2)9/C301 2[(1/C27r/C281)9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27r/C281)2/C284r/C281(1/C27r/C281)q ] /C301 2[(1/C27r/C281)9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C272r/C281/C27r/C282/C284r/C281/C284r/C282p ] /C301 2[(1/C27r/C281)9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C282r/C281/C283r/C282p ] /C301 2[(1/C27r/C281)9r/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (r/C283)(r/C271)p ]: (10) These solutions are only REAL forr]3;so this is where the 2- CYCLE begins. Note that the 2-cycle can also be found by computing the DISCRIMINANT of f2(x)/C28x f(x)/C28x/C30r2x2/C28r(1/C27r)x/C27(1/C27r)/C300; (11) which is (1/C27r)(3/C28r) r2: (12) When this equals 0, two roots coincide, so r2/C303 is the onset of period doubling. Now look for the onset of the 3- CYCLE . To eliminate the 1- CYCLES , consider f3(x)/C28x f(x)/C28x/C300: (13) This gives 1/C27r/C27r2/C28(r4/C272r3/C272r2/C27r)x /C27(2r5/C273r4/C273r3/C27r2)x2 /C28(r6/C275r5/C273r4/C27r3)x3/C27(3r6/C274r5/C27r4)x4 /C28(3r6/C28r5)x5/C27r6x5/C300: (14) The ROOTS of this equation are all IMAGINARY forr less than some cutoff r3;at which point two of them convert to REAL roots. The value of r3can be found by computing the DISCRIMINANT of (14), D/C30(r2/C285r/C277)2(r2/C282r/C287)3(1/C27r/C27r2)2 r30:(15) When the DISCRIMINANT is zero, two roots coincide. This happens at r3/C301/C272ffiffiffi 2p ;so the 3- CYCLE starts at r3:/ To find the onset of the 4- CYCLE , eliminate the 2- and 1-CYCLES by considering f4(x)/C28x f2(x)/C28x/C300: (16) This gives 1/C27r2/C27(/C28r2/C28r3/C28r4/C28r5)x /C27(2r3/C27r4/C274r5/C27r6/C272r7)x2 /C27(/C28r3/C285r5/C284r6/C285r7/C284r8/C28r9)x3 /C27(2r5/C276r6/C274r7/C2714r8/C275r9/C273r10)x4 /C27(/C284r6/C28r7/C2818r8/C2812r9/C2812r10/C283r11)x5 /C27(r6/C2710r8/C2717r9/C2718r10/C2715r11/C27r12)x6 /C27(/C282r8/C2814r9/C2812r10/C2830r11/C286r12)x7 /C27(6r9/C273r10/C2730r11/C2715r12)x8 /C27(/C28r9/C2815r11/C2820r12)x9/C27(3r11/C2715r12)x10(17) The value of r4can be found by computing the DISCRIMINANT of (17), D/C30(r2/C271)3(r2/C284r/C275)3 r132 /C29(r6/C286r5/C273r4/C2728r3/C289r2/C2854r/C28135) ; (18) which has roots at r4/C301/C27ffiffiffi 6p ;as well as at the 2nd root of r6/C286r5/C273r4/C2728r3/C289r2/C2854r/C28135/C300: The 4- CYCLE therefore starts at r4/C301/C27ffiffiffi6p /C303:449489 . . . : / The onset of 5-cycles can be found analogously, and gives a messy 22nd-order polynomial in rwhose real positive roots are 3.73817, 3.90557, and 3.99026. In general, the set of n/C271 equations which can be solved to give the onset of an arbitrary n-cycle (Saha and Strogatz 1995) is x2/C30rx1(1/C28x1) x3/C30rx2(1/C28x2) n xn/C30rxn/C281(1/C28xn/C281) x1/C30rxn(1/C28xn) rnQn k/C301(1/C282xk)/C301:8 >>>>>>< >>>>>>:(19) The first nof these give f(x);f 2(x);...,fn(x);and the last uses the fact that the onset of period noccurs by aTANGENT BIFURCATION , so the nthDERIVATIVE is 1. For small n, these can be solved exactly, but the complexity rapidly increases with n Forn/C302, the solutions ( x1;x2;r) are given by (0, 0, 91) and ( /2=3;2=3;3), so the first BIFURCATION occurs atr2/C303:/ Forn/C303, d[f3(x)] dx/C30d[f3(x)] d[f2(x)]d[f2(x)] d[f(x)]d[f(x)] dx /C30d[f(z)] dzd[f(y)] dyd[f(x)] dx /C30r3(1/C282z)(1/C282y)(1/C282x): (20)Solving the resulting CUBIC EQUATION using compu- ter algebra gives r/C301/C272ffiffiffi 2p (21) andx1;x2;x3the 2nd, 4th, and 5th roots of the sextic 343x6/C28980x5/C27868x4/C28134x3/C28161x2/C2770x/C287 /C300; (22) giving numerical roots x1:0:514355 (23) x2:0:956318 (24) x3:0:159929 (25) r:3:828427 : (26) Saha and Strogatz (1995) give a simplified algebraic treatment for the 3-cycle which involves solving r3(1/C282a/C274b/C288g)/C301; (27) together with three other simultaneous equations, where a/C13x1/C27x2/C27x3 (28) b/C13x1x2/C27x1x3/C27x2x3 (29) g/C13x1x2x3: (30) Further simplifications still are provided in Bech-hoeffer (1996) and Gordon (1996), but neither of thesetechniques generalizes easily to higher CYCLES . Bech- hoeffer (1996) expresses the three additional equa-tions as 2a/C303/C27r /C281(31) 4b/C303 2/C275r/C281/C2732r/C282(32) 8g/C30/C281 2/C2772r/C281/C2752r/C282/C2752r/C283; (33) giving r2/C282r/C287/C300: (34) This has the positive solution found previously, r3/C301/C272ffiffiffi 2p :/ Gordon (1996) derives not only the value for the onset of the 3- CYCLE , but also an upper bound for the r- values supporting stable period-3 orbits. This value isobtained by solving the CUBIC EQUATION s3/C2811s2/C2737s/C28108/C300 (35) fors, then r?/C301/C27ffiffiffisp(36) /C301/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 3/C271915 54/C275 2ffiffiffiffiffiffiffiffi 201p/C(%/C(r1=3 /C271915 54/C285 2ffiffiffiffiffiffiffiffi 201p/C(%/C(r1=3r /C303:841499007543 . . . (37) The illustration above shows the logistic map. A table of the CYCLE type and value of rn at which the cycle 2n appears is given below. n cycle (/2n)// rn/ 12 3 2 4 3.449490 3 8 3.544090 4 16 3.564407 5 32 3.568750 6 64 3.56969 7 128 3.56989 8 256 3.569934 9 512 3.569943 10 1024 3.5699451 11 2048 3.569945557 //C12/ ACC. PT. 3.569945672 For additional values, see Rasband (1990, p. 23). Note that the table in Tabor (1989, p. 222) is incorrect, as is the n /C302 entry in Lauwerier (1991). The period doubling BIFURCATIONS come faster and faster (8, 16, 32, ...), then suddenly break off. Beyond a certain point known as the ACCUMULATION POINT , periodicity gives way to CHAOS , as illustrated below. In the middle of the complexity, a window suddenly appears with a regular period like 3 or 7 as a result of MODE LOCKING . The period-3 BIFURCATION occurs at r /C30 1 /C272ffiffiffi 2p /C303:828427 ; and PERIOD DOUBLINGS then begin again with CYCLES of 6, 12, ...and 7, 14, 28, ..., and then once again break off to CHAOS .It is relatively easy to show that the logistic map is chaotic on an invariant Cantor set for r > 2 /C27ffiffiffi 5p : 4:236 (Devaney 1989, pp. 31 /C1/0; Gulik 1992, pp. 112 /C1/ 26; Holmgren 1996, pp. 69 /C1/5), but in fact, it is also chaotic for all r /C214 (Robinson 1995, pp. 33 /C1/7; Kraft 1999). The logistic equation has CORRELATION EXPONENT 0.50090.005 (Grassberger and Procaccia 1983), CA- PACITY DIMENSION 0.538 (Grassberger 1981), and INFORMATION DIMENSION 0.5170976 (Grassberger and Procaccia 1983). See also BIFURCATION ,FEIGENBAUM CONSTANT ,LO- GISTIC DISTRIBUTION ,LOGISTIC EQUATION R /C304,LO- GISTIC GROWTH CURVE ,P ERIOD THREE THEOREM , QUADRATIC MAP References Bechhoeffer, J. "The Birth of Period 3, Revisited." Math. Mag. 69, 115/C1/18, 1996. Beck, C.; and Schlo ¨gl, F. Thermodynamics of Chaotic Systems. Cambridge, England: Cambridge University Press, 1993. Bogomolny, A. "Chaos Creation (There is Order in Chaos)." http://www.cut-the-knot.com/blue/chaos.html. Costa, U. M. S. and Lyra, M. L. Phys. Rev. E 56, 245, 1997. Devaney, R. An Introduction to Chaotic Dynamical Systems, 2nd ed. Redwood City, CA: Addison-Wesley, 1989. Dickau, R. M. "Bifurcation Diagram." http://forum.swarth- more.edu/advanced/robertd/bifurcation.html. Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 69 /C1/0, 1988. Gordon, W. B. "Period Three Trajectories of the Logistic Map." Math. Mag. 69, 118/C1/20, 1996. Grassberger, P. "On the Hausdorff Dimension of Fractal Attractors." J. Stat. Phys. 26, 173/C1/79, 1981. Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189/C1/08, 1983. Gulick, D. Encounters with Chaos. New York: McGraw-Hill, 1992. Holmgren, R. A First Course in Discrete Dynamical Systems, 2nd ed. New York: Springer-Verlag, 1996. Kraft, R. L. "Chaos, Cantor Sets, and Hyperbolicity for the Logistic Maps." Amer. Math. Monthly 106, 400/C1/08, 1999. Latora, V.; Rapisarda, A.; Tsallis, C.; and Baranger, M. The Rate of Entropy Increase at the Edge of Chaos. 1999. http://xxx.lanl.gov/abs/cond-mat/9907412/. Lauwerier, H. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press, pp. 119 /C1/22, 1991. May, R. M. "Simple Mathematical Models with Very Com- plicated Dynamics." Nature 261, 459/C1/67, 1976. Peitgen, H.-O.; Ju ¨rgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer- Verlag, pp. 585 /C1/53, 1992. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 23, 1990. Robinson, C. Stability, Symbolic Dynamics, and Chaos. Boca Raton, FL: CRC Press, 1995. Russell, D. A.; Hanson, J. D.; and Ott, E. "Dimension of Strange Attractors." Phys. Rev. Let. 45, 1175 /C1/178, 1980. Saha, P. and Strogatz, S. H. "The Birth of Period Three." Math. Mag. 68,4 2/C1/7, 1995. Strogatz, S. H. Nonlinear Dynamics and Chaos. Reading, MA: Addison-Wesley, 1994. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Tsallis, C.; Plastino, A. R.; and Zheng, W.-M. Chaos, Solitons & Fractals 8, 885, 1997. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Wagon, S. "The Dynamics of the Quadratic Map." §4.4 in Mathematica in Action. New York: W. H. Freeman, pp. 117 /C1/40, 1991. Logistic Equation r /C304 With r /C304, the LOGISTIC EQUATION becomes xn/C271 /C304xn(1 /C28xn); (1) which is equivalent to the TENT MAP with m /C301 : Now let x /C13sin2(1 2 py) /C3012[1 /C28cos(py)] (2) ffiffiffixp/C30sin1 2 py/C(%/C(r (3) y /C302 psin/C281ffiffiffixp/CP/C( ; (4) so dy dx /C302p 1ffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 xp1 2 x/C281 =2 /C301 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x(1 /C28 x)p : (5) Manipulating (2) gives sin21 2 pyn/C271/C(%/C(r /C30412[1 /C28cos(pyn)] 1 /C28121 /C2812(1 /C28cos(pyn)hino /C302[1 /C28cos(py /C301 /C28cos2( pyn)sin2( pyn) ; (6) so 1 2 pyn/C271 /C309yn /C27sp (7) yn/C271 /C3092yn /C271 2 s : (8) But y /C23 [0; 1] : Taking yn /C23 [0; 1=2]; then s /C300 and yn/C271 /C302yn : (9) For y /C23 [1=2 ; 1]; s /C301 and yn/C271 /C302 /C282yn : (10) Combining gives yn/C271 /C302yn for yn /C23 0 ;12hi 2 /C282ynfor yn /C2312 ; 1hi ;8 < : (11) which can be written yn /C271 /C301 /C282 xn /C281 2/C()/C()/C()/C()/C()/C(); (12) which is just the TENT MAP with m /C301; whose NATURAL INVARIANT in y isr(y) /C301 : (13) Transforming back to x therefore gives r(x) /C30dy dx/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()r(y(x)) /C30 2 p1ffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 xp1 2 x/C281 =2 /C301 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x(1 /C28 x)p : (14) This can also be derived from r(x) /C30 lim N 0/C121 NXN i /C301d(xi /C28x) /C301 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix(1 /C28 x)p ; (15) where d(x) is the DELTA FUNCTION . See also LOGISTIC EQUATION ,TENT MAP References Jaffe, S. "The Logistic Equation: Computable Chaos." http:// www.mathsource.com/cgi-bin/msitem?0204 /C1/13. Whittaker, J. V. "An Analytical Description of Some Simple Cases of Chaotic Behavior." Amer. Math. Monthly 98, 489 /C1/04, 1991. Logistic Growth Curve The POPULATION GROWTH law which arises frequently in biology and is given by the differential equation dN dt/C30r(K /C28 N) K; (1) where r is the MALTHUSIAN PARAMETER and K is the so-called CARRYING CAPACITY (i.e., the maximum sustainable population). Rearranging and integrating both sides gives gN N0dN K /C28 N /C30r K gt 0dt (2) lnN0 /C28 K N/C28K ! /C30r Kt (3) N(t)/C30K/C27(N0/C28K)e/C28rt=K: (4) The curve y/C30a 1/C27bqx(5) is sometimes also known as the logical curve. See also GOMPERTZ CURVE ,LAW OF GROWTH ,LIFE EXPECTANCY ,LOGISTIC EQUATION ,MAKEHAM CURVE , MALTHUSIAN PARAMETER ,POPULATION GROWTH References Pearl, R. Ch. 18 in The Biology of Population Growth. New York: Knopf, 1978. Logistic Map LOGISTIC EQUATION Logit Transformation The function z /C30f(x) /C30lnx 1 /C28 x ! : This function has an inflection point at x /C301=2 ; where f ƒ(x) /C302x /C28 1 x2(x /C28 1)2 /C300: Applying the logit transformation to values obtained by iterating the LOGISTIC EQUATION generates a sequence of RANDOM NUMBERS having distribution Pz /C301 p(ex=2 /C27 e /C28x=2) ; which is very close to a GAUSSIAN DISTRIBUTION . References Collins, J.; Mancilulli, M.; Hohlfeld, R.; Finch, D.; Sandri, G.; and Shtatland, E. "A Random Number Generator Based on the Logit Transform of the Logistic Variable." Computers in Physics 6, 630 /C1/32, 1992. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 244 /C1/ 45, 1995. Logos A generalization of a HEYTING ALGEBRA which re- places BOOLEAN ALGEBRA in "intuitionistic" LOGIC . See also TOPOS Log-Series Distribution The terms in the series expansion of ln(1 /C28 u) about u /C300 are proportional to this distribution. P(n) /C30/C28un n ln(1 /C28 u) (1) D(n) /C13Xn i /C301P(i) /C30u1 /C27n F( u; 1; 1 /C27 n) /C27 ln(1 /C28 u) ln(1 /C28 u) ; (2) where F is the LERCH TRANSCENDENT . The MEAN , VARIANCE , SKEWNESS , and KURTOSISm /C30u (u /C28 1) ln(1 /C28 u) (3) s2 /C30/C28u[ u /C27 ln(1 /C28 u)] ( u /C28 1)2[ln(1 /C28 u)]2 (4) g1 /C302 u2 /C27 3u ln(1 /C28 u) /C27 (1 /C27 u)ln2(1 /C28 u) ln(1 /C28 u)[ u /C27 ln(1 /C28 u)]ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28u[u /C27 ln(1 /C28 u)]p /C2ln(1 /C28 u) (5) g2 /C306 u3 /C27 12 u2 ln(1 /C28 u) /C27 u(7 /C27 4u)ln2(1 /C28 u) u[ u /C27 ln(1 /C28 u)]2 /C27(1 /C27 4 u /C27 u2)ln3(1 /C28 u) u[ u /C27 ln(1 /C28 u)]2: (6) Log-Weibull Distribution FISHER- TIPPETT DISTRIBUTION Lommel Differential Equation A generalization of the BESSEL DIFFERENTIAL EQUA- TION z2d2y dz2 /C27zdy dz /C28(z2 /C27 n2)y /C30kz m/C271 (Watson 1966, p. 345; Zwillinger 1997, p. 125; Gradshteyn and Ryzhik 2000, p. 986). A further generalization gives z2d2y dz2 /C27zdy dz /C28(z2 /C27 n2)y /C309kz m/C271 : The solutions are L OMMEL FUNCTIONS . See also LOMMEL FUNCTION References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 986, 2000. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 125, 1997. Lommel Function There are several functions called "Lommel func- tions." One type of Lommel function is the solution to the L OMMEL DIFFERENTIAL EQUATION with a PLUS SIGN, given by y/C30ksm;n(z); (1) where s(/C27) m; n(z) /C131 2pYn(z)gz 0z mJn(z) dz /C28Jn(z)gz 0z mYn(z) dz/C)P/C)( : (2) Here, Jn(z) and Y n(z) are BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS (Watson 1966, p. 346). If a minus sign precedes k, then the solution is s /C28 m; n /C13I n(z)gc1 zz mKn(z) dz /C28J n(z)gz c2z mIn(z) dz; (3) where Kn(z) and In(z) are MODIFIED BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS . Lommel functions of two variables are related to the BESSEL FUNCTION OF THE FIRST KIND and arise in the theory of diffraction and, in particular, Mie scattering (Watson 1966, p. 537), Un(w ; z) /C30X/C12 m/C300(/C281)mw z !n/C272m Jn/C272m(z) (4) Vn(w; z) /C30X/C12 m/C300(/C281)mw z !/C28n/C282m J/C28n /C282m(z) : (5) See also LOMMEL DIFFERENTIAL EQUATION ,LOMMEL POLYNOMIAL References Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 369, 1960. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Lommel Functions sm ; n(x) and Sm; n(x):/" §1.5 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 28 /C1/9, 1990. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Lommel Polynomial Rm; n(z) /C30 G( n /C27 m) G( n)(z =2)m 2 F3(1 2(1 /C28m);/C2812 m; n ;/C28m; 1 /C28 n /C28m; z2) /C2pz 2 sin( np)[J n/C27m(z)J/C28 n/C271(z) /C27(/C281)mJ/C28 n/C28m(z)Jn /C281(z)]; where G(z)isa GAMMA FUNCTION , Jn(x)isaB ESSEL FUNCTION OF THE FIRST KIND , and2F3(a; b; c ; d; e; z) is a GENERALIZED HYPERGEOMETRIC FUNCTION . See also LOMMEL FUNCTION References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980.Lommel’s Integrals ( b2 /C28 a2)g xJn( ax)Jn( bx) dx /C30x[ aJ ?n(ax)Jn( bx) /C28 bJ ?n(bx)Jn(ax)] g xJ2 n( ax) dx /C301 2 x2[J2 n( ax) /C27Jn /C281( ax)Jn /C271( ax)] ; where Jn(x)isaB ESSEL FUNCTION OF THE FIRST KIND . References Bowman, F. Introduction to Bessel Functions. New York: Dover, p. 101, 1958. Long Cross DAGGER Long Division Long division is an algorithm for dividing two numbers, obtaining the QUOTIENT one DIGIT at a time. The above example shows how the division of / 123456 =17/ is performed to obtain the result 7262.11.... See also DIVISION References Beck, G. "Long Multiplication and Division." M ATHEMATICA NOTEBOOK LONGDIVISION.NB . Longest Increasing Scattered Subsequence The longest increasing scattered subsequence is the longest subsequence of increasing terms, where inter-vening nonincreasing terms may be dropped. Findingthe largest scattered subsequence is a much harder problem. The longest increasing scattered subse- quence of a PARTITION can be found using Long- estIncreasingSubsequence [p] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). For example, the long- est increasing scattered subsequence of the PERMUTATION f6; 3; 4; 8; 10; 5; 7; 1; 9; 2 g is f3; 4; 5; 7; 9g; whereas the longest contiguous sub- sequence is f3; 4 ; 8 ; 10 g:/ Any sequence of n2 /C271 distinct integers must contain either an increasing or decreasing scattered subse- quence of length n /C271 (Erdos and Szekeres 1935; Skiena 1990, p. 75). See also LONGEST INCREASING SUBSEQUENCE ,PER- MUTATION References Erdos, P. and Szekeres, G. "A Combinatorial Problem in Geometry." Compos. Math. 2, 464 /C1/70, 1935. Schensted, C. "Longest Increasing and Decreasing Subse- quences." Canad. J. Math. 13, 179 /C1/91, 1961. Skiena, S. "Longest Increasing Subsequences." §2.3.6 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 73 /C1/5, 1990. Longest Increasing Subsequence The longest increasing subsequence of a given se- quence is the subsequence of increasing terms con- taining the largest number of elements. For example, the longest increasing subsequence of the PERMUTA- TION f6 ; 3 ; 4 ; 8; 10 ; 5 ; 7 ; 1; 9; 2g is f3; 4; 8; 10g:/ See also LONGEST INCREASING SCATTERED SUBSE- QUENCE References Skiena, S. "Longest Increasing Subsequences." §2.3.6 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 73 /C1/5, 1990. Long Exact Sequence See also LONG EXACT SEQUENCE OF A PAIR AXIOM Long Exact Sequence of a Pair Axiom One of the EILENBERG- STEENROD AXIOMS . It states that, for every pair (X, A), there is a natural long exact sequence ... 0 Hn(A) 0 Hn(X) 0 Hn(X ; A) 0 Hn /C281(A) 0 ...; (1) where the MAP Hn(A) 0 Hn(X) is induced by the INCLUSION MAP A 0 X and Hn(X) 0 Hn(X ; A)is induced by the INCLUSION MAP (X ; f) 0 (X ; A) : The MAP Hn(X ; A) 0 Hn/C281(A) is called the BOUNDARY MAP. See also EILENBERG- STEENROD AXIOMS Longimeter A longimeter is a transparent sheet of plastic with a regular grid of lines inclined at an angle of 308 to thesides of the sheet. By counting the number of squares occupied by a linear feature on a map (such as a river) for six different rotations of the sheet, the length of the feature can be determined. See also COASTLINE PARADOX References Steinhaus, H. Mitteilungen der Sa¨chsischen Akad. 82, 120 /C1/ 30, 1930. Steinhaus, H. Przeglad Geogr. 21, 1947. Steinhaus, H. Comptes Rendus Soc. des Sciences et des Lettres de Wroc /l/aw, Se´r. B, 1949. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 105 /C1/10, 1999. Longitude The azimuthal coordinate on the surface of a SPHERE (/ u in SPHERICAL COORDINATES )orona SPHEROID (in PROLATE or OBLATE SPHEROIDAL COORDINATES ). Long- itude is defined such that 0/C14/C30360/C14: Lines of constant longitude are generally called MERIDIANS . The other angular coordinate on the surface of a SPHERE is called the LATITUDE . The shortest distance between any two points on a SPHERE is the so-called GREAT CIRCLE distance, which can be directly computed from the LATITUDE and longitudes of two points. See also GREAT CIRCLE ,LATITUDE ,MERIDIAN ,OBLATE SPHEROIDAL COORDINATES ,P ROLATE SPHEROIDAL COORDINATES Longitudinal Data Data resulting from the observation of a population on a number of variables over time. Wheneverobservations are made more than once, the data is considered to be longitudinal. References Bijleveld, C. C. J. H.; van der Kamp, L. J. T.; Mooijaart, A.; van der Kloot, W. A.; van der Leeden, R.; and van der Burg, E. Longitudinal Data Analysis: Designs, Models and Methods. London: Sage, 1998. Long Prime FULLREPTEND PRIME Look and Say Sequence The INTEGER SEQUENCE beginning with a single digit in which the next term is obtained by describing the previous term. Starting with 1, the sequence would be defined by "1, one 1, two 1s, one 2 one 1," etc., and theresult is 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... (Sloane’s A005150). Starting the sequence instead with the digit dfor 25 d59 gives d,1d, 111 d, 311 d, 13211 d, 111312211 d, 31131122211 d, 1321132132211 d, ... The sequences ford/C302 and 3 are Sloane’s A006751 and A006715. n terms of the look and say sequence (given as lists of digits) starting with digit d can be implemented in Mathematica as follows. RunLengthEncode[x_List] : /C30 (Through[{First, Length}[#]] &) /@ Split[x] LookAndSay[n_Integer?Positive, d_:1] : /C30 NestList[Flatten[Reverse /@ RunLengthEncode[#]] &, {d}, n - 1] The number of DIGITS in the nth term the sequence for 1 5d 59 is given by the sequence 1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, ... (Sloane’s A005341), which is asymptotic to C ln ; where C is a constant and l /C301 :303577269034296... (Sloane’s A014715) is CONWAY’S CONSTANT , given by the unique positive real root of the POLYNOMIAL 0 /C30x71 /C28x69 /C282x68 /C28x67 /C272x66 /C272x65 /C27x64 /C28x63 /C28x62 /C28x61 /C28x60 /C28x59 /C272x58 /C275x57 /C273x56 /C282x55 /C2810x54 /C283x53 /C282x52 /C276x51 /C276x50 /C27x49 /C279x48 /C283x47 /C287x46 /C288x45 /C288x44 /C2710x43 /C276x42 /C278x41 /C284x40 /C2812x39 /C277x38 /C287x37 /C277x36 /C27x35 /C283x34 /C2710x33 /C27x32 /C286x31 /C282x30 /C2810x29 /C283x28 /C272x27 /C279x26 /C283x25 /C2714x24 /C288x23 /C287x21 /C279x20 /C283x19 /C284x18 /C2810x17 /C287x16 /C2712x15 /C277x14 /C272x13 /C2812x12 /C284x11 /C282x10 /C285x9 /C27x7 /C287x6 /C277x5 /C284x4 /C2712x3 /C286x2 /C273x /C286 : In fact, the constant is even more general than this, applying to all starting sequences (i.e., even those starting with arbitrary starting digits), with the exception of 22, a result which follows from the COSMOLOGICAL THEOREM . Conway discovered that strings sometimes factor as a concatenation of two strings whose descendants never interfere with one another. A string with no nontrivial splittings is called an "element," and other strings are called "compounds." Every string of 1s, 2s, and 3s eventually "decays" into a compound of 92 special elements, named after the chemical elements. See also CONWAY’S CONSTANT ,COSMOLOGICAL THEO- REM,RUN-LENGTH ENCODING References Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay." Eureka 45,5/C1/8, 1985. Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay." §5.11 in Open Problems in Commu- nications and Computation. (Ed. T. M. Cover and B. Go- pinath). New York: Springer-Verlag, pp. 173 /C1/88, 1987. Conway, J. H. and Guy, R. K. "The Look and Say Sequence." In The Book of Numbers. New York: Springer-Verlag, pp. 208 /C1/09, 1996. Hilgemeier, M. "Die Gleichniszahlen-Reihe." Bild der Wis- sensch. 12, 19, 1986. Hilgemeier, M. "‘One Metaphor Fits All’: A Fractal Voyage with Conway’s Audioactive Decay." Ch. 7 in Pickover,C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York: St. Martin’s Press, 1996. Sloane, N. J. A. Sequences A005150/M4780, A005341/ M0321, A006715/M2965, and A006751/M2052 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 13 /C1/4, 1991. Loop A path whose initial and final points coincide in a fixed point p known as the BASEPOINT . Loop (Algebra) A QUASIGROUP with an IDENTITY ELEMENT e such that xe /C30x and ex /C30x for any x in the QUASIGROUP . All GROUPS are loops. See also GROUP ,QUASIGROUP References Albert, A. A. (Ed.). Studies in Modern Algebra. Washington, DC: Math. Assoc. Amer., 1963. Loop (Graph) A degenerate edge of a graph which joins a vertex to itself, also called a self-loop. A SIMPLE GRAPH cannot contain any loops, but a PSEUDOGRAPH can contain both multiple edges and loops. See also PSEUDOGRAPH ,SIMPLE GRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 82, 1990. Loop (Knot) AKNOT orHITCH which holds its form rigidly. References Owen, P. Knots. Philadelphia, PA: Courage, p. 35, 1993. Loop Space LetYXbe the set of continuous mappings f:X0Y: Then the TOPOLOGICAL SPACE forYXsupplied with a compact-open topology is called a MAPPING SPACE , and if Y /C30I is taken as the interval (0 ; 1); then YI /C30V(Y) is called a loop space (or SPACE OF CLOSED PATHS ). See also MACHINE ,M APPING SPACE ,M AY-THOMASON UNIQUENESS THEOREM References Brylinski, J.-L. Loop Spaces, Characteristic Classes and Geometric Quantization. Boston, MA: Birkha ¨user, 1993. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 658, 1980. Lopez Minimal Surface See also MINIMAL SURFACE Lorentz Group The Lorentz group is the GROUP L of time-preserving linear ISOMETRIES of MINKOWSKI SPACE R4 with the pseudo-Riemannian metric dr2 /C30/C28dt2 /C27dx2 /C27dy2 /C27dz2 : It is also the GROUP of ISOMETRIES of 3-D HYPERBOLIC SPACE . It is time-preserving in the sense that the unit time VECTOR (1; 0; 0; 0) is sent to another VECTOR (t; x; y; z) such that t /C210. A consequence of the definition of the Lorentz group is that the full GROUP of time-preserving isometries of MINKOWSKI R4 is the GROUP DIRECT PRODUCT of the group of translations of R4(i.e., R4itself, with addition as the group operation), with the Lorentz group, and that the full isometry group of the MINKOWSKI R4 is a group extension of Z2by the product L /C156R4 :/ The Lorentz group is invariant under space rotations and LORENTZ TRANSFORMATIONS . See also LORENTZ TENSOR ,LORENTZ TRANSFORMA- TION References Arfken, G. "Homogeneous Lorentz Group." §4.13 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 271 /C1/75, 1985. Lorentz Tensor The TENSOR in the LORENTZ TRANSFORMATION given by L /C13g /C28gb 00 /C28gb g 00 001 0 000 12 6643 775; (1) where beta and gamma are defined by b /C13v c (2)g /C131ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2q : (3) See also LORENTZ GROUP ,LORENTZ TRANSFORMATION Lorentz Transformation A 4-D transformation satisfied by all FOUR-VECTORS an; a?m/C30Lm nan: (1) In the theory of special relativity, the Lorentz transformation replaces the G ALILEAN TRANSFORMA- TION as the valid transformation law between refer- ence frames moving with respect to one another atconstant VELOCITY . Let xnbe the POSITION FOUR- VECTOR with x0/C30ct;and let the relative motion be along the x1axis with VELOCITY v. Then (1) becomes x?m/C30Lm nxn; (2) where the L ORENTZ TENSOR is given by L/C30L00L01L02L03 L10L11L12L13 L20L21L22L23 L30L31L32L332 6643 775/C13g/C28gb00 /C28gb g 00 00 1 0 00 0 12 6643 775:(3) Here, b/C13 v c(4) g/C131ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2q : (5) Written explicitly, the transformation between xnand xn?coordinate is x0?/C30g(x0/C28bx1) (6) x1?/C30g(x1/C28bx0) (7) x2?/C30x2(8) x3?/C30x3: (9) The DETERMINANT of the upper left 2 /C292MATRIX in (3) is D/C30(g)2/C28(/C28gb)2/C30g2(1/C28b2)/C30g2 g2/C301; (10) so L/C281/C30(L/C281)0 0(L/C281)01(L/C281)02(L/C281)03 (L/C281)10(L/C281)11(L/C281)12(L/C281)13 (L/C281)20(L/C281)21(L/C281)22(L/C281)23 (L/C281)30(L/C281)31(L/C281)32(L/C281)332 66643 7775 /C13gg b 00 gb g 00 0010 00012 6643 775: (11) A Lorentz transformation along the x 1 /-axis can also be written x0 ? x1 ? x2 ? x3 ?2 6643 775cosh u /C28sinh u 00 /C28sinh u cosh u 00 00 1 0 00 0 12 6643 775x0 x1 x2 x32 6643 775: (12) where u is called the rapidity, x0 /C13ct ; (13) and tanh u /C13 b /C13v c (14) cosh u /C13 g /C131ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 b2q (15) sinh u /C30 gb: (16) See also HYPERBOLIC ROTATION ,LORENTZ GROUP , LORENTZ TENSOR References Fraundorf, P. "Accel-1D: Frame-Dependent Relativity at UM-StL." http://www.umsl.edu/~fraundor/a1toc.html. Griffiths, D. J. Introduction to Electrodynamics. Englewood Cliffs, NJ: Prentice-Hall, pp. 412 /C1/14, 1981. Morse, P. M. and Feshbach, H. "The Lorentz Transforma- tion, Four-Vectors, Spinors." §1.7 in Methods of Theore- tical Physics, Part I. New York: McGraw-Hill, pp. 93 /C1/07, 1953. Lorentzian Distribution CAUCHY DISTRIBUTION Lorentzian Function The Lorentzian function is the singly peaked functiongiven by L(x) /C301 p1 2 G (x /C28 x0)2 /C2712 G/C(%/C(r2 : (1) It is normalized to that g/C12 /C28/C12L(x) /C301: (2) It has a maximum at x /C30x0 ; where L?(x) /C30/C2816(x /C28 x0) G p[4(x /C28 x0)2 /C27G2] /C300 : (3) Its value at the maximum is L(x0) /C302 p G: (4) It is equal to half its maximum at x /C30 x0 91 2 G/C(%/C(r ; (5) and so has FULL WIDTH AT HALF MAXIMUM G: The function has inflection points at Lƒ(x) /C3016 G12(x /C28 x0)2 /C28G2 p[4(x /C28 x0)2 /C27G2] /C300 ; (6) giving x1 /C30x0 /C281 6ffiffiffi 3p G; (7) where L(x1) /C303 2p G: (8) The Lorentzian function gives the shape of certain types of spectral lines and is the distribution function in the C AUCHY DISTRIBUTION . The Lorentzian func- tion has F OURIER TRANSFORM F1 p1 2G (x/C28x0)2/C27(1 2G)2"# /C30e/C282pikx0/C28Gpkjj: (9) See also CAUCHY DISTRIBUTION ,DAMPED EXPONEN- TIAL COSINE INTEGRAL ,FOURIER TRANSFORM– LOR- ENTZIAN FUNCTION Lorentzian Inner Product The standard Lorentzian inner product on R4is given by /C28dx2 0/C27dx21/C27dx22/C27dx23; (1) i.e, for vectors vandw, /C142v;w/C143/C30/C28v0w0/C27v1w1/C27v2w2/C27v3w3: (2) The Lorentzian inner product is used in special relativity as a measurement, replacing distances, which is independent of reference frame. The vari- ables x1 ; x2 ; and x3can be thought of as space variables, and the x0variable as the time variable. Sometimes, the time variable is labelled t instead of x0 and when used in special relativity, x0 /C30ct; where c is the speed of light. The formula (1) uses the convention that units are chosen so that the speed of light has the value c /C301 in order to simplify formulas. For a vector v, the sign of /C142v; v/C143 determines the type of v. If it is positive, then v is a space-like vector. If it is zero, then v is called a null vector, or light-like vector. If it is negative, then v is called a time-like vector. After a change of variables, it is possible to rewrite the Lorentzian inner product as above where t is in the direction of a given time-like vector v with /C142v; v/C143/C30/C281 : Such a change of variables corresponds to a change in reference frame. Altogether, these form the LORENTZ GROUP , also called the ORTHOGONAL GROUP O(3; 1):/ See also ORTHOGONAL GROUP Lorenz Asymmetry Coefficient This entry contributed by CHRISTIAN DAMGAARD The Lorenz asymmetry coefficient is a summary statistic of the Lorenz curve that measures the degree of asymmetry of a LORENZ CURVE . The Lorenz asymmetry coefficient is defined as S /C13F( m) /C27L( m) ; (1) where the functions F and L are defined as for the Lorenz curve. If S /C211, then the point where the LORENZ CURVE is parallel with the line of equality is above the axis of symmetry. Correspondingly, if S B1, then the point where the LORENZ CURVE is parallel to the line of equality is below the axis of symmetry.The sample statistic S can be calculated from ordered size data using the following equations d/C30m/C28x?m x?m/C271/C28x?m(2) F(m)/C30m/C27d n(3) L(m)/C30Lm/C27dx?m/C271 Ln; (4) where mis the number of individuals with a size less than m:/ See also GINI COEFFICIENT ,LORENZ CURVEReferences Damgaard, C. and Weiner, J. "Describing Inequality in Plant Size or Fecundity." Ecology 81, 1139 /C1/142, 2000. Lorenz Attractor The Lorenz attractor is a STRANGE ATTRACTOR that arises in a simplified system of equations describing the 2-D flow of fluid of uniform depth H, with an imposed temperature difference DT;under gravity g, with buoyancy a;thermal diffusivity k;and kinematic viscosity n:The full equations are @ @t(92f)/C30@c @z@ @x(92c)/C28@c @x@ @z(92c)/C27n92(92c) /C27gadT dx(1) @T @t/C30@T @z@c @x/C28@u @x@c @z/C27k92T/C27DT H@c @x: (2) Here, cis the "stream function," as usual defined such that u/C30@c @x;v/C30@c @x: (3) In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified system, periodic solutions OF THE FORM c/C30c0sinpax H ! sinpz H ! (4) u/C30u0cospax H ! sinpz H ! (5) grew for Rayleigh numbers larger than the criticalvalue, Ra>Ra c:Furthermore, vastly different re- sults were obtained for very small changes in theinitial values, representing one of the earliest dis- coveries of the so-called BUTTERFLY EFFECT . Lorenz included the following terms in his system of equations, X/C13c118convective intensity (6) Y/C13T11 8DTbetween descending and ascending currents (7) Z/C13T02 8Dvertical temperature profile from linearity ; (8) and obtained the simplified equations ˙X/C30s(Y/C28X) (9) ˙Y /C30/C28XZ /C27rX /C28Y (10) ˙Z /C30XY /C28bZ; (11) now known as the LORENZ EQUATIONS , where ˙X /C30 dX =dt; ˙Y /C30dY =dt; ˙Z /C30dZ=dt; and s /C13n k /C30Prandtl number (12) r /C13Ra Rac/C30normalized Rayleigh number (13) b /C134 1 /C27 a2 /C30geometric factor : (14) Lorenz took b /C138=3 and s /C1310:/ The CRITICAL POINTS at (0, 0, 0) correspond to no convection, and the CRITICAL POINTS at ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b(r /C281)p ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib(r /C281)p ; r /C281/C(%/C(r (15) and /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib(r /C281)p ;/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib(r /C281)p ; r /C281/C(%/C(r (16) correspond to steady convection. This pair is stable only if r /C30 s(s /C27 b /C27 3) s /C28 b /C28 1; (17) which can hold only for POSITIVE r if s > b /C271: The Lorenz attractor has a CORRELATION EXPONENT of 2.05 9 0.01 and CAPACITY DIMENSION 2.06 9 0.01 (Grassberger and Procaccia 1983). For more details, see Lichtenberg and Lieberman (1983, p. 65) and Tabor (1989, p. 204). See also BUTTERFLY EFFECT ,L ORENZ EQUATIONS , RO¨ SSLER MODEL References Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 27 /C1/1, 1988.Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/08, 1983. Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, 1983. Lorenz, E. N. "Deterministic Nonperiodic Flow." J. Atmos. Sci. 20, 130 /C1/41, 1963. Lorenz, E. N. "On the Prevalence of Aperiodicity in Simple Systems." In Global Analysis: Proceedings of the Biennial Seminar of the Canadian Mathematical Congress Held at the University of Calgary, Alberta., June 12 /C1/7 (Ed. M. Grmela and J. E. Marsden). New York: Springer-Ver- lag, pp. 53 /C1/5, 1979. Peitgen, H.-O.; Ju¨rgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer- Verlag, pp. 697 /C1/08, 1992. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 0821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Provi- dence, RI: Amer. Math. Soc., 2000. Sparrow, C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. New York: Springer-Verlag, 1982. Stewart, I. "The Lorenz Attractor Exists." Nature 406, 948 /C1/ 49, 2000. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Viana, M. "What’s New on Lorenz Strange Attractors." Math. Intell. 22,6/C1/9. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 142 /C1/43, 1991. Lorenz Curve This entry contributed by CHRISTIAN DAMGAARD The Lorenz curve is used in economics and ecology to describe inequality in wealth or size. The Lorenz curve is a function of the cumulative proportion of ordered individuals mapped onto the corresponding cumulative proportion of their size. Given a sample of n ordered individuals with x?ithe size of individual i and x?1 Bx?2 B...Bx?n ; then the sample Lorenz curve is the polygon joining the points (h=n; Lh =Ln) ; where h /C30 0, 1, 2, ...n, L0 /C300; and Lh /C30ah i/C301 x?i : Alternatively, the Lorenz curve can be expressed as L(y) /C30gy 0xdF (x) m; where F(y) is the cumulative distribution function of ordered individuals and m is the average size. If all individuals are the same size, the Lorenz curve is a straight diagonal line, called the line of equality. If there is any inequality in size, then the Lorenz curve falls below the line of equality. The total amount of inequality can be summarized by the GINI COEFFICIENT (also called the Gini ratio), which is the ratio between the area enclosed by the line of equality and the Lorenz curve, and the total trian- gular area under the line of equality. The degree ofasymmetry around the axis of symmetry is measured by the so-called L ORENZ ASYMMETRY COEFFICIENT . See also GINI COEFFICIENT ,L ORENZ ASYMMETRY COEFFICIENT References Dagum, C. "The Generation and Distribution of Income, the Lorenz Curve and the Gini Ratio." E´ con. Appl. 33, 327 /C1/ 67, 1980. Kotz, S.; Johnson, N. L.; and Read, C. B. Encyclopedia of Statistical Science. New York: Wiley, 1983. Lorenz, M. O. "Methods for Measuring the Concentration of Wealth." Amer. Stat. Assoc. 9, 209 /C1/19, 1905. Weiner, J. and Solbrig, O. T. "The Meaning and Measure- ment of Size Hierarchies in Plant Populations." Oecologia 61, 334 /C1/36, 1984. Lorenz Equations The system of ordinary differential equations ˙X /C30 s(Y /C28X) (1) ˙Y /C30rX /C28Y /C28XZ (2) ˙Z /C30XY /C28bZ ; (3) See also LORENZ ATTRACTOR References Sparrow, C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. New York: Springer-Verlag, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997. Lorenz System LORENZ ATTRACTOR ,LORENZ EQUATIONS Lorraine Cross GAULLIST CROSS Lo Shu The unique MAGIC SQUARE of order three. The Lo Shu is an ASSOCIATIVE MAGIC SQUARE , but not a PANMAGIC SQUARE . See also ASSOCIATIVE MAGIC SQUARE ,MAGIC SQUARE , PANMAGIC SQUARE References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 19 and 24, 1984. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 23 /C1/4, 1975. Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 146 /C1/47, 1942. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 75 /C1/6, 1986.Lossnitsch’s Triangle 1 11 111 1221 12421 136631 13910931 1 4 12 19 19 12 4 1 1 4 16 28 38 28 16 4 1 1 5 20 44 66 66 44 20 5 1 1 5 25 60 110 126 110 60 25 5 1 AP ASCAL’S TRIANGLE -like array of numbers for which each term is the sum of the two numbers immediately above it, except that, numbering the rows by n /C300, 1, 2, ... and the entries in each row by k /C300, 1, 2, ..., if n is EVEN and k is ODD, subtractn =2/C281 (k/C281)=2/C(%/C(r : Analytically, a(n ; k) /C30a(n /C281; k /C281) /C27a(n /C281; k) /C28n=2 /C281 (k /C281)=2/C(*/C(+ ; where the last term is present only if n is EVEN and k is ODD. References Lossnitsch, S. M. "Die Isometrie-Arten ... Paraffin-Reihe." Chem. Ber. 30, 1917 /C1/926, 1897. Sloane, N. J. A. http://www.research.att.com/~njas/se- quences/classic.html#LOSS. Sloane, N. J. A. Sequences A034851 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Los’ Theorem Let I be a set, and let U be an ULTRAFILTER on I, let f be a formula of a given language L, and let fAi : i /C23 I g be any collection of structures which is indexed by the setI. Denote by [ x]Uthe EQUIVALENCE CLASS ofx under U;for any element xof the productQ i/C23IAi: Then the ULTRAPRODUCTQ i/C23IA/CP/C( =Usatisfies fvia a valuation s/C30[(xi)i/C23I]UinQ i/C23IA/CP/C( =Uif and only if Tarski’s recursive definition of SATISFACTION holds, i/C23I:Aiffixifno /C23U: See also NONSTANDARD ANALYSIS ,TRANSFER PRINCI- PLE References Bell, J. L. and Slomson, A. B. Models and Ultraproducts: An Introduction. Amsterdam, Netherlands: North-Holland, 1971. Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985. Lost in a Forest Problem The problem of finding the strategy to guarantee reaching the boundary of a given region ("forest") in the shortest distance (i.e., a strategy having the best worst-case performance). For example, one simple strategy would consist of walking in a straight line in a random direction until encountering a boundary. Although this straightforward approach is indeed the best for some simple geometries, other approaches (e.g., walking in a spiral, alternating left and right turns after traveling some fixed distance, etc.) might be optimal for forests with more complicated bound- aries. References Bellman, R. "Minimization Problem." Bull. Amer. Math. Soc. 62, 270, 1956. Berzsenyi, G. "Lost in a Forest (A Problem Area Initiated by the Late Richard E. Bellman)." Quantum , p. 41, Nov./Dec. 1995. Finch, S. "Unsolved Mathematics Problems: Lost in a Forest." http://www.mathsoft.com/asolve/forest/for- est.html. Lotka-Volterra Equations An ecological model which assumes that a population x increases at a rate dx /C30Ax dt; but is destroyed at a rate dx /C30/C28Bxy dt: Population y decreases at a rate dy /C30/C28Cy dt; but increases at dy /C30Dxy dt; giving the coupled differential equations dx dt /C30Ax /C28Bxy (1) dy dt /C30/C28Cy /C27Dxy : (2) Critical points occur when dx=dt /C30dy=dt /C300 ; so A /C28By /C300 (3) /C28C /C27Dx /C300 : (4)The sole STATIONARY POINT is therefore located at (x; y) /C30(C=D ; A=B) :/ References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, p. 494, 1992. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 135, 1997. Lova´sz Number Let q(G) be the Lova´sz number of a GRAPH of G. Then v(G) 5q( ¯G) 5 x(G) ; where v(G) is the CLIQUE NUMBER and x(G) is the minimum number of colors needed to color the VERTICES of G. This is the SANDWICH THEOREM . See also CLIQUE NUMBER ,C OLORING ,S ANDWICH THEOREM References Knuth, D. E. "The Sandwich Theorem." Electronic J. Com- binatorics 1,A11 /C1/8, 1994. http://www.combinatorics.org/ Volume_1/volume1.html#A1. Love Transform The INTEGRAL TRANSFORM (Kf)(x) /C30g/C12 /C28/C12(x /C28 t)c/C281 /C27 G(c)2 F1a ; b; c;1/C28t x ! f(t) dt; where G(x) is the GAMMA FUNCTION ,2F1(a ; b; c; z)is a HYPERGEOMETRIC FUNCTION , where ya /C27denotes the TRUNCATED POWER FUNCTION . References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993. Low-Dimensional Topology Low-dimensional topology usually deals with objects that are 2-, 3-, or 4-dimensional in nature. Properly speaking, low-dimensional topology should be part of DIFFERENTIAL TOPOLOGY , but the general machinery of ALGEBRAIC and DIFFERENTIAL TOPOLOGY gives only limited information. This fact is particularly notice- able in dimensions three and four, and so alternative specialized methods have evolved. See also ALGEBRAIC TOPOLOGY ,DIFFERENTIAL TOPOL- OGY,HIGHER DIMENSIONAL GROUP THEORY ,TOPOL- OGY References Boroczky, K. Jr.; Neumann, W.; and Stipsicz, A. (Eds.). Low Dimensional Topology. Budapest, Hungary: Ja ´nos Bolyai Mathematical Society, 1999. Brown, R. and Thickstun, T. L. (Eds.). Low-Dimensional Topology: Proceedings of a Conference on Topology in Low Dimension, Bangor, 1979. Cambridge, England: Cam- bridge University Press, 1982. Stillwell, J. Classical Topology and Combinatorial Group Theory, 2nd ed. New York: Springer-Verlag, 1993. Lo¨wenheim-Skolem Theorem A fundamental result in MODEL THEORY which states that if a countable theory has a model, then it has a countable model. Furthermore, it has a model of every CARDINALITY greater than or equal to /C2100 (ALEPH-0 ). This theorem established the existence of "nonstandard" models of arithmetic. See also ALEPH-0 ,CARDINALITY ,GO¨ DEL’S COMPLETE- NESS THEOREM ,MODEL THEORY References Berry, G. D. W. Symposium on the Ontological Significance of the Lo¨wenheim-Skolem Theorem, Academic Freedom, Logic, and Religion. Philadelphia, PA: Amer. Philos. Soc., pp. 39 /C1/5, 1953. Beth, E. W. "A Topological Proof of the Theorem of Lo¨w- enheim-Skolem-Go ¨del." Nederl. Akad. Wetensch., Ser. A 54, 436 /C1/44, 1951. Beth, E. W. "Some Consequences of the Theorem of Lo¨w- enheim-Skolem-Go ¨del-Malcev." Nederl. Akad. Wetensch., Ser. A 56,66/C1/1, 1953. Chang, C. C. and Keisler, H. J. Model Theory, 3rd enl. ed. New York: Elsevier, 1990. Church, A. §45 and 49 in Introduction to Mathematical Logic. Princeton, NJ: Princeton University Press, 1996. Curry, H. B. Foundations of Mathematical Logic, 2nd rev. ed. New York: Dover, pp. 6 /C1/,95/C1/6, and 121, 1977. Fraenkel, A. A. and Bar-Hillel, Y. Foundations of Set Theory. Amsterdam, Netherlands, p. 105, 1958. Myhill, J. Symposium on the Ontological Significance of the Lo¨wenheim-Skolem Theorem, Academic Freedom, Logic, and Religion. Philadelphia, PA: Amer. Philos. Soc., pp. 57 /C1/0, 1953. Quine, W. V. "Completeness of Quantification Theory: Lo¨w- enheim’s Theorem." Appendix to Methods of Logic, rev. ed. New York: pp. 253 /C1/60, 1959. Quine, W. V. "Interpretation of Sets of Conditions." J. Symb. Logic 19,97/C1/02, 1954. Rasiowa, H. and Sikorski, R. "A Proof of the Lo¨wenheim- Skolem Theorem." Fund. Math. 38, 230 /C1/32, 1952. Skolem, T. "Sur la porte´e du the´ore`me de Lo¨wenheim- Skolem." Les Entretiens de Zurich sur les fondements et la me´thode des sciences mathe ´matiques (December 6 /C1/, 1938), pp. 25 /C1/2, 1941. Vaught, R. L. "Applications of the Lo¨wenheim-Skolem- Tarski Theorem to Problems of Completeness and Decid- ability." Nederl. Akad. Wetensch., Ser. A 57, 467 /C1/72, 1954. Lower Bound A function f is said to have a lower bound c if c 5f(x) for all x in its DOMAIN . The GREATEST LOWER BOUND is called the INFIMUM . See also INEQUALITY ,INFIMUM ,SUPREMUM ,U PPER BOUND Lower Central Series (Lie Algebra) The lower central series of a LIE ALGEBRA g is the sequence of subalgebras recursively defined bygk /C271 /C30[g;gk]; (1) with g0 /C30g: The sequence of subspaces is always decreasing with respect to inclusion or dimension, and becomes stable when g is finite dimensional. The notation [ a;b] means the linear span of elements of the form [A, B], where A /C23a and B /C23b:/ When the lower central series ends in the zero subspace, the Lie algebra is called NILPOTENT . For example, consider the LIE ALGEBRA of strictly UPPER TRIANGULAR MATRICES , then g0/C300a12a13a14a15 00 a23a24a25 00 0 a34a35 0 000 a45 0 00002 666643 77775(2) g 1/C3000 a13a14a15 00 0 a24a25 00 0 0 a35 0 0 000 0 0 0002 666643 77775(3) g 2/C30000 a14a15 000 0 a25 000 0 0 000 0 0 000 0 02 666643 77775(4) g 3/C300000 a15 0000 0 0000 00000 00000 02 666643 77775; (5) andg 4/C300:By definition, gkƒgk;where gkis the term in the COMMUTATOR SERIES , as can be seen by the example above. In contrast to the NILPOTENT LIE ALGEBRAS , the SEMISIMPLE LIE ALGEBRAS have a constant lower central series. Others are in between, e.g., [gln;gln]/C30sln; (6) which is semisimple, because the TRACE satisfies Tr(AB)/C30Tr(BA): (7) Here, glnis a general linear Lie algebra and slnis the SPECIAL LINEAR LIE ALGEBRA . Here are some Mathematica functions for determin- ing the lower central series, when given a list of matrices which is a basis for g:/ MatrixBasis[a_- List]: /C30Partition[#1,Length[a[[1]]]]&/@ LatticeReduce[Flatten/@a] LieCommutator[a_,b_]: /C30a.b-b.a NextLCS[gold_List,{}] /C30{}; NextLCS[gold_List,g_List]: /C30 MatrixBasis[Flatten[Outer[LieCommutator,gold,- g,1],1]] kthLCS[g_List, k_Integer]: /C30Nest[NextLCS[g,#1]&,g,k] For example, gl5 /C30Flatten[Table[ReplacePart[ Ta- ble[0,{i,5},{j,5}],1,{k,l}],{k,5},{l,5}],1]; sl5 /C30kthLCS[gl5, 1] See also COMMUTATOR SERIES (LIE ALGEBRA ), LIE ALGEBRA ,L IE GROUP ,L OWER CENTRAL SERIES (GROUP ), NILPOTENT LIE GROUP ,R EPRESENTATION (LIE ALGEBRA ), REPRESENTATION (NILPOTENT LIE GROUP ), UNIPOTENT Lower Denjoy Sum LOWER SUM Lower Factorial FALLING FACTORIAL Lower Half-Disk The unit lower half-disk is the portion of the COMPLEX PLANE satisfying zjj51;I z½/C138B0 fg :/ See also DISK,REAL AXIS,SEMICIRCLE ,U NIT DISK, LOWER HALF-PLANE ,UPPER HALF-DISK Lower Half-Plane The portion of the COMPLEX PLANE fx /C27iy : x ; y /C23(/C28/C12;/C12) g satisfying y /C30I[z] B0 ; i.e., fx /C27iy :/ /x /C23 (/C28/C12;/C12) ; y /C23 ( /C12; 0)g:/ See also COMPLEX PLANE ,HALF-PLANE ,LEFT HALF- PLANE ,L OWER HALF-DISK,R IGHT HALF-PLANE , UPPER HALF-PLANE Lower Integral The limit of a LOWER SUM, when it exists, as the MESH SIZE approaches 0. See also LOWER SUM,R IEMANN INTEGRAL ,U PPER INTEGRAL Lower Limit Let the least term h of a SEQUENCE be a term which is smaller than all but a finite number of the terms which are equal to h. Then h is called the lower limit of the SEQUENCE . A lower limit of a SERIES lower lim n 0/C12Sn /C30lim n0/C12Sn /C30h is said to exist if, for every e> 0 ;½Sn /C28h½Be for infinitely many values of n and if no number less than hhas this property. See also INFIMUM LIMIT,LIMIT,SUPREMUM LIMIT, UPPER LIMIT References Bromwich, T. J. I’a and MacRobert, T. M. "Upper and Lower Limits of a Sequence." §5.1 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 40 1991. Lower Sum For a given function f(x) over a partition of a given interval, the lower sum is the sum of box areas fx/C31 kðÞDxkusing the smallest value of the function fx/C31kðÞ) in each subinterval Dxk :/ See also LOWER INTEGRAL ,R IEMANN INTEGRAL , UPPER SUM Lower Triangular Matrix A TRIANGULAR MATRIX L OF THE FORM Lij /C30aijfor i ]j 0 for i Bj:/C)% Written explicitly, L /C30a11 0 /C1/C1/C1 0 a21a22 /C1/C1/C1 0 nn::: 0 an1an2/C1/C1/C1 ann2 6643 775 A lower triangular matrix with elements f[i,j] below the diagonal can be formed using LowerDia- gonalMatrix [f, n] in the Mathematica add-on pack- age LinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also TRIANGULAR MATRIX ,U PPER TRIANGULAR MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 10, 1962. Lower-Trimmed Subsequence The lower-trimmed subsequence of x /C30fxn g is the sequence V(x) obtained by subtracting 1 from each xn and then removing all 0s. If x is a FRACTAL SEQUENCE , then V(x)isa FRACTAL SEQUENCE .Ifx is a SIGNATURE SEQUENCE , then V(x) /C30x:/ See also SIGNATURE SEQUENCE ,U PPER- TRIMMED SUBSEQUENCE References Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157 /C1/68, 1997.Lowest Divisor Function LEAST PRIME FACTOR Lowest Terms Fraction REDUCED FRACTION Lo¨wner’s Differential Equation The ORDINARY DIFFERENTIAL EQUATION y?/C30/C28 y1 /C27 k(x)y 1 /C28 k(x)y : References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1345, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Loxodrome A path, also known as a RHUMB LINE, which cuts a MERIDIAN on a given surface at any constant ANGLE but a RIGHT ANGLE . If the surface is a SPHERE , the loxodrome is a SPHERICAL SPIRAL . The loxodrome is the path taken when a compass is kept pointing in a constant direction. It is a straight line on a MERCATOR PROJECTION or a LOGARITHMIC SPIRAL on a polar projection (Steinhaus 1983, pp. 218 /C1/19). The loxo- drome is not the shortest distance between two points on a sphere. See also GREAT CIRCLE ,SPHERE ,SPHERICAL SPIRAL References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 217 /C1/21, 1999. Lozenge An equilateral PARALLELOGRAM whose ACUTE ANGLES are 45 8. Sometimes, the restriction to 458 is dropped, and it is required only that two opposite angles are acute and the other two obtuse. The term RHOMBUS is commonly used for an arbitrary equilateral parallelo- gram. See also KITE,P ARALLELOGRAM ,Q UADRILATERAL , RHOMBUS Lozenge Method A method for constructing MAGIC SQUARES ofODD order. See also MAGIC SQUARE Lozi Map A 2-D map similar to the HE´ NON MAP which is given by the equations xn/C271 /C301 /C28 a½xn ½/C27yn yn/C271 /C30 bxn : See also HE´ NON MAP References Dickau, R. M. "Lozi Attractor." http://forum.swarthmor- e.edu/advanced/robertd/lozi.html. Peitgen, H.-O.; Ju¨rgens, H.; and Saupe, D. §12.1 in Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, p. 672, 1992. Lp’-Balance Theorem If every component L of X =Op ?(X) satisfies the "Schreler property," then Lp ?(Y) 5Lp?(X) for every p-local SUBGROUP Y of X, where Lp ? is the P- LAYER . See also P-LAYER ,SUBGROUP L-Polyomino The order n ]2 L-polyomino consists of a vertical line of n SQUARES with a single additional SQUARE attached at the bottom. See also L-POLYOMINO ,SKEW POLYOMINO ,SQUARE , SQUARE POLYOMINO ,STRAIGHT POLYOMINO Lp-Space The set of Lp /-functions generalizes L2-SPACE . Instead of SQUARE INTEGRABLE , the MEASURABLE FUNCTION f must be p-integrable for f to be in Lp :/ On a MEASURE SPACE X, the Lp norm of a function f is fkkLp/C30gXfjjp/C(*/C(+ 1 =p : The Lp /-functions are the functions for which this integral converges. For p "2; the space of Lp/-func- tions is a BANACH SPACE which is not a HILBERT SPACE .The Lp/-space on Rn ; and in most other cases, is the COMPLETION of the continuous functions with COM- PACT SUPPORT using the Lp norm. As in the case of an L2-SPACE ,an Lp/-function is really an equivalence class of functions which agree ALMOST EVERYWHERE . It is possible for a sequence of functions fn to converge in Lp but not in Lp ? for some other p ?; e.g., fn /C30 (1 /C27x2)/C281 =2 /C281 =nconverges in L2(R) but not L1(R): However, if a sequence converges in Lp and in Lp ?; then its limit must be the same in both spaces. For p /C211, the DUAL SPACE to Lp is given by integrat- ing against functions in Lq ; where 1=p /C271 =q /C301: This makes sense because of HO¨ LDER’S INEQUALITY FOR INTEGRALS . In particular, the only Lp /-space which is SELF-DUAL is L2 :/ While the use of Lp functions is not as common as L2 ; they are very important in ANALYSIS and PARTIAL DIFFERENTIAL EQUATIONS . For instance, some OPERA- TORS are only BOUNDED in Lp for some p /C212. See also BANACH SPACE ,C OMPLETION ,H ILBERT SPACE ,L EBESGUE INTEGRAL , LP-SPACE , L2-SPACE , MEASURE ,MEASURE SPACE LQ Decomposition The orthogonal decomposition of a matrix into lower trapezoidal matrices. References Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351/C1/69, 1999. L-Series DIRICHLET L-SERIES ,ROGERS L-FUNCTION L-System LINDENMAYER SYSTEM Lubbock’s Formula f0/C27f1=m/C27f2=m/C27.../C27fr mf0/C27f1/C27.../C27fr ðÞ /C281 2(m/C282)fr/C27f0 ðÞ /C28m2/C281 12m(Dfr/C281/C28Df0)/C28m2/C281 24m(D2fr/C282/C28D2f0) /C28(m2/C281)(19 m2/C281) 720m3(D3fr/C283/C28D3f0) /C28(m2/C281)(9m2/C281) 480m3(D4fr/C284/C28D4f0): References Lubbock, J. W. Cambridge Philos. Trans. 3, 323, 1829. Whittaker, E. T. and Robinson, G. "Lubbock’s Formula of Summation." §74 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 149 /C1/50, 1967. Lucas Correspondence The correspondence which relates the HANOI GRAPH to the ISOMORPHIC GRAPH of the ODD BINOMIAL COEFFICIENTS in PASCAL’S TRIANGLE , where the ad- jacencies are determined by adjacency (either hor- izontal or diagonal) in PASCAL’S TRIANGLE . The proof of the correspondence is given by the LUCAS CORRE- SPONDENCE THEOREM . See also BINOMIAL COEFFICIENT ,H ANOI GRAPH , PASCAL’S TRIANGLE References Poole, David G. "The Towers and Triangles of Professor Claus (or, Pascal Knows Hanoi)." Math. Mag. 67, 323 /C1/44, 1994. Lucas Correspondence Theorem Let p be PRIME and r /C30rmpm /C27.../C27r1p /C27r0(0 5ri Bp) (1) k /C30kmpm /C27.../C27k1p /C27k0(0 5ki Bp); (2) then r k/C(*/C(+ /C30Ym i/C300ri ki/C(*/C(+ (mod p) : (3) This is proved in Fine (1947). References Fine, N. J. "Binomial Coefficients Modulo a Prime." Amer. Math. Monthly 54, 589 /C1/92, 1947. Lucas-Lehmer Residue LUCAS- LEHMER TEST Lucas-Lehmer Test AM ERSENNE NUMBER Mpis prime IFF Mpdivides sp /C282 ; where s0 /C134 and si /C13s2 i/C281 /C282(mod 2p /C281) (1) for i ]1: The first few terms of this series are 4, 14, 194, 37634, 1416317954, ... (Sloane’s A003010). The remainder when sp /C282is divided by Mpis called the LUCAS- LEHMER RESIDUE for p. The LUCAS- LEHMER RESIDUE is 0 IFF Mpis PRIME . This test can also be extended to arbitrary INTEGERS . A generalized version of the Lucas-Lehmer test lets N /C271 /C30Yn j /C301q bj j; (2)with qjthe distinct PRIME FACTORS , and bjtheir respective POWERS . If there exists a LUCAS SEQUENCE Un such that GCD( U(N /C271)=qj; N) /C301 (3) for j /C301, ..., n and UN /C271 /C130 (mod N); (4) then N is a PRIME . The test is particularly simple for MERSENNE NUMBERS , yielding the conventional Lu- cas-Lehmer test. See also LUCAS SEQUENCE ,M ERSENNE NUMBER , RABIN- MILLER STRONG PSEUDOPRIME TEST References Sloane, N. J. A. Sequences A003010/M3494 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Lucas’ Married Couples Problem MARRIED COUPLES PROBLEM Lucas Number The numbers produced by the Vrecurrence in the LUCAS SEQUENCE with ( P;Q)/C30(1;/C281) are called Lucas numbers. They are the companions to the FIBONACCI NUMBERS Fnand satisfy the same recur- rence Ln/C30Ln/C281/C27Ln/C282; (1) where L1/C301;L2/C303:The first few are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (Sloane’s A000204). The analog of B INET’S FIBONACCI NUMBER FORMULA for Lucas numbers is Ln/C301/C27ffiffiffi 5p 2 !n /C271/C28ffiffiffi5p 2 ! n : (2) Another formula is Ln/C30[fn]; (3) where fis the GOLDEN RATIO and [ x] denotes the NINT function. Given Ln; Ln/C271/C30Ln1/C27ffiffiffi5p/CP/C( /C271 2$% ; (4) where xbcis the FLOOR FUNCTION , L2 n/C28Ln/C281Ln/C271/C305(/C281)n; (5) and Xn k/C300L2k/C30LnLn/C271/C282: (6) The Lucas numbers obey the negation formula L/C28n/C30(/C281)nLn; (7) the addition formula Lm/C27n/C301 2(5FmFn/C27LmLn); (8) where Fnis a F IBONACCI NUMBER , the subtraction formula Lm/C28n/C301 2(/C281)LmLn/C285FmFn ðÞ ; (9) the fundamental identity L2 n/C285F2 n/C304(/C281)n; (10) conjugation relation Ln/C30Fn/C281/C27Fn/C271; (11) successor relation Ln/C271/C301 25Fn/C27Ln ðÞ ; (12) double-angle formula L2n/C301 2(5F2 n/C27L2 n); (13) multiple-angle recurrence Lkn/C30LkLk(n/C281)/C28(/C281)kLk(n/C282); (14) multiple-angle formulas Lkn/C301 2k/C281Xk=2bc i/C300k 2i/C(*/C(+ 5iF2i nLk/C282i n (15) /C30Xk=2bc i/C300k k/C28ik/C28i i/C(*/C(+ (/C281)i(n/C271)Lk/C282i n (16) /C30Pk=2 i/C300k k/C28ik/C28i i/CP/C( (/C281)in5k=2/C28iFk/C282i n forkeven LnPk=2bc i/C300k/C281/C28i i/CP/C( (/C281)in5k=2bc/C28iFk/C281/C282i n forkodd( (17) /C30Xk i/C300k i/C(*/C(+ LiFi nFk/C28i n/C281; (18) product expansions FmLn/C30Fm/C27n/C27(/C281)nFm/C28n (19) and FmFn/C301 5[Lm/C27n/C28(/C281)nLm/C28n]; (20) square expansion, L2 n/C30L2n/C282(/C281)n; (21) and power expansion Lkn/C301 2Xk i/C300k i/C(*/C(+ (/C281)inL(k/C282i)n: (22) The Lucas numbers satisfy the power recurrenceXt/C271 j/C300(/C281)j(j/C271)=2t/C271 j/C)P/C)( FLt n/C28j/C300; (23) wherea b/C)/Cn Fis a F IBONACCI COEFFICIENT , the reciprocal sum Xn k/C301(/C281)k LkLk/C27a/C30Fn FaXa k/C301(/C281)k LkLk/C27n; (24) the convolution Xn k/C300LkLn/C28k/C30(n/C272)Ln/C27Fn; (25) the partial fraction decomposition /C285 Ln/C27aLn/C27bLn/C27c/C30A Ln/C27a/C27B Ln/C27b/C27C Ln/C27c; (26) where A/C30(/C281)n/C28a Fb/C28aFc/C28a(27) B/C30(/C281)n/C28b Fc/C28bFa/C28b(28) C/C30(/C281)n/C28c Fa/C28cFb/C28c; (29) and the summation formula Xn k/C300xkLak/C27b/C30g(n/C271)/C28g(0) 1/C28Lax/C27(/C281)ax2; (30) where g(n)/C30(/C281)aLa(n/C281)/C27bxn/C271/C28Lan/C27bxn: (31) Letpbe a PRIME >3 and kbe a POSITIVE INTEGER . Then L2pkends in a 3 (Honsberger 1985, p. 113). Analogs of the Cesa `ro identities for F IBONACCI NUM- BERS are Xn k/C300n k/C(*/C(+ Lk/C30L2n (32) Xn k/C300n k/C(*/C(+ 2kLk/C30L3n; (33) wheren k/CP/C( is a BINOMIAL COEFFICIENT . /LnFmj(/LnDIVIDES Fm)IFFnDIVIDES into manEVEN number of times. LnLmj IFFndivides into manODD number of times. 2nLnalways ends in 2 (Honsberger 1985, p. 137). Defining Dn /C133 i 00 /C1/C1/C1 00 i 1 i 0 /C1/C1/C1 00 0 i 1 i /C1/C1/C1 00 00 i 1 /C1/C1/C1 00 nnnn ::: nn 0000 /C1/C1/C1 1 i 0000 0 i 1/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C30L n/C271 (34) gives Dn /C30Dn/C281 /C27Dn/C282 (35) (Honsberger 1985, pp. 113 /C1/14). The number of ways of picking a set (including the EMPTY SET) from the numbers 1, 2, ..., n without picking two consecutive numbers (where 1 and n are now consecutive) is Ln (Honsberger 1985, p. 122). The only SQUARE NUMBERS in the Lucas sequence are 1 and 4, as proved by John H. E. Cohn (Alfred 1964). The only TRIANGULAR Lucas numbers are 1, 3, and 5778 (Ming 1991). The only Lucas CUBIC NUMBER is 1. The first few Lucas PRIMES Ln occur for n /C302, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, ... (Dubner and Keller 1999, Sloane’s A001606). See also FIBONACCI NUMBER References Alfred, Brother U. "On Square Lucas Numbers." Fib. Quart. 2,11/C1/2, 1964. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 94 /C1/01, 1987. Brillhart, J.; Montgomery, P. L.; and Solverman, R. D. "Tables of Fibonacci and Lucas Factorizations." Math. Comput. 50, 251 /C1/60 and S1-S15, 1988. Brown, J. L. Jr. "Unique Representation of Integers as Sums of Distinct Lucas Numbers." Fib. Quart. 7, 243 /C1/52, 1969. Dubner, H. and Keller, W. "New Fibonacci and Lucas Primes." Math. Comput. 68, 417 /C1/27 and S1-S12, 1999. Guy, R. K. "Fibonacci Numbers of Various Shapes." §D26 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 194 /C1/95, 1994. Hilton, P.; Holton, D.; and Pedersen, J. "Fibonacci and Lucas Numbers." Ch. 3 in Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 61 /C1/ 5, 1997. Hilton, P. and Pedersen, J. "Fibonacci and Lucas Numbers in Teaching and Research." J. Math. Informatique 3,36/C1/ 7, 1991 /C1/992. Hoggatt, V. E. Jr. The Fibonacci and Lucas Numbers. Boston, MA: Houghton Mifflin, 1969. Honsberger, R. "A Second Look at the Fibonacci and Lucas Numbers." Ch. 8 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/lucas.Z. Ming, L. "On Triangular Lucas Numbers." Applications of Fibonacci Numbers, Vol. 4 (Ed. G. E. Bergum, A. N. Phi- lippou, and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 231 /C1/40, 1991. Sloane, N. J. A. Sequences A000204/M2341 and A001606/ M0961 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Lucas Polynomial The w POLYNOMIALS obtained by setting p(x) /C30x and q(x) /C301 in the LUCAS POLYNOMIAL SEQUENCE . The first few are F1(x) /C30x F2(x) /C30x2 /C272 F3(x) /C303x3 /C273x F4(x) /C30x4 /C274x2 /C272 F5(x) /C30x5 /C275x3 /C275x: The corresponding W POLYNOMIALS are called FIBO- NACCI POLYNOMIALS . The Lucas polynomials satisfy Ln(1)/C30Ln; where the Ln/s are L UCAS NUMBERS . See also FIBONACCI POLYNOMIAL ,LUCAS NUMBER , LUCAS POLYNOMIAL SEQUENCE Lucas Polynomial Sequence A pair of generalized POLYNOMIALS which generalize the L UCAS SEQUENCE toPOLYNOMIALS is given by Wk n(x)/C30Dk(x)[an(x)/C28(/C281)kbn(x)] D(x)(1) wk n(x)/C30Dk(x)an(x)/C27(/C281)kbn(x)hi ; (2) where a(x)/C27b(x)/C30p(x) (3) a(x)b(x)/C30/C28q(x) (4) a(x)/C28b(x)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2(x)/C274q(x)p /C13D(x) (5) (Horadam 1996). Setting n/C300 gives Wk 0(x)/C30Dk(x)1/C28(/C281)k D(x)(6) wk 0(x)/C30Dk(x)[1/C27(/C281)k]; (7) giving W0 0(x)/C300 (8) w00(x)/C302: (9) The sequences most commonly considered have k/C300, giving Wn(x)/C13W0 n(x)/C30an(x)/C28bn(x) a(x)/C28b(x)(10) /C30p(x) /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2(x) /C27 4q(x)phin /C27 p(x) /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2(x) /C27 4q(x)phin 2nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2(x) /C27 4q2(x)p (11) wn(x) /C13w0 n(x) /C30an(x) /C27bn(x) (12) p(x) /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2(x) /C27 4q(x)phin /C27 p(x) /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2(x) /C27 4q(x)phin 2n : (13) The w polynomials satisfy the RECURRENCE RELATION wn(x) /C30p(x)wn/C281(x) /C27q(x)wn/C282(x) : (14) Special cases of the W and w polynomials are given in the following table. /p(x)//q(x)/ Polynomial 1 Polynomial 2 x 1F IBONACCI Fn(x)/ LUCAS Ln(x)/ /2x/ 1P ELL Pn(x)/ PELL-LUCAS Qn(x)/ 1 /2x/ JACOBSTHAL Jn(x)/ JACOBSTHAL jn(x)/ /3x/ /C282F ERMAT Fn(x)/ FERMAT- LUCAS fn(x)/ /2x/ /C281C HEBYSHEV POLY- NOMIAL OF THE SECOND KIND Un/C281(x)/CHEBYSHEV POLY- NOMIAL OF THE FIRST KIND 2Tn(x)/ See also CHEBYSHEV POLYNOMIAL OF THE FIRST KIND, CHEBYSHEV POLYNOMIAL OF THE SECOND KIND, FERMAT POLYNOMIAL ,FIBONACCI POLYNOMIAL ,JA- COBSTHAL POLYNOMIAL ,LUCAS POLYNOMIAL ,LUCAS SEQUENCE ,PELL POLYNOMIAL References Horadam, A. F. "Extension of a Synthesis for a Class of Polynomial Sequences." Fib. Quart. 34,68/C1/4, 1996. Lucas Pseudoprime When P and Q are INTEGERS such that D /C30P2 /C284Q " 0; define the LUCAS SEQUENCE Ukfg by Uk /C30ak /C28 bk a /C28 b for k ]0; with a and b the two ROOTS of x2 /C28Px /C27Q /C30 0: Then define a Lucas pseudoprime as an ODD COMPOSITE number n such that n¶Q; the JACOBI SYMBOL (D=n) /C30/C281; and nUn/C271 ::/C()/C()/ There are no EVEN Lucas pseudoprimes (Bruckman 1994). The first few Lucas pseudoprimes are 705, 2465, 2737, 3745, ... (Sloane’s A005845). See also EXTRA STRONG LUCAS PSEUDOPRIME ,LUCASSEQUENCE ,PSEUDOPRIME ,STRONG LUCAS PSEUDO- PRIME References Bruckman, P. S. "Lucas Pseudoprimes are Odd." Fib. Quart. 32, 155/C1/57, 1994. Ribenboim, P. "Lucas Pseudoprimes (lpsp( P, Q ))."§2.X.B in The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, p. 129, 1996. Sloane, N. J. A. Sequences A005845/M5469 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Lucas Sequence LetP,QbePOSITIVE INTEGERS . The ROOTS of x2/C28Px/C27Q/C300 (1) are a/C131 2P/C27ffiffiffiffi Dp/C(%/C(r (2) b/C131 2P/C28ffiffiffiffi Dp/C(%/C(r ; (3) where D/C13P2/C284Q; (4) so a/C27b/C30P (5) ab/C301 4(P2/C28D)/C30Q (6) a/C28b/C30ffiffiffiffi Dp : (7) Then define Un(P;Q)/C13an/C28bn a/C28b(8) Vn(P;Q)/C13an/C27bn: (9) The first few values are therefore U0(P;Q)/C300 (10) U1(P;Q)/C301 (11) V0(P;Q)/C302 (12) V1(P;Q)/C30P: (13) The sequences U(P;Q)/C30fUn(P;Q):n]1g (14) V(P;Q)/C30fVn(P;Q):n]1g (15) are called Lucas sequences, where the definition is usually extended to include U/C281/C30a/C281/C28b/C281 a/C28b/C30/C281 ab/C30/C281 Q: (16) For ( P;Q)/C30(1;/C281);theUnare the F IBONACCI NUM- BERS and Vnare the LUCAS NUMBERS . For (P; Q) /C30 (2;/C281); the PELL NUMBERS and Pell-Lucas numbers are obtained. (P; Q) /C30(1;/C282) produces the JA- COBSTHAL NUMBERS and Pell-Jacobsthal Numbers. The Lucas sequences satisfy the general RECURRENCE RELATIONS Um/C27n /C30am/C27n /C28 bm/C27n a /C28 b /C30(am /C28 bm)(an /C27 bn) a /C28 b/C28anbn(am/C28n /C28 bm/C28n) a /C28 b /C30UmVn /C28anbnUm/C28n (17) Vm/C27n /C30am/C27n /C27bm/C27n /C30(am /C27bm)(an /C27bn) /C28anbn(am/C28n /C27bm/C28n) /C30VmVn /C28anbnVm/C28n : (18) Taking n /C301 then gives Um(P ; Q) /C30PUm/C281(P ; Q) /C28QUm/C282(P; Q) (19) Vm(P; Q) /C30PVm/C281(P ; Q) /C28QVm/C282(P; Q) : (20) Other identities include U2n /C30UnVn (21) U2n/C271 /C30Un/C271Vn /C28Qn (22) V2n /C30V2 n /C282(ab)n /C30V2 n /C282Qn (23) V2n/C271 /C30Vn /C271Vn /C28PQn : (24) These formulas allow calculations for large n to be decomposed into a chain in which only four quantities must be kept track of at a time, and the number of steps needed is /C2lg n : The chain is particularly simple if n has many 2s in its factorization. The Us in a Lucas sequence satisfy the CONGRUENCE Upn/C281[p/C28(D=p)] /C130 (mod pn) (25) if GCD(2 QcD ; p) /C301 ; (26) where P2 /C284Q2 /C30c2D: (27) This fact is used in the proof of the general LUCAS- LEHMER TEST . See also FIBONACCI NUMBER ,JACOBSTHAL NUMBER , LUCAS- LEHMER TEST,LUCAS NUMBER ,LUCAS POLY- NOMIAL SEQUENCE ,P ELL NUMBER ,R ECURRENCE SEQUENCE ,SYLVESTER CYCLOTOMIC NUMBER References Dickson, L. E. "Recurring Series; Lucas’ un ; vn :/" Ch. 17 in History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 393 /C1/11, 1952.Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, pp. 35 /C1/3, 1991. Lucas’s Theorem Let n ]3bea SQUAREFREE integer, and Fn(z)a CYCLOTOMIC POLYNOMIAL . Then Fn(z) /C30U2 n(z) /C28(/C281)(n /C281)=2nzV2 n(z) ; (1) where Un(z) and Vn(z) are INTEGER POLYNOMIALS of degree f(n) =2 and f(n) =2 /C281 ; respectively. This identity can be expressed as Fn((/C281)(n /C281)=2z) /C30C2 n(z) /C28nzD2n(z) for n odd Fn=2(/C28z2) /C30C2n(z) /C28nzD2n(z) n /C304k /C272 F1(/C28z2) /C30C22(z) /C282zD22(z) for n /C302;8 < : (2) with Cn(z) and Dn(z) SYMMETRIC POLYNOMIALS . The following table gives the first few Cn(z) and Dn(z)/s (Riesel 1994, pp. 443 /C1/56). n /Cn(z)// Dn(z)/ 2 /z /C271/ 1 3 /z /C271/ 1 5 /z2 /C273z /C271// z /C271/ 6 /z2 /C273z /C271// z /C271/ 7 /z3 /C273z2 /C273z /C271// z2/C27z/C271/ 10 /z4/C275z3/C277z2/C275z/C271//z3/C272z2/C272z/C271/ See also CYCLOTOMIC POLYNOMIAL ,GAUSS’S CYCLO- TOMIC FORMULA References Brent, R. P. "On Computing Factors of Cyclotomic Polyno- mials." Math. Comput. 61, 131/C1/49, 1993. Kraitchik, M. Recherches sue la the ´orie des nombres, tome I. Paris: Gauthier-Villars, pp. 126 /C1/28, 1924. Riesel, H. "Lucas’s Formula for Cyclotomic Polynomials." In tables at end of Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 443 /C1/56, 1994. Lucky Number Write out all the ODD numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first ODD number >1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first ODD number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, .... Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (Sloane’s A000959). Many asymptotic properties ofthe PRIME NUMBERS are shared by the lucky numbers. The asymptotic density is 1 =lnN;just as the PRIME NUMBER THEOREM , and the frequency of TWIN PRIMES and twin lucky numbers are similar. A version of the GOLDBACH CONJECTURE also seems to hold. It therefore appears that the SIEVING process ac- counts for many properties of the PRIMES . See also GOLDBACH CONJECTURE ,LUCKY NUMBER OF EULER ,PRIME NUMBER ,PRIME NUMBER THEOREM , SIEVE References Gardner, M. "Mathematical Games: Tests Show whether a Large Number can be Divided by a Number from 2 to 12." Sci. Amer. 207, 232, Sep. 1962. Gardner, M. "Lucky Numbers and 2187." Math. Intell. 19, 26, 1997. Guy, R. K. "Lucky Numbers." §C3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108 /C1/09, 1994. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 100 /C1/02, 1988. Peterson, I. "MathTrek: Martin Gardner’s Luck Number." http://www.sciencenews.org/sn_arc97/9_6_97/math- land.htm. Sloane, N. J. A. Sequences A000959/M2616 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Ulam, S. M. A Collection of Mathematical Problems. New York: Interscience Publishers, p. 120, 1960. Wells, D. G. The Penguin Dictionary of Curious and Inter- esting Numbers. London: Penguin, p. 32, 1986. Lucky Number of Euler A number p such that the PRIME-GENERATING POLY- NOMIAL n2 /C28n /C27p is PRIME for n /C300, 1, ..., p /C282 : Such numbers are related to the COMPLEX QUADRATIC FIELD in which the RING of INTEGERS is factorable. Specifically, the Lucky numbers of Euler (excluding the trivial case p /C303) are those numbers p such that the QUADRATIC FIELD Qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C284pp/CP/C( has CLASS NUMBER 1 (Rabinowitz 1913, Le Lionnais 1983, Conway and Guy 1996). As established by Stark (1967), there are only nine numbers /C28d such that h(/C28d) /C301 (the HEEGNER NUMBERS /C282, /C283, /C287, /C2811, /C2819, /C2843, /C2867, and /C28163), and of these, only 7, 11, 19, 43, 67, and 163 are of the required form. Therefore, the only Lucky numbers of Euler are 2, 3, 5, 11, 17, and 41 (Le Lionnais 1983, Sloane’s A014556), and there does not exist a better PRIME-GENERATING POLYNOMIAL of Euler’s form. See also CLASS NUMBER ,HEEGNER NUMBER ,PRIME- GENERATING POLYNOMIAL References Conway, J. H. and Guy, R. K. "The Nine Magic Discrimi- nants." In The Book of Numbers. New York: Springer- Verlag, pp. 224 /C1/26, 1996.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983. Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahl- faktoren in quadratischen Zahlko ¨rpern." Proc. Fifth Inter- nat. Congress Math. (Cambridge) 1, 418 /C1/21, 1913. Sloane, N. J. A. Sequences A014556 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Stark, H. M. "A Complete Determination of the Complex Quadratic Fields of Class Number One." Michigan Math. J. 14,1/C1/7, 1967. LUCY A nonlinear DECONVOLUTION technique used in de- convolving images from the Hubble Space Telescope before corrective optics were installed. See also DECONVOLUTION ,MAXIMUM ENTROPY METH- OD LU Decomposition A procedure for decomposing an N/C29Nmatrix Ainto a product of a LOWER TRIANGULAR MATRIX Land an UPPER TRIANGULAR MATRIX U; LU/C30A: (1) LU decomposition is implemented in Mathematica as LUDecomposition [m]. Written explicitly for a 3 /C293MATRIX , the decomposi- tion is l1100 l21l220 l31l32l332 435u 11u12u13 0u22u23 00 u332435/C30a 11a12a13 a21a22a23 a31a32a332435(2) l 11u11 l11u12 l11u13 l21u11l21u22/C27l22u22 l21u13/C27l22u23 l31u11l31u12/C27l32u22l31u13/C27l32u23/C27l33u232435 /C30a 11a12a13 a21a22a23 a31a32a332 435: (3) This gives three types of equations iBjl i1u1j/C27li2u2j/C27.../C27liiuij/C30aij (4) i/C30jli1u1j/C27li2u2j/C27.../C27liiujj/C30aij (5) i>jli1u1j/C27li2u2j/C27.../C27lijujj/C30aij: (6) This gives N2equations for N2/C27Nunknowns (the decomposition is not unique), and can be solved using CROUT’S METHOD . To solve the MATRIX equation Ax/C30(LU)x/C30L(Ux)/C30b; (7) first solve Ly/C30bfory. This can be done by forward substitution y1/C30b1 l11(8) yi /C301 liibi /C28Xi /C281 j /C301lijyj ! (9) for i /C302, ..., N. Then solve Ux /C30y for x. This can be done by back substitution xN /C30yN uNN(10) xi /C301 uiiyi /C28XN j/C30i/C271uijxj ! (11) for i /C30N /C281; ..., 1:/ See also LOWER TRIANGULAR MATRIX ,M ATRIX DE- COMPOSITION ,C HOLESKY DECOMPOSITION ,QRD E- COMPOSITION ,T RIANGULAR MATRIX ,U PPER TRIANGULAR MATRIX References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "LU Decomposition and Its Applications." §2.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 34 /C1/2, 1992. Ludolph’s Constant PI Ludwig’s Inversion Formula Expresses a function in terms of its RADON TRANS- FORM , f(x; y) /C30R/C281(Rf)(x; y) /C301 p1 2p g/C12 /C28/C12@ @p(Rf)(p; a) x cos a /C27 y sin a /C28 pdp d a: See also RADON TRANSFORM Ludwig’s Law FIBONACCI NUMBER Luka ´cs Theorem Let r(x)bean mth degree POLYNOMIAL which is NONNEGATIVE in [/C281 ;1]: Then r(x) can be represented in the form [A(x)]2 /C27(1 /C28x2)[B(x)]2for m even (1 /C27x)[C(x)]2 /C27(1 /C28x)[D(x)]2for m odd ;/C)% where A(x) ; B(x) ; C(x); and D(x) are REAL POLYNO- MIALS whose degrees do not exceed m. References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 4, 1975.Lune A figure bounded by two circular ARCS of unequal RADII . Hippocrates of Chios SQUARED the above left lune, as well as two others, in the fifth century BC. Two more SQUARABLE lunes were found by T. Clausen in the 19th century (Dunham 1990 attributes these discoveries to Euler in 1771). In the 20th century, N. G. Tschebatorew and A. W. Dorodnow proved that these are the only five squarable lunes (Shenitzer and Steprans 1994). The left lune above is squared as follows, Ahalf small circle /C301 2 prffiffiffi 2p !2 /C301 4 pr2 Alens /C30Aquarter big circle /C28Atriangle /C3014pr2/C2812r2 Alune/C30Ahalf small circle /C28Alens/C3012r2 /C30Atriangle ; so the lune and TRIANGLE have the same AREA . In the right figure, A1/C27A2/C30AD:/ For the above lune, Alune/C302ADOBC: See also ANNULUS ,ARC,CIRCLE ,SALINON ,SPHERICAL LUNE References Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 inJourney through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 1 /C1/0, 1990. Heath, T. L. A History of Greek Mathematics, Vol. 1: From Thales to Euclid. New York: Dover, p. 185, 1981. Pappas, T. "Lunes." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 72 /C1/3, 1989. Shenitzer, A. and Steprans, J. "The Evolution of Integra- tion." Amer. Math. Monthly 101,6 6/C1/2, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 143 /C1/44, 1991. Lunule LUNE Lu¨ roth’s Theorem If x and y are nonconstant rational functions of a parameter, the curve so defined has GENUS 0. Furthermore, x and y may be expressed rationally in terms of a parameter which is rational in them. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 246, 1959. Lusin Area Integral If V⁄C is a DOMAIN and 8 : V0 C is a ONE-TO-ONE ANALYTIC FUNCTION , then 8( V)isa DOMAIN , and area(8( V)) /C30gV8?(z) jj2dx dy (Krantz 1999, p. 150). See also AREA INTEGRAL References Krantz, S. G. "The Lusin Area Integral." §12.1.3 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, p. 150, 1999. Lusin’s Theorem Let f(x) be a finite and MEASURABLE FUNCTION in (/C28/C12;/C12); and let e be freely chosen. Then there is a function g(x) such that 1. g(x) is continuous in (/C28/C12;/C12) ;/ 2. The MEASURE of fx : f(x) "g(x)g is Be;/ 3. Mgjj; R1 ðÞ 5Mfjj; R1 ðÞ ;/ where M(f; S) denotes the upper bound of the aggregate of the values of f(P)as P runs through all values of S. References Kestelman, H. §4.4 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 30 and 109 /C1/12, 1960. Lusternik-Schnirelmann Theorem LYUSTERNIK- SCHNIRELMANN THEOREM LUX Method A method for constructing MAGIC SQUARES ofSINGLY EVEN order n]6:/ See also MAGIC SQUARE Lyapunov Characteristic Exponent The Lyapunov characteristic exponent [LCE] gives the rate of exponential divergence from perturbed initial conditions. To examine the behavior of an orbitaround a point X/C31(t);perturb the system and write X(t)/C30X/C31(t)/C27U(t); (1) where U(t) is the average deviation from the unper- turbed trajectory at time t.I na CHAOTIC region, the LCE sis independent of X/C31(0):It is given by the OSEDELEC THEOREM , which states that si/C30lim t0/C121 tlnU(t) jj : (2) For an n-dimensional mapping, the Lyapunov char- acteristic exponents are given by si/C30lim N0/C12lnli(N) jj (3) fori/C301, ..., n, where liis the L YAPUNOV CHARACTER- ISTIC NUMBER . One Lyapunov characteristic exponent is always 0, since there is never any divergence for a perturbed trajectory in the direction of the unperturbed trajec-tory. The larger the LCE, the greater the rate of exponential divergence and the wider the correspond- ing SEPARATRIX of the CHAOTIC region. For the STANDARD MAP , an analytic estimate of the width of the CHAOTIC zone by Chirikov (1979) finds dI/C30Be/C28AK/C281=2: (4) Since the Lyapunov characteristic exponent increaseswith increasing K, some relationship likely exists connecting the two. Let a trajectory (expressed as a MAP) have initial conditions ( x0;y0) and a nearby trajectory have initial conditions ( x?;y?)/C30 (x0/C27dx;y0/C27dy):The distance between trajectories at iteration kis then dk/C30x?/C28x0;y?/C28y0 ðÞkk ; (5) and the mean exponential rate of divergence of thetrajectories is defined by s 1/C30lim k0/C121 klndk d0 ! : (6) For an n-dimensional phase space ( MAP), there are n Lyapunov characteristic exponents s1]s2]...> sn::However, because the largest exponent s1will dominate, this limit is practically useful only for finding the largest exponent. Numerically, since dk increases exponentially with k, after a few steps the perturbed trajectory is no longer nearby. It is there- fore necessary to renormalize frequently every t steps. Defining rkr/C13dkr d0; (7) one can then compute s1 /C30lim k 0/C121 nrXn k /C301ln rkr : (8) Numerical computation of the second (smaller) Lya- punov exponent may be carried by considering the evolution of a 2-D surface. It will behave as e(s1/C27s2)t ; (9) so s2 can be extracted if s1 is known. The process may be repeated to find smaller exponents. For HAMILTONIAN SYSTEMS , the LCEs exist in addi- tive inverse pairs, so if s is an LCE, then so is /C28s: One LCE is always 0. For a 1-D oscillator (with a 2-D phase space), the two LCEs therefore must be s1 /C30 s2 /C300; so the motion is QUASIPERIODIC and cannot be CHAOTIC . For higher order HAMILTONIAN SYSTEMS , there are always at least two 0 LCEs, but other LCEs may enter in plus-and-minus pairs l and /C28l: If they, too, are both zero, the motion is integrable and not CHAOTIC . If they are NONZERO , the POSITIVE LCE l results in an exponential separation of trajectories, which corresponds to a CHAOTIC region. Notice that it is not possible to have all LCEs NEGATIVE , which explains why convergence of orbits is never observed in HAMILTONIAN SYSTEMS . Now consider a dissipative system. For an arbitrary n-D phase space, there must always be one LCE equal to 0, since a perturbation along the path results in no divergence. The LCEs satisfy ai si B0 : There- fore, for a 2-D phase space of a dissipative system, s1 /C300; s2 B0: For a 3-D phase space, there are three possibilities: 1. (Integrable): s1 /C300 ; s2 /C300; s3 B0 ;/ 2. (Integrable): s1 /C300 ; s2 ; s3 B0 :;/ 3. (CHAOTIC ): s1 /C300 ; s2 > 0; s3 B/C28s2 B0:/ See also CHAOS ,H AMILTONIAN SYSTEM ,LYAPUNOV CHARACTERISTIC NUMBER ,OSEDELEC THEOREM References Chirikov, B. V. "A Universal Instability of Many-Dimen- sional Oscillator Systems." Phys. Rep. 52, 264 /C1/79, 1979. Ramasubramanian, K. and Sriram, M. S. A Comparative Study of Computation of Lyapunov Spectra with Different Algorithms 1999. http://xxx.lanl.gov/abs/chao-dyn/ 9909029/. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Lyapunov Characteristic Number Given a LYAPUNOV CHARACTERISTIC EXPONENT si ; the corresponding Lyapunov characteristic number liis defined as li /C13esi : (1) For an n-dimensional linear MAP,Xn/C271 /C30MX n: (2) The Lyapunov characteristic numbers l1 ; ..., lnare the EIGENVALUES of the MAP MATRIX . For an arbitrary MAP xn/C271 /C30f1(xn ; yn) (3) yn /C271 /C30f2(xn ; yn) ; (4) the Lyapunov numbers are the EIGENVALUES of the limit lim n 0/C12[J(xn ; yn)J(xn/C281 ; yn/C281) /C1/C1/C1J(x1 ; y1)]1=n ; (5) where J(x; y) is the JACOBIAN J(x; y) /C13@f1(x; y) @x@f1(x; y) @y @f2(x; y) @x@f2(x; y) @y/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C()/C(): (6) If l i for all i, the system is not CHAOTIC .If l "0 and the MAP is AREA-PRESERVING (HAMILTONIAN ), the product of EIGENVALUES is 1. See also ADIABATIC INVARIANT ,C HAOS ,LYAPUNOV CHARACTERISTIC EXPONENT Lyapunov Condition If the third MOMENT exists for a STATISTICAL DISTRI- BUTION of xi and the LEBESGUE INTEGRAL is given by r3 n /C30Xn i /C301g/C12 /C28/C12xjj3dFi(x) ; then if lim n0/C12rn sn/C300; the CENTRAL LIMIT THEOREM holds. See also CENTRAL LIMIT THEOREM Lyapunov Dimension For a 2-D MAP with s2 > s1 ; dLya /C301 /C28s1 s2; where snare the LYAPUNOV CHARACTERISTIC EXPO- NENTS . See also CAPACITY DIMENSION ,KAPLAN- YORKE CON- JECTURE References Frederickson, P.; Kaplan, J. L.; Yorke, E. D.; and Yorke, J. A. "The Liapunov Dimension of Strange Attractors." J. Diff. Eq. 49, 185/C1/07, 1983. Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, p. 549, 1995. Lyapunov Function This entry contributed by MARTIN KELLER- RESSEL A Lyapunov function is a SCALAR FUNCTION V(y) defined on a region D that is continuous, positive definite (i.e., V(0) /C300; V(y) > 0 for all y "0); and has continuous first-order PARTIAL DERIVATIVES at every point of D. The derivative of V with respect to the system y?/C30f(y) ; written as V /C31(y) is defined as the DOT PRODUCT V /C31(y) /C309V(y) /C215 F(y) : The existence of a Lyapunov function for which V /C31(y) 50 on some region D containing the origin, guarantees the stability of the zero solution of y?/C30 f(y) ; while the existence of a Lyapunov function for which V /C31(y) is negative definite (i.e., V /C31(0) /C300; V /C31(y) B0 for all y "0) on some region D containing the origin guarantees the asymptotical stability of the zero solution of y?/C30f(y)/ For example, given the system y?/C30z z?/C30/C28 y /C282z and the Lyapunov function V(y; z) /C30(y2 /C27z2) =2; we obtain V /C31(y; z) /C30yz /C27z(/C28y /C282z) /C30/C282z2 ; which is nonnegative on every region containing the origin, and thus the zero solution is stable. See also LINEAR STABILITY ,NONLINEAR STABILITY References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, pp. 502 /C1/12, 1992. Brauer, F. and Nohel, J. A. The Qualitative Theory of Ordinary Differential Equations: An Introduction. New York: Dover, 1989. Hahn, W. Theory and Application of Liapunov’s Direct Method. Englewood Cliffs, NJ: Prentice-Hall, 1963. Jordan, D. W. and Smith, P. Nonlinear Ordinary Differen- tial Equations. Oxford, England: Clarendon Press, p. 283, 1977. Kalman, R. E. and Bertram, J. E. "Control System Analysis and Design Via the ‘Second Method’ of Liapunov, I. Continuous-Time Systems." J. Basic Energ. Trans. ASME 82, 371 /C1/93, 1960. Oguzto ¨reli, M. N.; Lakshmikantham, V.; and Leela, S. "An Algorithm for the Construction of Liapunov Functions." Nonlinear Anal. 5, 1195 /C1/212, 1981. Zwillinger, D. "Liapunov Functions." §120 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 429 /C1/32, 1997.Lyapunov’s First Theorem A NECESSARY and SUFFICIENT condition for all the EIGENVALUES of a REAL n /C29n matrix A to have NEGATIVE REAL PARTS is that the equation ATV /C27VA /C30/C28 1 has as a solution where V is an n /C29n matrix and (x; Vx)isa POSITIVE DEFINITE QUADRATIC FORM . References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1122, 2000. Lyapunov’s Second Theorem If all the EIGENVALUES of a REAL MATRIX A have REAL PARTS , then to an arbitrary negative definite quad- ratic form (x; Wx) with x /C30x(t) there corresponds a positive definite quadratic form (x; Vx) such that if one takes dx dt /C30AAx; then (x; Vx) and (x; Wx) satisfy d dt (x; Vx) /C30(x; Wx): References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1122, 2000. Lyndon Word A Lyndon word is an aperiodic notation for represent- ing a NECKLACE . See also DE BRUIJN SEQUENCE ,IRREDUCIBLE POLY- NOMIAL ,NECKLACE References Ruskey, F. "Information on Necklaces, Lyndon Words, de Bruijn Sequences." http://www.theory.csc.uvic.ca/~cos/inf/ neck/NecklaceInfo.html. Sloane, N. J. A. Sequences A001037/M0116 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Lyons Group The SPORADIC GROUP Ly. See also SPORADIC GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/Ly.html. Lyusternik-Schnirelmann Theorem If a sphere is covered by three closed sets, then one of them must contain a pair of ANTIPODAL POINTS .References Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 122 and 284, 1997. M MacDonald Function A modified HANKEL FUNCTION . Macdonald Polynomial See also N! THEOREM References Haiman, M. "Macdonald Polynomials and Geometry." In New Perspectives in Algebraic Combinatorics (Ed. L. J. Billera, A. Bjo¨rner, C. Greene, R. E. Simion, and R. P. Stanley). Cambridge, England: Cambridge Univer- sity Press, pp. 207 /C1/54, 1999. Macdonald, I. G. Symmetric Functions and Hall Polyno- mials, 2nd ed. Oxford, England: Oxford University Press, 1995. Zabrocki, M. "Macdonald Polynomials." http://www.lacim.u- qam.ca/~zabrocki/MPWP.html. Macdonald’s Constant-Term Conjecture Macdonald’s constant term conjectures are related to ROOT SYSTEMS of LIE ALGEBRAS (Macdonald 1982, Andrews 1986). They can be regarded as general- izations of DYSON’S CONJECTURE (Dyson 1962), its q- analog due to Andrews, and Mehta’s conjecture (Mehta 1991). The simplest of these states that if R is a ROOT SYSTEM , then the constant term in Pa /C23R1 /C28e aðÞk; where k is a NONNEGATIVE INTEGER ,is Pl i/C301kdl kfflC{fflCz ; where the dl are fixed integer parameters of the ROOT SYSTEM R corresponding to the fundamental invariants of the WEYL GROUP W of R (Andrews 1986, p. 41). Opdam (1989) proved the q /C301 case for all root systems. The general conjecture had remained "al- most proved" for some time, since the infinite families were accomplished by Zeilberger-Bressoud (/An) ; Ka- dell (/Bn ; Dn) Gustafson (/BCn ; Cn) ; while the excep- tional cases were done by Zeilberger and (independently) Habsieger (/G2) ; Zeilberger (/G2dual), and Garvan and Gonnet (/F4and F4dual), using Zeilberger’s method. This left only the three root systems (/E6 ; E7 ; E8) which were infeasible to address using existing computers. In the meanwhile, how- ever, Cherednik (1993) proved the constant term conjectures for all root systems using a methodology not dependent on classification. A special case of the constant-term conjecture is given by the assertion that the constant term in Y 1Bi"j5n1/C28xi xj !k (1) is (nk)!=(k!)n:Another special case asserts that the constant term inY i55n(xi;q)a(q=xi;q)a"# /C29Y 15i5j5n(xixj;q)bq xixj;q ! bxi xj;q ! bqxj xi;q ! b (2) is (q;q)nb [(q;q)b]nY 15j5n/C281(q;q)2a/C272jb(q;q)2jb (q;q)a/C27(n/C27j/C281)n(q;q)a/C27jb(3) (Andrews 1986, p. 41). See also DYSON’S CONJECTURE ,ROOT SYSTEM ,W EYL GROUP References Andrews, G. E. "The Macdonald Conjectures." §4.5 in q- Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and ComputerAlgebra. Providence, RI: Amer. Math. Soc., pp. 40 /C1 /2, 1986. Cherednik, I. "The Macdonald Constant-Term Conjecture." Duke Math. J. 70, 165/C1/77, 1993 and Internat. Math. Res. Not., No. 6, 165 /C1/77, 1993. Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. I." J. Math. Phys. 3, 140/C1/56, 1962. Macdonald, I. G. "Some Conjectures for Root Systems." SIAM J. Math. Anal. 13, 988/C1/007, 1982. Mehta, M. L. Random Matrices, 2nd ref. enl. ed. New York: Academic Press, 1991. Opdam, E M. "Some Applications of Hypergeometric Shift Operators." Invent. Math. 98,1/C1/8, 1989. Macdonald’s Plane Partition Conjecture Macdonald’s plane partition conjecture proposes a formula for the number of CYCLICALLY SYMMETRIC PLANE PARTITIONS (CSPPs) of a given integer whose YOUNG DIAGRAMS fit inside an n/C29n/C29nbox. Macdo- nald gave a product representation for the power series whose coefficients qnwere the number of such partitions of n. LetD(p) be the set of all integer points ( i;j;k) in the first OCTANT such that a PLANE PARTITION p/C30(aij)i s defined and 1 5k5aij:Then pis said to be cyclically symmetric if D(p) is invariant under the mapping (i;j;k)0(j;k;i):Let M(m;n) be the number of cyclically symmetric partitions of nsuch that none ofi;j;aijexceed m. LetBmbe the box containing all integer points ( i;j;k) such that 1 5i;j;k5m;then M(m;n) is the number of cyclically symmetric plane partitions of nsuch that D(p)⁄Bm:Now, let Cmbe the set of all the orbits in Bm:Finally, for each point p/C30(i;j;k)i nBm;let its height ht(p)/C30i/C27j/C27k/C282 (1) and for each jinCm;let½j½be the number of points in j(either 1 or 3) and write ht( j) /C30X p /C23 jht(p): (2) Then Macdonald conjectured that X n]0M(m; n)qn /C30Y j /C23Cm1 /C28 q ½j ½/C27ht(j) 1 /C28 qht(j) (3) /C30Ym i/C3011 /C28 q3i/C281 1 /C28 q3i/C282Ym j/C30i1 /C28 q3(m/C27i/C27j/C281) 1 /C28 q3(2i/C27j/C281)"# ; (4) (Mills et al. 1982, Macdonald 1995), where the latter form is due to Andrews (1979). Andrews (1979) proved the q /C301 case, giving the total number of CSPPs fitting inside an n /C29n /C29n box. The general case was proved by Mills et al. (1982). See also CYCLICALLY SYMMETRIC PLANE PARTITION , DYSON’S CONJECTURE ,PLANE PARTITION ,ROOT SYS- TEM,ZEILBERGER- BRESSOUD THEOREM References Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math. 53, 193 /C1/25, 1979. Andrews, G. E. "Macdonald’s Conjecture and Descending Plane Partitions." In Combinatorics, Representation The- ory and Statistical Methods in Groups (Ed. T. V. Nar- ayana, R. M. Mathsen, and J. G. Williams). New York: Dekker, pp. 91 /C1/06, 1980. Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999. Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/46. Macdonald, I. G. "Some conjectures for Root Systems." SIAM J. Math. Anal. 13, 988 /C1/007, 1982. Macdonald, I. G. Symmetric Functions and Hall Polyno- mials, 2nd ed. Oxford, England: Oxford University Press, 1995. Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr. "Proof of the Macdonald Conjecture." Invent. Math. 66,73/C1/7, 1982. Morris, W. G. Constant Term Identities for Finite and Affine Root Systems: Conjectures and Theorems. Ph.D. thesis. Madison, WI: University of Wisconsin, 1982. Machine A method for producing infinite LOOP SPACES and spectra. See also GADGET ,L OOP SPACE ,M AY-THOMASON UNIQUENESS THEOREM ,TURING MACHINE Machin-Like Formulas Machin-like formulas have the form mcot/C281u/C27ncot/C281v/C301 4kp; (1) where u,v, and kare POSITIVE INTEGERS andmand nare NONNEGATIVE INTEGERS . Some such FORMULAS can be found by converting the INVERSE TANGENT decompositions for which cn"0 in the table of Todd (1949) to INVERSE COTANGENTS . However, this givesonly Machin-like formulas in which the smallest term is91. Machin-like formulas can be derived by writing cot/C281z/C301 2ilnz/C27i z/C28i ! (2) and looking for akanduksuch that X kakcot/C281uk/C301 4p; (3) so Y kuk/C27i uk/C28i !ak /C30e2pi=4/C30i: (4) Machin-like formulas exist IFF(4) has a solution in INTEGERS . This is equivalent to finding INTEGER values such that (1/C28i)k(u/C27i)m(v/C27i)n(5) isREAL (Borwein and Borwein 1987, p. 345). An equivalent formulation is to find all integral solutions to one of 1/C27x2/C302yn(6) 1/C27x2/C30yn(7) forn/C303, 5, .... There are only four such FORMULAS , 1 4p/C304 tan/C28115fflCz6fflCz7 /C28tan/C2811 239fflCz6fflCz7 (8) 14p/C30tan/C28112fflCz6fflCz7 /C27tan/C28113fflCz6fflCz7 (9) 1 4p/C302 tan/C28112fflCz6fflCz7 /C28tan/C28117fflCz6fflCz7 (10) 14p/C302 tan/C28113fflCz6fflCz7 /C27tan/C28117fflCz6fflCz7 ; (11) known as M ACHIN’S FORMULA ,EULER’S MACHIN-LIKE FORMULA ,HERMANN’S FORMULA , and H UTTON’S FOR- MULA . These follow from the identities 5/C27i 5/C28i !4239/C27i 239/C28i !/C281 /C30i (12) 2/C27i 2/C28i ! 3/C27i 3/C28i ! /C30i (13) 2/C27i 2/C28i ! 7/C27i 7/C28i !/C281 /C30i (14) 3/C27i 3/C28i ! 7/C27i 7/C28i ! /C30i: (15) Machin-like formulas with two terms can also be generated which do not have integral arc cotangent arguments such as Euler’s 1 4p/C305 tan/C28117fflCz6fflCz7 /C272 tan/C2813 79fflCz6fflCz7 (16) (Wetherfield 1996), and which involve inverse SQUARE ROOTS , such as p 2/C302 tan/C2811ffiffiffi 2p ! /C27tan/C2811ffiffiffi8p ! : (17) Three-term Machin-like formulas include G AUSS’S MACHIN-LIKE FORMULA 1 4p/C3012 cot/C28118/C278 cot/C28157/C285 cot/C281239;(18) STRASSNITZKY’S FORMULA 1 4p/C30cot/C2812/C27cot/C2815/C27cot/C2818; (19) and the following, 1 4p/C306 cot/C2818/C272 cot/C28157/C27cot/C281239 (20) 1 4p/C304 cot/C2815/C281 cot/C28170/C27cot/C28199 (21) 1 4p/C301 cot/C2812/C271 cot/C2815/C27cot/C2818 (22) 1 4p/C308 cot/C28110/C281 cot/C281239/C284 cot/C281515 (23) 1 4p/C305 cot/C2817/C274 cot/C28153/C272 cot/C2814443 : (24) The first is due to Størmer, the second due to Rutherford, and the third due to Dase. Using trigonometric identities such as cot/C281x/C302 cot/C281(2x)/C28cot/C2814x3/C273xfflC{fflCz ; (25) it is possible to generate an infinite sequence of Machin-like formulas. Systematic searches therefore most often concentrate on formulas with particularly "nice" properties (such as "efficiency"). The efficiency of a FORMULA is the time it takes to calculate pwith the POWER SERIES for arctangent p/C30a1cotb1ðÞ/C27a2cotb2ðÞ/C27...; (26) and can be roughly characterized using Lehmer’s "measure" formula e/C13X 1 log10bi: (27) The number of terms required to achieve a given precision is roughly proportional to e, so lower e- values correspond to better sums. The best currently known efficiency is 1.51244, which is achieved by the 6-term series1 4p/C30183 cot/C281239/C2732 cot/C2811023/C2868 cot/C2815832 /C2712 cot/C281110443 /C2812 cot/C2814841182 /C28100 cot/C2816826318 (28) discovered by C.-L. Hwang (1997). Hwang (1997) also discovered the remarkable identities 1 4p/C30Pcot/C2812/C28Mcot/C2813/C27Lcot/C2815/C27Kcot/C2817 /C27(N/C27K/C27L/C282M/C273P/C285) cot/C2818 /C27(2N/C27M/C28P/C272/C28L) cot/C28118 /C28(2P/C283/C28M/C27L/C27K/C28N) cot/C28157/C28Ncot/C281239; (29) where K,L,M,N, and Pare POSITIVE INTEGERS , and 1 4p/C30(N/C272) cot/C2812/C28Ncot/C2813 /C28(N/C271) cot/C281N: (30) The following table gives the number N(n) of Machin- like formulas of nterms in the compilation by Wetherfield and Hwang. Except for previously known identities (which are included), the criteria for inclu- sion are the following: 1. first term B8 digits: measure B1:8:/ 2. first term /C308 digits: measure B1:9:/ 3. first term /C309 digits: measure B2:0:/ 4. first term /C3010 digits: measure B2:0:/ n /N(n)//min e/ 11 0 2 4 1.851133 106 1.78661 4 39 1.58604 5 90 1.634856 120 1.512447 113 1.54408 8 18 1.65089 9 4 1.72801 10 78 1.63086 11 34 1.6305 12 188 1.67458 13 37 1.7193414 5 1.7516115 24 1.77957 16 51 1.81522 17 5 1.90938 18 570 1.87698 19 1 1.94899 20 11 1.95716 21 1 1.98938 Total 1500 1.51244 See also EULER’S MACHIN- LIKE FORMULA ,G AUSS’S MACHIN- LIKE FORMULA ,G REGORY NUMBE R,H ER- MANN’S FORMULA ,HUTTON’S FORMULA ,INVERSE CO- TANGENT ,MACHIN’S FORMULA ,PI,STøRMER NUMBER , STRASSNITZKY’S FORMULA References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 347 /C1/59, 1987. Berstel, J.; Pin, J.-E.; and Pocchiola, M. Mathe ´matiques et Informatique. New York: McGraw-Hill, 1991. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67 /C1/8, 1988. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 241 /C1/48, 1996. Hwang, C.-L. "More Machin-Type Identities." Math. Gaz. 81, 120 /C1/21, 1997. Lehmer, D. H. "On Arccotangent Relations for p:/" Amer. Math. Monthly 45, 657 /C1/64, 1938. Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981. Lewin, L. Structural Properties of Polylogarithms. Provi- dence, RI: Amer. Math. Soc., 1991. Nielsen, N. Der Euler’sche Dilogarithms. Leipzig, Germany: Halle, 1909. Se´roul, R. "Machin Formulas." §9.3 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 240 /C1/52, 2000. Størmer, C. "Sur l’Application de la The´orie des Nombres Entiers Complexes a` la Solution en Nombres Rationnels x1 ; x2 ; ..., c1 ; c2 ; ..., k de l’Equation...." Archiv for Mathematik og Naturvidenskab B19 ,75/C1/5, 1896. Todd, J. "A Problem on Arc Tangent Relations." Amer. Math. Monthly 56, 517 /C1/28, 1949. Weisstein, E. W. "Machin-Like Formulas." MATHEMATICA NOTEBOOK MACHIN FORMULAS.M . Wetherfield, M. "The Enhancement of Machin’s Formula by Todd’s Process." Math. Gaz. 80, 333 /C1/44, 1996. Wetherfield, M. "Machin Revisited." Math. Gaz. 81 121 /C1/23, 1997. Machin’s Formula 1 4 p /C304 tan /C28115fflCz6fflCz7 /C28tan /C2811 239fflCz6fflCz7 : There are a whole class of MACHIN-LIKE FORMULAS with various numbers of terms (although only four such formulas with only two terms). The properties ofthese formulas are intimately connected with COTAN- GENT identities. See also 239,GREGORY NUMBER ,M ACHIN- LIKE FOR- MULAS ,PI Mackey’s Theorem Let E and F be paired spaces with S a family of absolutely convex bounded sets of F such that the sets of S generate F and, if B1 ; B2 /C23 S ; there exists a B3 /C23 S such that B3 ‡B1and B3 ‡B2 : Then the dual space of ESis equal to the union of the weak completions of lB ; where l > 0 and B /C23 S:/ See also GROTHENDIECK’S THEOREM References Iyanaga, S. and Kawada, Y. (Eds.). "Mackey’s Theorem." §407M in Encyclopedic Dictionary of Mathematics. Cam- bridge, MA: MIT Press, p. 1274, 1980. Mac Lane’s Theorem A theorem which treats constructions of FIELDS of CHARACTERISTIC p. See also CHARACTERISTIC (FIELD), FIELD Maclaurin-Be ´zout Theorem The Maclaurin-Be ´zout theorem says that two curves of degree n intersect in n2 points, so two CUBICS intersect in nine points. This means that n(n /C273)=2 points do not always uniquely determine a single curve of order n. See also CRAME ´ R-EULER PARADOX Maclaurin-Cauchy Theorem Iff(x) is positive and decreases to 0, then an E ULER CONSTANT gf/C30lim n0/C12Xn k/C301f(k)/C28gn af(x)dx"# can be defined. If f(x)/C301=x;then g/C30lim n0/C12Xn k/C3011 k/C28gn 1dx x ! /C30lim n0/C12Xn k/C3011k/C28lnn ! ; where gis the E ULER- MASCHERONI CONSTANT . Maclaurin Integral Test INTEGRAL TEST Maclaurin Polynomial MACLAURIN SERIES Maclaurin Series A series expansion of a function about 0, f(x)/C30f(0)/C27f?(0)x/C27fƒ(0) 2!x2/C27f(3)(0) 3!x3/C27... /C27f(n)(0) n!xn/C27...; (1) named after the Scottish mathematician Maclaurin. Maclaurin series for common functions include 1 1/C28x/C301/C27x/C27x2/C27x3/C27x4/C27x5/C27... for/C281BxB1 (2) cn(x;k)/C301/C281 2x2/C271 241/C274k2fflC{fflCz x4/C27... ( 3 ) cosx/C301/C2812x2/C271 24x4/C281 720x6/C28... for/C28/C12B xB/C12 (4) cos/C281x/C3012p/C28x/C2816x3/C283 40x5/C285 112x7/C28... for/C281BxB1 (5) cosh x/C301/C271 2x2/C271 24x4/C271 720x6/C271 40;320x8/C27. . . (6) cosh/C281(1/C27x)/C30ffiffiffiffiffiffi 2xp 1/C281 2x/C273 160x2/C285 896x3/C27...fflCz6fflCz7 (7) cotx/C30x/C281/C281 3x/C281 45x3/C282 945x5/C281 4725x7/C28... ( 8 ) cot/C281x/C301 2p/C28x/C2713x3/C2815x5/C2717x7/C2819x9/C27... ( 9 ) cot/C2811 x ! /C30x/C2813x3/C2715x5/C2817x7/C2719x9/C27. . . (10) coth x/C30x/C281/C2713x/C281 45x4/C272 945x5/C281 4725x7/C27. . . (11) coth/C281(1/C27x)/C3012ln 2/C2812lnx/C2714x/C281 16x2/C27. . . (12) cscx/C30x/C281/C271 6x/C277 360x3/C2731 15120x5/C27. . . (13) csch x/C30x/C281/C2816x/C277 360x3/C2731 15120x5/C27. . . (14) csch/C281x/C30ln 2/C28lnx/C271 4x2/C283 32x4/C275 96x6/C28. . . (15) dn(x;k)/C301/C281 2k2x2/C271 24k24/C27k2fflC{fflCz x4/C27. . . (16) erfx/C301ffiffiffipp 2x/C282 3x3/C2715x5/C281 21x7/C27...fflCz6fflCz7 (17) ex/C301/C27x/C271 2x2/C2716x3/C271 24x4/C27... for/C28/C12B xB/C12 (18)2F1(a;b;g;x) /C301/C27ab 1gx/C27a(a/C271)b(b/C271) 2g(g/C271)x2/C27. . . (19) ln(1/C27x)/C30x/C2812x2/C2713x3/C2814x4/C27... for/C281BxB1 (20) ln1/C27x 1/C28x ! /C302x/C2723x3/C2725x5/C2727x7/C27... for/C281BxB1 (21) secx/C301/C271 2x2/C275 24x4/C2761 720x6/C27277 8064x8/C27. . . (22) sech x/C301/C2812x2/C275 24x4/C2861 720x6/C27277 8064x8/C27. . . (23) sech/C281x/C30ln 2/C28lnx/C2814x2/C283 32x4/C28. . . (24) sinx/C30x/C281 6x3/C271 120x5/C281 5040x7/C27... for/C28/C12B xB/C12 (25) sin/C281x/C30x/C271 6x3/C273 40x5/C275 112x7/C2735 112x9/C27. . . (26) sinh x/C30x/C271 6x3/C271 120x5/C271 5040x7/C271 362;880x9/C27. . . (27) sinh/C281x/C30x/C2816x3/C273 40x5/C285 112x7/C2735 1152x9/C28. . . (28) sn(x;k)/C30x/C28161/C27k2fflC{fflCz x3/C271 1201/C2714k2/C27k4fflC{fflCz x5/C27... (29) tanx/C30x/C2713x3/C272 15x5/C2717 315x7/C2762 2835x9/C27. . . (30) tan/C281x/C30x/C2813x3/C2715x5/C2817x7/C27... for/C281BxB1 (31) tan/C281(1/C27x)/C3014p/C2712x/C2814x2/C271 12x3/C271 40x5/C27. . . (32) tanh x/C30x/C2813x3/C272 15x5/C2817 315x7/C2762 2835x9/C27. . . (33) tanh/C281x/C30x/C2713x3/C2715x5/C2717x7/C2719x9/C27. . . (34) The explicit forms for some of these are 1 1/C28x/C30X/C12 n/C300xn(35) cosx/C30X/C12 n/C300(/C281)n (2n)!x2n(36) cosh x/C30X/C12 n/C3001 (2n)!x2n(37) cscx/C30X/C12 n/C300(/C281)n/C2712(22n/C281/C281)B2n (2n)!x2n/C281(38) ex /C30X/C12 n/C3001 n!xn (39) ln (1 /C27x) /C30X/C12 n/C301(/C281)n/C271 nxn (40) ln1 /C27 x 1 /C28 x ! /C30X/C12 n/C3012 (2n /C28 1)x2n/C281 (41) sec x /C30X/C12 n/C300( /C281)nE2n (2n)!x2n (42) sin x /C30X/C12 n/C300(/C281)n (2n /C27 1)!x2n/C271 (43) sinh x /C30X/C12 n/C3001 (2n /C27 1)!x2n/C271 (44) tan x /C30X/C12 n/C300( /C281)n22n/C272(22n/C272 /C28 1)B2n/C272 (2n /C27 2)! x2n/C271 (45) tan/C281x /C30X/C12 n/C301(/C281)n /C271 (2n /C28 1)x2n/C281 (46) tanh /C281x /C30X/C12 n/C3011 2n /C28 1x2n /C281 ; (47) where Bn are BERNOULLI NUMBERS and En are EULER NUMBERS . See also ALCUIN’S SEQUENCE ,LAGRANGE EXPANSION , LAGRANGE REMAINDER ,LEGENDRE SERIES ,TAYLOR SERIES References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 299 /C1/00, 1987. Maclaurin Trisectrix A curve first studied by Colin Maclaurin in 1742. It was studied to provide a solution to one of the GEOMETRIC PROBLEMS OF ANTIQUITY , in particular TRISECTION of an ANGLE , whence the name trisectrix.The Maclaurin trisectrix is an ANALLAGMATIC CURVE , and the origin is a CRUNODE . The Maclaurin trisectrix has CARTESIAN equation y2 /C30x2(x /C27 3a) a /C28 x; (1) or the PARAMETRIC EQUATIONS x /C30at2 /C28 3 t2 /C27 1 (2) y /C30at(t2 /C28 3) t2 /C27 1: (3) The ASYMPTOTE has equation x /C30 a, and the center of the loop is at (/C282a ; 0): If P is a point on the loop so that the line CP makes an ANGLE of 3a with the negative Y-AXIS , then the line OP will make an ANGLE of a with the negative Y-AXIS . The Maclaurin trisectrix is sometimes defined in- stead as xx2 /C27y2fflC{fflCz /C30ay2 /C283x2fflC{fflCz (4) y2 /C30x2(3a /C27 x) a /C28 x (5) r /C302a sin(3u) sin(2u): (6) Another form of the equation is the POLAR EQUATION r /C30a sec1 3 ufflCz6fflCz7 ; (7) where the origin is inside the loop and the crossing point is on the NEGATIVE X-AXIS . The tangents to the curve at the origin make angles of 960/C14 with the X-AXIS . The AREA of the loop is Aloop/C303ffiffiffi 3p a2; (8) and the NEGATIVE x-intercept is ( /C283a;0) (MacTutor Archive). The Maclaurin trisectrix is the PEDAL CURVE of the PARABOLA where the PEDAL POINT is taken as the reflection of the FOCUS in the DIRECTRIX . See also RIGHT STROPHOID ,TSCHIRNHAUSEN CUBIC References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 103 /C1/06, 1972. MacTutor History of Mathematics Archive. "Trisectrix of Maclaurin." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Trisectrix.html. Maclaurin Trisectrix Inverse Curve The INVERSE CURVE of the MACLAURIN TRISECTRIX with INVERSION CENTER at the NEGATIVE x-intercept is a TSCHIRNHAUSEN CUBIC . MacMahon’s Prime Number of Measurement PRIME NUMBER OF MEASUREMENT MacRobert’s E-Function Ep; ar : rs : x ðÞ /C13G aq /C271fflC{fflCz G r1 /C28 a1 ðÞ G r2 /C28 a2 ðÞ/C1/C1/C1 G rq /C28 aqfflC{fflCz /C2Yq m/C301g/C12 0l rm/C28a m/C281 m 1 /C27 l mfflC{fflCz/C28rmdlm /C2Yp /C28q/C281 n/C302g/C12 0e /C28lq/C27n laq/C27n /C281 q /C27 ndlq/C27 n /C2g/C12 0e /C28 lp l ap/C281 p 1 /C27lq/C272 lq /C273 /C1/C1/C1lp 1 /C27 l1 ðÞ/C1/C1/C1 1 /C27 lqfflC{fflCz x"#/C28aq/C271 dlp ; where G(z) is the GAMMA FUNCTION and other details are discussed by Gradshteyn and Ryzhik (2000). See also FOX’S H-FUNCTION ,K AMPE ´DE FE´ RIET FUNCTION ,MEIJER’S G-FUNCTION References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Definition of the E-Function." §5.2 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 203 /C1/06, 1981. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 896 /C1/03 and 1071 /C1/072, 2000. MacRobert, T. M. "Induction Proofs of the Relations between Certain Asymptotic Expansions and Corresponding Gen- eralised Hypergeometric Series." Proc. Roy. Soc. Edin- burgh 58,1/C1/3, 1937 /C1/8. MacRobert, T. M. "Some Formulæ for the E-Function." Philos. Mag. 31, 254 /C1/60, 1941. Macron A macron is a BAR placed over a single symbol or character, such as ¯x: The symbol ¯z is sometimes used to denote the following operations.1. The COMPLEX CONJUGATE . 2. NEGATION of a logical expression. 3. Infrequently, ADJOINT operator. A bar placed over multiple symbols or characters is called a VINCULUM . See also BAR,HAT,VINCULUM References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 281, 1997. Madelung Constants The quantities obtained from cubic, hexagonal, etc., LATTICE SUMS , evaluated at s /C301, are called Made- lung constants. For cubic LATTICE SUMS , they are expressible in closed form for EVEN indices, b2(2) /C30/C284b(1)h(1) /C30/C284p 4ln 2 /C30/C28p ln 2 (1) b4(2) /C30/C288h(1)h(0) /C30/C288ln2 /C2151 2 /C30/C284ln2 ; (2) where b(n) is the DIRICHLET BETA FUNCTION and h(n) is the DIRICHLET ETA FUNCTION . b3(1) is given by BENSON’S FORMULA , /C28b3(1) /C30X ?/C12 i; j; k /C30/C28/C12( /C281)i/C27j/C27k /C271 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 /C27 j2 /C27 k2p /C3012pX/C12 m; n /C301 ; 3 ; ...sech21 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 /C27n2pfflCz6fflCz7 ; (3) where the prime indicates that summation over (0, 0, 0) is excluded. b3(1) is sometimes called "the" Made- lung constant, corresponds to the Madelung constant for a 3-D NaCl crystal, and is numerically equal to /C281:74756 . . . :/ For hexagonal LATTICE SUM ,h2(2) is expressible in closed form as h2(2)/C30pln 3ffiffiffi 3p : (4) See also BENSON’S FORMULA ,LATTICE SUM References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Buhler, J. and Wagon, S. "Secrets of the Madelung Con- stant." Mathematica in Education and Research 5,4 9/C1/5, Spring 1996. Crandall, R. E. and Buhler, J. P. "Elementary Function Expansions for Madelung Constants." J. Phys. Ser. A: Math. and Gen. 20, 5497 /C1/510, 1987. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/mdlung/mdlung.html. Maeder’s Owl Minimal Surface BOUR’S MINIMAL SURFACE Maehly’s Procedure A method for finding ROOTS which defines Pj(x) /C30P(x) (x /C28 x1) /C1/C1/C1(x /C28 xj) ; (1) so the derivative is P?j(x) /C30P ?(x) x /C28 x1 ðÞ /C1 /C1 /C1 x /C28 xjfflC{fflCz /C28P(x) x /C28 x1 ðÞ/C1/C1/C1 x /C28 xjfflC{fflCzXj i /C301x /C28xi ðÞ/C281(2) One step of NEWTON’S METHOD can then be written as xk /C271 /C30xk /C28PxkðÞ P? xkðÞ/C28 PxkðÞPj i /C301xk /C28 xi ðÞ/C281 : (3) Magic Circles A set of n magic circles is a numbering of the intersections of the n CIRCLES such that the sum over all intersections is the same constant for all circles. The above sets of three and four magic circles have magic constants 14 and 39 (Madachy 1979). Another type of magic circle arranges the number 1, 2, ..., n in a number of rings, which each ring containing the same number of elements and corre- sponding elements being connected with radial lines. One of the numbers (which is subsequently ignored) is placed at the center. In a magic circle arrangement, the rings have equal sums and this sum is also equal to the sum of elements along each diameter (exclud- ing the central number). Three magic circles using the numbers 1 to 33 are illustrated above. (Hung). See also MAGIC GRAPH ,MAGIC SQUARE References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 86, 1979. Magic Constant The number M2(n)/C301 nXn2 k/C301k/C301 2nn2/C271fflC{fflCz to which the nnumbers in any horizontal, vertical, or main diagonal line must sum in a MAGIC SQUARE . The first few values are 1, 5, 15, 34, 65, 111, 175, 260, ... (Sloane’s A006003). The magic constant for an nth order magic square starting with an INTEGER Aand with entries in an increasing ARITHMETIC SERIES with difference Dbetween terms is M2(n;A;D)/C301 2n2a/C27Dn2/C281fflC{fflCzfflC}fflC( (Hunter and Madachy 1975, Madachy 1979). In a PANMAGIC SQUARE , in addition to the main diagonals, the broken diagonals also sum to M2(n):/ For a MAGIC CUBE ,MAGIC TESSERACT , etc., the magic d-D constant is Md(n) /C301 nd/C281Xnd k /C301k /C301 2nnd /C271fflC{fflCz : The first few magic constants are summarized in the following table. n /M2(n)// M3(n)// M4(n)/ Sloane A006003 A027441 A021003 1111 2591 731 54 21 2 3 4 34 130 514 5 65 315 1565 There is a corresponding multiplicative magic con- stant for MULTIPLICATION MAGIC SQUARES . A similar magic constant M(j) nof degree k is defined for MAGIC SERIES and MULTIMAGIC SERIES as 1=n times the sum of the first n2 kth powers, M(k) n/C301 nXn2 i /C301ik /C30H(/C28p) n2 n; where H(k) nis a HARMONIC NUMBER of order k. The following table gives the first few values. nk /C301 k /C302 k /C303 k /C304 Sloane A006003 A052459 A052460 A052461 11111 2 5 15 50 177 3 15 95 675 5111 4 34 374 4624 60962 5 65 1105 21125 430729 See also MAGIC CUBE,MAGIC GEOMETRIC CONSTANTS , MAGIC HEXAGON ,M AGIC SERIES ,M AGIC SQUARE , MULTIMAGIC SERIES ,M ULTIPLICATION MAGIC SQUARE ,PANMAGIC SQUARE References Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, pp. 23 /C1/4, 1975. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 86, 1979. Sloane, N. J. A. Sequences A006003/M3849, A021003, A027441, A052459, A052460, and A052461 in "An On- Line Version of the Encyclopedia of Integer Sequences."http://www.research.att.com/~njas/sequences/eisonli- ne.html. Magic Cube An n /C29n /C29n 3-D version of the MAGIC SQUARE in which the n2 rows, n2 columns, n2 pillars (or "files"), and four space diagonals each sum to a single number M3(n) known as the MAGIC CONSTANT . If the CROSS SECTION diagonals also sum to M3(n); the magic cube is called a PERFECT MAGIC CUBE ; if they do not, the cube is called a SEMIPERFECT MAGIC CUBE , or some- times an ANDREWS CUBE (Gardner 1988). A pandia- gonal cube is a perfect or SEMIPERFECT MAGIC CUBE which is magic not only along the main space diagonals, but also on the broken space diagonals. A magic cube using the numbers 1, 2, ..., n3 ; if it exists, has MAGIC CONSTANT M3(n) /C301 2 nn3 /C271fflC{fflCz : For n /C301, 2, ..., the magic constants are 1, 9, 42, 130, 315, 651, ... (Sloane’s A027441). The above SEMIPERFECT MAGIC CUBES of orders three (Hunter and Madachy 1975, p. 31; Ball and Coxeter 1987, p. 218) and four (Ball and Coxeter 1987, p. 220) have magic constants 42 and 130, respectively. There is a trivial SEMIPERFECT MAGIC CUBE of order one, but no semiperfect cubes of orders two or three exist. Semiperfect cubes of ODD order with n ]5 and DOUBLY EVEN order can be constructed by extending the methods used for MAGIC SQUARES . Semiperfect pandiagonal cubes exist for all orders 8 n and all ODD n/C218 (Ball and Coxeter 1987). A perfect pandiagonal magic cube has been constructed by Planck (1950), cited in Gardner (1988). See also BIMAGIC CUBE,M AGIC CONSTANT ,M AGIC GRAPH ,M AGIC HEXAGON ,M AGIC SQUARE ,M AGIC TESSERACT ,P ERFECT MAGIC CUBE,S EMIPERFECT MAGIC CUBE References Adler, A. and Li, S.-Y. R. "Magic Cubes and Prouhet Sequences." Amer. Math. Monthly 84, 618/C1/27, 1977. Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New York: Dover, 1960. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 216 /C1/24, 1987. Barnard, F. A. P. "Theory of Magic Squares and Cubes." Mem. Nat. Acad. Sci. 4, 209/C1/70, 1888. Benson, W. H. and Jacoby, O. Magic Cubes: New Recrea- tions. New York: Dover, 1981. Gardner, M. Sci. Amer. , Jan. 1976. Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213 /C1/25, 1988. Hirayama, A. and Abe, G. Researches in Magic Squares. Osaka, Japan: Osaka Kyoikutosho, 1983. Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 inMathematical Diversions. New York: Dover, p. 31, 1975. Lei, A. "Magic Cube and Hypercube." http://www.cs.ust.hk/ ~philipl/magic/mcube2.html. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 99 /C1/00, 1979. Pappas, T. "A Magic Cube." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 77, 1989. Planck, C. Theory of Path Nasiks. Rugby, England: Pri- vately Published, 1905. Rosser, J. B. and Walker, R. J. "The Algebraic Theory of Diabolical Squares." Duke Math. J. 5, 705/C1/28, 1939. Sloane, N. J. A. Sequences A027441 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Trenkler, M. "A Construction of Magic Cubes." Math. Gaz. 84,3 6/C1/1, 2000. Wynne, B. E. "Perfect Magic Cubes of Order 7." J. Recr. Math. 8, 285/C1/93, 1975 /C1/976. Magic Geometric Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. LetEbe a compact connected subset of d-dimen- sional E UCLIDEAN SPACE . Gross (1964) and Stadje (1981) proved that there is a unique REAL NUMBER a(E) such that for all x1;x2;...,xn/C23E;there exists y/C23E with 1 nXn j/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xd k/C301xj;k/C28ykfflC{fflCz2vuut/C30a(E): (1) The magic constant m(E)o fEis defined by m(E)/C30a(E) diam( E); (2) where diam( E)/C13max u;v/C23Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xd k/C301uk/C28vk ðÞ2vuut: (3) These numbers are also called DISPERSION NUMBERS and RENDEZVOUS VALUES . For any E, Gross (1964) and Stadje (1981) proved that 1 25m(E)B1: (4) IfIis a subinterval of the LINE and Dis a circular DISK in the PLANE , thenm(I)/C30m(D)/C301 2: (5) IfCis a CIRCLE , then m(C)/C302 p/C300:6366 . . . (6) An expression for the magic constant of an ELLIPSE in terms of its SEMIMAJOR and SEMIMINOR AXES lengths is not known. Nikolas and Yost (1988) showed that for aR EULEAUX TRIANGLE T 0:6675276 5m(T)50:6675284 : (7) Denote the MAXIMUM value of m(E)i nn-D space by M(n):Then /M(1) //1 2/ /M(2) /m(T)5M(2)52/C27ffiffiffi 3p 3ffiffiffi3pB0:7182336 /M(d)/d d/C2715M(d)5[G(1 2d)]22d/C282ffiffiffiffiffiffi 2dp G(d/C281 2)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (d/C271)pp Bffiffiffiffiffiffiffiffiffiffiffiffiffi d d/C271s where G(z) is the GAMMA FUNCTION (Nikolas and Yost 1988). An unrelated quantity characteristic of a given MAGIC SQUARE is also known as a MAGIC CONSTANT . References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/magic/magic.html. Cleary, J.; Morris, S. A.; and Yost, D. "Numerical Geome- try--Numbers for Shapes." Amer. Math. Monthly 95, 260/C1/ 75, 1986. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994. Gross, O. The Rendezvous Value of Metric Space. Princeton, NJ: Princeton University Press, pp. 49 /C1/3, 1964. Nikolas, P. and Yost, D. "The Average Distance Property for Subsets of Euclidean Space." Arch. Math. (Basel) 50, 380/C1/ 84, 1988. Stadje, W. "A Property of Compact Connected Spaces." Arch. Math. (Basel) 36, 275/C1/80, 1981. Magic Graph An edge-magic graph is a LABELED GRAPH with e EDGES labeled with distinct elements /1;2;...;e fg /so that the sum of the EDGE labels at each VERTEX is the same. A vertex-magic graph labeled VERTICES which give the same sum along every straight line segment. No magic pentagrams can be formed with the number 1, 2, ..., 10 (Trigg 1960; Langman 1962, pp. 80 /C1/3; Dongre 1971; Richards 1975; Buckley and Rubin 1977 /C1/8; Trigg 1998), but 168 almost magic penta- grams (in which the sums are the same for four of the five lines) can. The figure above show a magic pentagram with sums 24 built using the labels 1, 2, 3, 4, 5, 6, 8, 9, 10, and 12 (Madachy 1979). See also ANTIMAGIC GRAPH ,LABELED GRAPH ,M AGIC CIRCLES ,M AGIC CONSTANT ,M AGIC CUBE,M AGIC HEXAGON ,MAGIC SQUARE References Buckley, M. R. W. and Rubin, F. Solution to Problem 385. "Do Pentacles Exists?" J. Recr. Math. 10, 288 /C1/89, 1977 /C1/8. Doob, M. "Characterization of Regular Magic Graphs." J. Comb. Th. B 25,94/C1/04, 1978. Dongre, N. M. "More About Magic Star Polygons." Amer. Math. Monthly 78, 1025, 1971. Gallian, J. A. "Graph Labeling." Elec. J. Combin. DS6, 1 /C1/2, Apr. 15, 1999. http://www.combinatorics.org/Surveys/. Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction. San Diego, CA: Academic Press, 1990. Heinz, H. "Magic Stars." http://www.geocities.com/CapeCa- naveral/Launchpad/4057/magicstar.htm. Jezny ´, S. and Trenkler, M. "Characterization of Magic Graphs." Czech. Math. J. 33, 435 /C1/38, 1983. Jeurissen, R. H. "Magic Graphs, a Characterization." Europ. J. Combin. 9, 363 /C1/68, 1988. Langman, H. Play Mathematics. New York: Hafner, 1962. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 98 /C1/9, 1979. Richards, I. "Impossibility." Math. Mag. 48, 249 /C1/62, Nov. 1975. Rivera, C. "Problems & Puzzles: Puzzle The Prime-Magical Pentagram.-013." http://www.primepuzzles.net/puzzles/ puzz_013.htm. Trigg, C. W. "Solution of Problem 113." Pi Mu Epsilon J. 3, 119 /C1/20, Fall 1960. Trigg, C. W. "Ten Elements on a Pentagram." Eureka (Canada) 3,5/C1/, Jan. 1977. Trigg, C. W. "Almost Magic Pentagrams." J. Recr. Math. 29, 8 /C1/1, 1998. Wynne, B. E. "Perfect Magic Icosapentacles." J. Recr. Math. 9, 241 /C1/48, 1976 /C1/7.Magic Hexagon An arrangement of close-packed HEXAGONS contain- ing the numbers 1, 2, ..., Hn /C303n(n /C281) /C271; where Hn is the nth HEX NUMBER , such that the numbers along each straight line add up to the same sum. In the above magic hexagon, each line (those of lengths 3, 4, and 5) adds up to 38. This is the only magic hexagon of the counting numbers for any size hexagon, as proved by Trigg (Gardner 1984, p. 24). It was dis- covered by C. W. Adams, who worked on the problem from 1910 to 1957. Trigg showed that the magic constant for an order n hexagon would be 9n4/C282n3/C272n2/C28n ðÞ /C272 2(2n/C281); which requires 5 =(2n/C281) to be an integer for a solution to exist. But this is an integer for only n/C301 (the trivial case of a single hexagon) and Adam’s n/C303 (Gardner 1984, p. 24). See also HEX NUMBER ,H EXAGON ,M AGIC GRAPH , MAGIC SQUARE ,TALISMAN HEXAGON References Abraham, K. Philadelphia Evening Bulletin. July 19, 1963, p. 18 and July 30, 1963. Beeler, M. et al. Item 49 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 18, Feb. 1972. Gardner, M. "Permutations and Paradoxes in Combinatorial Mathematics." Sci. Amer. 209, 112/C1/19, Aug. 1963. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 22 /C1/4, 1984. Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 69 /C1/6, 1973. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 100 /C1/01, 1979. Trigg, C. W. "A Unique Magic Hexagon." Recr. Math. Mag. , Jan. 1964. Vickers, T. Math. Gaz. , p. 291, 1958. Magic Integer References Sloane, N. J. A. Sequences A004210/M2728 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Magic Labeling It is conjectured that every TREE with e edges whose nodes are all trivalent or monovalent can be given a "magic" labeling such that the INTEGERS 1, 2, ..., e can be assigned to the edges so that the SUM of the three meeting at a node is constant. See also MAGIC CONSTANT ,M AGIC CUBE,M AGIC GRAPH ,MAGIC HEXAGON ,MAGIC SQUARE References Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/09, 1989. Magic Number DIGITAL ROOT,MAGIC CONSTANT Magic Pentagram MAGIC GRAPH Magic Series A set n distinct numbers taken from the interval 1; n2½/C138 form a magic series if their sum is the nth MAGIC CONSTANT Mn /C301 2 nn2 /C271fflC{fflCz (Kraitchik 1942, p. 143). The numbers of magic series of orders n /C301, 2, ..., are 1, 2, 8, 86, 1394, ... (Sloane’s A052456). The following table gives the first few magic series of small order. n magic series 1 / f1g/ 2 / f1; 4g;f2; 3g/ 3 / f1; 5; 9g;f1; 6; 8 g;f2 ; 4 ; 9 g;f2 ; 5 ; 8g; f2; 6; 7g;f3; 4; 8g;f3 ; 5 ; 7 g;f4 ; 5 ; 6g/ If the sum of the kth powers of these number is the MAGIC CONSTANT of degree k for all k /C23 [1; p]; then they are said to form a pth order MULTIMAGIC SERIES . Here, the magic constant M(j) nof degree k is defined as 1=n times the sum of the first n2 kth powers,M(k) n/C301 nXn2 i /C301ik /C30H(/C28p) n2 n; where H(k) nis a HARMONIC NUMBER of order k. See also MAGIC CONSTANT ,M AGIC SQUARE ,M ULTI- MAGIC SERIES References Kraitchik, M. "Magic Series." §7.13.3 in Mathematical Recreations. New York: W. W. Norton, pp. 143 and 183 /C1/ 86, 1942. Sloane, N. J. A. Sequences A052456 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Magic Square A (normal) magic square consists of the distinct POSITIVE INTEGERS 1, 2, ..., n2such that the sum of thennumbers in any horizontal, vertical, or main diagonal line is always the same MAGIC CONSTANT M2(n)/C301 nXn2 k/C301k/C301 2nn2/C271fflC{fflCz : The unique normal square of order three was known to the ancient Chinese, who called it the L OSHU.A version of the order 4 magic square with the numbers15 and 14 in adjacent middle columns in the bottomrow is called D U¨RER’S MAGIC SQUARE . Magic squares of order 3 through 8 are shown above.The MAGIC CONSTANT for an nth order magic square starting with an INTEGER Aand with entries in an increasing ARITHMETIC SERIES with difference D between terms is M2(n;A;D)/C301 2n2a/C27Dn2/C281fflC{fflCzfflC}fflC( (Hunter and Madachy 1975). If every number in a magic square is subtracted from n2/C271;another magic square is obtained called the complementarymagic square. Squares which are magic under multi- plication instead of addition can be constructed and are known as MULTIPLICATION MAGIC SQUARES .I n addition, squares which are magic under both addi- tion and multiplication can be constructed and are known as ADDITION-MULTIPLICATION MAGIC SQUARES (Hunter and Madachy 1975). A square that fails to be magic only because one or both of the main diagonal sums do not equal the MAGIC CONSTANT is called a SEMIMAGIC SQUARE .I fall diagonals (including those obtained by wrappingaround) of a magic square sum to the MAGIC CON- STANT , the square is said to be a PANMAGIC SQUARE (also called a DIABOLIC SQUARE or PANDIAGONAL SQUARE ). If replacing each number niby its square n2 iproduces another magic square, the square is said to be a BIMAGIC SQUARE (orDOUBLY MAGIC SQUARE ). If a square is magic for ni;n2i;and n3i;it is called a TREBLY MAGIC SQUARE . If all pairs of numbers symmetrically opposite the center sum to n2/C271;the square is said to be an ASSOCIATIVE MAGIC SQUARE . Kraitchik (1942) gives general techniques of con- structing EVEN and ODD squares of order n. For n ODD, a very straightforward technique known as the Siamese method can be used, as illustrated above(Kraitchik 1942, pp. 148 /C1 /49). It begins by placing a 1 in any location (in the center square of the top row inthe above example), then incrementally placing sub-sequent numbers in the square one unit above and tothe right. The counting is wrapped around, so thatfalling off the top returns on the bottom and falling offthe right returns on the left. When a square is encountered which is already filled, the next number is instead placed below the previous one and the method continues as before. The method, also calledde la Loubere’s method, is purported to have beenfirst reported in the West when de la Louberereturned to France after serving as ambassador toSiam. A generalization of this method uses an "ordinary vector" ( x, y) which gives the offset for each non- colliding move and a "break vector" ( u, v) which gives the offset to introduce upon a collision. The standardSiamese method therefore has ordinary vector (1, /C281) and break vector (0, 1). In order for this to produce a magic square, each break move must end up on anunfilled cell. Special classes of magic squares can be constructed by considering the absolute sums u/C27v jj ; (u/C28x)/C27(v/C28y) jj ;u/C28v jj ;and ( u/C28x)/C28(v/C28y) jj /C30 u/C27y/C28x/C28v jj :Call the set of these numbers the sumdiffs (sums and differences). If all sumdiffs are RELATIVELY PRIME tonand the square is a magic square, then the square is also a PANMAGIC SQUARE . This theory originated with de la Hire. The following table gives the sumdiffs for particular choices ofordinary and break vectors. OrdinaryVectorBreakVectorSumdiffs Magic SquaresPanmagicSquares (1, -1) (0, 1) (1, 3) /2k/C271/none (1, -1) (0, 2) (0, 2) /6k91/none (2, 1) (1, -2) (1, 2, 3, 4) /6k91/none (2, 1) (1, -1) (0, 1, 2, 3) /6k91//6k91/ (2, 1) (1, 0) (0, 1, 2) /2k/C271/none (2, 1) (1, 2) (0, 1, 2, 3) /6k91/none A second method for generating magic squares of ODD order has been discussed by J. H. Conway under the name of the "lozenge" method. As illustrated above, in this method, the ODD numbers are built up along diagonal lines in the shape of a DIAMOND in the central part of the square. The EVEN numbers which were missed are then added sequentially along thecontinuation of the diagonal obtained by wrapping around the square until the wrapped diagonal reaches its initial point. In the above square, the firstdiagonal therefore fills in 1, 3, 5, 2, 4, the second diagonal fills in 7, 9, 6, 8, 10, and so on. An elegant method for constructing magic squares of DOUBLY EVEN order n/C304mis to draw xs through each 4/C294 subsquare and fill all squares in sequence. Then replace each entry aijon a crossed-off diagonal by n2/C271 ðÞ /C28aijor, equivalently, reverse the order of the crossed-out entries. Thus in the above example for n/C308, the crossed-out numbers are originally 1, 4, ..., 61, 64, so entry 1 is replaced with 64, 4 with 61, etc. A very elegant method for constructing magic squaresof SINGLY EVEN order n/C304m/C272 with m]1 (there is no magic square of order 2) is due to J. H. Conway,who calls it the "LUX" method. Create an arrayconsisting of m/C271 rows of Ls, 1 row of Us, and m/C28 1 rows of Xs, all of length n=2/C302m/C271:Interchange the middle U with the L above it. Now generate themagic square of order 2 m/C271 using the Siamese method centered on the array of letters (starting inthe center square of the top row), but fill each set offour squares surrounding a letter sequentially accord-ing to the order prescribed by the letter. That order isillustrated on the left side of the above figure, and thecompleted square is illustrated to the right. The"shapes" of the letters L, U, and X naturally suggestthe filling order, hence the name of the algorithm. It is an unsolved problem to determine the number of magic squares of an arbitrary order, but the numberof distinct magic squares (excluding those obtained byrotation and reflection) of order n/C301, 2, ... are 1, 0, 1, 880, 275305224, ... (Sloane’s A006052; Madachy 1979, p. 87). The 880 squares of order four were enumer- ated by Frenicle de Bessy in the seventeenth century, and are illustrated in Berlekamp et al. (1982, pp. 778 /C1 /83). The number of 6 /C296 squares is not known, but Pinn and Wieczerkowski (1998) estimatedit to be (1 :774590:0016)/C2910 19using Monte Carlo simulation and methods from statistical mechanics. The above magic squares consist only of PRIMES and were discovered by E. Dudeney (1970) and A. W. Johnson, Jr. (Gardner 1984, p. 86; Dewdney 1988). Madachy (1979, pp. 93 /C1/6) and Rivera discuss other magic squares composed of PRIMES . Benjamin Franklin constructed the above 8 /C298PAN- MAGIC SQUARE having MAGIC CONSTANT 260. Any half- row or half-column in this square totals 130, and thefour corners plus the middle total 260. In addition,bent diagonals (such as 52 /C1 /5/C1/4/C1/0/C1/7/C1/3/C1/6) also total 260 (Madachy 1979, p. 87). In addition to other special types of magic squares, a 3/C293 square whose entries are consecutive PRIMES , illustrated above, has been discovered by H. Nelson (Rivera). According to a 1913 proof of J. N. Murray (cited in Gardner 1984, pp. 86 /C1/7), the smallest magic square composed of consecutive primes starting with 3 and including the number 1 is of order 12. Variations on magic squares can also be constructed using letters (either in defining the square or as entries in it), such as the ALPHAMAGIC SQUARE and TEMPLAR MAGIC SQUARE . Various numerological properties have also been associated with magic squares. Pivari associates the squares illustrated above with Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon, respectively. Attractive patterns are obtained by connecting con- secutive numbers in each of the squares (with the exception of the Sun magic square). See also ADDITION- MULTIPLICATION MAGIC SQUARE ALPHAMAGIC SQUARE ,ANTIMAGIC SQUARE ,ASSOCIA- TIVE MAGIC SQUARE ,B IMAGIC SQUARE ,B ORDER SQUARE ,D U¨ RER’S MAGIC SQUARE ,E ULER SQUARE , FRANKLIN MAGIC SQUARE ,GNOMON MAGIC SQUARE ,HETEROSQUARE ,L ATIN SQUARE ,M AGIC CIRCLES , MAGIC CONSTANT ,M AGIC CUBE,M AGIC HEXAGON , MAGIC LABELING ,M AGIC SERIES ,M AGIC TESSERACT , MAGIC TOUR,M ULTIMAGIC SQUARE ,MULTIPLICATION MAGIC SQUARE ,P ANMAGIC SQUARE ,S EMIMAGIC SQUARE ,T ALISMAN SQUARE ,T EMPLAR MAGIC SQUARE ,TRIMAGIC SQUARE References Abe, G. "Unsolved Problems on Magic Squares." Disc. Math. 127,3/C1/3, 1994. Alejandre, S. "Suzanne Alejandre’s Magic Squares." http:// forum.swarthmore.edu/alejandre/magic.square.html. Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New York: Dover, 1960. Andrews, W. S. and Sayles, H. A. "Magic Squares Made with Prime Numbers to have the Lowest Possible Summa- tions." Monist 23, 623/C1/30, 1913. Ball, W. W. R. and Coxeter, H. S. M. "Magic Squares." Ch. 7 inMathematical Recreations and Essays, 13th ed. New York: Dover, 1987. Barnard, F. A. P. "Theory of Magic Squares and Cubes." Memoirs Natl. Acad. Sci. 4, 209/C1/70, 1888. Benson, W. H. and Jacoby, O. New Recreations with Magic Squares. New York: Dover, 1976. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982. Chabert, J.-L. (Ed.). "Magic Squares." Ch. 2 in A History of Algorithms: From the Pebble to the Microchip. New York: Springer-Verlag, pp. 49 /C1/1, 1999. Danielsson, H. "Magic Squares." http://www.magic-squar- es.de/magic.html. Dewdney, A. K. "Computer Recreations: How to Pan for Primes in Numerical Gravel." Sci. Amer. 259, pp. 120 /C1/23, July 1988. Dudeney, E. Amusements in Mathematics. New York: Dover, 1970. Fults, J. L. Magic Squares. Chicago, IL: Open Court, 1974. Gardner, M. "Magic Squares." Ch. 12 in The Second Scien- tific American Book of Mathematical Puzzles & Diversions:A New Selection. New York: Simon and Schuster, pp. 130 /C1 /40, 1961. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984. Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213 /C1/25, 1988. Grogono, A. W. "Magic Squares by Grog." http://www.grogo- no.com/magic/. Hawley, D. "Magic Squares." http://www.nrich.maths.or- g.uk/mathsf/journalf/aug98/art1/. Heinz, H. "Magic Squares." http://www.geocities.com/Cape- Canaveral/Launchpad/4057/magicsquare.htm. Hirayama, A. and Abe, G. Researches in Magic Squares. Osaka, Japan: Osaka Kyoikutosho, 1983. Horner, J. "On the Algebra of Magic Squares, I., II., and III." Quart. J. Pure Appl. Math. 11,5 7/C1/5, 123 /C1/31, and 213 /C1/ 24, 1871. Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 inMathematical Diversions. New York: Dover, pp. 23 /C1/4, 1975. Kraitchik, M. "Magic Squares." Ch. 7 in Mathematical Recreations. New York: Norton, pp. 142 /C1/92, 1942. Lei, A. "Magic Square, Cube, Hypercube." http:// www.cs.ust.hk/~philipl/magic/. Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in Madachy’s Mathematical Recreations. New York: Dover, pp. 85 /C1/13, 1979. Moran, J. The Wonders of Magic Squares. New York: Vintage, 1982. Pappas, T. "Magic Squares," "The "Special" Magic Square," "The Pyramid Method for Making Magic Squares," "An- cient Tibetan Magic Square," "Magic "Line"," and "A Chinese Magic Square." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 82 /C1/7, 112, 133, 169, and 179, 1989. Peterson, I. "Ivar Peterson’s MathLand: More than Magic Squares." http://www.maa.org/mathland/math- land_10_14.html. Pinn, K. and Wieczerkowski, C. "Number of Magic Squares from Parallel Tempering Monte Carlo." Int. J. Mod. Phys. C 9, 541 /C1/47, 1998. http://xxx.lanl.gov/abs/cond-mat/ 9804109/ Pivari, F. "Nice Examples." http://www.geocities.com/Cape- Canaveral/Lab/3469/examples.html. Pivari, F. "Simple Magic Square Checker and GIF Maker." http://www.geocities.com/CapeCanaveral/Lab/3469/squar- emaker.html. Rivera, C. "Problems & Puzzles: Puzzle Magic Squares with Consecutive Primes.-003." http://www.primepuzzles.net/ puzzles/puzz_003.htm. Rivera, C. "Problems & Puzzles: Puzzle Prime-Magical Squares.-004." http://www.primepuzzles.net/puzzles/ puzz_004.htm. Sloane, N. J. A. Sequences A006052/M5482 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Suzuki, M. "Magic Squares." http://www.pse.che.toho- ku.ac.jp/~msuzuki/MagicSquare.html. Weisstein, E. W. "Magic Squares." MATHEMATICA NOTEBOOK MAGICSQUARES.M . Weisstein, E. W. "Books about Magic Squares." http:// www.treasure-troves.com/books/MagicSquares.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 75, 1986. Magic Star MAGIC GRAPH Magic Tesseract A magic tesseract is a 4-D generalization of the 2-D MAGIC SQUARE and the 3-D MAGIC CUBE . A magic tesseract has MAGIC CONSTANT M4(n) /C301 2 nn4 /C271fflC{fflCz ; so for n /C301, 2, ..., the magic tesseract constants are 1, 17, 123, 514, 1565, 3891, ... (Sloane’s A021003). Berlekamp et al. (1982, p. 783) give a magic TESSER- ACT. J. Hendricks has constructed magic tesseracts of orders three, four, five (Hendricks 1999a, pp. 128 /C1/ 29), and six (Heinz). M. Houlton has used Hendricks’ techniques to construct magic tesseracts of orders 5, 7, and 9. There are 58 distinct magic tesseracts of order three, modulo rotations and reflections (Heinz, Hendricks 1999), one of which is illustrated above. Each of the 27 rows (e.g., 1 /C1/2 /C1/0), columns (e.g., 1 /C1/0 /C1/2), pillars (e.g., 1 /C1/4 /C1/8), and files (e.g., 1 /C1/8 /C1/4) sum to the magic constant 123. Hendricks (1968) has constructed a pan-4-agonal magic tesseract of order 4. No pan-4-agonal magic tesseract of order five is known, and Andrews (1960) and Schroeppel (1972) state that no such tesseract can exist. The smallest perfect magic tesseract is of order 16, having MAGIC CONSTANT 524,296, and has been constructed by Hendricks (Peterson 1999). n-dimensional magic hypercubes of order 3 are known for n/C305, 6, 7, and 8 (Hendricks). Hendricks has also constructed a perfect 16th order magic tesseract (where perfect means that all hyperplanesare perfect). See also M AGIC CUBE,MAGIC SQUARE References Adler, A. "Magic N-Cubes Form a Free Monoid." Electronic J. Combinatorics 4, No. 1, R15, 1 /C1/, 1997. http://www.com- binatorics.org/Volume_4/v4i1toc.html#R15. Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New York: Dover, 1960. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982. Heinz, H. "John Hendricks: Inlaid Magic Tesseract." http:// www.geocities.com/~harveyh/Hendricks.htm#Inlaid Ma- gic Tesseract. Hendricks, J. R. "The Five and Six Dimensional Magic Hypercubes of Order 3." Canad. Math. Bull. 5, 171 /C1/89, 1952. Hendricks, J. R. "A Pan-4-agonal Magic Tesseract." Amer. Math. Monthly 75, 384, 1968. Hendricks, J. R. "Magic Tesseracts and N-Dimensional Magic Hypercubes." J. Recr. Math. 6, 193 /C1/01, 1973. Hendricks, J. R. Erratum to ‘Magic Tesseracts and N- Dimensional Magic Hypercubes." J. Recr. Math. 7, 80, 1974. Hendricks, J. R. "Ten Magic Tesseracts of Order Three." J. Recr. Math. 18, 125 /C1/34, 1985 /C1/986. Hendricks, J. R. Magic Squares to Tesseracts by Computer. Published by the author, 1999a. Hendricks, J. R. All Third Order Magic Tesseracts. Pub- lished by the author, 1999b. Hendricks, J. R. Perfect n-Dimensional Hypercubes of Order 2n :/ Published by the author, 1999c. Peterson, I. "Ivar Peterson’s MathTrek: Magic Tesseracts." http://www.maa.org/mathland/mathtrek_10_18_99.html . Schroeppel, R. Item 51 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 18, Feb. 1972. Sloane, N. J. A. Sequences A021003 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Trenkler, M. "Magic p-Dimensional Cubes of Order n f2 (mod 4)." Acta Arith. 92, 189 /C1/04, 2000. Trenkler, M. "A Construction of Magic Cubes." Math. Gaz. 84,36/C1/1, 2000. Trenkler, M. "Magic p-Dimensional Cubes." Submitted to Acta Arith. , 2000. Magic Tour Let a chess piece make a TOUR on an n /C29n CHESS- BOARD whose squares are numbered from 1 to n2 along the path of the chess piece. Then the TOUR is called a magic tour if the resulting arrangement of numbers is a MAGIC SQUARE . If the first and last squares traversed are connected by a move, the tour is said to be closed (or "re-entrant"); otherwise it is open. The MAGIC CONSTANT for the 8 /C298 CHESSBOARD is 260. Magic KNIGHT’S TOURS are not possible on n /C29n boards for n ODD, and are believed to be impossible for n /C308. The "most magic" knight tour known on the 8 /C298 board is the SEMIMAGIC SQUARE illustrated in the above left figure (Ball and Coxeter 1987, p. 185) having main diagonal sums of 348 and 168. Combin- ing two half-knights’ tours one above the other as in the above right figure does, however, give a MAGICSQUARE (Ball and Coxeter 1987, p. 185). The above illustration shows a 16 /C2916 closed magic KNIGHT’S TOUR (Madachy 1979). A magic tour for king moves is illustrated above (Coxeter 1987, p. 186). See also CHESSBOARD ,K NIGHT’S TOUR,M AGIC SQUARE ,SEMIMAGIC SQUARE ,TOUR References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 185 /C1/87, 1987. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 87 /C1/9, 1979. Magnetic Pole Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27g(y)y?2/C27f(x)y?/C300: References Goldstein, M. E. and Braun, W. H. Advanced Methods for the Solution of Differential Equations. NASA SP-316. Washington, DC: U.S. Government Printing Office, p. 98, 1973. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 124, 1997. The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C28m(m /C27 1) /C271 4 /C28 m /C2712fflCz6fflCz7 cos x sin2 x /C27 l /C2712fflCz6fflCz72 435y /C300 : References Infeld, L. and Hull, T. E. "The Factorization Method." Rev. Mod. Phys. 23,21/C1/8, 1951. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 125, 1997. Magog Triangle A NUMBER TRIANGLE of order n with entries 1 to n such that entries are nondecreasing across rows and down columns and all entries in column j are less than or equal to j. An example is 1 111111113 11245 : Magog triangles are in 1-to-1 correspondence with CYCLICALLY SYMMETRIC PLANE PARTITIONS . See also CYCLICALLY SYMMETRIC PLANE PARTITION , MONOTONE TRIANGLE References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/46. Mahler-Lech Theorem Let K be a FIELD of CHARACTERISTIC 0 (e.g., the rationals Q) and let unfg be a SEQUENCE of elements of K which satisfies a difference equation OF THE FORM 0 /C30c0un /C27c1un/C271 /C27.../C27ckun /C27k ; where the COEFFICIENTS ciare fixed elements of K. Then, for any c /C23 K ; we have either un /C30c for only finitely many values of n, or un /C30c for the values of n in some ARITHMETIC PROGRESSION . The proof involves embedding certain FIELDS inside the P-ADIC NUMBERS Qp for some PRIME p, and using properties of zeros of POWER SERIES over Qp (STRASS- MAN’S THEOREM ).See also ARITHMETIC PROGRESSION , P-ADIC NUMBER , STRASSMAN’S THEOREM Mahler Measure This entry contributed by KEVIN O’BRYANT For a polynomial Px1 ; x2 ; ...; xk ðÞ ; the Mahler mea- sure of P is defined by Mk(P) /C13expg1 0...g1 0ln Pe2pit1 ; ...; e2 pitkfflC{fflCzfflCz}fflCz}fflCz}fflCz}dt 1 /C1/C1/C1dtk"# : (1) Using JENSEN’S FORMULA , it can be shown that for P(x) /C30aQn i/C301x /C28 ai ðÞ ; M1(P) /C30 ajjYn i/C301max 1; aijj fg (2) (Borwein and Erde´lyi 1995, p. 271). Specific cases are given by M1(ax /C27b) /C30max ajj; bjj fg (3) M2(1 /C27x /C27y) /C30M1max 1; 1 /C27x jj fg ðÞ (4) M2(1 /C27x /C27y /C28xy) /C30M1max 1 /C28x jj ; 1 /C27x jj fg ðÞ (5) (Borwein and Erde´lyi 1995, p. 272). A product of CYCLOTOMIC POLYNOMIALS has Mahler measure 1. LEHMER’S MAHLER MEASURE PROBLEM conjectures that a particular univariate polynomial has the smallest possible Mahler measure other than 1. The Mahler measure for a univariate polynomial can be computed in Mathematica as follows. MahlerMeasure[p_, x_] : /C30 Module[ {roots /C30x /. {ToRules[Roots[p /C30/C300, x]]}}, Abs[Function[x, p][0]] Times @@ (Max[Abs[#], 1] & /@ roots) ] See also JENSEN’S FORMULA ,L EHMER’S MAHLER MEASURE PROBLEM References Borwein, P. and Erde ´lyi, T. "Mahler’s Measure." §5.3.E.4 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 271 /C1/72, 1995. Graham, E. Heights of Polynomials and Entropy in Alge- braic Dynamics. London: Springer-Verlag, 1999. Mahler Polynomial Polynomials sn(x) which form the SHEFFER SEQUENCE for f /C281(t) /C301 /C27t /C28et ; where f /C281(t) is the INVERSE FUNCTION of f(t) ; and have GENERATING FUNCTION X/C12 k /C300sk(x) k!tk /C30ex 1 /C27t/C28etðÞ: The first few are s0(x) /C301 s1(x) /C300 s2(x) /C30/C28x s3(x) /C30/C28x s4(x) /C303x2 /C28x s5(x) /C3010x2 /C28x: References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 254, 1981. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Mahler’s Measure For a POLYNOMIAL , c /C23 K It is related to JENSEN’S INEQUALITY . See also JENSEN’S INEQUALITY Mainardi-Codazzi Equations PETERSON- MAINARDI- CODAZZI EQUATIONS Main Diagonal DIAGONAL Majorant A function used to study ORDINARY DIFFERENTIAL EQUATIONS . Major Axis SEMIMAJOR AXISMajorization This entry contributed by SERGE BELONGIE Let x /C30 x1 ; x2 ; ...; xn ðÞ and y /C30 y1 ; y2 ; ...; yn ðÞ be nonincreasing sequences of real numbers. Then x majorizes y if, for each k /C301, 2, ..., n, Xk i /C301xi ]Xk i/C301yi ; with equality if k /C30n. Note that some caution is needed when consulting the literature, since the direction of the inequality is not consistent from reference to reference. An order-free characterization along the lines of HORN’S THEOREM is also readily available. If P/ is a doubly stochastic matrix, then y /C30Px iff y is majorized by x. Intuitively, if x majorizes y, then y is more "mixed" than x.H ORN’S THEOREM relates the eigenvalues of a HERMITIAN MATRIX A to its diagonal entries using majorization. Given two vectors l; v /C23 Rn ; then l majorizes v iff there exists a HERMITIAN MATRIX A with eigenvalues li and diagonal entries vi :/ See also BIRKHOFF’S THEOREM ,H ORN’S THEOREM , SCHUR CONVEXITY References Bhatia, R. Matrix Analysis. New York: Springer-Verlag, 1997. Horn, R. A. and Johnson, C. R. Matrix Analysis, Repr. with Corrections. Cambridge, England: Cambridge University Press, 1987. Marshall, A. W. and Olkin, I. Inequalities: The Theory of Majorizations and Its Applications. New York: Academic Press, 1979. Nielsen, M. A. "Conditions for a Class of Entanglement Transformations." Phys. Rev. Lett. 83, 436 /C1/39, 1999. Major Triangle Center A TRIANGLE CENTER a : b : g is called a major center if the TRIANGLE CENTER FUNCTION a /C30 f(a ; b; c; A; B; C) is a function of ANGLE A alone, and therefore b and g of B and C alone, respectively. See also REGULAR TRIANGLE CENTER ,T RIANGLE CENTER References Kimberling, C. "Major Centers of Triangles." Amer. Math. Monthly 104, 431/C1/38, 1997. Makeham Curve The function defined by y/C13ksxbqx which is used in actuarial science for specifying a simplified mortality law (Kenney and Keeping 1962, pp. 241 /C1/42). Using s(x) as the probability that a newborn will achieve age x, the Makeham law (1860) uses s(x) /C30exp /C28Ax /C28Bcx /C281 ðÞ ðÞ for B /C210, A ]/C28B ; c /C211, x ]0 :/ See also GOMPERTZ CURVE ,LAW OF GROWTH ,LIFE EXPECTANCY ,LOGISTIC GROWTH CURVE ,POPULATION GROWTH References Bowers, N. L. Jr.; Gerber, H. U.; Hickman, J. C.; Jones, D. A.; and Nesbitt, C. J. Actuarial Mathematics. Itasca, IL: Society of Actuaries, p. 71, 1997. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Makeham, W. M. "On the Law of Mortality and the Construction of Annuity Tables." J. Inst. Actuaries and Assur. Mag. 8, 301 /C1/10, 1860. Makeham, W. M. "On an Application of the Theory of the Composition of Decremental Forces." J. Inst. Actuaries and Assur. Mag. 18, 317 /C1/22, 1874. Malfatti Circles Three circles packed inside a RIGHT TRIANGLE which are each tangent to the other two and to two sides of the TRIANGLE . Although these circles were for many years thought to provide the solutions to MALFATTI’S RIGHT TRIANGLE PROBLEM , they were subsequently shown never to provide the solution. See also APOLLONIAN GASKET ,M ALFATTI’S RIGHT TRIANGLE PROBLEM ,SODDY CIRCLES Malfatti Points AJIMA- MALFATTI POINTS Malfatti’s Right Triangle Problem In 1803, Malfatti asked for the three columns (of possibly different sizes) which, when carved out of a right triangular prism, would have the largest possi- ble total CROSS SECTION . This is equivalent to finding the maximum total AREA of three CIRCLES which can be packed inside a RIGHT TRIANGLE of any shape without overlapping. Malfatti gave the solution as three CIRCLES (the MALFATTI CIRCLES ) tangent to each other and to two sides of the TRIANGLE .In 1930, it was shown that the MALFATTI CIRCLES were not always the best solution. Then Goldberg (1967) showed that, even worse, they are never the best solution. Wells (1991) illustrates specific cases where alternative solutions are clearly optimal. See also CIRCLE PACKING ,M ALFATTI’S TANGENT TRIANGLE PROBLEM References Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, p. 245, 1965. Goldberg, M. "On the Original Malfatti Problem." Math. Mag. 40, 241 /C1/47, 1967. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 145 /C1/47, 1990. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/1, May 1998.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Malfatti’s Tangent Triangle Problem Draw within a given TRIANGLE three CIRCLES , each of which is TANGENT to the other two and to two sides of the TRIANGLE . Denote the three CIRCLES so con- structed GA ;GB ; and GC : Then GAis tangent to AB and AC, GBis tangent to BC and BA, and GCis tangent to ACandBC. See also AJIMA- MALFATTI POINTS ,M ALFATTI’S RIGHT TRIANGLE PROBLEM References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., pp. 154 /C1 /55, 1888. Do¨rrie, H. "Malfatti’s Problem." §30 in 100 Great Problems of Elementary Mathematics: Their History and Solutions.New York: Dover, pp. 147 /C1 /51, 1965. Forder, H. G. Higher Course Geometry. Cambridge, Eng- land: Cambridge University Press, pp. 244 /C1/45, 1931. Fukagawa, H. and Pedoe, D. "The Malfatti Problem." Japanese Temple Geometry Problems (San Gaku). Winni- peg: The Charles Babbage Research Centre, pp. 106 /C1/20, 1989. F. Gabriel-Marie. Exercices de ge ´ome´trie. Tours, France: Maison Mame, pp. 710 /C1/12, 1912. Gardner, M. Fractal Music, Hypercards, and More Mathe- matical Recreations from Scientific American Magazine.New York: W. H. Freeman, pp. 163 /C1 /65, 1992. Goldberg, M. "On the Original Malfatti Problem." Math. Mag. 40, 241/C1/47, 1967. Hart. Quart. J. 1, p. 219. Lob, H. and Richmond, H. W. "On the Solution of Malfatti’s Problem for a Triangle." Proc. London Math. Soc. 2, 287/C1/ 04, 1930. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 145 /C1/47, 1990. Rouche ´, E. and de Comberousse, C. Traite ´de ge ´ome´trie plane. Paris: Gauthier-Villars, pp. 311 /C1/14, 1900. Woods, F. S. Higher Geometry. New York: Dover, pp. 206 /C1/ 09, 1961. Malliavin Calculus An infinite-dimensional DIFFERENTIAL CALCULUS on the W IENER SPACE . Also called STOCHASTIC CALCULUS OF VARIATIONS . Mallows’ Sequence An INTEGER SEQUENCE given by the RECURRENCE RELATION a(n) /C30a(a(n /C282)) /C27a(n /C28a(n /C282)) with a(1) /C30a(2) /C301: The first few values are 1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, ... (Sloane’s A005229). See also HOFSTADTER- CONWAY $10,000 SEQUENCE , HOFSTADTER’S Q-SEQUENCE References Mallows, C. L. "Conway’s Challenge Sequence." Amer. Math. Monthly 98,5/C1/0, 1991. Sloane, N. J. A. Sequences A005229/M0441 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Malmste ´n’s Differential Equation The ORDINARY DIFFERENTIAL EQUATION yƒ/C27r zy ?/C30 Azm /C27s z2 ! y: References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 99 /C1/00, 1966. Malmste ´n’s Formula The integral representation of ln[G(z)] by ln[(z)] /C30gz 1c0(z ?) dz? /C30g/C12 0(z /C281) /C281 /C28 e/C28(z/C281)t 1 /C28 e /C28t"# e/C28t tdt; where G(z) is the GAMMA FUNCTION and c0(z) is the DIGAMMA FUNCTION . See also BINET’S LOG GAMMA FORMULAS ,G AMMA FUNCTION References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 20 /C1/1, 1981. Maltese Cross An irregular DODECAHEDRON CROSS shaped like a /C27 sign but whose points flange out at the end: w: The conventional proportions as computed on a 5 /C295 grid as illustrated above. See also CROSS ,DISSECTION ,DODECAHEDRON ,M AL- TESE CROSS CURVEReferences Frederickson, G. "Maltese Crosses." Ch. 14 in Dissections: Plane and Fancy. New York: Cambridge University Press, pp. 157 /C1/62, 1997. Maltese Cross Curve The plane curve with Cartesian equation xy(x2 /C28y2) /C30x2 /C27y2 and polar equation r2 /C301 cos u sin u(cos2 u /C28 sin2 u) (Cundy and Rollett 1989, p. 71), so named for its resemblance to the MALTESE CROSS . See also MALTESE CROSS References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989. Malthusian Parameter The parameter a in the exponential POPULATION GROWTH equation N1(t)/C30N0eat: See also LIFE EXPECTANCY ,POPULATION GROWTH Maltitude A perpendicular drawn to a side of a QUADRILATERAL from the MIDPOINT Miof the opposite side. If the QUADRILATERAL is CYCLIC , then the maltitudes are concurrent in a point T, known as the ANTICENTER , which is on the line connecting the CIRCUMCENTER O an the centroid G of the vertices. Furthermore, OM /C302OGM :/ See also ALTITUDE ,A NTICENTER ,B RAHMAGUPTA’S THEOREM ,CYCLIC QUADRILATERAL ,MIDPOINT ,QUAD- RILATERAL References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 36 /C1/7, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 146, 1991. Mandelbar Set A FRACTAL set analogous to the MANDELBROT SET or its generalization to a higher power with the variable zreplaced by its COMPLEX CONJUGATE ¯z:/ See also MANDELBROT SET Mandelbrot Set The set obtained by the QUADRATIC RECURRENCE zn/C271/C30z2 n/C27C; (1) where points Cfor which the orbit z0/C300 does not tend to infinity are in the SET. It marks the set of points in the COMPLEX PLANE such that the corre-sponding J ULIA SET isCONNECTED and not COMPUTA- BLE. The Mandelbrot set was originally called a MU MOLECULE by Mandelbrot. J. Hubbard and A. Douady proved that the Mandel- brot set is CONNECTED . Shishikura (1994) proved that the boundary of the Mandelbrot set is a FRACTAL with HAUSDORFF DIMENSION 2. However, it is not yet known if the Mandelbrot set is pathwise-connected. If it is pathwise-connected, then Hubbard and Doua- dy’s proof implies that the Mandelbrot set is theimage of a CIRCLE and can be constructed from a DISK by collapsing certain arcs in the interior (Douady1986). The AREA of the set is known to lie between 1.5031 and 1.5702; it is estimated as 1.50659....Decomposing the COMPLEX coordinate z/C30x/C27iyand z0/C30a/C27ibgives x?/C30x2/C28y2/C27a (2) y?/C302xy/C27b: (3) In practice, the limit is approximated by lim n0/C12znjj:lim n0nmaxznjjBrmax: (4) Beautiful computer-generated plots can be created by coloring nonmember points depending on how quickly they diverge to rmax:A common choice is to define an INTEGER called the COUNT to be the largest nsuch that znjjBr;where ris usually taken as r/C302, and to color points of different COUNT different colors. The boundary between successive COUNTS defines a series of " LEMNISCATES ," called EQUIPOTENTIAL CURVES by Peitgen and Saupe (1988), Ln(C) jj /C30rwhich have distinctive shapes. The first few LEMNISCATES are L1(C)/C30C (5) L2(C)/C30C(C/C271) (6) L3(C)/C30C/C27C/C27C2fflC{fflCz2(7) L4(C)/C30C/C27C/C27C2fflC{fflCz2hi2 : (8) When written in C ARTESIAN COORDINATES , the first three of these are r2/C30x2/C27y2(9) r2/C30x2/C27y2fflC{fflCz x/C271 ðÞ2/C27y2hi (10) r2/C30x2/C27y2fflC{fflCz 1/C272x/C275x2/C276x3/C276x4/C274x5/C27x6fflC{ /C283y2/C282xy2/C278x2y2/C278x3y2/C273x4y2/C272y4/C274xy4 /C273x2y4/C27y6Þ (11) which are a CIRCLE ,a n OVAL , and a PEAR CURVE .I n fact, the second LEMNISCATE L2can be written in terms of a new coordinate system with x?/C13x/C281=2a s x?/C281 2fflCz6fflCz72 /C27y2fflC}{fflC}z x?/C2712fflCz6fflCz72 /C27y2fflCzrfflCzD /C30r2 ; (12) which is just a CASSINI OVAL with a /C301=2 and b2 /C30r: The LEMNISCATES grow increasingly convoluted with higher COUNT and approach the Mandelbrot set as the COUNT tends to infinity. The kidney bean-shaped portion of the Mandelbrot set is bordered by a CARDIOID with equations 4x /C302 cos t /C28cos(2 t) (13) 4y /C302 sin t /C28sin(2 t): (14) The adjoining portion is a CIRCLE with center at (/C281; 0) and RADIUS 1=4 : One region of the Mandelbrot set containing spiral shapes is known as SEA HORSE VALLEY because the shape resembles the tail of a sea horse. Generalizations of the Mandelbrot set can be con- structed by replacing z2 nwith zkn or (¯zn)k ; where k is a POSITIVE INTEGER and ¯z denotes the COMPLEX CON- JUGATE of z. The following figures show the FRACTALS obtained for k/C302, 3, and 4 (Dickau). The plots on the right have zreplaced with ¯zand are sometimes called "MANDELBAR SETS ." See also CACTUS FRACTAL ,F RACTAL ,JULIA SET, LEMNISCATE (MANDELBROT SET), MANDELBAR SET, QUADRATIC MAP,R ANDELBROT SET,S EA HORSE VALLEY References Alfeld, P. "The Mandelbrot Set." http://www.math.utah.edu/ ~alfeld/math/mandelbrot/mandelbrot.html. Branner, B. "The Mandelbrot Set." In Chaos and Fractals: The Mathematics Behind the Computer Graphics, Proc. Sympos. Appl. Math., Vol. 39 (Ed. R. L. Devaney and L. Keen). Providence, RI: Amer. Math. Soc., 75 /C1/05, 1989. Devaney, R. "The Mandelbrot Set and the Farey Tree, and the Fibonacci Sequence." Amer. Math. Monthly 106, 289/C1/ 02, 1999. Dickau, R. M. "Mandelbrot (and Similar) Sets." http://for- um.swarthmore.edu/advanced/robertd/mandelbrot.html. Douady, A. "Julia Sets and the Mandelbrot Set." In The Beauty of Fractals: Images of Complex Dynamical Systems(Ed. H.-O. Peitgen and D. H. Richter). Berlin: Springer-Verlag, p. 161, 1986. Eppstein, D. "Area of the Mandelbrot Set." http://www.ics.u- ci.edu/~eppstein/junkyard/mand-area.html. Fisher, Y. and Hill, J. "Bounding the Area of the Mandelbrot Set." Submitted. Hill, J. R. "Fractals and the Grand Internet Parallel Proces- sing Project." Ch. 15 in Fractal Horizons: The Future Use of Fractals. New York: St. Martin’s Press, pp. 299 /C1 /23, 1996. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 148 /C1/ 51 and 179 /C1/80, 1991. Lei, T. (Ed.) The Mandelbrot Set, Theme and Variations. Cambridge, England: Cambridge University Press, 2000. Munafo, R. "Mu-Ency--The Encyclopedia of the Mandelbrot Set." http://www.mrob.com/muency.html. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, pp. 178 /C1/79, 1988. Shishikura, M. "The Boundary of the Mandelbrot Set has Hausdorff Dimension Two." Aste´risque , No. 222, 7, 389/C1/ 05, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 146 /C1/48, 1991. Mandelbrot Tree The FRACTAL illustrated above. References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 71 /C1/3, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Mangoldt Function The function defined by L(n) /C13ln p if n /C30pk for p a prime 0 otherwise ;fflC}6 (1) sometimes also called the lambda function. exp( L(n)) is also given by [1, 2, ..., n]/[1, 2, ..., n /C281]; where [a; b; c ; ...] denotes the LEAST COMMON MULTIPLE . The first few values of exp(( n)) for n /C301, 2, ..., plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, ... (Sloane’s A014963). The Mangoldt function is related to the RIEMANN ZETA FUNCTION z(z)by /C28z?(s) z(s)/C30X/C12 n/C301L(n) ns; (2) where R[s] > 1 (Hardy 1999, p. 28; Krantz 1999,p. 161). The SUMMATORY Mangoldt function, illustrated above, is defined by c(x) /C13X n5xL(n) ; (3) where L(n) is the MANGOLDT FUNCTION , and is also known as the second CHEBYSHEV FUNCTION . c(x) has the explicit formula c(x) /C30x /C28X rxr r/C28ln(2 p) /C281 2ln(1 /C28x2) ; (4) where the second SUM is over all complex zeros r of the RIEMANN ZETA FUNCTION z(s) ; i.e., those in the CRITICAL STRIP so 0 BR r½/C138B1; and interpreted as lim t 0/C12X I( r) jjBtxr r: (5) Vardi (1991, p. 155) also gives the interesting formula ln x½/C138!ðÞ/C30 c(x) /C27 c1 2 xfflCz6fflCz7 /C27 c13 xfflCz6fflCz7 /C27...; (6) where [x] is the NINT function and n!isa FACTORIAL . Valle´e Poussin’s version of the PRIME NUMBER THEO- REM states that c(x) /C30x /C27O xe /C28affiffiffiffiffiffi ln xpfflCz6fflCz7 (7) for some a(Davenport 1980, Vardi 1991). The PRIME NUMBER THEOREM is equivalent to the statement that c(x)/C30x/C27o(x) (8) asx0/C12(Dusart 1999). The R IEMANN HYPOTHESIS is equivalent to c(x)/C30x/C27Offiffiffixp(lnx)2fflCz6fflCz7 (9) (Davenport 1980, p. 114; Vardi 1991). See also BOMBIERI’S THEOREM ,C HEBYSHEV FUNC- TIONS ,GREATEST PRIME FACTOR ,LAMBDA FUNCTION , LEAST COMMON MULTIPLE ,L EAST PRIME FACTOR , RIEMANN FUNCTION References Costa Pereira, N. "Estimates for the Chebyshev Function / cðxÞ/C28 uðxÞ/." Math. Comp. 44, 211 /C1/21, 1985. Costa Pereira, N. "Corrigendum: Estimates for the Cheby- shev Function / cðxÞ/C28 u ðxÞ/." Math. Comp. 48, 447, 1987. Costa Pereira, N. "Elementary Estimates for the Chebyshev Function c(x) and for the Mo¨bius Function M(x):/" Acta Arith. 52, 307 /C1/37, 1989. Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, p. 110, 1980. Dusart, P. "Ine´galite ´s explicites pour c(X) ; u(X); p(X) et les nombres premiers." C. R. Math. Rep. Acad. Sci. Canad 21, 53 /C1/9, 1999. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 28, 1999. Krantz, S. G. "The Lambda Function" and "Relation of the Zeta Function to the Lambda Function." §13.2.10 and 13.2.11 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 161, 1999. Rosser, J. B. and Schoenfeld, L. "Sharper Bounds for Cheby- shev Functions u(x) and c(x) :/" Math. Comput. 29, 243 /C1/69, 1975. Schoenfeld, L. "Sharper Bounds for Chebyshev Functions u(x) and c(x) : II," Math. Comput. 30, 337 /C1/60, 1976. Sloane, N. J. A. Sequences A014963 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 146 /C1/47, 152 /C1/53, and 249, 1991. Manhattan Distance The distance between two points ( x, y) and ( u, v) given by the METRIC d/C30x/C28u jj/C27y/C28v jj (Skiena 1990, p. 227). See also METRIC References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 172 and 227, 1990. Manifold A manifold is a TOPOLOGICAL SPACE which is LOCALLY EUCLIDEAN (i.e., around every point, there is a NEIGHBORHOOD which is topologically the same as the OPEN UNIT BALL inRn):To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. This discrepancy arises essentially from the fact thaton the small scales that we see, the Earth does indeed look flat (although the Greeks did notice that the last part of a ship to disappear over the horizon was themast). In general, any object which is nearly "flat" onsmall scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, asfirst codified by Poincare ´. More formally, any object that can be "charted" is a manifold. As a TOPOLOGICAL SPACE , a manifold can be COMPACT or not compact, and CONNECTED or disconnected. Typically, by "manifold," one means a manifold with- out boundary. However, an author will sometimes bemore precise and use the term OPEN MANIFOLD (for a noncompact manifold without boundary) or CLOSED MANIFOLD (for a COMPACT MANIFOLD without bound- ary). If a manifold contains its own boundary, it is called, not surprisingly, a " MANIFOLD WITH BOUNDARY ." The closed unit ball in Rnis a manifold with boundary, and its boundary is the unit sphere. The concept can be generalized to manifolds with corners. By defini-tion, every point on a manifold has a neighborhood together with a HOMEOMORPHISM of that neighbor- hood with an OPEN BALL inRn:In addition, a manifold must have a SECOND COUNTABLE TOPOLOGY . Unless otherwise indicated, a manifold is assumed to have finite DIMENSION n, for na positive integer. DIFFERENTIABLE MANIFOLDS are manifolds for which overlapping charts "relate smoothly" to each other,meaning that the inverse of one followed by the otheris an infinitely differentiable map from E UCLIDEAN SPACE to itself. Manifolds arise naturally in a variety of mathematical and physical applications as "global objects." For example, in order to precisely describe all the configurations of a robot arm or all the possiblepositions and momenta of a rocket, an object isneeded to store all of these parameters. The objectsthat crop up are manifolds. From the geometricperspective, manifolds represent the profound idea having to do with global versus local properties. The basic example of a manifold is E UCLIDEAN SPACE , and many of its properties carry over to manifolds. In addition, any smooth boundary of a subset of Eu-clidean space, like the circle or the sphere, is amanifold. Manifolds are therefore of interest in the study of GEOMETRY ,TOPOLOGY , and ANALYSIS . One of the goals of topology is to find ways of distinguishing manifolds. For instance, a circle istopologically the same as any closed loop, no matterhow different these two manifolds may appear.Similarly, the surface of a coffee mug with a handleis topologically the same as the surface of the donut, and this type of surface is called a (one-handled) TORUS . ASUBMANIFOLD is a subset of a manifold which is itself a manifold, but has smaller dimension. For example, the equator of a sphere is a submanifold.Many common examples of manifolds are submani- folds of Euclidean space. In fact, Whitney showed in the 1930s that any manifold can be EMBEDDED in RN ; where N /C302n /C271:/ A manifold may be endowed with more structure than a locally Euclidean topology. For example, it could be SMOOTH , COMPLEX , or even ALGEBRAIC (in order of specificity). A smooth manifold with a METRIC is called a RIEMANNIAN MANIFOLD , and one with a SYMPLECTIC STRUCTURE is called a SYMPLECTIC MANI- FOLD . Finally, a COMPLEX MANIFOLD with a KA¨ HLER STRUCTURE is called a KA¨ HLER MANIFOLD . See also ALGEBRAIC MANIFOLD ,COBORDANT MANI- FOLD ,C OMPACT MANIFOLD ,C OMPLEX MANIFOLD , CONNECTED SUM DECOMPOSITION ,C OORDINATE CHART ,D IFFERENTIABLE MANIFOLD ,E UCLIDEAN SPACE ,F LAG MANIFOLD ,G RASSMANN MANIFOLD , HEEGAARD SPLITTING ,ISOSPECTRAL MANIFOLDS , JACO-SHALEN- JOHANNSON TORUS DECOMPOSITION , KA¨ HLER MANIFOLD ,L IE GROUP ,M ANIFOLD WITH BOUNDARY ,POINCARE ´ CONJECTURE ,POISSON MANI- FOLD ,P RIME MANIFOLD ,R IEMANNIAN MANIFOLD , SET,SMOOTH MANIFOLD ,SPACE ,STIEFEL MANIFOLD , STRATIFIED MANIFOLD ,S UBMANIFOLD ,S URGERY , SYMPLECTIC MANIFOLD ,TANGENT BUNDLE ,TANGENT VECTOR (MANIFOLD ), THURSTON’S GEOMETRIZATION CONJECTURE ,TOPOLOGICAL MANIFOLD ,TOPOLOGICAL SPACE ,T RANSITION FUNCTION ,W HITEHEAD MANI- FOLD ,W IEDERSEHEN MANIFOLD References Conlon, L. Differentiable Manifolds: A First Course. Boston, MA: Birkha ¨user, 1993. Ferreiro ´s, J. "A New Fundamental Notion: Riemann’s Manifolds." Ch. 2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 39 /C1/0, 1999. Mannheim’s Theorem The four planes determined by the four altitudes of a TETRAHEDRON and the orthocenters of the corre- sponding faces pass through the MONGE POINT of the TETRAHEDRON . See also MONGE POINT ,TETRAHEDRON References Altshiller-Court, N. "The Monge Point." §4.2c in Modern Pure Solid Geometry. New York: Chelsea, pp. 69 /C1/1, 1979. Mannheim, A. J. de math. e´le´mentaires , p. 225, 1895. Thompson, H. F. "A Geometrical Proof of a Theorem Con- nected with the Tetrahedron." Proc. Edinburgh Math. Soc. 17,51/C1/3, 1908 /C1/909. Mann’s Theorem This entry contributed by KEVIN O’B RYANT A theorem widely circulated as the "/ a/-/b conjecture" and proved by Mann (1942). It states that if A and B are sets of integers each containing 0, thens(A /C154B) ]min f1; s(A) /C27 s(B) g: Here, A /C154B denotes the DIRECT SUM, i.e., A /C154B /C30 fa /C27b : a /C23 A; b /C23 Bg; and s is the SCHNIRELMANN DENSITY . Mann’s theorem is best possible in the sense that A /C30 B /C30f0 ; 1 ; 11 ; 12 ; 13; ...g satisfies s(A /C154B) /C30/ /s(A) /C27 s(B) :/ Mann’s theorem implies SCHNIRELMANN’S THEOREM as follows. Let P /C30f0 ; 1 g@fp : p prime g; then Mann’s theorem proves that s(P /C27P /C27P /C27P) > 2s(P /C27P) ; so as more and more copies of the primes are included, the SCHNIRELMANN DENSITY increases at least linearly, and so reaches 1 with at most 2 /C215 1=( s(P /C27P)) copies of the primes. Since the only sets with SCHNIRELMANN DENSITY 1 are the sets contain- ing all positive integers, SCHNIRELMANN’S THEOREM follows. See also SCHNIRELMANN DENSITY ,SCHNIRELMANN’S THEOREM References Garrison, B. K. "A Nontransformation Proof of Mann’s Density Theorem." J. reine angew. Math. 245,41/C1/6, 1970. Khinchin, A. Y. "The Landau-Schnirelmann Hypothesis and Mann’s Theorem." Ch. 2 in Three Pearls of Number Theory. New York: Dover, pp. 18 /C1/6, 1998. Mann, H. B. "A Proof of the Fundamental Theorem on the Density of Sets of Positive Integers." Ann. Math. 43, 523 /C1/ 27, 1942. MANOVA MANOVA ("multiple analysis of variance") is a procedure for testing the equality of mean vectors of more than two populations. The technique is analo- gous to ANOVA for univariate data, except that groups are compared on multiple response variables simultaneously. While F-tests can be used in the uniseriate case to assess the hypothesis under con- sideration, there is no single test statistic in the multivariate case that is optimal in all situations (Everitt and Wykes 1999, p. 125). See also ANOVA References Bijleveld, C. C. J. H.; van der Kamp, L. J. T.; Mooijaart, A.; van der Kloot, W. A.; van der Leeden, R.; and van der Burg, E. Longitudinal Data Analysis: Designs, Models and Methods. London: Sage, 1998. Everitt, B. S. and Wykes, T. Dictionary of Statistics for Psychologists. London: Arnold, p. 125, 1999. Mantissa For a REAL NUMBER x, the mantissa is defined as the POSITIVE FRACTIONAL PART x /C28 xbc/C30frac(x) ; where xbc denotes the FLOOR FUNCTION . See also CHARACTERISTIC (REAL NUMBER ), FLOOR FUNCTION ,SCIENTIFIC NOTATION Many-to-One A FUNCTION f which may (but does not necessarily) associate a given member of the RANGE of f with more than one member of the DOMAIN of f. For example, TRIGONOMETRIC FUNCTIONS such as sin x are many- to-one since sin x /C30sin(2p /C27x) /C30sin(4 p /C27x) /C30/C1/C1/C1:/ See also DOMAIN ,ONE-TO- ONE,RANGE (IMAGE ) Many Valued Logic References Rescher, N. Many Valued Logic. Ashgate, 1993. Map A way of associating unique objects to every point in a given SET. So a map from A /C2B is an object f such that for every A /C23 B ; there is a unique object f(a) /C23 B: The terms FUNCTION and MAPPING are synonymous with map. the following table gives several common types of complex maps. Mapping FORMULA Domain Inversion /f(z) /C301 z/ Magnification /f(z) /C30az// a /C23R "0/ Magnification /C27Rotation/f(z) /C30az// a /C23C "0/ MO¨ BIUS TRANSFORMATION/f(z) /C30az /C27 b cz /C27 d//a; b; c ; d /C23C/ ROTATION /f(z) /C30ei uz// u /C23R/ TRANSLATION /f(z) /C30z /C27a//a /C23C/ See also 2X MOD 1 MAP,ARNOLD’S CAT MAP,BAKER’S MAP,BOUNDARY MAP,CONFORMAL MAP,FUNCTION , GAUSS MAP,GINGERBREADMAN MAP,HARMONIC MAP, HE´ NON MAP,IDENTITY MAP,INCLUSION MAP,K A- PLAN- YORKE MAP,LOGISTIC MAP,M ANDELBROT SET, MAP PROJECTION ,PULLBACK MAP,QUADRATIC MAP,SYMPLECTIC MAP,TANGENT MAP,TENT MAP,TRANS- FORMATION ,ZASLAVSKII MAP References Arfken, G. "Mapping." §6.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 384 /C1/ 92, 1985. Map-Airy Distribution A probability distribution having density P(x) /C302e /C282x3 =3 x Ai x2fflC{fflCz /C28Ai? x2fflC{fflCz fflC}fflC( ; where Ai(x) is the AIRY FUNCTION and Ai?(x) /C30 dAi(x) =dx: The corresponding distribution function is D(x) /C301 3 /C282x52F276 ;53 ;73;83; /C2843 x3fflCz6fflCz7 15 /C215 32=3 G5 3fflCz6fflCz7 /C28x42F256 ;43;53 ;73; /C2843 x3fflCz6fflCz7 6 /C215 31 =34 3fflCz6fflCz7 /C27x22F21 6 ;23;13 ;53; /C2843 x3fflCz6fflCz7 32 =3 G2 3fflCz6fflCz7 /C272x2F2/C281 6 ;13; /C2813 ;43; /C2843 x3fflCz6fflCz7 31 =3 G1 3fflCz6fflCz7 (M. Trott). The density is normalized with g/C12 /C28/C12A(x) dx /C301: The MEAN is 0, but the second moment m2is undefined. See also AIRY FUNCTIONS References Banderier, C.; Flajolet, P.; Schaeffer, G.; and Soria, M. "Planar Maps and Airy Phenomena." Preprint. Map Coloring Given a map with GENUS g/C210, Heawood showed in 1890 that the maximum number Nuof colors neces- sary to color a map (the CHROMATIC NUMBER )o na n unbounded surface is Nu /C131 27 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 48g /C271pfflCz6fflCz7jk /C301 27 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 49 /C2824 xpfflCz6fflCz7jk ; where xbcis the FLOOR FUNCTION , g is the GENUS , and x is the EULER CHARACTERISTIC . This is the HEAWOOD CONJECTURE . In 1968, for any orientable surface other than the SPHERE (or equivalently, the PLANE ) and any nonorientable surface other than the KLEIN BOTTLE , Nu was shown to be not merely a maximum, but the actual number needed (Ringel and Youngs 1968). When the FOUR-COLOR THEOREM was proven, the Heawood FORMULA was shown to hold also for all orientable and nonorientable surfaces with the ex- ception of the KLEIN BOTTLE . For this case, the actual number of colors N needed is six–one less than Nu /C307 (Franklin 1934; Saaty 1986, p. 45). surface g /Nu/ N KLEIN BOTTLE 176 MO¨ BIUS STRIP /1 2/ 66 PLANE 044 PROJECTIVE PLANE /1 2/ 66 SPHERE 044 TORUS 177 See also CHROMATIC NUMBER ,FOUR- COLOR THEO- REM,H EAWOOD CONJECTURE ,SIX-COLOR THEOREM , TORUS COLORING References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 237 /C1/38, 1987. Barnette, D. Map Coloring, Polyhedra, and the Four-Color Problem. Washington, DC: Math. Assoc. Amer., 1983. Franklin, P. "A Six Colour Problem." J. Math. Phys. 13, 363 /C1/69, 1934. Franklin, P. The Four-Color Problem. New York: Scripta Mathematica, Yeshiva College, 1941. Ore, Ø. The Four-Color Problem. New York: Academic Press, 1967. Ringel, G. and Youngs, J. W. T. "Solution of the Heawood Map-Coloring Problem." Proc. Nat. Acad. Sci. USA 60, 438 /C1/45, 1968. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986. Mapes’ Method A method for computing the PRIME COUNTING FUNC- TION . Define the function Tk(x; a) /C30(/C281)b0/C27b1/C27.../C27ba /C281x p b0 1p b1 2/C1/C1/C1pba /C281a$% ; (1) where xbcis the FLOOR FUNCTION and the biare thebinary digits (0 or 1) in k /C302a /C281 ba /C281 /C272a /C282 ba /C282 /C27.../C2721 b1 /C2720 b0 : (2) The LEGENDRE SUM can then be written f(x; a) /C30X2a /C281 k/C300Tk(x ; a): (3) The first few values of Tk(x; a) are T0(x; 3) /C30 xbc (4) T1(x; 3) /C30/C28x p1$% (5) T2(x; 3) /C30/C28x p2$% (6) T3(x;3)/C30x p1p2$% (7) T4(x;3)/C30x p3$% (8) T5(x;3)/C30x p1p3$% (9) T6(x;3)/C30x p2p3$% (10) T7(x;3)/C30/C28x p1p2p3$% : (11) Mapes’ method takes time /C2x0:7;which is slightly faster than the L EHMER- SCHUR METHOD . See also LEHMER- SCHUR METHOD ,PRIME COUNTING FUNCTION References Mapes, D. C. "Fast Method for Computing the Number of Primes Less than a Given Limit." Math. Comput. 17, 179/C1/ 85, 1963. Riesel, H. "Mapes’ Method." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨u- ser, p. 23, 1994. Map Folding A general FORMULA giving the number of distinct ways of folding an N/C30m/C29nrectangular map is not known. A distinct folding is defined as a permutation ofNnumbered cells reading from the top down. Lunnon (1971) gives values up to n/C3028. n /1/C29n//2/C29n//3/C29n//4/C29n// 5/C29n/ 11 1 22 8 3 6 60 1368 4 16 1980 300608 5 59 19512 18698669 6 144 15552 The limiting ratio of the number of 1 /C29(n /C271) strips to the number of 1 /C29n strips is given by lim n0/C12[1 /C29 (n /C27 1)] [1 /C29 n]/C23 [3:3868 ; 3:9821] : See also STAMP FOLDING References Gardner, M. "The Combinatorics of Paper Folding." Ch. 7 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 60 /C1/3, 1983. Koehler, J. E. "Folding a Strip of Stamps." J. Combin. Th. 5, 135 /C1/52, 1968. Lunnon, W. F. "A Map-Folding Problem." Math. Comput. 22, 193 /C1/99, 1968. Lunnon, W. F. "Multi-Dimensional Strip Folding." Computer J. 14,75/C1/9, 1971. Mapping (Function) MAP Mapping Space Let YX be the set of continuous mappings f : X 0 Y : Then the TOPOLOGICAL SPACE for YX supplied with a compact-open topology is called a mapping space. See also LOOP SPACE References Iyanaga, S. and Kawada, Y. (Eds.). "Mapping Spaces." §204B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 658, 1980. Map Projection A projection which maps a SPHERE (or SPHEROID ) onto a PLANE . Map projections are generally classified into groups according to common properties (cylindrical vs. conical, conformal vs. area-preserving, etc.), although such schemes are generally not mutually exclusive. Early compilers of classification schemes include Tissot (1881), Close (1913), and Lee (1944). However, the categories given in Snyder (1987) remain the most commonly used today, and Lee’s terms authalic and aphylactic are not commonly encountered. No projection can be simultaneously CONFORMAL and AREA-PRESERVING . See also AIRY PROJECTION ,A LBERS EQUAL- AREACONIC PROJECTION ,AXONOMETRY ,AZIMUTHAL EQUI- DISTANT PROJECTION ,A ZIMUTHAL PROJECTION , BALTHASART PROJECTION ,B EHRMANN CYLINDRICAL EQUAL- AREA PROJECTION ,BONNE PROJECTION ,CAS- SINI PROJECTION ,CHROMATIC NUMBER ,CONIC EQUI- DISTANT PROJECTION ,C ONIC PROJECTION , CYLINDRICAL EQUAL- AREA PROJECTION ,CYLINDRICAL EQUIDISTANT PROJECTION ,CYLINDRICAL PROJECTION , ECKERT IV PROJECTION ,E CKERT VI PROJECTION , FOUR- COLOR THEOREM ,G ALL ISOGRAPHIC PROJEC- TION ,GALL ORTHOGRAPHIC PROJECTION ,GNOMONIC PROJECTION ,G UTHRIE’S PROBLEM ,H AMMER- AITOFF EQUAL- AREA PROJECTION ,L AMBERT AZIMUTHAL EQUAL- AREA PROJECTION ,L AMBERT CONFORMAL CONIC PROJECTION ,M AP COLORING ,M ERCATOR PRO- JECTION ,M ILLER CYLINDRICAL PROJECTION ,M OLL- WEIDE PROJECTION ,O RTHOGRAPHIC PROJECTION , PETERS PROJECTION ,POLYCONIC PROJECTION ,PSEU- DOCYLINDRICAL PROJECTION ,RECTANGULAR PROJEC- TION ,SINUSOIDAL PROJECTION ,SIX-COLOR THEOREM , STEREOGRAPHIC PROJECTION ,TRISTAN EDWARDS PRO- JECTION , VAN DER GRINTEN PROJECTION ,VERTICAL PERSPECTIVE PROJECTION References Anderson, P. B. "Reciprocal Links." http://www.series2000.- com/users/pbander/. Close, C. F. Text-Book of Topographical and Geographical Surveying, 2nd ed. London: H. M. Stationary Office, 1913. Craig, T. A Treatise on Projections. Washington, DC: U.S. Government Printing Office, 1882. Dana, P. H. "Map Projections." http://www.colorado.edu/ geography/gcraft/notes/mapproj/mapproj_f.html. Hinks, A. R. Map Projections, 2nd rev. ed. Cambridge, England: Cambridge University Press, 1921. Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Review 7, 190 /C1/00, 1944. Mulcahy, K. "The Map Projection Home Page." http://ever- est.hunter.cuny.edu/mp/. Maling, D. H. Coordinate Systems and Map Projections, 2nd ed, rev. Woburn, MA: Butterworth-Heinemann, 1993. Snyder, J. P. Flattening the Earth: Two Thousand Years of Map Projections. Chicago, IL: University of Chicago Press, 1993. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987. Tissot, A. Me´moir sur la repre´sentation des surfaces et les projections des cartes ge´ographiques. Paris: Gauthier- Villars, 1881. Weisstein, E. W. "Books about Cartography." http:// www.treasure-troves.com/books/Cartography.html. Marcus’s Theorem A COMPACT MANIFOLD admits a LORENTZIAN STRUC- TURE IFF its EULER CHARACTERISTIC vanishes. There- fore, every noncompact manifold admits a LORENTZIAN STRUCTURE . See also EULER CHARACTERISTIC ,LORENTZIAN STRUC- TURE References Dodson, C. T. J. and Parker, P. E. "Marcus’s Theorem." §9.5 in A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 289 /C1/91, 1997. Marginal Analysis Let R(x) be the revenue for a production x, C(x) the cost, and P(x) the profit. Then P(x) /C30R(x) /C28C(x) ; and the marginal profit for the x0/th unit is defined by P ? x0ðÞ/C30R? x0ðÞ/C28C ? x0ðÞ ; where P ?(x) ; R?(x); and C ?(x) are the DERIVATIVES of P(x) ; R(x) ; and C(x); respectively. See also DERIVATIVE Marginal Probability Let S be partitioned into r /C29s disjoint sets Ei and Fj where the general subset is denoted Ei S Fj : Then the marginal probability of Ei is PEiðÞ/C30Xs j/C301PEi S FjfflC{fflCz : See also CONDITIONAL PROBABILITY ,D ISTRIBUTION FUNCTION ,JOINT DISTRIBUTION FUNCTION ,PROBABIL- ITY FUNCTION Markoff Chain MARKOV CHAIN Markoff Number MARKOV NUMBER Markoff’s Formulas Formulas obtained from differentiating NEWTON’S FORWARD DIFFERENCE FORMULA , f ? a0 /C27ph ðÞ /C301 hfflC}{ D0 /C271 2(2p /C281)D2 0 /C271 63p2 /C286p /C272fflC{fflCz D3 0 /C27.../C27d dpp nfflCzrfflCzD Dn0fflC}z /C27R?n ; where R?n /C30hnf(n/C271)( j)d dpp n /C271fflCzrfflCzD /C27hn /C271 p n /C271fflCzrfflCzD /C2d dpf(n/C271)( j) ; (1) /n kfflC{fflCz is a BINOMIAL COEFFICIENT , and a0 B j Ban : Abramowitz and Stegun (1972) and Beyer (1987)give derivatives hnf(n) 0in terms of Dk and derivatives in terms of dk and 9k :/ See also FINITE DIFFERENCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 883, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 449 /C1/50, 1987. Markov Algorithm An ALGORITHM which constructs allowed mathema- tical statements from simple ingredients. Markov Chain A collection of random variables Xtfg (where the index t runs through 0, 1, ...) having the property that, given the present, the future is conditionally independent of the past. In other words, PXt /C30j ½X0 /C30i0 ; X1 /C30i1 ; ...Xt /C281 /C30it /C281 ðÞ /C30PXt /C30j½Xt/C281 /C30it /C281 ðÞ : If a MARKOV SEQUENCE of random variates xntake the discrete values a1 ; ..., aN ; then Pxn /C30ain½xn/C281 /C30ain /C281; ...; x1 /C30a1fflCz6fflCz7 /C30Pxn /C30ain½xn/C281 /C30ain/C281fflCz6fflCz7 ; and the sequence xnis called a Markov chain (Papoulis 1984, p. 532). ASIMPLE RANDOM WALK is an example of a Markov chain. See also MARKOV SEQUENCE ,MONTE CARLO METHOD , RANDOM WALK References Gamerman, D. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Boca Raton, FL: CRC Press, 1997. Gilks, W. R.; Richardson, S.; and Spiegelhalter, D. J. (Eds.). Markov Chain Monte Carlo in Practice. Boca Raton, FL: Chapman & Hall, 1996. Grimmett, G. and Stirzaker, D. Probability and Random Processes, 2nd ed. Oxford, England: Oxford University Press, 1992. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 6, 1994. Kallenberg, O. Foundations of Modern Probability. New York: Springer-Verlag, 1997. Kemeny, J. G. and Snell, J. L. Finite Markov Chains. New York: Springer-Verlag, 1976. Papoulis, A. "Brownian Movement and Markoff Processes." Ch. 15 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 515 /C1/53, 1984. Stewart, W. J. Introduction to the Numerical Solution of Markov Chains. Princeton, NJ: Princeton University Press, 1995. Markov Matrix STOCHASTIC MATRIX Markov Moves A type I move (CONJUGATION ) takes AB 0 BA for A, B /C23 Bn where Bn is a BRAID GROUP . A type II move (STABILIZATION ) takes A 0 Abn or A 0 Ab/C281 nfor A /C23 Bn and bn ; Abn ; and Ab /C281 n/C23 Bn/C271 :/ See also BRAID GROUP ,CONJUGATION ,KNOT MOVE, REIDEMEISTER MOVES ,STABILIZATION Markov Number The Markov numbers m are the union of the solutions (x; y; z) to the DIOPHANTINE EQUATION x2 /C27y2 /C27z2 /C303xyz; and are related to LAGRANGE NUMBERS Ln by Ln /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 /C284 n2s : The first few solutions are (x; y; z) /C30(1; 1; 1); (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), .... All solutions can be generated from the first two of these since the equation is a quadratic in each of the variables, so one integer solution leads to a second, and it turns out that all solutions (other than the first two singular ones) have distinct values of x, y, and z, and share two of their three values with three other solutions (Guy 1994, p. 166). The Markov numbers are then given by 1, 2, 5, 13, 29, 34, ... (Sloane’s A002559). The Markov numbers for triples (x; y; z) in which one term is 5 are 1, 2, 13, 29, 194, 433, ... (Sloane’s A030452), whose terms are given by the RECURRENCERELATION a(n) /C3015a(n /C282) /C28a(n /C284); (1) with a(0) /C301; a(1) /C302; a(2) /C3013; and a(3) /C3029 :/ The solutions can be arranged in an infinite tree with two smaller branches on each trunk. It is not known if two different regions can have the same label. Strangely, the regions adjacent to 1 have alternate FIBONACCI NUMBERS 1, 2, 5, 13, 34, ..., and the regions adjacent to 2 have alternate PELL NUMBERS 1, 5, 29, 169, 985, .... Let M(N) be the number of TRIPLES with x 5y 5z 5 N ; then M(n) /C30C(ln N)2 /C27O((ln N)1 /C27 e) ; where C :0:180717105 (Guy 1994, p. 166). See also HURWITZ EQUATION ,HURWITZ’S IRRATIONAL NUMBER THEOREM ,IRRATIONALITY MEASURE ,L A- GRANGE NUMBER (RATIONAL APPROXIMATION )LIOU- VILLE’S APPROXIMATION THEOREM ,ROTH’S THEOREM , SEGRE’S THEOREM ,THUE- SIEGEL- ROTH THEOREM References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187 /C1/89, 1996. Descombes, R. "Proble `mes d’approximation diophantienne." Enseign. Math. 6,18/C1/6, 1960. Guy, R. K. "Don’t Try to Solve These Problems." Amer. Math. Monthly 90,35/C1/1, 1983. Guy, R. K. "Markoff Numbers." §D12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 166 /C1/68, 1994. Sloane, N. J. A. Sequences A002559/M1432 and A030452 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Markov Process A random process whose future probabilities are determined by its most recent values. A STOCHASTIC PROCESS x(t) is called Markov if for every n and t1 Bt2 ...Btn we have P(x(tn) 5xn x(tn/C281) ; ...; x(t1)) j /C30P(x(tn)5xnx(tn/C281)): j This is equivalent to P(x(tn)5xnx(t) for all t5tn/C281) j /C30P(x(tn)5xnx(tn/C281)) j (Papoulis 1984, p. 535). See also DOOB’S THEOREM References Bharucha-Reid, A. T. Elements of the Theory of Markov Processes and Their Applications. New York: McGraw- Hill, 1960. Papoulis, A. "Brownian Movement and Markoff Processes." Ch. 15 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 515 /C1/53, 1984. Markov Sequence A sequence X1 ; X2 ; ... of random variates is called Markov (or Markoff) if, for any n, F(Xn Xn/C281 ; Xn/C282 ; ...; X1) /C30F(Xn Xn/C281) ; j j i.e., if the conditional distribution F of Xnassuming Xn/C281 ; Xn/C282 ; ..., X1equals the conditional distribution F of Xn assuming only Xn/C281 (Papoulis 1984, pp. 528 /C1/ 29). The transitional densities of a Markov sequence satisfy the CHAPMAN- KOLMOGOROV EQUATION . See also CHAPMAN- KOLMOGOROV EQUATION ,MARKOV CHAIN References Papoulis, A. "Markoff Sequences." §15 /C1/ in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 528 /C1/35, 1984. Markov’s Inequality If x takes only NONNEGATIVE values, then P(x ]a) 5xhi a: To prove the theorem, write xhi/C30g/C12 0xf(x) dx /C30ga 0xf(x) dx /C27g/C12 axf(x) dx: Since P(x) is a probability density, it must be ]0 : We have stipulated that x ]0 ; so xhi/C30ga 0xf(x) dx /C27g/C12 axf(x) dx ]g/C12 0xf(x) dx ]g/C12 0af(x) dx /C30ag/C12 0f(x) dx /C30aP(x ]a); Q.E.D. Markov Spectrum A SPECTRUM containing the REAL NUMBERS larger than FREIMAN’S CONSTANT . See also FREIMAN’S CONSTANT ,SPECTRUM SEQUENCE References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 188 /C1/89, 1996.Markov’s Theorem Published by A. A. Markov in 1935, Markov’s theo- rem states that equivalent BRAIDS expressing the same LINK are mutually related by successive appli- cations of two types of MARKOV MOVES . Markov’s theorem is difficult to apply in practice, so it is difficult to establish the equivalence or nonequiva- lence of LINKS having different BRAID representations. See also BRAID,LINK,MARKOV MOVES References Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Marriage Theorem If a group of men and women may date only if they have previously been introduced, then a complete set of dates is possible IFF every subset of men has collectively been introduced to at least as many women, and vice versa (Hall 1935; Chartrand 1985, p. 121; Skiena 1990, p. 240). See also MATCHING References Chartrand, G. Introductory Graph Theory. New York: Dover, 1985. Hall, P. "On Representatives of Subsets." J. London Math. Soc. 10,2 6/C1/0, 1935. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Married Couples Problem Also called the ME´NAGE PROBLEM . In how many ways cannmarried couples be seated around a circular table in such a manner than there is always one man between two women and none of the men is next to his own wife? The solution (Ball and Coxeter 1987,p. 50) uses DISCORDANT PERMUTATIONS and can be given in terms of L AISANT’S RECURRENCE FORMULA (n/C281)An/C271/C30(n2/C281)An/C27(n/C271)An/C281/C274(/C281)n;(1) with A1/C30A2/C301:A closed form expression due to Touchard (1934) is An/C30Xn k/C3002n 2n/C28k2n/C28k kfflCzrfflCzD (n/C28k)!(/C281)k; (2) wheren kfflC{fflCz is a BINOMIAL COEFFICIENT (Vardi 1991). The sum can be evaluated explicitly as An/C30npI/C28n(2) csc( np) e2 /C284(/C281)n n2/C2812F2(1;3 2;2/C28n;2/C28n;2/C27n;/C284); (3) where2F2(a; b; c ; d; x)isa GENERALIZED HYPERGEO- METRIC FUNCTION . The first few values of An are /C281, 1, 0, 2, 13, 80, 579, ... (Sloane’s A000179), which are sometimes called ME´ NAGE NUMBERS . The desired solution is then 2n!An : The numbers An can be considered a special case of a restricted ROOKS PROBLEM . See also DISCORDANT PERMUTATION ,LAISANT’S RE- CURRENCE FORMULA ,ROOKS PROBLEM References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 50, 1987. Comtet, L. "The ‘Proble `me des Me´nages’." §4.3 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 182 /C1/85, 1974. Do¨rrie, H. §8in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 27 /C1/3, 1965. Halmos, P. R.; Vaughan, H. E. "The Marriage Problem." Amer. J. Math. 72, 214 /C1/15, 1950. Lucas, E. The´orie des Nombres. Paris: A. Blanchard, pp. 215 and 491 /C1/95, 1979. MacMahon, P. A. Combinatory Analysis, Vol. 1. London: Cambridge University Press, pp. 253 /C1/56, 1915. Newman, D. J. "A Problem in Graph Theory." Amer. Math. Monthly 65, 611, 1958. Sloane, N. J. A. Sequences A000179/M2062 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Touchard, J. "Sur un proble `me de permutations." C. R. Acad. Sci. Paris 198, 631 /C1/33, 1934. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 123, 1991. Marshall-Edgeworth Index The statistical INDEX PME /C13Ppn(q0 /C27 qn)P(v0 /C27 vn); where pnis the price per unit in period n, qnis the quantity produced in period n, and vn /C13pnqnis the value of the n units. See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 66 /C1/7, 1962. Martingale A sequence of random variates X0 ; X1 ; ... with finite means such that the conditional expectation of Xn/C271 given X0 ; X1 ; X2 ; ..., Xn is equal to Xn ; i.e., xn/C271 X0 ; ...; Xn ji /C30XnfflCz{ (Feller 1971, p. 210). The term was first used to describe a type of wagering in which the bet is doubled or halved after a loss or win, respectively.The concept of martingales is due to Le´vy, and it was developed extensively by Doob. A 1-D RANDOM WALK with steps equally likely in either direction /(p /C30q /C301=2) is an example of a martingale. See also ABSOLUTELY FAIR,G AMBLER’S RUIN,RAN- DOM WALK–1- D, SAINT PETERSBURG PARADOX References Doob, J. L. Stochastic Processes. New York: Wiley, 1953. Feller, W. "Martingales." §6.12 in An Introduction to Prob- ability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 210 /C1/15, 1971. Le´vy, P. Calcul de probabilite ´s.Paris: Gauthier-Villars, 1925. Le´vy, P. The´orie de l’addition des variables ale ´atoires. Paris: Gauthier-Villars, 1954. Le´vy, P. Processus stochastiques et mouvement Brownien, 2nd ed. Paris: Gauthier-Villars, 1965. Loe`ve, M. Probability Theory, 3rd ed. Princeton, NJ: Van Nostrand, 1963. Mascheroni Constant EULER- MASCHERONI CONSTANT Mascheroni Construction A geometric construction done with a movable COM- PASS alone. All constructions possible with a COMPASS and STRAIGHTEDGE are possible with a movable COMPASS alone, as was proved by Mascheroni (1797). Mascheroni’s results are now known to have been anticipated largely by Mohr (1672). An example of a Mascheroni construction of themidpoint Mof a LINE SEGMENT specified by two points Aand Billustrated above (Steinhaus 1983, Wells 1991). Without loss of generality, take AB/C301. 1. Construct circles centered at Aand Bpassing through BandA. These are unit circles centered at (0, 0) and (1, 0).2. Locate C, the indicated intersection of circles A and B, and draw a circle centered on Cpassing through points AandB. This circle has center (1/ 2,ffiffiffiffiffiffi 3=p 2) and radius 1. 3. Locate D, the indicated intersection of circles B and C, and draw a circle centered on Cpassing through points B and C. This circle has center (3/ 2,ffiffiffiffiffiffi 3=p 2) and radius 1. 4. Locate E, the indicated intersection of circles B and D, and draw a circle centers on E passing through point C. This circle has center (2, 0) and radiusffiffiffi 3p :/ 5. Locate F and G, the intersections of circles AE and EC. These points are located at positions (5/4, 9ffiffiffiffiffiffi39p =4): / 6. Locate M, the intersection of circles F and G. This point has position (1/2, 0), and is therefore the desired MIDPOINT ofAB :/ Pedoe (1995, pp. xviii-xix) also gives a Mascheroni solution. See also COMPASS ,GEOMETRIC CONSTRUCTION ,NEU- SIS CONSTRUCTION ,S TEINER CONSTRUCTION , STRAIGHTEDGE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 96 /C1/7, 1987. Bogomolny, A. "Geometric Constructions with the Compass Alone." http://www.cut-the-knot.com/do_you_know/com- pass.html. Courant, R. and Robbins, H. "Constructions with Other Tools. Mascheroni Constructions with Compass Alone." §3.5 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 146 /C1/58, 1996. Do¨rrie, H. "Mascheroni’s Compass Problem." §33 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 160 /C1/64, 1965. Gardner, M. "Mascheroni Constructions." Ch. 17 in Mathe- matical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific Amer- ican. New York: Knopf, pp. 216 /C1/31, 1979. Hutt, E. Die Mascheroni’schen Konstruktionen fu¨r die zwecke ho¨herer Lehrenstalten und zum Selbstuterrichte. Halle, Germany: H. W. Schmidt, 1880. Mascheroni, L. Geometria del compasso. Pavia, Italy, 1797. Mohr, G. Euclides Danicus. Amsterdam, Netherlands, 1672. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Quemper de Lanascol, A. Ge´ome´trie du compas. Blanchard, pp. 74 /C1/7, 1925. Schwerin. Mascheronische Konstruktionen. 1898. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 141 /C1/42, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 148 /C1/49, 1991. Maschke’s Theorem If a MATRIX GROUP is reducible, then it is completely reducible, i.e., if the MATRIX GROUP is equivalent to the MATRIX GROUP in which every MATRIX has the reduced form D(1) i Xi 0 D(2)ifflC}{fflC}z ; then it is equivalent to the MATRIX GROUP obtained by putting Xi /C300 :/See also MATRIX GROUP References Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 49, 1987. Mason’s abc Theorem MASON’S THEOREM Mason’s Theorem Let there be three POLYNOMIALS a(x) ; b(x) ; and c(x) with no common factors such that a(x) /C27b(x) /C30c(x) : Then the number of distinct ROOTS of the three POLYNOMIALS is one or more greater than their largest degree. The theorem was first proved by Stothers (1981). Mason’s theorem may be viewed as a very special case of a Wronskian estimate (Chudnovsky and Chud- novsky 1984). The corresponding Wronskian identity in the proof by Lang (1993) is c3 + W(a; b; c) /C30W(W(a; c); W(b ; c)); so if a, b, and c are linearly dependent, then so are W(a ; c) and W(b; c) : More powerful Wronskian esti- mates with applications toward Diophantine approx- imation of solutions of linear differential equations may be found in Chudnovsky and Chudnovsky (1984) and Osgood (1985). The RATIONAL FUNCTION case of F ERMAT’S LAST THEOREM follows trivially from Mason’s theorem (Lang 1993, p. 195). See also ABC CONJECTURE References Chudnovsky, D. V. and Chudnovsky, G. V. "The Wronskian Formalism for Linear Differential Equations and Pade ´ Approximations." Adv. Math. 53,2 8/C1/4, 1984. Lang, S. "Old and New Conjectured Diophantine Inequal- ities." Bull. Amer. Math. Soc. 23,3 7/C1/5, 1990. Lang, S. Algebra, 3rd ed. Reading, MA: Addison-Wesley, 1993. Mason, R. C. Diophantine Equations over Functions Fields. Cambridge, England: Cambridge University Press, 1984. Osgood, C. F. "Sometimes Effective Thue-Siegel-Roth- Schmidt-Nevanlinna Bounds, or Better." J. Number Th. 21, 347/C1/89, 1985. Stothers, W. W. "Polynomial Identities and Hauptmodulen." Quart. J. Math. Oxford Ser. II 32, 349/C1/70, 1981. Masser-Gramain Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Letf(z)b ea n ENTIRE FUNCTION such that f(n)i sa n INTEGER for each POSITIVE INTEGER n. Then Po ´lya (1915) showed that if lim sup r0/C12ln Mr rBln 2 /C300:693... ; (1) where Mr /C30sup zjj5rf(x)jj (2) is the SUPREMUM , then f is a POLYNOMIAL . Further- more, ln 2 is the best constant (i.e., counterexamples exist for every smaller value). If f(z)isan ENTIRE FUNCTION with f(n)aG AUSSIAN INTEGER for each GAUSSIAN INTEGER n, then Gelfond (1929) proved that there exists a constant a such that lim sup r0/C12ln Mr r2B a (3) implies that f is a POLYNOMIAL . Gramain (1981, 1982) showed that the best such constant is a /C30p 2e /C300:578... (4) Maser (1980) proved the weaker result that f must be a POLYNOMIAL if lim sup r0/C12ln Mr r2B a0 /C301 2exp /C28d /C274c p ! ; (5) where c /C30 gb(1) /C27 b?(1) /C300:642454398948114... ; (6) /g is the EULER- MASCHERONI CONSTANT , b(z) is the DIRICHLET BETA FUNCTION , d /C13 lim n0/C12Xn k /C3021 prk2 /C28ln n ! ; (7) and rkis the minimum NONNEGATIVE r for which there exists a COMPLEX NUMBER z for which the CLOSED DISK with center z and radius r contains at least k distinct GAUSSIAN INTEGERS . Gosper gave c /C30 p /C28ln[G(1 4)] /C2734 p /C2712 ln 2 /C2712 gno : (8) Gramain and Weber (1985, 1987) have obtained 1:811447299 B d B1 :897327177 ; (9) which implies 0:1707339 B a0 B0:1860446 : (10) Gramain (1981, 1982) conjectured that a0 /C301 2e ; (11) which would implyd /C301 /C274c p/C301:822825249... : (12) References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/masser/masser.html. Gramain, F. "Sur le the´ore`me de Fukagawa-Gel’fond." Invent. Math. 63, 495 /C1/06, 1981. Gramain, F. "Sur le the´ore`me de Fukagawa-Gel’fond-Gru- man-Masser." Se´minaire Delange-Pisot-Poitou (The´orie des Nombres), 1980 /C1/981. Boston, MA: Birkha ¨user, 1982. Gramain, F. and Weber, M. "Computing and Arithmetic Constant Related to the Ring of Gaussian Integers." Math. Comput. 44, 241 /C1/45, 1985. Gramain, F. and Weber, M. "Computing and Arithmetic Constant Related to the Ring of Gaussian Integers." Math. Comput. 48, 854, 1987. Masser, D. W. "Sur les fonctions entie`res a` valeurs entie`res." C. R. Acad. Sci. Paris Se´r. A-B 291, A1-A4, 1980. Mastermind References Bewersdorff, J. Glu¨ck, Logik and Bluff: Mathematik im Spiel: Methoden, Ergebnisse und Grenzen. Wiesbaden, Germany: Vieweg, 1998. Bogomolny, A. and Greenwell, D. "Cut the Knot: Invitation to Mastermind." http://www.maa.org/editorial/knot/Mas- termind.html. Chvatal, V. "Mastermind." Combinatorica 3, 325 /C1/29, 1983. Erdos, P. and C. Re´nyi, C. "On Two Problems in Information Theory." Magyar Tud. Akad. Mat. Kut. Int. Ko¨zl. 8, 229 /C1/ 42, 1963. Greenwell, D. L. "Mastermind." Submitted to J. Recr. Math. Guy, R. "The Strong Law of Small Numbers." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994. Knuth, D. E. "The Computer as a Master Mind." J. Recr. Math. 9,1/C1/, 1976 /C1/7. Koyama, K. and Lai, T. W. "An Optimal Mastermind Strategy." J. Recr. Math. 25, 251 /C1/56, 1993. Mitchell, M. "MasterMind † Mathematics." Key Curriculum Press, 1999. Neuwirth, E. "Some Strategies for Mastermind." Z. fu¨r Operations Research 26, B257-B278, 1982. Matching A matching on a GRAPH G is a set of edges of G such that no two of them share a vertex in common. The largest possible matching consists of n=2 edges, and such a matching is called a perfect matching. Although not all graphs have perfect matchings, a maximum matching exists for each graph. The maximum matching in a BIPARTITE GRAPH can be found using BipartiteMatching [g] in the Mathe- matica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ ). The maximum matching on a general graph can be found using MaximalMatch- ing[g] in the same package. See also BERGE’S THEOREM ,M ARRIAGE THEOREM , PERFECT MATCHING ,STABLE MARRIAGE PROBLEM References Hopcroft, J. and Karp, R. "An n5=2 Algorithm for Maximum Matching in Bipartite Graphs." SIAM J. Comput. , 225 /C1/ 31, 1975. Lova´sz, L. and Plummer, M. D. Matching Theory. Amster- dam, Netherlands: North-Holland, 1986. Skiena, S. "Matching." §6.4 in Implementing Discrete Mathe- matics: Combinatorics and Graph Theory with Mathema- tica. Reading, MA: Addison-Wesley, pp. 240 /C1/46, 1990. Match Problem Given n matches (i.e., rigid unit line segments), find the number of topologically distinct planar arrange- ments which can be made (Gardner 1991). In this problem, two matches laid end-to-end with no third match at their meeting point are considered equiva- lent to a single match, so triangles are equivalent to squares, n-match tails are equivalent to 1-match tails, etc. Solutions to the match problem are PLANAR TOPOLO- GICAL GRAPHS on e edges, and the first few values for e /C301, 1, 3, 5, 10, 19, 39, ... (Sloane’s A003055). See also CIGARETTES ,M ATCHSTICK GRAPH ,PLANAR GRAPH ,POLYNEMA ,TOPOLOGICAL GRAPH References Gardner, M. "The Problem of the Six Matches." In The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 79 /C1/1, 1991. Sloane, N. J. A. Sequences A003055/M2464 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Matchstick Construction Every point which can be constructed with a STRAIGHTEDGE and COMPASS , and no other points, can be constructed using identical matchsticks (i.e., identical movable line segments). Wells (1991) gives matchstick constructions which bisect a line segment and construct a SQUARE . See also GEOMETRIC CONSTRUCTION ,M ASCHERONI CONSTRUCTION ,N EUSIS CONSTRUCTION ,S TEINER CONSTRUCTION References Dawson, T. R. "‘Match-Stick’ Geometry." Math. Gaz. 23, 161 /C1/68, 1939.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 149, 1991. Matchstick Graph A PLANAR GRAPH whose EDGES are all unit line segments. The minimal number of EDGES for match- stick graphs of various degrees are given in the table below. The minimal degree 1 matchstick graph is a single EDGE , and the minimal degree 2 graph is an EQUILATERAL TRIANGLE . ne v 11 2 23 3 312 8 4 /542 Mathematical Induction INDUCTION Mathematics Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined. Whereas mathematics began merely as a calculational tool for computation and tabulation of quantities, it has blossomed into an extremely rich and diverse set of tools, terminologies, and approaches which range from the purely abstract to the utilitarian. Bertrand Russell once whimsically defined mathe- matics as "The subject in which we never know what we are talking about nor whether what we are saying is true" (Bergamini 1969). The term "mathematics" is often shortened to "math" in informal American speech and, consistent with the British penchant for adding superfluous letters, "maths" in British English. See also METAMATHEMATICS References Bergamini, D. Mathematics. New York: Time-Life Books, p. 9, 1969. Mathematics Contests There are several regular mathematics competitions available to students. The International Mathemati- cal Olympiad is perhaps the largest, while the William Lowell Putnam Competition is another im-portant contest. The International Mathematical Olympiad (IMO) is the yearly world championship of mathematics for high school students and is held in a different country each year. The first IMO was held in 1959 in Romania, but the contest has gradually expanded to include students from more than 80 different coun- tries. The William Lowell Putnam Mathematics Competi- tion is a North American math contest for college students. Each year, on the first Saturday in Decem- ber, more than 2000 students spend six hours in two sittings trying to solve 12 problems. The majority of the problems are very difficult, in the sense that their solution may require a nonstandard and creative approach. It is very rare for students to be able to solve all the problems, let alone the majority of them. The test can be taken both by individual and by teams, and the winners or their schools receive a small monetary compensation. Results for a given exam usually become available in early April of the following year. The International Mathematical Contest in Modeling (MCM) is a competition that challenges teams of undergraduate students to clarify, analyze, and propose solutions to open-ended problems. Problems are chosen with the advice of experts in industry and government, and the best papers are submitted to be published in professional journals. See also MATHEMATICS PRIZES ,UNSOLVED PROBLEMS References COMAP: The Consortium for Mathematics and Its Applica- tions. "Abut MCM." http://www.comap.com/undergradu- ate/contests/mcm/about.html. "International Mathematics Olympiad." http://imo.math.ca/ and http://olympiads.win.tue.nl/imo/. "William Lowell Putnam Competition." http://www.unl.edu/ amc/putnam/. Mathematics Prizes Several prizes are awarded periodically for outstand- ing mathematical achievement. There is no Nobel Prize in mathematics, and the most prestigious mathematical award is known as the FIELDS MEDAL . In rough order of importance, other awards are the $100,000 Wolf Prize of the Wolf Foundation of Israel, the Leroy P. Steele Prize of the American Mathema- tical Society, followed by the Boˆcher Memorial Prize, Frank Nelson Cole Prizes in Algebra and Number Theory, and the Delbert Ray Fulkerson Prize, all presented by the American Mathematical Society. The Clay Mathematics Institute of Cambridge, Mas- sachusetts (CMI) has named seven "Millennium Prize Problems," selected by focusing on important classic questions in mathematics that have resisted solution over the years. A $7 million prize fund has been established for the solution to these problems, with $1 million allocated to each. The problems consist of theR IEMANN HYPOTHESIS ,P OINCARE ´ CONJECTURE , HODGE CONJECTURE ,S WINNERTON- DYER CONJEC-TURE , solution of the Navier-Stokes equation, formu- lation of Yang-Mills theory , and determination of whether NP -PROBLEMS are actually P -PROBLEMS . See also FIELDS MEDAL ,M ATHEMATICS CONTESTS , UNSOLVED PROBLEMS ,W OLFSKEHL PRIZE References American Mathematical Society. "AMS Funds and Prizes." http://www.ams.org/secretary/prizes.html. Clay Mathematics Institute. "Millennium Prize Problems." http://www.claymath.org/prize_problems/. MacTutor History of Mathematics Archives. "The Fields Medal." http://www-groups.dcs.st-and.ac.uk/~history/So- cieties/FieldsMedal.html. "Winners of the Bo ˆcher Prize of the AMS." http://www-groups.dcs.st-and.ac.uk/~history/Societies/AMSBocherPrize.html. "Winners of the Frank Nelson Cole Prize of the AMS." http://www-groups.dcs.st- and.ac.uk/~history/Societies/AMSColePrize.html. MacTutor History of Mathematics Archives. "Mathematical Societies, Medals, Prizes, and Other Honours." http://www-groups.dcs.st-and.ac.uk/~history/Societies/. Monastyrsky, M. Modern Mathematics in the Light of the Fields Medals. Wellesley, MA: A. K. Peters, 1997. "Wolf Prize Recipients in Mathematics." http://www.aqua- net.co.il/wolf/wolf5.html. Mathematics Problems HILBERT’S PROBLEMS ,LANDAU’S PROBLEMS ,PROBLEM MathieuC MATHIEU FUNCTION MathieuCharacteristicA MATHIEU CHARACTERISTIC EXPONENT MathieuCharacteristicB MATHIEU CHARACTERISTIC EXPONENT Mathieu Characteristic Exponent MATHIEU CHARACTERISTIC EXPONENT MathieuCPrime MATHIEU FUNCTION Mathieu Differential Equation d2V dv2/C27[a/C282qcos(2 v)]V/C300 (1) (Abramowitz and Stegun 1972; Zwillinger 1997, p. 125), having solution y/C30C1C(a;q;v)/C27C2S(a;q;v); (2) where C(a;q;v) and S(a;q;v) are M ATHIEU FUNC- TIONS . The equation arises in separation of variables of the H ELMHOLTZ DIFFERENTIAL EQUATION inELLIP- TIC CYLINDRICAL COORDINATES . Whittaker and Wat- son (1990) use a slightly different form to define the MATHIEU FUNCTIONS . The modified Mathieu differential equation d2U du2 /C28[a /C282q cosh(2 u)]U /C300 (3) (Iyanaga and Kawada 1980, p. 847; Zwillinger 1997, p. 125) arises in SEPARATION OF VARIABLES of the HELMHOLTZ DIFFERENTIAL EQUATION in ELLIPTIC CYLINDRICAL COORDINATES , and has solutions y /C30C1C(a ; q;/C28iu) /C27C2S(a; q /C28iu) : (4) The associated Mathieu differential equation is given by yƒ/C27[(1 /C282r) cot x]y?/C27(a /C27k2 cos2 x)y /C300 (5) (Ince 1956, p. 403; Zwillinger 1997, p. 125). See also HILL’S DIFFERENTIAL EQUATION ,M ATHIEU FUNCTION ,W HITTAKER- HILL DIFFERENTIAL EQUA- TION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 722, 1972. Campbell, R. The´orie ge´ne´rale de l’e´quation de Mathieu et de quelques autres e´quations diffe´rentielles de la me´canique. Paris: Masson, 1955. Ince, E. L. Ordinary Differential Equations. New York: Dover, 1956. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 847, 1980. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 556 /C1/57, 1953. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 125, 1997. Mathieu Function The Mathieu functions are the solutions to the MATHIEU DIFFERENTIAL EQUATION d2V dv2 /C28[a /C282q cos(2 v)]V /C300: (1) Even solutions are denoted C(a ; q ; z) and odd solu- tions by S(a ; q ; z): These are returned by the Math- ematica functions MathieuC [a, q, z] and MathieuS [a, q, z], respectively. These functions appear in physical problems involving elliptical shapes or periodic potentials. The Mathieu functions have the special values C(a; 0; z) /C30cos(ffiffiffiapz) (2)S(a ; 0; z) /C30sin(ffiffiffiapz) : (3) For nonzero q, the Mathieu functions are only periodic in z for certain values of a. Such character- istic values are given by the Mathematica functions MathieuCharacteristicA [r, q] and Mathieu- CharacteristicB [r, q] with r an integer or rational number. These values are often denoted ar and br : For integer r, the even and odd Mathieu functions with characteristic values arand brare often denoted cer(z ; q) and ser(z; q) ; respectively (Abramowitz and Stegun 1972, p. 725). The left plot above shows ar for r /C300, 1, ..., 4 and the right plot shows br for r /C301, ..., 4. Whittaker and Watson (1990, p. 405) define the Mathieu function based on the equation d2u dz2 /C27[a /C2716q cos(2 z)]u /C300: (4) This equation is closely related to HILL’S DIFFEREN- TIAL EQUATION . For an EVEN Mathieu function, G(h)/C30lgp /C28pekcoshcosuG(u)du; (5) where k/C13ffiffiffiffiffiffiffiffi32qp:For an ODD Mathieu function, G(h)/C30lgp /C28psin(ksinhsinu)G(u)du: (6) Both EVEN and ODD functions satisfy G(h)/C30lgp /C28peiksinhsinuG(u)du: (7) Letting z/C13cos2ztransforms the M ATHIEU DIFFEREN- TIAL EQUATION to 4z(1/C28z)d2u dz2/C272(1/C282z)du dz/C27(a/C2816q/C2732qz)u/C300: (8) See also MATHIEU CHARACTERISTIC EXPONENT , MATHIEU DIFFERENTIAL EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). "Mathieu Func- tions." Ch. 20 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 721 /C1/46, 1972. Gradshteyn, I. S. and Ryzhik, I. M. "Mathieu Functions." §6.9 and 8.6 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 800 /C1/04 and 1006/C1/013, 2000. Humbert, P. Fonctions de Lame ´ et Fonctions de Mathieu. Paris: Gauthier-Villars, 1926. Mechel, F. P. Mathieu Functions: Formulas, Generation, Use. Stuttgart, Germany: Hirzel, 1997. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 562 /C1/68 and 633 /C1/42, 1953. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Mathieu Groups The first SIMPLE SPORADIC GROUPS discovered. M11 ; M12 ; M22 ; M23 ; M24 were discovered in 1861 and 1873 by Mathieu. Frobenius showed that all the Mathieu groups are SUBGROUPS of M24 :/ The Mathieu groups are most simply defined as AUTOMORPHISM GROUPS of STEINER SYSTEMS , as sum- marized in the following table. Mathieu group Steiner system /M11// S(4; 5; 11) / /M12// S(5; 6; 12) / /M22// S(3; 6; 22) / /M23// S(4; 7; 23) / /M24// S(5; 8; 24) / /M11 and M23 are TRANSITIVE PERMUTATION GROUPS of 11 and 23 elements. The ORDERS of the Mathieu groups are M11jj/C3024/C21532/C2155/C21511 M12jj/C3026/C21533/C2155/C21511 M22jj/C3027/C21532/C2155/C2157/C21511 M23jj/C3027/C21532/C2155/C2157/C21511 /C21523 M24jj/C30210/C21533/C2155/C2157/C21511 /C21523: See also AUTOMORPHISM GROUP ,S IMPLE GROUP , SPORADIC GROUP ,S TEINER SYSTEM ,T RANSITIVE GROUP ,W ITT GEOMETRYReferences Conway, J. H. and Sloane, N. J. A. "The Golay Codes and the Mathieu Groups." Ch. 11 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 299 /C1/30, 1993. Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Rotman, J. J. Ch. 9 in An Introduction to the Theory of Groups, 4th ed. New York: Springer-Verlag, 1995. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#spo. MathieuS MATHIEU FUNCTION MathieuSPrime MATHIEU FUNCTION Matrix The TRANSFORMATION given by the system of equa- tions x?1/C30a11x1/C27a12x2/C27.../C27a1nxn x?2/C30a21x1/C27a22x2/C27.../C27a2nxn n x?m/C30am1x1/C27am2x2/C27.../C27amnxn is denoted by the MATRIX EQUATION x?1 x?2 n x?m2 6643 775/C30a11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n nn:::n am1am2/C1/C1/C1 amn2 6643 775x1 x2 n xn2 6643 775: In concise notation, this could be written x?/C30Ax; where x?andxare VECTORS andAis called an m/C29n matrix. An m/C29nmatrix consists of mrows and n columns, and the set of m/C29nmatrices with real coefficients is sometimes denoted Rm/C29n:To remember which index refers to which direction, identify the indices of the last (i.e., lower right) term, so the indices m, n of the last element in the above matrix identifies it as an m/C29nmatrix. A matrix is said to be SQUARE ifm/C30n, and RECTAN- GULAR ifm"n:Anm/C291 matrix is called a COLUMN VECTOR , and a 1 /C29nmatrix is called a ROW VECTOR . Special types of SQUARE MATRICES include the IDEN- TITY MATRIX /I;with A2A3(where dijis the K RONECKER DELTA ) and the DIAGONAL MATRIX aij/C30cidij(where ci are a set of constants). For every linear transformation there exists one and only one corresponding matrix. Conversely, every matrix corresponds to a unique linear transforma- tion. The matrix is an important concept in mathe-matics, and was first formulated by Sylvester and Cayley. Two matrices may be added (MATRIX ADDITION )or multiplied (MATRIX MULTIPLICATION ) together to yield a new matrix. Other common operations on a single matrix are diagonalization, inversion (MATRIX IN- VERSE ), and transposition (matrix TRANSPOSE ). The DETERMINANT det(A)or½A½ of a matrix A is a very important quantity which appears in many diverse applications. Matrices provide a concise notation which is extremely useful in a wide range of problems involving linear equations (e.g., LEAST SQUARES FIT- TING ). See also ADJACENCY MATRIX ,A DJUGATE MATRIX , ALTERNATING SIGN MATRIX ,ANTISYMMETRIC MATRIX , BLOCK MATRIX ,BOHR MATRIX ,BOURQUE- LIGH CON- JECTURE ,CARTAN MATRIX ,CIRCULANT MATRIX ,CON- DITION NUMBER ,C RAMER’S RULE,D ETERMINANT , DIAGONAL MATRIX ,DIRAC MATRICES ,EIGENVECTOR , ELEMENTARY MATRIX ,ELEMENTARY ROW AND COL- UMN OPERATIONS ,E QUIVALENT MATRIX ,F OURIER MATRIX ,GRAM MATRIX ,HILBERT MATRIX ,HYPERMA- TRIX,IDENTITY MATRIX ,ILL-CONDITIONED MATRIX , INCIDENCE MATRIX ,IRREDUCIBLE MATRIX ,KAC MA- TRIX,LEAST COMMON MULTIPLE MATRIX ,LUD ECOM- POSITION ,M ARKOV MATRIX ,M ATRIX ADDITION , MATRIX DECOMPOSITION THEOREM ,MATRIX INVERSE , MATRIX MULTIPLICATION ,M CCOY’S THEOREM ,M INI- MAL MATRIX ,N ORMAL MATRIX ,P AULI MATRICES , PERMUTATION MATRIX ,POSITIVE DEFINITE MATRIX , RANDOM MATRIX ,RATIONAL CANONICAL FORM,RE- DUCIBLE MATRIX ,R OTH’S REMOVAL RULE,S HEAR MATRIX ,SINGULAR MATRIX ,SKEW SYMMETRIC MA- TRIX,SMITH NORMAL FORM,SPARSE MATRIX ,SPECIAL MATRIX ,SQUARE MATRIX ,STOCHASTIC MATRIX ,SUB- MATRIX ,SYMMETRIC MATRIX ,TOURNAMENT MATRIX References Arfken, G. "Matrices." §4.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 176 /C1/ 91, 1985. Bapat, R. B. Linear Algebra and Linear Models, 2nd ed. New York: Springer-Verlag, 2000. Frazer, R. A.; Duncan, W. J.; and Collar, A. R. Elementary Matrices and Some Applications to Dynamics and Differ- ential Equations. Cambridge, England: Cambridge Uni- versity Press, 1955. Lu¨tkepohl, H. Handbook of Matrices. New York: Wiley, 1996. Meyer, C. D. Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000. Zhang, F. Matrix Theory: Basic Results and Techniques. New York: Springer-Verlag, 1999. Matrix Addition Denote the sum of two MATRICES A and B (of the same dimensions) by C /C30A /C27B : The sum is defined by adding entries with the same indices cij /C13aij /C27bij over all i and j. For example,a11a12 a21a22fflC}{fflC}z /C27b11b12 b21b22fflC}{fflC}z /C30a11 /C27b11a12 /C27b12 a21 /C27b21a22 /C27b22fflC}{fflC}z : Matrix addition is therefore both COMMUTATIVE and ASSOCIATIVE . See also MATRIX ,MATRIX MULTIPLICATION Matrix Decomposition Matrix decomposition refers to the transformation of a given matrix (often assumed to be a SQUARE MATRIX ) into a given canonical form. See also CHOLESKY DECOMPOSITION ,JORDAN MATRIX DECOMPOSITION ,M ATRIX DECOMPOSITION THEOREM , LQ DECOMPOSITION ,LUD ECOMPOSITION ,ORTHOGO- NAL DECOMPOSITION ,QRD ECOMPOSITION ,S CHUR DECOMPOSITION ,SINGULAR VALUE DECOMPOSITION Matrix Decomposition Theorem Let P be a MATRIX of EIGENVECTORS of a given MATRIX A and D a MATRIX of the corresponding EIGENVALUES . Then A can be written A /C30PDP /C281 ; (1) where D is a DIAGONAL MATRIX and the columns of P are ORTHOGONAL VECTORS .If P is not a SQUARE MATRIX , then it cannot have a MATRIX INVERSE . However, if P is m /C29n (with m /C21n), then A can be written using a so-called SINGULAR VALUE DECOMPO- SITION OF THE FORM A /C30UDVT ; (2) where U and V are n /C29n SQUARE MATRICES with ORTHOGONAL columns so that UTU /C30VTV /C301: (3) See also SINGULAR VALUE DECOMPOSITION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Singular Value Decomposition." §2.6 in Nu- merical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 51 /C1/3, 1992. Matrix Diagonalization Diagonalizing a MATRIX is equivalent to finding the EIGENVECTORS and EIGENVALUES . The EIGENVALUES make up the entries of the diagonalized MATRIX , and the EIGENVECTORS make up the new set of axes corresponding to the DIAGONAL MATRIX . See also DIAGONAL MATRIX ,EIGENVALUE ,EIGENVEC- TOR References Arfken, G. "Diagonalization of Matrices." §4.6 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 217 /C1/29, 1985. Matrix Direct Product The matrix direct product gives the MATRIX of the LINEAR TRANSFORMATION induced by the TENSOR PRODUCT of the original VECTOR SPACES . More pre- cisely, suppose that S : V1 0 W1 (1) and T : V2 0 W2 (2) are given by S(x) /C30Ax and T(y) /C30By: Then S /C156T : V1 /C156V2 0 W1 /C156W2 (3) is determined by S /C156T(x /C156y) /C30(Ax) /C156(By) /C30(A /C156B)(x /C156y): (4) Given an m /C29n MATRIX A and a p /C29q MATRIX B; their direct product C /C30A /C156B is an (mp) /C29(nq) MATRIX with elements defined by cab /C30aijbkl ; (5) where a /C13p(i /C281) /C27k (6) b /C13q(j /C281) /C27l: (7) In Mathematica , the matrix direct product can be formed using the following code. BBLinearAlgebra‘MatrixManipulation‘; MatrixDirectProduct[a_List?MatrixQ, b_List?MatrixQ] : /C30 BlockMatrix[Outer[Times, a, b]] ] For example, the matrix direct product of the 2 /C292 MATRIX A and the 3 /C292 MATRIX B is given by the following 6 /C294 MATRIX , A /C156B /C30a11B a12B a21B a22BfflC}{fflC}z (8) /C30a11b11a11b12a12b11a12b12 a11b21a11b22a12b21a12b22 a11b31a11b32a12b31a12b32 a21b11a21b12a22b11a22b12 a21b21a21b22a22b21a22b22 a21b31a21b32a22b31a22b322 66666643 7777775: (9) See also D IRECT PRODUCT ,M ATRIX MULTIPLICATION , TENSOR DIRECT PRODUCTReferences Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 12, 1996. Matrix Direct Sum The construction of a BLOCK MATRIX from a set of SQUARE MATRICES , i.e., /C156n i /C301 Ai/C30diag( A1 ; A2 ; ...; A n) /C30A1 A2::: An2 6643 775: See also B LOCK MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 13 /C1/4, 1962. Matrix Equality Two MATRICES A and B are said to be equal IFF aij /C13bij for all i, j. Therefore, 12 34fflC}{fflC}z /C301234fflC}{fflC}z ; while 1234fflC}{fflC}z "0234fflC}{fflC}z : See also E QUIVALENT MATRIX Matrix Equation Nonhomogeneous matrix equations OF THE FORM Ax/C30b (1) can be solved by taking the MATRIX INVERSE to obtain x/C30A/C281b: (2) This equation will have a nontrivial solution IFFthe DETERMINANT det(A)"0:In general, more numeri- cally stable techniques of solving the equation include GAUSSIAN ELIMINATION ,L U DECOMPOSITION , or the SQUARE ROOT METHOD . For a homogeneous n/C29nMATRIX equation a11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n nn:::n an1an2/C1/C1/C1 ann2 6643 775x 1 x2 n xn2 6643 775/C300 0 n 02 6643 775(3) to be solved for the x i/s, consider the DETERMINANT a11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n nn::: n an1an2/C1/C1/C1 annfflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}: (4) Now multiply by x 1 ; which is equivalent to multi- plying the first column (or any column) by x1 ; x1a11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n nn::: n an1an2/C1/C1/C1 annfflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}/C30a 11x1a12 /C1/C1/C1 a1n a21x1a22 /C1/C1/C1 a2n nn::: n an1x1an2/C1/C1/C1 annfflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}: (5) The value of the DETERMINANT is unchanged if multiples of columns are added to other columns. So add x2times column 2, ..., and xntimes column n to the first column to obtain x1a11a12 /C1/C1/C1 a1n a21a22 /C1/C1/C1 a2n nn::: n an1an2/C1/C1/C1 annfflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz} /C30a 11x1 /C27a12x2 /C27.../C27a1nxna12 /C1/C1/C1 a1n a21x1 /C27a22x2 /C27.../C27a2nxna22 /C1/C1/C1 a2n nn::: n an1x1 /C27an2x2 /C27.../C27annxnan2/C1/C1/C1 annfflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}: (6) But from the original MATRIX , each of the entries in the first columns is zero since ai1x1 /C27ai2x2 /C27.../C27ainxn /C300; (7) so 0 a12 /C1/C1/C1 a1n 0 a22 /C1/C1/C1 a2n nn ::: n 0 an2/C1/C1/C1 annfflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}/C30 0: (8) Therefore, if there is an x 1 "0 which is a solution, the DETERMINANT is zero. This is also true for x2 ; ..., xn ; so the original homogeneous system has a nontrivial solution for all xi/s only if the DETERMINANT is 0. This approach is the basis for CRAMER’S RULE . Given a numerical solution to a matrix equation, the solution can be iteratively improved using the follow- ing technique. Assume that the numerically obtained solution to Ax /C30b (9) is x1 /C30x /C27 dx1 ; where dx1is an error term. The first solution therefore gives Ax1/C30A(x/C27dx1)/C30b/C27db (10) Adx1/C30db; (11) where dbis found by solving (10) db/C30Ax1/C28b: (12) Combining (11) and (12) then givesdx1/C30A/C281db/C30A/C281(Ax1/C28b)/C30x1/C28A/C281b: (13) See also CRAMER’S RULE,GAUSSIAN ELIMINATION ,LU DECOMPOSITION ,MATRIX ,M ATRIX ADDITION ,M ATRIX INVERSE ,M ATRIX MULTIPLICATION ,N ORMAL EQUA- TION ,SQUARE ROOT METHOD MatrixExp MATRIX EXPONENTIAL Matrix Exponential The POWER SERIES that defines the EXPONENTIAL MAP exalso defines a map between MATRICES . In particu- lar, exp(A)/C13eA/C30X/C12 n/C300An n!(1) /C30I/C27A/C27AA 2!/C27AAA 3!/C27...; (2) converges for any SQUARE MATRIX A, where Iis the IDENTITY MATRIX . The matrix exponential is imple- mented in Mathematica asMatrixExp [m]. In some cases, it is a simple matter to express the exponent. For example, when Ais a DIAGONAL MATRIX , exponentiation can be performed simply by exponentiating each of the diagonal elements. Forexample, given a diagonal matrix A/C30a 10 /C1/C1/C1 0 0a2/C1/C1/C1 0 nn:::n 00 /C1/C1/C1 ak2 6643 775; (3) The matrix exponential is given by exp(A)/C30e a10 /C1/C1/C1 0 0ea2/C1/C1/C1 0 nn:::n 00 /C1/C1/C1 eak2 6643 775: (4) Since most matrices are DIAGONALIZABLE , it is easiest to diagonalize the matrix before exponentiating it. When Ais a NILPOTENT MATRIX , the exponential is given by a MATRIX POLYNOMIAL because some power ofAvanishes. For example, when A/C300xz 00 y 0002 435; (5) then exp(A)/C301xz/C27 1 2xy 01 y 00 12 435 (6) andA 3/C300:/ For the ZERO MATRIX A /C300; e0 /C30I ; (7) i.e., the IDENTITY MATRIX . In general, eAe/C28A /C30e0 /C30I ; (8) so the exponential of a matrix is always invertible, with inverse the exponent of the negative of the matrix. However, in general, the formula eAeB /C30eA /C27B (9) holds only when A and B COMMUTE , i.e., [A; B] /C30AB /C28BA /C300: (10) For example, exp0 /C28x 00fflC}{fflC}z /C2700 x 0fflC}{fflC}zfflCzrfflCzD /C30cos x /C28sin x sin x cos xfflC}{fflC}z ; (11) while exp0 /C28x 00fflC}{fflC}zfflCzrfflCzD exp00 x 0fflC}{fflC}zfflCzrfflCzD /C301 /C28x 01fflC}{fflC}z 10 x 1fflC}{fflC}z /C301 /C28x2/C28x x 1fflC}{fflC}z : (12) See also EXPONENTIAL FUNCTION ,EXPONENTIAL MAP, MATRIX ,MATRIX POWER Matrix Fraction A pair of matrices ND/C281 or D /C281N ; where N is the matrix NUMERATOR and D is the DENOMINATOR . See also FRACTION Matrix Group A GROUP in which the elements are SQUARE MATRICES , the group multiplication law is MATRIX MULTIPLICA- TION , and the group inverse is simply the MATRIX INVERSE . Every matrix group is equivalent to a unitary matrix group (Lomont 1987, pp. 47 /C1/8). See also MASCHKE’S THEOREM References Lomont, J. S. "Matrix Groups." §3.1 in Applications of Finite Groups. New York: Dover, pp. 46 /C1/2, 1987. Matrix Inverse The inverse of a SQUARE MATRIX A; sometimes called a reciprocal matrix, is a matrix A/C281 such that AA /C281 /C30I; (1) where I is the IDENTITY MATRIX . Courant and Hilbert (1989, p. 10) use the notation ˘A to denote the inverse matrix.A SQUARE MATRIX A has an inverse IFF the DETERMI- NANT ½A ½"0 (Lipschutz 1991, p. 45) A matrix posses- sing an inverse is called NONSINGULAR , or invertible. The matrix inverse of a SQUARE MATRIX m may be taken in Mathematica using the function Inver- se[m]. For a 2 /C292 MATRIX A /C13ab cdfflC}{fflC}z ; (2) the inverse is A/C281 /C301 ½A½d /C28b /C28cafflC}{fflC}z /C301 ad/C28bcd /C28b /C28cafflC}{fflC}z : (3) For a 3 /C293 MATRIX , A/C281 /C301 ½A½a22a23 a32a33fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}a 13a12 a33a32fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}a 12a13 a22a23fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz} a 23a21 a33a31fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}a 11a13 a31a33fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}a 13a11 a23a21fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz} a 21a22 a31a32fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}a 12a11 a32a31fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}a 11a12 a21a22fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}2 666666643 77777775: (4) A general n /C29n matrix can be inverted using methods such as the G AUSS- JORDAN ELIMINATION ,GAUSSIAN ELIMINATION ,orLU DECOMPOSITION . The inverse of a PRODUCT AB of MATRICES A and B can be expressed in terms of A /C281 and B/C281 : Let C /C13AB : (5) Then B/C30A/C281AB/C30A/C281C (6) and A/C30ABB/C281/C30CB/C281: (7) Therefore, C/C30AB/C30(CB/C281)(A/C281C)/C30CB/C281A/C281C; (8) so CB/C281A/C281/C30I; (9) where Iis the IDENTITY MATRIX , and B/C281A/C281/C30C/C281/C30(AB)/C281: (10) See also GAUSS- JORDAN ELIMINATION ,G AUSSIAN ELIMINATION ,LUD ECOMPOSITION ,M ATRIX ,M ATRIX ADDITION ,M ATRIX MULTIPLICATION ,M OORE- PEN- ROSE GENERALIZED MATRIX INVERSE ,NONSINGULAR MATRIX ,SINGULAR MATRIX ,STRASSEN FORMULAS References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 11, 1962. Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications. New York: Wiley, 1977. Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989. Lipschutz, S. "Invertible Matrices." Schaum’s Outline of Theory and Problems of Linear Algebra, 2nd ed. New York: McGraw-Hill, pp. 44 /C1/5, 1991. Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 24 /C1/6, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Is Matrix Inversion an /N3/ Process?" §2.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 95 /C1/8, 1992. Rosser, J. B. "A Method of Computing Exact Inverses of Matrices with Integer Coefficients." J. Res. Nat. Bur. Standards Sect. B. 49, 349 /C1/58, 1952. Matrix Multiplication The product C of two MATRICES A and B is defined by cik /C30aijbjk ; (1) where j is summed over for all possible values of i and k. Therefore, in order for multiplication to be defined, the dimensions of the MATRICES must satisfy (n /C29m)(m /C29p) /C30(n /C29p) ; (2) where (a /C29b) denotes a MATRIX with a rows and b columns. Writing out the product explicitly, c11c12 /C1/C1/C1 c1p c21c22 /C1/C1/C1 c2p nn::: n cn1cn2/C1/C1/C1 cnp2 6643 775 /C30a 11a12 /C1/C1/C1 a1m a21a22 /C1/C1/C1 a2m nn::: n an1an2/C1/C1/C1 anm2 6643 775b 11b12 /C1/C1/C1 b1p b21b22 /C1/C1/C1 b2p nn::: n bm1bm2/C1/C1/C1 bmp2 6643 775; (3) where c 11 /C30a11b11 /C27a12b21 /C27.../C27a1mbm1 c12 /C30a11b12 /C27a12b22 /C27.../C27a1mbm2 c1p /C30a11b1p /C27a12b2p /C27.../C27a1mbmp c21 /C30a21b11 /C27a22b21 /C27.../C27a2mbm1 c22 /C30a21b12 /C27a22b22 /C27.../C27a2mbm2 c2p /C30a21b1p /C27a22b2p /C27.../C27a2mbmp cn1 /C30an1b11 /C27an2b21 /C27.../C27anmbm1 cn2 /C30an1b12 /C27an2b22 /C27.../C27anmbm2 cnp /C30an1b1p /C27an2b2p /C27.../C27anmbmp : Matrix multiplication is ASSOCIATIVE , as can be seen by taking [(ab)c]ij /C30(ab)ikckj /C30(ailblk)ckj : (4) Now, since ail ; blk ; and ckjare SCALARS , use the ASSOCIATIVITY of SCALAR MULTIPLICATION to write (ailblk)ckj /C30ail(blkckj) /C30ail(bc)lj /C30[a(bc)]ij : (5)Since this is true for all i and j, it must be true that (ab)c /C30a(bc) : (6) That is, matrix multiplication is ASSOCIATIVE . How- ever, matrix multiplication is not, in general, COM- MUTATIVE (although it is COMMUTATIVE ifAandBare DIAGONAL and of the same dimension). The product of two BLOCK MATRICES is given by multiplying each block oo oo o ooo ooo ooo2 66666643 7777775xx xx x xxx xxx xxx2 66666643 7777775 /C30oo oofflC}{fflC}z xxxxfflC}{fflC}z [o][x] ooo oooooo2 435xxx xxxxxx2 4352 6666643 777775: (7) See also L INEAR TRANSFORMATION ,M ATRIX ,M ATRIX ADDITION ,MATRIX INVERSE ,STRASSEN FORMULAS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 178 /C1/79, 1985. Higham, N. "Exploiting Fast Matrix Multiplication within the Level 3 BLAS." ACM Trans. Math. Soft. 16, 352/C1/68, 1990. Matrix Norm Given a SQUARE MATRIX Awith COMPLEX (or REAL ) entries, a MATRIX NORM ½A½is a NONNEGATIVE number associated with Ahaving the properties 1.½½A½½>0when A"0and½½A½½/C300IFFA/C300;/ 2.½½kA½½/C30½k½½½A½½for any SCALAR k, 3.½½A/C27B½½5½½A½½/C27½½B½½;/ 4.½½AB½½5½½A½½½½B½½/ For an n/C29nMATRIX Aand an n/C29nUNITARY MATRIX U; ½½AU½½/C30½½UA½½/C30½½A½½: Letl1;...,lnbe the EIGENVALUES ofA;then 1 ½½A/C281½½5½l½5½½A½½: The MAXIMUM ABSOLUTE COLUMN SUM NORM ½½A½½1; SPECTRAL NORM ½½A½½2;and MAXIMUM ABSOLUTE ROW SUM NORM ½½A½½/C12satisfy ½½A½½2 25½½A½½15½½A½½/C12: Matrix norms are implemented as MatrixNorm [m, p] in the Mathematica add-on package LinearAl- gebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ), where p /C30 1, 2, or /C12:/ For a SQUARE MATRIX , the SPECTRAL NORM , which is the SQUARE ROOT of the maximum EIGENVALUE of A /C31A (where A/C31 is the ADJOINT MATRIX ), is often referred to as "the" matrix norm. See also COMPATIBLE ,HILBERT- SCHMIDT NORM,MAX- IMUM ABSOLUTE COLUMN SUM NORM,M AXIMUM ABSOLUTE ROW SUM NORM,NATURAL NORM,NORM, POLYNOMIAL NORM,S PECTRAL NORM,S PECTRAL RADIUS ,VECTOR NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1114 /C1/125, 2000. Matrix p-Norm MATRIX NORM Matrix Polynomial A polynomial with matrix coefficients. An nth order matrix polynomial in a variable t is given by P(t) /C30A0 /C27A1 t/C27A2 t2/C27.../C27A ntn; (1) where Ak are p /C29p square matrices. If the entries of the matrices are real independent variates with a standard normal distribution, then the expected number of real solutions is given by En; p /C30ffiffiffippEG(1 2(p /C27 1)) G(1 2 p); (2) where En /C30ffiffiffi 2pPn=2 /C281 k /C300(4k /C28 1)!! (4k)!!for n even 1 /C27ffiffiffi 2pP(n/C281)=2 k /C301(4k /C28 3)!! (4k /C28 2)!!for n odd8 >>>< >>>:(3) (Edelman and Kostlan 1995). See also C AYLEY- HAMILTON THEOREM ,M ATRIX POWER ,NILPOTENT MATRIX ,POLYNOMIAL MATRIX References Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32,1/C1/7, 1995. Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 13, 1958. Matrix Polynomial Identity CAYLEY- HAMILTON THEOREMMatrix Power The power An of a MATRIX A for n a nonnegative integer is defined as the MATRIX PRODUCT of n copies of A ; An/C30A /C1/C1/C1A|fflfflffl{zfflfflffl} n: A matrix to the zeroth power is defined to be the IDENTITY MATRIX of the same dimensions, A0 /C30I : The MATRIX INVERSE is commonly denoted A /C281 ; which should not be interpreted to mean 1=A :/ See also MATRIX EXPONENTIAL ,M ATRIX MULTIPLICA- TION ,M ATRIX POLYNOMIAL ,N ILPOTENT MATRIX , PERIODIC MATRIX Matrix Product The result of a MATRIX MULTIPLICATION . See also PRODUCT Matrix Transpose TRANSPOSE Matrix Tree Theorem The number of nonidentical SPANNING TREES of a GRAPH G is equal to any COFACTOR of the DEGREE MATRIX of G minus the ADJACENCY MATRIX of G (Skiena 1990, p. 235). See also SPANNING TREE References Chaiken, S. "A Combinatorial Proof of the All-Minors Matrix Tree Theorem." SIAM J. Alg. Disc. Methods 3, 319/C1/29, 1982. Kirchhoff, G. "U ¨ber die Auflo ¨sung der Gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer Stro ¨me gefu ¨hrt wird." Ann. Phys. Chem. 72, 497/C1/08, 1847. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 235, 1990. Matroid Roughly speaking, a matroid is a finite set together with a generalization of a concept from linear algebra that satisfies a natural set of properties for that concept. For example, the finite set could be the rowsof a MATRIX , and the generalizing concept could be linear dependence and independence of any subset of rows of the MATRIX . Formally, a matroid consists of a finite set Mof elements together with a family C/C30fC1;C1;...gof nonempty subsets of M, called circuits, which satisfy the axioms 1. No PROPER SUBSET of a circuit is a circuit, 2. If x /C23 C1 S C2and C1 "C2 ; then C1 @ C2 /C28fxg contains a circuit. (Harary 1994, p. 40). An equivalent definition considers a matroid as a finite set M of elements together with a family of subsets of M, called independent sets, such that 1. The EMPTY SET is independent, 2. Every SUBSET of an independent set is indepen- dent, 3. For every subset A of M, all maximal indepen- dent sets contained in A have the same number of elements. (Harary 1994, pp. 40 /C1/1). The number of simple matroids (or COMBINATORIAL GEOMETRIES ) with n /C300, 1, ... points are 1, 1, 2, 4, 9, 26, 101, 950, ... (Sloane’s A002773), and the number of matroids on n /C300, 1, ... points are 1, 2, 4, 8, 17, 38, 98, 306, 1724, ... (Sloane’s A055545; Oxley 1993, p. 473). (The value for n /C305 given by Oxley 1993, p. 42, is incorrect.) See also COMBINATORIAL GEOMETRY ,G RAPHOID , ORIENTED MATROID References Bjo¨rner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; and Ziegler, G. Oriented Matroids, 2nd ed. Cambridge, Eng- land: Cambridge University Press, 1999. Blackburn, J. E.; Crapo, H. H.; and Higgs, D. A. "A Catalo- gue of Combinatorial Geometries." Math. Comput. 27, 155 /C1/66, 1973. Crapo, H. H. and Rota, G.-C. "On the Foundations of Combinatorial Theory. II. Combinatorial Geometries." Cambridge, MA: MIT Press, 109 /C1/33, 1970. Harary, F. "Matroids." Graph Theory. Reading, MA: Addi- son-Wesley, pp. 40 /C1/1, 1994. Minty, G. "On the Axiomatic Foundations of the Theories of Directed Linear Graphs, Electric Networks, and Network- Programming." J. Math. Mech. 15, 485 /C1/20, 1966. Oxley, J. G. Matroid Theory. Oxford, England: Oxford University Press, 1993. Papadimitriou, C. H. and Steiglitz, K. Combinatorial Opti- mization: Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall, 1982. Richter-Gebert, J. and Ziegler, G. M. In Handbook of Discrete and Computational Geometry (Ed. J. E. Good- man and J. O’Rourke). Boca Raton, FL: CRC Press, pp. 111 /C1/12, 1997. Sloane, N. J. A. Sequences A002773/M1197 and A055545 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M1197 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Tutte, W. T. "Lectures on Matroids." J. Res. Nat. Bur. Stand. Sect. B 69,1/C1/7, 1965. Whitely, W. "Matroids and Rigid Structures." In Matroid Applications, Encyclopedia of Mathematics and Its Appli- cations (Ed. N. White), Vol. 40. New York: Cambridge University Press, pp. 1 /C1/3, 1992.Whitney, H. "On the Abstract Properties of Linear Depen- dence." Amer. J. Math. 57, 509 /C1/33, 1935. Maurer Rose /n /C304; d /C30120; n /C306 ; d /C3072: A Maurer rose is a plot of a "walk" along an n- (or 2n/-) leafed ROSE in steps of a fixed number d degrees, including all cosets. See also STARR ROSE References Maurer, P. "A Rose is a Rose..." Amer. Math. Monthly 94, 631 /C1/45, 1987. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 96 /C1/02, 1991. Max MAXIMUM Maximal Ideal A maximal ideal of a RING R is an IDEAL I, not equal to R, such that there are no IDEALS "in between" I and R. In other words, if J is an IDEAL which contains I as a SUBSET , then either J /C30I or J /C30R. For example, nZ is a maximal ideal of Z IFF n is PRIME , where Z is the RING of INTEGERS . Only in a LOCAL RING is there just one maximal ideal. For instance, in the integers, a/C30 phiis a maximal ideal whenever p is prime. A maximal ideal m is always a PRIME IDEAL , and the QUOTIENT RING A=m is always a FIELD . In general, not all prime ideals are maximal. See also IDEAL ,M AXIMAL IDEAL THEOREM ,PRIME IDEAL ,QUOTIENT RING,REGULAR LOCAL RING,RING Maximal Ideal Theorem The proposition that every PROPER IDEAL of a BOO- LEAN ALGEBRA can be extended to a MAXIMAL IDEAL .It is equivalent to the BOOLEAN REPRESENTATION THE- OREM , which can be proved without using the AXIOM OF CHOICE (Mendelson 1997, p. 121). See also BOOLEAN REPRESENTATION THEOREM References Lo´s, J. "Sur la the´ore`me de Go¨del sur les theories inde´- nombrables." Bull. de l’Acad. Polon. des Sci. 3, 319 /C1/20, 1954. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 121, 1997. Rasiowa, H. and Sikorski, R. "A Proof of the Completeness Theorem of Go¨del." Fund. Math. 37, 193 /C1/00, 1951. Rasiowa, H. and Sikorski, R. "A Proof of the Skolem- Lo¨wenheim Theorem." Fund. Math. 38, 230 /C1/32, 1952. Maximally Linearly Independent A set of VECTORS is maximally linearly independent if including any other VECTOR in the VECTOR SPACE would make it LINEARLY DEPENDENT (i.e., if any other VECTOR in the SPACE can be expressed as a LINEAR COMBINATION of elements of a maximal set–the BASIS ). See also BASIS,L INEARLY DEPENDENT VECTORS , VECTOR ,VECTOR SPACE Maximal Sum-Free Set A maximal sum-free set is a set fa1 ; a2 ; ...; an g of distinct NATURAL NUMBERS such that a maximum l of them satisfy aij/C27aik"am for 1 5j Bk 5l ; 1 5m 5n :/ See also MAXIMAL ZERO-SUM-FREE SET References Guy, R. K. "Maximal Sum-Free Sets." §C14 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 128 /C1/29, 1994. Maximal Tori Theorem Let T be a maximal torus of a group G, then T intersects every CONJUGACY CLASS of G, i.e., every element g /C23 G is conjugate to a suitable element in T. The theorem is due to E´ . Cartan. References Hsiang, W. Y. Lectures on Lie Groups. Singapore: World Scientific, p. 42, 2000. Maximal Zero-Sum-Free Set A set having the largest number k of distinct residue classes modulo m so that no SUBSET has zero sum. See also MAXIMAL SUM-FREE SET References Guy, R. K. "Maximal Zero-Sum-Free Sets." §C15 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 129 /C1/31, 1994. Maximum The largest value of a set, function, etc. The max- imum value of a set of elements A /C30fai gN i /C301 is denoted max A or maxi ai ; and is equal to the last element of a sorted (i.e., ordered) version of A. For example, given the set f3; 5; 4; 1g; the sorted version is f1 ; 3; 4; 5g;so the maximum is 5. The maximum and MINIMUM are the simplest ORDER STATISTICS . A continuous FUNCTION may assume a maximum at a single point or may have maxima at a number of points. A GLOBAL MAXIMUM of a FUNCTION is the largest value in the entire RANGE of the FUNCTION , and a LOCAL MAXIMUM is the largest value in some local neighborhood. For a function f(x) which is CONTINUOUS at a point x0 ; a NECESSARY but not SUFFICIENT condition for f(x)to have a RELATIVE MAXIMUM at x /C30x0is that x0be a CRITICAL POINT (i.e., f(x) is either not DIFFERENTIABLE at x0or x0is a STATIONARY POINT , in which case f ?(x0) /C300):/ The FIRST DERIVATIVE TEST can be applied to CON- TINUOUS FUNCTIONS to distinguish maxima from MINIMA . For twice differentiable functions of one variable, f(x) ; or of two variables, f(x; y) ; the SECOND DERIVATIVE TEST can sometimes also identify the nature of an EXTREMUM . For a function f(x) ; the EXTREMUM TEST succeeds under more general condi- tions than the SECOND DERIVATIVE TEST . See also CRITICAL POINT ,E XTREMUM ,E XTREMUM TEST,FIRST DERIVATIVE TEST,G LOBAL MAXIMUM , INFLECTION POINT ,L OCAL MAXIMUM ,M IDRANGE , MINIMUM ,ORDER STATISTIC ,SADDLE POINT (FUNC- TION ), SECOND DERIVATIVE TEST,STATIONARY POINT References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Niven, I. Maxima and Minima without Calculus. Washing- ton, DC: Math. Assoc. Amer., 1982. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Minimization or Maximization of Functions."Ch. 10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 387 /C1 /48, 1992. Tikhomirov, V. M. Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., 1991. Maximum Absolute Column Sum Norm The NATURAL NORM induced by the L1-NORM is called the maximum absolute column sum norm and is defined by Akk1/C30max jXn i/C301½aij½ for a MATRIX A:This MATRIX NORM is implemented as MatrixNorm [m, 1] in the Mathematica add-on pack- ageLinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also L1-NORM,M ATRIX NORM,M AXIMUM ABSO- LUTE ROW SUM NORM,SPECTRAL NORM Maximum Absolute Row Sum Norm The NATURAL NORM induced by the L-INFINITY-NORM is called the maximum absolute row sum norm and is defined by Akk/C12/C30max iXn j /C301½aij ½ for a MATRIX A: This MATRIX NORM is implemented as MatrixNorm [m, Infinity] in the Mathematica add-on package LinearAlgebra‘MatrixMultiplica- tion‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also L-INFINITY- NORM,MATRIX NORM,MAXIMUM ABSOLUTE COLUMN SUM NORM,SPECTRAL NORM Maximum Clique Problem PARTY PROBLEM Maximum Entropy Method A DECONVOLUTION ALGORITHM (sometimes abbre- viated MEM) which functions by minimizing a smoothness function ("ENTROPY ") in an image. Max- imum entropy is also called the ALL-POLES MODEL or AUTOREGRESSIVE MODEL . For images with more than a million pixels, maximum entropy is faster than the CLEAN algorithm. MEM is commonly employed in astronomical synth- esis imaging. In this application, the resolution depends on the signal-to-noise ratio, which must be specified. Therefore, resolution is image dependent and varies across the map. MEM is also biased, since the ensemble average of the estimated noise is NONZERO . However, this bias is much smaller than the NOISE for pixels with a SNR /C271 : It can yield super-resolution, which can usually be trusted to an order of magnitude in SOLID ANGLE . Two definitions of "ENTROPY " normalized to the flux in the image are H1 /C13X klnIk Mk ! (1) H2 /C13/C28X kIk lnIk Mke ! ; (2) where Mk is a "default image" and Ik is the smoothed image. Several unnormalized entropy measures (Cornwell 1982, p. 3) are given by H3 /C13/C28X fi ln(fi) (3)H4 /C13X ln(fi) (4) H5 /C13/C28X 1 ln(fi) (5) H6 /C13/C28X 1 [ln(fi)]2 (6) H7 /C13Xffiffiffiffiffiffiffiffiffiffiffi ln(fi)p : (7) See also DECONVOLUTION , LUCY References Cornwell, T. J. "Can CLEAN be Improved?" VLA Scientific Memorandum No. 141, March 1982. Cornwell, T. and Braun, R. "Deconvolution." Ch. 8 in Synthesis Imaging in Radio Astronomy: Third NRAO Summer School, 1988 (Ed. R. A. Perley, F. R. Schwab, and A. H. Bridle). San Francisco, CA: Astronomical So- ciety of the Pacific, pp. 167 /C1/83, 1989. Christiansen, W. N. and Ho¨gbom, J. A. Radiotelescopes, 2nd ed. Cambridge, England: Cambridge University Press, pp. 217 /C1/18, 1985. Narayan, R. and Nityananda, R. "Maximum Entropy Re- storation in Astronomy." Ann. Rev. Astron. Astrophys. 24, 127 /C1/70, 1986. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Power Spectrum Estimation by the Maximum Entropy (All Poles) Method" and "Maximum Entropy Image Restoration." §13.7 and 18.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 565 /C1/69 and 809 /C1/17, 1992. Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. §3.2 in Interferometry and Synthesis in Radio Astronomy. New York: Wiley, pp. 349 /C1/52, 1986. Maximum Flow, Minimum Cut Theorem The maximum flow between vertices viand vjin a GRAPH Gis exactly the weight of the smallest set of edges to disconnect Gwith viand vjin different components (Ford and Fulkerson 1962; Skiena 1990, p. 178). See also NETWORK FLOW References Ford, L. R. and Fulkerson, D. R. Flows in Networks. Princeton, NJ: Princeton University Press, 1962. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Maximum Independent Set Problem This problem is NP -COMPLETE (Garey and Johnson 1983). References Garey, M. R. and Johnson, D. S. Computers and Intract- ability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman, 1983. Skiena, S. "Maximum Independent Set." §5.6.3. in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 218 /C1/19, 1990. Maximum Likelihood The procedure of finding the value of one or more parameters for a given statistic which makes theknown LIKELIHOOD distribution a MAXIMUM . The maximum likelihood estimate for a parameter mis denoted ˆm:/ For a B ERNOULLI DISTRIBUTION , d duN NpfflCzrfflCzD uNp(1/C28u)NqfflC}{fflC}z /C30Np(1/C28u)/C28uNq/C300;(1) so maximum likelihood occurs for u/C30p:Ifpis not known ahead of time, the likelihood function is f(x1;...;xn½p)/C30P(X1/C30x1;...;Xn/C30xn½p) /C30px1(1/C28p)1/C28x1/C1/C1/C1pxn(1/C28p)1/C28x1n/C30pSxi(1/C28p)S(1/C28xi) /C30pSxi(1/C28p)n/C28Sxi; (2) where x/C300 or 1, and i/C301, ..., n. lnf/C30X xilnp/C27n/C28X xifflCz6fflCz7 ln(1/C28p) (3) d(lnf) dp/C30Pxi p/C28n/C28Pxi 1/C28p/C300 (4) X xi/C28pX xi/C30np/C28pX xi (5) ˆp/C30Pxi n: (6) For a G AUSSIAN DISTRIBUTION , f(x1;...;xn½m;s)/C30Y 1 sffiffiffiffiffiffi 2pp e/C28(xi/C28m)2=2s2 /C30(2p)/C28n=2 snexp/C28P(xi/C28m)2 2s2"# (7) lnf/C30/C281 2nln(2p)/C28nlns/C28P(xi/C28m)2 2s2(8) @(lnf) @m/C30P(xi/C28m) s2/C300 (9) gives ˆm/C30Pxi n: (10) @(lnf) @s/C30/C28n s/C27P(xi/C28m)2 s3(11) givesˆs/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP(xi/C28ˆm)2 ns : (12) Note that in this case, the maximum likelihood STANDARD DEVIATION is the sample STANDARD DEVIA- TION , which is a BIASED ESTIMATOR for the population STANDARD DEVIATION . For a weighted G AUSSIAN DISTRIBUTION , f(x1;...;xn½m;s)/C30Y 1 siffiffiffiffiffiffi 2ppe/C28(xi/C28m)2=2s2 i /C30(2p)/C28n=2 snexp/C28P(xi/C28m)2 2s2"# (13) lnf/C30/C281 2nln(2p)/C28nX lnsi/C28X(xi/C28m)2 2s2 i(14) @(lnf) @m/C30X(xi/C28m) s2i/C30Xxi s2i/C28mX 1 s2i/C300 (15) gives ˆm/C30Pxi s2i P1 s2i: (16) The VARIANCE of the MEAN is then s2 m/C30X s2i@m @xi !2 : (17) But @m @xi/C30@ @xiP(xi=s2 i)P(1=s2 i)/C301=s2 iP(1=s2 i): (18) so s2 m/C30X s2i1=s2 iP(1=s2 i) !2 X 1=s2 iP(1=s2 i)fflC}fflC(2/C301P(1=s2i): (19) For a P OISSON DISTRIBUTION , f(x1;...;xn½l)/C30e/C28llx1 x1!/C1/C1/C1e/C28llxn xn!/C30e/C28nllP xi x1!/C1/C1/C1xn!(20) lnf/C30/C28nl/C27(lnl)X xi/C28lnY xi!fflCz6fflCz7 (21) d(lnf) l/C30/C28n/C27Pxi l/C300 (22) ˆl /C30Pxi n: (23) See also BAYESIAN ANALYSIS References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Least Squares as a Maximum Likelihood Estimator." §15.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 651 /C1/55, 1992. Maximum Modulus Principle Let U ⁄C be a DOMAIN , and let f be an ANALYTIC FUNCTION on U. Then if there is a point z0 /C23 U such that ½f(z0) ½]½f(z) ½ for all z /C23 U ; then f is constant. The following slightly sharper version can also be formu- lated. Let U ⁄C be a DOMAIN , and let f be an ANALYTIC FUNCTION on U. Then if there is a point z0 /C23 U at which ½f ½ has a LOCAL MAXIMUM , then f is constant. Furthermore, let U ⁄C be a bounded domain, and let f be a continuous function on the CLOSED SET ¯U that is analytic on U. Then the maximum value of ½f ½ on ¯U (which always exists) occurs on the boundary @U : In other words, max ¯U½f ½/C30max @U½f ½: The maximum modulus theorem is not always true on an unbounded domain. See also MINIMUM MODULUS PRINCIPLE ,M ODULUS (COMPLEX NUMBER ) References Krantz, S. G. "The Maximum Modulus Principle" and "Boundary Maximum Modulus Theorem." §5.4.1 and 5.4.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 76 /C1/7, 1999. Max Sequence A sequence defined from a FINITE sequence a0 ; a1 ; ..., an by defining an /C271 /C30maxi(ai /C27an/C28i):/ See also MEX SEQUENCE References Guy, R. K. "Max and Mex Sequences." §E27 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 227 /C1/28, 1994.Maxwell Distribution The distribution of speeds of molecules in thermal equilibrium as given by statistical mechanics. The probability and cumulative distributions over the range x /C23 [0;/C12) are P(x) /C30ffiffiffi 2 ps a3 =2x2e/C28ax2 =2 (1) D(x) /C302 g(3 2 ;12ax2) ffiffiffipp (2) /C30erf xffiffiffi a 2s ! /C28e/C28ax2 =2ffiffiffiffiffiffi 2a ps ; (3) where g(a ; x) is an incomplete GAMMA FUNCTION and erf(x)is ERF. The RAW MOMENTS are m?n /C3021/C27n=2a /C28n=2 G(1 2(3 /C27 n)) ffiffiffipp : (4) m ?/C302ffiffiffiffiffiffi 2 pas (5) m?2 /C303 a (6) m?3 /C308ffiffiffiffiffiffiffiffi 2 a3 ps (7) m?4 /C3015 2 (8) (Papoulis 1984, p. 149), and the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are given by m/C302ffiffiffiffiffiffi 2 pas (9) s2/C303p/C288 pa(10) g1/C308 3ffiffiffiffiffiffi 2 3ps (11) g2/C30/C284 3: (12) See also EXPONENTIAL DISTRIBUTION ,GAUSSIAN DIS- TRIBUTION ,RAYLEIGH DISTRIBUTION References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 104 and 149, 1984. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 252, 1993. Maxwell Equations The system of PARTIAL DIFFERENTIAL EQUATIONS describing classical electromagnetism and therefore of central importance in physics. In the so-called cgs system of units, the Maxwell equations are given by 9 /C215 D /C304pr (1) 9/C29E /C30/C281 c@B @t (2) 9 /C215 B /C300 (3) 9/C29H /C304p cJ /C271 c@D @t; (4) where D is the effective electric field in a dielectric , r is the charge density, E is the electric field, c is the speed of light, B is the imposed magnetic field, H is the effective magnetic field in a dielectric, and J is the current density. As usual, 9 /C215 V is the DIVERGENCE and 9/C29V is the CURL . In the MKS system of units, the equations are written 9 /C215D/C30r e0(5) 9/C29E/C30/C28@B @t(6) 9 /C215B/C300 (7) 9/C29H/C30m0J/C27e0m0@D @t; (8) where e0is the permittivity of free space and m0is the permeability of free space. See also DIRAC EQUATION References Jackson, J. D. Classical Electrodynamics, 3rd ed. New York: Wiley, p. 177, 1998. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Maxwell’s Equations The system of PARTIAL DIFFERENTIAL EQUATIONS describing electromagnetism. In the so-called cgs system of units, they are given by 9 /C215D (1)4pr (2) 9/C29E (3) /C281 c@B @t(4) where Dis the electric induction, ris the charge density, Bis the magnetic field, His the magnetic induction, cis the speed of light, Jis the current density, and Eis the electric field. References Jackson, J. D. Classical Electrodynamics, 3rd ed. New York: Wiley, p. 177, 1998. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. May’s Theorem Simple majority vote is the only procedure which is ANONYMOUS ,DUAL , and MONOTONIC . References May, K. "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision." Econometrica 20, 680/C1/84, 1952. May-Thomason Uniqueness Theorem For every infinite LOOP SPACE MACHINE E, there is a natural equivalence of spectra between EX and Segal’s spectrum BX:/ References May, J. P. and Thomason, R. W. "The Uniqueness of Infinite Loop Space Machines." Topology 17, 205/C1/24, 1978. Weibel, C. A. "The Mathematical Enterprises of Robert Thomason." Bull. Amer. Math. Soc. 34,1/C1/3, 1996. Maze A maze is a drawing of impenetrable line segments(or curves) with "paths" between them. The goal ofthe maze is to start at one given point and find a path which reaches a second given point. References Bellman, R.; Cooke, K. L.; and Lockett, J. A. Algorithms, Graphs, and Computers. New York: Academic Press, pp. 94 /C1/00, 1970. Dantzig, G. B. "All Shortest Routes in a Graph." Operations Res. Techn. Rep. 66 /C1/.Stanford, CA: Stanford University, pp. 346 /C1/65, Sept. 1961. Gardner, M. "Mazes." Ch. 10 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 112 /C1/ 18, 1961. Gardner, M. "Three-Dimensional Maze." §6.3 in The Sixth Book of Mathematical Games from Scientific American.Chicago, IL: University of Chicago Press, pp. 49 /C1 /0, 1984. Hu, T. C. and Torres, W. T. "Shortcut in the Decomposition Algorithm for Shortest Paths in a Network." IBM J. Res. Devel. 13, 387/C1/90, Jul. 1969. Jablan, S. "Roman Mazes." http://members.tripod.com/ ~modularity/mazes.htm. Lee, C. Y. "An Algorithm for Path Connections and Its Applications." IRE Trans. Elec. Comput. EC-10 , 346 /C1/65, 1961. Matthews, W. H. Mazes and Labyrinths: Their History and Development. New York: Dover, 1970. Moore, E. F. "The Shortest Path through a Maze." Ann. Comput. Lab. Harvard University 30, 285 /C1/92, 1959. Pappas, T. "Mazes." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 192 /C1/94, 1989. Phillips, A. "The Topology of Roman Mazes." Leonardo 25, 321 /C1/29, 1992. Shepard, W. Mazes and Labyrinths: A Book of Puzzles. New York: Dover, 1961. Weisstein, E. W. "Books about Mazes." http://www.treasure- troves.com/books/Mazes.html. Mazur’s Theorem The generalization of the SCHO¨ NFLIES THEOREM to n- D. A smoothly embedded n-HYPERSPHERE in an (n /C271)/-HYPERSPHERE separates the (n /C271)/-HYPER- SPHERE into two components, each HOMEOMORPHIC to (n /C271)/-BALLS . It can be proved using MORSE THEORY . See also BALL,HYPERSPHERE ,MORSE THEORY M’Cay Circle MCCAY CIRCLE McCay Circle The three circumcircles through the CENTROID G of a given triangle DA1A2A3 and the pairs of the vertices of the second BROCARD TRIANGLE are called the McCay circles (Johnson 1929, p. 306). If the VERTEX A1 of a TRIANGLE describes a NEUBERG CIRCLE N1 ; then its CENTROID G describes one of the McCay circles (Johnson 1929, p. 290), which has RADIUS ,r /C301 6a1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cot2 v /C283p ; 1/3 that of the NEUBERG CIRCLE , where a1is the length of the edge A2A3 and v is the BROCARD ANGLE (Johnson 1929, p. 307). In the above figure, the inner triangle is the second BROCARD TRIANGLE of DA1A2A3 ; whose two indicated edges are concyclic with G on the McCay circle. See also BROCARD TRIANGLES ,CIRCLE ,CONCURRENT , MEDIAN POINT ,NEUBERG CIRCLE References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 83 /C1/4 and 128 /C1/29, 1971. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 290 and 306 /C1/07, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 145 and 222, 1893. M’Cay, W. S. "On Three Circles Related to a Triangle." Trans. Roy. Irish Acad. 28, 453/C1/70, 1885. McCoy’s Theorem If two SQUARE n/C29nMATRICES Aand Bare simulta- neously upper triangularizable by similarity trans- forms, then there is an ordering a1;...,anof the EIGENVALUES ofAandb1;...,bnof the EIGENVALUES of Bso that, given any POLYNOMIAL p(x;y) in noncom- muting variables, the EIGENVALUES ofp(A;B) are the numbers p(ai;bi) with i/C301, ..., n. McCoy’s theorem states the converse: If every POLYNOMIAL exhibits the correct EIGENVALUES in a consistent ordering, then A andBare simultaneously triangularizable. References Luchins, E. H. and McLoughlin, M. A. "In Memoriam: Olga Taussky-Todd." Not. Amer. Math. Soc. 43, 838/C1/47, 1996. McGee Graph The unique 7- CAGE GRAPH (right figure) consisting of the union of the two leftmost subgraphs illustrated above. It has 24 nodes, 36 edges, and all nodes have degree 3. Its AUTOMORPHISM GROUP is of size 32. The graph is not vertex-transitive, having orbits of length 8 and 16. It was discovered by McGee (1960) and proven unique by Tutte (1966) (Wong 1982). An alternative embedding is illustrated above. See also CAGE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 174 /C1/75, 1994. McGee, W. F. "A Minimal Cubic Graph of Girth Seven." Canad. Math. Bull. 3, 149 /C1/52, 1960. Royle, G. "Cubic Cages." http://www.cs.uwa.edu.au/~gordon/ cages/. Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/2, 1982. McLaughlin Group The SPORADIC GROUP McL. References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/McL.html. McMahon’s Theorem PRICE’S THEOREM McNugget Number A number which can be obtained by adding together orders of McDonald’s † Chicken McNuggetsTM (prior to consuming any), which originally came in boxes of 6, 9, and 20. All integers are McNugget numbers except 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43. Since the Happy MealTM-sized nugget box (4 to a box) can now be purchased separately, the modern McNugget num- bers are LINEAR COMBINATIONS of 4, 6, 9, and 20. These new-fangled numbers are much less interest- ing than before, with only 1, 2, 3, 5, 7, and 11 remaining as non-McNugget numbers. The GREEDY ALGORITHM can be used to find a McNugget expansion of a given INTEGER .See also COMPLETE SEQUENCE ,GREEDY ALGORITHM References Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 19 /C1/0 and 233 /C1/34, 1991. Wilson, D. rec.puzzles newsgroup posting, March 20, 1990. Mean A mean is HOMOGENEOUS and has the property that a mean m of a set of numbers xi satisfies min( x1 ; ...; xn) 5 m 5max( x1 ; ...; xn) : There are several statistical quantities called means, e.g., ARITHMETIC-GEOMETRIC MEAN , GEOMETRIC MEAN , HARMONIC MEAN , QUADRATIC MEAN , ROOT-MEAN- SQUARE . However, the quantity referred to as "the" mean is the ARITHMETIC MEAN , also called the AVERAGE . An interesting empirical relationship between the mean, median, and mode which appears to hold for unimodal curves of moderate asymmetry is given by mean /C28mode :3(mean /C28median) (Kenney and Keeping 1962, p. 53), which is the basis for the definition of the PEARSON MODE SKEWNESS . See also ARITHMETIC- GEOMETRIC MEAN,A VERAGE , GENERALIZED MEAN,G EOMETRIC MEAN,H ARMONIC MEAN,PEARSON MODE SKEWNESS ,QUADRATIC MEAN, REVERSION TO THE MEAN,ROOT-MEAN-SQUARE References Kenney, J. F. and Keeping, E. S. "Averages," "Relation Between Mean, Median, and Mode," and "Relative Merits of Mean, Median, and Mode." §3.1 and §4.8 /C1/.9 in Mathe- matics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 32 and 52 /C1/4, 1962. Mean Absolute Deviation The mean absolute deviation (often inaccurately called the MEAN DEVIATION ), is defined by M :A :D /C301 NXN i/C301fi ½xi /C28 ¯x½; where the SAMPLE SIZE is N, the samples have values xi;the MEAN is¯x;andfiis an ABSOLUTE FREQUENCY . See also MEAN DEVIATION References Kenney, J. F. and Keeping, E. S. "Mean Absolute Devia- tion." §6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 76 /C1/7 1962. Mean Caliper Diameter MEAN TANGENT DIAMETER Mean Cluster Count Per Site S-CLUSTER Mean Cluster Density S-CLUSTER Mean Curvature Let k1and k2be the PRINCIPAL CURVATURES , then their MEAN H /C301 2( k1 /C27 k2) (1) is called the mean curvature. Let R1and R2be the radii corresponding to the PRINCIPAL CURVATURES , then the MULTIPLICATIVE INVERSE of the mean curva- ture H is given by the MULTIPLICATIVE INVERSE of the HARMONIC MEAN , H /C131 21 R1/C271 R2 ! /C30R1 /C27 R2 2R1R2: (2) In terms of the GAUSSIAN CURVATURE K, H /C301 2(R1 /C27R2)K : (3) The mean curvature of a REGULAR SURFACE in R3 at a point p is formally defined as H(p) /C301 2Tr(S(p)) (4) where S is the SHAPE OPERATOR and Tr(S) denotes the TRACE . For a MONGE PATCH with z /C30h(x; y) ; H /C30(1 /C27 h2 v)huu /C28 2huhvhuv /C27 (1 /C27 h2u)hvv 2(1 /C27 h2 u /C27 h2v)3=2 (5) (Gray 1997, p. 399). If x : U 0 R3 is a REGULAR PATCH , then the mean curvature is given by H /C30eG /C28 2fF /C27 gE 2(EG /C28 F2); (6) where E, F, and G are coefficients of the first FUNDAMENTAL FORM and e, f, and g are coefficients of the second FUNDAMENTAL FORM (Gray 1997, p. 377). It can also be written H /C30det(xuuxuxv)½xu ½2 /C28 2 det(xuvxuxv)(xu /C215 xv) 2[½xu ½2 ½xv ½/C28 (xu /C215 xv)2]3=2 /C27det(xvvxuxv)½xu ½2 2[½xu ½2 ½xv ½2 /C28 (xu /C215 xv)2]3 =2 (7) Gray (1997, p. 380). The GAUSSIAN and mean curvature satisfy H2 ]K ; (8) with equality only at UMBILIC POINTS , sinceH2 /C28K /C301 4( k1 /C28 k2)2 : (9) If p is a point on a REGULAR SURFACE M ƒR3 and vp and wp are tangent vectors to M at p, then the mean curvature of M at p is related to the SHAPE OPERATOR S by S(vp) /C29wp /C27vp /C29S(wp) /C302H(p)vp /C29wp (10) Let Z be a nonvanishing VECTOR FIELD on M which is everywhere PERPENDICULAR to M, and let V and W be VECTOR FIELDS tangent to M such that V /C29W /C30Z; then H /C30/C28Z /C215 (DvZ /C29 W /C27 V /C29 DWZ) 2 ½Z ½3 (11) (Gray 1997, p. 410). Wente (1985, 1986, 1987) found a nonspherical finite surface with constant mean curvature, consisting of a self-intersecting three-lobed toroidal surface. A fa- mily of such surfaces exists. See also GAUSSIAN CURVATURE ,LAGRANGE’S EQUA- TION ,M INIMAL SURFACE ,P RINCIPAL CURVATURES , SHAPE OPERATOR References Gray, A. "The Gaussian and Mean Curvatures." §16.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 373 /C1/80, 1997. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, p. 108, 1992. Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 69 /C1/0, 1988. Wente, H. C. "A Counterexample in 3-Space to a Conjecture of H. Hopf." In Workshop Bonn 1984, Proceedings of the 25th Mathematical Workshop Held at the Max-Planck Institut fu¨r Mathematik, Bonn, June 15 /C1/2, 1984 (Ed. F. Hirzebruch, J. Schwermer, and S. Suter). New York: Springer-Verlag, pp. 421 /C1/29, 1985. Wente, H. C. "Counterexample to a Conjecture of H. Hopf." Pac. J. Math. 121, 193 /C1/43, 1986. Wente, H. C. "Immersed Tori of Constant Mean Curvature in R3 :/"InVariational Methods for Free Surface Interfaces, Proceedings of a Conference Held in Menlo Park, CA, Sept. 7 /C1/2, 1985 (Ed. P. Concus and R. Finn). New York: Springer-Verlag, pp. 13 /C1/4, 1987. Mean Deviation The MEAN of the ABSOLUTE DEVIATIONS , MD/C131 NXN i/C301½xi/C28¯x½; where ¯xis the MEAN of the distribution. See also ABSOLUTE DEVIATION Mean Distribution For an infinite population with MEAN m;VARIANCE s2; SKEWNESS g1;and KURTOSIS g2;the corresponding quantities for the distribution of means are m¯x /C30 m (1) s2 ¯x /C30s2 N (2) g1 ; ¯x /C30g1ffiffiffiffiffi Np (3) g2 ; ¯x /C30g2 N: (4) For a population of M (Kenney and Keeping 1962, p. 181), m(M) ¯x/C30 m (5) s2(M) /C30s2 NM /C28 N M /C28 1: (6) References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Mean Run Count Per Site S-RUN Mean Run Density S-RUN Mean Square Error ROOT-MEAN-SQUARE Mean Tangent Diameter This entry contributed by ROD MACKERT The mean tangent diameter of a solid, also known as the mean caliper diameter, is the caliper dimension obtained by averaging over all orientations. See also INNER QUERMASS ,STEREOLOGY References Hilliard J. E. "The Calculation of the Mean Caliper Dia- meter of a Body for Use in the Analysis of the Number of Particles per Unit Volume." In Stereology (Ed. H. Elias). New York: Springer-Verlag, pp. 211 /C1/15, 1967. Russ, J. C. "Size Distributions." In Practical Stereology. New York: Plenum, pp. 53 /C1/2, 1986. Mean-Value Property Let a function h : U 0 R be continuous on an OPEN SET U ⁄C : Then h is said to have the ez0/-property if, for each z0 /C23 U ; there exists an ez0> 0 such that ¯D(z0 ; ez0) ⁄U ; where ¯D is a closed disk, and for every 0 B e B ez0;h(z0) /C301 2p g2p 0h(z0 /C27 eeiu) du: If h has the mean-value property, then h is harmonic. See also HARMONIC FUNCTION References Krantz, S. G. "The Mean Value Property on Circles." §7.4.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨u- ser, p. 94, 1999. Mean-Value Theorem Let f(x)be DIFFERENTIABLE on the OPEN INTERVAL (a, b) and CONTINUOUS on the CLOSED INTERVAL [a, b]. Then there is at least one point c in (a, b) such that f ?(c) /C30f(b) /C28 f(a) b/C28a: See also EXTENDED MEAN-VALUE THEOREM ,GAUSS’S MEAN-VALUE THEOREM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1097 /C1/098, 2000. Jeffreys, H. and Jeffreys, B. S. "Mean-Value Theorems." §1.13 in Methods of Mathematical Physics, 3rd ed. Cam- bridge, England: Cambridge University Press, pp. 49 /C1/0, 1988. Measurable Function A function f:X0Ris measurable if, for every real number a, the set fx/C23Xsuch that f(x)>ag isMEASURABLE . When X/C30Rwith L EBESGUE MEA- SURE , or more generally any B OREL MEASURE , then all CONTINUOUS functions are measurable. In fact, prac- tically any function that can be described is measur- able. Measurable functions are CLOSED under addition and multiplication, but not composition. The measurable functions form one of the most general classes of REAL FUNCTIONS . They are one of the basic objects of study in ANALYSIS , both because of their wide practical applicability and the aestheticappeal of their generality. Whether a function f:X0 Ris measurable depends on the MEASURE monX, and, in particular, it only depends on the SIGMA ALGEBRA ofMEASURABLE SETS inX. Sometimes, the MEASURE onXmay be assumed to be a standard measure. For instance, a measurable function on Ris usually measurable with respect to L EBESGUE MEASURE . From the point of view of MEASURE THEORY , subsets with measure zero do not matter. Often, instead ofactual real-valued functions, EQUIVALENCE CLASSES of functions are used. Two functions are equivalent if the subset of the domain X where they differ has MEASURE ZERO . See also BOREL MEASURE ,L EBESGUE MEASURE , MEASURE ,M EASURE SPACE ,M EASURE THEORY , REAL FUNCTION ,SIGMA ALGEBRA Measurable Set If F is a SIGMA ALGEBRA and A is a SUBSET of X, then A is called measurable if A is a member of F. X need not have, a priori, a topological structure. Even if it does, there may be no connection between the open sets in the topology and the given SIGMA ALGEBRA . See also MEASURABLE SPACE ,SIGMA ALGEBRA Measurable Space A SET considered together with the SIGMA ALGEBRA on the SET. See also MEASURABLE SET,M EASURE SPACE ,SIGMA ALGEBRA Measure The terms "measure," "measurable," etc., have very precise technical definitions (usually involving SIGMA ALGEBRAS ) which makes them a little difficult to understand. However, the technical nature of the definitions is extremely important, since it gives a firm footing to concepts which are the basis for much of ANALYSIS (including some of the slippery under- pinnings of CALCULUS ). For example, every definition of an INTEGRAL is based on a particular measure: the RIEMANN INTEGRAL is based on JORDAN MEASURE , and the LEBESGUE INTEGRAL is based on LEBESGUE MEASURE . The study of measures and their application to INTEGRATION is known as MEASURE THEORY . A measure is formally defined as a NONNEGATIVE MAP m : F 0 R (the reals) such that m(¥) /C300 and, if An is a COUNTABLE SEQUENCE in F and the An are pairwise DISJOINT , then m @ nAnfflCzrfflCzD /C30X nm(An) If, in addition, m(X) /C301 for X a MEASURE SPACE , then m is said to be a PROBABILITY MEASURE . A measure m may also be defined on SETS other than those in the SIGMA ALGEBRA F. By adding to F all sets to which m assigns measure zero, we again obtain a SIGMA ALGEBRA and call this the "completion" of F with respect to m. Thus, the completion of a SIGMA ALGEBRA is the smallest SIGMA ALGEBRA containing F and all sets of measure zero. See also ALMOST EVERYWHERE ,B OREL MEASURE , ERGODIC MEASURE ,EULER MEASURE ,G AUSS MEA- SURE ,H AAR MEASURE ,H AUSDORFF MEASURE ,H EL-SON-SZEGO MEASURE ,INTEGRAL ,JORDAN MEASURE , LEBESGUE MEASURE ,LIOUVILLE MEASURE ,M AHLER MEASURE ,M EASURABLE SPACE ,M EASURE ALGEBRA , MEASURE SPACE ,M INKOWSKI MEASURE ,N ATURAL MEASURE ,PROBABILITY MEASURE ,RADON MEASURE , WIENER MEASURE References Czyz, J. Paradoxes of Measures and Dimensions Originating in Felix Hausdorff’s Ideas. Singapore: World Scientific, 1994. Measure Algebra A Boolean SIGMA ALGEBRA which possesses a MEA- SURE . Measure Polytope HYPERCUBE Measure-Preserving Transformation ENDOMORPHISM Measure Space A measure space is a MEASURABLE SPACE possessing a NONNEGATIVE MEASURE . Examples of measure spaces include n-D EUCLIDEAN SPACE with LEBESGUE MEA- SURE and the unit interval with LEBESGUE MEASURE (i.e., probability). See also LEBESGUE MEASURE ,MEASURABLE SPACE Measure Theory The mathematical theory of how to perform INTEGRA- TION in arbitrary MEASURE SPACES . See also ALMOST EVERYWHERE CONVERGENCE ,CAN- TOR SET,FATOU’S LEMMA ,FRACTAL ,INTEGRAL ,IN- TEGRATION ,LEBESGUE’S DOMINATED CONVERGENCE THEOREM ,M EASURABLE FUNCTION ,M EASURABLE SET,M EASURABLE SPACE ,M EASURE ,M EASURE SPACE ,M ONOTONE CONVERGENCE THEOREM ,POINT- WISE CONVERGENCE References Doob, J. L. Measure Theory. New York: Springer-Verlag, 1994. Evans, L. C. and Gariepy, R. F. Measure Theory and Finite Properties of Functions. Boca Raton, FL: CRC Press, 1992. Gordon, R. A. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Providence, RI: Amer. Math. Soc., 1994. Halmos, P. R. Measure Theory. New York: Springer-Verlag, 1974. Henstock, R. The General Theory of Integration. Oxford, England: Clarendon Press, 1991. Kestelman, H. Modern Theories of Integration, 2nd rev. ed. New York: Dover, 1960. Kingman, J. F. C. and Taylor, S. J. Introduction to Measure and Probability. Cambridge, England: Cambridge Uni- versity Press, 1966. Rao, M. M. Measure Theory And Integration. New York: Wiley, 1987. Strook, D. W. A Concise Introduction to the Theory of Integration, 2nd ed. Boston, MA: Birkha ¨user, 1994. Weisstein, E. W. "Books about Measure Theory." http:// www.treasure-troves.com/books/MeasureTheory.html. Measure Zero A set of points capable of being enclosed in intervals whose total length is arbitrarily small. See also ALMOST EVERYWHERE References Jeffreys, H. and Jeffreys, B. S. " "Measure Zero": "Almost Everywhere"." §1.1013 in Methods of Mathematical Phy- sics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 29 /C1/0, 1988. Mechanical Quadrature GAUSSIAN QUADRATURE Mecon Buckminster Fuller’s term for the TRUNCATED OCTA- HEDRON . See also DYMAXION Medial Axis The boundaries of the cells of a VORONOI DIAGRAM . Medial Circle The CIRCUMCIRCLE of the MEDIAL TRIANGLE DM1M2M3 of a given triangle DA1A2A3 :/ See also CIRCUMCIRCLE ,M EDIAL TRIANGLE ,M EDIAN (TRIANGLE ), SPIEKER CIRCLEMedial Deltoidal Hexecontahedron The DUAL of the RHOMBIDODECADODECAHEDRON U38 and Wenninger dual W76 :/ See also DUAL POLYHEDRON ,RHOMBIDODECADODECA- HEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 84, 1983. Medial Disdyakis Triacontahedron The 30-faced DUAL of the TRUNCATED DODECADODE- CAHEDRON and Wenninger dual W98 :/ See also ARCHIMEDEAN SOLID,ICOSIDODECAHEDRON , TRUNCATED DODECADODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 96, 1983. Medial Hexagonal Hexecontahedron The DUAL of the SNUB ICOSIDODECADODECAHEDRON U44 and Wenninger dual W112 :/ See also DUAL POLYHEDRON ,SNUB ICOSIDODECADO- DECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 121, 1983. Medial Icosacronic Hexecontahedron The DUAL of the ICOSIDODECADODECAHEDRON and Wenninger dual /W83/. References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 85, 1983. Medial Inverted Pentagonal Hexecontahedron The DUAL of the INVERTED SNUB DODECADODECAHE- DRON U60 and Wenninger dual W114 :/ See also DUAL POLYHEDRON ,INVERTED SNUB DODE- CADODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 124, 1983. Medial Pentagonal Hexecontahedron The DUAL of the SNUB DODECADODECAHEDRON U40 and Wenninger dual W111 :/ See also DUAL POLYHEDRON ,SNUB DODECADODECA- HEDRONReferences Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 120, 1983. Medial Rhombic Triacontahedron A ZONOHEDRON which is the DUAL of the DODECADO- DECAHEDRON U36and Wenninger dual W73 : The medial rhombic triacontahedron contains interior pentagrammic vertices which are, however, hidden from view (Wenninger 1983, p. 41). The solid is also called the SMALL STELLATED TRIACONTAHEDRON . The CONVEX HULL of the DODECADODECAHEDRON is an ICOSIDODECAHEDRON and the dual of the ICOSIDODE- CAHEDRON is the RHOMBIC TRIACONTAHEDRON , so the dual of the DODECADODECAHEDRON (i.e., the medial rhombic triacontahedron) is one of the RHOMBIC TRIACONTAHEDRON STELLATIONS (Wenninger 1983, p. 41). See also DUAL POLYHEDRON ,DODECADODECAHEDRON , RHOMBIC TRIACONTAHEDRON STELLATIONS References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Cundy, H. and Rollett, A. "Small Stellated Triacontahedron. V( 5 /C2155 2)2:/"§3.9.3 in Mathematical Models, 3rd ed. Strad- broke, England: Tarquin Pub., p. 125, 1989. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 41 and 46, 1983. Medial Triambic Icosahedron The DUAL of the DITRIGONAL DODECADODECAHEDRON U41and Wenninger dual W80;whose outward appear- ance is the same as the GREAT TRIAMBIC ICOSAHEDRON (the dual of the GREAT DITRIGONAL ICOSIDODECAHE- DRON ), since the internal vertices are hidden from view. The medial triambic icosahedron has hidden pentagrammic faces, while the GREAT TRIAMBIC ICO- SAHEDRON has hidden triangular faces (Wenninger 1983, pp. 45 and 47 /C1/0). The CONVEX HULL of the SMALL DITRIGONAL ICOSIDO- DECAHEDRON is a regular DODECAHEDRON , whose dual is the ICOSAHEDRON , so the dual of the SMALL DITRIGONAL ICOSIDODECAHEDRON (i.e., the medial triambic icosahedron) is one of the ICOSAHEDRON STELLATIONS (Wenninger 1983, p. 42). See also DUAL POLYHEDRON ,DITRIGONAL DODECADO- DECAHEDRON ,GREAT TRIAMBIC ICOSAHEDRON ,ICOSA- HEDRON STELLATIONS ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 41 and 46, 1983. Wenninger, M. J. "Ninth Stellation of the Icosahedron." §34 in Polyhedron Models. New York: Cambridge University Press, p. 55, 1989. Medial Triangle The TRIANGLE DM1M2M3formed by joining the MID- POINTS of the sides of a TRIANGLE DA1A2A3 : The medial triangle is sometimes also called the AUXILI- ARY TRIANGLE (Dixon 1991). The medial triangle has TRILINEAR COORDINATES A?/C300:b/C281 : c /C281 B ?/C30a/C281 :0:c/C281 C?/C30a /C281 : b /C281 :0: The medial triangle DM ?1M ?2M ?3 of the medial triangleDM1M2M3of a TRIANGLE DA1A2A3is similar to DA1A2A3 :/ The INCIRCLE of the medial triangle is called the SPIEKER CIRCLE , and its INCENTER is called the SPIEKER CENTER . The CIRCUMCIRCLE of the medial triangle is called the MEDIAL CIRCLE . See also ANTICOMPLEMENTARY TRIANGLE ,CLEAVANCE CENTER ,CLEAVER ,SPIEKER CENTER ,SPIEKER CIRCLE References Coxeter, H. S. M. and Greitzer, S. L. "The Medial Triangle and Euler Line." §1.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 18 /C1/0, 1967. Dixon, R. Mathographics. New York: Dover, p. 56, 1991. Medial Triangle Locus Theorem Given an original triangle (thick line), find the MEDIAL TRIANGLE (outer thin line) and its INCIRCLE . Take the PEDAL TRIANGLE (inner thin line) of the MEDIAL TRIANGLE with the INCENTER as the PEDAL POINT . Now pick any point on the original triangle, and connect it to the point located a half- PERIMETER away (gray lines). Then the locus of the MIDPOINTS of these lines (the /C147s in the above diagram) is the PEDAL TRIANGLE . References Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 261 /C1/67, 1991. Tsintsifas, G. "Solution to Problem 674." Crux Math. 8, 256/C1/ 57, 1982. Median (Statistics) The middle value of a distribution (if the sample size Nis odd) or average of the two middle items (if Nis even), denoted m1=2or˜x:For a normal population, the mean mis the most efficient (in the sense that no other unbiased statistic for estimating mcan have smaller VARIANCE ) estimate (Kenney and Keeping 1962, p. 211). The efficiency of the median, measuredas the ratio of the variance of the mean to thevariance of the median, depends on the sample size N/C132n/C271a s 4n p(2n /C27) ; (1) which tends to the value 2=p :0:637 as N becomes large (Kenney and Keeping 1962, p. 211). Although, the median is less efficient than the MEAN , it is less sensitive to outliers than the MEAN For large N samples with population median ˜x0 ; m¯x /C30 ˜x0 (2) s2 ¯x /C301 8Nf2(˜x0) : (3) The median is an L-ESTIMATE (Press et al. 1992). An interesting empirical relationship between the mean, median, and mode which appears to hold for unimodal curves of moderate asymmetry is given by mean /C28mode :3(mean /C28median) (4) (Kenney and Keeping 1962, p. 53), which is the basis for the definition of the PEARSON MODE SKEWNESS . See also MEAN,MIDRANGE ,MODE,ORDER STATISTIC , PEARSON MODE SKEWNESS References Huang, J. S. "Third-Order Expansion of Mean Squared Error of Medians." Stat. Prob. Let. 42, 185 /C1/92, 1999. Kenney, J. F. and Keeping, E. S. "The Median," "Relation Between Mean, Median, and Mode," "Relative Merits of Mean, Median, and Mode," and "The Median." §3.2, 4.8 /C1/.9, and 13.13 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 32 /C1/5, 52 /C1/4, 211 /C1/12, 1962. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 694, 1992. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995. Median (Tetrahedron) The lines joining the vertices of a TETRAHEDRON to the centroids of the opposite faces are called medians. See also COMMANDINO’S THEOREM ,TETRAHEDRON References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 51, 1979.Median (Triangle) The median of a triangle is the CEVIAN from one of its VERTICES to the MIDPOINT of the opposite side. The three medians of any TRIANGLE are CONCURRENT (Casey 1888, p. 3), meeting in the TRIANGLE’S CEN- TROID (Durell 1928), which has TRILINEAR COORDI- NATES 1=a :1=b :1=c: In addition, the medians of a TRIANGLE divide one another in the ratio 2:1 (Casey 1888, p. 3). A median also bisects the AREA of a TRIANGLE . Let mi denote the length of the median of the ith side ai : Then m2 1 /C301 4(2a2 2 /C272a23 /C28a21) (1) m21 /C27m22 /C27m23 /C303 4(a2 1 /C27a22 /C27a23) (2) (Casey 1888, p. 23; Johnson 1929, p. 68). The AREA of a TRIANGLE can be expressed in terms of the medians by A /C304 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sm(sm /C28m1)(sm /C28m2)(sm /C28m3)p ; (3) where sm /C131 2(m1 /C27m2 /C27m3) : (4) A median triangle is a TRIANGLE whose sides are equal and PARALLEL to the medians of a given TRIANGLE . The median triangle of the median triangle is similar to the given TRIANGLE in the ratio 3/4. See also BIMEDIAN ,COMEDIAN TRIANGLES ,COMMAN- DINO’S THEOREM ,EXMEDIAN ,EXMEDIAN POINT ,HER- ONIAN TRIANGLE ,MEDIAL TRIANGLE References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 7 /C1/, 1967. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 20 /C1/1, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 68, 173 /C1/75, 282 /C1/83, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 62, 1893. Median Point CENTROID (TRIANGLE ) Mediant Given a FAREY SEQUENCE with consecutive terms h=k and h?=k?; then the mediant is defined as the reduced form of the fraction (h /C27h?) =(k /C27k?) :/ See also FAREY SEQUENCE References Conway, J. H. and Guy, R. K. "Farey Fractions and Ford Circles." The Book of Numbers. New York: Springer- Verlag, pp. 152 /C1/54, 1996. Mediating Plane MEDIATOR Mediator The PLANE through the MIDPOINT of a LINE SEGMENT and perpendicular to that segment, also called a mediating plane. The term "mediator" was introduced by J. Neuberg (Altshiller-Court 1979, p. 298). See also MIDPOINT ,PLANE References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 1, 1979. Meeussen Sequence A Meeussen sequence is an increasing sequence of positive integers (/m1 ; m2 ; ...) such that m1 /C301; every nonnegative integer is the sum of a subset of the fmi g; and each integer mi /C281 is the sum of a unique such subset. Cook and Kleber (2000) show that Meeussen sequences are isomorphic to TOURNAMENT SE- QUENCES . See also TOURNAMENT SEQUENCE References Cook, M. and Kleber, M. "Tournament Sequences and Meeussen Sequences." Electronic J. Combinatorics 7, No. 1, R44, 1 /C1/6, 2000. http://www.combinatorics.org/Vo- lume_7/v7i1toc.html#R44. Mega A LARGE NUMBER defined as where the CIRCLE NOTATION denotes "n in n squares," and triangles and squares are expanded in terms of STEINHAUS- MOSER NOTATION (Steinhaus1983, pp. 28 /C1/9). Here, the typographical error of Steinhaus has been corrected. See also CIRCLE NOTATION ,LARGE NUMBER ,M EGIS- TRON ,MOSER ,STEINHAUS- MOSER NOTATION References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 28 /C1/9, 1999. Megistron A very LARGE NUMBER defined in terms of CIRCLE NOTATION by Steinhaus (1983) as . See also MEGA,MOSER References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 28 /C1/9, 1999. Mehler-Dirichlet Integral Pn(cosa)/C30ffiffiffi 2p pga 0cos[(n/C271 2)f] ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosf/C28cosap df; where Pn(x)i saL EGENDRE POLYNOMIAL . Mehler-Fock Transform The integral transform defined by g(x)/C30g/C12 1t1=4/C28n=2(t/C281)1=4/C28n=2Pn/C281=2 /C281=2/C27ix(2t/C281)f(t)dt (Samko et al. 1993, p. 761) or g(x)/C30g/C12 1Pk /C281=2/C27ix(t)f(t)dt (Samko et al. 1993, p. 24), where /Pn(z)/is a L EGENDRE POLYNOMIAL . References Marichev, O. I. Eqn. 8.42 in Handbook of Integral Trans- forms of Higher Transcendental Functions: Theory and Algorithmic Tables. Chichester, England: Ellis Horwood, 1982. Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, pp. 24 and 761, 1993. Mehler Quadrature JACOBI- GAUSS QUADRATURE Mehler’s Bessel Function Formula J0(x)/C302 pg/C12 0sin(xcosh t)dt; where J0(x) is a zeroth order BESSEL FUNCTION OF THE FIRST KIND . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1472, 1980. Mehler’s Hermite Polynomial Formula X/C12 n /C300Hn(x)Hn(y) n!1 2 wfflCz6fflCz7n /C30(1 /C274w2) /C281=2exp2xyw /C28 (x2 /C27 y2)w2 1 /C28 w2"# ; where Hn(x)isaH ERMITE POLYNOMIAL . References Almqvist, G. and Zeilberger, D. "The Method of Differentiat- ing Under the Integral Sign." J. Symb. Comput. 10, 571 /C1/ 91, 1990. Foata, D. "A Combinatorial Proof of the Mehler Formula." J. Comb. Th. Ser. A 24, 250 /C1/59, 1978. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, pp. 194 /C1/95, 1996. Rainville, E. D. Special Functions. New York: Chelsea, p. 198, 1971. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 380, 1975. Meijer’s G-Function A very general function which reduces to simpler special functions in many common cases. Meijer’s G- function is defined by Gm;n p;qx a1 ; ...; ap b1 ; ...; bpfflCz}fflCz}fflCz}fflCz}fflCzD /C13fflCzr 1 2pi g gLQm j/C301G(bj /C28 z)Qnj /C301G(1 /C28 aj /C27 z)Qq j/C30m/C271G(1 /C28 bj /C27 z)Qqj/C30n/C271G(qj /C28 z)xz dz ; (1) where G(z) is the GAMMA FUNCTION . The CONTOUR gL lies between the POLES of G(1 /C28ai /C28z) and the POLES of G(bi /C27z) (Wolfram 1999, p. 772; Gradshteyn and Ryzhik 2000, pp. 896 /C1/03 and 1068 /C1/071). Prudnikov et al. (1990) contains an extensive nearly 200-page listing of formulas for the Meijer G-function. The function is built into Mathematica 4.0 as Mei- jerG [{{a1, ..., an}, {a(n/C271), ..., ap}}, {{b1, ..., bm}, {b(m/C271), ...,bq}},z]. Special cases include G21 12z1;1 1;0fflCz}fflCz}fflCz}fflCz}fflCzD /C30ln(z/C271)fflCzr (2) G21 12fflCzr z1;1 1;1fflCz}fflCz}fflCz}fflCz}fflCzD /C30z z/C271(3)G02 101 2zj01 2fflCzrfflCzD /C30cos(ffiffiffiffiffi 2zp )ffiffiffipp (4) G10 01(z½1/C28a)/C30e/C281=zz/C28a: (5) See also BARNES’ G-FUNCTION ,FOX’S H-FUNCTION , G- TRANSFORM ,KAMPE DE FERIET FUNCTION ,M ACRO- BERT’S E-FUNCTION ,R AMANUJAN G- AND G-FUNC- TIONS References Adamchik, V. "The Evaluation of Integrals of Bessel Func- tions via G-Function Identities." J. Comput. Appl. Math. 64, 283/C1/90, 1995. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Definition of the G-Function" et seq. §5.3/C1/.6 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 206 /C1/22, 1981. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Luke, Y. L. The Special Functions and Their Approxima- tions, 2 vols. New York: Academic Press, 1969. Mathai, A. M. A Handbook of Generalized Special Functions for Statistical and Physical Sciences. New York: Oxford University Press, 1993. Meijer, C. S. "Multiplikationstheoreme fu ¨r di Funktion Gm;n p;q(z):/"Proc. Nederl. Akad. Wetensch. 44, 1062 /C1/070, 1941. Meijer, C. S. "On the G-Function. II." Proc. Nederl. Akad. Wetensch. 49, 344/C1/56, 1946. Meijer, C. S. "On the G-Function. III." Proc. Nederl. Akad. Wetensch. 49, 457/C1/69, 1946. Meijer, C. S. "On the G-Function. IV." Proc. Nederl. Akad. Wetensch. 49, 632/C1/41, 1946. Meijer, C. S. "On the G-Function. V." Proc. Nederl. Akad. Wetensch. 49, 765/C1/72, 1946. Meijer, C. S. "On the G-Function. VI." Proc. Nederl. Akad. Wetensch. 49, 936/C1/43, 1946. Meijer, C. S. "On the G-Function. VII." Proc. Nederl. Akad. Wetensch. 49, 1063 /C1/072, 1946. Meijer, C. S. "On the G-Function. VIII." Proc. Nederl. Akad. Wetensch. 49, 1165 /C1/175, 1946. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. "Evaluation of Integrals and the Mellin Transform." Itogi Nauki i Tekhniki, Seriya Matemat. Analiz 27,3/C1/46, 1989. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, 1990. Wolfram, S. The Mathematica Book, 4th ed. Cambridge, England: Cambridge University Press, 1999. Meijer Transform The INTEGRAL TRANSFORM (Kf)(x)/C30g/C12 /C28/C12ffiffiffiffiffi xtp Kn(xt)f(t)dt where Kn(x)i sa MODIFIED BESSEL FUNCTION OF THE SECOND KIND . References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993. Meissel’s Formula A modification of LEGENDRE’S FORMULA for the PRIME COUNTING FUNCTION p(x) : It starts with xbc/C301 /C27X 15i 5ax pi$% /C28X 1 5i5j5ax pipj$% /C27X 15i5j5k 5ax pipjpk$% /C28.../C27 p(x) /C28a /C27P2(x; a) /C27P3(x; a) /C27...; (1) where xbcis the FLOOR FUNCTION , P2(x; a) is the number of INTEGERS pipj 5x with a /C271 5j 5j; and P3(x; a) is the number of INTEGERS pipjpk Bx with a /C27 1 5i 5j 5k: Identities satisfied by the Ps include P2(x; a) /C30X px pi ! /C28(i /C281)"# (2) for pa Bpi 5ffiffiffixpand P3(x; a) /C30X i>aP2x pi; a ! /C30Xc i/C30a /C271Xp(ffiffiffiffiffiffi x=pip ) j/C30ipx pipj ! /C28(j /C281)"# : (3) Meissel’s formula is p(x) /C30 xbc/C28Xc i /C301x pi$% /C27X 15i5j5cx pipj$% /C28... /C271 2(b /C27c /C282)(b /C28c /C271) /C28X c5i5bpx pi ! ; (4) where b/C13p(x1=2) (5) c/C13p(x1=3): (6) Taking the derivation one step further yields L EH- MER’S FORMULA . See also LEGENDRE’S FORMULA ,LEHMER’S FORMULA , PRIME COUNTING FUNCTION References Gram. Acta Math. 17, 301/C1/14, 1893. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 46, 1999. Mathews, G. B. Ch. 10 in Theory of Numbers. New York: Chelsea, 1961. Meissel. Math. Ann. 25, 251/C1/57, 1885. Riesel, H. "Meissel’s Formula." Prime Numbers and Com- puter Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, p. 12, 1994. Se´roul, R. "Meissel’s Formula." §8.7.3 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 179 /C1/81, 2000.Meixner-Pollaczek Polynomial The hypergeometric orthogonal polynomial defined by P(l) n(x;f)/C30(2l)n n!einf 2F1(/C28n;l/C27ix;2l;1/C28e/C282if); where ( x)nis the P OCHHAMMER SYMBOL . The first few are given by P(l) 0(x;f)/C301 P(l) 1(x;f)/C302(lcosf/C27xsinf) P(l) 2(x;f)/C30x2/C27l2/C27(l2/C27l/C28x2) cos(2 f) /C27(1/C272l)xsin (2 f): References Koekoek, R. and Swarttouw, R. F. "Meixner-Pollaczek." §1.7 inThe Askey-Scheme of Hypergeometric Orthogonal Poly- nomials and its q -Analogue. Delft, Netherlands: Tech- nische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, pp. 37 /C1/8, 1998. ftp://www.twi.tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1 /7.ps.gz. Meixner Polynomial of the First Kind Polynomials mk(x;b;c) which form the S HEFFER SEQUENCE for g(t)/C301/C28c 1/C28cet !b (1) f(t)/C301/C28et c/C281/C28et(2) and have GENERATING FUNCTION Xmk(x;b;c) k!tk/C301/C28t c ! (1/C28t)/C28x/C28b: (3) The are given in terms of the HYPERGEOMETRIC SERIES by m(g;m) n(x)/C30(g)n2F1(/C28n;/C28x;g;1/C28m/C281); (4) where ( x)nis the P OCHHAMMER SYMBOL (Koepf 1998, p. 115). The first few are m0(x;b;c)/C301 m1(x;b;c)/C30b/C27x1/C281 c ! m2(x;b;c) /C30b(b/C271)c2/C27(c/C281)(2bc/C27c/C271)x/C27(c/C281)2x2 c2: Koekoek and Swarttouw (1998) defined the Meixner polynomials without the P OCHHAMMER SYMBOL as M ?n(x; b; c) /C30 2 F1(/C28n ;/C28x; b;1/C281=c) : (5) The KRAWTCHOUK POLYNOMIALS are a special case of the Meixner polynomials of the first kind. See also KRAWTCHOUK POLYNOMIAL ,M EIXNER POLY- NOMIAL OF THE SECOND KIND,SHEFFER SEQUENCE References Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, p. 175, 1978. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. New York: Krieger, pp. 224 /C1/25, 1981. Koekoek, R. and Swarttouw, R. F. "Meixner." §1.9 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Uni- versiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, pp. 45 /C1/6, 1998. ftp://www.twi.- tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1/ 7.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 115, 1998. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 35, 1975. Meixner Polynomial of the Second Kind The polynomials Mk(x; d; h) which form the SHEFFER SEQUENCE for g(t) /C30f[1 /C27 df(t)]2 /C27[f(t)]2 gh =2 (1) f(t) /C30tant 1 /C27 dt ! (2) which have GENERATING FUNCTION X/C12 k /C300Mk(x; d; h) k!tk /C30[(1 /C27 dt)2] /C28 h =2expx tan/C281 t 1 /C28 d tan/C281 t ! : (3) The first few are M0(x; d ; h) /C301 M1(x; d ; h) /C30x /C28 dh M2(x; d ; h) /C30x2 /C272d(1 /C28 h)x /C27 h[( h /C271)d2 /C281]: See also MEIXNER POLYNOMIAL OF THE FIRST KIND, SHEFFER SEQUENCE References Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, p. 179, 1978. Roman, S. The Umbral Calculus. New York: Academic Press, 1984.Mellin-Barnes Integral A type of integral containing gamma functions in their integrands. A typical such integral is given by f(z) /C301 2pi g g/C27i/C12 g/C28i /C12G(a1 /C27 A1s)...G(an /C27 Ans) G(c1 /C27 C1s)...G(cp /C27 Cps) /C29G(b1 /C28 B1s)...G(bn /C28 Bns) G(d1 /C28 D1s)...G(dq /C28 Dqs)zs ds ; where g is real, Aj ; Bj ; Cj ; and Dj are positive, and the CONTOUR is a straight line parallel to the IMAGINARY AXIS with indentations if necessary to avoid poles of the integrand. References Barnes, E. W. "A New Development in the Theory of the Hypergeometric Functions." Proc. London Math. Soc. 6, 141 /C1/77, 1908. Dixon, A. L. and Ferrar, W. L. "A Class of Discontinuous Integrals." Quart. J. Math. (Oxford Ser.) 7,81/C1/6, 1936. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Mellin-Barnes Integrals." §1.19 in Higher Trans- cendental Functions, Vol. 1. New York: Krieger, pp. 49 /C1/0, 1981. Mellin, H. "Om Definita Integraler." Acta Societatis Scien- tiarum Fennicae 20, No. 7, 1 /C1/9, 1895. Mellin, H. "Abrißeiner einheitlichen Theorie der Gamma- und der hypergeometrischen Funktionen." Math. Ann. 68, 305 /C1/37, 1909. Pincherle, S. Atti d. R. Academia dei Lincei, Ser. 4, Rendiconti 4, 694 /C1/00 and 792 /C1/99, 1888. Ramanujan, S. Collected Papers. New York: Chelsea, p. 216, 1962. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 289, 1990. Mellin’s Formula eyc0(x) G(x) G(x /C27 g)/C30Y/C12 n /C3001 /C27g n /C27 x ! e /C28y =(n/C27x) ; (1) where c0(x) is the DIGAMMA FUNCTION , G(x) is the GAMMA FUNCTION , and gis the E ULER- MASCHERONI CONSTANT . See also DIGAMMA FUNCTION ,GAMMA FUNCTION References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 6, 1981. Mellin Transform The INTEGRAL TRANSFORM defined by f(z)/C30g/C12 0tz/C281f(t)dt (1) f(t)/C301 2pigc/C27i/C12 c/C28i/C12t/C28zf(z)dz: (2) The transform f(z) exists if the integral g/C12 0½f(x) ½xk /C281 dx (3) is bounded for some k /C210, in which case the inverse f(t) exists with c /C21k. The functions f(z) and f(t) are called a Mellin transform pair, and either can be computed if the other is known. The following table gives Mellin transforms of com- mon functions (Bracewell 1999, p. 255). Here, d is the DELTA FUNCTION , H(x) is the HEAVISIDE STEP FUNC- TION , G(z) is the GAMMA FUNCTION , B(z; a; b) is the INCOMPLETE BETA FUNCTION , erfc z is the complemen- tary error function ERFC , and Si(z) is the SINE INTEGRAL . /f(t)// f(z)/ convergence / d(t /C28a)// az /C281/ /H(t /C28a)// /C28az z// a > 0; z B0/ /H(a /C28t)//az z// a > 0; z > 0/ /tnH(t /C28a)// /C28an/C27z n /C27 z// a > 0;/ / R[z /C27n] B0/ /tnH(a /C28t)//an/C27z n /C27 z// a > 0;/ / R[n /C27z] > 0/ /e/C28at// a/C28z G(z)// R[a] ;R[z] > 0/ /e/C28t2 //1 2 G12 zfflCz6fflCz7 // R[z] > 0/ /sin t// G(z) sin12 pzfflCz6fflCz7 // /C281 BR[z] B1/ /cos t// G(z) cos12 pzfflCz6fflCz7 // 0 BR[z] B1/ /1 1 /C27 t// p csc(pz)// 0 BR[z] B1/ /1 (1 /C27 t)a//G(a /C28 z) G(z) G(a)// R[a /C28z] > 0;/ / R[z] > 0/ /1 1 /C27 t2//12 p csc12 pzfflCz6fflCz7 // 0 BR[z] B2/ /(1 /C28t)a /C281H(1 /C28t)//G(a) G(z) G(a /C27 z)// R[a] ;R[z] > 0//(t /C281)/C28aH(t /C281)//G(1 /C28 a) G(a /C28 z) G(1 /C28 x)//R[a /C28z] > 0;/ / R[a] B1/ /ln(1/C27t)//pcsc(pz) z// /C281BR[z]B0/ /12p/C28tan/C281t//psec(12pz) 2z// 0BR[z]B1/ /erfct//G(12(1/C27z)) ffiffiffippz// R[z]>0/ /Si(t)// /C281 zG(z) sin(1 2pz)// R[z]>/C281/ /ta 1/C28tH(t/C28a)///C28B(a/C281;1/C28a/C28z;0 ) //a>1;R[a/C27z]B1/ See also FOURIER TRANSFORM ,INTEGRAL TRANSFORM , STRASSEN FORMULAS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 795, 1985. Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 254 /C1/57, 1999. Gradshteyn, I. S. and Ryzhik, I. M. "Mellin Transform." §17.41 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1193 /C1/197, 2000. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 469 /C1/71, 1953. Oberhettinger, F. Tables of Mellin Transforms. New York: Springer-Verlag, 1974. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. "Evaluation of Integrals and the Mellin Transform." Itogi Nauki i Tekhniki, Seriya Matemat. Analiz 27,3/C1/46, 1989. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 567, 1995. Melnikov-Arnold Integral Am(l)/C30g/C12 /C28/C12cos1 2mf(t)/C28lthi dt; where the function f(t)/C134 tan/C281(et)/C28p describes the motion along the pendulum SEPARA- TRIX. Chirikov (1979) has shown that this integral has the approximate value Am(l):4p(2l)m/C281 G(m)e/C28pl=2forl>0 4e/C28p½l½=2 (2½l½)m/C271G(m/C271) sin( pm) for lB0:8 >>>< >>>: References Chirikov, B. V. "A Universal Instability of Many-Dimen- sional Oscillator Systems." Phys. Rep. 52, 264/C1/79, 1979. Melodic Sequence If a1 ; a2 ; a3 ; ... is an ARTISTIC SEQUENCE , then 1=a1 ; 1=a2 ; 1=a3 ; ... is a melodic sequence. The RECURRENCE RELATION obeyed by melodic series is bi/C273 /C30bib2 i/C272 b2 i/C271/C27b2 i/C272 bi/C271/C28bi/C272 : See also ARTISTIC SEQUENCE References Duffin, R. J. "On Seeing Progressions of Constant Cross Ratio." Amer. Math. Monthly 100,38/C1/7, 1993. MEM MAXIMUM ENTROPY METHOD Memoryless A variable x is memoryless with respect to t if, for all s with t "0; P(x > s /C27t½x > t) /C30P(x > s): (1) Equivalently, P(x > s /C27 t; x > t) P(x > t)/C30P(x > s) (2) P(x > s /C27t) /C30P(x > s)P(x > t) : (3) The EXPONENTIAL DISTRIBUTION , which satisfies P(x > t) /C30e /C28 lt (4) P(x > s /C27t) /C30e /C28 l(s/C27t) ; (5) and therefore P(x > s /C27t) /C30P(x > s)P(x > t) /C30e /C28 lse /C28 lt /C30e /C28l(s/C27t) ; (6) is the only memoryless random distribution. See also EXPONENTIAL DISTRIBUTION Me´nage Number MARRIED COUPLES PROBLEM Me´nage Problem MARRIED COUPLES PROBLEM Menasco’s Theorem For a BRAID with M strands, R components, P positive crossings, and N negative crossings, P /C28N 5U/C27/C27M /C28R if P ]N P /C28N 5U/C28/C27M /C28R if P 5N ;fflC}6 where U9are the smallest number of positive and negative crossings which must be changed to cross- ings of the opposite sign. These inequalities implyBENNEQUIN’S CONJECTURE . Menasco’s theorem can be extended to arbitrary knot diagrams. See also BENNEQUIN’S CONJECTURE ,BRAID,UNKNOT- TING NUMBER References Cipra, B. "From Knot to Unknot." What’s Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Soc., pp. 8 /C1/3, 1994. Menasco, W. W. "The Bennequin-Milnor Unknotting Con- jectures." C. R. Acad. Sci. Paris Se´r. I Math. 318, 831 /C1/36, 1994. Menelaus’ Theorem For TRIANGLES in the PLANE , AD /C215 BE /C215 CF /C30BD /C215 CE /C215 AF : (1) For SPHERICAL TRIANGLES , sin AD /C215 sin BE /C215 sin CF /C30sin BD /C215 sin CF /C215 sin AF (2) This can be generalized to n-gons P /C30[V1 ; ... ; Vn]; where a transversal cuts the side ViVi/C271in Wifor i /C301, ..., n,by Yn i/C301ViWi WiVi/C271"# /C30(/C281)n : (3) Here, ADICD and AB CD"# (4) is the ratio of the lengths [A, B] and [C, D] with a PLUS or MINUS SIGN depending if these segments have the same or opposite directions (Gru ¨nbaum and Shepard 1995). The case n/C303i sP ASCH’S AXIOM . See also CEVA’S THEOREM ,H OEHN’S THEOREM , PASCH’S AXIOM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 122, 1987. Coxeter, H. S. M. and Greitzer, S. L. "Menelaus’s Theorem." §3.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 66 /C1/7, 1967. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 42 /C1/4, 1928. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 81, 1930. Gru¨nbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254/C1/68, 1995. Honsberger, R. "The Theorem of Menelaus." Ch. 13 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 147 /C1/ 54, 1995. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., p. xxi, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 150, 1991. Menger’s n-Arc Theorem Let G be a GRAPH with A and B two disjoint n-tuples of VERTICES . Then either G contains n pairwise disjoint AB-paths, each connecting a point of A and a point of B, or there exists a set of fewer than n VERTICES that separate A and B. Harary (1994, pp. 47) states the theorem as "the minimum number of points separating two nonadja- cent points s and t is the maximum number of disjoint s /C28t paths." Skiena (1990, p. 178) states the theorem as "a graph is K-CONNECTED GRAPH IFF every pair of vertices is joined by at least k vertex-disjoint paths" (Menger 1927, Whitney 1932). See also K-CONNECTED GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Menger, K. "Zur allgemeinen Kurventheorie." Fund. Math. 10,95/C1/15, 1927. Menger, K. Kurventheorie. Leipzig, Germany: Teubner, 1932. Whitney, H. "Congruent Graphs and the Connectivity of Graphs." Amer. J. Math. 54, 150 /C1/68, 1932. Menger Sponge A FRACTAL which is the 3-D analog of the SIERPINSKI CARPET . Let Nnbe the number of filled boxes, Lnthe length of a side of a hole, and Vnthe fractional VOLUME after the nth iteration. Nn /C3020n (1) Ln /C301 3fflCz6fflCz7n /C303 /C28n (2) Vn /C30L3 nNn /C3020 27fflCz6fflCz7n : (3) The CAPACITY DIMENSION is thereforedcap /C30/C28 lim n0/C12ln Nn ln Ln/C30/C28 lim n0/C12ln (20n) ln (3/C28n) /C30ln 20 ln 3 /C30ln(25 /C215 5) ln 3/C302ln2 /C27 ln 5 ln 3/C302:726833028... (4) J. Mosely is leading an effort to construct a large Menger sponge out of old business cards. See also SIERPINSKI CARPET ,TETRIX References Dickau, R. "Sierpinski-Menger Sponge Code and Graphic." http://www.mathsource.com/cgi-bin/msitem22?0206 /C1/10. Dickau, R. M. "Menger (Sierpinski) Sponge." http://forum.s- warthmore.edu/advanced/robertd/sponge.html. Mosely, J. "Menger’s Sponge (Depth 3)." http://world.std.- com/~j9/sponge/. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Werbeck, S. "A Journey into Menger’s Sponge." http:// pages.hotbot.com/arts/werbeck/. Menger’s Theorem MENGER’S N-ARCTHEOREM Menn’s Surface A surface given by the PARAMETRIC EQUATIONS x(u;v)/C30u y(u;v)/C30v x(u;v)/C30au4/C27u2v/C28v2: References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 956, 1997. Mensuration Formula A mensuration formula is simply a formula for computing the length-related properties of an object (such as AREA ,CIRCUMRADIUS , etc., of a POLYGON ) based on other known lengths, areas, etc. Beyer (1987) gives a collection of such formulas for various plane and solid geometric figures. References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 121 /C1/33, 1987. Mercator Projection The following equations place the X-AXIS of the projection on the equator and the Y-AXIS atLONG- ITUDE l0;where lis the LONGITUDE andfis the LATITUDE . x/C30l/C28l0 (1) y/C30ln[tan(1 4p/C2712f)] (2) /C3012ln1/C27sinf 1/C28sinf ! (3) /C30sinh/C281(tanf) (4) /C30tanh/C281(sinf) (5) /C30ln(tan f/C27secf): (6) The inverse FORMULAS are f/C302 tan/C281(ey)/C2812p (7) /C30tan/C281(sinh y) (8) /C30gdy (9) l/C30x/C27l0; (10) where gd yis the G UDERMANNIAN FUNCTION .LOXO- DROMES are straight lines and GREAT CIRCLES are curved. An oblique form of the Mercator projection is illu- strated above. It has equations x/C30tan/C281[tanfcosfp/C27sinfpsin(l/C28l0)] cos(l/C28l0)(11) y/C301 2ln1/C27A 1/C28A ! /C30tanh/C281A; (12) where lp/C30 tan/C281cosf1sinf2cosl1/C28sinf1cosf2cosl2 sinf1cosf2sinl2/C28cosf1sinf2sinl1 ! (13) fp/C30tan/C281/C28cos(lp/C28l1) tanf1 ! (14) A/C30sinfpsinf/C28cosfpcosfsin(l/C28l0): (15) The inverse FORMULAS are f/C30sin/C281sinfptanh y/C27cosfpsinx cosh y ! (16) l/C30l0/C27tan/C281sinfpsinx/C28cosfpsinh y cosx ! :(17) There is also a transverse form of the Mercator projection, illustrated above (Deetz and Adams 1934, Snyder 1987). It is given by the equations x /C301 2 ln1 /C27 B 1 /C28 B ! /C30tanh /C281 B (18) y /C30tan /C281 tan f cos(l /C28 l0)"# /C28 f0 (19) f /C30sin/C281sin D cosh x ! (20) l /C30 l0 /C27tan /C281sinh x cos D ! ; (21) where B /C13cos f sin( l /C28 l0) (22) D /C13y /C27 f0 : (23) Finally, the "universal transverse Mercator projec- tion" is a MAP PROJECTION which maps the SPHERE into 60 zones of 68 each, with each zone mapped by a transverse Mercator projection with central MERIDIAN in the center of the zone. The zones extend from 80 8 S to 848 N (Dana). See also GUDERMANNIAN FUNCTION ,SPHERICAL SPIR- AL References Dana, P. H. "Map Projections." http://www.colorado.edu/ geography/gcraft/notes/mapproj/mapproj_f.html. Deetz, C. H. and Adams, O. S. Elements of Map Projection with Applications to Map and Chart Construction, 4th ed. Washington, DC: U. S. Coast and Geodetic Survey Special Pub. 68, 1934. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 38 /C1/5, 1987. Mercator Series The TAYLOR SERIES for the NATURAL LOGARITHM ln(1 /C27x) /C30x /C2812 x2 /C2713 x3 /C28...which was found by Newton, but independently discovered and first published by Mercator in 1668. See also LOGARITHMIC NUMBER ,N ATURAL LOGA- RITHM Mercer’s Theorem RIEMANN- LEBESGUE LEMMA Meredith Graph A counterexample to the conjecture that every 4- regular 4-connected graph is HAMILTONIAN . See also HAMILTONIAN GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 236 /C1/39, 1976. Meredith, G. H. J. "Regular n-valent n-connected nonha- miltonian non- n-edge-colorable Graphs." J. Combin. Th. B14,5 5/C1/0, 1973. Mergelyan’s Theorem Mergelyan’s theorem can be stated as follows (Krantz 1999). Let K⁄Cbe compact and suppose C/C31_Khas only finitely many connected components. If f/C23C(K) is holomorphic on the interior of Kand if e>0;then there is a RATIONAL FUNCTION r(z) with poles in C/C31_K such that max z/C23K½f(z)/C28r(z)½Be: (1) A consequence is that if P/C30fD1;D2;...gis an infinite set of disjoint OPEN DISKS Dnof radius rn such that the union is almost the unit DISK. Then X/C12 n/C301rn/C30/C12: (2) Define Mx(P)/C13X/C12 n/C301rx n: (3) Then there is a number e(P) such that Mx(P) diverges forxBe(P) and converges for x>e(P):The above theorem gives 1 Be(P) B2: (4) There exists a constant which improves the inequal- ity, and the best value known is S /C301:306951... : (5) See also RUNGE’S THEOREM References Krantz, S. G. "Mergelyan’s Theorem." §11.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 146 /C1/47, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 36 /C1/7, 1983. Mandelbrot, B. B. Fractals. San Francisco, CA: W. H. Free- man, p. 187, 1977. Melzack, Z. A. "On the Solid Packing Constant for Circles." Math. Comput. 23, 1969. Mergelyan-Wesler Theorem MERGELYAN’S THEOREM Meridian A line of constant LONGITUDE on a SPHEROID (or SPHERE ). More generally, a meridian of a SURFACE OF REVOLUTION is the intersection of the surface with a PLANE containing the axis of revolution. See also LATITUDE ,LONGITUDE ,PARALLEL (SURFACE OF REVOLUTION ), SURFACE OF REVOLUTION References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 238, 1997. Meromorphic Function A meromorphic function is a single-valued function that is ANALYTIC in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a POLYNOMIAL (i.e., these exceptional points must be POLES and not ESSENTIAL SINGULA- RITIES ). A simpler definition states that a mero- morphic function is a function f(z) OF THE FORM f(z) /C30g(z) h(z) where /g(z)/ and /h(z)/ are ENTIRE FUNCTIONS with / h(z) "0/ (Krantz 1999, p. 64). A meromorphic function therefore has only possibly finite, isolated POLES and zeros and no ESSENTIAL SINGULARITIES in its domain. A meromorphic function with an infinite number of poles is exemplified by / csc(1 =z)/ on the PUNCTURED /U /C30D_ f0g/, where D is the open unit disk. An equivalent definition of a meromorphic function is a complex analytic MAP to the RIEMANN SPHERE .The word derives from the Greek /mo ro&/ (meros ), meaning "part," and /mo r8 h/ (morphe ), meaning "form" or "appearance." See also ANALYTIC FUNCTION ,E NTIRE FUNCTION , ESSENTIAL SINGULARITY ,H OLOMORPHIC FUNCTION , POLE,REAL ANALYTIC FUNCTION ,RIEMANN SPHERE References Knopp, K. "Meromorphic Functions." Ch. 2 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 34 /C1/7, 1996. Krantz, S. G. "Meromorphic Functions and Singularities at Infinity." §4.6 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 63 /C1/8, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 382 /C1/83, 1953. Mersenne Number A number OF THE FORM Mn /C132n /C281 (1) for n an INTEGER is known as a Mersenne number. The Mersenne numbers are therefore 2-REPDIGITS , and also the numbers obtained by setting x /C301ina FERMAT POLYNOMIAL . The first few are 1, 3, 7, 15, 31, 63, 127, 255, ... (Sloane’s A000225). The number of digits D in the Mersenne number Mn is D /C30 log 2n /C281 ðÞ /C271 bc ; (2) where xbcis the FLOOR FUNCTION , which, for large n, gives D : n log 2 /C271 bc : 0 :301029 n /C271 bc /C30 0:301029 n bc /C271: (3) In order for the Mersenne number Mn to be PRIME , n must be PRIME . This is true since for COMPOSITE n with factors r and s, n /C30rs. Therefore, 2n /C281 can be written as 2rs /C281 ; which is a BINOMIAL NUMBER and can be factored. Since the most interest in Mersenne numbers arises from attempts to factor them, many authors prefer to define a Mersenne number as a number of the above form Mp/C302p/C281 (4) but with prestricted to PRIME values. The search for M ERSENNE PRIMES is one of the most computationally intensive and actively pursued areas of advanced and distributed computing. See also CUNNINGHAM NUMBER ,DOUBLE MERSENNE NUMBER ,E BERHART’S CONJECTURE ,F ERMAT NUM- BER,LUCAS- LEHMER TEST,M ERSENNE PRIME ,PER- FECT NUMBER ,REPUNIT ,RIESEL NUMBER ,SIERPINSKI NUMBER OF THE SECOND KIND,SOPHIE GERMAIN PRIME ,SUPERPERFECT NUMBER ,W HEAT AND CHESS- BOARD PROBLEM ,W IEFERICH PRIME References Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 13, 1952. Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150/C1/56, May 1957. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 15 /C1/6 and 22, 1979. Pappas, T. "Mersenne’s Number." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 211, 1989. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 14, 18 /C1/9, 22, and 29 /C1/0, 1993. Sloane, N. J. A. Sequences A000225/M2655 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 23 /C1 /4, 1999. Mersenne Prime AM ERSENNE NUMBER which is PRIME is called a Mersenne prime. In order for the Mersenne number Mndefined by Mn/C132n/C281 fornanINTEGER to be PRIME ,nmust be PRIME . This is true since for COMPOSITE nwith factors rand s, n/C30rs. Therefore, 2n/C281 can be written as 2rs/C281; which is a BINOMIAL NUMBER and can be factored. Every M ERSENNE PRIME gives rise to a PERFECT NUMBER . The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (Sloa- ne’s A000668) corresponding to n/C302, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (Sloane’s A000043). Ifn/C133 (mod 4) is a PRIME , then 2 n/C271DIVIDES Mn IFF2n/C271i s PRIME . It is also true that PRIME divisors of 2p/C281 must have the form 2 kp/C271 where kis a POSITIVE INTEGER and simultaneously of either the form 8 n/C271o r8 n/C281 (Uspensky and Heaslet). A PRIME factor pof a Mersenne number Mq/C302q/C281i s aW IEFERICH PRIME IFF p2½2q/C281;Therefore, M ERS- ENNE PRIMES arenotWIEFERICH PRIMES . All known Mersenne numbers Mpwith pPRIME are SQUARE- FREE . However, Guy (1994) believes that there are Mp which are not SQUAREFREE . TRIAL DIVISION is often used to establish the COMPO- SITENESS of a potential Mersenne prime. This test immediately shows Mpto be COMPOSITE forp/C3011, 23, 83, 131, 179, 191, 239, and 251 (with small factors 23, 47, 167, 263, 359, 383, 479, and 503, respectively). A much more powerful primality test for Mpis the LUCAS- LEHMER TEST . It has been conjectured that there exist an infinitenumber of Mersenne primes, although finding them is computationally very challenging. The table below gives the index pof known Mersenne primes (Sloa- ne’s A000043) M p;together with the number of digits, discovery years, and discoverer. A similar table hasbeen compiled by C. Caldwell. Note that the region after the 35th known Mersenne prime has not beencompletely searched, so identification of "the" 36thand larger Mersenne primes are tentative. L. Welsh maintains an extensive bibliography and history of Mersenne numbers. G. Woltman has organized adistributed search program via the Internet in whichhundreds of volunteers use their personal computersto perform pieces of the search. # p Digits Year Discoverer (Reference) 1 2 1 Antiquity 2 3 1 Antiquity3 5 2 Antiquity4 7 3 Antiquity5 13 4 1461 Reguis 1536, Cataldi 16036 17 6 1588 Cataldi 16037 19 6 1588 Cataldi 16038 31 10 1750 Euler 17729 61 19 1883 Pervouchine 1883, Seelhoff 1886 10 89 27 1911 Powers 191111 107 33 1913 Powers 1914 12 127 39 1876 Lucas 1876 13 521 157 1952 Lehmer 1952 /C1 /, Robinson 1952 14 607 183 1952 Lehmer 1952 /C1/, Robinson 1952 15 1279 386 1952 Lehmer 1952 /C1/, Robinson 1952 16 2203 664 1952 Lehmer 1952 /C1/, Robinson 1952 17 2281 687 1952 Lehmer 1952 /C1/, Robinson 1952 18 3217 969 1957 Riesel 195719 4253 1281 1961 Hurwitz 196120 4423 1332 1961 Hurwitz 196121 9689 2917 1963 Gillies 196422 9941 2993 1963 Gillies 196423 11213 3376 1963 Gillies 196424 19937 6002 1971 Tuckerman 197125 21701 6533 1978 Noll and Nickel 198026 23209 6987 1979 Noll 198027 44497 13395 1979 Nelson and Slowinski 197928 86243 25962 1982 Slowinski 198229 110503 33265 1988 Colquitt and Welsh 199130 132049 39751 1983 Slowinski 198831 216091 65050 1985 Slowinski 198932 756839 227832 1992 Gage and Slowinski 199233 859433 258716 1994 Gage and Slowinski 199434 1257787 378632 1996 Slowinski and Gage35 1398269 420921 1996 Armengaud, Woltman, et al. 36? 2976221 895832 1997 Spence (Devlin 1997) 37? 3021377 909526 1998 Clarkson, Woltman, et al. 38? 6972593 2098960 1999 Hajratwala 1999 See also CUNNINGHAM NUMBER ,DOUBLE MERSENNE NUMBER ,FERMAT- LUCAS NUMBER ,FERMAT NUMBER , FERMAT NUMBER (LUCAS ), FERMAT POLYNOMIAL , LUCAS- LEHMER TEST,M ERSENNE NUMBER ,PERFECT NUMBER ,REPUNIT ,SUPERPERFECT NUMBER References Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. "The New Mersenne Conjecture." Amer. Math. Monthly 96, 125/C1/28, 1989. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 66, 1987. Beiler, A. H. Ch. 3 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Bell, E. T. Mathematics: Queen and Servant of Science. Washington, DC: Math. Assoc. Amer., 1987. Caldwell, C. "Mersenne Primes: History, Theorems and Lists." http://www.utm.edu/research/primes/mersen- ne.shtml. Caldwell, C. K. "The Top Twenty: Mersenne Primes." http:// www.utm.edu/research/primes/lists/top20/Mersen-ne.html. Caldwell, C. "GIMPS Finds a Prime! 2 1398269/C281 is Prime." http://www.utm.edu/research/primes/notes/1398269/. Caldwell, C. "GIMPS Finds a Multi-Million Digit Prime!." http://www.utm.edu/research/primes/notes/6972593/. Colquitt, W. N. and Welsh, L. Jr. "A New Mersenne Prime." Math. Comput. 56, 867/C1/70, 1991. Conway, J. H. and Guy, R. K. "Mersenne’s Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 135 /C1/37, 1996. Devlin, K. "World’s Largest Prime." FOCUS: Newsletter Math. Assoc. Amer. 17, 1, Dec. 1997. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 13, 1952. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 85, 1984. Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243,1 8/C1/8, Dec. 1980. Gillies, D. B. "Three New Mersenne Primes and a Statistical Theory." Math Comput. 18,9 3/C1/7, 1964. Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape k/C2152n/C272 [sic]." §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8/C1/3, 1994. Haghighi, M. "Computation of Mersenne Primes Using a Cray X-MP." Intl. J. Comput. Math. 41, 251/C1/59, 1992. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 14 /C1/6, 1979. Kraitchik, M. "Mersenne Numbers and Perfect Numbers." §3.5 in Mathematical Recreations. New York: W. W. Nor- ton, pp. 70 /C1/3, 1942. Kravitz, S. and Berg, M. "Lucas’ Test for Mersenne Numbers 6000BpB7000 :/"Math. Comput. 18, 148/C1/49, 1964. Lehmer, D. H. "On Lucas’s Test for the Primality of Mersenne’s Numbers." J. London Math. Soc. 10, 162/C1/ 65, 1935. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/mersenne. Mersenne, M. Cogitata Physico-Mathematica. 1644.Mersenne Organization. "GIMPS Discovers 36th Known Mersenne Prime, 22976221/C281 is Now the Largest Known Prime." http://www.mersenne.org/2976221.htm. Mersenne Organization. "GIMPS Discovers 37th Known Mersenne Prime, 23021377/C281 is Now the Largest Known Prime." http://www.mersenne.org/3021377.htm. Mersenne Organization. "GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award. 26;972;593/C281i s Now the Largest Known Prime." http://www.mersen- ne.org/6972593.htm. Noll, C. and Nickel, L. "The 25th and 26th Mersenne Primes." Math. Comput. 35, 1387 /C1/390, 1980. Powers, R. E. "The Tenth Perfect Number." Amer. Math. Monthly 18, 195/C1/96, 1911. Powers, R. E. "Note on a Mersenne Number." Bull. Amer. Math. Soc. 40, 883, 1934. Sloane, N. J. A. Sequences A000043/M0672 and A000668/ M2696 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Slowinski, D. "Searching for the 27th Mersenne Prime." J. Recreat. Math. 11, 258/C1 /61, 1978 /C1/979. Slowinski, D. Sci. News 139, 191, 9/16/1989. Tuckerman, B. "The 24th Mersenne Prime." Proc. Nat. Acad. Sci. USA 68, 2319 /C1/320, 1971. Uhler, H. S. "A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes."Scripta Math. 18, 122/C1 /31, 1952. Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory . New York: McGraw-Hill, 1939. Weisstein, E. W. "Mersenne Numbers." M ATHEMATICA NOTE- BOOK MERSENNE.M . Welsh, L. "Marin Mersenne." http://www.scruznet.com/ ~luke/mersenne.htm. Welsh, L. "Mersenne Numbers & Mersenne Primes Biblio- graphy." http://www.scruznet.com/~luke/biblio.htm. Woltman, G. "The GREAT Internet Mersenne Prime Search." http://www.mersenne.org/prime.htm. Mertens Conjecture Given M ERTENS FUNCTION defined by M(n)/C13Xn k/C301m(k); (1) where m(n) is the M O¨BIUS FUNCTION , Mertens (1897) conjecture states that M(x) jjBx1=2(2) forx/C211. The conjecture has important implications, since the truth of any equality OF THE FORM M(x) jj5cx1=2(3) for any fixed c(the form of Mertens conjecture with c/C301) would imply the R IEMANN HYPOTHESIS . In 1885, Stieltjes claimed that he had a proof that M(x)x/C281=2 always stayed between two fixed bounds. However, it seems likely that Stieltjes was mistaken. Mertens conjecture was proved false by Odlyzko and te Riele (1985). Their proof is indirect and does not produce a specific counterexample, but it does show that lim sup x0/C12M(x)x/C281=2>1:06 (4) lim inf x0/C12M(x)x/C281 =2 B/C281:009: (5) Odlyzko and te Riele (1985) believe that there are no counterexamples to Mertens conjecture for x 51020 ; or even 1030. Pintz (1987) subsequently showed that at least one counterexample to the conjecture occurs for x 51065 ; using a weighted integral average of M(x) =x and a discrete sum involving nontrivial zeros of the RIEMANN ZETA FUNCTION . It is still not known if lim sup x0/C12M(x) jj x/C281=2 /C30/C12; (6) although it seems very probable (Odlyzko and te Riele 1985). See also MERTENS FUNCTION ,M O¨ BIUS FUNCTION , RIEMANN HYPOTHESIS References Anderson, R. J. "On the Mertens Conjecture for Cusp Forms." Mathematika 26, 236 /C1/49, 1979. Anderson, R. J. "Corrigendum: ‘On the Mertens Conjecture for Cusp Forms."’ Mathematika 27, 261, 1980. Devlin, K. "The Mertens Conjecture." Irish Math. Soc. Bull. 17,29/C1/3, 1986. Grupp, F. "On the Mertens Conjecture for Cusp Forms." Mathematika 29, 213 /C1/26, 1982. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 64, 1999. Jurkat, W. and Peyerimhoff, A. "A Constructive Approach to Kronecker Approximation and Its Application to the Mertens Conjecture." J. reine angew. Math. 286/287 , 322 /C1/40, 1976. Mertens, F. "Uuml;ber eine zahlentheoretische Funktion." Sitzungsber. Akad. Wiss. Wien IIa 106, 761 /C1/30, 1897. Odlyzko, A. M. and te Riele, H. J. J. "Disproof of the Mertens Conjecture." J. reine angew. Math. 357, 138 /C1/ 60, 1985. Pintz, J. "An Effective Disproof of the Mertens Conjecture." Aste´rique 147 /C1/48, 325 /C1/33 and 346, 1987. te Riele, H. J. J. "Some Historical and Other Notes About the Mertens Conjecture and Its Recent Disproof." Nieuw Arch. Wisk. 3, 237 /C1/43, 1985. Mertens Constant N.B. Portions of this entry based on a detailed online essay by S. Finch. A constant related to the TWIN PRIMES CONSTANT which appears in HARMONIC SERIES for the SUM of reciprocal PRIMES Xx p prime1 p /C30ln ln x /C27B1 /C27o(1) ; (1) which is given by B1 /C30 g /C27X p primeln 1 /C28p /C281fflC{fflCz /C271 p"# :0:2614972128 ; (2) where g is the EULER- MASCHERONI CONSTANT (Rosserand Schoenfeld 1962; Le Lionnais 1983; Ellison and Ellison 1985; Hardy and Wright 1985). According to Lindqvist and Peetre (1997), this was shown inde- pendently by Meissel in 1866 and Mertens (1874). (2) is equivalent to Y p 5x1 /C281 p ! /C2e/C28 g ln x ; (3) where g is the EULER- MASCHERONI CONSTANT (Hardy 1999, p. 57). Knuth (1998) gives 40 digits of B1 ; and Gourdon and Sebah give 100 digits. The constant is sometimes known as Kronecker’s constant (Schroeder 1997). A rapidly converging series for B1 is given by B1 /C30 g /C27X/C12 m/C302m(m) mln z(m) ½/C138 ; (4) where g is the EULER- MASCHERONI CONSTANT , z(n)is the RIEMANN ZETA FUNCTION , and m(n) is the MO¨ BIUS FUNCTION (Flajolet and Vardi 1996, Schroeder 1997, Knuth 1998). The constant B1also occurs in the SUMMATORY FUNCTION of the number of DISTINCT PRIME FACTORS v(k) ; Xn k /C302v(k) /C30n ln ln n /C27B1n /C27o(n) (5) (Hardy and Wright 1979, p. 355).The related constant B 2/C30g/C27X pprimeln 1/C28p/C281fflC{fflCz /C271 p/C281"# :1:034653 (6) appears in the SUMMATORY FUNCTION of the DIVISOR FUNCTION s0(n)/C30V(n); Xn k/C302V(k)/C30nln ln n/C27B2/C27o(n) (7) (Hardy and Wright 1979, p. 355).Another related series is lim n0/C12Xp(n) k/C301lnpk pk/C28lnn ! /C30/C28g/C28X/C12 j/C302X/C12 k/C301lnpk pj k /C13/C28C2/C30/C281:3325822757 . . . (8) (Rosser and Schoenfeld 1962, Montgomery 1971, Finch). See also BRUN’S CONSTANT ,HARMONIC SERIES ,PRIME FACTORS ,PRIME NUMBER ,TWIN PRIMES CONSTANT References Ellison, W. J. and Ellison, F. Prime Numbers. New York: Wiley, 1985. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hdmrd/hdmrd.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Gourdon, X. and Sebah, P. "Some Constants from Number Theory." http://xavier.gourdon.free.fr/Constants/Miscella- neous/constantsNumTheory.html. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. "Mertens’s Theorem." §22.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 351 /C1/53 and 355, 1979. Ingham, A. E. The Distribution of Prime Numbers. London: Cambridge University Press, pp. 22 /C1/4, 1990. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, pp. 100 /C1/02, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 24, 1983. Lindqvist, P. and Peetre, J. "On the Remainder in a Series of Mertens." Expos. Math. 15, 467 /C1/78, 1997. Mertens, F. J. fu¨r Math. 78,46/C1/2, 1874. Montgomery, H. L. Topics in Multiplicative Number Theory. New York: Springer-Verlag, 1971. Rosser, J. B. and Schoenfeld, L. "Approximate Formulas for Some Functions of Prime Numbers." Ill. J. Math. 6,64/C1/4, 1962. Schroeder, M. R. Number Theory in Science and Commu- nication, with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. New York: Springer-Verlag, 1997. Mertens Function The summary function M(n) /C13Xn k/C301m(k) ; (1) where m(n) is the MO¨ BIUS FUNCTION . The first few values are 1, 0, -1, -1, -2, -1, -2, -2, -2, -1, -2, -2, ... (Sloane’s A002321). The first few values of n at which M(n) /C300 are 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, ... (Sloane’s A028442). The Mertens function is related to the number of SQUAREFREE integers up to n, which is the sum from 1 to n of the absolute value of m(k) ;Xn k /C301m(k) jj/C26 p2n /C27OffiffiffinpfflC{fflCz : (2) The Mertens function obeys Xx n/C301Mx n ! /C301 (3) (Lehman 1960). The analytic form is unsolved, although MERTENS CONJECTURE that M(x) jjB x1 =2 (4) has been disproved. Lehman (1960) gives an algorithm for computing M(x) with O x2=3 /C27efflC{fflCz operations, while the Lagarias- Odlyzko (1987) algorithm for computing the PRIME COUNTING FUNCTION p(x) can be modified to give M(x) in O x3=5/C27 efflC{fflCz operations. See also MERTENS CONJECTURE ,M O¨ BIUS FUNCTION , SQUAREFREE References Lagarias, J. and Odlyzko, A. "Computing p(x) : An Analytic Method." J. Algorithms 8, 173 /C1/91, 1987. Lehman, R. S. "On Liouville’s Function." Math. Comput. 14, 311 /C1/20, 1960. Lehmer, D. H. Guide to Tables in the Theory of Numbers. Bulletin No. 105. Washington, DC: National Research Council, pp. 7 /C1/0, 1941. Odlyzko, A. M. and te Riele, H. J. J. "Disproof of the Mertens Conjecture." J. reine angew. Math. 357, 138 /C1/ 60, 1985. Sloane, N. J. A. Sequences A002321/M0102 and A028442 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sterneck, R. D. von. "Empirische Untersuchung u¨ber den Verlauf der zahlentheoretischer Function s(n) /C30an x/C301 m(x) im Intervalle von 0 bis 150 000." Sitzungsber. der Kaiserlichen Akademie der Wissenschaften Wien, Math.- Naturwiss. Klasse 2a 106, 835/C1/024, 1897. Mertens Theorem lim x0/C12Q 25p5x pprime1/C281 p ! e/C28g lnx/C301; where gis the E ULER- MASCHERONI CONSTANT and e/C28g/C300:56145 . . . :/ See also EULER PRODUCT References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, p. 351, 1979. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 66 /C1/ 7, 1994. Mertz Apodization Function An asymmetrical APODIZATION FUNCTION defined by M(x; b; d) /C300 for x B/C28b (x /C28b) =(2b) for /C28b Bx Bb 1 for b Bx Bb /C272d 0 for x Bb /C272d;8 >>< >>: where the two-sided portion is 2b long (total) and the one-sided portion is b /C272d long (Schnopper and Thompson 1974, p. 508). The APPARATUS FUNCTION is MA(k ; b; d) /C30sin[2pk(b /C27 2d)] 2pk /C27icos[2 pk(b /C27 2d)] 2pk/C28sin(2 b) 4p2k2b() : References Schnopper, H. W. and Thompson, R. I. "Fourier Spectro- meters." In Methods of Experimental Physics 12A. New York: Academic Press, pp. 491 /C1/29, 1974. Mesh See also FINITE ELEMENT METHOD ,LATTICE POINT , MESH SIZE References Bern, M. and Plassmann, P. "Mesh Generation." Ch. 6 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 291 /C1/32, 2000. Mesh Size When a CLOSED INTERVAL [a, b] is partitioned by points a Bx1 Bx2 B...Bxn/C281 Bb ; the lengths of the resulting intervals between the points are denoted Dx1 ;Dx2 ; ..., Dxn ; and the value max Dxk is called the mesh size of the partition. See also INTEGRAL ,LOWER SUM,RIEMANN INTEGRAL , UPPER SUM Mesokurtic A distribution with zero KURTOSIS g2 /C300 ðÞ :/ See also KURTOSIS ,LEPTOKURTICM-Estimate A ROBUST ESTIMATION based on maximum likelihood argument. See also L-ESTIMATE , R-ESTIMATE References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Robust Estimation." §15.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 694 /C1/00, 1992. Metabiaugmented Dodecahedron JOHNSON SOLID J60:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Metabiaugmented Hexagonal Prism JOHNSON SOLID J56:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Metabiaugmented Truncated Dodecahedron JOHNSON SOLID J70 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Metabidiminished Icosahedron JOHNSON SOLID J62 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Metabidiminished Rhombicosidodecahedron JOHNSON SOLID J81 :/References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Metabigyrate Rhombicosidodecahedron JOHNSON SOLID J74 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Metacyclic Group See also CYCLIC GROUP References Mac Lane, S. and Birkhoff, G. Algebra. New York: Macmil- lan, p. 462, 1967. Metadrome A metadrome is a number whose HEXADECIMAL digits are in strict ascending order. The first few are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, ... (Sloane’s A023784). The first few numbers which are not metadromes are 16, 17, 32, 33, 34, ..., correspond- ing to 1016;1116;2016;2116;2216;.... See also DIGIT,H EXADECIMAL ,K ATADROME ,N IALP- DROME ,PLAINDROME References Sloane, N. J. A. Sequences A023784 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." M ATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Metagyrate Diminished Rhombicosidodecahedron JOHNSON SOLID J78 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Metalogic METAMATHEMATICS Metamathematics The branch of LOGIC dealing with the study of the combination and application of mathematical sym- bols, sometimes called METALOGIC . Metamathematics is the study of MATHEMATICS itself, and one of its primary goals is to determine the nature of mathe- matical reasoning (Hofstadter 1989). See also LOGIC ,MATHEMATICS References Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, p. 326, 1996. Chaitin, G. J. The Unknowable. New York: Springer-Verlag, 1999. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 23, 1989. Meteorology Theorem Somewhere on the Earth, there is a pair of ANTIPODAL POINTS having simultaneously the same temperature and pressure. References Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 121 and 284, 1997. Method A particular way of doing something, sometimes also called an ALGORITHM or PROCEDURE . (According to Petkovsek et al. (1996), "a method is a trick that has worked at least twice.")References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 117, 1996. Method of Exclusions A method used by Gauss to solve the quadratic DIOPHANTINE EQUATION OF THE FORM mx2 /C27ny2 /C30A (Dickson 1992, pp. 391 and 407). References Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, p. 407, 1992. Method of False Position An ALGORITHM for finding ROOTS which retains that prior estimate for which the function value has opposite sign from the function value at the current best estimate of the root. In this way, the method of false position keeps the root bracketed (Press et al. 1992). Using the two-point form of the line y/C28y1/C30fxn/C281 ðÞ /C28fx1ðÞ xn/C281/C28x1xn/C28x1 ðÞ with y/C300, using y1/C30fx1ðÞ ;and solving for xnthere- fore gives the iteration xn/C30x1/C28xn/C281/C28x1 fxn/C281 ðÞ /C28fx1ðÞfx1ðÞ : See also BRENT’S METHOD ,RIDDERS’ METHOD ,SE- CANT METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 18, 1972. Chabert, J.-L. (Ed.). "Methods of False Position." Ch. 3 in A History of Algorithms: From the Pebble to the Microchip.New York: Springer-Verlag, pp. 83 /C1 /12, 1999. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Secant Method, False Position Method, andRidders’ Method." §9.2 in Numerical Recipes in FOR- TRAN: The Art of Scientific Computing, 2nd ed. Cam- bridge, England: Cambridge University Press, pp. 347 /C1 / 52, 1992. Whittaker, E. T. and Robinson, G. "The Rule of False Position." §49 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 92 /C1/4, 1967. Method of Reduction METHOD OF EXCLUSIONS Metric A NONNEGATIVE function g(x; y) describing the "DIS- TANCE " between neighboring points for a given SET.A metric satisfies the TRIANGLE INEQUALITY g(x; y) /C27g(y; z) ]g(x; z) (1) and is SYMMETRIC ,so g(x ; y) /C30g(y; x): (2) A metric also satisfies g(x; x) /C300: (3) A SET possessing a metric is called a METRIC SPACE . When viewed as a TENSOR , the metric is called a METRIC TENSOR . See also CAYLEY- KLEIN- HILBERT METRIC ,DISTANCE , FRENCH METRO METRIC ,FUNDAMENTAL FORMS ,HY- PERBOLIC METRIC ,METRIC ENTROPY ,METRIC EQUIVA- LENCE PROBLEM ,M ETRIC SPACE ,M ETRIC TENSOR , PART METRIC ,RIEMANNIAN METRIC ,ULTRAMETRIC References Gray, A. "Metrics on Surfaces." Ch. 15 in Modern Differ- ential Geometry of Curves and Surfaces with Mathema- tica, 2nd ed. Boca Raton, FL: CRC Press, pp. 341 /C1/58, 1997. Metric Entropy Also known as KOLMOGOROV ENTROPY ,KOLMOGOROV- SINAI ENTROPY , or KS Entropy. The metric entropy is 0 for nonchaotic motion and > 0 for CHAOTIC motion. References Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 138, 1993. Metric Equivalence Problem 1. Find a complete system of invariants, or 2. decide when two METRICS differ only by a coordinate transformation. The most common statement of the problem is, "Given METRICS g and g?; does there exist a coordinate transformation from one to the other?" Christoffel and Lipschitz (1870) showed how to decide this question for two RIEMANNIAN METRICS . The solution by E´ . Cartan requires computation of the 10th order COVARIANT DERIVATIVES . The demon- stration was simplified by A. Karlhede using theTETRAD formalism so that only seventh order COVAR- IANT DERIVATIVES need be computed. however, in many common cases, the first or second-order DERI- VATIVES are SUFFICIENT to answer the question. References Karlhede, A. and Lindstro ¨m, U. "Finding Space-Time Geo- metries without Using a Metric." Gen. Relativity Gravita- tion 15, 597 /C1/10, 1983. Metric Space A SET S with a global distance FUNCTION (the METRIC g) which, for every two points x, y in S, gives the DISTANCE between them as a NONNEGATIVE REAL NUMBER g(x; y) : A metric space must also satisfy 1. g(x; y) /C300 IFF x /C30y, 2. g(x; y) /C30g(y; x) ;/ 3. The TRIANGLE INEQUALITY g(x; y)/C27/ /g(y;z)]g(x;z):/ See also UNIVERSAL METRIC SPACE References Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975. Rudin, W. Principles of Mathematical Analysis. New York: McGraw-Hill, 1976. Metric Tensor ATENSOR , also called a R IEMANNIAN METRIC , which is symmetric and POSITIVE DEFINITE . Very roughly, the metric tensor gijis a function which tells how to compute the distance between any two points in a given SPACE . Its components can be viewed as multi- plication factors which must be placed in front of the differential displacements dxiin a generalized P YTHA- GOREAN THEOREM ds2/C30g11dx2 1/C27g12dx1dx2/C27g22dx22/C27...: (1) In E UCLIDEAN SPACE ,gij/C30dijwhere dis the K RO- NECKER DELTA (which is 0 for i"jand 1 for i/C30j), reproducing the usual form of the P YTHAGOREAN THEOREM ds2/C30dx21/C27dx22/C27...: (2) The metric tensor is defined abstractly as an INNER PRODUCT of every TANGENT SPACE of a MANIFOLD such that the INNER PRODUCT is a symmetric, nondegene- rate, BILINEAR FORM on a VECTOR SPACE . This means that it takes two VECTORS v;was arguments and produces a REAL NUMBER v;w hi such that kv;w hi /C30kv;w hi /C30v;kw hi (3) v/C27w;x hi /C30v;x hi/C27w;x hi (4) v;w/C27x hi /C30v;w hi /C27v;x hi (5) v;w hi /C30w;v hi (6) v; v hi]0 ; (7) with equality IFF v /C300:/ In coordinate NOTATION (with respect to the basis), g ab /C30 /C0e a/C215 /C0eb (8) gab /C30 /C0ea/C215 /C0e b : (9) gmn /C13@ ja @xm@ jb @xnhab ; (10) where hab is the MINKOWSKI METRIC . This can also be written g /C30DT hD ; (11) where Dam /C13@ ja @xm (12) DT am /C13D ma : (13) @ @xmgilglk /C30@ @xmdk i (14) gives gil@glk @xm /C30/C28glk@gil @xm : (15) The metric is POSITIVE DEFINITE , so a metric’s DISCRIMINANT is POSITIVE . For a metric in 2-space, g /C13g11g22 /C28g2 12 > 0: (16) The ORTHOGONALITY of CONTRAVARIANT and COVAR- IANT metrics stipulated by gikgij /C30 dj k (17) for i /C301, ..., n gives n linear equations relating the 2n quantities gijand gij : therefore, if n metrics are known, the others can be determined. in 2-space, g11 /C30g22 g (18) g12 /C30g21 /C30/C28g12 g (19) g22 /C30g11 g: (20) if g is symmetric, then gab /C30g ba (21) g ab /C30g ba : (22) in EUCLIDEAN SPACE (and all other symmetricSPACES ), g b a /C30g ba /C30 db a ; (23) so gaa /C301 gaa : (24) The ANGLE f between two parametric curves is given by cos f /C30ˆr1/C215 ˆr2 /C30r1 g1/C215r2 g2/C30g12 g1g2; (25) so sin f /C30ffiffiffigp g1g2(26) and r1 /C29r2 jj /C30g1g2 sin f /C30ffiffiffigp: (27) The LINE ELEMENT can be written ds2 /C30dxi dxi /C30gij dqi dqj (28) where EINSTEIN SUMMATION has been used. But dxi /C30@xi @q1dq1 /C27@xi @q2dq2 /C27@xi @q3dq3 /C30@xi @qjdqj ; (29) so gij /C30X k@2xk @qi @qj: (30) For ORTHOGONAL coordinate systems, gij /C300 for i "j; and the LINE ELEMENT becomes (for 3-space) ds2 /C30g11 dq2 1 /C27g22 dq22 /C27g33 dq23 /C30 h1 dq1 ðÞ2/C27 h2 dq2 ðÞ2/C27 h3 dq3 ðÞ2; (31) where hi /C13ffiffiffiffiffigiipare called the SCALE FACTORS . See also CURVILINEAR COORDINATES ,DISCRIMINANT (METRIC ), LICHNEROWICZ CONDITION S,L INE ELE- MENT ,M ETRIC ,M ETRIC EQUIVALENCE PROBLEM , MINKOWSKI SPACE ,SCALE FACTOR ,SPACE Metropolis Algorithm SIMULATED ANNEALING Mex The MINIMUM excluded value. The mex of a SET S of NONNEGATIVE INTEGERS is the least NONNEGATIVE INTEGER notin the set. See also MEX SEQUENCE References Guy, R. K. "Max and Mex Sequences." §E27 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 227 /C1/28, 1994. Mex Sequence A sequence defined from a FINITE sequence a0 ; a1 ; ..., anby defining an/C271 /C30mexiai /C27an/C28i ðÞ ; where mex is the MEX (minimum excluded value). See also MAX SEQUENCE ,MEX References Guy, R. K. "Max and Mex Sequences." §E27 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 227 /C1/28, 1994. Mian-Chowla Sequence The sequence produced by starting with a1 /C301 and applying the GREEDY ALGORITHM in the following way: for each k ]2; let akbe the least INTEGER exceeding ak /C281for which aj /C27akare all distinct, with 1 5j 5k: This procedure generates the sequence 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, ... (Sloane’s A005282). The RECIPROCAL sum of the sequence, S /C13X/C12 i/C3011 ai satisfies 2 :158435 5S 52:158677 (R. Lewis). See also A-SEQUENCE ,B2-SEQUENCE References Mian, A. M. and Chowla, S. D. "On the B2/-Sequences of Sidon." Proc. Nat. Acad. Sci. India A14,3/C1/, 1944. Guy, R. K. "/B2/-Sequences." §E28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 228 /C1/29, 1994. Sloane, N. J. A. Sequences A005282/M1094 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Mice Problem n mice start at the corners of a regular n-gon of unit side length, each heading towards its closest neigh- boring mouse in a counterclockwise direction atconstant speed. The mice each trace out a LOGARITH- MIC SPIRAL , meet in the center of the POLYGON , and travel a distance dn /C301 1 /C28 cos2p n ! : The first few values for n /C302, 3, ..., are 1 2 ;23 ; 1;155 /C27ffiffiffi 5pfflCz6fflCz7 ; 2;1 1 /C28 cos2p 7 ! ; 2 /C27ffiffiffi2p ; 1 1 /C28 cos2 p 9 ! ; 3 /C27ffiffiffiffi 5;p ... ; giving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a WHIRL . The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be general- ized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) con- sidered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original. See also APOLLONIUS PURSUIT PROBLEM ,P URSUIT CURVE ,SPIRAL ,TRACTRIX ,W HIRL References Bernhart, A. "Polygons of Pursuit." Scripta Math. 24,2 3/C1/0, 1959. Brocard, H. "Solution of Lucas’s Problem." Nouv. Corresp. Math. 3, 280, 1877. Clapham, A. J. Rec. Math. Mag. , Aug. 1962. Gardner, M. The Scientific American Book of Mathematical Puzzles and Diversions. New York: NY: Simon and Schuster, 1959. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 240 /C1/43, 1984. Good, I. J. "Pursuit Curves and Mathematical Art." Math. Gaz. 43,3 4/C1/5, 1959. Lucas, E. "Problem of the Three Dogs." Nouv. Corresp. Math. 3, 175/C1/76, 1877. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 201 /C1/04, 1979. Miller, R. K. Problem 16. Cambridge Math. Tripos Exam. January 5, 1871. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 136, 1999. Weisstein, E. W. "Mice Problem." M ATHEMATICA NOTEBOOK MICEPROBLEM.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 201 /C1/02, 1991. Wilson, J. "Problem: Four Dogs." http://jwilson.coe.uga.edu/ emt725/Four.Dogs/four.dogs.html. Microlocal Analysis References Demuth, M.; Schrohe, E.; Schulze, B.-E.; and Sjo¨strand, J. (Eds.). Spectral Theory, Microlocal Analysis, Singular Manifolds. Berlin: Akademie Verlag, 1997. Grigis, A. and Sjo¨strand, J. Microlocal Analysis for Differ- ential Operators: An Introduction. Cambridge, England: Cambridge University Press, 1994. Sjo¨strand, J. "Singularite ´s analytiques microlocales." Aste´r- isque 95,1/C1/66, 1982. Mid-Arc Points The mid-arc points MAB ; MAC ; and MBC of a TRIANGLE DABC are the points on the CIRCUMCIRCLE of the triangle which lie half-way along each of the three ARCS determined by the vertices (Johnson 1929). These points arise in the definition of the FUHRMANN CIRCLE and FUHRMANN TRIANGLE , and lie on the extensions of the PERPENDICULAR BISECTORS of the triangle sides drawn from the CIRCUMCENTER O. Kimberling (1988, 1994) and Kimberling and Veld- kamp (1987) define the mid-arc points as the POINTS which have TRIANGLE CENTER FUNCTIONS a1 /C30 cos1 2 BfflCz6fflCz7 /C27cos12 CfflCz6fflCz7 hi sec12 AfflCz6fflCz7 a2 /C30 cos1 2 BfflCz6fflCz7 /C27cos12 CfflCz6fflCz7 hi csc12 AfflCz6fflCz7 : See also ARC,C YCLIC QUADRILATERAL ,FUHRMANN CIRCLE ,FUHRMANN TRIANGLE References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 228 /C1/29, 1929. Kimberling, C. "Problem 804." Nieuw Archief voor Wiskunde 6, 170, 1988. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994.Kimberling, C. and Veldkamp, G. R. "Problem 1160 and Solution." Crux Math. 13, 298 /C1/99, 1987. Midcircle The midcircle of two given CIRCLES is the CIRCLE which would INVERT the circles into each other. Dixon (1991) gives constructions for the midcircle for four of the five possible configurations. In the case of the two given CIRCLES tangent to each other, there are two midcircles. See also INVERSION ,INVERSION CIRCLE References Dixon, R. Mathographics. New York: Dover, pp. 66 /C1/8, 1991. Middlespoint MITTENPUNKT Midpoint The point on a LINE SEGMENT dividing it into two segments of equal length. The midpoint of a line segment is easy to locate by first constructing a LENS using circular arcs, then connecting the cusps of the LENS . The point where the cusp-connecting line intersects the segment is then the midpoint (Pedoe1995, p. xii). It is more challenging to locate themidpoint using only a COMPASS (i.e., a M ASCHERONI CONSTRUCTION ). In a RIGHT TRIANGLE , the midpoint of the HYPOTE- NUSE is equidistant from the three VERTICES (Dun- ham 1990). Given a TRIANGLE da1a2a3with AREA d ; locate the midpoints mi : now inscribe two triangles dp1p2p3 and dq1q2q3with VERTICES Piand Qiplaced so that PiMi /C30QiMi : Then DP1P2P3 and DQ1Q2Q3 have equal areas DP /C30DQ /C30D 1 /C28m1 a1/C27m2 a2/C27m3 a3 ! /C27m2m2 a2a3/C27m3m1 a3a1/C27m1m2 a1a2"# ; where ai are the sides of the original triangle and mi are the lengths of the MEDIANS (Johnson 1929). See also ANTICENTER ,ARCHIMEDES’ MIDPOINT THEO- REM,BIMEDIAN ,BRAHMAGUPTA’S THEOREM ,BROCARD MIDPOINT ,CIRCLE- POINT MIDPOINT THEOREM ,CLEA- VER,D ROZ-FARNY THEOREM ,LINE SEGMENT ,M AL- TITUDE ,M ASCHERONI CONSTRUCTION ,M EDIAN (TRIANGLE ), MEDIATOR ,MIDPOINT ELLIPSE References Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120 /C1/21, 1990. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 80, 1929. Midpoint Ellipse The unique ELLIPSE tangent to the MIDPOINTS of a TRIANGLE’S LEGS . The midpoint ellipse has the max- imum AREA of any INSCRIBED ELLIPSE (Chakerian 1979). Under an AFFINE TRANSFORMATION , the mid- point ellipse can be transformed into the INCIRCLE of an EQUILATERAL TRIANGLE . See also AFFINE TRANSFORMATION ,ELLIPSE ,INCIR- CLE,MIDPOINT ,TRIANGLE References Central Similarities. University of Minnesota College Geo- metry Project. Distributed by International Film Bureau, Inc. Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 135 /C1/36 and 145 /C1/46, 1979.Pedoe, D. "Thinking Geometrically." Amer. Math. Monthly 77, 711 /C1/21, 1970. Midpoint Polygon A DERIVED POLYGON with side ratios chosen as r /C301=2 so that inscribed polygons are constructed by con- necting the midpoints of the base polygon. For a TRIANGLE P, the midpoint-inscribed polygons P1 ; P2 ; ... are similar triangles. For a QUADRILATERAL P, the midpoint-inscribed polygon P1is a PARALLELOGRAM known as the VARIGNON PARALLELOGRAM , and P1 ; P3 ; P5 ; ... are similar parallelograms, as are P2 ; P4 ; P6 ; .... See also DERIVED POLYGON ,M IDPOINT ,V ARIGNON PARALLELOGRAM ,VARIGNON’S THEOREM References Tischel, G. "Ein Konvergenzsatz fu ¨r Mittenpolygone." Mitt. Math. Ges. Hamburg 18, 169/C1/84, 1999. Midradius The RADIUS rof the MIDSPHERE of a POLYHEDRON , also called the interradius. Let Pbe a point on the original polyhedron and P?the corresponding point P on the dual. Then because Pand P?are INVERSE POINTS , the radii r/C30OP?;R/C30OP, and r/C30OQsatisfy rR/C30r2: The above figure shows a plane section of a mid- sphere. Let rbe the INRADIUS the dual polyhedron, R CIRCUMRADIUS of the original polyhedron, and athe side length of the original polyhedron. (For a P LA- TONIC SOLID or A RCHIMEDEAN SOLID ,ris not only the INRADIUS of the dual polyhedron, but also the INRA- DIUS of the original polyhedron.) For a REGULAR POLYHEDRON with S CHLA ¨FLI SYMBOL fq;pg;the DUAL POLYHEDRON isfp;qg:Then r2/C30acscp p !"#2 /C27R2/C30a2/C27r2(1) r2 /C30 a cotp p !"#2 /C27R2 : (2) Furthermore, let u be the ANGLE subtended by the EDGE of an ARCHIMEDEAN SOLID . Then r /C301 2 a cos12 ufflCz6fflCz7 cot12 ufflCz6fflCz7 (3) r /C301 2 a cot12 ufflCz6fflCz7 (4) R /C301 2 a csc12 ufflCz6fflCz7 ; (5) so r : r : R /C30cos12 ufflCz6fflCz7 :1:sec12 ufflCz6fflCz7 (6) (Cundy and Rollett 1989). Expressing the midradius in terms of the INRADIUS r and CIRCUMRADIUS R gives r /C3012ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27a2pq /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C281 4 a2q (7) for an ARCHIMEDEAN SOLID . References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 126 /C1/27, 1989. Midrange midrange[ f(x)] /C131 2 fmax[ f(x)] /C27min[ f(x)]g: See also MAXIMUM ,M EAN,M EDIAN (STATISTICS ), MINIMUM References Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995. Midsphere The SPHERE with respect to which the VERTICES of a POLYHEDRON are the POLES of the planes of the faces of the DUAL POLYHEDRON (and vice versa), also called the intersphere, reciprocating sphere, or INVERSION SPHERE . The midsphere touches all EDGES of a SEMIREGULAR or REGULAR POLYHEDRON , as well as the edges of the dual of that solid (Cundy and Rollett 1989, p. 117). The radius r of the midsphere is calledthe MIDRADIUS . The figure above shows the Platonic solids and their duals, with the CIRCUMSPHERE of the solid, MIDSPHERE , and INSPHERE of the dual super- posed. See also CIRCUMSPHERE ,D UAL POLYHEDRON ,IN- SPHERE ,MIDRADIUS ,POLE (INVERSION ) References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 16, 1973. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Midvalue CLASS MARK Midy’s Theorem If the period of a REPEATING DECIMAL for a =p has an EVEN number of digits, the sum of the two halves is a string of 9s, where p is PRIME and a =p is a REDUCED FRACTION . See also DECIMAL EXPANSION ,REPEATING DECIMAL References Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 158 /C1/60, 1957. Mikusinski’s Problem Is it possible to cover completely the surface of a SPHERE with congruent, nonoverlapping arcs of GREAT CIRCLES ? Conway and Croft (1964) proved that it can be covered with half-open arcs, but not with open arcs. They also showed that the PLANE can be covered with congruent closed and half-open segments, but not with open ones. References Conway, J. H. and Croft, H. T. "Covering a Sphere with Great-Circle Arcs." Proc. Cambridge Phil. Soc. 60, 787 /C1/ 00, 1964. Gardner, M. "Point Sets on the Sphere." Ch. 12 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 145 /C1/54, 1986. Milin Conjecture An INEQUALITY which IMPLIES the correctness of the ROBERTSON CONJECTURE (Milin 1971). de Branges (1985) proved this conjecture, which led to the proof of the full B IEBERBACH CONJECTURE . See also BIEBERBACH CONJECTURE ,ROBERTSON CON- JECTURE References de Branges, L. "A Proof of the Bieberbach Conjecture." Acta Math. 154, 137/C1/52, 1985. Milin, I. M. "The Area Method in the Theory of Univalent Functions." Dokl. Acad. Nauk SSSR 154, 264 /C1/67, 1964. Milin, I. M. Univalent Functions and Orthonormal Systems. Providence, RI: Amer. Math. Soc., 1977. Stewart, I. From Here to Infinity: A Guide to Today’s Mathematics. Oxford, England: Oxford University Press, p. 165, 1996. Mill Curve The n-roll mill curve is given by the equation xn /C28n 2fflCzrfflCzD xn/C282y2 /C27n 4fflCzrfflCzD xn/C284y4 /C28/C1/C1/C1/C30an ; wheren kfflC{fflCz is a BINOMIAL COEFFICIENT . References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 86, 1993. Miller-As ˇkinuze Solid ELONGATED SQUARE GYROBICUPOLA Miller Cylindrical Projection A MAP PROJECTION given by the following transforma- tion, x /C30 l /C28 l0 (1) y /C305 4 ln tan14 p /C2725 ffflCz6fflCz7hi (2) /C305 4sinh/C281 tan45 ffflCz6fflCz7hi : (3) Here x and y are the plane coordinates of a projected point, l is the longitude of a point on the globe, l0 iscentral longitude used for the projection, and f is the latitude of the point on the globe. The inverse FORMULAS are f /C305 2tan/C281 e4y=5fflC{fflCz /C2858 p /C3054tan /C281 sinh45 yfflCz6fflCz7hi (4) l /C30 l0 /C27x: (5) See also EQUIDISTANT PROJECTION ,M ILLER EQUIDI- STANT PROJECTION References Miller, O. M. "Notes on a Cylindrical World Map Projection." Geograph. Rev. 32, 424 /C1/30, 1942. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 86 /C1/9, 1987. United States Geological Survey. National Atlas of the United States. Washington, DC: USGS, pp. 330 /C1/31, 1970. Miller Equidistant Projection Several CYLINDRICAL EQUIDISTANT PROJECTIONS were devised by R. Miller. Miller’s projections have stan- dard parallels of f1 /C3037/C1430 ? (giving minimal overall scale distortion), f1/C3043/C14(giving minimal scale dis- tortion over continents), and f1/C3050/C1428?(Miller 1949). See also CYLINDRICAL EQUIDISTANT PROJECTION , MILLER CYLINDRICAL PROJECTION References Miller, R. "An Equi-Rectangular Map Projection." Geogra- phy Rev. 34, 196 /C1/01, 1949. Miller, R. "Correction to: An Equi-Rectangular Map Projec- tion." Geography 36, 270, 1951. Snyder, J. P. Flattening the Earth: Two Thousand Years of Map Projections. Chicago, IL: University of Chicago Press, 1993. Miller’s Algorithm For a catastrophically unstable recurrence in one direction, any seed values for consecutive xj and xj/C271 will converge to the desired sequence of functions in the opposite direction times an unknown normal- ization factor. Miller’s Primality Test If a number fails this test, it is not a PRIME . If the number passes, it may be a PRIME . A number passing Miller’s test is called a STRONG PSEUDOPRIME to base a. If a number n does not pass the test, then it is called a WITNESS for the COMPOSITENESS of n.Ifn is an ODD, POSITIVE COMPOSITE NUMBER , then n passes Miller’s test for at most (n /C281)=4 bases with 1 5a 5 /C281 (Long 1995). There is no analog of CARMICHAEL NUMBERS for STRONG PSEUDOPRIMES . The only COMPOSITE NUMBER less than 2:5 /C291013 which does not have 2, 3, 5, or 7 as a WITNESS is 3215031751. Miller showed that any composite n has a WITNESS less than 70(ln n)2 if the RIEMANN HYPOTH- ESIS is true. See also ADLEMAN- POMERANCE- RUMELY PRIMALITY TEST,STRONG PSEUDOPRIME References Long, C. T. Th. 4.21 in Elementary Introduction to Number Theory, 3rd ed. Prospect Heights, IL: Waveland Press, 1995. Miller’s Solid ELONGATED SQUARE GYROBICUPOLA Milliard In British, French, and German usage, one milliard equals 109. American usage does not have a number called the milliard, instead using the term BILLION to denote 109. See also BILLION ,LARGE NUMBER ,MILLION ,TRILLION Millin Series The series with sum S?/C13X/C12 n/C3001 F2n/C301 27 /C28ffiffiffi 5pfflCz6fflCz7 ; where /Fk/ is a FIBONACCI NUMBER (Honsberger 1985).See also FIBONACCI NUMBER References Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 135 /C1/37, 1985. Million The number 1,000,000 /C30106. While one million in the "American" system of numbers means the same thing as one million in the "British" system, the words BILLION , TRILLION , etc., refer to different numbers in the two naming systems. Fortunately, in recent years, the "American" system has become common in both the United States and Britain. While Americans may say "Thanks a million" to express gratitude, Norwegians offer "Thanks a thou- sand" ("tusen takk"). See also BILLION ,LARGE NUMBER ,M ILLIARD ,THOU- SAND ,TRILLION Mills’ Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Mills (1947) proved the existence of a constant u/C301:306377883863080690 . . . (1) (Sloane’s A051021) such that f(n)/C30u3nfflC)fflCq (2) isPRIME for all n/]1;where xbcis the FLOOR FUNC- TION . It is not, however, known if uisIRRATIONAL . The first few values of f(n) are 2, 11, 1361, 2521008887, ... (Sloane’s A051254).Mills’ proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let p nbe the nth PRIME , then there exists a constant Ksuch that pn/C271/C28pnBKp5=8 n (3) for all n. This has more recently been strengthened to pn/C271/C28pnBKp1051 =1920 n (4) (Mozzochi 1986). If the R IEMANN HYPOTHESIS is true, then Crame ´r (1937) showed that pn/C271/C28pn/C30OlnpnffiffiffiffiffipnpfflC{fflCz (5) (Finch). Hardy and Wright (1979) and Ribenboim (1996) point out that, despite the beauty of such PRIME FORMULAS , they do not have any practical consequences. In fact,unless the exact value of uis known, the PRIMES themselves must be known in advance to determine u: The numbers generated by f(n) grow very rapidly, with the first few being 2, 11, 1361, .... A generalization of Mills’ theorem to an arbitrary sequence of POSITIVE INTEGERS is given as an exercise by Ellison and Ellison (1985). Consequently, infi- nitely many values for u other than the number 1:3063 ... are possible. See also CEILING FUNCTION ,PRIME FORMULAS ,PRIME NUMBER References Caldwell, C. "Mills’ Theorem--A Generalization." http:// www.utm.edu/research/primes/notes/proofs/A3n.html. Ellison, W. and Ellison, F. Prime Numbers. New York: Wiley, pp. 31 /C1/2, 1985. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/mills/mills.html. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Mills, W. H. "A Prime-Representing Function." Bull. Amer. Math. Soc. 53, 604, 1947. Mozzochi, C. J. "On the Difference Between Consecutive Primes." J. Number Th. 24, 181 /C1/87, 1986. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 65, 1951. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 186 /C1/87, 1996. Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, pp. 109 /C1/10, 1991. Sloane, N. J. A. Sequences A051021 and A051254 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Mills-Robbins-Rumsey Determinant Formula deti /C27j /C27 m 2i /C28jfflCzrfflCzDn/C281 i; j /C300/C302/C28nYn/C281 k/C300D2k(2 m); where m is an indeterminate, D0( m) /C302; D2j( m) /C30(m /C27 2j /C27 2)j1 2 m2j /C2732fflCz6fflCz7 j/C281 (j)j12 m /C27 j /C2732fflCz6fflCz7 j/C281; for j /C301, 2, ..., and (x)j /C30x(x /C271) /C1/C1/C1(x /C27j /C281) is the RISING FACTORIAL (Mills et al. 1987, Andrews and Burge 1993). References Andrews, G. E. and Burge, W. H. "Determinant Identities." Pacific J. Math. 158,1/C1/4, 1993. Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr. "Enumera- tion of a Symmetry Class of Plane Partitions." Discrete Math. 67,43/C1/5, 1987. Petkovsek, M. and Wilf, H. S. "A High-Tech Proof of the Mills-Robbins-Runsey Determinant Formula." Electronic J. Combinatorics 3, No. 2, R19, 1 /C1/, 1996. http://www.com- binatorics.org/Volume_3/volume3_2.html. Milne’s Method A PREDICTOR-CORRECTOR METHOD for solution of ORDINARY DIFFERENTIAL EQUATIONS . The third-order equations for predictor and corrector areyn/C271 /C30yn/C283 /C2743 h(2y?n /C28y?n /C281 /C272y?n /C282) /C27O(h5) yn/C271 /C30yn/C281 /C2713 h(y?n/C281 /C284y?n /C27y?n /C271) /C27O(h5): Abramowitz and Stegun (1972) also give the fifth order equations and formulas involving higher deri- vatives. See also ADAMS’ METHOD ,GILL’S METHOD ,PREDIC- TOR-CORRECTOR METHODS ,RUNGE- KUTTA METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 896 /C1/97, 1972. Milnor’s Conjecture The UNKNOTTING NUMBER for a TORUS KNOT (p, q)is (p /C281)(q /C281)=2: This 40-year-old CONJECTURE was proved (Adams 1994) in Kronheimer and Mrowka (1993, 1995). See also TORUS KNOT,UNKNOTTING NUMBER References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 113, 1994. Kronheimer, P. B. and Mrowka, T. S. "Gauge Theory for Embedded Surfaces. I." Topology 32, 773 /C1/26, 1993. Kronheimer, P. B. and Mrowka, T. S. "Gauge Theory for Embedded Surfaces. II." Topology 34,37/C1/7, 1995. Milnor’s Theorem If a COMPACT MANIFOLD M has NONNEGATIVE RICCI CURVATURE , then its FUNDAMENTAL GROUP has at most POLYNOMIAL growth. On the other hand, if M has NEGATIVE curvature, then its FUNDAMENTAL GROUP has exponential growth in the sense that n( l) grows exponentially, where n( l) is (essentially) the number of different "words" of length l which can be made in the FUNDAMENTAL GROUP . References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Min MINIMUM Mincut Let G /C30(V ; E) be a (not necessarily simple) UNDIR- ECTED edge-weighted graph with nonnegative weights. A cut C of G is any nontrivial subset of V, and the weight of the cut is the sum of weights of edges crossing the cut. A mincut is then defined as a cut of Gof minimum weight. The problem is NP- complete for general graphs, but polynomial-timesolvable for trees. See also B OOLEAN FUNCTION ,W EIGHTED GRAPH References Stoer, M. and Wagner, F. "A Simple Min Cut Algorithm." Algorithms--ESA ’94, LNCS 855, 141 /C1/47, 1994. Minimal Cover A minimal cover is a COVER for which removal of any single member destroys the covering property. For example, of the five COVERS of f1; 2g; namely ff1g;f2 gg;ff1; 2gg;ff1 g;f1 ; 2 gg;ff2g;f1; 2gg; and ff1g;f2g;f1; 2gg; only ff1 g;f2 gg and ff1; 2gg are minimal covers. Similarly, the minimal covers of f1; 2; 3g are given by ff1g;f2g;f3gg;ff1 ; 2 g;f3gg; ff1; 3g;f2gg; ff1; 2g;f2; 3gg; ff1; 2g;f2; 3gg; ff1; 2; 3gg;ff1 ; 2 g;f1 ; 3 gg;ff1 ; 2 g;f2 ; 3 gg: The number of minimal covers of n members for n /C301, 2, ..., are 1, 2, 8, 49, 462, 6424, 129425, ... (Sloane’s A046165). Let m(n ; k) be the number of minimal covers of f1; ...; ng with k members. Then m(n; k) /C301 k!Xak m/C30k2k /C28k /C281 m /C28kfflCzrfflCzD m!s(n; m) ; wheren kfflC{fflCz is a BINOMIAL COEFFICIENT , s(n; m)isa STIRLING NUMBER OF THE SECOND KIND , and ak /C30min( n; 2k /C281): Special cases include m(n ; 1) /C301 and m(n; 2) /C30s(n /C27 1; 3): The table below gives the a triangle of m(n ; k) (Sloane’s A035348). nk/C30 1 k /C30 2 k /C30 3 k /C30 4 k /C30 5 k /C30 6 k /C30 7 Sloane Sloane’s A000392Sloane’s A003468Sloane’s A016111Sloane’s A046166Sloane’s A046167Sloane’s A057668 11 21 1 31 6 1 4 1 25 22 1 5 1 90 305 65 1 6 1 301 3410 2540 171 1 7 1 966 33621 77350 17066 420 1 8 1 3925 305382 2022951 1298346 100814 988 See also COVER ,LEW K-GRAM ,STIRLING NUMBER OF THE SECOND KIND References Hearne, T. and Wagner, C. "Minimal Covers of Finite Sets." Disc. Math. 5, 247 /C1/51, 1973. Macula, A. J. "Covers of a Finite Set." Math. Mag. 67, 141 /C1/ 44, 1994. Macula, A. J. "Lewis Carroll and the Enumeration of Minimal Covers." Math. Mag. 68, 269 /C1/74, 1995. Sloane, N. J. A. Sequences A000392, A003468, A016111, A035348, A046165, A046166, A046167, A046168, andA057668 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Minimal Discriminant FREY CURVE Minimal Matrix A MATRIX with 0 DETERMINANT whose DETERMINANT becomes NONZERO when any element on or below the diagonal is changed from 0 to 1. An example is M /C301 /C28100 00 /C2810 111 /C281 00102 6643 775: There are 2 n /C281 minimal SPECIAL MATRICES of size n/C29n:/ See also SPECIAL MATRIX References Knuth, D. E. "Problem 10470." Amer. Math. Monthly 102, 655, 1995. Minimal Polynomial (Matrix) The minimal polynomial of a matrix Ais the poly- nomial in Aof smallest degree nsuch that p(A)/C30Xn i/C300ciAi/C300: (1) The minimal polynomial divides any polynomial q with q(A)/C300and, in particular, it divides the CHAR- ACTERISTIC POLYNOMIAL . If the CHARACTERISTIC POLY- NOMIAL factors as char( A)(x)/C30(x/C28l1)n1...(x/C28lk)nk; (2) then its minimal polynomial is p(x)/C30(x/C28l1)m1...(x/C28lk)mk(3) with 15mi5ni:/ For example, the CHARACTERISTIC POLYNOMIAL of the n/C29nZERO MATRIX is (/C281)nxn;and its minimal poly- nomial is x. The CHARACTERISTIC POLYNOMIAL and minimal polynomial of 01 00fflC}{fflC}z (4) are the same (up to scalar multiple), x2:/ The following Mathematica command will find the minimal polynomial for the SQUARE MATRIX ain the variable x. MinPolyMatrix[a_List,x_]: /C30 Modu- le[{i,n /C301,qu /C30{},mnm /C30{Flatten[IdentityMatr- {Flatten[IdentityMatrix[Length[a]]]}}, While[Length[qu] /C30/C300, AppendTo[mnm,Flatten[MatrixPower[a,n]]]; qu /C30NullSpace[Transpose[mnm]]; n/C27/C27 ];First[qu].Table[x^i,{i,0,n-1}] ] See also CAYLEY- HAMILTON THEOREM ,CHARACTERIS- TIC POLYNOMIAL ,M INIMAL POLYNOMIAL (ALGEBRAIC NUMBER ), RATIONAL CANONICAL FORM References Dummit, D. and Foote, R. Abstract Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991. Herstein, I. §6.7 in Topics in Algebra, 2nd ed. New York: Wiley, 1975. Jacobson, N. §3.10 in Basic Algebra I. New York: W. H. Freeman, 1985. Minimal Residue The value b or b /C28m; whichever is smaller in ABSOLUTE VALUE , where a /C13b (mod m) :/ See also RESIDUE (CONGRUENCE ) Minimal Set A SET for which the dynamics can be generated by the dynamics on any SUBSET . Minimal Surface Minimal surfaces are defined as surfaces with zero MEAN CURVATURE . A minimal surface parametrized as x /C30(u ; v ; h(u; v)) therefore satisfies LAGRANGE’S EQUATION , 1 /C27f2 vfflC{fflCz fuu /C272fufvfuv /C27 1 /C27f2 ufflC{fflCz fvv /C300 : Finding a minimal surface of a boundary with specified constraints is a problem in the CALCULUS OF VARIATIONS and is sometimes known at PLATEAU’S PROBLEM . Minimal surfaces may also be character- ized as surfaces of minimal SURFACE AREA for given boundary conditions. A PLANE is a trivial MINIMAL SURFACE , and the first nontrivial examples (the CATENOID and HELICOID ) were found by Meusnier in 1776 (Meusnier 1785). The problem of finding the minimum bounding surface of a SKEW QUADRILAT- ERAL was solved by Schwarz (1890). Note that while a SPHERE is a "minimal surface" in the sense that it minimizes the surface area-to- volume ratio, it does not qualify as a minimal surface in the sense used by mathematicians. Euler proved that a minimal surface is planar IFF its GAUSSIAN CURVATURE is zero at every point so that it is locally SADDLE -shaped. The EXISTENCE of a solutionto the general case was independently proven by Douglas (1931) and Rado´ (1933), although their analysis could not exclude the possibility of singula- rities. Osserman (1970) and Gulliver (1973) showed that a minimizing solution cannot have singularities. The only known complete (boundaryless), embedded (no self-intersections) minimal surfaces of finite topology known for 200 years were the CATENOID , HELICOID , and PLANE . Hoffman discovered a three- ended GENUS 1 minimal embedded surface, and demonstrated the existence of an infinite number of such surfaces. A four-ended embedded minimal sur- face has also been found. L. Bers proved that any finite isolated SINGULARITY of a single-valued para- meterized minimal surface is removable. A surface can be parameterized using a ISOTHERMAL PARAMETERIZATION . Such a parameterization is mini- mal if the coordinate functions xk are HARMONIC , i.e., fk( z) are ANALYTIC . A minimal surface can therefore be defined by a triple of ANALYTIC FUNCTIONS such that fk fk /C300: The REAL parameterization is then obtained as xk/C30Rgfk(z)dz: (1) But, for an ANALYTIC FUNCTION fand a MEROMORPHIC FUNCTION g, the triple of functions f1(z)/C30f(1/C28g2) (2) f2(z)/C30if(1/C27g2) (3) f3(z)/C302fg (4) are ANALYTIC as long as fhas a zero of order ]mat every POLE ofgof order m. This gives a minimal surface in terms of the E NNEPER- WEIERSTRASS PARA- METERIZATION Rgf(1/C28g2) if(1/C27g2) 2fg2 435dz: (5) See also B ERNSTEIN MINIMAL SURFACE THEOREM , BOUR’S MINIMAL SURFACE ,B UBBLE ,C ALCULUS OF VARIATIONS ,C ATALAN’S SURFACE ,C ATENOID ,C OM- PLETE MINIMAL SURFACE ,COSTA MINIMAL SURFACE , DOUBLE BUBBLE ,ENNEPER’S MINIMAL SURFACE ,EN- NEPER- WEIERSTRASS PARAMETERIZATION ,FLAT SUR- FACE ,G YROID ,H ELICOID ,H ENNEBERG’S MINIMAL SURFACE ,HOFFMAN’S MINIMAL SURFACE ,IMMERSED MINIMAL SURFACE ,LICHTENFELS MINIMAL SURFACE , LOPEZ MINIMAL SURFACE ,M EAN CURVATURE ,N IR- ENBERG’S CONJECTURE ,O LIVEIRA’S MINIMAL SUR- FACE ,PARAMETERIZATION ,PLANE ,PLATEAU’S LAWS, PLATEAU’S PROBLEM ,SCHERK’S MINIMAL SURFACES , SCHWARZ’S MINIMAL SURFACE ,SURFACE AREA,TRI- NOID References Darboux, G. Lec¸ons sur la the´orie ge´ne´rale des surfaces. Paris: Gauthier-Villars, 1941. Dickson, S. "Minimal Surfaces." Mathematica J. 1,38/C1/0, 1990. Dierkes, U.; Hildebrandt, S.; Ku¨ster, A.; and Wohlraub, O. Minimal Surfaces, Vol. 1: Boundary Value Problems. New York: Springer-Verlag, 1992. Dierkes, U.; Hildebrandt, S.; Ku¨ster, A.; and Wohlraub, O. Minimal Surfaces, Vol. 2: Boundary Regularity. New York: Springer-Verlag, 1992. do Carmo, M. P. "Minimal Surfaces." §3.5 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 41 /C1/3, 1986. Douglas, J. "Solution of the Problem of Plateau." Trans. Amer. Math. Soc. 33, 263 /C1/21, 1931. Fischer, G. (Ed.). Plates 93 and 96 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Vo- lume. Braunschweig, Germany: Vieweg, pp. 89 and 96, 1986. Gray, A. "Minimal Surfaces" and "Minimal Surfaces and Complex Variables." Ch. 30 and 31 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 681 /C1/34, 1997. Gulliver, R. "Regularity of Minimizing Surfaces of Pre- scribed Mean Curvature." Ann. Math. 97, 275 /C1/05, 1973. Hoffman, D. "The Computer-Aided Discovery of New Em- bedded Minimal Surfaces." Math. Intell. 9,8/C1/1, 1987. Hoffman, D. and Meeks, W. H. III. The Global Theory of Properly Embedded Minimal Surfaces. Amherst, MA: University of Massachusetts, 1987. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, 1992. Lagrange. "Essai d’une nouvelle me´thode pour de´terminer les maxima et les minima des formules inte´grales inde´- finies." 1776. Meusnier, J. B. "Me´moire sur la courbure des surfaces." Me´m. des savans e´trangers 10 (lu 1776), 477 /C1/10, 1785. Nitsche, J. C. C. Introduction to Minimal Surfaces. Cam- bridge, England: Cambridge University Press, 1989. Osserman, R. A Survey of Minimal Surfaces. New York: Dover, 1986. Osserman, R. "A Proof of the Regularity Everywhere of the Classical Solution to Plateau’s Problem." Ann. Math. 91, 550 /C1/69, 1970. Osserman, R. (Ed.). Minimal Surfaces. Berlin: Springer- Verlag, 1997. Rado´, T. "On the Problem of Plateau." Ergeben. d. Math. u. ihrer Grenzgebiete. Berlin: Springer-Verlag, 1933. Schwarz, H. A. Gesammelte Mathematische Abhandlungen, 2nd ed. New York: Chelsea. Weisstein, E. W. "Books about Minimal Surfaces." http:// www.treasure-troves.com/books/MinimalSurfaces.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 185 /C1/87, 1991. Minimax Approximation A minimization of the MAXIMUM error for a fixed number of terms. See also REMEZ ALGORITHM Minimax Polynomial The approximating POLYNOMIAL which has the smal- lest maximum deviation from the true function. It is closely approximated by the CHEBYSHEV POLYNO- MIALS OF THE FIRST KIND .Minimax Theorem The fundamental theorem of GAME THEORY which states that every FINITE , ZERO-SUM , two-person GAME has optimal MIXED STRATEGIES . It was proved by John von Neumann in 1928. Formally, let X and Y be MIXED STRATEGIES for players A and B. Let A be the PAYOFF MATRIX . Then max Xmin YXTAY /C30min Ymax XXTAY /C30v; where v is called the VALUE of the GAME and X and Y are called the solutions. It also turns out that if there is more than one optimal MIXED STRATEGY , there are infinitely many. See also GAME,GAME THEORY ,MIXED STRATEGY References Willem, M. Minimax Theorem. Boston, MA: Birkha ¨user, 1996. Minimize INFIMUM Minimum The smallest value of a set, function, etc. The minimum value of a set of elements A/C30faigN i/C301is denoted min Aor miniai;and is equal to the first element of a sorted (i.e., ordered) version of A. For example, given the set f3;5;4;1g;the sorted version isf1;3;4;5g;so the minimum is 1. The MAXIMUM and minimum are the simplest ORDER STATISTICS . A continuous FUNCTION may assume a minimum at a single point or may have minima at a number of points. A GLOBAL MINIMUM of a FUNCTION is the smallest value in the entire RANGE of the FUNCTION , while a LOCAL MINIMUM is the smallest value in some local neighborhood. For a function f(x) which is CONTINUOUS at a point x0; aNECESSARY but not SUFFICIENT condition for f(x)t o have a RELATIVE MINIMUM atx/C30x0is that x0be a CRITICAL POINT (i.e., f(x) is either not DIFFERENTIABLE atx0orx0is a STATIONARY POINT , in which case f?(x0)/C300):/ The FIRST DERIVATIVE TEST can be applied to CON- TINUOUS FUNCTIONS to distinguish minima from MAXIMA . For twice differentiable functions of one variable, f(x);or of two variables, f(x;y);the SECOND DERIVATIVE TEST can sometimes also identify the nature of an EXTREMUM . For a function f(x);the EXTREMUM TEST succeeds under more general condi- tions than the SECOND DERIVATIVE TEST . See also CONJUGATE GRADIENT METHOD ,CRITICAL POINT ,EXTREMUM ,FIRST DERIVATIVE TEST,GLOBAL MAXIMUM ,INFLECTION POINT ,L OCAL MAXIMUM , MAXIMUM ,M IDRANGE ,O RDER STATISTIC ,S ADDLE POINT (FUNCTION ), SECOND DERIVATIVE TEST,STA- TIONARY POINT ,STEEPEST DESCENT METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Brent, R. P. Algorithms for Minimization Without Deriva- tives. Englewood Cliffs, NJ: Prentice-Hall, 1973. Nash, J. C. "Descent to a Minimum I-II: Variable Metric Algorithms." Chs. 15 /C1/6in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisa- tion, 2nd ed. Bristol, England: Adam Hilger, pp. 186 /C1/06, 1990. Niven, I. Maxima and Minima without Calculus. Washing- ton, DC: Math. Assoc. Amer., 1982. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Minimization or Maximization of Functions." Ch. 10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 387 /C1/48, 1992. Tikhomirov, V. M. Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., 1991. Minimum Clique CLIQUE Minimum Gossip Graph GOSSIPING Minimum Modulus Principle Let f be ANALYTIC on a DOMAIN U ⁄C; and assume that f never vanishes. Then if there is a point z0 /C23 U such that ½fz0ðÞ½5½f(z)½ for all z /C23 U ; then f is constant. Let U ⁄C be a bounded domain, let f be a continuous function on the closed set ¯U that is analytic on U, and assume that f never vanishes on ¯U : Then the minimum value of ½f ½ on ¯U (which always exists) must occur on @U : In other words, min ¯U½f ½/C30min @U½f ½: See also MAXIMUM MODULUS PRINCIPLE ,M ODULUS (COMPLEX NUMBER ) References Krantz, S. G. "The Minimum Principle." §5.4.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 77, 1999. Minimum Spanning Tree The minimum spanning tree of a WEIGHTED GRAPH is a set of n /C281 edges of minimum total weight which form a SPANNING TREE of the graph. When a graph isunweighted, any SPANNING TREE is a minimum spanning tree. The minimum spanning tree can be found in poly- nomial time. Common algorithms include those due to Prinn (1957) and Kruskal (1956). The problem can also be formulated using MATROIDS (Papadimitriou and Steiglitz 1982). The minimum spanning tree can be found using the command MinimumSpanning- Tree [g] in the Mathematica add-on package Dis- creteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also SPANNING TREE References Fredman, M. L. and Tarjan, R. E. "Fibonacci Heaps and Their Uses in Network Optimization." J. ACM 34, 596 /C1/ 15, 1987. Graham, R. L. and Hell, P. "On the History of the Minimum Spanning Tree Problem." Ann. History Comput. 7,43/C1/7, 1985. Kruskal, J. B. "On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem." Proc. Amer. Math. Soc. 7,48/C1/0, 1956. Papadimitriou, C. H. and Steiglitz, K. Combinatorial Opti- mization: Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall, 1982. Prinn, R. C. "Shortest Connection Networks and Some Generalizations." Bell System Tech. J. 36, 1389 /C1/401, 1957. Skiena, S. "Minimum Spanning Tree." §6.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 232 /C1/36, 1990. Minimum Vertex Cover VERTEX COVER Minkowski-Bouligand Dimension In many cases, the HAUSDORFF DIMENSION correctly describes the correction term for a resonator with FRACTAL PERIMETER in Lorentz’s conjecture. How- ever, in general, the proper dimension to use turns out to be the Minkowski-Bouligand dimension (Schroeder 1991). Let F(r) be the AREA traced out by a small CIRCLE with RADIUS r following a fractal curve. Then, providing the LIMIT exists, DM /C13lim r00lnF(r) /C28lnr/C272 (Schroeder 1991). It is conjectured that for all strictly self-similar fractals, the Minkowski-Bouligand di- mension is equal to the H AUSDORFF DIMENSION D; otherwise DM>D:/ See also HAUSDORFF DIMENSION ,MINKOWSKI COVER , MINKOWSKI SAUSAGE References Berry, M. V. "Diffractals." J. Phys. A12, 781/C1/97, 1979. Hunt, F. V.; Beranek, L. L.; and Maa, D. Y. "Analysis of Sound Decay in Rectangular Rooms." J. Acoust. Soc. Amer. 11,80/C1/4, 1939. Lapidus, M. L. and Fleckinger-Pelle ´, J. "Tambour fractal: vers une re´solution de la conjecture de Weyl-Berry pour les valeurs propres du laplacien." Compt. Rend. Acad. Sci. Paris Math. Se´r1306, 171 /C1/75, 1988. Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 41 /C1/ 5, 1991. Minkowski Convex Body Theorem A bounded plane convex region symmetric about a LATTICE POINT and with AREA > 4 must contain at least three LATTICE POINTS in the interior. In n-D, the theorem can be generalized to a region with AREA //C212n ; which must contain at least three LATTICE POINTS . The theorem can be derived from BLICH- FELDT’S THEOREM . See also BLICHFELDT’S THEOREM References Hilbert, D. and Cohn-Vossen, S. "Minkowski’s Theorem." §6.3 in Geometry and the Imagination. New York: Chel- sea, pp. 41 /C1/4, 1999. Minkowski, H. Geometrie der Zahlen. Leipzig, Germany: Teubner, 1912. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 99, 1999. Warmus, W. Colloq. Math. I 1,45/C1/6, 1947. Minkowski Cover The covering of a PLANE CURVE with disks of radius e whose centers lie on the curve. See also MINKOWSKI- BOULIGAND DIMENSION ,M IN- KOWSKI SAUSAGE Minkowski Geometry MINKOWSKI SPACE Minkowski-Hlawka Theorem There exist lattices in n-D having HYPERSPHERE PACKING densities satisfying h ]z(n) 2n/C281 ; where z(n) is the RIEMANN ZETA FUNCTION . However, the proof of this theorem is nonconstructive and it is still not known how to actually construct packings that are this dense. See also HERMITE CONSTANTS ,HYPERSPHERE PACK- ING References Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lat- tices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 14 /C1/6, 1993.Pach, J. and Agarwal, P. K. Combinatorial Geometry. New York: Wiley, 1995. Minkowski Integral Inequality If p /C211, then gb af(x) /C27g(x) jjpdx"#1 =p 5gb af(x)jjpdx"#1=p /C27gb ag(x) jjpdx"#1=p : See also MINKOWSKI SUM INEQUALITY References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000. Hardy, G. H.; Littlewood, J. E.; and Po ´lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 146 /C1/50, 1988. Minkowski, H. Geometrie der Zahlen, Vol. 1. Leipzig, Germany: pp. 115 /C1/17, 1896. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 33, 1991. Minkowski Measure The Minkowski measure of a bounded, CLOSED SET is the same as its L EBESGUE MEASURE . References Ko, K.-I. "A Polynomial-Time Computable Curve whose Interior has a Nonrecursive Measure." Theoret. Comput. Sci. 145, 241/C1/70, 1995. Minkowski Metric In C ARTESIAN COORDINATES , ds2/C30dx2/C27dy2/C27dz2(1) dr2/C30/C28c2dt2/C27dx2/C27dy2/C27dz2; (2) and gab/C13hab/C30/C281000 0100001000012 6643 775: (3) In SPHERICAL COORDINATES , ds2/C30dr2/C27r2du/C27r2sin2udf2(4) dr2/C30/C28c2dt2/C27dr2/C27r2du/C27r2sin2udf2; (5) and g /C30/C28100 0 010 0 00 r2 0 000 r2 sin2 u2 6643 775: (6) See also L ORENTZ TRANS FORMATION ,M INKOWSKI SPACE Minkowski Sausage A FRACTAL curve created from the base curve and motif illustrated above (Lauwerier 1991, p. 37). The number of segments after the nth iteration is Nn /C308n ; (1) and en /C301 4 !n ; (2) so the CAPACITY DIMENSION is D /C13/C28lim n0/C12ln Nn ln en/C30/C28 lim n0/C12ln 8n ln 4n /C30ln 8 ln 4 /C303ln22ln2/C3032 : (3) The term Minkowski sausage is also used to refer to the M INKOWSKI COVER of a curve. See also MINKOWSKI- BOULIGAND DIMENSION ,M IN- KOWSKI COVER References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 37 /C1/8 and 42, 1991. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, p. 283, 1988. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Minkowski’s Inequalities Ifp/C211, then Minkowski’s integral inequality states thatgb af(x)/C27g(x) jjpdx"#1=p 5gb af(x)jjpdx"#1=p /C27gb ag(x) jjpdx"#1=p : Similarly, if p/C211 and ak;bk>0;then Minkowski’s sum inequality states that Xn k/C301ak/C27bk ðÞp"# 1=p 5Xn k/C301ap k ! 1=p /C27Xn k/C301bpk ! 1=p : Equality holds IFFthe sequences a1;a2;... and b1;b2; ... are proportional. References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1092 and 1099, 2000. Hardy, G. H.; Littlewood, J. E.; and Po ´lya, G. ‘Minkowski’s Inequality" and "Minkowski’s Inequality for Integrals."§2.11, 5.7, and 6.13 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 30 /C1 /2, 123, and 146/C1/50, 1988. Minkowski, H. Geometrie der Zahlen, Vol. 1. Leipzig, Germany: pp. 115 /C1/17, 1896. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 33, 1991. Minkowski Space A 4-D space with the M INKOWSKI METRIC . Alterna- tively, it can be considered to have a E UCLIDEAN METRIC , but with its VECTORS defined by x0 x1 x2 x32 6643 775/C30ict x y z2 6643 775; (1) where cis the speed of light and Iis the IMAGINARY NUMBERffiffiffiffiffiffi /C281p :Minkowski space unifies Euclidean 3- space plus time (the "fourth dimension") in Einstein’s theory of special relativity. The METRIC of Minkowski space is DIAGONAL with gaa/C301 gaa; (2) so hbd/C30hbd: (3) LetLbe the TENSOR for a L ORENTZ TRANSFORMATION . Then hbdLg d/C30Lbg(4) hagLbg/C30Lb a (5) L b a /C30 hag Lbg /C30 h ag h bd Lg d : (6) The NECESSARY and SUFFICIENT conditions for a metric gmnto be equivalent to the Minkowski metric habare that the RIEMANN TENSOR vanishes every- where (/Rl mnk /C300) and that at some point g mn has three POSITIVE and one NEGATIVE EIGENVALUES . See also LORENTZ TRANS FORMATION ,M INKOWSKI METRIC ,TWISTOR ,TWISTOR SPACE References Thompson, A. C. Minkowski Geometry. New York: Cam- bridge University Press, 1996. Minkowski’s Question Mark Function The function y /C30?(x) defined by Minkowski for the purpose of mapping the rational numbers in the OPEN INTERVAL (0; 1) into the QUADRATIC IRRATIONAL NUM- BERS of (0; 1) in a continuous, order-preserving manner. ?(x) takes a number having BINARY expan- sion x /C300:a1a2a3 ...2 to the number ?(x) /C30X k(/C281)k /C281 2(a1 /C27.../C27ak)/C281 : (1) The function satisfies the following properties (Salem 1943). 1. ?(x) is strictly increasing. 2. If x is rational, then ?(x) is of the form k=2s ; with k and s integers. 3. If x is a QUADRATIC IRRATIONAL NUMBER , then the continued fraction is periodic, and hence ?(x)is rational. 4. The function is purely singular (Denjoy 1938). /?(x) can also be constructed as ?p /C27 p? q /C27 q? ! /C30?(p=q) /C27 ?(p ?=q?) 2; (2) where p=q and p?=q? are two consecutive irreducible fractions from the FAREY SEQUENCE . At the nth stage of this definition, ?(x) is defined for 2n /C271 values of x, and the ordinates corresponding to these values are x /C30k=2n for k /C300, 1, ..., 2n (Salem 1943).The function satisfies the identity ?1 kn ! /C301 2kn/C281: (3) A few special values include ?(0)/C300 ?1 3fflCz6fflCz7 /C3014 ?1 2fflCz6fflCz7 /C3012 ?(f/C281)/C302 3 ?2 3fflCz6fflCz7 /C3034 ?1 2ffiffiffi 2pfflCz6fflCz7 /C304 5 ?12ffiffiffi 3pfflCz6fflCz7 /C3084 85 ?(1)/C301; where fis the GOLDEN RATIO . See also DEVIL’S STAIRCASE ,FAREY SEQUENCE References Conway, J. H. "Contorted Fractions." On Numbers and Games. New York: Academic Press, pp. 82 /C1/6, 1976. Denjoy, A. "Sur une fonction re ´elle de Minkowski." J. Math. Pures Appl. 17, 105/C1/55, 1938. Girgensohn, R. "Constructing Singular Functions via Farey Fractions." J. Math. Anal. Appl. 203, 127/C1/41, 1996. Kinney, J. R. "Note on a Singular Function of Minkowski." Proc. Amer. Math. Soc. 11, 788/C1/94, 1960. Minkowski, H. "Zur Geometrie der Zahlen." In Gesammelte Abhandlungen, Vol. 2. New York: Chelsea, pp. 50 /C1/1, 1991. Salem, R. "On Some Singular Monotone Functions which Are Strictly Increasing." Trans. Amer. Math. Soc. 53, 427/C1/39, 1943. Tichy, R. and Uitz, J. "An Extension of Minkowski’s Singular Functions." Appl. Math. Lett. 8,3 9/C1/6, 1995. Viader, P.; Paradis, J.; and Bibiloni, L. "A New Light on Minkowski’s ?( x) Function." J. Number Th. 73, 212/C1/27, 1998. Minkowski Sum The sum of sets AandBin a VECTOR SPACE , equal to fa/C27b:a/C23A;b/C23Bg:/ References Skiena, S. S. "Minkowski Sum." §8.6.16 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 395 /C1/96, 1997. Minkowski Sum Inequality If p /C211 and ak ; bk > 0 ; then Xn k /C301ak /C27bk ðÞp"# 1 =p 5Xn k/C301ap k ! 1 =p /C27Xn k /C301bpk ! 1 =p : Equality holds IFF the sequences a1 ; a2 ; ... and b1 ; b2 ; ... are proportional. See also MINKOWSKI INTEGRAL INEQUALITY References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1092, 2000. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 24 /C1/6, 1988. Minor The reduced DETERMINANT of a DETERMINANT EXPAN- SION, denoted Mij ; which is formed by omitting the ith row and jth column. The minor can be computed in Mathematica using Minor[m_List,{i_Integer,j_Integer}] : /C30 Drop[Transpose[Drop[Transpose[m],{j}]],{i}] Minors [m] gives the minors of a matrix m, while Minors [m, k] gives the kth minors of m. See also COFACTOR ,D ETERMINANT ,D ETERMINANT EXPANSION BY MINORS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 169 /C1/70, 1985. Muir, T. "Minors and Expansion." Ch. 4 in A Treatise on the Theory of Determinants. New York: Dover, pp. 53 /C1/37, 1960. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 235, 1990. Minor Axis SEMIMINOR AXIS Minor Graph A "minor" is a sort of SUBGRAPH and is what Kuratowski means when he says "contain." It is roughly a small graph which can be mapped into the big one without merging VERTICES . Minuend A quantity from which another (the SUBTRAHEND )is subtracted. See also MINUS ,SUBTRACTION ,SUBTRAHENDMinus The operation of SUBTRACTION , i.e., a minus b. The operation is denoted a /C28b : The MINUS SIGN "//C28/" is also used to denote a NEGATIVE number, i.e., /C28x:/ See also MINUS SIGN,N EGATIVE ,P LUS,P LUS OR MINUS ,TIMES Minus or Plus PLUS OR MINUS Minus Sign The symbol "//C28/" which is used to denote a NEGATIVE number or SUBTRACTION . See also MINUS ,PLUS SIGN,SIGN,SUBTRACTION Minute ARC MINUTE Miquel Circles For a TRIANGLE DABC and three points FrðÞ; B ?; and C ?; one on each of its sides, the three Miquel circles are the circles passing through each VERTEX and its neighboring side points (i.e., AC ?B?; BA ?C?; and CB?A?) : According to MIQUEL’S THEOREM , the Miquel circles are CONCURRENT in a point M known as the MIQUEL POINT . Similarly, there are n Miquel circles for n lines taken (n /C281) at a time. See also CLIFFORD’S CIRCLE THEOREM ,M IQUEL POINT ,MIQUEL’S THEOREM ,MIQUEL TRIANGLE References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 81, 1995. Miquel Equation /C140A2MA3 /C30/C140A2A1A3 /C27/C140P2P1P3 ; where /C140 is a DIRECTED ANGLE . See also DIRECTED ANGLE ,MIQUEL’S THEOREM ,PIVOT THEOREM References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 131 /C1/44, 1929. Miquel Five Circles Theorem Let five circles with CONCYCLIC centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. By joining adjacent pairs of the intersection points which do not lie on the circle of center, an (irregular) PENTAGRAM is obtained whose five vertices lie on the circle of centers. Let the circle of centers have radius r and let the five circles be centered and angular positions ui along this circle. The radii riof the circles and their angular positions fialong the circle of centers can then be determined by solving the ten simultaneous equa- tions cos fi /C28cos ui ðÞ2/C27 sin fi /C28sin ui ðÞ2/C30r2 i r2 cos fi /C281 /C28cos ui ðÞ2/C27 sin fi/C281 /C28sin ui ðÞ2/C30r2 i r2 for i /C30 1, ..., 5, where f0 /C13 f5 and r0 /C13r5 :/ See also FIVE DISKS PROBLEM ,PENTAGRAM References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 151 /C1/52, 1888. Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, p. 79, 1991. Miquel Point The point of CONCURRENCE of the MIQUEL CIRCLES .See also MIQUEL CIRCLES ,M IQUEL’S THEOREM , MIQUEL TRIANGLE References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 87 /C1/0, 1971. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 81, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 151, 1991. Miquel’s Theorem If points A?;B?;andC?are marked on each side of a TRIANGLE DABC ;one on each side (or on a side’s extension), then the three M IQUEL CIRCLES (each through a VERTEX and the two marked points on the adjacent sides) are CONCURRENT at a point Mcalled the M IQUEL POINT . This result is a slight general- ization of the so-called PIVOT THEOREM . IfMlies in the interior of the triangle, then it satisfies /C218P2MP3/C30180/C14/C28a1 /C218P3MP1/C30180/C14/C28a2 /C218P1MP2/C30180/C14/C28a3: The lines from the M IQUEL POINT to the marked points make equal angles with the respective sides. (This is a by-product of the M IQUEL EQUATION .) A generalized version of Miquel’s theorem states that given four lines L1 ; ..., L4 each intersecting the other three, the four MIQUEL CIRCLES passing through each subset of three intersection points of the lines meet in a point known as the 4-Miquel point M. Furthermore, the centers of these four MIQUEL CIRCLES lie on a CIRCLE C4 (Johnson 1929, p. 139). The lines from M to given points on the sides make equal ANGLES with respect to the sides. Moreover, given n lines taken by (n /C281)/s yield n MIQUEL CIRCLES like C4passing through a point Pn ; and their centers lie on a CIRCLE Cn/C271 :/ See also CLIFFORD’S CIRCLE THEOREM ,M IQUEL CIRCLES ,M IQUEL FIVE CIRCLES THEOREM ,M IQUEL EQUATION ,M IQUEL TRIANGLE ,N INE-POINT CIRCLE , PEDAL CIRCLE ,PIVOT THEOREM References Honsberger, R. "The Miquel Theorem." Ch. 8 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 79 /C1/6, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 131 /C1/44, 1929. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 151 /C1/52, 1991. Miquel Triangle Given a point P and a triangle DABC ; the Miquel triangle is the triangle DPAPBPCconnecting the sidepoints PA ; PB ; and PC of DABC with respect to which M is the MIQUEL POINT . All Miquel triangles of a given point M are directly similar, and M is the SIMILITUDE CENTER in every case. See also MIQUEL CIRCLES ,M IQUEL POINT ,M IQUEL’S THEOREM References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 81, 1995. Mira Fractal A FRACTAL based on the map F(x) /C30ax /C272(1 /C28 a)x2 1 /C27 x2: References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, p. 136, 1991. Mirimanoff’s Congruence If the first case of FERMAT’S LAST THEOREM is false for the PRIME exponent p, then 3p/C281 /C131 mod p2ðÞ :/ See also FERMAT’S LAST THEOREM Mirror Image An image of an object obtained by reflecting it in a mirror so that the signs of one of its coordinates are reversed. AMPHICHIRAL ,C HIRAL ,E NANTIOMER ,H ANDEDNESS , REFLECTION ,SYMMETRY References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 87, 1967. Mirror Plane The SYMMETRY OPERATION (x; y; z) 0 (x; y;/C28z); etc., which is equivalent to ¯2; where the bar denotes an IMPROPER ROTATION . See also MIRROR IMAGE Mise`re Form A version of NIM-like GAMES in which the player taking the last piece is the loser. For most IMPARTIAL GAMES , this form is much harder to analyze, but it requires only a trivial modification for the game of NIM. Mitchell Index The statistical INDEX PM /C13PpnqaPp0qa; where pn is the price per unit in period n and qn is the quantity produced in period n. See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 66 /C1/7, 1962. Miter Surface A QUARTIC SURFACE named after its resemblance to the liturgical headdress worn by bishops and given by the equation 4x2 x2 /C27y2 /C27z2fflC{fflCz /C28y2 1 /C28y2 /C28z2fflC{fflCz /C300 : See also QUARTIC SURFACE References Nordstrand, T. "Surfaces." http://www.uib.no/people/nfytn/ surfaces.htm. Mittag-Leffler Function En(x) /C13X/C12 k /C300xk G(nk /C27 1) : (1) It is related to the GENERALIZED HYPERBOLIC FUNC- TIONS F a n; r(x)by F1 n; 0(x) /C30EnxnðÞ : (2)Special values for integer n are E0(x) /C301 1 /C28 x (3) E1(x) /C30ex (4) E2(x) /C30coshffiffiffixpfflC{fflCz (5) E3(x) /C301 3ex1=3 /C272e /C28x1=3 =2cos12ffiffiffi 3p x1=3fflCz6fflCz7 hi (6) E4(x) /C301 2cos x1 =4fflC{fflCz /C27cosh x1 =4fflC{fflCz fflC}fflC( ; (7) and special values of half-integer n are E1 =2(x) /C30ex2 (1 /C27erf x) (8) E3=2(x) /C3013fflC}{ ex2=3 /C272e /C28x2=3 =2cos12ffiffiffi 3p x2 =3fflCz6fflCz7 /C274x1F31;5 6 ;76;32;1 27 x2fflCz6fflCz7 ffiffiffippfflC}z (9) E5 =2(x) /C30 0 F4;1 5 ;25 ;35;45;1 3125 x2fflCz6fflCz7 /C278x1F51;7 10;9 10;1110;1310;32;1 3125 x2fflCz6fflCz7 15ffiffiffipp ; (10) wherepFqare generalized hypergeometric functions, and0Fqis a generalized confluent hypergeometric function. As can be seen, E1 =2(x) is closely related to DAWSON’S INTEGRAL D/C28(x) :/ The more general Mittag-Leffler function Em;n /C30X/C12 k/C300xk G(mk /C27 n) (11) can also be defined (Wiman 1905, Agarwal 1953, Gorenflo 1987, Miller 1993, Mainardi and Gorenflo 1995, Gorenflo 1998, Sixdeniers et al. ). See also DAWSON’S INTEGRAL ,GENERALIZED HYPER- BOLIC FUNCTIONS References Agarwal, R. P. "A propos d’une note de M. Pierre Humbert." C. R. Acad. Sci. Paris 236, 2031 /C1/032, 1953. Gorenflo, R. "Newtonsche Aufheizung, Abelsche Integralgle- ichungen zweiter Art und Mittag-Leffler-Funktionen." Z. Naturforsch. A 42, 1141 /C1/146, 1987. Gorenflo, R.; Kilbas, A. A.; and Rogosin, S. V. "On the Generalized Mittag-Leffler Type Functions." Integral Transform. Spec. Funct. 7, 215/C1/24, 1998. Humbert, P. "Quelques re ´sultats relatifs a `la fonction de Mittag-Leffler." C. R. Acad. Sci. Paris 236, 1467 /C1/468, 1953. Humbert, P. and Agarwal, R. P. "Sur la fonction de Mittag- Leffler et quelques-unes de ses ge ´ne´ralisations." Bull. Sci. Math. Ser. 2 77, 180/C1/85, 1953. Humbert, P. and Delerue, P. "Sur une extension a `deux variables de la fonction de Mittag-Leffler." C. R. Acad. Sci. Paris 237, 1059 /C1/060, 1953. Mainardi, F. and Gorenflo, R. "The Mittag-Leffler Function in the Riemann-Liouville Fractional Calculus." In Pro- ceedings of the International Conference Dedicated to the Memory of Academician F. D. Gakhov; Held in Minsk, February 16 /C1/0, 1996 (Ed. A. A. Kilbas). Minsk, Beloruss: Beloruss. Gos. Univ., Minsk, pp. 215 /C1/25, 1996. Miller, K. S. "The Mittag-Leffler and Related Functions." Integral Transform. Spec. Funct. 1,41/C1/9, 1993. Mittag-Leffler, M. G. C. R. Acad. Sci. Paris Ser. 2 137, 554, 1903. Muldoon, M. E. and Ungar, A. A. "Beyond Sin and Cos." Math. Mag. 69,3/C1/4, 1996. Sixdeniers, J.-M.; Penson, K. A.; and Solomon, A. I. "Mittag- Leffler Coherent States." J. Phys. A: Math. Gen. 32, 7543 /C1/ 563, 1999. Wiman, A. "Uuml;ber den Fundamentalsatz in der Teorie der Funktionen Ea(x):/" Acta Math. 29, 191 /C1/01, 1905. Mittag-Leffler Polynomial Polynomials Mk(x) which form the associated SHEF- FER SEQUENCE for f(t) /C30et /C28 1 et /C27 1 (1) and have the GENERATING FUNCTION X/C12 k /C300Mk(x) k!tk /C301 /C27 t 1 /C28 t !x : (2) An explicit formula is given by Mn(x) /C30Xn k /C300n kfflCzrfflCzD (n /C281)n /C28k2k(x)k ; (3) where (x)nis a FALLING FACTORIAL , which can be summed in closed form in terms of the HYPERGEO- METRIC FUNCTION , GAMMA FUNCTION , and POLY- GAMMA FUNCTION . The binomial identity associated with the SHEFFER SEQUENCE is Mn(x /C27y) /C30Xn k /C300n kfflCzrfflCzD Mk(x)Mn/C28k(y): (4) The Mittag-Leffler polynomials satisfy the recurrence formula Mn/C271(x) /C301 2 xMn(x /C271) /C272Mn(x) /C27Mn(x /C281) ½/C138 : (5) The first few Mittag-Leffler polynomials are M0(x)/C301 M1(x)/C302x M2(x)/C304x2 M3(x)/C308x3/C274x M4(x)/C3016x4/C2732x2: The Mittag-Leffler polynomials Mn(x) are related to the P IDDUCK POLYNOMIALS by Pn(x)/C301 2(et/C271)Mn(x) (6) (Roman 1984, p. 127).See also PIDDUCK POLYNOMIAL References Bateman, H. "The Polynomial of Mittag-Leffler." Proc. Nat. Acad. Sci. USA 26, 491/C1/96, 1940. Roman, S. "The Mittag-Leffler Polynomials." §4.1.6 in The Umbral Calculus. New York: Academic Press, pp. 75 /C1/8 and 127, 1984. Mittag-Leffler’s Partial Fractions Theorem Let any finite or infinite set of points having no finite LIMIT POINT be prescribed and associate with each of its points a principal part, i.e., a RATIONAL FUNCTION of the special form hn(z)/C30a(n) /C281 z/C28zn/C27a(n) /C282 (z/C28zn)2/C27.../C27a(n) /C28anu (z/C28zn)an forn/C301;2, ..., k. Then there exists a MEROMORPHIC FUNCTION which has poles with the prescribed prin- cipal parts at precisely the prescribed points, and is otherwise regular. It can be represented in the form of a partial fraction decomposition from which onecan read off again the poles, along with their principal parts. Further, if M 0(z) is one such function, then M(z)/C30M0(z)/C27G(z) is the most general function satisfying the conditions of the problem, where G(z) denotes an arbitrary ENTIRE FUNCTION . References Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 37 /C1/9, 1996. Krantz, S. G. "The Mittag-Leffler Theorem." §8.3.6 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, pp. 112 /C1/13, 1999. Mittag-Leffler’s Theorem If a function analytic at the origin has no SINGULA- RITIES other than POLES for finite x, and if we can choose a sequence of contours Cmabout z/C300 tending to infinity such that ½f(z)½never exceeds a given quantity Mon any of these contours and f½dz=z½is uniformly bounded on them, then f(z)/C30f(0)/C27limPm(z)/C28Pm(0) ½/C138 ; where Pm(z) is the sum of the principal parts of f(z)a t allPOLES awithin Cm:If there is a POLE atz/C300, then we can replace f(0) by the negative powers and the constant term in the L AURENT SERIES off(z) about z/C300. References Jeffreys, H. and Jeffreys, B. S. "Mittag-Leffler’s Theorem." §12.006 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 383 /C1/86, 1988. Mittenpunkt The SYMMEDIAN POINT of the EXCENTRAL TRIANGLE , i.e., the point of concurrence M of the lines from the EXCENTERS Jithrough the corresponding TRIANGLE side MIDPOINT Mi : It is also called the MIDDLESPOINT and has TRIANGLE CENTER FUNCTION a /C30b /C27c /C28a /C301 2cot A: See also EXCENTER ,E XCENTRAL TRIANGLE ,N AGEL POINT References Baptist, P. Die Entwicklung der Neueren Dreiecksgeometrie. Mannheim: Wissenschaftsverlag, p. 72, 1992. Eddy, R. H. "A Generalization of Nagel’s Middlespoint." Elem. Math. 45,14/C1/8, 1990. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Kimberling, C. "Mittenpunkt." http://cedar.evansville.edu/ ~ck6/tcenters/class/mitten.html. Mixed Fraction An IMPROPER FRACTION p =q > 1 written in the form n /C27r =s : In common usage such as cooking recipes, n /C27 r =s is often written as nr s(e.g., 112); much to the chagrin of mathematicians, to whom nr smeans nr =s; not n /C27r =s : (The author of this work discovered this fact early in his mathematical career after having points marked off a CALCULUS exam for using the recipe-like notation. Future mathematicians are therefore encouraged to avoid mixed fractions, except perhaps in the kitchen.) See also FRACTION ,IMPROPER FRACTION ,P ROPER FRACTION Mixed Indices MIXED TENSOR Mixed Partial Derivative A PARTIAL DERIVATIVE of second or greater order with respect to two or more different variables, for examplefxy /C30@2f @x @y : If the mixed partial derivatives exist and are contin- uous at a point x0 ; then they are equal at x0 regardless of the order in which they are taken. See also PARTIAL DERIVATIVE Mixed Strategy A collection of moves together with a corresponding set of weights which are followed probabilistically in the playing of a GAME . The MINIMAX THEOREM of GAME THEORY states that every finite, zero-sum, two- person game has optimal mixed strategies. See also GAME THEORY ,M INIMAX THEOREM ,STRAT- EGY Mixed Tensor A TENSOR having CONTRAVARIANT and COVARIANT indices. See also CONTRAVARIANT TENSOR ,COVARIANT TEN- SOR,TENSOR Mnemonic A mental device used to aid memorization. Common mnemonics for mathematical constants such as E and PI consist of sentences in which the number of letters in each word give successive digits. See also E,JOSEPHUS PROBLEM ,PI References Luria, A. R. The Mind of a Mnemonist: A Little Book about a Vast Memory. Cambridge, MA: Harvard University Press, 1987. Weisstein, E. W. "Books about Calculating Prodigies." http:// www.treasure-troves.com/books/CalculatingProdi- gies.html. Moat-Crossing Problem There are two versions of the moat-crossing problem, one geometric and one algebraic. The geometric moat problems asks for the widest moat Rapunzel can cross to escape if she has only two unit-length boards (and no means to nail or otherwise attach them together)?More generally, what is the widest moat which can becrossed using nboards? Matthew Cook has conjec- tured that the asymptotic solution to this problem isOn 1=3fflC{fflCz (Finch). The algebraic moat-crossing problem asks if it ispossible to walk to infinity on the REAL LINE using only steps of bounded lengths and steps on the primenumbers. The answer is negative (Gethner et al. 1998). However, the Gaussian moat problem thatasks whether it is possible to walk to infinity in theG AUSSIAN INTEGERS using the G AUSSIAN PRIMES as stepping stones and taking steps of bounded length is unresolved. Gethner et al. (1998) show that a moat of widthffiffiffiffiffiffi 26p exists. References Finch, S. "Unsolved Mathematics Problems: Moat Crossing Optimization Problem." http://www.mathsoft.com/asolve/ moat/moat.html. Gethner, E. and Stark, H. M. "Periodic Gaussian Moats." Experiment. Math. 6, 251/C1/54, 1997. Gethner, E.; Wagon, S.; and Wick, B. "A Stroll Through the Gaussian Primes." Amer. Math. Monthly 105, 327/C1/37, 1998. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Haugland, J. K. "A Walk on Complex Primes." [Norwegian.] Normat 43, 168/C1/70, 1995. Jordan, J. H. and Rabung, J. R. "A Conjecture of Paul Erdos Concerning Gaussian Primes." Math. Comput. 24, 221/C1/ 23, 1970. Montgomery, H. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. Provi- dence, RI: Amer. Math. Soc., 1994. Vardi, I. "Prime Percolation." Experiment. Math. 7, 275/C1/89, 1998. Wagon, S. Mathematica in Action, 2nd ed. New York: Springer-Verlag, 1999. Moat Problem MOAT-CROSSING PROBLEM Mo¨bius Band MO¨BIUS STRIPMo¨bius Function A number theoretic function defined by m(n)/C13 0i f nhas one or repeated prime factors 1i f n/C301 (/C281)kifnis a product of kdistinct primes ;8 < : (1) som(n)"0 indicates that nisSQUAREFREE . The first few values are 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, ... (Sloane’s A008683). The SUMMATORY FUNCTION of the Mo¨bius function is called M ERTENS FUNCTION . The Mo ¨bius function has GENERATING FUNCTIONS X/C12 n/C301m(n) ns/C301 z(s)(2) forR[s]>1 (Nagell 1951, p. 130), and X/C12 n/C301m(n)xn 1/C28xn/C30x (3) for½x½B1:It also obeys the infinite sums X/C12 n/C301m(n) n/C300 (4) X/C12 n/C301m(n)l nn n/C30/C281 (5) and the INFINITE PRODUCT Y/C12 n/C301(1/C28xn)m(n)=n/C30e/C28x(6) for½x½B1 (Bellman 1943; Buck 1944;, Po ´lya and Szego 1976, p. 126; Robbins 1999). (2) is as "deep" as the PRIME NUMBER THEOREM (Landau 1909, pp. 567 /C1/74; Landau 1911; Hardy 1999, p. 24), and behavesasymptotically as X n5xm(n)/C30O(xe/C28cffiffiffiffiffiffi lnxp ) (7) The Mo ¨bius function is MULTIPLICATIVE , m(mn) /C30m(m) m(n)i f( m; n) /C301 0i f ( m; n) > 1;fflC}6 (8) and satisfies X d½nm(d) /C30 dn1 ; (9) where dij is the KRONECKER DELTA , as well as X dm(d)s0n d ! /C301; (10) where s0(n) is the number of divisors (i.e., DIVISOR FUNCTION of order zero; Nagell 1951, p. 281). See also BRAUN’S CONJECTURE ,M ERTENS FUNCTION , MO¨ BIUS INVERSION FORMULA ,M O¨ BIUS PERIODIC FUNCTION ,PRIME ZETA FUNCTION ,RIEMANN FUNC- TION ,SQUAREFREE References Abramowitz, M. and Stegun, C. A. (Eds.). "The Mo¨bius Function." §24.3.1 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 826, 1972. Bellman, R. "Problem 4072." Amer. Math. Monthly 50, 124 /C1/ 25, 1943. Buck, R. C. "Solution to Problem 4072." Amer. Math. Monthly 51, 410, 1944. Dele´glise, M. and Rivat, J. "Computing the Summation of the Mo¨bius Function." Experiment. Math. 5, 291 /C1/95, 1996. Hardy, G. H. "A Note on the Mo¨bius Function." §4.9 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 64 /C1/5, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford: Clarendon Press, p. 236, 1979. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig, Germany: Teubner, 1909. Landau, E. Prac. Matematyczno-Fizycznych 21,97/C1/77, 1910. Landau, E. Wiener Sitzungsber. 120, 973 /C1/88, 1911. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 27, 1951. Po´lya, G. and Szego, G. Problems and Theorems in Analysis, Vol. 2. New York: Springer-Verlag, 1976. Robbins, N. "Some Identities Connecting Partition Func- tions to Other Number Theoretic Functions." Rocky Mtn. J. Math. 29, 335 /C1/45, 1999. Rota, G.-C. "On the Foundations of Combinatorial Theory I. Theory of Mo¨bius Functions." Z. fu¨r Wahrscheinlich- keitsth. 2, 340 /C1/68, 1964. Se´roul, R. "The Moebius Function." §2.12 and 8.5 in Programming for Mathematicians. Berlin: Springer-Ver- lag, pp. 19 /C1/1 and 167 /C1/69, 2000. Sloane, N. J. A. Sequences A008683 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 7 /C1/ and 223 /C1/25, 1991.Mo¨bius Group The equation x2 1 /C27x22 /C27.../C27x2n /C282x0x/C12/C300 represents an n-D HYPERSPHERE Sn as a quadratic hypersurface in an (n /C271)/-D real projective space Pn/C271 ; where xa are homogeneous coordinates in Pn/C271 : Then the GROUP M(n) of projective transformations which leave Sn invariant is called the Mo¨bius group. See also MODULAR GROUP GAMMA References Iyanaga, S. and Kawada, Y. (Eds.). "Mo¨bius Geometry." §78A in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 265 /C1/66, 1980. Mo¨bius Inversion Formula The transform inverting the sequence g(n) /C13X djnf(d) (1) into f(n) /C30X djnm(d)gn d ! ; (2) where the sums are over all possible INTEGERS d that DIVIDE n and m(d) is the MO¨ BIUS FUNCTION . The LOGARITHM of the CYCLOTOMIC POLYNOMIAL Fn(x) /C30Y djn(1 /C28xn=d) m(d) (3) is closely related to the Mo ¨bius inversion formula. See also CYCLOTOMIC POLYNOMIAL ,M O¨ BIUS FUNC- TION ,MO¨ BIUS TRANSFORM References Hardy, G. H. and Wright, W. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 91 /C1/3, 1979. Hunter, J. Number Theory. London: Oliver and Boyd, 1964. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, pp. 577 /C1/80, 1974. Nagell, T. Introduction to Number Theory. New York: Wiley, pp. 28 /C1/9, 1951. Schroeder, M. R. Number Theory in Science and Commu- nication, 3rd ed. New York: Springer-Verlag, 1997. Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. 19 /C1/0, 2000. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 7 /C1/and 223 /C1/25, 1991. Mo¨bius Periodic Function A function periodic with period 2 psuch that p(u/C27p)/C30/C28p(u) for all uis said to be Mo ¨bius periodic. See also PERIODIC FUNCTION Mo¨bius Problem Let A /C30fa1 ; a2 ; ...g be a free Abelian SEMIGROUP , where a1 is the IDENTITY ELEMENT , and let m(n) be the MO¨ BIUS FUNCTION . Define m(an) on the elements of the semigroup analogously to the definition of m(n) (as (/C281)r if n is the product of r distinct primes) by regarding generators of the semigroup as primes. Then the Mo¨bius problem asks if the properties 1. a Bb IMPLIES ac Bbc for a ; b ; c /C23 A; where A has the linear order a1 Ba2 B...;/ 2. m(an) /C30 m(n) for all n, imply that am;n /C30aman for all m; n ]1: Informally, the problem asks "Is the multiplication law on the positive integers uniquely determined by the values of the Mo¨bius function and the property that multiplication respects order? The problem is known to be true for all mn 574 if m(an) /C30 m(n) for all n 5240 (Flath and Zulauf 1995). See also BRAUN’S CONJECTURE ,MO¨ BIUS FUNCTION References Flath, A. and Zulauf, A. "Does the Mo¨bius Function Determine Multiplicative Arithmetic?" Amer. Math. Monthly 102, 354 /C1/56, 1995. Mo¨bius Shorts A one-sided surface reminiscent of the MO¨ BIUS STRIP , attributed to Gourmalin (Bouvier and George 1979, p. 477; Boas 1995). This surface is topologically equivalent to a KLEIN BOTTLE with a hole in it, and is topologically distinct from the MO¨ BIUS STRIP (Gramain 1984, Stewart 2000b). See also KLEIN BOTTLE ,MO¨ BIUS STRIP References Boas, R. P. Jr. "Mo ¨bius Shorts." Math. Mag. 68, 127, 1995. Bouvier, A. and George, M. Dictionaire des mathe ´matiques. Paris: Presses Universitaires de France, 1979. Gramain, A. Topology of Surfaces. Moscow, ID: BCS Associates, 1984.Stewart, I. "Mathematical Recreations: Reader Feedback." Sci. Amer. 282, 111, May 2000a. Stewart, I. "Mathematical Recreations: Reader Feedback." Sci. Amer. 283, 101, Sep. 2000b. Mo¨bius Strip n/C23N A one-sided NONORIENTABLE SURFACE obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then re-attaching the two ends. According to Madachy (1979), the B. F. Goodrich Company patented a conveyor belt in the form of a Mo ¨bius strip which lasts twice as long as conventional belts. AM o ¨bius strip of half-width wwith midcircle of radius Rand at height z/C300 can be represented parametrically by x/C30R/C27scos 1 2tfflCz6fflCz7hi cost (1) y/C30R/C27scos1 2tfflCz6fflCz7hi sint (2) z/C30ssin1 2tfflCz6fflCz7 ; (3) fors/C23[/C28w;w] and t/C23[0;2p]:/ The coefficients of the FIRST FUNDAMENTAL FORM for this surface are E/C301 (4) F/C300 (5) G/C30R2/C272Rscos1 2tfflCz6fflCz7 /C2712s2(3/C272 cos t); (6) the SECOND FUNDAMENTAL FORM coefficients are e/C300 (7) f/C30Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4R2/C273s2/C272s4Rcos1 2tfflCz6fflCz7 /C27scosthir (8) g/C302R2/C27s2ðÞ /C274Rscos1 2tfflCz6fflCz7 /C27s2costhi sin12tfflCz6fflCz7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4R2/C273s2/C272s4Rcos1 2tfflCz6fflCz7 /C27scosthir ; (9) the AREA ELEMENT is dS /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C272 Rs cos1 2 tfflCz6fflCz7 /C27s234 /C2712cos tfflCz6fflCz7r ds ffl dt; (10) and the GAUSSIAN and MEAN CURVATURES are K /C30/C284R2 4R2 /C27 3s2 /C27 2s 4 R cos1 2 tfflCz6fflCz7 /C27 s cos thi no2 (11) H /C3022R2 /C27 s2ðÞ /C27 4 Rs cos1 2 tfflCz6fflCz7 /C27 s2 cos thi sin12 tfflCz6fflCz7 4R2 /C27 3s2 /C27 2s 4 R cos1 2 tfflCz6fflCz7 /C27 s costhi no2 : (12) The perimeter of the Mo¨bius strip is given by integrating the complicated function ds /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x?2 /C27y ?2q /C30fflC}{ 1 16 w4 cos41 2 tfflCz6fflCz7 /C27 R /C27w cos(12 t)hi cos t /C2812 w sin12 tfflCz6fflCz7 sin tno4 /C27 R sin t /C2714 w sin12 tfflCz6fflCz7 /C273 sin32 tfflCz6fflCz7 hi no4fflC}z1 =2 (13) from 0 to 4 p; which can unfortunately not be done in closed form. Note that although the surface closes at t /C302p; this corresponds to the bottom edge connecting with the top edge, as illustrated above, so an addi- tional 2p must be traversed to comprise the entire arc length of the bounding edge. Cutting a Mo¨bius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called PARADROMIC RINGS (Listing and Tait 1847, Ball and Coxeter 1987) which are summarized in the table below. half-twists cuts divs. result 1 1 2 1 band, length 2 1 1 3 1 band, length 2 1Mo¨bius strip, length 11 2 4 2 bands, length 2 1 2 5 2 bands, length 2 1Mo¨bius strip, length 1 1 3 6 3 bands, length 2 1 3 7 3 bands, length 2 1Mo¨bius strip, length 1 2 1 2 2 bands, length 1 2 2 3 3 bands, length 1 2 3 4 4 bands, length 1 A TORUS can be cut into a Mo¨bius strip with an EVEN number of half-twists, and a KLEIN BOTTLE can be cut in half along its length to make two Mo¨bius strips. In addition, two strips on top of each other, each with a half-twist, give a single strip with four twists when disentangled. There are three possible SURFACES which can be obtained by sewing a Mo ¨bius strip to the edge of a DISK: the B OY SURFACE ,CROSS-CAP , and R OMAN SUR- FACE . The Mo ¨bius strip has E ULER CHARACTERISTIC x/C301 (or genus g/C301=2);so the H EAWOOD CONJECTURE shows that any set of regions on it can be colored using only six colors, as illustrated above. See also BOY SURFACE ,CROSS- CAP,M AP COLORING , MO¨ BIUS STRIP DISSECTION ,N ONORIENTABLE SUR- FACE ,PARADROMIC RINGS,PRISMATIC RING,ROMAN SURFACE ,TIETZE GRAPH References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 127 /C1/28, 1987. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 243, 1976. Bogomolny, A. "Mo ¨bius Strip." http://www.cut-the-knot.com/ do_you_know/moebius.html. Gardner, M. "Mo ¨bius Bands." Ch. 9 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific Amer-ican. New York: Vintage, pp. 123 /C1 /36, 1978. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 10, 1984. Gray, A. "The Mo¨bius Strip." §14.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 325 /C1/26, 1997. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 41 /C1/5, 1975. JavaView. "Classic Surfaces from Differential Geometry: Moebius Strip." http://www-sfb288.math.tu-berlin.de/vgp/ javaview/demo/surface/common/PaSurface_Moebius- Strip.html. Kraitchik, M. §8.4.3 in Mathematical Recreations. New York: W. W. Norton, pp. 212 /C1/13, 1942. Listing and Tait. Vorstudien zur Topologie, Go¨ttinger Stu- dien , Pt. 10, 1847. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 7, 1979. Mo¨bius, A. F. Werke, Vol. 2. p. 519, 1858. Nordstrand, T. "Moebiusband." http://www.uib.no/people/ nfytn/moebtxt.htm. Pappas, T. "The Moebius Strip & the Klein Bottle," "A Twist to the Moebius Strip," "The ‘Double’ Moebius Strip." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 207, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 269 /C1/74, 1999. Wagon, S. "Rotating Circles to Produce a Torus or Mo¨bius Strip." §7.4 in Mathematica in Action. New York: W. H. Freeman, pp. 229 /C1/32, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 152 /C1/53 and 164, 1991. Mo¨bius Strip Dissection Tiling of a Mo¨bius strip can be performed immedi- ately by carrying over a tiling of a rectangle with the same two-sided SURFACE AREA . However, additional tilings are possible by cutting tiles across glued edges. An example of such a tiling is the strip constructed from a 5 /C291 RECTANGLE consisting of two halves of a width 2 square (which are rejoined when edges are connected) separated by a 1 /C291 square (Stewart 1997). Unfortunately, since the long top and bottom edges must be glued together, this example is not constructible out of paper. It also suffers from having the unit square share a boundary with itself. In 1993, S. J. Chapman found a tiling free of the latter defect (although still suffering from the former) which can be constructed using five squares. No similar tiling is possible using fewer tiles (Stewart 1997). See also CYLINDER DISSECTION ,M O¨ BIUS STRIP,PER- FECT SQUARE DISSECTION ,TORUS DISSECTION References Stewart, I. "Squaring the Square." Sci. Amer. 277,94/C1/6, July 1997.Mo¨bius Transform The transformation of a sequence a1 ; a2 ; ... with an /C30X d ½nbd (1) into the sequence b1 ; b2 ; ... via the MO¨ BIUS INVERSION FORMULA , bn /C30X d ½nmn d ! ad : (2) The transformation of bn to an is sometimes called the sum-of-divisors transform. Two other equivalent for- mulations are given by X/C12 n/C301anxn /C30X/C12 n/C301bnxn 1 /C28 xn ; (3) the right side of which is called a LAMBERT SERIES , and X/C12 n/C301an ns /C30 z(s)X/C12 n/C301bn n2 ; (4) where z(s) is the RIEMANN ZETA FUNCTION (Sloane and Plouffe 1995, p. 21). Example Mo¨bius transformations (Sloane and Plouffe 1995, p. 22) include bn /C301 for all n, giving the inverse transform as an /C301 ; 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ... (Sloane’s A000005), the DIVISOR FUNCTION s0(n)ofn. The Mo¨bius transform of an /C30n gives bn /C301; 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, ... (Sloane’s A000010), the TOTIENT FUNCTION of n. The inverse Mo¨bius trans- form of the sequence b2n /C300 and b2n/C271 /C304(/C281)n gives an /C304 ; 4, 0, 4, 8, 0, 0, 4, 4, ... (Sloane’s A004018), the number of ways r(n) of writing n as a sum of two squares. The inverse Mo¨bius transform of bn /C301 for n prime and bn /C300 for n composite gives the sequence an/C300;1, 1, 1, 1, 2, 1, 1, 1, ... (Sloane’s A001221), the number of DISTINCT PRIME FACTORS ofn. See also BINOMIAL TRANSFORM ,DIVISOR FUNCTION , EULER TRANSFORM ,LAMBERT SERIES ,MO¨ BIUS INVER- SION FORMULA ,M O¨ BIUS TRANSFORMATION ,STIRLING TRANSFORM References Bender, E. A. and Goldman, J. R. "On the Applications of Mo¨bius Inversion in Combinatorial Analysis." Amer. Math. Monthly 82, 789/C1/03, 1975. Bernstein, M. and Sloane, N. J. A. "Some Canonical Se- quences of Integers." Linear Algebra Appl. 226//228 ,5 7/C1/ 2, 1995. Gessel, I. and Rota, C.-G. (Eds.). Classic Papers in Combi- natorics. Boston, MA: Birkha ¨user, 1987. Hardy, G. H. and Wright, E. M. §17.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clar- endon Press, 1979. Rota, G.-C. "On the Foundations of Combinatorial Theory I. Theory of Mo ¨bius Functions." Z. fu ¨r Wahrscheinlich- keitsth. 2, 340/C1/68, 1964. Sloane, N. J. A. Sequences A000005/M0246, A000010/ M0299, A001221/M0056, and A004018/M3218 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, p. 259, 1999. Mo¨bius Transformation Let a /C23C and ajjB 1; then 8a(z) /C30z /C28 a 1 /C28 ¯az is a Mo¨bius transformation, where ¯a is the COMPLEX CONJUGATE of a. 8ais a CONFORMAL TRANSFORMA- TION SELF-MAP of the UNIT DISK D for each a, and specifically of the boundary of the unit disk to itself. The same holds for (8a) /C281 /C308/C28a :/ Any conformal self-map of the UNIT DISK to itself is a composition of a Mo¨bius transformation with a ROTA- TION , and any conformal self-map f of the unit disk can be written in the form f(z) /C308b(wz) for some Mo¨bius transformation 8b and some complex number w with wjj/C301 (Krantz 1999, p. 81). See also LINEAR FRACTIONAL TRANSFORMATION References Krantz, S. G. "Mo¨bius Transformations." §6.2.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 81, 1999. Mo¨bius Triangles SPHERICAL TRIANGLES into which a SPHERE is divided by the planes of symmetry of a UNIFORM POLYHE- DRON . See also SPHERICAL TRIANGLE ,U NIFORM POLYHE- DRON Mock Theta Function In his last letter to Hardy, Ramanujan defined 17 JACOBI THETA FUNCTION -like functions F(q) with qjjB1 which he called "mock theta functions" (Wat- son 1936, Ramanujan 1988, pp. 127 /C1/31; Ramanujan 2000, pp. 354 /C1/55). These functions are Q-SERIES with exponential singularities such that the arguments terminate for some power tN:In particular, if f(q)i s notaJACOBI THETA FUNCTION , then it is a mock theta function if, for each ROOT OF UNITY r;there is an approximation OF THE FORM f(q)/C30XM m/C301tkmexpXN n/C30/C281cmntn ! /C27O(1) (1)ast00/C27with q/C30re/C28t(Gordon and McIntosh 2000b). If, in addition, for every ROOT OF UNITY rthere are modular forms h(r) j(q) and real numbers ajand 15 j5J(r) such that f(q)/C28XJ(r) j/C301qajh(r) j(q) (2) is bounded as qradially approaches r;then f(q)i s said to be a strong mock theta function (Gordon and McIntosh 2000b). Ramanujan found an additional three mock theta functions in his "lost notebook" which were subse-quently rediscovered by Watson (1936). The first formula on page 15 of Ramanujan’s lost notebook relates the functions which Watson calls r(/C28q) and v(/C28q) (equivalent to the third equation on page 63 of Watson’s 1936 paper), and the last formula on page31 of the lost notebook relates what Watson callsn(/C28q) and vq 2ðÞ(equivalent to the fourth equation on page 63 of Watson’s paper). The orders of these andRamanujan’s original 17 functions were all 3, 5, or 7. Ramanujan’s "lost notebook" also contained several mock theta functions of orders 6 and 10, which, however, were not explicitly identified as mock theta functions by Ramanujan. Their properties have nowbeen investigated in detail (Andrews and Hickerson1991, Choi 1999). Examples of the mock theta functions found by Ramanujan include F 0(q)/C30X/C12 n/C300q2n2 q;q2 ðÞn(3) F1(q)/C30X/C12 n/C301q2n(n/C281) q;q2 ðÞn: (4) (Gordon and McIntosh 2000b). Gordon and McIntosh (2000b) found eight mock theta functions of order 8, S0(q)/C30X/C12 n/C300qn2(/C28q;q2)n (/C28q2;q2)n(5) S1(q)/C30X/C12 n/C300qn(n/C272)(/C28q;q2)n (/C28q2;q2)n(6) T0(q)/C30X/C12 n/C300q(n/C271)(n/C272)/C28q2;q2ðÞn /C28q;q2 ðÞn/C271(7) T1(q)/C30X/C12 n/C300qn(n/C271)/C28q2;q2ðÞn /C28q;q2 ðÞn/C271(8) U0(q) /C30X/C12 n/C300qn2/C28q; q2ðÞn /C28q4; q4 ðÞn(9) U1(q) /C30X/C12 n /C300q(n/C271)2/C28q; q2ðÞn /C28q2; q4 ðÞn/C271(10) V0(q) /C30/C281 /C272X/C12 n/C300qn2/C28q; q2ðÞn q; q2 ðÞn(11) /C30/C281 /C272X/C12 n/C300q2n2/C28q2; q4ðÞn q; q2 ðÞ2n/C271(12) V1(q) /C30X/C12 n/C300q(n /C271)2/C28q; q2ðÞn q; q4 ðÞn/C271(13) /C30X/C12 n/C300q2n2 /C272n/C271 /C28q4; q4ðÞn q; q2 ðÞ2n/C272: (14) See also JACOBI THETA FUNCTIONS ,M ORDELL INTE- GRAL , Q-SERIES References Andrews, G. E. "The Fifth and Seventh Order Mock Theta Functions." Trans. Amer. Soc. 293, 113 /C1/34, 1986. Andrews, G. E. "Mock Theta Functions." Proc. Sympos. Pure Math. 49, 283 /C1/98, 1989. Andrews, G. E. and Hickerson, D. "Ramanujan’s "Lost" Notebook VII: The Sixth Order Mock Theta Functions." Adv. Math. 89,60/C1/05, 1991. Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart, and Winston, p. 51, 1961. Choi, Y.-S. "Tenth Order Mock Theta Functions in Rama- nujan’s Lost Notebook." Invent. Math. 136, 497 /C1/69, 1999. Gordon, B. and McIntosh, R. J. "Modular Transformations of Ramanujan’s Fifth and Seventh Order Mock Theta Func- tions." Submitted to Invent. Math. 2000a. Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." To appear in J. London Math. Soc. 2000b. Ramanujan, S. The Lost Notebook and Other Unpublished Manuscripts. New Delhi, India: Narosa, 1988. Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara, and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000. Selberg, A. "U¨ ber die Mock-Thetafunktionen siebenter Ordnung." Arch. Math. og Naturvidenskab 41,3/C1/5, 1938. Watson, G. N. "The Final Problem: An Account of the Mock Theta Functions." J. London Math. Soc. 11,55/C1/0, 1936. Watson, G. N. "The Mock Theta Function (2)." Proc. London Math. Soc. 42, 274 /C1/04, 1937. Mod CONGRUENCE Mode The most common value obtained in a set of observa- tions. An interesting empirical relationship between the mean, median, and mode which appears to hold for unimodal curves of moderate asymmetry is given bymean /C28mode :3(mean /C28median) (Kenney and Keeping 1962, p. 53), which is the basis for the definition of the PEARSON MODE SKEWNESS . See also MEAN,MEDIAN (STATISTICS ), ORDER STATIS- TIC,PEARSON MODE SKEWNESS References Kenney, J. F. and Keeping, E. S. "The Mode," "Relation Between Mean, Median, and Mode," and "Relative Merits of Mean, Median, and Mode." §4.7 /C1/.9 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 50 /C1/4, 1962. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995. Model A well-formed formula B is said to be true for the interpretation M (written ffiM B) IFF every sequence in a (the set of all denumerable sequences of elements of the domain of D), satisfies B. B is said to be false for M IFF no sequence in a satisfies B. Then an interpretation M is said to be a model for a setGof well-formed formulas IFFevery well-formed formula in Gis true for M(Mendelson 1997, pp. 59 /C1/ 0). See also GENERALIZED COMPLETENESS THEOREM References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 59 /C1/0, 1997. Model Completion Model completion is a term employed when EXISTEN- TIAL CLOSURE is successful. The formation of the COMPLEX NUMBERS , and the move from affine to projective geometry, are successes of this kind. The theory of existential closure gives a theoretical basis of Hilbert’s "method of ideal elements." References Manders, K. L. "Interpretations and the Model Theory of the Classical Geometries." In Models and Sets . Berlin: Springer-Verlag, pp. 297 /C1/30, 1984. Manders, K. L. "Domain Extension and the Philosophy of Mathematics." J. Philos. 86, 553/C1/62, 1989. Mode Locking A phenomenon in which a system being forced at an IRRATIONAL period undergoes rational, periodic mo- tion which persists for a finite range of forcing values.It may occur for strong couplings between natural and forcing oscillation frequencies. The phenomenon can be exemplified in the CIRCLE MAP when, after qiterations of the map, the new angle differs from the initial value by a RATIONAL NUMBER un/C27q /C30 un /C27p q : This is the form of the unperturbed CIRCLE MAP with the WINDING NUMBER V/C30p q : For V not a RATIONAL NUMBER , the trajectory is QUASIPERIODIC . See also CHAOS ,QUASIPERIODIC FUNCTION Model Theory Model theory is a general theory of interpretations of AXIOMATIC SET THEORY . It is the branch of LOGIC studying mathematical structures by considering first-order sentences which are true of those struc- tures and the sets which are definable in those structures by first-order FORMULAS (Marker 1996). Mathematical structures obeying axioms in a system are called "models" of the system. The usual axioms of ANALYSIS are second order and are known to have the REAL NUMBERS as their unique model. Weakening the axioms to include only the first-order ones leads to a new type of model in what is called NONSTANDARD ANALYSIS . See also KHOVANSKI’S THEOREM ,NONSTANDARD ANA- LYSIS ,W ILKIE’S THEOREM References Doets, K. Basic Model Theory. New York: Cambridge University Press, 1996. Hodges, W. A Shorter Model Theory. New York: Cambridge University Press, 1997. Manzano, M. Model Theory. Oxford, England: Oxford Uni- versity Press, 1999. Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753 /C1/59, 1996. Stewart, I. "Non-Standard Analysis." In From Here to Infinity: A Guide to Today’s Mathematics. Oxford, Eng- land: Oxford University Press, pp. 80 /C1/1, 1996. Modified Bernoulli Number The numbers /b2n/ having GENERATING FUNCTION X/C12 n/C300b2nx2n /C301 2 lnex =2 /C28 e /C28x=2 1 2 x ! /C3012 ln 2 /C271 48 x2 /C281 5760 x4 /C271 362880 x6 /C28...: For n/C301, 2, ..., the denominators are 48, 5760, 362880, 19353600, ... (Sloane’s A057868). See also BERNOULLI NUMBER ,KONTSEVICH INTEGRAL References Sloane, N. J. A. Sequences A057868 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html.Modified Bessel Differential Equation The second-order ordinary differential equation x2d2y dx2/C27xdy dx/C28(x2/C27n2)y/C300: (1) The solutions are the MODIFIED BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS , and can be written y/C30a1Jn(/C28ix)/C27a2Yn(/C28ix) (2) /C30c1In(x)/C27c2Kn(x); (3) where Jn(x)i saB ESSEL FUNCTION OF THE FIRST KIND , Yn(x)i saB ESSEL FUNCTION OF THE SECOND KIND , In(x)i sa MODIFIED BESSEL FUNCTION OF THE FIRST KIND , and Kn(x)i s MODIFIED BESSEL FUNCTION OF THE SECOND KIND . Ifn/C300, the modified Bessel differential equation becomes x2d2y dx2/C27xdydx/C28x 2y/C300; (4) which can also be written d dxxdy dx ! /C30xy: (5) References Abramowitz, M. and Stegun, C. A. (Eds.). §9.6.1 in Hand- book of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Modified Bessel Function of the First Kind A function In(x) which is one of the solutions to the MODIFIED BESSEL DIFFERENTIAL EQUATION and is closely related to the B ESSEL FUNCTION OF THE FIRST KIND Jn(x):The above plot shows In(x) for n/C301, 2, ..., 5. In terms of Jn(x); In(x)/C13i/C28nJn(ix)/C30e/C28npi=2Jnxeip=2fflC{fflCz : (1) For a REAL NUMBER n;the function can be computed using In(z) /C30(1 2 z) nX/C12 k /C3001 4 z2fflCz6fflCz7k k!G( n /C27 k /C27 1) ; (2) where G(z) is the GAMMA FUNCTION . An integral formula is In(z) /C301 p g p 0ez cos u cos(nu) d u /C28sin(np) p g/C12 0e /C28z cosh t/C28 nt dt ; (3) which simplifies for n an INTEGER n to In(z) /C301 p g p 0ez cos u cos(n u) du (4) (Abramowitz and Stegun 1972, p. 376). A derivative identity for expressing higher order modified Bessel functions in terms of I0(x)is In(x) /C30Tnd dx ! I0(x) ; (5) where Tn(x)isaC HEBYSHEV POLYNOMIAL OF THE FIRST KIND . The special case of n/C300 gives I0(z) as the series J0(z)/C30X/C12 k/C3001 4z2fflCz6fflCz7k (k!)2: (6) See also BESSEL FUNCTION OF THE FIRST KIND, MODIFIED BESSEL FUNCTION OF THE FIRST KIND, WEBER’S FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Bessel Functions Iand K."§9.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 374 /C1/77, 1972. Arfken, G. "Modified Bessel Functions, In(x) and Kn(x):/"§11.5 inMathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610 /C1/16, 1985.Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Bessel Functions of Fractional Order, AiryFunctions, Spherical Bessel Functions." §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing,2nd ed. Cambridge, England: Cambridge University Press, pp. 234 /C1 /45, 1992. Spanier, J. and Oldham, K. B. "The Hyperbolic Bessel Functions I0(x) and I1(x)/" and "The General Hyperbolic Bessel Function In(x):/" Chs. 49 /C1/0i nAn Atlas of Functions. Washington, DC: Hemisphere, pp. 479 /C1/87 and 489 /C1/97, 1987. Modified Bessel Function of the Second Kind The function Kn(x) which is one of the solutions to the MODIFIED BESSEL DIFFERENTIAL EQUATION . The mod- ified Bessel functions of the second kind are some- times called the Basset functions (Spanier andOldham 1987, p. 499) or Macdonald functions (Spa- nier and Oldham 1987, p. 499; Samko et al. 1993, p. 20). K n(x) is closely related to the MODIFIED BESSEL FUNCTION OF THE FIRST KIND In(x) and H ANKEL FUNCTION Hn(x); Kn(x)/C131 2pin/C271H(1) n(ix) (1) /C3012pin/C271[Jn(ix)/C27iNn(ix)] (2) /C30p 2I/C28n(x)/C28In(x) sin(np)(3) (Watson 1966, p. 185). A sum formula for Kn(x)i s Kn(z)/C3012(12z)/C28nXn/C281 k/C300(n/C28k/C281)! k!(/C2814z2)k /C27(/C281)n/C271ln(1 2z)In(z)/C27(/C281)n12(12z)n /C2X/C12 k/C300[c(k/C271)/C27c(n/C27k/C271)](1 4z2)k k!(n/C27k)!;(4) where cis the DIGAMMA FUNCTION (Abramowitz and Stegun 1972). An integral formula is Kn(z)/C30G(n/C2712)(2z)n ffiffiffipp g/C12 0costd t (t2/C27z2)n/C271=2(5) which, for n/C300;simplifies to K0(x) /C30g/C12 0cos(x sinh t) dt /C30g/C12 0cos(xt) dtffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 /C27 1p : (6) Other identities are Kn(z) /C30ffiffiffipp (n /C281 2)!(1 2 z)ng/C12 1e /C28zx(x2 /C281)n/C281=2 dx (7) for n >/C281=2 and Kn(z) /C30ffiffiffiffiffi p 2zs e /C28z (n /C2812)! g/C12 0e /C28ttn/C281 =21 /C28t 2z !n/C281 =2 dt (8) /C30ffiffiffiffiffiffiffi p 2zs e /C28z n /C281 2fflCz6fflCz7 !X/C12 r/C300n /C281 2fflCz6fflCz7 ! r! n /C28 r /C281 2fflCz6fflCz7 !(2z)/C28r /C2g/C12 0e /C28ttn/C27r/C281 =2 dt: (9) The special case of n /C300 gives K0(z) as the integrals K0(z) /C30g/C12 0cos(x sinh t) dt (10) /C30g/C12 0cos(xt)ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 /C27 1p dt (11) (Abramowitz and Stegun 1972, p. 376). References Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Bessel Functions I and K." §9.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 374 /C1/77, 1972. Arfken, G. "Modified Bessel Functions, In(x) and Kn(x) :/" §11.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610 /C1/16, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Modified Bessel Functions of Integral Order" and "Bessel Functions of Fractional Order, Airy Func- tions, Spherical Bessel Functions." §6.6 and 6.7 in Numer- ical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 229 /C1/45, 1992. Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 20, 1993. Spanier, J. and Oldham, K. B. "The Basset Kn(x):/" Ch. 51 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 499 /C1/07, 1987.Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Modified Emden Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27 a(x)y?/C27x2yn /C300: See also EMDEN DIFFERENTIAL EQUATION References Leach, P. G. L. "First Integrals for the Modified Emden Equation ¨q /C27 a(t)˙q /C27qn /C300:/" J. Math. Phys. 26, 2510 /C1/514, 1985. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997. Modified Spherical Bessel Differential Equation The modified spherical Bessel differential equation is given by the SPHERICAL BESSEL DIFFERENTIAL EQUA- TION with a NEGATIVE separation constant, r2d2R dr2 /C272rdR dr/C28rr2/C27n(n/C271)fflC}fflC( R/C300: The solutions are called MODIFIED SPHERICAL BESSEL FUNCTIONS . See also MODIFIED SPHERICAL BESSEL FUNCTION , SPHERICAL BESSEL DIFFERENTIAL EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). §10.2.1 in Hand- book of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 374 /C1/77, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Modified Spherical Bessel Function Solutions to the MODIFIED SPHERICAL BESSEL DIFFER- ENTIAL EQUATION , given by in(x)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2xIn/C271=2(x)s (1) i0(x)/C30sinh x x(2) kn(x)/C13ffiffiffiffiffiffi 2p xs Kn/C271=2(x) (3) k0(x)/C30e/C28x x; (4) where In(x)i sa MODIFIED BESSEL FUNCTION OF THE FIRST KIND and Kn(x)isa MODIFIED BESSEL FUNCTION OF THE SECOND KIND . See also MODIFIED BESSEL FUNCTION OF THE FIRST KIND,M ODIFIED BESSEL FUNCTION OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Sphe- rical Bessel Functions." §10.2 in Handbook of Mathema- tical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 443 /C1/45, 1972. Modified Struve Function Ln(z) /C301 2 zfflCz6fflCz7n/C271 X/C12 k/C30012 zfflCz6fflCz72k G k /C2732fflCz6fflCz7 G k /C27 n /C2732fflCz6fflCz7 /C30212 zfflCz6fflCz7n ffiffiffippG n /C271 2fflCz6fflCz7g p =2 0sinh( z cos u) sin2n u du; where G(z) is the GAMMA FUNCTION . For integer n, the function is related to the ordinary STRUVE FUNCTION Hn(z)by Ln(iz) /C30/C28ie /C28npi =2Hn(z) : The Struve function Ln(z) is built into Mathematica 4.0 asStruveL [n, z]. See also ANGER FUNCTION ,STRUVE FUNCTION ,W E- BER FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Struve Function Ln(x) :/" §12.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 498, 1972. Apelblat, A. "Derivatives and Integrals with Respect to the Order of the Struve Functions Hn(x) and Ln(x) :/" J. Math. Anal. Appl. 137,17/C1/6, 1999. Modul MODULE Modular Angle Given a MODULUS k in an ELLIPTIC INTEGRAL , the modular angle is defined by k /C13sin a: An ELLIPTIC INTEGRAL is written I( f½m) when the PARAMETER is used, I( f; k) when the MODULUS is used, and I(f_a) when the modular angle is used. See also AMPLITUDE ,CHARACTERISTIC (ELLIPTIC IN- TEGRAL ), ELLIPTIC INTEGRAL ,H ALF-PERIOD RATIO, MODULUS (ELLIPTIC INTEGRAL ), NOME,PARAMETER References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. New York: Dover, p. 590, 1972. Modular Discriminant Define q /C13e2pit (cf. the usual NOME ), where t is in the UPPER HALF-PLANE . Then the modular discriminant is defined by D( t) /C13qY/C12 r/C3011 /C28qrðÞ24 (Rankin 1977, p. 196; Berndt 1988, p. 326; Milne 2000). If g2( v1 ; v2) and g3( v1 ; v2) are the INVARIANTS of a WEIERSTRASS ELLIPTIC FUNCTION / /C212(z j v1 ; v2)/ //C30/C212(z; g2 ; g3)/ with periods v1and v2 ; then the discriminant is defined by D( v1 ; v2) /C30g3 2 /C2827g23 : (1) Letting t /C13 v2 =v1 ; then D( t) /C13D(1; t) /C30 v121D( v1 ; v2) (2) /C30g32(t) /C2827g23(t) : (3) The FOURIER SERIES of D(t) for t /C23 H ; where H is the UPPER HALF-PLANE ,is D(t) /C30(2p)12 X/C12 n /C301t(n)e2 pint ; (4) where t(n) is the TAU FUNCTION , and t(n) are integers (Apostol 1997, p. 20). The discriminant can also be expressed in terms of the DEDEKIND ETA FUNCTION h(t)by D( t) /C30(2p)12[h( t)]24 (5) (Apostol 1997, p. 51). See also DEDEKIND ETA FUNCTION ,INVARIANT (EL- LIPTIC FUNCTION ), KLEIN’S ABSOLUTE INVARIANT , NOME,TAU FUNCTION ,W EIERSTRASS ELLIPTIC FUNC- TION References Apostol, T. M. "The Discriminant D/" and "The Fourier Expansions of D(t) and J(t):/"§1.11 and 1.15 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 14 and 20 /C1/2, 1997. Berndt, B. C. Ramanujan’s Notebooks, Part II. New York: Springer-Verlag, p. 326, 1988. Milne, S. C. Hankel Determinants of Eisenstein Series. 13 Sep 2000. http://xxx.lanl.gov/abs/math.NT/0009130/. Nesterenko, Yu. V. §1.2 in A Course on Algebraic Indepen- dence: Lectures at IHP 1999. http://www.math.jussieu.fr/ ~nesteren/. Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, p. 196, 1977. Modular Equation The modular equation of degree ngives an algebraic connection OF THE FORM K?(l) K(l)/C30nK?(k) K(k)(1) between the TRANSCENDENTAL COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND with moduli kand l. When kandlsatisfy a modular equation, a relation- ship OF THE FORM M(l;k)dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28y2 ðÞ 1/C28l2y2 ðÞp /C30dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 ðÞ 1/C28k2x2 ðÞp (2) exists, and Mis called the multiplier. In general, if p is an ODD PRIME , then the modular equation is given by Vp(u;v)/C30v/C28u0 ðÞ v/C28u1 ðÞ /C1 /C1 /C1 v/C28upfflC{fflCz ; (3) where up/C13(/C281)(p2/C281)=8l(qp) ½/C1381=8/C13(/C281)(p2/C281)=8u(qp); (4) /lis a ELLIPTIC LAMBDA FUNCTION , and q/C13eipt(5) (Borwein and Borwein 1987, p. 126). An ELLIPTIC INTEGRAL identity gives K?(k) K(k)/C302K?2ffiffiffi kp 1/C27k ! K2ffiffiffi kp 1/C27k ! ; (6) so the modular equation of degree 2 is l/C302ffiffiffikp 1/C27k(7) which can be written as l21/C27k2fflC{fflCz /C304k: (8) A few low order modular equations written in terms ofkandlare V2/C30l2(1/C27k)2/C284k/C300 (9) V7/C30(kl)1=4/C27(k?l?)1=4/C281/C300 (10) V23/C30(kl)1=4/C27(k?l?)1=4/C2722=3(klk?l?)1=12/C281/C300: (11) In terms of uandv, V3(u;v)/C30u4/C28v4/C272uv1/C28u2v2fflC{fflCz /C300 (12)V5(u;v)/C30v6/C28u6/C275u2v2v2/C28u2fflC{fflCz /C274uv u4v4/C281fflC{fflCz /C30u v !3 /C27v u !3 /C302u2v2/C281 u2v2 ! /C300 (13) V7(u;v)/C301/C28u8fflC{fflCz 1/C28v8fflC{fflCz /C28(1/C28uv)8/C300; (14) where u2/C13ffiffiffi kp /C30q2(q) q3(q)(15) and v2/C13ffiffilp /C30q2qpðÞ q3qpðÞ: (16) Here, qiare J ACOBI THETA FUNCTIONS . A modular equation of degree 2rforr]2 can be obtained by iterating the equation for 2r/C281:Modular equations for PRIME pfrom 3 to 23 are given in Borwein and Borwein (1987). Quadratic modular identities include q3(q) q3q4ðÞ/C281/C30q2 3q2ðÞ q23q4ðÞ/C281"#1=2 : (17) Cubic identities include 3q2q9ðÞ q2(q)/C281"#3 /C309q4 2q3ðÞ q4 2(q)/C281 (18) 3q3q9ðÞ q3(q)/C281"#3 /C309q43q3ðÞ q43(q)/C281 (19) 3q4q9ðÞ q4(q)/C281"#3 /C309q44q3ðÞ q44(q)/C281: (20) A seventh-order identity is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q3(q)q3q7ðÞp /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q4(q)q4q7ðÞp /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2(q)q2q7ðÞp :(21) From Ramanujan (1913 /C1/914), (1/C27q)1/C27q3fflC{fflCz 1/C27q5fflC{fflCz /C1/C1/C1/C3021=6q1=24(kk?)/C281=12(22) (1/C28q)1/C28q3fflC{fflCz 1/C28q5fflC{fflCz /C1/C1/C1/C3021=6q1=24k/C281=12k?1=6:(23) When kand lsatisfy a MODULAR EQUATION ,a relationship OF THE FORM M(l;k)dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28y2 ðÞ 1/C28l2y2 ðÞp /C30dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 ðÞ 1/C28k2x2 ðÞp (24) exists, and Mis called the multiplier. The multiplier of degree ncan be given by Mn(l;k)/C13q2 3(q) q23(q1=p)/C30K(k) K(l); (25) where qiis a JACOBI THETA FUNCTION and K(k)isa complete ELLIPTIC INTEGRAL OF THE FIRST KIND . The first few multipliers in terms of l and k are M2(l ; k) /C301 1 /C27 k /C301 /C27 l ? 2 (26) M3(l ; k) /C301 /C28ffiffiffiffi l3 ks 1 /C28ffiffiffiffiffi k3 ls : (27) In terms of the u and v defined for MODULAR EQUATIONS , M3 /C30v v /C27 2u3 /C302v3 /C28 u 3u (28) M5 /C30v(1 /C28 uv3) v /C28 u5/C30u /C27 v5 5u(1 /C27 u3v) (29) M7/C30v(1/C28uv)(1/C28uv/C27(uv)2)] v/C28u2 /C30v7/C28u 7u(1/C28uv)(1/C28uv/C27(uv)2)]: (30) See also MODULAR FORM,M ODULAR FUNCTION , SCHLA ¨ FLI’S MODULAR FORM References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 127 /C1/32, 1987. Hanna, M. "The Modular Equations." Proc. London Math. Soc. 28,4 6/C1/2, 1928. Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/72, 1913 /C1/914. Modular Form A function fis said to be an entire modular form of weight kif it satisfies 1.fis analytic in the UPPER HALF-PLANE H, 2.fat/C27b ct/C27dfflCz6fflCz7 /C30(ct/C27d)kf(t) wheneverab cdfflC}fflC( is a mem- ber of the MODULAR GROUP GAMMA , 3. The F OURIER SERIES offhas the form f(t)/C30X/C12 n/C300c(n)e2pint(1) Care must be taken when consulting the literature because some authors use the term "dimension /C28k/"o r "degree /C28k/" instead of "weight k," and others write k instead of k(Apostol 1997, pp. 114 /C1/15). More general types of modular forms (which are not "entire"rpar;can also be defined which allow poles in Hor at i/C12: Since K LEIN’S ABSOLUTE INVARIANT J, which is a MODULAR FUNCTION , has a pole at i/C12;it is a nonentire modular form of weight 0. The set of all entire forms of weight kis denoted Mk; which is a linear space over the complex field. The dimension of Mkis 1 for k/C304, 6, 8, 10, and 14 (Apostol 1997, p. 119). /c(0) is the value of fati/C12;and if c(0)/C300;the function is called a CUSP FORM . The smallest rsuch that c(r)" 0 is called the order of the zero of fati/C12:An estimate forc(n) states that c(n)/C30O(n2k/C281) (2) iff/C23M2kand is not a CUSP FORM (Apostol 1997, p. 135). Iff"0 is an entire modular form of weight k, let f have Nzeros in the closure of the FUNDAMENTAL REGION RG(omitting the vertices). Then k/C3012N/C276N(i)/C274N(r)/C2712N(i/C12); (3) where N(p) is the order of the zero at a point p (Apostol 1997, p. 115). In addition, 1. The only entire modular forms of weight k/C300 are the constant functions. 2. If kisODD,kB0, or k/C302, then the only entire modular form of weight kis the zero function. 3. Every nonconstant entire modular form for weight k]4;where kisEVEN . 4. The only entire CUSP FORM of weight kB12 is the zero function. (Apostol 1997, p. 116). Forfan entire modular form of EVEN weight k]0; define E0(t)/C301 for all t:Then fcan be expressed in exactly one way as a sum f/C30Xk=12bc r/C300 k/C3012r"2arEk/C2812rDr; (4) where arare complex numbers, Enis an E ISENSTEIN SERIES , and Dis the MODULAR DISCRIMINANT of the WEIERSTRASS ELLIPTIC FUNCTION .CUSP FORMS of EVEN weight kare then those sums for which a0/C300 (Apostol 1997, pp. 117 /C1/18). Even more amazingly, every entire modular form fof weight kis a POLY- NOMIAL inE4andE6given by f/C30X a;bca;bEa 4Ea6; (5) where the ca;bare complex numbers and the sum is extended over all integers a;b]0 such that 4 a/C27 6b/C30k(Apostol 1998, p. 118). Modular forms satisfy rather spectacular and special properties resulting from their surprising array of internal symmetries. Hecke discovered an amazing connection between each modular form and a corre- sponding DIRICHLET L-SERIES . A remarkable connec- tion between rational ELLIPTIC CURVES and modular forms is given by the TANIYAMA- SHIMURA CONJEC- TURE , which states that any rational ELLIPTIC CURVE is a modular form in disguise. This result was the one proved by Andrew Wiles in his celebrated proof of FERMAT’S LAST THEOREM . See also CUSP FORM,D IRICHLET SERIES ,E LLIPTIC CURVE ,ELLIPTIC FUNCTION ,FERMAT’S LAST THEO- REM,HECKE ALGEBRA ,HECKE OPERATOR ,M ODULAR FUNCTION ,SCHLA ¨ FLI’S MODULAR FORM,TANIYAMA- SHIMURA CONJECTURE References Apostol, T. M. "Modular Forms with Multiplicative Coeffi- cients." Ch. 6 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 113 /C1/41, 1997. Hecke, E. "U¨ ber Modulfunktionen und die Dirichlet Reihen mit Eulerscher Produktentwicklungen. I." Math. Ann. 114,1/C1/8, 1937. Knopp, M. I. Modular Functions in Analytic Number The- ory. New York: Chelsea, 1993. Koblitz, N. Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993. Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, 1977. Sarnack, P. Some Applications of Modular Forms. Cam- bridge, England: Cambridge University Press, 1993. Modular Function A function is said to be modular (or "elliptic modular") if it satisfies: 1. f is MEROMORPHIC in the UPPER HALF-PLANE H, 2. f(A t) /C30f( t) for every MATRIX A in the MODULAR GROUP GAMMA , 3. The LAURENT SERIES of f has the form f( t) /C30Xm n/C30/C28ma(n)e2pin t (Apostol 1997, p. 34). Every RATIONAL FUNCTION of KLEIN’S ABSOLUTE INVARIANT J is a modular function, and every modular function can be expressed as a RATIONAL FUNCTION of J (Apostol 1997, p. 40). An important property of modular functions is that if f is modular and not identically 0, then the number of zeros of f is equal to the number of poles of f in the closure of the FUNDAMENTAL REGION RG(Apostol 1997, p. 34). See also DIRICHLET SERIES ,E LLIPTIC FUNCTION , ELLIPTIC LAMBDA FUNCTION ,E LLIPTIC MODULAR FUNCTION ,KLEIN’S ABSOLUTE INVARIANT ,M ODULAREQUATION ,M ODULAR FORM,M ODULAR GROUP GAM- MA,M ODULAR GROUP GAMMA0 ,M ODULAR GROUP LAMBDA References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, 1997. Askey, R. In Ramanujan International Symposium (Ed. N. K Thakare). pp. 1 /C1/3. Borwein, J. M. and Borwein, P. B. "Elliptic Modular Func- tions." §4.3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 112 /C1/16, 1987. Rademacher, H. "Zur Theorie der Modulfunktionen." J. reine angew. Math. 167, 312 /C1/36, 1932. Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, 1977. Schoeneberg, B. Elliptic Modular Functions: An Introduc- tion. Berlin: New York: Springer-Verlag, 1974. Weisstein, E. W. "Books about Modular Functions." http:// www.treasure-troves.com/books/ModularFunctions.html. Modular Group MODULAR GROUP GAMMA ,MODULAR GROUP GAMMA0 , MODULAR GROUP LAMBDA Modular Group Gamma The GROUP G of all MO¨ BIUS TRANSFORMATIONS OF THE FORM t ?/C30at /C27 b c t /C27 d ; (1) where a, b, c, and d are integers with ab /C28bc /C301: The group can be represented by the 2 /C292 matrix A /C30ab cdfflC}{fflC}z ; (2) where det(A) /C301: Every A /C23G can be expressed in the form A /C30Tn1 ST n2 S /C1/C1/C1ST nk; (3) where S /C300 /C281 10fflC}{fflC}z (4) T /C3011 01fflC}{fflC}z ; (5) although the representation is not unique (Apostol 1997, pp. 28 /C1/9). See also KLEIN’S ABSOLUTE INVARIANT ,M O¨ BIUS TRANSFORMATION ,M ODULAR GROUP GAMMA0 ,M OD- ULAR GROUP LAMBDA ,THETA FUNCTIONS ,UNIMODU- LAR TRANSFORMATION References Apostol, T. M. "The Modular Group and Modular Func- tions." Ch. 2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 17 and 26 /C1/6, 1997. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 113, 1987. Modular Group Gamma0 Let q be a POSITIVE INTEGER , then G0(q) is defined as the set of all matricesab cdfflC}fflC( in the MODULAR GROUP GAMMA G with c /C130 (mod q):G0(q)isa SUBGROUP of G: For any PRIME p, the set RG@@p /C281 k /C300STk(RG) is a FUNDAMENTAL REGION of the subgroup G0(q); where S t /C30/C281 =t and T t /C30 t /C271 (Apostol 1997). See also MODULAR GROUP GAMMA0 ,MODULAR GROUP LAMBDA References Apostol, T. M. "The Subgroup G0(q)/" and "Fundamental Region G0(q) :/" §4.2 /C1/.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 75 /C1/8, 1997. Modular Group Lambda The set l of linear MO¨ BIUS TRANSFORMATIONS w which satisfy w(t) /C30at /C27 b ct /C27 d ; where a and d are ODD and b and c are EVEN . l is a SUBGROUP of the MODULAR GROUP GAMMA , and is also called the THETA SUBGROUP . The FUNDAMENTAL RE- GION of the modular lambda group is illustrated above. See also MODULAR GROUP GAMMA References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 113 /C1/14, 1987.ModularLambda ELLIPTIC LAMBDA FUNCTION Modular Lattice A LATTICE which satisfies the identity (x ffly) /C150(x fflz) /C30x ffl(y /C150(x fflz)) is said to be modular. See also DISTRIBUTIVE LATTICE References Gra¨tzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, pp. 35 /C1/6, 1971. Modular System A set M of all POLYNOMIALS in s variables, x1 ; ..., xs such that if P, P1 ; and P2are members, then so are P1 /C27P2 and QP, where Q is any POLYNOMIAL in x1 ; ..., xs :/ See also HILBERT’S THEOREM ,M ODULE ,M ODULAR SYSTEM BASIS Modular System Basis A basis of a MODULAR SYSTEM M is any set of POLYNOMIALS B1 ; B2 ; ...of M such that every POLY- NOMIAL of M is expressible in the form R1B1 /C27R2B2 /C27...; where R1 ; R2 ; ...are POLYNOMIALS . Modular Transformation MODULAR EQUATION Modulation Theorem The important property of FOURIER TRANSFORMS that F[cos(2 pk0x)f(x)] can be expressed in terms of F[f(x)] /C30F(k) as follows, F[cos(2 pk0x)f(x)] /C301 2[F(k /C28k0) /C27F(k /C27k0)]: See also FOURIER TRANSFORM References Bracewell, R. "Modulation Theorem." The Fourier Trans- form and Its Applications, 3rd ed. New York: McGraw- Hill, p. 108, 1999. Module A mathematical object in which things can be added together COMMUTATIVELY by multiplying COEFFI- CIENTS and in which most of the rules of manipulat- ing VECTORS hold. A module is abstractly very similar to a VECTOR SPACE , although in modules, COEFFI- CIENTS are taken in RINGS which are much more general algebraic objects than the FIELDS used in VECTOR SPACES . A module taking its coefficients in a RING R is called a module over R,ora R-MODULE . Modules are the basic tool of HOMOLOGICAL ALGEBRA . Examples of modules include the set of INTEGERS Z; the cubic lattice in d dimensions Zd ; and the GROUP RING of a GROUP . /Z is a module over itself. It is CLOSED under ADDITION and SUBTRACTION (although it is SUFFICIENT to require closure under SUBTRACTION ). Numbers OF THE FORM for n /C23Z and a a fixed integer form a submodule since, for all (n; m) /C23Z; na 9ma /C30(n 9m) a and (n 9m) is still in Z:/ Given two INTEGERS a and b, the smallest module containing a and b is the module for their GREATEST COMMON DIVISOR , a /C30GCD( a ; b) :/ See also DIFFERENT ,D IRECT SUM,D ISCRIMINANT (MODULE ), FIELD,G RADED MODULE ,G ROUP RING, HOMOLOGICAL ALGEBRA ,M ODULAR SYSTEM , R-MOD- ULE,R ING,SUBMODULE ,V ERMA MODULE ,V ECTOR SPACE References Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, 1999. Berrick, A. J. and Keating, M.E An Introduction to Rings and Modules with K-Theory in View. Cambridge, Eng- land: Cambridge University Press, 2000. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillian, p. 390, 1996. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. Herstein, I. N. "Modules." §1.1 in Noncommutative Rings. Washington, DC: Math. Assoc. Amer., pp. 1 /C1/, 1968. Nagell, T. "Moduls, Rings, and Fields." §6inIntroduction to Number Theory. New York: Wiley, pp. 19 /C1/1, 1951. Riesel, H. "Modules." Prime Numbers and Computer Meth- ods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 239 /C1/40, 1994. Module Direct Sum The direct sum of modules A and B is the module A /C154B /C30fa /C154b ½ a /C23 A; b /C23 B g; (1) where all algebraic operations are defined compo- nentwise. In particular, suppose that A and B are left R-modules, then a1 /C154b1 /C27a2 /C154b2 /C30(a1 /C27a2) /C154(b1 /C27b2) (2) and r(a /C154b) /C30(ra /C154rb) ; (3) where r is an element of the RING R. The direct sum of an arbitrary family of MODULES over the same RING is also defined. If J is the indexing set for the family of MODULES , then the direct sum is represented by the collection of functions with finite support from J tothe union of all these MODULES such that the function sends j /C23 J to an element in the MODULE indexed by j. The dimension of a direct sum is the sum of the dimensions of the quantities summed. The significant property of the direct sum is that it is the COPRODUCT in the CATEGORY of MODULES . This general definition gives as a consequence the definition of the direct sum A /C154B of ABELIAN GROUPS A and B (since they are Z/-modules, i.e., MODULES over the INTEGERS ) and the direct sum of VECTOR SPACES (since they are MODULES over a FIELD ). Note that the direct sum of Abelian groups is the same as the GROUP DIRECT PRODUCT , but that the term direct sum is not used for groups which are NON- ABELIAN . Whenever C is a MODULE , with module homomorph- isms fA : A 0 C and fB : B 0 C ; then there is a module homomorphism fA : A /C154B 0 C; given by f(a /C154b) /C30 fA(a) /C27fB(b) : Note that this map is well-defined because addition in modules is commutative. Some- times direct sum is preferred over direct product when the coproduct property is emphasized. See also COPRODUCT ,D IRECT SUM,G ROUP DIRECT PRODUCT ,MODULE References Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, pp. 11 and 80, 1999. Moduli Space This entry contributed by EDGAR VAN TUYLL In ALGEBRAIC GEOMETRY classification problems, an ALGEBRAIC VARIETY (or other appropriate space in other parts of geometry) whose points correspond to the equivalence classes of the objects to be classified in some natural way. Moduli space can be thought of as the space of EQUIVALENCE CLASSES of COMPLEX STRUCTURES on a fixed surface of GENUS g, where two COMPLEX STRUCTURES are deemed "the same" if they are equivalent by CONFORMAL MAPPING . See also ALGEBRAIC VARIETY ,COMPLEX STRUCTURE References Kirwan, F. "Introduction to Moduli Spaces." In Proceedings of the EWM Workshop on Moduli Spaces, Oxford, EWM. 1999. Naber, G. L. Topology, Geometry and Gauge Fields: Founda- tions. New York: Springer-Verlag, 1997. Polchinski, J. G. String Theory: An Introduction to the Bosonic String. Cambridge, England: Cambridge Univer- sity Press, 1998. Modulo CONGRUENCE Modulo Multiplication Group AFINITE GROUP MmofRESIDUE CLASSES prime to m under multiplication mod m.Mmis A BELIAN ofORDER f(m);where f(m) is the TOTIENT FUNCTION . The following table gives the modulo multiplication groups of small orders, where Zndenotes the CYCLIC GROUP of order n. /Mm/Group /f(m)/Elements /M2///C142e/C143/ 11 /M3//Z2/ 21 , 2 /M4//Z2/ 21 , 3 /M5//Z4/ 4 1 ,2 ,3 ,4 /M6//Z2/ 21 , 5 /M7//Z6/ 6 1 ,2 ,3 ,4 ,5 ,6 /M8//Z2/C29Z2/ 4 1 ,3 ,5 ,7 /M9//Z6/ 6 1 ,2 ,4 ,5 ,7 ,8 /M10//Z4/ 4 1 ,3 ,7 ,9 /M11//Z10/ 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 /M12//Z2/C29Z2/ 4 1 ,5 ,7 ,1 1 /M13//Z12/ 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 /M14//Z6/ 6 1, 3, 5, 9, 11, 13 /M15//Z2/C29Z4/ 8 1, 2, 4, 7, 8, 11, 13, 14 /M16//Z2/C29Z4/ 8 1, 3, 5, 7, 9, 11, 13, 15 /M17//Z16/ 16 1, 2, 3, ..., 16 /M18//Z6/ 6 1, 5, 7, 11, 13, 17 /M19//Z18/ 18 1, 2, 3, ..., 18 /M20//Z2/C29Z4/ 8 1, 3, 7, 9, 11, 13, 17, 19 /M21//Z2/C29Z6/ 12 1, 2, 4, 5, 7, 8, 10, 11, 13, 16, 17, 19 /M22//Z10/ 10 1, 3, 5, 7, 9, 13, 15, 17, 19, 21 /M23//Z22/ 22 1, 2, 3, ..., 22 /M24//Z2/C29Z2/C29Z2/8 1, 5, 7, 11, 13, 17, 19, 23 /Mmis a CYCLIC GROUP (which occurs exactly when m has a PRIMITIVE ROOT )IFFmis of one of the forms m/C302, 4, pn;or 2pn;where pis an ODD PRIME and n]1 (Shanks 1993, p. 92). ISOMORPHIC modulo multiplication groups can be determined using a particular type of factorizationoff(m) as described by Shanks (1993, pp. 92 /C1 /3). To perform this factorization (denoted fm);factor min the standard form m/C30pa1 1pa2 2/C1/C1/C1pann: (1) Now write the factorization of the TOTIENT FUNCTION involving each power of an ODD PRIME fpai ifflC{fflCz /C30(pi/C281)pai/C281 i (2) as fpai ifflC{fflCz /C30qb1 1DE qb2 2DE /C1/C1/C1qbssfflCz{fflCzz pai/C281 iDE ; (3) where pi/C281/C30qb1 1qb2 2/C1/C1/C1qbss; (4) /qbfflCz{fflCzz denotes the explicit expansion of qb(i.e., 52/C3025); and the last term is omitted if ai/C301:Ifp1/C302;write f(2a1)/C302 for a1/C302 22a1/C282hi fora1>2:fflC}6 (5) Now combine terms from the odd and even primes.For example, consider m/C30104/C302 3/C21513:The only odd prime factor is 13, so factoring gives 13 /C281/C3012/C30 22hi 3hi/C303/C2154:The rule for the powers of 2 gives 23/C30 223/C282hi /C3022hi/C302/C2152:Combining these two gives f104/C302/C2152/C2153/C2154:Other explicit values of fmare given below. f3/C302 f4/C302 f5/C304 f6/C302 f15/C302/C2154 f16/C302/C2154 f17/C3016 f104/C302/C2152/C2153/C2154 f105/C302/C2152/C2153/C2154: /Mmand Mnare isomorphic IFFfmand fnare identical. More specifically, the abstract GROUP cor- responding to a given Mmcan be determined expli- citly in terms of a GROUP DIRECT PRODUCT ofCYCLIC GROUPS of the so-called CHARACTERISTIC FACTORS , whose product is denoted Fn:This representation is obtained from fmas the set of products of largest powers of each factor of fm:For example, for f104;the largest power of 2 is 4 /C3022and the largest power of 3 is 3/C3031;so the first characteristic factor is 4 /C293/C3012; leaving 2 /C2152 (i.e., only powers of two). The largest power remaining is 2 /C3021;so the second CHARACTER- ISTIC FACTOR is 2, leaving 2, which is the third and last CHARACTERISTIC FACTOR . Therefore, F104/C302/C2152/C215 4;and the group Mmis isomorphic to Z2/C29Z2/C29Z4:/ The following table summarizes the isomorphic mod- ulo multiplication groups Mnfor the first few nand identifies the corresponding abstract GROUP .N o Mm isISOMORPHIC toZ8;Q8;orD4:However, every finite ABELIAN GROUP is isomorphic to a SUBGROUP ofMm for infinitely many different values of m(Shanks 1993, p. 96). C YCLE GRAPHS corresponding to Mnfor small nare illustrated above, and more complicated CYCLE GRAPHS are illustrated by Shanks (1993, pp. 87 /C1/2). Group Isomorphic Mm/ //C142e/C143// M2/ /Z2// M3;M4;M6/ /Z4// M5;M10/ /Z2/C29Z2// M8;M12/ /Z6// M7;M9;M14;M18/ /Z2/C29Z4// M15;M16;M20;M30/ /Z2/C29Z2/C29Z2//M24/ /Z10// M11;M22/ /Z12// M13;M26/ /Z2/C29Z6// M21;M28;M36;M42/ /Z16// M17;M34/ /Z2/C29Z8// M32/ /Z2/C29Z2/C29Z4//M40;M48;M60/ /Z18// M19;M27;M38;M54//Z20// M25;M50/ /Z2/C29Z10// M33;M44;M66/ /Z22// M23;M46/ /Z2/C29Z12// M35;M39;M45;M52;M70;M78;M90/ /Z28// M29;M58/ /Z30// M31;M62/ /Z36// M37;M74/ The number of CHARACTERISTIC FACTORS rofMmfor m/C301, 2, ... are 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, ... (Sloane’s A046072). The number of QUADRATIC RESI- DUES inMmform/C212 are given by f(m)=2r(Shanks 1993, p. 95). The first few for m/C301, 2, ... are 0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 6, ... (Sloane’s A046073). In the table below, f(n) is the TOTIENT FUNCTION (Sloane’s A000010) factored into CHARACTERISTIC FACTORS ,l(n) is the C ARMICHAEL FUNCTION (Sloane’s A011773), and giare the smallest generators of the group Mn(of which there is a number equal to the number of CHARACTERISTIC FACTORS ). n /f(n)//l(n)// gi/n /f(n)//l(n)// gi/ 3 2 2 2 27 18 18 2 42 23 2 8 /2/C2156/6 13, 3 5 4 2 2 29 28 28 262 25 3 0 /2/C2154/4 11, 7 7 6 6 3 31 30 30 38 /2/C2152/27 , 3 3 2 /2/C2158/8 31, 3 96 62 3 3 /2/C21510/10 10, 2 10 4 4 3 34 16 16 311 10 10 2 35 /2/C21512/12 6, 2 12 /2/C2152/25 , 7 3 6 /2/C2156/6 19,5 13 12 12 2 37 36 36 214 6 6 3 38 18 18 3 15 /2/C2154/4 14, 2 39 /2/C21512/12 38, 2 16 /2/C2154/4 15, 3 40 /2/C2152/C2154/4 39, 11, 3 17 16 16 3 41 40 40 6 18 6 6 5 42 /2/C2156/6 13, 5 19 18 18 2 43 42 42 320 /2/C2154/4 19, 3 44 /2/C21510/10 43, 3 21 /2/C2156/6 20, 2 45 /2/C21512/12 44, 2 22 10 10 7 46 22 22 5 23 22 22 5 47 46 46 5 24 /2 /C215 2 /C215 2/ 25,7,1348 /2 /C215 2 /C215 4/ 4 47, 7, 5 25 20 20 2 49 42 42 3 26 12 12 7 50 20 20 3 See also CHARACTERISTIC FACTOR ,C YCLE GRAPH , FINITE GROUP ,RESIDUE CLASS References Riesel, H. "The Structure of the Group Mn :/" Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 270 /C1/72, 1994. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 61 /C1/2 and 92, 1993. Sloane, N. J. A. Sequences A000010/M0299, A011773, A046072, and A046073 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Groups." MATHEMATICA NOTEBOOK GROUPS.M . Modulus The word modulus has several different meanings in mathematics with respect to complex numbers, con- gruences, elliptic integrals, quadratic invariants, sets, etc. See also MODULUS (COMPLEX NUMBER ), MODULUS (CONGRUENCE ), MODULUS (ELLIPTIC INTEGRAL ), MOD- ULUS (QUADRATIC INVARIANTS ), MODULUS (SET) Modulus (Complex Number) The modulus of a COMPLEX NUMBER z is denoted ½z ½: jx /C27iy j/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p (1) rei ffflCz}fflCz}fflCz}fflCz}/C30½r½: (2) Let c 1 /C13Aeif1 and c2 /C13Beif2 be two COMPLEX NUM- BERS . Then c1 c2fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}/C30 Aeif1 Beif2fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}/C30 A Bei( f1/C28f2)fflCz}fflCz}fflCz}fflCz}/C30A B (3) ½c1 ½ ½c2 ½/C30Aeif1 jj Beif2 jj/C30A Bei f1jj ei f2jj/C30A B ; (4) so c1 c2fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}/C30 ½c1 ½ ½c2 ½: (5) Also, ½c1c2 ½/C30½(Aei f1 )(Beif2 ) ½/C30AB ½ei( f1/C27f2) ½/C30AB (6) ½c1 ½½c2 ½/C30½Aeif1 ½½Beif2 ½/C30AB ½ei f1 ½½eif2 ½/C30AB ; (7)so ½c1c2 ½/C30½c1 ½½c2 ½ (8) and, by extension, ½zn ½/C30½z ½n : (9) The only functions satisfying identities OF THE FORM ½f(x /C27iy) ½/C30½f(x) /C27f(iy) ½ (10) are f(z) /C30Az; f(z) /C30A sin(bz); and f(z) /C30A sinh( bz) (Robinson 1957). See also ABSOLUTE SQUARE ,A RGUMENT (COMPLEX NUMBER ), COMPLEX NUMBER ,IMAGINARY PART,MAX- IMUM MODULUS PRINCIPLE ,M INIMUM MODULUS PRINCIPLE ,REAL PART References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Krantz, S. G. "Modulus of a Complex Number." §1.1.4 n Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 2 /C1/, 1999. Robinson, R. M. "A Curious Mathematical Identity." Amer. Math. Monthly 64,83/C1/5, 1957. Modulus (Congruence) The modulus of a CONGRUENCE a /C13b (mod m) is the number m. It is the "base" with respect to which a CONGRUENCE is computed (i.e., m gives the number of multiples of a that are "thrown out"). For example, when computing the time of day using a 12-hour clock obtained by adding four hours to 9:00, the answer, 1:00, is obtained by taking 9 /C274 /C131 (mod 12) (i.e., adding the hours with modulus 12). In many computer languages (such as FORTRAN or Mathematica ), the COMMON RESIDUE ofb(mod m)i s writtenmod( b,m)(FORTRAN )o rMod[ b,m](Mathe- matica ). See also CONGRUENCE Modulus (Elliptic Integral) A parameter kused in ELLIPTIC INTEGRALS and ELLIPTIC FUNCTIONS defined to be k/C13ffiffiffiffiffimp;where m is the PARAMETER .A n ELLIPTIC INTEGRAL is written I(f;k) when the modulus is used. It can be computed explicitly in terms of J ACOBI THETA FUNCTIONS of zero argument: k/C30q2 2(0;q) q2 3(0;q): (1) The REAL period K(k) and IMAGINARY period K?(k)/C30 K(k?)/C30K(ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28k2p ) are given by 4K(k)/C302pq2 3(0½t) (2) 2iK ?(k) /C30 pt q2 3(0½ t); (3) where K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND and the complementary modulus is de- fined by k ?2 /C131 /C28k2 ; (4) with k the modulus. See also AMPLITUDE ,CHARACTERISTIC (ELLIPTIC IN- TEGRAL ), COMPLEMENTARY MODULUS ,ELLIPTIC FUNC- TION ,E LLIPTIC INTEGRAL ,E LLIPTIC INTEGRAL SINGULAR VALUE ,HALF-PERIOD RATIO,JACOBI THETA FUNCTIONS ,MODULAR ANGLE ,NOME,PARAMETER References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 590, 1972. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 35, 1987. To¨lke, F. "Parameterfunktionen." Ch. 3 in Praktische Funk- tionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83 /C1/15, 1966. Modulus (Quadratic Invariants) The quantity ps /C28rq obtained by letting x /C30pX /C27qY (1) y /C30rX /C27sY (2) in ax2 /C272bxy /C27cy2 (3) so that A /C30ap2 /C272bpr /C27cr2 (4) B /C30apq /C27b(ps /C27qr) /C27crs (5) C /C30aq2 /C272bqs /C27cs2 (6) and B2 /C28AC /C30(ps /C28rq)2(b2 /C28ac) ; (7) is called the modulus. Modulus (Set) The name for the SET of INTEGERS modulo m, denoted Z_mZ : If m is a PRIME p, then the modulus is a FINITE FIELD Fp /C30Z_pZ :/ Moebius MO¨ BIUS FUNCTION ,M O¨ BIUS GROUP ,M O¨ BIUS INVER- SION FORMULA ,MO¨ BIUS PERIODIC FUNCTION ,MO¨ BIUS PROBLEM ,M O¨ BIUS SHORTS ,M O¨ BIUS STRIP,M O¨ BIUS STRIP DISSECTION ,M O¨ BIUS TRANSFORMATION ,M O¨ - BIUS TRIANGLESMoebiusMu MO¨ BIUS FUNCTION Moessner’s Theorem Write down the POSITIVE INTEGERS in row one, cross out every k1th number, and write the partial sums of the remaining numbers in the row below. Now cross off every k2th number and write the partial sums of the remaining numbers in the row below. Continue. For every POSITIVE INTEGER k /C211, if every kth number is ignored in row 1, every (k /C281)/th number in row 2, and every (k /C271 /C28i)/th number in row i, then the kth row of partial sums will be the kth POWERS 1k ; 2k ; 3k ; .... References Conway, J. H. and Guy, R. K. "Moessner’s Magic." In The Book of Numbers. New York: Springer-Verlag, pp. 63 /C1/5, 1996. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 268 /C1/77, 1991. Long, C. T. "On the Moessner Theorem on Integral Powers." Amer. Math. Monthly 73, 846 /C1/51, 1966. Long, C. T. "Strike it Out--Add it Up." Math. Mag. 66, 273 /C1/ 77, 1982. Moessner, A. "Eine Bemerkung u¨ber die Potenzen der natu¨rlichen Zahlen." S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 29, 1952. Paasche, I. "Ein neuer Beweis des moessnerischen Satzes." S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 ,1/C1/, 1953. Paasche, I. "Ein zahlentheoretische-logarithmischer ‘Re- chenstab’." Math. Naturwiss. Unterr. 6,26/C1/8, 1953 /C1/4. Paasche, I. "Eine Verallgemeinerung des moessnerschen Satzes." Compositio Math. 12, 263 /C1/70, 1956. Mohammed Sign A curve consisting of two mirror-reversed intersect- ing crescents. This curve can be traced UNICURSALLY . See also UNICURSAL CIRCUIT Moire ´ Pattern An interference pattern produced by overlaying similar but slightly offset templates. Møire ´ patterns can also be created by plotting series of curves on a computer screen. Here, the interference is provided by the discretization of the finite-sized pixels. See also CIRCLES-AND- SQUARES FRACTAL References Amidror, I. The Theory of the Møire ´Phenomenon. Dor- drecht, Netherlands: Kluwer, 1999. Cassin, C. Visual Illusions in Motion with Møire ´ Screens: 60 Designs and 3 Plastic Screens. New York: Dover, 1997. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 229 /C1/30, 1984. Grafton, C. B. Optical Designs in Motion with Møire ´ Over- lays. New York: Dover, 1976. Oster, G. and Nishijima, Y. "Møire ´ Patterns." Sci. Amer. , May 1963. Strong, C. L. "The Amateur Scientist." Sci. Amer. , Nov. 1964. Molenbroek’s Equation The PARTIAL DIFFERENTIAL EQUATION 92 f /C30M2 /C12fflC}6 f2 x fxx /C272fx fy fxy /C27 f2y fyy /C271 2(g /C281)( f2 x /C27 f2y /C281) fxx /C27 fyy /C27 efy y !fflC}7 (Cole and Cook 1986, p. 34; Zwillinger 1997, p. 134). References Cole, J. D. and Cook, P. Transonic Aerodynamics. New York: North-Holland, p. 34, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 134, 1997. Mollweide Projection A MAP PROJECTION also called the ELLIPTICAL PROJEC- TION or HOMOLOGRAPHIC EQUAL-AREA PROJECTION . The forward transformation is x /C302ffiffiffi 2p ( l /C28 l0) cos u p (1) y /C3021 =2 sin u; (2) where u is given by 2u /C27sin(2u) /C30 p sin f : (3) NEWTON’S METHOD can then be used to compute u? iteratively from Du?/C30/C28u ?/C27sin u ?/C28p sin f 1 /C27 cos u ?; (4) where u?/C301 2u ? (5)or, better yet, u?/C302 sin /C2812f p ! (6) can be used as a first guess. The inverse FORMULAS are f /C30sin/C2812u /C27 sin(2u) p"# (7) l /C30 l0 /C27px 2ffiffiffi 2p cos u (8) where u /C30sin/C281yffiffiffi 2p ! : (9) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 249 /C1/52, 1987. Mollweide’s Formulas b/C28c a/C30sin[1 2(B/C28C)] cos(1 2A) c/C28a b/C30sin[12(C/C28A)] cos(1 2B) a/C28b c/C30sin[1 2(A/C28B)] cos(12C): See also NEWTON’S FORMULAS ,TRIANGLE ,TRIGONO- METRY References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 146, 1987. Moment ThenthRAW MOMENT m?n(i.e., moment about zero) of a distribution P(x) is defined by m?n/C30xnhi ; (1) where f(x) hi/C30Pf(x)P(x) discrete distribution gf(x)P(x)dx continuous distribution8 < :(2) /m?1;the MEAN , is usually simply denoted m/C30m1:If the moment is instead taken about a point a, mn(a) /C30 (x /C28a)nhi /C30X (x /C28a)nP(x) : (3) A STATISTICAL DISTRIBUTION is not uniquely specified by its moments, although it is by its CHARACTERISTIC FUNCTION . The moments are most commonly taken about the MEAN . These so-called CENTRAL MOMENTS are denoted mn and are defined by mn /C13 (x /C28 m)nhi ; (4) /C30g(x /C28 m)nP(x) dx; (5) with m1 /C300: The second moment about the MEAN is equal to the VARIANCE m2 /C30 s2 ; (6) where s /C30ffiffiffiffiffim2pis called the STANDARD DEVIATION . The related CHARACTERISTIC FUNCTION is defined by f(n)(0) /C13dn f dtn"# t /C300/C30in m(0): (7) The moments may be simply computed using the MOMENT-GENERATING FUNCTION , m?n /C30M(n)(0) : (8) See also ABSOLUTE MOMENT , CHARACTERISTIC FUNC- TION ,C HARLIER’S CHECK ,C UMULANT- GENERATING FUNCTION ,F ACTORIAL MOMENT ,K URTOSIS ,M EAN, MOMENT- GENERATING FUNCTION ,MOMENT PROBLEM , MOMENT SEQUENCE ,SKEWNESS ,STANDARD DEVIA- TION ,STANDARDIZED MOMENT ,VARIANCE References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 145 /C1/49, 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Moments of a Distribution: Mean, Variance, Skewness, and So Forth." §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 604 /C1/09, 1992. Momental Skewness a(m)/C131 2g1/C30m3 2s3; where g1is the F ISHER SKEWNESS . See also FISHER SKEWNESS ,SKEWNESS Moment-Generating Function Given a RANDOM VARIABLE x/C23R;if there exists an h/C210 such thatfor ½t½Bh;thenM(t)/C13etxhi /C30P RetxP(x) for a discrete distribution g/C12 /C28/C12etxP(x)dx for a continuous distribution8 < : (1) is the moment-generating function. M(t)/C30g/C12 /C28/C121/C27tx/C271 2!t2x2/C27... ! P(x)dx /C30/C27tm1/C271 2!t2m2/C27/C1/C1/C1; (3) where mris the rthMOMENT about zero. The moment- generating function satisfies Mx/C27y(t)/C30et(x/C27y)fflCz{fflCzz /C30etxetyhi /C30etxhi etyhi/C30Mx(t)My(t):(4) IfM(t) is differentiable at zero, then the nthMO- MENTS about the ORIGIN are given by M(n)(0) M(t)/C30etxhi M(0)/C301 (5) M?(t)/C30xetxhi M?(0)/C30xhi (6) M??(t)/C30x2etxfflCz{fflCzz M??(0)/C30x2fflCz{fflCzz (7) M(n)(t)/C30xnetxhi M(n)(0)/C30xnhi : (8) The MEAN and VARIANCE are therefore m/C13xhi/C30M?(0) (9) s2/C13x2fflCz{fflCzz /C28xhi2/C30M??(0)/C28M?(0)½/C1382: (10) It is also true that mn/C30Xn j/C300n jfflCzrfflCzD (/C281)n/C28jm?j(m?1)n/C28j; (11) where m?0/C301 and m?jis the jth moment about the origin. It is sometimes simpler to work with the LOGARITHM of the moment-generating function, which is also called the CUMULANT-GENERATING FUNCTION , and is defined by R(t)/C13ln[M(t)] (12) R?(t)/C30M?(t) M(t)(13) R??(t)/C30M(t)M??(t)/C28M?(t) ½/C1382 M(t) ½/C1382 (14) ButM(0)/C301hi/C301;so m/C30M?(0)/C30R?(0) (15) s2/C30M??(0)/C28M?(0)½/C1382/C30R??(0) (16) See also CHARACTERISTIC FUNCTION (PROBABILITY ), CUMULANT ,CUMULANT- GENERATING FUNCTION ,M O- MENT References Kenney, J. F. and Keeping, E. S. "Moment-Generating and Characteristic Functions," "Some Examples of Moment- Generating Functions," and "Uniqueness Theorem for Characteristic Functions." §4.6 /C1/.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 72 /C1/7, 1951. Moment Problem The moment problem, also called "Hausdorff’s mo- ment problem "or the "little moment problem," may be stated as follows. Given a sequence of numbers mnfg/C12 n /C300 ; under what conditions is it possible to determine a function a(t) of bounded variation in the interval (0; 1) such that mn /C30g1 0tn d a(t) for n /C300, 1, .... Such a sequence is called a MOMENT SEQUENCE , and Hausdorff (1921) was the first to obtain necessary and sufficient conditions for a sequence to be a MOMENT SEQUENCE . See also MOMENT ,MOMENT SEQUENCE References Hausdorff, F. "Summationsmethoden und Momentfolgen. I." Math. Z. 9,74/C1/09, 1921. Hausdorff, F. "Summationsmethoden und Momentfolgen. II." Math. Z. 9, 280 /C1/99, 1921. Leviatan, D. "A Generalized Moment Problem." Israel J. Math. 5,97/C1/03, 1967. Widder, D. V. "The Moment Problem." Ch. 3 in The Laplace Transform. Princeton, NJ: Princeton University Press, pp. 100 /C1/01, 1941. Moment Sequence A moment sequence is a sequence mnfg/C12n/C300 defined for n /C300, 1, ... by mn /C30g1 0tn da(t) ; where a(t) is a function of bounded variation in the interval (0; 1):/ See also MOMENT ,MOMENT PROBLEM Monad A mathematical object which consists of a set of a single element. The YIN-YANG is also known as the monad. See also HEXAD ,QUARTET ,QUINTET ,TETRAD ,TRIAD, YIN-YANGMoney-Changing Problem COINPROBLEM Monge-Ampe `re Differential Equation A second-order PARTIAL DIFFERENTIAL EQUATION OF THE FORM Hr/C272Ks/C27Lt/C27M/C27N(rt/C28s2)/C300; (1) where H,K,L,M, and Nare functions of x,y,z,p, andq, and r,s,t,p, and qare defined by r/C30@2z @x2(2) s/C30@2z @x@y(3) t/C30@2z @y2(4) p/C30@z @x(5) q/C30@z @y: (6) The solutions are given by a system of differential equations given by Iyanaga and Kawada (1980). Other equations called the Monge-Ampe `re equation are u2 xy/C28uxuy/C30f(x;y;u;ux;uy) (7) (Moon and Spencer 1969, p. 171; Zwillinger 1997, p. 134) and ux1x1ux1x2/C1/C1/C1 ux1xnux2x1ux2x2/C1/C1/C1 ux2xn/C1/C1/C1 /C1/C1/C1::: /C1/C1/C1 uxnx1uxnx2/C1/C1/C1 uxnxnfflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}fflCz}/C30f(u;x;9u) (8) (Gilberg and Trudinger 1983, p. 441; Zwillinger 1997, p. 134). References Caffarelli, L. A. and Milman, M. Monge Ampe `re Equation: Applications to Geometry and Optimization. Providence, RI: Amer. Math. Soc., 1999. Fairlie, D. B. and Leznov, A. N. The General Solution of the Complex Monge-Ampe `re Equation in a Space of Arbitrary Dimension. 16 Sep 1999. http://xxx.lanl.gov/abs/solv-int/ 9909014/. Gilberg, D. and Trudinger, N. S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, p. 441, 1983. Iyanaga, S. and Kawada, Y. (Eds.). "Monge-Ampe `re Equa- tions." §276 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 879 /C1/80, 1980. Moon, P. and Spencer, D. E. Partial Differential Equations. Lexington, MA: Heath, p. 171, 1969. Monge Patch A Monge patch is a PATCH x : U 0 R3 OF THE FORM x(u; v) /C30(u; v; h(u; v)); (1) where U is an OPEN SET in R2 and h : U 0 R is a differentiable function. The coefficients of the first FUNDAMENTAL FORM are given by E /C301 /C27h2 u (2) F /C30huhv (3) G /C301 /C27h2v (4) and the second FUNDAMENTAL FORM by e /C30huuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 h2 u /C27 h2vp (5) f /C30huvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 h2 u /C27 h2vp (6) g /C30gvvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 h2 u /C27 h2vp : (7) For a Monge patch, the GAUSSIAN CURVATURE and MEAN CURVATURE are K /C30huuhvv /C28 h2 uv 1 /C27 h2 u /C27 h2v ðÞ2 (8) H /C30(1 /C28 h2 v)huu /C28 2huhvhuv /C27 (1 /C28 h2u)hvv 21/C27 h2 u /C27 h2v ðÞ3 =2 : (9) See also MONGE’S FORM,PATCH References Gray, A. "A Monge Patch." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 398 /C1/01, 1997. Monge Point The point of concurrence of the six PLANES in MONGE’S TETRAHEDRON THEOREM . See also MANNHEIM’S THEOREM ,M ONGE’S TETRAHE- DRON THEOREM ,PLANE ,TETRAHEDRON References Altshiller-Court, N. "The Monge Point." §4.2c in Modern Pure Solid Geometry. New York: Chelsea, pp. 69 /C1/1, 1979. Forder, H. G. "Article 1006. A Theorem in Coolidge’s ‘Circle and Sphere."’ Math. Gaz. 15, pp. 470 /C1/71, 1930 /C1/931. Lez, H. and Dugrais, M. "Solution des questions propose ´es dans les Nouvelles Annales: Question 906." Nouvelles ann. de math. 8, 173, 1869. Monge, G. Corresp. sur l’E´ cole Polytech. 2, 266, 1795. Thompson, H. F. "A Geometrical Proof of a Theorem Con- nected with the Tetrahedron." Proc. Edinburgh Math. Soc. 17,51/C1/3, 1908 /C1/909.Monge’s Chordal Theorem RADICAL CENTER Monge’s Circle Theorem Draw three nonintersecting CIRCLES in the plane, and the common tangent line for each pair of two. The points of intersection of the three pairs of tangent lines lie on a straight line. Monge’s theorem has a 3-D analog which states that the apexes of the CONES defined by four SPHERES , taken two at a time, lie in a PLANE (when the CONES are drawn with the SPHERES on the same side of the apex; Wells 1991). See also CIRCLE TANGENTS References Coxeter, H. S. M. "The Problem of Apollonius." Amer. Math. Monthly 75,5/C1/5, 1968. Graham, L. A. Problem 62 in Ingenious Mathematical Problems and Methods. New York: Dover, 1959. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 115 /C1/ 17, 1990. Petersen, J. Methods and Theories for the Solution of Problems of Geometrical Constructions, Applied to 410 Problems. London: Sampson Low, Marston, Searle & Rivington, pp. 92 /C1/3, 1879. Walker, W. "Monge’s Theorem in Many Dimensions." Math. Gaz. 60, 185 /C1/88, 1976. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 153 /C1/54, 1991. Monge’s Form ASURFACE given by the form z/C30F(x;y):/ See also MONGE PATCH Monge’s Problem Draw a CIRCLE that cuts three given CIRCLES PER- PENDICULARLY . The solution is obtained by drawing the RADICAL CENTER R of the given three CIRCLES .Ifit lies outside the three CIRCLES , then the CIRCLE with center R and RADIUS formed by the tangent from R to one of the given CIRCLES intersects the given CIRCLES perpendicularly. Otherwise, if R lies inside one of the circles, the problem is unsolvable. See also CIRCLE TANGENTS ,RADICAL CENTER References Do¨rrie, H. "Monge’s Problem." §31 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 151 /C1/54, 1965. Monge’s Shuffle A SHUFFLE in which CARDS from the top of the deck in the left hand are alternatively moved to the bottom and top of the deck in the right hand. If the deck is shuffled m times, the final position xmand initial position x0 of a card are related by 2m/C271xm /C30(4p /C271) 2m/C281 /C27(/C281)m/C281 2m/C282 /C27/C1/C1/C1/C272 /C271fflC{fflCz hi /C27(/C281)m/C2812x0 /C272m /C27(/C281)m/C281 for a deck of 2p cards (Kraitchik 1942). See also CARDS ,SHUFFLE References Conway, J. H. and Guy, R. K. "Fractions Cycle into Deci- mals." In The Book of Numbers. New York: Springer- Verlag, pp. 157 /C1/63, 1996. Kraitchik, M. "Monge’s Shuffle." §12.2.14 in Mathematical Recreations. New York: W. W. Norton, pp. 321 /C1/23, 1942. Monge’s Tetrahedron Theorem The six PLANES through the midpoints of the edges of a TETRAHEDRON and perpendicular to the opposite edges CONCUR in a point known as the MONGE POINT . See also MONGE POINT ,PLANE ,TETRAHEDRONReferences Altshiller-Court, N. "The Monge Theorem." §228 in Modern Pure Solid Geometry. New York: Chelsea, p. 69, 1979. Forder, H. G. Math. Gaz. 15, p. 470, 1930 /C1/931. Lez, H. and Dugrais, M. "Solution des questions propose ´es dans les Nouvelles Annales: Question 906." Nouvelles ann. de math. 8, 173, 1869. Monge, G. Corresp. sur l’E´ cole Polytech. 2, 266, 1795. Thompson, H. F. "A Geometrical Proof of a Theorem Con- nected with the Tetrahedron." Proc. Edinburgh Math. Soc. 17,51/C1/3, 1908 /C1/909. Monge’s Theorem MONGE’S CIRCLE THEOREM ,M ONGE’S TETRAHEDRON THEOREM Monica Set The nth Monica set Mnis defined as the set of COMPOSITE NUMBERS x for which n½S(x) /C28Sp(x); where x /C30a0 /C27a1(101) /C27/C1/C1/C1/C27ad(10d) /C30p1p2 /C1/C1/C1pn ; (1) and S(x) /C30Xd j/C300aj (2) Sp(x) /C30Xm i/C301S(pi) (3) Every Monica set has an infinite number of elements. The Monica set Mn is a subset of the SUZANNE SET Sn : If x is a SMITH NUMBER , then it is a member of the Monica set Mn for all /n /C23N/. For any INTEGER k /C211, if x is a k-SMITH NUMBER , then x /C23 Mk/C281 :/ See also SUZANNE SET References Smith, M. "Cousins of Smith Numbers: Monica and Suzanne Sets." Fib. Quart. 34, 102 /C1/04, 1996. Monic Polynomial A POLYNOMIAL xn /C27an /C281xn/C281 /C27/C1/C1/C1/C27a1x /C27a0in which the COEFFICIENT of the highest ORDER term is 1. See also MONOMIAL Monkey and Coconut Problem AD IOPHANTINE problem (i.e., one whose solution must be given in terms of INTEGERS ) which seeks a solution to the following problem. Given nmen and a pile of coconuts, each man in sequence takes (1 =n)/th of the coconuts left after the previous man removed his (i.e., a1for the first man, a2;for the second, ..., an for the last) and gives mcoconuts (specified in the problem to be the same number for each man) which do not divide equally to a monkey. When all nmen have so divided, they divide the remaining coconuts n ways (i.e., taking an additional acoconuts each), and give the mcoconuts which are left over to the monkey. If m is the same at each division, then how many coconuts N were there originally? The solution is equivalent to solving the n /C271D IOPHANTINE EQUATIONS N /C30na1 /C27m N /C28a1 /C28m /C30na2 /C27m N /C28a1 /C28a2 /C282m /C30na3 /C27m (1) n N /C28a1 /C28a2 /C28a3 /C28/C1/C1/C1/C28an /C28nm /C30na /C27m; which can be rewritten as N /C30na1 /C27m (n /C281)a1 /C30na2 /C27m (n /C281)a1 /C30na3 /C27m (2) n (n /C281)an /C281 /C30nan /C27m (n /C281)aa /C30na /C27m: Since there are n /C271 equations in the n /C272 unknowns a1 ; a2 ; ..., an ; a, and N, the solutions span a 1- dimensional space (i.e., there is an infinite family of solution parameterized by a single value). The solu- tion to these equations can be given by N /C30knn/C271 /C28m(n /C281); (3) where k is an arbitrary INTEGER (Gardner 1961). For the particular case of n /C305 men and m /C301 left over coconuts, the 6 equations can be combined into the single DIOPHANTINE EQUATION 1;024N /C3015 ;625a /C2711;529; (4) where a is the number given to each man in the last division. The smallest POSITIVE solution in this case is N /C3015 ;621 coconuts, corresponding to k /C301 and a /C30 1;023; Gardner 1961). The following table shows how this rather large number of coconuts is divided under the scheme described above. Removed Given to Monkey Left 15,621 3,124 1 12,496 2,499 1 9,996 1,999 1 7,996 1,599 1 6,396 1,279 1 5,116 5 /C291,023 1 0If no coconuts are left for the monkey after the final n-way division (Williams 1926), then the original number of coconuts is (1 /C27nk)nn /C28(n /C281) n odd (n /C281 /C27nk)nn /C28(n /C281) n even :fflC}6 (5) The smallest POSITIVE solution for case n /C305 and m /C301is N /C303 ;121 coconuts, corresponding to k /C301 and 1,020 coconuts in the final division (Gardner 1961). The following table shows how these coconuts are divided. Removed Given to Monkey Left 3,121 624 1 2,496 499 1 1,996 399 1 1,596319 1 1,276255 1 1,020 /5/C29204 / 00 A different version of the problem having a solution of 79 coconuts is considered by Pappas (1989). See also DIOPHANTINE EQUATION ,PELL EQUATION References Anning, N. "Monkeys and Coconuts." Math. Teacher 54, 560/C1/62, 1951. Bowden, J. "The Problem of the Dishonest Men, the Monkeys, and the Coconuts." In Special Topics in Theore- tical Arithmetic. Lancaster, PA: Lancaster Press, pp. 203 /C1/12, 1936. Gardner, M. "The Monkey and the Coconuts." Ch. 9 in The Second Scientific American Book of Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 104 /C1/11, 1961. Kirchner, R. B. "The Generalized Coconut Problem." Amer. Math. Monthly 67, 516/C1/19, 1960. Moritz, R. E. "Solution to Problem 3,242." Amer. Math. Monthly 35,4 7/C1/8, 1928. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 52 /C1/4, 1988. Olds, C. D. Continued Fractions. New York: Random House, pp. 48 /C1/0, 1963. Pappas, T. "The Monkey and the Coconuts." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 226 /C1/27 and 234, 1989. Williams, B. A. "Coconuts." The Saturday Evening Post, Oct. 9, 1926. Monkey Saddle A SURFACE which a monkey can straddle with both his two legs and his tail. A simple Cartesian equation for such a surface is z /C30x(x2 /C283y2) ; (1) which can also be given by the PARAMETRIC EQUA- TIONS x(u; v) /C30u (2) y(u; v) /C30v (3) z(u; v) /C30u3 /C283uv2 : (4) The coefficients of the coefficients of the FIRST FUNDAMENTAL FORM of the monkey saddle are E /C301 /C279(u2 /C28v2)2 (5) F /C30/C2818uv(u2 /C28v2) (6) G /C301 /C2736u2v2 (7) and the SECOND FUNDAMENTAL FORM coefficients are e /C306uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 9(u2 /C27 v2)2p (8) f /C30/C286vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 9(u2 /C27 v2)2p (9) g /C30/C286uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 9(u 2 /C27 v2)2p ; (10) giving RIEMANNIAN METRIC ds2 /C30[1 /C27(3u2 /C283v2)2] du2 /C282[18uv(u2 /C28v2)] du dv /C27(1 /C2736u2v2) dv2 ; (11) AREA ELEMENT dA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C279(u2 /C27v2)2q du ffl dv ; (12) and GAUSSIAN and MEAN CURVATURES K /C30/C2836(u2 /C27 v2) [1 /C27 9(u2 /C27 v2)2]2 (13)H /C3027u( /C28u4 /C27 2u2v2 /C27 3v4) [1 /C27 9(u2 /C27 v2)2]3=2 (14) (Gray 1997). Every point of the monkey saddle except the origin has NEGATIVE GAUSSIAN CURVATURE . See also CROSSED TROUGH ,PARTIAL DERIVATIVE References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 365, 1969. Gray, A. "Monkey Saddle." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 299 /C1/01, 382 /C1/83, and 408, 1997. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 202, 1999. Monochromatic Forced Triangle Given a COMPLETE GRAPH Kn which is two-colored, the number of forced monochromatic TRIANGLES is at least 1 3u(u /C281)(u /C282) for n /C302u 23(u /C281)(4u /C271) for n /C304u /C271 23u(u /C271)(4u /C281) for n /C304u /C273:8 >< >: The first few numbers of monochromatic forced triangles are 0, 0, 0, 0, 0, 2, 4, 8, 12, 20, 28, 40, ... (Sloane’s A014557). See also COMPLETE GRAPH ,EXTREMAL GRAPH References Goodman, A. W. "On Sets of Acquaintances and Strangers at Any Party." Amer. Math. Monthly 66, 778 /C1/83, 1959. Sloane, N. J. A. Sequences A014557 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Monodromy A general concept in CATEGORY THEORY involving the globalization of local MORPHISMS . See also CATEGORY THEORY ,HOLONOMY ,MORPHISM Monodromy Group A technically defined GROUP characterizing a system of linear differential equations y?j /C30Xn k /C301ajk(x)yk for j /C301, ..., n, where ajkare COMPLEX ANALYTIC FUNCTIONS of x in a given COMPLEX DOMAIN . See also HILBERT’S 21ST PROBLEM ,RIEMANN P-SERIES References Iyanaga, S. and Kawada, Y. (Eds.). "Monodromy Groups." §253B in Encyclopedic Dictionary of Mathematics. Cam- bridge, MA: MIT Press, p. 793, 1980. Monodromy Theorem If a COMPLEX FUNCTION f is ANALYTIC in a DISK contained in a simply connected DOMAIN D and f can be ANALYTICALLY CONTINUED along every poly- gonal arc in D, then f can be ANALYTICALLY CON- TINUED to a single-valued ANALYTIC FUNCTION on all of D! See also ANALYTIC CONTINUATION References Flanigan, F. J. Complex Variables: Harmonic and Analytic Functions. New York: Dover, p. 234, 1983. Knopp, K. "The Monodromy Theorem." §25 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 105 /C1/11, 1996. Krantz, S. G. "The Monodromy Theorem." §10.3.5 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, p. 134, 1999. Monogenic Function If lim z0z0f(z) /C28 f(z0) z /C28 z0 is the same for all paths in the COMPLEX PLANE , then f(z) is said to be monogenic at z0 : Monogenic therefore essentially means having a single DERIVATIVE at a point. Functions are either monogenic or have infi- nitely many DERIVATIVES (in which case they are called POLYGENIC ); intermediate cases are not possi- ble. See also POLYGENIC FUNCTION References Newman, J. R. The World of Mathematics, Vol. 3. New York: Simon & Schuster, p. 2003, 1956. Monohedral Tiling A TILING in which all tiles are congruent. See also ANISOHEDRAL TILING ,ISOHEDRAL TILING , TILING References Berglund, J. "Is There a k-Anisohedral Tile for k ]5/?" Amer. Math. Monthly 100, 585 /C1/88, 1993. Gru¨nbaum, B. and Shephard, G. C. "The 81 Types of Isohedral Tilings of the Plane." Math. Proc. Cambridge Philos. Soc. 82, 177 /C1/96, 1977. Monoid A GROUP -like object which fails to be a GROUP because elements need not have an inverse within the object. A monoid S must also be ASSOCIATIVE and have an IDENTITY ELEMENT I /C23 S such that for all a /C23 S; 1a /C30 a1 /C30a: A monoid is therefore a SEMIGROUP with an IDENTITY ELEMENT . A monoid must contain at least one element.The numbers of free idempotent monoids on n letters are 1, 2, 7, 160, 332381, ... (Sloane’s A005345). See also BINARY OPERATOR ,GROUP ,SEMIGROUP References Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968. Sloane, N. J. A. Sequences A005345/M1820 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Monomial A POLYNOMIAL consisting of a product of powers of variables, e.g., x, xy2 ; x2y3z ; etc. Constant coefficients are sometimes also allowed in front of a monomial. One monomial is said to divide another if the powers of its variables are no greater than the corresponding powers in the second monomial. For example, x2y divides x3y but does not divide xy3 : A monomial m is said to reduce with respect to a polynomial if the leading monomial of that polynomial divides m. For example, x2y reduces with respect to 2xy /C27x /C273 because xy divides x2y; and te result of this reduction is x2y /C28x(2xy /C27x /C273)=2; or /C28x2 =2 /C283x=2 : A polyno- mial can therefore be reduced by reducing its mono- mials beginning with the greatest and proceeding downward. Similarly, a polynomial can be reduced with respect to a set of polynomials by reducing in turn with respect to each element in that set. A polynomial is fully reduced if none of its monomials can be reduced (Lichtblau 1996). See also BINOMIAL ,G RO¨ BNER BASIS,M ONIC POLY- NOMIAL ,POLYNOMIAL ,TRINOMIAL References Lichtblau, D. "Gro¨bner Bases in Mathematica 3.0." Mathe- matica J. 6,81/C1/8, 1996. Monomial Order "u Bv implies uw Bvw" for all monomials u, v, and w. Examples of monomial orders are the LEXICO- GRAPHIC ORDER and the total degree order. See also WELL ORDERED SET Monomino The unique 1-POLYOMINO , consisting of a single SQUARE . See also DOMINO ,TRIOMINO References Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer- ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 124 /C1/40, 1959. Monomorph An INTEGER which is expressible in only one way in the form x2 /C27Dy2 or x2 /C28Dy2 where x2 is RELATIVELY PRIME to Dy2 : If the INTEGER is expressible in more than one way, it is called a POLYMORPH . See also ANTIMORPH ,IDONEAL NUMBER ,PELL EQUA- TION ,POLYMORPH Monomorphism A MORPHISM f : Y 0 X in a CATEGORY is a mono- morphism if, for any two MORPHISMS u; v : Z 0 Y ; fu /C30fv implies that u /C30v. See also CATEGORY ,MORPHISM Monotone Another word for monotonic. See also MONOTONIC FUNCTION ,M ONOTONIC SE- QUENCE ,MONOTONIC VOTING Monotone Convergence Theorem If ffn g is a sequence of MEASURABLE FUNCTIONS , with 0 5fn 5fn/C271 for every n, then g lim n0/C12fn dm /C30 lim n0/C12g fn dm Monotone Decreasing Always decreasing; never remaining constant or increasing. Also called strictly decreasing. Monotone Increasing Always increasing; never remaining constant or decreasing. Also called strictly increasing. Monotone Triangle A monotone triangle (also called a strict Gelfand pattern or a gog triangle) of order n is a NUMBER TRIANGLE with n numbers along each side and the base containing entries between 1 and n such that there is strict increase across rows and weak increase diagonally up or down to the right. There is a bijection between monotone triangles of order n and ALTERNATING SIGN MATRICES of order n obtained by letting the kth row of the triangle equal the positions of 1s in the sum of the first k rows of an ALTERNATING SIGN MATRIX , as illustrated below. 00010 010 /C2811 1 /C2810 10 00100 010002 666643 77775l4 25 145 1345 12345 (0; 0; 0; 1; 0) 0 4(0; 0; 0; 1 ; 0) /C27(0; 1 ; 0;/C281; 1) /C30(0; 1 ; 0 ; 0 ; 1) 0 25 (0; 1; 0; 0 ; 1) /C27(1;/C281; 0; 1; 0; ) /C30(1; 0; 0; 1; 1) 0 145 (1; 0; 0; 1 ; 1) /C27(0; 0 ; 1; 0; 0) /C30(1; 0; 1; 1; 1) 0 1345 (1; 0; 1; 1 ; 1) /C27(0; 1 ; 0; 0; 0) /C30(1; 1 ; 1 ; 1; 1) 0 12345 References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/46. Monotonic Function A function which is either entirely NONINCREASING or NONDECREASING . A function is monotonic if its first DERIVATIVE (which need not be continuous) does not change sign. See also COMPLETELY MONOTONIC FUNCTION ,MONO- TONE ,MONOTONE DECREASING ,MONOTONE INCREAS- ING,N ONDECREASING FUNCTION ,N ONINCREASING FUNCTION Monotonic Sequence A SEQUENCE fan g such that either (1) ai /C271 ]aifor every i ]1 ; or (2) ai/C271 5ai for every i ]1:/ Monotonic Voting A term in SOCIAL CHOICE THEORY meaning a change favorable for X does not hurt X. See also ANONYMOUS ,DUAL VOTING ,VOTING Monster Group The highest order SPORADIC GROUP M. It has ORDER 246 /C215 320 /C215 59 /C215 76 /C215 112 /C215 133 /C215 17 /C215 19 /C215 23 /C215 29 /C215 31 /C215 41 /C215 47 /C215 59 /C215 71; and is also called the FRIENDLY GIANT GROUP . It was constructed in 1982 by Robert Griess as a GROUP of ROTATIONS in 196,883-D space. See also BABY MONSTER GROUP ,BIMONSTER ,LEECH LATTICE References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Sub- groups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. viii, 1985. Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308/C1/39, 1979. Conway, J. H. and Sloane, N. J. A. "The Monster Group and its 196884-Dimensional Space" and "A Monster Lie Alge- bra?" Chs. 29 /C1/0inSphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 554 /C1/71, 1993. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/M.html. Monte Carlo Integration In order to integrate a function over a complicated DOMAIN D, Monte Carlo integration picks random points over some simple DOMAIN D? which is a super- set of D, checks whether each point is within D, and estimates the AREA of D (VOLUME , n-D CONTENT , etc.) as the AREA of D ? multiplied by the fraction of points falling within D ?: Monte Carlo integration is imple- mented in Mathematica as NIntegrate [f, ..., Method- /C21MonteCarlo ]. An estimate of the uncertainty produced by this technique is given by g fdV:V /C142f /C1439ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f2hi/C28/C142f /C1432 Ns : See also MONTE CARLO METHOD ,NUMERICAL INTE- GRATION ,QUASI- MONTE CARLO INTEGRATION References Hammersley, J. M. "Monte Carlo Methods for Solving Multivariable Problems." Ann. New York Acad. Sci. 86, 844 /C1/74, 1960. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Simple Monte Carlo Integration" and "Adap- tive and Recursive Monte Carlo Methods." §7.6 and 7.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 295 /C1/99 and 306 /C1/19, 1992. Ueberhuber, C. W. "Monte Carlo Techniques." §12.4.4 in Numerical Computation 2: Methods, Software, and Ana- lysis. Berlin: Springer-Verlag, pp. 124 /C1/25 and 132 /C1/38, 1997. Weinzierl, S. Introduction to Monte Carlo Methods. 23 Jun 200. http://xxx.lanl.gov/abs/hep-ph/0006269/. Monte Carlo Method Any method which solves a problem by generating suitable random numbers and observing that fraction of the numbers obeying some property or properties. The method is useful for obtaining numerical solu- tions to problems which are too complicated to solve analytically. It is named by S. Ulam, who in 1946 became the first mathematician to dignify this ap- proach with a name, in honor of a relative having a propensity to gamble (Hoffman 1998, p. 239). The most common application of the Monte Carlo method is MONTE CARLO INTEGRATION . See also MARKOV CHAIN ,M ONTE CARLO INTEGRA- TION ,STOCHASTIC GEOMETRYReferences Gamerman, D. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Boca Raton, FL: CRC Press, 1997. Gilks, W. R.; Richardson, S.; and Spiegelhalter, D. J. (Eds.). Markov Chain Monte Carlo in Practice. Boca Raton, FL: Chapman & Hall, 1996. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 238 /C1/39, 1998. Manno, I. Introduction to the Monte Carlo Method. Buda- pest, Hungary: Akade ´miai Kiado ´, 1999. Mikhailov, G. A. Parametric Estimates by the Monte Carlo Method. Utrecht, Netherlands: VSP, 1999. Niederreiter, H. and Spanier, J. (Eds.). Monte Carlo and Quasi-Monte Carlo Methods 1998, Proceedings of a Con- ference held at the Claremont Graduate University, Clar- emont, California, USA, June 22 /C1/6, 1998. Berlin: Springer-Verlag, 2000. Sobol, I. M. A Primer for the Monte Carlo Method. Boca Raton, FL: CRC Press, 1994. Montel’s Theorem Let f(z)bean ANALYTIC FUNCTION of z, regular in the half-strip S defined by a Bx Bb and y /C210. If f(z)is bounded in S and tends to a limit l as y 0/C12 for a certain fixed value j of x between a and b, then f(z) tends to this limit l on every line x /C30x0inS, and f(z)0luniformly for a/C27d5x05b/C28d:/ See also VITALI’S CONVERGENCE THEOREM References Krantz, S. G. "Montel’s Theorem, First Version and Montel’s Theorem, Second Version." §8.4.3 and 8.4.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 114, 1999. Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 170, 1960. Monty Hall Dilemma MONTY HALLPROBLEM Monty Hall Problem The Monty Hall problem is named for its similarity to theLet’s Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assumethat a room is equipped with three doors. Behind twoare goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let’s say you pick door 1. Before the door isopened, however, someone who knows what’s behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do. The correct answer is that you dowant to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat. But after Monty has eliminated one ofthe doors for you, you obviously do not improve your chances of winning to better than 1/3 by sticking with your original choice. If you now switch doors, how- ever, there is a 2/3 chance you will win the car (counterintuitive though it seems). /d1//d2/ Winning Probability pick stick 1/3 pick switch 2/3 The problem can be generalized to four doors as follows. Let one door conceal the car, with goats behind the other three. Pick a door d1 : Then the host will open one of the nonwinners and give you the option of switching. Call your new choice (which could be the same as d1 if you don’t switch) d2 : The host will then open a second nonwinner, and you must decide for choice d3 if you want to stick to d2 or switch to the remaining door. The probabilities of winning are shown below for the four possible strategies. /d1//d2// d3/ Winning Probability pick stick stick 2/8 pick switch stick 3/8 pick stick switch 6/8 pick switch switch 5/8 The above results are characteristic of the best strategy for the n-stage Monty Hall problem: stick until the last choice, then switch. See also ALLAIS PARADOX References Barbeau, E. "The Problem of the Car and Goats." CMJ 24, 149, 1993. Bogomolny, A. "Monty Hall Dilemma." http://www.cut-the- knot.com/hall.html. Dewdney, A. K. 200% of Nothing. New York: Wiley, 1993. Donovan, D. "The WWW Tackles the Monty Hall Problem." http://math.rice.edu/~ddonovan/montyurl.html. Ellis, K. M. "The Monty Hall Problem." http://www.io.com/ ~kmellis/monty.html. Gardner, M. Aha! Gotcha: Paradoxes to Puzzle and Delight. New York: W. H. Freeman, 1982. Gillman, L. "The Car and the Goats." Amer. Math. Monthly 99, 3, 1992. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 233 /C1/40, 1998. Selvin, S. "A Problem in Probability." Amer. Stat. 29, 67, 1975. vos Savant, M. The Power of Logical Thinking. New York: St. Martin’s Press, 1996.Moore Graph A GRAPH of type (d, k)isa REGULAR GRAPH of vertex degree d /C212 and GRAPH DIAMETER k which contains the maximum possible number of nodes, n(d; k) /C301 /C27dXk r/C301(d /C281)r/C281 /C30d(d /C28 1)k /C28 2 d /C28 2 (Bannai and Ito 1973). Equivalently, it is a (d, g)- CAGE GRAPH , where d is the vertex degree and g is the GIRTH , with an EXCESS of zero (Wong 1982). Moore graphs are also called minimal (v, g)-graphs (Wong 1982), and are DISTANCE-REGULAR . Hoffman and Singleton (1960) first used the term "Moore graph," and showed that there is a unique Moore graph for types (3; 2) and (7; 2); but no other (d; 2) Moore graphs with the possible exception of (57; 2) (Bannai and Ito 1973). Bannai and Ito (1973) subsequently showed that there exist no Moore graphs of type (d, k) with GRAPH DIAMETER k ]4 and valence d /C212. Equivalently, a (v, g)-Moore graph exists only if (1) g /C305 and v /C303, 7, or (possibly) 57, or (2) g /C306, 8, or 12 (Wong 1982). This settled the existence and uniqueness problem from finite Moore graphs with the exception of the case (57; 2); which is still open. A proof of this theorem, sometimes called the H OFFMAN- SINGLETON THEOREM , is difficult (Hoff- man and Singleton 1960, Feit and Higman 1964, Damerell 1973, Bannai and Ito 1973), but can befound in Biggs (1993). The (3 ;5) /-Moore graph is the P ETERSEN GRAPH , and the (7 ;5)/-Moore graph is the H OFFMAN- SINGLETON GRAPH . The existence of a (57 ;5)/-graph remains an open question. See also CAGE GRAPH ,D ISTANCE- REGULAR GRAPH , GENERALIZED POLYGON ,G IRTH,G RAPH DIAMETER , HOFFMAN- SINGLETON GRAPH ,H OFFMAN- SINGLETON THEOREM ,PETERSEN GRAPH ,REGULAR GRAPH References Aschbacher, M. "The Non-Existence of Rank Three Permu- tation Group of Degree 3250 and Subdegree 57." J. Algebra 19, 538/C1/40, 1971. Bannai, E. and Ito, T. "On Moore Graphs." J. Fac. Sci. Univ. Tokyo Ser. A 20, 191 /C1/08, 1973. Biggs, N. L. Ch. 23 in Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993. Bosa´k, J. "Cubic Moore Graphs." Mat. Casopis Sloven. Akad. Vied 20,72/C1/0, 1970. Bosa´k, J. "Partially Directed Moore Graphs." Math. Slovaca 29, 181 /C1/96, 1979. Damerell, R. M. "On Moore Graphs." Proc. Cambridge Philos. Soc. 74, 227 /C1/36, 1973. Feit, W. and Higman, G. "The Non-Existence of Certain Generalized Polygons." J. Algebra 1, 114 /C1/31, 1964. Friedman, H. D. "On the Impossibility of Certain Moore graphs." J. Combin. Th. B 10, 245 /C1/52, 1971. Godsil, C. D. "Problems in Algebraic Combinatorics." Elec- tronic J. Combinatorics 2,F11/C1/0, 1995. http://www.com- binatorics.org/Volume_2/volume2.html#F1. Hoffman, A. J. and Singleton, R. R. "On Moore Graphs of Diameter 2 and 3." IBM J. Res. Develop. 4, 497 /C1/04, 1960. McKay, B. D. and Stanton, R. G. "The Current Status of the Generalised Moore Graph Problem." In Combinatorial Mathematics VI (Armidale 1978) . New York: Springer- Verlag, pp. 21 /C1/1, 1979. Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/2, 1982. Moore-Penrose Generalized Matrix Inverse Given an m /C29n MATRIX B ; the Moore-Penrose gen- eralized MATRIX INVERSE (sometimes called the pseu- doinverse) is a unique n /C29m MATRIX B /C27which satisfies BB /C27B /C30B (1) B /C27BB /C27/C30B /C27 (2) (BB /C27)T /C30BB /C27 (3) (B/C27B)T /C30B/C27B: (4) It is also true that z /C30B/C27c (5) is the shortest length LEAST SQUARES solution to the problem B /C30c: (6) If the inverse of (BTB) exists, then B/C27/C30(BTB)/C281BT ; (7) where BT is the matrix TRANSPOSE , as can be seen by premultiplying both sides of (7) by BT to create a SQUARE MATRIX which can then be inverted, BTBz /C30BTc ; (8) giving z /C30(BTB) /C281BTc /C13B/C27c: (9) See also LEAST SQUARES FITTING ,MATRIX INVERSEReferences Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications. New York: Wiley, 1977. Lawson, C. and Hanson, R. Solving Least Squares Problems. Englewood Cliffs, NJ: Prentice-Hall, 1974. Penrose, R. "A Generalized Inverse for Matrices." Proc. Cambridge Phil. Soc. 51, 406 /C1/13, 1955. Mordell Conjecture DIOPHANTINE EQUATIONS that give rise to surfaces with two or more holes have only finite many solutions in GAUSSIAN INTEGERS with no common factors. Fermat’s equation has (n /C281)(n /C282)=2 HOLES , so the Mordell conjecture implies that for each INTEGER n ]3 ; the FERMAT EQUATION has at most a finite number of solutions. This conjecture was proved by Faltings (1984). See also ABC CONJECTURE ,FERMAT EQUATION ,FER- MAT’S LAST THEOREM ,S AFAREVICH CONJECTUR E, SHIMURA- TANIYAMA CONJECTURE References Elkies, N. D. "ABC Implies Mordell." Internat. Math. Res. Not. 7,99/C1/09, 1991. Faltings, G. "Die Vermutungen von Tate und Mordell." Jahresber. Deutsch. Math.-Verein 86,1/C1/3, 1984. Ireland, K. and Rosen, M. "The Mordell Conjecture." §20.3 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 340 /C1/42, 1990. van Frankenhuysen, M. "The ABC Conjecture Implies Roth’s Theorem and Mordell’s Conjecture." Mat. Contemp. 16,45/C1/2, 1999. Mordell Integral The integral f(t; u) /C30ge pitx2 /C272 piux e2 pix /C28 1dx which is related to the JACOBI THETA FUNCTIONS , MOCK THETA FUNCTIONS ,RIEMANN ZETA FUNCTION , and SIEGEL THETA FUNCTION . See also JACOBI THETA FUNCTIONS ,M OCK THETA FUNCTION ,RIEMANN ZETA FUNCTION ,SIEGEL THETA FUNCTION Mordell-Weil Theorem For ELLIPTIC CURVES over the RATIONALS Q; the GROUP of RATIONAL POINTS is always FINITELY GEN- ERATED (i.e., there always exists a finite set of generators for the GROUP ). This theorem was proved by Mordell in 1921 and extended by Weil in 1928 to ABELIAN VARIETIES over NUMBER FIELDS . See also ELLIPTIC CURVE References Ireland, K. and Rosen, M. "The Mordell-Weil Theorem." Ch. 19 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 319 /C1/38, 1990. Nagell, T. "Rational Points on Plane Algebraic Curves. Mordell’s Theorem." §69 in Introduction to Number The- ory. New York: Wiley, pp. 253 /C1/60, 1951. Morera’s Theorem If f(z) is continuous in a region D and satisfies Ggfdz/C300 for all closed CONTOURS g in D, then f(z)is ANALYTIC in D. See also CAUCHY INTEGRAL THEOREM ,C ONTOUR INTEGRATION References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 373 /C1/74, 1985. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 26, 1999. Morgado Identity There are several results known as the Morgado identity. The first is FnFn /C271Fn/C272Fn/C274Fn/C275Fn/C276 /C27L2 n/C273 /C30[Fn/C273(2Fn/C272Fn/C274 /C28F2 n/C273)]2 ; (1) where Fnis a FIBONACCI NUMBER and Lnis a LUCAS NUMBER (Morgado 1987, Dujella 1995). An second Morgado identity is satisfied by GENERAL- IZED FIBONACCI NUMBERS wn ;/ 4wnwn/C271wn/C272wn/C274wn/C275wn/C276 /C27e2q2n(wnU4U5 /C28wn/C271U2U6 /C28wnU1U8)2 /C30(wn /C271wn /C272wn /C276 /C27wnwn/C274wn/C275)2 ; (2) where e/C13pab/C28qa2/C28b2(3) Un/C13wn(0;1;p;q) (4) (Morgado 1987, Dujella 1996). See also FIBONACCI NUMBER ,GENERALIZED FIBONAC- CI NUMBER References Dujella, A. "Diophantine Quadruples for Squares of Fibo- nacci and Lucas Numbers." Portugaliae Math. 52, 305/C1/ 18, 1995. Dujella, A. "Generalized Fibonacci Numbers and the Pro- blem of Diophantus." Fib. Quart. 34, 164/C1/75, 1996. Morgado, J. "Note on Some Results of A. F. Horadam and A. G. Shannon Concerning a Catalan’s Identity on Fibonacci Numbers." Portugaliae Math. 44, 243/C1/52, 1987.Morgan-Voyce Polynomial Polynomials related to the B RAHMAGUPTA POLYNO- MIALS . They are defined by the RECURRENCE RELA- TIONS bn(x)/C30xBn/C281(x)/C27bn/C281(x) (1) Bn(x)/C30(x/C271)Bn/C281(x)/C27bn/C281(x) (2) forn]1;with b0(x)/C30B0(x)/C301: (3) Alternative recurrences are bn(x)/C30(x/C272)bn/C281(x)/C28bn/C282(x) (4) Bn(x)/C30(x/C272)Bn/C281(x)/C28Bn/C282(x) (5) with b1(x)/C301/C27xandB1(x)/C302/C27x;and bn/C271bn/C281/C28b2 n/C30x: (6) Bn/C271Bn/C281/C28B2n/C30/C281 (7) The polynomials can be given explicitly by the sums Bn(x)/C30Xn k/C300n/C27k/C281 n/C28kfflCzrfflCzD xk(8) bn(x)/C30Xn k/C300n/C27k n/C28kfflCzrfflCzD xk: (9) Defining the MATRIX Q/C30x/C272/C281 10fflC}{fflC}z (10) gives the identities Qn/C30Bn/C28Bn/C281 Bn/C281/C28Bn/C282fflC}{fflC}z (11) Qn/C28Qn/C281/C30bn/C28bn/C281 bn/C281/C28bn/C282fflC}{fflC}z : (12) Defining cosu/C301 2(x/C272) (13) cosh f/C3012(x/C272) (14) gives Bn(x)/C30sin[(n/C271)u] sinu(15) Bn(x)/C30sinh[( n/C271)f] sinh f(16) and bn(x)/C30cos1 2(2n/C271)uhi cos1 2ufflCz6fflCz7 (17) bn(x) /C30cosh1 2(2n /C27 1)fhi cosh1 2 ufflCz6fflCz7 : (18) The Morgan-Voyce polynomials are related to the FIBONACCI POLYNOMIALS Fn(x)by bn(x2) /C30F2n/C271(x) (19) Bn(x2) /C301 xF2n/C272(x) (20) (Swamy 1968). /Bn(x) satisfies the ORDINARY DIFFERENTIAL EQUATION x(x /C274)yƒ/C273(x /C272)y?/C28n(n /C272)y /C300; (21) and bn(x) the equation x(x /C274)yƒ/C272(x /C271)y?/C28n(n /C271)y /C300 : (22) These and several other identities involving deriva- tives and integrals of the polynomials are given by Swamy (1968). See also BRAHMAGUPTA POLYNOMIAL ,F IBONACCI POLYNOMIAL References Lahr, J. "Fibonacci and Lucas Numbers and the Morgan- Voyce Polynomials in Ladder Networks and in Electric Line Theory." In Fibonacci Numbers and Their Applica- tions (Ed. G. E. Bergum, A. N. Philippou, and A. F. Hor- adam). Dordrecht, Netherlands: Reidel, 1986. Morgan-Voyce, A. M. "Ladder Network Analysis Using Fibonacci Numbers." IRE Trans. Circuit Th. CT-6 , 321 /C1/ 22, Sep. 1959. Swamy, M. N. S. "Properties of the Polynomials Defined by Morgan-Voyce." Fib. Quart. 4,73/C1/1, 1966. Swamy, M. N. S. "More Fibonacci Identities." Fib. Quart. 4, 369 /C1/72, 1966. Swamy, M. N. S. "Further Properties of Morgan-Voyce Polynomials." Fib. Quart. 6, 167 /C1/75, 1968. Morley Centers The CENTROID of MORLEY’S TRIANGLE is called Mor- ley’s first center. It has TRIANGLE CENTER FUNCTION a /C30cos1 3 AfflCz6fflCz7 /C272 cos13 BfflCz6fflCz7 cos13 CfflCz6fflCz7 : The PERSPECTIVE CENTER of MORLEY’S TRIANGLE with reference TRIANGLE ABC is called Morley’s second center. The TRIANGLE CENTER FUNCTION is a /C30sec1 3 AfflCz6fflCz7 : See also CENTROID (GEOMETRIC ), MORLEY’S THEO- REM,PERSPECTIVE CENTER References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994.Kimberling, C. "1st and 2nd Morley Centers." http://cedar.- evansville.edu/~ck6/tcenters/recent/morley.html. Oakley, C. O. and Baker, J. C. "The Morley Trisector Theorem." Amer. Math. Monthly 85, 737 /C1/45, 1978. Morley’s Formula X/C12 k /C300(m)k k!"#3 /C301 /C27m 1 !3 /C27m(m /C27 1) 1 /C215 2"#3 /C27... /C30G 1 /C2832 mfflCz6fflCz7 G 1 /C281 2 mfflCz6fflCz7hi3cos12 mpfflCz6fflCz7 ; where (m)k is a POCHHAMMER SYMBOL and G(z) is the GAMMA FUNCTION . This is a special case of the identity X/C12 k /C300(m)k k!"#n /C30n Fn/C281(m; ...; m|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n;1; ...; 1|fflfflfflfflfflffl{zfflfflfflfflfflffl} n/C281;1 ): See also GAMMA FUNCTION References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 104 and 111, 1999. Morley’s Theorem The points of intersection of the adjacent TRISECTORS of the ANGLES of any TRIANGLE DABC are the VERTICES of an EQUILATERAL TRIANGLE DDEF known as M ORLEY’S TRIANGLE . Taylor and Marr (1914) give two geometric proofs and one trigonometric proof. An even more beautiful result is obtained by taking the intersections of the exterior, as well as interior, angle trisectors, as shown above. In addition to the interior EQUILATERAL TRIANGLE formed by the inter- ior trisectors, four additional equilateral triangles are obtained, three of which have sides which are exten- sions of a central triangle (Wells 1991). A generalization of MORLEY’S THEOREM was discov- ered by Morley in 1900 but first published by Taylor and Marr (1914). Each ANGLE of a TRIANGLE DABC has six trisectors, since each interior angle trisector has two associated lines making angles of 1208 with it. The generalization of Morley’s theorem states that these trisectors intersect in 27 points (denoted Dij ; Eij ; Fij ; for i ; j /C300; 1, 2) which lie six by six on nine lines. Furthermore, these lines are in three triples of PARALLEL lines, (/D22E22 ; E12D21 ; F10F01) ; (/D22F22 ; F21D12 ; E01E10) ; and (/E22F22 ; F12E21 ; D10D01) ; making ANGLES of 60 8 with one another (Taylor and Marr 1914, Johnson 1929, p. 254). Let L, M, and N be the other trisector-trisector intersections, and let the 27 points Lij ; Mij ; Nijfor i ; j /C300 ; 1, 2 be the ISOGONAL CONJUGATES of D, E, and F. Then these points lie 6 by 6 on 9 CONICS through DABC :In addition, these CONICS meet 3 by 3 on the CIRCUMCIRCLE , and the three meeting points form an EQUILATERAL TRIANGLE whose sides are PARALLEL to those of DDEF :/ See also CONIC SECTION ,M ORLEY CENTERS ,TRISEC- TION References Child, J. M. "Proof of Morley’s Theorem." Math. Gaz. 11, 171, 1923. Coxeter, H. S. M. and Greitzer, S. L. "Morley’s Theorem." §2.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 47 /C1/0, 1967. Gardner, M. Martin Gardner’s New Mathematical Diver- sions from Scientific American. New York: Simon and Schuster, pp. 198 and 206, 1966. Honsberger, R. "Morley’s Theorem." Ch. 8 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 92 /C1/8, 1973. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 253 /C1/56, 1929. Kimberling, C. "Hofstadter Points." Nieuw Arch. Wiskunder 12, 109/C1/14, 1994. Lebesgue, H. "Sur les n-sectrices d’un triangle." L’enseign. math. 38,3 9/C1/8, 1939. Marr, W. L. "Morley’s Trisection Theorem: An Extension and Its Relation to the Circles of Apollonius." Proc. Edinburgh Math. Soc. 32, 136/C1/50, 1914. Morley, F. "On Reflexive Geometry." Trans. Amer. Math. Soc. 8,1 4/C1/4, 1907. Naraniengar, M. T. Mathematical Questions and Their Solutions from the Educational Times 15, 47, 1909. Oakley, C. O. and Baker, J. C. "The Morley Trisector Theorem." Amer. Math. Monthly 85, 737/C1/45, 1978. Pappas, T. "Trisecting & the Equilateral Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 174, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 6, 1999. Taylor, F. G. "The Relation of Morley’s Theorem to the Hessian Axis and Circumcentre." Proc. Edinburgh Math. Soc. 32, 132/C1/35, 1914. Taylor, F. G. and Marr, W. L. "The Six Trisectors of Each of the Angles of a Triangle." Proc. Edinburgh Math. Soc. 32, 119/C1/31, 1914. Weisstein, E. W. "Plane Geometry." M ATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 154 /C1/55, 1991. Morley’s Triangle An EQUILATERAL TRIANGLE considered by MORLEY’S THEOREM with side lengths 8R sin1 3 AfflCz6fflCz7 sin13 BfflCz6fflCz7 sin13 CfflCz6fflCz7 ; where R is the CIRCUMRADIUS of the original TRIAN- GLE. See also MORLEY’S THEOREM Morphism A morphism is a map between two objects in an abstract CATEGORY . 1. A general morphism is called a HOMOMORPHISM , 2. A morphism f : Y 0 X in a CATEGORY is a MONOMORPHISM if, for any two morphisms u; v : Z 0 Y ; fu /C30fv implies that u /C30v, 3. A morphism f : Y 0 X in a CATEGORY is an EPIMORPHISM if, for any two morphisms u; v : X 0 Z; uf /C30vf implies u /C30v, 4. A bijective morphism is called an ISOMORPHISM (if there is an isomorphism between two objects, then we say they are isomorphic), 5. A surjective morphism from an object to itself is called an ENDOMORPHISM , and 6. An ISOMORPHISM between an object and itself is called an AUTOMORPHISM . See also AUTOMORPHISM ,CATEGORY ,CATEGORY THE- ORY,EPIMORPHISM ,HOMEOMORPHISM ,HOMOMORPH- ISM,ISOMORPHISM ,MONOMORPHISM ,OBJECT Morrie’s Law cos(20/C14) cos(40/C14) cos(80/C14) /C301 8: An identity communicated to Feynman as a child by a boy named Morrie Jacobs (Gleick 1992, p. 47). Feyn- man remembered this fact all his life and referred to it in a letter to Jacobs in 1987 (Gleick 1992, p. 450). It is a special case of the general identity 2kYk/C281 j/C300cos(2ja) /C30sin(2ka) sin a; with k /C303 and a /C3020/C14 (Beyer et al. 1996). See also TRIGONOMETRY VALUES PI/9 References Anderson, E. C. "Morrie’s Law and Experimental Mathe- matics." To appear in J. Recr. Math. Beyer, W. A.; Louck, J. D.; Zeilberger, D. "A Generalization of a Curiosity that Feynman Remembered All His Life." Math. Mag. 69,43/C1/4, 1996. Gleick, J. Genius: The Life and Science of Richard Feynman. New York: Pantheon Books, pp. 47 and 450, 1992.Morse Function This entry contributed by SERGEI DUZHIN AND S. CHMUTOV A function for which all CRITICAL POINTS are non- degenerate and all CRITICAL LEVELS are different. See also KONTSEVICH INTEGRAL ,MORSE KNOT Morse Inequalities Topological lower bounds in terms of BETTI NUMBERS for the number of critical points form a smooth function on a smooth MANIFOLD . Morse Knot This entry contributed by SERGEI DUZHIN AND S. CHMUTOV A KNOT K embedded in R3 /C30Cz /C29Rt ; where the three- dimensional space R3is represented as a direct product of a complex line C with coordinate z and a real line R with coordinate t, in such a way that the coordinate t is a MORSE FUNCTION on K. See also KNOT,KONTSEVICH INTEGRAL ,MORSE FUNC- TION Morse-Rosen Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27a cosh2(ax)/C27btanh( ax)/C27g"# y/C300: References Barut, A. O.; Inomata, A.; and Wilson, R. "Algebraic Treat- ment of Second Po ¨schl-Teller, Morse-Rosen, and Eckart Equations." J. Phys. A: Math. Gen. 20, 4083 /C1/096, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 125, 1997. Morse Theory A generalization of CALCULUS OF VARIATIONS which draws the relationship between the stationary points of a smooth real-valued function on a MANIFOLD and the global topology of the MANIFOLD . For example, if a COMPACT MANIFOLD admits a function whose only stationary points are a maximum and a minimum, then the manifold is a SPHERE . Technically speaking, Morse theory applied to a FUNCTION gon a MANIFOLD Wwith g(M)/C300 and g(M?)/C301 shows that every COBORDISM can be realized as a finite sequence of SURGERIES . Conversely, a sequence of SURGERIES gives a COBORDISM . There are a number of classical applications of Morse theory, including counting geodesics on a R IEMANN SURFACE and determination of the topology of a L IE GROUP (Bott 1960, Milnor 1963). Morse theory has received much attention in the last two decades as a result of the paper by Witten (1982) which relates Morse theory to quantum field theory and also directly connects the stationary points of a smooth function to differential forms on the manifold. See also CALCULUS OF VARIATIONS ,C OBORDISM , MAZUR’S THEOREM ,SURGERY References Bott, R. Morse Theory and Its Applications to Homotopy Theory. Bonn, Germany: Universita ¨t Bonn, 1960. Chang, K. C. Infinite Dimensional Morse Theory and Multi- ple Solution Problems. Boston, MA: Birkha ¨user, 1993. Goresky, M. and MacPherson, R. Stratified Morse Theory. New York: Springer-Verlag, 1988. Milnor, J. W. Morse Theory. Princeton, NJ: Princeton University Press, 1963. Rassias, G. (Ed.). Morse Theory and Its Applications. Veverka, J. F. The Morse Theory and Its Application to Solid State Physics. Kingston, Ontario, Canada: Queen’s Uni- versity, 1966. Witten, E. "Supersymmetry and Morse Theory." J. Diff. Geom. 17, 661 /C1/92, 1982. Morse-Thue Sequence THUE- MORSE SEQUENCE Mortal A nonempty finite set of n /C29n INTEGER MATRICES for which there exists some product of the MATRICES in the set which is equal to the zero MATRIX . See also INTEGER MATRIX ,MORTALITY PROBLEM Mortality Problem For a given n, is the problem of determining if a set is MORTAL solvable? n /C301 is solvable, n /C302 is unknown, and n ]3 is unsolvable. See also MORTAL Morton-Franks-Williams Inequality Let E be the largest and e the smallest POWER of l in the HOMFLY POLYNOMIAL of an oriented LINK , and i be the BRAID INDEX . Then the MORTON- FRANKS- WILLIAMS INEQUALITY holds, i ]1 2(E /C28e) /C271 (Franks and Williams 1985, Morton 1985). The inequality is sharp for all PRIME KNOTS up to 10 crossings with the exceptions of 09 /C1/42, 09 /C1/49, 10 /C1/32, 10 /C1/ 50, and 10 /C1/56. See also BRAID INDEX References Franks, J. and Williams, R. F. "Braids and the Jones Polynomial." Trans. Amer. Math. Soc. 303,97/C1/08, 1987. Mosaic TESSELLATIONMoser The very LARGE NUMBER consisting of the number 2 inside a MEGA -gon. See also MEGA,MEGISTRON Moser-de Bruijn Sequence The sequence of numbers which are sums of distinct powers of 4. The first few are 0, 1, 4, 5, 16, 17, 20, 21, 64, 65, 68, 69, 80, 81, 84, ... (Sloane’s A000695). These numbers also satisfy the interesting properties that the sum of their BINARY digits equals the sum of their QUATERNARY digits, and that they have identical representations in BINARY and NEGABINARY . See also BINARY ,NEGABINARY ,QUATERNARY References Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences." Theor. Comput. Sci. 98, 163 /C1/97, 1992. de Bruijn, N. G. "Some Direct Decompositions of the Set of Integers." Math. Comput. 18, 537 /C1/46, 1964. Moser, L. "An Application of Generating Series." Math. Mag. 35,37/C1/8, 1962. Sloane, N. J. A. Sequences A000695/M3259 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Moser’s Circle Problem CIRCLE DIVISION BY CHORDS Moss’s Egg An OVAL whose construction is illustrated in the above diagram. See also EGG,OVAL References Dixon, R. Mathographics. New York: Dover, p. 5, 1991. Mott Polynomial Polynomials sk(x) which form the S HEFFER SEQUENCE for f(t)/C30/C282t 1/C28t2 and have GENERATING FUNCTION X/C12 k/C300sk(x) k!tk /C30expx 1 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 t2pfflC{fflCz t"# : The first few are s0(x) /C301 s1(x) /C30/C281 2 x s2(x) /C301 4 x2 s3(x) /C301 8(/C28x3 /C276x) s4(x) /C301 16(x4 /C2824x2) s5(x) /C301 32(/C28x5 /C2760x3 /C28240x): References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 251, 1981. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Motzkin Number The Motzkin numbers enumerate various combina- torial objects. Donaghey and Shapiro (1977) give 14 different manifestations of these numbers. In parti- cular, they give the number of paths from (0, 0) to (n, 0) which never dip below y /C300 and are made up onlyof the steps (1, 0), (1, 1), and (1, -1), i.e., 0;P; and o: The first are 1, 2, 4, 9, 21, 51, ... (Sloane’s A001006). The Motzkin number GENERATING FUNCTION M(z) satisfies M /C301 /C27xM /C27x2M2 (1) and is given by M(x) /C301 /C28 x /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 2x /C28 3x2p 2x2 /C301 /C27x /C272x2 /C274x3 /C279x4 /C2721x5 /C27...; (2) or by the RECURRENCE RELATION Mn /C30Mn/C281 /C27Xn/C282 k /C300MkMn/C282/C28k (3) with M0 /C301 : The Motzkin number Mn is also given by Mn/C30/C281 2X a/C27b/C30n/C272 a]0;b]0(/C283)a1 2 afflCzrfflCzD1 2 bfflCzrfflCzD (4) /C30(/C281)n/C271 22n/C275X a/C27b/C30n/C272 a]0;b]0(/C283)a (2a/C281)(2b/C281)2a afflCzrfflCzD 2b bfflCzrfflCzD ;(5) wheren kfflC{fflCz is a BINOMIAL COEFFICIENT . See also CATALAN NUMBER ,KING WALK,SCHRO ¨ DER NUMBER References Barcucci, E.; Pinzani, R.; and Sprugnoli, R. "The Motzkin Family." Pure Math. Appl. Ser. A 2, 249/C1/79, 1991. Dickau, R. M. "Delannoy and Motzkin Numbers." http:// www.prairienet.org/~pops/delannoy.html. Donaghey, R. "Restricted Plane Tree Representations of Four Motzkin-Catalan Equations." J. Combin. Th. Ser. B 22, 114/C1/21, 1977. Donaghey, R. and Shapiro, L. W. "Motzkin Numbers." J. Combin. Th. Ser. A 23, 291/C1/01, 1977. Kuznetsov, A.; Pak, I.; and Postnikov, A. "Trees Associated with the Motzkin Numbers." J. Combin. Th. Ser. A 76, 145/C1/47, 1996. Motzkin, T. "Relations Between Hypersurface Cross Ratios, and a Combinatorial Formula for Partitions of a Polygon, for Permanent Preponderance, and for NonassociativeProducts." Bull. Amer. Math. Soc. 54, 352/C1 /60, 1948. Sloane, N. J. A. Sequences A001006/M1184 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Moufang Identities For all x,y,ain an ALTERNATIVE ALGEBRA A; (xax)y/C30x[a(xy)] (1) y(xax)/C30[(yx)a]x (2) (xy)(ax) /C30x(ya)x (3) (Schafer 1996, p. 28). References Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, 1996. Moufang Plane A PROJECTIVE PLANE in which every line is a transla- tion line is called a Moufang plane. References Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 710, 1996. Mousetrap A PERMUTATION problem invented by Cayley. Let the numbers 1, 2, ..., n be written on a set of cards, and shuffle this deck of cards. Now, start counting using the top card. If the card chosen does not equal the count, move it to the bottom of the deck and continue counting forward. If the card chosen does equal the count, discard the chosen card and begin counting again at 1. The game is won if all cards are discarded, and lost if the count reaches n /C271:/ The number of ways the cards can be arranged such that at least one card is in the proper place for n /C301, 2, ... are 1, 1, 4, 15, 76, 455, ... (Sloane’s A002467). References Cayley, A. "A Problem in Permutations." Quart. Math. J. 1, 79, 1857. Cayley, A. "On the Game of Mousetrap." Quart. J. Pure Appl. Math. 15,8/C1/0, 1877. Cayley, A. "A Problem on Arrangements." Proc. Roy. Soc. Edinburgh 9, 338 /C1/42, 1878. Cayley, A. "Note on Mr. Muir’s Solution of a Problem of Arrangement." Proc. Roy. Soc. Edinburgh 9, 388 /C1/91, 1878. Guy, R. K. "Mousetrap." §E37 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 237 /C1/38, 1994. Guy, R. K. and Nowakowski, R. J. "Mousetrap." In Combi- natorics, Paul Erdos is Eighty, Vol. 1 (Ed. D. Miklo ´s, V. T. So´s, and T. Szonyi). Budapest: Ja´nos Bolyai Mathematical Society, pp. 193 /C1/06, 1993. Muir, T. "On Professor Tait’s Problem of Arrangement." Proc. Roy. Soc. Edinburgh 9, 382 /C1/87, 1878. Muir, T. "Additional Note on a Problem of Arrangement." Proc. Roy. Soc. Edinburgh 11, 187 /C1/90, 1882. Mundfrom, D. J. "A Problem in Permutations: The Game of ‘Mousetrap’." European J. Combin. 15, 555 /C1/60, 1994. Sloane, N. J. A. Sequences A002467/M3507, A002468/ M2945, and A002469/M3962 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Steen, A. "Some Formulae Respecting the Game of Mouse- trap." Quart. J. Pure Appl. Math. 15, 230 /C1/41, 1878. Tait, P. G. Scientific Papers, Vol. 1. Cambridge, England: University Press, p. 287, 1898.Mouth A PRINCIPAL VERTEX xiof a SIMPLE POLYGON P is called a mouth if the diagonal [xi/C281 ; xi /C271]isan extremal diagonal (i.e., the interior of [xi/C281 ; xi/C271] lies in the exterior of P). See also ANTHROPOMORPHIC POLYGON ,E AR,O NE- MOUTH THEOREM References Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122,31/C1/5, 1991. Moving Average Given a SEQUENCE fai gN i/C301an n-moving average is a new sequence fsi gN /C28n /C271 i/C301 defined from the ai by taking the AVERAGE of subsequences of n terms, si /C301 nXi/C27n/C281 j/C301aj : See also MEAN,SPENCER’S 15-POINT MOVING AVER- AGE,SPENCER’S FORMULA References Kenney, J. F. and Keeping, E. S. "Moving Averages." §14.2 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 221 /C1/23, 1962. Whittaker, E. T. and Robinson, G. "Graduation, or the Smoothing of Data." Ch. 11 in The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 285 /C1/16, 1967. Moving Ladder Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. What is the longest ladder which can be moved around a right-angled hallway of unit width? For a straight, rigid ladder, the answer is 2ffiffiffi 2p :For a smoothly-shaped ladder, the largest diameter is /]1(1/C27ffiffiffi2p ) (Finch). See also M OVING SOFA CONSTANT ,PIANO MOVER’S PROBLEM References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/sofa/sofa.html. Moving Sofa Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. What is the sofa of greatest AREA Swhich can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that S]p 2/C272 p/C302:2074 . . . : (1) Gerver (1992) found a sofa with larger AREA and provided arguments indicating that it is either optimal or close to it. The boundary of Gerver’s sofa is a complicated shape composed of 18 ARCS . Its AREA can be given by defining the constants A, B, f ; and u by solving A(cos u /C28cos f) /C282B sin f /C27(u /C28 f /C281) cos u /C28sin u /C27cos f /C27sin f /C300 (2) A(3 sin u /C27sin f) /C282B cos f /C273(u /C28 f /C281) sin u /C273 cos u /C28sin f /C27cos f /C300 (3) A cos f /C28(sin f /C271 2 /C2812cos f /C27B sin f) /C300 (4) (A /C271 2 p /C28 f /C28 u) /C28[B /C2812( u /C28 f)(1 /C27A) /C2814( u /C28 f)2] /C300 : (5) This gives A /C300:094426560843653... (6) B /C301:399203727333547... (7) f /C300 :039177364790084 ::: (8) u /C300:681301509382725... : (9) Now define r(a) /C131 2 for 0 5 a B f 1 2(1 /C27A /C27 a /C28 f) for f 5 a B u A /C27 a /C28 f for u 5 a B1 2 p /C28 u B /C281212 p /C28 a /C28 ffflCz6fflCz7 (1 /C27A) /C281412 p /C28 a /C28 ffflCz6fflCz72 for12 p /C28 u 5 a B12 p /C28 f;8 >>>>>>>>>>>>>< >>>>>>>>>>>>>: (10) where s( a) /C131 /C28r(a) (11) u( a) /C13B /C28 1 2(a /C28 f)(1 /C27A) for f 5 a B u /C281 4( a /C28 f)2 A /C2712 p /C28 f /C28 a for u 5 a B14 p8 >< >:(12) Du( a) /C30du da /C30/C281 2(1 /C27A) /C2812(a /C28 f) for f 5 a B u /C281i f u 5 a B14 p:( (13) Finally, define the functions y1(a) /C131 /C28g a 0r(t) sin tdt (14)y2(a) /C131 /C28g a 0s(t) sin tdt (15) y3( a) /C131 /C28g a 0s(t) sin tdt/C28u( a) sin a: (16) The AREA of the optimal sofa is given by A /C302g p =2/C28 f 0y1(a)r( a) cos a da /C272g u 0y2( a)s( a) cos a da /C272g p =4 fy3( a)[u(a) sin a /C28Du(a) cos a /C28s( a) cos a] da /C302:21953166887197 ... (17) (Finch). See also PIANO MOVER’S PROBLEM References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/sofa/sofa.html. Gerver, J. L. "On Moving a Sofa Around a Corner." Geome- triae Dedicata 42, 267 /C1/83, 1992. Stewart, I. Another Fine Math You’ve Got Me Into.... New York: W. H. Freeman, 1992. Mrs. Perkins’ Quilt The DISSECTION of a SQUARE of side n into a number Snof smaller squares. Unlike a PERFECT SQUARE DISSECTION , however, the smaller SQUARES need not be all different sizes. In addition, only prime dissec- tions are considered so that patterns which can be dissected on lower order SQUARES are not permitted. The smallest numbers of RELATIVELY PRIME dissec- tions of an n/C29nquilt for n/C301, 2, ..., are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (Sloane’s A005670). See also PERFECT SQUARE DISSECTION References Conway, J. H. "Mrs. Perkins’s Quilt." Proc. Cambridge Phil. Soc. 60, 363/C1/68, 1964. Croft, H. T.; Falconer, K. J.; and Guy, R. K. §C3 in Unsolved Problems in Geometry. New York: Springer-Verlag, 1991. Dudeney, H. E. Problem 173 in Amusements in Mathe- matics. New York: Dover, 1917. Dudeney, H. E. Problem 177 in 536 Puzzles & Curious Problems. New York: Scribner, 1967. Gardner, M. "Mrs. Perkins’ Quilt and Other Square-Packing Problems." Ch. 11 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage, 1977. Sloane, N. J. A. Sequences A005670/M3267 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Trustrum, G. B. "Mrs. Perkins’s Quilt." Proc. Cambridge Phil. Soc. 61,7/C1 /1, 1965. M-Tree A TREE not having the COMPLETE BIPARTITE GRAPH K1 ; 2with base at the vertex of degree two as a limb (Lu et al. 1993, Lu 1996). See also TREE References Lu, T. "The Enumeration of Trees with and without Given Limbs." Disc. Math. 154, 153 /C1/65, 1996. Lu, T. J.; Read, R. C.; and Palmer, E. M. "On the Enumera- tion of Trees with Certain Local Restrictions." Congr. Numer. 95, 183 /C1/02, 1993. Much Greater A strong INEQUALITY in which a is not only GREATER than b, but much greater (by some convention), is denoted a /C27b: For an astronomer, "much" may mean by a factor of 100 (or even 10), while for a mathema- tician, it might mean by a factor of 104 (or even much more). See also GREATER ,MUCH LESS Much Less A strong INEQUALITY in which a is not only LESS than b, but much less (by some convention) is denoted a /C10b :/ See also LESS,MUCH GREATER Mud Cracks RIGHT ANGLE Mu Function The 2-argument m/-function is defined by m(x; b) /C13g/C12 0xttb dt G( b /C27 1)G(t /C27 1) ; where G(z) is the GAMMA FUNCTION (Erde ´lyi et al. 1981, p. 388; Prudnikov et al. 1990, p. 798; Gradsh- teyn and Ryzhik 2000, p. 1109), while the 3-argument function is defined by m(x; b; a) /C13g/C12 0xa /C27ttb dt G( b /C27 1)G( a /C27 t /C27 1) (Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109). See also LAMBDA FUNCTION ,NU FUNCTION References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 388, 1981. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Ch. 18 in Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 217, 1981. Gradshteyn, I. S. and Ryzhik, I. M. "The Functions n(x); n(x; a); m(x; b) ; m(x; b; a) ; l(x; y) :/" §9.64 in Tables ofIntegrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1109, 2000. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, 1990. m Molecule MANDELBROT SET Muirhead’s Theorem A NECESSARY and SUFFICIENT condition that [a?] should be comparable with [a] for all POSITIVE values of the a is that one of/( a?) and (/ a) should be majorized by the other. If ( a?) )( a) ; then [ a?] 5[ a]; with equality only when (/( a?)) and (/ a) are identical or when all the a are equal. See Hardy et al. (1988) for a definition of notation. References Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "Muirhead’s Theorem" and "Proof of Muirhead’s Theorem." §2.18 and 2.19 in Inequalities, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 44 /C1/8, 1988. Muirhead, R. F. "Some Methods Applicable to Identities and Inequalities of Symmetric Algebraic Functions of n Let- ters." Proc. Edinburgh Math. Soc. 21, 144 /C1/57, 1903. Mu¨ller-Lyer Illusion An optical ILLUSION in which the orientation of arrowheads makes one LINE SEGMENT look longer than another. In the above figure, the LINE SEGMENTS on the left and right are of equal length in both cases. See also ILLUSION ,POGGENDORFF ILLUSION ,PONZO’S ILLUSION ,VERTICAL- HORIZONTAL ILLUSION References Fineman, M. The Nature of Visual Illusion. New York: Dover, p. 153, 1996. Luckiesh, M. Visual Illusions: Their Causes, Characteristics & Applications. New York: Dover, p. 93, 1965. Muller’s Method Generalizes the SECANT METHOD of root finding by using quadratic 3-point interpolation q/C13xn/C28xn/C281 xn/C281/C28xn/C282: (1) Then define A /C13qP(xn) /C28q(1 /C27q)P(xn/C281) /C27q2P(xn/C282) (2) B /C13(2q /C271)P(xn) /C28(1 /C27q)2P(xn/C281) /C27q2P(xn/C282) (3) C /C13(1 /C27q)P(xn) ; (4) and the next iteration is xn/C271 /C30xn /C28(xn /C28xn/C281)2C max B 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 /C28 4ACpfflCz6fflCz7 : (5) This method can also be used to find COMPLEX zeros of ANALYTIC FUNCTIONS . References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 364, 1992. Mulliken Symbols Symbols used to identify irreducible representations of GROUPS : /A /C30 singly degenerate state which is symmetric with respect to ROTATION about the principal Cn axis, /B /C30 singly DEGENERATE state which is antisym- metric with respect to ROTATION about the princi- pal Cn axis, /E /C30doubly DEGENERATE , /T /C30triply DEGENERATE , /Xg /C30(gerade, symmetric) the sign of the wavefunc- tion does not change on INVERSION through the center of the atom, /Xu /C30 (ungerade, antisymmetric) the sign of the wavefunction changes on INVERSION through the center of the atom, /X1 /C30(on a or b) the sign of the wavefunction does not change upon ROTATION about the center of the atom, /X2 /C30 (on a or b) the sign of the wavefunction changes upon ROTATION about the center of the atom, ?/C30 symmetric with respect to a horizontal sym- metry plane sh ;/ ƒ/C30 antisymmetric with respect to a horizontal symmetry plane sh :/ See also CHARACTER TABLE ,GROUP THEORY ,IRREDU- CIBLE REPRESENTATION References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, pp. 90 /C1/1, 1990. Multiamicable Numbers Two integers n and m Bn are (a; b)/-multiamicable if s(m) /C28m /C30 anand s(n) /C28n /C30 bm; where s(n) is the DIVISOR FUNCTION and a; b are POSITIVE INTEGERS .If a /C30 b /C301 ; (m, n)isan AMICABLE PAIR. m cannot have just one distinct prime factor, and if it has precisely two prime factors, then a /C301 and m is EVEN . Small multiamicable numbers for small a; b are given by Cohen et al. (1995). Several of these numbers are reproduced in the table below. /a//b/ mn 1 6 76455288 183102192 1 7 52920 152280 1 7 16225560 40580280 1 7 90863136 227249568 1 7 16225560 40580280 1 7 70821324288 177124806144 1 7 199615613902848 499240550375424 See also AMICABLE PAIR,DIVISOR FUNCTION References Cohen, G. L; Gretton, S.; and Hagis, P. Jr. "Multiamicable Numbers." Math. Comput. 64, 1743 /C1/753, 1995. Multichoose The number of MULTISETS of length k on n symbols is sometimes termed "n multichoose k," denotedn kfflC{fflCzfflC{fflCz by analogy with the BINOMIAL COEFFICIENT . n multi- choose k is given by the simple formula n kfflCzrfflCzDfflCzrfflCzD /C30nk ; giving the following array of numbers. /k_n/123 4 1 111 1 2 2481 6 33 92 7 8 1 4 4 16 64 256 See also BINOMIAL COEFFICIENT ,C HOOSE ,M ULTI- NOMIAL COEFFICIENT ,MULTISET References Schneiderman, E. R. Mathematics: A Discrete Introduction. Pacific Grove, CA: Brooks/Cole, 2000. Multidigital Number HARSHAD NUMBER Multidimensional Continued Fraction Algorithm INTEGER RELATION Multifactorial A generalization of the FACTORIAL and DOUBLE FACTORIAL , n! /C30n(n /C281)(n /C282) /C1/C1/C12 /C215 1 (1) n!! /C30n(n /C282)(n /C284) /C1/C1/C1 (2) n!!! /C30n(n /C283)(n /C286) /C1/C1/C1; (3) etc., where the products run through positive inte- gers. The FACTORIALS n! for n /C301, 2, ..., are 1, 2, 6, 24, 120, 720, ... (Sloane’s A000142); the DOUBLE FACTORIALS n!! are 1, 2, 3, 8, 15, 48, 105, ... (Sloane’s A006882); the triple factorials n!!! are 1, 2, 3, 4, 10, 18, 28, 80, 162, 280, ... (Sloane’s A007661); and the quadruple factor- ials n!!!! are 1, 2, 3, 4, 5, 12, 21, 32, 45, 120, ... (Sloane’s A007662). Letting fack(n) denote the k-multifactorial of n, fack(n) /C30Qn =k i /C301ik for (k; n) "1Qn =kbc i /C300n /C28ik for (k; n) /C301;( (4) Define r/C13n=kthen gives fack(n)/C30krr! for ( k;n)"1 (/C28k)1/C27rbc(/C28r)1/C27rfor ( k;n)/C301;fflC}6 (5) where ( x)nis the P OCHHAMMER SYMBOL . See also DOUBLE FACTORIAL ,F ACTORIAL ,G AMMA FUNCTION ,POCHHAMMER SYMBOL References Sloane, N. J. A. Sequences A000142/M1675, A006882/ M0876, A007661/M0596, and A007662/M0534 in "An On- Line Version of the Encyclopedia of Integer Sequences."http://www.research.att.com/~njas/sequences/eisonli- ne.html. Multifractal References Mandelbrot, B. B. Multifractals and /1=f/Noise: Wild Self- Affinity in Physics (1963 /C1/976). New York: Springer- Verlag, 1998.Multifractal Measure AMEASURE for which the Q-DIMENSION Dqvaries with q. References Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993. Multigrade Equation A(k, l)-multigrade equation is a D IOPHANTINE EQUA- TION OF THE FORM Xl i/C301nj i/C30Xl i/C301mji forj/C301, ..., k, where mand nare l-VECTORS . Multigrade identities remain valid if a constant is added to each element of mandn(Madachy 1979), so multigrades can always be put in a form where theminimum component of one of the vectors is 1. Moessner and Gloden (1944) give a bevy of multi- grade equations. Small-order examples are the (2, 3)- multigrade with m/C30f1;6;8gandn/C30f2;4;9g: X 3 i/C301m1 i/C30X3 i/C301n1i/C3015 X3 i/C301m2i/C30X3 i/C301n2i/C30101; the (3, 4)-multigrade with m/C30f1;5;8;12gandn/C30 f2;3;10;11g: X4 i/C301m1i/C30X4 i/C301n1i/C3026 X4 i/C301m2i/C30X4 i/C301n2i/C30234 X4 i/C301m3i/C30X4 i/C301n3i/C302366 ; and the (4, 6)-multigrade with m/C30 f1;5;8;12;18;19gandn/C30f2;3;9;13;16;20g: X6 i/C301m1i/C30X6 i/C301n1i/C3063 X6 i/C301m2i/C30X6 i/C301n2i/C30919 X6 i/C301m3i/C30X6 i/C301n3i/C3015057 X6 i/C301m3 i /C30X6 i/C301n4i /C30260755 (Madachy 1979). A spectacular example with k /C309 and l /C3010 is given by n /C30f912 ;911881 ;920231 ;920885 ;923738 g and m /C30f9436;911857 ;920499 ;920667 ;923750 g (Guy 1994), which has sums X9 i /C301m1i /C30X9 i /C301n1i /C300 X9 i/C301m2i /C30X9 i/C301n2i /C303100255070 X9 i /C301m3i /C30X9 i /C301n3i /C300 X9 i/C301m4i /C30X9 i/C301n4i /C301390452894778220678 X9 i /C301m5i /C30X9 i /C301n5i /C300 X9 i /C301m6i /C30X9 i/C301n6i /C30666573454337853049941719510 X9 i /C301m7i /C30X9 i /C301n7i /C300 X9 i/C301m8i /C30X9 i/C301n8i /C30330958142560259813821203262692838598 X9 i/C301m9i /C30X9 i/C301n9i /C300: Rivera considers multigrade equations involving primes, consecutive primes, etc. See also DIOPHANTINE EQUATION ,PROUHET- TARRY- ESCOTT PROBLEM References Chen, S. "Equal Sums of Like Powers: On the Integer Solution of the Diophantine System." http://www.nease.- net/~chin/eslp/ Gloden, A. Mehrgeradige Gleichungen. Groningen, Nether- lands: Noordhoff, 1944. Gloden, A. "Sur la multigrade A1 ; A2 ; A3 ; A4 ; A5/C30kB1 ; B2 ; B3 ; B4 ; B5 (k /C301, 3, 5, 7)." Revista Euclides 8, 383 /C1/84, 1948. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 143, 1994. Kraitchik, M. "Multigrade." §3.10 in Mathematical Recrea- tions. New York: W. W. Norton, p. 79, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 171 /C1/73, 1979.Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. E´ cole Polytech. de Timisoara 11, 196 /C1/19, 1944. Rivera, C. "Problems & Puzzles: Puzzle Multigrade Rela- tions.-065." http://www.primepuzzles.net/puzzles/ puzz_065.htm. Weisstein, E. W. "Like Powers." MATHEMATICA NOTEBOOK LIKEPOWERS.M . Multigraph A non- SIMPLE GRAPH in which no LOOPS are per- mitted, but multiple edges between any two nodes are. See also HYPERGRAPH ,PSEUDOGRAPH ,SIMPLE GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 10, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 89, 1990. Multilinear A basis, form, function, etc., in two or more variables is said to be multilinear if it is linear in each variable separately. See also BILINEAR FUNCTION ,L INEAR OPERATOR , MULTILINEAR BASIS,MULTILINEAR FORM Multilinear Basis See also BILINEAR BASIS Multimagic Series A set ndistinct numbers taken from the interval 1;n2½/C138 form a MAGIC SERIES if their sum is the nth MAGIC CONSTANT Mn/C301 2nn2/C271fflC{fflCz (Kraitchik 1942, p. 143). If the sum of the kth powers of these numbers is the MAGIC CONSTANT of degree k for all k/C23[1;p];then they are said to form a pth order MULTIMAGIC SERIES . Here, the magic constant M(j) nof degree kis defined as 1 =ntimes the sum of the first n2kth powers, M(k) n/C301 nXn2 i/C301ik/C30H(/C28p) n2 n; where H(k) nis a HARMONIC NUMBER of order k. For example f2; 8; 9; 15 g is bimagic since 2 /C278 /C279 /C27 15 /C3034 and 22 /C2782 /C2792 /C27152 /C30374:/ The numbers of magic series of various lengths n are gives in the following table for small orders k (Kraitchik 1942, p. 76). nk /C301 k /C302 k /C303 k /C304 Sloane A052456 A052457 A052458 1111 1 2200 0 3800 0 48 6 2 2 0 5 1,394 8 2 0 6 32,134 98 0 0 7 957,332 1,844 0 0 8 38,039 115 94 1 1011 961 See also M AGIC SERIES References Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical Recreations. New York: W. W. Norton, pp. 176 /C1/78, 1942. Sloane, N. J. A. Sequences A052456, A052457, and A052458 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Multimagic Square A MAGIC SQUARE is p-multimagic if the square formed by replacing each element by its kth power for k /C301, 2, ..., p is also magic. A 2-multimagic square is called a BIMAGIC SQUARE , and a 3-multimagic square is called a TRIMAGIC SQUARE . See also BIMAGIC SQUARE ,MAGIC SQUARE ,TRIMAGIC SQUARE References Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical Recreations. New York: W. W. Norton, pp. 176 /C1/78, 1942. Multinomial An algebraic expression containing more than one term (cf., BINOMIAL ). The term is also used to refer to a POLYNOMIAL .See also BINOMIAL ,M ULTINOMIAL COEFFICIENT , MULTINOMIAL SERIES ,POLYNOMIAL Multinomial Coefficient The multinomial coefficients n1 ; n2 ; ... ; nk ðÞ ! /C30(n1 /C27 n2 /C27/C1/C1/C1/C27 nk)! n1!n2! /C1/C1/C1n3! are the terms in the MULTINOMIAL SERIES expansion. The multinomial coefficient is returned by the Math- ematica function Multinomial [n1, n2, ...]. The number of distinct permutations in a MULTISET of k distinct elements of multiplicity ni(1 5i 5k)is n1 ; ...; nk ðÞ (Skiena 1990, p. 12). The multinomial coefficients satisfy n1 ; n2 ; n3 ; ... ðÞ /C30 n1 /C27n2 ; n3 ; ... ðÞ n1 ; n2 ðÞ /C30 n1 /C27n2 /C27n3 ;... ðÞ n1 ; n2 ; n3 ðÞ /C30... (Gosper 1972). The CONTENT V of the d-dimensional region ad k /C301 xkjjpkB1 is given by V /C302dXd k /C301p /C281 k; p /C281 1; p /C281 2;...;p/C281 d ! : See also BINOMIAL COEFFICIENT ,CHOOSE ,D YSON’S CONJECTURE ,M ULTICHOOSE ,M ULTINOMIAL SERIES , Q -MULTINOMIAL COEFFICIENT ,Z EILBERGER- BRES- SOUD THEOREM References Abramowitz, M. and Stegun, C. A. (Eds.). "Multinomial Coefficients." §24.1.2 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 823 /C1/24, 1972. Gosper, R. W. Item 42 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 16, Feb. 1972. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 113, 1992. Multinomial Distribution Let a set of random variates X1;X2;...,Xnhave a probability function PX1/C30x1;...;Xn/C30xn ðÞ /C30N!Qn i/C301xi!Yn i/C301uxi i (1) where xiare POSITIVE INTEGERS such that Xn i/C301xi/C30N; (2) anduiare constants with ui>0 and Xn i /C301ui /C301: (3) Then the joint distribution of X1 ; ..., Xnis a multi- nomial distribution and PX1 /C30x1 ; ...; Xn /C30xn ðÞ is given by the corresponding coefficient of the MULTI- NOMIAL SERIES u1 /C27 u2 /C27.../C27 un ðÞN: (4) In the words, if X1 ; X2 ; ..., Xnare mutually indepen- dent events with PX1ðÞ/C30 u1 ; ..., PxnðÞ/C30 un : Then the probability that X1occurs x1times, ..., Xnoccurs xn times is given by PNx1 ; x2 ; ... ; xn ðÞ /C30N! x1! /C1/C1/C1xn!ux1 1/C1/C1/C1uxn n : (5) (Papoulis 1984, p. 75). The MEAN and VARIANCE of Xi are mi /C30N ui (6) s2 i /C30N ui(1 /C28 ui) : (7) The COVARIANCE of Xi and Xj is s2ij /C30/C28N ui uj : (8) See also BINOMIAL DISTRIBUTION ,M ULTINOMIAL COEFFICIENT References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 532, 1987. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Multinomial Series A generalization of the BINOMIAL SERIES discovered by Johann Bernoulli and Leibniz. a1 /C27a2 /C27.../C27ak ðÞn /C30X n1 ; n2 ; ...; nkn! n1!n2!...nk!an1 1an2 2...ank k; where n /C13n1 /C27n2 /C27.../C27nk : The multinomial series arises in a generalization of the BINOMIAL DISTRIBU- TION called the MULTINOMIAL DISTRIBUTION . See also BINOMIAL SERIES ,M ULTINOMIAL DISTRIBU- TION Multinomial Theorem MULTINOMIAL SERIES Multinormal Distribution GAUSSIAN MULTIVARIATE DISTRIBUTIONMultiperfect Number A number nisk-multiperfect (also called a k-MULTI- PLY PERFECT NUMBER ork-PLUPERFECT NUMBER )i f s(n)/C30kn for some INTEGER k/C212, where s(n) is the DIVISOR FUNCTION . The value of kis called the CLASS . The special case k/C302 corresponds to PERFECT NUMBERS P2;which are intimately connected with M ERSENNE PRIMES (Sloane’s A000396). The number 120 was long known to be 3-multiply perfect ( /P3) since s(120)/C303/C215120: The following table gives the first few Pnforn/C302, 3, ..., 6. 2 A000396 6, 28, 496, 8128, ..., 3 A005820 120, 672, 523776, 459818240, 1476304896, 51001180160 4 A027687 30240, 32760, 2178540, 23569920, ... 5 A046060 14182439040, 31998395520, 518666803200, ... 6 A046061 154345556085770649600, 9186050031556349952000, ... In 1900 /C1/901, Lehmer proved that P3has at least three distinct PRIME FACTORS ,P4has at least four, P5 at least six, P6at least nine, and P7at least 14. As of 1911, 251 pluperfect numbers were known (Carmichael and Mason 1911). As of 1929, 334 pluperfect numbers were known, many of them foundby Poulet. Franqui and Garcı ´a (1953) found 63 additional ones (five P 5/s, 29 P6/s, and 29 P7/s), several of which were known to Poulet but had not beenpublished, bringing the total to 397. Brown (1954)discovered 110 pluperfects, including 31 discovered but not published by Poulet and 25 previously published by Franqui and Garcı ´a (1953), for a total of 482. Franqui and Garcı ´a (1954) subsequently discovered 57 additional pluperfects (3 P 6/s, 52 P7/s, and 2 P8/s), increasing the total known to 539. An outdated database is maintained by R. Schroep-pel, who lists 2,094 multiperfects, and up-to-date lists by J. L. Moxham (2000b) and A. Flammenkamp. It is believed that all multiperfect numbers of index 3, 4, 5,6, and 7 are known. The number of known n-multi- perfect numbers are 1, 37, 6, 36, 65, 245, 516, 1134, 1982, 183, 0, 0, ... (Moxham 2000b, Flammenkamp, Woltman 2000). Moxham (2000a) found the largestknown multiperfect number, approximately equal to 7:3/C2910 1345;on Feb. 13, 2000. If n is a P5number such that 3¶n; then 3n is a P4 number. If 3n is a P4k number such that 3¶n ; then n is a P3k number. If n is a P3 number such that 3 (but not 5 and 9) DIVIDES n, then 45n is a P4number. See also E-MULTIPERFECT NUMBER ,FRIENDLY PAIR, HYPERPERFECT NUMBER ,INFINARY MULTIPERFECT NUMBER ,M ERSENNE PRIME ,PERFECT NUMBER ,UNI- TARY MULTIPERFECT NUMBER References Beck, W. and Najar, R. "A Lower Bound for Odd Triperfects." Math. Comput. 38, 249/C1/51, 1982. Brown, A. L. "Multiperfect Numbers." Scripta Math. 20, 103/C1/06, 1954. Cohen, G. L. and Hagis, P. Jr. "Results Concerning Odd Multiperfect Numbers." Bull. Malaysian Math. Soc. 8, 23/C1/6, 1985. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 33 /C1/8, 1952. Flammenkamp, A. "Multiply Perfect Numbers." http:// www.uni-bielefeld.de/~achim/mpn.html. Franqui, B. and Garcı ´a, M. "Some New Multiply Perfect Numbers." Amer. Math. Monthly 60, 459/C1/62, 1953. Franqui, B. and Garcı ´a, M. "57 New Multiply Perfect Numbers." Scripta Math. 20, 169/C1/71, 1954. Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed.New York: Springer-Verlag, pp. 45 /C1/3, 1994. Helenius, F. W. "Multiperfect Numbers (MPFNs)." http:// home.netcom.com/~fredh/mpfn/. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 149 /C1/51, 1979. Moxham, J. L. "New Largest MPFN." [email protected] posting, 13 Feb. 2000a. Moxham, J. L. "New MPFNs for per3.6 server." [email protected] izona.edu posting, 19 Sep 2000b. Poulet, P. La Chasse aux nombres, Vol. 1. Brussels, pp. 9 /C1/7, 1929. Schroeppel, R. "Multiperfect Numbers-Multiply Perfect Numbers-Pluperfect Numbers-MPFNs." Rev. Dec. 13, 1995. ftp://ftp.cs.arizona.edu/xkernel/rcs/mpfn.html. Schroeppel, R. (moderator). mpfn mailing list. e-mail [email protected] to subscribe. Sloane, N. J. A. Sequences A000396/M4186, A005820/ M5376, A027687, A046060, and A046061 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Woltman, G. "5 new MPFNs." [email protected] posting, 23 Sep 2000. Multiple A multiple of a number xis any quantity y/C30nxwith nan integer. If xandyare integers, then xis called a FACTOR y. Multiple Analysis of Variance MANOVA Multiple-Angle Formulas Expressions OF THE FORM sin(nx);cos(nx);and tan( nx) can be expressed in terms of sin xand cos xonly using the E ULER FORMULA and BINOMIAL THEOREM . Forsin(nx); sin(nx)/C30einx/C28e/C28inx 2i/C30(eix)n/C28(e/C28ix)n 2i /C30(cosx/C27isinx)n/C28(cosx/C28isinx)n 2i /C30Xn k/C300n kfflCzrfflCzDcoskx(isinx)n/C28k/C28coskx(/C28isinx)n/C28k 2i /C30Xn k/C300n kfflCzrfflCzD coskxsinn/C28kxin/C28k/C28(/C28i)n/C28k 2i /C30Xn k/C300n kfflCzrfflCzD coskxsinn/C28kxsin[1 2(n/C28k)p]: (1) Particular cases for multiple angle formulas for sin x are given by sin(2 x)/C302 sin xcosx (2) sin(3 x)/C303 sin x/C284 sin3x (3) sin(4 x)/C304 sin xcosx/C288 sin3xcosx (4) sin(5 x)/C305 cos4sinx/C2810 cos2xsin3x/C27sin5x:(5) The function sin( nx) can also be expressed as a polynomial in sin x(for nodd) or cos xtimes a polynomial in sin xas sin(nx)/C30(/C281)(n/C281)=2Tn(sinx) for nodd (/C281)n=2/C281cosxUn(sinx) for neven ;fflC}6 (6) where Tnis a C HEBYSHEV POLYNOMIAL OF THE FIRST KIND and Unis a C HEBYSHEV POLYNOMIAL OF THE SECOND KIND . The first few cases are sin(2 x)/C302 cos xsinx (7) sin(3 x)/C303 sin x/C284 sin3x (8) sin(4 x)/C30cosx(4 sin x/C288 sin3x) (9) sin(5 x)/C305 sin x/C2820 sin3x/C2716 sin5x: (10) Similarly, sin( nx) can be expressed as sin xtimes a polynomial in cos xas sin(nx)/C30sinxUn/C281(cosx): (11) The first few cases are sin(2 x)/C302 cos xsinx (12) sin(3 x)/C30sinx(/C281/C274 cos2x) (13) sin(4 x)/C30sinx(/C284 cos x/C278 cos3x) (14) sin(5 x)/C30sinx(1/C2812 cos2x/C2716 cos4x): (15) Bromwich (1991) gave the formula sin(na)/C30 nx /C28n(n2 /C28 12)x3 3!/C27n(n2 /C28 12)(n2 /C28 32)x5 5!/C28... for n odd n cos ax/C28(n2 /C28 22)x3 3!/C27(n2 /C28 22)(n2 /C28 42)x5 5!/C28..."# for n even ;8 >>>>>>>< >>>>>>>: (16) where x /C30sin a: / For cos(nx); the multiple-angle formula can be de- rived as cos(nx) /C30einx /C27 e/C28inx 2i/C30(eix)n /C27 (e /C28ix)n 2 /C30(cos x /C27 i sin x)n /C27 (cos x /C28 i sin x)n 2 /C30Xn k /C300n kfflCzrfflCzDcosk x(i sin x)n/C28k /C27 cosk x( /C28i sin x)n/C28k 2 /C30Xn k /C300n kfflCzrfflCzD cosk x sinn/C28k xin/C28k /C27 ( /C28i)n /C28k 2 /C30Xn k /C300n kfflCzrfflCzD cosk x sinn/C28k x cos1 2(n /C28k) phi : (17) The first few values are cos(2 x) /C30cos2 x /C28sin2 x (18) cos(3 x) /C304 cos3 x /C283 cos x sin x (19) cos(4 x) /C30cos4 x /C286 cos2 x sin2 x /C27sin4 x (20) cos(5 x) /C30cos5 x /C2810 cos3 x sin2 x /C275 cos x sin4 x: (21) The function cos(nx) can also be expressed as a polynomial in sin x (for n even) or cos x times a polynomial in sin x as cos(nx) /C30(/C281)n/C281 =2 cos xUn/C281(sin x) for n odd (/C281)n=2Tn(sin x) for n even :fflC}6 (22) The first few cases are cos(2 x) /C301 /C282 sin2 x (23) cos(3 x) /C30cos x(1 /C284 sin2 x) (24) cos(4 x) /C30cos x(1 /C2812 sin2 x /C2716 sin4 x) (25) cos(5 x) /C301 /C288 sin2 x /C278 sin4 x: (26) Similarly, cos(nx) can be expressed as a polynomial in cos x as cos(nx) /C30Tn(cos x) (27) The first few cases are cos(2 x) /C30/C281 /C272 cos2 x (28)cos(3 x) /C30/C283 cos x /C274 cos3 x (29) cos(4 x) /C301 /C288 cos2 x /C278 cos4 x (30) cos(5 x) /C305 cos x /C2820 cos3 x /C2716 cos5 x : (31) Bromwich (1991) gave the formula cos(na) /C30 cos a 1 /C28(n2 /C28 12)x2 2!/C27(n2 /C28 12)(n2 /C28 32)x4 4!/C28/C1/C1/C1"# n odd 1 /C28n2x2 2!/C27n2(n2 /C28 22)x4 4!/C28/C1/C1/C1 n even ;8 >>>>>< >>>>>: (32) where x /C30sin a: / The first few multiple-angle formulas for tan(nx) are tan(2 x) /C302 tan x 1 /C28 tan2 x (33) tan(3 x) /C303 tan x /C28 tan3 x 1 /C28 3 tan2 x (34) tan(4 x)/C304 tan x/C284 tan3x 1/C286 tan2x/C27tan4x(35) are given by Beyer (1987, p. 139) for up to n/C306. Multiple angle formulas can also be written using the RECURRENCE RELATIONS sin(nx)/C302 sin[( n/C281)x] cos x/C28sin[(n/C282)x] (36) cos(nx)/C302 cos[( n/C281)x] cos x/C28cos[(n/C282)x] (37) tan(nx)/C30tan[( n/C281)x]/C27tanx 1/C28tan[( n/C281)x] tan x: (38) See also DOUBLE- ANGLE FORMULAS ,H ALF-ANGLE FORMULAS ,HYPERBOLIC FUNCTIONS ,PROSTHAPHAER- ESIS FORMULAS ,TRIGONOMETRIC ADDITION FORMU- LAS,TRIGONOMETRIC FUNCTIONS ,TRIGONOMETRY References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Bromwich, T. J. I’a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 202 /C1/07, 1991. Multiple-Free Set DOUBLE- FREESET,SUM-FREESET,TRIPLE- FREESET Multiple Integral A set of integrals taken over n /C211 variables g...g|fflfflfflffl{zfflfflfflffl} nf(x1 ;...;xn) dx1 ...dxn (1) is called a multiple integral. An nth order integral corresponds, in general, to an n-D VOLUME (CON- TENT ), with n /C302 corresponding to an AREA .Inan indefinite multiple integral, the order in which the integrals are carried out can be varied at will; for definite multiple integrals, care must be taken to correctly transform the limits if the order is changed. See also FUBINI THEOREM ,INTEGRAL ,M ONTE CARLO INTEGRATION ,REPEATED INTEGRAL References Kaplan, W. "Double Integrals" and "Triple Integrals and Multiple Integrals in General." §4.3 /C1/.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 228 /C1/35, 1991. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Multidimensional Integrals." §4.6 in Numer- ical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 155 /C1/58, 1992. Multiple Point MULTIPLE ROOT Multiple Regression A REGRESSION giving conditional expectation values of a given variable in terms of two or more other variables. See also LEAST SQUARES FITTING ,M ULTIVARIATE ANALYSIS ,NONLINEAR LEAST SQUARES FITTING References Chatterjee, S.; Hadi, A.; and Price, B. "Multiple Linear Regression." Ch. 3 in Regression Analysis by Example, 3rd ed. New York: Wiley, pp. 51 /C1/4, 2000. Edwards, A. L. Multiple Regression and the Analysis of Variance and Covariance. San Francisco, CA: W. H. Freeman, 1979. Multiple Root A ROOT with MULTIPLICITY n ]2 ; also called a multi- ple point. See also MULTIPLICITY ,ROOT,SIMPLE ROOT References Krantz, S. G. "Zero of Order n." §5.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 70, 1999. Multiple-Valued Function A function for which several distinct functional values correspond (as a result of different continua- tions) to one and the same point (Knopp 1996, p. 94).See also BRANCH CUT,RIEMANN SURFACE ,SINGLE- VALUED FUNCTION References Knopp, K. "Multiple-Valued Functions." Section II in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 93 /C1/46, 1996. Multiplicand A quantity that is multiplied by another (the MULTI- PLIER ). For example, in the expression a /C29b ; b is the multiplicand.See also M ULTIPLICATION ,MULTIPLIER Multiplication In simple algebra, multiplication is the process of calculating the result when a number a is taken b times. The result of a multiplication is called the PRODUCT of a and b, and each of the numbers a and b is called a FACTOR of the PRODUCT ab. Multiplication is denoted a /C29b; a /C215 b; (a)(b); or simply ab. The symbol /C29 is known as the MULTIPLICATION SIGN. Normal multiplication is ASSOCIATIVE , COMMUTATIVE , and DISTRIBUTIVE . More generally, multiplication can also be defined for other mathematical objects such as GROUPS , MA- TRICES , SETS , and TENSORS . Karatsuba and Ofman (1962) discovered that multi- plication of two n digit numbers can be done with a BIT COMPLEXITY of less than n2 using an algorithm now known as KARATSUBA MULTIPLICATION . Multiplication of numbers x and y carried out in base b can be implemented in Mathematica as Multiply[{x_,y_},b_]: /C30FromDigits[ ListConvolve[IntegerDigits[x, b], IntegerDigits[y, b], {1, -1}, 0], b] See also ADDITION ,BIT COMPLEXITY ,COMPLEX MUL- TIPLICATION ,DIVISION ,FACTOR ,KARATSUBA MULTI- PLICATION ,M ATRIX MULTIPLICATION ,M ULTIPLICAND , MULTIPLIER ,P RODUCT ,R USSIAN MULTIPLICATION , SCALAR MULTIPLICATION ,SUBTRACTION ,TIMES References Beck, G. "Long Multiplication and Division." M ATHEMATICA NOTEBOOK LONGDIVISION.NB . Cundy, H. M. "What Is /C29/?"Math. Gaz. 43, 101, 1959. Karatsuba, A. and Ofman, Yu. "Multiplication of Many- Digital Numbers by Automatic Computers." Doklady Akad. Nauk SSSR 145, 293/C1/94, 1962. Translation in Physics-Doklady 7, 595/C1/96, 1963. Multiplication Magic Square A square which is magic under multiplication instead of addition (the operation used to define a conven- tional MAGIC SQUARE ) is called a multiplication magic square. Unlike (normal) MAGIC SQUARES , the n2 entries for an nth order multiplicative magic square are not required to be consecutive. The above multi- plication magic square has a multiplicative magic constant of 4,096. See also ADDITION- MULTIPLICATION MAGIC SQUARE , MAGIC SQUARE References Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, pp. 30 /C1/1, 1975. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 89 /C1/1, 1979. Multiplication Principle If one event can occur in m ways and a second can occur independently of the first in n ways, then the two events can occur in mn ways. Multiplication Sign The symbol /C29 used to denote MULTIPLICATION , i.e., a /C29b denotes a times b. The symbol /C29 is also used to denote a GROUP DIRECT PRODUCT ,aC ARTESIAN PRODUCT , or a direct product in the appropriate category (such as a Cartesian product of manifolds when it is implied that the smooth structure is the natural product structure.) The similar symbol /C156is reserved for a tensor product, which may rear its head in several guises, represen- tations, bundles, modules. Multiplication Table A multiplication table is an array showing the result of applying a BINARY OPERATOR to elements of a given set S. 1234567891 0 11234567891 0 224681 0121416182 0 33691 215182124273 0 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 607 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100 See also BINARY OPERATOR ,TRUTH TABLE Multiplicative Character A continuous HOMEOMORPHISM of a GROUP into the NONZERO COMPLEX NUMBERS . A multiplicative char- acter v gives a REPRESENTATION on the 1-D SPACE C of COMPLEX NUMBERS , where the REPRESENTATION action by g /C23 G is multiplication by v(g): A multi- plicative character is UNITARY if it has ABSOLUTE VALUE 1 everywhere. See also GRO¨ SSENCHARAKTER ,UNITARY MULTIPLICA- TIVE CHARACTER References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537/C1/49, 1996. Multiplicative Digital Root Consider the process of taking a number, multiplying its DIGITS , then multiplying the DIGITS of numbers derived from it, etc., until the remaining number hasonly one DIGIT . The number of multiplications re- quired to obtain a single DIGIT from a number nis called the MULTIPLICATIVE PERSISTENCE ofn, and the DIGIT obtained is called the multiplicative digital root ofn. For example, the sequence obtained from the startingnumber 9876 is (9876, 3024, 0), so 9876 has a MULTIPLICATIVE PERSISTENCE of two and a multi- plicative digital root of 0. The multiplicative digitalroots of the first few positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, ... (Sloane’s A031347). nSloane numbers having multiplicative digital root n 0 A034048 0, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, ... 1 A002275 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, ... 2 A034049 2, 12, 21, 26, 34, 37, 43, 62, 73, 112, 121, 126, ... 3 A034050 3, 13, 31, 113, 131, 311, 1113, 1131, 1311, 3111, ... 4 A034051 4, 14, 22, 27, 39, 41, 72, 89, 93, 98, 114, 122, ... 5 A034052 5, 15, 35, 51, 53, 57, 75, 115, 135, 151, 153, 157, ... 6 A034053 6, 16, 23, 28, 32, 44, 47, 48, 61, 68, 74, 82, 84, ... 7 A034054 7, 17, 71, 117, 171, 711, 1117, 1171, 1711, 7111, ... 8 A034055 8, 18, 24, 29, 36, 38, 42, 46, 49, 63, 64, 66, 67, ... 9 A034056 9, 19, 33, 91, 119, 133, 191, 313, 331, 911, 1119, ... See also ADDITIVE PERSISTENCE ,D IGITADDITION , DIGITAL ROOT,MULTIPLICATIVE PERSISTENCE References Sloane, N. J. A. Sequences A002275, A031347, A034048, A034049, A034050, A034051, A034052, A034053, A034054, A034055, and A034056 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Multiplicative Function A function f(m) is called multiplicative if (m; m?) /C301 (i.e., the statement that m and m? are RELATIVELY PRIME ) implies f(mm?) /C30f(m)f(m?) : Examples of multiplicative functions are the MO¨ BIUS FUNCTION and TOTIENT FUNCTION . See also COMPLETELY MULTIPLICATIVE FUNCTION , MO¨ BIUS FUNCTION ,Q UADRATIC RESIDUE ,T OTIENT FUNCTION Multiplicative Inverse The multiplicative inverse of a REAL or COMPLEX NUMBER z is its RECIPROCAL 1=z: For complex z /C30 x /C27iy ; 1 z /C301 x /C27 iy /C30x x2 /C27 y2 /C28iy x2 /C27 y2 : Multiplicative Number Theory See also ADDITIVE NUMBER THEORY ,NUMBER THEORY References Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, p. 110, 1980. Montgomery, H. L. Topics in Multiplicative Number Theory. New York: Springer-Verlag, 1971.Multiplicative Order Let n be a positive number having PRIMITIVE ROOTS . If g is a PRIMITIVE ROOT of n, then the numbers 1, g, g2 ; ..., gf(n) /C281form a REDUCED RESIDUE SYSTEM modulo n, where f(n) is the TOTIENT FUNCTION .In this set, there are f( f(n)) PRIMITIVE ROOTS , and these are the numbers gc ; where c is RELATIVELY PRIME to f(n) : If a is an arbitrary integer RELATIVELY PRIME to n, then there exists among the numbers 0, 1, 2, ..., f(n /C281) exactly one number m such that a /C13g m (mod n) : (1) The number m is then called the generalized multi- plicative order of a with respect to the base g modulo n. Note that Nagell (1951, p. 112) instead uses the term "index" and writes m /C30indg a (mod n) : (2) For example, the number 7 in the least positive PRIMITIVE ROOT of n /C3041, and since 15 /C13 73 (mod 41); the number 15 has multiplicative order 3 with respect to base 7 (modulo 41) (Nagell 1951, p. 112). The generalized multiplicative order is im- plemented in Mathematica asMultiplicativeOr- der[a, n,{ g1}], or more generally as MultiplicativeOrder [a, n,{g1, g2, ...}]. If the PRIMITIVE ROOTS g1 /C30/C281 and g2 /C301 are chosen, the resulting function is called the SUBORDER FUNC- TION and is denoted sordn(a): If the single PRIMITIVE ROOT g1 /C301 is chosen, then the function reduces to "the" (i.e., ungeneralized) multiplicative order, de- noted ordn(a) ; implemented in Mathematica asMul- tiplicativeOrder [a, n]. This function is sometimes also known as the discrete logarithm (or, more confusingly, as the "index," a term which Nagell applied to the case of general g). See also CONGRUENCE ,H AUPT- EXPONENT ,O RDER (MODULO ), PRIMITIVE ROOT,SUBORDER FUNCTION References Nagell, T. "The Index Calculus." §33 in Introduction to Number Theory. New York: Wiley, pp. 111 /C1/15, 1951. Odlyzko, A. "Discrete Logarithms: The Past and the Future." http://www.research.att.com/~amo/doc/discrete.logs.fu- ture.ps. Multiplicative Perfect Number A number n for which the PRODUCT of DIVISORS is equal to n2 : The first few are 1, 6, 8, 10, 14, 15, 21, 22, ... (Sloane’s A007422). See also PERFECT NUMBER References Sloane, N. J. A. Sequences A007422/M4068 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Multiplicative Persistence Multiply all the digits of a number n by each other, repeating with the product until a single DIGIT is obtained. The number of steps required is known as the multiplicative persistence, and the final DIGIT obtained is called the MULTIPLICATIVE DIGITAL ROOT of n. For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has an multiplicative persistence of two and a MULTIPLICA- TIVE DIGITAL ROOT of 0. The multiplicative persis- tences of the first few positive integers are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, ... (Sloane’s A031346). The smallest numbers having multiplicative persis- tences of 1, 2, ... are 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (Sloane’s A003001; Wells 1986, p. 78). There is no number B1050 with multiplicative persistence > 11 (Wells 1986, p. 78). It is conjectured that the max- imum number lacking the DIGIT 1 with persistence 11 is 77777733332222222222222222222 There is a stronger conjecture that there is a max- imum number lacking the DIGIT 1 for each persistence ]2:/ The maximum multiplicative persistence in base 2 is 1. It is conjectured that all powers of 2 > 215 contain a 0 in base 3, which would imply that the maximum persistence in base 3 is 3 (Guy 1994). The multiplicative persistence of an n-DIGIT number is also called its LENGTH . The maximum lengths for n /C301-, 2-, 3-, ..., digit numbers are 0, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 10, ... (Sloane’s A014553; Beeler 1972, Gottlieb 1969 /C1/970). The numbers of n-digit numbers having maximal multiplicative persistence for n /C301, 2, ..., are 10 (which includes the number 0), 1, 9, 12, 20, 2430, ... (Sloane’s A046148). The smallest n-digit numbers with maximal multiplicative persistence are 0, 77, 679, 6788, 68889, 168889, ... (Sloane’s A046149). The largest n-digit numbers with maximal multiplicative persistence are 9, 77, 976, 8876, 98886, 997762, ... (Sloane’s A046150). The number of distinct n-digit numbers (except for 0s) are given by10/C27n/C281 nfflC{fflCz /C28 1 which, for n /C301, 2, 3, ..., gives 54, 219, 714, 2001, 5004, 11439, ... (Sloane’s A035927). The concept of multiplicative persistence can be generalized to multiplying the kth powers of the digits of a number and iterating until the result remains constant. All numbers other than REPUNITS , which converge to 1, converge to 0. The number of iterations required for the kth powers of a number’s digits to converge to 0 is called its k-multiplicative persistence. The following table gives the n-multi- plicative persistences for the first few positive inte- gers.n Sloane n-Persistences 2 Sloane’s A0313480, 7, 6, 6, 3, 5, 5, 4, 5, 1, ... 3 Sloane’s A0313490, 4, 5, 4, 3, 4, 4, 3, 3, 1, ... 4 Sloane’s A0313500, 4, 3, 3, 3, 3, 2, 2, 3, 1, ... 5 Sloane’s A0313510, 4, 4, 2, 3, 3, 2, 3, 2, 1, ... 6 Sloane’s A0313520, 3, 3, 2, 3, 3, 3, 3, 3, 1, ... 7 Sloane’s A0313530, 4, 3, 3, 3, 3, 3, 2, 3, 1, ... 8 Sloane’s A0313540, 3, 3, 3, 2, 4, 2, 3, 2, 1, ... 9 Sloane’s A0313550, 3, 3, 3, 3, 2, 2, 3, 2, 1, ... 10 Sloane’s A0313560, 2, 2, 2, 3, 2, 3, 2, 2, 1, ... Erdos suggested ignoring all zeros and showed that at most cln ln nsteps are needed to reduce nto a single digit, where cdepends on the base. The smallest primes with multiplicative persistences n/C301, 2, 3, ... are 2, 29, 47, 277, 769, 8867, 186889, 2678789, 26899889, 3778888999, 277777788888989,... (Sloane’s A046500). See also 196-ALGORITHM ,A DDITIVE PERSISTENCE , DIGITADDITION ,DIGITAL ROOT,KAPREKAR NUMBER , LENGTH (NUMBER ), MULTIPLICATIVE DIGITAL ROOT, NARCISSISTIC NUMBER ,RECURRING DIGITAL INVAR- IANT References Beeler, M. Item 56 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 22, Feb. 1972. Gottlieb, A. J. Problems 28 /C1/9 in "Bridge, Group Theory, and a Jigsaw Puzzle." Techn. Rev. 72, unpaginated, Dec. 1969. Gottlieb, A. J. Problem 29 in "Integral Solutions, Ladders, and Pentagons." Techn. Rev. 72, unpaginated, Apr. 1970. Guy, R. K. "The Persistence of a Number." §F25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 262 /C1/63, 1994. Rivera, C. "Problems & Puzzles: Puzzle Primes & Persis- tence.-022." http://www.primepuzzles.net/puzzles/ puzz_022.htm. Sloane, N. J. A. "The Persistence of a Number." J. Recr. Math. 6,9 7/C1/8, 1973. Sloane, N. J. A. Sequences A003001/M4687, A014553, A031346, and A046500 in "An On-Line Version of theEncyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 78, 1986. Multiplicative Primitive Residue Class Group MODULO MULTIPLICATION GROUP Multiplicity The word multiplicity is a general term meaning "the number of values for which a given condition holds." For example, the term is used to refer to the value of the TOTIENT VALENCE FUNCTION or the number of times a given polynomial equation has a ROOT at a given point. Let z0 be a ROOT of a function f, and let n be the least positive integer n such that f(n)(z0) "0: Then the POWER SERIES of f about z0 begins with the nth term, f(z) /C30X/C12 j/C30n1 j!@jf @zj j z/C30z0(z /C28z0)j ; and f is said to have a ROOT of multiplicity (or "order") n.Ifn /C301, the ROOT is called a SIMPLE ROOT (Krantz 1999, p. 70). See also DEGENERATE ,M ULTIPLE ROOT,N OETHER’S FUNDAMENTAL THEOREM ,ROOT,SIMPLE ROOT,TOTI- ENT VALENCE FUNCTION References Krantz, S. G. "Zero of Order n." §5.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 70, 1999. Multiplier A quantity by which another (the MULTIPLICAND )is multiplied. For example, in the expression a /C29b; a is the multiplier. The term "multiplier" also has a special meaning in the theory of MODULAR FUNCTION . See also MODULAR FUNCTION ,MULTIPLICAND ,MULTI- PLICATION Multiply Connected A set which is CONNECTED but not SIMPLY CONNECTED is called multiply connected. A SPACE is n-MULTIPLY CONNECTED if it is (n /C281)/-connected and if every MAP from the n-SPHERE into it extends continuously over the (n /C271)/-DISK A theorem of Whitehead says that a SPACE is infinitely connected IFF it is contractible.See also CONNECTIVITY ,L OCALLY PATHWISE- CON- NECTED ,SIMPLY CONNECTED Multiply Perfect Number MULTIPERFECT NUMBER Multipolynomial Quadratic Sieve QUADRATIC SIEVE Multisection SERIES MULTISECTION Multiset A SET-like object in which order is ignored, but multiplicity is explicitly significant. Therefore, multi- sets f1 ; 2 ; 3 g and f2; 1; 3g are equivalent, but f1; 1; 2; 3g and f1; 2; 3g differ. See also LIST,M ULTICHOOSE ,M ULTINOMIAL COEFFI- CIENT ,SET References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 12, 1990. Multistable A structure such as a polyhedron which can change form from one stable configuration to another with only a slight transient nondestructive elastic stretch (Goldberg 1978). The simplest example of a polyhe-dron having multistable forms is Wunderlich’s bis- table JUMPING OCTAHEDRON (Cromwell 1991, pp. 222 /C1/23). Goldberg (1978) give two tristable polyhedra: one having 12 faces and one having 20. Goldberg’sbistable icosahedron, illustrated above, consists of two adjoined PENTAGONAL DIPYRAMIDS , each with two adjacent triangles (one on top and one on bottom) omitted (Goldberg 1978; Wells 1991; Cromwell 1997, pp. 222 and 224). The variables in the schematic above are connected by the equations sin u /C301 2r x2 /C301 /C28r2 y /C30r sin(5u) /C30r(5 sin u /C2820 sin3 u /C2715 sin5 u) /C30r sin u(5 /C2820 sin2 u /C2716 sin4 u) /C301 25 /C285 r2 /C271 r4 ! : Plugging in r2 /C301 /C28x2 and setting y /C30x gives the QUINTIC EQUATION 2x5 /C284x2 /C284x3 /C275x2 /C272x /C281 /C300; which has smallest positive solution x :0 :327267 : Goldberg gives (x; y) /C30(0:071; 0:49) and (0:49; 0:071) as other solutions, although it’s not clear where these come from. See also JUMPING OCTAHEDRON References Efimow, N. W. "Flachenverbiegung im Grossen." Berlin: Akademie-Verlag, p. 130, 1957. Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165 /C1/70, 1978. Wunderlich, W. "Starre, kippende, wackelige und bewe- gliche Achtflache." Elem. Math. 20,25/C1/2, 1965. Multivalued Function A FUNCTION which assumes two or more distinct values at one or more points in its DOMAIN . See also BRANCH CUT,BRANCH POINT References Morse, P. M. and Feshbach, H. "Multivalued Functions." §4.4 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 398 /C1/08, 1953. Multivariate Analysis The study of random distributions involving more than one variable. See also GAUSSIAN JOINT VARIABLE THEOREM ,MULTI- PLE REGRESSION ,MULTIVARIATE FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 927 /C1/28, 1972. Feinstein, A. R. Multivariable Analysis. New Haven, CT: Yale University Press, 1996. Hair, J. F. Jr. Multivariate Data Analysis with Readings, 4th ed. Englewood Cliffs, NJ: Prentice-Hall, 1995. Schafer, J. L. Analysis of Incomplete Multivariate Data. Boca Raton, FL: CRC Press, 1997.Sharma, S. Applied Multivariate Techniques. New York: Wiley, 1996. Multivariate Distribution GAUSSIAN MULTIVARIATE DISTRIBUTION Multivariate Function A FUNCTION of more than one variable. See also MULTIVARIATE ANALYSIS ,UNIVARIATE FUNC- TION Multivariate Polynomial A POLYNOMIAL in more than one variable, e.g., P(x; y) /C30a22x2y2 /C27a21x2y /C27a12xy2 /C27a11xy /C27a10x /C27a01y /C27a00 : See also POLYNOMIAL ,UNIVARIATE POLYNOMIAL Multivariate Theorem GAUSSIAN JOINT VARIABLE THEOREM Mu Molecule MANDELBROT SET Mu¨ntz Space AMu ¨ntz space is a technically defined SPACE M( L) /C13span fx l0 ; x l1 ; ...g which arises in the study of function approximations. Mu¨ntz’s Theorem Mu¨ntz’s theorem is a generalization of the WEIER- STRASS APPROXIMATION THEOREM , which states that any continuous function on a closed and bounded interval can be uniformly approximated by POLYNO- MIALS involving constants and any INFINITE SE- QUENCE of POWERS whose RECIPROCALS diverge. In technical language, Mu¨ntz’s theorem states that the M U¨NTZ SPACE M(L) is dense in C[0;1]IFF X/C12 i/C3011 li/C30/C12: See also WEIERSTRASS APPROXIMATION THEOREM References Borwein, P. and Erde ´lyi, T. "Mu ¨ntz’s Theorem." §4.2 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 171 /C1/05, 1995. Mutant Knot Given an original KNOT K, the knots produced by MUTATIONS together with K itself are called mutant knots. Mutant knots are often difficult to distinguish. For instance, mutants have the same HOMFLY POLYNOMIALS and HYPERBOLIC KNOT volume. Many but not all mutants also have the same GENUS (KNOT ). See also KNOT,MUTATION Mutation Consider a KNOT as being formed from two TANGLES . The following three operations are called mutations. 1. Cut the knot open along four points on each of the four strings coming out of T2 ; flipping T2 over, and gluing the strings back together. 2. Cut the knot open along four points on each of the four strings coming out of T2 ; flipping T2 to the right, and gluing the strings back together. 3. Cut the knot, rotate it by 180 8, and reglue. This is equivalent to performing (1), then (2). Mutations applied to an alternating KNOT projection always yield an ALTERNATING KNOT . The mutation of a KNOT is always another KNOT (a opposed to a LINK ). See also KNOT,MUTANT KNOT,TANGLE References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 49, 1994. Mutual Energy Let V be a SPACE with MEASURE m ]0; and let F(P; Q) be a real function on the PRODUCT SPACE V/C29V: When ( m; n) /C30ggF(P; Q) d m(Q) dn(P) /C30gF(P ; m) dn(P) exists for measures m; n ]0 ; ( m; n) is called the mutual energy. ( m; m) is then called the ENERGY . See also ENERGY References Iyanaga, S. and Kawada, Y. (Eds.). "General Potential." §335.B in Encyclopedic Dictionary of Mathematics. Cam- bridge, MA: MIT Press, p. 1038, 1980. Mutual Information This entry contributed by ERIK G. MILLERThe mutual information between two discrete RAN- DOM VARIABLES X and Y is defined to be I(X; Y) /C30X x /C23 xX y /C23Yp(x; y)lnp(x; y) p(x)p(y) ! : (1) bits. Additional properties are I(X; Y) /C30I(Y; X); (2) I(X; Y) ]0; (3) and I(X; Y) /C30H(X) /C27H(Y) /C28H(X ; Y); (4) where H(X) is the ENTROPY of the RANDOM VARIABLE X and H(X ; Y) is the joint entropy of these variables. See also ENTROPY References Cover, T. M. and Thomas, J. A. Elements of Information Theory. New York: Wiley, pp. 18 /C1/6, 1991. Mutually Exclusive Events n events are said to be mutually exclusive if the occurrence of any one of them precludes any of the others. Therefore, for events X1 ; ..., Xn ; the CONDI- TIONAL PROBABILITY is P(Xi ½Xj) /C300 for all j "i :/ Mutually Exclusive Sets DISJOINT SETS Mutually Singular Let M be a SIGMA ALGEBRA M, and let l1and l2be MEASURES on M. If there EXISTS a pair of disjoint SETS A and B such that l1 is CONCENTRATED on A and l2 is CONCENTRATED on B, then l1and l2are said to be mutually singular, written l1/C222l2:/ See also ABSOLUTELY CONTINUOUS ,CONCENTRATED , SIGMA ALGEBRA References Rudin, W. Functional Analysis, 2nd ed. New York: McGraw- Hill, p. 121, 1991. Myriad The Greek word for 10,000. Myriagon A 10,000-sided POLYGON . Mystic Pentagram PENTAGRAM N N The SET of NATURAL NUMBERS (the POSITIVE INTEGERS Z/C27 1, 2, 3, ...; Sloane’s A000027), denoted N; also called the WHOLE NUMBERS . Like whole numbers, there is no general agreement on whether 0 should be included in the list of natural numbers. Due to lack of standard terminology, the following terms are recommended in preference to "COUNTING NUMBER ," "natural number," and "WHOLE NUMBER ." set name symbol ..., /C282, /C281, 0, 1, 2, ...INTEGERS Z 1, 2, 3, 4, ... POSITIVE INTEGERS Z/C27 0, 1, 2, 3, 4, ... NONNEGATIVE INTE- GERSZ* 0, /C281, /C282, /C283, /C284, ...NONPOSITIVE INTE- GERS /C281, /C282, /C283, /C284, ...NEGATIVE INTEGERS Z/C28 See also C, CARDINAL NUMBER ,COUNTING NUMBER ,I, INTEGER ,Q,R,W HOLE NUMBER ,Z,Z /C27 References Sloane, N. J. A. Sequences A000027/M0472 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Nabla DEL,LAPLACIAN Nagel Line The Nagel line is the term proposed for the first time in this work for the line on which the INCENTER I, CENTROID G,SPIEKER CENTER Sp, and NAGEL POINTNa lie. The points satisfy ISp /C30SpNa IG /C301 2GNa : See also CENTROID (TRIANGLE ), INCENTER ,N AGEL POINT ,SPIEKER CENTER References Honsberger, R. "The Nagel Point Mand the Spieker Circle." §1.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 5 /C1/13, 1995. Nagel Point LetT1be the point at which the J1/-EXCIRCLE meets the side A2A3of a TRIANGLE DA1A2A3;and define T2 and T3similarly. Then the lines T1;T2;and T3 CONCUR in the N AGEL POINT Na(sometimes denoted M) The points T1;T2;andT3can also be constructed as the points which bisect the PERIMETER ofDA1A2A3 starting at A1;A2;andA3:Then the lines A1T1;A2T2; andA3T3(sometimes called SPLITTERS ) concur in the Nagel point Na. For this reason, the Nagel point is sometimes known as the BISECTED PERIMETER POINT (Bennett et al. 1988, Chen et al. 1992, Kimberling 1994), although the CLEAVANCE CENTER is also a bisected perimeter point. The Nagel point has TRIANGLE CENTER FUNCTION a/C30b/C27c/C28a a: The Nagel point lies on the N AGEL LINE . The ORTHOCENTER and Nagel point form a DIAMETER of the F UHRMANN CIRCLE . The Nagel point Na is also the ISOTOMIC CONJUGATE POINT of the GERGONNE POINT Ge. See also CLEAVANCE CENTER ,EXCENTER ,EXCENTRAL TRIANGLE ,EXCIRCLE ,FUHRMANN CIRCLE ,GERGONNE POINT ,M ITTENPUNKT ,N AGEL LINE,SPLITTER ,TRI- SECTED PERIMETER POINT References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed. New York: Barnes and Noble, pp. 160 /C1/164, 1952. Bennett, G.; Glenn, J.; Kimberling, C.; and Cohen, J. M. "Problem E 3155 and Solution." Amer. Math. Monthly 95, 874, 1988. Chen, J.; Lo, C.-H.; and Lossers, O. P. "Problem E 3397 and Solution." Amer. Math. Monthly 99,70/C1/71, 1992. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 53, 1971. Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, p. 83, 1972. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 20, 1913. Honsberger, R. "The Nagel Point M and the Spieker Circle." §1.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 5 /C1/13, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 184 and 225 /C1/226, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Kimberling, C. "Nagel Point." http://cedar.evansville.edu/ ~ck6/tcenters/class/nagel.html. Nagel, C. H. Untersuchungen u¨ber die wichtigsten zum Dreiecke geho¨hrigen Kreise. Eine Abhandlung aus dem Gebiete der reinen Geometrie. Leipzig, Germany, 1836. Nahm’s Equation The system of PARTIAL DIFFERENTIAL EQUATIONS Ut /C30[V ;W] (1) Vt /C30[W ;U] (2) Wt /C30[U ;V]; (3) where [A, B] denotes the COMMUTATOR .References Steeb, W.-H. and Louw, J. A. "Nahm’s Equations, Singular Point Analysis, and Integrability." J. Math. Phys. 27, 2458 /C1/2460, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Naive Set Theory A branch of mathematics which attempts to formalize the nature of the SET using a minimal collection of independent axioms. Unfortunately, as discovered by its earliest proponents, naive set theory quickly runs into a number of PARADOXES (such as R USSELL’S PARADOX ), so a less sweeping and more formal theory known as AXIOMATIC SET THEORY must be used. See also AXIOMATIC SET THEORY ,RUSSELL’S PARA- DOX,SET THEORY NAND ACONNECTIVE inLOGIC equivalent to the composition NOT AND that yields TRUE if any condition is TRUE , and FALSE if all conditions are TRUE .ANAND Bis equivalent to ! AfflB ðÞ ;where ! Adenotes NOT and ffl denotes AND. In PROPOSITIONAL CALCULUS , the term ALTERNATIVE DENIAL is used to refer to the NAND connective. Notations for NAND include AfflBandAjB (Mendelson 1997, p. 26). The NAND operation is implemented in Mathematica 4.1 asNand [A,B, ...]. The circuit diagram symbol for an NAND gate isillustrated above.The BINARY NAND operator has the following TRUTH TABLE (Mendelson 1997, p. 27). AB /AfflB/ TTF TFTFTT FFT The NAND operation is the basic logical operation performed by the solid-state transistors ("NANDgates") that underlie virtually all integrated circuitsand modern computers. The first axiom system based on NAND was given by Henry Sheffer in 1913. In their landmark tome, Whitehead and Russell (1927)promoted NAND as the appropriate foundation for axiomatic logic. The AND function A fflB can be written in terms of NANDs as A fflB /C30 AfflBðÞffl AfflBðÞ : See also AND, BINARY OPERATOR ,C ONNECTIVE , INTER SECTION , NOR, NOT, OR, TRUTH TABLE , XNOR, XOR References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997. Simpson, R. E. "The NAND Gate." §12.5.5 in Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 548 /C1/550, 1987. Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, 1927. Napierian Logarithm Write a number N as N /C30107 1 /C2810/C287CC0CC1L; then L is the Napierian logarithm of N. This was the original definition of a LOGARITHM , and can be given in terms of the modern LOGARITHM as LNðÞ/C30/C28logn 107CC1:CC17 log107 107 /C281CC1:CC17 : The Napierian logarithm decreases with increasing numbers and does not satisfy many of the funda- mental properties of the modern LOGARITHM , e.g., N log(xy) "N logx /C27N logy: Napier’s Analogies Let a SPHERICAL TRIANGLE have sides a, b, and c with A, B, and C the corresponding opposite angles. Then sin1 2A /C28 B ðÞhi sin1 2A /C27 B ðÞhi /C30tan1 2a /C28 b ðÞhi tan1 2cCC1:CC17 (1)cos12A /C28 B ðÞhi cos1 2A /C27 B ðÞhi /C30tan1 2a /C27 b ðÞhi tan1 2cCC1:CC17 (2) sin1 2a /C28 b ðÞhi sin1 2a /C27 b ðÞhi /C30tan1 2A /C28 B ðÞhi cot1 2CCC1:CC17 (3) cos1 2a /C28 b ðÞhi cos1 2a /C27 b ðÞhi /C30tan1 2A /C27 B ðÞhi cot1 2CCC1:CC17 (4) (Smart 1960, p. 23). See also SPHERICAL TRIANGLE ,SPHERICAL TRIGONO- METRY References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 147 /C1/150, 1987. Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 109 /C1/110, 1998. Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, 1960. Zwillinger, D. (Ed.). "Spherical Geometry and Trigonome- try." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 468 /C1/471, 1995. Napier’s Bones Numbered rods which can be used to perform MULTI- PLICATION . This process is also called RABDOLOGY . See also GENAILLE RODS References Gardner, M. "Napier’s Bones." Ch. 7 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 85 /C1/93, 1986. Pappas, T. "Napier’s Bones." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 64 /C1/65, 1989. Napier’s Constant E Napier’s Inequality Forb>a>0; 1 bBlnb/C28lna b/C28aB1 a: References Nelsen, R. B. "Napier’s Inequality (Two Proofs)." College Math. J. 24, 165, 1993. Napier’s Rules NAPIER’S ANALOGIES Napkin Ring SPHERICAL RING Napoleon Points The inner Napoleon point N is the CONCURRENCE of lines drawn between VERTICES of a given TRIANGLE DABC and the opposite VERTICES of the corresponding inner NAPOLEON TRIANGLE DNABNACNBC : The TRIAN- GLE CENTER FUNCTION of the inner Napoleon point is a/C30csc A /C281 6 pCC1:CC17 : The outer Napoleon point N ? is the CONCURRENCE of lines drawn between VERTICES of a given TRIANGLE DABC and the opposite VERTICES of the corresponding outer NAPOLEON TRIANGLE DN ?ABN ?ACN ?BC : The TRIAN- GLE CENTER FUNCTION of the point is a/C30csc A /C271 6 pCC1:CC17 : See also FERMAT POINTS ,N APOLEON’S THEOREM , NAPOLEON TRIANGLES References Casey, J. Analytic Geometry, 2nd ed. Dublin: Hodges, Figgis, & Co., pp. 442 /C1/444, 1893. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994.Napoleon Triangles The inner Napoleon triangle is the TRIANGLE DNABNACNBCformed by the centers of internally erected EQUILATERAL TRIANGLES DABEAB ;DACEAC ; and DBCEBC on the sides of a given TRIANGLE DABC : It is an EQUILATERAL TRIANGLE . The outer Napoleon triangle is the TRIANGLE DN ?ABN ?ACN ?BCformed by the centers of externally erected EQUILATERAL TRIANGLES DABE ?AB ;DACE?AC ; and DBCE ?BC on the sides of a given TRIANGLE DABC : It is also an EQUILATERAL TRIANGLE . See also EQUILATERAL TRIANGLE ,NAPOLEON POINTS , NAPOLEON’S THEOREM References Belenkiy, I. "New Features of Napoleon’s Triangles." J. Geom. 66,17/C1/26, 1999. Coxeter, H. S. M. and Greitzer, S. L. "Napoleon Triangles." §3.3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 60 /C1/65, 1967. Rigby, J. F. "Napoleon Revisited." J. Geom. 33, 129 /C1/146, 1988. Yaglom, I. M. Geometric Transformations I. New York: Random House, pp. 38 and 93, 1962. Napoleon’s Problem Given the center of a CIRCLE , divide the CIRCLE into four equal arcs using a COMPASS alone (a M ASCHER- ONI CONSTRUCTION ). See also CIRCLE ,COMPASS ,M ASCHERONI CONSTRUC- TION References Mascheroni, L. Geometria del compasso. 1797. Quemper de Lanascol, A. Ge´ome´trie du compas. Blanchard, pp. 74 /C1/77, 1925. Schwerin. Mascheronische Konstruktionen. 1898. Napoleon’s Theorem If EQUILATERAL TRIANGLES are erected externally on the sides of any TRIANGLE , then the centers form an EQUILATERAL TRIANGLE (the outer NAPOLEON TRIAN- GLE). Furthermore, the inner NAPOLEON TRIANGLE is also EQUILATERAL , and the difference between the areas of the outer and inner Napoleon triangles equals the AREA of the original TRIANGLE (Wells 1991, p. 156). Drawing the centers of one EQUILATERAL TRIANGLE inwards and two outwards gives a 308-30 8-1208 TRIANGLE (Wells 1991, p. 156). Napoleon’s theorem has a very beautiful general- ization in the case of externally constructed triangles: If SIMILAR triangles of any shape are constructed externally on a triangle such that each is rotated relative to its neighbors and any three corresponding points of these triangles are connected, the result is a triangle which is SIMILAR to the external triangles (Wells 1991, pp. 156 /C1/157). See also EQUILATERAL TRIANGLE ,FERMAT POINTS , NAPOLEON POINTS ,NAPOLEON TRIANGLES ,SIMILAR References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 60 /C1/65, 1967. Pappas, T. "Napoleon’s Theorem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 57, 1989.Schmidt, F. "200 Jahre franzo ¨sische Revolution--Problem und Satz von Napoleon." Didaktik der Mathematik 19, 15 /C1/29, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 74 /C1/75 and 156 /C1/158, 1991. Wentzel, J. E. "Converses of Napoleon’s Theorem." Amer. Math. Monthly 99, 339 /C1/351, 1992. Nappe One of the two pieces of a DOUBLE CONE (i.e., two CONES placed apex to apex). See also BICONE ,CONE,DOUBLE CONE Narain G-Transform The INTEGRAL TRANSFORM defined by (Kf)(x)/C30g/C12 /C28/C12Gmn pqxtjapCC0CC1 bqCC0CC1 ! ftðÞdt; where Gmn pqis M EIJER’S G-FUNCTION . References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993. Narayana Polynomial References Sulanke, R. A. "Counting Lattice Paths by Narayana Poly- nomials." Electronic J. Combinatorics 7, No. 1, R40, 1 /C1/9, 2000. http://www.combinatorics.org/Volume_7/ v7i1toc.html. Narcissistic Number Ann-DIGIT number which is the SUM of the nth POWERS of its DIGITS is called an n-narcissistic number, or sometimes an A RMSTRONG NUMBER or PERFECT DIGITAL INVARIANT (Madachy 1979). The smallest example other than the trivial 1- DIGIT numbers is 153/C3013/C2753/C2733: (1) The series of smallest narcissistic numbers of ndigits are 0, (none), 153, 1634, 54748, 548834, ... (Sloane’s A014576). Hardy (1993) wrote, "There are just four numbers, after unity, which are the sums of the cubes of their digits: 153 /C3013 /C2753 /C2733 ; 370 /C3033 /C2773 /C2703 ; 371 /C3033 /C2773 /C2713 ; and 407 /C3043 /C2703 /C2773 : These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician." The following table gives the generalization of these "unappealing" numbers to other POWERS (Madachy 1979, p. 164). nn -narcissistic numbers 1 0,1,2,3,4,5,6,7,8,9 2 none 3 153, 370, 371, 407 4 1634, 8208, 9474 5 54748, 92727, 93084 6 548834 7 1741725, 4210818, 9800817, 9926315 8 24678050, 24678051, 88593477 9 146511208, 472335975, 534494836, 912985153 10 4679307774 A total of 88 NARCISSISTIC NUMBERS exist in base 10, as proved by D. Winter in 1985 and verified by D. Hoey. These numbers exist for only 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33, 34, 35, 37, 38, and 39 digits. It can easily be shown that base-10 n-narcissistic numbers can exist only for n 560 ; since n /C2159n B10n/C281 (2) for n /C2160. The largest base-10 narcissistic number is the 39-narcissistic 115132219018763992565095597973971522401 : (3) A table of the largest known narcissistic numbers in various BASES is given by Pickover (1995). A tabula- tion of narcissistic numbers in various bases is given by (Corning). A closely related set of numbers generalize the narcissistic number to n-DIGIT numbers which are the sums of any single POWER of their DIGITS . For example, 4150 is a 4-DIGIT number which is the sum of fifth POWERS of its DIGITS . Since the number of digits is not equal to the power to which they are taken for such numbers, they are not narcissistic numbers. The smallest numbers which are sums of any single positive power of their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151,8208, 9474, ... (Sloane’s A023052), with powers 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 4, 5, 5, 4, 4, ... (Sloane’s A046074). The smallest numbers which are equal to the nth powers of their digits for n /C303, 4, ..., are 153, 1634, 4150, 548834, 1741725, ... (Sloane’s A003321). The n- digit numbers equal to the sum of nth powers of their digits (a finite sequence) are called ARMSTRONG NUMBERS or plus perfect number and are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (Sloane’s A005188). If the sum-of- kth-powers-of-digits operation applied iteratively to a number neventually returns to n, the smallest number in the sequence is called a k- RECURRING DIGITAL INVARIANT . The numbers that are equal to the sum of consecutive powers of their digits are given by 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798 (Sloane’s A032799), e.g., 2646798 /C3021/C2762/C2743/C2764/C2775/C2796/C2787: (4) See also ADDITIVE PERSISTENCE ,D IGITAL ROOT, DIGITADDITION ,HARSHAD NUMBER ,KAPREKAR NUM- BER,MULTIPLICATIVE DIGITAL ROOT,MULTIPLICATIVE PERSISTENCE ,POWERFUL NUMBER ,RECURRING DIGI- TAL INVARIANT ,VAMPIRE NUMBER References Hardy, G. H. A Mathematician’s Apology. New York: Cam- bridge University Press, p. 105, 1993. Heinz, H. "Narcissistic Numbers." http://www.geocities.com/ CapeCanaveral/Launchpad/4057/Narciss.htm. Keith, M. "Wild Narcissistic Numbers." http://member.aol.- com/s6sj7gt/mikewild.htm. Madachy, J. S. "Narcissistic Numbers." Madachy’s Mathe- matical Recreations. New York: Dover, pp. 163 /C1/173, 1979. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 169 /C1/ 170, 1995. Rivera, C. "Problems & Puzzles: Puzzle Narcissistic and Handsome Primes.-015." http://www.primepuzzles.net/ puzzles/puzz_015.htm. Rumney, M. "Digital Invariants." Recr. Math. Mag. No. 12, 6/C1/8, Dec. 1962. Sloane, N. J. A. Sequences A005188/M0488, A003321/ M5403, A014576, A023052, A032799, and A046074 in"An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Weisstein, E. W. "Narcissistic Numbers." M ATHEMATICA NOTEBOOK NARCISSISTIC.DAT . Narumi Polynomial Polynomials skx;aðÞ which form the S HEFFER SE- QUENCE for gtðÞ/C30et/C281 t !/C28a (1) ftðÞ/C30et /C281 (2) which have GENERATING FUNCTION X/C12 k /C300skxðÞ k!tk /C30t ln 1 /C27 t ðÞ"#a 1 /C27t ðÞx: (3) The first few are s0x;aðÞ/C301 s1x;aðÞ/C301 22x /C27a ðÞ s2x;aðÞ/C301 12 12x2 /C2712 a /C281 ðÞ x /C27a 3a /C285 ðÞ ½/C138 : References Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 37, 1964. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 258, 1981. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Nash Equilibrium A set of MIXED STRATEGIES for finite, noncooperative GAMES of two or more players in which no player can improve his payoff by unilaterally changing strategy. See also FIXED POINT ,G AME,M IXED STRATEGY , NASH’S THEOREM Nash’s Embedding Theorem Two real algebraic manifolds are equivalent IFF they are analytically homeomorphic (Nash 1952). See also EMBEDDING References Kowalczyk, A. "Whitney’s and Nash’s Embedding Theorems for Differential Spaces." Bull. Acad. Polon. Sci. Se´r. Sci. Math. 28, 385 /C1/390, 1981. Masahiro, S. Nash Manifolds. Berlin: Springer-Verlag, 1987. Nash, J. "Real Algebraic Manifolds." Ann. Math. 56, 405 /C1/ 421, 1952. Nash’s Theorem A theorem in GAME THEORY which guarantees the existence of a NASH EQUILIBRIUM for MIXED STRATE- GIES in finite, noncooperative GAMES of two or more players. See also MIXED STRATEGY ,NASH EQUILIBRIUM Nasik Square PANMAGIC SQUARE Nasty Knot An UNKNOT which can only be unknotted by first increasing the number of crossings.Natural Boundary This entry contributed by JONATHAN DEANE Consider a POWER SERIES in a complex variable z gzðÞ/C30X/C12 n/C300anzn (1) that is convergent within the OPEN DISK C : zjjBR: Convergence is limited to within C by the presence of at least one SINGULARITY on the BOUNDARY @C of C: If the singularities on C are so densely packed that ANALYTIC CONTINUATION cannot be carried out on a path that crosses C ; then C is said to form a natural boundary for the function g(z) :/ As an example, consider the function fzðÞ/C30X/C12 n/C300z2n /C30z /C27z2 /C27z4 /C27... (2) Then fzðÞformally satisfies the FUNCTIONAL EQUA- TION fzðÞ/C30z /C27fz2CC0CC1 : (3) The series (2) clearly converges within C1 : zjjB1: Now consider z /C301. Equation (3) tells us that f(1) /C301 /C27 f(1) which can only be satisfied if f(1) /C30/C12: Consider- ing now z /C30/C28 1, equation (3) becomes f(/C281) /C30/C281 /C27/C12 and hence f(/C281) /C30/C12: Substituting z2 for z in equation (3) then gives fz2CC0CC1 /C30z2 /C27fz4CC0CC1 /C30f(z) /C28z : (4) from which it follows that fzðÞ/C30z /C27z2 /C27fz4CC0CC1 : (5) Now consider z equal to any of the fourth roots of unity, 9 1, 9i; for example z /C30/C28i : Then f(/C28i) /C30/C28i /C28 1 /C27f(1) /C30/C12: Applying this procedure recursively shows that fzðÞis infinite for any z such that z2n /C301 with n /C300, 1, 2, .... In any arc of the circle @C1 of finite length there will therefore be an infinite number of points for which fzðÞis infinite and so C1 constitutes a natural boundary for fzðÞ:/ A function that has a natural boundary is said to be a LACUNARY FUNCTION . See also BOUNDARY ,LACUNARY FUNCTION References Ash, R. B. Ch. 3 in Complex Variables. New York: Academic Press, 1971. Natural Density NATURAL INVARIANT Natural Equation A natural equation is an equation which specifies a curve independent of any choice of coordinates or parameterization. The study of natural equations began with the following problem: given two func- tions of one parameter, find the SPACE CURVE for which the functions are the CURVATURE and TORSION . Euler gave an integral solution for plane curves (which always have TORSION t /C300): Call the ANGLE between the TANGENT line to the curve and the X-AXIS f the TANGENTIAL ANGLE , then f /C30g k sðÞds ; (1) where k is the CURVATURE . Then the equations k /C30 k(s) (2) t /C300 ; (3) where t is the TORSION , are solved by the curve with PARAMETRIC EQUATIONS x /C30g cosf ds (4) y /C30g sinf ds : (5) The equations k /C30 k(s) and t /C30 t(s) are called the natural (or INTRINSIC ) equations of the space curve. An equation expressing a plane curve in terms of s and RADIUS OF CURVATURE R (or k) is called a CESA` RO EQUATION , and an equation expressing a plane curve in terms of s and f is called a WHEWELL EQUATION . Among the special planar cases which can be solved in terms of elementary functions are the CIRCLE , LOGARITHMIC SPIRAL , CIRCLE INVOLUTE , and EPICY- CLOID . Enneper showed that each of these is the projection of a HELIX on a CONIC surface of revolution along the axis of symmetry. The above cases corre- spond to the CYLINDER , CONE , PARABOLOID , and SPHERE . See also CESA` RO EQUATION ,INTRINSIC EQUATION , WHEWELL EQUATION References Cesa`ro, E. Lezioni di Geometria Intrinseca. Napoli, Italy, 1896. Euler, L. Comment. Acad. Petropolit. 8,66/C1/85, 1736. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 138 /C1/139, 1997. Melzak, Z. A. Companion to Concrete Mathematics, Vol. 2. New York: Wiley, 1976. Struik, D. J. Lectures on Classical Differential Geometry. New York: Dover, pp. 26 /C1/28, 1988. Natural Independence Phenomenon A type of mathematical result which is considered by most logicians as more natural than the METAMATHE- MATICAL incompleteness results first discovered by Go¨del. Finite combinatorial examples include GOOD-STEIN’S THEOREM , a finite form of RAMSEY’S THEOREM , and a finite form of KRUSKAL’S TREE THEOREM (Kirby and Paris 1982; Smorynski 1980, 1982, 1983; Gallier 1991). See also GO¨ DEL’S INCOMPLETENESS THEOREM ,GOOD- STEIN’S THEOREM ,KRUSKAL’S TREE THEOREM ,RAM- SEY’S THEOREM References Gallier, J. "What’s so Special about Kruskal’s Theorem and the Ordinal Gamma[0]? A Survey of Some Results in Proof Theory." Ann. Pure and Appl. Logic 53, 199/C1/260, 1991. Kirby, L. and Paris, J. "Accessible Independence Results for Peano Arithmetic." Bull. London Math. Soc. 14, 285/C1/293, 1982. Smorynski, C. "Some Rapidly Growing Functions." Math. Intell. 2, 149/C1/154, 1980. Smorynski, C. "The Varieties of Arboreal Experience." Math. Intell. 4, 182/C1/188, 1982. Smorynski, C. "‘Big’ News from Archimedes to Friedman." Not. Amer. Math. Soc. 30, 251/C1/256, 1983. Natural Invariant Letr(x)dxbe the fraction of time a typical dynamical ORBIT spends in the interval x;x/C27dx ½/C138 ;and let r(x)b e normalized such that g/C12 0rxðÞdx/C301 over the entire interval of the map. Then the fraction the time an ORBIT spends in a finite interval [ a, b], is given by gb arxðÞdx: The natural invariant is also called the INVARIANT DENSITY orNATURAL DENSITY . Natural Logarithm The LOGARITHM having base E, where e /C302:718281828... ; (1) which can be defined ln x /C13gx 1dt t (2) for x /C210. The natural logarithm can also be defined by ln x /C30lim x 0/C12x1 =n /C281CC0CC1 n: (3) The symbol ln x is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation log x: In this work, ln x /C30 loge x denotes a natural logarithm, whereas log x /C30 log10 x denotes the COMMON LOGARITHM . Common and natural logarithms can be expressed in terms of each other as ln x /C30log10 x log10 e (4) log10 x /C30ln x ln 10 : (5) The natural logarithm is especially useful in CALCU- LUS because its DERIVATIVE is given by the simple equation d dx ln x /C301 x ; (6) whereas logarithms in other bases have the more complicated DERIVATIVE d dxlogb x /C301 x ln b : (7) The natural logarithm can be analytically continued to COMPLEX NUMBERS as ln z /C13ln zjj/C27i arg(z) ; (8) where zjjis the MODULUS and arg(z) is the ARGUMENTThe MERCATOR SERIES ln 1 /C27x ðÞ /C30x /C281 2x2 /C2713x3 /C28... (9) gives a TAYLOR SERIES for the natural logarithm. CONTINUED FRACTION representations of logarithmic functions include ln 1 /C27x ðÞ /C30x 1 /C2712x 2 /C2712x 3 /C2722x 4 /C2722x 5 /C2732x 6 /C2732x 7 /C27 ...(10) ln1 /C27 x 1 /C28 x ! /C302x 1 /C28x2 3 /C284x2 5 /C289x2 7 /C2816x2 9 /C28 ... (11) For a COMPLEX NUMBER z, the natural logarithm satisfies ln z /C30ln rei u/C272np ðÞCC6CC7 /C30ln r /C27i u /C272np ðÞ (12) PV ln zðÞ/C30ln r /C27iu ; (13) where PV is the PRINCIPAL VALUE . Some special values of the natural logarithm are ln 1 /C300 (14) ln 0 /C30/C28/C12 (15) ln /C281ðÞ/C30pi (16) ln 9iðÞ/C30912 pi : (17) An identity for the natural logarithm of 2 discovered using the PSLQ ALGORITHM is ln 2ðÞ2¼2X/C12 i/C301pi 2ii2pifg/C302;/C2810;/C287;/C2810;2;/C281CCnCCo ;(18) where pifg is given by the periodic sequence obtained by appending copies of 2 ;/C2810;/C287;/C2810;2;/C281 fg (in other words, pi/C13pi/C281ðÞ mod 6ðÞ ½/C138 /C271fori/C216) (Bailey et al.1995, Bailey and Plouffe). See also COMMON LOGARITHM , E,LG,LOGARITHM References Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps. Bailey, D. and Plouffe, S. "Recognizing Numerical Con- stants." http://www.cecm.sfu.ca/organics/papers/bailey/. Gourdon, X. and Sebah, P. "The Constant ln2:/" http:// xavier.gourdon.free.fr/Constants/Log2/log2.html. Natural Measure /mieðÞ; sometimes denoted PieðÞ; is the probability that element i is populated, normalized such that XN i /C301mieðÞ/C301 : See also INFORMATION DIMENSION , Q-DIMENSION Natural Norm Let zkkbe a VECTOR NORM of a VECTOR z such that Ajjjj/C30max zjjjj/C301Azjjjj : Then Akk is a MATRIX NORM which is said to be the natural norm INDUCED (or SUBORDINATE ) to the VECTOR NORM zkk: For any natural norm, Ikk/C301 ; where I is the IDENTITY MATRIX . The natural matrix norms induced by the L1-NORM , L2-NORM , and L- INFINITY-NORM are called the MAXIMUM ABSOLUTE COLUMN SUM NORM , SPECTRAL NORM , and MAXIMUM ABSOLUTE ROW SUM NORM , respectively. See also L1-NORM, L2-NORM,M ATRIX NORM,M AX- IMUM ABSOLUTE COLUMN SUM NORM,S PECTRAL NORM,VECTOR NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1115, 2000. Natural Number A POSITIVE INTEGER 1, 2, 3, ... (Sloane’s A000027). The set of natural numbers is denoted N or Z/C27. Unfortu- nately, 0 is sometimes also included in the list of "natural" numbers (Bourbaki 1968, Halmos 1974), and there seems to be no general agreement about whether to include it. In fact, Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 /C23 P:/" Due to lack of standard terminology, the following terms are recommended in preference to "COUNTING NUMBER ," "natural number," and "WHOLE NUMBER ." set name symbol ..., -2, -1, 0, 1, 2, ...INTEGERS Z1, 2, 3, 4, ... POSITIVE INTEGERS Z/C27 0, 1, 2, 3, 4, ... NONNEGATIVE INTE- GERSZ* 0, -1, -2, -3, -4, ... NONPOSITIVE INTE- GERS -1, -2, -3, -4, ... NEGATIVE INTEGERS Z- See also COUNTING NUMBER ,INTEGER ,N,P OSITIVE , Z, Z-,Z/C27,Z* References Bourbaki, N. Elements of Mathematics: Theory of Sets. Paris, France: Hermann, 1968. Courant, R. and Robbins, H. "The Natural Numbers." Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 1 /C1/20, 1996. Halmos, P. R. Naive Set Theory. New York: Springer- Verlag, 1974. Ribenboim, P. "Catalan’s Conjecture." Amer. Math. Monthly 103, 529 /C1/538, 1996. Sloane, N. J. A. Sequences A000027/M0472 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Welbourne, E. "The Natural Numbers." http://www.chao- s.org.uk/~eddy/math/found/natural.html. Natural Perspective PERSPECTIVE Naught The British word for "ZERO ." It is often used to indicate 0 subscripts, so a0would be spoken as "a naught." See also ZERO Navier’s Equation The general equation of fluid flow l /C272m ðÞ 99 /C215uðÞ/C28 m9/C299/C29u ðÞ /C30 r@2u @t2 ; where m and l are coefficients of viscosity, u is the velocity of the fluid parcel, and r is the fluid density. See also NAVIER- STOKES EQUATION References Eringen, A. C. and Suhubi, E. S. Ch. 5 in Elastodynamics, Vol. 2. New York: Academic Press, 1975. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Navier-Stokes Equation The equation of incompressible fluid flow, @u @t/C27u /C2159u /C309P r/C27 n 92u ; where n is the kinematic viscosity, u is the velocity of the fluid parcel, P is the pressure, and r is the fluid density. See also NAVIER’S EQUATION References Landau, L. D. and Lifschitz, E. M. Fluid Mechanics, 2nd ed. Oxford, England: Pergamon Press, p. 15, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Navigation Problem A problem in the CALCULUS OF VARIATIONS . Let a vessel traveling at constant speed c navigate on a body of water having surface velocity u ¼ uðx;yÞ v /C30v(x;y) : The navigation problem asks for the course which travels between two points in minimal time. References Sagan, H. Introduction to the Calculus of Variations. New York: Dover, pp. 226 /C1/228, 1992. nc JACOBI ELLIPTIC FUNCTIONS N-Cluster A LATTICE POINT configuration with no three points COLLINEAR and no four CONCYCLIC . An example is the 6-cluster (0, 0), (132, /C28720), (546, /C28272), (960, /C28720), (1155, 540), (546, 1120). Call the RADIUS of the smallest CIRCLE centered at one of the points of an N-cluster which contains all the points in the N- cluster the EXTENT . Noll and Bell (1989) found 91 nonequivalent prime 6-clusters of EXTENT less than 20937 ; but found no 7-clusters. References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 187, 1994. Noll, L. C. and Bell, D. I. "n-clusters for 1 Bn B7:/" Math. Comput. 53, 439 /C1/444, 1989. n-Cube HYPERCUBE ,POLYCUBEnd JACOBI ELLIPTIC FUNCTIONS Near Noble Number A REAL NUMBER 0 B n B1 whose CONTINUED FRAC- TION is periodic, and the periodic sequence of terms is composed of a string of 1s followed by an INTEGER n /C211, n /C30[1 ;1;...;1;|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pn]: (1) This can be written in the form n/C30[1;1;...;1;|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pn;n/C281]; (2) which can be solved to give n/C301 2nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274nFp/C281/C27Fp/C282 n2Fps /C281 ! ; (3) where Fnis a F IBONACCI NUMBER . The special case n/C302 gives n/C30ffiffiffiffiffiffiffiffiffiffiffi Fp/C272 Fps /C281: (4) See also NOBLE NUMBER References Schroeder, M. R. Number Theory in Science and Commu- nication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity,2nd enl. ed., corr. printing. Berlin: Springer-Verlag, 1990. Schroeder, M. "Noble and Near Noble Numbers." In Frac- tals, Chaos, Power Laws: Minutes from an Infinite Para-dise. New York: W. H. Freeman, pp. 392 /C1 /394, 1991. Nearest Integer Function The nearest integer function nint(x) ofx, illu- strated above and also called nint or the round function, is defined such that [x] is the INTEGER closest to x. Since this definition is ambiguous for half-integers, the additional rule that half-integers are always rounded to even numbers is usually added in order to avoid statistical biasing. For example, [1:5] /C302; [2:5] /C302; [3:5] /C304; [4:4] /C304; etc. This con- vention is followed in the Cmath.h library function rint , as well as in Mathematica , where the nearest integer function is implemented asRound [x]. Although the notation /C26x /C29 is sometimes used to denote the nearest integer function (Hastad et al. 1989), this notation is rather cumbersome and is not recommended. Also note that while [x] is used to denote the nearest integer function in this work, [x]is also commonly used to denote the FLOOR FUNCTION xbc:/ The plots above illustrate x1 =n /C28[x1 =n] for small n. See also CEILING FUNCTION ,FLOOR FUNCTION ,NINT ZETA FUNCTION ,STAIRCASE FUNCTION References Hastad, J.; Just, B.; Lagarias, J. C.; and Schnorr, C. P. "Polynomial Time Algorithms for Finding Integer Rela- tions Among Real Numbers." SIAM J. Comput. 18, 859 /C1/ 881, 1988. Nearest Neighbor Problem The problem in COMPUTATIONAL GEOMETRY of identi- fying the point from a set of points which is nearest to a given point according to some measure of distance. The nearest neighborhood problem involves identify- ing the locus of points lying nearer to the query point than to any other point in the set. See also COMPUTATIONAL GEOMETRY References Martin, E. C. "Computational Geometry." http://www.math- source.com/cgi-bin/msitem22?0200 /C1/181. Smid, M. "Closest-Point Problems in Computational Geome- try." Ch. 20 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 877 /C1/935, 2000.Skiena, S. S. "Nearest Neighbor Search." §8.6.5 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 361 /C1/363, 1997. Near-Integer ALMOST INTEGER Nearly-Poised Let GENERALIZED HYPERGEOMETRIC FUNCTION pFqa1 ; a2 ;...; ap b1 ; b2 ;...; bq;zCC60CC61 (1) have p /C30q /C271: Then the generalized hypergeometric function is said to be nearly-poised of the first kind if b1 /C27 a2 /C30.../C30 bq /C27 aq /C271 : (2) (omitting the initial equality in the definition for WELL-POISED ), and nearly-poised of the second kind if 1 /C27 a1 /C30 b1 /C27 a2 /C30.../C30 bq /C281 /C27 aq : (3) See also GENERALIZED HYPERGEOMETRIC FUNCTION , K-BALANCED ,NEARLY- POISED ,SAALSCHU ¨ TZIAN References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, pp. 11 /C1/ 12, 1935. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 43, 1998. Whipple, F. J. W. "On Well-Poised Series, Generalized Hypergeometric Series Having Parameters in Pairs, Each Pair with the Same Sum." Proc. London Math. Soc. 24, 247 /C1/263, 1926. Near-Pencil An arrangement of n ]3 points such that n /C281of them are COLLINEAR . See also GENERAL POSITION ,ORDINARY LINE,PENCIL References Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/909, 1989. Kelly, L. M. and Moser, W. O. J. "On the Number of Ordinary Lines Determined by n Points." Canad. J. Math. 1, 210 /C1/219, 1958. Necessary A CONDITION which must hold for a result to be true, but which does not guarantee it to be true. If a CONDITION is both NECESSARY and SUFFICIENT , then the result is said to be true IFFthe CONDITION holds. See also SUFFICIENT References Jeffreys, H. and Jeffreys, B. S. "Necessary: Sufficient." §1.036 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 10/C1/11, 1988. Necker Cube An ILLUSION in which a 2-D drawing of an array of CUBES appears to simultaneously protrude from and intrude into the page. References Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 25 and 118, 1996. Jablan, S. "Impossible Figures." http://members.tripod.com/ ~modularity/impos.htm. Newbold, M. "Animated Necker Cube." http://dogfeathers.- com/java/necker.html. Necklace In the technical COMBINATORIAL sense, an a-ary necklace of length nis a string of ncharacters, each ofapossible types. Rotation is ignored, in the sense that b1b2...bnis equivalent to bkbk/C271...bnb1b2...bk-1 for any k. InFIXED necklaces, reversal of strings is respected, so they represent circular collections of beads in which the necklace may not be picked up out of the PLANE (i.e., opposite orientations are not considered equiva-lent). The number of fixed necklaces of length n composed of atypes of beads N(n;a) is given by N(n;a)/C301 nXn(n) i/C301f(di)an=di; (1)where diare the DIVISORS ofnwith d1/C131;d2;..., dn(n)/C13n;n(n) is the number of DIVISORS ofn, and f(x) is the TOTIENT FUNCTION . For FREE necklaces, opposite orientations ( MIRROR IMAGES ) are regarded as equivalent, so the necklace can be picked up out of the PLANE and flipped over. The number N?(n;a) of such necklaces composed of n beads, each of apossible colors, is given by N?(n;a)/C301 2n /C2Pn(n) i/C301f(di)an=di/C27na(n/C271)=2fornoddPn(n) i/C301f(di)an=di/C271 2n(1/C27a)an=2forneven :( Fora/C302 and n/C30panODD PRIME , this simplifies to N?(p;2)/C302p/C281/C281 p/C272(p/C281)=2/C271: A table of the first few numbers of necklaces for a/C302 and a/C303 follows. Note that N(n;2) is larger than N?(n;2) for n]6:Forn/C306, the necklace 110100 is inequivalent to its MIRROR IMAGE 0110100, account- ing for the difference of 1 between N(6;2) and N?(6;2): Similarly, the two necklaces 0010110 and 0101110 are inequivalent to their reversals, accounting for thedifference of 2 between N(7;2) and N?(7;2): / n /N(n;2)// N?(n;2)// N?(n;3)/ Sloane Sloane’s A000031Sloane’s A000029Sloane’s A027671 1223 2336344 1 0 466 2 1 588 3 961 41 39 27 20 18 198 8 36 30 498 9 60 46 1219 10 108 78 3210 11 188 126 8418 12 352 224 22913 13 632 380 62415 14 1182 687 173088 15 2192 1224 481598 Ball and Coxeter (1987) consider the problem of finding the number of distinct arrangements of n people in a ring such that no person has the same two neighbors two or more times. For 8 people, there are 21 such arrangements. See also ANTOINE’S NECKLACE , DE BRUIJN SEQUENCE , FIXED,FREE,IRREDUCIBLE POLYNOMIAL ,JOSEPHUS PROBLEM ,LYNDON WORD References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 49 /C1/50, 1987. Dudeney, H. E. Problem 275 in 536 Puzzles & Curious Problems. New York: Scribner, 1967. Gardner, M. Martin Gardner’s New Mathematical Diver- sions from Scientific American. New York: Simon and Schuster, pp. 240 /C1/246, 1966. Gilbert, E. N. and Riordan, J. "Symmetry Types of Periodic Sequences." Illinois J. Math. 5, 657 /C1/665, 1961. Riordan, J. "The Combinatorial Significance of a Theorem of Po´lya." J. SIAM 4, 232 /C1/234, 1957. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 162, 1980. Ruskey, F. "Information on Necklaces, Lyndon Words, de Bruijn Sequences." http://www.theory.csc.uvic.ca/~cos/inf/ neck/NecklaceInfo.html. Skiena, S. "Polya’s Theory of Counting." §1.2.6 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 25 /C1/26, 1990. Sloane, N. J. A. Sequences A000029/M0563, A000031/ M0564, A001869/M3860, and A027671 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Needle BUFFON- LAPLACE NEEDLE PROBLEM ,BUFFON’S NEE- DLE PROBLEM ,KAKEYA NEEDLE PROBLEM Negabinary The negabinary representation of a number n is given by the coefficients anan/C281 ...a1a0 in n /C30X i/C300ai(/C282)i /C30.../C27a2(/C282)2 /C27a1(/C282)1 /C27a0(/C282)0 ;where ai /C300;1 : Conversion of n to negabinary can be done using the Mathematica code Negabinary[n_Integer] : /C30 Module[{t /C30 (2/ 3)(4^Floor[Log[4, Abs[n] /C27 1] /C27 2] - 1)}, IntegerDigits[BitXor[n /C27 t, t], 2]] The following table gives the negabinary representa- tions for the first few integers (A039724). n negabinary n negabinary 1 1 11 11111 2 110 12 11100 3 111 13 11101 4 100 14 10010 5 101 15 10011 6 11010 16 10000 7 11011 17 10001 8 11000 18 10110 9 11001 19 10111 10 11110 20 10100 If these numbers are interpreted as binary numbers and converted to decimal, their values are 1, 6, 7, 4, 5, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, ... (Sloane’s A005351). The numbers having the same representa- tion in BINARY and negabinary are members of the MOSER-DE BRUIJN SEQUENCE , 0, 1, 4, 5, 16, 17, 20, 21, 64, 65, 68, 69, 80, 81, ... (Sloane’s A000695). See also BINARY ,M OSER-DE BRUIJN SEQUENCE , NEGADECIMAL References Gardner, M. Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, p. 101, 1986. Sloane, N. J. A. Sequences A000695/M3259, A005351/ M4059, and A039724 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Negadecimal The negadecimal representation of a number nis given by the coefficients anan/C281...a1a0in n/C30X i/C300ai(/C2810)i/C30...a2(/C2810)2/C27a1(/C2810)1/C27a0(/C2810)0; where ai/C300;1, ..., 9. The following table gives the negabinary representations for the first few integers (A039723). n negadecimal n negadecimal n negadecimal 1 1 11 191 21 181 2 2 12 192 22 182 3 3 13 193 23 183 4 4 14 194 24 184 5 5 15 195 25 185 6 6 16 196 26 186 7 7 17 197 27 187 8 8 18 198 28 188 9 9 19 199 29 189 10 190 20 180 30 170 The numbers having the same DECIMAL and negade- cimal representations are those which are sums of distinct powers of 100: 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, ... (Sloane’s A051022). See also DECIMAL ,NEGABINARY References Sloane, N. J. A. Sequences A039723 and A051022 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Negation The operation of interchanging true and false in a logical statement. The negation of A is often called "NOT- A," and can be denoted !A; or with the NEGA- TION SIGN /C15; so not-A is written /C15 A:/ Note that in computer languages such as C, perl , and Mathematica , not-A is denoted !A: In FORTRAN , not-A is written .not.A , where A is a variable of logical type. See also NEGATION SIGN, NOT Negation Sign The symbol /C15used to denote the NEGATION operation ("NOT") in symbolic logic, also called "logical not." See also NOT References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 281, 1997. Negative A quantity less than ZERO (/B0); denoted with a MINUS SIGN, i.e.,/C28x:/See also NONNEGATIVE ,N ONPOSITIVE ,N ONZERO , POSITIVE ,ZERO References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 20 /C1/ 21, 1986. Negative Binomial Distribution Also known as the P ASCAL DISTRIBUTION and P O´LYA DISTRIBUTION . The probability of r/C281 successes and x failures in x/C27r/C281 trials, and success on the ( x/C27r)/th trial is px/C27r/C281 r/C281CC1nCC1o pr/C281(1/C28p)[(x/C27r/C281)/C28(r/C281)]CC60CC61 /C30x/C27r/C281 r/C281CC1nCC1o pr/C281(1/C28p)xCC60CC61 p /C30x/C27r/C281 r/C281CC1nCC1o pr(1/C28p)x; (1) wheren kCC0CC1 is a BINOMIAL COEFFICIENT . Let P/C301/C28p p(2) Q/C301 p: (3) The CHARACTERISTIC FUNCTION is given by f(t)/C30Q/C28PeitCC0CC1 /C28r; (4) and the MOMENT-GENERATING FUNCTION by M(t)/C30etzhi/C30X/C12 x/C300etxx/C27r/C281 r/C281CC1nCC1o pr(1/C28p)x; (5) but, sinceN nCC0CC1 /C30N N/C28mCC0CC1 ; M(t)/C30prX/C12 x/C300x/C27r/C281 xCC1nCC1o 1/C28p ðÞ et½/C138x /C30pr1/C281/C28p ðÞ et½/C138/C28r(6) M?(t)/C30pr(/C28r)1/C281/C28p ðÞ et½/C138/C28r/C281p/C281 ðÞ et /C30pr(1/C28p)r1/C281/C28p ðÞ et½/C138/C28r/C281et(7) Mƒ(t)/C30(1/C28p)rpr(1/C28et/C27pet)/C28r/C282 /C2(/C281/C28etr/C27etpr)et(8) M§(t)/C30(1/C28p)rpr(1/C28et/C27etp)/C28r/C283 /C2[1/C27et(1/C28p/C273r/C283pr)/C27r2e2t(1/C28p)2]et:(9) The MOMENTS about zero K(u) are therefore m?1/C30m/C30r(1/C28p) p/C30rq p(10) m ?2 /C30r(1 /C28 p)[1 /C28 r(p /C28 1)] p2 /C30rq(1 /C28 rq) p2 (11) m?3 /C30(1 /C28 p)r(2 /C28 p /C27 3r /C28 3pr /C27 r2 /C28 2pr2 /C27 p2r2) p3 (12) m?4 /C30( /C281 /C27 p)r( /C286 /C27 6p /C28 p2 /C28 11r /C27 15pr /C28 4p2r /C28 6r2 p4 /C2712pr2 /C28 6p2r2 /C28 r3 /C27 3pr3 /C28 3p2r3 /C27 p3r3) p4 : (13) (Beyer 1987, p. 487, apparently gives the MEAN incorrectly.) The MOMENTS about the mean are m2 /C30 s2 /C30r(1 /C28 p) p2 (14) m3 /C30r 2 /C28 3p /C27 p2ðÞ p3 /C30rp/C28 1 ðÞ p /C28 2 ðÞ p3 (15) m4 /C30r(1 /C28 p)(6 /C28 6p /C27 p2 /C27 3r /C28 3pr) p4 : (16) The MEAN , VARIANCE , SKEWNESS and KURTOSIS are then m /C30r(1 /C28 p) p (17) g1 /C30m3 s3 /C30r(p /C28 1)(p /C28 2) p3p2 r(1 /C28 p)"#3 =2 /C30r(2 /C28 p)(1 /C28 p) p3p3 r(1 /C28 p)ffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28 pp /C302 /C28 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(1 /C28 p)p (18) g2 /C30m4 s4 /C283 /C30/C286 /C27 6p /C28 p2 /C28 3r /C27 3pr (p /C28 1)r; (19) which can also be written m /C30nP (20) m2 /C30nPQ (21) g1 /C30Q /C27 PffiffiffiffiffiffiffiffiffiffirPQp (22) g 2 /C301 /C27 6PQ rPQ/C283: (23) The first CUMULANT is k1 /C30nP ; (24)and subsequent CUMULANTS are given by the RECUR- RENCE RELATION kr/C271 /C30PQdkr dQ: (25) References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 533, 1987. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 118, 1992. Negative Binomial Series The SERIES which arises in the BINOMIAL THEOREM for NEGATIVE integer n, (x /C27a) /C28n /C30X/C12 k/C300/C28n kCC1nCC1o xka/C28n/C28k /C30X/C12 k /C300(/C281)k n /C27k /C281 kCC1nCC1o xka /C28n/C28k : For a /C301, the negative binomial series simplifies to (x /C271)/C28n /C301 /C28nx /C271 2n(n /C271)x2 /C2816n(n /C271)(n /C272) /C27...: See also BINOMIAL SERIES ,BINOMIAL THEOREM Negative Definite Matrix A negative definite matrix is a HERMITIAN MATRIX all of whose EIGENVALUES are negative. See also NEGATIVE SEMIDEFINITE MATRIX ,POSITIVE DEFINITE MATRIX ,POSITIVE SEMIDEFINITE MATRIX References Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 69, 1992. Negative Integer Z/C28 Negative Likelihood Ratio The term negative likelihood ratio is also used (especially in medicine) to test nonnested comple- mentary hypotheses as follows, NLR /C30[true negative rate] [false negative rate] /C30[specificity] 1/C28[sensitivity]: See also LIKELIHOOD RATIO,SENSITIVITY ,SPECIFICITY Negative Pedal Curve Given a curve C and O a fixed point called the PEDAL POINT , then for a point P on C, draw a LINE PERPENDICULAR to OP. The ENVELOPE of these LINES as P describes the curve C is the negative pedal of C. See also PEDAL CURVE References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 46 /C1/49, 1972. Lockwood, E. H. "Negative Pedals." Ch. 19 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 156 /C1/159, 1967. Negative Semidefinite Matrix A negative semidefinite matrix is a HERMITIAN MATRIX all of whose EIGENVALUES are nonpositive. See also NEGATIVE DEFINITE MATRIX ,P OSITIVE DEFINITE MATRIX ,POSITIVE SEMIDEFINITE MATRIX References Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 69, 1992. Neighborhood The word neighborhood is a word with many different levels of meaning in mathematics. One of the most general concepts of a neighborhood of a point x /C23Rn (also called an epsilon-neighborhood or infinitesimal OPEN SET) is the set of points inside an n-BALL with center x and RADIUS e > 0:/ See also BALL,OPEN SET Neile’s Parabola The solid curve in the above figure which is the EVOLUTE of the PARABOLA (dashed curve). In CARTE- SIAN COORDINATES , y /C303 4(2x)2 =3 /C2712/C215 Neile’s parabola is also called the SEMICUBICAL PARA- BOLA , and was discovered by William Neile in 1657. It was the first nontrivial ALGEBRAIC CURVE to have its ARC LENGTH computed. Wallis published the method in 1659, giving Neile the credit (MacTutor Archive). See also PARABOLA EVOLUTEReferences MacTutor History of Mathematics Archive. "Neile’s Semi- Cubical Parabola." http://www-groups.dcs.st-and.ac.uk/ ~history/Curves/Neiles.html. Nelder-Mead Method A direct search method of optimization that works moderately well for stochastic problems. It is based on evaluating a function at the vertices of a SIMPLEX , then iteratively shrinking the simplex as better points are found until some desired bound is obtained (Nelder and Mead 1965). See also STOCHASTIC OPTIMIZATION References Lagarias, J. C.; Reeds, J. A.; Wright, M. H.; and Wright, P. E. "Convergence Properties of the Nelder-Mead Algo- rithm in Low Dimensions." AT&T Bell Laboratories Tech.Rep. Murray Hill, NJ, 1995. Nelder, J. A. and Mead, R. "A Simplex Method for Function Minimization." Comput. J. 7, 308/C1 /313, 1965. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, 1989. Walters, F. H.; Parker, L. R. Jr.; Morgan, S. L.; and Deming, S. N. Sequential Simplex Optimization: A Technique for Improving Quality and Productivity in Research, Devel- opment, and Manufacturing. Boca Raton, FL: CRC Press, 1991. Woods, D. J. An Interactive Approach for Solving Multi- Objective Optimization Problems. Ph.D. thesis. Houston, TX: Rice University, 1985. Wright, M. H. "The Nelder-Mean Method: Numerical Ex- perimentation and Algorithmic Improvements." AT&T Bell Laboratories Techn. Rep. Murray Hill, NJ. Wright, M. H. "Direct Search Methods: Once Scorned, Now Respectable." In Numerical Analysis 1995. Papers from the Sixteenth Dundee Biennial Conference held at theUniversity of Dundee, Dundee, June 27 /C1 /30, 1995 (Ed. D. F. Griffiths and G. A. Watson). London: Longman, Harlow, pp. 191 /C1/208, 1996. Nephroid The 2-CUSPED EPICYCLOID is called a nephroid. Since n/C302,a/C30b=2;and the equation for r2in terms of the parameter fis given by EPICYCLOID equation r2 /C30a2 n2n2 /C272n /C272CC0CC1 /C282 n /C271 ðÞ cos(nf)CC6CC7 (1) with n /C302, r2 /C30a2 2222 /C272 /C215 2 /C272CC0CC1 /C2822/C271 ðÞ cos(2 f)CC6CC7 /C301 4a2 10 /C286 cos(2 f) ½/C138 /C3012a2 5 /C283 cos(2 f) ½/C138 ; (2) where tanu /C303 sinf /C28 sin(3f) 3 cos f /C28 cos(3 f) /C215 (3) This can be written r 2a !2 =3 /C30 sin12 uCC1:CC17hi2 =3 /C27 cos12 uCC1:CC17hi2 =3 /C215 (4) The PARAMETRIC EQUATIONS are x /C30a 3 cos t /C28cos(3 t) ½/C138 (5) y /C30a 3 sin t /C28sin(3 t) ½/C138 /C215 (6) The Cartesian equation is x2 /C27y2 /C284a2CC0CC13/C30108a4y2 /C215 (7) The name nephroid means "kidney shaped" and was first used for the two-cusped EPICYCLOID by Proctor in 1878 (MacTutor Archive). The nephroid has ARC LENGTH 24a and AREA 12p2a2 : The CATACAUSTIC for rays originating at the CUSP of a CARDIOID and reflected by it is a nephroid. Huygens showed in 1678 that the nephroid is the CATACAUSTIC of a CIRCLE when the light source is at infinity. He published this fact in Traite ´ de la lumine `re in 1690 (MacTutor Archive). The nephroid can be generated as the ENVELOPE of circles centered on a given circle and tangent to one of the circle’s diameters (Wells 1991). See also ASTROID ,DELTOID ,FREETH’S NEPHROIDReferences Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 221, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 169 /C1/173, 1972. Lockwood, E. H. "The Nephroid." Ch. 7 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 62/C1/71, 1967. MacTutor History of Mathematics Archive. "Nephroid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ne- phroid.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 158, 1991. Yates, R. C. "Nephroid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 152 /C1/154, 1952. Nephroid Evolute The EVOLUTE of the NEPHROID given by x/C301 23 cos t/C28cos(3 t) ½/C138 y/C301 23 sin t/C28sin(3 t) ½/C138 is given by x/C30cos3t y/C30143 sin t/C27sin(3 t) ½/C138 ; which is another NEPHROID . Nephroid Involute The INVOLUTE of the NEPHROID given by x/C30123 cos t/C28cos(3 t) ½/C138 y/C30123 sin t/C28sin(3 t) ½/C138 beginning at the point where the nephroid cuts the Y- AXIS is given by x /C304 cos3 t y /C303 sin t /C27sin(3 t) ; another NEPHROID . If the INVOLUTE is begun instead at the CUSP , the result is CAYLEY’S SEXTIC . Ne´ron-Severi Group Let V be a complete normal VARIETY , and write GVðÞ for the group of divisors, GnVðÞ for the group of divisors numerically equal to 0, and GaVðÞthe group of divisors algebraically equal to 0. Then the finitely generated QUOTIENT GROUP NS VðÞ/C30GVðÞ=GaVðÞis called the Ne´ron-Severi group. References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 75, 1980. Nerve The SIMPLICIAL COMPLEX formed from a family of objects by taking sets that have nonempty intersec- tions. See also DELAUNAY TRIANGULATION ,S IMPLICIAL COMPLEX Nested Hypothesis Let S be the set of all possibilities that satisfy HYPOTHESIS H, and let S ? be the set of all possibilities that satisfy HYPOTHESIS H ?: Then H ? is a nested hypothesis within H IFF S?ƒS; where ƒdenotes the PROPER SUBSET . See also LOG LIKELIHOOD PROCEDURE Nested Radical Expressions OF THE FORM lim k0/C12x0/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .../C27xkpqr /C215 Herschfeld (1935) proved that a nested radical of REAL NONNEGATIVE terms converges IFF /ðxnÞ2/C28n /is bounded. He also extended this result to arbitrary POWERS (which include continued square roots and CONTINUED FRACTIONS as well), a result is known as HERSCHFELD’S CONVERGENCE THEOREM . Nested radicals appear in the computation of PI, 2 p/C30ffiffi 1 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C2712ffiffi 12qrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C2712ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C2712ffiffi 12qrs ... ( 1 ) inTRIGONOMETRICAL values of COSINE and SINE for arguments OF THE FORM p=2n;e.g., sinp 8 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pq (2)cosp 8 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pq (3) sinp 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqr (4) cosp 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqr ; (5) and in the computation of the GOLDEN RATIO , f/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27...pqrs : (6) There are a number of general formula for nested radicals (Wong and McGuffin). For example, x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28q ðÞ xn/C27qxn/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28q ðÞ xn/C27qxn/C281ffiffiffiffiffiffi...pqr (7) which gives as special cases b/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C274ap 2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/C27bffiffiffiffiffiffi...pqrs (8) (n/C302,q/C301/C28a=x2;x/C30b=q); x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xn/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xn/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xn/C281ffiffiffiffiffiffi...pqrs (9) (q/C301), and x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xffiffiffiffiffiffiffiffiffiffiffiffi xffiffiffiffiffiffi...pqrsvuut(10) (/q/C301;n/C302):Equation (7) gives rise to q(nk/C281=n/C281ðÞxnj /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q(nk/C271/C28n)=(n/C281)1/C28q ðÞ xnj/C271/C27...q .../C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q(nk/C272/C28n)=(n/C281)1/C28q ðÞ xnj/C272/C27ffiffiffiffiffiffiffiffi...;pq ð11Þ which gives the special case for q/C301=2;n/C302,x/C301, andk/C30/C28 1, ffiffiffi 2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 220/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 221/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 222/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 223/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 224/C27...:svuutvuuutvuuuutvuuuuut (12) Ramanujan discovered x/C27n/C27a /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ax/C27n/C27a ðÞ 2/C27xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a(x/C27n)/C27n/C27a ðÞ2/C27...qr .../C27 x /C27n ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a(x /C272n) /C27 n /C27a ðÞ2/C27(x /C272n)ffiffiffiffiffiffi...pq ; which gives the special cases x /C271 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27(x /C271)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27(x /C272)ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27...pqrs ; (13) for a /C300, n /C301, and 3 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C273ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C274ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C275ffiffiffiffiffiffi...pqrsvuut(14) for a /C300, n /C301, and x /C302. For a nested radical OF THE FORM x /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin /C27...pqr (15) to be equal a given REAL NUMBER x, it must be true that x /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n /C27...pqr /C30ffiffiffiffiffiffiffiffiffiffiffin/C27xp ; (16) so x2/C30n/C27x (17) and x/C301 21/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4n/C271pCC1:CC17 /C215 (18) See also CONTINUED FRACTION ,G OLDEN RATIO , HERSCHFELD’S CONVERGENCE THEOREM ,PI,SQUARE ROOT References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 14 /C1/20, 1994. Herschfeld, A. "On Infinite Radicals." Amer. Math. Monthly 42, 419/C1/429, 1935. Landau, S. "A Note on ‘Zippel Denesting."’ J. Symb. Comput. 13,3 1/C1/45, 1992. Landau, S. "Simplification of Nested Radicals." SIAM J. Comput. 21,8 5/C1/110, 1992. Landau, S. "How to Tangle with a Nested Radical." Math. Intell. 16,4 9/C1/55, 1994. Landau, S. "ffiffiffi 2p /C27ffiffiffi 3p : Four Different Views." Math. Intell. 20,5 5/C1/60, 1998. Po´lya, G. and Szego, G. Problems and Theorems in Analysis, Vol. 1. New York: Springer-Verlag, 1997. Sizer, W. S. "Continued Roots." Math. Mag. 59,2 3/C1/27, 1986. Wong, B. and McGuffin, M. "The Museum of Infinite Nested Radicals." http://www.csclub.uwaterloo.ca/~mjmcguff/ math/nestedRadicals.html.Nested Square The black region in the nested square illustrated above, where the outer boundary is a unit square, has AREA 2. References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 165 /C1/166, 1984. Net The word net has several meanings in mathematics.It refers to a plane diagram in which the EDGES of a POLYHEDRON are shown. All convex POLYHEDRA have nets, but not all concave polyhedra do (the constitu-ent POLYGONS can overlap one another when a concave POLYHEDRON is flattened out). The GREAT DODECAHEDRON and STELLA OCTANGULA are examples of a concave polyhedron which have nonself-inter- secting nets. A corrected and concatenated version of the BellLaboratories netlib polyhedron database has been prepared by Weisstein, together with Mathematica code to access analytic vertex coordinates and plot nets for all Platonic and Archimedean solids and theirduals, as well as the Johnson solids. K. Fukuda has written routines which can unfold convex polyhedra into a planar net. The term net also has a technical meaning as a generalization of a SEQUENCE , in which context it is also known as a Moore-Smith sequence. In this context, nets is used in general topology and ANALYSIS to imbue non-metrizable topological spaces with convergence properties. This artifice is needed only in spaces which are not FIRST-COUNTABLE , since sequences alone provide an adequate way of dealing with CONTINUITY for FIRST-COUNTABLE SPACES . Nets are used in the study of the RIEMANN INTEGRAL . Formally, a net of a set S is a mapping from a DIRECTED SET D into S. See also DIRECTED SET,FIBER BUNDLE ,FIBER SPACE , FIBRATION ,UNFOLDING References Bell Laboratories. http://netlib.bell-labs.com/netlib/polyhe- dra/. Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Netto’s Conjecture The probability that two elements /P1/ and /P2/ of a SYMMETRIC GROUP generate the entire GROUP tends to 3u4as /n 0/C12/ (Netto 1964, p. 90). The conjecture was proven by Dixon (1969). See also PERMUTATION GROUP ,SYMMETRIC GROUP References Dixon, J. D. "The Probability of Generating the Symmetric Group." Math. Z. 110, 199 /C1/205, 1969. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 31, 1983. Netto, E. The Theory of Substitutions. New York: Chelsea, p. 90, 1964. Network A GRAPH or DIRECTED GRAPH together with a function which assigns a positive real number to each edge (Harary 1994, p. 52). See also GRAPH ,N ETWORK FLOW,SINK (DIRECTED GRAPH ), SMITH’S NETWORK THEOREM ,SOURCE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Network Flow The network flow problem considers a graph G with a set of sources S and sinks T and for which each edge has an assigned capacity (weight), and then asks to find the maximum flow that can be routed from S to T while respecting the given edge capacities. The network flow problem can be solved in time /O ðn3 Þ/ (Edmonds and Karp 1972; Skiena 1990, p. 237). It has been implemented as NetworkFlow [g, source , sink] in the Mathematica add-on package Discre- teMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ) and Net-workFlowEdges [g, source , sink] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also AUGMENTING PATH,M AXIMUM FLOW, MINI- MUM CUT THEOREM ,NETWORK References Edmonds, J. and Karp, R. M. "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems." J. ACM 19, 248 /C1/264, 1972. Even, S. and Tarjan, R. E. "Network Flow and Testing Graph Connectivity." SIAM J. Comput. 4, 507 /C1/518, 1975. Ford, L. R. and Fulkerson, D. R. Flows in Networks. Princeton, NJ: Princeton University Press, 1962. Gonery, R. E. and Hu, T. C. "Multiterminal Network Flows." J. SIAM 9, 551 /C1/570, 1961. Orlin, J. B. "A Faster Strongly Polynomial Minimum Cost Flow Algorithm." Proc. 20th ACM Symposium Theorem of Computing. pp. 377 /C1/387, 1988. Skiena, S. "Network Flow." §6.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 237 /C1/ 239, 1990. Skiena, S. S. "Network Flow." §8.4.9 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 297 /C1/ 300, 1997. Tarjan, R. E. Data Structures and Network Algorithms. Philadelphia, PA: SIAM Press, 1983. Neuberg Center The center of a NEUBERG CIRCLE . See also NEUBERG CIRCLE Neuberg Circle The LOCUS of the VERTEX A1of a TRIANGLE on a given base A2A3and with a given B ROCARD ANGLE vis a CIRCLE (actually two circles, one on either side of A2A3) known as the Neuberg circle. From the center N1;the base A2A3subtends the ANGLE 2v:The equation of the circle can be found by taking the base as (0, 0), (0, a1) and solving x2/C27y2/C30a2 3 (1) (x/C28a1)2/C27y2/C30a22 (2) while eliminating a2anda3using cosv/C30a2 1/C27a22/C27a33 4D; (3) where D is the area of the triangle DA1A2A3 : Solving for x gives x /C301 2a1 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 94a1y cot v /C284y2 /C283a2 1qCC1nCC1o ; (4) and squaring and completing the square results in x /C281 2 a1 !2 /C27 y 912a 1 cot v !2 /C3014 a 1 cot2 v /C283CC0CC1 (5) Therefore, the Neuberg circle N1on this edge has center N1 /C301 2a1 ;912 a 1 cot v ! (6) (sometimes called the NEUBERG CENTER ), and RADIUS r /C301 2 a1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cot2 v /C283p : The same procedure can be repeated for the other two sides of a TRIANGLE resulting in three Neuberg circles (with another corresponding three on opposite sides of the edges). The TRIANGLE connecting the three NEUBERG CENTERS is called the NEUBERG TRIANGLE . On one side of a given line taken as a base, it is possible to construct six triangles directly or inversely similar to a given SCALENE TRIANGLE , and the vertices of these triangles lie on their common Neuberg circles (Johnson 1929, p. 289). See also BROCARD ANGLE ,M CCAY CIRCLE ,NEUBERG TRIANGLE References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 79 /C1/80, 1971. Emmerich, A. Die Brocardschen Gebilde und ihre Beziehun- gen zu den verwandten merkwu ¨rdigen Punkten und Kreisen des Dreiecks. Berlin: Georg Reimer, 1891. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 287 /C1/290, 1929.Neuberg Triangle The TRIANGLE / DN1N2N3/ formed by joining a set of three NEUBERG CENTERS (i.e., centers of the NEUBERG CIRCLES ) obtained from the edges of a given triangle DA1A2A3(left figure). The CENTROID GNof / DN1N2N3/ is coincident with the CENTROID GAof DA1A2A3 (Johnson 1929, p. 288; right figure). The lines A1N1 ; A2N2 ; and A3N3are concurrent at a point T which Johnson (1929, p. 288) claims (appar- ently incorrectly) is the TARRY POINT . See also NEUBERG CIRCLE ,TARRY POINT References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Neumann Algebra VON NEUMANN ALGEBRA Neumann Boundary Conditions PARTIAL DIFFERENTIAL EQUATION BOUNDARY CONDI- TIONS which give the normal derivative on a surface. See also BOUNDARY CONDITIONS ,CAUCHY BOUNDARY CONDITIONS References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 679, 1953. Neumann Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION x2yƒ/C273xy?/C27 x2 /C271 /C28n2CC0CC1 y /C30x cos21 2npCC1:CC17 /C27n sin212npCC1:CC17 satisfied by the NEUMANN POLYNOMIALS /On ðxÞ/. See also NEUMANN POLYNOMIAL References Gradshteyn, I. S. and Ryzhik, I. M. "Neumann’s and Schla¨fli Polynomials: /On ðz Þ/ and /Sn ðxÞ/." §8.59 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 989 /C1/991, 2000. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 125, 1997. Neumann Function BESSEL FUNCTION OF THE SECOND KIND Neumann Polynomial Polynomials /On ðxÞ/ that can be defined by the sum On(x) /C301 4Xn=2bc k /C300n(n /C28 k /C28 1)! k!1 2xCC1:CC172k /C28n /C281 (1) for n ]1; where xbcis the FLOOR FUNCTION . They obey the RECURRENCE RELATION On(x) /C30/C28n n /C28 2 On/C282(x) /C272n xOn/C281(x) /C272(n /C28 1) (n /C28 2)xsin212(n /C281)phi (2) for n ]3 : They have the integral representation On(x) /C30g/C12 0 /C2u /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 /C27 x2pCC0CC1 n/C27 u /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 /C27 x2pCC0CC1 n 2xn/C271 e/C28udu ; (3) and the generating function 1 x /C28 j /C30J0( j)x/C281 /C272X/C12 n/C301Jn( j)On(x) (4) (Gradshteyn and Ryzhik 2000, p. 990), and obey the NEUMANN DIFFERENTIAL EQUATION . The first few Neumann polynomials are given by O0(x) /C301 x O1(x) /C301 x2O2(x) /C30x2 /C27 4 x3 O3(x) /C303x2 /C27 24 x4 O4(x) /C30x4 /C27 16x2 /C27 192 x5 (A057869). See also NEUMANN DIFFERENTIAL EQUATION ,SCHLA ¨ - FLI POLYNOMIAL References Erdelyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. Krieger, pp. 32 /C1/33, 1981. Gradshteyn, I. S. and Ryzhik, I. M. "Neumann’s and Schla ¨fli Polynomials: /On ðzÞ/ and /Sn ðzÞ/." §8.59 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 989 /C1/991, 2000. Sloane, N. J. A. Sequences A057869 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 196, 1993. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 298 /C1/305, 1966. Neumann Series (Bessel Function) A series OF THE FORM X/C12 n /C300anJn/C27n(z) ; (1) where n is a REAL and Jn/C27n(z)isaB ESSEL FUNCTION OF THE FIRST KIND . Special cases are z n /C302n G1 2v /C271CC1:CC17X/C12 n/C30012zCC1:CC17n=2 /C27n n!Jn=2/C27n(z) ; (2) where G(z) is the GAMMA FUNCTION , and X/C12 n/C300bnzn/C27n/C30X/C12 n/C300an12zCC1:CC17(n/C27n)=2 J(n/C27n)=2(z); (3) where an/C13Xn=2bc m/C3002n/C27n/C282mG1 2n/C2712n/C28m/C271CC1:CC17 m!bn/C282m; (4) and xbcis the FLOOR FUNCTION . See also KAPTEYN SERIES References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Neumann Series (Integral Equation) AF REDHOLM INTEGRAL EQUATION OF THE SECOND KIND f(x) /C30f(x) /C27gb aK(x;t) f(t)dt (1) may be solved as follows. Take f0(x) /C13f(x) (2) f1(x) /C30f(x) /C27 lgb aK(x; t)f(t)dt (3) f2(x) /C30f(x) /C27 lgb aKx; t1ðÞ ft1ðÞdt1 /C27 l2gb a gb aKx;t1ðÞ Kt1 ;t2 ðÞ ft2ðÞdt2dt1 (4) fn(x) /C30Xn i/C300liui(x) ; (5) where u0(x) /C30f(x) (6) u1(x) /C30gb aK(x;t)ft1ðÞdt1 (7) u2(x) /C30gb a gb aKx ;t1ðÞ Kt1 ;t2 ðÞ f ðt2 Þdt2dt1 : (8) un(x) /C30gb a gb a gb aKx;t1ðÞ Kt1 ;t2 ðÞ /C1 /C1 /C1 /C2Ktn/C281 ; tn ðÞ ftnðÞdtn /C1/C1/C1dt1 : (9) The Neumann series solution is then f(x) /C30 lim n0/C12fn(x) /C30 lim n0/C12Xn i/C300liui(x): (10) References Arfken, G. "Neumann Series, Separable (Degenerate) Ker- nels." §16.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 879 /C1/890, 1985. Neusis Construction A geometric construction, also called a VERGING CONSTRUCTION , which allows the classical GEOMETRIC CONSTRUCTION rules to be bent in order to permit sliding of a marked RULER . Using a Neusis construc- tion, CUBE DUPLICATION , angle TRISECTION , and con- struction of the regular HEPTAGON are soluble. The CONCHOID OF NICOMEDES can also be used to perform many Neusis constructions (Johnson 1975). Conway and Guy (1996) give Neusis constructions for the 7-, 9-, and 13-gons which are based on angle TRISECTION .See also CONCHOID OF NICOMEDES ,CUBE DUPLICA- TION ,G EOMETRIC CONSTRUCTION ,H EPTAGON , MASCHERONI CONSTRUCTION ,M ATCHSTICK CON- STRUCTION ,RULER ,STEINER CONSTRUCTION ,TRISEC- TION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 194 /C1/200, 1996. Johnson, C. "A Construction for a Regular Heptagon." Math. Gaz. 59,17/C1/21, 1975. Nevanlinna Theory An analytic refinement of results from COMPLEX analysis such as those codified by PICARD’S LITTLE THEOREM ,PICARD’S GREAT THEOREM , and the WEIER- STRASS- CASORATI THEOREM . See also PICARD’S GREAT THEOREM ,PICARD’S LITTLE THEOREM ,W EIERSTRASS- CASORATI THEOREM References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 141, 1999. Neville Theta Function The functions O2(x) /C30x2 /C27 4 x3 (1) O3(x) /C303x2 /C27 24 x4 (2) O4(x) /C30x4 /C27 16x2 /C27 192 x5 (3) X/C12 n/C300anJn/C27n(z)1 ¼ Jnþn ðz Þ (4) where zn /C302n G(1 2 n /C271)a/C12 n/C3001 2zCC1:CC17n =2/C27n n!Jn=2 /C27n(z) and G(z) are the JACOBI THETA FUNCTIONS and a/C12 n/C300bnz n/C27n /C30 a/C12n /C300an(1 2z)(n/C27n) =2J(n/C27n)=2(z) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND . See also JACOBI THETA FUNCTION ,THETA FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Neville’s Notation for Theta Functions." §16.36 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 578 /C1/579, 1972. Neville Theta Functions The functions qs(u) /C30H(u) H ?(0) (1) qd(u) /C30U(u /C27 K) U(k) (2) qs(u) /C30H(u) H(K) (3) qn(u) /C30U(u) U(0); (4) where H(u) and U(u) are the JACOBI THETA FUNC- TIONS and K(u) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND . See also JACOBI THETA FUNCTIONS ,THETA FUNC- TIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Neville’s Notation for Theta Functions." §16.36 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 578 /C1/579, 1972. Neville’s Algorithm An interpolation ALGORITHM which proceeds by first fitting a POLYNOMIAL Pkof degree 0 through the points (xk ;yk) for k /C300 ..., n, i.e., Pk /C30yk : A second iteration is then performed in which P12 is fit through pairs of points, yielding P12 ; P23 ; .... The procedure is repeated, generating a "pyramid" of approximations until the final result is reached P1 P2 P3 P4P12 P23 P34P123 P234P1234 : The final result is Pi(i/C271)/C1/C1/C1(i /C27m) /C30x /C28 xi/C27mCC0CC1 Pi(i/C271)/C1/C1/C1(i /C27m/C281) xi /C28 xi /C27m /C27xi /C28 x ðÞ P(i /C271)(i/C272)/C1/C1/C1(i /C27m) xi /C28 xi /C27m: See also BULIRSCH- STOER ALGORITHM NevilleThetaC NEVILLE THETA FUNCTIONS NevilleThetaD NEVILLE THETA FUNCTIONSNevilleThetaN NEVILLE THETA FUNCTIONS NevilleThetaS NEVILLE THETA FUNCTIONS Newcomb’s Paradox A paradox in DECISION THEORY . Given two boxes, B1 which contains $1000 and B2 which contains either nothing or a million dollars, you may pick either B2 or both. However, at some time before the choice is made, an omniscient Being has predicted what your decision will be and filled B2 with a million dollars if he expects you to take it, or with nothing if he expects you to take both. See also ALLAIS PARADOX References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 137 /C1/139, 1998. Gardner, M. The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: Chicago University Press, 1991. Gardner, M. "Newcomb’s Paradox." Ch. 13 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 155 /C1/161, 1986. Nozick, R. "Reflections on Newcomb’s Paradox." Ch. 14 in Gardner, M. Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986. Newman-Conway Sequence The sequence 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, ... (Sloane’s A004001) defined by P(1)/C30P(2)/C301 and the RECUR- RENCE RELATION P(n)/C30P(P(n/C281))/C27P(n/C28P(n/C281)) /C215 (1) It satisfies P2kCC0CC1 /C302k/C281(2) and P(2n)52P(n): (3) References Bloom, D. M. "Newman-Conway Sequence." Solution to Problem 1459. Math. Mag. 68, 400/C1/401, 1995. Sloane, N. J. A. Sequences A004001/M0276 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Newman’s Conjecture Ifmis an integer, then for every residue class r(mod m), there are infinitely many nonnegative integers n for which P(n)/C13r(mod m);where P(n) is the PARTI- TION FUNCTION P. See also ERDOS- IVIC CONJECTURE ,PARTITION FUNC- TION P References Newman, M. "Periodicity Modulo mand Divisibility Proper- ties of the Partition Function." Trans. Amer. Math. Soc. 97, 225/C1/236, 1960. Ono, K. "Distribution of the Partition Functions Modulo m." Ann. Math. 151, 293/C1/307, 2000. Newton Number KISSING NUMBER Newton-Bessel Formula BESSEL’S FINITE DIFFERENCE FORMULA Newton-Cotes Formulas The Newton-Cotes formulas are an extremely useful and straightforward family of NUMERICAL INTEGRA- TION techniques. To integrate a function f(x) over some interval [ a, b], divide it into nequal parts such that fn/C30fxnðÞand / h/C13ðb/C28aÞun/. Then find POLYNOMIALS which approx- imate the tabulated function, and integrate them toapproximate the AREA under the curve. To find the fitting POLYNOMIALS , use L AGRANGE INTERPOLATING POLYNOMIALS . The resulting formulas are called Newton-Cotes formulas, or QUADRATURE FORMULAS . Newton-Cotes formulas may be "closed" if the interval x1;xnCC6CC7 is included in the fit, "open" if the points x2;xn/C281 ½/C138 are used, or a variation of these two. If the formula uses npoints (closed or open), the COEFFI- CIENTS of terms sum to n/C281:/ If the function f(x) is given explicitly instead of simply being tabulated at the values xi;the best numerical method of integration is called G AUSSIAN QUADRA- TURE . By picking the intervals at which to sample the function, this procedure produces more accurate approximations (but is significantly more complicated to implement). The 2-point closed Newton-Cotes formula is called the TRAPEZOIDAL RULE because it approximates the area under a curve by a TRAPEZOID with horizontal base and sloped top (connecting the endpoints x1andx2):If the first point is x1;then the other endpoint will be located at x2/C30x1/C27h; (1)and the L AGRANGE INTERPOLATING POLYNOMIAL through the points x1;f1 ðÞ and x2;f2 ðÞ is P2ðxÞ/C30x/C28x2 x1/C28x2f1/C27x/C28x1 x2/C28x1f2 /C30x/C28x1/C28h /C28hf1/C27x/C28x1 hf2 x hf2/C28f1 ðÞ /C27f1/C27x1 hf1/C28x1 hf2 ! /C215 (2) Integrating over the interval (i.e., finding the area ofthe trapezoid) then gives gx2 x1f(x)dx/C30gx1/C27h x1P2(x)dx /C301 2hf2/C28f1 ðÞ x2CC6CC7x2 x1/C27f1/C27x1 hf1/C28x1 hf2 ! x½/C138x2 x1 /C301 2hf2/C28f1 ðÞ x2/C27x1 ðÞ x2/C28x1 ðÞ /C27x2/C28x1 ðÞ /C2f1/C27x1 hf1/C28x1 hf2 ! /C301 2f2/C28f1 ðÞ 2x1/C27h ðÞ /C27f1h/C27x1f1/C28f2 ðÞ /C30x1f2/C28f1 ðÞ /C2712hf2/C28f1 ðÞ /C27hf1/C28x1f2/C28f1 ðÞ /C301 2hf1/C27f2 ðÞ /C281 12h3fƒ(j): (3) This is the trapezoidal rule (Ueberhuber 1997, p. 100), with the final term giving the amount of error (which, since x15j5x2;is no worse than the maximum value of fƒ(j) in this range). The 3-point rule is known as S IMPSON’S RULE . The ABSCISSAS are x2/C30x1/C27h (4) x3/C30x1/C272h (5) and the L AGRANGE INTERPOLATING POLYNOMIAL is P3(x)/C30x/C28x2 ðÞ x/C28x3 ðÞ x1/C28x2 ðÞ x1/C28x3 ðÞf1/C27x/C28x1 ðÞ x/C28x3 ðÞ x2/C28x1 ðÞ x2/C28x3 ðÞf2 /C27x/C28x1 ðÞ x/C28x2 ðÞ x3/C28x1 ðÞ x3/C28x2 ðÞf3 /C30x2/C28xx2/C27x3 ðÞ /C27x2x3 h(2h)f1 /C27x2/C28xx1/C27x3 ðÞ /C27x1x3 h(/C28h)f2 /C27x2/C28xx1/C27x2 ðÞ /C27x1x2 2h(h)f3 /C301 h2fx21 2f1/C28f2/C2812f3CC1:CC17 /C27x/C28122x1/C273h ðÞ f1h /C272x1/C272h ðÞ f2/C281 22x1/C27h ðÞ /C138 /C2712x1/C27h ðÞ x1/C272h ðÞ f1h /C28x1x1/C272h ðÞ f2/C271 2x1x1/C27h ðÞ f3]g: (6) Integrating and simplifying gives gx2 x1f(x)dx/C30gx1/C272h x1P3(x)dx /C301 3hf1/C274f2/C27f3 ðÞ /C281 90h5f4ðÞjðÞ (7) (Ueberhuber 1997, p. 100). The 4-point closed rule is S IMPSON’S 3/8 RULE , gx4 x1f(x)dx/C303 8hf1/C273f2/C273f3/C27f4 ðÞ /C283 80h5f4ðÞ(j) (8) (Ueberhuber 1997, p. 100). The 5-point closed rule is BODE’S RULE , gx5 x1f(x)dx/C302 45h7f1/C2732f2/C2712f3/C2732f4/C277f5 ðÞ /C288 945h7f6ðÞ(j) (9) (Abramowitz and Stegun 1972, p. 886). Higher orderrules include the 6-point gx6 x1f(x)dx/C305 288h19f1/C2775f2/C2750f3/C2750f4/C2775f5 ð /C2719f6Þ/C28275 12096h7f6ðÞ(j); (10) 7-point gx7 x1f(x)dx/C301 140h41f1/C27216f2/C2727f3/C27272f4 ð /C2727f5/C27216f6/C2741f7Þ/C289 1400h9f8ðÞ(j); (11) 8-point gx8 x1f(x)dx/C307 17280h751f1/C273577 f2/C271323 f2/C272989 f3 ð /C272989 f5/C271323 f6/C273577 f7/C27751f8Þ /C288183 518400h9f8ðÞ(j); ð12Þ 9-point gx9 x1f(x)dx/C304 14175h989f1/C275888 f2/C28928f3 ð /C2710496 f4/C274540 f5/C2710496 f6/C28928f7/C275888 f8/C27989f9Þ/C282368 467775h11f10ðÞ(j) (13) (Ueberhuber 1997, p. 100), 10-point gx10 x1f(x)dx/C309 89600h2857 f1/C27f10 ðÞ ½ /C2715741 f2/C27f9 ðÞ /C271080 f3/C27f8Þ/C2719344 f4/C27f7 ðÞ ð /C275788 f5/C27f6 ðÞ /C138 /C28173 14620h11f10ðÞ(j); (14) and 11-point gx11 x1f(x)dx/C305 299376h16067 f1/C27f11 ðÞ ½ /C27106300 f2/C27f10 ðÞ /C138 /C2848525 f3/C27f9 ðÞ /C27272400 f4/C27f8 ðÞ /C28260550 f5/C27f7 ðÞ /C27427368 f6/C138/C281346350 326918592h13f12ðÞ(j)ð15Þ rules. In general, the n-point rule is given by the analytic expression gxn x1f(x)dx/C30hXn i/C301Hn;ifi; (16) where Hn;r/C271/C30/C281ðÞn/C28r r!n/C28r ðÞ !gn 0t(t/C281)/C1/C1/C1(t/C28r/C271) /C2(t/C28r/C281)/C1/C1/C1(t/C28n)dt (17) (Whittaker and Robinson 1967, p. 154).Closed "extended" rules use multiple copies of lower order closed rules to build up higher order rules. Byappropriately tailoring this process, rules with parti- cularly nice properties can be constructed. For n tabulated points, using the TRAPEZOIDAL RULE (n/C281) times and adding the results gives gxn x1f(x)dx/C30gx2 x1/C27gx3 x2/C27/C1/C1/C1gxn xn/C281 ! f(x)dx /C301 2hf1/C27f2 ðÞ /C27f2/C27f3 ðÞ /C27/C1/C1/C1/C27fn/C282/C27fn/C281 ðÞ ½ /C27fn/C281/C27fn ðÞ /C138 /C30h12f1/C27f2/C27f3/C27/C1/C1/C1/C27fn/C282/C27fn/C281/C2712fnCC1:CC17 /C281 12nh3fƒ(j) (18) (Ueberhuber 1997, p. 107). Using a series of refine- ments on the extended TRAPEZOIDAL RULE gives the method known as R OMBERG INTEGRATION . A 3-point extended rule for ODD nis gxn x1f(x)dx/C30h1 3f1/C2743f2/C2713f3CC1:CC17 /C2713f3/C2743f4/C2713f5CC1:CC17 h /C27/C1/C1/C1/C271 3fn/C284/C2743fn/C283/C2713fn/C282CC1:CC17 /C2713fn/C282/C2743fn/C281/C2713fnCC1:CC17 /C138 /C3013hf1/C274f2/C272f3/C274f4/C272f5/C27.../C274fn/C281/C27fn ðÞ /C28n/C281 21 90h5f4ðÞ(j): (19) Applying S IMPSON’S 3/8 RULE , then S IMPSON’S RULE (3- point) twice, and adding gives gx4 x1/C27gx6 x4/C27gx4 x1"# f(x)dx /C30h38f1/C2798f2/C2798f3/C2738f4CC1:CC17 /C2713f4/C2743f5/C2713f6CC1:CC17 h /C271 3f6/C2743f7/C2713f8CC1:CC17 /C138 /C30h3 8f1/C2798f2/C2798f3/C2738/C2713CC1:CC17 f4/C2743f5h /C271 3/C2713CC1:CC17 f6/C2743f7/C2713f8/C138 /C30h38f1/C2798f2/C2798f3/C271724f4/C2743f5/C2723f6/C2743f7/C2713f8CC1:CC17 :(20) Taking the next Simpson’s 3/8 step then gives gx11 x8f(x)dx/C30h38f8/C2798f9/C2798f10/C2738f11CC1:CC17 : (21) Combining with the previous result gives gx11 x1f(x)dx/C30h3 8f1/C2798f2/C2798f3/C271724f4/C2743f5h /C272 3f6/C2743f7/C2713/C2738CC1:CC17 f8/C2798f9/C2798f10/C2738f11/C138 /C30h38f1/C2798f2/C2798f3/C271724f4/C2743f5/C2723f6/C2743f7CC1: /C271724f8/C2798f9/C2798f10/C2738f11Þ; (22) where terms up to /f10/have now been completely determined. Continuing gives h3 8f1/C2798f2/C2798f3/C271724f4/C2743f5/C2723f6/C27...CC1: /C272 3fn/C285/C2743fn/C284/C271724fn/C283/C2798fn/C282/C2798fn/C281/C2738fnÞ:(23) Now average with the 3-point result h1 3f1/C2743f2/C2723f3/C2743f4/C2723f5/C2743fn/C281/C2713fnCC1:CC17 (24) to obtain h17 48f1/C275948f2/C274348f4/C274948f4/C27f5/C27f6/C27.../C27fn/C285/C27fn/C284 ðÞh /C2749 48fn/C283/C274338fn/C282/C275948fn/C281/C271748fn/C138/C27On/C284CC0CC1 : (25) Note that all the middle terms now have unityCOEFFICIENTS . Similarly, combining a 4-point with the (2/C274)-point rule gives h5 12f1/C2713 12f2/C27f3/C27f4/C27.../C27fn/C283/C27fn/C282/C271312fn/C281/C275 12CC1:CC17 /C27On/C283CC0CC1 : (26) Other Newton-Cotes rules occasionally encountered include D URAND’S RULE gxn x1f(x)dx/C30h2 5f1/C271110f2/C27f3/C27.../C27fn/C282/C271110fn/C281/C2725fnCC1:CC17 (27) (Beyer 1987), H ARDY’S RULE gx0/C273h x0/C283hf(x)dx /C301 100h28f/C283/C27162f/C282/C2722f0/C27162f2/C2728f3 ðÞ /C279 1400h72f(4)j2ðÞ/C28h2f(8)j1ðÞCC6CC7 ; (28) and W EDDLE’S RULE gx6n x1f(x)dx/C303 10hf1/C275f2/C27f3/C276f4/C275f5/C27f6 ð /C27.../C275f6n/C281/C27f6nÞ (29) (Beyer 1987). The open Newton-Cotes rules use points outside the integration interval, yielding the 1-point gx2 x0f(x)dx/C302hf1; (30) 2-point gx3 x0f(x)dx/C30gx1/C272h x1/C28hP2(x)dx /C301 2hf2/C28f1 ðÞ x2CC6CC7x1/C272h x1/C28h/C27f1/C27x1 hf1/C28x1 hf2 ! x½/C138x1/C272h x1/C28h /C303 2hf1/C27f2 ðÞ /C2714h3fƒ(j); (31) 3-point gx4 x0f(x)dx/C304 3h2f1/C28f2/C272f3 ðÞ /C272890h5f4ðÞ(j); (32) 4-point gx5 x0f(x)dx/C305 24h11f1/C27f2/C27f3/C2711f4 ðÞ /C2795 144h5f4ðÞ(j); (33) 5-point gx6 x0f(x) dx /C306 20h 11f1 /C2814f2 /C2726f3 /C2814f4 /C2711f5 ðÞ /C2841 140h7f 6ðÞ( j) ; (34) 6-point gx7 x0f(x)dx /C307 1440h 611f1 /C28453f2 /C27562f3 /C27562f4 ð /C28453f5 /C27611f6 Þ/C285257 8640h7f 6ðÞ(j) ; (35) and 7-point gx8 x0f(x)dx /C308 945h 460f1 /C28954f2 /C272196 f3 /C282459 f4 ð /C272196 f5 /C28954f6 /C27460f7 Þ/C283956 14175h9f(8)(j) (36) rules. A 2-point open extended formula is gxn x1f(x)dx /C30h1 2 f1 /C27f2 /C27.../C27fn /C281 /C2712 fnCC1:CC17h /C271 24 /C28f0 /C27f2 /C27fn/C281 /C28fn/C271CC0CC1 /C138/C2711(n/C271) 720h5f(4)(j) : ð37Þ Single interval extrapolative rules estimate the in- tegral in an interval based on the points around it. An example of such a rule is hf1 /C27O h2f ?CC0CC1 (38) 1 2h 3f1 /C28f2 ðÞ /C27O h3f ƒCC0CC1 (39) 1 12h 23f1 /C2816f2 /C275f3 ðÞ /C27O h4f(3)CC0CC1 (40) 1 24h 55f1 /C2859f2 /C2737f3 /C289f4 ðÞ /C27O h5f(4)CC0CC1 : (41) See also BODE’S RULE,DIFFERENCE EQUATION ,DUR- AND’S RULE,FINITE DIFFERENCE ,G AUSSIAN QUAD- RATURE ,H ARDY’S RULE,LAGRANGE INTERPOLATING POLYNOMIAL ,NUMERICAL INTEGRATION ,SHOVELTON’S RULE,SIMPSON’S RULE,SIMPSON’S 3/8 RULE,TRAPE- ZOIDAL RULE,W EDDLE’S RULE,W OOLHOUSE’S FOR- MULAS References Abramowitz, M. and Stegun, C. A. (Eds.). "Integration." §25.4 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 885 /C1/887, 1972. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987. Corbit, D. "Numerical Integration: From Trapezoids to RMS: Object-Oriented Numerical Integration." Dr. Dobb’s J., No. 252, 117 /C1/120, Oct. 1996. Daniell, P. J. "Remainders in Interpolation and Quadrature Formulae." Math. Gaz. 24, 238, 1940. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 160 /C1/161, 1956.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Classical Formulas for Equally Spaced Ab- scissas." §4.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 124 /C1/130, 1992. Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, 1997. Whittaker, E. T. and Robinson, G. "The Newton-Cotes Formulae of Integration." §76 in The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152 /C1/156, 1967. Newton-Gauss Backward Formula GAUSS’S BACKWARD FORMULA Newton-Gauss Forward Formula GAUSS’S FORWARD FORMULA Newton-Girard Formulas The identities between the elementary symmetric functionsQ kx1 ; ... ;xn ðÞ and the sums of nth powers of their variables Sk /C30an k/C301xk : For 1 5k 5n ; the identity is /C281ðÞnnY nx1 ;...;xk ðÞ /C27Xn/C281 k /C300/C281ðÞkSkx1 ;...;xk ðÞY kx1 ;...;xk ðÞ /C300; (1) the first few of which are S1 /C28Y n/C300 (2) S2 /C28S1Y 1/C272Y 2/C300 (3) S3/C28S2Y 1/C27S1Y 2/C283Y 3/C300: (4) See also SYMMETRIC POLYNOMIAL References Se´roul, R. "Newton-Girard Formulas." §10.12 in Program- ming for Mathematicians. Berlin: Springer-Verlag, pp. 278 /C1/279, 2000. Newtonian Form NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FOR- MULA Newton-Raphson Fractal NEWTON’S METHOD Newton-Raphson Method NEWTON’S METHOD Newton’s Backward Difference Formula fp /C30f0 /C27p 90 /C271 2!p(p /C271) 92 0 /C271 3!p(p /C271)(p /C272) 930 /C27...; for p /C23 [0;1]; where 9 is the BACKWARD DIFFERENCE . See also NEWTON’S FORWARD DIFFERENCE FORMULA References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Newton’s Diverging Parabolas Curves with CARTESIAN equation ay2 /C30xx2 /C282bx /C27cCC0CC1 with a /C210. The above equation represents the third class of Newton’s classification of CUBIC CURVES , which Newton divided into five species depending on the ROOTS of the cubic in x on the right-hand side of the equation. Newton described these cases as having the following characteristics: 1. "All the ROOTS are REAL and unequal. Then the Figure is a diverging Parabola OF THE FORM of a Bell, with an Oval at its Vertex. 2. Two of the ROOTS are equal. A PARABOLA will be formed, either Nodated by touching an Oval, or Punctate, by having the Oval infinitely small. 3. The three ROOTS are equal. This is the NEILIAN PARABOLA , commonly called SEMI-CUBICAL . 4. Only one REAL ROOT . If two of the ROOTS are impossible, there will be a Pure PARABOLA of a Bell-like Form" (MacTutor Archive). References MacTutor History of Mathematics Archive. "Newton’s Diver- ging Parabolas." http://www-groups.dcs.st-and.ac.uk/~his- tory/Curves/Newtons.html. Newton’s Divided Difference Interpolation Formula Let pn(x) /C13Yn i/C301x /C28xn ðÞ ; (1) then f(x) /C30f0 /C27Xn k /C301xk /C281(x) x0 ; x1 ...;xk ½/C138 /C27Rn ; (2) where x1 ;...½/C138 is a DIVIDED DIFFERENCE , and the remainder isRn(x) /C30pn(x) x0 ;...;xn ;x ½/C138 /C30pn(x)f n/C271 ðÞ( j) n /C27 1 ðÞ(3) for x0 B j Bxn :/ See also DIVIDED DIFFERENCE ,FINITE DIFFERENCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 43 /C1/44 and 62 /C1/63, 1956. Whittaker, E. T. and Robinson, G. "Newton’s Formula for Unequal Intervals." §13 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 24 /C1/26, 1967. Newton’s Formulas Let a TRIANGLE have side lengths a, b, and c with opposite angles A, B, and C. Then b /C27 c a/C30cos1 2(B /C28 C)hi sin1 2ACC1:CC17 c /C27 a b/C30cos1 2(C /C28 A)hi sin1 2BCC1:CC17 a /C27 b c/C30cos1 2(A /C28 B)hi sin1 2CCC1:CC17 : See also MOLLWEIDE’S FORMULAS ,TRIANGLE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 146, 1987. Newton’s Forward Difference Formula AFINITE DIFFERENCE identity giving an interpolated value between tabulated points /ffpg/in terms of the first value /f0/and the POWERS of the FORWARD DIFFERENCE D:For /a/C23½0;1/C138/, the formula states fa/C30f0/C27aD/C271 2!a(a/C281)D2/C271 3!a(a/C281)(a/C282)D3/C27... When written in the form fx/C27a ðÞ /C30X/C12 n/C300aðÞnDnfxðÞ n! with aðÞnthe P OCHHAMMER SYMBOL , the formula looks suspiciously like a finite analog of a T AYLOR SERIES expansion. This correspondence was one of the motivating forces for the development of UMBRAL CALCULUS . The DERIVATIVE of Newton’s forward difference for- mula gives MARKOFF’S FORMULAS . See also FINITE DIFFERENCE ,M ARKOFF’S FORMULAS , NEWTON’S BACKWARD DIFFERENCE FORMULA ,N EW- TON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 432, 1987. Whittaker, E. T. and Robinson, G. "The Gregory-Newton Formula of Interpolation" and "An Alternative Form of the Gregory-Newton Formula." §8 /C1/9in The Calculus of Ob- servations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 10 /C1/15, 1967. Newton’s Identities NEWTON’S RELATIONS Newton’s Iteration An algorithm for computing the SQUARE ROOT of a number n quadratically as limk 0/C12xk ; xk /C271 /C301 2xk /C27n xk ! ; where x0 /C301: The first few approximants toffiffiffinpare given by 1;1 2(1 /C27n);1 /C27 6n /C27 n2 4(n /C27 1); 1 /C27 28n /C27 70n2 /C27 28n3 /C27 n4 8(1 /C27 n)1/C27 6n /C27 n2 ðÞ;... Forffiffiffi 2p ; this gives the convergents as 1, 3/2, 17/12, 577/408, 665857/470832, ... (Sloane’s A051008 and A051009). See also SQUARE ROOT References Sloane, N. J. A. Sequences A051008 and A051008 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Newton’s Method AROOT -finding ALGORITHM which uses the first few terms of the T AYLOR SERIES of a function f(x) in the vicinity of a suspected ROOT to zero in on the root. It is also called the Newton-Raphson method. For f(x)a POLYNOMIAL , Newton’s method is essentially the same as H ORNER’S METHOD . The T AYLOR SERIES of f(x) about the point x/C27ois given by f(x/C27o)/C30f(x)/C27f?(x)o/C271 2fƒ(x)o2/C27...: (1)Keeping terms only to first order, f(x/C27o):f(x)/C27f?(x)o: (2) This expression can be used to estimate the amount of offset oneeded to land closer to the root starting from an initial guess x0:Setting fx0/C27o ðÞ /C300 and solving (2) forogives o0/C30/C28fx0ðÞ f?x0ðÞ; (3) which is the first-order adjustment to the ROOT ’s position. By letting x1/C30x0/C27o0;calculating a new o1; and so on, the process can be repeated until itconverges to a root. Unfortunately, this procedure can be unstable near a horizontal ASYMPTOTE or a LOCAL MINIMUM . However, with a good initial choice of the ROOT ’s position, the algorithm can by applied iteratively to obtain xn/C271/C30xn/C28fxnðÞ f?xnðÞ(4) forn/C301, 2, 3, .... An initial point x0that provides safe convergence of Newton’s method is called an APPROX- IMATE ZERO . The error on/C271after the ( n/C271)/st iteration is given by on/C271/C30on/C27xn/C271/C28xnCC0CC1 /C30on/C28fxnðÞ f?xnðÞ: (5) But fxnðÞ/C30f(x)/C27f?(x)on/C271 2fƒ(x)o2 n/C27... /C30f?(x)on/C271 2fƒ(x)o2 n/C27... ( 6 ) f?xnðÞ/C30f?(x)/C27fƒ(x)on/C27...; (7) so fxnðÞ f?xxðÞ/C30f?(x)on/C271 2fƒ(x)o2 n/C27... f?(x)fƒ(x)on/C27... :f?(x)o/C271 2fƒ(x)o2 n f?(x)/C27fƒ(x)on/C30onþfƒ(x) 2f?(x)o2 n; (8) and (5) becomes on/C271/C30on/C28on/C27fƒ(x) 2f?(x)o2n"# /C30/C28fƒ(x) 2f?(x)o2n: (9) Therefore, when the method converges, it does so quadratically. AFRACTAL is obtained by applying Newton’s method to finding a ROOT ofzn/C281/C300 (Mandelbrot 1983, Gleick 1988, Peitgen and Saupe 1988, Press et al. 1992, Dickau 1997). Iterating for a starting point z0 gives zi/C271 /C30zi /C28zn i/C28 1 nzn/C281 i: (10) Since this is an nth order POLYNOMIAL , there are n ROOTS to which the algorithm can converge. Coloring the BASIN OF ATTRACTION (the set of initial points z0which converge to the same ROOT ) for each ROOT a different color then gives the above plots, corresponding to n /C302, 3, 4, and 5. See also ALPHA- TEST,APPROXIMATE ZERO,HALLEY’S IRRATIONAL FORMULA ,HALLEY’S METHOD ,HORNER’S METHOD ,H OUSEHOLDER’S METHOD ,L AGUERRE’S METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 18, 1972. Acton, F. S. Ch. 2 in Numerical Methods That Work. Washington, DC: Math. Assoc. Amer., 1990. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 963 /C1/964, 1985. Boyer, C. B. and Merzbacher, U. C. A History of Mathe- matics, 2nd ed. New York: Wiley, 1991. Dickau, R. M. "Basins of Attraction for z5 /C301 Using New- ton’s Method in the Complex Plane." http://forum.swarth- more.edu/advanced/robertd/newtons.html. Dickau, R. M. "Variations on Newton’s Method." http:// forum.swarthmore.edu/advanced/robertd/newnew- ton.html. Dickau, R. M. "Compilation of Iterative and List Opera- tions." Mathematica J. 7,14/C1/15, 1997. Gleick, J. Chaos: Making a New Science. New York: Penguin Books, plate 6 (following pp. 114) and p. 220, 1988. Gourdon, X. and Sebah, P. "Newton’s Iteration." http:// xavier.gourdon.free.fr/Constants/Algorithms/new-ton.html. Householder, A. S. Principles of Numerical Analysis. New York: McGraw-Hill, pp. 135 /C1 /138, 1953.Mandelbrot, B. B. The Fractal Geometry of Nature. San Francisco, CA: W. H. Freeman, 1983. Newton, I. Methodus fluxionum et serierum infinitarum. 1664 /C1/1671. Ortega, J. M. and Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, PA: SIAM, 2000. Peitgen, H.-O. and Saupe, D. The Science of Fractal Images. New York: Springer-Verlag, 1988. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Newton-Raphson Method Using Derivatives" and "Newton-Raphson Methods for Nonlinear Systems of Equations." §9.4 and 9.6 in Numerical Recipes in FOR- TRAN: The Art of Scientific Computing, 2nd ed. Cam- bridge, England: Cambridge University Press, pp. 355 /C1/ 362 and 372 /C1/375, 1992. Ralston, A. and Rabinowitz, P. §8.4 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978. Raphson, J. Analysis aequationum universalis. London, 1690. Whittaker, E. T. and Robinson, G. "The Newton-Raphson Method." §44 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 84 /C1/87, 1967. Newton’s Parallelogram Approximates the possible values of y in terms of x if Xn i;j/C300aijxiyj /C300: Newton’s Relations Let si be the sum of the products of distinct ROOTS rj of the POLYNOMIAL equation of degree n anxn/C27an/C281xn/C281/C27.../C27a1x/C27a0/C300; (1) where the roots are taken iat a time (i.e., siis defined as the SYMMETRIC POLYNOMIALQ ir1;...;rn ðÞ )siis defined for i/C301, ..., n. For example, the first few values of siare s1/C30r1/C27r2/C27r3/C27r4/C27... ( 2 ) s2/C30r1r2/C27r1r3/C27r1r4/C27r2r3/C27. . . (3) s3/C30r1r2r3/C27r1r2r4/C27r2r3r4/C27...; (4) and so on. Then si/C30/C28 1ðÞian/C28i an: (5) This can be seen for a second DEGREE POLYNOMIAL by multiplying out, a2x2/C27a1x/C27a0/C30a2x/C28r1 ðÞ x/C28r2 ðÞ /C30a2x2/C28r1/C27r2 ðÞ x/C27r1r2CC6CC7 ; (6) so s1 /C30X2 i/C301ri /C30r1 /C27r2 /C30/C28a1 a2(7) s2 /C30X2 i;j/C301 i "jrirj /C30r1r2 /C30a0 a2; (8) and for a third DEGREE POLYNOMIAL , a3x3 /C27a2x2 /C27a1x /C27a0 /C30a3x /C28r1 ðÞ x /C28r2 ðÞ x /C28r3 ðÞ /C30a3x3 /C28 r1 /C27r2 /C27r3 ðÞ x2 /C27 r1r2 /C27r1r3 /C27r2r3 ðÞ x /C28r1r2r3CC6CC7 ; (9) so s1 /C30X3 i /C301ri /C30/C28a2 a3(10) s2 /C30X3 i;j i "jrirj /C30r1r2 /C27r1r3 /C27r2r3 /C30a1 a3(11) s3 /C30X3 i;j;k i"j"krirjrk /C30r1r2r3 /C30/C28a0 a3: (12) See also DISCRIMINANT (POLYNOMIAL ), SYMMETRIC POLYNOMIAL References Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 56, 1982. Borwein, P. and Erde´lyi, T. "Newton’s Identities." §1.1.E.2 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 5 /C1/6, 1995. Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 1 /C1/2, 1959. Newton’s Theorem If each of two nonparallel transversals with nonmi- nimal directions meets a given curve in finite points only, then the ratio of products of the distances from the two sets of intersections to the intersection of the lines is independent of the position of the latter point. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 189, 1959. Newton-Stirling Formula STIRLING’S FINITE DIFFERENCE FORMULANext Prime The next prime function NP(n) gives the smallest PRIME larger than n. The function can be given explicitly as NP(n) /C30p1/C27p(n) ; where piis the ith PRIME and p(n) is the PRIME COUNTING FUNCTION . For n /C301, 2, ... the values are 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, ... (Sloane’s A007918). See also FORTUNATE PRIME ,PRIME COUNTING FUNC- TION ,PRIME NUMBER References Sloane, N. J. A. Sequences A007918 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Nexus Number A FIGURATE NUMBER built up of the nexus of cells less than n steps away from a given cell. In k-D, the (n /C271)/th nexus number is given by Nn /C271(k) /C30Xk i/C300k iCC1nCC1o ni ; wheren nCC0CC1 is a BINOMIAL COEFFICIENT . The first few k- dimensional nexus numbers are given in the table below. k /Nn/C271/ name 0 1 unit 1 /1 /C272n/ ODD NUMBER 2 /1 /C273n /C273n2 / HEX NUMBER 3 /1 /C274n /C276n2 /C274n3 / RHOMBIC DODECAHEDRAL NUMBER See also BINOMIAL SUMS,HEX NUMBER ,ODD NUM- BER,RHOMBIC DODECAHEDRAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 53 /C1/54, 1996. Neyman-Pearson Lemma If there exists a critical region Cof size aand a NONNEGATIVE constant ksuch that Qn i/C301fxiðju1ÞQni/C301fxiðju0Þ]k for points in Cand Qn i/C301 fxiðj u1 ÞQni/C301 fxiðju0 Þ5k for points not in C, then C is a best critical region of size a:/ References Hoel, P. G.; Port, S. C.; and Stone, C. J. "Testing Hypoth- eses." Ch. 3 in Introduction to Statistical Theory. New York: Houghton Mifflin, pp. 56 /C1/67, 1971. Nialpdrome A nialpdrome is a number whose HEXADECIMAL digits are in nonincreasing order. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 48, 49, 50, ... (Sloane’s A023771), corresponding to 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 20, 21, 22, 30, 31, 32, .... See also DIGIT,H EXADECIMAL ,K ATADROME ,M ETA- DROME ,PLAINDROME References Sloane, N. J. A. Sequences A023771 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Nicholson’s Formula Let Jn(z)beaB ESSEL FUNCTION OF THE FIRST KIND , Yn(z)aB ESSEL FUNCTION OF THE SECOND KIND , and Kn(z)a MODIFIED BESSEL FUNCTION OF THE FIRST KIND . Also let R[z] > 0: Then J2 n (z) /C27Y2 n (z) /C308 p2 g/C12 0K0(2z sinh t) cos(2 nt)dt: See also DIXON- FERRAR FORMULA ,W ATSON’S FORMU- LA References Gradshteyn, I. S. and Ryzhik, I. M. Eqn. 6.664.4 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 727, 2000. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1476, 1980. Nicomachus’s Theorem The nth CUBIC NUMBER n3 is a sum of n consecutive ODD NUMBERS , for example 13 /C301 23 /C303 /C275 33 /C307 /C279 /C2711 43 /C3013 /C2715 /C2717 /C2719 ; etc. This identity follows fromXn i/C301n(n /C281) /C281 /C272i ½/C138 /C30n3 : It also follows from this fact that Xn k /C301k3 /C30Xn k /C301k ! 2 : See also CUBIC NUMBER ,ODD NUMBER ,ODD NUMBER THEOREM Nicomedes’ Conchoid CONCHOID OF NICOMEDES Nielsen Generalized Polylogarithm A generalization of the POLYLOGARITHM function defined by Sn;p(z) /C30/C281ðÞn/C27p /C281 (n /C28 1)!p! g1 0ln tðÞn/C281ln 1 /C28 zt ðÞ½/C138p t dt : The function reduces to the usual POLYLOGARITHM for the case Sn/C281 ;1(z) /C30Lin(z) : The function is implemented in Mathematica 4.0 as PolyLog [n, p, z]. See also POLYLOGARITHM Nielsen-Ramanujan Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. N. Nielsen (1909) and Ramanujan (Berndt 1985) considered the integrals ak/C30g2 1lnxðÞk x/C281dx: (1) They found the values for k/C301 and 2. The general constants for k/C213 were found by Levin (1950) and, much later, independently by V. Adamchik (Finch), ap/C30p!z(p/C271)/C28pln 2ðÞp/C271 p/C271/C28p!Xp/C281 k/C300 /C2Lip/C271/C28k1 2CC1:CC17 ln 2ðÞk k!; (2) where z(z) is the R IEMANN ZETA FUNCTION and Lin(x) is the POLYLOGARITHM . The first few values are a1/C3012z(2)/C301 12p2(3) a2/C301 4z(3) (4) a3 /C301 15p4 /C271 4 p2 ln 2ðÞ2/C2814ln 2ðÞ4/C286Li412CC1:CC17 /C2821 4(ln 2)z(3) (5) a4 /C302 3 p2 ln 2ðÞ3/C2845ln 2ðÞ5/C2824(ln 2)Li412CC1:CC17 /C2824Li512CC1:CC17 /C2821 2 ðln 2Þ2 zð3Þþ24 zð5Þ: ð6Þ See also POLYLOGARITHM ,RIEMANN ZETA FUNCTION References Berndt, B. C. Ramanujan’s Notebooks, Part I. New York: Springer-Verlag, 1985. Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Losinek, P. "Special Values of Multidimensional Polyloga- rithms." CECM-98:106, 14 May 1998. http://www.cecm.s- fu.ca/preprints/1998pp.html#98:106. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/nielram/nielram.html. Flajolet, P. and Salvy, B. "Euler Sums and Contour Integral Representation." Experim. Math. 7,15/C1/35, 1998. Levin, V. I. "About a Problem of S. Ramanujan" [Russian]. Uspekhi Mat. Nauk 5, 161 /C1/166, 1950. Nielsen’s Spiral The SPIRAL with PARAMETRIC EQUATIONS x(t) /C30a ci(t) (1) y(t) /C30a si(t); (2) where ci(t) is the COSINE INTEGRAL and si(t) is the SINE INTEGRAL . The CESA` RO EQUATION is k /C30es=a a: (3) See also CORNU SPIRAL ,C OSINE INTEGRAL ,S INE INTEGRALReferences Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 146 /C1/147, 1997. Nil Geometry The GEOMETRY of the LIE GROUP consisting of REAL MATRICES OF THE FORM 1 xy 01 z 0012 435; i.e., the H EISENBERG GROUP . See also HEISENBERG GROUP ,LIE GROUP ,THURSTON’S GEOMETRIZATION CONJECTURE Nilalgebra NILPOTENT ALGEBRA Nilmanifold Let N be a NILPOTENT , connected, SIMPLY CONNECTED LIE GROUP , and let D be a discrete SUBGROUP of N with compact right QUOTIENT SPACE . Then N =D is called a nilmanifold. Nilpotent Algebra An algebra, also called a nilalgebra, consisting only of NILPOTENT ELEMENTS . See also NILPOTENT ELEMENT References Schafer, R. D. "Nilpotent Algebras." §3.1 in An Introduction to Nonassociative Algebras. New York: Dover, pp. 27 /C1/32, 1996. Nilpotent Element An element B of a RING is nilpotent if there exists a POSITIVE INTEGER k for which Bk /C300 :/ See also ENGEL’S THEOREM Nilpotent Group A GROUP G for which the chain of groups I /C30Z0 ⁄Z1 ⁄...⁄Zn with Zk /C271 =Zk(equal to the CENTER of G =Zk) termi- nates finitely with 0 is called a nilpotent group. Here, Zn denotes a CYCLIC GROUP of order n. See also CENTER (GROUP ), NILPOTENT LIE GROUP Nilpotent Lie Algebra AL IE ALGEBRA is nilpotent when its LOWER CENTRAL SERIES gkvanishes for some k. Any nilpotent Lie algebra is also SOLVABLE . The basic example of a nilpotent Lie algebra is the VECTOR SPACE of strictly UPPER TRIANGULAR MATRICES , such as the Lie algebra of the HEISENBERG GROUP . The following Mathematica function tests whether a Lie algebra g is nilpotent, given a list of matrices which is a basis for g:/ MatrixBasis[a_- List]: /C30Partition[#1,Length[a[[1]]]]&/@ LatticeReduce[Flatten/@a] LieCommutator[a_,b_]: /C30a.b-b.a NextLCS[gold_List, {}] /C30{}; NextLCS[gold_List, g_List]: /C30 MatrixBasis[Flatten[Outer[LieCommutator,gold,- g,1],1]] NilpotentLieQ[g_List]: /C30 FixedPoint[NextLCS[g,#1]&,g] /C30/C30{} For example, borel5 /C30Flatten[Table[ReplacePart[ Ta- ble[0,{i,5},{j,5}],1,{k,l}],{k,5},{l,k,5}],1]; NilpotentLieQ[borel5] yieldsFalse , while uni5 /C30Flatten[Table[ReplacePart[ Ta- ble[0,{i,5},{j,5}],1,{k,l}],{k,5},{l,k- /C271,5}],1]; NilpotentLieQ[uni5] yieldsTrue . See also COMMUTATOR SERIES (LIE ALGEBRA ), LIE ALGEBRA ,LIE GROUP ,LOWER CENTRAL SERIES (LIE ALGEBRA ), NILPOTENT LIE GROUP ,REPRESENTATION (LIE ALGEBRA ), REPRESENTATION (NILPOTENT LIE GROUP ), SOLVABLE LIE GROUP ,UNIPOTENT Nilpotent Lie Group A nilpotent Lie group is a LIE GROUP G which is CONNECTED and whose LIE ALGEBRA is a NILPOTENT LIE ALGEBRA g: That is, its LOWER CENTRAL SERIES g1[ g;g] ;g2 /C30g;g1½/C138 ;... (1) eventually vanishes, gk /C300 for some k. So a nilpotent Lie group is a special case of a SOLVABLE LIE GROUP . The basic example is the GROUP of UPPER TRIANGULAR MATRICES with 1s on their diagonals, e.g., 1 a12a13 01 a23 00 12 435; (2) which is called the H EISENBERG GROUP . Its LOWER CENTRAL SERIES is given by g0 /C300 b12b13 00 b23 00 02435 (3)g 1 /C3000 c13 00 0 00 02435 (4) g 2 /C30000 0000002 435: (5) Any real nilpotent Lie group is DIFFEOMORPHIC to EUCLIDEAN SPACE . For instance, the group of ma- trices in the example above is diffeomorphic to R3 ; via the EXPONENTIAL MAPExponential Map (Lie Group). In general, the exponential map of a NILPOTENT LIE ALGEBRA is SURJECTIVE , in contrast to the more general SOLVABLE LIE GROUP . See also BOREL GROUP ,COMMUTATOR SERIES (LIE ALGEBRA ), FLAG (VECTOR SPACE ), LIE ALGEBRA ,LIE GROUP ,L OWER CENTRAL SERIES (LIE ALGEBRA ), MATRIX ,REPRESENTATION ,NILPO- TENT LIE GROUP , SOLVABLE LIE ALGEBRA ,SOLVABLE LIE GROUP ,SPLIT SOLVABLE LIE ALGEBRA ,U NIPO- TENT References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/549, 1996. Nilpotent Matrix There are two common definitions for a nilpotent matrix. 1. A SQUARE MATRIX whose EIGENVALUES are all 0. 2. A SQUARE MATRIX A such that An is the ZERO MATRIX 0 for some positive integer MATRIX POWER n, known as the index (Ayres 1962, p. 11). See also EIGENVALUE ,IDEMPOTENT MATRIX ,M ATRIX POLYNOMIAL ,SQUARE MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 11, 1962. Nilradical The set of NILPOTENT ELEMENTS in a COMMUTATIVE RING is an ideal, and it is called the nilradical. Another equivalent description is that it is the intersection of the prime ideals. It could be the zero ideal, as in the case of the integers. See also ALGEBRAIC GEOMETRY ,ALGEBRAIC NUMBER THEORY ,IDEAL ,JACOBSON RADICAL ,RADICAL (IDEAL ) Nim A game, also called TACTIX , which is played by the following rules. Given one or more piles ( NIM-HEAPS ), players alternate by taking all or some of the counters in a single heap. The player taking the last counter or stack of counters is the winner. Nim-like games are also called TAKE-AWAY GAMES and DISJUNCTIVE GAMES . If optimal strategies are used, the winner can be determined from any intermediate position by its associated NIM-VALUE . See also MISE` RE FORM,NIM-VALUE ,WYTHOFF’S GAME References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 36 /C1/38, 1987. Bogomolny, A. "The Game of Nim." http://www.cut-the- knot.com/bottom_nim.html. Bouton, C. L. "Nim, A Game with a Complete Mathematical Theory." Ann. Math. Princeton 3,35/C1/39, 1901 /C1/1902. Gardner, M. "Mathematical Games: Concerning the Game of Nim and Its Mathematical Analysis." Sci. Amer. 198, 104 /C1/111, Feb. 1958. Gardner, M. "Nim and Hackenbush." Ch. 14 in Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, pp. 142 /C1/151, 1983. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 117 /C1/120, 1990. Kraitchik, M. "Nim." §3.12.2 in Mathematical Recreations. New York: W. W. Norton, pp. 86 /C1/88, 1942. Nim-Heap A pile of counters in a game of NIM. Nim-Sum NIM-VALUE Nim-Value Every position of every IMPARTIAL GAME has a nim- value, making it equivalent to a NIM-HEAP . To find the nim-value (also called the SPRAGUE- GRUNDY NUM- BER), take the MEX of the nim-values of the possible moves. The nim-value can also be found by writing the number of counters in each heap in binary, adding without carrying, and replacing the digits with their values mod 2. If the nim-value is 0, the position is SAFE ; otherwise, it is UNSAFE . With two heaps, safe positions are (x, x) where x /C23 [1; 7]: With three heaps, (1, 2, 3), (1, 4, 5), (1, 6, 7), (2, 4, 6), (2, 5, 7), and (3, 4, 7). See also GRUNDY’S GAME,IMPARTIAL GAME,M EX, NIM,SAFE,UNSAFE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 36 /C1/38, 1987. Grundy, P. M. "Mathematics and Games." Eureka 2,6/C1/8, 1939. Sprague, R. "Uuml;ber mathematische Kampfspiele." Toˆ- hoku J. Math. 41, 438 /C1/444, 1936. n-in-a-Row TIC-TAC-TOENine Associated Points Theorem Any CUBIC CURVE that passes through eight of the nine intersections of two given cubic curves automa- tically passes through the ninth. References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. The Seven Circles Theorem and Other New Theorems. London: Stacey International, p. 15, 1974. Nine Circles Theorem Let A, B, and C be three circles in the plane, and let X be any circle touching B and C. Then build up a chain of circles such that Y : CAX ; Z : ABY ; X ? : BCZ ; Y ? : CAX ?; Z? : ABY ?; X ƒ : ABZ?; where C : C1C2C3 denotes a circle C tangent to circles C1 ; C2 ; and C3 : Although there are a number of choices for each successive tangent circle in the chain, if the choice at each stage is made appropriately, then the ninth and final circle X ƒ coincides with the first circle X (Evelyn et al. 1971, p. 58). See also CIRCLE ,S IX CIRCLES THEOREM ,S EVEN CIRCLES THEOREM References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Nine Circles Theorem." §3.4 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 58 /C1/68, 1974. Tyrrell, J. A. and Powell, M. T. "A Theorem in Circle Geometry." Bull. London Math. Soc. 3,70/C1/74, 1971. Nine-j Symbol WIGNER 9J-SYMBOL Nine-Point Center The center F (or N) of the NINE-POINT CIRCLE . It has TRIANGLE CENTER FUNCTION a/C30cos(B /C28C) /C30 cos A /C272 cos B cos C /C30bc a2b2 /C27a2c2 /C27 b2 /C28c2CC0CC12hi ; and is the MIDPOINT of the line between the CIRCUM- CENTER C and ORTHOCENTER H. It lies on the EULER LINE. See also EULER LINE,LESTER CIRCLE ,N INE-POINT CIRCLE ,NINE-POINT CONIC References Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 624, 1970. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. New York: Random House, p. 21, 1967. Dixon, R. Mathographics. New York: Dover, pp. 57 /C1/58, 1991. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 27 /C1/29, 1928. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/187, 1994. Kimberling, C. "Nine-Point Center." http://cedar.evansvil- le.edu/~ck6/tcenters/class/npcenter.html. Nine-Point Circle The CIRCLE , also called E ULER’S CIRCLE and the FEUERBACH CIRCLE , which passes through the feet of the PERPENDICULAR FA;FB;and FCdropped from the VERTICES of any TRIANGLE DABC on the sides opposite them. Euler showed in 1765 that it also passes through the MIDPOINTS MA;MB;MCof the sides of DABC :/ By F EUERBACH’S THEOREM , the nine-point circle also passes through the MIDPOINTS MHA;MHB;MHC(now called the E ULER POINTS ) of the segments which join the VERTICES and the ORTHOCENTER H. These three triples of points make nine in all, giving the circle itsname. The center Fof the nine-point circle is called the NINE-POINT CENTER . The RADIUS of the nine-point circle is R=2;where Ris the CIRCUMRADIUS . The center of K IEPERT’S HYPER- BOLA lies on the nine-point circle. The nine-point circle bisects any line from the ORTHOCENTER to a point on the CIRCUMCIRCLE . The nine-point circle of the INCENTER and EXCENTERS of a TRIANGLE is the CIRCUMCIRCLE . There are four CIRCLES that are tangent all three sides (or their extensions) of a given TRIANGLE : the INCIRCLE Iand three EXCIRCLES J1;J2;andJ3:These four circles are, in turn, all touched by the nine-point circle N. Given four arbitrary points, the four nine-points circles of the triangles formed by taking three pointsat a times are CONCURRENT (Lemoine 1904; Wells 1991, p. 209; Schro ¨der 1999). Moreover, if four points do not form an ORTHOCENTRIC SYSTEM , then there is a unique RECTANGULAR HYPERBOLA passing through them, and its center is given by the intersection ofthe nine-point circles of the points taken three at atime (Wells 1991, p. 209). Finally, the point of con- currence of the four nine-points circles is also the point of concurrence of the four circles determined bythe feet of the perpendiculars (Schro ¨der 1999). The sum of the powers of the VERTICES with regard to the nine-point circle is 1 4a2 1/C27a22/C27a23CC0CC1 : Also, FA12/C27FA22/C27FA32/C27FH2/C303R2; where Fis the NINE-POINT CENTER ,Aiare the VERTICES ,His the ORTHOCENTER , and Ris the CIRCUMRADIUS . All triangles inscribed in a given CIRCLE and having the same ORTHOCENTER have the same nine-point circle. See also COMPLETE QUADRILATERAL ,E IGHT- POINT CIRCLE THEOREM ,EULER POINT ,FEUERBACH’S THEO- REM,F ONTENE ´THEOREMS ,G RIFFITHS’ THEOREM , HART CIRCLE ,N INE-POINT CENTER ,N INE-POINT CONIC ,O RTHOCENTRIC SYSTEM ,R ECTANGULAR HY- PERBOLA References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 93 /C1/97, 1952. Brand, L. "The Eight-Point Circle and the Nine-Point Circle." Amer. Math. Monthly 51,84/C1/85, 1944. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 58 /C1/61, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 40 /C1/41, 1971. Coxeter, H. S. M. and Greitzer, S. L. "The Nine-Point Circle." §1.8 in Geometry Revisited. New York: Random House, pp. 20 /C1/22, 1967. Do¨rrie, H. "The Feuerbach Circle." §28 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 142 /C1/144, 1965. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 27 /C1/29, 1928. F. Gabriel-Marie. Exercices de ge´ome´trie. Tours, France: Maison Mame, pp. 306 /C1/314, 1912. Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, p. 59, 1977. Guggenbuhl, L. "Karl Wilhelm Feuerbach, Mathematician." Appendix to Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., pp. 89 /C1/100, 1995. Honsberger, R. "The Nine-Point Circle." §1.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 6 /C1/7, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 165 and 195 /C1/212, 1929. Lachlan, R. "The Nine-Point Circle." §123 /C1/125 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 70 /C1/71, 1893. Lange, J. Geschichte des Feuerbach’schen Kreises. Berlin, 1894. Lemoine, M. T. "Note de ge´ome´trie." Nouv. Ann. Math. 4, 400 /C1/402, 1904. Mackay, J. S. "History of the Nine-Point Circle." Proc. Edinburgh Math. Soc. 11,19/C1/61, 1892. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 119 /C1/120, 1990. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., pp. 1 /C1/4, 1995. Rouche ´, E. and de Comberousse, C. Traite ´ de ge´ome´trie plane. Paris: Gauthier-Villars, pp. 306 /C1/307, 1900. Schro ¨der, E. M. "Zwei 8-Kreise-Sa ¨tze fu¨r Vierecke." Mitt. Math. Ges. Hamburg 18, 105 /C1/117, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 73 /C1/ 74, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 158 /C1/159, 1991.Nine-Point Conic A CONIC SECTION on which the MIDPOINTS of the sides of any COMPLETE QUADRANGLE lie. The three diagonal points also lie on this conic. See also COMPLETE QUADRANGLE ,CONIC SECTION , NINE-POINT CIRCLE Nint NEAREST INTEGER FUNCTION Nint Zeta Function Let SN(s)/C30X/C12 n/C301n1=NCC0CC1CC6CC7 /C28s; (1) where [ x] denotes NEAREST INTEGER FUNCTION , i.e, the INTEGER closest to x. For s/C213, S2(s)/C302z(s/C281) (2) S3(s)/C303z(s/C282)/C274/C28sz(s) (3) S4(s)/C304z(s/C283)/C27z(s/C281): (4) /SN(n)i sa POLYNOMIAL inpwhose COEFFICIENTS are ALGEBRAIC NUMBERS whenever n/C28NisODD. The first few values are given explicitly by S3(4)/C30p2 2/C27p4 23046(5) S5(6)/C305p2 6/C27p4 36/C27p6 412 /C21 945/C28170912 /C2749928ffiffiffi 2p 25ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28ffiffiffi 1 2svuut0 @1A(6) S 6(7)/C30p2/C27p4 18/C27p6 2520/C27246013 /C27353664ffiffiffi 2p 45p7 227:(7) References Borwein, J. M.; Hsu, L. C.; Mabry, R.; Neu, K.; Roppert, J.; Tyler, D. B.; and de Weger, B. M. M. "Nearest Integer Zeta-Functions." Amer. Math. Monthly 101, 579/C1/580, 1994. Nirenberg’s Conjecture If the G AUSS MAP of a COMPLETE MINIMAL SURFACE omits a NEIGHBORHOOD of the SPHERE , then the surface is a PLANE . This was proven by Osserman (1959). Xavier (1981) subsequently generalized the result as follows. If the G AUSS MAP of a complete MINIMAL SURFACE omits]7 points, then the surface is aPLANE . See also COMPLETE MINIMAL SURFACE ,GAUSS MAP, MINIMAL SURFACE ,NEIGHBORHOOD References do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 42, 1986. Osserman, R. "Proof of a Conjecture of Nirenberg." Comm. Pure Appl. Math. 12, 229 /C1/232, 1959. Xavier, F. "The Gauss Map of a Complete Nonflat Minimal Surface Cannot Omit 7 Points on the Sphere." Ann. Math. 113, 211 /C1/214, 1981. Niven Number HARSHAD NUMBER Niven’s Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Given a POSITIVE INTEGER m /C211, let its PRIME FACTORIZATION be written m /C30pa1 1 pa2 2 pa3 3/C1/C1/C1pak k: (1) Define the functions h(n) and H(n)byh(1) /C301; H(1) /C30 1; and h(m) /C30min a1 ;a2 ...;ak ðÞ (2) H(m) /C30max a1 ;a2 ...;ak ðÞ (3) Then lim n0/C121 nXn m/C301h(m) /C301 (4) lim n0/C12Pn m/C301 h(m) /C28 nffiffiffinp /C30z3 2CC1:CC17 z(3); (5) where z(z) is the RIEMANN ZETA FUNCTION (Niven 1969). Niven (1969) also proved that lim n0/C121 nXn m/C301H(m) /C30C; (6) where C /C301 /C27X/C12 j/C3021 /C281 z(j)"#() /C301:705221... (7) (Sloane’s A033150). The CONTINUED FRACTION of Niven’s constant is 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, ... (Sloane’s A033151). The positions at which the digits 1, 2, ... first occur in the CONTINUED FRACTION are 1, 3, 10, 7, 47, 41, 34, 13, 140, 252, 20, ... (Sloane’s A033152). The sequence of largest terms in the CONTINUED FRACTION is 1, 2, 4, 8, 11, 14, 29, 372, 559, ... (Sloane’s A033153), which occur at positions 1, 3, 7, 13, 20, 35, 51, 68, 96, ... (Sloane’s A033154).References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/niven/niven.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 41, 1983. Niven, I. "Averages of Exponents in Factoring Integers." Proc. Amer. Math. Soc. 22, 356 /C1/360, 1969. Plouffe, S. "The Niven Constant." http://www.lacim.u- qam.ca/piDATA/niven.txt. Sloane, N. J. A. Sequences A033150, A033151, A033152, A033153, and A033154 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. n-Minex n-minex is defined as 10/C28n :/ See also N-PLEX References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 16, 1996. Nobbs Points Given a TRIANGLE DABC ; construct the CONTACT TRIANGLE DDEF : Then the Nobbs points are the three points D?; E ?; and F ? from which DABC and DDEF are PERSPECTIVE , as illustrated above. The Nobbs points are COLLINEAR and fall along the GERGONNE LINE. See also COLLINEAR ,C ONTACT TRIANGLE ,E VANS POINT ,FLETCHER POINT ,GERGONNE LINE,PERSPEC- TIVE TRIANGLES References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319/C1/329, 1996. Noble Number A noble number is defined as an IRRATIONAL NUMBER which has a CONTINUED FRACTION which becomes an infinite sequence of 1s at some point, n/C13a1;a2;...;an;¯1CC6CC7 : The prototype is the GOLDEN RATIO fwhose CONTIN- UED FRACTION is composed entirely of 1s, 1CC6CC7 :Any noble number can be written as n /C30An /C27 fAn /C281 Bn /C27 fBn/C271; where Akand Bkare the NUMERATOR and DENOMI- NATOR of the kth CONVERGENT of a1 ;a2 ;...;an ½/C138 : The noble numbers are a SUBFIELD of Qffiffiffi 5pCC0CC1 :/ See also NEAR NOBLE NUMBER References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 236, 1979. Schroeder, M. "Noble and Near Noble Numbers." In Frac- tals, Chaos, Power Laws: Minutes from an Infinite Para- dise. New York: W. H. Freeman, pp. 392 /C1/394, 1991. Node (Algebraic Curve) ORDINARY DOUBLE POINT Node (Fixed Point) A FIXED POINT for which the STABILITY MATRIX has both EIGENVALUES of the same sign (i.e., both are POSITIVE or both are NEGATIVE ). If l1 B l2 B0 ; then the node is called STABLE ;if l1 B l2 B0 ; then the node is called an UNSTABLE NODE . See also STABLE NODE,UNSTABLE NODE Node (Graph) A synonym for a VERTEX of a GRAPH , i.e., one of the points on which the graph may is defined and which may be connected by EDGES . The terms "point," "junction," and 0-simplex are also used (Harary 1994; Skiena 1990, p. 80). The following tables gives the total numbers of nodes for various classes of graphs on n /C301, 2, ... nodes. graph type Sloane total node count for n /C301, 2, ...nodes GRAPH A055543 1, 4, 12, 44, 170, 936, ... TREE A055544 1, 2, 3, 8, 15, 36, 77, 184 ... LABELED TREEA000169 1, 2, 9, 64, 625, ... ROOTEDTREE A055545 1, 2, 6, 16, 45, 120, ...See also EDGE (GRAPH ), GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A000169/M1946, A055543, A055544, and A055545 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Noetherian Module A MODULE M is Noetherian if every submodule is finitely generated. See also NOETHERIAN RING Noetherian Ring An abstract commutative RING satisfying the abstract chain condition. See also LOCAL RING,NOETHER- LASKER THEOREM Noether-Lasker Theorem Let Mbe a finitely generated MODULE over a commutative N OETHERIAN RING R. Then there exists a finite set Nij15i5l fg of submodules of Msuch that 1.Sl i/C301Ni/C300 andSi"i0Niis not contained in Ni0for all 15i05l:/ 2. Each quotient M=Niis primary for some prime Pi:/ 3. The Piare all distinct for 1 5i5l:/ 4. Uniqueness of the primary component Niis equivalent to the statement that Pidoes not contain Pjfor any j"i:/ Noether’s Fundamental Theorem If two curves fandcofMULTIPLICITIES ri"0 and si"0 have only ordinary points or ordinary singular points and CUSPS in common, then every curve which has at least MULTIPLICITY ri/C27si/C281 at every point (distinct or infinitely near) can be written f/C13fc?/C27cf?/C300; where the curves f?andc?have MULTIPLICITIES at least ri/C281 and si/C281:/ References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 29 /C1/30, 1959. Noether’s Symmetry Theorem An extremely powerful theorem in physics which states that each SYMMETRY of a system leads to a physically conserved quantity. SYMMETRY under TRANSLATION corresponds to momentum conserva- tion, SYMMETRY under ROTATION to angular momen- tum conservation, SYMMETRY in time to energy conservation, etc. See also SYMMETRY Noether’s Transformation Theorem Any irreducible curve may be carried by a factorable CREMONA TRANSFORMATION into one with none but ordinary singular points. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 207, 1959. Noise An error which is superimposed on top of a true signal. Noise may be random or systematic. Noise can be greatly reduced by transmitting signals digitally instead of in analog form because each piece of information is allowed only discrete values which are spaced farther apart than the contribution due to noise. CODING THEORY studies how to encode information efficiently, and ERROR-CORRECTING CODES devise methods for transmitting and reconstructing infor- mation in the presence of noise. See also ERROR ,STOCHASTIC FUNCTION References Abbott, D. and Kiss, L. B. (Eds.). Proc. 2nd Internat. Conf. Unsolved Problems of Noise and Fluctuations, 11 /C1/15 July, Adelaide Melville, NY: Amer. Inst. Physics Press,2000. Davenport, W. B. and Root, W. L. An Introduction to the Theory of Random Signals and Noise. New York: IEEE Press, 1987. McDonough, R. N. and Whalen, A. D. Detection of Signals in Noise, 2nd ed. Orlando, FL: Academic Press, 1995. Pierce, J. R. Symbols, Signals and Noise: The Nature and Process of Communication. New York: Harper & Row, 1961. Vainshtein, L. A. and Zubakov, V. D. Extraction of Signals from Noise. New York: Dover, 1970. van der Ziel, A. Noise: Sources, Characterization, Measure- ment. New York: Prentice-Hall, 1954. van der Ziel, A. Noise in Measurement. New York: Wiley, 1976. Wax, N. Selected Papers on Noise and Stochastic Processes. New York: Dover, 1954. Weisstein, E. W. "Books about Noise." http://www.treasure- troves.com/books/Noise.html. Noise Sphere A mapping of RANDOM NUMBER TRIPLES to points in SPHERICAL COORDINATES according tou ¼ 2pXn f ¼ pXnþ1 r ¼ffiffiffiffiffiffiffiffiffiffi Xnþ2q in order to detect unexpected structure indicating correlations between triples. When such structure is present (note that this does not include the expected bunching of points along the z-axis according to the factor sin f in the spherical volume element), num- bers may not be truly RANDOM . See also BALL POINT PICKING ,R ANDOM NUMBER , SPHERE POINT PICKING References Pickover, C. A. Computers and the Imagination. New York: St. Martin’s Press, 1991. Pickover, C. A. "Computers, Randomness, Mind, and In- finity." Ch. 31 in Keys to Infinity. New York: W. H. Freeman, pp. 233 /C1/247, 1995. Richards, T. "Graphical Representation of Pseudorandom Sequences." Computers and Graphics 13, 261 /C1/262, 1989. Nolid An assemblage of faces forming a POLYHEDRON of zero VOLUME (Holden 1991, p. 124). See also ACOPTIC POLYHEDRON References Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Nome Given a J ACOBI THETA FUNCTION , the nome is defined as qkðÞ/C13e pit /C30e /C28 pK ? kðÞ=KkðÞ/C30e /C28pKffiffiffiffiffiffiffiffiffi 1 /C28k2pðÞ =KkðÞ(1) (Borwein and Borwein 1987, pp. 41, 109 and 114), where t is the HALF-PERIOD RATIO , KkðÞ is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND , m /C30 k2 is the PARAMETER , and k is the MODULUS . The nome is implemented in Mathematica asElliptic- NomeQ [m]. Various notations for JACOBI THETA FUNCTIONS invol- ving the nome include qiz;qðÞ/C13q z tjÞ; ð (2) where t is the HALF-PERIOD RATIO (Whittaker and Watson 1972, p. 464) and qi /C13q 0; qðÞ : (3) See also AMPLITUDE ,CHARACTERISTIC (ELLIPTIC IN- TEGRAL ), ELLIPTIC INTEGRAL ,H ALF-PERIOD RATIO, INVERSE NOME,JACOBI THETA FUNCTIONS ,MODULAR ANGLE ,MODULAR DISCRIMINANT ,MODULUS (ELLIPTIC INTEGRAL ), PARAMETER References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 591, 1972. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. n-Omino POLYOMINO Nomogram A graphical plot which can be used for solving certain types of equations. According to Steinhaus (1983, p. 301), the Nomogram was invented by the French mathematicians Massau and M. P. Ocagne in 1889. References Iyanaga, S. and Kawada, Y. (Eds.). "Nomograms." §282 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 891 /C1/893, 1980. Menzel, D. (Ed.). Fundamental Formulas of Physics, Vol. 1. New York: Dover, p. 141, 1960. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 92 /C1/95 and 301, 1999. Whittaker, E. T. and Robinson, G. "Nomography." §128 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 128 /C1/130, 1967. Nomograph NOMOGRAMNon-Abelian A GROUP or other algebraic object is called non- Abelian is the law of commutativity does not always hold, i.e., if the object is not ABELIAN . For example, the group of INVERTIBLE MATRICES is non-Abelian, as can be seen by comparing 10 0 /C281CC60CC61 01 /C2810CC60CC61 /C300110CC60CC61 (1) and 01 /C2810CC60CC61 10 0 /C281CC60CC61 /C300 /C281 /C2810CC60CC61 : (2) See also A BELIAN ,ABELIANIZATION ,GROUP ,RING Nonadjacent Vertex Pairs The following table gives the number of nonadjacent vertex pairs k on graphs of n /C301, 2, ... vertices. k counts 1 0,1,1,1,1,1,1,... 2 0,0,1,2,2,2,2,... 3 0,0,1,3,4,5,5,... 4 0,0,0,2,6,9,10,... 5 0, 0, 0, 1, 6, 15, 21, ... See also ORE GRAPH Nonagon A 9-sided polygon, also known as an enneagon. Although the term "enneagon" is perhaps preferable (since it uses the Greek prefix and suffix instead ofthe mixed Roman/Greek nonagon), the term "nona-gon," which is simpler to spell and pronounce, is used in this work. The REGULAR POLYGON with nine sides and S CHLA ¨FLI SYMBOL 9fg:/ The nonagon cannot be constructed using the classi-cal Greek rules of GEOMETRIC CONSTRUCTION , but Conway and Guy (1996) give a NEUSIS CONSTRUCTION based on TRISECTION . Madachy (1979) illustrates how to construct a nonagon by folding and knotting a strip of paper. Although the regular nonagon is not a CONSTRUCTIBLE POLYGON , Dixon (1991) gives con- structions for several angles which are close approx- imations to the nonagonal angle 360/C14=9 /C302 p=9; including angles of tan/C281 5=6ðÞ:39 :805571 /C14and 2 tan/C281ffiffiffi 3p /C281CC0CC1 =2CC0CC1 :40:207819 /C14:/ Given a regular nonagon, let MAB be the MIDPOINT of one side, XBCbe the MID-ARC POINT of the arc connecting an adjacent side, and MOXthe MIDPOINT of OXBC : Then, amazingly, /C218OMABMOX /C3030/C14 (Karst, quoted in Bankoff and Garfunkel 1973). See also NONAGRAM ,TRIGONOMETRY VALUES PI/9 References Bankoff, L. and Garfunkel, J. "The Heptagonal Triangle." Math. Mag. 46,7/C1/19, 1973. Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 60 /C1/61, 1982. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 194 /C1/200, 1996. Dixon, R. Mathographics. New York: Dover, pp. 40 /C1/44, 1991. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 60 /C1/61, 1979. Nonagonal Heptagonal Number A number which is simultaneously a NONAGONAL NUMBER Nmand HEPTAGONAL NUMBER Hepnand therefore satisfies the DIOPHANTINE EQUATION 1 2m(7m /C285) /C3012n(5n /C284): (1) COMPLETING THE SQUARE and rearranging gives (14n /C285)2 /C287(10m /C283)2 /C3062 : (2) Defining x /C3014n /C285 and y /C3010m /C283 gives the Pell- like equation x2 /C287y2 /C3062 : (3) The first integral solutions in m and n are (m;n) /C30 (1;1); (88, 104), (12445, 14725), (1767052, 2090804), ... (Sloane’s A048919 and A048920), giving the nonago- nal heptagonal numbers 1, 26884, 542041975, 10928650279834, ... (Sloane’s A048921). See also HEPTAGONAL NUMBER ,NONAGONAL NUMBER References Sloane, N. J. A. Sequences A048919, A048920, and A048921 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Nonagonal Hexagonal Number A number which is simultaneously a NONAGONAL NUMBER Nmand HEXAGONAL NUMBER Hexnand therefore satisfies the DIOPHANTINE EQUATION12m(7m /C285) /C30n(2n /C281): (1) COMPLETING THE SQUARE and rearranging gives (14n /C285)2 /C287(4m /C281)2 /C3018: (2) Defining x /C3014n /C285 and y /C304m /C281 gives the Pell- like equation x2 /C287y2 /C3018 : (3) This has fundamental solutions (x; y) /C30(5;1); (9, 3), and (19, 17), giving the family of solutions (5, 1), (9, 3), (19, 17), (61, 23), (135, 51), (509, 193), .... These give solutions which are integers in m and n of (m;n) /C30(1; 1); (10, 13), (39025, 51625), ... (Sloane’s A048916 and A048917), giving the nonagonal hex- agonal numbers 1, 325, 5330229625,1353857339341, 22184715227362706161, ... (Sloane’s A048918). See also HEXAGONAL NUMBER ,NONAGONAL NUMBER References Sloane, N. J. A. Sequences A048916, A048917, and A048918 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. See also NONAGONAL NUMBER Nonagonal Number A FIGURATE NUMBER OF THE FORM n(7n/C285) =2; also called anENNEAGONAL NUMBER . The first few are 1, 9, 24, 46, 75, 111, 154, 204, ... (Sloane’s A001106). The first few odd nonagonal numbers are 1, 9, 75, 11, 261, 325, ... (Sloane’s A028991), and the first few even nonagonal numbers are 24, 46, 154, 204, 396, ... (Sloane’s A028992). See also FIGURATE NUMBER ,NONAGONAL HEPTAGO- NAL NUMBER ,N ONAGONAL HEXAGONAL NUMBER , NONAGONAL OCTAGONAL NUMBER ,NONAGONAL PEN- TAGONAL NUMBER ,N ONAGONAL SQUARE NUMBER , NONAGONAL TRIANGULAR NUMBER ,POLYGONAL NUM- BER References Sloane, N. J. A. Sequences A001106/M4604, A028991, and A028992 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Nonagonal Octagonal Number A number which is simultaneously a NONAGONAL NUMBER Nmand OCTAGONAL NUMBER Onand there- fore satisfies the DIOPHANTINE EQUATION 1 2m(7m /C285) /C30n(3n /C282): (1) COMPLETING THE SQUARE and rearranging gives (14n /C285)2 /C2856(3m /C281)2 /C3019 : (2) Defining x /C3014n /C285 and y /C303m /C281 gives the Pell- like equation 3x2 /C2856y2 /C3019: (3) The first integral solutions in m and n are (m;n) /C30 (1;1); (425, 459), (286209, 309141), (192904201, 208360351), ... (Sloane’s A048922 and A048923), giving the nonagonal octagonal numbers 1, 631125, 286703855361, 130242107189808901, ... (Sloane’s A048924). See also NONAGONAL NUMBER ,OCTAGONAL NUMBER References Sloane, N. J. A. Sequences A048922, A048923, and A048924 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Nonagonal Pentagonal Number A number which is simultaneously a NONAGONAL NUMBER Nm and PENTAGONAL NUMBER Pn and there- fore satisfies the DIOPHANTINE EQUATION 1 2m(7m /C285) /C3012n(3n /C281): (1) COMPLETING THE SQUARE and rearranging gives 3(14n /C285)2 /C287(6m /C281)2 /C3068 : (2) Defining x /C3014n /C285 and y /C306m /C271 gives the Pell- like equation 3x2 /C287y2 /C3068: (3) This has solutions in (x, y) corresponding to solutions which are integral in m and n of (m;n) /C30(1;1); (14, 21), (7189, 10981), (165026, 252081), (86968201, 132846121), ... (Sloane’s A048913 and A048914), giving the nonagonal pentagonal numbers 1, 651, 180868051, 95317119801, 26472137730696901, ... (Sloane’s A048915). See also NONAGONAL NUMBER ,PENTAGONAL NUMBER References Sloane, N. J. A. Sequences A048913, A048914, and A048915 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Nonagonal Square Number A number which is simultaneously a NONAGONAL NUMBER Nm and a SQUARE NUMBER Snand therefore satisfies the DIOPHANTINE EQUATION 1 2m(7m /C285) /C30n2 : (1) COMPLETING THE SQUARE and rearranging gives (14n /C285)2 /C2856m2 /C3025: (2) Defining x /C3014n /C285 and y /C302m2 gives the Pell-like equation x2 /C2814y2 /C3025: (3) This has unit solutions (x;y) /C30(9; 2); (23, 6), and (75, 20), which lead to the family of solutions (9, 2), (23, 6), (75, 20), (247, 66), (681, 182), (2245, 600), .... The corresponding integer solutions in n and m are (n;m) /C30(1; 1); (2, 3), (18, 33), (49, 91), (529, 989), ... (Sloane’s A048910 and A048911), giving the nonago- nal square numbers 1, 9, 1089, 8281, 978121, 7436529, ... (Sloane’s A048912). See also NONAGONAL NUMBER ,SQUARE NUMBER References Sloane, N. J. A. Sequences A048910, A048911, and A048912 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Nonagonal Triangular Number A number which is simultaneously a NONAGONAL NUMBER Nmand a TRIANGULAR NUMBER Tnand therefore satisfies the DIOPHANTINE EQUATION . 1 2m(7m /C285) /C3012n(1 /C27n) : (1) COMPLETING THE SQUARE and rearranging gives (14n /C285)2 /C287(2m /C271)2 /C3018: (2) Defining x /C3014n /C285 and y /C302m /C271 gives the Pell- like equation x2 /C287y2 /C3018 : (3) This has unit solutions (x;y) /C30(5; 1); (9, 3), and (19, 7), which lead to the family of solutions (5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), (971, 367), .... The corresponding integer solutions in n and m are (n;m) /C30(1; 1); (10, 25), (154, 406), (2449, 6478), ... (Sloane’s A048907 and A048908), giving the nonago- nal triangular numbers 1, 325, 82621, 20985481, 5330229625, 1353857339341, ... (Sloane’s A048909). See also NONAGONAL NUMBER ,TRIANGULAR NUMBER References Sloane, N. J. A. Sequences A048907, A048908, and A048909 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Nonagram The STAR FIGURE 9=3fg composed of three EQUILAT- ERAL TRIANGLES rotated at angles 08,408, and 80 8.It has been called the STAR OF GOLIATH by analogy with the STAR OF DAVID (HEXAGRAM ). See also HEXAGRAM ,NONAGON ,STAR FIGURE ,TRIGO- NOMETRY VALUES PI/9 Nonahedral Graph A POLYHEDRAL GRAPH having nine vertices. There are 2606 nonisomorphic nonahedral graphs, as first en- umerated by Federico (1969; Duijvestijn and Federico 1981). See also NONAHEDRON ,POLYHEDRAL GRAPH References Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral ( -Connected Planar) Graphs." Math. Com- put. 37, 523 /C1/532, 1981. Federico, P. J. "Enumeration of Polyhedra: The Number of 9-hedra." J. Combin. Th. 7, 155 /C1/161, 1969. Nonahedron A nine-faced POLYHEDRON . There are 2606 topologi- cally distinct convex nonahedra, corresponding to the 2606 nonisomorphic NONAHEDRAL GRAPHS . See also NONAHEDRAL GRAPH Nonalternating Knot A KNOT which is not ALTERNATING . Unlike alternating knots, FLYPE moves are not sufficient to pass between all minimal diagrams of a given nonalternating knot (Hoste et al. 1998). In fact, Thistlethwaite used 13 different moves in generating a list of 16-crossing alternating knots (Hoste et al. 1998), and still had 9,868 duplicates out of a list of 1,018,774 knots (Hoste et al. 1998). See also ALTERNATING KNOT,KNOTReferences Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Non-Archimedean Field See also HENSEL’S LEMMA ,NON-ARCHIMEDEAN GEO- METRY ,NON-ARCHIMEDEAN VALUATION ,VALUATION Non-Archimedean Geometry A geometry in which ARCHIMEDES’ AXIOM does not hold.See also A RCHIMEDES’ AXIOM ,H ORN ANGLE ,N ON- ARCHIMEDEAN FIELD,NON-ARCHIMEDEAN VALUATION References Itoˆ, K. (Ed.). §155D in Encyclopedic Dictionary of Mathe- matics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 611, 1986. Non-Archimedean Valuation See also NON-ARCHIMEDEAN FIELD,N ON-ARCHIME- DEAN GEOMETRY Nonarithmetic Progression Sequence Given two starting numbers a1;a2 ðÞ ;the following table gives the unique sequences aifg that contain no three-term arithmetic progressions. Sloane sequence A003278 1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, ... A033156 1, 3, 4, 6, 10, 12, 13, 15, 28, 30, 31, 33, ... A033157 1, 4, 5, 8, 10, 13, 14, 17, 28, 31, 32, 35, ... A033158 1, 5, 6, 8, 12, 13, 17, 24, 27, 32, 34, 38, ... A033159 2, 3, 5, 6, 11, 12, 14, 15, 29, 30, 32, 33, ... A033160 2, 4, 5, 7, 11, 13, 14, 16, 29, 31, 32, 34, ... A033161 2, 5, 6, 9, 11, 14, 15, 18, 29, 32, 33, 36, ... A033162 3, 4, 6, 7, 12, 13, 15, 16, 30, 31, 33, 34, ... A033163 3, 5, 6, 8, 12, 14, 15, 17, 30, 32, 33, 35, ... A033164 4, 5, 7, 8, 13, 14, 16, 17, 31, 32, 34, 35, ... See also ARITHMETIC SEQUENCE References Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences." Theor. Comput. Sci. 98, 163 /C1/197, 1992. Erdos, P. and Tura´n, P. "On Some Sequences of Integers." J. London Math. Soc. 11, 261 /C1/264, 1936. Gerver, J.; Propp, J.; and Simpson, J. "Greedily Partitioning the Natural Numbers into Sets Free of Arithmetic Pro- gressions." Proc. Amer. Math. Soc. 102, 765 /C1/772, 1988. Guy, R. K. "Theorem of van der Waerden, Szemere ´di’s Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 204 /C1/209, 1994. Iacobescu, F. "Smarandache Partition Type and Other Sequences." Bull. Pure Appl. Sci. 16E, 237 /C1/240, 1997. Ibstedt, H. "A Few Smarandache Sequences." Smarandache Notions J. 8, 170 /C1/183, 1997. Sloane, N. J. A. Sequences A003278/M0975, A033156, A033157, A033158, A033159, A033160, A033161, A033162, A033163, and A033164 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Nonassociative Algebra An ALGEBRA which does not satisfy a(bc) /C30(ab)c is called a nonassociative algebra. See also ALGEBRA ,CAYLEY NUMBER ,COMPLEX NUM- BER,DIVISION ALGEBRA ,QUATERNION ,REAL NUMBER References Kuz’min, E. N. and Shestakov, I. P. "Non-Associative Struc- tures." In Algebra VI. Combinatorial and Asymptotic Methods of Algebra: Nonassociative Structures (Ed. A. I. Kostrikin and I. R. Shafarevich). New York: Springer- Verlag, 1995. Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, 1996. Nonassociative Product The number of nonassociative n-products with k elements preceding the rightmost left parameter is F ðn;kÞ¼F ðn /C281; kÞþF ðn /C281 ;k /C281 Þ ¼n þ k /C282 kCC1nCC1o /C28n þ k /C281 k /C281CC1nCC1o wheren kCC0CC1 is a BINOMIAL COEFFICIENT . The number of n-products in a nonassociative algebra is FnðÞ/C30Cn /C30Xn/C282 j/C300Fn; jðÞ/C302n /C28 2 ðÞ ! n! n /C28 1 ðÞ ! ;where Cn is a CATALAN NUMBER , 1, 1, 2, 5, 14, 42, 132, ... (Sloane’s A000108). References Niven, I. M. Mathematics of Choice: Or, How to Count Without Counting. Washington, DC: Math. Assoc. Amer., pp. 140 /C1/152, 1965. Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Nonaveraging Sequence N.B. A detailed online essay by S. Finch was the starting point for this entry. A sequence of POSITIVE INTEGERS 1 5a1 Ba2 Ba3 ... is a nonaveraging sequence if it contains no three terms which are in an ARITHMETIC PROGRESSION , i.e., terms such that 1 2ai /C27ajCC0CC1 /C30ak for distinct ai ; aj ; ak : The EMPTY SET and sets of length one are therefore trivially nonaveraging. Consider all possible subsets on the integers Sn /C30 1; 2;...; n fg : There is one nonaveraging sequence on S0 (/¥); two on S1 (/¥ and 1fg) ; four on S2 ; and so on. For example, 13 of the 16 subjects of S4 are nonaver- aging, with 1; 2;3 fg ; 2 ;3; fg ; and 1 ;2;3 ;4 fg excluded. The numbers of nonaveraging subsets on S0 ; S1 ; ... are 1, 2, 4, 7, 13, 23, 40, ... (Sloane’s A051013). Wro´blewski (1984) showed that for infinite nonaver- aging sequences, SAðÞ/C13 sup all nonaveraging sequencesX/C12 k/C3011 ak>3:00849 : See also NONDIVIDING SET References Abbott, H. L. "On a Conjecture of Erdos and Straus on Non- Averaging Sets of Integers." In Proceedings of the Fifth British Combinatorial Conference (Es. C. St. J. A. Nash- Williams and J. Sheehan). Winnipeg, Manitoba, Canada: Utilitas Math. Pub., pp. 1 /C1/4, 1976. Abbott, H. L. "Extremal Problems on Non-Averaging and Non-Dividing Sets." Pacific J. Math. 91,1/C1/12, 1980. Abbott, H. L. "On the Erdos-Straus Non-Averaging Set Problem." Acta Math. Hungar. 47, 117/C1/119, 1986. Behrend, F. "On Sets of Integers which Contain no Three Terms in an Arithmetic Progression." Proc. Nat. Acad. Sci. USA 32, 331/C1/332, 1946. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/erdos/erdos.html. Gerver, J. L. "The Sum of the Reciprocals of a Set of Integers with No Arithmetic Progression of kTerms." Proc. Amer. Math. Soc. 62, 211/C1/214, 1977. Gerver, J. L. and Ramsey, L. "Sets of Integers with no Long Arithmetic Progressions Generated by the Greedy Algo-rithm." Math. Comput. 33, 1353/C1 /1360, 1979. Guy, R. K. "Nonaveraging Sets. Nondividing Sets." §C16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 131 /C1/132, 1994. Sloane, N. J. A. Sequences A051013 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Straus, E. G. "Non-Averaging Sets." Proc. Symp. Pure Math 19, 215 /C1/222, 1971. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Wro´blewski, J. "A Nonaveraging Set of Integers with a Large Sum of Reciprocals." Math. Comput. 43, 261 /C1/262, 1984. Noncentral Distribution CHI-SQUARED DISTRIBUTION , F-DISTRIBUTION ,STU- DENT’S T-DISTRIBUTION Noncommutative Group A group whose elements do not commute. The simplest noncommutative GROUP is the DIHEDRAL GROUP D3 of ORDER six. See also COMMUTATIVE ,FINITE GROUP D3,GROUP Noncommutative Ring This entry contributed by VIKTOR BENGTSSON A noncommutative ring R is a RING in which the law of multiplicative commutativity is not satisfied, i.e., a /C215 b "b /C215 a for any two elements a;b /C23 R: In such a case, the elements a and b of the ring R are said not to commute. An important example of a noncommuta- tive ring is the ring MnKðÞ consisting of all n /C29n matrices whose elements are members of the FIELD K. See also RING Nonconformal Map Let g be a path in C ; w /C30fzðÞ; and u and f be the tangents to the curves g and f gðÞat z0 and w0 : If there is an N such that f ðN Þðz0 Þ"0 ð1Þ f ðN Þðz0 Þ¼0 ð2Þ for all n BN (or, equivalently, if f ? zðÞhas a zero of order N /C281); then fzðÞ/C30fz0ðÞ/C27f NðÞz0ðÞ N! /C2 z /C28z0 ðÞN/C27f N /C271 ðÞz0ðÞ N /C27 1 ðÞ !z /C28z0 ðÞN /C271/C27/C1/C1/C1 (3) fzðÞ/C28fz0ðÞ /C30 z /C28z0 ðÞNfNðÞz0ðÞ N!/C27f N /C271 ðÞz0ðÞ N /C27 1 ðÞ !z /C28z0 ðÞ /C27/C1/C1/C1"# ; (4)so the ARGUMENT is arg fzðÞ/C28fz0ðÞ ½/C138 /C30N arg z /C28z0 ðÞ /C27argfNðÞz0ðÞ N!" /C27f N /C271 ðÞz0ðÞ N /C27 1 ðÞ !z /C28z0 ðÞ /C27...CC61 : (5) As z 0 z0 ; arg z /C28z0 ðÞ 0 u and arg fzðÞ/C28fz0ðÞ ½/C138 jj 0 f ; f /C30N u /C27argfNðÞz0ðÞ N!"# /C30N u /C27arg fNðÞz0ðÞ ½/C138 : (6) See also CONFORMAL MAPPING Nonconstructive Proof A PROOF which indirectly shows a mathematical object exists without providing a specific example or algorithm for producing an example. Nonconstructive proofs are also called existence proofs. See also EXISTENCE PROBLEM ,PROOF References Courant, R. and Robbins, H. "The Indirect Method of Proof." §2.4.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 86 /C1/87, 1996. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 229, 1998. Noncototient A POSITIVE value of n for which x /C28 f xðÞ/C30n has no solution, where f(x) is the TOTIENT FUNCTION . The first few are 10, 26, 34, 50, 52, ... (Sloane’s A005278). See also NONTOTIENT ,TOTIENT FUNCTION References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 91, 1994. Sloane, N. J. A. Sequences A005278/M4688 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Noncylindrical Ruled Surface A RULED SURFACE parameterization x u;vðÞ/C30b uðÞ/C27 vg uðÞis called noncylindrical if g /C29g? is nowhere 0: A noncylindrical ruled surface always has a parameter- ization OF THE FORM x u ;vðÞ/C30 s uðÞ/C27v d uðÞ; where djj/C301 and s?/C215 d ?/C300 ; where s is called the STRICTION CURVE of x and d the DIRECTOR CURVE . See also DISTRIBUTION PARAMETER ,RULED SURFACE , STRICTION CURVE References Gray, A. "Noncylindrical Ruled Surfaces." §19.4 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 445 /C1/ 448, 1997. Nondecreasing Function A function f(x) is said to be nondecreasing on an INTERVAL I if fbðÞ]faðÞfor all b /C21a, where a ;b /C23 I : Conversely, a function f(x) is said to be nonincreasing on an INTERVAL I if fbðÞ5faðÞfor all b /C21a with a ;b /C23 I :/ See also DECREASING FUNCTION ,M ONOTONE DE- CREASING ,M ONOTONE INCREASING ,N ONINCREASING FUNCTION References Jeffreys, H. and Jeffreys, B. S. "Increasing and Decreasing Functions." §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988. Nondividing Set A SET in which no element divides the SUM of any nonempty subset of the other elements. The EMPTY SET and sets of length one are therefore trivially nondividing. Also, any set other than 1fg which contains 1 is dividing. For example, 2 ;3;5 fg is dividing, since 2 3 /C275 ðÞj (and 5 2 /C273 ðÞ ) ; j but 4;6 ;7 fg is nondividing since 4 divides none of 6; 7;(6 þ 7) fg ; and similarly for 6 and 7. Consider all possible subsets on the integers Sn /C30 1; 2;...;n fg : Then the numbers of nondividing sub- sets on S0 ; S1 ; ... are 1, 2, 3, 5, 7, 12, 16, 28, 38, 60, ... (Sloane’s A051014). For example, the 12 nondividing sets in S6are ¥; 1fg; 2fg; 3fg; 4fg; 5fg; 6fg; 2;3fg ; 2; 5fg ; 3;4fg ; 3 ;5fg ; 4; 5fg ;f4;6 g; 5 ;6fg ; 3; 4;5 fg ; and 4; 5;6 fg :/ See also NONAVERAGING SEQUENCE ,PRIMITIVE SE- QUENCE References Abbott, H. L. "Extremal Problems on Non-Averaging and Non-Dividing Sets." Pacific J. Math. 91,1/C1/12, 1980. Guy, R. K. "Nonaveraging Sets. Nondividing Sets." §C16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 131 /C1/132, 1994. Sloane, N. J. A. Sequences A051014 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Straus, E. G. "Non-Averaging Sets." Proc. Symp. Pure Math 19, 215 /C1/222, 1971. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Nonequivalent If A [!B and B [!A (i.e., A [!B ðÞffl B [!A ðÞ ; where !A denotes NOT, [ denotes IMPLIES , and ffl denotes AND), then A and B are said to be inequivalent, arelationship which is written symbolically as A fB; AbB; A uXB Nonequivalence is implemented in Mathematica as Unequal [A, B, ...]. Binary none- quivalence has the same TRUTH TABLE as XOR (i.e., EXCLUSIVE DISJUNCTION ), reproduced below. AB /A fB/ TTF TFT FTT FFF See also CONNECTIVE ,EQUIVALENT ,EXCLUSIVE DIS- JUNCTION , XOR Nonessential Singularity REGULAR SINGULAR POINT Non-Euclidean Geometry In three dimensions, there are three classes of constant curvature GEOMETRIES . All are based on the first four of EUCLID’S POSTULATES , but each uses its own version of the PARALLEL POSTULATE . The "flat" geometry of everyday intuition is called EUCLIDEAN GEOMETRY (or PARABOLIC GEOMETRY ), and the non- Euclidean geometries are called HYPERBOLIC GEOME- TRY (or LOBACHEVSKY- BOLYAI- GAUSS GEOMETRY ) and ELLIPTIC GEOMETRY (or RIEMANNIAN GEOMETRY ). SPHERICAL GEOMETRY is a non-Euclidean 2-D geome- try. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as EUCLIDEAN GEOMETRY . See also ABSOLUTE GEOMETRY ,ELLIPTIC GEOMETRY , EUCLID’S POSTULATES ,E UCLIDEAN GEOMETRY ,H Y- PERBOLIC GEOMETRY ,PARALLEL POSTULATE ,SPHERI- CAL GEOMETRY References --. "Welcome to the Non-Euclidean Geometry Homepage." http://members.tripod.com/~noneuclidean/. Bolyai, J. "Scientiam spatii absolute veritam exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) indepentem: adjecta ad casum falsi-tatis, quadratura circuli geometrica." Reprinted as "TheScience of Absolute Space" in Bonola, R. Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Loba-chevski, with a Supplement Containing The Science of Absolute Space by John Bolyai. New York: Dover, 1955. Bonola, R. Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai.New York: Dover, 1955. Borsuk, K. Foundations of Geometry: Euclidean and Bolyai- Lobachevskian Geometry. Projective Geometry. Amster- dam, Netherlands: North-Holland, 1960. Carslaw, H. S. The Elements of Non-Euclidean Plane Geo- metry and Trigonometry. London: Longmans, 1916. Coxeter, H. S. M. Non-Euclidean Geometry, 6th ed. Wa- shington, DC: Math. Assoc. Amer., 1988. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 53 /C1/60, 1990. Greenberg, M. J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W. H. Freeman, 1994. Iversen, B. An Invitation to Hyperbolic Geometry. Cam- bridge, England: Cambridge University Press, 1993. Iyanaga, S. and Kawada, Y. (Eds.). "Non-Euclidean Geome- try." §283 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 893 /C1/896, 1980. Lobachevski, N. Reprinted as "Theory of Parallels" in Bonola, R. Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai. New York: Dover, 1955. Martin, G. E. The Foundations of Geometry and the Non- Euclidean Plane. New York: Springer-Verlag, 1975. Pappas, T. "A Non-Euclidean World." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 90 /C1/ 92, 1989. Ramsay, A. and Richtmeyer, R. D. Introduction to Hyper- bolic Geometry. New York: Springer-Verlag, 1995. Sommerville, D. Y. The Elements of Non-Euclidean Geome- try. London: Bell, 1914. Sommerville, D. Y. Bibliography of Non-Euclidean Geome- try, 2nd ed. New York: Chelsea, 1960. Sved, M. Journey into Geometries. Washington, DC: Math. Assoc. Amer., 1991. Trudeau, R. J. The Non-Euclidean Revolution. Boston, MA: Birkha ¨user, 1987. Weisstein, E. W. "Books about Non-Euclidean Geometry." http://www.treasure-troves.com/books/Non-EuclideanGeo- metry.html. Woods, F. S. "Non-Euclidean Geometry." Ch. 3 in Mono- graphs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 93 /C1/147, 1955. Nonhyperbolic Knot HYPERBOLIC KNOT,SATELLITE KNOT,TORUS KNOT Nonic Surface An ALGEBRAIC SURFACE of degree 9. See also ALGEBRAIC SURFACE Nonillion In the American system, 1030. See also LARGE NUMBER Nonincreasing Function A function f(x) is said to be nonincreasing on an INTERVAL I if fbðÞ5faðÞfor all b /C21a, where a ;b /C23 I : Conversely, a function f(x) is said to be nondecreasing on an INTERVAL I if fbðÞ]faðÞfor all b /C21a with a ;b /C23 I :/ See also INCREASING FUNCTION ,MONOTONE DECREAS- ING,MONOTONE INCREASING ,NONDECREASING FUNC- TIONReferences Jeffreys, H. and Jeffreys, B. S. "Increasing and Decreasing Functions." §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988. Noninvertible Knot INVERTIBLE KNOT Nonlinear Least Squares Fitting Given a function f(x) of a variable xtabulated at m values y1/C30fx1ðÞ;...,ym/C30fxmðÞ ;assume the function is of known analytic form depending on nparameters fx;l1;...;ln ðÞ ;and consider the overdetermined set of mequations y1/C30fx1;l1;l2;...;ln ðÞ (1) ym/C30fxm;l1;l2;...;ln ðÞ : (2) We desire to solve these equations to obtain the values l1;...,lnwhich best satisfy this system of equations. Pick an initial guess for the liand then define dbi/C30yi/C28fxi;l1;...;ln ðÞ /C215 (3) Now obtain a linearized estimate for the changes dli needed to reduce dbito 0, dbi/C30Xn j/C301@f @ljdljj xj;l(4) fori/C301, ..., n. This can be written in component form as dbi/C30Aijdli; (5) where Ais the m/C29nMATRIX Aij/C30@f dl1j x1;l@f dl1j x1;l/C1/C1/C1 @f dl2j x2;l@f dl2j x2;l/C1/C1/C1 nn::: @f dl1j xm;l@f dlnj xm;l/C1/C1/C10 BBBBBBBBBBB@1 CCCCCCCCCCCA/C215 (6) In more concise MATRIX form, db/C30Adl; (7) where dband dlare m-VECTORS . Applying the MATRIX TRANSPOSE ofAto both sides gives ATdb/C30ATACC0CC1 dl: (8) Defining a/C13ATA ð9Þ b /C13ATdb (10) in terms of the known quantities A and db then gives the MATRIX EQUATION adl /C30b; (11) which can be solved for dl using standard matrix techniques such as GAUSSIAN ELIMINATION . This off- set is then applied to l and a new d b is calculated. By iteratively applying this procedure until the elements of dl become smaller than some prescribed limit, a solution is obtained. Note that the procedure may not converge very well for some functions and also that convergence is often greatly improved by picking initial values close to the best-fit value. The sum of square residuals is given by R2 /C30db /C215 db after the final iteration. An example of a nonlinear least squares fit to a noisy GAUSSIAN FUNCTION fx;A; x0 ; s ðÞ /C30Ae /C28 x/C28x0 ðÞ2= 2 s2ðÞ(12) is shown above, where the thin solid curve is the initial guess, the dotted curves are intermediate iterations, and the heavy solid curve is the fit to which the solution converges. The actual parameters are A;x0 ; s ðÞ /C30 1;20 ;5 ðÞ ; the initial guess was (0.8, 15, 4), and the converged values are (1.03105, 20.1369, 4.86022), with R2 /C300:148461 : The PARTIAL DERIVA- TIVES used to construct the matrix A are @f @A /C30e /C28 x /C28x0 ðÞ2= 2 s2ðÞ(13) @f @x0/C30Ax/C28 x0 ðÞ s2e/C28 x/C28x0 ðÞ2= 2s2ðÞ(14) @f @ s0/C30Ax/C28 x0 ðÞ s3e /C28 x /C28x0 ðÞ2= 2 s2ðÞ/C215 (15) The technique could obviously be generalized to multiple Gaussians, to include slopes, etc., although the convergence properties generally worsen as the number of free parameters is increased.An analogous technique can be used to solve an overdetermined set of equations. This problem might, for example, arise when solving for the best-fit EULER ANGLES corresponding to a noisy ROTATION MATRIX ,in which case there are three unknown angles, but nine correlated matrix elements. In such a case, write the n different functions as fil1 ;...;ln ðÞ for i /C301, ..., n, call their actual values yi ; and define A /C30@f1 @ l1j li@f1 @ l2j li/C1/C1/C1@f1 @ lnj linn::: n @fm @ l1j li@fm @ l2j li/C1/C1/C1@fm @ lnj li0 BBBBB@1 CCCCCA; (16) and d b /C30y /C28f il1 ; ... ;ln ðÞ ; (17) where li are the numerical values obtained after the ith iteration. Again, set up the equations as Adl /C30db; (18) and proceed exactly as before. See also LEAST SQUARES FITTING ,LINEAR REGRES- SION ,M OORE- PENROSE GENERALIZED MATRIX IN- VERSE Nonlinear Stability See also LINEAR STABILITY ,LYAPUNOV FUNCTION Nonnegative A quantity which is either 0 (ZERO )or POSITIVE , i.e., ]0:/ See also NEGATIVE ,NONNEGATIVE INTEGER ,NONPO- SITIVE ,NONZERO ,POSITIVE ,ZERO Nonnegative Integer An INTEGER that is either 0 or positive, i.e., a member of the set Z+/C30 0fg@Z/C27; where Z/C27 denotes the POSITIVE INTEGERS . See also NEGATIVE INTEGER ,NONPOSITIVE INTEGER , POSITIVE INTEGER ,Z* Nonnegative Partial Sum The number of sequences with NONNEGATIVE partial sums which can be formed from n1s and n-1s (Bailey 1996, Brualdi 1992) is given by the C ATALAN NUM- BERS . Bailey (1996) gives the number of NONNEGATIVE partial sums of n1s and k/C281sa1;a2;...,an/C27k;so that a1/C27a2/C27.../C27ai]0 (1) for all 1 5i5n/C27k:The closed form expression is n 0CC6:CC67 /C301 (2) for n ]0 ; n 1CC6:CC67 /C30n (3) for n ]1 ; and n kCC6:CC67 /C30(n /C27 1 /C28 k)(n /C27 2)(n /C27 3) /C1/C1/C1(n /C27 k) k! ; (4) for n ]k ]2: Setting k /C30n then recovers the CATA- LAN NUMBERS Cn /C30n nCC6:CC67 /C301 n /C27 12n nCC1nCC1o : (5) See also CATALAN NUMBER References Bailey, D. F. "Counting Arrangements of 1’s and -1’s." Math. Mag. 69, 128 /C1/131, 1996. Brualdi, R. A. Introductory Combinatorics, 2nd ed. New York: Elsevier, 1992. Nonorientable Surface A surface such as the MO¨ BIUS STRIP or KLEIN BOTTLE (Gray 1997, pp. 322 /C1/323) on which there exists a closed path such that the directrix is reversed when moved around this path. The REAL PROJECTIVE PLANE is also a nonorientable surface, as are the BOY SURFACE , CROSS-CAP , and ROMAN SURFACE , all of which are homeomorphic to the REAL PROJECTIVE PLANE (Pinkall 1986). There is a general method for constructing nonorien- table surfaces which proceeds as follows (Banchoff 1984, Pinkall 1986). Choose three HOMOGENEOUS POLYNOMIALS of POSITIVE EVEN degree and consider the MAP f /C30 f1(x;y;z);f2(x; y;z) ;f3(x;y; z) ðÞ : R3 0 R3 /C215 (1) Then restricting x, y, and z to the surface of a sphere by writingx /C30cos u sin f (2) y /C30sin u sin f (3) z /C30cos f (4) and restricting u to 0;2p ½Þ and f to 0;p=2 ½/C138 defines a map of the REAL PROJECTIVE PLANE to R3 :/ In 3-D, there is no unbounded nonorientable surface which does not intersect itself (Kuiper 1961, Pinkall 1986). See also BOY SURFACE ,CROSS- CAP,KLEIN BOTTLE , MO¨ BIUS STRIP,ORIENTABLE SURFACE ,REAL PROJEC- TIVE PLANE ,ROMAN SURFACE References Banchoff, T. "Differential Geometry and Computer Gra- phics." In Perspectives of Mathematics: Anniversary of Oberwolfach (Ed. W. Jager, R. Remmert, and J. Moser). Basel, Switzerland: Birkha ¨user, 1984. Gray, A. "Nonorientable Surfaces." Ch. 14 in Modern Differ- ential Geometry of Curves and Surfaces with Mathema- tica, 2nd ed. Boca Raton, FL: CRC Press, pp. 317 /C1/340, 1997. Kuiper, N. H. "Convex Immersion of Closed Surfaces in E3 :/" Comment. Math. Helv. 35,85/C1/92, 1961. Pinkall, U. "Models of the Real Projective Plane." Ch. 6 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 63 /C1/67, 1986. Nonparametric Estimation This entry contributed by EDGAR VAN TUYLL Nonparametric estimation is a statistical method that allows the functional form of a fit to data to be obtained in the absence of any guidance or con- straints from theory. As a result, the procedures of nonparametric estimation have no meaningful asso- ciated parameters. Two types of nonparametric tech- niques are artificial neural networks and kernel estimation. Artificial neural networks model an unknown func- tion by expressing it as a weighted sum of several sigmoids, usually chosen to be logit curves, each of which is a function of all the relevant explanatory variables. This amounts to an extremely flexible functional form for which estimation requires a non- linear least-squares iterative search algorithm based on gradients. Kernel estimation specifies y /C30m(x) /C27e ; where m(x)is the conditional expectation of y with no parametric form whatsoever, and the density of the error e is completely unspecified. The N observations yi and xi are used to estimate a joint density function for y and x. The density at a point y0 ;x0 ðÞ is estimated by seeing what proportion of the N observations are "close to" y0 ;x0 ðÞ : This procedure involves the use of a function called a kernel to assign weights to nearby observa- tions. See also NONPARAMETRIC STATISTICS References Kennedy, P. A Guide to Econometrics. Cambridge, MA: MIT Press, 1998. Pagan, A. R. and Ullah, A. Non-Parametric Econometrics. Cambridge, England: Cambridge University Press, 1997. Nonparametric Statistics See also NONPARAMETRIC ESTIMATION ,PARAMETRIC STATISTICS References Brodsky, B. E. and Darkhovsky, B. S. Non-Parametric Statistical Diagnosis: Problems and Methods. Dordrecht, Netherlands: Kluwer, 2000. Sheskin, D. J. Handbook of Parametric and Nonparametric Statistical Procedures, 2nd ed. Boca Raton, FL: Chapman & Hall/CRC, 2000. Nonpositive A quantity which is either 0 (ZERO )or NEGATIVE , i.e., 5 0 :/ See also NEGATIVE ,N ONNEGATIVE ,N ONZERO ,POSI- TIVE,ZERO Nonpositive Integer An INTEGER that is either 0 or negative, i.e., a member of the set 0fg@Z/C28; where Z/C28 denotes the NEGATIVE INTEGERS . See also NEGATIVE INTEGER ,NONNEGATIVE INTEGER , POSITIVE INTEGER ,Z/C28 Nonseparable Graph BICONNECTED GRAPH Nonsingular Matrix A SQUARE MATRIX that is not SINGULAR , i.e., one that has a MATRIX INVERSE . Nonsingular matrices are sometimes also called regular matrices. A SQUARE MATRIX is nonsingular IFF its DETERMINANT is non- zero (Lipschutz 1991, p. 45). For example, there are 6 nonsingular 2 /C292(0,1)-MATRICES : 01 10CC60CC61 ;0111CC60CC61 ;1001CC60CC61 ;1011CC60CC61 ;1101CC60CC61 ;1110CC60CC61 : The following table gives the numbers of nonsingular n /C29n matrices for certain matrix classes. matrix type Sloane counts for n /C301, 2, ... /(/C281; 0;1)/-ma- tricesA056989 2, 48, 11808, ... /(/C281; 1)/-matrices A056990 2, 8, 192, 22272, .../(0 ;1)/-matrices A055165 1, 6, 174, 22560, ... See also DETERMINANT ,D IAGONALIZABLE MATRIX , MATRIX INVERSE ,SINGULAR MATRIX References Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 11, 1958. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, p. 51, 1996. Lipschutz, S. "Invertible Matrices." Schaum’s Outline of Theory and Problems of Linear Algebra, 2nd ed. New York: McGraw-Hill, pp. 44 /C1/45, 1991. Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 70, 1988. Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 3, 1992. Sloane, N. J. A. Sequences A055165, A056989, and A056990 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Nonsquarefree SQUAREFUL Nonstandard Analysis Nonstandard analysis is a branch of mathematical LOGIC which weakens the axioms of usual ANALYSIS to include only the first-order ones. It also introduces HYPERREAL NUMBERS to allow for the existence of "genuine INFINITESIMALS ," numbers which are less than 1u2, 1u3, 1u4, 1u5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as BANACH SPACES , differential equations, probability theory, microeconomic theory, and mathematical physics. See also AX-KOCHEN ISOMORPHISM THEOREM ,HYPER- FINITE SET,LOGIC ,LOS’ THEOREM ,M ODEL THEORY , SUPERSTRUCTURE ,T RANSFER PRINCIPLE ,U LTRA- POWER ,ULTRAPRODUCT References Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindst- røom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, 1986. Anderson, R. M. "Nonstandard Analysis with Applications to Economics." Ch. 39 in Handbook of Mathematical Economics, Vol. 4 (Ed. W. Hildenbrand and H. Son- nenschein). New York: Elsevier, pp. 2145 /C1/2208, 1991. Dauben, J. W. Abraham Robinson: The Creation of Non- standard Analysis, A Personal and Mathematical Odys- sey. Princeton, NJ: Princeton University Press, 1998. Davis, P. J. and Hersch, R. The Mathematical Experience. Boston, MA: Birkha ¨user, 1981. Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. New York: Academic Press, 1985. Keisler, H. J. Elementary Calculus: An Infinitesimal Approach. Boston, MA: PWS, 1986. Lindstrøom, T. "An Invitation to Nonstandard Analysis." In Nonstandard Analysis and Its Applications (Ed. N. Cut- land). New York: Cambridge University Press, 1988. Robinson, A. Non-Standard Analysis. Princeton, NJ: Prin- ceton University Press, 1996. Stewart, I. "Non-Standard Analysis." In From Here to Infinity: A Guide to Today’s Mathematics. Oxford, Eng- land: Oxford University Press, pp. 80 /C1/81, 1996. Nontotient A POSITIVE EVEN value of n for which f(x) /C30n; where f(x) is the TOTIENT FUNCTION , has no solution. The first few are 14, 26, 34, 38, 50, ... (Sloane’s A005277). See also NONCOTOTIENT ,TOTIENT FUNCTION References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 91, 1994. Sloane, N. J. A. Sequences A005277/M4927 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Nonwandering A point x in a MANIFOLD M is said to be nonwander- ing if, for every open NEIGHBORHOOD U of x, it is true that f/C28nU @ U "¥ for a MAP f for some n /C210. In other words, every point close to x has some iterate under f which is also close to x. The set of all nonwandering points is denoted V(f); which is known as the nonwandering set of f:/ See also ANOSOV DIFFEOMORPHISM ,AXIOM AD IFFEO- MORPHISM ,SMALE HORSESHOE MAP Nonzero A quantity which does not equal ZERO is said to be nonzero. A REAL nonzero number must be either POSITIVE or NEGATIVE , and a COMPLEX nonzero num- ber can have either REAL or IMAGINARY PART nonzero. See also NEGATIVE ,N ONNEGATIVE ,N ONPOSITIVE , POSITIVE ,ZERO NOR A PREDICATE in LOGIC equivalent to the composition NOT OR that yields FALSE if any condition is TRUE , and TRUE if all conditions are FALSE . A NOR B is equivalent to !(A /C150B); where !A denotes NOT and /C150 denotes OR. In PROPOSITIONAL CALCULUS , the term JOINT DENIAL is used to refer to the NOR connective. Notations for NOR include A/C150B and A ¡B (Mendelson 1997, p. 26). The NOR operation is implemented inMathematica 4.1 as Nor[A, B, ...]. The circuit diagram symbol for a NOR gate is illustrated above. The BINARY NOR operator has the following TRUTH TABLE (Simpson 1987, p. 547; Mendelson 1997, p. 26). AB /A/C150B/ TTF TFF FTFFFT See also AND, B INARY OPERATOR ,C ONNECTIVE , INTERSECTION ,NAND,NOT,OR,T RUTH TABLE , XNOR, XOR References Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 26, 1997. Simpson, R. E. "The NOR Gate." §12.5.4 in Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 547 /C1/548, 1987. Nordstrand’s Weird Surface An attractive CUBIC SURFACE defined by Nordstrand. It is given by the implicit equation 25 x3(y /C27z) /C27y3(x /C27z) /C27z3(x /C27y)CC6CC7 /C2750 x2y2 /C27x2z2 /C27y2z2CC0CC1 /C28125 x2yz /C27y2xz /C27z2xyCC0CC1 /C2760xyz /C284 xy /C27xz /C27yz ðÞ /C300: See also CUBIC SURFACE References Nordstrand, T. "Weird Cube." http://www.uib.no/people/ nfytn/weirdtxt.htm. Norm Given a n-D VECTOR x /C30x1 x2 n xn2 6643 775; a VECTOR NORM xkk is a NONNEGATIVE number satisfying 1. xkk > 0 when x "0 and xkk/C300 IFF x /C300;/ 2. kxkk/C30kjjxkkfor any SCALAR k, 3. x /C27y kk5 xkk/C27 ykk / The most common norm is the vector L2-NORM , defined by xkk2/C30 xjj/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 1 /C27x22 /C27/C1/C1/C1/C27x2 nq : Given a SQUARE MATRIX A; a MATRIX NORM Akk is a NONNEGATIVE number associated with A having the properties 1. Akk > 0 when A "0 and Akk/C300 IFF A /C300;/ 2. kAkk/C30kjjAkkfor any SCALAR k, 3. A /C27B kk 5 Akk/C27 Bkk;/ 4. ABkk5 Akk Bkk / See also BOMBIERI NORM,COMPATIBLE ,EUCLIDEAN NORM,HILBERT- SCHMIDT NORM,INDUCED NORM, L1- NORM, L2-NORM, L-INFINITY- NORM,M ATRIX NORM, MAXIMUM ABSOLUTE COLUMN SUM NORM,M AXIMUM ABSOLUTE ROW SUM NORM,N ATURAL NORM,N OR- MALIZED VECTOR ,N ORMED SPACE ,PARALLELOGRAM LAW,POLYNOMIAL NORM,SPECTRAL NORM,SUBORDI- NATE NORM,VECTOR NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1114 /C1/1125, 2000. Norm (Operator) The operator norm of a LINEAR OPERATOR T:V0W is the largest value by which Tstretches an element ofV, Tkk/C30sup vjjjj/C301T(v) kk : (1) It is necessary for Vand Wto be normed vector spaces. The operator norm of a composition is con- trolled by the norms of the operators, TSkk5Tkk Skk (2) When Tis given by a matrix, say /TðvÞ¼Av/, then /Tkk / is the SQUARE ROOT of the largest EIGENVALUE of the SYMMETRIC MATRIX /ATA/, all of whose eigenvalues are nonnegative. For instance, ifA/C30200 302CC60CC61 (3) then ATA/C3013 0 6 00 060 42 435; (4) which has eigenvalues 0 ;1;16 fg ;soAkk/C304: / The following Mathematica function will determine the operator norm of a matrix. OperatorNorm[a_List?MatrixQ] : /C30 Sqrt[Max[Eigenvalues[Transpose[a].a]]] Norm Theorem If a PRIME NUMBER divides a norm but not the bases of the norm, it is itself a norm. Normal NORMAL CURVE ,N ORMAL DISTRIBUTION ,N ORMAL DISTRIBUTION FUNCTION ,N ORMAL EQUATION ,N OR- MAL FORM,NORMAL GROUP ,NORMAL MAGIC SQUARE , NORMAL MATRIX ,NORMAL NUMBER ,NORMAL PLANE , NORMAL SUBGROUP ,NORMAL VECTOR Normal (Algebraically) GALOISIAN Normal Bundle This entry contributed by R YANBUDNEY The normal bundle of a submanifold N/C23Mis the VECTOR BUNDLE over Nthat consists of all pairs ( x, v), where xis in Nand vis a vector in the VECTOR QUOTIENT SPACE TxM =TxN :Provided Mhas a Riemann metric, TxM =TxN can be thought of as the orthogonal complement to Tx/C23TxM :/ Normal Curvature Letupbe a unit TANGENT VECTOR of a REGULAR SURFACE MƒR3:Then the normal curvature of Min the direction upis kupCC0CC1 /C30SupCC0CC1 /C215up; (1) where Sis the SHAPE OPERATOR . Let MƒR3be a REGULAR SURFACE ,p/C23M;xbe an injective REGULAR PATCH ofMwith p/C30xu0;v0 ðÞ ;and vp/C30axuu0;v0 ðÞ /C27bxvu0;v0 ðÞ ; (2) where vp/C23Mp:Then the normal curvature in the direction vpis k(vp)/C30ea2/C272fab/C27gb2 Ea2/C272Fab/C27Gb2; (3) where E, F, and G are the coefficients of the first FUNDAMENTAL FORM and e, f, and g are the coeffi- cients of the second FUNDAMENTAL FORM . The MAXIMUM and MINIMUM values of the normal curvature at a point on a REGULAR SURFACE are called the PRINCIPAL CURVATURES k1 and k2 :/ See also CURVATURE ,FUNDAMENTAL FORMS ,GAUS- SIAN CURVATURE ,MEAN CURVATURE ,PRINCIPAL CUR- VATURES ,SHAPE OPERATOR ,TANGENT VECTOR References Euler, L. "Recherches sur la courbure des surfaces." Me´m. de l’Acad. des Sciences, Berlin 16, 119 /C1/143, 1760. Gray, A. "Normal Curvature." §18.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 363 /C1/367, 1997. Meusnier, J. B. "Me´moire sur la courbure des surfaces." Me´m. des savans e´trangers 10 (lu 1776), 477 /C1/510, 1785. Normal Curve GAUSSIAN DISTRIBUTION Normal Developable A RULED SURFACE M is a normal developable of a curve y if M can be parameterized by x (u;v) /C30y(u) /C27 v ˆN(u) ; where N is the NORMAL VECTOR . See also BINORMAL DEVELOPABLE ,B OX-MULLER TRANSFORMATION ,TANGENT DEVELOPABLE References Gray, A. "Developables." §17.6 in Modern Differential Geo- metry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 352 /C1/354, 1993. Normal Deviates See also BOX-MULLER TRANSFORMATION ,G AUSSIAN DISTRIBUTION ,NORMAL DISTRIBUTION References Box, G. E. P. and Muller, M. E. "A Note on the Generation of Random Normal Deviates." Ann. Math. Stat. 28, 610 /C1/611, 1958. Muller, M. E. "Generation of Normal Deviates." Tech. Rep. No. 13. Statistical Techniques Research Group. Princeton, NJ: Princeton University. n.d. Muller, M. E. "An Inverse Method for the Generation of Random Normal Deviates on Large-Scale Computers." Math. Tables Aids Comput. 12, 167 /C1/174, 1958. Muller, M. E. "A Comparison of Methods for Generating Normal Deviates on Digital Computers." J. Assoc. Com- put. Mach. 6, 376 /C1/383, 1959.Normal Distribution Another name for a GAUSSIAN DISTRIBUTION . Given a normal distribution in a VARIATE x with MEAN m and VARIANCE s2 ; P(x)dx /C301 sffiffiffiffiffiffi 2pp e /C28(x /C28 m)2 =2 s2 dx; the so-called "STANDARD NORMAL DISTRIBUTION "is given by taking m /C300 and s2 /C301: An arbitrary normal distribution can be converted to a STANDARD NORMAL DISTRIBUTION by changing variables to z /C13(x /C28 m) =s; so dz /C30dx=s; yielding P(x)dx /C301ffiffiffiffiffiffi2pp e /C28z2 =2dz/C215 Feller (1968) uses the symbol 8(x) for P(x) in the above equation, but then switches to n(x) in Feller (1971). The FISHER- BEHRENS PROBLEM is the deter- mination of a test for the equality of MEANS for two normal distributions with different VARIANCES . See also FISHER- BEHRENS PROBLEM ,GAUSSIAN DIS- TRIBUTION ,HALF-NORMAL DISTRIBUTION ,KOLMOGOR- OV-SMIRNOV TEST,NORMAL DISTRIBUTION FUNCTION , STANDARD NORMAL DISTRIBUTION ,T ETRACHORIC FUNCTION References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 45, 1971. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 100 /C1/101, 1984. Normal Distribution Function A normalized form of the cumulative G AUSSIAN DISTRIBUTION function giving the probability that a variate assumes a value in the range [0;x] ; F(x) /C13Q(x) /C131ffiffiffiffiffiffi 2ppgx 0e /C28t2 =2dt: (1) It is related to the PROBABILITY INTEGRAL a(x) /C131ffiffiffiffiffiffi2ppgx /C28xe /C28t2 =2dt: (2) by F(x) /C301 2a(x) (3) Let u /C13t=ffiffiffi 2p so du /C30dt=ffiffiffi2p : Then F(x) /C301ffiffiffippgx=ffiffi 2p 0e /C28u2 du /C301 2erfxffiffiffi 2p ! /C215 (4) Here, ERF is a function sometimes called the error function. The probability that a normal variate assumes a value in the range x1 ;x2 ½/C138 is therefore given by F x1 ;x2 ðÞ /C301 2erfx2 ffiffiffi 2p ! /C28erfx1ffiffiffi2p ! "# /C215 (5) Neither F(z) nor ERF can be expressed in terms of finite additions, subtractions, multiplications, and ROOT EXTRACTIONS , and so must be either computed numerically or otherwise approximated. Note that a function different from F(x) is sometimes defined as "the" normal distribution function N(x) /C131ffiffiffiffiffiffi 2 ppgx /C28/C12e /C28t2 =2dt (6) /C30F(/C28/C12; x) (7) /C301 2 /C27F(x) (8) /C301 21 /C27erfxffiffiffi 2p !"# (9) (Feller 1968; Beyer 1987, p. 551), although this function is less widely encountered than the usual F(x) : The notation N(x) is due to Feller (1971). The value of a for which P(x) falls within the interval [/C28a;a] with a given probability P is a related quantity called the CONFIDENCE INTERVAL . For small values x /C101; a good approximation to F(x)is obtained from the MACLAURIN SERIES for ERF, F(x) /C301ffiffiffiffiffiffi 2 pp x /C281 6x3 /C271 40x5 /C281 336x7 /C271 3456x9 /C27...CC1:CC17 (10) (Sloane’s A014481). For large values x /C271 ; a good approximation is obtained from the asymptotic series for ERF,F(x) /C301 2 /C27e /C28x2 =2 2ffiffiffipp /C2 x/C281 /C28x/C283 /C273x/C285 /C2815x/C287 /C27105x/C289 /C27...CC0CC1 (11) (Sloane’s A001147). The value of F(x) for intermediate x can be computed using the CONTINUED FRACTION identity gx 0e /C28u2 du /C30ffiffiffipp 2/C281 2e/C28x2 x /C271 2x /C272 x /C273 2x /C274 x /C27 ...(12) A simple approximation of F(x) which is good to two decimal places is given by F1(x) :0:1x(4:4 /C28x) for 0 5x 52 :2 0:49 for 2 :2 Bx B2 :6 0:50 for x ]2 :6/C2158 < : (13) Abramowitz and Stegun (1972) and Johnson and Kotz (1970) give other functional approximations. An approximation due to Bagby (1995) is F2(x) /C301 2f1 /C281 30[7e /C28x2 =2 /C2716e /C28x2(2/C28ffiffi 2p ) /C27(7 /C2714px2 Þe /C28x2 /C138g1 =2 (14) The plots below show the differences between F and the two approximations. The first QUARTILE of a standard NORMAL DISTRIBU- TION occurs when gt 0F(z)dz /C301 4 /C215 (15) The solution is t /C300:6745... : The value of t giving14 is known as the PROBABLE ERROR of a normally dis- tributed variate. See also BERRY- ESSE´ EN THEOREM ,CONFIDENCE IN- TERVAL ,E RF,E RFC,F ISHER- BEHRENS PROBLEM , GAUSSIAN DISTRIBUTION ,G AUSSIAN INTEGRAL ,H H FUNCTION ,N ORMAL DISTRIBUTION ,PROBABILITY IN- TEGRAL ,TETRACHORIC FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 931 /C1/933, 1972. Bagby, R. J. "Calculating Normal Probabilities." Amer. Math. Monthly 102,4 6/C1/49, 1995. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 45, 1971. Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 1, 2nd ed. Boston, MA: Houghton Mifflin, 1994. Sloane, N. J. A. Sequences A001147/M3002 and A014481 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Whittaker, E. T. and Robinson, G. "Normal Frequency Distribution." Ch. 8 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 164 /C1/208, 1967. Normal Equation Given an overdetermined MATRIX EQUATION Ax /C30b; the normal equation is that which minimizes the sum of the square differences between left and right sides ATAx /C30ATb: See also LEAST SQUARES FITTING ,M OORE- PENROSE GENERALIZED MATRIX INVERSE ,N ONLINEAR LEAST SQUARES FITTING Normal Form A way of representing objects so that, although each may have many different names, every possible name corresponds to exactly one object (Petkovsek et al. 1996, p. 7). Koepf (1998, p. 2) defines normal form to mean the uniquely determined holonomic equation of lowest order up to multiplication by polynomials. See also CANONICAL FORM References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Normal Function A SQUARE INTEGRABLE function f(t) is said to be normal if g f(t)½/C1382dt /C301 However, the NORMAL DISTRIBUTION FUNCTION is also sometimes called "the normal function." See also NORMAL DISTRIBUTION FUNCTION ,SQUARE INTEGRABLEReferences Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 6, 1991. Normal Group NORMAL SUBGROUP Normal Line A LINE along a NORMAL VECTOR (i.e., perpendicular to some TANGENT LINE). If /K ƒRd / is a CENTROSYMMETRIC SET which has a twice differentiable boundary, then there are /2d þ 2/ normals through the center (Croft et al. 1991, p. 15). See also DOUBLE NORMAL ,N ORMAL VECTOR ,TAN- GENT LINE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1991. Normal Magic Square MAGIC SQUARE Normal Matrix A SQUARE MATRIX A is a normal matrix if A; A+½/C138/C300; where [a, b] is the COMMUTATOR and A + denotes the ADJOINT MATRIX . For example, the matrix i 0 03/C285iCC60CC61 is a normal matrix, but is not aH ERMITIAN MATRIX .A matrix mcan be tested to see if it is normal using the Mathematica function NormalQ[a_List?MatrixQ] : /C30Module[ {b/C30Conjugate@Transpose@a}, a. b /C30/C30/C30b. a ] The normal matrices are the matrices which are unitarily DIAGONALIZABLE . That is, Ais a normal matrix iff there exists a UNITARY MATRIX Usuch that / UA U/C281 /is a DIAGONAL MATRIX . All H ERMITIAN MA- TRICES are normal, but they are restricted to real eigenvalues. A normal matrix has no restriction on its eigenvalues. The following table gives the number of normal square matrices of given types for orders n/C301, 2, .... type Sloane counts /(0;1)/ A055547 2, 8, 68, 1124, ... /(/C281;1)/A055548 2, 12, 80, 2096, ... /(/C281;0 ;1)/ A055549 3, 33, 939, ... See also ADJOINT MATRIX ,DIAGONAL MATRIX ,HER- MITIAN MATRIX ,UNITARY MATRIX References Sloane, N. J. A. Sequences A055547, A055548, and A055549 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Normal Number An IRRATIONAL NUMBER for which any FINITE pattern of numbers occurs with the expected limiting fre- quency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0 /C1/9 would be expected to occur 1/10 of the time, each pair of digits 00 /C1/99 would be expected to occur 1/100 of the time, etc. Determining if numbers are normal is an unresolved problem. It is not even known if PI or E are normal. While tests offfiffiffinpfor n /C302, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 indicate that these SQUARE ROOTS may be normal (Beyer et al. 1970ab), normality of these numbers has also not been proven. Strangely enough, the only numbers known to be normal (in certain bases) are artificially constructed ones such as the CHAMPERNOWNE CONSTANT and the COPELAND- ERDOS CONSTANT . See also CHAMPERNOWNE CONSTANT ,COPELAND- ER- DOS CONSTANT , E,PI References Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent." Math. Comput. 23, 679, 1969. Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Statis- tical Study of Digits of Some Square Roots of Integers in Various Bases." Math. Comput. 24, 455 /C1/473, 1970a. Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases." Math. Comput. 24, 745 /C1/747, 1970b. Champernowne, D. G. "The Construction of Decimals Nor- mal in the Scale of Ten." J. London Math. Soc. 8, 254 /C1/260, 1933. Copeland, A. H. and Erdos, P. "Note on Normal Numbers." Bull. Amer. Math. Soc. 52, 857 /C1/860, 1946. Good, I. J. and Gover, T. N. "The Generalized Serial Test and the Binary Expansion offfiffiffi 2p :/" J. Roy. Statist. Soc. Ser. A 130, 102 /C1/107, 1967. Good, I. J. and Gover, T. N. "Corrigendum." J. Roy. Statist. Soc. Ser. A 131, 434, 1968. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 26, 1986. Normal Order A function f(n) has the normal order F(n)iff(n)is approximately F(n) for ALMOST ALL values of n. Moreprecisely, if (1 /C28 o)F(n) Bf(n) B(1 /C27 o)F(n) for every positive o and ALMOST ALL values of n, then the normal order of f(n)isF(n) :/ See also ALMOST ALL References Hardy, G. H. and Weight, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, p. 356, 1979. Normal Plane The PLANE spanned by the NORMAL VECTOR N and the BINORMAL VECTOR B. See also BINORMAL VECTOR ,NORMAL VECTOR ,PLANE Normal Polynomial In every RESIDUE CLASS modulo p, there is exactly one INTEGER POLYNOMIAL with COEFFICIENTS ]0 and 5 p /C281: This polynomial is called the normal polyno- mial modulo p in the class (Nagell 1951, p. 94). See also COEFFICIENT References Nagell, T. Introduction to Number Theory. New York: Wiley, p. 94, 1951. Normal Section Let M ƒR3 be a REGULAR SURFACE and upa unit TANGENT VECTOR to M, and letQup ;N(p)CC0CC1 be the PLANE determined by upand the normal to the surface N(p) : Then the normal section of M is defined as the intersection ofQup ;N(p)CC0CC1 and M. References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 365, 1997. Normal Series A normal series of a GROUP G is a finite sequence (A0 ;...;Ar)of SUBGROUPS such that I /C30A01A11...1Ar /C30G/C215 See also COMPOSITION SERIES ,INVARIANT SERIES , NORMAL SUBGROUP References Scott, W. R. Group Theory. New York: Dover, p. 36, 1987. Normal Subgroup LetHbe a SUBGROUP of a GROUP G. Then His a normal subgroup of G, written H1G;if xHx/C281 /C30H for every element x in G (Scott 1987, p. 25). Normal subgroups are also known as invariant subgroups. See also GROUP ,NORMAL SERIES ,QUOTIENT GROUP , SUBGROUP References Scott, W. R. Group Theory. New York: Dover, 1987. Normal to a Plane NORMAL VECTOR Normal Vector The normal to a PLANE specified by f(x;y;z) /C30ax /C27by /C27cz /C27d /C300 (1) is given by N /C309f /C30a b c2 435: (2) The normal vector at a point x 0 ;y0 ðÞ on a surface z /C30 f(x; y)is N /C30fxx0 ; y0 ðÞ fyx0 ; y0 ðÞ /C2812435: (3) In the PLANE , the unit normal vector is defined by ˆN /C13d ˆT d f ; (4) where ˆT is the unit TANGENT VECTOR and f is the polar angle. Given a unit TANGENT VECTOR ˆT /C13u1 ˆx /C27u2 ˆy (5) withffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 1 /C27u2 2p/C301 ; the normal is ˆN /C13u2 ˆx /C27u1 ˆy: (6) For a function given parametrically by (f(t) ;g(t)) ; the normal vector relative to the point (f(t) ;g(t)) is there- fore given by x(t) /C30/C28g?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ?2 /C27 g ?2p (7) y(t) /C30f ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ?2 /C27 g ?2p (8) To actually place the vector normal to the curve, it must be displaced by (f(t) ;g(t)) :/In 3-D SPACE , the unit normal is ˆN /C13d ˆT ds d ˆT dsCC16CC16CC16CC16CC16CC16CC16CC16CC16CC16/C30d ˆT dt d ˆT dtCC16CC16CC16CC16CC16CC16CC16CC16CC16CC16/C30 1 kd ˆT ds; (9) where k is the CURVATURE . Given a 3-D surface F(x;y;z) /C300; ˆn /C30Fx /C27 Fy /C27 Fzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F2x /C27 F2y /C27 F2zp : (10) If the surface is defined parametrically in the form x /C30x( f ; c) (11) y /C30y( f ; c) (12) z /C30z( f; c) (13) define the VECTORS a /C13xf yf zf2 435 (14) b /C13x f yf zf2435: (15) Then the unit normal vector is ˆN /C30 a /C29 bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ajj2bjj2/C28 a /C215 bjj2q (16) Let g be the discriminant of the METRIC TENSOR . Then N /C30r1/C29r2ffiffiffigp/C30oijrj: (17) See also BINORMAL VECTOR ,C URVATURE ,F RENET FORMULAS ,TANGENT VECTOR References Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5 inModern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 108 /C1/111, 1997. Normalized Laplacian Matrix LAPLACIAN MATRIX Normalized Vector The normalized vector of Xis a VECTOR in the same direction but with NORM (length) 1. It is denoted /ˆX/ and given by ˆX/C13X Xjj; where /j ˆXj/ is the NORM of X. It is also called a UNIT VECTOR . See also UNIT VECTOR Normalizer The set of elements g of a GROUP such that g /C281Hg /C30H ; is said to be the normalizer /NG ðH Þ/ with respect to a subset of group elements H.IfH is a SUBGROUP of G, /NG ðH Þ/ is also a SUBGROUP containing H. See also CENTRALIZER ,TIGHTLY EMBEDDED Normed Space A VECTOR SPACE possessing a NORM . Nosarzewska’s Inequality Given a convex PLANE region with AREA A and PERIMETER p, A /C281 2p BN 5A /C2712p /C271; where N is the number of enclosed LATTICE POINTS (Nosarzewska 1948). This improves on JARNICK’S INEQUALITY N /C28A jjBp : See also JARNICK’S INEQUALITY ,LATTICE POINT References Nosarzewska, M. "E´ valuation de la diffe´rence entre l’aire d’une re´gion plane convexe et le nombre des points aux coordonne ´es entie`res couverts par elle." Colloq. Math. 1, 305 /C1/311, 1948. NOT An CONNECTIVE in LOGIC which converts TRUE to FALSE and FALSE to TRUE . NOT A is denoted !A;/C15 A; ¯A (Simpson 1987, p. 537) or /C2A (Carnap 1958, p. 7; Mendelson 1997, p. 12). The NOT operation is im- plemented in Mathematica as Not[A], or !A. The circuit diagram symbol for a NOT gate is illustrated above. The NOT operation has the following TRUTH TABLE (Carnap 1958, p. 10; Simpson 1987, p. 546; Mendel- son 1997, p. 12). A /!A/TF FT See also AND, CONNECTIVE , NAND, NOR, OR, TRUTH TABLE , XNOR, XOR References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, pp. 7 and 10, 1958. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 12, 1997. Simpson, R. E. "The NOT Gate." §12.5.3 in Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 546 /C1/547, 1987. Not An operation in LOGIC which converts TRUE to FALSE and FALSE to TRUE . NOT [W ;U] is denoted dior N(n;a) /C301 na n(n) i /C301 f diðÞan =di :/ /[W ; U]//N(n;a) /C301 na n(n) i/C301 f diðÞan =di/ FT TF See also AND,OR,TRUTH TABLE , XOR Notation A NOTATION is a set of WELL DEFINED rules for representing quantities and operations with symbols. See also ARROW NOTATION ,CHAINED ARROW NOTA- TION ,CIRCLE NOTATION ,CLEBSCH- ARONHOLD NOTA- TION ,CONWAY’S KNOT NOTATION ,DOWKER NOTATION , DOWN ARROW NOTATION ,PETROV NOTATION ,SCIEN- TIFIC NOTATION ,STEINHAUS- MOSER NOTATION References Cajori, F. A History of Mathematical Notations, Vols. 1 /C1/2. New York: Dover, 1993. Miller, J. "Earliest Uses of Various Mathematical Symbols." http://members.aol.com/jeff570/mathsym.html. Miller, J. "Earliest Uses of Some of the Words of Mathe- matics." http://members.aol.com/jeff570/mathword.html. No¨ther NOETHER’S FUNDAMENTAL THEOREM ,N OETHER- LAS- KER THEOREM ,N OETHER’S TRANSFORMATION THEO- REM,NOETHERIAN MODULE ,NOETHERIAN RING Novemdecillion In the American system, 1060. See also LARGE NUMBER Nowhere Dense A SET X is said to be nowhere dense if the interior of the CLOSURE of X is the EMPTY SET. See also BAIRE CATEGORY THEOREM ,DENSE References Ferreiro ´s, J. "Lipschitz and Hankel on Nowhere Dense Sets and Integration." §5.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 154 /C1/156, 1999. Rudin, W. Functional Analysis, 2nd ed. New York: McGraw- Hill, p. 42, 1991. NP-Complete Problem A problem which is both NP (solvable in nondetermi- nistic POLYNOMIAL-TIME ) and NP-HARD (any other NP- PROBLEM can be translated into this problem). Ex- amples of NP-hard problems include the HAMILTO- NIAN CYCLE and TRAVELING SALESMAN PROBLEMS . In a landmark paper, Karp (1972) showed that 21 intractable combinatorial computational problems are all NP-complete. See also HAMILTONIAN CYCLE ,NP -HARD PROBLEM , NP-PROBLEM ,P -PROBLEM ,T RAVELING SALESMAN PROBLEM References Buckley, F. and Harary, F. Distances in Graphs. Redwood City, CA: Addison-Wesley, 1990. Garey, M. R. and Johnson, D. S. Computers and Intract- ability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman, 1983. Karp, R. M. "Reducibility Among Combinatorial Problems." In Complexity of Computer Computations, (Proc. Sympos. IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972). New York: Plenum, pp. 85 /C1/103, 1972. Levin, L. A. "Universal Searching Problems." Prob. Info. Transm. 9, 265 /C1/266, 1973. Papadimitriou, C. H. and Steiglitz, K. Combinatorial Opti- mization: Algorithms and Complexity. New York: Dover, 1998. NP-Hard Problem A problem is NP-hard if an ALGORITHM for solving it can be translated into one for solving any other NP- PROBLEM (nondeterministic POLYNOMIAL time) pro- blem. NP-hard therefore means "at least as hard as any NP-PROBLEM ," although it might, in fact, be harder. See also COMPLEXITY THEORY ,H ITTING SET,NP - COMPLETE PROBLEM ,NP -PROBLEM ,P-PROBLEM ,SA- TISFIABILITY PROBLEM n-Plex n-plex is defined as 10n :/ See also GOOGOLPLEX , N-MINEXReferences Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 16, 1996. NP-Problem A problem is assigned to the NP (nondeterministic POLYNOMIAL time) class if it is solvable in polynomial time by a nondeterministic TURING MACHINE .(A nondeterministic TURING MACHINE is a "parallel" TURING MACHINE which can take many computational paths simultaneously, with the restriction that the parallel Turing machines cannot communicate.) A P- PROBLEM (whose solution time is bounded by a polynomial) is always also NP. If a problem is known to be NP, and a solution to the problem is somehow known, then demonstrating the correctness of the solution can always be reduced to a single P (POLY- NOMIAL time) verification. LINEAR PROGRAMMING , long known to be NP and thought not to be P, was shown to be P by L. Kha- chian in 1979. It is an important UNSOLVED PROBLEM to determine if all apparently NP problems are actually P. A problem is said to be NP-HARD if an ALGORITHM for solving it can be translated into one for solving any other NP-problem. It is much easier to show that a problem is NP than to show that it is NP-HARD .A problem which is both NP and NP-HARD is called an NP-COMPLETE PROBLEM . See also COMPLEXITY THEORY ,NP -COMPLETE PRO- BLEM ,NP -HARD PROBLEM ,P-PROBLEM ,TURING MA- CHINE References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Clay Mathematics Institute. "The P vs. NP Problem." http:// www.claymath.org/prize_problems/p_vs_np.htm. Cook, S. "The P versus NP Problem." http://www.clay- math.org/prize_problems/p_vs_np.pdf. Greenlaw, R.; Hoover, H. J.; and Ruzzo, W. L. Limits to Parallel Computation: P-Completeness Theory. Oxford, England: Oxford University Press, 1995. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 0821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Provi-dence, RI: Amer. Math. Soc., 2000. ns JACOBI ELLIPTIC FUNCTIONS n-Sphere HYPERSPHERE NSW Number An NSW number is a side length of a SQUARE the square of whose diagonal is one more than a SQUARE NUMBER . Such numbers were called "rational diag- onals" by the Greeks (Wells 1986, p. 70). A formula for NSW numbers is given by S(m) /C301 /C27ffiffiffi 2pCC0CC1 m/C27 1 /C28ffiffiffi2pCC0CC1 m 2 for positive integers m.A RECURRENCE RELATION for / SðmÞ/ is given by S(n) /C306S(n /C281) /C28S(n /C282) (1) with S(1) /C301 and S(2) /C307 : The first few terms are 1, 7, 41, 239, 1393, ... (Sloane’s A002315). The lengths that are one more than the corresponding diagonals are 2, 50, 1682, 57122, .... The indices giving PRIME NSW numbers are 3, 5, 7, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, ... (Sloane’s A005850). References Ribenboim, P. "The NSW Primes." §5.9 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 367 /C1/369, 1996. Sloane, N. J. A. Sequences A002315/M4423 and A005850/ M2426 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 70, 1986. n! Theorem For any PARTITION m of n, define a polynomial in 2n variables x1 ; x2 ; ... and y1 ; y2 ; ... as Dm /C30det xpj iyqj iCC16CC16CC16CC16; (1) where p j ;qjCC0CC1 are the coordinates of the cells of the partition when it is placed in the coordinate plane with base cell at (0;0) and such that all other coordinates are nonnegative in x and y. Denote the linear span of all derivatives of this polynomial with respect to the variables by L @x @y DmCC6CC7 ; where @ represents a PARTIAL DERIVATIVE . This VECTOR SPACE is CLOSED under permutations acting on xiand yi simultaneously. Then the n! theorem states that dim L @x @y DmCC6CC7 /C30n! (2) (Zabrocki). The theorem was proven by M. Haiman in Dec. 1999. For example, consider the PARTITION m /C30(2;1): Then D(2;1) /C30det111 x1x2x3 y1y2y3CC16CC16CC16CC16CC16CC16CC16CC16CC16CC16CC16CC16(3) /C30x 2y3 /C28x3y2 /C28x1y3 /C27y1x3 /C27x1y2 /C28x2y1 (4) Then the five derivatives @x1D(2;1) /C30y2 /C28y3 (5)@x2D(2;1) /C30y3 /C28y1 (6) @y1D(2;1) /C30x3 /C28x2 (7) @y2D(2;1) /C30x1 /C28x3 (8) @x2@y2D(2;1) /C301 ; (9) together with D(2;1) ; 3! /C306 elements in all, form a basis for L @x @y D(2;1)CC6CC7 (Zabrocki). See also MACDONALD POLYNOMIAL References Zabrocki, M. "A Short Explanation of the n! Theorem." http:// www.lacim.uqam.ca/~zabrocki/nfactconj/nfactconj.html. Nu Function n(x) /C13g/C12 0xtdt G(t /C27 1) n(x; a) /C13g/C12 0xa /C27tdt G( a /C27 t /C27 1) ; where G(z) is the GAMMA FUNCTION (Erde ´lyi et al. 1981, p. 388; Prudnikov et al. 1990, p. 799; Gradsh- teyn and Ryzhik 2000, p. 1109). See also LAMBDA FUNCTION ,MU FUNCTION References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, 1981. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Ch. 18 in Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 217, 1981. Gradshteyn, I. S. and Ryzhik, I. M. "The Functions n(x); n(x;a);m(x;b);m(x;b;a);l(x;y):/"§9.64 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1109, 2000. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, 1990. Nucleus KERNEL (INTEGRAL ) Nugatory Crossing REDUCIBLE CROSSING Null Function A null function d0xðÞsatisfies gb ad0(x)dx/C300 (1) for all a, b,s o g/C12 /C28/C12d0(x)CC16CC16CC16CC16dx/C300: (2) Like a DELTA FUNCTION , they satisfy d0(x) /C300 x "0 1 x /C300:CC6: (3) See also DELTA FUNCTION ,LERCH’S THEOREM References Bracewell, R. "Null Functions." In The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 82 /C1/84, 1999. Null Graph The EMPTY GRAPH containing no VERTICES or EDGES . See also EMPTY GRAPH References Harary, F. and Read, R. "Is the Null Graph a Pointless Concept?" In Graphs and Combinatorics Conference, George Washington University. New York: Springer-Ver- lag, 1973. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 141, 1990. Null Hypothesis A hypothesis which is tested for possible rejection under the assumption that it is true (usually that observations are the result of chance). The concept was introduced by R. A. Fisher. Null Space NULLSPACE Null Tetrad gij /C3001 0 0 10 0 0 00 0 /C281 00/C28102 6643 775: It can be expressed as gab /C30lanb /C27lbna /C28ma ¯mb /C28mb ¯ma : See also TETRAD References d’Inverno, R. Introducing Einstein’s Relativity. Oxford, England: Oxford University Press, pp. 248 /C1/249, 1992. Null Vector The n-D null vector 0 is the n-D VECTOR of length 0. References Jeffreys, H. and Jeffreys, B. S. "Direction Vectors." §2.033 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 64, 1988.Nullspace Also called the kernel. If T is a LINEAR TRANSFORMA- TION of Rn ; then Null( T) is the set of all VECTORS X such that T(X) /C300; i.e., Null( T) /C13 X : T(X) /C300 fg : A list of vectors forming a BASIS for the nullspace of a set of vectors m is returned by the Mathematica command NullSpace [m]. See also BASIS (VECTOR SPACE ), FREDHOLM’S THEO- REM,L INEAR TRANSFORMATION ,S PAN (VECTOR SPACE ) Nullstellensatz HILBERT’S NULLSTELLENSATZ Number The word "number" is a general term which refers to a member of a given (possibly ordered) SET. The meaning of "number" is often clear from context (i.e., does it refer to a COMPLEX NUMBER ,INTEGER ,REAL NUMBER , etc.?). Wherever possible in this work, the word "number" is used to refer to quantities which are INTEGERS , and " CONSTANT " is reserved for nonintegral numbers which have a fixed value. Because terms such as REAL NUMBER ,B ERNOULLI NUMBER , and IRRATIONAL NUMBER are commonly used to refer to nonintegral quantities, however, it is not possible tobe entirely consistent in nomenclature. To indicate a particular numerical label, the abbre- viation "no." is sometimes used (deriving from "nu- mero," the ablative case of the Latin "numerus"), as is the less common "nr." References Barbeau, E. J. Power Play: A Country Walk through the Magical World of Numbers. Providence, RI: Amer. Math. Soc., 1997. Bogomolny, A. "What is a Number." http://www.cut-the- knot.com/do_you_know/numbers.html. Borwein, J. and Borwein, P. A Dictionary of Real Numbers. London: Chapman & Hall, 1990. Conway, J. H. On Numbers and Games. New York: Aca- demic Press, 1976. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996. Dantzig, T. Number: The Language of Science, 4th rev. ed. New York: Free Press, 1985. Davis, P. J. The Lore of Large Numbers. New York: Random House, 1961. Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A; Koecher, M.; Mainzer, M.; and Remmert, R. Numbers. New York: Springer-Verlag, 1990. Frege, G. Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd rev. ed. Evanston, IL: Northwestern University Press, 1980. Ifrah, G. From One to Zero: A Universal History of Numbers. New York: Viking, 1987. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. McLeish, J. Number: The History of Numbers and How They Shape Our Lives. New York: Fawcett Columbine, 1992. Phillips, R. Numbers: Facts, Figures & Fiction. Cambridge, England: Cambridge University Press, 1994. Rosenfelder, M. "Numbers from 1 to 10 in Over 4000 Languages." http://zompist.com/numbers.shtml. Russell, B. "Definition of Number." Introduction to Mathe- matical Philosophy. New York: Simon and Schuster, 1971. Smeltzer, D. Man and Number. Buchanan, NY: Emerson Books, 1974. Weisstein, E. W. "Books about Numbers." http://www.trea- sure-troves.com/books/Numbers.html. Wells, D. W. The Penguin Dictionary of Curious and Inter- esting Numbers. Harmondsworth, England: Penguin Books, 1986. Number Axis REAL LINE Number Field If r is an ALGEBRAIC NUMBER of degree n, then the totality of all expressions that can be constructed from r by repeated additions, subtractions, multi- plications, and divisions is called a number field (or an ALGEBRAIC NUMBER FIELD ) generated by r, and is denoted F[r] : Formally, a number field is a finite extension Q( a) of the FIELD Q of RATIONAL NUMBERS . The elements of a number field which are ROOTS of a POLYNOMIAL zn /C27an/C281zn/C281 /C27/C1/C1/C1/C27a0 /C300 with integer coefficients and leading coefficient 1 are called the ALGEBRAIC INTEGERS of that field. See also ALGEBRAIC INTEGER ,ALGEBRAIC NUMBER , FIELD,FINITE FIELD,FUNCTION FIELD,LOCAL FIELD, NUMBER FIELD SIEVE,Q,Q UADRATIC FIELD,SIGNA- TURE (NUMBER FIELD) References Cohen, H. A Course in Computational Algebraic Number Theory, 3rd. corr. ed. New York: Springer-Verlag, 1996. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 127, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 151 /C1/152, 1993. Number Field Sieve An extremely fast factorization method developed by Pollard which was used to factor the RSA -130 NUM- BER. This method is the most powerful known for factoring general numbers, and has complexity O exp c log n ðÞ1 =3log log n ðÞ2 =3hino ; (1) reducing the exponent over the CONTINUED FRACTION FACTORIZATION ALGORITHM and QUADRATIC SIEVE . There are three values of c relevant to different flavors of the method (Pomerance 1996). For the "special" case of the algorithm applied to numbersnear a large POWER , c /C3032 9CC1:CC171 =3 /C301 :526285 ... ; (2) for the "general" case applicable to any ODD POSITIVE number which is not a POWER , c /C3064 9CC1:CC171 =3 /C301 :922999 ... ; (3) and for a version using many POLYNOMIALS (Copper- smith 1993), c/C301 392/C2726ffiffiffiffiffiffi 13pCC1:CC171=3 /C301:901883 . . . (4) See also QUADRATIC SIEVE, RSA NUMBER References Coppersmith, D. "Modifications to the Number Field Sieve." J. Cryptology 6, 169/C1/180, 1993. Coppersmith, D.; Odlyzko, A. M.; and Schroeppel, R. "Dis- crete Logarithms in GF( p)."Algorithmics 1,1/C1/15, 1986. Cowie, J.; Dodson, B.; Elkenbracht-Huizing, R. M.; Lenstra, A. K.; Montgomery, P. L.; Zayer, J. A. "World Wide Number Field Sieve Factoring Record: On to Bits." InAdvances in Cryptology--ASIACRYPT ’96 (Kyongju) (Ed. K. Kim and T. Matsumoto.) New York: Springer-Verlag, pp. 382 /C1 /394, 1996. Elkenbracht-Huizing, R.-M. "A Multiple Polynomial General Number Field Sieve." Algorithmic Number Theory (Ta- lence, 1996). New York: Springer-Verlag, pp. 99 /C1/114, 1996. Elkenbracht-Huizing, R.-M. "An Implementation of the Number Field Sieve." Experiment. Math. 5, 231/C1/253, 1996. Elkenbracht-Huizing, R.-M. "Historical Background of the Number Field Sieve Factoring Method." Nieuw Arch. Wisk. 14, 375/C1/389, 1996. Elkenbracht-Huizing, R.-M. Factoring Integers with the Number Field Sieve. Doctor’s Thesis, Leiden University, 1997. Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673 /C1/715, 1990. Lenstra, A. K. and Lenstra, H. W. Jr. The Development of the Number Field Sieve. Berlin: Springer-Verlag, 1993. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473/C1/1485, 1996. Number Field Sieve Factorization Method An extremely fast factorization method developed by Pollard which was used to factor the RSA -130 NUM- BER. This method is the most powerful known for factoring general numbers, and has complexity reducing the exponent over the CONTINUED FRACTION FACTORIZATION ALGORITHM and QUADRATIC SIEVE FACTORIZATION METHOD . There are three values of c relevant to different flavors of the method (Pomer- ance 1996). For the "special" case of the algorithm applied to numbers near a large POWER , ˜A for the "general" case applicable to any ODD POSITIVE number which is not a POWER , /C2A and for a version using many POLYNOMIALS (Copper- smith 1993), 1060 See also RSA NUMBER References Coppersmith, D. "Modifications to the Number Field Sieve." J. Cryptology 6, 169 /C1/180, 1993. Coppersmith, D.; Odlyzko, A. M.; and Schroeppel, R. "Dis- crete Logarithms in GF(p)." Algorithmics 1,1/C1/15, 1986. Cowie, J.; Dodson, B.; Elkenbracht-Huizing, R. M.; Lenstra, A. K.; Montgomery, P. L.; Zayer, J. A. "World Wide Number Field Sieve Factoring Record: On to SmðÞ/C30 (1 /C27ffiffi 2p Þm /C27 1/C28ffiffi 2pðÞm 2 Bits." In Advances in Cryptology--ASIA- CRYPT ’96 (Kyongju) (Ed. K. Kim and T. Matsumoto.) New York: Springer-Verlag, pp. 382 /C1/394, 1996. Elkenbracht-Huizing, R.-M. "A Multiple Polynomial General Number Field Sieve." Algorithmic Number Theory (Ta- lence, 1996). New York: Springer-Verlag, pp. 99 /C1/114, 1996. Elkenbracht-Huizing, R.-M. "An Implementation of the Number Field Sieve." Experiment. Math. 5, 231 /C1/253, 1996. Elkenbracht-Huizing, R.-M. "Historical Background of the Number Field Sieve Factoring Method." Nieuw Arch. Wisk. 14, 375 /C1/389, 1996. Elkenbracht-Huizing, R.-M. Factoring Integers with the Number Field Sieve. Doctor’s Thesis, Leiden University, 1997. Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673 /C1/715, 1990. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/1485, 1996. Number Group FIELD Number Guessing By asking a small number of innocent-sounding questions about an unknown number, it is possible to reconstruct the number with absolute certainty (assuming that the questions are answered correctly). Ball and Coxeter (1987) give a number of sets of questions which can be used. One of the simplest algorithms uses only three queries that can be used to determine an unknown number n from an audience member.1. Ask the person to compute n?/C303n (i.e., three times the secret number n) and announce if the result is EVEN or ODD. 2. If you were told that n? is EVEN , ask the person to reveal the number nƒ which is half of n?: If you were told that n? is ODD, ask the person to reveal the number nƒ which is half of n?/C271 :/ 3. Ask the person to reveal the number of times k which 9 divides evenly into n§/C303nƒ:/ The original number n is then given by 2k if n? was EVEN ,or /2k þ 1/ if n? was ODD. For n /C302m even, n?/C30 6m; nƒ/C303m; n§/C309m; k /C30m,so2 k /C302m /C30n: For n /C30 2m /C271 odd, n?/C306m /C273; n ƒ/C303m /C272; n §/C309m /C276; k /C30m,so2 k /C271 /C302m /C271 /C30n:/ Another method asks: 1. Multiply the number n by 5. 2. Add 6 to the product. 3. Multiply the sum by 4. 4. Add 9 to the product. 5. Multiply the sum by 5 and reveal the result n?:/ The original number is then given by n /C30 n?/C28165 ðÞ =100; since the above steps give n?/C305(4(5 n /C276) /C279) /C30100n /C27165:/ See also NUMBER PICKING References Bachet, C. G. Proble `mes plaisans et de´lectables, 2nd ed. 1624. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 5 /C1/20, 1987. Kraitchik, M. "To Guess a Selected Number." §3.3 in Mathematical Recreations. New York: W. W. Norton, pp. 58 /C1/66, 1942. Number Pattern It is possible to construct simple functions which produce growing patterns. For example, the BAXTER- HICKERSON FUNCTION f(n) /C301 32 /C215 105n /C28104n /C272 /C215 103n /C27102n /C2710n /C271CC0CC1 produces the sequence 64037, 6634003367, 666334000333667, .... See also BAXTER- HICKERSON FUNCTION ,N UMBER PYRAMID Number Picking Place 2n balls in a bag and number them 1 to 2n; then pick half of them at random. The number of different possible sums for n /C301, 2, 3, ... are then 2, 5, 10, 17, 26, ... (Sloane’s A002522), or n2 /C271/ See also NUMBER GUESSING References Sloane, N. J. A. Sequences A002522 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Number Pyramid A set of numbers obeying a pattern like the following, 91 /C215 37 /C303367 9901 /C215 3367 /C3033336667 999001 /C215 333667 /C30333333666667 99990001 /C215 33336667 /C303333333366666667 42 /C3016 342 /C301156 3342 /C30111556 72 /C3049 672 /C304489 6672 /C30444889 : See also AUTOMORPHIC NUMBER ,NUMBER PATTERN References Heinz, H. "Miscellaneous Number Patterns." http:// www.geocities.com/CapeCanaveral/Launchpad/4057/mis- cnum.htm. Number Shape FIGURATE NUMBER Number Sign OCTOTHORPE Number System BASE (NUMBER ) Number Theoretic Transform Simplemindedly, a number theoretic transform is a generalization of a FAST FOURIER TRANSFORM ob- tained by replacing e /C282 pik=N with an nth PRIMITIVE ROOT OF UNITY . This effectively means doing a trans- form over the QUOTIENT RING Z=pZ instead of the COMPLEX NUMBERS C: The theory is rather elegant and uses the language of FINITE FIELDS and NUMBER THEORY . See also FAST FOURIER TRANSFORM ,FINITE FIELD References Arndt, J. "Numbertheoretic Transforms (NTTs)." Ch. 4 in "Remarks on FFT Algorithms." http://www.jjj.de/fxt/. Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993. Number Theory A vast and fascinating field of mathematics, some- times called "higher arithmetic," consisting of the study of the properties of whole numbers. PRIMES andPRIME FACTORIZATION are especially important in number theory, as are a number of functions such as the DIVISOR FUNCTION ,RIEMANN ZETA FUNCTION , and TOTIENT FUNCTION . Excellent introductions to number theory may be found in Ore (1988) and Beiler (1966). The classic history on the subject (now slightly dated) is that of Dickson (1952). The great difficulty required to prove relatively simple results in number theory prompted no less an authority than Gauss to remark that "it is just this which gives the higher arithmetic that magical charm which has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics." Gauss, often known as the "prince of mathematics," called mathematics the "queen of the sciences,"’ and considered number theory the "queen of mathematics" (Beiler 1966, Goldman 1997). See also ADDITIVE NUMBER THEORY ,A RITHMETIC , CONGRUENCE ,D IOPHANTINE EQUATION ,D IVISOR FUNCTION ,G O¨ DEL’S INCOMPLETENESS THEOREM , MULTIPLICATIVE NUMBER THEORY ,PEANO’S AXIOMS , PRIME COUNTING FUNCTION ,PRIME FACTORIZATION , PRIME NUMBER ,QUADRATIC RECIPROCITY THEOREM , RIEMANN ZETA FUNCTION ,TOTIENT FUNCTION References Andrews, G. E. Number Theory. New York: Dover, 1994. Andrews, G. E.; Berndt, B. C.; and Rankin, R. A. (Ed.). Ramanujan Revisited: Proceedings of the Centenary Con- ference, University of Illinois at Urbana-Champaign, June 1/C1/5, 1987. Boston, MA: Academic Press, 1988. Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976. Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed. New York: Dover, 1966. Bellman, R. E. Analytic Number Theory: An Introduction. Reading, MA: Benjamin/Cummings, 1980. Berndt, B. C. Ramanujan’s Notebooks, Part I. New York: Springer-Verlag, 1985. Berndt, B. C. Ramanujan’s Notebooks, Part II. New York: Springer-Verlag, 1988. Berndt, B. C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, 1997. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1993. Berndt, B. C. Ramanujan’s Notebooks, Part V. New York: Springer-Verlag, 1997. Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc, 1995. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.New York: Wiley, 1987. Brown, K. S. "Number Theory." http://www.seanet.com/ ~ksbrown/inumber.htm. Burr, S. A. The Unreasonable Effectiveness of Number Theory. Providence, RI: Amer. Math. Soc., 1992. Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989. Carmichael, R. D. The Theory of Numbers, and Diophantine Analysis. New York: Dover, 1959. Cohen, H. Advanced Topics in Computational Number Theory. New York: Springer-Verlag, 2000. Cohn, H. Advanced Number Theory. New York: Dover, 1980. Courant, R. and Robbins, H. "The Theory of Numbers." Supplement to Ch. 1 in What is Mathematics?: An Ele- mentary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 21 /C1/51, 1996. Davenport, H. The Higher Arithmetic: An Introduction to the Theory of Numbers, 6th ed. Cambridge, England: Cam- bridge University Press, 1992. Davenport, H. and Montgomery, H. L. Multiplicative Num- ber Theory, 2nd ed. New York: Springer-Verlag, 1980. Dickson, L. E. History of the Theory of Numbers, 3 vols. New York: Chelsea, 1952. Dudley, U. Elementary Number Theory. San Francisco, CA: W. H. Freeman, 1978. Friedberg, R. An Adventurer’s Guide to Number Theory. New York: Dover, 1994. Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT: Yale University Press, 1966. Goldman, J. R. The Queen of Mathematics: An Historically Motivated Guide to Number Theory. Natick, MA: A. K. Peters, 1997. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1959. Hasse, H. Number Theory. Berlin: Springer-Verlag, 1980. Herkommer, M. A. Number Theory: A Programmer’s Guide. New York: McGraw-Hill, 1999. Ireland, K. F. and Rosen, M. I. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, 1995. Kato, K.; Kurokawa, N.; and Saito, T. Number Theory 1: Fermat’s Dream. Providence, RI: Amer. Math. Soc., 2000. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., 1991. Koblitz, N. A Course in Number Theory and Cryptography. New York: Springer-Verlag, 1987. Landau, E. Elementary Number Theory, 2nd ed. New York: Chelsea, 1999. Lang, S. Algebraic Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Lenstra, H. W. and Tijdeman, R. (Eds.). Computational Methods in Number Theory, 2 vols. Amsterdam: Mathe- matisch Centrum, 1982. LeVeque, W. J. Fundamentals of Number Theory. New York: Dover, 1996. Mitrinovic, D. S. and Sandor, J. Handbook of Number Theory. Dordrecht, Netherlands: Kluwer, 1995. Mollin, R. A. Algebraic Number Theory. Boca Raton, FL: CRC Press, 1999. Mollin, R. A. Fundamental Number Theory with Applica- tions. Boca Raton, FL: CRC Press, 1998. Niven, I. M.; Zuckerman, H. S.; and Montgomery, H. L. An Introduction to the Theory of Numbers, 5th ed. New York: Wiley, 1991. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, 1988. Ore, Ø. Invitation to Number Theory. Washington, DC: Math. Assoc. Amer., 1967. Ore, Ø. Number Theory and Its History. New York: Dover, 1988. Rose, H. E. A Course in Number Theory, 2nd ed. Oxford, England: Clarendon Press, 1995.Rosen, K. H. Elementary Number Theory and Its Applica- tions, 3rd ed. Reading, MA: Addison-Wesley, 1993. Schroeder, M. R. Number Theory in Science and Commu- nication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. New York: Springer-Verlag, 1997. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993. Sierpinski, W. 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, 1939. Vinogradov, I. M. Elements of Number Theory, 5th rev. ed. New York: Dover, 1954. Weil, A. Basic Number Theory, 3rd ed. Berlin: Springer- Verlag, 1995. Weil, A. Number Theory: An Approach Through History From Hammurapi to Legendre. Boston, MA: Birkha ¨user, 1984. Weisstein, E. W. "Books about Number Theory." http:// www.treasure-troves.com/books/NumberTheory.html. Weyl, H. Algebraic Theory of Numbers. Princeton, NJ: Princeton University Press, 1998. Yildirim, C. Y. and Stepanov, S. A. (Eds.). Number Theory and Its Applications. New York: Dekker, 1998. Young, J. W. A. "The Theory of Numbers." Ch. 7 in Mono- graphs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 306 /C1/349, 1955. Number Triangle BELL TRIANGLE ,CLARK’S TRIANGLE ,EULER’S TRIAN- GLE,L EIBNIZ HARMONIC TRIANGLE ,L OSSNITSCH’S TRIANGLE ,M AGOG TRIANGLE ,M ONOTONE TRIANGLE , PASCAL’S TRIANGLE ,SEIDEL- ENTRINGER- ARNOLD TRI- ANGLE ,TRINOMIAL TRIANGLE Number Wall QUOTIENT- DIFFERENCE TABLE Numerator The number pin a FRACTION p=q:/ See also DENOMINATOR ,FRACTION ,RATIONAL NUM- BER Numeric Function AFUNCTION /f:A0B/such that Bis a SET of numbers. Numerical Derivative While it is usually much easier to compute a DERIVA- TIVE instead of an INTEGRAL (which is a little strange, considering that "more" functions have integrals than derivatives), there are still many applications wherederivatives need to be computed numerically. Thesimplest approach simply uses the definition of the DERIVATIVE f?xðÞ/C13lim h00f(x/C27h)/C28f(x) h for some small numerical value of h/C101:/ See also NUMERICAL INTEGRATION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Numerical Derivatives." §5.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 180 /C1/184, 1992. Weisstein, E. W. "Books about Numerical Methods." http:// www.treasure-troves.com/books/NumericalMethods.html. Numerical Integration The approximate computation of an INTEGRAL using numerical techniques. The numerical computation of an INTEGRAL is sometimes called QUADRATURE . Ue- berhuber (1997, p. 71) uses the word "QUADRATURE " to mean numerical computation of a univariate INTEGRAL , and "CUBATURE " to mean numerical com- putation of a MULTIPLE INTEGRAL . There are a wide range of methods available for numerical integration. A good source for such tech- niques is Press et al. (1992). The most straightforward numerical integration tech- nique uses the NEWTON- COTES FORMULAS (also called QUADRATURE FORMULAS ), which approximate a func- tion tabulated at a sequence of regularly spaced INTERVALS by various degree POLYNOMIALS . If the endpoints are tabulated, then the 2- and 3-point formulas are called the TRAPEZOIDAL RULE and SIMPSON’S RULE , respectively. The 5-point formula is called BODE’S RULE . A generalization of the TRAPE- ZOIDAL RULE is ROMBERG INTEGRATION , which can yield accurate results for many fewer function eva- luations. If the functions are known analytically instead of being tabulated at equally spaced intervals, the best numerical method of integration is called GAUSSIAN QUADRATURE . By picking the abscissas at which to evaluate the function, Gaussianquadrature produces the most accurate approximations possible. However, given the speed of modern computers, the additional complication of the GAUSSIAN QUADRATURE formalism often makes it less desirable than simply brute-force calculating twice as many points on a regular grid (which also permits the already computed values of the function to be re-used). An excellent reference for GAUSSIAN QUADRATURE is Hildebrand (1956). See also CUBATURE ,DOUBLE EXPONENTIAL INTEGRA- TION ,FILON’S INTEGRATION FORMULA ,GAUSS- KRON- ROD QUADRATURE ,GREGORY’S FORMULA ,INTEGRAL , INTEGRATION ,M ONTE CARLO INTEGRATION ,N UMER- ICAL DERIVATIVE ,QUADRATURE ,QUASI- MONTE CARLO INTEGRATION ,T-INTEGRATION References Corbit, D. "Numerical Integration: From Trapezoids to RMS: Object-Oriented Numerical Integration." Dr. Dobb’s J., No. 252, 117 /C1/120, Oct. 1996.Davis, P. J. and Rabinowitz, P. Methods of Numerical Integration, 2nd ed. New York: Academic Press, 1984. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319 /C1/323, 1956. Milne, W. E. Numerical Calculus: Approximations, Inter- polation, Finite Differences, Numerical Integration and Curve Fitting. Princeton, NJ: Princeton University Press, 1949. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, 1992. Ueberhuber, C. W. "Numerical Integration." Ch. 12 in Nu- merical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, pp. 65 /C1/169, 1997. Weisstein, E. W. "Books about Numerical Methods." http:// www.treasure-troves.com/books/NumericalMethods.html. Whittaker, E. T. and Robinson, G. "Numerical Integration and Summation." Ch. 7 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 132 /C1/163, 1967. Numerology The study of numbers for the supposed purpose of predicting future events or seeking connections with the occult. See also BEAST NUMBER ,NUMBER THEORY References Dudley, U. Numerology, or, What Pythagoras Wrought. Washington, DC: Math. Assoc. Amer., 1997. NURBS Curve A nonuniform rational B-SPLINE curve defined by C(t) /C30Pn i /C300 Ni;ptðÞwiPiPn i/C300 Ni ;ptðÞwi; where p is the order, Ni;pare the B-SPLINE basis functions, Pi are control points, and the weight wi of Piis the last ordinate of the homogeneous point Pw i : These curves are CLOSED under perspective transfor- mations and can represent CONIC SECTIONS exactly. See also B-SPLINE ,BE´ ZIER CURVE , NURBS SURFACE References Piegl, L. and Tiller, W. The NURBS Book, 2nd ed. New York: Springer-Verlag, 1997. NURBS Surface A nonuniform rational B-SPLINE surface of degree (p, q) is defined by Su;vðÞ/C30Pm i/C300Pnj/C300 Ni ;puðÞNj;qvðÞwi;jPi;jPm i /C300Pnj/C300 Ni;puðÞNj;qvðÞwi;j; where Ni ;pand Nj;qare the B-SPLINE basis functions, Pi;j are control points, and the weight wi;j of Pi;j is the last ordinate of the homogeneous point Pw i;j :/ See also B-SPLINE ,BE´ ZIER CURVE , NURBS CURVE Nyquist Frequency In order to recover all FOURIER components of a periodic waveform, it is necessary to sample more than twice as fast as the highest waveform frequency n ; i.e., fNyquist /C302 n: This cutoff frequency /fNyquist/ above which a signal must be sampled in order to be able to fully recon- struct it is called the Nyquist frequency.See also FOURIER SERIES ,F OURIER TRANSFORM , NYQUIST SAMPLING ,OVERSAMPLING ,SAMPLING THE- OREM Nyquist Sampling Sampling at the NYQUIST FREQUENCY . See also SAMPLING THEOREM O O The symbol O is sometimes used to represent CAYLEY NUMBERS (also commonly known as octonions). See also CAYLEY NUMBER Obelisk A polyhedron formed by two parallel rectangles, not congruent to each other, whose side faces are trape- zoids. The VOLUME is given by V /C301 6 h[(2a /C27a ?)b /C27(2a?/C27a)b?] /C3016 h[(ab /C27(a /C27a?)(b /C27b?) /C27a ?b ?]: The distance from the bottom base to the CENTROID is ¯z /C30h(ab /C27 ab ?/C27a ?b /C27 3a ?b?) 2(ab /C27 ab ?/C27a ?b /C27 2a ?b?): The term obelisk is sometimes also used to refer to the DAGGER symbol (Bringhurst 1997, p. 275). See also DAGGER References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, 1997. Harris, J. W. and Stocker, H. "Obelisk." §4.5.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 102, 1998. Obelus The symbol } used to indicate DIVISION . In typogra- phy, an obelus has a more general definition as any symbol, such as the DAGGER (/$) ; used to indicate a footnote (Bringhurst 1997, p. 225). See also DIVISION ,SOLIDUS References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, 1997.Object A mathematical structure (e.g., a GROUP , VECTOR SPACE ,or DIFFERENTIABLE MANIFOLD )ina CATEGORY . See also MORPHISM Oblate Ellipsoid OBLATE SPHEROID Oblate Spheroid A "squashed" SPHEROID for which the equatorial radius ais greater than the polar radius c,s oa/C21c (called an oblate ellipsoid by Tietze 1965, p. 27). An oblate spheroid is a SURFACE OF REVOLUTION obtained by rotating an ELLIPSE about its minor axis (Hilbert and Cohn-Vossen 1999, p. 10). To first approxima-tion, the shape assumed by a rotating fluid (includingthe Earth, which is "fluid" over astronomical time scales) is an oblate spheroid. The oblate spheroid can be specified parametrically by the usual SPHEROID equations (for a SPHEROID with Z-AXIS as the symme- try axis), x/C30asinvcosu (1) y/C30asinvsinu (2) z/C30ccosv; (3) with a/C21c,u/C230;2p ½Þ ;and v/C23[0;p]:Its Cartesian equation is x2/C27y2 a2/C27z2 c2/C301: (4) The ELLIPTICITY of an oblate spheroid is defined by e/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28c2 a2s ; (5) so that 1/C28e2/C30c2 a2: (6) The radial distance from center of the spheroid as a function of latitude dis given by r(d)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27c2/C27(a/C28c)(a/C27c) cos(2 d) 2s (7) /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2sin2dp : (8) The SURFACE AREA of an oblate spheroid can be computed as a SURFACE OF REVOLUTION about the Z- AXIS, S/C302pgr(z)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27[r?(z)]2q dz (9) with radius as a function of zgiven by r(z)/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28z c !2vuut: (10) Therefore S/C302pagc /C28cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28z2 c2 ! 1/C27a2z2 c2(c2/C28z2)"#vuutdz /C30pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28c2p /C22a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28c2p /C27c2alna/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28c2p a/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28c2p ! "# :(11) Using the identity ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28c2p /C30ae (12) gives S/C302pa2/C27pc2 eln1/C27e 1/C28e ! (13) (Beyer 1987, p. 131). Note that this is the conven- tional form in which the surface area of an oblate spheroid is written, although it is formally equivalent to the conventional form for the PROLATE SPHEROID via the identity c2p e(a;c)ln1/C27e(a;c) 1/C28e(a;c)"# /C302pac e(c;a)sin/C281[e(c;a)];(14) where e(x;y) is defined by e(x;y)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 y2s : (15) The VOLUME of an oblate spheroid can be computed from the formula for a general ELLIPSOID with b/C30a, V/C304 3pa2c (16) (Beyer 1987, p. 131). An oblate spheroid with its origin at a FOCUS has equation r/C30a(1/C28e2) 1/C27ecosf: (17) Define kand expand up to POWERS ofe6;k/C13e2(1/C28e2)/C281/C30e2(1/C27e2/C282e4/C276e6/C27... ) /C30e2/C27e4/C282e6/C27. . . (18) k2/C30e4/C27e6/C27. . . (19) k3/C30e6/C27. . . (20) Expanding rinPOWERS ofELLIPTICITY toe6therefore yields r a/C301/C281 2(e2/C27e4/C282e4/C276e6)sin2d/C2734(e4/C27e6)sin4d /C2815 8e6sin6d/C27...: (21) In terms of L EGENDRE POLYNOMIALS , r a/C301/C2816e2/C281120e4/C28103 1680e67C)67C)7 /C27/C281 3e2/C285 42e4/C283 56e67C)67C)7 P2 /C273 35e4/C2757 770e67C)67C)7 P4/C285 231e6P6/C27...: (22) The ELLIPTICITY may also be expressed in terms of the OBLATENESS (also called FLATTENING ), denoted eorf. e/C13a/C28c a(23) c/C30a(1/C28e) (24) c2/C30a2(1/C28e)2(25) (1/C28e)2/C301/C28e2; (26) so e/C301/C28ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2p (27) and e2/C301/C28(1/C28e)2/C301/C28(1/C282e/C27e2)/C302e/C28e2(28) r/C30a1/C272e/C28e2 (1/C28e)2sin2d"#/C281=2 : (29) Define kand expand up to POWERS ofe6 k/C13(2e/C28e)(1/C28e)/C282/C30(2e/C28e2)(1/C272e/C286e2/C27... ) /C302e/C274e4/C2812e3/C28e2/C282e3/C27... /C302e/C273e2/C2814e3/C27. . . (30) k2/C304e2/C276e3/C27. . . (31) k3/C308e3/C27. . . (32) Expanding rinPOWERS of the OBLATENESS toe3yields r a/C301/C281 2(2e/C273e2/C2814e3)sin2d/C2734(4e2/C276e3)sin4d /C278e3sin6d/C27...: (33) In terms of L EGENDRE POLYNOMIALS , r a/C301/C281 3e/C2825e2/C2813 105e37C)67C)7 /C27/C2823e/C2817e2/C281 21e37C)67C)7 P2 /C2712 35e2/C2896 385e37C)67C)7 P4/C2840 231e3P6/C27...: (34) To find the projection of an oblate spheroid onto a PLANE , set up a coordinate system such that the Z- AXIS is towards the observer, and the X-AXIS is in the PLANE of the page. The equation for an oblate spheroid is r(u)/C30a1/C272e/C28e2 (1/C28e)2cos2u"#/C281=2 : (35) Define k/C132e/C28e2 (1/C28e)2; (36) andx/C13sinu:Then r(u)/C30a[1/C27k(1/C28x2)]/C281=2/C30a(1/C27k/C28kx2)/C281=2:(37) Now rotate that spheroid about the X-AXIS by an ANGLE Bso that the new symmetry axes for the spheroid are x?/C13x;y?;andz?:The projected height of a point in the x/C300PLANE on the Y-AXIS is y/C30r(u) cos( u/C28B)/C30r(u)(cos ucosB/C28sinusinB) /C30r(u)ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p cosB/C27xsinB7C)67C)7 : (38) To find the highest projected point, dy du/C30asin(B/C28u) (1/C27kcos2u)1=2/C27akcos(B/C28u)cosusinu (1/C27kcos2u)3=2 /C300: (39) Simplifying, tan(B/C28u)(1/C27kcos2u)/C27kcosusinu/C300: (40) But tan(B/C28u) /C30tanB/C28tanu 1/C27tanBtanu/C30tanB/C28sinuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2up 1/C27tanBsinuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28sin2up /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2up tanB/C28sinuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2up /C27tanBsinu(41) Plugging (41) into (40),ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p tanB/C28xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x2p /C27xtanB[1/C27k(1/C28x2)]/C27kxffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C300 (42) and performing a number of algebraic simplifications ffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x 2p tanB/C28x7C)67C)7 (1/C27k/C28kx2)/C27kxffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x 2p /C2ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C27xtanB7C)67C)7 /C300 (43) (1/C27k)ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p tanB/C28kx2ffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x 2p tanB/C28x/C28kx/C27kx3hi /C27kx(1/C28x2)/C27kx2ffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x 2p tanBhi (44) (1/C27k) tan Bffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C28kx(1/C28x2)/C28x/C27kx(1/C28x2)/C300 (45) (1/C27k) tan Bffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C30x (46) (1/C27k)2tan2B(1/C28x2)/C30x2(47) x21/C27(1/C27k)2tan2Bhi /C30(1/C27k)2tan2B (48) finally gives the expression for xin terms of Bandk, x2/C30tan2B(1/C27k)2 1/C27(1/C27k)2tan2B: (49) Combine (37) and (38) and plug in for x, y/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p cosB/C27xsinBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27k/C28kx2p /C30acosB/C27(1/C27k)sin2B cosBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27k)[1/C27(1/C27k) tan2B]p /C30acos2B/C27(1/C27k) sin2B cosBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C27k)[1/C27(1/C27k) tan2B]p : (50) Now re-express kin terms of aand c, using e/C13 1/C28c=a; k/C13(2/C28e)e (1/C28e)2/C301/C27c a ! 1/C28c a ! c a !2 /C301/C28c a !2 c a !2 /C30a c !2 /C281; (51) so 1/C27k/C30a c !2 (52) Plug (51) and (52) into (50) to obtain the SEMIMINOR AXIS of the projected oblate spheroid, c?/C30acos2B/C27a c !2 sin2B cosBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c !2 1/C27a c !2 tan2B2 435vuuut /C30a cos2B/C27a c !2 sin2B a cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2B/C27a c !2 sin2Bvuut /C30cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2B/C27a c !2 sin2Bvuut/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2cos2B/C27a2sin2Bp /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C28e)2cos2B/C27sin2Bq : (53) We wish to find the equation for a spheroid which has been rotated about the x/C13x?/-axis by ANGLE B, then the Z-AXIS byANGLE P x? y? z?2 435/C3010 0 0 cos BsinB 0/C28sinBcosB2 435cosP0 sin P 01 0 /C28sinP0 cos P2435x y z2 435 /C30cosP 0 sin P /C28sinBsinPcosBsinBcosP /C28cosBsinP/C28sinBcosBcosP2435x y z2 435:(54) Now, in the original coordinates ( x?;y?;z?);the spher- oid is given by the equation x? a22 /C27y? c22 /C27z?2 a2/C301; (55) which becomes in the new coordinates, (xcosP/C27ysinP)2 a2 /C27(/C28xsinBsinP/C27zcosB/C27ysinBcosP)2 a2 /C27(/C28xcosBsinP/C28zsinB/C27ycosBcosP)2 c2/C301: (56) Collecting COEFFICIENTS , Ax2/C27By2/C27Cz2/C27Dxy/C27Exz/C27Fyz/C301; (57) where A/C13cos2P/C27sin2Bsin2P a2/C27cos2Bsin2P c2(58) B/C13sin2P/C27sin2Bcos2P a2/C27cos2Bcos2P c2(59)C/C13cos2B a2/C27sin2B c2(60) D/C132 cos PsinP1/C28sin2B a2/C28cos2B c2 ! /C302 cos PsinPcos2B1 a2/C281 c2 ! (61) E/C132 sin BcosBsinP1 b2/C281 a2 ! (62) F/C132 sin BcosBcosP1 a2/C281 b2 ! : (63) If we are interested in computing z, the radial distance from the symmetry axis of the spheroid ( y) corresponding to a point Cz2/C27(Ex/C27Fy)z/C27(Ax2/C27By2/C27Dxy/C281) /C30Cz2/C27G(x;y)z/C27H(x;y)/C300; (64) where G(x;y)/C13Ex/C27Fy (65) H(x;y)/C13Ax2/C27By2/C27Dxy/C281: (66) zcan now be computed using the quadratic equation when ( x, y) is given, z/C30/C28G(x;y)9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2(x;y)/C284CG(x;y)p 2C: (67) IfP/C300, then we have sin P/C300 and cos P/C301;so (58) to (63) and (65) to (66) become A/C131 a2(68) B/C13sin2B a2/C27cos2B b2(69) C/C13cos2B a2/C27sin2B b2(70) D/C130 (71) E/C130 (72) F/C132 sin BcosB1 a2/C281 b2 ! (73) G(x;y)/C13Fy/C302ysinBcosB1 a2/C281 b2 ! (74) H(x;y) /C13Ax2 /C27By2 /C281 /C30x2 a2 /C27y2sin2 B a2/C27cos2 B b2 ! /C281: (75) See also APPLE ,DARWIN-DE SITTER SPHEROID ,ELLIP- SOID,O BLATE SPHEROIDAL COORDINATES ,PROLATE SPHEROID ,SPHERE ,SPHEROID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 10, 1999. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, p. 27, 1965. Oblate Spheroid Geodesic The GEODESIC on an OBLATE SPHEROID can be com- puted analytically, although the resulting expression is much more unwieldy than for a simple SPHERE .A spheroid with equatorial radius aand polar radius c can be specified parametrically by x/C30asinvcosu (1) y/C30asinvsinu (2) z/C30ccosv; (3) where a/C21c. Using the first PARTIAL DERIVATIVES @x @u/C30/C28asinvsinu@x @v/C30acosvcosu (4) @y @u/C30asinvcosu@y @v/C30acosvsinu (5) @z @u/C300@z @v/C30/C28csinv; (6) and second PARTIAL DERIVATIVES @2x @u2/C30/C28asinvcosu@2x @v2/C30/C28asinvcosu (7) @2y @u2/C30/C28asinvsinu@2y @v2/C30/C28asinvsinu (8) @2z @u2/C300@2z @v2/C30/C28zcosv; (9) gives the GEODESICS functions as P/C13@x @u !2 /C27@y @u !2 /C27@z @u !2 /C30a2(sin2vcos2u/C27sin2vsin2u)/C30a2sin2v (10) Q/C13@x @u@x @v/C27@y @u@y @v/C27@z @u@z @v/C300 (11) R/C13@x @v !2 /C27@y @v !2 /C27@z @v !2 /C30a2/C27(c2/C28a2)sin2v/C30a2(1/C28e2sin2v); (12) where e/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28c2 a2s (13) is the ELLIPTICITY . Since Q/C300 and Pand Rare explicit functions of v only, we can use the special form of the GEODESIC equation u/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R P2/C28c2 1Ps dv/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2(1/C28e2sin2v) a4sin4v/C28c21a2sin2vs dv /C301 c1gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2sin2v a c17C)67C)72 sin2v/C281vuuutdv sinv: (14) Integrating gives u/C30 /C28e2Ff½(d2/C281)e2 d2/C28e2 ! /C28d2Pd2/C281;f½(d2/C281)e2 d2/C28e2 ! c1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2/C28e2p ; (15) where d/C13a c1(16) cosf/C13dcosvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2/C281p ; (17) /F(f½m)i sa n ELLIPTIC INTEGRAL OF THE FIRST KIND with PARAMETER m, and P(f½m;k)i sa n ELLIPTIC INTEGRAL OF THE THIRD KIND . GEODESICS other than MERIDIANS of an OBLATE SPHEROID undulate between two parallels with lati- tudes equidistant from the equator. Using the W EIER- STRASS SIGMA FUNCTION and W EIERSTRASS ZETA FUNCTION , the GEODESIC on the OBLATE SPHEROID can be written as x/C27iy/C30ks(a/C27u) s(u)s(a)eu[h/C28z(v/C27a)](18) x/C28iy/C30ks(a/C28u) s(u)s(a)e/C28u[h/C28z(v/C27a)](19) z2 /C30 l2s( vƒ/C27 u)(vƒ/C28 u) s2(u) s2(a) (20) (Forsyth 1960, pp. 108 /C1/109; Halphen 1886 /C1/1891). The equation of the GEODESIC can be put in the form df /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 e2 sin2 vp sin affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2 v /C28 sin2 ap sin vdv; (21) where a is the smallest value of v on the curve. Furthermore, the difference in longitude between points of highest and next lowest latitude on the curve is p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28 e 2 sin2 ap sin a g k 0dn u /C28 dn2 u 1 /C27 cot2 a sn2 udu; (22) where the MODULUS of the ELLIPTIC FUNCTION is k /C30e cos affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28 e 2 sin2 ap (23) (Forsyth 1960, p. 446). See also ELLIPSOID GEODESIC ,O BLATE SPHEROID , SPHERE GEODESIC References Forsyth, A. R. Calculus of Variations. New York: Dover, 1960. Halphen, G. H. Traite ´des fonctions elliptiques et de leurs applications fonctions elliptiques, Vol. 2. Paris: Gauthier- Villars, pp. 238 /C1/243, 1886 /C1/1891. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, pp. 28 /C1/29 and 40 /C1/ 41, 1965. Oblate Spheroidal Coordinates A system of CURVILINEAR COORDINATES in which two sets of coordinate surfaces are obtained by revolving the curves of the ELLIPTIC CYLINDRICAL COORDINATES about the Y-AXIS which is relabeled the Z-AXIS . The third set of coordinates consists of planes passing through this axis. x/C30acosh jcoshcosf (1) y/C30acosh jcoshsinf (2) z/C30asinh jsinh; (3)where j/C230;/C12½Þ ;h/C23[/C28p=2;p=2];andf/C230;2p ½Þ :Arf- ken (1970) uses ( u;v;8) instead of ( j;h;f):The SCALE FACTORS are hj/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh2j/C27sin2hq (4) hh/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisinh 2j/C27sin2hq (5) hf/C30acosh jcosh: (6) The L APLACIAN is 92f/C301 a3(sinh2j/C27sinh2h)cosh jcosh /C2@f @jacosh jcosh@f @h ! /C27@f @hacosh jcosh@f @h ! " /C27a2(sinh2j/C27sinh2h) acosh jcosh@2f @f27CP) /C301 a3(sinh2j/C27sinh2h)cosh jcosh /C2asinh jcosh@f @j/C27acosh jcosh@2f @j2" /C27asinh jcosh@f @h/C27acosh jcosh@2f @h27CP) /C271 a2(sinh2j/C27sinh2h)@2f @f2/C301 a2(sinh2j/C27sinh2h) /C21 cosh j@ @jcosh j@f @j ! /C271 cosh h@ @hcosh h@f @h ! "# /C271 a2(cosh2j/C27cos2h)@2f @f2(7) /C301 sinh2h/C27sinh2j /C2(sech2jtan2h/C27sec2tanh2j)@2 @f2/C27tanh j@ @j" /C27@2 @j2/C28tanh@ h/C27@2 h27CP) : (8) An alternate form useful for "two-center" problems is defined by j1/C30sinh j (9) j?1/C30cosh j (10) j2/C30cosh (11) j3/C30f; (12) where j1/C23[1;/C12];j2/C23[/C281;1];and j3/C23[0;2p):In these coordinates, y /C30a j?1 j2 sin j3 (13) z /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (j?12 /C281)(1 /C28 j2 2)q (14) x /C30a j?1 j2 cos j3 (15) (Abramowitz and Stegun 1972). The SCALE FACTORS are hj1/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j21 /C28 j22 j21 /C28 1s (16) hj2/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j21 /C28 j22 1 /C28 j22s (17) hj3/C30a jh; (18) and the LAPLACIAN is 92f /C301 a21 j2 1 /C27 j22@ @ j1(j2 1 /C271)@f @ j1"# ( /C271 j21 /C27 j22@ @ j2(1 /C28 j2 2)@f @ j2"# /C271 ( j2 1 /C28 1)(1 /C28 j22)@2f @ j237CP7 : (19) The HELMHOLTZ DIFFERENTIAL EQUATION is separ- able. See also HELMHOLTZ DIFFERENTIAL EQUATION– OB- LATE SPHEROIDAL COORDINATES ,L ATITUDE ,L ONG- ITUDE ,P ROLATE SPHEROIDAL COORDINATES , SPHERICAL COORDINATES References Abramowitz, M. and Stegun, C. A. (Eds.). "Definition of Oblate Spheroidal Coordinates." §21.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972. Arfken, G. "Prolate Spheroidal Coordinates (u, v, f) :/" §2.11 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 107 /C1/109, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 242, 1959. Moon, P. and Spencer, D. E. "Oblate Spheroidal Coordinates ( h; u ; c) :/" Table 1.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 31 /C1/34, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953. Oblate Spheroidal Wave Function The wave equation in OBLATE SPHEROIDAL COORDI- NATES is92 F/C27k2 F/C30@ @ j1( j2 1 /C271)@F @ j1"# /C27@ @ j2(1 /C28 j22)@F @ j2"# /C27j21 /C27 j22 (j21 /C27 1)(1 /C28 x2 2)@2 F @ f2 /C27c( j2 1 /C27 j22) F/C300 ; (1) where c /C131 2 ak : (2) Substitute in a trial solution F/C30Rmn(c ; j1)Smn(c ; j2)cos sin(mf): (3) The radial differential equation is d dj2(1 /C27 j2 2)d d j2Smn(c ; j2)"# /C28 lmn /C28c2 j22 /C27m2 1 /C27 j22 ! Rmn(c; j2) /C300; (4) and the angular differential equation is d dj2(1 /C28 j22)d d j2Smn(c ; j2)"# /C28 lmn /C28c2 j22 /C27m2 1 /C28 j22 ! Rmn(c ; j2) /C300 (5) (Abramowitz and Stegun 1972, pp. 753 /C1/755; Zwillin- ger 1997, p. 127). See also PROLATE SPHEROIDAL WAVE FUNCTION , SPHEROIDAL WAVE FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Spheroidal Wave Functions." Ch. 21 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 751 /C1/759, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Oblateness FLATTENING Oblique Angle An ANGLE which is not a RIGHT ANGLE . Oblique Cylinder CYLINDER Oblique Prism PRISM Oblique Triangle ATRIANGLE that is not a RIGHT TRIANGLE . See also RIGHT TRIANGLE ,TRIANGLE References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948. Oblong Number PRONIC NUMBER Obstruction Obstruction theory studies the extensibility of MAPS using algebraic GADGETS . While the terminology rapidly becomes technical and convoluted (as Iyanaga and Kawada note, "It is extremely difficult to discuss higher obstructions in general since they involve many complexities"), the ideas associated with ob- structions are very important in modern ALGEBRAIC TOPOLOGY . See also ALGEBRAIC TOPOLOGY ,CHERN CLASS ,EILEN- BERG- MAC LANE SPACE ,STIEFEL- WHITNEY CLASS References Iyanaga, S. and Kawada, Y. (Eds.). "Obstructions." §300 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 948 /C1/950, 1980. Obtuse Angle An ANGLE greater than p=2 RADIANS (908) and less than p RADIANS (180 8). See also ACUTE ANGLE ,FULL ANGLE ,OBTUSE TRIAN- GLE,REFLEX ANGLE ,RIGHT ANGLE ,STRAIGHT ANGLE Obtuse Triangle An obtuse triangle is a TRIANGLE in which one of the ANGLES is an OBTUSE ANGLE . (Obviously, only a single ANGLE in a TRIANGLE can be OBTUSE or it wouldn’t be aTRIANGLE .) A triangle must be either obtuse, ACUTE , orRIGHT . From the LAW OF COSINES , for a triangle with side lengths a,b, and c, cosC/C30a2/C27b2/C28c2 2ab; with Cthe angle opposite side C. For an angle to beobtuse, cos CB0:Therefore, an obtuse triangle satis- fies one of a2/C27b2Bc2;b2/C27c2Ba2;orc2/C27a2Bb2:/ An obtuse triangle can be dissected into no fewer than seven ACUTE TRIANGLES (Wells 1986, p. 71). A famous problem is to find the chance that three points picked randomly in a PLANE are the VERTICES of an obtuse triangle (Eisenberg and Sullivan 1996). Unfortunately, the solution of the problem depends on the procedure used to pick the "random" points(Portnoy 1994). In fact, it is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). Guy (1993)gives a variety of solutions to the problem. Woolhouse (1886) solved the problem by picking uniformly distributed points in the unit DISK, and obtained P2/C301/C284 p2/C281 8 ! /C3098/C284 p2/C300:719715 . . . : (1) The problem was generalized by Hall (1982) to n-D BALL TRIANGLE PICKING , and Buchta (1986) gave closed form evaluations for Hall’s integrals. Lewis Carroll (1893) posed and gave another solution to the problem as follows. Call the longest side of a TRIANGLE AB, and call the DIAMETER 2r:Draw arcs from AandBofRADIUS 2r:Because the longest side of the TRIANGLE is defined to be AB, the third VERTEX of the TRIANGLE must lie within the region ABCA .I f the third VERTEX lies within the SEMICIRCLE , the TRIANGLE is an obtuse triangle. If the VERTEX lieson the SEMICIRCLE (which will happen with probability 0), the TRIANGLE is a RIGHT TRIANGLE . Otherwise, it is an ACUTE TRIANGLE . The chance of obtaining an obtuse triangle is then the ratio of the AREA of the SEMICIRCLE to that of ABCA . The AREA ofABCA is then twice the AREA of a SECTOR minus the AREA of the TRIANGLE . Awhole figure /C3024pr2 6 ! /C28ffiffiffi 3p r2/C30r24 3p/C28ffiffiffi 3p7C)67C)7 :(2) Therefore, P/C301 2pr2 r24 3p/C28ffiffiffi 3p7C)67C)7 /C303p 8p/C286ffiffiffi 3p/C300:63938 . . . : (3) See also ACUTE ANGLE ,A CUTE TRIANGLE ,B ALL TRIANGLE PICKING ,OBTUSE ANGLE ,RIGHT TRIANGLE , TRIANGLE References Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653 /C1/659, 1986. Carroll, L. Pillow Problems & A Tangled Tale. New York: Dover, 1976. Eisenberg, B. and Sullivan, R. "Random Triangles n Dimensions." Amer. Math. Monthly 103, 308 /C1/318, 1996. Guy, R. K. "There are Three Times as Many Obtuse-Angled Triangles as There are Acute-Angled Ones." Math. Mag. 66, 175 /C1/178, 1993. Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob. 19, 712 /C1/715, 1982. Portnoy, S. "A Lewis Carroll Pillow Problem: Probability on at Obtuse Triangle." Statist. Sci. 9, 279 /C1/284, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 71, 1986. Wells, D. G. The Penguin Book of Interesting Puzzles. London: Penguin Books, pp. 67 and 248 /C1/249, 1992. Woolhouse, W. S. B. Solution to Problem 1350. Mathemati- cal Questions, with Their Solutions, from the Educational Times, 1. London: F. Hodgson and Son, 49 /C1/51, 1886. Ochoa Curve The ELLIPTIC CURVE 3Y2 /C302X3 /C27386X2 /C27256X /C2858195 ; given in WEIERSTRASS FORM as y2 /C30x3 /C28440067 x /C27106074110 : The complete set of solutions to this equation consists of (x; y)/ /C30(/C28761, 504), ( /C28745, 4520), ( /C28557, 13356), ( /C28446, 14616), (/C2817, 10656), (91, 8172), (227, 4228), (247, 3528), (271, 2592), (455, 200), (499, 3276), (523, 4356), (530, 4660), (599, 7576), (751, 14112), (1003, 25956), (1862, 75778), (3511, 204552), (5287, 381528), (23527, 3607272), (64507, 16382772), (100102, 31670478), and (1657891, 2134685628) (Stroeker and de Weger 1994). References Guy, R. K. "The Ochoa Curve." Crux Math. 16,65/C1/69, 1990. Ochoa Melida, J. "La ecuacion diofa´ntica b0y3 /C28b1y2 /C27b2y /C28b3 /C30z2 :/" Gaceta Math. 139 /C1/141, 1978. Stroeker, R. J. and de Weger, B. M. M. "On Elliptic Dio- phantine Equations that Defy Thue’s Method: The Case of the Ochoa Curve." Experiment. Math. 3, 209 /C1/220, 1994. Ockham Algebra References Blyth, T. S. and Varlet, C. Ockham Algebras. Oxford, England: Oxford University Press, 1994.Octacontagon An 80-sided POLYGON . Octadecagon An 18-sided POLYGON , sometimes also called an OCTAKAIDECAGON . See also POLYGON ,REGULAR POLYGON ,TRIGONOME- TRY VALUES PI/18 Octagon An octagon is an eight-sided POLYGON . The INRADIUS r,CIRCUMRADIUS R, and AREA Aof the regular octagon can be computed directly from the formulas for a general REGULAR POLYGON with side length s andn/C308 sides as r/C301 2scotp 8 ! /C301 21/C27ffiffiffi 2p7C)67C)7 s (1) R/C301 2scscp 8 ! /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C272ffiffiffi 2pq s (2) A/C301 4ns2cotp 8 ! /C3021/C27ffiffiffi 2p7C)67C)7 s2: (3) See also OCTAHEDRON ,POLYGON ,REGULAR POLYGON , TRIGONOMETRY VALUES PI/8 Octagonal Heptagonal Number A number which is simultaneously OCTAGONAL and HEPTAGONAL . Let Omdenote the mth OCTAGONAL NUMBER and Hn the nth HEPTAGONAL NUMBER , then a number which is both octagonal and hexagonal satisfies the equation Hn /C30Om ; or 1 2 n(5n /C283) /C30m(3m /C282): (1) COMPLETING THE SQUARE and rearranging gives 3(10n /C283)2 /C2840(3m /C281)2 /C30/C2813 : (2) Therefore, defining x /C13(10n /C283) (3) y /C132(3m /C281) (4) gives the second-order Diophantine equation 3x2 /C2810y2 /C30/C2813 (5) The first few solutions are (x;y)/ /C30(3, 2), (7, 4), (73, 40), (157, 86), .... These give the integer solutions (1, 1), (345, 315), (166145, 151669), ... (Sloane’s A048904 and A048905), corresponding to the octagonal hepta- gonal numbers 1, 297045, 69010153345, ... (Sloane’s A048906). See also HEPTAGONAL NUMBER ,OCTAGONAL NUMBER References Sloane, N. J. A. Sequences A048904, A048905, and A048906 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Octagonal Hexagonal Number A number which is simultaneously OCTAGONAL and HEXAGONAL . Let Ondenote the nth OCTAGONAL NUMBER and Hmthe mth HEXAGONAL NUMBER , then a number which is both octagonal and hexagonal satisfies the equation On /C30Hm ; or n(3n /C282) /C30m(2m /C281): (1) COMPLETING THE SQUARE and rearranging gives 8(3n /C281)2 /C283(4m /C281)2 /C305 : (2) Therefore, defining x /C132(3n /C281) (3) y /C134m /C281 (4) gives the second-order Diophantine equation 2x2 /C283y2 /C305 (5) The first few solutions are (x;y)/ /C30(2, 1), (4, 3), (16,13), (38, 31), (158, 129), (376, 307), .... These give the solutions (n;m) /C30(2=3; 1=2)/, (1, 1), (3, /7=2/), (/20 =3/, 8), (/80=3/,/65 =2/), (63, 77), ..., of which the integer solutions are (1, 1), (63, 77), (6141, 7521), (601723, 736957), ... (Sloane’s A046190 and A046191), corresponding to the octagonal hexagonal numbers 1, 11781, 113123361, 1086210502741, ... (Sloane’s A046192). See also HEXAGONAL NUMBER ,OCTAGONAL NUMBER , OCTAGONAL PENTAGONAL NUMBER ,O CTAGONAL SQUARE NUMBER ,OCTAGONAL TRIANGULAR NUMBER References Sloane, N. J. A. Sequences A046190, A046191, and A046192 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Octagonal Number A POLYGONAL NUMBER OF THE FORM n(3n /C282): The first few are 1, 8, 21, 40, 65, 96, 133, 176, ... (Sloane’s A000567). The GENERATING FUNCTION for the octago- nal numbers is x(5x /C27 1) (1 /C28 x)3 /C30x /C278x2 /C2721x3 /C2740x4 /C27... : See also OCTAGONAL HEPTAGONAL NUMBER ,OCTAGO- NAL HEXAGONAL NUMBER ,OCTAGONAL PENTAGONAL NUMBER ,OCTAGONAL SQUARE NUMBER ,OCTAGONAL TRIANGULAR NUMBER References Sloane, N. J. A. Sequences A000567/M4493 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Octagonal Pentagonal Number A number which is simultaneously OCTAGONAL and PENTAGONAL . Let Ondenote the nth OCTAGONAL NUMBER and Pmthe mth PENTAGONAL NUMBER , then a number which is both octagonal and pentago- nal satisfies the equation On/C30Pm;or n(3n/C282)/C301 2m(3m/C281): (1) COMPLETING THE SQUARE and rearranging gives (6m/C281)2/C288(3n/C281)2/C30/C287: (2) Therefore, defining x /C13(6m /C281) (3) y /C132(3n /C281) (4) gives the PELL EQUATION x2 /C282y2 /C30/C287 : (5) The first few solutions are (x;y)/ /C30(1, 2), (5, 4), (11, 8), (31, 22), (65, 46), .... These give the solutions (n;m) /C30(1=3; 2=3)/, (1, 1), (2, /5=3/), (/16=3/, 4), (11, 8), ..., of which the integer solutions are (1, 1), (11, 8), (1025, 725), (12507, 8844), ... (Sloane’s A046187 and A046188), corresponding to the octagonal pentagonal numbers 1, 176, 1575425, 234631320, 2098015778145, ... (Sloane’s A046189). See also OCTAGONAL NUMBER ,PENTAGONAL NUMBER References Sloane, N. J. A. Sequences A046187, A046188, and A046188 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Octagonal Prism A PRISM composed of octagonal faces. The regular right octagonal prism of unit edge length has SUR- FACE AREA and VOLUME S /C3043/C27ffiffiffi 2p7C)67C)7 V /C3021/C27ffiffiffi 2p7C)67C)7 : See also PRISM Octagonal Square Number A number which is simultaneously OCTAGONAL and SQUARE . Let Ondenote the nth OCTAGONAL NUMBER and Tmthe mth SQUARE NUMBER , then a number which is both octagonal and square satisfies the equation On /C30Sm ; or n(3n /C282) /C30m2 : (1) COMPLETING THE SQUARE and rearranging gives(3n /C281)2 /C283m2 /C301: (2) Therefore, defining x /C13(3n /C281) (3) y /C13m (4) gives the PELL EQUATION x2 /C283y2 /C301 (5) The first few solutions are (x;y)/ /C30(2, 1), (7, 4), (26, 15), (97, 56), (362, 209), (1351, 780), .... These give the solutions (n;m)/ /C30(1, 1), (/8=3/, 4), (9, 15), (/98 =3/, 56), (121, 209), ..., of which the integer solutions are (1, 1), (9, 15), (121, 209), (1681, 2911), ... (Sloane’s A046184 and A028230), corresponding to the octagonal square numbers 1, 225, 43681, 8473921, 1643897025, ... (Sloane’s A036428). See also OCTAGONAL NUMBER ,SQUARE NUMBER References Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Read- ing, MA: Addison-Wesley, p. 329, 1990. Konhauser, J. D. E.; Velleman, D.; and Wagon, S. Which Way Did the Bicycle Go? And Other Intriguing Mathema- tical Mysteries. Washington, DC: Math. Assoc. Amer., p. 104, 1996. Sloane, N. J. A. Sequences A028230, A036428, and A046184 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Octagonal Triangular Number A number which is simultaneously OCTAGONAL and TRIANGULAR . Let Ondenote the nth OCTAGONAL NUMBER andTmthemthTRIANGULAR NUMBER , then a number which is both octagonal and triangular satisfies the equation On/C30Tm;or n(3n/C282)/C301 2m(m/C271): (1) COMPLETING THE SQUARE and rearranging gives 8(3n/C281)2/C283(2m/C271)2/C305: (2) Therefore, defining x/C132(2n/C281) (3) y/C132m/C271 (4) gives the second-order Diophantine equation 2x2/C283y2/C305 (5) The first few solutions are ( x;y)//C30(2, 1), (4, 3), (16, 13), (38, 31), (158, 129), (376, 307), .... These give the solutions ( n;m)/C30(2=3;0)/, (1, 1), (3, 6), ( /20=3/, 15), (/80=3/, 64), (63, 153), ..., of which the integer solutions are (1, 1), (3, 6), (63, 153), (261, 638), (6141, 15041),(25543, 62566), (601723, 1473913), ... (Sloane’s A046181 and A046182), corresponding to the penta- gonal hexagonal numbers 1, 21, 11781, 203841, 113123361, ... (Sloane’s A046183). See also HEXAGONAL NUMBER ,OCTAGONAL HEXAGO- NAL NUMBER ,PENTAGONAL NUMBER References Sloane, N. J. A. Sequences A046181, A046182, and A046183 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Octagram The STAR POLYGON f8 =3g:/ Octahedral Graph AP LATONIC GRAPH on eight nodes. There are 257 topologically distinct octahedral graphs, as first en- umerated by Kirkman (1862) and Hermes (1899ab, 1900, 1901; Federico 1969; Duijvestijn and Federico 1981). Confusingly, the term "octahedral graph" is also used to refer to the 6-vertex POLYHEDRAL GRAPH having the connectivity of the OCTAHEDRON . It is isomorphic to the CIRCULANT GRAPH Ci1;2(6): Several circular em- beddings of this graph are illustrated above. The octahedral graph has 6 nodes, 12 edges, VERTEX CONNECTIVITY 4, EDGE CONNECTIVITY 4, GRAPH DIA- METER 2, GRAPH RADIUS 2, and GIRTH 3. It has CHROMATIC POLYNOMIAL pG(z) /C30z6 /C2812z5 /C2758z4 /C28137z3 /C27154z2 /C2864z ; and CHROMATIC NUMBER 3. See also CIRCULANT GRAPH ,CUBICAL GRAPH ,DODE- CAHEDRAL GRAPH ,ICOSAHEDRAL GRAPH ,O CTAHE- DRON ,P LATONIC GRAPH ,P OLYHEDRAL GRAPH , TETRAHEDRAL GRAPHReferences Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976. Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (/3/-Connected Planar) Graphs." Math. Comput. 37, 523 /C1/532, 1981. Federico, P. J. "Enumeration of Polyhedra: The Number of 9-Hedra." J. Combin. Th. 7, 155 /C1/161, 1969. Gru¨nbaum, B. Convex Polytopes. New York: Wiley, pp. 288 and 424, 1967. Hermes, O. "Die Formen der Vielflache. I." J. reine angew. Math. 120,27/C1/59, 1899a. Hermes, O. "Die Formen der Vielflache. II." J. reine angew. Math. 120, 305 /C1/353, 1899b. Hermes, O. "Die Formen der Vielflache. III." J. reine angew. Math. 122, 124 /C1/154, 1900. Hermes, O. "Die Formen der Vielflache. IV." J. reine angew. Math. 123, 312 /C1/342, 1901. Kirkman, T. P. "Application of the Theory of the Polyhedra to the Enumeration and Registration of Results." Proc. Roy. Soc. London 12, 341 /C1/380, 1862 /C1/1863. Octahedral Group The POINT GROUP of symmetries of the OCTAHEDRON having order 24 and denoted Oh : It is also the symmetry group of the CUBE , CUBOCTAHEDRON , and TRUNCATED OCTAHEDRON . It has symmetry opera- tions E,8C3 ; 6C4 ; 6C2 ; 3C2 /C30C2 4 ; i,6S4 ; 8S6 ; 3sh ; and 6 s4(Cotton 1990). See also CUBE,C UBOCTAHEDRON ,ICOSAHEDRAL GROUP ,OCTAHEDRON ,POINT GROUPS ,POLYHEDRAL GROUP ,TETRAHEDRAL GROUP ,TRUNCATED OCTAHE- DRON References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, pp. 47 /C1/49, 1990. Coxeter, H. S. M. "The Polyhedral Groups." §3.5 in Regular Polytopes, 3rd ed. New York: Dover, pp. 46 /C1/47, 1973. Lomont, J. S. "Octahedral Group." §3.10.D in Applications of Finite Groups. New York: Dover, p. 81, 1987. Octahedral Number AFIGURATE NUMBER which is the sum of two consecutive PYRAMIDAL NUMBERS , On/C30Pn/C281/C27Pn/C301 3n(2n2/C271): (1) The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, ... (Sloane’s A005900). The GENERAT- ING FUNCTION for the octahedral numbers is x(x/C271)2 (x/C281)4/C30x/C276x2/C2719x3/C2744x4/C27...: (2) A related set of numbers is the number of cubes in the HAUY CONSTRUCTION of the OCTAHEDRON . Each CROSS SECTION has area Sn/C30n/C272X i/C301;3;...;n/C282i/C301 2(n2/C271); (3) where nis an ODD NUMBER , and adding all CROSS SECTIONS gives HOk/C30Sk/C272X i/C301;3;...;k/C282Si/C3016k/C30(k2/C275); (4) forkanODD NUMBER . Re-indexing so that k/C302n/C281 gives HOn/C3013(2n/C281)(2n2/C282n/C273); (5) the first few values of which are 1, 7, 25, 63, 129, ... (Sloane’s A001845). These numbers have the GENER- ATING FUNCTION f(x)/C30(1/C27x)3 (1/C28x)4 /C301/C277x/C2725x2/C2763x3/C27129x4/C27...: (6) See also HAUY CONSTRUCTION ,OCTAHEDRON ,TRUN- CATED OCTAHEDRAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 50, 1996. Sloane, N. J. A. Sequences A001845/M4384 and A005900/ M4128 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Octahedron The P LATONIC SOLID P3with six VERTICES ,1 2 EDGES , and eight equivalent EQUILATERAL TRIANGULAR faces, 8f3g:It is also UNIFORM POLYHEDRON U5and Wen-ninger model W2:It is given by the S CHLA ¨FLI SYMBOL f3;4gand W YTHOFF SYMBOL 4½23:/ The octahedron of unit side length is the ANTIPRISM of n/C303 sides with height h/C30ffiffiffi 6p =3:The DUAL POLYHE- DRON of the octahedron is the CUBE . Like the CUBE ,i t has the OhOCTAHEDRAL GROUP of symmetries. The connectivity of the vertices is given by the OCTAHE- DRAL GRAPH . The octahedron has a single STELLATION : the STELLA OCTANGULA . The solid bounded by the two TETRAHE- DRA of the STELLA OCTANGULA (left figure) is an octahedron (right figure; Ball and Coxeter 1987). The following table gives polyhedra which can be constructed by CUMULATION of an octahedron by pyramids of given heights h. h /(r/C27h)=h/ Result /ffiffiffi 3p /C282 3ffiffiffi 6p // 5/C283ffiffiffi 2p / SMALL TRIAKIS OCTAHEDRON /1 3ffiffiffi 6p / 3 STELLA OCTANGULA In one orientation (left figure), the VERTICES are given by (91;0;0);(0;91;0);(0;0;91):In another orienta- tion (right figure), the vertices are ( 91;91;0) and 0;0;9ffiffiffi 2p7C07C) :/ The face planes are 9x9y9z/C301;so a solid octahe- dron is given by the equation ½x½/C27½y½/C27½z½51: (1) If the edges of an octahedron are divided in the GOLDEN RATIO such that the points of division for any face form an EQUILATERAL TRIANGLE , then the twelve points of division form an ICOSAHEDRON (Wells 1991). In fact, there are two ways in which the edges can be internally divided in the GOLDEN RATIO and two ways in which they can be externally divided, resulting infour possible icosahedra. Keeping the same connec-tivity, but reversing the long and short ends of the division gives J ESSEN’S ORTHOGONAL ICOSAHEDRON . A plane PERPENDICULAR to aC3axis of an octahedron cuts the solid in a regular HEXAGONAL CROSS SECTION (Holden 1991, pp. 22 /C1/23). Since there are four such axes, there are four possible HEXAGONAL CROSS SECTIONS . The centers of the faces of an octahedron form a CUBE , and the centers of the faces of a CUBE form an octahedron (Steinhaus 1983, pp. 194 /C1/195). Faceted forms of the octahedron include the CUBOCTATRUN- CATED CUBOCTAHEDRON and TETRAHEMIHEXAHEDRON . Let an octahedron be length aon a side. The height of the top VERTEX from the square plane is also the CIRCUMRADIUS R/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28d2p ; (2) where d/C301 2ffiffiffi 2p a (3) is the diagonal length, so R /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C281 2 a2q /C3012ffiffiffi 2p a :0 :70710 a: (4) Now compute the INRADIUS . l /C301 2ffiffiffi 3p a (5) b /C301 2 a (6) s /C301 2 a tan 30 /C14/C30a 2ffiffiffi 3p; (7) so s l /C301 2ffiffiffi3p 2ffiffiffi3p/C301 3 : (8) Use similar TRIANGLES to obtain b ?/C30s lb /C3016 a (9) z?/C30s lz /C30a 3ffiffiffi 2p (10) x /C30b /C28b?/C301 2 a /C2816 a /C3013 a; (11) so the INRADIUS is r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28z ?2p /C30affiffiffiffiffiffiffiffiffiffiffi 1 9 /C271 18q /C3016ffiffiffi 6p a :0:40824 a; (12) and twice the INRADIUS gives the height of the octahedron viewed as a 3-sided ANTIPRISM . The MIDRADIUS of the octahedron is r /C301 2 a /C300:5a : (13) The AREA of one face is the AREA of an EQUILATERAL TRIANGLE A /C3014ffiffiffi 3p a2 : (14) The volume is two times the volume of a square-base pyramid, V /C3021 3 a2R7C)67C)7 /C302137C)67C)7 a27C07C)12ffiffiffi 2p a7C)67C)7 /C301 3ffiffiffi 2p a3 : (15) The DIHEDRAL ANGLE is a /C30cos/C281/C281 37C)67C)7 :109:47/C14: (16) The octahedron can be built using a HAUY CONSTRUC- TION . The Hauy octahedral numbers HOn /C301 3(2n /C281)(2n2 /C282n /C273) (17) give another method for calculating the VOLUME of the octahedron, V /C30 lim n0/C12HOna nffiffiffi 2p !3 /C301 3ffiffiffi 2p a3 ; (18) in agreement with the result derived above. See also ANTIPRISM ,DU¨ RER’S SOLID ,HAUY CONSTRUC- TION ,ICOSAHEDRON ,JUMPING OCTAHEDRON ,OCTAHE- DRAL GRAPH ,O CTAHEDRAL GROUP ,O CTAHEDRON 3- COMPOUND ,O CTAHEDRON 5-COMPOUND ,P LATONIC SOLID ,S TELLA OCTANGULA ,T RUNCATED OCTAHE- DRON References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 228, 1987. Cundy, H. and Rollett, A. "Octahedron. 34." §3.5.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 64, 1989. Davie, T. "The Octahedron." http://www.dcs.st-and.ac.uk/ ~ad/mathrecs/polyhedra/octahedron.html. Harris, J. W. and Stocker, H. "Octahedron." §4.4.4 in Hand- book of Mathematics and Computational Science. New York: Springer-Verlag, p. 100, 1998. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 193 /C1/195, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 163, 1991. Wenninger, M. J. "The Octahedron." Model 2 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 15, 1989. Octahedron 3-Compound APOLYHEDRON COMPOUND consisting of three octahe- dra. See also OCTAHEDRON ,OCTAHEDRON 5-COMPOUND Octahedron 5-Compound A POLYHEDRON COMPOUND composed of five OCTAHE- DRA occupying the VERTICES of an ICOSAHEDRON . The 30 VERTICES of the compound form an ICOSIDODECA- HEDRON (Ball and Coxeter 1987), and the solid is one of the ICOSAHEDRON STELLATIONS (Wenninger 1983). The octahedron 5-compound is the dual of the CUBE 5- COMPOUND . Constructing the octahedra as the duals of the CUBE 5- COMPOUND where the cubes have unit edge lengths give a solid with edge lengths s1 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 53 /C28ffiffiffi 5p7C)67C)7r (1) s2 /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 57 /C283ffiffiffi 5p7C)67C)7r (2) s3 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C283ffiffiffi 5pq (3) s4 /C303 /C28ffiffiffi 5p : (4) The CIRCUMRADIUS is R /C301 ; (5) and the SURFACE AREA and VOLUME are S /C3020ffiffiffi 3p (6) V /C3020 3 : (7) The CONVEX HULL of the octahedron 5-compound is the ICOSIDODECAHEDRON . See also CUBE 5-COMPOUND ,C UBE 5-COMPOUND– OCTAHEDRON 5-COMPOUND ,ICOSAHEDRON STELLA- TIONS ,ICOSIDODECAHEDRON ,OCTAHEDRON ,OCTAHE- DRON 3-COMPOUND ,O CTAHEDRON 6-COMPOUND , POLYHEDRON COMPOUND ,STELLA OCTANGULAReferences Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 135 and 137, 1987. Cundy, H. and Rollett, A. "Five Octahedra About in Icosahedron." §3.10.7 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 137 /C1/138, 1989. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 55, 1983. Wenninger, M. J. "Compound of Five Octahedra." §23 in Polyhedron Models. New York: Cambridge University Press, p. 43, 1989. Octahedron 6-Compound See also OCTAHEDRON ,O CTAHEDRON 3-COMPOUND , OCTAHEDRON 5-COMPOUND Octahedron Stellation STELLA OCTANGULA Octahemioctacron The DUAL POLYHEDRON of the OCTAHEMIOCTAHEDRON U3and Wenninger dual W68 : When rendered, the octahemioctacron and HEXAHEMIOCTACRON appear the same. See also DUAL POLYHEDRON ,H EXAHEMIOCTACRON , OCTAHEMIOCTAHEDRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 104, 1983. Octahemioctahedron The UNIFORM POLYHEDRON U3 ; also called the OCTA- TETRAHEDRON , whose DUAL POLYHEDRON is the OCTA- HEMIOCTACRON . It has WYTHOFF SYMBOL3 23½3: Its faces are 8 f3g/C274 f6g: It is a FACETED CUBOCTAHE- DRON . For unit edge length, its CIRCUMRADIUS is R /C301 : The CONVEX HULL of the octahemioctahedron is the CUBOCTAHEDRON . References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 103, 1989. Octakaidecagon OCTADECAGON Octal The base 8 notational system for representing REAL NUMBERS . The digits used are 0, 1, 2, 3, 4, 5, 6, and 7, so that 810 (8 in base 10) is REPRESENTED AS 108 (10 /C30 1 /C215 81 /C270 /C215 80) in base 8. The following table gives the octal equivalents of the first few decimal numbers. 1 1 11 13 21 25 2 2 12 14 22 26 3 3 13 15 23 27 4 4 14 16 24 305 5 15 17 25 31 6 6 16 20 26 32 7 7 17 21 27 33 81018222834 91119232935 10 12 20 24 30 36 See also BASE (NUMBER ), BINARY ,DECIMAL ,HEXADE- CIMAL ,QUATERNARY ,TERNARY References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 9 /C1/10, 1991. Weisstein, E. W. "Bases." MATHEMATICA NOTEBOOK BASES.M . Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 72 /C1/ 73, 1986. Octant One of the eight regions of SPACE defined by the eight possible combinations of SIGNS (9;9;9) forx,y, and z. See also QUADRANT Octatetracontagon A 48-faced POLYGON . See also DISDYAKIS DODECAHEDRON ,GREAT RHOMBI- CUBOCTAHEDRON (ARCHIMEDEAN ) Octatetrahedron OCTAHEMIOCTAHEDRON Octave A multiple of 2. The word should really be something like "bicade" (by analogy with DECADE ) but the "oct" embedded in the stem of the word derives historically to the fact that eight notes correspond to a factor of two in frequency. See also DECADE Octiamond An 8-POLYIAMOND . See also OCTIAMOND TILING ,POLYIAMOND Octiamond Tiling See also HEPTIAMOND TILING ,H EXIAMOND TILING , OCTIAMOND ,PENTIAMOND TILING References Vichera, M. "Polyiamonds." http://alpha.ujep.cz/~vicher/puz- zle/polyform/iamond/iamonds.htm. Octic Reciprocity Theorem The RECIPROCITY THEOREM for x8 /C13q (mod p) : See also RECIPROCITY THEOREM References Aigner, A. "Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und /C282." Deutsche Math. 4,44/C1/52, 1939.Hasse, H. "Der 2n/-te Potenzcharakter von 2 im Koerper der 2n/-ten Einheitswurzeln." Rend. Circ. Matem. Palermo 7, 185 /C1/243, 1958. Whiteman, A. L. " The Sixteenth Power Residue Character of 2." Canad. J. Math. 6, 364 /C1/373, 1954. Octic Surface An ALGEBRAIC SURFACE of degree eight. The max- imum number of ORDINARY DOUBLE POINTS known to exist on an octic surface is 168 (the ENDRAß OCTICS ), although the rigorous upper bound is 174. See also ALGEBRAIC SURFACE ,ENDRAß OCTIC,ORDIN- ARY DOUBLE POINT Octillion In the American system, 1027. See also LARGE NUMBER Octodecillion In the American system, 1057. See also LARGE NUMBER Octomino An 8-POLYOMINO . There are 369 FREE , 2725 FIXED , and 704 one-sided octominoes. See also POLYOMINO Octonion CAYLEY NUMBER Octothorpe The number sign # sometimes used in mathematics to indicate the number of a quantity satisfying somecondition, e.g., # fn:n>1):The symbol is also used to denote a PRIMORIAL . References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 282, 1997. Odd Divisor Function The sum of powers of ODD DIVISORS of a number. It is the analog of the DIVISOR FUNCTION for odd divisors only and is written soðÞ k(n):For the case k/C301, soðÞ 1(n)/C30s1(n)/C282s1(n=2); where sk(n=2) is defined to be 0 if nisODD. The following table gives the first few soðÞ k(n):/ kSloane /soðÞ k(n)/ 0 A001227 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, ... 1 A000593 1, 1, 4, 1, 6, 4, 8, 1, 13, 6, ... 2 A050999 1, 1, 10, 1, 26, 10, 50, 1, 91, 26, ... 3 A051000 1, 1, 28, 1, 126, 28, 344, 1, 757, 126, ... 4 A051001 1, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, ... 5 A051002 1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, ... This function arises in Ramanujan’s EISENSTEIN SERIES L(q) and in a RECURRENCE RELATION for the PARTITION FUNCTION P. See also DIVISOR FUNCTION ,EVEN DIVISOR FUNCTION References Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 306, 1952. Hirzebruch, F. Manifolds and Modular Forms, 2nd ed. Braunschweig, Germany: Vieweg, p. 133, 1994. Riordan, J. Combinatorial Identities. New York: Wiley, p. 187, 1979. Sloane, N. J. A. Sequences A000593/M3197, A001227, A050999, A051000, A051001, and A051002 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Verhoeff, T. "Rectangular and Trapezoidal Arrangements." J. Integer Sequences 2, #99.1.6, 1999. Odd Function An odd function is a function for which f(x) /C30/C28f(/C28x): An EVEN FUNCTION times an odd function is odd. Odd Graph An odd graph On is a graph having vertices given by the n /C281/-subsets of f1;...;2n /C281g such that two vertices are connected by an edge IFF the associated subsets are disjoint (Biggs 1974). The number of nodes in On is therefore 2n/C281 n/C2817C07C) ; wheren k7C07C) is a BINOMIAL COEFFICIENT . For n /C301, 2, ..., the first few values are 1, 3, 10, 35, 126, ... (Sloane’s A001700). /O2 is isomorphic to the COMPLETE GRAPH K3 ; and O3 is the PETERSEN GRAPH (Skiena 1990, p. 162). See also COMPLETE GRAPH ,O DD NODE,PETERSEN GRAPH References Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993.Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Sloane, N. J. A. Sequences A001700/M2848 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Odd Node A NODE in a GRAPH is said to be an odd node if its VERTEX DEGREE is ODD. See also EVEN NODE,G RAPH ,N ODE (GRAPH ), ODD GRAPH ,VERTEX DEGREE Odd Number An INTEGER OF THE FORM N /C302n /C271; where n is an INTEGER . The odd numbers are therefore ..., /C283, /C281, 1, 3, 5, 7, ... (Sloane’s A005408), which are also the GNOMONIC NUMBERS . The GENERATING FUNCTION for the odd numbers is x(1 /C27 x) (x /C28 1)2 /C30x /C273x2 /C275x3 /C277x4 /C27...: Since the odd numbers leave a remainder of 1 when divided by two, N /C131 (mod 2) for odd N. Integers which are not odd are called EVEN . See also EVEN NUMBER ,GNOMONIC NUMBER ,NICO- MACHUS’S THEOREM ,O DD NUMBER THEOREM ,O DD PRIME References Commission on Mathematics of the College Entrance Ex- amination Board. Informal Deduction in Algebra: Proper- ties of Odd and Even Numbers. Princeton, NJ, 1959. Sloane, N. J. A. Sequences A005408/M2400 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Odd Number Theorem The sum of the first n ODD NUMBERS is a SQUARE NUMBER : Xn k /C301(2k /C281) /C302Xn k/C301k/C28Xn k/C3011/C302n(n/C271) 2"# /C28n /C30n(n/C271)/C28n/C30n2: See also NICOMACHUS’S THEOREM ,ODD NUMBER Odd Order Theorem FEIT-THOMPSON THEOREM Odd Part The odd part Od(n) of a positive integer n is defined by Od(n) /C30n 2b(n) ; where b(n) is the exponent of the exact power of 2 dividing n. Od(n) is therefore the product of odd factors of n. The values for n /C301, 2, ..., are 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... (Sloane’s A000265). The odd part function can be implemented in Mathematica as OddPart[n_Integer] : /C30 n/ 2^IntegerExponent[n,2] See also EVEN PART,GREATEST DIVIDING EXPONENT References "Problem H-81." Fib. Quart. 6, 52, 1968. Sloane, N. J. A. Sequences A000265/M2222 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Odd Perfect Number In Book IX of The Elements, Euclid gave a method for constructing PERFECT NUMBERS (Dickson 1957, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid’s form, and stated that he saw no reason why an odd perfect number could not exist (Dickson 1957, p. 12). Descartes was therefore among the first to consider the existence off odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid’s construction comprised all possible perfect numbers (Dickson 1957, pp. 6 /C1/12). In 1657, Frenicle repeated Descartes’ belief that every even perfect number is of Euclid’s form and that there was noreason odd perfect could not exist. Like Frenicle, Euler also considered odd perfect numbers. To this day, it is not known if any odd perfect numbers exist, although numbers up to 10300 have been checked without success, making the existence of odd perfect numbers appear unlikely (Brent et al. 1991; Guy 1994, p. 44). The following table sum- marizes the development of ever-higher bounds for the smallest possible odd perfect number. author bound Kanold (1957) 1020 Tuckerman (1973) 1036 Hagis (1973) 1050 Brent and Cohen (1989) 10160 Brent et al. (1991) 10300 Euler showed that an odd perfect number, if it exists, must be OF THE FORM m /C30p4 l /C271Q2 ; (1) where p is a prime of the form 4n /C271; a result similar to that derived by Frenicle in 1657 (Dickson 1957, pp. 14 and 19). In 1887, Sylvester conjectured and in 1925, Gradshtein proved that any odd perfect number must have at least six different prime aliquot factors (Ball and Coxeter 1987). If it is not divisible by 3, an odd perfect number must then have at least 11 different prime factors (Hagis 1983). Catalan (1888) proved that if an ODD perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime aliquot factors. Stuyvaert (1896) proved that an odd perfect number must be a sum of squares. See also ODD NUMBER ,PERFECT NUMBER References Brent, R. P. and Cohen, G. L. "A New Bound for Odd Perfect Numbers." Math. Comput. 53, 431/C1/437 and S7-S24, 1989. Brent, R. P.; Cohen, G. L.; te Riele, H. J. J. "Improved Techniques for Lower Bounds for Odd Perfect Numbers." Math. Comput. 57, 857/C1/868, 1991. Buxton, M. and Elmore, S. "An Extension of Lower Bounds for Odd Perfect Numbers." Not. Amer. Math. Soc. 22,A - 55, 1976. Buxton, M. and Stubblefield, B. "On Odd Perfect Numbers." Not. Amer. Math. Soc. 22, A-543, 1975. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3 /C1/33, 1952. Guy, R. K. "Perfect Numbers." §B1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 44 /C1/45, 1994. Hagis, P. Jr. "A Lower Bound for the Set of Odd Perfect Numbers." Math. Comput. 27, 951/C1/953, 1973. Hagis, P. Jr. "An Outline of a Proof that Every Odd Perfect Number has at Least Eight Prime Factors." Math. Comput. 34, 1027 /C1/1032, 1980. Hagis, P. Jr.; and Cohen, G. L. "Every Odd Perfect Number Has a Prime Factor Which Exceeds 106." Math. Comput. 67, 1323 /C1/1330, 1998. Heath-Brown, D. R. "Odd Perfect Numbers." Math. Proc. Cambridge Philos. Soc. 115, 191 /C1/196, 1994. Iannucci, D. E. "The Second Largest Prime Divisor of an Odd Perfect Number Exceeds Ten Thousand." Math. Comput. 68, 1749 /C1/1760, 1999. Iannucci, D. E. "The Third Largest Prime Divisor of an Odd Perfect Number Exceeds One Hundred." Math. Comput. 69, 867 /C1/879, 2000. Kanold, H.-J. "U¨ ber mehrfach vollkommene Zahlen. II." J. reine angew. Math. 197,82/C1/96, 1957. Subbarao, M. V. "Odd Perfect Numbers: Some New Issues." Period. Math. Hungar. 38, 103 /C1/109, 1999. Tuckerman, B. "Odd Perfect Numbers: A Search Procedure, and a New Lower Bound of 1036." Not. Amer. Math. Soc. 15, 226, 1968. Tuckerman, B. "A Search Procedure and Lower Bound for Odd Perfect Numbers." Math. Comp. 27, 943 /C1/949, 1973. Odd Prime Any PRIME NUMBER other than 2 (which is the unique EVEN PRIME ). See also EVEN PRIME ,PRIME NUMBER References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 44, 1986. Odd Sequence A SEQUENCE of n 0s and 1s is called an odd sequence if each of the n SUMS an/C28k i/C301aiai/C27k for k /C300, 1, ..., n /C281 is odd. References Guy, R. K. "Odd Sequences." §E38 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 238 /C1/239, 1994. Odd Triple TWO-GRAPH Odds Betting odds are written in the form r : s ( and correspond to the probability of winning P /C30s =(r /C27 s) : Therefore, given a probability P, the odds of winning are (1=P) /C281:1 :/ See also FRACTION ,RATIO,RATIONAL NUMBER References Kraitchik, M. "The Horses." §6.17 in Mathematical Recrea- tions. New York: W. W. Norton, pp. 134 /C1/135, 1942. ODE ORDINARY DIFFERENTIAL EQUATIONOesterle ´-Masser Conjecture ABC CONJECTURE Of Order ASYMPTOTIC NOTATION Of Shape OF THE FORM Of the Form An expression that is of a given type. For example, all primes p /C213 are "of the form" 6n 91 : The term "of shape" is sometimes also used. See also REPRESENTED AS References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 13, 1986. Offset Curves PARALLEL CURVES Offset Rings SURFACE OF REVOLUTION Ogive Any continuous cumulative frequency curve, such as the one illustrated above in the right figure. See also FREQUENCY POLYGON ,HISTOGRAM References Kenney, J. F. and Keeping, E. S. "Ogive Curves." §2.7 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 29 /C1/31, 1962. Oldknow Points The PERSPECTIVE CENTERS of a triangle and the TANGENTIAL TRIANGLES of its inner and outer SODDY CIRCLES , given by Ol /C30I /C272Ge Ol ?/C30I /C282Ge; where I is the INCENTER and Ge is the GERGONNE POINT . See also GERGONNE POINT ,INCENTER ,PERSPECTIVE CENTER ,SODDY CIRCLES ,TANGENTIAL TRIANGLE References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/329, 1996. Oliveira’s Minimal Surface See also MINIMAL SURFACE Oloid References Capocasa, C. "Oloid." http://www.blackpoint.net/capocssa/ oloid.html. Schatz, P. "Das Oloid als Wa¨lzko¨rper." §14 in Rythmus- forschung und Technik. Stuttgart: Verlag Freies Geiste- sleben, p. 122, 1975. Omega Constant W(1) /C130:5671432904. . . ; (1) where W(x)isL AMBERT’S W-FUNCTION . It is available in Mathematica using the function ProductLog [1]. W(1) can be considered a sort of "GOLDEN RATIO " for exponentials since exp[/C28W(1)] /C30W(1) ; (2) giving ln1 W(1)"# /C30W(1) : (3) See also GOLDEN RATIO,LAMBERT’S W-FUNCTION References Plouffe, S. "The Omega Constant or W(1):/" http://www.laci- m.uqam.ca/piDATA/omega.txt. Omega Function LAMBERT’S W-FUNCTION Omino POLYOMINO Omnific Integer The appropriate notion of INTEGER for SURREAL NUMBERS . See also SURREAL NUMBER O’Nan Group The SPORADIC GROUP O’N. References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/ON.html.Onduloid UNDULOID One 1 One-Form A linear real-valued FUNCTION v1 of VECTORS v such that v1(v) /C2R : VECTORS (i.e., CONTRAVARIANT VEC- TORS or "KETS " cji) and one-forms (i.e., COVARIANT VECTORS or "BRAS " fhj) are DUAL to each other. Therefore v1(v) /C13v v17C07C) /C13 v1 ; v7C)07C)) /C30 fcji: h The operation of applying the one-form to a VECTOR v1(v) is called CONTRACTION . See also ANGLE BRACKET ,B RA,C ONTRAVARIANT VECTOR ,COVARIANT VECTOR ,DIFFERENTIAL K-FORM, KET,MEROMORPHIC ONE-FORM,TWO-FORM,VECTOR , ZERO-FORM One-Mouth Theorem Except for convex polygons, every SIMPLE POLYGON has at least one MOUTH . See also MOUTH ,P RINCIPAL VERTEX ,T WO-EARS THEOREM References Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122,3 1/C1/35, 1991. One-Ninth Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let lm;nbe C HEBYSHEV CONSTANTS . Scho ¨nhage (1973) proved that lim n0/C12l0;n7C07C)1=n/C301 3: (1) It was conjectured that L/C13lim n0/C12ln;n7C07C)1=n/C3019: (2) Carpenter et al. (1984) obtained L/C300:1076539192 . . . (3) numerically. Gonchar and Rakhmanov (1980) showed that the limit exists and disproved the /1=9/conjecture, showing that Lis given by L/C30exp/C28pKffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28c2p7C)67C)7 K(c)2 435; (4) where Kis the complete ELLIPTIC INTEGRAL OF THE FIRST KIND , and c/C300:9089085575485414 . . . is the PARAMETER which solves K(k) /C302E(k) ; (5) and E is the complete ELLIPTIC INTEGRAL OF THE SECOND KIND . This gives the value for L computed by Carpenter et al. (1984) L is also given by the unique POSITIVE ROOT of f(z) /C301 8 ; (6) where f(z) /C13X/C12 j/C301ajzj (7) and aj /C30X d j j(/C281)dd7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P (8) (Gonchar and Rakhmanov 1980). a jmay also be computed by writing j as j /C302mpm1 1pm2 2/C1/C1/C1pmk k; (9) where m ]0 and mi ]1; then aj /C30 2m/C271 /C2837C)P7C)P7C)P7C)P /C2pm1 /C271 1 /C28 1 p1 /C28 1pm2 /C271 2 /C28 1 p2 /C28 1/C1/C1/C1pmk /C271 k /C28 1 pk /C28 1(10) (Gonchar 1990). Yet another equation for L is due to Magnus (1986). L is the unique solution with x /C23 (0; 1) of X/C12 k/C300(2k /C271)2(/C28x)k(k /C271)=2 /C300 ; (11) an equation which had been studied and whose root had been computed by Halphen (1886). It has there- fore been suggested (Varga 1990) that the constant be called the HALPHEN CONSTANT .1=L is sometimes called VARGA’S CONSTANT . See also CHEBYSHEV CONSTANTS ,H ALPHEN CON- STANT ,VARGA’S CONSTANT References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/onenin/onenin.html. Carpenter, A. J.; Ruttan, A.; and Varga, R. S. "Extended Numerical Computations on the ‘/1=9/’ Conjecture in Ra- tional Approximation Theory." In Rational Approximation and Interpolation (Tampa, FL, 1983) (Ed. P. R. Graves- Morris, E. B. Saff, and R. S. Varga). New York: Springer- Verlag, pp. 383 /C1/411, 1984. Cody, W. J.; Meinardus, G.; and Varga, R. S. "Chebyshev Rational Approximations to e/C28x in 0;/C27/C12 ½Þ and Applica- tions to Heat-Conduction Problems." J. Approx. Th. 2,50/C1/ 65, 1969. Dunham, C. B. and Taylor, G. D. "Continuity of Best Reciprocal Polynomial Approximation on 0 ;/C12½Þ :/" J. Ap- prox. Th. 30,71/C1/79, 1980. Gonchar, A. A. "Rational Approximations of Analytic Func- tions." Amer. Math. Soc. Transl. Ser. 2 147,25/C1/34, 1990.Gonchar, A. A. and Rakhmanov, E. A. "Equilibrium Distri- butions and Degree of Rational Approximation of Analytic Functions." Math. USSR Sbornik 62, 305 /C1/348, 1980. Magnus, A. P. "On Freud’s Equations for Exponential Weights, Papers Dedicated to the Memory of Ge´za Freud." J. Approx. Th. 46,65/C1/99, 1986. Rahman, Q. I. and Schmeisser, G. "Rational Approximation to the Exponential Function." In Pade´ and Rational Approximation, (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) (Ed. E. B. Saff and R. S. Varga). New York: Academic Press, pp. 189 /C1/194, 1977. Scho¨nhage, A. "Zur rationalen Approximierbarkeit von e /C28x u¨ber 0 ;/C12½Þ :/" J. Approx. Th. 7, 395 /C1/398, 1973. Varga, R. S. Scientific Computations on Mathematical Pro- blems and Conjectures. Philadelphia, PA: SIAM, 1990. One-Sheeted Hyperboloid A HYPERBOLOID consisting of a single sheet. See also HYPERBOLOID One-to-One Let f be a FUNCTION defined on a SET A and taking values in a set B. Then f is said to be one-to-one (a.k.a. an injection or embedding) if, whenever f(x) /C30 f(y) ; it must be the case that x /C30y. In other words, f is one-to-one if it MAPS distinct objects to distinct objects. If the function is a linear OPERATOR which assigns a unique MAP to each value in a VECTOR SPACE ,itis called one-to-one. Specifically, given a VECTOR SPACE V with X ; Y /C23V; then a TRANSFORMATION T defined on V is one-to-one if T(X) "T(Y) for all X "Y :/ A function which is both one-to-one and ONTO is said to be a BIJECTION . See also BIJECTION ,D OMAIN ,M ANY-TO- ONE,O NTO, RANGE (IMAGE ) One-Way Function Informally, a function f is a one-way function if 1. The description of f is publicly known and does not require any secret information for its opera- tion. 2. Given x, it is easy to compute f(x) :/ 3. Given y, in the range of f, it is hard to find an x such that f(x) /C30y: More precisely, any efficient algorithm (solving a P-PROBLEM succeeds in in- verting f with negligible probability. The existence of one-way functions is not proven. If true, it would imply P "NP : Therefore, it would answer the COMPLEXITY THEORY NP-PROBLEM ques- tion of whether all apparently NP-problems are actually P-problems. Yet a number of conjectured one-way functions are routinely used in commerce and industry. For example, it is conjectured, but not proved, that the following are one-way functions: 1. Factoring problem: f(p ; q) /C30pq ; for randomly chosen primes p, q. 2. Discrete logarithm problem: f(p ; g; x) /C30 p ; g; gx (mod p) hi ; for g a generator of Zp /C31; for some prime p. 3. Discrete root extraction problem: f(p ; q; e; y) /C30 pq ; e ; ye (mod pq) hi ; for y in Zpq /C31; e in Zpqand relatively prime to (p /C281)(q /C281); and p, q primes. This is the function commonly known as RSA ENCRYPTION . 4. SUBSET SUM PROBLEM : f(a; b) /C30an i /C301 aibi ; b7C)07C)) ; for ai /C23f0; 1g; and n-bit integers bi :/ 5. QUADRATIC RESIDUE problem. See also NP-PROBLEM ,ONE-WAY HASH FUNCTION ,P- PROBLEM ,Q UADRATIC RESIDUE , RSA ENCRYPTION , SUBSET SUM PROBLEM References Luby, M. Pseudorandomness and Cryptographic Applica- tions. Princeton, NJ: Princeton University Press, 1996. Ziv, J. "In Search of a One-Way Function" §4.1 in Open Problems in Communication and Computation (Ed. T. M. Cover and B. Gopinath). New York: Springer-Ver- lag, pp. 104 /C1/105, 1987. One-Way Hash Function A function H that maps an arbitrary length message M to a fixed length message digest MD is a one-way hash function if 1. It is a ONE-WAY FUNCTION . 2. Given M and H(M) ; it is hard to find a message M ?"M such that H(M ?) "H(M) :/ See also HASH FUNCTION ,ONE-WAY FUNCTION ,TRAP-DOOR ONE-WAY FUNCTION References Bakhtiari, S.; Safavi-Naini, R.; and Pieprzyk, J. Crypto- graphic Hash Functions: A Survey. Technical Report 95 /C1/ 09, Department of Computer Science, University of Wol- longong, July 1995. ftp://ftp.cs.uow.edu.au/pub/papers/ 1995/tr-95 /C1/09.ps.Z. Only Critical Point in Town Test If a univariate REAL FUNCTION f(x) has a single CRITICAL POINT and that point is a LOCAL MAXIMUM , then f(x) has its GLOBAL MAXIMUM there (Wagon 1991, p. 87). The test breaks downs for bivariate functions, but does hold for bivariate polynomials of degree 54: Such exceptions include z /C303xey /C28x3 /C28e3y (1) z /C30x2(1 /C27y)3 /C27y2 (2) z /C30xy x2 /C28 y2ðÞ x2 /C27 y2for (x ; y) "(0; 0) 0 for (x ; y) /C30(0; 0)8 < : (3) (Rosenholtz and Smylie 1985, Wagon 1991). Note that equation (3) has discontinuous PARTIAL DERIVATIVES zxy and zyx ; and zyx(0; 0) /C301 and zxy(0; 0) /C301 :/ See also CRITICAL POINT ,GLOBAL MAXIMUM ,LOCAL MAXIMUM ,PARTIAL DERIVATIVE References Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999. Apostol, T. M.; Mugler, D. H.; Scott, D. R.; Sterrett, A. Jr.; and Watkins, A. E. A Century of Calculus, Part II: 1969 /C1/ 1991. Washington, DC: Math. Assoc. Amer., 1992. Ash, A. M. and Sexton, H. "A Surface with One Local Minimum." Math. Mag. 58, 147/C1/149, 1985. Calvert, B. and Vamanamurthy, M. K. "Local and Global Extrema for Functions of Several Variables." J. Austral. Math. Soc. 29, 362/C1/368, 1980. Davies, R. "Solution to Problem 1235." Math. Mag. 61, 59, 1988. Rosenholtz, I. and Smylie, L. "The Only Critical Point in Town Test." Math. Mag. 58, 149/C1/150, 1985. Wagon, S. "Failure of the Only-Critical-Point-in-Town Test." §3.4 in Mathematica in Action. New York: W. H. Freeman, pp. 87 /C1/91 and 228, 1991. Ono Inequality Ono (1914) conjectured that the inequality 27 b2 /C27c2 /C28a27C07C)2a2 /C27c2 /C28b27C07C)2a2 /C27b2 /C28c27C07C)25(4K)6 holds true for all TRIANGLES , where a, b, and c are the lengths of the sides and K is the AREA of the TRIANGLE . This conjecture was shown to be false by Quijano (1915), although it was subsequently proved to be true for ACUTE TRIANGLES by Balitrand (1916). A simple counterexample is provided by the triangle with a /C303=4 ; b /C301 =2; and c /C301. See also ACUTE TRIANGLE References Balitrand, F. "Problem 4417." Intermed. Math. 23,86/C1/87, 1916. Mitrinovic, D. S.; Pecaric, J. E.; and Volenec, V. "A Question of Ono." §10.2.1 in Recent Advances in Geometric Inequal- ities. Dordrecht, Netherlands: Kluwer, pp. 240 /C1/241, 1989. Ono, T. "Problem 4417." Intermed. Math. 21, 146, 1914. Quijano, G. "Problem 4417." Intermed. Math. 22, 66, 1915. Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525 /C1/541, 2000. Onsager Differential Equation The ordinary Onsager equation is the sixth-order ORDINARY DIFFERENTIAL EQUATION d3 dx3exd2 dx2exdy dx !"# /C30f(x) (Vicelli 1983; Zwillinger 1997, p. 128), while the partial Onsager equation is given by the PARTIAL DIFFERENTIAL EQUATION ex exuxx ðÞxx7C07C) xx/C27B2uyy /C30F(x; y) (Wood and Martin 1980; Zwillinger 1997, p. 129). References Vicelli, J. A. "Exponential Difference Operator Approxima- tion for the Sixth Order Onsager Equation." J. Comput. Phys. 50, pp. 162 /C1/170, 1983. Wood, H. G. and Morton, J. B. "Onsager’s Pancake Approx- imation for the Fluid Dynamics of a Gas Centrifuge." J. Fluid Mech. 101,1/C1/31, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 128 /C1/129, 1997.Onto Let f be a FUNCTION defined on a SET A and taking values in a set B. Then f is said to be onto (a.k.a. a surjection) if, for any b /C23 B ; there exists an a /C23 A for which b /C30f(a) :/ Let the function be an OPERATOR which MAPS points in the DOMAIN to every point in the RANGE and let V be a VECTOR SPACE with X ; Y /C23V: Then a TRANSFOR- MATION T defined on V is onto if there is an X /C23V such that T(X) /C30Y for all Y. See also BIJECTION ,DOMAIN ,M ANY-TO- ONE,ONE-TO- ONE,RANGE (IMAGE ) Open Ball An n-D open ball of RADIUS r is the collection of points of distance less than r from a fixed point in EUCLI- DEAN n-space. Explicitly, the closed ball with center x and radius r is defined by Br(x) /C30fy : ½y /C28x½Br g: The open ball for n /C301 is called an OPEN INTERVAL , and the term OPEN DISK is sometimes used for n/C302 and sometimes as a synonym for open ball. See also BALL,C LOSED DISK,O PEN DISK,O PEN INTERVAL ,OPEN SET References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 1, 1991. Open Disk Ann-D open disk of RADIUS ris the collection of points of distance less than r from a fixed point in EUCLIDEAN n-space. Krantz (1999, p. 3) uses the symbol D(x; r) to denote the open disk, and D /C30 D(0;1) to denote the unit open disk centered at the origin. The open disk for n /C301 is called an OPEN INTERVAL , and the term OPEN BALL is often used for n ]3:/ See also CLOSED DISK,D ISK,O PEN BALL,O PEN INTERVAL ,OPEN SET,PERFORATION References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 3, 1999. Open Interval An INTERVAL which does not include its LIMIT POINTS , denoted (a, b). The non-standard notation ]a ; b[is sometimes also used. See also CLOSED INTERVAL ,HALF-CLOSED INTERVAL , INTERVAL ,OPEN DISK,OPEN SET References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 1, 1991. Open Manifold A noncompact manifold without boundary. See also CLOSED MANIFOLD Open Map A MAP which sends OPEN SETS to OPEN SETS. See also OPEN MAPPING THEOREM ,OPEN SET Open Mapping Theorem The two flavors of the open mapping theorem state: 1. A continuous surjective linear mapping between BANACH SPACES is an OPEN MAP. 2. A nonconstant ANALYTIC FUNCTION on a DOMAIN D is an OPEN MAP. See also ANALYTIC FUNCTION ,BANACH SPACE ,OPEN MAP References Krantz, S. G. "The Open Mapping Theorem." §5.2.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 73 /C1/74, 1999. Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.Open Problems UNSOLVED PROBLEMS Open Set A SET is open if every point in the set has a NEIGHBORHOOD lying in the set. An open set of RADIUS r and center x0is the set of all points x such that x /C28x0 jjBr ; and is denoted Drx0ðÞ : In 1-space, the open set is an OPEN INTERVAL . In 2-space, the open set is a DISK. In 3-space, the open set is a BALL . More generally, given a TOPOLOGY (consisting of a SET X and a collection of SUBSETS T), a SET is said to be open if it is in T. Therefore, while it is not possible for a set to be both finite and open in the TOPOLOGY of the REAL LINE (a single point is a CLOSED SET), it is possible for a more general topological SET to be both finite and open. The complement of an open set is a CLOSED SET.Itis possible for a set to be neither open nor CLOSED , e.g., the HALF-CLOSED INTERVAL 0; 1 ð/C138 :/ See also BALL,BOREL SET,CLOSED SET,EMPTY SET, OPEN BALL,OPEN DISK,OPEN INTERVAL References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 3, 1999. Operad A system of parameter chain complexes used for MULTIPLICATION on differential GRADED ALGEBRAS up to HOMOTOPY . Operand A mathematical object upon which an OPERATOR acts. For example, in the expression 1 /C292; the MULTIPLICA- TION OPERATOR acts upon the operands 1 and 2. See also OPERAD ,OPERATOR Operational Mathematics The theory and applications of L APLACE TRANSFORMS and other INTEGRAL TRANSFORMS . References Churchill, R. V. Operational Mathematics, 3rd ed. New York: McGraw-Hill, 1958. Operations Research A branch of mathematics which encompasses many diverse areas of minimization and optimization. Bronson (1982) describes operations research as being "concerned with the efficient allocation of scarce resources." The more modern term for opera- tions research is OPTIMIZATION THEORY . See also OPTIMIZATION ,OPTIMIZATION THEORY References Bronson, R. Schaum’s Outline of Theory and Problems of Operations Research. New York: McGraw-Hill, 1982. Hiller, F. S. and Lieberman, G. J. Introduction to Opera- tions Research, 5th ed. New York: McGraw-Hill, 1990. Marlow, W. H. Mathematics for Operations Research. New York: Dover. Singh, J. Great Ideas of Operations Research. New York: Dover, 1972. Trick, M. "Michael Trick’s Operations Research Page." http://mat.gsia.cmu.edu Weisstein, E. W. "Books about Operations Research." http:// www.treasure-troves.com/books/OperationsRe- search.html. Operator An operator A : f(n)(I) /C2f(I) assigns to every function f /C23 f(n)(I) a function A(f) /C23 f(I): It is therefore a map- ping between two FUNCTION SPACES . If the range is on the REAL LINE or in the COMPLEX PLANE , the mapping is usually called a FUNCTIONAL instead. See also ABSTRACTION OPERATOR ,BIHARMONIC OP- ERATOR ,B INARY OPERATOR ,C ASIMIR OPERATOR , CONVECTIVE OPERATOR , D’ALEMBERTIAN ,DELTA OP- ERATOR ,D IFFERENCE OPERATOR ,FUNCTIONAL ANA- LYSIS ,H ECKE OPERATOR ,H ERMITIAN OPERATOR , IDENTITY OPERATOR ,LAPLACIAN ,LAPLACE- BELTRAMI OPERATOR ,LINEAR OPERATOR ,OPERAND ,OPERATOR THEORY ,PERRON- FROBENIUS OPERATOR ,PROJECTION OPERATOR ,ROTATION OPERATOR ,SCATTERING OPERA- TOR,SHIFT- INVARIANT OPERATOR ,SHIFT OPERATOR , SPECTRUM (OPERATOR ), THETA OPERATOR ,U MBRAL OPERATOR ,V ECTOR LAPLACIAN ,W AVE OPERATOR , WEIERSTRASS OPERATOR Operator Theory A broad area of mathematics connected with FUNC- TIONAL ANALYSIS , DIFFERENTIAL EQUATIONS , index theory, representation theory, and mathematical physics. See also C*-ALGEBRA ,OPERATOR References Conway, J. H. A Course in Operator Theory. Providence, RI: Amer. Math. Soc., 2000. Gohberg, I.; Lancaster, P.; and Shivakuar, P. N. (Eds.). Recent Developments in Operator Theory and Its Applica- tions. Boston, MA: Birkha ¨user, 1996.Hutson, V. and Pym, J. S. Applications of Functional Analysis and Operator Theory. New York: Academic Press, 1980. Optimal Golomb Ruler GOLOMB RULER Optimization See also OPTIMIZATION THEORY ,STOCHASTIC OPTIMI- ZATION Optimization Theory A branch of mathematics which encompasses many diverse areas of minimization and optimization. Optimization theory is the more modern term for OPERATIONS RESEARCH . Optimization theory includes the CALCULUS OF VARIATIONS , CONTROL THEORY , CONVEX OPTIMIZATION THEORY , DECISION THEORY , GAME THEORY , LINEAR PROGRAMMING ,M ARKOV CHAINS , network analysis, OPTIMIZATION THEORY , queuing systems, etc. See also CALCULUS OF VARIATIONS ,CONTROL THEORY , CONVEX OPTIMIZATION THEORY ,D ECISION THEORY , DIFFERENTIAL EVOLUTION ,EVOLUTION STRATEGIES , GAME THEORY ,G ENETIC ALGORITHM ,LINEAR PRO- GRAMMING ,M ARKOV CHAIN ,NELDER- MEAD METHOD , OPERATIONS RESEARCH ,OPTIMIZATION ,QUEUE ,STO- CHASTIC OPTIMIZATION References Bhati, M. A. Practical Optimization Methods with Mathe- matica Applications. New York: Springer-Verlag, 2000. Bronson, R. Schaum’s Outline of Theory and Problems of Operations Research. New York: McGraw-Hill, 1982. Hiller, F. S. and Lieberman, G. J. Introduction to Opera- tions Research, 5th ed. New York: McGraw-Hill, 1990. Marlow, W. H. Mathematics for Operations Research. New York: Dover, 1993. Papadimitriou, C. H. and Steiglitz, K. Combinatorial Opti- mization: Algorithms and Complexity. New York: Dover, 1998. Polak, E. Computational Methods in Optimization. New York: Academic Press, 1971. Singh, J. Great Ideas of Operations Research. New York: Dover, 1972. Trick, M. "Michael Trick’s Operations Research Page." http://mat.gsia.cmu.edu Optimum EXTREMUM Or A term in LOGIC which yields TRUE if any one of a sequence conditions is TRUE , and FALSE ifallcondi- tions are FALSE . b OR /C2712 35e2/C2896 385e37C)67C)7 P4/C2840 231e3P6/C27...:/ is denoted / 27(b2/C27c2/C28a2)2(a2/C27c2/C28b2)2(a2/C27b2/C28c2)25(4K)6;// a /C303 =4; or b /C301=2: The symbol /C150 derives from the first letter of the Latin word "vel" meaning "or." The BINARY OR operator has the following TRUTH TABLE . /b///C2712 35 e2 /C2896 385 e37C)67C)7 P4 /C2840 231 e3P6 /C27...://b /C301=2/ FF F FT T TF T TT T A product of ORs is called a DISJUNCTION and is denoted d3 dx3exd2 dx2exdy dx !"# /C30f(x) Two BINARY numbers can have the operation OR performed bitwise. This operation is sometimes de- noted /27(b2 /C27c2 /C28a2)2(a2 /C27c2 /C28b2)2(a2 /C27b2 /C28c2)2 / /5(4K)6 :/ See also AND,BINARY OPERATOR ,LOGIC ,NOT,PRE- DICATE ,TRUTH TABLE ,UNION , XOR OR A CONNECTIVE in LOGIC which yields TRUE if any one of a sequence conditions is TRUE , and FALSE if all conditions are FALSE . In formal logic, the term DISJUNCTION (or, more specifically, inclusive disjunc- tion) is commonly used to describe the OR operator. A OR B is denoted A/C150B (Mendelson 1997, p. 13), AB;j A /C27B (Simpson 1987, p. 539), or A @ B (Simpson 1987, p. 539). The circuit diagram symbol for an OR gate is illustrated above. The symbol /C150derives from the first letter of the Latin word "vel," meaning "or," and the expression A/C150B is voiced either "A or B"or" A vel B." The way to distinguish the similar symbols ffl(AND) and /C150(OR) is to note that the symbol for AND is oriented in the same direction as the capital letter ‘A." The OR operation is implemented in Mathematica as Or[A, B, ...]. The OR operation can be written in terms of NOT and AND as A/C150B /C30!(!Affl!B) (Mendelson 1997, p. 26).The BINARY OR operator has the following TRUTH TABLE (Carnap 1958, p. 10; Simpson 1987, p. 542; Mendelson 1997, p. 13). AB /A/C150B/ TTT TFTFTTFFF A product of ORs is called a DISJUNCTION and is denoted /C150n k /C301Ak : For example, the TRUTH TABLE for the ternary OR operator is shown below (Simpson 1987, p. 543). ABC /A/C150B/C150C/ TTTT TTFTTFTT TFFT FTTTFTFTFFTT FFFF Two BINARY numbers can have the operation OR performed bitwise. This operation is sometimes de- noted AB:j / See also AND, BINARY OPERATOR ,C ONNECTIVE , DISJUNCTION ,E XCLUSIVE DISJUNCTION ,INCLUSIVE DISJUNCTION ,L OGIC , NAND, NOR, NOT, TRUTH TABLE ,UNION ,VEE, XNOR, XOR References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, pp. 7 and 10, 1958. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 13, 1997. Simpson, R. E. "The OR Gate." §12.5.1 in Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 542 /C1/544, 1987. Orbifold The object obtained by identifying any two points of a MAP which are equivalent under some symmetry of the MAP’S GROUP . Orbison’s Illusion The illusion illustrated above in which the bounding RECTANGLE and inner SQUARE both appear distorted. See also ILLUSION ,M U¨ LLER- LYER ILLUSION ,PONZO’S ILLUSION ,VERTICAL- HORIZONTAL ILLUSION References Fineman, M. The Nature of Visual Illusion. New York: Dover, p. 153, 1996. Orbit (Group) In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a GROUP G acts on a set X (this process is called a GROUP ACTION ), it permutes the elements of X. Any particular element X moves around in a fixed path, which is called its orbit. In the notation of set theory, a group orbit can be defined as G(x) /C30fgx /C23 X : g /C23 G g: Note that if y /C23 G(x) then x /C23 G(y) ; because y /C30 gx IFF x /C30g/C281y: Consequently, the orbits PARTITION X and, given a PERMUTATION GROUP G on a set S, the orbit of an element s /C23 S is the subset of S consisting of elements to which some element G can send s. Note that a FIXED POINT is an orbit consisting of a single element. For example, consider the action by the circle group S1 on the SPHERE S2 by rotations along its axis. Then the north pole is an orbit, as is the south pole. The equator is a one-dimensional orbit, as is a general orbit, corresponding to a line of latitude. Orbits of a LIE GROUP action may look different from each other. For example, O(1; 1); the ORTHOGONAL GROUP of SIGNATURE (1; 1); acts on the plane. It hasthree different kinds of orbits: the origin (a FIXED POINT , the four rays f(9t;9t); t > 0g; and the hyper- bolas such as y2 /C28x2 /C301 : In general, an orbit may be of any dimension, up to the dimension of the LIE GROUP . If the LIE GROUP G is COMPACT , then its orbits are SUBMANIFOLDS . The group’s action on the orbit through x is TRANSI- TIVE, and so is related to its ISOTROPY GROUP .In particular, the cosets of the isotropy subgroup corre- spond to the elements in the orbit, G(x) /C2G =Gx : See also EFFECTIVE ACTION ,FREE ACTION ,GROUP , ISOTROPY GROUP ,M ATRIX GROUP ,QUOTIENT SPACE (LIE GROUP ), REPRESENTATION ,TOPOLOGICAL GROUP , TRANSITIVE References Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 4, 35 /C1/41, 49 /C1/52, and 169 /C1/221, 1987. Orbit (Map) The SEQUENCE generated by repeated application of a MAP. The MAP is said to have a closed orbit if it has a finite number of elements. See also DYNAMICAL SYSTEM ,SINK (MAP) Orbit (Permutation) CYCLE (PERMUTATION ) Orchard Visibility Problem A tree is planted at each LATTICE POINT in a circular orchard which has CENTER at the ORIGIN and RADIUS r. If the radius of trees exceeds 1 =r units, one is unable to see out of the orchard in any direction. However, if the RADII of the trees are B1=ffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C271p ;one can see out at certain ANGLES . See also LATTICE POINT ,O RCHARD- PLANTING PRO- BLEM ,VISIBILITY References Honsberger, R. "The Orchard Problem." Ch. 4 in Mathema- tical Gems I. Washington, DC: Math. Assoc. Amer., pp. 43 /C1/52, 1973. Orchard-Planting Problem Also known as the TREE-PLANTING PROBLEM . Plant n trees so that there will be r straight rows with k trees in each row. The following table gives max( r) for various k. k /C303 is Sloane’s A003035 and k /C304is Sloane’s A006065. nk/C30 3 k /C30 4 k /C30 5 31 – – 41 1 – 52 1 1 64 1 1 76 2 1 87 2 1 91 0 3 2 10 12 5 2 11 16 6 2 12 19 7 3 13 /[22; 24] //]9/ 3 14 /[26; 27] //]10/ 4 15 /[31; 32] //]12//]6/ 16 37 /]15//]6/ 17 /[40; 42] //]15//]7/ 18 /[46; 48] //]18//]9/ 19 /[52; 54] //]19//]10/ 20 /[57; 60] //]21//]11/ 21 /[64; 67] /22 /[70; 73] / 23 /[77; 81] / 24 /[85; 88] / 25 /[92; 96] / Sylvester showed that r(k /C303) ]1 6 (n /C281)(n /C282)jk ; where xbcis the FLOOR FUNCTION (Ball and Coxeter 1987). Burr, Gru¨nbaum and Sloane (1974) have shown using cubic curves that r(k /C303) 51 /C2716 n(n /C283)jk ; except for n /C307, 11, 16, and 19, and conjecture that the inequality is an equality with the exception of the preceding cases. For n]4; r(k/C303)]1 312n(n/C281)/C2837nlmhijk ; where xdeis the CEILING FUNCTION . See also CONFIGURATION ,EUCLID’S ORCHARD ,ORCH- ARD VISIBILITY PROBLEM References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 104 /C1/105 and 129, 1987. Burr, S. A. "Planting Trees." In The Mathematical Gardner (Ed. David Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 90 /C1/99, 1981. Dudeney, H. E. Problem 435 in 536 Puzzles & Curious Problems. New York: Scribner, 1967. Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, p. 175, 1949. Dudeney, H. E. §213 in Amusements in Mathematics. New York: Dover, 1970. Friedman, E. "Tree Planting Problems." http://www.stetso- n.edu/~efriedma/trees/. Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, pp. 18 /C1/20 and 26, 1977. Gardner, M. "Tree-Plant Problems." Ch. 22 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 277 /C1/290, 1988. Gru¨nbaum, B. "New Views on Some Old Questions of Combinatorial Geometry." Teorie Combin. 1, 451/C1/468, 1976. Gru¨nbaum, B. and Sloane, N. J. A. "The Orchard Problem." Geom. Dedic. 2, 397/C1/424, 1974. Jackson, J. Rational Amusements for Winter Evenings. London, 1821. Macmillan, R. H. "An Old Problem." Math. Gaz. 30, 109, 1946. Sloane, N. J. A. Sequences A003035/M0982 and A006065/ M0290 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0982 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Order (Algebraic Curve) The order of the POLYNOMIAL defining an ALGEBRAIC CURVE . Order (Algebraic Surface) The order n of an ALGEBRAIC SURFACE is the order of the POLYNOMIAL defining a surface, which can be geometrically interpreted as the maximum number of points in which a line meets the surface. Order Surface 3 CUBIC SURFACE 4 QUARTIC SURFACE 5 QUINTIC SURFACE 6 SEXTIC SURFACE 7 Heptic Surface 8 OCTIC SURFACE 9 Nonic Surface 10 DECIC SURFACE See also ALGEBRAIC SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 8, 1986. Order (Conjugacy Class) The number of elements of a GROUP in a given CONJUGACY CLASS . Order (Difference Set) Let G be GROUP of ORDER h and D be a set of k elements of G. If the set of differences di /C28dj contains every NONZERO element of G exactly l times, then D is a (h ; k; l)/-difference set in G of order n /C30k /C28 l:/ Order (Field) The number of elements in a FINITE FIELD . Order (Function) The INFIMUM of all number a for which ½f(z)½5exp ½z ½aðÞ holds for all ½z ½> r and f an ENTIRE FUNCTION ,is called the ORDER of f, denoted l /C30 l(f) (Krantz 1999, p. 121).See also ENTIRE FUNCTION ,FINITE ORDER References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 121, 1999. Order (Graph) The number of nodes in a graph is called its order. See also GRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 82, 1990. Order (Group) The number of elements in a GROUP G, denoted ½G½: If the order of a GROUP is a finite number, the group is said to be a FINITE GROUP . The order of an element g of a FINITE GROUP G is the smallest POWER of n such that gn /C30I ; where I is the IDENTITY ELEMENT . In general, finding the order of the element of a group is at least as hard as factoring (Meijer 1996). However, the problem becomes signifi- cantly easier if ½G½ and the factorization of ½G½ are known. Under these circumstances, efficient ALGO- RITHMS are known (Cohen 1993). See also ABELIAN GROUP ,FINITE GROUP References Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993. Meijer, A. R. "Groups, Factoring, and Cryptography." Math. Mag. 69, 103 /C1/109, 1996. Order (Modulo) For an INTEGER n that is RELATIVELY PRIME to a number a, there exists a smallest exponent k ]1 such that ak /C131 (mod n); and k is called the order (or HAUPT-EXPONENT )ofa modulo n. For example, the order of 2 modulo 7 is 3, since 21/C132;22/C134;and 23/C30 8/C131 (mod 7). See also CARMICHAEL FUNCTION ,COMPLETE RESIDUE SYSTEM ,HAUPT- EXPONENT ,M ULTIPLICATIVE ORDER , ORDER (POLYNOMIAL ), PRIMITIVE ROOT References Burton, D. M. "The Order of an Integer Modulo n."§8.1 in Elementary Number Theory, 4th ed. Dubuque, IA: William C. Brown Publishers, pp. 184 /C1/190, 1989. Nagell, T. "Exponent of an Integer Modulo n."§31 in Introduction to Number Theory. New York: Wiley, pp. 102 /C1/106, 1951. Order (Ordering) A method for choosing the order in which elements are placed (i.e., a sorting function). See also LEXICOGRAPHIC ORDER ,M ONOMIAL ORDER , PARTIAL ORDER ,TOTAL ORDER ,TRANSPOSITION OR- DER,W ELL ORDER Order (Ordinary Differential Equation) An ORDINARY DIFFERENTIAL EQUATION of order n is an equation OF THE FORM Fx; y; y?; ...; y(n)7C07C) /C300 : Order (Permutation) PERMUTATION Order (Polynomial) The highest order POWER in a UNIVARIATE POLYNO- MIAL is known as its order (or, more properly, its DEGREE ). For example, the POLYNOMIAL P(x) /C30anxn /C27.../C27a2x2 /C27a1x /C27a0 is of order n, denoted deg P(x) /C30n : The order of a polynomial is implemented in Mathematica as Ex- ponent [poly, x]. It is preferable to use the word "degree" for the highest exponent in a polynomial, since a completely different meaning is given to the word "order" in polynomials taken modulo some integer (where this meaning is the one used in the ORDER of a modulus). In particular, the order of a polynomial P(x) with P(0) "0 is the smallest integer e for which P(x) divides xe /C271 : For example, in the FINITE FIELD GF(2), the order of x5 /C27x2 /C271 is 31, since x31 /C27 1 x5 /C27 x2 /C27 1 /C301 /C27x2 /C27x4 /C27x5 /C27x6 /C27x8 /C27x9 /C27x13 /C27x14 /C27x15 /C27x16 /C27x17 /C27x20 /C27x21 /C27x23 /C27x26 (mod 2) : This concept is closely related to that of the HAUPT- EXPONENT . See also DEGREE (POLYNOMIAL ), HAUPT- EXPONENT , IRREDUCIBLE POLYNOMIAL ,ORDER (MODULO ), PRIMI- TIVE POLYNOMIAL Order (Root) MULTIPLICITY Order (Tensor) RANK (TENSOR ) Order (Vertex) The number of EDGES meeting at a given node in a GRAPH is called the order of that VERTEX .Order (Zero) MULTIPLICITY Order Isomorphic Two TOTALLY ORDERED SETS (A;5) and (B ;5) are order isomorphic IFF there is a BIJECTION f from A to B such that for all a1 ; a2 /C23 A; a1 5a2iff fa1ðÞ5fa2ðÞ (Ciesielski 1997, p. 38). In other words, A and B are EQUIPOLLENT ("the same size") and there is an order preserving mapping between the two. Dauben (1979) and Suppes (1972) call this property "similar." The definition works equally well on PAR- TIALLY ORDERED SETS . See also AVOIDED PATTERN ,C ONTAINED PATTERN , PARTIALLY ORDERED SET,P ERMUTATION PATTERN , TOTALLY ORDERED SET References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Mansour, T. Permutations Avoiding a Pattern from Skand at Least Two Patterns from S3 : 31 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007194/. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Order of Magnitude Physicists and engineers use the phrase "order of magnitude" to refer to the smallest power of ten needed to represent a quantity. Two quantities which are within about a factor of 10 of each other are then said to be "of the same order of magnitude." Hardy and Wright (1979, p. 7) use the term to mean ASYMPTOTIC to. See also ASYMPTOTIC ,ASYMPTOTIC NOTATION References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Jeffreys, H. and Jeffreys, B. S. "Orders of Magnitude." §1.08 inMethods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 23 /C1/24, 1988. Order Statistic Given a sample of nvariates X1;...,Xn;reorder them so that X?1BX?2B...BX?n:Then the ith order statistic X/C142i/C143is defined as X?i;with the special cases mn/C30X/C1421/C143/C30min jXj7C07C) Mn/C30X/C142n/C143/C30max jXj7C07C) : AROBUST ESTIMATION technique based on LINEAR COMBINATIONS of order statistics is called an L- ESTIMATE . See also EXTREME VALUE DISTRIBUTION ,H INGE , MAXIMUM ,MEDIAN (STATISTICS ), MINIMUM References Balakrishnan, N. and Chen, W. W. S. Handbook of Tables for Order Statistics from Lognormal Distributions with Applications. Amsterdam, Netherlands: Kluwer, 1999. Balakrishnan, N. and Cohen, A. C. Order Statistics and Inference. New York: Academic Press, 1991. David, H. A. Order Statistics, 2nd ed. New York: Wiley, 1981. Gibbons, J. D. and Chakraborti, S. (Eds.). Nonparametric Statistic Inference, 3rd ed. exp. rev. New York: Dekker, 1992. Order Type Every TOTALLY ORDERED SET (A;5) is associated with a so-called order type. Two sets A and B are said to have the same order type IFF they are ORDER ISOMORPHIC (Ciesielski 1997, p. 38; Dauben 1990, pp. 184 and 199; Moore 1982, p. 52; Suppes 1972, pp. 127 /C1/129). Thus, an order type categorizes TO- TALLY ORDERED SETS in the same way that a CARDI- NAL NUMBER categorizes sets. The term is due to Georg Cantor, and the definition works equally well on PARTIALLY ORDERED SETS. The order type of the negative integers is called /C31v (Moore 1982, p. 62), although Suppes (1972, p. 128) calls it v/C31: The order type of the rationals is called h (Dauben 1990, p. 152; Moore 1982, p. 115; Suppes 1972, p. 128). Some sources call the order type of the reals u (Dauben 1990, p. 152), while others call it l (Suppes 1972, p. 128). In general, if a is any order type, then /C31a is the same type ordered backwards (Dauben 1990, p. 153). See also CARDINAL NUMBER ,O RDER ISOMORPHIC , ORDINAL NUMBER ,TOTALLY ORDERED SET References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Ordered Factorization An ordered factorization is a factorization (not neces- sarily into prime factors) in which a /C29b is considered distinct from b /C29a : The number of ordered factoriza- tions of n is equal to the number of PERFECT PARTITIONS of n /C281 (Goulden and Jackson 1983, p. 94). See also PERFECT PARTITIONReferences Goulden, I. P. and Jackson, D. M. Problem 2.5.12 in Combi- natorial Enumeration. New York: Wiley, p. 94, 1983. Ordered Geometry A GEOMETRY constructed without reference to mea- surement. The only primitive concepts are those of points and intermediacy. There are 10 AXIOMS under- lying ordered GEOMETRY . See also ABSOLUTE GEOMETRY ,AFFINE GEOMETRY , GEOMETRY Ordered List The number of nondecreasing lists a1 ; a2 ; ...; an fg consisting of n elements 1 5ai 5k is given by the binomial coefficient N(n; k) /C30n /C27k /C281 n /C2817C)87C)9 : For example, there are six nondecreasing lists of length 2 for elements chosen from 1 to 3: (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), and (3,3). Ordered Pair A PAIR of quantities (a, b) where ordering is sig- nificant, so (a, b) is considered distinct from (b, a) for a"b:/ See also LIST,M ULTISET ,ORDERED PAIRS REPRESEN- TATION ,PAIR,SET,VECTOR Ordered Pairs Representation A representation of a GRAPH in which edges are specified as ordered pairs (for a DIRECTED GRAPH ), or unordered pairs (for an UNDIRECTED GRAPH ). The ordered pairs representation of a graph gmay be computed using ToOrderedPairs [g] in the Mathe- matica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ )o rToUnorderedPairs [g]. A graph may be constructed from ordered pairs using FromOrderedPairs [l], or from unordered pairs usingFromUnorderedPairs [l]. References Skiena, S. "Ordered Pairs." §3.1.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 87 /C1/88, 1990. Ordered Set An ambiguous term which is sometimes used to mean aPARTIALLY ORDERED SET and sometimes to mean a TOTALLY ORDERED SET . Ordered Tree A ROOTED TREE in which the order of the subtrees is significant. There is a ONE-TO-ONE correspondence between ordered FORESTS with n nodes and BINARY TREES with n nodes. See also BINARY TREE,FOREST ,ROOTED TREE Ordering The number of "ARRANGEMENTS " in an ordering of n items is given by either a COMBINATION (order is ignored) or a PERMUTATION (order is significant). See also ARRANGEMENT ,C OMBINATION ,C UTTING , DERANGEMENT ,PARTIAL ORDER ,PERMUTATION ,SORT- ING,TOTAL ORDER Ordering Axioms The four of HILBERT’S AXIOMS which concern the arrangement of points. See also CONGRUENCE AXIOMS ,CONTINUITY AXIOMS , HILBERT’S AXIOMS ,INCIDENCE AXIOMS ,P ARALLEL POSTULATE References Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: Open Court, 1980. Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert’s System of Axioms." §163B in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 544 /C1/545, 1980. Ordinal ORDINAL NUMBER Ordinal Addition Let (A;5) and (B ;5) be disjoint TOTALLY ORDERED SETS with ORDER TYPES a and b: Then the ordinal sum is defined at set (C /C30A @ B;5) where, if c1 and c2 are both from the same SUBSET , the order is the same as in the subset, but if c1 is from A and c2 is from B, then c1 Bc2has ORDER TYPE a /C27 b (Ciesielski 1997, p. 48; Dauben 1990, p. 104; Moore 1982, p. 40). One should note that in the infinite case, ORDER TYPE addition is not commutative, although it is associa- tive. For example, 1 /C27 v /C30 v " v /C271: In addition, fa g@f0 ; 1 ; 2 ; 3 ; ...g; with a the least element, is ORDER ISOMORPHIC to f0 ; 1 ; 2 ; 3 ; ...g; but not to f0; 1; 2; 3; ...g@fag; with a the greatest element, since it has a greatest element and the other does not. An inductive definition for ordinal addition states that for any ORDINAL NUMBER a; a /C270 /C30 a; (1) anda /C27(successor to b) /C30the successor to ( a /C27 b) : (2) If b is a LIMIT ORDINAL , then a /C27 b is the least ordinal greater than any ordinal in the set fa /C27 g : g B bg (Rubin 1967, p. 188; Suppes 1972, p. 205). See also ORDINAL EXPONENTIATION ,ORDINAL MULTI- PLICATION ,ORDINAL NUMBER References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Ordinal Comparison Let (A;5) and (B ;5)be WELL ORDERED SETS with ORDINAL NUMBERS a and b: Then a B b IFF A is ORDER ISOMORPHIC to an INITIAL SEGMENT of B (Dauben 1990, p. 199). From this, it can easily be shown that the ORDINAL NUMBERS are TOTALLY ORDERED by the relation. In fact, they are WELL ORDERED by the relation. See also WELL ORDERED SET References Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Ordinal Exponentiation Letaandbbe any ORDINAL NUMBERS , then ordinal exponentiation is defined so that if b/C300 then ab/C301: Ifbis not a LIMIT ORDINAL , then choose gsuch that g/C271/C30b; alpha(successor of b)ab7C07C) +a: Ifbis a LIMIT ORDINAL , then if a/C300;ab/C300:Ifa"0 then, abis the least ordinal greater than any ordinal in the set ag:gBb fg (Rubin 1967, p. 204; Suppes 1972, p. 215). Note that this definition is not analogous to the definition for cardinals, since ½a½½b½may not equal abjj;even though ½a½/C27½b½/C30½a/C27b½and ½a½+½b½/C30 ½a+b½:Note also that 2v/C30v:/ A familiar example of ordinal exponentiation is thedefinition of Cantor’s first epsilon number. e 0is the least ordinal such that ve0/C30e0:It can be shown that it is the least ordinal greater than any ordinal in v;vv;vvv;... fg :/ References Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Ordinal Multiplication Let (A;5) and (B ;5)be TOTALLY ORDERED SETS . Let C /C30A /C29B be the CARTESIAN PRODUCT and define order as follows. For any a1 ; a2 /C23 A and b1 ; b2 /C23 B; 1. If a1 Ba2 ; then a1 ; b1 ðÞB a2 ; b2 ðÞ ;/ 2. If a1 /C30a2 ; then a1 ;b1 ðÞ and a2 ;b2 ðÞ compare the same way as b1 ; b2 (i.e., lexicographical order) (Ciesielski 1997, p. 48; Rubin 1967; Suppes 1972). However, Dauben (1990, p. 104) and Moore (1982, p. 40) define multiplication in the reverse order. Like addition, multiplication is not commutative, but it is associative, 2 + v /C30 v " v + 2 : (1) An inductive definition for ordinal multiplication states that for any ORDINAL NUMBER a; a + 0 /C300 (2) a + (successor to beta) /C30 a + b /C27 a: (3) Ifbis a LIMIT ORDINAL , then a/C27bis the least ordinal greater than any ordinal in the set fa+g:gBbg (Suppes 1972, p. 212). See also ORDINAL ADDITION ,ORDINAL EXPONENTIA- TION ,ORDINAL NUMBER ,SUCCESSOR References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Ordinal Number In common usage, an ordinal number is an adjective which describes the numerical position of an object,e.g., first, second, third, etc. In formal SET THEORY , an ordinal number (sometimes simply called an "ordinal" for short) is one of the numbers in Georg Cantor’s extension of the WHOLE NUMBERS . An ordinal number is defined as the ORDER TYPE of a WELL ORDERED SET (Dauben 1990, p. 199; Moore 1982, p. 52; Suppes 1972, p. 129). Finiteordinal numbers are commonly denoted using arabicnumerals, while transfinite ordinals as denoted using lower case Greek letters. It is easy to see that every finite TOTALLY ORDERED SET isWELL ORDERED . Any two TOTALLY ORDERED SETS with kelements (for ka nonnegative integer) are ORDER ISOMORPHIC , and therefore have the sameORDER TYPE (which is also an ordinal number). The ordinals for finite sets are denoted 0, 1, 2, 3, ..., i.e., the integers one less than the corresponding non-negative integers. The first transfinite ordinal, denoted v;is the ORDER TYPE of the set of nonnegative integers (Dauben 1979, p 152; Moore 1982, p. viii; Rubin 1967, pp. 86 and 177; Suppes 1972, p. 128). This is the "smallest" ofCantor’s TRANSFINITE NUMBERS , defined to be the smallest ordinal number greater than the ordinalnumber of the WHOLE NUMBERS . Conway and Guy (1996) denote it with the notation v/C30f0;1;...½g:/ From the definition of ORDINAL COMPARISON ,i s follows that the ordinal numbers are a WELL ORDERED SET. In order of increasing size, the ordinal numbers are 0, 1, 2, ..., v;v/C271;v/C272;...,v/C27v;v/C27v/C271;.... The notation of ordinal numbers can be a bit counter- intuitive, e.g., even though 1 /C27v/C30v;v/C271>v:The CARDINALITY of the set of countable ordinal numbers is denoted A LEPH-1 . If (A;5)i sa WELL ORDERED SET with ordinal number a;then the set of all ordinals BaisORDER ISOMORPHIC toA. This provides the motivation to define an ordinal as the set of all ordinals less that itself. John von Neumann defined a set ato be an ordinal number IFF 1. Ifbis a member of a;then bis a PROPER SUBSET ofa/ 2. If bandgare members of athen one of the following is true: b/C30g;bis a member of g;orgis a member of b:/ 3. If Bis a nonempty PROPER SUBSET ofa;then there exists a gmember of Bsuch that the intersection gSBis empty. (Rubin 1967, p. 176; Ciesielski 1997, p. 44). This isthe standard representation of ordinals. In thisrepresentation, symbol elements description 0 /fg/ empty set 1 /f0g/ set of one element 2 /f0;1g/ set of two elements 3 /f0;1;2g/ set of three elements /n/ /v// f0;1;2;...g/ set of all finite ordi- nals /v/C271// f0;1;2;...;vg/ /n/ / v1/ set of all countable ordinals /n/ / v2/ set of all countable and /C2101 ordinals /n/ / vv/ set all finite ordinals and /C210k ordinals for all nonnegative integers k /n/ Rubin (1967, p. 272) provides a nice definition of the va ordinals. Since for any ordinal a; the union a @ a is a bigger ordinal a /C271; there is no largest ordinal, and the class of all ordinals is therefore a PROPER CLASS (as shown by the BURALI- FORTI PARADOX ). Ordinal numbers have some other rather peculiar properties. The sum of two ordinal numbers can take on two different values, the sum of three can take on five values. The first few terms of this sequence are 2, 5, 13, 33, 81, 193, 449, 332,33 /C215 81; 812,81 /C215 193; 1922, ... (Conway and Guy 1996, Sloane’s A005348). The sum of n ordinals has either 193a81bor 33 /C215 81a possible answers for n ]15 (Conway and Guy 1996). /r /C29 v is the same as v; but v /C29r is equal to v /C27.../C27 v|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}: v2 is larger than any number OF THE FORM v /C29r ; v3 is larger than v2 ; and so on. There exist ordinal numbers which cannot be con- structed from smaller ones by finite additions, multi- plications, and exponentiations. These ordinals obey CANTOR’S EQUATION . The first such ordinal is e0 /C30 vvUv |ffl{zffl} v/C301 /C27 v /C27 vv /C27 vvv /C27...: The next is e1 /C30(1 /C27e0) /C27 ve0/C271 /C27 vve0/C271 /C27...; then follow e2 ;e3 ; ..., ev ;ev/C271 ; ..., ev/C292 ; ..., ev2 ;evv ; ..., ee0;ee0/C271 ; ..., ee0/C27v ; ..., ee0/C27v ; ..., ee0/C292 ; ..., ee1; ..., ee2; ..., eev; ..., eee0; ..., eee1; ..., eeev; ..., eeee0; ... (Conway and Guy 1996). ORDINAL ADDITION , ORDINAL MULTIPLICATION , and ORDINAL EXPONENTIATION can all be defined. Although these definitions also work perfectly well for ORDER TYPES , this does not seem to be commonly done. There are two methods common used to define operations on the ordinals: one is using sets, and the other is inductively. See also ALEPH-1 ,AXIOM OF CHOICE ,BURALI- FORTI PARADOX ,CANTOR’S EQUATION ,CARDINALITY ,CARDI- NAL NUMBER ,INITIAL ORDINAL ,O RDER STATISTIC , ORDER TYPE,POWER SET,SURREAL NUMBER ,W ELLORDERED SET References Cantor, G. U¨ber unendliche, lineare Punktmannigfa ¨ltigkei- ten, Arbeiten zur Mengenlehre aus dem Jahren 1872 /C1/ 1884. Leipzig, Germany: Teubner-Archiv zur Mathema- tik, 1884. Conway, J. H. and Guy, R. K. "Cantor’s Ordinal Numbers." InThe Book of Numbers. New York: Springer-Verlag, pp. 266 /C1/267 and 274, 1996. Dauben, J. W. Georg Cantor: His Mathematics and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1990. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Sloane, N. J. A. Sequences A005348/M1435 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Ordinary Differential Equation An ordinary differential equation (frequently abbre- viated ODE) is an equality involving a function and itsDERIVATIVES . An ODE of order nis an equation OF THE FORM F(x;y;y?;/C1/C1/C1;y(n))/C300; (1) where y?/C30dy=dxis a first DERIVATIVE with respect to xand y(n)/C30dny=dxnis an nth DERIVATIVE with respect to x. An ODE of order nis said to be linear if it is OF THE FORM an(x)y(n)/C27an/C281(x)y(n/C281)/C27/C1/C1/C1/C27a1(x)y?/C27a0(x)y /C30Q(x): (2) A linear ODE where Q(x)/C300 is said to be homo- geneous. Confusingly, an ODE OF THE FORM dy dx/C30fyx ! (3) is also sometimes called "homogeneous." In general, an nth-order ODE has nlinearly inde- pendent solutions. Furthermore, any LINEAR COMBI- NATION of LINEARLY INDEPENDENT FUNCTIONS solutions is also a solution.Simple theories exist for first-order ( INTEGRATING FACTOR ) and second-order (S TURM- LIOUVILLE THE- ORY) ordinary differential equations, and arbitrary ODEs with linear constant COEFFICIENTS can be solved when they are of certain factorable forms. Integral transforms such as the L APLACE TRANSFORM can also be used to solve classes of linear ODEs. Morse and Feshbach (1953, pp. 667 /C1/674) give cano- nical forms and solutions for second-order ODEs. While there are many general techniques for analy- tically solving classes of ODEs, the only practical solution technique for complicated equations is to usenumerical methods (Milne 1970, Jeffreys and Jeffreys 1988). The most popular of these is the R UNGE- KUTTA METHOD , but many others have been developed, including the COLLOCATION METHOD and G ALERKIN METHOD . A vast amount of research and huge numbers of publications have been devoted to the numerical solution of differential equations, both ordinary and PARTIAL (PDEs) as a result of their importance in fields as diverse as physics, engineer- ing, economics, and electronics. The solutions to an ODE satisfy EXISTENCE and UNIQUENESS properties. These can be formally estab- lished by P ICARD’S EXISTENCE THEOREM for certain classes of ODEs. Let a system of first-order ODE be given by dxi dt/C30fi(x1;...;xn;t); (4) fori/C301, ..., nand let the functions fi(x1;...;xn;t); where i/C301, ..., n, all be defined in a DOMAIN Dof the (n/C271)/-D space of the variables x1;...,xn;t. Let these functions be continuous in Dand have continuous first PARTIAL DERIVATIVES @fi=@xjfori/C301, ..., nand j/C301, ..., ninD. Let ( x0 1;...;x0n)b ei n D. Then there exists a solution of (4) given by x1/C30x1(t);...;xn/C30xn(t) (5) fort0/C28dBtBt0/C27d(where d>0) satisfying the initial conditions x1(t0)/C30x0 1;...;xn(t0)/C30x0n: (6) Furthermore, the solution is unique, so that if x1/C30x1/C31(t);...;xn/C30xn/C31(t) (7) is a second solution of (4) for t0/C28dBtBt0/C27dsatisfy- ing (6), then xi(t)/C13xi/C31(t) fort0/C28dBtBt0/C27d:Because every nth-order ODE can be expressed as a system of nfirst-order differential equations, this theorem also applies to the single nth-order ODE. An exact FIRST-ORDER ODES is one OF THE FORM p(x;y)dx/C27q(x;y)dy/C300; (8) where @p @y/C30@q @x: (9) An equation OF THE FORM (8) with @p @y"@q @x(10) is said to be nonexact. If @p @y/C28@q @x q/C30f(x) (11) in (8), it has an x-dependent integrating factor. If@q @x/C28@p @y xp/C28yq/C30f(xy) (12) in (8), it has an xy-dependent integrating factor. If @q @x/C28@p @y p/C30f(y) (13) in (8), it has a y-dependent integrating factor. Other special first-order types include cross multiple equations yf(xy)dx/C27xg(xy)dy/C300; (14) homogeneous equations dy dx/C30fyx ! ; (15) linear equations dydx/C27p(x)y/C30q(x); (16) and separable equations dydx/C30X(x)Y(y): (17) Special classes of SECOND-ORDER ODES include d2y dx2/C30f(y;y?) (18) (xmissing) and d2y dx2/C30f(x;y?) (19) (ymissing). A second-order linear homogeneous ODE d2y dx2/C27P(x)dydx/C27Q(x)y/C300 (20) for which Q?(x)/C272P(x)Q(x) 2[Q(x)]3=2/C30[constant] (21) can be transformed to one with constant coefficients. The undamped equation of SIMPLE HARMONIC MOTION is d2y dx2/C27v2 0y/C300; (22) which becomes d2y dx2/C27bdy dx/C27v2 0y/C300 (23) when damped, and d2y dx2/C27bdy dx/C27v2 0y/C30Acos(vt) (24) when both forced and damped. SYSTEMS WITH CONSTANT COEFFICIENTS are of the form dx dt/C30Ax(t)/C27p(t): (25) The following are examples of important ordinary differential equations which commonly arise in pro- blems of mathematical physics. ABEL’S DIFFERENTIAL EQUATION y?/C30f0(x)/C27f1(x)y/C27f2(x)y2/C27f3(x)y3/C27. . . (26) g0(x)/C27g1(x)y ½/C138 y?/C30f0(x)/C27f1(x)y/C27f2(x)y2/C27f3(x)y3:(27) AIRY DIFFERENTIAL EQUATION d2y dx2/C28xy/C300: (28) ANGER DIFFERENTIAL EQUATION yƒ/C27y? x/C271/C28n2 x2 ! y/C30x/C28n px2sin(nx): (29) BAER DIFFERENTIAL EQUATIONS x/C28a1 ðÞ x/C28a2 ðÞ yƒ/C271 22x/C28a1/C27a2 ðÞ ½/C138 y?/C28p2x/C27q27C07C) y /C300; (30) x/C28a1 ðÞ x/C28a2 ðÞ yƒ/C271 22x/C28a1/C27a2 ðÞ ½/C138 y? /C28k2x2/C28p2x/C27q27C07C) y/C300: (31) BERNOULLI DIFFERENTIAL EQUATION dy dx/C27p(x)y/C30q(x)yn: (32) BESSEL DIFFERENTIAL EQUATION x2d2y dx2/C27xdydx/C27l 2x2/C28n27C07C) y/C300: (33) BINOMIAL DIFFERENTIAL EQUATION (y?)m/C30f(x;y): (34) BOˆCHER EQUATION yƒ/C271 2m1 x/C28a1/C27.../C27mn/C281 x/C28an/C281"# y? /C271 4A0/C27A1x/C27.../C27Alxl x/C28a1 ðÞm1x/C28a2 ðÞm2/C1/C1/C1x/C28an/C281 ðÞmn/C281"# y/C300:(35) BRIOT- BOUQUET EQUATIONxmdy dx/C30f(x;y): (36) CHEBYSHEV DIFFERENTIAL EQUATION 1/C28x27C07C) d2y dx2/C28xdy dx/C27a2y/C300: (37) CLAIRAUT’S DIFFERENTIAL EQUATION y/C30xdydx/C27fdydx ! : (38) C ONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUA- TION xd2y dx2/C27c/C28x ðÞdydx/C28ay/C300: (39) D’ALEMBERT’S EQUATION . y/C30xf(y?)/C27g(y?): (40) DUFFING DIFFERENTIAL EQUATION ¨x/C27v2 0x/C27bx3/C300: (41) ECKART DIFFERENTIAL EQUATION yƒ/C27ah 1/C27h/C27bh (1/C27n)2/C27g"# y/C300; (42) where h/C30edx:/ EMDEN- FOWLER DIFFERENTIAL EQUATION xpy? ðÞ?9xsyn/C300: (43) EULER DIFFERENTIAL EQUATION x2d2y dx2/C27axdy dx/C27by/C30S(x): (44) HALM’S DIFFERENTIAL EQUATION 1/C27x27C07C)2/C27yƒ/C27ly/C300: (45) HERMITE DIFFERENTIAL EQUATION d2y dx2/C282xdy dx/C27ly/C300: (46) HEUN’S DIFFERENTIAL EQUATION d2w dx2/C27g x/C27d x/C281/C27o x/C28a ! dw dx/C27abx/C28q x(x/C281)(x/C28a)w /C300: (47) HILL’S DIFFERENTIAL EQUATION d2y dx2u0/C272X/C12 n/C301uncos(2 nz)"# /C300: (48) HYPERGEOMETRIC DIFFERENTIAL EQUATION x(x/C281)d2y dx2/C27[(1/C27a/C27b)x/C28g]dy dx/C27aby/C300: (49) JACOBI DIFFERENTIAL EQUATION 1/C28x27C07C) yƒ/C27[b/C28a/C28(a/C27b/C272)x]y?/C27n(n/C27a/C27b/C271)y /C300: (50) LAGUERRE DIFFERENTIAL EQUATION xd2y dx2/C27(1/C28x)dydx/C27ly/C300: (51) L AME´’S DIFFERENTIAL EQUATION x2/C28b27C07C) x2/C28c27C07C) d2z dx2/C27x(x2/C28b2/C27x2/C28c2)dz dx /C28m(m/C271)x2/C28b2/C27c27C07C) p7CP7C3 z/C300: (52) LANE-EMDEN DIFFERENTIAL EQUATION 1 j2d djj2du dj ! /C27un/C300: (53) LEGENDRE DIFFERENTIAL EQUATION (1/C28x2)d2y dx2/C282xdydx/C27a(a/C271)y/C300: (54) L INEAR CONSTANT COEFFICIENTS a0dny dxn/C27.../C27an/C281dy dx/C27any/C30p(x): (55) LOMMEL DIFFERENTIAL EQUATION z2d2y dz2/C27zdy dz/C28(z2/C27n2)y/C30kzm/C271: (56) LO¨WNER’S DIFFERENTIAL EQUATION y?/C30/C28 y1/C27k(x)y 1/C28k(x)y: MALMSTE ´N’S DIFFERENTIAL EQUATION d2y dx2/C27r zdydx/C30Az m/C27s z2 ! y: (57) MATHIEU DIFFERENTIAL EQUATION d2V dv2/C27[a/C282qcos(2 v)]V/C300: (58) MODIFIED BESSEL DIFFERENTIAL EQUATION x2d2y dx2/C27xdy dx/C28(x2/C27n2)y/C300: (59)MODIFIED SPHERICAL BESSEL DIFFERENTIAL EQUA- TION r2d2R dr2/C272rdR dr/C28k2r2/C27n(n/C271)7CP7C3 R/C300: (60) RAYLEIGH DIFFERENTIAL EQUATION yƒ/C28m1/C281 3y?27C)67C)7 y?/C27y/C300: (61) RICCATI DIFFERENTIAL EQUATION dw dx/C30q0(x)/C27q1(x)w/C27q2(x)w2: (62) RIEMANN P-DIFFERENTIAL EQUATION d2u dz2/C271/C28a/C28a? z/C28a/C271/C28b/C28b? z/C28b/C271/C28g/C28g? z/C28c"# du dz /C27aa?(a/C28b)(a/C28c) z/C28a/C27bb?(b/C28c)(b/C28a) z/C28b/C27gg?(c/C28a)(c/C28b) z/C28c"# /C29u (z/C28a)(z/C28b)(z/C28c)/C300: (63) SHARPE’S DIFFERENTIAL EQUATION zyƒ/C27y?/C27(z/C27A)y/C300: (64) SPHERICAL BESSEL DIFFERENTIAL EQUATION r2d2R dr2/C272rdR dr/C27k2r2/C28n(n/C271)7CP7C3 R/C300: (65) STRUVE DIFFERENTIAL EQUATION z2yƒ/C27zy?/C27z2/C28n27C07C) y/C30412z7C)67C)7n/C271 ffiffiffippGn/C271 27C)67C)7 : (66) STURM- LIOUVILLE EQUATION d dxp(x)dy dx"# /C27[lw(x)/C28q(x)]y/C300: (67) ULTRASPHERICAL DIFFERENTIAL EQUATION 1/C28x27C07C) yƒ/C28(2a/C271)xy?/C27n(n/C272a)y/C300: (68) VAN DER POL EQUATION yƒ/C28m1/C28y27C07C) y?/C27y/C300: (69) WEBER DIFFERENTIAL EQUATION d2y dz2/C27n/C271 2/C2814z27C)67C)7 y/C300: (70) WHITTAKER DIFFERENTIAL EQUATION d2u dz2 /C27du dz /C27k z /C271 4 /C28 m2 z2 ! u /C300 : (71) See also ADAMS’ METHOD ,GREEN’S FUNCTION ,ISO- CLINE ,LAPLACE TRANSFORM ,LEADING ORDER ANALY- SIS,M AJORANT ,ORDINARY DIFFERENTIAL EQUATION– FIRST- ORDER ,O RDINARY DIFFERENTIAL EQUATION– SECOND- ORDER ,P ARTIAL DIFFERENTIAL EQUATION , RELAXATION METHODS ,RUNGE- KUTTA METHOD ,SIM- PLE HARMONIC MOTION References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, 1992. Braun, M. Differential Equations and Their Applications, 4th ed. New York: Springer-Verlag, 1993. Carroll, J. "A Composite Integration Scheme for the Numer- ical Solution of Systems of Ordinary Differential Equa- tions." J. Comput. Appl. Math. 25,1/C1/13, 1989. Coddington, E. A. An Introduction to Ordinary Differential Equations. New York: Dover, 1989. Forsyth, A. R. Theory of Differential Equations, 6 vols. New York: Dover, 1959. Forsyth, A. R. A Treatise on Differential Equations. New York: Dover, 1997. Fulford, G.; Forrester, P.; and Jones, A. Modelling with Differential and Difference Equations. New York: Cam- bridge University Press, 1997. Guterman, M. M. and Nitecki, Z. H. Differential Equations: A First Course, 3rd ed. Philadelphia, PA: Saunders, 1992. Hull, T. E.; Enright, W. H.; Fellen, B. M.; and Sedgwick, A. E. "Comparing Numerical Methods for Ordinary Dif- ferential Equations." SIAM J. Numer. Anal. 9, 603/C1/637, 1972. Hull, T. E.; Enright, W. H.; Fellen, B. M.; and Sedgwick, A. E. "Erratum to ‘Comparing Numerical Methods forOrdinary Differential Equations."’ SIAM J. Numer. Anal. 11, 681, 1974. Ince, E. L. Ordinary Differential Equations. New York: Dover, 1956. Jeffreys, H. and Jeffreys, B. S. "Numerical Solution of Differential Equations." Methods of Mathematical Phy- sics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 290 /C1 /301, 1988. Kamke, E. Differentialgleichungen: Lo ¨sungsmethoden und Lo¨sungen, Bd. 1: Gewo ¨hnliche Differentialgleichungen, 9. Aufl. Stuttgart, Germany: Teubner, 1983. Milne, W. E. Numerical Solution of Differential Equations. New York: Dover, 1970. Morse, P. M. and Feshbach, H. "Ordinary Differential Equations." Ch. 5 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 492 /C1/675, 1953. Moulton, F. R. Differential Equations. New York: Dover, 1958. Polyanin, A. D. and Zaitsev, V. F. Handbook of Exact Solutions for Ordinary Differential Equations. Boca Ra- ton, FL: CRC Press, 1995. Postel, F. and Zimmermann, P. "A Review of the ODE Solvers of Axiom, Derive, Macsyma, Maple, Mathematica,MuPad, and Reduce." Submitted to The 5th Rhine Work- shop on Computer Algebra. July 26, 1996. http://www.lor- ia.fr/~zimmerma/ComputerAlgebra/ode_comp.ps.gz. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Integration of Ordinary Differential Equa-tions." Ch. 16 in Numerical Recipes in FORTRAN: The Artof Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 701 /C1 /744, 1992. Simmons, G. F. Differential Equations, with Applications and Historical Notes, 2nd ed. New York: McGraw-Hill, 1991. Weisstein, E. W. "Books about Ordinary Differential Equa- tions." http://www.treasure-troves.com/books/Ordinary-DifferentialEquations.html. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997. Ordinary Differential Equation * /First- Order Given a first-order ORDINARY DIFFERENTIAL EQUATION dy dx/C30F(x;y); (1) ifF(x;y) can be expressed using SEPARATION OF VARIABLES as F(x;y)/C30X(x)Y(y); (2) then the equation can be expressed as dy Y(y)/C30X(x)dx (3) and the equation can be solved by integrating both sides to obtain gdy Y(y)/C30gX(x)dx: (4) Any first-order ODE OF THE FORM dy dx/C27p(x)y/C30q(x) (5) can be solved by finding an INTEGRATING FACTOR m/C30 m(x) such that d dx(my)/C30mdydx/C27ydm dx/C30mq(x): (6) Dividing through by myyields 1 ydydx/C271 mdm dx/C30q(x) y: (7) However, this condition enables us to explicitly determine the appropriate mfor arbitrary pand q. To accomplish this, take p(x)/C301 mdm dx(8) in the above equation, from which we recover the original equation (5), as required, in the form 1 ydy dx/C27p(x)/C30q(x) y: (9) But we can integrate both sides of (8) to obtain g p(x) dx /C30gdm m/C30ln m /C27c (10) m /C30e g p(x) dx : (11) Now integrating both sides of (6) gives my /C30g mq(x) dx /C27c (12) (with m now a known function), which can be solved for y to obtain y /C30g mq(x) dx /C27 c m/C30g e gx p(x ?) dx ?q(x) d(x) /C27 c e gx p(x?) dx?; (13) where c is an arbitrary constant of integration. Given an nth-order linear ODE with constant COEF- FICIENTS dny dxn /C27an/C281dn/C281y dxn/C281 /C27...a1dy dx /C27a0y /C30Q(x) ; (14) first solve the characteristic equation obtained by writing y /C13erx (15) and setting Q(x) /C300 to obtain the n COMPLEX ROOTS . rnerx /C27an/C281rn/C281erx /C27.../C27a1rerx /C27a0erx /C300 (16) rn /C27an/C281rn/C281 /C27.../C27a1r /C27a0 /C300: (17) Factoring gives the ROOTS ri ; (r /C28r1)(r /C28r2) /C1/C1/C1(r /C28rn) /C300 : (18) For a nonrepeated REAL ROOT r, the corresponding solution is y /C30erx : (19) If a REAL ROOT r is repeated k times, the solutions are degenerate and the linearly independent solutions are y /C30erx ;y /C30xerx ;/C1/C1/C1;y /C30xk /C281erx : (20) Complex ROOTS always come in COMPLEX CONJUGATE pairs, r9/C30a 9ib: For nonrepeated COMPLEX ROOTS , the solutions are y /C30eax cos(bx) ;y /C30eax sin(bx) : (21) If the COMPLEX ROOTS are repeated k times, the linearly independent solutions are y /C30eax cos(bx) ;y /C30eax sin(bx) ;/C1/C1/C1; y /C30xk /C281eax cos(bx) ;y /C30xk/C281eax sin(bx) : (22) Linearly combining solutions of the appropriate types with arbitrary multiplicative constants then gives the complete solution. If initial conditions are specified,the constants can be explicitly determined. For example, consider the sixth-order linear ODE ( ˜D /C281)( ˜D /C282)3( ˜D2 /C27 ˜D /C271)y /C300; (23) which has the characteristic equation (r /C281)(r /C282)3(r2 /C27r /C271) /C300: (24) The roots are 1, 2 (three times), and (/C281 9ffiffiffi 3p i) =2; so the solution is y /C30Aex /C27Be2x /C27Cxe2x /C27Dx2e3x /C27Ee /C28x =2 cos1 2ffiffiffi 3p x7C)67C)7 /C27Fe/C28x sin1 2ffiffiffi 3p x7C)67C)7 : (25) If the original equation is nonhomogeneous /(Q(x)"0); now find the particular solution y/C31by the method of VARIATION OF PARAMETERS . The general solution is then y(x)/C30Xn i/C301ciyi(x)/C27y/C31(x); (26) where the solutions to the linear equations are y1(x); y2(x);...,yn(x);andy/C31(x) is the particular solution. See also INTEGRATING FACTOR ,ORDINARY DIFFEREN- TIAL EQUATION– FIRST- ORDER EXACT ,SEPARATION OF VARIABLES ,VARIATION OF PARAMETERS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 440 /C1/445, 1985. Ordinary Differential Equation * /First- Order Exact Consider a first-order ODE in the slightly different form p(x;y)dx/C27q(x;y)dy/C300: (1) Such an equation is said to be exact if @p @y/C30@q @x: (2) This statement is equivalent to the requirement that aCONSERVATIVE FIELD exists, so that a scalar poten- tial can be defined. For an exact equation, the solution is g(x;y) (x0;y0)p(x;y)dx/C27q(x;y)dy/C30c; (3) where cis a constant. A first-order ODE (1) is said to be inexact if @p @y"@q @x: (4) For a nonexact equation, the solution may be ob- tained by defining an INTEGRATING FACTOR mof (6) so that the new equation mp(x;y)dx/C27mq(x;y)dy/C300 (5) satisfies @ @y(mp)/C30@ @x(mq); (6) or, written out explicitly, p@m @y/C27m@p @y/C30q@m @x/C27m@p @x: (7) This transforms the nonexact equation into an exact one. Solving (7) for mgives m/C30q@m @x/C28p@m @y @p @y/C28@q @x: (8) Therefore, if a function msatisfying (8) can be found, then writing P(x;y)/C30mp (9) Q(x;y)/C30mq (10) in equation (5) then gives P(x;y)dx/C27Q(x;y)dy/C300; (11) which is then an exact ODE. Special cases in which m can be found include x-dependent, xy-dependent, and y-dependent integrating factors. Given an inexact first-order ODE, we can also look for anINTEGRATING FACTOR m(x) so that @m @y/C300: (12) For the equation to be exact in mpand mq;the equation for a first-order nonexact ODE p@m @y/C27m@p @y/C30q@m @x/C27m@p @x(13) becomes m@p @y/C30q@m @x/C27m@p @x: (14) Solving for @m=@xgives @m @x/C30m(x)@p @y/C28@q @x q/C13f(x;y)m(x); (15) which will be integrable iff(x;y)/C13@p @y/C28@q @x q/C30f(x); (16) in which case dm m/C30f(x)dx; (17) so that the equation is integrable m(x)/C30egf(x)dx; (18) and the equation [mp(x;y)]dx/C27[mq(x;y)]dy/C300 (19) with known m(x) is now exact and can be solved as an exact ODE. Given in an exact first-order ODE, look for an INTEGRATING FACTOR m(x;y)/C30g(xy):Then @m @x/C30@g @xy: (20) @m @y/C30@g @yx: (21) Combining these two, @m @x/C30y x@m @y: (22) For the equation to be exact in mpand mq;the equation for a first-order nonexact ODE p@m @y/C27m@p @y/C30q@m @x/C27m@p @x(23) becomes @m @yp/C28y xq ! /C30@p @x/C28@p @y ! m: (24) Therefore, 1 x@m @y/C30@q @x/C28@p @y xp/C28yqm: (25) Define a new variable t(x;y)/C13xy; (26) then @t=@y/C30x;so @m @t/C30@m @y@y @t/C30@q @x/C28@p @y xp/C28yqm(t)/C13f(x;y)m(t): (27) Now, if f(x;y)/C13@q @x/C28@p @y xp/C28yq/C30f(xy)/C30f(t); (28) then @m @t/C30f(t)m(t); (29) so that m/C30egf(t)dt(30) and the equation [mp(x;y)]dx/C27[mq(x;y)]dy/C300 (31) is now exact and can be solved as an exact ODE. Given an inexact first-order ODE, assume there exists an integrating factor m/C30f(y); (32) so@m=@x/C300:For the equation to be exact in mpand mq;equation (7) becomes @m @y/C30@q @x/C28@p @y pm/C30f(x;y)m(y): (33) Now, if f(x;y)/C13@q @x/C28@p @y p/C30f(y); (34) then dm m/C30f(y)dy; (35) so that m(y)/C30egf(y)dy; (36) and the equation mp(x;y)dx/C27mq(x;y)dy/C300 (37) is now exact and can be solved as an exact ODE. Given a first-order ODE OF THE FORM yf(xy)dx/C27xg(xy)dy/C300; (38) define v/C13xy: (39) Then the solution is lnx/C30gg(v)dv c[g(v)/C28f(v)]/C27cfor g(v)"f(v) xy/C30c for g(v)/C30f(v):8 < :(40) Ifdy dx/C30F(x;y)/C30G(v); (41) where v/C13yx; (42) then letting y/C13xv (43) gives dydx/C30xd v =dx/C27v (44) xdv dx/C27v/C30G(v): (45) This can be integrated by quadratures, so lnx/C30gdv f(v)/C28v/C27cforf(v)"v (46) y/C30cxforf(v)/C30v: (47) References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, 1986. Ordinary Differential Equation * /Second- Order An ODE yƒ/C27P(x)y?/C27Q(x)y/C300 (1) has singularities for finite x/C30x0under the following conditions: (a) If either P(x)o rQ(x) diverges as x0 x0;but x/C28x0 ðÞ P(x) and x/C28x0 ðÞ2Q(x) remain finite as x0x0;then x0is called a regular or nonessential singular point. (b) If P(x) diverges faster than x/C28x0 ðÞ/C281so that x/C28x0 ðÞ P(x)0/C12asx0x0;orQ(x) diverges faster than x/C28x0 ðÞ/C282so that x/C28x0 ðÞ2Q(x)0 /C12asx0x0;then x0is called an irregular or essential singularity. Singularities of equation (1) at infinity are investi- gated by making the substitution x/C13z/C281;sodx/C30 /C28z/C282dz;giving dy dx/C30/C28z2dy dz(2) d2y dx2/C30/C28z2d dz/C28z2dy dz ! /C30/C28z2/C282zdy dz/C28z2d2y dz2 ! /C302z3dy dz/C27z4d2y dz2: (3) Then (1) becomes z4d2y dz2/C272z3/C28z2P(z)7CP7C3 dy dz/C27Q(z)y/C300: (4) Case (a): If a(z)/C132z/C28P(z) z2(5) b(z)/C13Q(z) z4(6) remain finite at x/C309/C12 (y/C300), then the point is ordinary. Case (b): If either a(z) diverges no more rapidly than 1 =zorb(z) diverges no more rapidly than 1=z2;then the point is a regular singular point. Case (c): Otherwise, the point is an irregular singular point. Morse and Feshbach (1953, pp. 667 /C1/674) give the canonical forms and solutions for second-order ODEs classified by types of singular points. For special classes of second-order linear ordinary differential equations, variable COEFFICIENTS can be transformed into constant COEFFICIENTS . Given a second-order linear ODE with variable COEFFICIENTS d2y dx2/C27p(x)dy dx/C30q(x)y/C300: (7) Define a function z/C13y(x); dy dx/C30dz dxdy dz(8) d2y dx2/C30dz dx !2d2y dz2/C27d2z dx2dy dz(9) dz dx !2d2y dz2/C27d2z dx2/C27p(x)dz dx"# dy dz/C27q(x)y/C300 (10) d2y dz2/C27d2z dx2/C27P(x)dz dx dz dx !22 6666643 777775dy dz/C27q(x) dz dx !22 666643 77775y /C13 d2y dz2/C27Ady dz/C27By/C300: (11) This will have constant COEFFICIENTS ifAandBare not functions of x. But we are free to set Bto an arbitrary POSITIVE constant for q(x)]0 by defining z as z/C13B/C281=2g[q(x)]1=2dx: (12) Thendz dx/C30B/C281=2[q(x)]1=2(13) d2z dx2/C301 2B/C281=2[q(x)]/C281=2q?(x); (14) and A/C3012B/C281=2[q(x)]/C281=2q?(x)/C27B/C281=2p(x)[q(x)]1=2 B/C281q(x) /C30q?(x)/C272p(x)q(x) 2[q(x)]3=2B1=2: (15) Equation (11) therefore becomes d2y dz2/C27q?(x)/C272p(x)q(x) 2[q(x)]3=2B1=2dy dz/C27By/C300; (16) which has constant COEFFICIENTS provided that A/C13q?(x)/C272p(x)q(x) 2[q(x)]3=2B1=2/C30[constant] : (17) Eliminating constants, this gives A?/C13q?(x)/C272p(x)q(x) 2[q(x)]3=2/C30[constant] : (18) So for an ordinary differential equation in which A?is a constant, the solution is given by solving the second-order linear ODE with constant COEFFICIENTS d2y dz2/C27Ady dz/C27By/C300 (19) forz, where zis defined as above. A linear second-order homogeneous differential equa-tion of the general form yƒ/C27P(x)y?/C27Q(x)y/C300 (20) can be transformed into standard form zƒ/C27q(x)z/C300 (21) with the first-order term eliminated using the sub-stitution lny/C13lnz/C28 1 2gP(x)dx: (22) Then y? y/C30z? z/C281 2P(x) (23) yyƒ/C28y?2 y2/C30zzƒ/C28z?2 z2/C281 2P?(x) (24) yƒ y/C28y? y !2 /C30zƒ z/C28z?2 z/C28z?2 z2/C2812P?(x) (25) yƒ y/C28z? z/C281 2P(x)"#2 /C27zƒ z/C28z?2 z/C2812P?(x) /C30z?2 z2/C28z? zP(x)/C2714P2(x)/C27zƒ z/C28z?2 z2/C2812P?(x); (26) so yƒ y/C27P(x)y? y/C27Q(x) /C30/C28z? zP(x)/C2714P2(x)/C27zƒ z/C2812P?(x)/C27P(x)z? z/C2812P(x)"# /C27Q(x): (27) Therefore, zƒ/C27Q(x)/C2812P?(x)/C2814P2(x)hi z/C13zƒ(x)/C27q(x)z/C300;(28) where q(x)/C13Q(x)/C281 2P?(x)/C2814P2(x): (29) IfQ(x)/C300;then the differential equation becomes yƒ/C27P(x)y?/C300; (30) which can be solved by multiplying by expgx P(x?)dx?7CP07CP) (31) to obtain 0/C30d dxexpgx P(x?)dx?7CP07CP)dy dx() (32) c1/C30expgx P(x?)dx?7CP07CP)dy dx(33) y/C30c1gxdx expgx P(x?)dx?7CP07CP) /C27c2: (34) If one solution /y1ðÞto a second-order ODE is known, the other /y2ðÞmay be found using the REDUCTION OF ORDER method. From A BEL’S DIFFERENTIAL EQUATION IDENTITY dW W/C30/C28P(x)dx; (35) where W/C13y1y?2/C28y?1y2 (36) gx adW W/C30gx aP?(x?)dx? (37) lnW(x) W(a)"# /C30gx aP(x?)dx? (38)W(x)/C30W(a)exp/C28gx aP(x?)dx?7CP07CP) : (39) But W/C13y1y?2/C28y?1y2/C30y2 1d dxy2 y1 ! : (40) Combining (39) and (40) yields d dxy2 y1 ! /C30W(a)exp/C28gx aP(x?)dx?7CP07CP) y2 1(41) y2(x)/C30y1(x)W(a)gx bexp/C28gx? aP(xƒ)dxƒ"# y1(x?) ½/C1382dx?:(42) Disregarding W(a);since it is simply a multiplicative constant, and the constants aand b, which will contribute a solution which is not linearly indepen- dent of y1ðÞ; y2(x)/C30y1(x)gxexp/C28gx? P(xƒ)dxƒ"# y1(x?) ½/C1382 dx?: (43) IfP(x)/C300;this simplifies to y2(x)/C30y1(x)gxdx? y1(x?) ½/C1382: (44) For a nonhomogeneous second-order ODE in whichthexterm does not appear in the function f(x;y;y?); d2y dx2/C30f(y;y?) (45) letv/C13y?;then dv dx/C30f(v;y)/C30dv dydy dx/C30vdv dy: (46) So the first-order ODE vdv dy/C30f(y;v); (47) if linear, can be solved for vas a linear first-order ODE. Once the solution is known, dydx/C30v(y) (48) gdy v(y)/C30gdx: (49) On the other hand, if yis missing from f(x;y;y?); d2y dx2/C30f(x;y?); (50) let v /C13y?; then v?/C30yƒ; and the equation reduces to v?/C30f(x; v) ; (51) which, if linear, can be solved for v as a linear first- order ODE. Once the solution is known, y/C30gv(x)dx: (52) See also ABEL’S DIFFERENTIAL EQUATION IDENTITY , ADJOINT References Arfken, G. "A Second Solution." §8.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 467 /C1/480, 1985. Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, 1986. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 667 /C1/674, 1953. Ordinary Differential Equation * /System with Constant Coefficients To solve the system of differential equations dx dt/C30Ax(t)/C27p(t); (1) where Ais a MATRIX andxandpare VECTORS , first consider the homogeneous case with p/C300:Then the solutions to dx dt/C30ax(t) (2) are given by x(t)/C30eatx(t): (3) But, by the MATRIX DECOMPOSITION THEOREM , the MATRIX EXPONENTIAL can be written as eAt/C30uDu/C281; (4) where the EIGENVECTOR MATRIX is u/C30[u1/C1/C1/C1un] (5) and the EIGENVALUE MATRIX is D/C30el1t0 /C1/C1/C1 0 0el2t/C1/C1/C1 0 nn:::0 00 /C1/C1/C1 elnt2 6643 775: (6) Now considereAtu/C30uDu/C281u/C30uD /C30u11u21 /C1/C1/C1 un1 u12u22 /C1/C1/C1 un2 nn:::n u1nu2n/C1/C1/C1 unn2 6643 775el1t0 /C1/C1/C1 0 0el2t/C1/C1/C1 0 nn:::0 00 /C1/C1/C1 elnt2 6643 775 /C30u11el1t/C1/C1/C1 un1elnt u11el1t/C1/C1/C1 un2elnt n:::n un1el1t/C1/C1/C1 un2elnt2 6643 775: (7) The individual solutions are then x i/C30eAtu7C07C) /C215ˆei/C30uielit; (8) so the homogeneous solution is x/C30Xn i/C301ciuielit; (9) where the ci/s are arbitrary constants. The general procedure is therefore 1. Find the EIGENVALUES of the MATRIX A(/l1;...,ln) by solving the CHARACTERISTIC EQUATION . 2. Determine the corresponding EIGENVECTORS u1; ...,un:/ 3. Compute xi/C13elitui (10) fori/C301, ..., n. Then the VECTORS xiwhich are REAL are solutions to the homogeneous equation. If Ais a 2 /C292 matrix, the COMPLEX vectors xj correspond to REAL solutions to the homogeneous equation given by Rxj7C07C) andIxj7C07C) :/ 4. If the equation is nonhomogeneous, find the particular solution given by x/C31(t)/C30X(t)gX/C281(t)p(t)dt; (11) where the MATRIX Xis defined by X(t)/C13x1/C1/C1/C1xn ½/C138 : (12) If the equation is homogeneous so that p(t)/C300; then look for a solution OF THE FORM x/C30jelt: (13) This leads to an equation (A /C28 lI)j /C300 ; (14) so j is an EIGENVECTOR and l an EIGENVALUE . 5. The general solution is x(t) /C30x/C31(t) /C27Xn i/C301cixi : (15) Ordinary Double Point Portions of this entry contributed by SERGEI DUZHIN Let f : R 0 R3 (or f : S1 0 R3)bea SPACE CURVE . Then a point p /C23 im(f) ƒR3 (where im(f) denotes the IMMERSION of f) is an ordinary double point if its PREIMAGE under f consists of two values t1 and t2 ; and the two TANGENT VECTORS f ?(t1) and f ?(t2) are noncol- linear. Geometrically, this means that, in a NEIGH- BORHOOD of p, the curve consists of two transverse branches. Ordinary double points are ISOLATED SIN- GULARITIES having COXETER- DYNKIN DIAGRAM of type A1 ; and also called "nodes" or "simple double points." The above plot shows the curve x3 /C28x2 /C27y2 /C300; which has an ordinary double point at the ORIGIN . A surface in complex 3-space admits at most finitely many ordinary double points. The maximum possible number of ordinary double points m(d) for a surface of degree d /C301, 2, ..., are 0, 1, 4, 16, 31, 65, 93 5 m(7) 5 104; 168 5 m(8) 5174; 216 5 m(8) 5246; 345 5 m(10) 5 360; 425 5 m(11) 5480; 576 5 m(12) 5645 ... (Sloane’s A046001; Chmutov 1992, Endraß 1995). /m(4) /C3016 was known to Kummer in 1864 (Chmutov 1992), the fact that m(5) /C3031 was proved by Beauville (1980), and m(6) /C3065 was proved by Jaffe and Ruber- man (1994). For d ]3; the following inequality holds:m(d) 51 2[d(d /C281) /C283] (Endraß 1995). Examples of ALGEBRAIC SURFACES having the maximum (known) number of ordinary double points are given in the following table. d /m(d)/Surface 34 C AYLEY CUBIC 41 6 K UMMER SURFACE 53 1 DERVISH 66 5 B ARTH SEXTIC 79 3 C HMUTOV SURFACE 8 168 E NDRAß OCTIC 9 216 C HMUTOV SURFACE 10 345 B ARTH DECIC 11 425 C HMUTOV SURFACE 12 600 S ARTI DODECIC See also ALGEBRAIC SURFACE ,BARTH DECIC,BARTH SEXTIC ,C AYLEY CUBIC ,C HMUTOV SURFACE ,C USP, DERVISH ,DOUBLE POINT ,ENDRAß OCTIC,ISOLATED SINGULARITY ,KUMMER SURFACE ,RATIONAL DOUBLE POINT ,SARTI DODECIC References Basset, A. B. "The Maximum Number of Double Points on a Surface." Nature 73, 246, 1906. Beauville, A. "Sur le nombre maximum de points doubles d’une surface dans P3(/m(5)/C3031):/"Journe ´es de ge ´ome´trie alge´brique d’Angers (1979). Sijthoff & Noordhoff, pp. 207 /C1/ 215, 1980. Chmutov, S. V. "Examples of Projective Surfaces with Many Singularities." J. Algebraic Geom. 1, 191/C1/196, 1992. Endraß, S. "Surfaces with Many Ordinary Nodes." http:// enriques.mathematik.uni-mainz.de/kon/docs/Eflae- chen.shtml. Endraß, S. "Fla ¨chen mit vielen Doppelpunkten." DMV- Mitteilungen 4,1 7/C1/20, Apr. 1995. Endraß, S. Symmetrische Fla ¨che mit vielen gewo ¨hnlichen Doppelpunkten. Ph.D. thesis. Erlangen, Germany, 1996. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 12 /C1/13, 1986. Jaffe, D. B. and Ruberman, D. "A Sextic Surface Cannot have 66 Nodes." J. Algebraic Geom. 6, 151/C1/168, 1997. Kreiss, H. O. "U ¨ber syzygetische Fla ¨chen." Ann. Math. 41, 105/C1/111, 1955. Miyaoka, Y. "The Maximal Number of Quotient Singula- rities on Surfaces with Given Numerical Invariants." Math. Ann. 268, 159/C1/171, 1984. Sloane, N. J. A. Sequences A046001 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Togliatti, E. G. "Sulle superficie algebriche col massimo numero di punti doppi." Rend. Sem. Mat. Torino 9,4 7/C1 / 59, 1950. Varchenko, A. N. "On the Semicontinuity of Spectrum and an Upper Bound for the Number of Singular Points on a Projective Hypersurface." Dokl. Acad. Nauk SSSR 270, 1309 /C1/1312, 1983. Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 56 /C1/57, 1978. Ordinary Generating Function GENERATING FUNCTION Ordinary Line Given an arrangement of n ]3 points, a LINE contain- ing just two of them is called an ordinary line. Kelly and Moser (1958) proved that at least 3n =7 lines must be ordinary (Guy 1989, p. 903). See also COLINEAR ,G ENERAL POSITION ,INCIDENT , NEAR-PENCIL ,ORDINARY POINT ,SPECIAL POINT ,SYL- VESTER GRAPH References Coxeter, H. S. M. "A Problem of Collinear Points." Amer. Math. Monthly 55,26/C1/28, 1948. Coxeter, H. S. M. The Real Projective Plane, 3rd ed. Cam- bridge, England: Cambridge University Press, 1993. de Bruijn, N. G. and Erdos, P. "On a Combinatorial Problem." Hederl. Adad. Wetenach. 51, 1277 /C1/1279, 1948. Dirac, G. A. "Collinearity Properties of Sets of Points." Quart. J. Math. 2, 221 /C1/227, 1951. Erdos, P. "Problem 4065." Amer. Math. Monthly 51, 169, 1944. Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/909, 1989. Kelly, L. M. and Moser, W. O. J. "On the Number of Ordinary Lines Determined by n Points." Canad. J. Math. 1, 210 /C1/219, 1958. Lang, D. W. "The Dual of a Well-Known Theorem." Math. Gaz. 39, 314, 1955. Motzkin, T. "The Lines and Planes Connecting the Points of a Finite Set." Trans. Amer. Math. Soc. 70, 451 /C1/463, 1951. Sylvester, J. J. "Mathematical Question 11851." Educa- tional Times 59, 98, 1893. Ordinary Point A POINT which lies on at least one ORDINARY LINE is called an ordinary point, or sometimes a REGULAR POINT . See also ORDINARY LINE,REGULAR POINT ,SPECIAL POINT ,SYLVESTER GRAPH References Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/909, 1989. Ordinary Surface A surface which is homeomorphic to a finite collection of spheres, each with a finite number of HANDLES , cross-handles, CROSS-CAPS , and PERFORATIONS .A preliminary version of the CLASSIFICATION THEOREM OF SURFACES states that every surface is ordinary.References Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/399, 1999. Ordinate The y- (vertical) coordinate of a point in a two dimensional coordinate system. Physicists and as- tronomers sometimes use the term to refer to the axis itself instead of the distance along it. See also ABSCISSA , X-AXIS, Y-AXIS, Z-AXIS Ore Graph A GRAPH G in which the sums of the degrees of nonadjacent vertices is greater than the number of nodes n for all subsets of nonadjacent vertices (Ore 1960; Skiena 1990, p. 197). Ore graphs are always HAMILTONIAN , and a HAMILTONIAN CIRCUIT in such a graph can be constructed in polynomial time (Bondy and Chva´tal 1976; Skiena 1990, p. 197). The numbers of Ore graphs on n /C305, 6, ... nodes are 2, 6, 32, ..., the first few of which are illustrated above. See also HAMILTONIAN CIRCUIT ,HAMILTONIAN GRAPH References Bondy, J. A. and Chva´tal, V. "A Method in Graph Theory." Disc. Math. 15, 111 /C1/136, 1976. Ore, O. "A Note on Hamiltonian Circuits." Amer. Math. Monthly 67, 55, 1960. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Ore Number HARMONIC DIVISOR NUMBER Ore’s Conjecture Define the HARMONIC MEAN of the DIVISORS of n H(n) /C13t(n) P djn1 d; where t(n) is the TAU FUNCTION (the number of DIVISORS of n). If n is a PERFECT NUMBER , H(n)is an INTEGER . Ore conjectured that if n is ODD, then H(n) is not an INTEGER . This implies that no ODD PERFECT NUMBERS exist. See also HARMONIC DIVISOR NUMBER ,H ARMONIC MEAN,PERFECT NUMBER ,TAU FUNCTION Ore’s Theorem If a GRAPH G has n ]3 VERTICES such that every pair of the n VERTICES which are not joined by an EDGE has a sum of VALENCES which is ]n ; then G is HAMILTONIAN . See also HAMILTONIAN GRAPH Orientable Surface A REGULAR SURFACE M ƒRn is called orientable if each TANGENT SPACE Mphas a COMPLEX STRUCTURE Jp : Mp 0 Mpsuch that p 0 Jpis a continuous func- tion. See also NONORIENTABLE SURFACE ,REGULAR SUR- FACE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 318, 1997. Orientation (Bundle) A real VECTOR BUNDLE p : E 0 M has an orientation if there exists a covering by TRIVIALIZATIONS Ui /C29Rk such that the TRANSITION FUNCTIONS are ORIENTA- TION preserving. Alternatively, there exists a section of the PROJECTIVIZATION of the top exterior power of the bundle, PR(fflkE) : A bundle is called orientable if there exists an orientation. Hence a bundle E of RANK k is orientable iff fflkE is a TRIVIAL LINE BUNDLE . An orientation of the TANGENT BUNDLE is equivalent to an orientation on the BASE MANIFOLD . Not all bundles are orientable, as can be seen by the TANGENT BUNDLE of the MO¨ BIUS STRIP . The nontrivial LINE BUNDLE on the circle is also not orientable. See also BUNDLE ,ORIENTATION (MANIFOLD ), ORIEN- TATION (VECTOR SPACE ), VECTOR BUNDLE References Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 2nd ed. Houston, TX: Publish or Perish, pp. 273 /C1/383, 1999. Orientation (Graph) An orientation of an UNDIRECTED GRAPH G is an assignment of exactly one direction to each of the edges of G. Only connected, bridgeless graphs can have a strong orientation (Robbins 1939; Skiena 1990, p. 174). An oriented COMPLETE GRAPH is called a TOURNAMENT . See also DIRECTED GRAPH ,TOURNAMENT References Robbins, H. E. "A Theorem on Graphs with an Application to a Problem of Traffic Control." Amer. Math. Monthly 46, 281 /C1/283, 1939.Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Orientation (Manifold) An orientation on an n-dimensional MANIFOLD is given by a nowhere vanishing DIFFERENTIAL N- FORM . Alternatively, it is an ORIENTATION for the TANGENT BUNDLE . If an orientation exists on M, then M is called orientable. Not all MANIFOLDS are orientable, as exemplified by the MO¨ BIUS STRIP and the KLEIN BOTTLE , illustrated above. However, an (n /C281)/-dimensional SUBMANIFOLD of Rn is orientable IFF it has a unit normal vector field. The choice of unit determines the orientation of the submanifold. For example, the SPHERE S2 is orienta- ble. Some types of manifolds are always orientable. For instance, COMPLEX MANIFOLDS , including VARIETIES , and also SYMPLECTIC MANIFOLDS are orientable. Also, any unoriented manifold has a double COVER which is oriented.A map f : M 0 N between oriented manifolds of the same dimension is called orientation preserving if the volume form on Npulls back to a positive volume form on M. Equivalently, the differential dfmaps an ORIENTED BASIS inTMto an ORIENTED BASIS inTN. See also DIFFERENTIAL FORM,O RIENTATION (BUN- DLE), ORIENTATION (VECTOR SPACE ), VOLUME FORM References Berger, M. Differential Geometry. New York: Springer- Verlag, pp. 146 /C1/237, 1988. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 2nd ed. Houston, TX: Publish or Perish, pp. 273 /C1/383, 1999. Sternberg, S. Differential Geometry. New York: Chelsea, pp. 14 /C1/30, 1983. Orientation (Plane Curve) A curve has positive orientation if a region R is on the left when traveling around the outside of R, or on the right when traveling around the inside of R. Orientation (Vector Space) An ordered BASIS v1 ; ...; vnfor a finite-dimensional VECTOR SPACE V defines an orientation. Another basis wi /C30Avigives the same orientation if the matrix A has a positive determinant, in which case the basis wi is called oriented. Any VECTOR SPACE has two possible orientations since the DETERMINANT of an INVERTIBLE MATRIX is either positive or negative. For example, in R2 ;fe1 ; e2 g is one orientation and fe2 ; e1 g/C2fe1 ;/C28e2 g is the other orientation. In three dimensions, the CROSS PRODUCT uses the RIGHT-HAND RULE by convention, reflecting the use of the canonical orientation fe1 ; e2 ;e3 g as e1 /C29e2 /C30e3 :/ An orientation can be given by a nonzero element in the top exterior power of V, i.e. fflnV : For example, e1ffle2ffle3gives the canonical orientation on R3 and /C28e1ffle2ffle3 gives the other orientation. Some special vector space structures imply an orien- tation. For example, if v is a SYMPLECTIC FORM on V, of dimension 2n; then vn gives an orientation. Also, if V is a COMPLEX VECTOR SPACE , then as a real vector space of dimension 2n; the COMPLEX STRUCTURE gives an orientation. See also ORIENTATION (MANIFOLD ), ORIENTATION (VECTOR BUNDLE ) Orientation (Vectors) Let u be the ANGLE between two VECTORS .If0B u B p; the VECTORS are positively oriented. If p B u B2 p; the vectors are negatively oriented. Two vectors in the plane x1 x27CP07CP) andy1 y27CP07CP) are positively oriented IFF the DETERMINANT D /C13 x1y1 x2y27C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P> 0 ; and are negatively oriented IFF the DETERMINANT D B0.Orientation-Preserving A nonsingular linear MAP A : Rn 0 Rn is orientation- preserving if (A) > 0 :/ See also ORIENTATION- REVERSING ,ROTATION Orientation-Reversing A nonsingular linear MAP A : Rn 0 Rn is orientation- reversing if det(A) B0 :/ See also ORIENTATION- PRESERVING Oriented Graph A DIRECTED GRAPH having no symmetric pair of directed edges. See also DIRECTED GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 10, 1994. Oriented Knot See also KNOT,ORIENTED LINK References Cerf, C. "Atlas of Oriented Knots and Links." Topology Atlas Invited Contributions 3, No. 2, 1 /C1/32, 1998. http://at.yor- ku.ca/t/a/i/c/31.htm. Oriented Link See also LINK,ORIENTED KNOT References Cerf, C. "Atlas of Oriented Knots and Links." Topology Atlas Invited Contributions 3, No. 2, 1 /C1/32, 1998. http://at.yor- ku.ca/t/a/i/c/31.htm. Oriented Matroid The oriented matroid of a finite CONFIGURATION of points extracts relative position and orientation information from the CONFIGURATION . An oriented matroid can be described roughly as a MATROID in which every basis is equipped with an orientation (Richter-Gebert and Ziegler 1997, p. 112). See also CONFIGURATION ,MATROID References Bjo¨rner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; and Ziegler, G. Oriented Matroids, 2nd ed. Cambridge, Eng- land: Cambridge University Press, 1999. Richter-Gebert, J. and Ziegler, G. M. "Oriented Matroids." Ch. 6 in Handbook of Discrete and Computational Geo- metry (Ed. J. E. Goodman and J. O’Rourke). Boca Raton, FL: CRC Press, pp. 111 /C1/132, 1997. Origami The Japanese art of paper folding. CUBE DUPLICATION and TRISECTION of an ANGLE can be solved using origami, although they cannot be solved using the traditional rules for GEOMETRIC CONSTRUCTIONS . There are a number of recent very powerful results in origami mathematics. A very general result states that any planar straight-line drawing may be cut out of one sheet of paper by a single straight cut, after appropriate folding (Demaine, Demaine, and Lubiw, 1998, 1999, O’Rourke 1999). Another result is that any polyhedron may be wrapped with a sufficiently large square sheet of paper. This implies that any connected, planar, polygonal region may be covered by a flat origami folded from a single square of paper. Moreover, and 2-coloring of the faces may be realized with paper whose two sides are those colors (De- maine, Demaine, and Mitchell 1999, O’Rourke 1999). See also FOLDING ,GEOMETRIC CONSTRUCTION ,M AP FOLDING ,STAMP FOLDING ,STOMACHION ,TANGRAM References Andersen, E. "Origami on the Web." http://www.netspa- ce.org/users/ema/oriweb.html. Biddle, S. and Biddle, M. The New Origami. New York: St. Martin’s Press, 1993. Brill, D. Brilliant Origami: A Collection of Original Designs. Japan Pub., 1996. Cerceda, A. and Palacios, V. Fascinating Origami: 101 Models by Adolfo Cerceda. New York: Dover, 1997. Demaine, E. D.; Demaine, M. L.; and Lubiw, A. "Folding and Cutting Paper." In Proc. Japan Conf. Discrete Comput. Geom. New York: Springer-Verlag, 1998. Demaine, E. D.; Demaine, M. L.; and Lubiw, A. "Folding and One Straight Cut Suffice." In Proc. 10th Ann. ACM-SIAM Sympos. Discrete Alg. (SODA’99). Baltimore, MD, pp. 891 /C1/892, Jan. 1999. Demaine, E. D.; Demaine, M. L.; and Mitchell, J. S. B. "Folding Flat Silhouettes and Wrapping Polyhedral Packages: New Results in Computation Origami." In Proc. 15th Ann. ACM Sympos. Comput. Geom. Miami Beach, FL, pp. 105 /C1/114, June 1999. Eppstein, D. "Origami." http://www.ics.uci.edu/~eppstein/ junkyard/origami.html. Fuse, T. Unit Origami: Multidimensional Transformations. Japan Pub., 1990. ISBN: 0870408526. Geretschla ¨ger, R. "Euclidean Constructions and the Geome- try of Origami." Math. Mag. 68, 357 /C1/371, 1995. Gurkewitz, R. "Rona’s Modular Origami Polyhedra Page." http://www.wcsu.ctstateu.edu/~gurkewitz/homepa- ge.html. Gurkewitz, R. and Arnstein, B. 3-D Geometric Origami. New York: Dover, 1996. Harbin, R. Origami Step-By-Step. New York: Dover, 1998.Harbin, R. Secrets of Origami: The Japanese Art of Paper Folding. New York: Dover, 1997. Kasahara, K. Origami Omnibus: Paper-Folding for Every- one. Tokyo: Japan Publications, 1988. Kasahara, K. and Takahara, T. Origami for the Connois- seur. Tokyo: Japan Publications, 1987. Montroll, J. Origami Inside-Out. New York: Dover, 1993. Montroll, J. Origami Sculptures, 2nd ed. Antroll Pub., 1991. O’Rourke, J. "Computational Geometry Column 36." SI- GACT News 30,35/C1/38, Sep. 1999. Palacios, V. Fascinating Origami: 101 Models by Alfredo Cerceda. New York: Dover, 1997. Pappas, T. "Mathematics & Paperfolding." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 48 /C1/50, 1989. Row, T. S. Geometric Exercises in Paper Folding. New York: Dover, 1966. Simon, L.; Arnstein, B.; and Gurkewitz, R. Modular Origami Polyhedra. New York: Dover, 1999. by Takahama, T. The Complete Origami Collection. Japan Pub., 1997. Tomoko, F. Unit Origami. Tokyo: Japan Publications, 1990. Wu, J. "Joseph Wu’s Origami Page." http://www.origami.- vancouver.bc.ca/. Origin The central point (r /C300) in POLAR COORDINATES ,or the point with all zero coordinates (0, ..., 0) in CARTESIAN COORDINATES . In 3-D, the X-AXIS , Y-AXIS , and Z-AXIS meet at the origin. See also OCTANT ,QUADRANT , X-AXIS, Y-AXIS, Z-AXIS Ornstein’s Theorem An important result in ERGODIC THEORY . It states that any two "Bernoulli schemes" with the same MEASURE-THEORETIC ENTROPY are MEASURE-THEORE- TICALLY ISOMORPHIC . See also ERGODIC THEORY ,ISOMORPHISM ,M EASURE THEORY Orr’s Theorem If (1/C28z)a/C27b/C27g/C281=2 2F1(2a;2b;2g;z)/C30X anzn;(1) where2F1(a;b;c;z)i sa HYPERGEOMETRIC FUNC- TION , then 2F1(a;b;g;z)2F1g/C28a/C271 2;g/C28b/C2712;g/C271;z7C)67C)7 /C30X (g/C271=2)n=(g/C271)nanzn: (2) Furthermore, if (1/C28z)a/C27b/C28g/C281=2 2F1(2a/C281;2b;2g/C281;z) /C30X anzn; (3) then 2F1( a; b; g; z) G g /C28 a /C271 2 ; g /C28 b /C2812; g; z7C)67C)7 /C30X (g/C281 =2)n =( g)nanzn ; (4) where G(z) is the GAMMA FUNCTION (Bailey 1935, p. 84). References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, 1935. Cayley, A. "On a Theorem Relating to Hypergeometric Series." Philos. Mag. 16, 356 /C1/357, 1858. Reprinted in Collected Papers, Vol. 3, pp. 268 /C1/269. Edwards, D. "An Expansion in Factorials Similar to Van- dermonde’s Theorem, and Applications." Messenger Math. 52, 129 /C1/136, 1923. Orr, W. M. "Theorems Relating to the Product of Two Hypergeometric Series." Trans. Cambridge Philos. Soc. 17,1/C1/15, 1899. Watson, G. N. "The Theorems of Clausen and Cayley on Products of Hypergeometric Functions." Proc. London Math. Soc. 22, 163 /C1/170, 1924. Whipple, F. J. W. "Algebraic Proofs of the Theorems of Cayley and Orr Concerning the Products of Certain Hypergeometric Series." J. London Math. Soc. 2,85/C1/90, 1927. Whipple, F. J. W. "On a Formula Implied in Orr’s Theorems Concerning the Products of Hypergeometric Series." J. London Math. Soc. 4,48/C1/50, 1929. Orr-Sommerfeld Differential Equation The ORDINARY DIFFERENTIAL EQUATION 1 i aRd2 dx2 /C28 a2 !2 y /C27 [f(x) /C28c]d2 dx2 /C28 a2 ! /C28f ??(x)() y /C300: References Herron, I. H. "The Orr-Sommerfeld Equations on Infinite Intervals." SIAM Rev. 29, 597 /C1/620, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.Orthic Axis TheDHAHBHCbe the ORTHIC TRIANGLE of a TRIANGLE DABC:Then each side of each triangle meets the three sides of the other triangle, and the points of intersec- tion lie on a line O1O2O3called the orthic axis. See also ORTHIC TRIANGLE References Honsberger, R. §13.2 (ii) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 151, 1995. Orthic Triangle Given a TRIANGLE DA1A2A3;the TRIANGLE DH1H2H3 with VERTICES at the feet of the ALTITUDES (perpen- diculars from a point to the sides) is called the orthictriangle. The three lines A iHiare CONCURRENT at the ORTHOCENTER HofDA1A2A3:The orthic triangle is therefore the PEDAL TRIANGLE with respect to H. Given a triangle DA1A2A3 ; construct the orthic trian- gle DH1H2H3and determine the SYMMEDIAN POINTS K1 ; K2 ; and K3 of DA1H2H3 ;DH1A2H3 ; and DH1H2A3 ; respectively. Then the SYMMEDIANS K1 ; K2 ; and K3 of each corner triangle pass through the MIDPOINTS M1 ; M2 ; and M3 of the corresponding sides of the original triangle DA1A2A3 (Honsberger 1995, p. 75). Moreover, the lines K1M1 ; K2M2 ; and K3M3CONCUR in the CENTROID of DA1A2A3 :/ The sides of the orthic triangle are parallel to the tangents to the CIRCUMCIRCLE at the vertices (John- son 1929, p. 172). The centroid of the orthic triangle has TRIANGLE CENTER FUNCTION a /C30a2 cos(B /C28C) (Casey 1893, Kimberling 1994). The ORTHOCENTER of the orthic triangle has TRIANGLE CENTER FUNCTION a /C30cos(2 A)cos(B /C28C) (Casey 1893, Kimberling 1994). The SYMMEDIAN POINT of the orthic triangle has TRIANGLE CENTER FUNCTION a /C30tan A cos(B /C28C) (Casey 1893, Kimberling 1994). See also ALTITUDE ,FAGNANO’S PROBLEM ,ORTHOCEN- TER,P EDAL TRIANGLE ,SCHWARZ’S TRIANGLE PRO- BLEM ,SYMMEDIAN POINT References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 9, 1893. Coxeter, H. S. M. and Greitzer, S. L. "The Orthic Triangle." §1.6 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 16 /C1/18, 1967. Honsberger, R. "The Orthic Triangle." §2.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 21 /C1/25, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Orthobicupola A BICUPOLA in which the bases are in the same orientation. See also PENTAGONAL ORTHOBICUPOLA ,S QUARE ORTHOBICUPOLA ,TRIANGULAR ORTHOBICUPOLA Orthobirotunda ABIROTUNDA in which the bases are in the same orientation. Orthocenter The intersection Hof the three ALTITUDES of a TRIANGLE is called the orthocenter. The name was invented by Besant and Ferrers in 1865 while walk- ing on a road leading out of Cambridge, England inthe direction of London (Satterly 1962). The TRI- LINEAR COORDINATES of the orthocenter are cosBcosC: cos CcosA: cos AcosB: (1) If the TRIANGLE is not a RIGHT TRIANGLE , then (1) can be divided through by cos AcosBcosCto give secA: sec B: sec C: (2) If the triangle is ACUTE , the orthocenter is in the interior of the triangle. In a RIGHT TRIANGLE , the orthocenter is the VERTEX of the RIGHT ANGLE . When the vertices of a triangle are combined with itsorthocenter, any one of the points is the orthocenter of the other three, as first noted by Carnot (Wells 1991). These four points therefore form an ORTHO- CENTRIC SYSTEM . The CIRCUMCENTER O and orthocenter H are ISO- GONAL CONJUGATES . The orthocenter lies on the EULER LINE. The orthocenter and NAGEL POINT form a DIAMETER of the FUHRMANN CIRCLE . Relationships involving the orthocenter include the following: a2 1 /C27a22 /C27a23 /C27A1H2 /C27A2H2 /C27A3H2 /C3012R2 (3) A1H /C27A2H /C27A3H /C302(r /C27R) ; (4) A1H2 /C27A2H2 /C27A3H2 /C304R2 /C284Rr ; (5) where r is the INRADIUS and R is the CIRCUMRADIUS (Johnson 1929, p. 191). Any HYPERBOLA circumscribed on a TRIANGLE and passing through the orthocenter is RECTANGULAR , and has its center on the NINE-POINT CIRCLE (Falisse 1920, Vandeghen 1965). See also CENTROID (TRIANGLE ), CIRCUMCENTER , DROZ-FARNY CIRCLES ,EULER LINE,FUHRMANN CIR- CLE,INCENTER ,O RTHIC TRIANGLE ,O RTHOCENTRIC COORDINATES ,O RTHOCENTRIC QUADRILATERAL , ORTHOCENTRIC SYSTEM ,POLAR CIRCLE References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed. New York: Barnes and Noble, pp. 165 /C1/172, 1952. Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 622, 1970. Coxeter, H. S. M. and Greitzer, S. L. "More on the Altitudes and Orthocenter of a Triangle." Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 36 /C1/40, 1967. Dixon, R. Mathographics. New York: Dover, p. 57, 1991. Falisse, V. Cours de ge´ome´trie analytique plane. Brussels, Belgium: Office de Publicite ´, 1920. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 165 /C1/172 and 191, 1929.Honsberger, R. "The Orthocenter." Ch. 2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 17 /C1/26, 1995. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Kimberling, C. "Orthocenter." http://cedar.evansville.edu/ ~ck6/tcenters/class/orthocn.html. Satterly, J. "2997. Relations Between the Portions of the Altitudes of a Plane Triangle." Math. Gaz. 45,50/C1/51, 1962. Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091 /C1/1094, 1965. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 165, 1991. Orthocentric Coordinates Coordinates defined by an ORTHOCENTRIC SYSTEM . See also TRILINEAR COORDINATES Orthocentric Line The common axis of the three altitude planes of a TRIHEDRON . See also TRIHEDRON References Altshiller-Court, N. "The Orthocentric Line." §2.1 in Modern Pure Solid Geometry. New York: Chelsea, pp. 27 /C1/30, 1979. Orthocentric Quadrangle Given four points, A, B, C, and H, let H be the ORTHOCENTER of DABC : Then A is the ORTHOCENTER DHBC ; B is the ORTHOCENTER of DHAC ; and C is the ORTHOCENTER of DHAB : The configuration ABCH is called an orthocentric quadrangle. See also ORTHOCENTER ,ORTHOCENTRIC QUADRILAT- ERAL ,ORTHOCENTRIC SYSTEM References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 39, 1967. Orthocentric Quadrilateral If two pairs of opposite sides of a COMPLETE QUAD- RILATERAL are pairs of PERPENDICULAR lines, the QUADRILATERAL is said to be orthocentric. In such a case, the remaining sides are also PERPENDICULAR . See also ORTHOCENTRIC QUADRANGLE ,O RTHO- CENTRIC SYSTEM Orthocentric System A set of four points, one of which is the ORTHOCENTER of the other three. In an orthocentric system, each point is the ORTHOCENTER of the TRIANGLE of the other three, as illustrated above (Coxeter and Greit- zer 1967, p. 39). The INCENTER and EXCENTERS of a TRIANGLE are an orthocentric system. The centers of the CIRCUMCIRCLES of the points in an orthocentric system form another orthocentric system congruent to the first, and are the reflection of the original points in their common NINE-POINT CENTER (Wells 1991). The centroids of the points in an orthocentric system form another orthocentric system similar to the first, but one third the size (Wells 1991). The sum of the squares of any nonadjacent pair of connectors of an orthocentric system equals the square of the DIAMETER of the CIRCUMCIRCLE . Ortho- centric systems are used to define ORTHOCENTRIC COORDINATES . The four CIRCUMCIRCLES of points in an orthocentric system taken three at a time (illustrated above) have equal RADIUS (Wells 1991). The four triangles of an orthocentric system have a common NINE-POINT CIRCLE , illustrated above. Furthermore, this circle is tangent to the 16 incircles and excircles of the four triangles (Wells 1991). See also ANGLE BISECTOR ,C IRCUMCIRCLE ,C YCLIC QUADRANGLE ,NINE-POINT CIRCLE ,ORTHIC TRIANGLE , ORTHOCENTER ,ORTHOCENTRIC QUADRANGLE ORTHO- CENTRIC QUADRILATERAL ,POLAR CIRCLE ,RIGHT HY- PERBOLA References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed. New York: Barnes and Noble, pp. 109 /C1/114, 1952. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 165 /C1/176, 1929. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 165, 1991. Orthocupolarotunda A CUPOLAROTUNDA in which the bases are in the same orientation. See also GYROCUPOLAROTUNDA ,PENTAGONAL ORTHO- CUPOLARONTUNDA Orthodrome GREAT CIRCLE Orthogonal Array An orthogonal array OA(k, s)isa k /C29s2 ARRAY with entries taken from an s-set S having the property that in any two rows, each ordered pair of symbols from S occurs exactly once. References Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 111, 1996. Hedayat, A. S.; Sloane, N. J. A.; and Stufken, J. Orthogonal Arrays: Theory and Applications. New York: Springer- Verlag, 1999. Orthogonal Basis A BASIS of vectors x which satisfy xjxk /C30Cjk djk xmxn /C30C m n dm n ; where Cjk ; Cm nare constants (not necessarily equal to 1) and djk is the KRONECKER DELTA . See also BASIS,O RTHONORMAL BASIS,S PECTRUM (OPERATOR ) Orthogonal Circles Orthogonal circles are ORTHOGONAL CURVES , i.e., theycut one another at RIGHT ANGLES . Two CIRCLES with equations x2 /C27y2 /C272gx /C272fy /C27c /C300 (1) x2 /C27y2 /C272g ?x /C272f ?y /C27c ?/C300 (2) are orthogonal if 2gg?/C272ff ?/C30c /C27c ?: (3) The RADICAL LINES of three given circles concur in the RADICAL CENTER R. If a circle with center R cuts any one of the three circles orthogonally, it cuts all three orthogonally. This circle is called the orthogonal circle (or RADICAL CIRCLE ) of the system. The orthogonal circle is the LOCUS of a point whose POLARS with respect to the three given circles are concurrent (Lachlan 1893, p. 237). A theorem of Euclid states that, for the orthogonal circles in the above diagram, OP /C29OQ /C30OT2 (4) (Dixon 1991, p. 65). References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 42, 1888. Dixon, R. Mathographics. New York: Dover, pp. 65 /C1/66, 1991. Durell, C. V. "Orthogonal Circles." Ch. 8 in Modern Geome- try: The Straight Line and Circle. London: Macmillan, pp. 88 /C1/92, 1928. Euclid. The Thirteen Books of the Elements, 2nd ed. unabridged, Vol. 3: Books X-XIII. New York: Dover, p. 36, 1956. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, 1893. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., p. xxiv, 1995. Orthogonal Complement The orthogonal complement of a SUBSPACE WofRnis denoted W/C222:/ See also FREDHOLM’S THEOREM ,O RTHOGONAL DE- COMPOSITION Orthogonal Coordinate System A system of CURVILINEAR COORDINATES in which each family of surfaces intersects the others at right angles. Orthogonal CURVILINEAR COORDINATES satisfy the additional constraint that ˆui/C215 ˆuj /C30 dij : (1) Therefore, the LINE ELEMENT becomes ds2 /C30dr /C215 dr /C30h2 1 du21 /C27h22 du22 /C27h23 du23 (2) and the VOLUME ELEMENT is dV /C30 (h1 ˆu1 du1) /C215 (h2 ˆu2 du2) /C29(h3 ˆu3 du3) jj /C30h1h2h3 du1 du2 du3 /C30@r @u1/C215@r @u2/C29@r @u37C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P du 1 du2 du3 /C30@x @u1@x @u2@x @u3 @y @u1@y @u2@y @u3 @z @u1@z @u2@z @u37C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)Pdu 1 du2 du3 /C30@(x; y; z) @(u1 ; u2 ; u3)7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P du 1 du2 du3 ; (3) where the latter is the JACOBIAN . For surfaces of first degree, the only 3-D coordinate system of surfaces having orthogonal intersections is CARTESIAN COORDINATES (Moon and Spencer 1988, p. 1). Including degenerate cases, there are 11 sets of quadratic surfaces having orthogonal coordinates. Furthermore, LAPLACE’S EQUATION and the HELM- HOLTZ DIFFERENTIAL EQUATION are separable in all of these coordinate systems (Moon and Spencer 1988, p. 1). Planar orthogonal curvilinear coordinate systems of degree two or less include 2-D CARTESIAN COORDI- NATES and POLAR COORDINATES . 3-D orthogonal curvilinear coordinate systems of degree two or less include BIPOLAR CYLINDRICAL COORDINATES , BISPHERICAL COORDINATES , 3-D CARTE- SIAN COORDINATES , CONFOCAL ELLIPSOIDAL COORDI- NATES , CONFOCAL PARABOLOIDAL COORDINATES , CONICAL COORDINATES , CYCLIDIC COORDINATES , CY- LINDRICAL COORDINATES , ELLIPSOIDAL COORDINATES , ELLIPTIC CYLINDRICAL COORDINATES , OBLATE SPHER- OIDAL COORDINATES , PARABOLIC COORDINATES , PARA- BOLIC CYLINDRICAL COORDINATES , PARABOLOIDAL COORDINATES , PROLATE SPHEROIDAL COORDINATES , SPHERICAL COORDINATES , and TOROIDAL COORDI- NATES . These are degenerate cases of the CONFOCAL ELLIPSOIDAL COORDINATES .Orthogonal coordinate systems can also be built from fourth-order (in particular, CYCLIDIC COORDINATES ) and higher surfaces (Boˆcher 1894), but are generally less important in solving physical problems than are quadratic surfaces (Moon and Spencer 1988, p. 1). See also CHANGE OF VARIABLES THEOREM ,C URL, CURVILINEAR COORDINATES ,CYCLIDIC COORDINATES , DIVERGENCE ,GRADIENT ,JACOBIAN ,LAPLACIAN ,SKEW COORDINATE SYSTEM References Arfken, G. "Curvilinear Coordinates" and "Differential Vector Operators." §2.1 and 2.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 86 /C1/90 and 90 /C1/94, 1985. Boˆcher, M. U¨ ber die Reihenentwicklungen der Potentialthe- orie. Leipzig, Germany: Teubner, 1894. Darboux, G. Sur une classe remarquable de courbes et de surfaces alge´briques et sur la the´orie des imaginaires. Paris: Hermann, 1896. Darboux, G. Lec¸ons sur les systemes orthogonaux et les coordonne ´es curvilignes. Paris: Gauthier-Villars, 1910. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1084 /C1/1088, 2000. Lame ´,G. Lec¸ons sur les coordonne ´es curvilignes et leurs diverses applications. Paris: Mallet-Bachelier, 1859. Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1 /C1/48, 1988. Morse, P. M. and Feshbach, H. "Curvilinear Coordinates" and "Table of Properties of Curvilinear Coordinates." §1.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 21 /C1/31 and 115 /C1/117, 1953. Mu¨ller, E. "Die verschiedenen Koordinatensysteme." S. 596 inEncyk. Math. Wissensch., Bd. III.1.1. Leipzig, Ger- many: Teubner, 1907 /C1/1910. See also CURVILINEAR COORDINATES Orthogonal Curves Two intersecting curves which are PERPENDICULAR at their INTERSECTION are said to be orthogonal. Orthogonal Decomposition This entry contributed by V IKTOR BENGTSSON The orthogonal decomposition of a VECTOR yinRnis the sum of a vector in a SUBSPACE WofRnand a vector in the ORTHOGONAL COMPLEMENT W/C222toW. The orthogonal decomposition theorem states that if Wis a SUBSPACE ofRn;then each vector yinRncan be written uniquely in the form y/C30ˆy/C27x; where ˆyis in Wand zis in W/C222:In fact, if fu1;u2;...;upgis any ORTHOGONAL BASIS ofW, then ˆy /C30y /C215 u1 u1/C215 u1u1 /C27y /C215 u2 u2/C215 u2u2 /C27.../C27y /C215 up up /C215 upup ; and z /C30y /C28ˆy:/ Geometrically, ˆy is the ORTHOGONAL PROJECTION of y onto the SUBSPACE W and z is a vector orthogonal to ˆy/ See also FREDHOLM’S THEOREM ,LUD ECOMPOSITION , QR DECOMPOSITION References Golub, G. and van Loan, C. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996. Orthogonal Functions Two functions f(x) and g(x) are orthogonal on the interval a 5x 5b if /C142f(x) ½g(x)/C143/C13gb af(x)g(x)dx/C300: See also ORTHOGONAL POLYNOMIALS ,ORTHONORMAL FUNCTIONS Orthogonal Group For every DIMENSION n/C210, the orthogonal group O(n) is the GROUP ofn/C29nORTHOGONAL MATRICES . These matrices form a GROUP because they are CLOSED under multiplication and taking inverses. Thinking of a matrix as given by n2coordinate functions, the set of matrices is identified with Rn2: The orthogonal matrices are the solutions to the n2 equations AAT/C30I; (1) where Iis the IDENTITY MATRIX , which are redundant. Only n(n/C271)=2 of these are independent, leaving n(n/C281)=2 "free variables." In fact, the orthogonal group is a smooth n(n/C281)=2 dimensional SUBMANI- FOLD . Because the orthogonal group is a group and a manifold, it is a L IE GROUP .O(n) has a TANGENT SPACE at the identity that is the L IE ALGEBRA ofSKEW SYMMETRIC MATRICES o(n):In fact, the orthogonal group is a COMPACT LIE GROUP . The DETERMINANT of an ORTHOGONAL MATRIX is either 1 or /C281, and so the orthogonal group has two COMPONENTS . The component containing the identity is a the SPECIAL ORTHOGONAL GROUP SO(n):For example, The GROUP O(2) has GROUP ACTION on the plane that is a rotation: O(2)/C30cosu/C28sinu sinucosu7CP07CP)7CP67CP7 @/C28cosusinu sinucosu7CP07CP)7CP67CP7 ;(2) where uis any real number in 0 ;2p ½Þ :These matrices preserve the QUADRATIC FORM x2/C27y2;and so theyalso preserve CIRCLES x2/C27y2/C30r2;which are the ORBITS . As a manifold, O(2) is a one dimensional, two disjoint copies of the CIRCLE . The SUBGROUP SO(2) is not a NORMAL SUBGROUP ,s o O(2) is the SEMIDIRECT PRO- DUCT of the circle SO(2) and Z2:/ There are several generalizations of the orthogonalgroup. First, it is possible to define the orthogonalgroup for any SYMMETRIC QUADRATIC FORM Qwith SIGNATURE (p, q). The group of matrices Awhich preserve Q, that is, Q(v;w)/C30Q(Av;Aw); (3) is denoted O(p;q):The L ORENTZ GROUP isO(3;1): For example, the matrices A/C30cosh tsinh t sinh tcosh t7CP07CP) (4) are elements of O(1;1):They preserve the QUADRATIC FORM x2/C28y2so they preserve the HYPERBOLAS x2/C28y2/C30c:/ Instead of using real numbers for the coefficients, it ispossible to use coefficients from any FIELD F;in which case it is denoted O(n;F):The orthogonal matrices still satisfy AAt/C30I:For example, O(2;F23) contains 11 15 15 127CP07CP) ; (5) and has 48 elements in total. Of course, O(p;q;F) denotes the group of matrices which preserve the SYMMETRIC QUADRATIC FORM of SIGNATURE (p, q), with coefficients in the field F: When Fis not RorC;these are called L IE-TYPE GROUPS . When the coefficients are COMPLEX NUMBERS ,i ti s called the complex orthogonal group, which is much different from the UNITARY GROUP . For example, matrices OF THE FORM A /C30cos z /C28sin z sin z cos z7CP07CP) (6) are in O(2; C) : In particular, O(n; C) is not COMPACT . The equations defining O(n)in AFFINE SPACE are polynomials of degree two. Consequently, O(n)isa LINEAR ALGEBRAIC GROUP . The numbers of subgroups s(n) of orders n /C30 1, 2, 3, ... in the orthogonal group O(3) are 1, 3, 1, 5, 1, 5, 1, 7, 1, 5, 1, 8, ... (Sloane’s A001051), i.e., a repeating sequence of copies of f1; 5; 1; 7g with the exceptions s(2) /C303 ; s(4) /C305 ; s(12) /C308; s(24) /C3010 ; and s(48) /C30s(60) /C30s(120) /C308:/ See also DETERMINANT ,G ENERAL ORTHOGONAL GROUP ,G ROUP ,F IELD,L APLACIAN ,L IE ALGEBRA , LIE GROUP ,L IE-TYPE GROUP ,L INEAR ALGEBRAIC GROUP ,O RTHOGONAL GROUP REPRESENTATIONS , ORTHOGONAL MATRIX ,O RTHOGONAL TRANSFORMA - TION ,O RTHONORMAL BASIS,P ROJECTIVE GENERAL ORTHOGONAL GROUP ,PROJECTIVE SPECIAL ORTHOGO- NAL GROUP ,RIEMANNIAN METRIC ,SPECIAL ORTHO- GONAL GROUP ,SUBMANIFOLD ,SYMMETRIC QUADRATIC FORM,UNITARY GROUP ,VECTOR SPACE References Arfken, G. "Orthogonal Group, O/C27 3 :/" Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 252 /C1/253, 1985. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#orth. Orthogonal Group Representations Two representations of a GROUP xi and xj are said to be orthogonal if X Rxi(R) xj(R) /C300 for i "j; where the sum is over all elements R of the representation. See also GROUP Orthogonal Lines Two or more LINES or LINE SEGMENTS which are PERPENDICULAR are said to be orthogonal. See also ORTHOGONAL CURVES ,P ERPENDICULAR , RIGHT ANGLE Orthogonal Matrix A n /C29n matrix A is an orthogonal matrix if AAT /C30I ; (1) where AT is the TRANSPOSE of A and I is the IDENTITYMATRIX . In particular, an orthogonal matrix is always invertible, and A /C281 /C30AT (2) (Note that transpose is a much simpler computation than inverse.) For example, A /C301ffiffiffi 2p11 1 /C2817CP07CP) (3) A /C301 32 /C2821 12 2 21 /C2822 435 (4) are orthogonal matrices. A matrix m can be tested to see if it is orthogonal using the Mathematica function OrthogonalQ[m_List?MatrixQ] : /C30 (Transpose[m].m /C30/C30 IdentityMatrix@Length@m) The rows of an orthogonal matrix are an ORTHONOR- MAL BASIS . That is, each row has length one, and are mutually perpendicular. Similarly, the columns are also an orthonormal basis. In fact, given any ortho- normal basis, the matrix whose rows are that basis is an orthogonal matrix. It is automatically the case that the columns are another orthonormal basis. The orthogonal matrices are precisely those matrices which preserve the INNER PRODUCT /C142v; w /C143/C30/C142Av; Aw/C143: (5) Also, the determinant of A is either 1 or /C281. As a subset of Rn2 ; the orthogonal matrices are not CON- NECTED since the determinant is a CONTINUOUS FUNCTION . Instead, there are two COMPONENTS corre- sponding to whether the determinant is 1 or /C281. The orthogonal matrices with A /C301 are rotations, and such a matrix is called a SPECIAL ORTHOGONAL MATRIX . The product of two orthogonal matrices is another orthogonal matrix. In addition, the inverse of an orthogonal matrix is an orthogonal matrix, as is the IDENTITY MATRIX . Hence the set of orthogonal ma- trices form a GROUP , called the ORTHOGONAL GROUP O(n):/ See also EULER’S ROTATION THEOREM ,INNER PRO- DUCT ,ORTHOGONAL GROUP ,ORTHOGONAL TRANSFOR- MATION ,ORTHOGONALITY CONDITION ,ORTHONORMAL BASIS,R OTATION ,R OTATION MATRIX ,R OTOINVER- SION,SKEW SYMMETRIC MATRIX ,SPECIAL ORTHOGO- NAL MATRIX ,SPIN GROUP ,UNITARY MATRIX References Arfken, G. "Orthogonal Matrices." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 191 /C1/205, 1985. Goldstein, H. "Orthogonal Transformations." §4/C1/2i n Clas- sical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 132/C1/137, 1980. Orthogonal Polynomials Orthogonal polynomials are classes of POLYNOMIALS fpn(x)gover a range [ a, b] which obey an ORTHOGON- ALITY relation gb aw(x)pm(x)pn(x)dx/C30dmncn; (1) where w(x)i sa WEIGHTING FUNCTION anddis the KRONECKER DELTA .I fcn/C301;then the POLYNOMIALS are not only orthogonal, but orthonormal. Orthogonal polynomials have very useful properties in the solution of mathematical and physical pro-blems. Just as F OURIER SERIES provide a convenient method of expanding a periodic function in a series oflinearly independent terms, orthogonal polynomialsprovide a natural way to solve, expand, and interpret solutions to many types of important DIFFERENTIAL EQUATIONS . Orthogonal polynomials are especially easy to generate using G RAM- SCHMIDT ORTHONORMA- LIZATION . Abramowitz and Stegun (1972, pp. 774 /C1/ 775) give a table of common orthogonal polynomials. Type Interval /w(x)// cn/ CHEBYSHEV POLYNOMIAL OF THE FIRST KIND/[/C281;1]//(1/C28x2)/C281=2 //1 2pforn/C300 potherwise7CP6 CHEBYSHEV POLYNOMIAL OF THE SECOND KIND/[/C281;1]//ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p //1 2p/ HERMITE POLY- NOMIAL/(/C28/C12;/C12)//e/C28x2 //ffiffiffiffiffipp2nn!/ JACOBI POLYNO- MIAL/(/C281;1)//(1/C28x)a(1/C27x)b //hn/ LAGUERRE POLY- NOMIAL/0;/C12½Þ //e/C28x/ 1 LAGUERRE POLY- NOMIAL (Asso- ciated)/0;/C12½Þ //xke/C28x//(n/C27k)! n!/ LEGENDRE POLY- NOMIAL/[/C281;1]/1 /2 2n/C271/ ULTRASPHERICAL POLYNOMIAL/[/C281;1]//(1/C28x2)a/C281=2 //21/C282apG(n/C272a) n!(n/C27a)[G(a)]2fora"0 2p n2fora"0:( / In the above table, the normalization constant is the value of cn/C13gw(x)[pn(x)]2dx (2) andhn/C132a/C27b/C271 2n/C27a/C27b/C271G(n/C27a/C271)G(n/C27b/C271) n!G(n/C27a/C27b/C271);(3) where G(z)i sa GAMMA FUNCTION . The ROOTS of orthogonal polynomials possess many rather surprising and useful properties. For instance, letx1Bx2B:::Bxnbe the ROOTS of the pn(x) with x0/C30aand xn/C271/C30b:Then each interval [ xn;xn/C271] for n/C300;1, ..., ncontains exactly one ROOT ofpn/C271(x): Between two ROOTS ofpn(x) there is at least one ROOT ofpm(x) for m/C21n. Let cbe an arbitrary REAL constant, then the POLYNOMIAL pn/C271(x)/C28cpn(x) (4) hasn/C271 distinct REAL ROOTS .I fc/C210(cB0), these ROOTS lie in the interior of [ a, b], with the exception of the greatest (least) ROOT which lies in [ a, b] only for c5pn/C271(b) pn(b)c]pn/C271(a) pn(a) ! : (5) The following decomposition into partial fractions holds pn(x) pn/C271(x)/C30Xn n/C300ln x/C28j; (6) where fjngare the ROOTS ofpn/C271(x) and ln/C30pn(jn) p?n/C271(jn) /C30p?n/C271(jn)pn(jn)/C28p?n(jn)0pn/C271(jn) [p?n/C271(jn)]2>0: (7) Another interesting property is obtained by lettingfp n(x)gbe the orthonormal set of POLYNOMIALS associated with the distribution da(x)o n[ a, b ]. Then the CONVERGENTS Rn=Snof the CONTINUED FRACTION 1 A1x/C27B1/C28C2 A2x/C27B2/C28C3 A3x/C27B3/C28.../C28Cn Anx/C27Bn /C27. . . (8) are given by Rn/C30Rn(x) /C30c/C283=2 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0c2/C28c2 1qgb apn(x)/C28pn(t) x/C28tda(t) (9) Sn/C30Sn(x)/C30ffiffiffiffiffic0ppn(x); (10) where n/C300, 1, ...and cn/C30gb axnda(x): (11) Furthermore, the ROOTS of the orthogonal polyno- mials pn(x) associated with the distribution da(x)on the interval [a, b] are REAL and distinct and are located in the interior of the interval [a, b]. See also CHEBYSHEV POLYNOMIAL OF THE FIRST KIND, CHEBYSHEV POLYNOMIAL OF THE SECOND KIND, GRAM- SCHMIDT ORTHONORMALIZATION ,H ERMITE POLYNOMIAL ,JACOBI POLYNOMIAL ,K RAWTCHOUK POLYNOMIAL ,L AGUERRE POLYNOMIAL ,L EGENDRE POLYNOMIAL ,O RTHOGONAL FUNCTIONS ,SPHERICAL HARMONIC ,U LTRASPHERICAL POLYNOMIAL ,ZERNIKE POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771 /C1/802, 1972. Arfken, G. "Orthogonal Polynomials." Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 520 /C1/521, 1985. Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978. Gautschi, W.; Golub, G. H.; and Opfer, G. (Eds.) Applica- tions and Computation of Orthogonal Polynomials, Con- ference at the Mathematical Research Institute Oberwolfach, Germany, March 22 /C1/28, 1998. Basel, Swit- zerland: Birkha ¨user, 1999. Iyanaga, S. and Kawada, Y. (Eds.). "Systems of Orthogonal Functions." Appendix A, Table 20 in Encyclopedic Dic- tionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980. Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analo- gue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/17, 1 /C1/168, 1998. ftp://www.twi.tudelft.nl/publications/ tech-reports/1998/DUT-TWI-98 /C1/17.ps.gz. Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992. Sansone, G. Orthogonal Functions. New York: Dover, 1991. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 44 /C1/47 and 54 /C1/55, 1975. Orthogonal Projection A PROJECTION of a figure by parallel rays. In such a projection, tangencies are preserved. Parallel lines project to parallel lines. The ratio of lengths of parallel segments is preserved, as is the ratio of areas. Any TRIANGLE can be positioned such that its shadow under an orthogonal projection is EQUILATERAL . Also, the MEDIANS of a TRIANGLE project to the MEDIANS of the image TRIANGLE .ELLIPSES project to ELLIPSES , and any ELLIPSE can be projected to form a CIRCLE . The center of an ELLIPSE projects to the center of the image ELLIPSE . The CENTROID of a TRIANGLE projects to the CENTROID of its image. Under an ORTHOGONAL TRANSFORMATION , the MIDPOINT ELLIPSE can be transformed into a CIRCLE INSCRIBED in an EQUILAT- ERAL TRIANGLE .SPHEROIDS project to ELLIPSES (or CIRCLE in the DEGENERATE case). In an orthogonal projection, any vector v can be written v /C30vW /C27vW /C222; so v; Pw hi /C30 vW ; Pw hi /C30 Pv; w hi ; and the PROJECTION MATRIX is a SYMMETRIC MATRIX IFF the PROJECTION is orthogonal. The following Mathematica function will test whether a PROJEC- TION MATRIX is an orthogonal projection. OrthogProjectionMatrixQ[a_List?MatrixQ] : /C30 (a.a /C30/C30 a && Transpose[a] /C30/C30 a) The following Mathematica function gives the PROJECTION MATRIX for orthogonal projection onto a subspace spanned by a given basis. BBLinearAlgebra‘Orthogonalization‘; OrthogProjectMatrixOntoBasis[a_List?MatrixQ] : /C30 Module[{a1 /C30 GramSchmidt[a]}, Transpose[a1].a1] ] For instance, OrthogProjectMatrixOntoBa- sis[{{1, 2, 3}}] yields ff1=14 ; 1=7; 3=14 g;f1=7; 2=7 ; 3 =7g;f3=14 ; 3 =7;/ /9=14 gg::/ See also PROJECTION ,PROJECTION MATRIX Orthogonal Rotation Group ORTHOGONAL GROUP Orthogonal Set A subset fv1 ; ...; vk g of a VECTOR SPACE V, with the INNER PRODUCT ;hi; is called orthogonal if vi ; vj7C)07C)) /C300 when i "j: That is, the vectors are mutually PERPEN- DICULAR . Note that there is no restriction on the lengths of the vectors. If the vectors in an orthogonal set all have length one, then they are ORTHONORMAL . The notion of orthogonal makes sense for an abstract VECTOR SPACE over any field as long as there is a SYMMETRIC QUADRATIC FORM . The usual orthogonal sets and groups in EUCLIDEAN SPACE can be general- ized, with applications to special relativity, DIFFER- ENTIAL GEOMETRY , and ABSTRACT ALGEBRA . See also CLIFFORD ALGEBRA ,HOMOGENEOUS SPACE , HYPERBOLIC SPACE ,LIE GROUP ,LORENTZIAN INNER PRODUCT ,ORTHOGONAL GROUP ,ORTHOGONAL TRANS- FORMATION ,ORTHONORMAL BASIS,SYMMETRIC QUAD- RATIC FORM Orthogonal Subspaces Two SUBSPACES S1and S2ofRnare said to be orthogonal if v1/C215v2/C300 for all v1/C23S1and all v2/C23S2:/ Orthogonal Surfaces Families of surfaces which are mutually orthogonal. Up to three families of surfaces may be orthogonal in 3-D. The simplest example of three orthogonal sur- faces in 3-D are orthogonal planes, but three confocal conic surfaces are also mutually orthogonal. References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 166, 1991. Orthogonal Tensors Orthogonal CONTRAVARIANT and COVARIANT satisfy gikgij /C30 dj k ; where dk jis the KRONECKER DELTA . See also CONTRAVARIANT TENSOR ,COVARIANT TEN- SOR Orthogonal Transformation An orthogonal transformation is a LINEAR TRANSFOR- MATION T : V 0 V which preserves a SYMMETRIC INNER PRODUCT . In particular, an orthogonal trans- formation (technically, an orthonormal transforma- tion) preserves lengths of vectors and angles between vectors, /C142v ; w /C143/C30/C142Tv; Tw/C143: (1) In addition, an orthogonal transformation is either a rigid ROTATION or a ROTOINVERSION (a rotation followed by a flip). (Flipping and then rotating can be realized by first rotating in the reverse direction and then flipping). Orthogonal transformations cor- respond to and may be represented using ORTHOGO- NAL MATRICES . The set of orthonormal transformations forms the ORTHOGONAL GROUP , and an orthonormal transfor- mation can be realized by an ORTHOGONAL MATRIX . Any linear transformation in 3-D x?1 /C30a11x1 /C27a12x2 /C27x13x3 (2) x?2 /C30a21x1 /C27a22x2 /C27a23x3 (3) x?3 /C30a31x1 /C27a32x2 /C27a33x3 (4) satisfying the ORTHOGONALITY CONDITION aijaik /C30 djk ; (5) where EINSTEIN SUMMATION has been used and dij is the KRONECKER DELTA , is an orthogonal transforma- tion. If A : Rn 0 Rn is an orthogonal transformation, then det(A) /C3091:/ See also INNER PRODUCT ,LIE GROUP ,LINEAR TRANS- FORMATION ,L ORENTZ TRANSFORMATION ,M ATRIX , ORTHOGONAL MATRIX ,ORTHOGONAL GROUP ,ORTHO- GONALITY CONDITION ,SPIN GROUP ,ROTATION ,RO-TOINVERSION ,SYMMETRIC QUADRATIC FORM References Goldstein, H. "Orthogonal Transformations." §4 /C1/2in Clas- sical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 132 /C1/137, 1980. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 128 /C1/129, 1997. Orthogonal Vectors Two vectors u and v whose DOT PRODUCT is u /C215 v /C300 (i.e., the vectors are PERPENDICULAR ) are said to be orthogonal. In 3-space, three vectors can be mutually perpendicular. See also DOT PRODUCT ,O RTHONORMAL VECTORS , PERPENDICULAR Orthogonality Condition A linear transformation x?1 /C30a11x1 /C27a12x2 /C27x13x3 x?2 /C30a21x1 /C27a22x2 /C27a23x3 x?3 /C30a31x1 /C27a32x2 /C27a33x3 ; is said to be an ORTHOGONAL TRANSFORMATION if it satisfies the orthogonality condition aijaik /C30 djk ; where EINSTEIN SUMMATION has been used and dij is the KRONECKER DELTA . See also ORTHOGONAL TRANSFORMATION References Goldstein, H. "Orthogonal Transformations." §4/C1/2i n Clas- sical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 132 /C1/137, 1980. Orthogonality Theorem GROUP ORTHOGONALITY THEOREM Orthographic Projection A projection from infinity which preserves neither AREA nor angle. x /C30cos f sin( l /C28 l0) (1) y /C30cos f1 sin f /C28sin f1 cos f cos(l /C28 l0) : (2) The inverse FORMULAS are f /C30sin/C281cos c sin f1 /C27y sin c cos f1 /C27 r ! (3) l /C30 l0 /C27tan/C281 x sin c r cos f1 cos c /C28 y sin f1 sin c ! ; (4) where r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p (5) c /C30sin/C281 r: (6) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 145 /C1/153, 1987. Orthologic Triangles Two TRIANGLES A1B1C1and A2B2C2are orthologic if the perpendiculars from the VERTICES A1 ; B1 ; C1on the sides B2C2 ; A2C2 ; and A2B2pass through one point. This point is known as the orthology center of TRIANGLE 1 with respect to TRIANGLE 2. Orthomorphic Projection CONFORMAL PROJECTION Orthonormal Basis A subset fv1 ; ...; vk g of a VECTOR SPACE V, with the INNER PRODUCT ;hi; is called orthonormal if vi ; vj7C)07C)) /C30 0 when i "j: That is, the vectors are mutually PERPENDICULAR . Moreover, they are all required to have length one: /C142vi ; vi /C143/C301 :/An orthonormal set must be linearly independent, and so it is a BASIS for the space it SPANS . Such a basis is called an orthonormal basis. The simplest example of an orthonormal basis is the standard basis ei for EUCLIDEAN SPACE Rn : The vector eiis the vector with all 0s except for a 1 in the ith coordinate. For example, e1 /C30(1; 0 ; ...; 0): A rotation (or flip) through the origin will send an orthonormal set to another orthonormal set. In fact, given any orthonormal basis, there is a rotation, or rotation combined with a flip, which will send the orthonormal basis to the standard basis. These are precisely the transformations which preserve the inner product, and are called ORTHOGONAL TRANSFORMATIONS . Usually when one needs a basis to do calculations, it is convenient to use an orthonormal basis. For example, the formula for a PROJECTION is much simpler with an orthonormal basis. The savings in effort make it worthwhile to find an orthonormal basis before doing such a calculation. GRAM- SCHMIDT ORTHONORMALIZATION is a popular way to find an orthonormal basis. Another instance when orthonormal bases arise is as a set of EIGENVECTORS for a SYMMETRIC MATRIX . For a general matrix, the set of eigenvectors may not be orthonormal, or even be a basis. See also BASIS (VECTOR SPACE ), DOT PRODUCT ,INNER PRODUCT ,KRONECKER DELTA ,LIE GROUP ,LORENT- ZIAN INNER PRODUCT ,M ATRIX ,O RTHOGONAL BASIS ORTHOGONAL MATRIX ,ORTHOGONAL GROUP ,ORTHO- GONAL TRANSFORMATION ,P ROJECTION (VECTOR SPACE ), SYMMETRIC QUADRATIC FORM Orthonormal Functions A pair of functions fi(x) and fj(x) are orthonormal if they are ORTHOGONAL and each normalized. These two conditions can be succinctly written as gb afi(x) fj(x)w(x) dx /C30 dij ; where w(x)isa WEIGHTING FUNCTION and dijis the KRONECKER DELTA . See also ORTHOGONAL POLYNOMIALS Orthonormal Transformation ORTHOGONAL TRANSFORMATION Orthonormal Vectors UNIT VECTORS which are ORTHOGONAL are said to be orthonormal. See also ORTHOGONAL VECTORS Orthoplex CROSS POLYTOPE Orthopole If perpendiculars A?; B ?; and C? are dropped on any line L from the vertices of a TRIANGLE DABC ; then the perpendiculars to the opposite sides from their FEET Aƒ; B ƒ; and C ƒ are CONCURRENT at a point P called the orthopole. The orthopole of a line lies on the SIMSON LINE which is PERPENDICULAR to it (Honsberger 1995, p. 130). If a line crosses the CIRCUMCIRCLE of a triangle, the SIMSON LINES of the points of intersec- tion meet at the orthopole of the line. Also, the orthopole of a line through the CIRCUMCENTER O of a triangle DABC lies on that triangle’s NINE-POINT CIRCLE (Honsberger 1995, p. 127). If the line L is displaced PARALLEL to itself, the orthopole moves along a line PERPENDICULAR to L a distance equal to the displacement. If L is the SIMSON LINE of a point P, then P is called the POLE of L (Honsberger 1995, p. 128). See also NINE-POINT CIRCLE ,POLE (SIMSON LINE), RIGBY POINTS ,SIMSON LINE References Honsberger, R. "The Orthopole." Ch. 11 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 125 /C1/136, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 247, 1929. Ramler, O. J. "The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle." Amer. Math. Monthly 37, 130 /C1/136, 1930. Orthoptic Curve An ISOPTIC CURVE formed from the locus of TANGENTS meeting at RIGHT ANGLES . The orthoptic of a PARA- BOLA is its DIRECTRIX . The orthoptic of a central CONIC was investigated by Monge and is a CIRCLE concentric with the CONIC SECTION . The orthoptic of an ASTROID is a CIRCLE . Curve Orthoptic ASTROID QUADRIFOLIUM CARDIOID CIRCLE or LIMAC ¸ ON DELTOID CIRCLELOGARITHMIC SPIRAL equal LOGARITHMIC SPIRAL PARABOLA DIRECTRIX References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 58 and 207, 1972. Orthosymmetric Matrix PERSYMMETRIC MATRIX Orthotomic Given a source S and a curve g ; pick a point on g and find its tangent T. Then the LOCUS of reflections of S about tangents T is the orthotomic curve (also known as the secondary CAUSTIC ). The INVOLUTE of the orthotomic is the CAUSTIC . For a parametric curve (f(t) ; g(t)) with respect to the point (x0 ; y0) ; the orthotomic is x /C30x0 /C282g?[f ?(g /C28 y0) /C28 g?(f /C28 x0)] f ?2 /C27 g ?2 y /C30y0 /C272f ?[f ?(g /C28 y0) /C28 g?(f /C28 x0)] f ?2 /C27 g ?2 See also CAUSTIC ,INVOLUTE References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 60, 1972. Orthotope A PARALLELOTOPE whose edges are all mutually PERPENDICULAR . The orthotope is a generalization of the RECTANGLE and RECTANGULAR PARALLELEPIPED . See also RECTANGLE ,RECTANGULAR PARALLELEPIPED References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 122 /C1/123, 1973. Osborne’s Rule The prescription that a TRIGONOMETRY identity can be converted to an analogous identity for HYPERBOLIC FUNCTIONS by expanding, exchanging trigonometric functions with their hyperbolic counterparts, and then flipping the sign of each term involving theproduct of two HYPERBOLIC SINES . For example, given the identity cos(x/C28y)/C30cosxcosy/C27sinxsiny; Osborne’s rule gives the corresponding identity cosh( x /C28y) /C30cosh x cosh y /C27sinh x sinh y: See also HYPERBOLIC FUNCTIONS ,T RIGONOMETRIC FUNCTIONS Oscillation The variation of a FUNCTION which exhibits SLOPE changes, also called the SALTUS of a function. A series may also oscillate, causing it not to converge. References Jeffreys, H. and Jeffreys, B. S. "Bounded, Unbounded, Convergent, Oscillatory." §1.041 in Methods of Mathema- tical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 11 /C1/12 and 22, 1988. Oscillation Land CAROTID- KUNDALINI FUNCTION Osculating Circle The CIRCLE which shares the same TANGENT as a curve at a given point. Given a plane curve with PARAMETRIC EQUATIONS (f(t) ; g(t)) and parameterized by a variable t, the RADIUS OF CURVATURE of the osculating circle is r(t) /C301 k(t)jj; (1) where k(t) is the CURVATURE , and the center is x /C30f /C28(f ?2 /C27 g ?2)g? f ?g ƒ/C28 f ƒg? (2) y /C30g /C27(f ?2 /C27 g ?2)f ? f ?gƒ/C28 f ƒg ?: (3) Here, derivatives are taken with respect to the parameter t. Note that the centers of the osculatingcircles to a curve form the EVOLUTE to that curve. In addition, let C(t1 ; t2 ; t3) denote the CIRCLE passing through three points on a curve (f(t) ; g(t)) with t1 B t2 Bt3 : Then the osculating circle C is given by C /C30 lim t1 ; t2 ; t3 0tC(t1 ; t2 ; t3) (4) (Gray 1997). See also CURVATURE ,EVOLUTE ,OSCULATING CURVES , RADIUS OF CURVATURE ,TANGENT References Gardner, M. "The Game of Life, Parts I-III." Chs. 20 /C1/22 in Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, pp. 221, 237, and 243, 1983. Gray, A. "Osculating Circles to Plane Curves." §5.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 111 /C1/115, 1997. Osculating Curves An curve y(x) is osculating to f(x)atx0 if it is TANGENT at x0 and has the same CURVATURE there. Osculating curves therefore satisfy y(k)(x0) /C30f(k)(x0) for k /C300, 1, 2. The point of tangency is called a TACNODE . One of simplest examples of a pairs of osculating curves is x2and x2/C28x4;which osculate at the point x0/C300 since for k/C300, 1, 2, y(k)(0)/C30f(k)(0) is equal to 0, 0, and 2. See also OSCULATING CIRCLE ,TACNODE ,TANGENT CURVES Osculating Interpolation HERMITE’S INTERPOLATING POLYNOMIAL Osculating Plane The PLANE spanned by the three points x(t);x(t/C27h1); andx(t/C27h2) on a curve as h1;h200:Letzbe a point on the osculating plane, then [(z /C28x); x?; x ƒ] /C300; where [A ; B; C] denotes the SCALAR TRIPLE PRODUCT . The osculating plane passes through the tangent. The intersection of the osculating plane with the NORMAL PLANE is known as the PRINCIPAL NORMAL VECTOR . The VECTORS T and N (TANGENT VECTOR and NORMAL VECTOR ) span the osculating plane. See also NORMAL VECTOR ,O SCULATING SPHERE , SCALAR TRIPLE PRODUCT ,TANGENT VECTOR Osculating Sphere The center of any SPHERE which has a contact of (at least) first-order with a curve C at a point P lies in the normal plane to C at P. The center of any SPHERE which has a contact of (at least) second-order with C at point P, where the CURVATURE k > 0 ; lies on the polar axis of C corresponding to P. All these SPHERES intersect the OSCULATING PLANE of C at P along a circle of curvature at P. The osculating sphere has center a /C30x /C27 r ˆN /C27˙r tˆB where ˆN is the unit NORMAL VECTOR , ˆB is the unit BINORMAL VECTOR , r is the RADIUS OF CURVATURE , and t is the TORSION , and RADIUS R /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27˙r t !2vuut; and has contact of (at least) third order with C. See also CURVATURE ,OSCULATING PLANE ,RADIUS OF CURVATURE ,SPHERE ,TORSION (DIFFERENTIAL GEO- METRY ) References Kreyszig, E. Differential Geometry. New York: Dover, pp. 54 /C1/55, 1991. Osedelec Theorem For an n-D MAP, the LYAPUNOV CHARACTERISTIC EXPONENTS are given by si /C30 lim N 0/C12ln li(N) jj for i /C301, ..., n, where li is the LYAPUNOV CHARACTER- ISTIC NUMBER . See also LYAPUNOV CHARACTERISTIC EXPONENT , LYAPUNOV CHARACTERISTIC NUMBER Ostrowski-Hadamard Gap Theorem Let 0 Bp1 Bp2 B... be integers and suppose that there exists a l > 1 such that pj/C271 =pj > l for j /C301, 2, .... Suppose that for some sequence of complexnumbers faj g the POWER SERIES f(z) /C30X/C12 j/C301ajzpj has radius of convergence 1, then no point of @D is a REGULAR POINT for f (Krantz 1999, p. 120). See also REGULAR POINT References Krantz, S. G. "The Ostrowski-Hadamard Gap Theorem." §9.2.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 119 /C1/120, 1999. Ostrowski’s Inequality If f(x) is a monotonically increasing integrable func- tion on [a, b] with f(b) 50; then if g is a REAL function integrable on [a, b], gb af(x)g(x) dx7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P5 f(a) jj max a 5 j 5b g j ag(x) dx7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P7C)P: References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1100, 2000. Ostrowski’s Theorem Let A /C30aijbe a MATRIX with POSITIVE COEFFICIENTS and l0 be the POSITIVE EIGENVALUE in the FROBENIUS THEOREM , then the n /C281 EIGENVALUES lj " l0satisfy the INEQUALITY lj7C)P7C)P7C)P7C)P5 l 0M2 /C28 m2 M2 /C27 m2 ; where M /C30max i; jaij m /C30min i; jaij and i; j /C301 ; 2, ..., n. See also FROBENIUS THEOREM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000. Otter’s Theorem In any TREE , the number of dissimilar points minus the number of dissimilar lines plus the number of symmetry lines equals 1. See also TREE References Harary, F. and Prins, G. "The Number of Homeomorphically Irreducible Trees, and Other Species." Acta Math. 101, 141 /C1/162, 1959. Otter, R. "The Number of Trees." Ann. Math. 49, 583 /C1/599, 1948. Oudor References Moon, P. and Spencer, D. E. Theory of Holors: A General- ization of Tensors. Cambridge, England: Cambridge Uni- versity Press, 1986. Oui-Ja Board Curve COCHLEOID Outcome An outcome is a subset of a PROBABILITY SPACE . Experimental outcomes are not uniquely determined from the description of an experiment, and must be agreed upon to avoid ambiguity (Papoulis 1984, pp. 24 /C1/25). See also EVENT ,EXPERIMENT ,TRIAL References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Outdegree The number of outward directed EDGES from a given VERTEX in a DIRECTED GRAPH . See also DIRECTED GRAPH ,INDEGREE ,LOCAL DEGREE Outer Automorphism Group A particular type of AUTOMORPHISM GROUP which exists only for GROUPS . For a GROUP G, the outer automorphism group is the QUOTIENT GROUP Aut(G) =Inn(G); which is the AUTOMORPHISM GROUP of G modulo its INNER AUTOMORPHISM GROUP . See also AUTOMORPHISM GROUP ,INNER AUTOMORPH- ISM GROUP ,QUOTIENT GROUP Outer Product TENSOR DIRECT PRODUCT ,TENSOR PRODUCT (VECTOR SPACE ) Outer Quermass BRIGHTNESS Outplanar Graph A graph that can be embedded in the plane such that all vertices lie on the outer face (Skiena 1990, p. 251). See also PLANAR GRAPHReferences Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Oval An oval is a curve resembling a squashed CIRCLE but, unlike the ELLIPSE , without a precise mathematical definition. The word oval derived from the Latin word "ovus" for egg. Unlike ellipses, ovals sometimes have only a single axis of reflection symmetry (instead of two). Ovals can be constructed with a COMPASS by joining together arcs of different radii such that the centers of the arcs lie on a line passing through the join point (Dixon 1991). Albrecht Du¨rer used this method to design a Roman letter font. See also CARTESIAN OVALS ,C ASSINI OVALS ,E GG, ELLIPSE ,LEMON ,OVOID ,SUPERELLIPSE References Critchlow, K. Time Stands Still. London: Gordon Fraser, 1979. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Dixon, R. Mathographics. New York: Dover, pp. 3 /C1/11, 1991. Dixon, R. "The Drawing Out of an Egg." New Sci. , July 29, 1982. Pedoe, D. Geometry and the Liberal Arts. London: Peregrine, 1976. Oval of Descartes CARTESIAN OVALS Ovals of Cassini CASSINI OVALS Overbar MACRON Overdamping DAMPED SIMPLE HARMONIC MOTION– OVERDAMPING Overdot An "overdot" is a raised DOT appearing above a symbol most commonly used in mathematics to indicate a DERIVATIVE taken with respect to time (e.g., ˙x/C13dx=dt):The expression ˙ais voiced " adot," and was Newton’s notation for derivatives (which he called "FLUXIONS "). See also DERIVATIVE ,DOT,DOUBLE DOT References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 282, 1997. Overlapfree Word N.B. A detailed online essay by S. Finch was the starting point for this entry. A word is said to be overlapfree if it has no subwords OF THE FORM xyxyx .A SQUAREFREE WORD is overlap- free, and an overlapfree word is CUBEFREE . The number t(n) of binary overlapfree words of length n /C30 1, 2, ... are 2, 4, 6, 10, 14, 20, ... (Sloane’s A007777). t(n) satisfies p /C215 n1:155 5t(n) 5q /C215 n1 :587 (1) for some constants p and q (Restivo and Selemi 1985, Kobayashi 1988). In addition, while lim n0/C12ln t(n) ln n (2) does not exist, 1:155 BTL B1:276 B1:332 BTU B1:587; (3) where TL /C13lim inf n0/C12ln t(n) ln n (4) TU /C13lim sup n0/C12ln t(n) ln n (5) (Cassaigne 1993). See also CUBEFREE WORD,SQUAREFREE WORD,WORD References Cassaigne, J. "Counting Overlap-Free Binary Words." STACS ’93: Tenth Annual Symposium on Theoretical Aspects of Computer Science, Wu¨rzburg, Germany, Febru- ary 25 /C1/27, 1993 Proceedings (Ed. G. Goos, J. Hartmanis, A. Finkel, P. Enjalbert, K. W. Wagner). New York: Springer-Verlag, pp. 216 /C1/225, 1993. Cassaigne, J. Motifs e´vitables et re´gularite ´s dans les mots (The`se de Doctorat). Tech. Rep. LITP-TH 94 /C1/04. Paris: Institut Blaise Pascal, 1994. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/words/words.html. Kobayashi, Y. "Enumeration of Irreducible Binary Words." Discrete Appl. Math. 20, 221 /C1/232, 1988. Se´e´bold, P. "Overlap-Free Sequences." In Combinatorics on Words (Ed. L. J. Cummings). Toronto: Academic Press, pp. 207 /C1/215, 1983.Sloane, N. J. A. Sequences A007777 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Overlapping Rectangles See also RECTANGLE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Overlapping Convex Bodies." §A12 in Unsolved Problems in Geometry. New York: Springer-Verlag, p. 25, 1991. Overlapping Resonance Method RESONANCE OVERLAP METHOD Overline MACRON ,VINCULUM Oversampling A signal sampled at a frequency higher than the NYQUIST FREQUENCY is said to be oversampled b times, where the oversampling ratio is defined as b /C13nsampling nNyquist: See also NYQUIST FREQUENCY ,NYQUIST SAMPLING Ovoid An egg-shaped curve. Lockwood (1967) calls the NEGATIVE PEDAL CURVE of an ELLIPSE with ECCEN- TRICITY e 51=2 an ovoid. See also OVAL References Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967. P p (Prime) Group X is a /p?/-group if p does not divide the ORDER of X. Paasche’s Index The statistical INDEX PP /C13PpnqnPp0qn; where pn is the price per unit in period n and qn is the quantity produced in period n. See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 65, 1962. Packing The placement of objects so that they touch in some specified manner, often inside a container with specified properties. For example, one could consider a SPHERE PACKING , ELLIPSOID PACKING , POLYHEDRON PACKING , etc. See also BARLOW PACKING ,BOX-PACKING THEOREM , CIRCLE PACKING ,C OVERING ,E LLIPSOID PACKING , GROEMER PACKING ,HYPERSPHERE PACKING ,KEPLER PROBLEM ,KISSING NUMBER PACKING DENSITY ,POLY- HEDRON PACKING ,S PACE- FILLING POLYHEDRON , SPHERE PACKING ,SPHERICAL COVERING ,SPHERICAL DESIGN ,TRIANGLE PACKING References Eppstein, D. "Covering and Packing." http://www.ics.u- ci.edu/~eppstein/junkyard/cover.html. Friedman, E. "Erich’s Packing Center." http://www.stetso- n.edu/~efriedma/packing.html. Packing Density The fraction of a volume filled by a given collection of solids. See also HYPERSPHERE PACKING ,PACKING ,SPHERE PACKING Pade´ Approximant Approximants derived by expanding a function as a ratio of two POWER SERIES and determining both the NUMERATOR and DENOMINATOR COEFFICIENTS . Pade ´ approximations are usually superior to T AYLOR EX- PANSIONS when functions contain POLES , because the use of RATIONAL FUNCTIONS allows them to be well- represented. The Pade ´approximant RL=0corresponds to the MACLAURIN SERIES . When it exists, the RL=M/C13[L=M] Pade ´approximant to any POWER SERIES A(x)/C30X/C12 j/C300ajxj(1) is unique. If A(x)i sa TRANSCENDENTAL FUNCTION , then the terms are given by the T AYLOR SERIES about x0 an/C301 n!A(n)(x0): (2) The COEFFICIENTS are found by setting A(x)/C28PL(x) QM(x)/C300 (3) and equating COEFFICIENTS .QM(x) can be multiplied by an arbitrary constant which will rescale the other COEFFICIENTS , so an addition constraint can be applied. The conventional normalization is QM(0)/C301: (4) Expanding (3) gives PL(x)/C30p0/C27p1x/C27.../C27pLxL(5) QM(x)/C301/C27q1x/C27.../C27pMxM: (6) These give the set of equations a0/C30p0 (7) a1/C27a0q1/C30p1 (8) a2/C27a1q1/C27a0q2/C30p2 (9) n aL/C27aL/C281q1/C27.../C27a0qL/C30pL (10) aL/C271/C27aLq1/C27.../C27aL/C28M/C271qM/C300 (11) n qL/C27M/C27aL/C27M/C281q1/C27.../C27aLqM/C300; (12) where an/C300 for nB0 and qj/C300 for j/C21M. Solving these directly gives [L=M]/C30aL/C28m/C271 aL/C28m/C272 ... aL/C271 nn:::n aL aL/C271 /C1/C1/C1 aL/C27M XL j/C30Maj/C28MxjXL j/C30M/C281aj/C28M/C271xj/C1/C1/C1XL j/C300ajxj9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$ a L/C28m/C271aL/C28m/C272/C1/C1/C1 aL/C271 nn:::n aL aL/C271... aL/C27M xMxM/C281/C1/C1/C1 19+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$; (13) where sums are replaced by a zero if the lower index exceeds the upper. Alternate forms are [L=M]/C30XL/C28M j/C300ajxj/C27xL/C28M/C271wT L=MW/C281 L=MwL=M /C30XL/C27n j/C300ajxj/C27xL/C27n/C271wT(L/C27M)=MW/C281 L=Mw(L/C27n)=M for WL=M/C30aL/C28M/C271/C28xaL/C28M/C272/C1/C1/C1 aL/C28xaL/C271 n::: n aL/C28xaL/C271 /C1/C1/C1 aL/C27M/C281/C28xaL/C27M2 435 (14) w L=M/C30aL/C28M/C271 aL/C28M/C272 n aL26643 775; (15) and 05n5M: / For example, the first few Pade ´approximants for are exp0=0(x)/C301 exp0=1(x)/C301 1/C28x exp0=2(x)/C302 2/C282x/C27x2 exp0=3(x)/C306 6/C286x/C273x2/C28x3 exp1=0(x)/C301/C27x exp1=1(x)/C302/C27x 2/C28x exp1=2(x)/C306/C272x 6/C284x/C27x2 exp1=3(x)/C3024/C276x 24/C2818x/C276x2/C28x3 exp2=0(x)/C302/C272x/C27x2 2 exp2=1(x)/C306/C274x/C27x2 6/C282x exp2=2(x)/C3012/C276x/C27x2 12/C286x/C27x2 exp2=3(x)/C3060/C2724x/C273x2 60/C2836x/C279x2/C28x3 exp3=0(x)/C306/C276x/C273x2/C27x3 6 exp3=1(x)/C3024/C2718x/C2716x2/C27x3 24/C286x exp3=2(x)/C3060/C2736x/C279x2/C27x3 60/C2824x/C273x2 exp3=3(x)/C30120/C2760x/C2712x2/C27x3 120/C2860x/C2712x2/C28x3: Two-term identities includePL/C271(x) QM/C271(x)/C28P?L(x) Q?M(x)/C30C2 (L/C271)=(M/C271)xL/C27M/C271 QM/C271(x)Q?M(x)(16) PL/C271(x) QM(x)/C28P?L(x) Q?M(x)/C30C(L/C271)=MC(L/C271)=(M/C271)xL/C27M/C271 QM(x)Q?M(x)(17) PL(x) QM/C271(x)/C28P?L(x) Q?M(x)/C30CL=(M/C271)C(L/C271)=(M/C271)xL/C27M/C271 QM(x)Q?M(x)(18) PL(x) QM/C271(x)/C28P?L/C271(x) Q?M/C30C2(L/C271)=(M/C271)xL/C27M/C272 QM/C271Q?M(19) PL/C271 QM(x)/C28P?L/C281(x) Q?M(x)/C30 CL=(M/C271)C(L/C271)=MxL/C27M/C27CL=MC(L/C271)=(M/C271)xL/C27M/C271 QM(x)Q?M(x)ð20Þ PL(x) QM/C271(x)/C28P?L(x) Q?M/C281(x)/C30 CL=(M/C271)C(L/C271)=MxL/C27M/C28CL=MC(L/C271)=(M/C271)xL/C27M/C271 QM/C271(x)Q?M/C281(x);ð21Þ where Cis the C-DETERMINANT . Three-term identi- ties can be derived using the F ROBENIUS TRIANGLE IDENTITIES (Baker 1975, p. 32). A five-term identity is S(L/C271)=MS(L/C281)=M/C28SL=(M/C271)SL=(M/C281)/C30S2 L=M: (22) Cross ratio identities include RL=M/C28RL=(M/C271)9+;k9+;7 R(L/C271)=M/C28R(L/C271)=(M/C271)9+;k9+;7 RL=M/C28R(L/C271)=M9+;k9+;7 RL=(M/C271)/C28R(L/C271)=(M/C271)9+;k9+;7 /C30CL=(M/C271)C(L/C272)=(M/C271) C(L/C271)=MC(L/C271)=(M/C272)(23) RL=M/C28R(L/C271)=(M/C271)9+;k9+;7 R(L/C271)=M/C28RL=(M/C271)9+;k9+;7 RL=M/C28RL=(M/C271)9+;k9+;7 R(L/C271)=M/C28R(L/C271)=(M/C271)9+;k9+;7 /C30C2 (L/C271)=(M/C271)x CL=(M/C271)C(L/C272)=(M/C271)(24) RL=M/C28R(L/C271)=(M/C271)9+;k9+;7 R(L/C271)=M/C28RL=(M/C271)9+;k9+;7 RL=M/C28R(L/C271)=M9+;k9+;7 RL=(M/C271)/C28R(L/C271)=(M/C271)9+;k9+;7 /C30C2(L/C271)=(M/C271)x C(L/C271)=MC(L/C271)=(M/C272)(25) RL=M/C28R(L/C271)=(M/C271)9+;k9+;7 RL=(M/C271)/C28R(L/C271)=M9+;k9+;7 RL=M/C28/C28 RL=(M/C271)9+;k9+;7 R(L/C271)=(M/C271)/C28R(L/C271)=M9+;k9+;7 /C30C(L/C271)=MC(L/C271)=(M /C271)x CL =(M /C271)C(L /C272)=M(26) RL=M /C28 R(L/C281)=(M /C271)9+;k9+;7 R(L/C271)=M /C28 RL =(M /C271)9+;k9+;7 RL=M /C28 R(L/C271)=M9+;k9+;7 R(L/C281)=(M /C271) /C28 RL =(M /C271)9+;k9+;7 /C30CL =(M /C271)C(L /C271)=(M /C271)x C(L/C271)=MCL=(M /C272): (27) See also C-DETERMINANT ,E CONOMIZED RATIONAL APPROXIMATION ,FROBENIUS TRIANGLE IDENTITIES References Baker, G. A. Jr. "The Theory and Application of The Pade Approximant Method." In Advances in Theoretical Phy- sics, Vol. 1 (Ed. K. A. Brueckner). New York: Academic Press, pp. 1 /C1/58, 1965. Baker, G. A. Jr. Essentials of Pade´ Approximants in Theo- retical Physics. New York: Academic Press, pp. 27 /C1/38, 1975. Baker, G. A. Jr. and Graves-Morris, P. Pade´ Approximants. New York: Cambridge University Press, 1996. Brent, R. P.; Gustavson, F. G.; and Yun, D. Y. Y. "Fast Solution of Toeplitz Systems of Equations and Computa- tion of Pade´ Approximants." J. Algorithms 1, 259 /C1/295, 1980. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Pade ´ Approximants." §5.12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 194 /C1/197, 1992. Weisstein, E. W. "Books about Pade´ Approximants." http:// www.treasure-troves.com/books/PadeApproximants.html. Pade´ Conjecture If P(z)isa POWER SERIES which is regular for ½z ½51 except for m POLES within this CIRCLE and except for z /C30/C271; at which points the function is assumed continuous when only points ½z½51 are considered, then at least a subsequence of the [N, N]P ADE´ APPROXIMANTS are uniformly bounded in the domain formed by removing the interiors of small circles with centers at these POLES and uniformly continuous at z /C30/C271 for ½z ½51:/ See also PADE´ APPROXIMANT References Baker, G. A. Jr. "The Pade´ Conjecture and Some Conse- quences." §II.D in Advances in Theoretical Physics, Vol. 1 (Ed. K. A. Brueckner). New York: Academic Press, pp. 23 /C1/27, 1965. p-adic Absolute Value P-ADIC NORM p-adic Norm Any NONZERO RATIONAL NUMBER x can be represented byx /C30par s; (1) where p is a PRIME NUMBER , r and s are INTEGERS not DIVISIBLE by p, and a is a unique INTEGER . The p-adic norm of x is then defined by ½x½p /C30p /C28a : (2) Also define the p-adic value ½0½p /C300: (3) As an example, consider the FRACTION 140 297 /C3022 /C2153/C283 /C2155 /C2157 /C21511 /C281 : (4) It has p-adic absolute values given by 140 2979+;$9+;$9+;$9+;$9+;$9+;$ 2/C301 4 (5) 1402979+;$9+;$9+;$9+;$9+;$9+;$ 3/C3027 (6) 140 2979+;$9+;$9+;$9+;$9+;$9+;$ 5/C301 5 (7) 1402979+;$9+;$9+;$9+;$9+;$9+;$ 7/C301 7 (8) 140 2979+;$9+;$9+;$9+;$9+;$9+;$ 11/C3011: (9) The p-adic norm of a nonzero RATIONAL NUMBER x can be implemented in Mathematica as follows. PadicNorm[x_Integer, p_Integer?PrimeQ] : /C30 p^(-IntegerExponent[x, p]) PadicNorm[x_Rational, p_Integer?PrimeQ] : /C30 PadicNorm[Numerator[x], p]/ PadicNorm[Denominator[x], p] The p-adic norm satisfies the relations 1. ½x½p ]0 for all x, 2. ½x½p /C300 IFF x /C300, 3. ½xy½p /C30½x½p ½y½p for all x and y, 4. ½x /C27y½p 5½x½p /C27½y½pfor all x and y (the TRIANGLE INEQUALITY ), and 5. ½x /C27y½p 5max( ½x½p ;½y½p) for all x and y (the STRONG TRIANGLE INEQUALITY ). In the above, relation 4 follows trivially from relation 5, but relations 4 and 5 are relevant in the more general VALUATION THEORY . Thep-adic norm is the basis for the algebra of P-ADIC NUMBERS . See also P-ADIC NUMBER p-adic Number Ap-adic number is an extension of the FIELD of RATIONAL NUMBERS such that CONGRUENCES MODULO POWERS of a fixed PRIME pare related to proximity in the so called " p-adic metric." Any NONZERO RATIONAL NUMBER x can be represented by x /C30par s; (1) where p is a PRIME NUMBER , r and s are INTEGERS not DIVISIBLE by p, and a is a unique INTEGER . Then define the P-ADIC NORM of x by ½x½p /C30p /C28a : (2) Also define the p-adic norm ½0½p /C300: (3) The p-adics were probably first introduced by Hensel (1897) in a paper which was concerned with the development of algebraic numbers in POWER SERIES . p-adic numbers were then generalized to VALUATIONS by Ku¨rscha´k (1913). Hasse (1923) subsequently for- mulated the LOCAL-GLOBAL PRINCIPLE (now usually called the HASSE PRINCIPLE ), which is one of the chief applications of LOCAL FIELD theory. Skolem’s p-adic method, which is used in attacking certain DIOPHAN- TINE EQUATIONS , is another powerful application of p- adic numbers. Another application is the theorem that the HARMONIC NUMBERS Hnare never INTEGERS (except for H1) : A similar application is the proof of the VON STAUDT-CLAUSEN THEOREM using the p-adic valuation, although the technical details are some- what difficult. Yet another application is provided by the MAHLER- LECH THEOREM . Every RATIONAL x has an "essentially" unique p-adic expansion ("essentially" since zero terms can always be added at the beginning) x /C30X/C12 j/C30majpj ; (4) with m an INTEGER , ajthe INTEGERS between 0 and p /C281 inclusive, and where the sum is convergent with respect to p-adic valuation. If x "0 and am "0; then the expansion is unique. Burger and Struppeck (1996) show that for p a PRIME and n a POSITIVE INTEGER , ½n!½p /C30p/C28(n/C28Ap(n)) =(p/C281) ; (5) where the p-adic expansion of n is n /C30a0 /C27a1p /C27a2p2 /C27.../C27aLpL ; (6) and Ap(n) /C30a0 /C27a1 /C27a2 /C27.../C27aL : (7) For sufficiently large n, ½n! ½p 5p/C28n=(2p /C282) : (8) The p-adic valuation on Q gives rise to the p-adic metricd(x;y) /C30½x /C28y½p ; (9) which in turn gives rise to the p-adic topology. It can be shown that the rationals, together with the p-adic metric, do not form a COMPLETE METRIC SPACE . The completion of this space can therefore be constructed, and the set of p-adic numbers Qp is defined to be this completed space. Just as the REAL NUMBERS are the completion of the RATIONALS Q with respect to the usual absolute valuation ½x /C28y½; the p-adic numbers are the comple- tion of Q with respect to the p-adic valuation ½x /C28y½p : The p-adic numbers are useful in solving DIOPHAN- TINE EQUATIONS . For example, the equation X2 /C302 can easily be shown to have no solutions in the field of 2-adic numbers (we simply take the valuation of both sides). Because the 2-adic numbers contain the rationals as a subset, we can immediately see that the equation has no solutions in the RATIONALS .Sowe have an immediate proof of the irrationality offfiffiffi 2p :/ This is a common argument that is used in solving these types of equations: in order to show that an equation has no solutions in Q ; we show that it has no solutions in an EXTENSION FIELD . For another exam- ple, consider X2/C271/C300:This equation has no solu- tions in Qbecause it has no solutions in the reals R; andQis a subset of R:/ Now consider the converse. Suppose we have an equation that does have solutions in Rand in all theQpfor every PRIME p. Can we conclude that the equation has a solution in Q/? Unfortunately, in general, the answer is no, but there are classes ofequations for which the answer is yes. Such equations are said to satisfy the H ASSE PRINCIPLE . See also AX-KOCHEN ISOMORPHISM THEOREM ,D IO- PHANTINE EQUATION ,H ARMONIC NUMBER ,H ASSE PRINCIPLE ,LOCAL FIELD,LOCAL- GLOBAL PRINCIPLE , MAHLER- LECH THEOREM , P-ADIC NORM,P RODUCT FORMULA ,V ALUATION ,V ALUATION THEORY , VON STAUDT- CLAUSEN THEOREM References Burger, E. B. and Struppeck, T. "Does a/C12 n/C3001 n!Really Con- verge? Infinite Series and p-adic Analysis." Amer. Math. Monthly 103, 565/C1/577, 1996. Cassels, J. W. S. and Scott, J. W. Local Fields. Cambridge, England: Cambridge University Press, 1986. Gouve ˆa, F. Q. P-adic Numbers: An Introduction, 2nd ed. New York: Springer-Verlag, 1997. Hasse, H. "U ¨ber die Darstellbarkeit von Zahlen durch quadratische Formen im Ko ¨rper der rationalen Zahlen." J. reine angew. Math. 152, 129/C1/148, 1923. Hasses, H. "Die Normenresttheorie relativ-Abelscher Zahlk- o¨rper als Klassenko ¨rpertheorie in Kleinen." J. reine angew. Math. 162, 145/C1/154, 1930. Hensel, K. "U ¨ber eine neue Begru ¨ndung der Theorie der algebraischen Zahlen." Jahresber. Deutsch. Math. Verein 6,8 3/C1/88, 1897. Kakol, J.; De Grande-De Kimpe, N.; and Perez-Garcia, C. (Eds.). p-adic Functional Analysis. New York: Dekker, 1999. Koblitz, N. P-adic Numbers, P-adic Analysis, and Zeta- Functions, 2nd ed. New York: Springer-Verlag, 1984. Koch, H. "Valuations." Ch. 4 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 103 /C1/139, 2000. Mahler, K. P-adic Numbers and Their Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1981. Ostrowski, A. "U¨ ber sogennante perfekte Ko¨rper." J. reine angew. Math. 147, 191 /C1/204, 1917. Vladimirov, V. S. Tables of Integrals of Complex-Valued Functions of p.-adic Arguments 22 Nov 1999. http:// xxx.lanl.gov/abs/math-ph/9911027/. Weisstein, E. W. "Books about P-adic Numbers." http:// www.treasure-troves.com/books/P-adicNumbers.html. Padovan Sequence The INTEGER SEQUENCE defined by the RECURRENCE RELATION P(n) /C30P(n /C282) /C27P(n /C283) with the initial conditions P(0) /C30P(1) /C30P(2) /C301: The RECURRENCE RELATION can be solved explicitly, giving P(n) /C301 /C27 r1 rn /C272 1(2 /C27 3r1) /C271 /C27 r2 rn/C272 2(2 /C27 3r2) /C271 /C27 r3 rn/C272 3(2 /C27 3r3) ; where rn is the nth root of x3 /C27x2 /C281 /C300 : The first few terms are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, ... (Sloane’s A000931). The ratio limn0/C12 P(n) =P(n /C281) is called the PLASTIC CONSTANT . See also PERRIN SEQUENCE ,PLASTIC CONSTANT References Sloane, N. J. A. Sequences A000931/M0284 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stewart, I. "Tales of a Neglected Number." Sci. Amer. 274, 102 /C1/103, June 1996. Painleve ´ Property Following the work of Fuchs in classifying first-order ORDINARY DIFFERENTIAL EQUATIONS , Painleve ´ studied second-order ODEs OF THE FORM d2y dx2 /C30F(y0;y;x) ; where F is ANALYTIC in x and rational in y and y0: Painleve ´ found 50 types whose only movable SINGU- LARITIES are ordinary POLES . This characteristic is known as the Painleve ´ property. Six of the transcen- dents define new transcendents known as PAINLEVE ´ TRANSCENDENTS , and the remaining 44 can be inte- grated in terms of classical transcendents, quadra- tures, or the PAINLEVE ´ TRANSCENDENTS . See also PAINLEVE ´ TRANSCENDENTSPainleve ´ Transcendents There are six Painleve ´ transcendents, corresponding to second-order ordinary differential equations whose only movable singularities are ordinary poles and which cannot be integrated in terms of other known functions or transcendents. y 00/C306y2 /C27x (1) y /C302y3 /C27xy /C27 a (2) yƒ/C30y02 y/C28y0 x /C27ay2 /C27 b x/C27 gy3 /C27d y (3) yƒ/C30y02 2y /C273 2y3 /C274xy2 /C272(x2 /C28 a)y /C27b y(4) y /C301 2y /C271 y /C28 1 ! y02 /C28y0 x /C27(y /C28 1)2 x2ay /C27b y ! /C27gy x/C27dy(y /C27 1) y /C28 1 (5) y /C301 21 y /C271 y /C28 1 /C271 y /C28 x ! y02 /C271 x /C271 x /C28 1 /C271 y /C28 x ! y0 /C27y(y /C28 1)(y /C28 x) x2(x /C28 1)2 a /C27bx y2 /C27g(x /C28 1) (y /C28 1)2 /C27dx(x /C28 1) (y /C28 x)2"# (6) (Painleve ´ 1906; Ince 1956, p. 345; Zwillinger 1997, pp. 125 /C1/126). All Painleve ´ transcendents have first integrals for special values of their parameters except (2). Five of the transcendents were found by Painleve ´ and his students; the sixth transcendent was found by Gambier and contains the other five as limiting cases (Garnier 1916ab; Ince 1956, p. 345). See also PAINLEVE ´PROPERTY ,T RANSCENDENTAL FUNCTION References Garnier, R. "E ´tude de l’inte ´grale ge ´ne´rale de l’e ´quation (VI) de M. Painleve ´dans le voisinage de ses singularite ´s transcendantes." C. R. Acad. Sci. Paris 162, 939/C1/942, 1916a. Garnier, R. "E ´tude de l’inte ´grale ge ´ne´rale de l’e ´quation (VI) de M. Painleve ´dans le voisinage de ses singularite ´s transcendantes." C. R. Acad. Sci. Paris 163,8/C1/10, 1916b. Garnier, R. "E ´tude de l’inte ´grale ge ´ne´rale de l’e ´quation (VI) de M. Painleve ´dans le voisinage de ses singularite ´s transcendantes." C. R. Acad. Sci. Paris 163, 118, 1916c. Ince, E. L. "The Painleve ´Transcendents" and "The First Painleve ´Transcendent: Freedom from Movable Branch Points." §14.4 and 14.41 in Ordinary Differential Equa- tions. New York: Dover, pp. 345 /C1/347, 1956. Painleve ´, P. "Sur l’irre ´ducibilite ´des transcendantes uni- formes de ´finie par les e ´quations diffe ´rentielles du second ordre." C. R. Acad. Sci. Paris 135, 411/C1/415, 1902. Painleve ´, P. "De ´monstration de l’irre ´ducibilite ´absolue de l’e´quation y/C306y2/C27x:/"C. R. Acad. Sci. Paris 641/C1/647, 1902. Painleve ´, P. "Sur les transcendantes uniformed de´finies par l’e´quation y /C306y2 /C27x:/" C. R. Acad. Sci. Paris 135, 757 /C1/ 761, 1902. Painleve ´, P. "Sur l’irre´ducibilite ´ de l’e´quation: y /C306y2 /C27x:/" C. R. Acad. Sci. Paris 135, 1020 /C1/1025, 1902. Painleve ´, P. "Sur les e´quations diffe´rentielles du second ordre a` points critiques fixes." C. R. Acad. Sci. Paris 143, 1111--1117, 1906. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 125 /C1/126, 1997. Pair A SET of two numbers or objects linked in some way is said to be a pair. The pair a and b is usually denoted (a, b), and is generally considered to be ordered. In certain circumstances, pairs are also called BROTHERS or TWINS . See also AMICABLE PAIR,AUGMENTED AMICABLE PAIR, BROWN NUMBERS ,FRIENDLY PAIR,HEXAD ,HOMOGE- NEOUS NUMBERS ,IMPULSE PAIR,IRREGULAR PAIR, LAX PAIR,LONG EXACT SEQUENCE OF A PAIR AXIOM , MONAD ,ORDERED PAIR,PERKO PAIR,QUADRUPLET , QUASIAMICABLE PAIR,QUINTUPLET ,REDUCED AMIC- ABLE PAIR,SMITH BROTHERS ,TRIAD,TRIPLET ,TWIN PEAKS ,T WIN PRIMES ,T WINS ,U NITARY AMICABLE PAIR,W ILF-ZEILBERGER PAIR,ZIP-PAIR Pair Sum Given an AMICABLE PAIR (m, n), the quantity s(m) /C30 s(n) /C30s(m) /C27s(n) /C30m /C27n is called the pair sum, where s(n) is the DIVISOR FUNCTION and s(n) is the RESTRICTED DIVISOR FUNC- TION . See also AMICABLE PAIR Paired t-Test Given two paired sets Xi and Yi of n measured values, the paired t-test determines if they differ from each other in a significant way. Let ˆXi /C30(Xi /C28 ¯Xi) ˆYi /C30(Yi /C28 ¯Yi) ; then define t by t /C30( ¯X /C28 ¯Y)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n(n /C28 1)Pn i/C301( ˆXi /C28 ˆYi)2s : This statistic has n /C281 DEGREES OF FREEDOM . A table of STUDENT’S T-DISTRIBUTION confidence intervals can be used to determine the significance level at which two distributions differ. See also FISHER SIGN TEST,H YPOTHESIS TESTING , STUDENT’S T-DISTRIBUTION ,W ILCOXON SIGNED RANK TESTReferences Goulden, C. H. Methods of Statistical Analysis, 2nd ed. New York: Wiley, pp. 50 /C1/55, 1956. Paley Class The Paley class of a POSITIVE INTEGER m /C130 (mod 4) is defined as the set of all possible QUADRUPLES (k;e ;q; n) where m /C302e(qn /C271); q is an ODD PRIME , and k /C300i f q /C300 1i f qn /C283 /C130 (mod 4) 2i fqn /C281 /C130 (mod 4) undefined otherwise :8 >>< >>: See also HADAMARD MATRIX ,PALEY CONSTRUCTION Paley Construction HADAMARD MATRICES Hncan be constructed using FINITE FIELD GF /(pm) when p/C304l/C281 and misODD. Pick a representation rRELATIVELY PRIME top. Then by coloring white ( p/C281)=2 bc (where xbcis the FLOOR FUNCTION ) distinct equally spaced RESIDUES mod p (/r0;r,r2;...;r0;r2;r4;...; etc.) in addition to 0, a HADAMARD MATRIX is obtained if the POWERS ofr (mod p) run through B(p/C281)=2 bc :For example, n/C3012/C30111/C271/C302(5/C271)/C3022(2/C271) is of this form with p/C3011/C304/C293/C281 and m/C301. Since m/C301, we are dealing with GF(11), so pick p/C302 and compute its RESIDUES (mod 11), which are p0/C131 p1/C132 p2/C134 p3/C138 p4/C1316/C135 p5/C1310 p6/C1320/C139 p7/C1318/C137 p8/C1314/C133 p9/C136 p10/C1312/C131: Picking the first 11 =2 bc /C305RESIDUES and adding 0 gives: 0, 1, 2, 4, 5, 8, which should then be colored in the MATRIX obtained by writing out the RESIDUES increasing to the left and up along the border (0 through p /C281; followed by /C12); then adding horizontal and vertical coordinates to get the residue to place in each square. /C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12 1 00123456789 /C12 91 0012345678 /C12 891 001234567 /C12 7891 00123456 /C12 67891 0012345 /C12 567891 001234 /C12 4567891 00123 /C12 34567891 0012 /C12 234567891 001 /C12 1234567891 00 /C12 01234567891 0/C122 66666666666666666643 7777777777777777775 /H16can be trivially constructed from H4 /C156H4 : H20 cannot be built up from smaller MATRICES , so use n /C30 20 /C3019 /C271 /C302(32 /C271) /C3022(22 /C271) : Only the first form can be used, with p /C3019 /C304 /C295 /C281 and m /C301. We therefore use GF(19), and color 9 RESIDUES plus 0 white. H24 can be constructed from H2 /C156H12 :/ Now consider a more complicated case. For n /C3028 /C30 33 /C271 /C302(13 /C271); the only form having p /C304l /C281is the first, so use the GF(33) field. Take as the modulus the IRREDUCIBLE POLYNOMIAL x3 /C272x /C271; written 1021. A four-digit number can always be written using only three digits, since 1000 /C281021 /C130012 and 2000 /C282012 /C130021 : Now look at the moduli starting with 10, where each digit is considered separately. Then x0 /C131 x1 /C1310 x2 /C13100 x3 /C131000 /C1312 x4 /C13120 x5 /C131200 /C13212 x6 /C132120 /C13111 x7 /C131100 /C13122 x8 /C131220 /C13202 x9 /C132020 /C1311 x10 /C13110 x11 /C131100 /C13112 x12 /C131120 /C13102 x13 /C131020 /C132 x14 /C1320 x15 /C13200 x16 /C132000 /C1321 x17 /C13210 x18 /C132100 /C13121 x19 /C131210 /C13222 x20 /C132220 /C13211 x21 /C132110 /C13101 x22 /C13101 /C1322 x23 /C13220 x24 /C132200 /C13221 x25 /C132210 /C13201 x26 /C132010 /C131 Taking the alternate terms gives white squares as 000, 001, 020, 021, 022, 100, 102, 110, 111, 120, 121, 202, 211, and 221. References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 107 /C1/109 and 274, 1987. Beth, T.; Jungnickel, D.; and Lenz, H. Design Theory, 2nd ed. rev. Cambridge, England: Cambridge University Press, 1998. Geramita, A. V. Orthogonal Designs: Quadratic Forms and Hadamard Matrices. New York: Dekker, 1979. Kitis, L. "Paley’s Construction of Hadamard Matrices." http://www.mathsource.com/cgi-bin/msitem?0205 /C1/760. Paley’s Theorem Proved in 1933. If q is an ODD PRIME or q /C300 and n is any POSITIVE INTEGER , then there is a HADAMARDMATRIX of order m /C302e(qn /C271); where e is any POSITIVE INTEGER such that m /C13 0 (mod 4): If m is of this form, the matrix can be constructed with a PALEY CONSTRUCTION .If m is divisible by 4 but not OF THE FORM (1), the PALEY CLASS is undefined. However, HADAMARD MATRICES have been shown to exist for all m /C130 (mod 4) for m B428. See also HADAMARD MATRIX ,PALEY CLASS ,PALEY CONSTRUCTION Palindrome Number PALINDROMIC NUMBER Palindromic Number A symmetrical number which is written in some base b as a1a2 /C1/C1/C1a2a1 : The first few are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ... (Sloane’s A002113). The number of palindromic num- bers less than a given number are illustrated in the plot above. The number of palindromic numbers less than 10; 102,103, ... are 9, 18, 108, 198, 1098, 1998, 10998, ... (Sloane’s A050250). The sum of the reciprocals of the palindromic num- bers converges to a constant :3:36977 (Rivera), where this value has been computed using all palindromic numbers 5107 :/ The first few n for which the PRONIC NUMBER Pnis palindromic are 1, 2, 16, 77, 538, 1621, ... (Sloane’s A028336), and the first few palindromic numbers which are PRONIC are 2, 6, 272, 6006, 289982, ... (Sloane’s A028337). The first few numbers whose squares are palindromic are 1, 2, 3, 11, 22, 26, ... (Sloane’s A002778), and the first few palindromic squares are 1, 4, 9, 121, 484, 676, ... (Sloane’sA002779). There are no palindromic square n-digit numbers for n/C302, 4,8, 10, 14, 18, 20, 24, 30, ... (Sloane’s A034822). See also D EMLO NUMBER ,P ALINDROMIC NUMBER CONJECTURE ,PALINDROMIC PRIME ,REVERSAL References Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. New York: Dover, 1964. De Geest, P. "Palindromic Numbers and Other Recreational Topics." http://www.ping.be/~ping6758/index.shtml. De Geest, P. "Palindromic Products of Two Consecutive Integers." http://www.ping.be/~ping6758/consec.htm. De Geest, P. "Palindromic Squares." http://www.ping.be/ ~ping6758/square.htm. Dr. Pete. "The Math Forum. Ask Dr. Math: Questions & Answers from Our Archives. Palindromic Numbers." http://forum.swarthmore.edu/dr.math/problems/akyil- diz1.4.98.html. Dr. Rob. "The Math Forum. Ask Dr. Math: Questions & Answers from Our Archives. Palindromic Numbers." http://forum.swarthmore.edu/dr.math/problems/stang4.8.14.97.html. Keith, M. "On General Palindromic Numbers." http:// www.seanet.com/~ksbrown/kmath359.htm Pappas, T. "Numerical Palindromes." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 146, 1989. Rivera, C. "Problems & Puzzles: Puzzle The Honaker’s Constant.-056." http://www.primepuzzles.net/puzzles/ puzz_056.htm. Sloane, N. J. A. Sequences A002113/M0484, A002385/ M0670, A002778/M0907, A002779/M3371, A028336, A028337, A034822, and A050250 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Palindromic Number Conjecture Apply the 196-ALGORITHM , which consists of taking any POSITIVE INTEGER of two digits or more, reversing the digits, and adding to the original number. Now sum the two and repeat the procedure with the sum. Of the first 10,000 numbers, only 251 do not produce a PALINDROMIC NUMBER in 523 steps (Gardner 1979). It was therefore conjectured that all numbers will eventually yield a PALINDROMIC NUMBER . However, the conjecture has been proven false for bases which are a POWER of 2, and seems to be false for base 10 as well. Among the first 100,000 numbers, 5,996 num- bers apparently never generate a PALINDROMIC NUM- BER (Gruenberger 1984). The first few are 196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, ... (Sloane’s A006960). It is conjectured, but not proven, that there are an infinite number of palindromic PRIMES . With the exception of 11, palindromic PRIMES must have an ODD number of digits. See also 196-ALGORITHM ,DEMLO NUMBER References Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 242 /C1/245, 1979. Gruenberger, F. "How to Handle Numbers with Thousands of Digits, and Why One Might Want to." Sci. Amer. 250, 19 /C1/26, Apr. 1984. Sloane, N. J. A. Sequences A006960/M5410 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Palindromic Prime The first few palindromic PRIMES are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, ... (Sloane’s A002385; Beiler 1964, p. 228). The number of palindromic primes less than a given number are illustrated in the plot above. The number of palindromic numbers having n /C301, 2, 3, ... digits are 4, 1, 15, 0, 93, 0, 668, 0, 5172, ... (Sloane’s A016115; De Geest) and the total number of palin- dromic primes less than 10, 102,103, ... are 4, 5, 20, 20, 113, 113, 781, ... (Sloane’s A050251). The sum of the reciprocals of the palindromic primes converges to :1:32398 ; where this value has been computed using all palindromic primes 51011 (M. Keith). Palindromic primes formed from the reflected deci- mal expansion of PI include 3, 313, 31415926535897932384626433833462648323979853562951413 ; ... (Sloane’s A039954). The first few n such that both n and pnare palindromic (where pnis the nth prime) are given by 1, 2, 3, 4, 5, 8114118, ... (Sloane’s A046942; Rivera), corresponding to pnof 2, 3, 5, 7, 11, 143787341 (Sloane’s A046941; Rivera). Palindromic primes OF THE FORM ppn(x) /C30xn /C27(x /C271)n for n /C302 include 5, 181, 313, 3187813, ... (Sloane’s A050239; De Geest, Rivera), which occur for x/C301, 9, 12, 1262, ... (Sloane’s A050236; De Geest, Rivera), with no others for nB1020and xB2/C291010(De Geest). Dubner (1999) found P/C301035352/C272049402 +1017673/C271; which, at 35,353 digits is believed to be the largestknown prime that is not OF THE FORM ,abn91:/ See also PALINDROMIC NUMBER References Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. New York: Dover, 1964. De Geest, P. "Palindromic Numbers and Other Recreational Topics." http://www.ping.be/~ping6758/index.shtml. De Geest, P. "Palindromic Prime Statistics--The Table." http://www.ping.be/~ping6758/palprim1.htm. De Geest, P. "Palindromic Prime Page 3." http:// www.ping.be/~ping6758/palprim3.htm. De Geest, P. "Palindromic Sums of Squares of Consecutive Integers." http://www.ping.be/~ping6758/sumsquare.htm. Dubner, H. "Palindromic prime record: 35353 digits." [email protected] posting, 14 Nov 1999. Rivera, C. "Problems & Puzzles: Puzzle Pal-Primes and Sum of Powers.-014." http://www.primepuzzles.net/puzzles/ puzz_014.htm. Rivera, C. "Problems & Puzzles: Puzzle Pi Such that Pi is Palprime & i /C30Palindrome.-051." http://www.primepuz- zles.net/puzzles/puzz_051.htm. Rivera, C. "Problems & Puzzles: Puzzle The Honaker’s Constant.-056." http://www.primepuzzles.net/puzzles/ puzz_056.htm. Sloane, N. J. A. Sequences A002385/M0670, A016115, A039954, A046941, A046942, A050251, A050236, and A050239 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Palprime PALINDROMIC PRIME Pancake Cutting CIRCLE DIVISION BY LINES Pancake Sorting Problem Assume that n numbered pancakes are stacked, and that a spatula can be used to reverse the order of the top k pancakes for 2 5k 5n: Then the pancake sorting problem asks how many such "prefix rever- sals" are sufficient to sort an arbitrary stack (Skiena 1990, p. 48). See also PANCAKE THEOREM References Gates, W. and Papadimitriou, C. "Bounds for Sorting by Prefix Reversal." Discr. Math. 27,47/C1/57, 1979. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Pancake Theorem The 2-D version of the HAM SANDWICH THEOREM . See also HAM SANDWICH THEOREM ,PANCAKE SORT- ING PROBLEM Pancyclic Graph A simple unlabeled GRAPH on n vertices is called pancyclic if it contains cycles of all lengths, 3, 4, ..., n. Pandiagonal Square PANMAGIC SQUAREPandigital Fraction A FRACTION containing the digits 1 through 9 is called a pandigital fraction. The following table gives the number of pandigital fractions which represent sim- ple unit fractions. The numbers of pandigital frac- tions for 1/1, 1/2, 1/3, ... are 0, 12, 2, 4, 12, 3, 7, 46, 3, ... (Sloane’s A054383). f # fractions /1 2/ 12 /6729 13458;6792 13584 ;6927 13854 ;7269 14538 ;7293 14586 ;7329 14658 ;/ /7692 15384;7923 15846 ;7932 15864 ;9267 18534 ;9273 18546 ;9327 18654/ /1 3/ 2 /5823 17469;5832 17496/ /1 4/ 4 /3942 15768;4392 17568 ;5796 23184 ;7956 31824/ /15/ 12 /2697 13485;2769 13845 ;2937 14685 ;2967 14835 ;2973 14865 ;3297 16485 ;/ /3729 18645;6297 31485 ;7629 38145 ;9237 46185 ;9627 48135 ;9723 48615/ /16/ 3 /2943 17658;4653 27918 ;5697 34182/ /17/ 7 /2394 16758;2637 18459 ;4527 31689 ;5274 36918 ;5418 37926 ;5976 41832 ;/ /7614 53298/ /1 8/ 46 /3187 25496;4589 36712 ;4591 36728 ;4689 37512 ;4691 37528 ;4769 38152 ;/ /5237 41896;5371 42968 ;5789 46312 ;5791 46328 ;5839 46712 ;5892 47136 ;/ /5916 47328;5921 47368 ;6479 51832 ;6741 53928 ;6789 54312 ;6791 54328 ;/ /6839 54712;7123 56984 ;7312 58496 ;7364 58912 ;7416 59328 ;7421 59368 ;/ /7894 63152;7941 63528 ;8174 65392 ;8179 65432 ;8394 67152 ;8419 67352 ;/ /8439 67512;8932 71456 ;8942 71536 ;8953 71624 ;8954 71632 ;9156 73248 ;/ /9158 73264;9182 73456 ;9316 74528 ;9321 74568 ;9352 74816 ;9416 75328 ;/ /9421 75368;9523 76184 ;9531 76248 ;9541 76328/ /19/ 3 /6381 57429;6471 58239 ;8361 75249/ /1 10/ 0 /1 11/ 0 /1 12/ 4 /3816 45792;6129 73548 ;7461 89532 ;7632 91584/ See also PANDIGITAL NUMBER References Friedman, M. J. Scripta Math. 8. Sloane, N. J. A. Sequences A054383 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 27, 1986. Pandigital Number A decimal INTEGER which contains each of the digits from 0 to 9 (and whose leading digit must be nonzero). The first few pandigital numbers are 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, ... (Sloane’s A050278). A 10-digit pandigital number is always divisible by 9 since X9 i /C300i /C3045 : This passes the DIVISIBILITY TEST for 9 since 4 /C27 5 /C309. The smallest pandigital primes must therefore have 11 digits (no two of which can be 0). The first few pandigital primes are therefore 10123457689, 10123465789, 10123465897, 10123485679, ... (Sloa- ne’s A050288). If zeros are excluded, the first few "zeroless" pandi- gital numbers are 123456789, 123456798, 123456879, 123456897, 123456978, 123456987, ... (Sloane’s A050289), and the first few zeroless pandigital primes are 1123465789, 1123465879, 1123468597, 1123469587, 1123478659, ... (Sloane’s A050290). The sum of the first 32423 (a PALINDROMIC NUMBER ) consecutive PRIMES is 5897230146, which is pandigi- tal (Honaker). No other PALINDROMIC NUMBER shares this property. Numbers n that give zeroless pandigital numbers when the Fibonacci recurrence a(n) /C30a(n /C281) /C27a(n /C282) with a(1) /C301 and a(2) /C30n is applied are 718, 1790, 1993, 2061, 2259, 3888, 3960, 4004, 4396, 5093, 5832, 7031, 7310, 7712, 8039, 8955, 9236, .... See also PANDIGITAL FRACTION ,PERSISTENT NUMBER References De Geest, P. "The Nine Digits Page." http://www.ping.be/ ~ping6758/ninedigits.htm. Sloane, N. J. A. Sequences A050278, A050288, A050289, and A050290 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Panmagic Square If all the DIAGONALS –including those obtained by "wrapping around" the edges–of a MAGIC SQUARE sum to the same MAGIC CONSTANT , the square is said to be a panmagic square (Kraitchik 1942, pp. 143 and 189 /C1/ 191). (Only the rows, columns, and main diagonals must sum to the same constant for the usual type of magic square.) The terms DIABOLIC SQUARE (Hunter and Madachy 1975, p. 24; Madachy 1979, p. 87),PANDIAGONAL SQUARE (Hunter and Madachy 1975, p. 24), and NASIK SQUARE (Madachy 1979, p. 87) are sometimes also used. No panmagic squares exist of order 3 or any order 4k /C272 for k an INTEGER . The Siamese method for generating MAGIC SQUARES produces panmagic squares for orders 6k 91 with ordinary vector (2, 1) and break vector (1, /C281). The LO SHU is not panmagic, but it is an ASSOCIATIVE MAGIC SQUARE . Order four squares can be panmagic or ASSOCIATIVE , but not both. Order five squares are the smallest which can be both ASSOCIATIVE and panmagic, and 16 distinct ASSOCIATIVE panmagic squares exist, one of which is illustrated above (Gardner 1988). The number of distinct panmagic squares of order 1, 2, ... are 1, 0, 0, 384, 3600, 0, ... (Sloane’s A027567, Hunter and Madachy 1975). Panmagic squares arerelated to HYPERCUBES . See also ASSOCIATIVE MAGIC SQUARE ,H YPERCUBE , FRANKLIN MAGIC SQUARE ,LO SHU,MAGIC SQUARE References Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 135 /C1/137, 1961. Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213 /C1/225, 1988. Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 inMathematical Diversions. New York: Dover, pp. 24 /C1/25, 1975. Kraitchik, M. "Panmagic Squares." §7.9 in Mathematical Recreations. New York: W. W. Norton, pp. 143 and 174 /C1/ 176, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 87, 1979. Rosser, J. B. and Walker, R. J. "The Algebraic Theory of Diabolical Squares." Duke Math. J. 5, 705/C1/728, 1939. Sloane, N. J. A. Sequences A027567 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Pantograph A LINKAGE invented in 1630 by Christoph Scheiner for making a scaled copy of a given figure. The linkage is pivoted at O; hinges are denoted /C213: By placing a PENCIL at P (or P?) ; a DILATED image is obtained at P? (or P). See also HOMOTHETIC ,LINKAGE References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 232 /C1/233, 1989. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 69 /C1/70, 1969. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 5, 1928. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 167 /C1/168, 1991. Papal Cross See also CROSS Paper Folding FOLDING ,ORIGAMI Pappus Chain In the ARBELOS , construct a chain of TANGENT CIRCLES starting with the CIRCLE TANGENT to the two small interior semicircles and the large exterior one. This is called a Pappus chain (left figure). In a Pappus chain, the distance from the center of the first INSCRIBED CIRCLE P1to the bottom line is twice the CIRCLE’S RADIUS , from the second CIRCLE P2is four times the RADIUS , and for the nthCIRCLE Pnis 2ntimes the RADIUS . Furthermore, the centers of the circles Pilie on an ELLIPSE (right figure). Ifr/C13AB=AC;then the center and radius of the nth circle Pnin the Pappus chain are xn/C30r(1/C27r) 2[n2(1/C28r)2/C27r](1) yn/C30nr(1/C28r) n2(1/C28r)2/C27r(2) rn/C30(1/C28r)r 2n2(1/C28r)2/C27rhi : (3) This general result simplifies to rn/C301=(6/C27n2) for r/C30 2=3 (Gardner 1979). Further special cases when AC/C30 1/C27ABare considered by Gaba (1940). The positions of the points of tangency for the first circle are xA/C30r (1/C28r)2(4) yA/C30r(1/C28r) (1/C28r)2(5) xB/C30r(1/C27r) 1/C27r2(6) yB/C30r(1/C28r) 1/C27r2(7) xC/C30r2 1/C282r/C272r2(8) yC/C30r(1/C28r) 1/C282r/C272r2: (9) The centers of the CIRCLES lie on an ELLIPSE , and the DIAMETER of the nthCIRCLE Pnis ( /1=n)/thPERPENDI- CULAR distance to the base of the SEMICIRCLE . This result was known to Pappus, who referred to it as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981). The simplest proof is via INVERSIVE GEOMETRY . Eliminating nfrom the equa- tions for xnandyngives 4rx2 /C282r(1 /C27r)x /C27(1 /C27r)2y2 /C300 (10) 4rx/C281 4(1 /C27r)hi2 /C27 1 /C27r29+=9+; y2 /C3014 r(1 /C27r)2 (11) x /C281 4(1 /C27 r) 1 4(1 /C27 r)"#2 /C27y 12ffiffiffirp !2 /C301; (12) which is the equation of an ellipse with center ((1 /C27 r) =4; 0) and semimajor and semiminor axes (1 /C27r) =4 andffiffiffirp=2 respectively. The circles Tntangent to the first arbelos semicircle and adjacent Pappus circles Pn/C281and Pnhave posi- tions and sizes x?n /C30r(7 /C27 r) 2[4 /C27 4n(n /C28 1)(1 /C28 r)2 /C27 r(r /C28 1)](13) y?n /C302(2n /C28 1)r(1 /C28 r) 4 /C27 4n(n /C28 1)(1 /C28 r)2 /C27 r(r /C28 1)(14) r ?n /C30r(1 /C28 r) 2[4 /C27 4n(n /C28 1)(1 /C28 r)2 /C27 r(r /C28 1)] : (15) A special case of this problem with r /C301=2 (giving equal circles forming the arbelos) was considered in a Japanese temple tablet (Sangaku) problem from 1788 in the Tokyo prefecture (Rothman 1998). In this case, the solution simplifies to x ?n /C3015 215/C28 4n /C27 4n2 ðÞ(16) y?n /C302(2n /C28 1) 15 /C28 4n /C27 4n2 (17) r ?n /C301 215/C28 rn /C27 4n2 ðÞ: (18) Furthermore, the positions and radii of the three tangent circles surrounding this circle can also be found analytically, and are given by x(1) n/C30r(17 /C27 r) 212/C27 3n(3n /C28 4)(1 /C28 r)2 /C27 r(4r /C28 7)hi (19) y(1)n/C303(3n /C28 2)(1 /C28 r)r 12 /C27 3n(3n /C28 4)(1 /C28 r)2 /C27 r(4r /C28 7)(20) r(1)n/C30r(1 /C28 r) 212/C27 3n(3n /C28 4)(1 /C28 r)2 /C27 r(4r /C28 7)hi (21) x(2)n/C30r(17 /C27 r) 29/C27 3n(3n /C28 2)(1 /C28 r)2 /C28 r(1 /C28 r)hi (22) y(2)n/C303(3n /C28 1)(1 /C28 r)r 9 /C27 3n(3n /C28 2)(1 /C28 r)2 /C28 r(1 /C28 r)(23) r(2)n/C30r(1 /C28 r) 29/C27 3n(3n /C28 2)(1 /C28 r)2 /C28 r(1 /C28 r)hi (24) x(3)n/C30r(17/C277r) 29/C2712n(n/C281)(1/C28r)2/C27r(4r/C281)hi (25) y(3)n/C306(2n/C281)(1/C28r)r 9/C2712n(n/C281)(1/C28r)2/C27r(4r/C281)(26) rð3Þ n¼rð1/C28rÞ 2½9þ12nðn/C281Þð1/C28rÞ2þrð4r/C281Þ/C138: ð27Þ IfBdivides ACin the GOLDEN RATIO f;then the circles in the chain satisfy a number of other special properties (Bankoff 1955). See also ARBELOS ,COXETER’S LOXODROMIC SEQUENCE OF TANGENT CIRCLES ,SIX CIRCLES THEOREM ,SODDY CIRCLES ,STEINER CHAIN References Bankoff, L. "The Golden Arbelos." Scripta Math. 21,7 0/C1/76, 1955. Bankoff, L. "Are the Twin Circles of Archimedes Really Twins?" Math. Mag. 47, 214/C1/218, 1974. Bankoff, L. "How Did Pappus Do It?" In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 112 /C1/118, 1981. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 103, 1888. Gaba, M. G. "On a Generalization of the Arbelos." Amer. Math. Monthly 47,19/C1/24, 1940. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,18/C1/28, Jan. 1979. Hood, R. T. "A Chain of Circles." Math. Teacher 54, 134 /C1/ 137, 1961. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 117, 1929. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/91, May 1998. Steiner, J. Jacob Steiner’s gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 47, 1971. Pappus-Guldinus Theorem PAPPUS’S CENTROID THEOREM Pappus’s Centroid Theorem The SURFACE AREA S of a SURFACE OF REVOLUTION generated by the revolution of a curve about an external axis is equal to the product of the arc length s of the generating curve and the distance d1 traveled by the curve’s centroid ¯x1 ; S /C30sd1 /C302ps¯x1 : Similarly, the VOLUME V of a SOLID OF REVOLUTION generated by the revolution of a lamina about an external axis is equal to the product of the area A of the lamina and the distance d2traveled by the lamina’s centroid ¯x2 ; V /C30Ad2 /C302pA¯x2 : The following table summarizes the surface areas and volumes calculated using Pappus’s centroid theorem for various solids and surfaces of revolution. SOLID SECTION s / ¯x1/ SA / ¯x2/ V CONE RIGHT TRIANGLE/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27h2p //1 2 r//prffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27h2p //1 2 hr//1 3 hr//13 pr2/ CYLINDER CIRCLE h /1 2 r//2prh/ hr /12 r//pr2h/ SPHERE SEMI- CIRCLE/ pr//2r p//4pr2//12 pr2//4r 3p//43 pr3/ See also CENTROID (GEOMETRIC ), CROSS SECTION , PERIMETER ,SOLID OF REVOLUTION ,SURFACE AREA, SURFACE OF REVOLUTION ,TOROID ,TORUS References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987. Harris, J. W. and Stocker, H. "Guldin’s Rules." §4.1.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 96, 1998.Kern, W. F. and Bland, J. R. "Theorem of Pappus." §40 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 110 /C1/115, 1948. Pappus’s Harmonic Theorem AW, AB, and AY in the above figure are in a HARMONIC RANGE . See also CEVA’S THEOREM ,HARMONIC RANGE ,MENE- LAUS’ THEOREM References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 67 /C1/68, 1967. Pappus’s Hexagon Theorem If A, B, and C are three points on one LINE, D, E, and F are three points on another LINE, and AE meets BD at X, AF meets CD at Y, and BF meets CE at Z, then the three points X, Y, and Z are COLLINEAR . Pappus’s hexagon theorem is dual to DESARGUES’ THEOREM according to the DUALITY PRINCIPLE of PROJECTIVE GEOMETRY . See also BRIANCHON’S THEOREM ,CAYLEY- BACHARACH THEOREM ,D ESARGUES’ THEOREM ,D UALITY PRINCI- PLE,PASCAL’S THEOREM ,PROJECTIVE GEOMETRY References Coxeter, H. S. M. and Greitzer, S. L. "Pappus’s Theorem." §3.5 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 67 /C1/70, 1967. Eves, H. "Pappus’ Theorem." §6.2.6 in A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, pp. 79 and 250 /C1/251, 1965. Johnson, R. A. "Theorem of Pappus." §388 in Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 237 /C1/238, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 92 /C1/94, 1990. Pappas, T. "Pappus’ Theorem & the Nine Coin Puzzle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 163, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 168 /C1/169, 1991. Pappus’s Theorem There are several THEOREMS that generally are known by the generic name "Pappus’s Theorem." They include P APPUS’S CENTROID THEOREM , the P AP- PUS CHAIN ,PAPPUS’S HARMONIC THEOREM , and P AP- PUS’S HEXAGON THEOREM . Parabiaugmented Dodecahedron JOHNSON SOLID J59:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Parabiaugmented Hexagonal Prism JOHNSON SOLID J55:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Parabiaugmented Truncated Dodecahedron JOHNSON SOLID J69:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . ParabidiminishedRhombicosidodecahedron JOHNSON SOLID J80:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Parabigyrate Rhombicosidodecahedron JOHNSON SOLID J73:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Parabola The set of all points in the PLANE equidistant from a given LINE L(the DIRECTRIX ) and a given point Fnot on the line (the FOCUS ). The FOCAL PARAMETER (i.e., the distance between the directrix and focus) is therefore given by p/C302a;where ais the distance from the vertex to the directrix or focus. The parabola was studied by Menaechmus in an attempt to achieve CUBE DUPLICATION . Menaechmus solved the problem by finding the intersection of the two parabolas x2/C30yand y2/C302x:Euclid wrote about the parabola, and it was given its present name by Apollonius. Pascal considered the parabola as a projection of a CIRCLE , and Galileo showed that projectiles falling under uniform gravity follow para- bolic paths. Gregory and Newton considered the CATACAUSTIC properties of a parabola which bring parallel rays of light to a focus (MacTutor Archive), asillustrated above. For a parabola opening to the right with vertex at (0,0), the equation in C ARTESIAN COORDINATES is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C28a)2/C27y2q /C30x/C27a (1) (x/C28a)2/C27y2/C30(x/C27a)2(2) x2/C282ax/C27a2/C27y2/C30x2/C272ax/C27a2(3) y2/C304ax: (4) The quantity 4 ais known as the LATUS RECTUM . If the vertex is at ( x0;y0) instead of (0, 0), the equation ofthe parabola is (y/C28y0)2/C304a(x/C28x0): (5) If the parabola instead opens upwards, its equation is x2/C304ay: (6) InPOLAR COORDINATES , the equation of a parabola with parameter aand center (0, 0) is given by r/C30/C282a 1/C27cosu(7) (left figure). The equivalence with the Cartesian form can be seen by setting up a coordinate system(x?;y?)/C30(x/C28a;y) and plugging in r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x? 2/C27y?2p and u/C30tan/C281(y?=x?) to obtain ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C28a)2/C27y2q /C30/C282a 1/C27x/C28affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x/C28a)2/C27y2q: (8) Expanding and collecting terms, a/C27x/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C28x)2/C27y2q /C300; (9) so solving for y2gives (4). A set of confocal parabolas is shown in the figure on the right. InPEDAL COORDINATES with the PEDAL POINT at the FOCUS , the equation is p2/C30ar: (10) The parametric equations for the parabola are x/C30at2(11) y/C302at (12) or x/C30t2 4a(13) y/C30t: (14) A parabola may be generated as the envelope of two concurrent line segments by connecting opposite points on the two lines (Wells 1991). In the above figure, the lines SPA , SQB , and POQ are tangent to the parabola at points A, B, and O, respectively. Then SP =PA /C30QO=OP /C30BQ=QS (Wells 1991). Moreover, the CIRCUMCIRCLE of DPQS passes through the FOCUS F (Honsberger 1995, p. 47). In addition, the foot of the perpendicular to a tangent to a parabola from the FOCUS always lies on the tangent at the vertex (Honsberger 1995, p. 48). Given an arbitrary point P located "outside" a para- bola, the tangent or tangents to the parabola through P can be constructed by drawing the CIRCLE having PF as a DIAMETER , where F is the FOCUS . Then locate the points A and B at which the circle cuts the VERTICAL TANGENT through V. The points TA and TB (which can collapse to a single point in the degeneratecase) are then the points of tangency of the lines PA and PB and the parabola (Wells 1991). The CURVATURE , ARC LENGTH , and TANGENTIAL ANGLE are k(t) /C301 21/C27 t2 ðÞ3 =2 (15) s(t) /C30tffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27t2p /C27sinh/C281 t (16) f(t)/C30tan/C281t: (17) The TANGENT VECTOR of the parabola is xT(t)/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27t2p (18) yT(t)/C30tffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27t2p : (19) The plots below show the normal and tangent vectors to a parabola. See also CONIC SECTION ,E LLIPSE ,H YPERBOLA , QUADRATIC CURVE ,REFLECTION PROPERTY ,TSCHIRN- HAUSEN CUBIC PEDAL CURVE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 198 and 222 /C1/223, 1987. Casey, J. "The Parabola." Ch. 5 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 173 /C1/200, 1893. Coxeter, H. S. M. "Conics." §8.4 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115 /C1/119, 1969. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 4, 1999. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 47, 1995. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 67 /C1/72, 1972. Lockwood, E. H. "The Parabola." Ch. 1 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 2 /C1/ 12, 1967. MacTutor History of Mathematics Archive. "Parabola." http://www-groups.dcs.st-and.ac.uk/~history/Curves/ Parabola.html. Pappas, T. "The Parabolic Ceiling of the Capitol." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 22 /C1/23, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 169 /C1/172, 1991. Yates, R. C. "Conics." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 36 /C1/56, 1952. Parabola Caustic The CAUSTIC of a PARABOLA with rays PERPENDICULAR to the axis of the PARABOLA is TSCHIRNHAUSEN CUBIC . Parabola Evolute Given a PARABOLA y /C30x2 ; (1) the parametric equations of the parabola are x /C30t (2) y /C30t2 ; (3) and the derivatives are x?/C301 (4) xƒ/C300 (5) y?/C302t (6) yƒ/C302: (7) The RADIUS OF CURVATURE is therefore given by R ¼ðx?2 þ y ?2 Þ3 =2 x?yƒ/C28 x ƒy ?¼1 2 ð1 þ 4t2 Þ3 =2 : ð8Þ The TANGENT VECTOR is ˆT /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 4t2p 1 2t9+$=9+$; ; (9)so the parametric equations of the evolute are j /C30/C284t3 (10) h /C301 2 /C273t2 ; (11) and /C281 4 j /C30t3 (12) 1 3h /C28129+;k9+;7 /C30t2 (13) 1 3h /C28129+;k9+;7 /C30/C2814 j9+;k9+;72=3 (14) 1 3h /C28129+;k9+;7 /C30/C282 j 8 !2 =3 /C3014(2j)2 =3 : (15) The EVOLUTE is therefore h /C303 4(2j)2 =3 /C2712 : (16) This is known as NEILE’S PARABOLA and is a SEMI- CUBICAL PARABOLA . From a point above the evolute three normals can be drawn to the PARABOLA , while only one normal can be drawn to the PARABOLA from a point below the EVOLUTE . See also NEILE’S PARABOLA ,PARABOLA ,SEMICUBICAL PARABOLA Parabola Inverse Curve The INVERSE CURVE for a PARABOLA given by x/C30at2(1) y/C302at (2) with INVERSION CENTER (x0;y0) and INVERSION RA- DIUS kis x/C30x0/C27ka t2/C28x0 ðÞ at2/C27x0 ðÞ2/C27(2at/C28y0)2(3) y/C30y0/C27k(2at/C28y0) at2/C27x0 ðÞ2/C27(2at/C28y0)2: (4) For ( x0;y0)/C30(a;0) at the FOCUS , the INVERSE CURVE is the CARDIOID x/C30a/C27kt2/C281 ðÞ a1/C27t2 ðÞ2(5) y/C302kt a1/C27t2 ðÞ2: (6) For ( x0;y0)/C30(0;0) at the VERTEX , the INVERSE CURVE is the CISSOID OF DIOCLES x/C30k a4/C27t2 ðÞ(7) y/C302k at4/C27t2 ðÞ: (8) Parabola Involute dr dt/C301 2t9+$=9+$; (1) ˆT/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274t2p1 2t9+$=9+$; (2) ds2¼jdrj2¼ð1þ4t2Þdt2ð3Þ ds/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274t2p dt (4) s/C30gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C274t 2p dt/C301 2tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274t2p /C271 4sinh/C281(2t);(5) so the equation of the INVOLUTE is ri/C30r/C28sˆT/C30t t29+$=9+$; /C281 2tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274t2p /C271 4sinh/C281(2t) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274t2p1 2t9+$=9+$;/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C274t2pt/C281 2sinh/C281(2t) /C28sinh/C281(2t)"# : (6) Parabola Pedal Curve On the DIRECTRIX , the PEDAL CURVE of a PARABOLA is aSTROPHOID (top left). On the foot of the DIRECTRIX ,i t is a RIGHT STROPHOID (top middle). On reflection of the FOCUS in the DIRECTRIX ,i ti saM ACLAURIN TRISECTRIX (top right). On the VERTEX ,i ti sa CISSOID OFDIOCLES (bottom left). On the FOCUS ,i ti sa straight line (bottom right; Hilbert and Cohn-Vossen 1999, pp. 26 /C1/27). References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 94 /C1/97, 1972. Parabolic Coordinates A system of CURVILINEAR COORDINATES in which two sets of coordinate surfaces are obtained by revolving the parabolas of PARABOLIC CYLINDRICAL COORDI- NATES about the X-AXIS , which is then relabeled the Z-AXIS . There are several notational conventions. Whereas (u; v ; u) is used in this work, Arfken (1970) uses ( j; h;8) :/ The equations for the parabolic coordinates are x /C30uv cos u (1) y /C30uv sin u (2) z /C301 2u2 /C28v29+=9+; ; (3) where u /C23 [0;/C12) ; v /C23 [0;/C12); and u /C23 [0; 2 p): To solve for u, v, and u; examine x2 /C27y2 /C27z2 /C30u2v2 /C2714u4 /C282u2v2 /C27v49+=9+; /C301 4u4 /C272u2v2 /C27v49+=9+; /C3014u2 /C27v29+=9+;2; (4) so ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2 /C27z2p /C301 2u2 /C27v29+=9+; (5) and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2 /C27z2p /C27z /C30u2 (6) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C27y2 /C27z2p /C28z /C30v2 : (7) We therefore have u /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C27y2 /C27z2p /C27zq (8) v /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C27y2 /C27z2p /C28zq (9) u /C30tan/C281y x ! : (10) The SCALE FACTORS are hu /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 /C27v2p (11) hv /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu 2 /C27v2p (12) hu /C30uv: (13) The LINE ELEMENT is ds2 ¼ðu2 þ v2 Þðdu2 þ dv2 Þþu2v2 du2 ; (14) and the VOLUME ELEMENT is dV /C30uv u2 /C27v29+=9+; du dv du: (15) The LAPLACIAN is92f /C301 uv u2 /C27 v2 ðÞ@ @uuv@f @u ! /C27@ @vuv@f @v ! "# /C271 u2v2@2f @ u2 /C301 u2 /C27 v21 u@ @uu@f @u ! /C271 v@ @vv@f @v ! "# /C271 u2v2@2f @ u2 /C301 u2 /C27 v21 u@f @u /C27@2f @u2 /C271 v@f @v /C27@2f @v2 ! /C271 u2v2@2f @ u2 : (16) The HELMHOLTZ DIFFERENTIAL EQUATION is SEPAR- ABLE in parabolic coordinates. See also CONFOCAL PARABOLOIDAL COORDINATES , HELMHOLTZ DIFFERENTIAL EQUATION– PARABOLIC CO- ORDINATES ,PARABOLIC CYLINDRICAL COORDINATES References Arfken, G. "Parabolic Coordinates (/j; h; f) :/" §2.12 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109 /C1/112, 1970. Moon, P. and Spencer, D. E. "Parabolic Coordinates ( m; n ; c) :/" Table 1.08 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 34 /C1/36, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 660, 1953. Parabolic Cyclide ACYCLIDE formed by INVERSION of a STANDARD TORUS when INVERSION SPHERE is tangent to the TORUS . See also CYCLIDE ,INVERSION ,INVERSION SPHERE , PARABOLIC HORN CYCLIDE ,PARABOLIC RING CYCLIDE , PARABOLIC SPINDLE CYCLIDE Parabolic Cylinder AQUADRATIC SURFACE given by the equation x2 /C272rz /C300: References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 12, 1999. Parabolic Cylinder Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27 ax2 /C27bx /C27c9+=9+; /C300 (Abramowitz and Stegun 1972, p. 686; Zwillinger 1995, p. 414; Zwillinger 1997, p. 126) whose solutions are called P ARABOLIC CYLINDER FUNCTIONS . See also PARABOLIC CYLINDER FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Parabolic Cylin- der Function." Ch. 19 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 685 /C1/700, 1972. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997. Parabolic Cylinder Function These functions are sometimes called W EBER FUNC- TIONS . Whittaker and Watson (1990, p. 347) define the parabolic cylinder functions as solutions to the WEBER DIFFERENTIAL EQUATION yƒ(z)/C27n/C271 2/C2814z29+;k9+;7 y(z)/C300: (1) The two independent solutions are given by y/C30Dn(z) andD/C28n/C281zeip=29+=9+; ;where Dn(z)/C302n=2/C271=4z/C281=2Wn=2/C271=4;/C281=41 2z29+;k9+;7 (2) /C30G1 29+;k9+;7 2n=2/C271=4z/C281=2 G1 2/C2812n9+;k9+;71F11 2n/C2714;/C2814;12z29+;k9+;7 /C27G/C28129+;k9+;7 2n=2/C271=4z/C281=2 G/C281 2n9+;k9+;71F11 2n/C2714;14;12z29+;k9+;7 :(3) Here, Wa;b(z)i saW HITTAKER FUNCTION and 1F1(a;b;z)i sa CONFLUENT HYPERGEOMETRIC FUNC- TIONS . The solutions can also be written as y/C30e/C28z2=4C1Hnzffiffiffi 2p ! /C27C21F1/C281 2n;12;12z29+;k9+;7 ;" (4) where Hn(x)i saH ERMITE POLYNOMIAL . Abramowitz and Stegun (1972, p. 686) define the parabolic cylinder functions as solutions toyƒ/C27ax2/C27bx/C27c9+=9+; /C300; (5) sometimes called the PARABOLIC CYLINDER DIFFEREN- TIAL EQUATION (Zwillinger 1995, p. 414; Zwillinger 1997, p. 126). This can be rewritten by COMPLETING THE SQUARE , yƒ/C27ax/C27b 2a !2 /C28b2 4a/C27c2 435y/C300: (6) Now letting u/C30x/C27 b 2a(7) du/C30dx (8) gives d2y du2/C27au2/C27d9+=9+; y/C300 (9) where d/C13b2 4a/C27c: (10) Equation (5) has the two standard forms yƒ/C281 4x2/C27a9+;k9+;7 y/C300 (11) yƒ/C2714x2/C28a9+;k9+;7 y/C300: (12) For a general a, the EVEN and ODD solutions to (11) are y1(x)/C30e/C28x2=4 1f11 2a/C2714;12;12x29+;k9+;7 (13) y2(x)/C30xe/C28x2=4 1f112a/C2734;32;12x29+;k9+;7 ; (14) where1F1(a;b;z)i sa CONFLUENT HYPERGEOMETRIC FUNCTION .I fy(a;x) is a solution to (11), then (12) has solutions y9ia;xe/C14ip=49+=9+; ;y9ia;/C28xe/C14ip=49+=9+; : (15) Abramowitz and Stegun (1972, p. 687) define stan- dard solutions to (11) as U(a;x)/C30cosp1 4/C2712a9+;k9+;7hi Y1/C28sinp14/C2712a9+;k9+;7hi Y2(16) V(a;x)/C30sinp14/C2712a9+;k9+;7hi Y1/C27cosp14/C2712a9+;k9+;7hi Y2 G12/C28a9+;k9+;7 ; (17) where Y1/C131ffiffiffippG1 4/C2812a9+;k9+;7 2a=2/C271=4y1 /C301ffiffiffippG1 4 /C2812 a9+;k9+;7 2a =2 /C271=4e /C28x2 =4 1F112 a /C2714;12;12 x29+;k9+;7 (18) Y2 /C131ffiffiffippG3 4 /C2812 a9+;k9+;7 2a =2/C271 =4y2 /C301ffiffiffippG3 4 /C2812 a9+;k9+;7 2a =2 /C271 =4xe /C28x2 =4 1F112 a /C2734;32;12 x29+;k9+;7 ð19Þ In terms of Whittaker and Watson’s functions, U(a; x) /C30D/C28a /C281 =2(x) (20) V(a ; x) /C30G1 2 /C27 a9+;k9+;7 sin( pa)D/C28a /C281 =2(x) /C27 D /C28a /C281 =2( /C28x)hi p : (21) For NONNEGATIVE INTEGER n, the solution Dn reduces to Dn(x) /C302/C28n=2e /C28x2 =4Hnxffiffiffi 2p ! /C30e /C28x2 =4Hen(x) ; (22) where Hn(x)isaH ERMITE POLYNOMIAL and /Hen is a modified HERMITE POLYNOMIAL . The parabolic cylinder functions Dnsatisfy the RE- CURRENCE RELATIONS Dn/C271(z) /C28zDn(z) /C27 nD n/C281(z) /C300 (23) D?n(z) /C271 2 zDn(z) /C28 nD n/C281(z) /C300 : (24) The parabolic cylinder function for integral n can be defined in terms of an integral by Dn(z) /C301 p g p 0sin(nu /C28z sin u) d u (25) (Watson 1966, p. 308), which is similar to the ANGER FUNCTION . The result g/C12 /C28/C12Dm(x)Dn(x) dx /C30 dmnn!ffiffiffiffiffiffi 2pp ; (26) where dijis the KRONECKER DELTA , can also be used to determine the COEFFICIENTS in the expansion f(z) /C30X/C12 n/C300anDn (27) as an /C301 n!ffiffiffiffiffiffi 2 ppg/C12 /C28/C12Dn(t)f(t) dt : (28) For n real,g/C12 0Dn(t) ½/C1382dt /C30p1=22/C283=2f01 2/C2812n9+;k9+;7 /C28f0/C2812n9+;k9+;7 G(/C28n)(29) (Gradshteyn and Ryzhik 2000, p. 885, 7.711.3), where G(z) is the GAMMA FUNCTION andf0(z) is the POLY- GAMMA FUNCTION of order 0. See also ANGER FUNCTION ,BESSEL FUNCTION ,DAR- WIN’S EXPANSIONS ,HH FUNCTION ,STRUVE FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Parabolic Cylin- der Function." Ch. 19 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 685 /C1/700, 1972. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Iyanaga, S. and Kawada, Y. (Eds.). "Parabolic Cylinder Functions (Weber Functions)." Appendix A, Table 20.IIIinEncyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1479, 1980. Jeffreys, H. and Jeffreys, B. S. "The Parabolic Cylinder, Hermite, and Hh Functions" et seq. §23.08/C1 /23.081 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 620 /C1/627, 1988. Spanier, J. and Oldham, K. B. "The Parabolic Cylinder Function Dn(x):/" Ch. 46 in An Atlas of Functions. Wa- shington, DC: Hemisphere, pp. 445 /C1/457, 1987. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997. Parabolic Cylindrical Coordinates A system of CURVILINEAR COORDINATES . There are several different conventions for the orientation and designation of these coordinates. Arfken (1970) de- fines coordinates ( j; h ; z) such that x /C30 jh (1) y /C301 2h2 /C28 j29+=9+; (2) z /C30z: (3) In this work, following Morse and Feshbach (1953), the coordinates (u; v; z) are used instead. In this convention, the traces of the coordinate surfaces of the xy-PLANE are confocal PARABOLAS with a common axis. The u curves open into the NEGATIVE X-AXIS ; the v curves open into the POSITIVE X-AXIS . The u and v curves intersect along the Y-AXIS . x /C3012u2 /C28v29+=9+; (4) y /C30uv (5) z /C30z ; (6) where u /C23 [0;/C12) ; v /C23 [0;/C12) ; and z /C23 (/C28/C12;/C12): The SCALE FACTORS are h1 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 /C27v2p (7) h2 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 /C27v2p (8) h3 /C301 : (9) LAPLACE’S EQUATION is 92f /C301 u2 /C27 v2@2f @u2 /C27@2f @v2 ! /C27@2f @z2 : (10) The HELMHOLTZ DIFFERENTIAL EQUATION is SEPAR- ABLE in parabolic cylindrical coordinates. See also CONFOCAL PARABOLOIDAL COORDINATES , HELMHOLTZ DIFFERENTIAL EQUATION– PARABOLIC CY- LINDRICAL COORDINATES ,PARABOLIC COORDINATES References Arfken, G. "Parabolic Cylinder Coordinates (/j; h; z)." §2.8 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, p. 97, 1970. Moon, P. and Spencer, D. E. "Parabolic-Cylinder Coordi- nates ( m; n ; z) :/" Table 1.04 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 21 /C1/24, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 658, 1953. Parabolic Fixed Point A FIXED POINT of a LINEAR TRANSFORMATION for which the rescaled variables satisfy ( d /C28 a)2 /C274bg /C300: See also ELLIPTIC FIXED POINT (MAP), HYPERBOLIC FIXED POINT (MAP), LINEAR TRANSFORMATIONParabolic Geometry EUCLIDEAN GEOMETRY Parabolic Horn Cyclide A PARABOLIC CYCLIDE formed by INVERSION of a HORN TORUS when the INVERSION SPHERE is tangent to the TORUS . See also CYCLIDE ,INVERSION ,INVERSION SPHERE , PARABOLIC RING CYCLIDE ,PARABOLIC SPINDLE CY- CLIDE Parabolic Partial Differential Equation A PARTIAL DIFFERENTIAL EQUATION of second-order, i.e., one OF THE FORM Auxx /C272Buxy /C27Cuyy /C27Dux /C27Euy /C27F /C300; (1) is called parabolic if the MATRIX Z /C13AB BC9+$=9+$; (2) satisfies det(Z) /C300 : The HEAT CONDUCTION EQUATION and other diffusion equations are examples. Initial- boundary conditions are used to give u(x; t) /C30g(x; t) for x /C23@V; t > 0 (3) u(x ; 0) /C30v(x) for x /C23V; (4) where uxx /C30f(ux ; uy ; u; x; y) (5) holds in V:/ See also BOUNDARY CONDITIONS ,BOUNDARY VALUE PROBLEM ,E LLIPTIC PARTIAL DIFFERENTIAL EQUA- TION ,HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION , INITIAL VALUE PROBLEM ,P ARTIAL DIFFERENTIAL EQUATION Parabolic Point A point pon a REGULAR SURFACE M/C23R3is said to be parabolic if the G AUSSIAN CURVATURE K(p)/C300 but S(p)"0 (where Sis the SHAPE OPERATOR ), or equiva- lently, exactly one of the PRINCIPAL CURVATURES k1 and k2 is 0. See also ANTICLASTIC ,E LLIPTIC POINT ,G AUSSIAN CURVATURE ,H YPERBOLIC POINT ,P LANAR POINT , SYNCLASTIC References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 375, 1997. Parabolic Ring Cyclide A PARABOLIC CYCLIDE formed by INVERSION of a RING TORUS when the INVERSION SPHERE is tangent to the TORUS . See also CYCLIDE ,INVERSION ,INVERSION SPHERE , PARABOLIC HORN CYCLIDE ,PARABOLIC SPINDLE CY- CLIDE Parabolic Rotation The MAP x?/C30x /C271 (1) y?/C302x /C27y /C271; (2) which leaves the PARABOLA x?2 /C28y?/C30(x /C271)2 /C28(2x /C27y /C271) /C30x2 /C28y (3) invariant. See also PARABOLA ,ROTATION Parabolic Rule SIMPSON’S RULEParabolic Segment The ARC LENGTH of the parabolic segment y/C30h1/C28x2 a2 ! (1) illustrated above is given by s/C30ga /C28affiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q dx/C302ga 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27y? 2q dx (2) /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C274h2p /C27a2 4hln2h/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C274h2p a ! ; (3) and the AREA is given by A/C30ga /C28a/C30h1/C28x2 a2 ! dx/C304 3ah (4) (Kern and Bland 1948, p. 4). The weighted mean of y is /C142y/C143/C30inta /C28agh1/C28x2=a2ðÞ 0yd xd y /C308 15ah2; (5) so the CENTROID is then given by ¯y/C30/C142y/C143 A/C302 5h: (6) The AREA of the cut-off parabolic segment contained between the curves y/C30x2(7) y/C30ax/C27b (8) can be found by eliminating y, x2/C28ax/C28b/C300; (9) so the points of intersection are x9/C301 2a 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C274bp9+;k9+;7 ; (10) with corresponding y-coordinates y9/C30x2 9: The AREA is therefore given by A /C30ga /C27ffiffiffiffiffiffiffiffiffiffi a2 /C274bp a /C28ffiffiffiffiffiffiffiffiffiffi a2 /C274bp (ax /C27b) /C28x29+$9+% dx (11) /C301 6a2 /C274b9+=9+;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C274bp /C301 6a2 /C274b9+=9+;3 =2: (12) The maximum AREA of a TRIANGLE inscribed in this segment will have two of its VERTICES at the inter- sections x/C28; y/C28 ðÞ and x/C27; y/C279+=9+; ; and the third at a point x/C31; y/C31 ðÞ to be determined. From the general equation for a triangle, the AREA of the inscribed triangle is given by the DETERMINANT equation AD/C30x/C28y/C281 x/C27y/C271 x /C31 y/C31 19+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$: (13) Plugging in and using y/C31/C30x/C31 2gives AD/C301 2[b /C27(a /C28x/C31)x/C31]ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C274bp : (14) To find the maximum AREA , differentiable with respect to x/C31 and set to 0 to obtain @AD @x/C31/C301 2(a /C282x/C31)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C274bp /C300; (15) so x/C31/C301 2 a: (16) Plugging (16) into (14) then gives A /C301 8a2 /C274b9+=9+;3 =2: (17) This leads to the result known to Archimedes in the third century BC , namely A AD/C3016 1 8/C304 3 : (18) See also CENTROID (GEOMETRIC ), PARABOLA ,SEG- MENT References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 125, 1987. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 4, 1948.Parabolic Spindle Cyclide A PARABOLIC CYCLIDE formed by INVERSION of a SPINDLE TORUS when the INVERSION SPHERE is tan- gent to the TORUS . See also CYCLIDE ,INVERSION ,INVERSION SPHERE , PARABOLIC HORN CYCLIDE ,PARABOLIC RING CYCLIDE Parabolic Spiral FERMAT’S SPIRAL Parabolic Umbilic Catastrophe A CATASTROPHE which can occur for four control factors and two behavior axes. The parabolic umbilic catastrophe is given by the unfolding F(x; y; w; t; u; v) /C30y4 /C27x2y /C27ux2 /C27vy2 /C27wx /C27ty of f(x;y)/C30y4/C27x2y:/ See also CATASTROPHE THEORY References Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. Parabolic-Cylinder Coordinates PARABOLIC CYLINDRICAL COORDINATES Paraboloid The SURFACE OF REVOLUTION of the PARABOLA which is the shape used in the reflectors of automobile headlights (Steinhaus 1983, p. 242; Hilbert and Cohn-Vossen 1999). It is a QUADRATIC SURFACE which can be specified by the Cartesian equation z /C30bx2 /C27y29+=9+; : (1) The paraboloid which has radius a at height h is then given parametrically by x(u; v) /C30affiffiffiffiffiffiffiffiffi u=hp cos v (2) y(u; v) /C30affiffiffiffiffiffiffiffiffiu=hp sin v (3) z(u ; v) /C30u; (4) where u ]0 ; v /C23 [0; 2 p): / The coefficients of the FIRST FUNDAMENTAL FORM are given by E /C301 /C27a2 4hu (5) F /C300 (6) G /C30a2u h (7) and the SECOND FUNDAMENTAL FORM coefficients are e /C30a2 2uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 /C27 4a2hup (8) f /C300 (9) g /C302a2uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia4 /C27 4a2hup (10) The AREA ELEMENT is then dS /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 /C27 4a2hup 2hdu ffldv ; (11) giving SURFACE AREA S /C30g2 p 0gh 0dS /C30pa 6h2a2 /C274h29+=9+;3 =2/C28a3hi : (12) The GAUSSIAN CURVATURE is given byK /C304h2 a2 /C27 4hu ðÞ2 ; (13) and the MEAN CURVATURE H /C302ha2 /C27 2hu ðÞ a2 /C27 4hu ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia4 /C27 4a2hup : (14) The VOLUME of the paraboloid of height h is then V/C30pgh 0a2z hdz/C301 2pa2h: (15) The weighted mean of zover the paraboloid is /C142z/C143/C30pgh 0a2z hzd z/C3013pa2h2: (16) The CENTROID is then given by ¯z/C30/C142z/C143 V/C3023h (17) (Beyer 1987). See also ELLIPTIC PARABOLOID ,H YPERBOLIC PARA- BOLOID ,PARABOLA References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 133, 1987. Gray, A. "The Paraboloid." §13.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 307 /C1/308, 1997. Harris, J. W. and Stocker, H. "Paraboloid of Revolution." §4.10.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 10 /C1/11, 1999. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Paraboloid Geodesic AGEODESIC on a PARABOLOID has differential para- meters defined by P/C13@x @u !2 /C27@y @u !2 /C27@z @u !2 /C301/C27cos2v 4u/C27sin2v 4u/C301/C271 4u(1) Q/C13@2x @u@v/C27@2y @u@v/C27@2z @u@v /C300/C27ucos2v/C27usin2v/C30u (2) R/C130/C28sinv 2ffiffiffiup/C27cosv 2ffiffiffiup/C301 2ffiffiffiupcosv/C28sinv ðÞ : (3) The GEODESIC is then given by solving the E ULER- LAGRANGE DIFFERENTIAL EQUATION @P @v/C27 2v?@Q @v/C27 v?2@R @v 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P /C27 2Qv ?/C27Rv?2p /C28d duQ /C27 Rv ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP /C27 2Qv ?/C27Rv?2p ! /C300: (4) As given by Weinstock (1974), the solution simplifies to u /C28c2 /C30u(1 /C274c2) /C2sin2v /C282c ln k 2ffiffiffiffiffiffiffiffiffiffiffiffiffi u /C28c2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4u /C271p9+;k9+;7hino : (5) See also GEODESIC References Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, p. 45, 1974. Paraboloidal Coordinates CONFOCAL PARABOLOIDAL COORDINATES Paracompact Space A paracompact space is a HAUSDORFF SPACE such that every open COVER has a LOCALLY FINITE open REFINE- MENT . Paracompactness is a very common property that TOPOLOGICAL SPACES satisfy. Paracompactness is similar to the compactness property, but generalized for slightly "bigger" SPACES . All MANIFOLDS (e.g, second countable and Hausdorff) are paracompact. See also HAUSDORFF SPACE ,LOCALLY FINITE SPACE , MANIFOLD ,TOPOLOGICAL SPACE Paracycle ASTROID Paradox A statement which appears self-contradictory or contrary to expectations, also known as an ANTINOMY . Curry (1977, p. 5) uses the term PSEUDOPARADOX to describe an apparent paradox for which, however, there is no underlying actual contradiction. Bertrand Russell classified known logical paradoxes into seven categories. Ball and Coxeter (1987) give several examples of geometrical paradoxes. See also ALLAIS PARADOX ,ARISTOTLE’S WHEEL PARA- DOX,A RROW’S PARADOX ,B ANACH- TARSKI PARADOX , BARBER PARADOX ,B ERNOULLI’S PARADOX ,B ERRY PARADOX ,BERTRAND’S PARADOX ,BUCHOWSKI PARA- DOX,B URALI- FORTI PARADOX ,C ANTOR’S PARADOX , CATALOGUE PARADOX ,C OASTLINE PARADOX ,C OIN PARADOX ,E LEVATOR PARADOX ,E PIMENIDES PARA- DOX,E UBULIDES PARADOX ,G RELLING’S PARADOX , HAUSDORFF PARADOX ,HEMPEL’S PARADOX ,HETERO-LOGICAL PARADOX ,H YPERGAME ,LEONARDO’S PARA- DOX,L IAR’S PARADOX ,L OGICAL PARADOX ,P OTATO PARADOX ,P SEUDOPARADOX ,R ICHARD’S PARADOX , RUSSELL’S PARADOX ,SAINT PETERSBURG PARADOX , SIEGEL’S PARADOX ,S IMPSON’S PARADOX ,S KOLEM PARADOX ,SMARANDACHE PARADOX ,SOCRATES’ PARA- DOX,SORITES PARADOX ,THOMPSON LAMP PARADOX , UNEXPECTED HANGING PARADOX ,ZEEMAN’S PARADOX , ZENO’S PARADOXES References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 84 /C1/86, 1987. Bunch, B. Mathematical Fallacies and Paradoxes. New York: Dover, 1982. Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, 1958. Church, A. "Paradoxes, Logical." In The Dictionary of Philosophy, rev. enl. ed. (Ed. D. D Runes). New York: Rowman and Littlefield, p. 224, 1984. Curry, H. B. Foundations of Mathematical Logic. New York: Dover, 1977. Czyz, J. Paradoxes of Measures and Dimensions Originating in Felix Hausdorff’s Ideas. Singapore: World Scientific, 1994. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, 1998. Kasner, E. and Newman, J. R. "Paradox Lost and Paradox Regained." In Mathematics and the Imagination. Red- mond, WA: Tempus Books, pp. 193 /C1/222, 1989. Northrop, E. P. Riddles in Mathematics: A Book of Para- doxes. Princeton, NJ: Van Nostrand, 1944. O’Beirne, T. H. Puzzles and Paradoxes. New York: Oxford University Press, 1965. Quine, W. V. "Paradox." Sci. Amer. 206,84/C1/96, Apr. 1962. Paradromic Rings Rings produced by cutting a strip that has been given m half twists and been re-attached into n equal strips (Ball and Coxeter 1987, pp. 127 /C1/128). See also MO¨ BIUS STRIP References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 127 /C1/128, 1987. Paragyrate Diminished Rhombicosidodecahedron JOHNSON SOLID J77:/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Parallel Two lines in 2-dimensional EUCLIDEAN SPACE are said to be parallel if they do not intersect. In 3-dimen- sional EUCLIDEAN SPACE , parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. (Lines in 3-space which are not parallel but do not intersect are called SKEW LINES .) In a NON- EUCLIDEAN GEOMETRY , the concept of parallelism must be modified from its intuitive mean- ing. This is accomplished by changing the so-called PARALLEL POSTULATE . While this has counterintuitive results, the geometries so defined are still completely self-consistent. In a TRIANGLE DABC ; a MEDIAN BMBbisects all segments parallel to a given side AC (Honsberger 1995, p. 87). See also ABSOLUTE GEOMETRY ,ANTIPARALLEL ,H Y- PERPARALLEL ,L INE,N ON-EUCLIDE AN GEOMETRY , PARALLEL CURVES ,P ARALLEL LINE AND PLANE , PARALLEL LINES,PARALLEL PLANES ,PARALLEL POS- TULATE, PERPENDICULAR ,SKEW LINES References Honsberger, R. "Parallels and Antiparallels." §9.1 in Epi- sodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 87 /C1/ 88, 1995. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 9, 1948.Parallel (Surface of Revolution) A parallel of a SURFACE OF REVOLUTION is the intersection of the surface with a PLANE orthogonal to the axis of revolution. See also MERIDIAN ,SURFACE OF REVOLUTION References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 458, 1997. Parallel Axiom PARALLEL POSTULATE Parallel Class A set of blocks, also called a RESOLUTION CLASS , that partition the set V, where (V, B) is a balanced incomplete BLOCK DESIGN . See also BLOCK DESIGN ,RESOLVABLE References Abel, R. J. R. and Furino, S. C. "Resolvable and Near Resolvable Designs." §I.6 in The CRC Handbook of Combinatorial Designs (Ed. C. J. Colbourn and J. H. Di- nitz). Boca Raton, FL: CRC Press, pp. 87 /C1/94, 1996. Parallel Curves Parallel curves, frequently called "offset curves" in computer graphics applications, are curves which are displaced from a base curve by a constant offset,either positive or negative, in the direction of thecurve’s normal. The two branches of the parallel curve a distance kaway from a parametrically represented base curve ( f(t);g(t)) are x/C30f9 kg?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f?2/C27g?2p y/C30g/C14kf?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif?2/C27g?2p ; where f?/C30df=dtand g?/C30dg=dt:The above figure shows curves parallel to a CIRCLE ,ELLIPSE , and 3- petalled ROSE , where the base curves are indicated in red. See also PARALLEL ,PARALLEL LINES References Gray, A. "Parallel Curves." §5.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 115 /C1/117, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 42 /C1/43, 1972. Yates, R. C. "Parallel Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 155 /C1/ 159, 1952. Parallel Line and Plane A line and a plane are parallel if they do not intersect. See also PARALLEL ,P ARALLEL LINES ,P ARALLEL PLANES References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 9, 1948. Parallel Lines Two lines in 2-dimensional EUCLIDEAN SPACE are said to be parallel if they do not intersect. In 3-dimensional EUCLIDEAN SPACE , parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Therefore, parallel lines in 3-space lie in a single PLANE (Kern and Blank 1948, p. 9). Lines in 3- space which are not parallel but do not intersect are called SKEW LINES . See also PARALLEL ,P ARALLEL CURVES ,P ARALLEL LINE AND PLANE ,P ARALLEL PLANES ,P ARALLEL POSTULATE ,SKEW LINESReferences Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 9, 1948. Parallel Planes Two planes that do not intersect are said to be parallel. See also PARALLEL ,P ARALLEL LINES ,P ARALLEL PLANES ,PLANE References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 9, 1948. Parallel Postulate Portions of this entry contributed by MATTHEW SZUD- ZIK Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended. This statement is equivalent to the fifth of EUCLID’S POSTULATES , which Euclid himself avoided using until proposition 29 in the ELEMENTS . For centuries, many mathema- ticians believed that this statement was not a true postulate, but rather a theorem which could be derived from the first four of EUCLID’S POSTULATES . (That part of geometry which could be derived using only postulates 1 /C1/4 came to be known as ABSOLUTE GEOMETRY .) Over the years, many purported proofs of the parallel postulate were published. However, none were cor- rect, including the 28 "proofs" G. S. Klu¨gel analyzed in his dissertation of 1763 (Hofstadter 1989). The main motivation for all of this effort was that Euclid’s parallel postulate did not seem as "intuitive" as theother axioms, but it was needed to prove important results. John Wallis proposed a new axiom that implied the parallel postulate and was also intuitivelyappealing. His "axiom" states that any triangle can be made bigger or smaller without distorting its propor- tions or angles (Greenberg 1994, pp. 152 /C1 /153). How- ever, Wallis’s axiom never caught on. In 1823, Janos Bolyai and Lobachevsky indepen- dently realized that entirely self-consistent " NON- EUCLIDEAN GEOMETRIES " could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non- Euclidean geometries.) As stated above, the parallel postulate describes the type of geometry now known as PARABOLIC GEOME- TRY. If, however, the phrase "exists one and only one straight line which passes" is replaced by "exist no line which passes," or "exist at least two lines which pass," the postulate describes equally valid (though less intuitive) types of geometries known as ELLIPTIC and HYPERBOLIC GEOMETRIES , respectively. The parallel postulate is equivalent to the EQUI- DISTANCE POSTULATE ,PLAYFAIR’S AXIOM ,P ROCLUS’ AXIOM , the TRIANGLE POSTULATE , and the PYTHAGOR- EAN THEOREM . There is also a single parallel axiom in HILBERT’S AXIOMS which is equivalent to Euclid’s parallel postulate. S. Brodie has shown that the parallel postulate is equivalent to the PYTHAGOREAN THEOREM . See also ABSOLUTE GEOMETRY ,E UCLID’S AXIOMS , EUCLIDEAN GEOMETRY ,HILBERT’S AXIOMS ,NON-EU- CLIDEAN GEOMETRY ,PLAYFAIR’S AXIOM ,PYTHAGOR- EAN THEOREM ,TRIANGLE POSTULATE References Brodie, S. E. "The Pythagorean Theorem Is Equivalent to the Parallel Postulate." http://www.cut-the-knot.com/tri- angle/pythpar/PTimpliesPP.html. Dixon, R. Mathographics. New York: Dover, p. 27, 1991. Greenberg, M. J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W. H. Freeman, 1994. Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: Open Court, 1980. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 88 /C1/92, 1989. Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert’s System of Axioms." §163B in Encyclopedic Dictionary of Mathe- matics. Cambridge, MA: MIT Press, pp. 544 /C1/545, 1980. Parallelepiped In 3-D, a parallelepiped is a PRISM whose faces are all PARALLELOGRAMS . The volume of a 3-D parallelepiped is given by the SCALAR TRIPLE PRODUCT Vparallelepiped /C30½A /C215 (B /C29C)½ /C30½C /C215 (A /C29B) ½/C30½B /C215 (C /C29A) ½: In n-D, a parallelepiped is the POLYTOPE spanned by n VECTORS v1 ; ..., vn in a VECTOR SPACE over the reals, span v1 ; ...; vn ðÞ /C30t1v1 /C27.../C27tnvn ; where ti /C23 [0; 1] for i /C301, ..., n. In the usual inter- pretation, the VECTOR SPACE is taken as EUCLIDEAN SPACE , and the CONTENT of this parallelepiped is given by abs det v1 ; ... ; vn ðÞ ðÞ ; where the sign of the determinant is taken to be the "orientation" of the "oriented volume" of the paralle- lepiped.Given k vectors v1 ; ..., vkin n-dimensional space, their CONVEX HULL (along with the ZERO VECTOR ) 9+$kX tivij0 5ti 519+$7 (1) is called a parallelepiped, generalizing the notion of a parallelogram, or rather its interior, in the plane. If the number of vectors is equal to the dimension, then A /C30 v1 ...vk ðÞ (2) is a SQUARE MATRIX , and the volume of the paralle- lepiped is given by ½det A½; where the columns of A are given by the vectors v. More generally, a parallele- piped has k dimensional volume given by det ATA9+;$9+;$9+;$9+;$1 =2:/ When the vectors are TANGENT VECTORS , then the parallelepiped represents an infinitesimal k-dimen- sional VOLUME ELEMENT . Integrating this volume can give formulas for the volumes of k-dimensional objects in n-dimensional space. More intrinsically, the parallelepiped corresponds to a DECOMPOSABLE element of the EXTERIOR ALGEBRA LkRn :/ See also DETERMINANT ,DIFFERENTIAL K-FORM,EX- TERIOR ALGEBRA ,PARALLELOGRAM ,PRISMATOID ,REC- TANGULAR PARALLELEPIPED ,V OLUME ELEMENT , VOLUME INTEGRAL ,ZONOHEDRON References Phillips, A. W. and Fisher, I. Elements of Geometry. New York: Amer. Book Co., 1896. Parallelism ANGLE OF PARALLELISM Parallelizable A HYPERSPHERE Sn is parallelizable if there exist n cuts containing linearly independent tangent vectors. There exist only three parallelizable spheres: S1 ; S2 ; andS7(Adams 1962, Le Lionnais 1983). See also SPHERE References Adams, J. F. "On the Non-Existence of Elements of Hopf Invariant One." Bull. Amer. Math. Soc. 64, 279/C1/282, 1958. Adams, J. F. "On the Non-Existence of Elements of Hopf Invariant One." Ann. Math. 72,2 0/C1/104, 1960. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 49, 1983. Parallelogram A QUADRILATERAL with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a RHOMBUS , and a parallelogram whose ANGLES are all RIGHT ANGLES is called a RECTANGLE . The DIAGONALS of a parallelogram bisect each other (Casey 1888, p. 2). A parallelogram of base b and height h has AREA A /C30bh /C30ab sin A /C30ab sin B: (1) The height of a parallelogram is h /C30a sin A /C30a sin B ; (2) and the DIAGONALS p and q are p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2 /C282ab cos Ap (3) q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2 /C282ab cos Bp (4) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2 /C272ab cos Ap (5) (Beyer 1987). The sides a, b, c, d and diagonals p, q of a parallelogram satisfy p2 /C27q2 /C30a2 /C27b2 /C27c2 /C27d2 (6) (Casey 1888, p. 22). The AREA of the parallelogram with sides formed by the VECTORS (a, c) and (b, d)is A /C30detab cd9+$=9+$;9+;89+;9 /C30½ad /C28bc½: (7) Given a parallelogram P with area A(P) and linear transformation T, the AREA of T(P)is A(T(P)) /C30 ab cd9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$A(P) : (8) As shown by Euclid, if lines parallel to the sides are drawn through any point on a diagonal of a parallelo- gram, then the parallelograms not containing seg- ments of that diagonal are equal in AREA (andconversely), so in the above figure, A1/C30A2(Johnson 1929). The centers of four SQUARES erected either internally or externally on the sides of a parallelograms are the vertices of a SQUARE (Yaglom 1962, pp. 96 /C1/97; Cox- eter and Greitzer 1967, p. 84). See also DIAMOND ,LOZENGE ,PARALLELOGRAM ILLU- SION,PARALLELOGRAM LAW,Q UADRILATERAL ,R EC- TANGLE ,R HOMBUS ,S QUARE ,V ARIGNON PARALLELOGRAM ,W ITTENBAUER’S PARALLELOGRAM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., 1888. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 84, 1967. Harris, J. W. and Stocker, H. "Parallelogram." §3.6.3 in Handbook of Mathematics and Computational Science.New York: Springer-Verlag, p. 83, 1998. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 61, 1929. Yaglom, I. M. Geometric Transformations I. New York: Random House, pp. 96 /C1 /97, 1962. Parallelogram Illusion In the above figure, the sides aandbhave the same length, appearances to the contrary. In the related illusion illustrated above, the interior lines appear to be of different lengths, despite the fact that they are the same (Wells 1991). References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 86 /C1/87, 1991. Parallelogram Law The parallelogram law gives the rule for VECTOR ADDITION of vectors A and B. The sum A /C27B of the vectors is obtained by placing them head to tail and drawing the vector from the free tail to the free head. Let /C215jjdenote the NORM of a quantity. Then the quantities x and y are said to satisfy the parallelo- gram law if x /C27y kk2/C27 x /C28y kk2/C302 xkk2/C272 ykk2: If the NORM is defined as fjj/C30ffiffiffiffiffiffiffiffiffiffi f ½fhip (the so-called L2- NORM ), then the law will always hold. See also L2-NORM,NORM,VECTOR ,VECTOR ADDITION References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 1 /C1/2, 1985. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, p. 58, 1988. Parallelohedron A special class of ZONOHEDRON . There are five parallelohedra with an infinity of equal and similarly situated replicas which are SPACE-FILLING POLYHE- DRA: the CUBE , ELONGATED DODECAHEDRON , hexago- nal PRISM , RHOMBIC DODECAHEDRON , and TRUNCATED OCTAHEDRON . See also PARALLELOTOPE ,S PACE- FILLING POLYHE- DRON References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 29, 1973. Parallelotope Move a point P0 along a LINE from an initial point to a final point. It traces out a LINE SEGMENT P1 : When P1is translated from an initial position to a final position, it traces out a PARALLELOGRAM P2 : When P2is translated, it traces out a PARALLELEPIPED P3 : The generalization of Pnto n-D is then called a parallelotope. Pn has 2n vertices and Nk /C302n/C28k n k9+;89+;9 /Pk/s, where n k9+=9+; is a BINOMIAL COEFFICIENT and k /C300, 1, ..., n (Coxeter 1973). These are also the coefficients of (x /C272)n :/ See also HONEYCOMB ,HYPERCUBE ,ORTHOTOPE ,PAR- ALLELOHEDRON References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 122 /C1/123, 1973. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., 1991. Zaks, J. "Neighborly Families of Congruent Convex Poly- topes." Amer. Math. Monthly 94, 151 /C1/155, 1987. Paralogic Triangles At the points where a line cuts the sides of a TRIANGLE DA1A2A3 ; perpendiculars to the sides are drawn, forming a TRIANGLE DB1B2B3similar to the given TRIANGLE . The two triangles are also in perspective. One point of intersection of their CIRCUMCIRCLES is the SIMILITUDE CENTER , and the other is the PERSPEC- TIVE CENTER . The CIRCUMCIRCLES meet ORTHOGON- ALLY . See also CIRCUMCIRCLE ,ORTHOGONAL CIRCLES ,PER- SPECTIVE CENTER ,SIMILITUDE CENTER References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 258 /C1/262, 1929. Parameter A parameter mused in ELLIPTIC INTEGRALS defined to bem/C13k2;where kis the MODULUS .A n ELLIPTIC INTEGRAL is written I(f½m) when the parameter is used. The complementary parameter is defined by m?/C131/C28m; (1) where mis the parameter. Let qbe the NOME ,kthe MODULUS , and m/C13k2the PARAMETER . Then q(m)/C30e/C28pK?(m)=K(m)(2) where K(m) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND . Then the inverse of q(m) is given by m(q)/C30q4 2(q) q4 3(q); (3) where qiis a J ACOBI THETA FUNCTION . See also AMPLITUDE ,CHARACTERISTIC (ELLIPTIC IN- TEGRAL ), ELLIPTIC INTEGRAL ,ELLIPTIC INTEGRAL OF THE FIRST KIND,HALF-PERIOD RATIO,JACOBI THETA FUNCTIONS ,M ODULAR ANGLE ,M ODULUS (ELLIPTIC INTEGRAL ), NOME,PARAMETER References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 590, 1972. Parameter (Quadric) The number u in the QUADRIC x2 a2 /C27 u /C27y2 b2 /C27 u /C27z2 c2 /C27 u /C301 is called the parameter. See also QUADRIC Parameterization The specification of a curve, surface, etc., by means of one or more variables which are allowed to take on values in a given specified range. See also ISOTHERMAL PARAMETERIZATION ,P ARA- METRIC EQUATIONS ,R EGULAR PARAMETERIZATION , REPARAMETERIZATION ,SURFACE PARAMETERIZATION Parametric Equations Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as "para- meters." For example, while the equation of a CIRCLE in CARTESIAN COORDINATES can be given by r2 /C30x2 /C27 y2 ; one set of parametric equations for the circle are given by x /C30r cos t y /C30r sin t; illustrated above. Note that parametric representa- tions are generally nonunique, so the same quantities may be expressed by a number of different parame- terizations. A single parameter is usually represented with the parameter t, while the symbols u and v are commonly used for parametric equations in two parameters. Parametric equations provide a convenient way to represent curves and surfaces, as implemented, for example, in the Mathematica commands Parame-tricPlot [{x, y}, {t, t1, t2}] and Parametric- Plot3D [{x, y, z}, {u, u1, u2}, {v, v1, v2}]. Parametric Latitude An AUXILIARY LATITUDE also called the REDUCED LATITUDE and denoted h or u: It gives the LATITUDE on a SPHERE of RADIUS a for which the parallel has the same radius as the parallel of geodetic latitude f and the ELLIPSOID through a given point. It is given by h /C30tan/C281ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28e2p tan f9+;k9+;7 : In series form, h /C30 f /C28e1 sin(2f) /C271 2 e2 1 sin(4f) /C281 3 e3 1 sin(6 f) /C27...; where e1 /C131 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 e2p 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 e2p : See also AUXILIARY LATITUDE ,ELLIPSOID ,LATITUDE , SPHERE References Adams, O. S. "Latitude Developments Connected with Geo- desy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, 1921. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 18, 1987. Parametric Statistics See also NONPARAMETRIC STATISTICS References Sheskin, D. J. Handbook of Parametric and Nonparametric Statistical Procedures, 2nd ed. Boca Raton, FL: Chapman & Hall/CRC, 2000. Parametric Test ASTATISTICAL TEST in which assumptions are made about the underlying distribution of observed data. Parametrization PARAMETERIZATION Parenthesis One of the symbols ( or ) used to denote grouping. Parentheses have a great many specialized meanings in mathematics. A few of these are described below. 1. Parentheses are used in mathematical expres- sions to denote modifications to normal order of operations (precedence rules). In an expression like (3 /C275) /C297; the part of the expression within the parentheses, (3 /C275) /C308; is evaluated first, and then this result is used in the rest of the expres- sion. Nested parentheses work similarly, since parts of expressions within parentheses are also considered expressions. Parentheses are also used in this manner to clarify order of operations in confusing or abnormally large expressions. 2. A parenthesis can be used to denote an open end of an INTERVAL . For example, [0; 5) denotes the HALF-OPEN INTERVAL which includes all real num- bers from 0 to 5 except 5 itself. 3. Parentheses are used to enclose the variables of a FUNCTION in the form f(x) ; which means that values of the function f are dependent upon the values of x. 4. Large parentheses around two numbers, one above the other, denotes a BINOMIAL COEFFICIENT n k9+=9+; :/ 5. Parentheses around a set of two or more numbers, as in (a ; b; c) ; denote an n-tuple of numbers that are linked in some special way. 6. Large parentheses around an array of numbers, e.g.,a cb d9+=9+; indicate a MATRIX . (However, in this work, the symbola cb d9+$9+% is used instead.) 7. Parentheses may also be used to denote the GREATEST COMMON DIVISOR , e.g., (54 ;21)/C13/ /GCD(54 ;21)/C303:/ 8. Parenthesis are used to denote a CONGRUENCE , as in a/C13d(mod m):/ See also ANGLE BRACKET ,BRACE ,SQUARE BRACKET References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 282, 1997. Pareto Distribution The distribution with probability density function and distribution function P(x)/C30aba xa/C271(1) D(x)/C301/C28b x !a (2)defined over the interval x]b:The RAW MOMENTS are m?1/C30ab a/C281(3) m?2/C30ab2 a/C282(4) m?3/C30ab3 a/C283(5) m?4/C30ab4 a/C284(6) and the CENTRAL MOMENTS are m2/C30ab2 (a/C281)2(a/C282)(7) m3/C302a(a/C271)b3 (a/C281)3(a/C282)(a/C283)(8) m4/C303a(3a3/C27a/C272)b4 (a/C281)4(a/C282)(a/C283)(a/C284)(9) Giving MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS m/C30ab a/C281(10) s2/C30ab2 (a/C281)2(a/C282)(11) g1/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi a/C282 as 2(a/C271) a/C283(12) g2/C306(a3/C27a2/C286a/C282) a(a/C283)(a/C284): (13) References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 252, 1993. Parity The parity of an integer is its attribute of being EVEN orODD. Thus, it can be said that 6 and 14 have the same parity (since both are EVEN ), whereas 7 and 12 have opposite parity (since 7 is ODD and 12 is EVEN ). More specifically, the parity of an integer ncan be defined as the sum of the bits in BINARY representa- tion, computed modulo 2. The parities of the first fewintegers (starting with 0) are therefore 0, 1, 1, 0, 1, 0,0, 1, 1, 0, 0, ... (Sloane’s A010060), summarized in the following table. N Binary Parity N Binary Parity 1 1 1 11 1011 1 2 10 1 12 1100 0 3 11 0 13 1101 1 4 100 1 14 1110 1 5 101 0 15 1111 0 6 110 0 16 10000 1 7 111 1 17 10001 0 8 1000 1 18 10010 0 9 1001 0 19 10011 1 10 1010 0 20 10100 0 The parity function obeys the sum identity X2n /C271 /C281 k /C300(/C281)P(k)(k /C27r)n /C300 for any n. For example, for n /C302 and r /C300, 1 /C284 /C289 /C2716 /C2825 /C2736 /C2749 /C2864 /C300: The constant generated by the sequence of parity digits 0:011010011...2is called the THUE- MORSE CONSTANT . See also BINARY ,EVEN NUMBER ,ODD NUMBER ,THUE- MORSE CONSTANT References Commission on Mathematics of the College Entrance Ex- amination Board. Informal Deduction in Algebra: Proper- ties of Odd and Even Numbers. Princeton, NJ, 1959. Gardner, M. "Parity Checks." Ch. 8 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 71 /C1/78, 1984. Sloane, N. J. A. Sequences A010060 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Parity Constant THUE- MORSE CONSTANT Parking Constant RE´ NYI’S PARKING CONSTANTS Parodi’s Theorem The EIGENVALUES l satisfying P(l) /C300; where P( l)is the CHARACTERISTIC POLYNOMIAL , lie in the unions of the DISKS zjj51z /C27b1 jj5Xn j/C301bj9+;$9+;$9+;$9+;$: References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1119, 2000. Parrondo’s Paradox Two losing gambling games can be set up so that when they are played one after the other, they become winning. There are many ways to construct such scenarios, the simplest of which uses three biased coins (Harmer and Abbott 1999). References Doering, C. R. "Randomly Rattled Ratchets." Il Nuovo Cimento 17D, 685 /C1/697, 1995. Harmer, G. P. and Abbott, D. "Losing Strategies Can Win by Parrondo’s Paradox." Nature 402, 864, 1999. Harmer, G. P.; Abbott, D.; Taylor, P. G.; and Parrondo, J. M. R. "Parrondo’s Paradoxical Games and the Discrete Brownian Ratchet." In Proc. 2nd Internat. Conf. Unsolved Problems of Noise and Fluctuations, 11 /C1/15 July, Ade- laide (Ed. D. Abbott and L. B. Kiss). Melville, NY: Amer. Inst. Physics Press, pp. 189 /C1/200, 2000. Harmer, G. P.; Abbott, D.; Taylor, P. G.; Pearce, C. E. M.; and Parrondo, J. M. R. "Information Entropy and Parron- do’s Discrete-Time Ratchet." In Proc. Stochastic and Chaotic Dynamics in the Lakes, 16 /C1/20 August, Ambleside, UK (Ed. P. V. E. McClintock). Melville, NY: Amer. Inst. Physics Press, pp. 544 /C1/549, 2000. McClintock, P. V. E. "Unsolved Problems of Noise." Nature 401,23/C1/25, 1999. Pearce, C. E. M. "Entropy, Markov Information Sources and Parrondo Games." In Proc. 2nd Internat. Conf. Unsolved Problems of Noise and Fluctuations, 11 /C1/15 July, Ade- laide (Ed. D. Abbott and L. B. Kiss). Melville, NY: Amer. Inst. Physics Press, pp. 207 /C1/212, 2000. Pearce, C. E. M. "On Parrondo’s Paradoxical Games." In Proc. 2nd Internat. Conf. Unsolved Problems of Noise and Fluctuations, 11 /C1/15 July, Adelaide (Ed. D. Abbott and L. B. Kiss). Melville, NY: Amer. Inst. Physics Press,pp. 201 /C1 /206, 2000. Parry Circle The CIRCLE passing through the ISODYNAMIC POINTS and the CENTROID of a TRIANGLE (Kimberling 1998, pp. 227 /C1/228). See also CENTROID (TRIANGLE ), ISODYNAMIC POINTS , PARRY POINT References Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,1/C1/295, 1998. Parry Point The intersection of the P ARRY CIRCLE and the CIR- CUMCIRCLE of a TRIANGLE . The TRILINEAR COORDI- NATES of the Parry point are a 2a2 /C28 b2 /C28 c2 :b 2b2 /C28 c2 /C28 a2 :c 2c2 /C28 a2 /C28 b2 (Kimberling 1998, pp. 227 /C1/228). See also PARRY CIRCLE References Kimberling, C. "Parry Point." http://cedar.evansville.edu/ ~ck6/tcenters/recent/parry.html. Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,1/C1/295, 1998. Parseval’s Integral The POISSON INTEGRAL with n /C300, J0(z) /C301 G n /C271 29+;k9+;7hi2g p 0cos(z cos u) du; where J0(z)isaB ESSEL FUNCTION OF THE FIRST KIND and G(x)isa GAMMA FUNCTION . Parseval’s Relation Let F( n) and G( n) be the FOURIER TRANSFORMS of f(t) and g(t) ; respectively. Then g/C12 /C28/C12f(t)¯g(t) dt /C30g/C12 /C28/C12g/C12 /C28/C12F(n)e /C282 pint dn9+$=9+$;g/C12 /C28/C12¯G(n ?)e2pin?t dn ?9+$=9+$; dt /C30g/C12 /C28/C12F(n)g/C12 /C28/C12¯G( n?)g/C12 /C28/C12e2pit(n?/C28 n) dt9+$=9+$; dn ? d n /C30g/C12 /C28/C12F( n)g/C12 /C28/C12¯G(n ?) d( n ?/C28 n) dn ?9+$=9+$; dn /C30g/C12 /C28/C12F(n)¯G(n)dn; where ¯zdenotes the COMPLEX CONJUGATE . See also FOURIER TRANSFORM ,PARSEVAL’S THEOREM References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 425, 1985. Parseval’s Theorem LetE(t) be a continuous function and E(t) and Enbe FOURIER TRANSFORM pairs so that E(t)/C13g/C12 /C28/C12Ene/C282pintdn (1) ¯E(t)/C13g/C12 /C28/C12¯En?e2pin?tdn?; (2) where ¯zdenotes the COMPLEX CONJUGATE . Theng/C12 /C28/C12E(t)jj2dt/C30g/C12 /C28/C12E(t)¯E(t)dt /C30g/C12 /C28/C12g/C12 /C28/C12Ene/C282pintdng/C12 /C28/C12¯En?e2pin?tdn?9+$=9+$; dt /C30g/C12 /C28/C12g/C12 /C28/C12g/C12 /C28/C12En¯En?e2pit(n?/C28n)dndn?dt /C30g/C12 /C28/C12g/C12 /C28/C12g/C12 /C28/C12En¯En?e2pit(n?/C28n)dt dndn? /C30g/C12 /C28/C12g/C12 /C28/C12d(n?/C28n)En¯En?dndn? /C30g/C12 /C28/C12En¯En?dn/C30g/C12 /C28/C12Enjj2dn: (3) where d(x/C28x0) is the DELTA FUNCTION . For finite F OURIER TRANSFORM pairs hkandHn; XN/C281 k/C300hkjj2/C301 NXN/C281 n/C300Hnjj2: (4) If a function has a F OURIER SERIES given by f(x)/C301 2a0/C27X/C12 n/C301ancos(nx)/C27X/C12 n/C301bnsin(nx); (5) then B ESSEL’S INEQUALITY becomes an equality known as Parseval’s theorem. From (5), [f(x)]2/C301 4a2 0/C27a0X/C12 n/C301[ancos(nx)/C27bnsin(nx)] /C27X/C12 n/C301X/C12 m/C301[anamcos(nx) cos( mx) /C27anbmcos(nx) sin( mx) /C27ambnsin(nx) cos( mx) /C27bnbmsin(nx) sin( mx)]: (6) Integrating gp /C28p[f(x)]2dx /C301 4a2 0gp /C28pdx /C27a0gp /C28pX/C12 n/C301[ancos(nx)/C27bnsin(nx)]dx /C27gp /C28pX/C12 n/C301X/C12 m/C301[anamcos(nx) cos( mx) /C27anbmcos(nx) sin( mx)/C27ambnsin(nx) cos( mx) /C27bnbmsin(nx) sin( mx)]dx/C301 4a2 0(2p)/C270 /C27X/C12 n/C301X/C12 m/C301[anam pdnm /C270 /C270 /C27bnbm pdnm] ; (7) so 1 p g p /C28p[f(x)]2 dx /C301 2 a2 0 /C27X/C12 n/C301(a2n þ b2n): (8) For a generalized FOURIER SERIES with a COMPLETE BASIS ffi g/C12 i/C301 ; an analogous relationship holds. For a COMPLEX FOURIER SERIES , 1 2p g p /C28 pf(x)jj2dx /C30X/C12 n /C30/C28/C12anjj2: (9) References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1101, 2000. Part Metric A METRIC defined by d(z; w) /C30sup lnu(z) u(w)"#9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$ : u /C23 H /C27() ; where H /C27 denotes the POSITIVE HARMONIC FUNCTIONS on a DOMAIN . The part metric is invariant under CONFORMAL MAPS for any DOMAIN . References Bear, H. S. "Part Metric and Hyperbolic Metric." Amer. Math. Monthly 98, 109 /C1/123, 1991. Partial Derivative Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differ- entiation. @f @xm/C13 lim h00f(x1 ; ...; xm /C27 h; ...; xn) /C28 f(x1 ; ...; xm ; ...; xn) h : (1) The above partial derivative is sometimes denoted fxm for brevity. For a "nice" 2-D function f(x; y) (i.e., one for which f, fx ; fy ; fxy ; fyx exist and are continuous in a NEIGHBORHOOD (a, b)), then fxy(a; b) /C30fyx(a; b) : Par- tial derivatives involving more than one variable are called MIXED PARTIAL DERIVATIVES . For nice functions, mixed partial derivatives must be equal regardless of the order in which the differentia- tion is performed so, for example, fxy /C30fyx (2)fxxy /C30fxyx /C30fyxx : (3) For an EXACT DIFFERENTIAL , df /C30@f @x ! ydx /C27@f @y ! xdy; (4) so @y @x ! f/C30/C28@f @x ! y @f @y ! x: (5) A differential equation expressing one or more quan- tities in terms of partial derivatives is called a PARTIAL DIFFERENTIAL EQUATION . Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve. If the continuity requirement for MIXED PARTIALS is dropped, it is possible to construct functions for which MIXED PARTIALS arenot equal. An example is the function f(x;y)/C30xy(x2/C28y2) x2/C27y2for(x;y)"(0;0) 0 for( x;y)/C30(0;0);8 < :(6) which has fxy(0;0)/C30/C281 and fyx(0;0)/C301 (Wagon 1991). This function is depicted above and by Fischer (1986). Abramowitz and Stegun (1972) give FINITE DIFFER- ENCE versions for partial derivatives. See also ABLOWITZ- RAMANI- SEGUR CONJECTURE ,DE- RIVATIVE ,MIXED PARTIAL DERIVATIVE ,MONKEY SAD- DLE,PARTIAL DIFFERENTIAL EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 883 /C1/885, 1972. Fischer, G. (Ed.). Plate 121 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume.Braunschweig, Germany: Vieweg, p. 118, 1986. Thomas, G. B. and Finney, R. L. §16.8 in Calculus and Analytic Geometry, 9th ed. Reading, MA: Addison-Wesley, 1996. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 83 /C1/85, 1991. Partial Differential Equation A partial differential equation (PDE) is an equation involving functions and their PARTIAL DERIVATIVES ; for example, the WAVE EQUATION @2 c @x2 /C27@2 c @y2 /C27@2 c @z2 /C301 v2@2 c @t2 : (1) in general, partial differential equations are much more difficult to solve analytically than are ORDINARY DIFFERENTIAL EQUATIONS . They may sometimes be solved using a BA¨ CKLUND TRANSFORMATION , CHARAC- TERISTIC ,GREEN’S FUNCTION , INTEGRAL TRANSFORM , LAX PAIR, SEPARATION OF VARIABLES , or–when all else fails (which it frequently does)–numerical methods. Fortunately, partial differential equations of second- order are often amenable to analytical solution. Such PDEs are of the form Auxx /C272Buxy /C27Cuyy /C27Dux /C27Euy /C27F /C300 : (2) Second-order PDEs are then classified according to the properties of the MATRIX Z /C13AB BC9+$=9+$; (3) as ELLIPTIC , HYPERBOLIC ,or PARABOLIC . If Z is a POSITIVE DEFINITE MATRIX , i.e., det(Z) > 0 ; the PDE is said to be ELLIPTIC .LAPLACE’S EQUATION and POISSON’S EQUATION are examples. Boundary condi- tions are used to give the constraint u(x; y) /C30g(x; y) on @V; where uxx /C27uyy /C30f(ux ; uy ; u; x; y) (4) holds in V:/ If det /(Z) B0; the PDE is said to be HYPERBOLIC . The WAVE EQUATION is an example of a hyperbolic partial differential equation. Initial-boundary conditions are used to give u(x; y; t) /C30g(x; y; t) for x /C23@V; t > 0 (5) u(x; y; 0) /C30v0(x; y)in V (6) ut(x; y; 0) /C30v1(x; y)in V; (7) where uxy /C30f(ux ; ut ; x; y) (8) holds in V:/ If det /(Z) /C300; the PDE is said to be parabolic. The HEAT CONDUCTION EQUATION equation and other diffusion equations are examples. Initial-boundary conditions are used to giveu(x;t)/C30g(x;t) for x/C23@V;t>0 (9) u(x;0)/C30v(x) for x/C23V; (10) where uxx/C30f(ux;uy;u;x;y) (11) holds in V:/ See also BA¨ CKLUND TRANSFORMATION ,B OUNDARY CONDITION S,C HARACTERISTIC (PARTIAL DIFFEREN- TIAL EQUATION ), ELLIPTIC PARTIAL DIFFERENTIAL EQUATION ,GREEN’S FUNCTION ,HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION ,INTEGRAL TRANSFORM , JOHNSON’S EQUATION ,L AX PAIR,M ONGE- AMPE` RE DIFFERENTIAL EQUATION ,PARABOLIC PARTIAL DIF- FERENTIAL EQUATION ,SEPARATION OF VARIABLES References Arfken, G. "Partial Differential Equations of Theoretical Physics." §8.1 in Mathematical Methods for Physicists, 3rd ed.Orlando, FL: Academic Press, pp. 437 /C1/440, 1985. Bateman, H. Partial Differential Equations of Mathematical Physics. New York: Dover, 1944. Conte, R. Exact Solutions of Nonlinear Partial Differential Equations by Singularity Analysis. 13 Sep 2000. http:// xxx.lanl.gov/abs/nlin.SI/0009024/. Folland, G. B. Introduction to Partial Differential Equa- tions, 2nd ed. Princeton, NJ: Princeton University Press, 1996. Kevorkian, J. Partial Differential Equations: Analytical Solution Techniques, 2nd ed. New York: Springer-Verlag, 2000. Morse, P. M. and Feshbach, H. "Standard Forms for Some of the Partial Differential Equations of Theoretical Physics."Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 271 /C1 /272, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Partial Differential Equations." Ch. 19 inNumerical Recipes in FORTRAN: The Art of ScientificComputing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 818 /C1 /880, 1992. Sobolev, S. L. Partial Differential Equations of Mathemati- cal Physics. New York: Dover, 1989. Sommerfeld, A. Partial Differential Equations in Physics. New York: Academic Press, 1964. Taylor, M. E. Partial Differential Equations, Vol. 1: Basic Theory. New York: Springer-Verlag, 1996. Taylor, M. E. Partial Differential Equations, Vol. 2: Quali- tative Studies of Linear Equations. New York: Springer- Verlag, 1996. Taylor, M. E. Partial Differential Equations, Vol. 3: Non- linear Equations. New York: Springer-Verlag, 1996. Webster, A. G. Partial Differential Equations of Mathema- tical Physics, 2nd corr. ed. New York: Dover, 1955. Weisstein, E. W. "Books about Partial Differential Equa- tions." http://www.treasure-troves.com/books/PartialDif- ferentialEquations.html. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997. Partial Fraction Decomposition ARATIONAL FUNCTION P(x)=Q(x) can be rewritten using what is known as partial fraction decomposi- tion. This procedure often allows integration to beperformed on each term separately by inspection. For each factor of Q(x) the form ( ax/C27b) m;introduce terms A1 ax /C27 b /C27A2 (ax /C27 b)2 /C27.../C27Am (ax /C27 b)m : (1) For each factor OF THE FORM (ax2 /C27bx /C27c)m ; intro- duce terms A1x /C27 B1 ax2 /C27 bx /C27 c /C27A2x /C27 B2 (ax2 /C27 bx /C27 c)2 /C27... /C27Amx /C27 Bm (ax2 /C27 bx /C27 c)m : (2) Then write P(x) Q(x) /C30A1 ax /C27 b /C27.../C27A2x /C27 B2 ax2 /C27 bx /C27 c /C27... (3) and solve for the Ai/s and Bi/s. Partial fraction decomposition is implemented in Mathematica 4.0 asApart . References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 13 /C1/15, 1987. Partial Integration INTEGRATION BY PARTS Partial Latin Square In a normal n /C29n LATIN SQUARE , the entries in each row and column are chosen from a "global" set of n objects. Like a Latin square, a partial Latin square has no two rows or columns which contain the same two symbols. However, in a partial Latin square, each cell is assigned one of its own set of n possible "local" (and distinct) symbols, chosen from an overall set of more than three distinct symbols, and these symbols may vary from location to location. For example, given the possible symbols f1; 2; ...; 6g which must be arranged as f1; 2; 3g f2; 3; 5g f4; 3; 6gf1; 3; 4 g f1; 2; 3 g f3; 5; 6 gf2 ; 5 ; 6 g f4 ; 5 ; 6 g f2; 3; 5g; the 3 /C293 partial Latin square 132 245653 can be constructed. See also D INITZ PROBLEM ,LATIN SQUARE References Cipra, B. "Quite Easily Done." In What’s Happening in the Mathematical Sciences 2, pp. 41 /C1/46, 1994. Partial Order A RELATION "/5/" is a partial order on a SET S if it has:1. Reflexivity: a 5a for all a /C23 S:/ 2. Antisymmetry: a 5b and b 5a implies a /C30b. 3. Transitivity: a 5b and b 5c implies a 5c :/ For a partial order, the size of the longest CHAIN (ANTICHAIN ) is called the LENGTH (WIDTH ). A partially ordered set is also called a poset. A largest set of unrelated vertices in a PARTIAL ORDER can be found using MaximumAntichain [g] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). MinimumChain- Partition [g] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ) partitions a partial order into a minimum number of CHAINS . See also ANTICHAIN ,C HAIN ,F ENCE POSET ,IDEAL (PARTIAL ORDER ), LENGTH (PARTIAL ORDER ), LINEAR EXTENSION ,PARTIALLY ORDERED SET,TOTAL ORDER , WIDTH (PARTIAL ORDER ) References Ruskey, F. "Information on Linear Extension." http:// www.theory.csc.uvic.ca/~cos/inf/pose/LinearExt.html. Skiena, S. "Partial Orders." §5.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 203 /C1/ 209, 1990. Partial Quotient If the SIMPLE CONTINUED FRACTION of a REAL NUMBER x is given by x /C30a0 /C271 a1 /C271 a2 /C271 a3 /C27 ...; then the quantities ai are called partial quotients. See also CONTINUED FRACTION ,CONVERGENT ,SIMPLE CONTINUED FRACTION Partially Ordered Set A partially ordered set (or poset) is a SET taken together with a PARTIAL ORDER on it. Formally, a partially ordered set is defined as an ordered pair P /C30 (X ;5); where X is called the GROUND SET of P and 5is the PARTIAL ORDER ofP. See also CIRCLE ORDER ,C OVER RELATION ,D OMI- NANCE ,G ROUND SET,H ASSE DIAGRAM ,INTERVAL ORDER ,ISOMORPHIC POSETS ,O RDER ISOMORPHIC , PARTIAL ORDER ,POSET DIMENSION ,REALIZER ,RELA- TION References Dushnik, B. and Miller, E. W. "Partially Ordered Sets." Amer. J. Math. 63, 600 /C1/610, 1941. Fishburn, P. C. Interval Orders and Interval Sets: A Study of Partially Ordered Sets. New York: Wiley, 1985. Skiena, S. "Partial Orders." §5.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 203 /C1/ 209, 1990. Trotter, W. T. Combinatorics and Partially Ordered Sets: Dimension Theory. Baltimore, MD: Johns Hopkins Uni- versity Press, 1992. Particularly Well-Behaved Functions Functions which have DERIVATIVES of all orders at all points and which, together with their DERIVATIVES , fall off at least as rapidly as xjj/C28nas xjj0/C12; no matter how large n is. See also REGULAR SEQUENCE Partisan Game A GAME for which each player has a different set of moves in any position. Every position in an IMPARTIAL GAME has a NIM-VALUE . Partition A partition is a way of writing an INTEGER n as a sum of POSITIVE INTEGERS where the order of the sum- mands is not significant, possibly subject to one or more additional constraints. By convention, parti- tions are normally written from largest to smallest summands (Skiena 1990, p. 51), e.g., 10 /C303 /C272 /C272 /C27 2 /C271: PartitionsQ [p] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ) tests a list to determine that it consists of positive integers and therefore is a valid partition. Andrews (1998, p. 1) used the notation l /C159n to indicate "a sequence l is a partition of n," and the notationa12a2 /C1/C1/C1) to abbreviate the partition f1; ...; 1|fflfflfflfflfflffl{zfflfflfflfflfflffl} a1; 2; ...; 2|fflfflfflfflfflffl{zfflfflfflfflfflffl} a2; ...g:/ Particular types of partition functions include the PARTITION FUNCTION P, giving the number of parti- tions of a number as a sum of smaller integers without regard to order, and PARTITION FUNCTION Q, giving the number of ways of writing the INTEGER n as a sum of POSITIVE INTEGERS without regard to order and with the constraint that all INTEGERS in each sum are distinct. The PARTITION FUNCTION B, which gives the number of partitions of n in which no parts are multiples of k is sometimes also used (Gordon and Ono 1997). The EULER TRANSFORM bngives the number of partitions of n into integer parts of which there are a1different types of parts of size 1, a2of size 2, etc. For example, if an /C301 for all n, then bn is the number of partitions of n into integer parts. Similarly, if an /C301 for n prime and an /C300 for n composite, then bnis the number of partitions of n into prime parts (Sloane and Plouffe 1995, p. 21). A partition of a number n into a sum of elements of a list L can be determined using a GREEDY ALGORITHM . The following table gives the number of partitions of ninto a sum of positive powers pfor multiples of n. np /C301 p/C302 p/C303 p/C304 Sloane’s A000041Sloane’sA001156Sloane’sA003108Sloane’sA046042 10 42 4 2 1 50 204226 104 10 4 100 190569292 1116 39 9150 40853235313 6521 97 15 200 3972999029388 27482 208 24 250 /2:307/C291014/ 388 34 300 /9:253/C291015/ 683 49 See also AMENABLE NUMBER ,CONJUGATE PARTITION , DURFEE SQUARE ,ELDER’S THEOREM ,FERRERS DIA- GRAM ,GO¨ LLNITZ’S THEOREM ,GRAPHICAL PARTITION , GREEDY ALGORITHM ,PARTITION FUNCTION B,PARTI- TION FUNCTION P,PARTITION FUNCTION Q,PERFECT PARTITION ,P LANE PARTITION ,P RIME PARTITION , SELF-CONJUGATE PARTITION ,SET PARTITION ,SOLID PARTITION ,STANLEY’S THEOREM References Andrews, G. E. The Theory of Partitions. Cambridge, Eng- land: Cambridge University Press, 1998. Dickson, L. E. "Partitions." Ch. 3 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chel- sea, pp. 101 /C1/164, 1952. Gordon, B. and Ono, K. "Divisibility of Certain Partition Functions by Powers of Primes." Ramanujan J. 1,2 5/C1/34, 1997. Hardy, G. H. and Wright, E. M. "Partitions." Ch. 19 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 273 /C1/296, 1979. Savage, C. "Gray Code Sequences of Partitions." J. Algo- rithms 10, 577/C1/595, 1989. Skiena, S. "Partitions." §2.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 51 /C1/59, 1990. Sloane, N. J. A. Sequences A000041/M0663, A001156/ M0221, A003108/M0209, and A046042 in "An On-LineVersion of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Partition Function b The number of partitions of nin which no parts are multiples of kis sometimes denoted bk(n) (Gordon and Ono 1997). bk(n) is also the number of partitions ofninto at most k/C281 copies of each part. /b2(n)/C30Q(n);where Q(n) is the PARTITION FUNCTION Q, and bp(n) is the number of irreducible p-modular representations of the SYMMETRIC GROUP Sn:The generating function for bk(n) is given by X/C12 n/C300bk(n)xn/C30Y/C12 n/C3011/C28xkn 1/C28xn: (1) The following table gives the first few values of bk(n) for small k. kSloane /bk(n)/ 2 A000009 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, ... 3 A000726 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, ... 4 A001935 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, ... 5 A035959 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, ... Gordon and Ono (1997) show that b5(5n/C274)/C130 (mod 5) (2) b7(7n/C275)/C130 (mod 7) (3) b11(11n/C276)/C130 (mod 11) : (4) Defining Sk(N;M) as the number of positive integers n5Nfor which bk(n)/C130 (mod M);Gordon and Ono (1997) proved that if pai i]ffiffiffi kp ;then lim N0/C12Sk(N;pj i) N/C301 (5) for all j, where k/C30pa1 1pa2 2/C1/C1/C1pamm:/ References Andrews, G. E. The Theory of Partitions. Cambridge, Eng- land: Cambridge University Press, p. 109, 1998. Carlitz, L. "Generating Functions and Partition Problems." InTheory of Numbers (Ed. A. L. Whiteman). Providence, RI: Amer. Math. Soc., pp. 144 /C1/169, 1965. Cayley, A. "A Memoir on the Transformation of Elliptic Functions." Collected Mathematical Papers, Vol. 9. Lon- don: Cambridge University Press, p. 128, 1889 /C1/1897. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 241, 1985. Gordon, B. and Ono, K. "Divisibility of Certain Partition Functions By Powers of Primes." Ramanujan J. 1,2 5/C1/34, 1997.Sloane, N. J. A. Sequences A000009/M0281, A000726/ M0316, A001935/M0566, and A035959 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Partition Function P /P(n);denotes also denoted p(n);gives the number of ways of writing the INTEGER nas a sum of POSITIVE INTEGERS , where the order of summands is not considered significant. By convention, partitions are usually ordered from largest to smallest (Skiena1990, p. 51). For example, since 4 can be written 4/C304 /C303/C271 /C302/C272 /C302/C271/C271 /C301/C271/C271/C271 (1) it follows that P(4)/C305:The function P(n) is imple- mented in Mathematica asPartitionsP [n]. The values of P(n) for n/C301, 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (Sloane’s A000041). The following table gives the value of P(n) for selected small n. n /P(n)/ 50 204226 100 190569292 200 3972999029388300 9253082936723602400 6727090051741041926500 2300165032574323995027600 458004788008144308553622700 60378285202834474611028659800 5733052172321422504456911979900 415873681190459054784114365430 1000 24061467864032622473692149727991 6++++++ /C273+++ /C273+++ /C272++ /C271+ /C3015 When explicitly listing the partitions of a number n, the simplest form is the so-called natural representa- tion which simply gives the sequence of numbers in the representation (e.g., (2, 1, 1) for the number 4 /C30 2/C271/C271):The multiplicity representation instead gives the number of times each number occurs together with that number (e.g., (2, 1), (1, 2) for 4 /C30 2/C2151/C271/C2152):The F ERRERS DIAGRAM is a pictorial representation of a partition. For example, the dia- gram above illustrates the F ERRERS DIAGRAM of the partition 6 /C273/C273/C272/C271/C3015:/ Euler gave a GENERATING FUNCTION forP(n) using the Q-SERIES (q)/C12/C13Y/C12 m/C301(1/C28qm)/C30X/C12 n/C30/C28/C12(/C281)nq(3n/C271)=2(2) /C301/C28q/C28q2/C27q5/C27q7/C28q12/C28q15/C27q22/C27q26/C27...:(3) Here, the exponents are generalized PENTAGONAL NUMBERS 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (Sloane’s A001318) and the sign of the kth term (counting 0 as the 0th term) is ( /C281)(k/C271)=2 bc(with xbcthe FLOOR FUNCTION ). Then the partition numbers P(n) are given by the GENERATING FUNCTION 1 (q)/C12/C30X/C12 n/C300P(n)qn/C301/C27q/C272q2/C273q3/C275q4/C27... ( 4 ) (Hirschhorn 1999). Hirschhorn (1999) gives the addi- tional beautiful identity 1 (q)/C12/C30(q)9 /C12 (q)10/C12/C30((q)3/C12)3 ((q)5/C12)2: (5) Another GENERATING FUNCTION is given by X/C12 n/C300P(n)tn/C302t1=8 q?1(0;ffiffi tp ) !1=3 ; (6) where q?1(0;x) is the derivative of the J ACOBI THETA FUNCTION of the first kind. The number of partitions of a number nintomparts is equal to the number of partitions into parts of which the largest is m, and the number of partitions into at most mparts is equal to the number of partitions into parts which do not exceed m. Both these results follow immediately from noting that a FERRERS DIAGRAM can be read either row-wise or column-wise (although the default order is row-wise;Hardy 1999, p. 83). For example, if a n/C301 for all n, then the E ULER TRANSFORM bnis the number of partitions of ninto integer parts.Euler invented a GENERATING FUNCTION which gives rise to a POWER SERIES inP(n); P(n)/C30Xn k/C301(/C281)k/C271 /C2Pn/C281 2k(3k/C281)9+;k9+;7 /C27Pn/C2812k(3k/C271)9+;k9+;7 hi (7) (Skiena 1990, p. 57). Other recurrence formulas includeP(2n/C271)/C30P(n)/C27X/C12 k/C301Pn/C284k2/C283k9+=9+; /C27Pn/C284k2/C273k9+=9+; 9+$9+% /C28X/C12 k/C301(/C281)kP2n/C271/C283k2/C27k9+=9+;9+$ /C27P2n/C271/C283k2/C28k9+=9+; /C138 (8) and P(n)/C301 nXn/C281 k/C300s(n/C28k)P(k); (9) where s(n) is the DIVISOR FUNCTION (Skiena 1990, p. 77; Berndt 1994, p. 108), as well as the identity X(ffiffiffiffiffiffiffiffiffiffiffi 24n/C271p /C281)=6 bc k/C30/C28 (ffiffiffiffiffiffiffiffiffiffiffi 24n/C271p /C271)=6 de(/C281)kPn/C281 2k(3k/C271)9+;k9+;7 /C300;(10) where xbcis the FLOOR FUNCTION and xdeis the CEILING FUNCTION . ARECURRENCE RELATION involving the PARTITION FUNCTION Q is given by P(n)/C30Xn=2bc k/C300Q(n/C282k)P(k): (11) Atkin and Swinnerton-Dyer (1954) obtained the unexpected identities X/C12 n/C300P(5n)qn /C13Y/C12 n/C301(1/C28q5n/C283)(1/C28q5n/C282)(1/C28q5n) (1/C28q5n/C284)2(1/C28q5n/C281)2(mod 5) (12) X/C12 n/C300P(5n/C271)qn /C13Y/C12 n/C301(1/C28q5n) (1/C28q5n/C284)(1/C28q5n/C281)(mod 5) (13) X/C12 n/C300P(5n/C272)qn /C132Y/C12 n/C301(1/C28q5n) (1/C28q5n/C283)(1/C28q5n/C282)(mod 5) (14) X/C12 n/C300P(5n/C273)qn /C133Y/C12 n/C301(1/C28q5n/C284)(1/C28q5n/C281)(1/C28q5n) (1/C28q5n/C283)2(1/C28q5n/C282)2(mod 5) (15) (Hirschhorn 1999). MacMahon obtained the beautiful RECURRENCE RELA- TION P(n)/C28P(n/C281)/C28P(n/C282)/C27P(n/C285)/C27P(n/C287) /C28P(n/C2812)/C28P(n/C2815)/C27.../C300; (16) where the sum is over generalized PENTAGONAL NUMBERS 5nand the sign of the kth term is (/C281)(k/C271)=2 bc;as above. Ramanujan stated without proof the remarkable identities P(4)/C27P(9)x/C27P(14)x2/C27... /C305[(1/C28x5)(1/C28x10)(1/C28x15)/C1/C1/C1]5 [(1/C28x)(1/C28x2)(1/C28x3)/C1/C1/C1]6(17) (Darling 1921; Mordell 1922; Hardy 1999, pp. 89 /C1/90), and P(5)/C27P(12)x/C27P(17)x2/C27... /C3071/C28x7ðÞ 1/C28x14ðÞ 1/C28x21ðÞ /C1 /C1 /C1 ½/C1383 1/C28x ðÞ 1/C28x2 ðÞ 1/C28x3 ð Þ/C1/C1/C1 ½/C1384 /C2749x1/C28x7ðÞ 1/C28x14ðÞ 1/C28x21ð Þ/C1/C1/C1 ½/C1387 1/C28x ðÞ 1/C28x2 ðÞ 1/C28x3 ðÞ /C1 /C1 /C1 ½/C1388 (18) (Mordell 1922; Hardy 1999, pp. 89 /C1/90). Hardy and Ramanujan (1918) used the CIRCLE METHOD and MODULAR FUNCTIONS to obtain the asymptotic solution P(n)/C21 4nffiffiffi 3pepffiffiffiffiffiffiffi 2n=3p (19) (Hardy 1999, p. 116), which was also independently discovered by Uspensky (1920). Rademacher (1937)subsequently obtained an exact convergent seriessolution which yields the Hardy-Ramanujan formula (19) as the first term: P(n)/C30 1 pffiffiffi 2pX/C12 k/C301Ak(n)ffiffiffi kp /C2d dn?sinhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3n?/C281 249+;k9+;7r9+;89+;9 kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n?/C281 24q2 6666643 7777758 >>>>>< >>>>>:9 >>>>>= >>>>>; n?/C30n; (20) where Ak(n)/C30Xk h/C301dGCD( h;k);1 /C2exppiXk/C281 j/C301i khj k/C28hj k$% /C281 2 ! /C282pihn k"# ;(21) /dmnis the K RONECKER DELTA , and xbcis the FLOOR FUNCTION (Hardy 1999, pp. 120 /C1/121). The remainder after Nterms isR(N)BCN/C281=2/C27Dffiffiffiffiffi N ns sinhKffiffiffinp N ! ; (22) where Cand Dare fixed constants (Apostol 1997, pp. 104 /C1/110; Hardy 1999, pp. 121 and 128). Rather amazingly, the CONTOUR used by Rademacher in- volves F AREY SEQUENCES and F ORD CIRCLES (Apostol 1997, pp. 102 /C1/104; Hardy 1999, pp. 121 /C1/122). In 1942, Erdos showed that the formula of Hardy and Ramanujan could be derived by elementary means (Hoffman 1998, p. 91). With f(x) as defined above, Ramanujan also showed that 5(q5)5 /C12(x5) (q)6 /C12/C30X/C12 m/C300P(5m/C274)xm: (23) Ramanujan also found numerous PARTITION FUNC- TION PCONGRUENCES . Let fO(x) be the GENERATING FUNCTION for the number of partitions PO(n)o f ncontaining ODD numbers only and fD(x) be the GENERATING FUNCTION for the number of partitions PD(n)o f nwithout duplication, then fO(x)/C30fD(x)/C30Y/C12 k/C301;3;...X/C12 i/C280xik /C301Q/C12 k/C301;3;...1/C28xk /C30Y/C12 k/C301(1/C27xk)/C301/C27x/C27x2 /C272x3/C272x4/C273x5/C27...; (24) as discovered by Euler (Honsberger 1985; Andrews 1998, p. 5; Hardy 1999, p. 86), giving the first fewvalues of P O(n)/C30PD(n) forn/C300, 1, ... as 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (Sloane’s A000009). The identity Y/C12 k/C301(1/C27zk)/C30Y/C12 k/C301(1/C27z2k/C281)/C281; (25) /C301/C28x/C28x2/C27x5/C27x7/C28x12/C28x15/C27. . . (26) /C301/C27X/C12 k/C301ck; (27) where ck/C30(/C281)nforkof the form1 2n(3n91) 0 otherwise ;9+$k (28) which is the GENERATING FUNCTION for the difference between the number of partitions into an even number of unequal parts and the number of parti-tions in an odd number of unequal parts, is known as the E ULER IDENTITY (Hardy 1999, p. 84). Let PE(n) be the number of partitions of EVEN numbers only, and let PEO(n)(/PDO(n)) be the number of partitions in which the parts are all EVEN (ODD) and all different. Then the GENERATING FUNCTION of PDO(n) is given by fDO(n)/C30Y/C12 k/C301;3;...1/C27xk(29) (Hardy 1999, p. 86), and the first few values of are 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, ... (Sloane’s A000700). Some additional GENERATING FUNCTIONS are given by Honsberger (1985, pp. 241 /C1/242) X/C12 n/C301Pno even part repeated (n)xn /C30Y k/C301(1/C28x2k/C281)/C281(1/C27x2k) (30) X/C12 n/C301Pno part occurs more than 3 times (n)xn /C30Y k/C301(1/C27xk/C27x2k/C27x3k) (31) X/C12 n/C301Pno part divisible by 4 (n)xn/C30Y k/C3011/C28x4k 1/C28xk(32) X/C12 n/C301Pno part occurs more than dtimes(n)xn /C30Y k/C301Xd i/C300xik/C30Y k/C3011/C28x(d/C271)k 1/C28xk(33) X/C12 n/C301Pevery part occurs 2 ;3;or 5 times (n)xn /C30Y k/C301(1/C27x2k/C27x3k/C27x5k) /C30Y k/C301(1/C27x2k)(1/C27x3k)/C30Y k/C3011/C28x4k 1/C28x2k1/C28x6k 1/C28x3k(34) X/C12 n/C301Pno part occurs exactly once (n)xn /C30(1/C27x2k/C27x3k/C27... )/C30Y k1/C27x6k (1/C28x2k)(1/C28x3k):(35) Some additional interesting theorems following fromthese (Honsberger 1985, pp. 64 /C1 /68 and 143 /C1/146) are: 1. The number of partitions of nin which no EVEN part is repeated is the same as the number ofpartitions of nin which no part occurs more than three times and also the same as the number ofpartitions in which no part is divisible by four.2. The number of partitions of nin which no part occurs more often than dtimes is the same as thenumber of partitions in which no term is a multi-ple of d/C271: / 3. The number of partitions of nin which each part appears either 2, 3, or 5 times is the same as thenumber of partitions in which each part is CON- GRUENT mod 12 to either 2, 3, 6, 9, or 10. 4. The number of partitions of nin which no part appears exactly once is the same as the number of partitions of nin which no part is CONGRUENT to 1 or 5 mod 6. 5. The number of partitions in which the parts are all EVEN and different is equal to the absolute difference of the number of partitions with ODD and EVEN parts. /P(n) satisfies the inequality P(n)51 2(n/C271)/C27P(n/C281) ½/C138 (36) (Honsberger 1991). /P(n;k);also written Pk(n);is the number of ways of writing nas a sum of kterms or, equivalently, the number of partitions into parts of which the largest is k. The latter can be enumerated by Partitions [n, k] in the Mathematica add-on package Discrete- Math‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). For example, theP(5;3)/C305 partitions of 5 of which the largest member is 53 are f3;2g;f3;1;1g;f2;2;1g; f2;1;1;1g;and f1;1;1;1;1g:Similarly, the five partitions of 5 into three or fewer parts are f5g; f4;1g;f3;2g;f3;1;1g;and f2;2;1g:/ /P(n;k) is implemented as ConstrainedInteger- PartitionsP [n,k] in the Mathematica add-on package DiscreteMath‘IntegerPartitions‘ (which can be loaded with the command BBDiscreteMath‘ ), and can be computed from the RECURRENCE RELATION P(n;k)/C30P(n/C281;k/C281)/C27P(n/C28k;k) (37) (Skiena 1990, p. 58; Ruskey) with P(n;k)/C300 for k/C21n,P(n;n)/C301;and P(n;0)/C300:The triangle of P(k;n) is given by 1 11 111 1211 12211 133211 (Sloane’s A008284). The number of partitions of n with largest part kis the same as P(n;k):/ The RECURRENCE RELATION can be solved exactly to give P(n; 1) /C301 (38) P(n; 2) /C301 42n /C281 /C27(/C281)n½/C138 (39) P(n; 3) /C301 726n2 /C287 /C289(/C281)n /C2716 cos2 3 np9+;k9+;7 hi (40) P(n; 4) /C301 864 f3(n /C271) 2n(n /C272) /C2813 /C279(/C281)n½/C138 /C2896 cos2 3 np9+;k9+;7 /C27108(/C281)n=2 mod( n /C271; 2) /C2732ffiffiffi 3p sin(2 3 np) g; (41) where P(n; k) /C300 for n Bk. The functions P(n; k) can also be given explicitly for the first few values of k in the simple forms P(n; 2) /C301 2 njk (42) P(n ; 3) /C301 12 n2hi ; (43) where xbcis the FLOOR FUNCTION and [x] is the NINT function (Honsberger 1985, pp. 40 /C1/45). A similar treatment by B. Schwennicke defines tk(n) /C30n /C2714 k(k /C283) (44) and then yields P(n; 2) /C3012 t2(n)hi (45) P(n; 3) /C301 12 t3 2(n)hi (46) P(n; 4) /C301 144 t3 4(n) /C281 48 t4(n)hi for n even 1 144 t34(n) /C281 12 t4(n)hi for n odd:8 < : (47) Hardy and Ramanujan (1918) obtained the exact asymptotic formula P(n) /C30X kBaffiffinpPk(n)/C27O(n/C281=4); (48) where ais a constant. However, the sum X/C12 k/C301Pk(n) (49) diverges, as first shown by Lehmer (1937). See also ALCUIN’S SEQUENCE ,CONJUGATE PARTITION , ELDER’S THEOREM ,EULER IDENTITY ,FERRERS DIA- GRAM ,GO¨ LLNITZ’S THEOREM ,PARTITION FUNCTION P CONGRUENCES ,PARTITION FUNCTION Q,PENTAGONAL NUMBER ,P ENTAGONAL NUMBER THEOREM ,P LANE PARTITION ,RANDOM PARTITION ,ROGERS- RAMANUJAN IDENTITIES ,SELF-CONJUGATE PARTITION ,STANLEY’S THEOREM ,SUM OF SQUARES FUNCTION ,TAU FUNC- TIONReferences Abramowitz, M. and Stegun, C. A. (Eds.). "Unrestricted Partitions." §24.2.1 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 825, 1972. Adler, H. "Partition Identities--From Euler to the Present." Amer. Math. Monthly 76, 733/C1/746, 1969. Adler, H. "The Use of Generating Functions to Discover and Prove Partition Identities." Two-Year College Math. J. 10, 318/C1/329, 1979. Andrews, G. E. The Theory of Partitions. Cambridge, Eng- land: Cambridge University Press, 1998. Apostol, T. M. Ch. 4 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976. Apostol, T. M. "Rademacher’s Series for the Partition Func- tion." Ch. 5 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 94 /C1/112, 1997. Atkin, A. O. L. and Swinnerton-Dyer, P. "Some Properties of Partitions." Proc. London Math. Soc. 4,8 4/C1/106, 1954. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 307, 1974. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 94 /C1/96, 1996. David, F. N.; Kendall, M. G.; and Barton, D. E. Symmetric Function and Allied Tables. Cambridge, England: Cam- bridge University Press, p. 219, 1966. Gupta, H. "A Table of Partitions." Proc. London Math. Soc. 39, 142/C1/149, 1935. Gupta, H. "A Table of Partitions (II)." Proc. London Math. Soc. 42, 546/C1/549, 1937. Gupta, H.; Gwyther, A. E.; and Miller, J. C. P. Tables of Partitions. London: Royal Society Mathematical Tables, Vol. 4, 1958. Hardy, G. H. "Ramanujan’s Work on Partitions" and "Asymptotic Theory of Partitions." Chs. 6 and 8 in Rama- nujan: Twelve Lectures on Subjects Suggested by His Lifeand Work, 3rd ed. New York: Chelsea, pp. 83 /C1 /100 and 113/C1/131, 1999. Hardy, G. H. and Ramanujan, S. "Asymptotic Formulae in Combinatory Analysis." Proc. London Math. Soc. 17,7 5/C1/ 115, 1918. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 40 /C1/45 and 64 /C1/68, 1985. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 237 /C1/239, 1991. Jackson, D. and Goulden, I. Combinatorial Enumeration. New York: Academic Press, 1983. Lehmer, D. H. "On the Hardy-Ramanujan Series for the Partition Function." J. London Math. Soc. 12, 171/C1/176, 1937. Lehmer, D. H. "On a Conjecture of Ramanujan." J. London Math. Soc. 11, 114/C1/118, 1936. Lehmer, D. H. "The Series for the Partition Function." Trans. Amer. Math. Soc. 43, 271/C1/295, 1938. Lehmer, D. H. "On the Remainders and Convergence of the Series for the Partition Function." Trans. Amer. Math. Soc. 46, 362/C1/373, 1939. MacMahon, P. A. "Note of the Parity of the Number which Enumerates the Partitions of a Number." Proc. Cambridge Philos. Soc. 20, 281/C1/283, 1921. MacMahon, P. A. "The Parity of p(n);the Number of Partitions of n, when n51000 :/"J. London Math. Soc. 1, 225/C1/226, 1926. MacMahon, P. A. Combinatory Analysis. New York: Chel- sea, 1960. Rademacher, H. "Zur Theorie der Modulfunktionen." J. reine angew. Math. 167, 312/C1/336, 1932. Rademacher, H. "On the Partition Function p(n):/"Proc. London Math. Soc. 43, 241/C1/254, 1937. Rademacher, H. "On the Expansion of the Partition Func- tion in a Series." Ann. Math. 44, 416/C1/422, 1943. Ruskey, F. "Information of Numerical Partitions." http:// www.theory.csc.uvic.ca/~cos/inf/nump/NumParti- tion.html. Sloane, N. J. A. Sequences A000009/M0281, A000041/ M0663, A000700/M0217, A001318/M1336, and A008284in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Uspensky, J. V. "Asymptotic Formulae for Numerical Func- tions Which Occur in the Theory of Partitions.’ Bull. Acad. Sci. URSS 14, 199/C1 /218, 1920. Partition Function P Congruences The fraction of odd values of the PARTITION FUNCTION Pis roughly 50%, independent of n, whereas odd values of Q(n) occur with ever decreasing frequency asnbecomes large. Kolberg (1959) proved that there are infinitely many even and odd values of P(n):/ Leibniz noted that P(n) is prime for n/C302, 3, 4, 5, 6, but not 7. In fact, values of nfor which P(n)i s PRIME are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (Sloane’s A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (Sloane’s A049575). Num- bers which cannot be written as a PRODUCT ofP(n) are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (Sloane’sA046064), corresponding to numbers of noniso- morphic A BELIAN GROUPS which are not possible for any group order. Ramanujan conjectured a number of amazing and unexpected CONGRUENCES involving P(n):In particu- lar, he proved P(5m/C274)/C130 (mod 5) (1) using R AMANUJAN’S IDENTITY (Darling 1919; Hardy and Wright 1979; Drost 1997; Hardy 1999, pp. 87 /C1/88; Hirschhorn 1999). Ramanujan (1919) also showed thatP(25m/C2724)/C130 (mod 52); (2) and Krecmar (1933) proved that P(125m/C2799)/C130 (mod 53): (3) Watson (1938) then proved the general congruence P(n)/C130 (mod 5a)i f 2 4 n/C131 (mod 5a) (4) (Gordon and Hughes 1981; Hardy 1999, p. 89). For a/C301, 2, ..., the corresponding minimal values of nare 4, 24, 99, 599, 2474, 14974, 61849, ... (Sloane’s A052463). However, the even more general con- gruences P(125m/C2774;99;124)/C130 (mod 53) (5) P(3125 m/C271849 ;2474 ;3099)/C130 (mod 55) (6) seem also to hold. Ramanujan showed that P(7m/C275)/C130 (mod 7) (7) (Darling 1919), which can be derived using the E ULER IDENTITY and J ACOBI TRIPLE PRODUCT (Hardy 1999, pp. 87 /C1/88), and also that P(49m/C2747)/C130 (mod 72) (8) (Hardy 1999, p. 90). He conjectured that in general P(n)/C130 (mod 7b)i f 2 4 n/C131 (mod 7b) [incorrect](9) (Gordon and Hughes 1981, Hardy 1999), although Gupta (1936) showed that this is false when b/C303. Watson (1938) subsequently formulated and proved the modified relation P(n)/C130 (mod 7b)i f 2 4 n/C131 (mod 72b/C282) (10) forb]2:Forb/C301, 2, ..., the corresponding minimal values of nare 0, 47, 2301, 112747, ... (Sloane’s A052464). However, the even more general con-gruences P(49m/C2719;33;40;47)/C130 (mod 7 2) (11) appear to hold. Ramanujan showed that P(11m/C276)/C130 (mod 11) (12) holds (Gordon and Hughes 1981; Hardy 1999, pp. 87 /C1/ 88), and conjectured the general relation P(n)/C130 (mod 11c)i f 2 4 n/C131 (mod 11c):(13) This was finally proved by Atkin (1967). For c/C301, 2, ..., the corresponding minimal values of nare 6, 116, 721, 14031, ... (Sloane’s A052465). Atkin and O’Brien (1967) proved P(169n /C287) /C13 kdP(n) (mod 13d) if 24n /C131 (mod 13d) ;(14) where kdis an integer depending only on d (Gordon and Hughes 1981). For d /C301, 2, ..., the corresponding minimal values of n are 6, 162, 1007, 27371, ... (Sloane’s A052466). Subbarao (1966) conjectured that in every ARITH- METIC PROGRESSION r (mod t), there are infinitely many integers N /C13r (mod t) for which P(N)is EVEN , and infinitely many integer M /C13r (mod t) for which P(M)is ODD. See also CONGRUENCE ,E RDOS- IVIC CONJECTUR E, NEWMAN’S CONJECTURE ,P ARTITION FUNCTION P, PARTITION FUNCTION Q,P ARTITION FUNCTION Q, PARTITION FUNCTION Q CONGRUENCES References Atkin, A. O. L. "Proof of a Conjecture of Ramanujan." Glasgow Math. J. 8,14/C1/32, 1967. Atkin, A. O. L. and O’Brien, J. N. "Some Properties of p(n) and c(n) Modulo Powers of 13." Trans. Amer. Math. Soc. 126, 442 /C1/459, 1967. Chowla, S. "Congruence Properties of Partitions." J. London Math. Soc. 9, 247, 1934. Darling, H. B. C. "Proofs of Certain Identities and Con- gruences Enunciated by S. Ramanujan." Proc. London Math. Soc. 19, 350 /C1/372, 1921. Darling, H. B. C. "On Mr. Ramanujan’s Congruence Proper- ties of p(n):/" Proc. Cambridge Philos. Soc. 19, 217 /C1/218, 1919. Drost, J. L. "A Shorter Proof of the Ramanujan Congruence mod 5." Amer. Math. Monthly 104, 963 /C1/964, 1997. Getz, J. "On Congruence Properties of the Partition Func- tion." Internat. J. Math. Math. Sci. 23, 493 /C1/496, 2000. Gordon, B. and Hughes, K. "Ramanujan Congruences for q(n) :/"In Analytic Number Theory, Proceedings of the Conference Held at Temple University, Philadelphia, Pa., May 12 /C1/15, 1980 (Ed. M. I. Knopp). New York: Springer- Verlag, pp. 333 /C1/359, 1981. Gupta, H. "On a Conjecture of Ramanujan." Proc. Indian Acad. Sci. (A) 4, 625 /C1/629, 1936. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hirschhorn, M. D. "Another Short Proof of Ramanujan’s Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580 /C1/583, 1999. Kolberg, O. "Note on the Parity of the Partition Function." Math. Scand. 7, 377 /C1/378, 1959. Krecmar, W. "Sur les proprie ´te´s de la divisibilite ´ d’une fonction additive." Bull. Acad. Sci. URSS 7, 763 /C1/800, 1933. Lehmer, D. H. "An Application of Schla¨fli’s Modular Equa- tion to a Conjecture of Ramanujan." Bull. Amer. Math. Soc. 44,84/C1/90, 1938. Mordell, L. J. "Note on Certain Modular Relations Consid- ered by Messrs Ramanujan, Darling and Rogers." Proc. London Math. Soc. 20, 408 /C1/416, 1922. Ono, K. "Parity of the Partition Function in Arithmetic Progressions." J. reine. angew. Math. 472,1/C1/15, 1996.Ono, K. "The Partition Function in Arithmetic Progres- sions." Math. Ann. 312, 251 /C1/260, 1998. Ono, K. "Distribution of the Partition Function Modulo m." Ann. Math. 151, 293 /C1/307, 2000. Ramanujan, S. "Some Properties of p(n); the Number of Partitions of n." Proc. Cambridge Philos. Soc. 19, 207 /C1/ 210, 1919. Ramanujan, S. "Congruence Properties of Partitions." Math. Z. 9, 147 /C1/153, 1921. Sloane, N. J. A. Sequences A046063, A046064, A049575, A052462, A052463, A052464, A052465, and A052466 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Subbarao, M. V. "Some Remarks on the Partition Function." Amer. Math. Monthly 73, 851 /C1/854, 1966. Watson, G. N. "Ramanujans Vermutung u¨ber Zerfa¨llung- sanzahlen." J. fu¨r Math. 179,97/C1/128, 1938. Partition Function q The number of PARTITIONS ofnwith5ksummands is denoted q(n;k)o r qk(n):For example, q(10;2)/C306; since there are six partitions of 10 into two or fewer parts: f10g;f9;1g;f8;2g;f7;3g;f6;4g;and f5;5g: The q(n;k) satisfy the RECURRENCE RELATION q(n;k)/C30q(n;k/C281)/C27q(n/C28k;k); (1) with q(n;0)/C300;q(1;k)/C301;andq(n;k)/C30P(n) for k] n:The triangle of q(n;k) is given by 1 12 123 1345 13567 14791 01 1 (Sloane’s A026820). See also PARTITION FUNCTION P,PARTITION FUNC- TION Q References Sloane, N. J. A. Sequences A026820 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Partition Function Q /Q(n) gives the number of ways of writing the INTEGER nas a sum of POSITIVE INTEGERS without regard to order with the constraint that all INTEGERS in a given partition are distinct . For example, Q(10)/C3010;since the partitions of 10 into distinct parts are f1;2;3;4g;f2;3;5g;f1;4;5g;f1;3;6g;f4;6g; f1;2;7g;f3;7g;f2;8g;f1;9g;f10g:The Q(n) func- tion is implemented in Mathematica asParti- tionsQ [n].Q(0) is generally defined to be 1. The values for n/C301, 2, ... are 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (Sloane’s A000009). The GENERATING FUNCTION for Q(n)is G(x) /C30Y/C12 n/C301(1 /C27xn) (1) /C301Q/C12 n/C300(1 /C28 x2n/C271) (2) /C30Y/C12 n /C3011 /C28 x2n 1 /C28 xn (3) /C301 /C27x /C27x2 /C272x3 /C272x4 /C273x5 /C27...: (4) This can also be interpreted as another form of the JACOBI TRIPLE PRODUCT , written in terms of the Q- FUNCTIONS as Q1Q2Q3 /C301 (5) (Borwein and Borwein 1987, p. 64). A RECURRENCE RELATION is given by Q(0) /C30Q(1) /C301 and Q(n) /C301 nXn k /C301[s(k) /C282s(k=2)]Q(n /C28k) ; (6) where s(n) /C30s1(n) for n an integer 0 otherwise ;9+$k (7) and s1(n) /C13s(n) /C282s(n=2) (8) is the ODD DIVISOR FUNCTION giving the sum of odd divisors of n: 1, 1, 4, 1, 6, 4, 8, ... (Sloane’s A000593; Abramowitz and Stegun 1972, p. 826). /Q(n) satisfies the inequality Q(n) 51 2[Q(n /C271) /C27Q(n /C281)] (9) for n ]4 : Q(n) has the ASYMPTOTIC SERIES Q(n) /C2e pffiffiffiffiffiffi n=3p 4 /C215 31=4n3=4 (10) (Abramowitz and Stegun 1972, p. 826). A Rademacher-like convergent series for Q(n) is given by Q(n) /C301 2ffiffiffi 2pX/C12 k/C301A2k /C281(n) /C2d dn ?J0pi 2k /C28 1 ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3n?/C271 249+;k9+;7r !"#() n?/C30n; (11) where Ak(n) /C30Xk h/C301dGCD( h; k); 1/C2exp piXk /C281 j/C301i khj k/C28hj k$% /C281 2 ! /C282pihn k"# ; (12) where dmn is the KRONECKER DELTA , xbcis the FLOOR FUNCTION , and J0(x) is the zeroth order BESSEL FUNCTION OF THE FIRST KIND (Abramowitz and Ste- gun 1972, p. 825). (11) can also be written explicitly as Q(n) /C30p2ffiffiffi 2p 24X/C12 k /C301A2k/C281(n) (1 /C28 2k)2 0 F1 ;2;(1 /C27 24n) p2 288(1 /C28 2k)2 ! ; (13) where0F1(; a; b; z)isa GENERALIZED HYPERGEO- METRIC FUNCTION . Let Q(n; k) denote the number of ways of partitioning n into exactly k distinct parts. For example, Q(10; 3) /C304 since there are four partitions of 10 into three distinct parts: f1; 2 ; 7 g;f1; 3; 6g; f1; 4; 5g; and f2; 3; 5g: Q(n; k) is given by Q(n; k) /C30Pn/C28k 29+;89+;9 ;k9+;89+;9 ; (14) where P(n) is the PARTITION FUNCTION Pandn k9+=9+; is a BINOMIAL COEFFICIENT (Comtet 1974, p. 116). The following table gives the first few values of Q(n;k) (Sloane’s A008289; Comtet 1974, pp. 115 /C1/116). /n_k/1234 11 21 311411 512 612171318132 9143 1 01441 See also O DD DIVISOR FUNCTION ,PARTITION FUNC- TION P,PARTITION FUNCTION Q,PARTITION FUNCTION Q CONGRUENCES References Abramowitz, M. and Stegun, C. A. (Eds.). "Partitions into Distinct Parts." §24.2.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 825 /C1/826, 1972. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 114 /C1/115, 1974. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 58, 1990. Sloane, N. J. A. Sequences A000009/M0281, A000593/ M3197, and A008289 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Partition Function Q Congruences Odd values of Q(n) are 1, 1, 3, 5, 27, 89, 165, 585, ... (Sloane’s A051044), and occur with ever decreasing frequency as n becomes large (unlike P(n) ; for which the fraction of odd values remains roughly 50%). This follows from the PENTAGONAL NUMBER THEOREM which gives G(x) /C30Y/C12 n /C301(1 /C27xn) /C13Y/C12 n/C301(1 /C28xn) /C13X/C12 n/C30/C28/C12x(3n2/C27n)=2 (mod 2) (1) (Gordon and Ono 1997), so Q(n)is ODD IFF n is OF THE FORM k(3k 91)=2 ; i.e., 1, 5, 12, 22, 35, ... or 2, 7, 15, 26, 40, .... The values of n for which Q(n)is PRIME are 3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, ... (Sloane’s A035359), with no others for n 53; 015;000 (Weis- stein, May 6, 2000). These values correspond to 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (Sloa- ne’s A051005). It is not known if Q(n) is infinitely often prime, but Gordon and Ono (1997) proved that it is "almost always" divisible by any given power of 2 (1997). Gordon and Hughes (1981) showed that Q(n) /C130 (mod 5a)i f2 4n /C13/C281 (mod 52a /C271) (2) and Q(n) /C1349n /C272 (mod lbQ(n))7b if 24n /C13/C281 (mod 7b) ;(3) where lb is an integer depending only on b.See also PARTITION FUNCTION P,PARTITION FUNC- TION P CONGRUENCES ,PARTITION FUNCTION Q References Gordon, B. and Hughes, K. "Ramanujan Congruences for q(n) :/"In Analytic Number Theory, Proceedings of the Conference Held at Temple University, Philadelphia, Pa., May 12 /C1/15, 1980 (Ed. M. I. Knopp). New York: Springer- Verlag, pp. 333 /C1/359, 1981. Gordon, B. and Ono, K. "Divisibility of Certain Partition Functions by Powers of Primes." Ramanujan J. 1,25/C1/34, 1997. Sloane, N. J. A. Sequences A035359, A051005, and A051044 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Partition of Unity Given a SMOOTH MANIFOLD M with an OPEN COVER Ui ; a partition of unity is a collection of smooth, nonnegative functions ci ; such that the support of ci is contained in Ui and ai ci /C301 everywhere. Often one requires that the Uihave COMPACT CLOSURE , which can be interpreted as finite, or bounded, open sets. In the case that the Uiis a LOCALLY finite cover, any point x /C23 M has only finitely many i with ci(x) "0:/ A partition of unity can be used to patch together objects defined locally. For instance, there always exist smooth GLOBAL VECTOR FIELDS , possibly vanish- ing somewhere, but not identically zero. Cover M with coordinate charts Uisuch that only finitely many overlap at any point. On each coordinate chart Ui ; there are the local vector fields @=@xj : Label these vi ; jand, for each chart, pick the vector field vi; 1 /C30 @=@x1 : Then ai civi; 1 is a global vector field. The sum converges because at any x, only finitely many ci(x) "0 :/ Other applications require the objects to be inter- preted as functions, or a generalization of functions called SECTIONS , such as a RIEMANNIAN METRIC .By viewing such a metric as a section of a bundle, it is easy to show the existence of a smooth metric on any smooth manifold. The proof uses a partition of unity and is similar to the one used above. Strictly speaking, the sum aicidoesn’t have to be identically UNITY for the arguments to work. It goes with the name, because at every point the functions partition the value 1. Also, it is convenient when considered from the point of view of CONVEXITY . See also CONVEX SET,O PEN COVER RIEMANNIAN METRIC ,SECTION ,SMOOTH MANIFOLD ,VECTOR FIELD PartitionsP PARTITION FUNCTION P PartitionsQ PARTITION FUNCTION Q Party Problem Also known as the MAXIMUM CLIQUE PROBLEM . Find the minimum number of guests that must be invited so that at least m will know each other or at least n will not know each other. The solutions are known as RAMSEY NUMBERS . See also CLIQUE ,RAMSEY NUMBER References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 52, 1998. Parzen Apodization Function An APODIZATION FUNCTION similar to the BARTLETT FUNCTION . See also APODIZATION FUNCTION ,BARTLETT FUNC- TION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 547, 1992. Pascal Distribution NEGATIVE BINOMIAL DISTRIBUTION Pascal Lines The lines containing the three points of the intersec- tion of the three pairs of opposite sides of a (notnecessarily regular) HEXAGON . There are 6! (i.e., 6 FACTORIAL ) possible ways of taking all VERTICES in any order, but among these are six equivalent CYCLIC PERMUTATIONS and two possible orderings, so the total number of differenthexagons (not all simple) is 6! 2 /C2156/C30720 12/C3060: There are therefore a total of 60 Pascal lines createdby connecting VERTICES in any order. The 60 Pascal lines form a very complicated patternwhich can be visualized most easily in the degeneratecase of a regular hexagon inscribed in a circle, as illustrated above for magnifications ranging over five powers of 2. Only 45 lines are visible in this figuresince each of the three thick lines (located at 60 8 angles to each other) represents a degenerate groupof four Pascal lines, and six of the Pascal lines are LINES AT INFINITY (Wells 1991). The pattern for a general ellipse and hexagon (illu- strated above) is much more complicated, and is difficult to distinguish from a clutter of lines.The 60 Pascal lines intersect three at a time through20 S TEINER POINTS (some of which are shown as the filled circles in the above figures). In the symmetricalcase of the regular hexagon inscribed in a CIRCLE , the 20 Steiner points degenerate into seven distinct points arranged at the vertices and center of a regular hexagon centered at the origin of the circle. The 60 Pascal line also intersect three at a time in 60 KIRKMAN POINTS . Each Steiner point lines together with three Kirkman points on a total of 20 CAYLEY LINES . There is a dual relationship between the 60 Pascal lines and the 60 KIRKMAN POINTS . See also CAYLEY LINES,HEPTAGON THEOREM ,HEXA- GON,KIRKMAN POINTS ,PASCAL’S THEOREM ,PLU¨ CKER LINES,SALMON POINTS ,STEINER POINTS References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 75, 1967. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Heptagon Theorem." §2.1 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8 /C1/11, 1974. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 236, 1929. Lachlan, R. "Pascal’s Theorem." §181 /C1/191 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 113 /C1/119, 1893. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 172 /C1/173, 1991. Pascal’s Formula Each subsequent row of PASCAL’S TRIANGLE is ob- tained by adding the two entries diagonally above. This follows immediately from the BINOMIAL COEFFI- CIENT identity n r9+;89+;9 /C13n! (n /C28 r)!r! /C30(n /C28 1)!n (n /C28 r)!r! /C30(n /C28 1)!(n /C28 r) (n /C28 r)!r!/C27(n /C28 1)!r (n /C28 r)!r! /C30(n /C28 1)! (n /C28 r /C28 1)!r! /C27(n /C28 1)! (n /C28 r)!(r /C28 1)! /C30n /C281 r9+;89+;9 /C27n /C281 r /C2819+;89+;9 : See also BINOMIAL COEFFICIENT ,PASCAL’S TRIANGLE Pascal’s Hexagrammum Mysticum PASCAL’S THEOREM Pascal’s Limac ¸on LIMAC ¸ ON Pascal’s Rule PASCAL’S FORMULAPascal’s Theorem The dual of BRIANCHON’S THEOREM (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily REG- ULAR , or even CONVEX ) HEXAGON inscribed in a CONIC SECTION , the three pairs of the continuations of opposite sides meet on a straight LINE, called the PASCAL LINE . See also BRAIKENRIDGE- MACLAURIN CONSTRUCTION , BRIANCHON’S THEOREM ,C AYLEY- BACHARACH THEO- REM,CONIC SECTION ,DUALITY PRINCIPLE ,HEXAGON , PAPPUS’S HEXAGON THEOREM ,PASCAL LINES,STEI- NER POINTS ,STEINER’S THEOREM References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., pp. 129 /C1 /131, 1888. Casey, J. "Pascal’s Theorem." §255 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and ConicSections, Containing an Account of Its Most RecentExtensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 145, 328 /C1 /329, and 354, 1893. Cayley, A. Quart J. 9, p. 348. Coxeter, H. S. M. and Greitzer, S. L. "L’hexagramme de Pascal. Un essai pur reconstituer cette de ´couverte." Le Jeune Scientifique (Joliette, Quebec) 2,7 0/C1/72, 1963. Coxeter, H. S. M. and Greitzer, S. L. "Pascal’s Theorem." §3.8 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 74 /C1/76, 1967. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 44, 1928. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "Extensions of Pascal’s and Brianchon’s Theorems." Ch. 2 inThe Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8 /C1/30, 1974. Forder, H. G. Higher Course Geometry. Cambridge, Eng- land: Cambridge University Press, p. 13, 1931. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 260 /C1/261, 1930. Johnson, R. A. §386 in Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.Boston, MA: Houghton Mifflin, pp. 236 /C1 /237, 1929. Lachlan, R. "Pascal’s Theorem." §181/C1/191 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 113 /C1/119, 1893. Leibniz, G. Letter to M. Pe ´rier. In /Œ/uvres de B. Pascal, Vol. 5 (Ed. Bossut). p. 459. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 105 /C1/106, 1990. Pappas, T. "The Mystic Hexagram." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 118, 1989. Perfect, H. Topics in Geometry. London: Pergamon, p. 26, 1963. Salmon, G. §267 and "Notes: Pascal’s Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 245 /C1/246 and 379 /C1/382, 1960. Spieker, T. Lehrbuch der ebene Geometrie. Potsdam, Ger- many, 1888. Veronese. "Nuovi Teremi sull’ Hexagrammum Mysticum." Real. Accad. dei Lincei. 1877. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 69, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 173, 1991. Pascal’s Triangle ATRIANGLE of numbers arranged in staggered rows such that anr/C13n! r!(n/C28r)!/C13n r9+;89+;9 ; (1) wheren rðÞis a BINOMIAL COEFFICIENT . The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathemati- cian Yanghui (about 500 years earlier, in fact) and thePersian astronomer-poet Omar Khayya ´m. It is there- fore known as the Yanghui triangle in China. Start-ing with n/C300, the TRIANGLE is 1 11 121 1331 14641 1 5 10 10 5 1 1 6 15 20 15 6 1 (Sloane’s A007318). P ASCAL’S FORMULA shows that each subsequent row is obtained by adding the twoentries diagonally above, n r9+;89+;9 /C30n! (n/C28r)!r!/C30n/C281 r9+;89+;9 /C27n/C281 r/C2819+;89+;9 : (2) In addition, the " SHALLOW DIAGONALS " of Pascal’s triangle sum to F IBONACCI NUMBERS , Xn k/C301k n/C28k9+;89+;9 /C30(/C281)n 3F21;2;1/C28n;1 2(3/C28n);2/C2812n;/C2849+;k9+;7 p2/C283n/C27n2 ðÞ /C30Fn/C271; (3) where /3F2ða;b;c;d;e;zÞ/is a GENERALIZED HYPERGEO- METRIC FUNCTION . Pascal’s triangle contains the FIGURATE NUMBERS along its diagonals. It can be shown that Xn i/C301aij/C30n/C271 j/C271anj/C30a(n/C271);(j/C271) (4) and m/C271 19+;89+;9X km/C27m/C271 29+;89+;9X km/C281 /C27.../C27m/C271 m9+;89+;9X k/C30(n/C271) (n/C271)m/C281 ½/C138 : (5) The "shallow diagonals" sum to the F IBONACCI SE- QUENCE , i.e., 1/C301 1/C301 2/C301/C271 3/C302/C271 5/C301/C273/C271 8/C303/C274/C271: (6) In addition, Xi j/C301aij/C302i/C281: (7) It is also true that the first number after the 1 in each row divides all other numbers in that row IFFit is a PRIME .I fPnis the number of ODD terms in the first n rows of the Pascal triangle, then 0:812... BPnn /C28ln 2 =ln 3 B1 (8) (Harborth 1976, Le Lionnais 1983). The BINOMIAL COEFFICIENTm n9+=9+; mod 2 can be com- puted using the XOR operation n XOR m, making Pascal’s triangle mod 2 very easy to construct. Pascal’s triangle is unexpectedly connected with the construction of regular POLYGONS and with the SIERPINSKI SIEVE (Guy 1990). Starting at row 210, the numbers 120 /C3010 39+;89+;9 /C3010 79+;89+;9 /C3016 29+;89+;9 /C3016 149+;89+;9 /C30120 19+;89+;9 /C301201199+;89+;9 (9) 210 /C3010 49+;89+;9 /C3010 69+;89+;9 /C3021 29+;89+;9 /C3021199+;89+;9 /C30210 19+;89+;9 /C302102099+;89+;9 (10) 3003 /C3014 69+;89+;9 /C3014 89+;89+;9 /C3015 59+;89+;9 /C3015109+;89+;9 /C3078 29+;89+;9 /C3078769+;89+;9 (11) have appeared six times, more than any other number (excluding 1), and remain the most common numbers in the triangle up to at least row 1436. Guy (1990) gives another several unexpected proper- ties of Pascal’s triangle. See also B ELL TRIANGLE ,B INOMIAL COEFFICIENT , BINOMIAL THEOREM ,BRIANCHON’S THEOREM ,CATA- LAN’S TRIANGLE ,CLARK’S TRIANGLE ,EULER’S TRIAN- GLE,F IBONACCI NUMBER ,F IGURATE NUMBER TRIANGLE ,L EIBNIZ HARMONIC TRIANGLE ,L OSS- NITSCH’S TRIANGLE ,N UMBER TRIANGLE ,P ASCAL’S FORMULA ,POLYGON ,SEIDEL- ENTRINGER- ARNOLD TRI- ANGLE ,SIERPINSKI SIEVE,TRINOMIAL TRIANGLE References Conway, J. H. and Guy, R. K. "Pascal’s Triangle." In The Book of Numbers. New York: Springer-Verlag, pp. 68 /C1/70, 1996. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 17, 1996. Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63,3/C1/20, 1990. Harborth, H. "Number of Odd Binomial Coefficients." Not. Amer. Math. Soc. 23, 4, 1976. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 31, 1983. Pappas, T. "Pascal’s Triangle, the Fibonacci Sequence & Binomial Formula," "Chinese Triangle," and "Probability and Pascal’s Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 40 /C1/41 88, and 184 /C1/186, 1989. Sloane, N. J. A. Sequences A007318/M0082 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 86, 1984. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 284 /C1/285, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 174 /C1/175, 1991. Pascal’s Wager "God is or He is not...Let us weigh the gain and the loss in choosing...‘God is.’ If you gain, you gain all, if you lose, you lose nothing. Wager, then, unhesitat- ingly, that He is." References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 150 /C1/151, 1998. Pasch’s Axiom In the plane, if a line intersects one side of a TRIANGLE and misses the three VERTICES , then it must intersect one of the other two sides. This is a special case of the generalized MENELAUS’ THEOREM with n /C303. See also HELLY’S THEOREM ,M ENELAUS’ THEOREM , PASCH’S THEOREM Pasch’s Theorem A theorem stated in 1882 which cannot be derived from EUCLID’S POSTULATES . Given points a, b, c, and d on a LINE, if it is known that the points are ordered as (a ; b; c) and (b; c ; d) ; then it is also true that (a; b; d) :/ See also EUCLID’S POSTULATES ,LINE,PASCH’S AXIOM Pass Equivalent Two KNOTS are pass equivalent if there exists a sequence of pass moves taking one to the other. Every KNOT is either pass equivalent to the UNKNOT or TREFOIL KNOT . These two knots are not pass equivalent to each other, but the ENANTIOMERS of the TREFOIL KNOT are pass equivalent. A KNOT has ARF INVARIANT 0 if the KNOT is pass equivalent to the UNKNOT and 1 if it is pass equivalent to the TREFOIL KNOT . See also ARF INVARIANT ,KNOT,KNOT MOVE,PASS MOVE,TREFOIL KNOT,UNKNOT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 223 /C1/228, 1994. Pass Move A change in a knot projection such that a pair of oppositely oriented strands are passed through an- other pair of oppositely oriented strands. See also KNOT MOVE,PASS EQUIVALENT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 223 /C1/228, 1994. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Patch A patch (also called a LOCAL SURFACE ) is a differenti- able mapping x : U 0 Rn ; where U is an open subset of R2 : More generally, if A is any SUBSET of R2 ; then a map x : A 0 Rn is a patch provided that x can be extended to a differentiable map from U into Rn ; where U is an open set containing A. Here, x(U) (or more generally, x(A)) is called the TRACE of x. See also GAUSS MAP,INJECTIVE PATCH ,M ONGE PATCH ,REGULAR PATCH ,TRACE (MAP) References Gray, A. "Patches in Rn /" and "Patches in R3 :/" §12.1 and 12.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 269 /C1/278, 1997. Path A path g is a continuous mapping g :[a; b] /C2C; where g(a) is the initial point and g(b) is the final point. It is often written parametrically as s(t) :/ See also CHAIN (GRAPH ), CONTOUR ,CURVE ,EULERIAN CIRCUIT ,GRAPH CYCLE ,HAMILTONIAN CIRCUIT ,UNI- CURSAL CIRCUITPath Graph The path Pnis a TREE with two nodes of VERTEX DEGREE 1, and the other n /C282 nodes of VERTEX DEGREE 2. Path graphs Pnare always GRACEFUL for n /C214. See also CHAIN (GRAPH ), GRACEFUL GRAPH ,HAMIL- TONIAN PATH,TREE Path Integral Let g be a PATH given parametrically by s(t) : Let s denote ARC LENGTH from the initial point. Then ggf(s) ds /C30ggf( s(t)) s?(t) jj dt /C30ggf(x(t) ; y(t) ; z(t)) s?(t) jj dt: See also LINE INTEGRAL References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Evaluation of Functions by Path Integration." §5.14 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 201 /C1/204, 1992. Path Length EXTERNAL PATH LENGTH ,INTERNAL PATH LENGTH Path-Connected See also ARCWISE -CONNECTED ,C ONNECTED SET, LOCALLY PATHWISE- CONNECTED ,P ATHWISE- CON- NECTED Path-Connected Set See also ARCWISE- CONNECTED SET,CONNECTED SET Pathwise-Connected ATOPOLOGICAL SPACE Xis pathwise-connected IFFfor every two points x;y/C23X;there is a CONTINUOUS FUNCTION ffrom [0,1] to Xsuch that f(0)/C30xand f(1)/C30y:Roughly speaking, a SPACE Xis pathwise- connected if, for every two points in X, there is a path connecting them. For LOCALLY PATHWISE-CONNECTED SPACES (which include most "interesting spaces" such as MANIFOLDS and CW -COMPLEXES ), being CON- NECTED and being pathwise-connected are equiva- lent, although there are connected spaces which are not pathwise connected. Pathwise-connected spaces are also called 0-connected. See also CONNECTED SPACE ,CW -COMPLEX ,LOCALLY PATHWISE- CONNECTED ,PATH-CONNECTED ,TOPOLOGI- CAL SPACE Patriarchal Cross GAULLIST CROSS Patterson Quadrature GAUSS- KRONROD QUADRATURE Pauli Matrices Matrices which arise in Pauli’s treatment of spin in quantum mechanics. They are defined by s1 /C30 sx /C13P1 /C13 01 109+$=9+$; (1) s2 /C30 sy /C13P2 /C130 i /C28i 09+$=9+$; (2) s3 /C30 sz /C13P3 /C13 100 /C2819+$=9+$; : (3) The Pauli matrices plus the 2 /C292 IDENTITY MATRIX I form a complete set, so any 2 /C292 matrix A can be expressed as A /C30c0I /C27c1 s1 /C27c2 s2 /C27c3 s3 : (4) The associated matrices s/C27/C132 01009+$=9+$; (5) s /C28/C132 00109+$=9+$; (6) s 2 /C13310019+$=9+$; (7) can also be defined. The Pauli spin matrices satisfy the identities s i sj /C30Idij /C27 eijki sk (8) si sj /C30 sj si /C302sij (9) sxpx /C27 sypy /C27 szpz /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 x /C27p2y /C27p2zq : (10) See also DIRAC MATRICES ,QUATERNIONReferences Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211 /C1/212, 1985. Goldstein, H. "The Cayley-Klein Parameters and Related Quantities." Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 156, 1980. Pauli Spin Matrices PAULI MATRICES Payoff Matrix An m /C29n MATRIX which gives the possible outcome of a two-person ZERO-SUM GAME when player A has m possible moves and player B n moves. The analysis of the MATRIX in order to determine optimal strategies is the aim of GAME THEORY . The so-called "augmented" payoff matrix is defined as follows: G /C30P0P1 P2... PnPn/C271Pn /C272... Pn /C27m 011. . .0 0 0. . .0 /C281 a11a12... a1n 10. . .0 /C281 a21a22... a2n 01. . .0 nnn::: nnn::: n /C281 am1am2... amn 00. . .12 66666643 7777775: See also GAME THEORY ,ZERO-SUM GAME P-Circle SPIEKER CIRCLE PC-Point PEDAL- CEVIAN POINT Peacock’s Tail One name for the figure used by Euclid to prove the PYTHAGOREAN THEOREM . See also BRIDE’S CHAIR ,W INDMILL Peano Arithmetic The theory of NATURAL NUMBERS defined by the five PEANO’S AXIOMS . Paris and Harrington (1977) gave the first "natural" example of a statement which is true for the integers but unprovable in Peano arith- metic (Spencer 1983). See also KREISEL CONJECTURE ,N ATURAL INDEPEN- DENCE PHENOMENON ,N UMBER THEORY ,P EANO’S AXIOMS References Kirby, L. and Paris, J. "Accessible Independence Results for Peano Arithmetic." Bull. London Math. Soc. 14, 285/C1/293, 1982. Paris, J. and Harrington, L. "A Mathematical Incomplete- ness in Peano Arithmetic." In Handbook of Mathematical Logic (Ed. J. Barwise). Amsterdam, Netherlands: North- Holland, pp. 1133 /C1/1142, 1977. Spencer, J. "Large Numbers and Unprovable Theorems." Amer. Math. Monthly 90, 669 /C1/675, 1983. Peano Curve A FRACTAL curve which can be written as a LINDEN- MAYER SYSTEM . See also DRAGON CURVE ,H ILBERT CURVE ,LINDEN- MAYER SYSTEM ,SIERPINSKI CURVE References Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Hilbert, D. "Uuml;ber die stetige Abbildung einer Linie auf ein Flachenstu ¨ck." Math. Ann. 38, 459 /C1/460, 1891. Peano, G. "Sur une courbe, qui remplit une aire plane." Math. Ann. 36, 157 /C1/160, 1890. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 207, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Peano Surface The function f(x; y) /C30 2x2 /C28y9+=9+; y /C28x29+=9+; which does not have a LOCAL MAXIMUM at (0, 0), despite criteria commonly touted in the second half of the 1800s which indicated the contrary. See also LOCAL MAXIMUM References Fischer, G. (Ed.). Plate 122 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 119, 1986. Leitere, J. "Functions." §7.1.2 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 70 /C1/ 71, 1986.Peano-Gosper Curve A PLANE-FILLING CURVE originally called a FLOWS- NAKE by R. W. Gosper and M. Gardner. Mandelbrot (1977) subsequently coined the name Peano-Gosper curve. The GOSPER ISLAND bounds the space that the Peano-Gosper curve fills. See also DRAGON CURVE ,E XTERIOR SNOWFLAKE , GOSPER ISLAND ,HILBERT CURVE ,KOCH SNOWFLAKE , PEANO CURVE ,SIERPINSKI ARROWHEAD CURVE ,SIER- PINSKI CURVE References Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Mandelbrot, B. B. Fractals: Form, Chance, & Dimension. San Francisco, CA: W. H. Freeman, 1977. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Peano’s Axioms 1. Zero is a number. 2. If a is a number, the successor of a is a number. 3. ZERO is not the successor of a number. 4. Two numbers of which the successors are equal are themselves equal. 5. (INDUCTION AXIOM .) If a set S of numbers contains ZERO and also the successor of every number in S, then every number is in S. Peano’s axioms are the basis for the version of NUMBER THEORY known as PEANO ARITHMETIC . See also INDUCTION AXIOM ,PEANO ARITHMETIC Pear Curve The LEMNISCATE L3in the iteration towards the MANDELBROT SET .I nC ARTESIAN COORDINATES with a constant r, the equation is given by r2/C30x2/C27y29+=9+; (1/C272x/C275x2/C276x3/C276x4/C274x5/C27x6 /C283y2/C282xy2/C278x2y2/C278x3y2/C273x4y2/C272y4 /C274xy4/C273x2y4/C27y6): See also PEAR-SHAPED CURVE Pearls of Sluze ym /C30kxn(a /C28x)b : The curves with integer n, b, and m were studied by de Sluze between 1657 and 1698. The name "Pearls of Sluze" was given to these curves by Blaise Pascal (MacTutor Archive). References MacTutor History of Mathematics Archive. "Pearls of Sluze." http://www-groups.dcs.st-and.ac.uk/~history/Curves/ Pearls.html. Pear-Shaped Curve A curve given by the Cartesian equation b2y2 /C30x3(a /C28x) : See also PEAR CURVE ,TEARDROP CURVE References MacTutor History of Mathematics Archive. "Pear-Shaped Cubic." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Pearshaped.html. Pearson Kurtosis Let m4be the fourth CENTRAL MOMENT of random variable and m2 its second CENTRAL MOMENT (i.e., the VARIANCE ). Then the Pearson kurtosis is defined by b2 /C13m4 m2 2:See also CENTRAL MOMENT ,FISHER KURTOSIS ,KUR- TOSIS Pearson Mode Skewness Given a STATISTICAL DISTRIBUTION with measured MEAN , MODE , and STANDARD DEVIATION s, the Pearson mode skewness is mean /C28 mode s: See also MEAN,M ODE,PEARSON SKEWNESS ,PEAR- SON’S SKEWNESS COEFFICIENTS ,SKEWNESS Pearson Skewness Let a STATISTICAL DISTRIBUTION have third MOMENT m3and STANDARD DEVIATION s; then the Pearson skewness is defined by b1 /C30m3 s3 !2 : See also FISHER SKEWNESS ,PEARSON’S SKEWNESS COEFFICIENTS ,SKEWNESS Pearson System A system of equation types obtained by generalizing the differential equation for the G AUSSIAN DISTRIBU- TION dy dx/C30y(m/C28x) a; (1) which has solution y/C30Ce(2m/C28x)x=(2a); (2) to dy dx/C30y(m/C28x) a/C27bx/C27cx2; (3) which has solution y/C30Ca/C27bx/C27cx29+=9+;/C281=(2c) /C2exp(b/C272cm) tan/C281b/C272cxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ac/C28b2p ! cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac/C28b2p2 666643 77775: (4) Letc 1;c2be the roots of a/C27bx/C27cx2:Then the possible types of curves are 0.b/C30c/C300;a/C210. E.g., NORMAL DISTRIBUTION . I.b2=4acB0;c15x5c2:E.g., BETA DISTRIBUTION . II. b2=4ac/C300;cB0,/C28c15x5c1where c1/C13ffiffiffiffiffiffiffiffiffiffiffi /C28c=ap :/ III.b2=4ac/C30/C12;c/C300,c15xB/C12 where c1/C13/C28a=b: E.g., GAMMA DISTRIBUTION . This case is intermedi- ate to cases I and VI. IV. 0Bb2=4acB1;/C28/C12B xB/C12 :/ V.b2=4ac/C301;c15xB/C12 where c1/C13/C28b=2a:Inter- mediate to cases IV and VI.VI.b 2=4ac>1;c15xB/C12 where c1is the larger root. E.g., BETA PRIME DISTRIBUTION . VII.b2=4ac/C300;c/C210,/C28/C12B xB/C12 :E.g., S TUDENT’S T-DISTRIBUTION . Classes IX-XII are discussed in Pearson (1916). Seealso Craig (in Kenney and Keeping 1951). If a Pearson curve possesses a MODE , it will be at x/C30m. Let y(x)/C300a t c1and c2;where these may be /C28/C12or/C12:Ifyxr/C272also vanishes at c1;c2;then the rth MOMENT and ( r/C271)/thMOMENTS exist. gc2 c1dy dxaxr/C27bxr/C271/C27cxr/C2729+=9+; dx /C30gc2 c1ym xr/C28xr/C2719+=9+; dx; (5) giving ya xr/C27bxr/C271/C27cxr/C2729+=9+;9+$9+%c2 c1/C28gc2 c1y arxr/C281/C27b(r/C271)xr9+$ /C27c(r/C272)xr/C271/C138dx ¼gc2 c1y(mxr/C28xr/C271)dx (6) 0/C28gc2 c1y arxr/C281/C27b(r/C271)xr/C27c(r/C272)xr/C2719+$9+% dx /C30gc2 c1ym xr/C28xr/C2719+=9+; dx: (7) Now define the raw rth moment by nr/C30gc2 c1yxrdx; (8) so combining (7) with (8) gives arnr/C281/C27b(r/C271)nr/C27c(r/C272)nr/C271/C30/C28mnr/C27nr/C271:(9) Forr/C300, b/C272cn1/C30/C28m/C27n1; (10) so n1/C30m/C27b 1/C282c; (11) and for r/C301,a/C272bn1/C273cn2/C30/C28mn1/C27n2; (12) so n2/C30a/C27(m/C272b)n1 1/C283c: (13) Combining (11), (13), and the definitions n1/C300 (14) n2/C30m2/C301 (15) obtained by letting t/C13x/C28n1 ðÞ =sand solving simulta- neously gives b/C30/C28manda/C301/C283c:Writing ar/C30mr/C30nr (16) then allows the general recurrence to be written (1/C283c)rar/C281/C28mrar/C27[c(r/C272)/C281]ar/C271/C300: (17) For the special cases r/C302 and r/C303, this gives 2m/C27(1/C284c)a3/C300: (18) 3(1/C283c)/C283ma3/C28(1/C285c)a4/C300; (19) so the SKEWNESS and KURTOSIS are g1/C30a3/C302m 4c/C281(20) g2/C30a4/C283/C306m2/C284c2/C27c ðÞ (4c/C281)(5c/C281): (21) The parameters a,b, and ccan therefore be written a/C301/C283c (22) b/C30/C28m/C30g1 2(1/C272d)(23) c/C30d 2(1/C272d); (24) where d/C132g2/C283g2 1 g2/C276: (25) References Craig, C. C. "A New Exposition and Chart for the Pearson System of Frequency Curves." Ann. Math. Stat. 7,1 6/C1/28, 1936. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 107, 1951. Pearson, K. "Second Supplement to a Memoir on Skew Variation." Phil. Trans. A 216, 429/C1/457, 1916. Pearson Type III Distribution A skewed distribution which is similar to the BINO- MIAL DISTRIBUTION when p"q(Abramowitz and Ste- gun 1972, p. 930). y /C30k(t /C27A)A2/C281e /C28At ; (1) for t /C23 0 ;/C12½Þ where A /C132 =g (2) K /C13AA2 e /C28A2 G A2ðÞ; (3) /G(z) is the GAMMA FUNCTION , and t is a standardized variate. Another form is P(x) /C301 bG(p)x /C28 a b !p /C281 exp /C28x /C28 a b ! : (4) For this distribution, the CHARACTERISTIC FUNCTION is f(t) /C30eiat(1 /C28i bt)/C28p ; (5) and the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C30 a /C27p b (6) s2 /C30p b2 (7) g1 /C302 ffiffiffipp (8) g2 /C306 p : (9) See also PEARSON TYPE IV DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Pearson Type IV Distribution See also PEARSON TYPE III DISTRIBUTION References Nagahara, Y. "The PDF and CF of Pearson Type IV Distributions and the ML Estimation of the Parameters." Stat. Prob. Let. 43, 251 /C1/264, 1999. Pearson-Cunningham Function CUNNINGHAM FUNCTION Pearson’s Correlation CORRELATION COEFFICIENTPearson’s Function IX2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(k /C28 1)p ;k /C28 3 2 ! /C13G1 2 x2 s ;k /C28 1 2 ! Gk /C28 1 2 ! ; where G(x) is the GAMMA FUNCTION . See also CHI-SQUARED TEST,GAMMA FUNCTION Pearson’s Skewness Coefficients Given a STATISTICAL DISTRIBUTION with measured MEAN , MEDIAN , MODE , and STANDARD DEVIATION s, Pearson’s first skewness coefficient is 3[mean] /C28 [mode] s; and the second coefficient is 3[mean] /C28 [median] s : See also FISHER SKEWNESS ,P EARSON SKEWNESS , SKEWNESS References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 101 /C1/102, 1962. Peaucellier Cell PEAUCELLIER INVERSOR Peaucellier Inversor ALINKAGE with six rods which draws the inverse of a given curve. When a pencil is placed at P, the inverse is drawn at P?(or vice versa). If a seventh rod (dashed) is added (with an additional pivot), Pis kept on a circle and the locus traced out by P?is a straight line. It therefore converts circular motion to linear motion without sliding, and was discovered in1864. Another LINKAGE which performs this feat using hinged squares had been published by Sarrus in 1853 but ignored. Coxeter (1969, p. 428) shows that OP /C29OP?/C30OA2 /C28PA2 : See also HART’S INVERSOR ,KEMPE LINKAGE ,LINKAGE References Bogomolny, A. "Peaucellier Linkage." http://www.cut-the- knot.com/pythagoras/invert.html. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods. Oxford, England: Oxford University Press, p. 156, 1978. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 82 /C1/83, 1969. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 117, 1928. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 46 /C1/48, 1990. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 121 /C1/126, 1957. Sarrus. Comptes Rendus de l’Acade ´mie de Paris 36, 1036, 1853. Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 324, 1994. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 139, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 120 and 181 /C1/182, 1991. Peaucellier’s Linkage PEAUCELLIER INVERSOR Pedal PEDAL CURVE Pedal-Cevian Point If the PEDAL TRIANGLE of a point P in a TRIANGLE DABC is a CEVIAN TRIANGLE , then the point P is called the pedal-cevian point of DABC with respect to the PEDAL TRIANGLE . The CIRCUMCENTER O, ORTHOCENTER H, and INCEN- TER I of a triangle DA1A2A3 are always pedal-Cevian points, with corresponding pedal triangles given by the MEDIAL TRIANGLE DM1M2M3 ; ORTHIC TRIANGLE DH1H2H3 ; and CONTACT TRIANGLE DT1T2T3 ; respec- tively, and PEDAL POINTS the CENTROID G, ORTHO- CENTER H, and GERGONNE POINT Ge; respectively (Honsberger 1995, p. 142). If P is a pedal-Cevian point of a triangle, then so is its ISOTOMIC CONJUGATE POINT Q, as is its reflection P? in the CIRCUMCENTER (Honsberger 1995, p. 143). See also CEVIAN ,CEVIAN TRIANGLE ,PEDAL POINT , PEDAL TRIANGLEReferences Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 142 /C1/143, 1995. Pedal Circle The pedal circle with respect to a PEDAL POINT P of a TRIANGLE DA1A2A3 is the CIRCUMCIRCLE of the PEDAL TRIANGLE DP1P2P3 with respect to P. Amazingly, the vertices of the PEDAL TRIANGLE DQ1Q2Q3of the ISOGONAL CONJUGATE point Q of P also lie on the same circle (Honsberger 1995). If the PEDAL POINT is taken as the INCENTER , the pedal circle is given by the INCIRCLE . The radius of the pedal circle of a point P is r /C30A1P /C215A2P /C215A3P 2 R2 /C28OP29+;k9+;7 (Johnson 1929, p. 141). When P is on a side of the TRIANGLE , the line between the two perpendiculars is called the PEDAL LINE. Given four points, no three of which are COLLINEAR , then the four PEDAL CIRCLES of each point for the TRIANGLE formed by the other three have a common point through which the NINE-POINT CIRCLES of the four TRIANGLES pass. See also FONTENE ´ THEOREMS ,GRIFFITHS’ THEOREM , MIQUEL POINT ,N INE-POINT CIRCLE ,P EDAL LINE, PEDAL TRIANGLE References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 50, 1971. Fontene ´, G. "Sur le cercle pe ´dal." Nouv. Ann. Math. 65,5 5/C1/ 58, 1906. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., p. 54, 1991. Honsberger, R. "The Pedal Circle." §7.4 (viii) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 67 /C1/69, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Pedal Coordinates The pedal coordinates of a point Pwith respect to the curve Cand the PEDAL POINT Oare the radial distance r from O to P and the PERPENDICULAR distance p from O to the line L tangent to C at P. See also PEDAL CURVE ,PEDAL POINT References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 2 /C1/3, 1972. Yates, R. C. "Pedal Equations." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 166 /C1/ 169, 1952. Pedal Curve The pedal of a curve C with respect to a point O is the LOCUS of the foot of the PERPENDICULAR from P to the TANGENT to the curve. More precisely, given a curve C, the pedal curve P of C with respect to a fixed point O (called the PEDAL POINT ) is the locus of the point P of intersection of the PERPENDICULAR from O to a TANGENT to C. The parametric equations for a curve (f(t) ;g(t)) relative to the PEDAL POINT (x0 ; y0) are given by x /C30x0f ?2 /C27 fg ?2 /C27 y0 /C28 g ðÞ f ?g? f ?2 þ g ?2 y /C30gf ?2 /C27 y0g ?2 /C27 x0 /C28 f ðÞ f ?g ? f ?2 /C27 g ?22 : When a CLOSED CURVE rolls on a straight line, the AREA between the line and ROULETTE after a complete revolution by any point on the curve is twice the AREA of the pedal curve (taken with respect to the generat- ing point) of the rolling curve. The following table gives the pedal curves for a number of common special curves. Curve PEDAL POINT Pedal Curve ASTROID center QUADRIFOLIUM CARDIOID cusp CAYLEY’S SEXTIC CIRCLE any point LIMAC ¸ ON CIRCLE on CIRCUMFER- ENCECARDIOIDCIRCLE INVOLUTEcenter of CIRCLE ARCHIMEDEAN SPIRAL CISSOID OF DIOCLESFOCUS CARDIOID DELTOID center TRIFOLIUM DELTOID cusp simple FOLIUM DELTOID on curve unsymmetric double folium DELTOID vertex double folium ELLIPSE FOCUS CIRCLE EPICYCLOID center ROSE HYPERBOLA center LEMNISCATE HYPERBOLA FOCUS CIRCLE HYPOCYCLOID center ROSE LINE any point point LOGARITHMICSPIRAL pole LOGARITHMICSPIRAL PARABOLA FOCUS LINE PARABOLA foot of DIRECTRIXRIGHT STROPHOID PARABOLA onDIRECTRIX STROPHOID PARABOLA reflection of FOCUS by DIRECTRIXMACLAURIN TRISECTRIX PARABOLA vertex CISSOID OF DIOCLES SINUSOIDAL SPIRALpole SINUSOIDALSPIRAL TSCHIRNHAUSEN CUBICcenter PARABOLA See also NEGATIVE PEDAL CURVE References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 25, 1999. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 46 /C1/49 and 204, 1972. Lockwood, E. H. "Pedal Curves." Ch. 18 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 152 /C1/155, 1967. Yates, R. C. "Pedal Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 160 /C1/ 165, 1952. Pedal Line Mark a point Pon a side of a TRIANGLE and draw the perpendiculars from the point to the two other sides. The line between the feet of these two perpendiculars is called the pedal line. See also PEDAL TRIANGLE ,SIMSON LINE Pedal Point The fixed point with respect to which a PEDAL CURVE or PEDAL TRIANGLE is drawn. See also PEDAL- CEVIAN POINT ,PEDAL CURVE ,PEDAL TRIANGLE References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. New York: Random House, p. 22, 1967. Pedal Triangle Given a point P, the pedal triangle of P is the TRIANGLE whose VERTICES are the feet of the perpen- diculars from P to the side lines. The pedal triangle of a TRIANGLE with TRILINEAR COORDINATES a : b : g and angles A, B, and C has VERTICES with TRILINEAR COORDINATES 0:b /C27 a cos C : g /C27 a cos B (1) a /C27 b cos C :0:g /C27 b cos A (2) a /C27 g cos B : b /C27 g cos A :0: (3)The SYMMEDIAN POINT of a triangle is the CENTROID of its pedal triangle (Honsberger 1995, pp. 72 /C1/74). The third pedal triangle is similar to the original one. This theorem can be generalized to: the nth pedal n- gon of any n-gon is similar to the original one. It is also true that P2P3 /C30A1P sin a1 (4) (Johnson 1929, pp. 135 /C1/136; Stewart 1940; Coxeter and Greitzer 1967, p. 25). The AREA A of the pedal triangle of a point P is proportional to the POWER of P with respect to the CIRCUMCIRCLE , A /C301 2R2 /C28OP29+;k9+;7 sin a1 sin a2 sin a3 /C30R2 /C28OP2 4R2D (5) (Johnson 1929, pp. 139 /C1/141). The only closed BILLIARDS path of a single circuit in an ACUTE TRIANGLE is the pedal triangle. There are an infinite number of multiple-circuit paths, but all segments are parallel to the sides of the pedal triangle (Wells 1991). See also ANTIPEDAL TRIANGLE ,FAGNANO’S PROBLEM , ORTHIC TRIANGLE ,PEDAL CIRCLE ,PEDAL LINE References Coxeter, H. S. M. and Greitzer, S. L. "Pedal Triangles." §1.9 inGeometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 22 /C1/26, 1967. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 67 /C1/74, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Stewart, B. M. "Cyclic Properties of Miquel Polygons." Amer. Math. Monthly 47, 462/C1/466, 1940. Peg The answer to the question "which fits better, a round peg in a square hole, or a square peg in a round hole?"can be interpreted as asking which is larger, the ratioof the AREA of a CIRCLE to its circumscribed SQUARE , or the AREA of the SQUARE to its circumscribed CIRCLE ? In 2-D, the ratios are p=4 and 2 =p;respec- tively. Therefore, a round peg fits better into a square hole than a square peg fits into a round hole (Wells 1986, p. 74). However, this result is true only in dimensions n B9, and for n ]9; the unit n-hypersphere fits more closely into the 9-hypercube than vice versa (Singmaster; Wells 1986, p. 74). This can be demonstrated by noting that the formulas for the content V(n) of the unit n-ball, the content Vc(n) of its circumscribed HYPERCUBE , and the content Vi(n) of its inscribed HYPERCUBE are given by V(n) /C30pn=2 G1 2 n /C27 19+;k9+;7 (1) Vc(n) /C302n (2) Vi(n) /C302n nn=2 : (3) The ratios in question are then Rround peg /C30V(n) Vc(n) /C30pn=2 2n G12 n /C27 19+;k9+;7 (4) Rsquare peg /C30Vi(n) Vc(n) /C302 G12 n /C27 19+;k9+;7 nn=2nn=2 (5) (Singmaster 1964). As illustrated above, Rround B Rsquare only for n B9, with equality at n :8 :13785 :/ See also HOLE,H YPERSPHERE PACKING ,PEG SOLI- TAIRE References Singmaster, D. "On Round Pegs in Square Holes and Square Pegs in Round Holes." Math. Mag. 37, 335 /C1/339, 1964. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 74, 1986. Peg Knot CLOVE HITCHPeg Solitaire A game played on a cross-shaped board with 33 holes. All holes but the middle one are initially filled with pegs. The goal is to remove all pegs but one by jumping pegs from one side of an occupied peg hole to an empty space, removing the peg which was jumped over. Strategies and symmetries are discussed by Gosper et al. (1972). Berlekamp et al. (1982) give a complete solution of the puzzle. A triangular version called HI-Q also exists (Beeler et al. 1972, Item 76). Kraitchik (1942) considers a board with one additional hole placed at the vertices of the central right angles. See also HI-Q References Beasley, J. D. The Ins and Outs of Peg Solitaire. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Ch. 23 in Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982. Gardner, M. "Peg Solitaire." Ch. 11 in The Unexpected Hanging and Other Mathematical Diversions. New York: Simon and Schuster, pp. 122 /C1/135 and 250 /C1/251, 1969. Gosper, R. W.; Brown, S.; and Rayfield, M. Item 75 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cam- bridge, MA: MIT Artificial Intelligence Laboratory, MemoAIM-239, pp. 28 /C1 /29, Feb. 1972. Kraitchik, M. "Peg Solitaire." §12.19 in Mathematical Re- creations. New York: W. W. Norton, pp. 297 /C1/298, 1942. Peg Top PIRIFORM Peirce Decomposition LetAbe a finite-dimensional power-associative alge- bra, then Ais the vector space DIRECT SUM A/C30A11/C27A10/C27A01/C27A00; where Aij;with i;j/C300;1 is the subspace of Adefined by Aij/C30fxij:exij/C30ixij;xije/C30jxijg fori;j/C300;1;where eis an idempotent. References Schafer, R. D. "The Peirce Decomposition." §3.2 in An Introduction to Nonassociative Algebras. New York: Do- ver, pp. 32 /C1/37, 1996. Peirce’s Theorem The only linear associative algebra in which the coordinates are REAL NUMBERS and products vanish only if one factor is zero are the FIELD of REAL NUMBERS , the FIELD of COMPLEX NUMBERS , and the algebra of QUATERNIONS with REAL COEFFICIENTS . See also COMPLEX NUMBER ,Q UATERNION ,R EAL NUMBER ,W EIERSTRASS’S THEOREM References Schafer, R. D. "The Peirce Decomposition." §3.2 in An Introduction to Nonassociative Algebras. New York: Do- ver, pp. 32 /C1/37, 1996. p-Element SEMISIMPLE ELEMENT p-Elementary Subgroup A p-elementary subgroup of a FINITE GROUP G is a SUBGROUP H which is the GROUP DIRECT PRODUCT H /C30Cn /C29P ; where P is a P-GROUP , Cnis a cyclic group, and pdoes not divide n. See also GROUP ,GROUP DIRECT PRODUCT ,INDUCED REPRESENTATION , P-GROUP Pell Equation A special case of the quadratic D IOPHANTINE EQUA- TION having the form x2/C28Dy2/C301; (1) where D/C210 is a nonsquare NATURAL NUMBER (Dick- son 1952). The equation x2/C28Dy2/C3094 (2) arising in the computation of FUNDAMENTAL UNITS is sometimes also called the Pell equation (Do ¨rrie 1965, Itoˆ1987), and Do ¨rrie calls the positive form of (2) the FERMAT DIFFERENCE EQUATION . While Fermat de- serves credit for being the first to extensively study the equation, the erroneous attribution to Pell was perpetrated by none other than Euler himself (Nagell 1951, p. 197; Dickson 1957, p. 341; Burton 1989). ThePell equation was also solved by the Indian mathe-matician Bhaskara. Pell equations are extremely important in NUMBER THEORY , and arise in the investigation of numbers which are FIGURATE in more than one way, for example, simultaneouslysquare and triangular.The equation has an obvious generalization to the Pell-like equation ax 29by2/C30c; (3) as well as the general second-order bivariate Dio-phantine equation ax 2/C27bxy/C27cy2/C27dx/C27ey/C27f/C300: (4) However, several different technique are required to solve this equation for arbitrary values of a,b, and c. In a future release of Mathematica , the command Reduce will find solutions to the general equation (4), when they exist. Pell equations OF THE FORM (1), as well as certain cases of the analogous equation with a minus sign on the right, x2/C28Dy2/C30/C281; (5) can be solved by finding the CONTINUED FRACTION a0;a1;... ½/C138 offfiffiffiffi Dp :Note that although the equation (5) is solvable for only certain values of D, the continued fraction technique provides solutions when they exist, and always in the case of (1), forwhich a solution always exists. A necessary condition that (5) be solvable is that all odd prime factors of D be OF THE FORM 4n/C271;and that Dcannot be DOUBLY EVEN (i.e., divisible by 4). However, these conditions are not SUFFICIENT for a solution to exist, as demon- strated by the equation x2/C2834y2/C30/C281;which has no solutions in integers (Nagell 1951, pp. 201 and 204). In all subsequent discussion, ignore the trivial solu- tion x/C301,y/C300. Let pn=qndenote the nthCONVER- GENT a0;a1;...;an ½/C138 ;then we will have solved (1) or (5) if we can find a CONVERGENT which obeys the identity p2 n/C28Dq2n/C30(/C281)n/C271: (6) Amazingly, this turns out to always be possible as a result of the fact that the CONTINUED FRACTION of a QUADRATIC SURD always becomes periodic at some term ar/C271;where ar/C271/C302a0;i.e., ffiffiffiffi Dp /C30a0;a1;...;ar;2a09+$9+% : (7) To compute the CONTINUED FRACTION convergents toffiffiffiffi Dp ;use the usual RECURRENCE RELATIONS a0/C30ffiffiffiffi Dpjk p0/C30a0 (8) p1/C30a0a1/C271 (9) pn/C30anpn/C281/C27pn/C282 (10) q0/C301 (11) q1/C30a1 (12) qn/C30anqn/C281/C27qn/C282; (13) where xbcis the FLOOR FUNCTION . For reasons to be explained shortly, also compute the two additional quantities PnandQndefined by P0/C300 (14) P1/C30a0 (15) Pn/C30an/C281Qn/C281/C28Pn/C281 (16) Q0/C301 (17) Q1/C30D/C28a2 0 (18) Qn/C30D/C28P2 n Qn/C281(19) an/C30a0/C27Pn Qn$% : (20) Now, two important identities satisfied by CONTINUED FRACTION convergents are pnqn/C281/C28pn/C281qn/C30(/C281)n/C271(21) p2 n/C28Dq2n/C30(/C281)n/C271Qn/C271 (22) (Beiler 1966, p. 262), so both linear ax/C28by/C3091 (23) and quadratic x2/C28Dy2/C309c (24) equations are solved simply by finding an appropriate continued fraction. Letar/C271/C302a0be the term at which the continued fraction becomes periodic (which will always happen for a quadratic surd). For the Pell equation x2/C28Dy2/C301 (25) with rODD,(/C281)r/C271isPOSITIVE and the solution in terms of smallest INTEGERS isx/C30prandy/C30qr;where pr=qris the rth CONVERGENT .I f risEVEN , then (/C281)r/C271isNEGATIVE , but p2 2r/C271/C28Dq22r/C271/C301; (26) so the solution in smallest INTEGERS isx/C30p2r/C271;y/C30 q2r/C271:Summarizing, (x;y)/C30pr;qr ðÞ forrodd p2r/C271;p2r/C2719+=9+; forreven :9+$k (27) The equation x2/C28Dy2/C30/C281 (28) can be solved analogously to the equation with /C271o n the right side IFFrisEVEN , but has no solution if ris odd,(x;y)/C30pr;qr ðÞ forreven no solution for rodd:9+$k (29) Given one solution ( x;y)/C30(p;q) (which can be found as above), a whole family of solutions can be found by taking each side to the nthPOWER , x2/C28Dy2/C30p2/C28Dq29+=9+;n/C301: (30) Factoring gives x/C27ffiffiffiffi Dp y9+;k9+;7 x/C28ffiffiffiffiDp y9+;k9+;7 /C30p/C27ffiffiffiffiDp q9+;k9+;7 n p/C28ffiffiffiffiDp q9+;k9+;7 n (31) and x/C27ffiffiffiffiDp y/C30p/C27ffiffiffiffiDp q9+;k9+;7 n (32) x/C28ffiffiffiffi Dp y/C30p/C28ffiffiffiffiDp q9+;k9+;7 n ; (33) which gives the family of solutions x/C30p/C27qffiffiffiffi Dp9+;k9+;7n /C27p/C28qffiffiffiffiDp9+;k9+;7 n 2(34) y/C30p/C27qffiffiffiffiDp9+;k9+;7 n /C28p/C28qffiffiffiffiDp9+;k9+;7 n 2ffiffiffiffi Dp : (35) These solutions also hold for x2/C28Dy2/C30/C281; (36) except that ncan take on only ODD values. The following table gives the smallest integer solu- tions ( x, y) to the Pell equation with constant D5102 (Beiler 1966, p. 254). S QUARE D/C30d2are not included, since they would result in an equation OF THE FORM x2/C28d2y2/C30x2/C28(dy)2/C30x2/C28y?2/C301; (37) which has no solutions (since the difference of two SQUARES cannot be 1). Dx y D x y 2 3 2 54 485 66 32 1 5 5 8 9 1 259 4 5 6 1 5 2 6 5 2 57 151 20 7 8 3 58 19603 25748 3 1 59 530 69 10 19 6 60 31 4 11 10 3 61 1766319049 226153980 12 7 2 62 63 8 13 649 180 63 8 114 15 4 65 129 1615 4 1 66 65 8 17 33 8 67 48842 5967 18 17 4 68 33 419 170 39 69 7775 93620 9 2 70 251 30 21 55 12 71 3480 413 22 197 42 72 17 223 24 5 73 2281249 26700024 5 1 74 3699 430 26 51 10 75 26 3 27 26 5 76 57799 663028 127 24 77 351 4029 9801 1820 78 53 6 30 11 2 79 80 9 31 1520 273 80 9 132 17 3 82 163 1833 23 4 83 82 9 34 35 6 84 55 6 35 6 1 85 285769 3099637 73 12 86 10405 1122 38 37 6 87 28 3 39 25 4 88 197 2140 19 3 89 500001 5300041 2049 320 90 19 2 42 13 2 91 1574 165 43 3482 531 92 1151 12044 199 30 93 12151 126045 161 24 94 2143295 221064 46 24335 3588 95 39 4 47 48 7 96 49 548 7 1 97 62809633 637735250 99 14 98 99 10 51 50 7 99 10 1 52 649 90 101 201 2053 66249 9100 102 101 10The first few minimal values of xandyfor nonsquare Dare 3, 2, 9, 5, 8, 3, 19, 10, 7, 649, ... (Sloane’s A033313) and 2, 1, 4, 2, 3, 1, 6, 3, 2, 180, ... (Sloane’s A033317), respectively. The values of Dhaving x/C302, 3, ... are 3, 2, 15, 6, 35, 12, 7, 5, 11, 30, ... (Sloane’s A033314) and the values of Dhaving y/C301, 2, ... are 3, 2, 7, 5, 23, 10, 47, 17, 79, 26, ... (Sloane’s A033318). Values of the incrementally largest minimal xare 3, 9, 19, 649, 9801, 24335, 66249, ... (Sloane’s A033315)which occur at D/C302, 5, 10, 13, 29, 46, 53, 61, 109, 181, ... (Sloane’s A033316). Values of the incremen-tally largest minimal yare 2, 4, 6, 180, 1820, 3588, 9100, 226153980, ... (Sloane’s A033319), which occuratD/C302, 5, 10, 13, 29, 46, 53, 61, ... (Sloane’s A033320). The more complicated Pell-like equation x 2/C28Dy2/C30c (38) with cjjBffiffiffiffi Dp has solution IFFcis one of the values (/C281)kQkfork/C301, 2, ..., rcomputed in the process of finding the convergents toffiffiffiffiDp (where, as above, a r/C271/C302a0is the term at which the continued fraction becomes periodic). If cjj>ffiffiffiffi Dp ;the procedure is sig- nificantly more complicated (Beiler 1966, p. 265; Dickson 1992, pp. 387 /C1/388) and is discussed by Ge´rardin (1910) and Chrystal (1961). Regardless of how it is found, if a single solution x/C30p,y/C30qto (38) is known, other solutions can be found. Let pandqbe solutions to (38), and rands solutions to the "unit" form x2/C28Dy2/C301: (39) Then the identity p2/C28Dq29+=9+; r2/C28Ds29+=9+; /C30(pr9Dqs)2/C28D(ps9qr)2 /C30c (40) allows larger solutions ( x;y)/C30(pr9Dqs ;ps9qr)t o thecequation to be found by using incrementally larger values of the ( r, s), which can be easily computed using the standard technique for the Pell equation. Such a family of solutions does not neces- sarily generate allsolutions, however. For example, the equation x2/C2810y2/C309 (41) has three distinct sets of fundamental solutions, (x;y)/C30(7;2);(13, 4), and (57, 18). Using (40), these generate the solutions shown in the following table, from which the set of all solutions (7, 2), (13, 4), (57,18), (253, 80), (487, 154), (2163, 684), (9607, 3038), ... can be generated. fundamental generated solutions (7, 2) (253, 80), (9607, 3038), (364813, 115364), (13853287, 4380794), ... (13, 4) (487, 154), (18493, 5848), (702247, 222070), (26666893, 8432812), ... (57, 18) (2163, 684), (82137, 25974), (3119043, 986328), (118441497, 37454490), ... The case ax2 /C28by2 /C30c (42) can be reduced to the one above by multiplying through by a, (ax)2 /C28(ab)y2 /C30ac; (43) finding solutions in (x ?/C13ax ; y) ; and then selecting those for which x?=a is an integer. According to Dickson (1992, pp. 408 and 411), the equation ax2 /C27by2 /C30c (44) with a ; b; c > 0; which has either no solutions or a finite number of solutions, was solved by Gauss (1863) using the METHOD OF EXCLUSIONS and con- sidered by Euler (1773) and Nasimoff (1885), although Euler’s methods were incomplete (Dickson 1992, p. 378; Smith 1965). According to Itoˆ (1987), this equation can be solved completely using solutions to Pell’s equation. Nasimoff (1885) applied Jacobi elliptic functions to express the number of solutions of this equation for a, c ODD (Dickson 1992, p. 411). Additional discussion including the connection with elliptic functions is given in Dickson (1992, pp. 387 /C1/ 391). The special case of a /C301 and c prime was solved by Cornacchia (Cornacchia 1908, Cox 1989, Wagon 1990). Solution for a /C301, b ]1; and odd c is imple- mented in Mathematica as QuadraticRepresen- tation [b, c] in the Mathematica add-on pack- age NumberTheory‘NumberTheoryFunctions‘ (which can be loaded with the command BBNumberTheory‘ ). A deterministic algorithm for finding all primitive solutions to (44) for a ; b; c > 0 fixed relatively prime integers, c ]a /C27b /C271 ; and (c ; ab) /C301 was given by Hardy et al. (1990). This algorithm generalizes those of Hermite (1848), Ser- ret (1848), Brillhart (1972), Cornacchia (1908), and Wilker (1980). It requires factorization of c, and has worst case running time of Oc1=4(lnc)3(ln ln c))(ln ln ln c9+;k9+;7 ;independent of a andb.See also BINARY QUADRATIC FORM,D IOPHANTINE EQUATION ,D IOPHANTINE EQUATION–2ND POWERS , FUNDAMENTAL UNIT,H ILBERT SYMBOL ,LAGRANGE NUMBER (DIOPHANTINE EQUATION ), MONOMORPH , POLYMORPH References Beiler, A. H. "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Enter- tains. New York: Dover, pp. 248 /C1/268, 1966. Brillhart, J. "Note on Representing a Prime as a Sum of Two Squares." Math. Comput. 26, 1011/C1/1013, 1972. Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, pp. 379 /C1/382 and 392, 1989. Chrystal, G. Textbook of Algebra, 2nd ed., Vol. 2. New York: Chelsea, pp. 478 /C1/486, 1961. Cipolla, M. "Un metodo per la risoluzione della congruenza di secondo grado." Rend. Accad. Sci. Fis. Mat. Napoli 9, 154/C1/163, 1903. Cohn, H. "Pell’s Equation." §6.9 in Advanced Number Theory. New York: Dover, pp. 110 /C1/111, 1980. Cornacchia, G. "Su di un metodo per la risoluzione in numeri unteri dell’ equazione an h/C300chxn/C28hyh/C30P:/"Giornale di Matematiche di Battaglini 46,3 3/C1/90, 1908. Cox, D. A. Primes OF THE FORM x2/C27ny2:/New York: Wiley, 1989. Degan, C. F. Canon Pellianus. Copenhagen, Denmark, 1817. Dickson, L. E. "Pell Equation: ax2/C27bx/C27cMade Square." Ch. 12 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 341 /C1/400, 1952. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965. Euler, L. Novi Comm. Acad. Petrop. 18, 218, 1773. Rep- rinted in Opera Omnia, Vol. 3 , p. 310. Euler, L. Comm. Arith. 570. Reprinted in Opera Omnia, Vol. 3 , p. 310. Ge´rardin, A. "Formules de re ´currence." Sphinx-Oedipe 5, 17/C1/29, 1910. Hardy, K.; Muskat, J. B.; and Williams, K. S. "A Determi- nistic Algorithm for Solving n/C30fu2/C27gv2in Coprime Integers uandv."Math. Comput. 55, 327/C1/343, 1990. Hermite, C. "Note au sujet de l’article pre ´ce´dent." J. Math. Pures Appl. 13, 15, 1848. Itoˆ, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed, Vol. 1. Cambridge, MA: MIT Press, p. 450, 1987. Lagarias, J. C. "On the Computational Complexity of De- termining the Solvability or Unsolvability of the Equation X2/C28Dy2/C30/C281:/"Trans. Amer. Math. Soc. 260, 485/C1/508, 1980. Nagell, T. "The Diophantine Equation x2/C28Dy2/C301;/" "The Diophantine Equation x2/C28Dy2/C30/C281;/" and "The Diophan- tine Equation u2/C28Dv2/C30C:/"§56/C1/58 in Introduction to Number Theory. New York: Wiley, pp. 195 /C1/212, 1951. Nasimoff, P. S. Ch. 1 in Application of Elliptic Functions to the Theory of Numbers. Moscow, 1885. French summary inAnn. sci. de l’E ´cole normale supe ´r.5,2 3/C1/31, 1888. Serret, J. A. "Sur un the ´ore`me re ´latif aux nombres enti‘- eres." J. Math. Pures Appl. 13,1 2/C1/14, 1848. Sloane, N. J. A. Sequences A033313, A033314, A033315, A033316, A033317, A033318, A033319, and A033320 in"An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Smith, H. J. S. Collected Mathematical Papers I. New York: Chelsea, pp. 195 /C1 /202, 1965. Smarandache, F. "Un metodo de resolucion de la ecuacion diofantica." Gaz. Math. 1, 151/C1/157, 1988. Smarandache, F. " Method to Solve the Diophantine Equa- tion ax2 /C28by2 /C27c /C300 :/"InCollected Papers, Vol. 1. Lupton, AZ: Erhus University Press, 1996. Stillwell, J. C. Mathematics and Its History. New York: Springer-Verlag, 1989. Wagon, S. "The Euclidean Algorithm Strikes Again." Amer. Math. Monthly 97, 124 /C1/125, 1990. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Whitford, E. E. Pell Equation. New York: Columbia Uni- versity Press, 1912. Wilker, P. "An efficient Algorithmic Solution of the Dio- phantine Equation u2 /C275v2 /C30m:/" Math. Comput. 35, 1347 /C1/1352, 1980. Pell-Lucas Number PELL NUMBER Pell-Lucas Polynomial PELL POLYNOMIAL Pell Number The numbers obtained by the Un/s in the LUCAS SEQUENCE with P /C302 and Q /C30/C28 1. They and the Pell-Lucas numbers (the Vn/s in the LUCAS SEQUENCE ) satisfy the RECURRENCE RELATION Pn /C302Pn/C281 /C27Pn/C282 : (1) Using Pi to denote a Pell number and Qi to denote a Pell-Lucas number, Pm/C27n /C30PmPn/C271 /C27Pm/C281Pn (2) Pm/C27n /C302PmQn /C28(/C281)nPm/C28n ; (3) P2tm /C30Pm(2Qm)(2Q2m)(2Q4m) /C1/C1/C1 2Q2t/C281m ðÞ (4) Q2 m /C302P2m /C27(/C281)m (5) Q2m /C302Q2m /C28(/C281)m : (6) The Pell numbers have P0 /C300 and P1 /C301 and are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (Sloane’s A000129). The Pell-Lucas numbers have Q0 /C302 and Q1 /C302 and are 2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, ... (Sloane’s A002203). The only TRIANGULAR Pell number is 1 (McDaniel 1996). See also BRAHMAGUPTA POLYNOMIAL ,PELL POLYNO- MIAL References McDaniel, W. L. "Triangular Numbers in the Pell Se- quence." Fib. Quart. 34, 105 /C1/107, 1996. Ram, R. "Pell Numbers Formulae." http://users.tellurian.- net/hsejar/maths/pell/.Sloane, N. J. A. Sequences A000129/M1413 and A002203/ M0360 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Pell Polynomial The Pell polynomials P(x) and Lucas-Pell polynomials Q(x) are generated by a LUCAS POLYNOMIAL SE- QUENCE using generator (2x; 1): This gives recursive equations for P(x) from P0(x) /C30P1(x) /C301 and Pn/C272(x) /C302xPn/C271(x) /C27Pn(x) : (1) The first few are P1 /C301 P2 /C302x P3 /C304x2 /C281 P4 /C308x3 /C284x P5 /C3016x4 /C2812x2 /C271 : The Pell-Lucas numbers are defined recursively by q0(x) /C301; q1(x) /C30x and qn/C272(x) /C302xqn /C271(x) /C27qn(x); (2) together with Qn(x) /C132qn(x) : (3) The first few are Q1 /C302x Q2 /C304x2 /C282 Q3 /C308x3 /C286x Q4/C3016x4/C2816x2/C272 Q5/C3032x5/C2840x3/C2710x: See also LUCAS POLYNOMIAL SEQUENCE References Horadam, A. F. and Mahon, J. M. "Pell and Pell-Lucas Polynomials." Fib. Quart. 23,7/C1/20, 1985. Mahon, J. M. M. A. (Honors) thesis, The University of New England. Armidale, Australia, 1984. Sloane, N. J. A. Sequences A000129/M1413 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Pell Sequence PELLNUMBER Pencil The set of all LINES through a point. The term was first used by Desargues (Cremona 1960, p. x). The six angles of any pencils of four rays OfABCD g are connected by the relation sin BOC sin AOD /C27sin COA sin BOD /C27sin AOB sin COD /C300 and the lengths satisfy BC /C215 AD /C27CA /C215 BD /C27AB /C215 CD /C300 (Lachlan 1893). Woods (1961) uses the term pencil as a synonym for RANGE , and Altshiller-Court (1979, p. 12) uses the term to mean SHEAF OF PLANES . See also NEAR-PENCIL ,PERSPECTIVITY ,RANGE (LINE SEGMENT ), SECTION (PENCIL ), SHEAF OF PLANES References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979. Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960. Lachlan, R. "Relations Connecting the Angles of a Pencil." §29 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 16 /C1/18, 1893. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 36, 1930. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 8 and 11 /C1/12, 1961. Pencil of Coaxal Circles COAXAL CIRCLES Pencil of Planes SHEAF OF PLANESPeninsula Surface A QUINTIC SURFACE given by the equation x2 /C27y3 /C27z5 /C301: See also QUINTIC SURFACE Penrose Stairway An IMPOSSIBLE FIGURE (also called the SCHROEDER STAIRS ) in which a stairway in the shape of a square appears to circulate indefinitely while still possessing normal steps. The Dutch artist M. C. Escher included Penrose stairways in many of his mind-bending illustrations. See also IMPOSSIBLE FIGURE References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 15, 1989. Jablan, S. "Impossible Figures." http://members.tripod.com/ ~modularity/impos.htm. Pappas, T. "Optical Illusions and Computer Graphics." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 5, 1989. Robinson, J. O. and Wilson, J. A. "The Impossible Colonnade and Other Variations of a Well-Known Figure." Brit. J. Psych. 64, 363/C1/365, 1973. Penrose Tiles A pair of shapes which tile the plane only aperiodi- cally (when the markings are constrained to match at borders). The two tiles, illustrated above, are called the "KITE" and "DART ." To see how the plane may be tiled aperiodically using the kite and dart, divide the kite into acute and obtuse tiles, shown above. Now define "deflation" and "inflation" operations. The deflation operator takes an acute TRIANGLE to the union of two ACUTE TRIANGLES and one OBTUSE , and the OBTUSE TRIANGLE goes to an ACUTE and an OBTUSE TRIANGLE . These operations are illustrated below. When applied to a collection of tiles, the deflation operator leads to a more refined collection. The operators do not respect tile boundaries, but do respect the half tiles defined above. There are two ways to obtain aperiodic TILINGS with 5-fold symme- try about a single point. they are known an the "star" and "sun" configurations, and are show below. Higher order versions can then be obtained by deflation. For example, the following are third-order deflations: References Gardner, M. "Extraordinary Nonperiodic Tiling that En- riches the Theory of Tiles." Sci. Amer. 110 /C1/119, Dec. 1977. Gardner, M. "Penrose Tiling" and "Penrose Tiling II." Chs. 1 /C1/2in Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 1 /C1/29, 1989. Hurd, L. P. "Penrose Tiles." http://www.mathsource.com/cgi- bin/msitem?0206 /C1/772. Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 86 /C1/95, 1988. Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., pp. 2 and 34 /C1/36, 1999. Smith, T. "Penrose Tilings and Wang Tilings." http:// www.innerx.net/personal/tsmith/pwtile.html. Vichera, M. "Penrose Tiling." http://alpha.ujep.cz/~vicher/ puzzle/penrose/penr.htm. Wagon, S. "Penrose Tiles." §4.3 in Mathematica in Action. New York: W. H. Freeman, pp. 108 /C1/117, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 175 /C1/177, 1991. Penrose Triangle TRIBAR Penrose Tribar TRIBAR Pentabolo A5- POLYABOLO . Pentacle PENTAGRAM Pentacontagon A 50-sided POLYGON . Pentacube This entry contributed by RONALD M. AARTS A POLYCUBE composed of 5 cubes. There are 29 distinct three-dimensional pentacubes (Bouwkamp 1981). Of these, the 12 planar pentacubes (corre- sponding to solid pentominoes), are well known. Among the nonplanar pentacubes, there are fivethat have at least one plane of symmetry; each of them is its own mirror image. The remaining 12 pentacubes come in mirror image pairs. See also P OLYCUBE References Bouwkamp, C. J. "Packing Handed Pentacubes." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, 1981. Pentad A group of five elements. See also MONAD ,PAIR,Q UADRUPLET ,Q UINTUPLET , TETRAD ,TRIAD,TRIPLET ,TWINS Pentadecagon A 15-sided POLYGON , sometimes also called the PENTAKAIDECAGON . For a regular pentadecagon with side length 1, the INRADIUS r, CIRCUMRADIUS R, and AREA A are r /C301 2ffiffiffiffiffiffi 3/C27pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq9+;89+;9 R /C301 4ffiffiffi 3p /C27ffiffiffiffiffiffi15p /C27ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffi 5pq 9+;89+;9 A /C3015 8ffiffiffi3p /C27ffiffiffiffiffiffi15p /C27ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C27ffiffiffi 5pq 9+;89+;9 : See also P OLYGON ,REGULAR POLYGON ,TRIGONOME- TRY VALUES PI/15Pentaflake AFRACTAL with 5-fold symmetry. As illustrated above, five PENTAGONS can be arranged around an identical PENTAGON to form the first iteration of the pentaflake. This cluster of six pentagons has the shape of a pentagon with five triangular wedgesremoved. This construction was first noticed by Albrecht Du ¨rer (Dixon 1991). For a pentagon of side length 1, the first ring of pentagons has centers at RADIUS d1/C302r/C301 21/C27ffiffiffi 5p9+;k9+;7 R/C30fR; (1) where fis the GOLDEN RATIO . The INRADIUS rand CIRCUMRADIUS Rare related by r/C30Rcos1 5p9+;k9+;7 /C3014ffiffiffi 5p /C2719+;k9+;7 R; (2) and these are related to the side length sby s/C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2/C28r2p /C301 2Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pq : (3) The height his h/C30ssin2 5p9+;k9+;7 /C3014sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pq /C301 2ffiffiffi 5p R; (4) giving a RADIUS of the second ring as d2/C302R/C27h ðÞ /C302/C27ffiffiffi 5p9+;k9+;7 R/C30f3R: (5) Continuing, the nth pentagon ring is located at dn/C30f2n/C281: (6) Now, the length of the side of the first pentagon compound is given by s2/C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2r/C27R)2/C28(h/C27R)2q /C30Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pq ; (7) so the ratio of side lengths of the original pentagon to that of the compound is s2 s/C30Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27 2ffiffiffi 5pp 1 2 Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C28 2ffiffiffi 5pp /C301 /C27 f: (8) We can now calculate the dimension of the pentaflake fractal. Let Nn be the number of black pentagons and Lnthe length of side of a pentagon after the n iteration, Nn /C306n (9) Ln /C30 1 /C27 f ðÞ/C28n: (10) The CAPACITY DIMENSION is therefore dcap/C30/C28lim n0/C12lnNn lnLn/C30ln 6 ln(1/C27f)/C30ln 2/C27ln 3 ln(1/C27f)(11) See also PENTAGON References Dixon, R. Mathographics. New York: Dover, pp. 186 /C1/188, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 104, 1991. Pentagon The regular convex 5-gon is called the pentagon. By SIMILAR TRIANGLES in the figure on the left, d 1/C301 1 f/C30f; (1) where dis the diagonal distance. But the dashed vertical line connecting two nonadjacent VERTICES is the same length as the diagonal one, so f/C301/C271 f(2) f2/C28f/C281: (3) Solving the QUADRATIC EQUATION gives 1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C274p 2; (4)and taking the plus sign gives the GOLDEN RATIO f/C301 21/C27ffiffiffi 5p9+;k9+;7 : (5) (Taking the minus sign instead gives 1 =f:/) The coordinates of the VERTICES relative to the center of the pentagon with unit sides are given as shown in the above figure, with c1/C30cos2p 5 ! /C301 4ffiffiffi 5p /C2819+;k9+;7 (6) c2/C30cos4p 5 ! /C301 4ffiffiffi 5p /C2719+;k9+;7 (7) s1/C30sin2p 5 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pq (8) s2/C30sin4p 5 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pq : (9) For a REGULAR POLYGON , the CIRCUMRADIUS ,INRA- DIUS,SAGITTA , and AREA are given by Rn/C301 2acscp n ! (10) rn/C301 2acotp n ! (11) xn/C30Rn/C28rn/C3012atanp 2n ! (12) An/C301 4na2cotp n ! : (13) Plugging in n/C305 gives R/C3012acse15p9+;k9+;7 /C301 10affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2710ffiffiffi 5pq (14) r/C301 2acot15p9+;k9+;7 /C301 10affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pq (15) x/C301 2a1 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2810ffiffiffi 5pq (16) A/C301 4a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffiffi 5:pq (17) Five pentagons can be arranged around an identical pentagon to form the first iteration of the " PENTA- FLAKE ," which itself has the shape of a pentagon with five triangular wedges removed. For a pentagon of side length 1, the first ring of pentagons has centers at radius f; the second ring at f3 ; and the nth at f2n/C281 :/ In proposition IV.11, Euclid showed how to inscribe a regular pentagon in a CIRCLE . Ptolemy also gave a RULER and COMPASS construction for the pentagon in his epoch-making work The Almagest. While Ptole- my’s construction has a SIMPLICITY of 16, a GEO- METRIC CONSTRUCTION using CARLYLE CIRCLES can be made with GEOMETROGRAPHY symbol 2S1 /C27S2 /C27 8C1 /C270C2 /C274C3 ; which has SIMPLICITY 15 (De Temple 1991). The following elegant construction for the pentagon is due to Richmond (1893). Given a point, a CIRCLE may be constructed of any desired RADIUS , and a DIAMETER drawn through the center. Call the center O, and the right end of the DIAMETER P1 : The DIAMETER PERPEN- DICULAR to the original DIAMETER may be constructed by finding the PERPENDICULAR BISECTOR . Call the upper endpoint of this PERPENDICULAR DIAMETER B. For the pentagon, find the MIDPOINT of OB and call it D. Draw DP1 ; and BISECT /C218ODP1 ; calling the inter- section point with OP1N2 : Draw N2P2PARALLEL to OB, and the first two points of the pentagon are P1 and P2 ; and copying the angle /C218P1OP2 then gives the remaining points P3 ; P4 ; and P5 (Coxeter 1969, Wells 1991). Madachy (1979) illustrates how to construct a penta- gon by folding and knotting a strip of paper. See also CYCLIC PENTAGON ,DECAGON ,DISSECTION , FIVE DISKS PROBLEM ,H OME PLATE ,PENTAFLAKE , PENTAGRAM ,POLYGON ,TRIGONOMETRY VALUES PI/5References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 95 /C1/96, 1987. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 26 /C1/28, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97 /C1/108, 1991. Dickson, L. E. "Regular Pentagon and Decagon." §8.17 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 368 /C1/370, 1955. Dixon, R. Mathographics. New York: Dover, p. 17, 1991. Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 38, 1970. Fukagawa, H. and Pedoe, D. "Pentagons." §4.3 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Ca- nada: Charles Babbage Research Foundation, pp. 49 and 132 /C1/134, 1989. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 59, 1979. Pappas, T. "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188 /C1/189, 1989. Richmond, H. W. "A Construction for a Regular Polygon of Seventeen Sides." Quart. J. Pure Appl. Math. 26, 206 /C1/ 207, 1893. Wantzel, M. L. "Recherches sur les moyens de reconnaı ˆtre si un Proble `me de Ge´ome´trie peut se re´soudre avec la re`gle et le compas." J. Math. pures appliq. 1, 366 /C1/372, 1836. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 211, 1991. Pentagon Tiling There are at least 14 classes of convex PENTAGONAL tilings (Steinhaus 1983, p. 75; Wells 1991, pp. 177 /C1/ 179; Pegg), as illustrated above. It has not been proven whether these 14 cases exhaust all possibletilings, but no others are known. See also T ILING References Bowers, P. L. and Stephenson, K. "A ‘Regular’ Pentagonal Tiling of the Plane." Submitted to Conformal Geom. Dynamics . Pegg, E. Jr. "The 14 Different Types of Pentagons that Tile the Plane." http://www.mathpuzzle.com/tilepent.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 177 /C1/179, 208, and 211, 1991. Pentagonal Antiprism An ANTIPRISM and UNIFORM POLYHEDRON U77whose DUAL POLYHEDRON is the PENTAGONAL DELTAHEDRON . Pentagonal Cupola JOHNSON SOLID J5 : The bottom 10 VERTICES are 91 /C27ffiffiffi 5p9+=9+;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffi 5pp 4ffiffiffi 2p ;91 2 ; 0 ! ; 91 /C27ffiffiffi 5p9+=9+;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5pp 4ffiffiffi 2p ;93 /C27ffiffiffi 5p 2; 0 ! ; 0;91 21 /C27ffiffiffi 5p9+;k9+;7 ;09+;k9+;7 and the top five vertices areffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffi 5pp ffiffiffiffiffiffi10p ; 0;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5pp ffiffiffiffiffiffi10p ! ; ffiffiffi5p /C28 19+=9+;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffi 5pp 4ffiffiffiffiffiffi10p ;9 1 41 /C27ffiffiffi 5p9+;k9+;7 ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5pp ffiffiffiffiffiffi10p ! ; /C28ffiffiffi5p /C27 19+=9+;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C27ffiffiffi 5pp 4ffiffiffiffiffiffi10p ;9 1 2 ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5pp ffiffiffiffiffiffi10p ! : Pentagonal Deltahedron ATRAPEZOHEDRON which is the DUAL POLYHEDRON of the PENTAGONAL ANTIPRISM U77:/ See also DUAL POLYHEDRON ,P ENTAGONAL ANTI- PRISM ,TRAPEZOHEDRON Pentagonal Dipyramid The pentagonal dipyramid is one of the convex DELTAHEDRA , and J OHNSON SOLID J13:It is also the DUAL POLYHEDRON of the PENTAGONAL PRISM U76:The distance between two adjacent VERTICES on the base of the PENTAGON is d2 12 /C30 1 /C28cos2 5 p9+;k9+;7hi2 /C27sin225p9+;k9+;7 /C30 1 /C2814ffiffiffi 5p /C2819+;k9+;7hi2 /C271 /C27ffiffiffi 5p9+=9+;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5pp 4ffiffiffi 2p"# 2 /C301 25 /C28ffiffiffi 5p9+;k9+;7 ; (1) and the distance between the apex and one of the base points is d2 1h /C30 0 /C281 ðÞ2/C27 0 /C280 ðÞ2/C27 h /C280 ðÞ2/C301 /C27h2 : (2) But d212 /C30d212 (3) 1 25 /C28ffiffiffi 5p9+;k9+;7 /C301 /C27h2 (4) h2 /C301 23 /C28ffiffiffi 5p9+;k9+;7 ; (5) and h /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C28ffiffiffi 5p 2s : (6) This root is OF THE FORMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a /C27b /C27cp ; so applying SQUARE ROOT simplification gives h /C301 2ffiffiffi 5p /C2819+;k9+;7 /C13 f /C281; (7) where f is the GOLDEN MEAN . See also DELTAHEDRON ,DIPYRAMID ,GOLDEN MEAN, ICOSAHEDRON ,JOHNSON SOLID,RIGIDITY THEOREM , TRIANGULAR DIPYRAMID Pentagonal Gyrobicupola JOHNSON SOLID J31 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Pentagonal Gyrocupolarotunda JOHNSON SOLID J33 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Pentagonal Hexecontahedron The 60-faced DUAL POLYHEDRON of the SNUB DODECA- HEDRON A8and Wenninger dual W18:/ See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , HEXECONTAHEDRON ,SNUB DODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 29, 1983. Pentagonal Icositetrahedron The 24-faced DUAL POLYHEDRON of the SNUB CUBE A7 and Wenninger dual W17 : The mineral cuprite / Cu2O ðÞ forms in pentagonal icositetrahedral crystals (Steinhaus 1983, pp. 207 and 209). The dual formed from a SNUB CUBE with unit edge length has side lengths given by the unique positive real roots of 2s6 1 /C284s41 /C274s21 /C281 /C300 (1) 32s61 /C2832s41 /C278s21 /C281 /C300 : (2) The CIRCUMRADIUS R is given by the unique positive real root of 128r6 /C28224r4 /C2824r2 /C281 /C300 : (3) The SURFACE AREA S given by the positive real root of S6 /C28684S4 /C27142560 S2 /C289879408 /C300 ; (4) and VOLUME V given by the positive real root of 8V6 /C28452V4 /C27462V2 /C28121 /C300: (5) See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , ICOSITETRAHEDRON ,SNUB CUBE,SNUB CUBE-PENTA- GONAL ICOSITETRAHEDRON COMPOUND References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 28, 1983. Pentagonal Number A POLYGONAL NUMBER OF THE FORM n 3n /C281 ðÞ =2 : The first few are 1, 5, 12, 22, 35, 51, 70, ... (Sloane’sA000326). The GENERATING FUNCTION for the penta- gonal numbers is x 2x /C27 1 ðÞ 1 /C28 x ðÞ3/C30x /C275x2 /C2712x3 /C2722x4 /C27...: Every pentagonal number is 1/3 of a TRIANGULAR NUMBER . The so-called generalized pentagonal numbers are given by n 3n /C281 ðÞ =2 with n /C300, 9 1, 9 2, ..., the first few of which are 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (Sloane’s A001318). See also HEPTAGONAL PENTAGONAL NUMBER ,HEXA- GONAL PENTAGONAL NUMBER ,OCTAGONAL PENTAGO- NAL NUMBER ,PARTITION FUNCTION P,PENTAGONAL NUMBER THEOREM ,PENTAGONAL SQUARE NUMBER , PENTAGONAL TRIANGULAR NUMBER ,P OLYGONAL NUMBER ,TRIANGULAR NUMBER References Guy, R. K. "Sums of Squares." §C20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 /C1/138, 1994. Pappas, T. "Triangular, Square & Pentagonal Numbers." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989. Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996. Sloane, N. J. A. Sequences A000326/M3818 and A001318/ M1336 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Pentagonal Number Theorem Y/C12 k /C3011 /C28xk9+=9+; /C30X/C12 k /C30/C28/C12/C281ðÞkxk 3k/C271 ðÞ =2(1) /C301 /C27X/C12 k/C30/C281/C281ðÞkxk 3k/C281 ðÞ =2/C27xk 3k/C271 ðÞ =29+$9+% ; (2) where n 3n /C271 ðÞ =2 are generalized PENTAGONAL NUM- BERS . Related equalities are Y/C12 k/C3011/C28xkt9+=9+; /C30X/C12 n/C300/C281ðÞnxnn/C271 ðÞ =2tn Qn k/C3011/C28xk ðÞ(3) Y/C12 k/C3011/C28xkt9+=9+; /C281/C30X/C12 n/C300tn Qn k/C3011/C28xk ðÞ: (4) See also PARTITION FUNCTION P,PARTITION FUNC- TION Q,PENTAGONAL NUMBER ,RAMANUJAN THETA FUNCTIONS References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, p. 72, 1935. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 64, 1987. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 83 /C1/85, 1999. Pentagonal Orthobicupola JOHNSON SOLID J30 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Pentagonal Orthobirotunda JOHNSON SOLID J34 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Pentagonal Orthocupolarontunda JOHNSON SOLID J32 :/References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Pentagonal Prism A PRISM , HEPTAHEDRON , and UNIFORM POLYHEDRON U76whose DUAL POLYHEDRON is the PENTAGONAL DIPYRAMID . The SURFACE AREA and VOLUME for the pentagonal prism of unit edge length are S/C301 210/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 55/C272ffiffiffi 5p9+;k9+;7r9+;89+;9 V/C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 55/C272ffiffiffi 5p9+;k9+;7r : See also HEPTAHEDRON ,PENTAGRAMMIC PRISM Pentagonal Pyramid JOHNSON SOLID J2:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . APYRAMID with a PENTAGONAL base. The pentagonal pyramid having equilateral triangles as faces is J OHNSON SOLID J2:The SLANT HEIGHT of a regular pentagonal pyramid is a special case of the formula for a regular n- gonal PYRAMID with n/C305, given by s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C271 105/C27ffiffiffi 5p9+;k9+;7 a2;r (1) where h is the height and a is the length of a side of the base. See also PENTAGON ,PYRAMID Pentagonal Pyramidal Number A FIGURATE NUMBER corresponding to a PENTAGONAL PYRAMID . The first few are 1, 6, 18, 40, 75, ... (Sloane’s A002411). The GENERATING FUNCTION for the penta- gonal pyramidal numbers is x 2x /C27 1 ðÞ x /C28 1 ðÞ4/C30x /C276x2 /C2718x3 /C2740x4 /C27...: The odd pentagonal pyramidal numbers are given by 1, 75, 405, 1183, 2601, ... (Sloane’s A015223), having squares 1, 5625, 164025, ... (Sloane’s A014799), while the even pentagonal pyramidal numbers are given by 6, 18, 40, 126, 196, 288, ... (Sloane’s A015224), having squares 36, 324, 1600, 15876, ... (Sloane’s A014800). See also PENTAGONAL NUMBER ,PYRAMIDAL NUMBER References Sloane, N. J. A. Sequences A002411/M4116, A014799, A014800, A015223, and A015224 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Pentagonal Rotunda Half of an ICOSIDODECAHEDRON , denoted R5 : It has 10 triangular and five pentagonal faces separating a PENTAGONAL ceiling and a DODECAHEDRAL floor. It is JOHNSON SOLID J6 ; and the only true ROTUNDA . See also ICOSIDODECAHEDRON ,JOHNSON SOLID ,RO- TUNDA Pentagonal Square Number A number which is simultaneously a PENTAGONAL NUMBER Pn and a SQUARE NUMBER Sm : Such numbers exist when 1 2 n 3n /C281 ðÞ /C30m2 : (1) COMPLETING THE SQUARE gives12 n 3n /C281 ðÞ /C3032n2 /C2813 n9+;k9+;7 /C3032n /C28169+;k9+;72 /C283 72 /C30m2(2) 3 66n /C281 ðÞ2/C2832 /C3036m2 (3) 6n /C281 ðÞ2/C2824m2 /C301: (4) Substituting x /C306n /C281 and y /C302m gives the PELL EQUATION x2 /C286y2 /C301; (5) which has solutions x;yðÞ/C30 5;2ðÞ ; (49, 20), (495, 198), .... In terms of (n, m), these give (1,1), (25/3, 10), (81, 99), (2401/3, 980), (7921, 9701), ..., of which the whole number solutions are n ; mðÞ /C30 1; 1ðÞ ; (81, 99), (7921, 9701), (776161, 950599), ... (Sloane’s A046172 and A046173), corresponding to the pentagonal square numbers 1, 9801, 94109401, 903638458801, 8676736387298001, ... (Sloane’s A036353). Rathbun has searched for pentagonal square trian- gular numbers up to index 2000, but found none other than the trivial number 1. See also PENTAGONAL NUMBER ,SQUARE NUMBER References Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996. Sloane, N. J. A. Sequences A036353, A046172, and A046173 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Pentagonal Triangular Number A number which is simultaneously a PENTAGONAL NUMBER Pnand TRIANGULAR NUMBER Tm : Such numbers exist when 1 2 n 3n /C281 ðÞ /C3012 mm/C271 ðÞ : (1) COMPLETING THE SQUARE gives 6n /C281 ðÞ2/C2832m/C27ðÞ2/C30/C282: (2) Substituting x /C306n /C281 and y /C302m /C271 gives the Pell- like quadratic Diophantine equation x2 /C283y2 /C30/C282; (3) which has solutions x; yðÞ/C30 5; 3ðÞ ; (19, 11), (71, 41), (265, 153), .... In terms of (n, m), these give (1, 1), (10/ 3,5), (12, 20), (133/3, 76), (165, 285), ..., of which the whole number solutions are n; mðÞ /C30 1 ; 1ðÞ ; (12, 20), (165, 285), (2296, 3976), ... (Sloane’s A046174 and A046175), corresponding to the pentagonal triangular numbers 1, 210, 40755, 7906276, 1533776805, ... (Sloane’s A014979). Rathbun has searched for pentagonal square trian- gular numbers up to index 2000, but found none other than the trivial number 1. See also PENTAGONAL NUMBER ,TRIANGULAR NUMBER References Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996. Sloane, N. J. A. Sequences A014979, A046174, and A046175 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Pentagram The STAR POLYGON 5=2fg ; also called the PENTACLE , PENTALPHA ,or PENTANGLE . In the above figure, the pentagram has side length 1, and the indicated lengths are given by a /C30ffiffiffi 5p /C282 (1) b /C301 23 /C28ffiffiffi 5p9+;k9+;7 (2) r /C301 2 a cotp 5 ! /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 155 /C282ffiffiffi 5p9+;k9+;7r (3) R /C301 2 a cscp 5 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1025 /C2811ffiffiffi 5p9+;k9+;7r (4) h /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C281 2 a9+;k9+;72r /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C282ffiffiffi 5pq (5) x /C302 r /C27h ðÞ sinp 5 ! /C301 2ffiffiffi 5p /C2819+;k9+;7 (6) r ?/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h /C27r ðÞ2/C271 2 x9+;k9+;72r /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 105 /C27ffiffiffi 5p9+;k9+;7r (7) y /C30r ?/C28R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1225 /C2711ffiffiffi 5p9+;k9+;7r (8) L /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C281 4 x2q /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 125 /C27ffiffiffi 5p9+;k9+;7 :r (9) This gives the ratio b a /C30 f; (10) where f is the GOLDEN RATIO (Wells 1986, p. 36). A series of embedded pentagrams can be constructed to form a larger pentagram, as illustrated above (Williams 1979, p. 53). If the central pentagram has center (0, 0) and CIRCUMRADIUS 1, then the subse- quent pentagrams have radii rn/C30f/C28n and centers xn/C30/C281 4(1/C28f/C28n)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2722ffiffiffi 5pp yn/C301 2f(1/C28f/C28n) modulo rotation by 2 pk=5;where fis the GOLDEN RATIO . See also DISSECTION ,FIVE CIRCLES THEOREM ,GREAT DODECAHEDRON ,GREAT ICOSAHEDRON ,GREAT STEL- LATED DODECAHEDRON ,H EXAGRAM ,H OEHN’S THEO- REM,K EPLER- POINSOT SOLID ,P ENTAGON ,S MALL STELLATED DODECAHEDRON ,STAR FIGURE ,STAR OF LAKSHMI References Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 122 /C1/125, 1990. Pappas, T. "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188 /C1/189, 1989. Schwartzman, S. The Words of Mathematics: An Etymologi- cal Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 211, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 36, 1986. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Pentagrammic Antiprism An ANTIPRISM and UNIFORM POLYHEDRON U79whose DUAL POLYHEDRON is the PENTAGRAMMIC DELTAHE- DRON . Pentagrammic Concave Deltahedron The DUAL POLYHEDRON of the PENTAGRAMMIC CROSSED ANTIPRISM U80 :/ See also DUAL POLYHEDRON ,P ENTAGRAMMIC CROSSED ANTIPRISM Pentagrammic Crossed Antiprism An ANTIPRISM and UNIFORM POLYHEDRON U80whoseDUAL POLYHEDRON is the PENTAGRAMMIC CONCAVE DELTAHEDRON . Pentagrammic Deltahedron The DUAL POLYHEDRON of the PENTAGRAMMIC ANTI- PRISM U79 :/ See also DUAL POLYHEDRON ,PENTAGRAMMIC ANTI- PRISM Pentagrammic Dipyramid The DUAL POLYHEDRON of the PENTAGRAMMIC PRISM U78 :/ See also DUAL POLYHEDRON ,PENTAGRAMMIC PRISM Pentagrammic Prism A PRISM , self-intersecting HEPTAHEDRON , and UNI- FORM POLYHEDRON U78whose DUAL POLYHEDRON is the PENTAGRAMMIC DIPYRAMID . See also HEPTAHEDRON ,PENTAGONAL PRISM Pentagrammic Pyramid See also PYRAMID Pentahedral Graph A POLYHEDRAL GRAPH on five nodes. There are two topologically distinct pentahedral graphs, corre- sponding to the skeletons of the SQUARE PYRAMID (left figure) and TRIANGULAR DIPYRAMID (right figure). The pentahedral graphs were first enumerated by Steiner (1828; Duijvestijn and Federico 1981). See also POLYHEDRAL GRAPH ,S QUARE PYRAMID , TRIANGULAR DIPYRAMID . References Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (/3/-Connected Planar) Graphs." Math. Comput. 37, 523 /C1/532, 1981. Steiner, J. "Proble `me de situation." Ann. de Math 19, 36, 1828. Reprinted in Jacob Steiner’s gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 227, 1971. Pentahedron A POLYHEDRON having five faces. Common pentahe- dra include the SQUARE PYRAMID and the TRIANGULAR PRISM . Steiner (1828) was the first to enumerate the pentahedra (Duijvestijn and Federico 1981). See also PENTAHEDRAL GRAPH ,POLYHEDRON ,SQUARE PYRAMID ,TRIANGULAR PRISM References Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (/3/-Connected Planar) Graphs." Math. Comput. 37, 523 /C1/532, 1981.Steiner, J. "Proble `me de situation." Ann. de Math. 19, 36, 1828. Reprinted in Jacob Steiner’s gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 227, 1971. Pentakaidecagon PENTADECAGON Pentakis Dodecahedron The 60-faced DUAL POLYHEDRON of the TRUNCATED ICOSAHEDRON A11and Wenninger dual W9 : It can be constructed by CUMULATION of a unit edge-length DODECAHEDRON by a pyramid with height 1 19ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 565 /C2722ffiffiffi 5p9+=9+;q : Taking the dual of a TRUNCATED ICOSAHEDRON with unit edge lengths gives a pentakis dodecahedron with edge lengths s1 /C301 1918ffiffiffi 5p /C2899+;k9+;7 (1) s2 /C303 2ffiffiffi 5p /C2819+;k9+;7 : (2) Normalizing so that s1 /C301; the SURFACE AREA and VOLUME are S /C305 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12421 /C2763ffiffiffi 5p9+;k9+;7r (3) V /C305 3641 /C2725ffiffiffi5p9+;k9+;7 : (4) See also A RCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , DUAL POLYHEDRON ,HEXECONTAHEDRON ,TRUNCATED ICOSAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 18, 1983. Pentalpha PENTAGRAM Pentangle PENTAGRAM Pentaspherical Space The set of all points xthat can be put into one-to-one correspondence with sets of essentially distinct values of five homogeneous coordinates x0: x1: x2: x3: x4 ; not all simultaneously zero, which are connected by the relation x /C215 x /C30x2 0 /C27x21 /C27x22 /C27x23 /C27x24 /C300 : (1) See also TETRACYCLIC PLANE References Coolidge, J. L. "Pentaspherical Space." Ch. 7 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 282 /C1/305, 1971. Pentatope The simplest regular figure in 4-D, representing the 4-D analog of the solid TETRAHEDRON . It is also called the 5-cell, since it consists of five vertices. The pentatope is the 4-D SIMPLEX , and can be viewed as a regular TETRAHEDRON ABCD in which a point E along the fourth dimension through the center of ABCD is chosen so that EA /C30EB /C30EC /C30ED /C30AB: The pentatope has SCHLA ¨ FLI SYMBOL f3; 3; 3g: The pentatope is self-dual, has 5 3-D facets (each the shape of a TETRAHEDRON ), 10 ridges (faces), 10 edges, and 5 vertices. In the above figure, the pentatope is shown projected onto one of the four mutually perpendicular 3-spaces within the 4-space obtained by dropping one of the four vertex components (R. Towle). See also 16-CELL, 24-CELL, 120-CELL, 600-CELL,HYPER- CUBE ,POLYTOPE ,SIMPLEX ,TETRAHEDRON References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 187 /C1/188, 1984. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 179 /C1/180 and 210, 1991. Pentatope Number A FIGURATE NUMBER which is given by Ptopn /C301 4 Ten(n /C273) /C301 24 n(n /C271)(n /C272)(n /C273); where Ten is the nth TETRAHEDRAL NUMBER . The firstfew pentatope numbers are 1, 5, 15, 35, 70, 126, ... (Sloane’s A000332). The GENERATING FUNCTION for the pentatope numbers is x (1 /C28 x)5 /C30x /C275x2 /C2715x3 /C2735x4 /C27...: See also FIGURATE NUMBER ,TETRAHEDRAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55 /C1/57, 1996. Sloane, N. J. A. Sequences A000332/M3853 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Pentiamond One of the four 5-polyiamonds are called pentia- monds. See also PENTIAMOND TILING ,POLYIAMOND Pentiamond Tiling See also HEPTIAMOND TILING ,H EXIAMOND TILING , OCTIAMOND TILING ,PENTIAMOND References Vichera, M. "Polyiamonds." http://alpha.ujep.cz/~vicher/puz- zle/polyform/iamond/iamonds.htm. Pentomino The twelve 5-POLYOMINOES illustrated above and known by the letters of the alphabet they most closely resemble: f, I, L, N, P, T, U, V, W, X, y, Z (Gardner 1960, Golomb 1995). Another common naming con- vention replaces f, I, L, and N with R, O, Q, and S so that all letters from O to Z are used (Berlekamp et al. 1982). In particular, in the LIFE CELLULAR AUTOMA- TON, the f-pentomino is always known as the r- pentomino. The I, L, and T pentominoes can also be called the 5- STRAIGHT POLYOMINO ,L-POLYOMINO , andT-POLYOMINO , respectively. See also DOMINO ,H EXOMINO ,H EPTOMINO ,OCTOMI- NO,POLYOMINO ,TETROMINO ,TRIOMINO References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 110 /C1/111, 1987. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Games in General. London: Academic Press, 1982. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982. Dudeney, H. E. "The Broken Chessboard." Problem 74 in The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, pp. 119 /C1/120, 1949. Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150 /C1/156, May 1957. Gardner, M. "Mathematical Games: More About the Shapes that Can Be Made with Complex Dominoes." Sci. Amer. 203, 186 /C1/198, Nov. 1960. Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, 1995. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 80 /C1/86, 1975. Lei, A. "Pentominoes." http://www.cs.ust.hk/~philipl/omino/ pento.html. Madachy, J. S. "Pentominoes: Some Solved and Unsolved Problems." J. Rec. Math. 2, 181 /C1/188, 1969. O’Beirne, T. H. "Pentominoes and Hexiamonds." New Scien- tist 12, 379 /C1/380, 1961. Ruskey, F. "Information on Pentomino Puzzles." http:// www.theory.csc.uvic.ca/~cos/inf/misc/PentInfo.html. Smith, A. "Pentomino Relationships." http://www.snaffles.- demon.co.uk/pentanomes/. Pe´pin’s Test A test for the PRIMALITY of FERMAT NUMBERS Fn /C30 22n /C271; with n ]2 and k ]2: Then the two following conditions are equivalent: 1. Fnis PRIME and (k=Fn) /C30/C281; where (n=k) is the JACOBI SYMBOL , 2. k(Fn/C281)=2 /C13/C281 (mod Fn)::/ k is usually taken as 3 as a first test. See also FERMAT NUMBER ,PE´ PIN’S THEOREM References Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, p. 62, 1991. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 119 /C1/120, 1993. Pe´pin’s Theorem The FERMAT NUMBER Fn is PRIME IFF 322n /C281 /C13/C281 (mod Fn): See also FERMAT NUMBER ,PE´ PIN’S TEST,SELFRIDGE- HURWITZ RESIDUE Per Cent PERCENTPer Mil PERMIL Per Mille PERMIL Percent The use of percentages is a way of expressing RATIOS in terms of whole numbers. Given a RATIO or FRAC- TION , it is converted to a percentage by multiplying by 100 and appending a "percentage sign" %. For example, if an investment grows from a number P /C30 13 :00 to a number A /C3022 :50; then A is 22:50 =13:00 /C30 1:7308 times as much as P, or 173.08%, and the investment has grown by 73.08%. A change of a certain percent n is sometimes said to be a change of PERCENTAGE POINTS . See also PERCENTAGE ERROR ,PERCENTAGE POINT , PERMIL References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 282, 1997. Percent Sign The symbol % used to indicate PERCENT . References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 282, 1997. Percentage PERCENT ,PERCENTAGE ERROR ,PERCENTAGE POINT Percentage Error The percentage error is 100% times the RELATIVE ERROR . See also ABSOLUTE ERROR ,E RROR PROPAGATION , PERCENT ,RELATIVE ERROR References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Percentage Point 1%. See also BASIS POINT ,PERCENT Percentile The kth percentile Pk is that value of x, say xk ; which corresponds to a CUMULATIVE FREQUENCY of Nk =100:/ See also QUANTILE ,QUARTILE References Kenney, J. F. and Keeping, E. S. "Percentile Ranks." §3.6 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 38 /C1/39, 1962. Percolation Theory Percolation theory deals with fluid flow (or any other similar process) in random media. If the medium is a set of regular LATTICE POINTS , then there are two types of percolation. A SITE PERCOLATION considers the lattice vertices as the relevant entities; a BOND PERCOLATION considers the lattice edges as the relevant entities. See also BOND PERCOLATION ,CAYLEY TREE,CLUSTER , CLUSTER PERIMETER ,LATTICE ANIMAL ,PERCOLATION THRESHOLD ,POLYOMINO ,RANDOM WALK, S-CLUSTER , S-RUN,SITE PERCOLATION References Deutscher, G.; Zallen, R.; and Adler, J. (Eds.). Percolation Structures and Processes. Bristol: Adam Hilger, 1983. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/rndprc/rndprc.html. Grimmett, G. Percolation. New York: Springer-Verlag, 1989. Grimmett, G. Percolation and Disordered Systems. Berlin: Springer-Verlag, 1997. Kesten, H. Percolation Theory for Mathematicians. Boston, MA: Birkha ¨user, 1982. Stauffer, D. and Aharony, A. Introduction to Percolation Theory, 2nd ed. London: Taylor & Francis, 1992. Weisstein, E. W. "Books about Percolation Theory." http:// www.treasure-troves.com/books/PercolationTheory.html. Percolation Threshold The critical fraction of lattice points which must be filled to create a continuous path of nearest neighbors from one side to another. The following table is from Stauffer and Aharony (1992, p. 17). Lattice Site Bond Cubic (Body-Centered) 0.246 0.1803 Cubic (Face-Centered) 0.198 0.119 Cubic (Simple) 0.3116 0.2488 Diamond 0.43 0.388 Honeycomb 0.6962 0.65271 4-Hypercubic 0.197 0.16015-Hypercubic 0.141 0.1182 6-Hypercubic 0.107 0.0942 7-Hypercubic 0.089 0.0787 Square 0.592746 0.50000 Triangular 0.50000 0.34729 The square bond value is 1=2 exactly, as is the triangular site. pc /C302 sin( p=18) for the triangular bond and pc /C301 /C282 sin( p=18) for the honeycomb bond. An exact answer for the square site percolation threshold is not known. See also PERCOLATION THEORY References Essam, J. W.; Gaunt, D. S.; and Guttmann, A. J. "Percola- tion Theory at the Critical Dimension." J. Phys. A 11, 1983 /C1/1990, 1978. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/rndprc/rndprc.html. Kesten, H. Percolation Theory for Mathematicians. Boston, MA: Birkha ¨user, 1982. Stauffer, D. and Aharony, A. Introduction to Percolation Theory, 2nd ed. London: Taylor & Francis, 1992. Perfect Box EULER BRICK Perfect Code See also ERROR- CORRECTING CODE,HAMMING CODE References MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error- Correcting Codes. Amsterdam, Netherlands: North-Hol- land, 1977. Perfect Cubic Polynomial A perfect cubic POLYNOMIAL can be factored into a linear and a quadratic term, x3 /C27y3 /C30(x /C27y)(x2 /C28xy /C27y2) x3/C28y3/C30(x/C28y)(x2/C27xy/C27y2): See also CUBIC EQUATION ,PERFECT SQUARE ,POLY- NOMIAL Perfect Cuboid EULER BRICK Perfect Difference Set ASETofRESIDUES fa1;a2;...;ak/C271g(mod n) such that every NONZERO RESIDUE can be uniquely ex- pressed in the form ai/C28aj:Examples include f1;2;4g(mod 7) and f1;2;5;7g(mod 13). A NECESSARY condition for a difference set to exist is that n be OF THE FORM k2 /C27k /C271: A SUFFICIENT condition is that k be a PRIME POWER . Perfect sets can be used in the construction of PERFECT RULERS . See also PERFECT RULER References Guy, R. K. "Modular Difference Sets and Error Correcting Codes." §C10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 118 /C1/121, 1994. Perfect Digital Invariant NARCISSISTIC NUMBER Perfect Graph A GRAPH G such that for every INDUCED SUBGRAPH of G, the size of the largest CLIQUE equals the CHRO- MATIC NUMBER . A graph can be tested to see if it is perfect usingPerfectQ [g] in the Mathematica add- on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). Determining if a graph is perfect requires solving two NP-COMPLETE PROBLEMS (Skiena 1990, p. 219). The numbers of perfect graphs on n /C301, 2, ... nodes are 1, 2, 4, 11, 33, 148, 906, ... (Sloane’s A052431). The numbers of perfect CONNECTED GRAPHS on n /C301, 2, ... nodes are 1, 1, 2, 6, 20, 105, 724, ... (Sloane’s A052433). See also CHROMATIC NUMBER ,C LIQUE ,INDUCEDSUBGRAPH ,PERFECT GRAPH THEOREM ,STRONG PER- FECT GRAPH CONJECTURE References Golumbic, M. C. Algorithmic Graph Theory and Perfect Graphs. New York: Academic Press, 1980. Skiena, S. "Perfect Graphs." §5.6.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 219, 1990. Sloane, N. J. A. Sequences A052431 and A052433 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Perfect Graph Theorem The GRAPH COMPLEMENT of a PERFECT GRAPH is itself perfect (Fulkerson 1971; Lova´sz 1972; Skiena 1990, p. 219). See also PERFECT GRAPH ,STRONG PERFECT GRAPH CONJECTURE References Fulkerson, D. R. "Blocking and Anti-Blocking Pairs of Polyhedra." Math. Program. 1, 168 /C1/194, 1971. Lova´sz, L. "Normal Hypergraphs and the Perfect Graph Conjecture." Disc. Math. 2, 253 /C1/267, 1972. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Perfect Group References Holt, D. G. and Plesken, W. Perfect Groups. Oxford, Eng- land: Clarendon Press, 1989. Perfect Information A class of GAME in which players move alternately and each player is completely informed of previous moves. FINITE , ZERO-SUM , two-player GAMES with perfect information (including checkers and chess) have a SADDLE POINT , and therefore one or more optimal strategies. However, the optimal strategy may be so difficult to compute as to be effectively impossible to determine (as in the game of CHESS ). See also FINITE GAME,GAME,ZERO-SUM GAME Perfect Magic Cube A perfect magic cube is a MAGIC CUBE for which the CROSS SECTION diagonals, as well as the space diagonals, sum to the MAGIC CONSTANT . Perfect magic cubes are impossible for orders 3 and 4 (Schroeppel 1972, Gardner 1988), but it is not known if such cubes can exist for order 5 or 6 (Wells 1986, p. 72). Although no perfect magic cubes of order five are known, any such cube must have a central value of 63 (Schroeppel 1972; Gardner 1988). Langman (1962) constructed a perfect magic cube of order seven, and others were found by R. Schroeppel and Ernst Straus (Wells 1986, p. 72). An order-eight perfect magic cube was published anonymously in 1875 (Barnard 1888, Gardner 1976, Benson and Jacoby 1981, Gardner 1988). The construction of such a cube is discussed in Ball and Coxeter (1987). Rosser and Walker rediscovered the order-eight cube in the late 1930s (but did not publish it), and Myers independently discovered the cube illustrated above in 1970 (Wells 1986, p. 72; Gardner 1988). Order 9 and 11 magic cubes have also been discovered, but none of order 10 (Gardner 1988). See also MAGIC CUBE,SEMIPERFECT MAGIC CUBE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 216 /C1/224, 1987. Barnard, F. A. P. "Theory of Magic Squares and Cubes." Mem. Nat. Acad. Sci. 4, 209 /C1/270, 1888. Benson, W. H. and Jacoby, O. Magic Cubes: New Recrea- tions. New York: Dover, 1981. Gardner, M. Sci. Amer. , Jan. 1976. Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213 /C1/225, 1988.Langman, H. Play Mathematics. New York: Hafner, pp. 75 /C1/ 76, 1962. Schroeppel, R. Item 50 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 18, Feb. 1972. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 72, 1986. Perfect Matching A MATCHING of a GRAPH containing n=2 edges, the largest possible. Not all graphs have a perfect matching, although all graphs do have a maximal matching (Skiena 1990, p. 240). Every CUBIC GRAPH without BRIDGES has a perfect matching (Skiena 1990, p. 244). See also K-FACTOR ,MATCHING References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Perfect Number Perfect numbers are INTEGERS nsuch that n/C30s(n); (1) where s(n) is the RESTRICTED DIVISOR FUNCTION (i.e., the SUM ofPROPER DIVISORS ofn), or equivalently s(n)/C302n; (2) where s(n) is the DIVISOR FUNCTION (i.e., the SUM of DIVISORS ofnincluding nitself). The first few perfect numbers are 6, 28, 496, 8128, ... (Sloane’s A000396). This follows from the fact that 6/C301/C272/C273 28/C301/C272/C274/C277/C2714 496/C301/C272/C274/C278/C2716/C2731/C2762/C27124/C27248; etc. Perfect numbers were deemed to have importantnumerological properties by the ancients, and wereextensively studied by the Greeks, including Euclid. Perfect numbers are intimately connected with a class of numbers known as M ERSENNE PRIMES . This can be demonstrated by considering a perfect number POF THE FORM P/C30q2p/C281where qisPRIME . Then s(P)/C302P; (3) and using s(q)/C30q/C271 (4) forqprime, and s(2a)/C302a/C271/C281 (5) gives s(q2p/C281) /C30 s(q) s(2p /C281) /C30(q /C271)(2p /C281) /C302q2p /C281 /C30q2p (6) q(2p /C281) /C272p /C281 /C30q2p (7) q /C302p /C281: (8) Therefore, if Mp /C13q /C302p /C281is PRIME , then P /C301 2(Mp /C271)Mp /C302p /C281(2p /C281) (9) is a perfect number, as was stated in Proposition IX.36 of Euclid’s ELEMENTS (Dickson 1957, p. 3; Dunham 1990). The first few perfect numbers are summarized in the following table. # pP 12 6 23 2 8 3 5 496 4 7 8128 5 13 33550336 6 17 8589869056 7 19 137438691328 8 31 2305843008139952128 While many of Euclid’s successors implicitly assumed that all perfect numbers were of the form (9) (Dickson 1952, pp. 3 /C1/33), the precise statement that all even perfect numbers are of this form. This was considered in a 1638 letter from Descartes to Mersenne (Dickson 1957, p. 12), and proving or disproving that Euclid’s construction gives all possible even perfect numbers was prosed to Fermat in a 1658 letter from Frans van Schooten (Dickson 1957, p. 14). In a posthumous paper, Euler (Euler 1849) provided the first proof that Euclid’s construction gives all possible even perfect numbers (Dickson 1957, p. 19). It is known that all EVEN perfect numbers (except 6) end in 16, 28, 36, 56, 76, or 96 (Lucas 1891) and have DIGITAL ROOT 1. Every perfect number OF THE FORM 2p(2p /C271 /C281) can be written 2p(2p /C271 /C281) /C30Xp =2 k/C301(2k /C281)3 : (10) All EVEN perfect numbers P /C216 are OF THE FORM P /C301 /C279Tn ; (11) where Tn is a TRIANGULAR NUMBER Tn /C301 2 n(n /C271) (12)such that n /C308j /C272 (Eaton 1995, 1996). In addition, all even perfect numbers are HEXAGONAL NUMBERS ,so it follows that perfect numbers are always the sum of consecutive POSITIVE INTEGERS starting at 1, for example, 6 /C30X3 n/C301n (13) 28 /C30X7 n/C301n (14) 496 /C30X31 n /C301n (15) (Singh 1997). It is not known if any ODD PERFECT NUMBERS exist, although numbers up to 10300 have been checked (Brent et al. 1991; Guy 1994, p. 44) without success. The sum of reciprocals of all the divisors of a perfect number is 2, since n/C27.../C27c/C27b/C27a|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n/C302n (16) n a/C27n b/C27.../C302n (17) 1 a/C271 b/C27.../C302: (18) Ifs(n)>n;nis said to be an ABUNDANT NUMBER .I f s(n)Bn;nis said to be a DEFICIENT NUMBER . And if s(n)/C30knfor a POSITIVE INTEGER k/C211,nis said to be aMULTIPERFECT NUMBER of order k. The only even perfect number OF THE FORM x3/C271i s 28 (Makowski 1962). See also ABUNDANT NUMBER ,A LIQUOT SEQUENCE , AMICABLE NUMBERS ,D EFICIENT NUMBER ,D IVISOR FUNCTION , E-PERFECT NUMBER ,HARMONIC NUMBER , HYPERPERFECT NUMBER ,INFINARY PERFECT NUM- BER,M ERSENNE NUMBER ,M ERSENNE PRIME ,M ULTI- PERFECT NUMBE R,M ULTIPLICATIVE PERFECT NUMBER ,ODD PERFECT NUMBER ,PLUPERFECT NUM- BER,PSEUDOPERFECT NUMBER ,QUASIPERFECT NUM- BER,S EMIPERFECT NUMBER ,S MITH NUMBE R, SOCIABLE NUMBERS ,SUBLIME NUMBER ,SUPER UNI- TARY PERFECT NUMBER ,S UPERPERFECT NUMBER , UNITARY PERFECT NUMBER ,W EIRD NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 66 /C1/67, 1987. Brent, R. P.; Cohen, G. L. L.; and te Riele, H. J. J. "Im- proved Techniques for Lower Bounds for Odd Perfect Numbers." Math. Comput. 57, 857/C1/868, 1991. Conway, J. H. and Guy, R. K. "Perfect Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 136 /C1/ 137, 1996. Dickson, L. E. "Notes on the Theory of Numbers." Amer. Math. Monthly 18, 109 /C1/111, 1911. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3 /C1/33, 1952. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 75, 1990. Eaton, C. F. "Problem 1482." Math. Mag. 68, 307, 1995. Eaton, C. F. "Perfect Number in Terms of Triangular Numbers." Solution to Problem 1482. Math. Mag. 69, 308 /C1/309, 1996. Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math- ematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 160 /C1/171, 1978. Guy, R. K. "Perfect Numbers." §B1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 44 /C1/45, 1994. Iannucci, D. E. "The Second Largest Prime Divisor of an Odd Perfect Number Exceeds Ten Thousand." Math. Comput. 68, 1749 /C1/1760, 1999. Kraitchik, M. "Mersenne Numbers and Perfect Numbers." §3.5 in Mathematical Recreations. New York: W. W. Nor- ton, pp. 70 /C1/73, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 145 and 147 /C1/151, 1979. Makowski, A. "Remark on Perfect Numbers." Elemente Math. 17, 109, 1962. Powers, R. E. "The Tenth Perfect Number." Amer. Math. Monthly 18, 195 /C1/196, 1911. Se´roul, R. "Perfect Numbers." §8.3 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 163 /C1/165, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 1 /C1/13 and 25 /C1/29, 1993. Singh, S. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker, pp. 11 /C1/13, 1997. Sloane, N. J. A. Sequences A000396/M4186 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Smith, H. J. "Perfect Numbers." http://pweb.netcom.com/ ~hjsmith/Perfect.html. Souissi, M. Un Texte Manuscrit d’Ibn Al-Banna’ Al-Marra- kusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975. Wagon, S. "Perfect Numbers." Math. Intell. 7,66/C1/68, 1985. Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and Ore Numbers." Bull. Soc. Math. Gre`ce (New Ser.) 13,12/C1/ 22, 1972. Perfect Partition A PARTITION of n whose elements uniquely generate any number 1, 2, ..., n. The following table gives the first several perfect partitions for small n. n perfect partitions 1 /f1 g/ 2 /f1 ; 1 g/ 3 /f2 ; 1 g;f1 ; 1 ; 1g/4 /f1 ; 1 ; 1; 1g/ 5 /f3 ; 1 ; 1g;f2; 2; 1g;f1; 1; 1; 1; 1g/ 6 /f1 ; 1 ; 1; 1; 1; 1g/ The numbers of perfect partitions of n for n /C301, 2, ... are given by 1, 1, 2, 1, 3, 1, 4, 2, 3, ... (Sloane’s A002033). For pka PRIME POWER , the number of perfect partitions a(pk) is given by a(pk) /C302k/C281 : Let b(n) /C30a(n /C271) ; then b(n) is given by the RECUR- RENCE RELATION b(n) /C30X d j n d"nb(d): The number of perfect partitions of n is equal to the number of ordered factorizations of n /C271 (Goulden and Jackson 1983, p. 94). See also PARTITION References Cohen, D. I. A. Basic Techniques of Combinatorial Theory. New York: Wiley and Sons, p. 97, 1978. Goulden, I. P. and Jackson, D. M. Problem 2.5.12 in Combi- natorial Enumeration. New York: Wiley, 1983. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 140 /C1/143, 1985. Riordan, J. "An Introduction to Combinatorial Analysis." In (Ed. ). , pp. , . Sloane, N. J. A. Sequences A002033/M0131 and A035341 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Perfect Proportion Since 2a a /C27 b /C302ab (a /C27 b)b ; (1) it follows that a a/C27b 2/C302ab a/C27b b; (2) so a A/C30H b; (3) where Aand Hare the ARITHMETIC MEAN and HARMONIC MEAN ofaand b. This relationship was purportedly discovered by Pythagoras. See also ARITHMETIC MEAN,HARMONIC MEAN Perfect Rectangle A RECTANGLE which cannot be built up of SQUARES all of different sizes is called an imperfect rectangle. A RECTANGLE which can be built up of SQUARES all of different sizes is called perfect. The number of perfect rectangles of orders 8, 9, 10, ... are 0, 2, 6, 22, 67, 213, 744, 2609, ... (Sloane’s A002839) and the correspond- ing numbers of imperfect rectangles are 0, 1, 0, 0, 9, 34, 103, 283, ... (Sloane’s A002882). See also PERFECT SQUARE DISSECTION ,RECTANGLE TILING References Bouwkamp, C. J. "On the Dissection of Rectangles into Squares. I." Indag. Math. 8, 724 /C1/736, 1946. Bouwkamp, C. J. "On the Dissection of Rectangles into Squares. II." Indag. Math. 9,43/C1/56, 1947. Bouwkamp, C. J. "On the Dissection of Rectangles into Squares. III." Indag. Math. 9,57/C1/63, 1947. Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte, W. T. "The Dissection of Rectangles into Squares." Duke Math. J. 7, 312 /C1/340, 1940. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Squaring the Square." §C2 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 81 /C1/83, 1991. Descartes, B. "Division of a Square into Rectangles." Eur- eka, No. 34, 31 /C1/35, 1971. Duijvestijn, A. J. W. Electronic Computation of Squared Rectangles. Thesis. Eindhoven, Netherlands: Technische Hogeschool, 1962. Moron, Z. "O rozkl adach prostokato ´w na kwadraty." Prze- glad matematyczno-fizyczny 3, 152 /C1/153, 1925. Sloane, N. J. A. Sequences A002839/M1658 and A002882/ M4614 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Stewart, I. "Squaring the Square." Sci. Amer. 277,94/C1/96, July 1997. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 73, 1986. Perfect Ruler A type of RULER considered by Guy (1994) which hask distinct marks spaced such that the distances between marks can be used to measure all the distances 1, 2, 3, 4, ... up to some maximum distance n /C21k. Such a ruler can be constructed from a PERFECT DIFFERENCE SET by subtracting one from each element. For example, the PERFECT DIFFERENCE SET f1; 2; 5 ; 7 g gives 0, 1, 4, 6, which can be used to measure 1 /C1/0 /C301, 6 /C1/4 /C302, 4 /C1/1 /C303, 4 /C1/0 /C304, 6 /C1/1 /C305, 6 /C1/0 /C306 (so we get 6 distances with only four marks). See also GOLOMB RULER ,PERFECT DIFFERENCE SET, RULER References Guy, R. K. "Modular Difference Sets and Error Correcting Codes." §C10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 118 /C1/121, 1994. Perfect Set A SET P is called perfect if P /C30P ?; where P? is the DERIVED SET of P. See also DERIVED SET,SET Perfect Shuffle Gale (1992) considered the following problem. Take an infinite deck of cards labeled 1, 2, 3, 4, 5, 6, .... At step n, pick up the top n cards and interlace them with the next n cards. This is called a perfect n- shuffle. For example, after step two, we have 3, 2, 4, 1, 5, 6, 7, .... For step there, pick up 3, 2, 4 and shuffle them in, giving 1, 3, 5, 2, 6, 4, 7, 8, 9, .... Iterate this process. It is conjectured that eventually every number appears on top of the deck. The cards on top of deck at the nth step are 1, 2, 3, 1, 6, 5, 9, 1, 4, 2, 16, 10, 12, ... (Sloane’s A035485). The step at which card n first appears on top the deck is given by 0, 1, 2, 8, 5, 4, 78, 37, ... (Sloane’s A035490). The position of the first card after the nth shuffle is 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, 7, 14, 28, ... (Sloane’s A035492). The order in which new cards appear on top for the first time is 1, 2, 3, 6, 5, 9, 4, 16, 10, ... (Sloane’s A035493). The order in which record new high cards appear on top for the first time is 1, 2, 3, 6,9, 16, ... (Sloane’s A035494). See also K IMBERLING SHUFFLE ,SHUFFLE References Gale, D. "Mathematical Entertainments: Careful Card- Shuffling and Cutting Can Create Chaos." Math. Intell. 14,5 4/C1/56, 1992. Gale, D. Tracking the Automatic Ant and Other Mathema- tical Explorations, A Collection of Mathematical Enter- tainments Columns from The Mathematical Intelligencer.New York: Springer-Verlag, 1998. Sloane, N. J. A. Sequences A035485, A035490, A035492, A035493, and A035494 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Perfect Square The term perfect square is used to refer to a SQUARE NUMBER ,a PERFECT SQUARE DISSECTION , or a factor- able quadratic polynomial OF THE FORM a292ab/C27b2/C30(a9b)2:/ See also PERFECT SQUARE DISSECTION ,Q UADRATIC EQUATION ,SQUARE NUMBER ,SQUAREFREE Perfect Square Dissection ASQUARE which can be DISSECTED into a number of smaller SQUARES with no two equal is called a PERFECT SQUARE DISSECTION (or a SQUARED SQUARE ). Square dissections in which the squares need not be different sizes are called M RS.PERKINS’ QUILTS .I fn o subset of the SQUARES forms a RECTANGLE , then the perfect square is called "simple." Moroz (1925) constructed a 33 /C2932PERFECT RECTAN- GLE composed of nine squares of different sizes (Descartes 1971), but Lusin claimed that perfect squares were impossible to construct. This assertionwas proved erroneous when a 55- SQUARE perfect square was published by R. Sprague in 1939 (Wells 1991). Reichert and Toepkin (1940) proved that a RECTANGLE cannot be dissected into fewer than nine different SQUARES (Steinhaus 1983, p. 297). A 24- SQUARE perfect square was subsequently foundby Willcocks (Willcocks 1948, 1951; Steinhaus 1983, pp. 8/C1/9). There is a unique simple perfect square of order 21(the lowest possible order), discovered in 1978 by A. J. W. Duijvestijn (Bouwkamp and Duijvestijn 1992). It is composed of 21 squares with total side length 112, and is illustrated above. There is a simple notation (sometimes called Bouwkamp code) used to describe perfect squares. In this notation, brackets are used to group adjacent squares with flush tops, and then the groups are sequentially placed in the highest (and leftmost) possible slots. For example, the 21-square illustrated above is denoted [50, 35, 27], [8, 19], [15, 17, 11], [6, 24], [29, 25, 9, 2], [7, 18], [16], [42], [4, 37], [33]. A compound 26-perfect square having side length 608 was discovered in 1940 (Brooks et al. 1940; Kraitchik 1942, p. 198). Beiler (1966) illustrates a compound28-square and a simple 38-square. Gardner (1961, pp. 203 and 206) illustrates compound 39- and 24- squares. The number of simple perfect squares of order nfor n]21 are 1, 8, 12, 26, 160, 441, ... (Sloane’s A006983). Duijvestijn’s Table I gives a list of the 441 simple perfect squares of order 26, the smallest with side length 212 and the largest with side length 825.Skinner (1993) gives the smallest possible side length (and smallest order for each) as 110 (22), 112 (21), 120 (24), 139 (22), 140 (23), ... for simple perfect squared squares, and 175 (24), 235 (25), 288 (26), 324 (27), 325 (27), ... for compound perfect squared squares. There are actually three simple perfect squares having side length 110. They are [60, 50], [23, 27], [24, 22, 14], [7, 16], [8, 6], [12, 15], [13], [2, 28], [26], [4, 21, 3], [18], [17] (order 22; discovered by A. J. W. Duijvestijn); [60, 50], [27, 23], [24, 22, 14], [4, 19], [8, 6], [3, 12, 16], [9], [2, 28], [26], [21], [1, 18], [17] (order 22; discovered by T. H. Willcocks); and [44, 29, 37], [21, 8], [13, 32], [28, 16], [15, 19], [12,4], [3, 1], [2, 14], [5], [10, 41], [38, 7], [31] (order 23; discovered by A. J. W. Duijvestijn). D. Sleator has developed an efficient ALGORITHM for finding non-simple perfect squares using what he calls rectangle and "ell" grow sequences. This algo- rithm finds a slew of compound perfect squares of orders 24 /C1/32. Weisstein gives a partial list of known simple and compound perfect squares (where the number of simple perfect squares is exact for orders less than 27) as well as Mathematica algorithms for drawing them. Order # Simple # Compound 21 1 0 22 8 023 12 024 26 1 25 160 1 26 441 227 ? 228 ? 4 29 ? 2 30 ? 331 ? 232 ? 2 38 1 0 39 ? 169 1 0 See also B LANCHE’S DISSECTION ,CYLINDER DISSEC- TION ,DISSECTION ,EQUILATERAL TRIANGLE PACKING , FAULT- FREE RECTANGLE ,KLEIN BOTTLE DISSECTION , MO¨ BIUS STRIP DISSECTION ,M RS. PERKINS’ QUILT, PERFECT RECTANGLE ,P ROJECTIVE PLANE DISSEC- TION ,TORUS DISSECTION References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 115 /C1/116, 1987. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, pp. 157 /C1/161, 1966. Bouwkamp, C. J. and Duijvestijn, A. J. W. "Catalogue of Simple Perfect Squared Squares of Orders 21 Through25." Eindhoven Univ. Technology, Dept. Math, Report 92- WSK-03, Nov. 1992. Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte, W. T. "The Dissection of Rectangles into Squares." Duke Math. J.7, 312/C1/340, 1940. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Squaring the Square." §C2 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 81 /C1/83, 1991. Descartes, B. "Division of a Square into Rectangles." Eur- eka, No. 34, 31 /C1/35, 1971. Duijvestijn, A. J. W. "A Simple Perfect Square of Lowest Order." J. Combin. Th. Ser. B 25, 240/C1/243, 1978. Duijvestijn, A. J. W. "A Lowest Order Simple Perfect 2 /C291 Squared Rectangle." J. Combin. Th. Ser. B 26, 372/C1/374, 1979. Duijvestijn, A. J. W. ftp://ftp.cs.utwente.nl/pub/doc/dvs/Ta- bleI. Gardner, M. "Squaring the Square." Ch. 17 in The Second Scientific American Book of Mathematical Puzzles &Diversions: A New Selection. New York: Simon and Schuster, pp. 186 /C1 /209, 1961. Gardner, M. Fractal Music, Hypercards, and More: Mathe- matical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 172 /C1/174, 1992. Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 15 and 32 /C1/33, 1979. Mauldin, R. D. (Ed.). The Scottish Book: Math at the Scottish Cafe. Boston, MA: Birkha ¨user, 1982. Moron, Z. "O rozkl adach prostokato ´w na kwadraty." Prze- glad matematyczno-fizyczny 3, 152/C1/153, 1925. Reichert, H. and Toepken, H. Jahresber. deutschen math. Verein. 50, 1940. Skinner, J. D. II. Squared Squares: Who’s Who & What’s What. Published by the author, 1993. Sloane, N. J. A. Sequences A006983/M4482 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M4482 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Smith, C. A. B. and Tutte, W. T. "A Class of Self-Dual Maps." Canad. J. Math. 2, 179/C1/196, 1950. Sprague, R. "Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate." Math. Z. 45, 607/C1/608, 1939. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Stewart, I. "Squaring the Square." Sci. Amer. 277,9 4/C1/96, July 1997. Weisstein, E. W. "Perfect Squares." M ATHEMATICA NOTE- BOOK PERFECT SQUARE.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 241 /C1/242, 1991. Willcocks, T. H. Fairy Chess Review 7, 1948. Willcocks, T. H. "A Note on Some Perfect Squared Squares." Canad. J. Math. 3, 304/C1/308, 1951. Perfectly Weighted Tree IfGis a weighted tree with weights /wi/C211/assigned to each vertex vi;then Gis perfectly weighted if the matrix MG/C30w10 /C1/C1/C1 0 0w2/C1/C1/C1 0 n::::::n 00:::wn2 6643 775/C28adj(G); where akj(G) is the ADJACENCY MATRIX of G (Butske et al. 1999). See also ADJACENCY MATRIX References Brenton, L. and Drucker, D. "Perfect Graphs and Complex Surface Singularities with Perfect Local Fundamental Group." Toˆhoku Math. J. 41, 507 /C1/525, 1989. Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation ap =N1=p /C271 =N /C301 ; Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407 /C1/420, 1999. Perforation The portion of a SURFACE left when an OPEN DISK is removed from it. See also OPEN DISK References Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/399, 1999. Periapsis The smallest radial distance of an ELLIPSE as mea- sured from a FOCUS . Taking v /C300 in the equation of an ELLIPSE r /C30a 1 /C28 e2ðÞ 1 /C27 e cos v gives the periapsis distance r/C28/C30a(1 /C28e): Periapsis for an orbit around the Earth is called perigee, and periapsis for an orbit around the Sun is called perihelion. See also APOAPSIS ,ECCENTRICITY ,ELLIPSE ,FOCUS Perigon An ANGLE of 2p radians /C30360/C14 corresponding to the CENTRAL ANGLE of an entire CIRCLE . Perimeter The ARC LENGTH along the boundary of a closed 2-D region. The perimeter of a CIRCLE is called the CIRCUMFERENCE . See also CIRCUMFERENCE ,CLUSTER PERIMETER ,HON- EYCOMB CONJECTURE ,SEMIPERIMETER Perimeter Polynomial A sum over all CLUSTER PERIMETERS .Period Doubling A characteristic of some systems making a transition to CHAOS . Doubling is followed by quadrupling, etc. An example of a map displaying period doubling is the LOGISTIC MAP. See also CHAOS ,LOGISTIC MAP Period Ratio HALF-PERIOD RATIO Period Three Theorem Li and Yorke (1975) proved that any 1-D system which exhibits a regular CYCLE of period three will also display regular CYCLES of every other length as well as completely CHAOTIC CYCLES . See also CHAOS ,CYCLE (MAP) References Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." Amer. Math. Monthly 82, 985 /C1/92, 1975. Periodic Function A FUNCTION f(x) is said to be periodic with period p if f(x) /C30f(x /C27np) for n /C301, 2, .... For example, the SINE function sin x; illustrated above, is periodic with period 2p (as well as with period /C282 p; 4p; 6p; etc.). The CONSTANT FUNCTION f(x) /C300 is periodic with any period R for all NONZERO REAL NUMBERS R, so there is no concept analogous to the LEAST PERIOD for con- stant functions. See also ALMOST PERIODIC FUNCTION ,DOUBLY PER- IODIC FUNCTION ,L EAST PERIOD ,P ERIODIC POINT , PERIODIC SEQUENCE References Knopp, K. "Periodic Functions." Ch. 3 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 58 /C1/92, 1996. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 425 /C1/427, 1953. Spanier, J. and Oldham, K. B. "Periodic Functions." Ch. 36 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 343 /C1/349, 1987. Periodic Matrix A SQUARE MATRIX A such that the MATRIX POWER Ak/C271 /C30A for k a positive integer is called a periodic matrix. If k is the least such integer, then the matrix is said to have period k.Ifk /C301, then A2 /C30A and A is called IDEMPOTENT . See also MATRIX POWER References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 11, 1962. Periodic Point A point x0 is said to be a periodic point of a FUNCTION f of period n if fn(x0) /C30x0 ; where f0(x) /C30x and fn(x)is defined recursively by fn(x) /C30ffn/C281(x) ðÞ :/ See also LEAST PERIOD ,PERIODIC FUNCTION ,PERI- ODIC SEQUENCE Periodic Sequence A SEQUENCE aifg is said to be periodic with period p with if it satisfies ai /C30ai/C27npfor n /C301, 2, .... For example, f1; 2; 1; 2; 1; 2; 1; 2; 1; 2 ; 1 ; 2 ; 1 ; 2; ...g is a peri- odic sequence with LEAST PERIOD 2. See also EVENTUALLY PERIODIC ,PERIODIC FUNCTION , PERIODIC POINT Periodic Zeta Function F(x; s) /C30X/C12 m/C301e2pimx ms /C30 cse2pix9+=9+; ; where cs(x) is the POLYGAMMA FUNCTION . See also POLYGAMMA FUNCTION ,R IEMANN ZETA FUNCTION ,ZETA FUNCTION References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 55, 1997. Periodogram A graphical plot with ABSCISSA given by the number p of consecutive numbers constituting a single period and ORDINATE given by the correlation ratio h : The equation of the periodogram ish2 /C30a2 2m2sin2mpp T ! /C27s2 b m 1 2 a2 /C27 s2 b; where each of the terms of the sequence ux consists of a simple periodic part of period T, together with a part which does not involve this periodicity bx ; so ux /C30a sin2px T ! /C27bx ; /sbis the standard deviation of the bs, s is the standard deviation of the us, and m is the number of periods covered by the observations. See also TIME SERIES ANALYSIS References Schuster. Terrestrial Magnetism 3, 24, 1898. Whittaker, E. T. and Robinson, G. "The Periodogram in the Neighbourhood of a True Period" and "An Example of Periodogram Analysis." §174 /C1/175 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 346 /C1/362, 1967. Perko Pair The KNOTS 10 /C1/161 and 10 /C1/162 illustrated above. For many years, they were listed as separate knots in Little (1885) and all similar tables, including the pictorial enumeration of Rolfsen (1976, Appendix C). They were identified as identical by Perko (1974), who found that they are related to one another by the so-called PERKO MOVE (Perko 1974, Hoste et al. 1998). Although these knots are equivalent, their diagrams have different WRITHES (Hoste et al. 1998). See also PERKO MOVE References Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,3 3/C1/48, Fall 1998. Little, C. N. "On Knots, with a Census of Order Ten." Trans. Connecticut Acad. Sci. 18, 374/C1/378, 1885. Perko, K. A. Jr. "On the Classification of Knots." Proc. Amer. Math. Soc. 45, 262/C1/266, 1974. Rolfsen, D. "Table of Knots and Links." Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 280 /C1/287, 1976. Permanence of Algebraic Form All ELEMENTARY FUNCTIONS can be extended to the COMPLEX PLANE . Such definitions agree with the real definitions on the X-AXIS and constitute an ANALYTIC CONTINUATION . See also ANALYTIC CONTINUATION ,E LEME NTARY FUNCTION ,PERMANENCE OF MATHEMATICAL RELA- TIONS PRINCIPLE References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 380, 1985. Permanence of Mathematical Relations Principle CONTINUITY PRINCIPLE Permanent An analog of a DETERMINANT where all the signs in the expansion by MINORS are taken as POSITIVE . The permanent of a MATRIX A is the coefficient of x1 ...xn in Yn i/C301ai1x1 /C27ai2x2 /C27.../C27ainxn ðÞ (Vardi 1991). Another equation is the RYSER FORMULA perm aij9+=9+; /C30(/C281)nX a ⁄f1 ; ... ; ng(/C281)½s½Yn i/C301X j /C23saij ; where the SUM is over all SUBSETS of f1; ... ; ng; and ½s½ is the number of elements in s (Vardi 1991). Muir (1960, p. 19) uses the notation½/C27½/C27to denote a permanent. The permanent can be implemented in Mathematica as Permanent[m_List] : /C30 With[{v /C30 Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v] ] The computation of permanents has been studied fairly extensively in algebraic complexity theory. The complexity of the best-known algorithms grows as the exponent of the matrix size (Knuth 1998, p. 499), which would appear to be very surprising, given the permanent’s similarity to the tractable DETERMINANT . If M is a UNITARY MATRIX , then perm( M) jj 51 (Minc 1978, p. 25; Vardi 1991). The maximum per- manent for an n /C29n BINARY MATRIX is n! ; correspond- ing to all elements 1. See also DETERMINANT ,FROBENIUS- KO¨ NIG THEOREM , IMMANANT ,RYSER FORMULA ,SCHUR MATRIX References Borovskikh, Y. V. and Korolyuk, V. S. Random Permanents. Philadelphia, PA: Coronet Books, 1994.Comtet, L. "Permanents." §4.9 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 197 /C1/198, 1974. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, p. 51, 1997. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, pp. 499 and 515 /C1/516, 1998. Minc, H. Permanents. Reading, MA: Addison-Wesley, 1978. Muir, T. §27 in A Treatise on the Theory of Determinants. New York: Dover, p. 19 1960. Valiant, L. G. Theoret. Comp. Sci. 8, 189 /C1/201, 1979. Vardi, I. "Permanents." §6.1 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 108 and 110 /C1/112, 1991. Permil The use of permil (a.k.a. parts per thousand) is a way of expressing RATIOS in terms of whole numbers. Given a RATIO or FRACTION , it is converted to a permil-age by multiplying by 1000 and appending a "mil sign" %0. For example, if an investment grows from a number P /C3013 :00 to a number A /C3022 :50; then A is 22:50 =13:00 /C301 :7308 times as much as P,or 1730.8%0. See also PERCENT References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 283, 1997. Permutation The rearrangement of elements in an ordered list S into a ONE-TO-ONE correspondence with S itself, also called an "arrangement number" or "order." The number of permutations on a set of n elements is given by n!(n FACTORIAL ; Uspensky 1937, p. 18). For example, there are 2! /C302 /C215 1 /C302 permutations of f1; 2g; namely f1 ; 2g and f2; 1g; and 3! /C303 /C215 2 /C215 1 /C30 6 permutations of f1; 2; 3g; namely f1; 2; 3g; f1; 3; 2g;f2; 1 ; 3 g;f2; 3; 1g;f3; 1; 2 g; and f3;2;1g:The permutations of a list can be found in Mathematica using the command Permuta- tions [list]. A list of length ncan be tested to see if it is a permutation of 1, ..., nwith the command PermutationQ [list] in the Mathematica add-on packageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). Sedgewick (1977) summarized a number of algo- rithms for generating permutations, and identifiesthe minimum change permutation algorithm of Heap (1963) to be generally the fastest (Skiena 1990, p. 10). Another method of enumerating permutations wasgiven by Johnson (1963; Se ´roul 2000, pp. 213 /C1 /218). The number of ways of obtaining an ordered subset of kelements from a set of nelements is given by nPk /C13n! (n /C28 k)! (1) (Uspensky 1937, p. 18). For example, there are 4!=2! /C3012 2-subsets of f1; 2; 3; 4g; namely f1; 2g; f1; 3g;f1 ; 4g;f2 ; 1 g;f2; 3g;f2; 4g;f3; 1g;f3; 2g; f3; 4g;f4 ; 1 g;f4; 2g; and f4 ; 3 g: The unordered subsets containing k elements are known as the K- SUBSETS of a given set. A representation of a permutation as a product of CYCLES is unique (up to the ordering of the cycles). An example of a cyclic decomposition is (/ f1; 3; 4g;f2 g); corresponding to the permutations (/1 0 3; 3 0 4; 4 0 1) and (/2 0 2); which combine to give f4 ; 2; 1; 3g: Muir (1960, p. 8) uses the notation (1237)(4568) to denote the ordered permutation (12345678) ; and (1237)(4568) to denote (12374568) :/ Any permutation is also a product of TRANSPOSITIONS . Permutations are commonly denoted in LEXICO- GRAPHIC or TRANSPOSITION ORDER . There is a corre- spondence between a PERMUTATION and a pair of YOUNG TABLEAUX known as the SCHENSTED CORRE- SPONDENCE . The number of wrong permutations of n objects is [n!=e] where [x] is the NINT function. A permutation of n ordered objects in which no object is in its natural place is called a DERANGEMENT (or sometimes, a COMPLETE PERMUTATION ) and the number of such permutations is given by the SUBFACTORIAL !n:/ Using (x /C27y)n /C30Xn r/C300n r9+;89+;9 xn/C28ryr (2) with x /C30y /C301 gives 2n /C30Xn r/C300n r9+;89+;9 ; (3) so the number of ways of choosing 0, 1, ..., or n at a time is 2n :/ The set of all permutations of a set of elements 1, ..., n can be obtained using the following recursive proce- dure 12 = 21(4)12 3 = 132 = 31 2 ½ 32 1 _ 231 _ 21 3(5) Let the set of INTEGERS 1, 2, ..., n be permuted and the resulting sequence be divided into increasing RUNS .A s napproaches INFINITY , the average length of the nthRUN is denoted Ln:The first few values are L1/C30e/C281/C301:71828818 . . . (6) L2/C30e2/C282e/C301:9524 . . . (7) L3/C30e3/C283e2/C273 2e/C301:9957 . . . ; (8) where Eis the base of the NATURAL LOGARITHM (Knuth 1973, Le Lionnais 1983). See also ALTERNATING PERMUTATION ,B INOMIAL COEFFICIENT ,C IRCULAR PERMUTATION ,C OMBINA- TION ,C OMPLETE PERMUTATION ,C YCLE (PERMUTA- TION ), DERANGEMENT ,D ISCORDANT PERMUTATION , EULERIAN NUMBER , K-SUBSET ,LINEAR EXTENSION , PERMUTATION INVERSION ,P ERMUTATION MATRIX , PERMUTATION PATTERN ,PERMUTATION SYMBOL ,RAN- DOM PERMUTATION ,SUBFACTORIAL ,TRANSPOSITION References Bogomolny, A. "Graphs." http://www.cut-the-knot.com/ do_you_know/permutation.html. Conway, J. H. and Guy, R. K. "Arrangement Numbers." In The Book of Numbers. New York: Springer-Verlag, p. 66, 1996. Dickau, R. M. "Permutation Diagrams." http://forum.s- warthmore.edu/advanced/robertd/permutations.html. Heap, B. R. "Permutations by Interchanges." Computer J. 6, 293/C1/294, 1963. Johnson, S. M. "Generation of Permutations by Adjacent Transpositions." Math. Comput. 17, 282/C1/285, 1963. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1998. Kraitchik, M. "The Linear Permutations of nDifferent Things." §10.1 in Mathematical Recreations. New York: W. W. Norton, pp. 239 /C1/240, 1942. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 41 /C1/42, 1983. Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Ruskey, F. "Information on Permutations." http:// www.theory.csc.uvic.ca/~cos/inf/perm/PermInfo.html. Sedgewick, R. "Permutation Generation Methods." Comput. Surveys 9, 137/C1/164, 1977. Se´roul, R. "Permutations: Johnson’s’ [sic] Algorithm." §8.15 inProgramming for Mathematicians. Berlin: Springer- Verlag, pp. 213 /C1/218, 2000. Skiena, S. "Permutations." §1.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 3 /C1/16, 1990. Sloane, N. J. A. Sequences A000142/M1675 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Trotter, H. F. "Perm (Algorithm 115)." Comm. ACM 5, 434 /C1/ 435, 1962. Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, p. 18, 1937. Permutation Ascent An ascent is a pair of adjacent positions in a PERMUTATION which are out of order. k ascents imply k /C271 PERMUTATION RUNS (Skiena 1990, p. 31). See also PERMUTATION ,PERMUTATION RUN References Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. Mannila, H. "Measures of Presortedness and Optimal Sort- ing Algorithms." IEE Trans. Comput. 34, 318 /C1/325, 1985. Skiena, S. "Runs and Eulerian Numbers." §1.3.4 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 30 /C1/31, 1990. Permutation Graph For a PERMUTATION a in the SYMMETRIC GROUP Sp ; the a/-permutation graph of a LABELED GRAPH G is the GRAPH UNION of two disjoint copies of G (say, G1 and G2) ; together with the lines joining point vi of Gi with va(i) of G2(Harary 1994, p. 175). Skiena (1990, p. 28) defined a permutation graph Gpas a GRAPH whose edges vi ; vj9+89+9 correspond exactly to (i, j) being a PERMUTATION INVERSION is some PERMUTATION p, i.e., i Bj but j occurs before i in p. The above graph corresponds to the permutation f2; 1; 5; 6; 7; 10; 9; 4; 8; 3g; which has PERMUTA- TION INVERSION f2; 1; 10; 8; 3;4; 5; 9; 7; 6g:/ See also PERMUTATION ,PERMUTATION INVERSIONReferences Atallah, M. J.; Manacher, G. K.; and Urrutia, J. "Finding a Minimum Independent Dominating Set in a Permutation Graph." Discr. Appl. Math. 21, 177 /C1/183, 1988. Brandstadt, A. and Kratsch, D. "On Domination Problems for Permutation and Other Graphs." Theoret. Comput. Sci. 54, 181 /C1/198, 1987. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Permutation Group A FINITE GROUP of order n! consisting of substitutions of n elements for each other. For instance, the 24 PERMUTATIONS on four elements form a permutation group, and one the operations in this group is the permutation f4; 2; 1; 3g; which rearranges the ele- ments fA; B ; C ; D g in the order fD ; B; A; C g: A permutation group of two elements is called a TRANSPOSITION . Every SUBSTITUTION GROUP with > 2 elements can be written as a product of transpositions. For example, (abc) /C30(ab)(ac) (abcde ) /C30(ab)(ac)(ad)(ae) : CONJUGACY CLASSES of elements which are inter- changed are called CYCLES (in the above example, the CYCLES are ff1; 3; 4g;f2gg) :/ Two PERMUTATIONS form a group only if one is the identity element and the other is an INVOLUTION , i.e., a PERMUTATION which is its own inverse (Skiena 1990, p. 20). See also CAYLEY’S GROUP THEOREM ,CYCLE (PERMU- TATION ), GROUP ,INVOLUTION (PERMUTATION ), NET- TO’S CONJECTURE ,P ERMUTATION ,S UBSTITUTION GROUP ,TRANSPOSITION References Cameron, P. Permutation Groups. New York: Cambridge University Press, 1999. Furst, M.; Hopcroft, J.; and Luks, E. "Polynomial Time Algorithms for Permutation Groups." In Proc. Symp. Foundations Computer Sci. IEEE, pp. 36 /C1/41, 1980. Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Skiena, S. "Permutation Groups." §1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 17 /C1/26, 1990. Wielandt, H. Finite Permutation Groups. New York: Aca- demic Press, 1964. Permutation Index The index of a PERMUTATION pis defined as the sum of all subscripts jsuch that pj>pj/C271;for 15j5n: MacMahon (1960) proved that the number of permu- tations of size nhaving index kis the same as the number having exactly kinversions (Skiena 1990, p. 29). The permutation index can be computed as Index [p] in the Mathematica add-on package Dis- creteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also PERMUTATION References Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. MacMahon, P. A. Combinatory Analysis, 2 vols. New York: Chelsea, 1960. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Permutation Inversion A pair of elements (pi ; pj) is called an inversion in a permutation p if i /C21j and pi Bpj : For example, in the permutation a6a5a7a3a8contains the four inversions a7a3 ; a5a3 ; a6a3 ; and a6a5 : Inversions are pairs which are out of order, and are important in sorting algorithms (Skiena 1990, p. 27). The total number of inversions can be obtained by summing the elements of the INVERSION VECTOR , and is implemented as Inversions [p] in the Mathema- tica add-on package DiscreteMath‘Combinator- ica‘ (which can be loaded with the command BBDiscreteMath‘ ). The number of inversions in any PERMUTATION is the same as the number of interchanges of consecutive elements necessary to arrange them in their natural order (Muir 1960, p. 1). The value (/C281)i(p) can be found in Mathematica using Signature [p]. The number of inversions in a PERMUTATION is equal to that of its inverse permutation (Skiena 1990, p. 29; Knuth 1998). If, from any permutation, another is formed by interchanging two elements, then the difference between the number of inversions in the two is always an ODD NUMBER . See also INVERSE PERMUTATION ,INVERSION VECTOR , PERMUTATION ,PERMUTATION SYMBOL References Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. Mannila, H. "Measures of Presortedness and Optimal Sort- ing Algorithms." IEEE Trans. Comput. 34, 318 /C1/325, 1985. Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960. Skiena, S. "Encroaching Lists as a Measure of Presorted- ness." BIT 28, 775 /C1/784, 1988. Skiena, S. "Inversions and Inversion Vectors." §1.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 27 /C1/31, 1990.Permutation Matrix A MATRIX pijobtained by permuting the ith and jth rows of the IDENTITY MATRIX with i Bj. Every row and column therefore contain precisely a single 1, and every permutation corresponds to a unique permuta- tion matrix. A permutation matrix is nonsingular, so the DETERMINANT is always NONZERO . In addition, a permutation matrix satisfies p2 ij /C30I ; where I is the IDENTITY MATRIX . Applying to another MATRIX , pijA gives A with the ith and jth rows interchanged, and Apijgives A with the ith and jth columns interchanged. Interpreting the 1s in an n /C29n permutation matrix as ROOKS gives an allowable configuration of nonattack- ing ROOKS on an n/C29nCHESSBOARD . See also ALTERNATING SIGN MATRIX ,ELEMENTARY MATRIX ,IDENTITY ,PERMUTATION ,ROOK NUMBER Permutation Pattern LetF(n;s) denote the number of permutations on the SYMMETRIC GROUP Snwhich avoid s/C23Snas a sub- pattern, where " /tcontains sas a subpattern" is interpreted to mean that there exist 1 5x15x25 ...5xk5nsuch that for 1 5i;j5k; txiðÞBtxj9+=9+; (1) IFFs(i)Bs(j):/ For example, a permutation contains the pattern (123) IFFit has an ascending subsequence of length three. Here, note that members need not actually be consecutive, merely ascending (Wilf 1997). Therefore, of the 3! /C306 partitions of f1;2;3g;all but f3;2;1g (i.e., f1;2;3g;f1;3;2g;f2;1;3g;f2;3;1g;and f3;1;2g) contain the pattern (12) (i.e., an increasing subsequence of length two). The following table gives the numbers of pattern- matching permutations of k,k/C271;...,nnumbers for various patterns a1...ak ðÞ of length k. pattern Sloane number of pattern-matching permutations 1 A000142 1, 2, 6, 24, 120, 720, 5040, ... 12 A033312 1, 5, 23, 119, 719, 5039, 40319, ... /a3/ A056986 1, 10, 78, 588, 4611, 38890, ... 1234 A000000 1, 17, 207, ...1342 A000000 1, 17, 208, ... The following table gives the numbers of pattern- avoiding permutations of f1 ; ...; n g for various sets of patterns. Wilf class Sloane number of pattern-avoiding permutations / a3/ A000108 1, 2, 5, 14, 42, 132, ... 123, 132, 213A000027 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... 132, 231, 321A000027 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... 123, 132, 3214A000073 1, 2, 4, 7, 13, 24, 44, 81, 149, ... 123, 132, 3241A000071 1, 2, 7, 12, 20, 33, 54, 88, 143, ... 123, 132, 3412A000124 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ... 123, 231, a(1) 4 /A004275 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, ... 123, 231, a(2)4 /A000124 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ... 123, 231, 43211, 2, 4, 6, 3, 1, 0, ... 132, 213, 1234A000073 1, 2, 4, 7, 13, 24, 44, 81, 149, ... 213, 231, a(3) 4 /A000124 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ... Abbreviations used in the above table are summar- ized below. abbreviation patterns in class / a3/ 123, 132, 213, 232, 312, 321 / a(1)4 / 1432, 2143, 3214, 4132, 4213, 4312 / a(2)4 / 1234, 1243, 1324, 1342, 1423, 2134, 2314, 2341, 2413, 2431, 3124, 3142, 3241, 3412, 3421, 4123, 4231 / a(3)4 / 1234, 1243, 1423, 1432 See also CONTAINED PATTERN ,ORDER ISOMORPHIC , PERMUTATION ,P ERMUTATION PATTERN ,S TANLEY- WILF CONJECTURE ,W ILF CLASS ,W ILF EQUIVALENTReferences Arratia, R. "On the Stanley-Wilf Conjecture for the Number of Permutations Avoiding a Given Patter." Electronic J. Combinatorics 6, No. 1, N1, 1 /C1/4, 1999. http://www.com- binatorics.org/Volume_6/v6i1toc.html. Billey, S.; Jockusch, W.; and Stanley, R. P. "Some Combina- torial Properties of Schubert Polynomials." J. Alg. Com- bin. 2, 345 /C1/374, 1993. Guibert, O. "Permutations sans sous se´quence interdite." Me´moire de Diploˆ me d’Etudes Approfondies de L’Univer- site´ Bordeaux I. 1992. Mansour, T. Permutations Avoiding a Pattern from /Sk/ and at Least Two Patterns from /S3/. 31 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007194/. Simon, R. and Schmidt, F. W. "Restricted Permutations." Europ. J. Combin. 6, 383 /C1/406, 1985. Sloane, N. J. A. Sequences A000027/M0472, A000071/ M1056, A000073/M1074, A000108/M1459, A000124/ M1041, A000142/M1675, A004275, A033312, and A056986 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Stankova, Z. E. "Forbidden Subsequences." Disc. Math. 132, 291 /C1/316, 1994. West, J. "Generating Trees and Forbidden Subsequences." Disc. Math. 157, 363 /C1/372, 1996. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos’ 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press, pp. 557 /C1/562, 1997. Permutation Pseudotensor PERMUTATION TENSOR Permutation Run A set of ascending sequences in a PERMUTATION is called a run. A sorted permutation consists of a single run, whereas a reverse permutation consists of n runs, each of length 1. Runs are closely related to PERMUTATION ASCENTS , with n runs implying n /C281 ascents (Skiena 1990, p. 31). The number of runs in a permutation can be computed using Runs [p] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). The number of permutations of length n with exactly k runs is given by the EULERIAN NUMBERn k9+;=9+;; :/ Surprisingly, the expected length of the first run is shorter than the expected length of the second run (Gassner 1967; Skiena 1990, p. 30; Knuth 1998). See also EULERIAN NUMBER ,PERMUTATION ,PERMU- TATION ASCENT ,RUN References Gassner, B. J. "Sorting by Replacement Selection." Comm. ACM 10,8 9/C1/93, 1967. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1998. Mannila, H. "Measures of Presortedness and Optimal Sort- ing Algorithms." IEE Trans. Comput. 34, 318 /C1/325, 1985. Skiena, S. "Runs and Eulerian Numbers." §1.3.4 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 30 /C1/31, 1990. Permutation Symbol A three-index object sometimes called the Levi-Civita symbol or signature, and defined by eijk /C30 0 for i /C30j; j /C30k; or k /C30i /C271 for (i ; j; k) /C23f(1; 2; 3); (2; 3; 1); (3; 1; 2)g /C281 for (i ; j; k) /C23f(1; 3; 2); (3; 2; 1); (2; 1; 3)g:8 < : (1) The permutation symbol is implemented in Mathe- matica asSignature [list]. The permutation symbol satisfies dij eijk /C300 (2) eipq ejpq /C302dij (3) eijk eijk /C306 (4) eijk epqk /C30 dip djq /C28 diq djp ; (5) where dij is the KRONECKER DELTA . The symbol can be defined as the SCALAR TRIPLE PRODUCT of unit vectors in a right-handed coordinate system, eijk /C13ˆxi/C215 (ˆxj /C29ˆxk) : (6) The symbol can also be interpreted as a TENSOR ,in which case it is called the PERMUTATION TENSOR . The symbol can be generalized to an arbitrary number of elements, in which case the permutation symbol is (/C281)i(p) ; where i(p) is the number of transpositions of pairs of elements (i.e., PERMUTATION INVERSIONS ) that must be composed to build up the permutation p (Skiena 1990). This type of symbol arises in computation of determinants of n /C29n ma- trices. The number of permutations on n symbols having signature /C281is n!=2; which is also the number of permutations having signature /C271 :/ See also CYCLE (PERMUTATION ), PERMUTATION ,PER- MUTATION INVERSION ,PERMUTATION TENSOR ,TRANS- POSITION References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 132 /C1/133, 1985. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 69 /C1/74, 1988. Skiena, S. "Signature." §1.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory withMathematica. Reading, MA: Addison-Wesley, pp. 24 /C1/25, 1990. Permutation Tensor A PSEUDOTENSOR which is ANTISYMMETRIC under the interchange of any two slots. Recalling the definition of the PERMUTATION SYMBOL in terms of a SCALAR TRIPLE PRODUCT of the Cartesian unit vectors, eijk /C13ˆxi/C215 (ˆxj /C29ˆxk) /C30[ˆxi ; ˆxj ; ˆxk]; (1) the pseudotensor is a generalization to an arbitrary BASIS defined by eab /C1/C1/C1m /C30ffiffiffiffiffiffi gjjp [ a; b; ...; m] (2) eab/C1/C1/C1 m /C30[ a; b; ...; m]ffiffiffiffiffiffi gjjp ; (3) where [ a; b; ...; m] /C301 the arguments are an even permutation /C281 the arguments are an odd permutation 0 two or more arguments are equal ;8 < : (4) and g /C13det(gab) ; where g abis the METRIC TENSOR . e(x1 ; ...; xn)is NONZERO IFF the VECTORS are LINE- ARLY INDEPENDENT . See also KRONECKER DELTA ,PERMUTATION SYMBOL , SCALAR TRIPLE PRODUCT Permutation Tests See also BOOTSTRAP METHODS ,JACKKNIFE ,HYPOTH- ESIS TESTING ,RESAMPLING STATISTICS References Good, P. I. Permutation Tests: A Practical Guide to Resam- pling Methods for Testing Hypotheses, 2nd ed. New York: Springer-Verlag, 2000. Perpendicular Two lines, vectors, planes, etc., are said to be perpendicular if they meet at a RIGHT ANGLE .I nRn; two VECTORS AandBare PERPENDICULAR if their DOT PRODUCT A /C215 B /C300: In R2 ; a LINE with SLOPE m2 /C30/C281=m1 is PERPENDICU- LAR to a LINE with SLOPE m1 : Perpendicular objects are sometimes said to be "orthogonal." In the above figure, the LINE SEGMENT AB is perpen- dicular to the LINE SEGMENT CD. This relationship is commonly denoted with a small SQUARE at the vertex where perpendicular objects meet, as shown above, and is denoted AB /C222CD:/ See also ORTHOGONAL LINES,ORTHOGONAL VECTORS , PARALLEL ,PERPENDICULAR BISECTOR ,PERPENDICU- LAR FOOT,RIGHT ANGLE References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 10, 1948. Perpendicular Bisector The perpendicular bisectors of a TRIANGLE DA1A2A3 are lines passing through the MIDPOINT Miof each side which are PERPENDICULAR to the given side. A TRIANGLE’S three perpendicular bisectors meet (Casey 1888, p. 9) at a point C known as the CIRCUMCENTER (Durell 1928), which is also the center of the TRIAN- GLE’S CIRCUMCIRCLE . See also CIRCUMCENTER ,MIDPOINT ,PERPENDICULAR , PERPENDICULAR FOOT References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 19 /C1/20, 1928.Perpendicular Foot The FOOT of the PERPENDICULAR is the point on the leg opposite a given vertex of a TRIANGLE at which the PERPENDICULAR passing through that vertex inter- sects the side. The length of the LINE SEGMENT from the vertex to the perpendicular foot is called the ALTITUDE of the TRIANGLE . When a line is drawn from a POINT to a PLANE , its intersection with the PLANE is known as the foot. See also ALTITUDE ,FOOT,PERPENDICULAR ,PERPEN- DICULAR BISECTOR ,TAYLOR CIRCLE References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 9, 1967. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 9, 1948. Perrin Pseudoprime If p is PRIME , then p jP(p); where P(p) is a member of the PERRIN SEQUENCE 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (Sloane’s A001608). A Perrin pseudoprime is a COM- POSITE NUMBER n such that njP(n): Several "unrest- ricted" Perrin pseudoprimes are known, the smallest of which are 271441, 904631, 16532714, 24658561, ... (Sloane’s A013998). Adams and Shanks (1982) discovered the smallest unrestricted Perrin pseudoprime after unsuccessful searches by Perrin (1899), Malo (1900), Escot (1901), and Jarden (1966). (A 1996 article by Stewart’s stating that no Perrin pseudoprimes were then known was incorrect.) Grantham (1996) generalized the definition of Perrin pseudoprime with parameters (r, s)tobean ODD COMPOSITE NUMBER n for which either 1. ( D=n) /C301 and n has an S-SIGNATURE ,or 2. ( D=n) /C30/C281 and n has a Q-SIGNATURE , where (a=b) is the JACOBI SYMBOL . All the 55 Perrin pseudoprimes less than 50 /C29109 have been computed by Kurtz et al. (1986). All have S- SIGNATURE , and form the sequence Sloane calls "restricted" Perrin pseudoprimes: 27664033, 46672291, 102690901, ... (Sloane’s A018187). See also PERRIN SEQUENCE ,PSEUDOPRIME References Adams, W. W. "Characterizing Pseudoprimes for Third- Order Linear Recurrence Sequences." Math Comput. 48, 1 /C1/15, 1987. Adams, W. and Shanks, D. "Strong Primality Tests that Are Not Sufficient." Math. Comput. 39, 255 /C1/300, 1982. Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, p. 305, 1996. Escot, E.-B. "Solution to Item 1484." L’Interme ´diare des Math. 8,63/C1/64, 1901. Grantham, J. "Frobenius Pseudoprimes." http://www.clar- k.net/pub/grantham/pseudo/pseudo1.ps Holzbaur, C. "Perrin Pseudoprimes." http://ftp.ai.univie.a- c.at/perrin.html. Jarden, D. Recurring Sequences. Jerusalem: Riveon Lema- tematika, 1966. Kurtz, G. C.; Shanks, D.; and Williams, H. C. "Fast Prim- ality Tests for Numbers Less than 50 /C215 109 :/" Math. Comput. 46, 691 /C1/701, 1986. Perrin, R. "Item 1484." L’Interme ´diare des Math. 6,76/C1/77, 1899. Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, p. 135, 1996. Sloane, N. J. A. Sequences A001608/M0429, A013998, and A018187 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Stewart, I. "Tales of a Neglected Number." Sci. Amer. 274, 102 /C1/103, June 1996. Perrin Sequence The INTEGER SEQUENCE defined by the recurrence P(n) /C30P(n /C282) /C27P(n /C283) (1) with the initial conditions P(0) /C303; P(1) /C300 ; P(2) /C302: This RECURRENCE RELATION is the same as that for the PADOVAN SEQUENCE but with different initial conditions. The first few terms for n /C300, 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (Sloane’s A001608). P(n) is the solution of a third-order linear homogeneous DIFFERENCE EQUATION having characteristic equation x3 /C28x /C281 /C300; (2) discriminant -23, and ROOTS a :1:324717957 (3) b :/C280:6623589786 /C270:5622795121 i (4) g :/C280 :6623589786 /C280:5622795121 i : (5) The solution is then P(n) /C30 an /C27 bn /C27 gn ; (6) where P(n) /C2 an : (7) Perrin (1899) investigated the sequence and noticed that if n is PRIME , then njP(n): The first statement of this fact is attributed to E´ . Lucas in 1876 by Stewart (1996). Perrin also searched for but did not find any COMPOSITE NUMBER n in the sequence such that njP(n) : Such numbers are now known as PERRINPSEUDOPRIMES . Malo (1900), Escot (1901), and Jarden (1966) subsequently investigated the series and also found no PERRIN PSEUDOPRIMES . Adams and Shanks (1982) subsequently found that 271,441 is such a number. See also PADOVAN SEQUENCE ,PERRIN PSEUDOPRIME , SIGNATURE (RECURRENCE RELATION ) References Adams, W. and Shanks, D. "Strong Primality Tests that Are Not Sufficient." Math. Comput. 39, 255 /C1/300, 1982. Escot, E.-B. "Solution to Item 1484." L’Interme ´diare des Math. 8,63/C1/64, 1901. Jarden, D. Recurring Sequences. Jerusalem: Riveon Lema- tematika, 1966. Perrin, R. "Item 1484." L’Interme ´diare des Math. 6,76/C1/77, 1899. Stewart, I. "Tales of a Neglected Number." Sci. Amer. 274, 102 /C1/103, June 1996. Sloane, N. J. A. Sequences A001608/M0429 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Perron Integral An integral which is equivalent to the DENJOY INTEGRAL "in the restricted sense." See also DENJOY INTEGRAL Perron Tree A convex figure constructed by iteratively halving the base of an EQUILATERAL TRIANGLE and then sliding adjacent triangles so that they slightly overlap. Combining several Perron trees gives a region in which the needle in the KAKEYA NEEDLE PROBLEM can rotate, and can have arbitrarily small area. See also KAKEYA NEEDLE PROBLEM References Falconer, K. J. The Geometry of Fractal Sets, 1st pbk. ed., with corrections. Cambridge, England: Cambridge Uni- versity Press, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 128 /C1/129, 1991. Perron-Frobenius Operator An OPERATOR which describes the time evolution of densities in PHASE SPACE . The OPERATOR can be defined by rn /C271 /C30 ˜Lrn ; where rnare the NATURAL DENSITIES after the nth iteration of a map f. This can be explicitly written as ˜Lr(y)/C30X x/C23f/C281(y)r(x) f?(x) jj: See also FROBENIUS- PERRON EQUATION References Berman, A. and Plemmons, R. Nonnegative Matrices in the Mathematical Sciences. New York: Academic Press, 1979. Beck, C. and Schlo¨gl, F. "Transfer Operator Methods." Ch. 17 in Thermodynamics of Chaotic Systems. Cam- bridge, England: Cambridge University Press, pp. 190 /C1/ 203, 1995. Perron-Frobenius Theorem If all elements aijof an IRREDUCIBLE MATRIX A are NONNEGATIVE , then R /C30min Mlis an EIGENVALUE of A and all the EIGENVALUES of A lie on the DISK zjj5R; where, if l /C30( l1 ; ...; l2 ; ... ; ln) is a set of NONNE- GATIVE numbers (which are not all zero), Ml /C30inf m : mli >Xn j/C301aij9+;$9+;$9+;$9+;$l j ; 1 5i 5n() and R /C30min Ml : Furthermore, if A has exactly p EIGENVALUES (p 5n) on the CIRCLE zjj/C30R; then the set of all its EIGENVALUES is invariant under rotations by 2p=p about the ORIGIN . See also WIELANDT’S THEOREM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000. Perron’s Formula A /C31(x) /C30X? ln 5xan /C301 2pi gc/C27i /C12 c /C28i/C12f(s)esx sds ; where f(s) /C30X ane /C28 lns : References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Riesz. The General Theory of Dirichlet’s Series. p. 12. Perron’s Theorem If m /C30( m1 ; m2 ; ...; mn) is an arbitrary set of POSITIVE numbers, then all EIGENVALUES l of the n /C29n MATRIX a /C30aij lie on the DISK zjj5mm ; wheremm /C30 max 1 5i 5nXn j/C301mj miaij9+;$9+;$9+;$9+;$: References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000. MacCluer, C. R. "The Many Proofs and Applications of Perron’s Theorem." SIAM Rev. 42, 487 /C1/498, 2000. Perron, O. "Grundlagen fu¨r eine Theorie des Jacobischen Kettenbruchalgorithmus." Math. Ann. 64,11/C1/76, 1907. Persistence ADDITIVE PERSISTENCE ,M ULTIPLICATIVE PERSIS- TENCE ,PERSISTENT NUMBER ,PERSISTENT PROCESS Persistent Number An n-persistent number is a POSITIVE INTEGER k which contains the digits 0, 1, ..., 9 (i.e., is a PANDIGITAL NUMBER ), and for which 2k; ..., nk also share this property. No /C12/-persistent numbers exist. However, the number k /C301234567890 is 2-persistent, since 2k /C302469135780 but 3k /C303703703670 ; and the number k /C30526315789473684210 is 18-persistent. There exists at least one k-persistent number for each POSITIVE INTEGER k. n Sloane n-persistent 1 A051264 1023456798, 1023456897, 1023456978, 1023456987, ... 2 A051018 1023456789, 1023456879, 1023457689, 1023457869, ... 3 A051019 1052674893, 1052687493, 1052746893, 1052748693, ... 4 A051020 1053274689, 1089467253, 1253094867, 1267085493, ... See also ADDITIVE PERSISTENCE ,M ULTIPLICATIVE PERSISTENCE ,PANDIGITAL NUMBER References Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 15 /C1/18, 1991. Sloane, N. J. A. Sequences A051018, A051019, A051020, and A051264 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Persistent Process A FRACTAL PROCESS for which H > 1=2; so r /C210. See also ANTIPERSISTENT PROCESS ,FRACTAL PROCESS Perspective Perspective is the art and mathematics of realistically depicting 3-D objects in a 2-D plane, sometimes called CENTRIC or NATURAL PERSPECTIVE to distinguish it from BICENTRIC PERSPECTIVE . The study of the projec- tion of objects in a plane is called PROJECTIVE GEOMETRY . The principles of perspective drawing were elucidated by the Florentine architect F. Bru- nelleschi (1377 /C1/1446). These rules are summarized by Dixon (1991): 1. The horizon appears as a line. 2. Straight lines in space appear as straight lines in the image. 3. Sets of PARALLEL lines meet at a VANISHING POINT . 4. Lines PARALLEL to the picture plane appear PARALLEL and therefore have no VANISHING POINT . There is a graphical method for selecting vanishing points so that a CUBE or box appears to have the correct dimensions (Dixon 1991). See also BICENTRIC PERSPECTIVE ,LEONARDO’S PARA- DOX,PERSPECTIVE AXIS,PERSPECTIVE CENTER ,PER- SPECTIVE COLLINEATION ,P ERSPECTIVE TRIANGLES , PERSPECTIVITY ,PROJECTION ,PROJECTIVE GEOMETRY , VANISHING POINT ,ZEEMAN’S PARADOX References de Vries, V. Perspective. New York: Dover, 1968. Dixon, R. "Perspective Drawings." Ch. 3 in Mathographics. New York: Dover, pp. 79 /C1/88, 1991. Lambert, J. H. Freie Perspective, 2nd ed. Zu¨rich, 1774. Parramon, J. M. Perspective--How to Draw. Barcelona, Spain: Parramon Editions, 1984. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 157 /C1/159, 1999.Perspective Axis The line joining the three collinear points of inter- section of the extensions of corresponding sides in PERSPECTIVE TRIANGLES , sometimes also called the homology axis. See also PERSPECTIVE CENTER ,PERSPECTIVE TRIAN- GLES ,SONDAT’S THEOREM Perspective Center The point at which the three LINES connecting the VERTICES of PERSPECTIVE TRIANGLES (from a point) CONCUR , sometimes also called the homology center or pole. See also PERSPECTIVE AXIS,PERSPECTIVE TRIANGLES Perspective Collineation A perspective collineation with center O and axis o is a COLLINEATION which leaves all lines through O and points of o invariant. Every perspective collineation is a PROJECTIVE COLLINEATION . See also COLLINEATION ,ELATION ,HOMOLOGY (GEO- METRY ), PROJECTIVE COLLINEATION References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 247 /C1/248, 1969. Perspective Triangles Two TRIANGLES DABC and DA?B ?C? are perspective from a line if the extensions of their three pairs of corresponding sides meet in COLLINEAR points X, Y, and Z. The line joining these points is called the PERSPECTIVE AXIS. Two TRIANGLES are perspective from a point if their three pairs of corresponding VERTICES are joined by lines which meet in a point of CONCURRENCE O. This point is called the PERSPECTIVE CENTER , or sometimes the homology center or pole. DESARGUES’ THEOREM guarantees that if two TRIAN- GLES are perspective from a point, they are perspec- tive from a line (called the PERSPECTIVE AXIS). Triangles in perspective are sometimes said to be homologous or copolar. See also DESARGUES’ THEOREM ,D ILATION ,H OMO- THETIC TRIANGLES ,PARALOGIC TRIANGLES ,PERSPEC- TIVE AXIS,PERSPECTIVE CENTER References Coxeter, H. S. M. and Greitzer, S. L. "Perspective Triangles; Desargues’s Theorem." §3.6 in Geometry Revisited. Wa- shington, DC: Math. Assoc. Amer., pp. 70 /C1/72, 1967. Lachlan, R. "Triangles in Perspective" and "Relations Be- tween Two Triangles in Perspective." §160 /C1/180 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 100 /C1/113, 1893. Perspectivity A correspondence between two RANGES that are sections of one PENCIL by two distinct lines. See also PENCIL ,PROJECTIVITY ,RANGE (LINE SEG- MENT )Persymmetric Matrix A SQUARE MATRIX with constant SKEW DIAGONALS . Such matrices are sometimes known as orthosym- metric in older literature. See also DIAGONAL MATRIX ,SKEW DIAGONAL ,SKEW SYMMETRIC MATRIX ,SYMMETRIC MATRIX References Mays, M. E. and Wojciechowski, J. "A Determinant Property of Catalan Numbers." Disc. Math. 211, 125 /C1/133, 2000. Pesin Theory The theory of non-uniformly hyperbolic DIFFEO- MORPHISMS . See also DIFFEOMORPHISM References Katok, A. "Lyapunov Exponents, Entropy, and Periodic Orbits for Diffeomorphisms." Pub. Math. (IHS) 51, 137/C1/ 173, 1980. Katok, A. and Strelcyn, J.-M. Invariant Manifolds, Entropy and Billiards, Smooth Maps with Singularities. Berlin: Springer-Verlag, 1988. Newhouse, S. "Continuity Properties of Entropy." Ann. Math. 129, 215/C1/237, 1989. Newhouse, S. "Entropy and Volume." Ergodic Th. Dynam. Sys. 8, 283/C1/299, 1989. Pollicott, M. Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds. Cambridge, England: Cambridge University Press, 1993. Peters Polynomial Polynomials sk(x;l;m) which are a generalization of the B OOLE POLYNOMIALS , form the S HEFFER SE- QUENCE for g(t)/C30(1/C27elt)m(1) f(t)/C30et/C281 (2) and have GENERATING FUNCTION X/C12 k/C300sk(x;l;m) k!tk/C30[1/C27(1/C27t)l]/C28m(1/C27t)x: (3) The first few are s0(x;l;m)/C302/C28m s1(x;l;m)/C302/C28(m/C271)(2x/C28lm) s2(x;l;m)/C302/C28(m/C272)[4x(x/C281)/C27(2/C284x)lm /C27m(m/C281)l2]: References Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 37, 1964. Roman, S. "The Peters Polynomial." §4.6 in The Umbral Calculus. New York: Academic Press, p. 128, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/760, 1973. Peters Projection A CYLINDRICAL EQUAL-AREA PROJECTION that de- emphasizes the exaggeration of areas at high lati- tudes by shifting the standard LATITUDE to fs /C30 44 :138/C14 (or sometimes 458 or 478; Dana). See also BALTHASART PROJECTION ,BEHRMANN CY- LINDRICAL EQUAL- AREA PROJECTION ,C YLINDRICAL EQUAL- AREA PROJECTION ,CYLINDRICAL PROJECTION , EQUAL- AREA PROJECTION ,GALL ORTHOGRAPHIC PRO- JECTION ,LAMBERT AZIMUTHAL EQUAL- AREA PROJEC- TION ,PETERS PROJECTION References Dana, P. H. "Map Projections." http://www.colorado.edu/ geography/gcraft/notes/mapproj/mapproj_f.html. Petersen Graph "The" Petersen graph is the GRAPH illustrated above possessing ten nodes, all of whose nodes have DEGREE 3 (Saaty and Kainen 1986, Harary 1994, p. 89). The Petersen graph is the only smallest- GIRTH graph which has no Tait coloring, and is the unique 5-CAGE GRAPH (Harary 1994, p. 175). It is the complement of the LINE GRAPH of the COMPLETE GRAPH K5(Skiena 1990, p. 139), and the ODD GRAPH O3(Skiena 1990, p. 162). It is depicted on the cover of the journal Discrete Mathematics . The Petersen graph is thesmallest HYPOHAMILTONIAN GRAPH The Petersen graph provides a 6-color coloring of the PROJECTIVE PLANE . The seven graphs obtainable from the COMPLETE GRAPH K6by repeated triangle-Y exchanges are also called Petersen graphs, where the three EDGES form- ing the TRIANGLE are replaced by three EDGES and a new VERTEX that form a Y, and the reverse operation is also permitted. A GRAPH is intrinsically linked IFFit contains one of the seven Petersen graphs (Robertson et al. 1993). See also CAGE GRAPH ,GIRTH,HOFFMAN- SINGLETON GRAPH ,HYPOHAMILTONIAN GRAPH ,ODD GRAPH References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 221 /C1/222, 1994. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 236 and 243, 1976. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 89 and 112, 1994. Hoffman, A. J. and Singleton, R. R. "On Moore Graphs of Diameter Two and Three." IBM J. Res. Develop. 4, 497/C1/ 504, 1960. Holton, D A. and Sheehan, J. (Eds.). The Petersen Graph. Cambridge, England: Cambridge University Press, 1993. Robertson, N.; Seymour, P. D.; and Thomas, R. "Linkless Embeddings of Graphs in 3-Space." Bull. Amer. Math. Soc. 28,8 4/C1/89, 1993. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 102, 1986. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 139 and 191, 1990. Weisstein, E. W. "Graphs." M ATHEMATICA NOTEBOOK GRAPHS.M . Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/22, 1982. Petersen-Shoute Theorem A beautiful general theory of which the following two statements are special cases. 1. If DABC and DA?B?C ? are two DIRECTLY SIMILAR triangles, while DAA?Aƒ;DBB ?Bƒ; and DCC?Cƒ are three DIRECTLY SIMILAR triangles, then DAƒBƒC ƒ is directly similar to DABC :/ 2. When all the points P on AB are related by a SIMILARITY TRANSFORMATION to all the points P? on A?B?; the points dividing the segment PP? in a given ratio are distant and collinear, or else they coincide. See also DIRECTLY SIMILAR ,SIMILARITY TRANSFORMA- TION References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 95 /C1/100, 1967. Forder, H. G. Higher Course Geometry. Cambridge, Eng- land: Cambridge University Press, p. 53, 1931. Petersen, J. Methods and Theories for the Solution of Problems of Geometrical Constructions Applied to 410 Problems. New York: Stechert, p. 74, 1923. Reprinted in String Figures and Other Monographs. New York: Chel- sea, 1960. Peterson-Mainardi-Codazzi Equations @e @v /C28@f @u /C30e G1 12 /C27f( G212 /C28G111) /C28g G211 (1) @f @v /C28@g @u /C30e G1 22 /C27f( G222 /C28G112) /C28gG212 ; (2) where e, f, and g are coefficients of the second FUNDAMENTAL FORM and Gk ijare CHRISTOFFEL SYM- BOLS OF THE SECOND KIND . Therefore, @e @v /C301 2 Eve E /C27g G ! (3) @g @u /C301 2 Gue E /C27g G ! (4) @(ln f) @u/C30G1 11 /C28G212 (5) @(ln f) @v/C30G222 /C28G112 (6) @ @uln fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG /C28 F2p ! /C30/C282G2 12 (7) @ @vln fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG /C28 F2p ! /C30/C282G1 12 ; (8)where E, F, and G are coefficients of the first FUNDAMENTAL FORM . References Gray, A. "The Peterson-Mainardi-Codazzi Equations." §28.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 649 /C1/652, 1997. Green, A. E. and Zerna, W. Theoretical Elasticity, 2nd ed. New York: Dover, p. 37, 1992. Petersson Conjecture Petersson considered the absolutely converging DI- RICHLET L-SERIES f(s) /C30Y p1 1 /C28 c(p)p /C28s /C27 p2k /C281p /C282s : Writing the DENOMINATOR as 1 /C28c(p)x /C27p2k /C281x2 /C30(1 /C28r1x)(1 /C28r2x) ; where r1 /C27r2 /C30c(p) and r1r2 /C30p2k /C281 ; Petersson conjectured that r1and r2are always COMPLEX CONJUGATE , which implies r1jj/C30r2jj/C30pk/C281=2 and c(p)jj52pk/C281=2: This conjecture was proven by Deligne (1974), which also proved the TAU CONJECTURE as a special case. Deligne was awarded the F IELDS MEDAL for his proof. See also DIRICHLET L-SERIES ,TAU CONJECTURE References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 140, 1997. Deligne, P. "La conjecture de Weil. I." Inst. Hautes E ´tudes Sci. Publ. Math. 43, 273/C1/307, 1974. Deligne, P. "La conjecture de Weil. II." Inst. Hautes E ´tudes Sci. Publ. Math. 52, 137/C1/252, 1980. Peter-Weyl Theorem Establishes completeness for a group REPRESENTA- TION . References Huang, J.-S. "The Peter-Weyl Theorem." §8.5 in Lectures on Representation Theory. Singapore: World Scientific, pp. 99 /C1/103, 1999. Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537/C1/549, 1996. Petrie Polygon A SKEW POLYGON such that every two consecutive sides (but no three) belong to a face of a regular POLYHEDRON . Every REGULAR POLYHEDRON can be orthogonally projected onto a plane in such a way that one Petrie polygon becomes a REGULAR POLYGON with the remainder of the projection interior to it. The Petrie polygon of the POLYHEDRON fp ; q g has h sides, where cos2p h ! /C30cos2p p ! /C27cos2p q ! : The Petrie polygons shown above correspond to the PLATONIC SOLIDS . See also PLATONIC SOLID,REGULAR POLYGON ,REG- ULAR POLYHEDRON ,SKEW POLYGON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 135, 1987. Coxeter, H. S. M. "Petrie Polygons." §2.6 in Regular Poly- topes, 3rd ed. New York: Dover, pp. 24 /C1/25, 1973. Petrov Notation A TENSOR notation which considers the RIEMANN TENSOR Rlmnkas a matrix R(lm)(nk)with indices lm and nk :/ References Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 142, 1972. Petty Projection Inequality An affine isoperimetric inequality. References Lutwak, E. "Selected Affine Isoperimetric Inequalities." In Handbook of Convex Geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 151 /C1/176, 1993. Pfaff Transformation When xjjB1 =2; (1 /C28x) /C28a 2F1(a; b; c; /C28x=(1 /C28x)) /C30 2F1(a; c /C28b; c; x) :References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 39 /C1/40, 1998. Pfaffian An analog of the determinant for NUMBER TRIANGLES defined as a signed sum indexed by set partitions of f1; ...; ng into pairs of elements. The Pfaffian is the square root of the determinant of the corresponding skew symmetric matrix. References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/646. Pfaffian Form A 1-FORM v /C30Xn i/C301ai(x) dxi such that v /C300 : References Knuth, D. E. "Overlapping Pfaffians." Electronic J. Combi- natorics 3, No. 2, R5, 1 /C1/13, 1996. http://www.combinator- ics.org/Volume_3/volume3_2.html#R5. p-Form DIFFERENTIAL K-FORM p-Good Path A LATTICE PATH from one point to another is p-good if it lies completely below the line y/C30(p/C281)x: Hilton and Pederson (1991) show that the number of p-good paths from (1, q/C281) to ( k,n/C28k) under the condition 2 5k5n/C28p/C2715p(k/C281) is n/C28q k/C2819+;89+;9 /C28Xl j/C301pdqjn/C28pj k/C28j9+;89+;9 ; wherea b9+=9+; is a BINOMIAL COEFFICIENT , and l/C13n/C28k p/C281$% ; where xbcis the FLOOR FUNCTION . See also CATALAN NUMBER ,LATTICE PATH,SCHRO ¨ DER NUMBER References Hilton, P. and Pederson, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13,64/C1/75, 1991. p-Group When p is a PRIME NUMBER , then a p-group is a GROUP , all of whose elements have order some power of p. For a FINITE GROUP , the equivalent definition is that the number of elements in G is a power of p.In fact, every FINITE GROUP has subgroups which are p- groups by the SYLOW THEOREMS , in which case they are called SYLOW P-SUBGROUPS . Sylow proved that every GROUP of this form has a power-commutator representation on n generators defined by ap i /C30Yn k /C30i/C271ab(i; k) k (1) for 0 5 b(i ; k) Bp ; 1 5i 5n and [aj ; ai] /C30Yn k /C30j/C271a b(i ; j; k) k (2) for 0 5 b(i ; j; k) Bp; 1 5i Bj 5n: If (pm)isa PRIME POWER and f(pm) is the number of GROUPS of order (pm) ; then f(pm) /C30pAm3 ; (3) where lim m0/C12A /C302 27 (4) (Higman 1960ab). See also GROUP ,G ROUP DIRECT PRODUCT ,O RDER (GROUP ), SYLOW P-SUBGROUP ,SYLOW THEOREMS References Higman, G. "Enumerating p-Groups. I. Inequalities." Proc. London Math. Soc. 10,24/C1/30, 1960a. Higman, G. "Enumerating p-Groups. II. Problems Whose Solution is PORC." Proc. London Math. Soc. 10, 566 /C1/582, 1960b. Phase The angular position of a quantity. For example, the phase of a function cos( vt /C27 f0) as a function of time is f(t) /C30 vt /C27 f0 : The ARGUMENT of a COMPLEX NUMBER is sometimes also called the phase. See also ARGUMENT (COMPLEX NUMBER ), COMPLEX NUMBER ,PHASOR ,RETARDANCEPhase Space For a function or object with n DEGREES OF FREEDOM , the n-D SPACE which is accessible to the function or object is called its phase space. See also WORLD LINE Phase Transition Erdos and Re´nyi (1960) showed that for many mono- tone-increasing properties of RANDOM GRAPHS , graphs of a size slightly less than a certain threshold are very unlikely to have the property, whereas graphs with a few more EDGES are almost certain to have it. This is known as a PHASE TRANSITION (Janson et al. 2000, p. 103). See also RANDOM GRAPH References Erdos, P. and Re ´nyi, A. "On the Evolution of Random Graphs." Publ. Math. Inst. Hungar. Acad. Sci. 5,1 7/C1/61, 1960. Janson, S.; /uczak, T.; and Rucinski, A. "The Phase Transi- tion." Ch. 5 in Random Graphs. New York: Wiley, pp. 103 /C1/138, 2000. Phasor The representation, beloved of engineers and physi- cists, of a COMPLEX NUMBER in terms of a COMPLEX exponential x/C27iy/C30zjjeif; (1) where I(called Jby engineers) is the IMAGINARY NUMBER and the MODULUS and ARGUMENT (also called PHASE ) are zjj/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p (2) f/C30tan/C281y x ! : (3) Here, f(sometimes also denoted u) is called the ARGUMENT or the PHASE . It corresponds to the counterclockwise ANGLE from the POSITIVE REAL AXIS, i.e., the value of fsuch that x/C30cosfand y/C30 sinf:The special kind of INVERSE TANGENT used here takes into account the quadrant in which zlies and is returned by the FORTRAN command ATAN2(X,Y) and the Mathematica command ArcTan [x,y], and is often restricted to the range /C28pBu5p:In the degenerate case when x/C300, f/C30/C281 2p ifyB0 undefined if y/C300 12p ify>08 >< >:(4) It is trivially true that X iR[ci] /C30RX ici"# : (5) Now consider a SCALAR FUNCTION c /C13 c0eif : Then I /C13[ R( c)]2 /C301 2( c /C27 ¯c)hi2 /C3014( c /C27 ¯c)2 /C3014(c2 /C272c ¯c /C27 ¯c2) ; (6) where ¯c is the COMPLEX CONJUGATE . Look at the time averages of each term, c29+;=9+;; /C30 c2 0e2if9+;=9+;; /C30 c20e2if9+;=9+;; /C300 (7) /C142 c ¯c/C143/C30 c20ei f c0e /C28i f9+;=9+;; /C30 c20 /C30½ c½2 (8) ¯c29+;=9+;; /C30 c20e /C282if9+;=9+;; /C30 c20e /C282i f9+;=9+;; /C300: (9) Therefore, /C142I /C143/C301 2½ c½2 : (10) Consider now two scalar functions c1 /C13 c1 ; 0ei(kr1/C27f1) (11) c2 /C13 c2 ; 0ei(kr2/C27f2) : (12) Then I /C13[ R( c1) /C27R( c2)]2 /C3014[(c1 /C27 ¯c1) /C27(c2 /C27 ¯c2)]2 /C3014[(c1 /C27 ¯c1)2 /C27( c2 /C27 ¯c2)2 /C272(c1 c2 /C27 c1¯c2 /C27 ¯c1 c2 /C27 ¯c1¯c2)] (13) /C142I /C143/C3014[2c1 ¯c1 /C272c2 ¯c2 /C272 c1 ¯c2 /C272 ¯c1 c2] /C3012[ c1( ¯c1 /C27 ¯c2) /C27 c2( ¯c1 /C27 ¯c2)] /C301 2( c1 /C27 c2)( ¯c1 /C27 ¯c2) /C3012 ½c1 /C27 c2 ½2 : (14) In general, /C142I /C143/C301 2Xn i/C301ci9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$2 : (15) See also AFFIX,ARGUMENT (COMPLEX NUMBER ), CIS, COMPLEX MULTIPLICATION ,C OMPLEX NUMBER ,EX- PONENTIAL FUNCTION ,INVERSE TANGENT ,M ODULUS (COMPLEX NUMBER ), PHASE References Krantz, S. G. "Polar Form of a Complex Number." §1.2.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 8 /C1/10, 1999. Phi Curve An ADJOINT CURVE which bears a special relation to the base curve.References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 310, 1959. Phi Number System For every POSITIVE INTEGER n, there is a correspond- ing finite sequence of distinct INTEGERS k1 ; ..., km such that n /C30 fk1 /C27.../C27 fkm ; where fis the GOLDEN RATIO . See also GOLDEN RATIO References Bergman, G. "A Number System with an Irrational Base." Math. Mag. 31,9 8/C1/110, 1957. Knuth, D. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1997. Rousseau, C. "The Phi Number System Revisited." Math. Mag. 68, 283/C1/284, 1995. Phi-Four Equation The PARTIAL DIFFERENTIAL EQUATION uH/C28uxx/C28u/C27u3/C300: References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 60, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 134, 1997. Philo Line Given two intersecting lines OAandABforming an angle with vertex at Oand a point Xinside the angle /C218AOB ;the Philo line (or Philon line) is the shortest LINE SEGMENT ABtouching both lines and passing through X. The line is named for Philo of Byzantium who considered the line while attempting to duplicate the cube. The line can be constructed by finding OY/C222 AB such that AX /C30BY (Wells 1991). The distances along the angle edges x and h and the lengths along the Philo line l and dl can be computed by solving the simultaneous equations r2 sin2 f /C27x2 /C30l2 h2 /C28l2 /C30(r cos f /C27x)2 /C28(l /C27dl)2 (2l /C27dl)2 /C30h2 sin2 u /C27(r cos u /C27x /C28h cos u)2 (h2 /C28l2) /C27dl2 /C30r2 ; where u is the VERTEX ANGLE and the point X has POLAR COORDINATES (r ; f) :/ References Eves, H. "Philo’s Line." Scripta Math. 24, 141 /C1/148, 1959. Eves, H. W. A Survey of Geometry, Vol. 2. Boston, MA: Allyn and Bacon, pp. 39 and 234 /C1/238, 1965. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 182 /C1/183, 1991. Wells, D. G. You Are a Mathematician: A Wise and Witty Introduction to the Joy of Numbers. New York: Wiley, 1997. Philon Line PHILO LINE Phragme ´n-Linde ˆlo¨f Theorem Let f(z)bean ANALYTIC FUNCTION in an angular domain W : ½arg z½B ap=2: Suppose there is a constant M such that for each e > 0; each finite boundary point has a NEIGHBORHOOD such that ½f(z) ½BM /C27 e on the intersection of D with this NEIGHBORHOOD , and that for some POSITIVE number b > a for sufficiently large ½z½; the INEQUALITY ½f(z)½Bexp ½z½1 =b9+=9+; holds. Then ½f(z) ½5M in D. References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 160, 1980. Phyllotaxis The beautiful arrangement of leaves in some plants, called phyllotaxis, obeys a number of subtle mathe- matical relationships. For instance, the florets in the head of a sunflower form two oppositely directed spirals: 55 of them clockwise and 34 counterclock- wise. Surprisingly, these numbers are consecutive FIBONACCI NUMBERS . The ratios of alternate FIBO-NACCI NUMBERS are given by the convergents to f/C282 ; where f is the GOLDEN RATIO , and are said to measure the fraction of a turn between successive leaves on the stalk of a plant: 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). A similar phenomenon occurs for DAISIES , pineapples, pine- cones, cauliflowers, and so on. Lilies, irises, and the trillium have three petals; columbines, buttercups, larkspur, and wild rosehave five petals; delphiniums, bloodroot, and cosmos have eight petals; corn marigolds have 13 petals; asters have 21 petals; and daisies have 34, 55, or 89petals–all F IBONACCI NUMBERS . See also DAISY,FIBONACCI NUMBER ,SPIRAL References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 56 /C1/57, 1987. Church, A. H. The Relation of Phyllotaxis to Mechanical Laws. London: Williams and Norgate, 1904. Church, A. H. On the Interpretation of Phenomena of Phyllotaxis. Riverside, NJ: Hafner, 1968. Conway, J. H. and Guy, R. K. "Phyllotaxis." In The Book of Numbers. New York: Springer-Verlag, pp. 113 /C1/125, 1995. Cook, T. A. The Curves of Life, Being an Account of Spiral Formations and Their Application to Growth in Nature, To Science and to Art. New York: Dover, 1979. Coxeter, H. S. M. "The Golden Section and Phyllotaxis." Ch. 11 in Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. "The Role of Intermediate Convergents in Tait’s Explanation for Phyllotaxis." J. Algebra 10, 167/C1/ 175, 1972. Coxeter, H. S. M. "The Golden Section, Phyllotaxis, and Wythoff’s Game." Scripta Mathematica 19, 135/C1/143, 1953. Dixon, R. "The Mathematics and Computer Graphics of Spirals in Plants." Leonardo 16,8 6/C1/90, 1983. Dixon, R. Mathographics. New York: Dover, 1991. Douady, S. and Couder, Y. "Phyllotaxis as a Self-Organized Growth Process." In Growth Patterns in Physical Sciences and Biology (Ed. J. M. Garcia-Ruiz et al. ). New York: Plenum, 1993. Hargittai, I. and Pickover, C. A. (Eds.). Spiral Symmetry. New York: World Scientific, 1992. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 20 /C1/22, 1975. Jean, R. V. "Number-Theoretic Properties of Two-Dimen- sional Lattices." J. Number Th. 29, 206/C1/223, 1988. Jean, R. V. "On the Origins of Spiral Symmetry in Plants." InSpiral Symmetry. (Ed. I. Hargittai and C. A. Pickover). New York: World Scientific, pp. 323 /C1/351, 1992. Jean, R. V. Phyllotaxis: A Systematic Study in Plant Morphogenesis. New York: Cambridge University Press, 1994. Pappas, T. "The Fibonacci Sequence & Nature." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 222 /C1/225, 1989. Prusinkiewicz, P. and Lindenmayer, A. The Algorithmic Beauty of Plants. New York: Springer-Verlag, 1990. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 138, 1999. Stevens, P. S. Patterns in Nature. London: Peregrine, 1977. Stewart, I. "Daisy, Daisy, Give Me Your Answer, Do." Sci. Amer. 200,9 6/C1/99, Jan. 1995. Thompson, D. W. On Growth and Form. Cambridge, Eng- land: Cambridge University Press, 1952. Vogel, H. "A Better Way to Construct the Sunflower Head." Math. Biosci. 44, 179/C1/189, 1979. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 65 /C1/ 66, 1986. Pi AREAL NUMBER denoted pwhich is defined as the ratio of a CIRCLE ’sCIRCUMFERENCE Cto its DIAMETER p/C302r; p/C13C d/C30C 2r(1) It is equal to p/C30 3:141592653589793238462643383279502884197 . . . (2) (Sloane’s A000796). P I’S DIGITS have many interesting properties, although not very much is known about their analytic properties. P I’S CONTINUED FRACTION is given by [3, 7, 15, 1, 292, 1, 1, 1, ...] (Sloane’s A001203). /pis known to be IRRATIONAL (Lambert 1761, Legendre 1794, Hermite 1873, Nagell 1951, Niven 1956, Struik1969, Ko ¨nigsberger 1990, Schro ¨der 1993, Stevens 1999). In 1794, Legendre also proved that p 2is IRRATIONAL (Wells 1986, p. 76). pis also TRANSCEN- DENTAL (Lindemann 1882). An immediate conse- quence of Lindemann’s proof of the transcendence of palso proved that the GEOMETRIC PROBLEM OF ANTIQUITY known as CIRCLE SQUARING is impossible. A simplified, but still difficult, version of Lindemann’sproof is given by Klein (1955). It is also known that pis not a L IOUVILLE NUMBER (Mahler 1953). The following table summarizes pro- gress in computing upper bounds on the IRRATION- ALITY MEASURE forp:It is likely that the exponent can be reduced to 2 /C27e;where eis an infinitesimally small number (Borwein et al. 1989). upper boundreference 20 Mahler (1953), Le Lionnais (1983, p. 50)14.65 Chudnovsky and Chudnovsky (1984) 8.0161 Hata (1992) It is not known if p/C27e;p=e;or ln pare IRRATIONAL . However, it is known that they cannot satisfy any POLYNOMIAL equation of degree 58 with INTEGER COEFFICIENTS of average size 109(Bailey 1988, Borwein et al. 1989). J. H. Conway has shown that there is a sequence offewer than 40 FRACTIONS F1;F2;... with the property that if you start with 2nand repeatedly multiply by the first of the Fithat gives an integer result until a POWER of 2 (say, 2k) occurs, then kis the nth decimal digit of p:/ /pcrops up in all sorts of unexpected places in mathematics besides CIRCLES and SPHERES . For ex- ample, it occurs in the normalization of the G AUSSIAN DISTRIBUTION , in the distribution of PRIMES , in the construction of numbers which are very close to INTEGERS (the R AMANUJAN CONSTANT ), and in the probability that a pin dropped on a set of PARALLEL lines intersects a line (B UFFON’S NEEDLE PROBLEM ). Pi also appears as the average ratio of the actual lengthand the direct distance between source and mouth ina meandering river (Støllum 1996, Singh 1997). A brief history of NOTATION for pi is given by Castellanos (1988). pis sometimes known as L UDOL- PH’S CONSTANT after Ludolph van Ceulen (1539 /C1/ 1610), a Dutch pcalculator. The symbol pwas first used by English mathematician William Jones in 1706, and subsequently adopted by Euler. In Mea- surement of a Circle, Archimedes (ca. 225 BC ) obtained the first rigorous approximation by INSCRIB- ING and CIRCUMSCRIBING 6/C2152n /-gons on a CIRCLE using the A RCHIMEDES ALGORITHM . Using n/C304( a 96-gon), Archimedes obtained 3/C2710 71BpB3/C2717 (3) (Wells 1986, p. 49; Shanks 1993, p. 140). The Bible contains two references (I Kings 7:23 and Chronicles 4:2) which give a value of 3 for p(Wells 1986, p. 48). It should be mentioned, however, that both instances refer to a value obtained from physical measurements and, as such, are probably well within the bounds of experimental uncertainty. I Kings 7:23states, "Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits in height thereof; and a line thirty cubits did compass itround about." This implies p/C30C=d/C3030=10/C303:The Babylonians gave an estimate of pas 3/C271=8/C303:125: The Egyptians did better still, obtaining 2 8=34/C30 3:1605 . . . in the Rhind papyrus, and 22/7 elsewhere. The Chinese geometers, however, did best of all, rigorously deriving pto 6 decimal places. There are many, many FORMULAS FOR PI, from the simple to the very complicated. Ramanujan (1913 /C1/14) and Olds (1963) give geometric constructions for 355/113. Gardner (1966, pp. 92 /C1/93) gives a geometric construction for 3 /C2716 =113 /C30 3:1415929 ... : Dixon (1991) gives constructions for 6=5(1 /C27 f) /C303:141640... andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C27 3 /C28tan(30 /C14) ½/C1382q /C30 3:141533... : Constructions for approximations of p are approximations to CIRCLE SQUARING (which is itself impossible). See also ALMOST INTEGER ,ARCHIMEDES ALGORITHM , BRENT- SALAMIN FORMULA ,BUFFON- LAPLACE NEEDLE PROBLEM ,BUFFON’S NEEDLE PROBLEM ,CIRCLE ,CIR- CUMFERENCE ,DIAMETER ,DIRICHLET BETA FUNCTION , DIRICHLET ETA FUNCTION ,DIRICHLET LAMBDA FUNC- TION , E,E ULER- MASCHERONI CONSTANT ,G AUSSIAN DISTRIBUTION ,MACLAURIN SERIES ,MACHIN’S FORMU- LA,M ACHIN- LIKE FORMULAS ,PI APPROXIMATIONS ,PI CONTINUED FRACTION ,PI DIGITS ,PI FORMULAS ,PI WORDPLAY ,R ADIUS ,R ELATIVELY PRIME ,R IEMANN ZETA FUNCTION ,SPHERE ,TRIGONOMETRY References Almkvist, G. and Berndt, B. "Gauss, Landen, Ramanujan, and Arithmetic-Geometric Mean, Ellipses, p;and the Ladies Diary." Amer. Math. Monthly 95, 585/C1/608, 1988. Almkvist, G. "Many Correct Digits of p;Revisited." Amer. Math. Monthly 104, 351/C1/353, 1997. Arndt, J. "Cryptic Pi Related Formulas." http://www.jjj.de/ hfloat/pise.dvi. Arndt, J. and Haenel, C. Pi: Algorithmen, Computer, Arithmetik. Berlin: Springer-Verlag, 1998. Assmus, E. F. "Pi." Amer. Math. Monthly 92, 213/C1/214, 1985. Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving p;e, and Euler’s Constant." Math. Comput. 50, 275/C1/281, 1988a. Bailey, D. H. "The Computation of pto 29,360,000 Decimal Digit using Borwein’s’ Quartically Convergent Algorithm." Math. Comput. 50, 283/C1/296, 1988b. Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants."http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 55 and 274, 1987. Beck, G. and Trott, M. "Calculating Pi from Antiquity to 1996." http://library.wolfram.com/demos/v4/Calculating-Pi.nb. Beckmann, P. A History of Pi, 3rd ed. New York: Dorset Press, 1989. Beeler, M. et al. Item 140 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 69, Feb. 1972. Berggren, L.; Borwein, J.; and Borwein, P. Pi: A Source Book. New York: Springer-Verlag, 1997. Bellard, F. "Fabrice Bellard’s Pi Page." http://www-stu- d.enst.fr/~bellard/pi/. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Blatner, D. The Joy of Pi. New York: Walker, 1997. Blatner, D. "The Joy of Pi." http://www.joyofpi.com/. Borwein, P. B. "Pi and Other Constants." http://www.cecm.s- fu.ca/~pborwein/PISTUFF/Apistuff.html.Borwein, J. M. "Ramanujan Type Series." http:// www.cecm.sfu.ca/organics/papers/borwein/paper/html/lo- cal/omlink9/html/node1.html. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.New York: Wiley, 1987a. Borwein, J. M. and Borwein, P. B. "Ramanujan’s Rational and Algebraic Series for 1 =p: /"Indian J. Math. 51, 147/C1/ 160, 1987b. Borwein, J. M. and Borwein, P. B. "More Ramanujan-Type Series for 1 =p:/"I n Ramanujan Revisited. Boston, MA: Academic Press, pp. 359 /C1/374, 1988. Borwein, J. M. and Borwein, P. B. "Class Number Three Ramanujan Type Series for 1 =p:/"J. Comput. Appl. Math. 46, 281/C1/290, 1993. Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanu- jan, Modular Equations, and Approximations to Pi, orHow to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201/C1 /219, 1989. Brown, K. S. "Rounding Up to Pi." http://www.seanet.com/ ~ksbrown/kmath001.htm. Calvet, C. "First Communication. A) Secrets of Pi: Strange Things in a Mathematical Train." http://www.terravis-ta.pt/guincho/1219/1a_index_uk.html. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1 /98, 1988. Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148/C1/163, 1988. Chan, J. "As Easy as Pi." Math Horizons, Winter 1993, pp. 18 /C1/19, 1993. Choong, Daykin, and Rathbone. Math. Comput. 25, 387, 1971. Chudnovsky, D. V. and Chudnovsky, G. V. Pade´and Ra- tional Approximations to Systems of Functions and TheirArithmetic Applications. Berlin: Springer-Verlag, 1984. Chudnovsky, D. V. and Chudnovsky, G. V. "Approximations and Complex Multiplication According to Ramanujan." InRamanujan Revisited: Proceedings of the Centenary Con-ference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Ra- nkin). Boston, MA: Academic Press, pp. 375 /C1 /472, 1987. Conway, J. H. and Guy, R. K. "The Number p:/"I nThe Book of Numbers. New York: Springer-Verlag, pp. 237 /C1/239, 1996. David, Y. "On a Sequence Generated by a Sieving Process." Riveon Lematematika 11,2 6/C1/31, 1957. Dixon, R. "The Story of Pi ( /p):/"§4.3 in Mathographics. New York: Dover, pp. 44 /C1/49 and 98 /C1/101, 1991. Dunham, W. "A Gem from Isaac Newton." Ch. 7 in Journey through Genius: The Great Theorems of Mathematics.New York: Wiley, pp. 106 /C1 /112 and 155 /C1/183, 1990. Exploratorium. " /pPage." http://www.exploratorium.edu/ learning_studio/pi/. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/pi/pi.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996.http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Gardner, M. "Memorizing Numbers." Ch. 11 in The Scien- tific American Book of Mathematical Puzzles and Diver-sions. New York: Simon and Schuster, p. 103, 1959. Gardner, M. "The Transcendental Number Pi." Ch. 8 in Martin Gardner’s New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 91 /C1 /102, 1966. Gosper, R. W. Table of Simple Continued Fraction for pand the Derived Decimal Approximation. Stanford, CA: Arti- ficial Intelligence Laboratory, Stanford University, Oct. 1975. Reviewed in Math. Comput. 31, 1044, 1977. Gourdon, X. and Sebah, P. "The Constant p:/" http://xavier.- gourdon.free.fr/Constants/Pi/pi.html. Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press, 1952. Hata, M. "Improvement in the Irrationality Measures of p andp2:/"Proc. Japan. Acad. Ser. A Math. Sci. 68, 283/C1/286, 1992. Havermann, H. "Continued Fraction expansion of Pi: 20,000,000 terms." http://www.lacim.uqam.ca/piDATA/. Hermite, C. "Sur quelques approximations alge ´briques." J. reine angew. Math. 76, 342/C1/344, 1873. Reprinted in Oeuvres comple `tes, Tome III. Paris: Hermann, pp. 146 /C1/ 149, 1912. Hobsen, E. W. Squaring the Circle. New York: Chelsea, 1988. Kanada, Y. "New World Record of Pi: 51.5 Billion Decimal Digits." http://www.cecm.sfu.ca/personal/jborwein/Kana- da_50b.html. Klein, F. Famous Problems. New York: Chelsea, 1955. Knopp, K. §32, 136, and 138 in Theory and Application of Infinite Series. New York: Dover, p. 238, 1990. Ko¨nigsberger, K. Analysis 1. Berlin: Springer-Verlag, 1990. Laczkovich, M. "On Lambert’s Proof of the Irrationality of p:/" Amer. Math. Monthly 104, 439/C1/443, 1997. Lambert, J. H. "Me ´moire sur quelques proprie ´te´s remarqu- ables des quantite ´s transcendantes circulaires et logarith- miques." Me´moires de l’Academie des sciences de Berlin 17, 265/C1/322, 1761. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 22 and 50, 1983. Lindemann, F. "Uuml;ber die Zahl p:/"Math. Ann. 20, 213/C1/ 225, 1882. Lopez, A. "Indiana Bill Sets the Value of pto 3." http:// www.cs.unb.ca/~alopez-o/math-faq/mathtext/no-de18.html. MacTutor Archive. "Pi Through the Ages." http://www- groups.dcs.st-and.ac.uk/~history/HistToPi_-through_the_ages.html. Mahler, K. "On the Approximation of p: /"Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes Math. 15,3 0/C1/42, 1953. Nagell, T. "Irrationality of the numbers eandp:/"§13 in Introduction to Number Theory. New York: Wiley, pp. 38 /C1/ 40, 1951. Niven, I. M. Irrational Numbers. New York: Wiley, 1956. Ogilvy, C. S. "Pi and Pi-Makers." Ch. 10 in Excursions in Mathematics. New York: Dover, pp. 108 /C1/120, 1994. Olds, C. D. Continued Fractions. New York: Random House, pp. 59 /C1/60, 1963. Pappas, T. "Probability and p:/"The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 18 /C1/19, 1989. Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 178 /C1/186, 1990. Pickover, C. A. Keys to Infinity. New York: Wiley, p. 62, 1995. Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u-qam.ca/pi/records.html. Plouffe, S. "1 Billion Digits of Pi." http://www.lacim.uqam.ca/ piDATA/PI/. Plouffe, S. "PiHex: A Distributed Effort to Calculate Pi." http://www.cecm.sfu.ca/projects/pihex/. Plouffe, S. "Plouffe’s Inverter: A Few Approximations of Pi." http://www.lacim.uqam.ca/pi/approxpi.html. Plouffe, S. "The pPage." http://www.cecm.sfu.ca/pi/. Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u-qam.ca/pi/records.html. Plouffe, S. "Table of Computation of Pi from 2000 BC to Now." http://www.cecm.sfu.ca/projects/ISC/Pihistor-y.html.Preston, R. "Mountains of Pi." New Yorker 68,3 6/C1 /67, Mar. 2, 1992. http://www.lacim.uqam.ca/plouffe/Chudnovs- ky.html. Project Mathematics . "The Story of Pi." Videotape. http:// www.projmath.caltech.edu/storypi.htm. Rabinowitz, S. and Wagon, S. "A Spigot Algorithm for the Digits of p:/"Amer. Math. Monthly 102, 195/C1/203, 1995. Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/372, 1913 /C1/1914. Rivera, C. "Problems & Puzzles: Puzzle The Best Approx- imation to Pi with Primes.-050." http://www.primepuz-zles.net/puzzles/puzz_050.htm. Rudio, F. "Archimedes, Huygens, Lambert, Legendre." In Vier Abhandlungen u ¨ber die Kreismessung. Leipzig, Germany, 1892. Schro ¨der, E. M. "Zur Irrationalita ¨t von p 2und p:/"Mitt. Math. Ges. Hamburg 13, 249, 1993. Shanks, D. "Dihedral Quartic Approximations and Series for p:/"J. Number. Th. 14, 397/C1/423, 1982. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993. Singh, S. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker, pp. 17 /C1/18, 1997. Sloane, N. J. A. Sequences A000796/M2218, A001203/ M2646, A001901, A002485/M3097, A002486/M4456,A002491/M1009, A007509/M2061, A025547, A032510,A032523 A033089, A033090, A036903, and A046126 inin "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Smith, D. E. "The History and Transcendence of p: /" Ch. 9 in Monographs on Topics of Modern Mathematics Relevant tothe Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 388 /C1 /416, 1955. Stevens, J. "Zur Irrationalita ¨t von p:/"Mitt. Math. Ges. Hamburg 18, 151/C1/158, 1999. Støllum, H.-H. "River Meandering as a Self-Organization Process." Science 271, 1710/C1/1713, 1996. Stoschek, E. "Modul 33: Algames with Numbers" http:// marvin.sn.schule.de/~inftreff/modul33/task33.htm. Struik, D. A Source Book in Mathematics, 1200 /C1/1800. Cambridge, MA: Harvard University Press, 1969. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 159, 1991. Vie`te, F. Uriorum de rebus mathematicis responsorum, liber VIII, 1593. Wagon, S. "Is pNormal?" Math. Intel. 7,6 5/C1/67, 1985. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 48 /C1/55 and 76, 1986. Whitcomb, C. "Notes on Pi ( /p):/" http://witcombe.sbc.edu/ earthmysteries/EMPi.html. Woon, S. C. "Problem 1441." Math. Mag. 68,7 2/C1/73, 1995. Pi Approximations KOCHANSKY’S APPROXIMATION is the ROOT of 9x4/C28240x2/C271492 : (1) given by p:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40 3/C28ffiffiffiffiffiffi 12pq :3:141533 : (2) An approximation involving the GOLDEN MEAN is p:6 5f2/C306 5ffiffiffi 5p /C271 2 !2 /C303 53/C27ffiffiffi 5p9+;k9+;7 /C303:14164 . . . :(3) Some approximations due to Ramanujan include p:19ffiffiffi 7p 16ð4Þ :7 31/C2715ffiffiffi 3p9+;k9+;7 (5) :9 5/C27ffiffi 95q (6) :2143 229+;k9+;71=4 /C3092/C27192 22 !1=4 (7) ¼102/C282222 2222 !1=4 (8) /C3097/C271 2/C281 119+;k9+;71=4 (9) /C3097/C279 229+;k9+;71=4 (10) :63 2517/C2715ffiffiffi 5p 7/C2715ffiffiffi5p ! (11) :355 1131/C280:003 3533 ! (12) :12ffiffiffiffiffiffiffiffi 130p ln3/C27ffiffiffiffiffiffi13p9+=9+; ffiffiffi8p /C27ffiffiffiffiffiffi10p9+=9+; 2"# (13) : 24ffiffiffiffiffiffiffiffi 142p lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C2711ffiffiffi 2pp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C277ffiffiffi 2pp 2"# (14) :12ffiffiffiffiffiffiffiffi 190p ln 3/C27ffiffiffiffiffiffi 10p9+;k9+;7 ffiffiffi8p /C27ffiffiffiffiffiffi10p9+;k9+;7hi (15) : 12ffiffiffiffiffiffiffiffi 310p ln1 43/C27ffiffiffi 5p9+;k9+;7 2/C27ffiffiffi 2p9+;k9+;7h /C25/C272ffiffiffiffiffiffi10p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 61/C2720ffiffiffiffiffiffi 10pq 9+;89+;9 /C138 (16) :4ffiffiffiffiffiffiffiffi 522p ln5/C27ffiffiffiffiffiffi29p ffiffiffi2p ! 3 5ffiffiffiffiffiffi 29p /C2711ffiffiffi6p9+;k9+;72 4 /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9/C273ffiffiffi 6p 4s /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C273ffiffiffi6p 4s ! 69+$; ; (17) which are accurate to 3, 4, 4, 8, 8, 9, 14, 15, 15, 18, 23, 31 digits, respectively (Ramanujan 1913 /C1/1914; Hardy 1952, p. 70; Wells 1986, p. 54; Berndt 1994, pp. 48 /C1/49 and 88 /C1/89).S. Irvine noted that (0), giving an approximation to p good to 8 digits, can be written in a form using all digits 0 /C1/9, p:2143 22 !1=4 /C300/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 33/C27192 78/C2856svuut(18) (S. Plouffe). E. Pegg notes that 0/C273/C271/C28(9/C288/C285)/C286 7/C272/C284 /C30233546921420255777694970883318153571 74340293968115785654927455866388593(19) approximates pto 9 digits. Castellanos (1988) gives a slew of curious formulas: p:(2e3/C27e8)1=7(20) :553 311/C271 !2 (21) :3 149+;k9+;74 193 59+;k9+;72 (22) :296 1679+;k9+;72 (23) :663/C27862 553 !2 (24) :1:09999901 /C2151:19999911 /C2151:39999931 /C2151:69999961 ð25Þ :473/C27203 303/C281 (26) :2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27413 7509+;k9+;72r (27) :77729 2549+;k9+;71=5 (28) :31/C27622/C2714 284 !1=3 (29) :17003/C27823/C28103/C2893/C2863/C2833 695(30) :95/C27934/C27344/C27174/C2788 754 !1=4 (31) :100/C2821253/C272143/C27303/C27372 825 !1=4 ; (32) which are accurate to 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, and 13 digits, respectively. An extremely accurate approximation due to Shanks (1982) is p :6ffiffiffiffiffiffiffiffiffiffiffi 3502p ln(2u) /C277:37 /C2910/C2882 ; (33) where u is the product of four simple quartic units. A sequence of approximations due to Plouffe includes p :437 =23 (34) :ln 2198ffiffiffi6p (35) :13 49+;k9+;71181 =1216 (36) :689 396 ln689 3969+;k9+;7 (37) :2143 229+;k9+;71 =4 (38) :ffiffiffiffiffiffi 9 67s ln 5280 (39) :63023305109+;k9+;71=3 /C2714 /C2712ffiffiffi 5p /C2719+;k9+;7 (40) :48 23 ln60318 13387 ! (41) : 228 /C2716 13299+;k9+;71 =41 /C272 (42) :125 123 ln28102 1277 ! (43) :276694819753963 2265881 =158/C272 (44) :ln 262537412640768744ffiffiffiffiffiffiffiffi 163p ; (45) which are accurate to 4, 5, 7, 7, 8, 9, 10, 11, 11, 11, 23, and 30 digits, respectively. An approximation due to Stoschek using powers of two and the special number 163 (the largest HEEGNER NUMBER ) is given by p :29 163 /C30512 163 :3 :1411043 ; (46) which is good to 3 digits. A fraction with small numerator and denominator which gives is close approximation to p is 311 99/C303 :14141414... : (47) Some approximations involving the ninth roots of rational numbers includep :4297607660 144171 !1 =9 (48) p :4297607660 144171 !1 =9 ; (49) which are good to 12 and 15 digits, respectively (P. Galliani). J. Iuliano found p:1960/C271ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C215123449p !/C281 ; (50) which is good to 11 digits. Rivera gives other approximation formulas. See also PI References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1/98, 1988. Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148/C1/163, 1988. Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press, 1952. Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/372, 1913 /C1/1914. Rivera, C. "Problems & Puzzles: Puzzle The Best Approx- imation to Pi with Primes.-050." http://www.primepuz- zles.net/puzzles/puzz_050.htm. Shanks, D. "Dihedral Quartic Approximations and Series for p:/"J. Number. Th. 14, 397/C1/423, 1982. Pi Continued Fraction The SIMPLE CONTINUED FRACTION for PI, which gives the "best" approximation of a given order, is [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (Sloane’s A001203; Havermann). The very large term292 means that the CONVERGENT [3;7;15;1]/C30[3;7;16]/C30355 113/C303:1415929 . . . (1) is an extremely good approximation. The first few CONVERGENTS are 22/7, 333/106, 355/113, 103993/ 33102, 104348/33215, ... (Sloane’s A002485 and A002486). A nice expression for the third convergentofpis given by p:2[1;1;1;3;32]/C30 355 113:3:14159292 . . . (2) (Stoschek). Gosper has computed 17,001,303 terms of p/’sCON- TINUED FRACTION (Gosper 1977, Ball and Coxeter 1987), a record which was recently upped to 20,000,000 by H. Havermann in June 1999 (Plouffe). The first occurrences of nin the CONTINUED FRACTION are 4, 9, 1, 30, 40, 32, 2, 44, 130, 100, ... (Sloane’s A032523). The smallest integer which does not occur in the first 20,000,000 terms is 2297. The sequence of increasing terms in the CONTINUED FRACTION is 3, 7, 15, 292, 436, 20776, 78629, 179136, 528210, 12996958, 878783625, ... (Sloane’s A033089), occur- ring at positions 1, 2, 3, 5, 308, 432, 28422, 156382, 267314, 453294, 11504931 ... (Sloane’s A033090). The following table gives the first few occurrences of d-digit terms in the CONTINUED FRACTION of p; counting 3 as the 0th (e.g., Choong et al. 1971, Beeler et al. 1972). d Sloane Terms/Positions 1 Sloane’s A0482923, 7, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, ... Sloane’s A0482930, 1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, ... 2 Sloane’s A04829415, 14, 84, 15, 13, 99, 12, 16, 45, 22, ... Sloane’sA0489552, 12, 21, 25, 27, 33, 54, 77, 80, 82, ... 3 Sloane’s A048956292, 161, 120, 127, 436, 106, 141, ... Sloane’sA0489574, 79, 196, 222, 307, 601, 669, 725, ... 4 Sloane’s A0489581722, 2159, 8277, 1431, 1282, 2050, ... Sloane’s A0489593273, 3777, 3811, 4019, 4700, 6209, ... 5 Sloane’s A04896020776, 19055, 19308, 78629, 17538, ... Sloane’s A048961431, 15543, 23398, 28421, 51839, ... 6 Sloane’s A048962179136, 528210, 104293, 196030, ... Sloane’s A048963156381, 267313, 294467, 513205, ... 7 Sloane’s A0489648093211, 1811791, 3578547, ... Sloane’s A0489651118727, 2782369, 2899883, ... 8 Sloane’s A04896612996958, ... Sloane’sA048967453293, ... 9 Sloane’s A048968878783625, ... Sloane’sA04896911504930, ...The SIMPLE CONTINUED FRACTION for p does not show any obvious patterns, but clear patterns do emerge in the beautiful non-simple CONTINUED FRACTIONS 4 p /C301 /C2712 2 /C2732 2 /C2752 2 /C2772 2 /C27 ...(3) (Brouckner), giving convergents 1, 3/2, 15/13, 105/76, 315/263, ... (Sloane’s A025547 and A007509) and p 2/C301/C281 3/C282 /C2153 1/C281 /C2152 3/C284 /C2155 1/C283 /C2154 3/C286 /C2157 1/C285 /C2156 3/C28...(4) (Stern 1833), giving convergents 1, 2/3, 4/3, 16/15, 64/45, 128/105, ... (Sloane’s A001901 and A046126). See also P I References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 55 and 274, 1987. Beeler, M. et al. Item 140 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 69, Feb. 1972. Choong, Daykin, and Rathbone. Math. Comput. 25, 387, 1971. Gosper, R. W. Table of Simple Continued Fraction for pand the Derived Decimal Approximation. Stanford, CA: Arti- ficial Intelligence Laboratory, Stanford University, Oct. 1975. Reviewed in Math. Comput. 31, 1044, 1977. Havermann, H. "Simple Continued Fraction Expansion of Pi." http://members.home.net/hahaj/cfpi.html. Lochs, G. "Die ersten 968 Kettenbruchnenner von p:/" Monatsh. fu ¨r Math. 67, 311/C1/316, 1963. Stoschek, E. "Modul 33: Algames with Numbers." http:// marvin.sn.schule.de/~inftreff/modul33/task33.htm. Pi Digits The calculation of the p/’s digits has occupied mathe- maticians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his lifecalculating pto 35 places. Although he did not live to publish his result, it was inscribed on his grave-stone. Wells (1986, p. 48) discusses a number of othercalculations. The calculation of palso figures in the Star Trek episode "Wolf in the Fold," in which Captain Kirk and Mr. Spock force an evil entity(composed of pure energy and which feeds on fear)out of the starship Enterprise ’s computer by com- manding the computer to "compute to the last digit the value of pi," thus sending the computer into an infinite loop. /phas recently (Sep. 20, 1999) been computed to a world record 206 ;158;430;208:3/C215236DECIMAL DIGITS by Y. Kanada (Kanada, Plouffe). This calcula- tion was done using Borwein’s fourth-order conver-gent algorithm and required 46 hours on a massively parallel 1024-processor Hitachi SR8000 supercompu- ter. The largest number of digits of pcomputing using aP Ci s6 ;442;450;944:3/C21521 31DECIMAL DIGITS by S. Kondo on Jan. 13, 2000 (Gourdon). One billiondigits of pare accessible from Plouffe’s web site. Between April 19, 1998, and Feb. 9, 1999, 126computers from eighteen different countries set anew record for calculating specific bits of pusing a program written by C. Percival. The calculation tooka total of about 84,500 CPU hours and was done usingidle CPU cycles under Windows 95 and Windows NT. The answer, starting at the 39,999,999,999,997th bit ofpis 1010000011111001111111110011011100011101 0001011101011001001111100000 ; (1) so the 40 trillionth bit of pis 0 (Plouffe). In the following, the word "digit" refers to decimaldigit after the decimal point. The following table gives the starting positions for strings of ncopies of the digit d. dn Sloane Positions 0 1 Sloane’s A05020032, 50, 54, 65, 71, 77, 85, 97, ... 0 2 Sloane’s A050201307, 360, 601, 602, 855,856, 973, ... 0 3 Sloane’s A050202601, 855, 1598, 4255, 4793,7832, ... 0 4 Sloane’s A05020313390, 17534, 17535,37322, ... 0 5 17534, 211058, 215287, 652115, ... 0 6 1699927, 2328783, 2609392, ... 0 7 3794572, 13310436, 28970114, ... 1 1 Sloane’s A0502071, 3, 37, 40, 49, 68, 94, 95,... 1 2 Sloane’s A05020894, 153, 154, 174, 362, 395,427, ...1 3 Sloane’s A050209153, 983, 3503, 3992, 4508, 6116, ... 1 4 12700, 16732, 32788, 32789, ... 1 5 32788, 120459, 141899, 255945, ... 1 6 255945, 2645268, 3218870, ... 1 7 4657555, 42408103, 70787432, ... 2 1 Sloane’s A0502146, 16, 21, 28, 33, 53, 63, 73, 76, ... 2 2 Sloane’s A050215135, 185, 484, 535, 661,687, 824, ... 2 3 1735, 1889, 2278, 2376, 3434, ... 2 4 4902, 7964, 12486, 43405, 50271, ... 2 5 65260, 327074, 580735, 619398, ... 2 6 963024, 1637080, 1795773, ... 2 7 82599811, 88301507, ... 3 1 Sloane’s A0502219, 15, 17, 24, 25, 27, 43, 46, 64, ... 3 2 Sloane’s A05022224, 215, 230, 282, 364, 401,503, ... 3 3 1698, 4928, 6917, 7651, 8413, ... 3 4 28467, 28468, 66846, 79979, ... 3 5 28467, 89085, 146043, 335792, ... 3 6 710100, 710101, 1129019, ... 3 7 710100, 3204765, 12469058, ... 3 8 36488176, ... 4 1 Sloane’s A0502292, 19, 23, 36, 57, 59, 60, 70, 87, ... 4 2 Sloane’s A05023059, 125, 182, 201, 217, 453,511, ... 4 3 2707, 2928, 3476, 3809, 3866, ... 4 4 54525, 57609, 74544, 75558, ... 4 5 808650, 828499, 828500, ... 4 6 828499, 1264270, 1691163, ... 4 7 17893953, 22931745, 22931746, ... 4 8 22931745, 65122865, ... 5 1 Sloane’s A0502374, 8, 10, 31, 48, 51, 61, 90, ... 5 2 Sloane’s A050238130, 177, 178, 315, 809,914, ... 5 3 177, 1232, 1450, 2359, 2674, 7245, ... 5 4 24466, 24467, 33172, 39861, ... 5 5 24466, 39861, 205034, 205193, ... 5 6 244453, 253209, 419997, 3517236, ... 5 7 3517236, 9325203, 10519242, ... 6 1 Sloane’s A0502447, 20, 22, 41, 69, 72, 75, 82, ... 6 2 Sloane’s A050245117, 211, 257, 276, 309, 377, 516, ... 6 3 2440, 3151, 4000, 4435, 5403, 6840, ... 6 4 21880, 29868, 32427, 43523, 48439, ... 6 5 48439, 102387, 140744, 250129, ... 6 6 252499, 3813777, 4213896, ... 6 7 8209165, 18696860, 19715001, ... 6 8 45681781, 45681782, 55616210, ... 6 9 45681781, ... 7 1 Sloane’s A05025313, 29, 39, 47, 56, 66, 96, 99, 120, ... 7 2 Sloane’s A050254559, 621, 625, 633, 739, 742, 890, ... 7 3 1589, 1590, 4575, 5241, 5242, 5322, ... 7 4 1589, 5241, 5322, 5863, 29504, ...7 5 162248, 283693, 322347, 399579, ... 7 6 399579, 452071, 1006927, 2309218, ... 7 7 3346228, 3775287, 14233532, ... 7 8 24658601, 24658602, 82144203, ... 7 9 24658601, ... 8 1 Sloane’s A05026211, 18, 26, 34, 35, 52, 67, 74, 78, ... 8 2 Sloane’s A05026334, 204, 317, 322, 372, 472, 848, ... 8 3 4751, 4752, 4985, 5871, 6070, 6850, ... 8 4 4751, 30796, 59550, 60822, 62383, ... 8 5 213245, 222299, 222300, 493647, ... 8 6 222299, 2418533, 3019042, ... 8 7 4722613, 7820866, 19921876, ... 8 8 46663520, 46663521, ... 8 9 46663520, ... 9 1 Sloane’s A0502715, 12, 14, 30, 38, 42, 44, 45, 55, ... 9 2 Sloane’s A05027244, 79, 459, 705, 747, 762,763, ... 9 3 762, 763, 764, 765, 2949, 7759, ... 9 4 762, 763, 764, 17988, 19437, 19446, ... 9 5 762, 763, 19446, 56988, 161862, ... 9 6 762, 193034, 1722776, 1722777, ... 9 7 1722776, 3389380, 4313727, ... 9 8 36356642, 66780105, ... The following table gives the first few positions atwhich a digit doccurs ntimes. Note that the sequence 9999998 occurs at decimal 762 (which is sometimes called the F EYNMAN POINT ; Wells 1986, p. 51). This is the largest value of any seven digits in the first million decimals. d Sloane strings of 1, 2, ... ds first occur at 0 Sloane’s A05027932, 307, 601, 13390, 17534, 1699927, ... 1 Sloane’s A0502801, 94, 153, 12700, 32788, 255945, ... 2 Sloane’s A0502816, 135, 1735, 4902, 65260, 963024, ... 3 Sloane’s A0502829, 24, 1698, 28467, 28467, 710100, ... 4 Sloane’s A0502832, 59, 2707, 54525, 808650, 828499, ... 5 Sloane’s A0502844, 130, 177, 24466, 24466, 244453, ... 6 Sloane’s A0502857, 117, 2440, 21880, 48439, 252499, ... 7 Sloane’s A05028613, 559, 1589, 1589, 162248, 399579, ... 8 Sloane’s A05028711, 34, 4751, 4751, 213245, 222299, ... 9 Sloane’s A0502885, 44, 762, 762, 762, 762, 1722776, ... The first time the BEAST NUMBER 666 appears is decimal 2440. The digits 314159 appear at least six times in the first 10 million decimal places of p (Pickover 1995). The sequence 0123456789 occurs beginning at digits 17,387,594,880, 26,852,899,245, 30,243,957,439, 34,549,153,953, 41,952,536,161, and 43,289,964,000 (cf. Wells 1986, p. 51). The sequence 9876543210 occurs beginning at digits 21,981,157,633, 29,832,636,867, 39,232,573,648, 42,140,457,481, and 43,065,796,214. The sequence 27182818284 (the first few digits of E) occur begin- ning at digit 45,111,908,393. There are also interest- ing patterns for 1 =p: 0123456789 occurs at 6,214,876,462, 9876543210 occurs at 15,603,388,145 and 51,507,034,812, and 999999999999 occurs at 12,479,021,132 of 1=p:/ Scanning the decimal expansion of p until all n-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 0, 68, 483, 6716, 33394, 569540, ... (Sloane’s A032510). These end at digits 32, 606, 8555, 99849, 1369564, 14118312, ... (Sloane’s A036903). The last n-digit number seen in the decimal expan- sion of p for n /C301, 2, ... are 0, 68, 483, 6716, 33394,569540, 1075656, ... (Sloane’s A032150). The last digits of these numbers occur at positions 32, 606, 8555, 99849, ... (Sloane’s A036903). It is not known if p is NORMAL (Wagon 1985, Bailey and Crandall 2000), although the first 30 million DIGITS are very UNIFORMLY DISTRIBUTED (Bailey 1988). The following distribution is found for the first n DIGITS of p /C283: It shows no statistically SIGNIFICANT departure from a UNIFORM DISTRIBUTION (technically, in the CHI-SQUARED TEST , it has a value of x2 s /C305 :60 for the first 5 /C291010 terms). digit /1 /C29105 //1 /C29106 //6 /C29109 // 5 /C291010 / 0 9,999 99,959 599,963,005 5,000,012,647 1 10,137 99,758 600,033,260 4,999,986,263 2 9,908 100,026 599,999,169 5,000,020,237 3 10,025 100,229 600,000,243 4,999,914,405 4 9,971 100,230 599,957,439 5,000,023,598 5 10,026 100,359 600,017,176 4,999,991,499 6 10,029 99,548 600,016,588 4,999,928,368 7 10,025 99,800 600,009,044 5,000,014,860 8 9,978 99,985 599,987,038 5,000,117,637 9 9,902 100,106 600,017,038 4,999,990,486 The digits of 1 =pare also very uniformly distributed (x2 s/C307:04);as shown in the following table. digit /5/C291010/ 0 4,999,969,955 1 5,000,113,6992 4,999,987,893 3 5,000,040,906 4 4,999,985,8635 4,999,977,5836 4,999,990,916 7 4,999,985,552 8 4,999,881,1839 5,000,066,450 See also P I,PI FORMULAS References Plouffe, S. "Plouffe’s Inverter: Table of Current Records for the Computation of Constants." http://www.lacim.u- qam.ca/pi/records.html. Adamchik, V. and Wagon, S. "A Simple Formula for p:/" Amer. Math. Monthly 104, 852/C1/855, 1997. Bailey, D. H. "The Computation of pto 29,360,000 Decimal Digit using Borwein’s’ Quartically Convergent Algorithm."Math. Comput. 50, 283/C1 /296, 1988. Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants."http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps. Bailey, D. H. and Crandall, R. E. "On the Random Char- acter of Fundamental Constant Expansions." Manuscript, Mar. 2000. Caldwell, C. K. and Dubner, H. "Primes in Pi." J. Recr. Math. 29, 282/C1 /289, 1998. Gourdon, X. and Sebah, P. "PiFast: The Fastest Program to Compute Pi." http://xavier.gourdon.free.fr/Constants/Pi- Program/pifast.html. Kanada, Y. "Our Latest Record." Sep. 20, 1999. ftp:// www.cc.u-tokyo.ac.jp/README.our_latest_record. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 22 and 50, 1983. Pickover, C. A. Keys to Infinity. New York: Wiley, p. 62, 1995. Plouffe, S. "1 Billion Digits of Pi." http://www.lacim.uqam.ca/ piDATA/PI/. Rabinowitz, S. and Wagon, S. "A Spigot Algorithm for the Digits of p:/"Amer. Math. Monthly 102, 195/C1/203, 1995. Sloane, N. J. A. Sequences A032150 and A036903 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Smith, H. J. "Computing Pi." http://pweb.netcom.com/ ~hjsmith/Pi.html. Wagon, S. "Is pNormal?" Math. Intel. 7,6 5/C1 /67, 1985. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 46, 1986. Wrench, J. W. Jr. "The Evolution of Extended Decimal Approximations to p:/"Math. Teacher 53, 644/C1/650, 1960. Pi Formulas A method similar to Archimedes’ can be used to estimate pby starting with an n-gon and then relating the AREA of subsequent 2 n/-gons. Let bbe the ANGLE from the center of one of the POLYGON ’s segments, b/C301 4(n/C283)p; (1) then p/C302 sin(2 b) (n/C283)Q/C12 k/C300cos 2/C28kb ðÞ(2) (Beckmann 1989, pp. 92 /C1/94). Vie `te (1593) was the first to give an exact expression for pby taking n/C304 in the above expression, giving cosb/C30sinb/C301ffiffiffi 2p/C301 2ffiffiffi 2p ; (3) which leads to an INFINITE PRODUCT ofNESTED RADICALS ,2 p/C30ffiffi 1 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C2712ffiffi 12qrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C2712ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C2712ffiffi 12qrs /C1/C1/C1 (4) (Wells 1986, p. 50; Beckmann 1989, p. 95). However, this expression was not rigorously proved to convergeuntil Rudio (1892). A related formula is given by p/C30lim n0/C122nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27.../C27ffiffiffi 2pqrs |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} nvuuuut; (5) where the square root term can be written using the iteration p n/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2pn/C2819+;k9+;72 /C271/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C281 2pn/C2819+;k9+;72r"#2vuut; (6) where p0/C30ffiffiffi 2p (J. Munkhammer). The formula p/C302 lim m0/C12 /C2Xm n/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28n/C281 m !2vuut/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28n m !2vuut2 643 752 /C271 m2vuuuut (7) is also closely related. Another exact FORMULA is M ACHIN’S FORMULA , which is p 4/C304 tan/C2811 59+;k9+;7 /C28tan/C2811 2399+;k9+;7 : (8) There are three other M ACHIN-LIKE FORMULAS ,a s well as other FORMULAS with more terms. An inter- esting INFINITE PRODUCT formula due to Euler which relates pand the nthPRIME pnis p/C302 Q/C12 i/C30n1/C27sin1 2ppn9+;k9+;7 pn2 435(9) /C30 2 Q/C12 i/C30n1/C27(/C281)(pn/C281)=2 pn"# (10) (Blatner 1997, p. 119), plotted below as a function of the number of terms in the product. The AREA and CIRCUMFERENCE of the UNIT CIRCLE are given by A/C30p/C304g1 0ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p dx (11) /C30lim n0/C124 n2Xn k/C300ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin 2/C28k2p (12) and C/C302p/C304g1 0dxffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p (13) /C304g1 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27d xffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p !2vuutdx: (14) The SURFACE AREA and VOLUME of the unit SPHERE are S/C304p (15) V/C304 3p: (16) Beginning with any POSITIVE INTEGER n, round up to the nearest multiple of n/C281;then up to the nearest multiple of n/C282;and so on, up to the nearest multiple of 1. Let f(n) denote the result. Then the ratio lim n0/C12n2 f(n)/C30p (17) (Brown). David (1957) credits this result to Jabotinski and Erdos and gives the more precise asymptotic result f(n)/C30n2 p/C27On4=39+=9+; : (18) The first few numbers in the sequence ff(n)gare 1, 2, 4, 6, 10, 12, 18, 22, 30, 34, ... (Sloane’s A002491). A particular case of the W ALLIS FORMULA gives p 2/C30Y/C12 n/C301(2n)2 (2n/C281)(2n/C271)"# /C302 /C2152 1 /C21534 /C2154 3 /C21556 /C2156 5 /C2157/C1/C1/C1 (19)(Wells 1986, p. 50). This formula can also be written lim n0/C1224n n2n n9+;89+;92/C30plim n0/C12n[G(n)]2 G1 2/C27n9+;k9+;7hi2/C30p; (20) wheren k9+=9+; denotes a BINOMIAL COEFFICIENT andG(x)i s the GAMMA FUNCTION (Knopp 1990). Euler obtained p/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 61/C271 22/C271 32/C271 42/C27/C1/C1/C1 !vuut; (21) which follows from the special value of the R IEMANN ZETA FUNCTION z(2)/C30p2=6:Similar FORMULAS follow from z(2n) for all POSITIVE INTEGERS n. Gregory and Leibniz found p 4/C301/C2813/C2715/C27/C1/C1/C1 (22) (Wells 1986, p. 50), which is sometimes known as G REGORY’S FORMULA or the L EIBNIZ SERIES . The error after the nth term of this series in G REGORY’S FORMULA is larger than (2 n)/C281so this sum converges so slowly that 300 terms are not sufficient to calculate pcorrectly to two decimal places! However, it can be transformed to p/C30X/C12 k/C3013k/C281 4kz(k/C271); (23) where z(z) is the R IEMANN ZETA FUNCTION (Vardi 1991, pp. 157 /C1/158; Flajolet and Vardi 1996), so that the error after kterms is :(3=4)k:/ In 1666, Newton used p/C303 4ffiffiffi 3p /C2724g1=4 0ffiffiffiffiffiffiffiffiffiffiffiffiffi x/C28x2p dx (24) /C303ffiffiffi 3p 4/C27241 12/C281 5 /C21525/C281 28 /C21527/C281 72 /C21529/C28/C1/C1/C1 ! ð25Þ (Wells 1986, p. 50; Borwein et al. 1989). The coeffi- cients can be found from the integral I(x)/C30gffiffiffiffiffiffiffiffiffiffiffiffiffi x/C28x2p dx /C301 4(2x/C281)ffiffiffiffiffiffiffiffiffiffiffiffiffi x/C28x2p /C281 8sin/C281(1/C282x) (26) by taking the series expansion of I(x)/C28I(0) about 0, obtaining I(x)/C302 3x3=2/C2815x5=2/C281 28x7=2/C281 72x9=2/C285 704x11=2/C27/C1/C1/C1 (27) (Sloane’s A054387 and A054388). Using Euler’s CON- VERGENCE IMPROVEMENT transformation gives p 2/C301 2X/C12 n/C300(n!)22n/C271 (2n/C271)!/C30X/C12 n/C300n! (2n/C271)!! /C301/C2713/C271 /C2152 3 /C2155/C271 /C2152 /C2153 3 /C2155 /C2157/C27/C1/C1/C1 (28) /C301/C27131/C27251/C27371/C2749(1/C27... ) ! ! ! ð29Þ (Beeler et al. 1972, Item 120). This corresponds to plugging x/C301=ffiffiffi 2p into the POWER SERIES for the HYPERGEOMETRIC FUNCTION2F1(a;b;c;x); sin/C281xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p /C30X/C12 i/C300(2x)2i/C271(i!)2 2(2i/C271)!/C302F11;1;3 2;x29+;k9+;7 x:(30) Despite the convergence improvement, series (29) converges at only one bit/term. At the cost of a SQUARE ROOT , Gosper has noted that x/C301=2 gives 2 bits/term, 1 9ffiffiffi 3p p/C301 2X/C12 i/C300(i!)2 (2i/C271)!; (31) andx/C30sin(p=10) gives almost 3.39 bits/term, p 5ffiffiffiffiffiffiffiffiffiffiffiffiffif/C272p /C301 2X/C12 i/C300(i!)2 f2i/C271(2i/C271)!; (32) where fis the GOLDEN RATIO . Gosper also obtained p/C303/C271 609+;8 8/C272 /C2153 7 /C2158 /C21539+;8 13/C273 /C2155 10 /C21511 /C2153 /C29+;8 18/C274 /C2157 13 /C21514 /C2153(23/C27... )9+;99+;99+;9 : (33) An infinite sum due to Ramanujan is 1 p/C30X/C12 n/C3002n n9+;89+;9342n/C275 212n/C274(34) (Borwein et al. 1989). Further sums are given in Ramanujan (1913 /C1/14), 4 p/C30X/C12 n/C300(/C281)n(1123/C2721460 n)(2n/C281)!!(4 n/C281)!! 8822n/C27132n(n!)3 ð35Þ and 1p/C30ffiffiffi 8pX /C12 n/C300(1103/C2726390 n)(2n/C281)!!(4 n/C281)!! 994n/C27232n(n!)3 /C30ffiffiffi 8p 9801X/C12 n/C300(4n)!(1103 /C2726390 n) (n!)43964n(36) (Beeler et al. 1972, Item 139; Borwein et al. 1989). Equation (36) is derived from a modular identity of order 58, although a first derivation was not pre-sented prior to Borwein and Borwein (1987). The above series both give p:9801 2206ffiffiffi 2p/C303:14159273001 . . . (37) (Wells 1986, p. 54) as the first approximation and provide, respectively, about 6 and 8 decimal placesper term. Such series exist because of the rationality of various modular invariants. The general form of the series is X /C12 n/C300[a(t)/C27nb(t)](6n)! (3n)!(n!)31 [j(t)]n/C30ffiffiffiffiffiffiffiffiffiffiffiffi /C28j(t)p p; (38) where tis a QUADRATIC FORM DISCRIMINANT ,j(t) is the J-FUNCTION , b(t)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t[1728/C28j(t)]p (39) a(t)/C30b(t) 61/C28E4(t) E6(t)E2(t)/C286 pffiffi tp"#() ; (40) and the Eiare R AMANUJAN- EISENSTEIN SERIES .A CLASS NUMBER pfield involves pth degree ALGEBRAIC INTEGERS of the constants A/C30a(t);B/C30b(t);andC/C30 c(t):The fastest converging series that uses only INTEGER terms corresponds to the largest CLASS NUMBER 1 discriminant of d/C30/C28 163 and was formu- lated by the Chudnovsky brothers (1987). The 163 appearing here is the same one appearing in the fact that epffiffiffiffiffiffi 163p (the R AMANUJAN CONSTANT ) is very nearly anINTEGER . The series is given by 1 p/C3012X/C12 n/C300(/C281)n(6n)!(13591409 /C27545140134 n) (n!)3(3n)!(6403203)n/C271=2 /C30163 /C2158 /C21527 /C2157 /C21511 /C21519 /C215127 6403203=2 /C29X/C12 n/C30013591409 163 /C2152 /C2159 /C2157 /C21511 /C21519 /C215127/C27n ! /C29(6n)! (3n)!(n!)3(/C281)n 6403203n(41) (Borwein and Borwein 1993). This series gives 14 digits accurately per term. The same equation in another form was given by the Chudnovsky brothers (1987) and is used by Mathematica to calculate p (Vardi 1991), p/C30 426880ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10005p A3F21 6;12;56;1;1;B9+;k9+;7 /C28C3F276;32;11 6;2;2;B9+;k9+;7 hi ; (42) where A/C1313591409 (43) B/C13/C281 151931373056000(44) C/C1330285563 1651969144908540723200: (45) The best formula for CLASS NUMBER 2 (largest discriminant /C28427) is 1 p/C3012X/C12 n/C300(/C281)n(6n)!(A/C27Bn) (n!)3(3n)!Cn/C271=2; (46) where A/C13212175710912ffiffiffiffiffiffi 61p /C271657145277365 (47) B/C1313773980892672ffiffiffiffiffiffi 61p /C27107578229802750 (48) C/C135280 236674 /C2730303ffiffiffiffiffiffi61p 9+;k9+;7hi 3 (49) (Borwein and Borwein 1993). This series adds about 25 digits for each additional term. The fastest converging series for CLASS NUMBER 3 corresponds tod/C30/C28 907 and gives 37 /C1/38 digits per term. The fastest converging CLASS NUMBER 4 series corre- sponds to d/C30/C28 1555 and is ffiffiffiffiffiffiffiffiffiffiffi /C28C3p p/C30X/C12 n/C300(6n)! (3n)!(n!)3A/C27nB C3n; (50) where A/C3063365028312971999585426220 /C2728337702140800842046825600ffiffiffi 5p /C27384ffiffiffi5p 108917285511711782004674 . . .ð . . . 36212395209160385656017 /C27487902908657881022 . . . . . . 5077338534541688721351255040ffiffiffi 4p Þ1=2 (51) B/C307849910453496627210289749000 /C273510586678260932028965606400ffiffiffi 5p /C272515968ffiffiffiffiffiffiffiffiffiffiffi3110p 62602083237890016 . . .ð . . . 36993322654444020882161 /C272799650273060444296 . . . . . . 577206890718825190235ffiffiffi5p Þ 1=2(52) C/C30/C28214772995063512240 /C2896049403338648032ffiffiffi5p/C281296ffiffiffi5p 10985234579463550323713318473ð /C274912746253692362754607395912ffiffiffi 5p Þ 1=2; (53) This gives 50 digits per term. Borwein and Borwein (1993) have developed a general ALGORITHM for generating such series for arbitrary CLASS NUMBER . Bellard gives the exotic formula p/C301 740025X/C12 n/C3013P(n) 7n 2n9+;89+;9 2n/C281/C28203792802 6643 775; (54) where P(n)/C13/C28885673181 n 5/C273125347237 n4 /C282942969225 n3/C271031962795 n2 /C28196882274 n/C2710996648 : (55) A complete listing of Ramanujan’s series for 1 =p found in his second and third notebooks is given by Berndt (1994, pp. 352 /C1/354), 4 p/C30X/C12 n/C300(6n/C271)1 29+;k9+;73 n 4n(n!)3(56) 16 p/C30X/C12 n/C300(42n/C275)1 29+;k9+;73 n (64)n(n!)3(57) 32 p/C30X/C12 n/C30042ffiffiffi 5p n/C275ffiffiffi5p /C2730n/C2819+=9+; 1 29+;k9+;73 n (64)n(n!)3 /C2ffiffiffi 5p /C281 2 !8n (58) 27 4p/C30X/C12 n/C300(15n/C272)1 29+;k9+;7 n139+;k9+;7 n239+;k9+;7 n (n!)32 279+;k9+;7n (59) 15ffiffiffi 3p 2p/C30X/C12 n/C300(33n/C274)1 29+;k9+;7 n139+;k9+;7 n239+;k9+;7 n (n!)34 1259+;k9+;7n (60) 5ffiffiffi 5p 2pffiffiffi3p/C30X/C12 n/C300(11n/C271)1 29+;k9+;7 n169+;k9+;7 n569+;k9+;7 n (n!)34 1259+;k9+;7n (61) 85ffiffiffiffiffiffi 85p 18pffiffiffi3p/C30X/C12 n/C300(133n/C278)1 29+;k9+;7 n169+;k9+;7 n569+;k9+;7 n (n!)34 859+;k9+;7n (62) 4 p/C30X/C12 n/C300(/C281)n(20n/C273)1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)322n/C271(63) 4 pffiffiffi 3p/C30X/C12 n/C300(/C281)n(28n/C273)1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)33n4n/C271(64) 4 p/C30X/C12 n/C300(/C281)n(260n/C2723)1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)3(18)2n/C271(65) 4 pffiffiffi 5p/C30X/C12 n/C300(/C281)n(644n/C2741)1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)35n(72)2n/C271(66) 4 p/C30X/C12 n/C300(/C281)n(21460 n/C271123)129+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)3(882)2n/C271(67) 2ffiffiffi 3p p/C30X/C12 n/C300(8n/C271)n1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)39n(68) 1 2pffiffiffi 2p/C30X/C12 n/C300(10n/C271)n1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)392n/C271(69) 1 3pffiffiffi 3p/C30X/C12 n/C300(40n/C273)1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)3(49)2n/C271(70) 2 pffiffiffiffiffiffi 11p/C30X/C12 n/C300(280n/C2719)1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)3(99)2n/C271(71) 1 2pffiffiffi 2p/C30X/C12 n/C300(26390 n/C271103)1 29+;k9+;7 n149+;k9+;7 n349+;k9+;7 n (n!)3(99)4n/C272: (72) These equations were first proved by Borwein and Borwein (1987, pp. 177 /C1/187). Borwein and Borwein (1987b, 1988, 1993) proved other equations of thistype, and Chudnovsky and Chudnovsky (1987) foundsimilar equations for other transcendental constants. Another identity is p 2/C3036 Li21 29+;k9+;7 /C2836 Li2149+;k9+;7 /C2812 Li2189+;k9+;7 /C276L i21 649+;k9+;7 ; (73) where Lnis the POLYLOGARITHM . (73) is equivalent to p2 36/C30X/C12 i/C301ai 2ii2faig/C30[1;/C283;/C282;/C283;1;0] (74) and p2/C3012L21 29+;k9+;7 /C276(ln 2)2(75) (Bailey et al. 1995). ASPIGOT ALGORITHM forpis given by Rabinowitz and Wagon (1995). More amazingly still, a closed form expression giving a DIGIT-EXTRACTION ALGORITHM which produces digits of p(orp2) in base-16 was recently discovered by Bailey et al. (Bailey et al. 1995, Adamchik and Wagon 1997),p/C30X/C12 n/C3004 8n/C271/C282 8n/C274/C281 8n/C275/C281 8n/C276 ! 1 16 !n : (76) This formula, sometimes called the B AILEY- BORWEIN- PLOUFFE ALGORITHM can also be written using the shorthand notation p/C30X/C12 i/C301pi 16i=8bci fpig/C30f4;0;0;/C282;/C281;/C281;0;0g;(77) where fpigis given by the periodic sequence obtained by appending copies of f4;0;0;/C282;/C281;/C281;0;0g(in other words, pi/C13p[(i/C281) (mod 8)] /C271fori/C218) and xbcis the FLOOR FUNCTION . This expression was discovered using the PSLQ ALGORITHM (Ferguson et al. 1999) and is equivalent to p/C30g1 016y/C2816 y4/C282y3/C274y/C284dy: (78) A similar formula was subsequently discovered by Ferguson, leading to a 2-D lattice of such formulas which can be generated by these two formulas. Arelated integral is p/C30 22 7/C28g1 0x4(1/C28x)4 1/C27x2dx (79) (Le Lionnais 1983, p. 22). F. Bellard found the more rapidly converging DIGIT-EXTRACTION ALGORITHM (in HEXADECIMAL ) p/C301 26X/C12 n/C300(/C281)n 210n9+;8 /C2825 4n/C271/C281 4n/C273/C2728 10n/C271 /C2826 10n/C273/C282 10n/C275/C2822 10n/C277/C271 10n/C2799+;9 :(80) This formula can be generalized to p/C30X/C12 k/C3009+;84/C278r 8k/C271/C288r 8k/C272/C284r 8k/C273/C282/C278r 8k/C274 /C281/C272r 8k/C275/C281/C272r 8k/C276/C27r 8k/C2779+;99+;81 169+;9k (81) for any complex value of r(Adamchik and Wagon), giving the Bailey-Borwein-Plouffe algorithm as thespecial case r/C300. Related formulas are p 2/C301 8X/C12 k/C3001 64k9+$=144 (6k/C271)2/C28216 (6k/C272)2/C2872 (6k/C273)2 /C2854 (6k/C274)2/C279 (6k/C275)29+$; (82) and p2/C30X/C12 k/C3001 16k9+$=16 (8k/C271)2/C2816 (8k/C272)2/C288 (8k/C273)2 /C2816 (8k/C274)2/C284 (8k/C275)2/C284 (8k/C276)2/C272 (8k/C277)29+$; (83) (Bailey et al. 1995, Bailey and Plouffe). More amaz- ingly still, S. Plouffe has devised an algorithm to compute the nthDIGIT ofpin any base in O(n3(logn)3) steps. A slew of additional identities due to Ramanujan , Catalan, and Newton are given by Castellanos (1988, pp. 86 /C1/88), including several involving sums of F IBO- NACCI NUMBERS . Ramanujan found X/C12 k/C300(/C281)k(4k/C271)[(2k/C281)!!]3 [(2k)!!]3 /C30X/C12 k/C300(/C281)k(4k/C271)Gk/C271 29+;k9+;7hi3 p3=2[G(k/C271)]3/C302 p(84) (Hardy 1923; Hardy 1924; Hardy 1999, p. 7). Gasper quotes the result p/C3016 3lim x0/C12x1F21 2;2;3;/C28x29+;k9+;7hi/C281 ; (85) where1F2is a GENERALIZED HYPERGEOMETRIC FUNC- TION , and transforms it to p/C30lim x0/C124x1F212;32;32;/C28x29+;k9+;7 ; (86) Fascinating results due to Gosper include lim n0/C12Y2n i/C30np 2 tan/C281i/C3041=p/C301:554682275 . . . (87) and X/C12 n/C3011 n2cos9 np/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2p2/C289p ! /C30/C28p2 12e3 /C30/C280:040948222 . . . (88) Gosper also gives the curious identity 1 eY/C12 n/C3011 3n/C271 !3n/C271=2 /C303 /C21531=24ffiffiffiffiffiffiffiffiffi 1 39+;k9+;7 !r 25=6expg 3/C28pffiffiffi 3p 18/C27ffiffiffi3p 11 39+;k9+;7 12p/C282z?(2) p22 435p 5=6 /C301:01237855722912 . . . (89)Another curious fact is the ALMOST INTEGER ep/C28p/C3019:999099979 . . . ; (90) which can also be written as (p/C2720)i/C30/C280:9999999992 /C280:0000388927 i:/C281 (91) cos(ln( p/C2720)):/C280:9999999992 : (92) Applying COSINE a few more times gives cos(pcos(pcos(ln( p/C2720)))) :/C281/C273:9321609261 /C2910/C2835: (93) /pmay also be computed using iterative ALGORITHMS . A quadratically converging ALGORITHM due to Bor- wein is x0/C30ffiffiffi 2p (94) p0/C302/C27ffiffiffi 2p (95) y1/C3021=4(96) and xn/C271/C301 2ffiffiffiffiffixnp/C271 ffiffiffiffiffixnp ! (97) yn/C271/C30ynffiffiffiffiffixnp/C271 ffiffiffiffiffixnp yn/C271(98) pn/C30pn/C281xn/C271 yn/C271: (99) /pndecreases monotonically to pwith pn/C28pB10/C282/C271(100) forn]2:The B RENT- SALAMIN FORMULA is another quadratically converging algorithm which can be used to calculate p:A quadratically convergent algo- rithm for p=ln 2 based on an observation by Salamin is given by defining f(k)/C30k2/C28k=4X/C12 n/C3012/C28kn 2ðÞ"# 2 ; (101) then writing g0/C13f(n) f(2n): (102) Now iterate gk/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2gk/C281/C271 gk/C271 !vuut(103) to obtain p/C302(ln 2) f(n)Y/C12 k/C301gk: (104) A cubically converging ALGORITHM which converges to the nearest multiple of ptof0is the simple iteration fn/C30fn/C281/C27sin(fn/C281) (105) (Beeler et al. 1972). For example, applying to 23 gives the sequence f23;22:1537796 ;21:99186453 ;21:99114858 ;...g; (106) which converges to 7 p:21:99114858 :/ A quartically converging ALGORITHM is obtained by letting y0/C30ffiffiffi 2p /C281 (107) a/C306/C284ffiffiffiffi2;p (108) then defining y n/C271/C301/C28(1/C28y4 n)1=4 1/C27(1/C28y4 n)1=4(109) an/C271/C30(1/C27yn/C271)4an/C2822n/C273yn/C2711/C27yn/C271/C27y2 n/C2719+=9+; : (110) Then p/C30lim n0/C121 an(111) andanconverges to 1 =pquartically with an/C281 pB16 /C2154ne/C282p /C2154n(112) (Borwein and Borwein 1987, Bailey 1988, Borwein et al.1989). This ALGORITHM rests on a MODULAR EQUATION identity of order 4. A quintically converging ALGORITHM is obtained by letting s0/C305ffiffiffi 5p /C2829+;k9+;7 (113) a0/C301 2: (114) Then let sn/C271/C3025 z/C27x z/C271 !2 sn; (115) where x/C305 sn/C281 (116) y/C30(x/C281)2/C277 (117)z/C301 2xy/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2/C284x3p9+;k9+;7hi1=5 : (118) Finally, let an/C271/C30s2 nan/C285n1 2s2 n/C2859+=9+; /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sns2 n/C282sn/C275 ðÞq9+$=9+$; ;(119) then 0Ban/C281 pB16 /C2155ne/C28p5n(120) (Borwein et al. 1989). This ALGORITHM rests on a MODULAR EQUATION identity of order 5. Another ALGORITHM is due to Woon (1995). Define a(0)/C131 and a(n)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27Xn/C281 k/C300a(k)"# 2 :vuut(121) It can be proved by induction that a(n)/C30cscp 2n/C271 ! : (122) Forn/C300, the identity holds. If it holds for n5t;then a(t/C271)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27Xt k/C300cscp 2k/C271 !"# 2vuut; (123) but cscp 2k/C271 ! /C30cotp 2k/C272 ! /C28cotp 2k/C271 ! ; (124) so Xt k/C300cscp 2k/C271 ! /C30cotp 2t/C272 ! : (125) Therefore, a(t/C271)/C30cscp 2t/C272 ! ; (126) so the identity holds for n/C30t/C271 and, by induction, for all NONNEGATIVE n, and lim n0/C122n/C271 a(n)/C30lim n0/C122n/C271sinp 2n/C271 ! /C30lim n0/C122n/C271p 2n/C271sinp 2n/C271 ! p 2n/C271 /C30 plim u 00sin u u/C30 p: (127) Additional series in which p appears are 1 4 pffiffiffi 2p /C301 /C271 3 /C2815 /C2817 /C2719 /C271 11 /C28... (128) 14(p /C283) /C301 2 /C215 3 /C215 4 /C281 4 /C215 5 /C215 6 /C271 6 /C215 7 /C215 8 /C28... (129) p2 8/C301 /C271 32 /C271 52 /C271 72 /C27... (130) (Wells 1986, p. 53). Other iterative ALGORITHMS are the ARCHIMEDES ALGORITHM , which was derived by Pfaff in 1800, and the BRENT- SALAMIN FORMULA . Borwein et al. (1989) discuss pth order iterative algorithms. /psatisfies the INEQUALITY 1/C271 p !p/C271 :3:14097Bp: (131) See also PI References Adamchik, V. and Wagon, S. "A Simple Formula for p:/" Amer. Math. Monthly 104, 852/C1/855, 1997. Adamchik, V. and Wagon, S. "Pi: A 2000-Year Search Changes Direction." http://members.wri.com/victor/arti- cles/pi.html. Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving p;e, and Euler’s Constant." Math. Comput. 50, 275/C1/281, 1988a. Bailey, D. H. "The Computation of pto 29,360,000 Decimal Digit using Borwein’s’ Quartically Convergent Algorithm."Math. Comput. 50, 283/C1 /296, 1988b. Bailey, D. H.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants."Math. Comput. 66, 903/C1 /913, 1997. Beckmann, P. A History of Pi, 3rd ed. New York: Dorset Press, 1989. Beeler, M. et al. Item 140 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 69, Feb. 1972. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Blatner, D. The Joy of Pi. New York: Walker, 1997. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.New York: Wiley, 1987. Borwein, J. M. and Borwein, P. B. "Ramanujan’s Rational and Algebraic Series for 1 =p: /"Indian J. Math. 51, 147/C1/ 160, 1987b.Borwein, J. M. and Borwein, P. B. "More Ramanujan-Type Series for 1 =p:/"I n Ramanujan Revisited. Boston, MA: Academic Press, pp. 359 /C1/374, 1988. Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanu- jan, Modular Equations, and Approximations to Pi, orHow to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201/C1 /219, 1989. Borwein, J. M. and Borwein, P. B. "Class Number Three Ramanujan Type Series for 1 =p:/"J. Comput. Appl. Math. 46, 281/C1/290, 1993. Brown, K. S. "Rounding Up to Pi." http://www.seanet.com/ ~ksbrown/kmath001.htm. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1/98, 1988. Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148/C1/163, 1988. Chudnovsky, D. V. and Chudnovsky, G. V. "Approximations and Complex Multiplication According to Ramanujan." InRamanujan Revisited: Proceedings of the Centenary Con-ference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Ra- nkin). Boston, MA: Academic Press, pp. 375 /C1 /472, 1987. David, Y. "On a Sequence Generated by a Sieving Process." Riveon Lematematika 11,2 6/C1/31, 1957. Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351/C1/369, 1999. Finch, S. "Unsolved Mathematics Problems: The Miraculous Bailey-Borwein-Plouffe Pi Algorithm." http://www.math-soft.com/asolve/plouffe/plouffe.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996.http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Hardy, G. H. "Some Formulae of Ramanujan." Proc. London Math. Soc. (Records of Proceedings at Meetings) 22, xii- xiii, 1924. Hardy, G. H. "A Chapter from Ramanujan’s Note-Book." Proc. Cambridge Philos. Soc. 21, 492/C1 /503, 1923. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Ramanujan, S. "Modular Equations and Approximations to p:/"Quart. J. Pure. Appl. Math. 45, 350/C1/372, 1913 /C1/1914. Sloane, N. J. A. Sequences A054387 and A054388 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 159, 1991. Vie`te, F. Uriorum de rebus mathematicis responsorum, liber VIII, 1593. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. Woon, S. C. "Problem 1441." Math. Mag. 68,7 2/C1 /73, 1995. Pi Heptomino AHEPTOMINO in the shape of the Greek character PI. Pi Wordplay A short mnemonic for remembering the first eight DECIMAL DIGITS ofpis "May I have a large container of coffee?" giving 3.1415926 (Gardner 1959; Gardner 1966, p. 92; Eves 1990, p. 122, Davis 1993, p. 9). "But I must a while endeavour to reckon right" gives nine correct digits (3.1.4159265). A more substantial mne- monic giving 15 digits (3.14159265358979) is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics," originally due to Sir James Jeans (Gardner 1966, p. 92; Cas- tellanos 1988, p. 152; Eves 1990, p. 122; Davis 1993, p. 9; Blatner 1997, p. 112). A slight extension of this adds the phrase "All of thy geometry, Herr Planck, is fairly hard," giving 24 digits in all (3.14159265358979323846264). An even more extensive rhyming mnemonic giving 31 digits is "Now I will a rhyme construct, By chosen words the young instruct. Cunningly devised endea- vour, Con it and remember ever. Widths in circle here you see, Sketched out in strange obscurity." (Note that the British spelling of "endeavour" is required here.)The following stanzas are the first part of a poem written by M. Keith based on Edgar Allen Poe’s "The Raven." The entire poem gives 740 digits; the frag- ment below gives only the first 80 (Blatner 1997, p. 113). Words with ten letters represent the digit 0, and those with 11 or more digits are taken to represent two digits. Poe, E.: Near a Raven. Midnights so dreary, tired and weary. Silently pondering volumes extolling all by-now ob- solete lore. During my rather long nap-the weirdest tap! An ominous vibrating sound disturbing my chamber’s antedoor.‘This,’ I whispered quietly, ‘I ignore.’ Perfectly, the intellect remembers: the ghostly fires, a glittering ember.Inflamed by lightning’s outbursts, windows cast penumbras upon this floor. Sorrowful, as one mis- treated, unhappy thoughts I heeded: That inimitable lesson in elegance–Lenore– Is delighting, exciting... nevermore. An extensive collection of p mnemonics in many languages is maintained by A. P. Hatzipolakis. Other mnemonics in various languages are given by Cas- tellanos (1988) and Blatner (1997, pp. 112 /C1 /118). Keith (1999) considered the set of letters obtained by writing p to base 26 with digits 0 /C30A; 1 /C30B; ..., 25 /C30 Z; so that p /C30D :DRSQLOLYRTRODNLHNQTG ...: Then the sequence of the first Webster-sanctioned n- letter words in this expression is given by o, lo, rod, trod, steel, oxygen, subplot, .... Additional 6-letter words are: prinky, Libyan, and thingy. The positions of the starting letter of the first n-letter words are 6, 5, 11, 10, 6570, 11582, 115042, ....See also PI References Blatner, D. The Joy of Pi. New York: Walker, 1997. Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148 /C1/163, 1988. Davis, D. M. The Nature and Power of Mathematics. Princeton, NJ: Princeton University Press, 1993. Eves, H. An Introduction to the History of Mathematics, 6th ed. Philadelphia, PA: Saunders, 1990. Gardner, M. "Memorizing Numbers." Ch. 11 in The Scien- tific American Book of Mathematical Puzzles and Diver- sions. New York: Simon and Schuster, p. 103, 1959. Gardner, M. "The Transcendental Number Pi." Ch. 8 in Martin Gardner’s New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 91 /C1/102, 1966. Hatzipolakis, A. P. "PiPhilology." http://users.hol.gr/~xpola- kis/piphil.html. Keith, M. "The Pi Code." Word Ways 32, Nov. 1999. Sallows, L. "Base 27: The Key to a New Gematria." Word Ways 26,67/C1/77, May 1993. Piano Mover’s Problem N.B. A detailed online essay by S. Finch was the starting point for this entry. Given an open subset U in n-D space and two compact subsets C0 and C1 of U, where C1 is derived from C0 by a continuous motion, is it possible to move C0 to C1 while remaining entirely inside U? See also MOVING LADDER CONSTANT ,M OVING SOFA CONSTANT References Buchberger, B.; Collins, G. E.; and Kutzler, B. "Algebraic Methods in Geometry." Annual Rev. Comput. Sci. 3,85/C1/ 119, 1988. Feinberg, E. B. and Papadimitriou, C. H. "Finding Feasible Points for a Two-point Body." J. Algorithms 10, 109 /C1/119, 1989. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/sofa/sofa.html. Leven, D. and Sharir, M. "An Efficient and Simple Motion Planning Algorithm for a Ladder Moving in Two-Dimen- sional Space Amidst Polygonal Barriers." J. Algorithms 8, 192 /C1/215, 1987. Picard Variety Let V be a VARIETY , and write G(V) for the set of divisors, Gl(V) for the set of divisors linearly equiva- lent to 0, and Ga(V) for the group of divisors algebraically equal to 0. Then Ga(V)=Gl(V) is called the Picard variety. The A LBANESE VARIETY is dual to the Picard variety. See also ALBANESE VARIETY References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 75, 1980. Picard’s Existence Theorem If f is a continuous function that satisfies the LIPSCHITZ CONDITION ½f(x; t) /C28f(y; t) ½5L½x /C28y½ in a surrounding of (x0 ; t0) /C23VƒR /C29Rn /C30f(x; t):½x /C28 x0 ½Bb;½t /C28t0 ½Ba g; then the differential equation df dx /C30f(x; t) x(t0) /C30x0 has a unique solution x(t) in the interval ½t /C28t0 ½Bd; where d /C30min( a; b=B) ; min denotes the MINIMUM , B /C30sup ½f(t; x) ½; and sup denotes the SUPREMUM . See also LIPSCHITZ CONDITION ,ORDINARY DIFFEREN- TIAL EQUATION Picard’s Great Theorem Every nonconstant ENTIRE FUNCTION attains every complex value with at most one exception (Apostol 1997). Furthermore, every ANALYTIC FUNCTION as- sumes every complex value, with possibly one excep- tion, infinitely often in any NEIGHBORHOOD of an ESSENTIAL SINGULARITY . See also ANALYTIC FUNCTION ,ESSENTIAL SINGULAR- ITY,NEIGHBORHOOD ,PICARD’S LITTLE THEOREM References Apostol, T. M. "Application to Picard’s Theorem." §2.9 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 43 /C1/44, 1997. Krantz, S. G. "Picard’s Great Theorem." §10.5.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 140, 1999. Picard’s Little Theorem Any ENTIRE ANALYTIC FUNCTION whose RANGE omits two points must be a CONSTANT FUNCTION . Of course, an ENTIRE FUNCTION that omits a single point from its range need not be a constant, as illustrated by the function ez ; which is entire but omits the point z /C300 from its range. See also ENTIRE FUNCTION ,PICARD’S GREAT THEOREM References Krantz, S. G. "Picard’s Little Theorem." §10.5.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 140, 1999. Picard’s Theorem PICARD’S GREAT THEOREM Pick’s Formula PICK’S THEOREMPick’s Theorem Let A be the AREA of a simply closed LATTICE POLYGON . Let B denote the number of LATTICE POINTS on the EDGES and I the number of points in the interior of the POLYGON . Then A /C30I /C271 2 B /C281: The FORMULA has been generalized to 3-D and higher dimensions using EHRHART POLYNOMIALS . See also BLICHFELDT’S THEOREM ,EHRHART POLYNO- MIAL ,L ATTICE POINT ,M INKOWSKI CONVEX BODY THEOREM References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 209, 1969. DeTemple, D. "Pick’s Formula: A Retrospective." Math. Notes Washington State Univ. 32, Nov. 1989. Diaz, R. and Robins, S. "Pick’s Formula via the Weierstrass /C212/-Function." Amer. Math. Monthly 102, 431/C1/437, 1995. Ewald, G. Combinatorial Convexity and Algebraic Geome- try.New York: Springer-Verlag, 1996. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 215, 1984. Gru¨nbaum, B. and Shephard, G. C. "Pick’s Theorem." Amer. Math. Monthly 100, 150/C1/161, 1993. Haigh, G. "A ‘Natural’ Approach to Pick’s Theorem." Math. Gaz. 64, 173-, 1980. Hammer, J. Unsolved Problems Concerning Lattice Points. London: Pitman, 1977. Kelley, D. A. "Areas of Simple Polygons." Pentagon 20,3/C1/ 11, 1960. Khan, M. R. "A Counting Formula for Primitive Tetrahedra inZ3:/"Amer. Math. Monthly 106, 525/C1/533, 1999. Morelli, R. "Pick’s Theorem and the Todd Class of a Toric Variety." Adv. Math. 100, 183/C1/231, 1993. Niven, I. and Zuckerman, H. S. "Lattice Points and Poly- gonal Area." Amer. Math. Monthly 74, 1195, 1967. Pick, G. "Geometrisches zur Zahlentheorie." Sitzenber. Lotos (Prague) 19, 311/C1/319, 1899. Steinhaus, H. "O polu figur p //laskich." Przeglad Mat.-Fiz. , 1924. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 96 /C1/98, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 183 /C1/184, 1991. Picone’s Theorem Letf(x) be integrable in [ /C281;1];let (1/C28x2)f(x)b eo f bounded variation in [ /C281;1];letM?denote the least upper bound of ½f(x)(1/C28x2)½in [/C281;1];and let V? denote the total variation of f(x)(1/C28x2)i n[/C281;1]: Given the function F(x)/C30F(/C281)/C27gx 1f(x)dx; then the terms of its L EGENDRE SERIES F(x)/C2X/C12 n/C300anPn(x) an /C301 2(2n /C271)g1 /C281F(x)Pn(x) dx; where Pn(x)isaL EGENDRE POLYNOMIAL , satisfy the inequalities ½anPn(x) ½B8ffiffiffi 2 ps M ?/C27V ? (1 /C28 d2)1 =4 n/C283 =2 for ½x½5 d B1 2(M ?/C27V ?)n /C281 for ½x½518 >< >: for n ]1 (Sansone 1991). See also JACKSON’S THEOREM ,LEGENDRE SERIES References Picone, M. Appunti di Analise Superiore. Naples, Italy, p. 260, 1940. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 203 /C1/205, 1991. PID A popular acronym for "PRINCIPAL IDEAL DOMAIN ." In engineering circles, the acronym PID refers to the "PROPORTIONAL-INTEGRAL-DERIVATIVE METHOD " algo- rithm for controlling systems. See also PRINCIPAL IDEAL DOMAIN ,PRINCIPAL IDEAL RING,PROPORTIONAL- INTEGRAL- DERIVATIVE METHOD Pidduck Polynomial Polynomials /Pk ðx Þ/ which form the SHEFFER SEQUENCE for g(t) /C302t et /C28 1 (1) f ðtÞ¼et /C28 1 et þ 1 ð2Þ and have GENERATING FUNCTION X/C12 k /C300Pk(x) k!tk /C30t 1 /C28 t1 /C27 t 1 /C28 t !x : (3) The first few are P0(x) /C301 P1(x) /C302x /C271 P2(x) /C304x24x /C272 P3(x) /C308x3 /C2712x2 /C2716x /C276: The Pidduck polynomials are related to the MITTAG- LEFFLER POLYNOMIALS Mn(x)by Pn(x) /C301 2(et /C271)Mn(x) (4) (Roman 1984, p. 127). See also MITTAG- LEFFLER POLYNOMIAL ,S HEFFER SEQUENCEReferences Bateman, H. "The Polynomial of Mittag-Leffler." Proc. Nat. Acad. Sci. USA 26, 491 /C1/496, 1940. Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press, p. 38, 1964. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 248, 1981. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Pie Chart A chart made by plotting the numeric values of a set of quantities as a set of adjacent circular wedges with arc lengths proportional to the total amount. All wedges taken together comprise an entire disk. One or more segments are slightly separated from the disk center for emphasis in a so-called "exploded" pie chart. See also BAR CHART ,HISTOGRAM References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 23, 1962. Pie Cutting CIRCLE DIVISION BY LINES,CYLINDER CUTTING ,PAN- CAKE THEOREM ,PIZZA THEOREM Piecewise Circular Curve A curve composed exclusively of circular ARCS . See also ARC,FLOWER OF LIFE,LENS,R EULEAUX POLYGON ,REULEAUX TRIANGLE ,SALINON ,SEED OF LIFE,TRIANGLE ARCS,YIN-YANG References Banchoff, T. and Giblin, P. "On The Geometry Of Piecewise Circular Curves." Amer. Math. Monthly 101, 403 /C1/416, 1994. Piecewise Continuous A function or curve is piecewise continuous if it is CONTINUOUS on all but a finite number of points at which certain matching conditions are sometimes required. See also CONTINUOUS ,CONTINUOUS FUNCTION Pigeonhole Principle DIRICHLET’S BOX PRINCIPLE Pillai’s Conjecture For every k /C211, there exist only finite many pairs of POWERS (p ; p ?) with p and p ? NATURAL NUMBERS and k /C30p ?/C28p:/ References Ribenboim, P. "Catalan’s Conjecture." Amer. Math. Monthly 103, 529 /C1/538, 1996. Pillai’s Theorem Write the exact powers of 2 and 3 in sorted order as 1, 2, 3, 4, 8, 9, 16, 27, 32, ... (Sloane’s A006899), and let unbe the nth term in the sequence. Then un /C271 /C28un tends to infinity nearly as rapidly as un :/ References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Pillai. J. Indian Math. Soc. 19,1/C1/11, 1931. Sloane, N. J. A. Sequences A006899/M0588 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Pilot Vector VECTOR SPHERICAL HARMONIC Pinch Point A singular point such that every NEIGHBORHOOD of the point intersects itself. Pinch points are also called Whitney singularities or branch points. Pincherle Derivative Let x : p(x) 0 xp(x) ; then for any operator T, T ?/C30Tx /C28xT is called the Pincherle derivative of T.IfT is a SHIFT- INVARIANT OPERATOR , then its Pincherle derivative is also a SHIFT-INVARIANT OPERATOR . References Pincherle, S. "Operatori lineari e coefficienti di fattoriali." Alti Accad. Naz. Lincei, Rend. Cl. Fis. Mat. Nat. (6) 18, 417 /C1/519, 1933. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/760, 1973. Pinching Theorem Let g(x) 5f(x) 5h(x) for all x in some OPEN INTERVAL containing a.If lim x 0ag(x) /C30lim x0ah(x) /C30L; then limx0a f(x) /C30L :/ See also LIMIT,SQUEEZING THEOREMPine Cone Number FIBONACCI NUMBER Piriform A plane curve also called the PEG TOP and given by the CARTESIAN equation a4y2 /C30b2x3(2a /C28x) (1) and the parametric curves x /C30a(1 /C27sin t) (2) y /C30b cos t(1 /C27sin t) (3) for t /C23 [/C28p=2; p=2]: It was studied by G. de Long- champs in 1886. The generalization to a QUARTIC 3- D surface x4 /C28x39+=9+; /C27y2 /C27z2 /C300 ; (4) is shown below (Nordstrand). See also BUTTERFLY CURVE ,DUMBBELL CURVE ,EIGHT CURVE ,HEART SURFACE ,PEAR CURVE References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub. p. 71, 1989. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 148 /C1/150, 1972. Nordstrand, T. "Surfaces." http://www.uib.no/people/nfytn/ surfaces.htm. Pisot Constant PISOT- VIJAYARAGHAVAN CONSTANT Pisot-Vijayaraghavan Constant Letube a number greater than 1, laPOSITIVE number, and frac( x)/C13x/C28xbc (1) denote the FRACTIONAL PART ofx, where xbcis the FLOOR FUNCTION . Then for a given l ; the sequence of numbers frac lunðÞ for n /C301, 2, ... is an EQUIDISTRIB- UTED SEQUENCE in the interval (0, 1) when u does not belong to a l/-dependent exceptional set S of MEASURE ZERO (Koksma 1935). Pisot (1938) and Vijayaragha- van (1941) independently studied the exceptional values of u; and Salem (1943) proposed calling such values Pisot-Vijayaraghavan numbers. Pisot (1938) proved that if u is chosen such that there exists a l "0 for which the series X/C12 n/C300sin2( plu)n (2) converges, then u is an ALGEBRAIC INTEGER whose conjugates all (except for itself) have modulus B1; and l is an ALGEBRAIC INTEGER of the FIELD K(u): Vijayaraghavan (1940) proved that the set of Pisot- Vijayaraghavan numbers has infinitely many LIMIT POINTS . Salem (1944) proved that the set of Pisot-Vijayara- ghavan constants is closed. The proof of this theorem is based on the LEMMA that for a Pisot-Vijayaragha- van constant u; there always exists a number l such that 1 5 l B u and the following inequality is satisfied, X/C12 n/C300sin2 plunðÞ5p2(2u /C27 1)2 (u /C28 1)2: (3) The smallest Pisot-Vijayaraghavan constant is given by the POSITIVE ROOT u0 :1 :32372 of x3 /C28x /C281 /C300: (4) This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944). Siegel also identified the next smallest Pisot-Vijayaraghavan constant u1as the root of x4 /C28x3 /C281 /C300 : (5) showed that u1 and u2 are isolated in S, and showed that the roots of each POLYNOMIAL xn x2 /C28x /C2819+=9+; /C27x2 /C281 n /C301; 2; 3; ... (6) xn /C28xn /C271 /C28 1 x2 /C28 1n /C303; 5; 7; ... (7) xn /C28xn /C281 /C28 1 x /C28 1n /C303; 5; 7; ... (8) belong to S, where u0 /C30 f (the GOLDEN MEAN ) is the accumulation point of the set (in fact, the smallest; Le Lionnais 1983, p. 40). Some small Pisot-Vijayaraghavan constants and their POLYNOMIALS are given in the following table. The latter two entries are from Boyd (1977).k number order POLYNOMIAL 0 1.3247179572 3 1 0 -1 -1 1 1.3802775691 4 1 -1 0 0 -1 1.6216584885 16 1 -2 2 -3 2 -2 1 0 0 1 -1 2 - 22-21-1 1.8374664495 20 1 -2 0 1 -1 0 1 -1 0 1 0 -1 01-101-101-1 All the points in S less than f are known (Dufresnoy and Pisot 1955). Each point of S is a limit point from both sides of the set T of SALEM CONSTANTS (Salem 1945). Pisot-Vijayaraghavan constants give rise to ALMOST INTEGERS . For example, the larger the power to which u0is taken, the closer un 0/C28un09+Q9+j ;where xbcis the FLOOR FUNCTION , is to either 0 or 1 (Trott 2000). The powers of u0for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (Sloane’s A051016), and those for which it is closer to 1 are 2, 9, 10, 13, 15,16, 18, 20, 21, 23, ... (Sloane’s A051017). See also A LMOST INTEGER ,E QUIDISTRIBUTED SE- QUENCE ,SALEM CONSTANTS ,W EYL’S CRITERION References Bertin, M. J. and Pathiaux-Delefosse, A. Conjecture de Lehmer et petits nombres de Salem. Kingston: Queen’s Papers in Pure and Applied Mathematics, 1989. Bertin, M. J.; Decomps-Guilloux, A.; Grandet-Hugot, M.; Pathiaux-Delefosse, M.; and Schreiber, J. P. Pisot and Salem Numbers. Basel: Birkha ¨user, 1992. Borwein, P. and Hare, K. G. "Some Computations on Pisot and Salem Numbers." CECM-00:148, 18 May 2000. http:// www.cecm.sfu.ca/preprints/2000pp.html#00:148. Boyd, D. W. "Small Salem Numbers." Duke Math. J. 44, 315/C1/328, 1977. Boyd, D. W. "Pisot and Salem Numbers in Intervals of the Real Line." Math. Comput. 32, 1244 /C1/1260, 1978. Boyd, D. W. "Pisot Numbers in the Neighbourhood of a Limit Point. II." Math. Comput. 43, 593/C1/602, 1984. Boyd, D. W. "Pisot Numbers in the Neighbourhood of a Limit Point. I." J. Number Theory 21,1 7/C1/43, 1985. Dufresnoy, J. and Pisot, C. "E ´tude de certaines fonctions me´romorphes borne ´es sur le cercle unite ´, application a `un ensemble ferme ´d’entiers alge ´briques." Ann. Sci. E ´cole Norm. Sup. 72,6 9/C1/92, 1955. Erdos, P.; Joo´, M.; and Schnitzer, F. J. "On Pisot Numbers." Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 39,95/C1/99, 1997. Katai, I. and Kovacs, B. "Multiplicative Functions with Nearly Integer Values." Acta Sci. Math. 48, 221 /C1/225, 1985. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 38 and 148, 1983. Koksma, J. F. "Ein mengentheoretischer Satz u¨ber die Gleichverteilung modulo Eins." Comp. Math. 2, 250 /C1/ 258, 1935. Pisot, C. "La re´partition modulo 1 et les nombres alge´bri- ques." Annali di Pisa 7, 205 /C1/248, 1938. Salem, R. "Sets of Uniqueness and Sets of Multiplicity." Trans. Amer. Math. Soc. 54, 218 /C1/228, 1943. Salem, R. "A Remarkable Class of Algebraic Numbers. Proof of a Conjecture of Vijayaraghavan." Duke Math. J. 11, 103 /C1/108, 1944. Salem, R. "Power Series with Integral Coefficients." Duke Math. J. 12, 153 /C1/172, 1945. Siegel, C. L. "Algebraic Numbers whose Conjugates Lie in the Unit Circle." Duke Math. J. 11, 597 /C1/602, 1944. Sloane, N. J. A. Sequences A051016 and A051017 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Vijayaraghavan, T. "On the Fractional Parts of the Powers of a Number, II." Proc. Cambridge Phil. Soc. 37, 349 /C1/357, 1941. Pistol A4- POLYHEX . References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, p. 147, 1978. Pitchfork Bifurcation Let f : R /C29R 0 R be a one-parameter family of C3 maps satisfying f(/C28x; m) /C30/C28f(x; m) (1) @f @x"# m/C300; x/C300/C300 (2) @2f @x @ m"# 0 ; 0> 0 (3) @3f @ m3"# m/C300 ; x/C300B0: (4)(Actually, condition (1) can be relaxed slightly.) Then there are intervals having a single stable fixed point and three fixed points (two of which are stable and one of which is unstable). This BIFURCATION is called a pitchfork bifurcation. An example of an equation displaying a pitchfork bifurcation is ˙x /C30 mx /C28x3 (5) (Guckenheimer and Holmes 1997, p. 145). See also BIFURCATION ,TRANSCRITICAL BIFURCATION References Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. New York: Springer-Verlag, pp. 145 and 149 /C1/150, 1997. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 31, 1990. Pivot Theorem If the VERTICES A, B, and C of TRIANGLE DABC lie on sides QR, RP, and PQ of the TRIANGLE DPQR ; then the three CIRCUMCIRCLES CBP , ACQ , and BAR have a common point X. In extended form, this theorem becomes M IQUEL’S THEOREM . See also CIRCUMCIRCLE ,C LIFFORD’S CIRCLE THEO- REM,MIQUEL’S THEOREM References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. New York: Random House, pp. 61 /C1/62, 1967. Forder, H. G. Geometry. London: Hutchinson, p. 17, 1960. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 184, 1991. Pivoting The element in the diagonal of a matrix by which other elements are divided in an algorithm such as GAUSS- JORDAN ELIMINATION is called the pivot ele- ment. Partial pivoting is the interchanging of rows and full pivoting is the interchanging of both rows and columns in order to place a particularly "good" element in the diagonal position prior to a particular operation. See also GAUSS- JORDAN ELIMINATION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 29 /C1/30, 1992. Pizza Theorem If a circular pizza is divided into 8, 12, 16, ...slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal. There is also a second pizza theorem. This one gives the VOLUME of a pizza of thickness a and RADIUS z, pizza : Place (Digit) DIGIT Place (Field) A place n of a NUMBER FIELD k is an ISOMORPHISM class of field maps k onto a dense subfield of a nondiscrete locally compact FIELD kn :/ In the function field case, let F be a function field of algebraic functions of one variable over a FIELD K. Then by a place in F, we mean a subset p of F which is the IDEAL of nonunits of some VALUATION RING O over K. References Chevalley, C. Introduction to the Theory of Algebraic Func- tions of One Variable. Providence, RI: Amer. Math. Soc., p. 2, 1951. Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/549, 1996. van der Waerden, B. L. Algebra, 2 vols. New York: Springer- Verlag, 1991. Place (Game) For n players, n /C281 games are needed to fairly determine first place, and n /C281 /C271g(n /C281) are needed to fairly determine first and second place. Place (Riemann Sphere) The word "place" has a special meaning in complex variables, where it roughly corresponds to a point in the COMPLEX PLANE (except that it reflects the Riemann sheet structure imposed by whatever func- tion is under discussion). For example, if the function in question is ln z; then 1 and e2 pi are different places.Plaindrome A plaindrome is a number whose HEXADECIMAL digits are in nondecreasing order. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, ... (Sloane’s A023757). The first few which arenot plaindromes are 16, 32, 33, 48, 49, 50, 64, ...,corresponding to 10 16;2016;2116;3016;3116;3216;6416; .... See also DIGIT,H EXADECIMAL ,K ATADROME ,M ETA- DROME ,NIALPDROME References Sloane, N. J. A. Sequences A023757 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." M ATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Plaited Polyhedron There exist POLYHEDRA which can be plaited (braided). Examples include a plaited CUBE and plaited ICOSAHEDRON illustrated above (Pargetter 1959, Wells 1991). In the above figures, heavy lines indicate cuts, thin lines indicate folds, and polygons labeled " O" are placed over polygons labeled " U." References Gorham, J. Plaited Crystal Models. 1888. Pargetter, A. R. "Plaited Polyhedra." Math. Gaz. 43,8 8/C1/ 101, 1959. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 160, 1991. Planar Bubble Problem BUBBLE Planar Connected Graph A planar connected graph is a GRAPH which is both planar and connected. The numbers of planar con- nected graphs with n /C301, 2, ... nodes are 1, 1, 1, 2, 6, 20, 99, ... (Sloane’s A003094; Steinbach 1990, p. 131). A subset of planar 3-connected graphs are called POLYHEDRAL GRAPHS . The following table gives the numbers of planar connected graphs having minimal degrees of at least k. k Sloane n /C301, 2, 3, ... 2 A054381 0, 0, 1, 3, 10, 49, 332, ... The numbers of planar connected graphs with n /C301, 2, ... edges are 1, 1, 3, 5, 12, 30, 79, 227, 709, 2318, ... (Sloane’s A046091). See also CONNECTED GRAPH ,PLANAR GRAPH ,POLY- HEDRAL GRAPH ,POLYNEMA References Sloane, N. J. A. Sequences A003094/M1652, A046091, and A054381 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Steinbach, P. Field Guide to Simple Graphs. Albuquerque, NM: Design Lab, 1990. Planar Distance For n points in the PLANE , there are at least N1 /C30ffiffiffiffiffiffiffiffiffiffiffi n /C283 4q /C2812 different DISTANCES . The minimum DISTANCE can occur only 53n /C286 times, and the MAXIMUM DISTANCE can occur 5n times. Furthermore, no DISTANCE can occur as often as N2 /C301 4 n 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8n /C287p9+;k9+;7 Bn3 =2 ffiffiffi 2p/C28n 4 times. Finally, no set of n /C216 points in the PLANE can determine only ISOSCELES TRIANGLES . See also DISTANCE References Honsberger, R. "The Set of Distances Determined by n Points in the Plane." Ch. 12 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 111 /C1/135, 1976.Planar Graph AGRAPH is planar if it can be drawn in a PLANE without EDGES crossing (i.e., it has CROSSING NUMBER 0). The number of planar graphs with n/C301, 2, ... nodes are 1, 2, 4, 11, 33, 142, ... (Sloane’s A005470; Wilson 1975, p. 162).There are a number of efficient algorithms for planarity testing, which are unfortunately all difficult to implement. Most are based on the on 3ðÞ algorithm of Auslander and Parter (1961; Skiena 1990, p. 247). One implementation is given by PlanarQ [g] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mandBBDiscreteMath‘ ), which however should be trusted for only versions 4.1 and higher. Only planar graphs have DUALS and if Gis planar, then Ghas VERTEX DEGREE 55:A graph is planar IFF it has a COMBINATORIAL DUAL GRAPH (Harary 1994, p. 115). Any planar graph has a GRAPH EMBEDDING as aPLANAR STRAIGHT LINE GRAPH where edges do not intersect (Fa ´ry 1948; Bryant 1989; Skiena 1990, pp. 100 and 251; Scheinerman and Wilf 1994).C OMPLETE GRAPHS are planar only for n54:The complete BIPARTITE GRAPH K(3;3) is nonplanar. More generally, Kuratowski proved in 1930 that a graph is planar IFFit does not contain within it any graph which can be CONTRACTED to the pentagonal graph K(5) or the hexagonal graph K(3;3):K5can be decomposed into a union of two planar graphs, givingit a " DEPTH "o fE(K5)/C302:Simple CRITERIA for deter- mining the depth of graphs are not known. Beineke and Harary (1964, 1965) have shown that if nf4 (mod 6), then E(Kn)/C301 6(n/C277)jk : The DEPTHS of the graphs Knforn/C304, 10, 22, 28, 34, and 40 are 1, 3, 4, 5, 6, and 7 (Meyer 1970). All TREES are planar, as is a CYCLE GRAPH ,GRID GRAPH ,o r WHEEL GRAPH . Every planar graph on nine vertices has a nonplanar complement (Battle et al. 1962; Skiena 1990, p. 250).The following table gives the numbers of planar graphs having minimal degrees of at least k. k Sloane n /C301, 2, 3, ... 2 A049370 0, 0, 1, 3, 10, 50, 335, ... 3 A049371 0, 0, 0, 1, 2, 9, 46, 386, ... 4 A049372 0, 0, 0, 0, 0, 1, 1, 4, 14, 69, ... 5 A049373 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 5, ... See also BARNETTE’S CONJECTURE ,COMPLETE GRAPH , DUAL GRAPH ,FABRY IMBEDDING ,INTEGRAL DRAWING , KURATOWSKI REDUCTION THEOREM ,O UTPLANAR GRAPH ,P LANAR CONNECTED GRAPH ,P LANAR STRAIGHT LINE GRAPH ,POLYHEDRAL GRAPH ,STEI- NITZ’S THEOREM ,UTILITY GRAPH References Auslander, L. and Parter, S. "On Imbedding Graphs in the Sphere." J. Math. Mechanics 10, 517 /C1/523, 1961. Battle, J.; Harary, F.; and Kodama, Y. "Every Planar Graph with Nine Points has a Nonplanar Complement." Bull. Amer. Math. Soc. 68, 569 /C1/571, 1962. Beineke, L. W. and Harary, F. "On the Thickness of the Complete Graph." Bull. Amer. Math. Soc. 70, 618 /C1/620, 1964. Beineke, L. W. and Harary, F. "The Thickness of the Complete Graph." Canad. J. Math. 17, 850 /C1/859, 1965. Booth, K. S. and Lueker, G. S. "Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity using PQ-Tree Algorithms." J. Comput. System Sci. 13, 335 /C1/379, 1976. Bryant, V. W. "Straight Line Representation of Planar Graphs." Elem. Math. 44,64/C1/66, 1989. Cai, J.; Han, X.; and Tarjan, R. "New Solutions to Four Planar Graph Problems." Technical Report. New York University, 1990. Di Battista, G.; Eades, P.; Tamassia, R.; and Tollis, I. G. Graph Drawing: Algorithms for the Visualization of Graphs. Englewood Cliffs, NJ: Prentice-Hall, 1998. Eades, P. and Tamassia, R. "Algorithms for Drawing Graphs: An Annotated Bibliography." Technical Report CS-89 /C1/09. Department of Computer Science. Providence, RI: Brown University, Feb. 1989. Even, S. Graph Algorithms. Rockville, MD: Computer Science Press, 1979. Fa´ry, I. "On Straight Line Representations of Planar Graphs." Acta Sci. Math. (Szeged) 11, 229 /C1/233, 1948. Friedman, E. "Large Regular Graphs with Small Diameter." http://www.stetson.edu/~efriedma/planar/. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 91 /C1/94, 1984. Harary, F. "Planarity." Ch. 11 in Graph Theory. Reading, MA: Addison-Wesley, pp. 102 /C1/125, 1994. Hopcroft, J. and Tarjan, R. "Efficiency Planarity Testing." J. ACM 21, 549 /C1/568, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Meyer, J. "L’e´paisseur des graphes completes K34 et K40 :/" J. Comp. Th. 9, 1970. Schneinerman, E. and Wilf, H. S. "The Rectilinear Crossing Number of a Complete Graph and Sylvester’s ‘Four Point’Problem of Geometric Probability." Amer. Math. Monthly 101, 939 /C1/943, 1994. Skiena, S. "Planar Graphs." §6.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 247 /C1/ 253, 1990. Sloane, N. J. A. Sequences A005470/M1252, A049370, A049371, A049372, and A049373 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Steinbach, P. Field Guide to Simple Graphs. Albuquerque, NM: Design Lab, 1990. Stony Brook Algorithm Repository. §.4.12. "Detection and Embedding." http://www.cs.sunysb.edu/~algorith/files/pla- nar-drawing.shtml. Wagon, S. "Coloring Planar Maps and Graphs." Ch. 24 in Mathematica in Action, 2nd ed. New York: Springer- Verlag, pp. 507 /C1/537, 1999. Whitney, H. "Non-Separable and Planar Graphs." Trans. Amer. Math. Soc. 34, 339/C1/362, 1932. Whitney, H. "Planar Graphs." Fund. Math. 21,7 3/C1/84, 1933. Wilson, R. J. Introduction to Graph Theory. London: Long- man, 1975. Planar Point A point pon a REGULAR SURFACE M/C23R3is said to be planar if the G AUSSIAN CURVATURE K(p)/C300 and S(p)/C300 (where Sis the SHAPE OPERATOR ), or equiva- lently, both of the PRINCIPAL CURVATURES k1andk2 are 0. See also ANTICLASTIC ,E LLIPTIC POINT ,G AUSSIAN CURVATURE ,H YPERBOLIC POINT ,PARABOLIC POINT , SYNCLASTIC References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 375, 1997. Planar Polygon Flat polygons embedded in 3-D space can be trans- formed into a congruent planar polygon as follows. First, translate the starting vertex to (0, 0, 0) bysubtracting it from each vertex of the polygon. Thenfind the normal nto the polygon by taking the CROSS PRODUCT of the first and last vertices. Now, let Abe the rotation matrix for E ULER ANGLES c;u;andf;and solve Anx nyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28n2 x/C28n2yq2 643 75/C300 012 435 (1) for cos cand cos u(after first expressing sines in terms of cosines using cos x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin 2xp :The result is f¼9nyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 xþn2yq ð2Þ u /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28n2 x /C28n2yq : (3) The signs are chosen as follows: c /C30cos/C281 /C28sgn(nx)nyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 x /C27 n2yq2 435 (4) u /C30cos /C281 /C28sgn nxnz ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28n2 x /C28n2yqhi : (5) Plugging these back in and applying to the original polygon then gives a polygon whose vertices all have one component zero. This component can then be dropped. The only special cases which need to be taken into account are nzjj/C301 ; in which case the polygon is parallel to the xy-plane and the third components can be immediately dropped. The second occurs when nx /C300; in which case there is no component of the normal vector along the X-AXIS ,so the Euler rotation will not work. However, simply picking a different starting vertex from which to calculate the normal resolves this degenerate case. See also POLYGON Planar Space Let j1 ; j2 ðÞ be a locally EUCLIDEAN coordinate sys- tem. Then ds2 /C30dj2 1 /C27dj22 : (1) Now plug in d j1 /C30@ j1 @x1dx1 /C27@ j1 @x2dx2 (2) d j2 /C30@ j2 @x1dx1 /C27@ j2 @x2dx2 (3) to obtain ds2 /C30@ j1 @x1 !2 /C27@ j2 @x1 !22 435dx 2 1 /C272@ j1 @x1@ j1 @x2/C27@ j2 @x1@ j2 @x2"# dx1 dx2 /C27@ j1 @x2 !2 /C27@ j2 @x2 !22 435dx 2 2 : (4) Reading off the COEFFICIENTS from ds2 /C30g11 dx21 /C272g12 dx1 dx2 /C27g22(dx2)2 (5) gives g11 /C30@ j1 @x1 !2 /C27@ j2 @x1 !2 (6)g12 /C30@ j1 @x1@ j1 @x2/C27@ j2 @x1@ j2 @x2(7) g22 /C30@ j1 @x2 !2 /C27@ j2 @x2 !2 : (8) Making a change of coordinates x1 ; x2 ðÞ 0 x?1 ; x?2 ðÞ gives g?11 /C30@ j1 @x?1 !2 /C27@ j2 @x?1 !2 /C30@ j1 @x1@x1 @x ?1/C27@ j1 @x2@x2 @x ?1 !2 /C27@ j2 @x1@x1 @x?1/C27@ j2 @x2@x2 @x?1 !2 /C30g11@x1 @x?1 !2 /C272g12@x1 @x?1@x2 @x?1/C27g22@x2 @x ?1 !2 (9) g ?12 /C30@ j1 @x1@x1 @x?1@ j1 @x2@x2 @x?2/C27@ j2 @x1@x1 @x?1@ j2 @x2@x2 @x?2 /C30g12@x1 @x?1@x2 @x?2(10) g ?22 /C30g11@x1 @x?1 !2 /C272g12@x1 @x?2@x2 @x?2/C27g22@x2 @x?2 !2 : (11) Planar Straight Line Graph A GRAPH EMBEDDING of a PLANAR GRAPH in which only straight line segments are used to connect the VERTICES .Fa´ry (1948) showed that every PLANAR GRAPH has an EMBEDDING which is a planar straight line graph with noncrossing edges (Bryant 1989; Skiena 1990, pp. 100 and 251; Schneinerman and Wilf 1994). de Fraysseix et al. (1988) give an algorithm for constructing a planar straight line for a graph of order nby placing the vertices on a (2 n/C28 4)/C29(n/C282) grid (Skiena 1990, p. 251). See also PLANAR GRAPH ,R ECTILINEAR CROSSING NUMBER References Bryant, V. W. "Straight Line Representation of Planar Graphs." Elem. Math. 44,6 4/C1/66, 1989. de Fraysseix, H.; Pach, J; and Pollack, R. "Small Sets Supporting Fa ´ry Embeddings of Planar Graphs." Proc. of the 20th Symposium on the Theory of Computing. ACM, pp. 426 /C1/433, 1988. Fa´ry, I. "On Straight Line Representations of Planar Graphs." Acta Sci. Math. (Szeged( 11, 229/C1/233, 1948. Schneinerman, E. and Wilf, H. S. "The Rectilinear Crossing Number of a Complete Graph and Sylvester’s ‘Four Point’ Problem of Geometric Probability." Amer. Math. Monthly 101, 939/C1/943, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Plancherel’s Theorem g/C12 /C28/C12f(x)¯g(x) dx /C30g/C12 /C28/C12F(s) ¯G(s) ds ; where F(s) /C13F[f(x)] and F denotes a FOURIER TRANSFORM and ¯z is the COMPLEX CONJUGATE .Iff and g are real g/C12 /C28/C12f(x)g(/C28x) dx /C30g/C12 /C28/C12F(s)G(s)ds: See also FOURIER TRANSFORM ,PARSEVAL’S THEOREM Planck’s Radiation Function The function f(x)/C3015 p41 x5(e1=x/C281); (1) which is normalized so that g/C12 0f(x)dx/C301: (2) The first and second RAW MOMENTS are m?1/C3030z(3) p4(3) m?2/C305 2p2; (4) but higher order raw moments do not exist since the corresponding integrals do not converge.It has a MAXIMUM atx:0:201405 ;where f?(x)/C305x/C28e1=x(5x/C281) x7(e1=x/C281)2/C300; (5) and inflection points at x:0:11842 and x:0:283757 ; wherefƒ(x)/C30e1=x1/C27e1=x9+=9+; /C276xe1=x/C2819+=9+; e1=x(5x/C282)/C285x9+$9+% e1=x/C281 ðÞ3x9 /C300: (6) References Abramowitz, M. and Stegun, C. A. (Eds.). "Planck’s Radia- tion Function." §27.2 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 999, 1972. Plane A plane is a 2-D DOUBLY RULED SURFACE spanned by two linearly independent vectors. The generalization of the plane to higher DIMENSIONS is called a HYPER- PLANE . The angle between two intersecting planes is known as the DIHEDRAL ANGLE . In intercept form, a plane passing through the points(a;0;0);(0;b;0) and (0 ;0;c) is given by x a/C27y b/C27z c/C301: (1) The equation of a plane PERPENDICULAR to the NONZERO VECTOR ˆn/C30(a;b;c) through the point (x0;y0;z0)i s a b c2 435 /C215x/C28x 0 y/C28y0 z/C28z02435/C30a(x/C28x 0)/C27b(y/C28y0)/C27c(z/C28z0)/C300; (2) so ax/C27by/C27cz/C27d/C300: (3) where d/C13/C28ax0/C28by0/C28cz0: (4) A plane specified in this form therefore has x-,y-, and z-intercepts at x/C30/C28d a(5) y /C30/C28d b (6) z /C30/C28d c; (7) and lies at a DISTANCE h /C30djjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2 /C27 c2p (8) from the ORIGIN . The plane through P1 and parallel to (a1 ; b1 ; c1) and (a2 ; b2 ; c2)is x /C28x1y /C28y1z /C28z1 a1 b1 c1 a2 b2 c29+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C300: (9) The plane through points P 1and P2parallel to direction (a; b; c)is x /C28x1 y /C28y1 z /C28z1 x2 /C28x1y2 /C28y1z2 /C28z1 abc9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C300: (10) The three-point form is xyz 1 x 1y1z11 x2y2z21 x3y3z319+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C30x /C28x 1 y /C28y1 z /C28z1 x2 /C28x1y2 /C28y1z2 /C28z1 x3 /C28x1y3 /C28y1z3 /C28z19+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C300 : (11) The POINT-PLANE DISTANCE from a point (x0 ; y0 ; z0)to a plane ax /C27by /C27cz /C27d /C300 (12) is D /C30ax0 /C27 by0 /C27 cz0 /C27 d 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2 /C27 c2p : (13) The DIHEDRAL ANGLE between the planes A1x /C27B1y /C27C1z /C27D1 /C300 (14) A2x /C27B2y /C27C2z /C27D2 /C300 (15) which have normal vectors N1 /C30(A1 ; B1 ; C1) and N2 /C30(A2 ; B2 ; C2) is simply given via the DOT PRO- DUCT of the normals, cos u /C30N1/C215 N2 /C30A1A2 /C27 B1B2 /C27 C1C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 1 /C27 B21 /C27 C21pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A22 /C27 B22 /C27 C22p : (16) In order to specify the relative distances of n /C211 points in the plane, /1 þ 2ðn /C282Þ¼2n /C283/ coordinates are needed, since the first can always be placed at (0, 0) and the second at (x; 0); where it defines the X- AXIS. The remaining n /C282 points need two coordi- nates each. However, the total number of distances isnC2 /C30n 29+;89+;9 /C30n! 2!(n /C28 2)! /C301 2 n(n /C281); (17) wheren k9+=9+; is a BINOMIAL COEFFICIENT , so the distances between points are subject to m relationships, where m /C1312 n(n /C281) /C28(2n /C283) /C3012(n /C282)(n /C283): (18) For n /C302 and n /C303, there are no relationships. However, for a QUADRILATERAL (with n /C304), there is one (Weinberg 1972). It is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). In 4-D, it is possible for four planes to intersect in exactly one point. For every set of n points in the plane, there exists a point O in the plane having the property such that every straight line through O has at least 1/3 of the points on each side of it (Honsberger 1985). Every RIGID MOTION of the plane is one of the following types (Singer 1995): 1.ROTATION about a fixed point P. 2.TRANSLATION in the direction of a line l. 3.REFLECTION across a line l. 4. Glide-reflections along a line l. Every RIGID MOTION of the hyperbolic plane is one of the previous types or a 5. Horocycle rotation. See also ARGAND PLANE ,C OMPLEX PLANE ,C OX’S THEOREM ,D IHEDRAL ANGLE ,D IRECTOR ,D OUBLY RULED SURFACE ,E LLIPTIC PLANE ,F ANO PLANE , HYPERPLANE ,ISOCLINAL PLANE ,LINE-PLANE INTER- SECTION ,M EDIATOR ,M OUFANG PLANE ,NIRENBERG’S CONJECTURE ,N ORMAL SECTION ,POINT- PLANE DIS- TANCE ,PROJECTIVE PLANE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 208 /C1/209, 1987. Eisenberg, B. and Sullivan, R. "Random Triangles n Dimensions." Amer. Math. Monthly 103, 308/C1/318, 1996. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 189 /C1/191, 1985. Kern, W. F. and Bland, J. R. "Lines and Planes in Space." §4 inSolid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 9 /C1/12, 1948. Singer, D. A. "Isometries of the Plane." Amer. Math. Monthly 102, 628/C1/631, 1995. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972. Plane Chart EQUIRECTANGULAR PROJECTION Plane Curve ACURVE which lies in a single PLANE . A plane curve may be closed or open. Curves which are interesting for some reason and whose properties have therefore been investigates are called "special" curves (Lawr- ence 1972). Some of the most common open curves are the LINE, PARABOLA , and HYPERBOLA , and some of the most common closed curves are the CIRCLE and ELLIPSE . See also ALGEBRAIC CURVE ,CURVE ,SPACE CURVE , SPHERICAL CURVE References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 30, 1959. Gray, A. "Famous Plane Curves." Ch. 3 in Modern Differ- ential Geometry of Curves and Surfaces with Mathema- tica, 2nd ed. Boca Raton, FL: CRC Press, pp. 49 /C1/74, 1997. Hilbert, D. and Cohn-Vossen, S. "Plane Curves." §1in Geometry and the Imagination. New York: Chelsea, pp. 1 /C1/7, 1999. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, 1961. MacTutor History of Mathematics Archive. http://www- groups.dcs.st-and.ac.uk/~history/Curves/Curves.html. Weisstein, E. W. "Plane Curves." MATHEMATICA NOTEBOOK CURVES.M . Yates, R. C. A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, 1947. Plane Cutting PLANE DIVISION BY CIRCLES ,P LANE DIVISION BY ELLIPSES ,PLANE DIVISION BY LINES Plane Division by Circles Consider n intersecting CIRCLES . The maximal num- ber of regions into which these divide the PLANE are N(n) /C30n2 /C28n /C272 ; giving values for n /C301, 2, ... of 2, 4, 8, 14, 22, 32, 44, 58, ... (Sloane’s A014206). See also ARRANGEMENT ,CIRCLE ,CIRCLE DIVISION BY LINES,PLANE DIVISION BY ELLIPSES ,PLANE DIVISION BY LINES,SPACE DIVISION BY SPHERES References Indiana School Mathematics J. 14, No. 4, p. 4, 1979. Konhauser, J. D. E.; Velleman, D.; and Wagon, S. Which Way Did the Bicycle Go? And Other Intriguing Mathema- tical Mysteries. Washington, DC: Math. Assoc. Amer., p. 177, 1996. Problem Q736. Parabola 24, 22, 1988. Sloane, N. J. A. Sequences A014206 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html.Yaglom, A. M. and Yaglom, I. M. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, pp. 102 /C1/106, 1987. Plane Division by Ellipses Consider n intersecting ELLIPSES . The maximal num- ber of regions into which these divide the PLANE are N(n) /C302n2 /C282n /C272 /C302(n2 /C28n /C271); giving values for n /C301, 2, ... of 2, 6, 14, 26, 42, 62, 86, 114, .... See also ARRANGEMENT ,CIRCLE DIVISION BY LINES, ELLIPSE ,PLANE DIVISION BY CIRCLES ,PLANE DIVI- SION BY LINES References Problem Q607. Parabola 20, 27, 1984. Plane Division by Lines The maximal number of regions into which n lines divide a PLANE are N(n) /C301 2n2 /C27n /C2729+=9+; which, for n /C301, 2, ...gives 2, 4, 7, 11, 16, 22, ... (Sloane’s A000124), the same maximal number of regions into which a circle can be divided by n lines. See also ARRANGEMENT ,CIRCLE DIVISION BY LINES, LINE,PLANE DIVISION BY CIRCLES ,PLANE DIVISION BY ELLIPSES References Sloane, N. J. A. Sequences A000124/M1041 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Plane Geometry That portion of GEOMETRY dealing with figures in a PLANE , as opposed to SOLID GEOMETRY . Plane geome- try deals with the CIRCLE , LINE, POLYGON , etc. See also CONSTRUCTIBLE POLYGON ,GEOMETRIC CON- STRUCTION ,GEOMETRY ,SOLID GEOMETRY ,SPHERICAL GEOMETRY References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account ofIts Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. Cambridge, England, 1914. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., 1967. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Dixon, R. Mathographics. New York: Dover, 1991. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, 1928. Fuhrmann, W. Synthetische Beweise Planimetrische Sa ¨tze. Berlin, 1890. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, 1913. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 1: Books I and II. New York: Dover, 1956. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 2: Books III-IX. New York: Dover, 1956. Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 3: Books X-XIII. New York: Dover, 1956. Henderson, D. W. Experiencing Geometry: On Plane and Sphere. Englewood Cliffs, NJ: Prentice-Hall, 1995. Hilbert, D. The Foundations of Geometry. Chicago, IL: Open Court, 1980. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,1/C1 /295, 1998. Klee, V. "Some Unsolved Problems in Plane Geometry." Math. Mag. 52, 131/C1/145, 1979. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory, rev. ed. Washington, DC: Math. Assoc. Amer., 1991. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillan, 1893. McClelland, W. J. Geometry of the Circle. London, 1891. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Rouche ´, E. and de Comberousse, C. Traite ´de Ge ´ome´trie, nouv. e ´d., vol. 1: Ge ´ome´trie plane. Paris: Gauthier-Villars, 1922. Russell, J. W. Elementary Pure Geometry. Oxford, 1893. Simon, M. U¨ber die Entwicklung der Elementargeometrie im XIX Jahrhundert. Berlin, 1906. Weisstein, E. W. "Plane Geometry." M ATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Weisstein, E. W. "Books about Plane Geometry." http:// www.treasure-troves.com/books/PlaneGeometry.html. Plane Graph PLANAR GRAPHPlane Partition 54211 3222 A two-dimensional array of INTEGERS nonincreasing both left to right and top to bottom which add up to a given number, i.e., nij]ni(j/C271)and nij]n(i/C271)j:For example, a planar partition of 22 is illustrated above. The GENERATING FUNCTION for the number PL( n)o f planar partitions of nis X/C12 n/C300PL(n)xn/C301Q/C12 k/C301(1/C28xk)k /C301/C27x/C273x2/C276x3/C2713x4/C2724x5/C27... ( 1 ) (Sloane’s A000219, MacMahon 1912b, Speciner 1972, Bender and Knuth 1972, Bressoud and Propp 1999).MacMahon (1960) also showed that the number ofplane partitions PL( a;b;c) whose Y OUNG DIAGRAMS fit inside an a/C29b/C29cbox is given by PL(a;b;c)/C30Ya i/C301Yb j/C301Yc k/C301i/C27j/C27k/C281 i/C27j/C27k/C282(2) (Bressoud and Propp 1999, Fulmek and Krattentha- ler 2000). Expanding out the products gives PL(a;b;c)/C30Ya i/C301G(i)G(b/C27c/C27i) G(b/C27i)G(c/C27i)(3) /C30G(a/C271)G(b/C271)G(c/C271)G(a/C27b/C27c/C271) G(a/C27b/C271)G(a/C27c/C271)G(b/C27c/C271);(4) where G(n)i sB ARNES’ G-FUNCTION . Taking n/C30a/C30 b/C30cgives PL(n;n;n)/C30Yn i/C301G(i)G(i/C272n) [G(i/C27n)]2(5) /C30[G(n/C271)]3G(3n/C271) [G(2n/C271)]3; (6) the first few terms of which are 2, 20, 980, 232848, 267227532, 1478619421136, ... (Sloane’s A008793). Amazingly, PL(a ; b; c) also gives the number of HEXAGON TILINGS by RHOMBI for a hexagon of side lengths a, b, c, a, b, c (David and Tomei 1989, Fulmek and Krattenthaler 2000). The concept of planar partitions can also be general- ized to cubic partitions. See also CYCLICALLY SYMMETRIC PLANE PARTITION , DESCENDING PLANE PARTITION ,H EXAGON TILING , PARTITION ,M ACDONALD’S PLANE PARTITION CONJEC- TURE ,SOLID PARTITION ,TOTALLY SYMMETRIC SELF- COMPLEMENTARY PLANE PARTITION ,YOUNG DIAGRAM References Bender, E. A. and Knuth, D. E. "Enumeration of Plane Partitions." J. Combin. Theory Ser. A. 13,40/C1/54, 1972. Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999. Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/646. Cohn, H.; Larsen, M.; and Propp, J. "The Shape of a Typical Boxed Plane Partition." New York J. Math. 4, 137 /C1/166, 1998. David, G. and Tomei, C. "The Problem of the Calissons." Amer. Math. Monthly 96, 429 /C1/431, 1989. Fulmek, M. and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry Axes, II." Europ. J. Combin. 21, 601 /C1/640, 2000. Knuth, D. E. "A Note on Solid Partitions." Math. Comput. 24, 955 /C1/961, 1970. MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. V: Partitions in Two-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211,75/C1/110, 1912a. MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. VI: Partitions in Two-Dimensional Space, to which is Added an Adumbration of the Theory of Parti- tions in Three-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211, 345 /C1/373, 1912b. MacMahon, P. A. §429 and 494 in Combinatory Analysis, Vol. 2. New York: Chelsea, 1960. Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr. "Proof of the Macdonald Conjecture." Invent. Math. 66,73/C1/87, 1982. Sloane, N. J. A. Sequences A000219/M2566 and A008793 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Speciner, M. Item 18 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 10, Feb. 1972. Stanley, R. P. "Symmetry of Plane Partitions." J. Combin. Th. Ser. A 3, 103 /C1/113, 1986. Stanley, R. P. "A Baker’s Dozen of Conjectures Concerning Plane Partitions." In Combinatoire E´ nume ´rative (Ed. G. Labelle and P. Leroux). New York: Springer-Verlag, 285 /C1/293, 1986. Plane Symmetry Groups WALLPAPER GROUPS Plane-Filling Curve PLANE- FILLING FUNCTIONPlane-Filling Function A SPACE-FILLING FUNCTION which maps a 1-D INTER- VAL into a 2-D area. Plane-filling functions were thought to be impossible until Hilbert discovered the HILBERT CURVE in 1891. Plane-filling functions are often (imprecisely) defined to be the "limit" of an infinite sequence of specified curves which "fill" the PLANE without "HOLES ," hence the more popular term PLANE-FILLING CURVE . The term "plane-filling function" is preferable to "PLANE- FILLING CURVE " because "curve" informally connotes "GRAPH " (i.e., range) of some continuous function, but the GRAPH of a plane-filling function is a solid patch of 2-space with no evidence of the order in which it was traced (and, for a dense set, retraced). Actually, all that is needed to rigorously define a plane-filling function is an arbitrarily refinable correspondence between contiguous subintervals of the domain andcontiguous subareas of the range. True plane-filling functions are not ONE-TO-ONE .I n fact, because they map closed intervals onto closed areas, they cannot help but overfill, revisiting at leasttwice a dense subset of the filled area. Thus, every point in the filled area has at least one inverse image. See also H ILBERT CURVE ,P EANO CURVE ,P EANO- GOSPER CURVE ,S CHOENBERG CURVE ,S IERPINSKI CURVE ,S PACE- FILLING FUNCTION ,S PACE- FILLING POLYHEDRON References Bogomolny, A. "Plane Filling Curves." http://www.cut-the- knot.com/do_you_know/hilbert.html. Wagon, S. "A Space-Filling Curve." §6.3 in Mathematica in Action. New York: W. H. Freeman, pp. 196 /C1/209, 1991. Plane-Line Intersection LINE-PLANE INTERSECTION Planted Planar Tree A planted plane tree ( V;E;v;a) is defined as a vertex set V, edges set E,ROOT v, and order relation a onVwhich satisfies 1. For x; y /C23 V if r(x) B r(y); then x a y; where r(x) is the length of the path from v to x, 2. If fr ; sg;fx; yg/C23 E ; r(r) /C30 r(x) /C30 r(s) /C281 /C30 r(y) /C28 1 and r a x; then s a y/ (Klarner 1969, Chorneyko and Mohanty 1975). The CATALAN NUMBERS give the number of planar triva- lent planted trees. See also CATALAN NUMBER ,PLANTED TREE,TREE References Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration of Certain Sets of Planted Plane Trees." J. Combin. Th. Ser. B 18, 209 /C1/221, 1975. Harary, F.; Prins, G.; and Tutte, W. T. "The Number of Plane Trees." Indag. Math. 26, 319 /C1/327, 1964. Klarner, D. A. "A Correspondence Between Sets of Trees." Indag. Math. 31, 292 /C1/296, 1969. Planted Tree A planted tree is a ROOTED TREE whose ROOT NODE has VERTEX DEGREE 1. The number of planted trees of n nodes is Tn/C281 ; where Tn /C281 is the number of ROOTED TREES of n /C281 vertices (Harary 1994, pp. 188 /C1/190), so there are 1, 1, 1, 2, 4, 9, 20, ... (Sloane’s A000081) planted trees of n /C301, 2, 3, ... vertices. See also ROOTED TREE,TREE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Sloane, N. J. A. Sequences A000081/M1180 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Plastic Constant The limiting ratio of the successive terms of the PADOVAN SEQUENCE , P /C301 :32471795... : It is given exactly by the unique real root of x3 /C28x /C281 /C300:/ See also PADOVAN SEQUENCE References Stewart, I. "Tales of a Neglected Number." Sci. Amer. 274, 102 /C1/103, Jun. 1996. Plat A BRAID in which strands are intertwined in the center and are free in "handles" on either side of the diagram. Plate Carre EQUIRECTANGULAR PROJECTIONPlateau Curves A curve studied by the Belgian physicist and math- ematician Joseph Plateau. It has Cartesian equation x /C30a sin[(m /C27 n)t] sin[(m /C28 n)t] y /C302a sin(mt) sin(nt) sin[(m /C28 n)t]: If m /C302n; the Plateau curve degenerates to a CIRCLE with center (1; 0) and radius 2. References MacTutor History of Mathematics Archive. "Plateau Curves." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Plateau.html. Plateau’s Equation The PARTIAL DIFFERENTIAL EQUATION (1 /C27u2 x)uxx /C282uxuyuxy /C27(1 /C27u2y)uyy /C300: References Bateman, H. Partial Differential Equations of Mathematical Physics. New York: Dover, p. 501, 1944. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 134, 1997. Plateau’s Laws BUBBLES can meet only at ANGLES of 1208 (for two BUBBLES ) and 109/C1428 ?16ƒ (for three BUBBLES ), where the exact value of 109.5 8 is the TETRAHEDRAL DIHE- DRAL ANGLE . This was proved by Jean Taylor using MEASURE THEORY to study AREA minimization. The DOUBLE BUBBLE isAREA minimizing, but it is not known if the triple BUBBLE is also AREA minimizing. It is also unknown if empty chambers trapped inside can minimize AREA forn]3BUBBLES . See also BUBBLE ,CALCULUS OF VARIATIONS ,DOUBLE BUBBLE ,MINIMAL SURFACE ,PLATEAU’S PROBLEM References Morgan, F. "Mathematicians, including Undergraduates, Look at Soap Bubbles." Amer. Math. Monthly 101, 343/C1/ 351, 1994. Taylor, J. E. "The Structure of Singularities in Soap-Bubble- Like and Soap-Film-Like Minimal Surfaces." Ann. Math. 103, 489 /C1/539, 1976. Plateau’s Problem The problem in CALCULUS OF VARIATIONS to find the MINIMAL SURFACE of a boundary with specified con- straints (usually having no singularities on the sur- face). In general, there may be one, multiple, or no MINIMAL SURFACES spanning a given closed curve in space. The EXISTENCE of a solution to the general case was independently proven by Douglas (1931) and Rado´ (1933), although their analysis could not ex- clude the possibility of singularities. Osserman (1970) and Gulliver (1973) showed that a minimizing solu- tion cannot have singularities. The problem is named for the Belgian physicist who solved some special cases experimentally using soap films and wire frames (Isenberg 1992, Wells 1991). The illustration above shows the 13-polygon surface obtained for a cubical wire frame. See also BUBBLE ,CALCULUS OF VARIATIONS ,DOUBLE BUBBLE ,M INIMAL SURFACE ,PLATEAU’S LAWS,STEI- NER TREE,TRAVELING SALESMAN PROBLEM References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 48 /C1/49, 1989. Douglas, J. "Solution of the Problem of Plateau." Trans. Amer. Math. Soc. 33, 263 /C1/321, 1931. Gulliver, R. "Regularity of Minimizing Surfaces of Pre- scribed Mean Curvature." Ann. Math. 97, 275 /C1/305, 1973. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, 1992. Osserman, R. "A Proof of the Regularity Everywhere of the Classical Solution to Plateau’s Problem." Ann. Math. 91, 550 /C1/569, 1970. Osserman, R. "Plateau’s Problem." §1, Appendix in A Survey of Minimal Surfaces. New York: Dover, pp. 143 /C1/145, 1986. Rado´, T. "On the Problem of Plateau." Ergeben. d. Math. u. ihrer Grenzgebiete. Berlin: Springer-Verlag, 1933. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 119 /C1/121, 1999. Stuwe, M. Plateau’s Problem and the Calculus of Variations. Princeton, NJ: Princeton University Press, 1989.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 185 /C1/187, 1991. Platonic Graph A POLYHEDRAL GRAPH corresponding to the SKELETON of a PLATONIC SOLID . The five platonic graphs, the TETRAHEDRAL GRAPH , CUBICAL GRAPH , OCTAHEDRAL GRAPH , DODECAHEDRAL GRAPH , and ICOSAHEDRAL GRAPH , are illustrated above. They are special cases of S CHLEGEL GRAPHS . See also PLATONIC SOLID ,P OLYHEDRAL GRAPH , SCHLEGEL GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976. Platonic Solid The Platonic solids, also called the regular solids or regular polyhedra, are CONVEX POLYHEDRA with equivalent faces composed of congruent CONVEX REGULAR POLYGONS . There are exactly five such solids (Steinhaus 1983, pp. 252 /C1/256): the CUBE ,DODECAHE- DRON ,ICOSAHEDRON ,OCTAHEDRON , and TETRAHE- DRON , as was proved by Euclid in the last proposition of the ELEMENTS . The Platonic solids are sometimes also called "cosmic figures" (Cromwell 1997), although this term is sometimes used to refercollectively to both the Platonic solids and K EPLER- POINSOT SOLIDS (Coxeter 1973). The Platonic solids were known to the ancientGreeks, and were described by Plato in his Timaeus ca. 350 BC. In this work, Plato equated the TETRA- HEDRON with the "element" fire, the CUBE with earth, the ICOSAHEDRON with water, the OCTAHEDRON with air, and the DODECAHEDRON with the stuff of which the constellations and heavens were made (Cromwell 1997). IfPis a POLYHEDRON with congruent (convex) regular polygonal faces, then Cromwell (1997, pp. 77 /C1/78) shows that the following statements are equivalent. 1. The vertices of Pall lie on a SPHERE . 2. All the DIHEDRAL ANGLES are equal. 3. All the VERTEX FIGURES are REGULAR POLYGONS . 4. All the SOLID ANGLES are equivalent. 5. All the vertices are surrounded by the same number of FACES . Let v(sometimes denoted N0) be the number of VERTICES ,e(orN1) the number of EDGES , and f(or N2) the number of FACES . The following table gives the S CHLA ¨FLI SYMBOL ,W YTHOFF SYMBOL , and C&R symbol, the number of vertices v, edges e, and faces f, and the POINT GROUPS for the Platonic solids (Wen- ninger 1989). Solid S CHLA ¨FLI SYMBOLWYTHOFF SYMBOLC&R Symbolve f Group CUBE /f4;3g/3½224 4381 2 6 /Oh/ DODECA- HEDRON/f5;3g/3½225 5320 30 12 /Ih/ ICOSA- HEDRON/f3;5g/5½223 3512 30 20 /Ih/ OCTA-HEDRON/f3;4g/4½223 3461 2 8 /Oh/ TETRA- HEDRON/f3;3g/3½223 3346 4 /Td/ The duals of Platonic solids are other Platonic solids and, in fact, the dual of the TETRAHEDRON is another TETRAHEDRON . Let rbe the INRADIUS ,rthe MIDRA- DIUS, and Rthe CIRCUMRADIUS of a given Platonic solid. Then rR/C30r2: The following two tables give the analytic and numerical values of these distances for Platonic solidswith unit side length. Solid r /r/ R CUBE /1 2//12ffiffiffi 2p //1 2ffiffiffi 3p / DODECAHEDRON /1 20ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 250/C27110ffiffiffi 5pp //1 43/C27ffiffiffi 5p9+=9+; //1 4ffiffiffiffiffiffi 15p /C27ffiffiffi3p9+=9+; / ICOSAHEDRON /1 123ffiffiffi 3p /C27ffiffiffiffiffiffi15p9+=9+; //1 41/C27ffiffiffi 5p9+=9+; //1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pp / OCTAHEDRON /1 6ffiffiffi 6p //1 2//12ffiffiffi 2p /TETRAHEDRON /1 12ffiffiffi6p //1 4ffiffiffi 2p //1 4ffiffiffi 6p / Solid r /r/ R CUBE 0.5 0.70711 0.86603 DODECAHEDRON 1.11352 1.30902 1.40126 ICOSAHEDRON 0.75576 0.80902 0.95106 OCTAHEDRON 0.40825 0.5 0.70711 TETRAHEDRON 0.20412 0.35355 0.61237 Finally, let Abe the AREA of a single FACE ,Vbe the VOLUME of the solid, the EDGES be of unit length on a side, and abe the DIHEDRAL ANGLE . The following table summarizes these quantities for the Platonic solids. Solid AV /a/ Cube 1 1 /1 2p/ Dodecahedron /14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pp //1 415/C277ffiffiffi 5p9+=9+; //cos/C281/C281 5ffiffiffi 5p9+;k9+;7 / Icosahedron /1 4ffiffiffi 3p //5 123/C27ffiffiffi5p9+=9+; //cos/C281/C281 3ffiffiffi 5p9+;k9+;7 / Octahedron /1 4ffiffiffi 3p //1 3ffiffiffi 2p // cos/C281/C281 39+;k9+;7 / Tetrahedron /1 4ffiffiffi 3p //1 12ffiffiffi 2p // cos/C2811 39+;k9+;7 / The number of EDGES meeting at a VERTEX is 2e=v: The S CHLA ¨FLI SYMBOL can be used to specify a Platonic solid. For the solid whose faces are p-gons (denoted fpg);with qtouching at each VERTEX , the symbol is fp;qg:Given pand q, the number of VERTICES ,EDGES , and faces are given by N0/C304p 4/C28(p/C282)(q/C282) N1/C302pq 4/C28(p/C282)(q/C282) N2/C304q 4/C28(p/C282)(q/C282): The plots above show scaled duals of the Platonic solid embedded in a CUMULATED form of the original solid, where the scaling is chosen so that the dual edges lie at the incenters of the original faces (Wenninger 1983, pp. 8 /C1/9). Since the Platonic solids are convex, the CONVEX HULL of each Platonic solid is the solid itself. MINIMAL SURFACES for Platonic solid frames are illustrated in Isenberg (1992, pp. 82 /C1/83). See also ARCHIMEDEAN SOLID ,C ATALAN SOLID , JOHNSON SOLID,KEPLER- POINSOT SOLID,QUASIREGU- LAR POLYHEDRON ,UNIFORM POLYHEDRON References Artmann, B. "Symmetry Through the Ages: Highlights from the History of Regular Polyhedra." In In Eves’ Circles (Ed. J. M. Anthony). Washington, DC: Math. Assoc. Amer., pp. 139 /C1/148, 1994. Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 131 /C1/136, 1987. Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2: Geometry. Cam- bridge, MA: MIT Press, p. 272, 1974. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 128 /C1/129, 1987. Bogomolny, A. "Regular Polyhedra." http://www.cut-the- knot.com/do_you_know/polyhedra.html. Bourke, P. "Platonic Solids (Regular Polytopes in 3D)." http://www.swin.edu.au/astronomy/pbourke/geometry/ platonic/. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 1 /C1/17, 93, and 107 /C1/112, 1973. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 51 /C1/57, 66 /C1/70, and 77 /C1/78, 1997. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 78 /C1/81, 1990. Gardner, M. "The Five Platonic Solids." Ch. 1 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 13 /C1/23, 1961. Harris, J. W. and Stocker, H. "Regular Polyhedron." §4.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 99 /C1/101, 1998. Heath, T. A History of Greek Mathematics, Vol. 1: From Thales to Euclid. New York: Dover, p. 162, 1981. Hume, A. "Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals." Computing Science Tech. Rep. , No. 130. Murray Hill, NJ: AT&T Bell Laboratories, 1986. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, 1992. Kepler, J. Opera Omnia, Vol. 5. Frankfort, p. 121, 1864. Kern, W. F. and Bland, J. R. "Regular Polyhedrons." In Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 116 /C1/119, 1948. Meserve, B. E. Fundamental Concepts of Geometry. New York: Dover, 1983. Nooshin, H.; Disney, P. L.; and Champion, O. C. "Properties of Platonic and Archimedean Polyhedra." Table 12.1 in "Computer-Aided Processing of Polyhedric Configura- tions." Ch. 12 in Beyond the Cube: The Architecture of Space Frames and Polyhedra (Ed. J. F. Gabriel). New York: Wiley, pp. 360 /C1/361, 1997. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 129 /C1/131, 1990. Pappas, T. "The Five Platonic Solids." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 39 and 110 /C1/111, 1989.Pedagoguery Software. Poly . http://www.peda.com/poly/. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 191 /C1/201, 1999. Rawles, B. A. "Platonic and Archimedean Solids--Faces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios." http://www.intent.com/sg/polyhedra.html. Robertson, S. A. and Carter, S. "On the Platonic and Archimedean Solids." J. London Math. Soc. 2, 125 /C1/132, 1970. Sharp, A. Geometry Improv’d: 1. By a Large and Accurate Table of Segments of Circles, with Compendious Tables for Finding a True Proportional Part, Exemplify’d in Making out Logarithms from them, there Being a Table of them for all Primes to 1100, True to 61 Figures. 2. A Concise Treatise of Polyhedra, or Solid Bodies, of Many Bases. London: R. Mount, p. 87, 1717. Steinhaus, H. "Platonic Solids, Crystals, Bees’ Heads, and Soap." Ch. 8 in Mathematical Snapshots, 3rd ed. New York: Dover, pp. 199 /C1/201 and 252 /C1/256, 1983. Waterhouse, W. "The Discovery of the Regular Solids." Arch. Hist. Exact Sci. 9, 212 /C1/221, 1972 /C1/1973. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 60 /C1/ 61, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 187 /C1/188, 1991. Wenninger, M. "The Five Regular Convex Polyhedra and Their Duals." Ch. 1 in Dual Models. Cambridge, England: Cambridge University Press, pp. 7 /C1/13, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, 1971. Platonic Solid Stellations The only STELLATIONS of PLATONIC SOLIDS which are UNIFORM POLYHEDRA are the three DODECAHEDRON STELLATIONS and the GREAT ICOSAHEDRON . See also DODECAHEDRON STELLATIONS ,ICOSAHEDRON STELLATIONS ,STELLA OCTANGULA Plato’s Number A vaguely specified number appearing in The Repub- lic which involves 216 and 12,960,000. References Heath, T. L. Aristarchus of Samos: The Ancient Copernicus. New York: Dover, pp. 171 /C1/172, 1981. Plato. The Republic. New York: Oxford University Press, 1994. Wells, D. G. The Penguin Dictionary of Curious and Inter- esting Numbers. London: Penguin, p. 144, 1986. Platykurtic A distribution with FISHER KURTOSIS g2 B0 (and therefore having a flattened shape). See also FISHER KURTOSIS p-Layer The p-layer of H, Lp ?(H) is the unique minimal NORMAL SUBGROUP of H which maps onto E(H=Op?(H)):/ See also BP-THEOREM , LP’-BALANCE THEOREM ,SIG- NALIZER FUNCTOR THEOREM Playfair’s Axiom Through any point in space, there is exactly one straight line PARALLEL to a given straight line. This AXIOM is equivalent to the PARALLEL POSTULATE . See also PARALLEL POSTULATE References Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 inJourney through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 54, 1990. Henderson, D. W. Experiencing Geometry: On Plane and Sphere. Englewood Cliffs, NJ: Prentice-Hall, 1995. Playfair, J. Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Circle and the Geometry of Solids to which are added Elements of Plane and Spherical Trigonometry. New York: W. E. Dean. Plethysm A group theoretic operation which is useful in the study of complex atomic spectra. A plethysm takes aset of functions of a given symmetry type fmgand forms from them symmetrized products of a givendegree rand other symmetry type fng:A plethysm fmg/C156fng/C30X flg satisfies the rules A/C156(BC)/C30(A/C156B)(A/C156C)/C30A/C156BA/C156C; A/C156(B9C)/C30A/C156B9A/C156C (A/C156B)/C156C/C30A/C156(B/C156C) (A/C27B)/C156flg/C30X G mnl(A/C156fmg)(B/C156fng); where Gmnlis the coefficient of flginfmgfng; (A/C28B)/C156flg/C30X (/C281)rGmnl(A/C156fmg)(B/C156f˜ng); where f˜ngis the partition of rconjugate to fng;and (AB)/C156flg/C30X gmnl(A/C156fmg)(B/C156fng); where gmnlis the coefficient of flgin the inner product fmg/C14(ng(Wybourne 1970). References Littlewood, D. E. "Polynomial Concomitants and Invariant Matrices." J. London Math. Soc. 11,4 9/C1/55, 1936. Wybourne, B. G. "The Plethysm of S-Functions" and "Plethysm and Restricted Groups." Chs. 6 /C1/7i nSymmetry Principles and Atomic Spectroscopy. New York: Wiley, pp. 49 /C1/68, 1970. Plot GRAPH (FUNCTION ) Plot3D GRAPH (FUNCTION )Plouffe’s Constant N.B. A detailed online essay by S. Finch was thestarting point for this entry. Define the function r(x)/C131 for xB0 0 for x]0:9+$k (1) Let a n/C30sin(2n)/C30sin 1 for n/C300 2a0ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28a2 0p forn/C301 2an/C281(1/C282a2 n/C282) for n]2;8 < :(2) then X/C12 n/C300r(an) 2n/C271/C301 2p: (3) For bn/C30cos(2n)/C30cos 1 for n/C300 2b2 n/C281/C281 for n]1;9+$k (4) and X/C12 n/C300r(bn) 2n/C271/C300:4756260767 . . . : (5) Letting cn/C30tan(2n)/C30tan 1 for n/C300 2cn/C281 1/C28c2 n/C281forn]1;8 < :(6) then X/C12 n/C300r(cn) 2n/C271/C301 p: (7) Plouffe asked if the above processes could be "in- verted." He considered an/C30sin 2nsin/C2811 29+;k9+;7 /C3012 forn/C300 12ffiffiffi 3p forn/C301 2an/C2811/C282a2 n/C282 ðÞ forn]2;8 >< >:(8) giving X/C12 n/C300r(an) 2n/C271/C301 12; (9) and bn/C30cos 2ncos/C2811 29+;k9+;7 /C3012 forn/C300 2b2 n/C281/C281 for n]1;( (10) giving X/C12 n/C300r( bn) 2n/C271 /C301 2; (11) and gn /C30tan 2n tan/C281129+;k9+;7 /C3012 for n /C300 2 gn /C281 1 /C28 g2 n/C281for n ]1;8 >< >:(12) giving X/C12 n/C300r( an) 2n/C271 /C301 ptan /C2811 29+;k9+;7 : (13) The latter is known as Plouffe’s constant (Plouffe 1997). The positions of the 1s in the BINARY expansion of this constant are 3, 6, 8, 9, 10, 13, 21, 23, ... (Sloane’s A004715). Borwein and Girgensohn (1995) extended Plouffe’s gn to arbitrary REAL x, showing that if jn /C30tan(2n tan /C281 x) /C30x for n /C300 2jn/C281 1 /C28 j2 n/C281for n ]1 and jn/C281 jj"1 /C28/C12 for n ]1 and jn/C281 jj/C301;8 >>< >>:(14) then X/C12 n/C300r( jn) 2n/C271 /C30tan/C281 x pfor x ]0 1 /C27tan/C281 x pfor x B0:8 >>>< >>>:(15) Borwein and Girgensohn (1995) also give much more general recurrences and formulas. References Borwein, J. M. and Girgensohn, R. "Addition Theorems and Binary Expansions." Canad. J. Math. 47, 262 /C1/273, 1995. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/plff/plff.html. Plouffe, S.. "The Computation of Certain Numbers Using a Ruler and Compass." J. Integer Sequences 1, No. 98.1.3, 1998. http://www.research.att.com/~njas/sequences/JIS/ compass.html. Sloane, N. J. A. Sequences A004715 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Plu¨cker Characteristics The CLASS m, ORDER n, number of NODES d ; number of CUSPS k ; number of STATIONARY TANGENTS (INFLEC- TION POINTS ) i; number of BITANGENTS t ; and GENUS p. See also ALGEBRAIC CURVE ,BITANGENT ,CUSP,GENUS (SURFACE ), INFLECTION POINT ,N ODE (ALGEBRAIC CURVE ), STATIONARY TANGENTPlu¨cker Coordinates GRASSMANN COORDINATES Plu¨cker Lines The 60 PASCAL LINES of a HEXAGON inscribed in a CONIC SECTION intersect three at a time through 20 STEINER POINTS . There is a dual relationship between the 15 Plu¨cker lines and the 15 SALMON POINTS . See also KIRKMAN POINTS ,PASCAL LINES,PASCAL’S THEOREM ,SALMON POINTS ,STEINER POINTS References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 236 /C1/237, 1929. Plu¨cker, M. J. reine angew. Math. 5, p. 274. Salmon, G. "Notes: Pascal’s Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 379 /C1/ 382, 1960. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 172, 1991. Plu¨cker Relations PLU¨CKER’S EQUATIONS Plu¨cker’s Conoid ARULED SURFACE sometimes also called the CYLIN- DROID . von Seggern (1993) gives the general func- tional form as ax2/C27by2/C28zx2/C28zy2/C300; (1) whereas Fischer (1986) and Gray (1997) give z/C302xy (x2/C27y2: (2) A polar parameterization therefore gives x(r;u)/C30rcosu (3) y(r;u)/C30rsinu (4) z(r;u)/C302 cos usinu: (5) A generalization of Plu¨cker’s conoid to n folds is given by x(r ; u) /C30r cos u (6) y(r ; u) /C30r sin u (7) z(r ; u) /C30sin(nu) (8) (Gray 1997). The cylindroid is the inversion of the CROSS-CAP (Pinkall 1986). See also CROSS- CAP,RIGHT CONOID ,RULED SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 4 /C1/5, 1986. Gray, A. "Plu¨cker’s Conoid." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 435 /C1/437, 1997. Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 64, 1986. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 288, 1993. Plu¨cker’s Equations Relationships between the number of SINGULARITIES of plane algebraic curves. Given a PLANE CURVE , m /C30n(n /C281) /C282 d /C283k (1) n /C30m(m /C281) /C282t /C283 i (2) i /C303n(n /C282) /C286d /C288k (3) k /C303m(m /C282) /C286 t /C288i; (4) where m is the CLASS , n the ORDER , d the number of NODES , k the number of CUSPS , i the number of STATIONARY TANGENTS (INFLECTION POINTS ), and t the number of BITANGENTS . Only three of these equations are LINEARLY INDEPENDENT . See also ALGEBRAIC CURVE ,B IOCHE’S THEOREM , BITANGENT ,C USP,G ENUS (SURFACE ), INFLECTION POINT ,KLEIN’S EQUATION ,NODE (ALGEBRAIC CURVE ), STATIONARY TANGENT References Boyer, C. B. A History of Mathematics. New York: Wiley, pp. 581 /C1/582, 1968. Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 99 /C1/118, 1959. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 220 /C1/222, 1930.Plumbing The plumbing of a p-sphere and a q-sphere is defined as the disjoint union of Sp /C29Sq and Dp /C29Sq with their common Dp /C29Dq ; identified via the identity home- omorphism. See also HYPERSPHERE References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 180, 1976. Pluperfect Number MULTIPLY PERFECT NUMBER Plurisubharmonic Function An upper semicontinuous function whose restrictions to all complex lines are subharmonic (where defined). These functions were introduced by P. Lelong and Oka in the early 1940s. Examples of such a function are the logarithms of moduli of holomorphic func- tions. References Range, R. M. and Anderson, R. W. "Hans-Joachim Brem- mermann, 1926 /C1/1996." Not. Amer. Math. Soc. 43, 972 /C1/ 976, 1996. Plus The ADDITION of two quantities, i.e., a plus b. The operation is denoted a /C27b; and the symbol /C27is called the PLUS SIGN. Floating point ADDITION is sometimes denoted /C154:/ See also ADDITION ,MINUS ,PLUS OR MINUS ,TIMES Plus or Minus The symbol 9 is used to denote a quantity which should be both added and subtracted, as in a 9b: The symbol can be used to denote a range of uncertainty, or to denote a pair of quantities, such as the roots given by the QUADRATIC FORMULA x9/C30/C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp 2a: When order is relevant, the symbol a /C14b is also used, so an expression OF THE FORM x 9y /C14z is interpreted as x /C27y /C28z or x /C28y /C27z : In contrast, the expression x 9 y9zis interpreted to mean the set of four quantities x/C27y/C27z;x/C28y/C27z;x/C27y/C28z;andx/C28y/C28z:/ See also MINUS ,MINUS SIGN,PLUS,PLUS SIGN,SIGN Plus Perfect Number ARMSTRONG NUMBER Plus Sign The symbol "//C27/" which is used to denote a POSITIVE number or to indicate ADDITION . See also ADDITION ,MINUS SIGN,SIGN Plutarch Numbers In Moralia, the Greek biographer and philosopher Plutarch states "Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hip- parchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952.)" These numbers are known as the Plutarch numbers. 103,049 can be interpreted as the number s10of BRACKETINGS on ten letters (Stanley 1997, Habsieger et al. 1998). Similarly, Plutarch’s second number is given by s10 /C27s11 ðÞ =2 /C30310;954 (Habsieger et al. 1998). References Biermann, K.-R. and Mau, J. "U¨ berpru ¨fung einer fru¨hen Anwendung der Kombinatorik in der Logik." J. Symbolic Logic 23, 129 /C1/132, 1958. Biggs, N. L. "The Roots of Combinatorics." Historia Mathe- matica 6, 109 /C1/136, 1979. Habsieger, L.; Kazarian, M.; and Lando, S. "On the Second Number of Plutarch." Amer. Math. Monthly 105, 446, 1998. Heath, T. L. A History of Greek Mathematics, Vol. 2: From Aristarchus to Diophantus. New York: Dover, p. 256, 1981. Kneale, W. and Kneale, M. The Development of Logic. Oxford, England: Oxford University Press, p. 162, 1971. Neugebauer, O. A History of Ancient Mathematical Astron- omy. New York: Springer-Verlag, p. 338, 1975. Plutarch. §VIII.9 in Moralia, Vol. 9. Cambridge, MA: Har- vard University Press, p. 732, 1961. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, p. 63, 1996. Stanley, R. P. "Hipparchus, Plutarch, Schro ¨der, and Hough." Amer. Math. Monthly 104, 344 /C1/350, 1997. Pochhammer Symbol The Pochhammer symbol (x)n /C13G(x /C27 n) G(x)/C30x(x /C271) /C1/C1/C1(x /C27n /C281) /C30G(x /C27 n) G(x)(1) (Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for n ]0 is an unfortunate notation used in the theory of special functions for the RISING FACTORIAL , which is denoted x(n) (Roman 1984, p. 5) or /C142x/C143n(Comtet 1974, p. 6) in combinatorics. In combinatorial usage, (x)ndenotes the FALLING FAC- TORIAL . Extreme caution is therefore needed in interpreting the notations (x)nandx(n):/ The Pochhammer symbol ( x)nobeys the transforma- tion due to EulerX/C12 n/C300(a)n n!anzn/C30(1/C28z)/C28aX/C12 n/C300(a)n n!Dna0z 1/C28z !n ;(2) where Dis the FORWARD DIFFERENCE and Dka0/C30Xk m/C300(/C281)mk m9+;89+;9 ak/C28m (3) (Nørlund 1955). The sum of 1 =(k)pcan be done in closed form as Xn k/C3011 (k)p/C301 (p/C281)G(p)/C28nG(n) (p/C281)G(n/C27p)(4) forp/C211. See also FACTORIAL ,FALLING FACTORIAL ,GENERAL- IZED HYPERGEOMETRIC FUNCTION ,HANKEL’S SYMBOL , HARMONIC LOGARITHM ,HYPERGEOMETRIC FUNCTION , KRAMP’S SYMBOL References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 52, 1981. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities.Braunschweig, Germany: Vieweg, 1998. Nørlund, N. E. "Hypergeometric Functions." Acta Math. 94, 289/C1 /349, 1955. Roman, S. The Umbral Calculus. New York: Academic Press, p. 5, 1984. Spanier, J. and Oldham, K. B. "The Pochhammer Polyno- mials ( x)n:/" Ch. 18 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 149 /C1/165, 1987. Pocklington-Lehmer Test POCKLINGTON’S THEOREM Pocklington’s Criterion Letpbe an ODD PRIME ,kbe an INTEGER such that p¶kand 15k52(p/C271);and N/C132kp/C271: Then the following are equivalent 1.NisPRIME . 2. GCD ak/C271;N9+=9+; /C301;/ where GCD is the GREATEST COMMON DENOMINATOR . This is a modified version of the original theorem due to Lehmer. References Pocklington, H. C. "The Determination of the Prime or Composite Nature of Large Numbers by Fermat’s Theo- rem." Proc. Cambridge Phil. Soc. 18,29/C1/30, 1914/16. Pocklington’s Theorem Let n /C281 /C30FR where F is the factored part of a number F /C30pa1 1/C1/C1/C1parr; (1) where (R; F) /C301; and R Bffiffiffinp: If there exists a bifor i /C301, ..., r such that bn/C281 i/C131 (mod n) (2) GCD b(n/C281)=pi i /C281; n9+;k9+;7 /C301 ; (3) then n is a PRIME . Poggendorff Illusion The illusion that the two ends of a straight LINE SEGMENT passing behind an obscuring RECTANGLE are offset when, in fact, they are aligned. The Poggendorff illusion was discovered in 1860 by physicist and scholar J. C. Poggendorff, editor of Annalen der Physik und Chemie , after receiving a letter from astronomer F. Zo¨llner. In his letter, Zo¨llner described an illusion he noticed on a fabric design in which parallel lines intersected by a pattern of short diagonal lines appear to diverge (ZO¨ LLNER’S ILLUSION ). Pondering this illusion, Poggendorff no- ticed and described another illusion resulting from the apparent misalignment of a diagonal line; an illusion which today bears his name (IllusionWorks). See also ILLUSION ,M U¨ LLER- LYER ILLUSION ,PONZO’S ILLUSION ,V ERTICAL- HORIZONTAL ILLUSION ,Z O¨ LL- NER’S ILLUSION References Burmester, E. "Beitra ¨ge zu experimentellen Bestimmung geometrisch-optischer Ta¨uschungen." Z. Psychologie 12, 355 /C1/394, 1896. Day, R. H. and Dickenson, R. G. "The Components of the Poggendorff Illusion." Brit. J. Psychology 67, 537 /C1/552, 1976. Fineman, M. "Poggendorff’s Illusion." Ch. 19 in The Nature of Visual Illusion. New York: Dover, pp. 151 /C1/159, 1996. Gilliam, B. "A Depth Processing Theory of the Poggendorff Illusion." Perception & Psychophys. 10, 211 /C1/216, 1971. Gillam, B. "Geometrical Illusions." Sci. Amer. 242, 102 /C1/111, 1980.Greene, E. "The Corner Poggendorff." Perception 17,65/C1/70, 1988. IllusionWorks. "Poggendorf [sic]." http://www.illusion- works.com/html/poggendorf.html. Lucas, A. and Fisher, G. H. "Illusions in concrete situations: II. Experimental Studies of the Poggendorff Illusion." Ergonomics 12, 395 /C1/402, 1969. Robinson, J. O. The Psychology of Visual Illusion. London: Hutchinson, 1972. Rock, I. Perception. New York: W. H. Freeman, 1984. Schiffman, H. Sensation and Perception. New York: Wiley, 1995. Spivey-Knowlton, M. J. and Bridgeman, B. "Spatial Context Affects the Poggendorff Illusion." Perception & Psycho- phys. 53, 467 /C1/474, 1993. Pohlke’s Theorem The principal theorem of AXONOMETRY , first pub- lished without proof by Pohlke in 1860. It states that three segments of arbitrary length a ?x?; a?y?; and a ?z ? which are drawn in a PLANE from a point a? under arbitrary ANGLES form a parallel projection of three equal segments ax, ay, and az from the ORIGIN of three PERPENDICULAR coordinate axes. However, only one of the segments or one of the ANGLES may vanish. See also AXONOMETRY References Schwarz, H. A. J. reine angew. Math. 63, 309/C1/314, 1864. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 170 /C1/171, 1999. Pohlmeyer-Lund-Regge Equation The system of PARTIAL DIFFERENTIAL EQUATIONS uxx/C28uyy9sinucosu/C27cosu sin3u(v2 x/C28v2y)/C300 (1) (vxcot2u)x/C30(vycot2u)y: (2) References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 61, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Poincare ´ Conjecture The conjecture that every SIMPLY CONNECTED 3- MANIFOLD isHOMEOMORPHIC to the 3- SPHERE . This conjecture was first proposed in 1904 by H. Poincare ´ (Poincare ´1953, pp. 486 and 498), and subsequently generalized to the conjecture that every COMPACT n- MANIFOLD isHOMOTOPY -equivalent to the n-sphere IFF it is HOMEOMORPHIC to the n-SPHERE . The generalized statement reduces to the original con- jecture for n/C303. Then/C301 case of the generalized conjecture is trivial, then/C302 case is classical, n/C303 remains open, n/C304 was proved by Freedman (1982) (for which he was awarded the 1986 FIELDS MEDAL ), n /C305 by Zeeman (1961), n /C306 by Stallings (1962), and n ]7 by Smale in 1961. Smale subsequently extended his proof to include n ]5 :/ See also COMPACT MANIFOLD ,H OMEOMORPHIC , HOMOTOPY ,M ANIFOLD ,PROPERTY P,SIMPLY CON- NECTED ,SPHERE ,THURSTON’S GEOMETRIZATION CON- JECTURE References Adams, C. C. "The Poincare ´ Conjecture, Dehn Surgery, and the Gordon-Luecke Theorem." §9.3 in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 257 /C1/263, 1994. Batterson, S. Stephen Smale: The Mathematician Who Broke the Dimension Barrier. Providence, RI: Amer. Math. Soc., 2000. Bing, R. H. "Some Aspects of the Topology of 3- Manifolds Related to the Poincare ´ Conjecture." In Lectures on Modern Mathematics, Vol. II (Ed. T. L. Saaty). New York: Wiley, pp. 93 /C1/128, 1964. Birman, J. "Poincare ´’s Conjecture and the Homeotopy Group of a Closed, Orientable 2-Manifold." J. Austral. Math. Soc. 17, 214 /C1/221, 1974. Clay Mathematics Institute. "The Poincare ´ Conjecture." http://www.claymath.org/prize_problems/poincare.htm. Freedman, M. H. "The Topology of Four-Differentiable Manifolds." J. Diff. Geom. 17, 357 /C1/453, 1982. Gabai, D. "Valentin Poenaru’s Program for the Poincare ´ Conjecture." In Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topol., VI (Ed. S.-T. Yau). Cam- bridge, MA: International Press, pp. 139 /C1/166, 1995. Gillman, D. and Rolfsen, D. "The Zeeman Conjecture for Standard Spines is Equivalent to the Poincare ´ Conjec- ture." Topology 22, 315 /C1/323, 1983. Jakobsche, W. "The Bing-Borsuk Conjecture is Stronger than the Poincare ´ Conjecture." Fund. Math. 106, 127 /C1/ 134, 1980. Milnor, J. "The Poincare ´ Conjecture." http://www.clay- math.org/prize_problems/poincare.pdf. Papakyriakopoulos, C. "A Reduction of the Poincare ´ Con- jecture to Group Theoretic Conjectures." Ann. Math. 77, 250 /C1/205, 1963. Poincare ´,H. /Œ/uvres de Henri Poincare ´, tome VI. Paris: Gauthier-Villars, pp. 486 and 498, 1953. Rourke, C. "Algorithms to Disprove the Poincare ´ Conjec- ture." Turkish J. Math. 21,99/C1/110, 1997. Stallings, J. "The Piecewise-Linear Structure of Euclidean Space." Proc. Cambridge Philos. Soc. 58, 481 /C1/488, 1962. Smale, S. "Generalized Poincare ´’s Conjecture in Dimensions Greater than Four." Ann. Math. 74, 391 /C1/406, 1961. Smale, S. "The Story of the Higher Dimensional Poincare ´ Conjecture (What Actually Happened on the Beaches of Rio)." Math. Intell. 12,44/C1/51, 1990. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 0821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Provi- dence, RI: Amer. Math. Soc., 2000. Thickstun, T. L. "Open Acyclic 3-Manifolds, a Loop Theo- rem, and the Poincare ´ Conjecture." Bull. Amer. Math. Soc. 4, 192 /C1/194, 1981. Zeeman, E. C. "The Generalised Poincare ´ Conjecture." Bull. Amer. Math. Soc. 67, 270, 1961. Zeeman, E. C. "The Poincare ´ Conjecture for n ]5:/"In Topology of 3-Manifolds and Related Topics, Proceedings of the University of Georgia Institute, 1961. Englewood Cliffs, NJ: Prentice-Hall, pp. 198 /C1/204, 1961.Poincare ´ Disk POINCARE ´ HYPERBOLIC DISK Poincare ´ Duality The BETTI NUMBERS of a compact orientable n- MANIFOLD satisfy the relation bi /C30bn/C28i : See also BETTI NUMBER ,INTERSECTION (HOMOLOGY ) Poincare ´ Formula The POLYHEDRAL FORMULA generalized to a surface of GENUS g, V /C28E /C27F /C30 x(g) where V is the number of VERTICES , E is the number of EDGES , F is the number of faces, and x(g)/C132/C282g is called the E ULER CHARACTERISTIC . See also EULER CHARACTERISTIC ,GENUS (SURFACE ), POLYHEDRAL FORMULA References Coxeter, H. S. M. "Poincare ´’s Proof of Euler’s Formula." Ch. 9 in Regular Polytopes, 3rd ed. New York: Dover, pp. 165 /C1/172, 1973. Eppstein, D. "Fourteen Proofs of Euler’s Formula: V/C28E/C27F/C302:/" http://www.ics.uci.edu/~eppstein/junk- yard/euler/. Poincare ´ Group LORENTZ GROUP Poincare ´ Hyperbolic Disk A 2-D space having HYPERBOLIC GEOMETRY defined as the DISK x/C23R2:½x½B19+89+9 ;with HYPERBOLIC METRIC ds2/C30dx2/C27dy2 (1/C28r2)2: (1) The Poincare ´disk is a model for HYPERBOLIC GEOME- TRYin which a line is REPRESENTED AS an ARC of a CIRCLE whose ends are PERPENDICULAR to the DISK’s boundary (and DIAMETERS are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to PERPEN- DICULAR lines, and arcs which meet on the boundary are a pair of limits rays. The endpoints of any arc can be specified by two angles around the disk u1 and u2 : Define u /C131 2u1 /C27 u2 ðÞ (2) d u /C131 2 u1 /C28 u2 jj (3) Then trigonometry shows that in the above diagram, r /C30tan(du) (4) y /C30sin(du) tan(du) ; (5) so the radius of the circle forming the arc is R /C30cos(du) /C27y /C30sec(du) (6) and its center is located at R(cos u; sin u): The half- angle subtended by the arc is then sin f /C30sin(du) tan(du) /C30cos(du) ; (7) so f /C30sin/C281[cos(d u)]: (8) The Poincare ´ hyperbolic disk represents a CONFOR- MAL MAP, so angles between rays can be measured directly. There is an ISOMORPHISM between the Poincare ´ disk model and the KLEIN- BELTRAMI MODEL . See also ELLIPTIC PLANE ,H YPERBOLIC GEOMETRY , HYPERBOLIC METRIC ,KLEIN- BELTRAMI MODEL ,POIN- CARE ´ METRIC References Anderson, J. W. "The Poincare ´ Disc Model." §4.1 in Hyper- bolic Geometry. New York: Springer-Verlag, pp. 95 /C1/104, 1999. Goodman-Strauss, C. "Compass and Straightedge in the Poincare ´ Disk." To be submitted. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 188 /C1/189, 1991.Poincare ´ Manifold A nonsimply connected 3-manifold also called a DODECAHEDRAL SPACE . References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 245, 290, and 308, 1976. Poincare ´ Metric The METRIC ds2 /C30dx2 /C27 dy2 1 /C28 zjj29+;k9+;72 of the POINCARE ´ HYPERBOLIC DISK. See also POINCARE ´ HYPERBOLIC DISK Poincare ´ Separation Theorem Let yk9+89+9 be a set of orthonormal vectors with k/C301, 2, ...,K, such that the INNER PRODUCT yk;yk9+=9+; /C301:Then set x/C30XK k/C301ukyk(1) so that for any SQUARE MATRIX Afor which the product Axis defined, the corresponding QUADRATIC FORM is (x;Ax)/C30XK k;l/C301ukulyk;Ayl9+;k9+;7 (2) Then if Bk/C30yk;Ayl9+;k9+;7 (3) fork;l/C301;2, ..., K, it follows that liBKðÞ5l1(A) (4) lK/C28j(BK)]lN/C28j(A) (5) fori/C301, 2, ..., Kandj/C300, 1, ..., K/C281:/ References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1120, 2000. Poincare ´-Bertrand Theorem Fors1;s2/C3091; lim e100 e2001 x1/C28is1e11 x2/C28is2e2 /C30PV1 x1 ! /C27ips1d(x1)"# PV1 x2 ! /C27ips2d(x2)"# /C27p2 d x1ðÞd x2ðÞ; (1) where d(x) is the DELTA FUNCTION and PV denotes the CAUCHY PRINCIPAL VALUE . See also DELTA FUNCTION Poincare ´-Birkhoff Fixed Point Theorem For the rational curve of an unperturbed system with ROTATION NUMBER r =s under a map T (for which every point is a FIXED POINT of Ts) ; only an even number of FIXED POINTS 2ks (k /C301, 2, ...) will remain under perturbation. These FIXED POINTS are alter- nately stable (ELLIPTIC ) and unstable (HYPERBOLIC ). Around each elliptic fixed point there is a simulta- neous application of the Poincare ´-Birkhoff fixed point theorem and the KAM THEOREM , which leads to a self-similar structure on all scales. The original formulation was: Given a CONFORMAL ONE-TO-ONE transformation from an ANNULUS to itself that advances points on the outer edge positively and on the inner edge negatively, then there are at least two fixed points. It was conjectured by Poincare ´ from a consideration of the three-body problem in celestial mechanics and proved by Birkhoff. Poincare ´-Birkhoff-Witt Theorem Every LIE ALGEBRA L is isomorphic to a SUBALGEBRA of some LIE ALGEBRA A/C28; where the ASSOCIATIVE ALGEBRA A may be taken to be the linear operators over a VECTOR SPACE V. See also ASSOCIATIVE ,LIE ALGEBRA ,VECTOR SPACE References Jacobson, N. Lie Algebras. New York: Dover, pp. 159 /C1/160, 1979. Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 3, 1996. Poincare ´-Fuchs-Klein Automorphic Function f(z) /C30k (cz /C27 d)r faz /C27 b cz /C27 d ! where I[z] > 0 :/ See also AUTOMORPHIC FUNCTION Poincare ´-Hopf Index Theorem The index of a VECTOR FIELD with finitely many zeros on a compact, oriented MANIFOLD is the same as the EULER CHARACTERISTIC of the MANIFOLD . See also GAUSS- BONNET FORMULAPoincare ´’s Holomorphic Lemma Solutions to HOLOMORPHIC differential equations are themselves HOLOMORPHIC FUNCTIONS of time, initial conditions, and parameters. See also POINCARE ´ ’S LEMMA Poincare ´’s Lemma Poincare ´’s lemma says that on a CONTRACTIBLE MANIFOLD , all CLOSED FORMS are EXACT . While d2 /C30 0 implies that all exact forms are closed, it is not always true that all closed forms are exact. The Poincare ´ lemma is used to show that closed forms represent COHOMOLOGY CLASSES . See also COHOMOLOGY ,COHOMOLOGY CLASS ,CLOSED FORM, DE RHAM COHOMOLOGY ,DIFFERENTIAL FORM, EXACT FORM,E XTERIOR DERIVATIVE ,M ANIFOLD , POINCARE ´ ’S HOLOMORPHIC LEMMA ,STOKES’ THEO- REM,W EDGE PRODUCT Poincare ´’s Theorem If 9/C29F /C300 (i.e., F(x)isan IRROTATIONAL FIELD )ina simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the GRADIENT of a SCALAR FIELD f(x) ; F(x) /C30/C289 f(x) (1) for x /C23 U(x) ; where 9 is the gradient operator. Conse- quently, the GRADIENT THEOREM gives gsF /C215 ds /C30 f x1ðÞ/C28 f x2ðÞ (2) for any path s located completely within U(x) ; starting at x1 and ending at x2 :/ This means that if 9/C29F /C300; the LINE INTEGRAL of F is path-independent. See also CONSERVATIVE FIELD,GRADIENT THEOREM , IRROTATIONAL FIELD,LINE INTEGRAL Poinsot Solid KEPLER- POINSOT SOLID Poinsot’s Spirals References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 194, 1972. Point A0- DIMENSIONAL mathematical object which can be specified in n-D space using n coordinates. Although the notion of a point is intuitively rather clear, the mathematical machinery used to deal with points and point-like objects can be surprisingly slippery. This difficulty was encountered by none other than Euclid himself who, in his ELEMENTS , gave the vague definition of a point as "that which has no part." The basic geometric structures of higher DIMEN- SIONAL geometry–the LINE, PLANE , SPACE , and HYPER- SPACE –are all built up of infinite numbers of points arranged in particular ways. The DECIMAL POINT in a DECIMAL EXPANSION is voiced as "point" in the United States, e.g., 3.1415 is voiced "three point one four one five," whereas a COMMA is used for this purpose in continental Europe. See also ACCUMULATION POINT ,B OUNDARY POINT , BRANCH POINT ,C OMMA ,C ONCUR ,C ONCURRENT , CRITICAL POINT ,D OUBLE POINT ,E NDPOINT ,FIXEDPOINT ,ISOLATED POINT ,L IMIT POINT ,M IDPOINT , ORDINARY POINT ,S INGULAR POINT (ALGEBRAIC CURVE ), SINGULAR POINT (FUNCTION ) References Casey, J. "The Point." Ch. 1 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 1 /C1/29, 1893. Lachlan, R. "Special Points Connected with a Triangle." §112 /C1/117 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 62 /C1/66, 1893. Point at Infinity P is the point on the line AB such that PA=PB /C301 : It can also be thought of as the point of intersection of two PARALLEL lines. In 1639, Desargues (1864) became the first to consider the point at infinity (Cremona 1960, p. ix), although Poncelet was the first to systematically employ the point at infinity (Graus- tein 1930). The term point at infinity is also used for COMPLEX INFINITY (Krantz 1999, p. 82). See also COMPLEX INFINITY ,LINE AT INFINITY References Behnke, H.; Bachmann, F.; Fladt, K.; and Suss, W. (Eds.). Ch. 7 in Fundamentals of Mathematics, Vol. 3: Points at Infinity. Cambridge, MA: MIT Press, 1974. Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960. Desargues, G. "Brouillon-projet d’une atteinte aux e´ve´nements des recontres d’un coˆne avec un plan." Œuvres de Desargues, re´unies et analyse ´es par M. Pudra, tome 1. Paris, pp. 104, 105, and 205, 1864. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 38, 1928. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 30, 1930. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 82, 1999. Lachlan, R. "Point at Infinity." §9inAn Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 5 /C1/6, 1893. Point Circle Members of a COAXAL SYSTEM satisfy x2 /C27y2 /C272 lx /C27c /C30 x /C27 l ðÞ2/C27y2 /C27c /C28 l2 /C300 for values of l: Picking l2 /C30c then gives the two circles x9ffiffifficp9+=9+;2/C27y2/C300 of zero RADIUS , known as point circles. The two point circles 9ffiffifficp;0 ðÞ ;real or imaginary, are called the LIMITING POINTS of the COAXAL SYSTEM . See also COAXAL SYSTEM ,LIMITING POINT References Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 123, 1928. Point Connectivity VERTEX CONNECTIVITY Point Distances The maximum distance between n points in 3-D can occur no more than 2n /C282 times. Also, there exists a fixed number c such that no distance determined by a set of n points in 3-D space occurs more than cn5=3 times. The maximum distance can occur no more than1 4 n2jk times in 4-D, where xbcis the FLOOR FUNCTION . See also POINT- LINE DISTANCE–2- D, POINT- LINE DIS- TANCE–3- D, POINT- POINT DISTANCE–2- D, POINT- POINT DISTANCE–3- D, SPAN (GEOMETRY ) References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 122 /C1/123, 1976. Point Estimation Theory A theory of constructing initial conditions that pro- vides safe convergence of a numerical root-finding algorithm for an equation f(z) /C300 : Point estimation theory treats convergence conditions and the domain of convergence using only information about f at the initial point z0(Petkovic et al. 1997, p. 1). An initial point that provides safe convergence of NEWTON’S METHOD is called an APPROXIMATE ZERO . Point estimation theory should not be confusion with POINT ESTIMATORS of probability theory. See also ALPHA- TEST,APPROXIMATE ZERO,NEWTON’S METHOD ,POINT ESTIMATOR References Lehmann, E. L. and Casella, G. Theory of Point Estimation. New York: Springer-Verlag, 1998. Petkovic, M. S.; Herceg, D. D.; and Ilic, S. M. Point Estima- tion Theory and Its Applications. Novi Sad, Yugoslavia: Institute of Mathematics, 1997. Point Estimator An ESTIMATOR of the actual values of population. See also POINT ESTIMATION THEORY Point Groups A point group is a group of symmetry operations which all leave at least one point unmoved. Although an isolated object may have an arbitrary SCHO¨ NFLIES SYMBOL , the requirement that symmetry be present in a lattice requires that only 1, 2, 3, and 6-fold symmetry axes are possible (the CRYSTALLOGRAPHYRESTRICTION ), which restricts the number of possible so-called CRYSTALLOGRAPHIC POINT GROUPS to 32. See also CRYSTALLOGRAPHIC POINT GROUPS ,CRYSTAL- LOGRAPHY RESTRICTION ,SCHO¨ NFLIES SYMBOL ,SPACE GROUPS References Hahn, T. (Ed.). International Tables for Crystallography, vol. A, 4th ed. Dordrecht, Netherlands: Kluwer, p. 752, 1995. Point Lattice A regularly spaced array of points falling along regularly spaced line. The grid lines can be orientedto form unit cells in the shape of a square, rectangle, hexagon, etc. However, unless otherwise specified, point lattices are generally taken to refer to points ina square array, i.e., points with coordinates(m;n;/C1/C1/C1);where m,n, ... are INTEGERS . Such an array is often called a GRID or a MESH . Point lattices are frequently simply called "lattices," which unfor-tunately conflicts with the same term applied to ordered sets treated in LATTICE THEORY . Formally, a lattice is a DISCRETE SUBGROUP of EUCLIDEAN SPACE , assuming it contains the origin. That is, a lattice is closed under addition andinverses, and every point has a neighborhood in which it is the only lattice point. The commonexamples are ZƒRandZ 2ƒR2:Usually, a lattice is defined to have full rank, i.e., a lattice in Rnis the SUBGROUP a1v1/C27/C1/C1/C1anvn fg ; (1) where the aiare integers and viare LINEARLY INDEPENDENT vectors. Note that a lattice needs at most nelements to generate it. For example, the subgroup a1/C27a2ffiffiffi 2p9+89+9 ƒRrequires two generators but is not DISCRETE , and is not a lattice. The above illustration shows that the subgroup generated by 1 and 1 =ffiffiffi 2p is not a lattice by showing a/C27b=ffiffiffi2p for successive b/C23[0;1]: / The FRACTION of lattice points VISIBLE from the ORIGIN , as derived in Castellanos (1988, pp. 155 /C1/ 156), is N ?(r) N(r)/C3024 p2 r2 /C27 O(r ln r) 4r2 /C27 O(r) /C306 p2 /C27 Oln r r ! 1 /C27 O1 r ! /C306 p2 : (2) Therefore, this is also the probability that two randomly picked integers will be RELATIVELY PRIME to one another. For 2 5n 532 ; it is possible to select 2n lattice points with x; y /C23 [1; n] such that no three are in a straight LINE. The number of distinct solutions (not counting reflections and rotations) for n /C301, 2, ..., are 1, 1, 4, 5, 11, 22, 57, 51, 156 ... (Sloane’s A000769). For large n, it is conjectured that it is only possible to select at most (c /C27 e)n lattice points with no three COLLINEAR , where c /C30 2p2 =39+=9+;1 =3:1 :87 (3) (Guy and Kelly 1968; Guy 1994, p. 242). The number of the n2lattice points x; y /C23 [1; n] which can be picked with no four CONCYCLIC is O(n2 =3 /C28 e) (Guy 1994, p. 241). Any PARALLELOGRAM on the lattice in which two opposite sides each have length 1 has unit area (Hilbert and Cohn-Vossen 1999, pp. 33 /C1/34). A special set of POLYGONS defined on the regular lattice are the GOLYGONS .A NECESSARY and SUFFI- CIENT condition that a linear transformation trans- forms a lattice to itself is that it be UNIMODULAR . M. Ajtai has shown that there is no efficient ALGO- RITHM for finding any fraction of a set of spanning vectors in a lattice having the shortest lengths unless there is an efficient algorithm for all of them (of which none is known). This result has potential applications to cryptography and authentication (Cipra 1996).See also BARNES- WALL LATTICE ,BLICHFELDT’S THEO- REM,BROWKIN’S THEOREM ,CIRCLE LATTICE POINTS , COXETER- TODD LATTICE ,EHRHART POLYNOMIAL ,EL- LIPTIC CURVE ,GAUSS’S CIRCLE PROBLEM ,GOLYGON , INTEGRATION LATTICE ,JARNICK’S INEQUALITY ,LAT- TICE PATH,LATTICE SUM,LEECH LATTICE ,MINKOWS- KI CONVEX BODY THEOREM ,M ODULAR LATTICE ,N - CLUSTER ,NOSARZEWSKA’S INEQUALITY ,PICK’S THEO- REM,RANDOM WALK,SCHINZEL’S THEOREM ,SCHRO ¨ - DER NUMBER ,TORUS ,UNIT LATTICE ,VISIBLE POINT , VORONOI POLYGON References Apostol, T. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995. Castellanos, D. "The Ubiquitous Pi." Math. Mag. 61,67/C1/98, 1988. Cipra, B. "Lattices May Put Security Codes on a Firmer Footing." Science 273, 1047 /C1/1048, 1996. Eppstein, D. "Lattice Theory and Geometry of Numbers." http://www.ics.uci.edu/~eppstein/junkyard/lattice.html. Gardner, M. "The Lattice of Integer." Ch. 21 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 208 /C1/219, 1984. Guy, R. K. "Gauss’s Lattice Point Problem," "Lattice Points with Distinct Distances," "Lattice Points, No Four on a Circle," and "The No-Three-in-a-Line Problem." §F1, F2, F3, and F4 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 240 /C1/244, 1994. Guy, R. K. and Kelly, P. A. "The No-Three-in-Line-Pro- blem." Canad. Math. Bull. 11, 527 /C1/531, 1968. Hammer, J. Unsolved Problems Concerning Lattice Points. London: Pitman, 1977. Hilbert, D. and Cohn-Vossen, S. "Regular Systems of Points." Ch. 2 in Geometry and the Imagination. New York: Chelsea, pp. 32 /C1/93, 1999. Knupp, P. and Steinberg, S. Fundamentals of Grid Genera- tion. Boca Raton, FL: CRC Press, 1994. Nagell, T. "Lattice Points and Point Lattices." §11 in Introduction to Number Theory. New York: Wiley, pp. 32 /C1/34, 1951. Sloane, N. J. A. Sequences A000769/M3252 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Thompson, J. F.; Soni, B.; and Weatherill, N. Handbook of Grid Generation. Boca Raton, FL: CRC Press, 1998. Point Picking In finding the average area ¯ARof a triangle chosen from a closed, bounded, convex region Rof the plane, then ¯AT(R)/C30¯AR;forTany nonsingular affine trans- formation of the plane. See also 18-POINT PROBLEM ,BALL LINE PICKING ,BALL TRIANGLE PICKING ,CUBE LINE PICKING ,CUBE POINT PICKING ,CUBE TETRAHEDRON PICKING ,CUBE TRIAN- GLE PICKING ,D ISCREPANCY THEOREM ,D ISK LINE PICKING ,DISK POINT PICKING ,DISK TRIANGLE PICK- ING,HAPPY END PROBLEM ,PLANAR DISTANCE ,POINT- POINT DISTANCE–1- D, POINT- POINT DISTANCE–2- D, POINT- POINT DISTANCE–3- D, SIMPLEX POINT PICKING , SPHERE LINE PICKING ,S PHERE POINT PICKING , SPHERE TETRAHEDRON PICKING ,SYLVESTER’S FOUR- POINT PROBLEM ,TRIANGLE POINT PICKING References Pfiefer, R. E. "The Historical Development of J. J. Sylves- ter’s Four Point Problem." Math. Mag. 62, 309 /C1/17, 1989. Point Probability The portion of the probability distribution which has a P-VALUE equal to the observed P-VALUE . See also TAIL PROBABILITY Point-Line Distance * /2-D Given a line ax/C27by/C27c/C300 and a point x0;y0 ðÞ ;in slope-intercept form, the equation of the line is y/C30/C28a bx/C28c b; (1) so the line has SLOPE /C28a=b:Points on the line have the vector coordinates x /C28a bx/C28c d2 435/C300 /C28 c d2435/C28 1 b/C28b a9+$=9+$; x: (2) Therefore, the VECTOR /C28b a9+$=9+$; (3) isPARALLEL to the line, and the VECTOR v/C30a b9+$=9+$; (4) isPERPENDICULAR to it. Now, a VECTOR from the point to the line is given by r/C30x/C28x0 y/C28y09+$=9+$; (5)Projecting ronto v, d/C30projvr jj /C30v /C215r jj v/C30ˆv /C215r jj/C30a(x/C28x0)/C27b(y/C28y0) jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2p /C30ax/C27by/C28ax0/C28by0 jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2p /C30ax0/C27by0/C27c jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2p : (6) If the line is represented by the endpoints of a VECTOR x1;y1 ðÞ and x2;y2 ðÞ ;then the PERPENDICULAR VECTOR is v/C30y2/C28y1 /C28(x2/C28x1)9+$=9+$; (7) ˆv/C301 sy2/C28y1 /C28(x2/C28x1)9+$=9+$; ; (8) where s/C30vjj/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C28x1 ðÞ2/C27y2/C28y1 ðÞ2q ; (9) so the distance is d/C30ˆv /C215r jj/C30y2/C28y1 ðÞ x0/C28x1 ðÞ /C28x2/C28x1 ðÞ y0/C28y1 ðÞ jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C28x1 ðÞ2/C27y2/C28y1 ðÞ2q : (10) The distance from a point x0;y0 ðÞ to the line y/C30 a/C27bxcan also be computed using simple VECTOR algebra. Let Lbe a VECTOR in the same direction as the line L/C30x a/C27bx9+$=9+$; /C280 a9+$=9+$; /C30x bx9+$=9+$; (11) ˆL/C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C271p1 b9+$=9+$; : (12) A given point on the line is x/C30x0 y09+$=9+$; /C280 /C28a9+$=9+$; /C30x0 y0/C28a9+$=9+$; ; (13) so the point-line distance is r /C30 x /C215 ˆL9+=9+;ˆL /C28x /C301 1 /C27 b2x0 y0 /C28a9+$=9+$; /C2151 v9+$=9+$;9+;89+;9 1 b9+$=9+$; /C28x0 y0 /C28a9+$=9+$; /C30y0 /C28 a /C27 bx0 ðÞ 1 /C27 b2b /C2819+$=9+$; : (14) Therefore, d /C30 rjj/C30y0 /C28 a /C27 bx0 ðÞ jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 b2p : (15) This result can also be obtained much more simply by noting that the PERPENDICULAR distance is just cos u times the vertical distance y0 /C28 a /C27bx1 ðÞ jj : But the SLOPE b is just tan u; so sin2 u /C27cos2 u /C301 [tan2 u /C271 /C301 cos2 u ; (16) and cos u /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 tan2 up /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 b2p : (17) The PERPENDICULAR distance is then d /C30y0 /C28 a /C27 bx1 ðÞ jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 b2p ; (18) the same result as before. See also LINE,POINT ,POINT- LINE DISTANCE–3- D Point-Line Distance * /3-D Let a line in 3-D be specified by two points x1 and x2 lying on it, so a vector along the line is given by v /C30x1 /C27(x2 /C28x1)t y1 /C27(y2 /C28y1)t z1 /C27(z2 /C28z1)t2 435: (1) The distance between a point on the line withparameter t and a point (x 0 ; y0 ; z0) is therefore r2 /C30 x1 /C28x0 /C27(x2 /C28x1)t ½/C1382/C27y1 /C28y0 ½ /C27(y2 /C28y1)t/C1382 /C27 z1 /C28z0 /C27(z2 /C28z1)t ½/C1382: (2) To minimize the distance, set dr2ðÞ=dt /C300 and solve for t to obtain t /C30f =g; where f /C30 x1 /C28x0 ðÞ x2 /C28x1 ðÞ /C27 y1 /C28y0 ðÞ y2 /C28y1 ðÞ /C27 z1 /C28z0 ðÞ z2 /C28z1 ðÞ (3) g /C30 x2 /C28x1 ðÞ2/C27 y2 /C28y1 ðÞ2/C27 z2 /C28z1 ðÞ2; (4) and the minimum distance can then be found by plugging t into (2) and taking the SQUARE ROOT . This can be implemented in Mathematica as PointLineDistance[{x1_,x2_},x0_]: /C30Module[ {t /C30-(x1-x0).#/#.#&[x2-x1]}, Sqrt[#.#&[x1-x0 /C27t(x2-x1)]] ] See also LINE,POINT ,POINT- LINE DISTANCE–2- D Point-Plane Distance Given a PLANE ax/C27by/C27cz/C27d/C300 (1) and a point ( x0;y0;z0);the NORMAL to the PLANE is given by v/C30a b c2 435; (2) and a VECTOR from the plane to the point is given by w/C30/C28x/C28x0 y/C28y0 z/C28z02 435: (3) Projecting w onto v, D /C30 projv w jj /C30½v /C215 w½ ½v½ /C30a(x /C28 x0) /C27 b(y /C28 y0) /C27 c(z /C28 z0) jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2 /C27 c2p /C30ax /C27 by /C27 cz /C28 ax0 /C28 by0 /C28 cz0 jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2 /C27 c2p /C30ax0 /C27 by0 /C27 cz0 /C27 d jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2 /C27 c2p : (4) Given three points xi for i /C301, 2, 3, compute the unit normal ˆn /C30(x2 /C28 x1) /C29 (x3 /C28 x1) (x2 /C28 x1) /C29 (x3 /C28 x1) jj: (5) Then the distance from a point x0to the plane containing the three points is given by Di /C30ˆn /C215(xi /C28x0) ; (6) where xi is any of the three points. Expanding out the coordinates shows that D /C13D1 /C30D2 /C30D3 ; (7) as it must since all points are in the same plane, although this is far from obvious based on the abovevector equation. See also P ROJECTION THEOREM Point-Point Distance * /1-D Given a unit LINE SEGMENT [0;1];pick two points at random on it. Call the first point x1and the second point x2:Find the distribution of distances dbetween points. The probability of the points being a ( POSI- TIVE) distance dapart (i.e., without regard to order- ing) is given by P(d)/C30g1 0g1 0dd/C28x2/C28x1 jj ðÞ dx1dx2 g1 0g1 0dx1dx2 /C30(1/C28d)[H(1/C28d)/C28H(d/C281)/C27H(d)/C28H(/C28d)] /C302(1/C28d) for 0 5d51 0 otherwise ;9+$k (1) where dis the D IRAC DELTA FUNCTION and His the HEAVISIDE STEP FUNCTION . The MOMENTS are then m?m/C30g1 0dmp(d)dd/C302g1 0dm(1/C28d)dd/C302dm/C271 m/C271/C28dm/C272 m/C272"#1 0 /C3021 m/C271/C281 m/C272 ! /C302(m/C272)/C28(m/C271) (m/C271)(m/C272)"# /C302 (m/C271)(m/C272) /C301 (n/C271)(2n/C271)form/C302n 1 (n/C271)(2n/C273)form/C302n/C2718 >>>< >>>:(2) (Uspensky 1934, p. 257), giving RAW MOMENTS m?1/C301 3(3) m?2/C301 6(4) m?3/C301 10(5) m?4/C301 15: (6) The MOMENTS can also be computed directly without explicit knowledge of the distribution m?1/C30g1 0g1 0x2/C28x1 jj dx1dx2 g1 0g1 0dx1dx2 /C30g1 0g1 0x2/C28x1 jj dx1dx2 /C30g1 0g1 0 x2/C28x1>0x2/C28x1 ðÞ dx1dx2/C27g1 0g1 0 x2/C28x1B0x1/C28x2 ðÞ dx1dx2 /C30g1 0g1 x1x2/C28x1 ðÞ dx1dx2/C27g1 0gx1 01x2/C28x1 ðÞ dx1dx2 /C30g1 01 2x2 2/C28x1x2hi1 x1dx1/C27g1 0x1x2/C281 2x2 2hix1 0dx1 /C30g1 01 2/C28x19+;k9+;7 /C2812x2 1/C28x219+;k9+;7hi dx1 /C27g1 0x21/C281 2x2 19+;k9+;7 /C28(0/C280)hi dx1 /C30g1 01 2/C28x1/C27x2 19+;k9+;7 dx1/C301 2x1/C2812x2 1/C271 3x3 1hi1 0 /C301 2/C2812/C27139+;k9+;7 /C28(0/C280/C270)/C3013 (7) m?2/C30g1 0g1 0x2/C28x1 jjðÞ2dx2dx1 /C30g1 0g1 0x2 /C28x1 ðÞ2dx1 dx2 /C30g1 0g1 0x2 2 /C282x1x2 /C27x219+=9+; dx1 dx2 /C30g1 01 3 x3 2 /C28x1x22 /C27x21x2hi1 0dx1 /C30g1 01 3 /C28x1 /C27x2 19+;k9+;7 dx1 /C301 3 x3 1 /C281 2 x2 1 /C271 3 x1hi1 0 /C301 3 /C2812 /C2713 /C3016 : (8) The CENTRAL MOMENTS are therefore m2 /C30 m ?2 /C28 m ?12/C3016 /C28139+;k9+;72 /C301 18 (9) m3 /C30 m?3 /C283m ?2 m?1 /C272 m ?1ðÞ3/C301 135 (10) m4 /C30 m?4 /C284m?3 m?1 /C276m?2m?1ðÞ2/C283 m ?1ðÞ4/C301 135 ; (11) so the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C30 m?1 /C301 3 (12) s2 /C30 m2 /C301 18 (13) g1 /C30m3 s3 /C3025ffiffiffi 2p (14) g2 /C30m4 s4 /C283 /C30/C283 5 : (15) The probability distribution of the distance between two points randomly picked on a LINE SEGMENT is germane to the problem of determining the access time of computer hard drives. In fact, the average access time for a hard drive is precisely the time required to seek across 1/3 of the tracks (Benedict 1995). See also POINT- POINT DISTANCE–2- D, POINT- POINT DISTANCE–3- D, POINT- QUADRATIC DISTANCE ,SPHERE POINT PICKING References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 930 /C1/31, 1985. Benedict, B. Using Norton Utilities for the Macintosh. Indianapolis, IN: Que, pp. B-8-B-9, 1995. Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, p. 257, 1937. Point-Point Distance * /2-D Given two points in the PLANE , find the curve which minimizes the distance between them. The LINE ELEMENT is given by ds/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2/C27dy2p ; (1)so the ARC LENGTH between the points x1andx2is L/C30gds/C30gx2 x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q dx; (2) where y?/C13dy=dxand the quantity we are minimizing is f/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q : (3) Finding the derivatives gives @f @y/C300 (4) d dx@f @y?/C30d dx1/C27y?29+=9+; /C281=2y?hi ; (5) so the E ULER- LAGRANGE DIFFERENTIAL EQUATION becomes @f @y/C28d dx@f @y?/C30d dxy?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2p ! /C300: (6) Integrating and rearranging, y?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2p /C30c (7) y?2/C30c21/C27y?29+=9+; (8) y?21/C28c29+=9+; /C30c2(9) y?/C30cffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28c2p /C13a: (10) The solution is therefore y/C30ax/C27b; (11) which is a straight LINE. Now verify that the ARC LENGTH is indeed the straight-line distance between the points. aandbare determined from y1/C30ax1/C27b: (12) y2/C30ax2/C27b: (13) Writing (12) and (13) as a MATRIX EQUATION gives y1 y29+$=9+$; /C30x11 x219+$=9+$; a b9+$=9+$; (14) a b9+$=9+$; /C30x11 x219+$=9+$;/C281y1 y29+$=9+$; /C301 x1/C28x21/C281 /C28x2x19+$=9+$;/C281y1 y29+$=9+$; ; (15) so a/C30y1/C28y2 x1/C28x2/C30y2/C28y1 x2/C28x1(16) b /C30x1y2 /C28 x2y1 x1 /C28 x2(17) L /C30gx2 x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y?2q dy /C30 x2 /C28x1 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27a2p /C30 x2 /C28x1 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y2 /C28 y1 x2 /C28 x1 !2vuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 /C28x1 ðÞ2/C27 y2 /C28y1 ðÞ2q ; (18) as expected. The shortest distance between two points on a SPHERE is the so-called GREAT CIRCLE distance. See also CALCULUS OF VARIATIONS ,CIRCLE TRIANGLE PICKING ,GREAT CIRCLE ,POINT- POINT DISTANCE–1- D, POINT- POINT DISTANCE–3- D, POINT- QUADRATIC DIS- TANCE ,SPHERE POINT PICKING References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 930 /C1/31, 1985. Point-Point Distance * /3-D The LINE ELEMENT is ds /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 /C27dy2 /C27dz2p ; (1) so the ARC LENGTH between the points x1 and x2 is L /C30g ds /C30gx2 x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y ?2 /C27z ?2q dx (2) and the quantity we are minimizing is f /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27y? 2 /C27z?2q : (3) Finding the derivatives gives @f @y /C300 (4) @f @z /C300 (5) and @f @y?/C30y?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 y?2 /C27 z?2p (6) @f @z?/C30z?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 y?2 /C27 z?2p ; (7) so the EULER- LAGRANGE DIFFERENTIAL EQUATIONS become d dxy?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 y?2 /C27 z ?2p ! /C300 (8)d dxz?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 y?2 /C27 z?2p ! /C300: (9) These give y?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 y?2 /C27 z?2p /C30c1 (10) z ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 y?2 /C27 z?2p /C30c2 : (11) Taking the ratio, z?/C30c2 c1y? (12) y?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 y?2 /C27c2 c1 !2 y?2vuut/C30c1 (13) y?2 /C30c2 11 /C27y?2 /C27c2 c1 !2 y?22 435/C30c 2 1 /C27y?2 c21 /C27c229+=9+; ; (14) which gives y?2 /C30c2 1 1 /C28 c2 1 /C28 c22/C13a2 1 (15) z ?2 /C30c2 c1 !2 y?2 /C30c2 2 1 /C28 c2 1 /C28 c22/C13b2 1 : (16) Therefore, /y?¼ a1/ and /z?¼ b1/, so the solution is x y z2 435/C30x a 1x /C27a0 b1x /C27b02435; (17) which is the parametric representation of a straight line with parameter x /C23 x 1;x2 ½/C138 :Verifying the ARC LENGTH gives L/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27a2 1/C27b21q x2/C28x1 ðÞ (18) where y1 y29+$=9+$; /C30x11 x219+$=9+$; a1 a09+$=9+$; (19) z1 z29+$=9+$; /C30x11 x219+$=9+$; b1 b09+$=9+$; : (20) See also POINT- POINT DISTANCE–1- D, POINT- POINT DISTANCE–2- D, POINT- QUADRATIC DISTANCE Point-Quadratic Distance To find the minimum distance between a point in the plane x0 ; y0 ðÞ and a quadratic PLANE CURVE y /C30a0 /C27a1x /C27a2x2 ; (1) note that the square of the distance is r2 /C30 x /C28x0 ðÞ2/C27 y /C28y0 ðÞ2 /C30 x /C28x0 ðÞ2/C27 a0 /C27a1x /C27a2x2 /C28y09+=9+;2: (2) Minimizing the distance squared is equivalent to minimizing the distance (since r2 and ½r ½ have minima at the same point), so take @ðr2 Þ @x¼ 2ðx /C28x0 Þþ2ða0 þ a1x þ a2x2 /C28y0 Þða1 þ 2a2xÞ ¼ 0 ð3Þ x /C28x0 /C27a0a1 /C27a2 1 /C27a1a2x2 /C28a1y0 /C272a0a2x /C272a1a2x2 /C272a22x3 /C282a2y0x /C300 (4) 2a22x3 /C273a1a2x2 /C27 a21 /C272a0a2 /C282a2y0 /C2719+=9+; x /C27 a0a1 /C28a1y0 /C28x0 ðÞ /C300 : (5) Minimizing the distance to find the closest point (x/C31; y /C31) therefore requires solution of a CUBIC EQUA- TION . See also POINT- POINT DISTANCE–1- D, POINT- POINT DISTANCE–2- D, POINT- POINT DISTANCE–3- D Points Problem SHARING PROBLEMPoint-Set Topology The low-level language of TOPOLOGY , which is not really considered a separate "branch" of TOPOLOGY . Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or "closeness" on SPACES . Basic point-set topological notions are ones like CONTINUITY , DIMENSION , COMPACTNESS , and CON- NECTEDNESS . The INTERMEDIATE VALUE THEOREM (which states that if a path in the real line connects two numbers, then it passes over every point between the two) is a basic topological result. Others are that EUCLIDEAN n-space is HOMEOMORPHIC to EUCLIDEAN m-space IFF m /C30n, and that REAL valued functions achieve maxima and minima on COMPACT SETS. Foundational point-set topological questions are ones like "when can a topology on a space be derived from a metric?" Point-set topology deals with differing no- tions of continuity and compares them, as well as dealing with their properties. Point-set topology is also the ground-level of inquiry into the geometrical properties of spaces and continuous functions be- tween them, and in that sense, it is the foundation on which the remainder of topology (ALGEBRAIC , DIFFERENTIAL , and LOW-DIMENSIONAL ) stands. See also ALGEBRAIC TOPOLOGY ,DIFFERENTIAL TOPOL- OGY,LOW-DIMENSIONAL TOPOLOGY ,TOPOLOGY References Bing, R. H. "Elementary Point Set Topology." Amer. Math. Monthly 67, 1960. Ferreiro ´s, J. "Origins of the Theory of Point-Sets." Ch. 5 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 95 /C1/7, 1999. Sutherland, W. A. An Introduction to Metric & Topological Spaces. New York: Oxford University Press, 1975. Vaidyanathaswamy, R. Set Topology. New York: Dover, 1999. Pointwise Convergence The hypothesis is that, for X is a MEASURE SPACE , fn(x) 0 f(x) for each x /C23 X ; as n 0/C12: The hypothesis may be weakened to ALMOST EVERYWHERE CONVER- GENCE . See also ALMOST EVERYWHERE CONVERGENCE References Browder, A. Mathematical Analysis: An Introduction. New York: Springer-Verlag, 1996. Pointwise Dimension Dp(x)/C13lim e00lnmBe(x) ðÞ lne; where Be(x)isan n-D BALL of RADIUS e centered at x and m is the PROBABILITY MEASURE . See also BALL,PROBABILITY MEASURE References Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 541 /C1/45, 1995. Poised NEARLY- POISED ,W ELL-POISED Poisson Bracket Let F and G be infinitely differentiable functions of x and p. Then the Poisson bracket is defined by (F ; G) /C30Xn n/C301@F @pn@G @xn/C28@G @pn@F @xn ! : If F and G are functions of x and p only, then the LAGRANGE BRACKET [F, G] collapses the Poisson bracket ( F, G ). See also LAGRANGE BRACKET ,LIE BRACKET References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1004, 1980. Poisson Distribution Given a P OISSON PROCESS , the probability of k changes occurring in a given interval is given by the limit of the BINOMIAL DISTRIBUTION PB(k)/C30n! k!(n/C28k)!n n !k 1/C28n n !n/C28k : (1) As the number of trials becomes very large, (1)approaches the distribution P(k)/C30lim n0/C12PB(k) /C30lim n0/C12n(n/C281)/C1/C1/C1(n/C28k/C271) nknk k! /C21/C28n n !n 1/C28n n !/C28k /C301/C215nk k!/C215e/C28n/C2151/C30nke/C28n k!; (2) which is called the Poisson distribution (Papoulis1984, pp. 101 and 554; Pfeiffer and Schum 1973, p. 200). The Poisson distribution is normalized so that thesum of probabilities equals 1, since X /C12 k/C300P(k)/C30e/C28nX/C12 k/C300nk k!/C30e/C28nen/C301: (3) The ratio of probabilities is given by P(k/C30i/C271) P(k/C30i)/C30ni/C271e/C28n (i/C271)! i! e/C28nni/C30n i/C271: (4) The MOMENT-GENERATING FUNCTION of the Poisson distribution is given by M(t)/C30X/C12 k/C300etknke/C28n k!/C30e/C28nX/C12 k/C300netðÞk k! /C30e/C28nenet/C30enet/C281ðÞ(5) M?(t)/C30netenet/C281ðÞ(6) Mƒ(t)/C30netðÞ2enet/C281ðÞ/C27netenet/C281ðÞ(7) R(t)/C13lnM(t)/C30net/C281 ðÞ (8) R?(t)/C30net(9) Rƒ(t)/C30net; (10) so m/C30R?(0)/C30n (11) s2/C30Rƒ(0)/C30n (12) (Papoulis 1984, p. 554). The RAW MOMENTS can also be computed directly by summation, which yields an unexpected connection with S TIRLING NUMBERS OF THE SECOND KIND , X/C12 k/C300e/C28xxk k!kn/C30Xn k/C301xkS(n;k); (13) so m?2 /C30 n(1 /C27 n) (14) m ?3 /C30 n 1 /C273n /C27 n29+=9+; (15) m?4 /C30 n 1 /C277 n /C276n2 /C27 n39+=9+; : (16) The CENTRAL MOMENTS can then be computed as m2 /C30 n (17) m3 /C30 n (18) m4 /C30 n(1 /C273 n) ; (19) so the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C30 n (20) s2 /C30 n (21) g1 /C13m3 s3 /C30n n3 =2 /C30 n /C281 =2 (22) g2 /C13m4 s4 /C283 /C30n(1 /C27 3n) n/C283 /C30n /C27 3n2 /C28 3n2 n2/C30 n /C281 : (23) The CHARACTERISTIC FUNCTION for the Poisson dis- tribution is f(t) /C30en eit/C281ðÞ(24) (Papoulis 1984, pp. 154 and 554), and the CUMULANT- GENERATING FUNCTION is K(h) /C30 n eh /C2819+=9+; /C30 n h /C271 2!h2 /C271 3!h3 /C27... ! ; (25) so kr /C30 n : (26) The Poisson distribution can also be expressed in terms of l /C13n x ; (27) the rate of changes, so that P(k) /C30( lx)ke /C28 lx k!: (28) The MOMENT-GENERATING FUNCTION of a Poisson distribution in two variables is given by M(t) /C30e n1/C27n2 ðÞ et/C281ðÞ: (29) If the independent variables x1 ; x2 ; ..., xN have Poisson distributions with parameters m1 ; m2 ; ..., mN ; thenX /C30XN j/C301xj (30) has a Poisson distribution with parameter m /C30XN j/C301mj : (31) This can be seen since the CUMULANT-GENERATING FUNCTION is Kj(h) /C30 mjeh /C2819+=9+; ; (32) K /C13X jKj(h) /C30 eh /C2819+=9+;X jmj /C30 m eh /C2819+=9+; : (33) A generalization of the Poisson distribution has been used by Saslaw (1989) to model the observed cluster- ing of galaxies in the universe. The form of this distribution is given by fb(N) /C30¯N(1 /C28 b) N!¯N(1 /C28b) /C27Nb9+$9+%N /C281e ¯N(1/C28b)/C28Nb ; (34) where N is the number of galaxies in a volume V, ¯N /C30 ¯nV ; ¯n is the average density of galaxies, and b /C30 /C28W=(2K):0:7090:05;with 05bB1 is the ratio of gravitational energy to the kinetic energy of peculiar motions, Letting b/C300 gives f0(N)/C30e/C28¯N¯NN N!; (35) which is indeed a Poisson distribution with n/C30¯N: Similarly, letting b/C301 gives f1(N)/C300:/ See also BINOMIAL DISTRIBUTION ,POISSON PROCESS , POISSON THEOREM References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 532, 1987. Grimmett, G. and Stirzaker, D. Probability and Random Processes, 2nd ed. Oxford, England: Oxford University Press, 1992. Papoulis, A. "Poisson Process and Shot Noise." Ch. 16 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 554 /C1/76, 1984. Pfeiffer, P. E. and Schum, D. A. Introduction to Applied Probability. New York: Academic Press, 1973. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209 /C1/14, 1992. Saslaw, W. C. "Some Properties of a Statistical Distribution Function for Galaxy Clustering." Astrophys. J. 341, 588/C1/ 98, 1989. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 111 /C1/12, 1992. Poisson Integral There are at least two integrals called the Poisson integral. The first is also known as BESSEL’S SECOND INTEGRAL , Jn(z) /C301 29+;k9+;7n G n /C271 29+;k9+;7 G129+;k9+;7g p 0cos(z cos u) sin2n u du ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND and G(x)isa GAMMA FUNCTION . It can be derived from SONINE’S INTEGRAL . With n /C300, the integral becomes PARSEVAL’S INTEGRAL . In complex analysis, let u : U 0 R be a HARMONIC FUNCTION on a NEIGHBORHOOD of the CLOSED DISK ¯D(0; 1); then for any point z0in the OPEN DISK D(0; 1); uz0ðÞ/C301 2 p g2 p 0ueic9+=9+; 1 /C28 z0jj2 z0 /C28 eic jj2dc: In polar coordinates on ¯D(0; R); uz0ðÞ/C301 2p g2 p 0K(r ; u) f z0 /C27reiu9+=9+; du; (1) where R /C30 z0jjand K(r ; u) is the POISSON KERNEL . For a CIRCLE , u(x; y) /C301 2p g2 p 0u(a cos f ; a sin f) /C2a2/C28R2 a2/C27R2/C282arcos(u/C28f)df: (2) For a SPHERE , u(x;y;z)/C301 4paggSua2/C28R2 a2/C27R2/C282aRcosu ðÞ3=2dS; (3) where cosu/C13x /C215j: (4) See also BESSEL FUNCTION OF THE FIRST KIND, CIRCLE ,HARMONIC FUNCTION ,PARSEVAL’S INTEGRAL , POISSON KERNEL ,SONINE’S INTEGRAL ,SPHERE References Krantz, S. G. "The Poisson Integral." §7.3.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 92 /C1/3, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 373 /C1/74, 1953.Poisson Integral Representation jn(z)/C30zn 2n/C271n!gp 0cos(zcosu) sin2n/C271udu; where jn(z)i sa SPHERICAL BESSEL FUNCTION OF THE FIRST KIND . Poisson Kernel The KERNEL in the P OISSON INTEGRAL , given by K(c)/C301 2p1/C28z0jj2 z0/C28eic jj2(1) for the open UNIT DISK D(0;1):Writing z0/C30reiuand taking D(0;R) gives K(r;u)/C131 2pRR/C27reiu R/C28reiu"# /C301 2pRR/C27reiuðÞ R/C28re/C28iuðÞ R/C28reiu ðÞ R/C28re/C28iu ðÞ"# /C301 2pRR2/C28rR eiu/C28e/C28iuðÞ /C28r2 R2/C28rR eiu/C27e/C28iu ðÞ /C27r2"# ¼1 2pRR2/C272ir R sinu/C28r2 R2/C282Rrcosu/C27r2"# /C301 2pR2/C28r2 R2/C282Rrcosu/C27r2(2) (Krantz 1999, p. 93). In 3-D, u(y)/C30RR2/C28a2ðÞ 4pg2p 0gp 0f(u;f) sin ududf R2/C27a2/C282aRcosg ðÞ3=2; (3) where a/C30½y½and cosg/C30y /C215Rcosusinf Rsinusinf Rcosf2 435: (4) The Poisson kernel for the n- BALL is P(x;z)/C301 2/C28nDnvðÞ (z); (5) where Dnis the outward normal derivative at point z on a unit n-sphere and v(z)/C30½z/C28x½2/C28n/C28½x½2/C28nx ½x½29+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$2/C28n : (6) Letube harmonic on a neighborhood of the closed UNIT DISK ¯D(0;1);then the reproducing property of the Poisson kernal states that for z/C23D(0;1); u(z) /C301 2 p g2 p 0ueic9+=9+; 1 /C28½z½2 z /C28 eic jj2dc (7) (Krantz 1999, p. 94). See also DIRICHLET PROBLEM ,HARMONIC FUNCTION , MEAN-VALUE PROPERTY ,POISSON INTEGRAL ,POISSON KERNEL References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1090, 2000. Krantz, S. G. "The Poisson Kernel." §7.3.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 93, 1999. Poisson Manifold A smooth MANIFOLD with a POISSON BRACKET defined on its FUNCTION SPACE . Poisson Process A Poisson is a process satisfying the following proper- ties. 1. The numbers of changes in nonoverlapping intervals are independent for all intervals. 2. The probability of exactly one change in a sufficiently small interval h /C131=n is P /C30 nh /C13 n =n; where n is the probability of one change and n is the number of TRIALS . 3. The probability of two or more changes in a sufficiently small interval h is essentially 0. In the limit of the number of trials becoming large, the resulting distribution is called a POISSON DIS- TRIBUTION . See also POISSON DISTRIBUTION References Grimmett, G. and Stirzaker, D. Probability and Random Processes, 2nd ed. Oxford, England: Oxford University Press, 1992. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 548 /C1/49, 1984. Poisson Sum Formula A special case of the general result X/C12 n/C30/C28/C12f(x /C27n) /C30X/C12 k /C30/C28/C12e2 pikxg/C12 /C28/C12fx?ðÞe /C282 pikx? dx ? (1) with x /C300, yielding X/C12 n/C30/C28/C12f(n) /C30X/C12 k /C30/C28/C12g/C12 /C28/C12fx?ðÞe /C282 pikx ? dx?: (2) Given f a nonnegative, continuous, decreasing, and Riemann integrable function of [0 ;/C12) ; definec(x) /C30ffiffiffi 2 ps g/C12 0f(t) cos(xt) dt: (3) Then ffiffiffiap1 2 f(0) /C27X/C12 n/C301f(na)"# /C30ffiffiffi bp 1 2 g(0) /C27X/C12 n/C301g(nb)"# (4) whenever ab /C302p; from which it follows that ffiffiffiap1 2 /C27X/C12 n /C301e /C28 a2n2 =2"# /C30ffiffiffi bp 1 2 /C27X/C12 n /C301e /C28 b2n2 =2"# (5) (Apostol 1974, Borwein 1987). References Apostol, T. M. Mathematical Analysis. Reading, MA: Addi- son-Wesley, pp. 332 /C1/33, 1974. Borwein, J. M. and Borwein, P. B. "Poisson Summation." §2.2 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 36 /C1/ 0, 1987. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 14, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 466 /C1/67, 1953. Poisson Theorem Poisson’s theorem give the estimate n! k!(n/C28k)!pkqn/C28k/C2e/C28np(np)k k! for the probability of an event occurring ktimes in n trials with n/C271;p/C101;andnp:npq/C271:/ See also POISSON DISTRIBUTION References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 71, 1984. Poisson Trials A number sofTRIALS in which the probability of success pivaries from trial to trial. Let xbe the number of successes, then var(x)/C30spq/C28ss2 p; (1) where s2 pis the VARIANCE ofpiand q/C13(1/C28p): Uspensky has shown that P(s;x)/C30bmxe/C28m x!; (2) where b/C30[1/C28ug(x)]eh(x)(3) g(x) /C30(s /C28 x)m3 3(s /C28 m)3 /C27x3 2s(s /C28 x) (4) h(x) /C30mx s/C28m2 2s2 (s /C28x) /C28x(x /C28 1) 2s /C30px 21 /C271 m ! /C28(x /C28 m)2 2m"# (5) and u /C23 (0; 1) : The probability that the number of successes is at least x is given by Qm(x) /C30X/C12 r/C30xmre/C28m r!: (6) Uspensky gives the true probability that there are at least x successes in s trials as Pms(x) /C30Qm(x) /C27D; (7) where DjjBex /C281 ðÞ Qm(x /C271) for Qm(x /C271) ]1 2 ex /C281 ðÞ 1 /C28Qm(x /C271) ½/C138 for Qm(x /C271) 512( ð8Þ x /C30m /C271 4 /C27m3 s 2(s /C28 m): (9) See also TRIAL Poisson-Boltzmann Differential Equation The ORDINARY DIFFERENTIAL EQUATION yƒ/C27k xy?/C27 dey /C300: References Chambre ´, P. L. "On the Solution of the Poisson-Boltzmann Equation with Application to the Theory of Thermal Explosions." J. Chem. Phys. 20, 1795 /C1/797, 1952. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997. Poisson-Charlier Function rn( n ; x) /C13(1 /C27 n /C28 n)ffiffiffiffiffiffiffiffiffi n!xnp1 F1(/C28n;1/C27 n /C28n; x) ; where (a)n is a POCHHAMMER SYMBOL and 1F1(a; b; z) is a CONFLUENT HYPERGEOMETRIC FUNCTION . See also POISSON- CHARLIER POLYNOMIAL Poisson-Charlier Polynomial The Poisson-Charlier polynomials ck(x;a) form a SHEFFER SEQUENCE withg(t)/C30eaet/C281ðÞ(1) f(t)/C30aet/C281 ðÞ ; (2) giving the GENERATING FUNCTION X/C12 k/C300ck(x;a) k!tk/C30e/C28ta/C27t a !x : (3) The Sheffer identity is cn(x/C27y;a)/C30Xn k/C300n k9+;89+;9 ak/C28nck(y;a)(x)n/C28k; (4) where ( x)nis a FALLING FACTORIAL (Roman 1984, p. 121). The polynomials satisfy the RECURRENCE RELATION cn/C271(x;a)/C30a/C281xcn(x/C281;a)/C28cn(x;a): (5) These polynomials belong to the distribution da(x) where a(x)i sa STEP FUNCTION with JUMP j(x)/C30e/C28aax(x!)/C281(6) atx/C300, 1, ...for a/C210. They are given by the formulas cn(x;a)/C30Xn n/C300(/C281)n/C28nn n9+;89+;9 n!a/C28nx n9+;89+;9 (7) /C30Xn k/C300n k9+;89+;9 (/C281)n/C28ka/C28k(x)k (8) /C30an(/C281)n[j(x)]/C281Dnj(x/C28n) (9) /C30a/C28nn!Lx/C28n n(a) (10) /C30Xn j/C300xjXn k/C300n k9+;89+;9 (/C281)n/C28ka/C28ks(k;j) (11) wheren k9+=9+; is a BINOMIAL COEFFICIENT ,(x)nis a FALL- ING FACTORIAL ,Lk n(x) is an associated L AGUERRE POLYNOMIAL ,s(n;m)i saS TIRLING NUMBER OF THE FIRST KIND , and Df(x)/C30f(x/C271)/C28f(x) (12) Dnf(x)/C30DDn/C281f(x)9+$9+% /C30f(x/C27n)/C28n 19+;89+;9 f(x/C27n/C281)/C27.../C27(/C281)nf(x): (13) They are normalized so that X/C12 k/C300j(k)cn(k;a)cm(k;a)/C30a/C28nn!dnm; (14) where dmnis the DELTA FUNCTION . The first few polynomials are c0(x; a) /C301 c1(x; a) /C30/C28a /C28 x a c2(x; a) /C30a2 /C28 x /C28 2ax /C27 x2 a2 c3(x; a) /C30/C28a3 /C28 2x /C28 3ax /C28 3a2x /C27 3x2 /C27 3ax2 /C28 x3 a3 : See also LAGUERRE POLYNOMIAL ,POISSON- CHARLIER FUNCTION ,SHEFFER SEQUENCE References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. New York: Krieger, p. 226, 1981. Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, p. 473, 1965. Roman, S. "The Poisson-Charlier Polynomials." §4.3.3 in The Umbral Calculus. New York: Academic Press, pp. 119 /C1/ 22, 1984. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 34 /C1/5, 1975. Poisson’s Bessel Function Formula For R[ n] >/C281=2 ; Jn(z) /C30z 2 !n2 ffiffiffippG n /C271 29+;k9+;7g p =2 0cos(z cos t) sin2n tdt; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND , and G(z) is the GAMMA FUNCTION . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1472, 1980. Poisson’s Equation A second-order PARTIAL DIFFERENTIAL EQUATION aris- ing in physics, 92 c /C30/C284pr : If r /C300; it reduces LAPLACE’S EQUATION . It is also related to the HELMHOLTZ DIFFERENTIAL EQUATION 92 c /C27k2 c /C300 : See also HELMHOLTZ DIFFERENTIAL EQUATION ,LA- PLACE’S EQUATION ,VECTOR POISSON EQUATION References Arfken, G. "Gauss’s Law, Poisson’s Equation." §1.14 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 74 /C1/8, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 271, 1953.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997. Poke Move The REIDEMEISTER MOVE of type II. See also KNOT MOVE,REIDEMEISTER MOVES Poker Poker is a CARD game played with a normal deck of 52 CARDS . Sometimes, additional cards called "jokers" are also used. In straight or draw poker, each player is normally dealt a hand of five cards. Depending on the variant, players then discard and redraw CARDS , trying to improve their hands. Bets are placed at eachdiscard step. The number of possible distinct five-card hands is N/C3052 59+;89+;9 /C302;598;960; where n k9+=9+; is a BINOMIAL COEFFICIENT . There are special names for specific types of hands. Aroyal flush is an ace, king, queen, jack, and 10, all of one suit. A straight flush is five consecutive cards allof the same suit (but not a royal flush), where an ace may count as either high or low. A full house is three- of-a-kind and a pair. A flush is five cards of the samesuit (but not a royal flush or straight flush). A straight is five consecutive cards (but not a royal flush or straight flush), where an ace may again countas either high or low. The probabilities of being dealt five-card poker hands of a given type (before discarding and with no jokers) on the initial deal are given below (Packel 1981). As usual, for a hand with probability P, the ODDS against being dealt it are 1 =rðÞ/C281:1 :/ Hand Exact Probability Probability ODDS royal flush /4 N/C301 649;740// 1:54/C2910/C286/649,739.0:1 straight flush/4(10)/C284 N/C303 216;580// 1:39/C2910/C285/72,192.3:1 four of akind/13(48) N/C301 4;165// 2:40/C2910/C284/4,164.0:1 full house /134 39+=9+; 12429+=9+; N/C306 4;165// 1:44/C2910/C283/693.2:1 flush /413 59+=9+; /C2836/C284 N/C301;277 649;740//1:97/C2910/C283/507.8:1 straight /10 45ðÞ/C28 36 /C28 4 N/C305 1 ;274//3:92 /C2910 /C283/ 253.8:1 three of a kind/13 4 39+=9+;(48)(44) 2! N/C3088 4;165/ 0.0211 46.3:1 two pair /13 4 29+=9+; 12 429+=9+; 2!44 N/C30198 4;165/ 0.0475 20.0:1 one pair /13 4 29+=9+;(48)(44)(40) 3! N/C30352 833/ 0.423 1.366:1 Gadbois (1996) gives probabilities for hands if two jokers are included, and points out that it is impos- sible to rank hands in any single way which is consistent with the relative frequency of the hands. See also BRIDGE CARD GAME,CARDS References Cheung, Y. L. "Why Poker is Played with Five Cards." Math. Gaz. 73, 313 /C1/15, 1989. Conway, J. H. and Guy, R. K. "Choice Numbers with Repetitions." In The Book of Numbers. New York: Springer-Verlag, pp. 70 /C1/1, 1996. Friedman, E. "Erich’s Poker Page." http://www.stetson.edu/ ~efriedma/poker/. Gadbois, S. "Poker with Wild Cards--A Paradox?" Math. Mag. 69, 283 /C1/85, 1996. Jacoby, O. Oswald Jacoby on Poker. New York: Doubleday, 1981. Packel, E. W. The Mathematics of Games and Gambling. Washington, DC: Math. Assoc. Amer., 1981. Rubens, J. Win at Poker. New York: Dover. Sarrett, P. "Poker Game Variants." http://gamereport.com/ poker/. Polar If two points A and A? are INVERSE (sometimes called conjugate) with respect to a CIRCLE (the INVERSION CIRCLE ), then the straight LINE through A? which is PERPENDICULAR to the line of the points AA? is called the polar of A with respect to the CIRCLE , and A is called the POLE of the polar. An incidence-preserving transformation in which points and lines are transformed into their POLES and polars is called RECIPROCATION (a.k.a. construct- ing the dual). The concept of poles and polars can also be general- ized to arbitrary CONIC SECTIONS . If two tangents to a CONIC SECTION at points A and B meet at P, then P is called the POLE of the line AB with respect to the conic and AB is said to be the polar of the point P with respect to the conic (Wells 1991). In the above figure, let a line through the polar P meet a conic section at point X and Y, and let the line XY intersect the polar line AB and Q. Then fXPYQ g form a HARMONIC RANGE (Wells 1991). In the above figure, let two lines through the polar P meet a conic at points Pand Qand Sand T. Then QTandRSare concurrent on the polar (Wells 1991). The concept can be generalized even further to an arbitrary ALGEBRAIC CURVE so that every point has a polar with respect to the curve and every line has a pole (Wells 1991). See also APOLLONIUS’ PROBLEM ,DUAL POLYHEDRON , INVERSE POINTS ,INVERSION CIRCLE ,POLARITY ,POLE (INVERSION ), RECIPROCAL ,RECIPROCATION ,SALMON’S THEOREM ,TRILINEAR POLAR References Casey, J. "Theory of Poles and Polars, and Reciprocation." §6.7 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 141 /C1/48, 1888. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 157, 1965. Durell, C. V. "Poles and Polars." Ch. 9 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 93 /C1/ 7, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 100 /C1/06, 1929. Lachlan, R. "Poles and Polars." §243 /C1/57 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 151 /C1/57, 1893. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 190 /C1/91, 1991. Polar Angle The counterclockwise ANGLE from the X-AXIS at which a point lies. See also POLAR COORDINATES Polar Circle Given a TRIANGLE , the polar circle has center at the ORTHOCENTER H. Call Hithe FEET of the ALTITUDE . Then the RADIUS is r2 /C30HA1/C215HH1 /C30HA2/C215HH2 /C30HA2/C215HH2 (1) /C30/C284R2 cos a1 cos a2 cos a3 (2) /C301 2a2 1 /C27a22 /C27a239+=9+; /C284R2 ; (3) where R is the CIRCUMRADIUS , ai the VERTEX angles, and ai the corresponding side lengths. A TRIANGLE is self-conjugate with respect to its polar circle. Also, the RADICAL AXIS of any two polar circles is the ALTITUDE from the third VERTEX . Any two polar circles of an ORTHOCENTRIC SYSTEM are orthogonal. The polar circles of the triangles of a COMPLETE QUADRILATERAL constitute a COAXAL SYSTEM conju- gate to that of the circles on the diagonals. See also COAXAL SYSTEM ,O RTHOCENTRIC SYSTEM , POLAR ,POLE (INVERSION ), RADICAL AXIS References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 136 /C1/38, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 176 /C1/81, 1929.Polar Coordinates The polar coordinates r(the radial coordinate) and u (the angular coordinate) are defined in terms of CARTESIAN COORDINATES by x/C30rcosu (1) y/C30rsinu; (2) where ris the radial distance from the ORIGIN , and u is the counterclockwise angle from the X-AXIS .I n terms of xandy, r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p (3) u/C30tan/C281y x ! : (4) The ARC LENGTH of a polar curve given by r/C30r(u)i s s/C30gu2 u1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C27dr du !2vuutdu: (5) TheLINE ELEMENT is given by ds2/C30r2du2; (6) and the AREA element by dA/C30rd rd u: (7) The AREA enclosed by a polar curve r/C30r(u)i s A/C301 2gu2 u1r2du: (8) The SLOPE of a polar function r/C30r(u) at the point (r;u) is given by m/C30r/C27tanudr du /C28rtanu/C27dr du: (9) The ANGLE between the tangent and radial line at the point ( r;u)i s c/C30tan/C2819+;8 r dr du9+;9 : (10) A polar curve is symmetric about the X-AXIS if replacing u by /C28u in its equation produces an equivalent equation, symmetric about the Y-AXIS if replacing u by p /C28 u in its equation produces an equivalent equation, and symmetric about the origin if replacing r by /C28r in its equation produces an equivalent equation. In Cartesian coordinates, the POSITION VECTOR and its derivatives are r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p ˆr (11) ˙r /C30˙ˆrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p /C27ˆr(x2 /C27y2) /C281=2(x˙x /C27y˙y) (12) ˆr /C30xˆx /C27 yˆyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2p (13) ˙ˆr /C30˙xˆx /C27 ˙yˆyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2p /C281 2(x2 /C27y2) /C283=2(2)(x˙x /C27y˙y)(xˆx /C27yˆy) /C30(x˙y /C27 y˙x)(xˆy /C28 yˆx) (x2 /C27 y2)3 =2 : (14) In polar coordinates, the UNIT VECTORS and their derivatives are r /C13r cos u r sin u9+$=9+$; (15) ˆr /C13dr dr dr dr9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C30cos u sin u9+$=9+$; (16) ˆu /C13du du du du9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C30/C28sin u cos u9+$=9+$; (17) ˙ˆr /C30/C28sin u ˙u cos u ˙u9+$=9+$; /C30 ˙u ˆu (18) ˙ˆu /C30/C28cos u ˙u sin u ˙u9+$=9+$; /C30/C28 ˙uˆr (19) ˙r /C30/C28r sin u ˙u /C27cos u˙r r cos u ˙u /C27sin u˙r9+$=9+$; /C30r ˙u ˆu /C27 ˙rˆr (20) ¨r /C30 ˙r ˙u ˆu /C27r ¨u ˆu /C27r ˙u ˙ˆu /C27 ¨rˆr /C27 ˙r˙ˆr /C30 ˙r ˙u ˆu /C27r ¨u ˆu /C27r ˙u(/C28˙uˆr) /C27 ¨rˆr /C27 ˙r ˙u ˆu /C30(¨r /C28r ˙u 2)ˆr /C27(2˙r ˙u /C27r ¨u) ˆu /C30 ¨r /C28r ˙u29+=9+; ˆr /C271 rd dtr2 ˙˙u9+;89+;9 ˆu : (21) See also CARDIOID ,C IRCLE ,C ISSOID ,C ONCHOID ,CURVILINEAR COORDINATES ,C YLINDRICAL COORDI- NATES ,EQUIANGULAR SPIRAL ,LEMNISCATE ,LIMAC ¸ ON, ROSE Polar Line POLAR Polar Reciprocals INVERSE POINTS Polar Reciprocation INVERSE POINTS ,RECIPROCATION Polar Representation (Complex Number) PHASOR Polar Representation (Measure) A polar representation of a COMPLEX MEASURE m is analogous to the polar representation of a COMPLEX NUMBER as z /C30reiu ; where r /C30½z½; dm /C30eiud ½m½: (1) The analog of absolute value is the TOTAL VARIATION MEASURE ½ m½; and u is replaced by a MEASURABLE real- valued function u: Or sometimes one writes h with ½h½/C301 instead of eiu :/ More precisely, for any measurable set E, m(E) /C30gEeiu d½ m½; (2) where the integral is the LEBESGUE INTEGRAL .Itis natural to extend the definition of the Lebesgue integral to complex measures using the polar repre- sentation g fdm /C30g eiufd½ m½: (3) See also ABSOLUTELY CONTINUOUS ,COMPLEX MEA- SURE ,FUNDAMENTAL THEOREMS OF CALCULUS ,LE- BESGUE MEASURE ,P OLAR REPRESENTATION (MEASURE ), RADON- NIKODYM THEOREM References Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 124 /C1/25, 1987. Polarity A PROJECTIVE CORRELATION of period two. In a polarity, a is called the POLAR of A, and A the POLE a. See also CHASLES’S THEOREM ,CORRELATION (GEO- METRIC ), POLAR ,POLE (INVERSION ), PROJECTIVE COR- RELATION References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 248, 1969. Polarized Telephone GOSSIPING Pole A HOLOMORPHIC FUNCTION f has a pole of order m at a point z /C30z0if, in the LAURENT SERIES , an /C300 for n B /C28m and am "0 : Equivalently, f has a pole of order n at z0if n is the smallest POSITIVE INTEGER for which (z /C28z0)nf(z) is holomorphic at z0 : A holomorphic function f has a pole at infinity if lim z0/C12f(z) /C30/C12: A nonconstant polynomial P(z) has a pole at infinity of order deg P; i.e., the DEGREE of P. The basic example of a pole is f /C301=zn ; which has a single pole of order n at z /C300. A simple Mathematica function which finds the poles of a RATIONAL FUNC- TION is given by Poles[f_, z_] : /C30 Union[z /. {ToRules[Roots[Denominator[Together[D[f, z]]] /C30/C30 0, z]]}] A HOLOMORPHIC FUNCTION whose only singularities are poles is called a MEROMORPHIC FUNCTION . See also ARGUMENT PRINCIPLE ,ESSENTIAL SINGULAR- ITY,H OLOMORPHIC FUNCTION ,L AURENT SERIES , MEROMORPHIC FUNCTION ,POLE (INVERSION ), REMO- VABLE SINGULARITY ,RESIDUE (COMPLEX ANALYSIS ), SIMPLE POLE,SINGULAR POINT (FUNCTION ) References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 396 /C1/97, 1985. Knopp, K. "Essential and Non-Essential Singularities or Poles." §31 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 123 /C1/26, 1996. Krantz, S. G. "Removable Singularities, Poles, and Essential Singularities." §4.1.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 42, 1999. Pole (Inversion) If two points A and A? are INVERSE with respect to aCIRCLE (the INVERSION CIRCLE ), then the straight line through A? which is PERPENDICULAR to the line of the points AA? is called the POLAR of the POINT A with respect to the CIRCLE , and A is called the pole of the POLAR . An incidence-preserving transformation in which points and lines are transformed into their poles and POLARS is called a RECIPROCATION . The concept of poles and polars can also be general- ized to arbitrary CONIC SECTIONS . If two tangents to a CONIC SECTION at points A and B meet at P, then P is called the pole of the line AB with respect to the conic and AB is said to be the POLAR of the point P with respect to the conic (Wells 1991). Let a line through P meet a conic at points X and Y and its polar AB and Q. Then X, Y, P, and Q are a HARMONIC RANGE (Wells 1991). Furthermore, if two lines through a pole P meet a conic at points Q and R and points S and T, then the lines QTandSRmeet on the polar, as do the lines QSandRT. The concept can be generalized even further to an arbitrary ALGEBRAIC CURVE so that every point has a polar with respect to the curve and every line has apole (Wells 1991). See also D IAGONAL TRIANGLE ,INVERSE POINTS , INVERSION CIRCLE ,POLAR ,POLARITY ,R ECIPROCAL , RECIPROCATION ,TRILINEAR POLAR References Casey, J. "Theory of Poles and Polars, and Reciprocation." §6.7 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., pp. 141 /C1 /48, 1888. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 157, 1965. Durell, C. V. "Poles and Polars." Ch. 9 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 93 /C1/ 7, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 100 /C1/06, 1929. Lachlan, R. "Poles and Polars." §243/C1/57 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 151 /C1/57, 1893. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 190 /C1/91, 1991. Pole (Origin) ORIGIN Pole (Perspective) PERSPECTIVE CENTER Pole (Simson Line) If a line L is the SIMSON LINE of a point P on the CIRCUMCIRCLE of a TRIANGLE , then P is called the pole of L (Honsberger 1995, p. 128). See also SIMSON LINE References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 128, 1995. Policeman on Point Duty Curve CRUCIFORM Polignac’s Conjecture DE POLIGNAC’S CONJECTURE Polish Notation REVERSE POLISH NOTATION Polish Space The HOMEOMORPHIC image of a so-called "complete separable" METRIC SPACE . The continuous image of a Polish space is called a SOUSLIN SET. See also DESCRIPTIVE SET THEORY ,STANDARD SPACE Pollaczek Polynomial Let a >½b½; and write h(u) /C30a cos u /C27 b 2 sin u: (1) Then define Pn(x; a; b) by the GENERATING FUNCTION f(x; w) /C30f(cos u;w) /C30X/C12 n/C300Pn(x; a ; b)wn /C30(1 /C28wei u) /C281=2 /C27ih(u)(1 /C28wei u) /C281=2 /C28ih(u) : (2) The GENERATING FUNCTION may also be written f(x; w) /C30 1 /C282xw /C27w29+=9+;/C281 =2 /C2exp (ax /C27b)X/C12 m/C301wm mUm/C281(x)"# ; (3) where Um(x)isaC HEBYSHEV POLYNOMIAL OF THE SECOND KIND . Pollaczek polynomials satisfy the RECURRENCE RELA- TIONnPn(x; a ; b) /C30[(2n /C281 /C272a)x /C272b]Pn /C281(x; a; b) /C28(n /C281)Pn/C282(x; a ; b) (4) for n /C302, 3, ...with P0 /C301 (5) P1 /C30(2a /C271)x /C272b: (6) In terms of the HYPERGEOMETRIC FUNCTION 2F1(a ; b; c; x) ; Pn(cos u; a; b) /C30ein u 2F1/C28n;1 2 /C27ih( u); 1; 1 /C28e /C282iu9+;k9+;7 : (7) They obey the orthogonality relation g1 /C281Pn(x; a ; b)Pm(x; a ; b)w(x; a ; b) dx /C30 n /C2712(a /C271)hi/C281 dnm ; (8) where dmnis the KRONECKER DELTA , for n; m /C300; 1, ..., with the WEIGHT FUNCTION w(cos u; a ; b) /C30e(2u /C28 p)h(u) fcosh[ ph(u)]g/C281 : (9) References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 393 /C1/00, 1975. Pollard Monte Carlo Factorization Method POLLARD RHO FACTORIZATION METHOD Pollard p-1 Factorization Method A PRIME FACTORIZATION ALGORITHM which can be implemented in a single-step or double-step form. In the single-step version, PRIMES p are found if p /C281is a product of small PRIMES by finding an m such that m /C13cq (mod n) ; where p /C281½q; with q a large number and (c ; n) /C301: Then since p /C281½q; m /C131 (mod p); so p ½m /C281 : There is therefore a good chance that n¶m/C281;in which case GCD( m/C281;n) (where GCD is the GREATEST COMMON DIVISOR ) will be a nontrivial divisor of n. In the double-step version, a PRIMES pcan be factored ifp/C281 is a product of small PRIMES and a single larger PRIME . See also PRIME FACTORIZATION ALGORITHMS ,W IL- LIAMS P/C271 FACTORIZATION METHOD References Bressoud, D. M. Factorization and Prime Testing. New York: Springer-Verlag, pp. 67 /C1/9, 1989. Pollard, J. M. "Theorems on Factorization and Primality Testing." Proc. Cambridge Phil. Soc. 76, 521/C1/28, 1974. Pollard Rho Factorization Method A PRIME FACTORIZATION ALGORITHM also known as POLLARD MONTE CARLO FACTORIZATION METHOD . Let x0 /C302 ; then compute xi /C271 /C30x2 i /C28xi /C271 (mod n) : If GCD( x2i /C28xi ; n) > 1; then n is COMPOSITE and its factors are found. In modified form, it becomes BRENT’S FACTORIZATION METHOD . In practice, almost any unfactorable POLYNOMIAL can be used for the iteration (/x2 /C282; however, cannot). Under worst con- ditions, the ALGORITHM can be very slow. See also BRENT’S FACTORIZATION METHOD ,P RIME FACTORIZATION ALGORITHMS References Brent, R. P. "Some Integer Factorization Algorithms Using Elliptic Curves." Austral. Comp. Sci. Comm. 8, 149 /C1/63, 1986. Bressoud, D. M. Factorization and Prime Testing. New York: Springer-Verlag, pp. 61 /C1/7, 1989. Eldershaw, C. and Brent, R. P. "Factorization of Large Integers on Some Vector and Parallel Computers." Montgomery, P. L. "Speeding the Pollard and Elliptic Curve Methods of Factorization." Math. Comput. 48, 243 /C1/64, 1987. Pollard, J. M. "A Monte Carlo Method for Factorization." Nordisk Tidskrift for Informationsbehandlung (BIT) 15, 331 /C1/34, 1975. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 83 and 102 /C1/03, 1991. Poloidal Field A VECTOR FIELD resembling a magnetic multipole which has a component along the Z-AXIS of a SPHERE and continues along lines of LONGITUDE . See also DIVERGENCELESS FIELD,TOROIDAL FIELD References Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley, p. 239, 1977. Po´lya Conjecture Let n be a POSITIVE INTEGER and r(n) the number of (not necessarily distinct) PRIME FACTORS of n (with r(1) /C300): Let O(m) be the number of POSITIVE INTE- GERS 5m with an ODD number of PRIME FACTORS , and E(m) the number of POSITIVE INTEGERS 5m with an EVEN number of PRIME FACTORS .Po´lya conjectured that L(m) /C13E(m) /C28O(m) /C30Xm n /C301l(n) is 50 ; where l(n) is the LIOUVILLE FUNCTION . The conjecture was made in 1919, and disproven by Haselgrove (1958) using a method due to Ingham (1942). Lehman (1960) found the first explicit coun- terexample, L(906 ; 180; 359) /C301; and the smallestcounterexample m /C30/906,150,257 was found by Ta- naka (1980). The first n for which L(n) /C300 are n /C302, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Tanaka 1980, Sloane’s A028488). It is unknown if L(x) changes sign infinitely often (Tanaka 1980). See also ANDRICA’S CONJECTURE ,LIOUVILLE FUNC- TION ,PRIME FACTORS References Haselgrove, C. B. "A Disproof of a Conjecture of Po´lya." Mathematika 5, 141 /C1/45, 1958. Ingham, A. E. "On Two Conjectures in the Theory of Numbers." Amer. J. Math. 64, 313 /C1/19, 1942. Lehman, R. S. "On Liouville’s Function." Math. Comput. 14, 311 /C1/20, 1960. Sloane, N. J. A. Sequences A028488 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Tanaka, M. "A Numerical Investigation on Cumulative Sum of the Liouville Function" [sic]. Tokyo J. Math. 3, 187 /C1/89, 1980. Po´lya Distribution NEGATIVE BINOMIAL DISTRIBUTION Po´lya Enumeration Theorem A very general theorem which allows the number of discrete combinatorial objects of a given type to be enumerated (counted) as a function of their "order." The most common application is in the counting of the number of GRAPHS of n nodes, TREES and ROOTED TREES with n branches, GROUPS of order n, etc. The theorem is an extension of the CAUCHY- FROBENIUS LEMMA , which is sometimes also called BURNSIDE’S LEMMA , the PO´ LYA-BURNSIDE LEMMA , the CAUCHY- FROBENIUS LEMMA , or even "the LEMMA THAT IS NOT BURNSIDE’S !" Po´lya enumeration is implemented as[g, m], in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ) which returns the polynomial giving the number of colorings with M colors of a structure defined by a PERMUTATION GROUP g. See also CAUCHY- FROBENIUS LEMMA ,GRAPH ,GROUP , ROOTED TREE,TREE References Harary, F. "The Number of Linear, Directed, Rooted, and Connected Graphs." Trans. Amer. Math. Soc. 78, 445/C1/63, 1955. Harary, F. "Po ´lya’s Enumeration Theorem." Graph Theory. Reading, MA: Addison-Wesley, pp. 180 /C1/84, 1994. Po´lya, G. "Kombinatorische Anzahlbestimmungen fu ¨r Grup- pen, Graphen, und chemische Verbindungen." Acta Math. 68, 145/C1/54, 1937. Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Skiena, S. "Polya’s Theory of Counting." §1.2.6 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 25 /C1/6, 1990. Tucker, A. Applied Combinatorics, 3rd ed. New York: Wiley, 1995. Po´lya Polynomial The POLYNOMIAL giving the number of colorings with m colors of a structure defined by a PERMUTATION GROUP . See also PERMUTATION GROUP ,PO´ LYA ENUMERATION THEOREM Polyabolo An analog of the POLYOMINO composed of n ISOSCELES RIGHT TRIANGLES joined along edges of the same length. Polyaboloes are sometimes also called poly- tans. The number of fixed polyaboloes composed of n triangles are 1, 3, 4, 14, 30, 107, 318, 1106, 3671, ... (Sloane’s A006074). See also DIABOLO ,HEXABOLO ,PENTABOLO ,POLYABO- LO TILING ,POLYIAMOND ,TETRABOLO ,TRIABOLO References Sloane, N. J. A. Sequences A006074/M2379 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Vichera, M. "Polyforms." http://alpha.ujep.cz/~vicher/puzzle/ polyforms.htm. Polyabolo Tiling See also POLYABOLO References Vichera, M. "Polytans." http://alpha.ujep.cz/~vicher/puzzle/ polyform/tan/tan.htm. Po´lya-Burnside Lemma CAUCHY- FROBENIUS LEMMA ,P O´LYA ENUMERATION THEOREM Po´lya’s Random Walk Constants N.B. A detailed online essay by S. Finch was the starting point for this entry.Letp(d) be the probability that a RANDOM WALK on a d-D lattice returns to the origin. Po ´lya (1921) proved that p(1)/C30p(2)/C301; (1) but p(d)B1 (2) ford/C212. Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that p(3)/C301/C281 u(3)/C300:3405373296 . . . ; (3) where u(3)/C303 (2p)3gp /C28pgp /C28pgp /C28pdx dy dz 3/C28cosx/C28cosy/C28cosz (4) /C3012 p218/C2712ffiffiffi 2p /C2810ffiffiffi 3p /C287ffiffiffi6p 9+;k9+;7 /C2K2/C28ffiffiffi3p9+;k9+;7ffiffiffi3p /C28ffiffiffi 2p9+;k9+;7hino 2 (5) /C3031 8/C2712ffiffiffi 2p /C2810ffiffiffi 3p /C287ffiffiffi6p 9+;k9+;7 /C21/C272X /C12 k/C301exp/C28k2pffiffiffi6p9+;k9+;7"# 4 (6) /C30ffiffiffi 6p 32p3G1 249+;k9+;7 G5 249+;k9+;7 G7 249+;k9+;7 G11 249+;k9+;7 (7) /C301:5163860592 . . . : (8) Here, K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND andG(z) is the GAMMA FUNCTION . Closed forms for d/C213 are not known, but Montroll (1956) showed that p(d)/C301/C28[u(d)]/C281; (9) where u(d)/C30d (2p)dgp /C28pgp /C28p/C1/C1/C1gp /C28p|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} d /C2d/C28Xd k/C301cosxk ! /C281 dx1dx2/C1/C1/C1dxd /C30g/C12 0I0t d !"#d e/C28tdt; (10) and I0(z)i sa MODIFIED BESSEL FUNCTION OF THE FIRST KIND . Numerical values of p(d) from Montroll (1956) and Flajolet (Finch) are given in the following table. d /p(d)/ 3 0.3405086322 4 0.1932016706 5 0.1351786098 6 0.1047154956 7 0.0858449341 8 0.0729126492 See also RANDOM WALK References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/polya/polya.html. Domb, C. "On Multiple Returns in the Random-Walk Problem." Proc. Cambridge Philos. Soc. 50, 586 /C1/91, 1954. Glasser, M. L. and Zucker, I. J. "Extended Watson Integrals for the Cubic Lattices." Proc. Nat. Acad. Sci. U.S.A. 74, 1800 /C1/801, 1977. McCrea, W. H. and Whipple, F. J. W. "Random Paths in Two and Three Dimensions." Proc. Roy. Soc. Edinburgh 60, 281 /C1/98, 1940. Montroll, E. W. "Random Walks in Multidimensional Spaces, Especially on Periodic Lattices." J. SIAM 4, 241 /C1/60, 1956. Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 2 10, 266 /C1/76, 1939. Po´lya-Vinogradov Inequality Let x be a nonprincipal character (mod q). Then XM /C27N n/C30M /C271x(n) /C10ffiffiffiqpln q ; where /C10indicates MUCH LESS than. References Davenport, H. "The Po´lya-Vinogradov Inequality." Ch. 23 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 135 /C1/38, 1980. Po´lya, G. "Uuml;ber die Verteilung der quadratischen Reste und Nichtreste." Nachr. Ko¨nigl. Gesell. Wissensch. Go¨ttin- gen, Math.-phys. Klasse, 21 /C1/9, 1918. Vinogradov. Perm. Univ. Fiz.-Mat. ob.-vo Zh. 1,18/C1/4 and 94 /C1/8, 1918. Polychoron A POLYTOPE in 4-D. Polychora are bounded by poly- hedra. The NECESSARY condition for the polychoron with SCHLA ¨ FLI SYMBOL fp ; q; rg to be a finite polytope is cosp q ! Bsinp p ! sinp r ! : SUFFICIENCY can be established by consideration ofthe six figures satisfying this condition. Nine of the ten star polychora can be obtained by faceting f3; 3; 5g; in other words, they have the same vertices as f3; 3; 5g: The tenth, f5 =2; 3; 3g; can be obtained by faceting f5; 3 ; 3 g: In addition, of the ten regular star polychora, several share the same edges: f3; 3; 5g;f3 ; 5 ; 5 =2g;f5; 5=2 ; 5 g; and f5; 3; 5=2 g; f3;3;5=2g; f3;5=2;5g; f5=2;5;5=2g; and f5=2;3;5g; and f5=2;5;3gand f5;5=2;3g: f5=2;3;3gdoes not share edges with any other regular polychora. There are therefore only four different projections (into any given plane or 3-space)of the ten regular star polychora, illustrated above. See also P OLYTOPE ,REGULAR POLYCHORON ,UNIFORM POLYCHORON References Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z. 46, 380/C1/07, 1940. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Polyconic Projection A class of map projections in which the parallels are represented by a system of non-concentric circular arcs with centers lying on the straight line represent- ing the central meridian (Lee 1944). The term wasfirst applied by Hunt, and later extended by Tissot (1881). x/C30cotfsinE (1) y/C30(f/C28f 0)/C27cotf(1/C28cosE); (2) where E/C30(l/C28l0) sin f: (3) The inverse FORMULAS are l/C30sin/C281(xtanf) sinf/C27l0; (4) and f is determined from Df /C30/C28A(f tan f /C27 1) /C28 f /C281 2( f2 /C27 B) tan f f /C28 A tan f/C28 1; (5) where f0 /C30A and A /C30 f0 /C27y (6) B /C30x2 /C27A2 : (7) References Beaman, W. M. Topographic Mapping. Washington, DC: U. S. Geol. Survey Bull. 788-E, p. 167, 1928. Birdseye, C. H. Formulas and Tables for the Construction of Polyconic Projections. U. S. Geological Survey, Bulletin 809, 1929. Hunt. Appendix 39 in Report for the U.S. Coast and Geodetic Survey. 1853. Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190 /C1/00, 1944. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 124 /C1/37, 1987. Tissot, A. Me´moir sur la repre´sentation des surfaces et les projections des cartes ge´ographiques. Paris: Gauthier- Villars, 1881. Polycube 3-D generalization of the POLYOMINOES to n-D. The number of polycubes N(n) composed of n CUBES are 1, 1, 2, 8, 29, 166, 1023, ... (Sloane’s A000162, Ball and Coxeter 1987). There are 1390 distinct ways to pack the eight polycubes of order n /C304 into a 2 /C294 /C294 box (Beeler 1972). See also CONWAY PUZZLE ,CUBE DISSECTION ,DIABO- LICAL CUBE,P ENTACUBE ,S LOTHOUBER- GRAATSMA PUZZLE ,SOMA CUBE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 112 /C1/13, 1987. Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Bouwkamp, C. J. "Packing Handed Pentacubes." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, 1981. Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 76 /C1/7, 1961. Gardner, M. "Polycubes." Ch. 3 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 28 /C1/3, 1986. Keller, M. "Counting Polyforms." http://members.aol.com/ wgreview/polyenum.html. Sloane, N. J. A. Sequences A000162/M1845 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Polycyclic Group See also SOLVABLE GROUP References Roseblade, J. E.; Goldie, A. W.; and Wehrfritz, B. A. F. Three Lectures on Polycyclic Groups. London: Queen Mary College, 1973. Segal, D. Polycyclic Groups. Cambridge, England: Cam- bridge University Press, 1983. Polydisk Let c /C30(c1 ; ...; cn) be a point in Cn ; then the open polydisk is defined by S/C30fz:½zj/C28cj½B½z0 j/C28cj½g forj/C301, ..., n. See also DISK,OPEN DISK References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 100, 1980. Polyfrob POLYHEX Polygamma Function ASPECIAL FUNCTION which is given by the ( n/C271)/st DERIVATIVE of the LOGARITHM of the GAMMA FUNCTION G(z) (or, depending on the definition, of the FACTORIAL z!):This is equivalent to the nth normal derivative of the LOGARITHMIC DERIVATIVE ofG(z) (or z!) and, in the former case, to the nth normal derivative of the DIGAMMA FUNCTION c0(z):Because of this ambiguity in definition, two different notations are sometimes (but not always) used, namely cn(z)/C30dn/C271 dzn/C271ln[G(z)] /C30dn dznG?(z) G(z)/C30dn dznc0(z) (1) /C30(/C281)n/C271n!X/C12 k/C3001 (z/C27k)n/C271(2) /C30(/C281)n/C271n!z(n/C271;z); (3) where z(a;z) is the H URWITZ ZETA FUNCTION , and Fn(z)/C13dn/C271 dzn/C271lnz!: (4) The two notations are connected by cn(z)/C30Fn(z/C281): (5) Unfortunately, Morse and Feshbach (1953) adopt a notation no longer in standard use in which Morse and Feshbach’s " /cn(z)/" is equal to cn/C281(z) in the usual notation. Also note that the function c0(z) is equiva- lent to the DIGAMMA FUNCTION C(z):cn(z) is imple- mented in Mathematica asPolyGamma [n,z]. The polygamma function obeys the RECURRENCE RELATION cn(z/C271)/C30cn(z)/C27(/C281)nn!z/C28n/C281; (6) the reflection FORMULA cn(1/C28z)/C27(/C281)n/C271cn(z)/C30(/C281)npdn dzncot(pz);(7) and the multiplication FORMULA ,cn(mz)/C30dn0lnm/C271 mn/C271Xm/C281 k/C301cnz/C27k m ! ; (8) where dmnis the K RONECKER DELTA . In general, special values for integral indices are given by cn(1)/C30(/C281)n/C271n!z(n/C271) (9) cn1 29+;k9+;7 /C30(/C281)n/C271n!2n/C271/C2819+=9+; z(n/C271); (10) giving c1129+;k9+;7 /C3012p2(11) c1(1)/C30z(2)/C3016p2(12) c2(1)/C30/C282z(3); (13) c31 29+;k9+;7 /C30p4(14) and so on. The polygamma function can be expressed in terms of CLAUSEN FUNCTIONS for RATIONAL arguments and integer indices. Special cases are given by c11 39+;k9+;7 /C3023p2/C273ffiffiffi 3p Cl22 3p9+;k9+;7 (15) c12 39+;k9+;7 /C3023p2/C283ffiffiffi 3p Cl22 3p9+;k9+;7 (16) c11 49+;k9+;7 /C30p2/C278Cl212p9+;k9+;7 (17) /C30p2/C278K (18) c1349+;k9+;7 /C30p2/C288Cl212p9+;k9+;7 (19) /C30p2/C288K (20) c2129+;k9+;7 /C30/C288Cl3(0)/C28Cl3(p) ½/C138 (21) /C3014z(3) (22) c21 39+;k9+;7 /C30/C284p3 3ffiffiffi 3p/C2818Cl3(0)/C2718Cl32 3p9+;k9+;7 (23) c2239+;k9+;7 /C304p3 3ffiffiffi 3p/C2818Cl3(0)/C2718Cl32 3p9+;k9+;7 (24) c2149+;k9+;7 /C30/C282p3/C2832Cl3(0)/C28Cl3(p) ½/C138 (25) /C30/C282p3/C2856z(3) (26) c2349+;k9+;7 /C302p3/C2832Cl3(0)/C28Cl3(p) ½/C138 (27) /C302p3/C2856z(3) (28) c2169+;k9+;7 /C30/C28182z(3)/C284ffiffiffi 3p p3(29) c25 69+;k9+;7 /C30/C28182z(3) /C274ffiffiffi 3p p3 (30) c31 39+;k9+;7 /C3083 p4 /C27162ffiffiffi 3p Cl42 3 p9+;k9+;7 (31) c32 39+;k9+;7 /C3083 p4 /C28162ffiffiffi 3p Cl42 3 p9+;k9+;7 (32) c3149+;k9+;7 /C308 p4 /C27768 Cl412 p9+;k9+;7 (33) /C308 p4 /C27768b(4) (34) c33 49+;k9+;7 /C308 p4 /C28768 Cl412 p9+;k9+;7 (35) /C308p4 /C28768b(4) ; (36) where K is CATALAN’S CONSTANT , z(z) is the RIEMANN ZETA FUNCTION , and b(z) is the DIRICHLET BETA FUNCTION . See also CATALAN’S CONSTANT ,CLAUSEN FUNCTION , DIGAMMA FUNCTION ,D IRICHLET BETA FUNCTION , GAMMA FUNCTION ,PERIODIC ZETA FUNCTION ,RIE- MANN ZETA FUNCTION ,STIRLING’S SERIES References Abramowitz, M. and Stegun, C. A. (Eds.). "Polygamma Functions." §6.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 260, 1972. Adamchik, V. S. "Polygamma Functions of Negative Order." J. Comput. Appl. Math. 100, 191 /C1/99, 1999. Arfken, G. "Digamma and Polygamma Functions." §10.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 549 /C1/55, 1985. Davis, H. T. Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, 1933. Kolbig, V. "The Polygamma Function ck(x) for x /C301=4 and x /C303 =4:/" J. Comp. Appl. Math. 75,43/C1/6, 1996. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 422 /C1/24, 1953. Polygenic Function A function which has infinitely many DERIVATIVES at a point. If a function is not polygenic, it is MONO- GENIC . See also MONOGENIC FUNCTION References Newman, J. R. The World of Mathematics, Vol. 3. New York: Simon & Schuster, p. 2003, 1956. Polygon A closed plane figure with nsides. If all sides and angles are equivalent, the polygon is called REGULAR . Polygons can be CONVEX , concave, or STAR . The word "polygon" derives from the Greek poly(poly) meaning "many" and gvnia (gonia ) meaning "angle." The AREA of a planar CONVEX POLYGON with VERTICES x1;y1 ðÞ ;...,xn;yn ðÞ isA/C301 2x1x2 y1y29+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C27x 2x3 y2y39+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C27.../C27x nx1 yny19+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;89+;9 ; (1) which can be written A/C30 1 2x1y2/C28x2y1/C27x2y3/C28x3y2/C27.../C27xn/C281yn ð /C28xnyn/C281/C27xny1/C28x1ynÞ; (2) where the signs can be found from the following diagram. The AREA of a polygon is defined to be POSITIVE if the points are arranged in a counterclockwise order, and NEGATIVE if they are in clockwise order (Beyer 1987). The sum Iof interior angles in the top left diagram of a dissected polygon is I/C13Xn i/C301ai/C27bi ðÞ /C30Xn i/C301ai/C27bi/C27gi ðÞ /C28Xn i/C301gi: (3) But Xn i/C301gi/C30360/C14(4) and the sum of ANGLES of the nTRIANGLES is Xn i/C301ai/C27bi/C27gi ðÞ /C30Xn i/C301180/C14ðÞ/C30n180/C14ðÞ : (5) Therefore, I/C30n180/C14ðÞ/C28360/C14/C30(n/C282)180/C14: (6) The same equation can be derived using EXTERIOR ANGLES (top right figure) or a triangulation from a single vertex (bottom figure). The following table gives the names for polygons with nsides. The words for polygons with n]5 sides (e.g., PENTAGON ,HEXAGON ,HEPTAGON , etc.) can refer to either REGULAR or non-regular polygons, depending on context. It is therefore always best to specify "regular n-gon" explicitly. For some polygons, several different terms are used interchangeably, e.g., nona- gon and enneagon both refer to the polygon with n /C309 sides. n polygon 2 DIGON 3 TRIANGLE (trigon) 4 QUADRILATERAL (tetragon) 5 PENTAGON 6 HEXAGON 7 HEPTAGON 8 OCTAGON 9 NONAGON (enneagon) 10 DECAGON 11 UNDECAGON (hendecagon) 12 DODECAGON 13 TRIDECAGON (triskaidecagon) 14 TETRADECAGON (tetrakaidecagon) 15 PENTADECAGON (pentakaidecagon) 16 HEXADECAGON (hexakaidecagon) 17 HEPTADECAGON (heptakaidecagon) 18 OCTADECAGON (octakaidecagon) 19 ENNEADECAGON (enneakaidecagon) 20 ICOSAGON 30 TRIACONTAGON 40 TETRACONTAGON 50 PENTACONTAGON 60 HEXACONTAGON 70 HEPTACONTAGON 80 OCTACONTAGON 90 ENNEACONTAGON 100 HECTOGON 10000 MYRIAGON See also 257-GON , 65537-GON ,ANTHROPOMORPHIC POLY- GON,BICENTRIC POLYGON ,CARNOT’S POLYGON THEO- REM,C HAOS GAME,C ONVEX POLYGON ,C YCLIC POLYGON , DE MOIVRE NUMBER ,DERIVED POLYGON ,DIAGONAL (POLYGON ), EQUIANGULAR POLYGON ,EQUI- LATERAL POLYGON ,EQUILATERAL TRIANGLE ,EULER’S POLYGON DIVISION PROBLEM ,HEPTADECAGON ,HEXA- GON,H EXAGRAM ,ILLUMINATION PROBLEM ,JORDAN POLYGON ,LOZENGE ,OCTAGON ,PARALLELOGRAM ,PAS- CAL’S THEOREM ,P ENTAGO N,P ENTAGR AM,P ETRIE POLYGON ,PLANAR POLYGON ,POLYGON CIRCUMSCRIB- ING CONSTANT ,P OLYGON INSCRIBING CONSTAN T, POLYGONAL KNOT,POLYGONAL NUMBER ,POLYGONAL SPIRAL ,POLYGON TRIANGULATION ,POLYGRAM ,POLY- HEDRAL FORMULA ,POLYHEDRON ,POLYTOPE ,Q UAD- RANGLE ,Q UADRILATERAL ,R EGULAR POLYGON , REULEAUX POLYGON ,RHOMBUS ,ROTOR ,ROULETTE , SIMPLE POLYGON ,SIMPLICITY ,SQUARE ,STAR POLY- GON,TRAPEZIUM ,TRAPEZOID ,TRIANGLE ,VISIBILITY , VORONOI POLYGON ,W ALLACE- BOLYAI- GERWEIN THEOREM References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 124 /C1/25 and 196, 1987. Polygon Circumscribing Constant If a TRIANGLE isCIRCUMSCRIBED about a CIRCLE , another CIRCLE around the TRIANGLE ,aSQUARE out- side the CIRCLE , another CIRCLE outside the SQUARE , and so on. From POLYGONS , the CIRCUMRADIUS and INRADIUS for an n-gon are R/C301 2scscp n ! (1) r/C3012scotp n ! ; (2) where sis the side length. Therefore, R r/C301 cosp n !/C30secp n ! ; (3) and an infinitely nested set of circumscribed polygons and circles has K/C13rfinal circle rinitial circle/C30secp 3 ! secp 4 ! secp 5 ! ...: (4) Kasner and Newman (1989) and Haber (1964) state that K/C3012, but this is incorrect. Write K/C30Y/C12 n/C3031 cosp n ! (5) lnK/C30/C28X/C12 n/C303ln(cos x): (6) Define y0(x)/C13/C28ln(cos x)/C301 2x2/C271 12x4/C271 45x6/C2717 2520x8/C27... ( 7 ) Now define y1(x)/C301 2ax2; (8) with y1p 3 ! /C30y0p 3 ! (9) 1 2ap 3 !2 /C30ln 2 ; (10) so a/C3023 p !2 ln 2 ; (11) and y2(x)/C309l n2 p2x2: (12) Buty2(x)>y1(x) for x/C23(0;p=3);so X/C12 n/C303y2p n ! >/C28X/C12 n/C303ln cosp n !"# (13) lnKBX/C12 n/C303y2p n ! 9l n2 p2X/C12 n/C303p n !2 /C309l n2X/C12 n/C3031 n2 /C309l n2X/C12 n/C3011 n2/C28X2 n/C3011 n2 ! /C309l n2 z(2)/C285 4hi /C309l n2p2 6/C285 4 ! /C302:4637 (14) KBe2:4637/C3011:75: (15)If the next term is included, y2(x)/C30a1 2x2/C271 12x49+;k9+;7 : (16) As before, y2p 3 ! /C30y0p 3 ! (17) a/C30972 ln 2 p254/C27p2 ðÞ; (18) so y2(x)/C30972 ln 2 p254/C27p2 ðÞ1 2x2/C271 12x49+;k9+;7 (19) lnKB972 ln 2 p254/C27p2 ðÞX/C12 n/C3031 2p n !2 /C271 12p n !42 435 /C30 972 ln 2 p254/C27p2 ðÞ1 2z(2)/C2854"# /C27p2 12z(4)/C281/C281 24"# () /C30972 ln 2 p254/C27p2 ðÞ12p2 6/C2854 ! /C27p2 12p2 90/C281/C281 24 ! "# /C3098p6/C2845p2/C285400 ðÞ ln 2 80p2/C2754 ðÞ/C302:255; (20) and KBe2:255/C309:535: (21) The process can be automated using computer alge- bra, and the first few bounds are 11.7485, 9.53528, 8.98034, 8.8016, 8.73832, 8.71483, 8.70585, 8.70235, 8.70097, and 8.70042. In order to obtain this accuracyby direct multiplication of the terms, more than10,000 terms are needed. The limit is K/C308:700036625 . . . : (22) Bouwkamp (1965) produced the following INFINITE PRODUCT formulas K/C302 pY/C12 m/C301Y/C12 n/C3011/C281 m2n/C271 29+;k9+;722 643 75 (23) /C306 expX/C12 k/C301l(2k)/C281 ½/C138 22kz(2k)/C281/C282/C282k9+$9+% k() ;ð24Þ where z(x) is the R IEMANN ZETA FUNCTION andl(x)i s the D IRICHLET LAMBDA FUNCTION . Bouwkamp (1965) also produced the formula with accelerated conver- gence K /C301 12ffiffiffi 6p p4 1 /C281 2 p2 /C271 24 p49+;k9+;7 1 /C2818 p2 /C271 384 p49+;k9+;7 /C29cscp2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 /C27 2ffiffiffi 3pp ! cscp2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 /C28 2ffiffiffi 3pp ! B ; (25) where B /C13Y/C12 n/C3031 /C28p2 2n2 /C27p4 24n4 ! secp n ! (26) (cited in Pickover 1995). See also POLYGON INSCRIBING CONSTANT References Bouwkamp, C. "An Infinite Product." Indag. Math. 27,40/C1/ 6, 1965. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/infprd/infprd.html. Haber, H. "Das Mathematische Kabinett." Bild der Wis- senschaft 2, 73, Apr. 1964. Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Microsoft Press, pp. 311 /C1/12, 1989. Pappas, T. "Infinity & Limits." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 180, 1989. Pickover, C. A. "Infinitely Exploding Circles." Ch. 18 in Keys to Infinity. New York: W. H. Freeman, pp. 147 /C1/51, 1995. Pinkham, R. S. "Mathematics and Modern Technology." Amer. Math. Monthly 103, 539 /C1/45, 1996. Plouffe, S. "Product(cos(Pi/n),n /C303..infinity)." http://www.la- cim.uqam.ca/piDATA/productcos.txt. Polygon Construction GEOMETRIC CONSTRUCTION ,G EOMETROGRAPHY , POLYGON ,SIMPLICITY Polygon Division Problem EULER’S POLYGON DIVISION PROBLEM Polygon Fractal CHAOS GAME Polygon Inscribing Constant If a TRIANGLE is inscribed in a CIRCLE , another CIRCLE inside the TRIANGLE ,a SQUARE inside the CIRCLE , another CIRCLE inside the SQUARE , and so on, K ?/C13rfinal circle rinitial circle/C30cosp 3 ! cosp 4 ! cosp 5 ! ...: Numerically, K ?/C301 K /C301 8:7000366252 ... /C300 :1149420448... ; where K is the POLYGON CIRCUMSCRIBING CONSTANT . Kasner and Newman’s (1989) assertion that K /C301 =12 is incorrect.Let a convex POLYGON be inscribed in a CIRCLE and divided into TRIANGLES from diagonals from one VERTEX . The sum of the RADII of the CIRCLES inscribed in these TRIANGLES is the same independent of the VERTEX chosen (Johnson 1929, p. 193). See also POLYGON CIRCUMSCRIBING CONSTANT References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/infprd/infprd.html. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Microsoft Press, pp. 311 /C1/12, 1989. Pappas, T. "Infinity & Limits." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 180, 1989. Plouffe, S. "Product(cos(Pi/n),n /C303..infinity)." http://www.la- cim.uqam.ca/piDATA/productcos.txt. Polygon Tiling See also HEXAGON TILING ,PENTAGON TILING ,QUAD- RILATERAL TILING ,SQUARE TILING ,TILING ,TRIANGLE TILING References Laczkovich, M. "Tilings of Polygons with Similar Triangles." Combinatorica 10, 281/C1/06, 1990. Polygon Triangle Picking The mean area of a TRIANGLE picked inside a regular n-gon of unit area is ¯A/C309 cos2v/C2752 cos v/C2744 36n2sin2v; (1) where v/C132p=n(Alikoski 1939; Solomon 1978; Croft et al. 1991, p. 54). Prior to Alikoski’s work, only the special cases n/C303, 4, 6, 8, and /C12had been deter- mined. The first few cases are summarized in the following table, where ¯A7is the largest root of 784147392 x3/C2884015792 x2/C272125620 x/C2815289/C300; (2) and ¯A9is the largest root of 24794911296 x3/C282525407632 x2/C2755366092 x /C28312427 /C300: (3) n /¯An/ problem 3 /1 12/ TRIANGLE TRIANGLE PICKING 4 /11 144/ SQUARE TRIANGLEPICKING 5 /1 1809/C272ffiffiffi 5p9+=9+; / 6 /289 3888/ HEXAGON TRIANGLE PICKING 7 /¯A7/ 8 /1 230497 /C2752ffiffiffi 2p9+=9+; / 9 /¯A9/ 10 /1 18000745 /C27262ffiffiffi 5p9+=9+; / See also HEXAGON TRIANGLE PICKING ,SQUARE TRI- ANGLE PICKING ,SYLVESTER’S FOUR- POINT PROBLEM , TRIANGLE TRIANGLE PICKING References Alikoski, H. A. "Uuml;ber das Sylvestersche Vierpunktpro- blem." Ann. Acad. Sci. Fenn. 51, No. 7, 1 /C1/0, 1939. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1991. Kendall, M. G. "Exact Distribution for the Shape of Random Triangles in Convex Sets." Adv. Appl. Prob. 17, 308 /C1/29, 1985. Kendall, M. G. and Le, H.-L. "Exact Shape Densities for Random Triangles in Convex Polygons." Adv. Appl. Prob. 1986 Suppl. ,59/C1/2, 1986. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, pp. 109 /C1/14, 1978. Polygon Triangulation EULER’S POLYGON DIVISION PROBLEM ,TESSELLATION , TRIANGULATION Polygonal Knot A KNOT equivalent to a POLYGON in R3 ; also called a TAME KNOT . For a polygonal knot K, there exists a PLANE such that the orthogonal projection p on it satisfies the following conditions: 1. The image p(K) has no multiple points other than a FINITE number of double points. 2. The projections of the vertices of K are not double points of p(K):/ Such a projection p(K) is called a regular knot projection. References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 735, 1980. Polygonal Number A type of FIGURATE NUMBER which is a generalization of TRIANGULAR , SQUARE , etc., numbers to an arbitraryn-gonal number. The above diagrams graphically illustrate the process by which the polygonal num- bers are built up. Starting with the nth TRIANGULAR NUMBER Tn ; then n /C27Tn /C281 /C30Tn : (1) Now note that n /C272Tn /C281 /C30n2 /C30Sn (2) gives the nth SQUARE NUMBER , n /C273Tn/C281 /C301 2 n(3n /C281) /C30Pn ; (3) gives the nth PENTAGONAL NUMBER , and so on. The general polygonal number can be written in the form pn r /C301 2 r[(r /C281)n /C282(r /C282)] /C3012 r[(n /C282)r /C28(n /C284)]; (4) where pn ris the rth n-gonal number (Savin 2000). For example, taking n /C303 in (4) gives a TRIANGULAR NUMBER , n /C304 gives a SQUARE NUMBER , etc. Fermat proposed that every number is expressible as at most k k-gonal numbers (FERMAT’S POLYGONAL NUMBER THEOREM ). Fermat claimed to have a proof of this result, although this proof has never been found. Jacobi, Lagrange (1772), and Euler all proved the square case, and Gauss proved the triangular case in 1796. In 1813, Cauchy proved the proposition in its entirety. An arbitrary number N can be checked to see if it is a n-gonal number as follows. Note the identity 8(n/C282)pr n/C27(n/C284)2/C30(2rn/C284r/C28n/C274)2; (5) so 8( n/C282)N/C27(n/C284)2/C30S2must be a PERFECT SQUARE . Therefore, if it is not, the number cannot ben-gonal. If it is a PERFECT SQUARE , then solving S/C302rn/C284r/C28n/C274 (6) for the rank rgives r/C30S/C27n/C284 2(n/C282): (7) Ann-gonal number is equal to the sum of the ( n/C281)/- gonal number of the same RANK and the TRIANGULAR NUMBER of the previous RANK . See also CENTERED POLYGONAL NUMBER ,DECAGONAL NUMBER ,FERMAT’S POLYGONAL NUMBER THEOREM , FIGURATE NUMBER ,HEPTAGONAL NUMBER ,HEXAGO- NAL NUMBER ,N ONAGONAL NUMBER ,O CTAG ONAL NUMBER ,PENTAGONAL NUMBER ,PYRAMIDAL NUM- BER,SQUARE NUMBER ,TRIANGULAR NUMBER References Abramovich, S.; Fujii, T.; and Wilson, J. W. "Multiple- Application Medium for the Study of Polygonal Numbers." http://jwilson.coe.uga.edu/Texts.Folder/AFW/AFWarti- cle.html. Beiler, A. H. "Ball Games." Ch. 18 in Recreations in the Theory of Numbers: The Queen of Mathematics Enter- tains. New York: Dover, pp. 184 /C1/99, 1966. Cauchy, A. "De´monstration du the´ore`me ge´ne´ral de Fermat sur les nombres polygones." Oeuvres, 2e. serie, Vol. 6. pp. 320 /C1/53. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3 /C1/3, 1952. Guy, K. "Every Number is Expressible as a Sum of How Many Polygonal Numbers?" Amer. Math. Monthly 101, 169 /C1/72, 1994. Nathanson, M. B. "Sums of Polygonal Numbers." In Analytic Number Theory and Diophantine Problems: Proceedings of a Conference at Oklahoma State University, 1984 (Ed. A. Adolphson et al. ). Boston, MA: Birkha ¨user, pp. 305 /C1/ 16, 1987. Pappas, T. "Triangular, Square & Pentagonal Numbers." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989. Savin, A. "Shape Numbers." Quantum 11,14/C1/8, 2000. Sloane, N. J. A. Sequences A000217/M2535 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M2535 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Polygonal Spiral The length of the polygonal spiral is found by noting that the ratio of INRADIUS to CIRCUMRADIUS of a REGULAR POLYGON of n sides is r R /C30cotp n ! cscp n !/C30cosp n ! : (1) The total length of the spiral for an n-gon with sidelength s is therefore L /C301 2 sX/C12 k /C300coskp n ! /C30s 21/C28 cosp n !"# : (2) Consider the solid region obtained by filling in subsequent triangles which the spiral encloses. The AREA of this region, illustrated above for n-gons of side length s,i s A/C3014s2cotp n ! : (3) The shaded triangular polygonal spiral is a REP-4- TILE. See also REP-TILE References Sandefur, J. T. "Using Self-Similarity to Find Length, Area, and Dimension." Amer. Math. Monthly 103, 107/C1/20, 1996. Polygram A self-intersecting STAR POLYGON such as the PENTA- GRAM orHEXAGRAM . nsymbol polygram 5 /f5=2g/PENTAGRAM 6 /f6=2g/HEXAGRAM 7 /f7=2g/Heptagram 8 /f8=3g/OCTAGRAM /f8=2g/STAR OF LAKSHMI 9 /f9=3g/NONAGRAM 10 /f10=3g/DECAGRAM Lachlan (1893) defines polygram to be a figure consisting of n straight lines. See also DECAGRAM ,HEXAGRAM ,OCTAGRAM ,PENTA- GRAM ,STAR FIGURE ,STAR OF LAKSHMI ,STAR POLY- GON References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 83, 1893. Polyhedral Formula A formula relating the number of VERTICES V, FACES F, and EDGES E of a simply connected (i.e., GENUS 0) POLYHEDRON (or POLYGON ). It was discovered inde- pendently by Euler (1752) and Descartes, so it is also known as the Descartes-Euler polyhedral formula. Although the formula holds for some non- CONVEX POLYHEDRA , it does not hold for STELLATED POLYHE- DRA. The polyhedral formula states V /C27F /C28E /C302 ; (1) where V /C30N0is the number of VERTICES , E /C30N1is the number of EDGES , and F /C30N2is the number of FACES . For a proof, see Courant and Robbins (1978, pp. 239 /C1/40). The FORMULA was generalized to n-D POLYTOPES by Schla ¨fli (Coxeter 1968, p. 233), P1 : N0 /C302 (2) P2 : N0 /C28N1 /C300 (3) P3 : N0 /C28N1 /C27N2 /C302 (4) P4 : N0 /C28N1 /C27N2 /C28N3 /C300 (5) Pn : N0 /C28N1 /C27N2 /C28.../C27(/C281)n/C281Nn/C281 /C301 /C28(/C281)n : (6) and proved by Poincare ´ (Poincare ´ 1893; Coxeter 1973, pp. 166 /C1/71; Williams 1979, pp. 24 /C1/5). For GENUS g surfaces, the formula can be generalized to the POINCARE ´ FORMULA x /C13V /C28E /C27F /C30 x(g); (7) where x(g) /C302 /C282g; (8) is the EULER CHARACTERISTIC , sometimes also known as the EULER- POINCARE ´ CHARACTERISTIC . The poly- hedral formula corresponds to the special case g /C300. There exist polytopes which do not satisfy the poly- hedral formula, the most prominent of which are the GREAT DODECAHEDRON f5;5 2gand SMALL STELLATED DODECAHEDRON f52;5g;which no less than Schla ¨fli himself refused to recognize (Schla ¨fli 1901, p. 134) since for these solids,N0/C28N1/C27N2/C3012/C2830/C2712/C30/C286 (9) (Coxeter 1973, p. 172). See also DEHN INVARIANT ,EULER CHARACTERISTIC , DESCARTES TOTAL ANGULAR DEFECT ,G ENUS (SUR- FACE ), POINCARE ´FORMULA ,P OLYHEDRAL GRAPH , POLYTOPE References Aigner, M. and Ziegler, G. M. "Three Applications of Euler’s Formula." Ch. 10 in Proofs from the Book. Berlin: Springer-Verlag, 1998. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 128, 1987. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods. Oxford, England: Oxford University Press, 1978. Coxeter, H. S. M. The Beauty of Geometry: Twelve Essays. New York: Dover, 1999. Coxeter, H. S. M. "Euler’s Formula." and "Poincare ´’s Proof of Euler’s Formula." §1.6 and Ch. 9 in Regular Polytopes, 3rd ed. New York: Dover, pp. 9 /C1/1 and 165 /C1/72, 1973. Euler, L. "Elementa doctrine solidorum." Novi comm. acad. scientiarum imperialis petropolitanae 4, 109/C1/60, 1752 /C1/ 753. Reprinted in Opera, Vol. 26 , pp. 71 /C1/2. Poincare ´, H. "Sur la ge ´ne´ralisation d’un the ´ore`me d’Euler relatif aux polye `dres." Comptes rendus hebdomadaires des se´ances de l’Acade ´mie des Sciences 117, 144/C1/45, 1893. Schla¨fli, L. "Theorie der vielfachen Kontinuita ¨t."Denkschrif- ten der Schweizerischen naturforschenden Gessel. 38,1/C1/ 37, 1901. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 252 /C1/53, 1999. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Polyhedral Graph Ann-polyhedral graph (sometimes called a c-net) is a 3-CONNECTED SIMPLE PLANAR GRAPH onnnodes. Every CONVEX POLYHEDRON can be represented in the plane or on the surface of a sphere by a 3- connected PLANAR GRAPH . Conversely, by a theorem of Steinitz as restated by Gru ¨nbaum (1967, p. 235), every 3-connected planar graph can be realized as a CONVEX POLYHEDRON (Duijvestijn and Federico 1981). Polyhedral graphs are sometimes simply known as "polyhedra" (which is rather confusing since the term "polyhedron" more commonly refers to a solid with n faces , not nvertices). The number of distinct polyhedral graphs havingV/C301, 2, ... vertices (or equivalently F/C301, 2, ... faces) are 0, 0, 0, 1, 2, 7, 34, 257, 2606, ... (Sloane’s A000944; Gru¨nbaum 1967, p. 424; Duijvestijn and Federico 1981; Dillencourt 1992; Croft et al. 1994). There is therefore a single TETRAHEDRAL GRAPH , two PENTA- HEDRAL GRAPHS , etc. There is no known formula for enumerating the number of nonisomorphic polyhe- dral graphs by numbers of edges E, vertices V,o r faces F(Harary and Palmer 1973, p. 224; Duijvestijn and Federico 1981). V # graph name 41 TETRAHEDRAL GRAPH 52 PENTAHEDRAL GRAPH 67 HEXAHEDRAL GRAPH 73 4 HEPTAHEDRAL GRAPH 8 257 OCTAHEDRAL GRAPH 9 2606 NONAHEDRAL GRAPH 10 32300 DECAHEDRAL GRAPH Duijvestijn and Federico (1981) enumerated the polyhedral graphs on E edges, obtaining 1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, ... (Sloane’s A002840) for E /C306, 7, 8, .... See also CUBICAL GRAPH ,D ODECAHEDRAL GRAPH , ICOSAHEDRAL GRAPH , K-CONNECTED GRAPH ,OCTAHE- DRAL GRAPH ,PLANAR CONNECTED GRAPH ,PLANAR GRAPH ,P LATONIC GRAPH ,P OLYHEDRAL FORMULA , POLYHEDRAL GROUP ,POLYTOPAL GRAPH ,SCHLEGEL GRAPH ,S IMPLE GRAPH ,S KELETON ,T ETRAHEDRAL GRAPH References Bouwkamp, C. J.; Duijvestijn, A. J. W.; and Medema, P. Table of c-Nets of Orders 8 to 19, Inclusive, 2 vols. Unpublished manuscript. Eindhoven, Netherlands: Phi- lips Research Laboratories, 1960. Croft, H. T.; Falconer, K. J.; and Guy, R. K. §B15 in Unsolved Problems in Geometry. New York: Springer- Verlag, 1991. Dillencourt, M. B. "Polyhedra of Small Orders and Their Hamiltonian Properties." Tech. Rep. 92 /C1/1, Info. and Comput. Sci. Dept. Irvine, CA: Univ. Calif. Irvine, 1992. Duijvestijn, A. J. W. "List of 3-Connected Planar Graphs with 6 to 22 Edges." Unpublished computer tape. En- schede, Netherlands: Twente Univ. Technology, 1979. Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral ( -Connected Planar) Graphs." Math. Com- put. 37, 523 /C1/32, 1981. Federico, P. J. "Enumeration of Polyhedra: The Number of 9-Hedra." J. Combin. Th. 7, 155 /C1/61, 1969. Federico, P. J. "The Number of Polyhedra." Philips Res. Rep. 30, 220 /C1/31, 1975. Gru¨nbaum, B. Convex Polytopes. New York: Wiley, 1967. Gru¨nbaum, B. "Polytopal Graphs." In Studies in Graph Theory, Part II (Ed. D. R. Fulkerson). Washington, DC: Math. Assoc. Amer., pp. 201 /C1/24, 1975. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973. Sloane, N. J. A. Sequences A000944/M1796 and A002840/ M0339 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Tutte, W. T. "A Theory of 3-Connected Graphs." Indag. Math. 23, 451 /C1/55, 1961. Tutte, W. T. "On the Enumeration of Convex Polyhedra." J. Combin. Th. Ser. B 28, 105 /C1/26, 1980.Polyhedral Group One of the symmetry groups of the PLATONIC SOLIDS . There are three polyhedral groups: the TETRAHEDRAL GROUP of order 12, the OCTAHEDRAL GROUP of order 24, and the ICOSAHEDRAL GROUP of order 60. See also ICOSAHEDRAL GROUP ,OCTAHEDRAL GROUP , PLATONIC SOLID,POLYHEDRAL GRAPH ,TETRAHEDRAL GROUP References Coxeter, H. S. M. "The Polyhedral Groups." §3.5 in Regular Polytopes, 3rd ed. New York: Dover, pp. 46 /C1/7, 1973. Polyhedron The word polyhedron has slightly different meanings in geometry and ALGEBRAIC GEOMETRY . In geometry, a polyhedron is simply a 3-D solid which consists of a collection of POLYGONS , usually joined at their EDGES . The word derives from the Greek poly (many) plus the Indo-European hedron (seat). A polyhedron is the 3-D version of the more general POLYTOPE (in the geometric sense), which can be defined in arbitrarydimension. The plural of polyhedron is "polyhedra"(or sometimes "polyhedrons"). The term "polyhedron" is used somewhat differently in ALGEBRAIC TOPOLOGY , where it is defined as a space that can be built from such "building blocks" as line segments, triangles, tetrahedra, and their higherdimensional analogs by "gluing them together" along their faces (Munkres 1993, p. 2). More specifically, it can be defined as the UNDERLYING SPACE of a SIMPLICIAL COMPLEX (with the additional constraint sometimes imposed that the complex be finite;Munkres 1993, p. 9). In the usual definition, apolyhedron can be viewed as an intersection of half- spaces, while a POLYTOPE is abounded polyhedron. ACONVEX POLYHEDRON can be formally defined as the set of solutions to a system of linear inequalities mx5b; where mis a real s/C293MATRIX and bis a real s- VECTOR . Although usage varies, most authors addi- tional require that a solution be bounded for it to define a CONVEX POLYHEDRON . An example of a convex polyhedron is illustrated above. A polyhedron is said to be regular if its FACES and VERTEX FIGURES are REGULAR (not necessarily CON- VEX) polygons (Coxeter 1973, p. 16). Using this definition, there are a total of nine REGULAR POLY- HEDRA , five being the CONVEX PLATONIC SOLIDS and four being the CONCAVE (stellated) KEPLER- POINSOT SOLIDS . However, the term "regular polyhedra" is sometimes used to refer exclusively to the PLATONIC SOLIDS (Cromwell 1997, p. 53). The DUAL POLYHEDRA of the PLATONIC SOLIDS are not new polyhedra, but are themselves PLATONIC SOLIDS . A CONVEX polyhedron is called SEMIREGULAR if its FACES have a similar arrangement of nonintersecting regular plane CONVEX polygons of two or more different types about each VERTEX (Holden 1991, p. 41). These solids are more commonly called the ARCHIMEDEAN SOLIDS , and there are 13 of them. The DUAL POLYHEDRA of the ARCHIMEDEAN SOLIDS are 13 new (and beautiful) solids, sometimes called the CATALAN SOLIDS . A QUASIREGULAR POLYHEDRON is the solid region interior to two DUAL REGULAR POLYHEDRA (Coxeter 1973, pp. 17 /C1/0). There are only two CONVEX QUASIRE- GULAR POLYHEDRA : the CUBOCTAHEDRON and ICOSI- DODECAHEDRON . There are also infinite families of PRISMS and ANTIPRISMS . There exist exactly 92 CONVEX POLYHEDRA with REGULAR POLYGONAL faces (and not necessarily equivalent vertices). They are known as the JOHNSON SOLIDS . Polyhedra with identical VERTICES related by a symmetry operation are known as UNIFORM POLY- HEDRA . There are 75 such polyhedra in which only two faces may meet at an EDGE , and 76 in which any EVEN number of faces may meet. Of these, 37 were discovered by Badoureau in 1881 and 12 by Coxeter and Miller ca. 1930. Polyhedra can be superposed on each other (with the sides allowed to pass through each other) to yield additional POLYHEDRON COMPOUNDS . Those made from REGULAR POLYHEDRA have symmetries which are especially aesthetically pleasing. The graphs corresponding to polyhedra skeletons are calledS CHLEGEL GRAPHS . Behnke et al. (1974) have determined the symmetry groups of all polyhedra symmetric with respect totheir VERTICES . See also ACOPTIC POLYHEDRON ,APEIROGON ,ARCHI- MEDEAN SOLID ,CANONICAL POLYHEDRON ,CATALAN SOLID,C ONVEX POLYHEDRON ,C UBE,C UMULATION , DICE,D IGON ,D ODECAHEDRON ,D UAL POLYHEDRON , ECHIDNAHEDRON ,F LEXIBLE POLYHEDRO N,H AUY CONSTRUCTION ,HEXAHEDRON ,HOLYHEDRON ,HYPER- BOLIC POLYHEDRON ,ICOSAHEDRON ,ISOHEDRON ,JES- SEN’S ORTHOGONAL ICOSAHEDRON JOHNSON SOLID , KEPLER- POINSOT SOLID ,NOLID ,OCTAHEDRON ,PETRIE POLYGON ,PLAITED POLYHEDRON ,PLATONIC SOLID , POLYCHORON ,POLYHEDRON COLORING ,POLYHEDRONCOMPOU ND,P OLYTOPE ,P RISMATOID ,Q UADRICORN , QUASIREGULAR POLYHEDRON ,R IGID POLYHEDRON , RIGIDITY THEOREM ,SCHWARZ’S POLYHEDRON ,SHAKY POLYHEDRON ,S EMIREGULAR POLYHEDRON ,S KELE- TON,STELLATION ,TETRAHEDRON ,TRUNCATION ,UNI- FORM POLYHEDRON ,ZONOHEDRON References Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 130 /C1/61, 1987. Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2: Geometry. Cam- bridge, MA: MIT Press, 1974. Bulatov, V. "Polyhedra Collection." http://www.physics.or- st.edu/~bulatov/polyhedra/. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, 1997. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Davie, T. "Books and Articles about Polyhedra and Poly- topes." http://www.dcs.st-andrews.ac.uk/~ad/mathrecs/ polyhedra/polyhedrabooks.html. Davie, T. "The Regular (Platonic) and Semi-Regular (Archi- medean) Solids." http://www.dcs.st-andrews.ac.uk/~ad/ mathrecs/polyhedra/polyhedratopic.html. Eppstein, D. "Geometric Models." http://www.ics.uci.edu/ ~eppstein/junkyard/model.html. Eppstein, D. "Polyhedra and Polytopes." http://www.ics.u- ci.edu/~eppstein/junkyard/polytope.html. Gabriel, J. F. (Ed.). Beyond the Cube: The Architecture of Space Frames and Polyhedra. New York: Wiley, 1997. Hart, G. "Annotated Bibliography." http://www.georgehart.- com/virtual-polyhedra/references.html. Hart, G. "Virtual Polyhedra." http://www.georgehart.com/ virtual-polyhedra/vp.html. Hilton, P. and Pedersen, J. Build Your Own Polyhedra. Reading, MA: Addison-Wesley, 1994. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Kern, W. F. and Bland, J. R. "Polyhedrons." §41 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 115 /C1/19, 1948. Lyusternik, L. A. Convex Figures and Polyhedra. New York: Dover, 1963. Malkevitch, J. "Milestones in the History of Polyhedra." In Shaping Space: A Polyhedral Approach (Ed. M. Senechal and G. Fleck). Boston, MA: Birkha ¨user, pp. 80 /C1/2, 1988. Miyazaki, K. An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes. New York: Wiley, 1983. Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Paeth, A. W. "Exact Dihedral Metrics for Common Polyhe- dra." In Graphic Gems II (Ed. J. Arvo). New York: Academic Press, 1991. Pappas, T. "Crystals-Nature’s Polyhedra." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 38 /C1/ 9, 1989. Pearce, P. Structure in Nature Is a Strategy for Design. Cambridge, MA: MIT Press, 1990. Pedagoguery Software. Poly . http://www.peda.com/poly/. Pugh, A. Polyhedra: A Visual Approach. Berkeley: Univer- sity of California Press, 1976. Schaaf, W. L. "Regular Polygons and Polyhedra." Ch. 3, §4in A Bibliography of Recreational Mathematics. Washington, DC: National Council of Teachers of Math., pp. 57 /C1/0, 1978. Virtual Image. "Polytopia I" and "Polytopia II" CD-ROMs. http://ourworld.compuserve.com/homepages/vir_image/ html/polytopiai.html and http://ourworld.compuserve.- com/homepages/vir_image/html/polytopiaii.html. Weisstein, E. W. "Books about Solid Geometry." http:// www.treasure-troves.com/books/SolidGeometry.html. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Polyhedron Coloring Define a valid "coloring" to occur when no two faces with a common EDGE share the same color. Given two colors, there is a single way to color an OCTAHEDRON (Ball and Coxeter 1987, pp. 238 /C1/39). Given three colors, there is one way to color a CUBE (Ball and Coxeter 1987, pp. 238 /C1/39) and 144 ways to color an ICOSAHEDRON (Ball and Coxeter 1987, pp. 239 /C1/42). Given four colors, there are two distinct ways to color a TETRAHEDRON (Ball and Coxeter 1987, p. 238) and four ways to color a DODECAHEDRON , consisting of two enantiomorphous ways (Steinhaus 1983, pp. 196 /C1/98; Ball and Coxeter 1987, p. 238). Given five colors, there are four ways to color an ICOSAHEDRON . Given six colors, there are 30 ways to color a CUBE (Steinhaus 1983, p. 167). See also COLORING ,CUBE,DODECAHEDRON ,ICOSAHE- DRON ,OCTAHEDRON ,PLATONIC SOLID,POLYHEDRON , TETRAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, 238 /C1/42, 1987. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 82 /C1/3, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Polyhedron Compound A polyhedron compound is an arrangement of a number of interpenetrating polyhedra, either all the same or of several distinct types, usually having visually attractive symmetric properties. The follow- ing table gives some common polyhedron compounds. solid vertices CUBE 2-COMPOUND CUBE 3-COMPOUND CUBE 4-COMPOUND CUBE 5-COMPOUND DODECAHEDRON CUBE-OCTAHEDRON COMPOUNDbothDODECAHEDRON 2- COMPOUND DODECAHEDRON 3- COMPOUNDDODECAHEDRON 5- COMPOUND DODECAHEDRON-ICOSA- HEDRON COMPOUNDboth DODECAHEDRON-SMALLTRIAMBIC ICOSAHEDRON COMPOUNDboth GREAT DODECAHEDRON- SMALL STELLATED DO- DECAHEDRON COMPOUNDboth GREAT ICOSAHEDRON- GREAT STELLATED DO- DECAHEDRON COMPOUNDboth OCTAHEDRON 3-COMPOUND OCTAHEDRON 5-COMPOUND ICOSIDODECAHEDRON STELLA OCTANGULA CUBE TETRAHEDRON 4- COMPOUND TETRAHEDRON 5- COMPOUNDDODECAHEDRON TETRAHEDRON 10- COMPOUNDDODECAHEDRON In Coxeter’s NOTATION , d distinct VERTICES of fm; ng taken ctimes are denoted cfm;ng[dfp;qg]; (1) or faces of fs;tgetimes [dfp;qg]efs;tg; (2) or both cfm;ng[dfp;qg]efs;tg: (3) See also CUBE 2-COMPOUND ,C UBE 3-COMPOUND , CUBE 4-COMPOUND ,C UBE 5-COMPOUND ,C UBE 20- COMPOUND ,C UBE-OCTAHEDRON COMPOUND ,D ODE- CAHEDRON 2-COMPOUND ,D ODECAHEDRON 3-COM- POUND ,D ODECAHEDRON 5-C OMPOUND , DODECAHEDRON- ICOSAHEDRON COMPOUND ,DODECA- HEDRON- SMALL TRIAMBIC ICOSAHEDRON COMPOUND , OCTAHEDRON 3-COMPOUND ,O CTAHEDRON 5-COM- POUND ,STELLA OCTANGULA ,TETRAHEDRON 4-COM- POUND ,TETRAHEDRON 5-COMPOUND ,TETRAHEDRON 10-COMPOUND References Cundy, H. and Rollett, A. "Regular Compounds." §3.10 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 129 /C1/42, 1989. Hart, G. "Compounds of Cubes." http://www.georgehart.com/ virtual-polyhedra/compound-cubes-info.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 37 /C1/8, 1991. Wenninger, M. J. "Some Interesting Polyhedral Com- pounds." Ch. 5 in Dual Models. Cambridge, England: Cambridge University Press, pp. 143 /C1/48, 1983. Polyhedron Dissection A DISSECTION of one or more polyhedra into other shapes. See also CUBE DISSECTION ,DIABOLICAL CUBE,POLY- CUBE ,SOMA CUBE,W ALLACE- BOLYAI- GERWEIN THEO- REM References Bulatov, V. "Compounds of Uniform Polyhedra." http:// www.physics.orst.edu/~bulatov/polyhedra/uniform_com- pounds/. Coffin, S. T. The Puzzling World of Polyhedral Dissections. New York: Oxford University Press, 1990. Coffin, S. T. and Rausch, J. R. The Puzzling World of Polyhedral Dissections CD-ROM. Puzzle World Produc- tions, 1998. Polyhedron Dual DUAL POLYHEDRON Polyhedron Hinging RIGIDITY THEOREM Polyhedron Packing A packing of polyhedron in 3-D space. A polyhedron which can pack with no holes or gaps is said to be a SPACE-FILLING POLYHEDRON . Betke and Henk (1999) present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary polyhedron, and explicitly calculate the densities for the P LATONIC and A RCHIMEDEAN SOLIDS . See also KELVIN’S CONJECTURE ,P ACKING ,S PACE- FILLING POLYHEDRON References Betke, U. and Henk, M. "Densest Lattice Packings of 3- Polytopes." Preprint. Erwin Schro ¨dinger Institute for Mathematical Physics. Vienna, Austria, Sep. 7, 1999.ftp://ftp.esi.ac.at/pub/Preprints/esi747.ps.Polyhex An analog of the POLYOMINOES and POLYIAMONDS in which collections of regular hexagons are arranged with adjacent sides. They are also called HEXES , HEXAS ,o r POLYFROBS (Beeler 1972). For the 4-hexes (tetrahexes), the possible arrangements are known asthe BEE,BAR,PISTOL ,PROPELLER ,WORM ,ARCH , and WAVE . A simple connected polyhex is called a fusene. Let thenumber of internal vertices of a polyhex be denotedn i:Then catafusenes (or catacondensed fusenes) have ni/C300 (and are therefore also called "tree-like"), and perifusenes (or pericondensed fusenes) have ni/C301: The numbers of catafusenes composed of npolyhexes are sometimes called Harary-Read numbers, and have the impressive GENERATING FUNCTION H(x)/C301 24x/C282f12/C2724x/C2748x2/C2824x3 /C27[(1/C28x)(1/C285x)]3=2/C283(5x/C273) /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C28x2)(1/C285x2)p /C284ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(1/C28x 3)(1/C285x3)p g /C30x/C27x2/C272x3/C275x4/C2712x5/C2737x6/C27... (Harary and Read 1970, Cyvin et al. 1993). Polyhexes may also be classified on the basis of being geome- trically planar (called nonhelicenic) or geometrically nonplanar (called helicenic). Fusenes include thehelicenes. "One-sided" polyhexes are considered to be FIXED in the plane, and so mirror images are counted sepa- rately. The following table gives the numbers of n-polyhexes that are geometrically planar (Klarner 1967, Balaban and Harary 1968, Harary and Read 1970, Lunnon 1972, Gardner 1978, Knop et al. 1984, Gardner 1988), catafusenes (Harary and Read 1970, Beinecke and Pippert 1974, Knop et al. 1984, Cyvin et al. 1993), cata- and planar, cata- and simply connected, and one-sided. n planar cata- cata- planar cata- simpl. one-sided Sloane A000228 A002216 A038142 A018190 A006535 11 1 1 1 1 21 1 1 1 1 33 2 2 3 3 47 5 5 71 0 52 2 1 2 1 2 2 2 3 3 6 82 37 36 81 147 7 333 123 118 331 620 8 1448 446 411 1435 2821 9 6572 1689 1489 6505 12942 10 30490 6693 5572 30086 60639 11 143552 27034 141229 286190 12 683101 111630 669584 1364621 13 3274826 467262 3198256 6545430 14 1981353 15367577 15 8487400 74207910 16 36695369 359863778 17 159918120 175159464318 70195753919 310107205120 1377993543821 61557789660 22 276327463180 23 124593589192224 5640868033058 See also POLYHEX TILING ,POLYIAMOND ,POLYKING , POLYOMINOReferences Balaban, A. T. "Enumeration of Cyclic Graphs." In Chemical Applications of Graph Theory (Ed. A. T. Balaban). Lon- don: Academic Press, pp. 63 /C1/05, 1976. Balaban, A. T. and Harary, F. "Chemical Graphs V: Enu- meration and Proposed Nomenclature of Benzenoid Cata- Condensed Polycyclic Aromatic Hydrocarbons." Tetrahe- dron 24, 2505 /C1/506, 1968. Balasubramanian, K.; Kauffman, J. J.; Koski, W. S.; and Balaban, A. T. "Graph Theoretical Characterization andComputer Generation of Certain Carcinogenic BenzenoidHydrocarbons and Identification." J. Comput. Chem. 1, 149/C1 /57, 1980. Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Beineke, L. W. and Pippert, R. E. "On the Enumeration of Planar Trees of Hexagons." Glasgow Math. J. 15, 131/C1/47. Cyvin, S. J.; Brunvoll, J.; Xiaofeng, G.; and Fuji, Z. "Number of Perifusenes with One Internal Vertex." Rev. Roumaine Chem. 38,6 5/C1/7, 1993. Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydro- carbons. 1. Isomer Enumeration of Fused Polycyclic Aromatic Hydrocarbon." J. Chem. Inf. Comput. Sci. 22, 15/C1/2, 1982. Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydro- carbons. 2. Polycyclic Aromatic Hydrocarbons Containing Tetragonal, Pentagonal, Heptagonal, and Octagonal Rings." J. Chem. Inf. Comput. Sci. 22, 139/C1/52, 1982. Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydro- carbons. 3. Enumeration of All the Polycyclic Conjugated Isomers of Pyrene Having Ring Sizes Ranging from 3 to 9."Math. Chem (Mu ¨lheim/Ruhr) 14,8 3/C1 /38, 1983. Gardner, M. "Polyhexes and Polyaboloes." Ch. 11 in Math- ematical Magic Show: More Puzzles, Games, Diversions,Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 146 /C1 /59, 1978. Gardner, M. "Tiling with Polyominoes, Polyiamonds, and Polyhexes." Ch. 14 in Time Travel and Other Mathema- tical Bewilderments. New York: W. H. Freeman, pp. 175 /C1/ 87, 1988. Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, pp. 92 /C1/3, 1994. Harary, F. "Graphical Enumeration Problems." In Graph Theory and Theoretical Physics (Ed. F. Harary). London: Academic Press, pp. 1 /C1/1, 1967. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 178 /C1/97, 1994. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973. Harary, F. and Read, R. C. "The Enumeration of Tree-Like Polyhexes." Proc. Edinburgh Math. Soc. 17,1/C1/3, 1970. Keller, M. "Counting Polyforms." http://members.aol.com/ wgreview/polyenum.html. Klarner, D. A. "Cell Growth Problems." In Canad. J. Math 19, 851/C1/63, 1967. Knop, J. V.; Szymanski, K.; Jericevic, Z.; and Trinajstic, N. "On the Total Number of Polyhexes." Match: Commun. Math. Chem. , No. 16, 119 /C1/34, Aug. 1984. Lunnon, W. F. "Counting Hexagonal and Triangular Poly- ominoes." In Graph Theory and Computing (Ed. R. C. Read). New York: Academic Press, pp. 87 /C1/00, 1972. Palmer, E. M. "Variations of the Cell Growth Problem." In Graph Theory and Applications: Proceedings of the Con- ference at Western Michigan University, Kalamazoo,Mich., May 10 /C1 /3, 1972 (Ed. Y. Alavi, D. R. Lick, and A. T. White). New York: Springer-Verlag, pp. 214 /C1/23, 1972. Sloane, N. J. A. Sequences A000228/M2682, A002216/ M1426, A006535/M2846, A018190, and A038142 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Vichera, M. "Polyforms." http://alpha.ujep.cz/~vicher/puzzle/ polyforms.htm. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 342 /C1/43, 1993. Weisstein, E. W. "Polyominoes." MATHEMATICA NOTEBOOK POLYOMINO.M . Weisstein, E. W. "Books about Polyominoes." http:// www.treasure-troves.com/books/Polyominoes.html. Polyhex Tiling There are no tilings of the EQUILATERAL TRIANGLE of side length 7 by all the polyhexes of order n /C304. There are nine distinct solutions of all the polyhexes of order n /C304 which tile a PARALLELOGRAM of base length 7 and side length 4, one of which is illustrated above (Beeler 1972). See also POLYHEX ,POLYIAMOND TILING ,POLYOMINO TILING References Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Polyiamond A generalization of the POLYOMINOES using a collec- tion of equal-sized EQUILATERAL TRIANGLES (instead of SQUARES ) arranged with coincident sides. Polyia- monds are sometimes simply known as IAMONDS . The number of two-sided (i.e., can be picked up and flipped, so MIRROR IMAGE pieces are consideredidentical) polyiamonds made up of n triangles are 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, ... (Sloane’s A000577). The number of one-sided polyiamonds composed of n triangles are 1, 1, 1, 4, 6, 19, 43, 121, ... (Sloane’s A006534). One of the 160 9-polyiamonds has a hole (Gardner 1984, p. 174). The top row of HEXIAMONDS in the above figure are known as the BAR, CROOK , CROWN , SPHINX , SNAKE , and YACHT . The bottom row of 6-polyiamonds are known as the CHEVRON , SIGNPOST , LOBSTER , HOOK , HEXAGON , and BUTTERFLY . See also POLYABOLO ,POLYHEX ,POLYIAMOND TILING , POLYOMINO References Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Gardner, M. "Mathematical Games." Sci. Amer. 211, Dec. 1964. Gardner, M. "Polyiamond." Ch. 18 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 173 /C1/82, 1984. Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, pp. 90 /C1/2, 1994. Keller, M. "Counting Polyforms." http://members.aol.com/ wgreview/polyenum.html. O’Beirne, T. H. "Pentominoes and Hexiamonds." New Scien- tist12, 379/C1/80, 1961. Pegg, E. Jr. "Iamonds." http://www.mathpuzzle.com/ia- mond.htm. Reeve, J. E. and Tyrrell, J. A. "Maestro Puzzles." Math. Gaz. 45,9 7/C1/9, 1961. Sloane, N. J. A. Sequences A000577/M2374 and A006534/ M3287 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Torbijn, I. P. J. "Polyiamonds." J. Recr. Math. 2, 216/C1 /27, 1969. Vichera, M. "Polyforms." http://alpha.ujep.cz/~vicher/puzzle/ polyforms.htm. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 342 /C1/43, 1993. Weisstein, E. W. "Polyominoes." M ATHEMATICA NOTEBOOK POLYOMINO.M . Polyiamond Tiling HEPTIAMOND TILING ,H EXIAMOND TILING ,O CTIA- MOND TILING ,PENTIAMOND TILING Polyking POLYPLET PolyLog POLYLOGARITHM Polylogarithm The function Lin(z) /C13X/C12 k /C301zk kn ; (1) Also known as Jonquie `re’s function. (Note that the similar NOTATION Li(z) is used for the LOGARITHMIC INTEGRAL .) The polylogarithm is also denoted F(z ; n) and equal to Lin(z) /C30zF(z; n; 1); (2) where F(z ; n; a) is the LERCH TRANSCENDENT (Erde ´- lyi et al. 1981, p. 30). The polylogarithm arises in Feynman diagram integrals (and, in particular, in the computation of quantum electrodynamics corrections to the electrons gyromagnetic ratio ), and the special cases n /C302 and n /C303 are called the DILOGARITHM and TRILOGARITHM , respectively. The polylogarithm of NEGATIVE INTEGER order arises in sums OF THE FORM X/C12 k /C301knrk /C30Li/C28n(r) /C30r (1 /C28 r)n/C271Xn i/C301n i9+$89+$9 rn/C28i ; (3) wheren i9+;=9+;; is an EULERIAN NUMBER . Special forms of low-order polylogarithms include Li/C282(x) /C30x(x /C27 1) (1 /C28 x)3 (4) Li/C281(x) /C30x (1 /C28 x)2 (5) Li0(x) /C30x 1 /C28 x (6) Li xðÞ/C30/C28ln(1 /C28x) : (7) At arguments /C281 and 1, the general polylogarithms become Lin(/C281) /C30/C28h(n) (8) Lin(1) /C30 z(n) ; (9) where h(x) is the DIRICHLET ETA FUNCTION and z(x)isthe RIEMANN ZETA FUNCTION . The polylogarithm for argument 1=2 can also be evaluated analytically for small n, Li11 29+;k9+;7 /C30ln 2 (10) Li21 29+;k9+;7 /C301 12[ p2 /C286(ln 2)2] (11) Li31 29+;k9+;7 /C301 24[4(ln 2)3 /C282p2 ln 2 /C2721z(3)] : (12) No similar formulas of this type are known for higher orders (Lewin 1991, p. 2). Li4(1=2) appears in the third-order correction term in the gyromagnetic ratio of the electron. The derivative of a polylogarithm is itself a polyloga- rithm, d dxLin(x) /C301 xLin/C281(x) : (13) Bailey et al. showed that Lim1 649+;k9+;7 6m/C281/C28Lim1 89+;k9+;7 3m/C281/C282L im149+;k9+;7 2m/C281/C274L im129+;k9+;7 9/C285(/C28ln 2)m 9m! /C27p2(/C28ln 2)m/C282 54(m/C282)!/C28p4(/C28ln 2)m/C284 486(m/C284)!/C28403z(5)(/C28ln 2)m/C285 1296( m/C285)! /C300: (14) No general ALGORITHM is know for the integration of polylogarithms of functions. See also DILOGARITHM ,EULERIAN NUMBER ,LEGEN- DRE’S CHI-FUNCTION ,LOGARITHMIC INTEGRAL ,NIEL- SEN GENERALIZED POLYLOGARITHM ,N IELSEN- RAMANUJAN CONSTANTS ,TRILOGARITHM References Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps. Bailey, D. H. and Broadhurst, D. J. A Seventeenth-Order Polylogarithm Ladder. 20 Jun 1999. http://xxx.lanl.gov/abs/math.CA/9906134/. Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Losinek, P. "Special Values of Multidimensional Polyloga-rithms." CECM-98:106, 14 May 1998. http://www.cecm.s-fu.ca/preprints/1998pp.html#98:106. Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Losinek, P. Special Values of Multidimensional Polyloga-rithms. 8 Oct 1999. http://xxx.lanl.gov/abs/math.CA/9910045/. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 323 /C1 /26, 1994. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 30 /C1/1, 1981. Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, 1958. Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981. Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991. Nielsen, N. Der Euler’sche Dilogarithms. Leipzig, Germany: Halle, 1909. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Zeta Function z(s; x) ; Bernoulli Poly- nomials Bn(x); Euler Polynomials En(x) ; and Polyloga- rithms Lin(x) :/" §1.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 23 /C1/4, 1990. Truesdell, C. A. Ann. Math. 46, 114 /C1/57, 1945. Zagier, D. "Special Values and Functional Equations of Polylogarithms." Appendix A in Structural Properties of Polylogarithms (Ed. L. Lewin). Providence, RI: Amer. Math. Soc., 1991. Polymorph An INTEGER which is expressible in more than one way in the form x2 /C27Dy2 or x2 /C28Dy2 where x2 is RELATIVELY PRIME to Dy2 : If the INTEGER is expres- sible in only one way, it is called a MONOMORPH . See also ANTIMORPH ,IDONEAL NUMBE R,M ONO- MORPH ,PELL EQUATION Polymorph Tessellation TESSELLATION Polynema A polynema of order n is Kyrmse’s term for a CONNECTED GRAPH having n edges. An n-polynema must therefore have either n or n /C271 nodes. The numbers of n-polynemas for n /C301, 2 ... are 1, 1, 3, 5, 12, 30, 79, 227, ... (Sloane’s A002905). Polynemas are related to a graphical construction problem called the MATCH PROBLEM (Gardner 1991). See also CONNECTED GRAPH ,M ATCH PROBLEM , PLANAR CONNECTED GRAPH ,TREE References Gardner, M. "The Problem of the Six Matches." In The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 79 /C1/1, 1991. Kyrmse, R. http://users.sti.com.br/rkyrmse/POLIN-E.htm.Sloane, N. J. A. Sequences A002905/M2486 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html.Polynomial APOLYNOMIAL is a mathematical expression invol- ving a series of POWERS in one or more variables multiplied by COEFFICIENTS .A POLYNOMIAL in one variable (i.e., a univariate polynomial) with constant COEFFICIENTS is given by anxn/C27.../C27a2x2/C27a1x/C27a0: (1) The highest POWER in a univariate polynomial is called its ORDER .APOLYNOMIAL in two variables (i.e., a bivariate polynomial) with constant COEFFICIENTS is given by anmxnym/C27.../C27a22x2y2/C27a21x2y/C27a12xy2/C27a11xy/C27a10x /C27a01y/C27a00 (2) Exchanging the COEFFICIENTS of a univariate poly- nomial end-to-end produces a polynomial a0xn/C27a1xn/C281/C27.../C27an/C281x/C27an/C300 (3) whose ROOTS are RECIPROCALS 1=xiof the original ROOTS xi:/ HORNER’S RULE provides a computationally efficient method of forming a polynomial from a list of its coefficients, and can be implemented in Mathematica as follows. PolynomialFromCoefs[l_List, x_] : /C30Fold[x#1 /C27 #2 &, 0, l] The following table gives special names given topolynomials of low orders. ORDER Polynomial Type 1 LINEAR EQUATION 2 QUADRATIC EQUATION 3 CUBIC EQUATION 4 QUARTIC EQUATION 5 QUINTIC EQUATION 6 SEXTIC EQUATION Polynomials of fourth degree may be computed using three multiplications and five additions if a few quantities are calculated first (Press et al. 1989): a0/C27a1x/C27a2x2/C27a3x3/C27a4x4 /C30[(Ax/C27B)2/C27Ax/C27C][(Ax/C27B)2/C27D]/C27E; (4) where A/C13(a4)1=4(5) B /C13a3 /C28 A3 4A3 (6) D /C133B2 /C278B3 /C27a1A /C28 2a2B A2 (7) C /C13a2 A2 /C282B /C286B2 /C28D (8) E /C13a0 /C28B4 /C28B2 C /C27D ðÞ /C28CD: (9) Similarly, a POLYNOMIAL of fifth degree may be computed with four multiplications and five addi- tions, and a POLYNOMIAL of sixth degree may be computed with four multiplications and seven addi- tions. Polynomials of orders one to four are solvable using only rational operations and finite ROOT EXTRAC- TIONS . A first-order equation is trivially solvable. A second-order equation is soluble using the QUADRATIC EQUATION . A third-order equation is solvable using the CUBIC EQUATION . A fourth-order equation is solvable using the QUARTIC EQUATION . It was proved by Abel and Galois using GROUP THEORY that general equations of fifth and higher order cannot be solved rationally with finite ROOT EXTRACTIONS (ABEL’S IMPOSSIBILITY THEOREM ). However, the general QUINTIC EQUATION may be given in terms of the JACOBI THETA FUNCTIONS ,or HYPERGEOMETRIC FUNCTIONS in one variable. Her- mite and Kronecker proved that higher order POLY- NOMIALS are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the GROUP properties of the ICOSAHEDRON . Klein’s method of solving the quintic in terms of HYPERGEO- METRIC FUNCTIONS in one variable can be extended to the sextic, but for higher order POLYNOMIALS , either HYPERGEOMETRIC FUNCTIONS in several variables or "Siegel functions" must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincare ´ created functions which give the solution to the nth order POLYNOMIAL equation in finite form. These functions turned out to be "natural" generalizations of the ELLIPTIC FUNCTIONS . Given an nth degree polynomial, the ROOTS can be found by finding the EIGENVALUES of the MATRIX /C28a0 =an/C28a1 =an/C28a2 =an... /C281 1 0 0 ... 0 0 1 0 ... 0 nn 1::: 0 0 0 0 ... 02 666643 77775: (10) This method can be computationally expensive, but is fairly robust at finding close and multiple roots. Polynomial identities involving sums and differences of like POWERS include x2 /C28y2 /C30(x /C28y)(x /C27y) (11)x3 /C28y3 /C30(x /C28y)(x2 /C27xy /C27y2) (12) x3 /C27y3 /C30(x /C27y)(x2 /C28xy /C27y2) (13) x4 /C28y4 /C30(x /C28y)(x /C27y)(x2 /C27y2) (14) x5 /C28y5 /C30(x /C28y)(x4 /C27x3y /C27x2y2 /C27xy3 /C27y4) (15) x5 /C27y5 /C30(x /C27y)(x4 /C28x3y /C27x2y2 /C28xy3 /C27y4) (16) x6 /C28y6 /C30(x /C28y)(x /C27y)(x2 /C27xy /C27y2)(x2 /C28xy /C27y2) (17) x6/C27y6/C30(x2/C27y2)(x4/C28x2y2/C27y4): (18) Further identities include x2 1/C28Dy219+=9+; x22/C28Dy229+=9+; /C30(x1x2/C27Dy1y2)2/C28D(x1y2/C27x2y1)2(19) x21/C27Dy219+=9+; x22/C27Dy229+=9+; /C30(x1x29Dy1y2)2/C27D(x1y2/C14x2y1)2: (20) The identity (X/C27Y/C27Z)7/C28(X7/C27Y7/C27Z7)/C307(X/C27Y)(X/C27Z)(Y/C27Z) /C29[(X2/C27Y2/C27Z2/C27XY/C27XZ/C27YZ)2/C27XYZ(X/C27Y/C27Z)] ð21Þ was used by Lame ´in his proof that F ERMAT’S LAST THEOREM was true for n/C307. See also POLYNOMIAL EQUATION ,POLYNOMIAL FAC- TORIZATION References Barbeau, E. J. Polynomials. New York: Springer-Verlag, 1989. Belardinelli, G. "Fonctions hyperge ´ome´triques de plusieurs variables er re ´solution analytique des e ´quations alge ´bri- que ge ´ne´rales." Me´moral des Sci. Math. 145, 1960. Bini, D. and Pan, V. Y. Polynomial and Matrix Computa- tions, Vol. 1: Fundamental Algorithms. Boston, MA: Birkha ¨user, 1994. Borwein, P. and Erde ´lyi, T. Polynomials and Polynomial Inequalities. New York: Springer-Verlag, 1995. Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440/C1/48, 1999. Cockle, J. "Notes on the Higher Algebra." Quart. J. Pure Applied Math. 4,4 9/C1/7, 1861. Cockle, J. "Notes on the Higher Algebra (Continued)." Quart. J. Pure Applied Math. 5,1/C1/7, 1862. King, R. B. Beyond the Quartic Equation. Boston, MA: Birkha ¨user, 1996. Mignotte, M. and Stefanescu, D. Polynomials: An Algorith- mic Approach. Singapore: Springer-Verlag, 1999. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, 1989. Project Mathematics . "Polynomials." Videotape. http:// www.projmath.caltech.edu/polynom.htm. Ram, R. "Sums of Powers." http://users.tellurian.net/hsejar/ maths/sumsofpowers/. Weisstein, E. W. "Books about Polynomials." http:// www.treasure-troves.com/books/Polynomials.html. Polynomial Bar Norm POLYNOMIAL NORM Polynomial Bracket Norm BOMBIERI NORM Polynomial Curve A curve obtained by fitting POLYNOMIALS to each ordinate of an ordered sequence of points. The above plots show POLYNOMIAL curves where the order of the fitting POLYNOMIAL varies from p /C283top /C281; where p is the number of points. Polynomial curves have several undesirable features, including a nonintuitive variation of fitting curve with varying COEFFICIENTS , and numerical instability for high orders. SPLINES such as the BE´ ZIER CURVE are therefore used more commonly. See also BE´ ZIER CURVE ,POLYNOMIAL ,SPLINE Polynomial Equation An EQUATION of the form P(x) /C300; where P(x)isa POLYNOMIAL . See also POLYNOMIAL Polynomial Factorization A FACTOR of a POLYNOMIAL P(x) of degree n is a POLYNOMIAL Q(x) of degree less than n which can be multiplied by another POLYNOMIAL R(x) of degree less than n to yield P(x); i.e., a POLYNOMIAL Q(x) such that P(x) /C30Q(x)R(x): For example, since x2 /C281 /C30(x /C271)(x /C281); both x /C281 and x /C271 are FACTORS of x2 /C281: Polynomial factorization can be performed in Mathematica using Factor [poly]. The COEFFICIENTS of factor POLYNOMIALS are often required to be REAL NUMBERS or INTEGERS but could, in general, be COMPLEX NUMBERS . The FUNDAMENTAL THEOREM OF ALGEBRA states that a POLYNOMIAL P(z) of degree n has n values zi(some of which are possibly degenerate) for which P(zi) /C300 : Such values are called POLYNOMIAL ROOTS . See also FACTOR ,F ACTORIZATION ,F UNDAMENTAL THEOREM OF ALGEBRA ,K RONECKER’S ALGORITHM , POLYNOMIAL ROOTS ,PRIME FACTORIZATIONReferences Abbott, J.; Shoup, V.; and Zimmerman, P. "Factorization in Z[x] : The Searching Phase." To appear in ISSAC’2000 Proceedings. Kaltofen, E. "Polynomial Factorization." In Computer Alge- bra: Symbolic and Algebraic Computation, 2nd ed. (Ed. B. Buchberger, G. E.Collins, R. Loos, and R. Albrecht). Vienna: Springer-Verlag, pp. 95 /C1/13, 1983. Lenstra, A. K.; Lenstra, H. W.; and Lova´sz, L. "Factoring Polynomials with Rational Coefficients." Math. Ann. 261, 515 /C1/34, 1982. Se´roul, R. "Factoring a Polynomial with Integral Coeffi- cients." §10.14 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 286 /C1/95, 2000. van Hoeij, M. "Factoring Polynomials and the Knapsack Problem." Preprint. http://www.math.fsu.edu/~aluffi/ar- chive/paper124.ps.gz. Polynomial Height The l/C12/-POLYNOMIAL NORM defined for a polynomial P /C30akxk /C27.../C27a1x /C27a0 by ½½P½½/C12/C30max k½ak ½: Note that some authors (especially in the area of Diophantine analysis) use ½P ½ as a shorthand for ½½P ½½/C12; while others (especially in the area of computational complexity) used ½P½ to denote the l2/-norm ½½P ½½2 (Zippel 1993, p. 174). See also POLYNOMIAL NORM References Zippel, R. "Heights of Polynomials." §11.1 in Effective Polynomial Computation. Boston, MA: Kluwer, pp. 174 /C1/ 75, 1993. Polynomial Map A map OF THE FORM ff : Kn 0 Kn ff :(a1 ; ...; an) /C2(f1(a); ...; f1(a)) ; where f /C30(f1 ; ...; fn) /C23 (K[X1 ; ...; Xn])m in a FIELD K, and a /C30(a1 ; ...; an):/ See also INVERTIBLE POLYNOMIAL MAP,JACOBIAN CONJECTURE References Becker, T. and Weispfenning, V. Gro¨bner Bases: A Computa- tional Approach to Commutative Algebra. New York: Springer-Verlag, p. 330, 1993. Polynomial Matrix A MATRIX whose entries are POLYNOMIALS . See also MATRIX POLYNOMIAL References Pascoletti, A. "Polynomial Matrix Utilities." http:// www.mathsource.com/cgi-bin/msitem?0207 /C1/51. Polynomial Norm For a POLYNOMIAL P /C30Xn k /C300akzk ; (1) several classes of norms are commonly defined. The lp/-norm is defined as ½½P ½½p /C13Xn k /C300½ak ½p ! (2) for p ]1; giving the special cases ½½P½½1 /C13X j½ak ½ (3) ½½P½½2 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX k½ak ½2r (4) ½½P ½½/C12/C30max k½ak ½: (5) Here, ½½P½½/C12is called the POLYNOMIAL HEIGHT . Note that some authors (especially in the area of Diophan- tine analysis) use ½P½ as a shorthand for ½½P ½½/C12 and ½P ½ as a shorthand for ½½P2 ½½; while others (especially in the area of computational complexity) used ½P ½ to denote the l2/-norm ½½P½½2 and (Zippel 1993, p. 174). Another class of norms is the Lp/-norms, defined by ½½P½½Lp/C30g2 p 0½P(eiu) ½du 2p ! 1 =p (6) for p ]1; giving the special cases ½½P½½L1/C30g2 p 0½P(eiu) ½du 2p ½½P½½L2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2p 0½P(eiu) ½2du 2pvuut ½½P½½ L/C12/C30sup ½z½/C301½P(z) ½ (Borwein and Erde´lyi 1995, p. 6). See also BOMBIERI NORM,MATRIX NORM,NORM,UNIT CIRCLE ,VECTOR NORM References Borwein, P. and Erde´lyi, T. "Norms on Pn :/" §1.1.E.3 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 6 /C1/, 1995. Zippel, R. Effective Polynomial Computation. Boston, MA: Kluwer, 1993. Polynomial Remainder Theorem If the COEFFICIENTS of the POLYNOMIAL dnxn /C27dn /C281xn/C281 /C27.../C27d0 /C300 (1)are specified to be INTEGERS , then integral ROOTS must have a NUMERATOR which is a factor of d0 and a DENOMINATOR which is a factor of dn (with either sign possible). This follows since a POLYNOMIAL of ORDER n with k integral ROOTS can be expressed as (a1x /C27b1)(a2x /C27b2) /C1/C1/C1(akx /C27bk)(cn/C28kxn/C28k /C27.../C27c0) /C300 ; (2) where the ROOTS are x1 /C30/C28b1 =a1 ; x2 /C30/C28b2 =a2 ; ...; and xk /C30/C28bk =ak : Factoring out the ai/s, a1a2 ...akx /C28b1 a1 ! x /C28b2 a2 ! ... x /C28bk ak ! /C2(cn/C28kxn/C28k /C27.../C27c0) /C300 : (3) Now, multiplying through, a1a2 ...akcn/C28kxn /C27.../C27b1b2 ...bkc0 /C300; (4) where we have not bothered with the other terms. Since the first and last COEFFICIENTS are dnand d0 ; all the integral roots of (1) are OF THE FORM [factors of d0]//[factors of dn] :/ References Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 34, 1982. Niven, I. M. Numbers: Rational and Irrational. New York: Random House, 1961. Polynomial Ring The RING R[x]of POLYNOMIALS in a variable x. See also MODULE ,POLYNOMIAL ,RING Polynomial Roots A root of a polynomial P(z) is a number zisuch that P(zi)/C300:The FUNDAMENTAL THEOREM OF ALGEBRA states that a POLYNOMIAL P(z) of degree nhasnroots, some of which may be degenerate. For example, the roots of the polynomial x3/C282x2/C28x/C272/C30(x/C282)(x/C281)(x/C271) (1) are/C281, 1, and 2. Finding roots of a polynomial is therefore equivalent to POLYNOMIAL FACTORIZATION into factors of degree 1. The roots of a polynomial equation may be found in Mathematica using Roots [lhs/C30/C30 rhs,var]. Let the ROOTS of the polynomial P(x)/C13anxn/C27an/C281xn/C281/C27...a1x/C27a0 (2) be denoted r1;r2;...,rn:Then N EWTON’S RELATIONS are X ri/C30/C28an/C281 an(3) X rirj /C30an/C282 an(4) X r1r2 /C1/C1/C1rk /C30(/C281)kan /C28k an: (5) These can be derived by writing P(x) /C30an(x /C28r1)(x /C28r2) /C1/C1/C1(x /C28rn) ; (6) expanding, and then comparing the coefficients with (2). Any POLYNOMIAL can be numerically factored, although different ALGORITHMS have different strengths and weaknesses. If the COEFFICIENTS of the POLYNOMIAL dnxn /C27dn /C281xn/C281 /C27.../C27d0 /C300 (7) are specified to be INTEGERS , then integral roots must have a NUMERATOR which is a factor of d0and a DENOMINATOR which is a factor of dn (with either sign possible). This is known as the POLYNOMIAL REMAIN- DER THEOREM . If there are no NEGATIVE ROOTS of a POLYNOMIAL (as can be determined by DESCARTES’ SIGN RULE ), then the GREATEST LOWER BOUND is 0. Otherwise, write out the COEFFICIENTS , let n /C30/C28 1, and compute the next line. Now, if any COEFFICIENTS are 0, set them to minus the sign of the next higher COEFFICIENT , starting with the second highest order COEFFICIENT . If all the signs alternate, n is the greatest lower bound. If not, then subtract 1 from n, and compute another line. For example, consider the POLYNOMIAL y /C302x4 /C272x3 /C287x2 /C27x /C287 : (8) Performing the above ALGORITHM then gives 02 2 /C2871 /C287 /C2812 0 /C2878 /C2815 –2/C281 /C2878 /C2815 /C2822 /C282 /C2837 /C2821 /C2832 /C2845 /C2814 35 so the greatest lower bound is /C283. If there are no POSITIVE ROOTS of a POLYNOMIAL (as can be determined by DESCARTES’ SIGN RULE ), the LEAST UPPER BOUND is 0. Otherwise, write out the COEFFICIENTS of the POLYNOMIALS , including zeros as necessary. Let n /C301. On the line below, write the highest order COEFFICIENT . Starting with the second- highest COEFFICIENT , add n times the number just written to the original second COEFFICIENT , and write it below the second COEFFICIENT . Continue throughorder zero. If all the COEFFICIENTS are NONNEGATIVE , the least upper bound is n. If not, add one to x and repeat the process again. For example, take the POLYNOMIAL y /C302x4 /C28x3 /C287x2 /C27x /C287: (9) Performing the above ALGORITHM gives 02/C281 /C2871 /C287 12 1/C286 /C285 /C2812 22 3/C281 /C281 /C289 3 2 5 8 25 68 so the LEAST UPPER BOUND is 3. Plotting the roots in the complex plane of all poly- nomials up to some degree with integer coefficients less than some cutoff integer in absolute value shows the beautiful structure illustrated above (Trott 2000). See also BAIRSTOW’S METHOD ,D ESCARTES’ SIGN RULE,GRAEFFE’S METHOD ,JENKINS- TRAUB METHOD , LAGUERRE’S METHOD ,L EHMER- SCHUR METHOD , MAEHLY’S PROCEDURE ,M ULLER’S METHOD ,POLYNO- MIAL FACTORIZATION ,ROOT,ZASSENHAUS- BERLEKAMP ALGORITHM References Bharucha-Reid, A. T. and Sambandham, M. Random Poly- nomials. New York: Academic Press, 1986. Odlyzko, A. M.; and Poonen, B. L’Enseignement Math. 39, 317, 1993. Pan, V. Y. "Solving a Polynomial Equation: Some History and Recent Progress." SIAM Rev. 39, 187 /C1/20, 1997. Trott, M. "Numerical Computations." §1.2.1 in The Mathe- matica Guidebook, Vol. 1: Programming in Mathematica. New York: Springer-Verlag, 2000. Polynomial Sequence ASEQUENCE ofPOLYNOMIALS pi(x);fori/C300, 1, 2, ..., where pi(x) is exactly of degree ifor all i. See also BASIC POLYNOMIAL SEQUENCE ,POLYNOMIAL Polynomial Series MULTINOMIAL SERIES Polynomial-Time See also NP-PROBLEM ,P-PROBLEM Polyomino A generalization of the DOMINO , originally called "super-dominoes" by Gardner (1957). An n-polyomino (or "n-omino"rpar; is defined as a collection of n squares of equal size arranged with coincident sides. FREE polyominoes can be picked up and flipped, so mirror image pieces are considered identical, whereas FIXED polyominoes are distinct if they have different chirality or orientation. FIXED polyominoes are also called LATTICE ANIMALS . Redelmeier (1981) computed the number of FREE and FIXED polyominoes for n 524; and Mertens (1990) gives a simple computer program. The following table gives the number of FREE (Lunnon 1971, 1972; Read 1978; Redelmeier 1981; Ball and Coxeter 1987; Con- way and Guttmann 1995; Goodman and O’Rourke 1997, p. 229), FIXED (Redelmeier 1981), one-sided (i.e., chiral) polyominoes (Redelmeier 1981; Golomb 1994; Goodman and O’Rourke 1997, p. 229), as well as the number of possible holes (Parkin et al. 1967, Madachy 1969, Golomb 1994) for the first few n n FREE FIXED one-sided poss. holes Sloane A000105 A014559 A000988 A001419 111 1 0 212 1 0 326 2 0 451 9 7 0 51 25 3 18 0 6 35 216 60 0 7 108 760 196 1 8 369 2725 704 6 9 1285 9910 2500 37 10 4655 39446 9189 195 11 17073 125268 33896 979 12 63600 505861 126759 4663 13 238591 1903890 476270 21474 14 901971 7204874 1802312 96496 15 3426576 27394666 6849777 425365 16 13079255 104592937 26152418 17 50107909 400795844 100203194 18 192622052 1540820542 385221143 19 742624232 5940738676 1485200848 20 2870671950 22964779660 5741256764 21 11123060678 88983512783 22245940545 22 43191857688 345532572678 86383382827 23 168047007728 1344372335524 336093325058 24 654999700403 5239988770268 1309998125640The best currently known bounds on the number of n- polyominoes are 3:72n BP(n) B4 :65n (Eden 1961, Klarner 1967, Klarner and Rivest 1973, Ball and Coxeter 1987). There is a single unique 2-omino (the DOMINO ), and two distinct 3-ominoes (the straight- and L-TRIOMI- NOES ). The 4-ominoes (TETROMINOES ) are known as the STRAIGHT ,L,T,SQUARE , and SKEW TETROMINOES . The 5-ominoes ( PENTOMINOES ) are called f,I,L,N,P, T,U,V,W,X,y, and Z(Golomb 1995). Another common naming scheme replaces f,I,L, and Nwith R,O,Q, and Sso that all letters from O to Z are used (Berlekamp et al. 1982). See also COLUMN- CONVEX POLYOMINO ,CONVEX POLY- OMINO ,D OMINO ,H EXOMINO ,L ATTICE POLYGON , MONOMINO ,P ENTOMINO ,P OLYABOLO ,P OLYCUBE , POLYHEX ,POLYIAMOND ,POLYKING ,POLYPLET ,ROW- CONVEX POLYOMINO ,SELF-AVOIDING POLYGON ,TE- TROMINO ,TRIOMINO References Atkin, A. O. L. and Birch, B. J. (Eds.). Computers in Number Theory: Proc. Sci. Research Council Atlas Sym- posium No. 2 Held at Oxford from 18 /C1/3 Aug., 1969. New York: Academic Press, 1971. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 109 /C1/13, 1987. Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Beineke, L. W. and Wilson, R. J. (Eds.). Selected Topics in Graph Theory. New York: Academic Press, pp. 417 /C1/44, 1978. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Games in General. London: Academic Press, 1982. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games inParticular. London: Academic Press, 1982. Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly-ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/9908123/. Conway, A. R. and Guttmann, A. J. "On Two-Dimensional Percolation." J. Phys. A: Math. Gen. 28, 891/C1 /04, 1995. Eden, M. "A Two-Dimensional Growth Process." Proc. Fourth Berkeley Symposium Math. Statistics and Prob-ability, Held at the Statistical Laboratory, University of California, June 30-July 30, 1960. Berkeley, CA: Uni- versity of California Press, pp. 223 /C1/39, 1961. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/rndprc/rndprc.html. Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150 /C1/56, May 1957. Gardner, M. "Polyominoes and Fault-Free Rectangles." Ch. 13 in Martin Gardner’s New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 150 /C1/61, 1966. Gardner, M. "Polyominoes and Rectification." Ch. 13 in Mathematical Magic Show: More Puzzles, Games, Diver- sions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 172 /C1/87, 1978. Golomb, S. W. "Checker Boards and Polyominoes." Amer. Math. Monthly 61, 675 /C1/82, 1954. Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, 1995. Goodman, J. E. and O’Rourke, J. (Eds.). Handbook of Discrete & Computational Geometry. Boca Raton, FL: CRC Press, 1997. Keller, M. "Counting Polyforms." http://members.aol.com/ wgreview/polyenum.html. Klarner, D. A. "Cell Growth Problems." Can. J. Math. 19, 851 /C1/63, 1967. Klarner, D. A. and Riverst, R. "A Procedure for Improving the Upper Bound for the Number of n-ominoes." Can. J. Math. 25, 585 /C1/02, 1973. Lei, A. "Bigger Polyominoes." http://www.cs.ust.hk/~philipl/ omino/bigpolyo.html. Lei, A. "Polyominoes." http://www.cs.ust.hk/~philipl/omino/ omino.html. Lunnon, W. F. "Counting Polyominoes." In Computers in Number Theory (Ed. A. O. L. Atkin and B. J. Brich). London: Academic Press, pp. 347 /C1/72, 1971. Lunnon, W. F. "Counting Hexagonal and Triangular Poly- ominoes." In Graph Theory and Computing (Ed. R. C. Read). New York: Academic Press, 1972. Madachy, J. S. "Pentominoes: Some Solved and Unsolved Problems." J. Rec. Math. 2, 181 /C1/88, 1969. Martin, G. Polyominoes: A Guide to Puzzles and Problems in Tiling. Washington, DC: Math. Assoc. Amer., 1991. Marzetta, A. "List of Polyominoes of order 4..7." http:// wwwjn.inf.ethz.ch/ambros/polyo-list.html. Mertens, S. "Lattice Animals--A Fast Enumeration Algo- rithm and New Perimeter Polynomials." J. Stat. Phys. 58, 1095 /C1/108, 1990. Parkin, T. R.; Lander, L. J.; and Parkin, D. R. "Polyomino Enumeration Results." SIAM Fall Meeting. Santa Bar- bara, CA, 1967. Read, R. C. "Contributions to the Cell Growth Problem." Canad. J. Math. 14,1/C1/0, 1962. Read, R. C. "Some Applications of Computers in Graph Theory." In Selected Topics in Graph Theory (Ed. L. W. Beineke and R. J. Wilson). New York: Academic Press, pp. 417 /C1/44, 1978. Redelmeier, D. H. "Counting Polyominoes: Yet Another Attack." Discrete Math. 36, 191 /C1/03, 1981. Ruskey, F. "Information on Polyominoes." http://www.theor- y.csc.uvic.ca/~cos/inf/misc/PolyominoInfo.html. Schroeppel, R. Item 77 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 30, Feb. 1972. Sloane, N. J. A. Sequences A000105/M1425, A001419/ M4226, and A014559 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html.Vichera, M. "Polyforms." http://alpha.ujep.cz/~vicher/puzzle/ polyforms.htm. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 342 /C1/43, 1993. Weisstein, E. W. "Polyominoes." MATHEMATICA NOTEBOOK POLYOMINO.M . Weisstein, E. W. "Books about Polyominoes." http:// www.treasure-troves.com/books/Polyominoes.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 117, 1991. Wells, D. Recreations in Logic. New York: Dover, 1979. Polyomino Tiling A TILING of the PLANE by specified types of POLY- OMINOES . Interestingly, the FIBONACCI NUMBER Fn/C271 gives the number of ways for 2 /C291 DOMINOES to cover a2/C29n checkerboard. Each MONOMINO , DOMINO , TRIOMINO , TETROMINO , PENTOMINO , and HEXOMINO tiles the plane, with requiring flipping. In addition, each heptomino, with the exception of the four illustrated above, can tile the plane, also without flipping (Schroeppel 1972). Consider now those collections of all n -ominoes which form a RECTANGLE . The polynomials of orders n/C301 and n/C302 form only a SQUARE and RECTANGLE , respectively. The two polyominoes of order n/C303 cannot form a rectangle, nor can the five polyominoes of order n/C304 or the 35 polyominoes of order n/C306 (Beeler 1972). There are several rectangles formed bythe 12 polyominoes of order n/C305, as summarized in the following table (Beeler 1972). Size Solutions /3/C2920/ 2 /4/C2915/ 368 /5/C2912/ 1010 /6/C2910/ 2339 25/C296/ 2 /8/C298 with 2 /C292 hole 65 See also DOMINO ,F IBONACCI NUMBER ,P OLYHEX TILING ,POLYIAMOND TILING ,POLYOMINO References Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Friedman, E. "Puzzle of the Month (February 1999)." http:// www.stetson.edu/~efriedma/mathmagic/0299.html. Gardner, M. "Tiling with Polyominoes, Polyiamonds, and Polyhexes." Ch. 14 in Time Travel and Other Mathema- tical Bewilderments. New York: W. H. Freeman, pp. 177 /C1/ 87, 1988. Schroeppel, R. Item 109 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 48, Feb. 1972. Vichera, M. "Polyominoes." http://alpha.ujep.cz/~vicher/puz- zle/polyform/minio/polynom.htm. Weisstein, E. W. "Books about Polyominoes." http:// www.treasure-troves.com/books/Polyominoes.html. Polyplet A POLYOMINO -like object made by attaching squares joined either at sides or corners. Because neighboring squares can be in relation to one another as KINGS may move on a CHESSBOARD , polyplets are sometimes also called POLYKINGS . The number of n-polyplets (with holes allowed) are 1, 2, 5, 22, 94, 524, 3031, ... (Sloane’s A030222). The number of n-polyplets hav- ing bilateral symmetry are 1, 2, 4, 10, 22, 57, 131, ... (Sloane’s A030234). The number of n-polyplets not having bilateral symmetry are 0, 0, 1, 12, 72, 467, 2900, ... (Sloane’s A030235). The number of fixed n- polyplets are 1, 4, 20, 110, 638, 3832, ... (Sloane’s A030232). The number of one-sided n-polyplets are 1, 2, 6, 34, 166, 991, ... (Sloane’s A030233). See also POLYIAMOND ,POLYOMINO References Sloane, N. J. A. Sequences A030222, A030232, A030233, A030234, and A030235 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Polystigm Lachlan’s terms for a collection of n points. See also POLYGRAM ,TETRASTIGM References Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 83, 1893. Polytan POLYABOLO Polytopal Graph A GRAPH G is called d-polytopal if there exists a d- dimensional CONVEX POLYTOPE P such that the vertices and edges of G are in a one-to-one inci-dence-preserving correspondence with those of P.In other words G is d-polytopal IFF it is isomorphic to the 1-SKELETON of some convex d-polytopes P.If d /C303, the graph is called a POLYHEDRAL GRAPH . See also POLYHEDRAL GRAPH References Gru¨nbaum, B. "Polytopal Graphs." In Studies in Graph Theory, Part II (Ed. D. R. Fulkerson). Washington, DC: Math. Assoc. Amer., pp. 201 /C1/24, 1975. Polytope The word polytope is used to mean a number of related, but slightly different mathematica objects. A convex polytope may be defined as the CONVEX HULL of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces. Coxeter (1973, p. 118) defines polytope as the general term of the sequence "POINT , LINE SEGMENT , POLY- GON, POLYHEDRON , ...," or more specifically as a finite region of n-dimensional space enclosed by a finite number of hyperplanes. The special name POLY- CHORON is sometimes given to a 4-D polytope. How- ever, in ALGEBRAIC TOPOLOGY , the UNDERLYING SPACE of a SIMPLICIAL COMPLEX is sometimes called a polytope (Munkres 1993, p. 8). The word "polytope" was introduced by Alicia Boole, the somewhat colorful daughter of logician George Boole (MacHale 1985). The part of the polytope that lies in one of the bounding hyperplanes is called a cell. A 4-D polytope is sometimes called a POLYCHORON . Explicitly, a d- dimensional polytope may be specified as the set of solutions to a system of linear inequalities mx 5b; where m is a real s /C29d MATRIX and b is a real s- VECTOR . The positions of the vertices given by the above equations may be found using a process called VERTEX ENUMERATION . A regular polytope is a generalization of the PLATONIC SOLIDS to an arbitrary DIMENSION . The regular poly- topes were discovered before 1852 by the Swiss mathematician Ludwig Schla ¨fli. For n-D with n]5; there are only three regular convex polytopes: the HYPERCUBE ,CROSS POLYTOPE , and regular SIMPLEX , which are analogs of the CUBE ,OCTAHEDRON , and TETRAHEDRON (Coxeter 1969; Wells 1991, p. 210). See also 16-CELL, 24-CELL, 120-CELL, 600-CELL,CROSS POLYTOPE ,EDGE (POLYTOPE ), FACE,FACET ,H YPER- CUBE ,INCIDENCE MATRIX ,LINE SEGMENT ,PENTA- TOPE ,POINT ,POLYCHORON ,POLYGON ,POLYHEDRON , POLYTOPE STELLATIONS ,PRIMITIVE POLYTOPE ,RIDGE , SIMPLEX ,TESSERACT ,U NIFORM POLYCHORON ,VER- TEX (POLYHEDRON ) References Bisztriczky, T.; McMullen, P., Schneider, R.; and Weiss, A. W. (Eds.). Polytopes: Abstract, Convex, and Computa- tional. Dordrecht, Netherlands: Kluwer, 1994. Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z. 46, 380 /C1/07, 1940. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Eppstein, D. "Polyhedra and Polytopes." http://www.ics.u- ci.edu/~eppstein/junkyard/polytope.html. Fukuda, K. "Polytope Movie Page." http://www.ifor.- math.ethz.ch/~fukuda/polymovie/polymovie.html. MacHale, D. George Boole: His Life and Work. Dublin, Ireland: Boole, 1985. Munkres, J. R. Analysis on Manifolds. Reading, MA: Ad- dison-Wesley, 1991. Sullivan, J. "Generating and Rendering Four-Dimensional Polytopes." Mathematica J. 1,76/C1/5, 1991. Weisstein, E. W. "Books about Polyhedra." http://www.trea- sure-troves.com/books/Polyhedra.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Polytope Stellations There are 10 stellated regular 4-polytopes (Wells 1991, p. 209). See also POLYTOPE ,STELLATION References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Polytropic Differential Equation LANE-EMDEN DIFFERENTIAL EQUATION Poncelet Transform PONCELET TRANSVERSE Poncelet Transverse Let a CIRCLE C1lie inside another CIRCLE C2 : From any point on C2 ; draw a tangent to C1 and extend it to C2 : From the point, draw another tangent, etc. For n tangents, the result is called an n-sided Poncelet transverse. If, on the circle of circumscription there is one point of origin for which a four-sided Poncelet transverse is closed, then the four-sided transverse will also close for any other point of origin on the circle (Do¨rrie 1965). See also BICENTRIC POLYGON ,BICENTRIC QUADRILAT- ERAL ,PONCELET’S PORISMReferences Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 192, 1965. Poncelet’s Closure Theorem PONCELET’S PORISM Poncelet’s Coaxal Theorem If a CYCLIC QUADRILATERAL ABCD is inscribed in a circle c1 of a COAXAL SYSTEM such that one pair AC of connectors touches another circle c2of the system at P, then each pair of opposite connectors will touch a circle of the system (BD at P ? on c2 ; AB at Q on c3 ; CD at Q ? on c3 ; DA at R on c4 ; and CB at R? on c4) ; and the six points of contact P, P ?; Q, Q ?; R, and R? will be COLLINEAR . The general theorem states that if A1 ; A2 ; ..., Anare any number of points taken in order on a CIRCLE of a give COAXAL SYSTEM so that A1A2 ; A2A3 ; ..., An/C281An touch respectively n /C281 fixed circles X1 ; X2 ; ..., Xn/C281 of the system, then AnA1 must touch a fixed circle Xn of the system. Further, if A1A2 ; A2A3 ; ..., An/C281Antouch respectively any n /C281 of the circles X1 ; X2 ; ..., Xn ; then AnA1must touch the remaining CIRCLE . See also COAXAL SYSTEM References Lachlan, R. "Poncelet’s Theorem." §334/C1/42 in An Elemen- tary Treatise on Modern Pure Geometry. London: Macmil- lian, pp. 209 /C1/17, 1893. Poncelet’s Continuity Principle PERMANENCE OF MATHEMATICAL RELATIONS PRINCI- PLE Poncelet’s Porism If an n-sided P ONCELET TRANSVERSE constructed for two given CONIC SECTIONS is closed for one point of origin, it is closed for any position of the point of origin. Specifically, given one ELLIPSE inside another, if there exists one CIRCUMINSCRIBED (simultaneously inscribed in the outer and circumscribed on the inner)n-gon, then any point on the boundary of the outer ELLIPSE is the vertex of some CIRCUMINSCRIBED n- gon. If the conic is taken as a circle (Casey 1888,pp. 124 /C1 /26) , then a polygon which has both an incenter and a circumcenter (and for which thetransveRsals would therefore close) is called a BI- CENTRIC POLYGON . For an even-sided polygon, the diagonals are con-current at the LIMITING POINT of the two circles, whereas for an odd-sided polygon, the lines connect-ing the vertices to the opposite points of tangency areconcurrent at the LIMITING POINT . Inverting about either of the two LIMIT POINTS gives two concentric circles. However, the n-gonal sides become arcs of circles in the process, so this sort ofsimple INVERSION does not provide an automatic proof of the theorem (as happens in S TEINER’S PORISM , for example). Fuss (1792) derived formulas not only for the BI- CENTRIC QUADRILATERAL , but also the bicentric PEN- TAGON ,HEXAGON ,HEPTAGON , and OCTAGON , as did Steiner (Fuss 1792; Steiner 1827; Jacobi 1881; Do ¨rrie 1965, p. 192). Chaundy (1923) exhibited porisms for n/C303, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, as well as erroneous expressions for several other values (Ker-awala 1947). Richelot derived the expression forn/C3011. In fact, there is a general analytic expression relating the CIRCUMRADIUS R,INRADIUS r, and offset between the CIRCUMCENTER and INCENTER dfor a bicentric polygon. Given R,r, and d, define a/C301 R/C27d(1) b/C301 R/C28d(2)c/C301 r: (3) Now let l/C301/C272c2(a2/C28b2) a2(b2/C28c2)(4) v/C30cosh/C281l; (5) and define the MODULUS as k2/C301/C28e/C282v: (6) Then the condition for an n-gon to be bicentric is scK(k) n;k ! /C30cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C28a2p /C27bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2/C28a2p a(b/C27c); (7) where sc( x;k)i saJ ACOBI ELLIPTIC FUNCTION and K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND (Richelot 1830, Kerawala 1947). Kerawala (1947) was able to establish many porisms in simple explicit form without resorting to the use of elliptic functions. For the two circles illustrated above, the tangent on the inner circle can be determined by solving (x2/C28x1)/C215(x2/C28x0)/C300; (8) where x0/C30d 09+$=9+$; (9) x1/C30cosu sinu9+$=9+$; (10) x2/C30d/C27rcosf rsinf9+$=9+$; ; (11) ris the radius of the inner circle, xis the offset of the inner circle, uis the given position on the outer circle, andfis the angle around the inner circle at which the tangent occurs. Taking the DOT PRODUCT and simplifying gives r/C27dcosf/C28cos(f/C28u)/C300: (12) When this is solved for f;the point at which the extension of this line intersects the outer circle again can be found using the standard equation of a CIRCLE- LINE INTERSECTION . The degrees dnof the algebraic equations relating a, b, and cforn/C303, 4, ..., are 1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 21, 24, 24, 32, 36, ... (Sloane’s A002348; Kerawala 1947). Let the PRIME FACTORIZATION ofnbe written as n/C302a0Y ipai i; (13) then dnin general is given by dn/C304a0 8Y ip2(ai/C281) i p2 i/C2819+=9+; : (14) In the following expressions, write e0/C13a/C27b/C27c (15) e1/C13/C28a/C27b/C27c (16) e2/C13a/C28b/C27c (17) e3/C13a/C27b/C28c (18) E1/C13/C28a2/C27b2/C27c2(19) E2/C13a2/C28b2/C27c2(20) E3/C13a2/C27b2/C28c2(21) F1/C13/C28E2E3/C27E3E1/C27E1E2 (22) F2/C13E2E3/C28E3E1/C27E1E2 (23) F3/C13E2E3þE3E1/C28E1E2 (24) F0/C13E2E3/C27E3E1/C27E1E2/C13e0e1e2e3 (25) g0/C13E1E2E3/C272abE1E2/C272bcE2E3/C272caE3E1(26) g1/C13E1E2E3/C282abE1E2/C272bcE1E2/C282caE3E1(27) g2/C13E1E2E3/C282abE1E2/C282bcE2E3/C272caE3E1(28) g3/C13E1E2E3/C272abE1E2/C282bcE2E3/C282caE3E1(29) following Kerawala (1947), and p¼Rþd rð30Þ q/C30R/C28d r(31) following Richelot (1830). The equation for a bicentric triangle ( n/C303), i.e., any triangle, may be variously written as a/C27b/C30c (32)(R/C27d)/C281/C27(R/C28d)/C281/C30r/C281(33) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R/C28d/C28rp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R/C27d/C28rp /C30ffiffiffiffiffiffiffi 2Rp (34) (p/C281)(q/C281)/C301 (35) (Richelot 1830), R2/C282Rr/C28d2/C300 (36) (Steiner 1827; F. Gabriel-Marie 1912, pp. 497 /C1/01; Kerawala 1947; Altshiller-Court 1957, pp. 85 /C1/7; Wells 1991). The latter is sometimes known as the EULER TRIANGLE FORMULA . For a BICENTRIC QUADRILATERAL (n/C304), the radii and offset are connected by the equation a2/C27b2/C30c2; (37) (Kerawala 1947), which expands to 1 (R/C28d)2/C271 (R/C27d)2/C301 r2(38) ( Davis; Dure ´ge; Casey 1888, pp. 109 /C1/10; F. Gabriel- Marie 1912, pp. 321 and 814 /C1/16; Johnson 1929; Do ¨rie 1965). This can also be written (R2/C28d2)2/C302r2(R2/C27d2); (39) (R/C27r/C27d)(R/C27r/C28d)(R/C28r/C27d)(R/C28r/C28d)/C30r4(40) (Steiner 1827), or (p2/C281)(q2/C281)/C301 (41) (Richelot 1830). The relationship for a bicentric PENTAGON (n/C305) is rðR/C28dÞ¼ðRþdÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðR/C28rþdÞðR/C28r/C28dÞp /C27(R/C27d)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R(R/C28r/C28d)p (42) (Steiner 1827) or 4p2q2(p/C281)(q/C281)/C30(p2/C27q2/C28p2q2)2(43) (Richelot 1830). A number of alternative forms are given by ðaþbÞðbþcÞðcþaÞ¼a3þb3þc3ð44Þ (a/C27b/C27c)3/C304(a3/C27b3/C27c3) (45) (/C28a/C27b/C27c)(a/C28b/C27c)(a/C27b/C28c)/C274abc/C300 (46) e0e3e2 e3e0e1 e2e1e09+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$/C300 (47) a(/C28a 2/C27b2/C27c2)/C27b(a2/C28b2/C27c2)/C27c(a2/C27b2/C28c2) /C272abc/C300; (48) and e /C281 0/C27e /C281 1/C27e/C281 2/C27e /C281 3/C300 (49) (Kerawala 1947). For n /C306, 3(R2 /C28d2)4 /C304r2(R2 /C27d2)(R2 /C28d2) /C2716r4d2R2(50) (Steiner 1827), 4p2q2(p2 /C281)(q2 /C281) /C30(p2 /C27q2 /C28p2q2)2 (51) (Richelot 1830), F3 /C300; (52) or E /C281 1/C27E /C281 2/C27E /C281 3 (53) (Kerawala 1947). For n /C307, g3 /C300 (54) (Jacobi 1881, Kerawala 1947) For n /C308, E /C282 1/C27E /C282 2/C30E /C282 3 (55) (Kerawala 1947), which can also be written in the form 16p4q4(p2 /C281)(q2 /C281) /C30(p2 /C27q2 /C28p2q2)4 ; (56) (Richelot 1830, Jacobi 1881). The equation given by Steiner (1827) contains (at least one) typographical error. For n /C309, aF2F3 /C27bF3F1 /C28cF1F2 /C300: (57) for n /C3010, 16p2q2(p2 /C281)(q2 /C281)[p4q4 /C28(p2 /C28q2)2]2 /C30f[p4 /C28(p2q2 /C28q2)2] /C27[q4 /C28(p2q2 /C28p2)2]2 /C27[p4q4 /C28(p2 /C28q2)2] g2 (58) (Richelot 1989). For n /C3012, 64p4q4(p2 /C281)(q2 /C281)[p4q4 /C28(p2 /C28q2)2]2 /C30f[p4 /C28(p2q2 /C28q2)2] /C27[q4 /C28(p2q2 /C28p2)2]2 /C27[p4q4 /C28(p2 /C28q2)2] g2 (59) (Richelot 1989). For n /C3014, g1 /C300: (60) For n /C3016, E/C282 2/C27E/C282 3/C30E /C282 1; (61)(Kerawala 1947) or 64p4q4(p2 /C281)(q2 /C281)fp4q4 /C28(p2 /C28q2)2] /C29(p2 /C27q2 /C28p2q2) g4 /C30f[p4 /C28(p2q2 /C28q2)2] /C27[q4 /C28(p2q2 /C28p2)2]2 /C27[p4q4 /C28(p2 /C28q2)2]2 g4 (62) (Richelot 1830). Weill (1878) gives an algorithm for finding approx- imate solutions (d; r ; R) for porisms with even n. The following table gives the approximate relations for fixed R/C101:/ n /d=R//r=R/error 6 /1 2//34//243 128R8 / 8 /1 4//15 4r//2955538440751415296 6568408355712890625R16 / 10 /1 10ffiffiffiffiffiffi 10p //9 40ffiffiffiffiffiffi10p / See also BICENTRIC POLYGON ,BICENTRIC QUADRILAT- ERAL ,BILLIARDS ,CIRCLE- LINE INTERSECTION ,COLLI- NEAR ,C YCLIC QUADRILATERAL ,E ULER TRIANGLE FORMULA ,P ONCELET TRANSVERSE ,T RIQUETRA , WEILL’S THEOREM References Allanson, B. "Bicentric Polygons" java applet. http://www.a- delaide.net.au/~allanson/bimovie.html. Appell, P. and Lacour, E. Principes de la the ´orie des fonctions elliptiques et applications. Paris: Gauthier-Vil- lars, pp. 138 /C1/39 and 227 /C1/43, 1922. Barth, W. and Bauer, T. "Poncelet Theorems." Expos. Math. 14, 125/C1/44, 1996. Barth, W. and Michel, J. "Modular Curves and Poncelet Polygons." Math. Ann. 295,2 5/C1/9, 1993. Bos, H. J. M.; Kers, C.; Oort, F.; and Raven, D. W. "Ponce- let’s Closure Theorem, Its History, Its Modern Formula- tion, a Comparison of Its Modern Proof with Those byPoncelet and Jacobi, and Some Mathematical RemarksInspired by These Early Proofs." Expos. Math. 5, 289/C1 /64, 1987. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to ModernGeometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., 1888. Cayley, A. Philos. Mag. 5, 281/C1 /84, 1853. Cayley, A. Philos. Mag. 6,9 9/C1/02, 1853. Cayley, A. "Developments on the Porism of the In-and- Circumscribed Polygon." Philos. Mag. 7, 339/C1/45, 1854. Cayley, A. Phil. Trans. Roy. Soc. London 151, 225/C1/39, 1861. Chaundy, T. W. Proc. London Math. Soc. 22, 104/C1/23, 1923. Chaundy, T. W. Proc. London Math. Soc. 25,1 7/C1/4, 1926. Clifford, W. K. Proc. London Math. Soc. 7,2 9/C1/8. Clifford, W. K. Proc. London Math. Soc. 7, 225/C1/33. Clifford, W. K. Proc. Cambridge Phil. Soc. , 120/C1/23, 1868. Darboux, G. Comte Rendus de l’Acadamie de Sciences 90, 1880. Darboux, G. Principles de ge´ome´trie analytique, Vol. 3. Paris, pp. 250 /C1/87, 1917. Davis, M. A. Educ. Times 32. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 192 /C1/ 93, 1965. Dure´ge. Theorie der Elliptischen Functionen. p. 185. Fuss, N. Nova Acta Petropol. 10, 1792. Fuss, N. "De Polygonis symmetrice irregularibus circulo simul inscriptis et circumscriptis." Nova Acta Petropol. 13, 166 /C1/89, 1798. F. Gabriel-Marie. Exercices de Ge´ome´trie. Tours, France: Maison Mame, 1912. Griffiths, P. and Harris, J. "A Poncelet Theorem in Space." Comment. Math. Helv. 52, 145 /C1/60, 1977. Griffiths, P. and Harris, J. "On Cayley’s Explicit Solution to Poncelet’s Porism." Enseign. Math. 24,31/C1/0, 1978. Hart. Quart. J. Math. , 1857. Jacobi, C. G. J. "Ueber die Anwendung der elliptischen Transcendenten auf ein bekanntes Problem der Elemen- targeometrie." J. reine angew. Math. 3, 376 /C1/87, 1823. Reprinted in Gesammelte Werke, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 278 /C1/93, 1969. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 91 /C1/6, 1929. Kerawala, S. M. "Poncelet Porism in Two Circles." Bull. Calcutta Math. Soc. 39,85/C1/05, 1947. Lebesgue, H. "Polygones de Poncelet." Ch. 4 in Les Con- iques. Paris: Gauthier-Villars, pp. 115 /C1/49, 1955. Reprint of "Expose ´ ge´moe´trique d’un me´moire de Cayley sur les Polygones de Poncelet." Ann. de la Faculte ´ des Sci. de l’Universite ´ de Toulouse 14, 1922. Lelieuvre, A. "Sur les polygones de Poncelet." L’enseign. math. 2, 410 /C1/23, 1900. Lelieuvre, A. "Sur les polygones de Poncelet." L’enseign. math. 3, 115 /C1/17, 1901. Moutard, M. "Recherches analytiques sur les polygones simultane ´ment inscrits et circonscrits a` deux coniques." Appendix to Poncelet, J. V. Traite ´ des proprie ´te´s projec- tives des figures: ouvrage utile a` qui s’occupent des applications de la ge´ome´trie descriptive et d’ope´rations ge´ome´triques sur le terrain, Vol. 1, 2nd ed. Paris: Gau- thier-Villars, pp. 535 /C1/60, 1865 /C1/6. Poncelet, J. V. Traite ´ des proprie ´te´s projectives des figures: ouvrage utile a` qui s’occupent des applications de la ge´ome´trie descriptive et d’ope´rations ge´ome´triques sur le terrain, Vols. 1 /C1/, 2nd ed. Paris: Gauthier-Villars, 1865 /C1/ 6. Previato, E. "Poncelet’s Theorem in Space." Proc. Amer. Math. Soc. 127, 2547 /C1/556, 1999. Richelot, F. J. "Anwendung der elliptischen Transcendenten auf die spha¨rischen Polygone; welche zugleich einem kleinen Kreise der Kugel eingescrieben und einem andern umgeschrieben sind." J. reine angew. Math. 5, 250 /C1/67, 1830. Richelot. J. reine angew. Math. 38, p. 353. Rosanes, J. and Pasch, M. "U¨ ber das einem Kegelschnitte umbeschriebene und einem andern einbeschriebene Poly- gon." J. reine angew. Math. 64, 126 /C1/66, 1865. Rosanes, J. and Pasch, M. "U¨ ber eine algebraische Aufgabe, welche einer Gattung geometrischer Probleme zu Grunde liegt." J. reine angew. Math. 70, 169 /C1/73, 1869. Sloane, N. J. A. Sequences A002348/M0549 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Steiner, J. §26.57 in "Aufgaben und Lehrsa ¨tze, erstere aufzulo ¨sen, leztere zu beweisen." J. reine angew. Math. 2, 289, 1827. Titchmarsh, E. C. Messenger Math. 52, 42, 1922.Weill, M. and Bu¨tzberger. "Sur les polygones inscrits et circonscrits a` la fois a` deux cercle." Journal de Liouville, 3me se´rie 4,7/C1/2, 1878. Weill, M. "Sur une classe de polygones de Poncelet." Bull. de la Soc. Math. France 29, 199 /C1/08, 1901. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Viking Penguin, pp. 192 /C1/93, 1992. Poncelet-Steiner Theorem All Euclidean GEOMETRIC CONSTRUCTIONS can be carried out with a STRAIGHTEDGE alone if, in addition, one is given the RADIUS of a single CIRCLE and its center. The theorem was suggested by Poncelet in 1822 and proved by Steiner in 1833. A construction using STRAIGHTEDGE alone is called a STEINER CON- STRUCTION . See also GEOMETRIC CONSTRUCTION ,STEINER CON- STRUCTION References Do¨rrie, H. "Steiner’s Straight-Edge Problem." §34 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 165 /C1/70, 1965. Steiner, J. Geometric Constructions with a Ruler, Given a Fixed Circle with Its Center. New York: Scripta Mathe- matica, 1950. Pong Hau K’i A Chinese TIC-TAC-TOE -like game. See also TIC-TAC-TOE References Evans, R. "Pong Hau K’i." Games and Puzzles 53, 19, 1976. Straffin, P. D. Jr. "Position Graphs for Pong Hau K’i and Mu Torere." Math. Mag. 68, 382 /C1/86, 1995. Pons Asinorum An elementary theorem in geometry whose name means "asses’ bridge," perhaps in reference to the fact that fools would be unable to pass this point in their geometric studies. The theorem states that the ANGLES at the base of an ISOSCELES TRIANGLE (defined as a TRIANGLE with two legs of equal length) are equal and appears as the fifth proposition in Book I of Euclid’s ELEMENTS . See also ISOSCELES TRIANGLE ,PYTHAGOREAN THEO- REM References Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 38, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 193 /C1/94, 1991. Pontryagin Class Theith Pontryagin class of a VECTOR BUNDLE is (/C281)i times the ith C HERN CLASS of the complexification of the VECTOR BUNDLE . It is also in the 4 i/th cohomology group of the base SPACE involved. See also CHERN CLASS ,STIEFEL- WHITNEY CLASS Pontryagin Duality Let G be a locally compact ABELIAN GROUP . Let G /C31 be the group of all homeomorphisms G 0 R=Z ; in the compact open topology. Then G /C31 is also a locally compact ABELIAN GROUP , where the asterisk defines a contravariant equivalence of the category of locally compact Abelian groups with itself. The natural mapping G 0 (G/C31) /C31; sending g to G, where G(f) /C30 f(g) ; is an isomorphism and a HOMEOMORPHISM . Under this equivalence, compact groups are sent to discrete groups and vice versa. See also ABELIAN GROUP ,HOMEOMORPHISM Pontryagin Maximum Principle A result in CONTROL THEORY . Define H( c; x; u) /C13( c; f(x; u)) /C13Xn a /C300cafa(x; u) : Then in order for a control u(t) and a trajectory x(t)to be optimal, it is NECESSARY that there exist NONZERO absolutely continuous vector function c(t) /C30 ( c0(t) ; c1(t); ...; cn(t)) corresponding to the func- tions u(t) and x(t) such that 1. The function H(c(t) ; x(t) ; u) attains its max- imum at the point u /C30u(t) almost everywhere in the interval t0 5t 5t1 ; H( c(t) ; x(t) ; u(t)) /C30max u /C23UH( c(t); x(t) ; u) : 2. At the terminal time t1 ; the relations c0(t1) 50 and H( c(t1) ; x(t1); u(t1)) /C300 are satisfied. See also CONTROL THEORY References Iyanaga, S. and Kawada, Y. (Eds.). "Pontrjagin’s [sic] Maximum Principle." §88C in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 295 /C1/96, 1980. Pontryagin Number The Pontryagin number is defined in terms of the PONTRYAGIN CLASS of a MANIFOLD as follows. For any collection of PONTRYAGIN CLASSES such that their cup product has the same DIMENSION as the MANIFOLD , this cup product can be evaluated on the MANIFOLD ’s FUNDAMENTAL CLASS . The resulting number is called the Pontryagin number for that combination of Pontryagin classes. The most important aspect of Pontryagin numbers is that they are COBORDISM invariant. Together, Pontryagin and STIEFEL- WHIT- NEY NUMBERS determine an oriented manifold’s or- iented COBORDISM class. See also CHERN NUMBER ,STIEFEL- WHITNEY NUMBERPonzo’s Illusion The upper HORIZONTAL line segment in the above figure appears to be longer than the lower line segment despite the fact that both are the same length. See also ILLUSION ,M U¨ LLER- LYER ILLUSION ,POGGEN- DORFF ILLUSION ,VERTICAL- HORIZONTAL ILLUSION References Fineman, M. The Nature of Visual Illusion. New York: Dover, p. 153, 1996. Pop An action which removes a single element from the top of a QUEUE or STACK , turning the LIST (/a1 ; a2 ; ..., an) into (/a2 ; ..., an) and yielding the element a1 :/ See also PUSH,STACK Population The word population has a number of distinct but closely related meanings in statistics. 1. A finite and actually existing group of objects which, although possibly large, can be enumerated in theory (e.g., people living in the United States). 2. A generalization from experience which is indefinitely large (e.g., the total number of throws that might conceivably by made in unlimited time with a particular pair of dice). Any actual set of throws can then be regarded as a SAMPLE drawn from this practically infinite population. 3. A purely hypothetically population which can be completely described mathematically. See also SAMPLE References Kenney, J. F. and Keeping, E. S. "Populations and Sam- ples." §7.1 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 90 /C1/1, 1962. Population Comparison Letx1andx2be the number of successes in variates taken from two populations. Define ˆp1/C13x1 n1(1) ˆp2 /C13x2 n2(2) The ESTIMATOR of the difference is then ˆp1 /C28 ˆp2 : Doing a Z-TRANSFORM , z /C30ˆp1 /C28 ˆp2 ðÞ /C28 p1 /C28 p2 ðÞ sˆp1 /C28ˆp2; (3) where sˆp1/C28ˆp2/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 ˆp1/C28 s2 ˆp2q : (4) The STANDARD ERROR is SEˆp1/C28ˆp2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆp11 /C28 ˆp1 ðÞ n1/C27ˆp21 /C28 ˆp2 ðÞ n2s (5) SE¯x1/C28¯x2/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 1 n1/C27s22 n2s (6) s2 pool /C30n1 /C28 1 ðÞ s2 1 /C27 n2 /C28 1 ðÞ s22 n1 /C27 n2 /C28 2: (7) See also Z-TRANSFORM (POPULATION ) References Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, pp. 162 /C1/71, 1993. Population Growth The differential equation describing exponential growth is dN dt/C30N t: (1) This can be integrated directly gN N0dN N/C30gt 0dt t (2) lnN N0 ! /C30t t : (3) Exponentiating, N(t) /C30N0et=t : (4) Defining N(t /C301) /C30N0e a gives t /C301=a in (4), so N(t) /C30N0e at : (5) This equation is called the LAW OF GROWTH , and the quantity a in this equation is sometimes known as the MALTHUSIAN PARAMETER .Consider a more complicated growth law dN dt/C30at /C28 1 t ! N ; (6) where a > 1 is a constant. This can also be integrated directly dN N/C30 a /C281 t ! dt (7) ln N /C30 at /C28ln t /C27C (8) N(t) /C30Ce at t: (9) Note that this expression blows up at t /C300. We are given the INITIAL CONDITION that N(t /C301) /C30N0e a ; so C /C30N0 : N(t) /C30N0e at t: (10) The t in the DENOMINATOR of (10) greatly suppresses the growth in the long run compared to the simple growth law. The LOGISTIC GROWTH CURVE , defined by dN dt/C30r(K /C28 N) N (11) is another growth law which frequently arises in biology. It has a rather complicated solution for N(t):/ See also GOMPERTZ CURVE ,G ROWTH ,L AW OF GROWTH ,L IFE EXPECTANCY ,L OGISTIC GROWTH CURVE ,L OTKA- VOLTERRA EQUATIONS ,M AKEHAM CURVE ,M ALTHUSIAN PARAMETER ,S URVIVORSHIP CURVE References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 290 /C1/95, 1999. Porism An archaic type of mathematical proposition whose historical purpose is not entirely known. In modern usage, the term "porism" is used instead of "theorem" for a small number of results for historical reasons. See also AXIOM ,L EMMA ,P OSTULATE ,P ONCELET’S PORISM ,PRINCIPLE ,STEINER’S PORISM ,THEOREM Porous Medium Equation The PARTIAL DIFFERENTIAL EQUATION ut/C309 /C215um9u ðÞ : References Elliott, C. M.; Herrero, M. A.; King, J. R.; and Ockendon, J. R. "The Mesa Problem: Diffusion Patterns for ut /C309 /C215 um 9u ðÞ as m 0/C27/C12:/" IMA J. Appl. Math. 7, 147 /C1/54, 1986. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 134, 1997. Porter’s Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. The constant appearing in FORMULAS for the efficiency of the EUCLIDEAN ALGORITHM , C /C306ln2 p23ln2/C274g /C2824 p2z ?(2) /C282"# /C281 2 /C301:4670780794 ... ; where g is the EULER- MASCHERONI CONSTANT and z(z) is the RIEMANN ZETA FUNCTION . See also EUCLIDEAN ALGORITHM References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/porter/porter.html. Porter, J. W. "On a Theorem of Heilbronn." Mathematika 22,20/C1/8, 1975. Po´sa’s Conjecture Dirac (1952) proved that if the minimum VERTEX DEGREE d(G) ]n=2 for a graph G on n ]3 nodes, then G contains a HAMILTONIAN CIRCUIT (Bolloba ´s 1978, Komlo ´s et al. 1998). In 1962, Po´sa conjectured that G(V ; E) contains a square of a HAMILTONIAN CIRCUIT if d(G) ]2n=3 (Erdos 1964, p. 159; Komlo ´s et al. 1998), where a graph G(V ; E) contains the SQUARE of a HAMILTO- NIAN CIRCUIT if there is a HAMILTONIAN CIRCUIT H /C30 x1 ; x2 ;...; xn ; xn/C271 /C30x19+=9+; such that xi ; xi/C2729+=9+; /C23 E(G); for i /C301, 2, ..., n. Komlo ´s et al. (1996) proved that there exists a natural number n0such that if a graph G has order n ]n0and minimum degree at least 2n =3; then G contains the square of a Hamiltonian circuit. This proved Po´sa’s conjecture (Erdos 1964) for sufficiently large n. Kierstead and Quintana (1998) proved Po´sa’s conjecture for graphs G containing a 4-clique K4:/ The conjecture was generalized by Seymour (1974) to state that if d(G)]kn=(k/C271);then Gcontains the kth power of a H AMILTONIAN CIRCUIT (Komlo ´set al. 1998). See also HAMILTONIAN CIRCUIT ,PO´ SA’S CONJECTURE , SEYMOUR CONJECTURE References Dirac, G. A. "Some Theorems on Abstract Graphs." Proc. London Math. Soc. 2,6 9/C1/1, 1952.Erdos, P. "Problem 9." In Theory of Graphs and Its Applica- tions, Proceedings of the Symposium held in Smolenice in June 1963 (Ed. M. Fiedler). Prague, Czechoslovakia: Publishing House of the Czechoslovak Academy ofSciences, p. 159, 1964. Fan, G. and Kierstead, H. A. "Hamiltonian Square-Paths." J. Combin. Theory Ser. B 67, 167/C1 /82, 1996. Kierstead, H. A. and Quintana, J. "Square Hamiltonian Cycles in Graphs with Maximal 4 /-Cliques." Disc. Math. 178,8 1/C1/2, 1998. Komlo ´s, J.; Sa ´rkozy, G. N.; and Szemere ´di, E. "On the Square of a Hamiltonian Cycle in Dense Graphs." InRandom Structures Algorithms 9, 193/C1 /11, 1996. Seymour, P. Problem Section in Combinatorics: Proceedings of the British Combinatorial Conference, 1973 (Ed. T. P. McDonough and V. C. Mavron). Cambridge, Eng- land: Cambridge University Press, pp. 201 /C1/02, 1974. Po´sa’s Theorem There are several related theorems involving H AMIL- TONIAN CIRCUITS of graphs that are associated with Po´sa. LetGbe a SIMPLE GRAPH with nVERTICES . 1. If, for every kin 15kB(n/C281)=2;the number of VERTICES ofVERTEX DEGREE not exceeding kis less than k, and 2. If, for nODD, the number of VERTICES with VERTEX DEGREE not exceeding ( n/C281)=2 is less than or equal to ( n/C281)=2;/ then Gcontains a H AMILTONIAN CIRCUIT . Kronk (1969) generalized this result as follows. Let G be a SIMPLE GRAPH with nVERTICES , and let 0 5k5 n/C282:Then the following conditions are SUFFICIENT forGto be k-line Hamiltonian: 1. For all integers jwith k/C2715jB(n/C27k/C281)=2; the number of VERTICES ofVERTEX DEGREE not exceeding jis less than j/C28k;/ 2. The number of points of degree not exceeding (n/C27k/C281)=2 does not exceed ( n/C28k/C281)=2:/ Po´sa (1963) generalized a result of Dirac by proving that every FINITE SIMPLE GRAPH Gwith a sufficiently large valencies of all (or, in some cases, of ALMOST ALL) vertices and with a sufficiently large number of vertices satisfies one of the following conditions. 1.Ghas a Hamiltonian line containing all edges of given disjoint paths (Theorem 1),2.Ghas a circuit with a "large" number of vertices (Theorems 2 and 3), or3.Ghas a "small" number of disjoint circuits containing all vertices of the graph (Theorems 4and 5). References Bolloba ´s, B. Extremal Graph Theory. New York: Academic Press, 1978. Bondy, J. A. "Cycles in Graphs." In Combinatorial Struc- tures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969). New York: Gordon and Breach, pp. 15 /C1/8, 1970. Dirac, G. A. "Some Theorems on Abstract Graphs." Proc. London Math. Soc. 2,69/C1/1, 1952. Komlo ´s, J.; Sa´rkozy, G. N.; and Szemere ´di, E. "Proof of the Seymour Conjecture for Large Graphs." Ann. Comb. 2, 43 /C1/0, 1998. Kronk, H. V. "Variations on a Theorem of Po´sa." In The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968). Berlin: Springer-Verlag, pp. 193 /C1/97, 1969. Lick, D. R. "n-Hamiltonian Connected Graphs." Duke Math. J. 37, 387 /C1/92, 1970. Marshall, C. W. Applied Graph Theory. New York: Wiley, 1971. Nash-Williams, C. St. J. A. "Hamiltonian Lines in Graphs Whose Vertices Have Sufficiently Large Valencies." In Combinatorial Theory and Its Applications, III (Proc. Colloq., Balatonfu ¨red, 1969). Amsterdam, Netherlands: North-Holland, pp. 813 /C1/19, 1970. Nash-Williams, C. St. J. A. "Hamiltonian Lines in Infinite Graphs with Few Vertices of Small Valency." Aequationes Math. 7,59/C1/1, 1971. Po´sa, L. "On the Circuits of Finite Graphs." Magyar Tud. Akad. Mat. Kutato ´ Int. Kozl. 8, 355 /C1/61, 1963. Po¨schl-Teller Differential Equations The first and second Po¨schl-Teller differential equa- tions are given by yƒ/C28 a2k( k /C28 1) sin2(ax)/C27l( l /C28 1) cos2(ax)"# /C28b2() y /C300 and yƒ/C28 a2k( k /C28 1) sinh2(ax) /C27l( l /C28 1) cosh2(ax)"# /C28b2() y /C300 respectively. References Barut, A. O.; Inomata, A.; and Wilson, R. "Algebraic Treat- ment of Second Po¨schl-Teller, Morse-Rosen, and Eckart Equations." J. Phys. A: Math. Gen. 20, 4083 /C1/4096, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997. Poset PARTIALLY ORDERED SET Poset Dimension The DIMENSION of a POSET P /C30(X ;5) is the size of the smallest REALIZER of P. Equivalently, it is the smallest INTEGER d such that P is ISOMORPHIC to a DOMINANCE order in Rd :/ See also DIMENSION ,D OMINANCE ,ISOMORPHIC PO- SETS ,REALIZER References Dushnik, B. and Miller, E. W. "Partially Ordered Sets." Amer. J. Math. 63, 600 /C1/10, 1941.Trotter, W. T. Combinatorics and Partially Ordered Sets: Dimension Theory. Baltimore, MD: Johns Hopkins Uni- versity Press, 1992. Position Four-Vector The CONTRAVARIANT FOUR-VECTOR arising in special and general relativity, xm /C30x0 x1 x2 x32 6643 775/C13ct x y z2 6643 775; where c is the speed of light and t is time. Multi- plication of two four-vectors gives the spacetime interval I /C30g mnxmxv /C30(x0)2 /C28(x1)2 /C28(x2)2 /C28(x3)2 /C30(ct)2 /C28(x1)2 /C28(x2)2 /C28(x3)2 See also FOUR- VECTOR ,LORENTZ TRANSFORMATION , QUATERNION Position Vector RADIUS VECTOR Positive A quantity x /C210, which may be written with an explicit PLUS SIGN for emphasis, /C27x:/ See also NEGATIVE ,NONNEGATIVE ,PLUS SIGN,ZERO Positive Definite Function A positive definite FUNCTION fon a GROUP Gis a FUNCTION for which the MATRIX ff(xix/C281 j)gis always POSITIVE SEMIDEFINITE HERMITIAN . References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537/C1/49, 1996. Positive Definite Matrix AH ERMITIAN MATRIX Ais called positive definite if (Av)/C215v>0 (1) for all VECTORS v"0:This is equivalent to the requirement that all EIGENVALUES bePOSITIVE , and to the requirement that the DETERMINANTS associated with allupper-left SUBMATRICES are POSITIVE . The DETERMINANT of a positive definite matrix is POSITIVE , but the converse is not necessarily true (i.e., a matrix with a POSITIVE DETERMINANT is not neces- sarily positive definite). The numbers of positive definite n/C29nmatrices of given types are summarized in the following table. For example, the three positive definite 2 /C292(0,1)- MATRICES are 10 019+$=9+$; ;10119+$=9+$; ;11019+$=9+$; ; (2) all of which have eigenvalue 1 with degeneracy of two. /(0; 1)/-matrix A000000 0, 3, 25, 543, ... /(/C281 ; 0; 1)/-matrix A000000 0, 5, 133, ... A REAL SYMMETRIC MATRIX A is positive definite IFF there exists a REAL nonsingular MATRIX M such that A /C30MMT (3) where MTis the TRANSPOSE .A2 /C292 SYMMETRIC MATRIX ab bc9+$=9+$; (4) is positive definite if av2 1 /C272bv1v2 /C27cv22 > 0 (5) for all v /C30(v1 ; v2) "0:/ AH ERMITIAN MATRIX A is positive definite if 1. aii > 0 for all i, 2. aiiaij > aij9+;$9+;$9+;$9+;$2for i "j;/ 3. The element of largest modulus lies on the leading diagonal, 4. det(A) > 0:/ See also DETERMINANT ,E IGENVALUE ,H ERMITIAN MATRIX ,MATRIX ,NEGATIVE DEFINITE MATRIX ,NEGA- TIVE SEMIDEFINITE MATRIX ,POSITIVE SEMIDEFINITE MATRIX References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1106, 2000. Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 182, 1988. Marcus, M. and Minc, H. "Positive Definite Matrices." §4.12 in A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 69, 1992. Positive Definite Quadratic Form A QUADRATIC FORM Q(x) is said to be positive definite if Q(x) > 0 for x "0: A REAL QUADRATIC FORM in n variables is positive definite IFF its canonical form is Q(z) /C30z2 1 /C27z22 /C27.../C27z2n : (1) A BINARY QUADRATIC FORM F(x; y) /C30a11x2 /C272a12xy /C27a22y2 (2)of two REAL variables is positive definite if it is > 0 for any (x; y) "(0; 0); therefore if a11 > 0 and the DIS- CRIMINANT a /C13a11a22 /C28a2 12 > 0: A BINARY QUADRATIC FORM is positive definite if there exist NONZERO x and y such that ax2 /C272bxy /C27cy29+=9+;254 3ac /C28b29+;$9+;$9+;$9+;$ (3) (Le Lionnais 1983). A QUADRATIC FORM (x ; Ax) is positive definite IFF every EIGENVALUE of A is POSITIVE .A QUADRATIC FORM Q /C30(x ;Ax) with A aH ERMITIAN MATRIX is positive definite if all the principal minors in the top-left corner of A are POSITIVE , in other words a11 > 0 (4) a11a12 a21a229+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$> 0 (5) a 11a12a13 a21a22a23 a31a32a339+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$>0 (6) See also I NDEFINITE QUADRATIC FORM,LYAPUNOV’S FIRST THEOREM ,POSITIVE SEMIDEFINITE QUADRATIC FORM,QUADRATIC FORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1106, 2000. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 38, 1983. Positive Definite Sequence This entry contributed by R ONALD M.AARTS A sequence mnfg/C12 n/C300is positive definite if the moment of every nonnegative polynomial which is not identi- cally zero is greater than zero (Widder 1941, p. 132). Here, the moment of a polynomial Pn(x)/C30Xn m/C300amxm with respect to the sequence mnfg/C12 n/C300is defined as MPn(x) ðÞ /C30Xn m/C300ammm (Widder 1941, p. 102). References Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Positive Definite Tensor A TENSOR g whose discriminant satisfies g /C13g11g22 /C28g2 12 > 0: Positive Integer The positive integers are the numbers 1, 2, 3, ..., sometimes called the counting numbers or natural numbers. See also Z/C27 Positive Measure A positive measure is a MEASURE which is a function from the measurable sets of a MEASURE SPACE to the nonnegative real numbers. Sometimes, this is what is meant by MEASURE , while "positive" is used to distinguish it from an arbitrary COMPLEX MEASURE . See also COMPLEX MEASURE ,JORDAN MEASURE DECOMPOSITION ,L EBESGUE INTEGRAL ,M EASURE , MEASURE SPACE ,POLAR REPRESENTATION (MEASURE ) Positive Semidefinite Matrix A positive semidefinite matrix is a HERMITIAN MATRIX all of whose EIGENVALUES are nonnegative. See also NEGATIVE DEFINITE MATRIX ,N EGATIVE SEMIDEFINITE MATRIX ,POSITIVE DEFINITE MATRIX References Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 182, 1988. Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 69, 1992. Positive Semidefinite Quadratic Form A QUADRATIC FORM Q(x) is positive semidefinite if it is never B0 ; but is 0 for some x "0 : The QUADRATIC FORM , written in the form (x; Ax) ; is positive semi- definite IFF every EIGENVALUE of A is NONNEGATIVE . See also INDEFINITE QUADRATIC FORM,P OSITIVE DEFINITE QUADRATIC FORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1106, 2000. Postage Stamp Problem Consider a SET Ak /C30 a1 ; a2 ; ...ak fg of INTEGER de- nomination postage stamps with 1 /C30a1 Ba2 ...Bak : Suppose they are to be used on an envelope with room for no more than h stamps. The postage stamp problem then consists of determining the smallest INTEGER N(h ;Ak) which cannot be represented by a LINEAR COMBINATION ak i /C301 xiaiwith xi ]0 andaki /C301xi Bh: Exact solutions exist for arbitrary Akfor k /C302 and 3. The k /C302 solution is nh;A2 ðÞ /C30 h /C273 /C28a2 ðÞ a2 /C282 for h ]a2 /C282 : The general problem consists of finding n(h; k) /C30max Aknh;Ak ðÞ : It is known that n(h; 2) /C301 4h2 /C276h /C2719+=9+;jk ; (Sto¨hr 1955, Guy 1994), where xbcis the FLOOR FUNCTION , the first few values of which are 2, 4, 7, 10, 14, 18, 23, 28, 34, 40, ... (Sloane’s A014616). See also HARMONIOUS GRAPH ,INTEGER RELATION , STAMP FOLDING ,S TO¨ HR SEQUENCE ,S UBSET SUM PROBLEM References Guy, R. K. "The Postage Stamp Problem." §C12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 123 /C1/27, 1994. Mossige, S. "The Postage Stamp Problem: An Algorithm to Determine the h-Range on the h-Range Formula on the Extremal Basis Problem for k /C304." Math. Comput. 69, 325 /C1/37, 2000. Sloane, N. J. A. Sequences A014616 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sto¨hr, A. "Gelo¨ste und ungelo ¨ste Fragen u¨ber Basen der natu¨rlichen Zahlenreihe I, II." J. reine angew. Math. 194, 111 /C1/40, 1955. Posterior Distribution BAYESIAN ANALYSIS Postnikov System An iterated FIBRATION of EILENBERG- MAC LANE SPACES . Every TOPOLOGICAL SPACE has this HOMO- TOPY type. See also EILENBERG- MAC LANE SPACE ,FIBRATION , HOMOTOPY Postulate A statement, also known as an AXIOM , which is taken to be true without PROOF . Postulates are the basic structure from which LEMMAS and THEOREMS are derived. The whole of EUCLIDEAN GEOMETRY , for example, is based on five postulates known as EUCLID’S POSTULATES . See also ARCHIMEDES’ POSTULATE ,A XIOM ,B ER- TRAND’S POSTULATE ,C ONJECTURE ,E QUIDISTANCE POSTULATE ,E UCLID’S FIFTH POSTULATE ,E UCLID’S POSTULATES ,LEMMA ,PARALLEL POSTULATE ,PORISM , PROOF ,THEOREM ,TRIANGLE POSTULATE Potato Paradox You buy 100 pounds of potatoes and are told that they are 99% water. After leaving them outside, you discover that they are now 98% water. The weight of the dehydrated potatoes is then a surprising 50 pounds! References Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, p. 81, 1995. Potential Function The term used in physics and engineering for a HARMONIC FUNCTION . Potential functions are extre- mely useful, for example, in electromagnetism, where they reduce the study of a 3-component VECTOR FIELD to a 1-component SCALAR FUNCTION . See also HARMONIC FUNCTION ,LAPLACE’S EQUATION , SCALAR POTENTIAL ,VECTOR POTENTIAL Potential Theory The study of HARMONIC FUNCTIONS (also called POTENTIAL FUNCTIONS ). See also HARMONIC FUNCTION ,SCALAR POTENTIAL , VECTOR POTENTIAL References Kellogg, O. D. Foundations of Potential Theory. New York: Dover, 1953. MacMillan, W. D. The Theory of the Potential. New York: Dover, 1958. Weisstein, E. W. "Books about Potential Theory." http:// www.treasure-troves.com/books/PotentialTheory.html. Pothenot Problem SNELLIUS- POTHENOT PROBLEM Poulet Number AF ERMAT PSEUDOPRIME to base 2, denoted psp(2), i.e., a COMPOSITE ODD INTEGER n such that 2n/C281 /C131 (mod n) : The first few Poulet numbers are 341, 561, 645, 1105, 1387, ... (Sloane’s A001567). Pomerance et al. (1980) computed all 21,853 Poulet numbers less than 25 /C29 109 : The numbers less than 102,103, ..., are 0, 3, 22, 78, 245, ... (Sloane’s A055550). Pomerance has shown that the number of Poulet numbers less than x for sufficiently large x satisfy exp (ln x)5 =14hi BP2(x) Bx exp /C28ln x ln ln ln x 2lnln x ! (Guy 1994). A Poulet number all of whose DIVISORS d satisfy d 2d /C2829+;$9+;$ is called a SUPER- POULET NUMBER . There are an infinite number of Poulet numbers which are notSUPER- POULET NUMBERS . Shanks (1993) calls any integer satisfying 2n/C281 /C131 (mod n) (i.e., not limited to ODD composite numbers) a FERMATIAN . See also FERMAT PSEUDOPRIME ,PSEUDOPRIME ,ROT- KIEWICZ THEOREM ,SUPER- POULET NUMBER References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 28 /C1/9, 1994. Pinch, R. G. E. "The Pseudoprimes Up to 1013." ftp:// ftp.dpmms.cam.ac.uk/pub/PSP/. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimes to 25 /C215109:/"Math. Comput. 35, 1003 /C1/026, 1980. Available electronically from ftp://sable.ox.ac.uk/ pub/math/primes/ps2.Z. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 115 /C1/17, 1993. Sloane, N. J. A. Sequences A001567/M5441 and A055550 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Power The exponent to which a given quantity is raised is known as its POWER . The expression xais therefore known as " xto the athPOWER ." The power may be an integer, REAL NUMBER ,o r COMPLEX NUMBER . How- ever, the power of a real number to a non-integer power is not necessarily itself a real number. Forexample, x 1=2is real only for x]0:The rules for combining quantities containing powers are called the EXPONENT LAWS . While the simple equation ax/C30x cannot be solved for xusing traditional elementary functions, the solution can be given in terms of LAMBERT’S W-FUNCTION as x/C30/C28W(/C28lna) lna; where ln ais the NATURAL LOGARITHM ofa. Special names given to various powers are listed in the following table. Power Name /1=2/ SQUARE ROOT /1=3/ CUBE ROOT 2 SQUARED 3 CUBED The largest powers p which numbers n /C301, 2, 3, ... can be represented in the form n /C30ap are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, ... (Sloane’s A052409), with corresponding values of a given by 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (Sloane’s A052410). The POWER SUM of the first n POSITIVE INTEGERS is given by FAULHABER’S FORMULA , Xn k /C301kp /C301 p /C27 1Xp /C271 k /C301(/C281)dkpp /C271 k9+;89+;9 Bp /C271/C28knk ; where dkp is the KRONECKER DELTA , n k9+=9+; is a BINOMIAL COEFFICIENT , and Bk is a BERNOULLI NUMBER . Let snbe the largest INTEGER that is not the SUM of distinct nth powers of POSITIVE INTEGERS (Guy 1994). The first few values for n /C302, 3, ... are 128, 12758, 5134240, 67898771, ... (Sloane’s A001661). CATALAN’S CONJECTURE states that 8 and 9 (23 and 32) are the only consecutive POWERS (excluding 0 and 1), i.e., the only solution to CATALAN’S DIOPHANTINE PROBLEM . This CONJECTURE has not yet been proved or refuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the CONJECTURE not hold. It is also known that 8 and 9 are the only consecutive CUBIC and SQUARE NUM- BERS (in either order). Hyyro and Makowski proved that there do not exist three consecutive POWERS (Ribenboim 1996). Very few numbers OF THE FORM np 91 are PRIME (where composite powers p /C30kb need not be consid- ered, since n(kb) 91 /C30 nk9+=9+;b91): The only PRIME NUM- BERS OF THE FORM np /C281 for n 5100 and PRIME 2 5p 510 correspond to n /C302, i.e., 22 /C281 /C303; 23 /C281 /C30 7;25/C281/C3031;.... The only PRIME NUMBERS of the form np/C271 for n5100 and PRIME 25p510 correspond to p/C302 with n/C301, 2, 4, 6, 10, 14, 16, 20, 24, 26, ... (Sloane’s A005574). There are no nontrivial solutions to the equation 1n/C272n/C27.../C27mn/C30m/C271 ðÞn form5102;000;000(Guy 1994, p. 153). See also APOCALYPTIC NUMBER ,BIQUADRATIC NUM- BER,C ATALAN’S CONJECTURE ,C ATALAN’S DIOPHAN- TINE PROBLEM ,CUBE ROOT,CUBED ,CUBIC NUMBER , DIGIT-SHIFTING CONSTANTS ,E XPONENT ,E XPONENT LAWS,FAULHABER’S FORMULA ,FIGURATE NUMBER ,MOESSNER’S THEOREM ,N ARCISSISTIC NUMBER , POWER (CIRCLE ), POWER RULE,SQUARE NUMBER , SQUARE ROOT,SQUARED ,SUM,TRUNCATED POWER FUNCTION ,W ARING’S PROBLEM References Barbeau, E. J. Power Play: A Country Walk through the Magical World of Numbers. Washington, DC: Math. Assoc. Amer., 1997. Beyer, W. H. "Laws of Exponents." CRC Standard Mathe- matical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 158 and 223, 1987. Guy, R. K. "Diophantine Equations." Ch. D in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 137, 139 /C1/98, and 153 /C1/54, 1994. Ribenboim, P. "Catalan’s Conjecture." Amer. Math. Monthly 103, 529/C1/38, 1996. Sloane, N. J. A. Sequences A001661/M5393, A005574/ M1010, A052409, and A052410 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Integer Powers ( bx/C27c)n andxn/" and "The Noninteger Powers xn:/" Ch. 11 and 13 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 83 /C1/0 and 99 /C1/06, 1987. Power (Circle) The POWER of a fixed point Awith respect to a CIRCLE ofRADIUS rand center Ois defined by the product p/C13AP/C29AQ; (1) where PandQare the intersections of a line through Awith the circle. The term "power" was first used in this way by Jacob Steiner (Steiner 1826; Coxeter and Greitzer 1967, p. 30). Amazingly, p(sometimes writ- tenk2)i sindependent of the choice of the line APQ (Coxeter 1969, p. 81). Now consider a point Pnot necessarily on the circumference of the circle. If d/C30OPis the distance between Pand the circle’s center O, then the power of the point P relative to the circle is p /C30d2 /C28r2 : (2) If P is outside the CIRCLE , its power is POSITIVE and equal to the square of the length of the segment PQ from P to the tangent Q to the CIRCLE through P, p /C30PQ2 /C30d2 /C28r2 : (3) If OP lies along the X-AXIS , then the angle u around the circle at which Q lies is given by solving (d /C28cos u)2 /C27sin2 uhi /C271 /C30d2 (4) for u ; giving u /C309sec/C281 d (5) for coordinates (x; y) /C30r 91 d ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 /C28 1 d2s ! : (6) The points P and P ? are INVERSE POINTS , also called polar reciprocals, with respect to the INVERSION CIRCLE if OP /C215 OP?/C30OQ2 /C30r2 (7) (Wenninger 1983, p. 2). If P is inside the CIRCLE , then the power is NEGATIVE and equal to the product of the DIAMETERS through P. The LOCUS of points having POWER k with regard to a fixed CIRCLE of RADIUS r is a CONCENTRIC CIRCLE of RADIUSffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27kp : The CHORDAL THEOREM states that the LOCUS of points having equal POWER with respect to two given nonconcentric CIRCLES is a line called the RADICAL LINE (or CHORDAL ;Do¨rrie 1965). See also CHORDAL THEOREM ,COAXAL CIRCLES ,IN- VERSE POINTS ,INVERSION CIRCLE ,INVERSION RADIUS , INVERSIVE DISTANCE ,RADICAL LINE References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. and Greitzer, S. L. "The Power of a Point with Respect to a Circle." §2.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 27 /C1/1, 1967. Darboux, J. "Me´moir sur les Surfaces Cyclides." Ann. l’E´ cole Normale sup. 1, 273 /C1/92, 1872. Dixon, R. Mathographics. New York: Dover, p. 68, 1991. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 153, 1965. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 28 /C1/4, 1929. Lachlan, R. "Power of a Point with Respect to a Circle." §300 /C1/03 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 183 /C1/85, 1893. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., pp. xxii-xxiv, 1995.Steiner, J. "Einige geometrische Betrachtungen." J. reine angew. Math. 1, 161 /C1/84, 1826. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Power (Statistics) The probability of getting a positive result for a given test which should produce a positive result. See also PREDICTIVE VALUE ,SENSITIVITY ,SPECIFI- CITY,STATISTICAL TEST Power (Triangle) The total power of a TRIANGLE is defined by P /C131 2a2 1 /C27a22 /C27a239+=9+; ; (1) where ai are the side lengths, and the "partial power" is defined by p1 /C301 2a2 2 /C27a23 /C28a219+=9+; : (2) Then p1 /C30a2a3 cos a1 (3) P /C30p1 /C27p2 /C27p3 (4) P2 /C27p21 /C27p22 /C27p23 /C30a41 /C27a42 /C27a43 (5) D/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2p3 /C27p3p1 /C27p1p2p (6) p1 /C30A1H2/C215A1A3 (7) a1p1 cos a1/C30a1a2a3 /C304 DR (8) p1 tan a1 /C30p2 tan a2 /C30p3 tan a3 ; (9) where D is the AREA of the TRIANGLE and Hiare the FEET of the ALTITUDES . finally, if a side of the TRIANGLE and the value of any partial power are given, then the LOCUS of the third VERTEX is a CIRCLE or straight line. See also ALTITUDE ,FOOT,TRIANGLE References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 260 /C1/61, 1929. Power Associative Algebra An ALGEBRA in which the ASSOCIATOR (x; x; x) /C300: The SUBALGEBRA generated by one element is asso- ciative. See also ASSOCIATOR References Schafer, R. D. An Introduction to Non-Associative Algebras. New York: Dover, 1995. Power Center RADICAL CENTER Power Curve The curve with TRILINEAR COORDINATES at : bt : ct for a given POWER t. See also POWER POINT References Kimberling, C. "Major Centers of Triangles." Amer. Math. Monthly 104, 431 /C1/38, 1997. Power Line RADICAL AXIS Power Point Triangle centers with TRIANGLE CENTER FUNCTIONS OF THE FORM a /C30an are called nth power points. The 0th power point is the INCENTER , with TRIANGLE CENTER FUNCTION a /C301:/ See also INCENTER ,TRIANGLE CENTER FUNCTION References Groenman, J. T. and Eddy, R. H. "Problem 858 and Solu- tion." Crux Math. 10, 306 /C1/07, 1984. Kimberling, C. "Problem 865." Crux Math. 10, 325 /C1/27, 1984. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Power Polynomial The power polynomials xn are an associated SHEFFER SEQUENCE with f(t) /C30t; (1) giving GENERATING FUNCTION X/C12 k /C300xk k!tk /C30ext (2) and BINOMIAL IDENTITY (x /C27y)n /C30Xn k /C300n k9+;89+;9 xkyn/C28k : (3) See also SHEFFER SEQUENCE References Roman, S. "The Sequence xn :/" §4.1.1 in The Umbral Calcu- lus. New York: Academic Press, p. 55, 1984.Power Rule The DERIVATIVE of the POWER xn is given by d dxxnðÞ/C30nxn/C281: See also CHAIN RULE,DERIVATIVE ,EXPONENT LAWS, PRODUCT RULE References Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, p. 131, 1999. Power Series A power series in a variable zis an infinite SUM OF THE FORM X/C12 naizi; (1) where n]0 and aiare INTEGERS ,REAL NUMBERS , COMPLEX NUMBERS , or any other quantities of a given type. ACONJECTURE of Po´lya is that if a FUNCTION has a power series with INTEGER COEFFICIENTS and RADIUS OF CONVERGENCE 1, then either the FUNCTION is RATIONAL or the UNIT CIRCLE is a natural boundary. A generalized POWER sum a(h) for h/C300, 1, ... is given by a(h)/C30Xm i/C301Ai(h)ah i; (2) with distinct NONZERO ROOTS ai;COEFFICIENTS Ai(h) which are POLYNOMIALS of degree ni/C281 for POSITIVE INTEGERS ni;and i/C23[1;m]:The generalized POWER sum has order n/C13Xm i/C30mni: (3) For any power series, one of the following is true: 1. The series converges only for x/C300. 2. The series converges absolutely for all x. 3. The series converges absolutely for all xin some finite open interval ( /C28R;R) and diverges if xB/C28R orx/C21R. At the points x/C30Randx/C30/C28R;the series may converge absolutely, converge conditionally, or diverge. To determine the interval of convergence, apply the RATIO TEST for ABSOLUTE CONVERGENCE and solve for x. A power series may be differentiated or integrated within the interval of convergence. Convergent power series may be multiplied and divided (if there is no division by zero). X/C12 k /C301k /C28p (4) CONVERGES if p /C211 and DIVERGES if 0 Bp 51:/ See also BINOMIAL SERIES ,C ONVERGENCE TESTS , FORMAL POWER SERIES ,LAURENT SERIES ,M ACLAUR- IN SERIES ,M ULTINOMIAL SERIES , P-SERIES ,POLYNO- MIAL ,P OWER SET,Q UOTIENT- DIFFERENCE ALGORITHM ,RADIUS OF CONVERGENCE ,RECURRENCE SEQUENCE ,SERIES ,SERIES REVERSION ,TAYLOR SER- IES References Arfken, G. "Power Series." §5.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 313 /C1/ 21, 1985. Hanrot, G.; Quercia, M.; and Zimmerman, P. "Speeding Up the Division and Square Root of Power Series." Report RR- 3973. INRIA, Jul 2000. http://www.inria.fr.RRRT/RR- 3973.html. Myerson, G. and van der Poorten, A. J. "Some Problems Concerning Recurrence Sequences." Amer. Math. Monthly 102, 698 /C1/05, 1995. Niven, I. "Formal Power Series." Amer. Math. Monthly 76, 871 /C1/89, 1969. Po´lya, G. Mathematics and Plausible Reasoning, Vol. 2: Patterns of Plausible Inference. Princeton, NJ: Princeton University Press, p. 46, 1990. Power Set Given a SET S, the power set of S is the SET of all SUBSETS of S. The order of a POWER set of a SET of order n is 2n : Power sets are larger than the SETS associated with them. The power set of S is variously denoted 2S or P(S) :/ The power set of a given set s can be found using Subsets [s] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). A concise implementation in Mathematica is given by PowerSet[s_List] : /C30Distribute[Thread[{{}, List /@ s}, List, {2, 2}], List, List, List, Join] See also SET,SUBSET Power Spectrum For a given signal, the power spectrum gives a plot of the portion of a signal’s power (energy per unit time) falling within given frequency bins. The most com- mon way of generating a power spectrum is by usingaF OURIER TRANSFORM , but other techniques such as the MAXIMUM ENTROPY METHOD can also be used. References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Power Spectra Estimation Using the FFT" and "Power Spectrum Estimation by the Maximum Entropy (All Poles) Method." §13.4 and 13.7 in Numerical Recipesin FORTRAN: The Art of Scientific Computing, 2nd ed.Cambridge, England: Cambridge University Press,pp. 542 /C1 /51 and 565 /C1/69, 1992. Power Sum An analytic solution for a SUM ofPOWERS of integers is Sp(n)/C30Xn k/C301kp/C30z(/C28p)/C28z(/C28p;1/C27n)/C30H(/C28p) n; (1) where z(z) is the R IEMANN ZETA FUNCTION ,z(z;a)i s the H URWITZ ZETA FUNCTION , and H(k) nis a general- ized HARMONIC NUMBER . For the special case of pa POSITIVE INTEGER ,FAULHABER’S FORMULA gives the SUM explicitly as Sp(n)/C301 p/C271Xp/C271 k/C301(/C281)dkpp/C271 k9+;89+;9 Bp/C271/C28knk; (2) where dkpis the K RONECKER DELTA ,n k9+=9+; is a BINOMIAL COEFFICIENT , and Bkis a B ERNOULLI NUMBER . Writ- ten explicitly in terms of a sum of POWERS , Sp(n)/C30Bkp! k!(p/C28k/C271)!np/C28k/C271: (3) It is also true that the COEFFICIENTS of the terms in such an expansion sum to 1, as stated by Bernoulli without proof (Boyer 1943). Computing the sums for p/C301, ..., 10 gives Xn k/C301k/C301 2n2/C27n9+=9+; (4) Xn k/C301k2/C30162n3/C273n3/C27n9+=9+; (5) Xn k/C301k3/C301 4n4/C272n3/C27n29+=9+; (6) Xn k/C301k4/C301 306n5/C2715n4/C2710n3/C28n9+=9+; (7) Xn k/C301k5/C301 122n6/C276n5/C275n4/C28n29+=9+; (8) Xn k/C301k6/C301 426n7/C2721n6/C2721n5/C287n3/C27n9+=9+; (9) Xn k/C301k7/C301 243n8/C2712n7/C2714n6/C287n4/C272n29+=9+; (10) Xn k/C301k8/C301 9010n9/C2745n8/C2760n7/C2842n5/C2720n3/C283n9+=9+; (11) Xn k /C301k9 /C301 202n10 /C2710n9 /C2715n8 /C2814n6 /C2710n4 /C283n29+=9+; (12) Xn k /C301k10 /C301 666n11 /C2733n10 /C2755n9 /C2866n79+= /C2766n5 /C2833n3 /C275nÞ: (13) Xn k /C301k /C301 2 n(n /C271) (14) Xn k /C301k2 /C3016 n(n /C271)(2n /C271) (15) Xn k /C301k3 /C301 4 n2(n /C271)2 (16) Xn k/C301k4 /C301 30 n(n /C271)(2n /C271) 3n2 /C273n /C2819+=9+; (17) Xn k /C301k5 /C301 12 n2(n /C271)2(2n2 /C272n /C281) (18) Xn k /C301k6 /C301 42 n(n /C271)(2n /C271) 3n4 /C276n3 /C283n /C2719+=9+; (19) Xn k/C301k7 /C301 24 n2(n /C271)2 3n4 /C276n3 /C28n2 /C284n /C2729+=9+; (20) Xn k /C301k8 /C301 90 n(n /C271)(2n /C271) /C2 5n6 /C2715n5 /C275n4 /C2815n3 /C28n2 /C279n /C2839+=9+; (21) Xn k /C301k9 /C301 20 n2(n /C271)2 n2 /C27n /C2819+=9+; /C2 2n4 /C274n3 /C28n2 /C283n /C2739+=9+; (22) Xn k /C301k10 /C301 60 n(n /C271)(2n /C271) n2 /C27n /C2819+=9+; /C29 3n6 /C279n5 /C272n4 /C2811n3 /C2710n /C2859+=9+; : (23) A simple graphical proof of the special case of S1(n) /C30 n(n /C271)=2 can also be given by constructing a se- quence of stacks of boxes, each 1 unit across and k units high, where k /C301, 2, ..., n. Now add a rotated copy on top, as in the above figure. Note that theresulting figure has WIDTH n and HEIGHT n /C271; and so has AREA n(n /C271): The desired sum is half this, so the AREA of the boxes in the sum is n(n /C271)=2: Since the boxes are of unit width, this is also the value of the sum. The sum S1(n) /C30n(n /C271)=2 can also be computed using the first EULER- MACLAURIN INTEGRATION FOR- MULA Xn k /C301f(k) /C30gn 1f(x) dx /C2712 f(1) /C2712 f(n) /C271 2! B2[f ?(n) /C28f ?(1)] /C27... (24) with f(k) /C30k: Then Xn k /C301k /C30gn 1xdx/C2712/C215 1 /C2712/C215 n /C2716(1 /C281) /C27... /C3012n2 /C2819+=9+; /C2812 /C27h /C2712 n /C3012 n(n /C271): (25) The surprising identity S3(n) /C30Xn k /C301k3 /C30Xn k /C301k ! 2 ; (26) known as NICOMACHUS’S THEOREM , can also be illu- strated graphically (Wells 1991, pp. 198 /C1/99). Schultz (1980) showed that the sum Sk(n) can be found by writing Sk(n)/C30Ak/C271nk/C271/C27.../C27A1n (27) and solving the system of k/C271 equations Xk/C271 i/C30j/C271(/C281)i/C28j/C271i j9+;89+;9 Ai/C300 (28) for 05j5k(Guo and Qi 1999). /Si(n) is related to the BINOMIAL THEOREM by (1/C27n)k/C271/C301/C27Xk i/C300k/C271 i9+;89+;9 Si(n) (29) (Guo and Qi 1999). See also DIOPHANTINE EQUATION ,FAULHABER’S FOR- MULA ,MULTIGRADE EQUATION ,NICOMACHUS’S THEO- REM,SUM References Boyer, C. B. "Pascal’s Formula for the Sums of Powers of the Integers." Scripta Math. 9, 237/C1/44, 1943. Brualdi, R. A. Introductory Combinatorics, 3rd ed. New York: Elsevier, p. 119, 1997. Cao, J.-T. "A Method of Summing Series and Some Cor- ollaries" [Chinese]. Math. Pract. Th. 20,7 7/C1/4, 1990. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996. Guo, S.-L. and Qi, F. "Recursion Formulae for an m/C301mk:/"J. Anal. Appl. 18, 1123/C1/130, 1999. Schultz, H. J. "The Sums of the kth Powers of the First n Integers." Amer. Math. Monthly 87, 478/C1/81, 1980. Struik, D. A Source Book in Mathematics, 1200 /C1/800. Cam- bridge, MA: Harvard University Press, 1969. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 198 /C1/99, 1991. Yang, B.-C. "Formulae Related to Bernoulli Number and for Sums of the Same Power of Natural Numbers" [Chinese]. Math. Pract. Th. 24,5 2/C1/6 and 74, 1994. Zhang, N.-Y. "Euler’s Number and Some Sums Related to Zeta Function" [Chinese]. Math. Pract. Th. 20,6 2/C1/0, 1990. Power Tower N.B. A detailed online essay by S. Finch was the starting point for this entry. The power tower of order kis defined as a/C160/C160k/C13aaUa |fflffl{zfflffl} k; (1) where /C160is Knuth’s (1976) ARROW NOTATION , which in turn is defined by a/C160kn/C30a/C160k/C281a/C160k(n/C281)9+$9+% : (2) Rucker (1995, p. 74) uses the notation ka/C13aaUa |fflffl{zfflffl} n; (3) and refers to this operation as "tetration." A power tower can be implemented in Mathematica as PowerTower[a_, k_] : /C30Fold[Power[a, #] &, 1, Table[a, {k}]] The following table gives values of aaUa |fflffl{zfflffl} nfora/C301, 2, ... for small n. n /aaUa |fflffl{zfflffl} n/ 1 1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,1 0 ,. . . 2 1, 4, 27, 256, 3125, 46656, ... 3 1, 16, 7 :63/C291012;1:34/C2910154;... 4 1, 65536, ... The following table gives aaUa |fflffl{zfflffl} nforn/C301, 2, ... for small a.a /aaUa |fflffl{zfflffl} n/ 1 1 ,1 ,1 ,1 ,1 ,1 ,. . . 2 2, 4, 16, 65536, 2 :00/C291019728;... 3 3, 27, 7 :63/C291012;... 4 4, 256, 1 :34/C2910154;... The value of the infinite power tower h(x)/C30x/C160/C160/C12/C30 xxU;where xxxis an abbreviation for xxxðÞ;can be computed analytically by writing xxU/C30h(x) (4) taking the logarithm of both sides and plugging back in to obtain xxUlnx/C30h(x)l nx/C30ln[h(x)]: (5) Solving for h(x) gives h(x)/C30/C28W(/C28lnx) lnx; (6) where W(x)i sL AMBERT’S W-FUNCTION (Corless et al. ). h(x) converges IFFe/C28e5x5e1=e(0:06595x51:4446) ; as shown by Euler (1783) and Eisenstein (1844) (LeLionnais 1983, Wells 1986, p. 35). Knoebel (1981) gave the following series for h(z) h(z)/C301/C27lnx/C2732(lnz)2 3!/C2743(lnz)3 4!/C27... ( 7 ) (Vardi 1991), and a CONTINUED FRACTION due to Khovanskii (1963) is x1=x /C301 /C272(x /C28 1) x2 /C27 1 /C28x2 /C28 1 ðÞ (x /C28 1)2 3x(x /C27 1)4x2 /C28 1 ðÞ (x /C28 1)2 5x(x /C27 1) /C289x2 /C28 1 ðÞ (x /C28 1)2 7x(x /C27 1) /C28 ...: (8) The related function g(x) /C30x(1=x)(1=x) U (9) converges only for x ]e /C281=e ; that is, x ]0:692: The value it converges to is the inverse of xx which, for x Bee (i.e., x B15 :154) ; is given by g(x) /C30ln x W(ln x) (10) for e /C281 =e 5x 5ee :/ The function xx is plotted above along the real line and in the complex plane. It has a minimum where d dxxx /C30xx(1 /C27ln x) /C300 ; (11) which has solution x /C301=e: At this point, the function takes on the value e/C281 =e :/ Some interesting related integrals are g1 0xx dx /C30X/C12 n/C301(/C281)n/C271 nn/C300:7834305107 . . . (12) g1 0x/C28xdx/C30X/C12 n/C3011 nn/C301:2912859971 . . . (13) (Spiegel 1968, Abramowitz and Stegun 1972). See also ACKERMANN FUNCTION ,ARROW NOTATION , FERMAT NUMBER ,L AMBERT’S W-FUNCTION ,M ILLS’ CONSTANT ,STEINER’S PROBLEM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Ash, J. M. "The Limit of xxUxasxTends to Infinity." Math. Mag. 69, 207/C1/09, 1996. Baker, I. N. and Rippon, P. J. "Convergence of Infinite Exponentials." Ann. Acad. Sci. Fennicæ Ser. A. I. Math. 8, 179/C1/86, 1983. Baker, I. N. and Rippon, P. J. "Iteration of Exponential Functions." Ann. Acad. Sci. Fennicæ Ser. A. I. Math. 9, 49/C1/7, 1984. Baker, I. N. and Rippon, P. J. "A Note on Complex Itera- tion." Amer. Math. Monthly 92, 501/C1/04, 1985. Barrow, D. F. "Infinite Exponentials." Amer. Math. Monthly 43, 150/C1/60, 1936. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; and Knuth, D. E. "On the Lambert WFunction." Adv. Comput. Math. 5, 329/C1/59, 1996. Creutz, M. and Sternheimer, R. M. "On the Convergence of Iterated Exponentiation, Part I." Fib. Quart. 18, 341/C1/47, 1980. Creutz, M. and Sternheimer, R. M. "On the Convergence of Iterated Exponentiation, Part II." Fib. Quart. 19, 326/C1/35, 1981. de Villiers, J. M. and Robinson, P. N. "The Interval of Convergence and Limiting Functions of a HyperpowerSequence." Amer. Math. Monthly 93,1 3/C1 /3, 1986. Eisenstein, G. "Entwicklung von aaaU:/"J. reine angew. Math. 28,4 9/C1/2, 1844. Elstrodt, J. "Iterierte Potenzen." Math. Semesterber. 41, 167/C1/78, 1994. Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2,2 9/C1/1, 1783. Reprinted in Euler, L. Opera Omnia I6: Commenta- tiones Algebraicae. pp. 350 /C1/69. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/itrexp/itrexp.html. Ginsburg, J. "Iterated Exponentials." Scripta Math. 11, 340/C1/53, 1945. Khovanskii, A. N. The Application of Continued Fractions and Their Generalizations to Problems in ApproximationTheory. Groningen, Netherlands: P. Noordhoff, 1963. Knoebel, R. A. "Exponentials Reiterated." Amer. Math. Monthly 88, 235 /C1/52, 1981. Knuth, D. E. "Mathematics and Computer Science: Coping with Finiteness. Advances in our Ability to Compute are Bringing us Substantially Closer to Ultimate Limitations." Science 194 1235 /C1/242, 1976. La¨nger, H. "An Elementary Proof of the Convergence of Iterated Exponentials." Elem. Math. 51,75/C1/7, 1996. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 22 and 39, 1983. Mauerer, H. "U¨ ber die Funktion xx Ufu¨r ganzzahliges Argument (Abundanzen)." Mitt. Math. Gesell. Hamburg 4,33/C1/0, 1901. Meyerson, M. D. "The xx Spindle." Math. Mag. 69, 198 /C1/06, 1996. Rippon, P. J. "Infinite Exponentials." Math. Gaz. 67, 189 /C1/ 96, 1983. Rucker, R. Infinity and the Mind: The Science and Philoso- phy of the Infinite. Princeton, NJ: Princeton University Press, 1995. Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 11 /C1/2 and 226 /C1/29, 1991. Weber, R. O. and Roumeliotis, J. "i^i^i^i^...." Austral. Math. Soc. Gaz. 22, 182 /C1/84, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 35, 1986. Powerfree A POSITIVE INTEGER n is kth powerfree if there is no number d such that dk nj (/dk divides n), i.e., there are no kth powers or higher in the PRIME FACTORIZATION of n. A number which is free of all powers is therefore SQUAREFREE . See also BIQUADRATEFREE ,CUBEFREE ,PRIME NUM- BER,SQUAREFREE References Baake, M.; Moody, R. V.; and Pleasants, P. A. B. Diffraction from Visible Lattice Points and kth Power Free Integers. 19 Jun 1999. http://xxx.lanl.gov/abs/math.MG/9906132/. Powerful Number An INTEGER m such that if pm;j then p2 m;j is called a powerful number. The first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (Sloane’s A001694). Powerful numbers are always OF THE FORM a2b3 for a; b ]1:/ Not every NATURAL NUMBER is the sum of two powerful numbers, but Heath-Brown (1988) has shown that every sufficiently large NATURAL NUMBER is the sum of at most three powerful numbers. There are infinitely many pairs of consecutive powerful numbers, but Erdos has conjectured that there do not exist three consecutive powerful numbers. The CONJECTURE that there are no powerful number triples implies that there are infinitely many Wiefer- ich primes (Granville 1986, Vardi 1991). A separate usage of the term powerful number is for numbers which are the sums of any positive powers of their digits (not necessarily the same for each digit).The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, ... (Sloane’s A007532). These are also called hand- some numbers by Rivera, and are a special case of the NARCISSISTIC NUMBERS . Powerful numbers represen- table in two distinct ways (not counting different powers of duplicated digits as distinct) are 264, 373, 375, 2132, 2223, 2241, 2243, 2245, 2263, (Sloane’s A050240). Powerful numbers representable in two distinct ways (counting different powers of duplicated digits as distinct) are 224, 226, 264, 332, 334, 375, 377, 445, (Sloane’s A050241). See also NARCISSISTIC NUMBER References Granville, A. "Powerful Numbers and Fermat’s Last Theo- rem." C. R. Math. Rep. Acad. Sci. Canada 8, 215 /C1/18, 1986. Guy, R. K. "Powerful Numbers." §B16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 67 /C1/3, 1994. Heath-Brown, D. R. "Ternary Quadratic Forms and Sums of Three Square-Full Numbers." In Se´minaire de Theorie des Nombres, Paris 1986 /C1/7 (Ed. C. Goldstein). Boston, MA: Birkha ¨user, pp. 137 /C1/63, 1988. Ribenboim, P. "Catalan’s Conjecture." Amer. Math. Monthly 103, 529 /C1/38, 1996. Rivera, C. "Problems & Puzzles: Puzzle Narcissistic and Handsome Primes.-015." http://www.primepuzzles.net/ puzzles/puzz_015.htm. Sloane, N. J. A. Sequences A001694/M3325, A007532/ M0487, A050240, and A050241 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 59 /C1/2, 1991. P-Polynomial HOMFLY POLYNOMIAL P-Problem A problem is assigned to the P ( POLYNOMIAL time) class if the number of steps is bounded by a POLY- NOMIAL . See also COMPLEXITY THEORY ,NP -COMPLETE PRO- BLEM ,NP -HARD PROBLEM ,NP -PROBLEM References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Clay Mathematics Institute. "The P vs. NP Problem." http:// www.claymath.org/prize_problems/p_vs_np.htm. Cook, S. "The P versus NP Problem." http://www.clay- math.org/prize_problems/p_vs_np.pdf. Greenlaw, R.; Hoover, H. J.; and Ruzzo, W. L. Limits to Parallel Computation: P-Completeness Theory. Oxford, England: Oxford University Press, 1995. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 0821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Provi-dence, RI: Amer. Math. Soc., 2000. Practical Number A number n is practical if for all k 5n; k is the sum of distinct proper divisors of n. Defined in 1948 by A. K. Srinivasen. All even PERFECT NUMBERS are practical. The number m /C302n/C281 2n/C2819+=9+; is practical for all n /C302, 3, .... The first few practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, ... (Sloane’s A005153). G. Melfi has computed twins, triplets, and 5-tuples of practical numbers. The first few 5-tuples are 12, 18, 30, 198, 306, 462, 1482, 2550, 4422, .... References Melfi, G. "On Two Conjectures About Practical Numbers." J. Number Th. 56, 205 /C1/10, 1996. Sloane, N. J. A. Sequences A005153/M0991 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Prandtl’s Boundary Layer Equations The system of PARTIAL DIFFERENTIAL EQUATIONS ut /C27uux /C27vuy /C30Ut /C27UUx /C27m ruyy ux /C27vy /C300 : References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 672, 1980. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Pratt Certificate A primality certificate based on FERMAT’S LITTLE THEOREM CONVERSE . Although the general idea had been well-established for some time, Pratt became the first to prove that the certificate tree was of poly- nomial size and could also be verified in polynomial time. He was also the first to observe that the tree implies that PRIMES are in the complexity class NP. To generate a Pratt certificate, assume that n is a POSITIVE INTEGER and pifg is the set of PRIME FACTORS of n /C281: Suppose there exists an INTEGER x (called a "WITNESS ") such that xn/C281 /C131 (mod n) but xe f1 (mod n) whenever e is one of (n /C281)=pi : Then FERMAT’S LITTLE THEOREM CONVERSE states that n is PRIME (Wagon 1991, pp. 278 /C1/79). By applying FERMAT’S LITTLE THEOREM CONVERSE to n and recursively to each purported factor of n /C281; a certificate for a given PRIME NUMBER can be gener- ated. Stated another way, the Pratt certificate gives a proof that a number a is a PRIMITIVE ROOT of the multiplicative GROUP (mod p) which, along with the fact that a has order p /C281; proves that p is a PRIME . The figure above gives a certificate for the primality of n /C307919. The numbers to the right of the dashes are WITNESSES to the numbers to left. The set pifg for n /C281 /C307918 is given by f2; 37; 107g: Since 77918 /C13 1 (mod 7919) but 77918 =2 ; 77918 =37 ; 77918 =107 f1 (mod 7919), 7 is a WITNESS for 7919. The PRIME divisors of 7918 /C307919 /C1/ are 2, 37, and 107. 2 is a so-called "self- WITNESS " (i.e., it is recognized as a PRIME without further ado), and the remainder of the witnesses are shown as a nested tree. Together, they certify that 7919 is indeed PRIME . Because it requires the FACTOR- IZATION of n /C281; the METHOD of Pratt certificates is best applied to small numbers (or those numbers n known to have easily factorable n /C281):/ A Pratt certificate is quicker to generate for small numbers than are other types of primality certifi- cates. The Mathematica taskProvablePrimeQ [n]in the Mathematica add-on package NumberTheory‘- PrimeQ‘ (which can be loaded with the command BBNumberTheory‘ )therefore generates an ATKIN- GOLDWASSER-KILIAN-MORAIN CERTIFICATE only for numbers above a certain limit (1010 by default), and a Pratt certificate for smaller numbers. See also ATKIN- GOLDWASSER- KILIAN- MORAIN CERTI- FICATE ,FERMAT’S LITTLE THEOREM CONVERSE ,PRIM- ALITY CERTIFICATE ,W ITNESS References Pratt, V. "Every Prime Has a Succinct Certificate." SIAM J. Comput. 4, 214 /C1/20, 1975. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 278 /C1/85, 1991. Wilf, H. §4.10 in Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall, 1986. Pratt-Kasapi Theorem HOEHN’S THEOREM Precedes The relationship xprecedes yis written x)y:The relation xprecedes or is equal to yis written x/C19y:/ See also SUCCEEDS Precession CURVE OF CONSTANT PRECESSION Precisely Unless If A is true precisely unless B, then B implies not-A and not-B implies A. J. H. Conway has suggested the term "UNLESSS " for this state of affairs, by analogy with IFF. See also IFF,UNLESS Predecessor /a is called a predecessor if there is no ORDINAL NUMBER b such that b /C271 /C30 a:/ See also ORDINAL NUMBER ,SUCCESSOR Predicate An operator in LOGIC which returns either TRUE or FALSE . See also AND, FALSE , NAND, NOR, NOT, OR, PREDICATE CALCULUS ,TRUE, XNOR, XOR Predicate Calculus The branch of formal LOGIC , also called functional calculus, that deals with representing the logical connections between statements as well as the state- ments themselves. See also GO¨ DEL’S INCOMPLETENESS THEOREM ,LOGIC , PREDICATE ,PROPOSITIONAL CALCULUS Predictability Predictability at a time t in the future is defined by R(x(t) ; x(t /C27 t)) H(x(t)); and linear predictability by L(x(t) ; x(t /C27 t)) H(x(t)); where R and L are the REDUNDANCY and LINEAR REDUNDANCY , and H is the ENTROPY . Prediction Paradox UNEXPECTED HANGING PARADOX Prediction Theory The problem of forecasting future values Xt /C27t(/ t > 0) of a weakly stationary process Xtfg from the known values Xs (/s 5t):/ See also TIME SERIES ANALYSISReferences Itoˆ, K. (Ed.). "Prediction Theory." §395D in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 3. Cambridge, MA: MIT Press, pp. 1463 /C1/465, 1987. Predictive Value The positive predictive value is the probability that a test gives a true result for a true statistic. The negative predictive value is the probability that a test gives a false result for a false statistic. See also POWER (STATISTICS ), SENSITIVITY ,SPECIFI- CITY,STATISTICAL TEST Predictor-Corrector Methods A general set of methods for integrating ORDINARY DIFFERENTIAL EQUATIONS . Predictor-corrector meth- ods proceed by extrapolating a polynomial fit to the derivative from the previous points to the new point (the predictor step), then using this to interpolate the derivative (the corrector step). Press et al. (1992) opine that predictor-corrector methods have been largely supplanted by the BULIRSCH- STOER and RUNGE- KUTTA METHODS , but predictor-corrector schemes are still in common use. See also ADAMS’ METHOD ,GILL’S METHOD ,M ILNE’S METHOD ,RUNGE- KUTTA METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 896 /C1/97, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 493 /C1/94, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Multistep, Multivalue, and Predictor-Correc-tor Methods." §16.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 740 /C1 /44, 1992. Preimage Given f:X0Y;the image of xisf(x):The preimage ofyis then f/C281(y)/C30fx½f(x)/C30yg;or all xwhose image isy. Images are in the range, while preimages are in the domain (or they are empty). Present Value The present value vnof a single payment made at n periods in the future is vn/C30p (1/C27r)n; (1) where nis the number of periods until payment, pis the payment amount, and ris the periodic discount rate. The present value v/C12of equal payments made each successive period in perpetuity (a.k.a. the pre- sent value of a perpetuity) is given by v/C12/C30X/C12 n/C301p (1 /C27 r)n /C30p r: (2) The present value v? of equal payments made each successive period for n periods (a.k.a. the present value of an annuity) is given by v?/C30v/C12/C28vn /C30p r1 /C281 (1 /C27 r)n"# ; (3) where p is the periodic payment amount. See also INTEREST Pretzel Curve KNOT CURVE Pretzel Knot A KNOT obtained from a TANGLE which can be represented by a FINITE sequence of INTEGERS . See also TANGLE References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 48, 1994. Pretzel Transformation A topological transformation in which a surface is made out of an infinitely elastic material which, however, may not be torn or cut. Using this simple prescription gives the amazing two conversions illu- strated above, the first of which untangles two interlocked rings connected by a band, and the second of which unloops one of two rings connected by a band and threaded by a band (Wells 1991). See also TOPOLOGY References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 194, 1991.Price’s Theorem Consider a GAUSSIAN BIVARIATE DISTRIBUTION in variables x and y with COVARIANCE r /C30 r11 /C30 xyhi/C28 xhiyhi and an arbitrary function g(x; y): Then the expected value of the random variable g(x; y) g(x; y) hi /C30g/C12 /C28/C12g/C12 /C28/C12g(x; y)f(x; y) dx dy satisfies @n g(x ; y) hi @ rn/C30@2ng(x; y) @xn @yn*+ : See also COVARIANCE ,GAUSSIAN BIVARIATE DISTRIBU- TION References McMahon, E. L. "An Extension of Price’s Theorem." IEEE Trans. Inform. Th. 10, 168 /C1/71, 1964. Papoulis, A. "Price’s Theorem and Join Moments." Prob- ability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 226 /C1/28, 1984. Price, R. "A Useful Theorem for Non-Linear Devices Having Gaussian Inputs." IEEE Trans. Inform. Th. 4,69/C1/2, 1958. Primality Certificate A short set of data that proves the primality of a number. A certificate can, in general, be checked much more quickly than the time required to gen- erate the certificate. Varieties of primality certificates include the PRATT CERTIFICATE and ATKIN- GOLDWAS- SER-KILIAN-MORAIN CERTIFICATE . See also ATKIN- GOLDWASSER- KILIAN- MORAIN CERTI- FICATE ,COMPOSITENESS CERTIFICATE ,PRATT CERTI- FICATE References Wagon, S. "Prime Certificates." §8.7 in Mathematica in Action. New York: W. H. Freeman, pp. 277 /C1/85, 1991. Primality Test A test to determine whether or not a given number is PRIME . The RABIN- MILLER STRONG PSEUDOPRIME TEST is a particularly efficient ALGORITHM used by Math- ematica version 2.2. Like many such algorithms, it is a probabilistic test using PSEUDOPRIMES , and can potentially (although with very small probability) falsely identify a COMPOSITE NUMBER as PRIME (although not vice versa). Unlike PRIME FACTORIZA- TION , primality testing is believed to be a P-PROBLEM (Wagon 1991). In order to guarantee primality, an almost certainly slower algorithm capable of generat- ing a PRIMALITY CERTIFICATE must be used. See also ADLEMAN- POMERANCE- RUMELY PRIMALITY TEST,FERMAT’S LITTLE THEOREM CONVERSE ,FER- MAT’S PRIMALITY TEST,FERMAT’S THEOREM ,LUCAS- LEHMER TEST,M ILLER’S PRIMALITY TEST,P E´ PIN’S TEST,POCKLINGTON’S THEOREM ,PROTH’S THEOREM , PSEUDOPRIME ,RABIN- MILLER STRONG PSEUDOPRIME TEST,W ARD’S PRIMALITY TEST,W ILSON’S THEOREM References Beauchemin, P.; Brassard, G.; Cre´peau, C.; Goutier, C.; and Pomerance, C. "The Generation of Random Numbers that are Probably Prime." J. Crypt. 1,53/C1/4, 1988. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn 91 ; b /C302, 3; 5; 6; 7; 10; 11; 12 Up to High Powers, rev. ed. Provi- dence, RI: Amer. Math. Soc., pp. lviii-lxv, 1988. Cohen, H. and Lenstra, A. K. "Primality Testing and Jacobi Sums." Math. Comput. 42, 297 /C1/30, 1984. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, 1994. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 15 /C1/7, 1991. Williams, H. C. Edouard Lucas and Primality Testing. New York: Wiley, 1998. Primary Each factor p ai iin an INTEGER ’s PRIME FACTORIZATION is called a primary. Primary Pseudoperfect Number An integer N which is a product of distinct primes and which satisfies 1 N /C27X p½N1 p /C301 (Butske et al. 1999). The first few are 2, 6, 42, 1806, 47058, ... (Sloane’s A054377). The similar equation /C281 N /C27X p ½N1 p /C301 arises in the definition of GIUGA NUMBERS . See also GIUGA NUMBER ,SEMIPERFECT NUMBER References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga’s Conjecture on Primality." Amer. Math. Monthly 103,40/C1/0, 1996. Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation ap ½N 1 =p /C271=N /C301; Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407 /C1/20, 1999. Cao, Z.; Liu, R.; and Zhang, L. "On the Equation as j/C301 (1 xj) /C27 1 xj /C27/C1/C1/C1/C27xn ðÞand Zna´m’s Problem." J. Number Th. 27, 206 /C1/11, 1987. Ke, Z. and Sun, Q. "On the Representation of 1 by Unit Fractions." Sichuan Daxue Xuebao 1,13/C1/9, 1964.Sloane, N. J. A. Sequences A054377 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Primary Representation Let p be a UNITARY REPRESENTATION of a GROUP G on a separable HILBERT SPACE , and let R( p) be the smallest weakly closed algebra of bounded linear operators containing all p(g) for g /C23 G : Then p is primary if the center of R(p) consists of only scalar operations. See also REPRESENTATION References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Prime A symbol used to distinguish one quantity x? ("/x?/") from another related x. Primes are most commonly used to denote 1. Transformed coordinates, 2. Conjugate points, 3. DERIVATIVES , 4. The COMPLEMENT F ? of a set F, 5. As an alternate notation for TRANSPOSE . See also DOUBLE PRIME ,PRIME ALGEBRAIC NUMBER , PRIME NUMBER References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 283, 1997. Prime Algebraic Number An irreducible ALGEBRAIC INTEGER which has the property that, if it divides the product of two algebraic INTEGERS , then it DIVIDES at least one of the factors. 1 and -1 are the only INTEGERS which DIVIDE every INTEGER . They are therefore called the PRIME UNITS . See also ALGEBRAIC INTEGER ,PRIME UNIT Prime Arithmetic Progression An arithmetic progression of primes is a set of primes OF THE FORM mk/C27nfor fixed mand nand con- secutive k, i.e., fn;m/C27n;2m/C27n;...g:For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210. Let Pbe an increasing arithmetic progression of nPRIMES with minimal difference d/C210. If a PRIME p5ndoes not divide d, then the elements of Pmust assume all residues modulo p, specifically, some element of Pmust be divisible by p. Since Pcontains only primes, this element must be equal to p. Let the number of PRIMES OF THE FORM mk /C27n less than x be denoted pm; n(x) : Then lim x0/C12pa ; b(x) Li(x)/C301 f(a) ; where Li(x) is the LOGARITHMIC INTEGRAL and f(x)is the TOTIENT FUNCTION . If d Bn# (where n# is the PRIMORIAL of n), then some prime p 5n does not divide d, and that prime p is in P. Thus, in order to determine if P has d Bn# ; we need only check a finite number of possible P (those with d Bn# and containing prime p 5n) to see if they contain only primes. If not, then d ]n#: If d /C30n#; then the elements of P cannot be made to cover all residues of any prime p. The PRIME PATTERNS CON- JECTURE then asserts that there are infinitely many arithmetic progressions of primes with difference d. A computation shows that the smallest possible common difference for a set of n or more PRIMES in arithmetic progression for n /C301, 2, 3, ... is 0, 1, 2, 6, 6, 30, 150, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 510510, ... (Sloane’s A033188, Ribenboim 1989, Dubner and Nelson 1997, Wilson). The values up to n /C3013 are rigorous, while the remainder are lower bounds which assume the validity of the PRIME PATTERNS CONJECTURE and are simply given by pn/C287#; where piis the ith PRIME . The smallest first terms of arithmetic progressions of n primes with minimal differences are 2, 2, 3, 5, 5, 7, 7, 199, 199, 199, 60858179, 147692845283, 14933623, 856378247603, ... (Sloane’s A033189; Wilson). Smaller first terms are possible for nonminimal n- term progressions. Examples include the 8-term progression 11 /C271210230 k for k /C300, 1, ..., 7, the 12- term progression 23143 /C2730030 k for k /C300, 1, ..., 11 (Golubev 1969, Guy 1994), and the 13-term arith- metic progression 766439 /C27510510 k for k /C300, 1, ..., 12 (Guy 1994). The largest known set of primes in ARITHMETIC SEQUENCE is 22, 11 ; 410; 337; 580; 553 /C274 ; 609; 098; 694; 200k for k /C300, 1, ..., 21 (Pritchard et al. 1995, UTS School of Mathematical Sciences). The largest known sequence of consecutive PRIMES in ARITHMETIC PROGRESSION (i.e., all the numbers be- tween the first and last term in the progression, except for the members themselves, are composite) is ten, given by 100; 996 ; 972; 469 ; 714; 247; 637 ; 786; 655 ; 587 ; 969; 840 ; 329 509 ; 324; 689 ; 190; 041; 803 ; 603; 417 ; 758; 904 ; 341; 703 ; 348; 882; 159 ; 067; 229 ; 719 /C27210k for k /C300, 1, ..., 9 (Sloane’s A033290), discovered by Harvey Dubner, Tony Forbes, Manfred Toplic, et al.on March 2, 1998. This beats the record of nine consecutive primes set on January 15, 1998 by the same investigators, 99; 679; 432; 066 ; 701; 086 ; 484; 490; 653 ; 695; 853 ; 561; 638 ; 982; 364 ; 080; 991; 618 ; 395; 774 ; 048 ; 585; 529 ; 071; 475 ; 461; 114; 799 ; 677; 694 ; 651 /C27210k for k /C300, 1, ..., 8 (two sequences of nine are now known), the progression of eight consecutive primes given by 43; 804; 034; 644 ; 029; 893 ; 325; 717; 710 ; 709; 965 ; 599; 930 ; 101; 479 ; 007; 432; 825 ; 862; 862 ; 446 ; 333; 961 ; 919; 524 ; 977; 985; 103 ; 251; 510 ; 661 /C27210k for k /C300, 1, ..., 7, discovered by Harvey Dubner, Tony Forbes, et al. on November 7, 1997 (several are now known), and the progression of seven given by 1 ; 089; 533; 431 ; 247 ; 059; 310; 875 ; 780 ; 378; 922; 957 ; 732; 908; 036; 492; 993; 138; 195; 385; 213; 105; 561; 742 ; 150; 447; 308; 967; 213; 141; 717; 486 ; 151 /C27210k; for k /C300, 1, ..., 6, discovered by H. Dubner and H. K. Nelson on Aug. 29, 1995 (Peterson 1995, Dub- ner and Nelson 1997). The smallest sequence of six consecutive PRIMES in arithmetic progression is 121;174;811/C2730k fork/C300, 1, ..., 5 (Lander and Parkin 1967, Dubner and Nelson 1997). According to Dubner et al., a trillion-fold increase in computer speed is neededbefore the search for a sequence of 11 consecutiveprimes is practical, so they expect the ten-primes record to stand for a long time to come. It is conjectured that there are arbitrarily long sequences of PRIMES in ARITHMETIC PROGRESSION (Guy 1994). W. Roonguthai found the largest known arithmetic progression of three primes, (3, 1593 /C215 227757/C271;1593 /C215227758/C281);with common difference 1593 /C215227757/C282 (Roonguthai 1999). See also ARITHMETIC PROGRESSION ,C UNNINGHAM CHAIN ,D IRICHLET’S THEOREM ,L INNIK’S THEOREM , PRIME CONSTELLATION ,PRIME- GENERATING POLYNO- MIAL ,PRIME NUMBER THEOREM ,PRIME PATTERNS CONJECTURE ,PRIME QUADRUPLET References Abel, U. and Siebert, H. "Sequences with Large Numbers of Prime Values." Amer. Math. Monthly 100, 167/C1/69, 1993. Caldwell, C. K. "Cunningham Chain." http://www.utm.edu/ research/primes/glossary/CunninghamChain.html. Courant, R. and Robbins, H. "Primes in Arithmetical Progressions." §1.2b in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 26 /C1/7, 1996. Davenport, H. "Primes in Arithmetic Progression" and "Primes in Arithmetic Progression: The General Modu- lus." Chs. 1 and 4 in Multiplicative Number Theory, 2nd ed.New York: Springer-Verlag, pp. 1 /C1/1 and 27 /C1/4, 1980. Dubner, H. J. Recr. Math. 20, 211/C1/13, 1988. Dubner, H. and Nelson, H. "Seven Consecutive Primes in Arithmetic Progression." Math. Comput. 66, 1743 /C1/749, 1997. Forbes, T. "Searching for 9 Consecutive Primes in Arith- metic Progression." http://www.ltkz.demon.co.uk/ar2/9pri- mes.htm. Forman, R. "Sequences with Many Primes." Amer. Math. Monthly 99, 548/C1/57, 1992. Gardner, M. "Primes in Arithmetic Progression." In Press, R.Mathematical Sciences Calendar 1988. Golubev, V. A. "Faktorisation der Zahlen der Form x394x2/C273x91:/"Anz. O ¨sterreich. Akad. Wiss. Math.- Naturwiss. Kl. 184/C1/91, 1969. Guy, R. K. "Arithmetic Progressions of Primes" and "Con- secutive Primes in A.P." §A5 and A6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 15 /C1/7 and 18, 1994. Lander, L. J. and Parkin, T. R. "Consecutive Primes in Arithmetic Progression." Math. Comput. 21, 489, 1967. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 154 /C1/55, 1979. Nelson, H. L. "There Is a Better Sequence." J. Recr. Math. 8, 39/C1/3, 1975. Peterson, I. "Progressing to a Set of Consecutive Primes." Sci. News 148, 167, Sep. 9, 1995. Pritchard, P. A.; Moran, A.; and Thyssen, A. "Twenty-Two Primes in Arithmetic Progression." Math. Comput. 64, 1337/C1/339, 1995. Ramare ´, O. and Rumely, R. "Primes in Arithmetic Progres- sions." Math. Comput. 65, 397/C1/25, 1996. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 224, 1989. Roonguthai, W. "Record Arithmetic Progression of Primes." [email protected] mailing list posting. Feb. 4, 1999. Shanks, D. "Primes in Some Arithmetic Progressions and a General Divisibility Theorem." §104 in Solved and Un- solved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 104 /C1/09, 1993. Sloane, N. J. A. Sequences A033188, A033189, and A033290 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se-quences/eisonline.html. UTS School of Mathematical Sciences. "Primes in Arithmetic Progression." http://www.maths.uts.edu.au/numericon/prime2.html. Weintraub, S. "Consecutive Primes in Arithmetic Progres- sion." J. Recr. Math. 25, 169/C1 /71, 1993. Zimmerman, P. http://www.loria.fr/~zimmerma/records/ 8primes.announce. Prime Array Find the m/C29nARRAY of single digits which contains the maximum possible number of PRIMES , where allowable PRIMES may lie along any horizontal, vertical, or diagonal line. For m/C30n/C302;11 PRIMES are maximal and are contained in the two distinct arrays A(2;2)/C3013 479+$=9+$; ;13799+$=9+$; ; giving the PRIMES (3, 7, 13, 17, 31, 37, 41, 43, 47, 71, 73) and (3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97),respectively. For the 3 /C292 array, 18 PRIMES are maximal and are contained in the arrays A(3;2)/C30113 9749+$=9+$; ;1723599+$=9+$; ;1724399+$=9+$; ; 175 4399+$=9+$; ;1793259+$=9+$; ;1794329+$=9+$; ; 179 4349+$=9+$; ;3164799+$=9+$; ;3764199+$=9+$; : The best 3 /C293 array is A(3;3)/C30113 7549372 435; which contains 30 primes: 3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 97, 113, 157, 179, ... (Sloane’s A032529). This array was found by Riveraand Ayala and shown by Weisstein in May 1999 to be maximal and unique (modulo reflection and rotation). The best 4 /C294 arrays known are 1139 6451739739292 6643 775;1139 7692547917332 6643 775; 1733 9421659177392 6643 775;3167 7514929333732 6643 775; all of which contain 63 PRIMES . The first was found by C. Rivera and J. Ayala in 1998, and the other three by James Bonfield on April 13, 1999. The best 5 /C295 prime arrays known are 11933 995638941733731 329392 666643 77775;33199 839112745719673 979192 666643 77775 each of which contains 116 PRIMES . The first was found by C. Rivera and J. Ayala in 1998, and the second by Wilfred Whiteside on April 17, 1999. The best 6 /C296 prime arrays known are 139199 317234 9947939157139836179173332 66666643 7777775;139199 917234 6947937157139836179173332 66666643 7777775; 317333 995639118142136373 349199 3793792 66666643 7777775;317333 995639118142136373 349199 3793792 66666643 7777775; 317333 995639118142136373 349199 9793792 66666643 7777775;317333 995639118145136373 349199 9992332 66666643 7777775; each of which contain 187 primes. One was found by S. C. Root, and the others by M. Oswald in 1998. The best 7 /C297 prime array known is 3137339 9923333 6977894761591977342119947939 33719992 6666666643 777777775; which contains 281 primes and was found by Wilfred Whiteside on April 29, 1999. The best 8 /C298 prime array known is 33139133 69337397 79968571 979912491321139963919463 63853793 913139332 666666666643 77777777775 which contains 382 primes and was found by Wilfred Whiteside On Oct. 31, 1999. Heuristic arguments by Rivera and Ayala suggest that the maximum possible number of primes in 4 /C29 4; 5 /C295 ; and 6 /C296 arrays are 58 /C1 /3, 112 /C1/21, and 205 /C1/ 18, respectively. See also ARRAY ,PRIME ARITHMETIC PROGRESSION , PRIME CONSTELLATION ,PRIME STRING References Dewdney, A. K. "Computer Recreations: How to Pan for Primes in Numerical Gravel." Sci. Amer. 259, 120 /C1/23, July 1988. Lee, G. "Winners and Losers." Dragon User. May 1984. Lee, G. "Gordon’s Paradoxically Perplexing Primesearch Puzzle." http://www.geocities.com/MotorCity/7983/prime- search.html. Rivera, C. "Problems & Puzzles: Puzzle The Gordon Lee Puzzle.-061." http://www.primepuzzles.net/puzzles/ puzz_061.htm.Sloane, N. J. A. Sequences A032529 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Prime Arrays." MATHEMATICA NOTEBOOK PRIME ARRAY.M . Prime Circle A prime circle of order 2n is a free CIRCULAR PERMUTATION of the numbers from 1 to 2n with adjacent PAIRS summing to a PRIME . The number of prime circles for n /C301, 2, ..., are 1, 1, 1, 2, 48, 512, ... (Sloane’s A051252). The prime circles for the first few even orders are given in the table below. /2n/ prime circles 2 /f1; 2g/ 4 /f1; 2; 3; 4g/ 6 /f1; 4; 3; 2; 5; 6g/ 8 /f1; 2; 3; 8; 5; 6; 7 ; 4 g; f1;2;5;8;3;4;7;6g/ See also CIRCULAR PERMUTATION References Filz, A. "Problem 1046." J. Recr. Math. 14, 64, 1982. Filz, A. "Problem 1046." J. Recr. Math. 15, 71, 1983. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 105 /C1/06, 1994. Sloane, N. J. A. Sequences A051252 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Prime Cluster PRIME CONSTELLATION Prime Constant The characteristic function f(n)/C301nis prime 0notherwise9+$k (1) therefore has first few values 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, ... (Sloane’s A010051). The constant obtained by concatenating these digits in binary is therefore P /C130:011010100...2 /C300:4146825098511116602481... (2) (Sloane’s A051006), which has CONTINUED FRACTION [0, 2, 2, 2, 3, 12, 131, 1, ...] (Sloane’s A051007). See also PRIME NUMBER References Sloane, N. J. A. Sequences A010051, A051006, and A051007 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se-quences/eisonline.html. Prime Constellation A prime constellation, also called a prime k-tuple,- prime k-tuplet, or prime cluster, is a sequence of k consecutive numbers such that the difference be- tween the first and last is, in some sense, the least possible. More precisely, a prime k-tuplet is a sequence of consecutive PRIMES (/p1;p2;...,pk) with pk/C28p1/C30s(k);where s(k) is the smallest number sfor which there exist kintegers b1Bb2B...Bbk;bk/C28 b1/C30sand, for every PRIME q, not all the residues modulo qare represented by b1;b2;...,bk(Forbes). For each k, this definition excludes a finite number of clusters at the beginning of the prime number sequence. For example, (97, 101, 103, 107, 109)satisfies the conditions of the definition of a prime 5-tuplet, but (3, 5, 7, 11, 13) does not because all three residues modulo 3 are represented (Forbes). A prime double with s(2)/C302i s OF THE FORM (p,p/C272) and is called a pair of TWIN PRIMES . Prime doubles OF THE FORM (p,p/C274) are called COUSIN PRIMES , and prime doubles OF THE FORM (p,p/C276) are called SEXY PRIMES . A prime triplet has s(3)/C306:The constellation ( p,p/C27 2;p/C274) cannot exist, except for p/C303, since one of p, p/C272;andp/C274 must be divisible by three. However, there are several types of prime triplets which can exist: ( p,p/C272;p/C276);(p,p/C274;p/C276);(p,p/C276; p/C2712):/ APRIME QUADRUPLET is a constellation of four successive PRIMES with minimal distance s(4)/C308; and is of the form ( p,p/C272;p/C276;p/C278):The sequence s(n) therefore begins 2, 6, 8, and continues 12, 16, 20, 26, 30, ... (Sloane’s A008407). Another quadruplet constellation is ( p,p/C276;p/C2712;p/C2718):/ Hardy and Wright (1979, p. 5) conjecture, and it seems almost certain to be true, that there are infinitely many TWIN PRIMES (p,p/C272) and PRIME TRIPLETS OF THE FORM (p,p/C272;p/C276) and ( p,p/C274; p/C276):/The first FIRST HARDY-LITTLEWOOD CONJECTURE states that the numbers of constellations 5xare asymptotically given by Px(p;p/C272)/C22Y p]3p(p/C282) (p/C281)2gx 2dx? (lnx?)2 /C301:320323632 gx 2dx? (lnx?)2(1) Px(p;p/C274)/C22Y p]3p(p/C282) (p/C281)2gx 2dx? (lnx?)2 /C301:320323632 gx 2dx? (lnx?)2(2) Px(p;p/C276)/C24Y p]3p(p/C282) (p/C281)2gx 2dx? (lnx?)2 /C302:640647264 gx 2dx? (lnx?)2(3) Px(p;p/C272;p/C276)/C29 2Y p]5p2(p/C283) (p/C281)3gx 2dx? (lnx?)3 /C302:858248596 gx 2dx? (lnx?)3(4) Px(p;p/C274;p/C276)/C29 2Y p]5p2(p/C283) (p/C281)3gx 2dx? (lnx?)3 /C302:858248596 gx 2dx? (lnx?)3(5) Px(p;p/C272;p/C276;p/C278)/C227 2Y p]5p3(p/C284) (p/C281)4gx 2dx? (lnx?)4 /C304:151180864 gx 2dx? (lnx?)4(6) Px(p;p/C274;p/C276;p/C2710) /C227Y p]5p3(p/C284) (p/C281)4gx 2dx? (lnx?)4 /C308:302361728 gx 2dx? (lnx?)4(7) These numbers are sometimes called the H ARDY- LITTLEWOOD CONSTANTS . (1) is sometimes called the extended TWIN PRIME CONJECTURE , and Cp;p/C272/C302P2; (8) where P2is the TWIN PRIMES CONSTANT . Riesel (1994) remarks that the H ARDY- LITTLEWOOD CONSTANTS can be computed to arbitrary accuracy without needing the infinite sequence of primes. The integrals above have the analytic forms gx 2dx? (lnx?)2/C30Li(x)/C272 ln 2/C28n lnn(9) gx 2dx? (lnx?)4/C301 2Li(x)/C28x(1/C27lnx) (lnx)2/C271 ln 2/C271 (lnn)2(10) gx 2dx? (lnx?)3/C301 6Li(x)/C2722/C27ln 2/C27(ln 2)2hi (ln 2)38 < : /C28n½2/C27lnn/C27(lnn)2/C138 (lnn)39+$7 ; (11) where Li( x) is the LOGARITHMIC INTEGRAL . The following table gives the number of prime constellations 5108;and the second table gives the values predicted by the Hardy-Littlewood formulas. Count 105106107108 /(p;p/C272)/ 1224 8169 58980 440312 /(p;p/C274)/ 1216 8144 58622 440258 /(p;p/C276)/ 2447 16386 117207 879908 /(p;p/C272;p/C276)/ 259 1393 8543 55600 /(p;p/C274;p/C276)/ 248 1444 8677 55556 /(p;p/C272;p/C276;p/C278)/ 38 166 899 4768 /(p;p/C276;p/C2712;p/C2718) /75 325 1695 9330 Hardy-Littlewood 105106107108 /(p;p/C272)/ 1249 8248 58754 440368 /(p;p/C274)/ 1249 8248 58754 440368 /(p;p/C276)/ 2497 16496 117508 880736 /(p;p/C272;p/C276)/ 279 1446 8591 55491 /(p;p/C274;p/C276)/ 279 1446 8591 55491 /(p;p/C272;p/C276;p/C278)/ 53 184 863 4735 /(p;p/C276;p/C2712;p/C2718) / Consider prime constellations in which each term is OF THE FORM n2/C271:Hardy and Littlewood showed that the number of prime constellations of this form Bxis given by P(x)/C2Cffiffiffixp(lnx)/C281; (12)where C/C30Y p>2 pprime1/C28(/C281)(p/C281)=2 p/C281"# /C301:3727 . . . (13) (Le Lionnais 1983). Forbes gives a list of the "top ten" prime k-tuples for 25k517:The largest known 14-constellations are (11319107721272355839 /C270, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), ( 10756418345074847279 /C270, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (6808488664768715759 /C270, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), ( 6120794469172998449 /C270, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50),(5009128141636113611 /C270, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50). The largest known prime 15-constellations are (84244343639633356306067 /C270, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56),(8985208997951457604337 /C270, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56),(3594585413466972694697 /C270, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (3514383375461541232577 /C270, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (3493864509985912609487 /C270, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56). The largest known prime 16-constellations are (3259125690557440336637 /C270, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (1522014304823128379267 /C270, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (47710850533373130107 /C270, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (13, 17, 19, 23, 29, 31,37, 41, 43, 47, 53, 59, 61, 67, 71, 73). The largest known prime 17-constellations are (3259125690557440336631 /C270, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66), (17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83) (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79). Smith (1957) found 8 consecutive primes spaced like the cluster p nfg12 n/C305(Gardner 1980). K. Conrow and J. J. Devore have found 15 consecutive primes spaced like the cluster pnfg19 n/C305given by 1632373745527558118190 /C27pn fg19n/C305;the first mem- ber of which is 1632373745527558118201. Rivera tabulates the smallest examples of kconsecu- tive primes ending in a given digit d/C301, 3, 7, or 9 for k/C305 to 11. For example, 216401, 216421, 216431, 216451, 216481 is the smallest set of five consecutive primes ending in the digit 1. See also CLUSTER PRIME ,COMPOSITE RUNS,COUSIN PRIMES ,PRIME ARITHMETIC PROGRESSION , K-TUPLE CONJECTURE ,PRIME K-TUPLES CONJECTURE ,PRIME QUADRUPLET ,PRIME TRIPLET ,SEXY PRIMES ,TWIN PRIMES References Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. http://www.math.u-bordeaux.fr/~co- hen/hardylw.dvi. Forbes, T. "Prime k-tuplets." http://www.ltkz.demon.co.uk/ ktuplets.htm. Forbes, T. "Prime Clusters and Cunningham Chains." Math. Comput. 68, 1739/C1/748, 1999. Gardner, M. "Mathematical Games." Sci. Amer. 243, Dec. 1980. Guy, R. K. "Patterns of Primes." §A9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 23 /C1/5, 1994. Rivera, C. "Problems & Puzzles: Puzzle Consecutive Primes and Ending Digits.-016." http://www.primepuzzles.net/ puzzles/puzz_016.htm. Smith, H. F. "On a Generalization of the Prime Pair Problem." Math. Tables Aids Comput. 11, 249/C1/54, 1957. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 38, 1983. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 60 /C1/ 4, 1994. Sloane, N. J. A. Sequences A008407 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Prime Counting Function The function p(n) giving the number of PRIMES5n (Shanks 1993, p. 15). For example, there are no primes51;sop(1)/C300; there is a single prime (2) 52;sop(2)/C301; there are two primes (2 and 3) 53;so p(3)/C302; and so on. The first few values for n/C301, 2, ... are 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, ... (Sloane’s A000720). The following table gives the values of p(n) for powers of 10 (Sloane’s A006880; Hardy and Wright 1979, p. 4; Shanks 1993, pp. 242 /C1/43; Ribenboim 1996, p. 237). The value for p1020ðÞ comes from Deleglise and Rivat (1996). Note that p109ðÞ is incorrectly given as 50,847,478 in Hardy and Wright (1979) and Hardy (1999).n /p10nðÞ / 3 168 4 1,229 5 9,592 6 78,4987 664,579 8 5,761,455 9 50,847,534 10 455,052,51111 4,118,054,813 12 37,607,912,018 13 346,065,536,83914 3,204,941,750,80215 29,844,570,422,669 16 279,238,341,033,925 17 2,623,557,157,654,23318 24,739,954,287,740,86019 234,057,667,276,344,607 20 2,220,819,602,560,918,840 One of the most fundamental and important results in NUMBER THEORY is the asymptotic value of p(n)a s nbecomes large. The correct formula is p(n)/C2li(n); (1) where li( x) is the LOGARITHMIC INTEGRAL , which is known as the PRIME NUMBER THEOREM . The following table compares the prime countingfunction p(x); LOGARITHMIC INTEGRAL lix;and R IE- MANN PRIME NUMBER FORMULA R(x) for small x. Note that the values given by Hardy (1999, p. 26) for x/C30 109are incorrect. x /p(x)//lix/C28p(x)//R(x)/C28p(x)/ 100000 9592 38 //C285/ 1000000 78498 130 29 2000000 148933 122 //C289/ 3000000 216816 155 0 4000000 283146 206 335000000 348513 125 /C2864 6000000 412849 228 247000000 476648 179 /C2838 8000000 539777 223 //C286/ 9000000 602489 187 /C2853 10000000 664579 339 88 100000000 5761455 754 97 1000000000 50847534 1701 /C2879 The prime counting function can be expressed by LEGENDRE’S FORMULA ,LEHMER’S FORMULA ,M APES’ METHOD ,o rM EISSEL’S FORMULA . A brief history of attempts to calculate p(n) is given by Berndt (1994). The following table is taken from Riesel (1994), where O(x)i s ASYMPTOTIC NOTATION . Method Time Storage Legendre /O(x)// Ox1=29+=9+; / Meissel /Ox=(lnx)39+;k9+;7 //Ox1=2=lnx9+=9+; / Lehmer /Ox=(lnx)49+;k9+;7 //Ox1=3=lnx9+=9+; / Mapes’ /Ox0:7ðÞ // Ox0:7ðÞ / Lagarias-Miller- Odlyzko/Ox2=3/C27e9+=9+; //Ox1=3/C27e9+=9+; / Lagarias-Odlyzko 1 /Ox3=5/C27e9+=9+; //OxeðÞ / Lagarias-Odlyzko 2 /Ox1=2/C27e9+=9+; //Ox1=4/C27e9+=9+; / An approximate formula due to Locker-Ernst(Locker-Ernst 1959, Panaitopol 1999), illustrated above, is given by p(n):n hn; (2) where hnis related to the HARMONIC NUMBER Hnby hn/C30Hn/C283=2:This formula is within :2 of the actual value for 50 5n51000 :The values for which p/C28 n=hn>0 are 1, 109, 113, 114, 199, 200, 201, ...(Sloane’s A051046). Panaitopol (1999) shows that this quantity is positive for all n]1429 :/ An upper limit for p(n) is given by p(n)B2n/C286 lnn(3) (Rosser and Schoenfeld 1962). Hardy and Wright(1979, p. 414) give the formula p(n)/C30/C281/C27X n j/C303(j/C282)!/C28j(j/C282)! j$%"# ; (4) where xbcis the FLOOR FUNCTION . A modified version of the prime counting function isgiven by p 0(p)/C13p(p) for pcomposite p(p)/C281 2forpprime9+$k p0(p)/C30X/C12 n/C301mxðÞfx1=n9+=9+; n; where m(n) is the M O¨BIUS FUNCTION and f(x) is the RIEMANN FUNCTION . The notation pa;bis also used to denote the number of PRIMES OF THE FORM ak/C27b(Shanks 1993, pp. 21 /C1/2). Groups of EQUINUMEROUS values of pa;binclude ( /p3;1; p3;2);(/p4;1;p4;3);(/p5;1;p5;2;p5;3;p5;4);(/p6;1;p6;5);/ (/p7;1;p7;2;p7;3;p7;4;p7;5;p7;6);(/p8;1;p8;3;p8;5; p8;7);(/p9;1;p9;2;p9;4;p9;5;p9;7;p9;8);and so on. The values of /pn;k/for small nare given in the following table for the first few powers of ten (Shanks 1993). n /p3;1(n)//p3;2(n)//p4;1(n)//p4;3(n)/ 1011212 10211 13 11 13 10380 87 80 87 104611 617 609 619 1054784 4807 4783 4808 10639231 39266 39175 39322 107332194 332384 332180 332398 n /p5;1(n)//p5;2(n)//p5;3(n)//p5;4(n)/ 1010210 1025775 10340 47 42 38 104306 309 310 303 1052387 2412 2402 2390 10619617 19622 19665 19593 107166104 166212 166230 166032 n / p6; 1(n)//p6 ; 5(n)/ 101 11 102 11 12 103 80 86 104611 616 1054784 4806 10639231 39265 n /p7 ; 1//p7 ; 2//p7 ; 3//p7 ; 4//p7 ; 5//p7 ; 6/ 101 011010 102 345354 10328 27 30 26 29 27 104203 203 209 202 211 200 1051593 1584 1613 1601 1604 1596 10613063 13065 13105 13069 13105 13090 n /p8 ; 1(n)//p8; 3(n)//p8 ; 5(n)//p8 ; 7(n)/ 101 0111 102 5766 103 37 44 43 43 104295 311 314 308 1052384 2409 2399 2399 10619552 19653 19623 19669 107165976 166161 166204 166237 Note that since p8 ; 1(n) ; p8 ; 3(n) ; p8 ; 5(n) ; and p8 ; 7(n) are EQUINUMEROUS , p4; 1(n) /C30 p8; 1(n) /C27 p8 ; 5 p4; 3(n) /C30 p8; 3(n) /C27 p8 ; 7 are also equinumerous.Erdos proved that there exist at least one PRIME OF THE FORM 4k /C271 and at least one PRIME of the form 4k /C273 between n and 2n for all n /C216. The smallest x such that x ]np(x) for n /C302, 3, ... are 2, 27, 96, 330, 1008, ... (Sloane’s A038625), and the corresponding p(x) are 1, 9, 24, 66, 168, 437, ... (Sloane’s A038626). The number of solutions of x] np(x) for n/C302, 3, ... are 4, 3, 3, 6, 7, 6, ... (Sloane’s A038627). See also BERTELSEN’S NUMBER ,CHEBYSHEV’S THEO- REM,E QUINUMEROUS ,L EGENDRE’S CONSTANT ,L E- GENDRE’S FORMULA ,L EHMER- SCHUR METHOD , LOGARITHMIC INTEGRAL ,M APES’ METHOD ,P RIME ARITHMETIC PROGRESSION ,P RIME NUMBER ,P RIME NUMBER THEOREM ,RIEMANN PRIME NUMBER FOR- MULA References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 134 /C1/35, 1994. Brent, R. P. "Irregularities in the Distribution of Primes and Twin Primes." Math. Comput. 29,4 3/C1/6, 1975. Deleglise, M. and Rivat, J. "Computing p(x) : The Meissel, Lehmer, Lagarias, Miller, Odlyzko Method." Math. Com- put. 65, 235/C1/45, 1996. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html. Forbes, T. "Prime k-tuplets." http://www.ltkz.demon.co.uk/ ktuplets.htm. Guiasu, S. "Is There Any Regularity in the Distribution of Prime Numbers at the Beginning of the Sequence of Positive Integers?" Math. Mag. 68, 110/C1/21, 1995. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Lagarias, J.; Miller, V. S.; and Odlyzko, A. "Computing p(x): The Meissel-Lehmer Method." Math. Comput. 44, 537/C1/60, 1985. Lagarias, J. and Odlyzko, A. "Computing p(x) : An Analytic Method." J. Algorithms 8, 173/C1/91, 1987. Locker-Ernst, L. "Bemerkung u ¨ber die Verteilung der Primzahlen." Elemente Math. (Basel) 14,1/C1/, 1959. Mapes, D. C. "Fast Method for Computing the Number of Primes Less than a Given Limit." Math. Comput. 17, 179/C1/ 85, 1963. Meissel, E. D. F. "U ¨ber die Bestimmung der Primzahlmenge innerhalb gegebener Grenzen." Math. Ann. 2, 636/C1/42, 1870. Nagell, T. "The Function p(x):/"§16 in Introduction to Number Theory. New York: Wiley, pp. 54 /C1/7, 1951. Panaitopol, L. "Several Approximations of p(x):/"Math. Ineq. Appl. 2, 317/C1/24, 1999. Ribenboim, P. The New Book of Prime Number Records, 3rd ed.New York: Springer-Verlag, 1996. Riesel, H. "The Number of Primes Below x."Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 10 /C1/2, 1994. Rosser, J. B. and Schoenfeld, L. "Approximate Formulas for Some Functions of Prime Numbers." Illinois J. Math. 6, 64/C1/7, 1962. Se´roul, R. "The Function pi( x)." §8.7 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 175 /C1/81, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993. Sloane, N. J. A. Sequences A000720/M0256, A006880/ M3608, A038625, A038626, A038627, A052434, and A052435 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 74 /C1/6, 1991. Prime Cut Find two numbers such that x2 /C13y2 (mod n) : If you know the GREATEST COMMON DIVISOR of n and x /C28y; there exists a high probability of determining a PRIME factor. Taking small numbers x which additionally give small PRIMES x2 /C13p (mod n) further increases the chances of finding a PRIME FACTOR . See also GREATEST COMMON DIVISOR Prime Decomposition PRIME FACTORIZATION Prime Difference Function dn /C13pn/C271 /C28pn : The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (Sloane’s A001223). Rankin has shown that dn >c ln n ln ln n ln ln ln ln n (ln ln ln n)2 for infinitely many n and for some constant c (Guy 1994). An integer n is called a JUMPING CHAMPION if n is the most frequently occurring difference between conse- cutive primes n 5N for some N (Odlyzko et al. ). See also ANDRICA’S CONJECTURE ,GILBREATH’S CON- JECTURE ,GOOD PRIME ,JUMPING CHAMPION ,PO´ LYA CONJECTURE ,P RIME GAPS,S HANKS’ CONJECTURE , TWIN PEAKS References Bombieri, E. and Davenport, H. "Small Differences Between Prime Numbers." Proc. Roy. Soc. A 293,1/C1/8, 1966.Erdos, P.; and Straus, E. G. "Remarks on the Differences Between Consecutive Primes." Elem. Math. 35, 115 /C1/18, 1980. Guy, R. K. "Gaps between Primes. Twin Primes" and "Increasing and Decreasing Gaps." §A8 and A11 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19 /C1/3 and 26 /C1/7, 1994. Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping Cham- pions." http://www.research.att.com/~amo/doc/re- cent.html. Riesel, H. "Difference Between Consecutive Primes." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, p. 9, 1994. Sloane, N. J. A. Sequences A001223/M0296 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Prime Diophantine Equations /k /C272is PRIME IFF the 14 DIOPHANTINE EQUATIONS in 26 variables wz /C27h /C27j /C28q /C300 (1) (gk /C272g /C27k /C271)(h /C27j) /C27h /C28z /C300 (2) 16(k /C271)3(k /C272)(n /C271)2 /C271 /C28f2 /C300 (3) 2n /C27p /C27q /C27z /C28e /C300 (4) e3(e /C272)(a /C271)2 /C271 /C28o2 /C300 (5) a2 /C2819+=9+; y2 /C271 /C28x2 /C300 (6) 16r2y4 a2 /C2819+=9+; /C271 /C28u2 /C300 (7) n /C27l /C27v /C28y /C300 (8) a2 /C2819+=9+; l2 /C271 /C28m2 /C300 (9) ai /C27k /C271 /C28l /C28i /C300 (10) a /C27u2 u2 /C28a9+=9+;9+$9+%2/C281no (n /C274 dy)2 /C271 /C28(x /C27cu)2 /C300 (11) p /C27l(a /C28n /C281) /C27b(2an /C272a /C28n2 /C282n /C282) /C28m /C300 (12) q /C27y(a /C28p /C281) /C27s 2ap /C272a /C28p2 /C282p /C2829+=9+; /C28x /C300 (13) z /C27pl(a /C28p) /C27t(2ap /C28p2 /C281) /C28pm /C300 (14) have a solution in POSITIVE INTEGERS (Riesel 1994, p. 40). See also PRIME- GENERATING POLYNOMIAL References Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, 1994. Prime Divisor Iff(x) is a nonconstant INTEGER POLYNOMIAL andcis an integer such that f(c) is divisible by the prime p, that pis called a prime divisor of the polynomial f(x) (Nagell 1951, p. 81). Every INTEGER POLYNOMIAL f(x) which is not a constant has an infinite number of prime divisors (Nagell 1951, p. 82). See also BAUER’S THEOREM ,INTEGER POLYNOMIAL References Nagell, T. "Prime Divisors of Integral Polynomials." §25 in Introduction to Number Theory. New York: Wiley, pp. 81 /C1/ 3, 1951. Prime Factorization The FACTORIZATION of a numbers into its constituent PRIMES , also called prime decomposition. Given a POSITIVE INTEGER n ]2; the prime factorization is written n /C30p a1 1 p a2 2/C1/C1/C1p ak k ; where the pi/s are the k PRIME FACTORS , each of order ai : Each factor p ai iis called a PRIMARY . The first few prime factorizations (the number 1, by definition, has a prime factorization of "1") are given in the following table. 1 1 11 11 22 12 /22 /C215 3/ 3 3 13 13 42214 /2 /C215 7/ 55 15 /3 /C215 5/ 6 /2 /C215 3/ 16 24 7 7 17 17 82318 /2 /C215 9/ 93219 19 10 /2 /C215 5/ 20 /22 /C215 5/ The number of digits in the prime factorization of n /C301, 2, ..., are 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, (Sloane’s A050252). In general, prime factorization is a difficult problem, and many sophisticated PRIME FACTORIZATION ALGO- RITHMS have been devised for special types of num- bers. See also DISTINCT PRIME FACTORS ,E CONOMICAL NUMBER ,EQUIDIGITAL NUMBER ,FACTORIZATION ,PRI- MARY ,PRIME FACTORIZATION ,PRIME FACTORIZATION ALGORITHMS ,P RIME FACTORS ,P RIME NUMBER , ROUND NUMBER ,ROUNDNESS ,W ASTEFUL NUMBERPrime Factorization Algorithms Many ALGORITHMS have been devised for determining the PRIME FACTORS of a given number (a process called PRIME FACTORIZATION ). They vary quite a bit in sophistication and complexity. It is very difficult to build a general-purpose algorithm for this computa- tionally "hard" problem, so any additional informa- tion which is known about the number in question or its factors can often be used to save a large amount of time. The simplest method of finding factors is so-called "DIRECT SEARCH FACTORIZATION " (a.k.a. TRIAL DIVI- SION). In this method, all possible factors are system- atically tested using trial division to see if they actually DIVIDE the given number. It is practical only for very small numbers. The fastest-known fully proven deterministic algo- rithm is the Pollard-Strassen method (Pomerance 1987; Hardy et al. 1990). See also BRENT’S FACTORIZATION METHOD ,C LASS GROUP FACTORIZATION METHOD ,CONTINUED FRAC- TION FACTORIZATION ALGORITHM ,D IRECT SEARCH FACTORIZATIO N,D IXON’S FACTORIZATION METHOD , ELLIPTIC CURVE FACTORIZATION METHOD ,E ULER’S FACTORIZATION METHOD ,EXCLUDENT FACTORIZATION METHOD ,FERMAT’S FACTORIZATION METHOD ,LEGEN- DRE’S FACTORIZATION METHOD ,L ENSTRA ELLIPTIC CURVE METHOD ,NUMBER FIELD SIEVE,POLLARD P-1 FACTORIZATION METHOD ,POLLARD RHO FACTORIZA- TION METHOD ,PRIME FACTORIZATION ,PRIME NUM- BER,QUADRATIC SIEVE,QUITEPRIME ,TRIAL DIVISION , VERYPRIME ,W ILLIAMS P/C271 FACTORIZATION METHOD References Anderson, D. D. (Ed.). Factorization in Integral Domains. New York: Dekker, 1997. Bressoud, D. M. Factorization and Prime Testing. New York: Springer-Verlag, 1989. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn91;b/C302, 3;5;6;7;10;11;12 Up to High Powers, rev. ed. Provi- dence, RI: Amer. Math. Soc., liv-lviii, 1988. Dickson, L. E. "Methods of Factoring." Ch. 14 in History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 357 /C1/74, 1952. Hardy, K.; Muskat, J. B.; and Williams, K. S. "A Determi- nistic Algorithm for Solving n/C30fu2/C27gv2in Coprime Integers uandv."Math. Comput. 55, 327/C1/43, 1990. Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673 /C1/15, 1990. Odlyzko, A. M. "The Complexity of Computing Discrete Logarithms and Factoring Integers." §4.5 in Open Pro- blems in Communication and Computation (Ed. T. M. Co- ver and B. Gopinath). New York: Springer-Verlag,pp. 113 /C1 /16, 1987. Odlyzko, A. M. "The Future of Integer Factorization." CryptoBytes: The Technical Newsletter of RSA Labora-tories 1, No. 2, 5 /C1 /2, 1995. Pomerance, C. "Fast, Rigorous Factorization and Discrete Logarithm Algorithms." In Discrete Algorithms and Com- plexity (Ed. D. S. Johnson, T. Nishizeki, A. Nozaki, and H. S. Wilf). New York: Academic Press, pp. 119 /C1/43, 1987. Pomerance, C. "Analysis and Comparison of Some Integer Factorization Algorithms." In Computational Methods in Number Theory, Part 1 (Ed. H. W. Lenstra and R. Tijde- man). Amsterdam, Netherlands: Mathematisch Centrum, pp. 89 /C1/39, 1982. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473/C1/485, 1996. Riesel, H. "Algebraic Factors." Appendix 6 in Prime Num- bers and Computer Methods for Factorization, 2nd ed.Boston, MA: Birkha ¨user, pp. 304 /C1 /16, 1994. Weisstein, E. W. "Books about Prime Numbers." http:// www.treasure-troves.com/books/PrimeNumbers.html. Williams, H. C. and Shallit, J. O. "Factoring Integers Before Computers." In Mathematics of Computation 1943 /C1/993, Fifty Years of Computational Mathematics (Ed. W. Gautschi). Providence, RI: Amer. Math. Soc.,pp. 481 /C1 /31, 1994. Prime Factors The number of DISTINCT PRIME FACTORS of a number nis denoted v(n):v(n) therefore corresponds to a prime factorization OF THE FORM n/C30pa1 1pa2 2/C1/C1/C1pav(n) v(n): (1) The first few values for n/C301, 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, ... (Sloane’s A001221).The first few numbers u nwhich are products of an odd number of distinct prime factors (Hardy 1999, p. 64; Ramanujan 2000, pp. xxiv and 21) are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42 43, 47, ...(Sloane’s A030059). u nsatisfies X/C12 n/C3011 us n/C301 2z(s) z(2s)/C28z(s)"# (2) (Hardy 1999, pp. 64 /C1/5). In addition, if U(n) is the number of ukwith k5n;then U(x)/C23x p2(3) (Hardy 1999, pp. 64 /C1/5). The number of not necessarily distinct prime factors of a number nis denoted r(n):The first few values forn/C301, 2, ... are 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, ... (Sloane’s A001222). If nis chosen at random between 1 and x, then the probability that r(n)5 lnnlnn/C27cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ln xp approaches 1ffiffiffiffiffiffi 2ppgc /C28/C12e/C28u2=2du (4) (Knuth 1998, p. 384). In addition, the average value ¯t ofr(n)/C28ln ln xfor 15n5xapproaches ¯t/C30g/C27X pprimeln 1/C281 p ! /C271 p/C281"# (5) /C30g/C27X/C12 n/C302f(n) ln[z(n)] n(6) :1:0345638819 ; (7) where gis the E ULER- MASCHERONI CONSTANT ,f(n)i s the TOTIENT FUNCTION , and z(n) is the R IEMANN ZETA FUNCTION . The average orders of both v(n) and r(n) are v(n)/C2ln ln n (8) (Hardy 1999, p. 51). More precisely, X n5xv(n)/C30xln ln x/C27Ax/C27Ox lnx ! (9) X n5xr(n)/C30xln ln x/C27Bx/C27Ox lnx ! (10) for appropriate constants Aand B(Hardy and Ramanujan 1917; Hardy and Wright 1979, p. 355; Hardy 1999, p. 57), where O(x)i s ASYMPTOTIC NOTA- TION . The following table gives the prime factors for the positive integers 550:/ 1 1 11 11 21 /3/C2157/ 31 31 41 41 22 1 2 /22/C2153/22 /2/C21511/32 2542 2 /C2153 /C2157 3 3 13 13 23 23 33 /3/C21511/43 43 42214 /2/C2157/24 /23/C2153/34 /2/C21517/44 22/C21511 55 15/3 /C215 5/ 25 52 35 /5 /C215 7/ 45 33 /C215 5 6 /2 /C215 3/ 16 2426 /2 /C215 13/ 36 /22 /C215 32/ 46 2 /C215 23 7 7 17 17 27 33 37 37 47 47 82318 /2 /C215 32/ 28 /22 /C215 7/ 38 /2 /C215 19/ 48 24 /C215 3 93219 19 29 29 39 /3 /C215 13/ 49 72 10 /2/C2155/20 /22/C2155/30 /2/C2153/C2155/40 /23/C2155/50 2 /C21552 See also DICKMAN FUNCTION ,DISTINCT PRIME FAC- TORS ,DIVISOR FUNCTION ,GREATEST PRIME FACTOR , LEAST PRIME FACTOR ,LIOUVILLE FUNCTION ,M ER- TENS CONSTANT ,PO´ LYA CONJECTURE ,PRIME FACTOR- IZATION ALGORITHMS ,P RIMITIVE PRIME FACTOR , ROUND NUMBER References Erdos, P. and Kac, M. "The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions." Amer. J. Math. 26, 738/C1/42, 1940. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Ramanujan, S. Quart. J. Math. 48,7 6/C1/2, 1917. Hardy, G. H. and Wright, E. M. §22.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clar- endon Press, 1979. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, p. 384, 1998. Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara, and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000. Sloane, N. J. A. Sequences A001222/M0094, A001221/ M0056, and A030059 in "An On-Line Version of theEncyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Tura´n, P. "On a Theorem of Hardy and Ramanujan." J. London Math. Soc. 9, 274/C1 /76, 1934. Tura´n, P. "U ¨ber einige Verallgemeinerungen eines Satzes von Hardy und Ramanujan." J. London Math. Soc. 11, 125/C1/33, 1936. Prime Field AFINITE FIELD GF(p) where pisPRIME . Prime Formulas There exist a variety of formulas for producing either thenth prime as a function of n, or else taking on only prime values. However, all such formula require either extremely accurate knowledge of some un- known constant, or else effectively require knowledgeof the primes ahead of time in order to use the formula (Dudley 1969, Ribenboim 1996, p. 186). For example, there exists a CONSTANT /u¼1:3063 . . . / (Sloane’s A051021) known as M ILLS’ CONSTANT such that f(n)/C30u3n9+Q9+j ; (1)where xbcis the FLOOR FUNCTION , is prime for all n] 1 (Ribenboim 1996, p. 186). The first few values of f(n) are 2, 11, 1361, 2521008887, ... (Sloane’s A051254). It is not known if uisIRRATIONAL . There also exists a CONSTANT v:1:9287800 such that g(n)/C3022U2v |fflfflffl{zfflfflffl} n$% (2) (Wright 1951; Ribenboim 1996, p. 186) is prime for every n]1:The first few values of g(n) are 3, 13, 16381, .... In the case of both f(n) and g(n);the numbers at n/C304 grow so rapidly that an extremely precise value of uorvis needed in order to obtain the correct value. Values for n]5 are hopeless. Explicit FORMULAS exist for the nth prime both as a function of nand in terms of the primes 2, ..., pn/C281 (Hardy and Wright 1979, pp. 5 /C1/, 344/C1/45, and 414; Guy 1994, pp. 36 /C1/1). Let F(j)/C30cos2p(j/C281)!/C271 j"#$% (3) for integral j/C211, where xbcis again the FLOOR FUNCTION . Then pn/C301/C27X2n m/C301nPm j/C301F(j)$%1=n66647775 (4) /C301/C27X 2n m/C301n 1/C27p(m)$%1=n66647775; (5) where p(m) is the PRIME COUNTING FUNCTION . This formula conceals the prime numbers jas those for which F(j)/C301;i.e., the values of F(j) are 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, .... Gandhi gave the formula in which pn/C271is the unique integer such that 1B2pn/C271X djpn#m(d) 2d/C281/C281 2 ! B2; (6) where pn# is the PRIMORIAL function (Gandhi 1971, Eynden 1972, Golomb 1974) and m(n) is the M O¨BIUS FUNCTION . It is also true that pn/C271/C301/C27pn/C27Fpn/C271 ðÞ /C27Fpn/C271 ðÞ /C27Fpn/C272 ðÞ /C27Yp j/C301Fpn/C27j ðÞ (7) (Ribenboim 1996, pp. 180 /C1/82). Note that the number of terms in the summation to obtain the nth prime is 2n;so these formulas turn out not to be practical in the study of primes. An interesting INFINITE PRODUCT formula due to Euler which relates pand the nth PRIME pnis p /C302 Q/C12 i/C30n1 /C27sin1 2 ppn9+;k9+;7 pn2 435(8) /C30 2 Q/C12 i/C30n1 /C27( /C281) pn /C281 ðÞ =2 pn"# (9) (Blatner 1997). Hardy and Wright (1979, p. 414) give the formula pn /C301 /C27X2n j/C301f(n; p(j)); (10) for n /C213, where f(x; y) /C300 for x /C30y 1 21 /C27x /C28 y x /C28 y jj"# for x "y8 >< >:(11) and p(n) /C30/C281 /C27Xn j /C303(j /C282)! /C28j(j /C28 2)! j$%"# (12) (correcting a sign error), where xbcis the FLOOR FUNCTION . A double sum for the nth prime pn is pn /C301 /C27X2 n ln n bc /C271 ðÞ k /C3011 /C28Pk j/C3021 /C27 s(j)bc n$%"# ; (13) where s(j) /C13/C28Pj s/C301j s$% /C28j /C28 1 s$% ! /C28 2 j (14) (Ruiz 2000). B. M. Bredihin proved that f(x; y) /C30x2 /C27y2 /C271 (15) takes prime values for infinitely many integral pairs (x, y) (Honsberger 1976, p. 30). In addition, the function f(x; y) /C301 2(y /C281) B2(x; y) /C2819+;$9+;$9+;$9+;$/C28 B2(x; y) /C2819+=9+; 9+Q9+j /C272; (16) where B(x; y) /C30x(y /C271) /C28(y! /C271); (17) /y! is the FACTORIAL , and xbcis the FLOOR FUNCTION , generates only prime numbers for POSITIVE INTEGER arguments. It not only generates every prime num- ber, but generates ODD PRIMES exactly once each, with all other values being 2 (Honsberger 1976,p. 33). For example, f(1; 2) /C303 (18) f(5; 4) /C305 (19) f(103 ; 6) /C307; (20) with no new primes generated for x; y 51000 :/ Conway (Guy 1983, Conway and Guy 1996, p. 147) gives an algorithm for generating primes based on 14fractions, but it is actually just a concealed version of a SIEVE . See also MILLS’ CONSTANT ,PRIME NUMBER ,SIEVE References Blatner, D. The Joy of Pi. New York: Walker, p. 110, 1997. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 130, 1996. Dudley, U. "History of Formula for Primes." Amer. Math. Monthly 76,2 3/C1/8, 1969. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/mills/mills.html. Gandhi, J. M. "Formulae for the Nth Prime." Proc. Wa- shington State University Conferences on Number Theory. pp. 96 /C1/07, 1971. Gardner, M. "Patterns and Primes." Ch. 9 in The Sixth Book of Mathematical Games from Scientific American. Chi- cago, IL: University of Chicago Press, pp. 79 /C1/0, 1984. Guy, R. K. "Conway’s Prime Producing Machine." Math. Mag. 56,2 6/C1/3, 1983. Guy, R. K. "Prime Numbers," "Formulas for Primes," and "Products Taken Over Primes." Ch. A, §A17, and §B48 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 3 /C1/3, 36/C1/1 and 102 /C1/03, 1994. Hardy, G. H. and Wright, E. M. "Prime Numbers" and "The Sequence of Primes." §1.2 and 1.4 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 1 /C1/, 1979. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., 1976. Mills, W. H. "A Prime-Representing Function." Bull. Amer. Math. Soc. 53, 604, 1947. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Ruiz, S. M. "The General Term of the Prime Number Sequence and the Smarandache Prime Function." Smar- andache Notions J. 11,5 9/C1/1, 2000. Sloane, N. J. A. Sequences A051021 and A051254 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Wright, E. M. "A Prime-Representing Function." Amer. Math. Monthly 58, 616/C1/18, 1951. Prime Gaps Letting dn/C13pn/C271/C28pn (1) be the PRIME DIFFERENCE FUNCTION , Rankin has showed that dn >c ln n ln ln n ln ln ln ln n (ln ln ln n)2 (2) for infinitely many n and for some constant c (Guy 1994). Let p(d) be the smallest PRIME following d or more consecutive COMPOSITE NUMBERS . The largest known is p(804) /C3090 ;874;329;412;297: (3) The largest known prime gap is of length 4247, occurring following 10314 /C281929 (Baugh and O’Hara 1992), although this gap is almost certainly not maximal (i.e., there probably exists a smaller number having a gap of the same length following it). Crame ´r (1937) and Shanks (1964) conjectured that a maximal gap p(n) of length n first appears at approximately p(n) /C2expffiffiffinp9+=9+; : (4) Wolf conjectures a slightly different form p(n) /C2ffiffiffinpexpffiffiffinp9+=9+; ; (5) which agrees better with numerical evidence. Wolf conjectures that the maximal gap G(n) between two consecutive primes less than n appears approxi- mately at G(n) /C2n p(n)2ln p(n) /C28ln n /C27ln 2C2ðÞ ½/C138 /C13g(n); (6) where p(n) is the PRIME COUNTING FUNCTION and C2 is the TWIN PRIMES CONSTANT . Setting p(n) /C2n=ln n reduces to Cramer’s conjecture for large n, G(n) /C2(ln n)2 : (7) Let c(n) be the smallest starting INTEGER c(n) for a run of n consecutive COMPOSITE NUMBERS , also called a COMPOSITE RUN. No general method other than exhaustive searching is known for determining the first occurrence for a maximal gap, although arbitra- rily large gaps exist (Nicely 1998). The first few c(n) for n /C301, 2, ... are 4, 8, 8, 24, 24, 90, 90, 114, ... (Sloane’s A030296). The following table gives the sequence of maximal prime gaps, omitting degenerate runs which are part of a run with greater n. It is a complete list of smallest maximal runs up to 1016(Nicely, pers. comm., May 30, 2000). c(n) in this table is given by Sloane’s A008950, and n by Sloane’s A008996. The ending integers for the run corresponding to c(n) are given by Sloane’s A008995. Young and Potler (1989) determined the first occurrences of prime gaps up to 72,635,119,999,997, with all first occurrences found between 1 and 673. Nicely (1998) extended the list of maximal prime gaps to a length of 915, denoting gap lengths by the difference of bounding PRIMES , c(n) /C281:/n /c(n)/ n /c(n)/ 1 4 319 2,300,942,550 3 8 335 3,842,610,774 5 24 353 4,302,407,360 7 90 381 10,726,904,660 13 114 383 20,678,048,298 17 524 393 22,367,084,960 19 888 455 25,056,082,088 21 1,130 463 42,652,618,344 33 1,328 467 127,976,334,672 35 9,552 473 182,226,896,240 43 15,684 485 241,160,024,144 51 19,610 489 297,501,075,800 71 31,398 499 303,371,455,242 85 155,922 513 304,599,508,538 95 360,654 515 416,608,695,822 111 370,262 531 461,690,510,012 113 492,114 533 614,487,453,424 117 1,349,534 539 738,832,927,928 131 1,357,202 581 1,346,294,310,750 147 2,010,734 587 1,408,695,493,610 153 4,652,354 601 1,968,188,556,461 179 17,051,708 651 2,614,941,710,599 209 20,831,324 673 7,177,162,611,713 219 47,326,694 715 13,828,048,559,701221 122,164,748 765 19,581,334,192,423 233 189,695,660 777 42,842,283,925,352 247 191,912,784 803 90,874,329,411,493249 387,096,134 805 171,231,342,420,521281 436,273,010 905 218,209,405,436,543 287 1,294,268,492 915 1,189,459,969,825,483 291 1,453,168,142 923 1,686,994,940,955,803 1131 1,693,182,318,746,371 See also J UMPING CHAMPION ,PRIME CONSTELLATION , PRIME DIFFERENCE FUNCTION ,SHANKS’ CONJECTURE References Baugh, D. and O’Hara, F. "Large Prime Gaps." J. Recr. Math. 24, 186 /C1/87, 1992. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 133 /C1/34, 1994. Bombieri, E. and Davenport, H. "Small Differences Between Prime Numbers." Proc. Roy. Soc. A 293,1/C1/8, 1966. Brent, R. P. "The First Occurrence of Large Gaps Between Successive Primes." Math. Comput. 27, 959 /C1/63, 1973. Brent, R. P. "The Distribution of Small Gaps Between Successive Primes." Math. Comput. 28, 315 /C1/24, 1974. Brent, R. P. "The First Occurrence of Certain Large Prime Gaps." Math. Comput. 35, 1435 /C1/436, 1980. Crame ´r, H. "On the Order of Magnitude of the Difference Between Consecutive Prime Numbers." Acta Arith. 2,23/C1/ 6, 1937. Guy, R. K. "Gaps between Primes. Twin Primes" and "Increasing and Decreasing Gaps." §A8 and A11 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19 /C1/3 and 26 /C1/7, 1994. Lander, L. J. and Parkin, T. R. "On First Appearance of Prime Differences." Math. Comput. 21, 483 /C1/88, 1967. Nicely, T. R. "New Maximal Prime Gaps and First Occur- rences." Math. Comput. 68, 1311 /C1/315, 1999. Nicely, T. R. and Nyman, B. "First Occurrence of a Prime Gap of 1000 or Greater." Submitted to Math. Comput. Rivera, C. "Problems & Puzzles: Puzzle Distinct, Increasing & Decreasing Gaps.-011." http://www.primepuzzles.net/ puzzles/puzz_011.htm. Shanks, D. "On Maximal Gaps Between Successive Primes." Math. Comput. 18, 646 /C1/51, 1964. Sloane, N. J. A. Sequences A008950, A008995, A008996, and A030296 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Young, J. and Potler, A. "First Occurrence Prime Gaps." Math. Comput. 52, 221 /C1/24, 1989. Prime Group When the ORDER h of a finite GROUP is a PRIME NUMBER , there is only one possible GROUP of ORDER h. Furthermore, the GROUP is CYCLIC . See also P-GROUP Prime Ideal An IDEAL I such that if ab /C23 I ; then either a /C23 I or b /C23 I : For example, in the integers, the IDEAL a/C30 phi(i.e., the multiples of p) is prime whenever p is a PRIME NUMBER . Prime ideals are useful when the ring in question is not necessarily a PRINCIPAL IDEAL DOMAIN , e.g., a/C30 2;ffiffiffi 6p9+;=9+;; in Zffiffiffi6p9+$9+% : The general element of a can be written as 2a /C27bffiffiffi6p where a and b can be any integers. Suppose that x 1 /C27x2ffiffiffi 6p9+;k9+;7 y1 /C27y2ffiffiffi6p9+;k9+;7 /C302a /C27bffiffiffi6p ; then x 1y1 /C276x2y2 /C302a : So either x1or y1has to be even. The corresponding factor x1 /C27x2ffiffiffi 6p9+=9+; or y1 /C27y2ffiffiffi6p9+=9+; has to be in a/C30 2 ;ffiffiffi6p9+;=9+;; : Hence, the ideal a is prime. Note that this ring does not have UNIQUE FACTORIZATION since 2 /C215 3 /C306 /C30ffiffiffi6p /C215ffiffiffi6p : /One consequence of the definition is that the set of elements not in a prime ideal, R /C28p; is CLOSED under multiplication. This allows one to LOCALIZE at p by considering the RING OF FRACTIONS . This ring is analogous to the construction of the rationals as fractions of integers, except that the denominator must be in R /C28p: The only MAXIMAL IDEAL in this ring is the EXTENSION of p:/ From the perspective of ALGEBRAIC GEOMETRY , ideals correspond to VARIETIES . Because multiplication cor- responds to union (such as xy /C300 implies x /C300or y /C300), a prime ideal corresponds to an IRREDUCIBLE VARIETY . See also DEDEKIND RING,IDEAL ,IRREDUCIBLE VARI- ETY,K RULL DIMENSION ,M AXIMAL IDEAL ,STICKEL- BERGER RELATION ,STONE SPACE Prime Knot A KNOT other than the UNKNOT which cannot be expressed as a sum of two other KNOTS , neither of which is unknotted. A KNOT which is not prime is called a COMPOSITE KNOT . It is often possible to combine two prime knots to create two different COMPOSITE KNOTS , depending on the orientation of the two. Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a KNOT SUM of prime knots. There is no known FORMULA for giving the number of distinct prime knots as a function of the number of crossings. The numbers of distinct prime knots hav- ing n /C301, 2, ... crossings are 0, 0, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ... (Sloane’s A002863). Hoste et al. (1998) computed the number of distinct prime knots of n crossing up to n /C3016. Let N(n) be the number of distinct PRIME KNOTS of n crossings, counting CHIRAL versions of the same knot separately. Then 1 32n/C282 /C2819+=9+; 5N(n) +en (Ernst and Summers 1987). Welsh has shown that the number of knots is bounded by an exponential in n, and it is also known that lim sup[N(n)]1 =n B13:5 (Welsh 1991, Hoste et al. 1998, Thistlethwaite 1998). Menasco (1984) showed that a reduced alternating diagram represents a prime knot IFFthe diagram is itself prime ("an alternating knot is prime IFFit looks prime"; Hoste et al. 1998). See also COMPOSITE KNOT,KNOT References Ernst, C. and Sumners, D. W. "The Growth of the Number of Prime Knots." Math. Proc. Cambridge Philos. Soc. 102, 303/C1/15, 1987. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Menasco, W. "Closed Incompressible Surfaces in Alternating Knot and Link Complements." Topology 23,37/C1/4, 1984. Schubert, H. Sitzungsber. Heidelberger Akad. Wiss., Math.- Naturwiss. Klasse, 3rd Abhandlung. 1949. Sloane, N. J. A. Sequences A002863/M0851 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0851 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Thistlethwaite, M. "On the Structure and Scarcity of Alter- nating Links and Tangles." J. Knot Th. Ramifications 7, 981 /C1/004, 1998. Welsh, D. J. A. "On the Number of Knots and Links." Colloq. Math. Soc. J. Bolyai 60, 713 /C1/18, 1991. Prime k-Tuple PRIME CONSTELLATION Prime k-Tuples Conjecture K-TUPLE CONJECTURE Prime k-Tuplet PRIME CONSTELLATION Prime Manifold Ann-MANIFOLD which cannot be "nontrivially" de- composed into other n-MANIFOLDS . See also MANIFOLD Prime Number A prime number (or prime integer, often simply called a "prime" for short) is a POSITIVE INTEGER p/C211 that has no positive integer DIVISORS other than 1 and p itself. (More concisely, a prime number pis a POSITIVE INTEGER having exactly one positive divisor other than 1.) For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while thenumber 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization 24 /C302 3/C2153);mak- ing 24 nota prime number. P OSITIVE INTEGERS other than 1 which are not prime are called COMPOSITE NUMBERS . The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Lehmer 1909; Lehmer 1914; Hardy and Wright 1979, p. 11; Sloane and Plouffe 1995, p. 33; Hardy 1999,p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a classof its own. As noted by Tietze (1965, p. 2), "Why is thenumber 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable." The smal-lest prime is therefore 2. However, since 2 is the only EVEN PRIME , it is also somewhat special, the set of allprimes excluding 2 is called the " ODD PRIMES ." Note also that while 2 is considered a prime today, at one time it was not (Tietze 1965, p. 18; Tropfke 1921, p. 96). Excluding 1 and including 2, the first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ... (Sloane’s A000040; Hardy and Wright 1979, p. 3), and the SETof primes is sometimes denoted P:/ While the term "prime number" commonly refers toprime positive integers, other types of primes are also defined, such as the G AUSSIAN PRIMES . The function which gives the number of primes less than a number nis denoted p(n) and is called the PRIME COUNTING FUNCTION . The theorem giving an asymptotic form for p(n) is called the PRIME NUMBER THEOREM . Prime numbers can be generated by siev- ing processes (such as the E RATOSTHENES SIEVE ), and LUCKY NUMBERS , which are also generated by sieving, appear to share some interesting asymptotic proper-ties with the primes. Prime numbers satisfy manystrange and wonderful properties. Although there exist explicit PRIME FORMULAS (i.e., formulas which either generate primes for all values or else the nth prime as a function of n), they are contrived to such an extent that they are of little practical value. Many PRIME FACTORIZATION ALGORITHMS have been devised for determining the prime factors of a given INTEGER , a process known as factorization or prime factorization. They vary quite a bit in sophistication and complexity. It is very difficult to build a general- purpose algorithm for this computationally "hard"problem, so any additional information which is known about the number in question or its factors can often be used to save a large amount of time. Itshould be emphasized that although no efficient algorithms are known for factoring arbitrary primes, it has not been proved that no such algorithm exists. It is therefore conceivable that a suitably clever person could devise a general method of factoring which would render the vast majority of encryptionschemes in current widespread use, including those used by banks and governments, easily breakable. Because of their importance in encryption algorithms such as RSA ENCRYPTION , prime numbers can be important commercial commodities. In fact, RogerSchlafly has obtained U.S. Patent 5,373,560 (12/13/ 94) on the following two primes (expressed in hex- adecimal notation): 98A3DF52AEAE9799325CB258D767EBD1F4630E9B 9E21732A4AFB1624BA6DF911466AD8DA960586F4 A0D5E3C36AF099660BDDC1577E54A9F402334433 ACB14BCB and 93E8965DAFD9DFECFD00B466B68F90EA68AF5DC9 FED915278D1B3A137471E65596C37FED0C7829FF 8F8331F81A2700438ECDCC09447DC397C685F397 294F722BCC484AEDF28BED25AAAB35D35A65DB1FD62C9D7BA55844FEB1F9401E671340933EE43C54E4DC459400D7AD61248B83A2624835B31FFF2D95 95A5B90B276E44F9 : The FUNDAMENTAL THEOREM OF ARITHMETIC states that any POSITIVE INTEGER can be represented in exactly one way as a PRODUCT of primes. EUCLID’S SECOND THEOREM demonstrated that there are an infinite number of primes. However, it is not known if there are an infinite number of primes OF THE FORM n2 /C271 (Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206 /C1/08), whether there are an INFINITE number of TWIN PRIMES (the TWIN PRIME CONJEC- TURE ), or if a prime can always be found between n2 and (n /C271)2 (Hardy and Wright 1979, p. 415; Riben- boim 1996, pp. 397 /C1/98). The latter two of these are two of LANDAU’S PROBLEMS . The simplest method of finding factors is so-called "DIRECT SEARCH FACTORIZATION " (a.k.a. TRIAL DIVI- SION). In this method, all possible factors are system- atically tested using trial division to see if they actually DIVIDE the given number. It is practical only for very small numbers. More general (and complicated) methods include the ELLIPTIC CURVE FACTORIZATION METHOD and NUMBER FIELD SIEVE factorization method. It has been proven that the set of prime numbers is a DIOPHANTINE SET (Ribenboim 1991, pp. 106 /C1/07). Ramanujan also showed that d p(x) dx/C21 x ln xX/C12 n /C301m(n) nx1=n ; (1) where p(x) is the PRIME COUNTING FUNCTION and m(n) is the MO¨ BIUS FUNCTION (Berndt 1994, p. 117). With the exception of 2 and 3, all primes are of the form p /C306n 91; i.e., p /C136 (mod 1; 5): For n an INTEGER ]2; n is prime IFF n /C281 k9+;89+;9 /C13(/C281)k (mod n) (2) for k /C300, 1, ..., n /C281 (Deutsch 1996), wheren k9+=9+; is a BINOMIAL COEFFICIENT . In addition, an integer n is prime IFF f(n) /C27 s(n) /C302n: (3) The first few composite n for which n [f(n) /C27 s(n)] j are n /C30312, 560, 588, 1400, 23760, ... (Sloane’s A011774; Guy 1997), with a total of 18 such numbers less than 2 /C29107 :/ Cheng (1979) showed that for x sufficiently large, there always exist at least two prime factors betweenx /C28xaðÞ and x for a ]0 :477 ... (Le Lionnais 1983, p. 26). Let f(n) be the number of decompositions of n into two or more consecutive primes. Then lim x 0/C121 xXx n/C301f(n) /C30ln 2 (4) (Moser 1963, Le Lionnais 1983, p. 30). The probability that the GREATEST PRIME FACTOR of a RANDOM integer n is greater thanffiffiffinpis ln 2 (Schroeppel 1972). The probability that two INTEGERS picked at random are RELATIVELY PRIME is [ z(2)]/C281 /C30 6=p2 ; where z(x) is the RIEMANN ZETA FUNCTION (Cesaro and Sylvester 1883). Given three INTEGERS chosen at random, the probability that no common factor will divide them all is [z(3) /C281] :1:20206 /C281 :0:831907 ; (5) where z(3) is APE´ RY’S CONSTANT . In general, the probability that nrandom numbers lack a pthPOWER common divisor is [ z(np)]/C281(Beeler et al. 1972, Item 53). Large primes include the large M ERSENNE PRIMES , FERRIER’S PRIME , and 391581 /C2152216193/C281 (Cipra 1989). The largest known prime as of 1999 is the MERSENNE PRIME 26972593/C281:/ Primes consisting of consecutive DIGITS (counting 0 as coming after 9) include 2, 3, 5, 7, 23, 67, 89, 4567, 78901, ... (Sloane’s A006510). See also ADLEMAN- POMERANCE- RUMELY PRIMALITY TEST,ALMOST PRIME ,ANDRICA’S CONJECTURE ,BER- TRAND’S POSTULATE ,BROCARD’S CONJECTURE ,BRUN’S CONSTANT ,CARMICHAEL’S CONJECTURE ,CARMICHAEL FUNCTION ,CARMICHAEL NUMBER ,CHEBYSHEV FUNC- TIONS ,C HEBYSHEV- SYLVESTER CONSTANT ,C HEN’S THEOREM ,CHINESE HYPOTHESIS ,COMPOSITE NUM- BER,COMPOSITE RUNS,COPELAND- ERDOS CONSTANT , CRAMER CONJECTURE ,CUNNINGHAM CHAIN ,CYCLO- TOMIC POLYNOMIAL , DE POLIGNAC’S CONJECTUR E, DIRICHLET’S THEOREM ,D IVISOR ,E RDOS- KAC THEO- REM,EUCLID’S THEOREMS ,FEIT-THOMPSON CONJEC- TURE ,F ERMAT NUMBER ,F ERMAT QUOTIENT , FERRIER’S PRIME ,FORTUNATE PRIME ,FUNDAMENTAL THEOREM OF ARITHMETIC ,GIGANTIC PRIME ,GIUGA’S CONJECTURE ,GOLDBACH CONJECTURE ,GOOD PRIME , GRIMM’S CONJECTURE ,HARDY- RAMANUJAN THEOREM , HOME PRIME ,IRREGULAR PRIME ,KUMMER’S CONJEC- TURE ,L ANDAU’S PROBLEMS ,L EHMER’S PROBLEM , LINNIK’S THEOREM ,LONG PRIME ,M ERSENNE NUM- BER,M ERTENS FUNCTION ,M ILLER’S PRIMALITY TEST, MIRIMANOFF’S CONGRUENCE ,M O¨ BIUS FUNCTION ,PA- LINDROMIC NUMBER ,PE´ PIN’S TEST,PILLAI’S CONJEC- TURE ,P OULET NUMBER ,P RIMARY ,P RIME ARRAY , PRIME CIRCLE ,PRIME CONSTANT ,PRIME FACTORIZA- TION ALGORITHMS ,PRIME FORMULAS ,PRIME NUMBER OF MEASUREMENT ,PRIME NUMBER THEOREM ,PRIME POWER SYMBOL ,PRIME PRODUCTS ,PRIME STRING , PRIME SUMS,PRIME TRIANGLE ,PRIME ZETA FUNC- TION ,PRIMITIVE PRIME FACTOR ,PRIMORIAL ,PROB- ABLE PRIME ,P SEUDOPRIME ,R EGULAR PRIME , RIEMANN FUNCTION ,ROTKIEWICZ THEOREM ,SCHNIR- ELMANN’S THEOREM ,SELFRIDGE’S CONJECTURE ,SEMI- PRIME ,S HAH- WILSON CONSTANT ,S IERPINSKI’S COMPOSITE NUMBER THEOREM ,SIERPINSKI’S PRIME SEQUENCE THEOREM ,SMOOTH NUMBER ,SOLDNER’S CONSTANT ,SOPHIE GERMAIN PRIME ,TITANIC PRIME , TOTIENT FUNCTION ,T OTIENT VALENCE FUNCTION , TWIN PRIMES ,T WIN PRIMES CONSTANT ,V INOGRA- DOV’S THEOREM , VON MANGOLDT FUNCTION ,W AR- ING’S CONJECTURE ,W EAKLY PRIME ,W IEFERICH PRIME ,W ILSON PRIME ,W ILSON QUOTIENT ,W ILSON’S THEOREM ,W ITNESS ,W OLSTENHOLME’S THEOREM , ZSIGMONDY THEOREM References Berndt, B. C. "Ramanujan’s Theory of Prime Numbers." Ch. 24 in Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Caldwell, C. "Largest Primes." http://www.utm.edu/re- search/primes/largest.html. Caldwell, C. K. "The Top Twenty: Largest Known Primes." http://www.utm.edu/research/primes/lists/top20/Lar- gest.html. Cheng, J. R. "On the Distribution of Almost Primes in an Interval II." Sci. Sinica 22, 253/C1/75, 1979. Cipra, B. A. "Math Team Vaults Over Prime Record." Science 245, 815, 1989. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 130, 1996. Courant, R. and Robbins, H. "The Prime Numbers." §1i n Supplement to Ch. 1 in What is Mathematics?: An Ele- mentary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 21 /C1/1, 1996. Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980. Deutsch, E. "Problem 1494." Math. Mag. 69, 143, 1996. Dickson, L. E. "Factor Tables, Lists of Primes." Ch. 13 in History of the Theory of Numbers, Vol. 1: Divisibility andPrimality. New York: Chelsea, pp. 347 /C1 /56, 1952. Ellison, W. J. and Ellison, F. Prime Numbers. New York: Wiley, 1985. Eynden, C. V. "A Proof of Gandhi’s Formula for the nth Prime." Amer. Math. Monthly 79, 625, 1972. Giblin, P. J. Primes and Programming: Computers and Number Theory. New York: Cambridge University Press, 1994. Glaisher, J. Factor Tables for the Sixth Million: Containing the Least Factor of Every Number Not Divisible by 2, 3, or5 Between 5,000,000 and 6,000,000. London: Taylor and Francis, 1883. Golomb, S W. "A Direct Interpretation of Gandhi’s For- mula." Amer. Math. Monthly 81, 752/C1 /54. Guy, R. K. "Divisors and Desires." Amer. Math. Monthly 104, 359/C1/60, 1997. Guy, R. K. "Prime Numbers," "Formulas for Primes," and "Products Taken Over Primes." Ch. A, §A17, and §B48 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 3 /C1/3, 36/C1/1 and 102 /C1/03, 1994. Hardy, G. H. Ch. 2 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1978. Hardy, G. H. and Wright, E. M. "Prime Numbers" and "The Sequence of Primes." §1.2 and 1.4 in An Introduction to theTheory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 1 /C1/, 1979. Honaker, G. L. Jr. "Prime Curios!" http://www.utm.edu/ research/primes/curios/. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 30, 1976. Kraitchik, M. "Prime Numbers." §3.9 in Mathematical Recreations. New York: W. W. Norton, pp. 78 /C1/9, 1942. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 26, 30, and 46, 1983. Lehmer, D. N. Factor Table for the First Ten Millions. Washington, DC: Carnegie Institution, 1909. Lehmer, D. N. List of Prime Numbers from 1 to 10,006,721. Washington, DC: Carnegie Institution, 1914. Moser, L. "Notes on Number Theory III. On the Sum of Consecutive Primes." Can. Math. Bull. 6, 159/C1/61, 1963. Nagell, T. "Primes." §3i n Introduction to Number Theory. New York: Wiley, pp. 13 /C1/4, 1951. Ore, Ø.Number Theory and Its History. New York: Dover, 1988. Pappas, T. "Prime Numbers." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 100 /C1/01, 1989. Ramachandra, K. "Many Famous Conjectures on Primes; Meagre But Precious Progress of a Deep Nature." Proc. Indian Nat. Sci. Acad. Part A 64, 643/C1/50, 1998. Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, 1991. Ribenboim, P. "Prime Number Records." Coll. Math. J. 25, 280/C1/90, 1994. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, 1994. Schinzel, A. and Sierpinski, W. "Sur certains hypothe `ses concernant les nombres premiers." Acta Arith. 4, 185/C1/08, 1958. Schinzel, A. and Sierpinski, W. Erratum to "Sur certains hypothe `ses concernant les nombres premiers." Acta Arith. 5, 259, 1959. Schroeppel, R. Item 29 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 13, Feb. 1972. Sloane, N. J. A. Sequences A000040/M0652, A006510/ M0679, A010051, A011774, and A046024 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Tietze, H. "Prime Numbers and Prime Twins." Ch. 1 in Famous Problems of Mathematics: Solved and UnsolvedMathematics Problems from Antiquity to Modern Times.New York: Graylock Press, pp. 1 /C1 /0, 1965. Torelli, G. Sulla totalita `dei numeri primi fino ad un limite assegnato. Naples, Italy: Tip. della Reale accad. della scienze fisiche e matematiche, 1901. Tropfke, J. Geschichte der Elementar-Mathematik, Band 1. Berlin, Germany: p. 96, 1921. Wagon, S. "Primes Numbers." Ch. 1 in Mathematica in Action. New York: W. H. Freeman, pp. 11 /C1/7, 1991. Weisstein, E. W. "Books about Prime Numbers." http:// www.treasure-troves.com/books/PrimeNumbers.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 31, 1986. Zaiger, D. "The First 50 Million Prime Numbers." Math. Intel. 0, 221/C1/24, 1977. Prime Number of Measurement The set of numbers generated by excluding the SUMS of two or more consecutive earlier members is called the prime numbers of measurement, or sometimes the SEGMENTED NUMBERS . The first few terms are 1, 2, 4, 5, 8, 10, 14, 15, 16, 21, ... (Sloane’s A002048). Excluding two and three terms gives the sequence 1, 2, 4, 5, 8, 10, 12, 14, 15, 16, 19, 20, 21, ... (Sloane’s A005242). See also SUM-FREE SET References Guy, R. K. "MacMahon’s Prime Numbers of Measurement." §E30 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 230 /C1/31, 1994. Sloane, N. J. A. Sequences A002048/M0972 and A005242/ M0971 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Prime Number Theorem The theorem giving an asymptotic form for the PRIME COUNTING FUNCTION p(n);which counts the number of PRIMES less than some INTEGER n. Legendre (1808) suggested that, for large n, p(n)/C2n Alnn/C27B; (1) with A/C301 and B/C30/C281:08366 (where Bis sometimes called L EGENDRE’S CONSTANT ), a formula which is correct in the leading term only (Nagell 1951, p. 54; Wagon 1991, pp. 28 /C1/9). In 1791, Gauss became the first to suggest instead p(n)/C2n lnn: (2) Gauss later refined his estimate to p(n)/C2li(n); (3) where li( n) is the LOGARITHMIC INTEGRAL . This func- tion has n=lnnas the leading term and has been shown to be a better estimate than n=lnnalone. The statement (3) is often known as "the" prime numbertheorem and was proved independently by Hadamard (1896) and de la Valle ´e Poussin (1896). A plot of p(n) (lower curve) and li( n) is shown above for n51000 : / For small n, it has been checked and always found that p(n)Bli(n) :However, Skewes proved that thefirst crossing of p(n)Bli(n)/C300 occurs before 10101034 (the SKEWES NUMBER ). The upper bound for the crossing has subsequently been reduced to 10371. Littlewood (1914) proved that the INEQUALITY re- verses infinitely often for sufficiently large n(Ball and Coxeter 1987). Lehman (1966) proved that at least 10500reversals occur for numbers with 1166 or 1167 DECIMAL DIGITS . Chebyshev put limits on the RATIO 7 8Bp(n) n lnnB98(4) (Landau 1927; Nagell 1951, p. 55; Landau 1974; Hardy and Wright 1979, Ch. 22; Ingham 1990;Rubinstein and Sarnak 1994; Hardy 1999, p. 27),and showed that if the LIMIT lim n0/C12p(n) n lnn(5) existed, then it would be 1. Hadamard and Valle ´e Poussin proved the prime number theorem by showing that the R IEMANN ZETA FUNCTION z(z) has no zeros OF THE FORM 1/C27it;in the sense that no deeper properties of z(s) are required for the proof (Smith 1994, p. 128; Hardy 1999, pp. 58 /C1/0). Wiener (1951) allowed this somewhat vague state- ment to be interpreted literally (Hardy 1999, pp. 34 and 46), and this proof was simplified by Landau (1932) and Bochner (1933). Hadamard’s proof depends on the simple trigono- metric inequality 3/C274 cos u/C27cos(2 u)/C302(1/C27cosu)2]0 (6) (Hardy 1999, p. 58). Valle ´e Poussin (1899) showed that p(x)/C30li(x)/C27Ox lnxe/C28affiffiffiffiffiffi lnxp ! (7) for some constant a(Knuth 1997, p. 381), where O(x) isASYMPTOTIC NOTATION . A simplified proof was found by Erdos (1949) and Selberg (1950) (Ball andCoxeter 1987, p. 63), although an unfortunate prior-ity dispute over the joint work marred the otherwise beautiful proof (Hoffman 1998, pp. 39 /C1 /1). An elemen- tary proof of the prime number theorem, following Selberg, is the final section in Nagell’s 1951 textbook. The error term in (7) has subsequently improved to p(x)/C30li(x)/C27Oxexp/C28AlnxðÞ3=5 ln ln x ðÞ1=5 ! ! (8) (Walfisz 1963; Riesel 1994, p. 56; Knuth 1997, p. 382). Ingham (1930) proved the prime number theorem using the identity of Ramanujan X/C12 n/C301sa(n)sb(n) ns/C30z(s) z(s /C28 a) zðs /C28 bÞz(s /C28 a /C28 b) z(2s /C28 a /C28 b); (9) where sa(n) is the DIVISOR FUNCTION (Hardy 1999, pp. 59 /C1/0). Riemann estimated the PRIME COUNTING FUNCTION with p(n) /C2ln(n) /C281 2 li n1 =29+=9+; ; (10) which is a better approximation than li(n) for n B107 : Riemann (1859) also suggested the RIEMANN FUNC- TION R(x) /C30X/C12 n/C301m(n) nli x1 =n9+=9+; ; (11) where m is the MO¨ BIUS FUNCTION (Wagon 1991, p. 29). An even better approximation for small n (by a factor of 10 for n B109) is the GRAM SERIES . The prime number theorem is equivalent to either lim x0/C12u(x) x/C301 (12) or lim x 0/C12c(x) x/C301 ; (13) where u and c(x) are the CHEBYSHEV FUNCTIONS . Chebyshev showed that the only possible limit of these expressions was 1, but was not able to prove existence of the limit (Hardy 1999, p. 28). The RIEMANN HYPOTHESIS is equivalent to the asser- tion that Li(x) /C28 p(x) jj 5cffiffiffixpln x (14) for some value of c (Ingham 1990, p. 83; Landau 1974, pp. 378 /C1/88; Ball and Coxeter 1987; Hardy 1999, p. 26). Some limits obtained without assuming the RIEMANN HYPOTHESIS are pðxÞ¼Li ðxÞþO ½xe /C28ln x1 =2 =15 /C138ð 15Þ p(x) /C30Li(x) /C27O xe /C280 :009 ln x ðÞ3 =5= ln ln x ðÞ1=5hi : (16) Ramanujan showed that for sufficiently large x, p2(x)Bex lnxpx e ! : (17) The largest known PRIME for which the inequality fails is 38,358,837,677 (Berndt 1994, pp. 112 /C1/13). The related inequalityLi2(x)Bex lnxLix e ! (18) is true for x]2418 (Berndt 1994, p. 114). See also BERTRAND’S POSTULATE ,CHEBYSHEV FUNC- TIONS ,C HEBYSHEV’S THEOREM ,D IRICHLET’S THEO- REM,G RAM SERIES ,P RIME COUNTING FUNCTION , RIEMANN FUNCTION ,SELBERG’S FORMULA ,SKEWES NUMBER References Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 62 /C1/4, 1987. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Bochner. Math. Z. 37,1/C1/, 1933. Courant, R. and Robbins, H. "The Prime Number Theorem." §1.2c in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 27 /C1/0, 1996. Davenport, H. "Prime Number Theorem." Ch. 18 in Multi- plicative Number Theory, 2nd ed. New York: Springer- Verlag, pp. 111 /C1/14, 1980. de la Valle ´e Poussin, C.-J. "Recherches analytiques la the´orie des nombres premiers." Ann. Soc. scient. Bruxelles 20, 183/C1/56, 1896. Erdos, P. "De ´monstration e ´le´mentaire du the ´ore`me sur la distribution des nombres premiers." Scriptum 1, CentreMathe ´matique, Amsterdam, 1949. Hadamard, J. "Sur la distribution des ze ´ros de la fonction z(s) et ses conse ´quences arithme ´tiques (’)." Bull. Soc. math. France 24, 199/C1 /20, 1896. Hardy, G. H. "The Proof of the Prime Number Theorem" and "Second Approximation of the Proof." §2.5 and 2.6 in Ramanujan: Twelve Lectures on Subjects Suggested byHis Life and Work, 3rd ed. New York: Chelsea, pp. 16, 27, and 28 /C1 /3, 1999. Hardy, G. H. and Wright, E. M. "Statement of the Prime Number Theorem." §1.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 9/C1/0, 1979. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.New York: Hyperion, 1998. Ingham, A. E. "Note on Riemann’s z /-Function and Diri- chlet’s L-Functions." J. London Math. Soc. 5, 107/C1/12, 1930. Ingham, A. E. The Distribution of Prime Numbers. London: Cambridge University Press, p. 83, 1990. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Landau, E. Vorlesungen u ¨ber Zahlentheorie, Vol. 1. New York: Chelsea, pp. 79 /C1/6, 1970. Landau, E. Berliner Sitzungsber. , 514/C1/21, 1932. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, 1974. Legendre, A. M. Essai sur la The ´orie des Nombres. Paris: Duprat, 1808. Lehman, R. S. "On the Difference p(x)/C28li(x):/"Acta Arith. 11, 397/C1/10, 1966. Littlewood, J. E. "Sur les distribution des nombres pre- miers." C. R. Acad. Sci. Paris 158, 1869/C1/872, 1914. Lu, W. C. "On the Elementary Proof of the Prime Number Theorem with a Remainder Term." Rocky Mountain J. Math. 29, 979, 1999. Nagell, T. "The Prime Number Theorem." Ch. 8 in Introduc- tion to Number Theory. New York: Wiley, pp. 275 /C1/99, 1951. Riemann, G. F. B. "U¨ ber die Anzahl der Primzahlen unter einer gegebenen Gro¨sse." Monatsber. Ko¨nigl. Preuss. Akad. Wiss. Berlin , 671, 1859. Riesel, H. "The Remainder Term in the Prime Number Theorem." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, p. 6, 1994. Rubinstein, M. and Sarnak, P. "Chebyshev’s Bias." Experi- mental Math. 3, 173 /C1/97, 1994. Selberg, A. "An Elementary Proof of the Prime Number Theorem." Ann. Math. 50, 305 /C1/13, 1949. Shanks, D. "The Prime Number Theorem." §1.6 in Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 15 /C1/7, 1993. Smith, D. E. A Source Book in Mathematics. New York: Dover, 1994. Valle´e Poussin, C. Me´m. Couronne ´s Acad. Roy. Belgique 59, 1 /C1/4, 1899. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 25 /C1/5, 1991. Walfisz, A. Ch. 5 in Weyl’sche Exponentialsummen in der neueren Zahlentheorie. Berlin: Deutscher Verlag der Wissenschaften, 1963. Wiener, N. §19 et seq. in The Fourier Integral and Certain of Its Applications. New York: Dover, 1951. Prime Pairs TWIN PRIMES Prime Partition A prime partition of a POSITIVE INTEGER n ]2 is a set of PRIMES pi which sum to n. For example, there are three prime partitions of 7 since 7 /C307 /C302 /C275 /C302 /C272 /C273 : The number of prime partitions of n /C302, 3, ... are 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, ... (Sloane’s A000607). If an /C301 for n prime and an /C300 for n composite, then the EULER TRANSFORM bn gives the number of partitions of n into prime parts (Sloane and Plouffe 1995, p. 21). The minimum number of primes needed to sum to n /C302, 3, ... are 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, ... (Sloane’s A051034). The maximum number of primes needed to sum to n is just n=2bc ; 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... (Sloane’s A004526), corresponding to a representation in terms of all 2s for an even number or one 3 and the rest 2s for an odd number. The numbers which can be represented by a single prime are obviously the primes themselves. Compo- site numbers which can be REPRESENTED AS the sum of two primes are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ... (Sloane’s A051035), and composite numbers which are not the sum of fewer than three primes are 27, 35, 51, 57, 65, 77, 87, 93, 95, 117, 119, ..., (Sloane’s A025583). The conjecture that no numbers requirefour or more primes is called the GOLDBACH CON- JECTURE . See also GOLDBACH CONJECTURE ,PARTITION ,PARTI- TION FUNCTION P,SCHNIRELMANN’S THEOREM References Berndt, B.C. and Wilson, B. M. "Chapter 5 of Ramanujan’s Second Notebook." In Analytic Number Theory: Proceed- ings of the Conference Held at Temple University, Phila- delphia, Pa., May 12 /C1/5, 1980 (Ed. M. I. Knopp). Berlin: Springer-Verlag, pp. 49 /C1/8, 1981. Chawla, L. M. and Shad, S. A. "On a Trio-Set of Partition Functions and Their Tables." J. Natural Sciences and Mathematics 9,87/C1/6, 1969. Gupta, O. P. and Luthra, S. "Partitions into Primes." Proc. Nat. Inst. Sci. India. Part A 21, 181 /C1/84, 1955. Gupta, H. "Partitions into Distinct Primes." Proc. Nat. Inst. Sci. India. Part A 21, 185 /C1/87, 1955. Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697 /C1/12, 1988. Sloane, N. J. A. Sequences A000607/M0265, A004526, A025583, A051034, and A051035 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Prime Patterns Conjecture K-TUPLE CONJECTURE Prime Pi PRIME COUNTING FUNCTION Prime Polynomial PRIME- GENERATING POLYNOMIAL Prime Power A PRIME or integer power of a PRIME . The first few are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, ... (Sloane’s A000961). The first few prime powers with power ]2 are given by 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, ... (Sloane’s A025475). The number of prime powers (/]2) up to x does not exceed x1 =2 /C27x1 =3 /C27x1=4 /C27.../C30O x1 =2 ln x9+=9+; (Hardy 1999, p. 27).The following table gives prime kth powers. kSloane prime kth powers 1 A000040 2, 3, 5, 7, 11, 13, 17, 19, 23, ...2 A001248 4, 9, 25, 49, 121, 169, 289, 361, ...3 A030078 8, 27, 125, 343, 1331, 2197, 4913, ...4 A030514 16, 81, 625, 2401, 14641, 28561, 83521, ... 5 A050997 32, 243, 3125, 16807, 161051, 371293, ... See also P RIME NUMBER ,SOLITARY NUMBER References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Sloane, N. J. A. Sequences A000040/M0652, A000961/ M0517, A001248, A025475, A030078, A030514, and A050997 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Prime Power Conjecture An Abelian planar DIFFERENCE SET of order n exists only for n a PRIME POWER . Gordon (1994) has verified it to be true for n B2;000;000:/ See also DIFFERENCE SET References Gordon, D. M. "The Prime Power Conjecture is True for n B2 ;000; 000:/" Electronic J. Combinatorics 1,R61 /C1/, 1994. http://www.combinatorics.org/Volume_1/volu- me1.html#R6. Prime Power Symbol The symbol pe kn means, for p a PRIME , that pe kn; but pe/C271¶n :/ Prime Products The product of primes pn# /C13Yn k /C301pk ; (1) with pnthe nth prime, is called the PRIMORIAL function, by analogy with the FACTORIAL function. The EULER PRODUCT gives e g /C30 lim n 0/C121 ln nYn k /C3011 1 /C281 pk; (2) where g is the EULER- MASCHERONI CONSTANT . There is also an amazing infinite product formula for primes given by Y/C12 k/C301p2 k/C271 p2k/C281/C305 2: (3) (Ramanujan; Le Lionnais 1983, p. 46). See also EULER PRODUCT ,PRIME NUMBER ,PRIME SUMS,PRIMORIAL References Grosswald, E. "Some Number Theoretical Products." Rev. Columbiana Mat. 21231/C1/42, 1987. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983. Uchiyama, S. "On Some Products Involving Primes." Proc. Amer. Math. Soc. 28, 629/C1/30, 1971.Prime Quadratic Effect Letpm;n(x) denote the number of PRIMES5xwhich are congruent to nmodulo m. Then one might expect that D(x)/C13p4;3(x)/C28p4;1(x)/C21 2px1=29+=9+; >0 (Berndt 1994). Although this is true for small num- bers, Hardy and Littlewood showed that D(x) changes sign infinitely often. The effect was first noted by Chebyshev in 1853, and is sometimes called the CHEBYSHEV PHENOMENON . It was subsequently stu- died by Shanks (1959), Hudson (1980), and Bays and Hudson (1977, 1978, 1979). The effect was also noted by Ramanujan, who incorrectly claimed thatlim x0/C12D(x)/C30/C12(Berndt 1994). The values at which D(x)/C300 are x/C302946, 50378, 50380, 50382, 50392, 50414, ... (Sloane’s A051024),corresponding to p(x)/C3026861 ;616841, 616849, 616877, 617011, ... (Sloane’s A051025). References Bays, C. and Hudson, R. H. "The Mean Behavior of Primes in Arithmetic Progressions." J. reine angew. Math. 296, 80/C1/9, 1977. Bays, C. and Hudson, R. H. "On the Fluctuations of Little- wood for Primes of the Form 4 n91:/"Math. Comput. 32, 281/C1/86, 1978. Bays, C. and Hudson, R. H. "Numerical and Graphical Description of All Axis Crossing Regions for the Moduli 4 and 8 which Occur Before 1012."Internat. J. Math. Math. Sci. 2, 111/C1/19, 1979. Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 135 /C1/36, 1994. Hudson, R. H. "A Common Principle Underlies Riemann’s Formula, the Chebyshev Phenomenon, and Other SubtleEffects in Comparative Prime Number Theory. I." J. reine angew. Math. 313, 133/C1 /50, 1980. Shanks, D. "Quadratic Residues and the Distribution of Primes." Math. Comput. 13, 272/C1/84, 1959. Sloane, N. J. A. Sequences A051024 and A051025 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/eisonline.html. Prime Quadruplet APRIME CONSTELLATION of four successive PRIMES with minimal distance ( p;p/C272;p/C276;p/C278):The term was coined by Paul Sta ¨ckel (1892 /C1/919; Tietze 1965, p. 19). The quadruplet (2, 3, 5, 7) has smaller minimal distance, but it is an exceptional special case. With the exception of (5, 7, 11, 13), a prime quadruple must be OF THE FORM (/30n /C2711 ; 30n /C2713; 30n /C2717 ; 30n /C2719) : The first few values of n which give prime quadruples are n /C300, 3, 6, 27, 49, 62, 69, 108, 115, ... (Sloane’s A014561), and the first few values of p are 5 (the exceptional case), 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, ... (Sloane’s A007530). The number of prime quadruplets with largest member less than 101,102, ..., are 1, 2, 5, 12, 38, 166, 899, 4768, ... (Sloane’s A050258; Nicely 1999). The asymptotic FORMULA for the frequency of prime quadruples is analogous to that for other PRIME CONSTELLATIONS , Px ðp ;p þ 2;p þ 6; p þ 8 Þ/C227 2Y p]5p3 ðp /C28 4Þ ðp /C28 1Þ4 gx 2dx ðln xÞ4 /C304:151180864 gx 2dx ln xðÞ4 ; where c /C304:15118... is the Hardy-Littlewood con- stant for prime quadruplets. Roonguthai found the large prime quadruplets with p /C301099 /C27349781731 p /C3010199 /C2721156403891 p /C3010299 /C27140159459341 p /C3010399 /C2734993836001 p/C3010499/C27883750143961 p/C3010599/C271394283756151 p/C3010699/C27547634621251 (Roonguthai). Forbes found the large quadruplet with p/C3076912895956636885 23279/C28210939+=9+; /C286/C21521093/C287: See also PRIME ARITHMETIC PROGRESSION ,P RIME CONSTELLATION ,P RIME K -TUPLES CONJECTURE , PRIME TRIPLET ,SEXY PRIMES ,TWIN PRIMES References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. New York: Oxford University Press, 1979. Forbes, T. "Prime k-tuplets." http://www.ltkz.demon.co.uk/ ktuplets.htm. Forbes, T. "Large Prime Quadruplets." nmbrthry@list- serv.nodak.edu . Sep. 17, 1998. Nicely, T. R. "Enumeration to 1 :6/C291015of the Prime Quadruplets." Submitted to Math. Comput. Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 61 /C1/ 2, 1994. Roonguthai, W. "Large Prime Quadruplets." http:// www.mathsoft.com/asolve/constant/hrdyltl/roon- guth.html. Sloane, N. J. A. Sequences A007530/M3816, A014561, and A050258 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod-ern Times. New York: Graylock Press, p. 19, 1965. Prime Representation Let a"b;A, and Bdenote POSITIVE INTEGERS satisfying (a;b)/C301(A;B)/C301 (i.e., both pairs are RELATIVELY PRIME ), and suppose every PRIME p/C13B(mod A) with ( p;2ab)/C301 is ex- pressible if the form ax2/C28by2for some INTEGERS x andy. Then every PRIME qsuch that q/C13/C28B(mod A) and ( q;2ab)/C301 is expressible in the form bX2/C28aY2 for some INTEGERS Xand Y(Halter-Koch 1993, Williams 1991). Prime Form Representation /4n/C271// x2/C27y2 / /8n/C271;8n/C273// x2/C272y2/ /8n91// x2/C282y2/ /6n/C271// x2/C273y2 / /12n/C271// x2/C283y2 / /20n/C271;20n/C279// x2/C275y2/ /10n/C271;10n/C279// x2/C285y2/ /14n/C271;14n/C279;14n/C2725//x2/C277y2 / /28n/C271;28n/C279;28n/C2725//x2/C287y2 / /30n/C271;30n/C2749// x2/C2715y2/ /60n/C271;60n/C2749// x2/C2815y2/ /30n/C287;30n/C2717// 5x2/C273y2 / /60n/C287;60n/C2717// 5x2/C283y2 / /24n/C271;24n/C277// x2/C276y2/ /24n/C271;24n/C2719// x2/C286y2/ /24n/C275;24n/C2711// 2x2/C273y2 / /24n/C275;24n/C281// 2x2/C283y2/ References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 70 /C1/3, 1994. Halter-Koch, F. "A Theorem of Ramanujan Concerning Binary Quadratic Forms." J. Number. Theory 44, 209 /C1/ 13, 1993. Williams, K. S. "On an Assertion of Ramanujan Concerning Binary Quadratic Forms." J. Number Th. 38, 118 /C1/33, 1991. Prime Ring A RING for which the product of any pair of IDEALS is zero only if one of the two IDEALS is zero. All SIMPLE RINGS are prime. See also IDEAL ,RING,SEMIPRIME RING,SIMPLE RING Prime Sequence PRIME ARITHMETIC PROGRESSION ,P RIME ARRAY , PRIME- GENERATING POLYNOMIAL ,SIERPINSKI’S PRIME SEQUENCE THEOREM Prime Signature The prime signature of a positive integer n is a sorted list of exponents ai in the PRIME FACTORIZATION n /C30pa1 1 pa2 2/C1/C1/C1: The prime signature of n can therefore be computed in Mathematica as PrimeSignature[1] : /C30 {1} PrimeSignature[n_Integer?Positive] : /C30 Sort[Transpose[FactorInteger[n]][[2]]] See also PRIME FACTORIZATION Prime Spiral The numbers arranged in a SPIRAL 543 612789 with PRIMES indicated in black, as first drawn byS. Ulam. Unexpected patterns of diagonal lines are apparent in such a plot, as illustrated in the above 199 /C29199 grid. M. Charpentier has written a Post- Script file which can be downloaded to a printer and draws a prime spiral. See also PRIME- GENERATING POLYNOMIAL References Charpentier, M. "Prime Numbers in PostScript." http:// www.cs.unh.edu/~charpov/Programming/PostScript- primes/. Dewdney, A. K. "Computer Recreations: How to Pan for Primes in Numerical Gravel." Sci. Amer. 259, 120 /C1/23, July 1988. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 80 /C1/3 and 88 /C1/9, 1984. Goddard, T. "Ulam Spiral." http://www.d4maths.co.uk/mi- rage/ulam.htm. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 105 /C1/09, 1998. Lane, C. "Prime Spiral." http://www.best.com/~cdl/Prime- SpiralApplet.html. Leatherland, A. J. F. "The Mysterious Prime Spiral Phe- nomenon." http://yoyo.cc.monash.edu.au/~bunyip/primes/ #spiral. Morin, D. "Le Village Premier." http://platon.lacitec.on.ca/ ~dmorin/applet/village/. Stein, M. L.; Ulam, S. M.; and Wells, M. B. "A Visual Display of Some Properties of the Distribution of Primes." Amer. Math. Monthly 71, 516 /C1/20, 1964. Weisstein, E. W. "Prime Spiral." MATHEMATICA NOTEBOOK PRIME SPIRAL.M . Prime String TRUNCATABLE PRIME Prime Subfield The prime subfield of a FIELD F is the SUBFIELD of F generated by the multiplicative identity 1FofF.I ti s isomorphic to either Q(if the CHARACTERISTIC is 0), or the FINITE FIELD FP/C30Z=pZ(if the CHARACTERISTIC is p). See also SUBFIELD References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 423, 1998. Prime Sum 60n/C287;60n/C2717 Let 5x2/C283y2 be the sum of the first nPRIMES . The first few terms are 2, 5, 10, 17, 28, 41, 58, 77, ... (Sloane’s A007504). Bach and Shallit (1996) show that 24n/C271;24n/C277 and provide a general technique for estimating such sums. The first few values of n such that x2 /C276y2 are 1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, ... (Sloane’s A045345). The corresponding values of 24n /C271; 24n /C2719 are 2, 874, 5830, 2615298, 712377380, 86810649294, 794712005370, 105784534314378, 92542301212047102, ... (Sloane’s A050247; Rivera), and the values of x2 /C286y2 are 2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, ... (Sloane’s A050248; Rivera). See also PRIMORIAL References Bach, E. and Shallit, J. §2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, 1996. Rivera, C. "Problems & Puzzles: Puzzle The Average Prime number, 24 n/C275;24n/C2711:/-031." http://www.primepuz- zles.net/puzzles/puzz_031.htm. Sloane, N. J. A. Sequences A007504/M1370, A045345, A050247, and A050248 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Prime Sums Let X nðÞ/C13Xn i/C301pi (1) be the sum of the first nPRIMES (i.e., the sum analog of the PRIMORIAL function). The first few terms are 2, 5, 10, 17, 28, 41, 58, 77, ... (Sloane’s A007504). Bach and Shallit (1996) show that X nðÞ/C2n2 2log n; (2) and provide a general technique for estimating such sums.The first few values of nsuch that na(n) j are 1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287,... (Sloane’s A045345). The corresponding values ofa(n) are 2, 874, 5830, 2615298, 712377380, 86810649294, 794712005370, 105784534314378, 92542301212047102, ... (Sloane’s A050247; Rivera), and the values of n=a(n) are 2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, ... (Sloane’s A050248; Rivera).In 1737, Euler showed that the sum of the reciprocals of the primes diverges X /C12 k/C3011 pk/C30/C12 (3) (Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17and 22), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (Sloane’s A046024) primes, and its asymptotic equation is X x p/C302 pprime1 p/C30ln ln x/C27B1/C27o(1); (4) where B1is M ERTENS CONSTANT (Hardy and Wright 1979, p. 351). Dirichlet showed the even strongerresult that X prime p/C13bmod a ðÞ a;bðÞ/C3011 p/C30/C12 (5) (Davenport 1980, p. 34). Despite the divergence of thesum of reciprocal primes, the ALTERNATING SERIES X/C12 k/C301(/C281)k pk:/C280:2696065 (6) converges (Robinson and Potter 1971, Finch), but it isnot known if the sum X /C12 k/C301(/C281)kk pk(7) does (Guy 1994, p. 203; Erdos 1998; Finch). There are also classes of sums of reciprocal primes with sign determined by congruences on k, for example X/C12 k¼2ck pk:0:3349813253 ð8Þ where ck/C30/C281 for pk/C131 mod 4ðÞ 1 for pk/C133 mod 4ðÞ9+$k (9) (Glaisher 1891b, Finch) which, is not known to converge, while X/C12 k¼2ck p2 k:0:094619828 ð10Þ does converge (Glaisher 1893, Finch). It is not known if X/C12 k /C301dk pk:0:6419448385 (11) converges, where dk /C30/C281 for pk /C131 mod 3 ðÞ 1 for pk /C132 mod 3 ðÞ 0 for pk /C130 mod 3 ðÞ8 < : (12) (Glaisher 1891c, Finch). Although a 1=p diverges, Brun (1919) showed that X p p /C272 prime1 p /C30B B/C12 ; (13) where B is BRUN’S CONSTANT . The function defined by P(n) /C13X/C12 p /C3011 pn k(14) taken over the primes converges for n /C211 and is a generalization of the RIEMANN ZETA FUNCTION known as the PRIME ZETA FUNCTION . A rapidly converging series for the MERTENS CON- STANT B1 /C30 g /C27X/C12 k /C301ln 1 /C28p /C281 k9+=9+; /C271 pk"# :0 :2614972128 (15) is given by B1 /C30 g /C27X/C12 m/C302m(m) mln z(m) ½/C138 ; (16) where g is the EULER- MASCHERONI CONSTANT , z(n)is the RIEMANN ZETA FUNCTION , and m(n) is the MO¨ BIUS FUNCTION (Flajolet and Vardi 1996, Schroeder 1997, Knuth 1998). A similar formula gives the sum X/C12 k /C3011 p2 kX/C12 k /C301m(k) kln z(2k) ðÞ:0:45224742 (17) The sum X/C12 k /C3011 pk /C28 1 ðÞ2 :1:3750649947 (18) is also finite (Glaisher 1891a; Cohen; Finch). Some curious sums satisfied by primes p include Xp /C281 k /C301k3 p$% /C30(p /C28 2)(p /C28 1)(p /C27 1) 4 (19) Xp /C281 ðÞ p /C282 ðÞ k /C301kp9+=9+;1=3jk /C301 4(3p /C285)(p /C282)(p /C281) (20) (Doster 1993),X/C12 k/C301xklnk/C30X pprimeX/C12 k/C301xpk 1/C28xpk; (21) and X/C12 k/C301(/C281)k/C281e/C28kxlnk/C30/C28ln 2X/C12 k/C3011 e2kx/C281 /C27X pan odd primelnpX/C12 k/C3011 epkx/C271(22) (Berndt 1994, p. 114). See also MERTENS CONSTANT ,PRIME NUMBER ,PRIME PRODUCTS ,PRIME ZETA FUNCTION ,PRIMORIAL References Bach, E. and Shallit, J. §2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, 1996. Berndt, B. C. "Ramanujan’s Theory of Prime Numbers." Ch. 24 in Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Brun, V. "La serie 1 =5/C271=7/C27. . . est convergente ou finie." Bull. Sci. Math. 43, 124/C1/28, 1919. Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. http://www.math.u-bordeaux.fr/~co- hen/hardylw.dvi. Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980. Doster, D. "Problem 10346." Amer. Math. Monthly 100, 951, 1993. Erdos, P. "Some of My New and Almost New Problems and Results in Combinatorial Number Theory." In Number Theory: Diophantine, Computational and Algebraic As-pects. Proceedings of the International Conference Held inEger, July 29-August 2, 1996 (Ed. K. Gyory, A. Petho and V. T. So ´s). Berlin: de Gruyter, pp. 169 /C1 /80, 1998. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hdmrd/hdmrd.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996.http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Glaisher, J. W. L. "On the Sums of the Inverse Powers of the Prime Numbers." Quart. J. Pure Appl. Math. 25, 347/C1 /62, 1891a. Glaisher, J. W. L. "On the Series 1 =3/C281=5/C27/ /1=7/C271=11/C281=13/C28...:/"Quart. J. Pure Appl. Math. 25, 375/C1/83, 1891b. Glaisher, J. W. L. "On the Series 1 =2/C271=5/C28/ /1=7/C271=11/C281=13/C28...:/"Quart. J. Pure Appl. Math. 25, 48/C1/5, 1891c. Glaisher, J. W. L. "On the Series 1 =32/C281=52/ //C271=72/C271=112/C281=13/C28...:/"Quart. J. Pure Appl. Math. 26,3 3/C1/7, 1893. Guy, R. K. "A Series and a Sequence Involving Primes." §E7 inUnsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 203, 1994. Hardy, G. H. and Wright, E. M. "Prime Numbers" and "The Sequence of Primes." §1.2 and 1.4 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 1 /C1/, 17, 22, and 251, 1979. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Moree, P. "Approximation of Singular Series and Automata." Manuscripta Math. 101, 385 /C1/99, 2000. Nagell, T. Introduction to Number Theory. New York: Wiley, 1951. Rivera, C. "Problems & Puzzles: Puzzle 031.-The Average Prime Number, APN (k) /C30SpkðÞ =k:/" .htm" tar- get /C30"extwin">http://www.primepuzzles.net/puzzles/ puzz_The Average Prime Number, APN (k) /C30SpkðÞ =k :/htm. Robinson, H. P. and Potter, E. Mathematical Constants. Report UCRL-20418. Berkeley, CA: University of Califor- nia, 1971. Schroeder, M. R. Number Theory in Science and Commu- nication, with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. New York: Springer-Verlag, 1997. Sloane, N. J. A. Sequences A007504/M1370, A045345, A046024, A050247, and A050248 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Prime Theta Function CHEBYSHEV FUNCTIONS Prime Triangle A triangle with rows containing the numbers 1; 2; ...; n fg that begins with 1, ends with n, and such that the SUM of each two consecutive entries being a PRIME . Rows 2 to 6 are unique, + 12 123 1234 14325 143256 (Sloane’s A051237) but there are multiple possibili- ties starting with row 7. For example, the two possibilities for row 7 are 1 ; 4 ; 3 ; 2 ; 5; 6; 7; fg and 1; 6; 5; 2; 3; 4; 7 fg : The number of possible rows ending with n /C301, 2, ..., are 0, 1, 1, 1, 1, 1, 2, 4, 7, 24, 80, ... (Sloane’s A036440). See also PASCAL’S TRIANGLE References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994. Kenney, M. J. "Student Math Notes." NCTM News Bulletin. Nov. 1986. Sloane, N. J. A. Sequences A036440 and A051237 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Prime Triplet A prime triplet is a PRIME CONSTELLATION OF THE FORM (p, p /C272; p /C276); (p, p /C274; p /C276); etc. Hardy and Wright (1979, p. 5) conjecture, and it seems almost certain to be true, that there are infinitely manyprime triplets OF THE FORM (p, p /C272 ; p /C276) and (p, p /C274; p /C276):/ Triplet Sloane First Member (p, p /C272 ; p /C276)/Sloane’s A0220045, 11, 17, 41, 101, 107, ... (p, p /C272 ; p /C278)/Sloane’sA0461343, 5, 11, 29, 59, 71, 101, ... (p, p /C272 ; p /C2712) /Sloane’sA0461355, 11, 17, 29, 41, 59, 71, ... (p, p /C274 ; p /C276) /Sloane’s A0220057, 13, 37, 67, 97, 103, ... (p, p /C274 ; p /C2710) /Sloane’s A0461363, 7, 13, 19, 37, 43, 79, ... (p, p /C274 ; p /C2712) /Sloane’sA0463177, 19, 67, 97, 127, 229, ... (p, p /C276 ; p /C278) /Sloane’sA0461385, 11, 23, 53, 101, 131, ... (p, p /C276 ; p /C2710) /Sloane’sA0461397, 13, 31, 37, 61, 73, 97, ... (p, p /C276 ; p /C2712) /Sloane’sA0461405, 7, 11, 17, 31, 41, 47, ... (p, p /C278 ; p /C2712) /Sloane’s A0461415, 11, 29, 59, 71, 89, 101, ... See also PRIME CONSTELLATION ,PRIME QUADRUPLET , TWIN PRIMES References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Rivera, C. "Problems & Puzzles: Puzzle Prime Triplets in Arithmetic Progression.-034." http://www.primepuzzles.- net/puzzles/puzz_034.htm. Prime Unit 1 and /C281 are the only INTEGERS which divide every INTEGER . They are therefore called the prime units. See also INTEGER ,PRIME NUMBER ,UNIT Prime Zeta Function The prime zeta function P(n)/C13X p1 pn; (1) where the sum is taken over PRIMES is a general- ization of the R IEMANN ZETA FUNCTION z nðÞ/C13X k /C3011 kn ; (2) where the sum is over all integers. The prime zeta function can be expressed in terms of the RIEMANN ZETA FUNCTION by ln z(n) /C30/C28X p ]2ln 1 /C28p /C28nðÞ /C30X p ]2X/C12 k/C301p /C28kn k /C30X/C12 k /C3011 kX p ]2p/C28kn /C30X/C12 k/C301P(kn) k: (3) Inverting then gives P(n) /C30X/C12 k /C301m(k) kln z(kn); (4) where m(k) is the MO¨ BIUS FUNCTION (Cohen 2000). P(1) ; The analog of the HARMONIC SERIES , diverges, but convergence of the series for n /C211 is quadratic. ARTIN’S CONSTANT CArtin is connected with P(n)by ln CArtin /C30/C28X/C12 n /C302mn /C28 1 ðÞ P(n) n; (5) where un /C30un/C281 /C27un /C282 (6) with u1 /C301 ; u2 /C303 (Ribenboim 1998, Gourdon and Sebah). The values of P(n) for the first few integers n starting with two are P(2) :0:452247 (7) P(3) :0:174763 (8) P(4) :0:0769931 (9) P(5) :0 :035755 : (10) Merrifield (1881) computed P(n) for n up to 35 to 15 digits, and Lie´nard (1948) computed P(n)upto n /C30167 to 50 digits (Ribenboim 1996). Gourdon gives values to 60 digits for 2 ]n 58 :/ See also ARTIN’S CONSTANT ,H ARMONIC SERIES , MO¨ BIUS FUNCTION ,P RIME SUMS,R IEMANN ZETA FUNCTION ,ZETA FUNCTION References Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. http://www.math.u-bordeaux.fr/~co- hen/hardylw.dvi. Gourdon, X. and Sebah, P. "Some Constants from Number Theory." http://xavier.gourdon.free.fr/Constants/Miscella- neous/constantsNumTheory.html. Hardy, G. H. and Weight, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 355 /C1/56, 1979.Lie´nard, R. Tables fondamentales a` 50 de´cimales des sommes Sn ; un ;an :/ Paris: Centre de Docum. Univ., 1948. Merrifield, C. W. "The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers." Proc. Roy. Soc. London 33,4/C1/0, 1881. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Prime-Distance Graph A DISTANCE GRAPH with distance set given by the set of prime numbers. See also DISTANCE GRAPH References Eggleton, R. B.; Erdos, P.; and Skilton, D. K. "Coloring the Real Line." J. Combin. Th. B 39,8 6/C1/00, 1985. Eggleton, R. B.; Erdos, P.; and Skilton, D. K. "Research Problem 77." Discr. Math. 58, 323, 1986. Eggleton, R. B.; Erdos, P.; and Skilton, D. K. "Coloring Prime Distance Graphs." Graphs Combin. 6,1 7/C1/2, 1990. Maehara, H. "Distance Graphs in Euclidean Space." Ryukyu Math. J. 5,3 3/C1/1, 1992. Primefree Sequence A sequence whose terms are never prime. Graham proved that there exist primefree sequences gener- ated by Fibonacci-like recurrences OF THE FORM an/C30an/C281/C27an/C282 fora1;a2 ðÞ /C301;i.e., RELATIVELY PRIME . However, the purported example given by Hoffman (1998, p. 159) in fact contains prime terms for n/C30138, 163, 190, 523, .... References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Prime-Generating Polynomial Legendre showed that there is no RATIONAL algebraic function which always gives PRIMES . In 1752, Gold- bach showed that no POLYNOMIAL with INTEGER COEFFICIENTS can give a PRIME for all integer values (Nagell 1951, p. 65; Hardy and Wright 1979, pp. 18 and 22). However, there exists a POLYNOMIAL in 10 variables with INTEGER COEFFICIENTS such that the set of PRIMES equals the set of POSITIVE values of this POLYNOMIAL obtained as the variables run through all NONNEGATIVE INTEGERS , although it is really a set of DIOPHANTINE EQUATIONS in disguise (Ribenboim 1991). Polynomial Range Sloane Reference /36n2/C28810n/C272753 /[0, 44] A050268 Fung and Ruby /47n2/C281701 n/C2710181 /[0, 42] A050267 Fung and Ruby /n2 /C27n /C2741/ [0, 39] A005846 Euler /2n2 /C2729/ [0, 28] A033542 Legendre /n2 /C27n /C2717/ [0, 15] A033541 Legendre /4n2 /C274n /C2759/ [0, 13] A048988 /2n2 /C2711/ [0, 10] A050265 /n3 /C27n2 /C2717/ [0, 10] A050266 The above table gives some low-order polynomials which generate only PRIMES for the first few NON- NEGATIVE values (Mollin and Williams 1990). The best-known of these formulas is that due to Euler (Euler 1772; Nagell 1951, p. 65; Gardner 1984, p. 83; Ball and Coxeter 1987), n2 /C27n /C2741 : (1) which gives distinct primes for the 40 consecutive integers n /C300 to 39. (/n2 /C28n /C2741 gives the same 40 primes for n /C301 to 40.) By transforming the formula to n2 /C2879n /C271601 /C30(n /C2840)2 /C27(n /C2840) /C2741; (2) primes are obtained for 80 consecutive integers, corresponding to the 40 primes given by the above formula taken twice each (Hardy and Wright 1979, p. 18). Le Lionnais (1983) has christened numbers p such that the Euler-like polynomial n2 /C27n /C27p (3) is PRIME for n /C300, 1, ..., p /C282as LUCKY NUMBERS OF EULER (where the case p /C3041 corresponds to Euler’s formula). Rabinowitz (1913) showed that for a PRIME p /C210, Euler’s polynomial represents a PRIME for n /C23 [0; p /C282] (excluding the trivial case p /C303) IFF the FIELD Qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C284pp9+=9+; has CLASS NUMBER h /C301 (Rabino- witz 1913, Le Lionnais 1983, Conway and Guy 1996). As established by Stark (1967), there are only nine numbers /C28d such that h(/C28d) /C301 (the HEEGNER NUMBERS -2, -3, -7, -11, -19, -43, -67, and -163), and of these, only 7, 11, 19, 43, 67, and 163 are of the required form. Therefore, the only LUCKY NUMBERS OF EULER are 2, 3, 5, 11, 17, and 41 (le Lionnais 1983, Sloane’s A014556), and there does not exist a better prime-generating polynomial of Euler’s form. The connection between the numbers 163 and 43 and some of the prime-rich polynomials listed above can be seen explicitly by writing x2 /C27x /C2741 /C30 x /C271 29+;k9+;72 /C27163 4 (4) x2 /C27x /C2711 /C30 x /C271 29+;k9+;72 /C2743 4 ; (5) etc.Euler also considered quadratics OF THE FORM 2x2/C27p (6) and showed this gives PRIMES forx/C23[0;p/C281] for PRIME p/C210IFFQffiffiffiffiffiffiffiffiffi/C282pp9+=9+; has CLASS NUMBER 2, which permits only p/C303, 5, 11, and 29. Baker (1971) and Stark (1971) showed that there are no such FIELDS for p/C2129. Similar results have been found for POLYNO- MIALS OF THE FORM px2/C27px/C27n (7) (Hendy 1974). See also CLASS NUMBER ,HEEGNER NUMBER ,LUCKY NUMBER OF EULER ,PRIME ARITHMETIC PROGRESSION , PRIME DIOPHANTINE EQUATIONS ,S CHINZEL’S HY- POTHESIS References Abel, U. and Siebert, H. "Sequences with Large Numbers of Prime Values." Am. Math. Monthly 100, 167/C1/69, 1993. Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers." Mathematika 13, 204/C1/16, 1966. Baker, A. "Imaginary Quadratic Fields with Class Number Two." Ann. Math. 94, 139/C1/52, 1971. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 60, 1987. Boston, N. and Greenwood, M. L. "Quadratics Representing Primes." Amer. Math. Monthly 102, 595/C1/99, 1995. Conway, J. H. and Guy, R. K. "The Nine Magic Discrimi- nants." In The Book of Numbers. New York: Springer- Verlag, pp. 224 /C1/26, 1996. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 26, 1996. Dudley, U. "History of Formula for Primes." Amer. Math. Monthly 76,2 3/C1/8, 1969. Euler, L. Nouveaux Me ´moires de l’Acade ´mie royale des Sciences. Berlin, p. 36, 1772. Forman, R. "Sequences with Many Primes." Amer. Math. Monthly 99, 548/C1/57, 1992. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 83 /C1/4, 1984. Garrison, B. "Polynomials with Large Numbers of Prime Values." Amer. Math. Monthly 97, 316/C1/17, 1990. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hendy, M. D. "Prime Quadratics Associated with Complex Quadratic Fields of Class Number 2." Proc. Amer. Math. Soc. 43, 253/C1/60, 1974. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.New York: Hyperion, pp. 108 /C1 /09, 1998. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983. Mollin, R. A. and Williams, H. C. "Class Number Problems for Real Quadratic Fields." Number Theory and Cryptol- ogy; LMS Lecture Notes Series 154, 1990. Nagell, T. "Primes in Special Arithmetical Progressions." §44 inIntroduction to Number Theory. New York: Wiley, pp. 60 and 153 /C1/55, 1951. Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahl- faktoren in quadratischen Zahlko ¨rpern." Proc. Fifth Inter- nat. Congress Math. (Cambridge) 1, 418/C1/21, 1913. Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, 1991. Sloane, N. J. A. Sequences A005846/M5273, A014556, A033541, A033542, A048988, A050265, A050266, A050267, and A050268 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Stark, H. M. "A Complete Determination of the Complex Quadratic Fields of Class Number One." Michigan Math. J. 14,1/C1/7, 1967. Stark, H. M. "An Explanation of Some Exotic Continued Fractions Found by Brillhart." In Computers in Number Theory, Proc. Science Research Council Atlas Symposium No. 2 held at Oxford, from 18 /C1/3 August, 1969 (Ed. A. O. L. Atkin and B. J. Birch). London: Academic Press, 1971. Stark, H. M. "A Transcendence Theorem for Class Number Problems." Ann. Math. 94, 153 /C1/73, 1971. Primequad PRIME QUADRUPLET Primes The set of PRIME NUMBERS , sometimes denoted P ; and implemented in Mathematica as Primes .InMathe- matica , a quantity can be tested to determine if it is in the domain of prime numbers using Element[ n, Primes], which is equivalent toPrimeQ [n]. See also PRIME NUMBER Primitive Abundant Number An ABUNDANT NUMBER for which all PROPER DIVISORS are DEFICIENT is called a primitive abundant number (Guy 1994, p. 46). The first few ODD primitive abundant numbers are 945, 1575, 2205, 3465, ... (Sloane’s A006038). See also ABUNDANT NUMBER ,D EFICIENT NUMBER , HIGHLY ABUNDANT NUMBER ,SUPERABUNDANT NUM- BER,W EIRD NUMBER References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 46, 1994. Sloane, N. J. A. Sequences A006038/M5486 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Primitive Character See also CHARACTER (NUMBER THEORY ) Primitive Element Given algebraic numbers a1 ; ..., anit is always possible to find a single ALGEBRAIC NUMBER b such that each of a1 ; ..., ancan be expressed as a polynomial in b with rational coefficients. The num- ber b is then called a primitive element of the EXTENSION FIELD Qða1 ; ...; an Þ=Q: Stated differ- ently, an ALGEBRAIC NUMBER b is a primitive element of Qða1 ; ...; an Þ=Q IFF Q a1 ; ...; an ðÞ /C30Q(b) : Primi- tive elements are implemented in Mathematica asPrimitiveElement [z,{a1, ..., an}] in the Mathema- tica add-on package NumberTheory‘PrimitiveE- lement‘ (which can be loaded with the command BBNumberTheory‘ ). For example, a primitive element of Qffiffiffi 2p ;ffiffiffi 3p9+=9+; =Q is given by b /C30ffiffiffi2p /C27ffiffiffi3p ; with ffiffiffi 2p /C30 1 2bb2 /C289 ðÞffiffiffi 3p /C301 2b 11 /C28b2ðÞ : See also EXTENSION FIELD,PRIMITIVE POLYNOMIAL , PRIMITIVE ROOT References Loos, R. "Computing in Algebraic Extensions." Computing , Suppl. 4, 173 /C1/87, 1982. Primitive Function INTEGRAL Primitive Group A GROUP that has a PRIMITIVE GROUP ACTION . See also PRIMITIVE GROUP ACTION Primitive Group Action A primitive group action is TRANSITIVE and it has no nontrivial BLOCKS .A TRANSITIVE GROUP ACTION that is not primitive is called imprimitive. A group that has a primitive group action is called a PRIMITIVE GROUP . See also BLOCK (GROUP ACTION ), GROUP ,PRIMITIVE GROUP ,S OCLE ,T RANSITIVE GROUP ,T RANSITIVE GROUP ACTION References Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Primitive Polynomial A polynomial which generates all elements of an EXTENSION FIELD from a base field is called a primitive polynomial. Primitive polynomials are also IRREDUCIBLE POLYNOMIALS . For any PRIME orPRIME POWER qand any POSITIVE INTEGER n, there exists a primitive polynomial of order nover GF( q). There are fqn/C281 ðÞ =nprimitive polynomials over GF( q), where f(n) is the TOTIENT FUNCTION . Polynomials over the FINITE FIELD GF(2) (i.e., with coefficients either 0 or 1) are primitive if they have ORDER 2n/C281;where "order" is used in the specific sense of a HAUPT-EXPONENT orORDER of a modulo. For example, x2/C27x/C271/C30x2/C27x/C271 ðÞ (x/C271)/C30x3/C271 has order 3, and is therefore primitive (Ruskey). Amaz- ingly, primitive polynomials over GF(2) define a RECURRENCE RELATION which can be used to obtain a new RANDOM bit from the n preceding ones. The numbers of primitive polynomials over GF(2) for n /C301, 2, ... are 1, 1, 2, 2, 6, 6, 18, 16, 48, ... (Sloane’s A011260). The following table lists the primitive polynomials (mod 2) of orders 1 through 5. n primitive polynomials 1 x 2 /1 /C27x /C27x2 / 3 /1 /C27x /C27x3 ; 1 /C27x2 /C27x3/ 4 /1 /C27x /C27x4 ; 1 /C27x3 /C27x4/ 5 /1 /C27x2 /C27x5 ; 1 /C27x /C27x2 /C27x3 /C27x5 ; 1 /C27x3 /C27x5 ; /1 /C27x /C27x3 /C27x4 /C27x5 ; 1 /C27x2 /C27x3 /C27x4 /C27x5 ; 1 /C27x /C27x2 /C27x4 /C27x5 / See also FINITE FIELD,IRREDUCIBLE POLYNOMIAL , ORDER (POLYNOMIAL ), POLYNOMIAL ,PRIMITIVE ELE- MENT ,PRIMITIVE ROOT References Ruskey, F. "Information on Primitive and Irreducible Poly- nomials." http://www.theory.csc.uvic.ca/~cos/inf/neck/ PolyInfo.html. Sloane, N. J. A. Sequences A011260/M0107 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Zierler, N. and Brillhart, J. "On Primitive Trinomials." Inform. Control 13, 541 /C1/44, 1968. Zierler, N. and Brillhart, J. "On Primitive Trinomials (II)." Inform. Control 14, 566 /C1/69, 1969. Primitive Polytope A POLYTOPE in n-D Euclidean space Rnwhose vertices are integer lattice points but which does not contain any other lattice points in its interior or on its boundary (Khan 1999). See also HOWE’S THEOREM ,POLYTOPE References Khan, M. R. "A Counting Formula for Primitive Tetrahedra in ." Amer. Math. Monthly 106, 525 /C1/33, 1999. Primitive Prime Factor If n ]1 is the smallest INTEGER such that Pan /C28bnj (or an /C27bn) ; then p is a primitive prime factor. See also PRIME FACTORS ,PRIMITIVE ROOT Primitive Pseudoperfect Number PRIMITIVE SEMIPERFECT NUMBER Primitive Recursive Function For-loops (which have a fixed iteration limit) are a special case of while-loops. A function which can beimplemented using only for-loops is called primitive recursive. (In contrast, a COMPUTABLE FUNCTION can be coded using a combination of for- and while-loops, or while-loops only.) The ACKERMANN FUNCTION is the simplest example of a WELL DEFINED TOTAL FUNCTION which is COMPUTA- BLE but not primitive recursive, providing a counter- example to the belief in the early 1900s that every COMPUTABLE FUNCTION was also primitive recursive (Do¨tzel 1991). See also ACKERMANN FUNCTION ,COMPUTABLE FUNC- TION ,TOTAL FUNCTION References Do¨tzel, G. "A Function to End All Functions." Algorithm: Recreational Programming 2,1 6/C1/7, 1991. Primitive Root A primitive root of a PRIME pis an INTEGER g satisfying 1 5g5p/C281 such that the residue classes ofg,g2;g3;...,gp/C281/C301 are all distinct, i.e., g(mod p) has ORDER p/C281 (Ribenboim 1996, p. 22). If pis a PRIME NUMBER , then there are exactly f(p/C281) incon- gruent primitive roots of p(Burton 1989, p. 194). More generally, if ( g;n)/C301(gandnare RELATIVELY PRIME ) and gis of ORDER f(n) modulo n, where f(n)i s the TOTIENT FUNCTION , then gis a primitive root of n (Burton 1989, p. 187). In other words, nhasgas a primitive root if gf(n)/C131 (mod n);butgkf1 (mod n) for all positive integers kBf(n):A primitive root of a number n(but not necessarily the smallest primitive root for composite n) can be computed using the Mathematica routinePrimitiveRoot [n] in the Mathematica add-on package NumberTheory‘Num- berTheoryFunctions‘ (which can be loaded with the command BBNumberTheory‘ ). Ifnhas a primitive root, then it has exactly f(f(n)) of them (Burton 1989, p. 188). For n/C301, 2, ..., the first few values of f(f(n)) are 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 4, 8, ... (Sloane’s A010554). nhas a primitive root if it is OF THE FORM 2, 4, a power pa;or twice a power 2 pa;where pis an ODD PRIME and a]1 (Burton 1989, p. 204). The first few nfor which primitive roots exist are 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 22, ... (Sloane’s A033948), so the number of primitive root of order nforn/C301, 2, ... are 0, 1, 1, 1, 2, 1, 2, 0, 2, 2, 4, 0, 4, ... (Sloane’s A046144). The smallest primitive roots for the first few primes p are 1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, ... (Sloane’s A001918). Here is table of the primitiveroots for the first few nfor which a primitive root exists (Sloane’s A046147). n /g(n)/ 21 32 43 52 ,365 73 ,5 92 ,510 3, 7 11 2, 6, 7, 8 13 2, 6, 7, 11 The largest primitive roots for n /C301, 2, ..., are 0, 1, 2, 3, 3, 5, 5, 0, 5, 7, 8, 0, 11, ... (Sloane’s A046146). The smallest primitive roots for the first few INTEGERS n are given in the following table (Sloane’s A046145), which omits n when g(n) does not exist. 2 1 38 3 94 5 158 3 3 2 41 6 97 5 162 5 4 3 43 3 98 3 163 2 5 2 46 5 101 2 166 5 6 5 47 5 103 5 167 5 7 3 49 3 106 3 169 2 9 2 50 3 107 2 173 2 10 3 53 2 109 6 178 3 11 2 54 5 113 3 179 2 13 2 58 3 118 11 181 2 14 3 59 2 121 2 191 19 17 3 61 2 122 7 193 5 18 5 62 3 125 2 194 5 19 2 67 2 127 3 197 2 22 7 71 7 131 2 199 3 23 5 73 5 134 7 202 3 25 2 74 5 137 3 206 5 26 7 79 3 139 2 211 2 27 2 81 2 142 7 214 5 29 2 82 7 146 5 218 1131 3 83 2 149 2 223 3 34 3 86 3 151 6 226 3 37 2 89 3 157 5 227 2 Let p be any ODD PRIME k ]1; and let s /C13Xp /C281 j/C301jk : (1) Then s /C30/C281 (mod p) for p /C281 ½k 0 (mod p) for p /C281¶k9+$k (2) (Ribenboim 1996, pp. 22 /C1/3). For numbers m with primitive roots, all y satisfying (p ; y) /C301 are repre- sentable as y /C13gt (mod m); (3) where t /C300, 1, ..., f(m) /C281; t is known as the index, and y is an INTEGER . Kearnes (1984) showed that for any POSITIVE INTEGER m, there exist infinitely many PRIMES psuch that mBgpBp/C28m: (4) Call the least primitive root gp:Burgess (1962) proved that gp5Cp1=4/C27e(5) forCand ePOSITIVE constants and psufficiently large (Ribenboim 1996, p. 24). Matthews (1976) obtained a formula for the "two- dimensional" Artin’s constants for the set of primes for which mandnare both primitive roots. See also ARTIN’S CONJECTURE ,A RTIN’S CONSTANT , FULL REPTEND PRIME ,M ULTIPLICATIVE ORDER ,OR- DER (MODULO ), PRIMITIVE ELEMENT ,PRIMITIVE ROOT OF UNITY References Abramowitz, M. and Stegun, C. A. (Eds.). "Primitive Roots." §24.3.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, p. 827, 1972. Burgess, D. A. "On Character Sums and L-Series." Proc. London Math. Soc. 12, 193/C1/06, 1962. Burton, D. M. "The Order of an Integer Modulo n," "Primi- tive Roots for Primes," and "Composite Numbers Having Primitive Roots." §8.1/C1/.3 in Elementary Number Theory, 4th ed. Dubuque, IA: William C. Brown Publishers, pp. 184 /C1/05, 1989. Guy, R. K. "Primitive Roots." §F9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 248 /C1/49, 1994. Kearnes, K. "Solution of Problem 6420." Amer. Math. Monthly 91, 521, 1984. Lehmer, D. H. "A Note on Primitive Roots." Scripta Math. 26, 117/C1/19, 1961. Matthews, K. R. "A Generalization of Artin’s Conjecture for Primitive Roots." Acta Arith. 29, 113 /C1/46, 1976. Nagell, T. "Moduli Having Primitive Roots." §32 in Introduc- tion to Number Theory. New York: Wiley, pp. 107 /C1/11, 1951. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 22 /C1/5, 1996. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, p. 97, 1994. Sloane, N. J. A. Sequences A001918/M0242, A010554, and A033948 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Western, A. E. and Miller, J. C. P. Tables of Indices and Primitive Roots. Cambridge, England: Cambridge Uni- versity Press, pp. xxxvii-xlii, 1968. Primitive Root of Unity A number r is an nth ROOT OF UNITY if rn /C301 and a primitive nth root of unity if, in addition, n is the smallest INTEGER of k /C301, ..., n for which rk /C301:/ See also PRINCIPAL ROOT OF UNITY,ROOT OF UNITY References Nagell, T. Introduction to Number Theory. New York: Wiley, p. 157, 1951. Primitive Semiperfect Number A SEMIPERFECT NUMBER for which none of its PROPER DIVISORS are pseudoperfect (Guy 1994, p. 46). The first few are 6, 20, 28, 88, 104, 272, ... (Sloane’s A006036). Primitive semiperfect numbers are also called primitive pseudoperfect numbers (Guy 1994, p. 46) or irreducible semiperfect numbers. There are infinitely many primitive pseudoperfect numbers which are not HARMONIC DIVISOR NUMBERS , and infinitely many ODD primitive semiperfect numbers. See also HARMONIC DIVISOR NUMBER ,P RIMARY PSEUDOPERFECT NUMBER ,SEMIPERFECT NUMBER References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 46, 1994. Sloane, N. J. A. Sequences A006036/M4133 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Primitive Sequence A SEQUENCE in which no term DIVIDES any other. Let Snbe the set f1 ; ...; n g; then the number of primitive subsets of Snare 2, 3, 5, 7, 13, 17, 33, 45, 73, 103, 205, 253, ... (Sloane’s A051026). For example, the five primitive sequences in S4are ¥;f1g;f2g; f2; 3g;f3g;f3; 4g; and f4g:/ See also NONDIVIDING SET References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 202, 1994.Sloane, N. J. A. Sequences A051026 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Primorial For the nth PRIME pn ; primorial pnðÞ/C30pn# /C13Yn j/C301pj : The values of pn# for n /C301, 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (Sloane’s A002110). The primorial satisfies the unexpected limit lim n 0/C12pn#ðÞ1 =pn/C30e (Ruiz 1997), where E is the usual base of the NATURAL LOGARITHM . /p# /C281is PRIME for PRIMES p /C303, 5, 11, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, ... (Sloane’s A006794; Guy 1994), or pnfor n /C302, 3, 5, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, ..., up to a search limit of p /C3025000 (Caldwell 1995). /p# /C271 is known to be PRIME for the PRIMES p /C302, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, ... (Sloane’s A005234; Guy 1994, Mudge 1997), or pnfor n /C301, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, ... (Sloane’s A014545), up to a search limit of p /C3025000 (Caldwell 1995). The numbers En /C30 pn# /C271 for pnthe nth prime are known as EUCLID NUMBERS . It is not known if there are an infinite number of PRIMES for which p# /C271is PRIME or COMPOSITE (Ribenboim 1989, Guy 1994). See also EUCLID NUMBER ,F ACTORIAL ,F ACTORIAL PRIME ,F ORTUNATE PRIME ,P RIME SUMS, SMARAN- DACHE NEAR-TO- PRIMORIAL FUNCTION ,TWIN PEAKS References Borning, A. "Some Results for k!/C271 and 2 /C2153/C2155/C215p/C271:/" Math. Comput. 26, 567/C1/70, 1972. Buhler, J. P.; Crandall, R. E.; and Penk, M. A. "Primes of the Form M!/C271 and 3 /C2155/C215p/C271:/"Math. Comput. 38, 639/C1/43, 1982. Caldwell, C. K. "Prime Links /C27/C27: Resources in theory: special_forms: near_products: primorial." http://primes.ut- m.edu/links/theory/special_forms/near_products/primor-ial/. Caldwell, C. "On The Primality of n!91 and 2/C2153/C2155/C1/C1/C1p91: /"Math. Comput. 64, 889/C1/90, 1995. Caldwell, C. K. "The Top Twenty: Primorial and Factorial Primes." http://www.utm.edu/research/primes/lists/top20/ PrimorialFactorial.html. Dubner, H. "Factorial and Primorial Primes." J. Rec. Math. 19, 197 /C1/03, 1987. Dubner, H. "A New Primorial Prime." J. Rec. Math. 21, 276, 1989. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 7 /C1/, 1994. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/primorial- .Z and ftp://sable.ox.ac.uk/pub/math/factors/primorial /C27.Z. Mudge, M. "Not Numerology but Numeralogy!" Personal Computer World, 279 /C1/80, 1997. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 4, 1989. Rivera, C. "Problems & Puzzles: Puzzle Primes Associated to Primorials and Factorials.-010." http://www.primepuz- zles.net/puzzles/puzz_010.htm. Ruiz, S. M. "A Result on Prime Numbers." Math. Gaz. 81, 269 /C1/70, Jul. 1997. Sloane, N. J. A. Sequences A002110/M1691, A005234/ M0669, A006794/M2474, and A014545 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Temper, M. "On the Primality of k! /C271 and 3 /C215 5 /C1/C1/C1p /C271:/" Math. Comput. 34, 303 /C1/04, 1980. Prince Rupert’s Cube The largest CUBE which can be made to pass through a given CUBE . (In other words, the CUBE having a side length equal to the side length of the largest HOLE of a SQUARE CROSS SECTION which can be cut through a unit CUBE without splitting it into two pieces.) Prince Rupert’s cube cuts a HOLE of the shape indicated in the above illustration (Wells 1991). The Prince Rupert’s cube has side length 3ffiffiffi 2p =4 : 1:0606601 ... ; and any CUBE this size or smaller can be made to pass through the original CUBE . See also CUBE,HOLE,SQUARE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Prince Rupert’s Problem." §B4 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 53 /C1/4, 1991. Cundy, H. and Rollett, A. "Prince Rupert’s Cubes." §3.15.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 157 /C1/58, 1989. Schrek, D. J. E. "Prince Rupert’s Problem and Its Extension by Pieter Nieuwland." Scripta Math. 16,73/C1/0 and 261 /C1/ 67, 1950.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 33, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 195, 1991. Prince Rupert’s Problem PRINCE RUPERT’S CUBE Principal The original amount borrowed or lent on which INTEREST is then paid or given. See also INTEREST Principal Bundle A principal bundle is a special case of a FIBER BUNDLE where the FIBER is a GROUP G. More specifically, Gis usually a L IE GROUP . A principal bundle is a TOTAL SPACE Ealong with a SURJECTIVE map p:E0Bto a BASE MANIFOLD B. Any FIBER p/C281(b) is a space ISOMORPHIC toG. More specifically, Gacts FREELY without FIXED POINT on the fibers, and this makes a fiber into a HOMOGENEOUS SPACE . For example, in the case of a CIRCLE BUNDLE (i.e., when G/C30S1/C30eitfg); the fibers are circles, which can be rotated, although no point in particular corresponds to the identity. Near every point, the fibers can be given the GROUP structure of Gin the fibers over a NEIGHBORHOOD b/C23 Bby choosing an element in each fiber to be the IDENTITY ELEMENT . However, the fibers cannot be given a group structure globally, except in the case of aTRIVIAL BUNDLE . An important principal bundle is the FRAME BUNDLE on a R IEMANNIAN MANIFOLD . This bundle reflects the different ways to give an ORTHONORMAL BASIS for TANGENT VECTORS . Consider all of the unit tangent vectors on the sphere. This is a principal bundle Eon the SPHERE with FIBER the circle S1:Every TANGENT VECTOR projects to its base point in S2;giving the map p:E0S2:Over every point in S2;there is a circle of unit tangent vectors. No particular vector is singled out as theidentity, but the group S 1of rotations acts freely without fixed point on the fibers. In a similar way, any fiber bundle corresponds to a principal bundle where the group (of the principal bundle) is the group of isomorphisms of the fiber (of the fiber bundle). Given a principal bundle p : E 0 B and an action of G on a space F, which could be a REPRESENTATION , this can be reversed to give an ASSOCIATED FIBER BUNDLE . A TRIVIALIZATION of a principal bundle, an open set U in B such that the bundle over U, p/C281(U) ; is expressed as U /C29G ; has the property that the group G acts on the left. That is, g acts on (b, h)by( b, gh). Tracing through these definitions, it is not hard to see that the TRANSITION FUNCTIONS take values in G, acting on the fibers by right multiplication. This way the action of G on a fiber is independent of coordinate chart. See also ASSOCIATED FIBER BUNDLE ,A SSOCIATED VECTOR BUNDLE ,CECH COHOMOLOGY ,CIRCLE BUN- DLE,FIBER BUNDLE ,GROUP ,HOMOGENEOUS SPACE , LIE GROUP ,TRANSITION FUNCTION ,VECTOR BUNDLE Principal Curvatures The MAXIMUM and MINIMUM of the NORMAL CURVA- TURE k1 and k2 at a given point on a surface are called the principal curvatures. The principal curvatures measure the MAXIMUM and MINIMUM bending of a REGULAR SURFACE at each point. The GAUSSIAN CURVATURE K and MEAN CURVATURE H are related to k1 and k2 by K /C30 k1 k2 (1) H /C301 2k1 /C27 k2 ðÞ : (2) This can be written as a QUADRATIC EQUATION k2 /C282H k /C27K /C300 ; (3) which has solutions k1 /C30H /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2 /C28Kp (4) k2 /C30H /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2 /C28Kp : (5) See also GAUSSIAN CURVATURE ,M EAN CURVATURE , NORMAL CURVATURE ,N ORMAL SECTION ,PRINCIPAL DIRECTION ,PRINCIPAL RADIUS OF CURVATURE ,RO- DRIGUES’ CURVATURE FORMULA References Gray, A. "Normal Curvature." §16.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 363 /C1/67, 376, and 378, 1997. Principal Curve A curve on a REGULAR SURFACE M is a principal curve IFF the velocity always points in a PRINCIPALDIRECTION , i.e., S(a?) /C30 ki a?; where S is the SHAPE OPERATOR and ki is a PRINCIPAL CURVATURE .Ifa SURFACE OF REVOLUTION generated by a plane curve is a REGULAR SURFACE , then the MERIDIANS and PARALLELS are principal curves. References Gray, A. "Principal Curves" and "The Differential Equation for the Principal Curves of a Surface." §20.1 and 28.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 459 /C1/61 and 642 /C1/44, 1997. Principal Diagonal DIAGONAL Principal Direction The directions in which the PRINCIPAL CURVATURES occur. See also PRINCIPAL DIRECTION References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 364, 1997. Principal Ideal An IDEAL I of a RING R is called principal if there is an element a of R such that I/C30aR /C30far : r /C23 Rg: In other words, the IDEAL is generated by the element a. For example, the IDEALS nZ of the RING of INTEGERS Z are all principal, and in fact all IDEALS of Z are principal. See also IDEAL ,PRINCIPAL RING,RING Principal Ideal Domain A more common way to describe a PRINCIPAL IDEAL RING . See also ALGEBRAIC NUMBER THEORY ,P RINCIPAL IDEAL RING Principal Ideal Ring See also PRINCIPAL RING Principal Normal Vector NORMAL VECTOR Principal Part If a function fhas a POLE atz0;then the negative power part X/C281 j/C30/C28kajz /C28z0 ðÞj(1) of the LAURENT SERIES of f about z0 X/C12 j/C30/C28kajz /C28z0 ðÞj(2) is called the principal part of f at z0 : For example, the principal part of z2 /C27 1 sin z3ðÞ/C30z/C283 /C27z/C282 /C271 6 z3 /C2716 z4 /C27... (3) is z/C283 /C27z/C282 (Krantz 1999, pp. 46 /C1/7). See also LAURENT POLYNOMIAL ,LAURENT SERIES References Krantz, S. G. "Principal Part of a Function." §4.3.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 46 /C1/8, 1999. Principal Quintic Form A general QUINTIC EQUATION a5x5 /C27a4x4 /C27a3x3 /C27a2x2 /C27a1x /C27a0 /C300 (1) can be reduced to one OF THE FORM y5 /C27b2y2 /C27b1y /C27b0 /C300; (2) called the principal quintic form. NEWTON’S RELATIONS for the ROOTS yj in terms of the bj/s is a linear system in the bj ; and solving for the bj/s expresses them in terms of the POWER sums snyj9+=9+; : These POWER sums can be expressed in terms of the ajs/, so the bj/s can be expressed in terms of the aj/s. For a quintic to have no quartic or cubic term, the sums of the ROOTS and the sums of the SQUARES of the ROOTS vanish, so s1yj9+=9+; /C300 (3) s2yj9+=9+; /C300: (4) Assume that the ROOTS yjof the new quintic are related to the ROOTS xj of the original quintic by yj /C30x2 j /C27 axj /C27 b: (5) Substituting this into (1) then yields two equations for a and b which can be multiplied out, simplified by using NEWTON’S RELATIONS for the POWER sums in the xj ; and finally solved. Therefore, a and b can be expressed using RADICALS in terms of the COEFFI- CIENTS aj : Again by substitution into (4), we can calculate s3yj9+=9+; ; s4yj9+=9+; and s5yj9+=9+; in terms of a and b and the xj : By the previous solution for a and b and again by using NEWTON’S RELATIONS for the POWER sums in the xj ; we can ultimately express these POWER sums in terms of the aj :/See also BRING QUINTIC FORM,NEWTON’S RELATIONS , QUINTIC EQUATION Principal Radius of Curvature At each point on a given a 2-D SURFACE , there are two "principal" RADII OF CURVATURE . The larger is de- noted R1 ; and the smaller R2 : The "principal direc- tions" corresponding to the principal radii of curvature are PERPENDICULAR to one another. In other words, the surface normal planes at the point and in the principal directions are PERPENDICULAR to one another, and both are PERPENDICULAR to the surface tangent plane at the point. See also GAUSSIAN CURVATURE ,M EAN CURVATURE , RADIUS OF CURVATURE Principal Ring A principal ring (sometimes called a principal ideal ring) is a RING in which every IDEAL is PRINCIPAL , i.e. can be generated by a single element. Examples include the ring of integers Z; any FIELD , and any polynomial ring in one variable over a FIELD . Principal rings are very useful because in a principal ring, any two nonzero elements have a WELL DEFINED GREATEST COMMON DIVISOR . Furthermore each non- zero, nonunit element in a principal ring has a unique factorization into prime elements (up to unit ele- ments). While all EUCLIDEAN RINGS are principal rings, the converse is not true. See also EUCLIDEAN RING,PRINCIPAL IDEAL References Wilson, J. C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag. 34 /C1/8, 1973. Principal Root of Unity A principal nth root v of unity is a root satisfying the equations vn /C301 and Xn/C281 i/C300vij /C300 for j /C301, 2, ..., n. Therefore, every PRIMITIVE ROOT OF UNITY of fixed degree n over a field is a principal root of unity, although this is not in general true over rings (Bini and Pan 1994, p. 11). Informally, the term "principal root" is often used to refer to the ROOT OF UNITY having smallest positive ARGUMENT . See also PRIMITIVE ROOT OF UNITY ,P RINCIPAL SQUARE ROOT,ROOT OF UNITY References Bini, D. and Pan, V. Polynomial and Matrix Computations, Vol. 1: Fundamental Algorithms. Boston, MA: Birkha ¨u- ser, 1994. Principal Square Root The unique nonnegative SQUARE ROOT of a nonnega- tive REAL NUMBER . For example, the principal square root of 9 is 3, although both -3 and 3 are square roots of 9. The concept of principal square root cannot be extended to real negative numbers since the two square roots of a negative number cannot be distin- guished until one of the two is defined as the imaginary unit, at which point /C27i and /C28i can then be distinguished. Since either choice is possible, there is no ambiguity in defining i as "the" square root of -1. See also I,PRINCIPAL ROOT OF UNITY,SQUARE ROOT Principal Value CAUCHY PRINCIPAL VALUE Principal Vector A tangent vector vp /C30v1xu /C27v2xv is a principal vector IFF detv2 2/C28v1v2v21 EFG efg2 435/C300 ; where e, f, and g are coefficients of the first FUNDA- MENTAL FORM and E, F, G of the second FUNDAMEN- TAL FORM . See also FUNDAMENTAL FORMS ,PRINCIPAL CURVE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 364, 1997. Principal Vertex A VERTEX xiof a SIMPLE POLYGON P is a principal VERTEX if the diagonal xi/C281 ; xi/C2719+$9+% intersects the boundary of P only at xi/C281 and xi/C271 :/ See also EAR,MOUTH References Meisters, G. H. "Polygons Have Ears." Amer. Math. Monthly 82, 648 /C1/51, 1975. Meisters, G. H. "Principal Vertices, Exposed Points, and Ears." Amer. Math. Monthly 87, 284 /C1/85, 1980. Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 98,31/C1/5, 1991. Principle A loose term for a true statement which may be a POSTULATE , THEOREM , etc.See also AREA PRINCIPLE ,A RGUMENT PRINCIPLE , AXIOM ,CAVALIERI’S PRINCIPLE ,CONJECTURE ,CONTI- NUITY PRINCIPLE ,C OUNTING GENERALIZED PRINCI- PLE,DIRICHLET’S BOX PRINCIPLE ,DUALITY PRINCIPLE , DUHAMEL’S CONVOLUTION PRINCIPLE ,EUCLID’S PRIN- CIPLE ,FUBINI PRINCIPLE ,H ASSE PRINCIPLE ,INCLU- SION- EXCLUSION PRINCIPLE ,I NDIFFERENCE PRINCIPLE ,INDUCTION PRINCIPLE ,INSUFFICIENT REA- SON PRINCIPLE ,LEMMA ,LOCAL- GLOBAL PRINCIPLE , MULTIPLICATION PRINCIPLE ,PERMANENCE OF MATH- EMATICAL RELATIONS PRINCIPLE ,PONCELET’S CONTI- NUITY PRINCIPLE ,PONTRYAGIN MAXIMUM PRINCIPLE , PORISM ,POSTULATE ,SCHWARZ REFLECTION PRINCI- PLE,SUPERPOSITION PRINCIPLE ,SYMMETRY PRINCI- PLE,T HEOREM ,T HOMSON’S PRINCIPLE ,T RIANGLE TRANSFORMATION PRINCIPLE ,W ELL ORDERING PRIN- CIPLE Principle of Inclusion /C1/Exclusion If A1 ; ..., M(Pn(x)) /C30an m/C300 am mm are finite sets, then k! /C271 where 2 /C215 3 /C215 5 /C215 p /C271 is the sum of the CARDINALITIES of the INTERSECTIONS of the sets taken i at a time. The principle of inclusion-exclusion was used by Nicholas Bernoulli to solve the recontres problem of finding the number of DERANGEMENTS (Bhatnagar 1995, p. 8). References Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and Their p8 ; 1(n) /C27 p8 ; 5Extensions. Ph.D. thesis. Ohio State University, 1995. Principle of Strong Induction Let D be a subset of the nonnegative integers Z /C31 with the properties that (1) the integer 0 is in D and (2) any time that n is in D, one can show that n /C271is also in D. Under these conditions, D /C30Z /C31:/ See also INDUCTION ,P RINCIPLE OF TRANSFINITE INDUCTION ,PRINCIPLE OF WEAK INDUCTION ,Z* References Se´roul, R. "Reasoning by Induction." §2.14 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 22 /C1/5, 2000. Principle of Transfinite Induction Let E be a WELL ORDERED SET and D be a subset of the nonnegative integers Z/C31 with the properties that (1) the set D contains the least element 0 of E and (2) any time that [0 ; x) ƒD; one can show that x belongs toD. Under these conditions, D/C30E. See also INDUCTION ,PRINCIPLE OF STRONG INDUC- TION ,PRINCIPLE OF WEAK INDUCTION ,Z* References Se´roul, R. "Reasoning by Induction." §2.14 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 22 /C1/5, 2000. Principle of Weak Induction Let D be a subset of the nonnegative integers Z /C31 with the properties that (1) the integer 0 is in D and (2) any time that the interval [0; n] is contained in D, one can show that n /C271 is also in D. Under these conditions, D /C30Z/C31:/ See also INDUCTION ,PRINCIPLE OF STRONG INDUC- TION ,PRINCIPLE OF WEAK INDUCTION ,Z* References Se´roul, R. "Reasoning by Induction." §2.14 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 22 /C1/5, 2000. Pringle STEINMETZ SOLID Pringsheim’s Theorem Let Cv(I) be the set of real ANALYTIC FUNCTIONS on I. Then Cv(I)isa SUBALGEBRA of C /C12(I) : A NECESSARY and SUFFICIENT condition for a function f /C23 C /C12(I)to belong to C v(I) is that f(n)(x)9+;$9+;$9+;$9+;$5knn! for n /C300, 1, ... for a suitable constant k. See also ANALYTIC FUNCTION ,SUBALGEBRA References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 207, 1980. Printer’s Errors Typesetting "errors" in which exponents or multi- plication signs are omitted but the resulting expres- sion is equivalent to the original one. Examples include 2592 /C302592 (1) 34425 /C3034425 (2) 312325 /C30312325 (3) and 25 /C21525 31 /C30252531; (4) where a whole number followed by a fraction is interpreted as a MIXED FRACTION (e.g., 11 2 /C301 /C2712 /C3032): D. Wilson computed all possible errors obtained by dropping exponents in a product for bases 2 to 15 and numbers 5264 :24 /C30246 (5) 33 /C30338 (6) 51232874 /C30512328749 : (7) Wilson also gave 11292450 A0A812 and 372B9A83000000000012 ; where the two digit base- b satisfies pq /C30pb /C27q (8) and for which there exist an infinite number of examples. See also ANOMALOUS CANCELLATION ,PROOFREADING MISTAKES References Dudeney, H. E. Amusements in Mathematics. New York: Dover, 1970. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 174 /C1/75, 1979. Prior Distribution BAYESIAN ANALYSIS Priority Queue A data structure designed to allow repeated extrac- tion of the smallest remaining key (Skiena 1990, p. 38). See also HEAP,QUEUE References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Prism An oblique prism is a POLYHEDRON with two con- gruent POLYGONAL faces and all remaining faces PARALLELOGRAMS (left figure). A right prism is a prism in which the top and bottom polygons lie on top of each other so that the vertical polygonsconnecting their sides are not only PARALLELOGRAMS , but RECTANGLES (right figure). The prisms have particularly simple nets, given by two oppositely-oriented n-gonal bases connected by a ribbon of n squares. The VOLUME of a prism of height h and base area A is simply V /C30Ah: The above figure shows the first few regular right prisms, whose faces are regular n-gons. The 4-prism is simply the CUBE . The simple prisms and antiprisms include the decagonal antiprism, decagonal prism, hexagonal antiprism, hexagonal prism, octagonal antiprism, octagonal prism, pentagonal antiprism, pentagonal prism, square antiprism, and triangular prism. The DUAL POLYHEDRON of a simple (Archime- dean) prism is a DIPYRAMID . The unit regular right prism has volume given by Vn /C301 /C215 An /C301 4 n cotp n ! ; where Anis the AREA of the corresponding REGULAR POLYGON , and SURFACE AREA Sn /C302An /C27n /C215 12 /C30n 1 /C2712cotp n !"# : The triangular prism, square prism (cube), and hexagonal prism are all SPACE-FILLING POLYHEDRA . See also ANTIPRISM ,AUGMENTED HEXAGONAL PRISM , AUGMENTED PENTAGONAL PRISM ,AUGMENTED TRIAN- GULAR PRISM ,B IAUGMENTED PENTAGONAL PRISM , BIAUGMENTED TRIANGULAR PRISM ,CUBE,DIPYRAMID , HEXAGONAL PRISM ,M ETABIAUGMENTED HEXAGONAL PRISM ,OCTAGONAL PRISM ,PARABIAUGMENTED HEX- AGONAL PRISM ,P ENTAGONAL PRISM ,P RISMATOID ,PRISMOID ,T RAPEZOHEDRON ,T RIANGULAR PRISM , TRIAUGMENTED HEXAGONAL PRISM ,TRIAUGMENTED TRIANGULAR PRISM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 85 /C1/6, 1997. Harris, J. W. and Stocker, H. "Prism." §4.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 96 /C1/8, 1998. Kern, W. F. and Bland, J. R. "Prism." §13 in Solid Mensura- tion with Proofs, 2nd ed. New York: Wiley, pp. 28 /C1/2, 1948. Pedagoguery Software. Poly . http://www.peda.com/poly/. Weisstein, E. W. "SolidGeometry." MATHEMATICA NOTEBOOK SOLIDGEOMETRY.M . Prismatic Ring AM O¨ BIUS STRIP with finite thickness. See also MO¨ BIUS STRIP References Gardner, M. "Twisted Prismatic Rings." Ch. 5 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 76 /C1/7, 1992. Prismatoid A POLYHEDRON having two POLYGONS in PARALLEL planes as bases and TRIANGULAR or TRAPEZOIDAL lateral faces with one side lying in one base and the opposite VERTEX or side lying in the other base. Examples include the CUBE , PYRAMIDAL FRUSTUM , RECTANGULAR PARALLELEPIPED , PRISM , and PYRAMID . Let A1be the AREA of the lower base, A2 the AREA of the upper base, M the AREA of the midsection, and h the ALTITUDE . Then V/C3016hA1/C274M/C27A2 ðÞ : See also GENERAL PRISMATOID ,P ARALLELEPIPED , PRISMATOID THEOREM ,PRISMOID ,PYRAMIDAL FRUS- TUM,RECTANGULAR PARALLELEPIPED References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 128 and 132, 1987. Harris, J. W. and Stocker, H. "Prismoid, Prismatoid." §4.5.1 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 102, 1998. Kern, W. F. and Bland, J. R. "Prismatoid," "Prismatoid Theorem," "Proof of the Prismoidal Formula," and "Appli- cation of Prismatoid Theorem." §30 and 43 /C1/5in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 75 /C1/0 and 121 /C1/30, 1948. Prismatoid Theorem The VOLUME of a PRISMATOID is equal to the sum of the volumes of a PYRAMID ,aWEDGE , and a PARALLE- LEPIPED . See also GENERAL PRISMATOID ,PRISMOID References Kern, W. F. and Bland, J. R. "Prismatoid Theorem," "Proof of the Prismoidal Formula," and "Application of Prisma- toid Theorem." §43 /C1/5in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 121 /C1/30, 1948. Prismoid A PRISMATOID having planar sides and the same number of vertices in both of its parallel planes. The faces of a prismoid are therefore either TRAPEZOIDS or PARALLELOGRAMS . Ball and Coxeter (1987) use the term to describe an ANTIPRISM . See also ANTIPRISM ,PRISM ,PRISMATOID References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 130, 1987. Prisoner’s Dilemma A problem in GAME THEORY first discussed by A. Tucker. Suppose each of two prisoners A and B, who are not allowed to communicate with each other, is offered to be set free if he implicates the other. If neither implicates the other, both will receive the usual sentence. However, if the prisoners implicate each other, then both are presumed guilty and granted harsh sentences. A DILEMMA arises in deciding the best course of action in the absence of knowledge of the other prisoner’s decision. Each prisoner’s best strategy would appear to be to turn the other in (since if A makes the worst- case assumption that B will turn him in, then B will walk free and Awill be stuck in jail if he remains silent). However, if the prisoners turn each other in, they obtain the worst possible outcome for both. See also DILEMMA ,TIT-FOR- TATReferences Axelrod, R. The Evolution of Cooperation. New York: Basic- Books, 1985. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 164 /C1/65, 1998. Goetz, P. "Phil’s Good Enough Complexity Dictionary." http://www.cs.buffalo.edu/~goetz/dict.html. Prizes MATHEMATICS PRIZES Probability Probability is the branch of mathematics which studies the possible outcomes of given events togetherwith their relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events)will occur expressed on a linear scale from 0 (impos-sibility) to 1 (certainty), also expressed as a PERCEN- TAGE between 0 and 100%. The analysis of events governed by probability is called STATISTICS . There are several competing interpretations of theactual "meaning" of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation),while BAYESIANS treat probability more subjectively as a statistical procedure which endeavors to esti-mate parameters of an underlying distribution basedon the observed distribution. A properly normalized function which assigns a probability "density" to each possible outcome within some interval is called a PROBABILITY FUNCTION , and its cumulative value (integral for a continuous dis- tribution or sum for a discrete distribution) is called a DISTRIBUTION FUNCTION . Probabilities are defined to obey certain assumptions, called the PROBABILITY AXIOMS . Let a SAMPLE SPACE contain the UNION (/@) of all possible events Ei;so S/C13/C160N i/C301Ei9+;89+;9 ; (1) and let EandFdenote subsets of S. Further, let F?/C30 not-Fbe the complement of F, so that F@F?/C30S: (2) Then the set Ecan be written as E/C30ESS/C30ES(F@F?)/C30(ESF)@(ESF?); (3) whereSdenotes the intersection. Then P(E)/C30P(ESF)/C27P(ESF?)/C28P[(ESF)S(ESF?)] /C30P(ESF)/C27P(ESF?)/C28P[(FSF?)S(ESE)] /C30P(ESF)/C27P(ESF?)/C28P(¥SE) /C30P(ESF)/C27P(ESF?)/C28P(¥) /C30P(E S F) /C27P(E S F ?) ; (4) where ¥ is the EMPTY SET. Let P(E ½F) denote the CONDITIONAL PROBABILITY of E given that F has already occurred, then P(E) /C30P(E ½F)P(F) /C27P(E ½F ?)P(F ?) (5) /C30P(E½F)P(F) /C27P(E ½F ?)[1 /C28P(F)] (6) P(A S B) /C30P(A)P(B½A) (7) /C30P(B)P(A½B) (8) P(A?S B) /C30P(A?)P(B ½A?) (9) P(E½F) /C30P(E S F) P(F): (10) The relationship P(A S B) /C30P(A)P(B) (11) holds if A and B are independent events. A very important result states that P(E @ F) /C30P(E) /C27P(F) /C28P(E S F) ; (12) which can be generalized to P /C160n i/C301Ai9+;89+;9 /C30X iPAiðÞ/C28X ij? PAi @ Aj9+=9+; /C27X i; j; kƒ PAi S Aj S Ak9+=9+; /C28... /C27/C281ðÞn/C281P þn i/C301Ai ! : ð13Þ See also BAYES’ FORMULA ,CONDITIONAL PROBABIL- ITY,C OUNTABLE ADDITIVITY PROBABILITY AXIOM , DISTRIBUTION FUNCTION ,E QUALLY LIKELY OUT- COMES DISTRIBUTION ,INDEPENDENT STATISTICS , LIKELIHOOD ,P ROBABILITY AXIOMS ,P ROBABILITY FUNCTION ,P ROBABILITY INEQUALITY ,S TATISTICAL DISTRIBUTION ,STATISTICS Probability Axioms Given an event E in a SAMPLE SPACE S which is either finite with N elements or countably infinite with N /C30 /C12 elements, then we can write S /C13/C160N i/C301Ei9+;89+;9 ; and a quantity P(Ei) ; called the PROBABILITY of event Ei ; is defined such that1. 0 5PEiðÞB1:/ 2. P(S) /C301 :/ 3. Additivity: PE1 @ E2 ðÞ /C30PE1ðÞ/C27PE2ðÞ ; where E1 and E2 are mutually exclusive. 4. Countable additivity: P @ n i /C301Ei ðÞ /C30an i/C301 PEiðÞ for n /C301, 2, ..., N where E1 ; E2 ; ...are mutually exclusive (i.e., E1SE2/C30¥):/ See also EXPERIMENT ,OUTCOME ,PROBABILITY ,SAM- PLE SPACE ,TRIAL,UNION References Doob, J. L. "The Development of Rigor in Mathematical Probability (1900 /C1/950)." Amer. Math. Monthly 103, 586/C1/ 95, 1996. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 26 /C1/8, 1984. Probability Density Function PROBABILITY FUNCTION Probability Distribution Function PROBABILITY FUNCTION Probability Function The probability function P(x) (also called the prob- ability density or density function) of a continuous distribution is defined as the derivative of the (cumulative) DISTRIBUTION FUNCTION D(x); D?(x)/C30[P(x)]x /C28/C12/C30P(x)/C28P(/C28/C12)/C30P(x); (1) so D(x)/C30P(X5x)/C13gx /C28/C12P(y)dy: (2) A probability function satisfies P(x/C23B)/C30gBP(x)dx (3) and is constrained by the normalization condition, P(/C28/C12B xB/C12)/C30g/C12 /C28/C12P(x)dx/C131: (4) Special cases are P(a5x5b)/C30gb aP(x)dx (5) P(a5x5a/C27da)/C30ga/C27da aP(x)dx:P(a)da (6) P(x/C30a)/C30ga aP(x)dx/C300: (7) To find the probability function in a set of trans- formed variables, find the J ACOBIAN . For example, If u/C30u(x);then Pu du /C30Px dx ; (8) so Pu /C30Px@x @u9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$: (9) Similarly, if u /C30u(x; y) and v /C30v(x; y) ; then P u ; v /C30Px ; y@(x; y) @(u ; v)9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$9+;$: (10) Given the MOMENTS of a distribution (/m; s; and the GAMMA STATISTICS gr); the asymptotic probability function is given by P(x) /C30Z(x) /C281 6 g1Z(3)(x)hi /C271 24 g2Z(4)(x) /C271 72 g2 1Z(6)(x)hi /C281 120 g3Z(5)(x) /C271 144 g1 g2Z(7)(x) /C271 1296 g31Z(9)(x)hi /C271 720 g4Z(6)(x) /C271 1152 g22 /C271 720 g1 g39+;k9+;7 Z(8)(x)h /C271 1728 g21 g2Z(10)(x) /C271 31104 g41Z(12) ðxÞ/C138/C27... ; (11) where Z(x) /C301 sffiffiffiffiffiffi 2pp e /C28(x/C28 m)2 =2s2 (12) is the NORMAL DISTRIBUTION , and gr /C30kr sr/C272 (13) for r ]1 (with krCUMULANTS and s the STANDARD DEVIATION ; Abramowitz and Stegun 1972, p. 935). See also CONTINUOUS DISTRIBUTION ,CORNISH- FISHER ASYMPTOTIC EXPANSION ,D ISCRETE DISTRIBUTION , DISTRIBUTION FUNCTION ,JOINT DISTRIBUTION FUNC- TION References Abramowitz, M. and Stegun, C. A. (Eds.). "Probability Functions." Ch. 26 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 925 /C1/64, 1972. McLaughlin, M. "Common Probability Distributions." http:// www.geocities.com/~mikemclaughlin/math_stat/Dists/ Compendium.html. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 94, 1984. Probability Inequality If B ‡A (B is a SUPERSET of A), then P(A) 5P(B) :/Probability Integral a(x) /C131ffiffiffiffiffiffi 2ppgx /C28xe /C28t2 =2 dt (1) /C30ffiffiffi 2 ps gx 0e /C28t2 =2 dt (2) /C302 F(x) (3) /C30erfxffiffiffi 2p ! ; (4) where F(x) is the NORMAL DISTRIBUTION FUNCTION and ERF is the error function. See also ERF,NORMAL DISTRIBUTION FUNCTION Probability Measure Consider a PROBABILITY SPACE specified by the triple (S; S; P) ; where (S; S)isa MEASURABLE SPACE , with S the domain and S is its measurable subsets, and P is a MEASURE on S with P(S) /C301: Then the MEASURE P is said to be a probability measure. Equivalently, P is said to be normalized. See also MEASURABLE SPACE ,M EASURE ,PROBABIL- ITY,P ROBABILITY SPACE ,R ADON MEASURE ,STATE SPACE Probability Space A triple ( S;S;P) on the domain S, where ( S;S)i sa MEASURABLE SPACE ,Sare the measurable subsets of S, and Pis a MEASURE onSwith P(S)/C301:/ See also MEASURABLE SPACE ,M EASURE ,PROBABIL- ITY,P ROBABILITY MEASURE ,R ANDOM VARIABLE , STATE SPACE References Papoulis, A. "Probability Space." §2 /C1/ in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 24 /C1/3, 1984. Probable Error The first QUARTILE of a standard NORMAL DISTRIBU- TION occurs when gt 0F(z) dz /C301 4 : The solution is t /C300:6745... : The value of t giving 1=4 is known as the probable error of a NORMALLY DISTRIBUTED variate. However, the number d corre- sponding to the 50% CONFIDENCE INTERVAL , P( d) /C131 /C282g½d ½ 0f(t) dt /C3012 ; is sometimes also called the probable error. See also SIGNIFICANCE Probable Prime A number satisfying FERMAT’S LITTLE THEOREM (or some other primality test) for some nontrivial base. A probable prime which is shown to be COMPOSITE is called a PSEUDOPRIME (otherwise, of course, it is a PRIME ). See also PRIME NUMBER ,PSEUDOPRIME Problem A problem is an exercise whose solution is desired. Mathematical "problems" may therefore range from simple puzzles to examination and contest problems to propositions whose proofs require insightful ana- lysis. There are many UNSOLVED PROBLEMS in mathe- matics. Two famous problems which have recently been solved include FERMAT’S LAST THEOREM (by Andrew Wiles) and the KEPLER CONJECTURE (by T. C.Hales). Among the most prominent of remaining unsolved problems are the GOLDBACH CONJECTURE , RIEMANN HYPOTHESIS ,POINCARE ´CONJECTURE , the conjecture that there are an infinite number of TWIN PRIMES , as well as many more. K.S. Brown, D. Epp- stein, S. Finch, and C. Kimberling maintain exten- sive pages of unsolved problems in mathematics. See also UNSOLVED PROBLEMS References Artino, R. A.; Gaglione, A. M.; and Shell, N. The Contest Problem Book IV: Annual High School MathematicsExaminations 1973 /C1/982. Washington, DC: Math. Assoc. Amer., 1982. Alexanderson, G. L.; Klosinski, L.; and Larson, L. The William Lowell Putnam Mathematical Competition, Pro- blems and Solutions: 1965 /C1/984. Washington, DC: Math. Assoc. Amer., 1986. Barbeau, E. J.; Moser, W. O.; and Lamkin, M. S. Five Hundred Mathematical Challenges. Washington, DC: Math. Assoc. Amer., 1995. Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, 1964. Brown, K. S. "Most Wanted List of Elementary Unsolved Problems." http://www.seanet.com/~ksbrown/mwlist.htm. Chung, F. and Graham, R. Erdos on Graphs: His Legacy of Unsolved Problems. New York: A. K. Peters, 1998. Cover, T. M. and Gopinath, B. (Eds.). Open Problems in Communication and Computation. New York: Springer- Verlag, 1987. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991. Dixon, J. D. Problems in Group Theory. New York: Dover, 1973. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965. Dudeney, H. E. Amusements in Mathematics. New York: Dover, 1917. Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949. Dudeney, H. E. 536 Puzzles & Curious Problems. New York: Scribner, 1967. Eppstein, D. "Open Problems." http://www.ics.uci.edu/~epp- stein/junkyard/open.html. Erdos, P. "Some Combinatorial Problems in Geometry." In Geometry and Differential Geometry (Ed. R. Artzy and I. Vaisman). New York: Springer-Verlag, pp. 46 /C1/3, 1980. Fenchel, W. (Ed.). "Problems." In Proc. Colloquium on Convexity, 1965. Københavns Univ. Mat. Inst., pp. 308 /C1/ 25, 1967. Finch, S. "Unsolved Mathematical Problems." http:// www.mathsoft.com/asolve/. Gleason, A. M.; Greenwood, R. E.; and Kelly, L. M. The William Lowell Putnam Mathematical Competition, Pro-blems and Solutions: 1938 /C1 /964. Washington, DC: Math. Assoc. Amer., 1980. Graham, L. A. Ingenious Mathematical Problems and Meth- ods. New York: Dover, 1959. Graham, L. A. The Surprise Attack in Mathematical Pro- blems. New York: Dover, 1968. Greitzer, S. L. International Mathematical Olympiads, 1959/C1/977. Providence, RI: Amer. Math. Soc., 1978. Gruber, P. M. and Schneider, R. "Problems in Geometric Convexity." In Contributions to Geometry: Proceedings of the Geometry-Symposium Held in Siegen, June 28, 1978 toJuly 1, 1978 (Ed. J. To ¨lke and J. M. Wills.) Boston, MA: Birkha ¨user, pp. 255 /C1 /78, 1979. Guy, R. K. (Ed.). "Problems." In The Geometry of Metric and Linear Spaces. New York: Springer-Verlag, pp. 233 /C1/44, 1974. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 21, 1994. Halmos, P. R. Problems for Mathematicians Young and Old. Washington, DC: Math. Assoc. Amer., 1991. Hardy, K. and Williams, K. S. The Green Book of Mathema- tical Problems. New York: Dover, 1997. Hardy, K. and Williams, K. S. The Red Book of Mathema- tical Problems. New York: Dover, 1996. Herman, J.; Kucera Radan, K.; and Simsa, J. Equations and Inequalities: Elementary Problems and Theorems in Alge-bra and Number Theory. New York: Springer-Verlag, 2000. Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., 1973. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., 1976. Honsberger, R. Mathematical Morsels. Washington, DC: Math. Assoc. Amer., 1979. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., 1991. Honsberger, R. From Erdos to Kiev. Washington, DC: Math. Assoc. Amer., 1995. Honsberger, R. In Po´lya’s Footsteps: Miscellaneous Problems and Essays. Washington, DC: Math. Assoc. Amer., 1997. Honsberger, R. (Ed.). Mathematical Plums. Washington, DC: Math. Assoc. Amer., 1979. Inter-IREM Commission. History of Mathematics: Histories of Problems. Paris: Ellipses, 1997. Jacoby, O. and Benson, W. H. Intriguing Mathematical Problems. New York: Dover, 1998. Kimberling, C. "Unsolved Problems and Rewards." http:// cedar.evansville.edu/~ck6/integer/unsolved.html. Klee, V. "Some Unsolved Problems in Plane Geometry." Math. Mag. 52, 131 /C1/45, 1979. Klamkin, M. S. International Mathematical Olympiads, 1978 /C1/985 and Forty Supplementary Problems. Washing- ton, DC: Math. Assoc. Amer., 1986. Klamkin, M. S. U.S.A. Mathematical Olympiads, 1972 /C1/ 986. Washington, DC: Math. Assoc. Amer., 1988. Kordemsky, B. A. The Moscow Puzzles: 359 Mathematical Recreations. New York: Dover, 1992. Kurschak, J. and Hajos, G. Hungarian Problem Book, Based on the Eotvos Competitions, Vol. 1: 1894 /C1/905. New York: Random House, 1963. Kurschak, J. and Hajos, G. Hungarian Problem Book, Based on the Eotvos Competitions, Vol. 2: 1906 /C1/928. New York: Random House, 1963. Larson, L. C. Problem-Solving Through Problems. New York: Springer-Verlag, 1983. Meschkowski, H. Unsolved and Unsolvable Problems in Geometry. London: Oliver & Boyd, 1966. Mott-Smith, G. Mathematical Puzzles for Beginners and Enthusiasts, 2nd rev. ed. New York: Dover, 1954. Ogilvy, C. S. Tomorrow’s Math: Unsolved Problems for the Amateur. New York: Oxford University Press, 1962. Ogilvy, C. S. "Some Unsolved Problems of Modern Geome- try." Ch. 11 in Excursions in Geometry. New York: Dover, pp. 143 /C1/53, 1990. Posamentier, A. S. and Salkind, C. T. Challenging Problems in Algebra. New York: Dover, 1997. Posamentier, A. S. and Salkind, C. T. Challenging Problems in Geometry. New York: Dover, 1997. Rabinowitz, S. (Ed.). Index to Mathematical Problems 1980 /C1/ 984. Westford, MA: MathPro Press, 1992. Reid, L. "Southwest Missouri State University’s Problem Corner." http://www.math.smsu.edu/~les/POTW.html. Salkind, C. T. The Contest Problem Book I: Problems from the Annual High School Contests 1950 /C1/960. New York: Random House, 1961. Salkind, C. T. The Contest Problem Book II: Problems from the Annual High School Contests 1961 /C1/965. Washington, DC: Math. Assoc. Amer., 1966. Salkind, C. T. and Earl, J. M. The Contest Problem Book III: Annual High School Contests 1966 /C1/972. Washington, DC: Math. Assoc. Amer., 1973. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993. Shkliarskii, D. O.; Chentzov, N. N.; and Yaglom, I. M. The U.S.S.R. Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics. New York: Dover, 1993.Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964. Sierpinski, W. Problems in Elementary Number Theory. New York: Elsevier, 1980. Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993. Steinhaus, H. One Hundred Problems in Elementary Mathe- matics. New York: Dover, 1979. Tietze, H. Famous Problems of Mathematics. New York: Graylock Press, 1965. Trigg, C. W. Mathematical Quickies: 270 Stimulating Pro- blems with Solutions. New York: Dover, 1985. Ulam, S. M. A Collection of Mathematical Problems. New York: Interscience Publishers, 1960. Vakil, R. A Mathematical Mosaic: Patterns and Problem Solving. Washington, DC: Math. Assoc. Amer., 1997. van Mill, J. and Reed, G. M. (Eds.). Open Problems in Topology. New York: Elsevier, 1990. Weisstein, E. W. "Books about Mathematics Problems." http://www.treasure-troves.com/books/MathematicsPro- blems.html. Procedure A specific prescription for carrying out a task or solving a problem. Also called an ALGORITHM , METHOD ,o r TECHNIQUE See also BISECTION PROCEDURE ,M AEHLY’S PROCE- DURE Proclus’ Axiom If a LINE intersects one of two parallel lines, it must intersect the other also. This AXIOM is equivalent to the PARALLEL AXIOM . References Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 inJourney through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 54, 1990. Procrustian Stretch HYPERBOLIC ROTATION Product The term "product" refers to the result of one or more MULTIPLICATIONS . For example, the mathematical statement a/C29b/C30cwould be read " aTIMES bEQUALS c," where cis the product. The product symbol is defined by Yn i/C301fi/C13f1/C215f2/C1/C1/C1fn: Useful product identities include lnY/C12 i/C301fi ! /C30X/C12 i/C301lnfi Y/C12 i/C301fi/C30expX/C12 i/C301lnfi ! : For 0 5ai B1; then the productsQ/C12 i /C3011 /C27ai ðÞ andQ/C12 i /C3011 /C28ai ðÞ converge and diverge asQ/C12i/C301ai :/ See also CAUCHY PRODUCT ,C ROSS PRODUCT ,D OT PRODUCT ,INNER PRODUCT ,JORDAN PRODUCT ,M A- TRIX PRODUCT ,M ULTIPLICATION ,N ONASSOCIATIVE PRODUCT ,OUTER PRODUCT ,SUM,TENSOR PRODUCT , TIMES ,VECTOR TRIPLE PRODUCT References Guy, R. K. "Products Taken over Primes." §B87 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 102 /C1/03, 1994. Product Formula Let a be a NONZERO RATIONAL NUMBER a /C30 9pa1 1 pa2 2/C1/C1/C1paL L ; where p1 ; ..., pLare distinct PRIMES , al /C23Z and al "0: Then ½a ½Y p prime½ a½p /C30p a1 1 p a2 2/C1/C1/C1p aL L p /C28 a1 1p /C28a2 2/C1/C1/C1p/C28 aL L/C301 : References Burger, E. B. and Struppeck, T. "Does a/C12 n/C3001 nReally Con- verge? Infinite Series and p-adic Analysis." Amer. Math. Monthly 103, 565 /C1/77, 1996. Product Log Function LAMBERT’S W-FUNCTION Product Neighborhood TUBULAR NEIGHBORHOOD Product Rule The DERIVATIVE identity d dx [f(x)g(x)] /C30lim h00f(x /C27 h)g(x /C27 h) /C28 f(x)g(x) h /C30lim h00f(x /C27 h)g(x /C27 h) /C28 f(x /C27 h)g(x) h" /C27f(x /C27 h)g(x) /C28 f(x)g(x) h9+$; /C30lim h00f(x /C27h)g(x /C27 h) /C28 g(x) h" /C27g(x)f(x /C27 h) /C28 f(x) h/C138/C30f(x)g?(x) /C27g(x)f ?(x) : See also CHAIN RULE,EXPONENT LAWS,Q UOTIENT RULEReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Product Set CARTESIAN PRODUCT Product Space AC ARTESIAN PRODUCT equipped with a "product topology" is called a product space (or product topological space, or direct product). See also CARTESIAN PRODUCT References Iyanaga, S. and Kawada, Y. (Eds.). "Product Spaces." §408L Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1281 /C1/282, 1980. ProductLog LAMBERT’S W-FUNCTION Product-Moment Coefficient of Correlation CORRELATION COEFFICIENT Program A precise sequence of instructions designed to accom- plish a given task. The implementation of an ALGO- RITHM on a computer using a programming language is an example of a program. See also ALGORITHM Projection A projection is the transformation of POINTS and LINES in one PLANE onto another PLANE by connecting corresponding points on the two planes with PARAL- LEL lines. This can be visualized as shining a (point) light source (located at infinity) through a translu- cent sheet of paper and making an image of whatever is drawn on it on a second sheet of paper. The branch of geometry dealing with the properties and invar- iants of geometric figures under projection is called PROJECTIVE GEOMETRY . The projection of a VECTOR a onto a VECTOR u is given by projua /C30a /C215 u ½u ½2u ; where a /C215 u is the DOT PRODUCT , and the length of this projection is ½projua½/C30½a /C215 u ½ ½u ½: General projections are considered by Foley and VanDam (1983). The average projected area over all orientations of any ELLIPSOID is 1/4 the total SURFACE AREA . This theorem also holds for any convex solid. See also BICENTRIC PERSPECTIVE ,DOT PRODUCT ,MAP PROJECTION ,P OINT- PLANE DISTANCE ,P ROJECTION MATRIX ,PROJECTION OPERATOR ,PROJECTION THEO- REM,PROJECTION (VECTOR SPACE ), PROJECTIVE COL- LINEATION ,P ROJECTIVE GEOMETRY ,R EFLECTION , SHADOW ,STEREOLOGY ,TRIP-LET References Casey, J. "Theory of Projections." Ch. 11 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 349 /C1/67, 1893. Foley, J. D. and VanDam, A. Fundamentals of Interactive Computer Graphics, 2nd ed. Reading, MA: Addison- Wesley, 1990.Projection (Vector Space) If W is a k-dimensional subspace of a vector space V with inner product ;hi; then it is possible to project vectors from V to W. The most familiar projection is when W is the X-AXIS in the plane. In this case, P(x; y) /C30(x; 0) is the projection. This projection is an orthogonal projection. If the SUBSPACE W has an ORTHONORMAL BASIS fw1 ; ...; wk g then projW(v) /C30Xk i/C301v; wi hi wi is the orthogonal projection onto W. Any vector v /C23 V can be written uniquely as v /C30vW /C27vW /C222; where vW /C23 W and vW /C222 is in the ORTHOGONAL SUBSPACE W /C222:/ A projection is always a LINEAR TRANSFORMATION and can be represented by a PROJECTION MATRIX .In addition, for any projection, there is an inner product for which it is an orthogonal projection. See also IDEMPOTENT ,INNER PRODUCT ,PROJECTION MATRIX ,ORTHOGONAL SET,PROJECTION ,SYMMETRIC MATRIX ,VECTOR SPACE Projection Matrix A projection matrix Pis an n/C29nSQUARE MATRIX that gives a PROJECTION from Rnto a subspace W. The columns of Pare the projections of the standard basis vectors, and Wis the image of P:ASQUARE MATRIX P is a projection matrix iff P2/C30P:/ The following Mathematica function will test if a matrix is a projection matrix. ProjectionMatrixQ[a_List?MatrixQ] : /C30(a.a /C30/C30a) A projection matrix is a SYMMETRIC MATRIX iff the PROJECTION is orthogonal. In an orthogonal projec- tion, any vector vcan be written v/C30vW/C27vW/C222;so v;Pw hi /C30vW;Pw hi /C30Pv;w hi : (1) An example of a nonsymmetric projection matrix is P /C3001 019+$=9+$; ; (2) which projects onto the line y /C30x. The case of a COMPLEX VECTOR SPACE is analogous. A projection matrix is a HERMITIAN MATRIX iff the PROJECTION satisfies v ; Pw hi /C30 vW ; Pw hi /C30 Pv ; w hi ; (3) where the INNER PRODUCT is the HERMITIAN INNER PRODUCT . Projection operators play a role in quantum mechanics and quantum computing. The following Mathematica function gives the Hermitian projection matrix onto a complex subspace, given a basis. BBLinearAlgebra‘Orthogonalization‘; HermProjectMatrixOntoBasis[a_List?MatrixQ] : /C30 Module[{a1 /C30 GramSchmidt[a, InnerProduct - /C21 (#1.Conjugate[#2] &) ]}, Transpose[a1].a1] ] Any vector in W is fixed by the projection matrix Pw /C30w for any w in W. Consequently, a projection matrix P has norm equal to one, unless P /C300; ½½P½½/C30sup ½x ½/C301½Px½51: (4) See also IDEMPOTENT ,INNER PRODUCT ,PROJECTION (VECTOR SPACE ), ORTHOGONAL SET,SYMMETRIC MA- TRIX Projection Operator ˜p /C13 fi(x) ji fi(t) hj ˜pX jcj fj(t)9+;$9+;$9+;; /C30ci fi(x) ji X ifi(x) ji fi(x) hj/C301: See also BRA,KET Projection Theorem Let H be a HILBERT SPACE and M a closed subspace of H. Corresponding to any vector x /C23 H ; there is a unique vector m0 /C23 M such that ½½x /C28m0 ½½5½½x /C28m½½ for all m /C23 M : Furthermore, a necessary and sufficient condition that m0 /C23 M be the unique minimizing vector is that x /C28m0 be orthogonal to M (Luenberger 1997, p. 51). This theorem can be viewed as a formalization of the result that the closest POINT on a PLANE to a point noton the PLANE can be found by dropping a perpendi- cular. See also POINT- PLANE DISTANCE References Luenberger, D. G. Optimization by Vector Space Methods. New York: Wiley, 1997. Projective Algebraic Variety See also ALGEBRAIC VARIETY ,HODGE CONJECTURE Projective Collineation A COLLINEATION which transforms every 1-D form projectively. Any COLLINEATION which transforms one range into a projectively related range is a projective collineation. Every PERSPECTIVE COLLINEA- TION is a projective collineation. See also COLLINEATION ,ELATION ,HOMOLOGY (GEO- METRY ), PERSPECTIVE COLLINEATION References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 247 /C1/48, 1969. Projective Correlation Any CORRELATION which transforms one range into a projectively related PENCIL (or vice versa). See also CORRELATION (GEOMETRIC ), PENCIL References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 248, 1969. Projective General Linear Group The projective general linear group PGLn(q) is the GROUP obtained from the GENERAL LINEAR GROUP GLn(q) on factoring the scalar MATRICES contained in that group. See also GENERAL LINEAR GROUP ,PROJECTIVE GEN- ERAL ORTHOGONAL GROUP ,P ROJECTIVE GENERAL UNITARY GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GLn(q); SLn(q) ; PGLn(q) ; and PSLn(q) /C30Ln(q) :/" §2.1 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. Projective General Orthogonal Group The projective general orthogonal group PGOn(q)is the GROUP obtained from the GENERAL ORTHOGONAL GROUP GOn(q) on factoring the scalar MATRICES contained in that group.See also G ENERAL ORTHOGONAL GROUP ,PROJECTIVE GENERAL LINEAR GROUP ,PROJECTIVE GENERAL UNI- TARY GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GOn(q) ; SOn(q) ; PGOn(q); and PSOn(q) ; and On(q) :/" §2.4 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xi-xii, 1985. Projective General Unitary Group The projective general unitary group PGUn(q) is the GROUP obtained from the GENERAL UNITARY GROUP GUn(q) on factoring the scalar MATRICES contained in that group. See also GENERAL UNITARY GROUP ,P ROJECTIVE GENERAL LINEAR GROUP ,P ROJECTIVE GENERAL ORTHOGONAL GROUP ,PROJECTIVE GENERAL UNITARY GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GUn(q) ; SUn(q); PGUn(q); and PSUn(q) /C30Un(q) :/" §2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. Projective Geometry The branch of GEOMETRY dealing with the properties and invariants of geometric figures under PROJEC- TION . In older literature, projective geometry is some- times called "higher geometry," "geometry of position," or "descriptive geometry" (Cremona 1960, pp. v-vi). The most amazing result arising in projective geo- metry is the DUALITY PRINCIPLE , which states that a duality exists between theorems such as PASCAL’S THEOREM and BRIANCHON’S THEOREM which allows one to be instantly transformed into the other. More generally, all the propositions in projective geometry occur in dual pairs, which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "POINT " and "LINE." The AXIOMS of projective geometry are:1. If A and B are distinct points on a PLANE , there is at least one LINE containing both A and B. 2. If A and B are distinct points on a PLANE , there is not more than one LINE containing both A and B. 3. Any two LINES in a PLANE have at least one point of the PLANE (which may be the POINT AT INFINITY in common. 4. There is at least one LINE on a PLANE . 5. Every LINE contains at least three points of the PLANE . 6. All the points of the PLANE do not belong to the same LINE (Veblen and Young 1910 /C1/8, Kasner and Newman 1989). See also COLLINEATION ,DESARGUES’ THEOREM ,FUN- DAMENTAL THEOREM OF PROJECTIVE GEOMETRY , INVOLUTION (LINE), PENCIL ,PERSPECTIVITY ,PROJEC- TION ,PROJECTIVITY ,R ANGE (LINE SEGMENT ), SEC- TION (PENCIL ) References Birkhoff, G. and Mac Lane, S. "Projective Geometry." §9.14 inA Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 275 /C1/79, 1996. Casey, J. "Theory of Projections." Ch. 11 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most RecentExtensions, with Numerous Examples, 2nd ed., rev. enl.Dublin: Hodges, Figgis, & Co., pp. 349 /C1 /67, 1893. Chasles, M. Aperc ¸u historique. Chasles, M. Traite ´de Ge ´ome´trie supe ´rieure. Paris, 1852. Coxeter, H. S. M. Projective Geometry, 2nd ed. New York: Springer-Verlag, 1987. Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960. Kadison, L. and Kromann, M. T. Projective Geometry and Modern Algebra. Boston, MA: Birkha ¨user, 1996. Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Microsoft Press, pp. 150 /C1/51, 1989. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 119 /C1/27, 1893. Ogilvy, C. S. "Projective Geometry." Ch. 7 in Excursions in Geometry. New York: Dover, pp. 86 /C1/10, 1990. Pappas, T. "Art & Projective Geometry." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 66 /C1/ 7, 1989. Pedoe, D. and Sneddon, I. A. An Introduction to Projective Geometry. New York: Pergamon, 1963. Poncelet, J.-V. Traite ´des Proprie ´te´s Projectives. Paris, 1822. Reye. Geometrie der Lage, 2nd ed. Hannover, Germany, 1877. Semple, J. G. Algebraic Projective Geometry. Oxford, Eng- land: Oxford University Press, 1998. Seidenberg, A. Lectures in Projective Geometry. Princeton, NJ: Van Nostrand, 1962. Staudt, K. G. C. von. Geometrie der Lage. Nu¨rnberg, Ger- many, 1847. Steiner, J. Systematische Entwicklung der Abha ¨ngigkeit geometrischer Gestalten von einander. Berlin, 1832. Struik, D. Lectures on Projected Geometry. Reading, MA: Addison-Wesley, 1998. Veblen, O. and Young, J. W. Projective Geometry, 2 vols. Boston, MA: Ginn, 1910 /C1/8. Weisstein, E. W. "Books about Projective Geometry." http:// www.treasure-troves.com/books/ProjectiveGeome- try.html. Whitehead, A. N. The Axioms of Projective Geometry. New York: Hafner, 1960. Projective Plane A projective plane is derived from a usual PLANE by addition of a LINE AT INFINITY . Just as a straight line in projective geometry contains of single POINT AT INFINITY at which the endpoints meet, a plane in projective geometry contains a single LINE AT INFI- NITY at which the edges of the PLANE meet. A projective plane can be constructed by gluing both pairs of opposite edges of a RECTANGLE together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in 3-D space without crossing itself. A finite projective plane of order n is formally defined as a set of n2 /C27n /C271 POINTS with the properties that: 1. Any two POINTS determine a LINE, 2. Any two LINES determine a POINT , 3. Every POINT has n /C271 LINES on it, and 4. Every LINE contains n /C271 POINTS . (Note that some of these properties are redundant.) A projective plane is therefore a SYMMETRIC (/n2 /C27n /C271; n /C271 ; 1) BLOCK DESIGN .An AFFINE PLANE of order n exists IFF a projective plane of order n exists. A finite projective plane exists when the order n is a POWER of a PRIME , i.e., n /C30pa for a ]1: It is con- jectured that these are the only possible projective planes, but proving this remains one of the most important unsolved problems in COMBINATORICS . The first few orders which are powers of primes are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, ... (Sloane’s A000961). The first few orders which are not of this form are 6, 10, 12, 14, 15, ... (Sloane’s A024619). The smallest finite projective plane is of order n /C302, and consists of the 73CONFIGURATION known as the FANO PLANE . The remarkable BRUCK- RYSER-CHOWLA THEOREM says that if a projective plane of order n exists, and n /C301 or 2 (mod 4), then n is the sum of two SQUARES . This rules out n /C306. By answering LAM’S PROBLEM in the negative using massive computer calculations on top of some mathematics, it has been proved that there are no finite projective planes of order 10 (Lam 1991). The status of the order 12 projective plane remains open. The projective plane of order 2, also known as the FANO PLANE , is denoted PG(2, 2). It has INCIDENCE MATRIX1110000 10011001000011 0101010 0100101001100100101102 6666666643 777777775: Every row and column contains 3 1s, and any pair of rows/columns has a single 1 in common. The projective plane has EULER CHARACTERISTIC 1, and the HEAWOOD CONJECTURE therefore shows that any set of regions on it can be colored using six colors only (Saaty 1986). The Petersen graph provides a 6- color coloring of the PROJECTIVE PLANE . See also AFFINE PLANE ,B LOCK DESIGN ,B RUCK- RYSER- CHOWLA THEOREM ,C ONFIGURATION ,F ANO PLANE ,LAM’S PROBLEM ,M AP COLORING ,M OUFANG PLANE ,PROJECTIVE PLANE PK2,PROJECTIVE SPACE , REAL PROJECTIVE PLANE ,SYMMETRIC BLOCK DESIGN References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 281 /C1/87, 1987. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 243, 1976. Bruck, R. H. and Ryser, H. J. "The Nonexistence of Certain Finite Projective Planes." Canad. J. Math. 1,88/C1/3, 1949. Lam, C. W. H. "The Search for a Finite Projective Plane of Order 10." Amer. Math. Monthly 98, 305 /C1/18, 1991. Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. Pinkall, U. "Models of the Real Projective Plane." Ch. 6 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 63 /C1/7, 1986. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 45, 1986. Sloane, N. J. A. Sequences A000961/M0517 and A024619 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 72 and 195 /C1/97, 1991. Projective Plane Dissection Virtually nothing is known about dissection of a PROJECTIVE PLANE using unequal squares. See also CYLINDER DISSECTION ,KLEIN BOTTLE DIS- SECTION ,M O¨ BIUS STRIP DISSECTION ,P ERFECT SQUARE DISSECTION ,TORUS DISSECTION References Stewart, I. "Squaring the Square." Sci. Amer. 277,94/C1/6, July 1997. Projective Plane PK2 The 2-D SPACE consisting of the set of TRIPLES f(a ; b ; c):a ; b ; c /C23 K ; not all zero g; where triples which are SCALAR multiples of each other are identified. See also PROJECTIVE PLANE Projective Space A SPACE which is invariant under the GROUP G of all general LINEAR homogeneous transformation in the SPACE concerned, but not under all the transforma- tions of any GROUP containing G as a SUBGROUP . A projective space is the space of 1-D VECTOR SUBSPACES of a given VECTOR SPACE . For REAL VECTOR SPACES , the NOTATION RPn or Pn denotes the REAL projective space of dimension n (i.e., the SPACE of 1-D VECTOR SUBSPACES of Rn/C271) and CPndenotes the COMPLEX projective space of COMPLEX dimension n (i.e., the space of 1-D COMPLEX VECTOR SUBSPACES of Cn/C271) : Pn can also be viewed as the set consisting of Rn together with its POINTS AT INFINITY . See also PROJECTIVE SPACE Projective Special Linear Group The projective special linear group PSLn(q) is the GROUP obtained from the SPECIAL LINEAR GROUP SLn(q) on factoring by the SCALAR MATRICES con- tained in that GROUP .Itis SIMPLE for n ]2 except for PSL2(2) /C30S3 ; PSL3(3) /C30A4 ; and is therefore also denoted Ln(Q):/ See also PROJECTIVE SPECIAL ORTHOGONAL GROUP , PROJECTIVE SPECIAL UNITARY GROUP ,SPECIAL LINE- AR GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GLn(q); SLn(q) ; PGLn(q); and PSLn(q) /C30Ln(q) :/" §2.1 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. Projective Special Orthogonal Group The projective special orthogonal group PSOn(q)is the GROUP obtained from the SPECIAL ORTHOGONAL GROUP SOn(q) on factoring by the SCALAR MATRICES contained in that GROUP . In general, this GROUP is not SIMPLE .See also PROJECTIVE SPECIAL LINEAR GROUP ,PRO- JECTIVE SPECIAL UNITARY GROUP ,SPECIAL ORTHO- GONAL GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GOn(q) ; SOn(q) ; PGOn(q) ; and PSOn(q) ; and On(q) :/" §2.4 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xi-xii, 1985. Projective Special Unitary Group The projective special unitary group PSUn(q) is the GROUP obtained from the SPECIAL UNITARY GROUP SUn(q) on factoring by the SCALAR MATRICES con- tained in that GROUP . PSUn(q)is SIMPLE except for PSU2(2) /C30S3 PSU2(3) /C30A4 PSU3(2) /C3032 : Q8 ; so it is given the simpler name Un(q) ; with U2(q) /C30L2(q) :/ See also PROJECTIVE SPECIAL LINEAR GROUP ,PRO- JECTIVE SPECIAL ORTHOGONAL GROUP ,SPECIAL UNI- TARY GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GUn(q);SUn(q);PGUn(q); and PSUn(q)/C30Un(q):/"§2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. Projective Symplectic Group The projective symplectic group PSpn(q) is the GROUP obtained from the SYMPLECTIC GROUP Spn(q)o n factoring by the SCALAR MATRICES contained in that GROUP .PSp2m(q)i s SIMPLE except for psp2(2)/C30s3 psp2(3)/C30a4 psp4(2)/C30s6; so it is given the simpler name s2m(q);with / s2ðqÞ¼l2ðqÞ/. References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups spn(q) and pspnðqÞ¼snq:/" §2.3 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. x-xi, 1985. Projective Variety PROJECTIVE ALGEBRAIC VARIETY Projectivity The product of any number of PERSPECTIVITIES . See also INVOLUTION (TRANSFORMATION ), PERSPEC- TIVITY Projectivization Given a VECTOR SPACE V, its projectivization P(V); sometimes written P(V /C280); is the set of EQUIVA- LENCE CLASSES x /C2 lx for any l "0in V /C280: For example, COMPLEX PROJECTIVE SPACE has HOMOGE- NEOUS COORDINATES [x0 ; ... ; xn]; with not all xi /C300 :/ The projectivization is a MANIFOLD with one less dimension than V. In fact, it is covered by the n /C271 affine COORDINATE CHARTS , U0 /C30f[1; x1 ; ...; xn]g; ...; Un /C30f[x0 ; ...; xn /C281 ; 1]g: See also COMPLEX PROJECTIVE SPACE ,M ANIFOLD , VECTOR SPACE Prolate Cycloid The path traced out by a fixed point at a RADIUS b /C21a, where a is the RADIUS of a rolling CIRCLE , also sometimes called an EXTENDED CYCLOID . The prolate cycloid contains loops, and has PARAMETRIC EQUA- TIONS x /C30af /C28b sin f (1) y /C30a /C28b cos f : (2) The ARC LENGTH from f /C300is s/C302(a/C27b)E(u); (3) where sin1 2f9+;k9+;7 /C30snu (4) k2/C304ab (a/C27c)2: (5) See also CURTATE CYCLOID ,CYCLOID ,TROCHOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 216, 1987.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 325, 1998. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 194 /C1/97, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 146, 1967. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 147 /C1/48, 1999. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 292, 1995. Prolate Cycloid Evolute The EVOLUTE of the PROLATE CYCLOID is given by x¼a½/C282bfþ2afcosf/C282asinf/C27bsinð2fÞ/C138 2ðacosf/C28bÞ y/C30a(a/C28bcosf)2 b(acosf/C28b): Prolate Spheroid ASPHEROID which is "pointy" instead of "squashed," i.e., one for which the polar radius cis greater than the equatorial radius a,s o c/C21a(called "spindle- shaped ellipsoid" by Tietze 1965, p. 27). A symme- trical egg (i.e., with the same shape at both ends) would approximate a prolate spheroid. A prolatespheroid is a SURFACE OF REVOLUTION obtained by rotating an ELLIPSE about its major axis (Hilbert and Cohn-Vossen 1999, p. 10), and has Cartesian equa-tions x2/C27y2 a2/C27z2 c2/C301: (1) The ELLIPTICITY of the prolate spheroid is defined by e /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28 a2 c2s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28 a2p c/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28a2 c2s ; (2) so that 1 /C28e2 /C30a2 c2 : (3) Then r /C30a 1 /C27e2 1 /C28 e2sin2 d !/C281 =2 : (4) The SURFACE AREA of a prolate spheroid can be computed as a SURFACE OF REVOLUTION about the Z- AXIS, S /C302 pg r(z)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27[r ?(z)]2q dz (5) with radius as a function of z given by r(z) /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28z c !2vuut: (6) The INTEGRAND is then rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27r ?2p /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27(a /C28 c)(a /C27 c)z2 c4s ; (7) and the integral is given by S ¼ 2pagc /C28cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þða /C28 c Þða þ c Þz2 c4s dz ¼ 2 pa2 þ2pac2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28 a2p sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28 a2p c ! : ð8Þ Using the identity ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28a2p /C30ce (9) gives S /C302pa2 /C272 pac esin/C281 e (10) (Beyer 1987, p. 131). Note that this is the conven- tional form in which the surface area of an prolate spheroid is written, although it is formally equivalent to the conventional form for the OBLATE SPHEROID via the identity c2 p e(a ; c)ln1 /C27 e(a ; c) 1 /C28 e(a ; c)"# /C302 pac e(c ; a)sin/C281[e(c ; a)]; (11) where e(x; y) is defined bye(x;y)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 y2s : (12) The VOLUME of an prolate spheroid can be computed from the formula for a general ELLIPSOID with b/C30a, V/C304 3pa2c (13) (Beyer 1987, p. 131). See also DARWIN-DE SITTER SPHEROID ,E LLIPSOID , LEMON ,O BLATE SPHEROID ,P ROLATE SPHEROIDAL COORDINATES ,SPHERE ,SPHEROID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 10, 1999. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, p. 27, 1965. Wrinch, D. M. "Inverted Prolate Spheroids." Philos. Mag. 280, 1061/C1/070, 1932. Prolate Spheroidal Coordinates A system of CURVILINEAR COORDINATES in which two sets of coordinate surfaces are obtained by revolving the curves of the ELLIPTIC CYLINDRICAL COORDINATES about the X-AXIS , which is relabeled the Z-AXIS . The third set of coordinates consists of planes passingthrough this axis. x/C30asinh jsinhcosf (1) y/C30asinh jsinhsinf (2) z/C30acosh jcosh; (3) where j/C23[0;/C12);h/C23[0;p];andf/C23[0;2p):Note that several conventions are in common use; Arfken (1970)uses ( u;v; 8) instead of ( j;h;f);and Moon and Spencer (1988, p. 28) use ( h;u;c):/ In this coordinate system, the SCALE FACTORS are hj/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh2j/C27sin2hq (4) hh/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh2j/C27sin2hq (5) hf /C30a sinh j sin h : (6) The LAPLACIAN is 92f /C301 sin h sinh j(sin2 h /C27 sinh2 j) /C29@ @ jsin h sinh j@f @ j ! /C27@ @ hsin h sinh j@f @ h ! ( /C27@ @ f9+$= (csch j sin h /C27csc h sinh j)@f @ f9+$;9+$7 : (7) ¼1 sin2 h /C27 sinh2 j(csc2 h /C27csch2 j)@2f @ j2 /C27cot h@f @ h" /C27@2f @ h2 /C27coth j@f @ j /C27@2f @ j29+$; (8) An alternate form useful for "two-center" problems is defined by j1 /C30cosh j (9) j2 /C30cos h (10) j3 /C30 f; (11) where j1 /C23 [1;/C12] ; j2 /C23 [/C281 ; 1]; and j3 /C23 [0; 2p) (Abra- mowitz and Stegun 1972). In these coordinates, z /C30aj1 j2 (12) x /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 1 /C2819+=9+; 1 /C28 j229+=9+;q cos j3 (13) y /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j21 /C2819+=9+; 1 /C28 j229+=9+;q sin j3 : (14) In terms of the distances from the two FOCI, j1 /C30r1 /C27 r2 2a (15) j2 /C30r1 /C28 r2 2a (16) 2a /C30r12 : (17) The SCALE FACTORS are hj1/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j21 /C28 j22 j21 /C28 1s (18) hj2/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j21 /C28 j22 1 /C28 j22s (19) hj3/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j21 /C2819+=9+; 1 /C28 j229+=9+;q ; (20) and the LAPLACIAN is92f /C301 a21 j21 /C28 j22@ @ j1j2 1/C2819+=9+; @f @j1"# ( /C271 j2 1/C28j22@ @j21/C28j229+=9+; @f @"# /C271 j21/C2819+=9+; 1/C28j229+=9+;@2f dj229+$7 : (21) The H ELMHOLTZ DIFFERENTIAL EQUATION is separable in prolate spheroidal coordinates. See also HELMHOLTZ DIFFERENTIAL EQUATION– PRO- LATE SPHEROIDAL COORDINATES ,L ATITUDE ,L ONG- ITUDE ,O BLATE SPHEROIDAL COORDINATES , SPHERICAL COORDINATES References Abramowitz, M. and Stegun, C. A. (Eds.). "Definition of Prolate Spheroidal Coordinates." §21.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972. Arfken, G. "Prolate Spheroidal Coordinates ( u,v,f):/"§2.10 inMathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103 /C1/07, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics,with Applications to Problems in Mathematical Physics.New York: Dover, pp. 243 /C1 /44, 1959. Moon, P. and Spencer, D. E. "Prolate Spheroidal Coordi- nates ( h;u;c):/" Table 1.06 in Field Theory Handbook, Including Coordinate Systems, Differential Equations,and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 28 /C1 /0, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953. Wrinch, D. M. "Inverted Prolate Spheroids." Philos. Mag. 280, 1061/C1/070, 1932. Prolate Spheroidal Wave Function The WAVE EQUATION inPROLATE SPHEROIDAL COORDI- NATES is 92F/C27k2F/C30@ @j1j2 1/C2819+=9+; @F @j1"# /C27@ @j21/C28j229+=9+; @F @j2"# /C27j21/C28j22 j21/C2819+=9+; 1/C28x2 2 ðÞ@2F @f2/C27c2j2 1/C28j229+=9+; F/C300;(1) where c/C131 2ak: (2) Substitute in a trial solution F/C30Rmn(c;j1)Smn(c;j2)cos sin(mf) (3) d dj1j2 1/C2819+=9+; d dj1Rmn(c;j1)"# /C28 lmn /C28c2 j2 1 /C27m2 j21 /C28 1 ! Rmn(c ; j1) /C300 : (4) The radial differential equation is d dj2j22 /C2819+=9+; d dj2Smn(c; j2)"# /C28 lmn /C28c2 j22 /C27m2 j22 /C28 1 ! Rmn(c ; j2) /C300: (5) and the angular differential equation is d dj21 /C28 j229+=9+; d dj2Smn(c ; j2)"# /C28 lmn /C28c2 j22 /C27m2 1 /C28 j22 ! Rmn(c ; j2) /C300: (6) Note that these are identical (except for a sign change). The prolate angular function of the first kind is given by S(1) mn /C30P/C12 r/C301 ; 3 ; ...dr(c)Pm m/C27r(h) for n /C28m oddP/C12 r/C300 ; 2 ; ... dr(c)Pm m/C27r( h) for n /C28m even ;9+$k (7) where Pk k( h) is an associated LEGENDRE POLYNOMIAL . The prolate angular function of the second kind is given by S(2) mn /C30P/C12 r/C30...;/C281 ; 1 ; 3 ; ...dr(c)Qm m/C27r( h) for n /C28m oddP/C12 r/C30...;/C282 ; 0 ; 2 ; ... dr(c)Qm m/C27r(h) for n /C28m even ;9+$k (8) where Qm k ( h) is an associated LEGENDRE FUNCTION OF THE SECOND KIND and the COEFFICIENTS dr satisfy the RECURRENCE RELATION akdk /C272 /C27( bk /C28 lmn)dk /C27 gkdk /C282 /C300; (9) with ak /C30(2m /C27 k /C27 2)(2m /C27 k /C27 1)c2 (2m /C27 2k /C27 3)(2m /C27 2k /C27 3)(10) bk /C30(m /C27k)(m /C27k /C271) /C272(m /C27 k)(m /C27 k /C27 1) /C28 2m2 /C28 1 (2m /C27 2k /C28 1)(2m /C27 2k /C27 3)c2(11) gk /C30k(k /C28 1)c2 (2m /C27 2k /C28 3)(2m /C27 2k /C28 1) : (12) Various normalization schemes are used for the ds (Abramowitz and Stegun 1972, p. 758). Meixner and Scha¨fke (1954) use g1 /C281[Smn(c;h)]2dh/C302 2n/C271(n/C27m)! (n/C28m)!: (13) Stratton et al. (1956) use(n/C27m)! (n/C28m)!/C30P/C12 r/C301;3;/C1/C1/C1(r/C272m)! r!drforn/C28modd P/C12r/C300;2;...(r/C272m)! r!drforn/C28meven :8 >>>< >>>: (14) Flammer (1957) uses S mn(c;0)/C30Pm/C271 n(0) for n/C28modd Pm n(0) for n/C28meven :9+$k (15) See also OBLATE SPHEROIDAL WAVE FUNCTION , SPHEROIDAL WAVE FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Spheroidal Wave Functions." Ch. 21 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 751 /C1/59, 1972. Flammer, C. Spheroidal Wave Functions. Stanford, CA: Stanford University Press, 1957. Meixner, J. and Scha ¨fke, F. W. Mathieusche Funktionen und Spha ¨roidfunktionen. Berlin: Springer-Verlag, 1954. Rhodes, D. R. "On the Spheroidal Functions." J. Res. Nat. Bur. Standards--B. Math. Sci. 74B, 187/C1/09, Jul.-Sep. 1970. Stratton, J. A.; Morse, P. M.; Chu, L. J.; Little, J. D. C.; and Corbato ´,F . J . Spheroidal Wave Functions. New York: Wiley, 1956. Pronic Number AFIGURATE NUMBER OF THE FORM Pn/C302Tn/C30n(n/C27 1);where Tnis the nthTRIANGULAR NUMBER . The first few are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, ... (Sloane’s A002378). The GENERATING FUNCTION of the pronic numbers is 2x (1/C28x)3/C302x/C276x2/C2712x3/C2720x4/C27... Kausler (1805) was one of the first to tabulate pronicnumbers, creating a list up to n/C301000 (Dickson 1952, Vol. 1, p. 357; Vol. 2, p. 233). Pronic numbersare also known as oblong or heteromecic numbers. McDaniel (1998ab) proved that the only pronic Fibonacci numbers are F 0/C300 and F3/C302;and the only pronic Lucas number is L0/C302;rediscovering a result first published by Ming (1995). The first few nfor which Pnare PALINDROMIC are 1, 2, 16, 77, 538, 1621, ... (Sloane’s A028336), and the first few PALINDROMIC NUMBERS which are pronic are 2, 6, 272, 6006, 289982, ... (Sloane’s A028337). References De Geest, P. "Palindromic Products of Two Consecutive Integers." http://www.ping.be/~ping6758/consec.htm. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 357, 1952. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 6, 232 /C1/33, 350, and 407, 1952. Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag 63,3/C1/0, 1990. McDaniel, W. L. "Pronic Fibonacci Numbers." Fib. Quart. 36,56/C1/9, 1998. McDaniel, W. L. "Pronic Lucas Numbers." Fib. Quart. 36, 60 /C1/2, 1998. Ming, L. "Nearly Square Numbers in the Fibonacci and Lucas Sequences" [Chinese]. J. Chongqing Teachers Col- lege, No. 4, 1 /C1/, 1995. Sloane, N. J. A. Sequences A002378/M1581, A028336, and A028337 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Kausler, C. F. Nova Acta Acad. Petrop. 14, 268 /C1/89, ad annos 1797 /C1/, 1805. Proof A rigorous mathematical argument which unequivo- cally demonstrates the truth of a given PROPOSITION . A mathematical statement which has been proven is called a THEOREM . According to Hardy (1999, pp. 15 /C1/6), "all physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite unin- teresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply." There is some debate among mathematicians as to just what constitutes a proof. The FOUR-COLOR THEO- REM is an example of this debate, since its "proof" relies on an exhaustive computer testing of many individual cases which cannot be verified "by hand." While many mathematicians regard computer-as- sisted proofs as valid, some purists do not. There are several computer systems currently under devel- opment for automated theorem proving, among them, TH //C215/OREM //C214:/ See also DEEP THEOREM ,P ARADOX ,P ROPOSITION , Q.E.D, REDUCTIO AD ABSURDUM THEOREM ,TRIVIAL References Aigner, M. and Ziegler, G. M. Proofs from the Book. New York: Springer-Verlag, 1999. Allenby, R. Numbers and Proofs. Oxford, England: Oxford University Press, 1997. Benson, D. C. The Moment of Proof: Mathematical Epipha- nies. Oxford, England: Oxford University Press, 1999. Garnier, R. and Taylor, J. 100% Mathematical Proof. New York: Wiley, 1996. Hardy, G. H. "Mathematical Proof." Mind 38,1/C1/5, 1929. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. OMEGA. "Welcome to Omega, the Mathematical Proof Assistant." http://www.ags.uni-sb.de/~omega/primer/.Po´lya, G. How to Solve It: A New Aspect of Mathematical Method, 2nd ed. Princeton, NJ: Princeton University Press, 1988. Po´lya, G. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, 2 vols. in One. New York: Wiley, 1981. Po´lya, G. Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics. Princeton, NJ: Princeton University Press, 1990. Po´lya, G. Mathematics and Plausible Reasoning, Vol. 2: Patterns of Plausible Inference. Princeton, NJ: Princeton University Press, 1990. Krantz, S. G. Techniques of Problem Solving. Providence, RI: Amer. Math. Soc., 1997. Solow, D. How to Read and Do Proofs: An Introduction to Mathematical Thought Process, 2nd ed. New York: Wiley, 1990. TH //C215/OREM //C214 Computer-Supported Mathematical Theorem Proving. http://www.theorema.org. Vakil, R. A Mathematical Mosaic: Patterns and Problem Solving. Washington, DC: Math. Assoc. Amer., 1997. Wickelgren, W. A. How to Solve Mathematical Problems: Elements of a Theory of Problems and Problem Solving. New York: Dover, 1995. Proofreading Mistakes If proofreader A finds a mistakes and proofreader B finds b mistakes, c of which were also found by A, how many mistakes were missed by both A and B? Assume there are a total of m mistakes, so proof- reader A finds a FRACTION a=m of all mistakes, and also a FRACTION c =b of the mistakes found by B. Assuming these fractions are the same, then solving for m gives m /C30ab c: The number of mistakes missed by both is therefore approximately N/C30m/C28a/C28b/C27c/C30(a/C28c)(b/C28c) c: See also PRINTER’S ERRORS References Po´lya, G. "Probabilities in Proofreading." Amer. Math. Monthly ,83, 42, 1976. Propeller A4 - POLYHEX . References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, p. 147, 1978. Proper Class A CLASS which is not a SET. See also CLASS (SET), ORDINAL NUMBER ,SET Proper Cover Proper covers are defined as COVERS of a set X which do not contain the entire set X itself as a subset (Macula 1994). Of the five covers of f1; 2 g; namely ff1g;f2 gg;ff1; 2gg;ff1 g;f1 ; 2 gg;ff2g;f1; 2gg; and ff1 g;f2g;f1; 2gg; only ff1g;f2gg does not contain the subset f1; 2g and so is the unique proper cover of two elements. In general, the number of proper covers for a set of N elements is ½C?(N) ½/C30½C(N) ½/C281 4 22N /C301 2XN k/C300(/C281)k N k9+;89+;9 22N /C28k"# /C2822N 4; the first few of which are 0, 1, 45, 15913, 1073579193, ... (Sloane’s A007537). See also COVER ,MINIMAL COVER References Macula, A. J. "Covers of a Finite Set." Math. Mag. 67, 141 /C1/ 44, 1994. Sloane, N. J. A. Sequences A007537/M5287 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Proper Divisor A positive proper divisor is a positive DIVISOR of a number n, excluding n itself. For example, 1, 2, and 3 are positive proper divisors of 6, but 6 itself is not. The number of proper divisors of n is therefore given by s0(n) /C13 s0(n) /C281; where sk(n) is the DIVISOR FUNCTION . For n /C301, 2, ..., s0(n) is therefore given by 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, ... (Sloane’s A032741). The largest proper divisors of n /C302, 3, ... are 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, ... (Sloane’s A032742). The term "proper divisor" is sometimes includes negative integer divisors of a number n excluding /C28n: Using this definition, -3, -2, -1, 1, 2, and 3 are the proper divisors of 6, while /C286 and 6 are the IMPROPER DIVISORS . To make matters even more confusing, the proper divisor is often defined so that -1 and 1 are also excluded. Using this alternative definition, the proper divisors of 6 would then be -3, -2, 2, and 3, and the IMPROPER DIVISORS would be /C286; -1, 1, and 6.See also ALIQUANT DIVISOR ,ALIQUOT DIVISOR ,DIVI- SOR,IMPROPER DIVISOR References Sloane, N. J. A. Sequences A032741 and A032742 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Proper Fraction A FRACTION p=q B1 : A fraction p=q /C211 is called an IMPROPER FRACTION . See also FRACTION ,M IXED FRACTIO N,IMPROPER FRACTION ,REDUCED FRACTION Proper Integral An INTEGRAL which has neither limit INFINITE and from which the INTEGRAND does not approach INFI- NITY at any point in the range of integration. See also IMPROPER INTEGRAL ,INTEGRAL Proper k-Coloring K-COLORING Proper Subfield See also FIELD,SUBFIELD Proper Subset A SUBSET which is not the entire SET. For example, consider a SET f1; 2; 3; 4; 5g: Then f1; 2; 4g and f1g are proper subsets, while f1; 2; 6g and f1; 2; 3; 4; 5g are not. See also SET,SUBSET Proper Superset A SUPERSET which is not the entire SET. See also SET,SUPERSET Proper Value EIGENVALUE Proper Vector EIGENVECTOR Property P A KNOT having the property that no surgery could possibly yield a counterexample to the POINCARE ´ CONJECTURE is said to satisfy Property P (Adams 1994, p. 262). See also POINCARE ´ CONJECTURE References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. Proportional If a is (directly) proportional to b, then a =b is a constant. The relationship is written a 8b ; which implies a /C30cb; for some constant c. See also DIRECTLY PROPORTIONAL ,INVERSELY PRO- PORTIONAL Proportional-Integral-Derivative Method A very useful active feedback method for controlling things like temperature control systems, servo mo- tors, and flow control valves. Proposition A statement which is to be proved. Propositional Calculus The formal basis of LOGIC dealing with the notion and usage of words such as "NOT," "OR," "AND," and "IMPLIES ." Many systems of propositional calculus have been devised which attempt to achieve consis- tency, completeness, and independence of AXIOMS . The term "sentential calculus" is sometimes used as a synonym for propositional calculus. See also CONNECTIVE ,LOGIC , P-SYMBOL ,PREDICATE CALCULUS References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 254 /C1/55, 1989. Mendelson, E. "The Propositional Calculus." Ch. 1 in Intro- duction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 12 /C1/4, 1997. Nidditch, P. H. Propositional Calculus. New York: Free Press of Glencoe, 1962. Propositional Connective CONNECTIVE Prosthaphaeresis Formulas TRIGONOMETRY formulas which convert a product of functions into a sum or difference. The Prosthaphaer- esis formulas are sin a /C27sin b /C302 sin1 2(a /C27 b)hi cos12(a /C28 b)hi (1) sin a /C28sin b /C302 cos1 2( a /C27 b)hi sin12(a /C28 b)hi (2) cos a /C27cos b /C302 cos12(a /C27 b)hi cos12(a /C28 b)hi (3)cos a /C28cos b /C30/C282 sin12(a /C27 b)hi sin12( a /C28 b)hi : (4) Related formulas are sin a sin b /C3012sin( a /C28 b) /C27sin(a /C27 b) ½/C138 (5) cos a cos b /C301 2cos(a /C28 b) /C27cos(a /C27 b) ½/C138 (6) cos a sin b /C3012[sin(a /C27 b) /C28sin( a /C28 b)] (7) sin a sin b /C301 2[cos( a /C28 b) /C28cos(a /C27 b)]: (8) Multiplying both sides by 2 gives the equations sometimes known as the W ERNER FORMULAS . See also TRIGONOMETRIC ADDITION FORMULAS ,TRI- GONOMETRIC PRODUCT FORMULAS Proth’s Theorem For N/C30h /C2152n/C271 with ODD hand /2n/C21h/, if there exists an INTEGER asuch that a(N/C281)=2/C13/C281 (mod N); then NisPRIME . Protractor A ruled SEMICIRCLE used for measuring and drawing ANGLES . Prouhet’s Problem PROUHET- TARRY- ESCOTT PROBLEM Prouhet-Tarry-Escott Problem Find two distinct sets of integers fa1;...;angand fb1;...;bng;such that for k/C301, ..., m, Xn i/C301ak i/C30Xn i/C301bk i: The Prouhet-Tarry-Escott problem is therefore a special case of a MULTIGRADE EQUATION . A solution with n/C30m/C271 is said to be "ideal," and are of interest because they are minimal solutions of the problem (Borwein and Ingalls 1994). The smallest symmetric ideal solutions for m/C309 was found by Borwein et al. (Lisonek 2000), (/C28313)k/C27(/C28301)k/C27(/C28188)k/C27(/C28100)k/C27(/C2899)k /C2799k/C27100k/C27188k/C27301k/C27313k /C30(/C28308)k/C27(/C28307)k/C27(/C28180)k/C27(/C28131)k/C27(/C2871)k /C2771k/C27131k/C27180k/C27307k/C27308k; (1) as well as the second solution (/C28515)k/C27(/C28452)k/C27(/C28366)k/C27(/C28189)k/C27(/C28103)k /C27103k/C27189k/C27366k/C27452k/C27515k /C30(/C28508)k /C27(/C28417)k /C27(/C28331)k /C27(/C28245)k /C27(/C2818)k /C2718k /C27245k /C27331k /C27471k /C27508k : (2) The previous smallest known symmetric ideal solu- tion, found by Letac in the 1940s, is (/C2823750)k /C27(/C2820667)k /C27(/C2820499)k /C27(/C2811857)k /C27(/C28436)k /C27436k /C2711857k /C2720449k /C2720667k /C2723750k /C30(/C2823738)k /C27(/C2820855)k /C27(/C2820231)k /C27(/C2811881)k /C27(/C2812)k /C2712k /C2711881k /C2720231k /C2720885k /C2723738k : (3) In 1999, S. Chen found the first ideal solution with m ]10; 0k /C2711k /C2724k /C2765k /C2790k /C27129k /C27173k /C27212k /C27237k /C27278k /C27291k /C27302k /C303k /C275k /C2730k /C2757k /C27104k /C27116k /C27186k /C27198k /C27245k /C27272k /C27297k /C27299k ; (4) which is true for k /C301, 2, ..., 11. See also MULTIGRADE EQUATION References Borwein, P. and Ingalls, C. "The Prouhet-Tarry-Escott Problem Revisited." Enseign. Math. 40,3/C1/7, 1994. http://www.cecm.sfu.ca/~pborwein/PAPERS/P98.ps. Chen, S. "The Prouhet-Tarry-Escott Problem." http://mem- ber.netease.com/~chin/eslp/TarryPrb.htm. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 709 /C1/10, 1971. Dorwart, H. L. and Brown, O. E. "The Tarry-Escott Pro- blem." Amer. Math. Monthly 44, 613 /C1/26, 1937. Hahn, L. "The Tarry-Escott Problem." Problem 10284. Amer. Math. Monthly 102, 843 /C1/44, 1995. Hardy, G. H. and Wright, E. M. "The Four-Square Theorem" and "The Problem of Prouhet and Tarry: The Number P(k ; j) :/" §20.5 and 21.9 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 302 /C1/06 and 328 /C1/29, 1979. Lisonek, P. "New size 10 solutions of the Prouhet-Tarry- Escott Problem." [email protected] posting, 21 Jun 2000. Wright, E. M. "On Tarry’s Problem (I)." Quart. J. Math. Oxford Ser. 6, 216 /C1/67, 1935. Wright, E. M. "The Tarry-Escott and the ‘Easier’ Waring Problem." J. reine angew. Math. 311/312 , 170 /C1/73, 1972. Wright, E. M. "Prouhet’s 1851 Solution of the Tarry-Escott Problem of 1910." Amer. Math. Monthly 102, 199 /C1/10, 1959.Pru¨ fer Code An encoding which provides a bijection between the nn/C282 LABELED TREES on n nodes and strings of /n /C282/ integers chosen from an alphabet of the numbers 1 to n.A LABELED TREE can be converted to a Pru¨fer code using LabeledTreeToCode [g] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ), and a code can be converted to a LABELED TREE usingCodeToLabeledTree [g]. Pru¨fer’s bijection is based on the fact that every tree has at least two nodes of degree 1 (i.e., LEAVES ). Therefore, the node v which is incident to the lowest labeled leaf is uniquely determined, and vis then taken as the first symbol in the code. This node is then deleted and the procedure is repeated until asingle edge is left, giving a total of /n/C282/integers between 1 and n(Skiena 1990). This is demonstrated in the LABELED TREE shown above. See also LABELED TREE References Pru¨fer, H. "Neuer Beweis eines Satzes u ¨ber Permutationen." Arch. Math. Phys. 27, 742/C1/44, 1918. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Pru¨ fer Ring A metric space ˆZ in which the closure of a congruence class B(j ; m) is the corresponding congruence class fx /C23 ˆZ ½x /C13j (mod m)g:/ References Fontana, M.; Huckaba, J. A.; and Papick, I. J. Pru¨fer Domains. New York: Dekker. Fried, M. D. and Jarden, M. Field Arithmetic. New York: Springer-Verlag, pp. 7 /C1/1, 1986. Postnikov, A. G. Introduction to Analytic Number Theory. Providence, RI: Amer. Math. Soc., 1988. p-Series A shorthand name for a POWER SERIES with a NEGATIVE exponent, a/C12 k/C301 k/C28p ; where p /C210. See also POWER SERIES ,RIEMANN ZETA FUNCTION Pseudoanalytic Function A pseudoanalytic function is a function defined using generalized CAUCHY- RIEMANN EQUATIONS . Pseudoa- nalytic functions come as close as possible to having COMPLEX DERIVATIVES and are nonsingular "quasire- gular" functions. See also ANALYTIC FUNCTION ,SEMIANALYTIC ,SUB- ANALYTIC Pseudocircle A simple closed curve on a SPHERE that is not necessarily a GREAT CIRCLE but merely intersects as a GREAT CIRCLE would (Billera et al. 1999). See also GREAT CIRCLE References Billera, L. J.; Brown, K. S.; and Diaconis, P. "Random Walks and Plane Arrangements in Three Dimensions." Amer. Math. Monthly 106, 497 /C1/01, 1999. Bjo¨rner, A; Las Vargnas, M.; Sturmfels, B.; White, N.; and Ziegler, G. M. Oriented Manifolds. Cambridge, England: Cambridge University Press, 1993. Gru¨nbaum, B. Arrangements and Spreads. Providence, RI: Amer. Math. Soc., 1972. Ziegler, G. M. Lectures on Polytopes. New York: Springer- Verlag, 1995. Pseudoconic Projection A MAP PROJECTION in which the parallels are repre- sented by concentric circular arcs and the meridians by concurrent curves. References Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190 /C1/00, 1944.Pseudocrosscap A surface constructed by placing a family of figure- eight curves into R3 such that the first and last curves reduce to points. The surface has PARAMETRIC EQUA- TIONS x(u; v) /C30(1 /C28u2) sin v y(u; v) /C30(1 /C28u2) sin(2 v) z(u; v) /C30u : References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 337, 1997. Pseudocylindrical Projection A projection in which latitude lines are parallel but meridians are curves. See also CYLINDRICAL PROJECTION ,ECKERT IV PRO- JECTION ,ECKERT VI PROJECTION ,M OLLWEIDE PRO- JECTION ,R OBINSON PROJECTION ,S INUSOIDAL PROJECTION References Dana, P. H. "Map Projections." http://www.colorado.edu/ geography/gcraft/notes/mapproj/mapproj_f.html. Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Rev. 7, 190/C1/00, 1944. Pseudodifferential Operator References Folland, G. B. Introduction to Partial Differential Equa- tions, 2nd ed. Princeton, NJ: Princeton University Press, 1996. Hormander, L. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed. New York: Springer-Verlag, 1990. Hormander, L. The Analysis of Linear Partial Differential Operators II. New York: Springer-Verlag, 1983. Hormander, L. The Analysis of Linear Partial Differential Operators III. New York: Springer-Verlag, 1985. Hormander, L. The Analysis of Linear Partial Differential Operators IV. New York: Springer-Verlag, 1994. Saint Raymond, X. Elementary Introduction to the Theory of Pseudodifferential Operators. Boca Raton, FL: CRC Press, 1991. Taylor, M. E. Partial Differential Equations, Vol. 1: Basic Theory. New York: Springer-Verlag, 1996. Taylor, M. E. Partial Differential Equations, Vol. 2: Quali- tative Studies of Linear Equations. New York: Springer- Verlag, 1996. Taylor, M. E. Partial Differential Equations, Vol. 3: Non- linear Equations. New York: Springer-Verlag, 1996. Wloka, J. T.; Rowley, B.; and Lawruk, B. Boundary Value Problems for Elliptic Systems. Cambridge, England: Cam- bridge University Press, 1995. Pseudo-Euclidean Space A Euclidean-like space having LINE ELEMENT ds2 /C30(dz1)2 /C27.../C27(dzp)2 /C28(dzp /C271)2 /C28.../C28(dzp/C27q)2 ; having dimension m /C30p /C27q (Rosen 1965). In con- trast, the signs would be all be positive for a EUCLIDEAN SPACE . See also CAMPBELL’S THEOREM ,EUCLIDEAN SPACE References Rosen, J. "Embedding of Various Relativistic Spaces in Pseudo-Euclidean Spaces." Rev. Mod. Phys. 37, 204 /C1/14, 1965. Pseudograph A non- SIMPLE GRAPH in which both LOOPS and multi- ple edges are permitted. See also HYPERGRAPH ,LOOP (GRAPH ), MULTIGRAPH , SIMPLE GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 10, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 89, 1990. Pseudogroup An algebraic structure whose elements consist of selected HOMEOMORPHISMS between open subsets of a SPACE , with the composition of two transformations defined on the largest possible domain. The "germs" of the elements of a pseudogroup form a GROUPOID (Weinstein 1996).See also GROUP ,GROUPOID ,INVERSE SEMIGROUP References Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744 /C1/52, 1996. Pseudoinverse MOORE- PENROSE GENERALIZED MATRIX INVERSE Pseudolemniscate Case The case of the WEIERSTRASS ELLIPTIC FUNCTION with invariants g2 /C30/C281 and g3 /C300:/ See also EQUIANHARMONIC CASE,LEMNISCATE CASE, WEIERSTRASS ELLIPTIC FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Pseudo-Lemnis- cate Case (/g2 /C30/C281; g3 /C300):/" §18.15 in Handbook of Math- ematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 662 /C1/63, 1972. Pseudoparadox Curry (1977, p. 5) uses the term pseudoparadox to describe an apparent PARADOX , such as the CATALO- GUE PARADOX , for which there is no underlying actual contradiction. See also HYPERGAME ,PARADOX References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 5, 1977. Pseudoperfect Number SEMIPERFECT NUMBER Pseudoprime A pseudoprime is a COMPOSITE NUMBER which passes a test or sequence of tests which fail for most COMPOSITE NUMBERS . Unfortunately, some authors drop the " COMPOSITE " requirement, calling any num- ber which passes the specified tests a pseudoprime even if it is PRIME . Pomerance, Selfridge, and Wag- staff (1980) restrict their use of "pseudoprime" to ODD COMPOSITE NUMBERS . "Pseudoprime" used without qualification means F ERMAT PSEUDOPRIME . CARMICHAEL NUMBERS are ODD COMPOSITE numbers which are pseudoprimes to every base; they are sometimes called ABSOLUTE PSEUDOPRIMES . The fol- lowing table gives the number of F ERMAT PSEUDO- PRIMES psp(2), E ULER- JACOBI PSEUDOPRIMES ejpsp(2), and STRONG PSEUDOPRIMES spsp(2) to the base 2, as well as C ARMICHAEL NUMBERS CN which are less the first few powers of 10 (Guy 1994). /10n/ psp(2) ejpsp(2) spsp(2) CN Sloane A055550 A055551 A055552 A055553 Sloane CountsA001567 A047713 A001262 A002997 101 0000 102 0000 103 3101 104 22 12 5 7 105 78 36 16 16 106 245 114 46 43 107 750 375 162 105 108 2057 1071 488 255 109 5597 2939 1282 646 1010 14884 7706 3291 1547 1011 38975 20417 8607 3605 1012 101629 53332 22407 8241 1013 264239 124882 58897 19279 See also CARMICHAEL NUMBER ,E LLIPTIC PSEUDO- PRIME ,EULER PSEUDOPRIME ,EULER- JACOBI PSEUDO- PRIME ,EXTRA STRONG LUCAS PSEUDOPRIME ,FERMAT PSEUDOPRIME ,FIBONACCI PSEUDOPRIME ,FROBENIUS PSEUDOPRIME ,LUCAS PSEUDOPRIME ,PERRIN PSEU- DOPRIME ,PROBABLE PRIME ,SOMER- LUCAS PSEUDO- PRIME ,S TRONG ELLIPTIC PSEUDOPRIME ,S TRONG FROBENIUS PSEUDOPRIME ,STRONG LUCAS PSEUDO- PRIME ,STRONG PSEUDOPRIME References Caldwell, C. K. "Prime Links/C27/C27: Resources in theory: finding_and_proving: probable_primality." http://prime- s.utm.edu/links/theory/finding_and_proving/probable_- primality/. Grantham, J. "Frobenius Pseudoprimes." http://www.clar- k.net/pub/grantham/pseudo/pseudo1.ps Grantham, J. "Pseudoprimes/Probable Primes." http:// www.clark.net/pub/grantham/pseudo/. Guy, R. K. "Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes." §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 27 /C1/0, 1994. Pinch, R. G. E. "The Pseudoprimes Up to 1013." ftp:// ftp.dpmms.cam.ac.uk/pub/PSP/. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. "The Pseudoprimes to 25 /C215 109 :/" Math. Comput. 35, 1003 /C1/026, 1980. Available electronically from ftp://sable.ox.ac.uk/ pub/math/primes/ps2.Z. Sloane, N. J. A. Sequences A001262, A001567/M5441, A002997/M5462, A047713, A055550, A055551, A055552, and A055553 in "An On-Line Version of the Encyclopediaof Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Pseudorandom Number A slightly archaic term for a computer-generated RANDOM NUMBER . The prefix pseudo- is used to distinguish this type of number from a "truly" RAN- DOM NUMBER generated by a random physical process such as radioactive decay. See also RANDOM NUMBER References Luby, M. Pseudorandomness and Cryptographic Applica- tions. Princeton, NJ: Princeton University Press, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 266, 1992. Pseudorhombicuboctahedron ELONGATED SQUARE GYROBICUPOLA Pseudo-Riemannian Manifold A pseudo-Riemannian manifold is a manifold which has a metric that is of the signature diag(/C28;/C27; ...;/C27); as compared to a RIEMANNIAN MANIFOLD , which has a signature of all positive signs. See also CAMPBELL’S THEOREM ,RIEMANNIAN MANI- FOLD Pseudoscalar A SCALAR which reverses sign under inversion is called a pseudoscalar. The SCALAR TRIPLE PRODUCT A /C215 (B /C29C) is a pseudoscalar. Given a transformation MATRIX A; S?/C30det AjjS; where det is the DETERMINANT . See also PSEUDOTENSOR ,PSEUDOVECTOR ,SCALAR References Arfken, G. "Pseudotensors, Dual Tensors." §3.4 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 128 /C1/37, 1985. Pseudosmarandache Function The pseudosmarandache function Z(n) is the smallest integer such that XZ(n) k/C301k/C301 2Z(n)[Z(n)/C271] is divisible by n. The values for n/C301, 2, ... are 1, 3, 2, 7, 4, 3, 6, 15, 8, 4, ... (Sloane’s A011772; Kashihara 1996; Russo 2000, p. 4). See also SMARANDACHE FUNCTION References Ashbacher, C. "Problem 514." Pentagon 57, 36, 1997. Kashihara, K. "Comments and Topics on Smarandache Notions and Problems." Vail: Erhus University Press, 1996. Russo, F. A Set of New Smarandache Functions, Sequences, and Conjectures in Numer Theory. Lupton, AZ: American Research Press, 2000. Sloane, N. J. A. Sequences A011772 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Pseudosphere Half the SURFACE OF REVOLUTION generated by a TRACTRIX about its ASYMPTOTE to form a TRACTROID . The surfaces is sometimes also called the ANTISPHERE or TRACTRISOID (Steinhaus 1983, pp. 251). The Carte- sian PARAMETRIC EQUATIONS are x /C30sech u cos v (1) y /C30sech u sin v (2) z /C30u /C28tanh u (3) for u ]0 and v /C23 [0; 2 p):/ The coefficients of the FIRST FUNDAMENTAL FORM are E /C30tanh2 u (4) F /C300 (5) G /C30sech2 u; (6) the SECOND FUNDAMENTAL FORM coefficients are e /C30/C28sech u tanh u (7) f /C300 (8) g /C30sech u tanh u; (9) and the surface area element is dS /C30sech u tanh u: (10) The SURFACE AREA is S /C30g2p 0g/C12 0sech u tanh ududv /C302p: (11) The GAUSSIAN and MEAN CURVATURES areK /C30/C281 (12) H /C301 2(sinh u /C28csch u) : (13) The pseudosphere therefore has constant NEGATIVE GAUSSIAN CURVATURE , justifying the name "pseudo- sphere" (i.e., an analog of the SPHERE , which has constant POSITIVE curvature). Its constant NEGATIVE CURVATURE also makes it a model of HYPERBOLIC GEOMETRY . An equation for the GEODESICS on a pseudosphere is given by cosh2 u /C27(v /C27c)2 /C30k2 : (14) See also FUNNEL ,G ABRIEL’S HORN,H YPERBOLIC GEOMETRY ,TRACTRIX References Fischer, G. (Ed.). Plate 82 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 77, 1986. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 487 and 489 /C1/90, 1997. JavaView. "Classic Surfaces from Differential Geometry: Pseudo Sphere." http://www-sfb288.math.tu-berlin.de/ vgp/javaview/demo/surface/common/PaSurface_Pseudo- Sphere.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 251, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 199 /C1/00, 1991. Pseudosquare Given an ODD PRIME p,aSQUARE NUMBER n satisfies (n=p) /C300 or 1 for all p Bn, where (n=p) is the LEGENDRE SYMBOL . A number n /C212 which satisfies this relationship but is not a SQUARE NUMBER is called a pseudosquare. The only pseudosquares less than 109 are 3 and 6. See also LEGENDRE SYMBOL ,SQUARE NUMBER Pseudotensor A TENSOR -like object which reverses sign under inversion. Given a transformation MATRIX A ; A0 ij /C30det AjjaikajlAkl ; where det is the DETERMINANT . A pseudotensor is sometimes also called a TENSOR DENSITY . See also PSEUDOSCALAR ,P SEUDOVECTOR ,S CALAR , TENSOR DENSITY References Arfken, G. "Pseudotensors, Dual Tensors." §3.4 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 128 /C1/37, 1985. Pseudovector A typical VECTOR is transformed to its NEGATIVE under inversion. A VECTOR which is invariant under inversion is called a pseudovector, also called an AXIAL VECTOR in older literature (Morse and Fes- hbach 1953). The CROSS PRODUCT A /C29B (1) is a pseudovector, whereas the VECTOR TRIPLE PRO- DUCT A /C29(B /C29C) (2) is a VECTOR . [pseudovector] /C29[pseudovector] /C30[pseudovector] (3) [vector] /C29[pseudovector] /C30[vector] : (4) Given a transformation MATRIX A; C?i /C30det AjjaijCj : (5) See also PSEUDOSCALAR ,TENSOR ,VECTOR References Arfken, G. "Pseudotensors, Dual Tensors." §3.4 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 128 /C1/37, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 46 /C1/7, 1953. Psi Function C(z ; s ; v) /C13X/C12 n/C300zn (v /C27 n)s for zjjB1 and v "0;/C281; ... (Gradshteyn and Ryzhik 2000, pp. 1075 /C1/076). See also HURWITZ ZETA FUNCTION ,JACOBI THETA FUNCTIONS ,RAMANUJAN PSI SUM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. p-Signature Diagonalize a form over the rationals to diag[ pa /C215 A; pb /C215 B ; ...]; where all the entries are INTEGERS and A, B, ...are RELATIVELY PRIME to p. Then the p-signature OF THE FORM (for p "/C281 ; 2) is pa /C27pb /C27.../C274k (mod 8); where k is the number of ANTISQUARES . For p /C30/C281, the p-signature is SYLVESTER’S SIGNATURE .See also SIGNATURE (QUADRATIC FORM) PSLQ Algorithm An algorithm which can be used to find INTEGER RELATIONS between real numbers x1 ; ..., xn such that a1x1 /C27a2x2 /C27.../C27anxn /C300 ; with not all ai /C300: Although the algorithm operates by manipulating a lattice, it does not reduce it to a short vector basis, and is therefore not a LATTICE REDUCTION algorithm. PSLQ is based on a partial sum of squares scheme (like the PSOS ALGORITHM ) implemented using QR DECOMPOSITION . It was devel- oped by Ferguson and Bailey (1992). A much simpli- fied version of the algorithm was subsequently developed by Ferguson et al. (1999), which also extends the algorithm to complex numbers and quaternions. Ferguson et al. (1999) also demon- strated that PSLQ is distinct from the HJLS ALGO- RITHM . The PSLQ algorithm terminates after a number of iterations bounded by a polynomial in n and uses a numerically stable matrix reduction procedure (Fer- guson and Bailey 1992). PSLQ tends to be faster than the FERGUSON- FORCADE ALGORITHM and LLL ALGO- RITHM because of clever techniques that allow ma- chine arithmetic to be used at many intermediate steps. The LLL ALGORITHM , by comparison, must use moderate precision, although generally not as much as the HJLS ALGORITHM . While the LLL ALGORITHM is a more general LATTICE REDUCTION algorithm than PSLQ, using LLL to obtain integer relations is in some sense a "trick," whereas with PSLQ one gets either a relation or lower bounds on degrees of polynomials and sizes ofcoefficients for which such a relation must satisfy. See also F ERGUSON- FORCADE ALGORITHM ,INTEGER RELATION , LLL ALGORITHM , PSOS ALGORITHM References Bailey, D. H.; Borwein, J. M.; and Girgensohn, R. "Experi- mental Evaluation of Euler Sums." Exper. Math. 3,1 7/C1/0, 1994. Bailey, D. and Plouffe, S. "Recognizing Numerical Con- stants." http://www.cecm.sfu.ca/organics/papers/bailey/. Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899/C1/09, 1999. Crandall, R. E. Topics in Advanced Scientific Computation. New York: Springer-Verlag, 1996. Ferguson, H. R. P. and Bailey, D. H. "A Polynomial Time, Numerically Stable Integer Relation Algorithm." RNR Techn. Rept. RNR-91 /C1/32, Jul. 14, 1992. Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351/C1/69, 1999. PSOS Algorithm An INTEGER-RELATION algorithm which is based on a partial sum of squares approach, from which the algorithm takes its name. See also FERGUSON- FORCADE ALGORITHM ,HJLS ALGORITHM ,INTEGER RELATION , LLL ALGORITHM , PSLQ ALGORITHM References Bailey, D. H. and Ferguson, H. R. P. "Numerical Results on Relations Between Numerical Constants Using a New Algorithm." Math. Comput. 53, 649 /C1/56, 1989. Ferguson, H. "PSOS: A New Integral Relation Finding Algorithm Involving Partial Sums of Squares and No Square Roots." Abs. Papers Presented to Amer. Math. Soc. 9, No. 56 88T-11 /C1/5, 214, Mar. 1988. P-Symbol A symbol employed in a formal PROPOSITIONAL CALCULUS . References Nidditch, P. H. Propositional Calculus. New York: Free Press of Glencoe, p. 1, 1962. p-System A p-system of a SET S is a sequence of SUBSETS A1 ; A2 ; ..., Apof S, among which some may be empty or coinciding with each other. See also INCLUSION- EXCLUSION PRINCIPLE , K-SUBSET , SUBSET References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 176 /C1/77, 1974. Ptolemy Inequality For a QUADRILATERAL which is not CYCLIC ,PTOLEMY’S THEOREM becomes an INEQUALITY : AB /C29CD /C27BC /C29DA > AC /C29BD: See also PTOLEMY’S THEOREM ,QUADRILATERAL Ptolemy’s Theorem For a CYCLIC QUADRILATERAL , the sum of the productsof the two pairs of opposite sides equals the product of the diagonals AB /C29CD /C27BC /C29DA /C30AC /C29BD: This fact can be used to derive the TRIGONOMETRY addition formulas. See also CYCLIC QUADRILATERAL ,FUHRMANN’S THEO- REM,PTOLEMY INEQUALITY References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 38, 1971. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 42 /C1/3, 1967. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 17, 1928. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 200 /C1/01, 1991. Public-Key Cryptography A type of CRYPTOGRAPHY in which the encoding key is revealed without compromising the encoded message. The two best-known methods are the KNAPSACK PROBLEM and RSA ENCRYPTION . See also KNAPSACK PROBLEM , RSA ENCRYPTION References Diffie, W. and Hellman, M. "New Directions in Cryptogra- phy." IEEE Trans. Info. Th. 22, 644/C1/54, 1976. Flannery, S. and Flannery, D. In Code: A Mathematical Journey. Profile Books, 2000. Hellman, M. E. "The Mathematics of Public-Key Cryptogra- phy." Sci. Amer. 241, 130/C1/39, Aug. 1979. Rivest, R.; Shamir, A.; and Adleman, L. "A Method for Obtaining Digital Signatures and Public-Key Cryptosys- tems." MIT Memo MIT/LCS/TM-82, 1982. Wagon, S. "Public-Key Encryption." §1.2 in Mathematica in Action. New York: W. H. Freeman, pp. 20 /C1/2, 1991. Puiseux Diagram A diagram used in the solution of ordinary differen- tial equations OF THE FORM dw dz/C30g(z;w) h(z;q) which vanish when z/C300, where g(0;0)/C30h(0;0)/C300 (Ince 1956, pp. 298 and 427). The diagram is named in order of French mathematician Vicrot Puiseux. References Fine, H. B. "On the Functions Defined by Differential Equations, with an Extension of the Puiseux Polygon Construction to these Equations." Amer. J. Math. 11, 317/C1/28, 1889. Ince, E. L. Ordinary Differential Equations. New York: Dover, 1956. Puiseux Series A power series containing fractional exponents (Da- venport et al. 1993, p. 91). See also POWER SERIES References Davenport, J. H.; Siret, Y.; and Tournier, E. Computer Algebra: Systems and Algorithms for Algebraic Computa- tion, 2nd ed. San Diego: Academic Press, pp. 90 /C1/2, 1993. Siegel, C. L. Topics in Complex Function Theory, Vol. 1: Elliptic Functions and Uniformization Theory. New York: Wiley, p. 98, 1988. Puiseux’s Theorem The whole neighborhood of any point yiof an ALGEBRAIC CURVE may be uniformly represented by a certain finite number of convergent developments in POWER SERIES , xi /C30 rnyi /C27a ni1tn /C27ani2t2 n /C27...: References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 207, 1959. Puiseux, V. "Recherches sur les fonctions alge´briques." J. de math. pures et appl. 15, 207, 1850. Pullback Map A pullback is a general CATEGORICAL operation appearing in a number of mathematical contexts, sometimes going under a different name. If T : V 0 W is a linear transformation between VECTOR SPACES , then T /C31 : W /C310 V /C31 (usually called TRANSPOSE MAP or DUAL MAP because its associated matrix is the MATRIX TRANSPOSE of T) is an example of a pullback map. In the case of a DIFFEOMORPHISM and DIFFERENTI- ABLE MANIFOLD , a very explicit definition can be formulated. Given an r-form a on a MANIFOLD M2 ; define the r-form T /C31(a)on M1by its action on an r- tuple of tangent vectors (X1 ; ...; Xr) as the number T /C31( a)(X1 ; ... ; Xr) /C30 a(TX1 ; ...; TXr) : This defines a map on r-forms and is the pullback map. See also CATEGORY ,PUSHFORWARD MAP Pulse Function RECTANGLE FUNCTIONPunctured Set A SET S with a single point P removed is called a punctured set, written S_fPg:/ References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 41 /C1/2, 1999. Purser’s Theorem Let t, u, and v be the lengths of the tangents to a CIRCLE C from the vertices of a TRIANGLE with sides of lengths a, b, and c. Then the condition that C is tangent to the CIRCUMCIRCLE of the TRIANGLE is that 9at 9bu 9cv /C300: The theorem was discovered by Casey prior to Purser’s independent discovery. See also CASEY’S THEOREM ,CIRCUMCIRCLE Pursuit Curve IfAmoves along a known curve, then Pdescribes a pursuit curve if Pis always directed toward AandA and Pmove with uniform velocities. Pursuit curves were considered in general by the French scientist Pierre Bouguer in 1732, and subsequently by theEnglish mathematician Boole. The case restricting A to a straight line was studied by Arthur Bernhart (MacTutor Archive). It has CARTESIAN COORDINATES equation y /C30cx /C28ln x: The problem of n mice (or dogs) starting at the corners of a regular polygon and running towards each other is called the MICE PROBLEM . See also APOLLONIUS PURSUIT PROBLEM ,M ICE PRO- BLEM ,W HIRL References Barton, J. C. and Eliezer, C. J. "On Pursuit Curves." J. Austral. Math. Soc. Ser. B 41, 358 /C1/71, 2000. Bernhart, A. "Curves of Pursuit." Scripta Math. 20, 125 /C1/41, 1954. Bernhart, A. "Curves of Pursuit-II." Scripta Math. 23,49/C1/5, 1957. Bernhart, A. "Polygons of Pursuit." Scripta Math. 24,23/C1/0, 1959. Bernhart, A. "Curves of General Pursuit." Scripta Math. 24, 189 /C1/06, 1959. MacTutor History of Mathematics Archive. "Pursuit Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Pur- suit.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 201 /C1/02, 1991. Yates, R. C. "Pursuit Curve." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 170 /C1/ 71, 1952. Push An action which adds a single element to the top of a STACK , turning the STACK (/a1 ; a2 ; ..., an) into (/a0 ; a1 ; a2 ; ..., an) :/ See also POKE MOVE,POP,STACK Pushforward Map See also PULLBACK MAP Puzzle A mathematical PROBLEM , usually not requiring advanced mathematics, to which a solution is desired. Puzzles frequently require the rearrangement of existing pieces (e.g., 15 PUZZLE ) or the filling in of blanks (e.g., crossword puzzle). See also 15 PUZZLE ,BAGUENAUDIER ,CALIBAN PUZZLE , CONWAY PUZZLE ,CRYPTARITHMETIC ,DISSECTION PUZ- ZLES,ICOSIAN GAME,PYTHAGOREAN SQUARE PUZZLE , RUBIK’S CUBE,SLOTHOUBER- GRAATSMA PUZZLE ,T - PUZZLE References Bogomolny, A. "Interactive Mathematics Miscellany and Puzzles." http://www.cut-the-knot.com. Clessa, J. J. Math and Logic Puzzles for PC Enthusiasts. New York: Dover. Costello, M. J. The Greatest Puzzles of All Time. New York: Dover. Dudeney, H. E. Amusements in Mathematics. New York: Dover, 1917.Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949. Dudeney, H. E. 536 Puzzles & Curious Problems. New York: Scribner, 1967. Friedman, E. "Erich’s Puzzle Palace." http://www.stetso- n.edu/~efriedma/puzzle.html. Fujii, J. N. Puzzles and Graphs. Washington, DC: National Council of Teachers, 1966. Pegg, E. Jr. "Mathpuzzle." http://www.mathpuzzle.com/. Weisstein, E. W. "Books about Recreational Mathematics." http://www.treasure-troves.com/books/Recreational-Mathematics.html. Slocum, J. and Botermans, J. Puzzles Old and New: How to Make and Solve Them. Seattle, WA: University of Washington Press, 1988. P-Value The PROBABILITY that a variate would assume a value greater than or equal to the observed value strictly by chance: /Pðz]zobserved Þ/. See also ALPHA VALUE ,SIGNIFICANCE Pyramid APOLYHEDRON with one face (known as the "base") a POLYGON and all the other faces TRIANGLES meeting at a common VERTEX (known as the "apex"). A right pyramid is a pyramid for which the line joining the centroid of the base and the apex is perpendicular to the base. A regular pyramid is a pyramid whose bases is a REGULAR POLYGON .A n n-gonal regular pyramid (denoted Yn) having EQUILATERAL TRIANGLES as sides is possible only for n/C303, 4, 5. These correspond to the TETRAHEDRON ,SQUARE PYRAMID , and PENTAGONAL PYRAMID , respectively. An arbitrary pyramid has a single cross-sectional shape whose lengths scale linearly with height. Therefore, the AREA of a CROSS SECTION scales quadratically with height, decreasing from Abat the base ( z/C300) to 0 at the apex (assumed to lie at a height z/C30h). The AREA at a height zabove the base is therefore given by A(z)/C30Ab(h/C28z)2 h2: (1) As a result, the VOLUME of a pyramid, regardless of base shape or position of the apex relative to the base,is given by V/C30gh 0A(z)dz/C30Abgh 0(z/C28h)2 h2dz/C301 3Abh: (2) These results also hold for the CONE , ELLIPTIC CONE , TRIANGULAR PYRAMID , SQUARE PYRAMID , etc. The CENTROID is the same as for the CONE , given by ¯z /C301 4 h : (3) The SURFACE AREA of a pyramid is S /C3012 ps; (4) where s is the SLANT HEIGHT and p is the base PERIMETER . For a right pyramid with a regular n- gonal base of side length a, sn /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C27R2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C271 4 a2 csc2p n !vuut: (5) This gives the special cases s3 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C271 3 a2q (6) s4 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C271 2 a2q (7) s5 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C271 105 /C27ffiffiffi 5p9+;k9+;7 a2r (8) s6 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C27a2p : (9) Joining two PYRAMIDS together at their bases gives a BIPYRAMID , also called a DIPYRAMID . See also BIPYRAMID ,CUMULATION ,ELEVATUM ,ELON- GATED PYRAMID ,G YROELONGATED PYRAMID ,H EXA- GONAL PYRAMID ,INVAGINATUM ,P ENTAGONAL PYRAMID ,PYRAMID ,PYRAMIDAL FRUSTUM ,SQUARE PYRAMID ,T ETRAHEDRON ,T RIANGULAR PYRAMID , TRUNCATED SQUARE PYRAMID References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 128, 1987. Harris, J. W. and Stocker, H. "Pyramid." §4.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 98 /C1/9, 1998. Hart, G. "Pyramids, Dipyramids, and Trapezohedra." http:// www.georgehart.com/virtual-polyhedra/pyramids-in- fo.html. Kern, W. F. and Bland, J. R. "Pyramid" and "Regular Pyramid." §20 /C1/1in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 50 /C1/3, 1948. Pyramidal Frustum A pyramidal frustum is a FRUSTUM made by chopping the top off a PYRAMID . It is a special case of a PRISMATOID . Let s be the SLANT HEIGHT , p1 the bottom base PERIMETER , p2the top base PERIMETER , A1the bottom AREA , and A2 the top AREA . Then the SURFACE AREA (of the sides) and VOLUME of a pyramidal frustum are given by S /C301 2(p1 /C27p2)s (1) V /C301 3 hA1 /C27A2 /C27ffiffiffiffiffiffiffiffiffiffiffi A1A2p9+;k9+;7 : (2) The CENTROID of a right pyramidal frustum occurs at a height ¯z /C30hA1 /C27 2ffiffiffiffiffiffiffiffiffiffiffi A1A2p /C27 3A29+=9+; 4 A1 /C27ffiffiffiffiffiffiffiffiffiffiffiA 1A2p /C27 A29+=9+; (3) above the bottom base (Harris and Stocker 1998). The bases of a right n-gonal frustum are regular polygons of side lengths a and b with circumradii Rn /C301 2 c cscp n ! ; (4) where c is the side length, so the diagonal connecting corresponding vertices on top and bottom has length xn /C301 2(a /C28b) cscp n ! ; (5) and the SLANT HEIGHT is sn /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 /C27h2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4cscp n ! (a /C28b)2 /C27h2vuut: (6) The triangular (n /C303) and square (n /C304) right pyr- amidal frustums therefore have side surface areas S3 /C303 2(a /C27b)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13(a/C28b)2/C27h2q (7) S4/C302(a/C27b)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12(a/C28b)2/C27h2q : (8) The area of a regular n-gon is An/C301 4nc2cotp n ! ; (9) so the volumes of these frustums are V3/C301 12ffiffiffi 3p (a2/C27ab/C27b2)h (10) V4/C301 3(a2/C27ab/C27b2)h: (11) See also CONICAL FRUSTUM ,F RUSTUM ,H ERONIAN MEAN,PYRAMID ,SPHERICAL SEGMENT ,TRUNCATED SQUARE PYRAMID References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 128, 1987. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 3 /C1/, 1990. Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, p. 7, 1965. Harris, J. W. and Stocker, H. "Frustum of a Pyramid." §4.3.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 99, 1998. Kern, W. F. and Bland, J. R. "Frustum of Regular Pyramid." §28 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 67 /C1/1, 1948. Pyramidal Number A FIGURATE NUMBER corresponding to a configuration of points which form a pyramid with r-sided REGULAR POLYGON bases can be thought of as a generalized pyramidal number, and has the form Pr n /C301 6(n /C271) 2pr n /C27n ðÞ /C301 6 n(n /C271)[(r /C282)n /C27(5 /C28r)]: (1) The first few cases are therefore P3 n /C301 6 n(n /C271)(n /C272) (2) P4 n /C301 6 n(n /C271)(2n /C271) (3) P5 n /C301 2 n2(n /C271); (4) so r /C303 corresponds to a TETRAHEDRAL NUMBER Ten ; and r /C304toa SQUARE PYRAMIDAL NUMBER Pn :/ The pyramidal numbers can also be generalized to 4- D and higher dimensions (Sloane and Plouffe 1995).See also HEPTAGONAL PYRAMIDAL NUMBER ,HEXAGO- NAL PYRAMIDAL NUMBER ,PENTAGONAL PYRAMIDAL NUMBER ,S QUARE PYRAMIDAL NUMBER ,T ETRAHE- DRAL NUMBER References Conway, J. H. and Guy, R. K. "Tetrahedral Numbers" and "Square Pyramidal Numbers" The Book of Numbers. New York: Springer-Verlag, pp. 44 /C1/9, 1996. Sloane, N. J. A. and Plouffe, S. "Pyramidal Numbers." Extended entry for sequence M3382 in The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Pyritohedron An irregular DODECAHEDRON composed of identical irregular PENTAGONS . See also DODECAHEDRON ,RHOMBIC DODECAHEDRON , TRIGONAL DODECAHEDRON References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, p. 63, 1990. Pythagoras Tree AFRACTAL with symmetric and asymmetric forms. References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 67 /C1/7 and 111 /C1/13, 1991. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Pythagoras’s Constant The number ffiffiffi 2p /C301 :4142135623 ... ; which the Pythagoreans proved to be IRRATIONAL . This number is the length of the HYPOTENUSE of an ISOSCELES TRIANGLE with legs of length one, and the statement that it is IRRATIONAL means that it cannot be expressed as a ratio p =q of integers p and q. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality offfiffiffi 2p while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pytha- goreans . Theodorus subsequently proved that the square roots of the numbers from 3 to 17 (excluding 4, 9, and 16) are also irrational (Wells 1986, p. 34). The Babylonians gave the impressive approximation ffiffiffi 2p :1 /C2724 60 /C2751 602 /C2710 603 /C301:41421296296296... (Wells 1986, p. 35; Guy 1990; Conway and Guy 1996, pp. 181 /C1/82). See also IRRATIONAL NUMBER ,OCTAGON ,PYTHAGOR- AS’S THEOREM ,SQUARE References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 25 and 181 /C1/82, 1996. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/pythag/pythag.html. Good, I. J. and Gover, T. N. "The Generalized Serial Test and the Binary Expansion offfiffiffi 2p :/" J. Roy. Statist. Soc. Ser. A 130, 102 /C1/07, 1967. Good, I. J. and Gover, T. N. "Corrigendum." J. Roy. Statist. Soc. Ser. A 131, 434, 1968. Gourdon, X. and Sebah, P. "Pythagore’s Constant:ffiffiffi 2p :/" http://xavier.gourdon.free.fr/Constants/Sqrt2/sqrt2.html. Guy, R. K. "Review: The Mathematics of Plato’s Academy." Amer. Math. Monthly 97, 440 /C1/43, 1990. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 34, 1951. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 126, 1993. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 34 /C1/5, 1986. Pythagoras’s Theorem Proves that the DIAGONAL d of a SQUARE with sides of integral length s cannot be RATIONAL . Assume d=s is rational and equal to p =q where p and q are INTEGERSwith no common factors. Then d2 /C30s2 /C27s2 /C302s2 ; so d s !2 /C30p q !2 /C302; and p2 /C302q2 ; so p2 is even. But if p2 is EVEN , then p is EVEN . Since p=q is defined to be expressed in lowest terms, q must be ODD; otherwise p and q would have the common factor 2. Since p is EVEN , we can let p /C13 2r ; then 4r2 /C302q2 : Therefore, q2 /C302r2 ; and q2 ; so q must be EVEN . But q cannot be both EVEN and ODD,so there are no d and s such that d =s is RATIONAL , and d=s must be IRRATIONAL . In particular, PYTHAGORAS’S CONSTANTffiffiffi2p is IRRA- TIONAL . Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar proofs for f (the GOLDEN RATIO ) andffiffiffi 3p using a PENTAGON and HEXAGON . See also IRRATIONAL NUMBER ,PYTHAGORAS’S CON- STANT ,PYTHAGOREAN THEOREM References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 183 /C1/86, 1996. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 70, 1984. Pappas, T. "Irrational Numbers & the Pythagoras Theorem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 98 /C1/9, 1989. Pythagorean Extension An EXTENSION of an arbitrary FIELD F of the form Fffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 l2p9+;k9+;7 ; where l /C23 F :/ See also EXTENSION FIELD,PYTHAGOREAN FIELD References Itoˆ, K. (Ed.). §155B in Encyclopedic Dictionary of Mathe- matics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 611, 1986. Pythagorean Field A FIELD F in which any PYTHAGOREAN EXTENSION of Fcoincides with F. See also PYTHAGOREAN EXTENSION References Itoˆ, K. (Ed.). §155B in Encyclopedic Dictionary of Mathe- matics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 611, 1986. Pythagorean Fraction Given a P YTHAGOREAN TRIPLE (a;b;c);the fractions a=band b=aare called Pythagorean fractions. Dio- phantus showed that the Pythagorean fractions con- sist precisely of fractions OF THE FORM p2 /C28q2ðÞ =(2pq):/ References Conway, J. H. and Guy, R. K. "Pythagorean Fractions." In The Book of Numbers. New York: Springer-Verlag, pp. 171 /C1/73, 1996. Pythagorean Quadruple POSITIVE INTEGERS a, b, c, and d which satisfy a2 /C27b2 /C27c2 /C30d2 : (1) For POSITIVE EVEN a and b, there exist such INTEGERS c and d; for POSITIVE ODD a and b, no such INTEGERS exist (Oliverio 1996). Oliverio (1996) gives the follow- ing generalization of this result. Let S /C30 a1 ; ...; an /C282 ðÞ ; where aiare INTEGERS , and let T be the number of ODD INTEGERS in S. Then IFF T f2 (mod 4), there exist INTEGERS an/C281 and an such that a2 1 /C27a22 /C27.../C27a2n/C281 /C30a2n : (2) A set of Pythagorean quadruples is given by a /C302mp (3) b /C302np (4) c /C30p2 /C28 m2 /C27n29+=9+; (5)d /C30p2 /C27 m2 /C27n29+=9+; ; (6) where m, n, and p are INTEGERS , m /C27n /C27p /C131 (mod 2); (7) and (m; n; p) /C301 (8) (Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37). Another set of solutions can be obtained from a/C302mp/C272nq (9) b/C302np/C282mq (10) c/C30p2/C27q2/C28m2/C27n29+=9+; (11) d/C30p2/C27q2/C27m2/C27n29+=9+; (12) (Carmichael 1915). See also EULER BRICK,PYTHAGOREAN TRIPLE References Carmichael, R. D. Diophantine Analysis. New York: Wiley, 1915. Mordell, L. J. Diophantine Equations. London: Academic Press, 1969. Oliverio, P. "Self-Generating Pythagorean Quadruples and N-tuples." Fib. Quart. 34,9 8/C1/01, 1996. Q q-Abel’s Theorem Xm y /C300/C281ðÞm/C28yqm/C28y 2ðÞ m y1C2C1C2A q1 /C28 wqm q /C28 wqy /C2 1 /C28wqyðÞm/C281 /C28 z 1 /C28 wqy ; q ! y /C30 1 /C28z ðÞmqm 2ðÞ; wheren yhi qis a Q-BINOMIAL COEFFICIENT . See also ABEL’S BINOMIAL THEOREM References Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 105, 1995. Chu, W. C. and Hsu, L. C. "Some New Applications of Gould-Hsu Inversions." J. Combin. Inform. System Sci. 14,1/C1/4, 1990. q-Analog A q-analog, also called a Q-EXTENSION or Q-GENERAL- IZATION , is a mathematical expression parameterized by a quantity q which generalizes a known expres- sion and reduces to the known expression in the limit q 0 1 /C27: There are q-analogs of the FACTORIAL , BINOMIAL COEFFICIENT , DERIVATIVE , INTEGRAL ,FIBO- NACCI NUMBERS , and so on. Koornwinder, Suslov, and Bustoz, have even managed some kind of q-Fourier analysis. q-analogs are based on the observation that lim q01 /C281 /C28 qa 1 /C28 q/C30a; so that the quantity 1 /C28qaðÞ = 1 /C28q ðÞ is sometimes written a½/C138(Koekoek and Swarttouw 1998, p. 7). q-analogs also have a combinatorial interpretation based on the fact that one can count the elements of some set S to get the number #S : A so-called "statistic" f : S 0 Z can then be defined which is an integer-valued function on S and separates the elements of S into classes based on what value f takes on the elements. This relationship can be summarized by writing a polynomial in a new vari- able, usually taken as q, where the coefficient of qn is # s /C23 S : f(s) /C30n fg : Evaluating the polynomial at q /C301 then adds the coefficients together, returning the original S:/ The q-analog of a mathematical object is generally called the "q-object", hence Q-BINOMIAL COEFFICIENT , Q-FACTORIAL , etc. There are generally several q- analogs if there is one, and there is sometimes even a multibasic analog with independent q1 ; q2 ; ....See also D-ANALOG , Q-BETA FUNCTION , Q-BINOMIAL COEFFICIENT , Q-BINOMIAL THEOREM , Q-COSINE , Q- DERIVATIVE , Q-FACTORIAL , Q-GAMMA FUNCTION , Q- POCHHAMMER SYMBOL , Q-SERIES , Q-SINE, Q-VANDER- MONDE SUM References Exton, H. q-Hypergeometric Functions and Applications. New York: Halstead Press, 1983. Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analo- gue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/17, p. 7, 1998. ftp://www.twi.tudelft.nl/publications/ tech-reports/1998/DUT-TWI-98 /C1/17.ps.gz. Q-Bar The algebraic closure of the RATIONAL NUMBERS Q; denoted Q: This is equivalent to the set of ALGEBRAIC NUMBERS , sometimes denoted A :/ See also ALGEBRAIC NUMBER ,ALGEBRAICS ,Q References Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. http://www.math.jussieu.fr/~neste- ren/. q-Beta Function A Q-ANALOG of the BETA FUNCTION B(a;b) /C30g1 0ta /C281 1 /C28t ðÞq /C281dt /C30G(a) G(b) G(a /C27 b) ; where G(z)isa GAMMA FUNCTION , is given by Bq(a; b) /C13g1 0tb /C281 qt;q ðÞa /C281d(a ;t) /C30Gq(b) Gq(a) Gq(a /C27 b) ; where Gq(a)isa Q-GAMMA FUNCTION and (a;q)n is a Q- SERIES coefficient (Andrews 1986, pp. 11 /C1/12). See also Q-FACTORIAL , Q-GAMMA FUNCTION References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986. q-Binomial Coefficient AQ-ANALOG for the BINOMIAL COEFFICIENT , also called a G AUSSIAN COEFFICIENT or a Gaussian poly- nomial. a q-binomial coefficient is given by n m1C2C1C2A q/C13qðÞn qðÞmqðÞn/C28m/C30Ym/C281 i/C3001/C28qn/C28i 1/C28qi/C271; (1) where qðÞk/C13Y/C12 m/C3011 /C28 qm 1 /C28 qk /C27m (2) is a Q-SERIES (Koepf 1998, p. 26). For k;n /C23N; n k1C2C1C2A q/C30[n]q! [k]q![n /C28 k]q! ; (3) where [n]q!isa Q-FACTORIAL (Koepf 1998, p. 30). The q-binomial coefficient can also be defined in terms of the Q-BRACKETS by n k1C2C1C2A q/C13Yk i/C301[n /C28 i /C27 1]q [i]qfor 0 5k 5n 0 otherwise :8 >< >:(4) For q 0 1/C28; the q-binomial coefficients turn into the usual BINOMIAL COEFFICIENT . The first few q-bino- mial coefficients are 2 11C2C1C2A q/C301 /C28 q2 1 /C28 q/C301 /C27q (5) 311C2C1C2A q/C30321C2C1C2A q/C301 /C28 q3 1 /C28 q/C301 /C27q /C27q2 (6) 4 11C2C1C2A q/C30431C2C1C2A q/C301 /C28 q4 1 /C28 q/C301 /C27q /C27q2 /C27q3 (7) 421C2C1C2A q/C301 /C28 q3ðÞ 1 /C28 q4ðÞ 1 /C28 q ðÞ 1 /C28 q2 ðÞ/C301 /C27q /C272q2 /C27q3 /C27q4 : (8) From the definition, it follows that n 11C2C1C2A q/C30n n /C2811C2C1C2A q/C30Xn /C281 i/C300qi (9) Additional identities include n /C27 1 k /C27 11C2C1C2A q n k /C27 11C2C1C2A q/C301 /C28 qn/C271 1 /C28 qn/C28k (10) n /C27 1 k /C27 11C2C1C2A q n /C27 1 k1C2C1C2A q/C301 /C28 qn /C28k /C271 1 /C28 qk /C271: (11) The q-binomial coefficientm/C27n m1C21C3 qcan be interpreted as a polynomial in q whose coefficient qk counts the number of distinct partitions of k elements which fit inside an m /C29n rectangle. For example, the partitions of 1, 2, 3, and 4 are given in the following table.n partitions 0{} 1 {{1}} 2 {{2}, {1, 1}} 3 {{3}, {2, 1}, {1, 1, 1}} 4 {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1},} Of these, { }, f1g;f2g;f1;1g;f2;1g;and f2;2gfit inside a 2 /C292 box. The counts of these having 0, 1, 2, 3, and 4 elements are 1, 1, 2, 1, and 1, so the (4, 2)- binomial coefficient is given by 4 21C2C1C2A q/C301/C27q/C272q2/C27q3/C27q4; (12) as above. See also BINOMIAL COEFFICIENT ,CAUCHY BINOMIAL THEOREM , Q-SERIES References Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Koekoek, R. and Swarttouw, R. F. "The q-Gamma Function and the q-Binomial Coefficient." §0.3 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q - Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and InformaticsReport 98 /C1 /17, pp. 10 /C1/11, 1998. ftp://www.twi.tudelft.nl/ publications/tech-reports/1998/DUT-TWI-98 /C1/17.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 26, 1998. q-Binomial Theorem The Q-ANALOG of the BINOMIAL THEOREM 1/C28z ðÞn /C301/C28nz/C27nn/C281 ðÞ 1/C2152z2/C28nn/C281 ðÞ n/C282 ðÞ 1/C2152/C2153z3/C27... is given by 1/C28z qn ! 1/C28z qn/C281 ! /C1/C1/C11/C28z q ! /C301/C281/C28qn 1/C28qz qn/C271/C28qn 1/C28q1/C28qn/C281 1/C28q2z2 qn/C27n/C281 ðÞ /C28...9zn qnn/C271 ðÞ =2: Written as a Q-SERIES , the identity becomes X/C12 n/C300a;qðÞn q;qðÞnzn /C30az;q ðÞ/C12 z;qðÞ/C12; where a;qðÞn/C30Y/C12 m/C3001 /C28 aqmðÞ 1 /C28 aqm/C27n ðÞ (Heine 1847, p. 303; Andrews 1986). The CAUCHY BINOMIAL THEOREM is a special case of this general theorem. See also BINOMIAL SERIES ,B INOMIAL THEOREM , CAUCHY BINOMIAL THEOREM ,RAMANUJAN PSI SUM References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 10, 1986. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 24, 1995. Gasper, G. "Elementary Derivations of Summation and Transformation Formulas for q-Series." In Fields Inst. Comm. 14 (Ed. M. E. H. Ismail et al. ), pp. 55 /C1/70, 1997. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 7, 1990. Heine, E. "Untersuchungen u¨ber die Reihe 1 /C27(1/C28q a)(1/C28qb) (1 /C28q)(1/C28q g)/C215 x /C27(1 /C28qa)(1/C28qa/C271)(1/C28q b)(1/C28qb /C271) (1/C28q)(1/C28q2)(1/C28qg)(1/C28qg/C271)/C215 x2 /C27...": J. reine angew. Math. 34, 285 /C1/328, 1847. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 26, 1998. q-Bracket The function defined by k½/C138q/C131 /C28 qk 1 /C28 q (1) for integral k. The q-bracket satisfies lim q 01 /C28k½/C138q/C30k : (2) See also Q-BINOMIAL COEFFICIENT , Q-FACTORIAL References Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 26, 1998. q-Chu-Vandermonde Identity A Q-ANALOG of the CHU-VANDERMONDE IDENTITY given by2 f1q /C28n ;b;c;q ;cqn =b ðÞ /C30cqn;q ðÞ/C12c =b;q ðÞ/C12 c;qðÞ/C12cqn =b;q ðÞ/C12/C30c =b;q ðÞn c;qðÞn; where2 f1a;b;c;q ;z ðÞ is the Q-HYPERGEOMETRIC FUNCTION . The identity can also be written as 2 f1q/C28n ;b;c;q; q ðÞ /C30c=b;q ðÞn c;qðÞnbn See also CHU-VANDERMONDE IDENTITY , Q-HYPERGEO- METRIC FUNCTION References Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 18, 1995. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 236, 1990. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 43, 1998. q-Cosine A Q-ANALOG of the COSINE function, as advocated by R. W. Gosper, is defined by cosqz;qðÞ/C30q2z ;pðÞ q20;pðÞ; (1) where q2(z ;p)isaJ ACOBI THETA FUNCTION and p is defined via (ln p)(ln q) /C30p2 : (2) This is a period 2p; EVEN FUNCTION of unit amplitude with double and triple angle formulas and addition formulas which are analogous to ordinary SINE and COSINE . For example, cosq2z ;q ðÞ /C30cos2 qz ;q21CC1CA /C28sin2 qz;q21CC1CA ; (3) where sinqz;aðÞ is the Q-SINE , and pqisQ-PI. The q- cosine also satisfies cosq(pa)/C30P/C12 n/C30/C28/C12(/C281)nqn/C27a ðÞ2 P/C12n/C30/C28/C12(/C281)nqn2: (4) See also Q-FACTORIAL , Q-SINE References Gosper, R. W. "Experiments and Discoveries in q-Trigono- metry." Unpublished manuscript. q-Derivative The Q-ANALOG of the DERIVATIVE , defined by d dx ! qf(x)/C30f(x)/C28f(qx) x/C28qx: For example, d dx ! qsin x /C30sin x /C28 sin(qx) x /C28 qx d dx ! qln x /C30ln x /C28 ln(qx) x /C28 qx/C30ln1 q1CAr1CA7 (1 /C28 q)x d dx ! qx2 /C30x2 /C28 q2x2 x /C28 qx/C30(1 /C27q)x d dx ! qx3 /C30x3 /C28 q3x3 x /C28 qx/C30 1 /C27q /C27q21CC1CA x2 : In the LIMIT q 0 1; the q-derivative reduces to the usual DERIVATIVE . See also DERIVATIVE q-Dimension Dq /C131 1 /C28 qlim o 00ln I(q; o) ln1 o1CAr1CA7 (1) where Iq; oðÞ/C13XN i/C301mq i ; (2) /o is the box size, and mi is the NATURAL MEASURE . The CAPACITY DIMENSION (a.k.a. box-counting dimen- sion) is given by q /C300, D0 /C301 1 /C28 0lim o 00lnPN oðÞ i/C301 11CAr1CA7 /C28ln o/C30/C28lim o 00ln N oðÞ½/C138 ln o(3) If all mi/s are equal, then the CAPACITY DIMENSION is obtained for any q. The INFORMATION DIMENSION corresponds to q /C301 and is given by D1 /C30lim q01Dq /C30lim q 01limo 00lnPN oðÞ i/C301 mqihi /C28ln o 1 /C28 q /C30lim o 00lim q 01lnPN oðÞ i/C301 mqihi q /C28 1 ðÞ ln o: (4) But for the numerator, lim q01lnXN oðÞ i/C301mqi ! /C30lnXN oðÞ i/C301mi ! /C30ln1 /C300; (5) and for the denominator, limq 01q /C281 ðÞ /C300; so use L’HOSPITAL’S RULE to obtainD1 /C30lim o 001 ln olim q 01Pmqiln mi 1 ! : (6) Therefore, D1 /C30lim o 00PN oðÞ i/C301 mi ln mi ln o ! (7) (Ott 1993, p. 79). /D2 is called the CORRELATION DIMENSION . If q1 > q2 ; then Dq15Dq2(8) (Ott 1993, p. 79). See also CAPACITY DIMENSION ,CORRELATION DIMEN- SION,FRACTAL DIMENSION ,INFORMATION DIMENSION References Grassberger, P. "Generalized Dimensions of Strange Attrac- tors." Phys. Lett. A 97, 227, 1983. Hentschel, H. G. E. and Procaccia, I. "The Infinite Number of Generalized Dimensions of Fractals and Strange At- tractors." Physica D 8, 435, 1983. Ott, E. "Measure and the Spectrum of Dq Dimensions." §3.3 in Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 78 /C1/81, 1993. Re´nyi, A. Probability Theory. Amsterdam, Netherlands: North-Holland, 1970. q-Dougall Sum 8 f7a;qa1 =2 ;/C28qa1 =2 ; b;c ;d; e; q/C28N a1 =2 ;/C28a1=2 ;aq b;aq c;aq d;aq e;aqN /C271;q;q2 435 /C30aq bd;q ! Naq ed;q ! Naq;q ðÞNaq be;q ! N aq bd;q ! Naq bed ;q ! Naq b;q ! Naq e;q ! N; where8 f7 is a Q-HYPERGEOMETRIC SERIES . References Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 36, 1995. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 35, 1990. Q.E.D. An abbreviation for the Latin phrase "quod erat demonstrandum" ("that which was to be demon- strated"), a NOTATION which is often placed at the end of a mathematical PROOF to indicate its comple- tion. See also PROOF q-Extension Q-ANALOG q-Factorial The Q-ANALOG of the FACTORIAL (by analogy with the Q-GAMMA FUNCTION ). For a an integer, the q-factorial is defined by [k]q! /C30faq(k; q) /C301(1 /C27q)1/C27q /C27q21CC1CA /C1/C1/C1 1 /C27q /C27.../C27qk/C2811CC1CA (1) /C30(q;q)k (1 /C28 q)k (2) (Koepf 1998, p. 26). For k /C23N; [k]q! /C30Gq(k /C271); (3) where Gq(k /C271) is the Q-GAMMA FUNCTION . The first few values are [1]q! /C301 [2]q! /C301 /C27q [3]q! /C30(1 /C27q)1/C27q /C27q21CC1CA /C301 /C272q /C272q2 /C27q3 [4]q! /C30(1 /C27q)1/C27q /C27q21CC1CA 1 /C27q /C27q2 /C27q31CC1CA /C301 /C273q /C275q2 /C276q3 /C275q4 /C273q5 /C27q6 : A reflection formula analogous to the GAMMA FUNC- TION reflection formula is given by cosq( pa) /C30sinqp1 2 /C28a1CAr1CA7hi /C30pqq a /C281 =2 ðÞ a /C271 =2 ðÞ faq a /C281 2 ;q21CAr1CA7 faq /C28 a /C27121CAr1CA7 ;q21CAr1CA7 ; (4) where cosq(z) is the Q-COSINE , sinq(z) is the Q-SINE , and pq is Q-PI. See also Q-BETA FUNCTION , Q-BINOMIAL COEFFI- CIENT , Q-BRACKET , Q-COSINE , Q-GAMMA FUNCTION , Q-PI, Q-SINE References Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Gosper, R. W. "Experiments and Discoveries in q-Trigono- metry." Unpublished manuscript. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 26 and 30, 1998. Q-Function Let q /C30e /C28pK ?=K /C30e /C28ip t ; (1) thenQ0 /C13Y/C12 n/C3011 /C28q2n1CC1CA (2) Q1 /C13Y/C12 n/C3011 /C27q2n1CC1CA (3) Q2 /C13Y/C12 n /C3011 /C27q2n /C2811CC1CA (4) Q3 /C13Y/C12 n/C3011 /C28q2n/C2811CC1CA : (5) The Q-functions are sometimes written using a lower-case q instead of a capital Q. The Q-functions also satisfy the identities Q0Q1 /C30Q0q21CC1CA (6) Q0Q3 /C30Q0q1 =21CC1CA (7) Q2Q3 /C30Q3q21CC1CA (8) Q1Q2 /C30Q1q1 =21CC1CA : (9) The NORMAL DISTRIBUTION FUNCTION F(x) is some- times also denoted Q(x) :/ See also HOFSTADTER’S Q-SEQUENCE ,JACOBI IDENTI- TIES,N ORMAL DISTRIBUTION FUNCTION ,PARTITION FUNCTION Q, Q-SERIES References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 55 and 63 /C1/85, 1987. Tannery, J. and Molk, J. Elements de la The ´orie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars et fils, 1893 /C1/1902. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 469 /C1/473 and 488 /C1/489, 1990. q-Gamma Function AQ-ANALOG of the GAMMA FUNCTION defined by Gq(x)/C13(q;q)/C12 qx;q ðÞ/C121/C28q ðÞ1/C28x; (1) where x;qðÞ/C12is a Q-SERIES (Koepf 1998, p. 26; Koekoek and Swarttouw 1998). The q-gamma func- tion satisfies lim q01/C28Gq(x)/C30G(x) (2) where G(z) is the GAMMA FUNCTION , (Andrews 1986). The q-gamma function satisfies the functional equa- tion Gq(z/C271)/C301/C28qz 1/C28qGq(z) (3) with Gq(1) (Koekoek and Swarttouw 1998), which simplifies to G(z /C271) /C30z G(z) (4) as q 0 1/C28: A curious identity for the functional equation f(a /C28b)f(a /C28c)f(a /C28d)f(a /C28e) /C28f(b)f(c)f(d)f(e) /C30qbf(a)f(a /C28b /C28c)f(a /C28b /C28d)f(a /C28b /C28e); (5) where b /C27c /C27d /C27e /C302a (6) is given by f( a) /C30sin(ka) for q /C301 1 Gq( a) Gq(1 /C28 a)for 0 Bq B1;8 < : (7) for any k. See also GAMMA FUNCTION , Q-BETA FUNCTION , Q- FACTORIAL References Andrews, G. E. "W. Gosper’s Proof that limq01 /C28Gq(x) /C30G(x):/" Appendix A in q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 11 and 109, 1986. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Koekoek, R. and Swarttouw, R. F. "The q-Gamma Function and the q-Binomial Coefficient." §0.3 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q- Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/17, pp. 10 /C1/11, 1998. ftp://www.twi.tudelft.nl/ publications/tech-reports/1998/DUT-TWI-98 /C1/17.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Wenchang, C. Problem 10226 and Solution. "A q-Trigono- metric Identity." Amer. Math. Monthly 103, 175 /C1/177, 1996. q-Gauss Identity A Q-ANALOG of Gauss’s theorem due to Jacobi and Heine, 2 f1a ;b;c;q;c =(ab) ðÞ /C30c =a;q ðÞ/C12c =b;q ðÞ/C12 c;qðÞ/C12c = abðÞ;q ðÞ/C12(1) for c=(ab) jjB1 (Gordon and McIntosh 1997; Koepf 1998, p. 40), where2 f1a ;b;c;q;z ðÞ is a Q-HYPERGEO- METRIC SERIES . A special case for /a/C30q/C28n /is given by Xn k/C300qk2n k1C2C1C2A2 q/C30ffiffiffiqp;q1CC1CA n/C28ffiffiffiqp;q1CC1CA n/C28q;q ðÞn q;qðÞn; wheren k1C21C3 qis a Q-BRACKET (Koepf 1998, p. 43). See also Q-CHU-VANDERMONDEC IDENTITY , Q-HYPER- GEOMETRIC SERIESReferences Bhatnagar, G. Inverse Relations, Generalized Bibasic Ser- ies, and their U (n) Extensions. Ph.D. thesis. Ohio State University, p. 31, 1995. Gasper, G. and Rahman, M. Basic Hypergeometric Series. - Cambridge, England: Cambridge University Press, pp. 10 and 236, 1990. Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431/C1/448, 1997. Koepf, W. Hypergeometric Summation: An Algorithmic Ap- proach to Summation and Special Function Identities. - Braunschweig, Germany: Vieweg, 1998. q-Generalization Q-ANALOG q-Harmonic Series The series hq/C28rðÞ/C30X/C12 n/C3011 qn/C27r(1) forqanINTEGER other than 0 and 91 which is the Q- ANALOG of Hn/C30X/C12 n/C3011 n: (2) /hqand the related series Lnq(/C28r/C271)/C30X/C12 n/C301(/C281)n qn/C27r; (3) which is a q-extension of the NATURAL LOGARITHM ln 2 ;are irrational for raRATIONAL NUMBER other than 0 or /C28qn(Guy 1994). In fact, Amdeberhan and Zeilberger (1998) showed that the IRRATIONALITY MEASURES of both hq(1) and Lnq(2) are 4.80, improv- ing the value of 54.0 implied by Borwein (1991, 1992). Amdeberhan and Zeilberger (1998) also show that the q-harmonic series and q-extension of ln 2 can be written in the more quickly converging forms hq1ðÞ/C30X/C12 n/C301qn 1/C28qn ðÞ (q)n(4) /C30X/C12 n/C3011/C28qn/C28q2n qn/C281 ðÞ2n n1CA81CA9 q(q)n(5) Lnq(2)/C30X/C12 n/C301qn(q)n 1/C28qn ðÞ q2ðÞn(6) /C30X/C12 n/C301(/C281)n/C281qðÞn1/C28q3nðÞ 1/C28qn ðÞ22n n1CA81CA9 qq2ðÞn; (7) wheren k1CC1CA qis a Q-BINOMIAL COEFFICIENT and (q)n /C30(1 /C28q)1/C28q21CC1CA /C1/C1/C1 1 /C28qnðÞ (8) for n ]1 :/ See also HARMONIC SERIES ,IRRATIONALITY MEASURE References Amdeberhan, T. and Zeilberger, D. "q-Ape´ry Irrationality Proofs by q-WZ Pairs." Adv. Appl. Math. 20, 275 /C1/283, 1998. Borwein, P. B. "On the Irrationality of a1 = qn /C27r ðÞ :/" J. Number Th. 37, 253 /C1/259, 1991. Borwein, P. B. "On the Irrationality of Certain Series." Math. Proc. Cambridge Philos. Soc. 112, 141 /C1/146, 1992. Breusch, R. "Solution to Problem 4518." Amer. Math. Monthly 61, 264 /C1/265, 1954. Erdos, P. "On Arithmetical Properties of Lambert Series." J. Indian Math. Soc. 12,63/C1/66, 1948. Erdos, P. "On the Irrationality of Certain Series: Problems and Results." In New Advances in Transcendence Theory. Cambridge, England: Cambridge University Press, pp. 102 /C1/109, 1988. Erdos, P. and Kac, M. "Problem 4518." Amer. Math. Monthly 60, 47, 1953. Guy, R. K. "Some Irrational Series." §B14 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 69, 1994. q-Hypergeometric Function The modern definition of the q-hypergeometric func- tion is r fsa1 ;a2 ;...;ar b1 ;...bs;q ;z1C2C1C2A /C13X/C12 n/C300a1;q ðÞna2;q ðÞn... ar;q ðÞn b1;q ðÞn... bs;q ðÞnzn (q;q)n /C2 (/C281)nqn 21CA81CA92 643 751 /C27s/C28r ; (1) wheren 21CC1CA /C301 2nn/C281 ðÞ is a BINOMIAL COEFFICIENT and (a;q)n is a Q-POCHHAMMER SYMBOL (a;q)n /C30(1 /C28a)(1 /C28aq)1/C28aq21CC1CA /C1/C1/C1 1 /C28aqn /C2811CC1CA (2) (a;q)0 /C301 (3) (Gasper and Rahman 1990; Bhatnagar 1995, p. 21; Koepf 1998, p. 25). An old-fashioned definition omits the factor [(/C281)kqn 2ðÞ]1 /C27s/C28r ; r f?sa1 ;a2 ;...;ar b1 ;...; bs;q;z1C2C1C2A /C13X/C12 n /C300a1;q ðÞna2;q ðÞn... ar;q ðÞn b1;q ðÞn... bs;q ðÞnzn q;qðÞn; (4) This is the q-hypergeometric function as defined by Bailey (1935), Slater (1966), Andrews (1986), and Hardy (1999).A particular case ofr f ?s is given by 2 c?1(a ;b;c;q;z) /C30X/C12 n/C300(a;q)n(b;q)nzn (q;q)n(c;q)n(5) (Andrews 1986, p. 10). A q-analog of Gauss’s theorem (the Q-GAUSS IDENTITY ) due to Jacobi and Heine is given by 2 f?1a ;b;c;q;c =(ab) ðÞ /C30c=a;q ðÞ/C12c =b;q ðÞ/C12 c;qðÞ/C12c= abðÞ;q ðÞ/C12(6) for c =(ab) jjB1 (Koepf 1998, p. 40). Heine proved the transformation formula 2 f ?1(a ;b;c;q;z) /C30(b;q)/C12(az;q)/C12 (c;q)/C12(z;q) /C122 f1c=b2a;az;q;b ðÞ ; (7) (Andrews 1986, pp. 10 /C1/11). Rogers (1893) obtained the formulas 2 f ?1(a ;b;c;q;z) /C30c =b;q ðÞ/C12(bz;q) /C12 (z;q)/C12(c;q) /C122 f1b;abz =c;bz;q;c =b ðÞ (8) 2 f?1(a ;b; c;q ;z) /C30 abz=c;q ðÞ/C12(z;q) /C12 2 f1 c =a ;c =b;c;q ;abz=c ðÞ (9) (Andrews 1986, pp. 10 /C1/11). The functionrfshas the simple confluent identity lim ar0/C12rfsa1;a2;...;ar b1;...;bs;q;z ar"# /C30a1;a2;...;ar/C281 b1;...;bs;q;z1C2C1C2A : (10) In the limit q01/C28; lim q01/C28rfsqa1qa2;...;qar qb1;...;qbs;q;(q/C281)1/C27s/C28rz1C2C1C2A /C30rFsa1;a2;...;ar b2;...;bs;z1C2C1C2A ; (11) whererFsis a GENERALIZED HYPERGEOMETRIC FUNC- TION (Koepf 1998, p. 25). See also GENERALIZED HYPERGEOMETRIC FUNCTION , Q-POCHHAMMER SYMBOL , Q-SAALSCHUETZ SUM, Q- SERIES References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc.,p. 10, 1986. Bailey, W. N. "Basic Hypergeometric Series." Ch. 8 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 65 /C1 /72, 1935. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 21, 1995. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Gasper, G. "Elementary Derivations of Summation and Transformation Formulas for q-Series." In Fields Inst. Comm. 14 (Ed. M. E. H. Ismail et al. ), pp. 55 /C1/70, 1997. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 107 /C1/111, 1999. Heine, E. "U¨ ber die Reihe 1 /C27q a/C281 ðÞ qb /C281 ðÞ q/C281 ðÞ q g /C281 ðÞx /C27q a/C281 ðÞ qa/C271 /C281 ðÞ qb /C281 ðÞ qb /C271 /C281 ðÞ q/C281 ðÞ q2 /C281 ðÞ q g /C281 ðÞ q g/C271 /C281 ðÞx2 /C27...": J. reine angew. Math. 32, 210 /C1/212, 1846. Heine, E. "Untersuchungen u¨ber die Reihe 1 /C271/C28qaðÞ 1/C28q bðÞ 1/C28q ðÞ 1/C28qg ðÞ/C215 x /C271 /C28qaðÞ 1/C28qa/C271ðÞ 1/C28q bðÞ 1/C28q b/C271ðÞ 1/C28q ðÞ 1/C28q2 ðÞ 1 /C28qg ðÞ 1 /C28qg/C271 ðÞ/C215 x2 /C27...": J. reine angew. Math. 34, 285 /C1/328, 1847. Heine, E. Theorie der Kugelfunctionen und der verwandten Functionen, Bd. 1. Berlin: Reimer, pp. 97 /C1/125, 1878. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 25 /C1/26, 1998. Krattenthaler, C. "HYP and HYPQ." J. Symb. Comput. 20, 737 /C1/744, 1995. Rogers, L. J. "On a Three-Fold Symmetry in the Elements of Heine’s Series." Proc. London Math. Soc. 24, 171 /C1/179, 1893. Slater, L. J. Generalized Hypergeometric Functions. Cam- bridge, England: Cambridge University Press, 1966. q-Hypergeometric Series Q-HYPERGEOMETRIC FUNCTION q-Integral A q-analog of integration gqF(x)d(qx) which reduces to gF xðÞdx in the case q /C301. A specific case gives g/C12 0qxa /C281 1 /C28 xd(qx) /C30Gq1 21CAr1CA7hi sq(a)2 ; where Gq is the q-Gamma function and sq is a doubly periodic sigma function. If q /C301, the integral reduces to g/C12 0xa/C281 1 /C28 xdx /C30p sin( pa) : References Jackson, F. H. "q-Definite Integrals." Quart. J. Math. 41, 163, 1910.Jackson, F. H. "The q-Integral Analogous to Borel’s Inte- gral." Mess. Math. 47,5 7/C1/64, 1917. Q-Matrix FIBONACCI Q-MATRIX q-Multinomial Coefficient AQ-ANALOG of the MULTINOMIAL COEFFICIENT , de- fined as a1/C27.../C27an ½/C138 ! a1½/C138!...an½/C138!; where n½/C138!/C13(1)(1/C27q)/C1/C1/C11/C27q/C27.../C27qn/C2811CC1CA : See also MULTINOMIAL COEFFICIENT ,ZEILBERGER- BRESSOUD THEOREM Q-Number HOFSTADTER’S Q-SEQUENCE q-Pfaff-Saalschuetz Sum Q-SAALSCHUETZ SUM q-Pi The Q-ANALOG ofPIpqcan be defined by taking a/C300 in the Q-FACTORIAL faq(a;q)/C301(1/C27q)1/C27q/C27q21CC1CA /C1/C1/C11/C27q/C27.../C27qa/C2811CC1CA ; giving 1/C30sinq1 2p1CAr1CA7 /C30pq faq2/C281 2;q21CAr1CA7 q1=4; where sinq(z) is the Q-SINE . Gosper has developed an iterative algorithm for computing pqbased on the algebraic RECURRENCE RELATION 4pq4 q4/C271q2/C271 ðÞ2p2 q pq2/C28q4/C271 ðÞ p2 q2 pq4 q-Pochhammer Symbol The Q-ANALOG of the P OCHHAMMER SYMBOL defined by (a;q)k/C30Qk/C281 j/C3001/C28aqjðÞ ifk>0 1i f k/C300Qk j/C3001/C28aq/C28jðÞ/C281ifkB0Q/C12 j/C3001/C28aqjðÞ ifk/C30/C128 >>< >>:(1) (Koepf 1998, p. 25). q-Pochhammer symbols are frequently calledQ-SERIES and, for brevity, a;qðÞkis often simply written aðÞk:/ For q 0 1 /C28; lim q 01 /C28q a;q ðÞk (1 /C28 q)k /C30( a)k (2) gives the normal POCHHAMMER SYMBOL ( a)n (Koekoek and Swarttouw 1998, p. 7). The q-Pochhammer sym- bols are also called q-shifted factorials (Koekoek and Swarttouw 1998, pp. 8 /C1/9). The q-Pochhammer symbol satisfies (a;q)n /C30(a;q)/C12 aqn;q ðÞ/C12(3) 1 /C28 aq2n 1 /C28 a/C30qffiffiffiffiffia;pq1CC1CA n/C28qffiffiffiffiffia;pq1CC1CA nffiffiffiffiffia;pq1CC1CA n/C28ffiffiffiffiffia;pq1CC1CA n(4) (a;q)n(/C28a;q)n /C30 a2;q21CC1CA n (a;q)n /C30 q1 /C28n =a;q1CC1CA n(/C28a)nq n 2ðÞ (5) a;q /C2811CC1CA n/C30 a /C281;q1CC1CA n(/C28a)nq/C28 n 2ðÞ (6) (a;q)/C28n /C301 aq /C28n;q ðÞn/C30/C28q=a ðÞn q=a;q ðÞnqn 2ðÞ; (7) wheren 21CC1CA is a BINOMIAL COEFFICIENT and n 21CA81CA9 /C301 2n(n /C281); (8) as well as many other identities, some of which are given by Koekoek and Swarttouw (1998, p. 9). A generalized q-Pochhammer symbol can be defined using the concise notation a1 ;a2 ;...;ar;q ðÞ/C12/C30 a1;q ðÞ/C12a2;q ðÞ/C12... ar;q ðÞ/C12 (9) (Gordon and McIntosh 2000). See also POCHHAMMER SYMBOL , Q-SERIES References Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." To appear in J. London Math. Soc. 2000. Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analo- gue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/17, p. 7, 1998. ftp://www.twi.tudelft.nl/publications/ tech-reports/1998/DUT-TWI-98 /C1/17.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 25 and 30, 1998. Q-Polynomial BLM /HO POLYNOMIAL q-Product Q-FUNCTIONQR Decomposition Given a MATRIX A ; its QR-decomposition is OF THE FORM A /C30QR; where R is an upper TRIANGULAR MATRIX and Q isan ORTHOGONALMATRIX ,i.e.,onesatisfying QTQ /C30I where I is the IDENTITY MATRIX . This matrix decom- position can be used to solve linear systems of equations. QR decomposition is implemented in Mathematica asQRDecomposition [m]. See also CHOLESKY DECOMPOSITION ,LUD ECOMPOSI- TION ,M ATRIX DECOMPOSITION , PSLQ ALGORITHM , SINGULAR VALUE DECOMPOSITION References Gentle, J. E. "QR Factorization." §3.2.2 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer- Verlag, pp. 95 /C1/97, 1998. Householder, A. S. The Numerical Treatment of a Single Non-Linear Equations. New York: McGraw-Hill, 1970. Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 26 /C1/28, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "QR Decomposition." §2.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 91 /C1/95, 1992. Stewart, G. W. "A Parallel Implementation of the QR Algorithm." Parallel Comput. 5, 187 /C1/196, 1987. ftp:// thales.cs.umd.edu/pub/reports/piqra.ps. q-Saalschuetz Sum A q-analog of the Saalschu ¨tz theorem due to Jackson is given by 3 f2q /C28n ; a;b;c;ab = cqn/C2811CC1CA ;q; q1CC1CA /C30c=a;q ðÞnc=b;q ðÞn c;qðÞnc=abðÞ;q ðÞn(1) where3f2is the Q-HYPERGEOMETRIC FUNCTION (Koepf 1998, p. 40; Schilling and Warnaar 1999). See also Q-HYPERGEOMETRIC FUNCTION References Andrews, G. E. Encyclopedia of Mathematics and Its Appli- cations, Vol. 2: The Theory of Partitions. Cambridge, England: Cambridge University Press, 1984. Bailey, W. N. "The Analogue of Saalschu ¨tz’s Theorem." §8.4 inGeneralised Hypergeometric Series. Cambridge, Eng- land: University Press, p. 68, 1935. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U (n) Extensions. Ph.D. thesis. Ohio State University, p. 30, 1995. Carlitz, L. "Remark on a Combinatorial Identity." J. Com- bin. Th. Ser. A 17, 256/C1/257, 1974. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 13, 1990. Gould, H. W. "A New Symmetrical Combinatorial Identity." J. Combin. Th. Ser. A 13, 278/C1/286, 1972. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities.Braunschweig, Germany: Vieweg, pp. 25 /C1 /26, 1998. Schilling A. and Warnaar, S. O. A Generalization of the q.- Saalschu ¨tz Sum and the Burge Transform 8 Sep 1999. http://xxx.lanl.gov/abs/math.QA/9909044/. Watson, G. N. "A New Proof of the Rogers-Ramanujan Identities." J. London Math. Soc. 4,4/C1/9, 1929. q-Series ASERIES involving coefficients OF THE FORM (a;q)n/C13(a)n/C30Yn/C281 k/C3001/C28aqk1CC1CA (1) /C30Y/C12 k/C3001/C28aqk1CC1CA 1/C28aqk/C27n ðÞ(2) /C30(a;q)/C12 aqn;q ðÞ/C12(3) forn]1;also called a Q-POCHHAMMER SYMBOL (Andrews 1986, p. 10). The notation (q)n/C13(q;q)n/C30Yn/C281 k/C3011/C28qk1CC1CA (4) is also used (Hirschhorn 1999). The symbol for n0/C12 is defined as (a)/C12/C13(a;q)/C12/C30Y/C12 k/C3001/C28aqk1CC1CA ; (5) giving the special case h(t)/C30(q;q)/C12/C30q1=24Y/C12 k/C3001/C28q/C215qk1CC1CA /C30q1=24Y/C12 k/C3011/C28qk1CC1CA ; (6) where q/C13e2pirandh(t) is called the D EDEKIND ETA FUNCTION . Identities involving ( q)/C12include (q)3 /C12/C30X/C12 n/C300(/C281)n(2n/C271)qnn/C271 ðÞ =2(7) /C30X/C272qY (8) (Hardy and Wright 1979, Hirschhorn 1999), where X/C30Y/C12 n/C3011/C28q25n/C28151CC1CA 1/C28q25n/C28101CC1CA 1/C28q25n1CC1CA /C30X/C12 /C28/C12(/C281)nq25n2/C285n ðÞ =2(9)Y/C30Y/C12 n/C3011/C28q25n/C28301CC1CA 1/C28q25n/C2851CC1CA 1/C28q25n1CC1CA /C30X/C12 /C28/C12(/C281)nq25n2/C2815n ðÞ =2(10) (Hirschhorn 1999) The symbols [n]/C131/C27q/C27q2/C27.../C27qn/C281(11) [n]!/C13[n][n/C281]/C1/C1/C1[1] (12) are sometimes also used when discussing q-series. There are a great many other beautiful identities involving q-series, some of which follow directly by taking the Q-ANALOG of standard combinatorial iden- tities, e.g., the Q-BINOMIAL THEOREM X/C12 n/C300(a;q)nzn (q;q)n/C30(az;q)/C12 (z;q)/C12(13) (/jzjB1;jqjB1; Andrews 1986, p. 10), a special case of an identity due to Euler (aq;q)/C12/C30X/C12 k/C300(/C281)kqkk/C271 ðÞ =2ak (c;q)k(14) (Gasper and Rahman 1990, p. 9; Leininger and Milne1997), and Q-VANDERMONDE SUM 2f1a;q/C28n;c;q;q ðÞ /C30anc=a;q ðÞn (c;q)n; (15) where2f1a;b;c;q;z ðÞ is a Q-HYPERGEOMETRIC SERIES . Other q-series identities, e.g., the J ACOBI IDENTITIES , ROGERS- RAMANUJAN IDENTITIES , and Q-HYPERGEO- METRIC identity 2f1(a;b;c;q;z) /C30(b;q)/C12(az;q)/C12 (c;q)/C12(z;q)/C122f1c=b;a;az;q;b ðÞ ; (16) seem to arise out of the blue. Another such example is X/C12 n/C300/C28q;q2ðÞnqnn/C281 ðÞzn z;q2 ðÞn/C30X/C12 n/C300/C28zq;q4ðÞnqn2n/C281 ðÞzn z;q2 ðÞ2n/C271(17) (Gordon and McIntosh 2000). Asymptotic results for q-series include (q)/C12/C30ffiffiffiffiffiffi 2p ts exp/C28p2 6t/C27t 24 ! /C27X1ðÞ (18) q2;q21CC1CA /C12/C30ffiffiffi p ts exp/C28p2 12t/C27t 12 ! /C27X1ðÞ (19) q;q21CC1CA /C12/C30(q)/C12 q2;q2 ðÞ/C12/C30ffiffiffi 2p exp /C28p2 12t /C28t 24 ! /C27X 1ðÞ(20) (Watson 1936, Gordon and McIntosh 2000). See also BORWEIN CONJECTURES ,D EDEKIND ETA FUNCTION ,FINE’S EQUATION ,GAUSSIAN COEFFICIENT , JACKSON’S IDENTITY ,JACOBI IDENTITIES ,M OCK THE- TA FUNCTION , Q-ANALOG , Q-BINOMIAL THEOREM , Q- COSINE , Q-FACTORIAL , Q-FUNCTION , Q-GAMMA FUNC- TION , Q-HYPERGEOMETRIC FUNCTION , Q-MULTINO- MIAL COEFFICIENT , Q-POCHHAMMER SYMBOL , Q-SINE, RAMANUJAN PSI SUM,RAMANUJAN THETA FUNCTIONS , ROGERS- RAMANUJAN IDENTITIES References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986. Berndt, B. C. "q-Series." Ch. 27 in Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 261 /C1/286, 1994. Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan’s Lost Notebook." To appears in Trans. Amer. Math. Soc. Bhatnagar, G. "A Multivariable View of One-Variable q- Series." In Special Functions and Differential Equations. Proceedings of the Workshop (WSSF97) held in Madras, January 13 /C1/24, 1997) (Ed. K. S. Rao, R. Jagannathan, G. van den Berghe, and J. Van der Jeugt). New Delhi, India: Allied Pub., pp. 60 /C1/72, 1998. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Gasper, G. "Elementary Derivations of Summation and Transformation Formulas for q-Series." In Fields Inst. Comm. 14 (Ed. M. E. H. Ismail et al. ), pp. 55 /C1/70, 1997. Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." To appear in J. London Math. Soc. 2000. Gosper, R. W. "Experiments and Discoveries in q-Trigono- metry." Unpublished manuscript. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hirschhorn, M. D. "Another Short Proof of Ramanujan’s Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580 /C1/583, 1999. Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analo- gue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/17, 1 /C1/168, 1998. ftp://www.twi.tudelft.nl/publications/ tech-reports/1998/DUT-TWI-98 /C1/17.ps.gz. Leininger, V. E. and Milne, S. C. "Some New Infinite Families of Eta Function Identities." Preprint. http:// www.math.ohio-state.edu/~milne/preprints.html. Watson, G. N. "The Final Problem: An Account of the Mock Theta Functions." J. London Math. Soc. 11,55/C1/80, 1936. Weisstein, E. W. "Books about q-Series." http://www.trea- sure-troves.com/books/q-Series.html. q-Shifted Factorial Q-POCHHAMMER SYMBOLQ-Signature SIGNATURE (RECURRENCE RELATION ) q-Sine The Q-ANALOG of the SINE function, as advocated by R. W. Gosper, is defined by sinq(z ;q) /C30q1(z; p) q11 2 p;p1CAr1CA7 ; where q1(z ;p)isaJ ACOBI THETA FUNCTION and p is defined via (ln p)(ln q) /C30p2 : This is a period 2p; ODD FUNCTION of unit amplitude with double and triple angle formulas and addition formulas which are analogous to ordinary SINE and COSINE . For example, sinq(2z ;q) /C30(q /C271)pq Pq2cosqz ;q21CC1CA sinqz ;q21CC1CA ; where cosq(z; a) is the Q-COSINE , and pq is Q-PI. See also Q-COSINE , Q-FACTORIAL References Gosper, R. W. "Experiments and Discoveries in q-Trigono- metry." Unpublished manuscript. Quadrable A plane figure for which QUADRATURE is possible is said to be quadrable. Quadrangle A plane figure consisting of four points, each of which is joined to two other points by a LINE SEGMENT (where the line segments may intersect). A quadran- gle may therefore be CONCAVE orCONVEX ;i fi ti s CONVEX , it is called a QUADRILATERAL . See also COMPLETE QUADRANGLE ,CYCLIC QUADRAN- GLE,QUADRILATERAL ,TETRASTIGM References Coxeter, H. S. M. and Greitzer, S. L. "Collinearity and Concurrence." Ch. 3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 51 /C1/79, 1967. Durell, C. V. "The Quadrilateral and Quadrangle." Ch. 7 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 77 /C1/87, 1928. Quadrant One of the four regions of the PLANE defined by the four possible combinations of SIGNS (/C27;/C27) ; (/C27;/C28); (/C28;/C27); and (/C28;/C28) for (x, y). See also OCTANT , X-AXIS, Y-AXIS References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 73, 1996. Quadratfrei SQUAREFREE Quadratic Congruence Equation A CONGRUENCE OF THE FORM ax2 /C27bx /C27c /C130 (mod m); where a, b, and c are INTEGERS . A general quadratic congruence can be reduced to the congruence x2 /C13q (mod p) and can be solved using EXCLUDENTS , although solution of the general polynomial congruence amxm /C27.../C27a2x2 /C27a1x /C27a0 /C130 (mod n) is intractable. See also CONGRUENCE ,CONGRUENCE EQUATION ,EX- CLUDENT ,LINEAR CONGRUENCE EQUATION Quadratic Curve The general bivariate quadratic curve can be written ax2 /C272bxy /C27cy2 /C272dx /C272fy /C27g /C300: (1) Define the following quantities: D/C30abd bcf df g1CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA2(2)J /C30ab bc1CA21CA21CA21CA21CA21CA21CA21CA2 (3) I /C30a /C27c (4) K /C30ad dg1CA21CA21CA21CA21CA21CA21CA21CA2/C27 cf fg1CA21CA21CA21CA21CA21CA21CA21CA2: (5) Then the quadratics are classified into the types summarized in the following table (Beyer 1987). The real (nondegenerate) quadratics (the ELLIPSE , HYPER- BOLA , and PARABOLA ) correspond to the curves which can be created by the intersection of a PLANE with a (two- NAPPES ) CONE , and are therefore known as CONIC SECTIONS . Curve / D/ J / D=I/ K Coincident Lines 0 0 0 Ellipse (Imaginary) /"0//> 0//> 0/ ELLIPSE (Real) /"0//> 0//B0/ HYPERBOLA /"0//B0/ Intersecting Lines (Imaginary)0 /> 0/ Intersecting Lines (Real) 0 /B0/ PARABOLA /"0/0 Parallel Lines (Imaginary) 0 0 />0/ Parallel Lines (Real) 0 0 /B0/ It is always possible to eliminate the xycross term by a suitable ROTATION of the axes. To see this, consider rotation by an arbitrary angle u:The ROTATION MATRIX is x y1C2C1C2A /C30cosusinu /C28sinucosu1C2C1C2A x? y?1C2C1C2A /C30x?cosu/C27y?sinu /C28x?sinu/C27y?cosu1C2C1C2A ;(6) so x/C30x?cosu/C27y?sinu (7) y/C30/C28x?sinu/C27y?cosu (8) xy/C30/C28x?2cosusinu/C27x?y?cos2u/C28sin2u1CC1CA /C27y?2cosusinu (9) x2/C30x?2cos2u/C272x?y?cosusinu/C27y?2sin2u (10) y2/C30/C28x?2sin2u/C282x?y?sinucosu/C27y?2cos2u:(11) Plugging these into (1) gives ax?2cos2u/C272x?y?cosu/C27y?2sin2u1CC1CA /C272b(x?cosu/C27y?sinu)/C28(/C28x?sinu/C27y?cosu) /C27cx?2sin2u/C282x?y?cosusinu/C27y?2cos2u1CC1CA /C272d(x?cosu/C27y?sinu) /C272f(/C28x?sinu/C27y?cosu)/C27g/C300: (12) ax?2cos2u/C272x?y?cosu/C27y?2sin2u1CC1CA /C272b/C28x2cos2usinu/C28xysin2u/C27xycos2u/C27y2cosusinu1CC1CA /C27cx?2sin2u/C282x?y?cosusinu/C27y?2cos2u1CC1CA /C272d(x?cosu/C27y?sinu) /C272f(/C28x?sinu/C27y?cosu)/C27g/C300: (13) Grouping terms, x?2acos2u/C27csin2u/C282bcosusinu1CC1CA /C27x?y?2acosusinu/C282csinucosu/C272bcos2u/C28sin2u1CC1CA 1C21C3 /C27y?2asin2u/C27ccos2u/C272bcosusinu1CC1CA /C27x?(2dcosu/C282fsinu)/C27y?(/C282dsinu/C272fcosu) /C27g/C300: (14) Comparing the COEFFICIENTS with (1) gives an equa- tion OF THE FORM a?x?2/C272b?x?y?/C27c?y?2/C272d?x?/C272f?y?/C27g?/C300; (15) where the new COEFFICIENTS are a?/C30acos2u/C282bcosusinu/C27csin2u (16) b?/C30bcos2u/C28sin2u1CC1CA /C27a/C28c ðÞ sinucosu (17) c?/C30asin2u/C272bsinucosu/C27ccos2u (18) d?/C30dcosu/C28fsinu (19) f?/C30/C28 dsinu/C27fcosu (20) g?/C30g: (21) The cross term 2 b?x?y?can therefore be made to vanish by setting b?/C30b(cos2u/C28sin2u)/C28(c/C28a) sin ucosu /C30bcos(2 u)/C281 2(c/C28a) sin(2 u)/C300: (22) Forb?to be zero, it must be true that cos(2 u)/C30c/C28a 2b/C13K: (23) The other components are then given with the aid of the identity cos cot/C281(x)1C21C3 /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27x2p (24) by definingL/C13Kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27K2p ; (25) so sinu/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28L 2s (26) cosu/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27L 2s : (27) Rotating by an angle u/C301 2cot/C281c/C28a 2b ! (28) therefore transforms (1) into a?x?2/C27c?y?2/C272d?x?/C272f?y?/C27g?/C300: (29) COMPLETING THE SQUARE , a?x?2/C272d? a?x ! /C27c?y?2/C272f? c?y? ! /C27g?/C300 (30) a?x?d? a? !2 /C27c?y?/C27f? c? !2 /C30/C28g?/C27d?2 a?/C27f?2 c?: (31) Defining xƒ/C13x?/C27d?=a?;yƒ/C13y?/C27f?=c?;and gƒ/C13/C28g?/C27 d?2=a?/C27f?2=c?gives a?x?2/C27c?yƒ2/C30gƒ: (32) Ifgƒ"0;then divide both sides by gƒ:Defining aƒ/C13 a?=gƒandcƒ/C13c?=gƒthen gives aƒxƒ2/C27cƒyƒ2/C301: (33) Therefore, in an appropriate coordinate system, the general CONIC SECTION can be written (dropping the primes) as ax2/C27cy2/C301a;c;g"0 ax2/C27cy2/C300a;c"0;g/C300:1C2r (34) Consider an equation OF THE FORM ax2/C272bxy/C27cy2/C30 1 where b"0:Re-express this using t1and t2in the form ax2/C272bxy/C27cy2/C30t1x?2/C27t2y?2: (35) Therefore, rotate the COORDINATE SYSTEM x? y?1C2C1C2A /C30cosusinu /C28sinucosu1C2C1C2A x y1C2C1C2A ; (36) so ax2/C272bxy/C27cy2/C30t1x?2/C27t2y?2 /C30t1x2cos2u/C272xycosusinu/C27y2sin2u1CC1CA /C27t2x2sin2u/C282xysinucosu/C27y2cos2u1CC1CA /C30x2t1cos2u/C27t2sin2u1CC1CA /C272xycosusinut1/C28t2 ðÞ /C27y2t1sin2u/C27t2cos2u1CC1CA (37) and a/C30t1cos2u/C27t2sin2u (38) b/C30t1/C28t2 ðÞ cosusinu/C301 2t1/C28t2 ðÞ sin 2 uðÞ (39) c/C30t1sin2u/C27t2cos2u: (40) Therefore, a/C27c/C30t1cos2u/C27t2sin2u1CC1CA /C27t1sin2u/C27t2cos2u1CC1CA /C30t1/C27t2 (41) a/C28c/C30t1cos2u/C27t2sin2u/C28t1sin2u/C27t2cos2u /C30t1/C28t2 ðÞ cos2u/C28sin2u1CC1CA /C30t1/C28t2 ðÞ cos 2 uðÞ :(42) From (41) and (42), a/C28c b/C30t1/C28t2 ðÞ cos(2 u) 1 2t1/C28t2 ðÞ sin(2u)/C302 cot(2 u); (43) the same angle as before. But cos(2 u)/C30cos cot/C281a/C28c 2b !"# /C30cos tan/C2812b a/C28c !"# /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C272b a/C28c !2vuut; (44) so a/C28c/C30t1/C28t2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C272b a/C28c !2vuut: (45) Rewriting and copying (41), t 1/C28t2/C30(a/C28c)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C272b a/C28c !2vuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C28c)2/C274b2q (46) t1/C27t2/C30a/C27c: (47) Adding (46) and (47) gives t1/C301 2a/C27c/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C28c)2/C274b2q1C2C1C2A (48)t2/C30a/C27c/C28t1/C301 2a/C27c/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C28c)2/C274b2q1C2C1C2A : (49) Note that these ROOTS can also be found from t/C28t1 ðÞ t/C28t2 ðÞ /C30t2/C28tt1/C27t2 ðÞ /C27t1t2/C300 (50) t2/C28t(a/C27c)/C271 4(a/C27c)2/C28(a/C28c)2/C274b2hino /C30t2/C28t(a/C27c)/C2714a2/C272ac/C27c2/C28a2/C272ac/C28c2/C284b21C21C3 /C30t2/C28t(a/C27c)/C27ac/C28b21CC1CA /C30(a/C28t)(c/C28t)/C28b2 /C30a/C28tb bc/C28t1CA21CA21CA21CA21CA21CA21CA21CA2/C30(a/C28t)(c/C28t)/C28b 2/C300: (51) The original problem is therefore equivalent to look- ing for a solution to ab bc1C2C1C2A x y1C2C1C2A /C30txy1C2C1C2A (52) ax bx by cy1C2C1C2A x y1C2C1C2A /C30tx 2 y21C2C1C2A ; (53) which gives the simultaneous equations ax2/C27bxy/C30tx2 bxy/C27cy2/C30ty2:1C2r (54) LetXbe any point ( x, y) with old coordinates and (x?;y?) be its new coordinates. Then ax2/C272bxy/C27cy2/C30t/C27x?2/C27t/C28y?2/C301 (55) and x?/C30ˆX/C27/C215x y1C2C1C2A (56) y?/C30ˆX/C28/C215x y1C2C1C2A : (57) Ift/C27andt/C28are both >0;the curve is an ELLIPSE .I f t/C27andt/C28are both B0;the curve is empty. If t/C27and t/C28have opposite SIGNS , the curve is a HYPERBOLA .I f either is 0, the curve is a PARABOLA . To find the general form of a quadratic curve in POLAR COORDI- NATES (as given, for example, in Moulton 1970), plug x/C30rcosuandy/C30rsinuinto (1) to obtain ar2cos2u/C272br2cosusinu/C27cr2sin2u/C272drcosu /C272frsinu/C27g/C300 (58) acos2u/C272bcosusinu/C27csin2u1CC1CA /C272 r /C2(dcosu/C27fsinu)/C27g r2/C300: (59) Define u/C131=r:Forg"0;/we can divide through by 2 g; 1 2u2 /C271 g (d cos u /C27f sin u)u /C271 2g /C2 a cos2 u /C272b cos u sin u /C27c sin2 u1CC1CA /C300: (60) Applying the QUADRATIC FORMULA gives u /C30/C28d gcos u /C28f gsin u 9ffiffiffiffi Rp ; (61) where R /C13d cos u /C27 f sin u ðÞ2 g2 /C2841 2 ! 1 2g ! a cos2 u /C272b cos u sin u /C27c sin2 u1CC1CA /C30d2 g2cos2 u /C272df g2cos u sin u /C27f2 g2sin2 u /C281 ga cos2 u /C272b cos u sin u /C27c sin2 u1CC1CA : (62) Using the trigonometric identities sin2 u /C301 /C28cos2 u (63) sin(2u) /C302 sin u cos u; (64) it follows that R /C30d2 g2 /C28a g /C28f2 g2 /C27c g ! cos2 u /C27df g2 /C28bg ! sin 2uðÞ /C27f2 g2 /C28c g ! /C301 2 1 /C27cos 2uðÞ ½/C138d2 /C28 ag /C28 f2 /C27 cg g2 /C27sin(2u) /C2df /C28 bg g2 ! /C27f2 /C28 cg g2d2 /C28 ag /C28 f2 /C27 cg 2g2 cos(2 u) /C27df /C28 db g2sin(2u) /C27d2 /C28 ag /C28 f2 /C27 cg /C27 2f2 /C28 2cg 2g2 : (65) Defining A /C13/C28f g (66) B /C13/C28d g (67) C /C13df /C28 bg g2 (68)D /C13d2 /C28 f2 /C27 cg /C28 ag 2g2 (69) E /C13d2 /C27 f2 /C28 ag /C28 cg 2g2 (70) then gives the equation u /C131 r /C30A sin u /C27B cos u 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C sin(2u) /C27D cos(2 u) /C27Ep (71) (Moulton 1970). If g /C300, then (0) becomes instead u /C131 r /C30/C28a cos2 u /C27 2b cos u sin u /C27 c sin2 u 2(d cos u /C27 f sin u)/C215 (72) Therefore, the general form of a quadratic curve in polar coordinates is given by u/C30Asinu/C27Bcosu 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Csin(2u)/C27Dcos(2 u)/C27Ep forg"0 /C28acos2u/C272bcosusinu/C27csin2u 2(dcosu/C27fsinuforg/C300:8 >>< >>: (73) See also CONIC SECTION ,DISCRIMINANT (QUADRATIC CURVE ), ELLIPTIC CURVE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 200 /C1/201, 1987. Casey, J. "The General Equation of the Second Degree." Ch. 4 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with NumerousExamples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 151 /C1 /172, 1893. Moulton, F. R. "Law of Force in Binary Stars" and "Geome- trical Interpretation of the Second Law." §58 and 59 in An Introduction to Celestial Mechanics, 2nd rev. ed. New York: Dover, pp. 86 /C1/89, 1970. Quadratic Effect PRIME QUADRATIC EFFECT Quadratic Equation A quadratic equation is a second-order POLYNOMIAL ax2/C27bx/C27c/C300; (1) with a"0:The roots xcan be found by COMPLETING THE SQUARE : x2/C27b ax/C30/C28c a(2) x/C27b 2a !2 /C30/C28c a/C27b2 4a2/C30b2/C284ac 4a2(3) x /C27b 2a /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp 2a: (4) Solving for x then gives x /C30/C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp 2a: (5) This is the QUADRATIC FORMULA . An alternate form is given by dividing (1) through by x2 : a /C27b x /C27c x2 /C300 (6) c1 x2 /C27b cx ! /C27a /C300 (7) c1 x /C27b 2c !2 /C30cb 2c !2 /C28a /C30b2 4c /C284ac 4c/C30b2 /C28 4ac 4c: (8) Therefore, 1 x /C27b 2c /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp 2c (9) 1 x /C30/C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp 2c (10) x /C302c /C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp : (11) This form is helpful if b2 /C274ac ; in which case the usual form of the QUADRATIC FORMULA can give inaccurate numerical results for one of the ROOTS . This can be avoided by defining q /C13/C281 2b /C27sgn(b)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284acpjk (12) so that b and the term under the SQUARE ROOT sign always have the same sign. Now, if b /C210, then q /C30/C281 2b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284acp1CAr1CA7 (13) 1 q /C30/C282 b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acpb /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp /C30/C282 b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp1CAr1CA7 b2 /C28 b2 /C28 4ac ðÞ /C30/C282 b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp1CAr1CA7 4ac/C30/C28b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp 2ac; (14) so x1 /C13q a /C30/C28b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp 2a (15)x2 /C13c q /C30/C28b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp 2a (16) Similarly, if b B0, then q /C30/C281 2b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284acp1CAr1CA7 /C301 2/C28b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C284acp1CAr1CA7 (17) 1 q /C30/C282 /C28b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acpb /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp /C302 b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp1CAr1CA7 /C28b2 /C27 b2 /C28 4ac ðÞ /C30b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp /C282ac/C30/C28b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp 2ac; (18) so x1 /C13q a /C30/C28b /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C28 4acp 2a (19) x2 /C13c q /C30/C28b /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp 2a (20) Therefore, the ROOTS are always given by x1 /C30q=a and x2 /C30c =q:/ Now consider the equation expressed in the form a2x2 /C27a1x /C27a0 /C300 ; (21) with solutions z1and z2 : These solutions satisfy NEWTON’S RELATIONS z1/C27z2/C30/C28a1 a2(22) z1z2/C30a0 a2: (23) The properties of the SYMMETRIC POLYNOMIALS ap- pearing in N EWTON’S RELATIONS then give z2 1/C27z22/C30a2 1/C282a0a2 a2 2(24) z3 1/C27z32/C30/C28a3 1/C283a0a1a2 a3 2(25) z4 1/C27z42/C30a4 1/C284a0a21a2/C272a20a22 a4 2/C215 (26) See also CARLYLE CIRCLE ,C ONIC SECTION ,C UBIC EQUATION ,D ISCRIMINANT (POLYNOMIAL ), QUARTIC EQUATION ,QUINTIC EQUATION ,SEXTIC EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 9, 1987. Borwein, P. and Erde´lyi, T. "Quadratic Equations." §1.1.E.1a in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 4, 1995. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 91 /C1/92, 1996. King, R. B. Beyond the Quartic Equation. Boston, MA: Birkha ¨user, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Quadratic and Cubic Equations." §5.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 178 /C1/180, 1992. Spanier, J. and Oldham, K. B. "The Quadratic Function ax2 /C27bx /C27c and Its Reciprocal." Ch. 16 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 123 /C1/131, 1987. Quadratic Field An ALGEBRAIC INTEGER OF THE FORM a /C27bffiffiffiffi Dp where D is SQUAREFREE forms a quadratic field and is denoted Q(ffiffiffiffi Dp ) : If D /C210, the field is called a REAL QUADRATIC FIELD , and if D B0, it is called an IMAGINARY QUADRATIC FIELD . The integers in Qffiffiffi 1p1CC1CA are simply called "the" INTEGERS . The integers in Qffiffiffiffiffiffi /C281p1CC1CA are called GAUSSIAN INTEGERS , and the integers in Qffiffiffiffiffiffi /C283p1CC1CA are called EISENSTEIN INTEGERS . The ALGEBRAIC INTEGERS in an arbitrary quadratic field do not necessarily have unique factorizations. For example, the fields Qffiffiffiffiffiffi /C285p1CC1CA and Qffiffiffiffiffiffi /C286p1CC1CA are not uniquely factorable, since 21 /C303 /C215 7 /C30 1 /C272ffiffiffiffiffiffi /C285p1CAr1CA7 1 /C282ffiffiffiffiffiffi /C285p1CAr1CA7 (1) 6 /C30/C28ffiffiffi 6pffiffiffiffiffiffi /C286p1CAr1CA7 /C302 /C215 3 ; (2) although the above factors are all primes within these fields. All other quadratic fields Qffiffiffiffi Dp1CAr1CA7 with Djj57 are uniquely factorable. Quadratic fields obey the identities a /C27bffiffiffiffi Dp1CAr1CA7 9 c /C27dffiffiffiffiDp1CAr1CA7 /C30 a 9c ðÞ /C27 b 9d ðÞffiffiffiffiDp ; (3) a /C27bffiffiffiffiDp1CAr1CA7 c /C27dffiffiffiffiDp1CAr1CA7 /C30 ac /C27bdD ðÞ /C27 ad /C27bc ðÞffiffiffiffiDp ; (4) and a /C27 bffiffiffiffi Dp c /C27 dffiffiffiffiDp/C30ac /C28 bdD c2 /C28 d2D/C27bc /C28 ad ðÞ c2 /C28 d2Dffiffiffiffi Dp (5) The INTEGERS in the real field Qffiffiffiffi Dp1CAr1CA7 are of the form r /C27sp; wherer /C30ffiffiffiffi Dp for D /C132 or D /C133 (mod 4) 1 2/C281 /C27ffiffiffiffi Dp1CAr1CA7 for D /C131 (mod 4):8 < : (6) There are exactly 21 quadratic fields in which there is aE UCLIDEAN ALGORITHM , corresponding to /Q(m)/for SQUAREFREE integers /C2811,/C287,/C283,/C282,/C281, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and 73 (Sloane, N. J. A. Sequences048981). This list waspublished by Inkeri (1947), but erroneously included the spurious additional term 97 (Barnes and Swin- nerton-Dyer 1952; Hardy and Wright 1979, p. 217). See also A LGEBRAIC INTEGER ,EISENSTEIN INTEGER , GAUSSIAN INTEGER ,IMAGINARY QUADRATIC FIELD, INTEGER ,NUMBER FIELD,REAL QUADRATIC FIELD References Barnes, E. S. and Swinnerton-Dyer, H. P. F. "The Inhomo- geneous Minima of Binary Quadratic Forms. I." Acta Math 87, 259/C1/323, 1952. Berg, E. Fysiogr. Sa ¨llsk. Lund. Fo ¨hr.5,1/C1/6, 1935. Chatland, H. "On the Euclidean Algorithm in Quadratic Number Fields." Bull. Amer. Math. Soc. 55, 948/C1/953, 1949. Chatland, H. and Davenport, H. "Euclid’s Algorithm in Real Quadratic Fields." Canad. J. Math. 2, 289/C1/296, 1950. Hardy, G. H. and Wright, E. M. "Real Euclidean Fields" and "Real Euclidean Fields (Continued)." §14.8 and 14.9 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 213 /C1/217, 1979. Inkeri, K. "U ¨ber den Euklidischen Algorithmus in quad- ratischen Zahlko ¨rpern." Ann. Acad. Sci. Fennicae Ser. A. 1. Math.-Phys. , No. 41, 1 /C1/35, 1947. Koch, H. "Quadratic Number Fields." Ch. 9 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 275 /C1/314, 2000. LeVeque, W. J. Topics in Number Theory, Vol. 2. Reading, MA: Addison-Wesley, p. 57, 1956. Oppenheim. Math. Ann. 109, 349/C1/352, 1934. Samuel, P. "Unique Factorization." Amer. Math. Monthly 75, 945/C1/952, 1968. Stark, H. M. An Introduction to Number Theory. Chicago: Markham, p. 294, 1970. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 153 /C1/154, 1993. Sloane, N. J. A. Sequences A048981 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Quadratic Form A quadratic form involving nREAL variables x1;x2;..., xnassociated with the n/C29nMATRIX A/C30aijis given by Qx1;x2;...xn ðÞ /C30aijxixj; (1) where E INSTEIN SUMMATION has been used. Letting x be a VECTOR made up of x1;...,xnand xTthe TRANSPOSE , then Q(x)/C30xTAx; (2) equivalent to Q(x)/C30(x;Ax) (3) inINNER PRODUCT notation. A BINARY QUADRATIC FORM is a quadratic form in two variables and has the form Q(x;y) /C30a11x2 /C272a12xy /C27a22y2 : (4) It is always possible to express an arbitrary quadratic form Q(x) /C30aijxixj ; (5) in the form Q(x) /C30(x;Ax) ; (6) where A /C30aii is a SYMMETRIC MATRIX given by aij /C30aii i /C30j 1 2aij /C27aji1CC1CA i "j:8 < : (7) Any REAL quadratic form in n variables may be reduced to the diagonal form Q(x) /C30l1x2 1 /C27l2x22 /C27.../C27lnx2n (8) with /l1 ] l2 ]/C1/C1/C1] ln/ by a suitable orthogonal point- transformation. Also, two real quadratic forms are equivalent under the group of linear transformations IFF they have the same RANK and SIGNATURE . See also DISCONNECTED FORM,INDEFINITE QUADRA- TIC FORM,INNER PRODUCT ,INTEGER- MATRIX FORM, POSITIVE DEFINITE QUADRATIC FORM,POSITIVE SEMI- DEFINITE QUADRATIC FORM,R ANK (QUADRATIC FORM), SIGNATURE (QUADRATIC FORM), SYLVESTER’S INERTIA LAW,SYMMETRIC QUADRATIC FORM References Buell, D. A. Binary Quadratic Forms: Classical Theory and Modern Computations. New York: Springer-Verlag, 1989. Conway, J. H. and Fung, F. Y. The Sensual (Quadratic) Form. Washington, DC: Math. Assoc. Amer., 1997. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1104 /C1/106, 2000. Kitaoka, Y. Arithmetic of Quadratic Forms. Cambridge, England: Cambridge University Press, 1999. Lam, T. Y. The Algebraic Theory of Quadratic Forms. Reading, MA: W. A. Benjamin, 1973. Weisstein, E. W. "Books about Quadratic Forms." http:// www.treasure-troves.com/books/QuadraticForms.html. Quadratic Formula The formula giving the ROOTS of a QUADRATIC EQUATION ax2 /C27bx /C27c /C300 (1) as x /C30/C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C28 4acp 2a: (2) An alternate form is given byx /C302c /C28b 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2/C284acp : (3) See also QUADRATIC EQUATION Quadratic Integral To compute an integral OF THE FORM gdx a/C27bx/C27cx2; (1) COMPLETE THE SQUARE in the DENOMINATOR to obtain gdx a/C27bx/C27cx2/C301 cgdx x/C27b 2c !2 /C27a c/C28b2 4c2 ! /C215(2) Letu/C13x/C27b=2c:Then define /C28A2/C13a c/C28b2 4c2/C301 4c24ac/C28b21CC1CA /C131 4c2q; (3) where q/C134ac/C28b2(4) is the NEGATIVE of the DISCRIMINANT .I fqB0, then A/C301 2cffiffiffiffiffiffi/C28qp/C215 (5) Now use PARTIAL FRACTION DECOMPOSITION , 1 cgdu (u/C27A)(u/C28A)/C301 cgA1 u/C27A/C27A2 u/C28A ! du (6) A1 u/C27A/C27A2 u/C28A ! /C30A1u/C28A ðÞ /C27A2(u/C27A) u2/C28A2 /C30A1/C27A2 ðÞ u/C27AA2/C28A1 ðÞ u2/C28A2; (7) soA2/C27A1/C300[A2/C30/C28A1and AA2/C28A1 ðÞ /C30/C282AA1/C30 1[A1/C30/C281=(2A):Plugging these in, 1 cg/C281 2A1 u/C27A/C271 2A1 u/C28A ! du /C301 2Ac/C28In(u/C27A)/C27In(u/C28A) ½/C138 /C301 2AcInu/C28A u/C27A ! /C301 21 2c ! ffiffiffiffiffiffiffiffi/C28qpcInx /C27b 2c/C281 2cffiffiffiffiffiffiffiffi/C28qp x /C27b 2c/C271 2cffiffiffiffiffiffiffiffi/C28qp0 BBB@1 CCCA /C30 1 ffiffiffiffiffiffiffiffi/C28qp In2cx /C27 b /C28ffiffiffiffiffiffiffiffi/C28qp 2cx /C27 b /C27ffiffiffiffiffiffiffiffi/C28qp ! (8) for q B0. Note that this integral is also tabulated in Gradshteyn and Ryzhik (2000, equation 2.172), where it is given with a sign flipped. References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Quadratic Invariant Given the BINARY QUADRATIC FORM ax2 /C272bxy /C27cy2 (1) with DISCRIMINANT b2 /C28ac; let x /C30pX /C27qY (2) y /C30rX /C27sY /C215 (3) Then apX/C27qY ðÞ2/C272b(pX /C27qY)(rX /C27sY) /C27c(rX /C27sY)2 /C30AX2 /C272BXY /C27CY2 ; (4) where A /C30ap2 /C272bpr /C27cr2 (5) B /C30apq /C27b(ps /C27qr) /C27crs (6) C /C30aq2 /C272bqs /C27cs2 ; (7) so B2 /C28AC /C30 a2p2q2 /C27b2(ps /C27qr)2 /C27c2r2s2h /C272abpq (ps /C27qr) /C272acpqrs /C272bcrs(ps /C27qr)/C138 /C28 ap2 /C272bpr /C27cr21CC1CA aq2 /C272bqs /C27cs21CC1CA /C30a2p2q2 /C27b2p2s2 /C272b2pqrs /C27b2q2r2 /C27c2r2s2 /C272abp2qs /C272abpq2r /C272acpqrs /C272bcprs2 /C272bcqr2s /C28a2p2q2 /C282abp2qs /C28acp2s2 /C282abpq2r /C284b2pqrs /C282bcprs2 /C28acq2r2 /C282bcqr2s /C28c2r2s2 /C30b2p2s2 /C282b2pqrs /C27b2q2r2 /C272acprs /C28acp2s2 /C28acp2r2 /C30p2s2 b2 /C28ac1CC1CA /C27q2r2 b2 /C28ac1CC1CA /C282pqrs b2 /C28ac1CC1CA/C30 b2 /C28ac1CC1CA p2s2 /C282pqrs /C27q2r21CC1CA /C30(ps /C28rq)2 b2 /C28ac1CC1CA /C215 (8) Surprisingly, this is the same discriminant as before, but multiplied by the factor (ps /C28rq)2 : The quantity ps /C28rq is called the MODULUS . See also ALGEBRAIC INVARIANT Quadratic Irrational Number An IRRATIONAL NUMBER OF THE FORM P 9ffiffiffiffi Dp Q; where P and Q are INTEGERS and D is a SQUAREFREE INTEGER . Quadratic irrational numbers are some- times also called quadratic surds. In 1770, Lagrange proved that any quadratic irrational has a CONTIN- UED FRACTION which is periodic after some point. See also CONTINUED FRACTION ,M INKOWSKI’S QUES- TION MARK FUNCTION Quadratic Map A 1-D MAP often called "the" quadratic map is defined by xn/C271/C30x2 n/C27c/C215 (1) This is the real version of the complex map defining the M ANDELBROT SET . The quadratic map is called attracting if the J ACOBIAN JB1, and repelling if J/C211. F IXED POINTS occur when x(1)/C30[x(1)]2/C27c (2) x(1)1CC1CA 2/C28x(1)/C27c/C300 (3) x(1) 9/C301 219ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C284cp1CAr1CA7 : (4) Period two FIXED POINTS occur when xn/C272/C30x2 n/C271/C27c/C30x2n/C27c1CC1CA2/C27c /C30x4n/C272cx2n/C27(c2/C27c)/C30xn (5) x4/C272x2/C28x/C27cx2/C27c1CC1CA /C30x2/C28x/C27c1CC1CA x2/C27x/C271/C27c1CC1CA /C300 (6) x(2)9/C301 219ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C284(1/C27c)phi /C301 219ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C283/C284cp1CAr1CA7 :(7) Period three FIXED POINTS occur when x6/C27x5/C27(3c/C271)x4/C27(2c/C271)x3/C27(c2/C273c/C271)x2 /C27c/C271 ðÞ2x/C27c3/C272c2/C27c/C2711CC1CA /C300/C215 (8) The most general second-order 2-D MAP with an elliptic fixed point at the origin has the form x?/C30x cos a /C28y sin a /C27a20x2 /C27a11xy /C27a02y2 (9) y?/C30x sin a /C27y cos a /C27b20x2 /C27b11xy /C27b02y2 : (10) The map must have a DETERMINANT of 1 in order to be AREA -preserving, reducing the number of indepen- dent parameters from seven to three. The map can then be put in a standard form by scaling and rotating to obtain x?/C30x cos a /C28y sin a /C27x2 sin a (11) y?/C30x sin a /C27y cos a /C28x2 cos a: (12) The inverse map is x /C30x? cos a /C27y? sin a (13) y /C30/C28x? sin a /C27y ? cos a /C27 x? cos a /C27y? sin a ðÞ2/C215 (14) The FIXED POINTS are given by x2 isin a /C272xi cos a /C28xi/C281 /C28xi/C271 /C300 (15) for i /C300, ..., n /C281:/ See also BOGDANOV MAP,HE´ NON MAP,LOGISTIC MAP, LOZI MAP,MANDELBROT SET Quadratic Mean ROOT-MEAN-SQUARE Quadratic Nonresidue QUADRATIC RESIDUE Quadratic Phase Array A method to obtain a signal Cl(z) with a flat spectrum c( u;z) (such as a pulse), but having a smaller amplitude than the pulse. c( u; z) /C13eiz f( u) /C30X/C12 l/C30/C28/C12eiluCl(z); (1) whence Cl(z) /C301 =(2p)gp - peizf( u) /C28l u ðÞdu; (2) where f( u) /C30 1 /C28 ujj=p ðÞ u=p; (3) with / j uj5 p/. Thus c( u;z) and Cl(z) are a Fourier pair, and since / jc(u ; z)j/C301/, it is guaranteed that the sequence /Cl/ has a flat spectrum. The sequence /Cl/ is called the "quadratic phase array." References Aarts, R. M. and Janssen, A. J. E. M. "On Analytic Design of Loudspeaker Arrays with Uniform Radiation Character- istics." J. Acoust. Soc. Amer. 107, 287 /C1/292, 2000.Quadratic Reciprocity Law QUADRATIC RECIPROCITY THEOREM Quadratic Reciprocity Theorem Also called the AUREUM THEOREMA (GOLDEN THEO- REM) by Gauss. If p and q are distinct ODD PRIMES , then the CONGRUENCES x2 /C13q (mod p) x2 /C13p (mod q) are both solvable or both unsolvable unless both p and q leave the remainder 3 when divided by 4 (in which case one of the CONGRUENCES is solvable and the other is not). Written symbolically, p q ! q p ! /C30/C28 1ðÞ(p /C281)(q /C281)=4; where p q ! /C131 for x2 /C13p (mod q) solvable for x /C281 for x2 /C13p (mod q) not solvable for x1C2r is known as a LEGENDRE SYMBOL . Euler stated the theorem in 1783 without proof. Legendre was the first to publish a proof, but it was fallacious. In 1796, Gauss became the first to publish a correct proof (Nagell 1951, p. 144). The quadratic reciprocity theorem was Gauss’s favorite theorem from NUMBER THEORY , and he devised no fewer than eight different proofs of it over his lifetime. The GENUS THEOREM states that the D IOPHANTINE EQUATION x2/C27y2/C30p can be solved for paPRIME IFF p/C131 (mod 4) or p/C302. See also GENUS THEOREM ,JACOBI SYMBOL ,KRONECK- ER SYMBOL ,LEGENDRE SYMBOL ,QUADRATIC RESIDUE , RECIPROCITY THEOREM References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 39, 1996. Ireland, K. and Rosen, M. "Quadratic Reciprocity." Ch. 5 in A Classical Introduction to Modern Number Theory, 2nded.New York: Springer-Verlag, pp. 50 /C1 /65, 1990. Nagell, T. "The Quadratic Reciprocity Law." §41 in Introduc- tion to Number Theory. New York: Wiley, pp. 141 /C1/145, 1951. Riesel, H. "The Law of Quadratic Reciprocity." Prime Numbers and Computer Methods for Factorization, 2nded.Boston, MA: Birkha ¨user, pp. 279 /C1 /281, 1994. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 42 /C1/49, 1993. Quadratic Recurrence N.B. A detailed online essay by S. Finch was the starting point for this entry. A quadratic recurrence is a RECURRENCE RELATION on a SEQUENCE of numbers xnfg expressing xnas a second degree polynomial in xkwith k Bn. For example, xn /C30xn/C281xn/C282 (1) is a quadratic recurrence. Another simple example is xn /C30 xn/C281 ðÞ2(2) with x0 /C302 ; which has solution xn /C3022n : Another example is the number of "strongly" binary trees of height 5n; given by yn /C30 yn/C281 ðÞ2/C271 (3) with y0 /C301: This has solution yn /C30 c2n1C(1C) ; (4) where c /C30expX/C12 j/C3002/C28j/C281ln 1 /C27y/C282 j1CAr1CA7"# /C301:502836801... (5) and xbcis the FLOOR FUNCTION (Aho and Sloane 1973). A third example is the closest strict under- approximation of the number 1, sn /C30Xn i/C3011 zi; (6) where 1 Bz1 B...Bznare integers. The solution is given by the recurrence zn /C30 zn/C281 ðÞ2/C28zn/C281 /C271; (7) with z1 /C302: This has a closed solution as zn /C30 d2n /C271 2jk (8) where d /C301 2ffiffiffi 6p expX/C12 j/C3012 /C28j /C281ln 1 /C27 2zj /C2811CC1CA/C282hi() /C301:2640847353 . . . (9) (Aho and Sloane 1973). A final example is the well- known recurrence cn/C30cn/C281 ðÞ2/C28m (10) with c0/C300 used to generate the M ANDELBROT SET . See also MANDELBROT SET,RECURRENCE RELATION References Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential Sequences." Fib. Quart. 11, 429/C1/437, 1973.Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/quad/quad.html. Quadratic Representation SUM OF SQUARES FUNCTION Quadratic Residue If there is an INTEGER xsuch that x2/C13q(mod p); (1) then qis said to be a quadratic residue (mod p). If not, qis said to be a quadratic nonresidue (mod p). Hardy and Wright (1979, pp. 67 /C1/68) use the short- hand notations qRpandqNp;to indicated that qis a quadratic residue or nonresidue, respectively. For example, 42/C136;so 6 is a quadratic residue (mod 10). The entire set of quadratic residues (mod 10) are given by 1, 4, 5, 6, and 9, since 12/C131 (mod 10) 22/C134 (mod 10) 32/C139 (mod 10) 42/C136 (mod 10) 52/C135 (mod 10) 62/C136 (mod 10) 72/C139 (mod 10) 82/C134 (mod 10) 92/C131 (mod 10) making the numbers 2, 3, 7, and 8 the quadraticnonresidues (mod 10). A list of quadratic residues for p529 is given below (Sloane’s A046071), with those numbers Bpnot in the list being quadratic nonresidues of p. pQuadratic Residues 1 (none) 2131 41 51 , 461 , 3 , 471 , 2 , 4 81 , 4 91 , 4 , 7 10 1, 4, 5, 6, 9 11 1, 3, 4, 5, 9 12 1, 4, 9 13 1, 3, 4, 9, 10, 1214 1, 2, 4, 7, 8, 9, 1115 1, 4, 6, 9, 10 16 1, 4, 9 17 1, 2, 4, 8, 9, 13, 15, 16 18 1, 4, 7, 9, 10, 13, 16 19 1, 4, 5, 6, 7, 9, 11, 16, 17 20 1, 4, 5, 9, 16 Given an ODD PRIME p and an INTEGER a, then the LEGENDRE SYMBOL is given by a p ! /C301 if a is a quadratic residue mod p /C281 otherwise :1C2r (2) If r p /C281 ðÞ =2/C1391 (mod p) ; (3) then r is a quadratic residue ( /C27) or nonresidue /(/C28): This can be seen since if r is a quadratic residue of p, then there exists a square x2 such that r /C13x2 (mod p); so r p/C281 ðÞ =2/C13 x21CC1CAp /C281 ðÞ =2/C13xp /C281 (mod p) ; (4) and xp /C281is congruent to 1 (mod p)byF ERMAT’S LITTLE THEOREM . Given p and q in the congruence x2 /C13q (mod p) ; (5) x can be explicitly computed for p and q of certain special forms: x /C30qk /C271 (mod p) for p /C304k /C273 qk /C271 (mod p) for p /C308k /C275 and q2k /C271 /C131 (mod p) 1 24qðÞk /C271(p /C271) (mod p) for p /C308k /C275 and q2k /C271 /C13/C281 (mod p) :8 >>>>>>< >>>>>>:(6) For example, the first form can be used to find x given the quadratic residues q /C301, 3, 4, 5, and 9 (mod p /C3011, having k /C302), whereas the second and third forms determine x given the quadratic residues q /C301, 3, 4, 9, 10, and 12 (mod p /C3013, having k /C301), and q /C301, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36 (mod p /C3037, having k /C304). More generally, let q be a quadratic residue modulo an ODD PRIME p. Choose h such that the LEGENDRE SYMBOL h2 /C284q=p ðÞ /C30/C281 : Then defining V1 /C30h (7) V2 /C30h2 /C282q (8) Vi /C30hVi /C281 /C28qVi /C282for i ]3; (9) gives V2i /C30V2 i /C282qi (10)V2i/C271 /C30ViVi /C271 /C28hni ; (11) and a solution to the quadratic CONGRUENCE is x /C301 2(p /C271)V p /C271 ðÞ =2(mod p): (12) Schoof (1985) gives an algorithm for finding x with running time O ln nðÞ10(Hardy et al. 1990). The congruence is solved by the Mathematica command SqrtMod [q, p] in the Mathematica add-on package NumberTheory‘NumberTheoryFunctions‘ (which can be loaded with the command BBNumberTheory‘ ). The following table gives the PRIMES which have a given number d as a quadratic residue. d Primes /C286 24k/C271,5,7,11 /C285 20k/C271,3,7,9 /C2836 k /C271 /C2828 k /C271,3 /C2814 k /C271 28 k 91 3 12k91 5 10k91 6 24k91,5 Finding the CONTINUED FRACTION of a SQUARE ROOTffiffiffiffi Dp and using the relationship Qn /C30D /C28 P2 n Qn/C281(13) for the nth CONVERGENT Pn =Qn gives P2 n /C13/C28QnQn/C281(mod D) : (14) Therefore, /C28QnQn/C281is a quadratic residue of D. But since Q1 /C301;/C28Q2is a quadratic residue, as must be /C28Q2Q3:But since /C28Q2is a quadratic residue, so is Q3; and we see that /C281ðÞn/C281Qnare all quadratic residues ofD. This method is not guaranteed to produce all quadratic residues, but can often produce several small ones in the case of large D, enabling Dto be factored. The number of SQUARES s(n)i nZnis related to the number q(n) of quadratic residues in Znby qpnðÞ/C30spnðÞ/C28spn/C2821CC1CA (15) forn]3 (Stangl 1996). Both qand sare MULTI- PLICATIVE FUNCTIONS . See also ASSOCIATE ,E ULER’S CRITERION ,JACOBI SYMBOL ,K RONECKER SYMBOL ,LEGENDRE SYMBOL , MULTIPLICATIVE FUNCTION ,QUADRATIC RECIPROCITY THEOREM ,RIEMANN HYPOTHESIS References Burgess, D. A. "The Distribution of Quadratic Residues and Non-Residues." Mathematika 4, 106 /C1/112, 1975. Burton, D. M. Elementary Number Theory, 4th ed. New York: McGraw-Hill, p. 201, 1997. Courant, R. and Robbins, H. "Quadratic Residues." §2.3 in Supplement to Ch. 1 in What is Mathematics?: An Ele- mentary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 38 /C1/40, 1996. Guy, R. K. "Quadratic Residues. Schur’s Conjecture" and "Patterns of Quadratic Residues." §F5 and F6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 244 /C1/248, 1994. Hardy, G. H. and Wright, E. M. "Quadratic Residues." §6.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 67 /C1/68, 1979. Hilton, P.; Holton, D.; and Pedersen, J. Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, p. 43, 1997. Nagell, T. "Theory of Quadratic Residues." Ch. 4 in Intro- duction to Number Theory. New York: Wiley, pp. 115 and 132 /C1/155, 1951. Niven, I. and Zuckerman, H. An Introduction to the Theory of Numbers, 4th ed. New York: Wiley, p. 84, 1980. Rosen, K. H. Ch. 9 in Elementary Number Theory and Its Applications, 3rd ed. Reading, MA: Addison-Wesley, 1993. Schoof, R. "Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p." Math. Comput. 44, 483 /C1/494, 1985. Se´roul, R. "Quadratic Residues." §2.10 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 17 /C1/18, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 63 /C1/66, 1993. Sloane, N. J. A. Sequences A046071 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Stangl, W. D. "Counting Squares in Zn :/" Math. Mag. 69, 285 /C1/289, 1996. Tonelli, A. "Bemerkung u¨ber die Auflo¨sung quadratischer Congruenzen." Go¨ttingen Nachr. , 344 /C1/346, 1891. Wagon, S. "Quadratic Residues." §9.2 in Mathematica in Action. New York: W. H. Freeman, pp. 292 /C1/296, 1991. Quadratic Sieve A procedure used in conjunction with DIXON’S FAC- TORIZATION METHOD to factor large numbers n. Pick values of r given by ffiffiffinp1C(1C) /C27k; (1) where k /C301, 2, ... and xbcis the FLOOR FUNCTION .We are then looking for factors p such that n /C13r2(mod p) ; (2) which means that only numbers with LEGENDRE SYMBOL n=pðÞ/C301 (less than N /C30p(d) for TRIAL DIVI- SOR d, where p(d) is the PRIME COUNTING FUNCTION ) need be considered. The set of PRIMES for which this is true is known as the FACTOR BASE . Next, the CON- GRUENCESx2 /C13n (mod p) (3) must be solved for each p in the FACTOR BASE . Finally, a sieve is applied to find values of f(r) /C30r2 /C28n which can be factored completely using only the FACTOR BASE .G AUSSIAN ELIMINATION is then used as in DIXON’S FACTORIZATION METHOD in order to find a product of the f(r)/s, yielding a PERFECT SQUARE . The method requires about expffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnnln ln np1CAr1CA7 steps, improving on the CONTINUED FRACTION FAC- TORIZATION ALGORITHM by removing the 2 under the SQUARE ROOT (Pomerance 1996). The use of multiple POLYNOMIALS gives a better chance of factorization, requires a shorter sieve interval, and is well suited to parallel processing. See also NUMBER FIELD SIEVE,PRIME FACTORIZATION ALGORITHMS ,SMOOTH NUMBER References Alford, W. R. and Pomerance, C. "Implementing the Self Initializing Quadratic Sieve on a Distributed Network." In Number Theoretic and Algebraic Methods in ComputerScience, Proc. Internat. Moscow Conf., June-July 1993 (Ed. A. J. van der Poorten, I. Shparlinksi, and H. G. Zi- mer). Singapore: World Scientific, pp. 163 /C1 /174, 1995. Boender, H. and te Riele, H. J. J. "Factoring Integers with Large Prime Variations of the Quadratic Sieve." Preprint.Centrum voor Wiskunde en Informatica, No. NM-R9513,1995. Brent, R. P. "Parallel Algorithms for Integer Factorisation." InNumber Theory and Cryptography (Ed. J. H. Loxton). New York: Cambridge University Press, 26 /C1 /37, 1990. Bressoud, D. M. Ch. 8 in Factorization and Prime Testing. New York: Springer-Verlag, 1989. Gerver, J. "Factoring Large Numbers with a Quadratic Sieve." Math. Comput. 41, 287/C1/294, 1983. Lenstra, A. K. and Manasse, M. S. "Factoring by Electronic Mail." In Advances in Cryptology--Eurocrypt ’89 (Ed. J.- J. Quisquarter and J. Vandewalle). Berlin: Springer-Ver-lag, pp. 355 /C1 /371, 1990. Pomerance, C. "The Quadratic Sieve Factoring Algorithm." InAdvances in Cryptology: Proceedings of EUROCRYPT 84(Ed. T. Beth, N. Cot, and I. Ingemarsson). New York: Springer-Verlag, pp. 169 /C1/182, 1985. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/1485, 1996. Pomerance, C.; Smith, J. W.; and Tuler, R. "A Pipeline Architecture for Factoring Large Integers with the Quad- ratic Sieve Method." SIAM J. Comput. 17, 387/C1/403, 1988. Silverman, R. D. "The Multiple Polynomial Quadratic Sieve." Math. Comput. 48, 329/C1/339, 1987. Quadratic Surd QUADRATIC IRRATIONAL NUMBER Quadratic Surface A second-order ALGEBRAIC SURFACE given by the general equation ax2/C27by2/C27cz2/C272fyz/C272gzx/C272hxy/C272px/C272py/C272rz /C27d/C300: (1) Quadratic surfaces are also called quadrics, and there are 17 standard-form types. A quadratic surface intersects every plane in a (proper or degenerate) CONIC SECTION . In addition, the CONE consisting of all tangents from a fixed point to a quadratic surface cuts every plane in a CONIC SECTION , and the points of contact of this CONE with the surface form a CONIC SECTION (Hilbert and Cohn-Vossen 1999, p. 12). Define e /C30ahg hbf gfc2 435 (2) E /C30ahgp hbf q gfcr pqrd26643 775 (3) r 3 /C30rank e (4) r4 /C30rank E (5) D/C30det E ; (6) and k1 ; k2 ; as k3 are the roots of a /C28xh g hb /C28xf gfc /C28x1CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA2/C300: (7) Also define k /C131 if the signs of nonzero ks are the same 0 otherwise :1C2r (8) Then the following table enumerates the 17 quadrics and their properties (Beyer 1987). Surface Equation / r3//r4//sgn( D)/ k Coincident PLANES/x2 /C300/ 11 Ellipsoid (Ima- ginary)/x2 a2 /C27y2 b2 /C27z2 c2 /C30/C281/ 34 //C27/ 1 ELLIPSOID (Real)/x2 a2 /C27y2 b2 /C27z2 c2 /C301/ 34 /(/C28)/ 1 Elliptic Cone (Imaginary)/x2 a2 /C27y2 b2 /C27z2 c2 /C300/ 33 1 ELLIPTIC CONE (Real)/z2 /C30x2 a2 /C27y2 b2/ 33 0 Elliptic Cylin- der (Imagin- ary)/x2 a2 /C27y2 b2 /C30/C281/ 23 1 ELLIPTIC CYLIN- DER (Real)/x2 a2 /C27y2 b2 /C301/ 23 1 ELLIPTIC PARA- BOLOID/z /C30x2 a2 /C27y2 b2/ 24 /(/C28)/ 1HYPERBOLIC CYLINDER/x2 a2 /C28y2 b2 /C30/C281/ 23 0 HYPERBOLICPARABOLOID/z /C30y2 a2 /C28x2 b2/ 24 //C27/ 0 HYPERBOLOID of one Sheet/x2 a2 /C27y2 b2 /C28z2 c2 /C301/ 34 //C27/ 0 HYPERBOLOID of two Sheets/x2 a2 /C27y2 b2 /C28z2 c2 /C30/C281/ 34 /(/C28)/ 0 Intersecting Planes (Ima- ginary)/x2 a2 /C27y2 b2 /C300/ 22 1 Intersecting PLANES (Real)/x2 a2 /C28y2 b2 /C300/ 22 0 PARABOLIC CY- LINDER/x2 /C272rz /C300/ 13 Parallel Planes (Imaginary)/x2 /C30/C28a2/ 12 Parallel PLANES (Real)/x2 /C30a2/ 12 Of the non-degenerate quadratic surfaces, the ELLIP- TIC (and usual) CYLINDER , HYPERBOLIC CYLINDER , ELLIPTIC (and usual) CONE are RULED SURFACES , while the one-sheeted HYPERBOLOID and HYPERBOLIC PARABOLOID are DOUBLY RULED SURFACES . A curve in which two arbitrary quadratic surfaces inarbitrary positions intersect cannot meet any plane inmore than four points (Hilbert and Cohn-Vossen 1999, p. 24). See also C ONE,CONFOCAL QUADRICS ,CUBIC SURFACE , CYLINDER ,D OUBLY RULED SURFACE ,E LLIPSOID , ELLIPTIC CONE,ELLIPTIC CYLINDER ,ELLIPTIC PARA- BOLOID ,H YPERBOLIC CYLINDER ,H YPERBOLIC PARA- BOLOID ,H YPERBOLOID ,P LANE ,Q UARTIC SURFACE , RULED SURFACE ,SURFACE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 210 /C1/211, 1987. Hilbert, D. and Cohn-Vossen, S. "The Second-Order Sur- faces." §3i n Geometry and the Imagination. New York: Chelsea, pp. 12 /C1/19, 1999. Mollin, R. A. Quadrics. Boca Raton, FL: CRC Press, 1995. Quadratrix of Hippias The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for ANGLE TRISEC- TION or, more generally, division of an ANGLE into any integral number of equal parts, and CIRCLE SQUAR- ING.In POLAR COORDINATES , pr /C302r u csc u; so r /C30rp sin u u; which is proportional to the COCHLEOID . References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 223, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 198, 1972. MacTutor History of Mathematics Archive. "Quadratrix of Hippias." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Quadratrix.html. Quadrature The word quadrature has (at least) three incompa- tible meanings. Integration by quadrature either means solving an INTEGRAL analytically (i.e., symbo- lically in terms of known functions), or solving of an integral numerically (e.g., GAUSSIAN QUADRATURE , QUADRATURE FORMULAS ). Ueberhuber (1997, p. 71) uses the word "quadrature" to mean numerical computation of a univariate INTEGRAL , and "CUBA- TURE " to mean numerical computation of a MULTIPLE INTEGRAL . The word quadrature is also used to mean SQUARING : the construction of a square using only COMPASS and STRAIGHTEDGE which has the same AREA as a given geometric figure. If quadrature is possible for a PLANE figure, it is said to be QUADRABLE . For a function tabulated at given values xi(so the ABSCISSAS cannot be chosen at will), write the func- tion f as a sum of ORTHONORMAL FUNCTIONS pj satisfyinggb api(x)pj(x)W(x)dx /C30 dij (1) as f(x) /C30X/C12 j/C300ajpj(x); (2) and plug into gb af(x)W(x) dx /C30gb aXm j/C301p(x)W(x) x /C28 xj ðÞ p? xj1CC1CAdx f xj1CC1CA /C13Xm j/C301wjfxj1CC1CA ; (3) giving gb aX/C12 j/C300ajpj(x)W(x)dx /C30Xn i/C301wiX/C12 j/C300ajpjxj1CC1CA"# : (4) But we wish this to hold for all degrees of approxima- tion, so ajgb apj(x)W(x)dx /C30ajXn i/C301wipjxiðÞ (5) gb apj(x)W(x)dx /C30Xn i /C301wipjxiðÞ: (6) Setting i /C300 in (1) gives gb ap0(x)pj(x)W(x)dx /C30 d0j : (7) The zeroth order orthonormal function can always be taken as p0(x) /C301; so (7) becomes gb apj(x)W(x)dx/C30d0j (8) /C30Xn i/C301wipjxiðÞ; (9) where (6) has been used in the last step. We therefore have the MATRIX equation p0x1ð Þ /C1/C1/C1 p0xnðÞ p0x1ð Þ /C1/C1/C1 p1xnðÞ n:::n pn/C281x1ð Þ /C1/C1/C1 pn/C281xnðÞ2 6643 775w 1 w2 n wn2 6643 775/C301 0 n 02 6643 775(10) which can be inverted to solve for the w i/s (Press et al. 1992). See also CALCULUS ,CHEBYSHEV- GAUSS QUADRATURE , CHEBYSHEV QUADRATURE ,C UBATURE ,D ERIVATIVE , DOUBLE EXPONENTIAL INTEGRATION ,FUNDAMENTAL THEOREM OF GAUSSIAN QUADRATURE ,GAUSS- JACOBI MECHANICAL QUADRATURE ,G AUSS- KRONROD QUAD- RATURE ,G AUSSIAN QUADRATURE ,H ERMITE- GAUSS QUADRATURE ,HERMITE QUADRATURE ,JACOBI- GAUSS QUADRATURE ,J ACOBI QUADRATURE ,L AGUERRE- GAUSS QUADRATURE ,L AGUERRE QUADRATURE ,L E- GENDRE- GAUSS QUADRATURE ,L EGENDRE QUADRA- TURE ,L OBATTO QUADRATURE ,M ECHANICAL QUADRATURE ,MEHLER QUADRATURE ,NEWTON- COTES FORMULAS ,NUMERICAL INTEGRATION ,RADAU QUAD- RATURE ,RECURSIVE MONOTONE STABLE QUADRATURE References Abramowitz, M. and Stegun, C. A. (Eds.). "Integration." §25.4 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 885 /C1/897, 1972. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 365 /C1/366, 1992. Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, p. 71, 1997. Quadrature Formulas NEWTON- COTES FORMULAS Quadri-Amicable Number AMICABLE QUADRUPLE Quadric A quadric is a QUADRATIC SURFACE . A surface OF THE FORM x2 a2 /C27 u /C27y2 b2 /C27 u /C27z2 c2 /C27 u /C301 is also called a quadric, and u is said to be the parameter of the quadric. See also QUADRATIC SURFACE References Takahashi, H. "Quadrica Page." http://www2.kawase- h.ed.jp/Teachers/~Takahashi/Quadrica.html. Quadricorn A FLEXIBLE POLYHEDRON due to C. Schwabe (with the appearance of having four horns) which flexes from one totally flat configuration to another, passing through intermediate configurations of positive VO- LUME . See also FLEXIBLE POLYHEDRONQuadrifolium The ROSE with n /C302. It has polar equation r /C30a sin 2uðÞ ; and Cartesian form x2 /C27y21CC1CA3/C304a2x2y2: See also BIFOLIUM ,FOLIUM ,ROSE,TRIFOLIUM Quadrilateral A four-sided POLYGON sometimes (but not very often) also known as a tetragon. If not explicitly stated, all four VERTICES are generally taken to lie in a PLANE .I f the points do not lie in a PLANE , the quadrilateral is called a SKEW QUADRILATERAL . There are three topological types of quadrilaterals (Wenninger 1983,p. 50): convex quadrilaterals (left figure), concave quadrilaterals (middle figure), and crossed quadrilat- erals (or butterflies, or bow-ties; right figure). For a planar convex quadrilateral (left figure above), let the lengths of the sides be a,b,c, and d, the SEMIPERIMETER s, and the DIAGONALS pand q. The DIAGONALS are PERPENDICULAR IFF a2/C27c2/C30b2/C27d2:: Given any five points in the plane, four will alwaysform a convex quadrilateral. This result is a specialcase of the so-called HAPPY END PROBLEM (Hoffman 1998, pp. 74 /C1/78). The centroid of the vertices of a quadrilateral occurs at the point of intersection of the BIMEDIANS (i.e., the lines MABMCDand MADMBCjoining pairs of opposite MIDPOINTS ) (Honsberger 1995, pp. 36 /C1/37). In addi- tion, it is the MIDPOINT of the line MACMBD connecting the midpoints of the diagonals AC and BD (Honsber- ger 1995, pp. 39 /C1/40). An equation for the sum of the squares of side lengths is a2 /C27b2 /C27c2 /C27d2 /C30p2 /C27q2 /C274x2 ; (1) where x is the length of the line joining the MIDPOINTS of the DIAGONALS (Casey 1888, p. 22). The AREA of a quadrilateral is given by K /C301 2pq sin u (2) /C301 4b2 /C27d2 /C28a2 /C28c21CC1CA tan u (3) /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p2q2 /C28 b2 /C27d2 /C28a2 /C28c2 ðÞ2q (4) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s /C28a)(s /C28b)(s /C28c)(s /C28d) /C28abcd cos21 2A /C27B ðÞhir ; (5) where (4) is known as BRETSCHNEIDER’S FORMULA (Beyer 1987). The four ANGLE BISECTORS of a quadrilateral intersect adjacent bisectors in four CONCYCLIC points (Hon- sberger 1995, p. 35). Any non-self-intersecting quadrilateral tiles the plane.There is a relationship between the six distances d12 ; d13 ; d14 ; d23 ; d24 ; and d34 between the four points of a quadrilateral (Weinberg 1972): 0 /C30d4 12d234 /C27d413d224 /C27d414d223 /C27d423d214 /C27d424d213 /C27d434d212 /C27d212d223d231 /C27d212d224d241 /C27d213d234d241 /C27d223d234d242 /C28d212d223d234 /C28d213d232d224 /C28d212d224d243 /C28d214d242d223 /C28d213d234d242 /C28d214d243d232 /C28d223d231d214 /C27d221d213d234 /C28d224d241d213 /C28d221d214d243 /C28d231d212d224 /C28d232d221d214 : (6) This can be most simply derived by setting the left side of the CAYLEY- MENGER DETERMINANT 288V2 /C3001 1 1 1 10 d2 12d213d214 1 d221 0 d223d224 1 d231d232 0 d234 1 d2 41d242d243 01CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA21CA2(7) equal to 0 (corresponding to a TETRAHEDRON of volume 0), thus giving a relationship between the DISTANCES between vertices of a planar quadrilateral (Uspensky 1948, p. 256). A special type of quadrilateral is the CYCLIC QUAD- RILATERAL , for which a CIRCLE can be circumscribed so that it touches each VERTEX . For BICENTRIC QUAD- RILATERALS , the CIRCUMCIRCLE and INCIRCLE satisfy 2r2R2/C28s21CC1CA /C30R2/C28s21CC1CA /C284r2s2; (8) where Ris the CIRCUMRADIUS ,rin the INRADIUS , and sis the separation of centers. A quadrilateral with two sides PARALLEL is called a TRAPEZOID . See also ANTICENTER ,B ICENTRIC QUADRILATERAL , BIMEDIAN ,B RAHMAGUPTA’S FORMULA ,B RETSCHNEI- DER’S FORMULA ,BUTTERFLY THEOREM ,CAYLEY- MEN- GER DETERMINANT ,C OMPLETE QUADRILATERAL , CYCLIC QUADRILATERAL ,DIAMOND ,EIGHT- POINT CIR- CLE THEOREM ,E QUILIC QUADRILATERAL ,F ANO’S AXIOM ,L E´ ON ANNE’S THEOREM ,L OZENGE ,M ALTI- TUDE ,O RTHOCENTRIC QUADRILATERAL ,PARALLELO- GRAM ,P TOLEMY’S T HEOREM ,R ATIONAL QUADRILATERAL ,RECTANGLE ,RHOMBUS ,SKEW QUAD- RILATERAL ,S QUARE ,T ANGENTIAL QUADRILATERAL , TRAPEZOID ,VARIGNON’S THEOREM , VON AUBEL’S THE- OREM ,W ITTENBAUER’S PARALLELOGRAM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Durell, C. V. "The Quadrilateral and Quadrangle." Ch. 7 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 77 /C1/87, 1928. Fukagawa, H. and Pedoe, D. "Circles and Quadrilaterals" and "Quadrilaterals." §3.5 and 4.2 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 43 /C1/45, 47 /C1/ 48, and 125 /C1/132, 1989. Harris, J. W. and Stocker, H. "Quadrilaterals." §3.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 82 /C1/86, 1998. Honsberger, R. "On Quadrilaterals." Ch. 4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 35 /C1/41, 1995. Routh, E. J. "Moment of Inertia of a Quadrilateral." Quart. J. Pure Appl. Math. 11, 109 /C1/110, 1871. Uspensky, J. V. Theory of Equations. New York: McGraw- Hill, p. 256, 1948. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Quadrilateral of Chords CYCLIC QUADRILATERAL Quadrilateral Tiling Any nonself-intersecting QUADRILATERAL (Wells 1991, p. 208) tiles the plane, as illustrated above. References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 177 /C1/179, 208, and 211, 1991. Quadrillion In the American system, 1015. See also LARGE NUMBER Quadriplanar Coordinates The analog of TRILINEAR COORDINATES for TETRAHE- DRA. See also TETRAHEDRON ,TRILINEAR COORDINATES References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979. Mitrinovic, D. S.; Pecaric, J. E.; and Volenec, V. Ch. 19 in Recent Advances in Geometric Inequalities. Dordrecht, Netherlands: Kluwer, 1989. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 193 /C1/196, 1961.Quadrivium A word derived from the Latin roots quad- (four) and via (ways, roads), therefore a crossing of four roads. In medieval universities, the quadrivium consisted of the four subjects in the upper division of the seven liberal arts: ARITHMETIC , astronomy, GEOMETRY , and music. See also TRIVIUM Quadruple A group of four elements, also called a QUADRUPLET or TETRAD . See also AMICABLE QUADRUPLE ,DIOPHANTINE QUAD- RUPLE ,M ONAD ,PAIR,PRIME QUADRUPLET ,PYTHA- GOREAN QUADRUPLE ,Q UADRUPLET ,Q UINTUPLET , TETRAD ,TRIAD,TRIPLE ,TWINS ,VECTOR QUADRUPLE PRODUCT Quadruple Point A point where a curve intersects itself along four arcs. The above plot shows the quadruple point at the ORIGIN of the QUADRIFOLIUM /(x2 /C27y2)3 /C284x2y2 /C300/. See also DOUBLE POINT ,TRIPLE POINT References Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 57 /C1/58, 1978. Quadruplet QUADRUPLE Quadtree ATREE having four branches at each node. Quadtrees are used in the construction of some multidimen- sional databases (e.g., cartography, computer gra-phics, and image processing). For a d-D tree, the expected number of comparisons over all pairs ofintegers for successful and unsuccessful searches aregiven analytically for d/C302 and numerically for /d]3/ by Finch. References de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. "Quadtrees: Non-Uniform Mesh Generation." Ch. 14 in Computational Geometry: Algorithms and Ap- plications, 2nd rev. ed. Berlin: Springer-Verlag, pp. 291 /C1/ 306, 2000. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/infprd/infprd.html. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/qdt/qdt.html. Finkel, R. A. and Bentley, J. L. "Quad Trees, a Data Structure for Retrieval on Composite Keys." Acta Infor- matica 4,1/C1/9, 1974. Flajolet, P.; Gonnet, G.; Puech, C.; and Robson, J. M. "Analytic Variations on Quadtrees." Algorithmica 10, 473 /C1/500, 1993. Flajolet, P.; Labelle, G.; Laforest, L.; and Salvy, B. "Hyper- geometrics and the Cost Structure of Quadtrees." Random Structure Alg. 7, 117 /C1/144, 1995. http://pauillac.inria.fr/ algo/flajolet/Publications/publist.html. Gonnet, G. H. and Baeza-Yates, R. Ch. 3 in Handbook of Algorithms and Data Structures in Pascal and C. Read- ing, MA: Addison-Wesley, 1991. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 11 /C1/ 13, 1991. Samet, H. Applications of Spatial Data Structures: Compu- ter Graphics, Image Processing and GIS. Reading, MA: Addison-Wesley, 1989. Samet, H. The Design and Analysis of Spatial Data Structures. Reading, MA: Addison-Wesley, 1990. Quantic An m-ary n-ic polynomial (i.e., a HOMOGENEOUS POLYNOMIAL with constant COEFFICIENTS of degree n in m independent variables). See also ALGEBRAIC INVARIANT ,FUNDAMENTAL SYS- TEM, P-ADIC NUMBER ,SYZYGIES PROBLEM Quantified System A quantified system of real algebraic equations and inequalities in variables / fx1 ; ...; xn g/ is an expression QS /C30Q1(y1) Q2ðÞ y2ðÞ/C1/C1/C1 QmymðÞSx1 ;...; xn;y1 ;...;ym ðÞ ; where Q is a QUANTIFIER ( /C215 or /C214) and S is a system of real algebraic equations and inequalities in x1 ...;xn; y1 ;...ym fg : By TARSKI’S THEOREM , the solu- tion set of a quantified system of real algebraic equations and inequalities is a SEMIALGEBRAIC SET. See also QUANTIFIER ,SEMIALGEBRAIC SET,TARSKI’S THEOREM References Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525 /C1/541, 2000. Quantifier One of the operations EXISTS /C215or FOR ALL /C214. However, there also exist more exotic branches of logic which use quantifiers other than these two. See also BOUND VARIABLE ,EXISTS ,FOR ALL,FREE, QUANTIFIED SYSTEM ,QUANTIFIER ELIMINATION ,UNI- VERSAL QUANTIFIERReferences Hall, C. and O’Donnell, J. "Computing with Quantifiers." §3.2 in Discrete Mathematics Using a Computer. London: Springer-Verlag, pp. 98 /C1/100, 2000. Quantifier Elimination Quantifier elimination is the removal of all QUANTI- FIERS ( /C214 and /C215) from a quantified system. A first- order theory allows quantifier elimination if, for each quantified formula, there exists an equivalent quan- tifier-free formula. Examples of such theories include the real numbers with /C27;/C31;/C30; and >; and the theory of complex numbers with /C27;/C31; and /C30: Quantifier elimination is implemented in Mathematica as Re- solve [expr]. Unfortunately, it has been proven that the worst-case time complexity for real quantifier elimination is doubly exponential in the number of QUANTIFIER blocks (Weispfenning 1985, Davenport and Heintz 1988, Heintz et al. 1989, Caviness and Johnson 1998). See also CYLINDRICAL ALGEBRAIC DECOMPOSITION , TARSKI’S THEOREM References Caviness, B. F. and Johnson, J. R. (Eds.). Quantifier Elim- ination and Cylindrical Algebraic Decomposition. New York: Springer-Verlag, 1998. Collins, G. E. "Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition." In Proc. 2nd GI Conf. Automata Theory and Formal Languages. New York: Springer-Verlag, pp. 134 /C1/183, 1975. Collins, G. E. "Quantifier Elimination by Cylindrical Alge- braic Decomposition--Twenty Years of Progress." In Quan- tifier Elimination and Cylindrical Algebraic Decomposition (Ed. B. F. Caviness and J. R. Johnson). New York: Springer-Verlag, pp. 8 /C1/23, 1998. Collins, G. E. and Hong, H. "Partial Cylindrical Algebraic Decomposition for Quantifier Elimination." J. Symb. Comput. 12, 299/C1/328, 1991. Davenport, J. H. "Computer Algebra for Cylindrical Alge- braic Decomposition." Report TRITA-NA-8511, NADA,KTH, Stockholm, Sept. 1985. Davenport, J. and Heintz, J. "Real Quantifier Elimination if Doubly Exponential." J. Symb. Comput. 5,2 9/C1 /35, 1988. Dolzmann, A. and Sturm, T. "Simplification of Quantifier- Free Formulae over Ordered Fields." J. Symb. Comput. 24, 209/C1/231, 1997. Dolzmann, A. and Weispfenning, V. "Local Quantifier Elimination." http://www.fmi.uni-passau.de/~dolzmann/refs/MIP-0003.ps.Z. Heintz, J.; Roy, R.-F.; and Solerno, P. "Complexite ´du principe de Tarski-Seidenberg." C. R. Acad. Sci. Paris Se´r. I Math. 309, 825/C1 /830, 1989. Loos, R. and Weispfenning, V. "Applying Lattice Quantifier Elimination." Comput. J. 36, 450/C1/461, 1993. Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525/C1/541, 2000. Weispfenning, V. "The Complexity of Linear Problems in Fields." J. Symb. Comput. 5,3/C1/27, 1988. Quantile The kthn-tile Pkis that value of x, say xk;which corresponds to a CUMULATIVE FREQUENCY ofNk=n:If n /C304, the quantity is called a QUARTILE , and if n /C30100, it is called a PERCENTILE . See also PERCENTILE ,QUARTILE References Kenney, J. F. and Keeping, E. S. "Quantiles." §3.5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 37 /C1/38, 1962. Quantity See also EXPRESSION Quantization Efficiency Quantization is a nonlinear process which generates additional frequency components (Thompson et al. 1986). This means that the signal is no longer band- limited, so the SAMPLING THEOREM no longer holds. If a signal is sampled at the NYQUIST FREQUENCY , information will be lost. Therefore, sampling faster than the NYQUIST FREQUENCY results in detection of more of the signal and a lower signal-to-noise ratio [SNR]. Let b be the OVERSAMPLING ratio and define hQ /C13SNRquant SNRunquant: Then the following table gives values of / hQ/ for a number of parameters. Quantization Levels /hQ( b /C301)//hQ( b /C302)/ 2 0.64 0.74 3 0.81 0.89 4 0.88 0.94 The Very Large Array of 27 radio telescopes in Socorro, New Mexico uses three-level quantization at b /C301; so hQ /C300:81 :/ See also OVERSAMPLING References Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Fig. 8.3 in Interferometry and Synthesis in Radio Astron- omy. New York: Wiley, p. 220, 1986. Quantum Chaos The study of the implications of CHAOS for a system in the semiclassical (i.e., between classical and quantum mechanical) regime. References Ott, E. "Quantum Chaos." Ch. 10 in Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 334 /C1/362, 1993.Quarter The UNIT FRACTION 1/4, also called one-fourth. See also HALF,KO¨ BE’S ONE-FOURTH THEOREM ,QUAR- TILE Quarter Squares Rule a /C27 b 2 !2 /C28a /C28 b 2 !2 /C30ab : Quartet A SET of four, also called a TETRAD . See also HEXAD ,MONAD ,QUINTET ,TETRAD ,TRIAD Quartic Curve A general plane quartic curve is a curve OF THE FORM Ax4/C27By4/C27Cx3y/C27Dx2y2/C27Exy3/C27Fx3/C27Gy3 /C27Hx2y/C27Ixy2/C27Jx2/C27Ky2/C27Lxy/C27Mx/C27Ny/C27O/C300: (1) The incidence relations of the 28 bitangents of the general quartic curve can be put into a ONE-TO-ONE correspondence with the vertices of a particular POLYTOPE in 7-D space (Coxeter 1928, Du Val 1931). This fact is essentially similar to the discovery bySchoutte (1910) that the 27 S OLOMON’S SEAL LINES on aCUBIC SURFACE can be connected with a POLYTOPE in 6-D space (Du Val 1931). A similar but lesscomplete relation exists between the tritangent planes of the canonical curve of genus 4 and an 8-D POLYTOPE (Du Val 1931). The maximum number of DOUBLE POINTS for a nondegenerate quartic curve is three. A quartic curve OF THE FORM y2/C30(x/C28a)(x/C28j)(x/C28g)(x/C28d) (2) can be written y x/C28a !2 /C301/C28b/C28a x/C28a ! 1/C28g/C28a x/C28a ! 1/C28d/C28a x/C28a ! ; (3) and so is CUBIC in the coordinates X/C301 x/C28a(4) Y/C30y x/C28a2: (5) This transformation is a BIRATIONAL TRANSFORMA- TION . Let P and Q be the INFLECTION POINTS and R and S the intersections of the line PQ with the curve in Figure (a) above. Then A /C30C (6) B /C302A: (7) In Figure (b), let UV be the double tangent, and T the point on the curve whose x coordinate is the average of the x coordinates of U and V. Then UV PQkk RS and D /C30F (8) E /C30ffiffiffi 2p D : (9) In Figure (c), the tangent at P intersects the curve at W. Then G /C308B: (10) Finally, in Figure (d), the intersections of the tan- gents at PandQareWandX. Then H/C3027B (11) (Honsberger 1991). See also CUBIC SURFACE ,P EAR-SHAPED CURVE , SOLOMON’S SEAL LINES References Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc. 24, 7/C1/9, 1928. Du Val, P. "On the Directrices of a Set of Points in a Plane." Proc. London Math. Soc. Ser. 2 35,2 3/C1/74, 1933. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 114 /C1/118, 1991. Schoutte, P. H. "On the Relation Between the Vertices of a Definite Sixdimensional Polytope and the Lines of a CubicSurface." Proc. Roy. Akad. Acad. Amsterdam 13, 375/C1/383, 1910. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 49, 1991. Quartic Equation A general quartic equation (also called a BIQUADRATIC EQUATION ) is a fourth-order POLYNOMIAL OF THE FORM z4/C27a3z3/C27a2z2/C27a1z/C27a0/C300: (1) The ROOTS of this equation satisfy N EWTON’S RELA- TIONS : x1/C27x2/C27x3/C27x4/C30/C28a3 (2) x1x2/C27x1x3/C27x1x4/C27x2x3/C27x2x4/C27x3x4/C30a2 (3) x1x2x3/C27x2x3x4/C27x1x2x4/C27x1x3x4/C30/C28a1 (4) x1x2x3x4/C30a0; (5) where the denominators on the right side are all a4/C13 1:Writing the quartic in the standard form x4/C27px2/C27qx/C27r/C300; (6) the properties of the SYMMETRIC POLYNOMIALS ap- pearing in N EWTON’S RELATIONS then give z2 1/C27z22/C27z23/C27z24/C30/C282p (7) z31/C27z32/C27z33/C27z34/C30/C283p (8) z41/C27z42/C27z43/C27z44/C302p2/C284r (9) z51/C27z52/C27z53/C27z54/C305pq: (10) Eliminating p,q, and r, respectively, gives the relations z1z2p/C27z21/C27z1z2/C27z221CC1CA /C28r/C300 (11) z21z2z1/C27z2 ðÞ /C28qz1/C28r/C300 (12) q/C27pz2/C27z32/C300; (13) as well as their cyclic permutations. Ferrari was the first to develop an algebraic techni- que for solving the general quartic. He applied his technique (which was stolen and published by Car-dano) to the equation x 4/C276x2/C2860x/C2736/C300 (14) (Smith 1994, p. 207). The x3term can be eliminated from the general quartic (1) by making a substitution OF THE FORM z/C13x/C28l; (15) so x4/C27a3/C284l ðÞ x3/C27a2/C283a3l/C276l21CC1CA x2 /C27 a1 /C282a2 l /C273a3 l2 /C284l31CC1CA x /C27 a0 /C28a1 l /C27a2 l2 /C28a3 l3 /C27 l41CC1CA : (16) Letting l /C30a3 =4so z /C13x /C281 4a3 (17) then gives the standard form x4 /C27px2 /C27qx /C27r /C300 ; (18) where p /C13a2 /C283 8a2 3 (19) q /C13a1 /C281 2a2a3 /C2718a3 3 (20) r /C13a0 /C281 4a1a3 /C271 16a2a2 3 /C283 256a43 : (21) Adding and subtracting x2u /C27u2 =4 to (6) gives x4 /C27x2u /C271 4u21CAr1CA7 /C28x2u /C2814u2 /C27px2 /C27qx /C27r /C300; (22) which can be rewritten x2 /C2712u1CAr1CA72 /C28 (u /C28p)x2 /C28qx /C2714u2 /C28r1CAr1CA7 hi /C300 (23) (Birkhoff and Mac Lane 1965). The first term is a perfect square P2 ; and the second term is a perfect square Q2 for those u such that q2 /C304(u /C28p)14u2 /C28r1CAr1CA7 : (24) This is the resolvent CUBIC , and plugging a solution u1 back in gives P2 /C28Q2 /C30(P /C27Q)(P /C28Q) ; (25) so (23) becomes x2 /C271 2u1 /C27Q1CAr1CA7 x2 /C2712u1 /C28Q1CAr1CA7 ; (26) where Q /C13Ax /C28B (27) A /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiu1 /C28pp(28) B /C13/C28q 2A : (29) Let y1be a REAL ROOT of the resolvent CUBIC EQUATION y3 /C28a2y2 /C27 a1a3 /C284a0 ðÞ y /C27 4a2a0 /C28a2 1 /C28a23a01CC1CA /C300: (30) The four ROOTS are then given by the ROOTS of the equationx2 /C271 2a3 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 3 /C284a2q /C274y11CA81CA9 /C271 2y1 /C14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2 1 /C284a0q1CA81CA9 /C300; (31) which are z1 /C30/C281 4a3 /C2712R /C2712D (32) z2 /C30/C281 4a3 /C2712R /C2812D (33) z3 /C30/C2814a3 /C2812R /C2712E (34) z4 /C30/C281 4a3 /C2812R /C2812E ; (35) where R /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4a2 3 /C28a2 /C27y1q (36) D /C13 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4a2 3 /C28R2 /C282a2 /C281 44a3a2 /C288a1 /C28a3 3 ðÞ R/C281q R "0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4a2 3 /C282a2 /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2 1 /C284a0p q R /C3008 < : (37) E /C13 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4a2 3 /C28R2 /C282a2 /C281 44a3a2 /C288a1 /C28a3 3 ðÞ R/C281q R "0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4a2 3 /C282a2 /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2 1 /C284a0p q R /C3008 < : (38) Another approach to solving the quartic (6) defines a /C13 x1 /C27x2 ðÞ x3 /C27x4 ðÞ /C30/C28 x1 /C27x2 ðÞ2(39) b /C13 x1 /C27x3 ðÞ x2 /C27x4 ðÞ /C30/C28 x1 /C27x3 ðÞ2(40) g /C13 x1 /C27x4 ðÞ x2 /C27x3 ðÞ /C30/C28 x2 /C27x3 ðÞ2; (41) where the second forms follow from x1 /C27x2 /C27x3 /C27x4 /C30/C28a3 /C300; (42) and defining h(x) /C13(x /C28 a)(x /C28 b)(x /C28 g) (43) /C30x3 /C28 a /C27 b /C27 g ðÞ x2/C27ab/C27ag/C27bg ðÞ x/C28abg: (44) This equation can be written in terms of the original coefficients p,q, and ras h(x)/C30x3/C282px2/C27(p2/C284r)x/C27q2: (45) The roots of this CUBIC EQUATION then give a;b;and g;and the equations (39) to (41) can be solved for the four roots xiof the original quartic (Faucette 1996). See also CUBIC EQUATION ,DISCRIMINANT (POLYNO- MIAL ), QUINTIC EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 17 /C1/18, 1972. Berger, M. §16.4.1 /C1/16.4.11.1 in Geometry I. New York: Springer-Verlag, 1987. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 12, 1987. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 107 /C1/108, 1996. Borwein, P. and Erde´lyi, T. "Quartic Equations." §1.1.E.1e in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 4, 1995. Brown, K. S. "Reducing Quartics to Cubics." http://www.sea- net.com/~ksbrown/kmath296.htm. Ehrlich, G. §4.16 in Fundamental Concepts of Abstract Algebra. Boston, MA: PWS-Kent, 1991. Faucette, W. M. "A Geometric Interpretation of the Solution of the General Quartic Polynomial." Amer. Math. Monthly 103,51/C1/57, 1996. Smith, D. E. A Source Book in Mathematics. New York: Dover, 1994. van der Waerden, B. L. §64 in Algebra, Vol. 1. New York: Springer-Verlag, 1993. Quartic Graph A quartic graph is a GRAPH which is 4-REGULAR . The unique quartic graph on five nodes is the COMPLETE GRAPH K5 ; and the unique quartic graph on six nodes is the CIRCULANT GRAPH Ci1;2(6): There are two quartic graphs on seven nodes, one of which is the CIRCULANT GRAPH Ci1 ;3(7) : The numbers of connected quartic graphs on n /C301, 2, ... nodes are 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, ... (Sloane’s A006820), the numbers of not necessarily connected quartic graphs are 0, 0, 0, 0, 1, 1, 2, 6, 16, 60, ... (Sloane’s A033301), and the numbers of disconnected quartic graphs for n /C3010, 11, ... are 1, 1, 3, 8, 25, 88, ... (Sloane’s A033483; Read and Wilson 1998). The following tables gives polyhedra whose SKELE- TONS are quartic. POLYHEDRON nodes OCTAHEDRON 6 CUBOCTAHEDRON 12 SMALL RHOMBICUBOCTAHEDRON 24 ICOSIDODECAHEDRON 30 SMALL RHOMBICOSIDODECAHEDRON 60 See also CUBIC GRAPH ,Q UINTIC GRAPH ,R EGULAR GRAPHReferences Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, p. 648, 1996. Faradzev, I. A. "Constructive Enumeration of Combinatorial Objects." In Proble `mes combinatoires et the´orie des graphes (Orsay, 9 /C1/13 Juillet 1976). Paris: Centre Nat. Recherche Scient., pp. 131 /C1/135, 1978. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Sloane, N. J. A. Sequences A006820/M1617, A033301, and A033483 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Quartic Reciprocity Theorem BIQUADRATIC RECIPROCITY THEOREM Quartic Residue QUARTIC RECIPROCITY THEOREM Quartic Surface An ALGEBRAIC SURFACE of ORDER 4. Unlike CUBIC SURFACES , quartic surfaces have not been fully classified. See also BOHEMIAN DOME,B URKHARDT QUARTIC , CASSINI SURFACE ,CUSHION ,CYCLIDE ,DESMIC SUR- FACE ,F RESNEL’S ELASTICITY SURFACE ,G OURSAT’S SURFACE ,KUMMER SURFACE ,M ITER SURFACE ,PIRI- FORM ,R OMAN SURFACE ,S YMME TROID ,T ETRAHE- DROID ,TOOTH SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 9, 1986. Fischer, G. (Ed.). Plates 40 /C1/41, 45 /C1/49, and 52 /C1/56 in Mathematische Modelle/Mathematical Models, Bild- band/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 40 /C1/41, 45 /C1/49, and 52 /C1/56, 1986. Hunt, B. "Some Quartic Surfaces." Appendix B.5 in The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 310 /C1/319, 1996. Jessop, C. Quartic Surfaces with Singular Points. Cam- bridge, England: Cambridge University Press, 1916. Quartile One of the four divisions of observations which have been grouped into four equal-sized sets based on their RANK . The quartile including the top RANKED mem- bers is called the first quartile and denoted Q1 : The other quartiles are similarly denoted Q2 ; Q3 ; and Q4 : For N data points with N OF THE FORM 4n /C275 (for n /C300, 1, ...), the HINGES are identical to the first and third quartiles. See also HINGE ,INTERQUARTILE RANGE ,PERCENTILE , QUANTILE ,Q UARTILE DEVIATION ,Q UARTILE VARIA- TION COEFFICIENT References Kenney, J. F. and Keeping, E. S. "Quartiles." §3.3 in Mathe- matics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 35 /C1/37, 1962. Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 184 /C1/186, 1967. Quartile Deviation QD /C301 2Q3 /C28Q1 ðÞ ; where Q1and Q3are the first and third QUARTILES and Q3 /C28Q1 is the INTERQUARTILE RANGE . See also INTERQUARTILE RANGE ,QUARTILE ,QUARTILE VARIATION COEFFICIENT References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 36, 1962. Quartile Range INTERQUARTILE RANGE Quartile Skewness Coefficient BOWLEY SKEWNESS Quartile Variation Coefficient V /C13100Q3 /C28 Q1 Q3 /C27 Q1; where Q1and Q3are the first and third QUARTILES and Q3 /C28Q1 is the INTERQUARTILE RANGE . See also INTERQUARTILE RANGE ,QUARTILE ,QUARTILE DEVIATION Quasiamicable Pair Let s(m) be the DIVISOR FUNCTION of m. Then two numbers m and n are a quasiamicable pair if s(m) /C30 s(n) /C30m /C27n /C271 : The first few are (48, 75), (140, 195), (1050, 1575), (1648, 1925), ... (Sloane’s A005276). Quasiamicable numbers are sometimes called BETROTHED NUMBERS or REDUCED AMICABLE PAIRS . See also AMICABLE PAIR References Beck, W. E. and Najar, R. M. "More Reduced Amicable Pairs." Fib. Quart. 15, 331 /C1/332, 1977. Guy, R. K. "Quasi-Amicable or Betrothed Numbers." §B5 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 59 /C1/60, 1994. Hagis, P. and Lord, G. "Quasi-Amicable Numbers." Math. Comput. 31, 608 /C1/611, 1977.Sloane, N. J. A. Sequences A005276/M5291 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Quasiconformal Map A generalized CONFORMAL MAP. See also BELTRAMI DIFFERENTIAL EQUATION References Iyanaga, S. and Kawada, Y. (Eds.). "Quasiconformal Map- pings." §347 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1086 /C1/1088, 1980. Quasigroup A GROUPOID S such that for all a ;b /C23 S; there exist unique x;y /C23 S such that ax /C30b ya /C30b: No other restrictions are applied; thus a quasigroup need not have an IDENTITY ELEMENT , not be associa- tive, etc. Quasigroups are precisely GROUPOIDS whose multiplication tables are LATIN SQUARES . A qua- sigroup can be empty. See also BINARY OPERATOR ,G ROUPOID ,L ATIN SQUARE ,LOOP (ALGEBRA ), MONOID ,SEMIGROUP References Albert, A. A. (Ed.). Studies in Modern Algebra. Washington, DC: Math. Assoc. Amer., 1963. van Lint, J. H. and Wilson, R. M. A Course in Combinato- rics. New York: Cambridge University Press, 1992. Quasi-Monte Carlo Integration A method of NUMERICAL INTEGRATION based on equidistributed sequences (Ueberhuber 1997, p. 125). A quasi-Monte Carlo method known as the Halton-Hammersley-Wozniakowski algorithm is im- plemented in Mathematica as NIntegrate [f, ..., Method- /C21QuasiMonteCarlo ]. See also CUBATURE ,N UMERICAL INTEGRATION , MONTE CARLO INTEGRATION References Hammersley, J. M. "Monte Carlo Methods for Solving Multivariable Problems." Ann. New York Acad. Sci. 86, 844/C1/874, 1960. Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, pp. 124 /C1/125, 1997. Wozniakowski, H. "Average Case Complexity of Multivari- ate Integration." Bull. Amer. Math. Soc. 24, 185/C1/194, 1991. Quasiperfect Number A least ABUNDANT NUMBER , i.e., one such that s(n)/C302n/C271: Quasiperfect numbers are therefore the sum of their nontrivial DIVISORS . No quasiperfect numbers are known, although if any exist, they must be greater than 1035 and have seven or more DIVISORS . Singh (1997) called quasiperfect numbers SLIGHTLY EXCES- SIVE NUMBERS . See also ABUNDANT NUMBER ,A LMOST PERFECT NUMBER ,PERFECT NUMBER References Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45 /C1/53, 1994. Singh, S. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker, p. 13, 1997. Quasiperiodic Function WEIERSTRASS SIGMA FUNCTION ,W EIERSTRASS ZETA FUNCTION Quasiperiodic Motion The type of motion executed by a DYNAMICAL SYSTEM containing two incommensurate frequencies. Quasirandom Sequence A sequence of n-tuples that fills n-space more uniformly than uncorrelated random points. Such a sequence is extremely useful in computational pro- blems where numbers are computed on a grid, but it is not known in advance how fine the grid must be to obtain accurate results. Using a quasirandom se- quence allows stopping at any point where conver- gence is observed, whereas the usual approach of halving the interval between subsequent computa- tions requires a huge number of computations be- tween stopping points. See also PSEUDORANDOM NUMBER ,RANDOM NUMBER References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Quasi- (that is, Sub-) Random Sequences." §7.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 299 /C1/306, 1992. Quasiregular Polyhedron A quasiregular polyhedron is the solid region interior to two DUAL REGULAR POLYHEDRA with SCHLA ¨ FLI SYMBOLS p ;qfg : and q;pfg : Quasiregular polyhedra are denoted using a SCHLA ¨ FLI SYMBOL OF THE FORM p qno ; with p q1C2r1C27 /C30q p1C2r1C27 : (1) Quasiregular polyhedra have two kinds of regular faces with each entirely surrounded by faces of theother kind, equal sides, and equal dihedral angles. They must satisfy the Diophantine inequality 1 p /C271 q /C271 r> 1 : (2) But p ;q ]3 ; so r must be 2. This means that the possible quasiregular polyhedra have symbols3 31C81C9 ; 341C81C9 ; and351C81C9 : Now 3 31C2r1C27 /C30 3 ;4fg (3) is the OCTAHEDRON , which is a regular PLATONIC SOLID and not considered quasiregular. This leaves only two convex quasiregular polyhedra: the CUBOC- TAHEDRON3 41C81C9 and the ICOSIDODECAHEDRON351C81C9 :/ If nonconvex polyhedra are allowed, then additional quasiregular polyhedra the DODECADODECAHEDRON f5;5 2g GREAT ICOSIDODECAHEDRON f3;52 g; as well as 12 others (Hart). For faces to be equatorial hfg; h /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4N1 /C271p /C281: (4) The EDGES of quasiregular polyhedra form a system of GREAT CIRCLES : the OCTAHEDRON forms three SQUARES , the CUBOCTAHEDRON four HEXAGONS , and the ICOSIDODECAHEDRON six DECAGONS . The VERTEX FIGURES of quasiregular polyhedra are RECTANGLES (Hart). The EDGES are also all equivalent, a property shared only with the completely regular PLATONIC SOLIDS . See also CUBOCTAHEDRON ,D ODECADODECAHEDRON , GREAT ICOSIDODECAHEDRON ,ICOSIDODECAHEDRON , PLATONIC SOLID References Coxeter, H. S. M. "Quasi-Regular Polyhedra." §2 /C1/3in Reg- ular Polytopes, 3rd ed. New York: Dover, pp. 17 /C1/20, 1973. Fejes To´th, L. Ch. 4 in Regular Figures. Oxford, England: Pergamon Press, 1964. Hart, G. "Quasi-Regular Polyhedra." http://www.george- hart.com/virtual-polyhedra/quasi-regular-info.html. Robertson, S. A. and Carter, S. "On the Platonic and Archimedean Solids." J. London Math. Soc. 2, 125 /C1/132, 1970. Quasirhombicosidodecahedron GREAT RHOMBICOSIDODECAHEDRON (UNIFORM ) Quasirhombicuboctahedron GREAT RHOMBICUBOCTAHEDRON (UNIFORM ) Quasisimple Group A FINITE GROUP L is quasisimple if L /C30 L ;L½/C138 and L=Z(L)i sa SIMPLE GROUP . See also COMPONENT ,FINITE GROUP ,SIMPLE GROUP Quasithin Theorem In the classical quasithin case of the QUASI-UNIPO- TENT PROBLEM , if a group G does not have a "strongly embedded" SUBGROUP , then G is a GROUP of LIE-TYPE in characteristic 2 of Lie RANK 2 generated by a pair of parabolic SUBGROUPS P1 and P2 ; or G is one of a short list of exceptions. See also LIE-TYPE GROUP ,Q UASI- UNIPOTENT PRO- BLEM Quasitruncated Cuboctahedron GREAT TRUNCATED CUBOCTAHEDRON Quasitruncated Dodecadocahedron TRUNCATED DODECADODECAHEDRON Quasitruncated Dodecahedron TRUNCATED DODECAHEDRON Quasitruncated Great Stellated Dodecahedron GREAT STELLATED TRUNCATED DODECAHEDRON Quasitruncated Hexahedron STELLATED TRUNCATED HEXAHEDRON Quasitruncated Small Stellated Dodecahedron SMALL STELLATED TRUNCATED DODECAHEDRON Quasi-Unipotent Group A GROUP G is quasi-unipotent if every element of G of order p is UNIPOTENT for all PRIMES p such that G has p-RANK ]3:/ Quasi-Unipotent Problem QUASITHIN THEOREM Quaternary The BASE 4 method of counting in which only the DIGITS 0, 1, 2, and 3 are used. The following table gives the quaternary equivalents of the first few decimal numbers. 1 1 11 23 21 111 2 2 12 30 22 112 3 3 13 31 23 113 41014 3224120 51115 3325121 6 12 16 100 26 122 7 13 17 101 27 1238 20 18 102 28 130 9 21 19 103 29 131 10 22 20 110 30 132 These DIGITS have the following MULTIPLICATION TABLE . //C29/01 2 3 000 0 0 101 2 3 2021 01 2 3031 22 1 See also BASE (NUMBER ), BINARY ,DECIMAL ,HEXADE- CIMAL ,M OSER-DE BRUIJN SEQUENCE ,OCTAL ,TERN- ARY References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 9 /C1/10, 1991. Weisstein, E. W. "Bases." M ATHEMATICA NOTEBOOK BASES.M . Quaternary Tree QUADTREE Quaternion A member of a noncommutative DIVISION ALGEBRA first invented by William Rowan Hamilton. The idea for quaternions occurred to him while be was walkingalong the Royal Canal on his way to a meeting of the Irish Academy, and Hamilton was so pleased with his discovery that he scratched the fundamental formulaof quaternion algebra, i 2/C30j2/C30k2/C30ijk/C30/C281; (1) into the stone of the Brougham bridge (Mishchenkoand Solovyov 2000). The set of quaternions is denotedH;and the quaternions are a single example of a more general class of HYPERCOMPLEX NUMBERS discovered by Hamilton. While the quaternions are not commu-tative, they are associative, and they form a GROUP known as the QUATERNION GROUP . The quaternions can be represented using complex 2/C292MATRICES H/C30zw /C28¯w ¯z1C2C1C2A /C30a/C27ib c/C27id /C28c/C27id a/C28ib1C2C1C2A ; (2) where zandware COMPLEX NUMBERS ,a,b,c, and d are REAL , and ¯zis the COMPLEX CONJUGATE ofz.A quaternion can be represented using Quaternion [a, b,c,d] in the Mathematica add-on package Algeb- ra‘Quaternions‘ (which can be loaded with the command BBAlgebra‘ ), where a,b,c, and dare explicit real numbers. By analogy with the COMPLEX NUMBERS being repre- sentable as a sum of REAL and IMAGINARY PARTS ,a/C215 1/C27bi;a quaternion can also be written as a linear combination H/C30aU/C27bI/C27cJ/C27dK (3) of the four matrices U/C1310 011C2C1C2A (4) I/C13i0 0/C28i1C2C1C2A (5) J/C1301 /C28101C2C1C2A (6) K/C130i i01C2C1C2A : (7) (Note that here, Uis used to denote the IDENTITY MATRIX , not I:/) The matrices are closely related to the PAULI SPIN MATRICES sx;sy;sz;combined with the IDENTITY MATRIX . From the above definitions, it follows that I2/C30/C28U (8) J2/C30/C28U (9) K2/C30/C28U (10) Therefore I;J;and Kare three essentially different solutions of the matrix equation X2/C30/C28U; (11) which could be considered the square roots of the negative identity matrix. A LINEAR COMBINATION of basis quaternions with integer coefficients is some- times called a H AMILTONIAN INTEGER . InR4;the basis of the quaternions can be given by i/C130100 /C281 000 0001 00 /C28102 6643 775(12) j/C1300 0 /C281 00 /C2810 01 0 0 10 0 02 6643 775(13)k/C1300 /C2810 00 01 10 000/C281002 6643 775(14) 1/C131000 0100 001000012 6643 775: (15) The quaternions satisfy the following identities, sometimes known as H AMILTON’S RULES , i2/C30j2/C30k2/C30/C281 (16) ij/C30/C28ji/C30k (17) jk/C30/C28kj/C30i (18) ki/C30/C28ik/C30j: (19) They have the following multiplication table. 1 ij k 11 ij k ii/C281 k //C28j/ jj //C28k//C281 i kk j //C28i//C281 The quaternions 91,9i;9j;and9kform a NON- ABELIAN GROUP of order eight (with multiplication as the group operation) known as Q8ofH:/ The quaternions can be written in the form a/C30a1/C27a2i/C27a3j/C27a4k: (20) The conjugate quaternion is given by ¯a/C30a1/C28a2i/C28a3j/C28a4k: (21) The sum of two quaternions is then a/C27b/C30a1/C27b1 ðÞ /C27a2/C27b2 ðÞ i/C27a3/C27b3 ðÞ j /C27a4/C27b4 ðÞ k; (22) and the product of two quaternions is ab/C30a1b1/C28a2b2/C28a3b3/C28a4b4 ðÞ /C27a1b2/C27a2b1/C27a3b4/C28a4b3 ðÞ i /C27a1b3/C28a2b4/C27a3b1/C27a4b2 ðÞ j /C27a1b4/C27a2b3/C28a3b2/C27a4b1 ðÞ k; (23) so the norm is n(a)/C30ffiffiffiffiffiffi a¯ap /C30ffiffiffiffiffiffi ¯aap /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 1/C27a22/C27a23/C27a24q : (24) In this notation, the quaternions are closely related to FOUR-VECTORS . Quaternions can be interpreted as a SCALAR plus a VECTOR by writing a /C30a1 /C27a2i /C27a3j /C27a4k /C30 a1 ;a ðÞ ; (25) where a /C13 a2a3a4 ½/C138 :: In this notation, quaternion multiplication has the particularly simple form q1q2 /C30 s1 ;v1 ðÞ /C215 s2 ;v2 ðÞ /C30 s1s2 /C28v1/C215 v2 ; s1v2 /C27s2v1 /C27v1 /C29v2 ðÞ : (26) Division is uniquely defined (except by zero), so quaternions form a DIVISION ALGEBRA . The inverse of a quaternion is given by a/C281 /C30¯a a¯a ; (27) and the norm is multiplicative n(ab) /C30n(a)n(b) : (28) In fact, the product of two quaternion norms imme- diately gives the EULER FOUR-SQUARE IDENTITY . A rotation about the UNIT VECTOR ˆn by an angle u can be computed using the quaternion q /C30(s ;v) /C30 cos1 2 u1CAr1CA7 ; ˆn sin12u1CAr1CA7 1CAr1CA7 (29) (Arvo 1994, Hearn and Baker 1996). The components of this quaternion are called EULER PARAMETERS . After rotation, a point p /C30(0;p) is then given by p?/C30qpq/C281 /C30qp¯q; (30) since n(q) /C301: A concatenation of two rotations, first q1 and then q2 ; can be computed using the identity q2q1p¯q1 ðÞ ¯q2 /C30 q2q1 ðÞ p ¯q1 ¯q2 ðÞ/C30 q2q1 ðÞ pq2q1 (31) (Goldstein 1980). See also BIQUATERNION ,CAYLEY- KLEIN PARAMETERS , COMPLEX NUMBER ,DIVISION ALGEBRA ,EULER PARA- METERS ,FOUR- VECTOR ,H AMILTONIAN INTEGER ,H Y- PERCOMPLEX NUMBER ,O CTONION ,Q UATERNION GROUP References Altmann, S. L. Rotations, Quaternions, and Double Groups. Oxford, England: Clarendon Press, 1986. Arvo, J. Graphics Gems II. New York: Academic Press, pp. 351 /C1/354 and 377 /C1/380, 1994. Baker, A. L. Quaternions as the Result of Algebraic Opera- tions. New York: Van Nostrand, 1911. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 230 /C1/234, 1996. Crowe, M. J. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. New York: Dover, 1994. Dickson, L. E. Algebras and Their Arithmetics. New York: Dover, 1960.Downs, L. "CS184: Using Quaternions to Represent Rota- tion." http://http.cs.berkeley.edu/~laura/cs184/quat/qua- ternion.html. Du Val, P. Homographies, Quaternions, and Rotations. Oxford, England: Oxford University Press, 1964. Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A; Koecher, M.; Mainzer, M.; and Remmert, R. Numbers. New York: Springer-Verlag, 1990. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 151, 1980. Hamilton, W. R. Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method. Dublin: Hodges and Smith, 1853. Hamilton, W. R. Elements of Quaternions. London: Long- mans, Green, 1866. Hamilton, W. R. The Mathematical Papers of Sir William Rowan Hamilton. Cambridge, England: Cambridge Uni- versity Press, 1967. Hardy, A. S. Elements of Quaternions. Boston, MA: Ginn, Heath, & Co., 1881. Hardy, G. H. and Wright, E. M. "Quaternions." §20.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 303 /C1/306, 1979. Hearn, D. and Baker, M. P. Computer Graphics: C Version, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 419 /C1/420 and 617 /C1/618, 1996. Joly, C. J. A Manual of Quaternions. London: Macmillan, 1905. Julstrom, B. A. "Using Real Quaternions to Represent Rotations in Three Dimensions." UMAP Modules in Undergraduate Mathematics and Its Applications, Module 652. Lexington, MA: COMAP, Inc., 1992. Kelland, P. and Tait, P. G. Introduction to Quaternions, 3rd ed. London: Macmillan, 1904. Kuipers, J. B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Vir- tual Reality. Princeton, NJ: Princeton University Press, 1998. Mishchenko, A. and Solovyov, Y. "Quaternions." Quantum 11,4/C1/7 and 18, 2000. Nicholson, W. K. Introduction to Abstract Algebra, 2nd ed. New York: Wiley, 1999. Salamin, G. Item 107 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 46 /C1/47, Feb. 1972. Shoemake, K. "Animating Rotation with Quaternion Curves." Computer Graphics 19, 245 /C1/254, 1985. Tait, P. G. An Elementary Treatise on Quaternions, 3rd ed., enl. Cambridge, England: Cambridge University Press, 1890. Tait, P. G. "Quaternions." Encyclopædia Britannica, 9th ed. ca. 1886. ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/ tait/Encyc-Brit.ps.gz. Weisstein, E. W. "Books about Quaternions." http:// www.treasure-troves.com/books/Quaternions.html. Quaternion Group The NON- ABELIAN GROUP of order eight formed by the QUATERNIONS 9 1, 9i ;9j ; and 9k , denoted Q8 or H:/ See also QUATERNION Quattuordecillion In the American system, 1045. See also LARGE NUMBER Queens Problem What is the maximum number of queens which can be placed on an n/C29nCHESSBOARD such that no two attack one another? The answer is nqueens, which gives eight queens for the usual 8 /C298 board (Madachy 1979; Steinhaus 1983, p. 29). The number of different ways the nqueens can be placed on an n/C29nchess- board so that no two queens may attack each other for the first few nare 1, 0, 0, 2, 10, 4, 40, 92, ... (Sloane’s A000170; Madachy 1979; Steinhaus 1983, p. 29). The number of rotationally and reflectively distinct solu-tions are 1, 0, 0, 1, 2, 1, 6, 12, 46, 92, ... (Sloane’sA002562; Dudeney 1970; p. 96). The 12 distinctsolutions for n/C308 are illustrated above, and the remaining 80 are generated by ROTATION and REFLEC- TION (Madachy 1979, Steinhaus 1983). The minimum number of queens needed to occupy orattack all squares of an 8 /C298 board is 5 (Steinhaus 1983, p. 29). Dudeney (1970, pp. 95 /C1 /96) gave the following results for the number of distinct arrange-ments N p(k;n)o f kqueens attacking or occupying every square of an n/C29nboard for which every queen is attacked ("protected") by at least one other, withthen/C308 value given by Steinhaus (1983, p. 29). The 4860 solutions in the n/C305 case may be obtained from 638 fundamental arrangements by ROTATION and REFLECTION .kQueens /n/C29n//Np(k;n)/ 24 3 35 3 7 36 1 47 55 8 4860 Dudeney (1970, pp. 95 /C1 /96) also gave the following results for the number of distinct arrangements Nu(k;n)o f kqueens attacking or occupying every square of an n/C29nboard for which no two queens attack one another (they are "not protected"). kQueens /n/C29n//Nu(k;n)/ 12 1 13 1 34 2 35 246 1 7 47 1 58 9 1 Vardi (1991) generalizes the problem from a square chessboard to one with the topology of the TORUS . The number of solutions for nqueens with nODD are 1, 0, 10, 28, 0, 88, ... (Sloane’s A007705). Vardi (1991) alsoconsiders the toroidal "semiqueens" problem, in which a semiqueen can move like a rook or bishop,but only on POSITIVE broken diagonals. The number of solutions to this problem for nqueens with nODD are 1, 3, 15, 133, 2025, 37851, ... (Sloane’s A006717), and0 for EVEN n. Velucchi gives the solution to the question, "Howmany different arrangements of kqueens are possible on an order nchessboard?" as /1=8/th of the COEFFI- CIENT ofakbn2/C28kin the POLYNOMIAL p(a;b;n)/C30a/C27b ðÞn2/C272a/C27b ðÞna2/C27b2ðÞn2/C28nðÞ =2 /C273a2/C27b2ðÞn2=2/C272a4/C27b4ðÞn2=4 neven a/C27b ðÞn2/C272a/C27b ðÞ a4/C27b4ðÞn2/C281ðÞ =4 /C27a/C27b ðÞ a2/C27b2ðÞn2/C281ðÞ =2 /C274a/C27b ðÞna2/C27b2ðÞn2/C28nðÞ =2 nodd:8 >>>>>>>>>>< >>>>>>>>>>: Velucchi also considers the nondominating queens problem, which consists of placing n queens on an order n chessboard to leave a maximum number U(n) of unattacked vacant cells. The first few values are 0, 0, 0, 1, 3, 5, 7, 11, 18, 22, 30, 36, 47, 56, 72, 82, ... (Sloane’s A001366). The results can be generalized to k queens on an n /C29n board. See also BISHOPS PROBLEM ,CHESS ,KINGS PROBLEM , KNIGHTS PROBLEM ,KNIGHT’S TOUR,ROOKS PROBLEM References Ahrens, W. "Das Achtko ¨niginnenproblem." Ch. 9 in Mathe- matische Unterhaltungen und Spiele, dritte, verbesserte, anastatisch gedruckte aufl., Bd. 1. Leipzig, Germany: Teubner, pp. 211 /C1/284, 1921. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 166 /C1/169, 1987. Campbell, P. J. "Gauss and the 8-Queens Problem: A Study in the Propagation of Historical Error." Historia Math. 4, 397 /C1/404, 1977. Dudeney, H. E. "The Eight Queens." §300 in Amusements in Mathematics. New York: Dover, p. 89, 1970. Erbas, C. and Tanik, M. M. "Generating Solutions to the N- Queens Problem Using 2-Circulants." Math. Mag. 68, 343 /C1/356, 1995. Erbas, C.; Tanik, M. M.; and Aliyzaicioglu, Z. "Linear Congruence Equations for the Solutions of the N-Queens Problem." Inform. Proc. Let. 41, 301 /C1/306, 1992. Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243,18/C1/28, Dec. 1980. Garey, M. R. and Johnson, D. S. Computers and Intract- ability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman, 1983. Ginsburg, J. "Gauss’s Arithmetization of the Problem of n Queens." Scripta Math. 5,63/C1/66, 1939. Guy, R. K. "The n Queens Problem." §C18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 133 /C1/135, 1994. Kraitchik, M. "The Problem of the Queens" and "Domination of the Chessboard." §10.3 and 10.4 in Mathematical Recreations. New York: W. W. Norton, pp. 247 /C1/256, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 34 /C1/36, 1979. Riven, I.; Vardi, I.; and Zimmerman, P. "The n-Queens Problem." Amer. Math. Monthly 101, 629 /C1/639, 1994. Riven, I. and Zabih, R. "An Algebraic Approach to Con- straint Satisfaction Problems." In Proc. Eleventh Internat. Joint Conference on Artificial Intelligence, Vol. 1, August 20 /C1/25, 1989. Detroit, MI: IJCAII, pp. 284 /C1/289, 1989. Ruskey, F. "Information on the n Queens Problem." http:// www.theory.csc.uvic.ca/~cos/inf/misc/Queen.html. Sloane, N. J. A. Sequences A000170/M1958, A001366, A002562/M0180, A006717/M3005, and A007705/M4691 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0180 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 29 /C1/30, 1999. Vardi, I. "The n-Queens Problems." Ch. 6 in Computational Recreations in Mathematica. Redwood City, CA: Addison- Wesley, pp. 107 /C1/125, 1991.Velucchi, M. "For Me, this Is the Best Chess-Puzzle: Non- Dominating Queens Problem." http://anduin.eldar.org/ ~problemi/papers.html. Velucchi, M. "Different Dispositions on the ChessBoard." http://anduin.eldar.org/~problemi/papers.html. Queens Tour A TOUR of a queen on a CHESSBOARD satisfying certain properties. References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 116 /C1/118 and 124 /C1/126, 1984. Quermass BRIGHTNESS ,OUTER QUERMASS Question Mark Function MINKOWSKI’S QUESTION MARK FUNCTION Queue A queue is a special kind of LIST in which elements may only be removed from the bottom by a POPaction or added to the top using a PUSH action. Examples of queues include people waiting in line, and submitted jobs waiting to be printed on a printer. The study of queues is called QUEUING THEORY . See also LIST,PRIORITY QUEUE ,QUEUING THEORY , STACK Queuing Theory The study of the waiting times, lengths, and other properties of QUEUES . References Allen, A. O. Probability, Statistics, and Queueing Theory with Computer Science Applications, 2nd ed. Orlando, FL: Academic Press, 1990. Bunday, B. D. An Introduction to Queueing Theory. Oxford, England: Oxford University Press, 1996. Gross, D. and Harris, C. M. Fundamentals of Queueing Theory, 3rd ed. New York: Wiley, 1998. Quicksort The fastest known SORTING ALGORITHM (on average, and for a large number of elements), requiringO(nlgn) steps. Quicksort is a recursive algorithm which first partitions an array a ifgn i/C301according to several rules (Sedgewick 1978): 1. Some key n is in its final position in the array (i.e., if it is the jth smallest, it is in position aj) :/ 2. All the elements to the left of aj are less than or equal to aj : The elements a1 ; a2 ; ..., aj/C281 are called the "left subfile." 3. All the elements to the right of ajare greater than or equal to aj : The elements aj /C271 ; ..., anare called the "right subfile." Quicksort was invented by Hoare (1961, 1962), has undergone extensive analysis and scrutiny (Sedge- wick 1975, 1977, 1978), and is known to be about twice as fast as the next fastest SORTING algorithm. In the worst case, however, quicksort is a slow n2 algorithm (and for quicksort, "worst case" corre- sponds to already sorted). See also HEAPSORT ,SORTING References Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Struc- tures and Algorithms. Reading, MA: Addison-Wesley, pp. 260 /C1/270, 1987. Hoare, C. A. R. "Partition: Algorithm 63," "Quicksort: Algo- rithm 64," and "Find: Algorithm 65." Comm. ACM 4, 321 /C1/ 322, 1961. Hoare, C. A. R. "Quicksort." Computer J. 5,10/C1/15, 1962. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Quicksort." §8.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 323 /C1/327, 1992. Sedgewick, R. Quicksort. Ph.D. thesis. Stanford Computer Science Report STAN-CS-75 /C1/492. Stanford, CA: Stanford University, May 1975. Sedgewick, R. "The Analysis of Quicksort Programs." Acta Informatica 7, 327 /C1/355, 1977. Sedgewick, R. "Implementing Quicksort Programs." Comm. ACM 21, 847 /C1/857, 1978. Quillen-Lichtenbaum Conjecture A technical CONJECTURE which connects algebraic K- THEORY to E´ tale cohomology. The conjecture was made more precise by Dwyer and Friedlander (1982). Thomason (1985) established the first half of this conjecture, but the entire conjecture has not yet been established. References Dwyer, W. and Friedlander, E. "E´ tale K-Theory and Arith- metic." Bull. Amer. Math. Soc. 6, 453 /C1/455, 1982. Thomason, R. W. "Algebraic K-Theory and E´ tale Cohomol- ogy." Ann. Sci. E´ cole Norm. Sup. 18, 437 /C1/552, 1985. Weibel, C. A. "The Mathematical Enterprises of Robert Thomason." Bull. Amer. Math. Soc. 34,1/C1/13, 1996. Quincunx The pattern of dots on the "5" side of a 6-sided DIE. The word derives from the Latin words for both one and five. See also DICEReferences Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 9 and 22, 1996. Quindecillion In the American system, 1048. See also LARGE NUMBER Quintet A SET of five. See also HEXAD ,MONAD ,QUARTET ,TETRAD ,TRIAD Quintic Equation Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of ADDITIONS ,SUBTRACTIONS , MULTIPLICATIONS ,DIVISIONS , and ROOT EXTRACTIONS , as rigorously demonstrated by Abel (A BEL’S IMPOSSI- BILITY THEOREM ) and Galois. However, certain classes of quintic equations can be solved in this manner. Irreducible quintic equations can be associated with aG ALOIS GROUP , which may be a SYMMETRIC GROUP Sn; METACYCLIC GROUP Mn;DIHEDRAL GROUP Dn;ALTER- NATING GROUP An;orCYCLIC GROUP Cn;as illustrated above. Euler reduced the general quintic to x5/C2810qx2/C28p/C300: (1) A quintic also can be algebraically reduced to PRINCI- PAL QUINTIC FORM x5/C27a2x2/C27a1x/C27a0/C300: (2) By solving a quartic, a quintic can be algebraically reduced to the B RING QUINTIC FORM x5/C28x/C28a/C300; (3) as was first done by Jerrard. Runge (1885) and Cadenhad and Young found a parameterization of solvable quintics in the form x5/C28ax/C27b/C300; (4) by showing that all irreducible solvable quintics with COEFFICIENTS ofx4;x3;and x2missing have the following form x5/C275m44n/C273 ðÞ n2/C271x/C275m52n/C271 ðÞ 4n/C273 ðÞ n2/C271/C300; (5) where mandnare RATIONAL . Spearman and Williams (1994) showed that an irreducible quintic OF THE FORM (4) having RATIONAL COEFFICIENTS is solvable by radicals IFFthere exist rational numbers o/C3091; c]0;ande"0 such that a/C305e4(3/C284oc) c2/C271(6) b/C30/C284e5(11o/C272c) c2/C271(7) The ROOTS are then xj/C30evju1/C27v2ju2/C27v3ju3/C27v4ju41CC1CA ; (8) where u1/C30v2 1v3 D2 !1=5 (9) u2/C30v23v4 D2 !1=5 (10) u3/C30v22v1 D2 !1=5 (11) u4/C30v24v2 D2 !1=5 (12) v1/C30ffiffiffiffi Dp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D/C28offiffiffiffi Dpq (13) v2/C30/C28ffiffiffiffiDp /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D/C27offiffiffiffi Dpq (14) v 3/C30/C28ffiffiffiffiDp /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D/C27offiffiffiffi Dpq (15) v 4/C30ffiffiffiffiDp /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D/C28offiffiffiffi Dpq (16) D/C30c2/C271: (17) In the case of a solvable quintic, the roots can be found using the formulas of Malfatti (1771), who wasthe first to "solve" the quintic using a resolvent of sixth degree (Pierpont 1895).The general quintic can be solved in terms of J ACOBI THETA FUNCTIONS , as was first done by Hermite in 1858. Kronecker subsequently obtained the same solution more simply, and Brioshi also derived theequation. To do so, reduce the general quintic a 5x5/C27a4x4/C27a3x3/C27a2x2/C27a1x/C27a0/C300 (18) into B RING QUINTIC FORM x5/C28x/C27r/C300: (19) Then define k/C13tan1 4sin/C281 16 25ffiffiffi 5p r2 !"# (20) s/C13/C28sgn(I[r]) for R[r]/C300 sgn(R[r]) for R[r]"01C2r (21) b/C30sk2ðÞ1=8 2/C21553=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1/C28k2 ðÞp (22) q/C30qk21CC1CA /C30eipK?k2ðÞ =Kk2ðÞ; (23) where kis the MODULUS ,m/C13k2is the PARAMETER , andqis the NOME . Solving qmðÞ/C30eipK?(m)=K(m)(24) formgives the INVERSE NOME m(q);and the roots of the original quintic are then given by x1/C30/C28 1ðÞ3=4bm e/C282pi=5q1=51CC1CA1C21C3 1=8/C27ime2pi=5q1=51CC1CA1C21C3 1=8no /C29me/C284pi=5q1=51CC1CA1C21C3 1=8/C27me4pi=5q1=51CC1CA1C21C3 1=8no /C29mq1=51CC1CA1C21C3 1=8/C27q5=8q51CC1CA/C281=8mq51CC1CA1C21C31=8no (25) x2/C30b/C28mq1=51CC1CA1C21C3 1=8/C27e3pi=4me2pi=5q1=51CC1CA1C21C3 1=8no /C29e/C283pi=4me/C282pi=5q1=51CC1CA1C21C3 1=8/C27ime4pi=5q1=51CC1CA1C21C3 1=8no /C29ime/C284pi=5q1=51CC1CA1C21C3 1=8/C27q5=8q51CC1CA/C281=8mq51CC1CA1C21C31=8no (26) x3/C30be/C283pi=4me/C282pi=5q1=51CC1CA1C21C3 1=8/C28ime/C284pi=5q1=51CC1CA1C21C3 1=8no /C29/C28 mq1=51CC1CA1C21C3 1=8/C28ime4pi=5q1=51CC1CA1C21C3 1=8no /C29e/C283pi=4me2pi=5q1=51CC1CA1C21C3 1=8/C27q5=8q51CC1CA/C281=8mq1=51CC1CA1C21C3 1=8no (27) x4/C30bm q1=51CC1CA1C21C3 1=8/C28ime/C284pi=5q1=51CC1CA1C21C3 1=8no /C29/C28 e/C283pi=4me2pi=5q1=51CC1CA1C21C3 1=8/C28ime4pi=5q1=51CC1CA1C21C3 1=8no /C29 e /C283 pi=4 me/C282pi=5q1=51CC1CA1C21C3 1 =8/C27q5 =8 q51CC1CA/C281 =8mq51CC1CA1C21C31 =8no (28) x5 /C30bmq1 =51CC1CA1C21C3 1=8/C28e /C283pi=4 me/C282 pi=5q1 =51CC1CA1C21C3 1 =8no /C29/C28 e3 pi=4 me2pi=5q1=51CC1CA1C21C3 1 =8/C27ime/C284 pi=5q1 =51CC1CA1C21C3 1 =8no /C29/C28 ime4 pi =5q1 =51CC1CA1C21C3 1=8/C27q5 =8 q51CC1CA/C281 =8mq51CC1CA1C21C31 =81CAro :n (29) Felix Klein used a TSCHIRNHAUSEN TRANSFORMATION to reduce the general quintic to the form z5 /C275az2 /C275bz /C27c /C300: (30) He then solved the related ICOSAHEDRAL EQUATION I(z;1 ;Z) /C30z5 /C281 /C2711z5 /C27z101CC1CA5 /C28 1 /C27z30 /C2810005 z10 /C27z201CC1CA /C27522 /C28z5 /C27z251CC1CA 1C21C32Z /C300; (31) where Z is a function of radicals of a, b, and c. The solution of this equation can be given in terms of HYPERGEOMETRIC FUNCTIONS as Z /C281 =60 2F1/C281 60 ;29 60;45 ;1728 Z1CAr1CA7 Z11=60 2F111 60;4160;65 ;1728 Z1CAr1CA7 : (32) Another possible approach uses a series expansion, which gives one root (the first one in the list below) of the BRING QUINTIC FORM t5 /C28t /C28 r: (33) All five roots can be derived using differential equations (Cockle 1860, Harley 1862). Let F1rðÞ/C30F2rðÞ (34) F2rðÞ/C304 F31 5 ;25 ;35 ;45;12;34;54;3125 256 r41CAr1CA7 (35) F3rðÞ/C304 F39 20;1320;1720 ;2120;34 ;54 ;32;3125 256 r41CAr1CA7 (36) F4rðÞ/C304 F37 10 ;9 10;11 10;1310;54;32;74;3125 256 r41CAr1CA7 ; (37) then the ROOTS are t1 /C30/C28r4F31 5 ;25 ;35 ;45;12;34;54;3125 256 r41CAr1CA7 (38) t2 /C30/C28F1( r) /C271 4 rF2(r) /C275 32r2F3( r) /C275 32r3F4( r) (39) t3 /C30/C28F1( r) /C271 4 rF2(r) /C285 32r2F3( r) /C275 32r3F4( r) (40) t4 /C30/C28iF1( r) /C2714 rF2(r) /C285 32i r2F3( r) /C285 32 r3F4(r) (41) t5 /C30/C28iF1( r) /C2714 rF2(r) /C275 32i r2F3( r) /C285 32 r3F4(r) (42) This technique gives closed form solutions in terms ofHYPERGEOMETRIC FUNCTIONS in one variable for any POLYNOMIAL equation which can be written in the form xp /C27bxq /C27c : (43) Consider the quintic Y4 j/C300x /C28 vju1 /C27 v4ju21CC1CA1C21C3 /C300; (44) where v /C30e2 pi=5 and u1 and u2 are COMPLEX NUMBERS . This is called DE MOIVRE’S QUINTIC . Generalize it to Y4 j/C300x /C28 vju1 /C27 v2ju2 /C27 v3ju3 /C27 v4ju41CC1CA1C21C3 /C300 (45) Expanding, vju1 /C27 v2ju2 /C27 v3ju3 /C27 v4ju41CC1CA 5 /C285U vju1 /C27 v2ju2 /C27 v3ju3 /C27 v4ju41CC1CA 4 /C285V vju1 /C27 v2ju2 /C27 v3ju3 /C27 v4ju41CC1CA 2 /C275W vju1 /C27 v2ju2 /C27 v3ju3 /C27 v4ju41CC1CA /C275 X /C28Y ðÞ /C28Z ½/C138 /C300; (46) where U /C30u1u4 /C27u2u3 (47) V /C30u1u2 2 /C27u2u24 /C27u3u21 /C27u4u23 (48) W /C30u21u24 /C27u22u23 /C28u31u2 /C28u32u4 /C28u33u1 /C28u34u3 /C28u1u2u3u4 (49) X /C30u31u3u4 /C27u32u1u3 /C27u33u2u4 /C27u34u1u2 (50) Y/C30u1u23u24/C27u2u21u23/C27u3u22u24/C27u4u21u22(51) Z/C30u51/C27u52/C27u53/C27u54 (52) The ui/s satisfy u1u4/C27u2u3/C300 (53) u1u22/C27u2u24/C27u3u21/C27u4u23/C300 (54) u21u24/C27u22u23/C28u31u2/C28u32u4/C28u33u1/C28u34u3/C28u1u2u3u4 /C301 5a (55) 5u3 1u3u4/C27u32u1u3/C27u33u3u4/C27u34u1u21CC1CA1C2 /C28u1u23u24/C27u2u21u23/C27u3u22u24/C27u4u21u221CC1CA /C138 /C28u51/C27u52/C27u53/C27u541CC1CA /C30b: (56) See also BRING QUINTIC FORM,B RING- JERRARD QUINTIC FORM,CUBIC EQUATION , DE MOIVRE’S QUIN- TIC,PRINCIPAL QUINTIC FORM,QUADRATIC EQUATION , QUARTIC EQUATION ,SEXTIC EQUATION References Birkhoff, G. and Mac Lane, S. "Insolvability of Quintic Equations." §15.8 in A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 418 /C1/421, 1996. Chowla, S. "On Quintic Equations Soluble by Radicals." Math. Student 13, 84, 1945. Cockle, J. "Sketch of a Theory of Transcendental Roots." Phil. Mag. 20, 145 /C1/148, 1860. Cockle, J. " On Transcendental and Algebraic Solution-- Supplemental Paper." Phil. Mag. 13, 135 /C1/139, 1862. Davis, H. T. Introduction to Nonlinear Differential and Integral Equations. New York: Dover, p. 172, 1960. Drociuk, R. J. On the Complete Solution to the Most General Fifth Degree Polynomial. 3 May 2000. http://xxx.lanl.gov/ abs/math.GM/0005026/. Dummit, D. S. "Solving Solvable Quintics." Math. Comput. 57, 387 /C1/401, 1991. Glashan, J. C. "Notes on the Quintic." Amer. J. Math. 8, 178 /C1/179, 1885. Green, M. L. "On the Analytic Solution of the Equation of Fifth Degree." Compos. Math. 37, 233 /C1/241, 1978. Harley, R. "On the Solution of the Transcendental Solution of Algebraic Equations." Quart. J. Pure Appl. Math. 5, 337 /C1/361, 1862. Harley, R. "A Contribution to the History of the Problem of the Reduction of the General Equation of the Fifth Degree to a Trinomial Form." Quart. J. Math. 6,38/C1/47, 1864. Hermite, C. "Sulla risoluzione delle equazioni del quinto grado." Annali di math. pura ed appl. 1, 256 /C1/259, 1858. King, R. B. Beyond the Quartic Equation. Boston, MA: Birkha ¨user, 1996. King, R. B. and Cranfield, E. R. "An Algorithm for Calculat- ing the Roots of a General Quintic Equation from Its Coefficients." J. Math. Phys. 32, 823 /C1/825, 1991. Klein, F. "Sull’ equazioni dell’ Icosaedro nella risoluzione delle equazioni del quinto grado [per funzioni ellittiche]." Reale Istituto Lombardo, Rendiconto, Ser. 2 10, 1877. Klein, F. "U¨ ber die Transformation der elliptischen Funk- tionen und die Auflo¨sung der Gleichungen fu¨nften Grades." Math. Ann. 14, 1878/79. Klein, F. Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. New York: Dover, 1956. Pierpont, J. "Zur Entwicklung der Gleichung V. Grades (bis 1858)." Monatsh. fu¨r Math. und Physik 6,15/C1/68, 1895. Rosen, M. I. "Niels Hendrik Abel and Equations of the Fifth Degree." Amer. Math. Monthly 102, 495 /C1/505, 1995. Runge, C. "Ueber die aufloesbaren Gleichungen von der Form x5 /C27ux /C27v /C300:/" Acta Math. 7, 173 /C1/186, 1885. Shurman, J. Geometry of the Quintic. New York: Wiley, 1997. Spearman, B. K. and Williams, K. S. "Characterization of Solvable Quintics x5 /C27ax /C27b:/" Amer. Math. Monthly 101, 986 /C1/992, 1994. Wolfram Research. "Solving the Quintic." Poster. Cham- paign, IL: Wolfram Research, 1995. http://library.wol- fram.com/examples/quintic/. Wolfram Research. "A Short History." From the Quintic Poster. Champaign, IL: Wolfram Research, 1995. http:// library.wolfram.com/examples/quintic/timeline.html. Young, G. P. "Solution of Solvable Irreducible Quintic Equations, Without the Aid of a Resolvent Sextic." Amer. J. Math. 7, 170 /C1/177, 1885.Quintic Graph A quintic graph is a GRAPH which is 5-REGULAR . The only quintic graph on n 57 nodes is the COMPLETE GRAPH K6 : The following tables gives polyhedra whose SKELETONS are quartic. POLYHEDRON nodes ICOSAHEDRON 12 SNUB CUBE 24 SNUB DODECAHEDRON 60 TRUNCATED DODECAHEDRON 60 See also CUBIC GRAPH ,Q UARTIC GRAPH ,REGULAR GRAPH Quintic Surface A quintic surface is an ALGEBRAIC SURFACE of degree 5. Togliatti (1940, 1949) showed that quintic surfaces having 31 ORDINARY DOUBLE POINTS exist, although he did not explicitly derive equations for such surfaces. Beauville (1978) subsequently proved that 31 double points was the maximum possible, and quintic surfaces having 31 ORDINARY DOUBLE POINTS are therefore sometimes called TOGLIATTI SURFACES . van Straten (1993) subsequently constructed a 3-D family of solutions and in 1994, Barth derived the example known as the DERVISH . See also ALGEBRAIC SURFACE ,D ERVISH ,K ISS SUR- FACE ,ORDINARY DOUBLE POINT ,PENINSULA SURFACE References Beauville, A. "Surfaces alge ´briques complexes." Aste´risque 54,1/C1/172, 1978. Endraß, S. "Togliatti Surfaces." http://enriques.mathemati- k.uni-mainz.de/kon/docs/Etogliatti.shtml. Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Togliatti, E. G. "Una notevole superficie de 5/C14ordine con soli punti doppi isolati." Vierteljschr. Naturforsch. Ges. Zu ¨rich 85, 127/C1/132, 1940. Togliatti, E. "Sulle superficie monoidi col massimo numero di punti doppi." Ann. Mat. Pura Appl. 30, 201/C1/209, 1949. van Straten, D. "A Quintic Hypersurface in P4with 130 Nodes." Topology 32, 857/C1/864, 1993. Quintillion In the American system, 1018. See also LARGE NUMBER Quintuple A group of five elements, also called a QUINTUPLET or PENTAD . See also MONAD ,PAIR,PENTAD ,QUADRUPLE ,QUAD- RUPLET ,Q UINTUPLET ,T ETRAD ,T RIAD ,T RIPLET , TWINS Quintuple Product Identity A.k.a. the WATSON QUINTUPLE PRODUCT IDENTITY , Y/C12 n/C3011 /C28qnðÞ 1 /C28zqnðÞ 1 /C28z /C281qn/C2811CC1CA 1 /C28z2q2n/C2811CC1CA /C29 1 /C28z/C282q2n/C2811CC1CA /C30X/C12 m/C30/C28/C12z3m /C28z/C283m/C2811CC1CA qm(2m/C271)=2 : (1) It can also be written Y/C12 n/C3011 /C28q2n1CC1CA 1 /C28q2n/C281z1CC1CA 1 /C28q2n /C281z/C2811CC1CA 1 /C28q4n/C283z21CC1CA /C2 1 /C28q4n/C284z /C2821CC1CA /C30X/C12 n/C30/C28/C12q3n2/C282nz3n /C27z/C283n1CC1CA /C28 z3n/C282 /C27z /C28(3n/C282)1CC1CA 1C21C3 (2) or X/C12 k /C30/C28/C12/C281ðÞkq 3k2/C28k ðÞ =2x3k 1 /C27zqk1CC1CA /C30Y/C12 j/C3011 /C28qj1CC1CA 1 /C27z /C281qj1CC1CA 1 /C27zqj/C2811CC1CA 1 /C27z/C282q2j/C2811CC1CA /C2 1 /C27z2q2j/C2811CC1CA : (3) The quintuple product identity can be written in Q- SERIES notation as X/C12 k /C30/C28/C12/C281ðÞkqk 3k /C281 ðÞ =2z3k 1 /C27zqk1CC1CA /C30 1 ;/C28z;/C28q =z;q ðÞ/C12qz2 ;q=z2;q21CC1CA /C12; (4) where 0 Bjq jB1 and z "0 (Gasper and Rahman 1990, p. 134; Leininger and Milne 1997). Using the NOTATION of the RAMANUJAN THETA FUNCTION (Berndt, p. 83),fB3 =q ;q5 =B31CC1CA /C28B2fq=B3 ;B3q51CC1CA /C30f /C28q21CC1CA f /C28B2 ;/C28q2 =B2ðÞ fBq ;q =B ðÞ(5) See also JACOBI TRIPLE PRODUCT ,RAMANUJAN THETA FUNCTIONS References Berndt, B. C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, 1985. Bhargava, S. "A Simple Proof of the Quintuple Product Identity." J. Indian Math. Soc. 61, 226 /C1/228, 1995. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 306 /C1/309, 1987. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990. Leininger, V. E. and Milne, S. C. "Some New Infinite Families of Eta Function Identities." Preprint. http:// www.math.ohio-state.edu/~milne/preprints.html. Quintuplet A group of five elements, also called a QUINTUPLE or PENTAD . See also MONAD ,PAIR,PENTAD ,QUADRUPLE ,QUAD- RUPLET ,Q UINTUPLET ,T ETRAD ,T RIAD ,T RIPLET , TWINS Quiteprime A POSITIVE INTEGER n /C211 is quiteprime IFF all PRIMES p 5ffiffiffinpsatisfy 2 n (mod p ½/C138 /C28p jj 5p /C271 /C28ffiffiffipp: Also define 2 and 3 to be quiteprimes. Then the first few quiteprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 137, ... (Sloane’s A050260), and the first few primes which are not quiteprimes are 131, 181, 197, 199, 233, 241, 263, 307, 311, 313, 331, 337, 353, 373, 379, ... (Sloane’s A050261). See also VERYPRIME References Ferry, J. "RE: Veryprimes defined." sci.math posting, 09 Sep 1999. Sloane, N. J. A. Sequences A050260 and A050261 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Weisstein, E. W. "Integer Sequences." M ATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Quota Rule ARECURRENCE RELATION between the function Q arising in QUOTA SYSTEMS , Qn ;rðÞ/C30Qn/C281;r /C281 ðÞ /C27Qn/C281;r ðÞ : References Young, S. C.; Taylor, A. D.; and Zwicker, W. S. "Counting Quota Systems: A Combinatorial Question from Social Choice Theory." Math. Mag. 68, 331 /C1/342, 1995. Quota System A generalization of simple majority voting in which a list of quotas q0 ;...;qn fg specifies, according to the number of votes, how many votes an alternative needs to win (Taylor 1995). The quota system declares a tie unless for some k, there are exactly k tie votes in the profile and one of the alternatives has at least qk votes, in which case the alternative is the choice. Let Q(n) be the number of quota systems for n voters and Q(n; r) the number of quota systems for which q0 /C30r /C271; so Q(n) /C30Xn r/C30 n=2bcQ(n;r) /C30n /C271 n 2jk /C271 ! ; where xbcis the FLOOR FUNCTION . This produces the sequence of CENTRAL BINOMIAL COEFFICIENTS 1, 2, 3, 6, 10, 20, 35, 70, 126, ... (Sloane’s A001405). It may be defined recursively by Q 0ðÞ/C301 and Q(n /C271) /C302Q(n) for n even 2Q(n) /C28Cn/C271 ðÞ =2for n odd;1C2r where Ckis a CATALAN NUMBER (Young et al. 1995). The function Q(n;r) satisfies Q(n;r) /C30n /C271 r /C2711CA81CA9 /C28n /C271 r /C2721CA81CA9 for r > n=2 /C281 (Young et al. 1995). Q(n; r) satisfies the QUOTA RULE . See also BINOMIAL COEFFICIENT ,CENTRAL BINOMIAL COEFFICIENT References Sloane, N. J. A. Sequences A001405/M0769 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Taylor, A. Mathematics and Politics: Strategy, Voting, Power, and Proof. New York: Springer-Verlag, 1995. Young, S. C.; Taylor, A. D.; and Zwicker, W. S. "Counting Quota Systems: A Combinatorial Question from Social Choice Theory." Math. Mag. 68, 331 /C1/342, 1995. Quotient The ratio q /C30r =s of two quantities r and s, where s " 0: Less commonly, the term quotient is also used to mean the INTEGER PART of such a ratio. In Mathema- tica, the command Quotient [r, s] is defined in this latter sense, returning r =s½/C138 ; where xbcis the FLOOR FUNCTION .See also DIVISION ,FRACTION ,INTEGER PART,QUOTI- ENT GROUP ,QUOTIENT RING,QUOTIENT SPACE ,RA- TIONAL NUMBER ,REMAINDER Quotient Group For a GROUP G and a NORMAL SUBGROUP N of G, the quotient group of N in G, written G=N and read "G modulo N", is the set of COSETS of N in G. Quotient groups are also called factor groups. The elements of G =N are written Na and form a GROUP under the normal operation on the group N on the coefficient a. Thus, NaðÞ NbðÞ/C30Nab : Since all elements of G will appear in exactly one COSET of the NORMAL SUBGROUP N, it follows that G=Njj/C30Gjj= Njj where Gjjdenotes the order of a group. The slash NOTATION conflicts with that for an EXTEN- SION FIELD , but the meaning can be determined based on context. See also ABHYANKAR’S CONJECTURE ,COSET ,EXTEN- SION FIELD,OUTER AUTOMORPHISM GROUP ,NORMAL SUBGROUP ,SUBGROUP References Herstein, I. N. Topics in Algebra, 2nd ed. New York: Springer-Verlag, 1975. Quotient Ring A quotient ring (also called a residue-class ring) is a RING which is the quotient of a RING A and one of its IDEALS a; denoted A=a: For example, when the RING A is Z (the integers) and the IDEAL is 6Z (multiples of 6), the quotient ring is Z6 /C30Z=6Z :/ In general, a quotient ring is a set of EQUIVALENCE CLASSES where x½/C138/C30 y½/C138IFF x /C28y /C23 a :/ The quotient ring is an INTEGRAL DOMAIN iff the IDEAL a is PRIME . A stronger condition occurs when the quotient ring is a FIELD , which corresponds to when the ideal a is MAXIMAL . The IDEALS in a quotient ring A=a are in a ONE-TO-ONE correspondence with ideals in A which contain the ideal a: In particular, the zero ideal in A=a corre- sponds to a in A. In the example above from the integers, the ideal of even integers contains the ideal of the multiples of 6. In the quotient ring, the evenscorrespond to the ideal 0 ;2;4 fg inZ 6/C30Z=6Z:/ See also FIELD,IDEAL ,INTEGER ,INTEGRAL DOMAIN , MAXIMAL IDEAL ,M ODULE ,P RIME IDEAL ,R ESIDUE FIELD,RING Quotient Rule The DERIVATIVE rule d dxf(x) g(x)"# /C30g(x)f ?(x) /C28 f(x)g ?(x) g(x) ½/C1382 See also CHAIN RULE,D ERIVATIVE ,P OWER RULE, PRODUCT RULE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Quotient Space The quotient space X =/C2 of a TOPOLOGICAL SPACE X and an EQUIVALENCE RELATION /C2 on X is the set of EQUIVALENCE CLASSES of points in X (under the EQUIVALENCE RELATION /C2) together with the following topology given to subsets of X =/C2: a subset U of X =/C2 is called open IFF @a½/C138/C23U a is open in X. Quotient spaces are also called factor spaces. This can be stated in terms of MAPS as follows: if q : X 0 X =/C2denotes the MAP that sends each point to its EQUIVALENCE CLASS in X =/C2; the topology on X =/C2can be specified by prescribing that a subset of X =/C2 is open IFF q/C281 [the set] is open. In general, quotient spaces are not well behaved, and little is known about them. However, it is known that any compact metrizable space is a quotient of the CANTOR SET, any compact connected n-dimensional MANIFOLD for n /C21 0 is a quotient of any other, and a function out of a quotient space f : X =/C20 Y is continuous IFF the function f(q : X 0 Y is continu- ous. Let Dn be the closed n-D DISK and Sn/C281 its boundary, the (n /C281)/-D sphere. Then Dn =Sn/C281 (which is home- omorphic to Sn); provides an example of a quotient space. Here, Dn =Sn/C281is interpreted as the space obtained when the boundary of the n-DISK is col- lapsed to a point, and is formally the "quotient space by the equivalence relation generated by the relations that all points in Sn/C281 are equivalent." See also EQUIVALENCE RELATION ,QUOTIENT SPACE (LIE GROUP ), TOPOLOGICAL SPACE References Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975. Quotient Space (Lie Group) The set of LEFT COSETS of a SUBGROUP H of a TOPOLOGICAL GROUP G forms a topological space. Its topology is defined by the quotient topology from p : G 0 G=H : Namely, the open sets in G =H are theimages of the open sets in G. Moreover, if H is CLOSED , then G=H is HAUSDORFF . See also EFFECTIVE ACTION ,F REE ACTION ,G EO- METRIC INVARIANT THEORY ,GROUP ,ISOTROPY GROUP , MATRIX GROUP ,O RBIT (GROUP ), QUOTIENT SPACE , REPRESENTATION ,TOPOLOGICAL GROUP ,TRANSITIVE References Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 7 /C1/14 and 41 /C1/49, 1987. Quotient Vector Space Suppose that V /C30 x1 ;x2 ;x3 ðÞfg and W /C30 x1 ; 0;0 ðÞfg : Then the quotient space V =W (read as "V mod W") is isomorphic to x2 ; x3 ðÞfg /C30R2 :/ In general, when W is a SUBSPACE of a VECTOR SPACE V, the quotient space V =W is the set of EQUIVALENCE CLASSES v½/C138where v1 /C2v2if v1 /C28v2 /C23 W : By "/v1is equivalent to v2modulo W," it is meant that v1 /C30 v2 /C27w for some w in W, and is another way to say v1 /C2v2 : In particular, the elements of W represent 0½/C138: Sometimes the equivalence classes v½/C138are written as COSETS v /C27W :/ The quotient space is an ABSTRACT VECTOR SPACE , not necessarily isomorphic to a subspace of V. However, if V has an INNER PRODUCT , then V =W is isomorphic to W /C222/C30 v : v; whi/C300 for all w /C23 W fg : In the example above, W /C222/C30 0; x2x3 ðÞfg : Here is a Mathematica function which finds a basis to W /C222 when given a basis for W. PerpVectorBasis[a_List?MatrixQ] : /C30 NullSpace[a] For example,PerpVectorBasis [{{1, 2, 0, 0, 3}, {4, 0, 5, 0, 6}}] yields {{-6, -3, 0, 0, 4}, {0, 0, 0, 1, 0}, {-10, 5, 8, 0, 0}}. Unfortunately, a different choice of inner product can change W /C222: Also, in the infinite-dimensional case, it is necessary for W to be a CLOSED SUBSPACE to realize the isomorphism between V =W and W /C222; as well as to ensure the quotient space is HAUSDORFF . See also COSET ,ORTHOGONAL SET,QUOTIENT SPACE , VECTOR SPACE Quotient-Difference Algorithm The ALGORITHM of constructing and interpreting a QUOTIENT-DIFFERENCE TABLE which allows intercon- version of CONTINUED FRACTIONS , POWER SERIES , and RATIONAL FUNCTIONS approximations. See also QUOTIENT- DIFFERENCE TABLE References Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 15 /C1/17, 1995. Quotient-Difference Table A quotient-difference table is a triangular ARRAY of numbers constructed by drawing a sequence of n numbers in a horizontal row and placing a 1 above each. An additional "1" is then placed at the begin- ning and end of the row of 1s, and the value of rows underneath the original row is then determined by looking at groups of adjacent numbers N WXE S and computing S /C30X2 /C28 EW N for the elements falling within a triangle formed by the diagonals extended from the first and last "1," as illustrated above. 0s in quotient-difference tables form square "win- dows" which are bordered by GEOMETRIC SEQUENCES . Quotient-difference tables eventually yield a row of 0s IFF the starting sequence is defined by a linear RECURRENCE RELATION . For example, continuing the above example generated by the FIBONACCI NUMBERS 1111111 11235 /C2811 /C281 0 11111111 112358 /C2811 /C2811 00 1111111 11 1123581 3 /C2811 /C2811 /C281 000 01111111 1 11 1123581 32 1 /C2811 /C2811 /C2811 0000 00 and it can be seen that a row of 0s emerges (and furthermore that an attempt to extend the table will result in division by zero). This verifies that the FIBONACCI NUMBERS satisfy a linear recurrence, which is in fact given by the well-known formula Fn /C30Fn/C281 /C27Fn/C282 : However, construction of a quotient-difference table for the CATALAN NUMBERS ,M OTZKIN NUMBERS , etc., does not lead to a row of zeros, suggesting that these numbers cannot be generated using a linear recur- rence. See also DIFFERENCE TABLE ,FINITE DIFFERENCE References Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 85 /C1/89, 1996. Getu, S.; Shapiro, L. W.; Woan, W. J.; and Woodson, L. C. "How to Guess a Generating Function." SIAM J. Disc. Math. 5, 497 /C1/499, 1992. Gragg, W. B. " The Pade´ Table and Its Relation to Certain Algorithms of Numerical Analysis." SIAM Rev. 14,1/C1/16, 1972. Henrici, P. "Quotient-Difference Algorithms." In Mathema- tical Methods for Digital Computers, Vol. 2 (Ed. A. Ral- ston and H. S. Wilf). New York: Wiley, pp. 35 /C1/62, 1967. Jones, W. B. and Thron, W. J. Continued Fractions: Analy- tical Theory and Applications. Reading, MA: Addison- Wesley, 1980. Lidl, R. and Niederreiter, H. §6.6 in Introduction to Finite Fields and Their Applications, rev. ed. Cambridge, Eng- land: Cambridge University Press, 1994. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 15 /C1/17, 1995. q-Vandermonde Sum 2f1a;q/C28n;c;q;q ðÞ /C30anc=a;q ðÞn a;qðÞn; where2f1a;b;c;q;z ðÞ is a Q-HYPERGEOMETRIC SERIES . See also CHU-VANDERMONDE IDENTITY References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 15 /C1/16, 1986. q-Whipple Transformation 8 f7a; qa1 =2 ;/C28qa1 =2 ;b;c ;d ;e ;q/C28N a1=2 ;/C28a1 =2 ;aq b;aq c;aq d;aq e;aqN /C271;q ;aqN /C272 bcde2 435 /C30aq de ;q ! N aq d;q ! Naq e;q ! N4 f3d;e ;aq bc;q /C28N aq b;aq c;deq /C28n =a;q;q266643 7775; where s fg is a Q-HYPERGEOMETRIC SERIES . References Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 35, 1995. Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 35, 1990. q-Zeilberger Algorithm A Q-ANALOG of ZEILBERGER’S ALGORITHM . See also ZEILBERGER’S ALGORITHMReferences Bo¨ing, H. and Koepf, W. "Algorithms for q-Hypergeometric Summation in Computer Algebra." J. Symb. Comput. 11, 1 /C1/23, 1999. Koornwinder, T. H. "On Zeilberger’s Algorithm and Its q- Analogue." J. Comp. Appl. Math. 48,91/C1/111, 1993. Le, H. Q. "On the q-Analogue of Zeilberger’s Algorithm to Rational Functions." ftp://cs-archive.uwaterloo.ca/cs-ar- chive/CS-2000 /C1/03/CS-2000 /C1/03.ps.Z. Riese, A. A Mathematica q-Analog of Zeilberger’s Algorithm for Proving q-Hypergeometric Identities. Diploma thesis. Linz, Austria: University of Linz, 1995. Wilf, H. and Zeilberger, D. "A Algorithmic Proof Theory for Hypergeometric (Ordinary and "q") Multisum/Integral Identities." Invent. Math. 108, 575 /C1/633, 1992. Q /C27 The POSITIVE RATIONAL NUMBERS , denoted Q/C27:/ See also Q, Q-BAR,RATIONAL NUMBER References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 1, 1998. R R The DOUBLESTRUCK letter R denotes the FIELD of REAL NUMBERS . See also C, I, N, Q, R-,R/C27,REAL NUMBER ,Z References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 1, 1998. R /C28 /R /C28 denotes the REAL NEGATIVE numbers. See also R, R/C27,REAL NUMBER R /C27 /R /C27 denotes the REAL POSITIVE numbers. See also R, R-,REAL NUMBER References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 1, 1998. Raabe’s Test Given a SERIES of POSITIVE terms ui and a SEQUENCE of POSITIVE constants aifg; use KUMMER’S TEST r ?/C13 lim n0/C12anun un/C271/C28an/C271 ! with an /C30n; giving r?/C13 lim n0/C12nun un/C271/C28(n /C271)"# /C30 lim n0/C12nun un/C271/C281 ! /C281"# : Defining r /C13 r?/C271 /C30 lim n 0/C12nun un/C271/C281 !"# ; then gives Raabe’s test: 1. If r > 1 ; the SERIES CONVERGES . 2. If r B1 ; the SERIES DIVERGES . 3. If r /C301 ; the SERIES may CONVERGE or DIVERGE . See also CONVERGENT SERIES ,CONVERGENCE TESTS , DIVERGENT SERIES ,KUMMER’S TEST References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 286 /C1/287, 1985.Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 39, 1991. Rabbit Constant The limiting RABBIT SEQUENCE written as a BINARY FRACTION 0:1011010110110...2(Sloane’s A005614), where b2denotes a BINARY number (a number in base-2). The DECIMAL value is R /C300:7098034428612913146... (Sloane’s A014565). Amazingly, the rabbit constant is also given by the CONTINUED FRACTION [0, 2F0 ; 2F1 ; 2F2 ; 2F3 ; ...], where Fnare FIBONACCI NUMBERS with F0taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered by S. Plouffe. Define the BEATTY SEQUENCE aifg by ai /C13 i fbc where xbcis the FLOOR FUNCTION and f is the GOLDEN RATIO . The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (Sloane’s A000201). Then R/C30X/C12 i/C3012/C28ai See also RABBIT SEQUENCE ,THUE CONSTANT ,THUE- MORSE CONSTANT References Anderson, P. G.; Brown, T. C.; and Shiue, P. J.-S. "A Simple Proof of a Remarkable Continued Fraction Identity." Proc. Amer. Math. Soc. 123, 2005/C1/2009, 1995. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html. Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Free- man, pp. 21 /C1/22, 1989. Plouffe, S. "The Rabbit Constant to 330 Digits." http:// www.lacim.uqam.ca/piDATA/rabbit.txt. Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 55, 1991. Sloane, N. J. A. Sequences A000201/M2322, A005614, and A014565 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Rabbit Sequence ASEQUENCE which arises in the hypothetical repro- duction of a population of rabbits. Let the SUBSTITU- TION MAP 001 correspond to young rabbits growing old, and 1 010 correspond to old rabbits producing young rabbits. Starting with 0 and iterating using STRING REWRITING gives the terms 1, 10, 101, 10110, 10110101, 1011010110110, .... Converted to binary, this sequence gives 1, 2, 5, 22, 181, ... (Sloane’s A005203), with the nth term given by the RECUR- RENCE RELATION a(n) /C30a(n /C281)2Fn/C281 /C27a(n /C282); with a(0) /C300; a(1) /C301 ; and Fnthe nth FIBONACCI NUMBER . The limiting sequence written as a BINARY FRACTION 0:1011010110110...2(Sloane’s A005614), where an ...a1a0 ðÞ2 denotes a BINARY NUMBER (i.e., a number written in base 2, so ai /C300 or 1), is called the RABBIT CONSTANT . See also FIBONACCI NUMBER ,R ABBIT CONSTANT , THUE- MORSE SEQUENCE References Davison, J. L. "A Series and Its Associated Continued Fraction." Proc. Amer. Math. Soc. 63,29/C1/32, 1977. Gould, H. W.; Kim, J. B.; and Hoggatt, V. E. Jr. "Sequences Associated with t-ary Coding of Fibonacci’s Rabbits." Fib. Quart. 15, 311 /C1/318, 1977. Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 55, 1991. Sloane, N. J. A. Sequences A005203/M1539 and A005614 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Rabbit-Duck Illusion A perception ILLUSION in which the brain switches between seeing a rabbit and a duck. See also YOUNG GIRL-OLD WOMAN ILLUSION Rabdology NAPIER’S BONES Rabin-Miller Strong Pseudoprime Test A PRIMALITY TEST which provides an efficient prob- abilistic ALGORITHM for determining if a given num- ber is PRIME . It is based on the properties of STRONG PSEUDOPRIMES . Given an ODD INTEGER n, let n /C30 2rs /C271 with s ODD. Then choose a random integer a with 1 5a 5n /C281: If as /C131 (mod n)o r a2js /C13 /C281 (mod n) for some 0 5j 5r /C281 ; then n passes the test. A PRIME will pass the test for all a. The test is very fast and requires no more than (1 /C27 o(1)) lg n multiplications (mod n), where LG is the LOGARITHM base 2. Unfortunately, a number whichpasses the test is not necessarily PRIME . Monier (1980) and Rabin (1980) have shown that a COMPO- SITE NUMBER passes the test for at most 1/4 of the possible bases a. The Rabin-Miller test (combined with a LUCAS PSEU- DOPRIME test) is the PRIMALITY TEST used by Mathe- matica versions 2.2 and later. As of 1991, the combined test had been proven correct for all n B 2:5 /C291010 ; but not beyond. The test potentially could therefore incorrectly identify a large COMPOSITE NUMBER as PRIME (but not vice versa). STRONG PSEUDOPRIME tests have been subsequently proved valid for every number up to 3:4 /C291014 :/ See also LUCAS- LEHMER TEST,M ILLER’S PRIMALITY TEST,PSEUDOPRIME ,STRONG PSEUDOPRIME References Arnault, F. "Rabin-Miller Primality Test: Composite Num- bers Which Pass It." Math. Comput. 64, 355/C1/361, 1995. Damga ˚rd, I.; Landrock, P.; and Pomerance, C. "Average Case Error Estimates for the Strong Probably Prime Test." Math. Comput. 61, 177/C1/194, 1993. Miller, G. "Riemann’s Hypothesis and Tests for Primality." J. Comp. Syst. Sci. 13, 300/C1/317, 1976. Monier, L. "Evaluation and Comparison of Two Efficient Probabilistic Primality Testing Algorithms." Theor. Com- put. Sci. 12,9 7/C1/108, 1980. Rabin, M. O. "Probabilistic Algorithm for Testing Primality." J. Number Th. 12, 128/C1/138, 1980. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 15 /C1/17, 1991. Rabinovich-Fabrikant Equation The 3-D MAP ˙x/C30yz/C281/C27x2})0})@ /C27gx ˙y/C30x3z/C271/C28x2})0})@ /C27gy ˙z/C30/C282z(a/C27xy) (Rabinovich and Fabrikant 1979). The parameters are most commonly taken as g/C300:87 and a/C301:1:It has a CORRELATION EXPONENT of 2.1990.01. References Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189/C1/208, 1983. Rabinovich, M. I. and Fabrikant, A. L. "Stochastic Self- Modulation of Waves in Nonequilibrium Media." Sov. Phys. JETP 50, 311/C1/317, 1979. Racah 6j-Symbol WIGNER 6 J-SYMBOL Racah Polynomial A hypergeometric class of orthogonal polynomialsdefined by R n(l(x);a;b;g;d) /C304F3/C28n; n /C27a/C27b/C271;/C28x; x /C27g/C27d/C271 a/C271;b/C27d/C271 ;g/C271;1})@*})@+ for n /C300, 1, ..., N, where4F3(a ; b; c ; d; e ; f ; g; x)is a GENERALIZED HYPERGEOMETRIC FUNCTION , l(x) /C30x(x /C27 g /C27 d /C271); and one of the following holds a /C271 /C30/C28N b /C27 d /C271 /C30/C28N g /C271 /C30/C28N ;8 < : with N a NONNEGATIVE INTEGER . References Koekoek, R. and Swarttouw, R. F. "Racah." §1.2 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Uni- versiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/17, pp. 26 /C1/29, 1998. ftp:// www.twi.tudelft.nl/publications/tech-reports/1998/DUT- TWI-98 /C1/17.ps.gz. Racah V-Coefficient The Racah V-COEFFICIENTS are written Vj1 j2 ; m1m2m ðÞ (1) and are sometimes expressed using the related CLEBSCH- GORDAN COEFFICIENTS Cj m1m2/C30 j1 j2m1m2j1 j2 jm jÞ ; ð (2) or WIGNER 3J-SYMBOLS . Connections among the three are ðj1 j2m1m2 j1 j2m j Þ/C30(/C281)/C28j1/C27j2/C28m /C2ffiffiffiffiffiffiffiffiffiffiffiffiffi 2j /C271pj1 j2 j m1m2/C28m})@*})@+ (3) (j1 j2m1m2j1 j2 jm jÞ /C30(/C281)j/C27m /C2ffiffiffiffiffiffiffiffiffiffiffiffiffi2j /C271p Vj 1 j2 j; m1m2 /C28m ðÞ (4) Vj1 j2 j; m1m2m ðÞ /C30(/C281)/C28j1/C27j2/C27j j1 j2 j1 m2m1m2})@*})@+ : (5) See also CLEBSCH- GORDAN COEFFICIENT ,RACAH W- COEFFICIENT ,W IGNER 3J-SYMBOL ,W IGNER 6J-SYM- BOL,W IGNER 9J-SYMBOL References Biedenharn, L. C. and Louck, J. D. The Racah-Wigner Algebra in Quantum Theory. Reading, MA: Addison- Wesley, 1981. Sobel’man, I. I. "Angular Momenta." Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer- Verlag, 1992.Racah W-Coefficient Related to the CLEBSCH- GORDAN COEFFICIENTS by (J1J2[J ?]J3 J1 ; J2J3[J ƒ] j Þ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2J ?/C271)(2J ƒ/C271)p W(J1J2JJ3; J ?J ƒ) and (J1J2[J ?]J3 J1 ; J3[J ƒ]J2 j Þ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(2J ?/C271)(2J ƒ/C271)p W(J ? 1J3J2J ƒ; JJ1): See also CLEBSCH- GORDAN COEFFICIENT ,RACAH V- COEFFICIENT ,W IGNER 3J-SYMBOL ,W IGNER 6J-SYM- BOL,W IGNER 9J-SYMBOL References Messiah, A. "Racah Coefficients and ‘ /6j/’ Symbols." Appendix C.II in Quantum Mechanics, Vol. 2. Amsterdam, Nether- lands: North-Holland, pp. 1061 /C1/1066, 1962. Sobel’man, I. I. "Angular Momenta." Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer- Verlag, 1992. Radau Quadrature AG AUSSIAN QUADRATURE -like formula for numerical estimation of integrals. It requires m/C271 points and fits all POLYNOMIALS to degree 2 m;so it effectively fits exactly all POLYNOMIALS of degree 2 m/C281:It uses a WEIGHTING FUNCTION W(x)/C301 in which the endpoint /C281 in the interval [ /C281;1] is included in a total of n ABSCISSAS , giving r/C30n/C281 free abscissas. The general formula is g1 /C281f(x)dx/C30w1f(/C281)/C27Xn i/C302wif(xi): (1) The free abscissas xifori/C302, ..., nare the roots of the POLYNOMIAL Pn/C281(x)/C27Pn(x) 1/C27x; (2) where P(x)i saL EGENDRE POLYNOMIAL . The weights of the free abscissas are wi/C301/C28xi n2Pn/C281(xi) ½/C1382/C301 1/C28xi ðÞ P?n/C281xiðÞ ½/C1382; (3) and of the endpoint w1/C302 n2: (4) The error term is given by E/C3022n/C281n(n/C281)! ½/C1384 [(2n/C281)!]3f(2n/C281)(j); (5) forj/C23(/C281;1):/ n /xi// wi/ 2 /C281 0.5 0.333333 1.5 3 /C281 0.222222 //C280:289898 / 1.02497 0.689898 0.752806 4 /C281 0.125 //C280:575319 / 0.657689 0.181066 0.776387 0.822824 0.440924 5 /C281 0.08 //C280:72148 / 0.446208 //C280:167181 / 0.623653 0.446314 0.562712 0.885792 0.287427 The ABSCISSAS and weights can be computed analy- tically for small n. n /xi// wi/ 2-1 /1 2/ /13//32/ 3-1 /2 9/ /151 /C28ffiffiffi 6p})0})@ //1 1816 /C27ffiffiffi6p})0})@ / /1 51 /C27ffiffiffi 6p})0})@ //1 1816 /C28ffiffiffi6p})0})@ / See also CHEBYSHEV QUADRATURE ,LOBATTO QUAD- RATURE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 888, 1972. Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 61, 1960. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 338 /C1/343, 1956. Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, p. 105, 1997. Rademacher Function SQUARE WAVERadial Curve Let C be a curve and let O be a fixed point. Let P be on C and let Q be the CURVATURE CENTER at P. Let P1 be the point with P1O a line segment PARALLEL and of equal length to PQ. Then the curve traced by P1 is the radial curve of C. It was studied by Robert Tucker in 1864. The PARAMETRIC EQUATIONS of a curve (f(t) ; g(t)) with RADIAL POINT x0 ; y0 ðÞ and parameter- ized by a variable t are given by x /C30x0 /C28g? f ?2 /C27 g ?2})0})@ f ?gƒ/C28 f ƒg ? y /C30y0 /C27f ? f ?2 /C27 g ?2})0})@ f ?g ƒ/C28 f ƒg ?: Here, derivatives are taken with respect to the parameter t. Curve Radial Curve ASTROID QUADRIFOLIUM CATENARY KAMPYLE OF EUDOXUS CYCLOID CIRCLE DELTOID TRIFOLIUM LOGARITHMIC SPIRAL LOGARITHMIC SPIRAL TRACTRIX KAPPA CURVE References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40 and 202, 1972. Yates, R. C. "Radial Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 172 /C1/ 174, 1952. Radial Point The point with respect to which a RADIAL CURVE is computed. See also RADIANT POINT Radian A unit of angular measure in which the ANGLE of an entire CIRCLE is 2p radians. There are therefore 3608 per 2p radians, equal to 180/C14=p or 57. 29577951 8/ radian. A RIGHT ANGLE isp=2 radians. See also ANGLE ,ARC MINUTE ,ARC SECOND ,DEGREE , GRADIAN ,STERADIAN Radiant Point The point of illumination for a CAUSTIC . See also CAUSTIC ,RADIAL POINT Radical The symbolffiffiffixpused to indicate a root is called a radical. The expressionffiffiffixpis therefore read "x radical n," or "the nth ROOT of x." In the radical symbol, the horizonal line is called the VINCULUM , the quantity under the VINCULUM is called the RADICAND , and the quantity n written to the left is called the INDEX . The special caseffiffiffixpis writtenffiffiffixpand is called the SQUARE ROOT of x.ffiffiffix3pis called the CUBE ROOT . Some interesting radical identities are due to Rama- nujan, and include the equivalent forms 21 =3 /C271})0})@ 21 =3 /C281})0})@ 1 =3/C3031 =3 and 21 =3 /C281})0})@ 1 =3/C301 9})@D})@E1 =3 /C2829})@D})@E1 =3 /C2749})@D})@E1=3 : Another such identity is 51=3 /C2841=3})0})@ 1 =2/C301321=3 /C27201 =3 /C28251 =3})0})@ : See also CUBE ROOT,INDEX ,N ESTED RADICAL , POWER ,RADICAL INTEGER ,RADICAND ,ROOT (RADI- CAL), SQUARE ROOT,SURD,VINCULUM Radical (Ideal) The radical of an IDEAL r( a)ina RING R is the ideal which is the intersection of all PRIME IDEALS contain- ing r(a) : Note that any ideal is contained in a MAXIMAL IDEAL , which is always prime. So the radical of an ideal is always at least as big as the original ideal. Naturally, if the ideal r(a) is prime then r( a) /C30 x : xn /C23a for some integer n > 0 fg :/ Another description of the radical C[x]is a/C30 x2})@0})@@ This explains the connection with the RADICAL sym- bol. For example, in r( a) /C30 xhi; consider the ideal C of all polynomials with degree at least 2. Thenffiffiffi 73p /C27 ffiffiffiffiffiffi /C282p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27ffiffiffi 2p4pq is like a square root of r( a): Notice that the zero set (VARIETY )ofr( a) and C[x]is the same (in r( a) /C30 xhibecause is ALGEBRAICALLY CLOSED ). Radicals are an important part of the statement of the NULLSTELLENSATZ . See also ALGEBRAIC GEOMETRY ,IDEAL ,JACOBSON RADICAL ,N ILRADICAL ,N ULLSTELLENSATZ ,P RIME IDEAL ,VARIETY Radical Axis RADICAL LINE Radical Center The RADICAL LINES of three CIRCLES are CONCURRENT in a point known as the radical center (also called the power center). This theorem was originally demon- strated by Monge (Do¨rrie 1965, p. 153). It is a special case of the THREE CONICS THEOREM (Evelyn et al. 1974, pp. 13 and 15). See also APOLLONIUS’ PROBLEM ,CONCURRENT ,M ON- GE’S PROBLEM ,RADICAL LINE,THREE CONICS THEO- REM References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., p. 43, 1888. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 35, 1967. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 125, 1928. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Three-Conics Theorem." §2.2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 11 /C1 /18, 1974. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 32, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 185, 1893. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 35, 1991. Radical Circle ORTHOGONAL CIRCLES Radical Denesting NESTED RADICAL Radical Integer A radical integer is a number obtained by closing the INTEGERS under ADDITION , MULTIPLICATION , SUBTRAC- TION , and ROOT EXTRACTION . An example of such a number isffiffiffi 73p /C27ffiffiffiffiffiffi /C282p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27ffiffiffi 2p4pq : The radical integers are a SUBRING of the ALGEBRAIC INTEGERS . There exist cubic ALGEBRAIC INTEGERS which are not radical integers, namely those which can’t be ex- pressed in terms of radicals. R. Schroeppel proved that these are the only ones; i.e., if an ALGEBRAIC INTEGER can be expressed in terms of radicals, then it can be done so without using division. See also ALGEBRAIC INTEGER ,ALGEBRAIC NUMBER , EUCLIDEAN NUMBER References Schroeppel, R. "radical & algebraic integers." math-fun@c- s.arizona.edu posting, May 11, 1997. Radical Line The LOCUS of points of equal POWER with respect to two nonconcentric CIRCLES which is PERPENDICULAR to the line of centers (the CHORDAL THEOREM ;Do¨rrie 1965). Let the circles have RADII r1and r2and their centers be separated by a distance d. If the CIRCLES intersect in two points, then the radical line is the line passing through the points of intersection. If not, then draw any two CIRCLES which cut each original CIRCLE twice. Draw lines through each pair of points ofintersection of each CIRCLE . The line connecting their two points of intersection is then the radical line. The radical line is located at distances d1 /C30d2 /C27 r2 1 /C28 r22 2d (1) d2 /C30/C28d2 /C27 r2 2 /C28 r21 2d (2) along the line of centers from C1 and C2 ; respectively, where d /C13d1 /C28d2 : (3) The radical line of any two POLAR CIRCLES is the ALTITUDE from the third vertex. See also CHORDAL THEOREM ,COAXAL CIRCLES ,IN- VERSE POINTS ,INVERSION ,POWER (CIRCLE ), RADICAL CENTER References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 43, 1888. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 86, 1969. Coxeter, H. S. M. and Greitzer, S. L. "The Radical Axis of Two Circles." §2.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 31 /C1/34, 1967. Dixon, R. Mathographics. New York: Dover, p. 68, 1991. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 153, 1965. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 121, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 28 /C1/34 and 176 /C1/177, 1929. Lachlan, R. "The Radical Axis of Two Circles." §304 /C1/312 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 185 /C1/189, 1893. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 35, 1991. Radicand The quantity under a RADICAL sign. See also CUBE ROOT,RADICAL ,ROOT,SQUARE ROOT, VINCULUM Radius The distance from the center of a CIRCLE to its PERIMETER , or from the center of a SPHERE to its surface. The radius is equal to half the DIAMETER . See also BERTRAND’S PROBLEM ,CIRCLE ,CIRCUMFER- ENCE ,DIAMETER ,EXTENT ,GRAPH RADIUS ,INVERSION RADIUS ,KINNEY’S SET,PI,RADIUS OF CONVERGENCE , RADIUS OF CURVATURE ,RADIUS OF GYRATION ,RADIUS OF TORSION ,RADIUS VECTOR ,SPHERE Radius of Convergence A POWER SERIES S/C12ckxk will converge only for certain values of x. For instance, S/C12 k¼0xk converges for /C281 B x B1: In general, there is always an interval ð/C28R; RÞ in which a POWER SERIES converges, and the number R is called the radius of convergence. The quantity R is called the radius of convergence because, in the case of a power series with complex coefficients, the values of x with jxjBR form an OPEN DISK with radius R. A POWER SERIES always CONVERGES ABSOLUTELY within its radius of convergence. This can be seen by fixing r ¼jxj and supposing that there exists a SUBSEQUENCE cnisuch that jcnijrniis UNBOUNDED . Then the POWER SERIES Scnxn does not CONVERGE (in fact, the terms are unbounded) because it fails the LIMIT TEST . Therefore, for x with r /C30jxj/C21R; the power series does not converge, where c ¼ lim sup cn (1) R ¼1 c; ð2Þ and lim sup denotes the SUPREMUM LIMIT . Conversely, suppose that r BR. Then for any radius s with r Bs BR; the terms cnxn satisfy jcnxn jBs R !n (3) for n large enough (depending on s). It is sufficient to fix a value for s in between r and R. Because s=R B1; the power series is dominated by a convergent GEOMETRIC SERIES . Hence, the POWER SERIES con- verges absolutely by the LIMIT COMPARISON TEST . See also CONVERGENT SERIES ,POWER SERIES ,ROOT TEST References Levinson, N. and Raymond, R. Complex Variables. New York: McGraw-Hill, pp. 349 /C1/352, 1970. Rudin, W. Principles of Mathematical Analysis. New York: McGraw-Hill, p. 69, 1976.Radius of Curvature The radius of curvature is given by R /C131 k ; (1) where k is the CURVATURE . At a given point on a curve, R is the radius of the OSCULATING CIRCLE . The symbol r is sometimes used instead of R to denote the radius of curvature. Let x and y be given parametrically by x /C30x(t) (2) y /C30y(t); (3) then R /C30x?2 /C27 y?2})0})@ 3=2 x?yƒ/C28 y?xƒ; (4) where x?/C30dx =dt and y ?/C30dy=dt : Similarly, if the curve is written in the form y /C30f(x) ; then the radius of curvature is given by R /C301 /C27dy dx !22 4353 =2 d2y dx2: (5) In POLAR COORDINATES r /C30r( u) ; the radius of curva- ture is given by R /C30(r2 /C27 r2 u)3 =2 r2/C272r2 u/C28rruu; (6) where ru/C30dr=du(Gray 1997, p. 89). See also BEND (CURVATURE ), CURVATURE ,OSCULAT- ING CIRCLE ,RADIUS OF GYRATION ,RADIUS OF TOR- SION,TORSION (DIFFERENTIAL GEOMETRY ) References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997. Kreyszig, E. Differential Geometry. New York: Dover, p. 34, 1991. Radius of Gyration A positive number ksuch that a lamina or solid body with moment of inertia about an axis Iand mass mis given by I/C30mk2: Pickover (1995) defines a generalization of kas a function Rgquantifying the spatial extent of the structure of a curve and given by Rg /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g/C12 0r2p(r) drs 2g/C12 0p(r) dr; where p(r) is the LENGTH DISTRIBUTION FUNCTION . Small compact patterns have small Rg :/ See also RADIUS OF CURVATURE ,RADIUS OF TORSION References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 204 /C1/ 206, 1995. Radius of Torsion s /C131 t; where t is the TORSION . The symbol f is also some- times used instead of s:/ See also RADIUS OF CURVATURE ,TORSION (DIFFER- ENTIAL GEOMETRY ) References Kreyszig, E. Differential Geometry. New York: Dover, p. 39, 1991. Radius Vector The VECTOR r from the ORIGIN to the current position. It is also called the position vector. The derivative of r satisfies r /C215dr dt /C301 2d dt(r /C215 r) /C3012d dtr2})0})@ /C30rdr dt /C30rv; where v is the magnitude of the VELOCITY (i.e., the SPEED ). See also RADIUS ,SPEED ,VELOCITY Radix The BASE of a number system, i.e., 2 for BINARY , 8 for OCTAL , 10 for DECIMAL , and 16 for HEXADECIMAL . The radix is sometimes called the BASE or SCALE . See also BASE (NUMBER ) Radon Measure See also PROBABILITY MEASURE Radon Transform An INTEGRAL TRANSFORM whose inverse is used to reconstruct images from medical CT scans. A techni- que for using Radon transforms to reconstruct a map of a planet’s polar regions using a spacecraft in apolar orbit has also been devised (Roulston and Muhleman 1997).The Radon transform can be defined by R(p;t)[f(x;y)]/C30g/C12 /C28/C12f(x;t/C27px)dx /C30g/C12 /C28/C12g/C12 /C28/C12f(x;y)d[y/C28(t/C27px)]dy dx/C13U(p;t);(1) where pis the SLOPE of a line and tis its intercept. The inverse Radon transform is f(x;y)/C301 2pg/C12 /C28/C12d dyH[U(p;y/C28px)]dp; (2) where His a H ILBERT TRANSFORM . The transform can also be defined by R?(r;a)[f(x;y)] /C30g/C12 /C28/C12g/C12 /C28/C12f(x;y)d(r/C28xcosa/C28ysina)dx dy ;(3) where ris the PERPENDICULAR distance from a line to the origin and ais the ANGLE formed by the distance VECTOR . Using the identity F[R[f(v;a)]]/C30F2[f(u;v)]; (4) where Fis the F OURIER TRANSFORM , gives the inversion formula f(x;y)/C30cgp 0g/C12 /C28/C12F[R[f(v;a)]] /C2vjjeiv(xcosa/C27ysina)dvda: (5) The F OURIER TRANSFORM can be eliminated by writ- ing f(x;y)/C30gp 0g/C12 /C28/C12R[f(r;a)]W(r;a;x;y)dr da;(6) where Wis a WEIGHTING FUNCTION such as W(r;a;x;y)/C30h(xcosa/C27ysina/C28r)/C30F/C281vjj½/C138:(7) Nievergelt (1986) uses the inverse formula f(x;y)/C301 plim c00gp 0g/C12 /C28/C12R[f(r/C27xcosa /C27ysina;a)]Gc(r)dr da; (8) where Gc(r)/C301 pc2forrjj5c 1 pc21/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28c2=r2p ! forrjj>c:8 >>>>< >>>>:(9) L UDWIG’S INVERSION FORMULA expresses a function in terms of its Radon transform. R?(r;a) and R(p;t) are related by p /C30cot at/C30r csc a (10) r /C30t 1 /C27 p2a /C30cot /C281 p : (11) The Radon transform satisfies superposition R(p ; t) f1(x ; y) /C27f2(x; y) ½/C138 /C30U1(p; t) /C27U2(p ; t); (12) linearity R(p; t)[af(x; y)] /C30aU(p ; t) ; (13) scaling R(p; t) fx a ;y b !"# /C30 ajjUpa b ;t b ! ; (14) ROTATION , with Rf ROTATION by ANGLE f R(p; t) Rff(x; y)})1})A /C301 cos f /C27 p sin f jjU /C2p /C28 tan f 1 /C27 p tan f ;t cos f /C27 p sin f ! ; (15) and skewing R(p ; t)[f(ax /C27by; cx /C27dy)] /C301 a /C27 bp jjUc /C27 dp a /C27 bp ; td /C28 b(c /C27 bd) a /C27 bp"# (16) (Durrani and Bisset 1984). The line integral along p ; t is I /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27p2p U(p ; t) : (17) The analog of the 1-D CONVOLUTION THEOREM is R(p ; t)[f(x; y) + g(y)] /C30U(p; t) + g( t); (18) the analog of PLANCHEREL’S THEOREM is g/C12 /C28/C12U(p; t) dt /C30g/C12 /C28/C12g/C12 /C28/C12f(x; y) dx dy ; (19) and the analog of PARSEVAL’S THEOREM is g/C12 /C28/C12R(p ; t)[f(x; y)]2 dt /C30g/C12 /C28/C12g/C12 /C28/C12f2(x; y) dx dy : (20) If f is a continuous function on C ; integrable with respect to a plane LEBESGUE MEASURE , and glfds/C300 (21) for every (doubly) infinite line l where s is the length measure, then fmust be identically zero. However, if the global integrability condition is removed, this result fails (Zalcman 1982, Goldstein 1993). See also HAMMER’S X-RAY PROBLEMS ,TOMOGRAPHYReferences Anger, B. and Portenier, C. Radon Integrals. Boston, MA: Birkha ¨user, 1992. Armitage, D. H. and Goldstein, M. "Nonuniqueness for the Radon Transform." Proc. Amer. Math. Soc. 117, 175/C1/178, 1993. Deans, S. R. The Radon Transform and Some of Its Applications. New York: Wiley, 1983. Durrani, T. S. and Bisset, D. "The Radon Transform and its Properties." Geophys. 49, 1180 /C1/1187, 1984. Esser, P. D. (Ed.). Emission Computed Tomography: Cur- rent Trends. New York: Society of Nuclear Medicine, 1983. Gindikin, S. (Ed.). Applied Problems of Radon Transform. Providence, RI: Amer. Math. Soc., 1994. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Helgason, S. The Radon Transform. Boston, MA: Birkha ¨u- ser, 1980. Hungerbu ¨hler, N. "Singular Filters for the Radon Back- projection." J. Appl. Analysis 5,1 7/C1/33, 1998. Kak, A. C. and Slaney, M. Principles of Computerized Tomographic Imaging. IEEE Press, 1988. Kunyansky, L. A. "Generalized and Attenuated Radon Transforms: Restorative Approach to the Numerical In- version." Inverse Problems 8, 809/C1/819, 1992. Nievergelt, Y. "Elementary Inversion of Radon’s Transform." SIAM Rev. 28,7 9/C1/84, 1986. Rann, A. G. and Katsevich, A. I. The Radon Transform and Local Tomography. Boca Raton, FL: CRC Press, 1996. Robinson, E. A. "Spectral Approach to Geophysical Inversion Problems by Lorentz, Fourier, and Radon Transforms."Proc. Inst. Electr. Electron. Eng. 70, 1039 /C1 /1053, 1982. Roulston, M. S. and Muhleman, D. O. "Synthesizing Radar Maps of Polar Regions with a Doppler-Only Method." Appl. Opt. 36, 3912/C1/3919, 1997. Shepp, L. A. and Kruskal, J. B. "Computerized Tomography: The New Medical X-Ray Technology." Amer. Math. Monthly 85, 420/C1/439, 1978. Strichartz, R. S. "Radon Inversion--Variation on a Theme." Amer. Math. Monthly 89, 377/C1/384 and 420 /C1/423, 1982. Weisstein, E. W. "Books about Radon Transforms." http:// www.treasure-troves.com/books/RadonTransforms.html. Zalcman, L. "Uniqueness and Nonuniqueness for the Radon Transform." Bull. London Math. Soc. 14, 241/C1/245, 1982. Radon Transform * /Cylinder Let the 2-D cylinder function be defined by f(x;y)/C131 for rBR 0 for r>R:})1D (1) Then the Radon transform is given by R(p;t)/C30g/C12 /C28/C12g/C12 /C28/C12f(x;y)d[y/C28(t/C27px)]dy dx ;(2) where d(x)/C301 2pg/C12 /C28/C12e/C28ikx(3) is the DELTA FUNCTION . R(p;t)/C301 2pg2p 0gR 0g/C12 /C28/C12e/C28ik(rsinu/C28prcosu)rd rd udk /C301 2pg/C12 /C28/C12eikrg2p 0gR 0e/C28ikr(sinu/C28pcosu)rd rd udk: (4) Now write sinu/C28pcosu/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27p2p cos(u/C27f)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p 2p cosu?; (5) with fa phase shift. Then R(p;t)/C301 2pg/C12 /C28/C12eikt /C2gR 0g2p 0e/C28ikffiffiffiffiffiffiffiffiffi 1/C27p2p rcosu?du? ! rd rd k /C301 2pg/C12 /C28/C12eiktgR 02pJ0kffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p 2p r})@D})@E rd rd k /C30g/C12 /C28/C12eiktgR 0J0kffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p 2p r})@D})@E rd rd k : (6) Then use gz 0tn/C271Jn(t)dt/C30zn/C271Jn/C271(z); (7) which, with n/C300, becomes gz 0tJ0(t)dt/C30zJ1(z): (8) Define t/C13kffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p 2p r (9) dt/C30kffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p 2p dr (10) rd r/C30td t k21/C27p2 ðÞ; (11)so the inner integral is gRffiffiffiffiffiffiffiffiffi 1/C27p2p 0J0(t)td t k2(1/C27p2) /C301 k2(1/C27p2)kRffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p 2p J1kRffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p 2p})@D})@E (12) /C30J1kRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27p2p})@D})@E kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27p2p R; (13) and the Radon transform becomes R(p;t)/C30Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27p2p g/C12 /C28/C12eiktJ1kRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p2p})@D})@E kdk /C302Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27p2p g/C12 0cos(kr)J1kRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27p2p})@D})@E kdk /C302 1/C27p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R21/C27p2 ðÞ /C28t2p fort2BR21/C27p2ðÞ 0 fort2]R2(1/C27p2):8 >>>>< >>>>:(14) Converting to R?using p/C30cota; R?(r;a)/C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27cot2apffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27cot2a})0})@ R2/C28r2csc2aq /C302 cscaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi csc2aR2/C28r2csc2ap /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2/C28r2p ; (15) which could have been derived more simply by R?(r;a)/C30gffiffiffiffiffiffiffiffiffiffi R2/C28r2p /C28ffiffiffiffiffiffiffiffiffiffi R2/C28r2pdy: (16) Radon Transform * /Delta Function For a DELTA FUNCTION atx0;y0 ðÞ ; R(p;t)/C30g/C12 /C28/C12g/C12 /C28/C12dx/C28x0 ðÞ dy/C28y0 ðÞ /C2d[y/C28(t/C27px)]dy dx /C301 2pg/C12 /C28/C12g/C12 /C28/C12g/C12 /C28/C12e/C28ik[y/C28(t/C27px)]d(x/C28x0) /C2d(y/C28y0)dk dy dx /C301 2pg/C12 /C28/C12eiktg/C12 /C28/C12e/C28ikydy/C28y0 ðÞ dy})10 /C2g/C12 /C28/C12eikpxdx/C28x0 ðÞ dx/C138dk /C301 2pg/C12 /C28/C12eikte/C28iky0eikpx0dk: /C301 2pg/C12 /C28/C12eikt/C27px0/C28y0 ðÞdk/C30dt/C27px0/C28y0 ðÞ : Radon Transform * /Gaussian R(p;t)/C30g/C12 /C28/C12g/C12 /C28/C121 sffiffiffiffiffiffi 2pp e/C28x2/C27y2ðÞ =2s2"# /C2d[y/C28(t/C27px)]dy dx /C301 sffiffiffiffiffiffi2ppg/C12 /C28/C12e/C28x2/C27(t/C27px)2½/C138 =2s2½dx /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27p2p e/C28t2=21/C27p2ðÞ s2½/C138: Radon Transform * /Square R(p;t)/C30g/C12 /C28/C12g/C12 /C28/C12f(x;y)d[y/C28(t/C27px)]dy dx ;(1) where f(x;y)/C131 for x;y/C23[/C28a;a] 0 otherwise})1D (2) and d(x)/C301 2pg/C12 /C28/C12e/C28ikx(3)is the DELTA FUNCTION . R(p;r)/C301 2pga /C28aga /C28ag/C12 /C28/C12e/C28ik[y/C28(r/C27px)]dk dy dx /C301 2pg/C12 /C28/C12eikrga /C28ae/C28kydyga /C28aeikpxdx})10})1@ dk /C301 2peikr1 /C28ike/C28iky})1})A a /C28a1 ikpe/C28ikpx})1})A a/C28adk /C301 2pg/C12 /C28/C12eikr1 k2p[/C282isin (ka)][2isin(kpa)]dk /C302 ppg/C12 /C28/C12sin(ka) sin( kpa)eikr k2dk /C304 ppg/C12 /C28/C12sin(ka) sin( kpa) cos( kt) k2dk /C302 ppg/C12 /C28/C12sin[k(t/C27a)]/C28sin[k(t/C28a)] k2sin(kpa)dk /C302 pp})1Dg/C12 0sin[k(t/C27a)] sin( kpa) k2dk /C28g/C12 0sin[k(t/C28a)] sin( kpa) k2dk})1E : (4) From Gradshteyn and Ryzhik (2000, equation 3.741.3), g/C12 0sin(ax) sin( bx) x2dx/C301 2psgn(ab) min ajj;bjj ðÞ ;(5) so R(p;t)/C301 psgn[( t/C27a)pa] min t/C27ajj ;pajj ðÞ f /C28sgn[( t/C28a)pa] min t/C28ajj ;pajj ðÞ g : (6) References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Radon-Nikodym Derivative When a MEASURE lisABSOLUTELY CONTINUOUS with respect to a positive measure m;then it can be written as l(E)/C30gEfdm: By analogy with the first FUNDAMENTAL THEOREM OF CALCULUS , the function fis called the Radon-Niko- dym derivative of lwith respect to m:Sometimes it is denoted dl=dmorDl=Dm:/ See also ABSOLUTELY CONTINUOUS ,COMPLEX MEA- SURE ,FUNDAMENTAL THEOREMS OF CALCULUS ,LE- BESGUE MEASURE ,P OLAR REPRESENTATION (MEASURE ), RADON- NIKODYM THEOREM References Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, p. 122, 1987. Radon-Nikodym Theorem The Radon-Nikodym theorem asserts that any ABSO- LUTELY CONTINUOUS measure l with respect to some positive measure m (which could be LEBESGUE MEA- SURE or HAAR MEASURE ) is given by the integral of some L1/-function f, l(E) /C30gEfdm: (1) The function f is like a density function for the measure. A closely related theorem says that any COMPLEX MEASURE l decomposes into an ABSOLUTELY CONTIN- UOUS measure laand a singular measure lc : This is the LEBESGUE DECOMPOSITION l /C30 la /C27 lc : (2) One consequence of the Radon-Nikodym theorem is that any complex measure has a POLAR REPRESENTA- TION , dm /C30hd mjj; (3) with hjj/C301:/ See also ABSOLUTELY CONTINUOUS ,COMPLEX MEA- SURE ,H AAR MEASURE ,L EBESGUE DECOMPOSITION (MEASURE ), LEBESGUE MEASURE ,POLAR REPRESEN- TATION (MEASURE ), SINGULAR MEASURE References Doob, J. L. "The Development of Rigor in Mathematical Probability (1900 /C1/1950)." Amer. Math. Monthly 103, 586 /C1/595, 1996. Rudin, W. Real and Complex Analysis. New York:McGraw- Hill, pp. 121 /C1/129, 1987. Radon’s Theorem Any set of n/C272 points in Rncan always be partitioned in two subsets V1andV2such that the CONVEX HULLS ofV1andV2intersect. See also CONVEX HULL References Eckhoff, J. "Helly, Radon, and Carathe ´odory Type Theo- rems." Ch. 2.1 in Handbook of Convex Geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 389 /C1/448, 1993. McMullen, P. and Shepard, G. C. Convex Polytopes and the Upper Bound Conjecture. London: Cambridge University Press, pp. 22 /C1/24, 1971.Peterson, B. B. "The Geometry of Radon’s Theorem." Amer. Math. Monthly 79, 949/C1/963, 1972. Peyerimhoff, N. "Areas and Intersections in Convex Do- mains." Amer. Math. Monthly 104, 697/C1/704, 1997. Rado, R. "Theorems on the Intersection of Convex Sets of Points." J. London Math. Soc. 27, 320/C1/328, 1952. Ziegler, G. M. Ex. 6.0 in Lectures on Polytopes. New York: Springer-Verlag, 1994. Rado’s Sigma Function BUSYBEAVER Railroad Track Problem Given a straight segment of track of length l, add a small segment Dlso that the track bows into a circular ARC. Find the maximum displacement dof the bowed track. The P YTHAGOREAN THEOREM gives R2/C30x2/C27(1 2l)2: (1) ButRis simply x/C27d;so R2/C30(x/C27d)2/C30x2/C30x2/C272xd/C27d2: (2) Solving (1) and (2) for xgives x/C301 4l2/C28d2 2d: (3) Expressing the length of the ARC in terms of the central angle, 1 2(l/C27Dl)/C30u(d/C27x)/C30ud/C2714l2/C28d2 2d ! /C30u2d2/C2714l2/C28d2 2d ! /C30ud2/C271 4l2 2d ! : (4) Butuis given by tanu/C3012l x/C3012l(2d) 1 4l2/C28d2/C30dl 14l2/C28d2; (5) so plugging uin gives 1 2(l/C27Dl)/C30d2/C2714l2 2d ! tan/C281 dl 1 4l2/C28d2 ! (6) d(l/C27Dl)/C30d2/C271 4l2})@D})@E tan/C281 dl 1 4l2/C28d2 ! : (7) Forl/C27d; dl 14l21/C28d2 4l2 ! /C304d l1/C284d2 l2 !/C281 :4d l1/C274d l2 ! :(8) Therefore, d(l/C27Dl):(d2/C271 4l2) /C24d l1/C274d2 l2 ! /C281 34d l1/C274d2 l2 !"#38 < :9 = ; :d2/C271 4l2})@D})@E})104d l/C2716d3 l3/C281 34d l !3 /C2})@* 1/C2734d2 l2})@+})1@ : (9) Keeping only terms to order ( d=l)3; dl/C27Dl:4d3 l/C27dl/C274d3 l/C2816 3d3 l(10) Dl:8/C2816 3})@D})@Ed3 l/C3024/C2816 3d3 l/C3083d3 l; (11) so d2/C303 8lDl (12) and d:1 2ffiffiffiffiffiffiffiffiffiffi 32lDlq /C3014ffiffiffiffiffiffiffiffiffiffi 6lDlp : (13) If we take l/C301 mile /C305280 feet and Dl/C301 foot, then d:44:50 feet. References Abbott, P. "In and Out: Acton’s Railroad Problem." Mathe- matica J. 7, 448/C1/450, 2000. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., 1990. Ramanujan 6 /C1/10/C1/8 Identity Letad/C30bc, then 64[(a/C27b/C27c)6/C27(b/C27c/C27d)6/C28(c/C27d/C27a)6 /C28(d/C27a/C27b)6/C27(a/C28b)6/C28(b/C28c)6] /C29[(a/C27b/C27c)10/C27(b/C27c/C27d)10/C28(c/C27d/C27a)10/C28(d/C27a/C27b)10/C27(a/C28d)10/C28(b/C28c)10] /C3045[(a/C27b/C27c)8/C27(b/C27c/C27d)8/C28(c/C27d/C27a)8 /C28(d/C27a/C27b)8/C27(a/C28d)8/C28(b/C28c)8]2: (1) This can also be expressed by defining F2m(a;b;c;d)/C30(a/C27b/C27c)2m/C27(b/C27c/C27d)2m /C28(c/C27d/C27a)2m/C28(d/C27a/C27b)2m/C27(a/C28d)2m/C28(b/C28c)2m (2) f2m(x;y)/C30(1/C27x/C27y)2m/C27(x/C27y/C27xy)2m/C28(y/C27xy/C271)2m /C28(xy/C271/C27x)2m/C27(1/C28xy)2m/C28(x/C28y)2m: (3) Then F2m(a;b;c;d)/C30a2mf2m(x;y); (4) and identity (1) can then be written 64f6(x;y)f10(x;y)/C3045f2 8(x;y): (5) Incidentally, f2(x;y)/C300 (6) f4(x;y)/C300: (7) References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 3 and 102 /C1/106, 1994. Berndt, B. C. and Bhargava, S. "A Remarkable Identity Found in Ramanujan’s Third Notebook." Glasgow Math. J.34, 341/C1/345, 1992. Berndt, B. C. and Bhargava, S. "Ramanujan--For Low- brows." Amer. Math. Monthly 100, 644/C1/656, 1993. Bhargava, S. "On a Family of Ramanujan’s Formulas for Sums of Fourth Powers." Ganita 43,6 3/C1/67, 1992. Hirschhorn, M. D. "Two or Three Identities of Ramanujan." Amer. Math. Monthly 105,5 2/C1/55, 1998. Nanjundiah, T. S. "A Note on an Identity of Ramanujan." Amer. Math. Monthly 100, 485/C1/487, 1993. Ramanujan, S. Notebooks. New York: Springer-Verlag, pp. 385 /C1/386, 1987. Ramanujan Constant The IRRATIONAL constant R/C13epffiffiffiffiffiffi 163p /C30262537412640768743 :999999999999925 . . . which is very close to an INTEGER . Numbers such as the Ramanujan constant can be found using the theory of MODULAR FUNCTIONS . In fact, the nine HEEGNER NUMBERS (which include 163) share a deep number theoretic property related to some amazing properties of the J-FUNCTION that leads to this sort of near-identity. Although Ramanujan (1913 /C1/14) gave few rather spectacular examples of almost integers (such epffiffiffiffi 58p ); he did not actually mention particular near-identity give above. In fact, the first to observe this property of 163 was Hermite (1859). The name "Ramanujan’s constant" seems to derive from an April Fool’s joke played by Martin Gardner (Apr. 1975) on the readers of Scientific American . In his column, Gardner claimed that e pffiffiffiffiffiffi 163p was exactly an INTEGER , and that Ramanujan had conjectured this in his 1914 paper. Gardner admitted his hoax a few months later (Gardner, July 1975). See also ALMOST INTEGER ,CLASS NUMBER ,HEEGNER NUMBER , J-FUNCTION References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 387, 1987. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67 /C1/98, 1988. Gardner, M. "Mathematical Games: Six Sensational Dis- coveries that Somehow or Another have Escaped Public Attention." Sci. Amer. 232, 127 /C1/131, Apr. 1975. Gardner, M. "Mathematical Games: On Tessellating the Plane with Convex Polygons." Sci. Amer. 232, 112 /C1/117, Jul. 1975. Good, I. J. "What is the Most Amazing Approximate Integer in the Universe?" Pi Mu Epsilon J. 5, 314 /C1/315, 1972. Hermite, C. "Sur la the´orie des e´quations modulaires." C. R. Acad. Sci. (Paris) 49,16/C1/24, 110 /C1/118, and 141 /C1/144, 1859 Oeuvres comple `tes, Tome II. Paris: Hermann, p. 61, 1912. Plouffe, S. " e pffiffiffiffiffiffi 163p ; the Ramanujan Number." http://www.la- cim.uqam.ca/piDATA/ramanujan.txt. Ramanujan, S. "Modular Equations and Approximations to p:/" Quart. J. Pure Appl. Math. 45, 350 /C1/372, 1913 /C1/1914. Wolfram, S. The Mathematica Book, 3rd ed. New York: Cambridge University Press, p. 52, 1996. Ramanujan Continued Fraction ROGERS- RAMANUJAN CONTINUED FRACTION Ramanujan Cos/Cosh Identity The amazing identity 1 /C272X/C12 n/C301cos(n u) cosh( np)"# /C282 /C27 1 /C272X/C12 n/C301cosh( nu) cosh( np)"# /C282 /C302G43 4})@D})@E p for all u ; where G(z) is the GAMMA FUNCTION . Equat- ing coefficients of u0 ; u4 ; and u8 gives some amazing identities for the HYPERBOLIC SECANT . See also HYPERBOLIC SECANT Ramanujan Function The two-argument Ramanujan function is defined by f(a ; n) /C131 /C272Xn k /C3011 (ak)3 /C28 ak (1)/C301 /C281 aH/C281=a /C27H1 =a /C272Hn /C28Hn/C281 =a /C28Hn/C271 =a})@D})@E : (2) The one-argument function f(a) is then defined as the limiting sum of f(a ; n)asn 0/C12; f(a) /C13 lim n0/C12f(a ; n) /C301 /C272X/C12 k /C3011 (ak)3 /C28 ak(3) /C30/C281 ac01 a ! /C27 c01 /C281 a ! /C272 g"# ; (4) /C301 /C281 aH/C281 =a /C27H1 =a})@D})@E (5) where c0(x) is the DIGAMMA FUNCTION , g is the EULER- MASCHERONI CONSTANT , and Hnis a HARMONIC NUMBER . The values of f(n) for n /C302, 3, ... are f(2) /C302ln2 f(3) /C30ln 3 f(4) /C3032 ln 2 f(5) /C301 5ffiffiffi 5p ln f /C271 2 ln 5 f(6) /C301 2ln 3/C2723ln 2; where fis the GOLDEN RATIO . See also HARMONIC NUMBER ,RAMANUJAN G- AND G- FUNCTIONS ,TAU FUNCTION Ramanujan g- and G-Functions Following Ramanujan (1913 /C1/14), write Y/C12 k/C301;3;5;...1/C27e/C28kpffiffinp})@D})@E /C3021=4e/C28pffiffinp=24Gn (1) Y/C12 k/C301;3;5;...1/C27e/C28kpffiffinp})@D})@E /C3021=4e/C28pffiffinp=24gn: (2) These satisfy the equalities g4n/C3021=4gnGn (3) Gn/C30G1=n (4) g/C281 n/C30g4=n (5) 1 4/C30gnGn ðÞ8G8 n/C28g8n})0})@ : (6) /Gnandgncan be derived using the theory of MODULAR FUNCTIONS and can always be expressed as roots of algebraic equations when nisRATIONAL . For simpli- city, Ramanujan tabulated gnfornEVEN andGnforn ODD. However, (6) allows Gnandgnto be solved for in terms of gnandGn;giving gn /C301 2G8 n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G16 n/C28G /C288 nq})@D})@E 1 =8 (7) Gn /C301 2g8 n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g16 n/C27Gg/C288 nq})@D})@E 1=8 : (8) Using (3) and the above two equations allows g4n to be computed in terms of gn or Gn g4n /C3021=8gng8 n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g16 n/C27g /C288 np})0})@ 1=8for n even 21=8GnG8 n /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G16 n/C27G/C288 np})0})@ 1 =8for n odd:( (9) In terms of the PARAMETER k and complementary PARAMETER k ?; Gn /C30 2knk ?n ðÞ/C281 =12(10) gn /C30k ?n2 2k !1=12 : (11) Here, kn /C30 l /C31(n) (12) is the ELLIPTIC LAMBDA FUNCTION , which gives the value of k for which K ?(k) K(k)/C30ffiffiffinp: (13) Solving for l /C31(n) gives l /C31(n) /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27G/C2812 nq /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28G /C2812 nq hi (14) l /C31(n) /C30g6 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g12 n/C27g /C2812 nq /C28g6 nhi : (15) Analytic values for small values of n can be found in Ramanujan (1913 /C1/1914) and Borwein and Borwein (1987), and have been compiled by Weisstein. Rama- nujan (1913 /C1/1914) contains a typographical error labeling G465 as G265 :/ See also BARNES’ G-FUNCTION References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987. Ramanujan, S. "Modular Equations and Approximations to p:/" Quart. J. Pure. Appl. Math. 45, 350 /C1/372, 1913 /C1/1914. Weisstein, E. W. "Elliptic Singular Values." MATHEMATICA NOTEBOOK ELLIPTIC SINGULAR.M . Ramanujan Psi Sum A sum which includes both the JACOBI TRIPLE PRODUCT and the Q-BINOMIAL THEOREM as special cases. Ramanujan’s sum is X/C12 n /C30/C28/C12(a)n (b)nxn /C30(ax)/C12(q=ax) /C12(q) /C12(b =a) /C12 (x)/C12(b=ax) /C12(b) /C12(q=a) /C12;where the NOTATION (q)k denotes Q-SERIES . For b /C30q, this becomes the Q-BINOMIAL THEOREM . See also JACOBI TRIPLE PRODUCT , Q-BINOMIAL THE- OREM , Q-SERIES Ramanujan Theta Functions Ramanujan’s one-variable theta function is defined by 8(q)/C13X/C12 m/C30/C28/C12qm2; (1) /C30q3(0;q) (2) where q3(0;q)i saJ ACOBI THETA FUNCTION , and is equal to the J ACOBI TRIPLE PRODUCT with z/C301. Special values include 8e/C28pffiffi 2p})@D})@E /C30G9 8})@D})@E G5 4})@D})@Effiffiffiffiffiffiffiffiffiffiffi G1 4})@D})@E 21=4pvuut(3) 8(e/C28p)/C30p1=4 G3 4})@D})@E; (4) where G(x)i sa GAMMA FUNCTION . Another function sometimes given the same symbol is 8(q)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2(0;q) q3(0;q)s ; (5) where qi(0;q) is again a J ACOBI THETA FUNCTION , which has special value 8/C28e/C28pffiffi 3p})@D})@E /C304ffiffiffi 3p /C287})@D})@E1=8 : (6) Ramanujan’s two-variable theta function is defined by f(a;b)/C13X/C12 n/C30/C28/C12an(n/C271)=2bn(n/C281)=2(7) forabjjB1 (Berndt et al. ). It is a generalization of the function 8(x) f(x;x)/C308(x) (8) and satisfies f(/C281;a)/C300 (9) f(a;b)/C30f(b;a)/C30(/C28a;ab)/C12(/C28b;ab)/C12(ab;ab)/C12(10) f(/C28q)/C13f(/C28q;/C28q2) (11) /C30X/C12 k/C300(/C281)kqk(2k/C281)=2X/C12 k/C301(/C281)kqk(2k/C271)=2(12) /C30(q;q)/C12 (13) (Berndt et al. ), where ( a;q)/C12is a Q-POCHHAMMER SYMBOL . (13) is equivalent to EULER’S PENTAGONAL NUMBER THEOREM . See also EULER’S PENTAGONAL NUMBER THEOREM , JACOBI TRIPLE PRODUCT , Q-SERIES ,ROGERS- RAMANU- JAN CONTINUED FRACTION ,SCHRO ¨ TER’S FORMULA References Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan’s Lost Notebook." To appears in Trans. Amer. Math. Soc. Ramanujan-Eisenstein Series EISENSTEIN SERIES Ramanujan-Petersson Conjecture A CONJECTURE for the EIGENVALUES of MODULAR FORMS under HECKE OPERATORS . Ramanujan’s Formula g/C12 0cos(2 zt) sech( pt) dt /C301 2sech z for TzjjB p=2: A related integral is g/C12 0cosh(2 zt) sech( pt) dt /C301 2sech z for RzjjB p=2 :/ References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 11, 1981. Ramanujan’s Hypergeometric Identity 1 /C281 2 !3 /C271 /C215 3 2 /C215 4 !3 /C27.../C303F21 2 ;12 ;12 1; 1; /C281})@*})@+ /C302F11 4 ;14 1; /C281})@*})@+})10})1@2 /C30G29 8})@D})@E G25 4})@D})@E G278})@D})@E; where2F1(a; b; c; x)isa HYPERGEOMETRIC FUNC- TION ,3F2(a; b; c; d; e; x)isa GENERALIZED HYPER- GEOMETRIC FUNCTION , and G(z)isa GAMMA FUNCTION . References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 106, 1999.Ramanujan’s Hypothesis TAUCONJECTURE Ramanujan’s Identity 5f5(x5) f6(x)/C30X/C12 m/C300P(5m/C274)xm; where f(x)/C30Y/C12 m/C301(1/C28xm) andP(n) is the PARTITION FUNCTION P. See also PARTITION FUNCTION P,RAMANUJAN’S SUM IDENTITY Ramanujan’s Integral g/C12 /C28/C12Jm/C27j(x) xm/C27jJn/C28j(y) yn/C28jeitjdj /C302 cos12t})@D})@E x2e/C28it=2/C27y2eit=22 435(m/C27n)=2 /C29Jm/C27nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos1 2t})@D})@E x2e/C28it=2/C27y2eit=2 ðÞr})10})1@ e/C28it(n/C28m)=2; where Jn(z)i saB ESSEL FUNCTION OF THE FIRST KIND . References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Ramanujan’s Interpolation Formula g/C12 0xs/C281X/C12 k/C300(/C281)kxkf(k)dx/C30pf(/C28s) sin(sp)(1) g/C12 0xs/C281X/C12 k/C300(/C281)kxk k!l(k)dx/C30G(s)l(/C28s); (2) where l(z) is the D IRICHLET LAMBDA FUNCTION and G(z) is the GAMMA FUNCTION . Equation (2) is obtained from (1) by defining f(u)/C30l(u) G(1/C27u): (3) These formulas give valid results only for certain classes of functions, and are connected with Mellin transforms (Hardy 1999, p. 15). References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 15 and 186 /C1/195, 1999. Ramanujan’s Master Theorem Suppose that in some NEIGHBORHOOD of x /C300, F(x) /C30X/C12 k /C300f(k)(/C28x)k k!: Then g/C12 0xn/C281F(x) dx /C30G(n) f(/C28n) : References Berndt, B. C. Ramanujan’s Notebooks: Part I. New York: Springer-Verlag, p. 298, 1985. Ramanujan’s Square Equation The DIOPHANTINE EQUATION 2n /C287 /C30x2 : It has been proved that the only solutions to this equation are n /C303, 4, 5, 7, and 15 (Beeler et al. 1972, Item 31). References Schroeppel, R. C. Item 31 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 14, Feb. 1972. Ramanujan’s Sum The sum cq(m) /C30X h/C31(q)e2pihm =q ; (1) where h runs through the residues RELATIVELY PRIME to q, which is important in the representation of numbers by the sums of squares. If (q; q?) /C301 (i.e., q and q ? are RELATIVELY PRIME ), then cqq?(m) /C30cq(m)cq?(m) : (2) For argument 1, cb(1)/C30m(b); (3) where mis the M O¨BIUS FUNCTION , and for general m, cb(m)/C30mb (b;m) ! f(b) fb (b;m) ! : (4) See also MO¨ BIUS FUNCTION ,W EYL’S CRITERION References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 137 /C1/143, 1999.Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, p. 254, 1991. Ramanujan’s Sum Identity If 1/C2753x/C279x2 1/C2882x/C2882x2/C27x3/C30X/C12 n/C301anxn(1) 2/C2826x/C2812x2 1/C2882x/C2882x2/C27x3/C30X/C12 n/C300bnxn(2) 2/C278x/C2810x2 1/C2882x/C2882x2/C27x3/C30X/C12 n/C300cnxn(3) (Sloane’s A051028, A051029, and A051030), then a3 n/C27b3n/C30c3n/C27(/C281)n: (4) Hirschhorn (1995) showed that an/C301 8564/C278ffiffiffiffiffiffi 85p})@D})@E an/C2764/C288ffiffiffiffiffiffi85p})@D})@E bn/C2843(/C281)nhi (5) bn/C301 8577/C277ffiffiffiffiffiffi 85p})@D})@E an/C2777/C287ffiffiffiffiffiffi85p})@D})@E bn/C2816(/C281)nhi (6) cn/C301 8593/C279ffiffiffiffiffiffi 85p})@D})@E an/C2793/C289ffiffiffiffiffiffi85p})@D})@E bn/C2816(/C281)nhi ; (7) where a/C301 283/C279ffiffiffiffiffiffi 85p})@D})@E (8) b/C301 283/C289ffiffiffiffiffiffi 85p})@D})@E : (9) Hirschhorn (1996) showed that checking the first seven cases n/C300 to 6 is sufficient to prove the result. References Hirschhorn, M. D. "An Amazing Identity of Ramanujan." Math. Mag. 68, 199/C1/201, 1995. Hirschhorn, M. D. "A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan." Math. Mag. 69, 267/C1/ 269, 1996. Sloane, N. J. A. Sequences A051028, A051029, and A051030 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Ramanujan’s Tau Function TAUFUNCTION Ramanujan’s Tau-Dirichlet Series TAU-DIRICHLET SERIES Ramification Group References Koch, H. "Decomposition Group and Ramification Group." §6.1 in Number Theory: Algebraic Numbers and Func- tions. Providence, RI: Amer. Math. Soc., pp. 172 /C1/176, 2000. Ramp Function R(x) /C13xH(x) (1) /C30gx /C28/C12H(x?) dx ? (2) /C30gx /C28/C12H(x?)H(x /C28x?) dx? (3) /C30H(x) + H(x) ; (4) where H(x) is the HEAVISIDE STEP FUNCTION and + is the CONVOLUTION . The DERIVATIVE is R?(x) /C30/C28H(x) : (5) The FOURIER TRANSFORM of the ramp function is given by F[R(x)] /C30g/C12 /C28/C12e /C282 pikxR(x) dx /C30 pi d?(2pk) /C281 4 p2k2 ; (6) where d(x) is the DELTA FUNCTION and d?(x) its DERIVATIVE . See also FOURIER TRANSFORM– RAMP FUNCTION ,HEA- VISIDE STEP FUNCTION ,RECTANGLE FUNCTION ,SGN, SQUARE WAVE Ramphoid Cusp A type of CUSP as illustrated above for the curve x4 /C27x2y2 /C282x2y /C28xy2 /C27y2 /C300::/ See also CUSP References Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 57 /C1/58, 1978. Ramsey Number The Ramsey number R(m;n) gives the solution to the PARTY PROBLEM , which asks the minimum number of guests R(m;n) that must be invited so that at least m will know each other or at least nwill not know each other. In the language of GRAPH THEORY , the Ramsey number is the minimum number of vertices / v¼Rðm;nÞ/such that all undirected simple graphs of order vcontain a CLIQUE of order mo r an INDEPENDENT SET of order n.R AMSEY’S THEOREM states that such a number exists for all mandn. By symmetry, it is true that R(m;n)/C30R(n;m): (1) It also must be true that R(m;2)/C30m: (2) A generalized Ramsey number is written R(m1;...;mk;n) (3) and is the smallest INTEGER rsuch that, no matter how each n-element SUBSET of an r-element SETis colored with kcolors, there exists an isuch that there is a SUBSET of size mi;all of whose n-element SUBSETS are color i. The usual Ramsey numbers are then equivalent to R(m;n)/C30R(m;n;2 ):/ Bounds are given by R(k;l)5R(k/C281;l)/C27R(k;l/C281)/C281 forR(k/C281;1) and R(k;l/C281) even R(k/C281;l)/C27R(k;l/C281) otherwise8 >>< >>:(4) and R(k;k)54R(k/C282;k)/C272 (5) (Chung and Grinstead 1983). Erdos proved that for diagonal Ramsey numbers R(k;k); k2k=2 effiffiffi 2pBR(k;k): (6) This result was subsequently improved by a factor of 2 by Spencer (1975). R(3;k) was known since 1980 to be bounded from above by c2k2=lnk;and Griggs (1983) showed that c2/C305=12 was an acceptable limit. J.-H. Kim (Cipra 1995) subsequently bounded R(3;k) by a similar expression from below, so c1k2 lnk5R(3;k)5c2k2 lnk: (7) Burr (1983) gives Ramsey numbers for all 113 graphs with no more than 6 EDGES and no isolated points. A summary of known results up to 1983 for R(m;n)i s given in Chung and Grinstead (1983). Radziszowski(1999) maintains an up-to-date list of the best currentbounds, reproduced in part in the following table for R(m;n;2 ): / mn /R(m;n)/ Reference 3 3 6 Greenwood and Gleason 1955 3 4 9 Greenwood and Gleason 19553 5 14 Greenwood and Gleason 19553 6 18 Graver and Yackel 19683 7 23 Kalbfleisch 19663 8 28 McKay and Min 19923 9 36 Grinstead and Roberts 19823 10 [40, 43] Exoo 1989, Radziszowski and Kreher 1988 3 11 [46, 51] Radziszowski and Kreher 1988 3 12 [52, 60] Exoo 1993, Radziszowski and Kreher 1988, Exoo 1998 3 13 [59, 69] Piwakowski 1996, Radziszowski and Kreher 1988 3 14 [66, 78] Exoo (unpub.), Radziszowski and Kreher 1988 3 15 [73, 89] Wang and Wang 1989, Radziszowski (unpub.) 31 6 /]79/ Wang and Wang 1989 31 7 /]92/ WWY 31 8 /]98/ WWY 31 9 /]106 / WWY 32 0 /]109 / WWY 32 1 /]122 / WWY 32 2 /]125 / WWY 32 3 /]136 / WWY 32 6 /]150 / 4 4 18 Greenwood and Gleason 1955 4 5 25 Mckay and Radziszowski 1995 4 6 [35, 41] Ex8, MR44 7 [49, 61]4 8 [55, 84] Exoo 19984 9 [69, 115]4 10 [80, 149] 4 11 [96, 191] 4 12 [128, 238]4 13 [131, 291]4 14 [136, 349]4 15 [145, 417]41 7 /]164 / 41 8 /]182 / 41 9 /]194 / 42 0 /]230 / 42 1 /]242 / 42 2 /]282 / 5 5 [43, 49] Ex4, MR45 6 [58, 87] Exoo 1993, Walker 19715 7 [80, 143]5 8 [95, 216]5 9 [116, 316] Exoo 19985 10 [141, 442]51 1 /]153 / 51 2 /]181 / 51 3 /]193 / 51 4 /]221 / 51 5 /]237 / 51 7 /]282 / 51 9 /]338 / 52 1 /]374 / 52 2 /]410 / 52 3 /]432 / 52 6 /]464 / 6 6 [102, 165] Kalbfleisch 1965, Mac6 7 [109, 298] Exoo 1998 6 8 [122, 495] Exoo 1998 6 9 [153, 780]6 10 [167, 1171]61 1 /]203 / 61 2 /]224 / 61 3 /]242 / 61 4 /]258 / 61 5 /]338 / 61 7 /]500 / 7 7 [205, 540] Hill and Irving 1982, Giraud 1973 7 8 [1, 1031]7 9 [1, 1713]7 10 [1, 2826]71 7 /]548 / 71 9 /]618 / 72 0 /]648 / 72 1 /]674 / 8 8 [282, 1870] 8 9 [1, 3583] 8 10 [1, 6090] 81 6 /]602 / 81 7 /]674 / 82 0 /]752 / 82 1 /]770 / 9 9 [565, 6588] 9 10 [1, 12677] 10 10 [798, 23581] Guldan and Tomasta ? 11 11 [522, [522, /C12]] Guldan and Tomasta ? Known bounds for generalized Ramsey numbers (multicolor graph numbers) are given in the following table. /R(...; 2)/ Bounds Reference /R(3 ; 3; 3; 2)/ 17 Greenwood and Gleason 1955 /R(3 ; 3; 3; 3; 2)/ [51, 64] Chung 1973, Sanchez- Flores 1995 /R(3 ; 3; 3; 3; 3; 2)/ [162, 317] /R(3 ; 3; 3; 3; 3; 3; 2)/ [500, 1898] Exoo 1994 /R(3 ; 3; 3; 4; 2)/ [91, 155] Robertson 1999, Exoo 1998 /R(3 ; 3; 3; 5; 2)// ]137 / Robertson 1999 /R(3 ; 3; 3; 6; 2)// ]165 / Robertson 1999 /R(3 ; 3; 3; 7; 2)// ]220 / Robertson 1999 /R(3 ; 3; 3; 9; 2)// ]336 / Robertson 1999 /R(3 ; 3; 3; 11; 2)// ]422 / Robertson 1999 /R(3 ; 3; 4; 2)/ [30, 31] /R(3 ; 3; 4; 4; 2)// ]144 / /R(3 ; 3; 5; 2)/ [45, 57] /R(3 ; 3; 6; 2)// ]60/ /R(3 ; 3; 7; 2)// ]72/ /R(3 ; 3; 9; 2)// ]110 / /R(3 ; 3; 11; 2)// ]141 / /R(3 ; 4; 5; 2)/ [80, 161] Exoo 1998 /R(3 ; 4; 4; 2)/ [55, 79] /R(4 ; 4; 4; 2)/ [128, 236] Hill and Irving 1982, Giraud 1973 /R(4 ; 4; 4; 4; 2)// ]458 / /R(4 ; 4; 4; 4; 4; 2)// ]942 / /R(5 ; 5; 5; 2)// ]242 /Robertson 1999 /R(6;6;6; 2) // ]692 /Robertson 1999 Known bounds for hypergraph Ramsey numbers are given in the following table./R(. . . ; 3) / Bounds /R(4;4; 3) / 13 /R(4;4;4; 3) //]56/ /R(4;5; 3) // ]33/ /R(5;5; 3) // ]63/ See also CLIQUE ,CLIQUE NUMBER ,COMPLETE GRAPH , EXTREMAL GRAPH ,INDEPENDENCE NUMBER ,INDE- PENDENT SET,IRREDUNDANT RAMSEY NUMBER ,RAM- SEY’S THEOREM ,RAMSEY THEORY ,SCHUR NUMBER References Burr, S. A. "Generalized Ramsey Theory for Graphs--A Survey." In Graphs and Combinatorics (Ed. R. A. Bari and F. Harary). New York: Springer-Verlag, pp. 52 /C1/75, 1964. Burr, S. A. "Diagonal Ramsey Numbers for Small Graphs." J. Graph Th. 7,5 7/C1/69, 1983. Chartrand, G. "The Problem of the Eccentric Hosts: An Introduction to Ramsey Numbers." §5.1 in Introductory Graph Theory. New York: Dover, pp. 108 /C1/115, 1985. Chung, F. R. K. "On the Ramsey Numbers N(3;3;...;3; 2) :/"Discrete Math. 5, 317/C1/321, 1973. Chung, F. and Grinstead, C. G. "A Survey of Bounds for Classical Ramsey Numbers." J. Graph. Th. 7,2 5/C1/37, 1983. Cipra, B. "A Visit to Asymptopia Yields Insights into Set Structures." Science 267, 964/C1/965, 1995. Exoo, G. "On Two Classical Ramsey Numbers of the Form R(3;n):/"SIAM J. Discrete Math. 2, 488/C1/490, 1989. Exoo, G. "Announcement: On the Ramsey Numbers R(4;6); R(5;6) and R(3;12):/"Ars Combin. 35, 85, 1993. Exoo, G. "A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of K3:/"Electronic J. Combinatorics 1, R8 1/C1/3, 1994. http://www.combinatorics.org/Volume_1/ volume1.html#R8. Exoo, G. "Some New Ramsey Colorings." Electronic J. Combinatorics 5, No. 1, R29, 1 /C1/5, 1998. http://www.com- binatorics.org/Volume_5/v5i1toc.html. Folkmann, J. "Notes on the Ramsey Number N(3;3;3;3):/" J. Combinat. Theory. Ser. A 16, 371/C1/379, 1974. Fredricksen, H. "Schur Numbers and the Ramsey Numbers N(3;3;...;3; 2) :/"J. Combin. Theory Ser. A 27, 376/C1/377, 1979. Gardner, M. "Mathematical Games: In Which Joining Sets of Points by Lines Leads into Diverse (and Diverting) Paths." Sci. Amer. 237,1 8/C1/28, 1977. Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Free- man, pp. 240 /C1/241, 1989. Giraud, G. "Une minoration du nombre de quadrangles unicolores et son application a la majoration des nombresde Ramsey binaires bicolors." C. R. Acad. Sci. Paris A 276, 1173/C1 /1175, 1973. Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey Theory, 2nd ed. New York: Wiley, 1990. Graver, J. E. and Yackel, J. "Some Graph Theoretic Results Associated with Ramsey’s Theorem." J. Combin. Th. 4, 125/C1/175, 1968. Greenwood, R. E. and Gleason, A. M. "Combinatorial Rela- tions and Chromatic Graphs." Canad. J. Math. 7,1/C1/7, 1955. Griggs, J. R. "An Upper Bound on the Ramsey Numbers R(3;k):/"J. Comb. Th. A 35, 145/C1/153, 1983. Grinstead, C. M. and Roberts, S. M. "On the Ramsey Numbers R(3; 8) and R(3; 9) :/" J. Combinat. Th. Ser. B 33,27/C1/51, 1982. Guldan, F. and Tomasta, P. "New Lower Bounds of Some Diagonal Ramsey Numbers." J. Graph. Th. 7, 149 /C1/151, 1983. Hanson, D. "Sum-Free Sets and Ramsey Numbers." Discrete Math. 14,57/C1/61, 1976. Harary, F. "Recent Results on Generalized Ramsey Theory for Graphs." In Graph Theory and Applications: Proceed- ings of the Conference at Western Michigan University, Kalamazoo, Mich., May 10 /C1/13, 1972 (Ed. Y. Alavi, D. R. Lick, and A. T. White). New York: Springer-Verlag, pp. 125 /C1/138, 1972. Hill, R. and Irving, R. W. "On Group Partitions Associated with Lower Bounds for Symmetric Ramsey Numbers." European J. Combin. 3,35/C1/50, 1982. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 52 /C1/53, 1998. Kalbfleisch, J. G. Chromatic Graphs and Ramsey’s Theo- rem. Ph.D. thesis, University of Waterloo, January 1966. McKay, B. D. and Min, Z. K. "The Value of the Ramsey Number R(3; 8):/" J. Graph Th. 16,99/C1/105, 1992. McKay, B. D. and Radziszowski, S. P. "/R(4; 5) /C3025:/" J. Graph. Th 19, 309 /C1/322, 1995. Piwakowski, K. "Applying Tabu Search to Determine New Ramsey Numbers." Electronic J. Combinatorics 3,R61/C1/4, 1996. http://www.combinatorics.org/Volume_3/volu- me3.html#R6. Radziszowski, S. P. "Small Ramsey Numbers." Electronic J. Combin. 1, DS1 1 /C1/29, Rev. Jul. 5, 1999. http://www.com- binatorics.org/Surveys/. Radziszowski, S. and Kreher, D. L. "Upper Bounds for Some Ramsey Numbers R(3; k) :/" J. Combinat. Math. Combin. Comput. 4, 207 /C1/212, 1988. Robertson, A. "New Lower Bounds for Some Multicolored Ramsey Numbers." Electronic J. Combinatorics 6, No. 1, R3, 1 /C1/6, 1999. http://www.combinatorics.org/Volume_6/ v6i1toc.html. Spencer, J. H. "Ramsey’s Theorem--A New Lower Bound." J. Combinat. Theory Ser. A 18, 108 /C1/115, 1975. Wang, Q. and Wang, G. "New Lower Bounds for the Ramsey Numbers R(3; q) :/" Beijing Daxue Xuebao 25, 117 /C1/121, 1989. Whitehead, E. G. "The Ramsey Number N(3; 3; 3; 3; 2):/" Discrete Math. 4, 389 /C1/396, 1973. Ramsey Theory The mathematical study of combinatorial objects in which a certain degree of order must occur as the scale of the object becomes large. Ramsey theory is named after Frank Plumpton Ramsey, who did seminal work in this area before his untimely death at age 26 in 1930. The theory was subsequently developed extensively by Erdos. The classical problem in Ramsey theory is the PARTY PROBLEM , which asks the minimum number of guests R(m; n) that must be invited so that at least m will know each other (i.e., there exists a CLIQUE of order m) or at least n will not know each other (i.e., there exists an INDEPENDENT SET of order n. Here, R(m; n) is called a RAMSEY NUMBER . A typical result in Ramsey theory states that if some mathematical object is partitioned into finitely many parts, then one of the parts must contain a subobjectof an interesting kind. For example, it is known that if n is large enough and V is an n-dimensional VECTOR SPACE over the FIELD of integers (mod p), then however V is partitioned into r pieces, one of the pieces contains an affine subspace of dimension d. See also EXTREMAL GRAPH THEORY ,GRAHAM’S NUM- BER,HAPPY END PROBLEM ,PARTY PROBLEM ,RAMSEY NUMBER ,STRUCTURAL RAMSEY THEORY References Burr, S. A. "Generalized Ramsey Theory for Graphs--A Survey." In Graphs and Combinatorics (Ed. R. A. Bari and F. Harary). New York: Springer-Verlag, pp. 52 /C1/75, 1964. Erdos, P. and Szekeres, G. "On Some Extremum Problems in Elementary Geometry." Ann. Univ. Sci. Budapest Eotvos Soc. Math. 3 /C1/4,53/C1/62, 1961. Graham, R. L. and Nesetril, J. "Ramsey Theory in the Work of Paul Erdos." In The Mathematics of Paul Erdos (Ed. R. L. Graham and J. Nesetril). Heidelberg, Germany: Springer-Verlag, 1996. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 51 /C1/57, 1998. Ramsey’s Theorem A generalization of DILWORTH’S LEMMA . For each m; n /C23N with m; n ]2; there exists a least INTEGER R(m; n) (the RAMSEY NUMBER ) such that no matter how the COMPLETE GRAPH KR(m; n)is two-colored, it will contain a green SUBGRAPH Km or a red SUBGRAPH Kn : Furthermore, Rðm;nÞ5Rðm /C281;nÞþRðm;n /C281Þ if m; n ]3 :/ The theorem can be equivalently stated that, for all m/C23N;there exists an n/C23Nsuch that any COMPLETE DIGRAPH onnVERTICES contains a COMPLETE TRAN- SITIVE SUBGRAPH ofmVERTICES . Ramsey’s theorem is a generalization of the PIGEON- HOLE PRINCIPLE since R(2;2;...;2|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} t)/C30t/C271: See also DILWORTH’S LEMMA ,E XTREMAL GRAPH THEORY ,GRAPH COLORING ,NATURAL INDEPENDENCE PHENOMENON ,PARTY PROBLEM ,PIGEONHOLE PRINCI- PLE,RAMSEY NUMBER ,RAMSEY THEORY References Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey Theory, 2nd ed. New York: Wiley, 1990. Spencer, J. "Large Numbers and Unprovable Theorems." Amer. Math. Monthly 90, 669/C1/675, 1983. Ramus Tree A type of BINARY TREE . See also BINARY TREE Randelbrot Set The FRACTAL -like figure obtained by performing the same iteration as for the MANDELBROT SET, but adding a random component R, zn/C271 /C30z2 n /C27c /C27R: In the above plot, R /C13Rx /C27iRy ; where Rx ; Ry /C23 [/C280:05 ; 0:05] :/ See also MANDELBROT SET References Dickau, R. M. "Randelbrot Set." http://forum.swarthmor- e.edu/advanced/robertd/randelbrot.html. Random Close Packing Random close packing of spheres in three dimensions gives a PACKING DENSITY of only h :0 :64 (Jaeger and Nagel 1992), significantly smaller than the optimal PACKING DENSITY for cubic or hexagonal close packing of 0.74048. See also SPHERE PACKING References --. Nature 239, 488, 1972. Jaeger, H. M. and Nagel, S. R. "Physics of Granular States." Science 255, 1524, 1992. Torquato, S.; Truskett, T. M.; and Debenedetti, P. G. "Is Random Close Packing of Spheres Well Defined?" Phys. Lev. Lett. 84, 2064 /C1/2067, 2000. Random Composition A random composition of a number n in k parts is one of then/C27k/C281 n})0})@ possible COMPOSITIONS of n, wheren k})0})@ isa BINOMIAL COEFFICIENT . A random composition can be given byRandomComposition [n, k] in the Math- ematica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also COMPOSITION References Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Skiena, S. "Random Partitions." §2.1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 58 /C1/59, 1990. Random Distribution A STATISTICAL DISTRIBUTION in which the variates occur with PROBABILITIES asymptotically matching their "true" underlying STATISTICAL DISTRIBUTION is said to be random. See also RANDOM NUMBER ,STATISTICAL DISTRIBU- TION Random Dot Stereogram STEREOGRAM Random Fibonacci Sequence Consider the Fibonacci-like recurrence an /C309an/C281 9an /C282 ; where a0 /C300; a1 /C301; and each sign is chosen inde- pendently and at random with probability 1/2. Sur- prisingly, Viswanath (2000) showed that lim n0/C12½an ½1 =n /C301:13198824... with probability one. This constant can be numeri- cally computed by computing the product of a certain set of RANDOM MATRICES , and taking the SPECTRAL NORM of the result (Viswanath 2000). See also FIBONACCI NUMBER ,RANDOM MATRIX References Viswanath, D. "Random Fibonacci Sequences and the Number 1.13198824...." Math. Comput. 69, 1131 /C1/1155, 2000. Random Graph A random graph is a GRAPH in which properties such as the number of NODES ,EDGES , and connections between them are determined in some random way. The graphs illustrated above are random graphs on 10 edges with edge probabilities distributed uni- formly in [0; 1]:/ Erdos and Re´nyi (1960) showed that for many mono- tone-increasing properties of random graphs, graphs of a size slightly less than a certain threshold are very unlikely to have the property, whereas graphs with a few more EDGES are almost certain to have it. This is known as a PHASE TRANSITION (Janson et al. 2000, p. 103). Almost all graphs are connected and non- planar (Skiena 1990, p. 156). See also GRAPH ,GRAPH THEORY ,PHASE TRANSITION References Bolloba ´s, B. Graph Theory: An Introductory Course. New York: Springer-Verlag, 1979. Bolloba ´s, B. Random Graphs. London: Academic Press, 1985. Erdos, P. and Re´nyi, A. "On the Evolution of Random Graphs." Publ. Math. Inst. Hungar. Acad. Sci. 5,17/C1/61, 1960. Erdos, P. and Spencer, J. Probabilistic Methods in Combi- natorics. New York: Academic Press, 1974. Janson, S.; L uczak, T.; and Rucinski, A. Random Graphs. New York: Wiley, 2000. Kolchin, V. F. Random Graphs. New York: Cambridge University Press, 1998. Palmer, E. M. Graphical Evolution: An Introduction to the Theory of Random Graphs. New York: Wiley, 1985. Skiena, S. "Random Graphs." Implementing Discrete Mathe- matics: Combinatorics and Graph Theory with Mathema- tica. Reading, MA: Addison-Wesley, pp. 154 /C1/160, 1990. Steele, J. M. "Gibbs’ Measures on Combinatorial Objects and the Central Limit Theorem for an Exponential Family of Random Trees." Prob. Eng. Inform. Sci. 1,47/C1/59, 1987. Random Matrix A random matrix is a MATRIX of given type and size whose entries consist of random numbers from some specified distribution. If n matrices Mi are chosen with probability 1/2 from one of M/C27/C30 01 11})10})1@ (1) M/C28/C30 011 /C281})10})1@ ; (2) then lim n0/C12ln M1 /C1/C1/C1Mn kk n/C30c ; (3) where ec /C301:13198824... and Mkk denotes the ma- trix SPECTRAL NORM (Bougerol and Lacroix 1985, pp. 11 and 157; Viswanath 2000). This is the same constant appearing in the RANDOM FIBONACCI SE- QUENCE . The following Mathematica code can be used to estimate this constant.n /C30 100000; m /C30 Fold[Dot, IdentityMatrix[2], {{0, 1}, {1, #}} & /@ ((- 1)^Table[Random[Integer], {n}]) ]//N; Log[Sqrt[Max[Eigenvalues[Transpose[m].m]]]]/ n See also COMPLEX MATRIX ,M ATRIX ,RANDOM FIBO- NACCI SEQUENCE ,REAL MATRIX References Bougerol, P. and Lacroix, J. Random Products of Matrices with Applications to Schro ¨dinger Operators. Basel, Swit- zerland: Birkha ¨user 1985. Chassaing, P.; Letac, G.; and Mora, M. "Brocot Sequences and Random Walks on SL2(R):/"I n Probability Measures on Groups VII (Ed. H. Heyer). New York Springer-Verlag, pp. 36 /C1/48, 1984. Furstenberg, H. "Non-Commuting Random Products." Trans. Amer. Math. Soc. 108, 377/C1/428, 1963. Furstenberg, H. and Kesten, H. "Products of Random Matrices." Ann. Math. Stat. 31, 457/C1/469, 1960. Katz, M. and Sarnak, P. Random Matrices, Frobenius Eigenvalues, and Monodromy. Providence, RI: Amer. Math. Soc., 1999. Mehta, M. L. Random Matrices, 2nd rev. enl. ed. New York: Academic Press, 1991. Viswanath, D. "Random Fibonacci Sequences and the Number 1.13198824...." Math. Comput. 69, 1131 /C1/1155, 2000. Random Normal Deviates NORMAL DEVIATES Random Number Computer-generated random numbers are sometimes called PSEUDORANDOM NUMBERS , while the term "random" is reserved for the output of unpredictablephysical processes. When used without qualification,the word "random" usually means "random with a UNIFORM DISTRIBUTION ." Other distributions are, of course possible. For example, the B OX-MULLER TRANSFORMATION allows random numbers with a 2- D uniform distribution to be transformed to corre- sponding random numbers with a 2-D Gaussian distribution. Similarly, in order to generate apower-law distribution P(x) from a uniform distribu- tion P(y);write P(x)/C30Cx nforx/C23[x0;x1]:Then nor- malization gives gx1 x0P(x)dx/C30c[xn/C271]x1 x0 n/C271/C301; (1) so C/C30n/C271 xn/C271 1/C28xn/C271 0: (2) Letybe a uniformly distributed variate on [0 ;1]: Then D(x) /C30gx x0P(x?) dx?/C30Cgx x0x?n dx ? /C30C n /C27 1xn /C271 /C28xn/C271 0})0})@ /C13y; (3) and the variate given by x /C30n /C27 1 Cy /C27xn/C271 0 !1 =(n/C271) /C30 xn/C271 1/C28xn/C271 0})0})@ y /C27xn/C271 0})1})A1 =(n /C271)(4) is distributed as P(x):/ It is impossible to produce an arbitrarily long string of random digits and prove it is random. Strangely, it is very difficult for humans to produce a string of random digits, and computer programs can be written which, on average, actually predict some of the digits humans will write down based on previous ones. The LINEAR CONGRUENCE METHOD is one algorithm for generating PSEUDORANDOM NUMBERS . The initial number used as the starting point in a random number generating algorithm is known as the SEED . The goodness of random numbers generated by a given ALGORITHM can be analyzed by examining its NOISE SPHERE . When generating random numbers over some speci- fied boundary, it is often necessary to normalize the distributions so that each differential area can is equally populated. For example, picking u and f from uniform distributions does not give a uniform dis- tribution for SPHERE POINT PICKING . See also BAYS’ SHUFFLE ,BOX-MULLER TRANSFORMA- TION ,CLIFF RANDOM NUMBER GENERATOR ,QUASIR- ANDOM SEQUENCE ,R ANDOM VARIABLE ,S CHRAGE’S ALGORITHM ,STOCHASTIC ,UNIFORM DISTRIBUTION References Bassein, S. "A Sampler of Randomness." Amer. Math. Monthly 103, 483 /C1/490, 1996. Bennett, D. J. Randomness. Cambridge, MA: Harvard Uni- versity Press, 1998. Bratley, P.; Fox, B. L.; and Schrage, E. L. A Guide to Simulation, 2nd ed. New York: Springer-Verlag, 1996. Dahlquist, G. and Bjorck, A. Ch. 11 in Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, 1974. Deak, I. Random Number Generators and Simulation. New York: State Mutual Book & Periodical Service, 1990. Forsythe, G. E.; Malcolm, M. A.; and Moler, C. B. Ch. 10 in Computer Methods for Mathematical Computations. Eng- lewood Cliffs, NJ: Prentice-Hall, 1977. Gardner, M. "Random Numbers." Ch. 13 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage, pp. 161 /C1/ 172, 1977. James, F. "A Review of Pseudorandom Number Generators." Computer Physics Comm. 60, 329 /C1/344, 1990. Kac, M. "What is Random?" Amer. Sci. 71, 405 /C1/406, 1983.Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 200 /C1/201 and 205 /C1/207, 1962. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 151 /C1/154, 1951. Knuth, D. E. Ch. 3 in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998. Marsaglia, G. "A Current View of Random Number Gen- erators." In Computer Science and Statistics: Proceedings of the Symposium on the Interface, 16th, Atlanta, Georgia, March 1984 (Ed. L. Billard). New York: Elsevier, 1985. Marsaglia, G. "DIEHARD: A Battery of Tests for Random Number Generators." http://stat.fsu.edu/~geo/die- hard.html. Mascagni, M. "Random Numbers on the Web." http:// www.ncsa.uiuc.edu/Apps/CMP/RNG/mascagni/www- rng.html. Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Park, S. and Miller, K. "Random Number Generators: Good Ones are Hard to Find." Comm. ACM 31, 1192 /C1/1201, 1988. Peterson, I. The Jungles of Randomness: A Mathematical Safari. New York: Wiley, 1997. Pickover, C. A. "Computers, Randomness, Mind, and In- finity." Ch. 31 in Keys to Infinity. New York: W. H. Freeman, pp. 233 /C1/247, 1995. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Random Numbers." Ch. 7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 266 /C1/306, 1992. Schrage, L. "A More Portable Fortran Random Number Generator." ACM Trans. Math. Software 5, 132 /C1/138, 1979. Schroeder, M. "Random Number Generators." In Number Theory in Science and Communication, with Applications in Cryptography, Physics, Digital Information, Computing and Self-Similarity, 3rd ed. New York: Springer-Verlag, pp. 289 /C1/295, 1990. Weisstein, E. W. "Books about Randomness." http:// www.treasure-troves.com/books/Randomness.html. Wilf, H. S. Combinatorial Algorithms: An Update. Philadel- phia, PA: SIAM, 1989. Random Partition A random partition of a number nis one of the P(n) possible PARTITIONS ofn, where P(n) is the PARTITION FUNCTION P. A random partition can be given by RandomPartition [n] in the Mathematica add-on packageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also PARTITION References Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Skiena, S. "Random Partitions." §2.1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 58 /C1/59, 1990. Random Percolation PERCOLATION THEORY Random Permutation A PERMUTATION containing a fixed number n of a random selection from a given set of elements. There are two main algorithms for constructing random permutations. The first constructs a vector of random real numbers and uses them as keys to records containing the integers 1 to n. The second starts with an arbitrary permutation and then exchanges the ith element with a randomly selected one from the first i elements for i /C301, ..., n (Skiena 1990). There are an average of n(n /C281) =4 PERMUTATION INVERSIONS in a PERMUTATION on n elements (Skiena 1990, p. 29). See also PERMUTATION ,PERMUTATION INVERSION References Moses, L. E. and Oakford, R. V. Tables of Random Permuta- tions. Stanford, CA: Stanford University Press, 1963. Skiena, S. "Random Permutations." §1.1.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 6 /C1/ 9 and 29, 1990. Random Polygon A random polygon is a POLYGON generated in some random way. Kendall conjectured that the shape of a random polygon is close to a DISK as the area of the polygon becomes large (Stoyan et al. 1987, Kovalenko 1999) See also CROFTON CELL References Kovalenko, I. N. "Proof of David Kendall’s Conjecture Con- cerning the Shape of Large Random Polygons." Cybern. Sys. Anal. 33, 461 /C1/467, 1997. Kovalenko, I. N. "A Simplified Proof of a Conjecture of D. G. Kendall Concerning Shapes of Random Polygons." J. Appl. Math. Stoch. Anal. 12, 301 /C1/310, 1999. Miles, R. E. "A Heuristic Proof of a Long-Standing Con- jecture of D. G. Kendall Concerning the Shapes of Certain Large Random Polygons." Adv. Appl. Prob. (SGSA) 27, 397 /C1/471, 1997. Stoyan, D.; Kendall, W. S.; and Mecke, J. Stochastic Geo- metry and Its Applications, with a Foreword by D. G. Ken- dall. New York: Wiley, 1987. Random Polynomial A POLYNOMIAL having random COEFFICIENTS . See also KAC FORMULA References Bharucha-Reid, A. T. and Sambandham, M. Random Poly- nomials. New York: Academic Press, 1986. Bloch, A. and Po´lya, G. "On the Zeros of Certain Algebraic Equations." Proc. London Math. Soc. 33, 102 /C1/114, 1932.Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32,1/C1/37, 1995. Erdos, P. and Tura´n, P. "On the Distribution of Roots of Polynomials." Ann. Math. 51, 105 /C1/119, 1950. Hammersley, J. "The Zeros of a Random Polynomial." Proc. Third Berkeley Symp. Math. Stat. Prob. 2,89/C1/111, 1956. Kac, M. "On the Average Number of Real Roots of a Random Algebraic Equation." Bull. Amer. Math. Soc. 49, 314 /C1/320, 1943. Kac, M. "A Correction to ‘On the Average Number of Real Roots of a Random Algebraic Equation’." Bull. Amer. Math. Soc. 49, 938, 1943. Kostan, E. "On the Distribution of Roots in a Random Polynomial." Ch. 38 in From Topology to Computation: Proceedings of the Smalefest (Ed. M. W. Hirsch, J. E. Marsden, and M. Shub). New York: Springer-Verlag, pp. 419 /C1/431, 1993. Littlewood, J. and Offord, A. "On the Number of Real Roots of a Random Algebraic Equation." J. London Math. Soc. 13, 288 /C1/295, 1938. Maslova, N. "On the Distribution of the Number of Reals Roots of a Random Polynomial" [In Russian]. Teor. Veroyatnost. i Primenen 19, 488 /C1/500, 1974. Rice, S. O. "The Distribution of the Maxima of a Random Curve." Amer. J. Math. 61, 409 /C1/416, 1939. Rice, S. O. "Mathematical Analysis of Random Noise." Bell Syst. Tech. J. 24,45/C1/156, 1945. Random Tableau AY OUNG TABLEAU chosen at random from those having a given shape. A random tableau can be generated by RandomTableau [shape ] in the Mathe- matica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ ). The figure above shows four random tableaux of the 21 distinct ones of shape f3;2;2g:/ See also YOUNG TABLEAU References Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Skiena, S. "Random Tableaux." §2.3.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 72 /C1/73, 1990. Random Variable A random variable is a measurable function from a PROBABILITY SPACE (S;S;P) into a MEASURABLE SPACE (S?;S?) known as the STATE SPACE (Doob 1996). Papoulis (1984, p. 88) gives the slightly differ- ent definition of a random variable Xas a REAL FUNCTION whose domain is the PROBABILITY SPACE S and such that: 1. The set fX 5xg is an EVENT for any real number x. 2. The probability of the events fX /C30/C27/C12g and fX /C30/C28/C12g equals zero. The abbreviation "r.v." is sometimes used to denote a random variable. See also PROBABILITY SPACE ,RANDOM DISTRIBUTION , RANDOM NUMBER ,STATE SPACE ,VARIATE References Doob, J. L. "The Development of Rigor in Mathematical Probability (1900 /C1/1950)." Amer. Math. Monthly 103, 586 /C1/595, 1996. Gikhman, I. I. and Skorokhod, A. V. Introduction to the Theory of Random Processes. New York: Dover, 1997. Papoulis, A. "The Concept of a Ransom Variable." Ch. 4 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 83 /C1/115, 1984. Random Walk A random process consisting of a sequence of discrete steps of fixed length. The random thermal perturba- tions in a liquid are responsible for a random walk phenomenon known as Brownian motion, and the collisions of molecules in a gas are a random walk responsible for diffusion. Random walks have inter- esting mathematical properties that vary greatly depending on the dimension in which the walk occurs and whether it is confined to a lattice. See also MARKOV CHAIN ,MARTINGALE ,PERCOLATION THEORY ,R ANDOM WALK–1- D, RANDOM WALK–2- D, RANDOM WALK–3- D, SELF-AVOIDING WALK,S ELF- AVOIDING WALK CONNECTIVE CONSTANT References Barber, M. N. and Ninham, B. W. Random and Restricted Walks: Theory and Applications. New York: Gordon and Breach, 1970. Chandrasekhar, S. In Selected Papers on Noise and Sto- chastic Processes (Ed. N. Wax). New York: Dover, 1954. Doyle, P. G. and Snell, J. L. Random Walks and Electric Networks. Washington, DC: Math. Assoc. Amer, 1984. Dykin, E. B. and Uspenskii, V. A. Random Walks. New York: Heath, 1963. Erdos, P. and Re ´ve´sz, P. "Three Problems on the Random Walk in Zd:/"Studia Sci. Math. Hung. 26, 309/C1/320, 1991. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971. Gardner, M. "Random Walks and Gambling" and "Random Walks on the Plane and in Space." Chs. 6 /C1/7i nMathema- tical Circus: More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments. Washington, DC: Math. Assoc. Amer., pp. 66 /C1/86, 1992. Hughes, B. D. Random Walks and Random Environments, Vol. 1: Random Walks. New York: Oxford University Press, 1995. Hughes, B. D. Random Walks and Random Environments, Vol. 2: Random Environments. New York: Oxford Uni- versity Press, 1996.Lawler, G. F. Intersections of Random Walks. Boston, MA: Birkha ¨user, 1996. Re´ve´sz, P. Random Walks in Random and Non-Random Environments. Singapore: World Scientific, 1990. Spitzer, F. Principles of Random Walk, 2nd ed. New York: Springer-Verlag, 1976. Weiss, G. Aspects and Applications of the Random Walk. Amsterdam, Netherlands: North-Holland, 1994. Weisstein, E. W. "Books about Random Walks." http:// www.treasure-troves.com/books/RandomWalks.html. Random Walk * /1-D LetNsteps of equal length be taken along a LINE. Let pbe the probability of taking a step to the right, qthe probability of taking a step to the left, n1the number of steps taken to the right, and n2the number of steps taken to the left. The quantities p,q,n1;n2;and N are related by p/C27q/C301 (1) and n1/C27n2/C30N: (2) Now examine the probability of taking exactly n1 steps out of Nto the right. There are ðN n1Þ/C30ðn1/C27n2 n1Þways of taking n1steps to the right and n2to the left, where n m})0})@ is a BINOMIAL COEFFICIENT . The probability of taking a particular ordered sequence of n1and n2 steps is pn1qn2:Therefore, P(n1)/C30(n1/C27n2)! n1!n2!pn1qn2/C30N! n1!(N/C28n1)!pn1qN/C28n1;(3) where n!i sa FACTORIAL . This is a BINOMIAL DIS- TRIBUTION and satisfies XN n1/C300P(n1)/C30(p/C27q)N/C301N/C301: (4) The MEAN number of steps n1to the right is then n1hi/C13XN n1/C300n1P(n1) /C30XN n1/C300N! n1!(N/C28n1)!pn1qN/C28n1n1; (5) but n1pn1/C30p@ @ppn1; (6) so n1hi/C30XN n1/C300N! n1!(N/C28n1)!p@ @ppn1 ! qN/C28n1 /C30p@ @pXN n1/C300N! n1!(N/C28n1)!pn1qN/C28n1 /C30p@ @p(p/C27q)N/C30pN(p/C27q)N/C281/C30pN: (7) From the BINOMIAL THEOREM , n2hi/C30N/C28n1hi/C30N(1/C28p)/C30qN: (8) The VARIANCE is given by s2 n1/C30n21})@0})@@ /C28n1hi2: (9) But n21})@0})@@ /C30XN n1/C300N! n1!(N/C28n1)!pn1qN/C28n1n21; (10) so n21pn1/C30n1p@ @p ! pn1/C30p@ @p !2 pn1; (11) and n21})@0})@@ /C30XN n1/C300N! n1!(N/C28n1)!p@ @p !2 pn1qN/C28n1 /C30p@ @p !2XN n1/C300N! n1!(N/C28n1)!pn1qN/C28n1 /C30p@ @p !2 (p/C27q)N /C30p@ @p[pN(p/C27q)N/C281] /C30p[N(p/C27q)N/C281/C27pN(N/C281)(p/C27q)N/C282] /C30p[N/C27pN(N/C281)] /C30pN[1/C27pN/C28p)]/C30(Np)2/C27Npq /C30n1hi2/C27Npq: (12) Therefore, s2n 1/C30n21})@0})@@ /C28n1hi2/C30Npq; (13) and the ROOT-MEAN-SQUARE deviation is sn1/C30ffiffiffiffiffiffiffiffiffiffi Npqp : (14) For a large number of total steps N, the BINOMIAL DISTRIBUTION characterizing the distribution ap- proaches a G AUSSIAN DISTRIBUTION . Consider now the distribution of the distances dN traveled after a given number of steps,dN/C13n1/C28n2/C302n1/C28N; (15) as opposed to the number of steps in a given direction. The above plots show dN(p) for N/C30200 and three values p/C300:1;p/C300:5;and p/C300:9;respectively. Clearly, weighting the steps toward one direction or the other influences the overall trend, but there is still a great deal of random scatter, as emphasized by the plot below, which shows three random walks allwith p/C300:5: / Surprisingly, the most probable number of signchanges in a walk is 0, followed by 1, then 2, etc. For a random walk with p/C301=2;the probability P N(d) of traveling a given distance dafter Nsteps is given in the following table. steps /C285/C284/C283/C282/C281012345 01 1 /1 2/0 /12/ 2 /1 4/0 /24/0 /14/ 3 /18/0 /38/0 /38/0 /18/ 4 /1 16/0 /4 16/0 /6 16/0 /4 16/0 /1 16/ 5 /1 32/0 /5 32/0 /10 32/0 /1032/0 /5 32/0 /1 32/ In this table, subsequent rows are found by adding HALF of each cell in a given row to each of the two cells diagonally below it. In fact, it is simply P ASCAL’S TRIANGLE padded with intervening zeros and with each row multiplied by an additional factor of 1/2. The COEFFICIENTS in this triangle are given by PN(d)/C301 2NN d/C27N 20 @1A (16) (Papoulis 1984, p. 291). The moments m p/C30X d/C30/C28N;/C28(N/C282);...;NdpPN(d) (17) of this distribution of signed distances are then given by m/C300 (18) m2/C30N (19) m3/C300 (20) m4/C30N(3N/C282); (21) so the MEAN ism/C300;the SKEWNESS isg1/C300;and the KURTOSIS is g2/C30m4 m2 2/C283/C30/C282 N: (22) The expectation value of the absolute distance after Nsteps is therefore given by dNhi/C30XN d/C30/C28N;/C28(N/C282);...½d½PN(d) /C301 2NXN d/C30/C28N;/C28(N/C282);...½d½N! N/C27d 2 ! !N/C28d 2 ! !:(23) This sum can be done symbolically by separatelyconsidering the cases N EVEN and NODD. First, consider EVEN Nso that N/C132J:Then d2;J})@0})@@ /C30N! 2N})10 X/C282 d/C30/C282J; /C282(J/C281);...½d½ 2J/C27d 2 ! !2J/C28d 2 ! ! /C27X d/C300½d½ 2J/C27d 2 ! !2J/C28d 2 ! ! /C27X2J d/C302;4;...½d½ 2J/C27d 2 ! !2J/C28d 2 ! !})1@ /C30N! 2N})10 X/C281 d/C30/C28J;/C28(J/C281);...½2d½ 2J/C272d 2 ! !2J/C282d 2 ! ! /C27XJ d/C301;2 ...½2d½ 2J/C272d 2 ! !2J/C282d 2 ! !})1@ /C30N! 2N2XJ d/C3012d (J/C27d)!(J/C28d)!"# /C30N! 2N/C282XJ d/C301d (J/C27d)!(J/C28d)!: (24) But this sum can be evaluated analytically as XJ d/C301d (J/C27d)!(J/C28d)!/C301 2G(J)G(1/C27J): (25)Writing J/C30N=2;plugging back in, and simplifying gives dNeven hi /C302ffiffiffippG1 2/C2712N})@D})@E G1 2N})@D})@E /C30(N/C281)!! (N/C282)!!; (26) where N!! is the DOUBLE FACTORIAL . Now consider NODD,s oN/C132J/C281:Then dNodd hi /C30d2J/C281 hi /C30N! 2N})10 X/C281 d/C30/C28(2J/C281); /C28(2J/C271);...½d½ 2J/C281/C27d 2 ! !2J/C281/C28d 2 ! ! /C27X2J/C281 d/C301;3;...½d½ 2J/C281/C27d 2 ! !2J/C281/C28d 2 ! !/C138 /C30N! 2N/C281X2J/C281 d/C301;3;...d 2J/C281/C27d 2 ! !2J/C281/C28d 2 ! !2 666643 77775 /C30N! 2N/C281X2J d/C302;4;...d/C281 2J/C282/C27d 2 ! !2J/C28d 2 ! !2 666643 77775 /C30N! 2N/C281XJ d/C3012d/C281 (J/C27d/C281)!(J/C28d)!"# : (27) But this sum can be evaluated analytically as XJ d/C3012d/C281 (J/C27d/C281)!(J/C28d)!/C301 [G(J)]2: (28) Writing J/C30(N/C271)=2;plugging back in, and simpli- fying gives dNodd hi /C30N! 2N/C281G1 2/C2712N})@D})@Ehi2 /C302ffiffiffippG1 2N/C271})@D})@E G1 2N/C2712})@D})@E /C30N!! (N/C281)!: (29) Both the EVEN and ODD solutions can be written in terms of Jas dJhi/C302ffiffiffippGJ/C271 2})@D})@E G(J)/C30(2J/C281)!! (2J/C282)!!; (30) or explicitly in terms of Nas dNhi/C30(N /C28 1)!! (N /C28 2)!!for N even N!! (N /C28 1)!!for N odd:8 >>>< >>>:(31) The first few values of d Nhi are therefore d0hi/C300 d1hi/C30 d2hi/C301 d3hi/C30 d4hi/C303 2 d5hi/C30 d6hi/C3015 8 d7hi/C30 d8hi/C3035 16 d9hi/C30 d10hi/C30315128 d11hi/C30 d12hi/C30693 256 d13hi/C30 d14hi/C3030031024 (Sloane’s A001803 and A046161; Abramowitz and Stegun 1972, Pre´vost 1933, Hughes 1995), which are also given by the GENERATING FUNCTION (1 /C28x)/C283 =2 /C301 /C273 2 x /C2715 8x2 /C273516 x3 /C27315128 x4 /C27...: (32) These numbers also arise in the HEADS-MINUS-TAILS DISTRIBUTION . Now, examine the asymptotic behavior of dNhi: The asymptotic expansion of the GAMMA FUNCTION ratio is G J /C271 2})@D})@E G(J)/C30ffiffiffiffi Jp 1 /C281 8J /C271 128J2 /C27... ! (33) (Graham et al. 1994), so plugging in the expression for dNhi gives the asymptotic series dNhi/C30ffiffiffiffiffiffiffi 2N ps /C2 1 /C141 4N /C271 32N2 95 128N3 /C2821 2048 N4 /C14... ! ; (34) where the top signs are taken for N EVEN and the bottom signs for N ODD. Therefore, for large N, dNhi/C2ffiffiffiffiffiffiffi 2N ps ; (35) which is also shown in Mosteller et al. (1961, p. 14). To´th (2000) has proven that there are no more than three most-visited sites in a simple symmetric ran- dom walk in 1-D with unit steps. See also BINOMIAL DISTRIBUTION ,CATALAN NUMBER , HEADS- MINUS- TAILS DISTRIBUTION , P-GOOD PATH,PO´ LYA’S RANDOM WALK CONSTANTS ,RANDOM WALK– 2-D, RANDOM WALK–3- D, SELF-AVOIDING WALK,W I- ENER PROCESS References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 798, 1972. Chandrasekhar, S. "Stochastic Problems in Physics and Astronomy." Rev. Modern Phys. 15,1/C1/89, 1943. Reprinted inNoise and Stochastic Processes (Ed. N. Wax). New York: Dover, pp. 3 /C1/91, 1954. Erdos, P. and Re ´ve´sz, P. "On the Favourite Points of Random Walks." Math. Structures--Comput. Math.-- Math. Model. (Sofia) 2, 152/C1/157, 1984. Erdos, P. and Re ´ve´sz, P. "Problems and Results on Random Walks." In Mathematical Statistics and Probability The- ory, Vol. B: Statistical Inference and Methods. Proceedingsof the Sixth Pannonian Symposium on MathematicalStatistics Held in Bad Tatzmannsdorf, September 14 /C1 / 20, 1986 (Ed. P. Bauer, F. Koneczny, and W. Wertz). Dordrecht, Netherlands: Reidel, pp. 59 /C1/65, 1987. Feller, W. Ch. 3 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., rev. printing. New York: Wiley, 1968. Gardner, M. "Random Walks and Gambling." Ch. 6 in Mathematical Circus: More Puzzles, Games, Paradoxes,and Other Mathematical Entertainments. Washington, DC: Math. Assoc. Amer., pp. 66 /C1 /74, 1992. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Answer to problem 9.60 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Hersh, R. and Griego, R. J. "Brownian Motion and Potential Theory." Sci. Amer. 220,6 7/C1/74, 1969. Hughes, B. D. Eq. (7.282) in Random Walks and Random Environments, Vol. 1: Random Walks. New York: Oxford University Press, p. 513, 1995. Kac, M. "Random Walk and the Theory of Brownian Motion." Amer. Math. Monthly 54, 369/C1/391, 1947. Rep- rinted in Noise and Stochastic Processes (Ed. N. Wax). New York: Dover, pp. 295 /C1/317, 1954. Mosteller, F.; Rourke, R. E. K.; and Thomas, G. B. Prob- ability and Statistics. Reading, MA: Addison-Wesley, 1961. Papoulis, A. "Random Walk." Probability, Random Vari- ables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 290 /C1/291, 1984. Pre´vost, G. Tables de Fonctions Sphe ´riques. Paris: Gau- thier-Villars, pp. 156 /C1/157, 1933. Re´ve´sz, P. Random Walk in Random and Non-Random Environment. Singapore: World Scientific, 1990. Sloane, N. J. A. Sequences A001803/M2986 and A046161 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se-quences/eisonline.html. To´th, B. No More than Three Favourite Sites for Simple Random Walk. 26 Apr 2000. http://xxx.lanl.gov/abs/math.PR/0004164/. To´th, B. and Werner, W. "Tied Favourite Edges for Simple Random Walk." Combin., Prob., Comput. 6, 359/C1 /369, 1997. Random Walk * /2-D In a PLANE , consider a sum of N 2-D VECTORS with random orientations. Use PHASOR notation, and let the phase of each VECTOR be RANDOM . Assume N unit steps are taken in an arbitrary direction (i.e., with the angle u uniformly distributed in [0; 2 p) and not on a LATTICE ), as illustrated above. The position z in the COMPLEX PLANE after N steps is then given by z /C30XN j/C301eiuj ; (1) which has ABSOLUTE SQUARE ½z½2 /C30XN j/C301eiujXN k /C301e /C28iuk /C30XN j/C301XN k /C301ei( uj/C28uk) /C30N /C27XN j; k /C301 k "jei(uj/C28uk) : (2) Therefore, zjj2DE /C30N /C27XN j; k /C301 k"jei(uj/C28uk)*+ : (3) Each step is equally likely to be in any direction, so both ujand ukare RANDOM VARIABLES with identical MEANS of zero, and their difference is also a random variable. Averaging over this distribution, which has equally likely POSITIVE and NEGATIVE values yields an expectation value of 0, so zjj2DE /C30N : (4) The root-mean-square distance after N unit steps is therefore zjjrms/C30ffiffiffiffiffi Np ; (5)so with a step size of l, this becomes drms /C30lffiffiffiffiffiNp : (6) In order to travel a distance d N :d l !2 (7) steps are therefore required. Amazingly, it has been proven that on a 2-D LATTICE , a random walk has unity probability of reaching any point (including the starting point) as the number ofsteps approaches INFINITY . See also PO´ LYA’S RANDOM WALK CONSTANTS ,RANDOM WALK–1- D, RANDOM WALK–3- D References McCrea, W. H. and Whipple, F. J. W. "Random Paths in Two and Three Dimensions." Proc. Roy. Soc. Edinburgh 60, 281/C1/298, 1940. Random Walk * /3-D On a 3-D LATTICE , a random walk has less than unity probability of reaching any point (including thestarting point) as the number of steps approaches infinity. The probability of reaching the starting point again is 0.3405373296.... This is one of PO´ LYA’S RANDOM WALK CONSTANTS . See also PO´ LYA’S RANDOM WALK CONSTANTS ,RANDOM WALK–1- D, RANDOM WALK–2- D References Glasser, M. L. and Zucker, I. J. "Extended Watson Integrals for the Cubic Lattices." Proc. Nat. Acad. Sci. U.S.A. 74, 1800 /C1/1801, 1977. McCrea, W. H. and Whipple, F. J. W. "Random Paths in Two and Three Dimensions." Proc. Roy. Soc. Edinburgh 60, 281 /C1/298, 1940. Random Young Tableau RANDOM TABLEAU Range (Image) If T is a MAP (a.k.a., FUNCTION , TRANSFORMATION ) over a DOMAIN D, then the range of T is defined as Range( T) /C30T(D) /C30fT(X):X /C23 Dg: The range T(D) is also called the IMAGE of D under T. See also DOMAIN ,MAP,TRANSFORMATION Range (Line Segment) A number of points on a LINE SEGMENT . The term was first used by Desargues (Cremona 1960, p. x). If the points A, B, C, ... lie on a LINE SEGMENT with the coordinates of the points such that A BB BC ; they are said to form a range, denoted fABC ...g: Let AB denote the signed distance B /C28A: Then the range fABC g satisfies the relation AB /C27BC /C27CA /C300 : The range fABCD g satisfies BC /C215 AD /C27CA /C215 BD /C27AB /C215 CD /C300 and BC /C215 AD2 /C27CA /C215 BD2 /C27AB /C215 CD2 /C27BC /C215 CA /C215 AB /C300 ; the latter of which holds even when D is not on the line ABC (Lachlan 1893). Graustein (1930) and Woods (1961) use the term "range" to refer to the totality of points on a straight LINE, making it the dual of a PENCIL . See also AXIS,HOMOGRAPHIC ,LINE,LINE SEGMENT , PENCIL ,PERSPECTIVITY ,SECTION (PENCIL ) References Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960.Durell, C. V. "Concurrency and Collinearity." Ch. 4 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 37 /C1/39, 1928. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 40, 1930. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 14 /C1/15, 1893. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 8, 1961. Range (Statistics) R/C13max( xi)/C28min( xi): (1) For small samples, the range is a good estimator of the population STANDARD DEVIATION (Kenney and Keeping 1962, pp. 213 /C1/214). For a continuous UNI- FORM DISTRIBUTION P(x)/C301 Cfor 0BxBC 0 for xjjBC;8 < :(2) the distribution of the range is given by D(R)/C30NR C !N/C281 /C28(N/C281)R C !N : (3) Given two samples with sizes mandnand ranges R1 andR2;letu/C13R1=R2:Then D(u)/C30m(m/C281)n(n/C281) (m/C27n)(m/C27n/C281)(m/C27n/C282) /C29(m/C27n)um/C282/C28(m/C27n/C282)um/C281½/C138 for 05u51 m(m/C281)n(n/C281) (m/C27n)(m/C27n/C281)(m/C27n/C282) /C29(m/C27n)u/C28n/C28(m/C27n/C282)u/C28n/C281½/C138 for 15u5/C12:8 >>>>>>>>>>>< >>>>>>>>>>>:(4) The MEAN is mu/C30(m/C281)n (m/C271)(n/C282); (5) and the MODE is ˆu/C30(m/C282)(m/C27n) (m/C281)(m/C27n/C282)form/C28n52 (n/C271)(m/C27n/C282) n(m/C27n)form/C28n]2:8 >>>< >>>:(6) References Kenney, J. F. and Keeping, E. S. "The Range." §6.2 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 75 /C1/76, 213 /C1/214, 1962. Rank The word "rank" refers to several unrelated concepts in mathematics involving groups, matrices, quadratic forms, sequences, set theory, statistics, and tensors. In SET THEORY , rank is a (class) function from SETS to ORDINAL NUMBERS . The rank of a SET is the least ORDINAL NUMBER greater than the rank of any member of the set (Mirimanoff 1917; Moore 1982, pp. 261 /C1/262; Rubin 1967, p. 214). The proof that rank is WELL DEFINED uses the AXIOM OF FOUNDA- TION . For example, the EMPTY SET fg has rank 0 (since it has no members and 0 is the least ORDINAL NUMBER ), fgfg has rank 1 (since fg; its only member, has rank 0), fgfgfg has rank 2, and ffg;fgfg;fgfgfg ; ...g has rank v: Every ORDINAL NUMBER has itself as its rank. Mirimanoff (1917) showed that, assuming the class of URELEMENTS is a set, for any ORDINAL NUMBER a; the class of all sets having rank a is a SET, i.e., not a PROPER CLASS (Rubin 1967, p. 216) The number of sets having rank k for k /C300, 1, ... are 1, 1, 2, 12, 65520, ... (Sloane’s A038081), and the number of sets having rank at most k is 22 U 2 |fflffl{zfflffl} k; 1, 2, 4, 16, 65536, ... (Sloane’s A014221). The rank of a mathematical object is defined when- ever that object is FREE . In general, the rank of a FREE object is the CARDINALITY of the FREE generating SUBSET G. See also ORDINAL NUMBER ,RANK (BUNDLE ), RANK (GROUP ), RANK (LIE ALGEBRA ), RANK (MATRIX ), RANK (QUADRATIC FORM), RANK (SEQUENCE ), RANK (STA- TISTICS ), RANK (TENSOR ) References Mirimanoff, D. "Les antinomies de Russell et de Burali-Forti et le proble `me fondamental de la the´orie des ensembles." Enseign. math. 19,37/C1/52, 1917. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Sloane, N. J. A. Sequences A014221 and A038081 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Rank (Bundle) The rank of a VECTOR BUNDLE is the DIMENSION of its FIBER . Equivalently, it is the maximum number of linearly independent LOCAL SECTIONS in a TRIVIALIZA- TION . Naturally, the dimension here is measured in the appropriate CATEGORY . For instance, a real line bundle has fibers isomorphic with R; and a complex line bundle has fibers isomorphic to C ; but in both cases their rank is 1:/ The rank of the TANGENT BUNDLE of a real MANIFOLD M is equal to the dimension of M. The rank of atrivial bundle M /C29Rk is equal to k. There is no upper bound to the rank of a vector bundle over a fixed manifold M. See also DIMENSION ,F IBER,M ANIFOLD ,S ECTION (BUNDLE ), TANGENT BUNDLE ,VECTOR BUNDLE Rank (Group) For an arbitrary finitely generated ABELIAN GROUP G, the rank of G is defined to be the rank of the FREE generating SUBSET G modulo its TORSION SUBGROUP . For a finitely generated GROUP , the rank is defined to be the rank of its "Abelianization." See also ABELIAN GROUP ,BETTI NUMBER ,BURNSIDE PROBLEM ,Q UASITHIN THEOREM ,Q UASI- UNIPOTENT GROUP ,TORSION (GROUP ) Rank (Matrix) The rank of a MATRIX or a linear map is the DIMENSION of the range of the matrix or the linear map, corresponding to the number of LINEARLY INDEPENDENT rows or columns of the matrix, or to the number of nonzero singular values of the map. Rank (Quadratic Form) For a QUADRATIC FORM Q in the canonical form Q /C30y2 1 /C27y22 /C27.../C27y2p /C28y2p /C271 /C28y2p /C272 /C28.../C28y2r ; the rank is the total number r of square terms (both POSITIVE and NEGATIVE ). See also SIGNATURE (QUADRATIC FORM) References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1105, 2000. Rank (Sequence) The position of a RATIONAL NUMBER in the SEQUENCE 1 1 ;12;21 ;13 ;31;14 ;23 ;32;41 ;15 ; ..., ordered in terms of increasing NUMERATOR /C27DENOMINATOR . See also ENCODING ,FAREY SERIES Rank (Statistics) The ORDINAL NUMBER of a value in a list arranged in a specified order (usually decreasing). See also RANK TEST,SPEARMAN RANK CORRELATION COEFFICIENT ,WILCOXON RANK SUM TEST,WILCOXON SIGNED RANK TEST,ZIPF’S LAW Rank (Tensor) The total number of CONTRAVARIANT and COVARIANT indices of a TENSOR . The rank of a TENSOR is independent of the number of DIMENSIONS of the SPACE . Rank Object 0 SCALAR 1 VECTOR /]2/ TENSOR See also CONTRAVARIANT TENSOR ,COVARIANT TEN- SOR,SCALAR ,TENSOR ,VECTOR Rank Test A STATISTICAL TEST making use of the RANKS of data points. Examples include the KOLMOGOROV- SMIRNOV TEST and WILCOXON SIGNED RANK TEST . See also KOLMOGOROV- SMIRNOV TEST, R-ESTIMATE , RANK (STATISTICS ), SPEARMAN RANK CORRELATION COEFFICIENT ,STATISTICAL TEST,W ILCOXON SIGNED RANK TEST Ranunculoid An EPICYCLOID with n /C305 cusps, named after the buttercup genus Ranunculus (Madachy 1979). See also CARDIOID ,EPICYCLOID ,NEPHROID References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 223, 1979. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 79 /C1/ 80, 1995. Rapid Rumor Ramification GOSSIPING RAT-Free Set A RAT-free ("right angle triangle-free") set is a set of points, no three of which determine a RIGHT TRIAN- GLE. Let f(n) be the largest integer such that a RAT- free subset of size f(n) is guaranteed to be contained in any set of n coplanar points. Then the function f(n) is bounded byffiffiffinp5f(n) 52ffiffiffiffiffin:p See also R IGHT TRIANGLE References Abbott, H. L. "On a Conjecture of Erdos and Silverman in Combinatorial Geometry." J. Combin. Th. A 29, 380 /C1/381, 1980. Chan, W. K. "On the Largest RAT-FREE Subset of a Finite Set of Points." Pi Mu Epsilon 8, 357 /C1/367, 1987. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 250 /C1/251, 1991. Seidenberg, A. "A Simple Proof of a Theorem of Erdos and Szekeres." J. London Math. Soc. 34, 352, 1959. Ratio The ratio of two numbers r and s is written r=s ; where r is the NUMERATOR and s is the DENOMINATOR . The ratio of r to s is equivalent to the QUOTIENT r=s: Betting ODDS written as r : s correspond to s =(r /C27s): A number which can be expressed as a ratio of INTEGERS is called a RATIONAL NUMBER . See also DENOMINATOR ,D IVISION ,F RACTION ,N U- MERATOR ,ODDS,QUOTIENT ,RATIONAL NUMBER Ratio Distribution Given two distributions Yand Xwith joint prob- ability density function f(x;y);letU/C30Y=Xbe the ratio distribution. Then the distribution function of u is D(u)/C30P(U5u) /C30P(Y5uX X >0)/C27P(Y]uX XB0) j j /C30g/C12 0gux 0f(x;y)dy dx/C27g0 /C28/C12g0 uxf(x;y)dy dx : ð1Þ The probability function is then P(u)/C30D?(u)/C30g/C12 0xf(x;ux)dx/C28g0 /C28/C12xf(x;ux)dx /C30g/C12 /C28/C12xjjf(x;ux)dx: (2) For variates with a standard NORMAL DISTRIBUTION , the ratio distribution is a C AUCHY DISTRIBUTION . For aUNIFORM DISTRIBUTION f(x;y)/C301 for x;y/C230;1½/C138 0 otherwise ;})1D (3) P(u) /C300 u B0 g1 0xdx/C301 2 x2hi /C3012 for 0 5u 51 g1 =u 0xdx/C3012 x2hi1 =u 0/C301 2u2for u > 1 :8 >>>>>< >>>>>:(4) See also C AUCHY DISTRIBUTION Ratio Test Let uk be a SERIES with POSITIVE terms and suppose r /C13lim k 0/C12uk /C271 uk: Then 1. If r B1 ; the SERIES CONVERGES . 2. If r > 1or r /C30/C12; the SERIES DIVERGES . 3. If r /C301 ; the SERIES may CONVERGE or DIVERGE . The test is also called the CAUCHY RATIO TEST or D’ALEMBERT RATIO TEST . See also CONVERGENCE TESTS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 282 /C1/283, 1985. Bromwich, T. J. I’a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 28, 1991. Rational Approximation If a is any number and m and n are INTEGERS , then there is a RATIONAL NUMBER m=n for which a/C28m n})@1})@1})@1})@1})@1})@1})@1})@1})@1})@15 1 n : (1) If a is IRRATIONAL and k is any WHOLE NUMBER , there is a FRACTION m=n with n 5k and for which a/C28m n})@1})@1})@1})@1})@1})@1})@1})@1})@1})@15 1 nk : (2) Furthermore, there are an infinite number of FRAC- TIONS m=n for which a/C28m n})@1})@1})@1})@1})@1})@1})@1})@1})@1})@15 1 n2 (3) (Hilbert and Cohn-Vossen 1999, pp. 40 /C1/44). Hurwitz has shown that for an IRRATIONAL NUMBER z z/C28h k})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1B 1 ck2 ; (4) there are infinitely RATIONAL NUMBERS h =k if 0 Bc 5ffiffiffi 5p ; but if c >ffiffiffi5p ; there are some z for which this approximation holds for only finitely many h=k : / See also DIRICHLET’S APPROXIMATION THEOREM , HURWITZ’S IRRATIONAL NUMBER THEOREM ,IRRATION- ALITY MEASURE ,KRONECKER’S APPROXIMATION THE- OREM ,L AGRANGE N UMBER (RATIONAL APPROXIMATION ), LIOUVILLE’S APPROXIMATION THEO- REM,M ARKOV NUMBER ,ROTH’S THEOREM ,SEGRE’S THEOREM ,THUE- SIEGEL- ROTH THEOREM References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 41, 1999. Rational Canonical Form Any SQUARE MATRIX Thas a canonical form without any need to EXTEND the FIELD of its coefficients. For instance, if the entries of Tare RATIONAL NUMBERS , then so are the entries of its rational canonical form. (The J ORDAN CANONICAL FORM may require complex numbers.) There exists an INVERTIBLE MATRIX Qsuch that Q/C281TQ/C30diag[ L(c1);L(c2);...;L(cs)]; (1) called the rational canonical form, where L(f) is the COMPANION MATRIX for the MONIC POLYNOMIAL f(l)/C30f0/C27f1l/C27.../C27fn/C281ln/C281/C27ln: (2) The POLYNOMIALS ciare called the "invariant factors" ofT;and satisfy cici/C271})@1})@1 fori/C301, ..., s/C281 (Hartwig 1996). The polynomial csis the MINIMAL POLYNOMIAL and the productQciis the CHARACTERISTIC POLY- NOMIAL ofT:/ The rational canonical form is unique, and shows the extent to which the minimal polynomial characterizes a matrix. For example, there is only one 6 /C296 matrix whose MINIMAL POLYNOMIAL is (x2/C271)2;which is 0/C281000 0 10 0 0 00 00 0 0 0 /C281 00 1 0 0000 0 1 0 /C282 00 0 0 102 66666643 7777775(3) in rational canonical form. Given a LINEAR TRANSFORMATION T:V0V;the VECTOR SPACE Vbecomes a F[x]/-MODULE , that is a MODULE over the RING of polynomials with coeffi- cients in the FIELD F. The VECTOR SPACE determines the field F, which can be taken to be the maximal field containing the entries of a matrix for T. The polynomial xacts on a vector vbyx(v)/C30T(v):The rational canonical form corresponds to writing Vas F[x]=(a1)/C154.../C156F[x]=(as); (4) where ( ai) is the IDEAL generated by the INVARIANT FACTOR ai in F[x] ; the canonical form for any finitely generated module over a PRINCIPAL IDEAL RING such as F[x] :/ More constructively, given a basis ei for V, there is a MODULE HOMOMORPHISM t : F[x]n 0 V (5) which is ONTO , given by tX pi(x)ei})@D})@E /C30X pi(T)ei : (6) Letting K be the KERNEL , V $F[x]n =K : (7) To construct a basis for the rational canonical form, it is necessary to write K as K $Mn/C28s i/C281F[x] /C154 F[x] =(a1) /C154.../C154 F[x](as); (8) and that is done by finding an appropriate basis for F[x]n and for K. Such a basis is found by determining matrices P and Q that are invertible n /C29n matrices having entries in F[x] (and whose inverses are also in F[x]) such that P xI /C28T ðÞ Q /C30diag(1 ; ...; 1 ; a1 ; ...; as); (9) where l is the IDENTITY MATRIX and (a1 ; ...; an) denotes a DIAGONAL MATRIX . They can be found by using ELEMENTARY MATRIX OPERATIONS . The above matrix sends a basis for K, written as an n-tuple, to an n-tuple using a new basis fifor F[x]n ; and P gives the linear transformation from the original basis to the one with the fi : In particular, K /C30 b1f1 /C27...bn/C28sfn/C28s /C27 bn/C28s/C271a1fn/C28s/C271 /C27.../C27 bnasfn})*})+ ; (10) where biis an arbitrary polynomial in F[x] : Setting zi /C30P /C281(T)en /C28s/C27i ; V /C30F[x]z1 /C154.../C154 F[x]zs : (11) In particular, F[x]ziis the SUBSPACE of V which is generated by zi ; xzi ; ... ; xn/C281zi ; where n is the degree of ai : Therefore, a basis that puts T into rational canonical form is given by fz1 ;Tz1 ;...;Tn1 z1 ;z2 ;...; Tn2 x2 ; ... ;Tns zs g: ð12Þ See also BLOCK DIAGONAL MATRIX ,CHARACTERISTIC POLYNOMIAL ,COMPANION MATRIX ,FIELD,INVARIANT FACTOR ,JORDAN CANONICAL FORM,MATRIX ,MINIMAL POLYNOMIAL (MATRIX ), PRINCIPAL IDEAL RING (PID), REDUCTION ALGORITHM , S IMILAR MATRICES , SMITH NORMAL FORMReferences Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 203, 1962. Dummit, D. and Foote, R. Abstract Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991. Gantmacher, F. R. The Theory of Matrices, Vol. 1. New York: Chelsea, 1960. Hartwig, R. E. "Roth’s Removal Rule and the Rational Canonical Form." Amer. Math. Monthly 103, 332 /C1/335, 1996. Herstein, I. N. Topics in Algebra, 2nd ed. New York: Springer-Verlag, p. 162, 1975. Hoffman, K. and Kunze, K. Linear Algebra, 3rd ed. Engle- wood Cliffs, NJ: Prentice-Hall, 1996. Jacobson, N. §3.10 in Basic Algebra I. New York: W. H. Freeman, 1985. Lancaster, P. and Tismenetsky, M. The Theory of Matrices, 2nd ed. New York: Academic Press, 1985. Turnbull, H. W. and Aitken, A. C. An Introduction to the Theory of Canonical Matrices, 2nd impression. New York: Blackie and Sons, 1945. Rational Cuboid EULER BRICK Rational Diagonal NSW NUMBER Rational Distances It is possible to find six points in the PLANE , no three on a LINE and no four on a CIRCLE (i.e., none of which are COLLINEAR or CONCYCLIC ), such that all the mutual distances are RATIONAL . An example is illu- strated by Guy (1994, p. 185). It is not known if a TRIANGLE with INTEGER sides, MEDIANS , and AREA exists (although there are incor- rect PROOFS of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have showed that there are infinitely many triangles with RATIONAL sides (HERONIAN TRIANGLES ) with two RATIONAL MEDIANS (Guy 1994, p. 188). See also COLLINEAR ,CONCYCLIC ,CYCLIC QUADRILAT- ERAL ,EQUILATERAL TRIANGLE ,EULER BRICK,HERO- NIAN TRIANGLE ,R ATIONAL QUADRILATERAL , RATIONAL TRIANGLE ,SQUARE ,TRIANGLE References Guy, R. K. "Six General Points at Rational Distances" and "Triangles with Integer Sides, Medians, and Area." §D20 and D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 185 /C1/190, 1994. Rational Domain FIELD Rational Double Point There are nine possible types of ISOLATED SINGULA- RITIES on a CUBIC SURFACE , eight of them rational double points. Each type of ISOLATED SINGULARITY has an associated normal form and COXETER- DYNKIN DIAGRAM (/A1 ; A2 ; A3 ; A4 ; A5 ; D4 ; D5 ; E6 and ˘E6) :/ The eight types of rational double points (the ˘E6 type being the one excluded) can occur in only 20 combina- tions on a CUBIC SURFACE (of which Fischer 1986 gives 19): A1 ; 2A1 ; 3A1 ; 4A1 ; A2 ; A2 ; A1 ðÞ ; 2A2 ; 2A2 ; A1 ðÞ ; 3A2 ; A3 ; A3 ; A1 ðÞ ; A3 ; 2A1 ðÞ ; A4 ; A4 ; A1 ðÞ ; A5 ; A5 ; A1 ðÞ ; D4 ; D5 ; and E6(Looijenga 1978, Bruce and Wall 1979, Fischer 1986). In particular, on a CUBIC SURFACE , precisely those configurations of rational double points occur for which the disjoint union of the COXETER- DYNKIN DIAGRAM is a SUBGRAPH of the COXETER- DYNKIN DIAGRAM ˘E6 : Also, a surface specializes to a more complicated one precisely when its graph is contained in the graph of the other one (Fischer 1986). See also COXETER- DYNKIN DIAGRAM ,CUBIC SURFACE , DOUBLE POINT ,ISOLATED SINGULARITY ,O RDINARY DOUBLE POINT References Bruce, J. and Wall, C. T. C. "On the Classification of Cubic Surfaces." J. London Math. Soc. 19, 245 /C1/256, 1979. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 13, 1986. Fischer, G. (Ed.). Plates 14 /C1/31 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 17 /C1/31, 1986. Looijenga, E. "On the Semi-Universal Deformation of a Simple Elliptic Hypersurface Singularity. Part II: The Discriminant." Topology 17,23/C1/40, 1978. Rodenberg, C. "Modelle von Fla¨chen dritter Ordnung." In Mathematische Abhandlungen aus dem Verlage Mathe- matischer Modelle von Martin Schilling. Halle a. S., 1904. Rational Function A QUOTIENT of two polynomials P(z) and Q(z) ; R(z) /C13P(z) Q(z) ; is called a rational function. More generally, if P and Q are POLYNOMIALS in multiple variables, their quotient is called a (multivariate) rational function. A rational function has no singularities other than poles in the EXTENDED COMPLEX PLANE . Conversely, if a single-values function has no singularities other than poles in the EXTENDED COMPLEX PLANE , than it is a rational function (Knopp 1996, p. 137). In addi- tion, a rational function can be decomposed into partial fractions (Knopp 1996, p. 139). See also ABEL’S CURVE THEOREM ,C LOSED FORM, FUNDAMENTAL THEOREM OF SYMMETRIC FUNCTIONS , INSIDE- OUTSIDE THEOREM ,Q UOTIENT- DIFFERENCE ALGORITHM ,RATIONAL INTEGER ,RATIONAL NUMBER , RIEMANN CURVE THEOREMReferences Knopp, K. "Rational Functions." §35 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 96 and 137 /C1/139, 1996. Rational Integer A synonym for INTEGER . The word "rational" is sometimes used for emphasis to distinguish it from other types of "integers" such as CYCLOTOMIC INTE- GERS ,EISENSTEIN INTEGERS ,GAUSSIAN INTEGERS , and HAMILTONIAN INTEGERS . See also CYCLOTOMIC INTEGER ,EISENSTEIN INTEGER , GAUSSIAN INTEGER ,HAMILTONIAN INTEGER ,INTEGER , RATIONAL NUMBER References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 1, 1979. Rational Number A number that can be expressed as a FRACTION p =q where p and q are INTEGERS and q "0; is called a rational number with NUMERATOR p and DENOMINA- TOR q. Numbers which are not rational are called IRRATIONAL NUMBERS . The FIELD of rational numbers is denoted Q. Any rational number is trivially also an ALGEBRAIC NUMBER . The set of rational numbers is denotedRationals in Mathematica , and a number x can be tested to see if it is rational using the command Element[ x, Rationals]. Between any two members of the set of rationals, it is always possible to find another rational number. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable. For a, b, and c any different rational numbers, then 1 (a /C28 b)2 /C271 (b /C28 c)2 /C271 (c /C28 a)2 is the SQUARE of a rational number (Honsberger 1991).The probability that a random rational number has an EVEN DENOMINATOR is 1/3 (Salamin and Gosper 1972).It is conjectured that if there exists a REAL NUMBER x for which both 2x and 3x are integers, then x is rational. This result would follow from the FOUR EXPONENTIALS CONJECTURE (Finch). See also ALGEBRAIC INTEGER ,ALGEBRAIC NUMBER , ANOMALOUS CANCELLATION ,DENOMINATOR ,DIRICH- LET FUNCTION ,FAREY SEQUENCE ,FOUR EXPONEN- TIALS CONJECTURE ,FRACTION ,INTEGER ,IRRATIONAL NUMBER ,NUMERATOR ,Q,Q UOTIENT ,TRANSCENDEN- TAL NUMBER References Courant, R. and Robbins, H. "The Rational Numbers." §2.1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 52 /C1/58, 1996. Finch, S. "Powers of 3/2 Modulo One." http://www.mathsoft.- com/asolve/pwrs32/pwrs32.html. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 52 /C1/53, 1991. Salamin, E. and Gosper, R. W. Item 54 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM- 239, p. 18, Feb. 1972. Rational Point A K-rational point is a point (X, Y)onan ALGEBRAIC CURVE f(X ; Y) /C300; where X and Y are in a FIELD K. For example, rational point in the FIELD Q of ordinary rational numbers is a point (X, Y) satisfying the given equation such that both X and Y are rational numbers. The rational point may also be a POINT AT INFINITY . For example, take the ELLIPTIC CURVE Y2 /C30X3 /C27X /C2742 and homogenize it by introducing a third variable Z so that each term has degree 3 as follows: ZY2 /C30X3 /C27XZ2 /C2742Z3 : Now, find the points at infinity by setting Z /C300, obtaining 0 /C30X3 : Solving gives X /C300, Y equal to any value, and (by definition) Z /C300. Despite freedom in the choice of Y, there is only a single POINT AT INFINITY because the two triples (/X1 ; Y1 ; Z1) ; (/X2 ; Y2 ; Z2) are considered to be equivalent (or identified) only if one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a valid point. The triples (a, b, 1) correspond to the ordinary points (a, b), and the triples (a, b,0) correspond to the POINTS AT INFINITY , usually called the LINE AT INFINITY . The rational points on ELLIPTIC CURVES over the FINITE FIELD GF(q) are 5, 7, 9, 10, 13, 14, 16, ... (Sloane’s A005523). See also ELLIPTIC CURVE ,LINE AT INFINITY ,POINT AT INFINITY References Sloane, N. J. A. Sequences A005523/M3757 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Rational Quadrilateral A rational quadrilateral is a QUADRILATERAL for which the sides, DIAGONALS , and AREA are RATIONAL . The simplest case has sides a /C3052, b /C3025, c /C3039, and d /C3060, DIAGONALS of length p /C3063 and q /C3056, and AREA 1764. See also AREA,D IAGONAL (POLYGON ), RATIONAL TRIANGLE Rational Triangle A rational triangle is a TRIANGLE all of whose sides are RATIONAL NUMBERS and all of whose ANGLES are RATIONAL numbers of DEGREES . The only such trian- gle is the EQUILATERAL TRIANGLE (Conway and Guy 1996). See also EQUILATERAL TRIANGLE ,FERMAT’S RIGHT TRIANGLE THEOREM ,R ATIONAL QUADRILATERAL , RIGHT TRIANGLE References Conway, J. H. and Guy, R. K. "The Only Rational Triangle." In The Book of Numbers. New York: Springer-Verlag, pp. 201 and 228 /C1/239, 1996. Rationals RATIONAL NUMBER RATS Sequence A sequence produced by the instructions "reverse, add, then sort the digits," where zeros are suppressed. For example, after 668 we get 668 /C27866 /C301534 ; so the next term is 1345. Applied to 1, the sequence gives 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, ... (Sloane’s A004000) See also 196-ALGORITHM ,KAPREKAR ROUTINE ,REVER- SAL,SORT-THEN- ADD SEQUENCE References Sloane, N. J. A. Sequences A004000/M1137 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Raw Moment A MOMENT mnof a probability function P(x) taken about 0, m ?n /C30g xnP(x) dx: (1) The raw moments m?n can be expressed as terms of the CENTRAL MOMENTS mn(i.e., those taken about the MEAN m) using the inverse BINOMIAL TRANSFORM m?n /C30Xn k /C300n k})@*})@+ mk m?1n/C28k; (2) with m0 /C301 and m1 /C300 (Papoulis 1984, p. 146). The first few values are therefore m ?2 /C30 m2 /C27 m?12(3) m?3 /C30 m3 /C273m2 m?12/C27 m?14(4) m?4 /C30 m4 /C274m3 m?1 /C276m2 m?12/C27 m?14(5) m?5 /C30 m5 /C275 m4 m?1 /C2710m3 m?12/C2710 m2 m?13/C27 m ?15: (6) See also ABSOLUTE MOMENT ,C ENTRAL MOMENT , MEAN,MOMENT References Kenney, J. F. and Keeping, E. S. "Moments About the Origin." §7.2 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 91 /C1/92, 1962. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Ray A VECTOR AB})@A@!from a point A to a point B.In GEOMETRY , a ray is usually taken as a half-infinite LINE with one of the two points A and B taken to be at INFINITY . See also LINE,VECTOR Rayleigh Differential Equation y ƒ/C28 m 1 /C281 3 y ?2})@D})@E y?/C27y /C300; where m > 0 : Differentiating and setting y /C30y? gives the VAN DER POL EQUATION . The equation yƒ/C28 m 1 /C28y?2})0})@ y?/C27y /C300 with the 1=3 replaced by 1 is sometimes also calledthe Rayleigh differential equation (Birkhoff and Rota 1978, p. 134; Zwillinger 1997, p. 126). See also RAYLEIGH WAVE EQUATION , VAN DER POL EQUATION References Birkhoff, G. and Rota, G.-C. Ordinary Differential Equa- tions, 3rd ed. New York: Wiley, p. 134, 1978. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997. Rayleigh Distribution The distribution with PROBABILITY FUNCTION P(r)/C30re/C28r2=2s2 s2(1) for /r/C23½0;/C12Þ/. The MOMENTS about 0 are given by m?m/C13g/C12 0rmP(r)dr/C30s/C282g/C12 0rm/C271e/C28r2=2s2dr /C30s/C282Im/C2711 2s2 ! ; (2) where I(x)i saG AUSSIAN INTEGRAL (Papoulis 1984, p. 148). The first few of these are I1a/C281})0})@ /C3012a (3) I2a/C281})0})@ /C3014affiffiffiffiffiffiapp(4) I3a/C281})0})@ /C301 2a2(5) I4a/C281})0})@ /C3038a2ffiffiffiffiffiffiapp(6) I5a/C281})0})@ /C30a3; (7) so the RAW MOMENTS are m?0/C30s/C2821 22s2})0})@ /C301 (8) m?1/C30s/C2821 42s2})0})@ffiffiffiffiffiffiffiffiffiffi 2s2pp /C301 2sffiffiffiffiffiffi 2pp /C30sffiffiffi p 2s (9) m?2/C30s/C2821 22s2})0})@2/C302s2(10) m?3/C30s/C282382s2})0})@2ffiffiffiffiffiffiffiffiffiffi 2s2pp /C303 2s3ffiffiffiffiffiffi 2pp /C303s3ffiffiffi p 2s (11) m?4/C30s/C2822s2})0})@ /C308s4: (12) The CENTRAL MOMENTS are therefore m2 /C30 m ?2 /C28 m ?1ðÞ2/C304 /C28 p 2s2 (13) m3 /C30 m?3 /C283m ?2 m?1 /C272 m ?1ðÞ3/C30ffiffiffi p 2s p /C283 ðÞ s3 (14) m4 /C30 m?4 /C284m ?3 m?1 /C276m ?2m?1ðÞ2/C283 m /C281? ðÞ4 /C3032 /C28 3p2 4s4 ; (15) so the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C30 m?1 /C30sffiffiffi p 2s (16) s2 /C30 m2 /C304 /C28 p 2s2 (17) g1 /C30m3 s3/C302(p /C28 3)ffiffiffipp (4 /C28 p)3 =2 (18) g2 /C30m4 s4 /C283 /C30/C286p2 /C28 24 p /C27 16 ( p /C28 4)2 : (19) The CHARACTERISTIC FUNCTION is f(t) /C301 /C28ffiffiffi p 2s ste/C28s2t2 =2 erfistffiffiffi 2p ! /C28i"# : (20) See also MAXWELL DISTRIBUTION References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 104 and 148, 1984. Rayleigh Function The Rayleigh functions sn(n) for n /C301, 2, ..., are defined as sn( n) /C30X/C12 k /C301j/C282n nk; where 9jnk are the zeros of the BESSEL FUNCTION OF THE FIRST KIND Jn(z) (Watson 1966, p. 502; Gupta and Muldoon 1999). They were used by Euler, Rayleigh, and others to evaluate zeros of Bessel functions. There is a convolution formula connecting Rayleigh functions of different orders, sn(n) /C301 n /C27 nXn/C281 k /C301sk( n) sn/C28k( n) (Kishore 1963, Gupta and Muldoon 1999).See also BESSEL FUNCTION OF THE FIRST KIND References Gupta, D. P. and Muldoon, M. E. Riccati Equations and Convolution Formulas for Functions of Rayleigh Type. 24 Oct 1999. http://xxx.lanl.gov/abs/math.CA/9910128/. Ismail, M. E. H. and Muldoon, M. E. "Bounds for the Small Real and Purely Imaginary Zeros of Bessel and Related Functions." Meth. Appl. Anal. 2,1/C1/21, 1995. Kishore, N. "The Rayleigh Function." Proc. Amer. Math. Soc. 14, 527 /C1/533, 1963. Obi, E. C. "The Complete Monotonicity of the Rayleigh Function." J. Math. Anal. Appl. 77, 465 /C1/468, 1980. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Rayleigh Wave Equation The PARTIAL DIFFERENTIAL EQUATION utt /C28uxx /C30e ut /C28u3 t})0})@ : See also RAYLEIGH DIFFERENTIAL EQUATION References Hall, W. S. "The Rayleigh Wave Equation--An Analysis." Nonlinear Anal. 2, 129/C1/156, 1978. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 134, 1997. Rayleigh-Ritz Variational Technique A technique for computing EIGENFUNCTIONS and EIGENVALUES . It proceeds by requiring J/C30gb ap(x)y2x/C28q(x)y2})1})A dx (1) to have a STATIONARY VALUE subject to the normal- ization condition gb ay2w(x)dx/C301 (2) and the boundary conditions pyxyjb a¼0: ð3Þ This leads to the S TURM- LIOUVILLE EQUATION d dxpdy dx ! /C27qy/C27lwy/C300; (4) which gives the stationary values of Fy(x)½/C138/C30gb apy2 x/C28qy2ðÞ dx gb ay2wd x(5) as Fyn(x) ½/C138/C30ln; (6) where lnare the EIGENVALUES corresponding to the EIGENFUNCTION yn :/ References Arfken, G. "Rayleigh-Ritz Variational Technique." §17.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 957 /C1/961, 1985. Rayleigh, J. W. "In Finding the Correction for the Open End of an Organ-Pipe." Phil. Trans. 161, 77, 1870. Ritz, W. "U¨ ber eine neue Methode zur Lo¨sung gewisser Variationsprobleme der mathematischen Physik." J. reine angew. Math. 135,1/C1/61, 1908. Whittaker, E. T. and Robinson, G. "The Rayleigh-Ritz Method for Minimum Problems." §184 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 381 /C1/382, 1967. Rayleigh’s Formulas The formulas jn(z) /C30/C281 zd dz !nsin z z yn(z) /C30/C28zn /C281 zd dz !ncos z z for n /C300, 1, 2, ..., where jn(z)isa SPHERICAL BESSEL FUNCTION OF THE FIRST KIND and yn(z)isa SPHERICAL BESSEL FUNCTION OF THE SECOND KIND . References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 439, 1972. Rayleigh’s Theorem PARSEVAL’S THEOREM R-Bar The set of affine EXTENDED REAL NUMBERS . See also EXTENDED REAL NUMBER (AFFINE ) Re REAL PART Real Analysis That portion of mathematics dealing with functions of real variables. While this includes some portions of TOPOLOGY , it is most commonly used to distinguish that portion of CALCULUS dealing with real as opposed to COMPLEX NUMBERS . Real Analytic Function A REAL FUNCTION is said to be analytic if it possesses derivatives of all orders and agrees with its TAYLOR SERIES in the neighborhood of every point. See also ANALYTIC FUNCTIONReal Axis The axis in the COMPLEX PLANE corresponding to zero IMAGINARY PART , I[z] /C300:/ See also COMPLEX PLANE ,IMAGINARY AXIS,R EAL LINE Real Function A FUNCTION whose RANGE is in the REAL NUMBERS is said to be a real function, also called a real-valued function. See also COMPLEX FUNCTION ,S CALAR FUNCTION , VECTOR FUNCTION Real Line A LINE with a fixed scale so that every REAL NUMBER corresponds to a unique POINT on the LINE. The generalization of the real line to 2-D is called the COMPLEX PLANE . The term "real line" is also used to distinguish an ordinary LINE from a so-called IMAGINARY LINE which can arise in algebraic geometry. See also ABSCISSA ,COMPLEX PLANE ,IMAGINARY AXIS, IMAGINARY LINE,L INE,M OAT-CROSSING PROBLEM , REAL AXIS,REAL SPACE References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 57, 1996. Real Manifold See also COMPLEX MANIFOLD ,MANIFOLD Real Matrix A real matrix is a MATRIX whose elements consist entirely of REAL NUMBERS . The set of m/C29nreal matrices is sometimes denoted Rm/C29n (Zwillinger 1995, p. 116). For a real n /C29n matrix, the expected number of real EIGENVALUES is given by En /C30ffiffiffi 2pPn=2 /C281 k /C300(4k /C28 1)!! (4k)!!for n even 1 /C27ffiffiffi 2pP(n/C281)=2 k /C301(4k /C28 3)!! (4k /C28 2)!!for n odd8 >>>< >>>:(1) (Edelman et al. 1994, Edelman and Kostlan 1994), which has asymptotic behavior E n /C2ffiffiffiffiffiffi 2n ps : (2) GIRKO’S CIRCULAR LAW considers EIGENVALUES l (possibly complex) of a set of random n /C29n REAL MATRICES with entries independent and taken from a standard normal distribution. Then as n 0/C12; l =ffiffiffinp is uniformly distributed on the UNIT DISK in the COMPLEX PLANE . Edelman (1997) proved that the density of a random complex pair of eigenvalues x 9iy of a real n /C29n matrix whose elements are taken from a standard normal distribution is rn(x; y) /C30ffiffiffi 2 ps yey2/C28x2 erfcffiffiffi 2p y})@D})@E en/C282(x2 /C27y2) ¼ffiffiffi 2 ps e2y2 y erfc ðffiffiffi 2p yÞGðn /C28 1 ;x2 þ y2 Þ Gðn /C28 1Þð3Þ for y ]0; where erfc(z) is the ERFC (complementary error) function, en(z) is the EXPONENTIAL SUM FUNC- TION , and G(a ; x) is the upper INCOMPLETE GAMMA FUNCTION . Integrating over the UPPER HALF-PLANE gives half the expected number of complex eigenva- lues g/C12 /C28/C12g/C12 0rn(x; y) dy dx /C301 /C282n(1/C28n)=4 : (4) See also COMPLEX MATRIX ,GIRKO’S CIRCULAR LAW, INTEGER MATRIX ,MATRIX References Edelman, A. "The Probability that a Random Real Gaussian Matrix has k Real Eigenvalues, Related Distributions, and the Circular Law." J. Multivariate Anal. 60, 203 /C1/232, 1997.Edelman, A.; Kostlan, E.; and Shub, M. "How Many Eigenvalues of a Random Matrix are Real?" J. Amer. Math. Soc. 7, 247 /C1/267, 1994. Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32,1/C1/37, 1995. Girko, V. L. Theory of Random Determinants. Boston, MA: Kluwer, 1990. Lehmann, N. and Sommers, H.-J. "Eigenvalue Statistics of Random Real Matrices." Phys. Rev. Let. 67, 941 /C1/944, 1991. Mehta, M. L. Random Matrices, 2nd rev. enl. ed. New York: Academic Press, 1991. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995. Real Measure A MEASURE that takes on real values. See also MEASURE Real Normed Algebra A finite dimensional ALGEBRA A containing a copy of the reals is a real algebra. Note that this implies that A must be a real VECTOR SPACE . A real normed algebra is a real algebra A with a norm that is preserved by multiplication, i.e., ½a + b½/C30½a½½b½:/ For example, the REAL NUMBERS , the COMPLEX NUM- BERS , the QUATERNIONS , and the OCTONIONS are real normed algebras. Multiplication need not be commu- tative in a real normed algebra (e.g., QUATERNIONS and OCTONIONS are noncommutative), nor does it even need to be associative (e.g., the OCTONIONS ). A real normed algebra A satisfies a number of algebraic restrictions. For example, if the dimension of A is greater than 1, it must contain a copy of the complex numbers. Similarly, if the dimension is greater than 2, it must contain a copy of the QUATERNIONS . And if it is greater than 4, it must contain the OCTONIONS . In fact, these are the only examples, as the OCTONIONS cannot be "doubled" to make a normed algebra. See also ALGEBRA ,C OMPLEX NUMBER ,O CTONION , QUATERNION ,REAL NUMBER ,VECTOR SPACE Real Number The FIELD of all RATIONAL and IRRATIONAL numbers is called the real numbers, or simply the "reals," and denoted R:The set of real numbers is also called the CONTINUUM , denoted C. The set of reals is called Reals inMathematica , and a number xcan be tested to see if it is a member of the reals using thecommand Element[ x, Reals]. The real numbers can be extended with the addition of the IMAGINARY NUMBER I, equal toffiffiffiffiffiffi /C281p :Numbers OF THE FORM x/C27iy;where xandyare both real, are called COMPLEX NUMBERS , which also form a FIELD . Another extension which includes both the real numbers and the infinite ORDINAL NUMBERS of Georg Cantor is the SURREAL NUMBERS . Plouffe’s "Inverse Symbolic Calculator" includes a huge database of 54 million real numbers which are algebraically related to fundamental mathematical constants and functions. See also COMPLEX NUMBER ,CONTINUUM ,EXTENDED REAL NUMBER (AFFINE ), EXTENDED REAL NUMBER (PROJECTIVE ), I,IMAGINARY NUMBER ,INTEGER RELA- TION ,R ATIONAL NUMBER ,R EAL NUMBER PICKING , REAL PART,SURREAL NUMBER References Jeffreys, H. and Jeffreys, B. S. "Real Numbers." §1.03 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 5 /C1/6, 1988. Plouffe, S. "Inverse Symbolic Calculator." http:// www.cecm.sfu.ca/projects/ISC/. Plouffe, S. "Plouffe’s Inverter." http://www.lacim.uqam.ca/pi/ . Real Number Picking Pick two real numbers x and y at random in (0; 1) with a UNIFORM DISTRIBUTION . What is the PROB- ABILITY Peventhat [x=y] ; where [r] denotes NEAREST INTEGER FUNCTION ,is EVEN ? The answer may be found as follows. PaBx y Bb ! /C30P(ay Bx Bby) for 0 5a Bb B1 Px b By Bx a ! for 1 Ba Bb8 >< >: /C30g1 0gby aydx dy /C301 2(b /C28a) for 0 5a Bb B1 g1 0gx =a x=bdy dx /C301 2a /C281 2bfor 1 Ba Bb8 >>>>< >>>>:(1) so P even /C30P 0 Bx y B1 2 ! /C27X/C12 n/C301P 2n /C2812 Bx y B2n /C271 2 ! /C301212 /C280})@D})@E /C27X/C12 n/C3011 22n /C281 2})@D})@E /C281 22n /C2712})@D})@E2 435 /C30 1 4 /C27X/C12 n/C3011 4n /C28 1 /C271 4n /C28 1 ! /C3014 /C2713 /C2815 /C2717 /C2819 /C27...})@D})@E /C3014 /C27(1 /C28tan /C2811) /C305 4 /C28p 4 /C301 4(5 /C28 p) :46 :460% (2) (Putnam Exam). References Putnam Exam. Problem B-3 in the 54th Putnam Exam.Real Part The real part R[z]ofa COMPLEX NUMBER z /C30x /C27iy is the REAL NUMBER not multiplying I,soR[x /C27iy] /C30x: In terms of z itself, R[z] /C301 2(z /C27 ¯z) ; where ¯z is the COMPLEX CONJUGATE of z. The real part is implemented in Mathematica asRe[z]. See also ABSOLUTE SQUARE ,A RGUMENT (COMPLEX NUMBER ), COMPLEX CONJUGATE ,C OMPLEX PLANE , IMAGINARY PART,MODULUS (COMPLEX NUMBER ) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 2, 1999. Real Polynomial A POLYNOMIAL having only REAL NUMBERS as COEFFI- CIENTS . A polynomial with real coefficients is a product of IRREDUCIBLE POLYNOMIALS of first and second degrees. See also POLYNOMIAL Real Projective Plane The closed topological MANIFOLD , denoted RP2;which is obtained by projecting the points of a plane Efrom a fixed point P(not on the plane), with the addition of the LINE AT INFINITY , is called the real projective plane. There is then a one-to-one correspondence between points in E and lines through P. Since each line through P intersects the sphere S2 centered at P and tangent to E in two ANTIPODAL POINTS , RP2 can be described as a QUOTIENT SPACE of S2 by identifying any two such points. The real projective plane is a NONORIENTABLE SURFACE . The BOY SURFACE , CROSS-CAP , and ROMAN SURFACE are all homeomorphic to the real projective plane and, because RP2 is nonorientable, these surfaces contain self-intersections (Kuiper 1961, Pinkall 1986). See also BOY SURFACE ,CROSS- CAP,CROSS SURFACE , HENNEBERG’S MINIMAL SURFACE ,N ONORIENTABLE SURFACE ,P ROJECTIVE PLANE ,R EAL PROJECTIVE SPACE ,ROMAN SURFACE References Ape´ry, F. Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces. Braunschweig, Germany: Vieweg, 1987. Coxeter, H. S. M. The Real Projective Plane, 3rd ed. Cam- bridge, England: Cambridge University Press, 1993. Gray, A. "Realizations of the Real Projective Plane." §14.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 330 /C1/335, 1997. Klein, F. §1.2 in Vorlesungen u¨ber nicht-euklidische Geome- trie. New York: Springer-Verlag, 1968. Kuiper, N. H. "Convex Immersion of Closed Surfaces in E3 :/" Comment. Math. Helv. 35,85/C1/92, 1961. Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, pp. 64 /C1/65, 1986. Real Projective Space See also COMPLEX PROJECTIVE SPACE ,REAL PROJEC- TIVE PLANE ,REAL SPACE Real Quadratic Field A QUADRATIC FIELD Qðffiffiffiffi Dp Þ with D /C210. See also IMAGINARY QUADRATIC FIELD,Q UADRATIC FIELD Real Space See also COMPLEX SPACE ,REAL LINE Real Vector A VECTOR whose elements are REAL NUMBERS . See also COMPLEX VECTOR ,REAL NUMBER ,VECTOR Real Vector Bundle See also VECTOR BUNDLEReal Vector Space See also COMPLEX VECTOR SPACE ,VECTOR SPACE Realizer A SET R of LINEAR EXTENSIONS of a POSET P /C30(X ;5)is a realizer of P (and is said to realize P) provided that for all x; y /C23 X ; x 5y IFF x is below y in every member of R. See also DOMINANCE ,LINEAR EXTENSION ,PARTIALLY ORDERED SET,POSET DIMENSION Reals REAL NUMBER Real-Valued Function REAL FUNCTION Rearrangement Theorem Each row and each column in the GROUP multi- plication table lists each of the GROUP elements once and only once. From this, it follows that no two elements may be in the identical location in two rows or two columns. Thus, each row and each column is a rearranged list of the GROUP elements. Stated other- wise, given a GROUP of n distinct elements (I ; a ; b; c; ...; n) ; the set of products (aI ; a2 ; ab ; ac ; ...; an) reproduces the n original distinct elements in a new order. See also GROUP Reciprocal The reciprocal of a REAL or COMPLEX NUMBER z "0is its MULTIPLICATIVE INVERSE 1 =z: The reciprocal of a COMPLEX NUMBER z /C30x /C27iy is given by 1 x /C27 iy /C30x /C28 iy x2 /C27 y2 /C30x x2 /C27 y2 /C28y x2 /C27 y2 i : Given a geometric figure consisting of an assemblage of points, the POLARS with respect to an INVERSION CIRCLE constitute another figure. These figures are said to be reciprocal with respect to each other. Then there exists a DUALITY PRINCIPLE which states that theorems for the original figure can be immediately applied to the reciprocal figure after suitable mod- ification (Lachlan 1893). See also INVERSION ,POLAR ,POLE (INVERSION ), RE- CIPROCAL CURVE ,RECIPROCATION Reciprocal Curve The reciprocal curve of a given circle is the LOCUS of a point which moves so that its distance from the center of reciprocation varies as its distance from the line which is the reciprocal of the center of the given circle. The reciprocal of a circle is therefore a CONIC SECTION whose FOCUS is the center of reciprocation and whose directrix is the line which corresponds to the center of reciprocation. The conic will be an ELLIPSE , HYPERBOLA ,or PARABOLA if the center of reciprocation lies inside, outside, or on the given circle, respectively (Lachlan 1893, p. 181). See also DUALITY PRINCIPLE ,POLAR ,POLE (INVER- SION), RECIPROCATION References Lachlan, R. "Reciprocation." Ch. 11 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 174 /C1/182, 1893. Reciprocal Difference The reciprocal differences are closely related to the DIVIDED DIFFERENCE . The first few are explicitly given by r(x0 ; x1) /C30x0 /C28 x1 f0 /C28 f1(1) r2(x0 ; x1 ; x2) /C30x0 /C28 x2 r(x0 ; x1) /C28 r(x1 ; x2) /C27f1 (2) r3(x0 ; x1 ; x2 ; x3) /C30x0 /C28 x3 r2(x0 ; x1 ; x2) /C28 r2(x1 ; x2 ; x3) /C27 r(x1 ; x2) (3) rn(x0 ; x1 ; ... ; xn) /C30x0 /C28 xn rn/C281(x0 ; ...; xn/C281) /C28 rn /C281(x1 ; ... xn) /C27rn/C28x(x1 ; ... ; xn/C281) : (4) See also BACKWARD DIFFERENCE ,CENTRAL DIFFER- ENCE ,D IVIDED DIFFERENCE ,F INITE DIFFERENCE , FORWARD DIFFERENCE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 878, 1972. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 443, 1987. Reciprocal Matrix MATRIX INVERSE Reciprocal Permutation INVERSE PERMUTATION Reciprocal Polyhedron DUAL POLYHEDRONReciprocal Polynomial Given a polynomial in a single complex variable with complex coefficients p(z) /C30anzn /C27an/C281zn/C281 /C27.../C27a0 ; the reciprocal polynomial is defined by p/C31(z) /C13 ¯a0zn /C27 ¯a1zn/C281 /C27.../C27 ¯an ; where ¯a denotes the COMPLEX CONJUGATE . See also SCHUR TRANSFORM References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, p. 492, 1988. Reciprocating Sphere MIDSPHERE Reciprocation An incidence-preserving transformation in which points are transformed into their POLARS .A PROJEC- TIVE GEOMETRY -like DUALITY PRINCIPLE holds for reciprocation which states that theorems for the original figure can be immediately applied to the RECIPROCAL figure after suitable modification (La- chlan 1893, pp. 174 /C1/182). Reciprocation (or "polar reciprocation") is the strictly proper term for duality. Bru¨ckner (1900) gave one the first exact definitions of polar reciprocation for constructing DUAL POLYHEDRA , although the plane geometric version (POLE , POLAR , and POWER of a circle) was considered by none less than Euclid (Wenninger 1983, pp. 1 /C1/2). Lachlan 1893 (pp. 257 /C1/265) discusses another type of reciprocation he terms "circular reciprocation." How- ever, the circular reciprocal figure is, in general, more complicated than the original, so the method is not as powerful as the usual polar reciprocation. See also DUALITY PRINCIPLE ,POLAR ,POLE (INVER- SION), RECIPROCAL References Bru¨ckner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900. Casey, J. "Theory of Poles and Polars, and Reciprocation." §6.7 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., pp. 141 /C1 /148, 1888. Coxeter, H. S. M. and Greitzer, S. L. "Reciprocation." §6.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 132 /C1/136, 1967. Lachlan, R. "Reciprocation" and "Circular Reciprocation." Ch. 11 and §405/C1/414 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 174 /C1/ 182 and 257 /C1/265, 1893. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 1 /C1/6, 1983. Reciprocity Law RECIPROCITY THEOREM Reciprocity Theorem If there exists a RATIONAL INTEGER x such that, when n, p, and q are POSITIVE INTEGERS , xn /C13q (mod p) ; then q is the n-adic residue of p, i.e., q is an n-adic residue of p IFF xn /C13q (mod p) is solvable for x. Reciprocity theorems relate statements OF THE FORM "p is an n-adic residue of q" with reciprocal statements of the form "q is an n-adic residue of p." The first case to be considered was n /C302 (the QUADRATIC RECIPROCITY THEOREM ), of which Gauss gave the first correct proof. Gauss also solved the case n /C303(CUBIC RECIPROCITY THEOREM ) using INTEGERS OF THE FORM a /C27br ; where r is a root of x2 /C27x /C271 /C300 and a, b are rational INTEGERS . Gauss stated the case n /C304(BIQUADRATIC RECIPROCITY THEOREM ) using the GAUSSIAN INTEGERS . Proof of n-adic reciprocity for PRIME n was given by Eisenstein in 1844 /C1/50 and by Kummer in 1850 /C1/61. In the 1920s, Artin formulated ARTIN’S RECIPROCITY THEOREM , a general reciprocity law for all orders. See also ARTIN RECIPROCITY ,CLASS FIELD THEORY , CLASS NUMBER ,CUBIC RECIPROCITY THEOREM ,LANG- LANDS PROGRAM ,L ANGLANDS RECIPROCITY ,O CTIC RECIPROCITY THEOREM ,Q UADRATIC RECIPROCITY THEOREM ,Q UARTIC RECIPROCITY THEOREM ,R OOK RECIPROCITY THEOREM References Lemmermeyer, F. Reciprocity Laws: Their Evolution from Euler to Artin. Draft. http://www.rzuser.uni-heidel- berg.de/~hb3/rec.html. Lemmermeyer, F. "Bibliography on Reciprocity Laws." http://www.rzuser.uni-heidelberg.de/~hb3/recbib.html. Nagell, T. "Power Residues. Binomial Congruences." §34 in Introduction to Number Theory. New York: Wiley, pp. 115 /C1/120, 1951. Wyman, B. F. "What Is a Reciprocity Law?" Amer. Math. Monthly 79, 571 /C1/586, 1972. Recognize LATTICE REDUCTION Recontres Problem DERANGEMENT Rectangle A closed planar QUADRILATERAL with opposite sides ofequal lengths a and b, and with four RIGHT ANGLES . The AREA of the rectangle is A /C30ab; and its DIAGONALS p and q are of length p /C30q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2p : A SQUARE is a degenerate rectangle with a /C30b. A number of important topological surfaces can be constructed from the rectangle. Gluing both pairs of opposite edges together with no twists gives a TORUS , gluing two opposite edges together after giving a half- twist gives a MO¨ BIUS STRIP , gluing both pairs of opposite edges together giving one pair a half-twist gives a KLEIN BOTTLE , and giving both pairs a half- twist gives a PROJECTIVE PLANE (Stewart 1997). See also BLANCHE’S DISSECTION ,FAULT- FREE REC- TANGLE ,G OLDEN RECTANGLE ,INCOMPARABLE REC- TANGLES ,K LEIN BOTTLE ,M O¨ BIUS STRIP , OVERLAPPING RECTANGLES ,P ERFECT RECTANGLE , PROJECTIVE PLANE ,R ECTANGLE TILING ,S QUARE , TORUS References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 122, 1987. Eppstein, D. "Rectilinear Geometry." http://www.ics.uci.edu/ ~eppstein/junkyard/rect.html. Fukagawa, H. and Pedoe, D. "Circle and Rectangles." §3.4 in Japanese Temple Geometry Problems. Winnipeg, Mani- toba, Canada: Charles Babbage Research Foundation, pp. 43 /C1/44 and 125, 1989. Harris, J. W. and Stocker, H. "Rectangle." §3.6.5 in Hand- book of Mathematics and Computational Science. New York: Springer-Verlag, p. 84, 1998. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 2, 1948. Rectangle Function The rectangle function P(x) is a function which is 0 outside the interval [ /C281=2;1=2] and unity inside it. It is also called the GATE FUNCTION ,PULSE FUNCTION ,o r WINDOW FUNCTION , and is defined by P(x) /C130 for ½x½>1 2 12for ½x½/C3012 1 for ½x½B12:8 >< >:(1) The function f(x) /C30h P((x /C28c) =b) has height h, center c, and full-width b. Identities satisfied by the rec- tangle function include P(x) /C30Hx/C271 2})@D})@E /C28Hx/C2812})@D})@E (2) /C30H1 2 /C27x})@D})@E /C27H12 /C28x})@D})@E /C281 (3) /C30H1 4 /C28x2})@D})@E (4) /C301 2sgn x /C2712})@D})@E /C28sgn x /C2812})@D})@E hi ; (5) where H(x) is the HEAVISIDE STEP FUNCTION . The FOURIER TRANSFORM of the rectangle function is given by F[P(x)] /C30g/C12 /C28/C12e /C282 pikx P(x) dx /C30sinc( pk) ; (6) where sinc( x) is the SINC FUNCTION . See also ABSOLUTE VALUE ,BOXCAR FUNCTION ,FOUR- IER TRANSFORM– RECTANGLE FUNCTION ,H EAVISIDE STEP FUNCTION ,R AMP FUNCTION ,SGN,T RIANGLE FUNCTION ,UNIFORM DISTRIBUTION References Bracewell, R. "Rectangle Function of Unit Height and Base, P(x) :/"InThe Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 52 /C1/53, 1999. Rectangle Squaring Given a RECTANGLE /C176BCDE ; draw EF /C30DE on an extension of BE. Bisect BF and call the MIDPOINT G. Now draw a SEMICIRCLE centered at G, and construct the extension of ED which passes through the SEMICIRCLE at H. Then /C176EKLH has the same AREAas /C176BCDE : This can be shown as follows: A(/C176BCDE ) /C30BE /C215 ED /C30BE /C215 EF (a /C27b)(a /C28b) /C30a2 /C28b2 /C30c2 : References Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 13 /C1/14, 1990. Rectangle Tiling The number of ways N(m; n) in which an m /C29n RECTANGLE can be tiled into subrectangles can be computed by counting the number of ways in which the upper right-hand corner can be selected for a given lower left-hand corner. For a lower left-hand corner with coordinates (i, j), there are (m /C28i)(n /C28j) possible upper right-hand corners, so N(m; n) /C30Xm/C281 i/C300Xn/C281 j/C300(m /C28i)(n /C28j) /C3014m(m /C271)n(n /C271): Equivalently, N(m; n) is the number of ways of picking two lines out of sets of m/C271 and n/C271 lines, giving N(m;n)/C30m/C271 2})@*})@+ n/C271 2})@*})@+ /C3014m(m/C271)n(n/C271); as before. Particular tilings are shown above for 2 /C292 and 2/C293 rectangles. See also PERFECT RECTANGLE ,RECTANGLE ,TRIANGLE TILING References Stewart, I. "Squaring the Square." Sci. Amer. 277,9 4/C1/96, July 1997. Rectangular Coordinates CARTESIAN COORDINATES Rectangular Distribution UNIFORM DISTRIBUTION Rectangular Hyperbola A HYPERBOLA for which the ASYMPTOTES are PERPEN- DICULAR , also called an EQUILATERAL HYPERBOLA or RIGHT HYPERBOLA . This occurs when the SEMIMAJOR and SEMIMINOR AXES are equal. This corresponds to taking a /C30b, giving eccentricity e /C30ffiffiffi 2p : Plugging a /C30b into the general equation of a HYPERBOLA with SEMIMAJOR AXIS parallel to the X-AXIS and SEMIMINOR AXIS parallel to the Y-AXIS (i.e., vertical DIRECTRIX ), (x /C28 x0)2 a2/C28(y /C28 y0)2 b2/C301 (1) therefore gives (x /C28x0)2 /C28(y /C28y0)2 /C30a2 : (2) The rectangular hyperbola opening to the left and right has polar equation r2 /C30a2 sec(2 u) ; (3) and the rectangular hyperbola opening in the first and third quadrants has the Cartesian equation xy /C30a2 : (4) The INVERSE CURVE of a rectangular hyperbola with INVERSION CENTER at the center of the hyperbola is a LEMNISCATE (Wells 1991). If the three vertices of a TRIANGLE DABC lie on arectangular hyperbola, then so does the ORTHOCEN- TER H (Wells 1991). Equivalently, if four points form an ORTHOCENTRIC SYSTEM , then there is a family of rectangular hyperbolas through the points. Moreover, the LOCUS of centers O of these hyperbolas is the NINE-POINT CIRCLE of the triangle (Wells 1991). If four points do not form an ORTHOCENTRIC SYSTEM , then there is a unique rectangular hyperbola passing through them, and its center is given by the inter- section of the NINE-POINT CIRCLES of the points taken three at a time (Wells 1991). See also HYPERBOLA ,LEMNISCATE ,NINE-POINT CIR- CLE,ORTHOCENTRIC SYSTEM References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 218 /C1/219, 1987. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 76 /C1/77, 1996. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 118, 1969. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 209, 1991. Rectangular Matrix A MATRIX for which horizontal and vertical dimen- sions are not the same (i.e., an m /C29n MATRIX with m "n) :/ See also MATRIX ,SQUARE MATRIX Rectangular Parallelepiped A closed box composed of 3 pairs of rectangular faces placed opposite each other and joined at RIGHT ANGLES to each other. This PARALLELEPIPED therefore corresponds to a rectangular "box." If the lengths of the sides are denoted a, b, and c, then the VOLUME is V /C30abc; (1) the total SURFACE AREA is S /C302(ab /C27bc /C27ca) (2) and the length of the "space" DIAGONAL is dabc /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2/C27c2p : (3) Ifa/C30b/C30c;then the rectangular parallelepiped is a CUBE . See also CUBE,EULER BRICK,PARALLELEPIPED References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987. Kern, W. F. and Bland, J. R. "Rectangular Parallelepiped." §10 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 21 /C1/25, 1948. Rectangular Projection EQUIRECTANGULAR PROJECTION Rectifiable Current The space of currents arising from rectifiable sets by integrating a differential form is called the space of 2- D rectifiable currents. For C a closed bounded rectifiable curve of a number of components in R3 ; C bounds a rectifiable current of least AREA . The theory of rectifiable currents generalizes to m-D surfaces in Rn :/ See also INTEGRAL CURRENT ,REGULARITY THEOREM References Morgan, F. "What is a Surface?" Amer. Math. Monthly 103, 369 /C1/376, 1996. Rectifiable Set The rectifiable sets include the image of any LIPSCHITZ FUNCTION f from planar domains into R3 : The full set is obtained by allowing arbitrary measur- able subsets of countable unions of such images of Lipschitz functions as long as the total AREA remains finite. Rectifiable sets have an "approximate" tangent plane at almost every point. References Morgan, F. "What is a Surface?" Amer. Math. Monthly 103, 369 /C1/376, 1996. Rectification The term rectification is sometimes used to refer to the determination of the length of a curve. Rectification also refers to the operation which con- verts the midpoints of the edges of a regular poly- hedron to the vertices of the related "rectified" polyhedron. Rectified forms are bounded by a combi- nation of rectified cells and VERTEX FIGURES . There-fore, a rectified polychoron rfp ; q; rg is bounded by r fp; qgs/ and fq; rgs/. For example, r f3; 3; 5g is bounded by 600 truncated tetrahedra (truncated cells) and 120 icosahedra (vertex figures). A rectified polyhedron is indicated by perpending an "r" to the Schla ¨fli symbol. POLYHEDRON SCHLA ¨ FLI SYMBOLrectified polygon SCHLA ¨ FLI SYMBOL TETRAHEDRON /f3; 3g/ OCTAHEDRON /rf3; 3g/ //C30f3; 4g/ OCTAHEDRON /f3; 4g/ CUBOCTAHEDRON /rf3; 4g/C303 4})*})+ / CUBE /f4; 3g/ CUBOCTAHEDRON /rf4; 3g/C3034})*})+ / ICOSAHEDRON /f3; 5g/ ICOSIDODECAHEDRON /rf3; 5g/C3035})*})+ / DODECAHEDRON /f5; 3g/ ICOSIDODECAHEDRON /rf5; 3g/C3035})*})+ / 16-CELL /f3;3;4g/24-CELL /rf3;3;4g/ //C30f3;4;3g/ Rectification of the six regular POLYCHORA gives five (not six) new POLYCHORA since the rectified 16-CELL rf3;3;4gis the 24-CELL f3;4;3g:/ See also QUADRABLE ,SQUARING ,STELLATION ,TRUN- CATION ,VERTEX FIGURE Rectifying Latitude An AUXILIARY LATITUDE which gives a sphere having correct distances along the meridians. It is denoted m (orv) and is given by m/C30pM 2Mp: (1) /Mpis evaluated for Mat the north pole ( /f/C3090/C14);and Mis given by M/C30a1/C28e2})0})@gf 0df 1/C28e2sin2f})0})@ 3=2 /C30agf 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2sin2fq df/C28e2sinfcosfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28e2sin2fq2 435: ð2Þ A series for Mis M¼a½ð1/C28 1 4e2/C283 64e4/C285 256e6/C28...Þf /C2838e2/C273 32e4/C2745 1024e6/C27...})@D})@E sin(2f) /C2715 256e4/C2745 1024e6/C27...})@D})@E sin(4f) /C2835 3072 e6 /C27...})@D})@E sin(6f) /C27.../C138; (3) and a series for m is m /C30 f /C283 2 e1 /C289 16 e3 1 /C27...})@D})@E sin(2f) /C2715 16 e2 1 /C2815 32 e4 1 /C27...})@D})@E sin(4f) /C2835 48 e3 1 /C28...})@D})@E sin(6f) /C27315 512 e4 1 /C28...})@D})@E sin(8f) /C27... ; ð4Þ where e1 /C131 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 e2p 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 e2p : (5) The inverse formula is f /C30 m /C273 2 e1 /C282732 e3 1 /C27...})@D})@E sin(2m) /C2721 16 e2 1 /C2855 32 e4 1 /C27...})@D})@E sin(4m) /C27151 96e31 /C28...})@D})@E sin(6m) /C281097 512e4 /C28...})@D})@E sin(8m) /C27... (6) See also LATITUDE References Adams, O. S. "Latitude Developments Connected with Geo- desy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, pp. 125 /C1/128, 1921. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 16 /C1/17, 1987. Rectifying Plane The PLANE spanned by the TANGENT VECTOR T and BINORMAL VECTOR B. See also BINORMAL VECTOR ,TANGENT VECTOR Rectilinear Crossing Number The minimum number ¯n(G) of crossings in a straight line drawing of a graph G in a plane. For a COMPLETE GRAPH of order n ]10; the rectilinear crossing num- ber is always larger than the general graph crossing number. For the COMPLETE GRAPH Knwith n /C301, 2, ..., ¯n(G) is 0, 0, 0, 0, 1, 3, 9, 19, 36, 62, ... (Sloane’s A014540; White and Beineke 1978, Schneinerman and Wilf 1994). Although it had long been known that ¯n K10ðÞ was either 61 or 62 (Singer 1971, Gardner 1986), it was finally proven to be 62 by Brodsky et al. (2000).Upper limits have been provided by Singer (1971), who showed that ¯n KnðÞ51 3125n4 /C2839n3 /C2791n2 /C2857n})0})@ ; (1) and Jensen (1971), who showed that ¯n KnðÞ57 432 n4 /C27O n3})0})@ : (2) Bounds for ¯n KnðÞ are given by 0:290 B61 210 5 r /C30 lim n0/C12¯n KnðÞ n 4})@*})@+55 13 B0:385; (3) wheren k})0})@ is a BINOMIAL COEFFICIENT and the exact value of r is not known (Finch). The rectilinear crossing number has an unexpected connection with SYLVESTER’S FOUR-POINT PROBLEM (Finch). See also CROSSING NUMBER (GRAPH ), PLANAR STRAIGHT LINE GRAPH ,S YLVESTER’S FOUR- POINT PROBLEM ,TOROIDAL CROSSING NUMBER References Brodsky, A.; Durocher, S.; and Gethner, E. "Toward the Rectilinear Crossing Number of Kn: New Drawings, Upper Bounds, and Asymptotics." http://www.cs.ubc.ca/ spider/abrodsky/papers/reccr_n.ps.gz. Brodsky, A.; Durocher, S.; and Gethner, E. The Rectilinear Crossing Number of K10is 62. 22 Sep 2000. http:// xxx.lanl.gov/abs/cs.DM/0009023/. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/crss/crss.html. Gardner, M. Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986. Guy, R. K. "Crossing Numbers of Graphs." In Graph Theory and Applications: Proceedings of the Conference at Wes-tern Michigan University, Kalamazoo, Mich., May 10 /C1 /13, 1972 (Ed. Y. Alavi, D. R. Lick, and A. T. White). New York: Springer-Verlag, pp. 111 /C1/124, 1972. Harary, F. and Hill, A. "On the Number of Crossings in a Complete Graph." Proc. Edinburgh Math. Soc. 13, 333/C1/ 338, 1962/1963. Jensen, H. F. "An Upper Bound for the Rectilinear Crossing Number of the Complete Graph." J. Combin. Th. B 10, 212/C1/216, 1971. Klee, V. "What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given ConvexBody." Amer. Math. Monthly 76, 286/C1 /288, 1969. Schneinerman, E. and Wilf, H. S. "The Rectilinear Crossing Number of a Complete Graph and Sylvester’s ‘Four Point’ Problem of Geometric Probability." Amer. Math. Monthly 101, 939/C1/943, 1994. Singer, D. "The Rectilinear Crossing Number of Certain Graphs." Unpublished manuscript, 1971. Quoted in Gard- ner, M. Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986. Sloane, N. J. A. Sequences A014540 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. White, A. T. and Beineke, L. W. "Topological Graph Theory." InSelected Topics in Graph Theory (Ed. L. W. Beineke and R. J. Wilson). New York: Academic Press, pp. 15 /C1 /49, 1978. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos’ 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press, pp. 557 /C1/562, 1997. Recurrence Relation A mathematical relationship expressing fnas some combination of fi with i Bn. The solutions to a linear recurrence can be computed straightforwardly, but QUADRATIC RECURRENCES are not so well understood. The sequence generated by a recurrence relation is called a RECURRENCE SEQUENCE . Perhaps the most famous example of a recurrence relation is the one defining the F IBONACCI NUMBERS , Fn/C30Fn/C282/C27Fn/C281 forn]3 and with F1/C30F2/C301:/ See also ARGUMENT ADDITION RELATION ,ARGUMENT MULTIPLICATION RELATION ,CLENSHAW RECURRENCE FORMULA ,Q UADRATIC RECURRENCE ,R ECURRENCE SEQUENCE ,REFLECTION RELATION ,TRANSLATION RE- LATION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Recurrence Relations and Clenshaw’s Recur- rence Formula." §5.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 172 /C1/178, 1992. Sloane, N. J. A. and Plouffe, S. "Recurrences and Generat- ing Functions" and "Other Methods for Hand Analysis."§2.4 and 2.6 in The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 9 /C1 /10 and 13 /C1/18, 1995. Recurrence Sequence A sequence of numbers generated by a RECURRENCE RELATION is called a recurrence sequence. Perhaps the most famous recurrence sequence is the F IBO- NACCI NUMBERS . For a finite linear recurrence sequence of functions si(x)/C30Ai(x)si/C271(x)/C27Bi(x) where i/C301, ..., r/C281;andsr(x)/C30h(x);then s1(x)/C30B1(x)/C28A1(x)0::: 0 B2(x)1 /C28A2(x)::: 0 B3(x)0 1:::n nn:::::: 0 Br/C281(x)0 0:::/C28Ar/C281(x) h(x)0 0::: 1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1(1) (Mansour 2000). If a sequence x nfg with x1/C30x2/C301 is described by a two-term linear RECURRENCE RELATION OF THE FORM xn/C30Axn/C281/C27Bxn/C282 (2) forn]3 and AandBconstants, then the closed form forxnis given byxn/C30an/C28bn a/C28b(3) where aand bare the ROOTS of the QUADRATIC EQUATION x2/C28Ax/C28B/C300; (4) a/C301 2A/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2/C274Bp})@D})@E (5) b/C301 2A/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2/C274Bp})@D})@E (6) For example, the F IBONACCI NUMBERS Fnwhich are equal to 1, 1, 2, 3, 5, 8, ... for n/C301, 2, ..., have A/C30 B/C301;soa/C301/C27ffiffiffi 5p})0})@ =2 and b/C301/C28ffiffiffi5p})0})@ =2;giving F n/C301 21/C27ffiffiffi 5p})0})@hin /C281 21/C28ffiffiffi 5p})0})@hin ffiffiffi5p /C301/C27ffiffiffi5p})0})@ n/C281/C28ffiffiffi5p})0})@ n 2nffiffiffi5p : (7) Grosjean (1993) discusses how to rewrite such "dif- ference of powers of roots" solutions in explicit integerform. The general second-order linear recurrence x n/C30Axn/C281/C27Bxn/C282 (8) for constants Aand Bwith arbitrary x1and x2has terms x1/C30x1 x2/C30x2 x3/C30Bx1/C27Ax2 x4/C30Bx2/C27ABx1/C27A2x2 x5/C30B2x1/C272ABx2/C27A2Bx1/C27A3x2 x6/C30B2x2/C272AB2x1/C273A2Bx2/C27A3Bx1/C27A4x2 x7/C30B3x1/C274A3Bx2/C273A2B2x1/C273AB2x2/C27A4Bx1/C27A5x2; so an arbitrary term can be written as xn/C30Xn/C282 k/C3001 2(n/C27k/C282)jk k ! AkB(n/C28k/C281)=2 bc /C2x[n/C27k(mod 2)] 1 x[n/C27k/C271 (mod 2)] 2 : (9) /C30/C28(Ax1/C28x2)Xn/C282 k/C300A2k/C28n/C272B/C28k/C27n/C282 k n/C28k/C282})@*})@+ /C27x1Xn/C281 k/C300A2k/C28n/C271B/C28k/C27n/C281 k n/C28k/C281})@*})@+ : (10) The general linear third-order recurrence xn/C30Axn/C281/C27Bxn/C282/C27Cxn/C283 (11) has solution xn /C30x1})@*a/C28n A /C27 2 aB /C27 3a2C /C27b/C28n A /C27 2bB /C27 3b2C /C27g /C28n A /C27 2gB /C27 3g2C})@+ /C28 Ax1 /C28x2 ðÞ /C2})@*a1 /C28n A /C27 2aB /C27 3 a2C /C27b1/C28n A /C27 2 bB /C27 3b2B /C27g1 /C28n A /C27 2gC /C27 3g2C})@+ /C28 Bx1 /C27Ax2 /C28x3 ðÞ /C2})@*a2 /C28n A /C27 2aB /C27 3 a2C /C27b2/C28n A /C27 2 bB /C27 3b2C /C27g2 /C28n A /C27 2gB /C27 3g2C})@+ ; (12) where a; b; and g are the roots of the polynomial Cx3 /C27Bx2 /C27Ax /C301 : (13) A QUOTIENT-DIFFERENCE TABLE eventually yields a line of 0s IFF the starting sequence is defined by a linear RECURRENCE RELATION . A linear second-order recurrence fn/C271 /C30xfn /C27yfn/C281 (14) can be solved rapidly using a "rate doubling," fn/C272 /C30 x2 /C272y})0})@ fn /C28y2fn/C282 ; (15) "rate tripling" fn/C273 /C30 x3 /C273xy})0})@ fn /C27y3fn/C283 ; (16) or in general, "rate k-tupling" formula fn/C27k /C30pkfn /C27qkfn/C28k ; (17) where p0 /C302 (18) p1 /C30x (19) pk /C302(/C28y)k =2Tkx = 2iffiffiffiypðÞðÞ (20) pk /C271 /C30xpk /C27ypk/C281 (21) (here, Tk(x)isaC HEBYSHEV POLYNOMIAL OF THE FIRST KIND ) and q0 /C30/C281 (22) q1 /C30y (23) qk /C30/C28(/C28y)k (24) qk /C271 /C30/C28yqk (25) (Gosper and Salamin 1972).Let s(X) /C30Ym i/C301(1 /C28 aiX)ni /C301 /C28s1X /C28.../C28snXn ; (26) where the generalized POWER sum a(h) for h /C300, 1, ... is given by a(h) /C30Xm i/C301Ai(h) ah i ; (27) with distinct NONZERO roots ai ; COEFFICIENTS Ai(h) which are POLYNOMIALS of degree ni /C281 for POSITIVE INTEGERS ni ; and i /C23 [1; m]: Then the sequence ahfg with ah /C30a(h) satisfies the RECURRENCE RELATION ah/C27n /C30siah/C27n/C281 /C27.../C27snah (28) (Meyerson and van der Poorten 1995). The terms in a general recurrence sequence belong to a finitely generated RING over the INTEGERS ,soitis impossible for every RATIONAL NUMBER to occur in any finitely generated recurrence sequence. If a recurrence sequence vanishes infinitely often, then it vanishes on an arithmetic progression with a common difference 1 that depends only on the roots. The number of values that a recurrence sequence can take on infinitely often is bounded by some INTEGER l that depends only on the roots. There is no recurrence sequence in which each INTEGER occurs infinitely often, or in which every GAUSSIAN INTEGER occurs (Myerson and van der Poorten 1995). Letm(n) be a bound so that a nondegenerate INTEGER recurrence sequence of order ntakes the value zero at least m(n) times. Then m(2)/C301;m(3)/C306;andm(4)]9 (Myerson and van der Poorten 1995). The maximal case for m(3) is an/C273/C302an/C272/C284an/C271/C274an (29) with a0/C30a1/C300 (30) a2/C301: (31) The zeros are a0/C30a1/C30a4/C30a6/C30a13/C30a52/C300 (32) (Beukers 1991). See also BINET FORMS ,BINET’S FIBONACCI NUMBER FORMULA ,FAST FIBONACCI TRANSFORM ,FIBONACCI NUMBER ,LUCAS SEQUENCE ,Q UOTIENT- DIFFERENCE TABLE ,SKOLEM- MAHLER- LERCH THEOREM References Batchelder, P. M. An Introduction to Linear Difference Equations. New York: Dover, 1967. Beukers, F. "The Zero-Multiplicity of Ternary Recurrences." Composito Math. 77, 165/C1/177, 1991. Gosper, R. W. and Salamin, E. Item 14 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM- 239, pp. 8 /C1/9, Feb. 1972. Greene, D. H. and Knuth, D. E. Mathematics for the Analysis of Algorithms, 3rd ed. Boston, MA: Birkha ¨user, 1990. Grosjean, C. C. In Topics in Polynomials of One and Several Variables and Their Applications: Volume Dedicated to the Memory of P.L. Chebyshev (1821 /C1/1894) (Ed. T. M. Rassias, H. M. Srivastava, and A. Yanushauskas). Singapore: World Scientific, 1993. Levy, H. and Lessman, F. Finite Difference Equations. New York: Dover, 1992. Mansour, T. Permutations Avoiding a Pattern from and at Least Two Patterns from S3 : 31 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007194/. Myerson, G. and van der Poorten, A. J. "Some Problems Concerning Recurrence Sequences." Amer. Math. Monthly 102, 698 /C1/705, 1995. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980. Wimp, J. Computations with Recurrence Relations. Boston, MA: Pitman, 1984. Recurring Decimal REPEATING DECIMAL Recurring Digital Invariant To define a recurring digital invariant of order k, compute the sum of the kth powers of the digits of a number n. If this number n ? is equal to the original number n, then n /C30n ? is called a k-NARCISSISTIC NUMBER . If not, compute the sums of the kth powers of the digits of n?; and so on. If this process eventually leads back to the original number n, the smallest number in the sequence fn; n?; nƒ; ...g is said to be a k-recurring digital invariant. For example, 55 : 53 /C2753 /C30250 250 : 23 /C2753 /C2703 /C30133 133 : 13 /C2733 /C2733 /C3055 ; so 55 is an order 3 recurring digital invariant. The following table gives recurring digital invariants of orders 2 to 10 (Madachy 1979). Order RDI Cycle Lengths 24 8 3 55, 136, 160, 919 3, 2, 3, 2 4 1138, 2178 7, 2 5 244, 8294, 8299, 9044, 9045, 10933,28, 10, 6, 10, 22, 4, 12, 2, 2 24584, 58618, 89883 6 17148, 63804, 93531, 239459, 28259530, 2, 4, 10, 37 80441, 86874, 253074, 376762,92, 56, 27, 30, 14, 21 922428, 982108, five more 8 6822, 7973187, 8616804 9 322219, 2274831, 20700388, eleven more 10 20818070, five more See also 196-ALGORITHM ,A DDITIVE PERSISTENCE , DIGITADDITION ,DIGITAL ROOT,HAPPY NUMBER ,KA- PREKAR NUMBER ,N ARCISSISTIC NUMBER ,V AMPIRE NUMBER References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 163 /C1/165, 1979. Recursion A recursive process is one in which objects are defined in terms of other objects of the same type. Using some sort of RECURRENCE RELATION , the entire class of objects can then be built up from a few initial values and a small number of rules. The FIBONACCI NUM- BERS are most commonly defined recursively. Care, however, must be taken to avoid SELF-RECURSION ,in which an object is defined in terms of itself, leading to an infinite nesting. See also ACKERMANN FUNCTION ,PRIMITIVE RECUR- SIVE FUNCTION ,R ECURRENCE RELATION ,R ECUR- RENCE SEQUENCE ,R ECURSIVE FUNCTION , REGRESSION ,RICHARDSON’S THEOREM ,SELF-RECUR- SION,SELF-SIMILARITY , TAK FUNCTION References Buck, R. C. "Mathematical Induction and Recursive Defini- tions." Amer. Math. Monthly 70, 128/C1/135, 1963. Gardner, M. "Infinite Regress." Ch. 22 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 220 /C1/229, 1984. Knuth, D. E. "Textbook Examples of Recursion." In Artificial Intelligence and Mathematical Theory of Computation, Papers in Honor of John McCarthy (Ed. V. Lifschitz). Boston, MA: Academic Press, pp. 207 /C1/229, 1991. Pe´ter, R. Rekursive Funktionen. Budapest: Akad. Kiado, 1951. Thompson, W. "Recursive Algorithms: A Mixed Blessing." Computers in Physics 10,2 5/C1/29, 1996. Recursive Function A recursive function is a function generated by (1) ADDITION , (2) MULTIPLICATION , (3) selection of an element from a list, and (4) determination of the truth or falsity of the INEQUALITY aBbaccording to the technical rules: 1. If F and the sequence of functions G1 ; ..., Gn are recursive, then so is F(G1 ; ... ; Gn) :/ 2. If F is a recursive function such that there is an x for each a with H(a; x) /C300; then the smallest x can be obtained recursively. AT URING MACHINE is capable of computing recursive functions. See also TURING MACHINE References Kleene, S. C. Introduction to Metamathematics. Princeton, NJ: Van Nostrand, 1952. Pe´ter, R. Rekursive Funktionen. Budapest: Akad. Kiado, 1951. Schnorr, C. P. Rekursive Funktionen und ihre Komplexita ¨t. Stuttgart, Germany: Teubner, 1974. Recursive Monotone Stable Quadrature A QUADRATURE (NUMERICAL INTEGRATION ) algorithm which has a number of desirable properties. References Favati, P.; Lotti, G.; and Romani, F. "Interpolary Integration Formulas for Optimal Composition." ACM Trans. Math. Software 17, 207 /C1/217, 1991. Favati, P.; Lotti, G.; and Romani, F. "Algorithm 691: Improving QUADPACK Automatic Integration Routines." ACM Trans. Math. Software 17, 218 /C1/232, 1991. Red Net The coloring red of two COMPLETE SUBGRAPHS of n=2 points (for EVEN n) in order to generate a BLUE-EMPTY GRAPH . See also BLUE-EMPTY GRAPH ,COMPLETE GRAPH Red-Black Tree An extended BINARY TREE satisfying the following conditions: 1. Every node has two CHILDREN , each colored either red or black. 2. Every LEAF node is colored black. 3. Every red node has both of its CHILDREN colored black. 4. Every path from the ROOT to a LEAF contains the same number (the "black-height") of black nodes. Let n be the number of internal nodes of a red-black tree. Then the number of red-black trees for n /C301, 2,... is 2, 2, 3, 8, 14, 20, 35, 64, 122, ... (Sloane’s A001131). The number of trees with black roots and red roots are given by Sloane’s A001137 and Sloane’s A001138, respectively. Let /Th/ be the GENERATING FUNCTION for the number of red-black trees of black-height h indexed by the number of LEAVES . Then Th /C271(x) /C30 Th(x) ½/C1382/C27Th(x) ½/C1384; (1rpar (1) where T1(x) /C30x /C27x2 : If T(x) is the GENERATING FUNC- TION for the number of red-black trees, then T(x) /C30x /C27x2 /C27Tx2(1 /C27x)2})@D})@E (2) (Ruskey). Let rb(n) be the number of red-black trees with n LEAVES , r(n) the number of red-rooted trees, and b(n) the number of black-rooted trees. All three of the quantities satisfy the RECURRENCE RELATION R(n) /C30X n=4 5n5n=22m n /C282m})@*})@+ R(m) ; (3) wheren k})0})@ is a BINOMIAL COEFFICIENT , rb(1) /C301; rb(2) /C302 for R(n) /C30rb(n) ; r(1) /C30r(3) /C300; r(2) /C301 for R(n) /C30r(n) ; and b(1) /C301 for R(n) /C30b(n) (Ruskey). See also B-TREE References Beyer, R. "Symmetric Binary B-Trees: Data Structures and Maintenance Algorithms." Acta Informat. 1, 290 /C1/306, 1972. Binstock, A.; and Rex, J. Practical Algorithms for Program- mers. Reading, MA: Addison-Wesley, 1995. Cormen, T.; Leiserson, C.; and Rivest, R. Introduction to Algorithms. Cambridge MA: MIT Press, 1990. Guibas, L. and Sedgewick, R. "A Dichromatic Framework for Balanced Trees." In Proc. 19th IEEE Symp. Foundations of Computer Science, pp. 8 /C1/21, 1978. Rivest, R. L.; Leiserson, C. E.; and Cormen, R. H. Introduc- tion to Algorithms. New York: McGraw-Hill, 1990. Ruskey, F. "Information on Red-Black Trees." http:// www.theory.csc.uvic.ca/~cos/inf/tree/RedBlackTree.html. Skiena, S. S. The Algorithm Design Manual. New York: Springer-Verlag, pp. 177 and 179, 1997. Sloane, N. J. A. Sequences A001131, A001137, and A001138 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Wood, D. Data Structures, Algorithms, and Performance. Reading, MA: Addison-Wesley, 1993. Reduced Amicable Pair QUASIAMICABLE PAIR Reduced Fraction A FRACTION a=b written in lowest terms, i.e., by dividing NUMERATOR and DENOMINATOR through by their GREATEST COMMON DIVISOR (a, b). For example, 2/3 is the reduced fraction of 8/12. See also FRACTION ,IMPROPER FRACTION ,M IXED FRACTION ,PROPER FRACTION Reduced Knot Diagram A KNOT DIAGRAM in which none of the crossings are REDUCIBLE . See also KNOT DIAGRAM ,REDUCIBLE CROSSING References Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Reduced Latitude PARAMETRIC LATITUDE Reduced Maxwell-Bloch Equations The system of PARTIAL DIFFERENTIAL EQUATIONS Et /C28v /C300 (1) rx /C27 vv /C300 (2) qx /C27Ev /C300 (3) vx /C28 vr /C28Eq /C300: (4) References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 59, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Reduced Residue System Any system of f(n) integers, where f(n) is the TOTIENT FUNCTION , representing all the RESIDUE CLASSES RELATIVELY PRIME to n is called a reduced residue system (Nagell 1951, p. 71). See also COMPLETE RESIDUE SYSTEM ,RESIDUE CLASS References Nagell, T. "Residue Classes and Residue Systems." §20 in Introduction to Number Theory. New York: Wiley, pp. 69 /C1/ 71, 1951. Reduced Root System A ROOT SYSTEM R satisfying the additional property that, if a /C23 R; then the only multiples of a in R are 9a:/ See also ROOT SYSTEM References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 40, 1986. Humphrey, J. E. Introduction to Lie Algebras and Repre- sentation Theory. New York: Springer-Verlag, p. 42, 1972.Reducible Crossing A crossing in a KNOT DIAGRAM for which there exists a circle in the projection plane meeting the diagram transversely at that crossing, but not meeting the diagram at any other point. Removable crossings can be removed by twisting, and so cannot occur in a KNOT DIAGRAM of minimal CROSSING NUMBER . Redu- cible crossings are also called nugatory crossings (Tait 1898, Hoste et al. 1998) or removable crossings. See also ALTERNATING KNOT,K NOT DIAGRAM ,RE- DUCED KNOT DIAGRAM References Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Tait, P. G. "On Knots I, II, and III." Scientific Papers, Vol. 1. Cambridge, England: University Press, pp. 273 /C1/347, 1898. Reducible Matrix A SQUARE n /C29n matrix A /C30aij is called reducible if the indices 1, 2, ..., n can be divided into two disjoint nonempty sets i1 ; i2 ; ..., i m and j1 ; j2 ; ..., jn (with m /C27 n /C30 n) such that aiajb /C300 for a /C301; 2, ..., m and b /C301 ; 2, ..., n : A SQUARE MATRIX which is not reducible is said to be IRREDUCIBLE . See also SQUARE MATRIX References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1103, 2000. Reducible Representation IRREDUCIBLE REPRESENTATION Reductio ad Absurdum A method of PROOF which proceeds by stating a proposition and then showing that it results in a contradiction, thus demonstrating the proposition to be false. In the words of G. H. Hardy , "Reductio ad absurdum , which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any CHESS gambit: a CHESS player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game" (Coxeter and Greit- zer 1967, p. 16; Hardy 1993, p. 34). See also PROOF References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 16, 1967. Hardy, G. H. A Mathematician’s Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 34, 1993. Reduction of Order ORDINARY DIFFERENTIAL EQUATION– SECOND- ORDER Reduction Theorem If a fixed point is added to each group of a special complete series, then the resulting series is complete. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 253, 1959. Redundancy R(X1 ;...Xn) /C13Xn i/C301HXiðÞ/C28HX1 ; ...; Xn ðÞ ; where H(xi) is the ENTROPY and HX1 ; ...; Xn ðÞ is the joint ENTROPY . Linear redundancy is defined as LX1 ; ...; Xn ðÞ /C13/C281 2Xn i/C301ln si ; where si are EIGENVALUES of the correlation matrix. See also PREDICTABILITY References Fraser, A. M. "Reconstructing Attractors from Scalar Time Series: A Comparison of Singular System and Redundancy Criteria." Phys. D 34, 391 /C1/404, 1989. Palus, M. "Identifying and Quantifying Chaos by Using Information-Theoretic Functionals." In Time Series Pre- diction: Forecasting the Future and Understanding the Past (Ed. A. S. Weigend and N. A. Gerschenfeld). Proc. NATO Advanced Research Workshop on Comparative Time Series Analysis held in Sante Fe, NM, May 14 /C1/17, 1992. Reading, MA: Addison-Wesley, pp. 387 /C1/413, 1994. Ree Group The Ree group R(q) is the AUTOMORPHISM GROUP of a S 2 ; q /C271; q3 /C271 ðÞ STEINER SYSTEM . See also STEINER SYSTEMReferences Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Reeb Foliation The Reeb foliation of the HYPERSPHERE S3is a FOLIATION constructed as the UNION of two solid TORI with common boundary. See also FOLIATION References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 287 /C1/288, 1976. Reed-Sloane Algorithm An extension to the BERLEKAMP- MASSEY ALGORITHM which applies when the terms of the sequences are integers modulo some given modulus m. See also BERLEKAMP- MASSEY ALGORITHM References Reed and Sloane, N. J. A. SIAM J. Comput. 14, 505, 1985. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 26, 1995. Reef Knot SQUARE KNOT Re-Entrant Circuit A GRAPH CYCLE which terminates at the starting point. See also EULERIAN CIRCUIT ,GRAPH CYCLE ,HAMILTO- NIAN CYCLE Refined Alternating Sign Matrix Conjecture The fact that the numerators and denominators obtained by taking the ratios of adjacent terms in the triangular array of the number of /C271 "bordered" ALTERNATING SIGN MATRICES An with a 1 at the top of column k are respectively the numbers in the (2, 1)- and (1, 2)-Pascal triangles which are different from 1. This conjecture was proven by Zeilberger (1996). See also ALTERNATING SIGN MATRIX ,ALTERNATING SIGN MATRIX CONJECTURE References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637/C1/646. Zeilberger, D. "Proof of the Refined Alternating Sign Matrix Conjecture." New York J. Math. 2,5 9/C1/68, 1996. Refinement A refinement Xof a COVER Yis a COVER such that every element x/C23Xis a SUBSET of an element y/C23Y:/ See also COVER Reflection The operation of exchanging all points of a mathe- matical object with their MIRROR IMAGES (i.e., reflec- tions in a mirror). Objects which do not change HANDEDNESS under reflection are said to be AMPHI- CHIRAL ; those that do are said to be CHIRAL . If the PLANE of reflection is taken as the yz-PLANE , the reflection in 2- or 3-D SPACE consists of making the transformation x 0/C28x for each point. Consider an arbitrary point x0and a PLANE specified by the equation ax /C27by /C27cz /C27d /C300: (1) This PLANE has NORMAL VECTOR n /C30a b c2 435; (2) and the POINT-PLANE DISTANCE is D /C30ax0 /C27 by0 /C27 cz0 /C27 d jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 b2 /C27 c2p : (3) The position of the point reflected in the given plane is therefore given by x?0/C30x0/C282Dˆn /C30x0 y0 z02 435/C28 2jax0þby0þcz0þdj a2þb2þc2a b c2435: ð4Þ See also A MPHICHIRAL ,CHIRAL ,DILATION ,ENANTIO- MER,E XPANSION ,G LIDE,H ANDEDNESS ,IMPROPER ROTATION ,INVERSION OPERATION ,M IRROR IMAGE , PROJECTION ,R EFLECTION PROPERTY ,R EFLECTION RELATION ,REFLEXIBLE ,ROTATION ,ROTOINVERSION , TRANSLATION References Addington, S. "The Four Types of Symmetry in the Plane." http://forum.swarthmore.edu/sum95/suzanne/symsu- san.html. Coxeter, H. S. M. and Greitzer, S. L. "Reflection." §4.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 86 /C1/87, 1967. Voisin, C. Mirror Symmetry. Providence, RI: Amer. Math. Soc., 1999. Yaglom, I. M. Geometric Transformations I. New York: Random House, 1962. Reflection Formula REFLECTION RELATIONReflection Property In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73 /C1/77): 1. The LOCUS of the center of a variable CIRCLE , tangent to a fixed CIRCLE and passing through a fixed point inside that CIRCLE ,i sa n ELLIPSE . 2. If a variable CIRCLE is tangent to a fixed CIRCLE and also passes through a fixed point outside the CIRCLE , then the LOCUS of its moving center is a HYPERBOLA . 3. If a variable CIRCLE is tangent to a fixed straight line and also passes through a fixed point not onthe line, then the LOCUS of its moving center is a PARABOLA . Leta:I0R2be a smooth regular parameterized curve in R2defined on an OPEN INTERVAL I, and let F1 and F2be points in P2_a(I);where Pnis an n-D PROJECTIVE SPACE . Then ahas a reflection property with FOCI F1andF2if, for each point P/C23a(I); 1. Any vector normal to the curve aatPlies in the SPAN of the vectors F1P})@A@})@A@!andF2P})@A@})@A@!. 2. The line normal to aatPbisects one of the pairs of opposite ANGLES formed by the intersection of the lines joining F1andF2toP. A smooth connected plane curve has a reflectionproperty IFFit is part of an ELLIPSE ,HYPERBOLA , PARABOLA ,CIRCLE , or straight LINE. Foci Sign Both foci finite One focus finiteBoth foci infinite distinct POSITIVE confocal ellipses confocal parabolasparallel lines distinct NEGATIVE confocal hyper- bola and perpendicularconfocal parabolasparallellines bisector of inter- foci line segment equal concentric circles parallel lines LetS/C23R3be a smooth CONNECTED SURFACE , and let F1and F2be points in P3_S;where Pnis an n-D PROJECTIVE SPACE . Then Shas a reflection property with FOCI F1andF2if, for each point P/C23S; 1. Any vector normal to SatPlies in the SPAN of the vectors F1P})@A@})@A@!andF2P})@A@})@A@!. 2. The line normal to SatPbisects one of the pairs of opposite angles formed by the intersection of the lines joining F1andF2toP. A smooth CONNECTED SURFACE has a reflection property IFFit is part of an ELLIPSOID of revolution, a HYPERBOLOID of revolution, a PARABOLOID of revolu- tion, a SPHERE ,ora PLANE . Foci Sign Both foci finite One focus finiteBoth foci infinite distinct POSITIVE confocalellipsoidsconfocalparaboloidsparallelplanes distinct NEGATIVE confocal hyper- boloids and plane perpendicularconfocalparaboloidsparallelplanes bisector of inter- foci line segment equal concentric spheresparallelplanes See also BILLIARDS References Drucker, D. "Euclidean Hypersurfaces with Reflective Prop- erties." Geometrica Dedicata 33, 325 /C1/329, 1990. Drucker, D. "Reflective Euclidean Hypersurfaces." Geome- trica Dedicata 39, 361 /C1/362, 1991. Drucker, D. "Reflection Properties of Curves and Surfaces." Math. Mag. 65, 147 /C1/157, 1992. Drucker, D. and Locke, P. "A Natural Classification of Curves and Surfaces with Reflection Properties." Math. Mag. 69, 249 /C1/256, 1996. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 73 /C1/77, 1990. Wegner, B. "Comment on ‘Euclidean Hypersurfaces with Reflective Properties’." Geometrica Dedicata 39, 357 /C1/359, 1991. Reflection Relation A mathematical relationship relating f(/C28x)tof(x) ; or more generally, f(a /C28x)tof(x) as in the case of the GAMMA FUNCTION identity G(z)G(1 /C28z) /C30p sin( pz) : See also ARGUMENT ADDITION RELATION ,ARGUMENT MULTIPLICATION RELATION ,RECURRENCE RELATION , TRANSLATION RELATION Reflex Angle An ANGLE more than 1808. See also ACUTE ANGLE ,ANGLE ,FULL ANGLE ,OBTUSEANGLE ,RIGHT ANGLE ,STRAIGHT ANGLE Reflexible An object is reflexible if it is superposable with its image in a plane mirror. Also called AMPHICHIRAL . See also AMPHICHIRAL ,CHIRAL ,ENANTIOMER ,HAND- EDNESS ,MIRROR IMAGE ,REFLECTION References Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 130, 1987. Reflexible Map An AUTOMORPHISM which interchanges the two ver- tices of a regular map at each edge without inter- changing the vertices. See also EDMONDS’ MAP Reflexive Closure The reflexive closure of a BINARY RELATION R on a SET X is the minimal REFLEXIVE RELATION R? on X that contains R. Thus aR ?a for every element a of X and aR ?b for distinct elements a and b, provided that aRb:/ See also REFLEXIVE REDUCTION ,R EFLEXIVE RELA- TION ,RELATION ,TRANSITIVE CLOSURE Reflexive Graph DIRECTED GRAPH Reflexive Polyhedron References Skarke, H. Reflexive Polyhedra and Their Applications in String and F-Theory. 29 Feb 2000. http://xxx.lanl.gov/abs/ hep-th/0002246/. Reflexive Reduction The reflexive reduction of a BINARY RELATION R on a SET X is the minimum relation R? on X with the same REFLEXIVE CLOSURE as R. Thus aR?b for any elements a and b of X, provided that a and b are distinct and aRb :/ See also REFLEXIVE CLOSURE ,RELATION ,TRANSITIVE REDUCTION Reflexive Relation A RELATION R on a SET S is reflexive provided that xRx for every x in S. See also RELATION Reflexivity AREFLEXIVE RELATION . Region An OPEN CONNECTED SET is called a region (some- times also called a DOMAIN ). Regression A method for fitting a curve (not necessarily a straight line) through a set of points using some goodness-of-fit criterion. The most common type of regression is LINEAR REGRESSION . The term regression is sometimes also used to refer to RECURSION . See also FRACTAL ,LEAST SQUARES FITTING ,LINEAR REGRESSION ,M ULTIPLE REGRESSION ,N ONLINEAR LEAST SQUARES FITTING ,R ECURSION ,R EGRESSION COEFFICIENT ,SELF-RECURSION References Chatterjee, S.; Hadi, A.; and Price, B. Regression Analysis by Example, 3rd ed. New York: Wiley, 2000. Gardner, M. "Infinite Regress." Ch. 22 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 220 /C1/229, 1984. Kleinbaum, D. G. and Kupper, L. L. Applied Regression Analysis and Other Multivariable Methods. North Scitu- ate, MA: Duxbury Press, 1978. Passmore, J. "The Infinite Regress." In Philosophical Rea- soning. New York: Scribner’s, 1961. Regression Coefficient The slope b of a line obtained using linear LEAST SQUARES FITTING is called the regression coefficient. See also CORRELATION COEFFICIENT ,LEAST SQUARES FITTING References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 254, 1951. Regula Falsi FALSE POSITION METHOD Regular Function ANALYTIC FUNCTION ,HOLOMORPHIC FUNCTION ,REG- ULAR RATIONAL FUNCTION Regular Graph AGRAPH is said to be regular of degree rif all LOCAL DEGREES are the same number r. A 0-regular graph is an EMPTY GRAPH , a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of disconnected cycles. The first interesting case is therefore 3-regular graphs, which are called CUBIC GRAPHS (Harary 1994, pp. 14 /C1/15). Similarly, 4- and 5- regular graphs are called QUARTIC and QUINTIC GRAPHS , respectively.For an r-regular graph on nnodes. E/C301 2nr; where Eis the number of EDGES .n-UNITRANSITIVE GRAPHS are sometimes called n-regular (Harary 1994, p. 174). LetN(n;r) be the number of r-regular graphs with n points. Then 0 5r5n/C281;N(n;r)/C30N(n;n/C281;/C28r); and N(n;r)/C300 when both nand rare ODD. Zhang and Yang give N(p;r) for p512:The numbers of nonisomorphic regular graphs with nnodes are 1, 2, 2, 4, 3, 8, 6, 22, 26, 176, ... (Sloane’s A005176; Steinbach 1990). The numbers of nonisomorphic CONNECTED regular graphs of order n/C301, 2, ... are 1, 1, 1, 2, 2, 5, 4, 17, 22, 167, ... (Sloane’s A005177;Steinbach 1990) The following table gives the numbers N(n;r)o fr- regular graphs for small numbers of nodes n(Sloane’s A051031). n /N(n;0)//N(n;1)//N(n;2)//N(n;3)//N(n;4)//N(n;5)//N(n;6)/ 11 21 1 31 0141 11 151 01 01 61 12 21 1 71 02 02 01 The following table gives the number of connected regular graphs of degree ronn/C30r/C271;r/C272;... nodes for n even, and n /C30r /C271 ; r /C273; r /C275 ; ... nodes for n odd. r Sloane Numbers 4 A006820 1, 1, 2, 6, 16, 59, 265, 1544, ... 5 A006821 1, 3, 60, 7848, 3459383, ... 6 A006822 1, 1, 4, 21, 266, 7849, 367860, ... 7 A014377 1, 5, 1547, ... 8 A014378 1, 1, 6, 94, 10786, 3459386, ... 9 A014381 1, 9, 88193, ... 10 A014382 1, 1, 10, 540, 805579, ... 11 A014384 1, 13, 8037796, ... See also CAGE GRAPH ,COMPLETE GRAPH ,COMPLE- TELY REGULAR GRAPH ,C ONFIGURATION ,C UBIC GRAPH ,DISTANCE- REGULAR GRAPH ,LOCAL DEGREE , MOORE GRAPH ,Q UARTIC GRAPH ,Q UINTIC GRAPH , SUPERREGULAR GRAPH References Chartrand, G. Introductory Graph Theory. New York: Dover, p. 29, 1985. Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, p. 648, 1996. Comtet, L. "Asymptotic Study of the Number of Regular Graphs of Order Two on N." §7.3 in Advanced Combina- torics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 273 /C1/279, 1974. Faradzev, I. A. "Constructive Enumeration of Combinatorial Objects." In Proble `mes combinatoires et the´orie des graphes (Orsay, 9 /C1/13 Juillet 1976). Colloq. Internat. du C.N.R.S. Paris: Centre Nat. Recherche Scient., pp. 131 /C1/ 135, 1978. Gropp, H. "Enumeration of Regular Graphs 100 Years Ago." Discrete Math. 101,73/C1/85, 1992. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 14 and 62, 1994. Petersen, J. "Die Theorie der regula ¨ren Graphs." Acta Math. 15, 193 /C1/220, 1891. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Sachs, H. "On Regular Graphs with Given Girth." In Theory of Graphs and Its Applications: Proceedings of the Sym- posium, Smolenice, Czechoslovakia, 1963 (Ed. M. Fiedler). New York: Academic Press, 1964. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 159, 1990. Sloane, N. J. A. Sequences A005176/M0303, A005177/ M0347, A006820/M1617, A006821/M3168, A006822/ M3579, A014377, A014378, A014381, A014382, A014384, A051031 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Steinbach, P. Field Guide to Simple Graphs. Albuquerque, NM: Design Lab, 1990.Wormald, N. "Generating Random Regular Graphs." J. Algorithms 5, 247 /C1/280, 1984. Zhang, C. X. and Yang, Y. S. "Enumeration of Regular Graphs." J. Dailan Univ. Tech. 29, 389 /C1/398, 1989. Regular Isotopy The equivalence of MANIFOLDS under continuous deformation within the embedding space. KNOTS of opposite CHIRALITY have AMBIENT ISOTOPY , but not regular isotopy. See also AMBIENT ISOTOPY Regular Isotopy Invariant BRACKET POLYNOMIAL Regular Local Ring A regular local ring is a LOCAL RING R with MAXIMAL IDEAL m so that m can be generated with exactly d elements where d is the KRULL DIMENSION of the RING R. Equivalently, R is regular if the VECTOR SPACE m=m2 has dimension d. See also KRULL DIMENSION ,LOCAL RING,REGULAR RING,RING References Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry. New York: Springer-Verlag, p. 242, 1995. Regular Matrix NONSINGULAR MATRIX Regular Number A number which has a finite DECIMAL expansion. A number such as 1=3 /C300:33333... which is not regular is said to be nonregular. See also DECIMAL EXPANSION ,REPEATING DECIMAL Regular Parameterization A parameterization of a SURFACE x(u;v)i nuandvis regular if the TANGENT VECTORS @x @uand@x @v are always LINEARLY INDEPENDENT . Regular Patch A regular patch is a PATCH x:U0Rnfor which the JACOBIAN J(x)(u;v) has rank 2 for all ( u;v)/C23U:A PATCH is said to be regular at a point ( u0;v0)/C23U provided that its J ACOBIAN has rank 2 at ( u0;v0):For example, the points at f/C309p=2 in the standard parameterization of the SPHERE (cosusinf;sinusinf;cosf) are not regular. An example of a PATCH which is regular but not INJECTIVE is the CYLINDER defined parametrically by (cos u; sin u; v) with u /C23 (/C28/C12;/C12) and v /C23 (/C282 ; 2): However, if x : U 0 Rn is an injective regular patch, then x maps U diffeomorphically onto x(U) :/ See also INJECTIVE PATCH ,PATCH ,REGULAR SURFACE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 273, 1997. Regular Point If f is ANALYTIC on a DOMAIN U, then a point z0 on the boundary @U is called regular if f extends to be a ANALYTIC FUNCTION on an OPEN SET containing U and also the point z0 (Krantz 1999, p. 119). See also ORDINARY POINT References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 119, 1999. Regular Polychoron There are sixteen regular polychora, six of which are convex (Wells 1986, p. 68) and ten of which are stellated (Wells 1991, p. 209). The regular convex polychora have four principal types of symmetry axes, and the projections into 3-spaces orthogonal to these may be called the "canonical" projections (R. Towle). Of the six regular convex polychora, five are typically regarded as being analogous to the Platonic solids: the 4-simplex (a hyper-tetrahedron), the 4-cross polytope (a hyper-octahedron), the 4-cube (a hyper- cube), the 600-cell (a hyper-icosahedron), and the 120-cell (a hyper-dodecahedron). The 24-cell, how- ever, has no perfect analogy in higher or lower spaces (R. Towle). The PENTATOPE and 24-CELL are self-dual, the 16-CELL is the dual of the TESSERACT , and the 600- and 120-CELLS are dual to each other. The convex regular polychora are listed in the following table (Coxeter 1969, p. 414; Wells 1991, p. 210). Name Schla ¨fli SymbolClass /N0//N1//N2//N3/ PENTATOPE / f3; 3; 3g/ SIMPLEX 51 01 05 16-CELL / f3; 3; 4g/ CROSS POLY- TOPE82 43 21 6 TESSERACT / f4; 3; 3g/ HYPERCUBE 16 32 24 8 24-CELL / f3; 4; 3g/ 24 96 96 24 120-CELL /(5 ; 3; 3g/ 600 1200 720 120 600-CELL / f3; 3; 5g/ 120 720 1200 600Here, N0 is the number of VERTICES , N1 the number of EDGES , N2 the number of FACES , and N3 the number of cells. These quantities satisfy the identity N0 /C28N1 /C27N2 /C28N3 /C300; which is a version of the POLYHEDRAL FORMULA . See also POLYCHORON ,REGULAR POLYGON ,REGULAR POLYHEDRON References Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z. 46, 380/C1/407, 1940. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 68, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Regular Polygon Ann-sided POLYGON in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both EQUIANGULAR and EQUILATERAL ). The sum of PERPENDICULARS from any point to the sides of a regular polygon of nsides is ntimes the APOTHEM . Only certain regular polygons are " CONSTRUCTIBLE " with RULER and STRAIGHTEDGE . The terms EQUILATERAL TRIANGLE and SQUARE refer to the regular 3- and 4-polygons, respectively. Thewords for POLYGONS with n]5 sides (e.g., PENTAGON , HEXAGON ,HEPTAGON , etc.) can refer to either regular or non-regular POLYGONS , although the terms gen- erally refer to regular polygons in the absence ofspecific wording. Letsbe the side length, rbe the INRADIUS , and Rthe CIRCUMRADIUS of a regular polygon. Then s/C302rtanp n ! (1) /C302Rsinp n ! (2) r/C301 2scotp n ! (3) /C30Rcosp n ! (4) R/C3012scscp n ! (5) /C30rsecp n ! (6) A/C301 4ns2cotp n ! (7) /C30nr2tanp n ! (8) /C3012nR2sin2p n ! : (9) If the number of sides is doubled, then s2n/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R2/C28Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4R2/C28s2 nqr (10) A2n/C304rAn 2r/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2/C27s2 np : (11) Furthermore, if pkandPkare the PERIMETERS of the regular polygons inscribed in and circumscribed around a given CIRCLE andakandAktheir areas, then P2n/C302pnPn pn/C27Pn(12) p2n/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi pnP2np ; (13) and a2n/C30ffiffiffiffiffiffiffiffiffiffiffi anAnp (14) A2n/C302a2nAn a2n/C27An(15) (Beyer 1987, p. 125). The following table gives parameters for the first few regular polygons, where ais the vertex angle, bis the central angle, ris the INRADIUS ,Ris the CIRCUMRA- DIUS, and Ais the area (Williams 1979, p. 33). /fng//a// b/ rR A /f3g//1 3p/C3060(//23p/C30120(//16ffiffiffi 3p //1 3ffiffiffi 3p //1 4ffiffiffi 3p / /f4g//1 2p/C3090(//12p/C3090(//12//12ffiffiffi 2p / 1 /f5g//3 5p/C30108(//25p/C3072(//1 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pp //1 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2710ffiffiffi 5pp //1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pp //f6g//2 3p/C30120(//13p/C3060(//12ffiffiffi 3p / 1 /3 2ffiffiffi 3p / /f7g//5 7p/C30900 7( //27p/C30360 7( //12cot17p})@D})@E //12csc17p})@D})@E //74cot17p})@D})@E / /f8g//3 4p/C30135(//14p/C3045(//121/C27ffiffiffi 2p})0})@ //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C272ffiffiffi 2pp // 2ð1þffiffiffi2p /) /f9g//7 9p/C30140(//29p/C3040(//12cot19p})@D})@E //12csc19p})@D})@E //94cot19p})@D})@E / /f10g//4 5p/C30144(//15p/C3036(//12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pp //1 21/C27ffiffiffi 5p})0})@ //5 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pp / /f11g//9 11p/C301620 11( //2 11p/C30360 11( //1 2cot1 11p})@D})@E //12csc1 11p})@D})@E //11 4cot1 11p})@D})@E / /f12g//5 6p/C30150/C1416/308 /122/C27ffiffiffi 3p})0})@ //1 2ffiffiffi 2p /C27ffiffiffi 6p})0})@ // 3ð2þffiffiffi3p /) COMPASS and STRAIGHTEDGE constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, ..., sides. However, this listing is not a completeenumeration of "constructible" polygons. In fact, a regular n-gon is constructible only if f(n)i sa POWER of 2, where fis the TOTIENT FUNCTION (this is a NECESSARY but not SUFFICIENT condition). More specifically, a regular n-gon ( /n]3) can be con- structed by STRAIGHTEDGE and COMPASS (i.e., can have trigonometric functions of its ANGLES expressed in terms of finite SQUARE ROOT extractions) IFF n/C302kp1p2/C1/C1/C1ps; (16) where kis in INTEGER ]0 and the piare distinct FERMAT PRIMES .FERMAT NUMBERS are OF THE FORM Fm/C3022m/C271; (17) where mis an INTEGER ]0:The only known PRIMES of this form are 3, 5, 17, 257, and 65537. The fact that this condition was SUFFICIENT was first proved by Gauss in 1796 when he was 19 years old, and it relies on the property of IRREDUCIBLE POLY- NOMIALS that ROOTS composed of a finite number of SQUARE ROOT extractions exist only if the order of the equation is OF THE FORM 2h:That this condition was also NECESSARY was not explicitly proven by Gauss, and the first proof of this fact is credited to Wantzel (1836). Constructible values of nfornB300 were given by Gauss (Smith 1994), and the first few are 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, ... (Sloane’s A003401). Gardner (1977) and indepen-dently Watkins (Conway and Guy 1996) noticedthat the number of sides for constructible polygons with an ODD number of sides are given by the first 32 rows of P ASCAL’S TRIANGLE (mod 2) interpreted as BINARY numbers, giving 1, 3, 5, 15, 17, 51, 85, 255, ... (Sloane’s A004729, Conway and Guy 1996, p. 140). Although constructions for the regular TRIANGLE , SQUARE , PENTAGON , and their derivatives had been given by Euclid, constructions based on the FERMAT PRIMES ]17 were unknown to the ancients. The first explicit construction of a HEPTADECAGON (17-gon) was given by Erchinger in about 1800. Richelot and Schwendenwein found constructions for the 257-GON in 1832, and Hermes spent 10 years on the construc- tion of the 65537-GON at Go¨ttingen around 1900 (Coxeter 1969). Constructions for the EQUILATERAL TRIANGLE and SQUARE are trivial (top figures below). Elegant constructions for the PENTAGON and HEPTA- DECAGON are due to Richmond (1893) (bottom figures below). Given a point, a CIRCLE may be constructed of any desired RADIUS , and a DIAMETER drawn through the center. Call the center O, and the right end of the DIAMETER P0 : The DIAMETER PERPENDICULAR to the original DIAMETER may be constructed by finding the PERPENDICULAR BISECTOR . Call the upper endpoint of this PERPENDICULAR DIAMETER B. For the PENTAGON , find the MIDPOINT of OB and call it D. Draw DP0 ; and BISECT /C218ODP0 ; calling the intersection point with OP0 N1 : Draw N1P1 PARALLEL to OB, and the first two points of the PENTAGON are P0andP1:The construc- tion for the HEPTADECAGON is more complicated, but can be accomplished in 17 relatively simple steps. The construction problem has now been automated (Bishop 1978). See also 257-GON , 65537-GON ,CHAOS GAME,CONSTRUC- TIBLE POLYGON , DE MOIVRE NUMBER ,EQUILATERAL TRIANGLE ,H EPTADECAGON ,H EXAGON ,H EXAGRAM , OCTAGON ,PENTAGON ,PENTAGRAM ,POLYGON ,POLY- GON CIRCUMSCRIBING CONSTANT ,POLYGON INSCRIB-ING CONSTANT ,SQUARE ,STAR POLYGON References Bishop, W. "How to Construct a Regular Polygon." Amer. Math. Monthly 85, 186/C1/188, 1978. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 140 and 197 /C1/202, 1996. Courant, R. and Robbins, H. "Regular Polygons." §3.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 122 /C1/125, 1996. Coxeter, H. S.M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97/C1/108, 1991. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field(Ed. J. W. A. Young). New York: Dover, pp. 352 /C1 /386, 1955. Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, p. 207, 1977. Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. Translated by A. A Clarke. NewHaven, CT: Yale University Press, 1965. Harris, J. W. and Stocker, H. "Regular n-gons (Polygons)." §3.7 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 86 /C1 /89, 1998. Math Forum. "Naming Polygons and Polyhedra." http:// forum.swarthmore.edu/dr.math/faq/faq.polygon.na-mes.html. Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 238, 1997. Richmond, H. W. "A Construction for a Regular Polygon of Seventeen Sides." Quart. J. Pure Appl. Math. 26, 206/C1 / 207, 1893. Sloane, N. J. A. Sequences A003401/M0505 and A004729 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 350, 1994. Tietze, H. Ch. 9 in Famous Problems of Mathematics. New York: Graylock Press, 1965. Wantzel, M. L. "Recherches sur les moyens de reconnaı ˆtre si un Proble `me de Ge ´ome´trie peut se re ´soudre avec la re `gle et le compas." J. Math. pures appliq. 1, 366/C1 /372, 1836. Williams, R. "Polygons." §2/C1/1i n The Geometrical Founda- tion of Natural Structure: A Source Book of Design. New York: Dover, pp. 31 /C1/33, 1979. Regular Polyhedron A polyhedron is said to be regular if its FACES and VERTEX FIGURES are REGULAR (not necessarily CON- VEX) polygons (Coxeter 1973, p. 16). Using this definition, there are a total of nine regular polyhedra, five being the CONVEX PLATONIC SOLIDS and four being the CONCAVE (stellated) K EPLER- POINSOT SO- LIDS. However, the term "regular polyhedra" is some- times used to refer exclusively to the CONVEX PLATONIC SOLIDS . It can be proven that only nine regular solids (in theCoxeter sense) exist by noting that a possible regular polyhedron must satisfy cos2p p ! /C27cos2p q ! /C27cos2p r ! /C301: Gordon showed that the only solutions to 1 /C27cos f1 /C27cos f2 /C27cos f3 /C300 OF THE FORM fi /C30 pmi =niare the permutations of (2 3 p;23 p;13 p) and (23 p;25 p;45 p) : This gives three per- mutations of (3, 3, 4) and six of (3, 5,5 3) as possible solutions to the first equation. Plugging back in gives the SCHLA ¨ FLI SYMBOLS of possible regular polyhedra as f3; 3g;f3; 4g;f4 ; 3 g;f3; 5 g;f5; 3g;f3;52 g;f52; 3g; f5;52 g; and f52 ; 5 g (Coxeter 1973, pp. 107 /C1/109). The first five of these are the PLATONIC SOLIDS and the remaining four the KEPLER- POINSOT SOLIDS . Every regular polyhedron has e /C271 axes of symmetry, where e is the number of EDGES , and 3h=2 PLANES of symmetry, where h is the number of sides of the corresponding PETRIE POLYGON . See also CONVEX POLYHEDRON ,K EPLER- POINSOT SOLID,PETRIE POLYGON ,PLATONIC SOLID ,POLYHE- DRON ,P OLYHEDRON COMPOUND ,S PONGE ,V ERTEX FIGURE References Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z. 46, 380 /C1/407, 1940. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 1 /C1/17, 93, and 107 /C1/112, 1973. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, pp. 85 /C1/86, 1997. Regular Polytope REGULAR POLYCHORON Regular Prime A PRIME which does not DIVIDE the CLASS NUMBER h(p) of the CYCLOTOMIC FIELD obtained by adjoining a PRIMITIVE PTH ROOT OF UNITY to the FIELD of rationals. A PRIME p is regular IFF p does not divide the NUMERATORS of the BERNOULLI NUMBERS B0 ; B2 ; ..., Bp /C283 : A PRIME which is not regular is said to be an IRREGULAR PRIME . In 1915, Jensen proved that there are infinitely many IRREGULAR PRIMES . It has not yet been proven that there are an INFINITE number of regular primes (Guy 1994, p. 145). Of the 283,145 PRIMES B4 /C29106 ; 171,548 (or 60.59%) are regular (the conjectured FRACTION is e/C281 =2 :60:65%) : The first few are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, ... (Sloane’s A007703). See also BERNOULLI NUMBER ,FERMAT’S THEOREM , IRREGULAR PRIMEReferences Buhler, J.; Crandall, R. Ernvall, R.; and Metsankyla, T. "Irregular Primes and Cyclotomic Invariants to Four Million." Math. Comput. 61, 151 /C1/153, 1993. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 145, 1994. Ribenboim, P. "Regular Primes." §5.1 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 323 /C1/329, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 153, 1993. Sloane, N. J. A. Sequences A007703/M2411 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Regular Pyramid PYRAMID Regular Ring In the sense of von Neumann, a regular ring is a RING R such that for all a /C23 R; there exists a b /C23 R satisfying a /C30aba. See also REGULAR LOCAL RING,RING References Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, p. 196, 1989. Regular Sequence Let there be two PARTICULARLY WELL-BEHAVED FUNC- TIONS F(x) and pt(x) : If the limit lim t00 g/C12 /C28/C12pt(x)F(x) dx exists, then pt(x) is a regular sequence of PARTICU- LARLY WELL-BEHAVED FUNCTIONS . References Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences." Theoret. Comput. Sci. 98,16/C1/197, 1992. Regular Singular Point Consider a second-order ORDINARY DIFFERENTIAL EQUATION yƒP(x)y?/C27Q(x)y /C300: If P(x) and Q(x) remain FINITE at x /C30x0 ; then x0is called an ORDINARY POINT . If either P(x)or Q(x) diverges as x 0 x0 ; then x0 is called a singular point. If either P(x)or Q(x) diverges as x 0 x0but x /C28x0 ðÞ P(x) and x /C28x0 ðÞ2Q(x) remain FINITE asx0 x0;then x/C30x0is called a regular singular point (or NONESSENTIAL SINGULARITY ). See also IRREGULAR SINGULARITY ,SINGULAR POINT (DIFFERENTIAL EQUATION ) References Arfken, G. "Singular Points." §8.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 451 /C1/453 and 461 /C1/463, 1985. Regular Singularity REGULAR SINGULAR POINT Regular Skew Polyhedron A regular skew polyhedron is a polyhedron whose faces and VERTEX FIGURES are regular SKEW POLY- GONS . There are only three regular skew polyhedra in Euclidean 3-space (Coxeter 1937, Garner 1967), the simplest of which is f4; 6½4g:/ Garner (1967) considered regular skew polyhedra in hyperbolic space H3 ; and shows that there are exactly 32 which are derived from honeycombs whose cells and vertex figures are derived from honeycombs whose cells and vertex figures are not inscribed in equidistant surfaces. See also REGULAR POLYHEDRON References Coxeter, H. S. M. "Regular Skew Polyhedra in Three and Four Dimensions." Proc. London Math. Soc. 43,33/C1/62, 1937. Garner, C. W. L. "Regular Skew Polyhedra in Hyperbolic Three-Space." Canad. J. Math. 19, 1179 /C1/1186, 1967. Regular Surface A SUBSET M ƒRn is called a regular surface if for each point p /C23 M ; there exists a NEIGHBORHOOD V of p in Rn and a MAP x : U 0 Rn of an OPEN SET U ƒR2 onto V S M such that 1. x is differentiable, 2. x : U 0 V S M is a HOMEOMORPHISM , and 3. Each map x : U 0 M is a REGULAR PATCH . Any open subset of a regular surface is also a regular surface. See also REGULAR PATCH References Gray, A. "The Definition of a Regular Surface in Rn :/" §12.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 281 /C1/286, 1997. Regular Triangle Center A TRIANGLE CENTER is regular IFF there is a TRIANGLE CENTER FUNCTION which is a POLYNOMIAL in D; a, b, and c (where D is the AREA of the TRIANGLE ) such that the TRILINEAR COORDINATES of the center are f(a; b; c):f(b; c ; a):f(c ; a; b) : The ISOGONAL CONJUGATE of a regular center is a regular center. Furthermore, given two regular cen-ters, any two of their HARMONIC CONJUGATE POINTS are also regular centers. See also ISOGONAL CONJUGATE ,TRIANGLE CENTER , TRIANGLE CENTER FUNCTION Regular Variation References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 275 /C1/ 276, 1971. Regularity Axiom AXIOM OF FOUNDATION Regularity Lemma SZEMERE ´ DI’S REGULARITY LEMMA Regularity Theorem An AREA -minimizing surface (RECTIFIABLE CURRENT ) bounded by a smooth curve in R3is a smooth submanifold with boundary. See also MINIMAL SURFACE ,RECTIFIABLE CURRENT References Morgan, F. "What is a Surface?" Amer. Math. Monthly 103, 369 /C1/376, 1996. Regularized Beta Function The regularized beta function is defined by I(z; a ; b) /C30B(z; a ; b) B(a; b); where B(z; a; b) is the incomplete BETA FUNCTION and B(a; b) is the complete BETA FUNCTION . The regularized beta function is sometimes also denoted Iz(a; b) and is implemented in Mathematica as BetaRegularized [z, a, b]. The four-argument ver- sionBetaRegularized [z1, z2, a, b] is equivalent to Iz2;a;b ðÞ /C28Iz1;a;b ðÞ :/ See also BETA FUNCTION ,R EGULARIZED GAMMA FUNCTION Regularized Gamma Function The regularized gamma functions are defined by P(a;z)/C301/C28Q(a;z)/C13g(a;z) G(a)(1) and Q(a;z)/C301/C28P(a;z)/C13G(a;z) G(a); where g(a;z) and G(a;z) are INCOMPLETE GAMMA FUNCTIONS and G(a) is a complete GAMMA FUNCTION . The function Q(a ; z) is implemented in Mathematica asGammaRegularized [a, z]. The derivatives of P(a ; z) and Q(a ; z) are d dzP(a ; z) /C30e /C28zza /C281 G(a) (2) d dzQ(a; z) /C30e /C28zza /C281 G(a); (3) and the second derivatives are d2 dz2P(a; z) /C30e /C28z(a /C28 z /C28 1)za /C282 G(a) (4) d2 dz2Q(a; z) /C30e /C28z(1 /C27 z /C28 a)za /C282 G(a) (5) The integrals are g P(a; z) dz /C30zG(a) /C28 z G(a; z) /C27G(a /C27 1; z) G(a) (6) g Q(a ; z) dz /C30z G(a; z) /C28G(a /C27 1; z) G(a) (7) See also GAMMA FUNCTION ,INCOMPLETE GAMMA FUNCTION ,REGULARIZED BETA FUNCTION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 160 /C1/161, 1992. Regularized Long-Wave Equation The PARTIAL DIFFERENTIAL EQUATION ut /C27ux /C286uux /C28utxx /C300: See also KORTEWEG-DE VRIES EQUATION References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 49, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997. Regulus The locus of lines meeting three given SKEW LINES . ("Regulus" is also the name of the brightest star in the constellation Leo.)Reidemeister Moves In the 1930s, Reidemeister first rigorously proved that KNOTS exist which are distinct from the UNKNOT . He did this by showing that all KNOT deformations can be reduced to a sequence of three types of "moves," called the (I) TWIST MOVE , (II) POKE MOVE , and (III) SLIDE MOVE . These moves are most com- monly called Reidemeister moves, although the term "equivalence moves" is sometimes also used (Aneziris 1999, p. 29). REIDEMEISTER’S THEOREM guarantees that moves I, II, and III correspond to AMBIENT ISOTOPY (moves II and III alone correspond to REGULAR ISOTOPY ). He then defined the concept of COLORABILITY , which is invariant under Reidemeister moves. See also AMBIENT ISOTOPY ,COLORABLE ,KNOT MOVE, MARKOV MOVES ,REGULAR ISOTOPY ,UNKNOT References Aneziris, C. N. "The Equivalence Moves." Ch. 4 in The Mystery of Knots: Computer Programming for Knot Tabu- lation. Singapore: World Scientific, pp. 29 /C1/33, 1999. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Reidemeister, K. "Knotten und Gruppen." Abh. Math. Sem. Univ. Hamburg 5,7/C1/23, 1927. Reidemeister’s Theorem Two LINKS can be continuously deformed into each other IFF any diagram of one can be transformed into a diagram of the other by a sequence of R EIDEMEISTER MOVES . See also REIDEMEISTER MOVES Reinhardt Domain A Reinhardt domain with center cis a DOMAIN Din Cnsuch that whenever Dcontains z0;the DOMAIN D also contains the closed POLYDISK . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 101, 1980. Relation A relation is any SUBSET of a CARTESIAN PRODUCT . For instance, a SUBSET of A /C29B ; called a "BINARY RELATION from A to B," is a collection of ORDERED PAIRS (a, b) with first components from A and second components from B, and, in particular, a SUBSET of A /C29A is called a "relation on A." For a BINARY RELATION R, one often writes aRb to mean that (a, b)isin R. See also ADJACENCY RELATION ,ANTISYMMETRIC RE- LATION ,ARGUMENT ADDITION RELATION ,ARGUMENT MULTIPLICATION RELATION ,BINARY RELATION ,CLO- SURE RELATION ,C OVER RELATION ,E QUIVALENCE RELATION ,IRREFLEXIVE ,P ARTIAL ORDER ,R ECUR- RENCE RELATION ,REFLECTION RELATION ,REFLEXIVE RELATION ,S YMMETRIC RELATION ,T RANSITIVE , TRANSLATION RELATION Relational System This entry contributed by VIKTOR BENGTSSON A relational system is a structure R /C30 S; Pi : i /C23 I fg ; fj : j /C23 J})*})+ })0})@ consisting of a set S,a collection of relations Pi(i /C23 I)on S, and a collection of functions fj(j /C23 J)on S. Relative Cumulative Frequency The CUMULATIVE FREQUENCY in a FREQUENCY DIS- TRIBUTION divided by the total number of data points. See also ABSOLUTE FREQUENCY ,CUMULATIVE FRE- QUENCY ,FREQUENCY DISTRIBUTION ,RELATIVE FRE- QUENCY References Kenney, J. F. and Keeping, E. S. "Frequency Distributions." §1.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 12 /C1/19, 1962. Relative Degree DEGREE (EXTENSION FIELD Relative Entropy Let a DISCRETE DISTRIBUTION have probability func- tion pk ; and let a second DISCRETE DISTRIBUTION have probability function qk : Then the relative entropy of p with respect to q, also called the Kullback-Leibler distance, is defined by d /C30X kpk lnpk qk ! : Although relative entropy does not satisfy the trian- gle inequality and is therefore not a true metric, it satisfies many important mathematical properties. For example, it is a convex function of pk ; is always nonnegative, and equals zero only if pk /C30qk :/Relative entropy is a very important concept in quantum information theory, as well as statistical mechanics (Qian 2000). See also ENTROPY References Cover, T. M. and Thomas, J. A. Elements of Information Theory. New York: Wiley, 1991. Qian, H. Relative Entropy: Free Energy Associated with Equilibrium Fluctuations and Nonequilibrium Deviations. 8 Jul 2000. http://xxx.lanl.gov/abs/math-ph/0007010/. Relative Error Let the true value of a quantity be x and the measured or inferred value x0 : Then the relative error is defined by dx /C30Dx x/C30x0 /C28 x x/C30x0 x/C281; where Dx is the ABSOLUTE ERROR . The relative error of the QUOTIENT or PRODUCT of a number of quantities is less than or equal to the SUM of their relative errors. The PERCENTAGE ERROR is 100% times the relative error. See also ABSOLUTE ERROR ,E RROR PROPAGATION , PERCENTAGE ERROR References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Relative Extremum A RELATIVE MAXIMUM or RELATIVE MINIMUM , also called a LOCAL EXTREMUM . See also EXTREMUM ,GLOBAL EXTREMUM ,RELATIVE MAXIMUM ,RELATIVE MINIMUM Relative Frequency The ratio of the ABSOLUTE FREQUENCY to the total number of data points in a FREQUENCY DISTRIBUTION . See also ABSOLUTE FREQUENCY ,CUMULATIVE FRE- QUENCY ,FREQUENCY DISTRIBUTION ,RELATIVE CUMU- LATIVE FREQUENCY References Kenney, J. F. and Keeping, E. S. "Frequency Distributions." §1.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 12 /C1/19, 1962. Relative Maximum A MAXIMUM within some NEIGHBORHOOD which need not be a GLOBAL MAXIMUM . See also GLOBAL MAXIMUM ,M AXIMUM ,R ELATIVE MINIMUM Relative Minimum A MINIMUM within some NEIGHBORHOOD which need not be a GLOBAL MINIMUM . See also GLOBAL MINIMUM ,M INIMUM ,R ELATIVE MAXIMUM Relative Topology If A ƒB and B has a topology of open sets Ua then the relative topology on A is given by the collection of open sets Ua S A:/ Relatively Prime Two integers are relatively prime if they share no common positive factors (divisors) except 1. Using the notation (m, n) to denote the GREATEST COMMON DIVISOR , two integers m and n are relatively prime if (m; n) /C301 : Relatively prime integers are sometimes also called STRANGERS or COPRIME and are denoted m /C222n:/ The probability that two INTEGERS picked at random are relatively prime is [ z(2)] /C281 /C306=p2 ; where z(z)is the RIEMANN ZETA FUNCTION (Wells 1986, p. 28). This result is related to the fact that the GREATEST COMMON DIVISOR of m and n,(m; n) /C30k; can be interpreted as the number of LATTICE POINTS in the PLANE which lie on the straight LINE connecting the VECTORS (0; 0) and (m, n) (excluding (m, n) itself). In fact, 6=p2 is the fractional number of LATTICE POINTS VISIBLE from the ORIGIN (Castellanos 1988, pp. 155 /C1/ 156). Given three INTEGERS chosen at random, the prob- ability that no common factor will divide them all is [z(3)]/C281 :1:20206 /C281 :0:831907 ; (1) where z(3) is APE´ RY’S CONSTANT (Wells 1986, p. 29). This generalizes to k random integers (Schoenfeld 1976). See also DIVISOR ,G REATEST COMMON DIVISOR , HAFNER- SARNAK- MCCURLEY CONSTANT ,VISIBILITY References Castellanos, D. "The Ubiquitous Pi." Math. Mag. 61,67/C1/98, 1988. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 3 /C1/4, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 38 /C1/39, 1998. Nagell, T. "Relatively Prime Numbers. Euler’s 8/-Function." §8in Introduction to Number Theory. New York: Wiley, pp. 23 /C1/26, 1951. Schoenfeld, L. "Sharper Bounds for the Chebyshev Func- tions u(x) and c(x) ; II." Math. Comput. 30, 337 /C1/360, 1976. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 28 /C1/ 29, 1986.Relaxation Methods Methods of solving an ORDINARY DIFFERENTIAL EQUA- TION by replacing it with a FINITE DIFFERENCE equation on a regular grid spanning the domain of interest. The finite difference equations are then solved using an n-D NEWTON’S METHOD or other similar algorithm. References Jeffreys, H. and Jeffreys, B. S. "Relation Methods." §9.18 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 307 /C1/312, 1988. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Richardson Extrapolation and the Bulirsch- Stoer Method." §17.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 753 /C1/763, 1992. Remainder In general, a remainder is a quantity "left over" after performing a particular algorithm. The term is most commonly used to refer to the number left over when two integers are divided by each other in INTEGER DIVISION . For example, 55_7 /C307; with a remainder of 6. Of course in real division, there is no such thing as a remainder since, for example, 55=7 /C307 /C276=7 :/ The term remainder is also sometimes applied to the RESIDUE of a CONGRUENCE . See also DIVISION ,INTEGER DIVISION ,Q UOTIENT , RESIDUE (CONGRUENCE ) References Nagell, T. "Remainders." §2i n Introduction to Number Theory. New York: Wiley, pp. 12 /C1/13, 1951. Remainder Theorem POLYNOMIAL REMAINDER THEOREM Rembs’ Surface A surface of constant G AUSSIAN CURVATURE that can be given parametrically by x/C30a(Ucosu/C28U?sinu) (1) y/C30/C28a(Usinu/C28U?cosu) (2) z/C30v/C28aV?; (3) where U /C13cosh uffiffiffiffi Cp})@D})@E ffiffiffiffiCp (4) V /C13cos vffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C27 1p})0})@ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiC /C27 1p (5) a /C13 2V (C /C27 1) U2 /C28 V2 ðÞ; (6) and U ?/C30dU =du; and V ?/C30dV =dv : The value of v is restricted to ½v ½5v0 /C13p 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C27 1p (7) (Reckziegel 1986), and the values v /C309v0 correspond to the ends of the cleft in the surface. The surface illustrated above corresponds to C /C301. Rembs’ surface has FIRST FUNDAMENTAL FORM coeffi- cients E /C3016C(1 /C27 C) cos2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C27 1p})0})@ cosh2 uffiffiffiffi Cp})@D})@E 1 /C28 C cos 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C27 1p})0})@ /C27 (C /C27 1) cosh 2uffiffiffiffi Cp})@D})@E hi2 (8) F /C300 (9) G /C30 1 /C27 2C /C27 C cos 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C27 1p})0})@ /C27 (C /C27 1) cosh 2uffiffiffiffi Cp})@D})@E hi2 1 /C28 C cos 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C27 1p})0})@ /C27 (C /C27 1) cosh 2uffiffiffiffi Cp})@D})@E hi2 ; (10) SECOND FUNDAMENTAL FORM coefficients by similar, rather complicated expressions. The GAUSSIAN CUR- VATURE is K /C301; (11) with the MEAN CURVATURE given by a rather compli- cated expression. See also KUEN SURFACE ,SIEVERT’S SURFACE References Fischer, G. (Ed.). Plate 88 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 84, 1986. Reckziegel, H. "Sievert’s Surface." §3.4.4.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 39 /C1/40, 1986. Rembs, E. "Enneper’sche Fla¨chen konstanter positiver Kru¨mmung und Hazzidakissche Transformationen." Jahrber. DMV 39, 278 /C1/283, 1930. Remes Algorithm REMEZ ALGORITHMRemez Algorithm Portions of this entry contributed by CHARLES BOND Portions of this entry contributed by RONALD M. AARTS An algorithm for determining optimal coefficients for digital FILTERS . The Remez algorithm in effect goes a step beyond the MINIMAX APPROXIMATION algorithm to give a slightly finer solution to an approximation problem. The Remez exchange algorithm (Remez 1957) was first studied by Parks and McClellan (1972). The algorithm is an iterative procedure consisting of two steps. One step is the determination of candidate FILTER coefficients h(n) from candidate "alternation frequencies," which involves solving a set of linear equations. The other step is the determination of candidate alternation frequencies from the candidate FILTER coefficients (Lim and Oppenheim 1988). Ex- perience has shown that the algorithm converges very fast, and is widely used in practice to design optimal FILTERS . A FORTRAN implementation is given by Rabiner (1975). A description emphasizing the mathematical foundations rather than digital signal processing applications is given by Cheney (1999), who also spells Remez as Remes (Cheney 1966, p. 96). See also FILTER ,MINIMAX APPROXIMATION References Cheney, E. W. Introduction to Approximation Theory, 2nd ed. Providence, RI: Amer. Math. Soc., 1999. Lim, J S. and Oppenheim, A V. (Eds). Advanced Topics in Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1988. Parks, T. W. and McClellan, J. J. "Chebyshev Approxima- tion for Nonrecursive Digital Filters with Linear Phase." IEEE Trans. Circuit Th. 19, 189 /C1/194, 1972. Rabiner, L. W. and Gold, B. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Pre- ntice-Hall, 1975. Remez, E. Ya. General Computational Methods of Cheby- shev Approximation. Atomic Energy Translation 4491. Kiev, 1957. Removable Crossing REDUCIBLE CROSSING Removable Singularity A SINGULAR POINT z0 of a FUNCTION f(z) for which it is possible to assign a COMPLEX NUMBER in such a way that f(z) becomes ANALYTIC . A more precise way of defining a removable singularity is as a singularity z0 of a function f(z) about which the function f(z)is bounded. For example, the point x0/C300 is a removable singularity in the SINC FUNCTION sinc x/C30sinx=x; since this function satisfies sinc 0 /C301:/ See also ESSENTIAL SINGULARITY ,POLE,R IEMANN REMOVABLE SINGULARITY THEOREM ,SINGULAR POINT (FUNCTION ) References Krantz, S. G. "Removable Singularities, Poles, and Essential Singularities." §4.1.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 42, 1999. Rencontres Number DERANGEMENT ,SUBFACTORIAL Rendezvous Values MAGIC GEOMETRIC CONSTANTS Re´nyi’s Parking Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. Given the CLOSED INTERVAL [0;x] with x/C211, let 1-D "cars" of unit length be parked randomly on the interval. The MEAN number M(x) of cars which can fit (without overlapping!) satisfies M(x)/C300 for 0 5xB1 1/C272 x/C281gx/C281 0M(y)dy forx]1:8 < :(1) The mean density of the cars for large xis m/C13lim x0/C12MðxÞ x¼g/C12 0exp})@* /C282gx 01/C28e/C28v y})@+ dx /C300:7475979202 . . . (2) (Sloane’s A050996). While the inner integral can be done analytically, f(x)/C30g/C27G(0;x)/C27lnx; (3) where gis the E ULER- MASCHERONI CONSTANT and G(0;x) is the incomplete GAMMA FUNCTION , it is not known how to do the outer one m/C30g/C12 0exp[/C282f(x)]dx (4) /C30e/C282gg/C12 0e/C282G(0;x) x2(5) /C302/C282gg/C12 0e/C282ei(/C28x) x2; (6) where ei( x) is the EXPONENTIAL INTEGRAL . The slowly converging series expansion for the integrand is given by e/C282ei(/C28x) x2/C301/C282x/C275 2x2/C2822 9x3/C27293144x4/C2827111800x5/C27... ( 7 ) (Sloane’s A050994 and A050995). In addition, M(x)/C30mx/C27m/C281/C27O(x/C28n) (8)for all n(Re´nyi 1958), which was strengthened by Dvoretzky and Robbins (1964) to M(x)/C30mx/C27m/C281/C27O2e x !x/C283=22 435 (9) Dvoretzky and Robbins (1964) also proved that inf x5t5x/C271M(t)/C271 t/C2715m5sup x5t5x/C271M(t)/C271 t/C271: (10) LetV(x) be the variance of the number of cars, then Dvoretzky and Robbins (1964) and Mannion (1964) showed that v/C13lim z0/C12V(x) x /C302g/C12 0xg1 0e/C28xyR2(y)dy/C27x2g/C12 0e/C28xyR1(y)dy})10})1@ 2() /C29exp/C282gx 01/C28e/C28y ydy ! dx/C300:038156 . . . ;(11) where R1(x)/C30M(x)/C28mx/C28m/C271 (12) R2(x)/C30 (1/C28m/C28mx)2 for 05x51 4(1/C28m)2 forx/C301 2 x/C281gx/C281 0R2(y)dy/C27gx/C281 0R1(y)R1(x/C28y/C281)dy"# forx>18 >>>>>>>>>< >>>>>>>>>: (13) and the numerical value is due to Blaisdell and Solomon (1970). Dvoretzky and Robbins (1964) also proved that inf x5t5x/C271V(t) t/C2715v5sup x5t5x/C271V(t) t/C271; (14) and that V(x)/C30vx/C27v/C27O4e x !x/C2842 435: (15) Palasti (1960) conjectured that in 2-D, lim x;y0/C12M(x;y) xy/C30m2; (16) but this has not yet been proven or disproven (Finch). References Blaisdell, B. E. and Solomon, H. "On Random Sequential Packing in the Plane and a Conjecture of Palasti." J. Appl. Prob. 7, 667 /C1/698, 1970. Dvoretzky, A. and Robbins, H. "On the Parking Problem." Publ. Math. Inst. Hung. Acad. Sci. 9, 209 /C1/224, 1964. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/renyi/renyi.html. Mannion, D. "Random Space-Filling in One Dimension." Publ. Math. Inst. Hung. Acad. Sci. 9, 143 /C1/154, 1964. Palasti, I. "On Some Random Space Filling Problems." Publ. Math. Inst. Hung. Acad. Sci. 5, 353 /C1/359, 1960. Re´nyi, A. "On a One-Dimensional Problem Concerning Random Space-Filling." Publ. Math. Inst. Hung. Acad. Sci. 3, 109 /C1/127, 1958. Sloane, N. J. A. Sequences A050994, A050995, and A050996 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Solomon, H. and Weiner, H. J. "A Review of the Packing Problem." Comm. Statist. Th. Meth. 15, 2571 /C1/2607, 1986. Repartition ADE´ LE Repdigit A number composed of a single digit is called a repdigit. If the digits are all 1s, the repdigit is called a REPUNIT . The BEAST NUMBER 666 is a repdigit. See also KEITH NUMBER ,REPUNIT Repeated Integral A repeated integral is an integral taken multiple times over a single variable (as distinguished from a MULTIPLE INTEGRAL , which consists of a number of integrals taken with respect to different variables). The first FUNDAMENTAL THEOREM OF CALCULUS states that if F(x) /C30D /C281 f(x) is the INTEGRAL of f(x) ; then gx 0f(t) dt /C30F(x) /C28F(0) : (1) Now, if F(0) /C300 ; then F(x) /C30g f(x) dx /C30gx 0f(t) dt: It follows by induction that if F(0) /C30F(F(0)) /C30.../C300; then the n-fold integral of f(x) is given by D/C28n f(x) /C30g/C1/C1/C1gx 0|fflfflfflfflffl{zfflfflfflfflffl} nf(x) dx /C30gx 0f(t)(x /C28 t)n/C281 (n /C28 1)!dt: (2) Similarly, if Fx0ðÞ/C30FFx0ðÞðÞ /C30.../C300; then g/C1/C1/C1gx x0|fflfflfflfflffl{zfflfflfflfflffl} nf(x) dx /C30gx x0f(t)(x /C28 t)n/C281 (n /C28 1)!dt: (3) See also FRACTIONAL INTEGRAL ,FUBINI THEOREM ,INTEGRAL ,MULTIPLE INTEGRAL Repeating Decimal A number whose decimal representation eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely) is called a repeating decimal. Numbers such as 0.5 can be regarded as repeating decimals since 0 :5 /C300:5000... /C300:4999 ... : All RA- TIONAL NUMBERS have repeating decimals, e.g., 1=11 /C300:09: However, TRANSCENDENTAL NUMBERS , such as p /C303 :141592... do not. If 1=m is a repeating decimal and 1=n is a terminating decimal, them 1=(mn) has a nonperiodic part whose length is that of 1 =n and a repeating part whose length is that of 1 =m(Wells 1986, p. 60). See also CYCLIC NUMBER ,D ECIMAL EXPANSION , EULER’S TOTIENT RULE,FULL REPTEND PRIME ,IRRA- TIONAL NUMBER ,M IDY’S THEOREM ,RATIONAL NUM- BER,REGULAR NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 53 /C1/54, 1987. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 167 /C1/168, 1996. Courant, R. and Robbins, H. "Rational Numbers and Periodic Decimals." §2.2.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 66 /C1/68, 1996. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 60, 1986. Repfigit Number KEITH NUMBER Replicate One out of a set of identical observations in a given experiment under identical conditions. Replicating Symbol SHAH FUNCTION Representation A representation of a GROUP Gis a GROUP ACTION of Gon a VECTOR SPACE VbyINVERTIBLE LINEAR MAPS . For example, the group of two elements Z2/C30f0;1g has a representation fbyf(0)v/C30vandf(1)v/C30/C28v:A representation is a GROUP HOMOMORPHISM f:G0GL(V):/ Most groups have many different representations,possibly on different vector spaces. For example, the SYMMETRIC GROUP S3/C30fe;(12);(13);(23);(123) ;(132) g has a representation on Rby f1(s)v/C30sgn(s)v; (1) where sgn( s) is the SIGNATURE of the PERMUTATION s: It also has a representation on R3 by f2(s)(x1 ; x2 ; x3) /C30 x s(1) ; xs(2) ; xs(3)})0})@ : (2) A representation gives a matrix for each element, and so another representation of S3is given by the matrices 10 01})10})1@ ;0110})10})1@ ;/C2810 /C2811})10})1@ ; 1 /C281 0 /C281})10})1@ ;/C2811 /C2810})10})1@ ;0 /C281 1 /C281})10})1@ : (3) Two representations are considered equivalent if they are conjugates. For example, CONJUGATING the above matrices by 119 01})10})1@ gives the following equivalent representation of S3 ; 10 01})10})1@ ;/C2819 /C28360 11 9})10})1@ ;18 323 /C281 /C2818})10})1@ 13 7 0 /C281})10})1@ ;18 343 /C281 /C2819})10})1@ ;/C2819 /C28343 11 8})10})1@ (4) Any representation V of G can be RESTRICTED to a representation of any subgroup H, in which case, it is denoted ResG H : More surprisingly, any representation W on H can be extended to a representation of G,ona larger VECTOR SPACE V, called the INDUCED REPRE- SENTATION . Representations have applications to many branches of mathematics, aside from applications to physics and chemistry. The name of the theory depends on the GROUP G and on the VECTOR SPACE V. Different approaches are required depending on whether G is a FINITE GROUP , an infinite DISCRETE GROUP ,oraL IE GROUP . Another important ingredient is the field of scalars for V. The vector space V can be infinite dimensional such as a HILBERT SPACE . Also, special kinds of representations may require that a vector space structure is preserved. For instance, a UNITARY REPRESENTATION is a GROUP HOMOMORPHISM f : G 0 U(V) into the group of UNITARY TRANSFORMATIONS which preserve a HERMITIAN INNER PRODUCT on V. In favorable situations, such as a finite group, an arbitrary representation will break up into IRREDUCI- BLE REPRESENTATIONS , i.e., V /C30/C154Vi where the Vi are irreducible. For many groups, the irreducible repre- sentations have been classified. See also GROUP ,IRREDUCIBLE REPRESENTATION , MULTIPLICATIVE CHARACTER ,O RTHOGONAL GROUP REPRESENTATIONS ,PETER- WEYL THEOREM ,PRIMARY REPRESENTATION ,R EPRESENTATION (LIE ALGEBRA ), REPRESENTATION RING,R EPRESENTATION THEORY , SCHUR’S LEMMA ,SEMISIMPLE LIE GROUP ,T ENSORPRODUCT (REPRESENTATION ), UNITARY REPRESENTA- TION ,VECTOR SPACE References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups: Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/549, 1996. Representation (Lie Algebra) A representation of a LIE ALGEBRA g is a LINEAR MAP c : g0 M(V) ; where M(V) is the set of all linear transformations of a VECTOR SPACE V. In particular, if V /C30Rn ; then M(V) is the set of n /C29n square matrices. The map c is required to be a map of LIE ALGEBRAS so that c([A; B]) /C30 c(A)c(B) /C28 c(B) c(A) for all A; B /C23g: Note that the expression AB only makes sense as a MATRIX PRODUCT in a representa- tion. For example, if A and B are SKEW SYMMETRIC MATRICES , then AB /C28BA is skew-symmetric, but AB may not be skew symmetric. The possible IRREDUCIBLE REPRESENTATIONS of com- plex Lie algebras are determined by the classification of the SEMISIMPLE LIE ALGEBRAS . Any IRREDUCIBLE REPRESENTATION V of a complex LIE ALGEBRA g is the TENSOR PRODUCT V /C30V0 /C156L ; where V0is an IRREDU- CIBLE REPRESENTATION of the quotient gss =Rad(g)of the algebra g and its RADICAL , and L is a one- dimensional representation. AL IE ALGEBRA may be associated with a LIE GROUP , in which case it reflects the local structure of the LIE GROUP . Whenever a LIE GROUP G has a REPRESENTA- TION on V, its TANGENT SPACE at the identity, which is aLIE ALGEBRA , has a LIE ALGEBRA representation on V given by the differential at the identity. Conver- sely, if a CONNECTED LIE GROUP G corresponds to the Lie algebra g; and g has a LIE ALGEBRA representation onV, then Ghas a REPRESENTATION onVgiven by the MATRIX EXPONENTIAL . See also IRREDUCIBLE REPRESENTATION ,LIE ALGE- BRA,LIE GROUP ,MATRIX EXPONENTIAL ,REPRESENTA- TION ,SIMPLE LIE ALGEBRA ,VECTOR SPACE References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Representation Theory See also REPRESENTATION References Huang, J.-S. Lectures on Representation Theory. Singapore: World Scientific, 1999. Represented As An expression describing a form in which a quantity can be written. For example, all primes p /C213 can be "represented as" 6n 91 :/ See also OF THE FORM References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 13, 1986. Reptend Prime FULL REPTEND PRIME Reptile REP-TILE Rep-Tile A POLYGON which can be DISSECTED into n smaller copies of itself is called a rep-n-tile. The triangular POLYGONAL SPIRAL is a rep-4-tile. See also DISSECTION ,POLYGONAL SPIRAL References Gardner, M. "Rep-Tiles: Replicating Figures on the Plane." Ch. 19 in The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: Chicago University Press, pp. 222 /C1/233, 1991. Langford, C. D. "Uses of a Geometric Puzzle." Math. Gaz., No. 260, 1940. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 213 /C1/214, 1991.Repunit A (generalized) repunit to the base bis a number OF THE FORM Mb n/C30bn/C281 b/C281: The term "repunit" was coined by Beiler (1966), who also gave the first tabulation of known factors. Repunits Mn/C30M2 n/C302n/C281 with b/C302 are called MERSENNE NUMBERS .I fb/C3010, the number is called a repunit (since the digits are all 1s). A number OF THE FORM Rn/C3010n/C281 10/C281/C30Rn/C3010n/C281 9 is therefore a (decimal) repunit of order n. bSloane b-Repunits 2 Sloane’s A0002251, 3, 7, 15, 31, 63, 127, ... 3 Sloane’s A0034621, 4, 13, 40, 121, 364, ... 4 Sloane’s A0024501, 5, 21, 85, 341, 1365, ... 5 Sloane’s A0034631, 6, 31, 156, 781, 3906, ... 6 Sloane’s A0034641, 7, 43, 259, 1555, 9331, ... 7 Sloane’s A0230001, 8, 57, 400, 2801, 19608, ... 8 Sloane’s A0230011, 9, 73, 585, 4681, 37449, ... 9 Sloane’s A0024521, 10, 91, 820, 7381, 66430, ... 10 Sloane’s A0022751, 11, 111, 1111, 11111, ... 11 Sloane’s A0161231, 12, 133, 1464, 16105,177156, ... 12 Sloane’s A0161251, 13, 157, 1885, 22621,271453, ... Williams and Seah (1979) factored generalized repu-nits for 3 5b512 and 2 5n51000 :A (base-10) repunit can be PRIME only if nisPRIME , since otherwise 10ab/C281i sa BINOMIAL NUMBER which can be factored algebraically. In fact, if n/C302aisEVEN , then 102a/C281/C30(10a/C281)(10a/C271):/ The number of factors for the base-10 repunits for n /C301, 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (Sloane’s A046053). The only known base-10 repunit primes Rnare for n /C302, 19, 23, 317, 1031, 49081, (Sloane’s A004023; Madachy 1979, Williams and Dubner 1986, Ball and Coxeter 1987, Granlund, Dubner 1999). Williams and Dubner (1986) proved R1031 to be prime. T. Granlund completed a search up to 45,000 in 1998 using two months of CPU time on a parallel computer. The search was extended by H. Dubner in 1999, culminating in the discovery of the probable prime R49 ;081 :/ b Sloane n of Prime b-Repunits 2 Sloane’s A0000432, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, ... 3 Sloane’s A0284913, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, ... 5 Sloane’s A0040613, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, ... 6 Sloane’s A0040622, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, ... 7 Sloane’s A0040635, 13, 131, 149, 1699, ... 10 Sloane’s A0040232, 19, 23, 317, 1031, ... 11 Sloane’s A00580817, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, ... 12 Sloane’s A0040642, 3, 5, 19, 97, 109, 317, 353, 701, 9739, ... Yates (1982) published all the repunit factors for n 5 1000 ; a portion of which are reproduced in the Mathematica notebook by Weisstein. Brillhart et al. (1988) gave a table of repunit factors which cannot be obtained algebraically, and a continuously updated version of this table is now maintained on-line. These tables include factors for 10n /C281 (with n 5209 odd) and 10n /C271 (for n 5210 EVEN and ODD) in the files ftp://sable.ox.ac.uk/pub/math/cunningham/10- and ftp://sable.ox.ac.uk/pub/math/cunningham/10 /C27. After algebraically factoring Rn ; these types of factors are sufficient for complete factorizations. AS MITH NUMBER can be constructed from every factored repunit. See also CUNNINGHAM NUMBER ,FERMAT NUMBER , MERSENNE NUMBER ,REPDIGIT ,SMITH NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 66, 1987.Beiler, A. H. "11111...111." Ch. 11 in Recreations in the Theory of Numbers: The Queen of Mathematics Enter- tains. New York: Dover, 1966. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of bn 91; b /C302, 3; 5; 6; 7; 10; 11; 12 Up to High Powers, rev. ed. Provi- dence, RI: Amer. Math. Soc., 1988. Updates are available electronically from ftp://sable.ox.ac.uk/pub/math/cunning- ham. Dubner, H. "Generalized Repunit Primes." Math. Comput. 61, 927 /C1/930, 1993. Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 85 /C1/86, 1984. Granlund, T. "Repunits." http://www.swox.com/gmp/repu- nit.html. Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape k /C215 2n /C272 :/" §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8 /C1/13, 1994. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 152 /C1/153, 1979. Ribenboim, P. "Repunits and Similar Numbers." §5.5 in The New Book of Prime Number Records. New York: Springer- Verlag, pp. 350 /C1/354, 1996. Sloane, N. J. A. Sequences A000043/M0672, A000225/ M2655, A002275, A002450/M3914, A002452/M4733, A003462/M3463, A003463/M4209, A003464/M4425, A004023/M2114, A004023/M2114, A004061/M2620, A004062/M0861, A004063/M3836, A004064/M0744, A005808/M5032, A016123, A016125, A023000, A023001, A028491/M2643, and A046053 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Snyder, W. M. "Factoring Repunits." Am. Math. Monthly 89, 462 /C1/466, 1982. Weisstein, E. W. "Repunits." MATHEMATICA NOTEBOOK RE- PUNIT.M . Williams, H. C. and Dubner, H. "The Primality of R1031 :/" Math. Comput. 47, 703 /C1/711, 1986. Williams, H. C. and Seah, E. "Some Primes of the Form (an /C281)=(a /C281): Math. Comput. 33, 1337 /C1/1342, 1979. Yates, S. "Peculiar Properties of Repunits." J. Recr. Math. 2, 139 /C1/146, 1969. Yates, S. "Prime Divisors of Repunits." J. Recr. Math. 8,33/C1/ 38, 1975. Yates, S. "The Mystique of Repunits." Math. Mag. 51,22/C1/ 28, 1978. Yates, S. Repunits and Reptends. Delray Beach, FL: S. Yates, 1982. Resampling Statistics A set of methods that are generally superior to ANOVA for small data sets or where sample distribu- tions are non-normal. See also BAGGING ,BOOSTING ,BOOTSTRAP METHODS , HYPOTHESIS TESTING ,J ACKKNIFE ,P ERMUTATION TESTS References Good, P. I. Resampling Methods: A Practical Guide to Data Analysis. New York: Springer-Verlag, 1999. Good, P. I. Permutation Tests: A Practical Guide to Resam- pling Methods for Testing Hypotheses, 2nd ed. New York: Springer-Verlag, 2000. Residual The residual is the sum of deviations from a best-fit curve of arbitrary form. R /C13X yi /C28fxi ; a1 ; ...; an ðÞ ½/C1382: The residual should not be confused with the CORRE- LATION COEFFICIENT . Residual vs. Predictor Plot A plot of yi versus the ESTIMATOR ei /C13 ˆyi /C28yi : Random scatter indicates the model is probably good. A pattern indicates a problem with the model. If the spread in eiincreases as yiincreases, the errors are called HETEROSCEDASTIC . See also ESTIMATOR Residue BIQUADRATIC RESIDUE ,COMMON RESIDUE ,COMPLETE RESIDUE SYSTEM ,CUBIC RESIDUE ,MINIMAL RESIDUE , QUADRATIC RESIDUE ,RESIDUE CLASS ,RESIDUE (COM- PLEX ANALYSIS ), R ESIDUE (CONGRUENCE ), R ESIDUE INDEX ,RESIDUE THEOREM Residue (Complex Analysis) The constant a/C281in the L AURENT SERIES f(z)/C30X/C12 n/C30/C28/C12an(z/C28z0)n(1) off(z) about a point z0is called the residue of f(z): Unless z0is a POLE off, its residue is zero. The residue of a function fat a point z0may be denoted Resz/C30z(f(z)):Two basic examples of residues are given by Resz/C3001=z/C301 and Resz/C3001=zn/C300 for n/C211. The residue is implemented in Mathematica asResi- due[f,{z,z0}]. The residue is also defined by ggfd z; (2) where gis clockwise simple closed CONTOUR , smallenough to avoid any other poles of f. In fact, any clockwise path with WINDING NUMBER 1 which does not contain any other poles gives the same result by the C AUCHY INTEGRAL FORMULA . The above diagram shows a suitable CONTOUR for which to define the residue of function, where the poles are indicated asblack dots. It is more natural to consider the residue of a MEROMORPHIC ONE-FORM because it is independent of the choice of coordinate. On a R IEMANN SURFACE , the residue is defined for a MEROMORPHIC ONE-FORM a at a point pby writing a/C30fd z in a coordinate z around p. Then Res pa/C30Res z/C30pf: (3) The sum of the residues of ffd z is zero on the RIEMANN SPHERE . More generally, the sum of the residues of a MEROMORPHIC ONE-FORM on a compact RIEMANN SURFACE must be zero. The residues of a function f(z) may be found without explicitly expanding into a L AURENT SERIES as fol- lows. If f(z) has a POLE of order matz0;then an/C300 for nB/C28manda/C28m"0:Therefore, f(z)/C30X/C12 n/C30/C28man(z/C28z0)n/C30X/C12 n/C300a/C28m/C27n(z/C28z0)/C28m/C27n(4) (z/C28z0)mf(z)/C30X/C12 n/C300a/C28m/C27n(z/C28z0)n(5) d dzz/C28z0 ðÞmf(z) ½/C138 /C30X/C12 n/C300na/C28m/C27n(z/C28z0)n/C281 /C30X/C12 n/C301na/C28m/C27n(z/C28z0)n/C281 /C30X/C12 n/C300(n/C271)a/C28m/C27n/C271(z/C28z0)n(6) d2 dz2z/C28z0 ðÞmf(z) ½/C138 /C30X/C12 n/C300n(n/C271)a/C28m/C27n/C271z/C28z0 ðÞn/C281 /C30X/C12 n/C301n(n/C271)a/C28m/C27n/C271z/C28z0 ðÞn/C281 /C30X/C12 n/C300(n/C271)(n/C272)a/C28m/C27n/C272z/C28z0 ðÞn: (7) Iterating, dm/C281 dzm/C281z /C28z0 ðÞmf(z) ½/C138 /C30X/C12 n/C300(n /C271)(n /C272)(n /C27m /C281)an/C281(z /C28z0)n /C30(m /C281)!a/C281 /C27X/C12 n/C301(n /C271)(n /C272) /C2(n /C27m /C281)an/C281(z /C28z0)n/C281 : (8) So lim x0z0dm/C281 dzm/C281z /C28z0 ðÞmf(z) ½/C138 /C30lim z0z0(m /C281)!a/C281 /C270 /C30(m /C281)!a/C281 ; (9) and the residue is a/C281 /C301 (m /C28 1)!dm/C281 dzm/C281z /C28z0 ðÞmf(z) ½/C138z/C30z0: (10) The residues of a HOLOMORPHIC FUNCTION at its POLES characterize a great deal of the structure of a function, appearing for example in the amazing RESIDUE THEOREM of CONTOUR INTEGRATION . See also CONTOUR INTEGRATION ,LAURENT SERIES , MEROMORPHIC ONE-FORM,POLE,RESIDUE THEOREM , WINDING NUMBER (CONTOUR ) References Arfken, G. "Calculus of Residues." §7.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 400 /C1/421, 1985. Krantz, S. G. "The Calculus of Residues." §4.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 48 /C1/51, 1999. Residue (Congruence) The number b in the CONGRUENCE a /C13b (mod m)is called the residue of a (mod m). The residue of large numbers can be computed quickly using CON- GRUENCES . For example, to find 3713 (mod 17), note that 37 /C133 372 /C1332 /C139 /C13/C288 374 /C1381 /C13/C284 378 /C1316 /C13/C281 ; so 3713 /C13371/C274 /C278 /C133(/C284)(/C281) /C1312 (mod 17): See also COMMON RESIDUE ,CONGRUENCE ,M INIMAL RESIDUEReferences Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 55 /C1/56, 1993. Residue Class The residue classes of a function f(x) mod n are all possible values of the RESIDUE f(x) ðmod nÞ: For example, the residue classes of x2(mod 6) are f0; 1; 3; 4g; since 02 /C130 (mod 6) 12 /C131 (mod 6) 22 /C134 (mod 6) 32 /C133 (mod 6) 42 /C134 (mod 6) 52 /C131 (mod 6) are all the possible residues. A COMPLETE RESIDUE SYSTEM is a set of integers containing one element from each class, so f0 ; 1 ; 9; 16 g would be a COM- PLETE RESIDUE SYSTEM for x2(mod 6), as would f0; 5; 3; 4g; etc. The f(m) residue classes prime to m form a GROUP under the binary multiplication operation (mod m), where f(m) is the TOTIENT FUNCTION (Shanks 1993) and the GROUP is classed a MODULO MULTIPLICATION GROUP . See also COMPLETE RESIDUE SYSTEM ,CONGRUENCE , CUBIC NUMBER ,QUADRATIC RECIPROCITY THEOREM , QUADRATIC RESIDUE ,R EDUCED RESIDUE SYSTEM , RESIDUE (CONGRUENCE ), SQUARE NUMBER References Nagell, T. "Residue Classes and Residue Systems." §20 in Introduction to Number Theory. New York: Wiley, pp. 69 /C1/ 71, 1951. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 56 and 59 /C1/63, 1993. Residue Field In a LOCAL RING R, there is only one MAXIMAL IDEAL m: Hence, R has only one QUOTIENT RING R=m which is a FIELD . This field is called the residue field. See also ALGEBRAIC GEOMETRY ,ALGEBRAIC NUMBER THEORY ,LOCAL RING Residue Index MULTIPLICATIVE ORDER Residue System COMPLETE RESIDUE SYSTEM Residue Theorem Given an ANALYTIC FUNCTION f(z) whose LAURENT SERIES is given by f(z) /C30X/C12 n/C30/C28/C12anz /C28z0 ðÞn; (1) and integrate term by term using a closed CONTOUR g encircling z0 ; ggf(z) dz /C30X/C12 n/C30/C28/C12anggz /C28z0 ðÞndz /C30X/C282 n/C30/C28/C12anggz /C28z0 ðÞndz /C27a/C281ggdz z /C28 z0 /C27X/C12 n/C300anggz /C28z0 ðÞndz : (2) The CAUCHY INTEGRAL THEOREM requires that the first and last terms vanish, so we have ggf(x) dz /C30a/C281ggdz z /C28 z0; (3) where a/C281is the RESIDUE . Using the CONTOUR z /C30 g(t) /C30eit /C27z0 gives ggdz z /C28 z0/C30g2 p 0ieit dt eit/C302 pi; (4) so we have ggf(z) dz /C302pia/C281 : (5) If the contour g encloses multiple poles, then the theorem gives the general result ggf(z) dz /C302 piX a /C23ARes z/C30aif(z); (6) where A is the set of poles contained inside the contour. This amazing theorem therefore says that the value of a CONTOUR INTEGRAL for any contour in the COMPLEX PLANE depends only on the properties of a few very special points inside the contour. The diagram above shows an example of the residuetheorem applied to the illustrated CONTOUR g and the function g(z) /C303 z /C28 1 ðÞ2 /C272 z /C28 i /C282 z /C27 i /C27i z /C27 3 /C28 2i /C275 z /C27 1 /C27 2i : (7) Only the poles at 1 and iare contained in the contour, which have residues of 0 and 2, respectively. The values of the CONTOUR INTEGRAL is therefore given by ggg(z)dz/C302pi(0/C272)/C304pi: See also CAUCHY INTEGRAL FORMULA ,CAUCHY INTE- GRAL THEOREM ,CONTOUR ,CONTOUR INTEGRAL ,CON- TOUR INTEGRATION ,G ROUP RESIDUE THEOREM , LAURENT SERIES ,POLE,RESIDUE (COMPLEX ANALY- SIS) References Knopp, K. "The Residue Theorem." §33 in Theory of Func- tions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 129 /C1/134, 1996. Krantz, S. G. "The Residue Theorem." §4.4.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 48 /C1/49, 1999. Resistor Network Consider a network of nresistors Riso that R2may be connected in series or parallel with R1;R3may be connected in series or parallel with the network consisting of R1andR2;and so on. The resistance of two resistors in series is given by Rnet;series/C30R1/C27R2; and of two resistors in parallel by Rnet;parallel/C301 1 R1/C271 R2: The possible values for two resistors with resistancesaandbare therefore a/C27b;1 1 a/C271 b; for three resistances a,b, and care a/C27b/C27c;a/C271 1b/C271 c;b/C271 1 a/C271 c;c/C271 1 a/C271b 1 1 a/C271 b /C27 c;1 1 b/C271 a /C27 c;1 1 c/C271 a /C27 b;1 1 a/C271b/C271 c; and so on. These are obviously all rational numbers, and the numbers of distinct arrangements for n /C301, 2, ..., are 1, 2, 8, 46, 332, 2874, ... (Sloane’s A005840), which also arises in a completely different context (Stanley 1991). If the values are restricted to a /C30b /C30.../C301; then there are 2n/C281 possible resistances for n 1-/V resistors, ranging from a minimum of 1=n to a maximum of n. Amazingly, the largest denominators for n /C301, 2, ... are 1, 2, 3, 5, 8, 13, 21, ..., which are immediately recognizable as the FIBONACCI NUMBERS (Sloane’s A000045). The following table gives the values possi- ble for small n. n Possible resistances 11 2 /1 2 ; 2/ 3 /13 ;23 ;32 ; 3/ 4 /1 4 ;25;35 ;34 ;43 ;53 ;52; 4/ If the n resistors are given the values 1, 2, ..., n, then the numbers of possible net resistances for 1, 2, ... resistors are 1, 2, 8, 44, 298, 2350, ... (Sloane’s A051045). The following table gives the values possi- ble for small n. n Possible resistances 11 2 /2 3 ; 3/ 3 /6 11 ;3 2 ;11 3 ; 6/ 4 /1225 ;1211 ;4423 ;12 5 ;5011 ;11 2 ;23 3 ; 10/ See also FIBONACCI NUMBER References Amengual, A. "The Intriguing Properties of the Equivalent Resistances of n Equal Resistors Combined in Series and in Parallel." Amer. J. Phys. 68, 175 /C1/179, 2000. Sloane, N. J. A. Sequences A000045/M0692, A005840/ M1872, and A051045 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Stanley, R. P. "A Zonotope Associated with Graphical Degree Sequences." In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift (Ed. P. Gritz- mann and B. Sturmfels). Providence, RI: Amer. Math. Soc., pp. 555 /C1/570, 1991.Resolution Resolution is a widely used word with many different meanings. It can refer to resolution of equations, resolution of singularities (in ALGEBRAIC GEOMETRY ), resolution of modules or more sophisticated struc- tures, etc. In a BLOCK DESIGN ,a PARTITION R of a BIBD’s set of blocks B into PARALLEL CLASSES , each of which in turn partitions the set V, is called a resolution (Abel and Furino 1996). A resolution of the MODULE M over the RING R is a complex of R-modules Ciand morphisms diand a MORPHISM e such that /C1/C1/C10 Ci 0di Ci/C281 0/C1/C1/C10 C0 0e M 0 0 satisfying the following conditions: 1. The composition of any two consecutive morph- isms is the zero map, 2. For all i, ker di ðÞ = im di/C271})0})@ /C300;/ 3. C0 =(ker e) #M ;/ where ker is the kernel and im is the image. Here, the quotient ker di ðÞ im di /C271})0})@ is the ith HOMOLOGY GROUP . If all modules Ciare projective (free), then the resolution is called projective (free). There is a similar concept for resolutions "to the right" of M, which are called injective resolutions. See also HOMOLOGY GROUP ,M ODULE ,M ORPHISM , RING References Abel, R. J. R. and Furino, S. C. "Resolvable and Near Resolvable Designs." §I.6 in The CRC Handbook of Combinatorial Designs (Ed. C. J. Colbourn and J. H. Di- nitz). Boca Raton, FL: CRC Press, pp. 4 and 87 /C1/94, 1996. Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, p. 339, 1989. Resolution Class PARALLEL CLASS Resolution Modulus The least POSITIVE INTEGER m/C31 with the property that x(y) /C301 whenever y /C131 ðmod m/C31Þ and (y; m) /C301:/ Resolvable A balanced incomplete BLOCK DESIGN (B, V) is called resolvable if there exists a PARTITION R of its set of blocks B into PARALLEL CLASSES , each of which in turn partitions the set V. The partition Ris called a RESOLUTION . See also BLOCK DESIGN ,PARALLEL CLASS References Abel, R. J. R. and Furino, S. C. "Resolvable and Near Resolvable Designs." §I.6 in The CRC Handbook of Combinatorial Designs (Ed. C. J. Colbourn and J. H. Di- nitz). Boca Raton, FL: CRC Press, pp. 4 and 87 /C1/94, 1996. Furino, S.; Miao, Y.; and Yin, J. Frames and Resolvable Designs: Uses, Constructions, ad Existence. Boca Raton, FL: CRC Press, 1996. Resolve QUANTIFIER ELIMINATION Resolving Tree A tree of LINKS obtained by repeatedly choosing a crossing, applying the SKEIN RELATIONSHIP to obtain two simpler LINKS , and repeating the process. The DEPTH of a resolving tree is the number of levels of links, not including the top. The DEPTH of the LINK is the minimal depth for any resolving tree of that LINK . Resonance Overlap Isolated resonances in a DYNAMICAL SYSTEM can cause considerable distortion of preserved TORI in their NEIGHBORHOOD , but they do not introduce any CHAOS into a system. However, when two or more resonances are simultaneously present, they will render a system nonintegrable. Furthermore, if they are sufficiently "close" to each other, they will result in the appearance of widespread (large-scale) CHAOS . To investigate this problem, Walker and Ford (1969) took the integrable Hamiltonian H0I1 ; I2 ðÞ /C30I1 /C27I2 /C28I2 1 /C283I1I2 /C27I2 2 and investigated the effect of adding a 2:2 resonance and a 3:2 resonance H(I; u) /C30H0(I) /C27 aI1I2 cos 2u1 /C282u2 ðÞ /C27 bI3 =2 1I2 cos 2 u1 /C283u2 ðÞ : At low energies, the resonant zones are well-sepa- rated. As the energy increases, the zones overlap and a "macroscopic zone of instability" appears. When the overlap starts, many higher-order resonances are also involved so fairly large areas of PHASE SPACE have their TORI destroyed and the ensuing CHAOS is "wide- spread" since trajectories are now free to wander between regions that previously were separated by nonresonant TORI. Walker and Ford (1969) were able to numerically predict the energy at which the overlap of the resonances first occurred. They plotted the u2/-axis intercepts of the inner 2:2 and the outer 2:3 separa- trices as a function of total energy. The energy at which they crossed was found to be identical to that at which 2:2 and 2:3 resonance zones began to overlap. See also CHAOS ,RESONANCE OVERLAP METHODReferences Walker, G. H. and Ford, J. "Amplitude Instability and Ergodic Behavior for Conservative Nonlinear Oscillator Systems." Phys. Rev. 188, 416 /C1/432, 1969. Resonance Overlap Method A method for predicting the onset of widespread CHAOS . See also GREENE’S METHOD References Chirikov, B. V. "A Universal Instability of Many-Dimen- sional Oscillator Systems." Phys. Rep. 52, 264 /C1/379, 1979. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 154 /C1/163, 1989. R-Estimate A ROBUST ESTIMATION based on a RANK TEST . See also L-ESTIMATE , M-ESTIMATE ,R ANK TEST, ROBUST ESTIMATION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Robust Estimation." §15.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 694 /C1/700, 1992. Restricted Divisor Function The sum of the ALIQUOT DIVISORS of n, given by s(n) /C13 s(n) /C28n; where s(n) is the DIVISOR FUNCTION . The first few values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... (Sloane’s A001065). See also DIVISOR FUNCTION References Sloane, N. J. A. Sequences A001065/M2226 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Restricted Growth Function RESTRICTED GROWTH STRING Restricted Growth String For a SET PARTITION of n elements, the n-character string a1a2 ...anin which each character gives the BLOCK (B0 ; B1 ; ...) in which the corresponding element belongs is called the restricted growth string (or sometimes the RESTRICTED GROWTH FUNCTION ). For example, for the SET PARTITION ff1g;f2 g;f3 ; 4gg; the restricted growth string would be 0122. If the BLOCKS are "sorted" so that a1 /C300; then the restricted growth string satisfies the INEQUALITY ai /C271 51 /C27max fa1 ; a2 ; ... ; ai g for i /C301, 2, ..., n /C281 :/ References Ruskey, F. "Info About Set Partitions." http://www.theor- y.csc.uvic.ca/~cos/inf/setp/SetPartitions.html. Restriction (Representation) A REPRESENTATION of a GROUP G on a VECTOR SPACE V can be restricted to a SUBGROUP H. For example, the SYMMETRIC GROUP on three letters has a repre- sentation f on R2 by f(e) /C3010 01})10})1@ (1) f(12) /C3001 10})10})1@ (2) f(13) /C30/C2810 /C2811})10})1@ (3) f(23) /C301 /C281 0 /C281})10})1@ (4) f(123) /C30/C2811 /C2810})10})1@ (5) f(132) /C300 /C281 1 /C281})10})1@ (6) that can be restricted to the subgroup of ORDER 3, f(e) /C3010 01})10})1@ (7) f(123) /C30/C2811 /C2810})10})1@ (8) fð132Þ¼0 /C281 1 /C281})10})1@ (9) See also FROBENIUS RECIPROCITY ,REPRESENTATION , VECTOR SPACEResultant Given a POLYNOMIAL p(x) of degree n with roots ai ; i /C301, ..., n and a POLYNOMIAL q(x) of degree m with roots bj ; j /C301, ..., m, the resultant is defined by r(p ; q) /C30Yn i /C301Ym j/C301( bj /C28 ai) : The notation R(p; q) is also used. There exists an ALGORITHM similar to the EUCLIDEAN ALGORITHM for computing resultants (Pohst and Zassenhaus 1989). The resultant of two polynomials can be computed using the Mathematica command Resultant [poly1 , poly2 , var]. Resultants for a few simple pairs of polynomials include r(x /C28a; x /C28b) /C30a /C28b r((x /C28a)(x /C28b) ; x /C28c) /C30(a /C28c)(b /C28c) r((x /C28a)(x /C28b) ; (x /C28c)(x /C28d)) /C30(a /C28c)(b /C28c)(a /C28d)(b /C28d) : The resultant is the DETERMINANT of the correspond- ing SYLVESTER MATRIX . Given p and q, then h(x) /C30 r(q(t); p(x /C28t)) is a POLYNOMIAL of degree mn, having as its roots all sums OF THE FORM /ai þ bj/. See also DISCRIMINANT (POLYNOMIAL ), SUBRESUL- TANT ,SYLVESTER MATRIX References Apostol, T. M. "Resultants of Cyclotomic Polynomials." Proc. Amer. Math. Soc. 24, 457 /C1/462, 1970. Apostol, T. M. "The Resultant of the Cyclotomic Polynomials /Fm ðax Þ/ and Fn(bx):/" Math. Comput. 29,1/C1/6, 1975. Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num- ber Theory. Cambridge, England: Cambridge University Press, 1989. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 348, 1991. Retardance A shift in PHASE . See also PHASE Reuleaux Polygon A curvilinear polygon built up of circular ARCS . The Reuleaux polygon is a generalization of the R EU- LEAUX TRIANGLE and, for an ODD NUMBER of sides, is a CURVE OF CONSTANT WIDTH (Gray 1997). See also CURVE OF CONSTANT WIDTH ,DELTA CURVE , REULEAUX TRIANGLE References Gray, A. "Reuleaux Polygons." §7.8 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 176 /C1/177, 1997. Reuleaux, F. The Kinematics of Machinery. New York: Dover, 1963. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 52 /C1/54, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 219 /C1/220, 1991. Reuleaux Tetrahedron The Reuleaux tetrahedron is the 3-dimensional solid common to four SPHERES of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular TETRA- HEDRON , and the solid consists of an "inflated" tetrahedron with four curved edges. To analyze the Reuleaux tetrahedron, fix a TETRA- HEDRON of unit edge length with its vertices at 0;0;/C28ffiffiffi 6p =4})0})@ ;ffiffiffi3p =3;0;ffiffiffi6p =12})0})@ ;/C28ffiffiffi3p =6;1=2;})0 / /ffiffiffi6p =12Þ;and/C28ffiffiffi3p =6;/C281=2;ffiffiffi6p =12})0})@ :Simultaneously solving the equations of three of four spheres for x andyas a function of zthen gives x/C30 1 2ffiffiffi 2p z/C271 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 2/C286z(ffiffiffi 6p /C276z)q (1) y/C304ffiffiffi 3p z/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C284z(ffiffiffi 6p /C276z)p 4ffiffiffi 2p : (2) Half an arc is traced out as zpasses fromffiffiffi 6p =12 to 6/C28ffiffiffi6p})0})@ =12;and ds/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx dz !2 /C27dy dz !2 /C271vuutdz /C303ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 5/C284zffiffiffi 6p /C276z})0})@s dz; (3)so the ARC LENGTH of the curves connecting the vertices is given by s/C30gds /C306ffiffiffi 2pg6/C28ffiffi 6pðÞ =12 ffiffi 6p =125/C284zffiffiffi 6p /C276z})@D})@Ehi/C281=2 dz: (4) Making a change of coordinates, s/C30ffiffiffi3p gffiffi 6p 2(6/C28u2)/C281=2du/C30ffiffiffi3p cot/C281ffiffiffi 2p})@D})@E (5) :1:06604 : The VOLUME is significantly trickier to calculate analytically. Set up SPHERICAL COORDINATES from thecentroid of the TETRAHEDRON , so that the distance from the bottom vertex to the radius vector is 1, i.e., r2cos2sin2f/C27r2sin2usin2f/C27r/C271 4ffiffiffi 6p})@D})@E2 /C301;(6) giving r(u;f)/C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 cos(2 f)/C2713p /C28ffiffiffi 6p cosfhi : (7) By symmetry, the volume of the Reuleaux tetrahe- dron is given by V/C3024gp=3 0gfðuÞ 0grðu;fÞ 0r2sinfdr dfdu: (8) The integral over rcan be done immediately, V/C30 1 8gp=3 0gf(u) 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 cos(2 f)/C2713p /C28ffiffiffi 6p cosfhi3 sinfdfdu: (9) Now parameterize the top right edge as a function of the azimuthal coordinate uas x/C30cosuffiffiffi 3p cosu/C273 sin u(10) y/C30sinuffiffiffi3p cosu/C273 sin u(11) z/C301 12ffiffiffi 6p : (12) The polar angle fcan then be solved for as a function ofuas f(u)/C30cos/C281 zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2/C27z2p ! /C30tan/C281 2ffiffiffi 6p ffiffiffi 3p cosu/C273 sin u ! : (13) The integral over fcan be done by making the change of coordinates u /C302ffiffiffi 6p ffiffiffi3p cos u /C27 3 sin u ; (14) giving V /C30g p =3 01 32})10 256 /C2845ffiffiffi 6p /C2742ffiffiffi6p cos(2 tan /C281 u) /C2858ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 /C27 3 cos(2 tan/C281 u)p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 u2p /C276ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 /C273 cos(2 tan /C281 u)p cos(3 tan/C281 u) /C273ffiffiffi 6p cos(4 tan /C281 u)})1@ du : (15) Making the change of variables u /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6(1 /C27 3t2)p ffiffiffi 3p /C27 3ffiffiffi3p t (16) then gives the volume as V /C30g1 08ffiffiffi3p 1 /C27 3t2 /C2816ffiffiffi 2p (3t /C27 1)(4t2 /C27 t /C27 1)3 =2 (3t2 /C27 1)(11 t2 /C27 2t /C27 3)2 ! /C28ffiffiffi2p (249t2 /C27 54t /C27 65) (11t2 /C27 2t /C27 3)2Þ dt : (17) This integral can be done analytically, but the analytic form returned by symbolic algebra programs is an extremely complicated expression involving logarithms and inverse tangent functions. After arduous simplification of the expression by hand, the final solution V /C301 246ffiffiffi 2p /C2716p /C2757 cos /C28117 81})@D})@E /C28132 tan/C281ffiffiffi 2p})@D})@E hi (18) :0:422157733 (19) is obtained. This solution appears not to have been published previously.See also HYPERBOLIC TETRAHEDRON ,REULEAUX TRI- ANGLE ,S PHERE ,S PHERE- SPHERE INTERSECTION , SPHERICAL TRIANGLE ,STEINMETZ SOLID,TETRAHE- DRON Reuleaux Triangle ACURVE OF CONSTANT WIDTH constructed by drawing arcs from each VERTEX of an EQUILATERAL TRIANGLE between the other two VERTICES . The Reuleaux triangle has the smallest AREA for a given width of any CURVE OF CONSTANT WIDTH . Let the arc radius be r. Since the AREA of each meniscus-shaped portion of the Reuleaux triangle is a circular SEGMENT with opening angle u/C30p=3; As/C301 2r2(u/C28sinu)/C30p 6/C28ffiffiffi 3p 4 ! r2: (1) But the AREA of the central EQUILATERAL TRIANGLE with a/C301=ffiffiffi3p is A t/C301 4ffiffiffi 3p r2; (2) so the total AREA is then A/C303As/C27At/C301 2p/C28ffiffiffi 3p})@D})@E r2: (3) Because it can be rotated inside a SQUARE ,a s illustrated above, it is the basis for the Harry Watt square drill bit. When rotated inside a square of side length 2 having corners at (91;91); the envelope of the Reuleaux triangle is a region of the square with rounded corners. At the corner (/C281;/C281); the envelope of the boundary is given by the segment of the ellipse with PARAMETRIC EQUATIONS x /C301 /C28cos b /C28ffiffiffi 3p sin b (4) y /C301 /C28sin b /C28ffiffiffi 3p cos b (5) for b /C23 [p=6 ; p=3]; extending a distance 2 /C28ffiffiffi 3p from the corner (Gleißner and Zeitler 2000). The ellipse has center (1; 1); semimajor axis a /C301 /C27ffiffiffi3p ; semimi- nor axis b /C301 /C28ffiffiffi3p ; and is rotated by 45 8, which has Cartesian equation x 2 /C27y2 /C28ffiffiffi 3p xy /C28 2 /C28ffiffiffi3p})@D})@E x /C28 2 /C28ffiffiffi3p})@D})@E y /C271 /C28ffiffiffi3p /C300: (6) The fractional AREA covered as the Reuleaux triangle rotates is Acovered /C302ffiffiffi 3p /C271 6 p /C283 /C300:9877003907... : (7) Note that Gleißner and Zeitler (2000) fail to simplify their equivalent equation, and then proceed to assert that (7) is erroneous. The CENTROID does not stay fixed as the TRIANGLE is rotated, nor does it move along a CIRCLE . In fact, the path consists of a curve composed of four arcs of an ELLIPSE (Wagon 1991). For a bounding square of side length 2, the ellipse in the lower-left quadrant hasPARAMETRIC EQUATIONS x /C301 /C27cos b /C271 3ffiffiffi 3p sin b (8) y /C301 /C27sin b /C271 3ffiffiffi 3p cos b (9) for b /C23 [ p=6; p=3]: The ellipse has center (1; 1); semi- major axis a /C301 /C271=ffiffiffi 3p ; semiminor axis b /C30 1 /C281 =ffiffiffi3p ; and is rotated by 458, which has Cartesian equation 3x2 /C273y2 /C283ffiffiffi 3p xy /C283x 2 /C27ffiffiffi3p})@D})@E /C283y 2 /C27ffiffiffi3p})@D})@E /C275 /C283ffiffiffi3p /C300 : (10) The area enclosed by the locus of the centroid is given by A centroid/C304/C288 3ffiffiffi 3p /C272 9p (11) (Gleißner and Zeitler 2000; who again fail to simplify their expression). Note that the CENTROID ’s path can be closely approximated by a SUPERELLIPSE x a})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1r /C27y a})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1r /C301 (12) with a/C302ffiffiffi 3p =3/C281 and r:2:36185 :/ See also CURVE OF CONSTANT WIDTH ,DELTA CURVE , EQUILATERAL TRIANGLE ,FLOWER OF LIFE,PIECEWISE CIRCULAR CURVE ,R EULEAUX POLYGON ,R EULEAUX TETRAHEDRON ,ROTOR ,ROULETTE References Blaschke, W. "Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts." Math. Ann. 76, 504/C1/513, 1915. Bogomolny, A. "Shapes of Constant Width." http://www.cut- the-knot.com/do_you_know/cwidth.html. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 8, 1991. Dark, H. E. The Wankel Rotary Engine: Introduction and Guide. Bloomington, IN: Indiana University Press, 1974. Eppstein, D. "Reuleaux Triangles." http://www.ics.uci.edu/ ~eppstein/junkyard/reuleaux.html. Gardner, M. "Mathematical Games: Curves of Constant Width, One of which Makes it Possible to Drill Square Holes." Sci. Amer. 208, 148/C1/156, Feb. 1963. Gardner, M. "Curves of Constant Width." Ch. 18 in The Unexpected Hanging and Other Mathematical Diversions.Chicago, IL: University of Chicago Press, pp. 212 /C1 /221, 1991. Gleißner, W. and Zeitler, H. "The Reuleaux Triangle and Its Center of Mass." Result. Math. 37, 335/C1/344, 2000. Gray, A. "Reuleaux Polygons." §7.8 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nded.Boca Raton, FL: CRC Press, pp. 176 /C1 /177, 1997. Kunkel, P. "Reuleaux Triangle." http://www.nas.com/~kun- kel/reuleaux/reuleaux.htm. Math Forum. "Reuleaux Triangle, Reuleaux Drill." http:// mathforum.com/~sarah/HTMLthreads/articletocs/reu- leaux.triangle.html. Peterson, I. "Ivar Peterson’s MathLand: Rolling with Re- uleaux." Oct. 21, 1996. http://www.maa.org/mathland/ mathland_10_21.html. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, 1957. Reuleaux, F. The Kinematics of Machinery: Outlines of a Theory of Machines. London: Macmillan, 1876. Reprinted as The Kinematics of Machinery. New York: Dover, 1963. Smith, S. "Drilling Square Holes." Math. Teacher 86, 579 /C1/ 583, Oct. 1993. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 52 /C1/54 and 381 /C1/383, 1991. Yaglom, I. M. and Boltyansky, B. G. Convex Shapes. Mos- cow: Nauka, 1951. Reversal The reversal of a decimal number abc /C1/C1/C1 is /C1/C1/C1cba: Ball and Coxeter (1987) consider numbers whose reversals are integral multiples of themselves. PALIN- DROMIC NUMBERS and numbers ending with a ZERO are trivial examples. The first few nontrivial examples are 8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901, 87128712, 87999912, 98019801, 98999901, ... (Sloa- ne’s A031877). The pattern continues for large num- bers, with numbers OF THE FORM 879 /C1/C1/C19|fflffl{zfflffl} 12 equal to 4 times their reversals and numbers OF THE FORM 989 /C1/C1/C19|fflffl{zfflffl} 01 equal to 9 times their reversals. In addition, runs of numbers of either of these forms can be concatenated to yield numbers OF THE FORM 879 /C1/C1/C19|fflffl{zfflffl} 12 /C1/C1/C1879 /C1/C1/C19|fflffl{zfflffl} 12; equal to 4 times their reversals, and 989 /C1/C1/C19|fflffl{zfflffl} 01 /C1/C1/C1989 /C1/C1/C19|fflffl{zfflffl} 01 ; equal to 9 times their reversals. The product of a 2-digit number and its reversal is never a SQUARE NUMBER except when the digits are the same (Ogilvy 1988). Numbers whose product is the reversal of the products of their reversals include (221, 312) and (122, 213), since 312 /C29221 /C3068952 213 /C29122 /C3025986 (Ball and Coxeter 1987, p. 14). See also EMIRP , RATS SEQUENCE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 14 /C1/15, 1987. Edalj, J. Problem 1622. L’Interme ´d. Math. 16, 34, 1909. Jonesco, J. Problem 1622. L’Interme ´d. Math. 15, 128, 1908. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 88 /C1/89, 1988. Sloane, N. J. A. Sequences A031877 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Welsch. Problem 1622. L’Interme ´d. Math. 15, 278, 1908. Reverse Greedy Algorithm An algorithm for computing a UNIT FRACTION .See also GREEDY ALGORITHM ,UNIT FRACTION References Eppstein, D. Egypt.ma Mathematica notebook. http:// www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma. Reverse-Then-Add Sequence An integer sequence produced by the 196-ALGORITHM . See also 196-ALGORITHM ,SORT-THEN- ADD SEQUENCE Reversible Knot INVERTIBLE KNOT Reversible Prime EMIRP Reversion of Series SERIES REVERSION Reversion to the Mean This entry contributed by ANTON E. WEISSTEIN Reversion to the mean is the statistical phenomenon that a random variate which deviates strongly from the mean in a particular direction is likely to be succeeded by an event (independent of the first) that deviates less far in this direction. In other words, an extreme event is likely to be followed by a less extreme event. Although this phenomenon appears to violate the definition of INDEPENDENT EVENTS , it simply reflects the fact that there are more values from which to choose on the side of the probability distribution closer to the mean than there are on the side corresponding to even more extreme values. See also MEAN Reye’s Configuration A configuration of 12 planes and 12 points such that six points lie in every plane and six planes pass through every point. Alternatively, the configurationconsists of 16 lines and the same 12 points such thatfour lines pass through every point and three points lie on every line. The points consist of the eight vertices of a CUBE together with its center and the three POINTS AT INFINITY where parallel edges of the CUBE meet. The 12 planes are the six faces of the cube and the six planes passing through diagonally opposite edges.The 16 lines consist of the 12 edges and four space diagonals of the cube. Reye’s configuration can be realized without any points at infinity by squashing the cube and bringing the points at infinity to finite positions, as illustrated above. See also CONFIGURATION References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 214 /C1/215, 1991. Reznik’s Identity For P and Q POLYNOMIALS in n variables, ½P /C215 Q½2 2 /C30X i1 ; ... ; in ]0 /C2½P(i1 ; ... ; in)(D1 ; ...; Dn)Q(x1 ; ...; xn) ½22 i1! /C1/C1/C1in! ; where Di /C13@=@xi ;½X ½2 is the BOMBIERI NORM , and P(i1 ; ... ; in) /C30Di1 1/C1/C1/C1Din n P : BOMBIERI’S INEQUALITY follows from this identity. See also BEAUZAMY AND DE´ GOT’S IDENTITY Rhodonea ROSE Rhomb RHOMBUS Rhombic Dodecahedral Number A FIGURATE NUMBER which is constructed as a centered CUBE with a SQUARE PYRAMID appended to each face, RhoDodn /C30CCubn /C276Pn/C281 /C30(2n /C281)(2n2 /C282n /C271); (1) where CCubn is a CENTERED CUBE NUMBER and Pn is a PYRAMIDAL NUMBER . The first few are 1, 15, 65, 175, 369, 671, ... (Sloane’s A005917). The GENERATING FUNCTION of the rhombic dodecahedral numbers isx(1 /C27 11x /C27 11x2 /C27 x3) (x /C28 1)4 /C30x /C2715x2 /C2765x3 /C27175x4 /C27...: (2) A related set of numbers is the number of cubes in the HAUY CONSTRUCTION of the RHOMBIC DODECAHEDRON , given by HauyRhoDodk /C30k3 /C276X i/C301 ; 3 ; ... ; k /C282i2 ; (3) for k an ODD NUMBER . Re-indexing with k /C302n /C281 then gives HauyRhoDodn /C30(2n /C281)(8n2 /C2814n /C277); (4) giving the first few values 1, 33, 185, 553, 1233, ... (Sloane’s A046142). See also ESCHER’S SOLID ,H AUY CONSTRUCTION , OCTAHEDRAL NUMBER ,RHOMBIC DODECAHEDRON References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 53 /C1/54, 1996. Sloane, N. J. A. Sequences A005917/M4968 and A046142 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Rhombic Dodecahedron The DUAL POLYHEDRON of the CUBOCTAHEDRON A1 and Wenninger dual W11:Its sometimes also called the RHOMBOIDAL DODECAHEDRON (Cotton 1990). Its 14 vertices are joined by 12 RHOMBUSES of the dimensions shown in the figure below, where a/C302 cot/C281ffiffiffi 2p /C30cos/C2811 3})@D})@E :70:53/C14(1) b/C302 tan/C281ffiffiffi 2p :109:47/C14: (2) The rhombic dodecahedron can be built up by a placing six cubes on the faces of a seventh, in the configuration of a metal "jack." Joining the centers of the outer cubes with the vertices of the central cube then gives the rhombic dodecahedron. Affixing a SQUARE PYRAMID of height 1/2 on each face of a CUBE having unit edge length results in a rhombic dodeca- hedron (Bru¨ckner 1900, p. 130; Steinhaus 1983, p. 185). If the rhombic dodecahedron is hinged into six square pyramids along three consecutive face diagonals, the resulting model can be folded into a cube (Wells 1991). One possible construction for the rhombic dodecahedron is known as the BAUSPIEL . It can also be constructed by CUMULATION of a unit edge-length CUBE by a pyramid with height 1/2. The rhombic dodecahedron is a ZONOHEDRON and a SPACE-FILLING POLYHEDRON (Steinhaus 1983, p. 185). The vertices are given by (91, 91, 91), ( 92, 0, 0), (0, 92, 0), (0, 0, 92). The edges of the CUBE-OCTAHEDRON COMPOUND inter- secting in the points plotted above are the diagonals of RHOMBUSES , and the 12 RHOMBUSES form a rhombic dodecahedron (Ball and Coxeter 1987). There are three stellations of the rhombic dodecahedron. The rhombic dodecahedron can be built using a HAUY CONSTRUCTION . The Hauy RHOMBIC DODECAHEDRAL NUMBERS HRhoDodn /C30(2n /C281)(8n2 /C2814n /C277) (3) give a method for calculating the VOLUME of the rhombic dodecahedron, V/C30lim n0/C12HRhoDodna nffiffiffi 3p !3 /C3016 9ffiffiffi 3p a3(4) (Steinhaus 1983). The SURFACE AREA of a rhombic dodecahedron with unit edge length is S/C308ffiffiffi 2p : (5) See also BAUSPIEL ,CUBE-OCTAHEDRON COMPOUND , DODECAHEDRON ,H AUY CONSTRUCTION ,P YRITOHE- DRON ,R HOMBIC DODECAHEDRON STELLATIONS , RHOMBIC TRIACONTAHEDRON ,R HOMBUS ,S PHERE PACKING ,STEINMETZ SOLID ,TRIGONAL DODECAHE- DRON ,ZONOHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 137, 1987. Bru¨ckner, M. Vielecke under Vielflache. Leipzig, Germany, 1900. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, p. 62, 1990. Cundy, H. and Rollett, A. "Rhombic Dodecahedron. V(3:4)2:/" §3.8.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 120, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185 /C1/186, 1999. Weisstein, E. W. "Polyhedra." M ATHEMATICA NOTEBOOK POLYHEDRA.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 215 /C1/216, 1991. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 19, 21, and 34, 1983. Rhombic Dodecahedron Stellations There are three STELLATIONS of the RHOMBIC DODE- CAHEDRON (Wells 1991), two of which are illustrated above. The first stellation can be constructed by drawing diagonals across the square faces of a CUBOCTAHEDRON and connecting centers of these diagonals with the vertices of neighboring squares. The outer edges of the second stellation correspond with those of the TRUNCATED OCTAHEDRON . See also CUBOCTAHEDRON ,RHOMBIC DODECAHEDRON , STELLATION ,TRUNCATED OCTAHEDRON References Cundy, H. and Rollett, A. "The Stellated Rhombic Dodeca- hedron." §3.9.5 in Mathematical Models, 3rd ed. Strad- broke, England: Tarquin Pub., pp. 127 /C1/128, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 215 /C1/216, 1991. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 36, 1983. Rhombic Icosahedron A ZONOHEDRON which can be derived from the RHOMBIC TRIACONTAHEDRON by removing any one of the zones and bringing together the two pieces into which the remainder of the surface is thereby divided. See also RHOMBIC TRIACONTAHEDRON ,ZONOHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 143, 1987. Bilinski, S. "U¨ ber die Rhombenisoeder." Glasnik Mat.-Fiz. Astron. Drustro Mat. Fiz. Hrvatske Ser. II 15, 251 /C1/263, 1960. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Rhombic Polyhedron A POLYHEDRON with extra square faces, given by the SCHLA ¨FLI SYMBOL rfp qg:/ See also RHOMBIC DODECAHEDRON ,RHOMBIC ICOSA- HEDRON ,RHOMBIC TRIACONTAHEDRON ,SNUB POLY- HEDRON ,TRUNCATED POLYHEDRON Rhombic Spirallohedron A beautiful class of polyhedra composed of rhombic faces discovered accidentally by R. Towle while at- tempting to develop a function to create a rhombic hexahedron from a triple of vectors. References Towle, R. "Rhombic Spirallohedra." http://www.mathsour- ce.com/cgi-bin/msitem?0208 /C1/718. Rhombic Triacontahedron AZONOHEDRON which is the DUAL POLYHEDRON of the ICOSIDODECAHEDRON A4and Wenninger dual W12:It is composed of 30 RHOMBI joined at 32 vertices. The intersecting edges of the DODECAHEDRON-ICOSAHE- DRON COMPOUND form the diagonals of 30 RHOMBI which comprise the TRIACONTAHEDRON . The CUBE 5- COMPOUND has the 30 facial planes of the rhombic triacontahedron (Wenninger 1983, p. 36; Ball and Coxeter 1987). The short diagonals of the faces of the rhombic triacontahedron give the edges of a DODECAHEDRON , while the long diagonals give the edges of the ICOSAHEDRON (Steinhaus 1983, pp. 209 /C1/210). Taken together, the DODECAHEDRON and ICOSAHEDRON give a DODECAHEDRON-ICOSAHEDRON COMPOUND . The rhombic triacontahedron generated from an ICOSIDODECAHEDRON of unit edge lengths has edge lengths s /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 525 /C27ffiffiffi 5p})@D})@Er : (1) and INRADIUS r /C301 85 /C273ffiffiffi 5p})@D})@E : (2) Normalizing so that s /C301, the solid has SURFACE AREA and VOLUME given by S /C3012ffiffiffi 5p (3) V /C304ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq : (4) See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , CUBE 5-COMPOUND ,D ODECAHEDRON ,D ODECAHE- DRON- ICOSAHEDRON COMPOUND ,ICOSAHEDRON ,ICO- SIDODECAHEDRON ,R HOMBIC DODECAHEDRON , RHOMBIC TRIACONTAHEDRON STELLATIONS ,R HOM- BUS,TRIACONTAHEDRON ,ZONOHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 137, 1987. Bulatov, V. "Stellations of Rhombic Triacontahedron." http:// www.physics.orst.edu/~bulatov/polyhedra/rtc/. Cundy, H. and Rollett, A. "Rhombic Triacontahedron." §3.8.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 121 /C1/122 and 127, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 207 and 209 /C1/210, 1999. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 22, 1983.Rhombic Triacontahedron Stellations Ede (1958) enumerates 13 basic series of stellations of the rhombic triacontahedron, the total number of which is extremely large. Pawsey (1973) gave a set of restrictions upon which a complete enumeration of stellations can be achieved (Wenninger 1983, p. 36). Messer (1995) describes 226 stellations, some of which are illustrated above. The CONVEX HULL of the DODECADODECAHEDRON is an ICOSIDODECAHEDRON and the dual of the ICOSIDODE- CAHEDRON is the RHOMBIC TRIACONTAHEDRON , so the dual of the DODECADODECAHEDRON (the MEDIAL RHOMBIC TRIACONTAHEDRON ) is one of the rhombic triacontahedron stellations (Wenninger 1983, p. 41). Another is the GREAT RHOMBIC TRIACONTAHEDRON . See also GREAT RHOMBIC TRIACONTAHEDRON ,MEDIAL RHOMBIC TRIACONTAHEDRON ,RHOMBIC TRIACONTA- HEDRON ,STELLATION References Ede, J. D. "Rhombic Triacontahedra." Math. Gazette 42,9 8/C1/ 100, 1958. Messer, P. W. "Stellations of the Rhombic Triacontahedron and Beyond." Structural Topology 21,2 5/C1/46, 1995. Pawley, G. S "The 227 Triacontahedra." Geom. Dedicata 4, 221/C1/232, 1975. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 36, 1983. Rhombicosacron The DUAL POLYHEDRON of the RHOMBICOSAHEDRON U56 and Wenninger dual W96 :/ See also DUAL POLYHEDRON ,RHOMBICOSAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 85, 1983. Rhombicosahedron The UNIFORM POLYHEDRON U56 and Wenninger model W96whose DUAL POLYHEDRON is the RHOMBICOSA- CRON . It has WYTHOFF SYMBOL 25 2 3½: Its faces are 10 f6g/C2715 f4g/C2715 f43 g/C2710 f65 g: The CIRCUMRADIUS for unit edge length is R /C301 2ffiffiffi 7p : References Wenninger, M. J. "Rhombicosahedron." Model 96 in Poly- hedron Models. Cambridge, England: Cambridge Univer- sity Press, pp. 149 /C1/150, 1971. Rhombicosidodecahedron BIGYRATE DIMINISHED RHOMBICOSIDODECAHEDRON , DIMINISHED RHOMBICOSIDODECAHEDRON ,G REAT RHOMBICOSIDODECAHEDRON (ARCHIMEDEAN ), GREAT RHOMBICOSIDODECAHEDRON (UNIFORM ), GYRATE BI- DIMINISHED RHOMBICOSIDODECAHEDRON ,G YRATE RHOMBICOSIDODECAHEDRON ,M ETABIDIMINISHED RHOMBICOSIDODECAHEDRON ,M ETABIGYRATE RHOM- BICOSIDODECAHEDRON ,M ETAGYRATE DIMINISHED RHOMBICOSIDODECAHEDRON ,P ARABIDIMINISHED RHOMBICOSIDODECAHEDRON ,PARABIGYRATE RHOMBI- COSIDODECAHEDRON ,P ARAGYRATE DIMINISHED RHOMBICOSIDODECAHEDRON ,SMALL RHOMBICOSIDO-DECAHEDRON ,TRIDIMINISHED RHOMBICOSIDODECAHE- DRON ,TRIGYRATE RHOMBICOSIDODECAHEDRON Rhombicuboctahedron GREAT RHOMBICUBOCTAHEDRON (ARCHIMEDEAN ), GREAT RHOMBICUBOCTAHEDRON (UNIFORM ), SMALL RHOMBICUBOCTAHEDRON Rhombidodecadodecahedron The UNIFORM POLYHEDRON U38whose DUAL POLYHE- DRON is the MEDIAL DELTOIDAL HEXECONTAHEDRON .It has SCHLA ¨ FLI SYMBOL r f5 2 g and WYTHOFF SYMBOL 5 252:j Its faces are 12 f52 g/C273f4 g/C2712 f5g: The CIRCUM- RADIUS for unit edge length is R /C301 2ffiffiffi 7p : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 116 /C1/117, 1989. Rhombihexacron GREAT RHOMBIHEXACRON ,SMALL RHOMBIHEXACRON Rhombihexahedron GREAT RHOMBIHEXAHEDRON ,SMALL RHOMBIHEXAHE- DRON Rhombitruncated Cuboctahedron GREAT RHOMBICUBOCTAHEDRON (ARCHIMEDEAN ) Rhombitruncated Icosidodecahedron GREAT RHOMBICOSIDODECAHEDRON (ARCHIMEDEAN ) Rhombohedron A PARALLELEPIPED bounded by six congruent RHOMBS . See also PARALLELEPIPED ,RHOMB References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 142 and 161, 1987. Rhomboid A PARALLELOGRAM in which angles are oblique and adjacent sides are of unequal length. See also BAR (POLYIAMOND ), DIAMOND ,K ITE,L O- ZENGE ,PARALLELOGRAM ,QUADRILATERAL ,RHOMBUS , SKEW QUADRILATERAL ,TRAPEZIUM ,TRAPEZOID References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 176, 1984. Rhomboidal Dodecahedron RHOMBIC DODECAHEDRON Rhombus A QUADRILATERAL with both pairs of opposite sides PARALLEL and all sides the same length, i.e., an equilateral PARALLELOGRAM . The word RHOMB is sometimes used instead of rhombus, and a rhombus is sometimes also called a diamond. A rhombus with 2u /C3045 /C14 is sometimes called a LOZENGE . The DIAGONALS p and q of a rhombus are PERPENDI- CULAR and satisfy p2 /C27q2 /C304a2 : The AREA of a rhombus is given by A /C301 2 pq : See also DIAMOND ,H ARBORTH’S TILING ,K ITE,LO- ZENGE ,PARALLELOGRAM ,QUADRILATERAL ,RHOMBIC DODECAHEDRON ,R HOMBIC ICOSAHEDRON ,RHOMBIC TRIACONTAHEDRON ,R HOMBOID ,S KEW QUADRILAT- ERAL ,TRAPEZIUM ,TRAPEZOID References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Harris, J. W. and Stocker, H. "Rhombus." §3.6.4 in Hand- book of Mathematics and Computational Science. New York: Springer-Verlag, pp. 83 /C1/84, 1998. Rhumb Line LOXODROME Ribbon Knot If the KNOT K is the boundary K /C30f S1})0})@ of a singular disk f : D 0 S3 which has the property that each self- intersecting component is an arc A ƒf D2})0})@ for which f /C281(A) consists of two arcs in D2 ; one of which isinterior, then K is said to be a ribbon knot. Every ribbon knot is a SLICE KNOT , and it is conjectured that every SLICE KNOT is a ribbon knot. See also SLICE KNOT References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 225, 1976. Ribet’s Theorem If the TANIYAMA- SHIMURA CONJECTURE holds for all semistable ELLIPTIC CURVES , then FERMAT’S LAST THEOREM is true. Before its proof by Ribet in 1986, the theorem had been called the EPSILON CONJEC- TURE . It had its roots in a surprising result of G. Frey. See also ELLIPTIC CURVE ,E PSILON CONJECTURE , FERMAT’S LAST THEOREM ,MODULAR FORM,MODULAR FUNCTION ,TANIYAMA- SHIMURA CONJECTURE Riccati Differential Equation y?/C30P(z)/C27Q(z)y/C27R(z)y2; (1) where y?/C13dy=dz:The transformation w/C13/C28y? yR(z)(2) leads to the second-order linear homogeneous equa- tion R(z)yƒ/C28[R?(z)/C27Q(z)R(z)]y?/C27[R(z)]2P(z)y/C300:(3) Another equation sometimes called the Riccati differ-ential equation is z 2wƒ/C27z2/C28n(n/C271)})1})A w/C300 (4) (Zwillinger 1997, p. 126), which has solutions w/C30Azjn(z)/C27Bzyn(z); (5) where jn(z) and yn(z) are SPHERICAL BESSEL FUNC- TIONS OF THE FIRST and SECOND KINDS . Yet another form of "the" Riccati differential equation is dy dz/C30azn/C27by2; (6) which is solvable by algebraic, exponential, and logarithmic functions only when n/C30/C284m=(2m91); form/C300, 1, 2, .... References Abramowitz, M. and Stegun, C. A. (Eds.). "Riccati-Bessel Functions." §10.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 445, 1972. Bender, C. M. and Orszag, S. A. §1.6 in Advanced Mathe- matical Methods for Scientists and Engineers. New York: McGraw-Hill, 1978. Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 142 /C1/143, 1986. Glaisher, J. W. L. "On Riccati’s Equation." Quart. J. Pure Appl. Math. 11, 267 /C1/273, 1871. Goldstein, M. E. and Braun, W. H. Advanced Methods for the Solution of Differential Equations. NASA SP-316. Washington, DC: U.S. Government Printing Office, pp. 45 /C1/46, 1973. Ince, E. L. Ordinary Differential Equations. New York: Dover, pp. 23 /C1/35 and 295, 1956. Reid, W. T. Riccati Differential Equations. New York: Academic Press, 1972. Simmons, G. F. Differential Equations with Applications and Historical Notes. New York: McGraw-Hill, pp. 62 /C1/63, 1972. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995. Zwillinger, D. "Riccati Equation--1 and Riccati Equation--2." §II.A.75 and II.A.76 in Handbook of Differential Equa- tions, 3rd ed. Boston, MA: Academic Press, pp. 121 and 288 /C1/291, 1997. Riccati-Bessel Functions Sn(z) /C13zjn(z) /C30ffiffiffiffiffi pz 2s Jn/C271=2(z) Cn(z) /C13/C28znn(z) /C30/C28ffiffiffiffiffi pz 2s Nn/C271=2(z) ; where jn(z) and nn(z) are SPHERICAL BESSEL FUNC- TIONS OF THE FIRST and SECOND KIND . References Abramowitz, M. and Stegun, C. A. (Eds.). "Riccati-Bessel Functions." §10.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 445, 1972. Ricci Curvature RICCI CURVATURE TENSOR Ricci Curvature Tensor Rmk /C13R l ml k ; where Rl mlk is the RIEMANN TENSOR . Topologically, the Ricci curvature is the mathemati- cal object which controls the growth rate of the volume of metric balls in a MANIFOLD . See also BISHOP’S INEQUALITY ,CAMPBELL’S THEOREM , CURVATURE SCALAR ,E INSTEIN TENSOR ,M ILNOR’S THEOREM ,RIEMANN TENSOR References Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravita- tion. San Francisco: W. H. Freeman, 1973. Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, p. 40, 1984. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, pp. 135 and 142, 1972.Ricci Tensor RICCI CURVATURE TENSOR Rice Distribution P(Z) /C30Z s2exp /C28Z2 /C27 Vjj2 2 s2 ! I0ZVjj s2 ! ; where I0(z)isa MODIFIED BESSEL FUNCTION OF THE FIRST KIND and Z /C210. For a derivation, see Papoulis (1962). For Vjj/C300 /C300; this reduces to the RAYLEIGH DISTRIBUTION . See also RAYLEIGH DISTRIBUTION References Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. Richard’s Paradox It is possible to describe a set of POSITIVE INTEGERS that cannot be listed in a book containing a set of counting numbers on each consecutively numbered page. Another form of the paradox states that the set of all numerical functions is nondenumerable (Curry 1977). References Church, A. "A Bibliography of Symbolic Logic." J. Symb. Logic 1, 121 /C1/218, 1936. Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 6, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 172 /C1/173, 1998. Richardson Extrapolation The consideration of the result of a numerical calculation as a function of an adjustable parameter (usually the step size). The function can then be fitted and evaluated at h /C300 to yield very accurate results. Press et al. (1992) describe this process as turning lead into gold. Richardson extrapolation is one of the key ideas used in the popular and robust BULIRSCH- STOER ALGORITHM of solving ORDINARY DIFFERENTIAL EQUATIONS . See also BULIRSCH- STOER ALGORITHM References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 106, 1990. Jeffreys, H. and Jeffreys, B. S. "L. F. Richardson’s Method." §9.091 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 288, 1988. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Richardson Extrapolation and the Bulirsch-Stoer Method." §16.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 718 /C1 /725, 1992. Richardson’s Theorem Let R be the class of expressions generated by 1. The RATIONAL NUMBERS and the two REAL NUMBERS p and ln 2;/ 2. The variable x, 3. The operations of ADDITION , MULTIPLICATION , and composition, and 4. The SINE, EXPONENTIAL , and ABSOLUTE VALUE functions. Then if E /C23 R; the predicate "E /C300" is recursively UNDECIDABLE . See also INTEGER RELATION ,RECURSION ,U NDECID- ABLE References Caviness, B. F. "On Canonical Forms and Simplification." J. Assoc. Comp. Mach. 17, 385 /C1/396, 1970. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, 1996. Richardson, D. "Some Unsolvable Problems Involving Ele- mentary Functions of a Real Variable." J. Symbolic Logic 33, 514 /C1/520, 1968. Riddell’s Formula Riddell’s formula for unlabeled graphs is the EULER TRANSFORM relating the number of unlabeled CON- NECTED GRAPHS on n nodes satisfying some property with the corresponding total number (not necessarily connected) of GRAPHS on n nodes. Riddell’s formula for labeled graphs is the EXPONEN- TIAL TRANSFORM relating the number of labeled CONNECTED GRAPHS on n nodes satisfying some property with the corresponding total number (not necessarily connected) of labeled GRAPHS on n nodes. See also CONNECTED GRAPH ,E ULER TRANSFORM , EXPONENTIAL TRANSFORM ,GRAPH ,LABELED GRAPH , UNLABELED GRAPH References Cadogan, C. C. "The Mo¨bius Function and Connected Graphs." J. Combin. Th. B 11, 193 /C1/200, 1971. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, p. 90, 1973. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 20, 1995. Ridders’ Method A variation of the FALSE POSITION METHOD for finding ROOTS which fits the function in question with an exponential. See also FALSE POSITION METHOD ,ROOTReferences Ostrowski, A. M. Ch. 12 in Solutions of Equations and Systems of Equations, 2nd ed. New York: Academic Press, 1966. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Secant Method, False Position Method, and Ridders’ Method." §9.2 in Numerical Recipes in FOR- TRAN: The Art of Scientific Computing, 2nd ed. Cam- bridge, England: Cambridge University Press, pp. 347 /C1/ 352, 1992. Ralston, A. and Rabinowitz, P. §8.3 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978. Ridders, C. F. J. "A New Algorithm for Computing a Single Root of a Real Continuous Function." IEEE Trans. Circuits Systems 26, 979 /C1/980, 1979. Ridge An (n /C282)/-D FACE of an n-D POLYTOPE . See also POLYTOPE Riemann Curve Theorem If two algebraic plane curves with only ordinary singular points and CUSPS are related such that the coordinates of a point on either are RATIONAL FUNC- TIONS of a corresponding point on the other, then the curves have the same GENUS (CURVE ). This can be stated equivalently as the GENUS of a curve is unaltered by a BIRATIONAL TRANSFORMATION . References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 120, 1959. Riemann Differential Equation RIEMANN P-DIFFERENTIAL EQUATION Riemann Formula The solution u(x; y) /C30gx 0djgy 1R( j; h; x; y)f( j; h) dh ; (1) where R(x; y; j; h) is the RIEMANN FUNCTION of the linear GOURSAT PROBLEM with characteristics f /C30 c /C300 according to the RIEMANN METHOD . See also GOURSAT PROBLEM ,R IEMANN FUNCTION , RIEMANN METHOD References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 289, 1988. Riemann Function There are a number of functions in various branches of mathematics known as Riemann functions. Exam-ples include the R IEMANN P-SERIES ,RIEMANN- SIEGEL FUNCTIONS ,R IEMANN THETA FUNCTION ,R IEMANN ZETA FUNCTION , XI FUNCTION , the function F(x) obtained by Riemann in studying FOURIER SERIES , the function R(x; y; j; h) appearing in the applica- tion of the RIEMANN METHOD for solving the GOURSAT PROBLEM , the function R(n) in the RIEMANN PRIME NUMBER FORMULA , and the function f(x) related to the PRIME COUNTING FUNCTION defined below. The Riemann function F(x) for a FOURIER SERIES 1 2 a0 /C27X/C12 n/C301an cos(nx) /C27bn sin(nx) ½/C138 (1) is obtained by integrating twice term by term to obtain F(x) /C301 4 a0x2 /C28X/C12 n /C3011 n2ancos(nx) /C27bn sin(nx) ½/C138 /C27Cx /C27D; (2) where C and D are constants (Riemann 1957; Hazewinkel 1988, vol. 8, p. 118). The Riemann function R(x; y; j; h) arises in the solution of the linear case of the GOURSAT PROBLEM of solving the HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION ˜Lu /C30uxy /C27aux /C27buy /C27cu /C30f (3) with BOUNDARY CONDITIONS u(0; t) /C30 f(t) (4) u(t; 1) /C30 c(t) (5) f(1) /C30 f(0) : (6) Here, R(x; y; j; h) is defined as the solution of the equation Rxy /C28(aR)x /C28(bR)y /C27cR /C300 (7) which satisfies the conditions R( j; y; j; n) /C30expgy ha( j; t) dt"# (8) R(x ; h; j; h) /C30expgx jb(t; h) dt})10})1@ (9) on the characteristics x /C30 j and y /C30 h; where (j; h)isa point on the domain V on which (8) is defined (Hazewinkel 1988). The solution is then given by the RIEMANN FORMULA u(x; y) /C30gx 0d jgy 1R(j; h; x ; y)f(j; h) dh : (10) This method of solution is called the RIEMANN METHOD .Riemann defined the function f(x)by f(x) /C13X/C12 n/C301p x1 =n})0})@ n /C30 p(x) /C2712 p x1 =2})0})@ /C2713 p x1=3})0})@ /C27... (11) (Hardy 1999, p. 30), then the PRIME COUNTING FUNC- TION p(x) is related to f(x)by p(x) /C30X/C12 n/C301m(n) nfx1=n})0})@ ; (12) where m(n) is the MO¨ BIUS FUNCTION (Riesel 1994, p. 49). Riemann (1859) proposed that f(x) /C30li(x) /C28X rli(xr) /C28ln 2 /C27g/C12 xdt t ln tt2 /C28 1 ðÞ; (13) where li(x) is the LOGARITHMIC INTEGRAL and the sum is over all nontrivial zeros r of the RIEMANN ZETA FUNCTION z(z) (Mathews 1892, Ch. 10; Landau 1974, Ch. 19; Ingham 1990, Ch. 4; Hardy 1999, p. 40). This formula was subsequently proved by Mangoldt in 1895 (Riesel 1994, p. 47). A function related to f(x) is given by J(x)/C13p(x)/C271 2px1=2})0})@ /C2713px1=3})0})@ /C27.../C281 2m forpmwith pprime p(x)/C271 2px1=2})0})@ /C2713px1=3})0})@ /C27... otherwise8 >>< >>:(14) /C30lim t0/C121 2pig2/C27iT 2/C28iTxs slnz(s)ds; (15) where z(z) is the R IEMANN ZETA FUNCTION . This function satisfies lnz(s) s/C30g/C12 1J(x)x/C28s/C281dx (16) (Riesel 1994, p. 47). See also CRITICAL STRIP,GOURSAT PROBLEM ,LOGA- RITHMIC INTEGRAL ,M ANGOLDT FUNCTION ,RIEMANN METHOD ,PRIME NUMBER THEOREM ,RIEMANN PRIME NUMBER FORMULA ,RIEMANN ZETA FUNCTION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 144 /C1/145, 1996. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, Vol. 4, p. 289 and Vol. 8, p. 125, 1988. Ingham, A. E. The Distribution of Prime Numbers. London: Cambridge University Press, p. 83, 1990. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, 1974. Mathews, G. B. Ch. 10 in Theory of Numbers. New York: Chelsea, 1961. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 224 /C1/225, 1996. Riemann, G. F. B. "U¨ ber die Anzahl der Primzahlen unter einer gegebenen Gro¨sse." Monatsber. Ko¨nigl. Preuss. Akad. Wiss. Berlin , 671, 1859. Riemann, B. "U¨ ber die Darstellbarkeit einer Function durch eine trigonometrische Reihe." In Gesammelte math. Ab- handlungen. New York: Dover, pp. 227 /C1/264, 1957. Riesel, H. "The Riemann Prime Number Formula." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 50 /C1/52, 1994. Riesel, H. and Go¨hl, G. "Some Calculations Related to Riemann’s Prime Number Formula." Math. Comput. 24, 969 /C1/983, 1970. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 28 /C1/29 and 362 /C1/372, 1991. Riemann Hypothesis First published in Riemann (1859), the Riemann hypothesis states that the nontrivial ROOTS of the RIEMANN ZETA FUNCTION z(s) /C13X/C12 n/C3011 ns ; (1) where x /C23C (the COMPLEX NUMBERS ), all lie on the "CRITICAL LINE" R[s] /C301=2; where R[z] denotes the REAL PART of z. The Riemann hypothesis is also known as ARTIN’S CONJECTURE . Wiener showed that the PRIME NUMBER THEOREM is literally equivalent to the assertion that z(s) has no zeros on s /C301 (Hardy 1999, pp. 34 and 58 /C1/60). In 1914, Hardy proved that an INFINITE number of values for s can be found for which z(s) /C300 and R[s] /C30 1=2 : However, it is not known if all nontrivial roots s satisfy R[s] /C301=2; so the conjecture remains open. Andre ´ Weil proved the Riemann hypothesis to be true for field functions (Weil 1948, Eichler 1966, Ball and Coxeter 1987). In 1974, Levinson (1974ab) showed that at least 1/3 of the ROOTS must lie on the CRITICAL LINE (Le Lionnais 1983), a result which has since been sharpened to 40% (Vardi 1991, p. 142). It is known that the zeros are symmetrical placed about the line I[s] /C300:/ The Riemann hypothesis is equivalent to L50; where L is the DE BRUIJN- NEWMAN CONSTANT (Csordas et al. 1994). It is also equivalent to the assertion that for some constant c, Li(x) /C28 p(x) jj 5cffiffiffixpln x; (2) where Li(x) is the LOGARITHMIC INTEGRAL and p is the PRIME COUNTING FUNCTION (Wagon 1991). Another equivalent form states that spanL2(0; 1)ra ; 0 B a B1 fg /C30L2(0; 1); (3)where ra(t) /C13fraca t ! /C28 a frac1 t ! ; (4) where frac( x) is the FRACTIONAL PART (Balazard and Saias 2000). By modifying a criterion of Robin (1984), Lagarias (2000) showed that the Riemann hypothesis is equivalent to the statement that s(n) 5Hn /C27exp HnðÞ ln Hn ; (5) for all n ]1; with equality only for n /C301, where Hn is a HARMONIC NUMBER and s(n) is the DIVISOR FUNC- TION . There is also a finite analog of the Riemann hypoth- esis concerning the location of zeros for function fields defined by equations such as ayl /C27bzm /C27c /C300: (6) This hypothesis, developed by Weil, is analogous to the usual Riemann hypothesis. The number of solu- tions for the particular cases l;mðÞ/C30(2;2);(3, 3), (4, 4), and (2, 4) were known to Gauss. The hypothesis has thus far resisted all attempts to prove it, although it has been computationally tested and found to be true for the first 200 ;000;001 zeros by Brent et al. (1982). Brent’s calculation covered zeros s/C27itin the region 0 BtB81;702;130:19:In 2000, Clay Mathematics Institute offered a $1 million prize for proof of the Riemann hypothesis. See also BERRY CONJECTURE ,CRITICAL LINE,CRITI- CAL STRIP,EXTENDED RIEMANN HYPOTHESIS ,GRON- WALL’S THEOREM ,M ERTENS CONJECTURE ,M ILLS’ CONSTANT ,P RIME NUMBER THEOREM ,R IEMANN ZETA FUNCTION References Balazard, M. and Saias, E. "The Nyman-Beurling Equiva- lent Form for the Riemann Hypothesis." Expos. Math. 18, 131/C1/138, 2000. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 75, 1987. Bombieri, E. "Problems of the Millennium: The Riemann Hypothesis." http://www.claymath.org/prize_problems/rie- mann.pdf. Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361/C1/1372, 1979. Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. II." Math. Comput. 39, 681/C1/688, 1982. Caldwell, C. K. "Prime Links /C27/C27: Resources in theory: conjectures: Riemann." http://primes.utm.edu/links/the-ory/conjectures/Riemann/. Clay Mathematics Institute. "The Riemann Hypothesis." http://www.claymath.org/prize_problems/riemann.htm. Csordas, G.; Smith, W.; and Varga, R. S. "Lehmer Pairs of Zeros, the de Bruijn-Newman Constant and the RiemannHypothesis." Constr. Approx. 10, 107/C1 /129, 1994. Eichler, M. Introduction to the Theory of Algebraic Numbers and Functions. New York: Academic Press, 1966. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Krantz, S. G. "The Riemann Hypothesis." §13.2.9 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, p. 161, 1999. Lagarias, J. C. An Elementary Problem Equivalent to the Riemann Hypothesis 22 Aug 2000. http://xxx.lanl.gov/abs/ math.NT/0008177/. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983. Levinson, N. "More than One Third of Zeros of Riemann’s Zeta-Function Are on s/C301=2:/"Adv. Math. 13, 383/C1/436, 1974. Levinson, N. "At Least One Third of Zeros of Riemann’s Zeta-Function Are on s/C301=2:/"Proc. Nat. Acad. Sci. USA 71, 1013/C1/1015, 1974. Odlyzko, A. "The 1020th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Riemann, B. "U ¨ber die Anzahl der Primzahlen unter einer gegebenen Gro ¨sse," Mon. Not. Berlin Akad., pp. 671 /C1/680, Nov. 1859. Robin, G. "Grandes valeurs de la fonction somme des diviseurs er hypothe `se de Riemann." J. Math. Pures Appl. 63, 187/C1/213, 1984. Sloane, N. J. A. Sequences A002410/M4924 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 0821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Provi- dence, RI: Amer. Math. Soc., 2000. te Riele, H. J. J. "Corrigendum to: On the Zeros of the Riemann Zeta Function in the Critical Strip. II." Math. Comput. 46, 771, 1986. van de Lune, J. and te Riele, H. J. J. "On The Zeros of the Riemann Zeta-Function in the Critical Strip. III." Math. Comput. 41, 759/C1 /767, 1983. van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. IV." Math. Comput. 46, 667/C1/681, 1986. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 33, 1991. Weil, A. Sur les courbes alge ´briques et les varie ´te`s qui s’en de´duisent. Paris, 1948. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 28, 1986. Riemann Integral The Riemann integral is the INTEGRAL normally encountered in CALCULUS texts and used by physicists and engineers. Other types of integrals exist (e.g., the LEBESGUE INTEGRAL ), but are unlikely to be encoun- tered outside the confines of advanced mathematicstexts. In fact, according to Jeffreys and Jeffreys (1988,p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann’s [definition of the integral]is not are too rare in physics to repay the extra difficulty." The Riemann integral is based on the J ORDAN MEASURE , and defined by taking a limit of a R IEMANN SUM, ga bf(x)dx/C13 lim max Dxk00Xn k/C301fx/C31kðÞDxk (1) ggf(x;y)dA/C13 lim max DAk00Xn k/C301fx/C31k;y/C31k ðÞ DAk (2) gggf(x;yz)dV/C13 lim max DVk00Xn k/C301fx/C31k;y/C31k;z/C31k ðÞ DVk;(3) where a5x5bandx/C31k;y/C31k;andz/C31kare arbitrary points in the intervals Dxk;Dyk;andDzk;respectively. The value max Dxkis called the MESH SIZE of a partition of the interval [ a, b] into subintervals Dxk:/ As an example of the application of the Riemann integral definition, find the AREA under the curve y/C30 xrfrom 0 to a. Divide ( a, b) into nsegments, so Dxk/C30 b/C28a n/C13h;then f(x1)/C30f(0)/C300 (4) f(x2)/C30f(Dxk)/C30hr(5) f(x3)/C30f2Dxk ðÞ /C30(2h)r: (6) By induction fxkðÞ/C30f[k/C281]Dxk ðÞ /C30[(k/C281)h]r/C30hr(k/C281)r; (7) so f(xk)Dxk/C30hr/C271(k/C281)r(8) Xn k/C301f(xk)Dxk/C30hr/C271Xn k/C301(k/C281)r: (9) For example, take r/C302. Xn k/C301f(xk)Dxk/C30h3Xn k/C301(k/C281)2 /C30h3Xn k/C301k2/C282Xn k/C301k/C27Xn k/C3011 ! /C30h3n(n/C271)(2n/C271) 6/C282n(n/C271) 2/C27n"# ; (10) so I /C13 lim n0/C12Xn k /C301fx/C31kðÞDxk /C30 lim n0/C12Xn k/C301fxkðÞDxk /C30 lim n0/C12h3n(n /C27 1)(2n /C27 1) 6/C282n(n /C27 1) 2/C27n"# /C30a3 lim n0/C12n(n /C27 1)(2n /C27 1) 6n3 /C28n(n /C27 1) n3/C27n n3"# /C301 3 a3 : (11) See also INTEGRAL ,RIEMANN SUM References Ferreiro ´s, J. "The Riemann Integral." §5.1.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 150 /C1/ 153, 1999. Jeffreys, H. and Jeffreys, B. S. "Integration: Riemann, Stieltjes." §1.10 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 26 /C1/36, 1988. Kestelman, H. "Riemann Integration." Ch. 2 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 33 /C1/66, 1960. Riemann Mapping Theorem Let z0 be a point in a simply connected region R "C: Then there is a unique ANALYTIC FUNCTION w /C30f(z) mapping R one-to-one onto the DISK wjjB1 such that fz0ðÞ/C300 and f ? z0ðÞ/C300: The COROLLARY guarantees that any two simply connected regions except R2 can be mapped CONFORMALLY onto each other. References Krantz, S. G. "The Riemann Mapping Theorem." §6.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 86 /C1/87, 1999. Riemann Method The method for solving the GOURSAT PROBLEM and CAUCHY PROBLEM for linear HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS using a RIEMANN FUNC- TION . See also GREEN’S FUNCTION ,RIEMANN FUNCTION References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathe- matics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, Vol. 4, p. 289 and Vol. 8, pp. 125 /C1/126, 1988. Riemann P-Differential Equation The differential equationd2u dz2 /C271 /C28 a /C28 a? z /C28 a/C271 /C28 b /C28 b? z /C28 b/C271 /C28 g /C28 g 0 z /C28 c"# du dz /C27})10aa?(a /C28 b)(a /C28 c) z /C28 a/C27bb?(b /C28 c)(b /C28 a) z /C28 b /C27gg?(c /C28 a)(c /C28 b) z /C28 c})1@u (z /C28 a)(z /C28 b)(z /C28 c) /C300; where a /C27 a?/C27b /C27 b?/C27g /C27 g ?/C301; first obtained in the form by Papperitz (1885; Bares 1908). Solutions are RIEMANN P-SERIES (Abramowitz and Stegun 1972, pp. 564 /C1/565). Zwillinger (1995, p. 414) confusingly calls this equation the "hypergeo- metric equation." See also HEUN’S DIFFERENTIAL EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). "Riemann’s Differential Equation." §15.6 in Handbook of Mathemati- cal Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 564 /C1/565, 1972. Barnes, E. W. "A New Development in the Theory of the Hypergeometric Functions." Proc. London Math. Soc. 6, 141/C1/177, 1908. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 541 /C1/543, 1953. Papperitz. Math. Ann. 25, 213, 1885. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997. Riemann Prime Number Formula Riemann considered R(x)/C30X/C12 n/C301m(n) nlix1=n})0})@ ; (1) obtained by replacing fx1=n})0})@ in the R IEMANN FUNC- TION with the LOGARITHMIC INTEGRAL lix1=n})0})@ :;where z(z) is the R IEMANN ZETA FUNCTION andm(n) is the MO¨BIUS FUNCTION (Hardy 1999, pp. 16 and 23). This series is identical to the GRAM SERIES (Hardy 1999, pp. 24 /C1/25). The quantity R(x) /C28 p(x) is plotted above. In addition, p(x) /C30R(x) /C28X rR(xr) ; (2) where p(x) is the PRIME COUNTING FUNCTION and the SUM is over all complex (nontrivial) zeros r of z(s); i.e., those in the CRITICAL STRIP so 0 BR[r] B1 ; inter- preted to mean X rRxrðÞ/C30lim t 0/C12X I( r) jjBtRxrðÞ: (3) Riemann conjectured that R(n) /C30 p(n) (Knuth 1998, p. 382), but this was disproved by Littlewood in 1914 (Hardy and Littlewood 1918). Ramanujan independently derived the formula for R(n) ; but nonrigorously (Berndt 1994, p. 123; Hardy 1999, p. 23). The following table compares p(x) ; li x; and R(x) for small x. Note that the values given by Hardy (1999, p. 26) for x /C30109 are incorrect. x /p(x)//li(x) /C28 p(x)//R(x) /C28 p(x)/ 100000 9592 38 /C285 1000000 78498 130 29 2000000 148933 122 //C289/ 3000000 216816 155 0 4000000 283146 206 33 5000000 348513 125 /C2864 6000000 412849 228 24 7000000 476648 179 /C2838 8000000 539777 223 //C286/ 9000000 602489 187 /C2853 10000000 664579 339 88 100000000 5761455 754 97 1000000000 50847534 1701 /C2879 See also GRAM SERIES ,PRIME COUNTING FUNCTION , PRIME NUMBER THEOREM ,RIEMANN FUNCTION ,SOLD- NER’S CONSTANT References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Hardy, G. H. and Littlewood, J. E. Acta Math. 41, 119/C1/196, 1918. Hardy, G. H. "The Series R(x):/"§2.3 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.New York: Chelsea, 1999.Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, 1998. Riesel, H. "The Riemann Prime Number Formula." Prime Numbers and Computer Methods for Factorization, 2nded.Boston, MA: Birkha ¨user, pp. 50 /C1 /52, 1994. Riemann P-Series The solutions to the R IEMANN P-DIFFERENTIAL EQUA- TION z/C13Pabc abg a?b?g?;z8 < :9 = ;: Solutions are given in terms of the HYPERGEOMETRIC FUNCTION by u1/C30z/C28a z/C28b !az/C28c z/C28b !g 2F1(a/C27b/C27g;a/C27b?/C27g; 1/C27a/C28a?;l) u2/C30z/C28a z/C28b !a?z/C28c z/C28b !g 2F1(a?/C27b/C27g;a?/C27b?/C27g; 1/C27a?/C28a;l) u3/C30z/C28a z/C28b !az/C28c z/C28b !g? 2F1(a/C27b/C27g?;a/C27b?/C27g?; 1/C27a/C28a?;l) u4/C30z/C28a z/C28b !a?z/C28c z/C28b !g? 2F1(a?/C27b/C27g?;a?/C27b?/C27g?; 1/C27a?/C28a;l) where l/C30(z/C28a)(c/C28b) (z/C28b)(c/C28a): References Abramowitz, M. and Stegun, C. A. (Eds.). "Riemann’s Differential Equation." §15.6 in Handbook of Mathemati- cal Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 564 /C1/565, 1972. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 541 /C1/543, 1953. Riemann, B. Abh. d. Ges. d. Wiss. zu Go ¨ttingen 7, 1857. Reprinted in Mathematisch Werke , p. 67, 1892. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 283 /C1/284, 1990. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995. Riemann Removable Singularity Theorem Let f : Dz0 ; r ðÞ _ z0fg0 C be ANALYTIC and bounded on a PUNCTURED OPEN DISK Dz0 ; r ðÞ ; then limz0z0f(z) exists, and the function defined by ˜f : D(z0 ; r) 0 C ˜f(z) /C30f(z) for z "z0 limz?0z0f(z?) for z /C30z0})1D is ANALYTIC . See also REMOVABLE SINGULARITY References Krantz, S. G. "The Riemann Removable Singularity Theo- rem." §4.1.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 42 /C1/43, 1999. Riemann Series Theorem By a suitable rearrangement of terms, a CONDITION- ALLY CONVERGENT SERIES may be made to converge to any desired value, or to DIVERGE . See also CONDITIONAL CONVERGENCE ,D IVERGENT SERIES References Bromwich, T. J. I’a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 74, 1991. Gardner, M. Martin Gardner’s Sixth Book of Mathematical Games from Scientific American. New York: Scribner’s, p. 171, 1971. Riemann Space METRIC SPACE Riemann Sphere A 1-D COMPLEX MANIFOLD C*, which is the one-point COMPACTIFICATION of the COMPLEX NUMBERS C/C31/C30C @ f/C12g; together with two charts. (Here [522;/C12] de- noted COMPLEX INFINITY ). For all points in the COMPLEX PLANE , the chart is the IDENTITY MAP from the SPHERE (with infinity removed) to the COMPLEX PLANE . For the POINT AT INFINITY , the chart neighbor- hood is the sphere (with the ORIGIN removed), and the chart is given by sending infinity to 0 and all other points z to 1=z :/ See also C*, COMPLEX INFINITY ,C OMPLEX PLANE , EXTENDED COMPLEX PLANE References Anderson, J. W. "The Riemann Sphere C¯ ." §1.2 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 7 /C1/16, 1999. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 4, 1996. Krantz, S. G. "The Riemann Sphere." §6.3.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 83 /C1/84, 1999.Riemann Sum Let a CLOSED INTERVAL [a, b] be partitioned by points a Bx1 Bx2 B...Bxn/C281 Bb; where the lengths of the resulting intervals between the points are denoted Dx1 ;Dx2 ; ..., Dxn : Let x/C31kbe an arbitrary point in the kth subinterval. Then the quantity Xn k/C301f(x/C31k) Dxk is called a Riemann sum for a given function f(x) and partition, and the value max Dxkis called the MESH SIZE of the partition. If the LIMIT max Dxk 0 0 exists, this limit is known as the Riemann integral of f(x) over the interval [a, b]. The shaded areas in the above plots show the LOWER and UPPER SUMS for a constant MESH SIZE . See also INTEGRAL ,LOWER SUM,MESH SIZE,RIEMANN INTEGRAL ,UPPER SUM References Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, pp. 324 /C1/327, 1999. Riemann Surface A surface-like configuration which covers the COM- PLEX PLANE with several, and in general infinitely many, "sheets." These sheets can have very compli- cated structures and interconnections (Knopp 1996,pp. 98 /C1 /99). Riemann surfaces are one way of repre- senting MULTIPLE-VALUED FUNCTIONS ; another is BRANCH CUTS . The above plot shows Riemann sur- faces for solutions of the equation w(z)½/C138d/C27w(z)/C27zd/C281/C300 with d/C302, 3, 4, and 5, where w(z)i sL AMBERT’S W- FUNCTION (M. Trott). The Riemann surface Sof the FUNCTION FIELD Kis the set of nontrivial discrete valuations on K. Here, the set Scorresponds to the IDEALS of the RING Aof INTEGERS ofKoverC(z):(Aconsists of the elements of K that are ROOTS of MONIC POLYNOMIALS over C[z] :/) Riemann surfaces provide a geometric visualization of FUNCTIONS ELEMENTS and their ANALYTIC CONTI- NUATIONS . See also BRANCH CUT,FUNCTION FIELD,IDEAL ,RING References Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899 /C1/909, 1999. Corless, R. M. and Jeffrey, D. J. "Graphing Elementary Riemann Surfaces." ACM Sigsam Bulletin: Commun. Comput. Algebra 32,11/C1/17, 1998. Fischer, G. (Ed.). Plates 123 /C1/126 in Mathematische Mod- elle/Mathematical Models, Bildband/Photograph Vo- lume. Braunschweig, Germany: Vieweg, pp. 120 /C1/123, 1986. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 99 /C1/118, 1996. Krantz, S. G. "The Idea of a Riemann Surface." §10.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 135 /C1/139, 1999. Mathews, J. H. and Howell, R. W. Complex Analysis for Mathematics and Engineering, 4th ed. Boston, MA: Jones and Bartlett, 2000. Monna, A. F. Dirichlet’s Principle: A Mathematical Comedy of Errors and Its Influence on the Development of Analysis. Utrecht, Netherlands: Osothoek, Scheltema, and Holk- ema, 1975. Trott, M. "Visualization of Riemann Surfaces of Algebraic Functions." Mathematica J. 6,15/C1/36, 1997. Trott, M. "Visualization of Riemann Surfaces IIa." Mathe- matica J. 7, 465 /C1/496, 2000. Trott, M. "Visualization of Riemann Surfaces." http://librar- y.wolfram.com/examples/riemannsurface/. Riemann Tensor A TENSOR sometimes known as the RIEMANN- CHRIS- TOFFEL TENSOR . Let ˜Ds /C13@ @xs /C28X lsu l})1D})1E ; (1) where the quantity inside thesu l})*})+ is a CHRISTOFFEL SYMBOL OF THE SECOND KIND . Then Rpqrs /C13 ˜Dqpr s})1D})1E /C28 ˜Drrq s})1D})1E : (2) Broken down into its simplest decomposition in N-D, Rlmnk /C301 N /C28 2glnRmk /C28g lkRmn /C28g mnRlk /C27g mkRln})0})@ /C28R (N /C28 1)(N /C28 2)glngmk /C28g lkg mn})0})@ /C27Clmnk : (3) Here, Rmnis the RICCI TENSOR , R is the CURVATURE SCALAR , and Clmnk is the WEYL TENSOR . In terms of the JACOBI TENSOR J m nab ; Rm anb /C302 3J m nabJ m ban})0})@ : (4)The Riemann tensor is the only tensor that can be constructed from the METRIC TENSOR and its first and second derivatives, Ra bgd /C30Ga bd; g /C28Gabg; d /C27G m bd Ga mg /C28Gm bg Ga md ; (5) where Gg ab are CONNECTION COEFFICIENTS and A;k is a COMMA DERIVATIVE (Schmutzer 1968, p. 108). In 1-D, R1111 /C300 :/ The number of independent coordinates in n-D is given by Cn /C131 12n2 n2 /C281})0})@ ; (6) the "4-D pyramidal numbers," the first few values of which are 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, ... (Sloane’s A002415). The number of SCALARS which can be constructed from Rlmnk and gmn is Sn /C131 for n /C302 1 12 n(n /C281)(n /C282)(n /C273) for n /C301; n > 2})1D (7) (Weinberg 1972). The first few values are then 0, 1, 3, 14, 40, 90, 175, 308, 504, 780, ... (Sloane’s A050297). See also BIANCHI IDENTITIES ,CHRISTOFFEL SYMBOL OF THE SECOND KIND,COMMUTATION COEFFICIENT , CONNECTION COEFFICIENT ,C URVATURE SCALAR , GAUSSIAN CURVATURE ,JACOBI TENSOR ,PETROV NO- TATION ,RICCI TENSOR ,W EYL TENSOR References Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravita- tion. San Francisco: W. H. Freeman, pp. 220 /C1/221, 1973. Schmutzer, E. Relativistische Physik (Klassische Theorie). Leipzig, Germany: Akademische Verlagsgesellschaft, 1968. Sloane, N. J. A. Sequences A002415/M4135 and A050297 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972. Riemann Theta Function Let the IMAGINARY PART of a g /C29g MATRIX F be POSITIVE DEFINITE , and m /C30 m1;...;mg})0})@ be a row VECTOR with coefficients in Z:Then the Riemann theta function is defined by q(u)/C30X mexp 2 pimTu/C271 2mFTm})@D})@Ehi : See also JACOBI THETA FUNCTIONS ,R AMANUJAN THETA FUNCTIONS ,SIEGEL THETA FUNCTION ,THETA FUNCTIONS References Itoˆ, K. (Ed.). "Abelian Integrals." §3.L in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, p. 9, 1987. Riemann Xi Function XIFUNCTION Riemann Zeta Function The Riemann zeta function is an extremely important SPECIAL FUNCTION of mathematics and physics which arises in definite integration and is intimately related with very deep results surrounding the PRIME NUM- BER THEOREM . While many of the properties of this function have been investigated, there remain im-portant fundamental conjectures (most notably theR IEMANN HYPOTHESIS ) which remain unproved to this day.On the REAL LINE with x/C211, the Riemann zeta function can be defined by the integral z(x)/C131 G(x)g/C12 0ux/C281 eu/C281du; (1) where G(n) is the GAMMA FUNCTION .I fxis an INTEGER n, then we have the identity un/C281 eu/C281/C30e/C28uun/C281 1/C28e/C28u/C30e/C28uun/C281X/C12 k/C300e/C28ku /C30X/C12 k/C301e/C28kuun/C281; (2) so g/C12 0un/C281 eu/C281du/C30X/C12 k/C301g/C12 0e/C28kuun/C281du: (3) To evaluate z(n);lety/C13kuso that dy/C30kd u and plug in the above identity to obtainz(n)/C301 G(n)X/C12 k/C301g/C12 0e/C28kuun/C281du /C301 G(n)X/C12 k/C301g/C12 0e/C28yy k !n/C281dy k /C301 G(n)X/C12 k/C3011 kng/C12 0e/C28yyn/C281dy: (4) Integrating the final expression in (4) gives G(n); which cancels the factor 1 =G(n) and gives the most common form of the Riemann zeta function, z(n)/C30X/C12 k/C3011 kn: (5) The Riemann zeta function can also be defined interms of MULTIPLE INTEGRALS by z(n)/C30g1 0/C1/C1/C1g1 0|fflfflfflfflfflffl{zfflfflfflfflfflffl} nQn i/C301dxi 1/C28Qn i/C301xi; (6) and as a M ELLIN TRANSFORM by g/C12 0frac1 t ! tn/C281dt/C30/C28z(s) s(7) for 0BR[s]B1;where frac( x) is the FRACTIONAL PART (Balazard and Saias 2000). Note that the zeta function has a singularity at n/C301, where it reduces to the divergent HARMONIC SERIES . The Riemann zeta function satisfies the functional equation z(1/C28s)/C302(2p)/C28scos1 2sp})@D})@E G(s)z(s) (8) (Hardy 1999, p. 14; Krantz 1999, p. 160). As defined above, the zeta function z(s) with s/C30s/C27it aCOMPLEX NUMBER is defined for R[s]>1:However, z(s) has a unique ANALYTIC CONTINUATION to the entire COMPLEX PLANE , excluding the point s/C301, which corresponds to a SIMPLE POLE with RESIDUE 1 (Krantz 1999, p. 160). In particular, as s01;z(s) obeys lim s01z(s)/C281 s/C281/C30g; (9) where gis the E ULER- MASCHERONI CONSTANT (Whit- taker and Watson 1990, p. 271).To perform the ANALYTIC CONTINUATION forR[s]>0; write X/C12 n/C301(/C281)nn/C28s/C27X/C12 n/C301n/C28s/C302X/C12 n/C302;4;...n/C28s /C302X/C12 k/C301(2k)/C28s/C3021/C28sX/C12 n/C301k/C28s(10) X/C12 n/C301(/C281)nn/C28s/C27z(s)/C3021/C28sz(s): (11) Therefore, z(s)/C301 1/C2821/C28sX/C12 n/C301(/C281)n/C281n/C28s: (12) While this form defines z(s) for only the UPPER HALF- PLANE R[s]>0;equation (8) can be used to analyti- cally continue it to the rest of the COMPLEX PLANE . Analytic continuation can also be performed using HANKEL FUNCTIONS . A globally convergent series for the Riemann zeta function is given by z(z)/C301 1/C2821/C28zX/C12 n/C3001 2n/C271Xn k/C300(/C281)kn k})@*})@+ (k/C271)/C28z;(13) wheren k})0})@ is a BINOMIAL COEFFICIENT . A generalized Riemann zeta function z(s;a) known as the H URWITZ ZETA FUNCTION can also be defined such that z(s)/C13z(s;0): (14) In the COMPLEX PLANE , trivial zeros of z(s) occur at s/C30/C282,/C284,/C286;..., and nontrivial zeros at s/C13s/C27it (15) for 05s51:The figures below show the structure of the complex z(z) by plotting z(z)jj and 1 =z(z)jj :/ The R IEMANN HYPOTHESIS asserts that the nontrivial ROOTS ofz(s) all have REAL PART s/C30R[s]/C301=2;a line called the " CRITICAL LINE ." This is known to be true for the first 200 ;000;001 roots (Brent et al. 1982). The above plot shows z(1=2/C27it) jj fortbetween 0 and 60. As can be seen, the first few nontrivial zeros occur at t/C3014:134725 ;21.022040, 25.010858, 30.424876, 32.935062, 37.586178, ... (Wagon 1991, pp. 361 /C1/362 and 367 /C1/368; Odlyzko). Wiener showed that the PRIME NUMBER THEOREM is literally equivalent to the assertion that z(s) has no zeros on s/C301 (Hardy 1999, p. 34). The Riemann zeta function can be factored over its nontrivial zeros ras z(s)/C30eln(2p)/C281/C28g=2)s 2(s/C281)G1/C271 2s})@D})@EY r1/C28s r ! es=r(16) (Voros 1987). The Riemann zeta function can be split up into z1 2/C27it})@D})@E /C30z(t)e/C28iq(t); (17) where z(t) and q(t) are the R IEMANN- SIEGEL FUNC- TIONS . The Riemann zeta function is related to the DIRICHLET LAMBDA FUNCTION l(n) and D IRICHLET ETA FUNCTION h(n)b y z(n) 2n/C30l(n) 2n/C281/C30h(n) 2n/C282(18) and z(n)/C27h(n)/C302l(n) (19) (Spanier and Oldham 1987). It is related to the LIOUVILLE FUNCTION l(v)b y z(2s) z(s)/C30X/C12 n/C301l(n) ns(20) (Lehman 1960, Hardy and Wright 1979). Further- more, z2(s) z(2s)/C30X/C12 n/C3012v(n) ns; (21) where v(n) is the number of DISTINCT PRIME FACTORS ofn(Hardy and Wright 1979, p. 254). Two sum identities involving z(n) are X/C12 n/C302[z(n)/C281]/C301 (22) X/C12 n/C302(/C281)n[z(n)/C281]/C301 2: (23) The Riemann zeta function is related to the GAMMA FUNCTION G(z)b y Gs 2 ! p/C28s=2z(s)/C30G1/C28s 2 ! p/C28(1/C28s)=2z(1/C28s): (24) The DERIVATIVE of the Riemann zeta function is defined by z?(s)/C30/C28sX/C12 k/C301k/C28slnk/C30/C28X/C12 k/C302lnk ks: (25) Ass00; z?(0)/C30/C281 2ln(2p): (26) /z(n) is known to be transcendental for all EVEN n, but the study of the function at ODD nis significantly more difficult. Ape ´ry (1979) finally proved that z(3) to beIRRATIONAL , but no similar results are known for other ODD n. However, Rivoal (2000) recently proved that there are infinitely many integers nsuch that z(2n/C271) is irrational. As a result of Ape ´ry’s impor- tant discovery, z(3) is sometimes called A PE´RY’S CONSTANT . A number of interesting sums for z(n); with naPOSITIVE INTEGER , can be written in terms of binomial coefficients as the BINOMIAL SUMS z(2)/C303X/C12 k/C3011 k22k k})@*})@+ (27) z(3)/C305 2X/C12 k/C301(/C281)k/C281 k32k k})@*})@+ (28) z(4)/C3036 17X/C12 k/C3011 k42k k})@*})@+ (29) (Guy 1994, p. 257). Ape ´ry arrived at his result with the aid of the k/C283sum formula above. A relation OF THE FORM z(5)/C30Z5X/C12 k/C301(/C281)k/C281 k52k k})@*})@+ (30) has been searched for with Z5aRATIONAL orALGE-BRAIC NUMBER , but if Z5is a ROOT of a POLYNOMIAL of degree 25 or less, then the Euclidean norm of the coefficients must be larger than 2 /C291037(Bailey and Plouffe). Therefore, no such sums for z(n) are known forn]5:/ The Riemann zeta function may be computed analy-tically for EVEN nusing either CONTOUR INTEGRATION or P ARSEVAL’S THEOREM with the appropriate F OUR- IER SERIES . An unexpected and important formula involving the product of PRIMES was first discovered by Euler in 1737, z(x)(1/C282/C28x)/C301/C271 2x/C271 3x/C27... ! 1/C281 2x ! /C301/C271 2x/C271 3x/C27... ! /C281 2x/C271 4x/C271 6x/C27... ! (31) z(x)1/C282/C28xðÞ 1/C283/C28xðÞ /C301/C271 3x/C271 5x/C271 7x/C27... ! /C281 3x/C271 9x/C271 15x/C27... ! (32) z(x)1/C282/C28xðÞ 1/C283/C28xðÞ /C1 /C1 /C1 1/C28p/C28zð Þ/C1/C1/C1 /C30z(x)Y/C12 n/C302(1/C28p/C28x)/C301: (33) Here, each subsequent multiplication by the next PRIME pleaves only terms which are POWERS of /p/C28x /. Therefore, z(x)/C30Y/C12 p/C302(1/C28p/C28x)"#/C281 ; (34) where pruns over all PRIMES (Hardy 1999, p. 18; Krantz 1999, p. 159). Euler’s product formula can also be written z(s)/C301/C282/C28sðÞ/C281Y q/C301 (mod 4)1/C28q/C28sðÞ/C281Y r/C303 (mod 4)1/C28r/C28sðÞ/C281: (35) For EVEN n/C132k; z(n)/C302n/C281Bnjjpn n!; (36) where Bnis a B ERNOULLI NUMBER . Another intimate connection with the B ERNOULLI NUMBERS is provided by Bn/C30(/C281)n/C271nz(1/C28n) (37) forn]1;which can be written Bn/C30/C28nz(1/C28n) (38) forn]2:Although no analytic form for z(n) is known for ODD n, z(3)/C301 2X/C12 k/C3011 k21/C2712/C27.../C271k ! /C3012X /C12 k/C301hk k2;(39) where hkis a HARMONIC NUMBER (Stark 1974). In addition, z(n) can be expressed as the sum limit z(n)/C30lim x0/C121 (2x/C271)nXx k/C301cotk 2x/C271 !"#n (40) forn/C303, 5, ... (Apostol 1973, given incorrectly in Stark 1974). Form(n) the M O¨BIUS FUNCTION , 1 z(s)/C30X/C12 n/C301m(n) ns: (41) The values for small integral arguments are z(1)/C30/C12 z(2)/C30p2 6 z(3)/C301:2020569032 . . . z(4)/C30p4 90 z(5)/C301:0369277551 . . . z(6)/C30p6 945 z(7)/C301:0083492774 . . . z(8)/C30p8 9450 z(9)/C301:0020083928 . . . z(10)/C30p10 93;555: Euler gave z(2) to z(26) for EVEN n(Wells 1986, p. 54), and Stieltjes (1993) determined the values of z(2);..., z(70) to 30 digits of accuracy in 1887. The denomi- nators of z(2n) for n/C301, 2, ... are 6, 90, 945, 9450, 93555, 638512875, ... (Sloane’s A002432).The value at n/C300 is given by z(0)/C30/C28 1 2(42) The value z(/C281)/C30/C281=12 is a deep result of renorma- lization theory (Elizalde et al. 1994, Elizalde 1995). In general,z(/C28n)/C30/C28Bn/C271 n/C271(43) forn/C301, 3, ... where Bnis a B ERNOULLI NUMBER , the first few values of which are /C281=12;1/120, /C281=252;/ 1/240, ... (Sloane’s A001067 and A006953). Rapidly converging series for z(n) for nodd were first discovered by Ramanujan (Zucker 1979, Zucker 1984, Berndt 1988, Bailey et al. 1997, Cohen 2000). For n/C211 and n/C133 (mod 4) ; z(n)/C302n/C281pn (n/C271)!X(n/C271)=2 k/C300(/C281)k/C281n/C271 2k})@*})@+ Bn/C271/C282kB2k /C282X/C12 k/C3011 kn(e2pk/C281); (44) where Bkis again a B ERNOULLI NUMBER andn k})0})@ is a BINOMIAL COEFFICIENT . The first few for n/C303, 7, 11, ... are 7/180, 19/56700, 1453/425675250, 13687/ 390769879500, 7708537/21438612514068750, ... (Sloane’s A057866 and A057867). For n]5 and n/C13 1 (mod 4) ;the corresponding formula is slightly mes- sier, z(n)/C30(2p)n (n/C271)!(n/C281) /C29X(n/C271)=4 k/C300(/C281)k(n/C271/C284k)n/C271 2k})@*})@+ Bn/C271/C282kB2k /C282X/C12 k/C301e2pk1/C274pk k/C281 ! /C281 kn(e2pk/C281)2: (45) Defining S9(n)/C13X/C12 k/C3011 kne2pk91 ðÞ; (46) the first few values can then be written z(3)/C307 180p3/C282S/C28(3) (47) z(5)/C301 294p5/C2872 35S/C28(5)/C282 35S/C27(5) (48) z(7)/C3019 56700p7/C282S/C28(7) (49) z(9)/C30125 3704778p9/C28992 495S/C28(9)/C282 495S/C27(9) (50) z(11)/C301453 425675250p11/C282S/C28(11) (51) z(13)/C3089 257432175p13/C2816512 8255S/C28(13)/C282 8255S/C27(13) (52) z(15)/C3013687 390769879500p15/C282S/C28(15) (53) z(17)/C30397549 112024529867250p17/C28261632130815 S/C28(17) /C282 130815S/C27(17) (54) zð19Þ¼7708537 21438612514068750 p19 /C282S /C28ð19 Þð 55Þ z(21) /C3068529640373 1881063815762259253125p21 /C284196352 2098175 S/C28(21) /C282 2098175 S /C27(21) (56) (Plouffe). The inverse of the RIEMANN ZETA FUNCTION 1=z(p); plotted above, is the asymptotic density of pth-power- free numbers (i.e., SQUAREFREE numbers, CUBEFREE numbers, etc.). The following table gives the number Qp(n)of pth-powerfree numbers 5n for several values of n. p /1= z(p)//Qp(10) //Qp(100) //Qp(103)//Qp(104)//Qp(105)//Qp(106)/ 2 0.607927 7 61 608 6083 60794 607926 3 0.831907 9 85 833 8319 83190 831910 4 0.923938 10 93 925 9240 92395 923939 5 0.964387 10 97 965 9645 96440 964388 6 0.982953 10 99 984 9831 98297 982954 See also ABEL’S FUNCTIONAL EQUATION ,BERRY CON- JECTURE ,C RITICAL LINE,C RITICAL STRIP,D EBYE FUNCTIONS ,DIRICHLET BETA FUNCTION ,D IRICHLET ETA FUNCTION ,DIRICHLET LAMBDA FUNCTION ,EULER PRODUCT ,H ARMONIC SERIES ,H URWITZ ZETA FUNC- TION ,KHINTCHINE’S CONSTANT ,LEHMER’S PHENOM- ENON ,P ERIODIC ZETA FUNCTION ,P RIME NUMBER THEOREM ,P SI FUNCTION ,R IEMANN HYPOTHESIS , RIEMANN P-SERIES ,R IEMANN- SIEGEL FUNCTIONS , RIEMANN ZETA FUNCTION ZETA(2), STIELTJES CON- STANTS ,XI FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Riemann Zeta Function and Other Sums of Reciprocal Powers." §23.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807 /C1/808, 1972. Adamchik, V. S. and Srivastava, H. M. "Some Series of the Zeta and Related Functions." Analysis 18, 131/C1/144, 1998.Aizenberg, L.; Adamchik, V.; and Levit, V. E. "Approaching the Riemann Hypothesis with Mathematica ." http://librar- y.wolfram.com/demos/v4/Riemann.nb. Ape´ry, R. "Irrationalite ´dez(2) et z(3):/"Aste´risque 61,1 1/C1/13, 1979. Apostol, T. M. "Another Elementary Proof of Euler’s For- mula for z(2n):/"Amer. Math. Monthly 80, 425/C1/431, 1973. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 332 /C1/335, 1985. Ayoub, R. "Euler and the Zeta Function." Amer. Math. Monthly 81, 1067/C1/1086, 1974. Bailey, D. H. "Multiprecision Translation and Execution of Fortran Programs." ACM Trans. Math. Software. To appear. Bailey, D. and Plouffe, S. "Recognizing Numerical Con- stants." http://www.cecm.sfu.ca/organics/papers/bailey/. Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khintchine Constant." Math. Comput. 66, 417/C1/431, 1997. Balazard, M. and Saias, E. "The Nyman-Beurling Equiva- lent Form for the Riemann Hypothesis." Expos. Math. 18, 131/C1/138, 2000. Balazard, M.; Saias, E.; and Yor, M. "Notes sur la fonction z de Riemann, 2." Adv. Math. 143, 284/C1/287, 1999. Berndt, B. C. Ch. 14 in Ramanujan’s Notebooks, Part II. New York: Springer-Verlag, 1988. Borwein, D. and Borwein, J. "On an Intriguing Integral and Some Series Related to z(4):/"Proc. Amer. Math. Soc. 123, 1191/C1/1198, 1995. Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Com- putational Strategies for the Riemann Zeta Function." CECM-98:118, 23 Jun 1999. http://www.cecm.sfu.ca/pre-prints/1999pp.html#98:118. Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361/C1 /1372, 1979. Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in theCritical Strip. II." Math. Comput. 39, 681/C1 /688, 1982. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67/C1/98, 1988. Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. http://www.math.u-bordeaux.fr/~co-hen/hardylw.dvi. Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980. Edwards, H. M. Riemann’s Zeta Function. New York: Academic Press, 1974. Elizalde, E. Ten Physical Applications of Spectral Zeta Functions. Berlin: Springer-Verlag, 1995. Elizalde, E.; Odintsov, S. D.; Romeo, A.; Bytsenko, A. A.; and Zerbini, S. Zeta Regularization Techniques With Applications. River Edge, NJ: World Scientific, 1994. Farmer, D. W. "Counting Distinct Zeros of the Riemann Zeta-Function." Electronic J. Combinatorics 2,R 11 /C1 /5, 1995. http://www.combinatorics.org/Volume_2/volu-me2.html#R1. Guy, R. K. "Series Associated with the z /-Function." §F17 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 257 /C1/258, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. "The Zeta Function." §17.2 inAn Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 245 /C1/247 and 255, 1979. Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergenteReihen fu ¨r Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995. Howson, A. G. "Addendum to: ‘Euler and the Zeta Function’ (Amer. Math. Monthly 81(1974), 1067 /C1/1086) by Raymond Ayoub." Amer. Math. Monthly 82, 737, 1975. Ivic, A. A. The Riemann Zeta-Function. New York: Wiley, 1985. Ivic, A. A. Lectures on Mean Values of the Riemann Zeta Function. Berlin: Springer-Verlag, 1991. Karatsuba, A. A. and Voronin, S. M. The Riemann Zeta- Function. Hawthorne, NY: De Gruyter, 1992. Katayama, K. "On Ramanujan’s Formula for Values of Riemann Zeta-Function at Positive Odd Integers." Acta Math. 22, 149/C1/155, 1973. Keiper, J. "The Zeta Function of Riemann." Mathematica Educ. Res. 4,5/C1/7, 1995. Knopp, K. "4th Example: The Riemann z/-Function." Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 51 /C1/57, 1996. Krantz, S. G. "Riemann’s Zeta Function." §13.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 158 /C1/ 159, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 35, 1983. Lehman, R. S. "On Liouville’s Function." Math. Comput. 14, 311/C1/320, 1960. Odlyzko, A. "Andrew Odlyzko: Tables of Zeros of the Riemann Zeta Function." http://www.research.att.com/ ~amo/zeta_tables/. Odlyzko, A. M. "The 1020th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint. Patterson, S. J. An Introduction to the Theory of the Riemann Zeta-Function. New York: Cambridge Univer- sity Press, 1988. Plouffe, S. "Identities Inspired from Ramanujan Notebooks." http://www.lacim.uqam.ca/plouffe/identities.html. Rivoal, T. "La fonction Zeta de Riemann prend une infinite ´ de valeurs irrationnelles aux entiers impairs." C. R. Acad. Sci. 331, 267/C1/270, 2000. Sloane, N. J. A. Sequences A001067, A002432/M4283, A006953/M2039, A057866, and A057867 in "An On-LineVersion of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25 /C1 /33, 1987. Stieltjes, T. J. Oeuvres Comple `tes, Vol. 2 (Ed. G. van Dijk.) New York: Springer-Verlag, p. 100, 1993. Titchmarsh, E. C. The Zeta-Function of Riemann, 2nd ed. Oxford, England: Oxford University Press, 1987. Titchmarsh, E. C. and Heath-Brown, D. R. The Theory of the Riemann Zeta-Function, 2nd ed. Oxford, England: Oxford University Press, 1986. Vardi, I. "The Riemann Zeta Function." Ch. 8 in Computa- tional Recreations in Mathematica. Reading, MA: Addi- son-Wesley, pp. 141 /C1/174, 1991. Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439/C1/ 465, 1987. Wagon, S. "The Evidence: Where Are the Zeros of Zeta of s?" Math. Intel. 8,5 7/C1/62, 1986. Wagon, S. "The Riemann Zeta Function." §10.6 in Mathe- matica in Action. New York: W. H. Freeman, pp. 353 /C1/ 362, 1991. Weisstein, E. W. "Books about Riemann Zeta Function." http://www.treasure-troves.com/books/RiemannZetaFunc-tion.html. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Woon, S C. Generalization of a Relation Between the Riemann Zeta Function and Bernoulli Numbers. 24 Dec 1998. http://xxx.lanl.gov/abs/math.NT/9812143/.Zucker, I. J. "The Summation of Series of Hyperbolic Functions." SIAM J. Math. Anal. 10, 192/C1 /206, 1979. Zucker, I. J. "Some Infinite Series of Exponential and Hyperbolic Functions." SIAM J. Math. Anal. 15, 406/C1/ 413, 1984. Riemann Zeta Function Zeta(2) The value for z(2) can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970, Kimble 1987, Knopp and Schur1918, Kortram 1996, Matsuoka 1961, Papadimitriou 1973, Simmons 1992, Stark 1969, Stark 1970, Yaglom and Yaglom 1987). The problem of finding this valueanalytically is sometimes known as the B ASLER PROBLEM (Castellanos 1988). Yaglom and Yaglom (1987), Holme (1970), and Papadimitriou (1973) allderive the result, p 2=6 from DEMOIVRE’S IDENTITY or related identities. One derivation for z(2) considers the F OURIER SERIES off(x)/C30x2n f(x)/C301 2a0/C27X/C12 m/C301amcos(mx)/C27X/C12 m/C301bmsin(mx);(1) which has coefficients given by a0/C301 pgp /C28pf(x)dx/C302 pgp 0x2ndx /C302 px2n/C271 2n/C271"#p 0/C302p2n 2n/C271(2) am/C301 pgp /C28px2ncos(mx)dx /C302 pgp 0x2ncos(mx)dx (3) bm/C301 pgp /C28px2nsin(mx)dx/C300; (4) where the latter is true since the integrand is ODD. Therefore, the F OURIER SERIES is given explicitly by x2n/C30p2n 2n/C271/C27X/C12 m/C301amcos(mx): (5) Now, amis given by the COSINE INTEGRAL am/C302 p(/C281)n/C271(2n)!})10 sin(mx)Xn k/C300(/C281)k (2k)!m2n/C282k/C271x2k /C27cos(mx)Xn k/C301(/C281)k/C271 (2k/C283)!m2n/C282k/C272x2k/C281})1@p 0: (6) But cos( mp)/C30(/C281)m;and sin( mp)/C30sin 0/C300;so am/C302 p(/C281)n/C271(2n)!(/C281)mXn k/C301(/C281)k/C271 (2k/C283)!m2n/C282k/C272p2k/C281 /C30(/C281)m/C27n2(2n)!Xn k/C301(/C281)k (2k/C283)!m2n/C282k/C272p2k/C282: (7) Now, if n/C301, am/C30(/C281)m/C2712(2!)X1 k/C301(/C281)k (2k/C283)!m4/C282kp2k/C282 /C304(/C281)m/C271(/C281) (/C281)!m2p0/C304(/C281)m m2; (8) so the F OURIER SERIES is x2/C30p2 3/C274X/C12 m/C301(/C281)mcos(mx) m2: (9) Letting m/C13pgives cos( mp)/C30(/C281)m;so p2/C30p2 3/C274X/C12 m/C3011 m2; (10) and we have z(2)/C30X/C12 m/C3011 m2/C30p2 6: (11) Higher values of ncan be obtained by finding amand proceeding as above. The value z(2) can also be found simply using the ROOT LINEAR COEFFICIENT THEOREM . Consider the equation sin z/C300 and expand sin in a M ACLAURIN SERIES sinz/C30z/C28z3 3!/C27z5 5!/C27.../C300 (12) 0/C301/C28z2 3!/C27z4 5!/C27.../C301/C28w 3!/C27w2 5!/C27...; (13) where w/C13z2:But the zeros of sin( z) occur at p;2p;3p; ..., so the zeros of sin w/C30sinffiffiffizpoccur at p2;(2p)2;.... Therefore, the sum of the roots equals the COEFFI- CIENT of the leading term 1 p2/C271 (2p)2/C271 (3p)2/C27.../C301 3!/C301 6; (14) which can be rearranged to yield z(2)/C30p2 6: (15) Yet another derivation (Simmons 1992) evaluates the integral using the integral I/C30g1 0g1 0dx dy 1/C28xy/C30g1 0g1 0(1/C27xy/C27x2y2/C27... )dx dy /C30g1 0[(x/C271 2x2y/C2713x3y2/C27. . .)]1 0dy/C30g1 0(1/C271 2y/C2713y2/C27... )dy /C30y/C27y2 22/C27y3 32/C27..."#1 0/C301/C271 22/C271 32/C27...: (16) To evaluate the integral, rotate the coordinate system byp=4s o x/C30ucosu/C28vsinu/C301 2ffiffiffi 2p (u/C28v) (17) y/C30usinu/C27vcosu/C301 2ffiffiffi 2p (u/C27v) (18) and xy/C301 2(u2/C28v2) (19) 1/C28xy/C301 2(2/C28u2/C27v2): (20) Then I/C304gffiffi 2p =2 0gu 0du dv 2/C28u2/C27v2 /C274gffiffi 2p ffiffi 2p =2gffiffi 2p /C28u 0du dv 2/C28u2/C27v2 /C13I1/C27I2: (21) Now compute the integrals I1andI2: I1/C304gffiffi 2p =2 0gu 0dv 2/C28u2/C27v2"# du /C304gffiffi 2p =2 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28u2p tan/C281 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2/C28u2p ! "#u 0du /C304gffiffi 2p =2 01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28u2p tan/C281 uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2/C28u2p ! du: (22) Make the substitution u/C30ffiffiffi 2p sinu (23) ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28u2p /C30ffiffiffi 2p cosu (24) du/C30ffiffiffi2p cosudu; (25) so tan/C281 uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28u2p ! /C30tan/C281ffiffiffi 2p sinuffiffiffi2p cosu ! /C30u (26) and I 1/C304gp=6 01ffiffiffi 2p cosuuffiffiffi 2p cosudu/C302[u2]p=6 0 /C30p2 18: (27) /I2 can also be computed analytically, I2 /C304gffiffi 2p ffiffi 2p =2gffiffi 2p /C28u 0dv 2 /C28 u2 /C27 v2"# du /C304gffiffi 2p ffiffi 2p =21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28 u2p tan/C281 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 /C28 u2p ! "#ffiffi 2p /C28u 0du /C304gffiffi 2p ffiffi 2p =21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 /C28 u2p tan /C281ffiffiffi 2p /C28 uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28 u2p ! du : (28) But tan/C281ffiffiffi 2p /C28 uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28 u2p ! /C30tan/C281ffiffiffi 2p /C28ffiffiffi2p sin uffiffiffi2p cos u ! /C30tan 1 /C28 sin u cos u ! /C30tan/C281 cos u 1 /C27 sin u ! /C30tan/C281sin1 2 p /C28 u})@D})@E 1 /C27 cos1 2 p /C28 u})@D})@E2 435 /C30tan /C2812 sin1 212 p /C28 u})@D})@Ehi cos1212 p /C28 u})@D})@Ehi 2 cos21 212 p /C28 u})@D})@Ehi8 < :9 = ; /C301 212 p /C28 u})@D})@E ; (29) so I2 /C304g p =2 p=61ffiffiffi 2p cos u1 4 p /C2812 u})@D})@Effiffiffi 2p cos u du /C3041 4 pu /C2814 u2hip=2 p=6 /C304p2 8/C28p2 16 ! /C28p2 24 /C28p2 144 ! "# /C30p2 9: (30) Combining I1 and I2 gives z(2) /C30I1 /C27I2 /C30p2 18 /C27p2 9/C30p2 6: (31) See also RIEMANN ZETA FUNCTION References Apostol, T. M. "A Proof That Euler Missed: Evaluating z(2) the Easy Way." Math. Intel. 5,59/C1/60, 1983. Choe, B. R. "An Elementary Proof of a/C12 n/C3011 n2 /C30p2 6 :/" Amer. Math. Monthly 94, 662 /C1/663, 1987. Giesy, D. P. "Still Another Proof That a 1 =k2 /C30 p2 =6 :/" Math. Mag. 45, 148 /C1/149, 1972. Holme, F. "Ein enkel beregning av a/C12 k/C3011 k2 :/" Nordisk Mat. Tidskr. 18,91/C1/92 and 120, 1970. Kimble, G. "Euler’s Other Proof." Math. Mag. 60, 282, 1987.Knopp, K. and Schur, I. "Uuml;ber die Herleitug der Gleichung a/C12n/C3011 n2 /C30p2 6 :/" Archiv der Mathematik u. Physik 27, 174 /C1/176, 1918. Kortram, R. A. "Simple Proofs for a/C12k/C3011 k2 /C30p2 6and sin x /C30xQ/C12 k/C301 ð1 /C28x2 k2 p2 Þ:/" Math. Mag. 69, 122 /C1/125, 1996. Matsuoka, Y. "An Elementary Proof of the Formula a/C12 k /C3011 k2 /C30p2 6 :/" Amer. Math. Monthly 68, 486 /C1/487, 1961. Papadimitriou, I. "A Simple Proof of the Formula a/C12 k/C3011 k2 /C30p2 6 :/ " Amer. Math. Monthly 80, 424 /C1/425, 1973. Simmons, G. F. "Euler’s Formula a/C1211=n2 /C30 p2 =6 by Double Integration." Ch. B. 24 in Calculus Gems: Brief Lives and Memorable Mathematics. New York: McGraw-Hill, 1992. Stark, E. L. "Another Proof of the Formula a/C12 k/C3011 k2 /C30p2 6 :/" Amer. Math. Monthly 76, 552 /C1/553, 1969. Stark, E. L. " 1 /C281 4 /C2719 /C281 16 /C27.../C30p2 12:/" Praxis Math. 12,1/C1/3, 1970. Stark, E. L. "The Series a/C12 k/C301 k/C28s s /C302, 3, 4, ..., Once More." Math. Mag. 47, 197 /C1/202, 1974. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986. Yaglom, A. M. and Yaglom, I. M. Problem 145 in Challen- ging Mathematical Problems with Elementary Solutions, Vol. 2. New York: Dover, 1987. Riemann-Christoffel Tensor RIEMANN TENSOR Riemann-Finsler Geometry References Bao, D.; Chern, S.-S.; and Shen, Z. An Introduction to Riemann-Finsler Geometry. New York: Springer-Verlag, 2000. Riemannian Geometry The study of MANIFOLDS having a complete RIEMAN- NIAN METRIC . Riemannian geometry is a general space based on the LINE ELEMENT ds/C30Fx1;...;xn;dx1;...;dxn})0})@ ; with F(x;y)>0 for y"0 a function on the TANGENT BUNDLE TM. In addition, Fis homogeneous of degree 1i n yand OF THE FORM F2/C30gij(x)dxidxj (Chern 1996). If this restriction is dropped, the resulting geometry is called F INSLER GEOMETRY . See also NON-EUCLIDEAN GEOMETRY References Besson, G.; Lohkamp, J.; Pansu, P.; and Petersen, P. Riemannian Geometry. Providence, RI: Amer. Math. Soc., 1996. Buser, P. Geometry and Spectra of Compact Riemann Surfaces. Boston, MA: Birkha ¨user, 1992. Chavel, I. Eigenvalues in Riemannian Geometry. New York: Academic Press, 1984. Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Chern, S.-S. "Finsler Geometry is Just Riemannian Geome- try without the Quadratic Restriction." Not. Amer. Math. Soc. 43, 959 /C1/963, 1996. do Carmo, M. P. Riemannian Geometry. Boston, MA: Bir- kha¨user, 1992. Riemannian Geometry (Non-Euclidean) ELLIPTIC GEOMETRY Riemannian Manifold A MANIFOLD possessing a METRIC TENSOR . For a complete Riemannian manifold, the METRIC d(x; y) is defined as the length of the shortest curve (GEO- DESIC ) between x and y. See also BISHOP’S INEQUALITY ,CAMPBELL’S THEOREM , CHEEGER’S FINITENESS THEOREM ,PSEUDO- RIEMAN- NIAN MANIFOLD Riemannian Metric Suppose for every point x in a COMPACT MANIFOLD M, an INNER PRODUCT /C215;/C215hixis defined on a TANGENT SPACE TxM of M at x. Then the collection of all these INNER PRODUCTS is called the Riemannian metric. In 1870, Christoffel and Lipschitz showed how to decide when two Riemannian metrics differ by only a coordinate transformation. See also COMPACT MANIFOLD ,LINE ELEMENT ,METRIC TENSOR Riemannian Submersion See also SUBMERSION Riemann-Lebesgue Lemma Sometimes also called MERCER’S THEOREM . lim n0/C12gb aK( l; z)C sin(nz) dz /C300 for arbitrarily large C and "nice" K(l ; z) : Gradshteyn and Ryzhik (2000) state the lemma as follows. If f(x) is integrable on [/C28p; p]; then lim t0/C12g p /C28 pf(x) sin(tx) dx 0 0 and lim t0/C12g p /C28 pf(x) cos(tx) dx 0 0 : References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1101, 2000.Riemann-Roch Theorem The dimension of a complete series is equal to the sum of the order and index of specialization of any group, less the GENUS of the base curve r /C30N /C27i /C27p : References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 261, 1959. Koch, H. "The Riemann-Roch Theorem." §5.6 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 160 /C1/164, 2000. Riemann, B. Grundlagen fu¨r eine allgemeine Theorie der Funktionen einer vera¨ndlichen komplexen Gro¨sse. Ph.D. dissertation. Go¨ttingen, Germany: University of Go¨ttin- gen, 1851. Riemann’s Integral Theorem Associated with an irreducible curve of GENUS (CURVE ) p, there are p LINEARLY INDEPENDENT integrals of the first sort. The ROOTS of the integrands are groups of the canonical series, and every such group will give rise to exactly one integral of the first sort. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 274, 1959. Riemann’s Moduli Problem Find an ANALYTIC parameterization of the compact RIEMANN SURFACES in a fixed HOMOMORPHISM class. The AHLFORS-BERS THEOREM proved that RIEMANN’S MODULI SPACE gives the solution. See also AHLFORS- BERS THEOREM ,RIEMANN’S MOD- ULI SPACE Riemann’s Moduli Space Riemann’s moduli space Rpis the space of ANALYTIC EQUIVALENCE CLASSES of RIEMANN SURFACES of fixed GENUS p. See also AHLFORS- BERS THEOREM ,RIEMANN’S MOD- ULI PROBLEM ,RIEMANN SURFACE Riemann-Siegel Functions For a REAL POSITIVE t, the Riemann-Siegel Zfunction is defined by Z(t)/C13eiu(t)z(1 2/C27it): This function is sometimes also called the Hardy function or Hardy Z-function (Karatsuba and Vor- onin 1992, Borwein et al. 1999). The top plot super- poses Z(t) (thick line) on z1 2/C27it})@D})@E})@1})@1})@1})@1})@1})@1;where z(z) is the RIEMANN ZETA FUNCTION . It has an ASYMPTOTIC SERIES given "approximately" by Z(t)/C22Xn(t) k/C3011ffiffiffi kpcos[q(t)/C28tlnk]/C27R(t); (1) where n(t)/C30ffiffiffiffiffiffi t 2ps$% (2) R(t)/C30(/C281)n(t)/C281t 2p !/C281=4 /C29X/C12 k/C300ckffiffiffiffiffiffi t 2ps /C28n(t) ! t 2p !/C28k=2 (3) ck(p)/C30})10 vk})1@})1D exp})10 i})@* ln})@*t 2p})@+ /C281 2t/C2818p/C28q(t)})@+})1@ /C29y0})1})A})10})@*X/C12 j/C300Aj(y)vj})@+})@*X/C12 j/C300c(j)(p) j!yj})@+})1@})1E ð4ÞA0(y)/C30e2piy2(5) Aj(y)/C30/C281 2yAj/C281(y)/C281 32p2@2 @y2Aj/C281(y) y(6) cðpÞ¼cos½2pðp2/C28p/C281 16Þ/C138 cosð2ppÞð7Þ /xbcis the FLOOR FUNCTION (Edwards 1974), and yk})1})A isCOEFFICIENT NOTATION . The first few terms ck(p) are given by c0(p)/C30c(p) (8) c1(p)/C30/C28c(3)(p) 96p2(9) c2(p)/C30cƒ(p) 64p2/C27c(6)(p) 18432 p4(10) c3(p)/C30c?(p) 64p2/C27c(5)(p) 3840p4/C28c(9)(p) 5308416 p6(11) c4(p)/C30c(p) 128p2/C2719c(4)(p) 24576 p4/C2711c(8)(p) 5898240 p6 /C27c(12)(p) 2038431744 p8(12) c5(p)/C30/C285c(3)(p) 3072p4/C28901c(7)(p) 82575360 p6 /C287c(11)(p) 849346560 p8/C28c(15)(p) 978447237120 p10: (13) The numerators and denominators are 1, /C281, 1, 1, /C281,/C281,/C281, 1, 19, 11, 1, /C285,/C28901, ... (Sloane’s A050276) and 1, 96, 64, 18432, 64, 3840, 5308416, 128, ... (Sloane’s A050277), respectively. The Riemann-Siegel theta function appearing above is defined by q(t) /C13I ln G1 4 /C2712 it})@D})@E /C2812 t ln phi /C30arg G1 4 /C2712 it})@D})@Ehi /C2812 t ln p: These functions are implemented in Mathematica as RiemannSiegelZ [z] and RiemannSiegelTheta [z], illustrated above. See also RIEMANN ZETA FUNCTION ,XI FUNCTION References Berry, M. V. "The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders." Proc. Roy. Soc. London A 450, 439 /C1/462, 1995. Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Com- putational Strategies for the Riemann Zeta Function." CECM-98:118, 23 Jun 1999. http://www.cecm.sfu.ca/pre- prints/1999pp.html#98:118. Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361 /C1/1372, 1979. Edwards, H. M. Riemann’s Zeta Function. New York: Academic Press, 1974. Karatsuba, A. A. and Voronin, S. M. The Riemann Zeta- Function. Hawthorn, NY: de Gruyter, 1992. Odlyzko, A. M. "The 1020th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint. Sloane, N. J. A. Sequences A050276 and A050277 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987. van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. IV." Math. Comput. 46, 667 /C1/681, 1986. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 143, 1991. RiemannSiegelTheta RIEMANN- SIEGEL FUNCTIONS RiemannSiegelZ RIEMANN- SIEGEL FUNCTIONS Riemann-Stieltjes Integral STIELTJES INTEGRAL Riemann-Volterra Method RIEMANN METHOD Riesel Number There exist infinitely many ODD INTEGERS k such that k /C215 2n /C281is COMPOSITE for every n ]1: Numbers k with this property are called RIESEL NUMBERS , and analogous numbers with the minus sign replaced by a plus are called SIERPINSKI NUMBERS OF THE SECOND KIND . The smallest known Riesel number is k /C30 509; 203; but there remain 963 smaller candidates (the smallest of which is 659) which generate onlycomposite numbers for all n which have been checked (Ribenboim 1996, p. 358). Let a(k) be smallest n for which (2k /C281) /C215 2n /C281is PRIME , then the first few values are 2, 0, 2, 1, 1, 2, 3, 1, 2, 1, 1, 4, 3, 1, 4, 1, 2, 2, 1, 3, 2, 7, ... (Sloane’s A046069), and second smallest n are 3, 1, 4, 5, 3, 26, 7, 2, 4, 3, 2, 6, 9, 2, 16, 5, 3, 6, 2553, ... (Sloane’s A046070). See also CUNNINGHAM NUMBER ,MERSENNE NUMBER , SIERPINSKI’S COMPOSITE NUMBER THEOREM ,SIER- PINSKI NUMBER OF THE SECOND KIND,THAˆ BIT IBN KURRAH RULE References Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 357, 1996. Riesel, H. "Na ˚gra stora primtal." Elementa 39, 258/C1/260, 1956. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkha ¨user, pp. 394 /C1/398, 1994. Sloane, N. J. A. Sequences A046067, A046068, A046069, and A046070 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Riesz Representation Theorem There are a couple of versions of this theorem. Basically, it says that any bounded linear FUNC- TIONAL Ton the space of compactly supported continuous functions on Xis the same as integration against a measure m; Tf/C30gfdm: Here, the integral is the L EBESGUE INTEGRAL . Because linear functionals form a VECTOR SPACE , and are not "positive," the measure mmay not be a POSITIVE MEASURE . But if the functional Tis positive, in the sense that f]0 implies that Tf]0;then the measure mis also positive. In the generality of complex linear functionals, the measure mis a COMPLEX MEASURE . The measure mis uniquely de- termined by Tand has the properties of a regular BOREL MEASURE . It must be a finite measure, which corresponds to the boundedness condition on the functional. In fact, the NORM ofT,Tkk;is the TOTAL VARIATION MEASURE ofX,mjj(X):/ Naturally, there are some hypotheses necessary for this to make sense. The space Xhas to be LOCALLY COMPACT and H AUSDORFF , which is not a strong restriction. In fact, for unbounded spaces X, the theorem also applies to functionals on continuous functions which vanish at infinity, in the sense that for any e>0;there is a compact set Ksuch that for any xnot in K,f(x)jjBe(which is the notion from calculus of limx0/C12f(x)/C300):/ The Riesz representation theorem is useful in de- scribing the DUAL SPACE to any space which contains the compactly supported continuous functions as a DENSE subspace. Roughly speaking, a linear func- tional is modified, usually by convolving with a bump function, to a bounded linear functional on the compactly supported continuous functions. Then it can be realized as integration against a measure. Often the measure must be ABSOLUTELY CONTINUOUS , and so the dual is integration against a function. See also ABSOLUTELY CONTINUOUS ,COMPLEX MEA- SURE ,D UAL SPACE ,F UNCTIONAL ,H ILBERT SPACE , LEBESGUE MEASURE ,MEASURE SPACE ,POLAR REPRE- SENTATION (MEASURE ), RADON- NIKODYM THEOREM , SINGULAR MEASURE References Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990. Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 40 /C1/47 and 129 /C1/132, 1987. Riesz-Fischer Theorem A function is L2/- (square-) integrable IFF its FOURIER SERIES is L2/-convergent. The application of this theorem requires use of the LEBESGUE INTEGRAL . See also LEBESGUE INTEGRAL Riesz’s Theorem Every continuous linear functional U[f] for f /C23 C[a ; b] can be expressed as a STIELTJES INTEGRAL U[f] /C30gb af(x) dw(x) ; where w(x) is determined by U and is of bounded variation on [a, b]. See also STIELTJES INTEGRAL References Kestelman, H. "Riesz’s Theorem." §11.5 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 265 /C1/ 269, 1960. Riffle Shuffle A SHUFFLE , also called a FARO SHUFFLE , in which a deck of 2n cards is divided into two HALVES which are then alternatively interleaved from the left and right hands (an "in-shuffle") or from the right and left hands (an "out-shuffle"). Using an "in-shuffle," a deck originally arranged as 1 2 3 4 5 6 7 8 would become 5 1 6 2 7 3 8 4. Using an "out-shuffle," the deck order would become 1 5 263748. Riffle shuffles are used in card tricks (Marlo 1958ab, Adler 1973), and also in the theory of parallel processing (Stone 1971, Chen et al. 1981).In general, card k moves to the position originally occupied by the 2k/th card (mod 2n /C271): Therefore, in- shuffling 2n cards 2n times (where 2n /C271is PRIME ) results in the original card order. Similarly, out- shuffling 2n cards 2n /C282 times (where 2n /C281is PRIME ) results in the original order (Diaconis et al. 1983, Conway and Guy 1996). Amazingly, this means that an ordinary deck of 52 cards is returned to its original order after 8 out-shuffles. Morris (1994) further discusses aspects of the perfect riffle shuffle (in which the deck is cut exactly in half and cards are perfectly interlaced). Ramnath and Scully (1996) give an algorithm for the shortest sequence of in- and out-shuffles to move a card from arbitrary position i to position j. This algorithm works for any deck with an EVEN number of cards and is O(logn):/ See also CARDS ,SHUFFLE References Adler, I. "Make Up Your Own Card Tricks." J. Recr. Math. 6, 87/C1/91, 1973. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 323 /C1/325, 1987. Chen, P. Y.; Lawrie, D. H.; Yew, P.-C.; and Padua, D. A. "Interconnection Networks Using Shuffles." Computer 33, 55/C1/64, Dec. 1981. Conway, J. H. and Guy, R. K. "Fractions Cycle into Deci- mals." In The Book of Numbers. New York: Springer- Verlag, pp. 163 /C1/165, 1996. Diaconis, P.; Graham, R. L.; and Kantor, W. M. "The Mathematics of Perfect Shuffles." Adv. Appl. Math. 4, 175/C1/196, 1983. Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. Wa- shington, DC: Math. Assoc. Amer., 1989. Herstein, I. N. and Kaplansky, I. Matters Mathematical. New York: Harper & Row, 1974. Mann, B. "How Many Times Should You Shuffle a Deck of Cards." UMAP J. 15, 303/C1/332, 1994. Marlo, E. Faro Notes. Chicago, IL: Ireland Magic Co., 1958a. Marlo, E. Faro Shuffle. Chicago, IL: Ireland Magic Co., 1958b. Medvedoff, S. and Morrison, K. "Groups of Perfect Shuffles." Math. Mag. 60,3/C1/14, 1987. Morris, S. B. and Hartwig, R. E. "The Generalized Faro Shuffle." Discrete Math. 15, 333/C1/346, 1976. Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 240 /C1/244, 1990. Ramnath, S. and Scully, D. "Moving Card ito Position jwith Perfect Shuffles." Math. Mag. 69, 361/C1/365, 1996. Stone, H. S. "Parallel Processing with the Perfect Shuffle." IEEE Trans. Comput. 2, 153/C1/161, 1971. Rigby Points The PERSPECTIVE CENTERS of the TANGENTIAL and CONTACT TRIANGLES of the inner and outer S ODDY POINTS . The inner Riand outer Ri?Rigby points are given by Ri/C30I/C274 3Ge Ri ?/C30I /C284 3 Ge; where I is the INCENTER and Ge is the GERGONNE POINT . Honsberger (1995) defines a different point which he calls the "Rigby point" X. Let QR be an arbitrary CHORD of the CIRCUMCIRCLE of a given TRIANGLE DABC ; and let P be the POLE of the SIMSON LINE SP with respect to DABC which is PERPENDICULAR to QR. Then it also turns out that SQ /C222PR and SR /C222PQ: In addition, SA /C222BC; SB /C222AC ; and SC /C222AB with respect to DPQR :/ As a result of these remarkable facts, it can be shown that the SIMSON LINES SP ; SQ ; and SR with respect to DABC meet in the Rigby point X. Moreover, the SIMSON LINES SA ; SB ; and SCwith respect to DPQR also meet in X, and X is the ORTHOPOLE of AB, BC, and AC with respect to DPQR ; and of PQ, QR, and PR with respect to DABC : Finally, X is the MIDPOINT of the ORTHOCENTERS of DABC and DPQR (Honsber- ger 1996, p. 136). See also CONTACT TRIANGLE ,G ERGONNE POINT , GRIFFITHS POINTS ,INCENTER ,O LDKNOW POINTS , ORTHOPOLE ,SIMSON LINE,SODDY POINTS ,TANGEN- TIAL TRIANGLE References Honsberger, R. "The Rigby Point." §11.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 132 /C1/136, 1995.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319 /C1/329, 1996. Right Angle An ANGLE equal to half the ANGLE from one end of a line segment to the other. A right angle is p=2 radians or 90 8.ATRIANGLE containing a right angle is called a RIGHT TRIANGLE . However, a TRIANGLE cannot con- tain more than one right angle, since the sum of the two right angles plus the third angle would exceed the 1808 total possessed by a TRIANGLE . The patterns of cracks observed in mud which has been dried by the sun form curves which intersect in right angles (Williams 1979, p. 45; Steinhaus 1983, p. 88; Pearce 1990, p. 12). See also ACUTE ANGLE ,F ULL ANGLE ,O BLIQUE ANGLE ,O BTUSE ANGLE ,O RTHOGONAL LINES,PER- PENDICULAR ,R IGHT TRIANGLE ,S EMICIRCLE , STRAIGHT ANGLE ,THALES’ THEOREM References Pearce, P. Structure in Nature Is a Strategy for Design. Cambridge, MA: MIT Press, 1990. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Right Circular Cone A circular cone the centers of whose sections form a line perpendicular to the bases. When used without qualification, the term "cone" often refers to a right circular cone. See also CONE References Kern, W. F. and Bland, J. R. "Right Circular Cone." §25 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 60 /C1/64, 1948. Right Circular Cylinder A circular cylinder the centers of whose sections form a line perpendicular to the bases. When used without qualification, the term "cylinder" often refers to a right circular cylinder. See also CYLINDER References Kern, W. F. and Bland, J. R. "Right Circular Cylinder." §17 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 39 /C1/42, 1948. Right Cone CONE Right Conoid A RULED SURFACE is called a right conoid if it can be generated by moving a straight LINE intersecting a fixed straight LINE such that the LINES are always PERPENDICULAR (Kreyszig 1991, p. 87). Taking the PERPENDICULAR plane as the xy-plane and the line to be the X-AXIS gives the PARAMETRIC EQUATIONS x(u; v) /C30v cos q(u) y(u; v) /C30v sin q(u) z(u ; v) /C30h(u) (Gray 1997). Taking h(u) /C302u and q(u) /C30u gives the HELICOID . See also HELICOID ,P LU¨ CKER’S CONOID ,W ALLIS’S CONICAL EDGE References Dixon, R. Mathographics. New York: Dover, p. 20, 1991. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 450 /C1/452, 1997. Kreyszig, E. Differential Geometry. New York: Dover, 1991. Right Coset Consider a countable SUBGROUP H with ELEMENTS hi and an element x not in H, then hix for i /C301, 2, ... are the right cosets of the SUBGROUP H with respect to x. See also COSET ,LEFT COSET Right Cylinder CYLINDERRight Half-Plane The portion of the COMPLEX PLANE z /C30x /C27iy with REAL PART R[z] > 0:/ See also COMPLEX PLANE ,LEFT HALF-PLANE ,LOWER HALF-PLANE ,UPPER HALF-PLANE Right Hyperbola RECTANGULAR HYPERBOLA Right Line LINE Right Prism PRISM Right Strophoid The STROPHOID of a line Lwith pole Onot on Land fixed point O?being the point where the PERPENDI- CULAR from OtoLcuts Lis called a right strophoid. It is therefore a general STROPHOID with a/C30p=2:/ The right strophoid is given by the Cartesian equa- tion y2/C30c/C28x c/C27xx2; (1) or the polar equation r/C30ccos(2 u) sec u: (2) The parametric form of the strophoid is x(t)/C301/C28t2 t2/C271(3) y(t) /C30t(t2 /C28 1) t2 /C27 1: (4) The right strophoid has CURVATURE k(t) /C30/C284(1 /C27 3t2) (1 /C27 6t2 /C27 t4)3 =2 (5) and TANGENTIAL ANGLE f(t) /C30/C282 tan /C281 t /C28tan/C281 2t 1 /C27 t2 ! : (6) The right strophoid first appears in work by Isaac Barrow in 1670, although Torricelli describes the curve in his letters around 1645 and Roberval found it as the LOCUS of the focus of the conic obtained when the plane cutting the CONE rotates about the tangent at its vertex (MacTutor Archive). The AREA of the loop is Aloop /C301 2 c2(4 /C28 p) (7) (MacTutor Archive). Let C be the CIRCLE with center at the point where the right strophoid crosses the X-AXIS and radius the distance of that point from the origin. Then the right strophoid is invariant under inversion in the CIRCLE Cand is therefore an ANALLAGMATIC CURVE . See also STROPHOID ,TRISECTRIX References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 92, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 100 /C1/104, 1972. Lockwood, E. H. "The Right Strophoid." Ch. 10 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 90 /C1/97, 1967. MacTutor History of Mathematics Archive. "Right Stro- phoid." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Right.html. Right Strophoid Inverse Curve The INVERSE CURVE of a right strophoid is the same strophoid.Right Triangle ATRIANGLE with an ANGLE of 908(/p=2 radians). The sides a,b, and cof such a TRIANGLE satisfy the PYTHAGOREAN THEOREM . The largest side is conven- tionally denoted cand is called the HYPOTENUSE .A TRIANGLE that is not a right triangle is sometimes called an OBLIQUE TRIANGLE . For any three similar shapes on the sides of a right triangle, A1/C27A2/C30A3; (1) which is equivalent to the P YTHAGOREAN THEOREM . For a right triangle with sides a,b, and HYPOTENUSE c, let rbe the INRADIUS . Then 1 2ab/C3012ra/C2712rb/C2712rc/C3012r(a/C27b/C27c): (2) Solving for rgives r/C30ab a/C27b/C27c: (3) This can also be written in the equivalent forms r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ðc/C28aÞðc/C28bÞq ð4Þ /C3012(a/C27b/C28c): (5) Now, since any P YTHAGOREAN TRIPLE can be written a/C30m2/C28n2(6) b/C302mn (7) c/C30m2/C27n2; (8) (3) becomes r/C30(m2/C28n2)2mn m2/C28n2/C272mn/C27m2/C27n2/C30n(m/C28n); (9) which is an INTEGER when mand nare integers (Ogilvy and Anderson 1988, p. 68). The HYPOTENUSE of a right triangle is a DIAMETER of the triangle’s CIRCUMCIRCLE , so the CIRCUMRADIUS is given by R/C301 2c; (10) where cis the HYPOTENUSE . Given a right triangle DABC ;draw the ALTITUDE AH from the RIGHT ANGLE A. Then the triangles DAHC andDBHA are similar. In a right triangle, the MIDPOINT of the HYPOTENUSE is equidistant from the three VERTICES (Dunham 1990). This can be proved as follows. Given DABC ; letMbe the MIDPOINT ofAB(so that AM/C30BM). Draw DM½½CA;then since DBDM is similar to DBCA ; it follows that BD/C30DC. Since both DBDM and DCDM are right triangles and the corresponding legs are equal, the HYPOTENUSES are also equal, so we have AM/C30BM/C30CMand the theorem is proved. Fermat showed how to construct an arbitrary number of equiareal nonprimitive right triangles. An analysisof P YTHAGOREAN TRIPLES demonstrates that the right triangle generated by a triple ( m2 i/C28n2i;2mini;m2i/C27 n2i) has common AREA A/C30rs(2r/C27s)(r/C272s)(r/C27s)(r/C28s)(r2/C27rs/C27s2) (Beiler 1966, pp. 126 /C1/127). The only EXTREMUM of this function occurs at ( r;s)/C30(0;0):Since A(r;s)/C300 forr/C30s, the smallest AREA shared by three nonpri- mitive right triangles is given by ( r;s)/C30(1;2);which results in an area of 840 and corresponds to the triplets (24, 70, 74), (40, 42, 58), and (15, 112, 113) (Beiler 1966, p. 126). One can also find quartets ofright triangles with the same AREA . The QUARTET having smallest known area is (111, 6160, 6161),(231, 2960, 2969), (518, 1320, 1418), (280, 2442, 2458),with AREA 341,880 (Beiler 1966, p. 127). Guy (1994) gives additional information. The smallest known AREA shared by three primitive right triangles is 13123110, corresponding to the triples (4485, 5852, 7373), (1380, 19019, 19069), and(3059, 8580, 9109) (Beiler 1966, p. 127; Gardner 1984, p. 160).It is also possible to find sets of three and four Pythagorean triplets having the same PERIMETER (Beiler 1966, pp. 131 /C1/132). Lehmer (1900) showed that the number of primitive triples N(p) with PERIMETER less than pis lim p0/C12N(p)/C30pln 2 p2/C300:070230 . . . : (11) In a given right triangle, an infinite sequence ofsquares can be nested which alternately lie on the HYPOTENUSE and longest leg. These create a sequence of increasingly smaller similar right triangles. Let theoriginal triangle have legs of lengths aand band HYPOTENUSE of length c/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2p :Also define x/C13ac ab/C27c2(12) y/C131 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2[c2/C28(a/C27b)c/C27ab]p : (13) Then the sides of the nsquare are of length sn/C30bxn: (14) Number the upper left triangle as 1, and then the remainder by following the "strip" of triangles at adjoining vertices. Then the side lengths of these triangles are an/C30s(n/C271)=2fornodd ab cxn=2forneven8 >< >:(15) bn/C30b2 ax(n/C271)=2fornodd b2 cxn=2forneven8 >>>< >>>:(16) c n/C30bc ax(n/C271)=2fornodd sn=2 forneven :8 >< >:(17) The INRADII of the corresponding circles can be found from rn /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (bn /C27 cn /C28 an)(cn /C27 an /C28 bn)(an /C27 bn /C28 cn) an /C27 bn /C27 cns ; (18) giving rn /C30b ayx(n /C271)=2for n odd b cyxn=2 for n even :8 >>>< >>>:(19) AS ANGAKU PROBLEM from 1913 in the Miyagi Pre- fecture asks for the relationships between the first, third, and fifth inradii (Rothman 1998). This can be solved using elementary TRIGONOMETRY as well as the explicit equations given above, and has solution r3 /C30ffiffiffiffiffiffiffiffiffir1r5p: (20) See also ACUTE TRIANGLE ,ARCHIMEDES’ MIDPOINT THEOREM ,BROCARD MIDPOINT ,CIRCLE- POINT MID- POINT THEOREM ,FERMAT’S RIGHT TRIANGLE THEO- REM,ISOSCELES TRIANGLE ,M ALFATTI’S RIGHT TRIANGLE PROBLEM ,O BLIQUE TRIANGLE ,O BTUSE TRIANGLE ,P YTHAGOREAN TRIPLE ,Q UADRILATERAL , RAT -FREE SET,TRIANGLE ,TRIGONOMETRY References Beiler, A H. "The Eternal Triangle." Ch. 14 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987. Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120 /C1/121, 1990. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 160 /C1/161, 1984. Guy, R. K. "Triangles with Integer Sides, Medians, and Area." §D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188 /C1/190, 1994. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 2, 1948. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, p. 68, 1988. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/91, May 1998. Sierpinski, W. Pythagorean Triangles. New York: Academic Press, 1962. Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 155 /C1/161, 1943. Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 265 /C1/268, 1943.Right-Hand Rule The rule which determines the orientation of the CROSS PRODUCT u /C29v: The right-hand rule states that the orientation of the vectors’ cross product is determined by placing u and v tail-to-tail, flattening the right hand, extending it in the direction of u, and then curling the fingers in the direction that the angle v makes with u. The thumb then points in the direction of u /C29v :/ A three-dimensional COORDINATE SYSTEM in which the axes satisfy the right-hand rule is called a RIGHT- HANDED COORDINATE SYSTEM , while one that does not is called a LEFT-HANDED COORDINATE SYSTEM . See also CROSS PRODUCT ,LEFT-HANDED COORDINATE SYSTEM ,RIGHT- HANDED COORDINATE SYSTEM Right-Handed Coordinate System A three-dimensional COORDINATE SYSTEM in which the axes satisfy the RIGHT-HAND RULE . See also CROSS PRODUCT ,LEFT-HANDED COORDINATE SYSTEM ,RIGHT- HAND RULE Rigid Framework FRAMEWORK ,RIGID GRAPH Rigid Graph AFRAMEWORK (or GRAPH ) is rigid IFFcontinuous motion of the points of the configuration maintaining the bar constraints comes from a family of motions ofall E UCLIDEAN SPACE which are distance-preserving. AGRAPH that is not rigid is said to be FLEXIBLE (Maehara 1992). For example, the CYCLE GRAPH C3is rigid, while C4is flexible. An embedding of the BIPARTITE GRAPH K3;3 in the plane is rigid unless its six vertices lie on a CONIC (Bolker and Roth 1980, Maehara 1992). AGRAPH Gis (generically) d-rigid if, for almost all (i.e., an open dense set of) CONFIGURATIONS ofp, the FRAMEWORK G(p) is rigid in Rd:/ Cauchy (1813) proved the RIGIDITY THEOREM , one of the first results in rigidity theory. Although rigidity problems were of immense interest to engineers, intensive mathematical study of these types of pro- blems has occurred only relatively recently (Connelly 1993, Graver et al. 1993). See also BAR (EDGE), BRACED SQUARE ,F LEXIBLE GRAPH ,FLEXIBLE POLYHEDRON ,FRAMEWORK ,JUST RIGID,L AMAN’S THEOREM ,L IEBMANN’S THEOREM , RIGID POLYHEDRON ,RIGIDITY THEOREM ,TENSEGRITY References Asimov, L. and Roth, B. "The Rigidity of Graphs." Trans. Amer. Math. Soc. 245, 279 /C1/289, 1978. Bolker, E. D. and Roth, B. "When is a Bipartite Graph a Rigid Framework?" Pacific J. Math. 90,27/C1/44, 1980. Cauchy, A. L. "Sur les polygones et les polye`dres." XVIe Cahier IX,87/C1/89, 1813. Connelly, R. "Rigidity." Ch. 1.7 in Handbook of Convex Geometry, Vol. A (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 223 /C1/271, 1993. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 56, 1967. Crapo, H. and Whiteley, W. "Statics of Frameworks and Motions of Panel Structures, A Projective Geometry Introduction." Structural Topology 6,43/C1/82, 1982. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Rigidity of Frameworks." §B14 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 63 /C1/65, 1991. Dehn, M. "U¨ ber die Strakheit knovexer Polyeder." Math. Ann. 77, 466 /C1/473, 1916. Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165 /C1/170, 1978. Graver, J.; Servatius, B.; and Servatius, H. Combinatorial Rigidity. Providence, RI: Amer. Math. Soc., 1993. Maehara, H. "Distance Graphs in Euclidean Space." Ryukyu Math. J. 5,33/C1/51, 1992. Pegg, E. Jr. "Rigid Nonagon." http://www.mathpuzzle.com/ riginona.gif. Roth, B. "Rigid and Flexible Frameworks." Amer. Math. Monthly 88,6/C1/21, 1981. Rigid Motion A transformation consisting of ROTATIONS and TRANS- LATIONS which leaves a given arrangement un- changed. See also EUCLIDEAN MOTION ,PLANE ,ROTATION References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 141, 1996. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 84 /C1/85 and 89 /C1/91, 1930. Rigid Polyhedron A POLYHEDRON is rigid if it cannot be continuously deformed into another configuration. A rigid polyhe- dron may have two or more stable forms which cannot be continuously deformed into each other without bending or tearing (Wells 1991).A structure such as a polyhedron which can change form from one stable configuration to another with only a slight transient nondestructive elastic stretch is called MULTISTABLE (Goldberg 1978). A non-rigid polyhedron may be "SHAKY " (infinitesi- mally movable) or FLEXIBLE . An example of a concave FLEXIBLE POLYHEDRON with 18 triangular faces was given by Connelly (1978), and a FLEXIBLE POLYHE- DRON with only 14 triangular faces was subsequently found by Steffen (Mackenzie 1998). JESSEN’S ORTHOGONAL ICOSAHEDRON is an example of a SHAKY POLYHEDRON . See also FLEXIBLE POLYHEDRON ,JESSEN’S ORTHOGO- NAL ICOSAHEDRON ,JUMPING OCTAHEDRON ,M ULTI- STABLE ,P ENTAGONAL DIPYRAMID ,R IGID GRAPH , SHAKY POLYHEDRON References Cauchy, A. L. "Sur les polygons et le polyhe ´ders." XVIe Cahier IX,87/C1/89, 1813. Connelly, R. "A Flexible Sphere." Math. Intel. 1, 130 /C1/131, 1978. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Rigidity of Polyhedra." §B13 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 61 /C1/63, 1991. Cromwell, P. R. "Equality, Rigidity, and Flexibility." Ch. 6 in Polyhedra. New York: Cambridge University Press, pp. 219 /C1/247, 1997. Gluck, H. Almost All Simply Connected Closed Surfaces are Rigid. Heidelberg, Germany: Springer-Verlag, pp. 225 /C1/ 239, 1975. Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165 /C1/170, 1978. Graver, J.; Servatius, B.; and Servatius, H. Combinatorial Rigidity. Providence, RI: Amer. Math. Soc., 1993. Mackenzie, D. "Polyhedra Can Bend But Not Breathe." Science 279, 1637, 1998. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 161 /C1/162, 1991. Wunderlich, W. "Starre, kippende, wackelige und bewe- gliche Achtflache." Elem. Math. 20,25/C1/32, 1965. Rigidity Theorem If the faces of a convex POLYHEDRON were made of metal plates and the EDGES were replaced by hinges, the POLYHEDRON would be RIGID . The theorem was stated by Cauchy (1813), although a mistake in this paper went unnoticed for more than 50 years. See also FLEXIBLE POLYHEDRON ,RIGID POLYHEDRON , SHAKY POLYHEDRON References Cauchy, A. L. "Sur les polygons et le polyhe ´ders." XVIe Cahier IX,8 7/C1/89, 1813. Cromwell, P. R. "Cauchy’s Rigidity Theorem." In Polyhedra. New York: Cambridge University Press, pp. 228 /C1/233, 1997. Dehn, M. "U ¨ber die Strakheit knovexer Polyeder." Math. Ann. 77, 466/C1/473, 1916. Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165/C1/170, 1978. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 161 /C1/162, 1991. Rigorous A proof or demonstration is said to be rigorous if the validity of each step and the connections between the steps is explicitly made clear is such a way that the result follows with certainty. "Rigorous" proofs often rely on the postulates and results of formal systems that are themselves considered rigorous under stated conditions. Ring A ring (in the mathematical sense) is a SET S together with two BINARY OPERATORS /C27 and + (commonly interpreted as addition and multiplication, respec- tively) satisfying the following conditions: 1. Additive associativity: For all a; b; c /C23 S; (a /C27b) /C27c /C30a /C27(b /C27c) ;/ 2. Additive commutativity: For all a ; b /C23 S; a /C27b /C30b /C27a ;/ 3. Additive identity: There exists an element 0 /C23 S such that for all a /C23 S ; 0 /C27a /C30a /C270 /C30a ;/ 4. Additive inverse: For every a /C23 S there exists /C28a /C23 S such that a /C27(/C28a) /C30(/C28a) /C27a /C300;/ 5. Multiplicative associativity: For all a; b; c /C23 S; a + (b + c) /C30a + (b + c) ;/ 6. Left and right distributivity: For all a; b; c /C23 S; a + (b /C27c) /C30(a + b) /C27(a + c) and (b /C27c) + a /C30/ /(b + a) /C27(c + a) :/ A ring is therefore an ABELIAN GROUP under addition and a SEMIGROUP under multiplication. The French word for a ring is anneau , and the German word is Ring , both meaning (not so surpris- ingly) "ring." A ring must contain at least one element, but need not contain a multiplicative identity or be commu- tative. The number of finite rings of n elements for n /C301, 2, ..., are 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, ... (Sloane’s A027623 and A037234; Fletcher 1980). In general, the number of rings of order p3 for p an ODD PRIME is 3p /C2750 and 52 for p /C302 (Ballieu 1947, Gilmer and Mott 1973). A ring with a multiplicative identity is sometimes called a UNIT RING . Fraenkel (1914) gave the first abstract definition of the ring, although this work did not have much impact. A ring that is COMMUTATIVE under multiplication, has a unit element, and has no divisors of zero is called an INTEGRAL DOMAIN . A ring which is also a COMMU- TATIVE multiplication group is called a FIELD . The simplest rings are the INTEGERS Z;POLYNOMIALS R[x] andR[x;y] in one and two variables, and SQUARE n/C29 nREAL MATRICES .Rings which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to names which give very little insight into therelevant properties of the associated rings. See also A BELIAN GROUP ,A RTINIAN RING,C HOW RING,D EDEKIND RING,D IVISION ALGEBRA ,FIELD, GORENSTEIN RING,G ROUP ,G ROUP RING,IDEAL , INTEGRAL DOMAIN ,M ODULE ,N ILPOTENT ELEMENT , NOETHERIAN RING,NONCOMMUTATIVE RING,NUMBER FIELD,PRIME RING,PRU¨ FER RING,QUOTIENT RING, REGULAR RING,RINGOID ,SEMIPRIME RING,SEMIRING , SEMISIMPLE RING,SIMPLE RING,U NIT RING,ZERO DIVISOR References Allenby, R. B. Rings, Fields, and Groups: An Introduction to Abstract Algebra, 2nd ed. Oxford, England: Oxford Uni- versity Press, 1991. Ballieu, R. "Anneaux finis; syste `mes hypercomplexes de rang trois sur un corps commutatif." Ann. Soc. Sci. Bruxelles. Se ´r. I61, 222/C1/227, 1947. Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, 1999. Berrick, A. J. and Keating, M.E An Introduction to Rings and Modules with K-Theory in View. Cambridge, Eng- land: Cambridge University Press, 2000. Ellis, G. Rings and Fields. Oxford, England: Oxford Uni- versity Press, 1993. Fletcher, C. R. "Rings of Small Order." Math. Gaz. 64,9/C1/22, 1980. Fraenkel, A. "U ¨ber die Teiler der Null und die Zerlegung von Ringen." J. reine angew. Math. 145, 139/C1/176, 1914. Gilmer, R. and Mott, J. "Associative Rings of Order p3:/"Proc. Japan Acad. 49, 795/C1/799, 1973. Kleiner, I. "The Genesis of the Abstract Ring Concept." Amer. Math. Monthly 103, 417/C1/424, 1996. Nagell, T. "Moduls, Rings, and Fields." §6i nIntroduction to Number Theory. New York: Wiley, pp. 19 /C1/21, 1951. Sloane, N. J. A. Sequences A027623 and A037234 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/eisonline.html. van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985. Ring Cyclide The INVERSION of a RING TORUS . If the INVERSION CENTER lies on the torus, then the ring cyclide degenerates to a PARABOLIC RING CYCLIDE . See also CYCLIDE ,INVERSION ,PARABOLIC CYCLIDE , RING CYCLIDE ,RING TORUS ,SPINDLE CYCLIDE ,TORUS Ring Direct Product The direct product of the RINGS Rg ; for g some INDEX SET I, is the set Y g /C23IRg /C30})1D f : I 0/C160 g /C23IRgf(g) /C23 Rgall g /C23 I})1E :})@1})@1})@1})@1 The ring direct product is confusingly also called the complete direct sum (Herstein 1968). X 0 G ¡ H[X 0 G /C154H 0 G ¡ H the universal property of a direct product ; X factors through G /C154H : The ring direct product, like the GROUP DIRECT PRODUCT , has the UNIVERSAL PROPERTY that if any ring X has a HOMOMORPHISM to G and a homomorph- ism to H, then these homomorphisms factor through G /C29H in a unique way. References Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., p. 52, 1968. Ring Function TOROIDAL FUNCTION Ring Homomorphism A ring homomorphism is a map f : R 0 S between two RINGS such that 1. Addition is preserved: f(r1 /C27r2) /C30f(r1) /C27f(r2) ;/ 2. The zero element is mapped to zero: f(0R) /C300S ; and 3. Multiplication is preserved: f(r1r2) /C30f(r1)f(r2);/ where the operations on the left-hand side is in R and on the right-hand side in S. Note that a homomorph- ism must preserve the additive inverse map because f(g) /C27f(/C28g) /C30f(g /C27/C28g) /C30f(0R) /C300S so /C28f(g) /C30f(/C28g) :/ See also GROUP HOMOMORPHISM ,H OMOMORPHISM , ISOMORPHISM ,RING Ring of Polynomial POLYNOMIAL RING Ring Torus One of the three STANDARD TORI given by the PARA- METRIC EQUATIONS x /C30(c /C27a cos v) cos u y /C30(c /C27a cos v) sin u z /C30a sin v with c /C21a. This is the TORUS which is generally meant when the term "torus" is used without quali- fication. The inversion of a ring torus is a RING CYCLIDE if the INVERSION CENTER does not lie on the torus and a PARABOLIC RING CYCLIDE if it does. The above left figure shows a ring torus, the middle a cutaway, and the right figure shows a CROSS SECTION of the ring torus through the xz-plane. See also CYCLIDE ,H ORN TORUS ,PARABOLIC RING CYCLIDE ,RING CYCLIDE ,SPINDLE TORUS ,STANDARD TORI,TORUS References Gray, A. "Tori." §13.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 304 /C1/306, 1997. Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28 /C1/30, 1986. Ringoid A ringoid is a set R with two binary operators, conventionally denoted addition (//C27) and multiplica- tion (//C29) ; where /C29 distributes over /C27 left and right: a(b /C27c) /C30ab /C27ac and (b /C27c)a /C30ba /C27ca: A ringoid can be empty. See also BINARY OPERATOR ,RING,SEMIRING References Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968. Risch Algorithm An ALGORITHM for indefinite integration. See also ELEMENTARY FUNCTION ,INDEFINITE INTE- GRAL References Geddes, K. O.; Czapor, S. R.; and Labahn, G. "The Risch Integration Algorithm." Ch. 12 in Algorithms for Compu- ter Algebra. Amsterdam, Netherlands: Kluwer, pp. 511 /C1/ 573, 1992. Risch, R. "On the Integration of Elementary Functions which are Built Up using Algebraic Operations." Report SP-2801/002/00. Santa Monica, CA: Sys. Dev. Corp., 1968. Risch, R. "The Problem of Integral in Finite Terms." Trans. Amer. Math. Soc. 139, 167 /C1/189, 1969. Risch, R. "The Solution of the Problem of Integration in Finite Terms." Bull. Amer. Math. Soc.,1/C1/76, 605 /C1/608, 1970. Risch, R. "Algebraic Properties of Elementary Functions of Analysis." Amer. J. Math. 101, 743 /C1/759, 1979. Rising Factorial There are two notations used for the falling and rising factorials, (x)n and x(n) ; which are unfortunately polar opposites of one another. The rising factorial x(n) (sometimes also denoted /C142x/C143n; Comtet 1974, p. 6), frequently called the POCHHAMMER SYMBOL in the theory of special functions, is defined by x(n) /C30x(x /C271) /C1/C1/C1(x /C27n /C281): (1) It is related to the GAMMA FUNCTION G(z)by x(n) /C30G(x /C27 n) G(x); (2) where x(0) /C131 ; (3) and is related to the FALLING FACTORIAL (x)n by x(n) /C30(/C28x)n(/C281)n : (4) The rising factorial is implemented in Mathematica asPochhammer [x, n]. Note that in combinatorial usage, the FALLING FAC- TORIAL is denoted (x)nand the rising factorial is denoted (x)(n) (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of FINITE DIFFERENCES and the theory of special functions, the FALLING FACTORIAL is denoted x(n)and the rising factorial is denoted (x)n(Roman 1984, p. 5; Abramo- witz and Stegun 1972, p. 256; Spanier 1987). Extreme caution is therefore needed in interpreting the mean- ings of the notations (x)nand x(n) : In this work, the notation x(n) is used for the rising factorial , despite the fact that POCHHAMMER SYMBOL , which is another name for the rising factorial, is universally denoted (x)n :/ The rising factorial arises in series expansions of HYPERGEOMETRIC FUNCTIONS and GENERALIZED HY- PERGEOMETRIC FUNCTIONS . The first few rising fac- torials are x(0) /C301 x(1) /C30x x(2) /C30x(x /C271) /C30x2 /C27x x(3) /C30x(x /C271)(x /C272) /C30x3 /C273x2 /C272x x(4) /C30x(x /C271)(x /C272)(x /C273) /C30x4 /C276x3 /C2711x2 /C276x: Additional identities ared dxx(n) /C30x(n)[F(x /C27n /C281) /C28F(x /C281)] (5) x(n/C27k) /C30(x /C27n)kx(n) ; (6) where F(z) is the DIGAMMA FUNCTION . See also CENTRAL FACTORIAL ,FACTORIAL ,FALLING FACTORIAL ,G ENERALIZED HYPERGEOMETRIC FUNC- TION ,H ARMONIC LOGARITHM ,H YPERGEOMETRIC FUNCTION ,POCHHAMMER SYMBOL References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 101, 1999. Roman, S. The Umbral Calculus. New York: Academic Press, p. 5, 1984. Spanier, J. and Oldham, K. B. "The Pochhammer Polyno- mials (x)n :/" Ch. 18 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 149 /C1/165, 1987. Rivest-Shamir-Adleman Number RSA NUMBER R-Module A MODULE taking its coefficients in a RING R is called a module over R or R-module. See also MODULE RMS ROOT-MEAN-SQUARE Robbin Constant R /C304 105 /C2717 105ffiffiffi 2p /C282 35ffiffiffi 3p /C271 5 ln 1 /C27ffiffiffi 2p})@D})@E /C272 3 ln 2 /C27ffiffiffi 3p})@D})@E /C281 15p/C300:661707182 . . . : See also TRANSFINITE DIAMETER References Plouffe, S. "The Robbin Constant." http://www.lacim.u- qam.ca/piDATA/robbin.txt. Robbins Algebra Building on work of Huntington (1933), Robbins conjectured that the equations for a Robbins algebra, commutativity, associativity, and the R OBBINS AXIOM !(!(x /C150y) /C150!(x /C150!y)) /C30x; where !x denotes NOT and x /C150y denotes OR, imply those for a BOOLEAN ALGEBRA . The conjecture was finally proven using a computer (McCune 1997). See also BOOLEAN ALGEBRA ,H UNTINGTON AXIOM , ROBBINS CONJECTURE ,R OBBINS AXIOM ,W INKLER CONDITIONS References Huntington, E. V. "New Sets of Independent Postulates for the Algebra of Logic, with Special Reference to Whitehead and Russell’s Principia Mathematica. " Trans. Amer. Math. Soc. 35, 274 /C1/304, 1933. Huntington, E. V. "Boolean Algebra. A Correction." Trans. Amer. Math. Soc. 35, 557 /C1/558, 1933. Kolata, G. "Computer Math Proof Shows Reasoning Power." New York Times , Dec. 10, 1996. McCune, W. "Solution of the Robbins Problem." J. Automat. Reason. 19, 263 /C1/276, 1997. McCune, W. "Robbins Algebras are Boolean." http://www- unix.mcs.anl.gov/~mccune/papers/robbins/. Nelson, E. "Automated Reasoning." http://www.math.prin- ceton.edu/~nelson/ar.html. Wolfram Research, Inc. "Proof of the Robbins Conjecture." http://library.wolfram.com/demos/v4/Robbins.nb. Robbins Axiom The logical axiom R(x ; y) /C13!(!(x /C150y) /C150!(x /C150!y)) /C30x; where !x denotes NOT and x /C150y denotes OR, that, when taken together with associativity and commu- tativity, is equivalent to the axioms of BOOLEAN ALGEBRA . The Robbins operator can be defined in Mathematica by Robbins : /C30 Function[{x, y}, ! (! (! y \[Or] x) \[Or] ! (x \[Or] y))] That the Robbins axiom is a true statement in BOOLEAN ALGEBRA can be verified by examining its TRUTH TABLE . xy /R(x; y)/ TTT TFT FTF FFF See also ROBBINS ALGEBRA ,ROBBINS CONJECTURE , WOLFRAM AXIOMRobbins Conjecture The conjecture that the equations for a Robbins algebra, commutativity, associativity, and the ROB- BINS AXIOM !(!(x /C150y) /C150!(x /C150!y)) /C30x; where !x denotes NOT and x /C150y denotes OR, imply those for a BOOLEAN ALGEBRA . The conjecture was finally proven using a computer (McCune 1997). See also BOOLEAN ALGEBRA ,R OBBINS ALGEBRA , ROBBINS AXIOM References Kolata, G. "Computer Math Proof Shows Reasoning Power." New York Times , Dec. 10, 1996. McCune, W. "Solution of the Robbins Problem." J. Automat. Reason. 19, 263 /C1/276, 1997. McCune, W. "Robbins Algebras Are Boolean." http://www- unix.mcs.anl.gov/~mccune/papers/robbins/. Robbins Equation h(u) /C302u See also ROBBINS ALGEBRA Robbin’s Inequality If the fourth MOMENT m4 "0 ; then P( ½¯x /C28 m4 ½] l) 5m4 /C27 3(N /C28 1)s4 N3 l4 ; where s2 is the VARIANCE . Robbins Number ALTERNATING SIGN MATRIX Robbins-Monro Stochastic Approximation A STOCHASTIC APPROXIMATION method that functions by placing conditions on iterative step sizes and whose convergence is guaranteed under mild condi- tions. However, the method requires knowledge of the analytical gradient of the function under considera- tion. Kiefer and Wolfowitz (1952) developed a finite differ- ence version of the Robbins-Monro method which maintains the nice convergence properties, while obviating the need for knowledge of the analytic form of the gradient. See also STOCHASTIC APPROXIMATION ,STOCHASTIC OPTIMIZATION References Kiefer, J. and Wolfowitz, J. "Stochastic Estimation of the Maximum of a Regression Function." Ann. Math. Stat. 23, 462/C1/466, 1952. Robbins, H. and Munro, S. "A Stochastic Approximation Method." Ann. Math. Stat. 22, 400 /C1/407, 1951. Robertson Condition For the HELMHOLTZ DIFFERENTIAL EQUATION to be SEPARABLE in a coordinate system, the SCALE FACTORS hi in the LAPLACIAN 92 /C30X3 i /C3011 h1h2h3@ @uih1h2h3 h2 i@ @ui ! (1) and the functions fi(ui) and Fij defined by 1 fn@ @unfn@Xn @un ! /C27 k2 1 Fn1 /C27k22 Fn2 /C27k23 Fn3})0})@ Xn /C300 (2) must be OF THE FORM of a STA¨ CKEL DETERMINANT S /C30½Fmn ½/C30F11F12F13 F21F22F23 F31F32F33})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1/C30 h1h2h3 f1(u1)f2(u2)f3(u3) : (3) See also HELMHOLTZ DIFFERENTIAL EQUATION ,LA- PLACE’S EQUATION ,S EPARATION OF VARIABLES , STA¨ CKEL DETERMINANT References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part 1. New York: McGraw-Hill, p. 510, 1953. Robertson Conjecture A conjecture due to M. S. Robertson (1936) which treats a UNIVALENT POWER SERIES containing only ODD powers within the UNIT DISK. This conjecture IMPLIES the BIEBERBACH CONJECTURE and follows in turn from the MILIN CONJECTURE . de Branges’ proof of the BIEBERBACH CONJECTURE proceeded by proving the MILIN CONJECTURE , thus establishing the Robert- son conjecture and hence implying the truth of the BIEBERBACH CONJECTURE . See also BIEBERBACH CONJECTURE ,M ILIN CONJEC- TURE References Stewart, I. From Here to Infinity: A Guide to Today’s Mathematics. Oxford, England: Oxford University Press, p. 165, 1996.Robertson Graph The unique (4; 5)/-CAGE GRAPH , which has 19 vertices. See also CAGE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976. Robertson, N. "The Smallest Graph of Girth 5 and Valency 4." Bull. Amer. Math. Soc. 70, 824 /C1/825, 1964. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Wong, P. K. "Cages--A Survey." J. Graph Th. 6,1/C1/22, 1982. Robertson-Seymour Theorem A generalization of the KURATOWSKI REDUCTION THEOREM by Robertson and Seymour, which states that the collection of finite GRAPHS is well-quasi- ordered by minor embeddability, from which it follows that Kuratowski’s "forbidden minor" embed- ding obstruction generalizes to higher genus surfaces. Formally, for a fixed INTEGER g ]0 ; there is a finite list of graphs L(g) with the property that a GRAPH C embeds on a surface of genus g IFF it does not contain, as a minor, any of the GRAPHS on the list L. References Fellows, M. R. "The Robertson-Seymour Theorems: A Sur- vey of Applications." Comtemp. Math. 89,1/C1/18, 1987. Robertson-Wegner Graph The unique (5 ;5)/-CAGE GRAPH , which has 30 vertices. See also CAGE GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 238, 1976. Wegner, G. "A Smallest Graph of Girth 5 and Valency 5." J. Combin. Th. B 14, 203 /C1/208, 1973. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Robin Boundary Conditions PARTIAL DIFFERENTIAL EQUATION BOUNDARY CONDI- TIONS which, for an elliptic partial differential equa- tion in a region V; specify that the sum of au and the normal derivative of u /C30f at all points of the boundary of V; a and f being prescribed. Robin’s Constant TRANSFINITE DIAMETER Robinson Projection A PSEUDOCYLINDRICAL MAP PROJECTION which dis- torts shape, AREA , scale, and distance to create attractive average projection properties. See also MAP PROJECTION ,PSEUDOCYLINDRICAL PRO- JECTION References Dana, P. H. "Map Projections." http://www.colorado.edu/ geography/gcraft/notes/mapproj/mapproj_f.html. Robust Estimation An estimation technique which is insensitive to small departures from the idealized assumptions which have been used to optimize the algorithm. Classes of such techniques include M-ESTIMATES (which follow from maximum likelihood considerations), L-ESTI- MATES (which are LINEAR COMBINATIONS of ORDER STATISTICS ), and R-ESTIMATES (based on RANK tests). See also L-ESTIMATE , M-ESTIMATE , R-ESTIMATE References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Robust Estimation." §15.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 694 /C1/700, 1992. Rodrigues’ Curvature Formula d ˆN /C27 ki dr /C300 ; where ˆN is the unit NORMAL VECTOR and ki is one of the two PRINCIPAL CURVATURES . See also NORMAL VECTOR ,PRINCIPAL CURVATURES Rodrigues Formula An operator definition of a function. A Rodrigues formula may be converted into a SCHLA ¨ FLI INTEGRAL . See also RODRIGUES’ CURVATURE FORMULA ,RODRI- GUES’ ROTATION FORMULA ,SCHLA ¨ FLI INTEGRALRodrigues’ Rotation Formula This entry contributed by SERGE BELONGIE Rodrigues’ rotation formula gives an efficient method for computing the ROTATION MATRIX R /C23 SO(3) corre- sponding to a rotation by an angle u /C23R about a fixed axis specified by the unit vector v /C30(v1 ; v2 ; v3) /C23R3 : R is given by eˆvu /C301 /C27 ˆv sin u /C27 ˆv2(1 /C28cos u); where ˆv denotes the SKEW SYMMETRIC MATRIX with entries ˆv/C300/C28v3v2 v3 0/C28v1 /C28v2v1 02 435: See also R OTATION FORMULA ,ROTATION MATRIX References Brockett, R. W. "Robotic Manipulators and the Product of Exponentials Formula." In Mathematical Theory of Net- works and Systems. Proceedings of the international symposium held at the Ben Gurion University of theNegev, Beer Sheva, June 20 /C1 /24, 1983 (Ed. P. A. Fuhr- mann). Berlin: Springer-Verlag, pp. 120 /C1/127, 1984. Murray, R. M.; Li, Z.; and Sastry, S. S. A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press, 1994. Rogers L-Function If Li2(x) denotes the usual DILOGARITHM , then there are two variants that are normalized slightly differ- ently, both called the Rogers L-function (Rogers 1907). Bytsko (1999) defines L(x)/C306 p2Li2(x)/C271 2lnxln(1/C28x)hi (1) /C306 p2X/C12 n/C301xn n2/C2712lnxln(1/C28x)"# ; (2) (which he calls "the" dilogarithm), while Gordon and McIntosh (1997) and Loxton (1991, p. 287) define the Rogers L-function as LR(x)/C30Li2(x)/C271 2lnxln(1/C28x) (3) /C30p2 6L(x) (4) /C30X/C12 n/C301xn n2/C2712lnxln(1/C28x)"# : (5) The function L(x) satisfies the concise identity L(x)/C27L(1/C28x)/C301 (6) (Euler 1768), as well as A BEL’S FUNCTIONAL EQUATION L(x)/C27L(y)/C30L(xy)/C27Lx(1/C28y) 1/C28xy ! /C27Ly(1/C28x) 1/C28xy ! (7) (Abel 1988, Bytsko 1999). The duplication formula for L(x) follows from A BEL’S FUNCTIONAL EQUATION and is given by 1 2L(x2)/C30L(x)/C28Lx 1/C27x ! : (8) The function has the nice INFINITE SERIES X/C12 k/C302L1 k2 ! /C3016p2(9) (Lewin 1982; Loxton 1991, p. 298). In terms of L(x);the well-known dilogarithm identi- ties become L(0)/C300 (10) L(1/C28r)/C302 5(11) L12})@D})@E /C3012 (12) L(r)/C3035 (13) Lð1Þ¼1 ð14Þ (Loxton 1991, pp. 287 and 289; Bytsko 1999), where r/C30ffiffiffi 5p /C281})0})@ =2:/ Numbers u/C23(0;1) which satisfy Xn k/C300ckL(uk)/C300 (15) for some value of nare called L-ALGEBRAIC NUMBERS . Loxton (1991, p. 289) gives a slew of identities having rational coefficients Xn k/C300ek kL(uk)/C30c (16) instead of integers, where cis a RATIONAL NUMBER ,a corrected and expanded version of which is summar-ized in the following table. In this table, polynomialsP(x) denote the real root of x. Many more similar identities can be found using INTEGER RELATION algorithms. /u// ek/ c 11 1 /1 2/ 1 /12/ /12// /C281;6;3;0;0;/C283//12/ /13/ 3,/C2811 /1 2ffiffiffi 5p /C281})0})@ / 1 /3 5/ /12ffiffiffi 5p /C281})0})@ // 1;/C281;/C2812;0;0;6///C283 5/ /ffiffiffi 5p /C282})0})@ 1=3 / 2,/C2811 /ffiffiffi 2p /C281/ 2,/C281 /3 4/ /ffiffiffi 2p /C281// 1;2;0;/C281//5 8/ /3/C282ffiffiffi 2p / 5,/C2821 /1 2ffiffiffi 3p /C281})0})@ // 2;1;/C281//5 6/ /ffiffiffi 3p /C281// 2;/C283;/C281;0;0;1//1 2/ /2/C28ffiffiffi 3p // 4;1;0;/C281//5 4/ /2/C28ffiffiffi 3p // 5;/C283;/C281;0;0;1//4 3/ /5/C282ffiffiffi 6p // 23;/C2815;/C283;0;0;3/3 /1 2ffiffiffiffiffiffi 13p /C283})0})@ // 4;/C282;/C282;0;0;1//7 6/ /1 6ffiffiffiffiffiffi 13p /C281})0})@ // 3;1;/C283;0;0;1//4 3/ /16ffiffiffiffiffiffi 13p /C271})0})@ // 3;/C284;/C283;0;0;2//2 3/ /4/C28ffiffiffiffiffiffi 15p // 15;2;/C283;/C282//5 2/ /1 25/C28ffiffiffiffiffiffi 21p})0})@ // 7;/C281;/C283;0;0;1//5 3/ /12sec27p})@D})@E ;/ 1,/C282 /17/ /12sec17p})@D})@E / 1, 1 /57/ /2 cos3 7p})@D})@E / 1, 1 /47/ /12sec19p})@D})@E // 1;2;/C281//79/ /12sec29p})@D})@E // 1;/C283;/C281;0;0;1///C2819/ /2 cos4 9p})@D})@E // 1;/C283;/C281;0;0;1//19/ /x3/C272x/C281// 1;5;0;/C284/ 1 /x3/C272x/C281// 3;1;12;0;0;/C286/ 2 /2x3/C27x/C281// 2;1;3;/C282//3 2/ /x3/C27x/C281// 2;6;3;0;0;/C283/ 3 /x3/C283x2/C274x/C281//5;/C289;/C286;0;0;6/ 1 /x3/C27x2/C281// 1;6;6;0;0;/C286/ 2 /x3/C27x2/C27x/C281//1;1;/C283//1 2/ /x3/C27x2/C27x/C281//2;3;0;/C282//3 2/ Bytsko (1999) gives the additional identities L l /C282})0})@ /C27L l2 /C281})0})@ /C282})@D})@E /C304 7 (17) L l/C282})0})@ /C27L 1 /C27 l ðÞ/C281})@D})@E /C305 7 (18) L 1 /C281ffiffiffi 2p ! /C27Lffiffiffi 2p /C281})@D})@E /C303 4 (19) Lffiffiffirp})0})@ /C27L1 1 /C27ffiffiffirp ! /C3013 11 (20) L1 2 /C2812 r})@D})@E /C27L 2r /C281 ðÞ /C3012 (21) L 1 /C281 2 r /C2812ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 r /C283p})@D})@E /C27L1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 28r /C2745p /C282r /C282 5})@D})@E /C3025 (22) L 1 /C28 d2})0})@ /C27L (1 /C27 d)/C282})@D})@E /C302 5 (23) L3 2 /C2812ffiffiffi 2p /C281 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffi 2p /C281q })@*})@+ /C27L3 2 /C27ffiffiffi 2p})@D})@Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffiffiffiffiffiffiffiffiffi 2 /C281pq /C283 2 /C2832ffiffiffi 2p})@*})@+ /C301 2 (24) L(n) /C28L m/C281})0})@ /C301 7 (25) where l /C302 cos(p=7) r /C30ffiffiffi 5p /C281})@D})@E =2 d /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C272ffiffiffi 5pq /C281})@*})@+ ; with d the positive root of d4 /C27 d3 /C28 d /C281 /C300 (26) and 0 B n B1 and m > 1 the real roots of t6 /C287t5 /C2719t4 /C2828t3 /C2720t2 /C287t /C271 /C300: (27) Here, (17) and (18) are special cases of the WATSON IDENTITIES and (19) is a special case of ABEL’S DUPLICATION FORMULA with x /C301 =ffiffiffi 2p (Gordon and McIntosh 1997, Bytsko 1999). Rogers (1907) obtained a dilogarithm identity in m variables with m2 /C271 terms which simplifies to Euler’s identity for m /C301 and ABEL’S FUNCTIONAL EQUATION for m /C302 (Gordon and McIntosh 1997). For m /C303, it is equivalent to L(a) /C27L(b) /C27L(c) /C28L(u) /C28L(v) /C30L(abc)/C27L(ac=u)/C27L(bc=v)/C28L(av=u)/C28L(bu=v); (28)with av(1/C28bc)/C27bu(1/C28ac)/C30uv(1/C28ab) (29) v(1/C28a)/C27u(1/C28b)/C301/C28abc (30) (Gordon and McIntosh 1997). See also ABEL’S DUPLICATION FORMULA ,A BEL’S FUNCTIONAL EQUATION ,DILOGARITHM , L-ALGEBRAIC NUMBER ,LANDEN’S IDENTITY References Abel, N. H. Oeuvres Completes, Vol. 2 (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 189 /C1/192, 1988. Bytsko, A. G. J. Physics A 32, 8045, 1999. Bytsko, A. G. Two-Term Dilogarithm Identities Related to Conformal Field Theory. 9 Nov 1999. http://xxx.lanl.gov/ abs/math-ph/9911012/. Euler, L. Institutiones calculi integralis, Vol. 1. pp. 110 /C1/ 113, 1768. Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431/C1/448, 1997. Lewin, L. "The Dilogarithm in Algebraic Fields." J. Austral. Math. Soc. (Ser. A) 33, 302/C1/330, 1982. Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991. Loxton, J. H. "Partition Identities and the Dilogarithm." Ch. 13 in Structural Properties of Polylogarithms (Ed. L. Lewin). Providence, RI: Amer. Math. Soc., pp. 287 /C1/299, 1991. Rogers, L. J. "On Function Sum Theorems Connected with the Series a/C12 1xn=n2:/"Proc. London Math. Soc. 4, 169/C1/189, 1907. Watson, G. N. Quart. J. Math. Oxford Ser. 8, 39, 1937. Rogers-Ramanujan Continued Fraction The Rogers-Ramanujan continued fraction is defined by R(q)/C13q1=5 1/C27q 1/C27q2 1/C27q3 1/C27/C1/C1/C1(1) (Rogers 1894, Ramanujan 1957, Berndt et al. ). The coefficients of qnin the M ACLAURIN SERIES of R(q)=q1=5forn/C300, 1, 2, ... are 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 2, -3, ... (Sloane’s A007325). The fraction can be given explicitly as R(q)/C30q1=5q;q5ðÞ/C12q4;q5ðÞ/C12 q2;q5 ðÞ/C12q3;q5 ðÞ/C12(2) /C30q1=5Y/C12 k/C3011/C28x5k/C281})0})@ 1/C28x5k/C284})0})@ 1/C28x5k/C282 ðÞ 1/C28x5k/C283 ðÞ(3) /C30q1=5f/C28q;/C28q4ðÞ f/C28q2;/C28q3 ðÞ; (4) where /ða;qÞn/is a Q-SERIES and f(a;b)i saR AMANU- JAN THETA FUNCTION . /R(q) satisfies the amazing equalities 1 R(q)/C281/C28R(q)/C30f/C28q1=5})0})@ q1=5f/C28q5 ðÞ(5) 1 R(q) ½/C1385/C2811/C28[R(q)]5/C30f/C28qðÞ½/C1386 qf/C28q5 ðÞ½/C1386(6) as well as X/C12 n/C30/C28/C12(/C281)n(10n/C273)q(5n/C273)n=2 /C303 [R(q)]2/C27[R(q)]3"# q2=5f/C28q5})0})@})1})A3(7) X/C12 n/C30/C28/C12(/C281)n(10n/C271)q(5n/C271)n=2 /C301 [R(q)]3/C273[R(q)]2"# q3=5f/C28q5})0})@})1})A3(8) (Watson 1929ab; Berndt 1991, pp. 265 /C1/267; Berndt et al., Son). Defining u/C30R(q) (9) u?/C30/C28 R(/C28q) (10) v/C30Rq2})0})@ (11) w/C30Rq4})0})@ ; (12) these quantities satisfy the modular equations uv2/C30v/C28u2 v/C27u2(13) uw/C30w2/C28u2v w/C27u2(14) vw2/C30w/C28v2 w/C27v2(15)uu?v2/C30uu?/C28v u?/C28u(16) u?w/C30u?2/C28w v2/C27w(17) /C28vw/C30u?(v2/C28w) u?2v/C28w(18) uu?v/C30u?/C28u v/C27uu?(19) vw/C30uv2/C28w ðÞ u2v/C28w(20) (Berndt et al. ). As discussed by Hardy (1962, pp. xxvii and xxviii),Berndt and Rankin (1995), and Berndt et al. , Rama- nujan also defined the generalized continued fraction R(a;q)/C131 1/C27aq 1/C27aq2 1/C27aq3 1/C27/C1/C1/C1(21) Ramanujan also considered F(a;q)/C131/C28aq 1/C28aq2 1/C28aq3 1/C28/C1/C1/C1; (22) /C30P/C12 k/C300(/C28a)kqk2 (q)k P/C12k/C300(/C28a)kqk(k/C271) (q)k: (23) (Berndt 1991, p. 30; Berndt et al. ), of which the special case F(q)/C30F(1;q) is plotted above. Terminat- ing the terms in the continued fraction at a term aqn gives P(n/C271)=2 bc k/C300( /C28a)kqk2 (q)n/C28k/C271 (q)k Pn =2bc k /C300( /C28a)kq(k /C271)(q)n /C28k (q)k(q)n/C282k /C301 /C28aq 1 /C28aq2 1 /C28aq3 1 /C28/C1/C1/C1/C28aqn 1; (24) (Berndt et al. ). The real roots of F(q) are 0.576149, 0.815600, 0.882493, 0.913806, 0.931949, 0.943785, 0.952125, ..., the smallest of which was found by Ramanujan (Berndt et al. ). F(q) and its smallest positive root are related to the enumeration of coins in a FOUNTAIN (Berndt 1991, Berndt et al. ) and the study of birth and death processes (Berndt et al., Parthasarathy et al. 1998). In general, the least positive root q0(a)ofF(a; q) is given as a 0/C12 by q0(a) /C21 a /C281 a2 /C272 a3 /C286 a4 /C2721 a5 /C2879 a6 /C27311 a7/C281266 a8 /C275289 a9/C2822553 a10/C2797753 a11/C28... (25) (Berndt et al.). Ramanujan gave the amazing approx- imations q0(a) /C22 a /C28 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a /C27 1)(a /C27 5)p /C27O a/C288})0})@ (26) /C21 a /C28 1 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a /C27 1)(a /C27 5)p 2/C27a/C273/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(a/C271)(a/C275)p a/C281/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(a/C271)(a/C275)p"# 3 /C27Oa/C2811})0})@ : (27) See also FOUNTAIN , Q-SERIES ,R AMANUJAN THETA FUNCTIONS ,ROGERS- RAMANUJAN IDENTITIES References Andrews, G. E.; Berndt, B. C.; Jacobsen, L.; and Lamphere, R. L. The Continued Fractions Found in the Unorganized Portion of Ramanujan’s Notebooks. Providence, RI: Amer. Math. Soc., 1992. Andrews, G. On the General Rogers-Ramanujan Theorem. Providence, RI: Amer. Math. Soc., 1974. Berndt, B. C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, 1991. Berndt, B. C. "Continued Fractions." Ch. 32 in Ramanujan’s Notebooks, Part V. New York: Springer-Verlag, pp. 9 /C1/88, 1998. Berndt, B.C. and Chan, H. H. "Some Values for the Rogers- Ramanujan Continued Fraction." Canad. J. Math. 47, 897/C1/914, 1995. Berndt, B. C.; Chan, H. H.; Huang, S.-S.; Kang, S.-Y.; Sohn, J.; and Son, S. H. "The Rogers-Ramanujan Continued Fraction."Berndt, B. C.; Chan, H. H.; and Zhang, L.-C. "Explicit Evaluations of the Rogers-Ramanujan Continued Frac- tion." J. reine angew. Math. 480, 141/C1/159, 1996. Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fractionin Ramanujan’s Lost Notebook." To appears in Trans. Amer. Math. Soc. Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc, 1995. Joyce, G. S. "Exact Results for the Activity and Isothermal Compressibility of the Hard-Hexagon Model." J. Phys. A: Math. Gen. 21, L983-L988, 1988. Parthasarathy, P. R.; Lenin, R. B.; Schoutens, W.; and van Assche, W. "A Birth and Death Process Related to theRogers-Ramanujan Continued Fraction." J. Math. Anal. Appl. 224, 297/C1 /315, 1998. Ramanathan, K. G. "On Ramanujan’s Continued Fraction." Acta Arith. 43, 209/C1/226, 1984. Ramanathan, K. G. "On the Rogers-Ramanujan Continued Fraction." Proc. Indian Acad. Sci. (Math. Sci.) 93,6 7/C1/77, 1984. Ramanathan, K. G. "Ramanujan’s Continued Fraction." Indian J. Pure Appl. Math. 16, 695/C1/724, 1985. Ramanathan, K. G. "Some Applications of Kronecker’s Limit Formula." J. Indian Math. Soc. 52,7 1/C1/89, 1987. Ramanujan, S. Notebooks (2 Volumes). Bombay, India: Tata Institute, 1957. Ramanujan, S. Collected Papers. New York: Chelsea, 1962. Rogers, L. J. "Second Memoir on the Expansion of Certain Infinite Products." Proc. London Math. Soc. 25, 318/C1/343, 1894. Rogers, L. J. "On a Type of Modular Equations." Proc. London Math. Soc. 19, 387/C1/397, 1920. Sloane, N. J. A. Sequences A007325/M0415 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Watson, G. N. "Theorems Stated by Ramanujan (VII): Theorems on Continued Fractions." J. London Math. Soc. 4,3 9/C1 /48, 1929. Watson, G. N. "Theorems Stated by Ramanujan (IX): Two Continued Fractions." J. London Math. Soc. 4, 231/C1/237, 1929. Rogers-Ramanujan Identities For /jqjB1/and using the NOTATION of the R AMANUJAN THETA FUNCTION , the Rogers-Ramanujan identities are fð/C28q5Þ fð/C28q2;/C28q4Þ¼X/C12 k¼0qk2 ðqÞkð1Þ fð/C28q5Þ fð/C28q2;/C28q3Þ¼X/C12 k¼0qkðkþ1Þ ðqÞk; ð2Þ where /ðqÞk/are Q-SERIES . Written out explicitly (Hardy 1999, pp. 13 and 90), 1þq 1/C28qþq4 ð1/C28qÞð1/C28q2Þþq9 ð1/C28qÞð1/C28q2Þð1/C28q3Þþ... ¼1 ð1/C28qÞð1/C28q6Þ...ð1/C28q4Þð1/C28q9Þ... ¼1þxþx2þx3þ2x4þ2x5þ3x6þ... ð3Þ (Sloane’s A003114), and 1 þq2 1 /C28 qþq6 ð1 /C28 q Þð1 /C28 q2 Þþq12 ð1 /C28 qÞð1 /C28 q2 Þð1 /C28 q3 Þþ ... ¼1 ð1 /C28 q2 Þð1 /C28 q7 Þ...ð1 /C28 q3 Þð1 /C28 q8 Þ... ¼ 1 þ x2 þ x3 þ x4 þ x5 þ 2x6 þ ... ð4Þ (Sloane’s A003106). These identities can also be written succinctly as 1 þX/C12 k ¼1qk2 þak ð1 /C28 qÞð1 /C28 q2 Þ...ð1 /C28 qk Þ ¼Y/C12 j ¼01 ð1 /C28 q5jþa þ1 Þð1 /C28 q5j/C28a þ4 Þð5Þ where a /C30 0, 1. Other forms of the Rogers-Ramanujan identities include X kqk2 ðq; qÞk ðq; qÞn/C28k¼X kð/C281Þkqð5k2 /C28k Þ=2 ðq; qÞn/C28k ðq; qÞnþkð6Þ and X k2qk2 ðq; q Þk ðq; qÞn/C28k¼X kð/C281Þk ð1 þ qk Þqð5k2 /C28k Þ=2 ðq; qÞn/C28k ðq; qÞnþkð7Þ (Petkovsek et al. 1996). The formulas have a curious history, having been proved by Rogers (1894) in a paper that was com- pletely ignored, then rediscovered (without proof) by Ramanujan sometime before 1913. The formulas were communicated to MacMahon, who published them in his famous text, still without proof. Then, in 1917, Ramanujan accidentally found Roger’s 1894 paper while leafing through a journal. In the meantime, Schur (1917) independently rediscovered and pub- lished proofs for the identities (Hardy 1999, p. 91). Garsia and Milne (1981ab) gave the first proof of the Rogers-Ramanujan identities to construct a BIJEC- TION between the relevant classes of partitions (Andrews 1986, p. 59). Schur showed that (3) has the combinatorial inter- pretation that the number of partitions of n with minimal difference /E2/ is equal to the number of partitions into parts OF THE FORMS /5m þ 1/ or /5m þ 4/ (Hardy 1999, p. 92). The following table gives the first few values. n /an/ min. diff. //C131; 4/ (mod 5) 11 1 1 21 2 1/C27131 3 /1 þ 1 þ 1/ 4 2 4, 3 /C2714 , /1 þ 1 þ 1 þ 1/ 5 2 5, 4 /C2714 /C271,/1 þ 1 þ 1 þ 1 þ 1/ 6 3 6, 5 /C271, 4 /C2725, /4 þ 1 þ 1/,/1 þ 1 þ 1 þ 1 þ 1 þ 1/ There is a similar combinatorial interpretation for (4). A generalization of the Rogers-Ramanujan identities is given by X n1;...;nk/C281E0xN2 1þ/C1/C1/C1þ N2 k¼1þNiþ/C1/C1/C1þ Nk/C281 ðxÞn1/C1/C1/C1ðxÞnk¼1 ¼Y r¼1 ru;9iðmod 2 kþ1Þ1 1/C28xnð8Þ where /10i0k/,/kE2/,xcomplex with /jxjB1/, and / Nj¼njþ/C1/C1/C1 nk/C281/(Andrews 1984, p. 111; Fulman 1999). These identities have a number of important applications in mathematical physics (Fulman 1999). See also ANDREWS- SCHUR IDENTITY ,DOUGALL- RAMA- NUJAN IDENTITY ,SLATER’S IDENTITY References Andrews, G. E. "The Hard-Hexagon Model and Rogers- Ramanujan Type Identities." Proc. Nat. Acad. Sci. U.S.A. 78, 5290/C1/5292, 1981. Andrews, G. E. Encyclopedia of Mathematics and Its Appli- cations, Vol. 2: The Theory of Partitions. Cambridge, England: Cambridge University Press, pp. 109 and 238, 1984. Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics,and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 17 /C1 /20, 1986. Andrews, G. E. and Baxter, R. J. "A Motivated Proof of the Rogers-Ramanujan Identities." Amer. Math. Monthly 96, 401/C1/409, 1989. Andrews, G. E.; Baxter, R. J.; and Forrester, P. J. "Eight- Vertex SOS Model and Generalized Rogers-Ramanujan-Type Identities." J. Stat. Phys. 35, 193/C1 /266, 1984. Bressoud, D. M. Analytic and Combinatorial Generaliza- tions of the Rogers-Ramanujan Identities. Providence, RI: Amer. Math. Soc., 1980. Fulman, J. "The Rogers-Ramanujan Identities, The Finite General Linear Groups, and the Hall-Littlewood Polyno- mials." Proc. Amer. Math. Soc. 128,1 7/C1/25, 1999. Garsia, A. M. and Milne, S. C. "A Method for Constructing Bijections for Classical Partition Identities." Proc. Nat. Acad. Sci. USA 78, 2026/C1/2028, 1981. Garsia, A. M. and Milne, S. C. "A Rogers-Ramanujan Bijec- tion." J. Combin. Th. Ser. A 31, 289/C1/339, 1981. Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697/C1/712, 1988. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 13 and 90 /C1/99, 1999. Hardy, G. H. and Wright, E. M. "The Rogers-Ramanujan Identities." §19.13 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 290 /C1/294, 1979. MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 33 /C1/36, 1960. Paule, P. "Short and Easy Computer Proofs of the Rogers- Ramanujan Identities and of Identities of Similar Type." Electronic J. Combinatorics 1, R10 1 /C1/9, 1994. http:// www.combinatorics.org/Volume_1/volume1.html#R10. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, p. 117, 1996. Ramanujan, S. Problem 584. J. Indian Math. Soc. 6, 199 /C1/ 200, 1914. Robinson, R. M. "Comment to: ‘A Motivated Proof of the Rogers-Ramanujan Identities."’ Amer. Math. Monthly 97, 214 /C1/215, 1990. Rogers, L. J. "Second Memoir on the Expansion of Certain Infinite Products." Proc. London Math. Soc. 25, 318 /C1/343, 1894. Rogers, L. J. "On Two Theorems of Combinatory Analysis and Some Allied Identities." Proc. London Math. Soc. 16, 315 /C1/336, 1917. Rogers, L. J. "Proof of Certain Identities in Combinatory Analysis." Proc. Cambridge Philos. Soc. 19, 211 /C1/214, 1919. Schur, I. "Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbru ¨che." Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Klasse , pp. 302 /C1/321, 1917. Sloane, N. J. A. Sequences A003106/M0261, A003114/ M0266, and A006141/M0260 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Watson, G. N. "A New Proof of the Rogers-Ramanujan Identities." J. London Math. Soc. 4,4/C1/9, 1929. Watson, G. N. "Theorems Stated by Ramanujan (VII): Theorems on Continued Fractions." J. London Math. Soc. 4,39/C1/48, 1929. Roller CURVE OF CONSTANT WIDTH Rolle’s Theorem Let f be differentiable on (a, b) and continuous on [a, b]. If f(a) /C30f(b) /C300; then there is at least one point c /C23 (a; b) where f ?(c) /C300:/ See also FIXED POINT THEOREM ,M EAN-VALUE THEO- REM Rolling Polygon ROULETTE Roman Coefficient A generalization of the BINOMIAL COEFFICIENT whose NOTATION was suggested by Knuth, n k})11})17 /C30/C28 n /C27 ! /C28 k /C27 ! /C28 n /C28 k /C27 ! : (1) The above expression is read "Roman n choose k." Whenever the BINOMIAL COEFFICIENT is defined (i.e., n ]k ]0or k ]0 > n) ; the Roman coefficient agrees with it. However, the Roman coefficients are defined for values for which the BINOMIAL COEFFICIENTS are not, e.g.,n /C281})11})17 /C301 /C28 n /C27 1 /C27 (2) 0 k})11})17 /C30( /C281)k /C27(k >0) /C28 k /C27; (3) where n B0 /C131 for n B0 0 for n ]0 :})1D (4) The Roman coefficients also satisfy properties like those of the BINOMIAL COEFFICIENT , n k})11})17 /C30n n /C28k})11})17 (5) n k})11})17 k r})11})17 ¼n r})11})17 n /C28r k /C28r})11})17 ð6Þ an analog of PASCAL’S FORMULA n k})11})17 /C30n /C281 k})11})17 /C27n /C281 k /C281})11})17 ; (7) and a curious rotation/reflection law due to Knuth (/C281)k /C27(k >0) /C28n k/C281})11})17 /C30(/C281)n/C27(n>0)/C28k n/C281})11})17 (8) (Roman 1992). See also BINOMIAL COEFFICIENT ,ROMAN FACTORIAL References Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641/C1/648, 1992. Roman Factorial /C28n/C27!/C13n! for n]0 (/C281)/C28n/C281 (/C28n/C281)!fornB0:8 < :(1) The Roman factorial arises in the definition of the HARMONIC LOGARITHM and R OMAN COEFFICIENT .I t obeys the identities /C28n/C27!/C30/C28n/C27/C28n/C281/C27! (2) /C28n/C27! /C28n/C28k/C27!/C30/C28n/C27/C28n/C281/C27/C1/C1/C1/C28n/C28k/C271/C27 (3) /C28n/C27!/C28/C28n/C281/C27!/C30(/C281)n/C27(nB0); (4) where /C28n/C27/C13nforn"0 1 for n/C300})1D (5) and n B0 /C131 for n B0 0 for n ]0 :})1D (6) See also HARMONIC LOGARITHM ,HARMONIC NUMBER , ROMAN COEFFICIENT References Loeb, D. and Rota, G.-C. "Formal Power Series of Logarith- mic Type." Advances Math. 75,1/C1/118, 1989. Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641/C1/648, 1992. Roman Numeral A system of numerical notations used by the Romans. It is an additive (and subtractive) system in which letters are used to denote certain "base" numbers, andarbitrary numbers are then denoted using combina- tions of symbols. Unfortunately, little is known about the origin of the Roman numeral system (Cajori 1993,p. 30). Character Numerical Value I1V5X1 0 L5 0 C 100D 500M 1000 For example, the number 1732 would be denoted MDCCXXXII. One additional rule states that, instead of using four symbols to represent a 4, 40, 9, 90, etc., such numbers are instead denoted by preceding thesymbol for 5, 50, 10, 100, etc., with a symbolindicating subtraction. For example, 4 is denoted IV, 9 as IX, 40 as XL, etc. However, this rule isgenerally notfollowed on the faces of clocks, where IIII is usually encountered instead of IV. Further- more, the practice of placing smaller digits before large ones to indicate subtraction of value was hardlyever used by Romans and came into popularity in Europe after the invention of the printing press (Wells 1986, p. 60; Cajori 1993, p. 31). For large numbers, the Romans placed a partialframe around numbers (open at the bottom), which indicated that the framed number was to be multi-plied by 100,000, as illustrated above (Menninger 1992, p. 44; Cajori 1993, p. 32). In more recentpractice, the strokes were sometimes written onlyon the sides, e.g., ½X½(Cajori 19993, p. 32). It should also be noted that the Romans themselves neverwrote M for 1000, but instead wrote (I) for 1,000,(I)(I) for 2,000, etc., and also occasionally wrote IM,IIM, etc. (Menninger 1992, p. 281; Cajori 1993, p. 32).However, in the Middle Ages, the use of M becamequite common. The Romans sometimes used multipleparentheses to denote nested multiplications by 10, so(I) for 1,000, ((I)) for 10,000, (((I))) for 100,000, etc.(Cajori 1993, p. 33). The Romans also occasionally used a VINCULUM (called a titulus in the Middle Ages) over a Roman numeral to indicate multiplication by 1000, so ¯I/C30 1000 ;II/C302000 ;etc. (Menninger 1992, p. 281; Cajori 1993, p. 32). Roman numerals are encountered in the release year for movies and occasionally on the numerals on thefaces of watches and clocks, but in few other moderninstances. They do have the advantage that ADDITION can be done "symbolically" (and without worryingabout the "place" of a given DIGIT ) by simply combin- ing all the symbols together, grouping, writing groupsof five Is as V, groups of two Vs as X, etc. The number of characters in the Roman numerals for1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... (i.e, I, II, III, IV, V, VI, VII,VIII, IX, X, ...) are 1, 2, 3, 2, 1, 2, 3, 4, 2, 1, 2, 3, 4, ...(Sloane’s A006968). This leads to a scale-invariant FRACTAL -like stairstep pattern which rises in steps then falls abruptly. References Cajori, F. A History of Mathematical Notations, 2 vols. Bound as One, Vol. 1: Notations in Elementary Mathe- matics. New York: Dover, pp. 30 /C1/37, 1993. Menninger, K. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover, pp. 44 /C1/ 45 and 281, 1992. Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 4 /C1/5, 1969. Sloane, N. J. A. Sequences A006968/M0417 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 60 and 79, 1986. Roman Surface A QUARTIC NONORIENTABLE SURFACE , also known as the STEINER SURFACE . The Roman surface is one of the three possible surfaces obtained by sewing a MO¨ BIUS STRIP to the edge of a DISK. The other two are the BOY SURFACE and CROSS-CAP , all of which are homeomorphic to the REAL PROJECTIVE PLANE (Pin- kall 1986). The center point of the Roman surface is an ordinary TRIPLE POINT with (91; 0; 0) /C30(0;91 ; 0) /C30(0; 0;91); and the six endpoints of the three lines of self- intersection are singular PINCH POINTS , also known as WHITNEY SINGULARITIES . The Roman surface is essentially six CROSS-CAPS stuck together and con- tains a double INFINITY of CONICS . The Roman surface can given by the equation x2 /C27y2 /C27z2 /C28k2})0})@2/C30 (z /C28k)2 /C282x2hi (z /C27k)2 /C282y2hi : (1) Solving for z gives the pair of equations z /C30ky2 /C28 x2ðÞ 9 x2 /C28 y2ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 /C28 x2 /C28 y2p 2(x2 /C27 y2) : (2) If the surface is rotated by 458 about the Z-AXIS via the ROTATION MATRIX Rz(45/C14) /C301ffiffiffi 2p110 /C28110 0012 435 (3) to give x? y? z ?2435/C30R z(45 /C14)x y z2 435; (4) then the simple equation x 2y2 /C27x2z2 /C27y2z2 /C272kxyz /C300 (5) results. The Roman surface can also be generated using the general method for NONORIENTABLE SUR- FACES using the polynomial function f(x ; y; z) /C30(xy ; yz ; zx) (6) (Pinkall 1986). Setting x /C30cos u sin v (7) y /C30sin u sin v (8)z /C30cos v (9) in the former gives x(u; v) /C301 2sin(2 u) sin2 v (10) y(u; v) /C301 2sin u cos(2 v) (11) z(u ; v) /C3012cos u sin(2 v) (12) for u /C23 [0; 2p) and v /C23 [/C28p=2; p=2]: Flipping sin v and cos v and multiplying by 2 gives the form shown by Wang. A HOMOTOPY (smooth deformation) between the Ro- man surface and BOY SURFACE is given by the equations x(u; v) /C30ffiffiffi 2p cos(2 u) cos2v/C27cosusin(2 v) 2/C28affiffiffi 2p sin(3 u) sin(2 v)(13) y(u;v)/C30ffiffiffi 2p sin(2 u) cos2v/C28sinusin(2 v) 2/C28affiffiffi2p sin(3 u) sin(2 v)(14) z(u;v)/C30 3 cos2v 2/C28affiffiffi2p sin(3 u) sin(2 v)(15) foru/C23[/C28p=2;p=2] and v/C23[0;p]a savaries from 0 to 1.a/C300 corresponds to the Roman surface and a/C301t o the B OY SURFACE (Wang). See also BOY SURFACE ,CROSS- CAP,H EPTAHEDRON , MO¨ BIUS STRIP,N ONORIENTABLE SURFACE ,Q UARTIC SURFACE ,STEINER SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 19, 1986. Fischer, G. (Ed.). Plates 42 /C1/44 and 108 /C1/114 in Mathema- tische Modelle/Mathematical Models, Bildband/Photo- graph Volume. Braunschweig, Germany: Vieweg, pp. 42 /C1/44 and 108 /C1/109, 1986. Gray, A. "Steiner’s Roman Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nded.Boca Raton, FL: CRC Press, pp. 331 /C1 /333, 1997. Nordstrand, T. "Steiner’s Roman Surface." http:// www.uib.no/people/nfytn/steintxt.htm. Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 64, 1986. Roman Symbol /C28n /C27/C13n for n "0 1 for n /C300:})1D See also ROMAN FACTORIAL ,HARMONIC LOGARITHM References Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641 /C1/648, 1992. Romberg Integration A powerful NUMERICAL INTEGRATION technique which uses k refinements of the extended TRAPEZOIDAL RULE to remove error terms less than order O N /C282k})0})@ : The routine advocated by Press et al. (1992) makes use of NEVILLE’S ALGORITHM . References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 106 /C1/107, 1990. Dahlquist, G. and Bjorck, A. §7.4.1 /C1/7.4.2 in Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, 1974. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Romberg Integration." §4.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 134 /C1/135, 1992. Ralston, A. and Rabinowitz, P. §4.10 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978. Stoer, J.; and Bulirsch, R. §3.4 /C1/3.5 in Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. Ueberhuber, C. W. "Romberg Formulas." §12.3.4 in Numer- ical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, pp. 110 /C1/111, 1997. Rook Number The rook numbers rB nof an n /C29n BOARD B are the number of subsets of size n such that no two elements have the same first or second coordinate. In other word, it is the number of ways of placing n rooks on B such that none attack each other. The rook numbers of a board determine the rook numbers of the complementary board ¯B ; defined to be d /C29d_B: This is known as the ROOK RECIPROCITY THEOREM . The first few rook numbers are 1, 2, 7, 23, 115, 694, 5282, 46066, ... (Sloane’s A000903). For an n/C29nboard, each n/C29nPERMUTATION MATRIX corresponds to an allowed configuration of rooks. See also ROOK RECIPROCITY THEOREM References Sloane, N. J. A. Sequences A000903/M1761 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Rook Reciprocity Theorem Xd k/C300rB k(d/C28k)!xk/C30Xd k/C300(/C281)krBk(d/C28k)!xk(x/C271)d/C28k: References Chow, T. Y. "The Path-Cycle Symmetric Function of a Digraph." Adv. Math. 118,7 1/C1/98, 1996. Chow, T. "A Short Proof of the Rook Reciprocity Theorem." Electronic J. Combinatorics 3, R10 1 /C1/2, 1996. http:// www.combinatorics.org/Volume_3/volume3.html#R10. Goldman, J. R.; Joichi, J. T.; and White, D. E. "Rook Theory I. Rook Equivalence of Ferrers Boards." Proc. Amer. Math. Soc. 52, 485/C1/492, 1975. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1958. Rooks Problem The rook is a CHESS piece which may move any number of spaces either horizontally or vertically per move. The maximum number of nonattacking rooks which may be placed on an n/C29nCHESSBOARD isn. This arrangement is achieved by placing the rooks along the diagonal (Madachy 1979). The total number of ways of placing nnonattacking rooks on an n/C29n board is n! (Madachy 1979, p. 47). The number of rotationally and reflectively inequivalent ways of placing nnonattacking rooks on an n/C29nboard are 1, 2, 7, 23, 115, 694, ... (Sloane’s A000903; Dudeney1970, p. 96; Madachy 1979, pp. 46 /C1 /54). The minimum number of rooks needed to occupy or attack all spaces on an 8 /C298CHESSBOARD is 8 (Madachy 1979), arranged in the same orientation as above. Consider an n/C29nchessboard with the restriction that, for every subset of f1;...;ng;a rook may not be put in column s/C27j(mod n) when on row j, where the rows are numbered 0, 1, ..., n/C281:Vardi (1991) denotes the number of rook solutions so restricted as rook( s;n):rook( f1g;n) is simply the number of DERANGEMENTS onnsymbols, known as a SUBFAC- TORIAL . The first few values are 1, 2, 9, 44, 265, 1854, ... (Sloane’s A000166). rook( f1;2g;n) is a solution to the MARRIED COUPLES PROBLEM , sometimes known as ME´NAGE NUMBERS . The first few ME´NAGE NUMBERS are -1, 1, 0, 2, 13, 80, 579, ... (Sloane’s A000179). Although simple formulas are not known for general f1; ...; p g; RECURRENCE RELATIONS can be used to compute rook( f1; ...; p g; n) in polynomial time for p /C303, ..., 6 (Metropolis et al. 1969, Minc 1978, Vardi 1991). See also CHESS ,M E´ NAGE NUMBER ,ROOK NUMBER , ROOK RECIPROCITY THEOREM References Dudeney, H. E. "The Eight Rooks." §295 in Amusements in Mathematics. New York: Dover, p. 88, 1970. Kraitchik, M. "The Problem of the Rooks" and "Domination of the Chessboard." §10.2 and 10.4 in Mathematical Recreations. New York: W. W. Norton, pp. 240 /C1/247 and 255 /C1/256, 1942. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 36 /C1/37, 1979. Metropolis, M.; Stein, M. L.; and Stein, P. R. "Permanents of Cyclic (0, 1) Matrices." J. Combin. Th. 7, 291 /C1/321, 1969. Minc, H. §3.1 in Permanents. Reading, MA: Addison-Wesley, 1978. Riordan, J. Chs. 7 /C1/8in An Introduction to Combinatorial Analysis. Princeton, NJ: Princeton University Press, 1978. Sloane, N. J. A. Sequences A000903/M1761, A000166/ M1937, and A000179/M2062 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 123 /C1/124, 1991. Room Square A Room square (named after T. G. Room) of order n (for n EVEN ) is an arrangement in an (n /C281) /C29(n /C281) SQUARE MATRIX of n objects such that each cell is either empty or holds exactly two different objects. Furthermore, each object appears once in each row and column and each unordered pair occupies exactly one cell. The Room square of order 2 is shown below. 1,2 The Room square of order 8 is 1,8 5,7 3,4 2,6 3,7 2,8 6,1 4,5 5,6 4,1 3,8 7,2 6,7 5,2 4,8 1,3 2,4 7,1 6,3 5,8 3,5 1,2 7,4 6,8 4,6 2,3 1,5 7,8References Dinitz, J. H. and Stinson, D. R. In Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, 1992. Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102 /C1/107, May 1975. Gardner, M. Time Travel and Other Mathematical Bewil- derments. New York: W. H. Freeman, pp. 146 /C1/147 and 151 /C1/152, 1988. Mullin, R. C. and Nemeth, E. "On Furnishing Room Squares." J. Combin. Th. 7, 266 /C1/272, 1969. Mullin, R. D. and Wallis, W. D. "The Existence of Room Squares." Aequationes Math. 13,1/C1/7, 1975. O’Shaughnessy, C. D. "On Room Squares of Order /6m þ 2/." J. Combin. Th. 13, 306 /C1/314, 1972. Room, T. G. "A New Type of Magic Square" (Note 2569). Math. Gaz. 39, 307, 1955. Wallis, W. D. "Solution of the Room Square Existence Problem." J. Combin. Th. 17, 379 /C1/383, 1974. Wallis, W. D.; Street, A. P.; and Wallis, J. S. Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices. New York: Springer-Verlag, 1972. Root The roots (sometimes also called "zeros") of an equation f(x) /C300 (1) are the values of x for which the equation is satisfied. The FUNDAMENTAL THEOREM OF ALGEBRA states that every POLYNOMIAL equation of degree n has exactly n roots, where some roots may have a multiplicity greater than 1 (in which case they are said to be degenerate). In Mathematica , the expression Root [f, k] represents the kth root of the POLYNOMIAL f(x) /C300:/ To find the nth roots of a COMPLEX NUMBER , solve the equation zn/C30w:Then zn/C30½z½n[cos(nu)/C27isin(nu)]/C30½w½(cosf/C27isinf);(2) so ½z½/C30½w½1=n(3) and arg(z)/C30f n: (4) Rolle proved that any number has nnth roots (Boyer 1968, p. 476). Householder (1970) gives an algorithm for constructing root-finding algorithms with anarbitrary order of convergence. Special root-finding techniques can often be applied when the function in question is a POLYNOMIAL . See also BAILEY’S METHOD ,B ERNOULLI’S METHOD , BISECTION PROCEDURE ,B RENT’S METHOD ,C ROUT’S METHOD ,D ESCARTES’ SIGN RULE,FALSE POSITION METHOD ,F UNDAMENTAL THEOREM OF SYMMETRIC FUNCTIONS ,G RAEFFE’S METHOD ,H ALLEY’S IRRA- TIONAL FORMULA ,H ALLEY’S METHOD ,H ALLEY’S RA- TIONAL F ORMULA ,H ORNER’S M ETHOD , HOUSEHOLDER’S METHOD ,H UTTON’S METHOD ,IN- SIDE- OUTSIDE THEOREM ,ISOGRAPH ,JENKINS- TRAUB METHOD ,LAGUERRE’S METHOD ,LAMBERT’S METHOD , LEHMER- SCHUR METHOD ,LIN’S METHOD ,M AEHLY’S PROCEDURE ,MULLER’S METHOD ,MULTIPLICITY ,NEW- TON’S METHOD ,P OLYNOMIAL ,P OLYNOMIAL ROOTS , RIDDERS’ METHOD ,ROOT DRAGGING THEOREM ,ROOT EXTRACTION ,ROUCHE ´ ’S THEOREM ,SCHRO ¨ DER’S METH- OD,SECANT METHOD ,SIMPLE ROOT,STURM FUNC- TION ,S TURM THEOREM ,T ANGENT HYPERBOLAS METHOD ,VANISH ,WEIERSTRASS APPROXIMATION THE- OREM ,ZERO SET References Arfken, G. "Appendix 1: Real Zeros of a Function." Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 963 /C1/967, 1985. Boyer, C. B. A History of Mathematics. New York: Wiley, 1968. Householder, A. S. The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, 1970. Kravanja, P. and van Barel, M. Computing the Zeros of Analytic Functions. Berlin: Springer-Verlag, 2000. McNamee, J. M. "A Bibliography on Roots of Polynomials." J. Comput. Appl. Math. 47, 391 /C1/392, 1993. McNamee, J. M. "A Bibliography on Roots of Polynomials." http://www.elsevier.com/homepage/sac/cam/mcnamee/. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Roots of Polynomials." §9.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 362 /C1/372, 1992. Whittaker, E. T. and Robinson, G. "The Numerical Solution of Algebraic and Transcendental Equations." Ch. 6 in The Calculus of Observations: A Treatise on Numerical Mathe- matics, 4th ed. New York: Dover, pp. 78 /C1/131, 1967. Root (Lie Algebra) The roots of a SEMISIMPLE LIE ALGEBRA g are the WEIGHTS occurring in its ADJOINT REPRESENTATION . The set of roots form the ROOT SYSTEM , and are completely determined by g: It is possible to choose a set of POSITIVE ROOTS , every root a is either positive or /C28a is positive. The SIMPLE ROOTS are the positive roots which cannot be written as a sum of positive roots. The simple roots can be considered as a LINEARLY INDEPENDENT finite subset of EUCLIDEAN SPACE , and they generate the ROOT LATTICE . For example, in the SPECIAL LIE ALGEBRA sl2C of two by two matrices with zero TRACE , has a basis given by the matrices H /C3010 0 /C281})10})1@ ; X /C300100})10})1@ ; Y /C300010})10})1@ : The ADJOINT REPRESENTATION is given by the BRACK- ETS ad(H(X)) /C30[H ; X] /C302X ad(H(Y)) /C30[H ; Y] /C30/C282Y ;so there are two roots of sl2given by a(H) /C302 and /C28a(H) /C30/C282: The RANK of sl2C is one, and it has one positive root. See also CARTAN MATRIX ,LIE ALGEBRA ,SEMISIMPLE LIE ALGEBRA ,W EIGHT (LIE ALGEBRA ), WEYL GROUP References Fulton, W. and Harris, J. Representation Theory. New York:Springer-Verlag, 1991. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Root (Radical) The nth root (or "nth RADICAL ") of a quantity z is a value r such that z /C30rn ; and therefore is the INVERSE FUNCTION to the taking of a POWER . The nth root is denoted r /C30ffiffiffizpor, using POWER notation, r /C30z1 =n : The special case of the SQUARE ROOT is denotedffiffiffizp: / The quantities for which a general FUNCTION equals 0 are also called ROOTS , or sometimes ZEROS . See also CUBE ROOT,RADICAL ,ROOT,SQUARE ROOT, VINCULUM Root (Tree) ROOT NODE Root Dragging Theorem If any of the ROOTS of a POLYNOMIAL are increased, then all of the critical points increase. References Anderson, B. "Polynomial Root Dragging." Amer. Math. Monthly 100, 864 /C1/866, 1993. Root Extraction The operation of taking an nth ROOT of a number. See also ADDITION ,DIVISION ,MULTIPLICATION ,ROOT (RADICAL ), SUBTRACTION Root Lattice The root lattice of a SEMISIMPLE LIE ALGEBRA is the DISCRETE LATTICE generated by the ROOTS in h/C31; the DUAL SPACE to the CARTAN SUBALGEBRA . See also CARTAN MATRIX ,LIE ALGEBRA ,ROOT (LIE ALGEBRA ), ROOT SYSTEM ,SEMISIMPLE LIE ALGEBRA , WEIGHT (LIE ALGEBRA ), WEIGHT LATTICE ,W EYL CHAMBER ,W EYL GROUP References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Root Linear Coefficient Theorem The sum of the reciprocals of ROOTS of an equation equals the NEGATIVE COEFFICIENT of the linear term in the MACLAURIN SERIES . See also NEWTON’S RELATIONS Root Node A special node which is designated to turn a TREE into a ROOTED TREE . The root is sometimes also called "EVE"oran" ENDPOINT " (Saaty and Kainen 1986, p. 30) and each of the nodes which is one EDGE further away from a given EDGE is called a CHILD . Nodes connected to the same node are then called SIBLINGS . See also CHILD ,ROOTED TREE,SIBLING ,TREE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 187, 1994. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986. Root of Unity The nth ROOTS of UNITY are ROOTS e2 pik=nof the CYCLOTOMIC EQUATION xn /C301 ; which are known as the DE MOIVRE NUMBERS . The notations zk ; ek ; and ekare variously used to denote the kth nth root of unity. //C271 is always an nth root of unity, but /C281 is such a root only if n is even. See also CYCLOTOMIC EQUATION ,CYCLOTOMIC POLY- NOMIAL , DE MOIVRE’S IDENTITY , DE MOIVRE NUMBER , PRIMITIVE ROOT OF UNITY ,P RINCIPAL ROOT OF UNITY,UNITY References Courant, R. and Robbins, H. "De Moivre’s Formula and the Roots of Unity." §5.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 98 /C1/100, 1996. Lam, T. Y. and Leung, K. H. "On Vanishing Sums of Roots of Unity." J. Algebra 224,91/C1/109, 2000. Nagell, T. "Arithmetical Properties of the Roots of Unity." Ch. 5 in Introduction to Number Theory. New York: Wiley, pp. 156 /C1/187, 1951. Root System Let E be a Euclidean space, (b; a) be the dot product, and denote the reflection in the hyperplane Pa /C30fb /C23 E ½( b; a) /C300g by sa( b) /C30 b /C282 b; aðÞ =( a; a) a /C30 b /C28/C142b; a/C143a; whereb; ahi/C302(b; a) ( a; a): Then a subset R of the Euclidean space E is called a root system in E if: 1. R is finite, SPANS E, and does not contain 0, 2. If a /C23 R; the reflection sa leaves R invariant, and 3. If a; b /C23 R; then /C142 b; a/C143/C23Z:/ The ROOTS of a SEMISIMPLE LIE ALGEBRA are a root system, in a real subspace of the DUAL SPACE to the CARTAN SUBALGEBRA . In this case, the reflections Wa generate the WEYL GROUP , which is the symmetry group of the root system. See also CARTAN MATRIX ,L IE ALGEBRA ,M ACDO- NALD’S CONSTANT- TERM CONJECTURE ,R EDUCED ROOT SYSTEM ,ROOT (LIE ALGEBRA ), SEMISIMPLE LIE ALGEBRA ,W EIGHT (LIE ALGEBRA ), WEYL CHAMBER , WEYL’S DENOMINATOR FORMULA ,W EYL GROUP References Andrews, G. E. q-Series: Their Development and Applica- tion in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 40, 1986. Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Humphrey, J. E. Introduction to Lie Algebras and Repre- sentation Theory. New York: Springer-Verlag, p. 42, 1972. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Root Test Let uk be a SERIES with POSITIVE terms, and let r /C13lim k 0/C12u1 =k k: 1. If r B1 ; the SERIES CONVERGES . 2. If r > 1or r /C30/C12; the SERIES DIVERGES . 3. Ifr/C301;the SERIES may CONVERGE orDIVERGE . This test is also called the Cauchy root test. See also CONVERGENCE TESTS References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 281 /C1/282, 1985. Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 31 /C1/39, 1991. Rooted Tree A TREE with a single special ("labeled"rpar; node called the "ROOT " or "eve." A tree which is not rooted is sometimes called a FREE TREE . Denote the number of rooted trees with n nodes by Tn ; then the GENERATING FUNCTION is T(x) /C13X/C12 n/C300Tnxn /C30x /C27x2 /C272x3 /C274x4 /C279x5 /C2720x6 /C2748x7 /C27115x8 /C27286x9 /C27719x10 /C27... (1) (Sloane’s A000081). This POWER SERIES satisfies T(x) /C30x expX/C12 r/C3011 rTxrðÞ"# (2) t(x) /C30T(x) /C281 2T2(x) /C28Tx2})0})@})1})A ; (3) where t(x) is the GENERATING FUNCTION for unrooted TREES .AGENERATING FUNCTION for Tn can be written using a product involving the sequence itself as xY/C12 n/C3011 1 /C28 xn ðÞTn/C30X/C12 n/C301Tnxn : (4) The number of rooted trees can also be calculated from the RECURRENCE RELATION Ti/C271 /C301 iXi j/C301X d ½jdTd ! Ti/C28j/C271 ; (5) with T0 /C300 and T1 /C301; where the second sum is over all d which DIVIDE j (Finch). See also ORDERED TREE,PLANTED TREE,RED-BLACK TREE,W EAKLY BINARY TREE References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/otter/otter.html. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 187 /C1/190 and 232, 1994. Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Ruskey, F. "Information on Rooted Trees." http://www.theor- y.csc.uvic.ca/~cos/inf/tree/RootedTree.html.Sloane, N. J. A. Sequences A000081/M1180 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wilf, H. S. Combinatorial Algorithms: An Update. Philadel- phia, PA: SIAM, 1989. Root-Mean-Square The root-mean-square (RMS) of a variate x, some- times called the QUADRATIC MEAN , is the SQUARE ROOT of the mean squared value of x: R(x) /C13ffiffiffiffiffiffiffiffiffi x2hip (1) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i/C301 x2 i ns for a discrete distribution ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g P(x)x2 dx g P(x) dxvuuuuut for a continuous distribution :8 >>>>>>>>< >>>>>>>>: (2) Hoehn and Niven (1985) show that Ra 1 /C27c ; a2 /C27c ; ...; an /C27c ðÞ Bc /C27Ra1 ; a2 ;...; an ðÞ (3) for any POSITIVE constant c. Physical scientists often use the term root-mean- square as a synonym for STANDARD DEVIATION when they refer to the SQUARE ROOT of the mean squared deviation of a signal from a given baseline or fit. See also ARITHMETIC- GEOMETRIC MEAN,ARITHMETIC- HARMONIC MEAN,G ENERALIZED MEAN,G EOMETRIC MEAN,H ARMONIC MEAN,H ARMONIC- GEOMETRIC MEAN,M EAN,M EDIAN (STATISTICS ), STANDARD DE- VIATION ,VARIANCE References Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151/C1/156, 1985. Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 inMathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 59 /C1/60, 1962. RootSum POLYNOMIAL ROOTS Rosatti’s Theorem There is a one-to-one correspondence between the sets of equivalent correspondences (not of value 0) onan irreducible curve of GENUS (CURVE )p, and the rational COLLINEATIONS of a projective space of 2 p/C281 dimensions which leave invariant a space of p/C281 dimensions. The number of linearly independentcorrespondences will be that of linearly independent COLLINEATIONS . References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 339, 1959. Rose A curve which has the shape of a petalled flower. This curve was named RHODONEA by the Italian mathe- matician Guido Grandi between 1723 and 1728 because it resembles a rose (MacTutor Archive). The polar equation of the rose is r /C30a sin(nu); or r /C30a cos(nu) : If n is ODD, the rose is n-petalled. If n is EVEN , the rose is 2n/-petalled. If n is IRRATIONAL , then there are an infinite number of petals. The QUADRIFOLIUM is the rose with n /C302. The rose is the RADIAL CURVE of the EPICYCLOID . See also DAISY,MAURER ROSE,STARR ROSE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 223 /C1/224, 1987. Hall, L. "Trochoids, Roses, and Thorns--Beyond the Spiro- graph." College Math. J. 23,20/C1/35, 1992. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 175 /C1/177, 1972. MacTutor History of Mathematics Archive. "Rhodonea Curves." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Rhodonea.html. Wagon, S. "Roses." §4.1 in Mathematica in Action. New York: W. H. Freeman, pp. 96 /C1/102, 1991. Rosenbrock Function The function f(x; y) /C30(1 /C28x)2 /C27105 y /C28x2})0})@2that is often used as a test problem for optimization algorithms. It has a global minimum of 0 at the point (1, 1). References Germundsson, R. "Mathematica Version 4." Mathematica J. 7, 497 /C1/524, 2000. Rosenbrock Methods A generalization of the RUNGE- KUTTA METHOD for solution of ORDINARY DIFFERENTIAL EQUATIONS , also called KAPS-RENTROP METHODS . See also RUNGE- KUTTA METHOD References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 730 /C1/735, 1992. Ro¨ssler Model The nonlinear 3-D MAP ˙X/C30(/C28Y/C27Z) ˙Y/C30X/C27aY ˙Z/C30b/C27XZ/C28cZ: See also LORENZ SYSTEM References Dickau, R. M. "Ro ¨ssler Attractor." http://forum.swarthmor- e.edu/advanced/robertd/rossler.html. Peitgen, H.-O.; Ju ¨rgens, H.; and Saupe, D. §12.3 in Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, pp. 686 /C1/696, 1992. RotateLeft CYCLIC PERMUTATION RotateRight CYCLIC PERMUTATION Rotation The turning of an object or coordinate system by an ANGLE about a fixed point. A rotation is an ORIENTA- TION-PRESERVING ORTHOGONAL TRANSFORMATION .EU- LER’S ROTATION THEOREM states that an arbitrary rotation can be parameterized using three para- meters. These parameters are commonly taken as the EULER ANGLES . Rotations can be implemented using ROTATION MATRICES . The rotation SYMMETRY OPERATION for rotation by 360/C14=n is denoted "n." For periodic arrangements of points (, the CRYSTALLOGRAPHY RESTRICTION gives the only allowable rotations as 1, 2, 3, 4, and 6. See also DILATION ,EUCLIDEAN GROUP ,EULER AN- GLES ,EULER PARAMETERS ,EULER’S ROTATION THEO- REM,EXPANSION ,H ALF-TURN,IMPROPER ROTATION , INFINITESIMAL ROTATION ,INVERSION OPERATION , MIRROR PLANE ,O RIENTATION- PRESERVING ,O RTHO- GONAL TRANSFORMATION ,R EFLECTION ,R OTATION FORMULA ,R OTATION GROUP ,R OTATION MATRIX , ROTATION OPERATOR ,ROTOINVERSION ,SHIFT,SPIRAL SIMILARITY ,TRANSLATION References Addington, S. "The Four Types of Symmetry in the Plane." http://forum.swarthmore.edu/sum95/suzanne/symsu- san.html. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 211, 1987. Coxeter, H. S. M. and Greitzer, S. L. "Rotation." §4.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 82 /C1/85, 1967. Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Rotations of Coordinate Systems." §1.4 in Quantum Theory of Angular Momentum. Singapore: World Scien- tific, pp. 21 /C1/35, 1988. Yates, R. C. "Instantaneous Center of Rotation and the Construction of Some Tangents." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 119 /C1/122, 1952.Rotation Formula A formula which transforms a given coordinate system by rotating it through a counterclockwise angle F about an axis ˆn : This formula is used implicitly to transform objects in VRML (virtual reality markup language) using the command Rota- tion {angle nx ny nz Phi}. Referring to the above figure (Goldstein 1980), the equation for the "fixed" vector in the transformed coordinate system (i.e., the above figure corresponds to an ALIAS TRANSFORMA- TION ), is r?/C30!ON /C27!NV /C27!VQ (1) ¼ ˆn(ˆn /C215 r) /C27[r /C28ˆn(ˆn /C215 r)] cos F/C27(r /C29ˆn) sin F (2) /C30r cos F/C27ˆn(ˆn /C215 r)(1 /C28cos F) /C27(r /C29ˆn) sin F (3) (Goldstein 1980; Varshalovich et al. 1988, p. 24). The ANGLE Fand unit normal ˆnmay also be expressed as EULER ANGLES . In terms of the E ULER PARAMETERS , r?/C30re2 0/C28e21/C28e22/C28e23})0})@ /C272e(e/C215r)/C272(r/C29e)e0:(4) See also ALIAS TRANSFORMATION ,ALIBI TRANSFORMA- TION ,E ULER ANGLES ,E ULER PARAMETERS ,R ODRI- GUES’ ROTATION FORMULA References Gibbs, J. W. and Wilson, E. B. Vector Analysis: A Text-Book for the use of Students of Mathematics and Physics, Founded Upon the Lectures of J. Willard Gibbs. New York: Dover, p. 338, 1960. Goldstein, H. "Finite Rotations." §4/C1/7i n Classical Me- chanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 164 /C1/166, 1980. Grubin, C. "Derivation of the Quaternion Scheme via the Euler Axis and Angle." J. Spacecraft 7, 1251/C1/1263, 1970. Hamel, G. Theoretische Mechanik: Eine Einheitliche Ein- fu¨hrung in die Gesamte Mechanik. Berlin: New York: Springer-Verlag, p. 103, 1949. Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Description of Rotations in Terms of Rotation Axisand Rotation Angle." §1.4.2 in Quantum Theory of Angular Momentum. Singapore: World Scientific, pp. 23 /C1 /24, 1988. Rotation Group There are three REPRESENTATIONS of the rotation groups, corresponding to EXPANSION /CONTRACTION , ROTATION , and SHEAR . See also ROTATION MATRIX ,SPECIAL ORTHOGONAL GROUP Rotation Matrix When discussing a ROTATION , there are two possible conventions: rotation of the axes and rotation of the object relative to fixed axes. InR2;let a curve be rotated by a clockwise ANGLE u;so that the original axes of the curve are ˆxand ˆy;and the new axes of the curve are ˆx?and ˆy?:The MATRIX transforming the original curve to the rotated curve, referred to the original ˆxand ˆyaxes, is Ru/C30cosusinu /C28sinucosu})10})1@ ; (1) i.e., x/C30Rux?: (2) On the other hand, let the axes with respect to which a curve is measured be rotated by a clockwise ANGLE u;so that the original axes are ˆx0and ˆy0;and the new axes are ˆxand ˆy:Then the MATRIX transforming the coordinates of the curve with respect to ˆxand ˆyis given by the MATRIX TRANSPOSE of the above matrix: R?u/C30cosu/C28sinu sinucosu})10})1@ ; (3) i.e., x/C30R?ux0: (4) InR3;rotations of the x-,y-, and Z-AXES give the matrices Rx(a)/C3010 0 0 cos asina 0/C28sinacosa2 435 (5)R y(b)/C30cosb0/C28sinb 01 0 sinb0 cos b2435 (6) R z(g)/C30cosgsing0 /C28singcosg0 00 12 435: (7) Any ROTATION can be given as a composition of rotations about three axes (E ULER’S ROTATION THEO- REM), and thus can be represented by a 3 /C293MATRIX operating on a VECTOR , x?1 x?2 x?32 435/C30a 11a12a13 a21a22a23 a31a32a332435x 1 x2 x32435: (8) We wish to place conditions on this matrix so that it is consistent with an ORTHOGONAL TRANSFORMATION (basically, a ROTATION orROTOINVERSION ). In a ROTATION ,a VECTOR must keep its original length, so it must be true that x?ix?i/C30xixi (9) fori/C301, 2, 3, where E INSTEIN SUMMATION is being used. Therefore, from the transformation equation, (aijxj)(aikxk)/C30xixi: (10) This can be rearranged to aij(xjaik)xk/C30aij(aikxj)xk /C30aijaikxjxk/C30xixi: (11) In order for this to hold, it must be true that aijaik/C30djk (12) forj;k/C301;2, 3, where dijis the K RONECKER DELTA . This is known as the ORTHOGONALITY CONDITION , and it guarantees that A/C281/C30AT; (13) and ATA/C30I; (14) where ATis the MATRIX TRANSPOSE and lis the IDENTITY MATRIX . Equation (14) is the identity which gives the orthogonal matrix its name. Orthogonalmatrices have special properties which allow them tobe manipulated and identified with particular ease. Let Aand Bbe two orthogonal matrices. By the ORTHOGONALITY CONDITION , they satisfy aijaik/C30djk; (15) and bijbik/C30djk; (16) where dijis the K RONECKER DELTA . Now cijcik /C30(ab)ij(ab)jk /C30aisbsjaitbtk /C30aisaitbsjbtk ¼ dstbsjbtk /C30btjbtk /C30 djk ; (17) so the product C /C13AB of two orthogonal matrices is also orthogonal. The EIGENVALUES of an orthogonal matrix must satisfy one of the following: 1. All EIGENVALUES are 1. 2. One EIGENVALUE is 1 and the other two are /C281. 3. One EIGENVALUE is 1 and the other two are COMPLEX CONJUGATES OF THE FORM eiu and e /C28iu :/ An orthogonal MATRIX A is classified as proper (corresponding to pure ROTATION )if det(A) /C301; (18) where det(A) is the DETERMINANT of A ; or improper (corresponding to inversion with possible rotation; ROTOINVERSION )if det(A) /C30/C281: (19) See also EULER ANGLES ,EULER PARAMETERS ,EU- LER’S ROTATION THEOREM ,R OTATION ,R OTATION FORMULA Rotation Number The period for a QUASIPERIODIC trajectory to pass through the same point in a SURFACE OF SECTION .If the rotation number is IRRATIONAL , the trajectory will densely fill out a curve in the SURFACE OF SECTION .If the rotation number is RATIONAL , it is called the WINDING NUMBER , and only a finite number of points in the SURFACE OF SECTION will be visited by the trajectory. See also QUASIPERI ODIC FUNCTION ,S URFACE OF SECTION ,W INDING NUMBER (MAP) Rotation Operator The rotation operator can be derived from examining an INFINITESIMAL ROTATION d dt ! space/C30d dt ! body/C27v/C29; where d=dt is the time derivative, v is the ANGULAR VELOCITY , and /C29 is the CROSS PRODUCT operator. See also ACCELERATION ,A NGULAR ACCELERATION , INFINITESIMAL ROTATION Roth’s Removal Rule If the matrices A ; X ; B; and C satisfy AX /C28XB /C30C;then IX 0I})10})1@ AC 0B})10})1@ I /C28X 0I})10})1@ /C30A0 0B})10})1@ ; where I is the IDENTITY MATRIX . References Roth, W. E. "The Equations AX /C28YB /C30C and AX /C28XB /C30C in Matrices." Proc. Amer. Math. Soc. 3, 392 /C1/396, 1952. Turnbull, H. W. and Aitken, A. C. An Introduction to the Theory of Canonical Matrices. New York: Dover, p. 422, 1961. Roth’s Theorem For ALGEBRAIC a a/C28p q})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1B 1 q2 /C27 e ; with e > 0; has finitely many solutions. Klaus Roth received a FIELDS MEDAL for this result. See also HURWITZ EQUATION ,HURWITZ’S IRRATIONAL NUMBER THEOREM ,IRRATIONALITY MEASURE ,L A- GRANGE NUMBER (RATIONAL APPROXIMATION ), LIOU- VILLE’S APPROXIMATION THEOREM ,MARKOV NUMBER , SEGRE’S THEOREM ,SIEGEL’S THEOREM ,THUE- SIEGEL- ROTH THEOREM References Davenport, H. and Roth, K. F. "Rational Approximations to Algebraic Numbers." Mathematika 2, 160 /C1/167, 1955. Roth, K. F. "Rational Approximations to Algebraic Num- bers." Mathematika 2,1/C1/20, 1955. Roth, K. F. "Corrigendum to ‘Rational Approximations to Algebraic Numbers’." Mathematika 2, 168, 1955. Rotkiewicz Theorem If n /C2119, there exists a POULET NUMBER between n and n2 : The theorem was proved in 1965. See also POULET NUMBER References Rotkiewicz, A. "Les intervalles contenants les nombres pseudopremiers." Rend. Circ. Mat. Palermo Ser. 2 14, 278/C1/280, 1965. Rotkiewicz, A. "Sur les nombres de Mersenne de ´pourvus de diviseurs carre ´s et sur les nombres naturels n, tel que n2/C282n/C282:/"Mat. Vesnik 2 (17) ,7 8/C1/80, 1965. Rotkiewicz, A. "Sur les nombres pseudopremiers carre ´s." Elem. Math. 20,3 9/C1/40, 1965. Rotoinversion IMPROPER ROTATION Rotor A convex figure that can be rotated inside a POLYGON (or POLYHEDRON ) while always touching every side (or face). The least AREA rotor in a SQUARE is the REULEAUX TRIANGLE . The least AREA rotor in an EQUILATERAL TRIANGLE is a LENS with two 608 ARCS of CIRCLES and RADIUS equal to the TRIANGLE ALTI- TUDE . There exist nonspherical rotors for the TETRAHEDRON , OCTAHEDRON , and CUBE , but not for the DODECAHE- DRON and ICOSAHEDRON . See also DELTA CURVE ,LENS,REULEAUX POLYGON , REULEAUX TRIANGLE ,ROULETTE ,TRIP-LET References Gardner, M. The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: Chicago University Press, p. 219, 1991. Goldberg, M. "Circular-Arc Rotors in Regular Polygons." Amer. Math. Monthly 55, 392 /C1/402, 1948. Goldberg, M. "Two-Lobed Rotors with Three-Lobed Stators." J. Mechanisms 3,55/C1/60, 1968. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 151 /C1/152, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 221 /C1/222, 1991. Rotunda A POLYHEDRON consisting of a n-gon, a parallel 2n/- gon rotated a half-edge turn, and a band of paired triangles separated by pentagons. The only true member giving a polyhedron consisting of all regular polygons with unit edge lengths is the PENTAGONAL ROTUNDA . It corresponds to half of an ICOSIDODECA- HEDRON . See also ELONGATED ROTUNDA ,G YROELONGATEDROTUNDA ,ICOSIDODECAHEDRON ,P ENTAGONAL RO- TUNDA ,TRIANGULAR HEBESPHENOROTUNDA References Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169 /C1/200, 1966. Rouche ´’s Theorem Given two functions f and g ANALYTIC in A with g a simple loop HOMOTOPIC to a point in A,if½g(z) ½B½f(z) ½ for all z on g ; then f and f /C27g have the same number of ROOTS inside g :/ A stronger version has been proved by Estermann (1962). The strong version also has a converse, as shown by Challener and Rubel (1982). See also ARGUMENT PRINCIPLE References Challener, D. and Rubel, L. "A Converse to Rouche ´’s Theorem." Amer. Math. Monthly 89, 302/C1/305, 1982. Estermann, T. Complex Numbers and Functions. London: Oxford University Press, p. 156, 1962. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, p. 111, 1996. Krantz, S. G. "Rouche ´’s Theorem." §5.3.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 74, 1999. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 22, 1975. Roulette The curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the FOCI of CONICS when rolled upon a line are sections of MINIMAL SURFACES (i.e., they yield MINIMAL SURFACES when revolved about the line) known as UNDULOIDS . R/C28 R/C27 A particularly interesting case of a roulette is a regular n-gon rolling on a "road" composed of a sequence of truncated catenaries, as illustratedabove. This motion is smooth in the sense that the CENTROID follows a straight line, although in the case of the rolling EQUILATERAL TRIANGLE , a physical model would be impossible to construct (Wagon 1991). For the rolling SQUARE , the shape of the road is the CATENARY y /C30/C28cosh x truncated at x /C30 9sinh/C281 1 (Wagon 1991). For a regular n-gon, the Cartesian equation of the corresponding CATENARY is y /C30/C28A coshx A ! ; (1) where A /C13R cosp n ! : (2) Curve 1 Curve 2 Pole Roulette CIRCLE exterior CIRCLEon CIR- CUM- FERENCEEPICYCLOID CIRCLE interior CIRCLEon CIR- CUM-FERENCEHYPOCYCLOID CIRCLE LINE on CIR- CUM- FERENCECYCLOID CIRCLE same CIRCLEany pointROSE CIRCLE INVOLUTELINE CENTER PARABOLA CYCLOID LINE center ELLIPSE ELLIPSE LINE FOCUS elliptic catenary HYPERBOLA LINE FOCUS hyperbolic ca- tenary HYPERBOLIC SPIRALLINE ORIGIN TRACTRIX LINE any curve on LINE INVOLUTE of the curve LOGARITHMIC SPIRALLINE any pointLINE PARABOLA equal PARABOLAVERTEX CISSOID OF DIOCLES PARABOLA LINE FOCUS CATENARY See also CATENARY ,D ELTA CURVE ,G LISSETTE ,RE- ULEAUX POLYGON ,R EULEAUX TRIANGLE ,R OTOR , UNDULOID References Besant, W. H. Notes on Roulettes and Glissettes, 2nd enl. ed. Cambridge, England: Deighton, Bell & Co., 1890.Cundy, H. and Rollett, A. "Roulettes and Involutes." §2.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 46 /C1/55, 1989. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 128, 1984. Hall, L. and Wagon, S. "Mathematical Roads and Wheels." Math. Mag. To appear. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 56 /C1/58 and 206, 1972. Lockwood, E. H. "Roulettes." Ch. 17 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 138 /C1/151, 1967. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 52, 1991. Yates, R. C. "Roulettes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 175 /C1/185, 1952. Zwillinger, D. (Ed.). "Roulettes (Spirograph Curves)." §8.2 in CRC Standard Mathematical Tables and Formulae, 3rd ed. Boca Raton, FL: CRC Press, 1996. Round NEAREST INTEGER FUNCTION ,R OUND NUMBER , ROUNDNESS Round Number A number which is the product of a considerable number of comparatively small factors (Hardy 1999, p. 48). Round numbers are very rare. As Hardy (1999, p. 48) notes, "Half the numbers are divisible by 2, one-third by 3, one-sixth by both 2 and 3, and so on. Surely, then we may expect most numbers to have a large number of factors. But the facts seem to show the opposite." See also HIGHLY COMPOSITE NUMBER ,PRIME FAC- TORS ,ROUNDNESS ,SMOOTH NUMBER References Hardy, G. H. "Round Numbers." Ch. 3 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 48 /C1/57, 1999. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 89 /C1/90, 1998. Rounding The process of approximating a quantity, be it for convenience or, as in the case of numerical computa- tions, of necessity. If rounding is performed on each of a series of numbers in a long computation, ROUNDING ERROR can become important, especially if division by a small number ever occurs. See also NEAREST INTEGER FUNCTION ,R OUNDING ERROR ,SHADOWING THEOREM References Mulliss, C. "Significant Figures and Rounding Rules." http:// www.angelfire.com/oh/cmulliss/. Wilkinson, J. H. Rounding Errors in Algebraic Processes. New York: Dover, 1994. Rounding Error The error produced in a computation by rounding results at one or more intermediate steps, resulting in a result different from that which would be obtained using exact numbers. The most common problems resulting from rounding error occur either when many steps are involved with rounding occurring at each step, when two quantities very close to each other are subtracted, or when a number is divided by a number which is close to zero. An egregious example of rounding error is provided by a short-lived index devised at the Vancouver stock exchange. At its inception in 1982, the index was given a value of 1000.000. After 22 months of recomputing the index and truncating to three decimal places at each change in market value, the index stood at 524.881, despite the fact that its "true" value should have been 1009.811. Other sorts of rounding error can also occur. A notorious example is the fate of the Ariane rocket launched on June 4, 1996. In the 37th second of flight, the inertial reference system attempted to convert a 64-bit floating point number to a 16-bit number, but instead triggered an overflow error which was inter- preted by the guidance system as flight data, causing the rocket to veer off course and be destroyed. The Patriot missile defense system used during the Gulf War was also rendered ineffective due to roundoff error. The system used an integer timing register which was incremented at intervals of 0.1 s. However, the integers were converted to decimal numbers by multiplying by the BINARY approximation of 0.1, 0 :000110011001100110011002 /C30209715 2097152 : As a result, after 100 hours (3:6 /C29106 ticks), an error of 1 10 /C28209715 2097152})@D})@E (3600 /C215100 /C21510) /C305625 16384 :0:3433 second had accumulated. This discrepancy caused the Pa- triot system to continuously recycle itself instead of targeting properly. As a result, an Iraqi Scud missile could not be targeted and was allowed to detonate on a barracks, killing 28 people. See also ROUNDING Roundness Hoffman (1998, p. 90) calls the sum of the exponents in the PRIME FACTORIZATION of a number its round- ness. The first few values for n /C301, 2, ... are 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, ... (Sloane’s A001222). See also HIGHLY COMPOSITE NUMBER ,PRIME FACTOR- IZATION ,ROUND NUMBERReferences Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 844, 1972. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 90, 1998. Kac, M. Statistical Independence in Probability, Analysis, and Number Theory. Buffalo, NY: Math. Assoc. Amer., p. 64, 1959. Sloane, N. J. A. Sequences A001222/M0094 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Route An n-route is defined as a WALK of length n with specified initial point in which no line succeeds itself. See also TRANSITIVE GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 173, 1994. Routh-Hurwitz Theorem Consider the CHARACTERISTIC EQUATION ½ lI /C28A ½/C30 ln /C27b1 ln/C281 /C27.../C27bn/C281 l /C27bn /C300 determining the n EIGENVALUES l of a REAL n /C29n MATRIX A ; where l is the IDENTITY MATRIX . Then the EIGENVALUES l all have NEGATIVE REAL PARTS if D1 > 0;D2 > 0;...;Dn > 0; where Dk /C30b1 10000 /C1/C1/C1 0 b3 b2 b1 100 /C1/C1/C1 0 b5 b4 b3 b2 b1 0 /C1/C1/C1 0 nnnnnn::: n b2k/C281b2k/C282b2k/C283b2k /C284b2k/C285bk/C286/C1/C1/C1 bk})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1})@1: See also STABLE POLYNOMIAL References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1119, 2000. Se´roul, R. "Stable Polynomials." §10.13 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 280 /C1/286, 2000. Routh’s Theorem If the sides of a TRIANGLE are divided in the ratios l:1;m:1;and n:1;the CEVIANS form a central TRIANGLE whose AREA is a/C30(lmn/C281)2 (lm/C27l/C271)(mn/C27m/C271)(nl/C27n/C271)d; (1) where dis the AREA of the original TRIANGLE . forl/C30 m /C30 n /C13n; a /C30(n /C28 1)2 n2 /C27 n /C27 1 d : (2) for n /C301, 2, 3, ..., the areas are 0, 1/7 (Steinhaus 1983, pp. 8 /C1/9), 4/13, 3/7, 16/31, 25/43, ... (Sloane’s A046162 and A046163). The AREA of the TRIANGLE formed by connecting the division points on each side is A?/C30lmn /C27 1 (l /C27 1)( m /C27 1)( n /C27 1) D: (3) Routh’s theorem gives CEVA’S THEOREM and MENE- LAUS’ THEOREM ( lmn /C30/C281) as special cases. See also CEVA’S THEOREM ,CEVIAN ,M ENELAUS’ THE- OREM References Bottema, O. "On the Area of a Triangle in Barycentric Coordinates." Crux. Math. 8, 228 /C1/231, 1982. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 211 /C1/212, 1969. Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 27, 1970. Klamkin, M. S. Crux. Math. p. 199, 1981. Mikusinski, J. G. Ann. Univ. M. Curie-Sklodowska 1,45/C1/ 50, 1946. Sloane, N. J. A. Sequences A046162 and A046163 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Row Space See also COLUMN SPACE Row Vector A1/C29n MATRIX a11a12/C1/C1/C1 a1n ½/C138 : See also COLUMN VECTOR ,MATRIX ,VECTOR Row-Convex Polyomino A row-convex polyomino is a self-avoiding CONVEX POLYOMINO such that the intersection of any horizon- tal line with the polyomino has at most two connected components. A row-convex polyomino is also called ahorizontally convex polyomino. A COLUMN-CONVEX POLYOMINO is similarly defined. See also COLUMN- CONVEX POLYOMINO ,CONVEX POLY- OMINO ,POLYOMINO RPN REVERSE POLISH NOTATION RSA Encryption APUBLIC-KEY CRYPTOGRAPHY ALGORITHM which uses PRIME FACTORIZATION as the TRAPDOOR ONE-WAY FUNCTION . Define n/C13pq (1) forpandqPRIMES . Also define a private key dand a public key esuch that de/C131 (mod f(n)) (2) (e;f(n))/C301; (3) where f(n) is the TOTIENT FUNCTION ,(a, b) denotes the GREATEST COMMON DIVISOR (so (a;b)/C301 means that aand bare RELATIVELY PRIME ), and a/C13 b(mod m)i sa CONGRUENCE . Let the message be converted to a number M. The sender then makes nandepublic and sends E/C30Me(mod n): (4) To decode, the receiver (who knows d) computes Ed/C13(Me)d/C13Med/C13MNf(n)/C271/C13M(mod n); (5) since Nis an INTEGER . In order to crack the code, d must be found. But this requires factorization of n since f(n)/C30(p/C281)(q/C281): (6) Both pandqshould be picked so that p91 and q91 are divisible by large PRIMES , since otherwise the POLLARD P-1FACTORIZATION METHOD or W ILLIAMS P/C271FACTORIZATION METHOD potentially factor n easily. It is also desirable to have f(f(pq)) large and divisible by large PRIMES . It is possible to break the cryptosystem by repeated encryption if a unit of Z=f(n)Zhas small ORDER (Simmons and Norris 1977, Meijer 1996), where Z=sZ is the RING ofINTEGERS between 0 and s/C281 under addition and multiplication (mod s). Meijer (1996) shows that "almost" every encryption exponent eis safe from breaking using repeated encryption for factors OF THE FORM p/C302p1/C271 (7) q/C302q1/C271; (8) where p1/C302p2/C271 (9) q1 /C302q2 /C271 ; (10) and p, p1 ; p2 ; q, q1 ; and q2 are all PRIMES . In this case, f(n) /C304p1q1 (11) f( f(n)) /C308p2q2 : (12) Meijer (1996) also suggests that p2 and q2 should be of order 1075. Using the RSA system, the identity of the sender can be identified as genuine without revealing his private code. See also CONGRUENCE ,PUBLIC- KEY CRYPTOGRAPHY References Coutinho, S. C. The Mathematics of Ciphers: Number Theory and RSA Cryptography. Natick, MA: A. K. Peters, 1999. Flannery, S. and Flannery, D. In Code: A Mathematical Journey. Profile Books, 2000. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 166 /C1/173, 1985. Meijer, A. R. "Groups, Factoring, and Cryptography." Math. Mag. 69, 103/C1/109, 1996. Rivest, R. L. "Remarks on a Proposed Cryptanalytic Attack on the MIT Public-Key Cryptosystem." Cryptologia 2,6 2/C1/ 65, 1978. Rivest, R.; Shamir, A.; and Adleman, L. "A Method for Obtaining Digital Signatures and Public Key Cryptosys- tems." Comm. ACM 21, 120/C1/126, 1978. RSA Laboratories.†"RSA Factoring Challenge." http:// www.rsasecurity.com/rsalabs/challenges/factoring/. RSA Laboratories.†"Factoring Challenge: Status." http:// www.rsasecurity.com/rsalabs/challenges/factoring/sta- tus.html. Simmons, G. J. and Norris, M. J. "Preliminary Comments on the MIT Public-Key Cryptosystem." Cryptologia 1, 406/C1/414, 1977. RSA Number Numbers contained in the "factoring challenge" of RSA Data Security, Inc. An additional number whichis not part of the actual challenge is the RSA-129number. The RSA numbers which have been factored are RSA-100 (Apr. 1991), RSA-110 (Apr. 1992), RSA- 120 (Jun. 1993), RSA-129 (Apr. 1994), RSA-130 (Apr.1996), RSA-140 (Feb. 1999), and RSA-155 (Aug. 1999; Peterson 1999). RSA-150 has not yet been factored. RSA-129 is a 129-digit number used to encrypt one of the first public-key messages. This message waspublished by R. Rivest, A. Shamir, and L. Adleman (Gardner 1977), along with the number and a $100 reward for its decryption. Despite belief that themessage encoded by RSA-129 "would take millions of years to break," RSA-129 was factored in 1994 using a distributed computation which harnessednetworked computers spread around the globe per-forming a multiple polynomial QUADRATIC SIEVE factorization method. The effort was coordinated byP. Leylad, D. Atkins, and M. Graff. They received112,011 full factorizations, 1,431,337 single partial factorizations, and 8,881,138 double partial factoriza-tions out of a factor base of 524,339 PRIMES . The final MATRIX obtained was 188,346 /C29188,346 square. The text of the message was "The magic words are squeamish ossifrage" (an ossifrage is a rare, preda- tory vulture found in the mountains of Europe), andthe FACTORIZATION (into a 64- DIGIT number and a 65- DIGIT number) is 114381625757888867669235779976146612010218296 /C1/C1/C1 /C1/C1/C17212423625625618429357069352457338978305971 /C1/C1/C1 /C1/C1/C123563958705058989075147599290026879543541 /C303490529510847650949147849619903898133417764 /C1/C1/C1 /C1/C1/C1638493387843990820577 /C2153276913299326 /C1/C1/C1 /C1/C1/C16709549961988190834461413177642967992 /C1/C1/C1 /C1/C1/C1942539798288533 (Leutwyler 1994, Cipra 1995). On Feb. 2, 1999, a group led by H. te Riele completed factorization of RSA-140 into two 70-digits primes.Primality of the factors was proved using two differ- ent methods. The factorization was found using the NUMBER FIELD SIEVE factorization method, and beat the 130-digit record (for RSA-130) set on April 10, 1996. The amount of computer time spent on this factorization is estimated to be equivalent to 2000 MIPS years. (For the old 130-digit NFS-record, thiseffort is estimated to be 1000 MIPS years; te Riele 1999.) Sieving was done on about 125 SGI and Sun workstations running at 175 MHz on average, and onabout 60 PCs running at 300 MHz on average. The total amount of CPU-time spent on sieving was 8.9 CPU years (te Riele 1999). Sieving started the daybefore Christmas 1998 and was completed one month later. The relations were collected and required 3.7 GB of memory (te Riele 1999) The filtering of the dataand the building of the matrix took one calendarweek. The resulting matrix had 4,671,181 rows and 4,704,451 columns, and weight 151,141,999 (32.36 nonzero entries per row). It took almost 100 CPUhours and 810 MB of central memory to find 64 dependencies among the rows of this matrix (te Riele 1999a). On Aug. 22, 1999, a group led by H. te Riele completed factorization of RSA-155 into two 78-digit primes (te Riele 1999b, Peterson 1999). Primality of the factors was proved with the help of two differentprimality proving codes. This factorization was found using the NUMBER FIELD SIEVE factoring algorithm. The amount of computer time spent on this new factoring world record is estimated to be equivalent to8000 MIPS years. Sieving was done on about 160 175/C1 /400 MHz SGI and Sun workstations, on 8 300 MHz SGI Origin 2000 processors, on about 120 300 /C1/ 450 MHz Pentium II PCs, and on 4 500 MHz Digital/Compaq boxes. The total amount of CPU-time spent on sieving was 35.7 CPU years estimated to be equivalent to approximately 8000 MIPS years. Ca- lendar time for sieving was 3 1/2 months. The filtering of the data and the building of the matrix were carried out at CWI and took one month. The resulting matrix had 6,699,191 rows, 6,711,336 col- umns, and weight 417,132,631 (62.27 nonzeros per row). It took 224 CPU hours and 2 GB of central memory on the Cray C916 at the SARA Amsterdam Academic Computer Center to find 64 dependencies among the rows of this matrix (te Riele 1999b). See also NUMBER FIELD SIEVE References Cipra, B. "The Secret Life of Large Numbers." What’s Happening in the Mathematical Sciences, 1995 /C1/1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 90 /C1/99, 1996. Cowie, J.; Dodson, B.; Elkenbracht-Huizing, R. M.; Lenstra, A. K.; Montgomery, P. L.; Zayer, J. A. "World Wide Number Field Sieve Factoring Record: On to 512 Bits." In Advances in Cryptology--ASIACRYPT ’96 (Kyongju) (Ed. K. Kim and T. Matsumoto.) New York: Springer- Verlag, pp. 382 /C1/394, 1996. Gardner, M. "Mathematical Games: A New Kind of Cipher that Would Take Millions of Years to Break." Sci. Amer. 237, 120 /C1/124, Aug. 1977. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory, rev. ed. Washington, DC: Math. Assoc. Amer., p. 223, 1991. Leutwyler, K. "Superhack: Forty Quadrillion Years Early, a 129-Digit Code is Broken." Sci. Amer. 271,17/C1/20, 1994. Leyland, P. ftp://sable.ox.ac.uk/pub/math/rsa129. Peterson, I. "Crunching Internet Security Codes." Sci. News 156, 221, Oct. 2, 1999. RSA Data Security. † "RSA Factoring Challenge." http:// www.rsasecurity.com/rsalabs/challenges/factoring/. RSA Data Security. † "What is the RSA Factoring Challenge and What is RSA-129?" http://www.rsasecurity.com/rsa- labs/faq/. Taubes, G. "Small Army of Code-breakers Conquers a 129- Digit Giant." Science 264, 776 /C1/777, 1994. te Riele, H. "Factorisation of RSA-140." NMBRTHRY@LIST- SERV.NODAK.EDU mailing list posting, Feb. 4, 1999a. te Riele, H. "New Factorization Record." NMBRTHRY@- LISTSERV.NODAK.EDU mailing list posting, Aug. 26, 1999b. Weisstein, E. W. "RSA Numbers." MATHEMATICA NOTEBOOK RSAN UMBERS.M . Rubber-Sheet Geometry ALGEBRAIC TOPOLOGY Rubik’s Clock A puzzle consisting of 18 small clocks. There are 1218 possible configurations, although not all are realiz- able. See also RUBIK’S CUBE References De´nes, J. and Mullen, G. L. "Rubik’s Clock and Its Solution." Math. Mag. 68, 378 /C1/381, 1995.Zeilberger, D. "Doron Zeilberger’s Maple Packages and Programs: RubikClock." http://www.math.temple.edu/ ~zeilberg/programs.html. Rubik’s Cube A3/C293 /C293 CUBE in which the 26 subcubes on the outside are internally hinged in such a way that rotation (by a quarter turn in either direction or a half turn) is possible in any plane of cubes. Each of the six sides is painted a distinct color, and the goal of the puzzle is to return the cube to a state in which each side has a single color after it has been randomized by repeated rotations. The PUZZLE was invented in the 1970s by the Hungarian Erno Rubik and sold millions of copies worldwide over the next decade. The number of possible positions of Rubik’s cube is 8!12!38212 2 /C215 3 /C215 2/C3043;252;003;274;489;856;000 (Turner and Gold 1985, Scho¨nert). Hoey showed using the PO´ LYA-BURNSIDE LEMMA that there are 901,083,404,981,813,616 positions up to conjugacy by whole-cube symmetries. Algorithms exist for solving a cube from an arbitrary initial position, but they are not necessarily optimal (i.e., requiring a minimum number of turns). The minimum number of turns required for an arbitrary starting position is still not known, although it is bounded from above. Michael Reid (1995) produced the best proven bound of 29 turns (or 42 "quarter- turns"). The proof involves large tables of "subrou- tines" generated by computer. However, Dik Winter has produced a program based on work by Kociemba which has solved each of millions of cubes in at most 21 turns. Recently, Richard Korf (1997) has produced a different algo- rithm which is practical for cubes up to 18 movesaway from solved. Out of 10 randomly generated cubes, one was solved in 16 moves, three required 17 moves, and six required 18 moves. See also R UBIK’S CLOCK References Helms, G. "Rubik’s Cube." http://webplaza.pt.lu/public/geo- helm/myweb/cubeold.htm. Hoey, D. "The Real Size of Cube Space." http:// www.math.rwth-aachen.de/~Martin.Schoenert/Cube- Lovers/Dan_Hoey__The_real_size_of_cube_space.html. Hofstadter, D. R. "Metamagical Themas: The Magic Cube’s Cubies are Twiddled by Cubists and Solved by Cubeme- isters." Sci. Amer. 244,20/C1/39, Mar. 1981. Larson, M. E. "Rubik’s Revenge: The Group Theoretical Solution." Amer. Math. Monthly 92, 381 /C1/390, 1985. Longridge, M. "Domain of the Cube." http://web.idirect.com/ ~cubeman/. Miller, D. L. W. "Solving Rubik’s Cube Using the ‘Bestfast’ Search Algorithm and ‘Profile’ Tables." http://www.sunyi- t.edu/~millerd1/RUBIK.HTM. Schoenert, M. "Cube Lovers: Index by Date." http:// www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/. Scho¨nert, M. "Analyzing Rubik’s Cube with GAP." http:// www-groups.dsc.st-and.ac.uk/~gap/Intro/rubik.html. Singmaster, D. Notes on Rubik’s ‘Magic Cube.’ Hillside, NJ: Enslow Pub., 1981. Taylor, D. Mastering Rubik’s Cube. New York: Holt, Rine- hart, and Winston, 1981. Taylor, D. and Rylands, L. Cube Games: 92 Puzzles & Solutions. New York: Holt, Rinehart, and Winston, 1981. Turner, E. C. and Gold, K. F. "Rubik’s Groups." Amer. Math. Monthly 92, 617 /C1 /629, 1985. Rudin-Shapiro Sequence Let a number n be written in BINARY as n /C30( ek ek/C281 ...e1 e0)2 ; (1) and define bn /C30Xk/C281 i/C300ei ei /C271 (2) as the number of DIGITS BLOCKS of 11s in the BINARY expansion of n. For n /C300, 1, ..., bn is given by 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ... (Sloane’s A014081). Now define an /C30(/C281)bn (3) as the parity of the number of pairs of consecutive 1s in the BINARY expansion of n. For n /C300, 1, ..., the first few values are 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, ... (Sloane’s A020985). The SUMMATORY sequence of an is the defined by sn /C13Xn i /C300ai ; (4) giving the first few terms 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, ... (Sloane’s A020986). For the special case n /C302k /C281 ; sn can be computed using the formula sn /C302k =2 /C271i f k is even 2(k/C281)=2 /C271i f k is odd})1D (5) (Blecksmith and Laud 1995), giving 2, 3, 3, 5, 5, 9, 9, 17, 17, 33, 33, 65, ... (Sloane’s A051032). See also BINARY ,DIGIT BLOCK ,FOLDING ,STOLARSKY- HARBORTH CONSTANTReferences Blecksmith, R. and Laud, P. W. "Some Exact Number Theory Computations via Probability Mechanisms." Amer. Math. Monthly 102, 893 /C1/903, 1995. Brillhart, J.; Erdos, P.; and Morton, P. "On the Sums of the Rudin-Shapiro Coefficients II." Pac. J. Math. 107,39/C1/69, 1983. Brillhart, J. and Morton, P. "U¨ ber Summen von Rudin- Shapiroschen Koeffizienten." Ill. J. Math. 22, 126 /C1/148, 1978. Mendes France, M. and van der Poorten, A. J. "Arithmetic and Analytic Properties of Paper Folding Sequences." Bull. Austral. Math. Soc. 24, 123 /C1/131, 1981. Sloane, N. J. A. Sequences A014081, A020985, A020986, and A051032 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Rudvalis Group The SPORADIC GROUP Ru. See also SPORADIC GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/Ru.html. Ruffini-Horner Method HORNER’S METHOD Rule A usually simple ALGORITHM or IDENTITY . The term is frequently applied to specific orders of NEWTON- COTES FORMULAS . See also ALGORITHM , BAC -CAB RULE,BODE’S RULE, CHAIN RULE,CRAMER’S RULE,DESCARTES’ SIGN RULE, DURAND’S RULE,ESTIMATOR ,EULER’S RULE,EULER’S TOTIENT RULE,GOLDEN RULE,HARDY’S RULE,HOR- NER’S RULE,IDENTITY ,L’HOSPITAL’S RULE,LEIBNIZ INTEGRAL RULE,METHOD ,OSBORNE’S RULE,PASCAL’S RULE,P OWER RULE,P RODUCT RULE,Q UARTER SQUARES RULE,Q UOTA RULE,Q UOTIENT RULE, ROTH’S REMOVAL RULE,R ULE OF 72,S IMPSON’S RULE,SLIDE RULE,SUM RULE,TRAPEZOIDAL RULE, WEDDLE’S RULE,ZEUTHEN’S RULE Rule of 72 The time required for a given PRINCIPAL to double (assuming n/C301CONVERSION PERIOD ) for COMPOUND INTEREST is given by solving 2P /C30P(1 /C27r)t ; (1) or t /C30ln 2 ln(n /C27 r) ; (2) where LN is the NATURAL LOGARITHM . This function can be approximated by the so-called "rule of 72": t :0:72 r: (3) The above plots show the actual doubling time t (left plot) and the difference between the actual doubling time and the doubling time calculated using the rule of 72 (right plot) as a function of the interest rate r. See also COMPOUND INTEREST ,INTEREST References Avanzini, J. F. Rapid Debt-Reduction Strategies. Fort Worth, TX: HIS Pub., 1990. Ruled Surface A SURFACE which can be swept out by a moving a LINE in space and therefore has a parameterization OF THE FORM x(v; v) /C30b(u) /C27v d(u) ; (1) where b is called the DIRECTRIX (also called the BASE CURVE ) and d is the DIRECTOR CURVE . The straight lines themselves are called RULINGS . The rulings of a ruled surface are ASYMPTOTIC CURVES . Furthermore, the GAUSSIAN CURVATURE on a ruled REGULAR SUR- FACE is everywhere NONPOSITIVE . Examples of ruled surfaces include the elliptic HY- PERBOLOID of one sheet (a DOUBLY RULED SURFACE ) a(cos u /C14v sin u) b(sin u 9v cos u) 9cv2 435/C30a cos u b sin u 024359v/C28a sin u b cos u c2435; (2) the HYPERBOLIC PARABOLOID (a DOUBLY RULED SUR- FACE ) a(u /C27v) 9bv u2 /C272uv2435/C30au 0 u 22435/C27va 9b 2u2435; (3) P LU¨ CKER’S CONOID r cos u r sin u 2 cos u sin u2435/C300 0 2 cos u sin u2 435/C27rcos u sin u 02435; (4) and the M O¨ BIUS STRIPacos u /C27v cos1 2 u})@D})@E cos u sin u /C27v cos12 u})@D})@E sin u v sin12 u})@D})@E2 66643 7775 /C30acos u sin u 02 435/C27aucos u 1 2u})@D})@E cos u cos12u})@D})@E sin u sin12u})@D})@E2 66643 7775(5) (Gray 1997). The only ruled MINIMAL SURFACES are the PLANE and HELICOID (Catalan 1842, do Carmo 1986). See also ASYMPTOTIC CURVE ,CAYLEY’S RULED SUR- FACE ,D EVELOPABLE SURFACE ,D IRECTOR CURVE , DIRECTRIX (RULED SURFACE ), DOUBLY RULED SUR- FACE ,GENERALIZED CONE,GENERALIZED CYLINDER , HELICOID ,NONCYLINDRICAL RULED SURFACE ,PLANE , RIGHT CONOID ,RULING References Catalan E. "Sur les surfaces re´gle´es dont l’aire est un minimum." J. Math. Pure. Appl. 7, 203 /C1/211, 1842. do Carmo, M. P. "The Helicoid." §3.5B in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 44 /C1/45, 1986. Fischer, G. (Ed.). Plates 32 /C1/33 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 32 /C1/33, 1986. Gray, A. "Ruled Surfaces." Ch. 19 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 431 /C1/456, 1993. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 15, 1999. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 242 /C1/243, 1999. Ruler A STRAIGHTEDGE with markings to indicate distances. Although GEOMETRIC CONSTRUCTIONS are sometimes said to be performed with a ruler and COMPASS , the term STRAIGHTEDGE is preferable to ruler since markings are not allowed by the classical Greek rules. See also COASTLINE PARADOX ,COMPASS ,GEOMETRIC CONSTRUCTION ,GEOMETROGRAPHY ,GOLOMB RULER , PERFECT RULER ,SIMPLICITY ,SLIDE RULE,STRAIGHT- EDGE References Smogorzhevskii, A. S. The Ruler in Geometrical Construc- tions. New York: Blaisdell, 1961. Ruler Function The exponent of the largest POWER of 2 which DIVIDES a given number 2 n:The values of the ruler function forn/C301, 2, ..., are 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, ... (Sloane’s A001511). See also 2 References Guy, R. K. "Cycles and Sequences Containing All Permuta- tions as Subsequences." §E22 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 224, 1994. Sloane, N. J. A. Sequences A001511/M0127 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Ruling One of the straight lines sweeping out a RULED SURFACE . The rulings on a ruled surface are ASYMP- TOTIC CURVES . See also ASYMPTOTIC CURVE ,D IRECTOR CURVE , DIRECTRIX (RULED SURFACE ), RULED SURFACE Rumors GOSSIPING Rumor Spreading GOSSIPING Run A run is a sequence of more than one consecutive identical outcomes, also known as a CLUMP . Given n BERNOULLI TRIALS (say, in the form of COIN TOS- SINGS ), the probability Pt(n) of a run of tconsecutive heads or tails is given by the RECURRENCE RELATION Pt(n)/C30Pt(n/C281)/C272/C28t[1/C28Pt(n/C28t)]; (1) where Pt(n)/C300 for nBtand Pt(t)/C3021/C28t(Bloom 1996). Let R(r;n) be the probability that a run of r consecutive heads appears in nindependent tosses of a COIN . There is a beautiful formula for R(r;n) given in terms of the coefficients of the GENERATING FUNCTION Fp(r;s)/C30prsr(1/C28ps) 1/C28s/C27(1/C28p)prsr/C271/C13X/C12 i/C30rcp isi(2) (Feller 1968, 2nd ed. p. 300), where 0 BpB1 is the probability of obtaining a head in a single toss. Then Rp(r;n)/C30Xn i/C30rcp i (3) The following table gives the triangle of numbers 2nR1=2(r;n) for r/C301, 2, ... and n/C30r,r/C271;...;... (Sloane’s A050227). /r_n/1 2 3456 7 8 1 1 3 7 15 31 63 127 2552 0 1 3 8 19 43 94 201 3 0 0 1 3 8 20 47 1074 0 0 01382 04 85 0 0 0013 82 0 6 0 0 0001 3 8 7 0 0 0000 1 38 0 0 0000 0 1 The special case r/C302 gives the sequence R 2(n)/C302n/C271/C28Fn/C273; (4) where Fnis a F IBONACCI NUMBER , the first few terms of which for n/C301, 2, ... are 0, 1, 3, 8, 19, 43, 94, 201, ... (Sloane’s A008466). The first few R3(n) are given by 0, 0, 1, 3, 8, 20, 47, 107, 238, ... Sloane’s A050231; the first few R4(n) are 0, 0, 0, 1, 3, 8, 20, 48, 111, 251, 558, ... (Sloane’s A050232); and the first few R5(n)0 ,0 ,0 , 0, 1, 3, 8, 20, 48, 112, 255, 571, 1262, ... (Sloane’sA050233). Given nB ERNOULLI TRIALS with a probability of success (heads) p, the expected number of tails is n(1/C28p);so the expected number of tail runs ]1i s :n(1/C28p)p:Continuing, NR/C30n(1/C28p)pR(5) is the expected number of runs ]R:The longest expected run is therefore given by R/C30log1=p[n(1/C28p)] (6) (Gordon et al. 1986, Schilling 1990). Given m0s and n1s, the number of possible arrangements with u runs is fu/C302m/C281 k/C281})@*})@+ n/C281 k/C281})@*})@+ u/C132k m/C281 k/C281})@*})@+ n/C281 k/C282})@*})@+ /C27m/C281 k/C282})@*})@+ n/C281 k/C281})@*})@+ u/C132k/C2718 >>< >>: (7) forkanINTEGER , wheren k})0})@ is a BINOMIAL COEFFI- CIENT . Then P(u5u?)/C30Xu? u/C302fu m/C27n m})@*})@+ : (8) Feller (1968, pp. 278 /C1/279) proved that for w(n)/C13 1/C28R1=2(3;n); lim n0/C12w(n)an/C271/C30b; (9) where a /C301 3136 /C2724ffiffiffiffiffiffi 33p})@D})@E1=3 /C288 136 /C2724ffiffiffiffiffiffi33p})@D})@E /C281 =3 /C282})10})1@ ¼ 1 :087378025... ð10Þ and b /C302 /C28 a 4 /C28 3a /C301:236839845... : (11) The corresponding constants for a RUN of k /C211 heads are ak ; the smallest POSITIVE ROOT of 1 /C28x /C271 2 x})@D})@Ek /C271 /C300; (12) and bk /C302 /C28 a k /C27 1 /C28 k ak: (13) These are modified for unfair coins with P(H) /C30p and P(T) /C30q /C301 /C28p to a?k ; the smallest POSITIVE ROOT of 1 /C28x /C27qpkxk/C271 /C300 ; (14) and b?k /C301 /C28 p a?k (k /C27 1 /C28 k a?k)p (15) (Feller 1968, pp. 322 /C1/325). Let Ct(m; k) denote the number of sequences of m indistinguishable objects of type A and k indistin- guishable objects of type B in which no t-run occurs. The probability that a t-run does occur is then given by Pt(m; k) /C301 /C28Ct(m; k) m /C27 k k})@*})@+ ; (16) wherea b})0})@ is a BINOMIAL COEFFICIENT . Bloom (1996) gives the following recurrence sequence for Ct(m; k) ; Ct(m; k) /C30Xt/C281 i/C300Ct(m /C281; k /C28i) /C28Xt /C281 i/C301Ct(m /C28t; k /C28i) /C27et(m; k) ; (17) where et(m; k) /C131 /C281 0if m /C300 and 0 5k Bt if m /C30t and 0 5k Bt otherwise :8 < : (18) Another recurrence which has only a fixed number of terms is given by Ct(m; k) /C30Ct(m /C281; k) /C27Ct(m; k /C281) /C28Ct(m /C28t; k /C281) /C28Ct(m /C281; k /C28t) /C27Ct(m /C28t; k /C28t) /C27e /C31t (m; k) ; (19) wheree /C31t (m; k) /C131 /C281 0if (m; k) /C30(0; 0) or (t; t) if (m; k) /C30(0; t)or( t; 0) otherwise8 < : (20) (Goulden and Jackson 1983, Bloom 1996). These formulas disprove the assertion of Gardner (1982) that "there will almost always be a clump of six or seven CARDS of the same color" in a normal deck of cards by giving /P6 ð26; 26Þ¼0:46424 /. Bloom (1996) gives the expected number of noncon- tiguous t-runs in a sequence of m0s and n1s as E(n;m;t)/C30(m/C271)(n)t/C27(n/C271)(m)t (m/C27n)t; (21) where ( a)nis the P OCHHAMMER SYMBOL . For m/C2110, uhas an approximately NORMAL DISTRIBUTION with MEAN and VARIANCE mu/C301/C272mn m/C27n(22) s2 u/C302mn(2mn/C28m/C28n) (m/C27n)2(m/C27n/C281): (23) See also COIN TOSSING ,EULERIAN NUMBER ,PERMU- TATION ,PERMUTATION RUN, S-RUN References Bloom, D. M. "Probabilities of Clumps in a Binary Sequence (and How to Evaluate Them Without Knowing a Lot)." Math. Mag. 69, 366/C1/372, 1996. Feller, W. An Introduction to Probability Theory and Its Application, Vol. 1, 3rd ed. New York: Wiley, 1968. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/feller/feller.html. Gardner, M. Aha! Gotcha: Paradoxes to Puzzle and Delight. New York: W. H. Freeman, p. 124, 1982. Godbole, A. P. "On Hypergeometric and Related Distribu- tions of Order k."Commun. Stat.: Th. and Meth. 19, 1291/C1/1301, 1990. Godbole, A. P. and Papastavridis, G. (Eds.). Runs and Patterns in Probability: Selected Papers. New York: Kluwer, 1994. Gordon, L.; Schilling, M. F.; and Waterman, M. S. "An Extreme Value Theory for Long Head Runs." Prob. Th. and Related Fields 72, 279/C1/287, 1986. Goulden, I. P. and Jackson, D. M. Combinatorial Enumera- tion. New York: Wiley, 1983. Mood, A. M. "The Distribution Theory of Runs." Ann. Math. Statistics 11, 367/C1/392, 1940. Philippou, A. N. and Makri, F. S. "Successes, Runs, and Longest Runs." Stat. Prob. Let. 4, 211/C1/215, 1986. Schilling, M. F. "The Longest Run of Heads." Coll. Math. J. 21, 196/C1/207, 1990. Schuster, E. F. In Runs and Patterns in Probability: Selected Papers (Ed. A. P. Godbole and S. Papastavridis). Boston, MA: Kluwer, pp. 91 /C1/111, 1994. Sloane, N. J. A. Sequences A008466, A050227, A050231, A050232, and A050233 in "An On-Line Version of theEncyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Runge-Kutta Method A method of numerically integrating ORDINARY DIF- FERENTIAL EQUATIONS by using a trial step at the midpoint of an interval to cancel out lower-order error terms. The second-order formula is k1 /C30hf(sn ; yn) k2 /C30hf xn /C271 2 h; yn /C2712 k1})@D})@E yn/C271 /C30yn /C27k2 /C27O(h3) ; and the fourth-order formula is k1 /C30hf(sn ; yn) k2 /C30hf xn /C2712 h; yn /C2712 k1})@D})@E k3 /C30hf xn /C2712 h; yn /C2712 k2})@D})@E k4 /C30hf(xn /C27h; yn /C27k3) yn/C271 /C30yn /C2716 k1 /C2713 k2 /C2713 k3 /C2716 k4 /C27O(h5) : (Press et al. 1992). This method is reasonably simple and robust and is a good general candidate for numerical solution of differential equations when combined with an intelligent adaptive step-size rou- tine. See also ADAMS’ METHOD ,GILL’S METHOD ,M ILNE’S METHOD ,O RDINARY DIFFERENTIAL EQUATION ,R O- SENBROCK METHODS References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 896 /C1/897, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 492 /C1/493, 1985. Cartwright, J. H. E. and Piro, O. "The Dynamics of Runge- Kutta Methods." Int. J. Bifurcations Chaos 2, 427 /C1/449, 1992. http://formentor.uib.es/~julyan/TeX/rkpaper/root/ root.html. Kutta, M. W. Z. fu¨r Math. u. Phys. 46, 435, 1901. Lambert, J. D. and Lambert, D. Ch. 5 in Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. New York: Wiley, 1991. Lindelo ¨f, E. Acta Soc. Sc. Fenn. 2, 1938. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Runge-Kutta Method" and "Adaptive Step Size Control for Runge-Kutta." §16.1 and 16.2 in Numer- ical Recipes in FORTRAN: The Art of Scientific Comput- ing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 704 /C1/716, 1992. Runge, C. Math. Ann. 46, 167, 1895. Runge’s Theorem Let K ⁄ C be compact, let f be analytic on a neighborhood of K, and let P ⁄C/C31_K contain at least one point from each connected component of C /C31_K : Then for any e > 0; there is a RATIONAL FUNCTION /r ðzÞ/ with poles in P such thatmax z /C23K½f(z) /C28r(z)½B e (Krantz 1999, p. 143). A polynomial version can be obtained by taking P /C30 f/C12g: Let f(x)bean ANALYTIC FUNCTION which is REGULAR in the interior of a JORDAN CURVE C and continuous in the closed DOMAIN bounded by C. Then f(x) can be approximated with arbitrary accuracy by POLYNOMIALS (Szego o 1975, p. 5; Krantz 1999, p. 144). See also ANALYTIC FUNCTION ,JORDAN CURVE ,M ER- GELYAN’S THEOREM References Krantz, S. G. "Runge’s Theorem." §11.1.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 143 /C1/144, 1999. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 7, 1975. Runge-Walsh Theorem RUNGE’S THEOREM Run-Length Encoding A specification of elements in a list as a list of pairs giving the element and number of times it occurs in a run. For example, given the list f1 ; 1; 1; 3;/ /3; 6; 6; 6; 2; 2; 2; 2 ; 3 ; 3 ; 1 ; 4; 4g; the run-length encoding is ff1; 3g;f3; 2g;f6; 3g;f2; 4g;f3; 2;g;/ /f1; 1g;f4; 2gg: Run-length encoding can be imple- mented in Mathematica as RunLengthEncode[x_List] : /C30(Through[{First, Length}[#1]] &) /@ Split[x] See also LOOK AND SAY SEQUENCE ,RUN Running Average MOVING AVERAGE Running Knot AKNOT which tightens around an object when strained but slackens when the strain is removed. Running knots are sometimes also known as slipknots or nooses. References Owen, P. Knots. Philadelphia, PA: Courage, p. 60, 1993. Russell’s Antinomy LetRbe the set of all sets which are not members of themselves. Then Ris neither a member of itself nor not a member of itself. Symbolically, let R/C30fx:xQxg: Then R/C23RIFFRQR:/ Bertrand Russell discovered this PARADOX and sent it in a letter to G. Frege just as Frege was completing Grundlagen der Arithmetik. This invalidated much of the rigor of the work, and Frege was forced to add a note at the end stating, "A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press." See also BARBER PARADOX ,C ATALOGUE PARADOX , GRELLING’S PARADOX References Courant, R. and Robbins, H. "The Paradoxes of the Infinite." §2.4.5 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 78, 1996. Curry, H. B. Foundations of Mathematical Logic, 2nd rev. ed. New York: Dover, p. 4, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 175 /C1/177, 1998. Frege, G. Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd rev. ed. Evanston, IL: Northwestern University Press, 1980. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 116, 1998. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 20 /C1/21, 1989. Mirimanoff, D. "Les antinomies de Russell et de Burali-Forti et le proble `me fondamental de la the´orie des ensembles." Enseign. math. 19,37/C1/52, 1917. Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, pp. 79 and 101, 1927. Russell’s Paradox RUSSELL’S ANTINOMY Russian Doll Prime PRIME STRING Russian Multiplication Also called "Ethiopian multiplication." To multiply two numbers a and b, write a0 /C13a and b0 /C13b in two columns. Under a0 ; write a0 =2 bc ; where xbcis the FLOOR FUNCTION , and under b0 ; write 2b0 : Continue until ai/C301:Then cross out any entries in the b column which are opposite an EVEN NUMBER in the a column and add the bcolumn. The result is the desired product. For example, for a/C3027;b/C3035 27 35 13 70 6 140/C3 3 280 1560 945 See also MULTIPLICATIONReferences Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 44, 1986. Russian Roulette Russian roulette is a GAME of chance in which one or more of the six chambers of a gun are filled with bullets, the magazine is rotated at random, and the gun is fired. The shooter bets on whether the chamberwhich rotates into place will be loaded. If it is, he loses not only his bet but his life. A modified version is considered by Blom et al. (1996) and Blom (1989). In this variant, the revolver is loaded with a single bullet, and two duelists alter- nately spin the chamber and fire at themselves until one is killed. The probability that the first duelist iskilled is then 6/11. References Blom, G. Probabilities and Statistics: Theory and Applica- tions. New York: Springer-Verlag, p. 32, 1989. Blom, G.; Englund, J.-E.; and Sandell, D. "General Russian Roulette." Math. Mag. 69, 293/C1/297, 1996. Ruth-Aaron Pair A pair of consecutive numbers ( n;n/C271) such that the sums of the prime factors of nand n/C271 are equal. They are so named because they were inspired by the pair (714, 715) corresponding to Hank Aaron’s record- breaking 715th home run in 1974, breaking BabeRuth’s earlier record of 714 (Hoffman 1998, pp. 179 /C1 / 181). The first few ns giving Ruth-Aaron pairs are 5, 8, 15, 77, 125, 714, 948, ... (Sloane’s A039752),corresponding to the sums 5, 6, 8, 18, 15, 29, 86, ...(Sloane’s A054378). Pomerance suspected there were an infinite number of such pairs, and this was almost immediately proved true by P. Erdos (Hoffman 1998, pp. 180 /C1 / 181). References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Nelson, C.; Penney, D. E.; and Pomerance, C. "714 and 715." J. Recr. Math. 7,8 7/C1/89, 1994. Peterson, I. "Ivars Peterson’s MathLand: Playing with Ruth- Aaron Pairs." http://www.maa.org/mathland/math-land_6_30.html. Sloane, N. J. A. Sequences A039752 and A054378 in "An On-Line Version of the Encyclopedia of Integer Se-quences." http://www.research.att.com/~njas/sequences/eisonline.html. Rutishauser’s Rule Letmandm/C27hbe two consecutive CRITICAL INDICES offand let Fbe (m/C27h)/-normal. If the polynomials ˜p(n) kare defined by ˜p(n) 0(u) /C131 (1) ˜p(n) k /C271(u) /C13u˜p(n/C281) k(u) /C28q(n) m/C27k /C271 ˜p(n) k(u) (2) for n /C300, 1, ... and k /C300, ..., h /C281; then, under the hypothesis below, there exists an infinite set N of positive integers such that lim n 0/C12 n /C23N˜p(n) h(u) /C30 ˜ph(u) ; (3) where ˜ph(u) /C13(u /C28um/C271)(u /C28um/C272) /C1/C1/C1(u /C28um/C27h) : (4) By hypothesis, if m /C300, the polynomials ˜p(n) kare identical to the Hadamard polynomials p(n) L; and if m /C210, the algorithm for constructing the ˜p(n) kis applied to the qd scheme suitably bounded by columns e(n) mand e(n) m/C27h(Henrici 1988, pp. 642 /C1/643). See also CRITICAL INDEX References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, pp. 642 /C1/643, 1988. Ryser Formula A formula for the PERMANENT of a MATRIX perm( aij) /C30(/C281)nX s⁄f1 ; ... ; n g(/C281) sjjYn i/C301X j /C23saij ;where the SUM is over all SUBSETS of f1; ...; ng; and sjjis the number of elements in s. The formula can be optimized by picking the SUBSETS so that only a single element is changed at a time (which is precisely a GRAY CODE ), reducing the number of additions from n2 to n. It turns out that the number of disks moved after the kth step in the TOWERS OF HANOI is the same as the element which needs to be added or deleted in the kth ADDEND of the Ryser formula (Gardner 1988, Vardi 1991, p. 111). See also DETERMINANT ,G RAY CODE,P ERMANENT , TOWERS OF HANOI References Gardner, M. "The Icosian Game and the Tower of Hanoi." Ch. 6 in The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 55 /C1/62, 1959. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, p. 515, 1998. Nijenhuis, A. and Wilf, H. Chs. 7 /C1/8i n Combinatorial Algorithms. New York: Academic Press, 1975. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 111, 1991. S Saalschu ¨ tzian A GENERALIZED HYPERGEOMETRIC FUNCTION pFqa1 ; a2 ; ...; ap b1 ; b2 ; ...; bq; zYrtvYrtu ; is said to be Saalschu ¨tzian if it is K-BALANCED with k /C301, Xq i/C301bi /C301 /C27Xp i/C301ai : See also GENERALIZED HYPERGEOMETRIC FUNCTION , K-BALANCED ,NEARLY- POISED ,W ELL-POISED References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, p. 11, 1935. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 43, 1998. Whipple, F. J. W. "Well-Poised Series and Other General- ized Hypergeometric Series." Proc. London Math. Soc. 25, 525 /C1/544, 1926. Saalschu ¨ tz’s Theorem Mathematics:Calculus and Analysis:Special Func- tions:Hypergeometric Functions:Generalized Hyper- geometric Functions 3F2/C28x;/C28y;/C28z n /C271 ;/C28x /C28y /C28zYrtvYrtu /C30G(n /C27 1)G(x /C27 y /C27 n /C27 1) G(x /C27 n /C27 1)G(y /C27 n /C27 1) /C29G(y /C27 z /C27 n /C27 1)G(z /C27 x /C27 n /C27 1) G(z /C27 n /C27 1)(x /C27 y /C27 z /C27 n /C27 1); (1) where3F2(a; b; c; d; e; z)isa GENERALIZED HYPER- GEOMETRIC FUNCTION and G(z) is the GAMMA FUNC- TION . It can be derived from the DOUGALL- RAMANUJAN IDENTITY and written in the symmetric form 3F2(a ; b; c; d; e;1)/C30(d /C28 a)½c½(d /C28 b) ½c½ d½c½(d /C28 a /C28 b)½c½(2) for d /C27e /C30a /C27b /C27c /C271 (3) with c a NONPOSITIVE INTEGER and (a)nthe POCH- HAMMER SYMBOL (Bailey 1935, p. 9; Petkovsek et al. 1996; Koepf 1998, p. 32). If one of a, b, and c is nonpositive but it is not known which, an alternative formulation due to W. Gosper gives the form 3F2(a; b; c; d; e;1)/C30G(d) G(d /C28 a) G(d /C28 b) G(d /C28 c)G(e) G(e /C28 a)(e /C28 b)(e /C28 c) /C29p2 cos(pd) cos(pe) /C27 cos( pa) cos( pb) cos( pc) : (4) which is symmetric in (a ; b; c) and (d, e). If instead a /C27b /C27c /C272 /C30d /C27e ; (5) then 3F2(a ; b; c; d; e;2) p2 de /C28 (a /C27 1)(b /C27 1)(c /C27 1) /C27 abc cos(dp) cos(ep) /C28 cos(ap) cos(b p) cos(c p) /C29G(d) G(d /C28 a) G(d /C28 b) G(d /C28 c)G(e) G(e /C28 a)G(e /C28 b) G(e /C28 c) (6) (W. Gosper). See also DOUGALL- RAMANUJAN IDENTITY ,GENERAL- IZED HYPERGEOMETRIC FUNCTION ,KUMMER’S THEO- REM References Bailey, W. N. "Saalschu ¨tz’s Theorem." §2.2 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 9, 1935. Dougall, J. "On Vandermonde’s Theorem and Some More General Expansions." Proc. Edinburgh Math. Soc. 25, 114 /C1/132, 1907. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 104, 1999. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A /C30B. Well- esley, MA: A. K. Peters, pp. 43 and 126, 1996. Saalschu ¨tz, L. "Eine Summationsformel." Z. fu¨r Math. u. Phys. 35, 186 /C1/188, 1890. Saalschu ¨tz, L. "U¨ ber einen Spezialfall der hypergeome- trischen Reihe dritter Ordnung." Z. fu¨r Math. u. Phys. 36, 278 /C1/295 and 321 /C1/327, 1891. Shepard, W. F. "Summation of the Coefficients of Some Terminating Hypergeometric Series." Proc. London Math. Soc. 10, 469 /C1/478, 1912. s-Additive Sequence A generalization of an ULAM SEQUENCE in which each term is the SUM of two earlier terms in exactly s ways. (s, t)-additive sequences are a further generalization in which each term has exactly srepresentations as the SUM oftdistinct earlier numbers. It is conjectured that 0-additive sequences ultimately have periodic differences of consecutive terms (Guy 1994, p. 233). See also GREEDY ALGORITHM ,STO¨ HR SEQUENCE ,SUM- FREE SET,ULAM SEQUENCE References Finch, S. R. "Conjectures about s-Additive Sequences." Fib. Quart. 29, 209 /C1/214, 1991. Finch, S. R. "Are 0-Additive Sequences Always Regular?" Amer. Math. Monthly 99, 671 /C1/673, 1992. Finch, S. R. "On the Regularity of Certain 1-Additive Sequences." J. Combin. Th. Ser. A. 60, 123 /C1/130, 1992. Finch, S. R. "Patterns in 1-Additive Sequences." Experiment. Math. 1,57/C1/63, 1992. Finch, S. "Unsolved Mathematics Problems: Ulam s-Addi- tive Sequences." http://www.mathsoft.com/asolve/sadd/ sadd.html. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 110 and 233, 1994. Ulam, S. M. Problems in Modern Mathematics. New York: Interscience, p. ix, 1964. Saddle A SURFACE possessing a SADDLE POINT . See also HYPERBOLIC PARABOLOID ,MONKEY SADDLE , SADDLE POINT (FUNCTION ) Saddle Point (Fixed Point) HYPERBOLIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), HYPERBOLIC FIXED POINT (MAP) Saddle Point (Function) A POINT of a FUNCTION or SURFACE which is a STATIONARY POINT but not an EXTREMUM . An example of a 1-D FUNCTION with a saddle point is f(x) /C30x3 ; which has f ?(x) /C303x2 f ƒ(x) /C306x f §(x) /C306: This function has a saddle point at x0 /C300 by the EXTREMUM TEST since f ƒ(x0) /C300 and f §(x0) /C306 "0: An example of a SURFACE with a saddle point is the MONKEY SADDLE . Saddle Point (Game) For a general two-player ZERO-SUM GAME , max i5mmin j5naij 5min j5nmax i5maij : If the two are equal, then write max i5mmin j 5naij 5min j5nmax i5maij /C13v ; where v is called the VALUE of the GAME . In this case, there exist optimal strategies for the first and second players. A NECESSARY and SUFFICIENT condition for a saddle point to exist is the presence of a PAYOFF MATRIX element which is both a minimum of its row and a maximum of its column. A GAME may have more than one saddle point, but all must have the same VALUE . See also GAME,PAYOFF MATRIX ,VALUEReferences Dresher, M. "Saddle Points." §1.5 in The Mathematics of Games of Strategy: Theory and Applications. New York: Dover, pp. 12 /C1/14, 1981. Llewellyn, D. C.; Tovey, C.; and Trick, M. "Finding Sad- dlepoints of Two-Person, Zero Sum Games." Amer. Math. Monthly 95, 912 /C1/918, 1988. Saddle Polygon SKEW POLYGON Saddle-Node Bifurcation FOLD BIFURCATION Safarevich Conjecture SHAFAREVICH CONJECTURE Safe A position in a GAME is safe for a player A if the person who plays next (player B) will lose. See also GAME,UNSAFE Sagitta The PERPENDICULAR distance s from an ARC’s MID- POINT to the CHORD across it, equal to the RADIUS r minus the APOTHEM a, s /C30r /C28a: (1) For a REGULAR POLYGON of side length a, s /C13R /C28r /C301 2 a cscp n ! /C28cotp n ! "# /C301 2 a tanp 2n ! (2) /C30r tanp n ! tanp 2n ! (3) /C302R sin2p 2n ! : (4) where R is the CIRCUMRADIUS , r the INRADIUS , a is the side length, and nis the number of sides. See also APOTHEM ,CHORD ,SECTOR ,SEGMENT Saint Andrew’s Cross AG REEK CROSS rotated by 45 8, also called the crux decussata. The MULTIPLICATION SIGN /C29 is based on Saint Andrew’s cross (Bergamini 1969). See also CROSS ,GREEK CROSS ,MULTIPLICATION SIGN References Bergamini, D. Mathematics. New York: Time-Life Books, p. 11, 1969. Saint Anthony’s Cross A CROSS also called the tau cross or crux commissa. See also CROSS Saint Petersburg Paradox Consider a game, first proposed by Daniel Bernoulli, in which a player bets on how many TOSSES of a COIN will be needed before it first turns up heads. The player pays a fixed amount initially, and then receives 2n dollars if the coin comes up heads on the nth toss. The expectation value of the gain is then 1 2(2) /C2714(4) /C2718(8) /C27.../C301 /C271 /C271 /C27.../C30/C12 dollars, so any finite amount of money can be wagered and the player will still come out ahead on average. Feller (1968) discusses a modified version of the game in which the player receives nothing if a trial takes more than a fixed number N of tosses. The classical theory of this modified game concluded that /C12 is a fair entrance fee, but Feller notes that "the modern student will hardly understand the mysterious dis- cussions of this ‘paradox’." In another modified version of the game, the player bets $2 that heads will turn up on the first throw, $4 that heads will turn up on the second throw (if it did not turn up on the first), $8 that heads will turn up on the third throw, etc. Then the expected payoff is 1 2(2) /C2714(4) /C2718(8) /C27.../C301 /C271 /C271 /C27.../C30/C12; so the player can apparently be in the hole by any amount of money and still come out ahead in the end. This paradox can clearly be resolved by making the distinction between the amount of the final payoffand the net amount won in the game. It is misleading to consider the payoff without taking into account the amount lost on previous bets, as can be shown as follows. At the time the player first wins (say, on the nth toss), he will have lost Xn/C281 k /C3012k /C302n /C282 dollars. In this toss, however, he wins 2n dollars. This means that the net gain for the player is a whopping $2, no matter how many tosses it takes to finally win. As expected, the large payoff after a long run of tails is exactly balanced by the large amount that the player has to invest. In fact, by noting that the probability of winning on the nth toss is 1=2n ; it can be seen that the probability distribution for the number of tosses needed to win is simply a GEO- METRIC DISTRIBUTION with p/C301=2:/ See also COIN TOSSING ,GAMBLER’S RUIN,GEOMETRIC DISTRIBUTION ,MARTINGALE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 201 /C1/202, 1987. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 13 /C1/15, 1998. Feller, W. "The Petersburg Game." §10.4 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 235 /C1/237, 1968. Gardner, M. The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 51 /C1/52, 1959. Kamke, E. Einfu ¨hrung in die Wahrscheinlichkeitstheorie. Leipzig, Germany, pp. 82 /C1/89, 1932. Keynes, J. M. K. "The Application of Probability to Con- duct." In The World of Mathematics, Vol. 2 (Ed. K. New- man). Redmond, WA: Microsoft Press, 1988. Kraitchik, M. "The Saint Petersburg Paradox." §6.18 in Mathematical Recreations. New York: W. W. Norton, pp. 138 /C1/139, 1942. Todhunter, I. §391 in History of the Mathematical Theory of Probability. New York: Chelsea, p. 221, 1949. Sal WALSH FUNCTION Salamin Formula BRENT- SALAMIN FORMULA Salem Constants Each point of a P ISOT- VIJAYARAGHAVAN CONSTANT S is a LIMIT POINT from both sides of a set Tknown as the Salem constants (Salem 1945). The Salem con- stants are ALGEBRAIC INTEGERS >1 in which one or more of the conjugates is on the UNIT CIRCLE with the others inside (Le Lionnais 1983, p. 150). The smallestknown Salem number was found by Lehmer (1933) as the largest REAL ROOT of x10 /C27x9 /C28x7 /C28x6 /C28x5 /C28x4 /C28x3 /C27x /C271 /C300 ; which is s1 /C301:176280818... (Le Lionnais 1983, p. 35). Boyd (1977) found the following table of small Salem numbers, and sug- gested that s1 ; s2 ; s3 ; and s4 are the smallest Salem numbers. The NOTATION 110/C281 /C281 /C281 is short for 110/C281 /C281 /C281 /C281 /C281 0 1 1, the coefficients of the above polynomial. k / sk// (/ POLYNOMIAL 1 1.1762808183 10 1 1 0 /C281 /C281 /C281 2 1.1883681475 18 1 /C2811/C28100/C2811/C2811 3 1.2000265240 14 1 0 0 /C281 /C281001 4 1.2026167437 14 1 0 /C2810000 /C281 5 1.2163916611 10 1000 /C281 /C281 6 1.2197208590 18 1 /C281000000 /C2811 7 1.2303914344 10 1 0 0 /C2810/C281 8 1.2326135486 20 1 /C281000 /C281100 /C2811 9 1.2356645804 22 1 0 /C281 /C281000110 /C281 /C281 10 1.2363179318 16 1 /C281000000 /C281 11 1.2375048212 26 1 0 /C28100/C28100/C28101 001 12 1.2407264237 12 1 /C2811/C28100/C281 13 1.2527759374 18 1 0 0000 /C281 /C281 /C281 /C281 14 1.2533306502 20 1 0 /C28100/C28100000 15 1.2550935168 14 1 0 /C281 /C281010 /C281 16 1.2562211544 18 1 /C28100/C2811000 /C281 17 1.2601035404 24 1 /C28100/C28110/C2811 /C281 01/C281 18 1.2602842369 22 1 /C2810/C2811000 /C2811 /C2811 19 1.2612309611 10 1 0 /C28100/C281 20 1.2630381399 26 1 /C2810000 /C281000000 1 21 1.2672964425 14 1 /C2810000 /C2811 22 1.2806381563 8 1 0 0 /C281 /C281 23 1.2816913715 26 1 0 0000 /C281 /C281 /C281 /C281 /C281 /C281 /C281 /C281 24 1.2824955606 20 1 /C2822/C2822/C28210/C2811 /C28125 1.2846165509 18 1 0 0 0 /C2810/C281 /C2810 /C281 26 1.2847468215 26 1 /C28211/C282100 /C28110 /C2811/C281 27 1.2850993637 30 10000 /C281 /C281 /C281 /C281 /C281 /C28100001 28 1.2851215202 30 1 /C2822/C28210/C2812/C2821 0 /C2811/C2811/C281 29 1.2851856708 30 1 /C281000000 /C281000 /C28100/C281 30 1.2851967268 26 1 0 /C281 /C28100010 /C281 /C281011 31 1.2851991792 44 1 /C28100000 /C281000 /C28100000001001 32 1.2852354362 30 1 0 /C28100/C281 /C2810001 0010 /C281 33 1.2854090648 34 1 /C28100/C2811/C28101/C281 10/C2811/C28101/C281 34 1.2863959668 18 1 /C2822/C2822/C2822/C2833 /C283 35 1.2867301820 26 1 /C28100/C2811/C28101/C281 10/C2811 36 1.2917414257 24 1 /C2810000 /C281000000 37 1.2920391602 20 1 0 /C28100/C28100/C28101 38 1.2934859531 10 1 0 /C281 /C28101 39 1.2956753719 18 1 /C28100/C2811/C28101/C281 See also PISOT- VIJAYARAGHAVAN CONSTANT References Boyd, D. W. "Small Salem Numbers." Duke Math. J. 44, 315 /C1/328, 1977. Boyd, D. W. "Pisot and Salem Numbers in Intervals of the Real Line." Math. Comput. 32, 1244 /C1/1260, 1978. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Lehmer, D. H. "Factorization of Certain Cyclotomic Func- tions." Ann. Math., Ser. 2 34, 461 /C1/479, 1933. Salem, R. "Power Series with Integral Coefficients." Duke Math. J. 12, 153 /C1/172, 1945. Stewart, C. L. "Algebraic Integers whose Conjugates Lie Near the Unit Circle." Bull. Soc. Math. France 106, 169 /C1/ 176, 1978. Salesman Problem TRAVELING SALESMAN PROBLEM Salient Point A point at which two noncrossing branches of a curve meet with different tangents. See also CUSP Salinon The above figure formed from four connected SEMI- CIRCLES . The word salinon is Greek for "salt cellar," which the figure resembles. In his Book of Lemmas , Archimedes proved that the salinon has an area equal to the CIRCLE having the line segment joining the top and bottom points as its DIAMETER (Wells 1991). See also ARBELOS ,L UNE,P IECEWISE CIRCULAR CURVE ,SEMICIRCLE References Schwartzman, S. The Words of Mathematics: An Etymologi- cal Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., p. 192, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 144, 1991. Sally Sequence The Sally sequence gives the sequence of lengths of the repetitions which are avoided in the LINUS SEQUENCE . The first few terms are 0, 1, 1, 2, 1, 3, 1, 1, 3, 2, 1, 6, 3, 2, ... (Sloane’s A006346). See also LINUS SEQUENCE References Sloane, N. J. A. Sequences A006346/M0126 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0126 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Salmon Points The 20 CAYLEY LINES generated by a HEXAGON inscribed in a CONIC SECTION pass four at a time though 15 points known as Salmon points (Wells 1991). There is a dual relationship between the 15 Salmon points and the 15 PLU¨ CKER LINES . See also CAYLEY LINES,K IRKMAN POINTS ,PASCAL LINES,PASCAL’S THEOREM ,PLU¨ CKER LINES,STEINER POINTS References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 172, 1991.Salmon’s Theorem There are at least two theorems known as Salmon’s theorem. This first states that if P and S are two points, PX and SY are the perpendiculars from P and S to the POLARS of S and P, respectively, with respect to a CIRCLE with center O, then OP=OS /C30PX =SY (Durell 1928). The second Salmon’s theorem states that, given a track bounded by two confocal ELLIPSES , if a ball is rolled so that its trajectory is tangent to the inner ELLIPSE , the ball’s trajectory will be tangent to the inner ELLIPSE following all subsequent caroms as well. See also BILLIARDS ,POLAR References Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 95, 1928. Salmon, G. A Treatise on Conic Sections. New York: Chelsea, p. 182, 1960. Saltus The word saltus has two different meanings: either a jump or an oscillation of a function. Sample See also POPULATION ,SAMPLE PROPORTION ,SAMPLE SIZE,SAMPLE SPACE ,SAMPLE VARIANCE ,SAMPLING References Kenney, J. F. and Keeping, E. S. "Populations and Sam- ples." §7.1 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 90 /C1/91, 1962. Sample Proportion Let there be xsuccesses out of nBERNOULLI TRIALS . The sample proportion is the fraction of samples which were successes, so ˆp/C30x n: (1) For large n,ˆphas an approximately NORMAL DIS- TRIBUTION . Let RE be the RELATIVE ERROR and SE the STANDARD ERROR , then /C142p/C143/C30p (2) SEˆpðÞ/C13sˆpðÞ/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p(1/C28p) ns (3) RE ˆpðÞ/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ˆp1/C28ˆp ðÞ ns erf/C281(CI) ; (4) where CI is the CONFIDENCE INTERVAL and erf xis the ERF function. The number of tries needed to deter- mine pwith RELATIVE ERROR RE and CONFIDENCE INTERVAL CI is n /C302 erf /C281(CI)YrtYrP 2ˆp 1 /C28 ˆp ðÞ (RE)2 : (5) Sample Size See also SAMPLE ,SAMPLE VARIANCE Sample Space Informally, the sample space for a given set of events is the set of all possible values the events may assume. Formally, the set of possible events for a given variate forms a SIGMA ALGEBRA , and sample space is defined as the largest set in the SIGMA ALGEBRA . See also PROBABILITY SPACE ,R ANDOM VARIABLE , SAMPLE ,SIGMA ALGEBRA ,STATE SPACE Sample Variance To estimate the population VARIANCE s2from a sample of N elements with a priori unknown MEAN (i.e., the MEAN is estimated from the sample itself), we need an unbiased ESTIMATOR for s2 : This ESTIMATOR is given by K-STATISTIC k2 ; where k2 /C30N N /C28 1m2 (1) and m2 /C13s2 is the sample variance s2 /C131 NXN i/C301xi /C28 ¯x ðÞ2: (2) Note that some authors prefer the definition s ?2 /C131 N /C28 1XN i/C301xi /C28 ¯x ðÞ2; (3) since this makes the sample variance an UNBIASED ESTIMATOR for the population variance. Also note that, in general,ffiffiffiffiffi ˆs2p in not an UNBIASED ESTIMATOR of s even if ˆs2 is an UNBIASED ESTIMATOR for s2/). See also K-STATISTIC ,SAMPLE ,UNBIASED ESTIMATOR , VARIANCE Sampling The selection and implementation of statistical ob- servations in order to estimate properties of an underlying population. Sampling is a vital part of modern polling, market research, and manufactur-ing, and its proper use is vital in the functioning of modern economies. For infinite precision sampling of a band-limited signal at the NYQUIST FREQUENCY , the SIGNAL-TO- NOISE RATIO after Nq samples is SNR /C30/C142r/C12/C143 s/C12/C30rs2 s2N /C281 =2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 r2p /C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 r2pffiffiffiffiffiffi Nqq ; (1) where r is the normalized CROSS-CORRELATION COEF- FICIENT r /C13/C142x(t) /C143/C142y(t) /C143ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2(t) hi y2(t) hip : (2) For r /C101 ; SNR : rffiffiffiffiffiffiN qq : (3) The identical result is obtained for oversampling. For undersampling, the SIGNAL-TO-NOISE RATIO decreases (Thompson et al. 1986). See also NYQUIST SAMPLING ,OVERSAMPLING ,QUAN- TIZATION EFFICIENCY ,SAMPLE ,SAMPLING FUNCTION , SHANNON SAMPLING THEOREM ,SINC FUNCTION References Feuer, A. Sampling in Digital Signal Processing and Control. Boston, MA: Birkha ¨user, 1996. Govindarajulu, Z. Elements of Sampling Theory and Meth- ods. Upper Saddle River, NJ: Prentice-Hall, 1999. Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, pp. 214 /C1/216, 1986. Sampling Function SHAH FUNCTION Sampling Theorem In order for a band-limited (i.e., one with a zero POWER SPECTRUM for frequencies n > B) baseband (/ n > 0) signal to be reconstructed fully, it must be sampled at a rate n ]2B: A signal sampled at n /C302B is said to be NYQUIST SAMPLED , and n /C302B is called the NYQUIST FREQUENCY . No information is lost if a signal is sampled at the N YQUIST FREQUENCY , and no additional information is gained by sampling faster than this rate. See also ALIASING ,N YQUIST FREQUENCY ,N YQUIST SAMPLING ,OVERSAMPLING Sampling Theory The study of SAMPLING San Marco Fractal The FRACTAL J(/C283 =4; 0); where J is the JULIA SET.It slightly resembles the MANDELBROT SET. See also DENDRITE FRACTAL ,D OUADY’S RABBIT FRACTAL ,JULIA SET,M ANDELBROT SET,S IEGEL DISK FRACTAL References Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 173, 1991. Sandwich Theorem The LOVA´ SZ NUMBER q(G)ofa GRAPH G satisfies v(G) 5q¯GYrvYru 5 x(G) : where v(G) is the CLIQUE NUMBER and x is the minimum number of colors needed to color the VERTICES of G. q(G) can be computed efficiently despite the fact that the computation of the two numbers it lies between is an NP-HARD PROBLEM . The SQUEEZING THEOREM is also sometimes known as the sandwich theorem. See also HAM SANDWICH THEOREM ,S QUEEZING THEOREM References Gro¨tschel, M.; Lova´sz, L.; and Schrijver, A. "The Ellipsoid Method and Its Consequences in Combinatorial Optimiza- tion." Combinatorica 1, 169 /C1/197, 1981. Knuth, D. E. "The Sandwich Theorem." Electronic J. Com- binatorics 1,A11 /C1/48, 1994. http://www.combinatoric- s.org/Volume_1/volume1.html#A1. Sangaku Problem A geometric problem found on a mathematical woo- den tablet ( in Japan. Such problems typically involve mutually TANGENT CIRCLES or TANGENT SPHERES . See also CASEY’S THEOREM ,C IRCLE INSCRIBING , CYLINDER- SPHERE INTERSECTION ,DESCARTES CIRCLE THEOREM ,E LLIPSE TANGENT ,H EXLET ,JAPANESE THEOREM ,RIGHT TRIANGLE ,TANGENT CIRCLES ,TAN- GENT SPHERESReferences Fukagawa, H. and Sokolowsky, D. Traditional Japanese Mathematics Problems from the 18th and 19th Centuries. Singapore: Science Culture Technology Press, in prepara- tion. Fukagawa, H. and Pedoe, D. Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, 1989. Mikami, Y. The Development of Mathematics in China and Japan, 2nd ed. New York: Chelsea, 1974. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/91, May 1998. Smith, D. E. and Mikami, Y. A History of Japanese Mathe- matics. Chicago: Open Court, 1914. Sard’s Theorem The set of "critical values" of a MAP u : Rn 0 Rn of CLASS C1 has LEBESGUE MEASURE 0inRn :/ See also CLASS (MAP), LEBESGUE MEASURE ,TRANS- VERSAL INTERSECTION References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 682, 1980. Sarkovskii’s Theorem Order the NATURAL NUMBERS as follows: 3 )5 )7 )9 )11 )13 )15 )...)2 /C215 3 )2 /C215 5 )2 /C215 7 )2 /C215 9 )...)2 /C215 2 /C215 3 )2 /C215 2 /C215 5 )2 /C215 2 /C215 7 )2 /C215 2 /C215 9 )...)2 /C215 2 /C215 2 /C215 3 )...)25 )24 )23 )22 )2 )1: Now let F be a CONTINUOUS FUNCTION from the REALS to the REALS and suppose p )q in the above ordering. Then if F has a point of LEAST PERIOD p, then F also has a point of LEAST PERIOD q. A special case of this general result, also known as Sarkovskii’s theorem, states that if a CONTINUOUS REAL function has a PERIODIC POINT with period 3, then there is a PERIODIC POINT of period n for every INTEGER n. A converse to Sarkovskii’s theorem says that if p )q in the above ordering, then we can find a CONTINUOUS FUNCTION which has a point of LEAST PERIOD q, but does not have any points of LEAST PERIOD p(Elaydi 1996). For example, there is a CONTINUOUS FUNCTION with no points of LEAST PERIOD 3 but having points of all other LEAST PERIODS . See also LEAST PERIOD References Conway, J. H. and Guy, R. K. "Periodic Points." In The Book of Numbers. New York: Springer-Verlag, pp. 207 /C1/208, 1996. Devaney, R. L. An Introduction to Chaotic Dynamical Systems, 2nd ed. Reading, MA: Addison-Wesley, 1989. Elaydi, S. "On a Converse of Sharkovsky’s Theorem." Amer. Math. Monthly 103, 386 /C1/392, 1996. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 49, 1993. Sharkovsky, A. N. "Co-Existence of Cycles of a Continuous Mapping of a Line onto Itself." Ukranian Math. Z. 16,61/C1/ 71, 1964. Stefan, P. "A Theorem of Sharkovsky on the Existence of Periodic Orbits of Continuous Endomorphisms of the Real Line." Comm. Math. Phys. 54, 237 /C1/248, 1977. Sa´rko¨zy’s Theorem A partial solution to the ERDOS SQUAREFREE CON- JECTURE which states that the BINOMIAL COEFFICIENT 2n nYrvYru is never SQUAREFREE for all sufficiently large n ] n0 : Sa´rkozy (1985) showed that if s(n) is the square part of the BINOMIAL COEFFICIENT2n nYrvYru ; then ln s(n) /C2ffiffiffi 2p /C282Yru*Yru+ z1 2Yru*Yru+ffiffiffinp: where z(z) is the RIEMANN ZETA FUNCTION . An upper bound on n0 of 28,000 has been obtained. See also BINOMIAL COEFFICIENT ,ERDOS SQUAREFREE CONJECTURE References Erdos, P. and Graham, R. L. Old and New Problems and Results in Combinatorial Number Theory. Geneva, Swit- zerland: L’Enseignement Mathe ´matique Universite ´ de Gene`ve, Vol. 28, 1980. Sander, J. W. "A Story of Binomial Coefficients and Primes." Amer. Math. Monthly 102, 802 /C1/807, 1995. Sa´rkozy, A. "On the Divisors of Binomial Coefficients, I." J. Number Th. 20,70/C1/80, 1985. Vardi, I. "Applications to Binomial Coefficients." Computa- tional Recreations in Mathematica. Reading, MA: Addi- son-Wesley, pp. 25 /C1/28, 1991. Sarrus Linkage ALINKAGE which converts circular to linear motion using a hinged square. See also HART’S INVERSOR ,LINKAGE ,PEAUCELLIER INVERSOR Sarrus Number POULET NUMBERSarti Dodecic The DODECIC SURFACE defined by X12/C30243S12/C2822Q12/C300; (1) where Q12/C30x2/C27y2/C27z2/C27w2YrvYru6(2) S12/C3033ffiffiffi 5p s/C28 2;3/C27s/C283;4/C27s/C284;2Yru*Yru+ /C2719s/C272;3/C27s/C273;4/C27s/C274;2Yru*Yru+ /C2710s2;3;4/C2814s1;0/C272s1;1/C286s1;2 /C28352s5;1/C27336l25l1/C2748l2l3l4 (3) l1/C30x4/C27y4/C27z4/C27w4(4) l2/C30x2y2/C27z2w2(5) l3/C30x2z2/C27y2w2(6) l4/C30x2w2/C27y2z2(7) l5/C30xyzw (8) s1;0/C30l1l2l3/C27l2l4/C27l3l4 ðÞ (9) s1;1/C30l21l2/C27l3/C27l4 ðÞ (10) s1;2/C30l1l22/C27l23/C27l24YrvYru (11) s5;1/C30l25l2/C27l3/C27l4 ðÞ (12) s2;3;4/C30l32/C27l33/C27l34 (13) s9 2;3/C30l2 2l39l2l23(14) s9 3;4/C30l2 3l49l3l24 (15) s94;2/C30l24l29l4l22: (16) /Q12 and S12 are both invariants of order 12. The Sarti surface is invariant under the BIPOLYHEDRAL GROUP and has exactly 600 ORDINARY POINTS (Endraß). It was discovered by A. Sarti in 1999. See also ALGEBRAIC SURFACE ,BIPOLYHEDRAL GROUP , DODECIC SURFACE References Endraß, S. "The Sarti Surface." http://enriques.mathemati- k.uni-mainz.de/kon/docs/Esarti.shtml. SAS Theorem Specifying two sides and the ANGLE between them uniquely determines a TRIANGLE . Let c be the base length and h be the height. Then the AREA is K /C301 2 ch /C3012 ac sin B : (1) The length of the third side is given by the LAW OF COSINES , b2 /C30a2 /C27c2 /C282ac cos B: so b /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27c2 /C282ac cos Bp : (2) Using the LAW OF SINES a sin A /C30b sin B /C30c sin C (3) then gives the two other ANGLES as A /C30sin/C281 a sin Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 c2 /C28 2ac cos Bp ! (4) C /C30sin/C281 c sin Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27 c2 /C28 2ac cos Bp ! (5) See also AAA THEOREM , AAS THEOREM , ASA THEO- REM, ASS THEOREM , SSS THEOREM ,TRIANGLESatellite Knot Let K1 be a knot inside a TORUS , and knot the TORUS in the shape of a second knot (called the COMPANION KNOT ) K2 ; with certain additional mild restrictions to avoid trivial cases. Then the new knot resulting from K1is called the satellite knot K3 : All satellite knots are PRIME (Hoste et al. 1998). The illustration above illustrates a satellite knot of the TREFOIL KNOT , which is the form all satellite knots of 16 or fewer crossings take (Hoste et al. 1998). Satellites of the trefoil share the trefoil’s chirality, and all have wrapping number 2. Any satellite knot having wrapping number > 2 must have at least 27 crossings, and any satellite of the FIGURE EIGHT KNOT must have at least 17 crossings (Hoste et al. 1998). The numbers of satellite knots with n crossings are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, ... (Sloane’s A051765), so the satellite knot of minimal crossing number occurs for 13 crossings. The only KNOTS which are not HYPERBOLIC KNOTS are TORUS KNOTS and satellite knots (including COMPO- SITE KNOTS ). No satellite knot is an ALMOST ALTER- NATING KNOT .Ifa COMPANION KNOT has crossing number k and the satellite ravels m times long- itudinally around the solid torus, then it is conjec- tured that the satellite cannot be projected with fewer than km2crossings (Hoste et al. 1998). See also ALMOST ALTERNATING KNOT,CABLE KNOT, COMPANION KNOT,C OMPOSITE KNOT,H YPERBOLIC KNOT,TORUS KNOT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 115 /C1/118, 1994. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,3 3/C1/48, Fall 1998. Sloane, N. J. A. Sequences A051765 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Satisfaction LetAbe a RELATIONAL SYSTEM , and let Lbe a language which is appropriate for A:Letfbe a well-formed formula of L, and let sbe a valuation in A: Then A ffis f is written provided that one of the following holds: 1. f is of the form x /C30y, for some variables x and y of L, and s maps x and y to the same element of the structure A:/ 2. f is of the form Rx1 /C1/C1/C1xn ; for some n-ary predicate symbol R of the language L, and some variables x1 /C1/C1/C1xnof L, and sx1ðÞ ; ...; sxnðÞ fg is a member of RA :/ 3. f is of the form ( c ffl g) ; for some formulas c and g of L such that A ffis c and A ffis g :/ 4. f is of the form (( /C215x) c); and there is an element a of A such that A ffis(x½a) c :/ In this case, A is said to satisfy f with the valuation s. See also LOS’ THEOREM References Bell, J. L. and Slomson, A. B. Models and Ultraproducts: An Introduction. Amsterdam, Netherlands: North-Holland, 1969. Enderton, H. E. A Mathematical Introduction to Logic. Boston, MA: Academic Press, 1972. Satisfiability Problem Deciding whether a given Boolean formula in con- junctive normal form has an assignment that makes the formula "true." In 1971, Cook showed that the problem is NP-COMPLETE . See also BOOLEAN ALGEBRA References Cook, S. A. and Mitchell, D. G. "Finding Hard Instances of the Satisfiability Problem: A Survey." In Satisfiability Problem: Theory and Applications (Piscataway, NJ, 1996) (Ed. D. Du, J. Gu, and P. M.Pardalos). Providence, RI: Amer. Math. Soc., pp. 1 /C1/17, 1997. Sausage Conjecture In n-D for n ]5 the arrangement of HYPERSPHERES whose CONVEX HULL has minimal CONTENT is always a "sausage" (a set of HYPERSPHERES arranged with centers along a line), independent of the number of n- spheres. The CONJECTURE was proposed by Fejes To´th, and solved for dimensions ]42 by Betke et al. (1994) and Betke and Henk (1998). See also CONTENT ,C ONVEX HULL,H YPERSPHERE , HYPERSPHERE PACKING ,SPHERE PACKING References Betke, U.; Henk, M.; and Wills, J. M. "Finite and Infinite Packings." J. reine angew. Math. 453, 165 /C1/191, 1994. Betke, U. and Henk, M. "Finite Packings of Spheres." Discrete Comput. Geom. 19, 197 /C1/227, 1998. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Problem D9 in Unsolved Problems in Geometry. New York: Springer- Verlag, 1991.Fejes To´th, L. "Research Problems." Periodica Methematica Hungarica 6, 197 /C1/199, 1975. Savitzky-Golay Filter A low-pass filter which is useful for SMOOTHING data. See also FILTER References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 183 and 644 /C1/645, 1992. Savoy Knot FIGURE-OF- EIGHT KNOT Sawada-Kotera Equation The PARTIAL DIFFERENTIAL EQUATION ut /C2745u2ux /C2715uxuxx /C2715uuxxx /C27uxxxxx /C300: See also CAUDREY- DODD- GIBBON- SAWADA- KOTERA EQUATION References Matsumo, Y. Bilinear Transformation Method. New York: Academic Press, p. 7, 1984. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 134, 1997. Sawtooth Wave The periodic function given by S(x) /C30A frac( x=T /C27 f): (1) where frac( x) is the FRACTIONAL PART frac x /C13x /C28 xbc; A is the amplitude, T is the period of the wave, and f is its phase. If f /C300; A /C301, and T /C302L ; then the FOURIER SERIES is given by f(x)/C301 2/C281 pX/C12 n/C2811 nsinnpx L ! : See also FOURIER SERIES– SAWTOOTH WAVE,FRAC- TIONAL PART,STAIRCASE FUNCTION References Spanier, J. and Oldham, K. B. An Atlas of Functions. Washington, DC: Hemisphere, p. 74, 1987. sc JACOBI ELLIPTIC FUNCTIONS Scalar A one-component quantity which is invariant under ROTATIONS of the coordinate system. See also PSEUDOSCALAR ,S CALAR FIELD ,S CALAR FUNCTION ,SCALAR MULTIPLICATION ,SCALAR POTEN- TIAL,SCALAR TRIPLE PRODUCT ,TENSOR ,VECTOR References Jeffreys, H. and Jeffreys, B. S. "Scalars and Vectors." Ch. 2 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 56 /C1/85, 1988. Scalar Curvature The scalar curvature (called the "curvature scalar" by Weinberg 1972, p. 135) is given by R /C13g mkRmk : where g mk is the METRIC TENSOR and Rmk is the RICCI TENSOR . See also CURVATURE ,EINSTEIN TENSOR ,G AUSSIAN CURVATURE ,M EAN CURVATURE ,M ETRIC TENSOR , RADIUS OF CURVATURE ,R ICCI TENSOR ,R IEMANN- CHRISTOFFEL TENSOR References Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravita- tion. San Francisco: W. H. Freeman, p. 222, 1973. Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, p. 40, 1984. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 135, 1972. Scalar Field A MAP f : Rn /C2R which assigns each x a SCALAR FUNCTION f(x) :/ See also VECTOR FIELD References Morse, P. M. and Feshbach, H. "Scalar Fields." §1.1 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 4 /C1/8, 1953. Scalar Function A function f(x1 ; ...; xn) of one or more variables whose RANGE is one-dimensional, as compared to a VECTOR FUNCTION , whose RANGE is three-dimensional (or, in general, n-dimensional). See also COMPLEX FUNCTION ,REAL FUNCTION ,VEC- TOR FUNCTIONScalar Multiplication Scalar multiplication refers to the multiplication of a VECTOR by a constant s, producing a vector in the same (for s /C21 0) or opposite (for s B 0) direction but of different length. Scalar multiplication is indicated in Mathematica by placing a scalar next to a vector (with or without an optional asterisk), s{a1, a2, ..., an}. See also MULTIPLICATION ,V ECTOR ,V ECTOR ADDI- TION ,VECTOR MULTIPLICATION Scalar Potential A conservative VECTOR FIELD (for which the CURL 9/C29 F /C300) may be assigned a scalar potential f(x; y; z) /C28 f(0; 0 ;0) /C13/C28gCF /C215 ds /C30/C28g(x; 0 ; 0) (0; 0 ; 0)F1(t; 0 ; 0) dt /C27g(x ; y; 0) (x; 0; 0)F2(x; t; 0) dt /C27gx ; y ; z (x ; y; 0)F3(x; y; t) dt: where fCF /C215 ds is a LINE INTEGRAL . See also LINE INTEGRAL ,P OTENTIAL FUNCTION , VECTOR POTENTIAL Scalar Product DOTPRODUCT Scalar Triple Product The scalar triple product of three VECTORS A,B, and Cis denoted [ A;B;C] and defined by [A;B;C]/C13A /C215(B/C29C) (1) /C30B /C215(C/C29A) (2) /C30C /C215(A/C29B) (3) /C30det(A(BC)) (4) /C30A1A2A3 A2C2B3 A3C2C3YrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut(5) where A /C215Bdenotes a DOT PRODUCT ,A/C29Bdenotes a CROSS PRODUCT , det( A)/C30½A½denotes a DETERMINANT , and Ai ; Bi ; and Ci are components of the vectors A, B, and C, respectively. The scalar triple product is a PSEUDOSCALAR (i.e., it reverses sign under inversion). The scalar triple product can also be written in terms of the PERMUTATION SYMBOL eijk as A /C215 (B /C29C) /C30eijkAiBjCk : (6) where EINSTEIN SUMMATION has been used to sum over repeated indices. Additional identities involving the scalar triple pro- duct are A /C215(B /C29C) /C30B /C215(C /C29A) /C30C /C215(A /C29B) (7) [A ; B; C]D /C30[D ; B; C]A /C27[A ; D ; C]B /C27[A ; B ; D]C (8) [q; q?; qƒ][r; r?; rƒ] /C30q /C215 rq /C215 r? q /C215 r ƒ q ?/C215 rq?/C215 r? q ?/C215 rƒ q ƒ/C215 rqƒ/C215 r? q ƒ/C215 rƒYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut: (9) The VOLUME of a PARALLELEPIPED whose sides are given by the vectors A, B, and C is given by the ABSOLUTE VALUE of the scalar triple product Vparallelepiped /C30:½A /C215(B /C29C)½: (10) See also CROSS PRODUCT ,DOT PRODUCT ,PARALLELE- PIPED ,V ECTOR MULTIPLICATION ,V ECTOR TRIPLE PRODUCT References Arfken, G. "Triple Scalar Product, Triple Vector Product." §1.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 26 /C1/33, 1985. Jeffreys, H. and Jeffreys, B. S. "The Triple Scalar Product." §2.091 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 74 /C1/75, 1988. Scale BASE (NUMBER ) Scale Factor For a diagonal METRIC TENSOR gij /C30gii dij ; where dij is the KRONECKER DELTA , the scale factor is defined by hi /C13ffiffiffiffiffigiip: (1) The LINE ELEMENT (first FUNDAMENTAL FORM ) is then given by ds2 /C30g11 dx2 11 /C27g22 dx222 /C27g33 dx233 (2) /C30h21 dx211 /C27h22 dx222 /C27h23 dx233 : (3) The scale factor appears in vector derivatives of coordinates in CURVILINEAR COORDINATES . See also CURVILINEAR COORDINATES ,FUNDAMENTAL FORMS ,LINE ELEMENTScale Invariance SELF-SIMILARITY Scalene Triangle A TRIANGLE with three unequal sides. See also ACUTE TRIANGLE ,EQUILATERAL TRIANGLE , ISOSCELES TRIANGLE ,OBTUSE TRIANGLE ,TRIANGLE Scaling Increasing a plane figure’s linear dimensions by a scale factor s increases the PERIMETER p ?0 sp and the AREA A?0 s2A:/ See also CONTRACTION (GEOMETRY ), EXPANSION , FRACTAL ,HOMOTHETIC ,SELF-SIMILARITY Scattering Operator An OPERATOR relating the past asymptotic state of a DYNAMICAL SYSTEM governed by the Schro ¨dinger equation id dtc(t) /C30H c(t) to its future asymptotic state. See also WAVE OPERATOR Scattering Theory The mathematical study of the SCATTERING OPERATOR and Schro ¨dinger equation. See also SCATTERING OPERATOR References Yafaev, D. R. Mathematical Scattering Theory: General Theory. Providence, RI: Amer. Math. Soc., 1996. Schaar’s Identity A generalization of the GAUSSIAN SUM. For p and q of opposite PARITY (i.e., one is EVEN and the other is ODD), Schaar’s identity states 1 ffiffiffiqpXq /C281 r /C300e/C28 pir2p =q /C30e/C28 pi=4 ffiffiffippXp /C281 r/C300epjr2q =p : Schaar’s identity can also be written so as to be valid for p, q with pq EVEN . See also GAUSSIAN SUM References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Evans, R. and Berndt, B. "The Determination of Gauss Sums." Bull. Amer. Math. Soc. 5, 107/C1/129, 1981. Schanuel’s Conjecture Let l1 ; ..., ln /C23C be linearly independent over the RATIONALS Q; then Q l1 ; ...; ln ; e l1 ; ...; e lnYrvYru has TRANSCENDENCE degree at least n over Q: Schanuel’s conjecture implies the LINDEMANN- WEIER- STRASS THEOREM and GELFOND’S THEOREM . If the conjecture is true, then it follows that e and p are ALGEBRAICALLY INDEPENDENT . Mcintyre (1991) proved that the truth of Schanuel’s conjecture also guarantees that there are no unexpected exponential- algebraic relations on the INTEGERS Z (Marker 1996). At present, a proof of Schanuel’s conjecture seems out of reach (Chow 1999). See also ALGEBRAICALLY INDEPENDENT ,C ONSTANT PROBLEM ,GELFOND’S THEOREM ,LINDEMANN- WEIER- STRASS THEOREM References Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440 /C1/448, 1999. Chudnovsky, G. V. "On the Way to Schanuel’s Conjecture." Ch. 3 in Contributions to the Theory of Transcendental Numbers. Providence, RI: Amer. Math. Soc., pp. 145 /C1/176, 1984. Lin, F.-C. "Schanuel’s Conjecture Implies Ritt’s Conjecture." Chinese J. Math. 11,41/C1/50, 1983. Macintyre, A. "Schanuel’s Conjecture and Free Exponential Rings." Ann. Pure Appl. Logic 51, 241 /C1/246, 1991. Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753 /C1/759, 1996. Schauder Fixed Point Theorem Let A be a closed convex subset of a BANACH SPACE and assume there exists a continuous MAP T sending A to a countably compact subset T(A)ofA. Then T has fixed points. References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 543, 1980. Schauder, J. "Der Fixpunktsatz in Funktionalra ¨umen." Studia Math. 2, 171 /C1/180, 1930. Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995. Scheme A local-ringed SPACE which is locally isomorphic to an AFFINE SCHEME . See also AFFINE SCHEMEReferences Itoˆ, K. (Ed.). "Schemes." §16D in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, p. 69, 1986. Schensted Correspondence A correspondence between a PERMUTATION and a pair of Y OUNG TABLEAUX . See also PERMUTATION ,YOUNG TABLEAU References Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1973. Stanton, D. W. and White, D. E. §3.6 in Constructive Combinatorics. New York: Springer-Verlag, pp. 85 /C1/87, 1986. Scherk’s Minimal Surfaces Scherk’s two MINIMAL SURFACES were discovered by Scherk in 1834. They were the first new surfaces discovered since Meusnier in 1776. Beautiful imagesof wood sculptures of Scherk surfaces are illustrated by Se ´quin. Scherk’s first surface is doubly periodic and is defined by the implicit equation ezcosy/C30cosx; (1) (Osserman 1986, Wells 1991, von Seggern 1993). It has been observed to form in layers of block copoly- mers (Peterson 1988). Scherk’s second surface can be written parametrically as x /C302R ln 1 /C27reiuYrvYru /C28ln 1 /C28reiuYrvYru YrtYrP (2) y /C30R 4i tan/C281 reiuYrvYruYrtYrP (3) z /C30R 2i /C28ln 1 /C28r2e2iuYrtYrP /C27ln 1 /C27r2e2iuYrtYrP YrvYruYr$Yr% (4) for u /C23 [0; 2p) ; and r /C23 (0; 1):/ References Dickson, S. "Minimal Surfaces." Mathematica J. 1,38/C1/40, 1990. do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 41, 1986. Meusnier, J. B. "Me´moire sur la courbure des surfaces." Me´m. des savans e´trangers 10 (lu 1776), 477 /C1/510, 1785. Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 18 and 101, 1986. Peterson, I. "Geometry for Segregating Polymers." Sci. News , 151, Sep. 3, 1988. Scherk, H. F. "Bemerkung u¨ber der kleinste Fla¨che inner- halb gegebener Grenzen." J. reine angew. Math. 13, 185 /C1/ 208, 1834. Se´quin, C. H. "Scherk-Collins Sculpture Generator." http:// www.cs.berkeley.edu/~sequin/SCULPTS/scherk.html. Thomas, E. L.; Anderson, D. M.; Henkee, C. S.; and Hoff- man, D. "Periodic Area-Minimizing Surfaces in Block Copolymers." Nature 334, 598 /C1/601, 1988. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 304, 1993. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 223, 1991. Wolfram Research. "Mathematica Version 2.0 Graphics Gallery." http://www.mathsource.com/cgi-bin/ msitem22?0207 /C1/155.Schiffler Point The CONCURRENCE S of the EULER LINES Enof the TRIANGLES DXBC ;DXCA ;DXAB ; and DABC where X is the INCENTER . The TRIANGLE CENTER FUNCTION is a /C301 cos B /C27 cos C /C30b /C27 c /C28 a b /C27 c: References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Kimberling, C. "Schiffler Point." http://cedar.evansville.edu/ ~ck6/tcenters/recent/schiff.html. Schiffler, K.; Veldkamp, G. R.; and van der Spek, W. A. "Problem 1018 and Solution." Crux Math. 12, 176 /C1/179, 1986. Schinzel Circle A CIRCLE having a given number of LATTICE POINTS on its CIRCUMFERENCE . The Schinzel circle having n lattice points is given by the equation x /C281 2Yru*Yru+2 /C27y2 /C3014 5k /C281forn/C302keven x/C2813Yru*Yru+2 /C27y2/C301952kforn/C302k/C271 odd :8 >< >: Note that these solutions do not necessarily have the smallest possible RADIUS . For example, while the Schinzel circle centered at ( /1=3/, 0) and with radius 625/3 has nine lattice points on its circumference, sodoes the circle centered at ( /1=3/, 0) with radius /65=3/. See also CIRCLE ,CIRCLE LATTICE POINTS ,KULIKOWS- KI’S THEOREM ,LATTICE POINT ,SCHINZEL’S THEOREM , SPHERE References Honsberger, R. "Circles, Squares, and Lattice Points." Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117 /C1/127, 1973. Kulikowski, T. "Sur l’existence d’une sphe `re passant par un nombre donne ´aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 5,8 9/C1/90, 1959. Schinzel, A. "Sur l’existence d’un cercle passant par un nombre donne ´de points aux coordonne ´es entie `res." L’Enseignement Math. Ser. 2 4,7 1/C1/72, 1958. Sierpinski, W. "Sur quelques proble `mes concernant les points aux coordonne ´es entie`res." L’Enseignement Math. Ser. 2 4,25/C1/31, 1958. Sierpinski, W. "Sur un proble `me de H. Steinhaus concernant les ensembles de points sur le plan." Fund. Math. 46, 191 /C1/194, 1959. Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964. Schinzel’s Hypothesis If f1(x); ..., fs(x) are IRREDUCIBLE POLYNOMIALS with INTEGER COEFFICIENTS such that no INTEGER n /C211 divides f1(x) ; ..., fs(x) for all INTEGERS x, then there should exist infinitely many x such that f1(x) ; ..., fs(x) are simultaneously PRIME . References Dickson, L. E. "A New Extension of Dirichlet’s Theorem on Prime Numbers." Messenger Math. 33, 155 /C1/161, 1904. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Schinzel, A. and Sierpinski, W. "Sur certaines hypothe `ses concernant les nombres premiers. Remarque." Acta Ar- ithm. 4, 185 /C1/208, 1958. Schinzel’s Theorem For every POSITIVE INTEGER n, there exists a CIRCLE in the plane having exactly n LATTICE POINTS on its CIRCUMFERENCE . The theorem is based on the num- ber r(n) of integral solutions (x, y) to the equation x2 /C27y2 /C30n; (1) given by r(n) /C304 d1 /C28d3 ðÞ ; (2) where d1 is the number of divisors of n OF THE FORM 4k /C271 and d3 is the number of divisors OF THE FORM 4k /C273: It explicitly identifies such circles (the SCHIN- ZEL CIRCLES )as x /C281 2Yru*Yru+2 /C27y2 /C3014 5k/C281for n /C302k x /C2813Yru*Yru+2 /C27y2 /C3019 52kfor n /C302k /C271 :8 >< >:(3) Note, however, that these solutions do not necessarily have the smallest possible radius. See also BROWKIN’S THEOREM ,KULIKOWSKI’S THEO- REM,SCHINZEL CIRCLE References Honsberger, R. "Circles, Squares, and Lattice Points." Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117 /C1/127, 1973. Kulikowski, T. "Sur l’existence d’une sphe`re passant par un nombre donne ´ aux coordonne ´es entie`res." L’Enseignement Math. Ser. 2 5,89/C1/90, 1959. Schinzel, A. "Sur l’existence d’un cercle passant par un nombre donne ´ de points aux coordonne ´es entie`res." L’Enseignement Math. Ser. 2 4,71/C1/72, 1958. Sierpinski, W. "Sur quelques proble `mes concernant les points aux coordonne ´es entie`res." L’Enseignement Math. Ser. 2 4,25/C1/31, 1958.Sierpinski, W. "Sur un proble `me de H. Steinhaus concernant les ensembles de points sur le plan." Fund. Math. 46, 191 /C1/194, 1959. Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964. Schisma The musical interval by which eight fifths and a major third exceed five octaves, R[z] > 0 See also COMMA OF DIDYMUS ,C OMMA OF PYTHA- GORAS ,DIESIS Schla ¨fli Double Sixes DOUBLE SIXES Schla ¨fli Function The function giving the VOLUME of the spherical quadrectangular TETRAHEDRON : V /C30p2 8fp p ;p q ;p r ! ; where 1 2 p2fp2 /C28x; y ;p 2 /C28zYru*Yru+ /C30X/C12 m/C301D /C28 sin x sin z D /C27 sin x sin z !m /C2cos(2 mx) /C28 cos(2 my) /C27 cos(2 mz) /C28 1 m2 /C28x2 /C28y2 /C28z2 ; and D /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 x cos2 z /C28cos2 yp : See also TETRAHEDRON Schla ¨fli Integral A definition of a function using a CONTOUR INTEGRAL . Schla ¨fli integrals may be converted into RODRIGUES FORMULAS . See also RODRIGUES FORMULA Schla ¨fli Polynomial A polynomial given in terms of the NEUMANN POLY- NOMIALS On(x)by Sn(x) /C302xOn(x) /C28 2 cos21 2npYru*Yru+ n: See also NEUMANN POLYNOMIAL References Erdelyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. Krieger, p. 34, 1981. Gradshteyn, I. S. and Ryzhik, I. M. "Neumann’s and Schla¨fli Polynomials: On(z) and Sn(z):/" §8.59 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 989 /C1/991, 2000. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 196, 1993. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 312 /C1/313, 1966. Schla ¨fli Symbol A symbol of the form fp ; q; r ; ...g used to describe regular polygons, polyhedra, and their higher-dimen- sional counterparts. The symbol fp g denotes a REGULAR POLYGON . The symbol fp ; q g denotes a TESSELLATION of regular p- gons, with q of them surrounding each VERTEX . The Schla ¨fli symbol can also be used to describe PLATONIC SOLIDS and KEPLER- POINSOT SOLIDS , and a general- ized version describes QUASIREGULAR POLYHEDRA and ARCHIMEDEAN SOLIDS . Higher dimensional symbols can be used to describe the REGULAR POLYCHORA and POLYTOPES . The symbol has the particularly nice property that its reversal gives the symbol of the DUAL POLYHEDRON . The following tables gives Schla ¨fli symbols for several polytopes. POLYHEDRON Symbol GREAT STELLATED DODECAHEDRON /5 2; 3no / SMALL STELLATED DODECAHEDRON /5 2; 5no / GREAT ICOSAHEDRON / 3;52no / TETRAHEDRON /f3 ; 3 g/ PENTATOPE /f3 ; 3 ; 3 g/ n-simplex f3 ; ...; 3|fflfflfflfflfflffl{zfflfflfflfflfflffl} n/C281g 16-CELL /f3 ; 3 ; 4 g/ n-cross polytope f3 ; ...; 3|fflfflfflfflfflffl{zfflfflfflfflfflffl} n/C282; 4 g 600-CELL /f3 ; 3 ; 5 g/ OCTAHEDRON /f3 ; 4 g/ 24-CELL /f3 ; 4 ; 3 g/ ICOSAHEDRON /f3 ; 5 g/ CUBE /f4 ; 3 g/TESSERACT /f4 ; 3 ; 3 g/ n-hypercube f4 ; 3 ; ...; 3|fflfflfflfflfflffl{zfflfflfflfflfflffl} n/C282g GREAT DODECAHEDRON / 5;5 2no / DODECAHEDRON /f5 ; 3 g/ 120-CELL /f5 ; 3 ; 3 g/ See also ARCHIMEDEAN SOLID ,P LATONIC SOLID , QUASIREGULAR POLYHEDRON ,R EGULAR POLYCHOR- ON,REGULAR POLYGON ,TESSELLATION Schla ¨fli’s Formula For /R½z/C138/C210/, Jn(z) /C301 p g p =2 0cos(z sin t /C28 nt) dt /C28sin(np) p g/C12 0e /C28z sinh te /C28 nt dt ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1472, 1980. Schla ¨fli’s Modular Form The MODULAR EQUATION of degree five can be written u v !3 /C27v u !3 /C302 u2v2 /C281 u2v2 ! : See also MODULAR EQUATION Schlegel Graph A GRAPH corresponding to POLYHEDRA skeletons. The POLYHEDRAL GRAPHS are special cases. See also POLYHEDRAL GRAPH ,SKELETON References Gardner, M. Wheels, Life, and Other Mathematical Amuse- ments. New York: W. H. Freeman, p. 158, 1983. Schlicht Function An ANALYTIC FUNCTION fon the UNIT DISK is called schlicht if 1. f is ONE-TO-ONE , 2. f(0) /C300 ; and 3. f ?(0) /C301 ;/ in which case it is written f /C23 S: Schlicht functions have power series of the form f(z) /C30Z /C27X/C12 j/C302ajzj : See also AREA PRINCIPLE ,BIEBERBACH CONJECTURE , KO¨ BE FUNCTION ,KO¨ BE’S ONE-FOURTH THEOREM References Krantz, S. G. "Schlicht Functions." §12.1.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 149, 1999. Schlo ¨milch Remainder AT AYLOR SERIES remainder formula that gives after n terms of the series Rn /C30f(n/C271)(x/C31) n!p(x /C28x/C31)n/C271 /C28p x /C28x0 ðÞn/C271 for x/C31/C23 x0 ; x ðÞ and any p /C210 (Blumenthal 1926, Beesack 1966), which Blumenthal (1926) ascribes to Roche (1858). The choices p /C30n /C271 and p /C301 give the LAGRANGE and CAUCHY REMAINDERS , respectively (Beesack 1966). See also CAUCHY REMAINDER ,LAGRANGE REMAINDER References Beesack, P. R. "A General Form of the Remainder in Taylor’s Theorem." Amer. Math. Monthly 73,64/C1/67, 1966. Blumenthal, L. M. "Concerning the Remainder Term in Taylor’s Formula." Amer. Math. Monthly 33, 424 /C1/426, 1926. Maak, W. An Introduction to Modern Calculus. New York: Holt, Rinehart, and Winston, p. 99, 1963. Roche. Mem. de l’Acad. de Montpellier. 1858. Schlo¨milch, O. Kompendium der ho¨heren Analysis. Braunschweig, Germany: Vieweg, 1923. Schlo ¨milch’s Function Mathematics:Calculus and Analysis:Special Func- tions:Hypergeometric Functions:Confluent Hyper- geometric Functions S( n; z) /C13g/C12 0(1 /C27t)/C28 ne /C28zt dt /C30z n/C281ezg/C12 zu/C28 ne /C28u du /C30zn=2 /C281ez=2W/C28 n=2;(1/C28 n)=2(z) ; where Wk ; m(z) is the WHITTAKER FUNCTION . Schlo ¨milch’s Series AF OURIER SERIES -like expansion of a twice continu- ously differentiable functionf(x) /C301 2 a0 /C27X/C12 n/C301anJ0(nx) for 0 Bx B p; where J0(x) is a zeroth order BESSEL FUNCTION OF THE FIRST KIND and a0 /C132f(0) /C272 p g p 0dug p =2 0f ?(u sin f) df an /C132 p g p 0dug p =2 0uf ?(u sin f)cos(np) df: A special case gives the amazing identity 1 /C30J0(z) /C272X/C12 n/C301J2n(z) /C30 J0(z) ½/C1382/C272X/C12 n/C301Jn(z) ½/C1382: See also BESSEL FUNCTION OF THE FIRST KIND, BESSEL FUNCTION FOURIER EXPANSION ,F OURIER SERIES References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1473, 1980. Schmitt-Conway Biprism A CONVEX POLYHEDRON which is SPACE-FILLING , but only aperiodically, was found by Conway in 1993. See also CONVEX POLYHEDRON ,SPACE- FILLING POLY- HEDRON Schnirelmann Constant The constant s0in S CHNIRELMANN’S THEOREM such that every INTEGER >1 is a sum of at most s0PRIMES . Of course, by V INOGRADOV’S THEOREM , it is known that 4 primes suffice for all sufficiently large num- bers, but this constant gives a sufficient number for allnumbers. The best current estimate is s0/C307 (Ramare ´1995), and a summary of progress on upper bounds for s0is summarized in the following table. /s0/author 7 Ramare ´(1995) 19 Riesel and Vaughan (1983) 26 Deshouillers (1977) 27 Vaughan (1977) 55 Klimov (1975) 115 Klimov et al. (1972) 159 Deshouillers (1973) See also SCHNIRELMANN’S THEOREM ,W ARING’S PRO- BLEM References Deshouillers, J.-M. No. 17 in "Ame´lioration de la constante de Snirelman dans le proble ´me de Goldbach." Se´minaire Delange-Pisot-Poitou (14e anne´e: 1972/73). The´orie des nombres: Fascicule 2: Expose ´s17a ` 26, et Groupe d’e´tude. Paris: Secre´tariat Mathe ´matique, pp. 1 /C1/4, 1973. Deshouillers, J.-M. "Sur la constante de Snirel’man." No. G16 in Se´minaire Delange-Pisot-Poitou, 17e anne´e (1975/76). The´orie des nombres: Fascicule 2: Expose ´s23 a` 31 et Groupe d’e´tude. Paris: Secre´tariat Math., pp. 1 /C1/6, 1977. Klimov, K. I. Naucn. Trudy Kuibysev Gos. Ped. Inst. 158, 14 /C1/30, 1975. Klimov, N. I.; Pil’tja / ; G. Z.; and Septickaja, T. A. "An Estimate of the Absolute Constant in the Goldbach- Snirel’man Problem." In Issledovaniya po teorii chisel, Vyp. 4. [Studies in number theory, No. 4] (Ed. N. Lensko /): Saratov: Izdat. Saratov. Univ., pp. 35 /C1/51, 1972. Ramare ´, O. "On Snirel’man’s Constant." Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 645 /C1/706, 1995. Riesel, H. and Vaughan, R. C. "On Sums of Primes." Ark. Mat. 21,46/C1/74, 1983. Vaughan, R. C. "On the Estimation of Schnirelman’s Con- stant." J. reine angew. Math. 290,93/C1/108, 1977. Schnirelmann Density The Schnirelmann density of a sequence of natural numbers is the GREATEST LOWER BOUND of the FRACTIONS A(n) =n where A(n) is the number of terms in the sequence 5n:/ See also MANN’S THEOREM ,SCHNIRELMANN’S THEO- REM References Khinchin, A. Y. "The Landau-Schnirelmann Hypothesis and Mann’s Theorem." Ch. 2 in Three Pearls of Number Theory. New York: Dover, pp. 18 /C1/36, 1998. Schnirelmann’s Theorem This entry contributed by KEVIN O’BRYANT There exists a POSITIVE INTEGER s such that every SUFFICIENTLY LARGE INTEGER is the sum of at most s PRIMES . It follows that there exists a POSITIVE INTEGER s0 ]s such that every INTEGER > 1isa sum of at most s0 PRIMES . The smallest proven value of s0 is known as the SCHNIRELMANN CONSTANT . Schnirelmann’s theorem can be proved using MANN’S THEOREM , although Schnirelmann used the weaker inequality s(A /C154B) ] s(A) /C27 s(B) /C28 s(A)s(B) ; where 0 /C23 A S B; A /C154B /C30fa /C27b : a /C23 A; b /C23 B g; and s is the SCHNIRELMANN DENSITY . Let P /C30 f0; 1; 2; 3; 5; ...g be the set of primes, together with 0 and 1, and let Q /C30P /C154P: Using a sophisticated version of the INCLUSION-EXCLUSION PRINCIPLE , Schnirelmann showed that although s(P) /C300; s(Q) > 0: By repeated applications of MANN’S THEOREM , thesum of k copies of Q satisfies s(Q /C27Q /C27.../C27Q) ] min f1; ks(Q) g: Thus, if k > 1=s(Q) ; the sum of k copies of Q has SCHNIRELMANN DENSITY 1, and so contains all positive integers. See also CHEN’S THEOREM ,GOLDBACH CONJECTURE , MANN’S THEOREM ,PRIME NUMBER ,PRIME PARTITION , SCHNIRELMANN CONSTANT ,SCHNIRELMANN DENSITY , WARING’S PRIME NUMBER CONJECTURE ,W ARING’S PROBLEM References Khinchin, A. Y. "The Landau-Schnirelmann Hypothesis and Mann’s Theorem." Ch. 2 in Three Pearls of Number Theory. New York: Dover, pp. 18 /C1/36, 1998. Schoenberg Curve A SPACE-FILLING CURVE . Scholz Conjecture Let the minimal length of an ADDITION CHAIN for a number n be denoted l(n): Then the Scholz conjecture states that l(2n /C281) 5n /C281 /C27l(n) : The conjecture has been proven for a variety of special cases but not in general. See also ADDITION CHAIN References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 111, 1994. Scho¨nemann’s Theorem If the integral COEFFICIENTS C0 ; C1 ; ..., CN /C281of the POLYNOMIAL f(x) /C30C0 /C27C1x /C27C2x2 /C27.../C27CN /C281xN /C281 /C27xN are divisible by a PRIME NUMBER p, while the free term C0 is not divisible by p2 ; then f(x) is irreducible in the natural rationality domain. See also ABEL’S IRREDUCIBILITY THEOREM ,A BEL’S LEMMA ,G AUSS’S POLYNOMIAL THEOREM ,K RONECK- ER’S POLYNOMIAL THEOREM References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 118, 1965. Scho¨nemann. "Grundzu ¨ge einer allgemeinen Theorie der ho¨hern Congruenzen, deren Modul eine reelle Primzahl ist." J. reine angew. Math. 32, 269/C1/325, 1846. Scho¨nflies Symbol One of the set of symbols Ci;Cs;C1;C2;C3;C4;C5;C6; C7;C8;C2h;C3h;C4h;C5h;C6h;C2v;C3v;C4v;C5v;C6v; C/C12v ; D1 ; D2 ; D3 ; D4 ; D5 ; D6 ; D2h ; D4h ; D5h ; D6h ; D8h ; D/C12h ; D2d ; D3d ; D4d ; D5d ; D6d ; I, Ih ; O, Oh ; S4 ; S6 ; S8 ; T, Td ; and Th used to identify POINT GROUPS . Cotton (1990), gives a table showing the translations between Scho¨nflies symbols and HERMANN- MAUGUIN SYMBOLS . Some of the Scho¨nflies symbols denote different sets of symmetry operations but correspond to the same abstract GROUP and so have the same CHARACTER TABLE . See also CHARACTER TABLE ,H ERMANN- MAUGUIN SYMBOL ,POINT GROUPS ,SPACE GROUPS ,SYMMETRY OPERATION References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 379, 1990. Scho¨nflies Theorem If J is a simple closed curve in R2 ; the closure of one of the components of R2 /C28J is HOMEOMORPHIC with the unit 2-BALL . This theorem may be proved using the RIEMANN MAPPING THEOREM , but the easiest proof is via MORSE THEORY . The generalization to n-D is called MAZUR’S THEO- REM. It follows from the Scho¨nflies theorem that any two KNOTS of S1 in S2 or R2 are equivalent. See also JORDAN CURVE THEOREM ,M AZUR’S THEO- REM,RIEMANN MAPPING THEOREM References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 9, 1976. Thomassen, C. "The Jordan-Scho ¨nflies Theorem and the Classification of Surfaces." Amer. Math. Monthly 99, 116 /C1/ 130, 1992. Schoof-Elkies-Atkin Algorithm An algorithm for determining the order of an ELLIPTIC CURVE E=Fp over the FINITE FIELD Fp :/ See also ELLIPTIC CURVE References Izu, T.; Kogure, J.; Noro, M.; and Yokoyama, K. "Efficient Implementation of Schoof’s Algorithm." Advances in Cryptology: ASIACRYPT’98: International Conference on the Theory and Application of Cryptology and Information Security, Beijing, China, October 18 /C1/22, 1998 (Ed. K. Ohta and D. Pei). New York: Springer-Verlag, pp. 66 /C1/79, 1998. Schoof, R. "Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p." Math. Comput. 44, 483 /C1/494, 1985. Schoof, R. "Counting Points on Elliptic Curves Over Finite Fields." J. The´or. Nombres Bordeaux 7, 219 /C1/264, 1995.Schoolgirl Problem KIRKMAN’S SCHOOLGIRL PROBLEM Schoute Coaxal System The CIRCUMCIRCLE ,BROCARD CIRCLE ,LEMOINE LINE, and ISODYNAMIC POINTS belong to a COAXAL SYSTEM orthogonal to the APOLLONIUS CIRCLES , called the Schoute coaxal system. In general, there are 12 points whose PEDAL TRIANGLES with regard to a given TRIANGLE have a given form. They lie six by six on two CIRCLES of the Schoute coaxal system. See also APOLLONIUS CIRCLES ,B ROCARD CIRCLE , CIRCUMCIRCLE ,COAXAL SYSTEM ,ISODYNAMIC POINTS , LEMOINE LINE,SCHOUTE’S THEOREM References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 297 /C1/299, 1929. Schoute’s Theorem In any TRIANGLE , the LOCUS of a point whose PEDAL TRIANGLE has a constant B ROCARD ANGLE and is described in a given direction is a CIRCLE of the SCHOUTE COAXAL SYSTEM . References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 297 /C1/299, 1929. Schrage’s Algorithm An algorithm for multiplying two 32-bit integers modulo a 32-bit constant without using any inter- mediates larger than 32 bits. It is also useful incertain types of RANDOM NUMBER generators. References Bratley, P.; Fox, B. L.; and Schrage, E. L. A Guide to Simulation, 2nd ed. New York: Springer-Verlag, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Random Numbers." Ch. 7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 269, 1992. Schrage, L. "A More Portable Fortran Random Number Generator." ACM Trans. Math. Software 5, 132/C1/138, 1979. Schro ¨der Number The Schro ¨der number Snis the number of LATTICE PATHS in the Cartesian plane that start at (0, 0), end at (n, n), contain no points above the line y /C30x, and are composed only of steps (0, 1), (1, 0), and (1, 1), i.e., 0;/C160; and P: The diagrams illustrating the paths generating S1 ; S2 ; and S3are illustrated above. The numbers Sn are given by the RECURRENCE RELATION Sn /C30Sn/C281 /C27Xn /C281 k/C300SkSn /C281 /C28k ; where S0 /C301; and the first few are 2, 6, 22, 90, ... (Sloane’s A006318). The Schro ¨der Numbers bear the same relation to the DELANNOY NUMBERS as the CATALAN NUMBERS do to the BINOMIAL COEFFICIENTS . See also BINOMIAL COEFFICIENT ,CATALAN NUMBER , DELANNOY NUMBER ,LATTICE PATH,M OTZKIN NUM- BER, P-GOOD PATH References Bonin, J.; Shapiro, L.; and Simion, R. "Some q-Analogs of the Schro ¨der Numbers Arising from Combinatorial Sta- tistics on Lattice Paths." J. Stat. Planning Inference 34, 35 /C1/55, 1993. Moser, L. and Zayachkowski, W. "Lattice Paths with Diagonal Steps." Scripta Math. 26, 223 /C1/229, 1963. Pergola, E. and Sulanke, R. A.. "Schro ¨der Triangles, Paths, and Parallelogram Polyominoes." J. Integer Sequences 1, No. 98.1.7, 1998. http://www.research.att.com/~njas/se- quences/JIS/PergolaSulanke/. Rogers, D. G. "A Schro ¨der Triangle." Combinatorial Mathe- matics V: Proceedings of the Fifth Australian Conference. New York: Springer-Verlag, pp. 175 /C1/196, 1977. Rogers, D. G. and Shapiro, L. "Some Correspondences involving the Schro ¨der Numbers." Combinatorial Mathe- matics: Proceedings of the International Conference, Can- berra, 1977. New York: Springer-Verlag, pp. 267 /C1/276, 1978. Schro ¨der, E. "Vier kombinatorische Probleme." Z. Math. Phys. 15, 361 /C1/376, 1870. Sloane, N. J. A. Sequences A006318/M1659 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Stanley, R. P. "Hipparchus, Plutarch, Schro ¨der, Hough." Amer. Math. Monthly 104, 344 /C1/350, 1997. Sulanke, R. A. "Bijective Recurrences Concerning Schro ¨der Paths." Electronic J. Combinatorics 5, No. 1, R47, 1 /C1/11, 1998. http://www.combinatorics.org/Volume_5/ v5i1toc.html#R47. Schro ¨der-Bernstein Theorem The Schro ¨der-Bernstein theorem for numbers states that if n 5m 5n: then m /C30n For SETS, the theorem states that if there are INJECTIONS of the SET A into the SET B and of B into A, then there is a BIJECTIVE correspondence between A and B (i.e., they are EQUIPOLLENT ). See also BIJECTION ,CARDINAL COMPARISON ,EQUI- POLLENT ,INJECTION ,TRICHOTOMY LAW Schro ¨der’s Equation The functional equation f(f(x)) /C30sf(x): with s "0 ; 1 :/ References Kuczma, M. Ch. 6 in Functional Equations in a Single Variable. Warsaw, Poland: Polska Akademia Nauk, 1968. Schro ¨der’s Method Two families of equations used to find roots of non- linear functions of a single variable. The "B" family is more robust and can be used in the neighborhood of degenerate multiple roots while still providing a guaranteed convergence rate. Almost all other root- finding methods can be considered as special cases of Schro ¨der’s method. Householder humorously claimed that papers on root-finding could be evaluated quickly by looking for a citation of Schro ¨der’s paper; if the reference were missing, the paper probably consisted of a rediscovery of a result due to Schro ¨der (Stewart 1993). One version of the "A" method is obtained by applying NEWTON’S METHOD tof=f?; xn/C271/C30xn/C28fxnðÞf?xnðÞ f?(xn) ½/C1382/C28fxnðÞfƒxnðÞ (Scavo and Thoo 1995). See also NEWTON’S METHOD References Householder, A. S. The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, 1970. Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley’s Method." Amer. Math. Monthly 102, 417/C1/426, 1995. Schro ¨der, E. "U ¨ber unendlich viele Algorithmen zur Au- flo¨sung der Gleichungen." Math. Ann. 2, 317/C1/365, 1870. Stewart, G. W. "On Infinitely Many Algorithms for Solving Equations." English translation of Schro ¨der’s original paper. College Park, MD: University of Maryland, Insti- tute for Advanced Computer Studies, Department of Computer Science, 1993. ftp://thales.cs.umd.edu/pub/re- ports/imase.ps. Schro ¨dinger Equation The Schro ¨dinger equation describes the motion of particles in nonrelativistic quantum mechanics, and was first written down by Erwin Schro ¨dinger. The time-dependent Schro ¨dinger equation is given by ih@ c(x; y; z ; t) @t /C28h2 2m92 /C27V(x)"# C(x; y; z; t) /C30 ¯H C(x; y; z; t); (1) where h is h-bar , C is the time-dependent wavefunc- tion, m is the mass of a particle, 92 is the LAPLACIAN , V is the potential, and ¯H is the Hamiltonian operator. The time-independent Schro ¨dinger equation is /C28h2 2m92 /C27V(x)"# c(x; y; z ; t) /C30Ec(x; y; z ; t): (2) The one-dimensional versions of these equations are then ih@C(x; t) @t/C30/C28h2 2m@2 @x2 /C27V(x)"# C(x; t) /C30 ¯H C(x; t); (3) and /C28h2 2md2 dx2 /C27V(x)"# c(x) /C30E c(x): (4) The logarithmic Schro ¨dinger equation is given by iut /C2792u /C27u ln½u½2 /C300 (5) (Cazenave 1983; Zwillinger 1997, p. 134), the non- linear Schro ¨dinger equation by iut /C27uxx 92½u½2u /C300 (6) (Calogero and Degasperis 1982, p. 56; Tabor 1989, p. 309; Zwillinger 1997, p. 134) or iut /C27uxx /C27au /C27b½u½2u /C300 (7) (Infeld and Rowlands 2000, p. 126), and the deriva- tive nonlinear Schro ¨dinger equation by iut /C27uxx 9i( ½u½2u)x /C300 (8) (Calogero and Degasperis 1982, p. 56; Zwillinger 1997, p. 134). See also DIRAC EQUATIONReferences Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 56, 1982. Cazenave, T. "Stable Solution of the Logarithmic Schro ¨din- ger Equation." Nonlinear Anal. 7, 1127 /C1/1140, 1983. Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge Univer- sity Press, 2000. Tabor, M. "The NLS Equation." §7.5.c in Chaos and Integr- ability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 309, 1989. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 134, 1997. Schroeder Stairs PENROSE STAIRWAY Schro ¨ter’s Formula Let a general THETA FUNCTION be defined as T(x; q) /C13X/C12 n /C30/C28/C12xnqn2 : then T(x; qa)T(y; qb) /C30Xa /C27b /C281 k/C300ykqbk2 T(xyq2bk ; qa /C27b)T(yax/C28bq2abk ; qab(a /C27b)) : See also BLECKSMITH- BRILLHART- GERST THEOREM , JACOBI TRIPLE PRODUCT ,RAMANUJAN THETA FUNC- TIONS ,THETA FUNCTIONS References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 111, 1987. Tannery, J. and Molk, J. Elements de la The ´orie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars et fils, 1893 /C1/1902. Schur Algebra An Auslander algebra which connects the representa- tion theories of the symmetric group of PERMUTA- TIONS and the GENERAL LINEAR GROUP GL(n;C): Schur algebras are "quasihereditary." References Martin, S. Schur Algebras and Representation Theory. New York: Cambridge University Press, 1993. Schur Decomposition The Schur decomposition of a numerical matrix Mis a pair of matrices QandTsuch that M/C30QTQ/C31; where Qis an ORTHOGONAL MATRIX ,Tis a BLOCK UPPER TRIANGULAR MATRIX , and Q/C31is the ADJOINT MATRIX . Schur decomposition is implemented in Mathematica asSchurDecomposition [m]. See also MATRIX DECOMPOSITION References Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, pp. 312 /C1/314, 1996. Schur, I. "On the Characteristic Roots of a Linear Substitu- tion with an Application to the Theory of Integral Equations." Math. Ann. 66, 488 /C1/510, 1909. Schur Functor A FUNCTOR which defines an equivalence of module CATEGORIES . References Martin, S. Schur Algebras and Representation Theory. New York: Cambridge University Press, 1993. Schur Matrix The p /C29p SQUARE MATRIX formed by setting sij /C30 jij ; where j is a pth ROOT OF UNITY . The Schur matrix has a particularly simple DETERMINANT given by det S /C30eppp=2; where p is an ODD PRIME and ep /C131i f p /C131 (mod 4) i if p /C133 (mod 4):Yrt* This determinant has been used to prove the QUAD- RATIC RECIPROCITY LAW (Landau 1958, Vardi 1991). The ABSOLUTE VALUES of the PERMANENTS of the Schur matrix of order 2p /C271 are given by 1, 3, 5, 105, 81, 6765, ... (Sloane’s A003112, Vardi 1991). Denote the Schur matrix Spwith the first row and first column omitted by S?p: Then perm Sp/C30p perm S?p; where perm denoted the PERMANENT (Vardi 1991). References Graham, R. L. and Lehmer, D. H. "On the Permanent of Schur’s Matrix." J. Austral. Math. Soc. 21, 487 /C1/497, 1976. Landau, E. Elementary Number Theory. New York: Chelsea, 1958. Sloane, N. J. A. Sequences A003112/M2509 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 119 /C1/122 and 124, 1991. Schur Multiplier A property of FINITE SIMPLE GROUPS which is known for all such GROUPS . See also FINITE GROUP ,SIMPLE GROUPSchur Number The Schur number S(k) is the largest integer n for which the interval [1; n] can be partitioned into k SUM-FREE SETS (Fredricksen and Sweet 2000). S(k)is guaranteed to exist for each k by SCHUR’S PROBLEM . Note the definition of the Schur number as the smallest number S?(k) /C30S(k) /C271 for which such a partition does not exist is also prevalent in the literature (Sloane’s A030126; Fredricksen and Sweet 2000). Schur (1916) gave the lower bound S(k) ]1 2(3n /C281) (1) which is sharp for n /C301, 2, and 3 (Guy 1994). The Schur numbers also satisfy the inequality S(k) ]c(321)k =5 > c(1:17176)k (2) for k /C215 and some constant c (Abbott and Moser 1966, Abbott and Hanson 1972, Exoo 1994). SCHUR’S THEOREM also shows that S(n) 5R(n) /C282 : (3) where R(n)isaR AMSEY NUMBER . The first few Schur numbers are 1, 4, 13, 44, 160 5S(5) 5315; S(6) ]536; S(7) ]1680 ; ... (Sloane’s A045652; Fredricksen and Sweet 2000). S(4) is due to Baumert (Baumert 1965, Abbott and Hanson 1972), the lower bound on S(5) is due to Exoo (1994), and the lower limits on S(6) and S(7) are due to Fredricksen and Sweet (2000). See also RAMSEY NUMBER ,R AMSEY’S THEOREM , SCHUR’S PROBLEM ,SCHUR’S THEOREM References Abbott, H. L. and Hanson, D. "A Problem of Schur ad its Generalizations." Acta Arith. 20, 175/C1/187, 1972. Abbott, H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith. 11, 392/C1/396, 1966. Baumert, L. D. and Golomb, S. W. "Backtrack Program- ming." J. Ass. Comp. Machinery 12, 516/C1/524, 1965. Beutelspacher, A. and Brestovansky, W. "Generalized Schur Numbers." In Combinatorial Theory. Proceedings of a Conference Held at Schloss Rauischholzhausen, May 6 /C1/ 9, 1982. Berlin: Springer-Verlag, pp. 30 /C1/38, 1982. Exoo, G. "A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of K3:/"Electronic J. Combinatorics 1, R8 1/C1/3, 1994. http://www.combinatorics.org/Volume_1/ volume1.html#R8. Fredricksen, H. "Schur Numbers and the Ramsey Numbers N(3;3;...;3; 2) :/"J. Combin. Theory Ser. A 27, 376/C1/377, 1979. Fredricksen, H. and Sweet, M. M. "Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers." Elec- tronic J. Combinatorics 7, No. 1, R32, 1 /C1/9, 2000. http:// www.combinatorics.org/Volume_7/v7i1toc.html#R32. Guy, R. K. "Schur’s Problem. Partitioning Integers into Sum-Free Classes" and "The Modular Version of Schur’s Problem." §E11 and E12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 209 /C1/212, 1994. Radziszowski, S. P. "Small Ramsey Numbers." Electronic J. Combin. 1, DS1 1 /C1/29, Rev. Jul. 5, 1999. http://www.com- binatorics.org/Surveys/. Schur, I. "U¨ ber die Kongruenz xm /C27ym /C13zm(mod p)." Jahresber. Deutsche Math.-Verein. 25, 114 /C1/116, 1916. Sloane, N. J. A. Sequences A030126 and A045652 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Whitehead, E. G. "The Ramsey Number N(3; 3; 3; 3; 2):/" Disc. Math. 4, 389 /C1/396, 1973. Schur Transform For p(z) /C30anzn /C27an/C281zn/C281 /C27.../C27a0 ; (1) polynomial of degree n ]1; the Schur transform is defined by the (n /C281)/-degree polynomial Tp(z) /C13 ¯a0p(z) /C28anp /C31(z) (2) /C30Xn/C281 k /C300(¯a0ak /C28an ¯an/C28k)zk (3) where p /C31 is the RECIPROCAL POLYNOMIAL . See also RECIPROCAL POLYNOMIAL References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, p. 493, 1988. Schur-Cohn Algorithm An algorithm that can always be used to decide whether a given polynomial is fere of zeros in the closed unit disk (or, using an entire linear transfor- mation, to any other disk in the complex plane). Under certain conditions, the algorithm can also be used to determine the exact number of zeros in a disk (Henrici 1988, p. 494). The method is also useful to control engineers, since it can be used to determine whether a dynamic control system is stable. References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, pp. 491 /C1/494, 1988. Schur-Jabotinsky Theorem Let P /C30a1x /C27a1x2 /C27... be an ALMOST UNIT in the INTEGRAL DOMAIN of FORMAL POWER SERIES (with a1 " 0) and define Pk /C13X/C12 n /C30ka(k) nxn (1) for k /C3091;9 2, .... If Q /C13P/C281 ; then for all positive integers m,Qm /C30X/C12 n /C30mb(m) nxn ; (2) where b(m) n/C13m na( /C28n) /C28m (3) for n ]m:/ See also LAGRANGE INVERSION THEOREM References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, pp. 55 /C1/56, 1988. Schur’s Hermitian Matrix Theorem HORN’S THEOREM Schur’s Inequalities Let A /C30aij be an n /C29n MATRIX with COMPLEX (or REAL ) entries and EIGENVALUES l1 ; l2 ; ..., ln ; then Xn i/C301½li ½2 5Xn i; j /C301½aij ½2 Xn i/C301½R[ li]½2 5Xn i ; j/C301aij /C27 ¯aji 2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut2 Xn i/C301½I[ li] ½2 5Xn i ; j/C301aij /C28 ¯aji 2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut 2 ; where ¯z is the COMPLEX CONJUGATE . References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1120, 2000. Schur’s Lemma The endomorphism ring of an irreducible module is a DIVISION ALGEBRA . Hsiang (2000, p. 3) calls the following result the Schur lemma. Let V,Wbe irreducible (linear) G- spaces and A:V0WaG-linear map. Then Ais either invertible or A/C300. See also DIVISION ALGEBRA ,SCHUR’S REPRESENTA- TION LEMMA References Herstein, I. N. Topics in Algebra, 2nd ed. New York: Springer-Verlag, 1975. Hsiang, W. Y. Lectures on Lie Groups. Singapore: World Scientific, p. 3, 2000. Schur’s Partition Theorem Schur’s partition theorem lets A(n) denote the num- ber of partitions of n into parts congruent to 91 (mod 6), B(n) denote the number of partitions of n into distinct parts congruent to 91 (mod 3), and C(n) the number of partitions of n into parts that differ by at least 3, with the added constraint that the difference between multiples of three is at least 6. Then A(n) /C30 B(n) /C30C(n) (Schur 1926; Bressoud 1980; Andrews 1986, p. 53). The values of A(n) /C30B(n) /C30C(n) for n /C301, 2, ... are 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, ... (Sloane’s A003105). For example, for n /C3015, there are nine partitions satisfying these conditions, as sum- marized in the following table (Andrews 1986, p. 54). /A(15) /C309// B(15) /C309// C(15) /C309/ /13 /C271 /C271/ 14/C2711 5 /11 /C271 /C271 /C271 /C271/ 13/C2721 4 /C271 /7 /C277 /C271/ 11/C2741 3 /C272 /7 /C275 /C271 /C271 /C271/ 10/C2751 2 /C273 /7 /C271 /C271 /C271 /C27.../C271//10 /C274 /C271/ 11 /C274 /5 /C275 /C275/ 8/C2771 0 /C275 /5 /C275 /C271 /C271 /C27.../C271//8 /C275 /C272// 10 /C274 /C271/ /5 /C271 /C271 /C27.../C271// 8 /C274 /C272 /C271//9 /C275 /C271/ /1 /C271 /C27.../C271// 7 /C275 /C272 /C271//8 /C275 /C272/ The identity A(n) /C30B(n) can be established using the identity X/C12 n/C300B(n)qn /C30(/C28q; q3)/C12(/C28q2; q3) /C12 (1) /C30(q2; q6)/C12(q4; q6)/C12 (q; q3)/C12(q2; q3) /C12(2) /C30(q2; q6)/C12(q4; q6)/C12 (q; q6)/C12(q4; q6) /C12(q2; q6) /C12(q5; q6)/C12(3) /C301 (q; q6)/C12(q5; q6) /C12(4) /C30X/C12 n/C300A(n)qn (5) (Andrews 1986, p. 54). The identity B(n) /C30C(n)is significantly trickier. See also GO¨ LLNITZ’S THEOREM ,R AMSEY NUMBER , SCHUR’S LEMMA ,SCHUR NUMBERReferences Andrews, G. E. "q-Series and Schur’s Theorem" and "Bres- soud’s Proof of Schur’s Theorem." §6.2 /C1/6.3 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 53 /C1/58, 1986. Bressoud, D. M. "Combinatorial Proof of Schur’s 1926 Partition Theorem." Proc. Amer. Math. Soc. 79, 338 /C1/ 340, 1980. Schur, I. "U¨ ber die Kongruenz xm /C27ym /C13zm(mod p)." Jahresber. Deutsche Math.-Verein. 25, 114 /C1/116, 1916. Schur, I. "Zur additiven Zahlentheorie." Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., pp. 488 /C1/495, 1926. Rep- rinted in Gesammelte Abhandlungen, Vol. 3. Berlin: Springer-Verlag, pp. 43 /C1/50, 1973. Sloane, N. J. A. Sequences A003105/M0254 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Schur’s Problem Schur (1916) proved that no matter how the set of POSITIVE INTEGERS less than or equal to n!ebc (where xbcis the FLOOR FUNCTION ) is partitioned into n classes, one class must contain INTEGERS x, y, z such that x /C27y /C30z; where x and y are not necessarily distinct. The least INTEGER S(n) with this property is known as the SCHUR NUMBER . The upper bound has since been slightly improved to n!(e/C281=24) bc :/ See also COMBINATORICS ,RAMSEY NUMBER ,SCHUR NUMBER ,SCHUR’S THEOREM ,SUM-FREE SET References Abbott, H. L. and Hanson, D. "A Problem of Schur and Its Generalizations." Acta Arith. 20, 175/C1/187, 1972. Abbott, H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith. 11, 393/C1/396, 1966. Beutelspacher, A. and Brestovansky, W. "Generalized Schur Numbers." In Combinatorial Theory: Proceedings of a Conference Held at Schloss Rauischholzhausen, May 6 /C1/ 9, 1982 (Ed. D. Jungnickel and K. Vedder). Berlin: Springer-Verlag, pp. 30 /C1/38, 1982. Choi, S. L. G. "The Largest Sum-free Subsequence from a Sequence of nNumbers." Proc. Amer. Math. Soc. 39,4 2/C1/ 44, 1973. Choi, S. L. G.; Komlo ´s, J.; and Szemere ´di, R. "On Sum-Free Subsequences." Trans. Amer. Math. Soc. 212, 307/C1/313, 1975. Erdos, P. "Some Problems and Results in Number Theory." InNumber Theory and Combinatorics: Japan 1984 (Ed. J. Akiyama). Singapore: World Scientific, pp. 65 /C1/87, 1985. Guy, R. K. "Schur’s Problem. Partitioning Integers into Sum-Free Classes" and "The Modular Version of Schur’s Problem." §E11 and E12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 209 /C1/212, 1994. Irving, R. W. "An Extension of Schur’s Theorem on Sum- Free Partitions." Acta Arith. 25,5 5/C1/63, 1973. Scho¨nheim, J. "On Partitions of the Positive Integers with no x,y,zBelonging to Distinct Classes Satisfying x/C27y/C30z:/" InNumber Theory: Proceedings of the First Conference of the Canadian Number Theory Association Held at the Banff Center, Banff, Alberta, April 17 /C1/27, 1988 (Ed. R. A. Mollin). Berlin: de Gruyter, pp. 515 /C1/528, 1990. Wallis, W. D.; Street, A. P.; and Wallis, J. S. Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices. New York: Springer-Verlag, 1972. Schur’s Ramsey Theorem As shown by Schur (1916), the SCHUR NUMBER S(n) satisfies S(n) 5R(n) /C282 (1) for n /C301, 2, ..., where R(n)isaR AMSEY NUMBER . References Fredricksen, H. and Sweet, M. M. "Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers." Elec- tronic J. Combinatorics 7, No. 1, R32, 1 /C1/9, 2000. http:// www.combinatorics.org/Volume_7/v7i1toc.html#R32. Schur, I. "U¨ ber die Kongruenz xm /C27ym /C13zm mod p." Jahres- ber. Deutsche Math.-Verein. 25, 114 /C1/116, 1916. Schur’s Representation Lemma If p on V and p? on V ? are irreducible representations and E : V /C2V ? is a linear map such that p?(g)E /C30 E p(g) for all g /C23 and GROUP G, then E /C300or E is invertible. Furthermore, if V /C30V ?; then E is a SCALAR . See also SCHUR’S LEMMA References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/549, 1996. Schur’s Theorem SCHUR’S PARTITION THEOREM ,SCHUR’S RAMSEY THE- OREM Schwarz Reflection Principle Suppose that f is a ANALYTIC FUNCTION which is defined in the UPPER HALF-DISK f½z½2 B1;I[z] > 0g: Assume that f extends to a continuous function on the REAL AXIS, and takes on real values on the REAL AXIS. Then f can be extended to an ANALYTIC FUNCTION on the whole disk by the formula f(¯z) /C30f(z) ; and the values for z reflected across the REAL AXIS are the reflections of f(z) across the REAL AXIS. It is easy to check that the above function is COMPLEX DIFFER- ENTIABLE in the interior of the LOWER HALF-DISK . What is remarkable is that the resulting function must be analytic along the REAL AXIS as well, despite no assumptions of differentiability. This is called the Schwarz reflection principle, and issometimes also known as the Schwarz’s symmetric principle (Needham 2000, p. 257). The diagram above shows the reflection principle applied to a function f defined for UPPER HALF-DISK (left figure; red) and its image (right figure; red). The function is real on the real axis, so it is possible to extend the function to the reflected domain (left and right figures; pink). For the reflected function to be continuous, it is necessary for the values at the boundary to be continuous and to fall on the line being reflected. The reflection principle also applies in the generality of reflecting along any line, not just the REAL AXIS,in which case the function f has to take values along a line in the range. In fact, any arc which has a neighborhood biholomorphic to a straight line can be reflected across. The basic example is the bound- ary of the UNIT CIRCLE which is mapped to the REAL AXIS by z 0 (iz /C271)=(z /C27i) :/ The reflection principle can also be used to reflect a HARMONIC FUNCTION which extends continuously to the zero function on its boundary. In this case, for negative y, defining v(x;y)/C30/C28v(x;/C28y) extends vto a harmonic function on the reflected domain. Again note that it is necessary for v(x;0)/C300: This result provides a way of extending a HARMONIC FUNCTION from a given OPEN SET to a larger OPEN SET (Krantz 1999, p. 95). See also ANALYTIC CONTINUATION ,HARMONIC FUNC- TION References Flanigan, F. J. Complex Variables: Harmonic and Analytic Functions. New York: Dover, p. 234, 1983. Krantz, S. G. "The Schwarz Reflection Principle." §7.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 95 /C1/97, 1999. Levinson, N. and Raymond, R. Complex Variables. New York: McGraw-Hill, pp. 318 /C1/320, 1970. Needham, T. "Analytic Continuation via Reflections." §5.XI.5 inVisual Complex Analysis. New York: Clarendon Press, pp. 252 /C1/257, 2000. Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 237 /C1/239, 1987. Schwarz, H. A. Gesammelte Mathematische Abhandlungen, Bd. II. New York: Chelsea, pp. 144 /C1/171, 1972. Schwarz Triangle The Schwarz triangles are SPHERICAL TRIANGLES which, by repeated reflection in their indices, lead to a set of congruent SPHERICAL TRIANGLES covering the SPHERE a finite number of times. Schwarz triangles are specified by triples of numbers(p;q;r):There are four "families" of Schwarz trian- gles, and the largest triangles from each of these families are 22n? ðÞ ;3 23232Yru*Yru+ ;324343Yru*Yru+ ;545454Yru*Yru+ : The others can be derived from (pqr ) /C30(pxr1) /C27(xqr2); where 1 r1/C271 r2/C301 r and cosp x ! /C30/C28cosp x? ! /C30cosp q ! sinp r1 ! /C28 cosp p ! sinp r2 ! sinp r ! See also COLUNAR TRIANGLE ,SPHERICAL TRIANGLE , WYTHOFF SYMBOL References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 112 /C1/113 and 296, 1973. Schwarz, H. A. "Zur Theorie der hypergeometrischen Re- ihe." J. reine angew. Math. 75, 292 /C1/335, 1873. Schwarz-Christoffel Mapping A CONFORMAL MAPPING from the UPPER HALF-PLANE to a POLYGON . See also CONFORMAL MAPPING ,SCHWARZ- CHRISTOF- FEL PARAMETER PROBLEM References Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Lo- cation of Zeros. New York: Wiley, pp. 396 /C1/431, 1988. Krantz, S. G. "Numerical Approximation of the Schwarz- Christoffel Mapping." §14.4.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 175 /C1/179, 1999. Schwarz-Christoffel Parameter Problem The problem of determining the vertices of a SCHWARZ- CHRISTOFFEL MAPPING (Krantz 1999, p. 176). See also CONFORMAL MAPPING ,SCHWARZ- CHRISTOF- FEL MAPPING References Krantz, S. G. in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 176, 1999.Schwarzian Derivative The Schwarzian derivative is defined by DSchwarzian /C13f §(x) f ?(x)/C283 2f(x) f ?(x)"#2 : The FEIGENBAUM CONSTANT is universal for 1-D MAPS if its Schwarzian derivative is NEGATIVE in the bounded interval (Tabor 1989, p. 220). See also FEIGENBAUM CONSTANT References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Schwarz-Pick Lemma Letfbe analytic on the UNIT DISK , and assume that 1.½f(z)½51 for all z, and 2.f(a)/C30bfor some a;b/C23D(0;1);the UNIT DISK . Then ½f?(a)½51/C28½b½2 1/C28½a½2: (1) Furthermore, if f(a1)/C30b1andf(a2)/C30b2;then b2/C28b1 1/C28b1/C31b2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut5 a2/C28a1 1/C28¯a1a2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut; (2) where ¯zis the COMPLEX CONJUGATE (Krantz 1999, p. 78). As a consequence, if either ½f?(a)½51/C28½b½2 1/C28½a½2(3) or b2/C28b1 1/C28¯b1b2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30a2/C28a1 1/C28¯a1a2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut(4) fora 1a2;then fis a conformal SELF-MAP ofD(0;1) to itself. Stated succinctly, the Schwarz-Pick lemma guaran- tees that if fis an analytic map of the DISKDintoD andfpreserves the hyperbolic distance between any two points, then fis a disk map and preserves all distances. References Busemann, H. The Geometry of Geodesics. New York: Academic Press, p. 41, 1955. Krantz, S. G. "The Schwarz-Pick Lemma." §5.5.2 in Hand- book of Complex Analysis. Boston, MA: Birkha ¨user, p. 78, 1999. Schwarz’s Inequality Let c1(x) and c2(x) by any two REAL integrable functions in [a, b], then Schwarz’s inequality, also called the Cauchy-Schwarz inequality (Gradshteyn and Ryzhik 2000, p. 1099) or Buniakowsky inequality (Hardy et al. 1952, p. 16), is given by c1 ½ c2 hijj25 c1 ½ c1 hi c2 ½c2 hi : (1) Written out explicitly gb ac1(x) c2(x) dx"#2 5gb ac1(x) ½/C1382dxgb ac2(x) ½/C1382dx; (2) with equality IFF g(x) /C30 af(x) with a a constant. To derive the inequality, let c(x)bea COMPLEX FUNCTION and l a COMPLEX constant such that c(x) /C13 f(x) /C27 lg(x) for some f and g. Since f ¯cc dx ]0; where ¯z is the COMPLEX CONJUGATE , g ¯cc dx /C30g ¯ff dx/C27 l g ¯fgdx/C27 ¯l g ¯gf dx /C27l ¯lg ¯gg dx ]0 ; (3) with equality when c(x) /C300: Writing this in compact notation, ¯f ; fYruvYruu /C27 l ¯f ; gYruvYruu /C27 ¯l ¯g ; fhi/C27 l ¯l ¯g; ghi]0 : (4) Now define l /C30/C28¯g ; fhi ¯g ; ghi (5) ¯l /C30/C28g ; ¯fYruvYruu ¯g ;ghidx: (6) Multiply (4) by ¯g ; ghi and then plus in (5) and (6) to obtain ¯f ; fYruvYruu ¯g ; ghi/C28 ¯f ; gYruvYruu ¯g ; fhi /C28 ¯g; ¯fYruvYruu g ; ¯fYruvYruu /C27 ¯g ; fhi g ; ¯fYruvYruu ; (7) which simplifies to ¯g ; fhi ¯f ; gYruvYruu 5 ¯f ; fYruvYruu ¯g ; ghi (8) so f ; ghijj25 f ; fhi g ; ghi : (9) BESSEL’S INEQUALITY follows from SCHWARZ’S IN- EQUALITY . See also BESSEL’S INEQUALITY ,H O¨ LDER’S INEQUAL- ITIES References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 527 /C1/529, 1985. Buniakowsky, V. "Sur quelques ine´galite ´s concernant les inte´grales ordinaires et les inte´grales aux diffe´rences finies." Me´moires de l’Acad. de St. Pe´tersbourg (VII) 1, No. 9, p. 4, 1959. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "Further Remarks on Method: The Inequality of Schwarz." §6.5 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 132 /C1/134, 1952. Schwarz, H. A. "U¨ ber ein die Fla¨chen kleinsten Fla¨chenin- halts betreffendes Problem der Variationsrechnung." Acta Soc. Scient. Fen. 15, 315 /C1/362, 1885. Reprinted in Gesam- melte Mathematische Abhandlungen, Vol. 1. New York: Chelsea, pp. 224 /C1/269, 1972. Schwarz’s Lemma Let f be analytic on the UNIT DISK, and assume that 1. ½f(z)½51 for all z and 2. f(0) /C300 :/ Then ½f(z) ½5½z½ and ½f ?(0) ½51 :/ If either ½f(z) ½/C30½z ½ for some z "0orif ½f ?(0) ½/C301 ; then f is a ROTATION , i.e., f(z) /C30az for some complex con- stant a with ½a½/C301:/ See also MO¨ BIUS TRANSFORMATION ,SCHWARZ- PICK LEMMA References Krantz, S. G. "Schwarz’s Lemma." §5.5.1 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 78, 1999. Schwarz’s Minimal Surface A periodic MINIMAL SURFACE constructed by Schwarz using the following two principles: 1. If part of the boundary of a MINIMAL SURFACE is a straight line, then the reflection across the line, when added to the original surface, makes another MINIMAL SURFACE . 2. If a MINIMAL SURFACE meets a PLANE at RIGHT ANGLES , then the mirror image of the PLANE , when added to the original surface, also makes a MINI- MAL SURFACE . See also MINIMAL SURFACE References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 224 /C1/225, 1991. Schwarz’s Polyhedron A polyhedron constructed by ruling 2n equally spaced vertical lines along the surface of a CYLINDER together with 2n3 circles around the cylinder at equally spaced heights. Amazingly, joining neighboring points in triangles and letting n 0/C12 gives a surface whose total SURFACE AREA approaches, not that of the cylinder, but infinity. See also CYLINDER References Ogilvy, C. S. Tomorrow’s Math, 2nd ed. Oxford, England: Oxford University Press, 1972. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 224 /C1/225, 1991. Schwarz’s Symmetry Principle SCHWARZ REFLECTION PRINCIPLE Schwarz’s Triangle Problem FAGNANO’S PROBLEM Schweins’s Theorem If we expand the determinant of a matrix A using DETERMINANT EXPANSION BY MINORS , first in terms of the MINORS of order r formed from any r rows, with their complementaries, and second in terms of the MINORS of order m formed from any m columns (r Bm), with their complementaries; then the sum of the (n /C28r)m/C28r terms of the second expansion which have in common the elements in the intersection of the selected r rows and m columns is equal to the sum of the mr terms of the first expansion which have for one factor the minors of the rth order formed from the elements in the intersection of the selected r rows and m columns. See also DETERMINANT ,DETERMINANT EXPANSION BY MINORS ,MINOR References Muir, T. "Schweins’s Theorem." §141 in A Treatise on the Theory of Determinants. New York: Dover, pp. 124 /C1/125, 1960. Schwenk’s Formula Let R /C27B be the number of MONOCHROMATIC FORCED TRIANGLES (where R and B are the number of red and blue TRIANGLES )inan EXTREMAL GRAPH . ThenR /C27B /C30n 3Yru$Yru% /C281 2 n14(n /C281)2jkjk ; wheren kYrvYru is a BINOMIAL COEFFICIENT and xbcis the FLOOR FUNCTION (Schwenk 1972). See also EXTREMAL GRAPH ,MONOCHROMATIC FORCED TRIANGLE References Schwenk, A. J. "Acquaintance Party Problem." Amer. Math. Monthly 79, 1113 /C1/1117, 1972. Scientific Notation Scientific notation is the expression of a number n in the form a /C2910p ; where p /C13 log10 ½n½ bc is the FLOOR of the base-10 LOGARITHM of n (the "order of magnitude"), and a /C13n 10p is a REAL NUMBER satisfying 1 5½a ½B10 : For example, in scientific notation, the number n /C30101; 325 has order of magnitude p /C30 log10101;325 bc /C30 5:00572bc /C305 ; so n would be written 1:01325 /C29105 : The special case of 0 does not have a unique representation in scientific notation, i.e., 0 /C300 /C29100 /C300 /C29101 /C30... :/ See also CHARACTERISTIC (REAL NUMBER ), FIGURES , MANTISSA ,SIGNIFICANT DIGITS s-Cluster N.B. A detailed online essay by S. Finch was the starting point for this entry. Let an /n/C29n/BINARY MATRIX have entries which are 1 (with probability p) or 0 (with probability /q/C301/C28p/). Ans-cluster is an isolated group of sadjacent (i.e., horizontally or vertically connected) 1s. Let /Cn/be the total number of these " SITE" clusters. Then the value KS(p)/C30lim n0/C12/C142Cn/C143 n2; (1) called the MEAN CLUSTER COUNT PER SITE orMEAN CLUSTER DENSITY , exists. Numerically, it is found that /KS(1=2):0:065770 . . . /(Ziff et al. 1997). Considering instead " BOND " clusters (where numbers are assigned to the edges of a grid) and letting /Cn/be the total number of bond clusters, then KB(p)/C13lim n0/C12/C142Cn/C143 n2; (2) exists. The analytic value is known for /p/C301=2/, KB(1 2) /C3032ffiffiffi 3p /C2841 16 (3) (Ziff et al. 1997). See also BOND PERCOLATION ,PERCOLATION THEORY , S-RUN,SITE PERCOLATION References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/rndprc/rndprc.html. Temperley, H. N. V. and Lieb, E. H. "Relations Between the ‘Percolation’ and ‘Colouring’ Problem and Other Graph- Theoretical Problems Associated with Regular Planar Lattices; Some Exact Results for the ‘Percolation’ Pro- blem." Proc. Roy. Soc. London A 322, 251 /C1/280, 1971. Ziff, R.; Finch, S.; and Adamchik, V. "Universality of Finite- Sized Corrections to the Number of Percolation Clusters." Phys. Rev. Let. To appear, 1998. Score Sequence The score sequence of a TOURNAMENT is a monotonic nondecreasing sequence of the OUTDEGREES of the VERTICES . The score sequences for n /C301, 2, ... are 1, 1, 2, 4, 9, 22, 59, 167, ... (Sloane’s A000571). See also DIRECTED GRAPH ,TOURNAMENT References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 207 /C1/208, 1994. Ruskey, F. "Information on Score Sequences." http:// www.theory.csc.uvic.ca/~cos/inf/nump/ScoreSequen- ce.html. Ruskey, F.; Cohen, R.; Eades, P.; and Scott, A. "Alley CATs in Search of Good Homes." Congres. Numer. 102,97/C1/110, 1994. Sloane, N. J. A. Sequences A000571/M1189 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Scrawny Cantor Set A Cantor set C in R3 is said to be scrawny if for each neighborhood U of an arbitrary point p in C, there is a neighborhood V of p such that every map f : S1 0 V ƒC extends to a map F : B2 0 U such that F /C281(C) is finite. Babich (1992) presents examples of wild Cantor sets of this type and provides a proof that such objects cannot be defined by solid tori. See also CANTOR SET References Babich, A. "Scrawny Cantor Sets are Not Definable by Tori." Proc. Amer. Math. Soc. 115, 829 /C1/836, 1992. Screw A TRANSLATION along a straight line L and a ROTA- TION about L such that the angle of ROTATION is proportional to the TRANSLATION at each instant. Also known as a TWIST . See also DINI’S SURFACE ,HELICOID ,ROTATION ,SCREW THEOREM ,SEASHELL ,TRANSLATIONScrew Theorem Any motion of a rigid body in space at every instant is a SCREW motion. This theorem was proved by Mozzi and Cauchy. See also SCREW Scruple An archaic UNIT FRACTION variously defined as /1=200 / (of an hour), /1 =10/ or /1=12/ (of an inch), /1 =12/ (of a celestial body’s angular diameter), or /1=60/ (of an hour or DEGREE ). See also CALCUS ,UNCIA Sea Horse Valley A portion of the MANDELBROT SET centered around /C281:25 /C270:047i with width approximately 0:009 /C270 :005i :/ See also MANDELBROT SET Search Tree TREE SEARCHING Searching Searching refers to locating a given element or an element satisfying certain conditions from some (usually ordered or partially ordered) table, list, TREE , etc. See also BINARY SEARCH ,SORTING ,TABU SEARCH , TREE SEARCHING References Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1973. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "How to Search an Ordered Table." §3.4 in Numerical Recipes in FORTRAN: The Art of ScientificComputing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 110 /C1 /113, 1992. Skiena, S. "Sorting and Searching." §1.1.6 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 14 /C1/16, 1990. Seashell A conical surface modeled after the shape of a seashell. One parameterization (left figure) is given by x /C302[1 /C28eu=(6p)]cos u cos21 2vYru*Yru+ (1) y /C302[/C281 /C27eu=(6p)]cos212 vYru*Yru+ sin u (2) z /C301 /C28eu=(3p) /C28sin v /C27eu=(6p) sin v; (3) where v /C23 0; 2p ½Þ ; and u /C23 0 ; 6 p ½Þ (Wolfram). Nord- strand gives the parameterization x /C30 1 /C28v 2p ! (1 /C27cos u) /C27c"# cos(nv) (4) x /C30 1 /C28v 2 p ! (1 /C27cos u) /C27c"# sin(nv) (5) z /C30bv 2p /C27a sin u 1 /C28v 2p ! (6) for u; v /C23 [0; 2p] (right figure with a /C300:2; b /C301, c /C30 0:1; and n /C302). See also CONICAL SPIRAL References Gray, A. "Sea Shells." §13.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 308 /C1/309, 1997. Nordstrand, T. "Conic Spiral or Seashell." http:// www.uib.no/people/nfytn/shelltxt.htm. Wolfram Research "Mathematica Version 2.0 Graphics Gallery." http://www.mathsource.com/cgi-bin/ msitem22?0207 /C1/155. Sec SECANTSecant The function defined by sec x /C131=cos x; where cos x is the COSINE . The MACLAURIN SERIES of the secant is sec x /C30( /C281)nE2n (2n)!x2n /C301 /C271 2 x2 /C275 24 x4 /C2761 720 x6 /C27277 8064 x8 /C27...: where E2nis an E ULER NUMBER . See also ALTERNATING PERMUTATION ,C OSECANT , COSINE ,EULER NUMBER ,EXSECANT ,INVERSE SECANT References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/79, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 224, 1987. Spanier, J. and Oldham, K. B. "The Secant sec( x) and Cosecant csc( x) Functions." Ch. 33 in An Atlas of Func- tions. Washington, DC: Hemisphere, pp. 311 /C1/318, 1987. Secant Line A line joining two points of a curve. As the two points are brought together (or, more precisely, as one isbrought towards the other), the secant line tends to a TANGENT LINE . In abstract mathematics, the points which a secant line connects can be either REAL or COMPLEX CONJUGATE IMAGINARY . See also BITANGENT ,TANGENT LINE,TRANSVERSAL LINE Secant Method A ROOT -finding algorithm which assumes a function to be approximately linear in the region of interest. Each improvement is taken as the point where the approximating line crosses the axis. The secant method retains only the most recent estimate, so the root does not necessarily remain bracketed. When the ALGORITHM does converge, its order of conver- gence is lim k0/C12½ ek /C271 ½:C ½ e½f : (1) where C is a constant and f is the GOLDEN MEAN . f ? xn/C281 ðÞ:fxn/C281 ðÞ /C28 fxn/C282 ðÞ xn/C281 /C28 xn/C282(2) fxnðÞ:fxn /C281 ðÞ/C27f ? xnðÞ xn /C28xn/C281 ðÞ /C300 (3) fxn/C281 ðÞ/C27fxn/C281 ðÞ /C28 fxn /C282 ðÞ xn/C281 /C28 xn/C282xn /C28xn /C281 ðÞ /C300 : (4) so xn /C30xn /C281 /C28fxn/C281 ðÞ xn/C281 /C28 xn/C282 ðÞ fxn/C281 ðÞ /C28 fxn/C282 ðÞ: (5) See also FALSE POSITION METHOD References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Secant Method, False Position Method, and Ridders’ Method." §9.2 in Numerical Recipes in FOR- TRAN: The Art of Scientific Computing, 2nd ed. Cam- bridge, England: Cambridge University Press, pp. 347 /C1/ 352, 1992. Secant Number A number, more commonly called an EULER NUMBER , giving the number of EVEN ALTERNATING PERMUTA-TIONS . The term ZIG NUMBER is sometimes also used. The first few are 1, 5, 61, 1385, ... (Sloane’s A000364). See also ALTERNATING PERMUTATION ,EULER NUM- BER,EULER ZIGZAG NUMBER ,TANGENT NUMBER References Sloane, N. J. A. Sequences A000364 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sech HYPERBOLIC SECANT Second ARC SECOND Second Countable Topology A TOPOLOGICAL SPACE is second countable if it has a countable TOPOLOGICAL BASIS . See also TOPOLOGICAL BASIS,TOPOLOGICAL SPACE Second Curvature TORSION (DIFFERENTIAL GEOMETRY ) Second Derivative Test Suppose f(x)isa FUNCTION of x which is twice DIFFERENTIABLE at a STATIONARY POINT x0 : 1. If f ƒ x0ðÞ > 0; then f has a RELATIVE MINIMUM at x0 :/ 2. If f ƒ(x0) B0 ; then f has a RELATIVE MAXIMUM at x0 :/ The EXTREMUM TEST gives slightly more general conditions under which a function with f ƒ(x0) /C300is a maximum or minimum. If f(x; y) is a 2-D FUNCTION which has a RELATIVE EXTREMUM at a point (x0 ; y0) and has CONTINUOUS PARTIAL DERIVATIVES at this point, then fx(x0 ; y0) /C300 and fy(x0 ; y0) /C300: The second PARTIAL DERIVATIVES test classifies the point as a MAXIMUM or MINIMUM . Define the DISCRIMINANT as D /C13fxxfyy /C28fxyfyx /C30fxxfyy /C28f2 xy : 1. If D /C210, fxx(x0 ; y0) > 0 and fxx(x0 ; y0) /C27 fyy(x0 ; y0) > 0; the point is a RELATIVE MINIMUM . 2. If D /C210, fxx(x0;y0)B0;and fxx(x0;y0)/C27 fyy(x0;y0)B0;the point is a RELATIVE MAXIMUM . 3. If DB0, the point is a SADDLE POINT . 4. If D/C300, higher order tests must be used. See also DISCRIMINANT (SECOND DERIVATIVE TEST), EXTREMUM ,E XTREMUM TEST,F IRST DERIVATIVE TEST,GLOBAL MAXIMUM ,GLOBAL MINIMUM ,HESSIAN DETERMINANT ,M AXIMUM ,M INIMUM ,RELATIVE MAX- IMUM ,R ELATIVE MINIMUM ,SADDLE POINT (FUNC- TION ) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Second Fundamental Form Let M be a REGULAR SURFACE with vp ; wppoints in the TANGENT SPACE Mpof M. For M /C23R3 ; the second fundamental form is the symmetric bilinear form on the TANGENT SPACE Mp ; II vp ; wpYrvYru /C30S vpYrvYru /C215 wp : (1) where S is the SHAPE OPERATOR . The second funda- mental form satisfies II axu /C27bxv ; axu /C27bxv ðÞ /C30ea2 /C272fab /C27gb2(2) for any nonzero TANGENT VECTOR . The second fundamental form is given explicitly by edu2 /C272fdudv /C27gdv2 (3) where e /C30X iXi@2xi @u2 (4) f /C30X iXi@2xi @u @v (5) g /C30X iXi@2xi @v2 (6) and Xiare the DIRECTION COSINES of the surface normal. The second fundamental form can also be written e /C30/C28Nu/C215 xv /C30N /C215 xuv (7) f /C30/C28Nv/C215 xu /C30N /C215 xuv /C30Nvu /C215 xvu /C30/C28Nu/C215 xv (8) g /C30/C28Nv/C215 xv /C30N /C215 xvv ; (9) where N is the NORMAL VECTOR , x : U 0 R3is a REGULAR PATCH , and xuand xvare the partial derivatives of x with respect to parameters u and v, respectively, or e /C30det xuvxuxv ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG /C28 F2p (10) f /C30det(xuvxuxv)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEG /C28 F2p (11)g /C30det xuvxuxv ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FG /C28 F2p : (12) See also FIRST FUNDAMENTAL FORM,FUNDAMENTAL FORMS ,S HAPE OPERATOR ,T HIRD FUNDAME NTAL FORM References Gray, A. "The Three Fundamental Forms." §16.6 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 380 /C1/ 382, 1997. Second Fundamental Tensor WEINGARTEN MAP Second Kind Special functions which arise as solutions to second order ordinary differential equations are commonly said to be "of the first kind" if they are nonsingular at the origin, while the linearly independent solutions which are singular are said to be "of the second kind." Common examples of functions of the second kind defined in this way include the BESSEL FUNCTION OF THE SECOND KIND ,CHEBYSHEV POLYNOMIAL OF THE SECOND KIND , CONFLUENT HYPERGEOMETRIC FUNC- TION OF THE SECOND KIND ,H ANKEL FUNCTION OF THE SECOND KIND , and so on. The term "second kind" is also used in a more general context to distinguish between two or more types of mathematical objects which, however, all satisfy some common overall property. Examples of objects of this kind include the CHRISTOFFEL SYMBOL OF THE SECOND KIND , ELLIPTIC INTEGRAL OF THE SECOND KIND ,FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND ,STIRLING NUMBER OF THE SECOND KIND ,VOL- TERRA INTEGRAL EQUATION OF THE SECOND KIND , and so on. See also BESSEL FUNCTION OF THE SECOND KIND, CHEBYSHEV POLYNOMIAL OF THE SECOND KIND, CONFLUENT HYPER GEOMETRIC FUNCTION OF THE SECOND KIND,ELLIPTIC INTEGRAL OF THE SECOND KIND,FIRST KIND,FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND,H ANKEL FUNCTION OF THE SECOND KIND,SPECIAL FUNCTION ,STIRLING NUMBER OF THE SECOND KIND,T HIRD KIND,V OLTERRA INTEGRAL EQUATION OF THE SECOND KIND Section A section of a solid is the plane figure cut from the solid by passing a plane through it (Kern and Bland 1948, p. 18). See also CONIC SECTION ,CROSS SECTION ,CUBICAL CONIC SECTION ,C YLINDRICAL SECTION ,D EDEKIND SECTION ,G RAPH SECTION ,M ULTISECTION ,N ORMAL SECTION ,SECTION (BUNDLE ), SECTION (PENCIL ), SEC- TION (TANGENT BUNDLE ), SPIRIC SECTION ,SURFACE OF SECTION ,TORIC SECTION References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948. Section (Bundle) A section of a FIBER BUNDLE gives an element of the fiber over every point in B. Usually it is described as a map s : B 0 E such that p(s is the identity on B.A real-valued function on a manifold M is a section of the trivial LINE BUNDLE M /C29R : Another common example is a VECTOR FIELD , which is a section of the TANGENT BUNDLE . See also FIBER BUNDLE ,TANGENT BUNDLE ,VECTOR BUNDLE ,ZERO SECTION Section (Pencil) The lines of a PENCIL joining the points of a RANGE to another POINT . See also PENCIL ,RANGE (LINE SEGMENT ) Section (Tangent Bundle) A VECTOR FIELD is a section of its TANGENT BUNDLE , meaning that to every point x in a MANIFOLD M,a VECTOR X(x) /C23 TxM is associated, where Txis the TANGENT SPACE . See also TANGENT BUNDLE ,TANGENT SPACE Sectional Curvature The mathematical object k which controls the rate of geodesic deviation. See also BISHOP’S INEQUALITY ,CHEEGER’S FINITENESS THEOREM ,GEODESIC Sector A WEDGE obtained by taking a portion of a DISK with CENTRAL ANGLE u B p radians (1808), illustrated above as the shaded region. A sector of p radians would be a SEMICIRCLE . Let R be the radius of the CIRCLE , c the CHORD length, s the ARC LENGTH , h the height of the arced portion, and d the height of the triangularportion. Then R /C30h /C27d (1) s /C30Ru (2) d /C30R cos1 2 uYru*Yru+ (3) /C301 2 c cot12 uYru*Yru+ (4) /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4R2 /C28c2p (5) c /C302R sin1 2 uYru*Yru+ (6) /C302d tan1 2 uYru*Yru+ (7) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28d2p (8) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h(2R /C28h)p : (9) The ANGLE u obeys the relationships u /C30s R /C302 cos/C281d R ! /C302 tan/C281c 2d ! /C302 sin/C281c 2R ! : (10) The AREA of the sector is A /C301 2 Rs /C3012 R2 u (11) (Beyer 1987). See also CIRCLE- CIRCLE INTERSECTION ,LENS,OBTUSE TRIANGLE ,SEGMENT References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 125, 1987. Harris, J. W. and Stocker, H. "Sector." §3.8.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 91 /C1/92, 1998. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948. Sectorial Harmonic A SPHERICAL HARMONIC OF THE FORM sin(mu)Pm m(cosf): or cos(mu)Pmm(cosf): See also SPHERICAL HARMONIC ,TESSERAL HARMONIC , ZONAL HARMONIC Secular Equation CHARACTERISTIC EQUATION Seed The initial number used as the starting point in a RANDOM NUMBER generating ALGORITHM . Seed of Life One of the beautiful arrangements of CIRCLES found at the Temple of Osiris at Abydos, Egypt (Rawles 1997). The CIRCLES are placed with 6-fold symmetry, forming a mesmerizing pattern of CIRCLES and LENSES . See also CIRCLE ,C IRCLE COVERING ,F IVE DISKS PROBLEM ,FLOWER OF LIFE,VENN DIAGRAM References Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 15, 1997. Weisstein, E. W. "Flower of Life." MATHEMATICA NOTEBOOK FLOWER OFLIFE.M . Seek Time POINT- POINT DISTANCE–1- D Segment A portion of a DISK whose upper boundary is a circular ARC and whose lower boundary is a CHORD making a CENTRAL ANGLE u B p radians (180 8), illu- strated above as the shaded region. Let R be the radius of the CIRCLE , c the CHORD length, s the ARC LENGTH , h the height of the arced portion, and d theheight of the triangular portion. Then R /C30h /C27d (1) s /C30Ru (2) d /C30R cos1 2 uYru*Yru+ (3) /C3012 c cot12 uYru*Yru+ (4) /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4R2 /C28c2p (5) c /C302R sin1 2 uYru*Yru+ (6) /C302d tan12 uYru*Yru+ (7) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28d2p (8) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h(2R /C28h)p : (9) The ANGLE u obeys the relationships u /C30s R /C302 cos/C281d R ! /C302 tan/C281c 2d ! /C302 sin/C281c 2R ! : (10) The AREA of the segment is then A /C30Asector /C28Aisosocles triangle (11) /C301 2 R2( u /C28sin u) (12) /C3012(Rs /C28cd) (13) /C30R2 cos/C281d R ! /C28dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28d2p (14) /C30R2 cos/C281R /C28 h R ! /C28(R /C28h)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Rh /C28h2p : (15) where the formula for the ISOSCELES TRIANGLE in terms of the VERTEX angle has been used (Beyer 1987). Approximate formulas for the ARC LENGTH and AREA are s :ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2/C2716 3h2q (16) accurate to within 0.3% for 0/C145u590/C14;and A:2 3ch/C27h3 2c: (17) accurate to within 0.1% for 0/C145u5150/C14and 0.8% for 150/C145u5180/C14(Harris and Stocker 1998). See also CHORD ,CIRCLE- CIRCLE INTERSECTION ,CY- LINDRICAL SEGMENT ,L ENS,P ARABOLIC SEGMENT , REULEAUX TRIANGLE ,SAGITTA ,SECTOR ,SPHERICAL SEGMENT References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 125, 1987. Fukagawa, H. and Pedoe, D. "Segments of a Circle." §1.6 in Japanese Temple Geometry Problems. Winnipeg, Mani- toba, Canada: Charles Babbage Research Foundation, pp. 14 /C1/15 and 88 /C1/92, 1989. Harris, J. W. and Stocker, H. "Segment of a Circle." §3.8.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 92 /C1/93, 1998. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 4, 1948. Segmented Number PRIME NUMBER OF MEASUREMENT Segner’s Recurrence Formula The RECURRENCE RELATION En /C30E2En/C281 /C27E3En/C282 /C27.../C27En /C281E2 which gives the solution to EULER’S POLYGON DIVI- SION PROBLEM . See also CATALAN NUMBER ,EULER’S POLYGON DIVI- SION PROBLEM Segre Characteristic A set of integers that give the orders of the blocks in a JORDAN CANONICAL FORM , with those integers corre- sponding to submatrices containing the same latent root bracketed together. For example, the Segre characteristic of a 1 a a b 1 b 1 b g d 1 d d2 6666666666666643 777777777777775 is [(21)31(21)] (Frazer et al. 1955, p. 94). References Frazer, R. A.; Duncan, W. J.; and Collar, A. R. Elementary Matrices and Some Applications to Dynamics and Differ- ential Equations. Cambridge, England: Cambridge Uni- versity Press, p. 94, 1955. Segre’s Theorem For any REAL NUMBER r ]0 ; an IRRATIONAL number a can be approximated by infinitely many RATIONAL fractions p =q in such a way that/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 4rp q2 Bp q /C28 a Brffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 4rp q2 : If r /C301, this becomes HURWITZ’S IRRATIONAL NUMBER THEOREM . See also HURWITZ’S IRRATIONAL NUMBER THEOREM Seiberg-Witten Equations DA c /C300 F /C27 A /C30/C28t( c; c); where /t/ is the sesquilinear map /t : W /C27/C29W /C27/ /0 A/C27/C156C :/ See also WITTEN’S EQUATIONS References Donaldson, S. K. "The Seiberg-Witten Equations and 4- Manifold Topology." Bull. Amer. Math. Soc. 33,45/C1/70, 1996. Marshakov, A. Seiberg-Witten Theory and Integrable Sys- tems. Singapore: World Scientific, 1999. Morgan, J. W. The Seiberg-Witten Equations and Applica- tions to the Topology of Smooth Four-Manifolds. Prince- ton, NJ: Princeton University Press, 1996. Seiberg-Witten Invariants WITTEN’S EQUATIONS Seidel-Entringer-Arnold Triangle The NUMBER TRIANGLE consisting of the ENTRINGER NUMBERS En; k arranged in "ox-plowing" order, E00 E10 0 E11 E22 1 E21 1 E20 E30 0 E30 0 E32 0 E33 E441E431E421E411E40 giving 1 001 11110 0010202 515141210 See also BELL NUMBER ,B OUSTROPHEDON TRANS- FORM ,CLARK’S TRIANGLE ,ENTRINGER NUMBER ,EU- LER’S TRIANGLE ,L EIBNIZ HARMONIC TRIANGLE , LOSSNITSCH’S TRIANGLE ,N UMBER TRIANGLE ,P AS- CAL’S TRIANGLE References Arnold, V. I. "Bernoulli-Euler Updown Numbers Associated with Function Singularities, Their Combinatorics, and Arithmetics." Duke Math. J. 63, 537 /C1/555, 1991. Arnold, V. I. "Snake Calculus and Combinatorics of Ber- noulli, Euler, and Springer Numbers for Coxeter Groups." Russian Math. Surveys 47,3/C1/45, 1992. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, 1996. Dumont, D. "Further Triangles of Seidel-Arnold Type and Continued Fractions Related to Euler and Springer Numbers." Adv. Appl. Math. 16, 275 /C1/296, 1995. Entringer, R. C. "A Combinatorial Interpretation of the Euler and Bernoulli Numbers." Nieuw. Arch. Wisk. 14, 241 /C1/246, 1966. Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Operation on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76,44/C1/54, 1996. Seidel, I. "U¨ ber eine einfache Entstehungsweise der Ber- noullischen Zahlen und einiger verwandten Reihen." Sitzungsber. Mu¨nch. Akad. 4, 157 /C1/187, 1877. Seifert Circle Eliminate each KNOT crossing by connecting each of the strands coming into the crossing to the adjacent strand leaving the crossing. The resulting strands no longer cross but form instead a set of nonintersecting CIRCLES called Seifert circles. References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 96, 1994. Seifert Conjecture Every smooth NONZERO VECTOR FIELD on the 3- SPHERE has at least one closed orbit. The conjecture was proposed in 1950, proved true for Hopf fibrations, but proved false in general by Kuperberg (1994). References Kuperberg, G. "A Volume-Preserving Counterexample to the Seifert Conjecture." Comment. Math. Helv. 71,70/C1/97, 1996. Kuperberg, G. and Kuperberg, K. "Generalized Counter- examples to the Seifert Conjecture." Ann. Math. 143, 547 /C1/576, 1996. Kuperberg, G. and Kuperberg, K. "Generalized Counter- examples to the Seifert Conjecture." Ann. Math. 144, 239 /C1/268, 1996. Kuperberg, K. "A Smooth Counterexample to the Seifert Conjecture." Ann. Math. 140, 723 /C1/732, 1994. Seifert Form For K a given KNOT in S3 ; choose a SEIFERT SURFACE M2 in S3 for K and a bicollar ˆM /C29[/C281; 1] in S3 /C28K : If x /C23 H1(M) is represented by a 1-cycle in ˆM ; let x/C27 denote the homology cycle carried by x /C291 in the bicollar. Similarly, let x/C28 denote x /C29/C281: The function f : H1( ˆM) /C29H1( ˆM) 0 Z defined by f(x; y) /C30lk(x; y/C27) :where lk denotes the LINKING NUMBER , is called a Seifert form for K. See also SEIFERT MATRIX References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200 /C1/201, 1976. Seifert Matrix Given a SEIFERT FORM f(x; y) ; choose a basis e1 ; ..., e2g for H1( ˆM)asa Z/-module so every element is uniquely expressible as n1e1 /C27.../C27n2ge2g (1) with niinteger. Then define the Seifert matrix V as the 2g /C292g INTEGER MATRIX with entries vij /C30lk ei ; e /C27 jYru*Yru+ : (2) For example, the right-hand TREFOIL KNOT has Seifert matrix V /C30/C2811 0 /C281YrtvYrtu : (3) A Seifert matrix is not a KNOT INVARIANT , but it can be used to distinguish between different SEIFERT SURFACES for a given knot. See also ALEXANDER MATRIX References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200 /C1/203, 1976. Seifert Surface An orientable surface with one boundary component such that the boundary component of the surface is a given KNOT K. In 1934, Seifert proved that such a surface can be constructed for any KNOT . The process of generating this surface is known as Seifert’s algorithm. Applying Seifert’s algorithm to an alter- nating projection of an alternating knot yields a Seifert surface of minimal GENUS . There are KNOTS for which the minimal genus Seifert surface cannot be obtained by applying Seifert’s algorithm to any projection of that KNOT , as proved by Morton in 1986 (Adams 1994, p. 105). See also GENUS (KNOT), SEIFERT MATRIX References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 95 /C1/106, 1994. Seifert, H. "U ¨ber das Geschlecht von Knotten." Math. Ann. 110, 571/C1/592, 1934. Seiffert’s Spherical Spiral The SPHERICAL CURVE obtained when moving along the surface of a sphere with constant speed, while maintaining a constant angular velocity with respect to a fixed diameter (Erdos 2000). This curve is given in CYLINDRICAL COORDINATES by the parametric equations r /C30sn(s ; k) u /C30ks z /C30cn(s ; k) ; where k is a POSITIVE constant and sn(s) and cn(s) are JACOBI ELLIPTIC FUNCTIONS (Whittaker and Watson 1990, pp. 527 /C1/528). Erdos (2000) provides a derivation of the equations of this curve, as well as an analysis of its properties, including conditions for obtaining periodic orbits. See also SPHERICAL CURVE ,SPHERICAL SPIRAL References Bowman, F. Introduction to Elliptic Functions, with Appli- cations. New York: Dover, p. 34, 1961. Erdos, P. "Spiraling the Earth with C. G. J. Jacobi." Amer. J. Phys. 68, 888 /C1/895, 2000. Seiffert. "U¨ ber eine neue geometrische Einfu ¨hrung in die Theorie der elliptischen Funktionen." Wissensch. Beitra ¨ge Jahresber. Sta¨dtischen Realschule zu Charlottenburg, Ostern. 1896. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Selberg Trace Formula Let p run over all distinct primitive ordered periodic geodesics, and let t(p) denote the positive length of p, then every EVEN FUNCTION h( r) analytic in ½I[ r] ½5 e/C271 =2 and such that ½h(r) ½5O ½ r½/C282/C28 dYrvYru for r 09/C12 satisfies the summation formulaX/C12 k /C300h(rk) /C30(g /C281)g/C12 /C28/C12/C28d ˆh d t ! dt sinh1 2 tYru*Yru+ /C27X fp gX/C12 n/C301t(p) 2 sinh1 2 nt(p)hi ˆh(nt(p)) : where g is the genus of the surface whose area is 4p(g /C281) by the GAUSS- BONNET THEOREM . See also SELBERG ZETA FUNCTION References Balazs, N. L. and Voros, A. "Chaos on the Pseudosphere." Phys. Rep. 143, 109 /C1/240, 1986. Elstrodt, J. Jahresber. d. Deutsche Math. Verein 83,45/C1/77, 1981. Hejhal, D. A. "The Selberg Trace Formula and the Riemann Zeta Function." Duke Math. J. 43, 441 /C1/482, 1976. Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439 /C1/ 465, 1987. Selberg Zeta Function Let p run over all distinct primitive ordered periodic geodesics, and let t(p) denote the positive length of p, then the Selberg zeta function is defined as Z(s) /C30Y fp gY/C12 k /C3001 /C28e /C28z(p)(s/C27k)YrtYrP : fors/C211. See also SELBERG TRACE FORMULA References d’Hoker, E. and Phong, D. H. "Multiloop Amplitudes for the Bosonic Polyakov String." Nucl. Phys. B 269, 205/C1/234, 1986. d’Hoker, E. and Phong, D. H. "On Determinants of Lapla- cians on Riemann Surfaces." Commun. Math. Phys. 104, 537/C1/545, 1986. Fried, D. Invent. Math. 84, 523/C1/540, 1986. Selberg, A. "Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applica- tions to Dirichlet Series." J. Indian Math. Soc. 20,4 7/C1/87, 1956. Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439/C1/ 465, 1987. Selberg’s Formula Letxbe a positive number, and define l(d)/C30m(d)l nx d !"#2 (1) f(n)/C30X dl(d): (2) where the sum extends over the divisors dofn, and m(n) is the M O¨BIUS FUNCTION . Then S /C30X n5xf(n) /C302x ln x /C27o(x ln x) (3) (Nagell 1951, p. 286). See also PRIME NUMBER THEOREM References Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976. Nagell, T. "Further Lemmata. Proofs of Selberg’s Formula." §73 in Introduction to Number Theory. New York: Wiley, pp. 279 /C1/280 and 283 /C1/286, 1951. Selberg, A. "An Elementary Proof of the Prime Number Theorem." Ann. Math. 50, 305 /C1/313, 1949. Selection Sort A SORTING algorithm which makes n passes over a set of n elements, in each pass selecting the smallest element and deleting it from the set. This algorithm has running time O(n2) ; compared to O(n ln n) for the best algorithms (Skiena 1990, p. 14). See also SORTING References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Self Number A number (usually base 10 unless specified other- wise) which has no GENERATOR . Such numbers were originally called COLUMBIAN NUMBERS (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the RECURRENCE RELATION Ck /C308 /C215 10k /C281 /C27Ck /C281 /C278 ; (1) for k /C302, 3, ..., where C1 /C309 : The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (Sloane’s A003052). An infinite number of 2-self numbers (i.e., base-2 self numbers) can be generated by the sequence Ck /C302j /C27Ck /C281 /C271 (2) for k /C301, 2, ..., where C1 /C301 and j is the number of digits in Ck /C281 : An infinite number of n-self numbers can be generated from the sequence Ck /C30(n /C282)nk /C281 /C27Ck /C281 /C27(n /C282) (3) for k /C302, 3, ..., and C1 /C30n /C281 for n even n /C282 for n odd :Yrt* (4) Joshi (1973) proved that if k is ODD, then m is a k-self number IFF m is ODD. Patel (1991) proved that 2k;4k /C272; and k2 /C272k /C271 are k-self numbers in every EVEN base k>4:/ See also DIGITADDITION References Cai, T. "On k-Self Numbers and Universal Generated Numbers." Fib. Quart. 34, 144/C1/146, 1996. Gardner, M. Time Travel and Other Mathematical Bewil- derments. New York: W. H. Freeman, pp. 115 /C1/117, 122, 1988. Joshi, V. S. Ph.D. dissertation. Gujarat University, Ahma- dabad, 1973. Kaprekar, D. R. The Mathematics of New Self-Numbers. Devaiali, pp. 19 /C1/20, 1963. Patel, R. B. "Some Tests for k-Self Numbers." Math. Stu- dent 56, 206/C1/210, 1991. S., B. R. "Solution to Problem E 2048." Amer. Math. Monthly 81, 407, 1974. Sloane, N. J. A. Sequences A003052/M2404 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Self-Adjoint Consider a second-order differential operator ˜Lu(x)/C13p0d2u dx2/C27p1du dx/C27p2u; (1) where u/C13u(x) and pi/C13pi(x) are REAL FUNCTIONS ofx on the region of interest [ a, b] with 2 /C28icontinuous derivatives and with p0(x)"0o n[ a, b]. This means that there are no singular points in [ a, b]. Then the ADJOINT operator ˜L/C31is defined by ˜L/C31u/C13d2 dx2p0uðÞ/C28d dxp1uðÞ/C27p2u (2) /C30p0d2u dx2/C272p?0/C28p1 ðÞdu dx/C27pƒ0/C28p?1/C27p2 ðÞ u: (3) In order for the operator to be self-adjoint, i.e., ˜L/C30˜L/C31: (4) the second terms in (1) and (3) must be equal, so p?0(x)/C30p1(x): (5) This also guarantees that the third terms are equal, since p?0(x)/C30p1(x)[pƒ0(x)/C30p?1(x): (6) so (3) becomes ˜Lu/C30˜L/C31u/C30p0d2u dx2/C27p?0du dx/C27p2u (7) /C30d dxp0du dx ! /C27p2u/C300: (8) The differential operators corresponding to the L E- GENDRE DIFFERENTIAL EQUATION and the equation of SIMPLE HARMONIC MOTION are self-adjoint, while those corresponding to the LAGUERRE DIFFERENTIAL EQUATION and HERMITE DIFFERENTIAL EQUATION are not. A nonself-adjoint second-order linear differential operator can always be transformed into a self-adjoint one using STURM- LIOUVILLE THEORY . In the special case p2(x) /C300; (8) gives d dxp0(x)du dx"# /C300 (9) p0(x)du dx /C30C (10) du /C30Cdx p0(x) (11) u /C30Cgdx p0(x) ; (12) where C is a constant of integration. A self-adjoint operator which satisfies the BOUNDARY CONDITIONS ¯vpU ?½x/C30a /C30 ¯vpU ?½x/C30b (13) is automatically a HERMITIAN OPERATOR . See also ADJOINT ,H ERMITIAN OPERATOR ,S TURM- LIOUVILLE THEORY References Arfken, G. "Self-Adjoint Differential Equations." §9.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 497 /C1/509, 1985. Self-Adjoint Matrix A MATRIX A for which A /C31/C13AT /C30A: where the ADJOINT MATRIX is denoted A/C31; AT is the MATRIX TRANSPOSE , and ¯z is the COMPLEX CONJUGATE . If a MATRIX is self-adjoint, it is said to be HERMITIAN . See also ADJOINT ,H ERMITIAN MATRIX ,M ATRIX TRANSPOSESelf-Avoiding Polygon A LATTICE POLYGON consisting of a closed SELF- AVOIDING WALK on a square lattice. The perimeter, horizontal perimeter, vertical perimeter, and AREA are all WELL DEFINED for self-avoiding polygons. Special classes of self-avoiding polygons include the BAR GRAPH POLYGON , CONVEX POLYGON ,F ERRERS GRAPH POLYGON , STACK POLYGON , and STAIRCASE POLYGON . Self-avoiding polygon are used in physics to model crystal growth and polymers (Bousquet- Me´lou 1992). Enumerating self-avoiding polygons according to perimeter or area is an unsolved problem (Bous- quet-Me ´lou et al. 1999). See also POLYOMINO ,SELF-AVOIDING WALK,STAIR- CASE POLYGON References Bousquet-Me ´lou, M. "Convex Polyominoes and Heaps of Segments." J. Phys. A: Math. Gen. 25, 1925 /C1/1934, 1992. Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/9908123/. Self-Avoiding Walk N.B. A detailed online essay by S. Finch was the starting point for this entry. A self-avoiding walk is a path from one point to another which never intersects itself. Such paths are usually considered to occur on lattices, so that steps are only allowed in a discrete number of directions and of certain lengths. Consider a self-avoiding walk on a 2-D n/C29nsquare grid (i.e., a lattice path which never visits the same lattice point twice) which starts at the origin, takes first step in the positive horizontal direction, and isrestricted to nonnegative grid points only. The num- ber of such paths of n/C301, 2, ... steps are 1, 2, 5, 12, 30, 73, 183, 456, 1151, ... (Sloane’s A046170). Similarly, consider a self-avoiding walk which starts at the origin, takes first step in the positive horizontal direction, is notrestricted to nonnegative grid points only, but which isrestricted to take an up step before taking the first down step. The number of such pathsofn/C301, 2, ... steps are 1, 2, 5, 13, 36, 98, 272, 740, 2034, ... (Sloane’s A046171). Self-avoiding rook walks are walks on an m/C29ngrid which start from (0 ;0);end at ( m, n ), and arecomposed of only horizontal and vertical steps. The following table gives the first few numbers R(m;n)o f such walks for small mand n. The values for m/C30 n/C301;2, ... are 2, 12, 184, 8512, 1262816, ... (Sloane’s A007764). /m/23 4 5 6 22 341 2 483 81 8 4 5 16 125 976 85126 32 414 5382 79384 1262816 There are a number of known formulas for computing R(m;n) for small m, n . For example, R(m;2)/C302 m/C281: There is a RECURRENCE RELATION forR(m;3);given byR(1;3)/C301;R(2;3)/C304;R(3;3)/C3012;R(4;3)/C3038; and R(m;3)/C304R(m/C281;3)/C283R(m/C282;3)/C272R(m/C283;3) /C27R(m/C283;4) form]5;as well as the GENERATING FUNCTION R(m;3) /C301 (m/C281)!dm/C281 dxm/C281(x/C281)(x/C271) x2/C273x/C281 ðÞ x2/C28x/C271 ðÞ j x/C300 (Abbott and Hanson 1978, Finch). A related sequence is the number of shapes which can be formed by bending a piece of wire of length nin the plane, where bends are of 0 or 990/C14and the wire may cross itself at right angles but not pass over itself. Thenumber of shapes for wires of length 1, 2, ... are 1, 2, 4, 10, 24, 66, 176, 493, ... (Sloane’s A001997). Consider a self-avoiding walk on a 2-D n/C29nsquare grid from one corner to another such that no twoconsecutive steps are in the same direction. The number of such paths for n/C301, 2, ... are 1, 2, 2, 4, 10, 36, 188, ... (Sloane’s A034165; counting the number of paths on the 1 /C291 point "lattice" as 1), and the maximum lengths of these paths are 0, 2, 4, 10, 12, 26, 36, ... (Sloane’s A034166). See also LATTICE PATH,RANDOM WALK,SELF-AVOID- ING POLYGON ,S ELF-AVOIDING WALK CONNECTIVE CONSTANT ,S TAIRCASE POLYGON ,T HREE- CHOICE WALK References Abbott, H. L. and Hanson, D. "A Lattice Path Problem." Ars Combinatoria 6, 163 /C1/178, 1978. Alm, S. E. "Upper Bounds for the Connective Constant of Self-Avoiding Walks." Combin. Prob. Comput. 2, 115 /C1/136, 1993. Domb, C. "On Multiple Returns in the Random-Walk Problem." Proc. Cambridge Philos. Soc. 50, 586 /C1/591, 1954. Domb, C. "Self-Avoiding Walks on Lattices." In Adv. Chem. Phys. 15, 1969. Finch, S. "Unsolved Mathematics Problems: Self-Avoiding Walks of a Rook on a Chessboard." http://www.mathsoft.- com/asolve/gammel/gammel.html. Hayes, B. "How to Avoid Yourself." Amer. Sci. 86, Jul./Aug. 1998. Kesten, H. "On the Number of Self-Avoiding Walks." J. Math. Phys. 4, 960 /C1/969, 1963. Lawler, G. F. Intersections of Random Walks. Boston, MA: Birkha ¨user, 1991. Sloane, N. J. A. Sequences A0019971206, A007764, A034165, A034166, A046170, and A046171 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Whittington, S. G. and Guttman, A. J. "Self-Avoiding Walks which Cross a Square." J. Phys. A 23, 5601 /C1/5609, 1990. Self-Avoiding Walk Connective Constant Let the number of RANDOM WALKS on a d-D hypercu- bic lattice starting at the ORIGIN which never land on the same lattice point twice in n steps be denoted cd(n): The first few values are cd(0) /C301 (1) cd(1) /C302d (2) cd(2) /C302d(2d /C281): (3) In general, dn 5cd(n) 52d(2d /C281)n /C281 (4) (Po¨nitz and Tittman 2000), with tighter bounds given by Madras and Slade (1993). Conway and Guttmann (1996) have enumerated walks of up to length 51. The so-called "connective constants" are defined by md /C13lim n0/C12[cd(n)]1=n (5) and are known to exist and be FINITE . The best ranges for these constants are m2 /C23 [2:62002 ; 2:679192495] (6)m3 /C23 [4:572140 ; 4:7476] (7) m4 /C23 [6:742945 ; 6:8179] (8) m5 /C23 [8:828529 ; 8:88602] (9) m6 /C23 [10:874038 ; 10 :8886] (10) (Beyer and Wells 1972, Noonan 1998, Finch). The upper bound of m2improves on the 2.6939 found by Noonan 1998 and was computed by Po¨nitz and Titt- man (2000). For the triangular lattice in the plane, m B4:278 (Alm 1993), and for the hexagonal planar lattice, it is conjectured that m /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq (11) (Madras and Slade 1993). The following limits are also believed to exist and to be FINITE : limn0/C12c(n) mnn g/C281 for d "4 limn0/C12c(n) mnn g/C281(ln n)1=4for d /C304:8>>>< >>>:(12) where the critical exponent g /C301 for d /C214 (Madras and Slade 1993) and it has been conjectured that g /C3043 32 for d /C302 1:162... for d /C303 1 for d /C304:8 < :(13) Define the mean square displacement over all n-step self-avoiding walks vas s(n)/C13½v(n)½2YruvYruu /C301 c(n)X v½v(n)½2: (14) The following limits are believed to exist and be FINITE : limn0/C12s(n) n2nford"4 limn0/C12s(n) n2n(lnn)1=4ford/C304:8 >>>< >>>:(15) where the critical exponent n/C301=2 for d/C214 (Madras and Slade 1993), and it has been conjectured that n/C303 4ford/C302 0:59 . . . for d/C303 12 ford/C304:8 >< >:(16) See also RANDOM WALK,SELF-AVOIDING WALK References Alm, S. E. "Upper Bounds for the Connective Constant of Self-Avoiding Walks." Combin. Probab. Comput. 2, 115 /C1/ 136, 1993. Beyer, W. A. and Wells, M. B. "Lower Bound for the Connective Constant of a Self-Avoiding Walk on a Square Lattice." J. Combin. Th. A 13, 176 /C1/182, 1972. Conway, A. R. and Guttmann, A. J. "Square Lattice Self- Avoiding Walks and Corrections to Scaling." Phys. Rev. Lett. 77, 5284 /C1/5287, 1996. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cnntv/cnntv.html. Madras, N. and Slade, G. The Self-Avoiding Walk. Boston, MA: Birkha ¨user, 1993. Noonan, J. "New Upper Bounds for the Connective Con- stants of Self-Avoiding Walks." J. Stat. Phys. 91, 871 /C1/888, 1998. Po¨nitz, A. and Tittman, P. "Improved Upper Bounds for Self- Avoiding Walks in Zd :/" Electronic J. Combinatorics 7, No. 1, R21, 1 /C1/19, 2000. http://www.combinatorics.org/ Volume_7/v7i1toc.html. Self-Complementary Graph A self-complementary graph is a GRAPH which is isomorphic to its GRAPH COMPLEMENT . The numbers of simple self-complementary graphs on n /C301, 2, ... nodes are 1, 0, 0, 1, 2, 0, 0, 10, ... (Sloane’s A000171). The first few of these compose to the trivial graph on one node, the PATH GRAPH P4 ; and the CYCLE GRAPH C5 :/ All self-complementary graphs have GRAPH DIAMETER 2 or 3 (Sachs 1962; Skiena 1990, p. 187). See also GRAPH COMPLEMENT ,ISOMORPHIC GRAPHS References Read, R. C. "On the Number of Self-Complementary Graphs and Digraphs." J. London Math. Soc. 38,99/C1/104, 1963. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Sachs, H. "U¨ ber selbstkomplementa ¨re Graphen." Publ. Math. Debrecen 9, 270 /C1/288, 1962. Skiena, S. "Self-Complementary Graphs." §5.2.3 in Imple- menting Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 187, 1990. Sloane, N. J. A. Sequences A000171/M0014 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html.Wille, D. "Enumeration of Self-Complementary Structures." J. Combin. Th. B 25, 143 /C1/150, 1978. Self-Conjugate Partition A PARTITION whose CONJUGATE PARTITION is equiva- lent to itself. The FERRERS DIAGRAMS corresponding to the self-conjugate partitions for 3 5n 510 are illustrated above. The numbers of self-conjugate partitions of n /C301, 2, ... are 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, ... (Sloane’s A000700). The number of self-conjugate partitions Sn of n is equal to the number of partitions of n into distinct odd parts, and has generating function Y/C12 k /C3001 /C27x2k /C271 /C30X/C12 k /C300Skxk ; and (/C281)nSnhas GENERATING FUNCTION Y/C12 k/C3011 1/C27xk/C30X/C12 k/C300(/C281)kSkxk: See also CONJUGATE PARTITION ,FERRERS DIAGRAM , PARTITION FUNCTION P References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 277, 1979. Osima, M. "On the Irreducible Representations of the Symmetric Group." Canad. J. Math. 4, 381/C1/384, 1952. Watson, G. N. "Two Tables of Partitions." Proc. London Math. Soc. 42, 550/C1/556, 1936. Self-Conjugate Permutation INVOLUTION (PERMUTATION ) Self-Conjugate Subgroup INVARIANT SUBGROUP Self-Descriptive Number A 10-DIGIT number satisfying the following property. Number the DIGITS 0 to 9, and let DIGIT n be the number of ns in the number. There is exactly one such number: 6210001000. References Pickover, C. A. "Chaos in Ontario." Ch. 28 in Keys to Infinity. New York: Wiley, pp. 217 /C1/219, 1995. Self-Dual A geometric proposition is said to be self-dual when application of the DUALITY PRINCIPLE of PROJECTIVE GEOMETRY results in a proposition equivalent to the original. DESARGUES’ THEOREM is an example of a self-dual proposition. See also SELF-DUAL GRAPH ,SELF-DUAL POLYHEDRON Self-Dual Graph A GRAPH that is DUAL to itself. WHEEL GRAPHS are self-dual, as are the examples illustrated above. Naturally, the SKELETON of a SELF-DUAL POLYHEDRON is a self-dual graph. See also DUAL GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 243, 1976. Smith, C. A. B. and Tutte, W. T. "A Class of Self-Dual Maps." Canad. J. Math. 2, 179 /C1/196, 1950. Self-Dual Polyhedron A POLYHEDRON that is DUAL to itself. For example, the TETRAHEDRON is self-dual. Naturally, the SKELETON of a self-dual polyhedron is a SELF-DUAL GRAPH . See also DUAL POLYHEDRON ,SELF-DUAL GRAPH . Self-Homologous Point SIMILITUDE CENTERSelf-Linking Number CALUGAREANU THEOREM ,GAUSS INTEGRAL ,LINKING NUMBER Self-Loop LOOP (GRAPH ) Self-Map A mapping of a DOMAIN F : U 0 U to itself. See also MO¨ BIUS TRANSFORMATION References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 78, 1999. Self-Reciprocating Property Let h be the number of sides of certain SKEW POLYGONS (Coxeter 1973, p. 15). Then h /C302(p /C27 q /C27 2) 10 /C28 p /C28 q: References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Self-Recursion SELF-RECURSION is a RECURSION which is defined in terms of itself, resulting in an ill-defined infinite regress. See also RECURSION ,REGRESSION ,SELF-RECURSION References Carroll, L. "‘What the Tortoise Said to Achilles." Mind 4, 278 /C1/280, 1895. Gardner, M. "Infinite Regress." Ch. 22 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 220 /C1/229, 1984. Selfridge-Hurwitz Residue Let the RESIDUE from PE´ PIN’S THEOREM be Rn /C133 Fn/C281 ðÞ =2mod Fn ðÞ ; where Fn is a FERMAT NUMBER . Selfridge and Hurwitz use Rnmod 235 /C281; 236 ; 236 /C281YrvYru : A nonvanishing Rnmod 236ðÞ indicates that Fnis COMPOSITE forn/C215. See also FERMAT NUMBER ,PE´ PIN’S THEOREM References Crandall, R.; Doenias, J.; Norrie, C.; and Young, J. "The Twenty-Second Fermat Number is Composite." Math. Comput. 64, 863 /C1/868, 1995. Selfridge’s Conjecture There exist infinitely many n /C210 with p2 n > pn/C28ipn/C27i for all i Bn, where pnis the nth PRIME . Also, there exist infinitely many n /C210 such that 2pn Bpn /C28i /C27pn/C27i for all i Bn. Self-Similarity An object is said to be self-similar if it looks "roughly" the same on any scale. FRACTALS are a particularly interesting class of self-similar objects. Self-similar objects with parameters N and s are described by a power law such as N /C30sd ; where d /C30ln N ln s is the "DIMENSION " of the scaling law, known as the HAUSDORFF DIMENSION . See also FRACTAL ,HAUSDORFF DIMENSION References Harris, J. W. and Stocker, H. "Scaling Invariance and Self- Similarity" and "Construction of Self-Similar Objects." §4.11.1 /C1/4.11.2 in Handbook of Mathematics and Compu- tational Science. New York: Springer-Verlag, p. 113, 1998. Hutchinson, J. "Fractals and Self-Similarity." Indiana Univ. J. Math. 30, 713 /C1/747, 1981. Self-Transversality Theorem Let j, r, and s be distinct INTEGERS (mod n), and let W be the point of intersection of the side or diagonal V ; Vi/C27j of the n-gon P /C30 V1 ...Vn ½/C138 with the transver- sal Vi/C27r Vi/C27s : Then a NECESSARY and SUFFICIENT condition for Yn i/C301ViWi WiVi /C27j"# /C30(/C281)n ; where AB ½½CD and AB CD"# ; is the ratio of the lengths [A, B] and [C, D] with a plus or minus sign depending on whether these segments have the same or opposite direction, is that1. n /C302m is EVEN with j /C13m (mod n) and s /C13r /C27m (mod n) ;/ 2. n is arbitrary and either s /C132r and j /C133r ; or 3. r /C132s (mod n) and j /C133s (mod n):/ References Gru¨nbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254 /C1/268, 1995. Sellke’s Self-Describing Sequence KOLAKOSKI SEQUENCE Selmer Group A GROUP which is related to the TANIYAMA- SHIMURA CONJECTURE . See also TANIYAMA- SHIMURA CONJECTURE Semialgebraic Set A subset of Rn which is a finite Boolean combination of sets OF THE FORM ¯x /C30 x1 ;...; xn ðÞ : f(¯x) > 0 fg and f¯x : g(¯x) /C300 g; where f ; g /C23R X1 ;...; Xn ½/C138 :/ By TARSKI’S THEOREM , the solution set of a QUANTI- FIED SYSTEM of real algebraic equations and inequal- ities is a semialgebraic set (Strzebonski 2000). See also TARSKI’S THEOREM References Bierstone, E. and Milman, P. "Semialgebraic and Subanaly- tic Sets." IHES Pub. Math. 67,5/C1/42, 1988. Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753 /C1/759, 1996. Strzebonski, A. "Solving Algebraic Inequalities." Mathema- tica J. 7, 525 /C1/541, 2000. Semianalytic /X ⁄Rn is semianalytic if, for all x /C23Rn ; there is an open neighborhood U of x such that X S U is a finite Boolean combination of sets f¯x /C23 U : f(¯x) /C300g and f¯x /C23 U:g(¯x)>0g;where f;g:U0Rare ANALYTIC . See also ANALYTIC FUNCTION ,P SEUDOA NALYTIC FUNCTION ,SUBANALYTIC References Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753/C1/759, 1996. Semicircle Half a CIRCLE . The AREA of a semicircle of radius r is given by A /C30gr 0gffiffiffiffiffiffiffiffiffi r2 /C28x2p /C28ffiffiffiffiffiffiffiffiffi r2 /C28x2p dx dy /C302gr 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28x2p dx /C301 2 pr2 : (1) The weighted mean of y is xhi2/C302gr 0xffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28x2p dx /C302 3 r3 : (2) The semicircle is the CROSS SECTION of a HEMISPHERE for any PLANE through the Z-AXIS . The perimeter of the curved boundary is given by s /C30gr /C28rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27x?2p dy: (3) With x /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28y2p ; this gives s /C30 pr : (4) The PERIMETER of the semicircular lamina is then L /C302r /C27 pr /C30r(2 /C27 p) : (5) The weighted value of x of the semicircular curve is given by xhi1/C30gr /C28rxffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27x?2p dy /C30gr /C28rrdy/C302r2 ; (6) so the CENTROID is ¯x1 /C30xhi1 A/C302r p: (7) The CENTROID of the semicircular lamina is given by¯x2 /C30xhi2 A/C304r 3p (8) (Kern and Bland 1948, p. 113). See also ARBELOS ,ARC,CIRCLE ,DISK,HEMISPHERE , LENS,R IGHT ANGLE ,SALINON ,THALES’ THEOREM , YIN-YANG References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948. Semicolon The symbol ; given special meanings in several mathematics contexts, the most common of which is the COVARIANT DERIVATIVE . See also COVARIANT DERIVATIVE References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 284, 1997. Semicolon Derivative COVARIANT DERIVATIVE Semiconvergent Series ASYMPTOTIC SERIES Semicubical Parabola APARABOLA -like curve with Cartesian equation y/C30ax3=2; (1) PARAMETRIC EQUATIONS x/C30t2(2) y/C30at3(3) and POLAR COORDINATES , r/C30tan2usecu a: (4) The semicubical parabola is the curve along which a particle descending under gravity describes equalvertical spacings within equal times, making it an ISOCHRONOUS CURVE . The problem of finding the curve having this property was posed by Leibniz in 1687 and solved by Huygens (MacTutor Archive). The ARC LENGTH , CURVATURE , and TANGENTIAL ANGLE are s(t) /C301 274 /C279t2YrvYru3 =2/C288 27 (5) k(t) /C306 t 4 /C27 9t2 ðÞ3 =2 (6) f(t) /C30tan/C2813 2 tYru*Yru+ : (7) See also NEILE’S PARABOLA ,PARABOLA INVOLUTE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 223 /C1/224, 1987. Gray, A. "The Semicubical Parabola." §1.8 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 21 /C1/22, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 85 /C1/87, 1972. MacTutor History of Mathematics Archive. "Neile’s Para- bola." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Neiles.html. Yates, R. C. "Semi-Cubic Parabola." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 186 /C1/187, 1952. Semiderivative A FRACTIONAL DERIVATIVE of order 1u2. The semider- ivative of tl is given by D1 =2tl /C30tl/C281 =2 G( l /C27 1) G l /C2712Yru*Yru+ ; so the semiderivative of the CONSTANT FUNCTION f(t) /C30c is given by D1=2c /C30c lim l00tl /C281 =2 G( l /C27 1) G l /C271 2Yru*Yru+ /C30cffiffiffiffiffi ptp : See also DERIVATIVE ,FRACTIONAL DERIVATIVE ,SEMI- INTEGRAL References Spanier, J. and Oldham, K. B. An Atlas of Functions. Washington, DC: Hemisphere, pp. 8 and 14, 1987. Semidirect Product A "split" extension G of GROUPS N and F which contains a SUBGROUP ¯F isomorphic to F with G /C30 ¯F ¯N and ¯F S ¯N /C30feg (Ito 1987, p. 710). Then the semi- direct product of a GROUP G by a group H, denoted H /C29G (or sometimes H : G) with homomorphism T is given by(g; h)(g?; h?) /C30(gg?;(h(g?T))h?); where g; g ?/C23 G ; h ; h?/C23 H ; and T /C23 Hom( F ; Aut(H)) (Suzuki 1982, p. 67; Scott 1987, p. 213). Note that the semidirect product of two groups is not uniquely defined. The semidirect product of a group G by a group H can also be defined as a group S /C30GH which is the product of its subgroups G and H, where H is normal in S and G S H /C30f1g: If G is also normal in S, then the semidirect product becomes a GROUP DIRECT PRODUCT (Shmel’kin 1988, p. 247). See also ACTION ,GROUP DIRECT PRODUCT ,SUBGROUP References Itoˆ, K. (Ed.). ‘Extensions." §190.N in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 710, 1987. Kurosh, A. G. The Theory of Groups, 2nd ed., 2 vols. New York: Chelsea, 1960. Scott, W. R. "Semi-Direct Products." §9.2 in Group Theory. New York: Dover, pp. 212 /C1/217, 1987. Shmel’kin, A. L. "Semi-Direct Product." In Vol. 8 of Ency- clopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia" (Managing Ed. M. Hazewinkel). Dordrecht, Netherlands: Reidel, p. 247, 1988. Suzuki, M. Group Theory, Vol. 1. New York: Springer- Verlag, 1982. Semiflow An ACTION with G /C30R/C27:/ See also FLOW Semigroup A mathematical object defined for a set and a BINARY OPERATOR in which the multiplication operation is ASSOCIATIVE . No other restrictions are placed on a semigroup; thus a semigroup need not have an IDENTITY ELEMENT and its elements need not have inverses within the semigroup. A semigroup is an ASSOCIATIVE GROUPOID . A semigroup can be empty. The total number of semigroups of order n are 1, 4, 18, 126, 1160, 15973, 836021, ... (Sloane’s A001423). The number of semigroups of order n with one IDEMPOTENT are 1, 2, 5, 19, 132, 3107, 623615, ... (Sloane’s A002786), and with two IDEMPOTENTS are 2, 7, 37, 216, 1780, 32652, ... (Sloane’s A002787). The number a(n) of semigroups having nIDEMPOTENTS are 1, 2, 6, 26, 135, 875, ... (Sloane’s A002788). See also ASSOCIATIVE ,B INARY OPERATOR ,F REE SEMIGROUP ,GROUPOID ,INVERSE SEMIGROUP ,M ONO- ID,QUASIGROUP References Birget, J.-C.; Margolis, S.; Meakin, J. and Sapir, M. (Eds.). Algorithmic Problems in Groups and Semigroups. Boston, MA: Birkha ¨user, 2000. Clifford, A. H. and Preston, G. B. The Algebraic Theory of Semigroups. Providence, RI: Amer. Math. Soc., 1961. Howie, J. H. Fundamentals of Semigroup Theory. Oxford, England: Oxford University Press, 1996. Sloane, N. J. A. Sequences A001423/M3550, A002786/ M1522, A002787/M1802, and A002788/M1679 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Semi-Integral A FRACTIONAL INTEGRAL of order 1u2. The semi- integral of tl is given by D /C281 =2tl /C30tl/C271 =2 G( l /C27 1) G l /C273 2Yru*Yru+ ; so the semi-integral of the CONSTANT FUNCTION f(t) /C30 c is given by D /C281 =2c /C30c lim l00tl/C271 =2 G( l /C27 1) G l /C273 2Yru*Yru+ /C302cffiffiffi t ps : See also FRACTIONAL INTEGRAL ,INTEGRAL References Spanier, J. and Oldham, K. B. An Atlas of Functions. Washington, DC: Hemisphere, pp. 8 and 14, 1987. Semilatus Rectum In general, the CHORD through a FOCUS parallel to the DIRECTRIX of a CONIC SECTION is called the LATUS RECTUM . Half this length is called the semilatus rectum (Coxeter 1969). Given an ELLIPSE , the semilatus rectum is the distance L measured from a FOCUS such that 1 L /C131 21 r/C27/C271 r/C28 ! ; (1) where r/C27/C30a(1 /C27e) and r /C28/C30a(1 /C28e) are the APOAPSIS and PERIAPSIS , and e is the ELLIPSE ’s ECCENTRICITY . Plugging in for r/C27 and r /C28 then gives 1 L /C301 2a1 1 /C28 e /C271 1 /C27 e ! /C301 2a(1 /C27 e) /C27 (1 /C28 e) 1 /C28 e2 /C301 a1 1 /C28 e2 ; (2) so L /C30a 1 /C28e2YrvYru : (3) See also CONIC SECTION ,DIRECTRIX (CONIC SECTION ), ECCENTRICITY ,E LLIPSE ,F OCUS ,L ATUS RECTUM , SEMIMAJOR AXIS,SEMIMINOR AXISReferences Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 116 /C1/118, 1969. Semimagic Square A square that fails to be a MAGIC SQUARE only because one or both of the main diagonal sums do not equal the MAGIC CONSTANT (Kraitchik 1942, p. 143). See also MAGIC SQUARE References Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, 1942. Semimajor Axis HALF the distance across an ELLIPSE along the longest of its three principal axes. See also ELLIPSE ,SEMIMINOR AXIS Semiminor Axis Half the distance across an ELLIPSE along its short principal axis. See also ELLIPSE ,SEMIMAJOR AXIS Seminorm A seminorm is a function on a VECTOR SPACE V, denoted ½½v ½½; such that the following conditions hold for all v and w in V, and any scalar c. 1. ½½v½½]0:;/ 2. ½½cv½½/C30½c ½½½v ½½; and 3. ½½v /C27w ½½5½½v½½/C27½½w½½:/ Note that it is possible for ½½v½½/C300 for nonzero v. For example, the FUNCTIONAL ½½f ½½/C30½f(0) ½ for continuous functions is a seminorm which is not a norm. A seminorm is a norm if ½½v½½/C300 is equivalent to v /C300. See also FRE´ CHET SPACE ,NORM,TOPOLOGICAL VEC- TOR SPACE Semiperfect Magic Cube A semiperfect magic cube, also called an "Andrews cube," is a MAGIC CUBE for which the CROSS SECTION diagonals do not sum to the MAGIC CONSTANT . See also MAGIC CUBE,PERFECT MAGIC CUBE References Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213 /C1/225, 1988. Semiperfect Number A number such as 20 /C301/C274/C275/C2710 which is the SUM of some (or all) of its PROPER DIVISORS is called a semiperfect number, or sometimes a pseudoperfect number (Butske et al. 1999). A semiperfect number which is the SUM of all its PROPER DIVISORS is called a PERFECT NUMBER . The first few semiperfect numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (Sloane’s A005835). Every multiple of a semiperfect number is semiperfect, as are all numbers 2mp for m > 1 and p a PRIME between 2m and 2m/C271 (Guy 1994, p. 47). A semiperfect number cannot be DEFICIENT . Rare ABUNDANT NUMBERS which are not semiperfect are called WEIRD NUMBERS . Semiperfect numbers are sometimes also called pseudoperfect numbers. See also ABUNDANT NUMBER ,D EFICIENT NUMBER , PERFECT NUMBER ,PRIMARY PSEUDOPERFECT NUM- BER,PRIMITIVE SEMIPERFECT NUMBER ,W EIRD NUM- BER References Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation ap =N1=p /C271 =N /C301 ; Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407 /C1/420, 1999. Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45 /C1/53, 1994. Sloane, N. J. A. Sequences A005835/M4094 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and Ore Numbers." Bull. Soc. Math. Gre´ce (New Ser.) 13,12/C1/ 22, 1972. Semiperimeter The semiperimeter on a figure is defined as s /C131 2 p; (1) where p is the PERIMETER . The semiperimeter of POLYGONS appears in unexpected ways in the compu- tation of their AREAS . The most notable cases are in the ALTITUDE , EXRADIUS , and INRADIUS of a TRIANGLE , the SODDY CIRCLES ,HERON’S FORMULA for the AREA of a TRIANGLE in terms of the legs a, b, and c AD/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s /C28a)(s /C28b)(s /C28c)p ; (2) and BRAHMAGUPTA’S FORMULA for the AREA of a QUADRILATERAL Aquadrilateral /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s /C28a)(s /C28b)(s /C28c)(s /C28d) /C28abcd cos2A /C27 B 2 !vuut: (3) The semiperimeter also appears in the beautiful L’HUILIER’S THEOREM about SPHERICAL TRIANGLES . For a TRIANGLE , the following identities hold, s /C28a /C301 2(/C28a /C27b /C27c) (4) s /C28b /C3012(/C27a /C28b /C27c) (5) s /C28c /C301 2(/C27a /C27b /C28c) : (6) Now consider the above figure. Let I be the INCENTER of the TRIANGLE DABC ; with D, E, and F the tangent points of the INCIRCLE . Extend the line BA with GA /C30CE. Note that the pairs of triangles (ADI, AFI), (BDI, BEI), (CFI, CEI) are congruent. Then BG /C30BD /C27AD /C27AG /C30BD /C27AD /C27CE /C301 2(2BD /C272AD /C272CE) /C301 2[(BD /C27BE) /C27(AD /C27AF) /C27(CE /C27CF)] /C3012[(BD /C27AD) /C27(BE /C27CE) /C27(AF /C27CF)] /C301 2(AB/C27BC/C27AC)/C3012(a/C27b/C27c)/C30s: (7) Furthermore, s/C28a/C30BG/C28BC /C30(BD/C27AD/C27AG)/C28(BE/C27CE) /C30(BD/C27AD/C27CE)/C28(BD/C27CE)/C30AD (8) s/C28b/C30BG/C28AC /C30(BD/C27AD/C27AG)/C28(AF/C27CF) /C30(BD/C27AD/C27CE)/C28(AD/C27CE)/C30BD (9) s/C28c/C30BG/C28AB/C30AG (10) (Dunham 1990). These equations are some of the building blocks of Heron’s derivation of H ERON’S FORMULA . See also PERIMETER References Dunham, W. "Heron’s Formula for Triangular Area." Ch. 5 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 113 /C1/132, 1990. Semiprime A COMPOSITE number which is the PRODUCT of two PRIMES (possibly equal). They correspond to the 2- ALMOST PRIMES . The first few are 4, 6, 9, 10, 14, 15, 21, 22, ... (Sloane’s A001358). See also ALMOST PRIME ,CHEN’S THEOREM ,COMPO- SITE NUMBER ,LANDAU’S PROBLEMS ,PRIME NUMBER References Sloane, N. J. A. Sequences A001358/M3274 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Semiprime Ring Given an IDEAL A, a semiprime ring is one for which An /C300 IMPLIES A /C300 for any POSITIVE n. Every PRIME RING is semiprime. See also PRIME RING Semiregular Polyhedron A POLYHEDRON or plane TESSELLATION is called semiregular if its faces are all REGULAR POLYGONS and its corners are alike (Walsh 1972; Coxeter 1973, pp. 4 and 58; Holden 1991, p. 41). The usual name for a semiregular polyhedron is an ARCHIMEDEAN SOLID , of which there are exactly 13. See also ARCHIMEDEAN SOLID,POLYHEDRON ,TESSEL- LATION References Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z. 46, 380 /C1/407, 1940. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods of Semi-Regular Polyhedra." Geometriae Dedicata 1, 117 /C1/ 123, 1972. Semiregular Tessellation TESSELLATION Semiring A semiring is a set together with two BINARY OPERATORS S(/C27;+) satisfying the following condi- tions:1. Additive associativity: For all a; b; c /C23 S; (a /C27b) /C27c /C30a /C27(b /C27c) ;/ 2. Additive commutativity: For all a ; b /C23 S; a /C27b /C30b /C27a ;/ 3. Multiplicative associativity: For all a; b; c /C23 S; (a+b)+c /C30a+(b+c) ;/ 4. Left and right distributivity: For all a; b; c /C23 S; a+(b /C27c) /C30(a+b) /C27(a +c)/ and /(b /C27c)+a /C30(b+a)/ //C27(c +a) :/ A semiring is therefore a commutative SEMIGROUP under addition and a SEMIGROUP under multiplica- tion. A semiring can be empty. See also BINARY OPERATOR ,RING,RINGOID ,SEMI- GROUP References Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968. Semisecant TRANSVERSAL LINE Semisimple Algebra An ALGEBRA with no nontrivial nilpotent IDEALS .In the 1890s, Cartan, Frobenius, and Molien indepen- dently proved that any finite-dimensional semisimple algebra over the REAL or COMPLEX numbers is a finite and unique DIRECT SUM of SIMPLE ALGEBRAS . This result was then extended to algebras over arbitrary fields by Wedderburn in 1907 (Kleiner 1996). See also IDEAL ,NILPOTENT ELEMENT ,SIMPLE ALGE- BRA References Kleiner, I. "The Genesis of the Abstract Ring Concept." Amer. Math. Monthly 103, 417 /C1/424, 1996. Semisimple Element A P-ELEMENT x of a GROUP G is semisimple if E(CG(x)) "1; where E(H) is the commuting product of all components of H and CG(x) is the CENTRALIZER of G. See also CENTRALIZER , P-ELEMENT Semisimple Lie Group AL IE GROUP which has a simply connected covering group HOMEOMORPHIC to Rn : The prototype is any connected closed subgroup of upper TRIANGULAR COMPLEX MATRICES . The HEISENBERG GROUP is such a group. See also HEISENBERG GROUP ,LIE GROUP References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537/C1/549, 1996. Semisimple Ring A SEMIPRIME RING which is also an ARTINIAN RING . See also ARTINIAN RING References Herstein, I. N. "Semisimple Rings." §1.2 in Noncommutative Rings. Washington, DC: Math. Assoc. Amer., pp. 52 /C1/56, 1968. Semistable When a PRIME l divides the DISCRIMINANT of a ELLIPTIC CURVE E, two or all three roots of E become congruent (mod l). An ELLIPTIC CURVE is semistable if, for all such PRIMES l, only two roots become CONGRUENT mod l (with more complicated definitions for p /C302 or 3). See also DISCRIMINANT (ELLIPTIC CURVE ), ELLIPTIC CURVE Sensitivity The probability that a STATISTICAL TEST will be positive for a true statistic. See also SPECIFICITY ,S TATISTICAL TEST,T YPE I ERROR ,TYPE II ERROR Sentence This entry contributed by MATTHEW SZUDZIK A sentence is a logic formula in which every variable is QUANTIFIED . The concept of a sentence is important because formulas with variables that are not quanti- fied are ambiguous. The concept of the sentence can be illustrated as follows (Enderton 1977). The formula //C215(x;/C214(y; y /C23 x)); in which each variable is quantified, can be trans- lated into English as the complete sentence "There exists a set which has every set as an element." However, the formula /C214(y;(y /C23 x)); in which x is not quantified, can only be translated as the sentence fragment "Every set is an element of ___," where "___" is unspecified because x is not quantified. Because a "quantified variable" is just a more de- scriptive name for a BOUND VARIABLE , a sentence can also be defined as a logic formula with no FREE VARIABLES . See also BOUND VARIABLE ,FREE VARIABLE ,QUANTI- FIER,THEORY References Enderton, H. B. Elements of Set Theory. New York: Aca- demic Press, 1977. Sentential Calculus PROPOSITIONAL CALCULUSSeparating Edge An EDGE of a GRAPH is separating if a path from a point A to a point B must pass over it. Separating EDGES can therefore be viewed as either bridges or dead ends. See also EDGE (GRAPH ) Separating Family A SEPARATING FAMILY is a SET of SUBSETS in which each pair of adjacent elements are found separated, each in one of two disjoint subsets. The 26 letters of the alphabet can be separated by a family of 9, (abcdefghi )( jklmnopqr )(stuvwxyz ) (abcjklstu )(defmnovwx )(ghipqryz ) (adgjmpsvy )(behknqtwz )( cfilorux ): The minimal size of the separating family for an n-set is 0, 2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, ... (Sloane’s A007600). See also KATONA’S PROBLEM References Honsberger, R. "Cai Mao-Cheng’s Solution to Katona’s Problem on Families of Separating Subsets." Ch. 18 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 224 /C1/239, 1985. Sloane, N. J. A. Sequences A007600/M0456 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Separation Two distinct point pairs ACand BDseparate each other if A,B,C, and Dlie on a CIRCLE (or line) in such order that either of the arcs (or the line segment AC) contains one but not both of Band D.I n addition, the point pairs separate each other if every CIRCLE through AandCintersects (or coincides with) every CIRCLE through Band D. If the point pairs separate each other, then the symbol AC==BDis used. Separation of Variables A method of solving partial differential equations in afunction /F(x;y;... ) /and variables x,y, ... by making a substitution OF THE FORM F(x;y;... )/C13X(x)Y(y)/C1/C1/C1; breaking the resulting equation into a set of indepen-dent ordinary differential equations, solving these forX(x);Y(y);..., and then plugging them back into the original equation. This technique works because if the product of functions of independent variables is a constant,each function must separately be a constant. Successrequires choice of an appropriate coordinate system and may not be attainable at all depending on the equation. Separation of variables was first used byL’Hospital in 1750. It is especially useful in solving equations arising in mathematical physics, such as LAPLACE’S EQUATION , the HELMHOLTZ DIFFERENTIAL EQUATION , and the Schro ¨dinger equation. See also HELMHOLTZ DIFFERENTIAL EQUATION ,LA- PLACE’S EQUATION ,PARTIAL DIFFERENTIAL EQUATION , STA¨ CKEL DETERMINANT References Arfken, G. "Separation of Variables" and "Separation of Variables--Ordinary Differential Equations." §2.6 and §8.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 111 /C1/117 and 448 /C1/451, 1985. Bateman, H. Partial Differential Equations of Mathematical Physics. New York: Dover, 1944. Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value Problems, 5th ed. New York: McGraw- Hill, 1993. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959. Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989. Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 2. New York: Wiley, 1989. Eisenhart, L. P. "Separable Systems in Euclidean 3-Space." Physical Review 45, 427 /C1/428, 1934. Eisenhart, L. P. "Separable Systems of Sta¨ckel." Ann. Math. 35, 284 /C1/305, 1934. Eisenhart, L. P. "Potentials for Which Schroedinger Equa- tions Are Separable." Phys. Rev. 74,87/C1/89, 1948. Frank, P. and Mises, R. von. Die Differential- und Integral- gleichungen der Mechanik und Physik, 8th ed. Braunsch- weig, Germany: Vieweg, 1930. Hildebrand, F. B. Advanced Calculus for Engineers. Engle- wood Cliffs, NJ: Prentice-Hall, 1949. Jeffreys, S. H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, 1988. Kellogg, O. D. Foundations of Potential Theory. New York: Dover, 1953. Lense, J. Reihenentwicklungen in der mathematischen Physik. Berlin: de Gruyter, 1933. Maxwell, J. C. A Treatise on Electricity and Magnetism, Vol. 1, unabridged 3rd ed. New York: Dover, 1954. Maxwell, J. C. A Treatise on Electricity and Magnetism, Vol. 2, unabridged 3rd ed. New York: Dover, 1954. Miller, W. Jr. Symmetry and Separation of Variables. Reading, MA: Addison-Wesley, 1977. Moon, P. and Spencer, D. E. "Separability Conditions for the Laplace and Helmholtz Equations." J. Franklin Inst. 253, 585 /C1/600, 1952. Moon, P. and Spencer, D. E. "Theorems on Separability in Riemannian n-Space." Proc. Amer. Math. Soc. 3, 635 /C1/642, 1952. Moon, P. and Spencer, D. E. "Recent Investigations of the Separation of Laplace’s Equation." Proc. Amer. Math. Soc. 4, 302 /C1/307, 1953. Moon, P. and Spencer, D. E. "Separability in a Class of Coordinate Systems." J. Franklin Inst. 254, 227 /C1/242, 1952. Moon, P. and Spencer, D. E. Field Theory for Engineers. Princeton, NJ: Van Nostrand, 1961. Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1 /C1/48, 1988. Morse, P. M. and Feshbach, H. "Separable Coordinates" and "Table of Separable Coordinates in Three Dimensions."§5.1 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 464 /C1/523 and 655 /C1/666, 1953. Murnaghan, F. D. Introduction to Applied Mathematics. New York: Wiley, 1948. Smythe, W. R. Static and Dynamic Electricity, 3rd ed, rev. pr. New York: Hemisphere, 1989. Sommerfeld, A. Partial Differential Equations in Physics. New York: Academic Press, 1964. Weber, E. Electromagnetic Field. New York: Wiley, 1950. Webster, A. G. Partial Differential Equations of Mathema- tical Physics, 2nd corr. ed. New York: Dover, 1955. Separation Theorem There exist numbers y1 By2 B...Bxn /C281 ; a Byn/C281 ; yn/C281 Bb ; such that ln /C30 a ynðÞ/C28 a yn/C281ðÞ : where n /C301; 2, ..., n, y0 /C30a and yn /C30b : Furthermore, the zeros x1 ; ..., xn ; arranged in increasing order, alternate with the numbers y1 ; .../yn/C281 ; so xn By n Bxn/C271 : More precisely, a xn /C27e ðÞ /C28 a(a) B a ynðÞ/C28 a(a) /C30 l1 /C27.../C27 l n B a xn/C271 /C28eYrvYru /C28 a(a) for n /C301; ..., n /C281:/ See also POINCARE ´ SEPARATION THEOREM ,STURMIAN SEPARATION THEOREM References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 50, 1975. Separatrix A phase curve (i.e., an invariant MANIFOLD ) which meets a HYPERBOLIC FIXED POINT (i.e., an intersection of a stable and an unstable invariant MANIFOLD )or connects the unstable and stable manifolds of a pair of hyperbolic or parabolic fixed points. A separatrix marks a boundary between phase curves with differ- ent properties. For example, the separatrix in the equation of motion for the pendulum occurs at the angular momentum where oscillation gives way to rotation. There are also many systems that have pairs of connected fixed points, e.g., the flow in an open cavity, which has a separatrix that connects two parabolic points. Septendecillion In the American system, 1054. See also LARGE NUMBER Septillion In the American system, 1024. See also LARGE NUMBER Sequence A sequence is an ordered set of mathematical objects which is denoted using braces. For example, the symbol f2ng/C12 n/C301denotes the infinite sequence of EVEN NUMBERS f2; 4; ...; 2n ; ...g:/ See also 196-ALGORITHM , A-SEQUENCE ,ALCUIN’S SE- QUENCE ,A PPELL CROSS SEQUENCE ,A PPELL SE- QUENCE , B2 -SEQUENCE ,B ASIC POLYNOMIAL SEQUENCE ,B EATTY SEQUENCE ,BINOMIAL- TYPE SE- QUENCE ,C ARMICHAEL SEQUENCE ,C AUCHY SE- QUENCE ,CONVERGENT SEQUENCE ,CROSS SEQUENCE , DECREASING SEQUENCE ,DEGREE SEQUENCE ,DENSITY (SEQUENCE ), FRACTAL SEQUENCE ,GIUGA SEQUENCE , INCREASING SEQUENCE ,INFINITIVE SEQUENCE ,INTE- GER SEQUENCE ,ITERATION SEQUENCE ,L IST,N ON- AVERAGING SEQUENCE ,P OLYNOMIAL SEQUENCE , PRIMITIVE SEQUENCE ,R EVERSE- THEN- ADD SE- QUENCE ,S CORE SEQUENCE ,S ERIES ,S HEFFER SE- QUENCE ,S IGNATURE SEQUENCE ,S ORT-THEN- ADD SEQUENCE ,STEFFENSEN SEQUENCE ,ULAM SEQUENCE References Hardy, G. H. A Course of Pure Mathematics, 10th ed. London: Cambridge University Press, 1952. Jeffreys, H. and Jeffreys, B. S. "Sequences." §1.04 in Meth- ods of Mathematical Physics, 3rd ed. Cambridge, Eng- land: Cambridge University Press, pp. 10 /C1/14, 1988. Knopp, K. Theory and Application of Infinite Series. New York: Dover, 1990. SequenceLimit WYNN’S EPSILON METHOD Sequency The sequency k of a WALSH FUNCTION is defined as half the number of zero crossings in the time base. See also WALSH FUNCTION Sequency Function WALSH FUNCTION Sequential Graph A CONNECTED GRAPH having e EDGES is said to be sequential if it is possible to label the nodes i with distinct INTEGERS fiin f0; 1 ; 2 ; ...; e /C281 g such that when EDGE ij is labeled fi /C27fj ; the set of EDGE labels is a block of e consecutive integers (Grace 1983, Gallian 1990). No HARMONIOUS GRAPH is known which cannot also be labeled sequentially. See also CONNECTED GRAPH ,HARMONIOUS GRAPH References Gallian, J. A. "Open Problems in Grid Labeling." Amer. Math. Monthly 97, 133/C1/135, 1990. Grace, T. "On Sequential Labelings of Graphs." J. Graph Th.7, 195/C1/201, 1983.Series A series is an (often infinite) sum of terms specified by some rule. If the difference between successive terms is a constant, then the series is said to be an ARITHMETIC SERIES . If each term equals the previous multiplied by a constant, it is said to be a GEOMETRIC SERIES . A series usually has an INFINITE number of terms, but the phrase INFINITE SERIES is sometimes used for emphasis or clarity. Let the terms in a series be denoted /ai/, let the kth partial sum be given by Sk/C30Xk i/C301ai (1) and let the sequence of partial sums be given by S1/C30a1;S2/C30a1/C27a2;S3/C30a1/C27a2/C27a3;... fg :If the se- quence of partial sums does not converge to a LIMIT (e.g., it oscillates or approaches 9/C12);the series is said to diverge. An example of a convergent series is the GEOMETRIC SERIES X/C12 n/C3001 2Yru*Yru+n /C302: (2) and an example of a divergent series is the HARMONIC SERIES X/C12 n/C3011 n/C30/C12: (3) A number of methods known as CONVERGENCE TESTS can be used to determine whether a given series converges. Although terms of a series can have either sign, convergence properties can often be computed in the "worst case" of all terms being POSITIVE , and then applied to the particular series at hand. A series of terms anis said to be ABSOLUTELY CONVERGENT if the series formed by taking the absolute values of the an; X nanjj; (4) converges. An especially strong type of convergence is called UNIFORM CONVERGENCE , and series which are uni- formly convergent have particularly "nice" properties. For example, the sum of a UNIFORMLY CONVERGENT series of continuous functions is continuous. A CON- VERGENT SERIES can be DIFFERENTIATED term by term, provided that the functions of the series havecontinuous derivatives and that the series of DERIVA- TIVES isUNIFORMLY CONVERGENT . Finally, a UNI- FORMLY CONVERGENT series of continuous functions can be INTEGRATED term by term. For a table listing the COEFFICIENTS for various series operations, see Abramowitz and Stegun (1972, p. 15). While it can be difficult to calculate analytical expressions for arbitrary convergent infinite series, many algorithms can handle a variety of common series types. The program Mathematica implements many of these algorithms. General techniques also exist for computing the numerical values of any but the most pathological series (Braden 1992). Ramanujan found the interesting series identity 1 /C283! (1!2!)3 x2 /C276! (2!4!)3 x4 /C28/C1/C1/C1 /C30 1 /C27x (1!)3 /C27x2 (2!)3 /C27..."# 1 /C28x (1!)3 /C27x2 (2!)3 /C28..."# (5) (Preece 1928; Hardy 1999, p. 7). See also ALTERNATING SERIES ,ARITHMETIC SERIES , ASYMPTOTIC SERIES ,B IAS (SERIES ), CONVERGENCE IMPROVEMENT ,C ONVERGENCE TESTS ,E ULER- MA- CLAURIN INTEGRATION FORMULAS ,G EOMETRIC SER- IES,H ARMONIC SERIES ,H YPERASYMPTOTIC SERIES , INFINITE SERIES , Q-SERIES ,RIEMANN SERIES THEO- REM,SEQUENCE ,SERIES EXPANSION ,SERIES REVER- SION,SUPERASYMPTOTIC SERIES References Abramowitz, M. and Stegun, C. A. (Eds.). "Infinite Series." §3.6 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Arfken, G. "Infinite Series." Ch. 5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 277 /C1/351, 1985. Boas, R. P. Jr. "Partial Sums of Infinite Series, and How They Grow." Amer. Math. Monthly 84, 237 /C1/258, 1977. Boas, R. P. Jr. "Estimating Remainders." Math. Mag. 51, 83 /C1/89, 1978. Borwein, J. M. and Borwein, P. B. "Strange Series and High Precision Fraud." Amer. Math. Monthly 99, 622 /C1/640, 1992. Braden, B. "Calculating Sums of Infinite Series." Amer. Math. Monthly 99, 649 /C1/655, 1992. Bromwich, T. J. I’a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991. Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975. Hardy, G. H. A Course of Pure Mathematics, 10th ed. London: Cambridge University Press, 1952. Hardy, G. H. Divergent Series. Oxford, England: Clarendon Press, 1949. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Jeffreys, H. and Jeffreys, B. S. "Series." §1.05 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cam- bridge University Press, pp. 14 /C1/17, 1988. Jolley, L. B. W. Summation of Series, 2nd rev. ed. New York: Dover, 1961. Knopp, K. Theory and Application of Infinite Series. New York: Dover, 1990. Mangulis, V. Handbook of Series for Scientists and Engi- neers. New York: Academic Press, 1965. Preece, C. T. "Theorems Stated by Ramanujan (III): Theo- rems on Transformation of Series and Integrals." J. London Math. Soc. 3 274 /C1/282, 1928. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Series and Their Convergence." §5.1 inNumerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 159 /C1/163, 1992. Rainville, E. D. Infinite Series. New York: Macmillan, 1967. Weisstein, E. W. "Books about Series." http://www.treasure- troves.com/books/Series.html. Series Expansion This entry contributed by DANIEL SCOTT UZNANSKI A series expansion is a representation of a particular function as a sum of powers in one of its variables, or by a sum of powers of another (usually elementary) function f(x):/ See also LAURENT SERIES ,MACLAURIN SERIES ,POWER SERIES ,SERIES ,SERIES REVERSION ,TAYLOR SERIES Series Inversion SERIES REVERSION Series Multisection If f(x) /C30f0 /C27f1x /C27f2x2 /C27.../C27fnxn /C27... then S(n ; j) /C30fjxj /C27fj/C27nxj /C27n /C27fj/C272nxj/C272n /C27... is given by S(n; j) /C301 nXn/C281 t /C300w/C28jtfwtxðÞ ; where w/C30e2pi=n:/ See also SERIES REVERSION References Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 210 /C1/214, 1985. Series Reversion Series reversion is the computation of the COEFFI- CIENTS of the inverse function given those of the forward function. For a function expressed in a series as y/C30a1x/C27a2x2/C27a3x3/C27...; (1) the series expansion of the inverse series is given by x/C30A1y/C27A2y2/C27A3y3/C27. . . (2) By plugging (2) into (1), the following equation isobtained y/C30a 1A1y/C27a2A2 1/C27a1A2YrvYru y2 /C27a3A31/C272a2A1A2/C27a1A3YrvYru y3 /C273a3A21A2/C27a2A22/C27a2A1A3YrvYru /C27... ( 3 ) Equating COEFFICIENTS then gives A1 /C30a /C281 1 (4) A2 /C30/C28a2 a1A2 1 /C30/C28a/C283 1a2 (5) A3 /C30a /C285 12a22 /C28a1a3YrvYru (6) A4 /C30a/C287 15a1a2a3 /C28a21a4 /C285a32YrvYru (7) A5 /C30a /C289 16a21a2a4 /C273a21a2a3 /C2714a42 /C28a31a5 /C2821a1a22a3YrvYru (8) A6 /C30a/C2811 1 7a31a2a5 /C277a31a3a4 /C2784a1a32a3Yrv /C28a41a6 /C2828a21a2a23 /C2842a52 /C2828a21a22a4 Þ (9) A7 /C30a /C2813 1 8a41a2a6 /C278a41a3a4 /C274a41a24YrvYru /C27120a21a3244 /C27180a21a22a23 /C27132a62 /C28a51a7 /C2836a31a22a5 /C2872a31a2a3a4 /C2812a31a33 /C28330a1a42a3 Þ (10) (Dwight 1961, Abramowitz and Stegun 1972, p. 16). A derivation of the explicit formula for the nth term is given by Morse and Feshbach (1953), An /C301 nan 1X s; t; u...(/C281)s/C27t /C27u/C27... /C2n(n /C27 1) /C1/C1/C1(n /C28 1 /C27 s /C27 t /C27 u ...) s!t!u! /C1/C1/C1a2 a1 !sa3 a1 !t /C1/C1/C1; (11) where s /C272t /C273u /C27.../C30n /C281: (12) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 316 /C1/317, 1985. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 297, 1987. Dwight, H. B. Table of Integrals and Other Mathematical Data, 4th ed. New York: Macmillan, 1961. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 411 /C1/413, 1953. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 22, 1995. Series-Reduced Tree A TREE in which all nodes have degree other than 2 (in other words, no node merely allows a single edge to "pass through"). Series-reduced trees are alsocalled homeomorphically irreducible or topological trees (Bergeron et al. 1998). The numbers of series- reduced trees with 1, 2, ... nodes are 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, ... (Sloane’s A000014). The numbers of series-reduced PLANTED TREES are 0, 1, 0, 1, 1, 2, 3, 6, 10, 19, 35, ... (Sloane’s A001678). The numbers of series-reduced ROOTED TREES are 1, 1, 0, 2, 2, 4, 6, 12, 20, 39, 71, ... (Sloane’s A001679). See also PLANTED TREE,ROOTED TREE,TREE References Bergeron, F.; Leroux, P.; and Labelle, G. Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, pp. 188, 283 /C1/284, 291, and 337, 1998. Cameron, P. J. "Some Treelike Objects." Quart. J. Math. Oxford 38, 155/C1/183, 1987. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 232, 1994. Harary, F. and Palmer, E. M. "Probability that a Point of a Tree Is Fixed." Math. Proc. Camb. Phil. Soc. 85, 407/C1/415, 1979. Harary, F. and Prins, G. "The Number of Homeomorphically Irreducible Trees, and Other Species." Acta Math. 101, 141/C1/162, 1959. Harary, F.; Robinson, R. W. and Schwenk, A. J. "Twenty- Step Algorithm for Determining the Asymptotic Number of Trees of Various Species." J. Austral. Math. Soc., Ser. A 20, 483/C1/503, 1975. Sloane, N. J. A. Sequences A000014/M0320, A001678/ M0768, and A001679/M0327 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Serpentine Curve A curve named and studied by Newton in 1701 and contained in his classification of CUBIC CURVES . It had been studied earlier by L’Hospital and Huygens in1692 (MacTutor Archive).The curve is given by the C ARTESIAN equation y(x)/C30abx x2/C27a2(1) and PARAMETRIC EQUATIONS x(t)/C30acott (2) y(t)/C30bsintcost: (3) The curve has a MAXIMUM atx/C30aand a MINIMUM at x/C30/C28a;where y?(x)/C30ab(a/C28x)(a/C27x) a2/C27x2 ðÞ2/C300; (4) and inflection points at x/C309ffiffiffi 3p a;where yƒ(x)/C302abx x2/C283a2ðÞ x2/C27a2 ðÞ3/C300: (5) The CURVATURE is given by k(x)/C302abx x2/C283a2ðÞ x2/C27a2 ðÞ31/C27a3b/C28abx2ðÞ2 x2/C27a2 ðÞ4"#3=2 (6) k(t)/C304ffiffiffi 2p ab[2 cos(2 t)/C281]cot tcsc2t b2[1/C27cos(4 t)]/C272a2csc4t fg3=2: (7) References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 111 /C1/112, 1972. MacTutor History of Mathematics Archive. "Serpentine." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ser- pentine.html. Serret-Frenet Formulas FRENET FORMULAS Set A set is a FINITE orINFINITE collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a LIST orMULTISET ). Older words for set include AGGREGATE and CLASS . Russell also uses the unfortunate term MANIFOLD to refer to a set. The study of sets and their properties isthe object of SET THEORY . Historically, a single horizontal overbar was used todenote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from theset and hence represented the cardinal number of the set. This practice was begun by SET THEORY founder Georg Cantor. Symbols used to operate on sets include S(which means "and" or INTERSECTION ), and@(which means "or" or UNION ). The symbol ¥is used to denote the set containing no elements, called the EMPTY SET . The NOTATION AB;where AandBare arbitrary sets, is used to denote the set of MAPS from BtoA. For example, an element of XNwould be a MAP from the NATURAL NUMBERS Nto the set X. Call such a function f, then f(1);f(2);etc., are elements of X,s o call them x1;x2;etc. This now looks like a SEQUENCE of elements of X, so sequences are really just func- tions from NtoX. This NOTATION is standard in mathematics and is frequently used in symbolic dynamics to denote sequence spaces.LetE,F, and Gbe sets. Then operation on these sets using the Sand@operators is COMMUTATIVE ESF/C30FSE (1) E@F/C30F@E: (2) ASSOCIATIVE (ESF)SG/C30ES(FSG) (3) (E@F)@G/C30E@(F@G): (4) and DISTRIBUTIVE (ESF)@G/C30(E@G)S(F@G) (5) (E@F)SG/C30(ESG)@(FSG): (6) More generally, we have the infinite distributive laws AS@ l/C23LBlYru$Yru% /C30@ l/C23LASBl ðÞ (7) A@S l/C23LBlYru$Yru% /C30S l/C23LA@Bl ðÞ (8) where lruns through any INDEX SET L:The proofs follow trivially from the definitions of union and intersection. Many classes of sets are denoted using DOUBLE- STRUCK characters. The table below gives symbols for some common sets in mathematics. symbol set /A/ ALGEBRAIC NUMBERS /B/ BOOLEANS /Bn / n-BALL /C/ COMPLEX NUMBERS /Cn;C(n) /n-differentiable functions /Dn / n-DISK /H/ QUATERNIONS /I/ INTEGERS /N/ NATURAL NUMBERS /O/ CAYLEY NUMBERS /P/ PRIME NUMBERS /Q/ RATIONAL NUMBERS /Rn / real n-tuples /Rm/C29n /real m/C29nmatrices /Sn / n-SPHERE /Tn / n-torus /Z/ INTEGERS /Zn/ integers (mod n) /Z /C28 / NEGATIVE INTEGERS /Z /C27 / POSITIVE INTEGERS /Z /C31/ NONNEGATIVE INTEGERS See also AGGREGATE ,ANALYTIC SET,BOREL SET,C, CAYLEY NUMBER ,C LASS (SET), COANALYTIC SET, DEFINABLE SET,D ERIVED SET,D OUBLE- FREE SET, EXTENSION (SET), GROUND SET,I,I NCLUSION- EXCLU- SION PRINCIPLE ,INTENSION ,INTERSECTION ,KINNEY’S SET,LIST,M ANIFOLD ,M ULTISET ,N,P ERFECT SET, POSET ,P ROPER CLASS ,Q,R,R EAL MATRIX ,S ET DIFFERENCE ,SET THEORY ,TRIPLE- FREE SET,UNION , VENN DIAGRAM ,W ELL ORDERED SET,Z,Z /C28,Z/C27 References Courant, R. and Robbins, H. "The Algebra of Sets." Supple- ment to Ch. 2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 108 /C1/116, 1996. Set Difference The set difference /A_B/ is defined by A_B /C30fx : x /C23 A and x QBg: The set difference is therefore equivalent to the COMPLEMENT SET, and is implemented in Mathema- tica asComplement [A, B]. Note that the symbol \ is also used to denote QUOTIENT GROUPS . The symbol A /C28B is sometimes also used to denote a set difference (Smith et al. 1997, p. 68). See also COMPLEMENT SET,SYMMETRIC DIFFERENCE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Smith, D.; Eggen, M.; and St. Andre, R. A Transition to Advanced Mathematics, 4th ed. New York: Brooks/Cole, 1997. Set Direct Product CARTESIAN PRODUCTSet Partition A set partition of a SET S is a collection of disjoint SUBSETS of S whose UNION is S. The number of partitions of the SET fkgn k /C301 is called a BELL NUMBER . See also BELL NUMBER ,B LOCK ,P ARTITION ,R E- STRICTED GROWTH STRING ,S TIRLING NUMBER OF THE SECOND KIND References Ruskey, F. "Info About Set Partitions." http://www.theor- y.csc.uvic.ca/~cos/inf/setp/SetPartitions.html. Set Theory The mathematical theory of SETS. Set theory is closely associated with the branch of mathematics known as LOGIC . There are a number of different versions of set theory, each with its own rules and AXIOMS . In order of increasing CONSISTENCY STRENGTH , several versions of set theory include PEANO ARITHMETIC (ordinary ALGEBRA ), second-order arithmetic (ANALYSIS ), ZER- MELO- FRAENKEL SET THEORY , Mahlo, weakly com- pact, hyper-Mahlo, ineffable, measurable, Ramsey, supercompact, huge, and n-huge set theory. See also ANALYSIS (LOGIC ), AXIOMATIC SET THEORY , CONSISTENCY STRENGTH ,C ONTINUUM HYPOTHESIS , DESCRIPTIVE SET THEORY ,IMPREDICATIVE ,K URA- TOWSKI’S CLOSURE- COMPONENT PROBLEM ,N AIVE SET THEORY ,PEANO ARITHMETIC ,SENTENCE ,SET, THEORY ,Z ERMELO- FRAENKEL AXIOMS ,Z ERMELO- FRAENKEL SET THEORY ,ZERMELO SET THEORY References Brown, K. S. "Set Theory and Foundations." http://www.sea- net.com/~ksbrown/ifoundat.htm. Courant, R. and Robbins, H. "The Algebra of Sets." Supple- ment to Ch. 2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 108 /C1/116, 1996. Devlin, K. The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd ed. New York: Springer-Verlag, 1993. Ferreiro ´s, J. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, 1999. Halmos, P. R. Naive Set Theory. New York: Springer- Verlag, 1974. MacTutor History of Mathematics Archive. "The Beginnings of Set Theory." http://www-groups.dcs.st-and.ac.uk/~his- tory/HistToBeginnings_of_set_theory.html. Stewart, I. The Problems of Mathematics, 2nd ed. Oxford: Oxford University Press, p. 96, 1987. Weisstein, E. W. "Books about Set Theory." http://www.trea- sure-troves.com/books/SetTheory.html. Seven Circles Theorem Draw an initial CIRCLE , and arrange six circles tangent to it such that they touch both the original circle and their two neighbors. Then the three lines joining opposite points of tangency are concurrent in a point. The figures above show several possible configurations (Evelyn et al. 1974, pp. 31 /C1/37). Letting the RADII of three of the circles approach infinity turns three of the CIRCLES into the straight sides of a triangle and the central circle into the triangle’s INCIRCLE . As illustrated above, the three lines connecting opposite points of tangency (with those along the triangle edges corresponding to the vertices of the CONTACT TRIANGLE ) concur (Evelyn et al. 1974, pp. 39 and 42). See also CIRCLE ,CONTACT TRIANGLE ,H EXLET ,IN- CIRCLE ,SIX CIRCLES THEOREM References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Seven Circles Theorem." §3.1 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 31 /C1/42, 1974. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 224 /C1/225, 1991. Sexagesimal The base-60 notational system for representing REAL NUMBERS . A base-60 number system was used by the Babylonians and is preserved in the modern mea-surement of time (hours, minutes, and seconds) and ANGLES (DEGREES , ARC MINUTES , and ARC SECONDS ). See also BASE (NUMBER ), BINARY ,DECIMAL ,HEXADE- CIMAL ,O CTAL ,Q UATERNARY ,S CRUPLE ,T ERNARY , VIGESIMAL References Bergamini, D. Mathematics. New York: Time-Life Books, pp. 16 /C1/17, 1969. Weisstein, E. W. "Bases." MATHEMATICA NOTEBOOK BASES.M . Sexdecillion In the American system, 1051. See also LARGE NUMBER Sextic Equation The general sextic polynomial equation x6 /C27a5x5 /C27a4x4 /C27a3x3 /C27a2x2 /C27a1x /C27a0 /C300 can be solved in terms of HYPERGEOMETRIC FUNC- TIONS in one variable using Klein’s approach to solving the QUINTIC EQUATION . See also CUBIC EQUATION ,Q UADRATIC EQUATION , QUARTIC EQUATION ,QUINTIC EQUATION References Coble, A. B. "The Reduction of the Sextic Equation to the Valentiner Form--Problem." Math. Ann. 70, 337 /C1/350, 1911a. Coble, A. B. "An Application of Moore’s Cross-ratio Group to the Solution of the Sextic Equation." Trans. Amer. Math. Soc. 12, 311 /C1/325, 1911b. Cole, F. N. "A Contribution to the Theory of the General Equation of the Sixth Degree." Amer. J. Math. 8, 265 /C1/286, 1886. Sextic Surface An ALGEBRAIC SURFACE which can be represented implicitly by a polynomial of degree six in x, y, and z. Examples are the BARTH SEXTIC and BOY SURFACE . See also ALGEBRAIC SURFACE ,BARTH SEXTIC ,BOY SURFACE ,CUBIC SURFACE ,DECIC SURFACE ,H UNT’S SURFACE ,QUADRATIC SURFACE ,QUARTIC SURFACE References Catanese, F. and Ceresa, G. "Constructing Sextic Surfaces with a Given Number of Nodes." J. Pure Appl. Algebra 23, 1 /C1/12, 1982. Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Sextillion In the American system, 1021. See also LARGE NUMBER Sexy Primes Since a PRIME NUMBER cannot be divisible by 2 or 3, it must be true that, for a PRIME p, p /C131; 5 (mod 6): This motivates the definition of sexy primes as a pair of primes (p, q) such that p /C28q /C306 ("sexy" since "sex" is the Latin word for "six."). The first few sexy prime pairs are (5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), ... (Sloane’s A023201 and A046117). Sexy constellations also exist. The first few sexy triplets (i.e., numbers such that each of (p ; p /C276; p /C27 12) is PRIME but p /C2718 is not PRIME ) are (7, 13, 19), (17, 23, 29), (31, 37, 43), (47, 53, 59), ... (Sloane’s A046118, A046119, and A046120). The first few sexy quadruplets are (11, 17, 23, 29), (41, 47, 53, 59), (61, 67, 73, 79), (251, 257, 263, 269), ... (Sloane’s A046121, A046122, A046123, and A046124). Sexy quadruplets can only begin with a PRIME ending in a "1." There is only a single sexy quintuplet, (5, 11, 17, 23, 29), since every fifth number of the form 6n 91 is divisible by 5, and therefore cannot be PRIME . See also PRIME CONSTELLATION ,PRIME QUADRUPLET , TWIN PRIMES References Sloane, N. J. A. Sequences A023201, A046117, A046118, A046119, A046120, A046121, A046122, A046123, and A046124 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Trotter, T. "Sexy Primes." http://www.geocities.com/Cape- Canaveral/Launchpad/8202/sexyprim.html. Seydewitz’s Theorem If a TRIANGLE is inscribed in a CONIC SECTION , any line conjugate to one side meets the other two sides in conjugate points. See also CONIC SECTION ,TRIANGLE Seymour Conjecture Seymour conjectured that a graph G of order n with minimum VERTEX DEGREE d(G) ]kn =(k /C271) contains the kth GRAPH POWER of a HAMILTONIAN CIRCUIT , generalizing PO´ SA’S CONJECTURE . Komlo ´s et al. (1998) proved the conjecture for sufficiently large n using SZEMERE ´ DI’S REGULARITY LEMMA and a techni- que called the BLOW-UP LEMMA . See also HAJNAL- SZEMERE ´ DI THEOREM ,HAMILTONIAN CIRCUIT ,PO´ SA’S CONJECTURE ,PO´ SA’S THEOREM ,SZE- MERE ´ DI’S REGULARITY LEMMA References Faudree, R. J.; Gould, R. J.; Jacobson, M. S.; and Schelp, R. H. "On a Problem of Paul Seymour." In Recent Advances in Graph Theory (Ed. V. R. Kulli). Vishwa International Publishers, pp. 197 /C1/215, 1991.Komlo ´s, J.; Sa´rkozy, G. N.; and Szemere ´di, E. "On the Square of a Hamiltonian Cycle in Dense Graphs." In Random Structures Algorithms 9, 193 /C1/211, 1996. Komlo ´s, J.; Sa´rkozy, G. N.; and Szemere ´di, E. "Proof of the Seymour Conjecture for Large Graphs." Ann. Comb. 2, 43 /C1/60, 1998. Seymour, P. Problem Section in Combinatorics: Proceedings of the British Combinatorial Conference, 1973 (Ed. T. P. McDonough and V. C. Mavron). Cambridge, Eng- land: Cambridge University Press, pp. 201 /C1/202, 1974. Sgn Also called SIGNUM . It can be defined as sgn /C13/C281 x B0 0 x /C300 1 x > 08 < : (1) or sgn(x) /C302H(x) /C281 : (2) where H(x) is the HEAVISIDE STEP FUNCTION . For x " 0; this can be written sgn(x) /C13x ½x½: (3) See also A BSOLUTE VALUE ,H EAVISIDE STEP FUNC- TION ,RAMP FUNCTION References Bracewell, R. "The Sign Function, sgn x:/"In The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 61 /C1/62, 1999. Sh HYPERBOLIC SINE Shadow The SURFACE corresponding to the region of obscura- tion when a solid is illuminated from a point light source (located at the RADIANT POINT ). A DISK is the SHADOW of a SPHERE on a PLANE perpendicular to the SPHERE -RADIANT POINT line. If the PLANE is tilted, the shadow can be the interior of an ELLIPSE or a PARABOLA . See also CORK PLUG,PROJECTION ,STEREOLOGY ,TRIP- LET References Croft, H. T.; Falconer, K. J.; and Guy, R. K. "What Can You Tell About a Convex Body from Its Shadows?" §A10 in Unsolved Problems in Geometry. New York: Springer- Verlag, pp. 23 /C1/24, 1991. Shadowing Theorem Although a numerically computed CHAOTIC trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically com- puted one. Therefore, the FRACTAL structure of chaotic trajectories seen in computer maps is real. References Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 18 /C1/19, 1993. Shafarevich Conjecture A conjecture which implies the MORDELL CONJEC- TURE , as proved in 1968 by A. N. Parshin. See also MORDELL CONJECTURE References Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, p. 45, 1987. Shah Function III(x) /C13X/C12 n/C30/C28/C12d(x /C28n) (1) where d(x) is the DELTA FUNCTION , so III(x) /C300 for x Q Z (i.e., x not an INTEGER ). The shah function is also called the sampling symbol or replicating symbol (Bracewell 1999, p. 77) and obeys the identities III(ax) /C301 ½a ½X/C12 n/C30/C28/C12d x /C28n a ! (2) III(/C28x) /C30III(x) (3) III(x /C27n) /C30III(x) (4) III x /C281 2Yru*Yru+ /C30III x /C2712Yru*Yru+ : (5) The shah function is normalized so that gn/C271 =2 n /C281 =2III(x) dx /C301: (6) The "sampling property" is III(x)f(x) /C30X/C12 n/C30/C28/C12f(n)d(x /C28n) (7)and the "replicating property" is III(x) +f(x) /C30X/C12 n/C30/C28/C12f(x /C28n) : (8) where + denotes CONVOLUTION . The 2-D sampling function, sometimes called the bed- of-nails function, is given by 2III(x; y) /C30X/C12 m/C30/C28/C12X/C12 n/C30/C28/C12d(x /C28m; y /C28n); (9) which can be adjusted using a series of weighted as v(x; y) /C30X RmnTmnDmn d x /C28mn ; y /C28n ðÞ ; (10) where Rmnis a reliability weight, Dmnis a density weight (WEIGHTING FUNCTION ), and Tmnis a taper. The 2-D shah function satisfies 2III(x; y) /C30III(x)III(y) (11) (Bracewell 1999, p. 85). See also CONVOLUTION ,DELTA FUNCTION ,IMPULSE PAIR,SINC FUNCTION References Bracewell, R. "The Sampling of Replicating Symbol III(x) :/" In The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 77 /C1/79, 1999. Shah-Wilson Constant TWIN PRIMES CONSTANT Shaky Polyhedron A shaky polyhedron is a non-rigid concave polyhedron which is only infinitesimally movable. JESSEN’S ORTHOGONAL ICOSAHEDRON is a shaky polyhedron (Wells 1991). See also FLEXIBLE POLYHEDRON ,JESSEN’S ORTHOGO- NAL ICOSAHEDRON ,M ULTISTABLE ,R IGID POLYHE- DRON ,RIGIDITY THEOREM References Blaschke, W. "Wackelige Achtflache." Math. Z. 6,8 5/C1/93, 1920. Cromwell, P. R. Polyhedra. New York: Cambridge Univer- sity Press, p. 222, 1997. Gluck, H. Almost All Simply Connected Closed Surfaces are Rigid. Heidelberg, Germany: Springer-Verlag, pp. 225 /C1/ 239, 1975. Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165/C1/170, 1978. Jessen, B. "Orthogonal Icosahedron." Nordisk Mat. Tidskr. 15,9 0/C1/96, 1967. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 161, 1991. Shallit Constant Define fx1 ; x2 ; ...; xn ðÞ with xi POSITIVE as fx1 ; x2 ; ...; xn ðÞ /C13Xn i /C301xi /C27X 1 5i 5k 5nYk j/C3011 xj: Then min f /C303n /C28C /C27o(1) as n increases, where the Shallit constant is C /C301:369451403937... (Shallit 1995). In their solution, Grosjean and De Meyer (quoted in Shallit 1995) reduced the complex- ity of the problem. References MacLeod, A. http://www.mathsoft.com/asolve/constant/sha- piro/macleod.html. Shallit, J. Solution by C. C. Grosjean and H. E. De Meyer. "A Minimization Problem." Problem 94 /C1/15 in SIAM Re- view 37, 451 /C1/458, 1995. Shallow Diagonal See also DIAGONAL ,PASCAL’S TRIANGLE Shanks’ Algorithm An ALGORITHM which finds the least NONNEGATIVE value offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a(mod p)p for given a and PRIME p. Shanks’ Conjecture Let p(n) be the first PRIME which follows a PRIME GAP of n between consecutive PRIMES . Shanks’ conjecture holds that p(n) /C2expffiffiffinpYrvYru : Wolf conjectures a slightly different form p(n) /C2ffiffiffinpexpffiffiffinpYrvYru ; which agrees better with numerical evidence. See also P RIME DIFFERENCE FUNCTION ,PRIME GAPS References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 21, 1994. Rivera, C. "Problems & Puzzles: Conjecture Shanks’ Con- jecture.-009." http://www.primepuzzles.net/conjectures/ conj_009.htm. Shanks, D. "On Maximal Gaps Between Successive Primes." Math. Comput. 18, 646 /C1/651, 1964. Shannon Entropy ENTROPY Shannon Sampling Theorem SAMPLING THEOREMShannon’s Noiseless Coding Theorem Let S be an information source with entropy H(S): Then H(S) 5m(S); where m(S) is the minimum average code-word length among all uniquely decipherable coding schemes for S References Casti, J. L. "The Shannon Coding Theorem." Ch. 1 in Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics. New York: Wiley, pp. 207 /C1/254, 2000. Shape Number FIGURATE NUMBER Shape Operator The negative derivative S(v) /C30/C28DvN (1) of the unit normal N vector field of a SURFACE is called the shape operator (or WEINGARTEN MAP or SECOND FUNDAMENTAL TENSOR ). The shape operator S is an EXTRINSIC CURVATURE , and the GAUSSIAN CURVATURE is given by the DETERMINANT of S.Ifx : U 0 R3 is a REGULAR PATCH , then S xuðÞ/C30/C28Nu (2) S xvðÞ/C30/C28Nv : (3) At each point p on a REGULAR SURFACE M ƒR3 ; the shape operator is a linear map S : Mp 0 Mp : (4) The shape operator for a surface is given by the WEINGARTEN EQUATIONS . See also CURVATURE ,FUNDAMENTAL FORMS ,W EIN- GARTEN EQUATIONS References Gray, A. "The Shape Operator," "Calculation of the Shape Operator," and "The Eigenvalues of the Shape Operator." §16.1, 16.3, and 16.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 360 /C1/363 and 367 /C1/372, 1997. Reckziegel, H. In Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunsch- weig, Germany: Vieweg, p. 30, 1986. Shapiro’s Cyclic Sum Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Consider the sum fn(x1 ; x2 ; ... ; xn) /C30x1 x2 /C27 x3/C27x2 x3 /C27 x4/C27.../C27xn/C281 xn /C27 x1/C27xn x1 /C27 x2; (1) where the xj/s are NONNEGATIVE and the DENOMINA- TORS are POSITIVE . Shapiro (1954) asked if fn(x1 ; x2 ; ...; xn) ]1 2 n (2) for all n. It turns out (Mitrinovic et al. 1993) that this INEQUALITY is true for all EVEN n 512 and ODD n 523: Ranikin (1958) proved that for f(n) /C30inf x]0fn(x1 ; x2 ; ...; xn) ; (3) l /C30 lim n0/C12f(n) n/C30inf n ]1f(n) nB12 /C287 /C2910 /C288 : (4) /l can be computed by letting f(x) be the CONVEX HULL of the functions y1 /C30e /C28x (5) y2 /C302 ex /C27 ex=2 : (6) Then l /C3012 f(0) /C300 :4945668... (7) (Drinfeljd 1971). A modified sum was considered by Elbert (1973): gn(x1 ; x1 ; ... ; xn) /C30x1 /C27 x3 x1 /C27 x2/C27x2 /C27 x4 x2 /C27 x3/C27.../C27xx /C281 /C27 x1 xn /C281 /C27 xn/C27xn /C27 x2 xn /C27 x1: (8) Consider m /C30 lim n0/C12g(n) n; (9) where g(n)/C30inf x]0gn(x1;x2;...;xn); (10) and let c(x) be the CONVEX HULL of y1/C301 2(1/C27ex) (11) y2/C301/C27ex 1/C27ex=2: (12) Then m/C30c(0)/C300:978012 . . . : (13) See also CONVEX HULLReferences Drinfeljd, V. G. "A Cyclic Inequality." Math. Notes. Acad. Sci. USSR 9,6 8/C1/71, 1971. Elbert, A. "On a Cyclic Inequality." Period. Math. Hungar. 4, 163/C1/168, 1973. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/shapiro/shapiro.html. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical and New Inequalities in Analysis. New York: Kluwer, 1993. Sharing Problem A problem also known as the POINTS PROBLEM or UNFINISHED GAME . Consider a tournament involving kplayers playing the same game repetitively. Each game has a single winner, and denote the number of games won by player iat some juncture wi:The games are independent, and the probability of the ith player winning a game is pi:The tournament is specified to continue until one player has won n games. If the tournament is discontinued before anyplayer has won ngames so that w iBnfori/C301, ..., k, how should the prize money be shared in order to distribute it proportionally to the players’ chances of winning? For player i, call the number of games left to win ri/C13 n/C28wi>0 the "quota." For two players, let p/C13p1and q/C13p2/C301/C28pbe the probabilities of winning a single game, and a/C13r1/C30n/C28w1and b/C13r2/C30n/C28w2be the number of games needed for each player to win the tournament. Then the stakes should be divided in the ratio m:n;where m/C30pa1/C27a 1q/C27a(a/C271) 2!q2/C27..." /C27a(a/C271)/C1/C1/C1(a/C27b/C282) (b/C281)!qb/C281Yrtu (1) n/C30qb1/C27b 1p/C27b(b/C271) 2!p2/C27..." /C27b(b/C271)/C1/C1/C1(b/C27a/C282) (a/C281)!pa/C281Yrtu (2) (Kraitchik 1942). Ifiplayers have equal probability of winning ("cell probability"), then the chance of player iwinning for quotas r1;...,rkis Wi/C30Dk/C281 1(r1;...;ri/C281;ri/C271;...;rk;ri): (3) where Dis the D IRICHLET INTEGRAL of type 2D. Similarly, the chance of player ilosing is Li/C30Ck/C281 1(r1;...;ri/C281;ri/C271;...;rk;ri); (4) where Cis the D IRICHLET INTEGRAL of type 2C. If the cell quotas are not equal, the general Dirichlet integral Dnmust be used, where ai /C30pi 1 /C28Pk /C281 i/C301pi: (5) If ri /C30r and ai /C301; then Wiand Lireduce to 1=k as they must. Let P(r1 ; ...; rk) be the joint probability that the players would be RANKED in the order of the ri/s in the argument list if the contest were completed. For k /C303, P(r1 ; r2 ; r3) /C30CD(1; 1) 1(r1 ; r2 ; r3) : (6) For k /C304 with quota vector r /C30(r1 ; r2 ; r3 ; r4) and D/C30 p2 /C27p3 /C27p4 ; P(r) /C30Xr3 /C281 i/C300Xr4 /C281 j/C300r2 /C281 /C27i /C27j r2 /C281 ; i ; jYru$Yru%p2 D !r2p3 D !ip4 D !j /C2C(1) p1 =D(r1 ; r2 /C27i /C27j)D(1)p4 =p3(r4 /C28j; r3 /C28i) : (7) An expression for k /C305 is given by Sobel and Frankowski (1994, p. 838). See also DIRICHLET INTEGRALS References Kraitchik, M. "The Unfinished Game." §6.1 in Mathematical Recreations. New York: W. W. Norton, pp. 117 /C1/118, 1942. Sobel, M. and Frankowski, K. "The 500th Anniversary of the Sharing Problem (The Oldest Problem in the Theory of Probability)." Amer. Math. Monthly 101, 833 /C1/847, 1994. Sharkovsky’s Theorem SARKOVSKII’S THEOREM Sharpe Ratio A risk-adjusted financial measure developed by Nobel Laureate William Sharpe. It uses a fund’s standard deviation and excess return to determine the reward per unit of risk. The higher a fund’s Sharpe ratio, the better the fund’s "risk-adjusted" performance. See also ALPHA ,BETA Sharpe’s Differential Equation A generalization of the BESSEL DIFFERENTIAL EQUA- TION for functions of order 0, given by zyƒ/C27y?/C27(z /C27A)y /C300 : Solutions are y /C30e 9iz 1F11 2 /C1412 iA;1;/C142izYru*Yru+ : where1F1(a; b; x)isa CONFLUENT HYPERGEOMETRIC FUNCTION . See also BESSEL DIFFERENTIAL EQUATION ,CONFLU- ENT HYPERGEOMETRIC FUNCTION Sheaf SHEAF OF PLANES ,SHEAF (TOPOLOGY )Sheaf (Topology) A topological GADGET related to families of ABELIAN GROUPS and MAPS . References Iyanaga, S. and Kawada, Y. (Eds.). "Sheaves." §377 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1171 /C1/1174, 1980. Sheaf of Planes The set of all PLANES through a LINE. The line is sometimes called the AXIS of the sheaf, and the sheaf itself is sometimes called a pencil (Altshiller-Court 1979, p. 12). See also LINE,PENCIL ,PLANE References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 13, 1948. Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 12, 1961. Shear A transformation in which all points along a given LINE L remain fixed while other points are shifted parallel to L by a distance proportional to their PERPENDICULAR distance from L. Shearing a plane figure does not change its AREA . The shear can also be generalized to 3-D, in which PLANES are translated instead of lines. See also SHEAR FACTOR ,SHEAR MATRIX Shear Factor The distance a point moves due to SHEAR divided by the perpendicular distance of a point from the invariant line. See also SHEAR ,SHEAR MATRIX References Pimentel, R. and Wall, T. IGCSE Mathematics. London: John Murray, p. 312, 1997. Shear Matrix The shear matrix e s ijis obtained from the IDENTITY MATRIX by inserting s at (i, j), e.g., e s 12 /C301 s 0 010 0012 435: (1) Bolt and Hobbs (1998) define a shear matrix as a matrix ab cdYrtvYrtu (2) such that a/C27b/C302 (3) ad/C28bc/C301: (4) See also E LEMENTARY MATRIX ,SHEAR ,SHEAR FACTOR References Bolt, B. and Hobbs, D. A Mathematical Dictionary for Schools. Cambridge, England: Cambridge University Press, 1998. Sheffer Sequence A sequence sn(x) is called a Sheffer sequence IFFits GENERATING FUNCTION has the form X/C12 k/C300sk(x) k!tk/C30A(t)exB(t); (1) where A(t)/C30A0/C27A1t/C27A2t2/C27... ( 2 ) B(t)/C30B1t/C27B2t2/C27...: (3) with A0;B1"0:/ Iff(t) is a delta series and g(t) is an invertible series, then there exists a unique sequence sn(x) of Sheffer polynomials sn(x) satisfying the orthogonality condi- tion g(t)[f(t)]kjsn(x)DE /C30n!dnk; (4) where dnkis the K RONECKER DELTA (Roman 1984, p. 17). Examples of general Sheffer sequences includethe ACTUARIAL POLYNOMIALS ,B ERNOULLI POLYNO- MIALS OF THE SECOND KIND ,BOOLE POLYNOMIALS , LAGUERRE POLYNOMIALS ,M EIXNER POLYNOMIALS OF THE FIRST and SECOND KINDS ,P OISSON- CHARLIER POLYNOMIALS , and S TIRLING POLYNOMIALS . The Sheffer sequence for (1 ;f(t)) is called the asso- ciated sequence for f(t);and Roman (1984, pp. 53 /C1/86) summarizes properties of the associated Sheffer sequences and gives a number of specific examples (ABEL POLYNOMIAL ,BELL POLYNOMIAL ,CENTRAL FAC- TORIAL ,EXPONENTIAL POLYNOMIAL ,FALLING FACTOR- IAL,G OULD POLYNOMIAL ,M AHLER POLYNOMIAL , MITTAG- LEFFLER POLYNOMIAL ,M OTT POLYNOMIAL , POWER POLYNOMIAL ). The Sheffer sequence for (g(t);t) is called the A PPELL SEQUENCE ofg(t);and Roman (1984, pp. 86 /C1/106) summarizes properties of Appell sequences and gives a number of specific examples. Ifsn(x) is a Sheffer sequence for ( g(t);f(t));then for any polynomial p(x); p(x)/C30X/C12 k/C300g(t)[f(t)]kjp(x)DE k!sk(x): (5) The sequence sn(x) is Sheffer for ( g(t);f(t))IFF 1 g(¯f(t))ey¯f(t)/C30X/C12 k/C300sk(y) k!tk(6) for all yin the field Cof characteristic 0, where ¯f(t)i s the compositional INVERSE FUNCTION off(t) (Roman 1984, p. 18). This formula immediately gives the GENERATING FUNCTION associated with a given Shef- fer sequence.A sequence is Sheffer for ( g(t);f(t)) for some inver- tible g(t) IFF f(t)sn(x)/C30nsn/C281(x) (7) for all n]0 (Roman 1984, p. 20). The Sheffer identity states that a sequence sn(x) is Sheffer for ( g(t);f(t)) for some invertible f(t)IFFit satisfies some BINOMIAL- TYPE SEQUENCE sn(x/C27y)/C30Xn k/C300n kYru$Yru% pk(y)sn/C28k(x) (8) for all yinC, where pn(x) is associated to f(t) (Roman 1984, p. 21). The RECURRENCE RELATION for Sheffer sequences is given by sn/C271(x)/C30x/C28g?(t) g(t)"# 1 f?(t)sn(x) (9) (Roman 1984, p. 50). A nontrivial RECURRENCE RELA- TION is given by sn/C271(x)/C30(x/C28bn)sn(x)/C28dnsn/C281(x) (10) fors/C281(x)/C300;s0(x)/C301;and n]0 (Meixner 1934; Sheffer 1939; Chihara 1978; Roman 1984, pp. 156 /C1/ 160). The connection coefficients cnk in the expression sn(x) /C30Xn k /C300cnkrk(n) (11) are given by cnk /C301 k!h(f /C281(t)) g(f /C281(t))[l(f /C281(t))]kj xn*+ ; (12) where sn(x) is Sheffer for (g(t) ; f(t)) and rn(x)is Sheffer for (h(t); l(t)) : This can also be written in terms of the polynomial of coefficients tn(x) /C30Xn k /C300cnkxk : (13) which is Sheffer for g(l /C281(t)) h(l /C281(t)) ; f(l/C281(t)) ! (14) (Roman 1984, pp. 132 /C1/138). A duplication formula OF THE FORM rn(ax) /C30Xn k /C300cnkrk(x) (15) is given by cnk /C301 k!h(al/C281(t)) h(l /C281(t))[l(al /C281(t))kj xn*+ ; (16) where rn(x) is Sheffer for (h(t) ; l(t)) (Roman 1984, pp. 132 /C1/138). See also APPELL CROSS SEQUENCE ,A PPELL SE- QUENCE ,B INOMIAL -TYPE SEQUENCE ,C ROSS SE- QUENCE ,STEFFENSEN SEQUENCE ,UMBRAL CALCULUS References Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978. Meixner, J. "Orthogonale Polynomsystem mit linern beson- deren Gestalt der eryengenden Funktion." J. London Math. Soc. 9,6/C1/13, 1934. Roman, S. "Sheffer Sequences." Ch. 2 and §4.3 in The Umbral Calculus. New York: Academic Press, pp. 2, 6 /C1/ 31, and 107 /C1/130, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/760, 1973. Sheffer, I. M. "Some Properties of Polynomial Sets of Type Zero." Duke Math. J. 5, 590 /C1/622, 1939. Sheffer Stroke NANDShephard’s Problem Measurements of a centered convex body in Eucli- dean n-space (for n ]3) show that its brightness function (the volume of each projection) is smaller than that of another such body. Is it true that its VOLUME is also smaller? C. M. Petty and R. Schnei- der showed in 1967 that the answer is yes if the body with the larger brightness function is a projection body, but no in general for every n. References Gardner, R. J. "Geometric Tomography." Not. Amer. Math. Soc. 42, 422 /C1/429, 1995. Sheppard’s Correction A correction which must be applied to the measured MOMENTS mkobtained from NORMALLY DISTRIBUTED data which have been BINNED in order to obtain correct estimators ˆmifor the population moments mi : The corrected versions of the second, third, and fourth moments are then ˆm2 /C30m2 /C281 12 c2 (1) ˆm3 /C30m3 (2) ˆm4 /C30m4 /C281 2 m2 /C277 240 c2 ; (3) where c is the CLASS INTERVAL . If k ?ris the rth CUMULANT of an ungrouped distribu- tion and krthe rth CUMULANT of the grouped distribution with CLASS INTERVAL c, the corrected cumulants (under rather restrictive conditions) are k?r/C30krfor r odd kr/C28Br rcrfor r even ;8 < :(4) where Bris the rth B ERNOULLI NUMBER , giving k?1/C30k1 (5) k?2/C30k2/C281 12c2(6) k?3/C30k3 (7) k?4/C30k4/C271 120c4(8) k?5/C30k5 (9) k?6/C30k6/C281 252c6: (10) For a proof, see Kendall et al. (1987). See also BIN,CLASS INTERVAL ,HISTOGRAM References Fisher, R. A. Statistical Methods for Research Workers, 14th ed., rev. and enl. Darien, CO: Hafner, 1970. Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall’s Ad- vanced Theory of Statistics, Vol. 1: Distribution Theory, 6th ed. New York: Oxford University Press, 1987. Kenney, J. F. and Keeping, E. S. "Sheppard’s Correction for Grouping Errors." §7.6 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 95 /C1/96, 1962. Kenney, J. F. and Keeping, E. S. "Sheppard’s Correction." §4.12 in Mathematics of Statistics, Pt. 2, 2nd ed. Prince- ton, NJ: Van Nostrand, pp. 80 /C1/82, 1951. Whittaker, E. T. and Robinson, G. "Sheppard’s Corrections." §99 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 194 /C1/196, 1967. Sherman-Morrison Formula A formula which allows a perturbed MATRIX to be computed for a small change to a given MATRIX A: If the change can be written in the form u /C156v for two vectors u and v, then the Sherman-Morrison formula is (A /C27u /C156v) /C281 /C30A/C281 /C28(A/C281u) /C156 (v /C215 A /C281) 1/C27 l; where l /C13v /C215 A/C281u: See also WOODBURY FORMULA References Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, p. 51, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Sherman-Morrison Formula." In Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 65 /C1/67, 1992. Shi Shi(z) /C30gz 0sinh t tdt: The function is given by the Mathematica command SinhIntegral [z]. See also CHI,COSINE INTEGRAL ,SINE INTEGRAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Cosine Integrals." §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231 /C1/233, 1972. Shidlovskii Theorem Let f1(z) ; ..., fm(z) for m ]1 be a set of E-FUNCTIONS that (1) form a solution of the system of differential equations y?k /C30qk0 /C27Xm j /C301qkjyj for qkj /C23C(z) and k /C301, ..., m and (2) are ALGEBRAI- CALLY INDEPENDENT over C(z) : Then for all a /C23A; where A denotes the set of ALGEBRAIC NUMBERS with a "0 and distinct from singularities of the differential equations, the numbers f1( a) ; ..., fm( a) are ALGEBRAI- CALLY INDEPENDENT (Nesterenko 1999). See also ALGEBRAICALLY INDEPENDENT ,E-FUNCTION References Nesterenko, Yu. V. §1.2 in A Course on Algebraic Indepen- dence: Lectures at IHP 1999. http://www.math.jussieu.fr/ ~nesteren/. Shidlovskii, A. B. Transcendental Numbers. New York: de Gruyter, 1989. Shift A TRANSLATION without ROTATION or distortion. See also DILATION ,EXPANSION ,ROTATION ,TRANSLA- TION ,TWIRL Shift Operator An operator E such that Eap(x) /C30p(x /C27a) : See also SHIFT- INVARIANT OPERATOR References Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/760, 1973. Shift Property DELTA FUNCTION Shift Transformation The transformation T(x) /C30frac1 x ! /C301 x /C281 x$% ; where frac( x) is the FRACTIONAL PART of x and xbcis the FLOOR FUNCTION , that takes a CONTINUED FRAC- TION [a1 ; a2 ; ...] to [a2 ; a3 ; ...]:/ See also GAUSS- KUZMIN- WIRSING CONSTANT References Viader, P.; Paradis, J.; and Bibiloni, L. "A New Light on Minkowski’s ?(x) Function." J. Number Th. 73, 212 /C1/227, 1998. Shifted Factorial POCHHAMMER SYMBOL ,RISING FACTORIAL Shift-Invariant Operator An operator T which commutes with all SHIFT OPERATORS Ea ; so TEa /C30EaT for all real a in a FIELD . Any two shift-invariant operators commute. See also DELTA OPERATOR ,H EAVISIDE CALCULUS , SHIFT OPERATOR References Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684/C1/760, 1973. Shimura-Taniyama Conjecture TANIYAMA- SHIMURA CONJECTURE Shimura-Taniyama-Weil Conjecture TANIYAMA- SHIMURA CONJECTUREShoe HOOK,SHOE SURFACE Shoe Surface A surface given by the PARAMETRIC EQUATIONS x(u;v)/C30u (1) y(u;v)/C30v (2) z(u;v)/C301 3u3/C2812v2: (3) The coefficients of the coefficients of the FIRST FUNDAMENTAL FORM are E/C301/C27u4(4) F/C30/C28u2v (5) G/C301/C27v2; (6) and the SECOND FUNDAMENTAL FORM coefficients are e/C302uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u4/C27v2p (7) f/C300 (8) g/C30/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u4/C27v2p ; (9) giving AREA ELEMENT dA/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C282uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u4/C27v2ps duffldv; (10) and G AUSSIAN and MEAN CURVATURES K/C30/C282u (1/C27u4/C27v2)2(11) H/C302u(1/C27v2)/C28u4/C281 2(1/C27u4/C27v2)3=2: (12) References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 965, 1997. Shoemaker’s Knife ARBELOS Short Exact Sequence A short exact sequence of groups A, B, and C is given by two maps a : A 0 B and b : B 0 C and is written 0 0 A 0 B 0 C 0 0: Because it is an EXACT SEQUENCE , a is INJECTIVE , and b is SURJECTIVE . Moreover, the KERNEL of b is the image of a: Hence, the group A can be considered as a (normal) subgroup of B, and C is isomorphic to B =A:/ A short exact sequence is said to split if there is a map g : C 0 B such that b(g is the identity on C. This only happens when B is the DIRECT PRODUCT of A and C. The notion of a short exact sequence also makes sense for MODULES and SHEAVES . See also EXACT SEQUENCE ,GROUP EXTENSION ,LONG EXACT SEQUENCE ,MODULE ,PRINCIPAL BUNDLE References Atiyah, M. F. and MacDonald, I. G. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, pp. 22 /C1/ 24, 1969. Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, p. 144, 1995. Hilton, P. and Stammbach, U. A Course in Homological Algebra. New York: Springer-Verlag, 1997. Munkres, J. Elements of Algebraic Topology. Reading, MA: Addison-Wesley, pp. 130 /C1/133, 1984. Shortening A KNOT used to shorten a long rope. See also BEND (KNOT) References Owen, P. Knots. Philadelphia, PA: Courage, p. 65, 1993. Shortest Path DIJKSTRA’S ALGORITHM ,GRAPH GEODESIC Shortness Exponent Let v(G) be the number of vertices in a GRAPH G and h(G) the length of the maximum cycle in G. Then the shortness exponent of a class of graphs G is defined by s(G) /C30lim inf G /C23Gln h(G) ln v(G):References Gru¨nbaum, B. and Walther, H. "Shortness Exponents of Families of Graphs." J. Combin. Th. A 14, 364 /C1/385, 1973. Owens, P. J. "Bipartite Cubic Graphs and a Shortness Exponent." Disc. Math. 44, 327 /C1/330, 1983. Shovelton’s Rule Let the values of a function f(x) be tabulated at points xiequally spaced by h /C30xi/C271 /C28xi ; so f1 /C30f(x1) ; f2 /C30 f(x2) ; ..., f11 /C30f(x11): Then Shovelton’s rule approxi- mating the integral of f(x) is given by the NEWTON- COTES -like formula gx11 x1f(x) dx /C305 126 h[8(f1 /C27f11) /C2735(f2 /C27f4 /C27f8 /C27f10) /C2715(f3/C27f5/C27f7/C27f9)/C2736f6]: See also BODE’S RULE,HARDY’S RULE,NEWTON- COTES FORMULAS ,S IMPSON’S 3/8 RULE,S IMPSON’S RULE, TRAPEZOIDAL RULE,W EDDLE’S RULE References King, A. E. "Approximate Integration. Note on Quadrature Formulae: Their Construction and Application to Actuar- ial Functions." Trans. Faculty of Actuaries 9, 218/C1/231, 1923. Sheppard, W. F. "Some Quadrature-Formulæ." Proc. Lon- don Math. Soc. 32, 258/C1/277, 1900. Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 151, 1967. Shuffle The randomization of a deck of CARDS by repeated interleaving. More generally, a shuffle is a rearrange- ment of the elements in an ordered list. Shuffling by exactly interleaving two halves of a deck is called a RIFFLE SHUFFLE . Normal shuffling leaves gaps of different lengths between the two layers of cards and so randomizes the order of the cards. A deck of 52 CARDS must be shuffled seven times for it to be randomized (Aldous and Diaconis 1986, Bayer and Diaconis 1992). This is intermediate between too few shuffles and the decreasing effectiveness of many shuffles. One of Bayer and Diaconis’s randomness CRITERIA , however, gives 3 lg k=2 shuffles for a k-card deck, yielding 11 /C1/12 shuffles for 52 CARDS . Amaz- ingly, if a deck of ncards is shuffled by successively exchanging the cards in position 1, 2, ..., nwith cards in randomly chosen positions (a so-called EXCHANGE SHUFFLE ), then for n]18;the identity permutation (i.e., the original state before the cards were shuffled)is the most likely (Goldstine and Moews 2000). Keller (1995) shows that roughly ln kshuffles are needed just to randomize the bottom card. See also BAYS’ SHUFFLE ,CARDS ,EXCHANGE SHUFFLE , FARO SHUFFLE ,M ONGE’S SHUFFLE ,PERFECT SHUF- FLE,RIFFLE SHUFFLE References Aldous, D. and Diaconis, P. "Shuffling Cards and Stopping Times." Amer. Math. Monthly 93, 333 /C1/348, 1986. Bayer, D. and Diaconis, P. "Trailing the Dovetail Shuffle to Its Lair." Ann. Appl. Probability 2, 294 /C1/313, 1992. Goldstein, D. ad Moews, D. The Identity Is the Most Likely Exchange Shuffle for Large n. 6 Oct 2000. http://xxx.lanl.- gov/abs/math.CO/0010066/. Keller, J. B. "How Many Shuffles to Mix a Deck?" SIAM Review 37,88/C1/89, 1995. Morris, S. B. "Practitioner’s Commentary: Card Shuffling." UMAP J. 15, 333 /C1/338, 1994. Morris, S. B. Magic Tricks, Card Shuffling, and Dynamic Computer Memories. Washington, DC: Math. Assoc. Amer., 1998. Rosenthal, J. W. "Card Shuffling." Math. Mag. 54,64/C1/67, 1981. Siamese Dodecahedron SNUB DISPHENOID Siamese Method A method for constructing MAGIC SQUARES of ODD order, also called DE LA LOUBERE’S METHOD . See also MAGIC SQUARE Sibling Two nodes connected to the same node which are same distance from the ROOT NODE in a ROOTED TREE are called siblings. See also CHILD ,ROOT NODE,ROOTED TREE,TREE Sicherman Dice A pair of DICE which have the same ODDS for throwing every number as a normal pair of 6-sided DICE. They are the only such alternate arrangement if face values are required to be positive. However, if faces are permitted to have zero value (i.e., to be blank), then two additional possible equal-odds pairs of dice are obtained by subtracting one from each face on either of the two dice and adding one to each face the other. If negative values are permitted, there are an infinite number of equal-odds dice. See also DICE,EFRON’S DICESici Spiral The spiral x /C30c ci(t) y /C30c si(t) /C281 2 p) :h where ci(t) and si(t) are the COSINE INTEGRAL and SINE INTEGRAL , respectively, and c is a constant. See also COSINE INTEGRAL ,SINE INTEGRAL ,SPIRAL References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 204 and 270, 1993. Side The edge of a POLYGON or face of a POLYHEDRON are sometimes called sides. Sidon Sequence B2-SEQUENCE Siegel Disk Fractal AJ ULIA SET with c /C30/C280 :390541 /C280 :586788 i : The FRACTAL somewhat resembles the better known MAN- DELBROT SET. See also DENDRITE FRACTAL ,D OUADY’S RABBIT FRACTAL ,JULIA SET,M ANDELBROT SET,SAN MARCO FRACTAL References Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 176, 1991. Siegel Modular Function SIEGEL THETA FUNCTION Siegel Theta Function A Gn/-invariant meromorphic function on the space of all p /C29p symmetric COMPLEX MATRICES Z /C30X /C27iY with positive definite IMAGINARY PART . It is defined by U(Z; s) /C30X te pitTZt/C272 p itTs ; where s is a complex p-vector, t is an integer p-vector that ranges over the entire p-D lattice of integers, and AT denotes a matrix (or vector) transpose. This function was investigated by many of the luminaries of nineteenth century mathematics, Rie- mann , Weierstrass , Frobenius , Poincare ´. Umemura has expressed the ROOTS of an arbitrary POLYNOMIAL in terms of Siegel theta functions (Mumford 1984). The Siegel theta functions is implemented in Math- ematica asSiegelTheta in the Mathematica add-on package NumberTheory‘SiegelTheta‘ (which can be loaded with the command BBNumberTheory‘ ). See also RIEMANN THETA FUNCTION References Iyanaga, S. and Kawada, Y. (Eds.). "Siegel Modular Func- tions." §34F in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 131 /C1/132, 1980. Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkha ¨user, 1984. Siegel, C. L. Topics in Complex Function Theory, Vol. 2: Automorphic Functions and Abelian Integrals. New York: Wiley, p. 163, 1988. Siegel’s Paradox If a fixed FRACTION x of a given amount of money P is lost, and then the same FRACTION x of the remaining amount is gained, the result is less than the original and equal to the final amount if a FRACTION x is first gained, then lost. This can easily be seen from the fact that [P(1 /C28x)](1 /C27x) /C30P(1 /C28x2) BP [P(1 /C27x)](1 /C28x) /C30P(1 /C28x2) BP: Siegel’s Theorem There are at least two Siegel’s theorems. The first states that an ELLIPTIC CURVE can have only a finite number of points with INTEGER coordinates. The second states that if j is an ALGEBRAIC NUMBER of degree r, then there is an A( j) depending only on j such thatj/C28p qYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C21 A(j) q2r1=2 for all integer p and q (Landau 1970, pp. 37 /C1/56; Hardy 1999, p. 79). See also ELLIPTIC CURVE ,ROTH’S THEOREM ,THUE- SIEGEL- ROTH THEOREM References Davenport, H. "Siegel’s Theorem." Ch. 21 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 126 /C1/125, 1980. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Landau, E. Vorlesungen u¨ber Zahlentheorie, Vol. 3. New York: Chelsea, 1970. Siegel’s Upper Half-Space See also HALF-SPACE Sierpinski Arrowhead Curve A FRACTAL which can be written as a LINDENMAYER SYSTEM with initial string "YF" , STRING REWRITING rules "X" - /C21 "YF/C27XF/C27Y", "Y" - /C21 "XF-YF-X" , and angle 60 8. See also DRAGON CURVE ,H ILBERT CURVE ,K OCH SNOWFLAKE ,LINDENMAYER SYSTEM ,PEANO CURVE , PEANO- GOSPER CURVE ,SIERPINSKI CURVE ,SIERPINS- KI SIEVE References Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Sierpinski Carpet AFRACTAL which is constructed analogously to the SIERPINSKI SIEVE , but using squares instead of trian- gles. Let Nnbe the number of black boxes, Lnthe length of a side of a white box, and Anthe fractional AREA of black boxes after the nth iteration. Then Nn /C308n (1) Ln /C30(1 3)n /C303/C28n (2) An /C30L2 nNn /C30(8 9)n : (3) The CAPACITY DIMENSION is therefore dcap /C30/C28 lim n0/C12ln Nn ln Ln/C30/C28 lim n0/C12ln(8n) ln(3/C28n) /C30ln 8 ln 3 /C303ln2 ln 3 /C301:892789260... : (4) See also MENGER SPONGE ,SIERPINSKI SIEVE References Dickau, R. M. "The Sierpinski Carpet." http://forum.swarth- more.edu/advanced/robertd/carpet.html. Peitgen, H.-O.; Ju¨rgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer- Verlag, pp. 112 /C1/121, 1992. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Sierpin ´ski Constant Let the SUM OF SQUARES FUNCTION /rk(n)/ denote the number of representations of n by k squares, then the SUMMATORY FUNCTION of /r2(k)=k/ has the ASYMPTOTIC expansion Xn k /C301r2(k) k/C30K /C27 p ln n /C27O(n /C281 =2) ; where /K /C302:5849817596 / is the Sierpinski constant. The above plot shows Xn k/C301r2 ðkÞ k"# /C28 p ln n; with the value of K indicated as the solid horizontal line. See also SUM OF SQUARES FUNCTIONReferences Sierpinski, W. Oeuvres Choisies, Tome 1. Editions Scienti- fiques de Pologne, 1974. Sierpinski Curve There are several FRACTAL curves associated with Sierpinski. The above curve is one example, and the SIERPINSKI ARROWHEAD CURVE is another. The limit of the curve illustrated above has AREA A /C305 12: The AREA for a related curve due to Sierpinski (1912) illustrated above is A /C301 3(7 /C284ffiffiffi 2p ): (Steinhaus 1983, pp. 102 /C1/103; Cundy and Rollett 1989; Wells 1991, p. 229). See also EXTERIOR SNOWFLAKE ,G OSPER ISLAND , HILBERT CURVE ,KOCH ANTISNOWFLAKE ,KOCH SNOW- FLAKE ,PEANO CURVE ,PEANO- GOSPER CURVE ,SIER- PINSKI ARROWHEAD CURVE References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 67 /C1/68, 1989. Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Free- man, p. 34, 1989. Sierpinski, W. Bull. l’Acad. des Sciences Cracovie A , 462/C1/ 478, 1912. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 207, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 229, 1991. Sierpinski Gasket SIERPINSKI SIEVE Sierpinski Number of the First Kind Numbers OF THE FORM Sn/C13nn/C271:The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (Sloane’s A014566). Sierpinski proved that if Snis PRIME with n ]2; then Sn /C30Fm/C272m ; where Fmis a FERMAT NUMBER with m ]0: The first few such numbers are F1 /C305 ; F3 /C30257; F6 ; F11 ; F20 ; and F37 : Of these, 5 and 257 are PRIME , and the first unknown case is F37 > 103 /C291010 :/ See also CULLEN NUMBER ,CUNNINGHAM NUMBER , FERMAT NUMBER ,W OODALL NUMBER References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 155, 1979. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 74, 1989. Sloane, N. J. A. Sequences A014566 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sierpin ´ski Number of the Second Kind A number k satisfying SIERPINSKI’S COMPOSITE NUM- BER THEOREM , i.e., such that k /C215 2n /C271is COMPOSITE for every n ]1 : The smallest known is k /C3078 ;557; but there remain 35 smaller candidates (the smallest of which is 4847) which are known to generate only composite numbers for n 518 ;000 or more (Riben- boim 1996, p. 358). Let a(k) be smallest n for which (2k /C281) /C215 2n /C271is PRIME , then the first few values are 0, 1, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, ... (Sloane’s A046067). The second smallest n are given by 1, 2, 3, 4, 2, 3, 8, 2, 15, 10, 4, 9, 4, 4, 3, 60, 6, 3, 4, 2, 11, 6, 9, 1483, ... (Sloane’s A046068). Quite large n can be required to obtain the first prime even for small k. For example, the smallest prime OF THE FORM 383 /C215 2n /C271 is 383 /C215 26393 /C271: There are an infinite number of Sierpinski numbers which are PRIME . The smallest odd k such that k /C272n is COMPOSITE for all n Bk are 773, 2131, 2491, 4471, 5101, .... See also MERSENNE NUMBER ,RIESEL NUMBER ,SIER- PINSKI’S COMPOSITE NUMBER THEOREM References Buell, D. A. and Young, J. "Some Large Primes and the Sierpinski Problem." SRC Tech. Rep. 88004, Supercom- puting Research Center, Lanham, MD, 1988. Jaeschke, G. "On the Smallest k such that k /C215 2N /C271 are Composite." Math. Comput. 40, 381 /C1/384, 1983. Jaeschke, G. Corrigendum to "On the Smallest k such that k /C215 2N /C271 are Composite." Math. Comput. 45, 637, 1985. Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k /C215 2n /C271 :/" Math. Comput. 41, 661 /C1/673, 1983. Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k /C215 2n /C271 ; II." In prep. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357 /C1/359, 1996. Sierpinski, W. "Sur un proble `me concernant les nombres k /C215 2n /C271:/" Elem. d. Math. 15,73/C1/74, 1960. Sloane, N. J. A. Sequences A046067 and A046068 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html.Sierpinski Sieve A FRACTAL described by Sierpinski in 1915. It is also called the SIERPINSKI GASKET or SIERPINSKI TRIAN- GLE. The curve can be written as a LINDENMAYER SYSTEM with initial string "FXF-FF-FF" , STRING REWRITING rules "F" - /C21 "FF", "X" - /C21 "- FXF /C27/C27FXF/C27/C27FXF-" , and angle 608. Let Nnbe the number of black triangles after iteration n, Lnthe length of a side of a triangle, and Anthe fractional AREA which is black after the nth iteration. Then Nn/C303n(1) Ln/C301 2Yru*Yru+n /C302/C28n(2) An/C30L2 nNn/C303 4Yru*Yru+n : (3) The CAPACITY DIMENSION is therefore dcap/C30/C28lim n0/C12lnNn lnLn/C30/C28lim n0/C12ln 3nðÞ ln 2/C28nðÞ/C30ln 3 ln 2 /C301:584962500 . . . : (4) In P ASCAL’S TRIANGLE , coloring all ODD numbers black and EVEN numbers white produces a Sierpinski sieve (Guy 1990). See also LINDENMAYER SYSTEM ,SIERPINSKI ARROW- HEAD CURVE ,SIERPINSKI CARPET ,TETRIX References Bulaevsky, J. "The Sierpinski Triangle Fractal." http:// www.best.com/~ejad/java/fractals/sierpinski.shtml. Crownover, R. M. Introduction to Fractals and Chaos. Sud- bury, MA: Jones & Bartlett, 1995. Dickau, R. M. "Two-Dimensional L-Systems." http://forum.s- warthmore.edu/advanced/robertd/lsys2d.html. Dickau, R. M. "Typeset Fractals." Mathematica J. 7, 15, 1997. Dickau, R. "Sierpinski-Menger Sponge Code and Graphic." http://www.mathsource.com/cgi-bin/msitem22?0206 /C1/110. Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63,3/C1/20, 1990. Harris, J. W. and Stocker, H. "Sierpinski Gasket." §4.11.7 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 115, 1998. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 13 /C1/ 14, 1991. Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983. Peitgen, H.-O.; Ju¨rgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer- Verlag, pp. 78 /C1/88, 1992. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, p. 282, 1988. Sved, M. "Divisibility--with Visibility." Math. Intell. 10,56/C1/ 64, 1988. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 108 and 151 /C1/153, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Sierpinski Sponge TETRIX Sierpinski Square Snowflake SIERPINSKI CURVE Sierpinski Tetrahedron TETRIX Sierpinski Triangle SIERPINSKI SIEVE Sierpinski-Menger Sponge MENGER SPONGE Sierpinski’s Composite Number Theorem There exist infinitely many ODD INTEGERS k such that k /C215 2n /C271is COMPOSITE for every n ]1: Numbers k with this property are called SIERPINSKI NUMBERS OF THE SECOND KIND , and analogous numbers with the plus sign replaced by a minus are called RIESEL NUMBERS . it is conjectured that the smallest SIER- PINSKI NUMBER OF THE SECOND KIND is k /C3078;557 and the smallest RIESEL NUMBER is k /C30509;203:/ See also CUNNINGHAM NUMBER ,SIERPINSKI NUMBER OF THE SECOND KIND References Buell, D. A. and Young, J. "Some Large Primes and the Sierpinski Problem." SRC Tech. Rep. 88004, Supercom- puting Research Center, Lanham, MD, 1988. Jaeschke, G. "On the Smallest k such that k /C215 2N /C271 are Composite." Math. Comput. 40, 381 /C1/384, 1983.Jaeschke, G. Corrigendum to "On the Smallest k such that k /C215 2N /C271 are Composite." Math. Comput. 45, 637, 1985. Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k /C215 2n /C271:/" Math. Comput. 41, 661 /C1/673, 1983. Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k /C215 2n /C271; II." In prep. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357 /C1/359, 1996. Riesel, H. "Na˚gra stora primtal." Elementa 39, 258 /C1/260, 1956. Sierpinski, W. "Sur un proble `me concernant les nombres k /C215 2n /C271:/" Elem. d. Math. 15,73/C1/74, 1960. See also COMPOSITE NUMBER ,SIERPINSKI NUMBERS OF THE SECOND KIND,SIERPINSKI’S PRIME SEQUENCE THEOREM Sierpinski’s Conjecture The conjecture that all integers > 1 occur as a value of the TOTIENT VALENCE FUNCTION (i.e., all integers > 1 occur as multiplicities). The conjecture was proved by Ford (1998ab). See also CARMICHAEL’S TOTIENT FUNCTION CONJEC- TURE References Ford, K. "The Distribution of Totients." Ramanujan J. 2, 67 /C1/151, 1998a. Ford, K. "The Distribution of Totients, Electron. Res. Announc. Amer. Math. Soc. 4,27/C1/34, 1998b. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 94, 1994. Schlafly, A. and Wagon, S. "Carmichael’s Conjecture on the Euler Function is Valid Below 1010 ;000;000 :/" Math. Comput. 63, 415 /C1/419, 1994. Sierpinski’s Prime Sequence Theorem For any M, there exists a /t?/ such that the sequence n2 /C27t?; where n /C301, 2, ...contains at least M PRIMES . See also DIRICHLET’S THEOREM ,FERMAT 4N /C271 THEO- REM,SIERPINSKI’S COMPOSITE NUMBER THEOREM References Abel, U. and Siebert, H. "Sequences with Large Numbers of Prime Values." Amer. Math. Monthly 100, 167/C1/169, 1993. Ageev, A. A. "Sierpinski’s Theorem is Deducible from Euler and Dirichlet." Amer. Math. Monthly 101, 659/C1/660, 1994. Forman, R. "Sequences with Many Primes." Amer. Math. Monthly 99, 548/C1/557, 1992. Garrison, B. "Polynomials with Large Numbers of Prime Values." Amer. Math. Monthly 97, 316/C1/317, 1990. Sierpinski, W. "Les bino ˆmes x2/C27net les nombres premiers." Bull. Soc. Roy. Sci. Liege 33, 259/C1/260, 1964. Sierpinski’s Theorem SIERPINSKI’S COMPOSITE NUMBER THEOREM ,S IER- PINSKI’S PRIME SEQUENCE THEOREM Sieve A process of successively crossing out members of a list according to a set of rules such that only some remain. The best known sieve is the ERATOSTHENES SIEVE for generating PRIME NUMBERS . In fact, num- bers generated by sieves seem to share a surprisingly large number of properties with the PRIME NUMBERS . See also BRUN’S SIEVE,H APPY NUMBER ,N UMBER FIELD SIEVE,P RIME NUMBER ,Q UADRATIC SIEVE, SIERPINSKI SIEVE,SIEVE OF ERATOSTHENES ,W ALLIS SIEVE References Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974. Hawkins, D. "Mathematical Sieves." Sci. Amer. , Dec. 1958. Huskey, H. D. "Derrick Henry Lehmer (1905 /C1/1991)." IEEE Ann. Hist. Comput. 17,64/C1/68, 1995. Lehmer, D. H. "The Sieve Problem for All-Purpose Compu- ters." Math. Tables and Other Aids to Comput. 7,6/C1/14, 1953. Lukes, R. F.; Patterson, C. D.; and Williams, H. C. "Numer- ical Sieving Devices: Their History and Some Applica- tions." Nieuw Arch. Wisk. 13, 113 /C1/139, 1995. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/1485, 1996. Williams, H. C. and Shallit, J. O. "Factoring Integers Before Computers." In Mathematics of Computation 1943 /C1/1993: A Half-Century of Computational Mathematics (Vancou- ver, BC, 1993) (Ed. W. Gautschi). Providence, RI: Amer. Math. Soc., pp. 481 /C1/531, 1994. Sieve Formula INCLUSION- EXCLUSION PRINCIPLE Sieve of Eratosthenes An ALGORITHM for making tables of PRIMES . Sequen- tially write down the INTEGERS from 2 to the highest number n you wish to include in the table. Cross out all numbers > 2 which are divisible by 2 (every second number). Find the smallest remaining number > 2: It is 3. So cross out all numbers > 3 which are divisible by 3 (every third number). Find the smallest remaining number > 3: It is 5. So cross out all numbers > 5 which are divisible by 5 (every fifth number). Continue until you have crossed out all numbers divisible byffiffiffinpbc ; where xbcis the FLOOR FUNCTION .The numbers remaining are PRIME . This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out PRIMES up toffiffiffiffiffiffi 50pYrDYrE /C307: If the procedure is then continued up to n, then the number of cross-outs gives the number of distinct PRIME FACTORS of each number. See also PRIME NUMBER ,SIEVE References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 127 /C1/130, 1996. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 79 /C1/80, 1984. Nagell, T. "General Remarks. The Sieve of Eratosthenes." §15 in Introduction to Number Theory. New York: Wiley, pp. 51 /C1/54, 1951. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 100 /C1/101, 1989. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 20 /C1/21, 1996. Se´roul, R. "The Sieve of Eratosthenes." §8.6 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 169 /C1/ 175, 2000. Sievert Integral The integral gu 0e/C28xsecfdf: References Abramowitz, M. and Stegun, C. A. (Eds.). "Sievert Integral." §27.4 in Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1000 /C1/1001, 1972. Sievert’s Surface A constant-curvature surface which can be given parametrically by x/C30rcosf (1) y/C30rsinf (2) z/C30ln tan1 2vYru*Yru+hi /C27a(C/C271)cos v ffiffiffiffi Cp ; (3) where f /C13/C28uffiffiffiffiffiffiffiffiffiffiffiffiffiffi C /C27 1p /C27tan/C281tan uffiffiffiffiffiffiffiffiffiffiffiffi C /C271pYru*Yru+ (4) a /C132 C /C27 1 /C28 C sin2 v cos2 u (5) r /C13affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (C /C27 1) 1 /C27 C sin2 uYrvYruq sin vffiffiffiffi Cp ; (6) with ½u ½B p=2 and 0 Bv B p (Reckziegel 1986). The coefficients of the FIRST FUNDAMENTAL FORM are E /C3064a cos2 u cos2 v 4 /C27 3a /C28 a cos(2 u) /C27 2a cos2 u cos2(2v) ½/C1382(7) F /C300 (8) G /C3064 (1 /C27 a) csc v /C27 a cos2 u sin v ½/C1382 4a 4 /C27 3a /C28 a cos(2 u) /C27 2a cos2 u cos2(2v) ½/C1382 ; (9) and the coefficients of the SECOND FUNDAMENTAL FORM are e /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi a a /C27 1s /C28a cos3 u sin(3 v) /C28 4 cos u[8 /C27 11a /C27 3a cos(2 u)] 4 /C27 3a /C28 a cos(2 u) /C27 2a cos2 u cos2(2v) ½/C1382 (10) f /C300 (11) g /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi a /C27 1 as /C24 /C27 5a /C27 a cos(2 u) /C28 2a cos2 u cos(2 v) ½/C138 csc1 2 vYru*Yru+ sec12 vYru*Yru+ 4 /C27 3a /C28 a cos(2 u) /C27 2a cos2 u cos2(2v) ½/C1382 : (12) The Sievert surface has GAUSSIAN and MEAN CURVA- TURES given by K /C301 (13) H /C301 1 /C27 (a /C27 1)tan2 u : (14) References Fischer, G. (Ed.). Plate 87 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 83, 1986. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 499 /C1/500, 1997. Reckziegel, H. "Sievert’s Surface." §3.4.4.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 38 /C1/39, 1986.Sievert, H. U¨ ber die Zentralfla ¨chen der Enneperschen Flachen konstanten Kru¨mmungsmaßes. Dissertation, Tu¨- bingen, 1886. Sifting Property The property g f(y) d(x /C28y)dy /C30f(x) obeyed by the DELTA FUNCTION d(x) :/ See also DELTA FUNCTION References Bracewell, R. "The Sifting Property." In The Fourier Trans- form and Its Applications, 3rd ed. New York: McGraw- Hill, pp. 74 /C1/77, 1999. Sigma Algebra Let X be a SET. Then a s/-algebra F is a nonempty collection of SUBSETS of X such that the following hold: 1. The EMPTY SET is in F. 2. If A is in F, then so is the complement of A. 3. If Anis a SEQUENCE of elements of F, then the UNION of the An/sisin F. If S is any collection of subsets of X, then we can always find a s/-algebra containing S, namely the POWER SET of X. By taking the INTERSECTION of all s/- algebras containing S, we obtain the smallest such s/- algebra. We call the smallest s/-algebra containing S thes/-algebra generated by S. See also BOREL SIGMA ALGEBRA ,B OREL SPACE , MEASURABLE SET,M EASURABLE SPACE ,M EASURE ALGEBRA ,STANDARD SPACE Sigma Function DIVISOR FUNCTION Sigmoid Curve SIGMOID FUNCTION Sigmoid Function The function y /C301 1 /C27 e /C28x which is the solution to the ORDINARY DIFFERENTIAL EQUATION dy dx /C30y(1 /C28y) : It has an inflection point at x /C300, where y??(x) /C30/C28ex(ex /C28 1) (ex /C28 1)3 /C300: See also EXPONENTIAL FUNCTION ,E XPONENTIAL RAMP References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 124, 1993.Sign The sign of a number, also called SGN,is/C281 for a NEGATIVE number (i.e., one with a MINUS SIGN "//C28/"), 0 for the number ZERO ,or/C271 for a POSITIVE number (i.e., one with a PLUS SIGN "//C27/"). See also ABSOLUTE VALUE ,M INUS SIGN,NEGATIVE , PLUS SIGN,POSITIVE ,SGN,ZERO Signalizer Functor Theorem U(G;A)/C30u(a):a/C23A/C281 hi is an A-invariant solvable p?/-subgroup of G. Signature PERMUTATION SYMBOL ,SIGNATURE (KNOT), S IGNA- TURE (MATRIX ), S IGNATURE (NUMBER FIELD), S IGNA- TURE (QUADRATIC FORM), S IGNATURE (RECURRENCE RELATION ), SIGNATURE SEQUENCE Signature (Knot) The signature s(K)o fa KNOT Kcan be defined using the SKEIN RELATIONSHIP s(unknot) /C300 s(K/C27)/C28s(K/C28)/C23f0;2g; and 4½s(K)l9(K)(2i)>0; where 9(K) is the A LEXANDER- CONWAY POLYNOMIAL and9(K)(2i)i sa n ODD NUMBER . Many UNKNOTTING NUMBERS can be determined using a knot’s signature. See also SKEIN RELATIONSHIP ,UNKNOTTING NUMBER References Gordon, C. M.; Litherland, R. A.; and Murasugi, K. "Signa- tures of Covering Links." Canad. J. Math. 33, 381 /C1/394, 1981. Murasugi, K. "On the Signature of Links." Topology 9, 283 /C1/ 298, 1970. Murasugi, K. "Signatures and Alexander Polynomials of Two-Bridge Knots." C. R. Math. Rep. Acad. Sci. Canada 5, 133 /C1/136, 1983. Murasugi, K. "On the Signature of a Graph." C. R. Math. Rep. Acad. Sci. Canada 10, 107 /C1/111, 1988. Murasugi, K. "On Invariants of Graphs with Applications to Knot Theory." Trans. Amer. Math. Soc. 314,1/C1/49, 1989. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976. Stoimenow, A. "Signatures." http://guests.mpim- bonn.mpg.de/alex/ptab/sig10.html. Signature (Matrix) A real, nondegenerate n /C29n SYMMETRIC MATRIX A; and its corresponding SYMMETRIC BILINEAR FORM Q(v ; w) /C30vTAw; has signature (p, q) if there is a nondegenerate matrix C such that CTAC is a diagonal matrix with p 1s and q /C281s. In this case, Q(Cv; Cw)is a DIAGONAL QUADRATIC FORM . For example, A /C30100 0 010 0 001 0 000 /C2812 6643 775 gives a SYMMETRIC BILINEAR FORM Q called the LORENTZIAN INNER PRODUCT , which has signature (3; 1): The following Mathematica function returns the signature of a SYMMETRIC MATRIX as a list of three elements, corresponding to 1s, 0s, and /C281s. SignatureMatrix[a_List?MatrixQ] : /C30 Module[ { q, ctr, diag, t2, signplus, signminus, v1 /C30 Prepend[Table[0, {Length[a] - 1}], 1] }, q[v_] : /C30 v.a.v; If[(t2 /C30 q[v1]) ! /C30 0, v1 / /C30 Sqrt[Abs[t2]]]; ctr /C30 {v1}; Do[ v1 /C30 NullSpace[ctr.a][[1]]; If[(t2 /C30 q[v1]) ! /C30 0, v1 / /C30 Sqrt[Abs[t2]]]; AppendTo[ctr, v1], {Length[a] - 1} ];diag /C30 ctr.a.Transpose[ctr]; signplus /C30 Count[diag, 1, 2]; signminus /C30 Count[diag, -1, 2];{signplus, Length[a] - signplus - signminus, signminus} ] See also DIAGONAL QUADRATIC FORM,ORTHOGONAL GROUP ,Q UADRATIC FORM,S YMMETRIC BILINEAR FORM,VECTOR SPACE Signature (Number Field) This entry contributed by KEVIN O’BRYANT The ordered pair (s, t), where s is the number of real embeddings of the NUMBER FIELD and t is the number of complex-conjugate pairs of embeddings. The degree of the number field is s /C272t:/ See also FUNDAMENTAL UNIT,NUMBER FIELD,UNIT References Cohen, H. A Course in Computational Algebraic Number Theory, 3rd. corr. ed. New York: Springer-Verlag, 1996. Signature (Permutation) PERMUTATION SYMBOL Signature (Quadratic Form) The signature of the QUADRATIC FORM Q /C30y2 1 /C27y22 /C27.../C27y2p /C28y2p /C271 /C28y2p /C272 /C28.../C28y2r is the number p of POSITIVE squared terms in the reduced form. (The signature is sometimes defined as 2p/C28r:/) See also P-SIGNATURE ,R ANK (QUADRATIC FORM), SYLVESTER’S INERTIA LAW,SYLVESTER’S SIGNATURE References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1105, 2000. Signature (Recurrence Relation) Let a sequence be defined by A/C281/C30s A0/C303 A1/C30r An/C30rAn/C281/C28sAn/C282/C27An/C283: Also define the associated POLYNOMIAL f(x)/C30x3/C28rx2/C27sx/C271; and let Dbe its discriminant. The P ERRIN SEQUENCE is a special case corresponding to An(0;/C281):The signature mod mof an INTEGER nwith respect to the sequence Ak(r;s) is then defined as the 6-tuple / (A/C28n/C281;A/C28n;A/C28n/C271;An/C281;An;An/C271) (mod m). 1. An INTEGER n has an S-signature if its signature (mod n)is( /A/C282 ; A/C281 ; A0 ; A1 ; A2) :/ 2. An INTEGER n has a Q-signature if its signature (mod n)is CONGRUENT to (/A; s ; B; B ; r ; C) where, for some INTEGER a with f(a) /C130(mod n); A /C13 a/C282 /C272a ; B /C13/C28ra2 /C27 r2 /C28s ðÞ a ; and C /C13a2 /C272a /C281 :/ 3. An INTEGER n has an I-signature if its signature (mod n)is CONGRUENT to ( r; s ; D?; D ; r; s) ; where D?/C27D /C13rs /C283 and D?/C27D ðÞ /C13D:/ See also PERRIN PSEUDOPRIME References Adams, W. and Shanks, D. "Strong Primality Tests that Are Not Sufficient." Math. Comput. 39, 255 /C1/300, 1982. Grantham, J. "Frobenius Pseudoprimes." http://www.clar- k.net/pub/grantham/pseudo/pseudo1.ps. Signature Sequence Let u be an IRRATIONAL NUMBER , define S( u) /C30fc /C27 du : c ; d /C23Ng; and let cn( u) /C27dn u(u) be the sequence obtained by arranging the elements of S(u) in in- creasing order. A sequence x is said to be a signature sequence if there EXISTS a POSITIVE IRRATIONAL NUMBER u such that x /C30 cn uðÞfg ; and x is called the signature of u :/ The signature of an IRRATIONAL NUMBER is a FRACTAL SEQUENCE . Also, if x is a signature sequence, then the LOWER-TRIMMED SUBSEQUENCE is V(x) /C30x:/ References Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157 /C1/168, 1997. Signed Deviation The signed deviation is defined by Dui /C13 ui /C28 ¯u ðÞ ; so the average deviation is Du /C30ui /C28u /C30ui /C28 ¯u /C300: See also ABSOLUTE DEVIATION ,DEVIATION ,DISPER- SION (STATISTICS ), MEAN DEVIATION ,QUARTILE DE- VIATION ,STANDARD DEVIATION ,VARIANCE Significance Let d /C13z 5zobserved : A value 0 5 a 51 such that P( d) 5 a is considered "significant" (i.e., is not simply due to chance) is known as an ALPHA VALUE . The PROBABIL- ITY that a variate would assume a value greater than or equal to the observed value strictly by chance, P(d); is known as a P-VALUE . Depending on the type of data and conventional practices of a given field of study, a variety of different alpha values may be used. One commonly used terminology takes P( d) ]5% as "not significant,"1% BP( d) B5%; as "significant" (sometimes denoted *), and P( d) B1% as "highly significant" (sometimes denoted **). Some authors use the term "almost significant" to refer to 5% BP(d) B10% ; although this practice is not recommended. See also ALPHA VALUE ,COINCIDENCE ,CONFIDENCE INTERVAL , P-VALUE ,PROBABLE ERROR ,SIGNIFICANCE TEST,STATISTICAL TEST Significance Test A test for determining the probability that a given result could not have occurred by chance (its SIG- NIFICANCE ). See also SIGNIFICANCE ,STATISTICAL TEST References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 491 /C1/492, 1987. Significant Digits When a number is expressed in SCIENTIFIC NOTATION , the number of significant figures is the number of DIGITS needed to express the number to within the uncertainty of measurement. For example, if a quantity had been measured to be 1.234 9 0.002, four figures would be significant. No more figures should be given than are allowed by the uncertainty. For example, a quantity written as 1.234 9 0.1 is incorrect; it should really be written as 1.2 9 0.1. The number of significant figures of a MULTIPLICA- TION or DIVISION of two or more quantities is equal to the smallest number of significant figures for the quantities involved. For ADDITION or SUBTRACTION , the number of significant figures is determined with the smallest significant figure of all the quantities involved. For example, the sum 10 :234 /C275:2 /C27 100:3234 is 115.7574, but should be written 115.8 (with rounding), since the quantity 5.2 is significant only to90.1. See also FRACTIONAL PART,INTEGER PART,N INT, ROUND ,TRUNCATE References Kenney, J. F. and Keeping, E. S. "Significant Figures." §1.5 inMathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 8 /C1/9, 1962. Mulliss, C. "Significant Figures and Rounding Rules." http:// www.angelfire.com/oh/cmulliss/. Significant Figures SIGNIFICANT DIGITS Signpost A6- POLYIAMOND . References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Signum SGN Silver Constant The REAL ROOT of the equation x3 /C285x2 /C276x /C281 /C300; given analytically by 2 /C272 cos2 7 pYru*Yru+ ; which is 3.2469.... It is the seventh BERAHA CON- STANT . See also BERAHA CONSTANTS ,SILVER RATIO,TRIGO- NOMETRY VALUES PI/7 References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 51 and 143, 1983. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 162, 1986. Silver Mean SILVER RATIO Silver Ratio The quantity defined by the CONTINUED FRACTION dS /C13[2; 2 ; 2 ; ...:] /C302 /C271 2 /C271 2 /C271 2 /C27/C1/C1/C1: It follows that dS /C281 ðÞ2/C302 ; so dS /C30ffiffiffi 2p /C271 /C302 :41421... : See also GOLDEN RATIO,GOLDEN RATIO CONJUGATE Silver Root SILVER CONSTANTSilverman Constant X/C12 n/C3011 f(n) s(n) /C30Y p prime1 /C27X/C12 k /C3011 p2k /C28 pk /C281 ! /C301 :786576459... : where f(n) is the TOTIENT FUNCTION and s(n) is the DIVISOR FUNCTION . References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/totient/totient.html. Zimmerman, P. http://www.mathsoft.com/asolve/constant/ totient/zimmermn.html. Silverman’s Sequence Let f(1) /C301 ; and let f(n) be the number of occurrences of n in a nondecreasing sequence of INTEGERS . then the first few values of f(n) are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, ... (Sloane’s A001462). the asymptotic value of the nth term is f2/C28 fnf /C281 ; where f is the GOLDEN RATIO . References Guy, R. K. "Silverman’s Sequences." §E25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 225 /C1/226, 1994. Sloane, N. J. A. Sequences A001462/M0257 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Similar Two figures are said to be similar when all corre- sponding ANGLES are equal. Two figures are DIRECTLY SIMILAR when all corresponding ANGLES are equal and described in the same rotational sense. This relationship is written A /C2B: (The symbol /C2 is also used to mean "is the same order of magnitude as" and "is ASYMPTOTIC to.") Two figures are INVERSELY SIMILAR when all corresponding ANGLES are equal and described in the opposite rotational sense. See also COINCIDENT ,CONGRUENT ,D IRECTLY SIMI- LAR,HOMOTHETIC ,INVERSELY SIMILAR ,NAPOLEON’S THEOREM ,SIMILAR MATRICES ,SIMILAR TRIANGLES , SIMILARITY TRANSFORMATION ,SPIRAL SIMILARITY References Durell, C. V. "Similar Figures." Ch. 1 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 1 /C1/ 9, 1928. Kern, W. F. and Bland, J. R. "Similar Figures." §22 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 4 and 53 /C1/57, 1948. Lachlan, R. "The Theory of Similar Figures." Ch. 9 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 128 /C1/147, 1893. Project Mathematics . "Similarity." Videotape. http:// www.projmath.caltech.edu/similar.htm. Similar Matrices Two SQUARE MATRICES A and B that are related by B /C30X/C281AX; (1) where X is a square NONSINGULAR MATRIX are said to be similar. A transformation of the form X/C281AX is called a SIMILARITY TRANSFORMATION , or conjugation by X: For example, 01 00YrtvYrtu (2) and 00 10YrtvYrtu (3) are similar under conjugation by C /C3001 10YrtvYrtu : (4) Similar matrices represent the same LINEAR TRANS- FORMATION after a change of basis (for the domain and range simultaneously). Recall that a matrix corresponds to a LINEAR TRANSFORMATION , and a LINEAR TRANSFORMATION corresponds to a matrix after choosing a basis bi ; TX libiYru*Yru+ /C30X aji libj (5) Changing the basis changes the coefficients of the matrix, TX gieiYru*Yru+ /C30X a?ji giej (6) If T(v) /C30Av uses the standard basis vectors, then T is the matrix CAC/C281 using the basis vectors bi /C30Cei :/ See also BASIS (VECTOR SPACE ), DIAGONAL MATRIX , DIAGONALIZABLE MATRIX ,G ROUP ,JORDAN CANONI- CAL FORM,LINEAR TRANSFORMATION ,RATIONAL CA- NONICAL FORM ,S IMILARITY TRANSFORMATION , SQUARE MATRIX ,VECTOR SPACE References Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 311, 1996.Similar Triangles Two triangles are similar if their triples of vertex angles are the same. See also DIRECTLY SIMILAR ,INVERSELY SIMILAR , SIMILAR Similarity Axis D’ALEMBERT’S THEOREM Similarity Dimension To multiply the size of a d-D object by a factor a, c /C13ad copies are required, and the quantity d /C30ln c ln a is called the similarity dimension. Similarity Point External (or positive) and internal (or negative) similarity points of two CIRCLES with centers C and C ? and RADII r and r ? are the points E and I on the lines CC ? such that CE C?E /C30r r?; or CI C ?I /C30/C28r r?: See also D’ALEMBERT’S THEOREM Similarity Transformation An ANGLE -preserving transformation. A similarity transformation has a transformation matrix A?of the form A?/C13BAB/C281; (1) where Aand Bare known as SIMILAR MATRICES (Golub and van Loan 1996, p. 311). IfAis an ANTISYMMETRIC MATRIX (aij/C30/C28aji) and Bis anORTHOGONAL MATRIX , then bab /C281YrvYru ij/C30bikaklb/C281 lj/C30/C28bikalkb/C281 lj /C30/C28b$ kialkb $YrvYru/C281 jl/C30/C28b/C281 kiakibjl /C30bjlalkb/C281 ki /C30/C28 bab/C281YrvYru ji : (2) The DETERMINANT of the similarity transformation of a MATRIX is equal to the determinant of the original MATRIX ½BAB/C281 ½/C30½B½½A½½B /C281 ½/C30½B½½A ½1 ½B ½/C30½A½: (3) The determinant of a similarity transformation minus a multiple of the unit MATRIX is given by ½B/C281AB /C28lI½/C30½B /C281AB /C28B /C281 lIB ½/C30½B/C281(A /C28lI)B ½ /C30½B/C281 ½½A /C28lI ½½B½/C30½A /C28lI ½: (4) Similarity transformations and the concept of SELF- SIMILARITY are important foundations of FRACTALS and ITERATED FUNCTION SYSTEMS . See also CONFORMAL MAPPING ,DETERMINANT ,DILA- TION ,ITERATED FUNCTION SYSTEM ,S IMILAR MA- TRICES References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 311, 1996. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 83 /C1/ 103, 1991. Similitude Center Also called a self-homologous point. If two SIMILAR figures lie in the plane but do not have parallel sides (they are not HOMOTHETIC ), there exists a center of similitude which occupies the same homologous posi- tion with respect to the two figures. The LOCUS of similitude centers of two nonconcentric circles is another circle having the line joining the two homo- thetic centers as its DIAMETER . There are a number of interesting theorems regard- ing three CIRCLES (Johnson 1929, pp. 151 /C1/152). 1. The external similitude centers of three circles are COLLINEAR . 2. Any two internal similitude centers are COLLI- NEAR with the third external one. 3. If the center of each circle is connected with the internal similitude center of the other three [sic], the connectors are CONCURRENT . 4. If one center is connected with the internal similitude center of the other two, the others withthe corresponding external centers, the connectors are CONCURRENT . The six centers of similitude of three circles taken by pairs are the vertices of a COMPLETE QUADRILATERAL (Evelyn et al. 1974, pp. 21 /C1/22). See also SIMILITUDE CENTER ,SIMILITUDE CIRCLE References --. Problem 2819. Amer. Math. Monthly 28, 229 /C1/230, 1921. Casey, J. "Centers of Similitude." §6.2 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Exam- ples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 82 /C1/86, 1888. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 21 /C1/22, 1974. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 19 /C1/27 and 151 /C1/153, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 130, 1893. Similitude Circle The LOCUS of the SIMILITUDE CENTER of two circles. See also INVARIABLE POINT ,SIMILITUDE CENTER References Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 135, 1928. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 307 /C1/310, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 192, 1893. Similitude Ratio Two figures are HOMOTHETIC if they are related by a DILATION (a dilation is also known as a HOMOTHECY ). This means that the connectors of corresponding points are CONCURRENT at a point which divides each connector in the same ratio k, known as the similitude ratio. See also CONCURRENT ,D ILATION ,H OMOTHECY , HOMOTHETIC Simon Newcomb’s Problem Given a set P with ½P ½/C30p elements consisting of c1 numbers 1, c2 numbers 2, ..., and cn numbers n and c1 /C27c2 /C27.../C27cn /C30p; find the number of permutations with k /C281 rises (Comtet 1974, p. 246). See also EULER NUMBER References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Dillon, J. F. and Roselle, D. P. "Simon Newcomb’s Problem." SIAM J. Appl. Math. 17, 1086 /C1/1093, 1969. Kreweras, G. "Sur une class de proble `mes de de´nombrement lie´s au treillis des partitions d’entiers." Cahiers Buro 6,2/C1/ 107, 1965. Kreweras, G. "Sur une extension du proble `me dir ‘de Simon Newcomb’." Comptes rendus 263,43/C1/45, 1966. Kreweras, G. "Traitement simultane ´ du ‘proble `me de Young’ et du ‘proble `me de Simon Newcomb’." Cahiers Buro 10, 23 /C1/31, 1967. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, pp. 216 and 265, 1958. Simple Algebra An ALGEBRA with no nontrivial IDEALS . See also ALGEBRA ,IDEAL ,SEMISIMPLE ALGEBRA Simple Continued Fraction A CONTINUED FRACTION s /C30a0 /C27b1 a1 /C27b2 a2 /C27b3 a3 /C27 ...(1) in which the bi/s are all unity, leaving a continued fraction OF THE FORM s /C30a0 /C271 a1 /C271 a2 /C271 a3 /C27 ...: (2) A simple continued fraction can be written in a compact abbreviated NOTATION as s /C30 a0 ; a1 ; a2 ; a3 ... ½/C138 : (3) Bach and Shallit (1996) show how to compute the JACOBI SYMBOL in terms of the simple continued fraction of a RATIONAL NUMBER a=b :/ See also CONTINUED FRACTION References Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, pp. 343 /C1/344, 1996.Simple Curve A curve is simple if it does not cross itself. See also CLOSED CURVE ,JORDAN CURVE References Krantz, S. G. "Closed Curves." §2.1.2 in Handbook of Com- plex Analysis. Boston, MA: Birkha ¨user, pp. 19 /C1/20, 1999. Simple Double Point ORDINARY DOUBLE POINT Simple Function A simple function is a finite sum ai ai xAi ; where the functions xAi are CHARACTERISTIC FUNCTIONS on a set A. Another description of a simple function is a function that takes on finitely many values in its range. The collection of simple functions is CLOSED under addition and multiplication. In addition, it is easy to integrate a simple function. By approximating a given function f by simple functions, the LEBESGUE INTEGRAL of f can be calculated. See also CHARACTERISTIC FUNCTION (SET), LEBESGUE INTEGRAL ,SET Simple Graph A GRAPH for which at most one EDGE connects any two nodes. Unless stated otherwise, the unqualified term "graph" usually refers to a simple graph. A non- simple graph with no loops but which can contain more than one edge between any two points is called a MULTIGRAPH . See also ADJACENCY MATRIX ,EDGE (GRAPH ), GRAPH , MULTIGRAPH ,REGULAR GRAPH ,STEINITZ’S THEOREM References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 89, 1990. Steinbach, P. Field Guide to Simple Graphs. Albuquerque, NM: Design Lab, 1990. Simple Group A simple group is a GROUP whose NORMAL SUBGROUPS (INVARIANT SUBGROUPS ) are ORDER one or the whole of the original GROUP . Simple groups include ALTERNAT- ING GROUPS , CYCLIC GROUPS ,LIE-TYPE GROUPS (five varieties), and SPORADIC GROUPS (26 varieties, includ- ing the MONSTER GROUP ). The CLASSIFICATION THEO- REM of finite simple groups states that such groups can be classified completely into the five types: 1. CYCLIC GROUPS of PRIME ORDER , 2. ALTERNATING GROUPS of degree at least five 3. LIE-TYPE CHEVALLEY GROUPS , 4. LIE-TYPE (TWISTED CHEVALLEY GROUPS or the TITS GROUP ), and 5. SPORADIC GROUPS . BURNSIDE’S CONJECTURE states that every non-A BE- LIAN SIMPLE GROUP has EVEN ORDER . See also ALTERNATING GROUP ,BURNSIDE’S CONJEC- TURE ,C HEVALLEY GROUPS ,C LASSIFICATION THEO- REM,C YCLIC GROUP ,F EIT-THOMPSON THEOREM , FINITE GROUP ,G ROUP ,INVARIANT SUBGROUP ,LIE- TYPE GROUP ,M ONSTER GROUP ,SCHUR MULTIPLIER , SPORADIC GROUP ,TITS GROUP ,TWISTED CHEVALLEY GROUPS Simple Harmonic Motion Simple harmonic motion refers to the periodic sinu- soidal oscillation of an object or quantity. Simple harmonic motion is executed by any quantity obeying the DIFFERENTIAL EQUATION ¨x /C27 v2 0x /C300: (1) where ¨x denotes the second DERIVATIVE of x with respect to t, and v0is the angular frequency of oscillation. This ORDINARY DIFFERENTIAL EQUATION has an irregular SINGULARITY at /C12: The general solution is x /C30A sin v0tðÞ/C27B cos(v0t) (2) /C30C cos v0t /C27 f ðÞ ; (3) where the two constants A and B (or C and f) are determined from the initial conditions. Many physical systems undergoing small displace- ments, including any objects obeying Hooke’s law, exhibit simple harmonic motion. This equation arises, for example, in the analysis of the flow of current in an electronic CL circuit (which contains a capacitor and an inductor ). If a damping force such as Friction is present, an additional term b˙x must be added to the DIFFERENTIAL EQUATION and motion dies out over time.See also DAMPED SIMPLE HARMONIC MOTION ,SIMPLE HARMONIC MOTION -QUADRATIC PERTURBATION Simple Harmonic Motion * /Quadratic Perturbation Given a simple harmonic oscillator with a quadratic perturbation ex2; ¨x/C27v2 0x/C28aex2/C300: (1) find the first-order solution using a perturbation method. Write x/C13x0/C27ex1/C27...: (2) so ¨x/C30˙x0/C27e˙x1/C27...: (3) Plugging (2) and (3) back into (1) gives ˙x0/C27e˙x1 ðÞ /C27v2 0x0/C27v20ex1YrvYru /C28aex0/C272x0x1e/C27... ðÞ /C300: (4) Keeping only terms of order eand lower and group- ing, we obtain ˙x/C27v20x0YrvYru /C27˙x1/C27v20x1/C28ax20YrvYru e/C300: (5) Since this equation must hold for all POWERS ofe;we can separate it into the two differential equations ˙x0/C27v2 0x0/C300 (6) ˙x1/C27v2 0x1/C30ax20: (7) The solution to (6) is just x0/C30Acos(v0t/C27f): (8) Setting our clock so that f/C300 gives x0/C30Acosv0tðÞ : (9) Plugging this into (7) then gives ˙x1/C27v20x1/C30aA2cos2v0tðÞ (10) The two homogeneous solutions to (10) are x1/C30cosv0tðÞ (11) x2/C30sinv0tðÞ : (12) The particular solution to (10) is therefore given by xp(t)/C30/C28x1(t)gx2(t)g(t) W(t)dt/C27x2(t)gx1(t)g(t) W(t)dt:(13) where g(t)/C30aA2cos2v0tðÞ : (14) and the W RONSKIAN is W/C13x1˙x2/C28˙x1x2/C30v0: (15) Plugging everything into (13), xp /C30 aA2 /C28cos v0tðÞgsin v0tðÞ cos2 v0tðÞ v0dt" /C27sin v0tðÞgcos3 v0tðÞ v0dtYrtu /C30aA2 v0sin v0tðÞg 1 /C28sin2 v0tðÞYrtYrP cos v0tðÞ dtYrt* /C28cos v0tðÞgsin v0tðÞ cos2 v0tðÞ dtYrt+ : (16) Now let u /C13sin v0tðÞ (17) du /C30 v0 cos v0tðÞ dt (18) v /C13cos v0tðÞ (19) dv /C30/C28v0sin v0tðÞ dt : (20) Then xp /C30aA2 v2 0sin v0tðÞg 1 /C28u2YrvYru du /C27cos v0tðÞgv2 dvYrtvYrtu /C30aA2 v20sin v0tðÞ 1 /C281 3 u3Yru*Yru+ /C27cos v0tðÞ13 v3hi /C30aA2 v2 0sin v0tðÞ 1 /C281 3sin3 v0tðÞhi /C2713cos v0tðÞ cos3 v0tðÞno /C30aA2 6v2 03 /C28cos 2v0t ðÞ ½/C138 : (21) Plugging x0(t) and (21) into (2), we obtain the solution x(t) /C30A cos v0tðÞ/C28aA2 6 v20e cos 2 v0t ðÞ /C283 ½/C138 : (22) As can be seen in the top figure above, this solution approximates x(t) only for e/C101 : As the lower figure shows, the differences from the unperturbed oscilla- tor grow stronger over time for even relatively small values of e:/Simple Harmonic Oscillator SIMPLE HARMONIC MOTION Simple Interest INTEREST which is paid only on the PRINCIPAL and not on the additional amount generated by previous INTEREST payments. A formula for computing simple interest is a(t) /C30a(0)(1 /C27rt) : where a(t) is the sum of PRINCIPAL and INTEREST at time t for a constant interest rate r. See also COMPOUND INTEREST ,INTEREST References Kellison, S. G. Theory of Interest, 2nd ed. Burr Ridge, IL: Richard D. Irwin, 1991. Simple Lie Algebra References Huang, J.-S. "Simple Lie Algebras." Part II in Lectures on Representation Theory. Singapore: World Scientific, pp. 27 /C1/70, 1999. Simple Pole A simple pole of a ANALYTIC FUNCTION f is a POLE of order one. That is, (z /C28z0)f(z)isan ANALYTIC FUNC- TION at the pole z /C30z0 : Alternatively, its PRINCIPAL PART is c =(z /C28z0) for some c "0: It is called simple because a function with a pole of order n at a can be written as the product of n functions with simple poles at z0 :/ See also DIVISOR (CURVE ), ESSENTIAL SINGULARITY , POLE Simple Polygon A POLYGON P is said to be simple (or JORDAN ) if the only points of the plane belonging to two EDGES of P are the VERTICES of P. Such a polygon has a WELL DEFINED interior and exterior. Simple polygons are topologically equivalent to a DISK. See also POLYGON ,REGULAR POLYGON ,SIMPLE POLY- HEDRON ,TWO-EARS THEOREM References Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122,3 1/C1/35, 1991. Simple Polyhedron A POLYHEDRON that is topologically equivalent to a sphere (i.e., if it were inflated, it would produce a sphere) and whose faces are SIMPLE POLYGONS . The simple polyhedra on n nodes correspond to the simple PLANAR GRAPHS with 3n /C286 edges, and are also called "simplicial polyhedra." The number of simple poly- hedra on n /C301, 2, ... nodes are 0, 0, 1, 1, 1, 2, 5, 15, 50, 233, 1249, ... (Sloane’s A000109). See also PLANAR GRAPH ,SIMPLE POLYGON References Bokowski, J. and Schuchert, P. "Equifacetted 3-Spheres as Topes of Nonpolytopal Matroid Polytopes." Disc. Comput. Geom. 13, 347 /C1/361, 1995. Bowen, R. and Fisk, S. "Generation of Triangulations of the Sphere." Math. Comput. 21, 250 /C1/252, 1967. Dillencourt, M. B. "Polyhedra of Small Orders and Their Hamiltonian Properties." Tech. Rep. 92 /C1/91, Info. and Comput. Sci. Dept., Univ. Calif. Irvine, 1992. Federico, P. J. "Enumeration of Polyhedra: The Number of 9-Hedra." J. Combin. Th. 7, 155 /C1/161, 1969. Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102 /C1/107, May 1975. Gru¨nbaum, B. Convex Polytopes. New York: Wiley, p. 424, 1967. Lederberg, J. "Hamilton Circuits of Convex Trivalent Poly- hedra (up to 18 Vertices)." Amer. Math. Monthly 74, 522 /C1/ 527, 1967. Sloane, N. J. A. Sequences A0001091469 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Simple Random Walk See also RANDOM WALKSimple Ring A NONZERO RING S whose only (two-sided) IDEALS are S itself and zero. Every commutative simple ring is a FIELD . Every simple ring is a PRIME RING . See also FIELD,IDEAL ,PRIME RING,RING Simple Root A ROOT having MULTIPLICITY n /C30 1 is called a simple root. For example, f(z) /C30(z /C281)(z /C282) has a simple root at z0 /C301; but g /C30(z /C281)2 has a root of MULTI- PLICITY 2at z0 /C301; which is therefore not a simple root. See also MULTIPLE ROOT,MULTIPLICITY ,ROOT References Krantz, S. G. "Zero of Order n."§5.1.3 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 70, 1999. Simple Zero SIMPLE ROOT Simplex The generalization of a tetrahedral region of space to n-D. The boundary of a k-simplex has k/C271 0-faces (VERTICES ),k(k/C271)=2 1-faces ( EDGES ), andk/C271 i/C271Yru*Yru+ i- faces, wheren kYrvYru is a BINOMIAL COEFFICIENT .A n n-D simplex can be denoted using the S CHLA ¨FLI SYMBOL f3;...;3|fflfflfflfflfflffl{zfflfflfflfflfflffl} n/C281g: The simplex named because it represents the sim- plest possible polytope in any given space. The CONTENT (i.e., hypervolume) of a simplex can be computed using the C AYLEY- MENGER DETERMINANT . In 1-D, the simplex is the LINE SEGMENT [/C281; 1]: In 2- D, the simplex f3g is the CONVEX HULL of the EQUILATERAL TRIANGLE . In 3-D, the simplex f3; 3g is the CONVEX HULL of the TETRAHEDRON . The simplex in 4-D (the PENTATOPE ) is a regular TETRAHEDRON ABCD in which a point E along the fourth dimension through the center of ABCD is chosen so that EA /C30 EB /C30EC /C30ED /C30AB : The regular simplex in n-D with n ]5 is denoted an :/ The above figures show the graphs for the n-sim- plexes with n /C302to7. See also CAYLEY- MENGER DETERMINANT ,COMPLEX , CROSS POLYTOPE ,E QUILATERAL TRIANGLE ,L INE SEGMENT ,M EASURE POLYTOPE ,NERVE ,PENTATOPE , POINT ,P OLYTOPE ,S IMPLEX METHOD ,S PHERICAL SIMPLEX ,TETRAHEDRON References Bourke, P. "Regular Polytopes (Platonic Solids) in 4D." http://www.swin.edu.au/astronomy/pbourke/geometry/ platonic4d/. Eppstein, D. "Triangles and Simplices." http://www.ics.u- ci.edu/~eppstein/junkyard/triangulation.html. Munkres, J. R. "Simplices." §1.1 in Elements of Algebraic Topology. Perseus Press, pp. 2 /C1/7, 1993. Simplex Method A method for solving problems in LINEAR PROGRAM- MING . This method, invented by G. B. Dantzig in 1947, runs along EDGES of the visualization SOLID to find the best answer. In 1970, Klee and Minty constructed examples in which the simplex method required an exponential number of steps, but such cases seem never to be encountered in practical applications. A much more efficient (POLYNOMIAL -time) ALGORITHM was found in 1984 by N. Karmarkar. This method goes through the middle of the SOLID and then transforms and warps. It offers many advantages over the simplex method (Nemirovsky and Yudin 1994). See also LINEAR PROGRAMMING References Nemirovsky, A. and Yudin, N. Interior-Point Polynomial Methods in Convex Programming. Philadelphia, PA: SIAM, 1994. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Downhill Simplex Method in Multidimen- sions" and "Linear Programming and the Simplex Method." §10.4 and 10.8 in Numerical Recipes in FOR- TRAN: The Art of Scientific Computing, 2nd ed. Cam- bridge, England: Cambridge University Press, pp. 402 /C1/ 406 and 423 /C1/436, 1992.Tokhomirov, V. M. "The Evolution of Methods of Convex Optimization." Amer. Math. Monthly 103,65/C1/71, 1996. Simplex Point Picking Given a SIMPLEX of unit CONTENT in Euclidean d- space, pick d /C271 points uniformly and independently at random, and denote the expected CONTENT of their CONVEX HULL by V(d; n): The special values V(1; n) /C301 /C282 n /C27 1 /C30n /C28 1 n /C27 1 (1) and V(2; n) /C301 /C282 n /C27 1Xn k /C3011 k /C301 /C282Hn n /C27 1 ; (2) where Hn is a HARMONIC NUMBER , are known (Buchta 1984, 1986). Not much is known about V(3; n); although V(3; 5) /C305 2 V(3; 4) (3) (Buchta 1983, 1986) and 1/C28V(3;n)/C23 4(lnn)2 n(4) (Buchta 1986). See also DISK TRIANGLE PICKING References Buchta, C. "U ¨ber die konvexe Hu ¨lle von Zufallspunkten in Eibereichen." Elem. Math. 38, 153/C1/156, 1983. Buchta, C. "Zufallspolygone in konvexen Vielecken." J. reine angew. Math. 347, 212/C1/220, 1984. Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653/C1/659, 1986. Klee, V. "What is the Expected Volume of a Simplex whose Vertices are Chosen at Random from a Given Convex Body." Amer. Math. Monthly 76, 286/C1/288, 1969. Simplicial Complex A simplicial complex is a SPACE with a TRIANGULA- TION . Formally, a simplicial complex KinRnis a collection of SIMPLICES inRnsuch that 1. Every face of a simplex of Kis in K, and 2. The intersection of any two simplices of Kis a face of each of them (Munkres 1993, p. 7). Objects in the space made up of only the simplices in the triangulation of the space are called SIMPLICIAL SUBCOMPLEXES . When only simplicial complexes and SIMPLICIAL SUBCOMPLEXES are considered, defining HOMOLOGY is particularly easy (and, in fact, combi- natorial because of its finite/counting nature). This kind of homology is called SIMPLICIAL HOMOLOGY . See also ABSTRACT SIMPLICIAL COMPLEX ,HOMOLOGY (TOPOLOGY ), NERVE ,SIMPLEX ,SIMPLICIAL SUBCOM- PLEX ,SIMPLICIAL HOMOLOGY ,SPACE ,TRIANGULATION References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 7, 1994. Munkres, J. R. "Simplicial Complexes and Simplicial Maps." §1.2 in Elements of Algebraic Topology. Perseus Press, pp. 7 /C1/14, 1993. Simplicial Homology The type of HOMOLOGY which results when the spaces being studied are restricted to SIMPLICIAL COMPLEXES and subcomplexes. See also SIMPLICIAL COMPLEX Simplicial Homomorphism Let f : K(0) 0 L(0) be a bijective correspondence such that the vertices v0 ; ..., vnof K span a SIMPLEX of K IFF f(v0) ; ..., f(vn) span a SIMPLEX of L. Then the induced SIMPLICIAL MAP g : Kjj0 Ljjis a HOMEO- MORPHISM , and the map g is called a simplicial homeomorphism (Munkres 1993, p. 13). References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Simplicial Map Let K and L be SIMPLICIAL COMPLEXES , and let f : K(0) 0 L(0)be a map. Suppose that whenever the vertices v0 ; ..., vn of K span a SIMPLEX of K, the points f(v0) ; ..., f(vn) are vertices of a SIMPLEX of L. Then f can be extended to a continuous map g : Kjj0 Ljj such that x /C30Xn i /C300tivi implies g(x) /C30Xn i/C300tifviðÞ: The map g is then called the linear simplicial map induced by the vertex map f (Munkres 1993, p. 12). References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993.Simplicial Polyhedron SIMPLE POLYHEDRON Simplicial Subcomplex If L is a subcollection of a SIMPLICIAL COMPLEX K that contains all faces of its elements, then L is another SIMPLICIAL COMPLEX called a simplicial subcomplex. See also SIMPLICIAL COMPLEX References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Simplicity The number of operations needed to effect a GEO- METRIC CONSTRUCTION as determined in GEOMETRO- GRAPHY . If the number of operations of the five GEOMETROGRAPHIC types are denoted m1 ; m2 ; n1 ; n2 ; and n3 ; respectively, then the simplicity is m1 /C27m2 /C27 n1 /C27n2 /C27n3and the symbol m1S1 /C27m2S2 /C27n1C1 /C27 n2C2 /C27n3C3 : It is apparently an unsolved problem to determine if a given GEOMETRIC CONSTRUCTION is of smallest possible simplicity. See also GEOMETRIC CONSTRUCTION ,GEOMETROGRA- PHY References De Temple, D. W. "Carlyle Circles and the Lemoine Simpli- city of Polygonal Constructions." Amer. Math. Monthly 98, 97 /C1/108, 1991. Eves, H. An Introduction to the History of Mathematics, 6th ed. New York: Holt, Rinehart, and Winston, 1976. Simply Connected A CONNECTED DOMAIN is said to be simply connected (also called 1-connected) if any simple closed curve can be shrunk to a point continuously in the set. If the domain is CONNECTED but not simply, it is said to be MULTIPLY CONNECTED . In particular, a SUBSET E of R2 is said to be simply connected if both E and R2_E; where F_E denotes a SET DIFFERENCE , are CON- NECTED . A SPACE S is simply connected if it is 0-connected and if every MAP from the 1-SPHERE to S extends con- tinuously to a MAP from the 2- DISK. In other words, every loop in the SPACE is contractible. See also CONNECTED SET,CONNECTED SPACE ,MULTI- PLY CONNECTED References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 27, 1999. Simpson’s 3/8 Rule Let the values of a function f(x) be tabulated at points xiequally spaced by h /C30xi/C271 /C28xi ; so f1 /C30f(x1) ; f2 /C30 f(x2) ; ..., f4 /C30f(x4): Then Simpson’s 3/8 rule approx- imating the integral of f(x) is given by the NEWTON- COTES -like formula gx4 x1f(x) dx /C303 8 hf1 /C273f2 /C273f3 /C27f4 ðÞ /C283 80 h5 f(4)( j): See also BODE’S RULE,N EWTON- COTES FORMULAS , SIMPSON’S RULE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 286 /C1/287, 1988. Whittaker, E. T. and Robinson, G. "The Trapezoidal and Parabolic Rules." The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 156 /C1/158, 1967. Simpson’s Formulas The TRIGONOMETRIC ADDITION FORMULAS sin a /C27sin b /C302 sina /C27 b 2 ! cosa /C28 b 2 ! (1) sin a /C28sin b /C302 sina /C28 b 2 ! cosa /C27 b 2 ! (2) cos a /C27cos b /C302 cosa /C27 b 2 ! cosa /C28 b 2 ! (3) cos a /C28cos b /C30/C282 sina /C28 b 2 ! sina /C27 b 2 ! : (4) Simpson’s Paradox It is not necessarily true that averaging the averages of different populations gives the average of the combined population. References Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, p. 135, 1995.Simpson’s Rule Let h /C13(b /C28a) =n; and assume a function f(x)is defined at points f(a /C27kh) /C30yk for k /C300, ..., n. Then gb af(x) dx /C3013 hy0 /C274y1 /C272y2 /C274y3 /C27... ð /C272yn /C282 /C274yn /C281 /C27yn Þ/C28Rn : where the remainder is Rn /C301 90(b/C28a)4f(4)(x/C31) for some x/C31/C23[a;b]:/ See also BODE’S RULE,N EWTON- COTES FORMULAS , SIMPSON’S 3/8 RULE,TRAPEZOIDAL RULE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, p. 286, 1988. Whittaker, E. T. and Robinson, G. "The Trapezoidal and Parabolic Rules." The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 156 /C1/158, 1967. Simson Line The Simson line is the LINE containing the feet P1;P2; andP3of the perpendiculars from an arbitrary point Pon the CIRCUMCIRCLE of a TRIANGLE to the sides or their extensions of the TRIANGLE . This line was attributed to Simson by Poncelet , but is now frequently known as the Wallace-Simson line sinceit does not actually appear in any work of Simson(Johnson 1929, p. 137; Coxeter and Greitzer 1967, p. 41; de Guzma ´n 1999). The inverse statement to that given above, namely that the locus of all points P in the plane of a TRIANGLE DABC such that the feet of perpendiculars from the three sides of the triangle is collinear is given by the CIRCUMCIRCLE of DABC ; is sometimes called the Wallace-Simson theorem (Guz- ma´n 1999). The Simson line bisects the line HP, where H is the ORTHOCENTER (Honsberger 1995, p. 46). Moreover, the MIDPOINT of HP lies on the NINE-POINT CIRCLE (Honsberger 1995, pp. 46 /C1/47). The Simson lines of two opposite point on the CIRCUMCENTER of a triangle are PERPENDICULAR and meet on the NINE-POINT CIRCLE . The ANGLE between the Simson lines of two points P and P ? is half the ANGLE of the arc PP ?: The Simson line of any VERTEX is the ALTITUDE through that VERTEX . The Simson line of a point opposite a VERTEX is the corresponding side. If T1T2T3 is the Simson line of a point T of the CIRCUMCIRCLE , then the triangles TT1T2 and TA2A1 are directly similar. The ENVELOPE of the Simson lines of a triangle is a DELTOID (Butchart 1939; Wells 1991, pp. 155 and 230). The area of the deltoid is half the area of the circumcircle (Wells 1991, p. 230), and MORLEY’S TRIANGLE of the starting triangle has the same orientation as the DELTOID . Each side of the triangle is tangent to the DELTOID at a point whose distance from the MIDPOINT of the side equals the chord of the NINE-POINT CIRCLE cut off by that side (Wells 1991, p. 231). If a line Lis the Simson line of a point Pon the CIRCUMCIRCLE of a TRIANGLE , then Pis called the POLE ofL(Honsberger 1995, p. 128).See also CIRCUMCIRCLE ,POLE (SIMSON LINE), RIGBY POINTS References Baker, H. F. An Introduction to Plane Geometry. London: Cambridge University Press, 1963. Butchart, J. H. "The Deltoid Regarded as the Envelope of Simson Lines." Amer. Math. Monthly 46,8 5/C1/86, 1939. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 164, 1888. Chou, S.-C. "Proving Elementary Geometry Theorems Using Wu’s Algorithm." Contemporary Math. 29, 243/C1/286, 1984. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 49, 1971. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. and Greitzer, S. L. "Simson Lines" and "More on Simson Lines." §2.5 and 2.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 40 /C1/ 41 and 43 /C1/45, 1967. de Guzma ´n, M. "An Extension of the Wallace-Simson Theorem: Projecting in Arbitrary Directions." Amer. Math. Monthly 106, 574/C1/580, 1999. Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965. Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 46 /C1/48, 1928. F. Gabriel-Marie. Exercices de Ge ´ome´trie. Tours, France: Maison Mame, p. 329, 1912. Honsberger, R. "The Simson Line" and "Simson Lines." §5.2 and 8.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 43 /C1/44 and 82 /C1/83, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 137 /C1/139, 1929. Patterson, B. C. "The Triangle: Its Deltoids and Foliates." Amer. Math. Monthly 47,1 1/C1/18, 1940. Ramler, O. J. "The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle." Amer. Math. Monthly 37, 130/C1/136, 1930. van Horn, C. E. "The Simson Quartic of a Triangle." Amer. Math. Monthly 45, 434/C1/437, 1938. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 155 and 230 /C1/231, 1991. Simson’s Formula CASSINI’S IDENTITY Sin SINE Sinc SINCFUNCTION Sinc Function A function also called the "sampling function" that arises frequently in signal processing. There are two definitions in common use. The one adopted in this work defines sinc( x)/C131 for x/C300 sinx xotherwise ;8 < :(1) where sin xis the SINE function, while Woodward (1953) and Bracewell (1999, p. 62) adopt the alter- native definition sincp(x)/C131 for x/C300 sin(px) (px)otherwise :8 < :(2) The latter definition is sometimes more convenient as a result of its simple normalization, g/C12 /C28/C12sincp(x)dx/C301: (3) LetP(x) be the RECTANGLE FUNCTION , then the FOURIER TRANSFORM ofP(x) is the sinc function F[P(x)]/C30sinc(pk): (4) The sinc function therefore frequently arises in physical applications such as Fourier transform spectroscopy as the so-called INSTRUMENT FUNCTION , which gives the instrumental response to a DELTA FUNCTION input. Removing the instrument functions from the final spectrum requires use of some sort of DECONVOLUTION algorithm. The sinc function can be written as a complex INTEGRAL by noting that, for x"0; sinc( nx)/C13sin(nx) nx/C301 nxeinx/C28e/C28inx 2i /C301 2inxeitxYrtYrP n /C28n/C301n 2ngn /C28neixtdt: (5) and that sinc( nx) and the integral both equal 1 for x/C300. The sinc function can also be written as the INFINITE PRODUCTsinc x/C30Y/C12 k/C301cosx 2k ! : (6) Definite integrals involving the sinc function include g/C12 0sinc( x)dx/C301 2p (7) g/C12 0sinc2(x)dx/C301 2p (8) g/C12 0sinc3(x)dx/C303 8p (9) g/C12 0sinc4(x)dx/C3013p (10) g/C12 0sinc5(x)dx/C30115 384p: (11) These are all special cases of the amazing general result g/C12 0sinax xbdx/C30p1/C28c(/C281)/C28(a/C28b)=2/C29 2a/C28c(b/C281)! /C2Xa=2bc/C28c k/C300(/C281)ka kðÞ(a/C282k)b/C281[ln(a/C282k)]c: (12) where aandbare POSITIVE INTEGERS such that a] b>c;c/C13a/C28b(mod 2) ;xbcis the FLOOR FUNCTION , and 00is taken to be equal to 1 (Kogan). This spectacular formula simplifies in the special case when nis a POSITIVE EVEN integer to g/C12 0sin2nx x2ndx/C30p 2(2n/C281)!2n/C281 n/C281Yrt$Yrt% : (13) wheren kYruvYruu is an E ULERIAN NUMBER (Kogan). The solution of the integral can also be written in terms of the RECURRENCE RELATION for the coefficients c(a;b)/C30p 2a/C271/C28ba/C281 1 2(a/C281)0 @1A forb/C301o r b/C302 a (b/C281)(b/C282)[(a/C281)c(a/C282;b/C282) /C28a /C215c(a;b/C282)] otherwise8 >>>>>>>>>>< >>>>>>>>>>:(14) (Zimmerman). The half-infinite integral of sinc( x) can be derived using CONTOUR INTEGRATION . In the above figure, consider the path g /C13 g1 /C27 g12 /C27 g2 /C27 g21 : Now write z /C30 Reiu : On an arc, dz /C30iReiu du and on the X-AXIS , dz /C30 eiu dR: Write g/C12 /C28/C12sinc xdx/C30Iggeiz zdx: (15) where I denotes the IMAGINARY POINT . Now define I /C13ggeiz zdz /C30 lim R1 00 g0 pexp(iR1eiu) R1eiui uR1eiu d u /C27lim R1 00lim R2 0/C12gR2 R1eiR RdR /C27 lim R2 0/C12g p 0exp(iz) zdx /C27 lim R1 00 gR1 R2e /C28iR /C28R(/C28dR) : (16) where the second and fourth terms use the identities ei0 /C301 and eip /C30/C281 : Simplifying, I /C30 lim R1 00 g0 pexp iR1eiuYrvYru iu du /C27g/C12 0 /C27eiR RdR /C27 lim R2 0/C12g p 0exp(iz) zdz /C27g0 /C27 /C12e /C28iR /C28R(/C28dR) /C30/C28g p 0i u du /C27g/C12 0 /C27eiR RdR /C270 /C27g0 /C28 /C28/C12eiR RdR: (17) where the third term vanishes by JORDAN’S LEMMA . Performing the integration of the first term and combining the others yield I /C30/C28i p /C27g/C12 /C28/C12eiz zdz /C300: (18) Rearranging gives g/C12 /C28/C12eiz zdz /C30i p: (19) sog/C12 /C28/C12sin z zdz /C30 p: (20) The same result is arrived at using the method of RESIDUES by noting I /C300 /C271 2 2pi Res z/C300f(z) /C30i p(z /C280)eiz z j z/C300/C30ip eizYrtYrP z/C300/C30i p; (21) so I(I) /C30 p: (22) Since the integrand is symmetric, we therefore have g/C12 0sinx xdx /C301 2 p; (23) giving the SINE INTEGRAL evaluated at 0 as si(0)/C30/C28g/C12 0sinx xdx/C30/C281 2p: (24) An interesting property of sinc( x) is that the set of LOCAL EXTREMA of sinc( x) corresponds to its intersec- tions with the COSINE function cos( x);as illustrated above. See also FOURIER TRANSFORM ,FOURIER TRANSFORM– RECTANGLE FUNCTION ,INSTRUMENT FUNCTION ,JINC FUNCTION ,KILROY CURVE ,SINE,SINE INTEGRAL References Bracewell, R. "The Filtering or Interpolating Function, sinc x:/"I n The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 62 /C1/64, 1999. Kogan, S. "A Note on Definite Integrals Involving Trigono- metric Functions." http://www.mathsoft.com/asolve/con- stant/pi/sin/sin.html. Morrison, K. E. "Cosine Products, Fourier Transforms, and Random Sums." Amer. Math. Monthly 102, 716/C1/724, 1995. Woodward, P. M. Probability and Information Theory with Applications to Radar. New York: McGraw-Hill, 1953. Sinclair’s Soap Film Problem Find the shape of a soap film (i.e., MINIMAL SURFACE ) which will fill two inverted conical FUNNELS facing each other is known as Sinclair’s soap film problem (Bliss 1925, p. 121). The soap film will assume the shape of a CATENOID . See also CATENOID ,FUNNEL ,MINIMAL SURFACE References Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 121 /C1/122, 1925. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, p. 81, 1992. Sinclair, M. E. "On the Minimum Surface of Revolution in the Case of One Variable End Point." Ann. Math. 8, 177/C1/ 188, 1907. Sine One of the basic TRIGONOMETRIC FUNCTIONS encoun- tered in TRIGONOMETRY . Let ube an ANGLE measured counterclockwise from the X-AXIS along the arc of the UNIT CIRCLE . Then sin uis the vertical coordinate of the arc endpoint. As a result of this definition, the sine function is periodic with period 2 p:By the PYTHAGOREAN THEOREM , sinualso obeys the identity sin2u/C27cos2u/C301: (1) The definition of the sine function can be extended to complex arguments zusing the definition sinz/C30eiz/C28e/C28iz 2i; (2) where Eis the base of the NATURAL LOGARITHM and I is the IMAGINARY NUMBER . A related function known as the HYPERBOLIC SINE is similarly defined, sinh z/C301 2ez/C28e/C28zðÞ ; (3) The sine function can be defined algebraically by the infinite sum sinx/C30X/C12 n/C301(/C281)n/C281 (2n/C281)!x2n/C281(4) and INFINITE PRODUCT sinx/C30xYx n/C3011/C28x2 n2p2 ! : (5) It is also given by the IMAGINARY PART of the complex exponential sinx/C30IeixYrtYrP (6) The multiplicative inverse of the sine function is the COSECANT , defined as cscx/C131 sinx: (7) The sine function is also given by the slowly con-vergent INFINITE SERIES sin(z)/C30/C28pX/C12 k/C301m(k)l nn k ! frackz 2p ! klnn(8) where m(k) is the M O¨BIUS FUNCTION and frac xis the FRACTIONAL PART (M. Trott). Using the results from the EXPONENTIAL SUM FOR- MULAS XN n/C300sin(nx)/C30IXN n/C300einx"# /C30Isin1 2NxYru*Yru+ sin1 2xYru*Yru+ ei(N/C271)x=22 435 /C30sin1 2 NxYru*Yru+ sin1 2 xYru*Yru+ sin1 2 x(N /C271)hi : (9) Similarly, X/C12 n /C300pn sin(nx) /C30IX/C12 n/C300pneinx"# /C30I1 /C28 pe/C28iz 1 /C28 2p cos x /C27 p2"# /C30p sin x 1 /C28 2p cos x /C27 p2 : (10) The sum of sin2(kx) can also be done in closed form, XN k /C300sin2(kx) /C3014 f1 /C272N /C28csc x sin[x(1 /C272N) g: (11) The sine function obeys the identity sin(nu) /C302 cos u sin[(n /C281)u] /C28sin[(n /C282)u] (12) and the MULTIPLE-ANGLE FORMULA sin(nx) /C30Xn k /C300n kYru$Yru% cosk x sinn/C28k x sin12(n /C28k) phi : (13) wheren kYrvYru is a BINOMIAL COEFFICIENT . Cvijovic and Klinowski (1995) show that the sum Sn( a) /C30X/C12 k/C300sin(2 k /C27 1)a (2k /C27 1)n (14) has closed form for n /C302n /C271; S2n/C271(a) /C30( /C281) 4(2n)! p2n/C271E2na p ! ; (15) where En(x)isanE ULER POLYNOMIAL . A CONTINUED FRACTION representation of sin x is sin x /C30x 1 /C27x2 2 /C215 3 /C28 x2 ðÞ2 /C215 3x2 4 /C215 5 /C28 x2 ðÞ /C274 /C215 5x2 6 /C215 7 /C28 x2 ðÞ /C27 ... (16) The value of sin(2 p=n)is IRRATIONAL for all n except 4 and 12, for which sin( p=2) /C301 and sin( p=6) /C301=2:/ The FOURIER TRANSFORM of sin 2pk0x ðÞ is given by F sin 2 pk0x ðÞ½/C138 /C30g/C12 /C28/C12e /C282 pikx sin 2pk0x ðÞ dx /C3012 i d k /C27k0 ðÞ /C28 d k /C28k0 ðÞ ½/C138 : (17)Definite integrals involving sin x include g/C12 0sinx2YrvYru dx/C301 4ffiffiffiffiffiffi 2pp (18) g/C12 0sinx3YrvYru dx/C301 6G13Yru*Yru+ (19) g/C12 0sinx4YrvYru dx/C30/C28cos5 8pYru*Yru+ G54Yru*Yru+ (20) g/C12 0sinx5YrvYru dx/C301 4ffiffiffi 5p /C281Yru*Yru+ G6 5Yru*Yru+ ; (21) where G(x) is the GAMMA FUNCTION . See also ANDREW’S SINE,COSECANT ,COSINE ,FOURIER TRANSFORM– SINE,HYPERBOLIC SINE,SINC FUNCTION , SINUSOID ,TANGENT ,TRIGONOMETRY References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/79, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987. Cvijovic, D. and Klinowski, J. "Closed-Form Summation of Some Trigonometric Series." Math. Comput. 64, 205/C1/210, 1995. Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975. Project Mathematics . "Sines and Cosines, Parts I-III." Videotape. http://www.projmath.caltech.edu/sincos1.htm. Spanier, J. and Oldham, K. B. "The Sine sin( x) and Cosine cos(x) Functions." Ch. 32 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 295 /C1/310, 1987. Sine Integral There are two types of "sine integrals" commonly defined, Si(x) /C13gz 0sin t tdt (1) and Si(x) /C13/C28gz 0sin t tdt (2) /C301 2i[ei(ix) /C28ei(/C28ix)] /C301 2ie1(ix) /C28e1(/C28ix) ½/C138 (3) Si(z) /C281 2 p; (4) where ei(x) is the EXPONENTIAL INTEGRAL and e1(x) /C13/C28ei(/C28x) : (5) /Si(x) is the function returned by the Mathematica command SinIntegral [x] and displayed above. The half-infinite integral of the SINC FUNCTION is given by si(0) /C30/C28g/C12 0sin x xdx /C30/C281 2 p: (6) To compute the integral of a sine function times a power I /C13g x2n sin(mx) dx ; (7) use INTEGRATION BY PARTS . Let u /C30x2ndv /C30sin(mx) dx (8) du /C302nx2n/C281 dx v /C301 mcos(mx); (9) so I /C30/C281 mx2n cos(mx) /C272n m g x2n/C281 cos(mx) dx: (10) Using INTEGRATION BY PARTS again, u /C30x2n/C281dv /C30cos(mx) dx (11) du /C30(2n /C281)x2n/C282 dx v1 msin(mx) (12)g x2n sin(mx) dx /C30/C281 mx2n cos(mx) /C272n m1 mx2n/C281 cos(mx) /C282n /C28 1 m g x2n/C282 sin(mx) dx"# /C30/C281 mx2n sin(mx) /C272n m2x2n/C281 sin(mx) /C28(2n)(2n /C28 1) m2 g x2n/C282 sin(mx) dx /C30/C281 mx2n cos(mx) /C272n m2x2n/C281 sin(mx) /C27... /C27(2n)! m2n g x0 sin(mx) dx /C30/C281 mx2n cos(mx) /C272n m2x2n/C281 sin(mx) /C27... /C28(2n)! m2n/C271cos(mx) /C30cos(mx)Xn k/C300(/C281)k /C271 (2n)! (2n /C28 2k)!m2k /C271x2n/C282k /C27sin(mx)Xn k/C301(/C281)k /C271 (2n)! (2k /C28 2n /C28 1)!m2kx2n /C282k /C271 (13) Letting k ?/C13n /C28k; so g x2n sin(mx) dx /C30cos(mx)Xn k /C301(/C281)n/C28k /C271 (2n)! (2k)!m2n/C282k/C271x2k /C27sin(mx)Xn /C281 k/C300(/C281)n/C28k /C271 (2n)! (2k /C28 1)!m2n/C282kx2k /C271 /C30(/C281)n/C271(2n)! cos(mx)Xn k /C300( /C281)k (2k)!m2n/C282k /C271x2k" /C27sin(mx)Xn k/C301(/C281)k/C271 (2k/C283)!m2n/C282k/C272x2k/C281/C138: (14) General integrals OF THE FORM I(k;l)/C30g/C12 0sinkx xldx (15) are related to the SINC FUNCTION and can be com- puted analytically. See also CHI,COSINE INTEGRAL ,EXPONENTIAL INTE- GRAL ,N IELSEN’S SPIRAL ,S HI,S ICI SPIRAL ,S INC FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Cosine Integrals." §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231 /C1/233, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 342 /C1/343, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Fresnel Integrals, Cosine and Sine Integrals."§6.79 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 248 /C1 /252, 1992. Spanier, J. and Oldham, K. B. "The Cosine and Sine Integrals." Ch. 38 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 361 /C1/372, 1987. Sine Surface The surface given by the PARAMETRIC EQUATIONS x/C30asinu (1) y/C30asinv (2) z/C30asin(u/C27v): (3) The coefficients of the FIRST FUNDAMENTAL FORM are E/C30a2cos2u/C27cos2(u/C27v)YrtYrP (4) F/C30a2cos2(u/C27v) (5) G/C30a2cos2v/C27cos2(u/C27v)YrtYrP ; (6) the SECOND FUNDAMENTAL FORM coefficients are e/C30/C28acosvsinvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2ucos2v/C27cos2u/C27cos2v ðÞ cos2(u/C27v)p (7) f/C30/C28acosucosvsin(u/C27v)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2ucos2v/C27cos2u/C27cos2v ðÞ cos(u/C27v)p (8) g/C30/C28acosusinuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2ucos2v/C27cos2u/C27cos2v ðÞ cos2(u/C27v)p (9)the AREA ELEMENT is dS/C30a2 /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2ucos2v/C27cos2u/C27cos2v ðÞ cos2(u/C27v)p ; (10) the Gaussian curvature is k/C30cosucosvsinusinv/C28cosucosvsin2(u/C27v)YrtYrP acos2ucos2v/C27acos2u/C27cos2v ðÞ cos2(u/C27v) ½/C1382; (11) and the MEAN CURVATURE is a complicated expres- sion. References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 315 /C1/316, 1997. Sine-Gordon Equation APARTIAL DIFFERENTIAL EQUATION which appears in differential geometry and relativistic field theory. Its name is a wordplay on its similar form to the K LEIN- GORDON EQUATION . The sine-Gordon equation is vtt/C28vxx/C27sinv/C300: (1) where vttand vxxare PARTIAL DERIVATIVES . The equation can be transformed by defining j/C131 2(x/C28t) (2) h/C1312(x/C27t): (3) Then, by the CHAIN RULE , @ @x/C30@j @x@ @j/C27@h @x@ @h(4) /C301 2@ @j/C27@ @h ! (5) @ @t/C30@j @t@ @j/C27@h @t@ @h(6) /C3012 @ @h/C27@ @j ! (7) This gives @2v @x2/C3014 @ @j/C27@ @h ! @v @j/C27@v @h ! /C301 4@2v @j2/C272@2v @j@h/C27@2v @h2 ! (8) @2v @t2/C301 4@ @h/C28@ @j ! @v @h/C28@v @j ! /C3014@2v @j2/C282@2v @j@h/C27@2v @h2 ! (9) Plugging in gives vjh/C30sinv: (10) Traveling wave analysis by setting v(x;t)/C30g(z) yields after one integration z/C28z0/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi c2/C281pgdfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2d/C282 sin21 2fYru*Yru+hir (11) where dis a constant of integration (Tabor 1989, p. 306). For the particular case d/C300, z/C28z0/C309ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28c2p ln9tan1 4fYru*Yru+hi ; (12) so integrating gives f(z)/C3094 tan/C281[e9z/C28z0 ðÞ =1/C28c2ðÞ 1=2]: (13) The solution with the plus sign is called the "kink solution," while that with the minus sign is called the"antikink solution" (Tabor 1989, pp. 306 /C1 /307). Another solution to the sine-Gordon equation is given by making the substitution v(j;h)/C30f(z);where z/C30 jh;giving the ORDINARY DIFFERENTIAL EQUATION zfƒ/C27f?/C30sinf: (14) However, this cannot be solved analytically, since letting g/C13eifgives gƒ/C28g?2 f/C272g?/C28g2/C271 2z/C300: (15) which is the third P AINLEVE ´TRANSCENDENT (Tabor 1989, p. 309). Now looking for a solution OF THE FORM v(x;t)/C304 tan/C281f(x) c(t)"# (16) givesc2 ffxx/C27f2 cctt /C30c2/C272ct/C28ccttYrvYru /C27/C28f2/C272fx/C28ffxxYrvYru :(17) Further differentiation gives fxxfðÞx ffx/C30/C28ctt=c ðÞt cct/C30/C284k2: (18) where kis a separation constant. Integrating twice then gives fxx/C30/C28k2f4/C27m2f2/C27n2(19) ctt/C30k2c4/C27m2/C281YrvYru c2/C28n2; (20) which can be solved in terms of ELLIPTIC FUNCTIONS (Infeld and Rowlands 2000, pp. 178 /C1/179). A single- SOLITON solution is obtained when k/C30n/C300; m/C211: v/C304 tan/C281exp9x/C28btffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28b2q0 @1A2435; (21) where b/C13 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2/C281p m; (22) with the plus and minus signs corresponding to the soliton and antisoliton solutions. A two- SOLITON solution exists with k/C300,m/C211: v/C304 tan/C281bsinh( bmx) cosh( bmt)"# : (23) A two-kink solution is given by v/C304 tan/C281msinhxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28m2p ! bcoshmtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28m2p !2 666643 77775(24) (Perring and Skyrme 1962; Drazin 1988; Tabor 1989, pp. 307 /C1 /308). A "breather" solution occurs for k"0;n/C300,m2B1: v/C30/C284 tan/C281 mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28m2psinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28m2tpYru*Yru+ cosh( mx)2 435: (25) For a fixed x, v, this is a periodic function of twith frequency 2 p=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28m 2p (Infeld and Rowlands 2000, p. 179). The so-called double sine-Gordon equation is given by uxt9sinu/C27hsin1 2uYru*Yru+ hi /C300 (26) (Calogero and Degasperis 1982, p. 135; Zwillinger 1997, p. 135). See also KLEIN- GORDON EQUATION ,S INH-GORDON EQUATION ,SOLITON References Baker, H. F. Abelian Functions: Abel’s Theorem and the Allied Theory, Including the Theory of the Theta Func- tions. New York: Cambridge University Press, p. xix, 1995. Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, 1982. Drazin, P. G. and Johnson, R. S. Solitons: An Introduction. Cambridge, England: Cambridge University Press, 1988. Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge Univer- sity Press, pp. 178 /C1/180, 2000. Lamb, G. L. Jr. Elements of Soliton Theory. New York: Wiley, 1980. Perring, K. K. and Skyrme, T. H. "A Model Uniform Field Equation." Nucl. Phys. 31, 550 /C1/555, 1962. Tabor, M. "The Sine-Gordon Equation." §7.5.b in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 305 /C1/309, 1989. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Sines Law LAW OF SINES Sine-Tangent Theorem If sin a sin b /C30m n; then tan1 2(a /C28 b)hi tan1 2(a /C27 b)hi /C30m /C28 n m /C27 n ; Single-Valued Function A function which has the same value at every point z0 independent of the path along which it is reached by ANALYTIC CONTINUATION (Knopp 1996, p. 93). See also SINGLE- VALUED FUNCTION References Knopp, K. "Multiple-Valued Functions." Section II in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 93 /C1/146, 1996. Singly Even Number An EVEN NUMBER OF THE FORM 4n /C272 (i.e., an INTEGER which is DIVISIBLE by 2 but not by 4). The first few for n /C300, 1, 2, ... are 2, 6, 10, 14, 18, ... (Sloane’s A016825)See also DOUBLY EVEN NUMBER ,EVEN NUMBER ,ODD NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 30, 1996. Sloane, N. J. A. Sequences A016825 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Singular Homology The general type of HOMOLOGY which is what math- ematicians generally mean when they say "homol- ogy." Singular homology is a more general version than Poincare ´’s original SIMPLICIAL HOMOLOGY . See also HOMOLOGY (TOPOLOGY ), SIMPLICIAL HOMOL- OGY Singular Knot This entry contributed by SERGEI DUZHIN A SMOOTH MAP f : S1 0 R3 whose IMAGE has singula- rities. In particular, in the theory of Vassiliev’s knot invariants, singular knots with a finite number of ORDINARY DOUBLE POINTS play an important role. See also ORDINARY DOUBLE POINT ,VASSILIEV INVAR- IANT Singular Matrix A SQUARE MATRIX that not have a MATRIX INVERSE .A matrix is singular IFF its DETERMINANT is 0. For example, there are 10 singular 2 /C292(0,1)-MATRICES : 00 00YrtvYrtu ;0001YrtvYrtu ;0010YrtvYrtu ;0011YrtvYrtu ;0100YrtvYrtu 0101YrtvYrtu ;1000YrtvYrtu ;1010YrtvYrtu ;1100YrtvYrtu ;1111YrtvYrtu : The following table gives the numbers of singular n /C29 n matrices for certain matrix classes. matrix type Sloane counts for n /C301, 2, ... /(/C281; 0; 1)/-ma- tricesA000000 1, 33, 7875, ... /(/C281; 1)/-matrices A000000 0, 8, 320, 43264, ... /(0;1)/-matrices A046747 1, 10, 338, 42976, ... See also DETERMINANT ,ILL-CONDITIONED MATRIX , MATRIX INVERSE ,N ONSINGULAR MATRIX ,SINGULAR VALUE DECOMPOSITION References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 39, 1962. Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 11, 1958. Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, p. 51, 1996. Kahn, J.; Komlo ´s, J.; and Szemeredi, E. "On the Probability that a Random 9 1 Matrix is Singular." J. Amer. Math. Soc. 8, 223 /C1/240, 1995. Komlo ´s, J. "On the Determinant of (0; 1)/-Matrices." Studia Math. Hungarica 2,7/C1/21 1967. Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 70, 1988. Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 3, 1992. Sloane, N. J. A. Sequences A046747 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Singular Measure Two COMPLEX MEASURES m and n on a MEASURE SPACE X, are mutually singular if they are supported on different subsets. More precisely, X /C30A @ B where A and B are two DISJOINT SETS such that the following hold for any MEASURABLE SET E, 1. The sets A S E and B S E are measurable. 2. The TOTAL VARIATION MEASURE of m is supported on A and that of n on B, i.e., mkk(B S E) /C300 /C30 nkk(A S E) : The relation of two measures being singular, written as m /C222 n; is plainly symmetric. Nevertheless, it is sometimes said that "/ n is singular with respect to m:/" A discrete singular measure (with respect to LEBES- GUE MEASURE on the reals) is a MEASURE l supported at 0 ; say l(E) /C301 iff 0 /C23 E : In general, a MEASURE l is concentrated on a SUBSET A if l(E) /C30 l(E S A): For instance, the measure above is concentrated at 0 :/ See also ABSOLUTELY CONTINUOUS ,COMPLEX MEA- SURE ,LEBESGUE DECOMPOSITION (MEASURE ), LEBES- GUE MEASURE References Halmos, P. Measure Theory, 2nd ed. New York: Springer- Verlag, p. 126, 1977. Reed, M. and Simon, B. Methods of Modern Mathematical Physics: Fourier Analysis, Self-Adjointness, Vol. 2. New York: Academic Press, 1975. Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 116 /C1/132, 1987. Singular Point (Algebraic Curve) A singular point of an ALGEBRAIC CURVE is a point where the curve has "nasty" behavior such as a CUSP or a point of self-intersection (when the underlying field K is taken as the REALS ). More formally, a point (a, b) on a curve f(x; y) /C300 is singular if the x and yPARTIAL DERIVATIVES of f are both zero at the point (a, b). (If the field K is not the REALS or COMPLEX NUMBERS , then the PARTIAL DERIVATIVE is computed formally using the usual rules of CALCULUS .) Consider the following two examples. For the curve x3 /C28y2 /C300: the CUSP at (0, 0) is a singular point. For the curve x2 /C27y2 /C30/C281: /(0; i) is a nonsingular point and this curve is nonsingular. See also ALGEBRAIC CURVE ,CUSP Singular Point (Differential Equation) Consider a second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27P(x)y?/C27Q(x)y /C300: If P(x) and Q(x) remain FINITE at x /C30x0 ; then x0is called an ORDINARY POINT . If either P(x)or Q(x) diverges as x 0 x0 ; then x0 is called a singular point. Singular points are further classified as follows: 1. If either P(x)or Q(x) diverges as x 0 x0but x /C28x0 ðÞ P(x) and x /C28x0 ðÞ2Q(x) remain FINITE as x 0 x0 ; then x /C30x0 is called a REGULAR SINGULAR POINT (or NONESSENTIAL SINGULARITY ). 2. If P(x) diverges more quickly than 1= x /C28x0 ðÞ ; so x /C28x0 ðÞ P(x) approaches INFINITY as x 0 x0 ; or Q(x) diverges more quickly than 1= x /C28x0 ðÞ2Q so that x /C28x0 ðÞ2Q(x) goes to INFINITY as x 0 x0 ; then x0is called an IRREGULAR SINGULARITY (or ESSENTIAL SINGULARITY ). See also IRREGULAR SINGULARITY ,REGULAR SINGU- LAR POINT ,SINGULARITY References Arfken, G. "Singular Points." §8.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 451 /C1/454, 1985. Singular Point (Function) Singular points (also simply called "singularities") are points z0in the DOMAIN of a FUNCTION fwhere ffails to be ANALYTIC .ISOLATED SINGULARITIES may be classified as ESSENTIAL SINGULARITIES ,POLES ,o r REMOVABLE SINGULARITIES . ESSENTIAL SINGULARITIES are POLES of INFINITE order. APOLE of order nis a singularity z0off(z) for which the function z/C28z0 ðÞnf(z) is nonsingular and for which z/C28z0 ðÞkf(z) is singular for k/C300, 1, ..., n/C281:/ REMOVABLE SINGULARITIES are singularities for which it is possible to assign a COMPLEX NUMBER in such a way that f(z) becomes ANALYTIC . For example, the function f(z) /C30z2 =z has a REMOVABLE SINGULAR- ITY at 0, since f(z) /C30z everywhere but 0, and f(z) can be set equal to 0 at z /C300. REMOVABLE SINGULARITIES are not POLES . The function f(z) /C30csc(1 =z) has POLES at z /C301=(2pn); and a nonisolated singularity at 0. See also ESSENTIAL SINGULARITY ,IRREGULAR SINGU- LARITY ,ORDINARY POINT ,POLE,REGULAR SINGULAR POINT ,R EMOVABLE SINGULARITY ,SINGULAR POINT (DIFFERENTIAL EQUATION ) References Arfken, G. "Singularities." §7.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 396 /C1/ 400, 1985. Singular Series r2s(n) /C30p8 G(s)ns/C281X p; qSp ; q q !2s e2np pi =q ; where Sp ; q is a GAUSSIAN SUM, and /G(s)/ is the GAMMA FUNCTION . Singular System A system is singular if its CONDITION NUMBER is INFINITE and ILL-CONDITIONED if it is too large. See also CONDITION NUMBER ,ILL-CONDITIONED MA- TRIX Singular Value There are two types of singular values, one in the context of elliptic integrals, and the other in linear algebra. For a MATRIX A ; the values ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lj(A /C31A) ;q (1) where ljis an EIGENVALUE and A/C31 is the ADJOINT MATRIX , are called singular values (Marcus and Minc 1992, p. 69). Singular values can be found using the Mathematica command SingularValues [m], which returns the so-called SINGULAR VALUE DECOMPOSI- TION as a list {u, w, v}, where u and v are matrices and w is the list of the singular values. If A /C30UH: (2) where U is a UNITARY MATRIX and H is a HERMITIAN MATRIX , then the EIGENVALUES of H are the singular values of A :/ For elliptic integrals, a MODULUS kr such that K ?(kr) K(kr) /C30ffiffiffirp; (3) where K(k) is a complete ELLIPTIC INTEGRAL OF THE FIRST KIND , and K ?(kr) /C13Kffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28k2 rpYrvYru : The ELLIPTICLAMBDA FUNCTION l /C31(r) gives the value of kr : Abel (quoted in Whittaker and Watson 1990, p. 525) proved that if r is an INTEGER , or more generally whenever K ?(k) K(k)/C30a /C27 bffiffiffinp c /C27 dffiffiffinp; (4) where a, b, c, d, and n are INTEGERS , then the MODULUS k is the ROOT of an algebraic equation with INTEGER COEFFICIENTS . See also ELLIPTIC INTEGRAL SINGULAR VALUE ,ELLIP- TIC INTEGRAL OF THE FIRST KIND,ELLIPTIC LAMBDA FUNCTION ,MODULUS (ELLIPTIC INTEGRAL ), SINGULAR VALUE DECOMPOSITION References Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 191, 1988. Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 69, 1992. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 524 /C1/528, 1990. Singular Value Decomposition A decomposition of a matrix A into the form A /C30U /C31DV ; where U is a UNITARY MATRIX , U /C31 is its ADJOINT MATRIX , and D is a DIAGONAL MATRIX whose elements are the SINGULAR VALUES of the original matrix. If A is a COMPLEX MATRIX , then there always exists such a decomposition with positive singular values (Golub and van Loan 1996, pp. 70 and 73). Singular value decomposition is implemented in Mathematica asSingularValues [m], which re- turns a list { u,w,v}, where uand vare matrices andwis a list of the singular values. See also CHOLESKY DECOMPOSITION ,LUD ECOMPOSI- TION ,M ATRIX DECOMPOSITION ,M ATRIX DECOMPOSI- TION THEOREM ,QRD ECOMPOSITION ,S INGULAR VALUE ,UNITARY MATRIX References Gentle, J. E. "Singular Value Factorization." §3.2.7 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer-Verlag, pp. 102 /C1/103, 1998. Golub, G. H. and van Loan, C. F. "The Singular Value Decomposition" and "Unitary Matrixes." §2.5.3 and 2.5.6 inMatrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, pp. 70 /C1/71 and 73, 1996. Nash, J. C. "The Singular-Value Decomposition and Its Use to Solve Least-Squares Problems." Ch. 3 in Compact Numerical Methods for Computers: Linear Algebra andFunction Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 30 /C1 /48, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Singular Value Decomposition." §2.6 in Nu- merical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 51 /C1/63, 1992. Singularity In general, a point at which an equation, surface, etc., blows up or becomes DEGENERATE . Singularities are often also called singular points. See also ESSENTIAL SINGULARITY ,ISOLATED SINGU- LARITY ,SINGULAR POINT (ALGEBRAIC CURVE ), SINGU- LAR POINT (DIFFERENTIAL EQUATION ), SINGULAR POINT (FUNCTION ), WHITNEY SINGULARITY References Knopp, K. "Singularities." Section IV in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 117 /C1/139, 1996. Sinh HYPERBOLIC SINE Sinh-Gordon Equation The PARTIAL DIFFERENTIAL EQUATION uxt /C30sinh u; which contains uxtinstead of uxx /C28uttand sinh u instead to sin u; as in the SINE- GORDON EQUATION (Grauel 1985; Zwillinger 1997, p. 135). See also SINE-GORDON EQUATION ,S INH-POISSON EQUATION References Grauel, A. "Sinh-Gordon Equation, Painleve ´ Property and Ba¨cklund Transformation." Physica A 12, 557 /C1/568, 1985. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 135, 1997. Sinh-Poisson Equation The PARTIAL DIFFERENTIAL EQUATION 92u /C27 l2 sinh u /C300; where 92 is the LAPLACIAN (Ting et al. 1987; Zwillin- ger 1997, p. 135). See also SINH-GORDON EQUATION References Ting, A. C.; Cheb, H. H.; and Lee, Y. C. "Exact Solutions of a Nonlinear Boundary Value Problem: The Vortices of the Two-Dimensional Sinh-Poisson Equation." Physica D,37/C1/ 66, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 135, 1997. SinIntegral SINE INTEGRALSink (Directed Graph) A local sink is a node of a DIRECTED GRAPH with no exiting edges, also called a TERMINAL (Borowski and Borwein 1991, p. 401; left figure). A global sink (often simply called a sink) is a node in a DIRECTED GRAPH which is reached by all directed edges (Harary 1994, p. 201; right figure). See also DIRECTED GRAPH ,NETWORK ,SOURCE References Borowski, E. J. and Borwein, J. M. (Eds.). The HarperCol- lins Dictionary of Mathematics. New York: HarperCollins, 1991. Cormen, T. H.; Leiserson, C. E.l and Rivest, R. L. Introduc- tion to Algorithms. Cambridge, MA: MIT Press, 1990. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Sink (Map) A stable fixed point of a MAP which, in a dissipative DYNAMICAL SYSTEM ,isan ATTRACTOR . See also ATTRACTOR ,DYNAMICAL SYSTEM Sinusoid A curve similar to the SINE function but possibly shifted in phase, period, amplitude, or any combina- tion thereof. The general sinusoid of amplitude a, angular frequency v (and period 2p=v) ; and phase c is given by f(x)/C30asin(vx/C27c): See also SINE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987. Sinusoidal Projection An equal AREA MAP PROJECTION . x /C30 l /C28 l0 ðÞ cos f (1) y /C30 f; (2) The inverse FORMULAS are f /C30y (3) l /C30 l0 /C27x cos f ; (4) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 243 /C1/248, 1987. Sinusoidal Spiral A curve OF THE FORM rn /C30an cos(nu) with n RATIONAL , which is not a true SPIRAL . Sinusoidal spirals were first studied by Maclaurin. Special cases are given in the following table. n Curve /C282 HYPERBOLA /C281 LINE //C281 2/ PARABOLA //C281 3/ TSCHIRNHAUSEN CUBIC /13/ CAYLEY’S SEXTIC /1 2/ CARDIOID 1 CIRCLE 2 LEMNISCATEReferences Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 184, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967. MacTutor History of Mathematics Archive. "Sinusoidal Spirals." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Sinusoidal.html. Sinusoidal Spiral Inverse Curve The INVERSE CURVE of a SINUSOIDAL SPIRAL r /C30a(1=n)[cos(nt)]1 =n with INVERSION CENTER at the origin and inversion radius k is another SINUSOIDAL SPIRAL r /C30ka(1=n)[cos(nt)]1 =n ; Sinusoidal Spiral Pedal Curve The PEDAL CURVE of a SINUSOIDAL SPIRAL r/C30a(1=n)[cos(nt)]1=n with PEDAL POINT at the center is another SINUSOIDAL SPIRAL x/C30cos1/C271=n(nt) cos[( n/C271)t] y/C30cos1/C271=n(nt) sin[( n/C271)t]: See also PEDAL CURVE ,SINUSOIDAL SPIRAL Sister Celine’s Method A method for finding RECURRENCE RELATIONS for hypergeometric polynomials directly from the series expansions of the polynomials. The method is effec- tive and easily implemented, but usually slower thanZ EILBERGER’S ALGORITHM . Given a sum f(n)/C30 akF(n;k);the method operates by finding a recur- rence of the form XI i/C300XJ j/C300aij(n)F(n /C28j ; k /C28i) /C300 by proceeding as follows (Petkovsek et al. 1996, p. 59): 1. Fix trial values of I and J. 2. Assume a recurrence formula of the above form where aij(n) are to be solved for. 3. Divide each term of the assumed recurrence by F(n ; k) and reduce every ratio F(n /C28j ; k /C28 i)=F(n; k) by simplifying the ratios of its constitu- ent factorials so that only RATIONAL FUNCTIONS in n and k remain. 4. Put the resulting expression over a common DENOMINATOR , then collect the numerator as a POLYNOMIAL in k. 5. Solve the system of linear equations that results after setting the coefficients of each power of k in the NUMERATOR to 0 for the unknown coefficients aij :/ 6. If no solution results, start again with larger I or J. Under suitable hypotheses, a "fundamental theorem" (Verbaten 1974, Wilf and Zeilberger 1992, Petkovsek et al. 1996) guarantees that this algorithm always succeeds for large enough I and J (which can be estimated in advance). The theorem also generalizes to multivariate sums and to q- and multi- q-sums (Wilf and Zeilberger 1992, Petkovsek et al. 1996). See also GENERALIZED HYPERGEOMETRIC FUNCTION , GOSPER’S ALGORITHM ,H YPERGEOMETRIC IDENTITY , HYPERGEOMETRIC SERIES ,ZEILBERGER’S ALGORITHM References Fasenmyer, Sister M. C. Some Generalized Hypergeometric Polynomials. Ph.D. thesis. University of Michigan, Nov. 1945. Fasenmyer, Sister M. C. "Some Generalized Hypergeometric Polynomials." Bull. Amer. Math. Soc. 53, 806 /C1/812, 1947. Fasenmyer, Sister M. C. "A Note on Pure Recurrence Relations." Amer. Math. Monthly 56,14/C1/17, 1949. Koepf, W. "Holonomic Recurrence Equations." Ch. 4 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunsch- weig, Germany: Vieweg, pp. 44 /C1/60, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Sister Celine’s Method." Ch. 4 in A /C30B. Wellesley, MA: A. K. Peters, pp. 55 /C1/72, 1996. Rainville, E. D. Chs. 14 and 18 in Special Functions. New York: Chelsea, 1971. Verbaten, P. "The Automatic Construction of Pure Recur- rence Relations." Proc. EUROSAM ’74, ACM-SIGSAM Bull. 8,96/C1/98, 1974. Wilf, H. S. and Zeilberger, D. "An Algorithmic Proof Theory for Hypergeometric (Ordinary and "q") Multisum/Integral Identities." Invent. Math. 108, 575 /C1/633, 1992.Site Percolation A PERCOLATION which considers the lattice vertices as the relevant entities (left figure). See also BOND PERCOLATION ,PERCOLATION THEORY Siteswap A siteswap is a sequence encountered in JUGGLING in which each term is a POSITIVE integer, encoded in BINARY . The transition rule from one term to the next consists of changing some 0 to 1, subtracting 1, and then dividing by 2, with the constraint that the DIVISION by two must be exact. Therefore, if a term is EVEN , the bit to be changed must be the units bit. In siteswaps, the number of 1-bits is a constant. Each transition is characterized by the bit position of the toggled bit (denoted here by the numeral on top of the arrow). For example, The second term is given from the first as follows: 000111 with bit 5 flipped becomes 100111, or 39. Subtract 1 to obtain 38 and divide by two to obtain 19, which is 10011. See also JUGGLING References Juggling Information Service. "Siteswaps." http://www.jug- gling.org/help/siteswap/. Smith, H. J. "Juggler Numbers." http://pweb.netcom.com/ ~hjsmith/Juggler.html. Six Circles Theorem Starting with a triangle, draw a circle touching two sides. Then draw a circle tangent to this circle and two other sides. Continue in the same direction. Then a chain is formed in which the sixth circle is tangent to the first. See also CIRCLE ,CONTACT TRIANGLE ,H EXLET ,IN- CIRCLE ,N INE CIRCLES THEOREM ,P APPUS CHAIN , SEVEN CIRCLES THEOREM References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "A Theorem about a Triangle and Six Circles." §3.3 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 49 /C1/58, 1974. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 231, 1991. Six Exponentials Theorem Let x1and x2be two linearly independent complex numbers, and let y1 ; y2 ; y3 be three linearly indepen- dent complex numbers. Then at least one of ex1y1 ; ex1y2 ; ex1y3 ; ex2y1 ; ex2y2 ; ex2y3 is TRANSCENDENTAL (Waldschmidt 1979, p. 3.5). This theorem is due to Siegel, Schneider, Lang, and Ramachandra. The corresponding statement ob- tained by replacing y1 ; y2 ; y3with y1 ; y2is called the FOUR EXPONENTIALS CONJECTURE and remains unproven. See also FOUR EXPONENTIALS CONJECTURE ,HERMITE- LINDEMANN THEOREM ,TRANSCENDENTAL NUMBER References Finch, S. "Powers of 3/2 Modulo One." http://www.mathsoft.- com/asolve/pwrs32/pwrs32.html. Ramachandra, K. "Contributions to the Theory of Transcen- dental Numbers. I, II." Acta Arith. 14,65/C1/78, 1967 /C1/68. Ramachandra, K. and Srinivasan, S. "A Note to a Paper: ‘Contributions to the Theory of Transcendental Numbers. I, II’ by Ramachandra on Transcendental Numbers." Hardy-Ramanujan J. 6,37/C1/44, 1983. Waldschmidt, M. Transcendence Methods. Queen’s Papers in Pure and Applied Mathematics, No. 52. Kingston, Ontario, Canada: Queen’s University, 1979. Waldschmidt, M. "On the Transcendence Method of Gelfond and Schneider in Several Variables." In New Advances in Transcendence Theory (Ed. A. Baker). Cambridge, Eng- land: Cambridge University Press, 1988. Six-Color Theorem To color any map on the SPHERE or the PLANE requires at most six-colors. This number can easily be reduced to five, and the FOUR-COLOR THEOREM demonstrates that the NECESSARY number is, in fact, four. See also FOUR- COLOR THEOREM ,HEAWOOD CONJEC- TURE ,MAP COLORING References Franklin, P. "A Six Colour Problem." J. Math. Phys. 13, 363 /C1/369, 1934. Hoffman, I. and Soifer, A. "Another Six-Coloring of the Plane." Disc. Math. 150, 427 /C1/429, 1996. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986.Six-j Symbol WIGNER 6J-SYMBOL SixJSymbol WIGNER 6J-SYMBOL Six-Sphere Coordinates 6-SPHERE COORDINATES Skein Relationship A relationship between KNOT POLYNOMIALS for links in different orientations (denoted below as L/C27; L0 ; and L/C28): J. H. Conway was the first to realize that the ALEXANDER POLYNOMIAL could be defined by a relationship of this type. See also ALEXANDER POLYNOMIAL , HOMFLY POLY- NOMIAL ,SIGNATURE (KNOT) Skeleton In ALGEBRAIC TOPOLOGY ,ap-skeleton is a SIMPLICIAL SUBCOMPLEX of K which is the collection of all SIMPLICES of K of dimension at most p, denoted K(p) :/ The GRAPH obtained by replacing the faces of a polyhedron with its edges and vertices is therefore the skeleton of the polyhedron. The polyhedral graphs corresponding to the skeletons of PLATONIC SOLIDS are illustrated above. The number of topolo- gically distinct skeletons N(n) with n VERTICES for n/C304, 5, 6, ... are 1, 2, 7, 18, 52, ... (Sloane’s A006869). See also POLYHEDRAL GRAPH ,SCHLEGEL GRAPH References Gardner, M. Martin Gardner’s New Mathematical Diver- sions from Scientific American. New York: Simon and Schuster, p. 233, 1966. Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Sloane, N. J. A. Sequences A006869/M1748 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Skeleton Division A LONG DIVISION in which most or all of the digits are replaced by a symbol (usually asterisks) to form a CRYPTARITHM . See also CRYPTARITHM Skew Conic Also known as a GAUCHE CONIC , SPACE CONIC , TWISTED CONIC ,or CUBICAL CONIC SECTION . A third- order SPACE CURVE having up to three points in common with a plane and having three points in common with the plane at infinity. A skew cubic is determined by six points, with no four of them COPLANAR . A line is met by up to four tangents to a skew cubic. A line joining two points of a skew cubic (REAL or conjugate imaginary) is called a SECANT of the curve, and a line having one point in common with the curve is called a SEMISECANT or TRANSVERSAL . Depending on the nature of the roots, the skew conic is classified as follows: 1. The three ROOTS are REAL and distinct (CUBICAL HYPERBOLA ). 2. One root is REAL and the other two are COMPLEX CONJUGATES (CUBICAL ELLIPSE ). 3. Two of the ROOTS coincide (CUBICAL PARABOLIC HYPERBOLA ). 4. All three ROOTS coincide (CUBICAL PARABOLA ). See also CONIC SECTION ,CUBICAL ELLIPSE ,CUBICAL HYPERBOLA ,CUBICAL PARABOLA ,CUBICAL PARABOLIC HYPERBOLA Skew Coordinate System A system of CURVILINEAR COORDINATES in which each family of surfaces intersects the others at angles other than right angles. See also CURVILINEAR COORDINATES ,O RTHOGONAL COORDINATE SYSTEM References Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 1, 1988.Skew Diagonal A diagonal of a SQUARE MATRIX which is traversed in the "northeast" direction. "The" skew diagonal (or "secondary diagonal") of an n /C29n square matrix is the skew diagonal from an1 to a1n :/ See also DIAGONAL Skew Field A FIELD in which the commutativity of multiplication is not required, more commonly called a DIVISION ALGEBRA . See also DIVISION ALGEBRA ,FIELD Skew Hermitian Matrix A SQUARE MATRIX A is skew Hermitian if is satisfies A /C31/C30/C28 A ; (1) where A /C31 is the ADJOINT MATRIX . For example, the matrix i 1 /C27i 2i /C281 /C27i 5i 3 2i /C28302 435 (2) is a skew Hermitian matrix. A matrix m can be tested to see if it is skew Hermitian using the Mathematica function SkewHermitianQ[m_List?MatrixQ] : /C30(m/C30/C30/C30- Conjugate@Transpose@m) The set of n/C29nskew Hermitian matrices is a VECTOR SPACE , and the COMMUTATOR A;B½/C138/C30AB/C28BA (3) of two skew Hermitian matrices is skew Hermitian. Hence, the skew Hermitian matrices are a L IE ALGEBRA , which is related to the L IE GROUP of UNITARY MATRICES . In particular, suppose A(t)i sa path of unitary matrices through A(0)/C30I;i.e., A(t)/C30A/C31(t)/C30I (4) for all t, where A/C31is the ADJOINT MATRIX andIis the IDENTITY MATRIX . The DERIVATIVE att/C300 of both sides must be equal so dA dtj t/C300/C27dA/C31 dtj t/C300/C300: (5) That is, the DERIVATIVE of A(t) at the identity must be a skew Hermitian matrix. The EXPONENTIAL MAP of a skew Hermitian matrix is a UNITARY MATRIX . See also ADJOINT MATRIX ,HERMITIAN MATRIX ,SKEW SYMMETRIC MATRIX ,UNITARY MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 13 and 118, 1962. Skew Lines Two or more LINES which have no intersections but are not PARALLEL , also called AGONIC LINES . Since two LINES in the PLANE must intersect or be PARALLEL , skew lines can exist only in three or more DIMEN- SIONS . Three skew lines always define a one-sheeted HYPER- BOLOID , except in the case where they are all parallel to a single PLANE but not to each other. In this case, they determine a HYPERBOLIC PARABOLOID (Hilbert and Cohn-Vossen 1999, p. 15). See also DIRECTOR ,GALLUCCI’S THEOREM ,REGULUS References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 1, 1979. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 15, 1999. Skew Polygon A polygon whose vertices do not all lie in a PLANE . See also REGULAR SKEW POLYHEDRON ,SKEW QUAD- RILATERAL References Williams, R. "Skew Polygons (Saddle Polygons)." §2.2 in The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, p. 34, 1979. Skew Polyhedron REGULAR SKEW POLYHEDRON Skew Polyomino See also L -POLYOMINO , SQUARE POLYOMINO , STRAIGHT POLYOMINO ,T-POLYOMINOSkew Quadrilateral A four-sided QUADRILATERAL not contained in a plane. The lines connecting the midpoints of opposite sides of a skew quadrilateral intersect (and bisect) each other (Steinhaus 1983). The problem of finding the minimum bounding sur- face of a skew quadrilateral was solved by Schwarz (Schwarz 1890, Wells 1991) in terms of ABELIAN INTEGRALS and has the shape of a SADDLE . It is given by solving 1 /C27f2 yYru*Yru+ fxy /C282fxfyfxy /C27 1 /C27f2 xYrvYru fyy/C300: See also HYPERBOLIC PARABOLOID ,QUADRILATERAL , SKEW POLYGON References Altshiller-Court, N. "The Skew Quadrilateral." Ch. 3 and §5.1 in Modern Pure Solid Geometry. New York: Chelsea, pp. 42 /C1/47 and 111 /C1/115, 1979. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 53, 1967. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, p. 81, 1992. Forsyth, A. R. Calculus of Variations. New York: Dover, p. 503, 1960. Schwarz, H. A. Gesammelte Mathematische Abhandlungen, 2nd ed. New York: Chelsea. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 242 and 244, 1999. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 186 /C1/187, 1991. Skew Symmetric Matrix ASQUARE MATRIX Ais skew symmetric if AT/C30/C28A; (1) with ATdenoting the matrix TRANSPOSE . For example, A/C300/C281 10YrtvYrtu (2) is a skew symmetric matrix. The set of n/C29nskew symmetric matrices is denoted o(n):A matrix mcan be tested to see if it is skew symmetric using the Mathematica function SkewSymmetricQ[l_List?MatrixQ] : /C30 (l /C30/C30/C30 - Transpose[l]) The set o(n)ofn /C29n skew symmetric matrices is a VECTOR SPACE , and the COMMUTATOR A ; B½/C138/C30AB /C28BA (3) of two skew symmetric matrices is skew symmetric. Hence, the skew symmetric matrices are a LIE ALGEBRA , which is related to the LIE GROUP of ORTHOGONAL MATRICES . In particular, suppose A(t) is a path of orthogonal matrices through A(0) /C30I ; i.e., A(t)At(t) /C30I for all t. The DERIVATIVE at t /C300 of both sides must be equal so dA=dt(0) /C27dAt =dt(0) /C300: That is, the DERIVATIVE of A(t) at the identity must be a skew symmetric matrix. The EXPONENTIAL MAP of a skew symmetric matrix is an ORTHOGONAL MATRIX . See also BISYMMETRIC MATRIX ,D IAGONAL MATRIX , PERSYMMETRIC MATRIX ,SKEW HERMITIAN MATRIX , SYMMETRIC MATRIX ,TRANSPOSE References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 12 and 117, 1962. Skewes Number The Skewes number (or first Skewes number) is the number Sk1above which p(n BLi(n)) must fail (as- suming that the RIEMANN HYPOTHESIS is true), where p(n) is the PRIME COUNTING FUNCTION and Li(n) is the LOGARITHMIC INTEGRAL . In 1912, Littlewood proved that Sk1exists (Hardy 1999, p. 17), and the upper bound Sk1 /C30eee79 :10101034 was subsequently found by Skewes. The Skewes number has since been reduced to / ee27=4 :8:185 /C2910370/ by te Riele (1987), although Con- way and Guy (1996) claim that the best current limit is 101167. In 1914, Littlewood proved that the inequal- ity must, in fact, fail infinitely often. The second Skewes number /Sk2/ is the number above which p(nBLi(n)) must fail (assuming that the RIEMANN HYPOTHESIS is false). It is much larger than the Skewes number Sk1; Sk2/C30101010103 : See also GRAHAM’S NUMBER ,RIEMANN HYPOTHESIS References Asimov, I. "Skewered!" Of Matters Great and Small. New York: Ace Books, 1976. Asimov, I. Magazine of Fantasy and Science Fiction, Nov. 1974.Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 63, 1987. Boas, R. P. "The Skewes Number." In Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., 1979. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 61, 1996. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 17 and 21, 1999. Lehman, R. S. "On the Difference p(x)/C28li(x):/"Acta Arith. 11, 397/C1/410, 1966. Skewes. J. London Math. Soc. 8, 277/C1/283, 1933. te Riele, H. J. J. "On the Sign of the Difference p(x)/C28Li(x):/" Math. Comput. 48, 323/C1/328, 1987. Wagon, S. Mathematica in Action. New York: W. H. Free- man, p. 30, 1991. Skewness The degree of asymmetry of a distribution. If the distribution has a longer tail less than the maximum, the function has NEGATIVE skewness. Otherwise, it has POSITIVE skewness. Several types of skewness are defined. The F ISHER SKEWNESS (the most common type of skewness, usually referred to simply as "the"skewness) is defined by g 1/C30m3 m3=2 2/C30m3 s3; (1) where m3is the third CENTRAL MOMENT , and m1=2 2/C13s is the STANDARD DEVIATION . The following table gives the skewness for a number of common distributions. distribution skewness BERNOULLI DISTRIBU- TION/1/C282pffiffiffiffiffiffiffiffiffiffiffi p(1/C28p)p/ BETA DISTRIBUTION /2(b/C28a) (2a/C27b)ffiffiffiffiffiffiffiffiffiffiffiffi 1/C27a/C27b abq / BINOMIAL DISTRIBU- TION/1/C282pffiffiffiffiffiffiffiffiffiffiffiffiffi np(1/C28p)p/ CHI-SQUARED DISTRI-BUTION/2ffiffi 2 rq / EXPONENTIAL DISTRI-BUTION 2 FISHER- TIPPETT DIS- TRIBUTION/12ffiffi 6p &(3) p3/ F-DISTRIBUTION /2(2n/C27m/C282) m/C286ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(m/C284) n(m/C27n/C282)q / GAMMA DISTRIBUTION /2ffiffinp/ GEOMETRIC DISTRIBU- TION/2/C28pffiffiffiffiffiffiffi 1/C28pp/ HALF-NORMAL DISTRI-BUTION/ffiffi 2p (4/C28p) (p/C282)3=2/ HYPERGEOMETRIC DIS-TRIBUTION/(m/C28n)(m/C27n/C282N) m/C27n/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m/C27n/C281 mnN (m/C27n/C28N)q / LAPLACE DISTRIBU- TION0 LOG NORMAL DISTRI- BUTION/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eS2 /C281p 2 /C27eS2YrvYru / MAXWELL DISTRIBU- TION/8 3ffiffiffiffi 2 3pq / NEGATIVE BINOMIAL DISTRIBUTION/2/C28pffiffiffiffiffiffiffiffiffiffiffi r(1/C28p)p/ NORMAL DISTRIBUTION 0 POISSON DISTRIBUTION /n /C281 =2/ RAYLEIGH DISTRIBU- TION/( p /C283)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 22/C281 2 pYru*Yru+3s / SNEDECOR’S F-DISTRI- BUTION/2(n/C272m/C282) (n/C286)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(n/C284) m(m/C27n/C282)q / STUDENT’S T-DISTRI- BUTION0 UNIFORM DISTRIBU- TION0 The PEARSON SKEWNESS is defined by b1 /C30m3 s3 !2 /C30 g2 1 : (2) The MOMENTAL SKEWNESS is defined by a(m) /C131 2 g1 : (3) The PEARSON MODE SKEWNESS is defined by mean½/C138 /C28 mode½/C138 s: (4) PEARSON’S SKEWNESS COEFFICIENTS are defined by 3 mean½/C138 /C28 mode½/C138 s (5) and 3 mean½/C138 /C28 median½/C138 s: (6) The BOWLEY SKEWNESS (also known as QUARTILE SKEWNESS COEFFICIENT ) is defined by (Q3 /C28 Q2) /C28 (Q2 /C28 Q1) Q3 /C28 Q1/C30Q1 /C28 2Q2 /C27 Q3 Q3 /C28 Q1; (7) where the Qs denote the INTERQUARTILE RANGES . The MOMENTAL SKEWNESS is a(m) /C131 2 g /C30m3 2s3 : (8)An ESTIMATOR for the FISHER SKEWNESS / g1/ is g1 /C30k3 k3=2 2; (9) where the ks are K-STATISTICS . For a normal popula- tion with a SAMPLE SIZE of N, the VARIANCE of /g1/ is var g1ðÞ:6 N (10) (Kendall et al. 1987). See also BOWLEY SKEWNESS ,F ISHER SKEWNESS , GAMMA STATISTIC , H-STATISTIC ,K URTOSIS ,M EAN, MOMENTAL SKEWNESS ,PEARSON SKEWNESS ,STAN- DARD DEVIATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972. Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall’s Ad- vanced Theory of Statistics, Vol. 1: Distribution Theory, 6th ed. New York: Oxford University Press, 1987. Kenney, J. F. and Keeping, E. S. "Skewness." §7.10 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 100 /C1/101, 1962. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Moments of a Distribution: Mean, Variance, Skewness, and So Forth." §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 604 /C1/609, 1992. Sklar’s Theorem Let H be a 2-D distribution function with marginal distribution functions F and G. Then there exists a COPULA C such that H(x ; y) /C30C(F(x); G(y)) : Conversely, for any univariate distribution functions FandGand any COPULA C, the function His a two- dimensional distribution function with marginals F andG. Furthermore, if FandGare continuous, then Cis unique. See also COPULA Skolem Paradox Even though real ARITHMETIC is uncountable, it possesses a countable "model." References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 6 /C1/7, 1977. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 191 /C1/192, 1998. Skolem Sequence A Skolem sequence of order n is a sequence S /C30 s1 ; s2 ; ...; s2n fg of 2n integers such that 1. For every k /C23f1; 2; ... ; ng; there exist exactly two elements si ; sj/C23 S such that si /C30sj /C30k; and 2. If si /C30sj /C30k with i B j, then j /C28i /C30k:/ References Colbourn, C. J. and Dinitz, J. H. (Eds.). "Skolem Sequences." Ch. 43 in CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, pp. 457 /C1/461, 1996. Skolem-Graceful Graph See also EDGE- GRACEFUL GRAPH ,S UPER- EDGE- GRACEFUL GRAPH Skolem-Mahler-Lerch Theorem If a0 ; a1 ; ... fg is a RECURRENCE SEQUENCE , then the set of all k such that ak /C300 is the union of a finite (possibly EMPTY ) set and a finite number (possibly zero) of full arithmetical progressions, where a full arithmetic progression is a set OF THE FORM fr ; r /C27 d; r /C272d; ...g with r /C23 0; d½Þ :/ References Myerson, G. and van der Poorten, A. J. "Some Problems Concerning Recurrence Sequences." Amer. Math. Monthly 102, 698 /C1/705, 1995. SL SPECIAL LINEAR GROUP Slant Height The height of an object (such as a CONE , FRUSTUM ,or PYRAMID ) measured along a side from the edge of the base to the apex. For a right PYRAMID with a regular n-gonal base of side length a, the slant height is given by sn /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C27R2p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C271 4 a2 csc2p n !vuut where R is the CIRCUMRADIUS of the base. Slater’s Identity The Q-SERIES Identity of ROGERS- RAMANUJAN -type given by X/C12 k /C300q2k2 (q; q)2k/C30q ; q7 ; q8; q8ðÞinfty q6 ; q10; q16ðÞ /C12 (q; q) /C12 (1) (Leininger and Milne 1997). See also ROGERS- RAMANUJAN IDENTITIESReferences Leininger, V. E. and Milne, S. C. "Some New Infinite Families of Eta Function Identities." Preprint. http:// www.math.ohio-state.edu/~milne/preprints.html. Slater, L. J. "Further Identities of the Rogers-Ramanujan Type." Proc. London Math. Soc. Ser. 2 54, 147 /C1/167, 1952. Slice Knot A KNOT K in S3 /C30@D4 is a slice knot if it bounds a DISK D2 in D4 which has a TUBULAR NEIGHBORHOOD D2 /C29D2whose intersection with S3is a TUBULAR NEIGHBORHOOD K /C29D2 for K. Every RIBBON KNOT is a slice knot, and it is con- jectured that every slice knot is a RIBBON KNOT . See also RIBBON KNOT,TUBULAR NEIGHBORHOOD References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 218, 1976. Slide Move The REIDEMEISTER MOVE of type III. See also KNOT MOVE,REIDEMEISTER MOVES Slide Rule A mechanical device consisting of a sliding portion and a fixed case, each marked with logarithmic axes. By lining up the ticks, it is possible to do MULTI- PLICATION by taking advantage of the additive prop- erty of LOGARITHMS . More complicated slide rules also allow the extraction of roots and computation of trigonometric functions. According to Steinhaus (1983, p. 301), the principle of the slide rule was first enumerated by E. Gunter in 1623, and in 1671, S. Partridge constructed an instrument similar to the modern slide rule. The slide rule was an indispensable tool for scientists and engineers through the 1960s, but the development of the desk calculator (and subsequently pocket calcu- lator) rendered slide rules largely obsolete beginning in the early 1970s. See also ABACUS ,RULER ,STRAIGHTEDGE References Electronic Teaching Laboratories. Simplify Math: Learn to Use the Slide Rule. New Augusta, IN: Editors and Engineers, 1966. Johnson, L. H. The Slide Rule. New York: Van Nostrand, 1949. Saffold, R. The Slide Rule. Garden City, NY: Doubleday, 1962. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 91 /C1/92 and 301, 1999. Slightly Defective Number ALMOST PERFECT NUMBER Slightly Excessive Number QUASIPERFECT NUMBER Slip Knot RUNNING KNOT Slope A quantity which gives the inclination of a curve or line with respect to another curve or line. For a LINE in the xy-PLANE making an ANGLE u with the X-AXIS , the slope m is a constant given by m /C13Dy Dx /C30tan u; (1) where Dx and Dy are changes in the two coordinates over some distance. It is meaningless to talk about the slope of a curve in 3-dimensional space unless the slope with respect to what is specified. J. Miller has undertaken a detailed study of the origin of the symbol m to denote slope. The consensus seems to be that it is not known why the letter m was chosen. One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is "monter." However, there is no evidence to make any such connection and in fact, Descartes, who was French, did not use m (Miller). Eves (1971) suggests "it just happened." The earliest known example of the symbol m appear- ing in print is O’Brien (1844). Salmon (1960) subse- quently used the symbols commonly employed today to give the slope-intercept form of a line y /C30mx /C27b (2) in his famous treatise published in several editions beginning in 1848. Todhunter (1888) also employed the symbol m, writing the slope-intercept form y /C30mx /C27c : (3) However, Webster’s New International Dictionary (1909) gives the "slope form" asy /C30sx /C27b: (4) (Miller). In Swedish textbooks, the slope-intercept equation is usually written as y /C30kx /C27m; (5) where k may derive from "koefficient" in the Swedish word for slope, "riktningskoefficient." In the Nether- lands, the equation is commonly written as one of y /C30ax /C27b (6) y /C30px /C27q (7) y /C30mx /C27n: (8) In Austria, k is used for the slope, and d for the y- intercept (Miller). See also LINE, X-INTERCEPT , Y-INTERCEPT References Eves, H. W. Mathematical Circles Revisited: A Second Collection of Mathematical Stories and Anecdotes. Prin- dle, Weber, and Schmidt, 1972. Miller, J. "Earliest Uses of Symbols from Geometry." http:// members.aol.com/jeff570/geometry.html. O’Brien, M. A Treatise on Plane Co-Ordinate Geometry, or, The Application of the Method of Co-Ordinates to the Solution of Problems in Plane Geometry. Cambridge, England: Deightons, 1844. Salmon, G. Conic Sections, 6th ed. New York: Chelsea, 1960. Todhunter, I. Treatise on Plane Co-Ordinate Geometry as Applied to the Straight Line and the Conic Sections. London: Macmillan, 1888. Slothouber-Graatsma Puzzle Assemble six 1 /C292 /C292 blocks and three 1 /C291 /C291 blocks into a 3 /C293 /C293 CUBE . See also BOX-PACKING THEOREM ,CONWAY PUZZLE , CUBE DISSECTION , DE BRUIJN’S THEOREM ,KLARNER’S THEOREM ,POLYCUBE References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 75 /C1/77, 1976. Slow Variation REGULAR VARIATION Slutzky-Yule Effect A MOVING AVERAGE may generate an irregular oscil- lation even if none exists in the original data. See also MOVING AVERAGE Sluze Pearls PEARLS OF SLUZE Smale Horseshoe Map The basic topological operations for constructing an ATTRACTOR consist of stretching (which gives sensi- tivity to initial conditions) and folding (which gives the attraction). Since trajectories in PHASE SPACE cannot cross, the repeated stretching and folding operations result in an object of great topological complexity. The Smale horseshoe map consists of a sequence of operations on the unit square. First, stretch by a factor of 2 in the x direction, then compress by 2a in the y direction. Then, fold the rectangle and fit it back into the square. Repeating this generates the horse- shoe attractor. If one looks at a CROSS SECTION of the final structure, it is seen to correspond to a CANTOR SET. See also ATTRACTOR ,CANTOR SET References Gleick, J. Chaos: Making a New Science. New York: Penguin, pp. 50 /C1/51, 1988. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 77, 1990. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Smale-Hirsch Theorem The SPACE of IMMERSIONS of a MANIFOLD in another MANIFOLD is HOMOTOPICALLY equivalent to the space of bundle injections from the TANGENT SPACE of the first to the TANGENT BUNDLE of the second. See also HOMOTOPY ,IMMERSION ,M ANIFOLD ,TAN- GENT BUNDLE ,TANGENT SPACE Small Circle A SECTION of a SPHERE which does not contain a DIAMETER of the SPHERE (Kern and Bland 1948, p. 87; Tietze 1965, p. 25). See also GREAT CIRCLE ,SPHERE References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948. Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Mod- ern Times. New York: Graylock Press, p. 25, 1965.Small Cubicuboctahedron UNIFORM POLYHEDRON U13whose DUAL POLYHEDRON is the SMALL HEXACRONIC ICOSITETRAHEDRON . It has WYTHOFF SYMBOL3 24j4;and is Wenninger model W69: Its faces are 8 f3g/C276f4g/C276f8g:The CIRCUMRADIUS for the solid with unit edge length is R/C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 2pq : FACETED versions include the uniform GREAT RHOM- BICUBOCTAHEDRON and SMALL RHOMBIHEXAHEDRON . The CONVEX HULL of the small cubicuboctahedron is the Archimedean SMALL RHOMBICUBOCTAHEDRON A6; whose dual is the DELTOIDAL ICOSITETRAHEDRON ,s o the dual of the small cubicuboctahedron (i.e., the SMALL HEXACRONIC ICOSITETRAHEDRON ) is one of the stellations of the DELTOIDAL ICOSITETRAHEDRON (Wenninger 1983, p. 57). References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 104 /C1/105, 1971. Small Ditrigonal Dodecacronic Hexecontahedron The DUAL POLYHEDRON of the SMALL DITRIGONAL DODECICOSIDODECAHEDRON U43and Wenninger dual W82:/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 74, 1983. Small Ditrigonal Dodecicosidodecahedron The UNIFORM POLYHEDRON U43whose DUAL POLYHE- DRON is the SMALL DITRIGONAL DODECACRONIC HEX- ECONTAHEDRON . It has WYTHOFF SYMBOL 35 3 ½5: Its faces are 20 f3g/C27125 2no /C2712 f10 g: Its CIRCUMRADIUS with a /C301is R /C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34 /C276ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 126 /C1/127, 1971. Small Ditrigonal Icosidodecahedron The UNIFORM POLYHEDRON U30whose DUAL POLYHE- DRON is the SMALL TRIAMBIC ICOSAHEDRON . It has WYTHOFF SYMBOL 3 j35 2 : Its faces are /20 f3g/C2712 f52 g/.A FACETED version is the DITRIGONAL DODECADODECA- HEDRON . Its CIRCUMRADIUS with a /C30 1is R /C301 2ffiffiffi 3p : The CONVEX HULL of the small ditrigonal icosidode- cahedron is a regular DODECAHEDRON , whose dual is the ICOSAHEDRON , so the dual of the great ditrigonal dodecicosidodecahedron (the SMALL TRIAMBIC ICOSA- HEDRON ) is one of the ICOSAHEDRON STELLATIONS (Wenninger 1983, p. 42). References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 106 /C1/107, 1971.Small Dodecacronic Hexecontahedron The DUAL POLYHEDRON of the SMALL DODECICOSIDO- DECAHEDRON U33 and Wenninger dual W72 :/ See also DUAL POLYHEDRON ,SMALL DODECICOSIDO- DECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 70, 1983. Small Dodecahemicosacron The DUAL POLYHEDRON of the SMALL DODECAHEMICO- SAHEDRON U62and Wenninger dual W100 : When rendered, the small dodecahemicosacron and GREAT DODECAHEMICOSACRON appear the same. See also DUAL POLYHEDRON ,SMALL DODECAHEMICO- SAHEDRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 107, 1983. Small Dodecahemicosahedron The UNIFORM POLYHEDRON U62whose DUAL POLYHE- DRON is the SMALL DODECAHEMICOSACRON . It has WYTHOFF SYMBOL5 352 ½3: Its faces are 10 f6g/C271252no : It is a FACETED version of the ICOSIDODECAHEDRON . Its CIRCUMRADIUS with unit edge length is R /C301 : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 155, 1971. Small Dodecahemidodecacron The DUAL POLYHEDRON of the SMALL DODECAHEMIDO- DECAHEDRON U51and Wenninger dual W91 : When rendered, the SMALL ICOSIHEMIDODECACRON and small dodecahemidodecacron appear the same. See also DUAL POLYHEDRON ,SMALL DODECAHEMIDO- DECAHEDRON ,S MALL ICOSIHEMIDODECACRON ,U NI- FORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 104, 1983.Small Dodecahemidodecahedron The UNIFORM POLYHEDRON U51whose DUAL POLYHE- DRON is the SMALL DODECAHEMIDODECACRON . It has WYTHOFF SYMBOL 2532 5 2: Its faces are 30 f4g/C2712f10 g: Its CIRCUMRADIUS with a /C301is R /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 /C274ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 113 /C1/114, 1971. Small Dodecicosacron The DUAL POLYHEDRON of the SMALL DODECICOSAHE- DRON U50and Wenninger dual W90:/ See also DUAL POLYHEDRON ,SMALL DODECICOSAHE- DRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 74, 1983. Small Dodecicosahedron The UNIFORM POLYHEDRON U50whose DUAL POLYHE- DRON is the SMALL DODECICOSACRON . It has WYTHOFF SYMBOL 353 2 5 4j: Its faces are 20 f6g/C2712f10 g: Its CIRCUMRADIUS with a /C301is R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34 /C276ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 141 /C1/142, 1971. Small Dodecicosidodecahedron The UNIFORM POLYHEDRON U33whose DUAL POLYHE- DRON is the SMALL DODECACRONIC HEXECONTAHE- DRON . It has WYTHOFF SYMBOL3 2 5½5: Its faces are 20 f3g/C2712 f5g/C2712 f10 g: It is a FACETED version of the SMALL RHOMBICOSIDODECAHEDRON . Its CIRCUMRA- DIUS with a /C30 1is R /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 /C274ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 110 /C1/111, 1971.Small Hexacronic Icositetrahedron The DUAL POLYHEDRON of the SMALL CUBICUBOCTA- HEDRON U13and Wenninger dual W69:/ See also DUAL POLYHEDRON ,SMALL CUBICUBOCTAHE- DRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 57, 1983. Small Hexagonal Hexecontahedron The DUAL POLYHEDRON of the SMALL SNUB ICOSICOSI- DODECAHEDRON U32and Wenninger dual W110:/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 119, 1983. Small Hexagrammic Hexecontahedron The DUAL POLYHEDRON of the SMALL RETROSNUB ICOSICOSIDODECAHEDRON and Wenninger dual W118:/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 135, 1983. Small Icosacronic Hexecontahedron The DUAL POLYHEDRON of the SMALL ICOSICOSIDODE- CAHEDRON U31 and Wenninger dual W71 :/ See also DUAL POLYHEDRON ,SMALL ICOSICOSIDODE- CAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 74, 1983. Small Icosicosidodecahedron The UNIFORM POLYHEDRON U31whose DUAL POLYHE- DRON is the SMALL ICOSACRONIC HEXECONTAHEDRON . It has WYTHOFF SYMBOL 35 2 ½3 : Its faces are 20f3g/C27 20 f6g/C271252no : Its CIRCUMRADIUS with a /C301is R /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17 /C27 3ffiffiffi 5p 2s : References Wenninger, M. J. "Small Icosicosidodecahedron." Solid 71 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 108, 1971.Small Icosihemidodecacron The DUAL POLYHEDRON of the SMALL ICOSIHEMIDODE- CAHEDRON U49and Wenninger dual W89 : When rendered, the small icosihemidodecacron and SMALL DODECAHEMIDODECACRON appear the same. See also DUAL POLYHEDRON ,SMALL DODECAHEMIDO- DECACRON ,S MALL ICOSIHEMIDODECAHEDRON ,U NI- FORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 104, 1983. Small Icosihemidodecahedron The UNIFORM POLYHEDRON U49whose DUAL POLYHE- DRON is the SMALL ICOSIHEMIDODECACRON . It has WYTHOFF SYMBOL3 23½5:Its faces are 20 f3g/C276f10g:It is a FACETED version of the ICOSIDODECAHEDRON . Its CIRCUMRADIUS with a/C301i s R/C30f/C301 21/C27ffiffiffi 5pYru*Yru+ : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 140, 1971. Small Inverted Retrosnub Icosicosidodecahedron SMALL RETROSNUB ICOSICOSIDODECAHEDRON Small Multiple Method An algorithm for computing a UNIT FRACTION . References Eppstein, D. Egypt.ma Mathematica notebook. http:// www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma. Small Number Guy’s "STRONG LAW OF SMALL NUMBERS " states that there aren’t enough small numbers to meet the many demands made of them. Guy (1988) also gives several interesting and misleading facts about small num- bers: 1. 10% of the first 100 numbers are SQUARE NUMBERS . 2. A QUARTER of the numbers B100 are PRIMES . 3. All numbers less than 10, except for 6, are PRIME POWERS . 4. Half the numbers less than 10 are F IBONACCI NUMBERS . See also LARGE NUMBER ,STRONG LAW OF SMALL NUMBERS References Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697/C1/712, 1988. Small Retrosnub Icosicosidodecahedron The UNIFORM POLYHEDRON U72also called the SMALL INVERTED RETROSNUB ICOSICOSIDODECAHEDRON whose DUAL POLYHEDRON is the SMALL HEXAGRAMMIC HEXECONTAHEDRON . It has W YTHOFF SYMBOL3 23252:YrutYrutYrut Its faces are 100 f3g/C2712 5 2no :It has CIRCUMRADIUS with a/C301 R/C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13/C273ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 102/C2746ffiffiffi 5pqr :0:580694800133921 : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 194 /C1/199, 1971.Small Rhombicosidodecahedron The 62-faced A RCHIMEDEAN SOLID A5with faces 20f3g/C2730f4g/C2712f5g:It is UNIFORM POLYHEDRON U27and Wenninger model W14:It has S CHLA ¨FLI SYMBOL r3 5Yr$Yr% and W YTHOFF SYMBOL A9:The SMALL DODECICOSIDODECAHEDRON and SMALL RHOMBIDODE- CAHEDRON are FACETED versions. Its DUAL POLYHEDRON is the DELTOIDAL HEXECONTA- HEDRON . The INRADIUS rof the dual, MIDRADIUS rof the solid and dual, and CIRCUMRADIUS Rof the solid fora/C301 are r/C301 41(15/C272ffiffiffi 5p )ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11/C274ffiffiffi 5pq /C302:12099 . . . r /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C274ffiffiffi 5pq /C302:17625... R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 /C274ffiffiffi 5pq /C302 :23295... See also ARCHIMEDEAN SOLID ,G REAT RHOMBICOSI- DODECAHEDRON (ARCHIMEDEAN ), GREAT RHOMBICO- SIDODECAHEDRON (UNIFORM ), HEXECONTAHEDRON , ZOME References Cundy, H. and Rollett, A. "lpar;Small) Rhombicosidodecahe- dron. Sk(n) :/" §3.7.11 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 111, 1989. Wenninger, M. J. "The Rhombicosidodecahedron." Model 14 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 28, 1989. Small Rhombicuboctahedron The 26-faced ARCHIMEDEAN SOLID /A6/ consisting of faces /8 f3g/C2718 f4g/. Although this solid is sometimes also called the truncated icosidodecahedron, this name is inappropriate since true TRUNCATION would yield rectangular instead of square faces. It is UNI- FORM POLYHEDRON /U10/ and Wenninger model /W13/.It has SCHLA ¨ FLI SYMBOL /r f3 4 g/ and WYTHOFF SYMBOL 34|2.Its DUAL POLYHEDRON is the DELTOIDAL ICOSITETRA- HEDRON , also called the TRAPEZOIDAL ICOSITETRAHE- DRON . The INRADIUS r of the dual, MIDRADIUS /r/ of the solid and dual, and CIRCUMRADIUS R of the solid for a /C301 are r /C301 17(6 /C27ffiffiffi 2p )ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 2pq /C301 :22026... r /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C272ffiffiffi 2pq /C301:30656... R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 2pq /C301 :39896... The distances between the solid center and centroids of the triangular and square faces are r3 /C301 2(1 /C27ffiffiffi 2p ) (1) r4 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13(11 /C276ffiffiffi 2p )q : (2) The SURFACE AREA and VOLUME are S /C3018 /C272ffiffiffi3p (3) V /C301 3(12 /C2710ffiffiffi 2p ): (4) The CONVEX HULL of the SMALL CUBICUBOCTAHEDRON is the small rhombicuboctahedron, whose dual is the DELTOIDAL ICOSITETRAHEDRON , so the dual of the SMALL CUBICUBOCTAHEDRON (i.e., the SMALL HEXA- CRONIC ICOSITETRAHEDRON ) is one of the stellations of the DELTOIDAL ICOSITETRAHEDRON (Wenninger 1983, p. 57). A version of the small rhombicuboctahedron in which the top and bottom halves are rotated with respect to each other is known as the ELONGATED SQUARE GYROBICUPOLA . See also ARCHIMEDEAN SOLID,ELONGATED SQUARE GYROBICUPOLA ,GREAT RHOMBICUBOCTAHEDRON (AR- CHIMEDEAN ), GREAT RHOMBICUBOCTAHEDRON (UNI- FORM ), ICOSITETRAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 137 /C1/138, 1987. Cundy, H. and Rollett, A. "lpar;Small) Rhombicuboctahe- dron. 3.42."§3.7.5 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 105, 1989. Wenninger, M. J. "The Rhombicuboctahedron." Model 13 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 27, 1989. Small Rhombidodecacron The DUAL POLYHEDRON of the SMALL RHOMBIDODECA- HEDRON and Wenninger model W74:/ References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 70, 1983. Small Rhombidodecahedron The UNIFORM POLYHEDRON U39whose DUAL POLYHE- DRON is the SMALL RHOMBIDODECACRON . It has WYTH- OFF SYMBOL 253 2 5 2: Its faces are 30f4 g/C2712 f10g: It is a FACETED version of the SMALL RHOMBICOSIDODECAHEDRON . Its CIRCUM- RADIUS with a /C301is R /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 /C274ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 113 /C1/114, 1971. Small Rhombihexacron The DUAL POLYHEDRON of the SMALL RHOMBIHEXAHE- DRON U18and Wenninger dual W86/ See also DUAL POLYHEDRON ,SMALL RHOMBIHEXAHE- DRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 57, 1983.Small Rhombihexahedron The UNIFORM POLYHEDRON U18whose DUAL POLYHE- DRON is the SMALL RHOMBIHEXACRON . It has W YTH- OFF SYMBOL 243 2 42j and is Wenninger model W86:Its faces are 12 f4g/C27 6f8g:It is a FACETED version of the SMALL RHOMBI- CUBOCTAHEDRON . Its CIRCUMRADIUS with a/C301i s R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 2pq : The CONVEX HULL of the small rhombihexahedron is the Archimedean SMALL RHOMBICUBOCTAHEDRON A6; whose dual is the DELTOIDAL ICOSITETRAHEDRON ,s o the dual of the small rhombihexahedron (i.e., the SMALL RHOMBIHEXACRON ) is one of the stellations of the DELTOIDAL ICOSITETRAHEDRON (Wenninger 1983, p. 57). References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 134, 1971. Small Snub Icosicosidodecahedron The UNIFORM POLYHEDRON U32whose DUAL POLYHE- DRON is the SMALL HEXAGONAL HEXECONTAHEDRON .It has WYTHOFF SYMBOL j335 2(Har’El 1993 gives the symbol as j52 33:/) Its faces are 100f3g/C271252no : Its CIRCUMRADIUS for a /C30 1is R /C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 /C273ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 102 /C2746ffiffiffi 5pqr /C301 :4581903307387 ... References Har’El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47,57/C1/110, 1993. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 172 /C1/173, 1971. Small Stellapentakis Dodecahedron The DUAL POLYHEDRON of the TRUNCATED GREAT DODECAHEDRON U37and Wenninger dual W75:/ See also DUAL POLYHEDRON ,T RUNCATED GREAT DODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 84, 1983.Small Stellated Dodecahedron One of the K EPLER- POINSOT SOLIDS whose DUAL POLYHEDRON is the GREAT DODECAHEDRON . It is also UNIFORM POLYHEDRON U34;Wenninger model W21; and is the first STELLATION of the DODECAHEDRON (Wenninger 1989). It was originally called the URCHIN by Kepler. The small stellated dodecahedron has SCHLA ¨FLI SYMBOL5 2;5no and W YTHOFF SYMBOL 5½25 2:It is composed of 12 PENTAGRAMMIC faces. Its faces are 125 2no :/ The easiest way to construct a small stellated dode- cahedron is by CUMULATION , i.e., building twelve PENTAGONAL PYRAMIDS and attaching them to the faces of a DODECAHEDRON . The height of the pyramids for a small stellated dodecahedron built on a DODE- CAHEDRON of unit edge length isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 55/C272ffiffiffi 5pYrvYruq :The CIRCUMRADIUS of the small stellated dodecahedron with pentagrammic edge length a/C301i s R/C301 251=4f/C281=2/C301451=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffiffiffi 5p /C281Yru*Yru+r : Schla ¨fli (1901, p. 134) did not recognize the small stellated dodecahedron because it, like the GREAT DODECAHEDRON , satisfies N0/C28N1/C27N2/C3012/C2830/C2712/C30/C286; (1) where N0is the number of vertices, N1the number of edges, and N2the number of faces (Coxeter 1973, p. 172), thus violating the POLYHEDRAL FORMULA . The CONVEX HULL of the small stellated dodecahedron is a regular DODECAHEDRON and the dual of the DODECAHEDRON is the ICOSAHEDRON , so the dual of the small stellated dodecahedron is one of the ICOSAHEDRON STELLATIONS (Wenninger 1983, p. 40) See also DODECAHEDRON ,G REAT DODECAHEDRON , GREAT ICOSAHEDRON ,GREAT STELLATED DODECAHE- DRON ,KEPLER- POINSOT SOLID ,STELLATION References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Cundy, H. and Rollett, A. "Small Stellated Dodecahedron. (5 2)5 :/" §3.6.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 90 /C1/91, 1989. Fischer, G. (Ed.). Plate 103 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 102, 1986. Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 219, 1997. Schla¨fli, L. "Theorie der vielfachen Kontinuita ¨t." Denkschrif- ten der Schweizerischen naturforschenden Gessel. 38,1/C1/ 237, 1901. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 211 /C1/212, 1999. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 39, 1983. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 35 and 38, 1989. Small Stellated Triacontahedron MEDIAL RHOMBIC TRIACONTAHEDRON Small Stellated Truncated Dodecahedron The UNIFORM POLYHEDRON U58 also called the QUASI- TRUNCATED SMALL STELLATED DODECAHEDRON whose DUAL POLYHEDRON is the GREAT PENTAKIS DODECAHE-DRON . It has SCHLA ¨ FLI SYMBOL t’5 2 ; 5no and WYTH- OFF SYMBOL 2553 :YrutYrutYrut Its faces are 12 f5g/C271210 3no : Its CIRCUMRADIUS with a /C30 1is R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34 /C2810ffiffiffi 5pq : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 151, 1971. Small Triakis Octahedron The 24-faced DUAL POLYHEDRON of the TRUNCATED CUBE A9and Wenninger dual W8 : It can be con- structed by CUMULATION of a unit edge-length OCTA- HEDRON by a pyramid with heightffiffiffi 3p /C282 3ffiffiffi 6p : For a TRUNCATED CUBE of unit side length the dual has edges of lengths s1/C302 (1) s2/C302/C27ffiffiffi 2p : (2) Normalizing so that s1/C301;the resulting small triakis octahedron has SURFACE AREA and VOLUME S/C303ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C274ffiffiffi 2pq (3) V/C301 23/C272ffiffiffi 2pYru*Yru+ : (4) See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , GREAT TRIAKIS OCTAHEDRON ,ICOSITETRAHEDRON , SMALL TRIAKIS OCTAHEDRON STELLATIONS ,T RUN- CATED CUBE References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 7, 1983. Small Triakis Octahedron Stellations R. Whorf found that there are probably several thousand stellations of the small triakis octahedron (Wenninger 1983, p. 36). In particular, the CONVEX HULLS of the GREAT CUBICUBOCTAHEDRON U14 ; the Archimedean GREAT RHOMBICUBOCTAHEDRON A3 /C30 U17 ; and GREAT RHOMBIHEXAHEDRON U21are all the Archimedean TRUNCATED CUBE A9 ; whose dual is the SMALL TRIAKIS OCTAHEDRON , so the duals of these solids (i.e., the GREAT HEXACRONIC ICOSITETRAHE- DRON , GREAT DELTOIDAL ICOSITETRAHEDRON , and GREAT RHOMBIHEXAHEDRON ) are all stellations of the small triakis octahedron (Wenninger 1983, p. 57). References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 36, 38, and 57 /C1/58, 1983.Small Triambic Icosahedron The DUAL POLYHEDRON of the SMALL DITRIGONAL ICOSIDODECAHEDRON U30and Wenninger model W70 : It can be constructed by CUMULATION of a unit edge- length ICOSAHEDRON by a pyramid with heightffiffiffiffiffiffi 15p =15: Wenninger (1989, p. 49) calls this solid the triakis octahedron (which is a term more commonly used for the dual of one of the Archimedean solids). The CONVEX HULL of the SMALL DITRIGONAL ICOSIDO- DECAHEDRON is a regular DODECAHEDRON , whose dual is the ICOSAHEDRON , so the dual of the SMALL DITRIGONAL ICOSIDODECAHEDRON (the small triambic icosahedron) is one of the ICOSAHEDRON STELLATIONS (Wenninger 1983, p. 42). See also DODECAHEDRON- SMALL TRIAMBIC ICOSAHE- DRON COMPOUND ,DUAL POLYHEDRON ,SMALL DITRI- GONAL ICOSIDODECAHEDRON ,TRIAKIS ICOSAHEDRON , TRIAKIS OCTAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 42 and 46 /C1/47 1983. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, p. 46, 1989. Small World Problem The small world problem asks for the probability that two people picked at random have at least one acquaintance in common. See also BIRTHDAY PROBLEM Smarandache Ceil Function AS MARANDACHE -like function which is defined where Sk(n) is defined as the smallest integer for which njSk(n)k:The Smarandache Sk(n) function can there- fore be obtained by replacing any factors which are kth powers in nby their kroots. Sk(n)/C30n Mk(n); where Mk(n) is the number of solutions to xk/C130 (mod n):/ The functions Sk(n) for k /C302, 3, ..., 6 for values such that Sk(n) "n are tabulated by Begay (1997). The following tables gives Sk(n) for small k and n /C301, 2, .... k Sloane /Sk(n)/ 1 A000027 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... 2 A019554 1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, ... 3 A019555 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, ... 4 A053166 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, ... See also PSEUDOSMARANDACHE FUNCTION ,SMARAN- DACHE FUNCTION ,SMARANDACHE- KUREPA FUNCTION , SMARANDACHE NEAR-TO- PRIMORIAL FUNCTION ,SMAR- ANDACHE SEQUENCES ,S MARANDACHE- WAGSTAFF FUNCTION References Begay, A. "Smarandache Ceil Functions." Bull. Pure Appl. Sci. 16E, 227/C1/229, 1997. http://www.gallup.unm.edu/ ~smarandache/smarceil.htm. Sloane, N. J. A. Sequences A000027/M0472, A019554, A019555, and A053166 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Smarandache, F. Collected Papers, Vol. 2. Kishinev, Mol- dova: Kishinev University Press, 1997. Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993. Smarandache Constants "The" Smarandache constant is the smallest solution to the generalized A NDRICA’S CONJECTURE , x:0:567148 :/ The first Smarandache constant is defined as S1/C13X/C12 n/C3021 [S(n)]!>1:093111 ; (1) where S(n) is the S MARANDACHE FUNCTION . Cojocaru and Cojocaru (1996a) prove that S1exists and is bounded by 0 :717BS1B1:253:The lower limit given above is obtained by taking 40,000 terms of the sum. Cojocaru and Cojocaru (1996b) prove that the second Smarandache constantS2/C13X/C12 n/C302S(n) n!:1:71400629359162 (2) is an IRRATIONAL NUMBER . Cojocaru and Cojocaru (1996c) prove that the series S3/C13X/C12 n/C3021Qn i/C302Si)ðÞ:0:719960700043708 (3) converges to a number 0 :71BS3B1:01;and that S4(a)/C13X/C12 n/C302na Qni/C302S(i)(4) converges for a fixed REAL NUMBER a]1:The values for small aare S4(1):1:72875760530223 (5) S4(2):4:50251200619297 (6) S4(3):13:0111441949445 (7) S4(4):42:4818449849626 (8) S4(5):158:105463729329 : (9) Sandor (1997) shows that the series S5/C13X/C12 n/C301(/C281)n/C281S(n) n!(10) converges to an IRRATIONAL . Burton (1995) and Dumitrescu and Seleacu (1996) show that the series S6/C13X/C12 n/C302S(n) (n/C271)!(11) converges. Dumitrescu and Seleacu (1996) show that the series S7/C13X/C12 n/C30rS(n) (n/C27r)!(12) and S8/C13X/C12 n/C30rS(n) (n/C28r)!(13) converge for ra natural number (which must be nonzero in the latter case). Dumitrescu and Seleacu(1996) show that S 9/C13X/C12 n/C3011Pn i/C302S(i) i!(14) converges. Burton (1995) and Dumitrescu and Se- leacu (1996) show that the series S10/C13X/C12 n/C3021 [S(n)]affiffiffiffiffiffiffiffiffiffiffi S(n)!p (15) and S11 /C13X/C12 n/C3021 [S(n)]affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [S(n) /C27 1]!p (16) converge for a > 1:/ See also ANDRICA’S CONJECTURE ,S MARANDACHE FUNCTION References Burton, E. "On Some Series Involving the Smarandache Function." Smarandache Notions J. 6,13/C1/15, 1995. Burton, E. "On Some Convergent Series." Smarandache Notions J. 7,7/C1/9, 1996. Cojocaru, I. and Cojocaru, S. "The First Constant of Smarandache." Smarandache Notions J. 7, 116 /C1/118, 1996a. Cojocaru, I. and Cojocaru, S. "The Second Constant of Smarandache." Smarandache Notions J. 7, 119 /C1/120, 1996b. Cojocaru, I. and Cojocaru, S. "The Third and Fourth Constants of Smarandache." Smarandache Notions J. 7, 121 /C1/126, 1996c. "Constants Involving the Smarandache Function." http:// www.gallup.unm.edu/~smarandache/CONSTANT.TXT. Dumitrescu, C. and Seleacu, V. "Numerical Series Involving the Function S." The Smarandache Function in Number Theory. Vail: Erhus University Press, pp. 48 /C1/61, 1996. Ibstedt, H. Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics. Lupton, AZ: Erhus University Press, pp. 27 /C1/30, 1997. Sandor, J. ‘On The Irrationality Of Certain Alternative Smarandache Series." Smarandache Notions J. 8, 143 /C1/ 144, 1997. Smarandache, F. Collected Papers, Vol. 1. Bucharest, Ro- mania: Tempus, 1996. Smarandache, F. Collected Papers, Vol. 2. Kishinev, Mol- dova: Kishinev University Press, 1997. Smarandache Function The smallest value S(n) for a given n for which /njS(n)!/ (n divides S(n) FACTORIAL ). For example, the number 8 does not divide 1!; 2!; 3!; but does divide 4! /C304 /C215 3 /C215 2 /C215 1 /C308 /C215 3 ; so S(8) /C304: For a PRIME p, S(p) /C30p ; and for an EVEN PERFECT NUMBER r, S(r)is PRIME (Ashbacher 1997). Sloane places the restriction S(n) > 0; while Ashbacher (1995) and Russo (2000, p. 4) take S(n) ]0:/ The Smarandache numbers for n /C301, 2, ... are 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (Sloane’s A002034; but,depending on the convention, S(1) may equal either 0 or 1). Letting a(n) denote the smallest value of n for which S(n) /C301 ; 2, ..., then a(n) is given by 1, 2, 3, 4, 5, 9, 7, 32, 27, 25, 11, 243, ... (Sloane’s A046021). Some values of S(n) first occur only for very large n, for example, S(59; 049) /C3024 ; S(177 ; 147) /C3027; S(134 ; 217; 728) /C3030 ; S(43; 046; 721) /C3036; and S(9; 765; 625) /C3045: D. Wilson points out that if we let I(n ; p) /C30n /C28P (n; p) p /C28 1; be the power of the PRIME p in n!; where a (n; p)is the sum of the base- p digits of n, then it follows that a(n) /C30min pI(n/C281 ; p)/C271 ; where the minimum is taken over the PRIMES p dividing n. This minimum appears to always be achieved when p is the GREATEST PRIME FACTOR of n.If n /C302k /C281 2k /C281YrvYru is an even PERFECT NUMBER (i.e., 2k/C281 is prime), then S(n)/C30p(Ruiz 1999a). If p is a prime number and n]2 an integer, then SppnðÞ/C30 pn/C271/C28pn/C27p:(Ruiz 1999b). The incrementally largest values of S(n) are 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (Sloane’s A046022), which occur for n/C301, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (Sloane’s A046023), i.e., the values where S(n)/C30n:/ Tutescu (1996) conjectures that the D IOPHANTINE EQUATION S(n)/C30S(n/C271) has no solution. See also FACTORIAL ,GREATEST PRIME FACTOR ,PSEU- DOSMARANDACHE FUNCTION ,S MARA NDACHE CEIL FUNCTION ,S MARANDACHE CONSTANTS ,S MARAN- DACHE- KUREPA FUNCTION ,SMARANDACHE NEAR-TO- PRIMORIAL FUNCTION ,S MARANDACHE- WAGSTAFF FUNCTION References Ashbacher, C. An Introduction to the Smarandache Func- tion. Cedar Rapids, IA: Decisionmark, 1995. Ashbacher, C. "Problem 4616." School Sci. Math. 97, 221, 1997. Begay, A. "Smarandache Ceil Functions." Bulletin Pure Appl. Sci. India 16E, 227/C1/229, 1997. Dumitrescu, C. and Seleacu, V. The Smarandache Function. Vail, AZ: Erhus University Press, 1996. Finch, S. "Unsolved Mathematics Problems: Questions In- volving the Smarandache Function." http://www.math- soft.com/asolve/smarand/smarand.html. "Functions in Number Theory." http://www.gallup.unm.edu/ ~smarandache/FUNCT1.TXT. Ibstedt, H. Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics. Lupton, AZ: Erhus University Press, pp. 27 /C1/30, 1997. Ruiz, S. M. "Smarandache Function Applied to Perfect Numbers." Smarandache Notions J. 10, 114/C1/155, 1999. Ruiz, S. M. "A Result Obtained Using Smarandache Func- tion." Smarandache Notions J. 10, 123/C1/124, 1999. Russo, F. A Set of New Smarandache Functions, Sequences, and Conjectures in Numer Theory. Lupton, AZ: American Research Press, 2000. Sandor, J. "On Certain Inequalities Involving the Smaran- dache Function." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996. Sloane, N. J. A. Sequences A002034/M0453, A046021, A046022, and A046023 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Smarandache, F. "A Function in Number Theory." Analele Univ. Timisoara, Ser. St. Math. 43,79/C1/88, 1980. Smarandache, F. Collected Papers, Vol. 1. Bucharest, Ro- mania: Tempus, 1996. Smarandache, F. Collected Papers, Vol. 2. Kishinev, Mol- dova: Kishinev University Press, 1997. Tutescu, L. "On a Conjecture Concerning the Smarandache Function." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996. Smarandache Near-to-Primorial Function /SNTP (n) is the smallest PRIME such that p# /C281 ; p#; or p# /C271 is divisible by n, where p# is the PRIMORIAL of p. Ashbacher (1996) shows that SNTP (n) only exists 1. If there are no square or higher powers in the factorization of n,or 2. If there exists a PRIME q Bp such that n (q# 91); j where p is the smallest power contained in the factorization of n. Therefore, SNTP (n) does not exist for the SQUAREFUL numbers n/C304, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, ... (Sloane’s A013929). The first few values of SNTP (n); where defined, are 2, 2, 2, 3, 3, 3, 5, 7, ... (Sloane’s A046026). See also PRIMORIAL ,SMARANDACHE FUNCTION References Ashbacher, C. "A Note on the Smarandache Near-To- Primordial Function." Smarandache Notions J. 7,4 6/C1/ 49, 1996. Mudge, M. R. "The Smarandache Near-To-Primorial Func- tion." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 585, 1996. Sloane, N. J. A. Sequences A013929 and A046026 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/eisonline.html. Smarandache Paradox Let Abe some attribute (e.g., possible, present, perfect, etc.). If all is A, then the non- Amust also beA. For example, "All is possible, the impossible too," and "Nothing is perfect, not even the perfect." References Le, C. T. "The Smarandache Class of Paradoxes." Bull. Transylvania Univ. Brasov 36,7/C1/8, 1994. Le, C. T. "The Smarandache Class of Paradoxes." Bull. Pure Appl. Sci. 14E, 109/C1/110, 1995. Le, C. T. "The Smarandache Class of Paradoxes." J. Indian Acad. Math. 18,5 3/C1/55, 1996.Mitroiescu, I. The Smarandache Class of Paradoxes. Glen- dale, AZ: Erhus University Press, 1994. Mitroiescu, I. "The Smarandache’s Class of Paradoxes Applied in Computer Science." Abstracts of Papers Pre- sented to the Amer. Math. Soc. 16, 651, 1995. Smarandache Sequences Smarandache sequences are any of a number of simply generated INTEGER SEQUENCES resembling those considered in published works by Smarandachesuch as the CONSECUTIVE NUMBER SEQUENCES and EUCLID NUMBERS (Iacobescu 1997). Some other "Smarandache" sequences are given below. 1. The concatenation of ncopies of the INTEGER n: 1, 22, 333, 4444, 55555, ... (Sloane’s A000461;Marimutha 1997), 2. The concatenation of the first nF IBONACCI NUMBERS : 1, 11, 112, 1123, 11235, ... (Sloane’s A019523; Marimutha 1997),3. The smallest number that is the sum of squares oftwodistinct earlier terms: 1, 2, 5, 26, 29, 677, ... (Sloane’s A008318, Bencze 1997), 4. The smallest number that is the sum of squares of any number of distinct earlier terms: 1, 1, 2, 4, 5,6, 16, 17, ... (Sloane’s A008319, Bencze 1997), 5. The smallest number that is notthe sum of squares of twodistinct earlier terms: 1, 2, 3, 4, 6, 7, 8, 9, 11, ... (Sloane’s A008320, Bencze 1997), 6. The smallest number that is notthe sum of squares of any number of distinct earlier terms: 1, 2, 3, 6, 7, 8, 11, ... (Sloane’s A008321, Bencze 1997), 7. The smallest number that is a sum of cubes of two distinct earlier terms: 1, 2, 9, 730, 737, ... (Sloane’s A008322, Bencze 1997), 8. The smallest number that is a sum of cubes of any number of distinct earlier terms: 1, 1, 2, 8, 9, 10, 512, 513, 514, ... (Sloane’s A019511, Bencze1997), 9. The smallest number that is nota sum of cubes oftwodistinct earlier terms: 1, 2, 3, 4, 5, 6, 7, 8, 10, ... (Sloane’s A031980, Bencze 1997), 10. The smallest number that is nota sum of cubes of any number of distinct earlier terms: 1, 2, 3, 4, 5, 6, 7, 10, 11, ... (Sloane’s A031981, Bencze 1997), 11. The number of PARTITIONS of a number n/C301, 2, ... into SQUARE NUMBERS :1 ,1 ,1 ,1 ,2 ,2 ,2 ,2 ,3 ,4 , 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, ... (Sloane’s A001156, Iacobescu 1997), 12. The number of PARTITIONS of a number n/C301, 2, ... into CUBIC NUMBERS :1 ,1 ,1 ,1 ,1 ,1 ,1 ,1 ,2 ,2 , 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, ... (Sloane’s A003108, Iacobescu 1997), 13. Two copies of the first nPOSITIVE INTEGERS : 11, 1212, 123123, 12341234, ... (Sloane’s A019524, Iacobescu 1997), 14. Numbers written in base of triangular num- bers: 1, 2, 10, 11, 12, 100, 101, 102, 110, 1000, 1001, 1002, ... (Sloane’s A000462, Iacobescu 1997), 15. Numbers written in base of double factorial numbers: 1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, ... (Sloane’s A019513, Iacobescu 1997), 16. Sequences starting with terms a1;a2 fg which contain no three-term arithmetic progressions starting with f1;2g: 1, 2, 4, 5, 10, 11, 13, 14, 28, ... (Sloane’s A003278, Iacobescu 1997, Mudge 1997,Weisstein), 17. Numbers OF THE FORM fn!g2/C271 : 2, 5, 37, 577, 14401, 518401, 25401601, 1625702401, 131681894401, ... (Sloane’s A020549, Iacobescu 1997), 18. Numbers OF THE FORM fn!g3/C271 : 2, 9, 217, 13825, 1728001, 373248001, 128024064001, ... (Sloane’s A019514, Iacobescu 1997), 19. Numbers OF THE FORM 1/C271!2!3! /C1/C1/C1n! : 2, 3, 13, 289, 34561, 24883201, 125411328001, 5056584744960001, ... (Sloane’s A019515, Iaco- bescu 1997), 20. Sequences starting with terms a1;a2 fg which contain no three-term geometric progressions starting with f1;2g: 1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, ... (Sloane’s A000452, Iacobescu 1997), 21. Numbers repeating the digit 1 pntimes, where pnis the nth prime: 11, 111, 11111, 1111111, ... (Sloane’s A031974, Iacobescu 1997). These are a subset of the REPUNITS , 22. Integers with all 2s, 3s, 5s, and 7s (prime digits) removed: 1, 4, 6, 8, 9, 10, 11, 1, 1, 14, 1, 16, 1, 18, 19, 0, ... (Sloane’s A019516, Iacobescu 1997), 23. Integers with all 0s, 1s, 4s, and 9s (square digits) removed: 2, 3, 5, 6, 7, 8, 2, 3, 5, 6, 7, 8, 2, 2, 22, 23, ... (Sloane’s A031976, Iacobescu 1997). 24. (Smarandache-Fibonacci triples) Integers n such that S(n)/C30S(n/C281)/C27S(n/C282);where S(k)i s the S MARANDACHE FUNCTION : 3, 11, 121, 4902, 26245, ... (Sloane’s A015047; Aschbacher and Mudge 1995; Ibstedt 1997, pp. 19 /C1/23; Begay 1997). The largest known is 19,448,047,080,036, 25. (Smarandache-Radu triplets) Integers nsuch that there are no primes between the smaller and larger of S(n) and S(n/C271) : 224, 2057, 265225, ... (Sloane’s A015048; Radu 1994/1995, Begay 1997,Ibstedt 1997). The largest known is270,329,975,921,205,253,634,707,051,822,848,570- ,391,313, 26. (Smarandache crescendo sequence): Integers obtained by concatenating strings of the first n/C271 integers for n/C300, 1, 2, ...: 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, ... (Sloane’s A002260; Brown 1997, Brown andCastillo 1997). The nth term is given by n/C28m(m/C27 1)=2/C271;where m/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8n/C271p /C281YrvYru =2YrDYrE ;with xbc the FLOOR FUNCTION (Hamel 1997), 27. (Smarandache descrescendo sequence): Inte- gers obtained by concatenating strings of the first nintegers for n/C30...;2, 1: 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, ... (Sloane’s A004736; Smarandache 1997, Brown 1997), 28. (Smarandache crescendo pyramidal sequence, a.k.a. Smarandache descrescendo symmetric se- quence): Integers obtained by concatenatingstrings of rising and falling integers: 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, ... (Sloane’s A004737; Brown 1997, Brown and Castillo 1997, Smaran-dache 1997), 29. (Smarandache descrescendo pyramidal se- quence): Integers obtained by concatenating strings of falling and rising integers: 1, 2, 1, 2, 3,2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, ... (Sloane’s A004738; Brown 1997), 30. (Smarandache crescendo symmetric sequence): 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 2, 1, ... (Sloane’s A004739, Brown 1997, Smarandache 1997), 31. (Smarandache permutation sequence): Num- bers obtained by concatenating sequences of in-creasing length of increasing ODD NUMBERS and decreasing EVEN NUMBERS :1 ,2 ,1 ,3 ,4 ,2 ,1 ,3 ,5 ,6 , 4, 2, ... (Sloane’s A004741; Brown 1997, Brown andCastillo 1997), 32. (Smarandache pierced chain sequence): Num- bers OF THE FORM c(n)/C30101 0101|fflffl{zfflffl}/C1/C1/C10101|fflffl{zfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} n forn/C300, 1, ...: 101, 1010101, 10101010101, ... (Sloane’s A031982; Ashbacher 1997). In addition, c(n)=101 contains no PRIMES (Ashbacher 1997), 33. (Smarandache symmetric sequence): 1, 11, 121, 1221, 12321, 123321, ... (Sloane’s A007907; Smar- andache 1993, Dumitrescu and Seleacu 1994, sequence 3; Mudge 1995), 34. (Smarandache square-digital sequence): square numbers all of whose digits are also squares: 1, 4, 9, 49, 100, 144, ... (Sloane’sA019544; Mudge 1997), 35. (Square-digits): numbers composed of digits which are squares: 0, 1, 4, 9, 10, 11, 14, 19, 40, 41, ... (Sloane’s A046030), 36. (Cube-digits): numbers composed of digits which are cubes: 1, 8, 10, 11, 18, 80, 81, 88, 100, 101, ... (Sloane’s A046031), 37. (Smarandache cube-digital sequence): cube- digit numbers which are themselves cubes: 1, 8, 1000, 8000, 1000000, ... (Sloane’s A019545; Mudge 1997), 38. (Prime-digits): numbers composed of digits which are primes: 2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, ... (Sloane’s A046034), 39. (Smarandache prime-digital sequence): prime- digit numbers which are themselves prime: 2, 3, 5, 7, 23, 37, 53, ... (Sloane’s A019546; Smith 1996, Mudge 1997). 40. (Smarandache deconstructive sequence): inte- gers constructed by sequentially repeating the digits 1 /C1/9 in the following way: 1, 23, 456, 7891, 23456, 789123, 4567891, ... (Sloane’s A007923; Smarandache 1993, Kashihara 1996, Ashbacher, Atanassov 1999ab). Of these, 23, 4567891, 23456789, 1234567891, ... (Sloane’s A050234) are prime (Kashihara 1996, Ashbacher). See also ADDITION CHAIN ,C ONSECUTIVE NUMBER SEQUENCES ,CUBIC NUMBER ,EUCLID NUMBER ,EVEN NUMBER ,FIBONACCI NUMBER ,INTEGER SEQUENCE , ODD NUMBER ,PARTITION ,SMARANDACHE FUNCTION , SQUARE NUMBER References Ashbacher, C. "Some Problems Concerning the Smaran- dache Deconstructive Sequence." J. Recr. Math. 29,82/C1/ 84, 1998. Ashbacher, C. Collection of Problems On Smarandache Notions. Vail, AZ: Erhus University Press, 1996. Ashbacher, C. Pluckings from the Tree of Smarandache Sequences and Functions. Lupton, AZ: American Re- search Press, 1998. Aschbacher, C. and Mudge, M. Personal Computer World. pp. 302, Oct. 1995. Atanassov, K. "On the 4th Smarandache Problem." Notes on Number Theory and Discrete Mathematics (Sophia, Bul- garia) 5,33/C1/35, 1999. Atanassov, K. T. On Some of the Smarandache’s Problems. Lupton, AZ: American Research Press, pp. 16 /C1/21, 1999. Begay, A. "Smarandache Ceil Functions." Bull. Pure Appl. Sci. 16E, 227 /C1/229, 1997. Bencze, M. "Smarandache Recurrence Type Sequences." Bull. Pure Appl. Sci. 16E, 231 /C1/236, 1997. Bencze, M. and Tutescu, L. (Eds.). Some Notions and Questions in Number Theory, Vol. 2. http://www.gallu- p.unm.edu/~smarandache/SNAQINT2.TXT. Brown, J. "Crescendo & Descrescendo." In Richard Henry Wilde: An Anthology in Memoriam (1789 /C1/1847) (Ed. M. Myers). Bristol, IN: Bristol Banner Books, p. 19, 1997. Brown, J. and Castillo, J. "Problem 4619." School Sci. Math. 97, 221 /C1/222, 1997. Dumitrescu, C. and Seleacu, V. (Eds.). Some Notions and Questions in Number Theory, 4th ed. Glendale, AZ: Erhus University Press, 1994. http://www.gallup.unm.edu/ ~smarandache/SNAQINT.TXT. Dumitrescu, C. and Seleacu, V. (Eds.). Proceedings of the First International Conference on Smarandache Type Notions in Number Theory. Lupton, AZ: American Re- search Press, 1997. Hamel, E. Solution to Problem 4619. School Sci. Math. 97, 221 /C1/222, 1997. Iacobescu, F. "Smarandache Partition Type and Other Sequences." Bull. Pure Appl. Sci. 16E, 237 /C1/240, 1997.Ibstedt, H. Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics. Lupton, AZ: Erhus University Press, 1997. Kashihara, K. Comments and Topics on Smarandache Notions and Problems. Vail, AZ: Erhus University Press, 1996. Mudge, M. "Top of the Class." Personal Computer World, 674 /C1/675, June 1995. Mudge, M. "Not Numerology but Numeralogy!" Personal Computer World, 279 /C1/280, 1997. Programs and the Abstracts of the First International Conference on Smarandache Notions in Number Theory. Craiova, Romania, Aug. 21 /C1/23, 1997. Radu, I. M. Mathematical Spectrum 27, 43, 1994/1995. Rivera, C. "Problems & Puzzles: Puzzle Primes by Listing.- 008." http://www.primepuzzles.net/puzzles/puzz_008.htm. Sloane, N. J. A. Sequences A000452, A000461, A000462, A001156/M0221, A002260, A003108/M0209, A003278/ M0975, A004736, A004737, A004738, A004739, A004741, A007907, A008318, A008319, A008320, A008321, A008322, A015047, A015048, A019524, A019511, A019513, A019514, A019515, A019516, A019523, A019544, A019545, A019546 A020549, A031974, A031976, A031980, A031981, A031982, A046030, A046031, A046034, and A050234 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Smarandache, F. "Properties of the Numbers." Tempe, AZ: Arizona State University Special Collection, 1975. Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993. Smarandache, F. Collected Papers, Vol. 2. Kishinev, Mol- dova: Kishinev University Press, 1997. Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101/C1/107, 1996. Smarandache-Kurepa Function Given the sum-of-factorials function X (n)/C30Xn k/C301k!; /SK(p) for pPRIME is the smallest integer nsuch that pj1/C27a(n/C281):The first few known values of SK( p) are 2, 4, 6, 6, 5, 7, 7, 12, 22, 16, 55, 54, 42, 24, ... for p/C302, 5, 7, 11, 17, 19, 23, 31, 37, 41, 61, 71, 73, 89, .... The function SK( p) doe not exists for p/C303, 13, 29, 43, 47, 53, 67, 79, 83, .... See also PSEUDOSMARANDACHE FUNCTION ,SMARAN- DACHE CEIL FUNCTION ,S MARANDACHE FUNCTION , SMARANDACHE- WAGSTAFF FUNCTION ,SMARANDACHE FUNCTION References Ashbacher, C. "Some Properties of the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Math. Infor- matics Quart. 7, 114/C1/116, 1997. Mudge, M. "Introducing the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Smarandache No- tions J. 7,5 2/C1/53, 1996. Mudge, M. "Introducing the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996. Smarandache-Wagstaff Function Given the sum-of- FACTORIALS function X (n) /C30Xn k /C301k!; /SW(p) is the smallest integer for p PRIME such that a[SW(p)] is divisible by p.Ifp¶ a(n) for all n Bp, then p never divides any sum for all n. Therefore, the values SW(p) do not exist for 2, 5, 7, 13, 19, 31, ... (Sloane’s A056985). The function is defined for p /C303, 11, 17, 23, 29, 37, 41, 43, 53, 67, 73, 79, 97, ... (Sloane’s A056983), with corresponding values 2, 4, 5, 12, 19, 24, 32, 19, 20, 20, 20, 7, 57, 6, ... (Sloane’s A056985). See also FACTORIAL ,SMARANDACHE FUNCTION References Ashbacher, C. "Some Properties of the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Math. Infor- matics Quart. 7, 114 /C1/116, 1997. "Functions in Number Theory." http://www.gallup.unm.edu/ ~smarandache/FUNCT1.TXT. Mudge, M. "Introducing the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Smarandache No- tions J. 7,52/C1/53, 1996. Mudge, M. "Introducing the Smarandache-Kurepa and Smarandache-Wagstaff Functions." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996. Sloane, N. J. A. Sequences A056983, A056984, and A056985 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Smith Brothers Consecutive SMITH NUMBERS . The first few Smith brothers are (728, 729), (2964, 2965), (3864, 3865), (4959, 4960), ... (Sloane’s A050219 and A050220). See also SMITH NUMBER References Sloane, N. J. A. Sequences A050219 and A050220 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Smith Conjecture The set of fixed points which do not move as a KNOT is transformed into itself is not a KNOT . The conjecture was proved in 1978 (Morgan and Bass 1984). Accord- ing to Morgan and Bass (1984), the Smith conjecture stands in the first rank of mathematical problemswhen measured by the amount and depth of newmathematics required to solve it. The generalized Smith conjecture states considers S n/C282to be a piecewise linear ( n/C282)/-dimensional sphere in Sn;andMnthek-fold cyclic covering of Sn branched along Sn/C282;and asks if Sn/C282is unknotted if Mnis anSn(Hartley 1983). This conjecture is true forn53;and false for n]4;with counterexamples in the latter case provided by Giffen (1966), Gordon (1974), and Sumners (1975). References Giffen, C. H. "The Generalized Smith Conjecture." Amer. J. Math. 88, 187/C1/198, 1966. Gordon, C. M. "On the Higher-Dimensional Smith Conjec- ture." Proc. London Math. Soc. 29,9 8/C1/110, 1974. Hartley, R. "Whitehead Torsion and the Smith Conjecture." Michigan Math. J. 30, 121/C1/128, 1983. Morgan, J. W. and Bass, H. (Eds.). The Smith Conjecture, Papers Presented at the Symposium Held at Columbia University, New York, 1979. Orlando, FL: Academic Press, 1984. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 350 /C1/351, 1976. Smith, P. A. "Transformations of Finite Period. II." Ann. Math. 40, 690/C1/711, 1939. Summers, D. W. "Smooth ZpActions on Spheres which Leave Knots Pointwise Fixed." Trans. Amer. Math. Soc. 205, 193/C1/203, 1975. Waldhausen, F. "U ¨ber Involutionen der 3-Spha ¨re." Topology 8,8 1/C1/91, 1969. Smith Normal Form LetAbe an n/C29nMATRIX over a FIELD F. Using the three ELEMENTARY ROW AND COLUMN OPERATIONS over elements in the field, the n/C29nmatrix xI/C28A with entries from F[x] can be put into the diagonal form 10 /C1/C1/C1 00 000 01:::00 000 n::::::::::::::::::n 0 001 0 0 0 0 0 000 a1(x)00 0 0 000 0 a2(x)0 0 n::::::::::::::::::n 0 000 0 0 0 am(x)2 666666666643 77777777775: called the Smith normal form, which that a 1(x);a2(x); ...,am(x) are monic nonzero elements of F[x] with degrees at least one and satisfying a1(x)/C2 a2(x) jj ...am(x) jj (Dummit and Foote 1998, pp. 390 /C1/ 391 and 414). The elements ai(x) are then called the INVARIANT FACTORS ofA:/ References Ayres, F. Jr. "Smith Normal Form." Ch. 24 in Theory and Problems of Matrices. New York: Schaum, pp. 188 /C1/195, 1962. Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. Jabon, D. "Smith Normal Forms." http://www.mathsource.- com/cgi-bin/msitem?0207 /C1/470. Smith Number ACOMPOSITE NUMBER the SUM of whose DIGITS is the sum of the DIGITS of its PRIME FACTORS (excluding 1). (The PRIMES are excluded since they trivially satisfy this condition). One example of a Smith number is the BEAST NUMBER 666 /C302 /C215 3 /C215 3 /C215 37 ; since 6 /C276 /C276 /C302 /C273 /C273 /C27(3 /C277) /C3018 : Another Smith number is 4937775 /C303 /C215 5 /C215 5 /C215 65837 ; since 4 /C279 /C273 /C277 /C277 /C277 /C275 /C303 /C275 /C275 /C27(6 /C275 /C278 /C273 /C277) /C3042: The first few Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, ... (Sloane’s A006753). The corresponding digits sums are 4, 4, 9, 13, 13, 13, 4, 13, 4, 13, 13, 13, 13, ... (Sloane’s A050218) McDaniel (1987a) showed that there are an infinite number of Smith numbers. A generalized k-Smith number can also be defined as a number m satisfying Sp(m) /C30kS(m) ; where Sp(m)is the sum of the digits of m’s prime factors and S(m)is the usual sum of m’s digits. The following table gives the first few k-Smith numbers for k ]2:/ k Sloane k-Smith numbers 2 A050224 88, 169, 286, 484, 598, 682, 808, 844, 897, ... 3 A050225 6969, 19998, 36399, 39693, 66099, 69663, ... A Smith number can be constructed from every factored REPUNIT Rn(Hoffman 1998, pp. 205 /C1/206). The largest known Smith number is 9 /C29R1031104594 /C273 /C29102297 /C271YrvYru1476/C29103913210 : See also HOAX NUMBER ,M ONICA SET,P ERFECT NUMBER ,REPUNIT ,SMITH BROTHERS ,SUZANNE SET References Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Free- man, pp. 99 /C1/100, 1989. Guy, R. K. "Smith Numbers." §B49 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 103 /C1/104, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 205 /C1/206, 1998. McDaniel, W. L. "The Existence of Infinitely Many k-Smith Numbers." Fib. Quart. , 25,76/C1/80, 1987a. McDaniel, W. L. "Powerful K-Smith Numbers." Fib. Quart. 25, 225 /C1/228, 1987b. Oltikar, S. and Weiland, K. "Construction of Smith Num- bers." Math. Mag. 56,36/C1/37, 1983.Sloane, N. J. A. Sequences A006753/M3582, A050218, A050224, and A050225 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Wilansky, A. "Smith Numbers." Two-Year College Math. J. 13, 21, 1982. Yates, S. "Special Sets of Smith Numbers." Math. Mag. 59, 293 /C1/296, 1986. Yates, S. "Smith Numbers Congruent to 4 (mod 9)." J. Recr. Math. 19, 139 /C1/141, 1987. Smith’s Markov Process Theorem Consider P2y1 ; t1 jy3 ; t3 ðÞ /C30g P2 y1 ; t1 y2 ; t2 j ÞP3y1 ; t1; y2 ; t2 y3 ; t3 j Þ dy2 : ð ð (1) If the probability distribution is governed by a MARKOV PROCESS , then P3y1 ; t1; y2 ; t2 y3 ; t3 j Þ/C30P2y2 ; t2 y3 ; t3 j Þ ð ð /C30P2y2 y3 ; t3 /C28t2 j Þ: ð (2) Assuming no time dependence, so t1 /C130; P2y1 jy3 ; t3 ðÞ /C30g P2 y1 y2 ;t2 j ÞP2y2 y3 ; t3 /C28t2 j Þ dy2 : ð ð (3) See also MARKOV PROCESS Smith’s Network Theorem In a NETWORK with three EDGES at each VERTEX , the number of HAMILTONIAN CIRCUITS through a specified EDGE is 0 or EVEN . See also EDGE (GRAPH ), HAMILTONIAN CIRCUIT ,NET- WORK Smooth Function A smooth function is a function that has continuous second-order derivatives over some domain. A func- tion can therefore be said to be smooth over a restricted interval such as (a, b)or[ a, b]. See also CONTINUOUS FUNCTION ,DERIVATIVE Smooth Manifold Another word for a C/C12(infinitely differentiable) MANIFOLD . A smooth manifold is a TOPOLOGICAL MANIFOLD together with its "functional structure" (Bredon 1995) and so differs from a TOPOLOGICAL MANIFOLD because the notion of differentiability exists on it. Every smooth manifold is a TOPOLOGICAL MANIFOLD , but not necessarily vice versa. (The first nonsmooth TOPOLOGICAL MANIFOLD occurs in 4-D.) In 1959, Milnor showed that a 7-D HYPERSPHERE can be made into a smooth manifold in 28 ways. See also DIFFERENTIABLE MANIFOLD ,HYPERSPHERE , MANIFOLD ,TOPOLOGICAL MANIFOLD References Bredon, G. E. Topology & Geometry. New York: Springer- Verlag, p. 69, 1995. Smooth Number An INTEGER is k-smooth if it has no PRIME FACTORS > k: The following table gives the first few k-smooth numbers for small k. Berndt (1994, p. 52) called the 7-smooth numbers "highly composite numbers." k Sloane k-smooth numbers 2 A000079 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... 3 A003586 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ... 5 A051037 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, ... 7 A002473 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, ... 11 A051038 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, ... The probability that a random POSITIVE INTEGER 5n is k-smooth is c(n; k) =n; where c(n; k) is the number of k-smooth numbers 5n: This fact is important in application of Kraitchik’s extension of FERMAT’S FACTORIZATION METHOD because it is related to the number of random numbers which must be examined to find a suitable subset whose product is a square. Since about p(k) k-smooth numbers must be found (where p(k) is the PRIME COUNTING FUNCTION ), the number of random numbers which must be examined is about p(k)n=c(n; k) : But because it takes about p(k) steps to determine if a number is k-smooth using TRIAL DIVISION , the expected number of steps needed to find a subset of numbers whose product is a square is /C2[ p(k)]2n=c(n; k) (Pomerance 1996). Canfield et al. (1983) showed that this function is minimized when k /C2exp1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln n ln ln npYru*Yru+ and that the minimum value is about exp 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln n ln ln npYru*Yru+ : In the CONTINUED FRACTION FACTORIZATION ALGO- RITHM , n can be taken as 2ffiffiffinp; but in FERMAT’S FACTORIZATION METHOD ,itis n1 =2 /C27 e : k is an estimate for the largest PRIME in the FACTOR BASE (Pomerance 1996). See also HIGHLY COMPOSITE NUMBER ,ROUND NUM- BERReferences Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994. Blecksmith, R.; McCallum, M.; and Selfridge, J. L. " 3- Smooth Representations of Integers." Amer. Math. Monthly 105, 529 /C1/543, 1998. Canfield, E. R.; Erdos, P.; and Pomerance, C. "On a Problem of Oppenheim Concerning ‘Factorisation Numerorum."’ J. Number Th. 17,1/C1/28, 1983. Mintz, D. J. "2, 3 Sequence as a Binary Mixture." Fib. Quart. 19, 351 /C1/360, 1981. Pomerance, C. "On the Role of Smooth Numbers in Number Theoretic Algorithms." In Proc. Internat. Congr. Math., Zu¨rich, Switzerland, 1994, Vol. 1 (Ed. S. D. Chatterji). Basel: Birkha ¨user, pp. 411 /C1/422, 1995. Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473 /C1/1485, 1996. Ramanujan, S. Collected Papers (Ed. G. H. Hardy et al. ) New York: Chelsea, p. xxiv, 1962. Sloane, N. J. A. Sequences A000079/M1129, A002473/ M0477, A003586, A051037, and A051038 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Smooth Structure A smooth structure on a TOPOLOGICAL MANIFOLD (also called a differentiable structure) is given by a smooth ATLAS of coordinate charts, i.e., the TRANSITION FUNCTIONS between the coordinate charts are C /C12 smooth. A manifold with a smooth structure is called a DIFFERENTIABLE MANIFOLD or a SMOOTH MANIFOLD . A smooth structure is used to define DIFFERENTIA- BILITY for real-valued functions on a manifold. This extends to a notion of when a map between two differentiable manifolds is smooth, and naturally to the definition of a DIFFEOMORPHISM . In addition, the smooth structure is used to define TANGENT VECTORS , the collection of which is the TANGENT BUNDLE . Two smooth structures are considered equivalent ifthere is a HOMEOMORPHISM of the manifold which pulls back one atlas to an atlas compatible to theother one, i.e., a DIFFEOMORPHISM . For instance, any two smooth structures on the circle S1are equivalent, as can be seen by integration. It is surprising that some manifolds admit more than one smooth structure. The first such example was an EXOTIC SPHERE ofS7;the 7-dimensional HYPER- SPHERE , found by Milnor (1956) using the calculus ofOCTONIONS . In the 1980s, several mathematicians, including Casson, Freedman, and Donaldson, showedthat 4-dimensional Euclidean space R 4has smooth structures that are distinct from the standard struc- ture. These are called EXOTIC R4, and some of their techniques involve D ONALDSON THEORY . Another approach to smooth structures is through SHEAF theory. Notice that a coordinate chart for an n- dimensional manifold is really an ordered collection of ncontinuous functions. Whenever two coordinate charts overlap on the manifold, the functions from one chart are infinitely differentiable with respect to those from the other chart. The collection of compa- tible real-valued continuous functions defines the sheaf of smooth functions. Conversely, one can define a smooth structure to be defined by a subsheaf of continuous functions which satisfies the mutually differentiable condition. See also ATLAS ,DIFFEOMORPHISM DONALDSON THEO- RY,EXOTIC R4,EXOTIC SPHERE ,M ANIFOLD ,O CTO- NION ,S HEAF (TOPOLOGY ), SMOOTH FUNCTION , SMOOTH SURFACE ,TANGENT BUNDLE ,TANGENT VEC- TOR (MANIFOLD ) References Milnor, J. "Topological Manifolds and Smooth Manifolds." In Proc. Internat. Congr. Mathematicians (Stockholm, 1962). Djursholm: Inst. Mittag-Leffler, pp. 132 /C1/138, 1963. Smooth Surface A surface PARAMETERIZED in variables u and v is called smooth if the TANGENT VECTORS in the u and v directions satisfy Tu /C27Tv "0 ; where A /C29B is a CROSS PRODUCT . Smoothing The modification of a set of data to make it smooth and nearly continuous and remove or diminish out- lying points. See also MOVING AVERAGE ,SAVITZKY- GOLAY FILTER References Lanczos, C. "Trigonometric Interpolation of Empirical and Analytic Functions." J. Math. Phys. 17, 123, 1938. Rhodes, E. C. Tract on Smoothing. No. 6 in Tracts for Computers (Ed. K. Pearson). London: Cambridge Univer- sity Press, 1921. Whittaker, E. T. and Robinson, G. "Graduation, or the Smoothing of Data." Ch. 11 in The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 285 /C1/316, 1967. sn JACOBI ELLIPTIC FUNCTIONS Snake A simple circuit in the d-HYPERCUBE which has no chords (i.e., for which all snake edges are edges of the HYPERCUBE ). Klee (1970) asked for the maximum length s(d)ofa d-snake. Klee (1970) gave the bounds 7 4(d /C28 1) 5s(d) 2d1 2 /C281 /C27 12=2 /C28d 7d(d /C28 1)2 /C27 2(1) for d ]6 (Danzer and Klee 1967, Douglas 1969), as well as numerous references. Abbott and Katchalski (1988) show s(d) ]77 /C215 2d/C288 ; (2)and Snevily (1994) showed that s(d) 52d /C2811 /C281 20d /C28 41 ! (3) for d 512 ; and conjectured s(d) 53 /C215 2d/C283 /C272 (4) for d 55: The first few values for s(d) for d /C301, 2, ..., are 2, 4, 6, 8, 14, 26, ... (Sloane’s A000937). See also HYPERCUBE References Abbott, H. L. and Katchalski, M. "On the Snake in the Box Problem." J. Combin. Th. Ser. B 44,12/C1/24, 1988. Danzer, L. and Klee, V. "Length of Snakes in Boxes." J. Combin. Th. 2, 258 /C1/265, 1967. Douglas, R. J. "Some Results on the Maximum Length of Circuits of Spread k in the d-Cube." J. Combin. Th. 6, 323 /C1/339, 1969. Evdokimov, A. A. "Maximal Length of a Chain in a Unit n- Dimensional Cube." Mat. Zametki 6, 309 /C1/319, 1969. Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/909, 1989. Guy, R. K. "Monthly Unsolved Problems." Amer. Math. Monthly 94, 961 /C1/970, 1989. Kautz, W. H. "Unit-Distance Error-Checking Codes." IRE Trans. Elect. Comput. 7, 177 /C1/180, 1958. Klee, V. "What is the Maximum Length of a d-Dimensional Snake?" Amer. Math. Monthly 77,63/C1/65, 1970. Sloane, N. J. A. Sequences A000937/M0995 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Snevily, H. S. "The Snake-in-the-Box Problem: A New Upper Bound." Disc. Math. 133, 307 /C1/314, 1994. Snake Eyes A roll of two 1s (the lowest roll possible) on a pair of six-sided DICE. The probability of rolling snake eyes is /1=36/, or 2.777...%. See also BOXCARS ,DICE Snake Oil Method The expansion of the two sides of a sum equality in terms of POLYNOMIALS inxmandyk;followed by closed form summation in terms of xandy. For an example of the technique, see Bloom (1995). References Bhatnagar, G. "A Multivariable View of One-Variable q- Series." In Special Functions and Differential Equations. Proceedings of the Workshop (WSSF97) held in Madras, January 13 /C1/24, 1997) (Ed. K. S. Rao, R. Jagannathan, G. van den Berghe, and J. Van der Jeugt). New Delhi,India: Allied Pub., pp. 60 /C1 /72, 1998. Bloom, D. M. "A Semi-Unfriendly Identity." Problem 10206. Solution by R. J. Chapman. Amer. Math. Monthly 102, 657 /C1/658, 1995. Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1993. Snake Polyiamond A6- POLYIAMOND . References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Snedecor’s F-Distribution If a random variable X has a CHI-SQUARED DISTRIBU- TION with m degrees of freedom / x2 mðÞ and a random variable Y has a CHI-SQUARED DISTRIBUTION with n degrees of freedom / x2 nðÞ ; and X and Y are indepen- dent, then F /C13X =m Y =n (1) is distributed as Snedecor’s F-distribution with m and n degrees of freedom f(F(m; n)) /C30Gm/C27n 2Yru*Yru+ m nYru*Yru+m=2 F(m/C282)=2 Gm 2Yru*Yru+ Gn 2Yru*Yru+ 1 /C27m nFYru*Yru+(m/C272)=2 (2) for 0 BF B/C12 : The RAW MOMENTS are m?1 /C30n n /C28 2 (3) m?2 /C30n2(m /C27 2) m(n /C28 2)(n /C28 4) (4) m?3 /C30n3(m /C27 2)(m /C27 4) m2(n /C28 2)(n /C28 4)(n /C28 6) (5) m ?4 /C30n4(m /C27 2)(m /C27 4)(m /C27 6) m3(n /C28 2)(n /C28 4)(n /C28 6)(n /C28 8) ; (6) so the CENTRAL MOMENTS are given by m2 /C302n2(m /C27 n /C28 2) m(n /C28 2)2(n /C28 4) (7) m3 /C308n3(m /C27 n /C28 2)(2m /C27 n /C28 2) m2(n /C28 2)3(n /C28 4)(n /C28 6) (8)m4 /C30 12n4(m /C27 n /C28 2) 4(n /C28 2)2 /C27 m2(n /C27 10) /C27 m(n /C28 2)(n /C27 10)hi m3(n /C28 2)4(n /C28 4)(n /C28 6)(n /C28 8) : (9) and the MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C30 m?1 /C30n n /C28 2 (10) s2 /C302n2(m /C27 n /C28 2) m(n /C28 2)2(n /C28 4) (11) g1 /C30m3 s3 /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(n /C28 4) m(m /C27 n /C28 2)s 2m /C27 n /C28 2 n /C28 6(12) g2 /C30m4 s4 /C283 /C30 3(n /C28 4) 4(n /C28 2)2 /C27 m2(n /C27 10) /C27 m(n /C28 2)(n /C27 10)hi m(m /C27 n /C28 2)(n /C28 6)(n /C28 8) : (13) The CHARACTERISTIC FUNCTION can be computed, but it is rather messy and involves the GENERALIZED HYPERGEOMETRIC FUNCTION /3F2(a ; b ; c; d ; e; z)/. Letting w /C13mF n 1 /C27mF n(14) gives a BETA DISTRIBUTION (Beyer 1987, p. 536). See also BETA DISTRIBUTION ,CHI-SQUARED DISTRIBU- TION ,STUDENT’S T-DISTRIBUTION References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 536, 1987. Snellius-Pothenot Problem A SURVEYING PROBLEM which asks: Determine the position of an unknown accessible point P by its bearings from three inaccessible known points A,B, andC. See also HANSEN’S PROBLEM References Do¨rrie, H. "Annex to a Survey." §40 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 193 /C1/197, 1965. Snowflake EXTERIOR SNOWFLAKE ,KOCH ANTISNOWFLAKE ,KOCH SNOWFLAKE ,PENTAFLAKE Snub Cube The 38-faced ARCHIMEDEAN SOLID A7 ; also called the SNUB CUBOCTAHEDRON , whose faces are 32 f3g/C276f4g: It has two ENANTIOMERS .Itis UNIFORM POLYHEDRON U12and Wenninger model W17 : It has SCHLA ¨ FLI SYMBOL s 3 4fgand WYTHOFF SYMBOL 234: j / Its DUAL POLYHEDRON is the PENTAGONAL ICOSITE- TRAHEDRON . The INRADIUS r of the dual, MIDRADIUS r of the dual and solid, and CIRCUMRADIUS R for unit edge length are given by the unique positive real roots of the equations 896r6 /C281248 r4 /C2764r2 /C281 /C300 (1) 64 r6 /C28112r4 /C2720 r2 /C281 /C300 (2) 32R6 /C2880R4 /C2744R2 /C287 /C300 (3) which given by r /C301:157661791... (4) r /C301:247223168... (5) R /C301:3437133737446... (6) The SURFACE AREA of the snub cube of side length 1 is S /C306 /C278ffiffiffi 3p (7) and the VOLUME V is given by the positive real solution to the equation 729V6 /C2845684 V4 /C2719386 V2 /C2812842 /C300; (8) which is given approximately by V :7:88948 : (9) The distances from the center to the centroids of the triangular and square faces are given by the unique positive roots to the equations864r6 3/C281296 r43/C2736r23/C281/C300 (10) 32r64/C2832r44/C2812r24/C281/C300; (11) which are given by r3/C301:213355800 . . . (12) r4/C301:142613508 . . . (13) See also ARCHIMEDEAN SOLID,ICOSITETRAHEDRON , PENTAGONAL ICOSITETRAHEDRON ,SNUB DODECAHE- DRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 139, 1987. Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401/C1/450, 1954. Cundy, H. and Rollett, A. "Snub Cube. 34:4:/"§3.7.7 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 107, 1989. Wenninger, M. J. "The Snub Cube." Model 17 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 31, 1989. Snub Cube-Pentagonal Icositetrahedron Compound The compound of the SNUB CUBE and its dual, the PENTAGONAL ICOSITETRAHEDRON . It can be con- structed from the snub cube with unit edge length by heights h3andh4;given by the unique positive real roots of 3456 h6 3/C28864h43/C27216h23/C281/C300 (1) 128h64/C2796h44/C2716h24/C281/C300: (2) The corresponding solid has edge lengths 128s61/C2864s41/C2716s21/C281/C300 (3) s2/C301=2 (4) 128s63/C286s33/C281/C300; (5) and s4/C301 2ffiffiffi 2p ; (6) where s1and s3are unique real roots of the above polynomials. The CIRCUMRADIUS is given by the root of 32R6 /C2880R4 /C2744R2 /C287 /C300; (7) the SURFACE AREA by the root of 1028869776 /C2735418062592 S /C2845028405440 S2 /C2722712607360 S3 /C285396081328 S4 /C27463818960 S5 /C2735732664 S6 /C287379424 S7 /C2723652 S8 /C2729160 S9 /C28576S10 /C2836S11 /C27S12 ; (8) and VOLUME by the root of 128V6 /C288864 V4 /C2719152 V2 /C2810609 /C300 : (9) See also COMPOUND POLYHEDRON ,P ENTAGONAL ICOSITETRAHEDRON ,SNUB CUBE Snub Cuboctahedron SNUB CUBE Snub Disphenoid The 12-faced convex DELTAHEDRA also known as the SIAMESE DODECAHEDRON , which is also JOHNSON SOLID J84 :/ The coordinates of the VERTICES of a snub disphenoid of unit side length may be found by solving the set of four simultaneous equations 1 2Yru*Yru+2 /C27x2 2 /C27z21 /C301 x2 /C281 2Yru*Yru+2 /C27 z3 /C28z1 ðÞ2/C301 12Yru*Yru+2 /C27x2 2 /C27 z3 /C28z2 ðÞ2/C301 x22 /C27x22 /C27 z2 /C28z1 ðÞ2/C301 for the four unknowns x2 ; z1 ; z2 ; and z3 : The analytic solution requires solving the CUBIC EQUATION , and the solutions are given by the unique positive real roots of 2x32 /C283x22 /C282x2 /C272 /C300 (1) 32z61 /C2764z41 /C2822z21 /C281 /C300 (2) 16z62 /C278z42 /C2815z22 /C288 /C300 (3) 2z6 3 /C28z43 /C288z23 /C284 /C300: (4) Numerically, x2 :0:644584 z1 :0 :578369 z2 :0 :989492 z3 :1:56786 : The SURFACE AREA of the unit snub disphenoid is S/C303ffiffiffi 3p ; (5) and the VOLUME Vis given by the positive real root of 5832 V6/C281377 V4/C282160 V2/C284/C300; (6) approximately V:0:859494 :/ See also DELTAHEDRON ,DISPHENOID ,JOHNSON SOLID Snub Dodecadodecahedron The UNIFORM POLYHEDRON U40whose DUAL POLYHE- DRON is the MEDIAL PENTAGONAL HEXECONTAHEDRON . It has WYTHOFF SYMBOL ½25 2 5: Its faces are 1252no /C27 60 f3g/C2712 f5g: It has CIRCUMRADIUS for a /C301of R /C301:27443994 : See also SNUB CUBE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 139, 1987. Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401 /C1/450, 1954. Cundy, H. and Rollett, A. "Snub Dodecahedron. 34 :5:/" §3.7.13 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 114 /C1/115, 1989. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 174 /C1/176, 1971. Snub Dodecahedron The 92-faced ARCHIMEDEAN SOLID A8consisting of faces 80f3 g/C2712 f5g which is also called the snub icosidodecahedron. It is UNIFORM POLYHEDRON U29 and Wenninger model W18 : It has SCHLA ¨ FLI SYMBOL s 3 5Yr$Yr% and WYTHOFF SYMBOL ½235:/The DUAL POLYHEDRON of the snub dodecahedron is the PENTAGONAL HEXECONTAHEDRON . The INRADIUS r of the dual, MIDRADIUS r of the solid and dual, and CIRCUMRADIUS R of the solid for a /C301 are r /C302 :039873155... r /C302:097053835... R /C302:15583737511564 ... : See also ARCHIMEDEAN SOLID,H EXECONTAHEDRON , SNUB CUBE References Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401/C1/450, 1954. Wenninger, M. J. "The Snub Dodecahedron." Model 18 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 32, 1989. Snub Icosidodecadodecahedron The UNIFORM POLYHEDRON U46whose DUAL POLYHE- DRON is the MEDIAL HEXAGONAL HEXECONTAHEDRON . It has W YTHOFF SYMBOL j35 35:Its faces are 1242no /C27 80f3g/C2712f5g:It has CIRCUMRADIUS fora/C301o f R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24=3/C2814x/C2722=3x2 24=3/C288x/C2722=3x2s /C301:12689791279994 . . . ; where x/C3025/C273ffiffiffiffiffiffi 69pYru*Yru+1=3 : References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 177 /C1/178, 1971. Snub Icosidodecahedron SNUB DODECAHEDRON Snub Polyhedron A polyhedron with extra triangular faces, given by the SCHLA ¨ FLI SYMBOL sp qno :/ See also RHOMBIC POLYHEDRON ,TRUNCATED POLY- HEDRON Snub Square Antiprism JOHNSON SOLID J85:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . SO SPECIAL ORTHOGONAL GROUP Soap Bubble BUBBLE Soccer Ball TRUNCATED ICOSAHEDRON Sociable Numbers Numbers which result in a periodic ALIQUOT SE- QUENCE , where an ALIQUOT SEQUENCE is the sequence of numbers obtained by repeatedly applying the restricted divisor function s(n)/C30s(n)/C28n (1) ton. Here s(n) is the usual DIVISOR FUNCTION . If the period is 1, the number is called a PERFECT NUMBER . If the period is 2, the two numbers are called an AMICABLE PAIR . In general, if the period is t]3; the number is called sociable of order t. Only two sociable numbers were known prior to 1970, the sets of orders 5 and 28 discovered by Poulet (1918). In 1970, Cohen discovered nine groups of order 4. For example, 1264460 is a sociable number of order four since its ALIQUOT SEQUENCE is 1264460, 1547860, 1727636, 1305184, 1264460, .... The firstfew sociable numbers are 12496, 14316, 1264460,2115324, 2784580, 4938136, ... (Sloane’s A003416), which have orders 5, 28, 4, 4, 4, 4, ... (Sloane’s A052470). The table below summarizes the numbersof sociable cycles known as a function of order as given in the compilation by Moews (1995). order known 30 45 35162 82 91 28 1 total 60 Y. Kohmoto has considered a generalization of the sociable numbers defined according to the generalized ALIQUOT SEQUENCE a(n)/C30s(a(n/C281)) m: (2) MULTIPERFECT NUMBERS are fixed points of this mapping, since if a(n)/C30a(n/C281);then ma(n)/C30s(a(n)); (3) which is the definition of an m-multiperfect number. If the sequence a(n) becomes cyclic after k/C211 terms, it is then called an 1 =m/-sociable number of order k. IfMmandMnare distinct M ERSENNE PRIMES , then 1 2s2m/C281MnYrvYru /C30122m/C281 ðÞ 2n/C302n/C281Mm (4) 1 2s(2(n/C281)Mm)/C302m/C281Mn; (5) so 2m/C281Mnand 2n/C281Mmare /1=2/-sociable numbers of order 2. The following table summarizes the smallest mem- bers of the generalized 1 =m/-aliquot sequences of order k, found by Kohmoto. mk starting numbers 3 2 14913024 4 2 2096640, 4226880004 12 3396556800 See also ALIQUOT SEQUENCE ,C ATALAN’S ALIQUOT SEQUENCE CONJECTURE ,PERFECT NUMBER ,UNITARY SOCIABLE NUMBERS References Borho, W. "U¨ ber die Fixpunkte der k-fach iterierten Teiler- ersummenfunktion." Mitt. Math. Gesellsch. Hamburg 9, 34 /C1/48, 1969. Cohen, H. "On Amicable and Sociable Numbers." Math. Comput. 24, 423 /C1/429, 1970. Creyaufmu ¨ller, W. "Aliquot Sequences." http://home.t-onli- ne.de/home/Wolfgang.Creyaufmueller/aliquote.htm. Devitt, J. S.; Guy, R. K.; and Selfridge, J. L. Third Report on Aliquot Sequences, Congr. Numer. XVIII, Proc. 6th Man- itoba Conf. Numerical Math, pp. 177 /C1/204, 1976. Flammenkamp, A. "New Sociable Numbers." Math. Comput. 56, 871 /C1/873, 1991. Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math- ematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 160 /C1/171, 1978. Guy, R. K. "Aliquot Cycles or Sociable Numbers." §B7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 62 /C1/63, 1994. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 145 /C1/146, 1979. Moews, D. and Moews, P. C. "A Search for Aliquot Cycles Below 1010." Math. Comput. 57, 849 /C1/855, 1991. Moews, D. and Moews, P. C. "A Search for Aliquot Cycles and Amicable Pairs." Math. Comput. 61, 935 /C1/938, 1993. Moews, D. "A List of Aliquot Cycles of Length Greater than 2." Rev. Dec. 18, 1995. http://xraysgi.ims.uconn.edu/socia- ble.txt. Pedersen, J. A. M. "Tables of Aliquot Cycles." http:// www.vejlehs.dk/staff/jmp/aliquot/tables.htm. Poulet, P. Question 4865. L’interme ´d. des Math. 25, 100 /C1/ 101, 1918. Root, S. Item 61 in Beeler, M.; Gosper, R. W.; and Schroep- pel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelli- gence Laboratory, Memo AIM-239, p. 23, Feb. 1972. Sloane, N. J. A. Sequences A003416 and A052470 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. te Riele, H. J. J. "Perfect Numbers and Aliquot Sequences." In Computational Methods in Number Theory, Part I. (Ed. H. W. Lenstra Jr. and R. Tijdeman). Amsterdam, Nether- lands: Mathematisch Centrum, pp. 141 /C1/157, 1982. Weisstein, E. W. "Sociable and Amicable Numbers." MATH- EMATICA NOTEBOOK SOCIABLE.M . Social Choice Theory The theory of analyzing a decision between a collec- tion of alternatives made by a collection of n voters with separate opinions. Any choice for the entire group should reflect the desires of the individual voters to the extent possible. Fair choice procedures usually satisfy ANONYMITY (invariance under permutation of voters), DUALITY (each alternative receives equal weight for a single vote), and MONOTONICITY (a change favorable for X does not hurt X). Simple majority vote is anonymous, dual, and monotone. MAY’S THEOREM states a stron- ger result.See also ANONYMOUS ,A RROW’S PARADOX ,D UAL VOTING ,M AY’S THEOREM ,M ONOTONIC VOTING ,VOT- ING References Taylor, A. Mathematics and Politics: Strategy, Voting, Power, and Proof. New York: Springer-Verlag, 1995. Young, S. C.; Taylor, A. D.; and Zwicker, W. S. "Counting Quota Systems: A Combinatorial Question from Social Choice Theory." Math. Mag. 68, 331 /C1/342, 1995. Socle The socle of a group G is the SUBGROUP generated by its minimal NORMAL SUBGROUPS . For example, the SYMMETRIC GROUP S4has two nontrivial normal subgroups: A4 and N /C30ff1; 2; 3; 4g;/ /f2; 1; 4; 3g;f3 ; 4 ; 1 ; 2g;f4; 3; 2; 1gg: But A4con- tains N,so N is the only minimal subgroup, and the socle of S4 is N. See also BLOCK (GROUP ACTION ), GROUP ,N ORMAL SUBGROUP ,PRIMITIVE GROUP ,TRANSITIVE GROUP References Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Socrates’ Paradox Socrates is reported to have stated: "One thing I know is that I know nothing." See also LIAR’S PARADOX References Pickover, C. A. Keys to Infinity. New York: Wiley, p. 134, 1995. Soddy Centers SODDY POINTS Soddy Circles Given three distinct noncollinear points A,B, and C, let three CIRCLES be drawn, one centered about each point and each one tangent to the other two. Call the RADII ri(/r3/C30a?;r1/C30b?;r2/C30c?):Then the CIRCLES satisfy a?/C27b?/C30c (1) a?/C27c?/C30b (2) b?/C27c?/C30a; (3) as shown in the diagram below. Solving for the RADII then gives a?/C301 2(b/C27c/C28a) (4) b?/C3012(a/C27c/C28b) (5) c?/C3012(a/C27b/C28c): (6) The TRIANGLE illustrated above has sides a,b, and c, and SEMIPERIMETER s/C1312(a/C27b/C27c): (7) Plugging in, 2s/C30a?/C27b? ðÞ /C27a?/C27c? ðÞ /C27b?/C27c? ðÞ /C302a?/C27b?/C27c? ðÞ ;(8) giving a?/C27b?/C27c?/C30s: (9) In addition, a/C30b?/C27c?/C30a?/C27b?/C27c?/C28a?/C30s/C28a?: (10) Switching aand a?to opposite sides of the equation and noting that the above argument applies equally well to b?andc?then gives a?/C30s/C28a (11) b?/C30s/C28b (12) c?/C30s/C28c: (13) As can be seen from the first figure, there exist exactly two nonintersecting CIRCLES which are TAN- GENT to all three CIRCLES . These are called the inner and outer Soddy circles ( SandS?;respectively), and their centers are called the inner and outer S ODDY POINTS . The inner Soddy circle is the solution to the FOUR COINS PROBLEM and its center S, the inner Soddypoint, is the EQUAL DETOUR POINT . The center of the outer Soddy circle, the outer Soddy point S?;is the ISOPERIMETRIC POINT (Kimberling 1994). Frederick Soddy (1936) gave the FORMULA for finding the RADII of the Soddy circles ( /r4) given the RADII ri (i/C301, 2, 3) of the other three. The relationship is 2e2 1/C27e22/C27e23/C27e24YrvYru /C30e1/C27e2/C27e3/C27e4 ðÞ2; (14) where ei/C139ki/C3091=riare the so-called BENDS , de- fined as the signed CURVATURES of the CIRCLES . If the contacts are all external, the signs are all taken as POSITIVE , whereas if one circle surrounds the other three, the sign of this circle is taken as NEGATIVE (Coxeter 1969). Using the QUADRATIC FORMULA to solve for e4;expressing in terms of radii instead of curvatures, and simplifying gives r9 4/C30r1r2r3 r2r3/C27r1r2/C27r3 ðÞ 92ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1r2r3r1/C27r2/C27r3 ðÞp : (15) Here, the NEGATIVE solution corresponds to the outer Soddy circle and the POSITIVE one to the inner Soddy circle. This FORMULA is called the D ESCARTES CIRCLE THEO- REM since it was known to Descartes. Soddy extended the result to TANGENT SPHERES , and Gosper has further extended the result to n/C272 mutually tangent n-DHYPERSPHERES . Bellew has derived a generalization applicable to a CIRCLE surrounded by nCIRCLES which are, in turn, circumscribed by another CIRCLE . The relationship is ncn/C281 ðÞ2/C271hi Xn/C271 i/C301k2 i/C27n3nc2n/C282n/C286YrvYru c2ncn/C281 ðÞ2/C30 f(n) nen/C281 ðÞ /C271"# ; (16) where kn/C271is the curvature of the central circle, f(n)/C30ncn/C281 ðÞ2/C271hi Xn/C271 i/C301ki/C27ncncn/C281 ðÞ /C2nc2n/C27(3/C28n)cn/C284YrtYrP (17) and cn/C13cscp n ! : (18) Forn/C303, this simplifies to the D ESCARTES CIRCLE THEOREM 2X4 i/C301k21/C30X4 i/C301ki ! 2 : (19) See also APOLLONIAN GASKET ,APOLLONIUS CIRCLES , APOLLONIUS’ PROBLEM ,A RBELOS ,B END (CURVA- TURE ), BOWL OF INTEGERS ,C IRCUMCIRCLE ,D ES- CARTES CIRCLE THEOREM ,E XCENTRAL TRIANGLE , FOUR COINS PROBLEM ,HART’S THEOREM ,M ALFATTI CIRCLES ,P APPUS CHAIN ,S ODDY POINTS ,S PHERE PACKING ,STEINER CHAIN ,TANGENT CIRCLES ,TAN- GENT SPHERES References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 13 /C1/4, 1969. Elkies, N. D. and Fukuta, J. "Problem E3236 and Solution." Amer. Math. Monthly 97, 529 /C1/31, 1990. Gosper, R. W. "Soddy’s Theorem on Mutually Tangent Circles, Generalized to n Dimensions." http://www.ippi.- com/rwg/Sodddy.htm. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, p. 181, 1994. "The Kiss Precise." Nature 139, 62, 1937. Soddy, F. "The Kiss Precise." Nature 137, 1021, 1936. Vandeghen, A. "Soddy’s Circles and the De Longchamps Point of a Triangle." Amer. Math. Monthly 71, 176 /C1/79, 1964. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 4 /C1/, 1991. Soddy Line A LINE on which the INCENTER I,GERGONNE POINT Ge, inner and outer SODDY POINTS S and S?; GRIF- FITHS POINTS Gr, Gr?; OLDKNOW POINTS Ol, Ol ?; RIGBY POINTS Ri, Ri?; and FLETCHER POINT Fl lie. The Soddy line can be given parametrically in TRI- LINEAR COORDINATES by I /C27 lGe; where l is a parameter (Oldknow 1996). The Soddy line is also given by X (f /C28e)a /C300; where cyclic permutations of the d, e, and f are taken and the sum is over TRILINEAR COORDINATES a; b; and g : The following table gives the values of l corre-sponding to a number of special points on the Soddy line. /l/ Center /C284 Outer GRIFFITHS POINT Gr?/ /C282 Outer OLDKNOW POINT Ol?/ //C284 3/ Outer RIGBY POINT Ri ?/ /C281 Outer SODDY POINT S?/ 0 INCENTER I 1 Inner SODDY POINT S /43/ Inner RIGBY POINT Ri 2 Inner OLDKNOW POINT Ol 4 Inner GRIFFITHS POINT Gr //C12/ GERGONNE POINT Ge /S?;I,S, and Geform a HARMONIC RANGE (Vandeghen 1964, Oldknow 1996). There are a total of 22 HARMONIC RANGES for sets of four points out of these 10 (Oldknow 1996). The Soddy line intersects the EULER LINE in the DELONGCHAMPS POINT , and the GERGONNE LINE in the F LETCHER POINT . See also DE LONGCHAMPS POINT ,E ULER LINE, FLETCHER POINT ,G ERGONNE POINT ,G RIFFITHS POINTS ,H ARMONIC RANGE ,INCENTER ,O LDKNOW POINTS ,RIGBY POINTS ,SODDY POINTS References Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319/C1/29, 1996. Vandeghen, A. "Soddy’s Circles and the De Longchamps Point of a Triangle." Amer. Math. Monthly 71, 176/C1/79, 1964. Soddy Points Given three mutually tangent CIRCLES , there exist exactly two nonintersecting circles which are TAN- GENT CIRCLES to all three original CIRCLES . These are called the inner and outer SODDY CIRCLES , and their centers S and S ? are called the inner and outer Soddy points, respectively. The inner Soddy point is the EQUAL DETOUR POINT , and the outer Soddy point S ? is the ISOPERIMETRIC POINT (Kimberling 1994). See also EQUAL DETOUR POINT ,ISOPERIMETRIC POINT ,SODDY CIRCLES References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, p. 181, 1994. Soddy’s Hexlet HEXLET Sofa Constant MOVING SOFA CONSTANT Sokhotskii’s Formula lim e001 x 9 ie /C30/C14i pd(x) /C27PV1 x ! ; where d(x) is the DELTA FUNCTION and PV denotes the CAUCHY PRINCIPAL VALUE . See also DELTA FUNCTION Sol Geometry The GEOMETRY of the LIE GROUP R SEMIDIRECT PRODUCT with R2 ; where R acts on R2by/ (t; (x; y)) 0 (etx; e/C28ty)/. See also THURSTON’S GEOMETRIZATION CONJECTURE Soldner’s Constant Consider the following formulation of the PRIME NUMBER THEOREM , p(x) /C30Xm(m) mgx edt ln t : where m(m) is the MO¨ BIUS FUNCTION and c (some- times also denoted m) is called Soldner’s constant. Ramanujan obtained c /C301:45136380... (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is 1.4513692346..., the root of li(x) /C300 (Soldner 1812; Nielsen 1965, p. 88). See also RIEMANN PRIME NUMBER FORMULA References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 123 /C1/24, 1994.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 23 and 45, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 39, 1983. Nielsen, N. "Theorie des Integrallograrithmus und Ver- wandter Transzendenten." Part II in Die Gammafunktion. New York: Chelsea, 1965. Soldner. Abhandlungen 2, 333, 1812. Solenoidal Field A solenoidal VECTOR FIELD satisfies 9 /C215 B /C300 (1) for every VECTOR B, where 9 /C215 B is the DIVERGENCE . If this condition is satisfied, there exists a vector A, known as the VECTOR POTENTIAL , such that B /C139/C29A ; (2) where 9/C29A is the CURL . This follows from the vector identity 9 /C215 B /C309 /C215( 9/C29A) /C300: (3) If A is an IRROTATIONAL FIELD , then A /C29r (4) is solenoidal. If u and v are irrotational, then u /C29v (5) is solenoidal. The quantity ( 9u) /C29( 9v); (6) where 9u is the GRADIENT , is always solenoidal. For a function f satisfying LAPLACE’S EQUATION 92 f /C300: (7) it follows that 9f is solenoidal (and also IRROTA- TIONAL ). See also BELTRAMI FIELD,CURL,DIVERGENCE ,DIVER- GENCELESS FIELD,G RADIENT ,IRROTATIONAL FIELD, LAPLACE’S EQUATION ,VECTOR FIELD References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1084, 2000. Solid A closed 3-D figure (which may, according to some terminology conventions, be self-intersecting). Kern and Bland (1948, p. 18) define a solid as any limited portion of space bounded by surfaces. Among the simplest solids are the SPHERE , CUBE , CONE , CYLIN- DER, and more generally, the POLYHEDRA . See also APPLE ,ARCHIMEDEAN SOLID ,BARREL ,CAT- ALAN SOLID,CONE,CORK PLUG,CUBE,CUBOCTAHE- DRON ,CYLINDER ,CYLINDRICAL HOOF,CYLINDRICAL WEDGE ,D ODECAHEDRON ,G EODESIC DOME,G OUR- SAT’S SURFACE ,GREAT DODECAHEDRON ,GREAT ICO- SAHEDRON ,G REAT RHOMBICOSIDODECAHEDRON (ARCHIMEDEAN ), GREAT RHOMBICUBOCTAHEDRON (ARCHIMEDEAN ), GREAT STELLATED DODECAHEDRON , ICOSAHEDRON ,ICOSIDODECAHEDRON ,JOHNSON SO- LID,KEPLER- POINSOT SOLID ,LEMON ,M O¨ BIUS STRIP, OCTAHEDRON ,PLATONIC SOLID ,POLYHEDRON ,PSEU- DOSPHERE ,R HOMBICOSIDODECAHEDRON ,R HOMBICU- BOCTAHEDRON ,SMALL STELLATED DODECAHEDRON , SNUB CUBE,SNUB DODECAHEDRON ,SOLID OF REVO- LUTION ,S PHERE ,S PHERICAL WEDGE ,S TEINMETZ SOLID,STELLA OCTANGULA ,SURFACE ,TETRAHEDRON , TORUS ,TRUNCATED CUBE,TRUNCATED DODECAHE- DRON ,TRUNCATED ICOSAHEDRON ,TRUNCATED OCTA- HEDRON ,T RUNCATED TETRAHEDRON ,U NIFORM POLYHEDRON ,W ULFF SHAPE References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948. Solid Angle Defined as the SURFACE AREA V of a UNIT SPHERE which is subtended by a given object S. Writing the SPHERICAL COORDINATES as f for the COLATITUDE (angle from the pole) and u for the LONGITUDE (azimuth), V/C13Aprojected /C30ggSsin f du df: Solid angle is measured in STERADIANS , and the solid angle corresponding to all of space being subtended is 4p STERADIANS . See also SPHERE ,STERADIAN Solid Geometry That portion of GEOMETRY dealing with SOLIDS ,as opposed to PLANE GEOMETRY . Solid geometry is con- cerned with POLYHEDRA , SPHERES , 3-D SOLIDS , lines in 3-space, PLANES , and so on. See also GEOMETRY ,PLANE GEOMETRY ,SPHERICAL GEOMETRY References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979. Bell, R. J. T. An Elementary Treatise on Coordinate Geome- try of Three Dimensions. London: Macmillan, 1926. Cohn, P. M. Solid Geometry. New York: Routledge, 1968. Dresden, A. Solid Analytical Geometry and Determinants. New York: Dover, 1964. Farin, G. E. and Hensford, D. The Geometry Toolbox for Graphics and Modeling. Natick, MA: A. K. Peters, 1997. Frost, P. Solid Geometry, 3rd ed. London: Macmillan, 1886. Harris, J. W. and Stocker, H. "Solid Geometry." Ch. 4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 95 /C1/16, 1998.Kenison, E. and Bradley, H. C. Descriptive Geometry. New York: Macmillan, 1935. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948. Lines, L. Solid Geometry. New York: Dover, 1965. Rouche ´, E. and de Comberousse, C. Traite ´ de Ge´ome´trie, nouv. e´d., vol. 2: Ge´ome´trie dans l’espace. Paris: Gauthier- Villars, 1922. Salmon, G. Treatise on the Analytic Geometry of Three Dimensions, 6th ed. London: Longmans Green, 1914. Shute, W. G.; Shirk, W. W.; and Porter, G. F. Solid Geome- try. New York: American Book Co., 1960. Weisstein, E. W. "Solid Geometry." MATHEMATICA NOTE- BOOK SOLID GEOMETRY.M . Weisstein, E. W. "Books about Solid Geometry." http:// www.treasure-troves.com/books/SolidGeometry.html. Wentworth, G. A. and Smith, D. E. Solid Geometry. Boston, MA: Ginn and Company, 1913. Solid Harmonic A SURFACE HARMONIC of degree l which is premulti- plied by a factor rl : Confusingly, solid harmonics are also known as "spherical harmonics" (Whittaker and Watson 1990, p. 392). See also SPHERICAL HARMONIC ,SURFACE HARMONIC References Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 198, 1959. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Solid of Revolution To find the VOLUME of a solid of rotation by adding up a sequence of thin cylindrical shells, consider a region bounded above by y /C30f(x); below by y /C30g(x) ; on the left by the LINE x /C30a, and on the right by the LINE x /C30b. When the region is rotated about the Y-AXIS , the resulting VOLUME is given by V /C302 pga bx[f(x) /C28g(x)] dx: To find the volume of a solid of rotation by adding up a sequence of thin flat disks, consider a region bounded above by y /C30f(x); below by y /C30g(x) ; on the left by the LINE x/C30a, and on the right by the LINE x/C30b. When the region is rotated about the X-AXIS , the resulting VOLUME is V/C30pga bf(x)½/C1382/C28g(x)½/C1382no dx: See also SURFACE OF REVOLUTION ,VOLUME References Harris, J. W. and Stocker, H. "Solids of Rotation." §4.10 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 111 /C1/13, 1998. Solid Partition Solid partitions are generalizations of PLANE PARTI- TIONS . MacMahon (1960) conjectured the GENERATING FUNCTION for the number of solid partitions was f(z) /C301 (1 /C28 z)1/C28 z2 ðÞ31 /C28 z3 ðÞ61 /C28 z4 ðÞ10/C1/C1/C1; but this was subsequently shown to disagree at n /C306 (Atkin et al. 1967). Knuth (1970) extended the tabulation of values, but was unable to find a correct generating function. The first few values are 1, 4, 10, 26, 59, 140, ... (Sloane’s A000293). See also PARTITION FUNCTION P References Atkin, A. O. L.; Bratley, P.; Macdonald, I. G.; and McKay, J. K. S. "Some Computations for m-Dimensional Parti- tions." Proc. Cambridge Philos. Soc. 63, 1097 /C1/100, 1967. Knuth, D. E. "A Note on Solid Partitions." Math. Comput. 24, 955 /C1/61, 1970. MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. VI: Partitions in Two-Dimensional Space, to which is Added an Adumbration of the Theory of Parti- tions in Three-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211, 345 /C1/73, 1912b. MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 75 /C1/76, 1960. Sloane, N. J. A. Sequences A000293/M3392 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Solid Spherical Harmonic SOLID HARMONIC Solidus The diagonal slash "/" used as the bar between NUMERATOR and DENOMINATOR of an in-line FRACTION (Bringhurst 1997, p. 284). The solidus is also called a DIAGONAL . See also DIVISION ,FRACTION ,O BELUS ,V INCULUM , VIRGULE References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 284, 1997. Solitary Number A number which does not have any FRIENDS . Solitary numbers include all PRIMES , PRIME POWERS , and numbers for which (n; s(n)) /C301 ; where (a, b) is the GREATEST COMMON DIVISOR of a and b and s(n) is the DIVISOR FUNCTION . The first few numbers satisfying (n; s(n)) /C301 are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ... (Sloane’s A014567). However, there exist numbers such as n /C3018, 45, 48, and 52 which are solitary but for which (n; s(n)) "1: It is believed that 10, 14, 15, 20, 22, and many othersare also solitary, although a proof appears to be extremely difficult. See also FRIEND ,FRIENDLY PAIR,PRIME POWER References Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly Integers." Amer. Math. Monthly 84,65/C1/6, 1977. Sloane, N. J. A. Sequences A014567 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Soliton A stable isolated (i.e., solitary) traveling wave solu- tion to a set of equations. See also KORTEWEG-DE VRIES EQUATION ,LAX PAIR, SINE-GORDON EQUATION References Bullough, R. K. and Caudrey, P. J. (Eds.). Solitons. Berlin: Springer-Verlag, 1980. Dodd, R. K.; Eilbeck, J. C.; and Morris, H. C. Solitons and Nonlinear Equations. London: Academic Press, 1984. Drazin, P. G. and Johnson, R. S. Solitons: An Introduction. Cambridge, England: Cambridge University Press, 1988. Filippov, A. The Versatile Solitons. Boston, MA: Birkha ¨user, 1996. Gu, C. H. Soliton Theory and Its Applications. New York: Springer-Verlag, 1995. Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge Univer- sity Press, 2000. Lamb, G. L. Jr. Elements of Soliton Theory. New York: Wiley, 1980. Makhankov, V. G.; Fedyann, V. K.; and Pashaev, O. K. (Eds.). Solitons and Applications. Singapore: World Scien- tific, 1990. Newell, A. C. Solitons in Mathematics and Physics. Phila- delphia, PA: SIAM, 1985. Olver, P. J. and Sattinger, D. H. (Eds.). Solitons in Physics, Mathematics, and Nonlinear Optics. New York: Springer- Verlag, 1990. Remoissent, M. Waves Called Solitons, 2nd ed. New York: Springer-Verlag, 1996. Russell, J. S. "Report on Waves." Report of the 14th Meeting of the British Association for the Advancement of Science. London: Jon Murray, pp. 311 /C1/90, 1844. Weisstein, E. W. "Books about Solitons." http://www.trea- sure-troves.com/books/Solitons.html. Solomon’s Seal Knot The (5,2) TORUS KNOT 05 /C1/01with BRAID WORD s5 1:/ Solomon’s Seal Lines The 27 REAL orIMAGINARY LINES which lie on the general CUBIC SURFACE and the 45 triple tangent PLANES to the surface. All are related to the 28 BITANGENTS of the general QUARTIC CURVE . Schoutte (1910) showed that the 27 lines can be put into a ONE-TO-ONE correspondence with the vertices of a particular POLYTOPE in 6-D space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the POLYTOPE and conversely (Du Val 1931). A similar correspondence can be made between the 28 bitangents and a 7-D POLYTOPE (Coxeter 1928) and between the tritangent planes of the canonical curve of genus four and an 8-D POLYTOPE (Du Val 1933). See also BRIANCHON’S THEOREM ,C UBIC SURFACE , DOUBLE SIXES,PASCAL’S THEOREM ,Q UARTIC SUR- FACE ,STEINER SET References Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, pp. 322 /C1/25, 1945. Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc. 24, 7 /C1/, 1928. Du Val, P. "On the Directrices of a Set of Points in a Plane." Proc. London Math. Soc. Ser. 2 35,23/C1/4, 1933. Schoutte, P. H. "On the Relation Between the Vertices of a Definite Sixdimensional Polytope and the Lines of a Cubic Surface." Proc. Roy. Akad. Acad. Amsterdam 13, 375 /C1/83, 1910. Solomon’s Seal Polygon HEXAGRAM Soluble Group SOLVABLE GROUP Solvable Congruence A CONGRUENCE that has a solution. Solvable Group A solvable group is a GROUP having a "normal series" such that each "normal factor" is ABELIAN . The special case of a solvable FINITE GROUP is a group whose composition indices are all PRIME NUMBERS . Solvable groups are sometimes called "soluble groups," a turn of phrase that is a source of possible amusement to chemists. The term "solvable" derives from this type of group’s relationship to GALOIS’S THEOREM , namely that the SYMMETRIC GROUP Snis unsolvable for n ]5 while it is solvable for n /C301, 2, 3, and 4. As a result, the POLYNOMIAL equations of degree ]5 are not solvable using finite additions, multiplications, divisions, and ROOT EXTRACTIONS . Every FINITE GROUP of order B60; every ABELIAN GROUP , and every SUBGROUP of a solvable group is solvable. Betten (1996) has computed a table of solvable groups of order up to 242 (Besche and Eick 1999).See also ABELIAN GROUP ,C OMPOSITION SERIES , GALOIS’S THEOREM ,S OLVABLE LIE GROUP ,S YM- METRIC GROUP References Besche, H.-U. and Eick, B. "The Groups of Order at Most 1000 Except 512 and 768." J. Symb. Comput. 27, 405 /C1/13, 1999. Betten, A. "Parallel Construction of Finite Soluble Groups." In Parallel Virtual Machine, Euro PVM ’96: Third European PVM Conference, Munich, Germany, October 7 /C1/, 1996 (Ed. A. Bode et al.). Berlin: Springer-Verlag, pp. 126 /C1/33, 1996. Doerk, K. and Hawkes, T. Finite Soluble Groups. Berlin: de Gruyter, 1992. Gruenberg, K. W. and Roseblade, J. E. (Eds.). Group Theory: Essays for Philip Hall. London: Academic Press, 1984. Laue, R. "Zur Konstruktion und Klassifikation endlicher auflo¨sbarer Gruppen." Bayreuther Mathemat. Schriften 9, 1982. Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993. Magnus, W. "Neuere Ergebnisse u¨ber auflo¨sbare Gruppen." Jahresber. der DMV 47, 69, 1937. Robinson, D. J. S. Finiteness Conditions and Generalized Soluble Groups, 2 vols. Berlin: Springer-Verlag, 1972. Scott, W. R. "Solvable Groups." §2.6 in Group Theory. New York: Dover, pp. 38 /C1/9, 1987. Segal, D. Polycyclic Groups. Cambridge, England: Cam- bridge University Press, 1983. Solvable Lie Algebra AL IE ALGEBRA g is solvable when its COMMUTATOR SERIES , or derived series, gk vanishes for some k. Any NILPOTENT LIE ALGEBRA is solvable. The basic exam- ple is the VECTOR SPACE of UPPER TRIANGULAR MATRICES , because every time two such matrices commute, their nonzero entries move further from the diagonal. The following Mathematica function tests whether a Lie algebra g is solvable, when given a list of matrices which form a basis for g:/ MatrixBasis[a_- List]: /C30Partition[#1,Length[a[[1]]]]&/@ LatticeReduce[Flatten/@a] LieCommutator[a_,b_]: /C30a.b-b.a NextDerived[{}] /C30{}; NextDerived[g_List]: /C30 MatrixBasis[Flatten[Outer[LieCommutator,g,- g,1],1]] SolvableLieQ[g_List]: /C30 FixedPoint[NextDerived,g] /C30/C30{} For example, borel5 /C30Flatten[Table[ReplacePart[ Ta- ble[0,{i,5},{j,5}],1,{k,l}],{k,5},{l,k,5}],1]; SolvableLieQ[borel5] yieldsTrue . See also BOREL SUBALGEBRA ,COMMUTATOR SERIES (LIE ALGEBRA ), LIE ALGEBRA ,LIE GROUP ,NILPOTENT LIE GROUP ,N ILPOTENT LIE ALGEBRA ,REPRESENTA- TION (LIE ALGEBRA ), REPRESENTATION (SOLVABLE LIE GROUP ), SOLVABLE LIE GROUP ,SPLIT SOLVABLE LIE ALGEBRA Solvable Lie Group A solvable Lie group is a LIE GROUP G which is CONNECTED and whose LIE ALGEBRA g is a SOLVABLE LIE ALGEBRA . That is, the COMMUTATOR SERIES g1 /C30[ g;g];g2 /C30g1 /C215g1YrtYrP ; ... (1) eventually vanishes, gk /C300 for some k. Since NILPO- TENT LIE ALGEBRAS are also SOLVABLE , any NILPO- TENT LIE GROUP is a solvable Lie group. The basic example is the GROUP of invertible UPPER TRIANGULAR MATRICES with positive DETERMINANT , e.g., a11a12a13 0 a22a23 00 a332 435 (2) such thatQ iaii > 0: The LIE ALGEBRA g of G is its TANGENT SPACE at the identity matrix, which is the VECTOR SPACE of all upper triangular matrices, and it is a SOLVABLE LIE ALGEBRA . Its COMMUTATOR SERIES is given by g1 /C300 b12b13 00 b23 00 02435 (3) g 2 /C3000 c13 00 0 00 02 435; (4) g 3 /C30000 000 0002 435: (5) Any real solvable Lie group is DIFFEOMORPHIC to EUCLIDEAN SPACE . For instance, the group of ma- trices in the example above is diffeomorphic to R6 ; via the EXPONENTIAL MAPExponential Map (Lie Group). However, in general, the exponential map in a SOLVABLE LIE ALGEBRA need not be SURJECTIVE . See also BOREL GROUP ,COMMUTATOR SERIES (LIE ALGEBRA ), FLAG (VECTOR SPACE ), LIE ALGEBRA ,LIE GROUP ,M ATRIX ,NILPOTENT LIE GROUP ,REPRESEN- TATION ,R EPRESENTATION (SOLVABLE LIE GROUP ), SOLVABLE GROUP ,S OLVABLE LIE ALGEBRA ,S PLIT SOLVABLE LIE ALGEBRA References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996.SOMA Let k ]0 and n ]2 be integers. A SOMA, or more specifically a SOMA( k, n), is an n /C29n array A, whose entries are k-subsets of a kn-set V; such that each element of V occurs exactly once in each row and exactly once in each column of A, and no 2-subset of V is contained in more than one entry of A (Soicher 1999). A SOMA( k, n) can be constructed by superposing k mutually orthogonal LATIN SQUARES of order n with pairwise disjoint symbol-sets, and so a SOMA( k, n) can be seen as a generalization of k mutually orthogonal LATIN SQUARES of order n. See also LATIN SQUARE References Soicher, L. H. "On the Structure and Classification of SOMAs: Generalizations of Mutually Orthogonal Latin Squares." Electronic J. Combinatorics 6, No. 1, R32, 1 /C1/5, 1999. http://www.combinatorics.org/Volume_6/ v6i1toc.html. Soma Cube A solid DISSECTION puzzle invented by Piet Hein during a lecture on Quantum Mechanics by Werner Heisenberg. There are seven soma pieces composed of all the irregular face-joined cubes (POLYCUBES ) with / 54/ cubes. The object is to assemble the pieces into a CUBE . There are 240 essentially distinct ways of doing so (Beeler 1972, Berlekamp et al. 1982), as first enumerated one rainy afternoon in 1961 by J. H. Conway and Mike Guy. A commercial version of the cube colors the pieces black, green, orange, white, red, and blue. When the 48 symmetries of the cube, three ways of assembling the black piece, and 25 ways of assembling the green, orange, white, red, and blue pieces are counted, the total number of solutions rises to 1,105,920. See also CUBE DISSECTION ,POLYCUBE References Albers, D. J. and Alexanderson, G. L. (Eds.). Mathematical People: Profiles and Interviews. Boston, MA: Birkha ¨user, p. 43, 1985. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 112 /C1/13, 1987. Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48 /C1/0, Feb. 1972. Berlekamp, E. R.; Conway, J. H.; and Guy, R. K. Ch. 24 in Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 203 /C1/05, 1989. Gardner, M. "Mathematical Games: A Game in Which Standard Pieces Composed of Cubes are Assembled into Larger Forms." Sci. Amer. , 185. Gardner, M. "The Soma Cube." Ch. 6 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 65 /C1/7, 1961. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 168 /C1/69, 1999. Somer-Lucas Pseudoprime An ODD COMPOSITE NUMBER N is called a Somer- Lucas d-pseudoprime (with d ]1) if there EXISTS a nondegenerate LUCAS SEQUENCE U(P; Q) with U0 /C30 0; U1 /C301 ; D /C30P2 /C284Q; such that (N ; D) /C301 and the rank appearance of N in the sequence U(P ; Q)is (1=a)(N /C28(D =N)); where (D=N) denotes the JACOBI SYMBOL . See also LUCAS SEQUENCE ,PSEUDOPRIME References Ribenboim, P. "Somer-Lucas Pseudoprimes." §2.X.D in The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 131 /C1/32, 1996. Sommerfeld’s Formula There are (at least) two equations known as Som- merfeld’s formula. The first is Jn(z) /C301 2p g2 p /C28h /C27i/C12 /C28h /C27i/C12eiz cos tein(t /C28p =2) dt; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . The second states that under appropriate restrictions, g/C12 0J0( tr)e /C28½x ½ffiffiffiffiffiffiffiffi t2/C28k2p t dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 /C28 k2p /C30eikffiffiffiffiffiffiffiffiffiffi t2 /C27k2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27 x2p : See also WEYRICH’S FORMULA References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1472 and 1474, 1980.Somos Sequence The Somos sequences are a set of related symmetrical RECURRENCE RELATIONS which, surprisingly, always give integers. The Somos sequence of order k is defined by an /C30Pk =2bc j/C301an/C28jan/C28(k /C28j) an/C28k; where xbcis the FLOOR FUNCTION and aj /C301 for j /C300, ..., k /C281 : The 2- and 3-Somos sequences consist entirely of 1s. The k-Somos sequences for k /C304, 5, 6, and 7 are an /C30an /C281an/C283 /C27 a2 n/C282 an/C284 an /C30an /C281an/C284 /C27 an/C282an/C283 an/C285 an /C301 an/C286an/C281an/C285 /C27an/C282an/C284 /C27a2 n/C283YrtYrP an /C301 an/C287an/C281an/C286 /C27an/C282an/C285 /C27an/C283an/C284 ½/C138 : giving 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, ... (Sloane’s A006720), 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, ... (Sloane’s A006721), 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, ... (Sloane’s A006722), 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, ... (Sloane’s A006723). Gale (1991) gives simple proofs of the integer-only property of the 4- Somos and 5-Somos sequences. Hickerson proved 6- Somos generates only integers using computer alge- bra, and empirical evidence suggests 7-Somos is also integer-only. However, the k-Somos sequences for k ]8 do not give integers. The values of n for which anfirst becomes nonintegral for the k-Somos sequence for k /C308, 9, ... are 17, 19, 20, 22, 24, 27, 28, 30, 33, 34, 36, 39, 41, 42, 44, 46, 48, 51, 52, 55, 56, 58, 60, ... (Sloane’s A030127). See also GO¨ BEL’S SEQUENCE ,HERONIAN TRIANGLE References Buchholz, R. H. and Rathbun, R. L. "An Infinite Set of Heron Triangles with Two Rational Medians." Amer. Math. Monthly 104, 107/C1/15, 1997. Gale, D. "Mathematical Entertainments: The Strange and Surprising Saga of the Somos Sequences." Math. Intel. 13, 40/C1/2, 1991. Malouf, J. L. "An Integer Sequence from a Rational Recur- sion." Disc. Math. 110, 257/C1/61, 1992. Robinson, R. M. "Periodicity of Somos Sequences." Proc. Amer. Math. Soc. 116, 613/C1/19, 1992. Sloane, N. J. A. Sequences A006720/M0857, A006721/ M0735, A006722/M2457, A006723/M2456, and A030127 in "An On-Line Version of the Encyclopedia of IntegerSequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sondat’s Theorem The PERSPECTIVE AXIS bisects the line joining the two ORTHOCENTERS . See also ORTHOCENTER ,PERSPECTIVE AXIS References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 259, 1929. Sonine Polynomial LAGUERRE POLYNOMIAL Sonine’s Integral Jm(x) /C302xm/C28n 2m/C28n G(m /C28 n) g1 0Jn(xt)tn/C271 /C2 1 /C28t2YrvYrum/C28n/C281dt; where Jm(x)isaB ESSEL FUNCTION OF THE FIRST KIND and G(x) is the GAMMA FUNCTION . See also HANKEL’S INTEGRAL ,POISSON INTEGRAL Sonine-Schafheitlin Formula g/C12 0Jm(at)Jn(bt)t/C28 l dt /C30am G[(m /C27 n /C28 l /C27 1)=2] 2 lbm/C28 l/C271 G[(/C28 m /C27 n /C27 l /C27 1)=2]G( m /C27 1) /C292F1( m /C27 n /C28 l /C271)=2 ;(m /C28 n /C28 l /C271)=2; m /C271; a2 =b2YrvYru ; where R[ m /C27 n /C28 l /C271] > 0;R[ l] >/C281; 0 Ba Bb ; Jn(x) is a BESSEL FUNCTION OF THE FIRST KIND , G(x) is the GAMMA FUNCTION , and2F1(a ; b; c; x)isa HYPERGEO- METRIC FUNCTION . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1474, 1980. Sophie Germain Prime A PRIME p is said to be a Sophie Germain prime if both p and 2p /C271 are PRIME . The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ... (Sloane’s A005384). Sophie Germain primes p OF THE FORM /p /C30k /C215 2n /C281/ (which makes 2p /C271a PRIME ) correspond to the indices of composite MERSENNE NUMBERS /Mp/. The largest known Sophie Germain prime is 92:305 /C29 216 :998 /C271; found in 1998 (Hoffman 1998, p. 190). It is not known if there are an infinite number of Sophie German primes (Hoffman 1998, p. 190). Around 1825, Sophie Germain proved that the first case of FERMAT’S LAST THEOREM is true for suchprimes, i.e., if p is a Sophie Germain prime, there do not exist INTEGERS x, y, and z different from 0 and not multiples of p such that xp /C27yp /C30zp : See also CUNNINGHAM CHAIN ,FERMAT’S LAST THEO- REM,MERSENNE NUMBER ,TWIN PRIMES References Caldwell, C. K. "The Top Twenty: Sophie Germain Primes." http://www.utm.edu/research/primes/lists/top20/Sophie- Germain.html. Dubner, H. "Large Sophie Germain Primes." Math. Comput. 65, 393 /C1/96, 1996. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 190, 1998. Indlekofer, K. H. and Ja´rai, A. "Largest Known Twin Primes and Sophie Germain Primes." Math. Comput. 68, 1317 /C1/ 324, 1999. Ribenboim, P. "Sophie Germane Primes." §5.2 in The New Book of Prime Number Records. New York: Springer- Verlag, pp. 329 /C1/32, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 154 /C1/57, 1993. Sloane, N. J. A. Sequences A005384/M0731 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sorites Paradox Sorites paradoxes are a class of paradoxical argu- ments also known as little-by-little arguments. The name "sorites" derives from the Greek word soros , meaning "pile" or "heap." Sorites paradoxes are exemplified by the problem that a single grain of wheat does not comprise a heap, nor do two grains of wheat, three grains of wheat, etc. However, at some point, the collection of grains becomes large enough tobe called a heap, but there is apparently no definite point where this occurs. See also U NEXPECTED HANGING PARADOX References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 196 /C1/99, 1998. Sorting Sorting is the rearrangement of numbers (or other orderable objects) in a list into their correct lexo- graphic order. Alphabetization is therefore a form of sorting. Because of the extreme importance of sortingin almost all database applications, a great deal ofeffort has been expended in the creation and analysis of efficient sorting algorithms. The minimum number of comparisons a(n) needed for a merge sort of nelements for n/C301, 2, ... are 0, 1, 3, 5, 7, 10, 13, 16, 19, 22, 26, 30, ... (Sloane’s A001768). An upper limit b(n) is given by the sequence a(n) 5b(n) /C301 /C27kn /C282k where k /C30 log2 n bc /C271 ; where xbcis the FLOOR FUNCTION (Steinhaus 1983, pp. 55 /C1/6), or equivalently, b(n) /C30Xn k /C301log2 k de ; giving 0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, ... (Sloane’s A001855). See also HEAPSORT ,O RDERING ,QUICKSORT ,SELEC- TION SORT,W EIGHING References Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1973. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Sorting." Ch. 8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 320 /C1/39, 1992. Skiena, S. "Sorting and Searching." §1.1.6 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 14 /C1/6, 1990. Sloane, N. J. A. Sequences A001768/M2408 and A001855/ M2433 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sort-Then-Add Sequence A sequence produced by sorting the digits of a number and adding them to the previous number. The algorithm terminates when a sorted number is obtained. For n /C301, 2, ..., the algorithm terminates on 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 33, ... (Sloane’s A033862). The first few numbers not known to terminate are 316, 452, 697, 1376, 2743, 5090, ... (Sloane’s A033861). The least numbers of sort-then-add persistence n /C301, 2, ..., are 1, 10, 65, 64, 175, 98, 240, 325, 302, 387, 198, 180, 550, ... (Sloane’s A033863). See also 196-ALGORITHM , RATS SEQUENCE References Sloane, N. J. A. Sequences A033861, A033862, and A033863 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html.Source A local source is a node of a DIRECTED GRAPH with no entering edges (Borowski and Borwein 1991, p. 401; left figure), and a global source (often simply called a source) is a node in a DIRECTED GRAPH which reaches all other nodes (Harary 1994, p. 201; right figure). See also DIRECTED GRAPH ,NETWORK ,SINK (DIREC- TED GRAPH ) References Borowski, E. J. and Borwein, J. M. (Eds.). The HarperCol- lins Dictionary of Mathematics. New York: HarperCollins, 1991. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Sous-Double A3- MULTIPERFECT NUMBER P3 : Six sous-doubles are known (120, 672, 523776, 459818240, 1476304896, and 51001180160; Sloane’s A005820), and these are believed to comprise all sous-doubles. See also MULTIPERFECT NUMBER ,SOUS-TRIPLE References Sloane, N. J. A. Sequences A005820/M5376 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Souslin Set The continuous image of a POLISH SPACE , also called an ANALYTIC SET. See also ANALYTIC SET,POLISH SPACE Souslin’s Hypothesis Every dense linear order complete set without end- points having at most vdisjoint intervals is order isomorphic to the CONTINUUM ofREAL NUMBERS , where vis the set of NATURAL NUMBERS . References Iyanaga, S. and Kawada, Y. (Eds.). "Souslin’s Hypothesis." §35E.4 in Encyclopedic Dictionary of Mathematics. Cam- bridge, MA: MIT Press, p. 137, 1980. Sous-Triple A4 - MULTIPERFECT NUMBER P4:36 sous-triples are known (30240, 32760, 2178540, 23569920, ...; Sloane’s A027687), and these are believed to comprise all sous- triples. See also MULTIPERFECT NUMBER ,SOUS-DOUBLE References Sloane, N. J. A. Sequences A027687 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Space The concept of a space is an extremely general and important mathematical construct. Members of the space obey certain addition properties. Spaces which have been investigated and found to be of interest are usually named after one or more of their investiga- tors. This practice unfortunately leads to names which give very little insight into the relevant properties of a given space. The everyday type of space familiar to most people is called EUCLIDEAN SPACE . In Einstein’s theory of Special Relativity, Euclidean 3-space plus time (the "fourth dimension") are unified into the so-called MINKOWSKI SPACE . One of the most general type of mathematical spaces is the TOPOLOGICAL SPACE . See also AFFINE SPACE ,BAIRE SPACE ,BANACH SPACE , BASE SPACE ,BERGMAN SPACE ,BESOV SPACE ,BOREL SPACE ,C ALABI- YAU SPACE ,C ELLULAR SPACE ,C HU SPACE ,D ODECAHEDRAL SPACE ,D RINFELD’S SYM- METRIC SPACE ,EILENBERG- MAC LANE SPACE ,EUCLI- DEAN SPACE ,FIBER SPACE ,FINSLER SPACE ,FIRST- COUNTABLE SPACE ,F RE´ CHET SPACE ,F UNCTION SPACE , G-SPACE ,GREEN SPACE ,HAUSDORFF SPACE , HEISENBERG SPACE ,H ILBERT SPACE ,H YPERBOLIC SPACE ,INNER PRODUCT SPACE ,L 2-SPACE ,L ENS SPACE ,LINE SPACE ,LINEAR SPACE ,LIOUVILLE SPACE , LOCALLY CONVEX SPACE ,L OCALLY FINITE SPACE , LOOP SPACE ,M APPING SPACE ,M EASURE SPACE , METRIC SPACE ,M INKOWSKI SPACE ,M U¨ NTZ SPACE , NON-EUCLIDEAN GEOMETRY ,N ORMED SPACE ,PARA- COMPACT SPACE ,P LANAR SPACE ,P OLISH SPACE , PROBABILITY SPACE ,PROJECTIVE SPACE ,Q UOTIENT SPACE ,RIEMANN’S MODULI SPACE ,RIEMANN SPACE , SAMPLE SPACE ,S TANDARD SPACE ,S TATE SPACE , STONE SPACE ,S YMPLECTIC SPACE ,T EICHMU ¨ LLER SPACE ,TENSOR SPACE ,TOPOLOGICAL SPACE ,TOPOLO- GICAL VECTOR SPACE ,TOTAL SPACE ,VECTOR SPACE Space Conic SKEW CONIC Space Curve A curve which may pass through any region of 3-D space, as contrasted to a PLANE CURVE which must lie in a single PLANE . Von Staudt (1847) classified space curves geometrically by considering the curve f : I 0 R3 (1)at t0 /C300 and assuming that the parametric functions fi(t) for i /C301, 2, 3 are given by POWER SERIES which converge for small t. If the curve is contained in no PLANE for small t, then a coordinate transformation puts the PARAMETRIC EQUATIONS in the normal form f1(t) /C30t1 /C27k1 /C27... (2) f2(t) /C30t2 /C27k1/C27k2 /C27... (3) f3(t) /C30t3 /C27k1/C27k2/C27k3 /C27... (4) for integers k1 ; k2 ; k3 ]0; called the local numerical invariants. See also CURVE ,CYCLIDE ,FUNDAMENTAL THEOREM OF SPACE CURVES ,HELIX,PLANE CURVE ,SEIFFERT’S SPHERICAL SPIRAL ,S KEW CONIC ,S PACE- FILLING FUNCTION ,S PHERICAL CURVE ,S PHERICAL SPIRAL , SURFACE ,VIVIANI’S CURVE References do Carmo, M.; Fischer, G.; Pinkall, U.; and Reckziegel, H. "Singularities of Space Curves." §3.1 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 24 /C1/5, 1986. Fine, H. B. "On the Singularities of Curves of Double Curvature." Amer. J. Math. 8, 156 /C1/77, 1886. Fischer, G. (Ed.). Plates 57 /C1/4in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 58 /C1/9, 1986. Gray, A. "Curves in Rn /" and "Curves in Space." §1.2 and Ch. 8 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 5 /C1/ and 181 /C1/06, 1997. Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, 1978. Saurel, P. "On the Singularities of Tortuous Curves." Ann. Math. 7,3/C1/, 1905. Staudt, C. von. Geometrie der Lage. Nu¨rnberg, Germany, 1847. Wiener, C. "Die Abha¨ngigkeit der Ru¨ckkehrelemente der Projektion einer unebenen Curve von deren der Curve selbst." Z. Math. & Phys. 25,95/C1/7, 1880. Space Diagonal The LINE SEGMENT connecting opposite VERTICES (i.e., two VERTICES which do not share a common face) in a PARALLELEPIPED or other similar solid. See also DIAGONAL (POLYGON ), DIAGONAL (POLYHE- DRON ), EULER BRICK Space Distances POINT DISTANCES Space Division by Planes The maximal number of regions into which space can be divided by nplanes is f(n)/C301 6n3/C275n/C276YrvYru (Yaglom and Yaglom 1987, pp. 102 /C1/06), giving the values 2, 4, 8, 15, 26, 42, ... (Sloane’s A000125) for n /C301, 2, ... planes. This is the same solution as for CYLINDER CUTTING . See also CIRCLE DIVISION BY LINES,CUBE DIVISION BY PLANES ,C YLINDER CUTTING ,P LANE DIVISION BY CIRCLES ,SPACE DIVISION BY SPHERES References Sloane, N. J. A. Sequences A000125/M1100 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 72, 1986. Yaglom, A. M. and Yaglom, I. M. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, pp. 102 /C1/06, 1987. Space Division by Spheres The number of regions into which space can be divided by n mutually intersecting SPHERES is N /C301 3 nn2 /C283n /C278YrvYru ; giving 2, 4, 8, 16, 30, 52, 84, ... (Sloane’s A046127) for n /C301, 2, .... See also PLANE DIVISION BY CIRCLES ,SPACE DIVISION BY PLANES ,SPHERE- SPHERE INTERSECTION References Sloane, N. J. A. Sequences A046127 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Yaglom, A. M. and Yaglom, I. M. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, pp. 102 /C1/06, 1987. Space Groups The space groups in 2-D are called WALLPAPER GROUPS . In 3-D, the space groups are the symmetry GROUPS possible in a crystal lattice with the transla- tion symmetry element. There are 230 space groups in R3 ; although 11 are MIRROR IMAGES of each other. They are listed by HERMANN- MAUGUIN SYMBOL in Cotton (1990). See also HERMANN- MAUGUIN SYMBO L,L ATTICE GROUPS ,POINT GROUPS ,W ALLPAPER GROUPS References Arfken, G. "Crystallographic Point and Space Groups." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 248 /C1/49, 1985.Buerger, M. J. Elementary Crystallography. New York: Wiley, 1956. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, pp. 250 /C1/51, 1990. Space of Closed Paths LOOP SPACE Space-Filling Curve SPACE- FILLING FUNCTION Space-Filling Function A"CURVE " (i.e., a continuous map of a 1-D INTERVAL ) into a 2-D area (a PLANE-FILLING FUNCTION ) or a 3-D volume. See also HILBERT CURVE ,P EANO CURVE ,P EANO- GOSPER CURVE ,PLANE- FILLING CURVE ,SIERPINSKI CURVE ,SPACE- FILLING POLYHEDRON References Pappas, T. "Paradoxical Curve-Space-Filling Curve." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 208, 1989. Platzman, L. K. and Bartholdi, J. J. "Spacefilling Curves and the Planar Travelling Salesman Problem." J. Assoc. Comput. Mach. 46, 719/C1/37, 1989. Wagon, S. "A Spacefilling Curve." §6.3 in Mathematica in Action. New York: W. H. Freeman, pp. 196 /C1/09, 1991. Space-Filling Polyhedron A space-filling polyhedron is a POLYHEDRON which can be used to generate a TESSELLATION of space. Although even Aristotle himself proclaimed in his work On the Heavens that the TETRAHEDRON fills space, it in fact does not (Hilbert and Cohn-Vossen 1999, p. 45). The CUBE is the only PLATONIC SOLID possessing this property (Gardner 1984, pp. 183 /C1/84). However, a combination of TETRAHEDRA and OCTAHE- DRA do fill space (Steinhaus 1983, p. 210; Wells 1991, p. 232). In addition, octahedra, truncated octahedron, and cubes, combined in the ratio 1:1:3, can also fill space (Wells 1991, p. 235). Of the Archimedean solids, the RHOMBIC DODECAHE- DRON and TRUNCATED OCTAHEDRON are space-fillers (Steinhaus 1983, pp. 185 /C1/90; Wells 1991, pp. 233 /C1/ 34). The ELONGATED DODECAHEDRON and hexagonal PRISM are also space-fillers. These five solids are all "primary" PARALLELOHEDRA (Coxeter 1973). In 1914, Fo¨ppl discovered a space-filling compound of tetra- hedra and truncated tetrahedra (Wells 1991, p. 234). The CUBOCTAHEDRON , TRIANGULAR ORTHOBICUPOLA , and squashed dodecahedron appearing in SPHERE PACKING also fill space (Steinhaus 1983, pp. 203 /C1/ 07), as does an arbitrary TRIANGULAR PRISM or any non-self-intersecting quadrilateral PRISM . There exists a tetrahedron with bevelled edges which fills space (Wells 1991, p. 234). There exists one 16- sided space-filling POLYHEDRON , but it is unknown if it is the unique 16-sided space-filler. There exists an 18-faced space-filler, as well space-fillers of up to 38 faces, as discovered by P. Engel in 1980 (Wells 1991, pp. 234 /C1/35). P. Schmitt discovered a nonconvex aper- iodic polyhedral space-filler around 1990, and a convex POLYHEDRON known as the SCHMITT- CONWAY BIPRISM which fills space only aperiodically was found by J. H. Conway in 1993 (Eppstein). See also CUBE,CUBOCTAHEDRON ,ELONGATED DODE- CAHEDRON ,KELLER’S CONJECTURE ,KELVIN’S CONJEC- TURE ,O CTAHEDRON ,P ARALLELOHEDRON ,P RISM , RHOMBIC DODECAHEDRON ,SCHMITT- CONWAY BIPR- ISM,SPHERE PACKING ,TESSELLATION ,TETRAHEDRON , TILING ,T RIANGULAR ORTHOBICUPOLA ,T RUNCATED OCTAHEDRON References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29 /C1/0, 1973. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970. Devlin, K. J. "An Aperiodic Convex Space-Filler is Discov- ered." Focus: The Newsletter of the Math. Assoc. Amer. 13, 1, Dec. 1993. Eppstein, D. "Re: Aperiodic Space-Filling Tile?." http:// www.ics.uci.edu/~eppstein/junkyard/biprism.html. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Holden, A. Shapes, Space, and Symmetry. New York: Dover, pp. 154 /C1/63, 1991. Kramer, P. "Non-Periodic Central Space Filling with Icosa- hedral Symmetry Using Copies of Seven Elementary Cells." Acta Cryst. A 38, 257 /C1/64, 1982. Pearce, P. Structure and Nature as a Strategy for Design. Cambridge, MA: MIT Press, 1978.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185 /C1/90, 1999. Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11,3/C1/ 4, 1910. Thompson, D’A. W. On Growth and Form, 2nd ed., compl. rev. ed. New York: Cambridge University Press, 1992. Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, pp. 567 and 723, 1964. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 232 /C1/36, 1991. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, 1979. Span (Geometry) The largest possible distance between two points drawn from a finite set of points. See also COMPUTATIONAL GEOMETRY ,CONVEX HULL, JUNG’S THEOREM ,POINT DISTANCES Span (Link) The span of an unoriented LINK diagram (also called the SPREAD ) is the difference between the highest and lowest degrees of its BRACKET POLYNOMIAL . The span is a topological invariant of a knot. If a KNOT K has a reduced alternating projection of ncrossings, then the span of Kis 4n:/ See also LINK Span (Polynomial) The difference between the highest and lowest de- grees of a POLYNOMIAL . Span (Set) For a SET S, the span is defined by /max S /C28min S/, where max is the MAXIMUM and min is the MINIMUM . References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 207, 1994. Span (Vector Space) The span of SUBSPACE generated by VECTORS v1and v2 /C23V is Span v1 ; v2 ðÞ /C13 rv1 /C27sv2 : r; s /C23R fg A set of vectors m /C30 v1 ; ...; vn fg can be tested to see if they span n-D space using the following Mathema- tica function. SpanningVectorsQ[m_List?MatrixQ] : /C30 (NullSpace[m] /C30/C30 {}) See also BASIS (VECTOR SPACE ), LINEAR COMBINA- TION ,NULLSPACE ,VECTOR SPACE Spanning Tree A spanning tree of a GRAPH is a subset of n /C281 edges which form a TREE . The shortest-path spanning tree is the tree have the smallest possible total distance, where the distance used is MANHATTAN DISTANCE (Skiena 1990, p. 227). The number of nonidentical spanning trees of a GRAPH G is equal to any COFACTOR of the DEGREE MATRIX of G minus the ADJACENCY MATRIX of G (Skiena 1990, p. 235). This result is known as the MATRIX TREE THEOREM .A TREE contains a unique spanning tree, a CYCLE GRAPH Cncontaining n spanning trees, and a COMPLETE GRAPH Kncontains nn/C282 spanning trees (Skiena 1990, p. 236). A count of the spanning trees of a graph can be found using thecommand NumberOfSpanningTrees [g] in the Math- ematica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also MATRIX TREE THEOREM ,MINIMUM SPANNING TREE,TREE References Colbourn, C. J.; Day, R. P. J.; and Nel, L. D. "Unranking and Ranking Spanning Trees of a Graph." J. Algorithms 10, 271/C1/86, 1989. Eppstein, D. "Spanning Trees and Spanners." Ch. 9 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Hol- land, pp. 425 /C1/61, 2000. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 224 /C1/27, 1990. Sparse Matrix AMATRIX which has only a small number of NONZERO elements. References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Sparse Linear Systems." §2.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing,2nd ed. Cambridge, England: Cambridge University Press, pp. 63 /C1 /2, 1992. Spearman Rank Correlation Coefficient A nonparametric (distribution-free) rank statistic proposed by Spearman in 1904 as a measure of the strength of the associations between two variables (Lehmann and D’Abrera 1998). The Spearman rankcorrelation coefficient can be used to give an R - ESTIMATE . The Spearman rank correlation coefficient is definedby r?/C131/C286X d2 NN2/C281 ðÞ; (1) where dis the difference in RANK of corresponding variables, and is an approximation to the exact CORRELATION COEFFICIENT r/C13PxyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPx2Py2p (2) computed from the original data. Because it uses ranks, the Spearman rank correlation coefficient is much easier to compute. The VARIANCE ,KURTOSIS , and higher order MOMENTS are s2/C301 N/C281(3) g2 /C30/C28114 25N /C286 5N2 /C28... (4) g3 /C30 g5 /C30.../C300: (5) Student was the first to obtain the VARIANCE . See also CORRELATION COEFFICIENT ,LEAST SQUARES FITTING ,LINEAR REGRESSION ,RANK (STATISTICS ) References Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, pp. 338 and 400, 1995. Lehmann, E. L. and D’Abrera, H. J. M. Nonparametrics: Statistical Methods Based on Ranks, rev. ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 292, 300, and 323, 1998. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 634 /C1/37, 1992. Special Curve PLANE CURVE ,SPACE CURVE ,SPHERICAL CURVE Special Function A function (usually named after an early investigator of its properties) having a particular use in mathe- matical physics or some other branch of mathematics. Prominent examples include the GAMMA FUNCTION , HYPERGEOMETRIC FUNCTION ,W HITTAKER FUNCTION , and MEIJER’S G-FUNCTION . See also ELEMENTARY FUNCTION ,FIRST KIND,FUNC- TION ,SECOND KIND,THIRD KIND References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. Andrews, G. E.; Askey, R.; and Roy, R. Special Functions. Cambridge, England: Cambridge University Press, 1999. Arscott, F. M. "The Land Beyond Bessel: A Survey of Higher Special Functions." In Ordinary and Partial Differential Equations (Ed. W. N. Everitt and B. D. Sleeman). New York: Springer-Verlag, pp. 26 /C1/5, 1981. Luke, Y. L. The Special Functions and their Approxima- tions, Vol. 1. New York: Academic Press, 1969. Luke, Y. L. The Special Functions and their Approxima- tions, Vol. 2. New York: Academic Press, 1969. Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. New York: Springer-Verlag, 1966. Nikiforov, A. F. and Uvarov, V. B. Special Functions of Mathematical Physics: A Unified Introduction with Appli- cations. Boston, MA: Birkha ¨user, 1988. National Institute of Standards. "Digital Library of Mathe- matical Functions." http://dlmf.nist.gov/. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon and Breach, 1986. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 2: Special Functions. New York: Gordon and Breach, 1990.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 3: More Special Functions. New York: Gordon and Breach, 1989. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 4: Direct Laplace Transforms. New York: Gordon and Breach, 1992. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 5: Inverse Laplace Transforms. New York: Gordon and Breach, 1992. Spanier, J. and Oldham, K. B. An Atlas of Functions. Washington, DC: Hemisphere, 1987. Weisstein, E. W. "Books about Special Functions." http:// www.treasure-troves.com/books/SpecialFunctions.html. Wolfram Research, Inc. "Wolfram Research’s Special Func- tions." http://functions.wolfram.com/. Special Jordan Algebra AJ ORDAN ALGEBRA which is isomorphic to a sub- algebra. See also EXCEPTIONAL JORDAN ALGEBRA ,JORDAN ALGEBRA References Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 4, 1996. Special Lie Algebra See also LIE ALGEBRA ,SPECIAL LINEAR LIE ALGEBRA Special Linear Group The special linear group SLn(q) is the MATRIX GROUP corresponding to the set of n /C29n COMPLEX MATRICES having DETERMINANT /C271: It is a SUBGROUP of the GENERAL LINEAR GROUP GLn(q) and is also a LIE GROUP . See also GENERAL LINEAR GROUP ,SPECIAL ORTHO- GONAL GROUP ,SPECIAL UNITARY GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GLn(q); SLn(q) ; PGLn(q) ; and PSLn(q) /C30Ln(q) :/" §2.1 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. Special Linear Lie Algebra Denoted sln:/ See also LIE ALGEBRA ,SPECIAL LIE ALGEBRA Special Matrix An INTEGER MATRIX whose entries satisfy aij/C300i f j>i/C271 /C281i f j/C30i/C271 0o r 1 i f j51:8 < : There are 2n/C281special MINIMAL MATRICES of size n /C29n :/ References Knuth, D. E. "Problem 10470." Amer. Math. Monthly 102, 655, 1995. Special Orthogonal Group The special orthogonal group SOn(q) is the SUBGROUP of the elements of GENERAL ORTHOGONAL GROUP GOn(q) with DETERMINANT 1. SO3(often written SO(3) is the ROTATION GROUP for 3-dimensional space. See also BIPOLYHEDRAL GROUP ,GENERAL ORTHOGO- NAL GROUP ,ICOSAHEDRAL GROUP ,ROTATION GROUP , SPECIAL LINEAR GROUP ,SPECIAL UNITARY GROUP References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GOn(q) ; SOn(q) ; PGSOn(q); and PSOn(q) ; and On(q) :/" §2.4 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xi-xii, 1985. Special Orthogonal Matrix A SQUARE MATRIX A is a special orthogonal matrix if AAT /C30I : (1) where I is the IDENTITY MATRIX , and the DETERMINANT satisfies det A /C301: (2) The first condition means that A is an ORTHOGONAL MATRIX , and the second restricts the determinant to /C271 (while a general ORTHOGONAL MATRIX may have determinant /C281or/C271): For example, 1ffiffiffi 2p1 /C281 11YrtvYrtu (3) is a special orthogonal matrix since 1ffiffi 2p/C281ffiffi 2p 1ffiffi 2p 1ffiffi 2p"#1ffiffi 2p 1ffiffi 2p /C281ffiffi 2p 1ffiffi 2p"# /C3010 01YrtvYrtu (4) and its DETERMINANT is 1 =2 /C28(/C281=2) /C301: A matrix m can be tested to see if it is a special orthogonal matrix using the Mathematica function SpecialOrthogonalQ[m_List?MatrixQ] : /C30 (Transpose[m].m /C30/C30 IdentityMatrix@Length@m && Det[m] /C30/C30 1) The special orthogonal matrices are CLOSED under multiplication and the inverse operation, and there- fore form a MATRIX GROUP called the SPECIAL ORTHO- GONAL GROUP SO(n) :/ See also INNER PRODUCT ,O RTHOGONAL GROUP , ORTHOGONAL MATRIX ,O RTHOGONAL TRANSFORMA-TION ,S KEW SYMMETRIC MATRIX ,S PECIAL LINEAR MATRIX ,SPECIAL ORTHOGONAL GROUP ,SPIN GROUP , UNITARY MATRIX Special Point A POINT which does not lie on at least one ORDINARY LINE. See also ORDINARY POINT References Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903/C1/09, 1989. Special Series Theorem If the difference between the order and the dimension of a series is less than the GENUS (CURVE ), then the series is special. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 253, 1959. Special Unitary Group The special unitary group SUn(q) is the set of n/C29n UNITARY MATRICES with DETERMINANT /C271 (having n2/C281 independent parameters). SU(2) is HOMEO- MORPHIC with the ORTHOGONAL GROUP O/C27 3(2):It is also called the UNITARY UNIMODULAR GROUP and is a LIE GROUP . Special unitary groups can be represented by ma- trices U(a;b)/C30ab /C28¯b¯aYrtvYrtu : (1) where ¯aa/C27¯bb/C301 and a, b are the C AYLEY- KLEIN PARAMETERS . The special unitary group may also be represented by matrices U(j;h;z)/C30eijcosh eizsinh /C28e/C28izsinhe/C28ijcoshYrtvYrtu : (2) or the matrices Ux1 2fYru*Yru+ /C30cos1 2fYru*Yru+ isin12fYru*Yru+ isin12fYru*Yru+ cos12fYru*Yru+2 435 (3) U y1 2bYru*Yru+ /C30cos1 2bYru*Yru+ sin12bYru*Yru+ /C28sin12bYru*Yru+ cos12bYru*Yru+2 435 (4) U z(j)/C30eij0 0e/C28ijYrtvYrtu (5) The order 2 j/C271 representation is U(j) p ; q( a; b; g) /C30X m(/C281)m/C28q /C28pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (j /C27 p)!(j /C28 p)!(j /C27 q)!(j /C28 q)!p (j /C28 p /C28 m)!(j /C27 q /C28 m)!(m /C27 p /C28 q)!m! /C29eiqa cos2j/C27q /C28p/C282m1 2 bYru*Yru+ sinp /C272m/C28q12 bYru*Yru+ eipg (6) The summation is terminated by putting 1 =(/C28N)! /C300: The CHARACTER is given by X(j)( a) /C301 /C272 cos a /C27.../C272 cos(ja) 2 cos12 aYru*Yru+ /C27cos32 aYru*Yru+ /C27.../C27cos(j a)hi( /C30sin j /C2712Yru*Yru+ ahi sin1 2 aYru*Yru+ for j /C300 ; 1 ; 2; ... sin j /C2712Yru*Yru+ ahi sin1 2 aYru*Yru+ for j /C3012 ;32; ...:8 >>>>>>< >>>>>>:(7) See also O RTHOGONAL GROUP ,S PECIAL LINEAR GROUP ,SPECIAL ORTHOGONAL GROUP References Arfken, G. "Special Unitary Group, SU(2) and SU(2)/-/O/C27 3 Homomorphism." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 253 /C1/59, 1985. Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GUn(q) ; SUn(q); PGUn(q); and PSUn(q) /C30Un(q) :/" §2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985. Special Unitary Matrix A SQUARE MATRIX U is a special unitary matrix if UU/C31/C30I: (1) where I is the IDENTITY MATRIX and U /C31 is the ADJOINT MATRIX , and the DETERMINANT is det U /C301: (2) The first condition means that U is a UNITARY MATRIX , and the second condition provides a restriction beyond a general UNITARY MATRIX , which may have determinant eiu for u any real number. For example, 1ffiffiffi 2pii i /C28iYrtvYrtu (3) is a special unitary matrix. A matrix m can be tested to see if it is a special unitary matrix using the Mathematica function SpecialUnitaryQ[m_List?MatrixQ] : /C30 (Conjugate@[email protected] /C30/C30 IdentityMa- trix@Length@m && Det[m] /C30/C30 1)The special unitary matrices are CLOSED under multiplication and the inverse operation, and there- fore form a MATRIX GROUP called the SPECIAL UNITARY GROUP SU(n) :/ See also HERMITIAN INNER PRODUCT ,SKEW HERMI- TIAN MATRIX ,S PECIAL LINEAR MATRIX ,S PECIAL UNITARY GROUP ,SPIN GROUP ,UNITARY GROUP UNI- TARY MATRIX Species A species of structures is a rule F which 1. Produces, for each finite set U, a finite set F[U];/ 2. Produces, for each bijection s : U 0 V ; a func- tion F[ s]:F[U] 0 F[V] : The functions F[ s] should further satisfy the follow- ing functorial properties: 1. For all bijections s : U 0 V and t : V 0 W ; F[t( s] /C30F[ t](F[s]: 2. For the IDENTITY MAP IdU : U 0 U ; F [Id] U/C30 Id F[U]: An element s /C23 F[U] is called an F-structure on U (or a structure of species F on U). The function F[ s]is called the transport of F-structures along s:/ References Bergeron, F.; Labelle, G.; and Leroux, P. Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, p. 5, 1998. Specificity The probability that a STATISTICAL TEST will be negative for a negative statistic. See also SENSITIVITY ,S TATISTICAL TEST,T YPE I ERROR ,TYPE II ERROR Spectral Graph Partitioning A GRAPHICAL PARTITIONING based on the eigenvalues and eigenvectors of the LAPLACIAN MATRIX of a graph. See also GRAPHICAL PARTITION ,LAPLACIAN MATRIX References Chung, F. R. K. Spectral Graph Theory. Providence, RI: Amer. Math. Soc., 1997. Demmel, J. "CS 267: Notes for Lecture 23, April 9, 1999. Graph Partitioning, Part 2." http://www.cs.berkeley.edu/ ~demmel/cs267/lecture20/lecture20.html. Spectral Norm The NATURAL NORM induced by the L2-NORM . Let A/C31 be the ADJOINT of the SQUARE MATRIX A; so that (aij) /C31/C30(¯aji) ; then Akk2/C30(maximum eigenvalue of A/C31A)1=2 /C30 max xkk2 "0Axkk2 xkk2: This MATRIX NORM is implemented as Matrix- Norm [m, 2] in the Mathematica add-on package LinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also L2-NORM,M ATRIX NORM,M AXIMUM ABSO- LUTE COLUMN SUM NORM,MAXIMUM ABSOLUTE ROW SUM NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1115, 2000. Strang, G. §6.2 and 7.2 in Linear Algebra and Its Applica- tions, 4th ed. New York: Academic Press, 1980. Spectral Power Density Py( n) /C13 lim T 0/C122 T gT =2 /C28T =2[y(t) /C28 ¯y]e /C282 pint dtYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut2 : so g/C12 0Py( n) dn /C13 lim T 0/C121 T gT =2 /C28T =2[y(t) /C28 ¯y]2 dt /C30 (y /C28 ¯y)2DE /C30 s2 y : See also POWER SPECTRUM Spectral Radius Let A be an n /C29n MATRIX with COMPLEX or REAL elements with EIGENVALUES l1 ; ..., ln : Then the spectral radius r(A)ofA is r(A) /C30max 15i5n½ li½: References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1115 /C1/116, 2000. Spectral Rigidity The mean square deviation of the best local fit straight line to a staircase cumulative spectral density over a normalized energy scale.References Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 341, 1993. Spectral Theorem Let H be a HILBERT SPACE , B(H) the set of BOUNDED linear operators from H to itself, T an OPERATOR on H, and s(T) the SPECTRUM of T. Then if T /C23 B(H) and T is normal, there exists a unique resolution of the identity E on the BOREL SUBSETS of s(T) which satisfies T /C30gs(T)l dE(l) : Furthermore, every projection E( v) COMMUTES with every S /C23 B(H) that COMMUTES with T. See also SPECTRUM (OPERATOR ) References Rudin, W. Theorem 12.23 in Functional Analysis, 2nd ed. New York: McGraw-Hill, 1991. Spectrum The word "spectrum" confusingly has a number of unrelated meanings in various branches of mathe- matics. See also GRAPH SPECTRUM ,S PECTRUM (MATRIX ), SPECTRUM (OPERATOR ), SPECTRUM (RING), SPECTRUM SEQUENCE Spectrum (Graph) GRAPH SPECTRUM Spectrum (Matrix) The EIGENVALUES of a MATRIX A are called its spectrum, and are denoted l(A): If l(A) /C30 fl1 ; ...; ln g; then the DETERMINANT of A is given by det(A)/C30l1l2...ln: See also CHARACTERISTIC POLYNOMIAL ,EIGENVALUE References Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 310, 1996. Spectrum (Operator) Let Tbe an OPERATOR on a H ILBERT SPACE . The spectrum s(T)o fTis the set of lsuch that ( T/C28lI)i s not invertible on all of the H ILBERT SPACE , where the l/s are COMPLEX NUMBERS and Iis the IDENTITY OPERATOR . The definition can also be stated in terms of the resolvent of an operator r(T) /C30fl :(T /C28 lI) is invertible g; and then the spectrum is defined to be the comple- ment of r(T) in the COMPLEX PLANE . It is easy to demonstrate that r(T)isan OPEN SET, which shows that the spectrum is closed (in fact, it is even compact). If V is a domain in Rd (i.e., a Lebesgue measurable subset of Rd with finite nonzero LEBESGUE MEASURE ), the Iosevich et al. (1999) say a set LƒRdis a spectrum of V is e2 pix lfgl /C23Lis an ORTHOGONAL BASIS of L2( V) :/ See also FUGLEDE’S CONJECTURE ,H ILBERT SPACE , ORTHOGONAL BASIS,SPECTRAL THEOREM References Iosevich, A.; Katz, N. H.; and Tao, T. Convex Bodies with a Point of Curvature Do Not Have Fourier Bases. 23 Nov 1999. http://xxx.lanl.gov/abs/math.CA/9911167/. Rudin, W. Functional Analysis, 2nd ed. New York: McGraw- Hill, 1991. Spectrum (Ring) The spectrum of a RING is the set of proper PRIME IDEALS , Spec( R) /C30fp : p is a prime ideal in R g: (1) The classical example is the spectrum of POLYNOMIAL RINGS . For instance, Spec(C[x]) /C30 x /C28a hi : a /C23C fg @ 0hifg : (2) and Spec(C[x ; y]) /C30 x /C28a; y /C28b hi ; (a; b) /C23C2Yr$Yr% @ f(x; y) hi : f is irreducable fg @ 0hifg : (3) The points are, in classical algebraic geometry, ALGEBRAIC VARIETIES . Note that x /C28a ; y /C28b hi are MAXIMAL IDEALS , hence also prime. The spectrum of a ring has a TOPOLOGY called the ZARISKI TOPOLOGY . The closed sets are of the form V(S) /C30 phi: S ƒ phi fg : (4) For example, Spec(Z) /C30 phi: p is prime fg @ 0hifg : (5) Every PRIME IDEAL is closed except for 0hi; whose closure is V(0) /C30Spec(Z) :/ See also AFFINE SCHEME ,CATEGORY THEORY ,COM- MUTATIVE ALGEBRA ,CONIC SECTION ,IDEAL ,PRIME IDEAL ,PROJECTIVE VARIETY ,SCHEME ,VARIETY ,ZAR- ISKI TOPOLOGY References Bump, D. Algebraic Geometry. Singapore: World Scientific, pp. 1 /C1/, 1998. Hartshorne, R. Algebraic Geometry. New York: Springer- Verlag, 1977.Spectrum Sequence A spectrum sequence is a SEQUENCE formed by successive multiples of a REAL NUMBER a rounded down to the nearest INTEGER sn /C30 nabc : If a is IRRATIONAL , the spectrum is called a BEATTY SE- QUENCE . See also BEATTY SEQUENCE ,LAGRANGE SPECTRUM , MARKOV SPECTRUM Speed The SCALAR ½v½/C30ds =dt; where s is the ARC LENGTH , equal to the magnitude of the VELOCITY v. See also ANGULAR VELOCITY ,VELOCITY Spencer’s 15-Point Moving Average A MOVING AVERAGE using 15 points having weights /C283, /C286, /C285, 3, 21, 46, 67, 74, 67, 46, 21, 3, /C285, /C286, and /C283. It is sometimes used by actuaries. See also MOVING AVERAGE ,SPENCER’S FORMULA References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 223, 1962. Spencer’s Formula Define the notation [n]f0 /C30f/C28(n/C281)=2 /C27.../C27f0 /C27.../C27f(n /C281)=2 (1) and let d be the central difference, so d2f0 /C30f1 /C282f0 /C27f/C281 : (2) Spencer’s 21-term moving average formula is then given by f ?0 /C30[5][5][7] 5 /C215 5 /C215 7 (1 /C284d2)f0 ; which, written explicitly, gives f ?0 /C301 350 60f0 /C2757(f /C281 /C27f1) /C2747(f /C282 /C27f2) /C2733(f /C283 /C27f3) ½ /C2718(f/C284 /C27f4) /C276(f /C285 /C27f5) /C282(f /C286 /C27f6) /C285(f/C287 /C27f7) /C285f/C288/C27f8 ðÞ /C283f/C289/C27f9 ðÞ /C28f/C2810/C27f10 ðÞ /C138 (3) See also MOVING AVERAGE ,SMOOTHING References Spencer, J. J. I. A. 38, 334, 1904. Spencer, J. J. I. A. 38, 339, 1904. Spencer, J. J. I. A. 41, 361, 1907. Whittaker, E. T. and Robinson, G. "Spencer’s Formula." §144 inThe Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 290 /C1/94, 1967. Spence’s Function F(x) /C30/C28Li2(/C28x) /C30gx 0ln(1 /C27 t) tdt: where Li2(x) is the DILOGARITHM . See also DILOGARITHM ,SPENCE’S INTEGRAL References Berestetskii, V. B.; Lifschitz, E. M.; and Ditaevskii, L. P. Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, p. 596, 1982. Spence’s Integral F(x) /C30Li2(1 /C28x) /C30g0 1 /C28xln(1 /C28 t) tdt: where Li2(x) is the DILOGARITHM . See also DILOGARITHM ,SPENCE’S FUNCTION Sperner System ANTICHAIN Sperner’s Theorem The MAXIMUM CARDINALITY of a collection of SUBSETS of a t-element SET T, none of which contains another, is the BINOMIAL COEFFICIENTt /C28t =2/C29Yru*Yru+ ; where xbcis the FLOOR FUNCTION . See also CARDINALITY Sphenocorona JOHNSON SOLID J86:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Sphenoid DISPHENOID Sphenomegacorona JOHNSON SOLID J88:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Sphere A sphere is defined as the set of all points in R3which are a distance r(the " RADIUS ") from a given point (the "CENTER "). Twice the RADIUS is called the DIAMETER , and pairs of points on opposite sides of a DIAMETER are called ANTIPODES . The term "sphere" technically refers to the outer surface of a " BUBBLE ," which is denoted S2:However, in common usage, the word sphere is also used to mean the UNION of a sphere and itsINTERIOR (a "solid sphere"), where the INTERIOR is called a BALL . The SURFACE AREA of the sphere and VOLUME of the BALL ofRADIUS Rare given by S/C304pR2(1) V/C304 3pR3(2) (Beyer 1987, p. 130). In On the Sphere and Cylinder (ca. 225 BC ), Archimedes became the first to derive these equations (although he expressed pin terms of the sphere’s circular CROSS SECTION ). The fact thatVsphere Vcircumscribed cylinder /C28Vsphere/C302 (3) was also known to Archimedes (Steinhaus 1983,p. 223; Wells 1991, pp. 236 /C1 /37). Any CROSS SECTION through a sphere is a CIRCLE (or, in the degenerate case where the slicing PLANE is tangent to the sphere, a point). The size of the CIRCLE is maximized when the PLANE defining the CROSS SECTION passes through a DIAMETER . The equation of a sphere of RADIUS ris given in CARTESIAN COORDINATES by x2/C27y2/C27z2/C30r2: (4) which is a special case of the ELLIPSOID x2 a2/C27y2 b2/C27z2 c2/C301 (5) and SPHEROID x2/C27y2 a2/C27z2 c2/C301: (6) A sphere may also be specified in SPHERICAL COORDI- NATES by x/C30rcosusinf (7) y/C30rsinusinf (8) z/C30rcosf: (9) where uis an azimuthal coordinate running from 0 to 2p(LONGITUDE ),fis a polar coordinate running from 0t op(COLATITUDE ), and ris the RADIUS . Note that there are several other notations sometimes used inwhich the symbols for uandfare interchanged or where ris used instead of r:Ifris allowed to run from 0 to a given RADIUS r, then a solid BALL is obtained. The volume of the sphere, V/C304=3pR3;can be found in Cartesian, cylindrical, and spherical coordinates, respectively, using the integrals V/C30gR /C28Rgffiffiffiffiffiffiffiffiffiffi R2/C28x2p /C28ffiffiffiffiffiffiffiffiffiffi R2/C28x2pgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2/C28x2/C28y2p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2/C28x2/C28y2p dz dy dx (10) /C30g2p 0gR 0gffiffiffiffiffiffiffiffiffiffi R2/C28x2p /C28ffiffiffiffiffiffiffiffiffiffi R2/C28x2prd zd rd u (11) /C30g2p 0gp 0gR 0r2sinfdrdfdu: (12) Converting to "standard" parametric variables a/C30r; u/C30u;and v/C30fgives the coefficients of the FIRST FUNDAMENTAL FORM E/C30a2sin2v (13) F /C300 (14) G /C30a2 : (15) SECOND FUNDAMENTAL FORM coefficients e /C30a sin2 v (16) f /C300 (17) g /C30a: (18) AREA ELEMENT dA /C30a sin vduffldv: (19) GAUSSIAN CURVATURE K /C301 a2 : (20) and MEAN CURVATURE H /C301 a : (21) A sphere may also be represented parametrically by letting u /C13r cos f ; so x /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28u2p cos u (22) y /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir 2 /C28u2p sin u (23) z /C30u ; (24) where u runs from 0 to 2p and u runs from /C28r to r. Given two points on a sphere, the shortest path on the surface of the sphere which connects them (the SPHERE GEODESIC )isan ARC of a CIRCLE known as a GREAT CIRCLE . The equation of the sphere with points fx1 ; y1 ; z1 g and fx2 ; y2 ; z2 g lying on a DIAMETER is given by (x /C28x1)(x /C28x2) /C27(y /C28y1)(y /C28y2) /C27(z /C28z1)(z /C28z2) /C300: (25) Four points are sufficient to uniquely define a sphere. Given the points fxi ; yi ; zi g with i /C301, 2, 3, and 4, the sphere containing them is given by the beautiful DETERMINANT equation x2 /C27y2 /C27z2xyz 1 x2 1 /C27y21 /C27z21x1y1z11 x22 /C27y22 /C27z22x2y2z21 x23 /C27y23 /C27z23x3y3z31 x24 /C27y24 /C27z24x4y4z41YrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C300 (26) (Beyer 1987, p. 210). The generalization of a sphere in n dimensions is called a HYPERSPHERE .An n-D HYPERSPHERE can be specified by the equation x2 1/C27x22/C27.../C27x2n/C30r2: (27) The distribution of ANGLES for random rotation of asphere is P(u)/C302 psin21 2uYru*Yru+ ; (28) giving a MEAN ofp=2/C272=p:/ See also BALL,BING’S THEOREM ,BOWL OF INTEGERS , BUBBLE ,C IRCLE ,C ONE- SPHERE INTERSECTION ,C Y- LINDER- SPHERE INTERSECTION ,D ANDELIN SPHERES , DIAMETER ,ELLIPSOID ,EXOTIC SPHERE ,FEJES TO´ TH’S PROBLEM ,G EODESIC DOME,G LOME ,H YPERSPHERE , LIEBMANN’S THEOREM ,LIOUVILLE’S SPHERE- PRESER- VING THEOREM ,M IKUSINSKI’S PROBLEM ,N OISE SPHERE ,O BLATE SPHEROID ,O SCULATING SPHERE , PARALLELIZABLE ,PROLATE SPHEROID ,RADIUS ,SPACE DIVISION BY SPHERES ,SPHERE PACKING ,SPHERE- SPHERE INTERSECTION ,TANGENT SPHERES ,TENNIS BALL THEOREM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971. Eppstein, D. "Circles and Spheres." http://www.ics.uci.edu/ ~eppstein/junkyard/sphere.html. Fukagawa, H. and Pedoe, D. "Spheres," "Spheres and Ellipsoids," and "Spheres, Pyramids and Prisms". §2.2/C1/.6 and 9.1 /C1/.3 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 26 /C1/7, 69/C1/6, 102 /C1/16, and 160 /C1/66, 1989. Harris, J. W. and Stocker, H. "Sphere." §4.8 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 106 /C1/08, 1998. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, p. 10, 1999. JavaView. "Classic Surfaces from Differential Geometry: Sphere." http://www-sfb288.math.tu-berlin.de/vgp/java-view/demo/surface/common/PaSurface_Sphere.html. Kenison, E. and Bradley, H. C. "The Intersection of a Sphere with Another Surface." §198 in Descriptive Geometry. New York: Macmillan, 1935. Kern, W. F. and Bland, J. R. "Sphere." §33 in Solid Men- suration with Proofs, 2nd ed. New York: Wiley, pp. 87 /C1 /3, 1948. Kiang, T. "An Old Chinese Way of Finding the Volume of a Sphere." Math. Gaz. 56,8 8/C1/1, 1972. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Sphere Embedding A 4-sphere has POSITIVE CURVATURE , with R2/C30x2/C27y2/C27z2/C27w2(1) 2xdx dw/C272ydy dw/C272zdz dw/C272w/C300: (2) Since r/C13xˆx/C27yˆy/C27zˆz: (3) dw/C30/C28xd x/C27yd y/C27zd z w/C30/C28r /C215drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2/C28r2p : (4) To stay on the surface of the sphere, ds2 /C30dx2 /C27dy2 /C27dz2 /C27dw2 /C30dx2 /C27dy2 /C27dz2 /C27r2 dr2 R2 /C28 r2 /C30dr2 /C27r2 d V2 /C27dr2 R2 r2/C28 1 /C30dr21 /C271 R2 r2 /C28 10 BBB@1 CCCA/C27r 2 dV2 /C30dr2R2 r2 R2 r2 /C28 10 BBB@1 CCCA/C27r 2 dV2 /C30dr2 1 /C28r2 R2/C27r2 dV2 : (5) With the addition of the so-called expansion para- meter, this is the Robertson-Walker line element. Sphere Eversion Smale (1958) proved that it is mathematically possi- ble to turn a SPHERE inside-out without introducing a sharp crease at any point. This means there is a regular homotopy from the standard embedding of the 2-SPHERE in EUCLIDEAN 3-space to the mirror- reflection embedding such that at every stage in the homotopy, the sphere is being IMMERSED in EUCLI- DEAN SPACE . This result is so counterintuitive and the proof so technical that the result remained contro- versial for a number of years. In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin’s eversion also produced explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1979). See also EVERSION ,SPHERE References Bulatov, V. "Sphere Eversion--Visualization of the Famous Topological Procedure." http://www.physics.orst.edu/~bu- latov/vrml/evert.wrl. Francis, G. K. Ch. 6 in A Topological Picturebook. New York: Springer-Verlag, 1987. Francis, G. K. and Morin, B. "Arnold Shapiro’s Eversion of the Sphere." Math. Intell. 2, 200 /C1/03, 1979.Levy, S.; Maxwell, D.; and Munzner, T. Making Waves: A Guide to the Ideas Behind Outside In. Wellesley, MA: A. K. Peters, 1995. Book and 22 minute Outside-In. videotape. Max, N. "Turning a Sphere Inside Out." Videotape. Chicago, IL: International Film Bureau, 1977. Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 240 /C1/44, 1990. Petersen, I. "Forging Links Between Mathematics and Art." Science News 141, 404 /C1/05, June 20, 1992. Phillips, A. "Turning a Surface Inside Out." Sci. Amer. 214, 112 /C1/20, Jan. 1966. Smale, S. "A Classification of Immersions of the Two- Sphere." Trans. Amer. Math. Soc. 90, 281 /C1/90, 1958. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Sphere Geodesic GREAT CIRCLE Sphere Inversion INVERSION in 3 dimensions with respect to an INVER- SION SPHERE . See also INVERSION ,INVERSION SPHERE Sphere Line Picking Pick two points at random on a unit sphere. The first one can be placed at the north pole, i.e., assigned the coordinate (0, 0, 1), without loss of generality. Thesecond point is then chosen at random using SPHERE POINT PICKING , and so can be assigned coordinates x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u2p cosu (1) y/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28u 2p sinu (2) z/C30u (3) with u/C23[/C281;1] and u/C23[0;2p):The distance lbe- tween first and second points is then l/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2/C27(z/C281)2q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C282up ; (4) and solving for ugives u/C301 22/C28l2YrvYru : (5) Now the probability function Plfor distance is then given by Pl/C30Pu@u @lYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30 1 2ldl (6) (Solomon 1978, p. 163), since Pu/C301=2 and du=dl/C30 /C28l:Here, l/C23[0;2]:/ Therefore, somewhat surprisingly, large distances are the most common, contrary to most people’s intuition. A plot of 15 random lines is shown above. The RAW MOMENTS are m?n /C30 lnhi/C30g2 0lnPl dl /C302n/C271 2 /C27 n : (7) giving the first few as m?1 /C304 3 (8) m?2 /C302 (9) m?3 /C3016 5 (10) m?4 /C3016 3 : (11) so the CENTRAL MOMENTS are m /C3043 (12) m2 /C30 s2 /C3029 (13) m3 /C30/C288 135 (14) m4 /C3016 135: (15) so the VARIANCE , SKEWNESS and KURTOSIS are s2/C302 9(16) g1/C304 5ffiffiffi 2p (17) g2/C30/C285 3(18) (Solomon 1978, p. 163). See also BALL LINE PICKING ,CIRCLE LINE PICKING , POINT- POINT DISTANCE–1- D, SPHERE POINT PICKING , SPHERE TETRAHEDRON PICKING References Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978.Sphere Packing In 2-D, there are two periodic CIRCLE PACKINGS for identical circles: square lattice and hexagonal lattice. Fejes To ´th (1940) proved that the hexagonal lattice is the densest of allpossible plane packings (Conway and Sloane 1993, pp. 8 /C1/). In 3-D, there are three periodic packings for identicalspheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal) is thedensest possible (has the greatest PACKING DENSITY h; which is the fraction of a VOLUME filled by identical packed SPHERES ), and this assertion is known as the KEPLER CONJECTURE . The problem of finding the densest packing of spheres (not necessarily periodic)is therefore known as the K EPLER PROBLEM . The KEPLER CONJECTURE is intuitively obvious, but the proof remained elusive until it was accomplished in aseries of papers by Hales culminating in 1998. Gauss(1831) did prove that the face-centered cubic is thedensest lattice packing in 3-D (Conway and Sloane 1993, p. 9). This result has since been extended to HYPERSPHERE PACKING . The maximum number of equivalent spheres (or n-D hyperspheres) which can touch an equivalent sphere (hypersphere) without intersections is called the n-D KISSING NUMBER . In 3-D, face-centered cubic close packing and hex- agonal close packing (which is distinct from hexago- nal lattice packing), both give hCCP/C30hHCP/C30p 3ffiffiffi 2p:74:048% (1) (Steinhaus 1983, p. 202; Wells 1986, p. 29; Wells 1991, p. 237). For packings in 3-D, C. A. Rogers (1958) showed that the maximum possible PACKING DENSITY hmaxsatisfies hmaxBffiffiffiffiffiffi 18p cos/C2811 3/C2813pYru*Yru+ :77:96355700% (2) (Le Lionnais 1983). This was subsequently improved to 77.844% (Lindsey 1986), then 77.836% (Muder 1988). However, Rogers (1958) remarks that "many mathematicians believe, and all physicists know" thatthe actual answer is 74.048% (Conway and Sloane 1993, p. 3). Hilbert and Cohn-Vossen (1999, pp. 48 /C1/0) consider a tetrahedral packing in which each sphere touched four neighbors and the density is pffiffiffi 3p =16:0:3401 :/ The rigid packing with lowest density known has h: 0:0555 (Gardner 1966), significantly lower than that reported by Hilbert and Cohn-Vossen (1999, p. 51). To be rigid, each SPHERE must touch at least four others, and the four contact points cannot be in a single HEMISPHERE or all on one equator. RANDOM CLOSE PACKING of spheres in 3-D gives packing densities in the range 0.06 to 0.65 (Jaeger and Nagel 1992, Torquato et al. 2000). The PACKING DENSITIES for several packing types are summarized in the following table. Packing /h/ (exact)/h/ reference loose packing – 0.0555 Gardner (1966) tetrahedral lattice/pffiffi 3p 16/ 0.3401 Hilbert and Cohn-Vossen (1999, pp. 48 /C1/0) cubic lattice /p 6/ 0.5236 hexagonal lattice/p 3ffiffi 3p/ 0.6046 random – 0.6400 Jaeger and Nagel 1992 face-centeredcubic lattice/p 3ffiffi 2p/ 0.7405 Steinhaus 1983, p. 202; Wells 1986, p. 29;Wells 1991, p. 237 square lattice (2-D)/p 4/ 0.7854 hexagonal lattice (2-D)/p 2ffiffi 3p/ 0.9069 Arranging layers of close-packed spheres such thatthe spheres of every third layer overlying one another gives cubic close packing. To see where the namecomes from, consider packing six SPHERES together in the shape of an EQUILATERAL TRIANGLE and place another SPHERE on top to create a TRIANGULAR PYRAMID . Now create another such grouping of seven SPHERES and place the two PYRAMIDS together facing in opposite directions. A CUBE emerges (Steinhaus 1983, pp. 203 /C1/04). Connecting the centers of these 14 spheres gives a STELLA OCTANGULA . Consider the CUBE defined by 14 spheres in cubic close packing, as illustrated above. This "unit cell" contains eight 1 =8/-spheres (one at each VERTEX ) and sixHEMISPHERES . The total VOLUME ofSPHERES in the unit cell is therefore Vspheres in unit cell /C308/C2151 8/C276/C21512Yru*Yru+4p 3r3 /C304/C2154p 3r3/C3016 3pr3: (3) The diagonal of the face is 4 r;so each side is 2ffiffiffi 2p r: The VOLUME of the unit cell is therefore Vunit cell /C302ffiffiffi 2p rYru*Yru+3 /C3016ffiffiffi2p r3: (4) and the PACKING DENSITY is hCCP/C3016 3pr2 16ffiffiffi 2p r3/C30p 3ffiffiffi2p (5) (Conway and Sloane 1993, p. 2). In cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 suchspheres gives the cluster illustrated above. Connect- ing the centers of the external 12 spheres gives a CUBOCTAHEDRON (Steinhaus 1983, pp. 203 /C1/05; Wells 1991, p. 237). In hexagonal close packing, layers of spheres are packed so that spheres in alternating layers overlie one another. As in cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives JOHNSON SOLID J27known as the TRIANGULAR ORTHOBICUPOLA (Steinhaus 1983, pp. 203 /C1/05; Wells 1991, p. 237). Hexagonal close packing must give the same packing density as cubic close packing, since sliding one sheet of SPHERES cannot affect the volume they occupy. To verify this, construct a 3-D diagram containing a hexagonal unit cell with three layers (Steinhaus 1983, pp. 203 /C1/04). Both the top and the bottom contain six 1=6/-SPHERES and one HEMISPHERE . The total number of spheres in these two rows is therefore 26 /C2151 6 /C271 /C21512Yru*Yru+ /C303: (6) The VOLUME of SPHERES in the middle row cannot be simply computed using geometry. However, symme- try requires that the piece of the SPHERE which is cut off is exactly balanced by an extra piece on the other side. There are therefore three SPHERES in the middle layer, for a total of six, and a total VOLUME Vspheres in unit cell /C306 /C2154p 3r3(3 /C273) /C308pr3 : (7) The base of the HEXAGON is made up of 6 EQUILAT- ERAL TRIANGLES with side lengths 2r : The unit cell base AREA is therefore Aunit cell /C3061 2(2r)ffiffiffi 3p rYru*Yru+hi /C306ffiffiffi3p r2 : (8) The height is the same as that of two TETRAHEDRA length 2r on a side, so hunit cell /C3022rffiffiffi 2 3s ! : (9) giving hHCP /C308 pr3 6ffiffiffi 3p r2YrvYru 4rffiffi 2 3qYru$Yru% /C30p 3ffiffiffi 2p (10) (Conway and Sloane 1993, pp. 7 and 9). If we had actually wanted to compute the VOLUME of SPHERE inside and outside the HEXAGONAL PRISM ,we could use the SPHERICAL CAP equation to obtain Vƒ/C301 3 ph2(3r /C28h) /C301 3pr3133 /C281ffiffiffi 3p ! /C301 9pr33 /C28ffiffiffi 3p 3 ! /C301 27 pr3 9 /C28ffiffiffi 3pYru*Yru+ (11)V‡/C30 pr34 3 /C281 27(9 /C28ffiffiffi 3p )hi /C301 27 pr3 36 /C289 /C27ffiffiffi3pYru*Yru+ /C301 27 pr3 27 /C27ffiffiffi 3pYru*Yru+ : (12) If spheres packed in a cubic lattice, face-centered cubic lattice, and hexagonal lattice are allowed to expand uniformly until running into each other, they form cubes, hexagonal prisms, and rhombic dodeca- hedra, respectively. In particular, if the spheres of cubic close packing are expanded until they fill up the gaps, they form a solid RHOMBIC DODECAHEDRON (left figure above), and if the spheres of hexagonal close packing are expanded, they form a second irregular dodecahedron consisting of six rhombi and six trape- zoids (right figure above; Steinhaus 1983, p. 206). The latter can be obtained from the former by slicing in half and rotating the two halves 60 8 with respect to each other. The lengths of the short and long edges of the rotated dodecahedron have lengths /2=3/and /4=3/ times the length of the rhombic faces. Both the RHOMBIC DODECAHEDRON and squashed dodecahe- dron are SPACE-FILLING POLYHEDRA . Compressing a random packing gives polyhedra with an average of 13.3 faces (Coxeter 1958, 1961). For sphere packing inside a CUBE , see Goldberg (1971), Schaer (1966), and Friedman. See also CANNONBALL PROBLEM ,C IRCLE PACKING , CUBOCTAHEDRON ,DODECAHEDRAL CONJECTURE ,EL- LIPSOID PACKING ,H EMISPHERE ,H ERMITE CON- STANTS ,H YPERSPHERE ,H YPERSPHERE PACKING , KEPLER CONJECTURE ,K EPLER PROBLEM ,K ISSING NUMBER ,LOCAL DENSITY ,LOCAL DENSITY CONJEC- TURE ,RANDOM CLOSE PACKING ,REULEAUX TETRAHE- DRON ,S PACE- FILLING POLYHEDRON ,S PHERE , SPHERICAL DESIGN ,SPHERICON ,STELLA OCTANGULA , TANGENT SPHERES ,T RIANGULAR ORTHOBICUPOLA , UNIT CELL References Barlow, W. "Probable Nature of the Internal Symmetry of Crystals." Nature 29, 186/C1/88, 1883. Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lat- tices, and Groups, 2nd ed. New York: Springer-Verlag, 1993. Coxeter, H. S. M. "Close-Packing and so Forth." Illinois J. Math. 2, 746/C1/58, 1958. Coxeter, H. S. M. "Close Packing of Equal Spheres." Section 22.4 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 405 /C1/11, 1961. Coxeter, H. S. M. "The Problem of Packing a Number of Equal Nonoverlapping Circles on a Sphere." Trans. New York Acad. Sci. 24, 320/C1/31, 1962. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 195 /C1/97, 1989. Eppstein, D. "Covering and Packing." http://www.ics.u- ci.edu/~eppstein/junkyard/cover.html. Fejes To ´th, G. "U ¨ber einen geometrischen Satz." Math. Z. 46,7 8/C1/3, 1940. Fejes To ´th, G. Lagerungen in der Ebene, auf der Kugel und in Raum, 2nd ed. Berlin: Springer-Verlag, 1972. Friedman, E. "Spheres in Cubes." http://www.stetson.edu/ ~efriedma/sphincub/. Gardner, M. "Packing Spheres." Ch. 7 in Martin Gardner’s New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 82 /C1/0, 1966. Gauss, C. F. "Besprechung des Buchs von L. A. Seeber: Intersuchungen u ¨ber die Eigenschaften der positiven terna¨ren quadratischen Formen usw." Go¨ttingsche Ge- lehrte Anzeigen (1831, July 9) 2, 188/C1/96, 1876. Goldberg, M. "On the Densest Packing of Equal Spheres in a Cube." Math. Mag. 44, 199/C1/08, 1971. Hales, T. C. "The Sphere Packing Problem." J. Comput. Appl. Math 44,4 1/C1/6, 1992. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, pp. 45 /C1/3, 1999. Jaeger, H. M. and Nagel, S. R. "Physics of Granular States." Science 255, 1524, 1992. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 31, 1983. Lindsey, J. H. II. "Sphere Packing in R3:/"Math. 33, 137/C1/47, 1986. Muder, D. J. "Putting the Best Face of a Voronoi Polyhe- dron." Proc. London Math. Soc. 56, 329/C1/48, 1988. Rogers, C. A. "The Packing of Equal Spheres." Proc. London Math. Soc. 8, 609/C1/20, 1958. Rogers, C. A. Packing and Covering. Cambridge, England: Cambridge University Press, 1964. Schaer, J. "On the Densest Packing of Spheres in a Cube." Can. Math. Bul. 9, 265/C1/70, 1966. Sigrist, F. "Sphere Packing." Math. Intell. 5,3 4/C1/8, 1983. Sloane, N. J. A. "The Packing of Spheres." Sci. Amer. 250, 116/C1/25, 1984. Sloane, N. J. A. "The Sphere Packing Problem." Proc. Inter- nat. Congress Math., Vol. 3 (Berlin, 1998). Doc. Math. Extra Volume ICM 1998, 387 /C1/96, 1998. http://www.re- search.att.com/~njas/doc/icm.ps. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 202 /C1/03, 1999. Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, pp. 69 /C1/2, 1987. Thompson, T. M. From Error-Correcting Codes Through Sphere Packings to Simple Groups. Washington, DC: Math. Assoc. Amer., 1984. Torquato, S.; Truskett, T. M.; and Debenedetti, P. G. "Is Random Close Packing of Spheres Well Defined?" Phys. Lev. Lett. 84, 2064 /C1/067, 2000. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 29, 1986. Weisstein, E. W. "Books about Sphere Packings." http:// www.treasure-troves.com/books/SpherePackings.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 237 /C1/38, 1991. Zong, C. and Talbot, J. Sphere Packings. New York: Springer-Verlag, 1999.Sphere Point Picking To pick a random point on the surface of a UNIT SPHERE , it is incorrect to select SPHERICAL COORDI- NATES uandffrom uniform distributions u/C23[0;2p) and f/C23[0;p];since the area element dV/C30 sinfdudfis a function of f;and hence points picked in this way will be "bunched" near the poles (left figure above). To obtain points such that any small area on the sphere is expected to contain the same number ofpoints (right figure above), choose uand vto be random variates on (0 ;1):Then u/C302pu (1) f/C30cos /C281(2v/C281) (2) gives the SPHERICAL COORDINATES for a set of points which are uniformly distributed over S2:This works since the differential element of SOLID ANGLE is given by dV/C30sinfdudf/C30dud(cosf): (3) Similarly, we can pick u/C30cosfto be uniformly distributed (so we have du/C30sinfdf) and obtain the points x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u2p cosu (4) y/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u2p sinu (5) z/C30u; (6) with u/C23[0;2p) and u/C23[/C281;1];which are also uni- formly distributed over S2:/ Marsaglia (1972) derived an elegant method that consists of picking x1and x2from independent uni- form distributions on ( /C281;1) and rejecting points for which x2 1/C27x22]1:From the remaining points, x/C302x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 1/C28x22q (7) y/C302x2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 1/C28x22q (8) z /C301 /C282(x2 1 /C27x22) (9) have a uniform distribution on the surface of a unit sphere. This method can also be extended to HYPER- SPHERE POINT PICKING . The plots above show the distribution of points for 100, 1000, and 5000 initial points (where the counts refers to the number of points before throwing away). Cook (1957) extended a method of von Neumann (1951) to give a simple method of picking points uniformly distributed on the surface of a UNIT SPHERE . Pick four numbers x0 ; x1 ; x2 ; and x3from a UNIFORM DISTRIBUTION on (/C281 ; 1); and reject pairs with x20 /C27x21 /C27x22 /C27x23 ]1 : (10) From the remaining points, the rules of QUATERNION transformation then imply that the points with CARTESIAN COORDINATES x /C302(x1x3 /C27 x0x2) x2 0 /C27 x21 /C27 x22 /C27 x23(11) y /C302(x2x3 /C28 x0x1) x20 /C27 x21 /C27 x22 /C27 x23(12) z /C30x2 0 /C27 x23 /C28 x21 /C28 x22 x2 0 /C27 x21 /C27 x22 /C27 x23(13) have the desired distribution (Cook 1957, Marsaglia 1972). The plots above show the distribution of points for 100, 1000, and 5000 initial points (where the counts refers to the number of points before throwing away). Another easy way to pick a random point on a SPHERE is to generate three Gaussian random variables x, y, andz. Then the distribution of the vectors 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2/C27z2px y z2 435 (14) is uniform over the surface S 2(Muller 1959, Marsa- glia 1972). See also BALL TRIANGLE PICKING ,C IRCLE POINT PICKING ,DISK POINT PICKING ,HYPERSPHERE POINT PICKING ,N OISE SPHERE ,S PHERE LINE PICKING , SPHERE TETRAHEDRON PICKING References Cook, J. M. "Technical Notes and Short Papers: Rational Formulae for the Production of a Spherically Symmetric Probability Distribution." Math. Tables Aids Comput. 11, 81/C1/2, 1957.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, pp. 130 /C1/31, 1998. Marsaglia, G. "Choosing a Point from the Surface of a Sphere." Ann. Math. Stat. 43, 645/C1/46, 1972. Muller, M. E. "A Note on a Method for Generating Points Uniformly on N-Dimensional Spheres" Comm. Assoc. Comput. Mach. 2,1 9/C1/0, 1959. Rusin, D. "N-Dim Spherical Random Number Drawing." in The Mathematical Atlas. http://www.math.niu.edu/~ru- sin/known-math/96/sph.rand. Stephens, M. A. "The Testing of Unit Vectors for Random- ness." J. Amer. Stat. Assoc. 59, 160/C1/67, 1964. von Neumann, J. "VArious Techniques Used in Connection with Random Digits." NBS Appl. Math. Ser. , No. 12. Washington, DC: U.S. Government Printing Office, pp. 36 /C1/8, 1951. Watson, G. S. and Williams, E. J. "On the Construction of Significance Tests on the Circle and Sphere." Biometrika 43, 344/C1/52, 1956. Sphere Tetrahedron Picking Pick four points on a sphere. What is the probability that the TETRAHEDRON having these points as VER- TICES contains the CENTER of the sphere? In the 1-D case, the probability that a second point is on theopposite side of /1=2/is /1=2/. In the 2-D case, pick two points. In order for the third to form a TRIANGLE containing the CENTER , it must lie in the quadrant bisected by a LINE SEGMENT passing through the center of the CIRCLE and the bisector of the two points. This happens for one QUADRANT , so the probability is /1=4/. Similarly, for a sphere the prob- ability is one OCTANT ,o r /1=8/. Pick four points at random on the surface of a unit SPHERE using x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u2p cosu (1) y/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28u 2p sinu (2) z/C30u (3) with u/C23[/C281;1] and u/C23[0;p):Now find the distribu- tion of possible VOLUMES of the (nonregular) TETRA- HEDRA determined by these points. Without loss of generality, the first point may be taken as u1/C301;or (0;0;1);while the second may be taken as (0 ;u2);orffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28u2 2p ;0;u2Yru*Yru+ :The average VOLUME is then ¯V/C30f1 /C281f1 /C281f1 /C281f1 /C281g2x 0g2x 0½V(xi)½du2du3du4du3du4 f1 /C281f1 /C281f1 /C281f1 /C281g2x 0g2x 0du2du3u4du3du4; (4) where the VERTICES are located at fxi;yi;zigwhere i/C301, ..., 4, and the (signed) VOLUME is given by the DETERMINANT V /C301 3!x1y1z11 x2y2z21 x3y3z31 x4y4z41YrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut: (5) The analytic result is difficult to compute, but is numerically given by ¯V :0 :120: / See also BALL TETRAHEDRON PICKING ,CUBE TETRA- HEDRON PICKING ,F EJES TO´ TH’S PROBLEM ,P OINT PICKING ,SPHERE LINE PICKING ,TETRAHEDRON References Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653/C1/59, 1986. Sphere with Tunnel Find the tunnel between two points Aand Bon a gravitating SPHERE which gives the shortest transit time under the force of gravity. Assume the SPHERE to be nonrotating, of RADIUS a, and with uniform density r:Then the standard form E ULER- LAGRANGE DIFFER- ENTIAL EQUATION in polar coordinates is rffr3/C28ra2YrvYru /C27r2 f2a2/C28r2YrvYru /C27a2r2/C300: (1) along with the boundary conditions r(f/C300)/C30r0; rf(f/C300)/C300;rf/C30fA ðÞ /C30a;and rf/C30fB ðÞ /C30a:Inte- grating once gives r2f/C30a2r2 r2 0r2/C28r2 0 a2/C28r2: (2) But this is the equation of a HYPOCYCLOID generated by a CIRCLE ofRADIUS1 2(a/C28r0) rolling inside the CIRCLE ofRADIUS a, so the tunnel is shaped like an arc of a HYPOCYCLOID . The transit time from point Ato point Bis T/C30pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28r2 0 ags ; (3) where g/C30GM a2/C304 3prGa (4) is the surface gravity with Gthe universal gravita- tional constant. Sphere-Cone Intersection CONE- SPHERE INTERSECTION Sphere-Cylinder Intersection CYLINDER- SPHERE INTERSECTIONSphere-Sphere Intersection Let two spheres of RADII Randrbe located along the X-AXIS centered at (0 ;0;0) and ( d;0;0);respec- tively. Not surprisingly, the analysis is very similar to the case of the CIRCLE-CIRCLE INTERSECTION . The equations of the two SPHERES are x2/C27y2/C27z2/C30R2(1) (x/C28d)2/C27y2/C27z2/C30r2: (2) Combining (1) and (2) gives (x/C28d)2/C27(R2/C28x2)/C30r2: (3) Multiplying through and rearranging give x2/C282dx/C27d2/C28x2/C30r2/C28R2: (4) Solving for xgives x/C30d2/C28r2/C27R2 2d: (5) The intersection of the SPHERES is therefore a curve lying in a PLANE parallel to the yz-plane at a single x- coordinate. Plugging this back into (1) gives y2/C27z2/C30R2/C28x2/C30R2/C28d2/C28r2/C27R2 2d !2 /C304d2B2/C28d2/C28r2/C27R2ðÞ2 4d2: (6) which is a CIRCLE with RADIUS a/C301 2dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4d2R2/C28(d2/C28r2/C27R2)2q /C301 2d[(/C28d/C27r/C28R)(/C28d/C28r/C27R)(/C28d/C27r/C27R) /C2(d/C27r/C27R)]1=2: (7) The VOLUME of the 3-D LENS common to the two spheres can be found by adding the two SPHERICAL CAPS . The distances from the SPHERES’ centers to the bases of the caps are d1/C30x (8) d2 /C30d /C28x; (9) so the heights of the caps are h1 /C30R /C28d1 /C30(r /C28 R /C27 d)(r /C27 R /C28 d) 2d (10) h2 /C30r /C28d2 /C30(R /C28 r /C27 d)(R /C27 r /C28 d) 2d : (11) The VOLUME of a SPHERICAL CAP of height h? for a SPHERE of RADIUS R? is V(R?; h?) /C301 3 ph?2(3R?/C28h?) : (12) Letting R1 /C30R and R2 /C30r and summing the two caps gives V /C30V(R1 ; h1) /C27V(R2 ; h2) /C30p(R /C27 r /C28 d)2 d2 /C27 2dr /C28 3r2 /C27 2dR /C27 6rR /C28 3R2ðÞ 12d : (13) This expression gives V /C300 for d /C30r /C27R as it must. In the special case r /C30R, the VOLUME simplifies to V /C301 12 p(4R /C27d)(2R /C28d)2 : (14) The SURFACE AREA of the sphere R that lies inside the sphere ris equal to the GREAT CIRCLE of the sphere r, provided that r52R(Kern and Blank 1948, p. 97). See also APPLE ,CIRCLE- CIRCLE INTERSECTION ,DOU- BLE BUBBLE ,L ENS,SPACE DIVISION BY SPHERES , SPHERE References Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 97, 1948. Spherical Bessel Differential Equation Take the H ELMHOLTZ DIFFERENTIAL EQUATION 92F/C27k2F/C300 (1) inSPHERICAL COORDINATES . This is just L APLACE’S EQUATION inSPHERICAL COORDINATES with an addi- tional term, d2R dr2FU/C302 rdR drFU/C271 r2sin2fd2U du2FR /C27cosf r2sinfdF dfUR/C271 r2d2F df2UR/C27k2RFU/C300: (2) Multiply through by r2=RFU; r2 Rd2R dr2/C272r RdR dr/C27k2r2/C271 Usin2fd2U du2/C27cosf FsinfdF df/C271 Fd2F df2/C300: (3) This equation is separable in R. Call the separation constant n(n/C271); r2 Rd2R dr2/C272r RdR dr/C27k2r2/C30n(n/C271): (4) Now multiply through by R, r2d2R dr2/C272rdR dr/C27k2r2/C28n(n/C271)YrtYrP R/C300: (5) This is the SPHERICAL BESSEL DIFFERENTIAL EQUA- TION . It can be transformed by letting x/C13kr;then rdR(r) dr/C30krdR(r) kd r/C30krdR(r) d(kr)/C30xdR(r) dx: (6) Similarly, r2d2R(r) dr2/C30x2d2R(r) dx2: (7) so the equation becomes x2d2R dx2/C302xdR dx/C27x2/C28n(n/C271)YrtYrP R/C300: (8) Now look for a solution OF THE FORM R(r)/C30Z(x)x/C281=2; denoting a derivative with respect to xby a prime, R?/C30Z?x/C281=2/C281 2Zx/C283=2(9) R??/C30Z??x/C281=2/C2812Z?x/C283=2/C2812Z?x/C283=2/C2812/C2832Yru*Yru+ Zx/C285=2 /C30Z??x/C281=2/C28Z?x/C283=2/C273 4Zx/C285=2(10) so x2Z??x/C281=2/C28Z?x/C283=2/C2734Zx/C285=2Yru*Yru+ /C272xZ?x/C281=2/C2812Zx/C283=2Yru*Yru+ /C27x2/C28n(n/C271)YrtYrP Zx/C281=2/C300 (11) x2Z??/C28Z?x/C281/C2734Zx/C282Yru*Yru+ /C272xZ?/C2812Zx/C281Yru*Yru+ /C27x2/C28n(n/C271)YrtYrP Z/C300 (12) x2Z??/C27(/C28x/C272x)Z?/C273 4/C281/C27x2/C28n(n/C271)hi Z/C300 (13) x2Z??/C27xZ?/C27x2/C28n2/C27n/C271 4Yru*Yru+hi Z/C300 (14) x2Z??/C27xZ?/C27x2/C28n/C2712Yru*Yru+2YrtvYrtu Z/C300: (15) But the solutions to this equation are B ESSEL FUNC- TIONS of half integral order, so the normalized solutions to the original equation are R(r) /C13AJn/C271 =2(kr)ffiffiffiffiffi krp /C27BYn/C271=2(kr)ffiffiffiffiffikrp (16) which are known as SPHERICAL BESSEL FUNCTIONS . The two types of solutions are denoted jn(x)(SPHERI- CAL BESSEL FUNCTION OF THE FIRST KIND )or nn(x) (SPHERICAL BESSEL FUNCTION OF THE SECOND KIND ), and the general solution is written R(r) /C30A?jn(kr) /C27B?nn(kr) : (17) where jn(z) /C13ffiffiffi p 2s Jn/C271 =2(z)ffiffiffizp (18) nn(z) /C13ffiffiffi p 2s Yn /C271 =2(z)ffiffiffizp : (19) See also S PHERICAL BESSEL FUNCTION ,SPHERICAL BESSEL FUNCTION OF THE FIRST KIND,SPHERICAL BESSEL FUNCTION OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 437, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Spherical Bessel Function A solution to the SPHERICAL BESSEL DIFFERENTIAL EQUATION . The two types of solutions are denoted jn(x)(SPHERICAL BESSEL FUNCTION OF THE FIRST KIND )ornn(x)(SPHERICAL BESSEL FUNCTION OF THE SECOND KIND ). See also SPHERICAL BESSEL DIFFERENTIAL EQUATION , SPHERICAL BESSEL FUNCTION OF THE FIRST KIND, SPHERICAL BESSEL FUNCTION OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437 /C1/42, 1972. Arfken, G. "Spherical Bessel Functions." §11.7 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 622 /C1/36, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions." §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 234 /C1/45, 1992.Spherical Bessel Function of the First Kind jn(x) /C13ffiffiffiffiffiffi p 2xs Jn /C271 =2(x) (1) /C302nxnX/C12 s/C300( /C281)s(s /C27 n)! s!(2s /C27 2n /C27 1)!x2s (2) /C30xn (2n /C27 1)!! /C2 1 /C281 2 x2 1!(2n /C27 3) /C2712 x2Yru*Yru+2 2!(2n /C27 3)(2n /C27 5) /C27...2 643 75 (3) /C30(/C281)nxnd xdx !nsin x x (4) where jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . The first few functions are j0(x)/C30sinx x(5) j1(x)/C30sinx x2/C28cosx x(6) j2(x)/C303 x3/C281 x ! sinx/C283 x2cosx: (7) Spherical Bessel functions are not explicitly imple- mented in Mathematica . See also SPHERICAL BESSEL DIFFERENTIAL EQUATION , BESSEL FUNCTION OF THE SECOND KIND,POISSON INTEGRAL REPRESENTATION ,RAYLEIGH’S FORMULAS , SPHERICAL BESSEL FUNCTION OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437 /C1/42, 1972. Arfken, G. "Spherical Bessel Functions." §11.7 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 622 /C1/36, 1985. Spherical Bessel Function of the Second Kind nn(x) /C13ffiffiffiffiffiffi p 2xs Yn/C271 =2(x) (1) /C30( /C281)n/C271 2nxn /C271X/C12 s/C300(/C281)s(s /C28 n)! s!(2s /C28 2n)!x2s (2) /C30( /C281)n/C271 2nxn/C271X/C12 s/C300( /C281)s4n/C28sffiffiffipp G(s /C27 1)G1 2 /C28 n /C27 sYru*Yru+ (3) /C30/C28(2n /C28 1)!! xn/C271 /C2 1 /C281 2 x2 1!(1 /C28 2n) /C2712 x2Yru*Yru+2 2!(1 /C28 2n)(3 /C28 2n) /C27...2 643 75 (4) /C30(/C281)n/C271ffiffiffiffiffiffi p 2xs J/C28n /C281 =2(x) : (5) where Yn(z)isaB ESSEL FUNCTION OF THE SECOND KIND and jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . The first few functions are n0(x) /C30/C28cos x x (6) n1(x) /C30/C28cos x x2/C28sin x x (7) n2(x) /C30/C283 x3 /C281 x ! cos x /C283 x2sin x: (8) Spherical Bessel functions are not explicitly imple- mented in Mathematica . See also SPHERICAL BESSEL DIFFERENTIAL EQUATION , BESSEL FUNCTION OF THE SECOND KIND,RAYLEIGH’S FORMULAS ,S PHERICAL BESSEL FUNCTION OF THE FIRST KINDReferences Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437 /C1/42, 1972. Arfken, G. "Spherical Bessel Functions." §11.7 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 622 /C1/36, 1985. Spherical Bessel Function of the Third Kind SPHERICAL HANKEL FUNCTION OF THE FIRST KIND, SPHERICAL HANKEL FUNCTION OF THE SECOND KIND Spherical Cap A spherical cap is the region of a SPHERE which lies above (or below) a given PLANE . If the PLANE passes through the CENTER of the SPHERE , the cap is a called aHEMISPHERE , and if the cap is cut by a second PLANE , it is called a SPHERICAL SEGMENT . However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for what is here called a spherical cap and "zone" for SPHERICAL SEGMENT . Let the SPHERE have RADIUS R, then the VOLUME of a spherical cap of height hand base RADIUS ais given by the equation of a SPHERICAL SEGMENT Vspherical segment /C301 6ph(3a2/C273b2/C27h2) (1) with b/C300, giving Vcap/C3016ph(3a2/C27h2): (2) Using the P YTHAGOREAN THEOREM gives (R/C28h)2/C27a2/C30R2; (3) which can be solved for a2as a2/C302Rh/C28h2: (4) so the radius of the base circle is a/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h(2R/C28h)p : (5) and plugging this in gives the equivalent formula Vcap/C301 3ph2(3R/C28h): (6) In terms of the so-called CONTACT ANGLE (the angle between the normal to the sphere at the bottom of the cap and the base plane) R /C28h /C30R sin a (7) a /C13sin/C281R /C28 h R ! ; (8) so Vcap /C301 3 pR3(2 /C283 sin a /C27sin3 a) : (9) The CENTROID occurs at a distance ¯z /C303(2R /C28 h)2 4(3R /C28 h) (10) above the center of the sphere (Harris and Stocker 1998, p. 107). Consider a cylindrical box enclosing the cap so that the top of the box is tangent to the top of the SPHERE . Then the enclosing box has VOLUME Vbox /C30 pa2h /C30 p(R cos a)[R(1 /C28sin a)] /C30 pR3(1 /C28sin a /C28sin2 a /C27sin3 a) ; (11) so the hollow volume between the cap and box is given by Vbox /C28Vcap /C301 3 pR3 1 /C283sin2 a /C272sin3 aYrvYru : (12) If a second PLANE cuts the cap, the resulting SPHE- RICAL FRUSTUM is called a SPHERICAL SEGMENT . The SURFACE AREA of the spherical cap is given by the same equation as for a general ZONE : Scap/C302pRh/C30p(a2/C27h2): (13) See also CONTACT ANGLE ,D OME,FRUSTUM ,H EMI- SPHERE ,SOLID OF REVOLUTION ,SPHERE ,SPHERICAL SEGMENT ,SPHERICAL WEDGE ,TORISPHERICAL DOME, ZONE References Harris, J. W. and Stocker, H. "Spherical Segment (Spherical Cap)." §4.8.4 in Handbook of Mathematics and Computa- tional Science. New York: Springer-Verlag, p. 107, 1998. Kern, W. F. and Bland, J. R. "Spherical Segment." §36 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 97 /C1/02, 1948. Spherical Code How can npoints be distributed on a UNIT SPHERE such that they maximize the minimum distance between any pair of points? This maximum distance is called the covering radius, and the configuration is called a spherical code (or spherical packing). In 1943,Fejes To ´th proved that for npoints, there always exist two points whose distance disd5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C28csc 2pn 6(n/C282)"#vuut; and that the limit is exact for n/C303, 4, 6, and 12. The problem of spherical packing is therefore sometimes known as the Fejes To ´th’s problem. The general problem has not been solved. For two points, the points should be at opposite ends of aDIAMETER . For four points, they should be placed at the VERTICES of an inscribed regular TETRAHE- DRON . There is no unique best solution for five points since the distance cannot be reduced below that for six points. For six points, they should be placed at the VERTICES of an inscribed regular OCTAHEDRON . For seven points, the best solution is four equilateralspherical triangles with angles of 80 8. For eight points, the best dispersal is notthe VERTICES of the inscribed CUBE , but of a SQUARE ANTIPRISM with equal EDGES . The solution for nine points is eight equilat- eral spherical triangles with angles of cos/C281(1=4):For 12 points, the solution is an inscribed regular ICOSA- HEDRON . A spherical packing corresponds to the placement of nspheres around a central unit sphere. From simple trigonometry, sin1 2uYru*Yru+ /C30r 1/C27r: so the radii of the nspheres are given by r/C301 csc1 2uYru*Yru+ /C281 for a minimum separation angle of u:Hardin and Sloane give tables of minimum separations and sphere positions for n5130 and d/C303, 4, 5. "Almost" 13 spheres can fit around a central sphere in the sense that there is a gap left over when 12 spheres are in place which is nearly big enough for an additional sphere (left figure). In fact, the radii of the spheres can be increased to 1.10851 (assuming a central unit sphere) before 12 spheres no longer fit (middle figure). In order to fit 13 spheres around a central unit sphere, their radius must be no larger than 0.916468 (right figure). These values correspond to Hardin and Sloane’s angles of 63.4349488 8 and 57.1367031 8, respectively. Pack eight unit spheres whose centers are at the vertices of a cube. Then the radius of the largest sphere which fits in the center hole (left figure) is given by r1 /C301 2d1 /C282R ðÞ with d1 /C30ffiffiffi 2p (2R); giving r1 /C30ffiffiffi 2p /C281Yru*Yru+ R: (1) Similarly, the radius of the largest sphere which can be passed through from one side to another (right figure) has d2 /C30ffiffiffi 3p (2R); giving r2 /C301 2d2 /C282R ðÞ /C30ffiffiffi 3p /C281Yru*Yru+ R: (2) See also KISSING NUMBER ,SPHERICAL COVERING , SPHERICAL DESIGN ,THOMSON PROBLEM References Friedman, E. "Points on a Sphere." http://www.stetson.edu/ ~efriedma/ptsphere/. Hardin, R. H.; Sloane, N. J. A. S.; and Smith, W. D. Sphe- rical Codes. In preparation. http://www.research.att.com/ ~njas/packings/. Hardin, R. H.; Sloane, N. J. A.; and Smith, W. D. Spherical Codes. In preparation. Ogilvy, C. S. Excursions in Mathematics. New York: Dover, p. 99, 1994. Ogilvy, C. S. Solved by L. Moser. "Minimal Configuration of Five Points on a Sphere." Problem E946. Amer. Math. Monthly 58, 592, 1951.Schu¨tte, K. and van der Waerden, B. L. "Auf welcher Ku¨gel haben 5, 6, 7, 8 oder 9 Pu¨nkte mit Mindestabstand Eins Platz?" Math. Ann. 123,96/C1/24, 1951. Whyte, L. L. "Unique Arrangement of Points on a Sphere." Amer. Math. Monthly 59, 606 /C1/11, 1952. Spherical Cone The SURFACE OF REVOLUTION obtained by cutting a conical "wedge" with vertex at the center of a SPHERE out of the SPHERE . A spherical cone is therefore a degenerate case of a SPHERICAL SECTOR . The volume of the spherical cone is V /C302 3 pR2h (1) (Kern and Bland 1948, p. 104). The SURFACE AREA of a closed spherical sector is S/C30pR(2h/C27r); (2) and the CENTROID is located at a height ¯z/C3038(2R/C28h) (3) above the sphere’s center (Harris and Stocker 1998). See also CONE,SPHERE ,SPHERICAL CAP,SPHERICAL SECTOR References Harris, J. W. and Stocker, H. "Spherical Sector." §4.8.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 106 /C1/07, 1998. Kern, W. F. and Bland, J. R. "Spherical Sector." §37 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 103 /C1/06, 1948. Spherical Coordinates A system of CURVILINEAR COORDINATES which is natural for describing positions on a SPHERE or SPHEROID . Define uto be the azimuthal ANGLE in thexy-PLANE from the X-AXIS with 05uB2p(denoted lwhen referred to as the LONGITUDE ),fto be the POLAR ANGLE from the Z-AXIS with 05f5p(COLATI- TUDE , equal to f/C3090/C14/C28dwhere dis the LATITUDE ), and rto be distance ( RADIUS ) from a point to the ORIGIN . Unfortunately, the convention in which the symbols u andfare reversed is frequently used, especially in physics, leading to unnecessary confusion. The sym- bolris sometimes also used in place of r. Arfken (1985) uses ( r;f;u);whereas Beyer (1987) uses (r;u;f):Be very careful when consulting the litera- ture. In this work, the symbols for the azimuthal, polar, and radial coordinates are taken as u;f;and r, respectively. Note that this definition provides a logical extension of the usual POLAR COORDINATES notation, with uremaining the ANGLE in the xy-PLANE andfbecoming the ANGLE out of the PLANE . r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2/C27z2p (1) u/C30tan/C281y x ! (2) f/C30sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p r ! /C30cos/C281z r ! ; (3) where r/C230;/C12½Þ ;u/C23[0;2p);andf/C23[0;p]:In terms of CARTESIAN COORDINATES , x/C30rcosusinf (4) y/C30rsinusinf (5) z/C30rcosf: (6) The SCALE FACTORS are hr/C301 (7) hu/C30rsinf (8)hf/C30r; (9) so the METRIC COEFFICIENTS are grr/C301 (10) guu/C30r2sin2f (11) gff/C30r2: (12) The LINE ELEMENT is ds/C30drˆr/C27rdfˆf/C27rsinfduˆu; (13) the AREA element da/C30r2sinfdudfˆr; (14) and the VOLUME ELEMENT dV/C30r2sinfdudfdr: (15) The J ACOBIAN is @(x;y;z) @(r;u;f)YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30r 2sinf jj : (16) The POSITION VECTOR is r/C13rcosusinf rsinusinf rcosf2 435; (17) so the UNIT VECTORS are ˆr/C13dr dr dr drYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30cosusinf sinusinf cosf2 435 (18) ˆu/C13 dr du dr duYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30/C28sinu cosu 02 435 (19) ˆf/C13 dr df dr dfYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30cosucosf sinucosf /C28sinf2 435: (20) Derivatives of the UNIT VECTORS are @ˆr @r/C300 (21) @ˆu @r/C300 (22) @ˆf @r/C300 (23) @ˆr @u/C30/C28sinusinf cosusinf 02 435/C30sinfˆu (24) @ˆu @u/C30/C28cosu /C28sinu 02 435/C30/C28cosfˆf/C28sinfˆr (25) @ˆf @u/C30/C28sinucosf cosucosf 02 435/C30cosfˆu (26) @ˆr @f/C30cosu sinucosf /C28sinf2435/C30ˆf (27) @ˆu @f/C300 (28) @ˆf @f/C30/C28cosusinf /C28sinusinf /C28cosf2435/C30/C28 ˆr: (29) The GRADIENT is 9/C30ˆr@ @r/C271 rˆf@ @f/C271 rsinfˆu@ @u; (30) so 9rˆr/C300 (31) 9rˆu/C300 (32) 9rˆf/C30ˆ0 (33) 9rˆr/C30sinfˆu rsinf/C301 rˆu (34) 9uˆu/C30/C28cosfˆf/C27sinfˆr rsinf/C30/C28cotf rˆf/C281 rˆr (35) 9uˆf/C30cosfˆf rsinf/C301 rcotfˆu: (36) Now, since the CONNECTION COEFFICIENTS are given byGi jk/C30ˆxi/C2159kˆxjYrvYru ; Gu/C3001 r0 000 0cotf r02 6666643 777775(37) G f/C30001 r 0/C28cotf r0 0002 6666643 777775(38)G r/C3000 0 0/C281 r0 00 /C281 r2 6666643 777775: (39) The DIVERGENCE is 9 /C215F/C30Ak ;k/C27Gk jkAj /C30Ar ;r/C27Gr rrAr/C27GrurAu/C27GrfrAfYru*ih /C27Au ;u/C27Gu ruAr/C27GuuuAu/C27GufuAfYru*Yru+hi /C27Af ;f/C27GfrfAr/C27GfufAu/C27GfffAfYru*Yru+hi /C301 gr@Ar @r/C271 gu@Au @u/C271 gf@Af @f/C27(0/C270/C270) /C271 rAr/C270/C27cotf rAf ! 1 rAr/C270/C270 ! /C30@ @rAr/C272 rAr/C271 rsinf@ @uAu/C271 r@ @fAf /C27cotf rAf; (40) or, in VECTOR notation, 9 /C215F/C302 r/C27@ @r ! Fr/C271 r@ @f/C27cotf r ! Ff /C271 sinf@Fu @u /C301 r2@ @rr2FrYrvYru /C271 rsinf@ @fsinfFfYrvYru /C271 rsinf@Fu @u: (41) The COVARIANT DERIVATIVES are given by Aj;k/C301 gkk@Aj @xk/C28Gi jkAi; (42) so Ar;r/C30@Ar @r/C28Gi rrAi/C30@Ar @r(43) Ar;u/C301 rsinf@Ar @u/C28Gitu/C301 rsinf@Ar @u/C28GruAu /C301 rsinf@Ar @f/C28Au r(44) Ar;f/C301 r@Ar @f/C28Gi rfAi/C301 r@Ar @f/C28Gf rfAf /C301 r@Ar @f/C28Af ! (45) Au;r/C30@Au @r/C28Gi urAi/C30@Au @r(46) Au;u/C301 rsinf@Au @u/C28GiuuAi /C301 rsinf@Au@u/C28Gf uuAf/C28Gr uuAr /C301 rsinf@Au @u/C27cotf rAf/C27Ar r(47) Au;f/C301 r@Au @r/C28GifrAi@Au @f(48) Af;r/C30@Af @r/C28GifrAi/C30@Af r(49) Af;u/C301 rsinf@Af @u/C28GifuAi/C301 rsinf@Af @u/C28Gufu /C301 rsinf@Af @u/C28cotf rAu (50) Af;f/C301 r@Af @f/C28GiffAi/C301 r@Af @f/C28GrffAr /C301 r@Af @f/C27Ar r: (51) The COMMUTATION COEFFICIENTS are given by cm ab /C0em/C30 /C0ea; /C0ebYrtYrP /C309a /C0eb/C289b /C0ea (52) ˆr;ˆr½/C138/C30ˆu;ˆuYrtYrP /C30ˆf;ˆfYrtYrP /C300; (53) soca rr/C30cauu/C30caff/C300;where a/C30r;u;f: ˆr;ˆuYrtYrP /C30/C28 ˆu;ˆrYrtYrP /C309rˆu/C289uˆr/C300/C281 rˆu/C30/C281 rˆu:(54) socuru/C30/C28cuur/C30/C281 r;crru/C30cf ru/C300: ˆr;ˆfYrtYrP /C30/C28 ˆf;ˆrYrtYrP /C300/C281 rˆf/C30/C281 rˆf; (55) socfrf/C30/C28cffr/C301 r: ˆu;ˆfYrtYrP /C30/C28 ˆf;ˆuYrtYrP /C301 rcotfˆu/C280/C301 rcotfˆu: (56) socu uf/C30/C28cufu/C301 rcotf: (57) Summarizing, cr/C30000 0000002 435 (58) c u/C300/C281 r0 1 r01 rcotf 0/C281 rcotf 02 643 75 (59) cf/C3000 /C281 r 00 0 1 r002 643 75: (60) Time derivatives of the POSITION VECTOR are ˙r/C30cosusinf˙r/C28rsinusinf˙u/C27rcosucosf˙f sinusinf˙r/C27rcosusinf˙u/C27rsinucosf˙f cosf˙r/C28rsinf˙f2 435 /C30cosusinf sinusinf cosf2435˙r/C27rsinf/C28sinu cosu 02435˙u /C27rcosucosf sinucosf /C28sinf2 435˙f /C30˙rˆr/C27rsinf˙uˆu/C27r˙fˆf: (61) The SPEED is therefore given by v/C13˙rjj/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˙r2/C27r2sin2f˙u2/C27r2˙f2q : (62) The ACCELERATION is ¨x/C30(/C28sinusinf˙u˙r/C27cosucosf˙r˙f/C27cosusinf¨r) /C28(sinusinf˙r˙u/C27rcosusinf˙u2/C27rsinucosf˙u˙f) /C27rsinusinf¨u)/C27(cosucosf˙r˙f/C28rsinucos˙u˙f /C28rcosusinf˙f2/C27rcosucosf¨f) /C30/C282 sin usinf˙u˙r/C272 cos ucosf˙r˙f /C282rsinucosf˙u˙f /C27cosusinf¨r/C28rsinusinf¨u/C27rcosucosf¨f /C28rcosusinf˙u2/C27˙f2YrvYru (63) ¨y/C30(sinusinf¨r/C27rcosusinf˙u/C27rcosfsinu˙f) /C27(cosusinf˙r˙u/C28rsinusinf˙u2/C27rcosucosf˙u˙f) /C27rcosusinf¨u)/C27(sinucosf˙r˙f/C27rcosucosf˙u˙f /C28rsinusinf˙f2/C27rsinucosf¨f /C302 cos usinf˙u˙r/C272 sin ucosf˙r˙f/C272rcosucosf˙u˙f /C27sinusinf¨r/C27rcosusinf¨u/C27rsinucosf¨f /C28rsinusinf˙u2/C27˙f2YrvYru (64) ¨z/C30(cosf¨r/C28sinf˙r˙f) /C28(˙rsinf˙f/C27rcosf˙f2/C27rsinf¨f) /C30/C28rcosf˙f2/C27cosf¨r/C282 sin f˙f˙r/C28rsinf¨f:(65) Plugging these in gives ¨r/C30¨r/C28r˙f2YrvYrucosusinf sinusinf cosf2 435 /C27(2rcosf˙u˙f/C27rsinf¨u)/C28sinu cosu 02435 /C27(2˙r˙f/C27r¨f)cosucosf sinucosf /C28sinf2435/C28rsinf˙u 2cosu sinu 02435:(66) but sinfˆr/C27cosfˆf/C30cosusin 2f/C27cosucos2f sinusin2f/C27sinucos2f 02435 /C30cosu sinu 02435 (67) so ¨r/C30¨r/C28r˙f 2YrvYru ˆr/C27(2rcosf˙u˙f/C272 sin f˙u˙r/C27rsinf¨u)ˆu /C28(2˙r˙f/C28r¨f)ˆf/C28rsinf˙u2(sinfˆr/C27cosfˆf) /C30(¨r/C28r˙f2/C28rsin2f˙u2)ˆr /C27(2 sin f˙u˙r/C272rcosf˙uf/C27rsinf¨u)ˆu /C27(2˙r˙f/C27r¨f/C28rsinfcosf˙u2)ˆf: (68) Time DERIVATIVES of the UNIT VECTORS are ˙ˆr/C30/C28sinusinf˙u/C27cosucosf˙f cosusinf˙u/C27sinucosf˙f /C28sinf˙f2 435 /C30sinf˙uˆu/C27˙fˆf (69) ˙ˆu/C30/C28cosu˙u /C28sinu˙u 02 435/C30/C28 ˙ucosu sinu 02435 /C30/C28 ˙u(sinfˆr/C27cosfˆf) (70) ˙ˆf/C30/C28sinucosf˙u/C28cosusinf˙f cosucosf˙u/C28sinusinf˙f /C28cosf˙f2 435 /C30/C28 ˙fˆr/C27cosf˙uˆu: (71) The CURL is9/C29F/C301 rsinf@ @fsinfFu ðÞ /C28@Ff @u"# ˆr /C271 r1 sinf@Fr @u/C28@ @rrFuðÞ"# ˆf/C271 r /C2@ @rrFfYrvYru /C28@Fr @f"# ˆu: (72) The L APLACIAN is 92/C131 r2@ @rr2@ @r ! /C271 r2sin2@2 @u2/C271 r2sin@ @f /C2sinf@ @f ! /C301 r2r2@2 @r2/C272r@ @r ! /C271 r2sin2@2 @u2/C271 r2sinf /C2cosf@ @f/C27sinf@2 @f2 ! /C30@2 @r2/C272 r@ @r/C271 r2sin2f@2 @u2/C27cosf r2sinf@ @f /C271 r2@2 @f2: (73) The vector L APLACIAN is 92v/C30 1 r@2rvrðÞ @r2/C271 r2@2vr @u2/C271 r2sin2u@2vr @f2/C27cotu r2@vr u/C282 r2@vu @u/C282 r2sinu@vf @f/C282vr r2/C282 cot u r2vu 1 r@2rvuðÞ @r2/C271 r2@2vu @u2/C271 r2sin2u@2vu @f2/C27cotu r2@vu u/C282 r22 cot u r2sinu@vf @f/C272 r22vr @u/C28vu r2sin2u 1 r@2rvfðÞ @r2/C271 r2@2vfa @u2/C271 r2sin2u@2vf @f2/C27cotu r2@vf @u/C272 r2@vr @f/C272 cot u r2sinu@vu @f/C28vf r2sin2u2 66643 7775: (74) To express PARTIAL DERIVATIVES with respect to Cartesian axes in terms of PARTIAL DERIVATIVES of the spherical coordinates, x y z2 435/C30rcosusinf rsinusinf rcosf2435 (75) dx dy dz2 435/C30 cosusinfdr/C28rsinusinfdu/C27rcosucosfdf sinusinfdr/C27rsinfcosudu/C27rsinucosfdf cosfdr/C28rsinfdf2 435 /C30cosusinf/C28rsinusinfrcosucosf sinusinfrsinfcosursinucosf cosf 0 /C28rsinf2 435 /C2dx dy dz2 435: (76) Upon inversion, the result is dr du d f2 435/C30cos u sin f sin u sin f cos f /C28 sin u r sin fcos u r sin f0 cos u cos f rsin u cos f r/C28sin f r26666643 777775 /C2dr dy dz2 435: (77) The Cartesian PARTIAL DERIVATIVES in spherical coordinates are therefore @ @x /C30@r @x@ @r /C27@ u @x@ @ u /C27@ f @x@ @ f /C30cos u sin f@ @r /C28sin u r sin f@ @ u /C27cos u cos f r@ @ f(78) @ @y /C30@ @y@ @r /C27@ u @y@ @ u /C27@ f @y@ @ f /C30sin u sin f@ @r /C27cos u r sin f@ @ u /C27sin u cos f r@ @ f(79) @ @z /C30@r @z@ @r /C27@ u @z@ @ u /C27@ f @z@ @ f /C30cos f@ @r /C28sin f r@ @ f (80) (Gasiorowicz 1974, pp. 167 /C1/68). The HELMHOLTZ DIFFERENTIAL EQUATION is separable in spherical coordinates. See also COLATITUDE ,G REAT CIRCLE ,H ELMHOLTZ DIFFERENTIAL EQUATION– SPHERICAL COORDINATES , LATITUDE ,LONGITUDE ,OBLATE SPHEROIDAL COORDI- NATES ,PROLATE SPHEROIDAL COORDINATES References Arfken, G. "Spherical Polar Coordinates." §2.5 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 102 /C1/11, 1985. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 212, 1987. Gasiorowicz, S. Quantum Physics. New York: Wiley, 1974. Moon, P. and Spencer, D. E. "Spherical Coordinates (r ; u; c) :/" Table 1.05 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 24 /C1/7, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 658, 1953. Spherical Covering The placement of n points on a SPHERE so as to minimize the maximum distance of any point on the sphere from the closest one of the n points.See also SPHERICAL CODE,SPHERICAL COVERING References Hardin, R. H.; Sloane, N. J. A. S.; and Smith, W. D. Sphe- rical Codes. In preparation. http://www.research.att.com/ ~njas/coverings/. Spherical Curve A CURVE on the surface of a SPHERE . Examples include the BASEBALL COVER ,SEIFFERT’S SPHERICAL SPIRAL , SPHERICAL HELIX , and SPHERICAL SPIRAL . See also BASEBALL COVER ,C URVE ,PLANE CURVE , SPACE CURVE ,TENNIS BALL THEOREM Spherical Defect Let a, b, and c be the sides of a SPHERICAL TRIANGLE , then the spherical defect is defined as D /C302 p /C28(a /C27b /C27c): See also ANGULAR DEFECT ,S PHERICAL EXCESS , SPHERICAL TRIANGLE References Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 109, 1998. Spherical Design Xis a spherical t-design in EIFFit is possible to exactly determine the average value on Eof any POLYNOMIAL fof degree at most tby sampling fat the points of X. In other words, 1 volume EgEf(j)dj/C301 XjjX x/C23Xf(x): Spherical t-designs give the placement of npoints on a sphere for use in numerical integration with equal weights. References Colbourn, C. J. and Dinitz, J. H. (Eds.). "Spherical t-De- signs." Ch. 44 in CRC Handbook of Combinatorial De- signs. Boca Raton, FL: CRC Press, pp. 462 /C1/66, 1996. Hardin, R. H. and Sloane, N. J. A. S. "McLaren’s Improved Snub Cube and Other New Spherical Designs in Three Dimensions." Disc. Comput. Geom. 15, 429/C1/31, 1996. Hardin, R. H.; Sloane, N. J. A. S.; and Smith, W. D. Sphe- rical Codes. In preparation. http://www.research.att.com/ ~njas/sphdesigns/. McLaren, A. D. "Optimal Numerical Integration on a Sphere." Math. Comput. 17, 361/C1/83, 1963. Spherical Excess The difference between the sum of the angles A,B, andCof a SPHERICAL TRIANGLE andpradians (180 8), E/C30A/C27B/C27C/C28p: The notation D is sometimes used for spherical excess instead of E, which can cause confusion since it is also frequently used to denote the SURFACE AREA of a SPHERICAL TRIANGLE (Zwillinger 1995, p. 469). The notation /C23 is also used (Gellert et al. 1989, p. 263). The equation for the spherical excess in terms of the side lengths a, b, and c is known as L’HUILIER’S THEOREM , tan1 4 EYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 2 sYru*Yru+ tan12s /C28a ðÞhi tan12s /C28b ðÞhi tan12s /C28c ðÞhir ; where s is the SEMIPERIMETER . See also ANGULAR DEFECT ,DESCARTES TOTAL ANGU- LAR DEFECT ,GIRARD’S SPHERICAL EXCESS FORMULA , L’HUILIER’S THEOREM ,SPHERICAL TRIANGLE References Gellert, W.; Gottwald, S.; Hellwich, M.; Ka¨stner, H.; and Ku¨nstner, H. (Eds.). VNR Concise Encyclopedia of Mathe- matics, 2nd ed. New York: Van Nostrand Reinhold, 1989. Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 109, 1998. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 469, 1995. Spherical Frustum SPHERICAL SEGMENT Spherical Geometry The study of figures on the surface of a SPHERE (such as the SPHERICAL TRIANGLE and SPHERICAL POLYGON ), as opposed to the type of geometry studied in PLANE GEOMETRY or SOLID GEOMETRY . In spherical geome- try, straight lines are GREAT CIRCLES , so any two lines meet in two points. There are also no parallel lines. The angle between two lines in spherical geometry is the angle between the planes of the corresponding great circles, and a SPHERICAL TRIANGLE is defined by its three angles. There is no concept of similar triangles in spherical geometry. See also GREAT CIRCLE ,H YPERBOLIC GEOMETRY , PLANE GEOMETRY ,S OLID GEOMETRY ,S PHERICAL TRIANGLE ,SPHERICAL TRIGONOMETRY ,T HURSTON’S GEOMETRIZATION CONJECTURE References Harris, J. W. and Stocker, H. "Spherical Geometry." §4.9 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 108 /C1/13, 1998. Henderson, D. W. Experiencing Geometry: On Plane and Sphere. Englewood Cliffs, NJ: Prentice-Hall, 1995. Zwillinger, D. (Ed.). "Spherical Geometry and Trigonome- try." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 468 /C1/71, 1995.Spherical Hankel Function of the First Kind h(1) n(x) /C13ffiffiffiffiffiffi p 2xs H(1) n/C271 =2(x) /C30jn(x) /C27inn(x); where H(1)(x) is the HANKEL FUNCTION OF THE FIRST KIND and jn(x) and nn(x) are the SPHERICAL BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS . Expli- citly, the first few are h(1) 0(x) /C301 x(sin x /C28i cos x) /C30/C28i xeix h(1)1(x) /C30eix/C281 x /C28i x2 ! h(1)2(x) /C30eixi x /C283 x2 /C283i x3 ! : References Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437 /C1/42, 1972. Spherical Hankel Function of the Second Kind h(2) n(x) /C13ffiffiffiffiffiffi p 2xs H(2) n/C271 =2(x) /C30jn(x) /C28inn(x); where H(2)(x) is the HANKEL FUNCTION OF THE SECOND KIND and jn(x) and nn(x) are the SPHERICAL BESSEL FUNCTIONS OF THE FIRST and SECOND KINDS . Explicitly, the first is h(2)0(x)/C301 x(sinx/C27icosx)/C30i xe/C28ix: See also SPHERICAL BESSEL FUNCTION OF THE FIRST KIND,SPHERICAL BESSEL FUNCTION OF THE SECOND KIND References Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437 /C1/42, 1972. Spherical Harmonic The spherical harmonics Ym l(u;f) are the angular portion of the solution to L APLACE’S EQUATION in SPHERICAL COORDINATES where azimuthal symmetry is not present. Some care must be taken in identifying the notational convention being used. In this entry, u is taken as the polar (colatitudinal) coordinate with u/C23[0;p];and fas the azimuthal (longitudinal) coordinate with f/C23[0;2p):This is the convention normally used in physics, as described by Arfken(1985) and Mathematica (in mathematical literature, uusually denotes the longitudinal coordinate and f the colatitudinal coordinate). Spherical harmonicsare implemented in Mathematica asSpherical- HarmonicY [l,m,theta ,phi]. Spherical harmonics satisfy the SPHERICAL HARMONIC DIFFERENTIAL EQUATION , which is given by the angular part of L APLACE’S EQUATION inSPHERICAL COORDINATES . Writing F/C30F(f)U(u) in this equation gives F(f) sinud dusinudU du ! /C27U(u) sin2ud2F(f) df2 /C27l(l/C271)U(u)F(f)/C300: (1) Multiplying by sin2u=UFðÞ gives sinu U(u)d dusinudU du ! /C27l(l/C271) sin2u"# /C271 F(f)d2F(f) df2 /C300: (2) Using SEPARATION OF VARIABLES by equating the f/- dependent portion to a constant gives 1 F(f)d2F(f) df2/C30/C28m2; (3) which has solutions F(f)/C30Ae/C28imf/C27Beimf; (4) Plugging in (3) into (2) gives the equation for the u/- dependent portion, whose solution is U(u)/C30Pm l(cosu); (5) where m/C30/C281;/C28(l/C281);..., 0, ..., l/C281;landPm l(z)i sa n associated L EGENDRE POLYNOMIAL . The spherical harmonics are then defined by combining F(f) and U(u); Ym l(u;f)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l/C271 4p(l/C28m)! (l/C27m)!s Pm l(cosu)eimf: (6) where the normalization is chosen such that g2p 0gp 0Ym l(u;f)¯Ym? l?(u;f)sinududf /C30g2p 0g1 /C281Ym l(u;f)¯Ym? l?(u;f)d(cosu)df/C30dmm;dll: (7) (Arfken 1985, p. 681). Here, ¯zdenotes the COMPLEX CONJUGATE anddmnis the K RONECKER DELTA . Some- times (e.g., Arfken 1985), the C ONDON- SHORTLEYPHASE (/C281)mis prepended to the definition of the spherical harmonics. The spherical harmonics are sometimes separated into their REAL and IMAGINARY PARTS , Yms l(u;f)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l/C271 4p(l/C28m)! (l/C27m)!s Pm l(cosu) sin( mf) (8) Ymc l(u;f)/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l/C271 4p(l/C28m)! (l/C27m)!s Pml(cosu) cos( mf):(9) The spherical harmonics obey Y/C28l l(u;f)/C301 2ll!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2l/C271)! 4ps sinlue/C28ilf(10) Y0 l(u;f)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l/C271 4ps Pl(cosu) (11) Y/C28m l(u;f)/C30(/C281)m¯Yml(u;f); (12) where Pl(x)i saL EGENDRE POLYNOMIAL . Integrals of the spherical harmonics are given by g2p 0gp 0Ym1 l1(u;f)Ym2 l2(u;f)Ym3 l3(u;f) sin ududf /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l1/C271 ðÞ 2l2/C271 ðÞ 2l3/C271 ðÞ 4ps l1l2l3 000Yru$Yru% /C2l1l2l3 m1m2m3Yru$Yru% ; (13) wherel1l2l3 m1m2m3Yru*Yru+ is a W IGNER 3 J-SYMBOL (which is related to the C LEBSCH- GORDAN COEFFICIENTS ). Spe- cial cases include g2p 0gp 0YM Lu;fðÞ Y0 0u;fðÞ ¯YMLu;fðÞ sinududf /C301ffiffiffiffiffiffi 2pp (14) g2p 0gp 0YM Lu;fðÞ Y0 1u;fðÞ ¯YM L/C271u;fðÞ sinududf /C30ffiffiffiffiffiffi 3 4ps ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (L/C27M/C271)(L/C28M/C271) (2L/C271)(2L/C273)s (15) g2p 0gp 0YM Lu;fðÞ Y1 1u;fðÞ ¯YM/C271 L/C271u;fðÞ sinududf /C30ffiffiffiffiffiffi 3 8ps ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (L/C27M/C271)(L/C27M/C272) (2L/C271)(2L/C273)s (16) g2p 0gp 0YM Lu;fðÞ Y1 1u;fðÞ ¯YM/C271 L/C281u;fðÞ sinududf /C30ffiffiffiffiffiffi 3 8ps ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (L/C28M)(L/C28M/C281) (2L/C281)(2L/C271)s (17) (Arfken 1985, p. 700). The above illustrations show Ym l(u;f)YrtYrP2(top), RYm l(u;f)YrtYrP2(bottom left), and IYm l(u;f)YrtYrP2(bottom right). The first few spherical harmonics are Y0 0(u;f)/C301 21ffiffiffipp Y/C281 1(u;f)/C301 2ffiffiffiffiffiffi 3 2ps sinue/C28if Y0 1(u;f)/C301 2ffiffiffi 3 ps cosu Y1 1(u;f)/C30/C281 2ffiffiffiffiffiffi 3 2ps sinueif Y/C282 2(u;f)/C3014ffiffiffiffiffiffi 15 2ps sin2ue/C282if Y/C281 2(u;f)/C3012ffiffiffiffiffiffi 15 2ps sinucosueif Y0 2(u;f)/C3014ffiffiffi 5ps 3 cos 2u/C281YrvYru Y1 2(u;f)/C30/C281 2ffiffiffiffiffiffi 152ps sinucosue if Y2 2(u;f)/C301 4ffiffiffiffiffiffi 15 2ps sin2ue2if Y/C283 3(u;f)/C3018ffiffiffiffiffiffi 35 ps sin3ue/C283ifY/C282 3(u;f)/C301 4ffiffiffiffiffiffiffiffi 105 2ps sin2ucosue/C282if Y/C281 3(u;f)/C301 8ffiffiffiffiffiffi 21 ps sinu5 cos2u/C281YrvYru e/C28if Y0 3(u;f)/C3014ffiffiffi 7 ps (5 cos3u/C283 cos u) Y1 3(u;f)/C30/C2818ffiffiffiffiffiffi 21 ps sinu5 cos2u/C281YrvYru eif Y2 3(u;f)/C3014ffiffiffiffiffiffiffiffi 105 2ps sin2ucosue2if Y3 3(u;f)/C30/C2818ffiffiffiffiffiffi 35 ps sin3ue3if: Written in terms of C ARTESIAN COORDINATES , eif/C30x/C27iyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p (18) u/C30sin/C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2 x2/C27y2/C27z2s ! (19) /C30cos/C281 zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2/C27y2/C27z2p ! ; (20) so Y0 0(u;f)/C301 21ffiffiffipp (21) Y0 1(u;f)/C301 2ffiffiffi 3 ps zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2/C27z2p (22) Y1 1(u;f)/C30/C281 2ffiffiffiffiffiffi 3 2ps x/C27iyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2/C27z2p (23) Y0 2(u;f)/C301 4ffiffiffi 5 ps 3z2 x2/C27y2/C27z2/C281 ! (24) Y1 2(u;f)/C30/C281 2ffiffiffiffiffiffi 15 2ps z(x/C27iy) x2/C27y2/C27z2(25) Y2 2(u;f)/C301 4ffiffiffiffiffiffi 15 2ps (x/C27iy)2 x2/C27y2/C27z2: (26) The ZONAL HARMONICS are defined to be those OF THE FORM P0 l(cosu)/C30Pl(cosu): (27) The TESSERAL HARMONICS are those OF THE FORM sin(mf)Pm l(cos u) (28) cos(mf)Pml(cos u) (29) for l "m: The SECTORIAL HARMONICS are OF THE FORM sin(mf)Pmm(cos u) (30) cos(mf)Pmm(cos u) : (31) The spherical harmonics form a COMPLETE ORTHO- NORMAL BASIS , so an arbitrary REAL FUNCTION f(u ; f) can be expanded in terms of complex spherical harmonics by f( u; f) /C13X/C12 l/C300Xl m/C30/C28lAmlYm l( u; f) : (32) or in terms of real spherical harmonics by f(u ; f) /C13X/C12 l/C300Xl m/C300CmlYmc l( u; f) /C27SmlYms l( u; f) ½/C138 : (33) The process of determining the coefficients Am lin (32) is analogous to that to determine the coefficients in a FOURIER SERIES , i.e., multiply both sides of (32) by ¯Ym? l?( u; f) ; integrate, and use the orthogonality rela- tionship (7) to obtain g2 p 0g p 0f( u; f) ¯Ym? l?( u; f) sin u du df /C30X/C12 l/C300Xl m/C30/C28lg2 p 0g p 0Am lYm l¯Ym? l?( u ; f) sin u( u; f) du df /C30X/C12 l/C301Xl m/C30/C28lAmldll? dmm?/C30Aml: (34) The following sequence of plots shows successive approximations to the function f(u;f)/C303/C27 cos3(2u)/C27(sinf)=2;which is illustrated in the final plot. See also CONDON- SHORTLEY PHASE ,C ORRELATION COEFFICIENT ,SECTORIAL HARMONIC ,SOLID HARMO- NIC,S PHERICAL HARMONIC ADDITION THEOREM , SPHERICAL HARMONIC DIFFERENTIAL EQUATION , SPHERICAL HARMONIC CLOSURE RELATIONS ,SPHERI- CAL VECTOR HARMONIC ,SURFACE HARMONIC ,TESS- ERAL HARMONIC ,ZONAL HARMONIC References Arfken, G. "Spherical Harmonics" and "Integrals of the Products of Three Spherical Harmonics." §12.6 and 12.9inMathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 680 /C1/85 and 698 /C1/00, 1985. Byerly, W. E. "Spherical Harmonics." Ch. 6 in An Elemen- tary Treatise on Fourier’s Series, and Spherical, Cylind- rical, and Ellipsoidal Harmonics, with Applications toProblems in Mathematical Physics. New York: Dover, pp. 195 /C1 /18, 1959. Ferrers, N. M. An Elementary Treatise on Spherical Har- monics and Subjects Connected with Them. London: Macmillan, 1877. Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York: Cambridge University Press, 1996. Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics. New York: Chelsea, 1955. MacRobert, T. M. and Sneddon, I. N. Spherical Harmonics: An Elementary Treatise on Harmonic Functions, withApplications, 3rd ed. rev. Oxford, England: Pergamon Press, 1967. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Spherical Harmonics." §6.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing,2nd ed. Cambridge, England: Cambridge University Press, pp. 246 /C1 /48, 1992. Sansone, G. "Harmonic Polynomials and Spherical Harmo- nics," "Integral Properties of Spherical Harmonics and theAddition Theorem for Legendre Polynomials," and "Com-pleteness of Spherical Harmonics with Respect to SquareIntegrable Functions." §3.18/C1 /.20 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 253 /C1/72, 1991. Sternberg, W. and Smith, T. L. The Theory of Potential and Spherical Harmonics, 2nd ed. Toronto: University of Toronto Press, 1946. Weisstein, E. W. "Books about Spherical Harmonics." http:// www.treasure-troves.com/books/SphericalHarmo-nics.html. Whittaker, E. T. and Watson, G. N. "Solution of Laplace’s Equation Involving Legendre Functions" and "The Solu-tion of Laplace’s Equation which Satisfies AssignedBoundary Conditions at the Surface of a Sphere." §18.31 and 18.4 in A Course in Modern Analysis, 4th ed. Cam- bridge, England: Cambridge University Press, pp. 391 /C1 / 95, 1990. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997. Spherical Harmonic Addition Theorem AFORMULA also known as the L EGENDRE ADDITION THEOREM which is derived by finding G REEN’S FUNC- TIONS for the SPHERICAL HARMONIC expansion and equating them to the generating function for L E- GENDRE POLYNOMIALS . When gis defined by cosg/C13cosu1cosu2/C27sinu1sinu2cosf1/C28f2 ðÞ ; The L EGENDRE POLYNOMIAL of argument gis given by Pl(cosg)/C304p 2l/C271Xl m/C30/C28l(/C281)mYm lu1;f1 ðÞ Y/C28m lu2;f2 ðÞ /C304p 2l/C271Xl m/C30/C28lYm lu1;f1 ðÞ ¯Ym lu2;f2 ðÞ /C30Plcosu1 ðÞ Plcosu2 ðÞ /C272Xl m/C301(l /C28 m)! (l /C27 m)!Pm lcos u1 ðÞ Pmlcos u2 ðÞ cos m f1 /C28 f2 ðÞ½/C138 : See also LEGENDRE POLYNOMIAL ,SPHERICAL HARMO- NIC References Arfken, G. "The Addition Theorem for Spherical Harmo- nics." §12.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 693 /C1/95, 1985. Spherical Harmonic Closure Relations The sum of the absolute squares of the SPHERICAL HARMONICS Ym l( u; f) over all values of m is Xl m/C30/C28lYm l( u; f)YrutYrutYrutYrut2/C302l /C27 1 4p: The double sum over m and l is given by X/C12 l/C300Xl m/C30/C28lYm lu1 ; f1 ðÞ ¯Ym lu2 ; f2 ðÞ /C301 sin u1du1 /C28 u2 ðÞ df1 /C28 f2 ðÞ /C30 d cos u1 /C28cos u2 ðÞ d cos f1 /C28cos f2 ðÞ ; where d(x) is the DELTA FUNCTION . Spherical Harmonic Differential Equation In three dimensions, the spherical harmonic differ- ential equation is given by 1 sin u@ @ usin u@ @ u ! /C271 sin2 u@2 @ f2 /C27l(l /C271)"# u /C300; and solutions are called SPHERICAL HARMONICS (Zwil- linger 1997, p. 130). In four dimensions, the spherical harmonic differential equation is uxx /C272ux cot x /C27csc2 xuyy /C27uy cot y /C27uzz csc2 yYrvYru /C27 n2 /C281YrvYru u /C300 (Humi 1987; Zwillinger 1997, p. 130). See also SPHERICAL HARMONIC References Humi, M. "Factorisation of Separable Partial Differential Equations." J. Phys. A: Math. Gen. 20, 4577 /C1/585, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, 1997. Spherical Harmonic Tensor A tensor defined in terms of the TENSORS which satisfy the DOUBLE CONTRACTION RELATION . See also DOUBLE CONTRACTION RELATION ,SPHERICAL HARMONICSpherical Helix The TANGENT INDICATRIX of a CURVE OF CONSTANT PRECESSION is a spherical helix. The equation of a spherical helix on a SPHERE with RADIUS r making an ANGLE u with the Z-AXIS is x( c) /C301 2 r(1 /C27cos u)cos c /C281 2 r(1 /C28cos u)cos1 /C27 cos u 1 /C28 cos uc ! (1) y( c) /C301 2 r(1 /C27cos u)sin c /C281 2 r(1 /C28sin u)sin1 /C27 cos u 1 /C28 cos uc ! (2) z(c) /C30r sin u coscos u 1 /C28 cos uc ! : (3) The projection on the xy-plane is an EPICYCLOID with RADII a/C30rcosu (4) b/C30rsin212uYru*Yru+ : (5) See also HELIX,LOXODROME ,SPHERICAL SPIRAL References Scofield, P. D. "Curves of Constant Precession." Amer. Math. Monthly 102, 531/C1/37, 1995. Spherical Lune A sliver of the surface of a SPHERE ofRADIUS rcut out by two planes through the azimuthal axis with DIHEDRAL ANGLE u:The SURFACE AREA of the lune is S/C302r2u; which is just the area of the SPHERE times u=(2p):The VOLUME of the associated SPHERICAL WEDGE has VOLUME V /C302 3 r3 u: See also LUNE,SPHERE ,SPHERICAL WEDGE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 130, 1987. Harris, J. W. and Stocker, H. "Spherical Wedge." §4.8.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 108, 1998. Gellert, W.; Gottwald, S.; Hellwich, M.; Ka¨stner, H.; and Ku¨nstner, H. (Eds.). VNR Concise Encyclopedia of Mathe- matics, 2nd ed. New York: Van Nostrand Reinhold, p. 262, 1989. Spherical Packing SPHERICAL CODE Spherical Polygon A closed geometric figure on the surface of a SPHERE which is formed by the ARCS of GREAT CIRCLES . The spherical polygon is a generalization of the SPHERICAL TRIANGLE .If u is the sum of the RADIAN ANGLES of a spherical polygon on a SPHERE ofRADIUS R, then the AREA is S/C30[u/C28(n/C282)p]R2: See also GREAT CIRCLE ,SPHERICAL TRIANGLE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 131, 1987. Spherical Ring ASPHERE with a CYLINDRICAL HOLE cut so that the centers of the CYLINDER and SPHERE coincide, also called a NAPKIN RING . Let the SPHERE be of RADIUS r and the CYLINDER ofRADIUS R. The VOLUME of the entire CYLINDER is Vcyl/C30pLR2; (1)and the VOLUME of the upper segment is Vseg/C3016ph3R2/C27h2YrvYru ; (2) where R/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C281 4L2q (3) h/C30r/C281 2L; (4) so the VOLUME removed upon drilling of a CYLINDRI- CALhole is Vrem/C30Vcyl/C272Vseg/C30pLR2/C271 3h3R2/C27h2YrvYruhi /C30pLR2/C27hR2/C2713h3Yru*Yru+ /C30pLr2/C281 4L2Yru*Yru+ /C27r/C2812LYru*Yru+ r2/C2814L2Yru*Yru+ /C2713r/C2812LYru*Yru+3YrtvYrtu /C30pYrtv Lr2/C281 4L3/C27r3/C2812r2L/C2814RL2/C2718L3Yru*Yru+ /C2713r3/C2832r2L/C2734rL2/C2818L3Yru*Yru+ Yrtu /C30pYrtv 43r3/C271/C2812/C2812Yru*Yru+ r2L/C27/C2814/C2714Yru*Yru+ RL2 /C27L3/C281 4/C2718/C281 24Yru*Yru+Yrtu /C304 3pr3/C2816pL3/C3016p8r3/C28L3YrvYru ; (5) so Vleft/C30Vsphere/C28Vrem/C304 3pr3/C2843pr3/C2816pL3Yru*Yru+ /C3016pL3: (6) Spherical Sector A spherical sector is a SOLID OF REVOLUTION enclosed by two radii from the center of a SPHERE . The spherical sector may either be "open" and have a conical HOLE (left figure; Beyer 1987), or may be a "closed" SPHERICAL CONE (right figure; Harris and Stocker 1998). The VOLUME of a spherical sector in either case is given by V /C302 3 pR2h ; where h is the vertical distance between where the upper and lower radii intersect the sphere and R is the sphere’s radius. See also CYLINDRICAL SEGMENT ,SPHERE ,SPHERICAL CAP,SPHERICAL CONE,SPHERICAL SEGMENT ,SPHE- RICAL WEDGE ,ZONE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 131, 1987. Harris, J. W. and Stocker, H. "Spherical Sector." §4.8.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 106 /C1/07, 1998. Kern, W. F. and Bland, J. R. "Spherical Sector." §37 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 103 /C1/06, 1948. Smith, D. E. "Spherical Sector." §542 in Essentials of Plane and Solid Geometry. Boston, MA: Ginn and Co., p. 542, 1923. Spherical Segment A spherical segment is the solid defined by cutting a SPHERE with a pair of PARALLEL PLANES . It can be thought of as a SPHERICAL CAP with the top truncated, and so it corresponds to a SPHERICAL FRUSTUM . The surface of the spherical segment (excluding the bases) is called a ZONE . However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for SPHERICAL CAP and "zone" for what is here called a spherical segment. Call the RADIUS of the SPHERE R and the height of the segment (the distance from the plane to the top of SPHERE ) h. Let the RADII of the lower and upper bases be denoted a and b, respectively. Call the distance from the center to the start of the segment d, and the height from the bottom to the top of the segment h. Call the RADIUS parallel to the segment r, and the height above the center y. Then r2 /C30R2 /C28y2 ; V /C30gd /C27h dpr2 dy /C30 pgd /C27h dR2 /C28y2YrvYru dy /C30 p R2y /C281 3 y3hid/C27h d/C30 p R2h /C2813(d /C27h)3 /C28d3hino/C30 p R2h /C2813d3 /C273d2h /C273h2d /C27h3 /C28d3YrvYruhi /C30 phR2/C28d2 /C28hd /C281 3 h2Yru*Yru+ ; (1) Using a2 /C30R2 /C28d2 (2) b2 /C30R2 /C28(d /C27h)2 /C30R2 /C28d2 /C282dh /C28h2 ; (3) gives a2 /C27b2 /C302R2 /C282d2 /C282dh /C28h2 (4) R2 /C28d2 /C28dh /C301 2a2 /C27b2 /C27h2YrvYru ; (5) so V /C30 ph12a2 /C27b2 /C27h2YrvYru /C2813 h2hi /C30 ph12 a2 /C2712 b2 /C2716 h2Yru*Yru+ /C3016 ph 3a2 /C273b2 /C27h2YrvYru : (6) The surface area of the ZONE (which excludes the top and bottom bases) is given by S /C302pRh : (7) See also ARCHIMEDES’ HAT-BOX THEOREM ,A RCHI- MEDES’ PROBLEM ,FRUSTUM ,H EMISPHERE ,SPHERE , SPHERICAL CAP,S PHERICAL SECTOR ,S PHERICAL WEDGE ,SURFACE OF REVOLUTION ,ZONE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 130, 1987. Harris, J. W. and Stocker, H. "Spherical Zone (Spherical Layer)." §4.8.5 in Handbook of Mathematics and Compu- tational Science. New York: Springer-Verlag, pp. 107 /C1/08, 1998. Kern, W. F. and Bland, J. R. "Spherical Segment." §36 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 97 /C1/02, 1948. Smith, D. E. "Spherical Segment." §541 in Essentials of Plane and Solid Geometry. Boston, MA: Ginn and Co., p. 542, 1923. Spherical Shell A generalization of an ANNULUS to 3-D. A spherical shell is the intersection of two concentric BALLS of differing RADII . See also ANNULUS ,BALL,CHORD ,SPHERE ,SPHERICAL HELIX Spherical Simplex The only irreducible spherical simplexes generated by reflection are An(/n]1);Bn(/n]4);Cn(/n]2);DP 2/ (/p]5);E6;E7;E8;F4;G3;andG4:The only irreducible Euclidean simplexes generated by reflection are W2; Pm(/m]3);Qm(/m]5);Rm(/m]3);Sm(/m]4);V3;T7; T8;T9;andU5:/ Spherical Spiral The SPHERICAL CURVE taken by a ship which travels from the south pole to the north pole of a SPHERE while keeping a fixed (but not RIGHT ) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles. It is given by the PARAMETRIC EQUATIONS x /C30cos t cos c y /C30sin t cos c z /C30/C28sin c ; where c /C13tan/C281(at) and a is a constant, and is a special case of a LOXODROME . See also HELIX,LOXODROME ,MERCATOR PROJECTION , SEIFFERT’S SPHERICAL SPIRAL ,SPHERICAL CURVE References Gray, A. "Loxodromes on Spheres." §10.6 in Modern Differ- ential Geometry of Curves and Surfaces with Mathema- tica, 2nd ed. Boca Raton, FL: CRC Press, pp. 238 /C1/40, 1997. Lauwerier, H. "Spherical Spiral." In Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 64 /C1/6, 1991. Spherical Symmetry Let A and B be constant VECTORS . Define Q /C133(A /C215 ˆr)(B /C215 ˆr) /C28A /C215 B : Then the average of Q over a spherically symmetric surface or volume is Qhi/C30 3 cos2 u /C281YruvYruu (A /C215 B) /C300; since 3 cos2 u /C281 hi /C300 over the sphere. Spherical Tessellation TRIANGULAR SYMMETRY GROUPSpherical Triangle A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar TRIANGLE , and is sometimes called EULER’S TRIANGLE (Harris and Stocker 1998). Let a spherical triangle have ANGLES A, B, and C (measured in radians at the vertices along the surface of the sphere) and let the sphere on which the spherical triangle sits have RADIUS R. Then the SURFACE AREA D of the spherical triangle is D/C30R2[(A /C27B /C27C) /C28 p] /C30R2E; where E is called the SPHERICAL EXCESS , with E /C300 in the degenerate case of a planar triangle. The sum of the angles of a spherical triangle is between p and 3p radians (1808 and 5408; Zwillinger 1995, p. 469). The amount by which it exceeds 1808 is called the SPHERICAL EXCESS and is denoted E or D; the latter of which can cause confusion since it also can refer to the SURFACE AREA of a spherical triangle. The difference between 2 pradians (360 8) and the sum of the side arc lengths a,b, and cis called the SPHERICAL DEFECT and is denoted Dord:/ The study of angles and distances of figures on a sphere is known as SPHERICAL TRIGONOMETRY . See also CIRCULAR TRIANGLE ,COLUNAR TRIANGLE , GEODESIC DOME,G EODESIC TRIANGLE ,G IRARD’S SPHERICAL EXCESS FORMULA ,L’HUILIER’S THEOREM , NAPIER’S ANALOGIES ,SPHERICAL DEFECT ,SPHERICAL EXCESS ,SPHERICAL POLYGON ,SPHERICAL TRIGONO- METRY References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 79, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 147 /C1/50, 1987. Gellert, W.; Gottwald, S.; Hellwich, M.; Ka ¨stner, H.; and Ku¨nstner, H. (Eds.). "The Spherical Triangle." §12.2 in VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, pp. 262 /C1/72, 1989. Green, R. M. Spherical Astronomy. New York: Cambridge University Press, 1985. Harris, J. W. and Stocker, H. "General Spherical Triangle." §4.9.1 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 108 /C1/09, 1998. Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, 1960. Zwillinger, D. (Ed.). "Spherical Geometry and Trigonome- try." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 468 /C1/71, 1995. Spherical Trigonometry Let a SPHERICAL TRIANGLE be drawn on the surface of aSPHERE of radius R, centered at a point /O/C30(0;0;0)/ , with vertices A,B, and C. The vectors from the center of the sphere to the vertices are therefore given bya/C13/OAYruPu!,b/C13/OBYruPu!, and c/C13/OCYruPu!. Now, the angular lengths of the sides of the triangle (in radians) are then a?/C13/C218BOC ;b?/C13/C218COA ;andc?/C13/C218AOB ;and the actual arc lengths of the side are a/C30Ra?;b/C30Rb?;and c/C30Rc?:Explicitly, a /C215b/C30R2cosc?/C30R2cosc R ! (1) a /C215c/C30R2cosb?/C30R2cosb R ! (2) b /C215c/C30R2cosa?/C30R2cosa R ! : (3) Now make use of A,B, and Cto denote both the vertices themselves and the angles of the spherical triangle at these vertices, so that the DIHEDRAL ANGLE between PLANES AOB and AOC is written A, the DIHEDRAL ANGLE between PLANES BOC and AOB is written B, and the DIHEDRAL ANGLE between PLANES BOC and AOC is written C. (These angles are sometimes instead denoted a;b;g; e.g., Gellert et al. 1989)Consider the DIHEDRAL ANGLE Abetween planes AOB and AOC , which can be calculated using the DOT PRODUCT of the normals to the planes. The normals are given by CROSS PRODUCTS of the vectors to the vertices, so ˆa/C29ˆbYrvYru /C215ˆa/C29ˆc ðÞ /C30½ˆa½½ˆb½sincYrvYru ½ˆa½½ˆc½sinb ðÞ cosA /C30sinbsinccosA: (4) However, using a well-known vector identity givesˆa/C29ˆbYrvYru /C215ˆa/C29ˆcYrvYru /C30ˆa /C215ˆb/C29ˆa/C29ˆc ðÞYrtYrP /C30ˆa /C215ˆaˆb /C215ˆcYrvYru /C28ˆcˆa /C215ˆbYrvYruYrtYrP /C30ˆb /C215ˆcYrvYru /C28ˆa /C215ˆc ðÞ ˆa /C215ˆbYrvYru /C30cosa/C28cosccosb: (5) Since these two expressions must be equal, we obtainthe identity (and its two analogous formulas) cosa/C30cosbcosc/C27sinbsinccosA (6) cosb/C30cosccosa/C27sincsinacosB (7) cosc/C30cosacosb/C27sinasinbcosC: (8) known as the cosine rules for sides (Smart 1960, pp. 7/C1 /; Gellert et al. 1989, p. 264; Zwillinger 1995, p. 469). The identity sinA/C30ˆa/C29ˆbYrvYru /C29ˆa/C29ˆc ðÞYrutYrutYrutYrut ˆa/C29ˆbYrutYrutYrutYrutˆa/C29ˆc jj /C30ˆaˆb;ˆa;ˆcYrtYrP /C27ˆbˆa;ˆa;ˆc ½/C138YrutYrutYrutYrut sinbsinc /C30ˆa;ˆb;ˆcYrtYrP sinbsinc; (9) where /[a;b;c]/is the SCALAR TRIPLE PRODUCT , gives sinA sina/C30ˆa;ˆb;ˆcYrtYrP sinasinbsinc; (10) so the spherical analog of the LAW OF SINES can be written sinA sina/C30sinB sinb/C30sinC sinc/C306 Vol( OABC ) sinasinbsinc(11) (Smart 1960, pp. 9 /C1/0; Gellert et al. 1989, p. 265; Zwillinger 1995, p. 469), where Vol( OABC ) is the VOLUME of the TETRAHEDRON . The analogs of the LAW OF COSINES for the angles of a SPHERICAL TRIANGLE are given by cosA/C30/C28cosBcosC/C27sinBsinCcosa (12) cosB/C30/C28cosCcosA/C27sinCsinAcosb (13) cosC/C30/C28cosAcosB/C27sinAsinBcosc (14) (Gellert et al. 1989, p. 265; Zwillinger 1995, p. 470). Finally, there are spherical analogs of the LAW OF TANGENTS , tan1 2(B/C28C)hi tan1 2(B/C27C)hi /C30tan1 2(b/C28c)hi tan1 2(b/C27c)hi (15) tan1 2(C/C28A)hi tan1 2(C/C27A)hi /C30tan1 2(c/C28a)hi tan1 2(c/C27a)hi (16) tan1 2(A/C28B)hi tan1 2(A/C27B)hi /C30tan1 2(a/C28b)hi tan1 2(a/C27b)hi (17) (Beyer 1987; Gellert et al. 1989; Zwillinger 1995, p. 470). Additional important identities are given by cosA/C30cscbcscc(cosa/C28cosbcosc): (18) (Smart 1960, p. 8), sinacosB/C30cosbsinc/C28sinbcosccosA (19) (Smart 1960, p. 10), and cosacosC/C30sinacotb/C28sinCcotB (20) (Smart 1960, p. 12).Let s/C13 1 2(a/C27b/C27c) (21) be the semiperimeter, then half-angle formulas for sines can be written as sin1 2AYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin(s/C28b)sin(s/C28c) sinbsincs (22) sin1 2BYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin(s/C28a)sin(s/C28c) sinasincs (23) sin12CYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin(s/C28a)sin(s/C28b) sinasinbs : (24) for cosines can be written as cos1 2AYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinssin(s/C28a) sinbsincs (25) cos1 2BYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinssin(s/C28b) sinasincs (26) cos1 2CYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinssin(s/C28c) sinasinbs : (27) and tangents can be written as tan1 2AYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin(s/C28b)sin(s/C28c) sinssin(s/C28a)s /C30k sin(s/C28a)(28) tan12BYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin(s/C28a)sin(s/C28c) sinssin(s/C28b)s /C30k sin(s/C28b)(29) tan1 2CYru*Yru+ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin(s/C28a)sin(s/C28b) sinssin(s/C28c)s /C30k sin(s/C28c);(30)where k2/C30sin(s/C28a)sin(s/C28b)sin(s/C28c) sins(31) (Smart 1960, pp. 8 /C1/; Gellert et al. 1989, p. 265; Zwillinger 1995, p. 470). Let S/C131 2(A/C27B/C27C) (32) be the sum of half-angles, then the half-side formulas are tan1 2aYru*Yru+ /C30Kcos(S/C28A) (33) tan1 2bYru*Yru+ /C30Kcos(S/C28B) (34) tan1 2cYru*Yru+ /C30Kcos(S/C28C): (35) where K2/C30/C28cosS cos(S/C28A)cos(S/C28B)cos(S/C28C)(36) (Gellert et al. 1989, p. 265; Zwillinger 1995, p. 470). The HAVERSINE formula for sides, where havx/C131 2(1/C28cosx)/C30sin212xYru*Yru+ ; (37) is given by hava/C30hav(b/C28c)/C27sinbsinchavA (38) (Smart 1960, pp. 18 /C1/9; Zwillinger 1995, p. 471), and the HAVERSINE formula for angles is given by havA/C30sin(s/C28b)sin(s/C28c) sinbsinc(39) /C30hava/C28hav(b/C28c) sinbsinc(40) /C30hav[p/C28(B/C27C)]/C27sinBsinChava (41) (Zwillinger 1995, p. 471). GAUSS’S FORMULAS (also called Delambre’s analogies) are sin1 2(a/C28b)hi sin1 2cYru*Yru+ /C30sin1 2(A/C28B)hi cos1 2CYru*Yru+ (42) sin12(a/C27b)hi sin1 2cYru*Yru+ /C30cos1 2(A/C28B)hi sin1 2CYru*Yru+ (43) cos1 2(a/C28b)hi cos1 2cYru*Yru+ /C30sin1 2(A/C27B)hi cos1 2CYru*Yru+ (44) cos1 2(a /C27 b)hi cos1 2 cYru*Yru+ /C30cos1 2(A /C27 B)hi sin1 2 CYru*Yru+ (45) (Smart 1960, p. 22; Zwillinger 1995, p. 470). NAPIER’S ANALOGIES are sin12(A /C28 B)hi sin1 2(A /C27 B)hi /C30tan1 2(a /C28 b)hi tan1 2 cYru*Yru+ (46) cos1 2(A /C28 B)hi cos1 2(A /C27 B)hi /C30tan1 2(a /C27 b)hi tan1 2 cYru*Yru+ (47) sin1 2(a /C28 b)hi sin1 2(a /C27 b)hi /C30tan1 2(A /C28 B)hi cot1 2 CYru*Yru+ (48) cos1 2(a /C28 b)hi cos1 2(a /C27 b)hi /C30tan1 2(A /C27 B)hi cot1 2 CYru*Yru+ (49) (Beyer 1987; Gellert et al. 1989, p. 266; Zwillinger 1995, p. 471). See also ANGULAR DEFECT ,DESCARTES TOTAL ANGU- LAR DEFECT ,GAUSS’S FORMULAS ,GIRARD’S SPHERICAL EXCESS FORMULA ,LAW OF COSINES ,LAW OF SINES, LAW OF TANGENTS ,L’HUILIER’S THEOREM ,NAPIER’S ANALOGIES ,SPHERICAL EXCESS ,SPHERICAL GEOME- TRY,SPHERICAL POLYGON ,SPHERICAL TRIANGLE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 147 /C1/50, 1987. Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed., rev. ed. Richmond, VA: Willmann-Bell, 1988. Gellert, W.; Gottwald, S.; Hellwich, M.; Ka¨stner, H.; and Ku¨nstner, H. (Eds.). "Spherical Trigonometry." §12 in VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, pp. 261 /C1/82, 1989. Green, R. M. Spherical Astronomy. New York: Cambridge University Press, 1985. Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, 1960. Zwillinger, D. (Ed.). "Spherical Geometry and Trigonome- try." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 468 /C1/71, 1995. Spherical Vector Harmonic VECTOR SPHERICAL HARMONIC Spherical Wedge The VOLUME of a spherical wedge is V /C3023 r3 u: The surface area of the corresponding SPHERICALLUNE is S/C302r2u: See also SPHERE ,SPHERICAL CAP,SPHERICAL LUNE, SPHERICAL SECTOR ,SPHERICAL SEGMENT ,W EDGE References Harris, J. W. and Stocker, H. "Spherical Wedge." §4.8.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 108, 1998. SphericalHarmonicY SPHERICAL HARMONIC Sphericon The solid formed from a BICONE with opening angle of 908. Slice the solid by a plane containing the rota- tional axes. The resulting CROSS SECTION is a SQUARE . Now rotate the two pieces by 90 8and reconnect them. The above net shows another way the sphericon can be constructed. In this figure u/C30pffiffiffi 2p =2 radians : 127:28/C14: This solid was discovered by C. J. Roberts, and is not as widely known as it should be! A sphericon has a single continuous face. A sphericon rolls by wobbling from one face to another, resulting in straight-line motion. In addition, one sphericon can roll around another. See also BICONE ,CONE,CONE NET,SPHERE References Stewart, I. "Cone with a Twist." Sci. Amer. 281, 116 /C1/17, Oct. 1999. Spheroid A spheroid is an ELLIPSOID r2 cos2 u sin2 f a2 /C27r2 sin2 u sin2 f b2 /C27r2 cos2 f c2/C301 (1) with two SEMIMAJOR AXES equal. Orient the ELLIPSE so that the a and b axes are equal, then r2 cos2 u sin2 f a2 /C27r2 sin2 u sin2 f a2 /C27r2 cos2 f c2/C301 (2) r2 sin2 f a2/C27r2 cos2 f c2/C301 : (3) where a is the equatorial RADIUS and c is the polar RADIUS . The PARAMETRIC EQUATIONS therefore be- come x /C30a cos u sin f (4) y /C30a sin u sin f (5) z /C30c cos f (6) for u /C23 [0; 2p) and f /C23 [0; p] :/ Here f is the colatitude, so take d /C13 p=2 /C28 f to express in terms of latitude. r2 cos2 d a2/C27r2 sin2 d c2/C301 : (7) Rewriting cos2 d /C301 /C28sin2 d gives r2 a2 /C27r2 sin2 d1 c2 /C281 a2 ! /C301 (8)r21 /C27a2 sin2 da2 /C28 c2 c2a2 ! /C30r21 /C27sin2 da2 /C28 c2 c2 ! /C30a2 : (9) so r /C30a 1 /C27sin2 da2 /C28 c2 c2 !/C281 =2 : (10) If a /C21c, the spheroid is OBLATE .Ifa Bc, the spheroid is PROLATE .Ifa /C30c, the spheroid degenerates to a SPHERE . See also DARWIN-DE SITTER SPHEROID ,E LLIPSOID , OBLATE SPHEROID ,PROLATE SPHEROID Spheroidal Coordinates OBLATE SPHEROIDAL COORDINATES ,PROLATE SPHER- OIDAL COORDINATES Spheroidal Function OBLATE SPHEROIDAL WAVE FUNCTION ,P ROLATE SPHEROIDAL WAVE FUNCTION ,S PHEROIDAL WAVE FUNCTION Spheroidal Harmonic A spheroidal harmonic is a special case of the ELLIPSOIDAL HARMONIC which satisfies the differen- tial equation d dx1/C28x2YrvYru ds dx"# /C27l/C28c2x2/C28m2 1/C28x2 ! S/C300 on the interval /C2815x51:/ See also ELLIPSOIDAL HARMONIC References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "A Worked Example: Spheroidal Harmonics." §17.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 764 /C1/73, 1992. Spheroidal Wave Function Whittaker and Watson (1990, p. 403) define the internal and external spheroidal wavefunctions as S(1) mn/C302p(n/C28m)! (n/C27m)!Pmn(ir)Pmn(cosu)cos sin(mf) S(2) mn/C302p(n/C28m)! (n/C27m)!Qmn(ir)Qmn(cosu)cos sin(mf); where Pm l(x)i saL EGENDRE POLYNOMIAL andQml(x)i s aLEGENDRE FUNCTION OF THE SECOND KIND . Stratton (1935), Chu and Stratton (1941), and Rhodes (1970) define the spheroidal functions as those solu- tions of the differential equation 1 /C28 h2YrvYru cƒan(c ; h) /C282(a /C271)hc?an(c; h) /C27 ban /C28c2 h2YrvYru can(c ; h) /C300 which remain finite at the singular points h /C3091: The condition of finiteness restricts the admissible values of the parameter ban(c) to a discrete set of eigenvalues indexed by n /C300, 1, 2, ... (Rhodes 1970). See also ELLIPSOIDAL HARMONIC ,OBLATE SPHEROI- DAL WAVE FUNCTION ,PROLATE SPHEROIDAL WAVE FUNCTION ,SPHERICAL HARMONIC References Abramowitz, M. and Stegun, C. A. (Eds.). "Spheroidal Wave Functions." Ch. 21 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 751 /C1/59, 1972. Chu, L. J. and Stratton, J. A. "Elliptic and Spheroidal Wave Functions." J. Math. and Phys. 20, 259 /C1/09, 1941. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 642 /C1/44, 1953. Rhodes, D. R. "On the Spheroidal Functions." J. Res. Nat. Bur. Standards--B. Math. Sci. 74B, 187 /C1/09, Jul.-Sep. 1970. Stratton, J. A. "Spheroidal Functions." Proc. Nat. Acad. Sci. 21,51/C1/6, 1935. Stratton, J. A.; Morse, P. M.; Chu, L. J.; Little, J. D. C.; and Corbato ´,F.J. Spheroidal Wave Functions. New York: Wiley, 1956. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Sphinx A6- POLYIAMOND named for its resemblance to the Great Sphinx of Egypt. References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Spider and Fly Problem In a rectangular room (a CUBOID ) with dimensions30 ?/C2912?/C2912 ?; a spider is located in the middle of one 12 ?/C2912? wall one foot away from the ceiling. A fly is in the middle of the opposite wall one foot away from the floor. If the fly remains stationary, what is the shortest distance the spider must crawl to capture the fly? The answer, 40 ?; can be obtained by "flattening" the walls as illustrated above. The puzzle was originally posed in an English newspaper by Dudeney in 1903 (Gardner 1958). References Gardner, M. "Mathematical Games: About Henry Ernest Dudeney, A Brilliant Creator of Puzzles." Sci. Amer. 198, 108 /C1/12, Jun. 1958. Pappas, T. "The Spider & the Fly Problem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 218 and 233, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 173 /C1/75, 1999. Spider Lines EPITROCHOID Spiegeldrieck FUHRMANN TRIANGLE Spieker Center The center of the S PIEKER CIRCLE . It is the CENTROID of the PERIMETER of the original TRIANGLE . The Spieker center is also the CLEAVANCE CENTER (Hon- sberger 1995). The Spieker center lies on the N AGEL LINE. The Spieker center, third B ROCARD POINT , and ISO- TOMIC CONJUGATE POINT of the INCENTER are COLLI- NEAR . See also BROCARD POINTS ,C ENTROID (TRIANGLE ), CLEAVANCE CENTER ,CLEAVER ,INCENTER ,ISOTOMIC CONJUGATE POINT ,NAGEL LINE,PERIMETER ,SPIEKER CIRCLE ,TAYLOR CENTER References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nded., rev. enl. Dublin: Hodges, Figgis, & Co., p. 81, 1893. Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 3 /C1 /, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 226 /C1/29 and 249, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/87, 1994. Spieker Circle The common INCIRCLE of the MEDIAL TRIANGLE DMAMBMCand the congruent triangle DQ1Q2Q3 illustrated above, where Qiare the MIDPOINTS of the line segment joining the NAGEL POINT Na with the vertices of the original triangle DABC : The center of the Spieker circle is called the SPIEKER CENTER Sp. See also INCIRCLE ,M EDIAL TRIANGLE ,M IDPOINT , NAGEL POINT ,SPIEKER CENTER References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 53, 1971. Honsberger, R. "The Nagel Point M and the Spieker Circle." §1.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 3 /C1/3, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 226 /C1/28, 1929. Spieker, T. "Ein merkwu ¨rdiger Kreis um den Schwerpunkt des Perimeters des geradlinigen Dreiecks als Analogen des Kreises der neun Punkte." Archiv Math. u. Phys. 51, 10 /C1/4, 1870. Spigot Algorithm An ALGORITHM which generates digits of a quantity one at a time without using or requiring previously computed digits. Amazingly, spigot ALGORITHMS are known for both PI and E. Spijker’s Lemma The image on the RIEMANN SPHERE of any CIRCLE under a COMPLEX rational mapping with NUMERATOR and DENOMINATOR having degrees no more than n has length no longer than 2np:/ References Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32,1/C1/7, 1995. Wegert, E. and Trefethen, L. N. "From the Buffon Needle Problem to the Kreiss Matrix Theorem." Amer. Math. Monthly 101, 132 /C1/39, 1994.Spindle LEMON ,SPINDLE CYCLIDE Spindle Cyclide The inversion of a SPINDLE TORUS . If the inversion center lies on the torus, then the spindle cyclide degenerates to a PARABOLIC SPINDLE CYCLIDE . See also CYCLIDE ,H ORN CYCLIDE ,PARABOLIC CY- CLIDE ,RING CYCLIDE ,SPINDLE TORUS ,TORUS Spindle Torus One of the three STANDARD TORI given by the PARA- METRIC EQUATIONS x /C30(c /C27a cos v)cos u y /C30(c /C27a cos v)sin u z /C30a sin v with c Ba. The exterior surface is called an APPLE and the interior surface a LEMON . The above left figure shows a spindle torus, the middle a cutaway, and the right figure shows a CROSS SECTION of the spindle torus through the xz-plane. See also APPLE ,C YCLIDE ,H ORN TORUS ,L EMON , PARABOLIC SPINDLE CYCLIDE ,RING TORUS ,SPINDLE CYCLIDE ,STANDARD TORI,TORUS References Gray, A. "Tori." §13.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 304 /C1/06, 1997. Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28 /C1/0, 1986. Spindle-Shaped Ellipsoid PROLATE SPHEROID Spinode A function f(x) has a spinode (also called a horizontal cusp) at a point x0 if f(x)is CONTINUOUS at x0 and lim x0x0f ?(x) /C30/C12 from one side while lim x0x0f ?(x) /C30/C28/C12 from the other side, so the curve is CONTINUOUS but the DERIVATIVE is not. See also ACNODE ,CRUNODE ,CUSP,TACNODE Spinor A two-component COMPLEX COLUMN VECTOR . Spinors are used in physics to represent particles with half- integral spin (i.e., fermions ). See also LIE DERIVATIVE (SPINOR ), MINKOWSKI SPACE , SPINOR FIELD,TWISTOR References Cartan, E` . The Theory of Spinors. New York: Dover, 1981. Corson, E. M. Introduction to Tensors, Spinors and Relati- vistic Wave-Equations. London: Blackie and Son, 1955. Lounesto, P. "Counterexamples to Theorems Published and Proved in Recent Literature on Clifford Algebras, Spinors, Spin Groups, and the Exterior Algebra." http://www.hit.fi/ ~lounesto/counterexamples.htm. Morse, P. M. and Feshbach, H. "The Lorentz Transforma- tion, Four-Vectors, Spinors." §1.7 in Methods of Theore- tical Physics, Part I. New York: McGraw-Hill, pp. 93 /C1/07, 1953. Penrose, R. and Rindler, W. Spinors and Space-Time, Vol. 1: Two-Spinor Calculus and Relativistic Fields. Cambridge, England: Cambridge University Press, 1987. Penrose, R. and Rindler, W. Spinors and Space-Time, Vol. 2: Spinor and Twistor Methods in Space-Time Geometry Cambridge, England: Cambridge University Press, 1987. Spinor Field See also SPINOR ,TWISTOR Spira Mirabilis LOGARITHMIC SPIRALSpiral In general, a spiral is a curve with t(s) =k(s) equal to a constant for all s, where t is the TORSION and k is the CURVATURE . See also ARCHIMEDES’ SPIRAL ,C IRCLE INVOLUTE , CONICAL SPIRAL ,C ORNU SPIRAL ,C OTES’ SPIRAL , DAISY,EPISPIRAL ,FERMAT’S SPIRAL ,H ELIX,H YPER- BOLIC SPIRAL ,LOGARITHMIC SPIRAL ,M ICE PROBLEM , NIELSEN’S SPIRAL ,PHYLLOTAXIS ,POINSOT’S SPIRALS , POLYGONAL SPIRAL ,SPHERICAL SPIRAL References Eppstein, D. "Spirals." http://www.ics.uci.edu/~eppstein/ junkyard/spiral.html. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 54 /C1/6, 1991. Lockwood, E. H. "Spirals." Ch. 22 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 172 /C1/75, 1967. Weisstein, E. W. "Books about Spirals." http://www.trea- sure-troves.com/books/Spirals.html. Yates, R. C. "Spirals." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 206 /C1/16, 1952. Spiral Point A FIXED POINT for which the EIGENVALUES are COM- PLEX CONJUGATES . See also STABLE SPIRAL POINT ,U NSTABLE SPIRAL POINT References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Spiral Similarity The combination of a CENTRAL DILATION and a ROTATION about the same center. However, the combination of a central dilation and a rotation whose centers are distinct is also a spiral symmetry. In fact, any two DIRECTLY SIMILAR figures are related either by a TRANSLATION or by a spiral symmetry (Coxeter and Greitzer 1967, p. 97). See also CENTRAL DILATION ,D ILATION ,R OTATION , SIMILAR References Coxeter, H. S. M. and Greitzer, S. L. "Spiral Similarity." §4.8 inGeometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 95 /C1/00, 1967. Spirallohedron RHOMBIC SPIRALLOHEDRON Spiral-Similarity Tessellation A tessellation constructed by placing a series of polygonal tiles of decreasing size on an equilateral spiral. Any ordinary TESSELLATION can be converted to such a form. See also TESSELLATION References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 239, 1991. Spiric Section The equation of the curve of intersection of a TORUS with a plane perpendicular to both the midplane of the torus and to the plane x /C300. (The general intersection of a TORUS with a plane is called a TORIC SECTION ). Let the tube of a torus have radius a, let its midplane lie in the z /C300 plane, and let the center of the tube lie at a distance c from the origin. Now cut the torus with the plane y /C30r. The equation of the TORUS with y /C30r gives the equationc /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27r2pYru*Yru+2 /C27z2 /C30a2 (1) c2 /C28a2 /C27x2 /C27z2 /C302cffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27r2p (2) r2 /C28a2 /C27c2 /C27x2 /C27z2YrvYru2/C304c2 x2 /C27r2YrvYru : (3) The above plots show a series of spiric sections for the RING TORUS , HORN TORUS , and SPINDLE TORUS , re- spectively. When r /C300, the curve consists of two CIRCLES of RADIUS a whose centers are at (c ; 0) and (/C28c; 0): If r /C30c /C27a ; the curve consists of one point (the origin), while if r > c /C27a ; no point lies on the curve. The spiric extensions are an extension of the CONIC SECTIONS constructed by Menaechmus around 150 BC by cutting a CONE by a PLANE , and were first considered around 50 AD by the Greek mathemati- cian Perseus (MacTutor). If r /C30a, then (3) simplifies to x2 /C27z2 /C27c2YrvYru2/C284c2x2 /C304c2a2 ; (4) which is the equation of CASSINI OVALS .C ASSINI OVALS are therefore SPIRIC SECTIONS . Furthermore, the surface having these curves as CROSS SECTIONS is the C ASSINI SURFACE illustrated above, with the modification that the vertical component is squared instead of to the fourth power (Gosper). See also TORIC SECTION ,TORUS References MacTutor History of Mathematics Archive. "Spiric Sections." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Spir- ic.html. Spirograph AHYPOTROCHOID generated by a fixed point on a CIRCLE rolling inside a fixed CIRCLE . It has parametric equations, x/C30(R/C27r) cos u/C28(r/C27r) cosR/C27r ru ! (1) y/C30(R/C27r) sin u/C28(r/C27r) sinR/C27r ru ! ; (2) where Ris the radius of the fixed circle, ris the radius of the rotating circle, and ris the offset of the edge of the rotating circle. The figure closes only if R, r, and rare RATIONAL . The equations can also be written x/C30x0[mcost/C27acos(nt)]/C28y0[msint/C28asin(nt)] (3) y/C30y0[mcost/C27acos(nt)]/C27x0[msint/C28asin(nt)]:(4) where the outer wheel has radius 1, the inner wheel a radius p=q;the pen is placed aunits from the center, the beginning is at uradians above the X-AXIS , and m/C13q/C28p q(5) n/C13q/C28p p(6) x0/C13cosu (7) y0/C13sinu: (8) The following curves are for a/C30i=10;with i/C301, 2, ..., 10, and u/C300:/ (p;q)/C30(1;3) (p;q)/C30(1;4) (p;q)/C30(1;5) (p;q)/C30(2;5) (p;q)/C30(2;7) (p; q) /C30(3; 7) Additional attractive designs such as the following can also be made by superposing individual spiro- graphs. See also EPITROCHOID ,H ARMONOGRAPH ,H YPOTRO- CHOID ,MAURER ROSE,SPIROLATERAL Spirolateral A figure formed by taking a series of steps of length 1, 2, ..., n, with an angle u turn after each step. The symbol for a spirolateral is a1 ; ... ; ak nu ; where the ai/s indicate that turns are in the /C28u direction for these steps. See also MAURER ROSE,SPIROGRAPH References Gardner, M. "Fantastic Patterns Traced by Programmed ‘Worms."’ Sci. Amer. , Nov 1973. Gardner, M. "Worm Paths." Ch. 17 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 205 /C1/21, 1986. Hall, L. "Trochoids, Roses, and Thorns--Beyond the Spiro- graph." College Math. J. 23,20/C1/5, 1992. Odds, F. C. "Spirolaterals." Math. Teacher 66, 121 /C1/24, 1973. Trott, M. "Spirographs with Mathematica ." http://library.- wolfram.com/demos/v4/Spirograph.nb. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 239 /C1/41, 1991. Splay Tree A self-organizing data structure which uses rotations to move any accessed key to the root. This leaves recently accessed nodes near the top of the tree, making them very quickly searchable (Skiena 1997, p. 177). See also TREE References Skiena, S. S. The Algorithm Design Manual. New York: Springer-Verlag, pp. 177 and 179, 1997. Sleator, D. and Tarjan, R. "Self-Adjusting Binary Search Trees." J. ACM 32, 652/C1/86, 1985. Tarjan, R. Data Structures and Network Algorithms. Phila- delphia, PA: SIAM Press, 1983. Wood, D. Data Structures, Algorithms, and Performance. Reading, MA: Addison-Wesley, 1993. Spline A piecewise polynomial function that can have a locally very simple form, yet at the same time be globally flexible and smooth. Splines are very useful for modeling arbitrary functions, and are used ex- tensively in computer graphics. See also B-SPLINE ,B E´ ZIER SPLINE ,C UBIC SPLINE , NURBS CURVE ,THIN PLATE SPLINE References Bartels, R. H.; Beatty, J. C.; and Barsky, B. A. An Introduc- tion to Splines for Use in Computer Graphics and Geometric Modelling. San Francisco, CA: Morgan Kauf- mann, 1998. de Boor, C. A Practical Guide to Splines. New York: Springer-Verlag, 1978. Dierckx, P. Curve and Surface Fitting with Splines. Oxford, England: Oxford University Press, 1993. Micula, G. and Micula, S. Handbook of Splines. Dordrecht, Netherlands: Kluwer, 1999. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Interpolation and Extrapolation." Ch. 3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 99 /C1/22, 1992. Spa¨th, H. One Dimensional Spline Interpolation Algo- rithms. Wellesley, MA: A. K. Peters, 1995. Weisstein, E. W. "Books about Splines." http://www.trea- sure-troves.com/books/Splines.html. Splitter A perimeter-bisecting line segment which originates at a vertex of a polygon. The three splitters of a TRIANGLE CONCUR in a point known as the NAGEL POINT Na. See also B-LINE,CLEAVER References Honsberger, R. "Cleavers and Splitters." Ch. 1 in Episodes in Nineteenth and Twentieth Century Euclidean Geome- try. Washington, DC: Math. Assoc. Amer., pp. 1 /C1/4, 1995.Splitting Splitting Algorithm A method for computing a UNIT FRACTION . This method always terminates (Beeckmans 1993). References Beeckmans, L. "The Splitting Algorithm for Egyptian Frac- tions." J. Number Th. 43, 173 /C1/85, 1993. Eppstein, D. Egypt.ma Mathematica notebook. http:// www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma. Splitting Field The EXTENSION FIELD K of a FIELD F is called a splitting field for the polynomial f(x) /C23 F[x]if f(x) factors completely into linear factors in K[x] and f(x) does not factor completely into linear factors over any PROPER SUBFIELD of K containing F (Dummit and Foote 1998, p. 448). See also ALGEBRAIC CLOSURE ,E XTENSION FIELD, FIELD,GALOIS EXTENSION FIELD References Dummit, D. S. and Foote, R. M. "Splitting Fields and Algebraic Closures." §13.4 in Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 425 and 448 /C1/58, 1998. Spoke WHEEL GRAPH Sponge HONEYCOMB Sporadic Group One of the 26 FINITE SIMPLE GROUPS . The most complicated is the MONSTER GROUP . A summary, as given by Conway et al. (1985), is given below. Symbol Name Order MA /M11/ MATHIEU /24/C21532/C2155/C21511/ 11 /M12/ MATHIEU /26/C21533/C2155/C21511/ 22 /M22/ MATHIEU /27/C21532/C2155/C2157/C21511/ 12 2 /M23/ MATHIEU /27/C21532/C2155/C2157/C21511 /C21523/ 11 /M24/ MATHIEU /210/C21533/C2155/C2157/C21511 /C21523/ 11 /J2/C30HJ/JANKO /27/C21533/C21552/C2157/ 22 Suz SUZUKI /213 /C215 37 /C215 52 /C215 7 /C215 11 /C215 13/ 62 HS HIGMAN- SIMS /29 /C215 32 /C215 53 /C215 7 /C215 11/ 22 McL MCLAUGHLIN /27 /C215 36 /C215 53 /C215 7 /C215 11/ 32 /Co3/ CONWAY /210 /C215 37 /C215 53 /C215 7 /C215 11 /C215 23/ 11 /Co2/ CONWAY /218 /C215 36 /C215 53 /C215 7 /C215 11 /C215 23/ 11 /Co1/ CONWAY /221 /C215 39 /C215 54 /C215 72 /C215 11 /C215 13 /C215 23/ 21 He HELD /210 /C215 33 /C215 52 /C215 73 /C215 17/ 12 /Fi22/ FISCHER /217 /C215 39 /C215 52 /C215 7 /C215 11 /C215 13/ 62 /Fi23/ FISCHER /218 /C215 313 /C215 52 /C215 7 /C215 11 /C215 13 /C215 17 /C215 23/ 11 /Fi?24/ FISCHER /221 /C215 316 /C215 52 /C215 73 /C215 11 /C215 13 /C215 17 /C215 23 /C215 29/ 32 HN HARADA- NOR- TON/214 /C215 36 /C215 56 /C215 7 /C215 11 /C215 19/ 12 Th THOMPSON /215 /C215 310 /C215 53 /C215 72 /C215 13 /C215 19 /C215 31/ 11 B BABY MON- STER/241 /C215 313 /C215 56 /C215 72 /C215 11 /C215 13 /C215 17 /C215 19 /C215 23/ / /C21531 /C215 47/21 M MONSTER /246 /C215 320 /C215 59 /C215 76 /C215 112 /C215 133 /C215 17 /C215 19 /C215 23/ / /C21529 /C215 31 /C215 41 /C215 47 /C215 59 /C215 71/11 /J1/ JANKO /23 /C215 3 /C215 5 /C215 7 /C215 11 /C215 19/ 11 O’N O’NAN /29 /C215 34 /C215 5 /C215 73 /C215 11 /C215 19 /C215 31/ 32 /J3/ JANKO /27 /C215 35 /C215 5 /C215 17 /C215 19/ 32 Ly LYONS /28 /C215 37 /C215 56 /C215 7 /C215 11 /C215 31 /C215 37 /C215 67/ 11 Ru RUDVALIS /214 /C215 33 /C215 53 /C215 7 /C215 13 /C215 29/ 21 /J4/ JANKO /221 /C215 33 /C215 5 /C215 7 /C215 113 /C215 23 /C215 29 /C215 31 /C215 37 /C215 43/ 11 See also BABY MONSTER GROUP ,CONWAY GROUPS , FISCHER GROUPS ,H ARADA- NORTON GROUP ,H ELD GROUP ,HIGMAN- SIMS GROUP ,JANKO GROUPS ,LYONS GROUP ,M ATHIEU GROUPS ,M CLAUGHLIN GROUP , MONSTER GROUP ,O’NAN GROUP ,RUDVALIS GROUP , SUZUKI GROUP ,THOMPSON GROUP References Aschbacher, M. Sporadic Groups. New York: Cambridge University Press, 1994. Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Sub- groups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. viii, 1985. Ivanov, A. A. Geometry of Sporadic Groups I: Petersen and Tilde Geometries. Cambridge, England: Cambridge Uni- versity Press, 1999. Math. Intell. Cover of volume 2, 1980. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#spo. Sports BASEBALL ,BOWLING ,CHECKERS ,CHESS ,GOSprague-Grundy Function NIM-VALUE Sprague-Grundy Number NIM-VALUE Sprague-Grundy Value NIM-VALUE Spread (Link) SPAN (LINK) Spread (Tree) A TREE having an infinite number of branches and whose nodes are sequences generated by a set of rules. See also FAN Spreading A Rumor GOSSIPING Springer Number References Arnold, V. I. "Springer Numbers and Morsification Spaces." J. Alg. Geom 1, 197 /C1/14, 1992. Spun Knot A 3-D KNOT spun about a plane in 4-D. Unlike SUSPENDED KNOTS , spun knots are smoothly em- bedded at the poles. See also SUSPENDED KNOT,TWIST- SPUN KNOT Spur TRACE (MATRIX ) Sqrt SQUARE ROOT Squarable An object which can be constructed by SQUARING is called squarable. Square The term square is sometimes used to mean SQUARE NUMBER . When used in reference to a geometric figure, however, it means a convex QUADRILATERAL with four equal sides at RIGHT ANGLES to each other, illustrated above. When used as a symbol, IABCD denotes a square with given vertices, while G1IG2is sometimes used to denote a GRAPH PRODUCT (Clark and Suen 2000). The PERIMETER of a square with side length ais L/C304a (1) and the AREA is A/C30a2: (2) The INRADIUS r,CIRCUMRADIUS R, and AREA Acan be computed directly from the formulas for a general REGULAR POLYGON with side length aandn/C304 sides, r/C301 2acotp 4 ! /C3012a (3) R/C301 2acscp 4 ! /C301 2ffiffiffi 2p a (4) A/C301 4na2cotp 4 ! /C30a2; (5) The length of the DIAGONAL of the UNIT SQUARE isffiffiffi 2p ; sometimes known as P YTHAGORAS’S CONSTANT . The AREA of a square constructed inside a UNIT SQUARE as shown in the above diagram can be found as follows. Label xandyas shown, then x2/C27y2/C30r2(6) ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27r2p /C28xYru*Yru+2 /C27y2/C301: (7)Plugging (6) into (7) gives ffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27r 2p /C28xYru*Yru+2 /C27r2/C28x2YrvYru /C301: (8) Expanding x2/C282xffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27r 2p /C271/C27r2/C27r2/C28x2/C301 (9) and solving for xgives x/C30r2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27r2p : (10) Plugging in for yyields y/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28x2p /C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27r2p : (11) The area of the shaded square is then A/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27r2p /C28x/C28yYru*Yru+2 /C30(1/C28r)2 1/C27r2(12) (Detemple and Harold 1996). The STRAIGHTEDGE and COMPASS construction of the square is simple. Draw the line P?OOP0and construct a circle having OP0as a radius. Then construct the perpendicular OBthrough O. Bisect P0OBandP?0OB to locate P1andP2;where P?0is opposite P0:Similarly, construct P3and P4on the other SEMICIRCLE . Con- necting P1P2P3P4then gives a square. An infinity of points in the interior of a square are known whose distances from three of the corners of a square are RATIONAL NUMBERS . Calling the distances a,b, and cwhere sis the side length of the square, these solutions satisfy s2/C27b2/C28a2YrvYru2/C27s2/C27b2/C28c2YrvYru2/C30(2bs)2(13) (Guy 1994). In this problem, one of a,b,c, and sis DIVISIBLE by 3, one by 4, and one by 5. It is not known if there are points having distances from all four corners RATIONAL , but such a solution requires the additional condition a2/C27c2/C30b2/C27d2: (14) In this problem, s is DIVISIBLE by 4 and a, b, c, and d are ODD.Ifs is not DIVISIBLE by 3 (5), then two of a, b, c, and d are DIVISIBLE by 3 (5) (Guy 1994). The centers of four squares erected either internally or externally on the sides of a PARALLELOGRAMS are the vertices of a square (Yaglom 1962, pp. 96 /C1/7; Coxeter and Greitzer 1967, p. 84). See also BROWKIN’S THEOREM ,DISSECTION ,DOUGLAS- NEUMANN THEOREM ,FINSLER- HADWIGER THEOREM , LOZENGE ,NESTED SQUARE ,PERFECT SQUARE DISSEC- TION ,P YTHAGORAS’S CONSTANT ,P YTHAGOREAN SQUARE PUZZLE ,RECTANGLE ,SQUARE DIVISION BY LINES ,S QUARE INSCRIBING ,S QUARE NUMBER , SQUARE PACKING ,S QUARE QUADRANTS ,U NIT SQUARE , VON AUBEL’S THEOREM References Clark, W. E. and Suen, S. "An Inequality Related to Vizing’s Conjecture." Electronic J. Combinatorics 7, No. 1, N4, 1 /C1/, 2000. http://www.combinatorics.org/Volume_7/ v7i1toc.html#N4. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 84, 1967. Detemple, D. and Harold, S. "A Round-Up of Square Problems." Math. Mag. 69,15/C1/7, 1996. Dixon, R. Mathographics. New York: Dover, p. 16, 1991. Eppstein, D. "Rectilinear Geometry." http://www.ics.uci.edu/ ~eppstein/junkyard/rect.html. Fukagawa, H. and Pedoe, D. "One or Two Circles and Squares," "Three Circles and Squares," and "Many Circles and Squares (Casey’s Theorem)." §3.1 /C1/.3 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Ca- nada: Charles Babbage Research Foundation, pp. 37 /C1/2 and 117 /C1/25, 1989. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 165 and 167, 1984. Guy, R. K. "Rational Distances from the Corners of a Square." §D19 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 181 /C1/85, 1994. Harris, J. W. and Stocker, H. "Square." §3.6.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 84 /C1/5, 1998. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 2, 1948. Yaglom, I. M. Geometric Transformations I. New York: Random House, pp. 96 /C1/7, 1962.Square Antiprism The ANTIPRISM with square bases. See also ANTIPRISM ,SQUARE PRISM Square Bracket One of the symbols [ and ] used in many different contexts in mathematics. 1. Square brackets are occasionally used in espe- cially complex expressions in place of (or in addition to) PARENTHESES , especially as a group symbol outside an inner set of parentheses, e.g., [3 /C274 /C29(5 /C276)] =7:/ 2. Large brackets around an array of numbers, e.g.,ab cdYrtYrP indicate a MATRIX . (The symbolab cdYrvYru is also commonly used.) 3. A square bracket at one end of an INTERVAL indicates that the INTERVAL is closed at that end, that is, it includes the number at that end. 4. Brackets may be used to denote the LEAST COMMON MULTIPLE , e.g., [10; 6] /C13LCM(10 ; 6) /C3030 :/ 5. Some authors (although this work does not) use [x] to denote the FLOOR FUNCTION xbc:/ See also ANGLE BRACKET ,BRACE ,PARENTHESIS References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 285, 1997. Square Bracket Polynomial APOLYNOMIAL which is not necessarily an invariant of a LINK . It is related to the DICHROIC POLYNOMIAL .I t is defined by the SKEIN RELATIONSHIP BL/C27/C30q/C281=2vBL0/C27BL/C12; (1) and satisfies Bunknot/C30q1=2(2) and BL@unknot/C30q1=2BL: (3) References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 235 /C1/41, 1994. Square Cupola JOHNSON SOLID J4 : The bottom eight VERTICES are 91 21 /C27ffiffiffi 2pYru*Yru+ ;91 2 ; 0Yru*Yru+ ;912 ;9121 /C27ffiffiffi 2pYru*Yru+ ; 0Yru*Yru+ ; and the top four VERTICES are 91ffiffiffi 2p; 0;1ffiffiffi2p ! ; 0;91ffiffiffi2p;1ffiffiffi2p ! : Square Curve SIERPINSKI CURVE Square Division by Lines The average number of regions N(n) into which n lines divide a SQUARE is N(n) /C301 16 n(n /C281)p /C27n /C271 (Santalo ´ 1976). See also CIRCLE DIVISION BY LINES References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/geom/geom.html. Santalo ´,L.A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Square Graph The CYCLE GRAPH C4 :/ See also CYCLE GRAPH ,TRIANGLE GRAPHReferences Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 144, 1990. Square Gyrobicupola JOHNSON SOLID J29 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Square Inscribing As shown by Schnirelman (1944), a SQUARE can be INSCRIBED in any closed convex curve, although it is not known if this holds true for every JORDAN CURVE (Steinhaus 1983, p. 104). However, a SQUARE can be CIRCUMSCRIBED about any JORDAN CURVE (Steinhaus 1999, p. 104). See also JORDAN CURVE ,SQUARE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Inscribing Polygons in Curves." §B2 in Unsolved Problems in Geo- metry. New York: Springer-Verlag, pp. 51 /C1/2, 1991. Schnirelman, L. G. "On Certain Geometrical Properties of Closed Curves." Uspehi Matem. Nauk 10,34/C1/4, 1944. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 104 and 302, 1999. Square Integrable A function f(x) is said to be square integrable if g/C12 /C28/C12f(x)jj2dx is finite. See also INTEGRABLE , L2-NORM,TITCHMARSH THEO- REM References Sansone, G. "Square Integrable Functions." §1.1 in Ortho- gonal Functions, rev. English ed. New York: Dover, pp. 1/C1/, 1991. Square Knot A composite KNOT of six crossings consisting of a KNOT SUM of a TREFOIL KNOT and its MIRROR IMAGE . The GRANNY KNOT has the same ALEXANDER POLYNOMIAL x2 /C28x /C271 ðÞ2as the square knot. The square knot is also called the REEF KNOT . See also GRANNY KNOT,M IRROR IMAGE ,T REFOIL KNOT References Owen, P. Knots. Philadelphia, PA: Courage, p. 50, 1993. Square Matrix A MATRIX for which horizontal and vertical dimen- sions are the same (i.e., an n /C29n MATRIX ). A matrix can be tested to see if it is square using SquareMa- trixQ [m] in the Mathematica add-on packageLin- earAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also MATRIX ,RECTANGULAR MATRIX Square Number A FIGURATE NUMBER OF THE FORM Sn /C30n2 ; where n is an INTEGER . A square number is also called a PERFECT SQUARE . The first few square numbers are 1, 4, 9, 16, 25, 36, 49, ... (Sloane’s A000290). The GENERATING FUNCTION giving the square numbers is x(x /C27 1) (1 /C28 x)3 /C30x /C274x2 /C279x3 /C2716x4 /C27...: (1) The (n /C271)/st square number Sn/C271 is given in terms ofthe nth square number Sn by Sn/C271 /C30Sn /C272n /C271: (2) since (n /C271)2 /C30n2 /C272n /C271; (3) which is equivalent to adding a GNOMON to the previous square, as illustrated above. The nth square number is equal to the sum of the (n/C281)/-st and nthTRIANGULAR NUMBERS , Sn/C301 2(n/C281)n/C2712n(n/C271)/C30n2: (4) as can seen in the above diagram, in which the (n/C281)/-st triangular number is represented by the white triangles, the nth triangular number is repre- sented by the black triangles, and the total number of triangles is the square number Sn/C30n2(R. Sobel). As a part of the study of W ARING’S PROBLEM ,i ti s known that every positive integer is a sum of no more than 4 positive squares ( /g(2)/C304; L AGRANGE’S FOUR- SQUARE THEOREM ), that every "sufficiently large" integer is a sum of no more than 4 positive squares (/G(2)/C304);and that every integer is a sum of at most 3 signed squares ( eg(2)/C303):Actually, the basis set for representing positive integers with positive squares is f1;1;4;9;16;25;36;64;81;100;...g;so 49 need never be used. Furthermore, since an infinite numberofnrequire four squares to represent them, the least INTEGER G(2) such that every POSITIVE INTEGER beyond a certain point requires G(2) squares is given byG(2)/C304:/ The number of representation of a number nbyk squares, distinguishing signs and order, is denoted rk(n) and called the SUM OF SQUARES FUNCTION . The minimum number of squares needed to represent the numbers 1, 2, 3, ... are 1, 2, 3, 1, 2, 3, 4, 2, 1, 2, ...(Sloane’s A002828), and the number of distinct ways to represent the numbers 1, 2, 3, ... in terms of squares are 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, ... (Sloane’sA001156). A brute-force algorithm for enumerating the square partitions of nis repeated application of the GREEDY ALGORITHM . However, this approach rapidly becomes impractical since the number ofrepresentations grows extremely rapidly with n,a s shown in the following table. nSquare Partitions 10 4 50 104 100 1116 150 6521200 27482 The kth nonsquare number a kis given by an/C30n/C271 2/C27ffiffiffinpjk ; (5) where xbcis the FLOOR FUNCTION , and the first few are 2, 3, 5, 6, 7, 8, 10, 11, ... (Sloane’s A000037). The only numbers which are simultaneously square and PYRAMIDAL (the CANNONBALL PROBLEM ) are P1/C30 1 and P24/C304900 ;corresponding to S1/C301 and S70/C30 4900 (Dickson 1952, p. 25; Ball and Coxeter 1987, p. 59; Ogilvy 1988), as conjectured by Lucas (1875, 1876) and proved by Watson (1918). The CANNONBALL PROBLEM is equivalent to solving the D IOPHANTINE EQUATION y2/C301 6x(x/C271)(2x/C271) (6) (Guy 1994, p. 147). The only numbers which are square and TETRAHE- DRAL areTe1/C301;Te2/C304;and Te48/C3019600 (giving S1/C301;S2/C304;and S140/C3019600) ;as proved by Meyl (1878; cited in Dickson 1952, p. 25; Guy 1994, p. 147). In general, proving that only certain numbers aresimultaneously figurate in two different ways is far from elementary. To find the possible last digits for a square number, write n/C3010a/C27bfor the number written in decimal NOTATION asab10(a,b/C300, 1, ..., 9). Then n2/C30100a2/C2720ab/C27b2: (7) so the last digit of n2is the same as the last digit of b2: The following table gives the last digit of b2forb/C300, 1, ..., 9 (where numbers with more that one digit haveonly their last digit indicated, i.e., 16 becomes _6). As can be seen, the last digit can be only 0, 1, 4, 5, 6, or 9. 0123 456789 0149_ 6_ 5_ 6_ 9_ 4_ 1 We can similarly examine the allowable last twodigits by writing abc 10as n/C30100a/C2710b/C27c; (8) so n2/C30(100a/C2710b/C27c)2 /C30104a2/C272(1000 ab/C27100ac/C2710bc)/C27100b2/C27c2/C30(104a2/C272000 ab/C27100ac/C27100b2)/C2720bc/C27c2 /C30100(100 a2/C2720ab/C27ac/C27b2)/C27(20bc/C27c2) (9) so the last two digits must have the last two digits of 20bc/C27c2:Furthermore, the last two digits can be obtained by considering only b/C300, 1, 2, 3, and 4, since 20(b/C275)c/C27c2/C30100c/C2720bc/C27c2YrvYru (10) has the same last two digits as 20 bc/C27c2(with the one additional possibility that c/C300 in which case the last two digits are 00). The following table (with theaddition of 00) therefore exhausts all possible last two digits. c b 123456789 00 10 40 91 62 53 64 96 48 1 1 _21 _44 _69 _96 _25 _56 _89 _24 _612 _41 _84 _29 _76 _25 _76 _29 _84 _41 3 _61 _24 _89 _56 _25 _96 _69 _44 _21 4 _81 _64 _49 _36 _25 _16 _09 _04 _01 The only 22 possibilities are therefore 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81,84, 89, and 96, which can be summarized succinctly as 00, e1;e4;25,o6;and e9;where estands for an EVEN NUMBER and ofor an ODD NUMBER . Addition- ally, a NECESSARY (but not SUFFICIENT ) condition for a number to be square is that its DIGITAL ROOT be 1, 4, 7, or 9. The digital roots of the first few squares are 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, ... (Sloane’s A056992),while the list of number having digital roots 1, 4, 7, or 9 is 1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, ... (Sloane’s A056991). The following table gives the possible residues mod n for square numbers for n/C301 to 20. The quantity s(n) gives the number of distinct residues for a given n. n /s(n)//x2(mod n)/ 2 2 0, 1 3 2 0, 1 4 2 0, 15 3 0, 1, 46 4 0, 1, 3, 4 7 4 0, 1, 2, 4 8 3 0, 1, 49 4 0, 1, 4, 7 1 0 6 0 ,1 ,4 ,5 ,6 ,9 1 1 6 0 ,1 ,3 ,4 ,5 ,9 1 2 4 0 ,1 ,4 ,913 7 0, 1, 3, 4, 9, 10, 121 4 8 0 ,1 ,2 ,4 ,7 ,8 ,9 ,1 1 1 5 6 0 ,1 ,4 ,6 ,9 ,1 0 1 6 4 0 ,1 ,4 ,917 9 0, 1, 2, 4, 8, 9, 13, 15, 1618 8 0, 1, 4, 7, 9, 10, 13, 16 19 10 0, 1, 4, 5, 6, 7, 9, 11, 16, 17 2 0 6 0 ,1 ,4 ,5 ,9 ,1 6 In general, the ODD squares are congruent to 1 (mod 8) (Conway and Guy 1996). Stangl (1996) gives an explicit formula by which the number of squares s(n) inZn(i.e., mod n) can be calculated. Let pbe an ODD PRIME . Then s(n) is the MULTIPLICATIVE FUNCTION given by s(2)/C302 (11) s(p)/C301 2(p/C271) ( p"2) (12) sp2YrvYru /C3012p2/C28p/C272YrvYru (p"2) (13) s2nðÞ/C301 32n/C281/C274 ðÞ forneven 1 32n/C281/C275 ðÞ fornodd( (14) spnðÞ/C30pn/C271/C27p/C272 2(p/C271)forn]3 even pn/C271/C272p/C271 2(p/C271)forn]3 odd :8 >>>< >>>:(15) /s(n) is related to the number q(n)o f QUADRATIC RESIDUES inZnby qpnðÞ/C30spnðÞ/C28spn/C282YrvYru (16) forn]3 (Stangl 1996). For a perfect square n,(n=p)/C300 or 1 for all ODD PRIMES pBnwhere ( n=p) is the L EGENDRE SYMBOL .A number nwhich is not a perfect square but which satisfies this relationship is called a PSEUDOSQUARE . In a Ramanujan conference talk, W. Gosper conjec- tured that every sum of four distinct odd squares isthe sum of four distinct even squares. This conjecture was proved by M. Hirschhorn using the identity(4a/C271) 2/C27(4b/C271)2/C27(4c/C271)2/C27(4d/C271)2 /C304[(a/C27b/C27c/C27d/C271)2/C27(a/C28b/C28c/C27d)2 /C27(a/C28b/C27c/C28d)2/C27(a/C27b/C28c/C28d)2]; (17) where a,b,c, and dare positive or negative integers. Hirschhorn also showed that every sum of four distinct oddly even squares is the sum of four distinct odd squares. APRIME NUMBER pcan be written as the sum of two squares IFFp/C271 is not divisible by 4 the (F ERMAT 4N/C271 THEOREM ). An arbitrary positive number nis expressible as the sum of two squares IFF, given its PRIME FACTORIZATION n/C30pa1 1pa2 2pa3 3/C1/C1/C1pak k; (18) none of pai i/C271 is divisible by 4 (Conway and Guy 1996, p. 147). This is equivalent the requirement that all the odd factors of the SQUAREFREE PART n?ofnare equal to 1 (mod 4) (Hardy and Wright 1979, Finch). The first few numbers which can be expressed as the sum of two squares are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17,18, 20, 25, 26, ... (Sloane’s A001481). Letting d(n)b e the fraction of numbers 5nwhich are expressible as the sum of two squares, lim n0/C12d(n)/C300; (19) and lim n0/C12d(n)ffiffiffiffiffiffiffiffiffi lnnp /C30K; (20) where Kis the L ANDAU- RAMANUJAN CONSTANT . Numbers expressible as the sum of three squares are those not OF THE FORM 4k(8l/C277) for k;l]0 (Nagell 1951, p. 194; Wells 1986, pp. 48 and 56; Hardy 1999, p. 12). The following table gives the first few numbers which require N /C301, 2, 3, and 4 squares to represent them as a sum (Wells 1986, p. 70). N Sloane Numbers 1 Sloane’s A0002901, 4, 9, 16, 25, 36, 49, 64, 81, ... 2 Sloane’s A0004152, 5, 8, 10, 13, 17, 18, 20, 26,29, ... 3 Sloane’s A0004193, 6, 11, 12, 14, 19, 21, 22, 24, 27, ... 4 Sloane’s A0042157, 15, 23, 28, 31, 39, 47, 55,60, 63, ... The F ERMAT 4 N/C271 THEOREM guarantees that every PRIME OF THE FORM 4n/C271 is a sum of two SQUARE NUMBERS in only one way. There are only 31 numbers which cannot be ex- pressed as the sum of distinct squares: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44,47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128 (Sloane’sA001422; Guy 1994; Savin 2000). The following numbers cannot be represented using fewer than five distinct squares: 55, 88, 103, 132, 172, 176, 192,240, 268, 288, 304, 368, 384, 432, 448, 496, 512, and 752, together with all numbers obtained by multi- plying these numbers by a power of 4. This gives allknown such numbers less than 10 5(Savin 2000). All numbers >188 can be expressed as the sum of at most five distinct squares, and only 124/C301/C274/C279/C2725/C2736/C2749 (21) and 188/C301/C274/C279/C2725/C2749/C27100 (22) require six distinct squares (Bohman et al. 1979; Guy 1994, p. 136; Savin 2000). In fact, 188 can also be represented using seven distinct squares: 188/C301/C274/C279/C2725/C2736/C2749/C2764: (23) The following table gives the numbers which can be represented in Wdifferent ways as a sum of S squares. For example, 50/C3012/C2772/C3052/C2752(24) can be represented in two ways ( W/C302) by two squares ( S/C302). SW Sloane Numbers 1 1 Sloane’s A0002901, 4, 9, 16, 25, 36, 49, 64, 81,100, 121, ... 2 1 Sloane’s A0252842, 5, 8, 10, 13, 17, 18, 20, 25,26, 29, 32, ... 2 2 Sloane’s A02528550, 65, 85, 125, 130, 145,170, 185, 200, ... 3 1 Sloane’s A0253213, 6, 9, 11, 12, 14, 17, 18, 19,21, 22, 24, ... 3 2 Sloane’s A02532227, 33, 38, 41, 51, 57, 59, 62,69, 74, 75, ... 3 3 Sloane’s A02532354, 66, 81, 86, 89, 99, 101,110, 114, 126, ... 3 4 Sloane’s A025324129, 134, 146, 153, 161, 171,189, 198, ... 4 1 Sloane’s A0253574, 7, 10, 12, 13, 15, 16, 18,19, 20, 21, 22, ... 4 2 Sloane’s A02535831, 34, 36, 37, 39, 43, 45, 47,49, 50, 54, ...4 3 Sloane’s A02535928, 42, 55, 60, 66, 67, 73, 75, 78, 85, 95, 99, ... 4 4 Sloane’s A02536052, 58, 63, 70, 76, 84, 87, 91,93, 97, 98, 103, ... The least numbers which are the sum of two squares in exactly ndifferent ways for n/C301, 2, ... are given by 2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, ... (Sloane’s A016032; Beiler 1966, pp. 140 /C1 /41; Culbertson; Hardy and Wright 1979; Rivera). The product of four distinct NONZERO INTEGERS in ARITHMETIC PROGRESSION is square only for ( /C283,/C281, 1, 3), giving ( /C283)(/C281)(1)(3) /C309 (Le Lionnais 1983, p. 53). It is possible to have three squares in ARITH- METIC PROGRESSION , but not four (Dickson 1952, pp. 435 /C1/40). If these numbers are r2;s2;andt2;there are POSITIVE INTEGERS pandqsuch that r/C30p2/C282pq/C28q2YrutYrutYrutYrut (25) s/C30p2/C27q2(26) t/C30p2/C272pq/C28q2; (27) where ( p;q)/C301 and one of r,s,o rtisEVEN (Dickson 1952, pp. 437 /C1/38). Every three-term progression of squares can be associated with a P YTHAGOREAN TRIPLE (X;Y;Z)) by X/C301 2(r/C27t) (28) Y/C3012(t/C28r) (29) Z/C30s (30) (Robertson 1996). CATALAN’S CONJECTURE states that 8 and 9 (23and 32) are the only consecutive POWERS (excluding 0 and 1), i.e., the only solution to C ATALAN’S DIOPHANTINE PROBLEM . This CONJECTURE has not yet been proved or refuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the CONJECTURE not hold. It is also known that 8 and 9 are the only consecutive CUBIC and square numbers (in either order). The numbers that are not the difference of two squares are 2, 6, 10, 14, 18, ... (Wells 1986, p. 76). A square number can be the concatenation of two squares, as in the case 16 /C3042and 9 /C3032giving 169/C30132. The first few numbers which are neither square nor the sum of a square and a PRIME are 10, 34, 58, 85, 91, 130, 214, ... (Sloane’s A020495). It is conjectured that, other than 102n,4/C29102nand 9/C29102n, there are only a FINITE number of squares n2having exactly two distinct NONZERO DIGITS (Guy 1994, p. 262). The first few such nare 4, 5, 6, 7, 8, 9, 11, 12, 15, 21, ... (Sloane’s A016070), corresponding to n2 of 16, 25, 36, 49, 64, 81, 121, ... (Sloane’s A018884). The following table gives the first few numbers which, when squared, give numbers composed of only certain digits. The values of n such that n2 contains exactly two different digits are given by 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, ... (Sloane’s A016069), whose squares are 16, 25 36, 49, 64, ... (Sloane’s A018885). The only known square number composed only of the digits 7, 8, and 9 is 9. Based on a discussion inrec.puzzles , Vardi (1991) considered numbers composed only of the square digits: 1, 4, and 9. It is conjectured that there are only finitely many, and the largest known is 6480702115891070212 /C30419994999149149944149149944191494441 (31) found by G. Jacobson and D. Applegate (rec.puz- zles FAQ). Digits Sloane n, n2 / 1, 2, 3 Sloane’s A0301751, 11, 111, 36361, 363639, ... Sloane’s A0301741, 121, 12321, 1322122321, ... 1, 4, 6 Sloane’s A0276771, 2, 4, 8, 12, 21, 38, 108, ... Sloane’sA0276761, 4, 16, 64, 144, 441, 1444, ... 1, 4, 9 Sloane’s A0276751, 2, 3, 7, 12, 21, 38, 107, ... Sloane’sA0067161, 4, 9, 49, 144, 441, 1444, 11449, ... 2, 4, 8 Sloane’s A0276792, 22, 168, 478, 2878, 210912978, ... Sloane’sA0276784, 484, 28224, 228484, 8282884, ... 4, 5, 6 Sloane’s A0301772, 8, 216, 238, 258, 738, 6742, ... Sloane’sA0301764, 64, 46656, 56644, 66564, ... B ROWN NUMBERS are pairs (m, n)of INTEGERS satisfying the condition of BROCARD’S PROBLEM , i.e., such that n! /C271 /C30m2 ; (32)where n!isa FACTORIAL . Only three such numbers are known: (5,4), (11,5), (71,7). Erdos conjectured that these are the only three such pairs. Either 5x2 /C274 /C30y2 or 5x2 /C284 /C30y2 has a solution in POSITIVE INTEGERS IFF, for some n,(x; y) /C30 Fn ; Ln ðÞ ; where Fnis a FIBONACCI NUMBER and Lnis a LUCAS NUMBER (Honsberger 1985, pp. 114 /C1/18). The smallest and largest square numbers containing the digits 1 to 9 are 11 ;8262 /C30139;854; 276; (33) 30;3842 /C30923; 187;456: (34) The smallest and largest square numbers containing the digits 0 to 9 are 32 ;0432 /C301; 026;753;849; (35) 99; 0662 /C309;814;072;356 (36) (Madachy 1979, p. 159). The smallest and largest square numbers containing the digits 1 to 9 twice each are 335;180;1362 /C30112; 345;723;568;978;496 (37) 999;390;4322 /C30998;781;235;573;146;624; (38) and the smallest and largest containing 1 to 9 three times are 10 ;546;200;195;3122 /C30111;222;338;559;598;866;946;777;344 (39) 31;621;017;808;1822 /C30999;888;767;225;363;175;346;145;124 (Madachy 1979, p. 159). Madachy (1979, p. 165) also considers number which are equal to the sum of the squares of their two"halves" such as 1233/C3012 2/C27332(40) 8833/C30882/C27332(41) 10100 /C30102/C271002(42) 5882353 /C305882/C2723532; (43) in addition to a number of others. See also ANTISQUARE NUMBER ,BIQUADRATIC NUM- BER,BROCARD’S PROBLEM ,BROWN NUMBERS ,CAN- NONBALL PROBLEM ,C ATALAN’S CONJECTURE , CENTERED SQUARE NUMBER ,C LARK’S TRIANGLE , CUBIC NUMBER ,D IOPHANTINE EQUATION ,F ERMAT 4N /C271 THEOREM ,GREEDY ALGORITHM ,GROSS ,HEPTA- GONAL SQUARE NUMBER ,LAGRANGE’S FOUR- SQUARE THEOREM ,LANDAU- RAMANUJAN CONSTANT ,OCTAGO- NAL SQUARE NUMBER ,P ARTITION ,P ENTAGONAL SQUARE NUMBER ,PSEUDOSQUARE ,PYRAMIDAL NUM- BER,S QUAREFREE ,S QUARE TRIANGULAR NUMBER , SUM OF SQUARES FUNCTION ,W ARING’S PROBLEM References Archibald, R. G. "Waring’s Problem: Squares." Scripta Math. 7,3 3/C1/8, 1940. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 59, 1987. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Bohman, J.; Fro ¨berg, C.-E.; and Riesel, H. "Partitions in Squares." BIT 19, 297/C1/01, 1979. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30 /C1/2 and 146 /C1/47, 1996. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952. Finch, S. "Unsolved Mathematics Problems: On a General- ized Fermat-Wiles Equation." http://www.mathsoft.com/ asolve/fermat/fermat.html. Grosswald, E. Representations of Integers as Sums of Squares. New York: Springer-Verlag, 1985. Guy, R. K. "Sums of Squares" and "Squares with Just Two Different Decimal Digits." §C20 and F24 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 /C1/38 and 262, 1994. Hajdu, L. and Pinte ´r, A´. "Square Product of Three Integers in Short Intervals." Math. Comput. 68, 1299 /C1/301, 1999. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. "The Representation of a Number by Two or Four Squares." Ch. 20 in An Introduc- tion to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 297 /C1/16, 1979. Honsberger, R. "A Second Look at the Fibonacci and Lucas Numbers." Ch. 8 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Lucas, E ´. Question 1180. Nouv. Ann. Math. Ser. 2 14, 336, 1875. Lucas, E ´. Solution de Question 1180. Nouv. Ann. Math. Ser. 215, 429/C1/32, 1876. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 159 and 165, 1979. Meyl, A.-J.-J. Solution de Question 1194. Nouv. Ann. Math. 17, 464/C1/67, 1878. Nagell, T. Introduction to Number Theory. New York: Wiley, 1951. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 77 and 152, 1988. Pappas, T. "Triangular, Square & Pentagonal Numbers." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989. Pietenpol, J. L. "Square Triangular Numbers." Amer. Math. Monthly 69, 168/C1/69, 1962. rec.puzzles FAQ3. http://www.cs.caltech.edu/~adam/ PUZZLES/rec.puz.faq3. Rivera, C. "Problems & Puzzles: Puzzle The qs-Sequence.- 062." http://www.primepuzzles.net/puzzles/puzz_062.htm. Robertson, J. P. "Magic Squares of Squares." Math. Mag. 69, 289/C1/93, 1996. Savin, A. "Shape Numbers." Quantum 11,1 4/C1/8, 2000. Sloane, N. J. A. Sequences A000037/M0613, A000290/ M3356, A000415, A000419, A001156/M0221, A001422/M,A001481/M0968, A002828/M0404, A004215/M4349,A006716/M3369, A016069, A016070, A016032, A018884, A018885, A020495, A025284, A025285, A025321, A025322, A025323, A025324, A025357, A025358,A025359, A025360, A027675, A027676, A027677,A027678, A027679, A030174, A030175, A030176,A030177, A056991, and A056992 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Stangl, W. D. "Counting Squares in Z n:/"Math. Mag. 69, 285/C1/89, 1996. Taussky-Todd, O. "Sums of Squares." Amer. Math. Monthly 77, 805/C1/30, 1970. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 20 and 234 /C1/37, 1991. Watson, G. N. "The Problem of the Square Pyramid." Messenger. Math. 48,1/C1/2, 1918. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 48 and 70, 1986. Square Orthobicupola JOHNSON SOLID J28:/ References Weisstein, E. W. "Johnson Solids." M ATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." M ATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Square Packing Find the minimum size SQUARE capable of bounding n equal SQUARES arranged in any configuration. The first few cases are illustrated above (Friedman). The only packings which have been proven optimal are 2, 3, 5, 6, 7, 8, 14, 15, 24, and 35, in addition to thetrivial cases of the SQUARE NUMBERS (Friedman). Ifn/C30a2/C28afor some a,i ti s CONJECTURED that the size of the minimum bounding square is afor small n. The smallest nfor which the CONJECTURE is known to be violated is n/C30272 (with a/C3017). The size is known to scale as kb;where 1 23 /C28ffiffiffi 3pYru*Yru+ Bb B1 2 : The following table gives the smallest known side lengths for a square into which n unit squares can be packed. n exact approx. n exact approx. 11 11 4 4 4 22 21 5 4 4 32 21 6 4 4 42 21 7 /4 /C271 2ffiffiffi 2p / 4.707... 5 /2 /C271 2ffiffiffi 2p / 2.707... 18 /1 27 /C27ffiffiffi 7pYrvYru / 4.822... 63 31 9 /3 /C274 3ffiffiffi 2p / 4.885... 73 32 0 5 5 83 32 1 5 5 93 32 2 5 5 10 /3 /C271 2ffiffiffi 2p / 3.707... 23 5 5 11 /s11/ 3.877... 24 5 5 12 4 4 25 5 5 13 4 4 26 5.650... Here, s11 is the larger of the two positive real roots of s4 /C2810s3 /C2735s2 /C2846s /C279: The best known packings of squares into a circle are illustrated above for the first few cases (Friedman). The best known packings of squares into an equilat-eral triangle are illustrated above for the first few cases (Friedman). The best packing of a SQUARE inside a PENTAGON , illustrated above, is 1.0673.... See also CIRCLE PACKING ,PACKING ,TRIANGLE PACK- ING References Erdos, P. and Graham, R. L. "On Packing Squares with Equal Squares." J. Combin. Th. Ser. A 19, 119 /C1/23, 1975. Friedman, E. "Erich’s Packing Center." http://www.stetso- n.edu/~efriedma/packing.html. Friedman, E. "Circles in Squares." http://www.stetson.edu/ ~efriedma/cirinsqu/. Friedman, E. "Squares in Squares." http://www.stetson.edu/ ~efriedma/squinsqu/. Friedman, E. "Triangles in Squares." http://www.stetso- n.edu/~efriedma/triinsqu/. Friedman, E. "Packing Unit Squares in Squares." Elec. J. Combin. DS7, 1 /C1/4, Mar. 5, 1998. http://www.combinator- ics.org/Surveys/. Gardner, M. "Packing Squares." Ch. 20 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 289 /C1/06, 1992. Go¨bel, F. "Geometrical Packing and Covering Problems." In Packing and Covering in Combinatorics (Ed. A. Schrijver). Amsterdam: Tweede Boerhaavestraat, 1979. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 174, 1998. Roth, L. F. and Vaughan, R. C. "Inefficiency in Packing Squares with Unit Squares." J. Combin. Th. Ser. A 24, 170/C1/86, 1978. Square Part The largest square dividing a POSITIVE INTEGER n. Forn/C301, 2, ..., the first few are 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, ... (Sloane’s A008833). See also CUBIC PART,SQUARE NUMBER ,SQUAREFREE PART References Sloane, N. J. A. Sequences A008833 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Square Polyomino See also L-POLYOMINO ,SKEW POLYOMINO ,STRAIGHT POLYOMINO ,T-POLYOMINO Square Prism CUBE,CUBOID Square Pyramid A square pyramid is a PENTAHEDRON consisting of a PYRAMID with a SQUARE base. If the top of the pyramid is cut off by a PLANE , a square PYRAMIDAL FRUSTUM is obtained. If the four TRIANGLES of the square pyramid are EQUILATERAL , the square pyra- mid is the "regular" POLYHEDRON known as J OHNSON SOLID J1and, for side length a, has height h/C301 2ffiffiffi 2p a: (1) Using the equation for a general PYRAMID , the VOLUME of the "regular" is therefore V/C301 3hAb/C3016ffiffiffi 2p a3: (2) The SLANT HEIGHT of a square pyramid is a special case of the formula for a regular n-gonal PYRAMID with n/C302, given by s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C271 2a2q ; (3) where his the height and ais the length of a side of the base. Consider a HEMISPHERE placed on the base of a square pyramid (having side lengths aand height h). Further, let the hemisphere be tangent to the four apex edges. Then what is the volume of the HEMI- SPHERE which is interior the pyramid (Cipra 1993)? From Fig. (a), the CIRCUMRADIUS of the base is a=ffiffiffi 2p : Now find hin terms of randa. Fig. (b) shows a CROSS SECTION cut by the plane through the pyramid’s apex, one of the base’s vertices, and the base center. Thisfigure gives b/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2a2/C28r2q (4) c/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C28r2p ; (5) so the SLANT HEIGHT is s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C271 2a2q /C30b/C27c/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12a2/C28r2q /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C28r2p :(6) Solving for hgives h/C30raffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C282r2p : (7) We know, however, that the HEMISPHERE must be tangent to the sides, so /r/C30a=2/, and h/C301 2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C281 2a2q a/C3012 ffiffi 12qa/C3012ffiffiffi 2p a: (8) Fig. (c) shows a CROSS SECTION through the center, apex, and midpoints of opposite sides. The P YTHA- GOREAN THEOREM once again gives l/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4a2/C27h2q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14a2/C2712a2q /C3012ffiffiffi 3p a: (9) We now need to find xandy. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4a2/C28x2q /C27d/C30l: (10) But we know landh, and dis given by d/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2/C28x2p : (11) so ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4a2/C28x2q /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12a2/C28x2q /C3012ffiffiffi 3p a: (12) Solving gives x/C301 6ffiffiffi 6p a; (13) so y/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28x2p /C30ffiffiffiffiffiffiffiffiffi 1 4/C2816q a/C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 3/C282 12s a/C30a 2ffiffiffi 3p: (14) We can now find the AREA of the SPHERICAL CAP as Vcap/C301 6pH3A2/C27H2YrvYru ; (15) where A/C13y/C30a 2ffiffiffi 3p (16) H/C13r/C28x/C301 2a/C28affiffiffi 6p/C30a1 2/C281ffiffiffi 6p ! ; (17) so Vcap /C301 6 pa3 31 12 ! /C271 2 /C281ffiffiffi 6p !22 435 1 2 /C281ffiffiffi 6p ! /C301 6 pa31 4 /C2714 /C2716 /C281ffiffiffi 6p !"# 1 2 /C281ffiffiffi 6p ! /C301 6 pa32 3 /C281ffiffiffi 6p ! 1 2 /C281ffiffiffi 6p ! /C301 6 pa31 3 /C281 2ffiffiffi 6p/C282 3ffiffiffi6p/C271 6 ! /C301 6 pa31 2 /C287 6ffiffiffi 6p ! : (18) Therefore, the volume within the pyramid is Vinside /C302 3 pr3 /C284Vcap /C3023 p18 a3 /C2823 pa31 2 /C287 6ffiffiffi 6p ! /C302 3 pa31 8 /C2812 /C277 6ffiffiffi 6p ! /C302 3 pa37 6ffiffiffi 6p/C283 8 ! /C30 pa37 9ffiffiffi 6p/C281 4 ! : (19) This problem appeared in the Japanese scholastic aptitude test (Cipra 1993). See also PENTAHEDRON ,PYRAMID ,SQUARE PYRAMI- DAL NUMBER References Cipra, B. "An Awesome Look at Japan Math SAT." Science 259, 22, 1993. Square Pyramidal Number A FIGURATE NUMBER OF THE FORM Pn /C301 6 n(n /C271)(2n /C271); (1) corresponding to a configuration of points which form a SQUARE PYRAMID , is called a square pyramidal number (or sometimes, simply a PYRAMIDAL NUMBER ). The first few are 1, 5, 14, 30, 55, 91, 140, 204, ...(Sloane’s A000330). They are sums of consecutive pairs of TETRAHEDRAL NUMBERS and satisfy Pn /C301 3(2n /C271)Tn ; (2) where Tn is the nth TRIANGULAR NUMBER . The only numbers which are simultaneously SQUARE Sm /C30m2 and square pyramidal Pn /C30n(n /C271)(2n /C27 1)=6 (the CANNONBALL PROBLEM ) are P1 /C301 and P24 /C30 4900 ; corresponding to S1 /C301 and S70 /C304900 (Dickson 1952, p. 25; Ball and Coxeter 1987, p. 59; Ogilvy 1988), as conjectured by Lucas (1875, 1876) and proved by Watson (1918). The proof is far from elementary, and requires solving the DIOPHANTINE EQUATION m2 /C301 6 n(n /C271)(2n /C271) (3) (Guy 1994, p. 147). However, an elementary proof has also been given by a number of authors. Numbers which are simultaneously TRIANGULAR Tm /C30m(m /C271)=2 and square pyramidal Pn /C30 n(n /C271)(2n /C271)=6 satisfy the DIOPHANTINE EQUATION 12 m(m /C271) /C3016 n(n /C271)(2n /C271): (4) COMPLETING THE SQUARE gives 1 2m /C2712Yru*Yru+2 /C2818 /C30162n3 /C273n2 /C27nYrvYru (5) 1 8(2m /C271)2 /C30162n3 /C273n2 /C27nYrvYru /C2718 (6) 3(2m /C271)2 /C308n3 /C2712n2 /C274n /C273: (7) The only solutions are (n; m) /C30(/C281; 0); (0, 0), (1, 1), (5, 10), (6, 13), and (85, 645) (Guy 1994, p. 147), corresponding to the nontrivial triangular square pyramidal numbers 1, 55, 91, 208335. Numbers which are simultaneously TETRAHEDRAL Tem /C30m(m /C271)(m /C272)=6 and square pyramidal Pn /C30 n(n/C271)(2n/C271)=6 satisfy the D IOPHANTINE EQUATION m(m/C271)(m/C272)/C30n(n/C271)(2n/C271): (8) Beukers (1988) has studied the problem of finding solutions via integral points on an ELLIPTIC CURVE and found that the only solution is the trivial Te1/C30P1/C301:/ See also PYRAMIDAL NUMBER ,TETRAHEDRAL NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 59, 1987. Beukers, F. "On Oranges and Integral Points on Certain Plane Cubic Curves." Nieuw Arch. Wisk. 6, 203/C1/10, 1988. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 47 /C1/0, 1996. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952. Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 147 /C1/50, 1994. Lucas, E ´. Question 1180. Nouvelles Ann. Math. Ser. 2 14, 336, 1875. Lucas, E ´. Solution de Question 1180. Nouvelles Ann. Math. Ser. 2 15, 429/C1/32, 1876. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 77 and 152, 1988. Sloane, N. J. A. Sequences A000330/M3844 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Watson, G. N. "The Problem of the Square Pyramid." Messenger. Math. 48,1/C1/2, 1918. Square Quadrants The areas of the regions illustrated above can be found from the equations A/C274B/C274C/C301 (1) A/C273B/C272C/C301 4p: (2) Since we want to solve for three variables, we need a third equation. This can be taken as A/C272B/C27C/C302E/C27D; (3) where D/C301 4ffiffiffi 3p (4) D/C27E/C301 6p; (5) leading to A/C272B/C27C/C30D/C272E/C302(D/C27E)/C28D/C3013p/C2814ffiffiffi 3p :(6) Combining the equations (1), (2), and (6) gives the matrix equation 144 132 1212 435A B C2 435/C301 1 4p 13p/C2814ffiffiffi 3p2 643 75; (7)which can be inverted to yield A/C301/C28ffiffiffi 3p /C281 3p (8) B/C30/C281/C2712ffiffiffi 3p /C271 12p (9) C/C301/C281 4ffiffiffi 3p /C271 6p: (10) References Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 67 /C1/9, 1991. Square Root A square root of xis a number rsuch that r2/C30x: Square roots are also called radicals or surds. Any positive real number has two square roots: one positive and one negative. For example, the squareroots of 9 are /C273 and /C283, since f/C273g 2/C30f/C283g2/C309: Any nonnegative real number xhas a unique non- negative square root r; this is called the PRINCIPAL SQUARE ROOT and is written r/C30x1=2orr/C30ffiffiffixp:For example, the PRINCIPAL SQUARE ROOT of 9 isffiffiffi 9p /C30/C273; while the other square root of 9 is /C28ffiffiffi9p /C30/C283:In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root. The functionffiffiffixpis the INVERSE FUNCTION off(x)/C30x2;forx]0:/ Any nonzero COMPLEX NUMBER zhas two square roots. For example, using the IMAGINARY UNIT I, the two square roots of /C289 are9ffiffiffiffiffiffi /C289p /C3093i:The PRINCI- PAL SQUARE ROOT of a number zis returned by the Mathematica Sqrt [x]. The square root of 2 is the IRRATIONAL NUMBERffiffiffi 2p /C30 1:41421356 (Sloane’s A002193), which has the simple periodic CONTINUED FRACTION 1, 2, 2, 2, 2, 2, ... (Sloane’s A040000). The square root of 3 is the IRRATIONAL NUMBERffiffiffi 3p :1:73205081 (Sloane’s A002194), which has the simple periodic CONTINUED FRACTION 1, 1, 2, 1, 2, 1, 2, ... (Sloane’s A040001). In general, the CONTINUED FRACTIONS of the square roots of all POSITIVE INTEGERS are periodic. The square roots of a COMPLEX NUMBER z /C30x /C27iy are given by ffiffiffiffiffiffiffiffiffiffiffiffi x /C27iyp /C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p /C2 cos1 2tan/C281yx !"# /C27i sin12tan /C281yx !"# () : (1) In addition, ffiffiffiffiffiffiffiffiffiffiffiffi x /C27iyp /C309 1 2ffiffiffi 2pYrtvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p /C27xq /C27i sgn(y)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C27y2p /C28xqYrtu : (2) As can be seen in the above figure, the IMAGINARY PART of the complex square root function has a BRANCH CUT along the NEGATIVE real axis. A NESTED RADICAL OF THE FORMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 9bffiffifficpp can some- times be simplified into a simple square root by equating ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 9bffiffifficpq /C30ffiffiffi dp 9ffiffiffiep: (3) Squaring gives a 9bffiffifficp/C30d /C27e 92ffiffiffiffiffiffi dep : (4) so a /C30d /C27e (5) b2c /C304de : (6) Solving for d and e gives d; e /C30a 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28 b2cp 2: (7) For example, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 6pq /C30ffiffiffi 2p /C27ffiffiffi 3p (8) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C282ffiffiffi 2pq /C30ffiffiffi2p /C281: (9) The Simplify command of Mathematica does not apply such simplifications, but FullSimplify does. In general, radical denesting is a difficult problem (Landau). A sequence of approximations a =b toffiffiffinpcan be derived by factoring a2 /C28nb2 /C3091 (10) (where /C281 is possible only if /C281isa QUADRATIC RESIDUE of n). Thena /C27bffiffiffinpYrvYru a /C28bffiffiffinpYrvYru /C3091 (11) a /C27bffiffiffinpYrvYru ka /C28bffiffiffinpYrvYru k/C30(91)k /C3091; (12) and 1 /C27ffiffiffinpYrvYru1/C301 /C27ffiffiffinp(13) 1 /C27ffiffiffinpYrvYru 2/C30(1 /C27n) /C272ffiffiffinp(14) 1 /C27ffiffiffinpYrvYru a /C27bffiffiffinpYrvYru /C30(a /C27bn) /C27ffiffiffinp(a /C27b) : (15) Therefore, a and b are given by the RECURRENCE RELATIONS ai /C30ai/C281 /C27bi/C281n (16) bi /C30ai/C281 /C27bi/C281 (17) with a1 /C30b1 /C301: The error obtained using this method is a b /C28ffiffiffinpYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30 1 ba/C27 bffiffiffinpðÞB1 2b2 : (18) The first few approximants toffiffiffinpare therefore given by 1;1 2(1 /C27n) ;1 /C27 3n 3 /C27 n;1 /C27 6n /C27 n2 4(n /C27 1);1 /C27 10n /C27 5n2 5 /C27 10n /C27 n2;::: (19) This ALGORITHM is sometimes known as the BHAS- KARA- BROUCKNER ALGORITHM . For the case n /C302, this gives the convergents toffiffiffi 2p as 1, /3=2/, /7=5/, /17=12/, / 41 =29/, /99 =70/, ... (Sloane’s A001333 and A000129; Wells 1986, p. 34). The numerators are given by the RECURRENCE RELATION a(n) /C302a(n /C281) /C27a(n /C282); (20) and the denominators are the PELL NUMBERS . Another general technique for deriving this sequence, known as NEWTON’S ITERATION , is obtained by letting x /C30ffiffiffinp:Then x/C30n=x;so the SEQUENCE xk/C301 2xk/C281/C27n xk/C281 ! (21) converges quadratically to the root. The first few approximants toffiffiffinpare therefore given by 1;1 2(1/C27n);1/C276n/C27n2 4(n/C271);1/C2728n/C2770n2/C2728n3/C27n4 8(1/C27n)(1/C276n/C27n2); ::: (22) Forffiffiffi 2p ;this gives the convergents 1, 3/2, 17/12, 577/ 408, 665857/470832, ... (Sloane’s A051008 and A051009). See also CUBE ROOT,N ESTED RADICAL ,N EWTON’S ITERATION ,P RINCIPAL SQUARE ROOT,Q UADRATIC SURD,R OOT OF UNITY,SQUARE NUMBER ,SQUARE TRIANGULAR NUMBER ,SURD References Sloane, N. J. A. Sequences A000129/M1314, A001333/ M2665, A002193/M3195, A002194/M4326, A040000, A040001, A051008, and A051009 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. "The Square-Root Function ffiffiffiffiffiffiffiffiffiffiffiffiffi bx /C27cp and Its Reciprocal," "The bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28x2p Function and Its Reciprocal," and "The bffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27ap Function." Chs. 12, 14, and 15 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 91 /C1/9, 107 /C1/15, and 115 /C1/22, 1987. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 34, 1986. Williams, H. C. "A Numerical Investigation into the Length of the Period of the Continued Fraction Expansion offfiffiffiffi Dp :/" Math. Comp. 36, 593 /C1/01, 1981. Square Root Inequality 2ffiffiffiffiffiffiffiffiffiffiffiffi n /C271p /C282ffiffiffinpB1ffiffiffinpB2ffiffiffinp/C282ffiffiffiffiffiffiffiffiffiffiffiffi n /C281p : Square Root Method The square root method is an algorithm which solves the MATRIX EQUATION Au /C30g (1) for u, with /A a p /C29p SYMMETRIC MATRIX and g a given VECTOR . Convert A to a TRIANGULAR MATRIX such that TTT /C30A; (2) where TT is the MATRIX TRANSPOSE . Then TTk /C30g (3) Tu /C30k; (4) so T /C30s11s12/C1/C1/C1/C1/C1/C1 0 s22/C1/C1/C1/C1/C1/C1 nn::: n 00 /C1/C1/C1 spp2 6643 775: (5) giving the equations s2 11 /C30a11 s11s12 /C30a12 s212 /C27s222 /C30a22 s21j /C27s22j /C27.../C27s2jj /C30ajj s1j /C27s2js2k /C27.../C27sjjsjk /C30ajk : (6) These give s11 /C30ffiffiffiffiffiffiffia11ps12 /C30a12 s11 s22 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a22 /C28s2 12q Sjj /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ajj /C28s2 ij /C28s22j /C28.../C28s2j/C281; jq sjk /C30ajk /C28 s1js1k /C28 s2js2k /C28 ... /C28 sj/C281 ; jsj/C281 ; k sjj; (7) giving T from A : Now solve for k in terms of the sij/ s and g, s11k1 /C30g1 s12k1 /C27s22k2 /C30g2 s1jk1 /C27s2jk2 /C27.../C27sjjkj /C30gj ; (8) which gives k1 /C30g1 s11 k2 /C30g2 /C28 s12k1 s22 kj /C30gj /C28 s1jk1 /C28 s2jk2 /C28 ... /C28 sj/C281; jkj/C281 sjj: (9) Finally, find ufrom the sij/s and k, s11u1/C27s12u2.../C27s1pup/C30k1 s22u2/C27.../C27s2pup/C30k2 sppup/C30kp; (10) giving the desired solution, up/C30kp spp up/C281/C30kp/C281/C28sp/C281;pup sp/C281;p/C281 uj/C30kj/C28sj;j/C271uj/C271/C28sj;j/C272uj/C272/C28.../C28sjpup sjj:(11) See also LU DECOMPOSITION References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 298 /C1/00, 1951. Square Tiling There are a number of interesting results related to the tiling of squares. For example, M. Laczkovich has shown that there are exactly three shapes of non- right triangles that tile the square with similar copies, corresponding to angles ( p=8; p=4; 5p=8); ( p=4; p=3; 5p=12); and ( p=12; p=4; 2p=3) (Stein and Szabo ´ 1994). In particular, given triangles of shape 1 /C282 /C28ffiffiffi 5p with no two the same size, tile the square. The best known solution has 8 triangles (Berlekamp 1999). See also TILING References Berlekamp, E. and Rodgers, T. (Eds.). The Mathemagician and the Pied Puzzler: A Collection in Tribute to Martin Gardner. Boston, MA: A. K. Peters, 1999. Laczkovich, M. "Tilings of Polygons with Similar Triangles." Combinatorica 10, 281 /C1/06, 1990. Schattschneider, D. "Unilateral and Equitransitive Tilings by Squares." Disc. Comput. Geom. 24, 519 /C1/25, 2000. Stein, S. and Szabo ´,S.Algebra and Tiling: Homomorphisms in the Service of Geometry. Washington, DC: Math. Assoc. Amer., 1994. Square Torus The square torus is the quotient of the plane by the integer lattice. Square Triangle Picking Given three points chosen at random inside a UNIT SQUARE , the average AREA of the TRIANGLE deter- mined by these points is given by ¯A /C30g1 0/C1/C1/C1g1 0|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} 6A xiðÞjj dx1 /C1/C1/C1dx3 dy1 /C1/C1/C1dy3 g1 0/C1/C1/C1g1 0|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} 6dx1 /C1/C1/C1dx3 dy1 /C1/C1/C1dy3; where the VERTICES are located at xi ; yi ðÞ where i /C301, ..., 3, and the (signed) AREA is given by the DETERMI- NANT A /C301 2!x1y11 x2y21 x3y31YrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut: The integral can be evaluated analytically to yield ¯A /C3011=144 (Ambartzumian 1987, Pfiefer, Trott 1998), and first calculated by Woolhouse (1867).Because attempting to do the integrals directly quickly results in intractable integrands, the best approach to accomplish the integration is to divide the 6-dimensional region of integration into subre- gions such that the sign of Adoes not change (Trott 1998). The distribution function for the area of a random triangle in a square is known exactly. See also C UBE TETRAHEDRON PICKING ,H EXAGON TRIANGLE PICKING ,P OLYGON TRIANGLE PICKING , TRIANGLE TRIANGLE PICKING ,UNIT SQUARE References Alagar, V. S. "On the Distribution of a Random Triangle." J. Appl. Prob. 14, 284/C1/97, 1977. Ambartzumian, R. V. (Ed.). Stochastic and Integral Geome- try.Dordrecht, Netherlands: Reidel, 1987. Buchta, C. "U ¨ber die konvexe Hu ¨lle von Zufallspunkten in Eibereichen." Elem. Math. 38, 153/C1/56, 1983. Buchta, C. "Zufallspolygone in konvexen Vielecken." J. reine angew. Math. 347, 212/C1/20, 1984. Henze, N. "Random Triangles in Convex Regions." J. Appl. Prob. 20, 111/C1/25, 1983. Klee, V. "What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body." Amer. Math. Monthly 76, 286/C1/88, 1969. Pfiefer, R. E. "The Historical Development of J. J. Sylves- ter’s Four Point Problem." Math. Mag. 62, 309/C1/17, 1989. Seidov, Z. F. "Letters: Random Triangle." Mathematica J. 7, 414, 2000. Trott, M. "The Area of a Random Triangle." Mathematica J. 7, 189/C1/98, 1998. Woolhouse, W. S. B. "Question 2471" Mathematical Ques- tions, with Their Solutions, from the Educational Times,Vol. 8. London: F. Hodgson and Son, pp. 100 /C1 /05, 1867. Square Triangular Number A number which is simultaneously SQUARE and TRIANGULAR . Let Tndenote the nth TRIANGULAR NUMBER and SmthemthSQUARE NUMBER , then a number which is both triangular and square satisfies the equation Tn/C30Sm;or 1 2n(n/C271)/C30m2: (1) COMPLETING THE SQUARE gives 12n2/C27nYrvYru /C3012n/C2712Yru*Yru+2 /C2812Yru*Yru+ 14Yru*Yru+ /C30m2(2) 18(2n/C271)2/C2818/C30m2(3) (2n/C271)2/C288m2/C301: (4) Therefore, defining x/C132n/C271 (5) y/C132m (6) gives the P ELL EQUATION x2/C282y2/C301 (7) (Conway and Guy 1996). The first few solutions are (x; y) /C30(3; 2); (17, 12), (99, 70), (577, 408), .... These give the solutions (n; m) /C30(1; 1); (8, 6), (49, 35), (288, 204), ... (Sloane’s A001108 and A001109), correspond- ing to the triangular square numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (Sloane’s A001110; Pietenpol 1962). In 1730, Euler showed that there are an infinite number of such solutions (Dickson 1952). The general FORMULA for a square triangular number STnis b2c2 ; where b=c is the nth convergent to the CONTINUED FRACTION offfiffiffi 2p (Ball and Coxeter 1987, p. 59; Conway and Guy 1996). The first few are 1 1 ;32 ;75 ;1712 ;4129 ;9970 ;239169 ;/C1/C1/C1; (8) The NUMERATORS and DENOMINATORS can also be obtained by doubling the previous FRACTION and adding to the FRACTION before that. A general FORMULA for square triangular numbers is STn /C301 /C27ffiffiffi 2pYrvYru 2n/C28 1 /C28ffiffiffi2pYrvYru 2n 4ffiffiffi 2p"# 2 (9) /C301 3217 /C272ffiffiffi 2pYru*Yru+n /C27 17 /C282ffiffiffi2pYru*Yru+ n /C282hi : (10) The square triangular numbers also satisfy the RECURRENCE RELATION STn /C3034STn /C281 /C28STn/C282 /C272 (11) un /C272 /C306un/C271 /C28un (12) with u0 /C300 ; u1 /C301 ; where STn /C13u2 n : A curious product formula for STn is given by STn /C3022n/C285Y2n k /C3013 /C27coskp n !"# : (13) An amazing GENERATING FUNCTION is f(x) /C301 /C27 x (1 /C28 x)1/C28 34x /C27 x2 ðÞ /C301 /C2736x /C271225 x2 /C27... (14) (Sloane and Plouffe 1995). Taking the square and triangular numbers together gives the sequence 1, 1, 3, 4, 6, 9, 10, 15, 16, 21, 25, ... (Sloane’s A005214; Hofstadter 1996, p. 15). See also SQUARE NUMBER ,SQUARE ROOT,TRIANGU- LAR NUMBER References Allen, B. M. "Squares as Triangular Numbers." Scripta Math. 20, 213 /C1/14, 1954. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, 1987. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 203 /C1/05, 1996.Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 10, 16, and 27, 1952. Guy, R. K. "Sums of Squares" and "Figurate Numbers." §C20 and §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 /C1/38 and 147 /C1/50, 1994. Hofstadter, D. R. Fluid Concepts & Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought. New York: Basic Books, 1996. Khatri, M. N. "Triangular Numbers Which are Also Squares." Math. Student 27,55/C1/6, 1959. Pietenpol, J. L. "Square Triangular Numbers." Problem E 1473. Amer. Math. Monthly 69, 168 /C1/69, 1962. Potter, D. C. D. "Triangular Square Numbers." Math. Gaz. 56, 109-, 1972. Sierpinski, W. Teoria Liczb, 3rd ed. Warsaw, Poland: Monografie Matematyczne t. 19, p. 517, 1950. Sierpinski, W. "Sur les nombres triangulaires carre´s." Pub. Faculte ´ d’E´ lectrotechnique l’Universite ´ Belgrade , No. 65, 1 /C1/, 1961. Sierpinski, W. "Sur les nombres triangulaires carre´s." Bull. Soc. Royale Sciences Lie`ge, 30 ann., 189 /C1/94, 1961. Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996. Sloane, N. J. A. Sequences A001108/M4536, A001109/ M4217, and A001110/M5259 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Walker, G. W. "Triangular Squares." Problem E 954. Amer. Math. Monthly 58, 568, 1951. Square Wave The square wave is a periodic waveform consisting of instantaneous transitions between two levels which can be denoted 91. The square wave is sometimes also called the R ADEMACHER FUNCTION . Let the square wave have period 2 L:The square wave function is ODD, so the F OURIER SERIES hasa0/C30an/C30 0 and b0/C302 LgL 0sinnpx L ! dx /C304 npsin21 2npYru*Yru+ /C304 np0neven 1nodd:Yrt* The F OURIER SERIES for the square wave is therefore f(x)/C304 pX/C12 n/C301;3;5;...1 nsinnpx L ! : See also HADAMARD MATRIX ,W ALSH FUNCTION References Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, p. 203, 1986. Squared A number to the POWER 2 is said to be squared, so that x2 is called "x squared." See also CUBED ,SQUARE ROOT Squared Square PERFECT SQUARE DISSECTION Squarefree A number is said to be squarefree (or sometimes QUADRATFREI ; Shanks 1993) if its PRIME decomposi- tion contains no repeated factors. All PRIMES are therefore trivially squarefree. The squarefree num- bers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, ... (Sloane’s A005117). The SQUAREFUL numbers (i.e., those that contain at least one square) are 4, 8, 9, 12, 16, 18, 20, 24, 25, ... (Sloane’s A013929). The asymptotic number Q(n) of squarefree numbers 5n is given by Q(n) /C306n p2 /C27OffiffiffinpYrvYru (1) (Landau 1974, pp. 604 /C1/09; Nagell 1951, p. 130; Hardy and Wright 1979, pp. 269 /C1/70; Hardy 1999, p. 65). Q(n) for n /C3010, 100, 1000, ... are 7, 61, 608, 6083, 60794, 607926, ..., while the asymptotic density is 1=z(2) /C306=p2 :0:607927 ; where z(n) is the RIE- MANN ZETA FUNCTION . The MO¨ BIUS FUNCTION is given by m(n) /C130i f n has one or more repeated prime factors 1i f n /C301 (/C281)kif n is the product of k distinct primes ;8 >>>>< >>>>: (2) so m(n) "0 indicates that n is squarefree. The asymptotic formula for Q(x) is equivalent to theformula X x n/C301½ m(n) ½/C306x p2 /C27OffiffiffixpYrvYru (3) (Hardy and Wright 1979, p. 270) There is no known polynomial-time algorithm for recognizing squarefree INTEGERS or for computing the squarefree part of an INTEGER . In fact, this problem may be no easier than the general problem of integer factorization (obviously, if an integer n can be factored completely, n is squarefree IFF it contains no duplicated factors). This problem is an important unsolved problem in NUMBER THEORY because com- puting the RING of integers of an algebraic number field is reducible to computing the squarefree part of an INTEGER (Lenstra 1992, Pohst and Zassenhaus 1997). The Mathematica function SquareFreeQ [n] in the Mathematica add-on package NumberTheor- y‘NumberTheoryFunctions‘ (which can be loaded with the command BBNumberTheory‘ ) deter- mines whether a number is squarefree. No SQUAREFUL FIBONACCI NUMBERS Fpare known with p PRIME . All numbers less than 2:5 /C291015 in SYLVESTER’S SEQUENCE are squarefree, and no SQUAREFUL numbers in this sequence are known (Vardi 1991). Every C ARMICHAEL NUMBER is square- free. The BINOMIAL COEFFICIENTS2n/C281 nYrvYru are square- free only for n/C302, 3, 4, 6, 9, 10, 12, 36, ..., with no others less than n/C301500. The CENTRAL BINOMIAL COEFFICIENTS are SQUAREFREE only for n/C301, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, ... (Sloane’s A046098), with no others less than 1500. See also BINOMIAL COEFFICIENT ,BIQUADRATEFREE , COMPOSITE NUMBER ,CUBEFREE ,ERDOS SQUAREFREE CONJECTURE ,FIBONACCI NUMBER ,KORSELT’S CRITER- ION,M O¨ BIUS FUNCTION ,PRIME NUMBER ,RIEMANN ZETA FUNCTION ,SA´ RKOZY’S THEOREM ,SQUARE NUM- BER,S QUAREFREE PART,S QUAREFUL ,S YLVESTER’S SEQUENCE References Bellman, R. and Shapiro, H. N. "The Distribution of Square- free Integers in Small Intervals." Duke Math. J. 21, 629/C1/ 37, 1954. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. "The Number of Squarefree Numbers." §18.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 269 /C1/70, 1979. Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, 1974. Lenstra, H. W. Jr. "Algorithms in Algebraic Number The- ory." Bull. Amer. Math. Soc. 26, 211/C1/44, 1992. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 130, 1951. Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num- ber Theory. Cambridge, England: Cambridge University Press, p. 429, 1997. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 114, 1993. Sloane, N. J. A. Sequences A005117/M0617, A013929, and A046098 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Vardi, I. "Are All Euclid Numbers Squarefree?" §5.1 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 7 /C1/,82/C1/5, and 223 /C1/24, 1991. Square-Free SQUAREFREE Squarefree Part That part of a POSITIVE INTEGER left after all square factors are divided out. For example, the squarefree part of 24 /C3023 /C215 3 is 6, since 6 /C215 22 /C3024 : For n /C301, 2, ..., the first few are 1, 2, 3, 1, 5, 6, 7, 2, 1, 10, ... (Sloane’s A007913). The squarefree part function can be implemented in Mathematica as SquarefreePart[n_Integer?Positive] : /C30 Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]) See also CUBEFREE PART,SQUARE PART,SQUAREFREE References Atanassov, K. "On the 22nd, 23rd, and the 24th Smaran- dache Problems. Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria 5,80/C1/2, 1999. Atanassov, K. On Some of the Smarandache’s Problems. Lupton, AZ: American Research Press, pp. 16 /C1/1, 1999. Sloane, N. J. A. Sequences A007913 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993. Squarefree Word N.B. A detailed online essay by S. Finch was the starting point for this entry. A "square" word consists of two identical adjacent subwords (for example, acbacb ). A squarefree word contains no square words as subwords (for example, abcacbabcb ). The only squarefree binary words are a, b, ab, ba, aba, and bab (since aa, bb, aaa, aab, abb, baa, bba, and bbb contain square identical adjacent subwords a, b, a, a, b, a, b, and b, respectively). However, there are arbitrarily long ternary square- free words. The number s(n) of ternary squarefree words of length n /C301, 2, ... are 1, 3, 6, 12, 18, 30, 42, 60, ... (Sloane’s A006156), and s(n) is bounded by 6 /C215 1 :032n 5s(n) 56 /C215 1:379n (1) (Brandenburg 1983). In addition, S /C13 lim n0/C12[s(n)]1=n /C301:302... (2) (Brinkhuis 1983, Noonan and Zeilberger 1997).The number of squarefree quaternary words of length n /C301, 2, ... are 4, 12, 36, 96, 264, 696, ... (Sloane’s A051041). See also ALPHABET ,CUBEFREE WORD,OVERLAPFREE WORD,W ORD References Baake, M.; Elser, V.; and Grimm, U. The Entropy of Square- Free Words. 8 Sep 1998. http://xxx.lanl.gov/abs/math-ph/ 9809010/. Bean, D. R.; Ehrenfeucht, A.; and McNulty, G. F. "Avoidable Patterns in Strings of Symbols." Pacific J. Math. 85, 261 /C1/ 94, 1979. Berstel, J. and Reutenauer, C. "Square-Free Words and Idempotent Semigroups." In Combinatorics on Words (Ed. M. Lothaire). Reading, MA: Addison-Wesley, pp. 18 /C1/8, 1983. Brandenburg, F.-J. "Uniformly Growing kth Power-Free Homomorphisms." Theor. Comput. Sci. 23,69/C1/2, 1983. Brinkhuis, J. "Non-Repetitive Sequences on Three Symbols." Quart. J. Math. Oxford Ser. 2 34, 145 /C1/49, 1983. Crochemore, M. "Sharp Characterizations of Squarefree Morphisms." Theor. Comput. Sic. 18, 221 /C1/26, 1982. Crochemore, M. "Tests sur les morphismes faiblement sans carre´." In Combinatorics on Words (Ed. L. J. Cummings). Toronto: Academic Press, pp. 63 /C1/9, 1983. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/words/words.html. Kobayashi, Y. "Repetition-Free Words." Theor. Comput. Sci. 44, 175 /C1/97, 1986. Leconte, M. "kth Power-Free Codes." In Automata on Infinite Words (Ed. M. Nivat and D. Perrin). Berlin: Springer-Verlag, pp. 172 /C1/78, 1985. Noonan, J. and Zeilberger, D. "The Goulden-Jackson Cluster Method: Extensions, Applications, and Implementations." 1997. Pleasants, P. A. B. "Nonrepetitive Sequences." Proc. Cam- bridge Philos. Soc. 68, 267 /C1/74, 1970. Sloane, N. J. A. Sequences A006156/M2550 and A051041 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Thue ,A. "U¨ ber unendliche Zeichenreihen." Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiana 7,1/C1/2, 1906. Reprinted in Nagell, T.; Selberg, A.; Selberg, S.; and Thalberg, K. (Eds.). Selected Mathematical Papers of Axel Thue. Oslo, Norway: Universitetsforlaget, pp. 139 /C1/ 58, 1977. Thue ,A. "U¨ ber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen." Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiana 1,1/C1/7, 1912. Reprinted in Nagell, T.; Selberg, A.; Selberg, S.; and Thalberg, K. (Eds.). Selected Mathe- matical Papers of Axel Thue. Oslo, Norway: Universitets- forlaget, pp. 413 /C1/77, 1977. Squareful A number is squareful, also called nonsquarefree, if it contains at least one SQUARE in its prime factoriza- tion. The first few are 4, 8, 9, 12, 16, 18, 20, 24, 25, ... (Sloane’s A013929). The greatest multiple prime factors for the squareful integers are 2, 2, 3, 2, 2, 3,2, 2, 5, 3, 2, 2, 3, ... (Sloane’s A046028). The least multiple prime factors for squareful integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, ... (Sloane’s A046027). See also G REATEST PRIME FACTOR ,L EAST PRIME FACTOR ,SMARANDACHE NEAR-TO- PRIMORIAL FUNC- TION ,SQUAREFREE References Sloane, N. J. A. Sequences A013929, A046027, and A046028 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Squaring Squaring is the GEOMETRIC CONSTRUCTION , using only COMPASS and STRAIGHTEDGE ,ofa SQUARE which has the same area as a given geometric figure. Squaring is also called QUADRATURE . An object which can be constructed by squaring is called SQUARABLE . See also CIRCLE SQUARING ,COMPASS ,CONSTRUCTI- BLE NUMBER ,G EOMETRIC CONSTRUCTION ,RECTAN- GLE SQUARING ,STRAIGHTEDGE ,TRIANGLE SQUARING Squaring the Circle CIRCLE SQUARING Squeezing Theorem Let there be two functions f/C28(x) and f /C27(x) such that f(x) is "squeezed" between the two, f/C28(x) 5f(x) 5f /C27(x) : If r /C30lim x0af/C28(x) /C30lim x 0af/C27(x) ; then limx0a f(x) /C30r : In the above diagram the func- tions f/C28(x) /C30/C28x2 and f /C27(x) /C30x2 "squeeze" x2 sin(cx)at 0, so limx0a x2 sin(cx) /C300: The squeezing theorem is also called the sandwich theorem. See also LIMIT,PINCHING THEOREM s-Run N.B. A detailed online essay by S. Finch was the starting point for this entry. Let v be a n-VECTOR whose entries are each 1 (with probability p) or 0 (with probability q /C301 /C28p) : An s- run is an isolated group of s consecutive 1s. Ignoringthe boundaries, the total number of runs /Rn/ satisfies Kn /C30Rnhi n/C30 1 /C28p ðÞ2Xn s/C301ps /C30p(1 /C28p)(1 /C28pn); so K(p) /C13 lim n0/C12Kn /C30p(1 /C28p) ; which is called the MEAN RUN COUNT PER SITE or MEAN RUN DENSITY in PERCOLATION THEORY . See also PERCOLATION THEORY , S-CLUSTER References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/rndprc/rndprc.html. S-Signature SIGNATURE (RECURRENCE RELATION ) SSS Theorem Specifying three sides uniquely determines a TRIAN- GLEwhose AREA is given by H ERON’S FORMULA , A/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s/C28a)(s/C28b)(s/C28c)p ; (1) where s/C131 2(a/C27b/C27c) (2) is the SEMIPERIMETER of the TRIANGLE . Let Rbe the CIRCUMRADIUS , then A/C30abc 4R: (3) Using the LAW OF COSINES a2/C30b2/C27c2/C282bccosA (4) b2/C30a2/C27c2/C282accosB (5) c2/C30a2/C27b2/C282abcosC (6) gives the three ANGLES as A /C30cos /C281b2 /C27 c2 /C28 a2 2bc ! (7) B /C30cos/C281a2 /C27 c2 /C28 b2 2ac ! (8) C /C30cos/C281a2 /C27 b2 /C28 c2 2ab ! : (9) See also AAA THEOREM , AAS THEOREM , ASA THEO- REM, ASS THEOREM ,HERON’S FORMULA , SAS THEO- REM,SEMIPERIMETER ,TRIANGLE St. Ives Problem A well-known nursery rhyme states, "As I was going to St. Ives, I met a man with seven wives. Every wife had seven sacks, every sack had seven cats, every cat had seven kitts. Kitts, cats, sacks, wives, how many were going to St. Ives?" Upon being presented with this conundrum, most readers begin furiously adding and multiplying numbers in order to calculate the total quantity of objects mentioned. However, the problem is a trick question. Since the man and his wives, sacks, etc. were met by the narrator on the way to St. Ives, they were in fact leaving–not going to– St. Ives. The number going to St. Ives is therefore "one," i.e., the narrator. Should a diligent reader nevertheless wish to calcu- late the sum total N of kitts, cats, sacks, and wives, the answer is easily given by the GEOMETRIC SERIES Xn k /C301rk /C30r 1 /C28 rnðÞ 1 /C28 r (1) with n /C304 and r /C307. Therefore, N /C30X4 i/C3017i /C3071/C28 74ðÞ 7 /C28 1/C302800 : (2) N /C3071 /C2772 /C2773 /C2774 /C307(1 /C277(1 /C277(1 /C277))) /C307(1 /C277(1 /C277 /C215 8)) /C307(1 /C277 /C215 57) /C307 /C215 400 /C302800 : (3) A similar question was given as problem 79 of the Rhind papyrus, dating from 1650 BC. This problem concerns 7 houses, each with 7 cats, each with 7 mice, each with 7 spelt, each with 7 hekat. The total number of items is then X5 i/C3017i /C3019607 (4) (Wells 1986, p. 71). In turn, the problem of the Rhindpapyrus is repeated in Fibonacci’s Liber Abaci (1202, 1228). References Eisele, C. "Liber Abaci." Scripta Math. 17. Gill, R. J. Mathematics in the Time of the Pharaohs. Cam- bridge, MA: MIT Press, 1972. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 71, 1986. Stability The robustness of a given outcome to small changes in initial conditions or small random fluctuations. CHAOS is an example of a process which is not stable. See also STABILITY MATRIX Stability Matrix Given a system of two ordinary differential equations ˙x /C30f(x; y) (1) ˙y /C30g(x; y) ; (2) let x0 and y0 denote FIXED POINTS with ˙x /C30 ˙y /C300 ; so fx0 ; y0 ðÞ /C300 (3) gx0 ; y0 ðÞ /C300: (4) Then expand about x0 ; y0 ðÞ so d˙x /C30fxx0 ; y0 ðÞ dx /C27fyx0 ; y0 ðÞ dy /C27fxyx0 ; y0 ðÞ dxdy /C27...: (5) d˙y /C30gxx0 ; y0 ðÞ dx /C27gyx0 ; y0 ðÞ dy /C27gxyx0 ; y0 ðÞ dxdy /C27...: (6) To first-order, this gives d dtdx dyYrtvYrtu /C28fxx0 ; y0 ðÞ fyx0 ; y0 ðÞ gxx0 ; y0 ðÞ gyx0 ; y0 ðÞYrtvYrtu dx dyYrtvYrtu ; (7) where the 2 /C292 MATRIX , or its generalization to higher dimension, is called the stability matrix. Analysis of the EIGENVALUES (and EIGENVECTORS )o f the stability matrix characterizes the type of FIXED POINT . See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), LINEAR STABILITY , STABLE IMPROPER NODE,S TABLE NODE,S TABLE SPIRAL POINT ,STABLE STAR,U NSTABLE IMPROPER NODE,U NSTABLE NODE,U NSTABLE SPIRAL POINT , UNSTABLE STAR References Tabor, M. "Linear Stability Analysis." §1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20 /C1/1, 1989. Stabilization A type II MARKOV MOVE . See also MARKOV MOVES Stable Equivalence Two VECTOR BUNDLES are stably equivalent IFF ISOMORPHIC VECTOR BUNDLES are obtained upon WHITNEY SUMMING each VECTOR BUNDLE with a trivial VECTOR BUNDLE . See also VECTOR BUNDLE ,W HITNEY SUM Stable Improper Node A FIXED POINT for which the STABILITY MATRIX has equal NEGATIVE EIGENVALUES . See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE NODE,STABLE SPIRAL POINT ,UNSTABLE IMPROPER NODE,UNSTABLE NODE,UNSTABLE SPIRAL POINT ,UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Stable Marriage Problem Given a set of n men and n women, marry them off in pairs after each man has ranked the women in order of preference from 1 to n, w1 ; ...; wn fg and each women has done likewise, m1 ; ...; mn fg : If the resulting set of marriages contains no pairs OF THE FORM mi ; wjYr$Yr% ; mk ; wl fg such that mi prefers wl to wj and wlprefers mito mk ; the marriage is said to be stable. Gale and Shapley (1962) showed that a stable marriage exists for any choice of rankings (Skiena 1990, p. 245). In the United States, the algorithm of Gale and Shapley (1962) is used to match hospitals to medical interns (Skiena 1990, p. 245). In the rankings illustrated above, the male-optimal stable marriage is 4, 2, 6, 5, 3, 1, 7, 9, 8, and thefemale-optimal stable marriage is 1, 2, 8, 9, 3, 4, 7, 6, 5. A stable marriage can be found using Stable- Marriage [m, w] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also DIVORCE DIGRAPH ,MATCHING References Gale, D. and Shapley, L. S. "College Admissions and the Stability of Marriage." Amer. Math. Monthly 69,9/C1/4, 1962. Gusfield, D. and Irving, R. W. The Stable Marriage Problem: Structure and Algorithms. Cambridge, MA: MIT Press, 1989. Skiena, S. "Stable Marriages." §6.4.4 in Implementing Dis- crete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 245 /C1/46, 1990. Stable Node A FIXED POINT for which the STABILITY MATRIX has both EIGENVALUES NEGATIVE ,sol1 B l2 B0 :/ See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,S TABLE SPIRAL POINT ,S TABLE STAR,U N- STABLE IMPROPER NODE,UNSTABLE NODE,UNSTABLE SPIRAL POINT ,UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Stable Polynomial AREAL POLYNOMIAL Pis said to be stable if all its ROOTS lie in the LEFT HALF-PLANE . The term "stable" is used to describe such a polynomial because, in the theory of linear servomechanisms, a system exhibits unforced time-dependent motion of the form est; where sis the root of a certain REAL POLYNOMIAL P(s)/C300:A system is therefore mechanically stable IFF Pis a stable polynomial. The polynomial x/C27ais stable IFFa/C210, and the IRREDUCIBLE POLYNOMIAL x2/C27ab/C27bis stable IFF both aand bare greater than zero. The R OUTH- HURWITZ THEOREM can be used to determine if a polynomial is stable. Given two real polynomials PandQ,i fPandQare stable, then so is their product PQ, and vice versa (Se´roul 2000, p. 280). It therefore follows that the coefficients of stable real polynomials are either all positive or all negative (although this is not a SUFFICIENT condition, as shown with the counter- example x3/C27x2/C27x/C271):Furthermore, the values of a stable polynomial are never zero for x]0 and have the same sign as the coefficients of the polynomial. It is possible to decide if a polynomial is stable without first knowing its roots using the following theorem due to Strelitz (1977). Let A /C30xn /C27 an/C281xn/C281 /C27.../C27a0be a real polynomial with roots a1 ; ..., an ; and construct B /C30xm /C27bm/C281xm/C281 /C27.../C27b0 as the monic real polynomial of degree m /C30n(n /C281)=2 having roots ai /C27 ajfor 1 5i 5j 5n: Then A is stable IFF all coefficients of A and B are positive (Se´roul 2000, p. 281). For example, given the third-order polynomial A /C30 x3 /C27ax2 /C27bx /C27c ; the sum-of-roots polynomial B is given by B /C30x3 /C272ax2 /C27 a2 /C27bYrvYru x /C27(ab /C28c) : (1) Resolving the inequalities given by requiring that each coefficient of A and B be greater than zero then gives the conditions for A to be stable as a /C21 0, b /C21 0, 0 Bc Bab :/ Similarly, for the fourth-order polynomial A /C30x4 /C27 ax3 /C27bx2 /C27cx /C27d; the sum-of-roots-polynomial is x6 /C273ax5 /C27 3a2 /C272bYrvYru x4 /C27 a3 /C274abYrvYru x3 /C27 2a2b /C27b2 /C27ac /C284dYrvYru x2 /C27 ab2 /C27a2c /C284adYrvYru /C27x /C27 abc /C28c2 /C28a2dYrvYru ; (2) so the condition for A to be stable can be resolved to a /C21 0, b /C21 0, 0 Bc Bab ; 0 Bd B abc /C28c2ðÞ =a2 :/ The fifth-order polynomial is x10 /C274ax9 /C27 6a2 /C273bYrvYru x8 /C27 4a3 /C279ab /C27cYrvYru x7 /C27 a4 /C279a2b /C273b2 /C274ac /C283dYrvYru x6 /C27 3a3b /C276ab2 /C275a2c /C272bc /C285ad /C2811eYrvYru x5 /C27 3a2b2 /C27b3 /C272a3c /C276abc /C28c2 /C282a2d /C282bd /C2822aeYrvYru x4 /C27(ab3 /C274a2bc /C27b2c /C284cd /C2816a2e /C284be)x3 /C27 2ab2c /C27a2c2 /C28bc2 /C27a2bd /C27b2d /C283acd /C284d2 /C284a3eYrv /C289abe /C277ceÞx2 /C27 abc2 /C28c3 /C27ab2d /C284ad2 /C284a2be /C28b2e /C274ace /C274deYrvYru x /C27 abcd /C28c2d /C28a2d2 /C28ab2e /C27bce /C272ade /C28e2YrvYru : (3) The following Mathematica code computes the sum- of-roots polynomial B and inequalities obtained from the coefficients, RootSumPolynomial[r_List,x_]: /C30Module[ {n /C30Length[r],i,j}, RootReduce@Collect[Expand[ Times@@((x-#)&/@Flatten[ Table[r[[i]] /C27r[[j]],{i,n},{j,i /C271,n}]]) ],x] ] RootSumPolynomial[p_?PolynomialQ,x_]: /C30 RootSumPolynomial[RootList[p,x],x] RootList[p_?PolynomialQ,x_]: /C30x/.{ToRules[Roots[p /C30/C300,x, Cubics- /C21False,Quartics- /C21False ]]} RootSumInequalities[p_?PolynomialQ,x_]: /C30 And@@(# /C210&/@Flatten[CoefficientList[#,x]&/@ {RootSumPolynomial[p,x],p}]) while the following reduces the inequalities to a minimal set in the cubic case. Resolve[Exists[x, (a | b |c|x) \[Element] Reals, RootSumInequalities[x^3 /C27 a x^2 /C27 bx/C27 c, x] ], {a, b, c}] See also LEFT HALF-PLANE ,ROUTH- HURWITZ THEO- REM References Se´roul, R. "Stable Polynomials." §10.13 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 280 /C1/86, 2000. Strelitz, S. "On the Routh-Hurwitz Problem." Amer. Math. Monthly 84, 542 /C1/44, 1977. Stable Spiral Point A FIXED POINT for which the STABILITY MATRIX has EIGENVALUES OF THE FORM l9/C30/C28a 9ib (with a; b > 0):/ See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,STABLE NODE,STABLE STAR,U NSTABLE IM- PROPER NODE,U NSTABLE NODE,U NSTABLE SPIRAL POINT ,UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Stable Star A FIXED POINT for which the STABILITY MATRIX has one zero EIGENVECTOR with NEGATIVE EIGENVALUE / lB0/. See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,STABLE NODE,STABLE SPIRAL POINT ,U N- STABLE IMPROPER NODE,UNSTABLE NODE,UNSTABLE SPIRAL POINT ,UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/25, 1989. Stable Type A POLYNOMIAL equation whose ROOTS all have NEGA- TIVE REAL PARTS . For a REAL QUADRATIC EQUATION z2 /C27Bz /C27C /C300: the stability conditions are B ; C > 0: For a REAL CUBIC EQUATION z3 /C27Az2 /C27Bz /C27C /C300: the stability conditions are A; B ; C > 0 and AB /C21 C. References Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 108 /C1/09, 1996. Stab-Werner Projection WERNER PROJECTION Stack A DATA STRUCTURE which is a special kind of LIST in which elements may be added to or removed from the top only. These actions are called a PUSH or a POP, respectively. Actions may be taken by popping one or more values, operating on them, and then pushing the result back onto the stack. Stacks are used as the basis for computer languages such as FORTH, PostScript † (Adobe Systems), and the RPN language used in Hewlett-Packard † pro- grammable calculators. See also LIST,POP,PUSH,QUEUE ,REVERSE POLISH NOTATION Stack Polygon A SELF-AVOIDING POLYGON containing two adjacent corners of its minimal bounding rectangle. The anisotropic area and perimeter generating function G(x; y) and partial generating functions Hm(y) ; con- nected by G(x; y; q) /C30X m]1Hm(y; q)xm : satisfy the self-reciprocity and inversion relations Hm(1=y; 1=q) /C30/C28y2m/C283qm2/C282mHm(y; q) and G(x;y)/C27y3Gx =y2;1=yYrvYru /C300 (Bousquet-Me ´louet al. 1999). See also LATTICE POLYGON ,SELF-AVOIDING POLYGON References Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly-ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/ 9908123/. Sta¨ckel Determinant ADETERMINANT used to determine in which coordi- nate systems the H ELMHOLTZ DIFFERENTIAL EQUA- TION is separable (Morse and Feshbach 1953). A determinant S/C30Fmnjj/C30F11F12F13 F21F22F23 F31F32F33YrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut(1) in which F mare functions of uialone is called a Sta¨ckel determinant. A coordinate system is separ- able if it obeys the R OBERTSON CONDITION , namely that the SCALE FACTORS hiin the L APLACIAN 92/C30X3 i/C3011 h1h2h3@ @uih1h2h3 h2 i@ @ui ! (2) can be rewritten in terms of functions fi(ui) defined by 1 h1h2h3@ @uih1h2h3 h2i@ @ui ! /C30g(ui/C271;ui/C272) h1h2h3@ @uifi(ui)@ @ui"# /C301 h2ifi@ @uifi@ @ui ! (3) such that Scan be written S/C30h1h2h3 f1(u1)f2(u2)f3(u3): (4) When this is true, the separated equations are OF THE FORM 1 fn@ @unfn@Xn @un ! /C27k2 1Fn1/C27k22Fn2/C27k23Fn3YrvYru Xn/C300 (5) TheFij/s obey the minor equations M1/C30F22F33/C28F23F32/C30S h2 1(6) M2/C30F13F32/C28F12F33/C30S h22(7) M3/C30F12F23/C28F13F22/C30S h23: (8) which are equivalent to M1F11/C27M2F21/C27M3F31/C30S (9) M1F12/C27M2F22/C27M3F32/C300 (10) M1F13/C27M2F23/C27M3F33/C300 (11) (Morse and Feshbach 1953, p. 509). This gives a total of four equations in nine unknowns. Morse and Feshbach (1953, pp. 655 /C1/66) give not only the Sta¨ckel determinants for common coordinate sys- tems, but also the elements of the determinant (although it is not clear how these are derived). See also HELMHOLTZ DIFFERENTIAL EQUATION ,LA- PLACE’S EQUATION ,POISSON’S EQUATION ,ROBERTSON CONDITION ,SEPARATION OF VARIABLES References Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 5 /C1/, 1988. Morse, P. M. and Feshbach, H. "Tables of Separable Co- ordinates in Three Dimensions." Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 509 /C1/11 and 655 /C1/66, 1953. Staircase Function A function composed of a set of equally spaced jumps of equal length, such as the CEILING FUNCTION f(x) /C30 xde; FLOOR FUNCTION f(x) /C30 xbc; or NEAREST INTEGER FUNCTION f(x) /C30 x½/C138:/ See also CEILING FUNCTION ,FLOOR FUNCTION ,NEAR- EST INTEGER FUNCTION ,SAWTOOTH WAVE References Spanier, J. and Oldham, K. B. An Atlas of Functions. Washington, DC: Hemisphere, p. 74, 1987. Staircase Polygon Define the minimal bounding rectangle as the smal- lest rectangle containing a given lattice polygon. If the perimeter of the lattice polygon is equal to that of its minimal bounding rectangle, it is said to be convex. (Note that a "convex" lattice polygon is not necessarily convex in the usual sense of the word.) A staircase polygon is then defined as a convex polygon which contains two opposite corners of its bounding rectangle (Bousquet-Me ´lou et al. 1999). The area generating function Hm(y; q) that counts polygons of width m for staircase polygons of width 4 is given by H4(q) /C30 q4 1 /C27 2q /C27 4q2 /C27 6q3 /C27 7q4 /C27 6q5 /C27 4q6 /C27 2q7 /C27 q8ðÞ (1 /C28 q)2 1 /C28 q2 ðÞ21 /C28 q3 ðÞ21 /C28 q4 ðÞ: (1) which satisfies H4(1=q) /C30/C28H4(q) (Bousquet-Me ´lou 1992, Bousquet-Me ´lou et al. 1999). The anisotropic area and perimeter generating func-tion G(x; y; q) and partial generating functions Hm(y; q) ; connected by G(x; y; q) /C30X m]1Hm(y; q)xm : satisfy the self-reciprocity and inversion relations Hm(1=y; 1=q) /C30/C28ym/C281Hm(y; q) for m ]2 and G(x; y; q) /C27yG(x=y; 1=y; 1=q) /C30/C28x (Bousquet-Me ´lou et al. 1999). The anisotropic area and perimeter generating func- tion G(x; y; q) of staircase polygon with a staircase hole satisfies an inversion relation OF THE FORM G(x; y; q) /C27y2G(x=y; 1=y ; 1=q) (Bousquet-Me ´louet al. 1999). See also SELF-AVOIDING POLYGON ,STAIRCASE WALK References Bousquet-Me ´lou, M. "Convex Polyominoes and Heaps of Segments." J. Phys. A: Math. Gen. 25, 1925 /C1/934, 1992. Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/9908123/. Staircase Walk The numbers of staircase walks on an m/C29ngrid are given by m/C27n/C282 m/C281Yru$Yru% /C30(m/C27n/C282)! (m/C281)!(n/C281)!(1) (Vilenkin 1971, Mohanty 1979, Narayana 1979, Finch). The first few values for m/C30n/C301;2, ..., are 1, 2, 6, 20, 70, 252, ... (Sloane’s A000984), which are the CENTRAL BINOMIAL COEFFICIENTS . The number of staircase walks on an n /C29n grid which remain below the diagonal is given by the CATALAN NUMBER Cn/C281 /C301 n /C27 12n nYru$Yru% : i.e., 1, 2, 5, 14, 42, 132, ... (Sloane’s A000108). See also CATALAN NUMBER ,C ENTRAL BINOMIAL COEFFICIENT ,STAIRCASE POLYGON References Finch, S. "Unsolved Mathematics Problems: Self-Avoiding Walks of a Rook on a Chessboard." http://www.mathsoft.- com/asolve/gammel/gammel.html. Mohanty, S. G. Lattice Path Counting and Applications. New York: Academic Press, 1979. Narayana, T. V. Lattice Path Combinatorics with Statistical Applications. Toronto, Ontario, Canada: University of Toronto Press, 1979. Sloane, N. J. A. Sequences A000108/M1459 and A000984/ M1645 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Vilenkin, N. Ya. Combinatorics. New York: Academic Press, 1971. Stamp Folding The number of ways of folding a strip of stamps has several possible variants. Considering only positions of the hinges for unlabeled stamps without regard to orientation of the stamps, the number of foldings is denoted U(n) : If the stamps are labeled and orienta- tion is taken into account, the number of foldings is denoted N(n) : Finally, the number of symmetric foldings is denoted S(n) : The following table sum- marizes these values for the first n. n /S(n)// U(n)// N(n)/ Sloane Sloane’s A001010Sloane’s A001011Sloane’s A000136 11112212 32264451 6 5 6 14 50 6 8 38 144 7 18 120 462 8 20 353 1392 9 56 1148 4536 10 48 3527 14060 See also M AP FOLDING ,POSTAGE STAMP PROBLEM References Gardner, M. "The Combinatorics of Paper-Folding." In Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 60 /C1/3, 1983. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 21 and 26 /C1/7, 1984. Koehler, J. E. "Folding a Strip of Stamps." J. Combin. Th. , Sep. 1968. Lunnon, W. F. "A Map-Folding Problem." Math. Comput. , Jan. 1968. Ruskey, F. "Information of Stamp Folding." http:// www.theory.csc.uvic.ca/~cos/inf/perm/StampFol- ding.html. Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, p. 22, 1973. Sloane, N. J. A. Sequences A000136/M1614, A001010/ M0323, and A001011/M1455 in "An On-Line Version ofthe Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Stamp Problem POSTAGE STAMP PROBLEM Standard Deviation The standard deviation stdv( x) is defined as the SQUARE ROOT of the VARIANCE s2; stdv( x)/C30s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2hi/C28xhi2q /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m?2/C28m2p : (1) where m/C30¯x/C30xhiis the MEAN ,m?2/C30x2hiis the second RAW MOMENT , and f(x)hi denotes an EXPECTATION VALUE . The variance s2is therefore equal to the second CENTRAL MOMENT (i.e., moment about the MEAN ), s2/C30m2: (2) The variate value producing a CONFIDENCE INTERVAL CI is often denoted xCI;and xCI/C30ffiffiffi 2p erf/C281(CI) : (3) The following table lists the CONFIDENCE INTERVALS corresponding to the first few multiples of the standard deviation. range CI /s/ 0.6826895 /2s/ 0.9544997 /3s/ 0.9973002 /4s/ 0.9999366 /5s/ 0.9999994 To find the standard deviation range corresponding to a given CONFIDENCE INTERVAL , solve (2) for n, giving n /C30ffiffiffi 2p erf /C281(CI) : (4) CI range 0.800 /91 :28155 s/ 0.900 /91 :64485 s/ 0.950 /91 :95996 s/ 0.990 /92 :57583 s/ 0.995 /92 :80703 s/ 0.999 /93 :29053 s/ The square root of the SAMPLE VARIANCE is the "sample" standard deviation, sN /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 NXN i/C301(xi /C28 ¯x)2vuut: (5) It is a BIASED ESTIMATOR of the population standard deviation. An unbiased ESTIMATOR is given by sN /C281 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N /C28 1XN i/C301(xi /C28 ¯x)2vuut: (6) Physical scientists often use the term ROOT-MEAN- SQUARE as a synonym for standard deviation when they refer to the SQUARE ROOT of the mean squared deviation of a signal from a given baseline or fit. See also CONFIDENCE INTERVAL ,M EAN,M OMENT , ROOT-MEAN-SQUARE ,SAMPLE VARIANCE ,STANDARD ERROR ,VARIANCE References Kenney, J. F. and Keeping, E. S. "The Standard Deviation" and "Calculation of the Standard Deviation." §6.5 /C1/.6 inMathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 77 /C1/0, 1962. Standard Error The square root of the ESTIMATED VARIANCE of a quantity, standard error /C30ffiffiffiffiffiffiffiffiffiffiffiffiˆvar xp : However, the standard error is sometimes also used to mean var( ¯x) /C30Xn i/C3011 n !2 s2 i /C30Xn i/C3011 n !2 s2/C30s2 n: See also ESTIMATOR ,STANDARD DEVIATION ,VARIANCE Standard Map A 2-D MAP also called the Taylor-Greene-Chirikov map in some of the older literature and defined by In/C271/C30In/C27Ksinun (1) un/C271/C30un/C30Inþ1/C27un/C27Ksinun; (2) where Iand uare computed mod 2 pand Kis a POSITIVE constant. The standard map can be implemented in Mathema- tica as StandardMap[k_, its_:100, cnt_:50] : /C30Mod- ule[{}, f[{t_, i_}] : /C30Mod[{i /C27t/C27k Sin[t], i /C27k Sin[t]}, 2Pi]; Graphics[{ PointSize[.01], Table[ Point /@ NestList[f, #, its] & [ Table[Random[Real, {0, 2Pi}], {2}]], {cnt}] }, AspectRatio- /C21Automatic] ] An analytic estimate of the width of the CHAOTIC zone (Chirikov 1979) finds dI /C30Be /C28AK /C281=2 : (3) Numerical experiments give A :5:26 and B :240: The value of K at which global CHAOS occurs has been bounded by various authors. GREENE’S METHOD is the most accurate method so far devised. Author Bound Fraction Decimal Hermann / >//1 34/ 0.029411764 Italians / >/ - 0.65 Greene /:/ - 0.971635406 MacKay and Pearson/B//63 64/ 0.984375000 Mather /B//43/ 1.333333333 FIXED POINTS are found by requiring that In/C271 /C30In (4) un/C271 /C30 un : (5) The first gives K sin un /C300 ; so sin un /C300 and un /C300; p: (6) The second requirement gives In /C27K sin un /C30In /C300: (7) The FIXED POINTS are therefore (I ; u) /C30(0; 0) and (0; p): In order to perform a LINEAR STABILITY analysis, take differentials of the variables dIn/C271 /C30dIn /C27K cos un dun (8) dun/C271 /C30dIn /C27 1 /C27K cos un ðÞ dun : (9) In MATRIX form, dIn/C271 dun/C271YrtvYrtu /C301 K cos un 11/C27K cos unYrtvYrtu dIn dunYrtvYrtu : (10) The EIGENVALUES are found by solving the CHARAC- TERISTIC EQUATION 1 /C28 l K cos un 11 /C27K cos un /C28 lYrtvYrtu /C300: (11) so l2 /C28 l K cos un /C272 ðÞ /C271 /C300 (12)l9/C3012K cos un /C272 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K cos un /C272 ðÞ2/C284q YrtvYrtu : (13) For the FIXED POINT (0; p) ; l(0; p) 9/C30122 /C28K 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28K ðÞ2/C284qYrtvYrtu /C30122 /C28K 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 /C284KpYru*Yru+ : (14) The FIXED POINT will be stable if R l(0; p)YrvYruYrutYrutYrutYrutB2: Here, that means 1 2 2 /C28K jjB1 (15) 2 /C28K jjB2 (16) /C282 B2 /C28K B2 (17) /C284 B/C28K B0 (18) so K /C23 0; 4½Þ : For the FIXED POINT (0, 0), the EIGENVA- LUES are l(0; 0) 9/C30122 /C27K 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (K /C272)2 /C284qYrtvYrtu 1 22 /C27K 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 /C274KpYru*Yru+ : (19) If the map is unstable for the larger EIGENVALUE ,itis unstable. Therefore, examine l(0; 0) 9 :We have 1 22/C27K/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2/C274KpYrutYrutYrutYrutYrutYrutB1: (20) so /C282B2/C27K/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2/C274Kp B2 (21) /C284/C28KBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK 2/C274Kp B/C28K: (22) ButK/C210, so the second part of the inequality cannot be true. Therefore, the map is unstable at the FIXED POINT (0, 0). See also HE´ NON- HEILES EQUATION References Chirikov, B. V. "A Universal Instability of Many-Dimen- sional Oscillator Systems." Phys. Rep. 52, 264/C1/79, 1979. Rasband, S. N. "The Standard Map." §8.5 in Chaotic Dy- namics of Nonlinear Systems. New York: Wiley, pp. 11 and 178 /C1/79, 1990. Tabor, M. "The He ´non-Heiles Hamiltonian." §4.2.r in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 134 /C1/35, 1989. Standard Normal Distribution ANORMAL DISTRIBUTION with zero MEAN (/m/C300) and unity STANDARD DEVIATION (/s2/C301);given by P(x) dx /C301ffiffiffiffiffiffi 2pp e /C28z2 =2 dz : See also NORMAL DISTRIBUTION ,TETRACHORIC FUNC- TION Standard Space A SPACE which is ISOMORPHIC to a BOREL SUBSET B of aP OLISH SPACE equipped with its SIGMA ALGEBRA of BOREL SETS. See also BOREL SET,POLISH SPACE ,SIGMA ALGEBRA Standard Tableau YOUNG TABLEAU Standard Tori One of the three classes of TORI illustrated above and given by the PARAMETRIC EQUATIONS x /C30(c /C27a cos v)cos u (1) y /C30(c /C27a cos v)sin u (2) z /C30a sin v: (3) The three different classes of standard tori arise from the three possible relative sizes of a and c. c /C21a corresponds to the RING TORUS shown above, c /C30a corresponds to a HORN TORUS which touches itself at the point (0, 0, 0), and c Ba corresponds to a self- intersecting SPINDLE TORUS (Pinkall 1986). If no specification is made, "torus" is taken to mean RING TORUS . The standard tori and their inversions are CYCLIDES . See also APPLE ,CYCLIDE ,HORN TORUS ,LEMON ,RING TORUS ,SPINDLE TORUS ,TORUS References Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28 /C1/0, 1986.Standard Unit References Kenney, J. F. and Keeping, E. S. "Standard Units." §7.7 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 96 /C1/8, 1962. Standardized Moment Defined for samples xi ; i /C301, ..., N by ar /C131 NXN i/C301zr i /C30mr sr : (1) where zi /C13xi /C28 ¯x sx: (2) The first few are a1 /C300 (3) a2 /C301 (4) a3 /C30m3 s3 (5) a4 /C30m4 s4 : (6) See also KURTOSIS ,MOMENT ,SKEWNESS References Kenney, J. F. and Keeping, E. S. "Moments in Standard Units." §7.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 98 /C1/9, 1962. Standardized Score Z-SCORE Stanley’s Identity X/C12 k/C30/C28/C12a m /C28kYru$Yru% b n/C28kYru$Yru% a/C27b/C27k kYru$Yru% /C30a/C27n mYru$Yru% b/C27m nYru$Yru% : See also BINOMIAL SUMS References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 41, 1998. Strehl, V. "Binomial Identities--Combinatorial and Algorith- mic Aspects." Discrete Math. 136, 309/C1/46, 1994. Stanley’s Theorem The total number of 1s that occur among all un- ordered PARTITIONS of a POSITIVE INTEGER is equal to the sum of the numbers of distinct members of those PARTITIONS . For example, the partitions of 5 are f5g; f1; 1g;f3 ; 2 g;f3; 1; 1g;f2; 2; 1g;f2 ; 1; 1; 1g; f1; 1; 1; 1; 1g: There are a total of 0 /C271 /C270 /C272 /C271 /C27 3 /C275 /C3012 1s in this list, which is equal to the sums of the numbers of unique terms in each partition: 1 /C272 /C272 /C272 /C272 /C272 /C271 /C3012 :/ The numbers of 1s occurring in all partitions of n /C301, 2, 3, ... are 1, 2, 4, 7, 12, 19, 30, 45, 67, ... (Sloane’s A000070). See also ELDER’S THEOREM ,PARTITION References Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer, pp. 6 /C1/, 1985. Sloane, N. J. A. Sequences A000070/M1054 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stanley-Wilf Conjecture Stanley and Wilf conjectured (Bona 1997, Arratia 1999), that for every PERMUTATION PATTERN s; there is a constant c( s) B/C12 such that for all n, F(n; s) 5[c( s)]n : (1) A related conjecture stated that for every s; the limit lim n0/C12[F(n; s)]1 =n (2) exists and is finite. Arratia (1999) showed that these two conjectures are equivalent. See also PERMUTATION PATTERN References Alon, N. and Friedgut, E. "On the Number of Permutations Avoiding a Given Pattern." To appear in J. Combin. Th. Ser. A. Arratia, R. "On the Stanley-Wilf Conjecture for the Number of Permutations Avoiding a Given Pattern." Electronic J. Combinatorics 6, No. 1, N1, 1 /C1/, 1999. http://www.combi- natorics.org/Volume_6/v6i1toc.html. Bona, M. "Exact and Asymptotic Enumeration of Permuta- tions with Subsequence Conditions." Ph.D. thesis. Cam- bridge, MA: MIT, 1997. Bona, M. "The Solution of a Conjecture of Stanley and Wilf for All Layered Patterns." J. Combin. Th. Ser. A 85,96/C1/ 04, 1999. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos’ 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press, pp. 557 /C1/62, 1997.Star The word "star" is used to voice an asterisk when appearing in a mathematical expression. For exam- ple, a/C31 is voiced "a-star". The "star" is used to denote the ADJOINT a /C31; or sometimes the COMPLEX CONJU- GATE . In common usage, a star is a STAR POLYGON or STAR FIGURE (i.e., regular convex polygon or polygon compound) such as the PENTAGRAM or HEXAGRAM In formal geometry, a star is a set of 2n VECTORS 9a1 ; ..., 9anwhich form a fixed center in EUCLIDEAN 3- SPACE . In ALGEBRAIC TOPOLOGY ,if v is a vertex of a SIMPLICIAL COMPLEX K, then the star of v in K, denoted St v or St(v ;K) ; is the union of the interiors of those SIMPLICES of K that have v as a vertex (Munkres 1993, p. 11). See also CLOSED STAR,C ROSS ,E UTACTIC STAR, HEXAGRAM ,L INK (SIMPLICIAL COMPLEX ), PENTA- GRAM ,STAR FIGURE ,STAR POLYGON References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Star (Fixed Point) A FIXED POINT which has one zero EIGENVECTOR . See also STABLE STAR,UNSTABLE STAR Star Figure A STAR POLYGON -like figure fp =qg for which p and q are not RELATIVELY PRIME . Examples include the HEXAGRAM f6=3 g; STAR OF LAKSHMI f8=2 g; and NON- AGRAM f9=3g:/ See also HEXAGRAM ,NONAGRAM ,STAR OF LAKSHMI , STAR POLYGON Star Fractal A FRACTAL composed of repeated copies of a PENTA- GRAM or other polygon. The above figure shows a generalization to different offsets from the center. References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 72 /C1/7, 1991. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Star Graph The n-star graph is a TREE on n /C271 nodes with one node having VERTEX DEGREE n and the others havingVERTEX DEGREE 1. Star graphs Sn are always GRACE- FUL. Star graphs can be constructed usingStar [n]in the Mathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). The COMPLETE BIPARTITE GRAPH K1 ; n /C281is the STAR GRAPH Sn(Skiena 1990, p. 146). The CHROMATIC POLYNOMIAL of Sn is given by psn(z) /C30z(z /C281)n/C281 : and the CHROMATIC NUMBER is 1 for n /C301, and x SnðÞ/C302 otherwise. See also CAYLEY TREE,TREE References Skiena, S. "Cycles, Stars, and Wheels." §4.2.3 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 83 and 144 /C1/47, 1990. Star Number The number of cells in a generalized Chinese checkers board (or "centered" HEXAGRAM ). Sn /C306n(n /C271) /C271 /C30Sn/C281 /C2712(n /C281): (1) The first few are 1, 13, 37, 73, 121, ... (Sloane’s A003154). Every star number has DIGITAL ROOT 1or 4, and the final digits must be one of: 01, 21, 41, 61, 81, 13, 33, 53, 73, 93, or 37. The first TRIANGULAR star numbers are 1, 253, 49141, 9533161, ... (Sloane’s A006060), and can be computed using TSn /C3037/C27 4ffiffiffi 3pYrvYru 2n /C281/C27 7 /C28 4ffiffiffi3pYrvYru 2n/C281hi /C28 10 32 (2) /C30194TSn/C281 /C2760 /C28TSn/C282 : (3) The first few SQUARE star numbers are 1, 121, 11881, 1164241, 114083761, ... (Sloane’s A006061). SQUARE star numbers are obtained by solving the DIOPHAN- TINE EQUATION 2x2/C271/C303y2(4) and can be computed using SSn/C30 5/C272ffiffiffi 6pYrvYru nffiffiffi6p /C282YrvYru /C285/C282ffiffiffi6pYrvYru nffiffiffi6p /C272YrvYru hi 2 4:(5) See also HEX NUMBER ,SQUARE NUMBER ,TRIANGULAR NUMBER References Gardner, M. "Hexes and Stars." Ch. 2 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 15 /C1/4, 1988. Hindin, H. "Stars, Hexes, Triangular Numbers, and Pytha- gorean Triples." J. Recr. Math. 16, 191/C1/93, 1983 /C1/984. Sloane, N. J. A. Sequences A003154/M4893, A006060/ M5425, and A006061/M5385 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Star of David HEXAGRAM Star of David Theorem As originally stated by Gould (1972), GCDn/C281 kYru$Yru% ;n k/C281Yru$Yru% ;n/C271 k/C271Yru$Yru% Yrt*Yrt+ /C30GCDn/C281 k/C281Yru$Yru% ;n k/C271Yru$Yru% ;n/C271 kYru$Yru% Yrt*Yrt+ ; (1) where GCD is the GREATEST COMMON DIVISOR andn kYrvYru is a BINOMIAL COEFFICIENT . This was subsequently extended by D. Singmaster to GCDn/C281 kYru$Yru% ;n k/C281Yru$Yru% ;n/C271 k/C271Yru$Yru% Yrt*Yrt+ /C30GCDn/C281 k/C281Yru$Yru% ;n k/C271Yru$Yru% ;n/C271 kYru$Yru% Yrt*Yrt+ /C30GCDn/C281 k/C282Yru$Yru% ;n/C281 k/C281Yru$Yru% ;n/C281 kYru$Yru% ;n/C281 k/C271Yru$Yru% Yrt*Yrt+ (2) (Sato 1975), and generalized by Sato (1975) to GCDYrt*n k/C272Yru$Yru% ;n/C281 kYru$Yru% ;n/C282 k/C282Yru$Yru% ;n k/C281Yru$Yru% ; n/C272 kYru$Yru% n/C271 k/C271Yru$Yru%Yrt+ /C30GCDYrt*n/C282 kYru$Yru% ;n/C281 k/C281Yru$Yru% ;n k/C282Yru$Yru% ;n/C271 kYru$Yru% ; n/C272 k/C272Yru$Yru% n k/C271Yru$Yru%Yrt+ (3) An even larger generalization was obtained by Hito- tumatu and Sato (1975), who definedMp/C30n/C28p/C271 k/C282p/C27j/C271Yru$Yru%Yrt* YrutYrutYrutYrutj/C301;2;...;3p/C282Yrt+ ; (p]1) (4) A p/C30n/C28p/C27j k/C27p/C281Yru$Yru%Yrt* YrutYrutYrutYrutj/C301;2;...;3p/C282Yrt+ (p]1) (5) R p/C30n/C28p/C27j k/C282p/C27j/C281Yru$Yru%Yrt* YrutYrutYrutYrutj/C301;2;...;3p/C282Yrt+ (p]1) (6) Dp/C30n/C28p/C272t/C271 k/C28p/C27t/C271Yru$Yru% ;n/C27p/C28t/C281 k/C27tYru$Yru% ;Yrt* n/C28t k/C27p/C282t/C281Yru$Yru% jt/C301;2;...;p/C281Yrt+ (p]2) (7) 9p/C30n/C28t k/C28p/C27t/C271Yru$Yru% ;n/C28p/C272t/C271 k/C27tYru$Yru% ;Yrt* n/C27p/C28t/C281 k/C27p/C282t/C281Yru$Yru% jt/C301;2;...;p/C281Yrt+ (p]2) (8) Up/C30@p r/C301Mr (9) Vp/C30@p r/C301Ar (10) Wp/C30@p r/C301Rr (11) Dp/C30@p r/C301Dr (12) Np/C30@p r/C3019r (13) Bp/C30Mp@Ap@Rp (14) Sp/C30@p r/C301Br (15) with D1/C3091/C30n kYru$Yru% : (16) and showed that each of the twelve BINOMIAL COEFFI- CIENTS Mp;Ap;Rp;Dp;9p;Up;Vp;Wp;9p;Np;Bp;and Sphas equal GREATEST COMMON DIVISOR . References Ando, S. and Sato, D. "Translatable and Rotatable Config- urations which Give Equal Product, Equal GCD and Equal LCM Properties Simultaneously." In Applications of Fibonacci Numbers, Vol. 3: Proceedings of the Third International Conference on Fibonacci Numbers and their Applications held at the University of Pisa, Pisa, July 25 /C1/ 9, 1988 (Ed. G. E. Bergum, A. N. Philippou and A. F. Horadam). Dordrecht, Netherlands: Kluwer,pp. 15 /C1 /6, 1990. Ando, S. and Sato, D. "A GCD Property on Pascal’s Pyramid and the Corresponding LCM Property of the ModifiedPascal Pyramid." In Applications of Fibonacci Numbers, Vol. 3: Proceedings of the Third International Conference on Fibonacci Numbers and their Applications held at the University of Pisa, Pisa, July 25 /C1/9, 1988 (Ed. G. E. Ber- gum, A. N. Philippou and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 7 /C1/4, 1990. Ando, S. and Sato, D. "On the Proof of GCD and LCM Equalities Concerning the Generalized Binomial and Multinomial Coefficients." In Applications of Fibonacci numbers, Vol. 4: Proceedings of the Fourth International Conference on Fibonacci Numbers and their Applications held at Wake Forest University, Winston-Salem, North Carolina, July 30-August 3, 1990 (Winston-Salem, NC, 1990) (Ed. G. E. Bergum, A. N. Philippou and A. F. Hor- adam). Dordrecht, Netherlands: Kluwer, 9 /C1/6, 1991. Ando, S. and Sato, D. "Multiple Color Version of the Star of David Theorems on Pascal’s Triangle and Related Arrays of Numbers." In Applications of Fibonacci Numbers, Vol. 6: Proceedings of the Sixth International Research Conference on Fibonacci Numbers and their Applications held at Washington State University, Pullman, Washing- ton, July 18 /C1/2, 1994 (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 31 /C1/5, 1996. Gould, H. W. Not. Amer. Math. Soc. 19, A-685, 1972. Hitotumatu, S. and Sato, D. "Expansion of the Star of David Theorem." Abstracts Amer. Math. Soc., p. A-377, 1975. Hitotumatu, S. and Sato, D. "Star of David Theorem. I." Fib. Quart. 13, 70, 1975. Sato, D. "Expansion of the Star of David Theorem of H. W. Gould and David Singmaster." Abstracts Amer. Math. Soc., p. A-377, 1975. Star of Goliath NONAGRAM Star of Lakshmi The STAR FIGURE f8=2g;which is used by Hindus to symbolize Ashtalakshmi, the eight forms of wealth. This symbol appears prominently in the Lugash national museum portrayed in the fictional filmReturn of the Pink Panther. See also D ISSECTION ,HEXAGRAM ,PENTAGRAM ,STAR FIGURE ,STAR POLYGON References Savio, D. Y. and Suryanaroyan, E. R. "Chebyshev Polyno- mials and Regular Polygons." Amer. Math. Monthly 100, 657/C1/61, 1993.Star Polygon A star polygon fp=qg;with p, q POSITIVE INTEGERS ,i s a figure formed by connecting with straight lines every qth point out of pregularly spaced points lying on a CIRCUMFERENCE . The number qis called the DENSITY of the star polygon. Without loss of general- ity, take qBp=2:The star polygons were first system- atically studied by Thomas Bradwardine. The usual definition (Coxeter 1969) requires pandq to be RELATIVELY PRIME . However, the star polygon can also be generalized to the STAR FIGURE (or "improper" star polygon) when pand qshare a common divisor (Savio and Suryanaroyan 1993). For such a figure, if all points are not connected after thefirst pass, i.e., if ( p;q)"1;then start with the first unconnected point and repeat the procedure. Repeatuntil all points are connected. For ( p;q)"1;the fp=qgsymbol can be factored as p q() /C30np? q?() ; (1) where p?/C30p n(2) q?/C30q n; (3) to give nfp0=q?gfigures, each rotated by 2 p=p radians, or 360/C14=p:/ Ifq/C301, a REGULAR POLYGON fpgis obtained. Special cases of fp=qginclude f5=2g(the PENTAGRAM ),f6=2g (the HEXAGRAM ,o r STAR OF DAVID),f8=2g(the STAR OFLAKSHMI ),f8=3g(the OCTAGRAM ),f10=3g(the DECAGRAM ), and f12=5g(the DODECAGRAM ). Superposing all distinct star polygons fp =qg for a given p gives beautiful patterns such as those illu- strated above. These figures can also be obtained by wrapping thread around p nails spaced equally around the circumference of a circle (Steinhaus 1983, pp. 259 /C1/60). See also DECAGRAM ,H EXAGRAM ,NONAGRAM ,OCTA- GRAM ,P ENTAGRAM ,R EGULAR POLYGON ,S TAR OF LAKSHMI ,STELLATED POLYHEDRON References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 93 /C1/4, 1973. Coxeter, H. S. M. "Star Polygons." §2.8 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 36 /C1/8, 1969. Fejes To´th, L. Regular Figures. Oxford, England: Pergamon Press, pp. 102 /C1/03, 1964. Frederickson, G. "Stardom." Ch. 16 in Dissections: Plane and Fancy. New York: Cambridge University Press, pp. 172 /C1/86, 1997. Savio, D. Y. and Suryanaroyan, E. R. "Chebyshev Polyno- mials and Regular Polygons." Amer. Math. Monthly 100, 657 /C1/61, 1993. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 211 and 259 /C1/60, 1999. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, p. 32, 1979. Star Polyhedron KEPLER- POINSOT SOLID Starr Rose See also MAURER ROSEReferences Wagon, S. "Variations of Circular Motion." §4.5 in Mathe- matica in Action. New York: W. H. Freeman, pp. 137 /C1/40, 1991. State Space The MEASURABLE SPACE (S?; S?) into which a RANDOM VARIABLE from a PROBABILITY SPACE is a measurable function. See also PROBABILITY SPACE ,RANDOM VARIABLE Stationary Point A point x0 at which the DERIVATIVE of a FUNCTION f(x) vanishes, f ?(x0) /C300: A stationary point may be a MINIMUM , MAXIMUM ,or INFLECTION POINT . See also CRITICAL POINT ,D ERIVATIVE ,E XTREMUM , FIRST DERIVATIVE TEST,INFLECTION POINT ,M AX- IMUM ,MINIMUM ,SECOND DERIVATIVE TEST Stationary Tangent INFLECTION POINT Stationary Value The value at a STATIONARY POINT . Statistic A quantity (such as a MEDIAN , QUARTILE DEVIATION , etc.), which is calculated from observed data. See also ANDERSON- DARLING STATISTIC , H-STATISTIC , K-STATISTIC ,KUIPER STATISTIC ,VARIATE References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 37, 1962. Statistical Distribution The distribution of a variable is a description of the relative numbers of times each possible outcome will occur in a number of trials. The function describingthe distribution is called the PROBABILITY FUNCTION , and the function describing the cumulative probabil- ity that a given value or any value smaller than it will occur is called the DISTRIBUTION FUNCTION . Formally, a distribution can be defined as a normal- ized MEASURE , and the distribution of a RANDOM VARIABLE xis the MEASURE PxonS?defined by setting Px(A?) /C30Ps/C23 S : x(s) /C23 A? fg ; where (S; S ; P)isa PROBABILITY SPACE ,(S; S)isa MEASURABLE SPACE , and P a MEASURE on S with P(S) /C301: If the MEASURE is a RADON MEASURE (which is usually the case), then the statistical distribution is a DISTRIBUTION in the sense of a generalized function. See also CONTINUOUS DISTRIBUTION ,DISCRETE DIS- TRIBUTION ,D ISTRIBUTION FUNCTION ,D ISTRIBUTION (GENERALIZED FUNCTION ), MEASURABLE SPACE ,MEA- SURE ,PROBABILITY ,PROBABILITY DENSITY FUNCTION , RANDOM VARIABLE ,STATISTICS References Doob, J. L. "The Development of Rigor in Mathematical Probability (1900 /C1/950)." Amer. Math. Monthly 103, 586 /C1/ 95, 1996. Evans, M.; Hastings, N.; and Peacock, B. Statistical Dis- tributions, 3rd ed. New York: Wiley, 2000. Statistical Index INDEX NUMBER Statistical Test A test used to determine the statistical SIGNIFICANCE of an observation. Two main types of error can occur: 1. A TYPE I ERROR occurs when a false negative result is obtained in terms of the NULL HYPOTHESIS by obtaining a false positive measurement. 2. A TYPE II ERROR occurs when a false positive result is obtained in terms of the NULL HYPOTHESIS by obtaining a false negative measurement. The probability that a statistical test will be positive for a true statistic is sometimes called the test’s SENSITIVITY , and the probability that a test will be negative for a negative statistic is sometimes called the SPECIFICITY . The following table summarizes the names given to the various combinations of the actual state of affairs and observed test results. result name true positive result SENSITIVITY false negative result 1-SENSITIVITY true negative result SPECIFICITY false positive result 1-SPECIFICITY Multiple-comparison corrections to statistical tests are used when several statistical tests are being performed simultaneously. For example, let’s suppose you were measuring leg length in eight different lizard species and wanted to see whether the MEANS of any pair were different. Now, there are 8!=2!6! /C3028pairwise comparisons possible, so even if all of the population means are equal, it’s quite likely that at least one pair of sample means would differ signifi- cantly at the 5% level. An ALPHA VALUE of 0.05 is therefore appropriate for each individual comparison, but not for the set of all comparisons. In order to avoid a lot of spurious positives, the ALPHA VALUE therefore needs to be lowered to account for the number of comparisons being performed. This is a correction for multiple comparisons. There are many different ways to do this. The simplest, and the most conservative, is the BONFERRONI CORRECTION .In practice, more people are more willing to accept false positives (false rejection of NULL HYPOTHESIS ) than false negatives (false acceptance of NULL HYPOTH- ESIS), so less conservative comparisons are usually used. See also ANOVA, BONFERRONI CORRECTION ,C HI- SQUARED TEST,FISHER’S EXACT TEST,FISHER SIGN TEST,KOLMOGOROV- SMIRNOV TEST,LIKELIHOOD RA- TIO,LOG LIKELIHOOD PROCEDURE , MANOVA, NEGA- TIVE LIKELIHOOD RATIO,PAIRED T-TEST,PARAMETRIC TEST,PREDICTIVE VALUE ,SENSITIVITY ,SIGNIFICANCE TEST,SPECIFICITY ,TYPE IE RROR ,TYPE II ERROR , WILCOXON RANK SUM TEST,WILCOXON SIGNED RANK TEST Statistics The mathematical study of the LIKELIHOOD and PROBABILITY of events occurring based on known information and inferred by taking a limited number of samples. Statistics plays an extremely important role in many aspects of economics and science, allowing educated guesses to be made with a mini- mum of expensive or difficult-to-obtain data.See also B OX-AND- WHISKER PLOT,BUFFON- LAPLACE NEEDLE PROBLEM ,B UFFON’S NEEDLE PROBLEM , CHERNOFF FACE,C OIN FLIPPING , DE MERE’S PRO- BLEM ,D ICE,G AMBLER’S RUIN,INDEX ,LIKELIHOOD , MOVING AVERAGE , P-VALUE ,POPULATION COMPAR- ISON,POWER (STATISTICS ), PROBABILITY ,R ESIDUAL VS. PREDICTOR PLOT,RUN,SHARING PROBLEM ,STA- TISTICAL DISTRIBUTION ,S TATISTICAL TEST,T AIL PROBABILITY References Brown, K. S. "Probability." http://www.seanet.com/ ~ksbrown/iprobabi.htm. Babu, G. and Feigelson, E. Astrostatistics. New York: Chapman & Hall, 1996. Bernstein, S. and Bernstein, R. Theory and Problems of Elements of Statistics I: Descriptive Statistics and Prob- ability. New York: McGraw-Hill, 1999. Dixon, W. J. and Massey, F. J. Introduction to Statistical Analysis, 4th ed. New York: McGraw-Hill, 1983. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. New York: Wiley, 1968. Fisher, N. I.; Lewis, T.; and Embleton, B. J. J. Statistical Analysis of Spherical Data. Cambridge, England: Cam- bridge University Press, 1987. Fisher, R. A. and Prance, G. T. The Design of Experiments, 9th ed. rev. New York: Hafner, 1974. Fisher, R. A. Statistical Methods for Research Workers, 14th ed., rev. and enl. Darien, CO: Hafner, 1970. Goldberg, S. Probability: An Introduction. New York: Dover, 1986. Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, 1993. Goulden, C. H. Methods of Statistical Analysis, 2nd ed. New York: Wiley, 1956. Hoel, P. G.; Port, S. C.; and Stone, C. J. Introduction to Statistical Theory. New York: Houghton Mifflin, 1971. Hogg, R. V. and Tanis, E. A. Probability and Statistical Inference, 5th ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. Keeping, E. S. Introduction to Statistical Inference. New York: Dover, 1995. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall’s Ad- vanced Theory of Statistics, Vol. 1: Distribution Theory, 6th ed. New York: Oxford University Press, 1987. Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall’s Ad- vanced Theory of Statistics, Vol. 2A: Classical Inferenceand Relationship, 6th ed. New York: Oxford University Press, 1987. Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall’s Ad- vanced Theory of Statistics, Vol. 2B: Bayesian Inference.New York: Oxford University Press, 1987. Keynes, J. M. A Treatise on Probability. London: Macmillan, 1921. Mises, R. von Mathematical Theory of Probability and Statistics. New York: Academic Press, 1964. Mises, R. von Probability, Statistics, and Truth, 2nd rev. English ed. New York: Dover, 1981. Mood, A. M. Introduction to the Theory of Statistics. New York: McGraw-Hill, 1950. Mosteller, F. Fifty Challenging Problems in Probability with Solutions. New York: Dover, 1987. Mosteller, F.; Rourke, R. E. K.; and Thomas, G. B. Prob- ability: A First Course, 2nd ed. Reading, MA: Addison- Wesley, 1970. Neyman, J. First Course in Probability and Statistics. New York: Holt, 1950. Ostle, B. Statistics in Research: Basic Concepts and Techni- ques for Research Workers, 4th ed. Ames, IA: Iowa State University Press, 1988. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Statistical Description of Data." Ch. 14 inNumerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 603 /C1 /49, 1992. Pugh, E. M. and Winslow, G. H. The Analysis of Physical Measurements. Reading, MA: Addison-Wesley, 1966. Re´nyi, A. Foundations of Probability. San Francisco, CA: Holden-Day, 1970. Robbins, H. and van Ryzin, J. Introduction to Statistics. Chicago, IL: Science Research Associates, 1975. Ross, S. M. A First Course in Probability, 5th ed. Englewood Cliffs, NJ: Prentice-Hall, 1997. Ross, S. M. Introduction to Probability and Statistics for Engineers and Scientists. New York: Wiley, 1987. Ross, S. M. Applied Probability Models with Optimization Applications. New York: Dover, 1992.Ross, S. M. Introduction to Probability Models, 6th ed. New York: Academic Press, 1997. Snedecor, G. W. Statistical Methods Applied to Experiments in Agriculture and Biology, 5th ed. Ames, IA: State College Press, 1956. Spiegel, M. R. and Stephens, L. J. Theory and Problems of Statistics, 3rd ed. New York: McGraw-Hill, 1998. Tippett, L. H. C. The Methods of Statistics: An Introduction Mainly for Experimentalists, 3rd rev. ed. London: Wil- liams and Norgate, 1941. Todhunter, I. A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace. New York: Chelsea, 1949. Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, 1977. Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, 1937. Weaver, W. Lady Luck: The Theory of Probability. New York: Dover, 1963. Weisstein, E. W. "Books about Statistics." http://www.trea- sure-troves.com/books/Statistics.html. Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, 1967. Young, H. D. Statistical Treatment of Experimental Data. New York: McGraw-Hill, 1962. Yule, G. U. and Kendall, M. G. An Introduction to the Theory of Statistics, 14th ed., rev. and enl. New York: Hafner, 1950. Staudt-Clausen Theorem VON STAUDT- CLAUSEN THEOREM Steenrod Algebra The Steenrod algebra has to do with the COHOMOLOGY operations in singular COHOMOLOGY with INTEGER mod 2 COEFFICIENTS . For every n/C23Zand i/C23 f0;1;2;3;...gthere are natural transformations ofFUNCTORS Sqi:Hn/C147;Z2 ðÞ 0Hn/C27i/C147;Z2 ðÞ satisfying: 1.Sqi/C300 for i/C21n. 2.Sqn(x)/C30x%xfor all x/C23HnX;A;Z2 ðÞ and all pairs ( X, A ). 3.Sq0/C30idHn/C147;Z2 ðÞ :/ 4. The Sqimaps commute with the coboundary maps in the long exact sequence of a pair. In other words, Sqi:H/C31/C147;Z2 ðÞ 0H/C31/C27i/C147;Z2 ðÞ is a degree i transformation of cohomology the- ories. 5. (CARTAN RELATION ) Sqi(x % y) /C30X j/C27k /C30iSqj(x) % Sqk(y): 6. (ADEM RELATIONS ) For i B2j; Sqi(Sqj(x) /C30Xibc k /C300j /C28k /C281 i /C282kYru$Yru% Sqi/C27j/C28k(Sqk(x) : 7. Sqi( a/C30a (Sqiwhere a is the cohomology suspension isomorphism. The existence of these cohomology operations endows the cohomology ring with the structure of a MODULE over the Steenrod algebra A; defined to be TFZ2Sqi : i /C23f0; 1; 2; 3; ...g fgYru*Yru+ =R; where FZ2/C147ðÞis the free module functor that takes any set and sends it to the free Z2module over that set. We think of FZ2Sqi : i /C23f0; 1; 2; ...g fg as being a graded Z2 mod- ule, where the i-th gradation is given by Z2/C215 Sqi : This makes the tensor algebra TFZ2Sqi : i /C23f0; 1; 2; 3; ...g fgYru*Yru+ into a GRADED AL- GEBRA over Z2 : R is the IDEAL generated by the elements SqiSqj /C27a ibc k /C300j/C28k/C281 i/C282kYrvYru Sqi /C27j/C28kSqk and 1 /C27Sq0 for 0 Bi B2j: This makes A into a graded Z2 algebra. By the definition of the Steenrod algebra, for any SPACE (X, A), H /C31 X ; A; Z2 ðÞ is a MODULE over the Steenrod algebra A; with multiplication induced by Sqi /C215 x /C13Sqi(x): With the above definitions, cohomol- ogy with COEFFICIENTS in the RING Z2 ; H /C31/C147; Z2 ðÞ is a FUNCTOR from the category of pairs of TOPOLOGICAL SPACES to graded modules over A:/ See also ADEM RELATIONS ,CARTAN RELATION ,COHO- MOLOGY ,GRADED ALGEBRA ,IDEAL ,M ODULE ,TOPO- LOGICAL SPACE Steenrod-Eilenberg Axioms EILENBERG- STEENROD AXIOMS Steenrod’s Realization Problem When can homology classes be realized as the image of fundamental classes of MANIFOLDS ? The answer is known, and singular BORDISM GROUPS provide insight into this problem. See also BORDISM GROUP ,MANIFOLD Steepest Descent Method An ALGORITHM for finding the nearest LOCAL MINI- MUM of a function which presupposes that the GRADIENT of the function can be computed. The steepest descent method, also called the gradient descent method, starts at a point P0and, as many times as needed, moves from Pi to Pi/C271 by minimizing along the line extending from Piin the direction of /C289f PiðÞ ; the local downhill GRADIENT .This method has the severe drawback of requiring a great many iterations for functions which have long, narrow valley structures. In such cases, a CONJUGATE GRADIENT METHOD is preferable. See also CONJUGATE GRADIENT METHOD ,GRADIENT , LOCAL MINIMUM ,MINIMUM References Arfken, G. "The Method of Steepest Descents." §7.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 428 /C1/36, 1985. Menzel, D. (Ed.). Fundamental Formulas of Physics, Vol. 2, 2nd ed. New York: Dover, p. 80, 1960. Morse, P. M. and Feshbach, H. "Asymptotic Series; Method of Steepest Descent." §4.6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 434 /C1/43, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 414, 1992. Steffensen Sequence A sequence s(l) n(x) /C30[h(t)] lsn(x); where sn(x)isaS HEFFER SEQUENCE , h(t) is invertible, and l ranges over the real numbers. If sn(x)isan associated SHEFFER SEQUENCE , then s(l) nis called a CROSS SEQUENCE .Ifsn(x) /C30xn ; then s( l) n(x) /C30[h(t)]lxn is called an APPELL CROSS SEQUENCE . An example is the LAGUERRE POLYNOMIAL . See also APPELL CROSS SEQUENCE ,CROSS SEQUENCE , SHEFFER SEQUENCE References Brown, J. W. "A Note on Generalized Appell Polynomials." Amer. Math. Monthly 75, 1968. Roman, S. "Cross Sequences and Steffensen Sequences." §5.3 inThe Umbral Calculus. New York: Academic Press, pp. 140 /C1/43, 1984. Rota, G.-C.; Kahaner, D.; and Odlyzko, A. "On the Founda- tions of Combinatorial Theory VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684/C1/60, 1973. Steffensen’s Inequality Letf(x)b ea NONNEGATIVE and monotonic decreasing function in [ a, b] and g(x) such that 0 5g(x)51i n[ a, b], then gb b/C28kf(x)dx5gb af(x)g(x)dx5ga/C27k af(x)dx: where k /C30gb ag(x) dx: References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000. Steffenson’s Formula fp /C30f0 /C271 2 p(p /C271)d1 =2 /C2812(p /C281)p d/C281=2 /C27 S3 /C27S4 ðÞ d3 1 =2 /C27 S3 /C28S4 ðÞ d3/C281 =2 /C27...; (1) for p /C23/C281 2 ;12hi ; where d is the CENTRAL DIFFERENCE and S2n/C271 /C301 2p /C27n 2n /C271Yru$Yru% (2) S2n/C272 /C30p 2n /C27 2p /C27n 2n /C271Yru$Yru% (3) S2n/C271 /C28S2n/C272 /C30p /C27n /C271 2n /C272Yru$Yru% (4) S2n/C271/C28S2n/C272/C30/C28p/C27n 2n/C272Yru$Yru% ; (5) wheren kYrvYru is a BINOMIAL COEFFICIENT . See also CENTRAL DIFFERENCE ,STIRLING’S FINITE DIFFERENCE FORMULA References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Steinbach Screw ASURFACE generated by the PARAMETRIC EQUATIONS x(u;v)/C30ucosv (1) y(u;v)/C30usinv (2) z(u;v)/C30vcosu: (3) The above image uses u/C23[/C284;4] and v/C23[0;6;25]:/The coefficients of the FIRST FUNDAMENTAL FORM are E/C301/C27v2sin2u (4) F/C30/C28vcosusinu (5) G/C301 21/C272u2/C27cos(2 u)YrtYrP ; (6) the coefficients of the SECOND FUNDAMENTAL FORM are e/C30ffiffiffi 2p uvcosuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27u22/C27v2 ðÞ /C271/C28u2v2 ðÞ cos(2 u)p (7) f/C30ffiffiffi2p (cosu/C27usinu)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27u 22/C27v2 ðÞ /C271/C28u2v2 ðÞ cos(2 u)p (8) g/C30ffiffiffi 2p u2vsinu)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u22/C27v2 ðÞ /C271/C28u2v2 ðÞ cos(2 u)p ; (9) the AREA ELEMENT is dA/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27u22/C27v2 ðÞ /C271/C28u2v2 ðÞ cos(2 u) 2s duffldv; (10) and the G AUSSIAN and MEAN CURVATURES are given by K/C304uu2v2/C282 ðÞ cosusinu/C28u2sin2u/C28cos2uYrtYrP 1/C27u22/C27v2 ðÞ /C271/C28u2v2 ðÞ cos(2 u) ½/C1382 (11) H/C30/C28vu5/C274u2ðÞ cosu/C28ucos(3 u) fg 2ffiffiffi 2p 1/C27u22/C27v2 ðÞ /C271/C28u2v2 ðÞ cos(2 u) ½/C1383=2 /C28v22/C27u22/C27v2/C272/C28u2v2ðÞ cos(2 u) ðÞ ½/C138 sinu fg 2ffiffiffi2p 1/C27u22/C27v2 ðÞ /C271/C28u2v2 ðÞ cos(2 u) ½/C1383=2: (12) References Pickover, C. A. Mazes for the Mind: Computers and the Unexpected. New York: St. Martin’s Press, 1992. Steiner Chain Given two nonconcentric CIRCLES with one interior to the other, if small TANGENT CIRCLES can be inscribed around the region between the two CIRCLES such that the final CIRCLE isTANGENT to the first, the CIRCLES form a Steiner chain. The simplest way to construct a Steiner chain is to perform an INVERSION on a symmetrical arrangement on n circles packed between a central circle of radius b and an outer concentric circle of radius a (Wells 1991). In this arrangement, sinp n ! /C30a /C28 b a /C27 b ; (1) so the ratio of the radii for the small and large circles is b a /C301 /C28 sinp nYru*Yru+ 1 /C27 sinp nYru*Yru+ : (2) In addition, the radii of the circles in the ring are c /C301 2(a /C28b) ; (3) and their centers are located at a distance r /C30b /C27c /C301 2(a /C27b) (4) from the origin. To transform the symmetrical arrangement into a Steiner chain, take an INVERSION CENTER which is a distance d from the center of the symmetrical figure. Then the radii a? and b ? of the outer and center circles become a ?/C30a d2 /C28 a2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30 a a2 /C28 d2 (5) b?/C30b d2 /C28 b2YrutYrutYrutYrutYrutYrutYrutYrutYrutYrut/C30 b b2 /C28 d2 ; (6) respectively. Equivalently, a Steiner chain results whenever the INVERSIVE DISTANCE between the two original circles is given by d /C302 ln secp n ! /C27tanp n ! "# (7) /C302 ln tanp 4 /C27p 2n !"# (8) (Coxeter and Greitzer 1967). The centers of the circles in a Steiner chain lie on an ELLIPSE (Ogilvy 1990, p. 57). The lines of tangency passing through the contact points of neighboring circles in the chain are concurrent in a point. Furthermore, this is the same point at which the lines through the contact points of the inner and outer circles also concur (Wells 1991, p. 245). STEINER’S PORISM states that if a Steiner chain is formed from one starting circle, then a Steiner chain is also formed from any other starting circle. A Steiner chain may also close after several loops around the central circle, in which case a Steiner chain will also be formed after the same number ofloops from any starting point. See also A RBELOS ,COXETER’S LOXODROMIC SEQUENCE OF TANGENT CIRCLES ,HEXLET ,PAPPUS CHAIN ,SEVEN CIRCLES THEOREM ,STEINER’S PORISM References Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta Math. 18, 113/C1/21, 1952. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 87, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 124 /C1/26, 1967. Forder, H. G. Geometry, 2nd ed. London: Hutchinson’s University Library, p. 23, 1960. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,1 8/C1/8, Jan. 1979. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 113 /C1/15, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 51 /C1/4, 1990. Weisstein, E. W. "Plane Geometry." M ATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 120 and 244 /C1/45, 1991. Steiner Construction A construction done using only a STRAIGHTEDGE . The PONCELET- STEINER THEOREM proves that all con- structions possible using a COMPASS and STRAIGHT- EDGE are possible using a STRAIGHTEDGE alone, as long as a fixed CIRCLE and its center, two intersecting CIRCLES without their centers, or three nonintersect- ing CIRCLES are drawn beforehand. For example, the centers of two intersecting circles can be found using a STRAIGHTEDGE alone (Steinhaus 1983, p. 42). See also GEOMETRIC CONSTRUCTION ,M ASCHERONI CONSTRUCTION ,MATCHSTICK CONSTRUCTION ,NEUSIS CONSTRUCTION ,P ONCELET- STEINER THEOREM , STRAIGHTEDGE References Do¨rrie, H. "Steiner’s Straight-Edge Problem." §34 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 165 /C1/70, 1965. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, p. 204, 1957. Steiner, J. Geometric Constructions with a Ruler, Given a Fixed Circle with Its Center. Translated from the first German ed. (1833). New York: Scripta Mathematica, 1950. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 142, 1999. Steiner Points There are two different types of points known as Steiner points. The point S of CONCURRENCE of the three lines drawn through the VERTICES of a TRIANGLE PARALLEL to the corresponding sides of the first BROCARD TRIANGLE is called the Steiner point (Honsberger 1995). It lies on the CIRCUMCIRCLE opposite the TARRY POINT T and has TRIANGLE CENTER FUNCTION a /C30bc a2 /C28b2YrvYru a2 /C28c2YrvYru : The BRIANCHON POINT for KIEPERT’S PARABOLA is also called the Steiner point. The SYMMEDIAN POINT K is the Steiner point of the first BROCARD TRIANGLE (Honsberger 1995, pp. 120 /C1/21). The SIMSON LINE of the Steiner point is PARALLEL to the line OK, when O is the CIRCUMCENTER and K is the SYMMEDIAN POINT (Honsberger 1995, p. 121). The Steiner point of a TRIANGLE is the CENTROID of the system obtained by placing a mass equal to the magnitude of the exterior angle at each vertex (Honsberger 1995, p. 120). If triplets of opposites sides on a CONIC SECTION in PASCAL’S THEOREM are extended for all permutations of VERTICES ,60P ASCAL LINES are produced. The 20 points of their three by three intersections are called Steiner points. STEINER’S THEOREM states that these points are generated by the hexagons 123456, 143652, and 163254 formed by interchanging the vertices at positions 2, 4, and 6 (where the numbers denote the order in which the vertices of the hexagon are taken). The configuration of PASCAL LINES for a general hexagon inscribed in a general ellipse are shown above, with Steiner points shown as filled circles. A blow-up of the region in the upper left figure is shown below, illustrating the concurrence of three Pascal lines at each Steiner point. Each Steiner point lies together with three K IRKMAN POINTS on a total of 20 lines known as C AYLEY LINES . The Steiner points also lie four at a time on 15 PLU¨CKER LINES (Wells 1991). There is a dual relation- ship between the 20 Steiner points and the 20 C AYLEY LINES . See also BRIANCHON POINT ,B ROCARD TRIANGLES , CAYLEY LINES,CIRCUMCIRCLE ,CONIC SECTION ,KIE- PERT’S PARABOLA ,K IRKMAN POINTS ,S YMMEDIAN POINT ,PASCAL LINES,PASCAL’S THEOREM ,PLU¨ CKER LINES,S ALMON POINTS ,S TEINER SET,S TEINER’S THEOREM ,STEINER TRIPLE SYSTEM ,TARRY POINT References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 66 and 329, 1893. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 77, 1971. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 102, 1913. Honsberger, R. "The Steiner Point and the Tarry Point." §10.5 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 119 /C1/24, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 236 /C1/37, 281 /C1/82, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 115, 1893. Salmon, G. "Notes: Pascal’s Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 379 /C1/82, 1960. Steiner. Gergonne Ann. Math. 18. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 172, 1991. Steiner Quadruple System A Steiner quadruple system is a STEINER SYSTEM S(t /C303; k /C304; v) ; where S is a v-set and B is a collection of k-sets of S such that every t-subset of S is contained in exactly one member of B. Barrau (1908) established the uniqueness of S(3; 4; 8); 1248 23583468 4578 156826781378356714671257 1236 234713452456 and /S(3; 4; 10) 12452356 3467 457856896790 1780 12892390134012372348 3459 456015672678 3789 48901590126013582469 3570 146825793680 1479 258013692470 Fitting (1915) subsequently constructed the cyclic systems S(3; 4 ; 26) and S(3; 4; 34); and Bays and de Weck (1935) showed the existence of at least one S(3; 4; 14): Hanani (1960) proved that a NECESSARYand SUFFICIENT condition for the existence of an S(3; 4; v) is that v /C132 or 4 (mod 6). The number of nonisomorphic steiner quadruple systems of orders 8, 10, 14, and 16 are 1, 1, 4 (Mendelsohn and Hung 1972), and at least 31,021 (Lindner and Rosa 1976). See also STEINER SYSTEM ,STEINER TRIPLE SYSTEM References Barrau, J. A. "On the Combinatory Problem of Steiner." K. Akad. Wet. Amsterdam Proc. Sect. Sci. 11, 352 /C1/60, 1908. Bays, S. and de Weck, E. "Sur les syste`mes de quadruples." Comment. Math. Helv. 7, 222 /C1/41, 1935. Fitting, F. "Zyklische Lo¨sungen des Steiner’schen Pro- blems." Nieuw. Arch. Wisk. 11, 140 /C1/48, 1915. Hanani, M. "On Quadruple Systems." Canad. J. Math. 12, 145 /C1/57, 1960. Lindner, C. L. and Rosa, A. "There are at Least 31,021 Nonisomorphic Steiner Quadruple Systems of Order 16." Utilitas Math. 10,61/C1/4, 1976. Lindner, C. L. and Rosa, A. "Steiner Quadruple Systems--A Survey." Disc. Math. 22, 147 /C1/81, 1978. Mendelsohn, N. S. and Hung, S. H. Y. "On the Steiner Systems S(3;4;14) and S(4;5;15) /."Utilitas Math. 1, 5/C1/5, 1972. Steiner Set Three sets of three LINES such that each line is incident with two from both other sets. See also SOLOMON’S SEAL LINES,STEINER POINTS , STEINER TRIPLE SYSTEM Steiner Surface A projection of the V ERONESE SURFACE into 3-D (which must contain singularities) is called a Steiner surface. A classification of Steiner surfaces allowing complex parameters and projective transformationswas accomplished in the 19th century. The surfacesobtained by restricting to real parameters and trans- formations were classified into 10 types by Coffman et al.(1996). Examples of Steiner surfaces include the R OMAN SURFACE (Coffman type 1) and CROSS-CAP (type 3). The Steiner surface of type 2 is given by the implicit equation x2y2/C28x2z2/C27y2z2/C28xyz/C300: and can be transformed into the R OMAN SURFACE or CROSS-CAP by a complex projective change of coordi- nates (but not by a real transformation). It has twopinch points and three double lines and, unlike the R OMAN SURFACE orCROSS-CAP , is not compact in any affine neighborhood. The Steiner surface of type 4 has the implicit equation y2/C282xy2/C28xz2/C27x2y2/C27x2z2/C28z4/C300: and two of the three double lines of surface 2 coincide along a line where the two noncompact "components" are tangent. See also CROSS- CAP,R OMAN SURFACE ,V ERONESE VARIETY References Ape´ry, F. Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces. Braunschweig, Germany: Vieweg, 1987. Coffman, A. "Steiner Surfaces." http://www.ipfw.edu/math/ Coffman/steinersurface.html. Coffman, A.; Schwartz, A.; and Stanton, C. "The Algebra and Geometry of Steiner and Other Quadratically Parametriz- able Surfaces." Computer Aided Geom. Design 13, 257 /C1/86, 1996. Nordstrand, T. "Steiner Relative." http://www.uib.no/people/ nfytn/stmtxt.htm. Nordstrand, T. "Steiner Relative [2]." http://www.uib.no/ people/nfytn/stm2txt.htm. Steiner System A Steiner system S(t; k; v) is a set X of v points, and a collection of subsets of X of size k (called blocks), such that any t points of X are in exactly one of the blocks. The special case t /C302 and k /C303 corresponds to a so-called STEINER TRIPLE SYSTEM . For a PROJECTIVE PLANE , v /C30n2 /C27n /C271; k /C30n /C271; t /C302, and the blocks are simply lines. The number r of blocks containing a point in a S(t; k; v) Steiner system is independent of the point. In fact, r /C30v /C28 1 t /C28 1Yru$Yru% k /C28 1 t /C28 1Yru$Yru% ; wheren kYrvYru is a BINOMIAL COEFFICIENT . The total number of blocks b is also determined and is given by b /C30vr k: These numbers also satisfy v 5b and k 5r :/ The PERMUTATIONS of the points preserving the blocks of a Steiner system S is the AUTOMORPHISM GROUP of S. For example, consider V the set of 9 points in the 2-dimensional VECTOR SPACE over theFIELD over 3 elements. The blocks are the 12 lines of the form fa /C27tbg/C30fa; a /C27b; a /C272bg; which have three elements each. The system is a S(2; 3; 9) because any two points uniquely determine a line. The AUTOMORPHISM GROUP of a Steiner system is the AFFINE GROUP which preserves the lines. For a vector space of dimension n over a field of q elements, this construction gives a Steiner system S 2; q; qdYrvYru :/ Several interesting groups arise as automorphism groups of Steiner systems. For example, the MATHIEU GROUPS are the AUTOMORPHISM GROUPS of Steiner systems, as summarized in the following table. These groups are unique up to ISOMORPHISM , and are not only SPORADIC SIMPLE GROUPS , but are also highly TRANSITIVE . Mathieu group Steiner system /M11// S(3; 4 ; 14) / /M12// S(5; 6 ; 12) / /M22// S(3; 6 ; 22) / /M23// S(4; 7 ; 23) / /M24// S(5; 8 ; 24) / See also AUTOMORPHISM GROUP ,C ONFIGURATION , MATHIEU GROUPS ,SIMPLE GROUP ,STEINER QUAD- RUPLE SYSTEM ,STEINER TRIPLE SYSTEM , T-DESIGN , TRANSITIVE GROUP ,W ITT GEOMETRY References Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, 1996. Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Gropp, H. "Enumeration of Regular Graphs 100 Years Ago." Discrete Math. 101,73/C1/5, 1992. Woolhouse, W. S. B. "Prize Question 1733." Lady’s and Gentleman’s Diary. 1844. Steiner Tree The Steiner tree of some subset of the vertices of a GRAPH G is a minimum-weight connected SUBGRAPH of G that includes all the vertices. It is always a tree. Steiner trees have practical applications, for example, in the determination of the shortest total length of wires needed to join some number of points (Hoffman 1998, pp. 164 /C1/65). See also PLATEAU’S PROBLEM ,TREE References Chopra, S. and Rao, M. R. "The Steiner Tree Problem 1: Formulations, Compositions, and Extension of Facets." Mathematical Programming 64, 209 /C1/29, 1994. Chopra, S. and Rao, M. R. "The Steiner Tree Problem 2: Properties and Classes of Facets." Mathematical Program- ming 64, 231 /C1/46, 1994. Chung, F. R. K.; Gardner, M.; and Graham, R. L. "Steiner Trees on a Checkerboard." Math. Mag. 62,83/C1/6, 1989. Du, D.-Z.; Smith, J. M.; and Rubinstein, J. H. Advances in Steiner Trees. Dordrecht, Netherlands: Kluwer, 2000. Ganley, J. "The Steiner Tree Page." http://ganley.org/stei- ner/. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Hwang, F.; Richards, D.; and Winter, P. The Steiner Tree Problem. Amsterdam, Netherlands: North-Holland, 1992. Ivanov, A. O. and Tuzhilin, A. A. Minimal Networks: The Steiner Problem and Its Generalizations. Boca Raton, FL: CRC Press, 1994. Skiena, S. S. "Steiner Tree." §8.5.10 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 339 /C1/42, 1997. Steiner Triple System Let X be a set of v ]3 elements together with a set B of 3-subset (triples) of X such that every 2-SUBSET of X occurs in exactly one triple of B. Then B is called a Steiner triple system and is a special case of a STEINER SYSTEM with t /C302 and k /C303. A Steiner triple system S(v) /C30S(v ; k /C303 ; l /C301) of order v exists IFF v /C131; 3(mod 6) (Kirkman 1847). In addition, if Stei- ner triple systems S1 and S2 of orders v1 and v2 exist, then so does a Steiner triple system S of order v1v2 (Ryser 1963, p. 101). Examples of Steiner triple systems S(v) of small orders v are S3 /C30ff1 ; 2 ; 3gg S7 /C30ff1; 2; 4g;f2 ; 3; 5g;f3; 4; 6g;f4; 5; 7g: f5; 6; 1g;f6; 7 ; 2 g;f7; 1; 3gg S9 /C30ff1; 2 ; 3 g;f4; 5; 6g;f7; 8 ; 9 g;f1; 4; 7g; f2; 5 ; 8 g;f3; 6; 9g;f1 ; 5 ; 9 gf2; 6; 7g; f3 ; 4 ; 8 g;f1; 6; 8g;f2 ; 4 ; 9g;f3; 5; 7gg: The number of nonisomorphic Steiner triple systems S(v) of orders v /C307, 9, 13, 15, 19, ... (i.e., 6k /C271:3) are 1, 1, 2, 80, > 1:1 /C29109 ; ... (Colbourn and Dinitz 1996, pp. 14 /C1/5; Sloane’s A030129). S(7) is the same as the finite PROJECTIVE PLANE of order 2. S(9) is a finite AFFINE PLANE which can be constructed from the array abc def ghi: One of the two S(13) /s is a finite HYPERBOLIC PLANE . The 80 Steiner triple systems S(15) have been studied by Tonchev and Weishaar (1997). There are morethan 1 :1 /C29109Steiner triple systems of order 19 (Stinson and Ferch 1985; Colbourn and Dinitz 1996, p. 15). See also HADAMARD MATRIX ,KIRKMAN TRIPLE SYS- TEM,STEINER QUADRUPLE SYSTEM ,STEINER SYSTEM References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 107 /C1/09 and 274, 1987. Colbourn, C. J. and Dinitz, J. H. (Eds.). "Steiner Triple Systems." §4.5 in CRC Handbook of Combinatorial De- signs. Boca Raton, FL: CRC Press, pp. 14 /C1/5 and 70, 1996. Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102 /C1/07, May 1975. Kirkman, T. P. "On a Problem in Combinatorics." Cam- bridge Dublin Math. J. 2, 191 /C1/04, 1847. Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 99 /C1/02, 1963. Sloane, N. J. A. Sequences A030129 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Stinson, D. R. and Ferch, H. "2000000 Steiner Triple Systems of Order 19." Math. Comput. 44, 533 /C1/35, 1985. Tonchev, V. D. and Weishaar, R. S. "Steiner Triple Systems of Order 15 and Their Codes." J. Stat. Plan. Inference 58, 207 /C1/16, 1997. Steinerian Curve The LOCUS of points whose first POLARS with regard to the curves of a linear net have a common point. It is also the LOCUS of points of CONCURRENCE of line POLARS of points of the JACOBIAN CURVE . It passes through all points common to all curves of the system and is of order /3(n /C281)2 /. See also CAYLEYIAN CURVE ,JACOBIAN CURVE References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 150, 1959. Steiner-Lehmus Theorem Any TRIANGLE that has two equal ANGLE BISECTORS (each measured from a VERTEX to the opposite sides) is an ISOSCELES TRIANGLE . This theorem is also called the "internal bisectors problem" and "Lehmus’ theo- rem." See also ISOSCELES TRIANGLE References Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 72 /C1/3, 1952. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 9, 1969. Coxeter, H. S. M. and Greitzer, S. L. "The Steiner-Lehmus Theorem." §1.5 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 14 /C1/6, 1967. Gardner, M. Martin Gardner’s New Mathematical Diver- sions from Scientific American. New York: Simon and Schuster, pp. 198 /C1/99 and 206 /C1/07, 1966. Henderson, A. "The Lehmus-Steiner-Terquem Problem in Global Survey." Scripta Math. 21, 223 /C1/32 and 309 /C1/12, 1955. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 72 /C1/3, 1975. Neuberg, J. Bibliographie du triangle et du te´trae`dre. p. 337, 1923. The´bault, V. "Sur le triangle isosce `le." Mathesis 44, 97, 1930. Steiner’s Ellipse Let a? : b? : g ? be the ISOTOMIC CONJUGATE POINT of a point with TRILINEAR COORDINATES a : b : g : The iso- tomic conjugate of the LINE AT INFINITY having trilinear equation a a /C27bb /C27c g /C300 is b? g? a/C27g ? a? b/C27a? b? c/C300 : known as Steiner’s ellipse (Vandeghen 1965). See also ISOTOMIC CONJUGATE POINT ,L INE AT INFINITY References Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091 /C1/094, 1965. Steiner’s Hypocycloid DELTOID Steiner’s Porism If a STEINER CHAIN is formed from one starting circle, then a STEINER CHAIN is formed from any other starting circle. In other words, given two noncon- centric CIRCLES , draw CIRCLES successively touching them and each other. If the last touches the first, this will also happen for any position of the first CIRCLE . See also HEXLET ,SEVEN CIRCLES THEOREM ,STEINER CHAINReferences Allanson, B. "Steiner’s Porism" java applet. http://www.ade- laide.net.au/~allanson/steiner.html. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 34, 1971. Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta Math. 18, 113 /C1/21, 1952. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 87, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 124 /C1/26, 1967. Forder, H. G. Geometry, 2nd ed. London: Hutchinson’s University Library, p. 23, 1960. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,18/C1/8, Jan. 1979. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 113 /C1/15, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 53 /C1/4, 1990. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 120 and 244 /C1/45, 1991. Steiner’s Problem For what value of x is f(x) /C30x1 =x a MAXIMUM ? The maximum occurs at x /C30e, where f ?(x) /C30x/C282 /C271 =x(1 /C28ln x) /C300: which is zero at x /C30e and gives a maximum of e1 =e /C301:444667861... : The function has an inflection point at x /C30 0:581933... ; where fƒ(x)/C30x/C284/C271=x[1/C283x/C27(lnx)(2x/C282/C27lnx)]/C300: See also FERMAT’S PROBLEM ,POWER TOWER References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 35, 1986. Steiner’s Segment Problem Given n points, find the line segments with the shortest possible total length which connect the points. The segments need not necessarily be straight from one point to another. For three points, if all ANGLES are less than 120 8, then the line segments are those connecting the three points to a central point P which makes the ANGLES AhiPB; BhiPC; and ChiPA all 1208. If one ANGLE is greater that 1208, then P coincides with the offending ANGLE . For four points, P is the intersection of the two diagonals, but the required minimum segments are not necessarily these diagonals. A modified version of the problem is, given two points, to find the segments with the shortest total length connecting the points such that each branch point may be connected to only three segments. There is no general solution to this version of the problem. Steiner’s Theorem The most common statement known as Steiner’s theorem (Casey 1893, p. 329) states that the PASCAL LINES of the HEXAGONS 123456, 143652, and 163254 formed by interchanging the vertices at positions 2, 4, and 6 are concurrent (where the numbers denote the order in which the vertices of the hexagon are taken). The 20 points of concurrence so generated are known as STEINER POINTS . Another theorem due to Steiner lets LINES x and y join a variable point on a CONIC SECTION to two fixed points on the same CONIC SECTION . Then x and y are PROJECTIVELY related. A third "Steiner’s theorem" states that if two opposite edges of a TETRAHEDRON move on two fixed SKEW LINES in any way whatsoever but remain fixed in length, then the volume of the TETRAHEDRON remains constant (Altshiller-Court 1979, p. 87). See also CONIC SECTION ,PROJECTION ,TETRAHEDRON References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 329, 1893. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 252 /C1/53, 1930. Steinhaus Dissection CUBE DISSECTIONSteinhaus Property References Kanemitsu, S. and Gyory, K. (Eds.). "A Problem of Steinhaus Concerning the Existence of a Plane Set with a Certain Property." In Number Theory and Its Applications. Dor- drecht, Netherlands: Kluwer, pp. 1 /C1/, 1999. Steinhaus-Moser Notation A NOTATION for LARGE NUMBERS defined by Steinhaus (1983, pp. 28 /C1/9). In this notation, denotes nn ; denotes "n in n TRIANGLES ," and denotes "n in n SQUARES ." A modified version due to Moser eliminates the circle notation, continuing instead with POLYGONS of ever increasing size, so n in a PENTAGON is n with n SQUARES around it, etc. See also CIRCLE NOTATION ,LARGE NUMBER ,M EGA, MOSER References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Steinitz’s Lemma If, in a plane or spherical convex polygon ABCDEFG , all of whose sides AB, BC, CD, ..., FG (with the exception of AG) have fixed lengths, one simulta- neously increases (decreases) the angles between these sides, then the length of the variable side increases (decreases). References Cromwell, P. R. "Steinitz’ Lemma." In Polyhedra. New York: Cambridge University Press, pp. 235 /C1/37, 1997. Steinitz’s Theorem A GRAPH G is the edge graph of a POLYHEDRON IFF G is a SIMPLE PLANAR GRAPH which is 3-connected. See also CONNECTED GRAPH ,PLANAR GRAPH ,SIMPLE GRAPH Steinmetz Solid The solid common to two (or three) right circular CYLINDERS of equal RADII intersecting at RIGHT ANGLES is called the Steinmetz solid. Two CYLINDERS intersecting at RIGHT ANGLES are called a bicylinder, and three intersecting CYLINDERS aTRICYLINDER . Half of a bicylinder is called a VAULT . For two cylinders of radius roriented long the z- and x-axes gives the equations x2/C27y2/C30r2(1) y2/C27z2/C30r2(2) which can be solved for xandygives the PARAMETRIC EQUATIONS of the edges of the solid, x/C309z (3) y/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p : (4) The SURFACE AREA can be found as fxd s ;where ds/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27dy dz !2vuutdz/C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p : (5) Taking the range of integration as a quarter or one face and then multiplying by 16 gives S2/C3016gr 0r2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p dz/C3016r2: (6) The VOLUME common to two cylinders is was known to Archimedes (Heath 1953, Gardner 1962) and the Chinese mathematician Tsu Ch’ung-Chih (Kiang1972), and does not require CALCULUS to derive. Using calculus provides a simple derivation, however. Noting that the solid has a square CROSS SECTION of side-half-lengthffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28z2p ;the volume is given by V2(r;r)/C30gr /C28r2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2pYru*Yru+2 dz/C3016 3r3(7) (Moore 1974). The VOLUME can also be found using CYLINDRICAL ALGEBRAIC DECOMPOSITION , which re- duces the inequalities x2/C27y2B1 /C28LBzBL y2/C27z2B1 /C28LBxBL8 >>< >>:(8) to /C281BxB1 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2p ByBffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28x2p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28y2p BzBffiffiffiffiffiffiffiffiffiffiffiffiffi1/C28y2p ;8 < :(9)giving the integral V2(1;1)/C30g1 /C281gffiffiffiffiffiffiffiffi 1/C28x2p /C28ffiffiffiffiffiffiffiffi 1/C28x2pgffiffiffiffiffiffiffiffi 1/C28y2p /C28ffiffiffiffiffiffiffiffi 1/C28y2p dx dy dz /C3016 3:(10) If the two right CYLINDERS are of different RADII a andbwith a/C21b, then the VOLUME common to them is V2(a;b)/C308 3aa2/C27b2YrvYru E(k)/C28a2/C28b2YrvYru K(k)YrtYrP ;(11) where K(k) is the complete ELLIPTIC INTEGRAL OF THE FIRST KIND ,E(k) is the complete ELLIPTIC INTEGRAL OF THE SECOND KIND , and k/C13b=ais the MODULUS . The curves of intersection of two cylinders of RADII a and b, shown above, are given by the parametric equations x(t)/C30acost (12) g(t)/C30asint (13) z(t)/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2/C28a2sin2tp (14) (Gray 1997). The VOLUME common to two ELLIPTIC CYLINDERS x2 a2/C27z2 c2/C301y2 b2/C27z2 c?2/C301 (15) with cBc?is V2(a;c;b;c?) /C308ab 3cc?2/C27c2YrvYru E(k)/C28c?2/C28c2YrvYru K(k)YrtYrP : (16) where k/C30c=c?(Bowman 1961, p. 34). For three CYLINDERS of RADII r intersecting at RIGHT ANGLES , The resulting solid has 12 curved faces. If tangent planes are drawn where the faces meet, the result is a RHOMBIC DODECAHEDRON (Wells 1991). The VOLUME of intersection can be computed in a number of different ways, V3(r ; r ; r) /C30g 16r3 g p =4 0g1 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28s2 cot2 tp ds dt (17) /C30ffiffiffi 2p rYru*Yru+3 /C276gr r=ffiffi 2p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28z2pYru*Yru+2 dz (18) /C3082/C28ffiffiffi 2pYru*Yru+ r3 (19) (Moore 1974). Four cylinders can also be placed with axes along the lines joining the vertices of a TETRAHEDRON with the centers of the opposite sides. The resulting solid of intersection has VOLUME V4 /C3012 2ffiffiffi 2p /C28ffiffiffi 6pYru*Yru+ (20) and 24 curved faces analogous to a CUBE-OCTAHEDRON COMPOUND (Moore 1974, Wells 1991). Six cylinders can be place with axes parallel to the face diagonals of a CUBE . The resulting solid of intersection has VOLUME V4/C3012 3/C272ffiffiffi 3p /C284ffiffiffi 2p Yru*Yru+ (21) and 36 curved faces, 24 of which are kite-shaped and 12 of which are rhombic (Moore 1974). See also BICYLINDER ,CYLINDER ,ELLIPTIC CYLINDER , REULEAUX TETRAHEDRON ,RHOMBIC DODECAHEDRON , RIGHT ANGLE ,VAULT References Angell, I. O. and Moore, M. "Symmetrical Intersections of Cylinders." Acta Cryst. Sect. A 43, 244/C1/50, 1987. Bowman, F. Introduction to Elliptic Functions, with Appli- cations. New York: Dover, 1961. Gardner, M. "Mathematical Games." Sci. Amer. 207, 164, 1962.Gardner, M. The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: Chicago University Press, pp. 183 /C1/85, 1991. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 204 /C1/04, 1997. Heath, T. L. The Method of Archimedes. New York: Dover, 1953. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 128, 1948. Kiang, T. "An Old Chinese Way of Finding the Volume of a Sphere." Math. Gaz. 56,8 8/C1/1, 1972. Moore, M. "Symmetrical Intersections of Right Circular Cylinders." Math. Gaz. 58, 181/C1/85, 1974. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 118 /C1/19, 1991. Wells, D. G. #555 in The Penguin Book of Curious and Interesting Puzzles. London: Penguin Books, 1992. Stella Octangula APOLYHEDRON COMPOUND composed of a TETRAHE- DRON and its DUAL (a second TETRAHEDRON rotated 1808with respect to the first). The stella octangula is also called a STELLATED TETRAHEDRON , and is the only STELLATION of the OCTAHEDRON . The stella octangula can be constructed using the following NET by cutting along the solid lines, folding back along the plain lines, and folding forward along the dotted lines. Another construction builds a single TETRAHEDRON , then attaches four tetrahedral caps, one to each face. This CUMULATION of a unit edge-length OCTAHEDRON uses pyramids with height1 3ffiffiffi 6p . A tetrahedron with edge length 1 produces a stella octangula with edge lengths /1 =2/. This solid has SURFACE AREA and VOLUME S /C303 2ffiffiffi 3p V /C301 8ffiffiffi 2p : The CONVEX HULL of the stella octangula is a CUBE . The above diagrams show two projections of the stella octangula. The edges lying on tetrahedral faces are represented using dashed lines, while the edges of the two large tetrahedron are showing using solid lines. The solid common to both tetrahedra is an OCTAHE- DRON (left figure; Ball and Coxeter 1987), which is another way of saying that the stella octangula is a STELLATION of the OCTAHEDRON (in fact, the only stellation). The edges of the two tetrahedra in the stella octangula form the 12 DIAGONALS of a CUBE (middle figure). Finally, the stella octangula can be constructed using eight of the 20 vertices of the DODECAHEDRON (right figure). See also CUBE,O CTAHEDRON ,P OLYHEDRON COM- POUND ,SPHERE PACKING ,STELLATION ,TETRAHEDRONReferences Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 135 /C1/37, 1987. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 158, 1969. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 48 /C1/1, 1973. Cundy, H. and Rollett, A. "Stella Octangula (Two Tetrahe- dra)." §3.10.1 in Mathematical Models, 3rd ed. Strad- broke, England: Tarquin Pub., p. 129, 1989. Kepler, J. "Harmonice Mundi." In Opera Omnia, Vol. 5. Frankfurt, 1864. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 212 /C1/13, 1999. Weisstein, E. W. "Polyhedra." M ATHEMATICA NOTEBOOK POLYHEDRA.M . Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, pp. 35 and 37, 1989. Stella Octangula Number AFIGURATE NUMBER OF THE FORM , StOctn/C30On/C278Tn/C281/C30n2n2/C281YrvYru : The first few are 1, 14, 51, 124, 245, ... (Sloane’sA007588). The GENERATING FUNCTION for the stella octangula numbers is x(x2/C2710x/C271) (x/C281)4/C30x/C2714x2/C2751x3/C27124x4/C27...: References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 51, 1996. Sloane, N. J. A. Sequences A007588/M4932 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stellated Octahedron STELLA OCTANGULA Stellated Polyhedron STELLATION Stellated Tetrahedron STELLA OCTANGULA Stellated Truncated Hexahedron The UNIFORM POLYHEDRON U19 ; also called the QUASI- TRUNCATED HEXAHEDRON , whose DUAL POLYHEDRON is the GREAT TRIAKIS OCTAHEDRON . It has SCHLA ¨ FLI SYMBOL t?f4; 3g; WYTHOFF SYMBOL 23½4 3; and is Wen- ninger model W92 : Its faces are 8f3 g/C276f83 g: For a /C301, its CIRCUMRADIUS is R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C284ffiffiffi 2pq : The CONVEX HULL of the stellated truncated hexahe- dron is the Archimedean SMALL RHOMBICUBOCTAHE- DRON A6, whose dual is the DELTOIDAL ICOSITETRAHEDRON , so the dual of the stellated truncated hexahedron (i.e., the GREAT TRIAKIS OCTA- HEDRON ) is one of the stellations of the DELTOIDAL ICOSITETRAHEDRON (Wenninger 1983, p. 57). References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 144, 1989. Stellation The process of constructing POLYHEDRA by extending the facial PLANES past the EDGES of a given POLY- HEDRON until they intersect (Wenninger 1989). The set of all possible EDGES of the stellations can be obtained by finding all intersections on the facial planes. Since the number and variety of intersections can become unmanageable for complicated polyhedra, additional rules are sometimes added to constrain allowable stellations. There exists a Mathematica function Stellate [poly, ratio ] in the Mathematica add-on package Graphics‘Polyhedra‘ (which can be loaded with the command BBGraphics‘ ),although it only replaces facial planes with pyramids and does not perform true stellation. There are no stellations of the CUBE or TETRAHEDRON (Wenninger 1989, p. 35). The only stellated form of the octahedron is the STELLA OCTANGULA , which is a compound of two TETRAHEDRA (Wenninger 1989, pp. 35 and 37). The DODECAHEDRON has three stella- tions: the SMALL STELLATED DODECAHEDRON , GREAT DODECAHEDRON , and GREAT STELLATED DODECAHE- DRON (Wenninger 1989, pp. 35 and 38 /C1/0). Coxeter (1982) shows that 59 ICOSAHEDRON STELLATIONS exist, subject to certain restrictions. The KEPLER- POINSOT SOLIDS , which consist of three DODECAHEDRON STELLATIONS and one of the ICOSAHE- DRON STELLATIONS . The only STELLATIONS of P LA- TONIC SOLIDS which are UNIFORM POLYHEDRA are the three DODECAHEDRON STELLATIONS and one of the ICOSAHEDRON STELLATIONS . There are three stellations of the RHOMBIC DODECA- HEDRON (Wells 1991, pp. 216 /C1/17). See also ARCHIMEDEAN SOLID STELLATION ,DELTOI- DAL ICOSITETRAHEDRON STELLATIONS ,D ODECAHE- DRON STELLATIONS ,F ACETING ,ICOSAHEDRON STELLATIONS ,KEPLER- POINSOT SOLID ,PLATONIC SO- LID STELLATIONS ,POLYHEDRON ,POLYTOPE STELLA- TIONS ,R ECTIFICATION ,R HOMBIC DODECAHEDRON STELLATIONS ,RHOMBIC TRIACONTAHEDRON STELLA- TIONS ,SMALL TRIAKIS OCTAHEDRON STELLATIONS , STELLA OCTANGULA ,STELLATED POLYHEDRON ,STEL- LATED TRUNCATED HEXAHEDRON ,TRIAKIS TETRAHE- DRON STELLATIONS ,T RUNCATION ,U NIFORM POLYHEDRON References Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra. Stradbroke, England: Tarquin Publications, 1999. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Publications, 1989. Fleurent, G. M. "Symmetry and Polyhedral Stellation Ia and Ib. Symmetry 2: Unifying Human Understanding, Part 1." Comput. Math. Appl. 17, 167/C1/93, 1989. Messer, P. W. "Stellations of the Rhombic Triacontahedron and Beyond." Structural Topology 21,2 5/C1/6, 1995. Messer, P. W. and Wenninger, M. J. "Symmetry and Poly- hedral Stellation. II. Symmetry 2: Unifying Human Understanding, Part 1." Comput. Math. Appl. 17, 195/C1/ 01, 1989. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. Wenninger, M. J. "Stellated Forms of Convex Duals." Ch. 3 inDual Models. Cambridge, England: Cambridge Uni- versity Press, pp. 36 /C1/8, 1983. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, 1989. Stem-and-Leaf Diagram The "stem" is a column of the data with the last digit removed. The final digits of each column are placed next to each other in a row next to the appropriate column. Then each row is sorted in numerical order. This diagram was invented by John Tukey. References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, pp. 7 /C1/6, 1977. Step 1.5 times the H-SPREAD . See also FENCE ,H-SPREAD References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 44, 1977. Step Function A function on the REALS R is a step function if it can be written as a finite linear combination of semi-open intervals [a; b) ⁄R: Therefore, a step function f can be written as f(x) /C30 a1f1(x) /C27/C1/C1/C1/C27 anfn(x) : where ai /C23R ; fi(x) /C301ifx /C23 ai ; bi ½Þ and 0 otherwise, for i /C301, ..., n. See also HEAVISIDE STEP FUNCTION Step Polynomial HERMITE’S INTERPOLATING POLYNOMIAL Stephens’ Constant Let a and b be nonzero integers such that ambn "1 except when m /C30n /C300; and let T(a ; b) be the set of PRIMES p for which pak /C28bYrvYru for some NONNEGATIVE INTEGER k. Then assuming the generalized RIEMANN HYPOTHESIS , Stephens (1976) showed that the density of T(a; b) relative to the primes is a rational multiple of CStephens /C30Y/C12 j/C3011 /C28pj p3 j/C28 1 ! /C300:5759599688... : where pj is the jth PRIME (Finch). See also ARTIN’S CONSTANT References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/artin/artin.html. Moree, P. "Approximation of Singular Series and Automata." Submitted to Manuscripta Math. 1999. Moree, P. and Stevenhagen, P. "A Two Variable Artin Conjecture." Submitted 1999. Stephens, P. J. "Prime Divisor of Second-Order Linear Recurrences, I." J. Number Th. 8, 313 /C1/32, 1976.Steradian The unit of SOLID ANGLE . The SOLID ANGLE corre- sponding to all of space being subtended is 4p steradians. See also RADIAN ,SOLID ANGLE Stereogram A plane image or pair of 2-D images which, when appropriately viewed using both eyes, produces an image which appears to be three-dimensional. By taking a pair of photographs from slightly different angles and then allowing one eye to view each image, a stereogram is not difficult to produce. Amazingly, it turns out that the 3-D effect can be produced by both eyes looking at a single image by defocusing the eyes at a certain distance. Such stereograms are called "random-dot stereograms." See also ANAGLYPH References Bar-Natan, D. "Random-Dot Stereograms." Math. J. 1,6 9/C1/ 1, 1991. Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 89 /C1/3, 1996. Julesz, B. Foundations of Cyclopean Perception. Chicago, IL: University of Chicago Press, 1971. Julesz, B. "Stereoscopic Vision." Vision Res. 26, 1601 /C1/611, 1986. Terrell, M. S. and Terrell, R. E. "Behind the Scenes of a Random Dot Stereogram." Amer. Math. Monthly 101, 715/C1/24, 1994. Tyler, C. "Sensory Processing of Binocular Disparity." In Vergence Eye Movements: Basic and Clinical Aspects. Boston, MA: Butterworth, pp. 199 /C1/95, 1983. Stereographic Projection AMAP PROJECTION obtained by projecting points p?on the surface of sphere from the sphere’s north pole N to point P in a plane tangent to the south pole S (Coxeter 1969, p. 93). In such a projection, GREAT CIRCLES are mapped to CIRCLES , and LOXODROMES become LOGARITHMIC SPIRALS . The transformation equations for a sphere of radius R are given by x /C30k cos f sin( l /C28 l0) (1) y /C30k cos f1 sin f /C28sin f1 cos f cos l /C28 l0 ðÞ ½/C138 : (2) where l0is the central longitude, f1is the central latitude, and k /C302R 1 /C27 sin f1sin f /C27 cos f1 cos f cos l /C28 l0 ðÞ: (3) The inverse FORMULAS for latitude f and longitude l are then given by f /C30sin/C281cos c sin f1 /C27y sin c cos f1 r ! (4) l /C30 l0 /C27tan/C281 x sin c r cos f1 cos c /C28 y sin f1 sin c ! ; (5) where r /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p (6) c /C302 tan/C281r 2R ! : (7) For an OBLATE SPHEROID , R can be interpreted as the "local radius," defined by R /C30Re cos f 1 /C28 e2 sin2 fYrvYru cos x ; (8) where Reis the equatorial radius and x is the CONFORMAL LATITUDE . See also GNOMONIC PROJECTION ,MAP PROJECTIONReferences Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 93 and 289 /C1/90, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 150 /C1/53, 1967. Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 154 /C1/63, 1987. Stereology The exploration of 3-D space from 2-D sections of PROJECTIONS of solid bodies. See also AXONOMETRY ,B RIGHTNESS ,C ORK PLUG, CROSS SECTION ,INNER QUERMASS ,M EAN TANGENT DIAMETER ,PROJECTION ,SHADOW ,TRIP-LET References Elias, H. and Hyde, D. M. (Eds.). Guide to Practical Stereology. S. Karger, 1983. Elias, H. (Ed.). Stereology. New York: Springer-Verlag, 1967. Stern-Brocot Tree A special type of BINARY TREE obtained by starting with the fractions0 1and10and iteratively inserting (m /C27m?)=(n /C27n?) between each two adjacent fractions m=n and m?=n ?: The result can be arranged in tree form as illustrated above. The FAREY SEQUENCE Fn defines a subtree of the Stern-Brocot tree obtained by pruning off unwanted branches (Vardi 1991, Graham et al. 1994). See also BINARY TREE,F AREY SEQUENCE ,F ORD CIRCLE References Brocot, A. "Calcul des rouages par approximation, nouvelle me´thode." Revue Chonome ´trique 6, 186/C1/94, 1860. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 116 /C1/17, 1994. Stern, M. A. "U ¨ber eine zahlentheoretische Funktion." J. reine angew. Math. 55, 193/C1/20, 1858. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, p. 253, 1991. Viswanath, D. "Random Fibonacci Sequences and the Number 1.13198824...." Math. Comput. 69, 1131 /C1/155, 2000. Stevedore’s Knot The 6-crossing KNOT 06 /C1/01 having CONWAY-ALEXAN- DER POLYNOMIAL D(t) /C302t2 /C285t /C272: References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 225, 1976. Stewart’s Theorem Let a CEVIAN A1P be drawn on a TRIANGLE DA1A2A3 ; and denote the lengths m /C30A2P and n /C30PA3 ; with a1 /C30m /C27n: Then ma2 2 /C27na23 /C30(m /C27n)A1P2 /C27mPA32/C27nPA22: This theorem is sometimes also called APOLLONIUS’ THEOREM . References Altshiller-Court, N. "Stewart’s Theorem." §6B in College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 152 /C1/53, 1952. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 6, 10, and 31, 1967. Stick Number Let the stick number s(K)ofa KNOT K be the least number of straight sticks needed to make a KNOT K. The smallest stick number of any KNOT is s(T) /C306; where T is the TREFOIL KNOT .IfJ and K are KNOTS , then s(J /C27K) 5s(J) /C27s(K) /C271: For a nontrivial KNOT K, let c(K) be the CROSSING NUMBER (i.e., the least number of crossings in any projection of K). Then 1 25 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 /C278(c(K) /C282)phi 5s(K) 52c(K) :The following table gives the stick number for some common knots. TREFOIL KNOT 6 WHITEHEAD LINK 8 See also CROSSING NUMBER (LINK), TRIANGLE COUNT- ING References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 27 /C1/0, 1994. Stickelberger Relation Let P be a PRIME IDEAL in Dm not containing m. Then ( F(P)) /C30PP ts/C281 t; where the sum is over all 1 5t Bm which are RELATIVELY PRIME to m. Here Dmis the RING of integers in Q zmðÞ ;F(P) /C30g(P)m ; and other quantities are defined by Ireland and Rosen (1990). See also PRIME IDEAL References Ireland, K. and Rosen, M. "The Stickelberger Relation and the Eisenstein Reciprocity Law." Ch. 14 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 203 /C1/27, 1990. Stiefel Manifold The Stiefel manifold of ORTHONORMAL k-frames in Rn is the collection of vectors (/v1 ; ..., vk) where vi is in Rn for all i, and the k-tuple (/v1 ; ..., vk)is ORTHONORMAL . This is a submanifold of Rnk ; having DIMENSION nk /C28(k /C271)k=2 :/ Sometimes the "orthonormal" condition is dropped in favor of the mildly weaker condition that the k-tuple (/ v1 ; ..., vk) is linearly independent. Usually, this does not affect the applications since Stiefel manifolds are usually considered only during HOMOTOPY THEORETIC considerations. With respect to HOMOTOPY THEORY , the two definitions are more or less equivalent since GRAM- SCHMIDT ORTHONORMALIZATION gives rise to a smooth deformation retraction of the second type of Stiefel manifold onto the first. See also GRASSMANN MANIFOLD Stiefel-Whitney Class The ith Stiefel-Whitney class of a REAL VECTOR BUNDLE (or TANGENT BUNDLE or a REAL MANIFOLD ) is in the ith cohomology group of the base SPACE involved. It is an OBSTRUCTION to the existence of ( n/C28 i/C271)REAL linearly independent VECTOR FIELDS on that VECTOR BUNDLE , where n is the dimension of the FIBER . Here, OBSTRUCTION means that the ith Stiefel- Whitney class being NONZERO implies that there do not exist (n /C28i /C271) everywhere linearly dependent VECTOR FIELDS (although the Stiefel-Whitney classes are not always the OBSTRUCTION ). In particular, the nth Stiefel-Whitney class is the obstruction to the existence of an everywhere NON- ZERO VECTOR FIELD , and the first Stiefel-Whitney class of a MANIFOLD is the obstruction to orientability. See also CHERN CLASS ,OBSTRUCTION ,PONTRYAGIN CLASS,STIEFEL- WHITNEY NUMBER Stiefel-Whitney Number The Stiefel-Whitney number is defined in terms of the STIEFEL- WHITNEY CLASS of a MANIFOLD as follows. For any collection of STIEFEL- WHITNEY CLASSES such that their cup product has the same DIMENSION as the MANIFOLD , this cup product can be evaluated on the MANIFOLD ’s FUNDAMENTAL CLASS . The resulting num- ber is called the PONTRYAGIN NUMBER for that combination of Pontryagin classes. The most important aspect of Stiefel-Whitney num- bers is that they are COBORDISM invariant. Together, PONTRYAGIN and Stiefel-Whitney numbers determine an oriented MANIFOLD ’sCOBORDISM class. See also CHERN NUMBER ,P ONTRYAGIN NUMBER , STIEFEL- WHITNEY CLASS Stieltjes Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. Expanding the R IEMANN ZETA FUNCTION about z/C301 gives z(z)/C301 z/C281/C27X/C12 n/C300(/C281)n n!gn(z/C281)n; (1) where gn/C13lim m0/C12Xm k/C301(lnk)n k/C28(lnm)n/C271 n/C271"# : (2) These constants are returned by the Mathematica functionStieltjesGamma [n]. An alternative defini- tion is given by absorbing the coefficient of gninto the constant, g?n/C13(/C281)n n!gn (3) (e.g., Hardy 1912, Kluyver 1927). The case n/C300 gives the usual E ULER- MASCHERONI CONSTANT g0/C13g:The first few numerical values are given in the following table.n /gn/ 0 0.5772156649 1 //C280:07281584548 / 2 //C280:009690363192 / 3 0.0020538344204 0.002325370065 5 0.0007933238173 Briggs (1955 /C1 /956) proved that there infinitely many gnof each SIGN. Berndt (1972) gave upper bounds of ½gn½B4(n/C281)! pnforneven 2(n/C281)! pnfornodd:8 >>>< >>>:(4) However, these bounds are extremely weak, so it is likely that better ones can be derived. Vacca (1910) proved that the EULER- MASCHERONI CONSTANT may be expressed as g/C30X/C12 k/C301(/C281)k klgkbc ; (5) where xbcis the FLOOR FUNCTION and the LGfunction lgx/C13log2xis the LOGARITHM to base 2. Hardy (1912) gave the FORMULA 2g1 ln 2/C30X/C12 k/C301(/C281)k k2l gk/C28lg(2k) bc ½/C138 lgkbc : (6) /g1is also given by the sum Xx n/C3011 nlnx n ! /C301 2(lnx)2/C27glnx/C28g1/C27Ox/C281YrvYru ;(7) where g1was called /C28Dand given incorrectly by Ellision and Mende `s-France (1975) and the error was reproduced by Le Lionnais (1983, p. 47). The exact form of (7) is given by Xx n/C3011 nlnx n ! /C30Hxlnx/C28z?(1;x/C271)/C27g1; (8) where Hxis a HARMONIC NUMBER ,QzmðÞ is the HURWITZ ZETA FUNCTION , and z?(1;a) denotes lims01dz(s;a)=dz½z/C30s:/ Kluyver (1927) gave similar series for gnvalid for all n/C211, gn /C30 n!(ln 2)n Xn/C271 m/C301( /C281)m/C281 m!X/C12 k /C301( /C281)k k lg kbcmB1/C27n/C28mln k ln 2 ! ; (9) where Bn(x)isaB ERNOULLI POLYNOMIAL . However, this series converges extremely slowly, requiring more than 104 terms to get two digits of g1and many more for higher order gn : gncan also be expressed as a single sum using gn /C30(ln 2)n n /C27 1X/C12 k /C301( /C281)k kBn/C271ln k ln 2 ! : (10) A set of constants related to gn is dn /C13 lim m0/C12Xm k /C301(ln k)n /C28gm 1(ln x)n dx /C281 2(ln m)n"# (11) (Sitaramachandrarao 1986, Lehmer 1988). See also BERNOULLI POLYNOMIAL ,EULER PRODUCT , RIEMANN ZETA FUNCTION References Berndt, B. C. "On the Hurwitz Zeta-Function." Rocky Mountain J. Math. 2, 151 /C1/57, 1972. Bohman, J. and Fro¨berg, C.-E. "The Stieltjes Function-- Definitions and Properties." Math. Comput. 51, 281 /C1/89, 1988. Briggs, W. E. "Some Constants Associated with the Riemann Zeta-Function." Mich. Math. J. 3, 117 /C1/21, 1955 /C1/956. Ellison, W. J. and Mende `s-France, M. Les nombres pre- miers. Paris: Hermann, 1975. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/stltjs/stltjs.html. Hardy, G. H. "Note on Dr. Vacca’s Series for g :/" Quart. J. Pure Appl. Math. 43, 215 /C1/16, 1912. Hardy, G. H. and Wright, E. M. "The Behavior of z(s) when s 0 1:/" §17.3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 246 /C1/47, 1979. Kluyver, J. C. "On Certain Series of Mr. Hardy." Quart. J. Pure Appl. Math. 50, 185 /C1/92, 1927. Knopfmacher, J. "Generalised Euler Constants." Proc. Edinburgh Math. Soc. 21,25/C1/2, 1978. Lammel, E. "Ein Beweis dass die Riemannsche Zetafunktion z(s)is½s /C281 ½51 keine Nullstelle besitzt." Univ. Nac. Tucma ´n Rev. Ser. A 16, 209 /C1/17, 1966. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 47, 1983. Lehmer, D. H. "The Sum of Like Powers of the Zeros of the Riemann Zeta Function." Math. Comput. 50, 265 /C1/73, 1988. Liang, J. J. Y. and Todd, J. "The Stieltjes Constants." J. Res. Nat. Bur. Standards--Math. Sci. 76B, 161 /C1/78, 1972. Sitaramachandrarao, R. "Maclaurin Coefficients of the Riemann Zeta Function." Abstracts Amer. Math. Soc. 7, 280, 1986. Vacca, G. "A New Series for the Eulerian Constant." Quart. J. Pure Appl. Math. 41, 363 /C1/68, 1910. Stieltjes Integral The Stieltjes integral is a generalization of the RIEMANN INTEGRAL . Let f(x) and a(x) be real-valuedbounded functions defined on a CLOSED INTERVAL [a, b]. Take a partition of the INTERVAL a /C30x0 Bx1 Bx2 ; ...Bxn/C281 Bxn /C30b; (1) and consider the Riemann sum Xn/C281 i/C300f jiðÞa xi/C271YrvYru /C28 a xiðÞYrtYrP (2) with ji /C23 xi ; xi/C271YrtYrP : If the sum tends to a fixed number I as max xi /C271 /C28xiYrvYru 0 0; then I is called the Stieltjes integral, or sometimes the RIEMANN- STIELTJES INTE- GRAL . The Stieltjes integral of f with respect to a is denoted g f(x) da(x) (3) or sometimes simply g fda: (4) If f and a have a common point of discontinuity, then the integral does not exist. However, if f is continuous and a? is Riemann integrable over the specified interval, then g f(x) da(x) /C30g f(x) a?(x) dx (5) (Kestelman 1960). For enumeration of many properties of the Stieltjes integral, see Dresher (1981, p. 105). See also CONVOLUTION ,RIEMANN INTEGRAL References Dresher, M. The Mathematics of Games of Strategy: Theory and Applications. New York: Dover, 1981. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 152 /C1/55, 1988. Jeffreys, H. and Jeffreys, B. S. "Integration: Riemann, Stieltjes." §1.10 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 26 /C1/6, 1988. Kestelman, H. "Riemann-Stieltjes Integration." Ch. 11 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 247 /C1/69, 1960. Pollard, S. Quart. J. Math. 49,73/C1/38, 1923. Stieltjes, T. J. Ann. d. fac. d. sciences Toulouse 8,68/C1/5, 1894J. Widder, D. V. Ch. 1 in The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Stieltjes’ Theorem The m /C271 ELLIPSOIDAL HARMONICS when k1;k2;and k3are given can be arranged in such a way that the rth function has r/C281 zeros between /C28a2and/C28b2and the remaining m/C27r/C281 zeros between /C28b2and/C28c2 (Whittaker and Watson 1990). See also ELLIPSOIDAL HARMONIC References Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 560 /C1/62, 1990. Stieltjes Transform The INTEGRAL TRANSFORM (Kf)(x)/C30g/C12 /C28/C12G(p)(x/C27t)/C28pf(t)dt: References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993. StieltjesGamma STIELTJES CONSTANTS Stieltjes-Wigert Polynomial Orthogonal POLYNOMIALS associated with WEIGHTING FUNCTION w(x)/C30p/C281=2kexp/C28k2ln2xYrvYru /C30p/C281=2kx/C28k2lnx(1) forx/C23(0;/C12) and k/C210. Using n nYrtvYrtu /C301/C28qnðÞ 1/C28qn/C281ðÞ /C1 /C1 /C1 1/C28qn/C28n/C271ðÞ (1/C28q)1/C28q2 ð Þ/C1/C1/C1 1/C28qn ðÞ(2) where 0 BnBn; n 0YrtvYrtu /C30n nYrtvYrtu /C301; (3) and q/C30exp/C282k2YrvYru/C281hi : (4) Then pn(x)/C30(/C281)nqn=2/C271=4(1/C28q)1/C28q2YrvYru /C1/C1/C11/C28qnðÞYrtYrP/C281=2 /C29Xn n/C300n nYrtvYrtu qn2/C28q1=2xYrvYru n(5) forn/C210 and p0(x)/C30q1=4: (6) References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 33, 1975.Stiff Differential Equation References Byrne, G. D. and Hindmarsh, A. C. "Stiff ODE Solvers: A Review of Current and Coming Attractions." J. Comput. Phys. 70,1/C1/2, 1987. Enright, W. H.; Hull, T. E.; and Lindberg, B. "Comparing Numerical Methods for Stiff Systems of ODEs." BIT 15, 10/C1/8, 1975. Enright, W. H. and Hull, T. E. "Comparing Numerical Methods for the Solution of Stiff Systems of ODEs Arising in Chemistry." In Numerical Methods for Differential Systems, Recent Developments in Algorithms, Softwareand Applications (Ed. L. Lapidus and W. E. Schiesser). New York: Academic Press, pp. 45 /C1 /6, 1976. Hairer, E. and Wanner, G. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems,2nd rev. ed. Berlin: Springer-Verlag, 1996. Shampine, L. F. "Ill-Conditioned Matrices and the Integra- tion of Stiff ODEs." J. Comput. Appl. Math. 48, 279/C1 /92, 1993. Stirling Cycle Number STIRLING NUMBER OF THE FIRST KIND Stirling Number of the First Kind The signed Stirling numbers of the first kind are variously denoted s(n;m) (Riordan 1980, Roman 1984), S(m) n(Fort 1958, Abramowitz and Stegun 1971), Sm n(Jordan 1950). Abramowitz and Stegun (1971, p. 822) summarize the various notational conventions, which can be a bit confusing (especially since an unsigned version S1(n;m)/C30½s(n;m)½is also in common use). The signed Stirling number of the first kind s(n;m) is are returned by StirlingS1 [n, m]i nMathematica . The signed Stirling numbers of the first kind s(n;m) are defined such that the number of PERMUTATIONS of nelements which contain exactly mCYCLES is the nonnegative number ½s(n;m)½/C30(/C281)n/C28ms(n;m): (1) This means that s(n;m)/C300 for m/C21nands(n;n)/C30 1:A related set of numbers is known as the associated Stirling numbers of the first kind. Both these are the usual Stirling numbers of the first kind are special cases of a general function dr(n;k) which is related to the number of cycles in a permutation. The triangle of signed Stirling numbers of the first kind is 1 /C2811 2/C2831 /C2861 1 /C2861 24/C2850 35 /C2810 1 (Sloane’s A008275). Special values include s(n;0)/C30dn0 (2) s(n;1)/C30(/C281)n/C281(n/C281)! (3) s(n;2)/C30(/C281)n(n/C281)!Hn/C281 (4) s(n;3)/C301 2(/C281)n/C281(n/C281)!H2 n/C281/C28H(2) n/C281YrtYrP (5) s(n;n/C281)/C30/C28n 2Yru$Yru% ; (6) where dmnis the K RONECKER DELTA ,Hnis a HARMO- NIC NUMBER ,H(r) nis a HARMONIC NUMBER of order r, andn kYrvYru is a BINOMIAL COEFFICIENT . The GENERATING FUNCTION for the Stirling numbers of the first kind is (x)n/C30x(x/C281)/C1/C1/C1(x/C28n/C271)/C30Xn m/C300s(n;m)xm; (7) where ( x)nis a FALLING FACTORIAL . Other generating functions are Xn k/C300s(n;k)xk/C30(1/C27x/C28n)n (8) Xn k/C300s(n;k)xk/C30(/C281)nn!n/C28x/C281 nYru$Yru% (9) X/C12 k/C30ms(k;m)xk/C30[ln(x/C271)]m m!(10) Yn k/C301(1/C27kx)/C30Xn/C271 k/C301(/C281)n/C271/C28ks(n/C271;k)xn/C271/C28k: (11) The Stirling numbers of the first kind satisfies the RECURRENCE RELATION s(n/C271;m)/C30s(n;m/C281)/C28ns(n;m) (12) for 15m5nand the sum identities s(n;m)/C30Xn k/C30mnk/C28ms(n/C271;k/C271) (13) form]1 and m rYru$Yru% s(n;m)/C30Xn/C28r k/C30m/C28rn kYru$Yru% s(n/C28k;r)(k;m/C28r) (14) for 05r5m;wheren kYrvYru is a BINOMIAL COEFFICIENT . The Stirling numbers of the first kind s(n;m) are connected with the S TIRLING NUMBERS OF THE SEC- OND KIND S(n;m) through the formulas s(n;i)/C30Xn k/C30iXk j/C300s(n;k)s(k;j)S(j;i) (15)S(n;i)/C30Xn k/C30iXk j/C300S(n;k)S(k;j)s(j;i) (16) (Roman 1984, p. 67), as well as S(n;m)/C30Xn/C28m k/C300(/C281)kk/C27n/C281 k/C27n/C28mYru$Yru% /C22n/C28m n/C28k/C28mYru$Yru% s(k/C28m/C27n;k) (17) s(n;m)/C30Xn/C28m k/C300(/C281)kk/C27n/C281 k/C27n/C28mYru$Yru% /C22n/C28m n/C28k/C28mYru$Yru% s(k/C28m/C27n;k) (18) Xmax ( k;j)/C271 l/C300s(l;j)S(k;1)/C30djk (19) Xmax ( k;j)/C271 l/C300s(k;l)S(l;j)/C30djm: (20) The NONNEGATIVE version simply gives the number of PERMUTATIONS ofnobjects having mCYCLES (with cycles in opposite directions counted as distinct) and is obtained by taking the ABSOLUTE VALUE of the signed version. The nonnegative Stirling numbers ofthe first kind are variously denoted S 1(n;m)/C13n mYrtvYrtu /C13½s(n;m)½ (21) (Graham et al. 1994). Diagrams illustrating S1(5;1)/C3024;S1(5;3)/C3035;S1(5;4)/C3010;and S1(5;5)/C301 (Dickau) are shown below. The nonnegative Stirling numbers of the first kind satisfy the curious identity X/C12 n/C301Xn/C282 k /C300ex /C28 x /C28 1 ðÞk /C271S1(n; n /C28 k) (k /C27 1)!"# e /C28xn /C30ln(x /C271) (22) (Gosper) and satisfy S1(n /C271 ; k) /C30nS1(n; k) /C27S1(n; k /C281): (23) The Stirling numbers can be generalized to noninte- gral arguments (a sort of "Stirling polynomial") using the identity G(j /C27 h) jh G(j)/C30X/C12 k /C300S1(h ; h /C28 k) jk /C301 /C27(h /C28 1)h 2j/C27(h /C28 2)(3h /C28 1)(h /C28 1)h 24j2 /C27(h /C28 3)(h /C28 2)(h /C28 1)2h2 48j3 /C27/C1/C1/C1 (24) which is a generalization of an ASYMPTOTIC SERIES for a ratio of GAMMA FUNCTIONS G(j /C271=2)=G(j) (Gosper). The associated Stirling numbers of the first kind d2(n ; k) /C30d(n; k) are defined as the number of per- mutations of a given number n having exactly k CYCLES , all of which are of length r /C302 or greater (Comtet 1974, p. 256; Riordan 1980, p. 75). They are a special case of the more general numbers dr(n; k); and have the RECURRENCE RELATION d2(n /C271; k) /C30nd2(n; k) /C27d2(n /C281; k /C281) ½/C138 (25) with initial conditions d2(n; k) /C300 for n 52k /C281; and d2(n ; 1) /C30(n /C281)! (Appell 1880; Tricomi 1951; Carlitz 1958; Comtet 1974, pp. 256, 293, and 295) with . The GENERATING FUNCTION for d2(n; k) is given by e /C28tu(1 /C28t)/C28u /C301 /C27Xn=2 k/C301d2(n; k) n!tnuk /C301 /C27t2 2 /C27t3 3 /C27t4 4 /C27t5 5 /C27t6 6 /C27... ! u /C27t4 8 /C27t5 6 /C2713t6 72/C27... ! u2 /C27t6 48 /C27... ! u3 /C27... (26) (Comtet 1974, p. 256). The associated Stirling num- bers of the first kind satisfy the sum identity Xn k /C301(/C281)k/C281d2(n; k) /C30n /C281: (27) For k ]2 and p a PRIME , d(p; k) /C130 (mod p(p /C281)): (28) For all integers l,X m(/C281)md2(l /C27m; m) /C30(/C281)l ; (29) and similarly, X m( /C281)md2(l /C27 m; m) l /C27 m /C28 1/C300 (30) (Comtet 1974, p. 256). Special cases of the associated Stirling numbers of the first kind are given by d2(n; 1) /C30(n /C281)! (31) d2(2k; k) /C30(2k /C281)!! (32) d2(2k /C271; k) /C30(2k /C27 1)ak! 3(k /C28 1)!2k (33) d2(2k/C272;k)/C30(4k/C275)(2k/C272)! 18(k/C281)!2k(34) (Comtet 1974, p. 256), where akis a coefficient in the expansion of (1 /C273x)=(1/C282x)7=2: 1, 10, 105, 1260, 17325, ... (Sloane’s A000457), omitted in Comtet (1974). The triangle of these numbers is given by 1 2 6;3 24;20 120;130;15 720;924;210 5040 ;7308 ;2380 ;105 (Sloane’s A008306). See also CYCLE (PERMUTATION ), HARMONIC NUMBER , PERMUTATION ,STIRLING NUMBER OF THE SECOND KIND,STIRLING POLYNOMIAL ,STIRLING TRANSFORM References Abramowitz, M. and Stegun, C. A. (Eds.). "Stirling Numbers of the First Kind." §24.1.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 824, 1972. Adamchik, V. "On Stirling Numbers and Euler Sums." J. Comput. Appl. Math. 79, 119/C1/30, 1997. Appell, P. "De ´veloppments en se ´rie entie `re de (1 /C27ax)1=x:/" Grunert Archiv 65, 171/C1/75, 1880. Butzer, P. L. and Hauss, M. "Stirling Functions of the First and Second Kinds; Some New Applications." Israel Math- ematical Conference Proceedings: Approximation, Interpo- lation, and Summability, in Honor of Amnon Jakimovski on his Sixty-Fifth Birthday (Ed. S. Baron and D. Levia- tan). Ramat Gan, Israel: IMCP, pp. 89 /C1/08, 1991. Carlitz, L. "On Some Polynomials of Tricomi." Boll. Un. M. Ital. 13,5 8/C1/4, 1958. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 91 /C1/2, 1996. David, F. N.; Kendall, M. G.; and Barton, D. E. Symmetric Function and Allied Tables. Cambridge, England: Cam- bridge University Press, p. 226, 1966. Dickau, R. M. "Stirling Numbers of the First Kind." http:// forum.swarthmore.edu/advanced/robertd/stirling1.html. Fort, T. Finite Differences. Oxford, England: Clarendon Press, 1948. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Stirling Numbers." §6.1 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 257 /C1/67, 1994. Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen fu ¨r Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995. Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965. Knuth, D. E. "Two Notes on Notation." Amer. Math. Monthly 99, 403/C1/22, 1992. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980. Roman, S. The Umbral Calculus. New York: Academic Press, pp. 59 /C1/3, 1984. Sloane, N. J. A. Sequences A000457/M4736, A008275, and A008306 in "An On-Line Version of the Encyclopedia ofInteger Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Stirling, J. Methodus differentialis, sive tractatus de sum- mation et interpolation serierum infinitarium. London, 1730. English translation by Holliday, J. The Differential Method: A Treatise of the Summation and Interpolation ofInfinite Series. 1749. Tricomi, F. G. "A Class of Non-Orthogonal Polynomials Related to those of Laguerre." J. Analyse M. 1, 209/C1 /31, 1951. Young, P. T. "Congruences for Bernoulli, Euler, and Stirling Numbers." J. Number Th. 78, 204/C1/27, 1999. Stirling Number of the Second Kind The number of ways of partitioning a set of n elements into mnonempty SETS (i.e., mBLOCKS ), also called a S TIRLING SET NUMBER . for example, the SET f1;2;3gcan be partitioned into three SUBSETS in one way: ff1g;f2g;f3gg; into two SUBSETS in three ways: ff1;2g;f3gg;ff1;3g;f2gg;and ff1g;f2;3gg; and into one SUBSET in one way: ff1;2;3gg:/ The Stirling numbers of the second kind are variously denoted S(n;m) (Riordan 1980, Roman 1984), S(m) n (Fort 1958, Abramowitz and Stegun 1971), Sm n(Jor- dan 1950), s(m) n;S2(n;m);orn mYr$Yr% (Graham et al. 1994). Abramowitz and Stegun (1971, p. 822) summarize the various notational conventions, which can be a bit confusing. The Mathematica command for a Stirling number of the second kind is StirlingS2 [n,m]. The Stirling numbers of the second kind for three ele-ments are S(3;1)/C301 (1) S(3;2)/C303 (2) S(3;3)/C301: (3) Since a set of nelements can only be partitioned in asingle way into 1 or n SUBSETS , S(n;1)/C30S(n;n)/C301: (4) Other special cases include S(n;0)/C30dn0 (5) S(n;2)/C302n/C281/C281 (6) S(n;n/C281)/C30n 2Yru$Yru% : (7) The triangle of Stirling numbers of the second kind is 1 11 131 1761 11 52 51 01 13 19 06 51 51 (Sloane’s A008277), the nth row of which corresponds to the coefficients of the EXPONENTIAL POLYNOMIAL fn(x):/ The Stirling numbers of the second kind can becomputed from the sum S(n;k)/C301 k!Xk/C281 i/C300(/C281)ik iYru$Yru% (k/C28i)n; (8) withn kYrvYru aBINOMIAL COEFFICIENT , or the GENERATING FUNCTIONS xn/C30Xn m/C300S(n;m)(x)m /C30Xn m/C300S(n;m)x(x/C281)/C1/C1/C1(x/C28m/C271); (9) where ( x)mis the FALLING FACTORIAL (Roman 1984, pp. 60 and 101), X n]kS(n;k)xn n!/C301 k!ex/C281 ðÞk; (10) and 1 (1/C28x)(1/C282x)/C1/C1/C1(1/C28kx)/C30Xk n/C301S(n;k)xn: (11) Other generating functions are Xn k/C281S(n;k)(k/C281)!zk/C30(/C281)nLi1/C28n(1/C271=z) (12) forn]2;where Lin(z) is the POLYLOGARITHM , and X/C12 k /C30mS(k; m)zk /C30zm Q/C12 k/C301(1 /C28 kz) : (13) Stirling numbers of the second kind are intimately connected with the POISSON DISTRIBUTION through the identity X/C12 k /C300e /C28xxk k!kn /C30Xn k /C301xkS(n; k) : (14) The above diagrams (Dickau) illustrate the definition of the Stirling numbers of the second kind S(n; m) for n /C303 and 4. Stirling numbers of the second kind obey the RECURRENCE RELATIONS S(n; k) /C30S(n /C281; k /C281) /C27kS(n /C281 ; k) (15) S(n; k) /C30Xn m/C30kkn/C28mS(m /C281; k /C281): (16) The STIRLING NUMBERS OF THE FIRST KIND s(n ; m) are connected with the Stirling numbers of the second kind S(n; m) through the formulas s(n; i) /C30Xn k /C30iXk j/C300s(n; k)s(k ; j)S(j ; i) (17) S(n; i) /C30Xn k /C30iXk j/C300S(n ; k)S(k ; j)s(j; i) (18) (Roman 1984, p. 67), as well as S(n; m) /C30Xn /C28m k /C300(/C281)k k /C27n /C281 k /C27n /C28mYru$Yru% /C22n /C28m n /C28k /C28mYru$Yru% s(k /C28m /C27n; k) (19) s(n; m) /C30Xn/C28m k/C300(/C281)k k /C27n /C281 k /C27n /C28mYru$Yru% /C22n /C28m n /C28k /C28mYru$Yru% s(k /C28m /C27n; k) (20) Xmax (k ; j)/C271 l/C300s(l; j)S(k ; 1) /C30 djk (21) Xmax (k ; j) /C271 l/C300s(k ; l)S(l ; j) /C30 djm : (22) Identities involving Stirling numbers of the second kind are given byXn m/C301(/C281)m(m /C281)!S(n; m) /C300 (23) Xm k /C300kn /C30Xn k /C300k!m /C271 k /C271Yru$Yru% S(n; k) (24) f(m; n) /C13X/C12 k /C301kn m m /C27 1 !l /C30(m /C271)Xm k /C301k!S(n; k)mk : (25) It turns out that f(1; n) can have only 0, 2, or 6 as a last DIGIT (Riskin 1995). The Stirling numbers of the second appear in the operator identity (x ˜D)n /C30Xn k/C300S(n; k)xkf(k) ; (26) where ˜Dis the differential operator d=dx(Roman 1984, p. 144), giving (x˜D)1/C30x˜D (27) (x˜D)2/C30x˜D/C27x2˜D2(28) (x˜D)3/C30x˜D/C273x2˜D2/C27x3˜D3(29) (x˜D)4/C30x˜D/C277x2˜D2/C276x3˜D3/C27x4˜D4(30) and so on. Similarly, [(x/C28a)˜D]n/C30Xn k/C300S(n;k)(x/C28a)k˜Dk(31) (Roman 1984, p. 146). See also BELL NUMBER ,COMBINATION LOCK,EXPO- NENTIAL POLYNOMIAL ,LENGYEL’S CONSTANT ,M INI- MAL COVER ,P OISSON DISTRIBUTION ,S TIRLING NUMBER OF THE FIRST KIND,STIRLING POLYNOMIAL , STIRLING TRANSFORM References Abramowitz, M. and Stegun, C. A. (Eds.). "Stirling Numbers of the Second Kind." §24.1.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 824 /C1/25, 1972. Butzer, P. L. and Hauss, M. "Stirling Functions of the First and Second Kinds; Some New Applications." Israel Math- ematical Conference Proceedings: Approximation, Interpo-lation, and Summability, in Honor of Amnon Jakimovskion his Sixty-Fifth Birthday (Ed. S. Baron and D. Levia- tan). Ramat Gan, Israel: IMCP, pp. 89 /C1 /08, 1991. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 91 /C1/2, 1996. Dickau, R. M. "Stirling Numbers of the Second Kind." http:// forum.swarthmore.edu/advanced/robertd/stirling2.html Dickau, R. "Visualizing Combinatorial Enumeration." Math- ematica in Educ. Res. 8,11/C1/8, 1999. Fort, T. Finite Differences. Oxford, England: Clarendon Press, 1948. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Stirling Numbers." §6.1 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, pp. 257 /C1/67, 1994. Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965. Knuth, D. E. "Two Notes on Notation." Amer. Math. Monthly 99, 403 /C1/22, 1992. Riordan, J. Combinatorial Identities. New York: Wiley, 1979. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980. Riskin, A. "Problem 10231." Amer. Math. Monthly 102, 175 /C1/ 76, 1995. Roman, S. The Umbral Calculus. New York: Academic Press, pp. 59 /C1/3, 1984. Sloane, N. J. A. Sequences A008277 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, 1997. Stirling, J. Methodus differentialis, sive tractatus de sum- mation et interpolation serierum infinitarium. London, 1730. English translation by Holliday, J. The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series. 1749. Young, P. T. "Congruences for Bernoulli, Euler, and Stirling Numbers." J. Number Th. 78, 204 /C1/27, 1999. Stirling Polynomial Polynomials Sk(x) which form the SHEFFER SEQUENCE for g(t) /C30e /C28t (1) f /C281(t) /C30ln1 1 /C28 e /C28t ! ; (2) where f /C281(t) is the INVERSE FUNCTION of f(t) ; and have GENERATING FUNCTION X/C12 k /C300Sk(x) k!tk /C30t 1 /C28 e /C28t !x /C271 : (3) The first few polynomials are S0(x) /C301 S1(x) /C301 2(x /C271) S2(x) /C301 12(3x /C272)(x /C271) S3(x) /C301 8 x(x /C271)2 : The Stirling polynomials are related to the STIRLING NUMBERS OF THE FIRST KIND s(n; m)by Sn(m) /C30( /C281)n m nYru$Yru% s(m /C271; m /C28n /C271); (4) wherem nYrvYru is a BINOMIAL COEFFICIENT and m is an integer with m ]n; and to STIRLING NUMBERS OF THE SECOND KIND S(n ; m)bySn(m) /C30(/C281)nn! (n /C28 m /C28 1)!S(n /C28m /C281;/C28m /C281) (5) for m a NEGATIVE INTEGER . See also STIRLING NUMBER OF THE FIRST KIND, STIRLING NUMBER OF THE SECOND KIND References Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 257, 1981. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Stirling Set Number STIRLING NUMBER OF THE SECOND KIND Stirling Transform The transformation of a sequence a1 ; a2 ; ... into a sequence b1 ; b2 ; ..., by the formula bn /C30Xn k /C300S(n; k)ak ; where S(n; k)isaS TIRLING NUMBER OF THE SECOND KIND . The inverse transform is given by an /C30Xn k/C300s(n; k)bk ; where s(n; k)isaS TIRLING NUMBER OF THE FIRST KIND (Sloane and Plouffe 1995, p. 23). The Stirling transform of an /C301 for all n gives the BELL NUMBERS 1, 2, 5, 15, 52, ... (Sloane’s A000110). The Stirling transform of an /C30n gives 1, 3, 10, 37, 151, 674, ... (Sloane’s A005493), which has EXPONENTIAL GENERATING FUNCTION g(x) /C30exp ex /C272x /C281 ðÞ : The Stirling transform of the sequence an /C301 for n prime and an /C300 for n composite is 0, 1, 4, 13, 41, 136, 505, .... The Stirling transform of the sequence an /C301 for n even and an /C300 for n odd is 0, 1, 3, 8, 25, 97, 434, 2095, ... (Sloane’s A024430). The Stirling transform of the sequence an/C300 for neven and an/C301 for nodd is 1, 1, 2, 7, 27, 106, 443, ... (Sloane’s A024429). The inverse Stirling transform of bn/C30nis given by the sequence of signed factorials 1, 1, -1, 2, -6, 24, -120, .... See also BINOMIAL TRANSFORM ,EULER TRANSFORM , EXPONENTIAL TRANSFORM ,MO¨ BIUS TRANSFORM ,STIR- LING NUMBER OF THE FIRST KIND,STIRLING NUMBER OF THE SECOND KIND References Bernstein, M. and Sloane, N. J. A. "Some Canonical Se- quences of Integers." Linear Algebra Appl. 226//228 ,5 7/C1/ 2, 1995. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Factorial Factors." §4.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 252, 1994. Riordan, J. Combinatorial Identities. New York: Wiley, p. 90, 1979. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 48, 1980. Sloane, N. J. A. Sequences A000110/M1483, A005493/ M2851, A024429, A024430, and A052437 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Stirling’s Approximation Stirling’s approximation gives an approximate value for the FACTORIAL function n! or the GAMMA FUNCTION G(n) for n /C271: The approximation can most simply be derived for n an INTEGER by approximating the sum over the terms of the FACTORIAL with an INTEGRAL ,so that ln n! /C30ln 1 /C27ln 2 /C27.../C27ln n /C30Xn k/C301ln k :gn 1ln xdx /C30[x ln x /C28x]n 1 /C30n ln n /C28n /C271 :n ln n /C28n : (1) The equation can also be derived using the integral definition of the FACTORIAL , n! /C30g/C12 0e /C28x xn dx: (2) Note that the derivative of the LOGARITHM of the integrand can be written d dxln e/C28xxnðÞ /C30d dx (n ln x /C28x) /C30n x /C281 : (3) The integrand is sharply peaked with the contribu- tion important only near x /C30n. Therefore, let x /C13 n /C27 j where j /C26n; and write ln(xne/C28x) /C30n ln x /C28x /C30n ln(n /C27 j) /C28(n /C27 j): (4) Now, ln(n /C27 j) /C30ln n 1 /C27j n !"# /C30ln n /C27ln 1 /C27j n ! /C30ln n /C27j n /C281 2j n2 /C27/C1/C1/C1; (5) so ln(xne /C28n) /C30n ln(n /C27 j) /C28(n /C27 j) /C30n ln n /C27 j /C2812j2 n/C28n /C28 j /C27... /C30n ln n /C28n /C28j2 2n /C27... (6)Taking the EXPONENTIAL of each side then gives xne /C28x :en ln ne /C28ne/C28 j2 =2n /C30nne /C28ne/C28 j2 =2n : (7) Plugging into the integral expression for n! then gives n! :g/C12 /C28nnne /C28ne /C28 j2 =2n dj :nne /C28ng/C12 /C28/C12e /C28 j2 =2n dj: (8) Evaluating the integral gives n! :nne/C28nffiffiffiffiffiffiffiffiffi 2pnp : (9) /C30ffiffiffiffiffiffi 2 pp nn /C271 =2e /C28n (10) (Wells 1986, p. 45). Taking the LOGARITHM of both sides then gives ln n! :n ln n /C28n /C271 2ln(2 pn) /C30 n /C2712Yru*Yru+ ln n /C28n /C2712ln(2 p) : (11) This is STIRLING’S SERIES with only the first term retained and, for large n, it reduces to Stirling’s approximation ln n! :n ln n /C28n: (12) Taking successive terms of //C28nn =n! /C29/, where xbcis the FLOOR FUNCTION , gives the sequence 1, 2, 4, 10, 26, 64, 163, 416, 1067, 2755, ... (Sloane’s A055775). Stirling’s approximation can be extended to the double inequality ffiffiffiffiffiffi 2 pp nn/C271 =2e /C28n/C271 =(12n/C271) Bn! Bffiffiffiffiffiffi2pp nn/C271=2e/C28n/C271=(12n)(13) (Robbins 1955, Feller 1968). Gosper has noted that a better approximation to n! (i.e., one which approximates the terms in S TIRLING’S SERIES instead of truncating them) is given by n!:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n/C271 3Yru*Yru+ pr nne/C28n: (14) This also gives a much closer approximation to the FACTORIAL of 0, 0! /C301;yieldingffiffiffiffiffiffiffiffi p=3p :1:02333 in- stead of 0 obtained with the conventional Stirling approximation. See also STIRLING’S SERIES References Feller, W. "Stirling’s Formula." §2.9 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 50 /C1/3, 1968. Robbins, H. "A Remark of Stirling’s Formula." Amer. Math. Monthly 62,2 6/C1/9, 1955. Sloane, N. J. A. Sequences A055775 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Stirling, J. Methodus differentialis. 1730. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 45, 1986. Whittaker, E. T. and Robinson, G. "Stirling’s Approximation to the Factorial." §70 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 138 /C1/40, 1967. Stirling’s Finite Difference Formula fp /C30f0 /C271 2 p d1=2 /C27 d/C281=2Yru*Yru+ /C2712 p2 d2 0 /C27S3d21=2 /C27 d2/C281=2Yru*Yru+ /C27S4 d40 /C27... for p /C23 [/C281=2 ; 1 =2]; where d is the CENTRAL DIFFER- ENCE and S2n /C271 /C301 2p /C27n 2n /C271Yru$Yru% S2n/C272 /C30p 2n /C27 2p /C27n 2n /C271Yru$Yru% : withn kYrvYru a BINOMIAL COEFFICIENT . See also CENTRAL DIFFERENCE ,STEFFENSON’S FOR- MULA References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Whittaker, E. T. and Robinson, G. "The Newton-Stirling Formula." §23 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 38 /C1/9, 1967. Stirling’s Formula STIRLING’S APPROXIMATION ,STIRLING’S SERIES Stirling’s Series The ASYMPTOTIC SERIES for the GAMMA FUNCTION is given by G(z) /C2e /C28zzz/C281 =2ffiffiffiffiffiffi 2pp /C2 1 /C271 12z /C271 288z2 /C28139 51840 z3 /C28571 2488320 z4 /C27... ! (1) (Sloane’s A001163 and A001164). The coefficient an of z /C28n can given explicitly by an /C30X2n k /C301(/C281)kd3(2n /C27 2k ; k) 2n/C27k(n /C27 k)!; (2) where d3(n; k) is the number of permutations of n with k CYCLES all of which are ]3 (Comtet 1974, p. 267). Another formula for the an/s is given by the recurrence relation bn /C301 n /C27 1bn/C281 /C28Xn/C281 k /C302kakan/C271/C28k ! ; (3) with b0 /C30b1 /C301; thenan /C30(2n /C271)!!b2n /C271 : (4) where x!! is the DOUBLE FACTORIAL (Borwein and Corless 1999). The series for z! is obtained by adding an additional factor of z, z! /C30G(z /C271) /C30e /C28zzz/C271=2ffiffiffiffiffiffi 2pp /C2 1 /C271 12z /C271 288z2 /C28139 51840 z3 /C28571 2488320 z4 /C27... ! : (5) The expansion of ln G(z) is what is usually called Stirling’s series. It is given by the simple analytic expression lnG(z)/C30X/C12 n/C301B2n 2n(2n/C281)z2n/C281(6) /C301 2ln(2p)/C27z/C2812Yru*Yru+ lnz/C28z/C271 12z/C281 360z3/C271 1260 z5 /C28. . . (7) where Bnis a B ERNOULLI NUMBER . See also BERNOULLI NUMBER ,CYCLE (PERMUTATION ), K-FUNCTION ,STIRLING’S APPROXIMATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 257, 1972. Arfken, G. "Stirling’s Series." §10.3 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 555 /C1/59, 1985. Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899/C1/09, 1999. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 267, 1974. Conway, J. H. and Guy, R. K. "Stirling’s Formula." In The Book of Numbers. New York: Springer-Verlag, pp. 260 /C1/ 61, 1996. Marsaglia, G. and Marsaglia, J. C. "A New Derivation of Stirling’s Approximation to n!:/"Amer. Math. Monthly 97, 826/C1/29, 1990. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 443, 1953. Sloane, N. J. A. Sequences A001163/M5400 and A001164/ M4878 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Uhler, H. S. "The Coefficients of Stirling’s Series for log G(z): / "Proc. Nat. Acad. Sci. U.S.A. 28,5 9/C1/2, 1942. Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput. 22, 617/C1/26, 1968. StirlingS1 STIRLING NUMBER OF THE FIRST KIND StirlingS2 STIRLING NUMBER OF THE SECOND KIND Stirrup Curve A plane curve given by the equation x2 /C281YrvYru2/C30y2(y /C281)(y /C282)(y /C275): References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Stochastic See also RANDOM VARIABLE ,STOCHASTIC APPROXIMA- TION ,STOCHASTIC CALCULUS ,STOCHASTIC GEOME- TRY,S TOCHASTIC GROUP ,S TOCHASTIC MATRIX , STOCHASTIC OPTIMIZATION ,S TOCHASTIC PROCESS , STOCHASTIC RESONANCE Stochastic Approximation A method of STOCHASTIC OPTIMIZATION including techniques such as gradient search or ROBBINS- MONRO STOCHASTIC APPROXIMATION . See also ROBBINS- MONRO STOCHASTIC APPROXIMA- TION ,STOCHASTIC OPTIMIZATION Stochastic Calculus References Durrett, R. Stochastic Calculus: A Practical Introduction. Boca Raton, FL: CRC Press, 1996. Stochastic Calculus of Variations MALLIAVIN CALCULUS Stochastic Function A function f(t) of one or more parameters containing a noise term e(t) f(t) /C30L(t) /C27 e(t) : where the noise is (without loss of generality) assumed to be additive. See also NOISE,STOCHASTIC OPTIMIZATIONStochastic Geometry The study of random geometric structures. Stochastic geometry leads to modelling and analysis tools such as MONTE CARLO METHODS . See also GEOMETRIC PROBABILITY ,INTEGRAL GEOME- TRY,MONTE CARLO METHOD ,RANDOM POLYGON References Kendall, W. S.; Barndorff-Nielson, O.; and van Lieshout, M. C. Current Trends in Stochastic Geometry: Likelihood and Computation. Boca Raton, FL: CRC Press, 1998. Stoyan, D.; Kendall, W. S.; and Mecke, J. Stochastic Geo- metry and Its Applications, with a Foreword by D. G. Ken- dall. New York: Wiley, 1987. Stochastic Group The GROUP of all nonsingular n /C29n STOCHASTIC MATRICES over a FIELD F. It is denoted S(n; F): If p is PRIME and F is the FINITE FIELD of ORDER q /C30pm ; S(n; q) is written instead of S(n; F): Particular examples include S(2; 2) /C30Z2 S(2; 3) /C30S3 S(2; A) /C30A4 S(3; 2) /C30S4 S(2; 5) /C30Z4 /C29u Z5 where Z2is an ABELIAN GROUP , Snare SYMMETRIC GROUPS on n elements, and /C29u denotes the semidirect product with u : Z4 0 Aut(Z5) (Poole 1995). See also STOCHASTIC MATRIX References Poole, D. G. "The Stochastic Group." Amer. Math. Monthly 102, 798/C1/01, 1995. Stochastic Matrix A stochastic matrix is the transition matrix for a finite M ARKOV CHAIN , also called a M ARKOV MATRIX . Elements of the matrix must be REAL NUMBERS in the CLOSED INTERVAL [0, 1]. A completely independent type of stochastic matrix isdefined as a SQUARE MATRIX with entries in a FIELD F such that the sum of elements in each column equals 1. There are two nonsingular 2 /C292STOCHASTIC MATRICES overZ2(i.e., the integers mod 2), 10 01YrtvYrtu and1001YrtvYrtu : There are six nonsingular stochastic 2 /C292 MATRICES overZ3; 0110YrtvYrtu ;0212YrtvYrtu ;1001YrtvYrtu ;1202YrtvYrtu ;2021YrtvYrtu ;2120YrtvYrtu ; In fact, the set S of all nonsingular stochastic n /C29n matrices over a FIELD F forms a GROUP under MATRIX MULTIPLICATION . This GROUP is called the STOCHASTIC GROUP . The following tables give the number of distinct stochastic matrices (and distinct nonsingular stochas- tic matrices) over Zm for small m. m stochastic n /C29n matrices over Zm / 2 1, 4, 64, 4096, ... 3 1, 9, 729, ... 4 1, 16, 4096, ... m stochastic nonsingular n /C29n matrices over Zm / 2 1, 2, 24, 1440, ... 3 1, 6, 450, ... 4 1, 12, 3108, ... See also DOUBLY STOCHASTIC MATRIX ,HORN’S THEO- REM,M AJORIZATION ,M ARKOV CHAIN ,S TOCHASTIC GROUP References Poole, D. G. "The Stochastic Group." Amer. Math. Monthly 102, 798 /C1/01, 1995. Stochastic Optimization Stochastic optimization refers to the minimization (or maximization) of a function in the presence of randomness in the optimization process. The random- ness may be present as either noise in measurements or Monte Carlo randomness in the search procedure, or both. Common methods of stochastic optimization include direct search methods (such as the NELDER- MEAD METHOD ), STOCHASTIC APPROXIMATION , stochastic pro- gramming, and miscellaneous methods such as SIMU- LATED ANNEALING and GENETIC ALGORITHMS . See also GENETIC ALGORITHM ,NELDER- MEAD METH- OD,OPTIMIZATION ,OPTIMIZATION THEORY ,ROBBINS- MONRO STOCHASTIC APPROXIMATION ,SIMULATED AN- NEALING ,STOCHASTIC APPROXIMATION Stochastic Process Doob (1996) defines a stochastic process is a family of RANDOM VARIABLES x(t;/C147) ; t /C23J fg from some PROB- ABILITY SPACE (S; S; P) into a STATE SPACE (S ?; S ?): Here, J is the INDEX SET of the process.Papoulis (1984, p. 312) describes a stochastic process x(t) as a family of functions. See also INDEX SET,PROBABILITY SPACE ,RANDOM VARIABLE ,STATE SPACE References Doob, J. L. "The Development of Rigor in Mathematical Probability (1900 /C1/950)." Amer. Math. Monthly 103, 586 /C1/ 95, 1996. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. Stochastic Resonance A stochastic resonance is a phenomenon in which a nonlinear system is subjected to a periodic modulated signal so weak as to be normally undetectable, but it becomes detectable due to resonance between the weak deterministic signal and stochastic NOISE . The earliest definition of stochastic resonance was the maximum of the output signal strength as a function of NOISE (Bulsara and Gammaitoni 1996). See also KRAMERS RATE,NOISE References Benzi, R.; Sutera, A.; and Vulpiani, A. "The Mechanism of Stochastic Resonance." J. Phys. A 14, L453-L457, 1981. Bulsara, A. R. and Gammaitoni, L. "Tuning in to Noise." Phys. Today 49,39/C1/5, March 1996. Gammaitoni, L. "Stochastic Resonance E-Print Server." http://www.umbrars.com/sr/. Sto¨hr Sequence Let a1 /C301 and define an/C271to be the least INTEGER greater than an which cannot be written as the SUM of at most h ]2 ADDENDS among the terms a1 ; a2 ; ..., an : This defines the h-Sto¨hr sequence. The first few of these are given in the following table. h Sloane h-Sto¨hr sequence 2 A033627 1, 2, 4, 7, 10, 13, 16, 19, 22, 25, ... 3 A026474 1, 2, 4, 8, 15, 22, 29, 36, 43, 50, ... 4 A051039 1, 2, 4, 8, 16, 31, 46, 61, 76, 91, ... 5 A051040 1, 2, 4, 8, 16, 32, 63, 94, 125, 156, ... See also GREEDY ALGORITHM ,INTEGER RELATION , POSTAGE STAMP PROBLEM , S-ADDITIVE SEQUENCE , SUBSET SUM PROBLEM ,SUM-FREE SET,U LAM SE- QUENCE References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 233, 1994. Mossige, S. "The Postage Stamp Problem: An Algorithm to Determine the h-Range on the h-Range Formula on the Extremal Basis Problem for k /C304." Math. Comput. 69, 325 /C1/37, 2000. Selmer, E. S. "On Sto¨hr’s Recurrent h-Bases for N." Kgl. Norske Vid. Selsk. Skrifter 3,1/C1/5, 1986. Selmer, E. S. and Mossige, S. "Sto¨hr Sequences in the Postage Stamp Problem." Bergen Univ. Dept. Pure Math. , No. 32, Dec. 1984. Sloane, N. J. A. Sequences A026474, A033627, A051039, and A051040 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Stokes Phenomenon The ASYMPTOTIC SERIES of the AIRY FUNCTION Ai(z) (and other similar functions) has a different form in different sectors of the COMPLEX PLANE . See also AIRY FUNCTIONS References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 609 /C1/11, 1953. Stokes’ Theorem For v a DIFFERENTIAL (K-1)-FORM with compact support on an oriented n-dimensional MANIFOLD WITH BOUNDARY M, gMdv /C30g@Mv; (1) where dv is the EXTERIOR DERIVATIVE of the differ- ential form v: When M is a COMPACT MANIFOLD without boundary, then the formula holds with the right hand side zero. Stokes’ theorem connects to the "standard" GRADIENT , CURL , and DIVERGENCE THEOREMS by the following relations. If f is a function on R3 ; grad (f) /C30c/C281 df : (2) where c : R3 0 R3 /C31 (the dual space) is the duality isomorphism between a VECTOR SPACE and its dual, given by the Euclidean INNER PRODUCT on R3 : If f is a VECTOR FIELD on a R3 ; div(f) /C30/C31d/C31c(f); (3) where /C31 is the HODGE STAR operator. If f is a VECTOR FIELD on R3 ; curl( f) /C30c /C281 /C31 dc(f) : (4) With these three identities in mind, the above Stokes’ theorem in the three instances is transformed into the GRADIENT , CURL , and DIVERGENCE THEOREMS respectively as follows. If f is a function on R3 and g is a curve in R3 ; then g0grad( f) /C215 dl /C30ggdf /C30f( g(1)) /C28f( g(0)); (5) which is the GRADIENT THEOREM .Iff : R3 0 R3 is aVECTOR FIELD and M an embedded compact 3-mani- fold with boundary in R3 ; then g@Mf /C215 dA /C30g@M/C31cf /C30gMd+cf /C30gMdiv(f) dV ; (6) which is the DIVERGENCE THEOREM .Iff is a VECTOR FIELD and M is an oriented, embedded, compact 2- MANIFOLD with boundary in R3 ; then g@Mfdl/C30g@Mcf /C30gMdc(f) /C30gMcurl( f) /C215 dA; (7) which is the CURL THEOREM . DE RHAM COHOMOLOGY is defined using DIFFEREN- TIAL K-FORMS . When N is a SUBMANIFOLD (without boundary), it represents a homology class. Two closed forms represent the same COHOMOLOGY CLASS if they differ by an EXACT FORM , v1 /C28 v2 /C30dh : Hence, gNv1 /C28 v2 /C30gNdh /C300: (8) Therefore, the evaluation of a COHOMOLOGY CLASS on a HOMOLOGY CLASS is WELL DEFINED . Physicists generally refer to the CURL THEOREM gS( 9/C29F) /C215 da /C30g@SF /C215 ds (9) as Stokes’ theorem. See also COHOMOLOGY ,CURL THEOREM ,D IFFEREN- TIAL K-FORM,DIVERGENCE THEOREM , See also EXTERIOR ALGEBRA ,EXTERIOR DERIVATIVE , GRADIENT THEOREM ,H ODGE STAR,INTEGRATION (FORM), JACOBIAN ,M ANIFOLD ,POINCARE ´ ’S LEMMA , TANGENT BUNDLE References Berger, M. Differential Geometry. New York: Springer- Verlag, pp. 195 /C1/03, 1988. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 2nd ed. Houston, TX: Publish or Perish, pp. 343 /C1/83, 1999. Sternberg, S. Differential Geometry. New York: Chelsea, p. 119, 1983. Stolarsky Array An INTERSPERSION array given by 1235 8 1 32 13 4 55 /C1/C1/C1 4 6 10 16 26 42 68 110 178 /C1/C1/C1 7 11 18 29 47 76 123 199 322 /C1/C1/C1 9 15 24 39 6 102 165 267 432 /C1/C1/C1 12 19 31 50 81 131 212 343 555 /C1/C1/C1 14 23 37 60 97 157 254 411 665 /C1/C1/C1 17 28 45 73 118 191 309 500 809 /C1/C1/C1 20 32 52 84 136 220 356 576 932 /C1/C1/C1 22 36 58 94 152 246 398 644 1042 /C1/C1/C1 nnnn nnnnn::: the first row of which is the FIBONACCI NUMBERS . See also INTERSPERSION ,W YTHOFF ARRAY References Kimberling, C. "Interspersions and Dispersions." Proc. Amer. Math. Soc. 117, 313 /C1/21, 1993. Morrison, D. R. "A Stolarsky Array and Wythoff Pairs." In A Collection of Manuscripts Related to the Fibonacci Se- quence. Santa Clara, CA: Fibonacci Assoc., pp. 134 /C1/36, 1980. Stolarsky-Harborth Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let b(k) be the number of 1s in the BINARY expression of k. Then the number of ODD BINOMIAL COEFFICIENTS k jYru*Yru+ where 0 5j 5k is 2b(k) (Glaisher 1899, Fine 1947). The number of ODD elements in the first n rows of PASCAL’S TRIANGLE is f(n) /C30Xn/C281 k/C3002b(k) : (1) This function is well approximated by nu ; where u /C13ln 3 ln 2 /C301:58496... : (2) Stolarsky and Harborth showed that 0:812556 5lim inf n0/C12f(n) nuB0 :812557 Blim sup n0/C12f(n) nu /C301 : (3) The value SH /C30lim inf n0/C12f(n) nu (4) is called the Stolarsky-Harborth constant. See also BINARY ,B INOMIAL COEFFICIENT ,R UDIN- SHAPIRO SEQUENCE References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/stlrsky/stlrsky.html.Fine, N. J. "Binomial Coefficients Modulo a Prime." Amer. Math. Monthly 54, 589 /C1/92, 1947. Wolfram, S. "Geometry of Binomial Coefficients." Amer. Math. Monthly 91, 566 /C1/71, 1984. Stolarsky’s Inequality If 0 5g(x) 51 and g is nonincreasing on the INTERVAL [0, 1], then for all possible values of a and b, g1 0g(x1=(a /C27b)) dx ]g1 0g(x1=a) dxg1 0g(x1 =b) dx: Stomachion A DISSECTION game similar to TANGRAMS described in fragmentary manuscripts attributed to Archimedes and was referred to as the LOCULUS OF ARCHIMEDES (Archimedes’ box) in Latin texts. The word Stoma- chion has as its root the Greek word for stomach. The game consisted of 14 flat pieces of various shapes arranged in the shape of a square. Like TANGRAMS , the object is to rearrange the pieces to form interest- ing shapes. See also DISSECTION ,TANGRAM References Rorres, C. "Stomachion Introduction." http://www.mcs.drex- el.edu/~crorres/Archimedes/Stomachion/intro.html. Rorres, C. "Stomachion Construction." http://www.mcs.drex- el.edu/~crorres/Archimedes/Stomachion/construc- tion.html. Stone Space Let P(L) be the set of all PRIME IDEALS of L, and define r(a) /C30fP½a QPg: Then the Stone space of L is the TOPOLOGICAL SPACE defined on P(L) by postulating that the sets OF THE FORM r(a) are a subbase for the open sets. See also PRIME IDEAL ,TOPOLOGICAL SPACE References Gra¨tzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, p. 119, 1971. Stone-von Neumann Theorem A theorem which specifies the structure of the generic unitary representation of the Weyl relations and thus establishes the equivalence of Heisenberg’s matrix mechanics and Schro ¨dinger’s wave mechanics formu- lations of quantum mechanics in Euclidean Rn space. References Neumann, J. von. "Die Eindeutigkeit der Schro ¨dingerschen Operationen." Math. Ann. 104, 570 /C1/78, 1931. Stone-Weierstrass Theorem If X is any COMPACT SPACE , let A be a subalgebra of the algebra C(X) over the reals R with binary operations /C27 and /C29: Then, if A contains the constant functions and separates the points of X, A is dense in (C(X) ; tn) ; where tn is a metrizable space as defined by Cullen (1968, p. 286). References Cullen, H. F. "The Stone-Weierstrass Theorem" and "The Complex Stone-Weierstrass Theorem." In Introduction to General Topology. Boston, MA: Heath, pp. 286 /C1/93, 1968. Stopper Knot A KNOT used to prevent the end of a string from slipping through a hole. References Owen, P. Knots. Philadelphia, PA: Courage, p. 11, 1993. Størmer Number A Størmer number is a POSITIVE INTEGER n for which the largest PRIME factor p of n2 /C271 is at least 2n: Every GREGORY NUMBER /tx/ can be expressed uniquely as a sum of tn/s where the ns are Størmer numbers. Conway and Guy (1996) give a table of Størmer numbers reproduced below (Sloane’s A005529). In a paper on INVERSE TANGENT relations, Todd (1949) gives a similar compilation. npnpnpnpn p 1 2 10 101 19 181 26 677 35 613 2 5 11 61 20 401 27 73 36 1297 4 17 12 29 22 97 28 157 37 137 5 13 14 197 23 53 29 421 39 761 6 37 15 113 24 577 33 109 40 1601 9 41 16 257 25 313 34 89 42 353 See also GREGORY NUMBER ,INVERSE TANGENTReferences Conway, J. H. and Guy, R. K. "Størmer’s Numbers." The Book of Numbers. New York: Springer-Verlag, pp. 245 /C1/ 48, 1996. Sloane, N. J. A. Sequences A005529/M1505 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Todd, J. "A Problem on Arc Tangent Relations." Amer. Math. Monthly 56, 517 /C1/28, 1949. Straight Angle An ANGLE of 180/C14/C30 p RADIANS . See also ACUTE ANGLE ,ANGLE ,DIGON ,FULL ANGLE , OBTUSE ANGLE ,REFLEX ANGLE ,RIGHT ANGLE Straight Line LINE Straight Polyomino The straight polyomino of order n is the n-POLY- OMINO in which all squares are placed along a line. See also L-POLYOMINO ,SKEW POLYOMINO ,SQUARE POLYOMINO ,T-POLYOMINO Straightedge An idealized mathematical object having a rigorously straight edge which can be used to draw a LINE SEGMENT . Although GEOMETRIC CONSTRUCTIONS are sometimes said to be performed with a RULER and COMPASS , the term straightedge is preferable to RULER since markings on the straightedge (usually assumed to be present on a RULER ) are not allowed by the classical Greek rules. See also COMPASS ,GEOMETRIC CONSTRUCTION ,GEO- METROGRAPHY ,M ASCHERONI CONSTANT ,P OLYGON , PONCELET- STEINER THEOREM ,R ULER ,S IMPLICITY , STEINER CONSTRUCTION Strange Attractor An attracting set that has zero MEASURE in the embedding PHASE SPACE and has FRACTAL dimension. Trajectories within a strange attractor appear to skip around randomly. See also CORRELATION EXPONENT ,FRACTAL References Benmizrachi, A.; Procaccia, I.; and Grassberger, P. "Char- acterization of Experimental (Noisy) Strange Attractors." Phys. Rev. A 29, 975 /C1/77, 1984. Grassberger, P. "On the Hausdorff Dimension of Fractal Attractors." J. Stat. Phys. 26, 173 /C1/79, 1981. Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/08, 1983a. Grassberger, P. and Procaccia, I. "Characterization of Strange Attractors." Phys. Rev. Let. 50, 346 /C1/49, 1983b. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 137 /C1/ 38, 1991. Sprott, J. C. Strange Attractors: Creating Patterns in Chaos. New York: Henry Holt, 1993. Viana, M. "What’s New on Lorenz Strange Attractors." Math. Intell. 22,6/C1/9. Strange Loop A phenomenon in which, whenever movement is made upwards or downwards through the levels of some hierarchical system, the system unexpectedly arrives back where it started. Hofstadter (1987) uses the strange loop as a paradigm in which to interpret paradoxes in logic (such as GRELLING’S PARADOX and RUSSELL’S PARADOX ) and calls a system in which a strange loop appears a TANGLED HIERARCHY . See also GRELLING’S PARADOX ,RUSSELL’S PARADOX , TANGLED HIERARCHY References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 10, 1989. Strangers Two numbers which are RELATIVELY PRIME . See also RELATIVELY PRIME References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 145, 1983. Strassen Formulas The usual number of scalar operations (i.e., the total number of additions and multiplications) required toperform n/C29n MATRIX MULTIPLICATION is M(n)/C302n3/C28n2(1) (i.e., n3multiplications and n3/C28n2additions). How- ever, Strassen (1969) discovered how to multiply two MATRICES in S(n)/C307/C2157lgn/C286/C2154lgn(2) scalar operations, where lg is the LOGARITHM to base 2, which is less than M(n) for n/C21654. For na power of two ( /n/C302k);the two parts of (2) can be written7/C2157lgn/C307/C2157lg 2k/C307/C2157k/C307/C2152klg 7 /C3072kYrvYru lg 7/C307nlg 7(3) 6/C2154lgn/C306/C2154lg 2k/C306/C2154klg 2/C306/C2154k /C3062kYrvYru 2/C306n2; (4) so (2) becomes S(2k)/C307nlg 7/C286n2: (5) Two 2 /C292 matrices can therefore be multiplied C/C30AB (6) c11c12 c21c22YrtvYrtu /C30a11a12 a21a22YrtvYrtu b11b12 b21b22YrtvYrtu (7) with only S(2)/C307/C2152lg 7/C286/C21522/C3049/C2824/C3025 (8) scalar operations (as it turns out, seven of them aremultiplications and 18 are additions). Define theseven products (involving a total of 10 additions) as Q 1/C13a11/C27a22 ðÞ b11/C27b22 ðÞ (9) Q2/C13a21/C27a22 ðÞ b11 (10) Q3/C13a11b12/C28b22 ðÞ (11) Q4/C13a22/C28b11/C27b21 ðÞ (12) Q5/C13a11/C27a12 ðÞ b22 (13) Q6/C13/C28a11/C27a21 ðÞ b11/C27b12 ðÞ (14) Q7/C13a12/C28a22 ðÞ b21/C27b22 ðÞ : (15) Then the matrix product is given using the remaining eight additions as c11/C30Q1/C27Q4/C28Q5/C27Q7 (16) c21/C30Q2/C27Q4 (17) c12/C30Q3/C27Q5 (18) c22/C30Q1/C27Q3/C28Q2/C27Q6 (19) (Strassen 1969, Press et al. 1989). Matrix inversion of a 2 /C292 matrix Ato yield C/C30A-1 can also be done in fewer operations than expected using the formulas R1/C13a/C281 11 (20) R2/C13a21R1 (21) R3/C13R1a12 (22) R4/C13a21R3 (23) R5/C13R4/C28a22 (24) R6/C13R/C281 5 (25) c12 /C30R3R6 (26) c21 /C30R6R2 (27) R7 /C30R3c21 (28) c11 /C30R1 /C28R7 (29) c22 /C30/C28R6 (30) (Strassen 1969, Press et al. 1989). The leading exponent for Strassen’s algorithm for a POWER of 2 is lg 7 :2:808: The best leading exponent currently known is 2.376 (Coppersmith and Winograd 1990). It has been shown that the exponent must be at least 2. Unfortunately, Strassen’s algorithm is not numeri- cally well-behaved. It is only weakly stable, i.e., the computed result C /C30AB satisfies the inequality ½½C /C28AB ½½B/C30nu½½A ½½½½B ½½/C27O u2YrvYru ; (31) where u is the unit roundoff error, while the corresponding strong stability inequality (obtained by replacing matrix norms with absolute values of the matrix elements) does not hold. See also COMPLEX MULTIPLICATION ,K ARATSUBA MULTIPLICATION References Coppersmith, D. and Winograd, S. "Matrix Multiplication via Arithmetic Programming." J. Symb. Comput. 9, 251 /C1/ 80, 1990. Douglas, C.; Heroux, M.; Slishman, G.; and Smith, R. "GEMMW: A Portable Level 3 BLAS Winograd Variant of Strassen’s Matrix-Matrix Multiply Algorithm." J. Com- put. Phys. 110,1/C1/0, 1994. Pan, V. How to Multiply Matrices Faster. New York: Springer-Verlag, 1982. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Is Matrix Inversion an N3 Process?" §2.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 95 /C1/8, 1989. Strassen, V. "Gaussian Elimination is Not Optimal." Nu- merische Mathematik 13, 354 /C1/56, 1969. Strassman’s Theorem Let (K ;½/C215½) be a complete non-A RCHIMEDEAN VALU- ATED FIELD , with VALUATION RING R, and let f(X)bea POWER SERIES with COEFFICIENTS in R. Suppose at least one of the COEFFICIENTS is NONZERO (so that f is not identically zero) and the sequence of COEFFI- CIENTS converges to 0 with respect to ½/C215½: Then f(X) has only finitely many zeros in R. See also ARCHIMEDEAN VALUATION ,M AHLER- LECH THEOREM ,VALUATION ,VALUATION RINGStrassnitzky’s Formula The MACHIN-LIKE FORMULA 1 4 p /C30cot /C281 2 /C27cot /C281 5 /C27cot /C281 8: See also MACHIN’S FORMULA ,M ACHIN- LIKE FORMU- LAS Strategy A set of moves which a player plans to follow while playing a GAME . See also GAME,MIXED STRATEGY Stratified Manifold A set that is a smooth embedded 2-D MANIFOLD except for a subset that consists of smooth embedded curves, except for a set of ISOLATED POINTS . References Morgan, F. "What is a Surface?" Amer. Math. Monthly 103, 369 /C1/76, 1996. Strehl Identities The sum identities X/C12 j/C300n jYru$Yru%3 /C30X/C12 k /C300n jYru$Yru%22(n /C28k) nYru$Yru% and Xn k/C300Xn j/C300n kYru$Yru% n /C27k kYru$Yru% k jYru$Yru%3 /C30Xn k/C300n kYru$Yru% n /C27k kYru$Yru%2 (Strehl 1993; Strehl 1994; Koepf 1998, p. 55), where n kYrvYru is a BINOMIAL COEFFICIENT . See also BINOMIAL COEFFICIENT References Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Strehl, V. "Binomial Sums and Identities." Maple Technical Newsletter 10,37/C1/9, 1993. Strehl, V. "Binomial Identities--Combinatorial and Algorith- mic Aspects." Discrete Math. 136, 309 /C1/46, 1994. Stretch A TRANSFORMATION characterized by an invariant line and a scale factor (one-way stretch) or two invariant lines and corresponding scale factors (two- way stretch). See also TRANSFORMATION Strict Gelfand Pattern MONOTONE TRIANGLE Strict Inequality An INEQUALITY is strict if replacing any "less than" and "greater than" signs with equal signs never gives a true expression. For example, a 5b is not strict, whereas a Bb is. See also EQUALITY ,INEQUALITY Striction Curve A NONCYLINDRICAL RULED SURFACE always has a parameterization OF THE FORM x(u; v) /C30 s(u) /C27vd(u) ; (1) where ½ d½/C301 ; s?/C215 d?/C300; and s is called the striction curve of x. Furthermore, the striction curve does not depend on the choice of the base curve. The striction and DIRECTOR CURVES of the HELICOID x(u; v) /C300 0 bu2 435/C27avcos u sin u 02435 (2) are s(u) /C300 0 bu2 435 (3) d(u) /C30a cos u a sin u 02 435: (4) For the HYPERBOLIC PARABOLOID x(u; v) /C30u 0 02 435/C27v0 1 u2 435; (5) the striction and DIRECTOR CURVES are s(u) /C30u 0 02 435 (6) d(u) /C300 1 u2 435: (7) See also D IRECTOR CURVE ,DISTRIBUTION PARAMETER , NONCYLINDRICAL RULED SURFACE ,RULED SURFACE References Gray, A. "Noncylindrical Ruled Surfaces" and "Examples of Striction Curves of Noncylindrical Ruled Surfaces." §19.3 and 19.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 445 /C1/49, 1997. Strictly Egyptian Number EGYPTIAN NUMBERString A string of length k on an ALPHABET l of m characters is an arrangement of k not necessarily distinct symbols from l. There are mk such distinct strings. For example, the strings of length k /C30 3 on the alphabet f1; 2; 3g are f1; 1; 1 g;f1 ; 1 ; 2 g;f1; 2; 1g; f1; 2; 2g;f2; 1 ; 1 g;f2; 1; 2g;f2; 2; 1 g; and f2; 2; 2g: In Mathematica , strings of length k in the ALPHABET consisting of the members in a list l can be enumerated using the following function. Strings[l_List,k_Integer?Positive] : /C30 Modu- le[{k}, Flatten[Outer[List, Sequence @@ Table[l, {k}]], k-1] ] See also ALPHABET References Skiena, S. "Strings." §1.5.1 in Implementing Discrete Mathe- matics: Combinatorics and Graph Theory with Mathema- tica. Reading, MA: Addison-Wesley, p. 40, 1990. String Rewriting A SUBSTITUTION MAP in which rules are used to operate on a string consisting of letters of a certain alphabet. String rewriting is a particularly useful technique for generating successive iterations of certain types of FRACTALS , such as the BOX FRACTAL , CANTOR DUST ,CANTOR SQUARE FRACTAL , and SIER- PINSKI CARPET . See also RABBIT SEQUENCE ,SUBSTITUTION MAP References Peitgen, H.-O. and Saupe, D. (Eds.). "String Rewriting Systems." §C.1 in The Science of Fractal Images. New York: Springer-Verlag, pp. 273 /C1/75, 1988. Wagon, S. "Recursion via String Rewriting." §6.2 in Math- ematica in Action. New York: W. H. Freeman, pp. 190 /C1/ 96, 1991. Strip CRITICAL STRIP,MO¨ BIUS STRIP Strombic Hexecontahedron DELTOIDAL HEXECONTAHEDRON Strombus A term meaning "spinning top" in Greek which was coined by J. H. Conway by e-mail in the Polyhedron Discussion List as a term for kite-shaped quadrilat- erals. Formally, a strombus is a QUADRILATERAL ABCD that has ACfor an axis of symmetry. See also DIAMOND ,KITE,LOZENGE ,PARALLELOGRAM , QUADRILATERAL ,RHOMBOID ,RHOMBUS ,SKEW QUAD- RILATERAL ,STROMBUS ,TRAPEZOID Strong Convergence Strong convergence is the type of convergence usually associated with convergence of a SEQUENCE . More formally, a SEQUENCE fxn g of VECTORS in a normed space (and, in particular, in an INNER PRODUCT SPACE E )is called convergent to a VECTOR x in E if xn /C28x kk 0 0a s n 0/C12: See also CONVERGENT SEQUENCE ,INNER PRODUCT SPACE ,W EAK CONVERGENCE Strong Elliptic Pseudoprime Let n be an ELLIPTIC PSEUDOPRIME associated with (E, P), and let n /C271 /C302sk with k ODD and s ]0 : Then n is a strong elliptic pseudoprime when either kP /C13 0(mod n)or2rkP /C130 (mod n) for some r with 1 5r Bs:/ See also ELLIPTIC PSEUDOPRIME References Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 132 /C1/34, 1996. Strong Frobenius Pseudoprime A PSEUDOPRIME which obeys an additional restriction beyond that required for a FROBENIUS PSEUDOPRIME . A number n with (n ;2a) /C301 is a strong Frobenius pseudoprime with respect to x /C28a IFF n is a STRONG PSEUDOPRIME with respect to f(x) : Every strong Frobenius pseudoprime with respect to x /C28a is an EULER PSEUDOPRIME to the base a. Every strong Frobenius pseudoprime with respect to f(x) /C30x2 /C28bx /C28c such that b2 /C274c ðÞ =n ðÞ /C30/C281isa STRONG LUCAS PSEUDOPRIME with parameters (b, c). Every strong Frobenius pseudoprime n with respect to x2 /C28bx /C271isan EXTRA STRONG LUCAS PSEUDO- PRIME to the base b. See also FROBENIUS PSEUDOPRIME References Grantham, J. "Frobenius Pseudoprimes." 1996. http:// www.clark.net/pub/grantham/pseudo/pseudo1.ps Strong Goldbach Conjecture GOLDBACH CONJECTURE Strong Law of Large Numbers The sequence of variates Xiwith corresponding means miobeys the strong law of large numbers if, to every pair e: d > 0 ; there corresponds an N such that there is probability 1 /C28 d or better that for every r /C210, all r /C271 inequalitiesSn /C28 mn jj nB e for n /C30N, N /C271; ..., N /C27r will be satisfied, where Sn /C13Xn i/C301Xn mn /C13 Snhi/C30 m1 /C27.../C27 mn (Feller 1968). Kolmogorov established that the con- vergence of the sequence Xs2 k k2 ; sometimes called the Kolmogorov criterion, is a sufficient condition for the strong law of large numbers to apply to the sequence of mutually independent random variables Xkwith variances sk (Feller 1968). See also FRIVOLOUS THEOREM OF ARITHMETIC ,LAW OF LARGE NUMBERS ,LAW OF TRULY LARGE NUMBERS , STRONG LAW OF SMALL NUMBERS References Feller, W. "The Strong Law of Large Numbers." §10.7 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 243 /C1/45, 1968. Feller, W. "Strong Laws for Martingales." §7.8 in An Introduction to Probability Theory and Its Applications,Vol. 2, 3rd ed. New York: Wiley, pp. 234 /C1 /38, 1971. Strong Law of Small Numbers The first law of strong numbers (Gardner 1980, Guy 1988ab, Guy 1990) states "There aren’t enough smallnumbers to meet the many demands made of them." The second law of strong numbers (Guy 1990) states that "When two numbers look equal, in ain’t necessa- rily so." Guy (1988a) gives 35 examples of this statement, and 40 more in Guy (1990). For example,example 35 notes that the first few values of the interpolating polynomial n 4/C286n3/C2723n2ð / //C2818n/C2724Þ=24 (erroneously given with a coefficient 24 instead of 23) for n/C301, 2, ... are 1, 2, 4, 8, 16, ..., appears to give the powers of 2 (but the continues 31, 57, 99, ...). Similar, example 41 notes the curious factthat e (n/C281)=2Yr*Yr+ forn/C300, 1, ... gives 1, 1, 2, 5, 8, 13, 21, 34, 55, ... (the F IBONACCI NUMBERS ), although it subsequently continues 91, 149, ... (Sloane’sA005181). References Gardner, M. "Mathematical Games: Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243,1 8/C1/8, Dec. 1980. Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697/C1/12, 1988a. Guy, R. K. "Graphs and the Strong Law of Small Numbers." InProc. 6th Internat. Conf. Theory Appl. Graphs. Kala- mazoo, MI: 1988. Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63,3/C1/0, 1990. Sloane, N. J. A. Sequences A005181/M0693 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Strong Lucas Pseudoprime Let U(P; Q) and V(P; Q)beL UCAS SEQUENCES generated by P and Q, and define D /C13P2 /C284Q : Let n be an ODD COMPOSITE NUMBER with (n ; D) /C301; and n /C28(D=n) /C302sd with d ODD and s ]0; where (a=b) is the LEGENDRE SYMBOL .If Ud /C130 (mod n) or V2rd /C130 (mod n) for some r with 0 5r Bs ; then n is called a strong Lucas pseudoprime with parameters (P, Q). A strong Lucas pseudoprime is a LUCAS PSEUDOPRIME to the same base. Arnault (1997) showed that any COMPOSITE NUMBER n is a strong Lucas pseudoprime for at most /4=15/ of possible bases (unless n is the PRODUCT of TWIN PRIMES having certain properties). See also EXTRA STRONG LUCAS PSEUDOPRIME ,LUCAS PSEUDOPRIME References Arnault, F. "The Rabin-Monier Theorem for Lucas Pseudo- primes." Math. Comput. 66, 869 /C1/81, 1997. Ribenboim, P. "Euler-Lucas Pseudoprimes (elpsp( P, Q)) and Strong Lucas Pseudoprimes (slpsp( P, Q))." §2.X.C in The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 130 /C1/31, 1996. Strong Perfect Graph Conjecture The conjecture that a graph is PERFECT IFF neither the graph nor its complement contains an odd cycle of length at least five as an INDUCED SUBGRAPH (Go- lumbic 1980; Skiena 1990, p. 221). See also PERFECT GRAPH ,PERFECT GRAPH THEOREM References Golumbic, M. C. Algorithmic Graph Theory and Perfect Graphs. New York: Academic Press, 1980. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Strong Pseudoprime A strong pseudoprime to a base ais an ODD COMPOSITE NUMBER nwith n/C281/C30d /C2152s(fordODD) for which either ad/C131 (mod n) (1) orad /C2152s/C13/C281 (mod n) (2) for some r/C300, 1, ..., s/C281 (Riesel 1994, p. 91). Note that Guy (1994, p. 27) restricts the definition of strong pseudoprimes to only those satisfying (1). The definition is motivated by the fact that a F ERMAT PSEUDOPRIME nto the base bsatisfies bn/C281/C281/C130 (mod n): (3) But since nisODD, it can be written n/C302m/C271;and b2m/C281/C30bm/C281 ðÞ bm/C271 ðÞ /C130 (mod n): (4) IfnisPRIME , it must DIVIDE at least one of the FACTORS , but can’t DIVIDE both because it would then DIVIDE their difference bm/C271 ðÞ /C28bm/C281 ðÞ /C302: (5) Therefore, bm/C1391 (mod n): (6) so write n/C302at/C271 to obtain bn/C281/C281/C30bt/C271 ðÞ bt/C281 ðÞ b2t/C271YrvYru /C1/C1/C1b2a/C281t/C271Yru*Yru+ :(7) IfnDIVIDES exactly one of these FACTORS but is COMPOSITE , it is a strong pseudoprime. A COMPOSITE number is a strong pseudoprime to at most /1=4/of all bases less than itself (Monier 1980, Rabin 1980). The strong pseudoprimes provide the basis for M ILLER’S PRIMALITY TEST and R ABIN- MILLER STRONG PSEUDO- PRIME TEST . A strong pseudoprime to the base ais also an E ULER PSEUDOPRIME to the base a(Pomerance et al. 1980). The strong pseudoprimes include some E ULER PSEU- DOPRIMES ,FERMAT PSEUDOPRIMES , and C ARMICHAEL NUMBERS . The first few strong pseudoprimes to the base 2 are2047, 3277, 4033, 4681, ... (Sloane’s A001262). The number of strong pseudoprimes less than 10 3,1 04, ... are 0, 5, 16, 46, 162, ... (Sloane’s A055552). Note that Guy’s (1994, p. 27) definition gives only the subset 2047, 4681, 15841, 42799, 52633, 90751, ..., giving counts inconsistent with those in Guy’s table. The strong k-pseudoprime test for k/C302, 3, 5 correctly identifies all PRIMES below 2 :5/C291010with only 13 exceptions, and if 7 is added, then the only exception less than 2 :5/C291010is 315031751. Jaeschke (1993) showed that there are only 101 strong pseudoprimes for the bases 2, 3, and 5 less than 1012, nine if 7 is added, and none if 11 is added. Also, the bases 2, 13, 23, and 1662803 have no exceptions up to 1012. IfnisCOMPOSITE , then there is a base for which nis not a strong pseudoprime. There are therefore no "strong C ARMICHAEL NUMBERS ." Let ckdenote the smallest strong pseudoprime to all of the first k PRIMES taken as bases (i.e, the smallest ODD NUMBER for which the R ABIN- MILLER STRONG PSEUDOPRIME TEST on bases less than or equal to k fails). Jaeschke (1993) computed ckfrom k /C305 to 8 and gave upper bounds for k /C309 to 11. c1 /C302047 c2 /C301373653 c3 /C3025326001 c4 /C303215031751 c5 /C302152302898747 c6 /C303474749660383 c7 /C30341550071728321 c8 /C30341550071728321 c9 541234316135705689041 c10 51553360566073143205541002401 c11 /C3056897193526942024370326972321 (Sloane’s A014233). A seven-step test utilizing these results (Riesel 1994) allows all numbers less than 3:4 /C291014 to be tested. Pomerance et al. (1980) have proposed a test based on a combination of STRONG PSEUDOPRIMES and LUCAS PSEUDOPRIMES . They offer a $620 reward for discov- ery of a COMPOSITE NUMBER which passes their test (Guy 1994, p. 28). See also CARMICHAEL NUMBER ,M ILLER’S PRIMALITY TEST,POULET NUMBER ,RABIN- MILLER STRONG PSEU- DOPRIME TEST,ROTKIEWICZ THEOREM ,STRONG EL- LIPTIC PSEUDOPRIME ,STRONG LUCAS PSEUDOPRIME References Baillie, R. and Wagstaff, S. "Lucas Pseudoprimes." Math. Comput. 35, 1391 /C1/417, 1980. Guy, R. K. "Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes." §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 27 /C1/0, 1994. Jaeschke, G. "On Strong Pseudoprimes to Several Bases." Math. Comput. 61, 915 /C1/26, 1993. Monier, L. "Evaluation and Comparison of Two Efficient Probabilistic Primality Testing Algorithms." Theor. Com- put. Sci. 12,97/C1/08, 1980. Pinch, R. G. E. "The Pseudoprimes Up to 1013." ftp:// ftp.dpmms.cam.ac.uk/pub/PSP/. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimes to 25 /C215 109 :/" Math. Comput. 35, 1003 /C1/026, 1980. Available electronically from ftp://sable.ox.ac.uk/ pub/math/primes/ps2.Z. Rabin, M. O. "Probabilistic Algorithm for Testing Primality." J. Number Th. 12, 128 /C1/38, 1980. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkha ¨user, p. 92, 1994. Sloane, N. J. A. Sequences A001262, A014233, and A055552 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html.Strong Pseudoprime Test RABIN- MILLER STRONG PSEUDOPRIME TEST Strong Subadditivity Inequality f(A) /C27 f(B) /C28 f(A @ B) ] f(A S B) : References Doob, J. L. "The Development of Rigor in Mathematical Probability (1900 /C1/950)." Amer. Math. Monthly 103, 586 /C1/ 95, 1996. Strong Triangle Inequality The p-adic norm satisfies x /C27y jjp5max xjjp ; xjjpYru*Yru+ for all x and y. See also P-ADIC NUMBER ,TRIANGLE INEQUALITY Strong Twin Prime Conjecture TWIN PRIME CONJECTURE Strongly Connected Component A maximal SUBGRAPH of a DIRECTED GRAPH such that for every pair of vertices u, v in the SUBGRAPH , there is a directed path from u to v and a directed path from v to u. Tarjan (1972) has devised an O(n) algorithm for determining strongly connected compo- nents, which is implemented in Mathematica as StronglyConnectedComponents [g] in the Mathe- matica add-on package DiscreteMath‘Combina- torica‘ (which can be loaded with the command BBDiscreteMath‘ ) (Skiena 1990, p. 172). See also BI-CONNECTED COMPONENT ,STRONGLY CON- NECTED DIGRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Tarjan, R. E. "Depth-First Search and Linear Graph Algo- rithms." SIAM J. Comput. 1, 146/C1/60, 1972. Strongly Connected Digraph ADIRECTED GRAPH in which it is possible to reach any node starting from any other node by traversing edges in the direction(s) in which they point. The nodes in a strongly connected digraph therefore must all have INDEGREE of at least 1. The numbers of nonisomorphic simple strongly connected digraphs on n /C301, 2, ... nodes are 1, 1, 5, 83, 5048, 1047008, ... (Sloane’s A035512). See also CONNECTED DIGRAPH ,W EAKLY CONNECTED DIGRAPH References Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, p. 218, 1973. Liskovec, V. A. "A Contribution to the Enumeration of Strongly Connected Digraphs." Dokl. AN BSSR 17, 1077 /C1/080, 1973. Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998. Skiena, S. "Strong and Weak Connectivity." §5.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 94 and 172 /C1/74, 1990. Sloane, N. J. A. Sequences A035512 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Strongly Connected Graph STRONGLY CONNECTED DIGRAPH Strongly Embedded Theorem The strongly embedded theorem identifies all SIMPLE GROUPS with a strongly 2-embedded SUBGROUP .In particular, it asserts that no SIMPLE GROUP has a strongly 2-embedded 2’-local SUBGROUP . See also SIMPLE GROUP ,SUBGROUP Strongly Independent An infinite sequence aifg of POSITIVE INTEGERS is called strongly independent if any relation a eiai ; with ei /C300 ;9 1, or 9 2 and ei /C300 except finitely often, IMPLIES ei /C300 for all i. See also WEAKLY INDEPENDENT References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 136, 1994. Strongly Triple-Free Set TRIPLE- FREE SET Strophoid Let C be a curve, let O be a fixed point (the POLE ), and let O? be a second fixed point. Let P and P ? be points on a line through O meeting C at Q such that P?Q /C30 QP /C30QO?: The LOCUS of P and P? is called the strophoid of C with respect to the POLE O and fixed point O ?: Let C be represented parametrically by (f(t) ; g(t)); and let O /C30 x0 ; y0 ðÞ and O ?/C30 x1 ; y1 ðÞ : Then the equation of the strophoid isx /C30f 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 /C28 f ðÞ2/C27 y1 /C28 g ðÞ2 1 /C27 m2s (1) x /C30g 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 /C28 f ðÞ2/C27 y1 /C28 g ðÞ2 1 /C27 m2s ; (2) where m /C13g /C28 y0 f /C28 x0: (3) The name strophoid means "belt with a twist," and was proposed by Montucci in 1846 (MacTutor Ar- chive). The polar form for a general strophoid is r /C30b sin(a /C28 2u) sin(a /C28 u): (4) If a /C30 p=2; the curve is a RIGHT STROPHOID . The following table gives the strophoids of some common curves. Curve Pole Fixed Point Strophoid line not on lineon line oblique strophoid line not on linefoot of PERPENDI- CULAR origin to lineRIGHT STRO- PHOID CIRCLE center on the circumfer- enceFREETH’S NEPHROID See also RIGHT STROPHOID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 51 /C1/3 and 205, 1972. Lockwood, E. H. "Strophoids." Ch. 16 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 134 /C1/37, 1967. MacTutor History of Mathematics Archive. "Right." http:// www-groups.dcs.st-and.ac.uk/~history/Curves/Right.html. Yates, R. C. "Strophoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 217 /C1 /20, 1952. Structural Ramsey Theory A generalization of R AMSEY THEORY to mathematical objects in which one would not normally expect structure to be found. For example, there exists agraph with very few triangles (more precisely, a graph which can always be constructed so that there is no "cycle" of triangles which are all distinct andT i/C271meets Tiin at least one vertex) and such that however it is colored with rcolors, one of the colors contains a triangle. The usual proof of RAMSEY’S THEOREM gives no insight on how to prove such a result. See also EXTREMAL GRAPH THEORY ,RAMSEY’S THEO- REM,RAMSEY THEORY Structurally Stable A MAP f : M 0 M where M is a MANIFOLD is Cr structurally stable if any Cr perturbation is TOPOLO- GICALLY CONJUGATE to f: Here, Crperturbation means a FUNCTION c such that c is close to f and the first r derivatives of c are close to those of f:/ See also TOPOLOGICALLY CONJUGATE Structure LATTICE Structure Constant The structure constant is defined as i eijk ; where eijk is the PERMUTATION SYMBOL . The structure constant forms the starting point for the development of LIE ALGEBRA . See also LIE ALGEBRA ,PERMUTATION SYMBOL Structure Factor The structure factor SGof a discrete set G is the FOURIER TRANSFORM of d/-scatterers of equal strengths on all points of G; SG(k) /C30gX x /C23Gd x?/C28x ðÞ e /C282 pikx? dx ?/C30X x /C23Ge /C282pikx : References Baake, M.; Grimm, U.; and Warrington, D. H. "Some Re- marks on the Visible Points of a Lattice." J. Phys. A: Math. General 27, 2669 /C1/674, 1994. Strut TENSEGRITY Struve Differential Equation The ORDINARY DIFFERENTIAL EQUATION z2yƒ/C27zy ?/C27 z2 /C28 n2YrvYru y /C3041 2 zYru*Yru+n/C271 ffiffiffippG n /C271 2Yru*Yru+ ; where G(z) is the GAMMA FUNCTION (Abramowitz and Stegun 1972, p. 496; Zwillinger 1997, p. 127). The solution is y /C30aJn(z) /C27bYn(z) /C27H n(z); where Jn(z) and Y n(z) are BESSEL FUNCTIONS OF THEFIRST and SECOND KINDS , and Hn(z)isaS TRUVE FUNCTION (Abramowitz and Stegun 1972). See also BESSEL FUNCTION OF THE FIRST KIND, BESSEL FUNCTION OF THE SECOND KIND,STRUVE FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 496, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Struve Function Abramowitz and Stegun (1972, pp. 496 /C1/99) define the Struve function as Hn(z) /C30(1 2z) n/C271 X/C12 k /C300( /C281)k(12z)2k G(k /C273 2) G(k /C27 n /C2732) ; (1) where G(z) is the GAMMA FUNCTION . Watson (1966, p. 338) defines the Struve function as Hn(z) /C1321 2 zYru*Yru+n G n /C271 2Yru*Yru+ G12Yru*Yru+g1 01 /C28t2YrvYrun/C281=2sin(zt) dt: (2) The series expansion is Hn(z) /C30X/C12 m/C300(/C281)m1 2 zYru*Yru+2m/C27 n/C271 G m /C273 2Yru*Yru+ G n /C27 m /C2732Yru*Yru+ : (3) For half integer orders, Hn/C271=2(z) /C30Yn/C271=2(z) /C271 pXn m/C300G m /C2712Yru*Yru+ 12 zYru*Yru+/C282m/C27n/C281 =2 G(n /C27 1 /C28 m)(4) H/C28(n/C271=2)(z) /C30(/C281)nJn/C271 =2(z): (5) The Struve function and its derivatives satisfy Hn/C281(z) /C28H n/C271(z) /C302H?n(z) /C281 2 zYru*Yru+n ffiffiffippG n /C273 2Yru*Yru+ : (6) For integer n, the Struve function gives the solution to z2yƒ/C27zy?/C27z2/C28n2YrvYru y/C302 pzn/C271 (2n/C281)!!; (7) where n!! is the DOUBLE FACTORIAL . The Struve function is built into Mathematica 4.0 as StruveH [n,z]. See also ANGER FUNCTION ,BESSEL FUNCTION ,M OD- IFIED STRUVE FUNCTION ,W EBER FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Struve Function Hn(x):/"§12.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 496 /C1/98, 1972. Apelblat, A. "Derivatives and Integrals with Respect to the Order of the Struve Functions Hn(x) and Ln(x):/"J. Math. Anal. Appl. 137,1 7/C1/6, 1999. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Struve Functions Hn(x) and Ln(x):/"§1.4 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 24 /C1/7, 1990. Spanier, J. and Oldham, K. B. "The Struve Function." Ch. 57 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 563 /C1/71, 1987. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Struve H-Function STRUVE FUNCTION Struve L-Function MODIFIED STRUVE FUNCTION StruveH STRUVE FUNCTION StruveL MODIFIED STRUVE FUNCTION Student’s t-Distribution ASTATISTICAL DISTRIBUTION published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose "Student." Given nindependent measurements xi;let t/C13¯x/C28m s=ffiffiffinp: (1) where mis the population MEAN ,¯xis the sample MEAN , and sis the ESTIMATOR for population STAN- DARD DEVIATION (i.e., the SAMPLE VARIANCE ) defined by s2/C131 N/C281Xn i/C301xi/C28¯x ðÞ2: (2)Student’s t-distribution is defined as the distribution of the random variable twhich is (very loosely) the "best" that we can do not knowing s:Ifs/C30s;t/C30zand the distribution becomes the NORMAL DISTRIBUTION . AsNincreases, Student’s t-distribution approaches the NORMAL DISTRIBUTION . Student’s t-distribution can be derived by transform- ing S TUDENT’S Z-DISTRIBUTION using z/C13¯x/C28m s; (3) and then defining t/C13zffiffiffiffiffiffiffiffiffiffiffiffi n/C281p : (4) The resulting probability and cumulative distribution functions are fr(t)/C30G1 2(r/C271)hi ffiffiffiffiffirppG1 2rYru*Yru+ 1/C27t2 r !(r/C271)=2/C30r r/C27t2 !(1/C27r)=2 ffiffiffirpB1 2r;12Yru*Yru+ (5) Fr(t)/C30gt /C28/C12G1 2(r/C271)hi ffiffiffiffiffirppG1 2rYru*Yru+ 1/C27t?2 r !(r/C271)=2dt? /C301 2/C2712I1; 1 2r;12Yru*Yru+ /C28Ir r/C27t2;12r;12 ! "# /C301/C281 2Ir r/C27t2;1 2r;12 ! ; (6) where r/C13n/C281 (7) is the number of DEGREES OF FREEDOM ,/C28/C12B tB/C12 ; G(z) is the GAMMA FUNCTION ,B(a;b) is the BETA FUNCTION , and I(z;a;b) is the REGULARIZED BETA FUNCTION defined by I(z;a;b)/C30B(z;a;b) B(a;b): (8) The MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS of Student’s t-distribution are m/C300 (9) s2/C30r r/C282(10) g1/C300 (11) g2/C306 r/C284: (12) The CHARACTERISTIC FUNCTIONS fn(t) for the first few values of nare f1(t)/C30e/C28tjj(13) f2(t)/C30ffiffiffi 2p tjjK1ffiffiffi2p tjjYru*Yru+ (14) f 3(t)/C30e/C28ffiffi 3p tjj1/C27ffiffiffi3p tjjYru*Yru+ (15) f 4(t)/C302t2K22tjjðÞ (16) f5(t)/C301 3e/C28ffiffi 5p tjj3/C273ffiffiffi 5p tjj/C275t2Yru*Yru+ ; (17) and so on, where Kn(x)i sa MODIFIED BESSEL FUNC- TION OF THE SECOND KIND . Beyer (1987, p. 571) gives 60%, 70%, 90%, 95%, 97.5%, 99%, 99.5%, and 99.95% confidence intervals,and Goulden (1956) gives 50%, 90%, 95%, 98%, 99%, and 99.9% confidence intervals. A partial table is given below for small rand several common con- fidence intervals. r90% 95% 97.5% 99.5% 1 3.07766 6.31371 12.7062 63.656 2 1.88562 2.91999 4.30265 9.92482 3 1.63774 2.35336 3.18243 5.84089 4 1.53321 2.13185 2.77644 4.603935 1.47588 2.01505 2.57058 4.03212 10 1.37218 1.81246 2.22814 3.16922 30 1.31042 1.69726 2.04227 2.74999 100 1.29007 1.66023 1.98397 2.62589 //C12/1.28156 1.64487 1.95999 2.57584 The so-called A(t½n) distribution is useful for testing if two observed distributions have the same MEAN . A(t½n) gives the probability that the difference in two observed MEANS for a certain statistic twith n DEGREES OF FREEDOM would be smaller than the observed value purely by chance: A(t½n)/C301 ffiffiffinpB1 2;12nYru*Yru+gt /C28t1/C27x2 n !/C28(1/C27n)=2 dx:(18) LetXbe a NORMALLY DISTRIBUTED random variablewith MEAN 0 and VARIANCE s2;letY2=s2have a CHI- SQUARED DISTRIBUTION with nDEGREES OF FREEDOM , and let XandYbe independent. Then t/C13Xffiffiffinp Y(19) is distributed as Student’s twith nDEGREES OF FREEDOM . The noncentral Student’s t-distribution is given by P(x)/C30nn=2n! 2nel2=2n/C27x2 ðÞn=2G1 2n ! /C2Yrt*ffiffiffi 2p lx1F11 2n/C271;32;l2z2 2n/C27z2 ðÞ ! n/C27x2 ðÞ G1 2(n/C271)"# /C271F112(n/C271);12;l2z2 2n/C27z2 ðÞ ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n/C27x2p G1 2n/C271 !Yrt+ ; (20) where G(z) is the GAMMA FUNCTION and1F1(a;b;z) is a CONFLUENT HYPERGEOMETRIC FUNCTION . The MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS are m/C30Lffiffiffi n 2s G1 2(n/C281)Yru*Yru+ G1 2nYru*Yru+ (21) s2/C30L2/C271 ðÞ n n/C282/C28L2nG1 2(n/C281)Yru*Yru+hi2 2G1 2nYru*Yru+hi2 (22) g1/C30g(n) 1 g(d) 1(23) g2/C30g(n) 2 g(d) 2: (24) where g(n) 1/C302lffiffiffinpG1 2(n/C281)hiYrt* l2(2n/C287)/C283YrtYrP G12nYru*Yru+hi2 /C28l2(n/C282)(n/C283)G12(n/C281)Yru*Yru+hi2Yrt+ (25) g(d) 1/C30(n/C283) /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nl2/C271YrvYru n/C282/C28l2nG1 2(n/C281)Yru*Yru+hi2 G1 2nYru*Yru+hi2vuuuut G 1 2nYru*Yru+ /C29 l2(n /C282) G1 2(n /C281)Yru*i2 /C282 l2 /C271YrvYru G12 nYru*Yru+hi2YrtvYrt+Yrt* (26) g(n) 2/C302 /C283l4(n /C282)2(n /C283)(n /C284) G12(n /C281)Yru*Yru+hi4Yrt* /C2726/C282n l2(n /C282)(n /C284) l2(2n /C287) /C283YrtYrP p G(n /C271) ½/C1382 /C284 l4(n /C285) /C286 l2 /C283YrtYrP (n /C283) G12 nYru*Yru+hi4Yrt+ (27) g(d) 2/C30(n /C283)(n /C284) l2(n /C282) G1 2(n /C281)Yru*Yru+hi2Yrt* /C282 l2 /C271YrvYru G1 2 nYru*Yru+hi2Yrt+2 : (28) See also BESSEL’S STATISTICAL FORMULA ,PAIRED T- TEST,STUDENT’S Z-DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 948 /C1/49, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 536, 1987. Fisher, R. A. "Applications of ‘Student’s’ Distribution." Me- tron 5,3/C1/7, 1925. Fisher, R. A. "Expansion of ‘Student’s’ Integral in Powers of n /C281:/" Metron 5,22/C1/2, 1925. Fisher, R. A. Statistical Methods for Research Workers, 10th ed. Edinburgh: Oliver and Boyd, 1948. Goulden, C. H. Table A-3 in Methods of Statistical Analysis, 2nd ed. New York: Wiley, p. 443, 1956. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Incomplete Beta Function, Student’s Distribu- tion, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 219 /C1/23, 1992. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 116 /C1/17, 1992. Student. "The Probable Error of a Mean." Biometrika 6,1/C1/5, 1908. Student’s z-Distribution The probability density function for Student’s z-distribution are given by fm; n(z) /C30Gn 2 ! ffiffiffippGn /C28 1 2 ! 1 /C27z2YrvYru/C28n=2: (1) Now define dm; n(z) /C13zjj1 /C28n G1 2 nYru*Yru+ 2F112(n /C28 1);12 n;12(n /C27 1); /C28z/C282Yru*Yru+ 2ffiffiffippG1 2(n /C27 1)hi ; (2) then the cumulative distribution functions is given by Dm; n(z) /C30dm; n(z) for z 50 1 /C28dm; n(z) for z ]0Yrt* (3) The MEAN is 0, so the MOMENTS are m1 /C300 (4) m2 /C301 n /C28 3 (5) m3 /C300 (6) m4 /C303 (n /C28 3)(n /C28 5) : (7) The MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C300 (8) s2 /C301 n /C28 3 (9) g1 /C300 (10) g2/C306 n/C285: (11) The CHARACTERISTIC FUNCTION is f(t)/C302(3/C28n)=2tjj(n/C281)=2K(1/C28n)=2tjjðÞ G1 2(n/C281)hi ; (12) where Kn(z)i sa MODIFIED BESSEL FUNCTION OF THE SECOND KIND . Letting z/C13¯x/C28m s; (13) where xis the sample MEAN andmis the population MEAN gives S TUDENT’S T-DISTRIBUTION . See also STUDENT’S T-DISTRIBUTION Study’s Theorem Given three curves f1 ; f2 ; f3 with the common group of ordinary points G (which may be empty), let their remaining groups of intersections g23 ; g31 ; and g12 also be ordinary points. If f?1is any other curve through g23 ; then there exist two other curves f?2 ; f?3 such that the three combined curves fi f ?i are of the same order and LINEARLY DEPENDENT , each curve f ?k contains the corresponding group gij ; and every intersection of fi or f?i with fj or f?j lies on fk or f ?k :/ References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 34, 1959. Sturm Chain The series of S TURM FUNCTIONS arising in application of the S TURM THEOREM . See also STURM FUNCTION ,STURM THEOREM Sturm Function Given a function f(x)/C13f0(x);write f1/C13f?(x) and define the Sturm functions by fn(x)/C30/C28 fn/C282(x)/C28fn/C281(x)fn/C282(x) fn/C281(x)"# () : (1) where [ P(x)=Q(x)] is a polynomial quotient. Then construct the following chain of Sturm functions, f0/C30q0f1/C28f2 f1/C30q1f2/C28f3 f2/C30q2f3/C28f4 n fs/C282/C30qs/C282fs/C281/C28fs; known as a S TURM CHAIN . The chain is terminated when a constant /C28fs(x) is obtained. Sturm functions provide a convenient way for finding the number of real roots of an algebraic equation withreal coefficients over a given interval. Specifically, the difference in the number of sign changes between the Sturm functions evaluated at two points x/C30aand x/C30bgives the number of real roots in the interval ( a, b). This powerful result is known as the S TURM THEOREM . However, when the method is applied numerically, care must be taken when computingthe polynomial quotients to avoid spurious results due to roundoff error. As a specific application of Sturm functions towardfinding POLYNOMIAL ROOTS , consider the function f0(x)/C30x5/C283x/C281;plotted above, which has roots /C281:21465 ;/C280:334734 ;0:0802951 91:32836 i;and 1.38879 (three of which are real). The DERIVATIVE is given by f?(x)/C305x4/C283;and the S TURM CHAIN is then given by f0/C30x5/C283x/C281 (3) f1/C305x4/C283 (4) f2/C301 5(12x/C275) (5) f3/C3059083 20736: (6) The following table shows the signs of fiand the number of sign changes Dobtained for points sepa- rated by Dx/C302:/ x /f0//f1//f2//f3//D/ /C282/C2811 /C28113 0/C281/C2811 1 1 2111 1 0 This shows that 3 /C1//C302 real roots lie in ( /C282;0);and 1/C1//C301 real root lies in (0 ;2):Reducing the spacing to D:r/C300:5 gives the following table. x /f0//f1//f2//f3//D/ //C282:0//C2811 /C28113 //C281:5//C2811 /C28113 //C281:0/11 /C28112 //C280:5/1/C281/C28112 0.0/C281/C2811 1 1 0.5/C281/C2811 1 1 1.0/C28111 1 1 1.5 1 1 1 1 0 2.0 1 1 1 1 0 This table isolates the three real roots and shows that they lie in the intervals (/C281:5;/C281 :0); (/C280 :5; 0:0); and (1:0; 1:5): If desired, the intervals in which the roots fall could be further reduced. The Sturm functions satisfy the following conditions: 1. Two neighboring functions do not vanish simul- taneously at any point in the interval. 2. At a null point of a Sturm function, its two neighboring functions are of different signs. 3. Within a sufficiently small interval surrounding a zero point of f0(x); f1(x) is everywhere greater than zero or everywhere smaller than zero. See also DESCARTES’ SIGN RULE,S TURM CHAIN , STURM THEOREM References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 334, 1990. Do¨rrie, H. "Sturm’s Problem of the Number of Roots." §24 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 112 /C1/16, 1965. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 469, 1992. Rusin, D. "Known Math." http://www.math.niu.edu/~rusin/ known-math/96/sturm. Sturm, C. "Me´moire sur la re´solution des e´quations nume ´r- iques." Bull. des sciences de Fe´russac 11, 1929. Sturm Theorem The number of REAL ROOTS of an algebraic equation with REAL COEFFICIENTS whose REAL ROOTS are simple over an interval, the endpoints of which are not ROOTS , is equal to the difference between the number of sign changes of the STURM CHAINS formed for the interval ends. See also STURM CHAIN ,STURM FUNCTION References Do¨rrie, H. "Sturm’s Problem of the Number of Roots." §24 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 112 /C1/16, 1965. Rusin, D. "Known Math." http://www.math.niu.edu/~rusin/ known-math/96/sturm. Sturmian Separation Theorem Let Ar/C30aij be a SEQUENCE of N SYMMETRIC MATRICES of increasing order with i:j /C301 ; 2, ..., r and r /C301, 2, ..., N. Let lkArðÞbe the kth EIGENVALUE of Ar for k /C301, 2, ..., r, where the ordering is given byl1ArðÞ] l2ArðÞ]...] lrArðÞ : Then it follows that lk /C271Ai/C271YrvYru 5 lkAiðÞ5 lkAi/C271YrvYru : References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000. Sturmian Sequence If a SEQUENCE has the property that the BLOCK GROWTH function B(n) /C30n /C271 for all n, then it is said to have minimal block growth, and the sequence is called a Sturmian sequence. An example of this is the sequence arising from the SUBSTITUTION MAP 0 0 01 1 0 0 yielding 0 0 01 0 010 0 01001 0 01001010 0 ...; which gives us the Sturmian sequence 01001010.... STURM FUNCTIONS are sometimes also said to form a Sturmian sequence. See also STURM FUNCTION ,STURM THEOREM Sturm-Liouville Equation A second-order ORDINARY DIFFERENTIAL EQUATION d dxp(x)dy dx"# /C27[lw(x) /C28q(x)]y /C300; where l is a constant and w(x) is a known function called either the density or WEIGHTING FUNCTION . The solutions (with appropriate boundary conditions) of l are called EIGENVALUES and the corresponding ul(x) EIGENFUNCTIONS . The solutions of this equation satisfy important mathematical properties under appropriate boundary conditions (Arfken 1985). See also ADJOINT ,SELF-ADJOINT References Arfken, G. "Sturm-Liouville Theory--Orthogonal Functions." Ch. 9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 497 /C1/38, 1985. Sturm-Liouville Theory STURM- LIOUVILLE EQUATION SU SPECIAL UNITARY GROUP Subalgebra AnALGEBRA S?which is part of a large ALGEBRA Sand shares its properties. See also ALGEBRA Subanalytic /X ⁄Rn is subanalytic if, for all x /C23Rn ; there is an open set U and a bounded SEMIANALYTIC set Y ƒRn/C27m such that X S U is the projection of Y into U. See also SEMIANALYTIC References Bierstone, E. and Milman, P. "Semialgebraic and Subanaly- tic Sets." IHES Pub. Math. 67,5/C1/2, 1988. Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753 /C1/59, 1996. Subdiagonal The subdiagonal of a SQUARE MATRIX is the set of elements directly under the elements comprising the DIAGONAL . For example, in the following matrix, the diagonal elements are denoted diand the subdiago- nals are denoted si ; d1a12a13... a1n s1 d2a23::: a2n a31s2 d3::: a3n n:::::::::::: an1an2an3/C1/C1/C1 dn2 666643 77775: See also C ANONICAL BOX MATRIX ,DIAGONAL ,SUPER- DIAGONAL ,TRIDIAGONAL MATRIX References Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 50, 1958. Subfactorial The number of PERMUTATIONS of n objects in which no object appears in its natural place (i.e., the number of so-called "DERANGEMENTS "). !n /C13n!Xn k /C300( /C281)k k! (1) or !n /C13n! e"# : (2) where k! is the usual FACTORIAL and [x] is the NINT function. The first few values are !1 /C300; !2 /C301; !3 /C302; !4 /C309; !5 /C3044 ; !6 /C30265; !7 /C301854 ; !8 /C3014833 ; ... (Sloane’s A000166). For example, the only DERANGE- MENTS of f1; 2; 3g are f2; 3; 1 g and f3 ; 1 ; 2 g; so !3 /C30 2: Similarly, the DERANGEMENTS of f1 ; 2 ; 3 ; 4g are f2; 1; 4; 3g;f2 ; 3 ; 4 ; 1 g;f2; 4; 1; 3g;f3 ; 1; 4; 2g; f3; 4; 1; 2g;f3 ; 4 ; 2 ; 1 g;f4; 1; 2; 3g;f4 ; 3; 1; 2g; and f4; 3; 2 ; 1 g; so !4 /C309: The only prime subfactor- ial is !3 /C302:/The subfactorials are also called the RENCONTRES NUMBERS and satisfy the RECURRENCE RELATIONS !n /C30n /C215!(n /C281) /C27(/C281)n (3) !(n /C271) /C30n[!n /C27!(n /C281)]: (4) The subfactorial can be considered a special case of a restricted ROOKS PROBLEM . The only number equal to the sum of subfactorials of its digits is 148; 349 /C30!1 /C27!4 /C27!8 /C27!3 /C27!4 /C27!9 (5) (Madachy 1979). See also DERANGEMENT ,FACTORIAL ,M ARRIED COU- PLES PROBLEM ,ROOKS PROBLEM ,SUPERFACTORIAL References Do¨rrie, H. §6in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 19 /C1/1, 1965. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 167, 1979. Sloane, N. J. A. Sequences A000166/M1937 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M1937 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, p. 67, 1997. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 27, 1986. Subfield If a subset S of the elements of a FIELD F satisfies the FIELD AXIOMS with the same operations of F, then S is called a subfield of F.Ina FINITE FIELD of ORDER pn ; with p a prime, there exists a subfield of ORDER pm for every m DIVIDING n. See also EXTENSION FIELD,FIELD,PRIME SUBFIELD , SUBMANIFOLD ,SUBSPACE Subgraph A GRAPH G ? whose VERTICES and EDGES form subsets of the VERTICES and EDGES of a given GRAPH G.IfG ? is a subgraph of G, then G is said to be a SUPERGRAPH of G?:/ See also GRAPH ,INDUCED SUBGRAPH ,SUPERGRAPH , SUBTREE ,ULAM’S CONJECTURE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 11, 1994. Subgroup A subset HofGROUP elements of a group Gthat satisfies the four GROUP requirements. " His a subgroup of G" is written HƒG:The ORDER of any subgroup of a GROUP of ORDER h must be a DIVISOR of h. See also CARTAN SUBGROUP ,COMPOSITION SERIES , FITTING SUBGROUP ,GROUP ,NORMAL SUBGROUP Subharmonic Function Let U ⁄C be an OPEN SET and f a real-valued continuous function on U. Suppose that for each CLOSED DISK D(P; r) ⁄U and every real-valued HAR- MONIC FUNCTION h defined on a NEIGHBORHOOD of D(P ; r) which satisfies f 5h on @D(P ; r) ; it holds that f 5h on the OPEN DISK D(P; r) : Then f is said to be subharmonic on U (Krantz 1999, p. 99). 1. If f1 ; f2 are subharmonic on U, then so is f1 /C27f2 :/ 2. If f1is subharmonic on U and a /C21 0isa constant, than af1 is subharmonic on U. 3. If f1 ; f2are subharmonic on U, then max f1(z) ; f2(z) fg is also subharmonic on U. See also BARRIER ,HARMONIC FUNCTION References Krantz, S. G. "The Dirichlet Problem and Subharmonic Functions." §7.7 in Handbook of Complex Analysis. Bos- ton, MA: Birkha ¨user, pp. 97 /C1/01, 1999. Sublime Number Let s0(n) and s1(n) denote the number and sum of the divisors of n, respectively (i.e., the zeroth- and first- order DIVISOR FUNCTIONS ). A number n is called sublime if s0(n) and s1(n) are both PERFECT NUMBERS . The only two known sublime numbers are 12 and 60865556702383789896703717342431696 /C1/C1/C1 /C1/C1/C122657830773351885970528324860512791691264 : It is not known if any ODD sublime number exists. See also DIVISOR FUNCTION ,PERFECT NUMBER References Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Submanifold A C /C12 (infinitely differentiable) MANIFOLD is said to be a submanifold of a C /C12 MANIFOLD M ? if M is a SUBSET of M ? and the IDENTITY MAP of M into M ? is an EMBEDDING . See also EMBEDDING ,M ANIFOLD ,S UBFIELD ,S UB- SPACESubmatrix A p /C29q submatrix of an m /C29n MATRIX (with p 5m; n 5q)isa p /C29q MATRIX formed by taking a block of the entries of this size from the original matrix. See also MATRIX Submersion A submersion is a SMOOTH MAP f : M 0 N when dim M ]dim N ; given that the DIFFERENTIAL ,orJ ACOBIAN ,is SURJEC- TIVE at every x in M. The basic example of a submersion is the canonical submersion a of Rn onto Rk when n ]k ; a x1 ; ...; xn ðÞ /C30 x1 ; ...; xk ðÞ : In fact, if f is a submersion, then it is possible to find coordinates around x in M and coordinates around f(x)in N such that f is the canonical submersion written in these coordinates. For example, consider the submersion of R2 /C28f(0; 0)g onto the circle S1 ; given by f(x; y) /C30(x; y) =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27y2p :/ See also IMMERSION ,RIEMANNIAN SUBMERSION Submodule A MODULE over a RING that is contained in and has the same addition as another MODULE over the same RING . See also MODULE Subnormal If the LEXIS RATIO L B1, a set of trials are said to be subnormal. See also LEXIS RATIO,SUBNORMAL SUBGROUP ,SUPER- NORMAL Subnormal Subgroup L is a subnormal SUBGROUP of H if there is a "normal series" (in the sense of Jordan-Holder) from L to H. Suborder Function A special case of the generalized MULTIPLICATIVE ORDER function taken with respect to the PRIMITIVE ROOTS /C281 and 1. This function is denoted sordn(a) and is implemented in Mathematica as Multipli- cativeOrder [a,n,{/C281, 1}]. See also MULTIPLICATIVE ORDER Subordinate Norm NATURAL NORM Subresultant Subresultants for a few simple pairs of polynomials include S(x /C28a; x /C28b) /C30fa /C28b; 1g S((x /C28a)(x /C28b) ; x /C28c) /C30f(a /C28c)(b /C28c) ; 1 g S((x /C28a)(x /C28b) ; (x /C28c)(x /C28d)) /C30f(a /C28c)(b /C28c)(a /C28d)(b /C28d) ; a /C27b /C28c /C28d; 1g: The principal subresultants of two polynomials can be computed using the Mathematica command Subre- sultants [poly1 , poly2 , var]. The first k subresul- tants of two polynomials p1 and p2 ; both with leading coefficient one, are zero when p1and p2have k common roots. See also DISCRIMINANT (POLYNOMIAL ), RESULTANT References J. Pure Appl. Algebra 145, 149, 2000. Hong, H. "Subresultants Under Composition." J. Symb. Comput. 23, 355 /C1/65, 1997. Hong, H. "Subresultants in Roots." Submitted 1999. Subring A subring of a RING R is a SUBGROUP of R that is CLOSED under multiplication. See also RING,SUBGROUP References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 230, 1998. Subscript A quantity displayed below the normal line of text (and generally in a smaller point size), as the "i"inai ; is called a subscript. Subscripts are commonly used to indicate indices (/aij is the entry in the ith row and jth column of a MATRIX A) ; partial differentiation (/yx is an abbreviation for @y=@x) ; and a host of other operations and notations in mathematics. See also SUPERSCRIPT Subselfsimilar Set Giving a set F /C30 f1 ; f2 ; ...; fn fg of contracting simili- tudes of R ?; the closed set E is said to be subselfsimi- lar for F if E ƒ@n i/C301fi(E) (Falconer 1995, Duvall and Keesling 1999).References Duvall, P. and Keesling, J. The Hausdorff Dimension of the Boundary of the Le´vy Dragon. 22 Jul 1999. http:// xxx.lanl.gov/abs/math.DS/9907145/. Falconer, K. J. "Sub-Self-Similar Sets." Trans. Amer. Math. Soc. 247, 3121 /C1/129, 1995. Subsequence A subsequence of a SEQUENCE S /C30 xifgn i/C301 is a derived sequence yifgNi/C301/C30 xi/C27jYr$Yr% for some j ]0 and N 5n /C28j: More generally, the word subsequence is sometimes used to mean a sequence derived from a sequence S by discarding some of its terms. See also LOWER- TRIMMED SUBSEQUENCE ,U PPER- TRIMMED SUBSEQUENCE Subset A portion of a SET. B is a subset of A (written B ⁄A) IFF every member of B is a member of A.IfB is a PROPER SUBSET of A (i.e., a subset other than the set itself), this is written B ƒA: If B is not a subset of A, this is written B ¢A: (The notation B /C148A is generally not used, since B ¢A automatically means that B and A cannot be the same.) The set of subsets of a set S is called the POWER SET of S, and a SET of n elements has 2n subsets (including both the set itself and the EMPTY SET). This follows from the fact that the total number of distinct K- SUBSET on a set of n elements is given by the BINOMIAL SUM Xn k /C300n kYru$Yru% /C302n : For sets of n /C301, 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ... (Sloane’s A000079). For example, the set f1g has the two subsets ¥ and f1g: Similarly, the set f1 ; 2g has subsets ¥ (the EMPTY SET, f1g;f2g; and f1; 2g: The subsets (i.e., POWER SET) of a given set can be found using Subsets [list] in the Mathematica add-on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also EMPTY SET,IMPLIES , K-SUBSET , P-SYSTEM , POWER SET,PROPER SUBSET ,SUPERSET ,VENN DIA- GRAM References Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 109, 1996. Ruskey, F. "Information of Subsets of a Set." http:// www.theory.csc.uvic.ca/~cos/inf/comb/SubsetInfo.html. Skiena, S. "Binary Representation and Random Sets." §1.5.2 inImplementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 41 /C1/2, 1990. Sloane, N. J. A. Sequences A000079/M1129 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Subset Sum Problem The problem of finding what subset of a list of integers has a given sum. The subset sum is an INTEGER RELATION problem where the relation coeffi- cients ai are 0 or 1. See also INTEGER RELATION ,LATTICE REDUCTION , KNAPSACK PROBLEM ,P OSTAGE STAMP PROBLEM , STO¨ HR SEQUENCE References Coster, M. J.; LaMacchia, B. A.; Odlyzko, A. M.; and Schnorr, C. P. "An Improved Low-Density Subset Sum Algorithm." In Advances in Cryptology: EUROCRYPT ’91 (Brighton, 1999) (Ed. D. W. Davis). New York: Springer- Verlag, pp. 54 /C1/7, 1992. Coster, M. J.; Joux, A.; LaMacchia, B. A.; Odlyzko, A. M.; Schnorr, C. P.; and Stern, J. "Improved Low-Density Subset Sum Algorithms." Comput. Complex. 2, 111 /C1/28, 1992. Ferguson, H. R. P. and Bailey, D. H. "A Polynomial Time, Numerically Stable Integer Relation Algorithm." RNR Techn. Rept. RNR-91 /C1/32, Jul. 14, 1992. Lagarias, L. C. and Odlyzko, A. M. "Solving Low-Density Subset Sum Problems." J. ACM 32, 229 /C1/46, 1985. Schnorr, C. P. and Euchner, M. "Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems." In Fundamentals of Computation Theory (Gosen 1991). Berlin: Springer-Verlag, pp. 68 /C1/5, 1991. Subspace Let V be a REAL VECTOR SPACE (e.g., the real continuous functions C(I)ona CLOSED INTERVAL I, 2-D EUCLIDEAN SPACE R2 ; the twice differentiable real functions C(2)(I)onI, etc.). Then W is a real SUBSPACE of V if W is a SUBSET of V and, for every w1 ; w1 /C23W and t /C23R (the REALS ), w1 /C27w2 /C23W and tw1 /C23W: Let (H) be a homogeneous system of linear equations in x1 ; ..., xn : Then the SUBSET S of Rn which consists of all solutions of the system (H) is a subspace of Rn :/ More generally, let Fq be a FIELD with q /C30pa ; where p is PRIME , and let Fq; ndenote the n-D VECTOR SPACE over Fq : The number of k-D linear subspaces of Fq ; n is NFq ; nYrvYru /C30n kYru$Yru% q; where this is the Q-BINOMIAL COEFFICIENT (Aigner 1979, Exton 1983). The asymptotic limit is NFq ; nYrvYru /C30ceqn2 =4[1 /C27o(1)] for n even coqn2 =4[1 /C27o(1)] for n odd ;Yrt* where ce /C30P/C12 k /C30/C28/C12q/C28k2 Q/C12j /C3011 /C28 q /C28j ðÞco /C30P/C12k/C30/C28/C12q /C28(k/C271 =2)2 Q/C12j/C3011 /C28 q/C28j ðÞ (Finch). The case q /C302 gives the Q-ANALOG of the WALLIS FORMULA . See also Q-BINOMIAL COEFFICIENT ,SUBFIELD ,SUB- MANIFOLD References Aigner, M. Combinatorial Theory. New York: Springer- Verlag, 1979. Exton, H. q-Hypergeometric Functions and Applications. New York: Halstead Press, 1983. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/dig/dig.html. Substitution Group PERMUTATION GROUP Substitution Map A MAP which uses a set of rules to transform elements of a sequence into a new sequence using a set of rules which "translate" from the original sequence to its transformation. For example, the substitution map f1 0 0 ; 0 0 11g would take 10 to 011. See also GOLDEN RATIO,M ORSE- THUE SEQUENCE , STRING REWRITING ,THUE CONSTANT Substitution Tensor KRONECKER DELTA ,PERMUTATION SYMBOL ,PERMU- TATION TENSOR Subtend Given a geometric object O in the PLANE and a point P, let A be the ANGLE from one edge of O to the other with VERTEX at P. Then O is said to subtend an ANGLE A from P. See also ANGLE ,VERTEX ANGLE Subtraction Subtraction is the operation of taking the DIFFERENCE x /C28y of two numbers x and y. Here, x is called the MINUEND , y is called the SUBTRAHEND , and the symbol between the x and y is called the MINUS SIGN. The expression "/x /C28y/" is read "x MINUS y." Subtraction is the inverse of ADDITION ,sox /C27y /C28y /C30x /C28y /C27y /C30x:/ The subtraction of a number from itself gives 0, while the subtraction of a real number from a smaller real number gives a negative real number. Subtraction of real numbers can be naturally extended to complex numbers. See also ADDITION ,D IVISION ,M INUEND ,M INUS , MINUS SIGN,MULTIPLICATION ,SUBTRAHEND Subtrahend A quantity which is subtracted from another (the MINUEND ). See also MINUEND ,SUBTRACTION Subtree A TREE G ? whose VERTICES and EDGES form subsets of the VERTICES and EDGES of a given TREE G. See also SUBGRAPH ,TREE Subvariety See also ALGEBRAIC VARIETY Succeeds The relationship x succeeds (or FOLLOWS ) y is written x cy: The relation x succeeds or is equal to y is written x Ty :/ See also PRECEDES Successes DIFFERENCE OF SUCCESSES Successor For any ORDINAL NUMBER a; the successor of a is a @ fag (Ciesielski 1997, p. 46). The successor of an ordinal number a is therefore the next ordinal, a /C271:/ See also LIMIT ORDINAL ,ORDINAL NUMBER References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Sufficient A CONDITION which, if true, guarantees that a result is also true. (However, the result may also be true if the CONDITION is not met.) If a CONDITION is both NECESSARY and SUFFICIENT , then the result is said to be true IFF ( the CONDITION holds. For example, the condition that a decimal number n end in the DIGIT 2 is a sufficient but not NECESSARY condition that n be EVEN . See also IFF,IMPLIES ,N ECESSARY ,S UFFICIENTLY LARGE References Jeffreys, H. and Jeffreys, B. S. "Necessary: Sufficient." §1.036 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 10 /C1/1, 1988. Suitable Number IDONEAL NUMBERSultan’s Dowry Problem A sultan has granted a commoner a chance to marry one of his n daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter’s dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must immediately decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will allow the marriage to take place only if the commoner picks the daughter with the overall high- est dowry. Then what is the commoner’s best strat- egy, assuming he knows nothing about the distribution of dowries (B. Elbows)? Since the commoner knows nothing about the dis- tribution of the dowries, the best strategy is to wait until a certain number x of daughters have been presented, then pick the highest dowry thereafter. The exact number to skip is determined by the condition that the odds that the highest dowry has already been seen is just greater than the odds that it remains to be seen and that if it is seen it will be picked. This amounts to finding the smallest x such that x n ]x n1 x /C27 1 /C27.../C271 n /C28 1 ! : (1) Computing the sum analytically gives the solution as the smallest x such that Hx ]Hn /C281 ; (2) where Hn is a HARMONIC NUMBER . Solving Hx /C30Hn /C281 (3) numerically and taking the CEILING FUNCTION xde then gives the solutions 0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, ... (Sloane’s A054382) for n /C30 1, 2, ... daughters. The problem is most commonly stated with n /C30100 daughters, which gives the result that the commoner should wait until he has seen 37 of the daughters, then pick the first daughter with a dowry that is bigger than any preceding one. With this strategy, his odds of choosing the daughter with the highest dowry are surprisingly high: about 37% (B. Elbows; Hon- sberger 1979, pp. 104 /C1/10, Mosteller 1987). See also BIRTHDAY PROBLEM References Elbows, B. http://xraysgi.ims.uconn.edu/rpa-output/decision/ dowry.s. Honsberger, R. "Some Surprises in Probability." Ch. 5 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 104 /C1/10, 1979. Mosteller, F. Problem 47 in Fifty Challenging Problems in Probability with Solutions. New York: Dover, 1987. Sloane, N. J. A. Sequences A054382 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sum A sum is the result of an ADDITION . For example, adding 1, 2, 3, and 4 gives the sum 10, written 1 /C272 /C273 /C274 /C3010: (1) The numbers being summed are called ADDENDS ,or sometimes SUMMANDS . The summation operation can also be indicated using a capital sigma with upper and lower limits written above and below, and the index indicated below. For example, the above sum could be written X4 k /C301k /C3010 : (2) A sum Xn i/C301ai (3) in which each term ai is given by some fixed rule (i.e., fai gn i/C301 is a well defined SEQUENCE ) is called a SERIES , and if the number of terms n is infinite, the sum is called an INFINITE SERIES . A sum of the form Xn k/C301rk (4) is called a GEOMETRIC SERIES . The general finite POWER SUM Xn k /C301kp (5) can be given by the expression Xn k /C301kp /C30(B /C27 n /C27 1)[p /C271] /C28 B[p /C271] p /C27 1 ; (6) which is equivalent to FAULHABER’S FORMULA , where the NOTATION B[k] means the quantity in question is raised to the appropriate POWER k and all terms OF THE FORM Bm are replaced with the corresponding BERNOULLI NUMBERS Bm :/ NICOMACHUS’S THEOREM gives as curious expression for the POWER SUM an k /C301 k3 :/ Other analytic sums include Xn k /C300xk ! p /C301 (p /C28 1)!Xnp k/C300(n /C28½n /C28 k½/C27 p /C28 1)! (n /C28½n /C28 k½)!xk(7) for p /C301; 2X/C12 n/C300anxn ! 2 /C30X/C12 n/C300a2 nx2n /C272X/C12 n/C301 i /C27j /C30n i Bjaiajxn ; (8) and X xy /C30x1y1 /C27x1y2 /C27.../C27x2y1 /C27x2y2 /C27... /C30 x1 /C27x2 /C27... ðÞ y1 /C27 x1 /C27x2 /C27... ðÞ y2 /C30X xYru*Yru+ y1 /C27y2 /C27... ðÞ /C30X xX y; (9) so Xm i/C301Xn j/C301xixj /C30Xm i/C301xi !Xn j/C301yj ! : (10) Xn j/C300jxj /C30nxn/C272 /C28 (n /C27 1)xn/C271 /C27 x (x /C28 1)2 (11) Xn j/C301xr j Yn k/C301 k "jxj /C28 xkYrvYru/C300 for 0 5r Bn /C281 1 for r /C30n /C281Pn j/C301xjfor r /C30n8 < : (12) Xn k /C301Yn r /C301 r "k(x /C27 k /C28 r) Yn r /C301 r"k(k/C28r)/C301 (13) (n/C271)Xn m/C301mk/C30Xn m/C301mk/C271/C27Xn p/C301Xp m/C301mk !"# :(14) To minimize the sum of a set of squares of numbers xifgabout a given number x0 S/C13X ixi/C28x0 ðÞ2/C30X ix2 i/C282x0X xi/C27Nx20:(15) take the DERIVATIVE . d dx0S/C30/C282X ixi/C272Nx0/C300: (16) Solving for x0gives x0/C13¯x/C301 NX ixi: (17) soSis minimized when x0is set to the MEAN . See also ARITHMETIC SERIES ,BERNOULLI NUMBER , BINOMIAL SUMS,CLARK’S TRIANGLE ,CONVERGENCE IMPROVEMENT ,DEDEKIND SUM,DOUBLE SUM,EULER SUM,FACTORIAL SUMS,FAULHABER’S FORMULA ,GAB- RIEL’S STAIRCASE ,GAUSSIAN SUM,GEOMETRIC SERIES , GOSPER’S METHOD ,H URWITZ ZETA FUNCTION ,INFI- NITE SERIES ,INFINITE PRODUCT ,K LOOSTERMAN’S SUM,LEGENDRE SUM,LERCH TRANSCENDENT ,NICO- MACHUS’S THEOREM ,O DD NUMBER THEOREM ,PAS- CAL’S TRIANGLE ,POWER SUM,PRODUCT ,RAMANUJAN’S SUM,R IEMANN ZETA FUNCTION ,SERIES ,W HITNEY SUM References Courant, R. and Robbins, H. "The Sum of the First n Squares." §1.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 14 /C1/5, 1996. Finch, S. "Unsolved Mathematics Problems: Sleeping Habits of Armadillos." http://www.mathsoft.com/asolve/glasser/ glasser.html. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A/C30B.Well- esley, MA: A. K. Peters, 1996. Sum of Squares Function The number of representations of nbyksquares, distinguishing signs and order, is denoted rk(n):For example, consider the number of ways of representing 5 as the sum of two squares. 5/C30(/C282)2/C27(/C281)2/C30(/C282)2/C2712/C3022/C27(/C281)2 /C3022/C2712/C30(/C281)2/C27(/C282)2/C30(/C281)2/C2722 /C3012/C27(/C282)2/C3012/C2722(1) sor2(5)/C308:TheMathematica functionSumOfSquar- esR[k,n] in the Mathematica add-on package Num- berTheory‘NumberTheoryFunctions‘ (which can be loaded with the command BBNumberTheory‘ ) gives rk(n):/ The function r2(n) is often written simply as r(n);and is intimately connected with the L EIBNIZ SERIES and with G AUSS’S CIRCLE PROBLEM (Hilbert and Cohn- Vossen 1999, pp. 27 /C1/9). It is also given by the inverse Mo¨bius transform of the sequence b2n/C300 and b2n/C271/C30 4(/C281)n(Sloane and Plouffe 1995, p. 22). The average order of r(n)i sp;but the normal order is 0 (Hardy 1999, p. 55). Jacobi gave analytic expressions for rk(n) for the cases k/C302, 4, 6, and 8 (Hardy 1999, p. 132). The cases k/C302, 4, and 6 were found by equating COEFFICIENTS of the J ACOBI THETA FUNCTIONS q3(x);q2 3(x);andq43(x): The solutions for k/C3010 and 12 were found by Liouville and Eisenstein, and Glaisher (1907) gives a table of rk(n) for k/C302s/C3018:r3(n) was found as a finite sum involving quadratic reciprocity symbols by Dirichlet. r5(n) and r7(n) were found by Eisenstein, Smith, and Minkowski. APOSITIVE INTEGER can be represented as the sum of two squares IFFeach of its prime factors of the form k/C273 occurs as an even power, as first established by Euler in 1738. In L AGRANGE’S FOUR-SQUARE THEO- REM, Lagrange proved that every POSITIVE INTEGERcan be written as the SUM of at most four SQUARES . where 4 may be reduced to 3 except for numbers OF THE FORM 4n(8k/C277);as proved by Legendre in 1798 (Nagell 1951, p. 194; Wells 1986, pp. 48 and 56; Hardy 1999, p. 12; Savin 2000). /r(n)/C30r2(n) is 0 whenever nhas a PRIME divisor OF THE FORM 4k/C273t oa n ODD POWER ; it doubles upon reaching a new PRIME OF THE FORM 4k/C271:It is given explicitly by r2(n)/C304X d/C301;3;...½n(/C281)(d/C281)=2(2) /C304d1(n)/C28d3(n) ½/C138 (3) /C304X d½nsin1 2pdYru*Yru+ ; (4) where dk(n) is the number of DIVISORS ofnOF THE FORM 4m/C27k(Hilbert and Cohn-Vossen 1999, pp. 37 /C1/ 8; Hardy 1999, p. 12). The first few values are 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, ... (Sloane’s A004018). r(n) obeys the un- expected identities X/C12 n/C300r(n)ffiffiffiffiffiffiffiffiffiffiffiffiffin/C27ap e/C282pffiffiffiffiffiffiffiffiffiffiffi (n/C27a)bp /C30X/C12 n/C300r(n)ffiffiffiffiffiffiffiffiffiffiffiffiffi n/C28bp e/C282pffiffiffiffiffiffiffiffiffiffiffi (n/C27b)ap (5) forRffiffiffiap½/C138 ;Rffiffiffi bphi >0; X 05n5xr(n)ffiffiffiffiffiffiffiffiffiffiffiffiffix/C28np /C302pffiffiffixp/C27X/C12 n/C301r(n)ffiffiffinpsin 2 pffiffiffiffiffiffinxpYrvYru (6) and X 05n5xr(n)/C30px/C27ffiffiffixpX/C12 n/C301r(n)ffiffiffinpJ12pffiffiffiffiffiffinxpYrvYru (7) (Hardy 1999, p. 82). The first few values of the summatory function R(n)/C30Xn k/C301r2(n) (8) are 0, 4, 8, 8, 12, 20, 20, 20, 24, 28, 36, ... (Sloane’s A014198). Shanks (1993) defines instead R?(n)/C301/C27 R(n);with R?(0)/C301:AL AMBERT SERIES forr2(n)i s X/C12 n/C3014(/C281)n/C271xn 1/C28xn/C30X/C12 n/C301r2(n)xn(9) (Hardy and Wright 1979). Explicit values of R?(n) for several powers of 10 are given in the following table (Mitchell 1966; Shanks 1993, pp. 165 and 234). n /R?(10n)/ 05 13 7 2 317 3 3149 4 31417 5 314197 6 3141549 8 314159053 10 31415925457 12 3141592649625 14 31415926535058 Asymptotic results include Xn k /C301r2(k) /C30 pn /C27OffiffiffinpYrvYru (10) Xn k /C301r2(k) k/C30K /C27 p ln n /C27O n/C281=2YrvYru ; (11) where K is a constant known as the SIERPINSKI CONSTANT . The left plot above shows Xn k /C301r2(k)"# /C28 pn; (12) with 9ffiffiffinpillustrated by curved envelope, and the right plot shows Xn k /C301r2(k) k"# /C28 p ln n; (13) with the value of K indicated as the solid horizontal line. The number of solutions of x2 /C27y2 /C27z2 /C30n (14) for a given n without restriction on the signs or relative sizes of x, y, and z is given by r3(n) : Gauss proved that if n is SQUAREFREE and n /C214, then r3(n) /C3024h(/C28n) for n /C133 (mod 8) 12h(/C284n) for n /C131; 2; 5; 6 (mod 8) 0 for n /C137 (mod 8)8 < : (15) (Arno 1992), where h(x) is the CLASS NUMBER of x.Additional higher-order identities are given by r4(n) /C308X d½nd /C308s(n) (16) /C3024X d/C301 ; 3 ; ... ½nd (17) /C3024 s0(n) (18) rs(n) /C2816X d ½n(/C281)n/C27dd3 (19) r16(n) /C30/C2832 3 (/C281)n s ?1(d) /C27 s?3(d) /C27 s?5(d) ½/C138 /C30(/C281)n256 3Xn /C281 k/C301s ?1(k) s?5(n /C28k) /C28 s?3(k) s?3(n /C28k) ½/C138 (20) r10(n) /C304 5E ?4(n) /C2716E?4(n) /C278x4(n) ½/C138 (21) r24(n) /C30 r24(24) /C27128 691(/C281)n/C281259t(n) /C28512t12 nYru*Yru+ hi ; (22) where s ?r(n) /C30X d½n(/C281)n/C27n=ddr (23) E4(n) /C30X d/C301 ; 3 ; ... ½n(/C281)(d/C281)=2d4 (24) E?4(n) /C30X d?/C301 ; 3 ; ... ½n(/C281)(d?/C281)=2d4 (25) x4(n) /C301 4X a2 /C27b2 /C30n(a /C27bi)4 ; (26) /d?/C30n=d; dk(n) is the number of divisors of n OF THE FORM 4m /C27k; r24(n)isa SINGULAR SERIES , s(n) is the DIVISOR FUNCTION , s0(n) is the DIVISOR FUNCTION of order 0 (i.e., the number of DIVISORS ), and t is the TAU FUNCTION . r24(n) may also be written in the alternate form r24(n) /C30(/C281)n16 917 s??3(d) /C278s ??5(d) /C272s??7(d) ð /C27(/C281)n512 9Xn/C281 k/C301s??3(k)s??7(n/C28k)/C28s??5(d)s??5(n/C28k) ½/C138 ;(27) where s??r(n)/C30X d½n(/C281)ndr: (28) Similar expressions exist for larger EVEN k, but they quickly become extremely complicated and can be written simply only in terms of expansions of modular functions. See also CLASS NUMBER ,D IOPHANTINE EQUATION– 2ND POWERS ,FERMAT’S POLYGONAL NUMBER THEO- REM,GAUSS’S CIRCLE PROBLEM ,LANDAU- RAMANUJAN CONSTANT ,LEIBNIZ SERIES ,PRIME FACTORS ,SIER- PINSKI CONSTANT ,TAU FUNCTION References Arno, S. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith. 60, 321 /C1/34, 1992. Boulyguine, M. B. "Sur la repre´sentation d’un nombre entier par une somme de carre´s." Comptes Rendus Hebdoma- daires de Se´ances de l’Acade ´mie des Sciences 161,28/C1/0, 1915. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, p. 317, 1952. Ewell, J. A. "New Representations of Ramanujan’s Tau Function." Proc. Amer. Math. Soc. 128, 723 /C1/26, 1999. Glaisher, J. W. L. "On the Numbers of a Representation of a Number as a Sum of 2r Squares, where 2r Does Not Exceed 18." Proc. London Math. Soc. 5, 479 /C1/90, 1907. Grosswald, E. Representations of Integers as Sums of Squares. New York: Springer-Verlag, 1985. Hardy, G. H. Quart. J. Math. 46, 283, 1915. Hardy, G. H. Proc. London Math. Soc. 15, 192 /C1/13, 1916. Hardy, G. H. "The Representation of Numbers as Sums of Squares." Ch. 9 in Ramanujan: Twelve Lectures on Sub- jects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. "The Function r(n);/" "Proof of the Formula for r(n) ;/" "The Generating Function of r(n);/" and "The Order of r(n);/" and "Representations by a Larger Number of Squares." §16.9, 16.10, 17.9, 18.7, and 20.13 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 241 /C1/43, 256 /C1/58, 270 /C1/71, and 314 /C1/15, 1979. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Milne, S. "Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions." In prep. http://www.math.ohio- state.edu/~milne/preprints.html. Mitchell, W. C. "The Number of Lattice Points in a k- Dimensional Hypersphere." Math. Comput. 20, 300 /C1/10, 1966. Nagell, T. Introduction to Number Theory. New York: Wiley, 1951. Savin, A. "Shape Numbers." Quantum 11,14/C1/8, 2000. Se´roul, R. "Prime Number and Sum of Two Squares." §2.11 in Programming for Mathematicians. Berlin: Springer- Verlag, pp. 18 /C1/9, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 162 /C1/53, 1993. Sloane, N. J. A. Sequences A004018/M3218 and A014198 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Wagon, S. "The Magic of Imaginary Factoring." Mathema- tica in Education and Res. 5,43/C1/7, 1996. Sum Rule d dx[f(x) /C27g(x)] /C30f ?(x) /C27g ?(x) : where d =dx denotes a derivative and f ?(x) and g ?(x) are the derivatives of f(x) and g(x) ; respectively. See also DERIVATIVESum-Free Set A set S of integers is called sum-free if x /C27y QS for all x; y /C23 S:/ See also A-SEQUENCE ,C AMERON’S SUM-FREE SET CONSTANT ,D OUBLE- FREE SET,H OFSTADTER SE- QUENCES ,P RIME NUMBER OF MEASUREMENT , S- ADDITIVE SEQUENCE ,SCHUR NUMBER ,SCHUR’S PRO- BLEM ,STO¨ HR SEQUENCE ,TRIPLE- FREE SET References Abbott, H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith. 11, 392 /C1/96, 1966. Exoo, G. "A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of K3 :/" Electronic J. Combinatorics 1, R8 1 /C1/, 1994. http://www.combinatorics.org/Volume_1/vo- lume1.html#R8. Finch, S. "Unsolved Mathematics Problems: Several Pro- blems Concerning Sum-Free Sets." http://www.mathsoft.- com/asolve/sf/sf.html. Fredricksen, H. and Sweet, M. M. "Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers." Elec- tronic J. Combinatorics 7, No. 1, R32, 1 /C1/, 2000. http:// www.combinatorics.org/Volume_7/v7i1toc.html#R32. Wallis, W. D.; Street, A. P.; and Wallis, J. S. Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices. New York: Springer-Verlag, 1972. Wang, E. T. H. "On Double-Free Sets of Integers." Ars Combin. 28,97/C1/00, 1989. Summand ADDEND Summation by Parts Summation by parts for discrete variables is the equivalent of INTEGRATION BY PARTS for continuous variables D/C281[v(x) D(x)] /C30u(x)v(x) /C28D/C281[Eu(x) Dv(x)] ; or X [v(x)Du(x)] /C30u(x)v(x) /C28X u(x /C27h)Dv(x)]; where /D/C281 / is the indefinite summation operator and the E-operator is defined by Ey(x) /C30y(x /C27h) ; where h is any constant. See also INTEGRATION BY PARTS Summatory Function For a discrete function f(n);the summatory function is defined by F(n)/C13Xn k/C23Df(k); where Dis the DOMAIN of the function. See also DIVISOR FUNCTION ,M ANGOLDT FUNCTION , MERTENS FUNCTION ,RUDIN- SHAPIRO SEQUENCE ,TAU FUNCTION ,TOTIENT FUNCTION Sum-of-Divisors Transform MO¨ BIUS TRANSFORM Sum-Product Number A sum-product number is a number n such that the sum of n’s digits times the product of n’s digit is n itself, for example 135 /C30(1 /C273 /C275)(1 /C215 3 /C215 5): Obviously, such a number must be divisible by its digits as well as the sum of its digits. There are only three sum-product numbers: 1, 135, 144, ... (Sloane’s A038369). This can be demonstrated using the follow- ing argument due to D. Wilson. Let n be a d-digit sum-product number, and let s and p be the sum and product of its digits. Because n is a d-digit number, we have 10d/C281 5n; s 59d; p 59d : Now, since n is a sum-product number, we have n /C30sp, giving 10d/C281 5n /C30sp 5(9d)9dYrvYru : The inequality 10d/C281 5(9d)9dYrvYru is fulfilled only by d 584 ; so a sum-product number has at most 84 digits. This gives s 59d 5756; p 5n B1085 : Now, since p is a product of digits, p must be OF THE FORM 2a3b5c7d : However, if 10 divides p, then it also divides n. This means that n ends in 0 so the product of its digit is p /C300, giving n /C30sp /C300: Hence we need not consider p divisible by 10, and can assume p is either OF THE FORM 2a3b7c or 3a5b7c : This reduces the search space for sum-product numbers to a tractable size, and allowed Wilson to verify that there are no further sum-product numbers. See also AMENABLE NUMBER ,DIGIT,HARSHAD NUM- BER References Sloane, N. J. A. Sequences A038369 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Sup SUPREMUM ,SUPREMUM LIMITSuper Catalan Number While the CATALAN NUMBERS are the number of P- GOOD PATHS from (n, n) to (0,0) which do not cross the diagonal line, the super Catalan numbers count the number of LATTICE PATHS with diagonal steps from (n, n) to (0,0) which do not touch the diagonal line x /C30y. the super catalan numbers are given by the RECUR- RENCE RELATION s(n) /C303(2n /C28 3)s(n /C28 1) /C28 (n /C28 3)s(n /C28 2) n (comtet 1974), with s(1) /C30s(2) /C301: (note that the expression in vardi (1991, p. 198) contains two errors.) a closed form expression in terms of LE- GENDRE POLYNOMIALS Pn(x)is S(n) /C303Pn/C281(3) /C28 Pn/C282(3) 4n (Vardi 1991, p. 199). The first few super Catalan numbers are 1, 1, 3, 11, 45, 197, ... (Sloane’s A001003). See also CATALAN NUMBER References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 56, 1974. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Exercise 7.50 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Motzkin, T. "Relations Between Hypersurface Cross Ratios and a Combinatorial Formula for Partitions of a Polygon for Permanent Preponderance and for Non-AssociativeProducts." Bull. Amer. Math. Soc. 54, 352/C1 /60, 1948. Schro ¨der, E. "Vier combinatorische Probleme." Z. Math. Phys. 15, 361/C1/76, 1870. Sloane, N. J. A. Sequences A001003/M2898 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 198 /C1 /99, 1991. Super-3 Number An INTEGER nsuch that h(x) contains three consecu- tive 3s in its DECIMAL representation. The first few super-3 numbers are 261, 462, 471, 481, 558, 753, 1036, ... (Sloane’s A014569). A. Anderson has shownthat all numbers ending in 471, 4710, or 47100 aresuper-3 (Pickover 1995). For a digit d;super-3 numbers can be generalized to super- /dnumbers nsuch that r4(n) contains dd /s in its DECIMAL representation. The following table gives the first few super- /dnumbers for small d:/ /d/Sloane Super- /dnumbers 2 Sloane’s A03274319, 31, 69, 81, 105, 106, 107, 119, 127, ... 3 Sloane’s A014569261, 462, 471, 481, 558, 753, 1036, 1046, ... 4 Sloane’s A0327441168, 4972, 7423, 7752, 8431, 10267, 11317, ... 5 Sloane’s A0327454602, 5517, 7539, 12955, 14555, 20137, 20379, ... 6 Sloane’s A03274627257, 272570, 302693, 323576, 364509, 502785, ... 7 Sloane’s A032747140997, 490996, 1184321, 1259609, 1409970, ... 8 Sloane’s A032748185423, 641519, 1551728, 1854230, 6415190, ... 9 Sloane’s A03274917546133, 32613656, 93568867, 107225764, ... References Pickover, C. A. Keys to Infinity. New York: Wiley, p. 7, 1995. Sloane, N. J. A. Sequences A014569 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Superabundant Number HIGHLY COMPOSITE NUMBER Superasymptotic Series See also ASYMPTOTIC SERIES ,H YPERASYMPTOTIC SERIES References Boyd, J. P. "The Devil’s Invention: Asymptotic, Superasymp- totic and Hyperasymptotic Series." Acta Appl. Math. 56, 1/C1/8, 1999. Super-d Number An INTEGER nsuch that 3 n3contains three consecu- tive 3s in its DECIMAL representation is called a super- 3 number. The first few super-3 numbers are 261, 462, 471, 481, 558, 753, 1036, ... (Sloane’s A014569).A. Anderson has shown that all numbers ending in 471, 4710, or 47100 are super-3 (Pickover 1995). In general, a super- dnumber is a number nsuch that dn dcontains dds in its DECIMAL representation. The following table gives the first few super- dnumbers for small d. dSloane super- dnumbers 2 A032743 19, 31, 69, 81, 105, 106, 107, 119, ...3 A014569 261, 462, 471, 481, 558, 753, 1036, ... 4 A032744 1168, 4972, 7423, 7752, 8431, 10267, ... 5 A032745 4602, 5517, 7539, 12955, 14555, 20137, ... 6 A032746 27257, 272570, 302693, 323576, ... 7 A032747 140997, 490996, 1184321, 1259609, ... 8 A032748 185423, 641519, 1551728, 1854230, ... 9 A032749 17546133, 32613656, 93568867, ... The following table gives the first few palindromic super- dnumbers for small d. dSloane palindromic super- dnumbers 2 A032750 131, 181, 333, 454, 919, 969, 1331, ... 3 A032751 4554, 6776, 17471, 22322, 22722, 28182, 43434, ... 4 A032752 83338, 1142411, 1571751, 1587851, 2013102, ... 5 A032753 3975793, 9799979, 39199193, 41299214, 65455456, ... 6 A032754 2023202, 374929473, 458353854, 499202994, 749858947, ... References Pickover, C. A. Keys to Infinity. New York: Wiley, p. 7, 1995. Sloane, N. J. A. Sequences A014569, A032743, A032744, A032745, A032746, A032747, A032748, A032749, A032750, A032751, A032752, A032753, A032754, A032755, and A032756 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.-att.com/~njas/sequences/eisonline.html. Superdiagonal The superdiagonal of a SQUARE MATRIX is the set of elements directly above the elements comprising the DIAGONAL . For example, in the following matrix, the diagonal elements are denoted diand the super- diagonal elements are denoted si; d1 s1a13... a1n a21d2 s2::: a2n a31a32d3::: a3n n:::::::::::: an1an2an3/C1/C1/C1 dn2 666643 77775: See also D IAGONAL ,S UBDIAGONAL ,T RIDIAGONAL MATRIX Super-Domino POLYOMINO Super-Edge-Graceful Graph See also EDGE-GRACEFUL GRAPH ,SKOLEM- GRACEFUL GRAPH Superegg A superegg is a solid described by the equation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C27 y2 a2sYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutn /C27z bYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutn /C301: Supereggs will balance on either end for any a, b, and n. See also EGG,SUPERELLIPSE ,SUPERELLIPSOID References Gardner, M. "Piet Hein’s Superellipse." Ch. 18 in Mathema- tical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage, pp. 240 /C1/54, 1977. Superellipse A curve with Cartesian equation x aYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutn /C27y bYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutn /C301: (1) where n/C212, first discussed in 1818 by Lame ´. The curves illustrated above correspond to a/C301,b/C302, andn/C302:5;3.0, and 3.5. Superellipses with a/C30bare also known as Lame ´curves. The AREA of the super-ellipse with a/C30b/C301 is given by A/C304g1 01/C28xnðÞ1=ndx (2) /C302G1 nYru*Yru+ G1/C271 nYru*Yru+ G2 nYru*Yru+ : (3) Ifnis a rational, then the curve is algebraic. However, for irrational n, the curve is transcenden- tal. For EVEN INTEGERS n, the curve becomes closer to a rectangle as nincreases. For ODD INTEGER values of n, the curve looks like the EVEN case in the POSITIVE quadrant but goes to infinity in both the second and fourth quadrants (MacTutor Archive). A special case of the superellipse is given by the ASTROID (/n/C302=3); (ax)2=3/C27(by)2=3/C30a2/C28b2YrvYru2=3(4) (left figure). Piet Hein called the curve with n/C305=2 anda/C30b"the" superellipse (right figure). The above plots show the function ½x½p/C27½y½q(5) for p /C301, ..., 4 and q /C301, ..., 4. A degenerate superellipse is a superellipse with r 52: The above curves are for a /C301, b /C302, and r /C300:5; 1.0, 1.5, and 2.0. See also ASTROID ,C HMUTOV SURFACE ,E LLIPSE , GOURSAT’S SURFACE ,SUPEREGG References Gardner, M. "Piet Hein’s Superellipse." Ch. 18 in Mathema- tical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage, pp. 240 /C1/54, 1977. MacTutor History of Mathematics Archive. "Lame ´ Curves." http://www-groups.dcs.st-and.ac.uk/~history/Curves/La- me.html. Superellipsoid A generalization of the ELLIPSOID , also called the superquadratic ellipsoid, defined by the equation ½x½2 =e /C27½y½2 =eYrvYru e=n/C27½z ½2 =n /C301: (1) where e and n are the east-west and north-south exponents, respectively. The superellipsoid can be rendered in POVRay † with the command superellipsoid{ Be,n /C21 } The generalization x aYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutn /C27y bYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutn /C27z cYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutn /C301 (2) of the surface considered by Gray (1997) might also be called a superellipsoid. The VOLUME of the solid with a /C30b /C30c /C301is Vn /C308g1 0g(1/C28xn)1 =n 01 /C28xn /C28ynðÞ1 =ndy dx (3) /C308G 1 /C271 nYru*Yru+ G 1 /C273 nYru*Yru+ : (4) As n 0/C12; the solid becomes a CUBE ,so lim n 0/C12Vn /C308 (5) as it must. This is a special case of the integral 3.2.2.2ggg x ]0; y ]0; z]0 x aYru*Yru+p /C27y bYru*Yru+q /C27z cYru*Yru+r 51xa/C281yb/C281z g/C281 dx dy dz /C30a abbc g pqrGa p ! Gb q ! Gg r ! Ga p/C27b q/C27gr ! (6) in Prudnikov et al. (1986, p. 583). See also E LLIPSOID ,G OURSAT’S SURFACE ,SUPEREL- LIPSE References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 292, 1997. POV-Ray Team. "Superquadratic Ellipsoid." §4.5.1.10 in Persistence of Vision Ray-Tracer Version 3.1g User’s Documentation, p. 199, May 1999. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon and Breach, 1986. Superfactorial The superfactorial of n is defined by Pickover (1995) as n$ /C13n!n!Un! |fflffl{zfflffl} n!: The first two values are 1 and 4, but subsequently grow so rapidly that 3$ already has a huge number of digits. Sloane and Plouffe (1995) define the superfactorial by n$ /C13Yn i/C301i!; which is equivalent to the integral values of the BARNES’ G-FUNCTION . The first few values are 1, 1, 2, 12, 288, 34560, ... (Sloane’s A000178). This function has an unexpected connection with B ELL NUMBERS . See also BARNES’ G-FUNCTION ,BELL NUMBER ,FAC- TORIAL ,L ARGE NUMBER ,S UBFACTORIAL ,V ANDER- MONDE DETERMINANT References Fletcher, A.; Miller, J. C. P.; Rosenhead, L.; and Comrie, L. J. An Index of Mathematical Tables, Vol. 1. Oxford, England: Blackwell, p. 50, 1962. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 231 1994. Pickover, C. A. Keys to Infinity. New York: Wiley, p. 102, 1995. Radoux, C. "Query 145." Not. Amer. Math. Soc. 25, 197, 1978. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., p. 53, 1963. Sloane, N. J. A. Sequences A000178/M2049 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Supergraph If G ? is a SUBGRAPH of G, then G is said to be a supergraph of G?:/ See also GRAPH ,SUBGRAPH Supernormal Trials for which the LEXIS RATIO L /C13s sB; satisfies L /C211, where s is the VARIANCE in a set of s LEXIS TRIALS and sBis the VARIANCE assuming BERNOULLI TRIALS . See also BERNOULLI TRIAL,LEXIS TRIALS ,SUBNORMAL Superperfect Number A number n such that s2(n) /C30 s( s(n)) /C302n : where s(n) is the DIVISOR FUNCTION is called a superperfect number. EVEN superperfect numbers are just 2p /C281 ; where Mp /C302p /C281isaM ERSENNE PRIME . If any ODD superperfect numbers exist, they are SQUARE NUMBERS and either n or s(n)is DIVISIBLE by at least three distinct PRIMES . More generally, an m-superperfect number is a number for which sm(n) /C302n; and an (m, k)-perfect number is a number n for which sm(n) /C302n: A number n can tested to see if it is (m, k)-perfect using the following Mathematica code. SuperperfectQ[m_, n_, k_:2] : /C30 Nest[DivisorSigma[1, #] &, n, m] /C30/C30 kn The first few (2,2)-perfect numbers are 2, 4, 16, 64, 4096, 65536, 262144, ... (Sloane’s A019279; Cohen and te Riele 1996). For m ]3 ; there are no EVEN m- superperfect numbers (Guy 1994, p. 65). There are no (3; 2)/-superperfect numbers n B2 /C215 108 for 4 5m 55:/ See also MERSENNE NUMBER ,PERFECT NUMBER References Cohen, G. L. and te Riele, J. J. "Iterating the Sum-of- Divisors Function." Experim. Math. 5,93/C1/00, 1996. Guy, R. K. "Superperfect Numbers." §B9 in Unsolved Pro- blems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 65 /C1/6, 1994. Kanold, H.-J. "U¨ ber ‘Super Perfect Numbers."’ Elem. Math. 24,61/C1/2, 1969. Lord, G. "Even Perfect and Superperfect Numbers." Elem. Math. 30,87/C1/8, 1975.Sloane, N. J. A. Sequences A019279 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Suryanarayana, D. "Super Perfect Numbers." Elem. Math. 20,16/C1/7, 1969. Suryanarayana, D. "There is No Odd Super Perfect Number of the Form p2 a :/" Elem. Math. 24, 148 /C1/50, 1973. Superposition Principle For a linear homogeneous ORDINARY DIFFERENTIAL EQUATION ,ify1(x) and y2(x) are solutions, then so is y1(x) /C27y2(x):/ Super-Poulet Number AP OULET NUMBER whose DIVISORS d all satisfy d½2d /C282: The first few are 341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, ... (Sloane’s A050218). See also POULET NUMBER References Sloane, N. J. A. Sequences A050218 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Superquadratic Ellipsoid SUPERELLIPSOID Superregular Graph For a VERTEX x of a GRAPH , let Gxand Dxdenote the SUBGRAPHS of G/C28x induced by the VERTICES adjacent to and nonadjacent to x, respectively. The empty graph is defined to be superregular, and G is said to be superregular if G is a REGULAR GRAPH and both Gx and Dx are superregular for all x. The superregular graphs are precisely C5 ; mKn// (m; n ]1); Gn(/n ]1); and the complements of these graphs, where Cnis a CYCLIC GRAPH , Knis a COMPLETE GRAPH and mKnis m disjoint copies of Kn ; and Gnis the Cartesian product of Knwith itself (the graph whose VERTEX set consists of n2 VERTICES arranged in an n /C29n square with two VERTICES adjacent IFF they are in the same row or column). See also COMPLETE GRAPH ,CYCLIC GRAPH ,REGULAR GRAPH References Vince, A. "The Superregular Graph." Problem 6617. Amer. Math. Monthly 103, 600/C1/03, 1996. West, D. B. "The Superregular Graphs." J. Graph Th. 23, 289/C1/95, 1996. Superscript A quantity displayed above the normal line of text (and generally in a smaller point size), as the " i"i nxi; is called a superscript. Superscripts are commonly used to indicate raising to a POWER (/x3means x/C215x/C215x orxCUBED ), multiple differentiation ( /f(3)(x)i sa n abbreviation for f §(x) /C30d3f =dx3); and a host of other operations and notations in mathematics. See also SUBSCRIPT Superset A SET containing all elements of a smaller SET.IfB is a SUBSET of A, then A is a superset of B, written /A –B: If A is a PROPER SUPERSET of B, this is written A ‡B:/ See also PROPER SUBSET ,PROPER SUPERSET ,SUBSET Superstructure In NONSTANDARD ANALYSIS , the limitation to first- order analysis can be avoided by using a construction known as a superstructure. Superstructures are constructed in the following manner. Let X be an arbitrary set whose elements are not sets, and call the elements of X "individuals." Define inductively a sequence of sets with S0(X) /C30X and, for each natural number k, Sk /C271(X) /C30Sk(X) @B Sk(X) ðÞ ; and let S(X) /C30@/C12 k /C300Sk(X) : (1) Then S(X) is called the superstructure over X.An element of S(X)isan ENTITY of S(X) :/ Using the definition of ordered pair provided by Kuratowski, namely (a; b) /C30ffa g;fa; bgg; it follows that (a; b) /C23 S2(X) for any a ; b /C23 X : Therefore, X /C29X ⁄ S2(X) ; and for any function f from X into X, we have f /C23 S3(X) : Now assume that the set X is (in one-to-one correspondence with) the set of real numbers R; and then the relation R which describes continuity of a function at a point is a member of S6(X): Careful consideration shows that, in fact, all the objects studied in classical analysis over R are entities of this superstructure. Thus, first-order formulas about S(X) are sufficient to study even what is normally done in classical analysis using second-order reason- ing. To do nonstandard analysis on the superstructure S(X); one forms an ULTRAPOWER of the relational structure (S(X) ;/C23): LOS’ THEOREM yields the TRANS- FER PRINCIPLE of nonstandard analysis. See also LOS’ THEOREM ,N ONSTANDARD ANALYSIS , ULTRAPOWER References Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindst- røom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, p. 16, 1986. Hurd, A. E. and Loeb, P. A. Ch. 3 in An Introduction to Nonstandard Real Analysis. New York: Academic Press, 1985.Supplementary Angle Two ANGLES a and p /C28 a which together form a STRAIGHT ANGLE are said to be supplementary. See also ANGLE ,C OMPLEMENTARY ANGLE ,D IGON , STRAIGHT ANGLE Support The CLOSURE of the SET of arguments of a FUNCTION f for which f is not zero. See also CLOSURE (SET) Support Function Let M be an oriented REGULAR SURFACE in R3 with normal N. Then the support function of M is the function h : M 0 R defined by h(p) /C30p /C215 N(p): References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 410 /C1/11, 1997. Supremum Portions of this entry contributed by JEROME R. BREITENBACH The supremum is the least upper bound of a set S, defined as a quantity M such that no member of the SET exceeds M, but if e is any POSITIVE quantity, however small, there is a member that exceeds M /C28e (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., sup R does not exist), is it denoted supS or supx /C23S :/ More formally, the supremum sup S for S a (none- mpty) SUBSET of the extended reals ¯R /C30R @f9/C12 g is the smallest value y /C23 ¯R such that for all x /C23 S we have x 5y: Using this definition, sup S always exists and, in particular, sup R /C30/C12:/ Whenever a supremum exists, its value is unique. On the REAL LINE, the supremum of a set is the same as the supremum of its CLOSURE . See also INFIMUM ,LIMIT,SUPREMUM LIMIT,U PPER BOUND References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Jeffreys, H. and Jeffreys, B. S. "Upper and Lower Bounds." §1.044 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 13, 1988. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 6, 1996. Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, p. 31, 1988. Rudin, W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, p. 7, 1987. Supremum Limit Given a sequence of real numbers an ; the supremum limit, also called the UPPER LIMIT , but more often simply called the supremum limit and pronounced ‘lim-soup’ and written lim sup; is the limit of An /C30sup k >nak as n 0/C12; where supSdenotes the SUPREMUM . Note that, by definition, An is nonincreasing and so either has a limit or tends to /C28/C12: For example, suppose an /C30 (/C281)n =n; then for n odd, An /C301 =(n /C271); and for n even, An /C301 =n: Another example is an /C30sin n ; in which case An is a constant sequence An /C301:/ When lim sup an /C30lim inf an ; the sequence converges to the real number lim an /C30lim sup an /C30lim inf an : Otherwise, the sequence does not converge. See also INFIMUM LIMIT,LIMIT,SUPREMUM ,U PPER LIMIT Surd An archaic term for an IRRATIONAL NUMBER . See also IRRATIONAL NUMBER ,QUADRATIC SURD Surface The word "surface" is an important term in mathe- matics and is used in many ways. The most common and straightforward use of the word is to denote a 2-D SUBMANIFOLD of 3-D EUCLIDEAN SPACE . Surfaces can range from the very complicated (e.g., FRACTALS such as the MANDELBROT SET) to the very simple (such as the PLANE ). More generally, the word "surface" can be used to denote an (n /C281)/-D SUBMANIFOLD of an n-D MANIFOLD , or in general, any CODIMENSION -1 subob- ject in an object (like a BANACH SPACE or an infinite- dimensional MANIFOLD ). Even simple surfaces can display surprisingly coun- terintuitive properties. For example, the SURFACE OF REVOLUTION of y /C301=x around the X-AXIS for x ]1 (called G ABRIEL’S HORN ) has FINITE VOLUME but INFINITE SURFACE AREA . See also ALGEBRAIC SURFACE ,C OMPACT SURFACE , COMPLETE SURFACE ,D EVELOPABLE SURFACE ,FLAT SURFACE ,H YPERSURFACE ,IMMERSED MINIMAL SUR- FACE ,M ANIFOLD ,M INIMAL SURFACE ,O RIENTABLE SURFACE ,O RTHOGONAL SURFACES ,R IEMANN SUR- FACE ,SMOOTH SURFACE ,SOLIDReferences Andrews, P. "The Classification of Surfaces." Amer. Math. Monthly 95, 861/C1/68, 1988. Endraß, S. "Home Page of S. Endraß." http://www.mathe- matik.uni-mainz.de/~endrass/. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, 1986. Francis, G. K. A Topological Picturebook. New York: Springer-Verlag, 1987. Gallier, J. H. Curves and Surfaces for Geometric Design: Theory and Algorithms. New York: Academic Press, 1999. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997. Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Javaview. "Classic Surfaces from Differential Geometry." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/ surface/common/PaSurface.html. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 135, 1999. Morgan, F. "What is a Surface?" Amer. Math. Monthly 103, 369/C1/76, 1996. Nordstrand, T. "Gallery." http://www.uib.no/people/nfytn/ mathgal.htm. Nordstrand, T. "Surfaces." http://www.uib.no/people/nfytn/ surfaces.htm. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, 1993. Wagon, S. "Surfaces." Ch. 3 in Mathematica in Action. New York: W. H. Freeman, pp. 67 /C1/1, 1991. Wilkinson, S. "Intersections of Surfaces." Mathematica in Educ. Res. 8,5/C1/0, 1999. Yamaguchi, F. Curves and Surfaces in Computer Aided Geometric Design. New York: Springer-Verlag, 1988. Surface Area Surface area is the AREA of a given surface. Roughly speaking, it is the "amount" of a surface (i.e., it is proportional to the amount of paint needed to coverit), and has units of distance squared. It is commonlydenoted Sfor a surface in 3-D, or Afor a region of the plane (in which case it is simply called "the" AREA ). If the surface is PARAMETERIZED using uandv, then S/C30gSTu/C29Tv jj du dv ; (1) where TuandTvare tangent vectors and a/C29bis the CROSS PRODUCT .I fz/C30f(x;y) is defined over a region R, then S/C30ggRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @z @x !n /C27@z @y !2 /C271vuutdA; (2) where the integral is taken over the entire surface (Kaplan 1992, 3rd ed. pp. 245 /C1/48). Writing x/C30 x(u;v);y/C30y(u;v);and z/C30z(u;v) then gives the symmetrical form S/C30ggR?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG/C28F2p du dv : (3) where R?is the transformation of R, and E /C30@x @u !2 /C27@y @u !2 /C27@z @u !2 (4) F /C30@x @u@x @v /C27@y @u@y @v /C27@z @u@z @v (5) G /C30@x @v !2 /C27@y @v !2 /C27@z @v !2 (6) are coefficients of the first FUNDAMENTAL FORM (Kaplan 1992, 3rd ed. pp. 245 /C1/46). The following tables gives lateral surface areas S for some common SURFACES . Here, r denotes the RADIUS , h the height, e the ELLIPTICITY of a SPHEROID , p the base PERIMETER , s the SLANT HEIGHT , a the tube radius of a torus, and c the radius from the rotation axis of the torus to the center of the tube (Beyer 1987). Note that many of these surfaces are SURFACES OF REVOLUTION . SURFACE S CONE / prffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27h2p / CONICAL FRUSTUM / p R1 /C27R2 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 /C28R2 ðÞ2/C27h2q / CUBE /6a2/ CYLINDER /2prh/ OBLATE SPHEROID /2pa2 /C27pe2 eln1 /C27 e 1 /C28 eYru$Yru% / PROLATE SPHEROID /2pa2 /C272 pae esin/C281 e/ PYRAMID /1 2 ps/ PYRAMIDAL FRUSTUM /1 2 ps/ SPHERE /4pr2/ SPHERICAL LUNE /2r2 u/ TORUS /4p2ac/ ZONE /2prh/ Even simple surfaces can display surprisingly coun- terintuitive properties. For instance, the surface of revolution of y /C301=x around the X-AXIS for x ]1is called GABRIEL’S HORN , and has FINITE VOLUME but INFINITE surface area. See also AREA,F UNDAMENTAL FORMS ,S URFACE INTEGRAL ,SURFACE OF REVOLUTION ,VOLUME References Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127 /C1/32, 1987.Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1992. Surface Harmonic Any LINEAR COMBINATION of real SPHERICAL HARMO- NICS AlPl(cos u) /C27Xl m/C301Am lcos(mf) /C27Bmlsin(mf) ½/C138 Pml(cos u) for l fixed whose sum is not premultiplied by a factor rl (Whittaker and Watson 1990, p. 392). See also SOLID HARMONIC ,SPHERICAL HARMONIC References Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 197, 1959. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Surface Integral For a SCALAR FUNCTION f over a surface parameter- ized by u and v, the surface integral is given by F/C30gSfda/C30gSf(u; v) Tu /C29Tv jj du dv : (1) where Tu and Tv are tangent vectors and a /C29b is the CROSS PRODUCT . For a VECTOR FUNCTION over a surface, the surface integral is given by F/C30gSF /C215 da /C30gS(F /C215 ˆn) da (2) /C30gSfx dy dz /C27fy dz dx /C27fz dx dy: (3) where a /C215 b is a DOT PRODUCT and ˆn is a unit NORMAL VECTOR .Ifz /C30f(x; y) ; then da is given explicitly by da/C309/C28@z @xˆx/C28@z @yˆy/C27ˆz ! dx dy : (4) If the surface is SURFACE PARAMETERIZED using uand v, then F/C30gSF /C215(Tu/C29Tv)du dv : (5) See also INTEGRAL ,PATH INTEGRAL ,SURFACE PARA- METERIZATION ,VOLUME INTEGRAL References Leathem, J. G. Volume and Surface Integrals Used in Physics. 1905. Surface of Revolution A surface of revolution is a SURFACE generated by rotating a 2-D CURVE about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the APPLE , CONE (excluding the base), CONICAL FRUSTUM (exclud- ing the ends), CYLINDER (excluding the ends), D AR- WIN-DE SITTER SPHEROID ,G ABRIEL’S HORN , HYPERBOLOID ,LEMON ,OBLATE SPHEROID ,PARABO- LOID ,PROLATE SPHEROID ,PSEUDOSPHERE ,SPHERE , SPHEROID , and TORUS (and its generalization, the TOROID ). The area element of the SURFACE OF REVOLUTION obtained by rotating the curve y/C30f(x) from x/C30ato x/C30babout the X-AXIS is dS/C302pyd s/C302pyffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q dx: (1) so the surface area is S/C302pgb af(x)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27f?(x) ½/C1382q dx: (2) (Anton 1999, p. 380). If we are interested instead in finding the area of the SURFACE OF REVOLUTION obtained by rotating the curve x/C30g(y) around the Y-AXIS from y/C30atoy/C30b (as opposed to rotating about the X-AXIS ), the area element is given by dS/C302pxd s/C302pxffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27x?2p dy: (3) so the surface area is S/C302pgb ag(y)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27g?(y) ½/C1382q dy (4) (Kaplan 1992, 3rd ed. p. 251; Anton 1999, p. 380). The following table gives the lateral surface areas S for some common surfaces of revolution where r denotes the RADIUS (of a cone, cylinder, sphere, or zone), R1and R2the inner and outer radii of a frustum, hthe height, ethe ELLIPTICITY of a SPHER- OID, and aandcthe equatorial and polar radii (for a spheroid) or the radius of a circular cross-section and rotational radius (for a torus). surface S CONE /prffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C27h2p / CONICAL FRUSTUM /pR1/C27R2 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1/C28R2 ðÞ2/C27h2q / CYLINDER /2prh/ OBLATE SPHEROID /2pa2/C27pe2 eln1/C27e 1/C28eYru$Yru% / PROLATE SPHEROID /2pa2/C272pae esin/C281e/ SPHERE /4pr2 / TORUS /4p2ac/ ZONE /2prh/ The standard parameterization of a surface of revolu- tion is given by x(u;v)/C30f(v)cosu (5) y(u;v)/C30f(v)sinu (6) z(u;v)/C30c(v): (7) For a curve so parameterized, the first FUNDAMENTAL FORM has E/C30c2(8) F/C300 (9) G/C30f?2/C27c?2: (10) Wherever fand f?2/C27c?2are nonzero, then the surface is regular and the second FUNDAMENTAL FORM has e/C30/C28½f½c?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f?2/C27c?2p (11) f/C300 (12) g/C30sgn(f)fƒc?/C28f?cƒ ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif?2/C27c?2p : (13) Furthermore, the unit NORMAL VECTOR is ˆN(u;v)/C30sgn(f)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f?2/C27c?2pf?cosu c?sinu f?2 435: (14) and the PRINCIPAL CURVATURES are k1/C30g G/C30sgn(f)(fƒc?/C28f?cƒ) (f?2/C27c?2)3=2(15) k2/C30e E/C30/C28c? ½f½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f?2/C27c?2p : (16) The G AUSSIAN and MEAN CURVATURES are K/C30/C28c?2fƒ/C27f?c?cƒ ff?2/C27c?2YrvYru 2 (17) H/C30ffƒc?/C28f?cƒ ðÞ /C28c?f?2/C27c?2YrvYru 2½f½f?2/C27c?2YrvYru 3=2 (18) (Gray 1997). PAPPUS’S CENTROID THEOREM gives the VOLUME of a solid of rotation as the cross-sectional AREA times the distance traveled by the centroid as it is rotated.C ALCULUS OF VARIATIONS can be used to find the curve from a point x1;y1 ðÞ to a point x2;y2 ðÞ which, when revolved around the X-AXIS , yields a surface of smallest SURFACE AREA A(i.e., the MINIMAL SURFACE ). This is equivalent to finding the MINIMAL SURFACE passing through two circular wire frames. The AREA element is dA/C302pyd s/C302pyffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q dx: (19) so the SURFACE AREA is A/C302pgyffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27y? 2q dx: (20) and the quantity we are minimizing is f/C30yffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27y? 2q : (21) This equation has fx/C300;so we can use the B ELTRAMI IDENTITY f/C28yx@f @yx/C30a (22) to obtain yffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q /C28y?yy?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2p /C30a (23) y1/C27y?2YrvYru /C28yy?2/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q (24) y/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27y? 2q (25)yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2p /C30a (26) y2 a/C281/C30y?2(27) dx dy/C301 y?/C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2/C28a2p (28) x/C30agdyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2/C28a2p /C30acosh/C281y a ! /C27b (29) y/C30acoshx/C28b a ! : (30) which is called a CATENARY , and the surface gener- ated by rotating it is called a CATENOID . The two constants aand bare determined from the two implicit equations y1/C30acoshx1/C28b a ! (31) y2/C30acoshx2/C28b a ! : (32) which cannot be solved analytically. The general case is somewhat more complicated than this solution suggests. To see this, consider the MINIMAL SURFACE between two rings of equal RADIUS y0:Without loss of generality, take the origin at the midpoint of the two rings. Then the two endpoints are located at /C28x0;y0 ðÞ and x0;y0 ðÞ ;and y0/C30acosh/C28x0/C28b a ! /C30acoshx0/C28b a ! : (33) But cosh( /C28x)/C30cosh( x);so cosh/C28x0/C28b a ! /C30cosh/C28x0/C27b a ! : (34) Inverting each side /C28x0/C28b/C30/C28x0/C27b: (35) sob/C300 (as it must by symmetry, since we have chosen the origin between the two rings), and the equation of the MINIMAL SURFACE reduces to y/C30acoshx a ! ; (36) At the endpoints y0/C30acoshx0 a ! : (37) but for certain values of x0andy0;this equation has no solutions. The physical interpretation of this fact is that the surface breaks and forms circular disks in each ring to minimize AREA .CALCULUS OF VARIATIONS cannot be used to find such discontinuous solutions(known in this case as G OLDSCHMIDT SOLUTIONS ). The minimal surfaces for several choices of endpoints areshown above. The first two cases are CATENOIDS , while the third case is a G OLDSCHMIDT SOLUTION . To find the maximum value of x0=y0at which CATENARY solutions can be obtained, let p/C131=a: Then (35) gives y0p/C30cosh px0ðÞ : (38) Now, denote the maximum value of x0asx/C31 0:Then it will be true that dx0=dp/C300:Take d=dpof (38), y0/C30sinh px0ðÞ x0/C27pdx0 dp ! : (39) Now set dx0=dp/C300 y0/C30x0sinh px0/C31 ðÞ : (40) From (38), py0/C31/C30cosh px0/C31 ðÞ : (41) Take (41) }(40), px0/C31/C30coth px0/C31 ðÞ : (42) Defining u/C13px0/C31; u/C30coth u: (43) This has solution u/C301:1996789403 . . . :From (40), y0p/C30cosh u:Divide this by (43) to obtain y0=x0/C30 sinh u;so the maximum possible value of x0=y0is x0 y0/C30csch u/C300:6627434193 . . . : (44) Therefore, only Goldschmidt ring solutions exist for x0=y0>0:6627 . . . :/ The SURFACE AREA of the minimal CATENOID surface is given by A/C302(2p)gx0 0yffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q dx; (45) but since y/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27y? 2q a (46)/C30acoshx a ! : (47) A/C304p agx0 0y2dx/C304pagx0 0cosh2x a ! dx /C304pagx0 01 2cosh2x a ! /C271"# dx /C302pagx0 0cosh2x a ! dx/C27gx0 0dx"# /C302paa 2sinh2x a ! /C27x"#x0 0 /C30pa2sinh2x a ! /C272x a"#x0 0 /C30pa2sinh2x0 a ! /C272x0 a"# : (48) Some caution is needed in solving (37) for a.I fw e take x0/C301=2 and y0/C301 then (37) becomes 1/C30acosh1 2a ! : (49) which has twosolutions: a1/C300:2350 . . . ("deep"), and a2/C300:8483 . . . /However, upon plugging these into (48) with x0/C301=2;we find A1/C306:8456 . . . and A2/C30 5:9917 . . . :SoA1isnot, in fact, a local minimum, and A2is the only true minimal solution. The SURFACE AREA of the CATENOID solution equals that of the G OLDSCHMIDT SOLUTION when (48) equals the AREA of two disks, pa2sinh2x0 a ! /C272x0 a"# /C302py2 0 (50) a22 sinhx0 a ! coshx0 a ! /C272x0 a"# /C282y20/C300 (51) a2coshx0 a !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh2x0 a ! /C281vuut/C27x0 a2 435/C28y 2 0/C300:(52) Plugging in y0 a/C30coshx0 a ! : (53) y0 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y0 a !2 /C281vuut/C27cosh/C281y0 a ! /C28y0 a !2 /C300: (54) Defining u/C13y0 a(55) gives uffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2/C281p /C27cosh/C281u/C28u2/C300: (56) This has a solution u/C301:2113614259 :The value of x0=y0for which Acatenary/C30A2 disks (57) is therefore x0 y0/C30x0 a y0 a/C30cosh/C281y0 a ! y0 a/C30cosh/C281u u /C300:5276973967 : (58) Forx0=y0/C23f0:52770 ;0:6627) ;the CATENARY solution has larger AREA than the two disks, so it exists only as aRELATIVE MINIMUM . There also exist solutions with a disk (of radius r) between the rings supported by two CATENOIDS of revolution. The AREA is larger than that for a simple CATENOID , but it is a RELATIVE MINIMUM . The equa- tion of the POSITIVE half of this curve is y/C30c1coshx c1/C27c3 ! : (59) At (0 ;r); r/C30c1cosh c3ðÞ: (60) Atx0;y0 ðÞ ; y0/C30c1coshx0 c1/C27c3 ! : (61) The AREA of the two CATENOIDS is Acatenoids /C302(2p)gx0 0yffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27y?2q dx/C304p c1gx0 0y2dx /C304pc1gx0 0cosh2x c1/C27c3 ! dx: (62) Now let u/C13x=c1/C27c3;sodu/C30dx=c1A/C304pc2 1gx0=x1/C27c3 c3cosh2ud u /C304pc211 2gx0=x1/C27c3 c3[cosh(2 u)/C271]du /C302pc2 11 2sinh(2 u)/C27uhix0=x1/C27c3 c3 /C302pc2 11 2sinh 2x0 c1/C27c3 !"# /C2812sinh(2 c3)/C27x0 c1() /C30pc2 1sinh 2x0 c1/C27c3 !"# /C28sinh(2 c3)/C272x0 c1() :(63) The AREA of the central DISK is Adisk/C30pr2/C30pc21cosh2c3; (64) so the total AREA is A/C30pc21 /C2sinh 2x0 c1/C27c3 !"# /C27cosh2c3/C28sinh(2 c3)YrtYrP /C272x0 c1() : (65) By P LATEAU’S LAWS , the CATENOIDS meet at an ANGLE of 120 8,s o tan 30/C14/C30dy dx"# x/C300/C30sinhx c1/C27c3 !"# x/C300 /C30sinh c3/C301ffiffiffi 3p (66) and c3/C30sinh/C2811ffiffiffi 3p ! : (67) This means that cosh2c3/C28sinh(2 c3) /C301/C27sinh2c3YrtYrP /C282 sinh c3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27sinh2c3q /C301/C271 3Yru*Yru+ /C2821ffiffiffi 3p !ffiffiffiffiffiffiffiffiffiffi 1/C271 3q /C304 3/C282ffiffiffi 3p2ffiffiffi3p/C300: (68) so A/C30pc2 1sinh 2x0 c1/C27c3 !"# /C272x0 c1() : (69) Now examine x0=y0; x0 y0/C30x0 c1 y0 c1/C30x0 c1 coshx0 c1/C27 c3 ! u sech( u /C27c3): (70) where u /C13x0 =c1 : Finding the maximum ratio of x0 =y0 gives d dux0 y0 ! /C30sech( u /C27c3) /C28u tanh( u /C27c3) sech( u /C27c3) /C300 (71) u tanh( u /C27c3) /C301: (72) with c3 /C30sinh /C281 1=ffiffiffi 3pYrvYru as given above. The solution is u /C301:0799632187 ; so the maximum value of x0 =y0 for two CATENOIDS with a central disk is y0 /C300 :4078241702 :/ If we are interested instead in finding the curve from a point x1 ; y1 ðÞ to a point x2 ; y2 ðÞ which, when revolved around the Y-AXIS (as opposed to the X- AXIS), yields a surface of smallest SURFACE AREA A,we proceed as above. Note that the solution is physically equivalent to that for rotation about the X-AXIS , but takes on a different mathematical form. The AREA element is dA /C302pxds/C302pxffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y?2q dx (73) A /C302pg xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y?2q dx : (74) and the quantity we are minimizing is f /C30xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27y? 2q : (75) Taking the derivatives gives @f @y /C300 (76) d dx@f @y?/C30d dxxy?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 y?2p ! : (77) so the EULER- LAGRANGE DIFFERENTIAL EQUATION becomes @f @y /C28d dx@f @y?/C30d dxxy ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 y?2p ! /C300: (78) xy?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C27 y?2p /C30a (79) x2y ?2 /C30a2 1 /C27y?2YrvYru (80) y?2 x2 /C28a2YrvYru /C30a2 (81) dy dx /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 a2p (82)y /C30agdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 /C28 a2p /C27b /C30a cosh /C281x a ! /C27b: (83) Solving for x then gives x /C30a coshy /C28 b a ! : (84) which is the equation for a CATENARY . The SURFACE AREA of the CATENOID product by rotation is A /C302 pg xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27y?2q dx /C302pg xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27a2 x2 /C28 a2s dx /C302pgxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 a2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28a2 ðÞ /C27a2p dx /C302pgx2 dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28 a2p /C30x 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 /C28a2p /C27a2 2ln x /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 /C28a2pYru*Yru+"#x2 x1 /C301 2x2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 2/C28a2q /C28x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 1/C28a2q /C27a2lnx2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 2/C28a2p x1/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 1/C28a2p ! "# :(85) Isenberg (1992, p. 80) discusses finding the MINIMAL SURFACE passing through two rings with axes offset from each other. See also APPLE ,CATENOID ,CONE CONICAL FRUSTUM , CYLINDER ,DARWIN-DE SITTER SPHEROID ,EIGHT SUR- FACE ,GABRIEL’S HORN,HYPERBOLOID ,LEMON ,M ER- IDIAN ,M INIMAL SURFACE ,O BLATE SPHEROID , PAPPUS’S CENTROID THEOREM ,PARABOLOID ,PARAL- LEL (SURFACE OF REVOLUTION ), PENINSULA SURFACE , PROLATE SPHEROID ,PSEUDOSPHERE ,SINCLAIR’S SOAP FILM PROBLEM ,S OLID OF REVOLUTION ,S PHERE , SPHEROID ,TOROID ,TORUS ,UNDULOID References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 931 /C1/37, 1985. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 42, 1980. Gray, A. "Surfaces of Revolution." Ch. 20 in Modern Differ- ential Geometry of Curves and Surfaces with Mathema- tica, 2nd ed. Boca Raton, FL: CRC Press, pp. 457 /C1/80, 1997. Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 inGeometry and the Imagination. New York: Chelsea, pp. 7/C1/1, 1999. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, pp. 79 /C1/0 and Appendix III, 1992. Surface of Section A surface (or "space"rpar; of section is a way of presenting a trajectory in n-D PHASE SPACE in an (n/C281)/-DSPACE . By picking one phase element con- stant and plotting the values of the other elements each time the selected element has the desired value, an intersection surface is obtained. If the equations of motion can be formulated as a MAP in which an explicit FORMULA gives the values of the other elements at successive passages through the selected element value, the time required to compute the surface of section is greatly reduced. See also HE´ NON- HEILES EQUATION ,PHASE SPACE References Tabor, M. "The Surface of Section." §4.1 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 121 /C1/26, 1989. Surface Parameterization A surface in 3-SPACE can be parameterized by two variables (or coordinates) u and v such that x /C30x(u; v) (1) y /C30y(u; v) (2) z /C30z(u; v) : (3) If a surface is parameterized as above, then the tangent VECTORS Tu /C30@x @uˆx /C27@y @uˆy /C27@z @uˆz (4) Tv /C30@x @vˆx /C27@y @vˆy /C27@z @vˆz (5) are useful in computing the SURFACE AREA and SURFACE INTEGRAL . See also SMOOTH SURFACE ,SURFACE AREA,SURFACE INTEGRAL Surface Spherical Harmonic SURFACE HARMONIC Surgery In the process of attaching a k-HANDLE to a MANIFOLD M, the BOUNDARY of M is modified by a process called (k /C281)/-surgery. Surgery consists of the removal of a TUBULAR NEIGHBORHOOD of a (k /C281)/-SPHERE S(k /C281) from the BOUNDARIES of M and the dim(M) /C281 standard SPHERE , and the gluing together of these two scarred-up objects along their common BOUND- ARIES . See also BOUNDARY ,DEHN SURGERY ,HANDLE ,MANI- FOLD ,SPHERE ,TUBULAR NEIGHBORHOOD References Cappell, S.; Ranicki, A.; and Rosenberg, J. (Eds.). Surveys on Surgery Theory, Vol. 1. Princeton, NJ: Princeton Univer- sity Press, 2000.Surjection An ONTO (a.k.a. surjective) MAP. See also BIJECTION ,D OMAIN ,O NE-TO- ONE,O NTO, RANGE (IMAGE ) Surjective ONTO Surprise Examination Paradox UNEXPECTED HANGING PARADOX Surreal Number The most natural collection of numbers which in- cludes both the REAL NUMBERS and the infinite ORDINAL NUMBERS of Georg Cantor. They were invented by John H. Conway in 1969. Every REAL NUMBER is surrounded by surreals, which are closer to it than any REAL NUMBER . Knuth (1974) describes the surreal numbers in a work of fiction. The surreal numbers are written using the NOTATION fa ½b g; where f½g/C300;f0½g/C301 is the simplest number greater than 0, f1½g/C302 is the simplest number greater than 1, etc. Similarly, f½0g/C30/C281 is the sim- plest number less than 1, etc. However, 2 can also be represented by f1½3g;f3=2 ½4 g;f1½ vg; etc. See also OMNIFIC INTEGER ,ORDINAL NUMBER ,REAL NUMBER References Berlekamp, E. R.; Conway, J. H.; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Games in General. London: Academic Press, 1982. Conway, J. H. On Numbers and Games. New York: Aca- demic Press, 1976. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 283 /C1/84, 1996. Conway, J. H. and Jackson, A. "Budding Mathematician Wins Westinghouse Competition." Not. Amer. Math. Soc. 43, 776/C1/79, 1996. Gonshor, H. An Introduction to Surreal Numbers. Cam- bridge, England: Cambridge University Press, 1986. Knuth, D. Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness.Reading, MA: Addison-Wesley, 1974. http://www-cs-facul- ty.stanford.edu/~knuth/sn.html. Surrogate Surrogate data are artificially generated data which mimic statistical properties of real data. Isospectral surrogates have identical POWER SPECTRA as real data but with randomized phases. Scrambled surrogates have the same probability distribution as real data, but with white noise POWER SPECTRA . See also POWER SPECTRUM Surveying Problems HANSEN’S PROBLEM ,SNELLIUS- POTHENOT PROBLEM Survivorship Curve Plotting lxfrom a LIFE EXPECTANCY table on a logarithmic scale versus x gives a curve known as a survivorship curve. There are three general classes of survivorship curves, illustrated above. 1. Type I curves are typical of populations in which most mortality occurs among the elderly (e.g., humans in developed countries). 2. Type II curves occur when mortality is not dependent on age (e.g., many species of large birds and fish). For an infinite type II population, e0 /C30 e1 /C30...; but this cannot hold for a finite popula- tion. 3. Type III curves occur when juvenile mortality is extremely high (e.g., plant and animal species producing many offspring of which few survive). In type III populations, it is often true that ei/C271 > eifor small i. In other words, life expectancy increases for individuals who survive their risky juvenile period. See also LIFE EXPECTANCY Suslin’s Theorem A SET in a POLISH SPACE is a BOREL SET IFF it is both ANALYTIC and COANALYTIC . For subsets of w, a set is d1 1IFF it is "hyperarithmetic." See also ANALYTIC SET,BOREL SET,COANALYTIC SET, POLISH SPACE Suspended Knot An ordinary KNOT in 3-D suspended in 4-D to create a knotted 2-sphere. Suspended knots are not smooth at the poles.See also SPUN KNOT,TWIST- SPUN KNOT Suspension The JOIN of a TOPOLOGICAL SPACE X and a pair of points S0 ;a(X) /C30X + S0 :/ See also JOIN (SPACES ), TOPOLOGICAL SPACE References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 6, 1976. Suzanne Set The nth Suzanne set Snis defined as the set of COMPOSITE NUMBERS x for which n ½S(x) and n½Sp(x); where x /C30a0 /C27a1101YrvYru /C27.../C27ad10dYrvYru /C30p1p2 /C1/C1/C1pn : and S(x) /C30Xd j/C300aj Sp(x) /C30Xm i /C301SpiðÞ: Every Suzanne set has an infinite number of ele- ments. The Suzanne set Snis a superset of the MONICA SET Mn:/ See also MONICA SET References Smith, M. "Cousins of Smith Numbers: Monica and Suzanne Sets." Fib. Quart. 34, 102/C1/04, 1996. Suzuki Group The SPORADIC GROUP Suz. References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/Suz.html. Swallowtail Catastrophe A CATASTROPHE which can occur for three control factors and one behavior axis. The swallowtail cata- strophe is the universal unfolding of singularity f(x) /C30x5 with codimension 3, i.e., in three unfolding parameters, and is of the form F(x; u; v; w) /C30x5 /C27 ux3 /C27vx2 /C27wx : The equations x /C30uv2 /C273v4 y /C30/C282uv /C284v3 z /C30u display such a catastrophe (von Seggern 1993, Nord- strand). The above surface uses u /C23 [/C282 ; 2] and v /C23 [/C280 :8 ; 0 :8]:/ References Nordstrand, T. "Swallowtail." http://www.uib.no/people/ nfytn/stltxt.htm. Sanns, W. Catastrophe Theory with Mathematica: A Geo- metric Approach. Germany: DAV, 2000. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 94, 1993. Swastika An irregular ICOSAGON , also called the gammadion or fylfot, which symbolized good luck in ancient Arabic and Indian cultures. In more recent times, it was adopted as the symbol of the Nazi Party in Hitler’s Germany and has thence come to symbolize anti- Semitism. See also CROSS ,DISSECTION References Gardner, M. "Form a Swastika." §20.6 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 198 and 203 /C1/04, 1984. Swastika Curve The plane curve with Cartesian equation y4 /C28x4 /C30xyand polar equation r2 /C30sin u cos u sin4 u /C28 cos4 u : References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989. Sweep Signal The general function y(a ; b; c ; d) /C30c sinp b /C28 a(b /C28a)x d /C27a !2 /C28a22 4358 < :9 = ;: References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 160, 1993. Swept Sine SWEEP SIGNAL Swinnerton-Dyer Conjecture In the early 1960s, B. Birch and H. P. F. Swinnerton- Dyer conjectured that if a given ELLIPTIC CURVE has an infinite number of solutions, then the associated L-series has value 0 at a certain fixed point. In 1976, Coates and Wiles showed that elliptic curves with COMPLEX multiplication having an infinite number of solutions have L-series which are zero at the relevant fixed point ( COATES- WILES THEOREM ), but they were unable to prove the converse. V. Kolyvagin extended this result to modular curves. See also COATES- WILES THEOREM ,ELLIPTIC CURVE References Birch, B. and Swinnerton-Dyer, H. "Notes on Elliptic Curves. II." J. reine angew. Math. 218,7 9/C1/08, 1965. Cipra, B. "Fermat Prover Points to Next Challenges." Science 271, 1668 /C1/669, 1996. Clay Mathematics Institute. "The Birch and Swinnerton- Dyer Conjecture." http://www.claymath.org/prize_pro- blems/birchsd.htm. Ireland, K. and Rosen, M. "New Results on the Birch- Swinnerton-Dyer Conjecture." §20.5 in A Classical Intro- duction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 353 /C1/57, 1990. Mazur, B. and Stevens, G. (Eds.). p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture. Providence, RI: Amer. Math. Soc., 1994. Wiles, A. "The Birch and Swinnerton-Dyer Conjecture." http://www.claymath.org/prize_problems/birchsd.pdf. Swinnerton-Dyer Polynomial The minimal POLYNOMIAL Sn(x) whose ROOTS are sums and differences of the SQUARE ROOTS of the first n PRIMES , Sn(x) /C30Y x 9ffiffiffi 2p 9ffiffiffi3p 9ffiffiffi5p 9...9ffiffiffiffiffip npYru*Yru+ : References Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 11 and 225 /C1/26, 1991. Swirl A swirl is a generic word to describe a function having arcs which double back around each other. The plots above correspond to the function f(r ; u) /C30sin(6 cos r /C28n u) for n /C300, 1, ..., 5. See also DAISY,W HIRL Switching Class TWO-GRAPH Swung Dash The symbol /C2 used to denote similarity, equivalence relations, or asymptosy. References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 285, 1997. Sylow p-Subgroup If pk is the highest POWER of a PRIME p dividing the ORDER of a FINITE GROUP G, then a SUBGROUP of G of ORDER pk is called a Sylow p-subgroup of G.See also ABHYANKAR’S CONJECTURE ,SUBGROUP ,SY- LOW THEOREMS Sylow Theorems Let p be a PRIME NUMBER , G a FINITE GROUP , and ½G ½ the order of G. 1. If p divides ½G½; then G has a SYLOW P- SUBGROUP . 2. In a FINITE GROUP , all the SYLOW P-SUBGROUPS are CONJUGATE for some fixed p. 3. The number of SYLOW P-SUBGROUPS for a fixed p is CONGRUENT to 1 (mod p). See also CONJUGATE SUBGROUP ,SYLOW P-SUBGROUP Sylvester Cyclotomic Number Given a LUCAS SEQUENCE with parameters P and Q, discriminant D "0; and roots a and b; the Sylvester cyclotomic numbers are Qn /C30Y ra /C28 zr b ðÞ : where z /C13cos2p n ! /C27i sin2p n ! is a PRIMITIVE ROOT OF UNITY and the product is over all exponents r RELATIVELY PRIME to n such that r /C23 1; n½Þ :/ See also LUCAS SEQUENCE References Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 69, 1989. Sylvester Graph The Sylvester graph of a configuration is the set of ORDINARY POINTS and ORDINARY LINES . See also ORDINARY LINE,ORDINARY POINT References Guy, R. K. "Monthly Unsolved Problems, 1969 /C1/987." Amer. Math. Monthly 94, 961 /C1/70, 1987. Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903 /C1/09, 1989. Sylvester Matrix For POLYNOMIALS of degree m and n, the Sylvester matrix is an (m /C27n) /C29(m /C27n) matrix whose DETERMI- NANT is the RESULTANT of the two POLYNOMIALS . See also DETERMINANT ,RESULTANT Sylvester’s Determinant Identity Given a MATRIX A ; let ½A½ denote its determinant. Then ½A½½Ars; pq ½/C30½Ar; p ½½As; q ½/C28½Ar; q ½½As; p ½; (1) where Au; wis the SUBMATRIX of A formed by the intersection of the subset w of columns and u of rows. Bareiss (1968) writes the identity as ½A½ a(k /C281) kkYrtYrP n/C28k /C281/C30a(k) k /C271 ; k /C281/C1/C1/C1 a(k) k /C271; n n::: n a(k) n; k /C271 /C1/C1/C1 a(k) n; nYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut: (2) where a (k) ij/C30a11a12/C1/C1/C1 a1kaij a21a22/C1/C1/C1 a2ka2j nn::: nn ak1ak2/C1/C1/C1 akkakj ai1ai2/C1/C1/C1 aikaijYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut(3) forkBi;j5n: / See also DETERMINANT References Bareiss, E. H. "Multistep Integer-Preserving Gaussian Elimination." Argonne National Laboratory Report ANL- 7213, May 1966. Bareiss, E. H. "Sylvester’s Identity and Multistep Integer- Preserving Gaussian Elimination." Math. Comput. 22, 565/C1/78, 1968. Sylvester’s Four-Point Problem Sylvester’s four-point problem asks for the probability q(R) that four points chosen at random in a planar region Rhave a CONVEX HULL which is a QUADRILAT- ERAL (Sylvester 1865). Depending on the method chosen to pick points from the infinite plane, anumber of different solutions are possible, promptingSylvester to conclude "This problem does not admit of a determinate solution" (Sylvester 1865; Pfiefer 1989).For points selected from an open, convex subset of the PLANE having finite AREA , the probability if given by P(R)/C301/C284¯AR A(R): where ¯ARis the expected area of a triangle over region RandA(R) is the area of region R. Note that¯ARis simply the value computed for an appropriate region, e.g., DISK TRIANGLE PICKING ,TRIANGLE TRIAN- GLE PICKING ,SQUARE TRIANGLE PICKING , etc. P(R) can range between 2 35q(R)51/C2835 12p2(1) (0:666665q(R)50:70448) depending on the shape of the region, as first proved by Blaschke (Blaschke 1923, Peyerimhoff 1997). The following table givesthe probabilities for various simple plane regions (Kendall and Moran 1963; Pfiefer 1989; Croft et al. 1991, pp. 54 /C1 /5; Peyerimhoff 1997). R /P(R)/approx. TRIANGLE /2 3/ 0.66667 SQUARE /25 36/ 0.69444 HEXAGON /683972/ 0.70267 ELLIPSE ,CIRCLE /1/C2835 12p2/0.70448 Sylvester’s problem can be generalized to ask for the probability that the CONVEX HULL ofn/C272 randomly chosen points in the UNIT BALL Bnhasn/C271 vertices. The solution is given by Pn/C30(n/C272)n/C271 1 2(n/C271)Yru$Yru% n/C271 2n(n/C271)2 1 2(n/C271)2 ! (2) (Kingman 1969, Groemer 1973, Peyerimhoff 1997), which is the maximum possible for any bounded convex domain K/C23Rn:The first few values are P1/C301 P2/C3035 12p2 P3/C309 143 P4/C30676039 648000 p4 P5/C3020000 12964479 (Sloane’s A051050 and A051051). Another generalization asks the probability that n randomly chosen points in a fixed bounded convex domain KƒR2are the vertices of a convex n-gon. The solution is Pn /C302n(3n /C28 3)! [(n /C28 1)!]3(2n)! (3) for a triangular domain, which has first few values 1, 1, 1, 2/3, 11/36, 91/900, 17/675, ... (Sloane’s A004677 and A004824), and Pn /C301 n!2n /C282 n /C281Yru$Yru%"#2 (4) for a parallelogram domain, which has first few values 1, 1, 1, /25 =36/, /49=144 /, /121=3600 /, ... (Sloane’s A004936 and A005017; Valtr 1996, Peyerimhoff 1997). Sylvester’s four-point problem has an unexpected connection with the RECTILINEAR CROSSING NUMBER of graphs (Finch). See also DISK TRIANGLE PICKING ,HEXAGON TRIANGLE PICKING ,RECTILINEAR CROSSING NUMBER ,SQUARE TRIANGLE PICKING ,TRIANGLE TRIANGLE PICKING References Alikoski, H. A. "U¨ ber das Sylvestersche Vierpunktproblem." Ann. Acad. Sci. Fenn. 51, No. 7, 1 /C1/0, 1939. Blaschke, W. "U¨ ber affine Geometrie XI: Lo¨sung des ‘Vier- punktproblems’ von Sylvester aus der Theorie der geome- trischen Wahrscheinlichkeiten." Leipziger Ber. 69, 436 /C1/ 53, 1917. Blaschke, W. §24 /C1/5in Vorlesungen u¨ber Differentialgeome- trie, II. Affine Differentialgeometrie. Berlin: Springer- Verlag, 1923. Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Random Polygons and Polyhedra." §B5 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 54 /C1/7, 1991. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/crss/crss.html. Groemer, H. "On Some Mean Values Associated with a Randomly Selected Simlpex in a Convex Set." Pacific J. Math. 45, 525 /C1/33, 1973. Kendall, M. G. and Moran, P. A. P. Geometric Probability. New York: Hafner, 1963. Kingman, J. F. C. "Random Secants of a Convex Body." J. Appl. Prob. 6, 660 /C1/72, 1969. Klee, V. "What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body." Amer. Math. Monthly 76, 286 /C1/88, 1969. Peyerimhoff, N. "Areas and Intersections in Convex Do- mains." Amer. Math. Monthly 104, 697 /C1/04, 1997. Pfiefer, R. E. "The Historical Development of J. J. Sylves- ter’s Four Point Problem." Math. Mag. 62, 309 /C1/17, 1989. Rottenberg, R. R. "On Finite Sets of Points in P3 :/" Israel J. Math. 10, 160 /C1/71, 1971. Santalo ´,L.A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976. Schneinerman, E. and Wilf, H. S. "The Rectilinear Crossing Number of a Complete Graph and Sylvester’s ‘Four Point’ Problem of Geometric Probability." Amer. Math. Monthly 101, 939 /C1/43, 1994. Sloane, N. J. A. Sequences A004677, A004824, A004936, A005017, A051050, and A051051 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Solomon, H. "Crofton’s Theorem and Sylvester’s Problem in Two and Three Dimensions." Ch. 5 in Geometric Prob- ability. Philadelphia, PA: SIAM, pp. 97 /C1/25, 1978.Sylvester, J. J. "Question 1491." The Educational Times (London). April 1864. Sylvester, J. J. "On a Special Class of Questions on the Theory of Probabilities." Birmingham British Assoc. Rept. , pp. 8 /C1/, 1865. Valtr, P. "Probability that n Random Points are in a Convex Position." Discrete Comput. Geom. 13, 637 /C1/43, 1995. Valtr, P. "The Probability that n Random Points in a Triangle are in Convex Position." Combinatorica 16, 567 /C1/73, 1996. Weil, W. and Wieacker, J. "Stochastic Geometry." Ch. 5.2 in Handbook of Convex Geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 1391 /C1/438, 1993. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos’ 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bolloba ´s and A. Thoma- son). Cambridge, England: Cambridge University Press, pp. 557 /C1/62, 1997. Woolhouse, W. S. B. "Some Additional Observations on the Four-Point Problem." Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 7. London: F. Hodgson and Son, p. 81, 1867. Sylvester’s Inertia Law The numbers of EIGENVALUES that are POSITIVE , NEGATIVE , or 0 do not change under a congruence transformation. Gradshteyn and Ryzhik (2000) state it as follows: when a QUADRATIC FORM Q in n variables is reduced by a nonsingular linear trans- formation to the form Q /C30y2 1 /C27y22 /C27.../C27y2p /C28p2p /C271 /C28y2p 2/C28.../C28y2r : the number p of POSITIVE SQUARES appearing in the reduction is an invariant of the QUADRATIC FORM Q and does not depend on the method of reduction. See also EIGENVALUE ,QUADRATIC FORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1105, 2000. Sylvester’s Line Problem It is not possible to arrange a finite number of points so that a LINE through every two of them passes through a third unless they are all on a single LINE. See also COLLINEAR ,SYLVESTER’S FOUR- POINT PRO- BLEM Sylvester’s Sequence The sequence defined by e0/C302 and the RECURRENCE RELATION en/C301/C27Yn/C281 i/C300ei/C30e2n/C281/C28en/C281/C271: (1) This sequence arises in Euclid’s proof that there are anINFINITE number of PRIMES . The proof proceeds by constructing a sequence of PRIMES using the RECUR- RENCE RELATION en/C271 /C30e0e1 /C1/C1/C1en /C271 (2) (Vardi 1991). Amazingly, there is a constant E :1:264084735306 (3) such that en /C30 E2n/C271 /C271 2jk (4) (Vardi 1991, Graham et al. 1994). The first few numbers in Sylvester’s sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... (Sloane’s A000058). The en satisfy X/C12 n/C3001 en/C301: (5) In addition, if 0 Bx B1isan IRRATIONAL NUMBER , then the nth term of an infinite sum of unit fractions used to represent x as computed using the GREEDY ALGORITHM must be smaller than 1=en :/ The n of the first few PRIME enare 0, 1, 2, 3, 5, ..., corresponding to 2, 3, 7, 43, 3263443, ... (Sloane’s A014546). Vardi (1991) gives a lists of factors less than 5 /C29107 of enfor n 5200 and shows that enis COMPOSITE for 6 5n 517 : Furthermore, all numbers less than 2:5 /C291015in Sylvester’s sequence are SQUAREFREE , and no SQUAREFUL numbers in this sequence are known (Vardi 1991). See also EUCLID’S THEOREMS ,G REEDY ALGORITHM , SQUAREFREE ,SQUAREFUL References Graham, R. L.; Knuth, D. E.; and Patashnik, O. Research problem 4.65 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Sloane, N. J. A. Sequences A000058/M0865 and A014546 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Vardi, I. "Are All Euclid Numbers Squarefree?" and "Power- Mod to the Rescue." §5.1 and 5.2 in Computational Recreations in Mathematica. Reading, MA: Addison-Wes- ley, pp. 82 /C1/9, 1991. Sylvester’s Signature Diagonalize a form over the RATIONALS to diag pa /C215 A; pb /C215 B ; ...YrtYrP : where all the entries are INTEGERS and A, B, ...are RELATIVELY PRIME to p. Then Sylvester’s signature is the sum of the /C281-parts of the entries. See also P-SIGNATURESylvester’s Triangle Problem The resultant of the vectors represented by the three RADII from the center of a TRIANGLE’S CIRCUMCIRCLE to its VERTICES is the segment extending from the CIRCUMCENTER to the ORTHOCENTER . See also CIRCUMCENTER ,CIRCUMCIRCLE ,ORTHOCEN- TER,TRIANGLE References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 142, 1965. Symbolic Calculus UMBRAL CALCULUS Symbolic Logic The study of the meaning and relationships of statements used to represent precise mathematical ideas. Symbolic logic is also called FORMAL LOGIC . See also FORMAL LOGIC ,LOGIC ,METAMATHEMATICS References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, 1958. Symmedian The lines AKA ; BKB ; and CKB which are ISOGONAL to the MEDIANS AMA ; BMB ; and CMC of a TRIANGLE are called the triangle’s symmedian. The symmedians are concurrent in a point K called the SYMMEDIAN POINT which is the ISOGONAL CONJUGATE of the CENTROID G. See also CENTROID (TRIANGLE ), ISOGONAL CONJU- GATE ,SYMMEDIAN POINT ,MEDIAN (TRIANGLE ) References Casey, J. "Theory of Isogonal and Isotomic Points, and of Antiparallel and Symmedian Lines." Supp. Ch. §1i n A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 165 /C1/73, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 65, 1971. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 213 /C1/18, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 62 /C1/3, 1893. Mackay, J. S. "Symmedians of a Triangle and Their Con- comitant Circles." Proc. Edinburgh Math. Soc. 14,3 7/C1/03, 1896. Symmedian Point The point of concurrence Kof the SYMMEDIANS , sometimes also called the L EMOINE POINT (in England and France) or the G REBE POINT (in Germany). Equivalently, the symmedian point is the ISOGONAL CONJUGATE of the CENTROID G. In other words, let G be the CENTROID of a TRIANGLE DABC ;AMA;BMB;and CMCthe medians of DABC ;ALA;BLB;and CLCthe ANGLE BISECTORS ofANGLES A,B,C, and AKA;BKB; andCKCthe reflections of AMA;BMB;andCMCabout ALA;BLB;and CLC:Then Kis the point of concur- rence of the lines AKA;BKB;and CKC:According to Honsberger (1995, p. 53), the symmedian point is "one of the crown jewels of modern geometry." The TRILINEAR COORDINATES of the symmedian point is a:b:c (1) (Honsberger 1995, p. 75), or sinA: sin B: sin C: (2) In AREAL COORDINATES (actual TRILINEAR COORDI- NATES ), the symmedian point is the point for which a2/C27b2/C27g2is a minimum (Honsberger 1995, pp. 75 /C1/ 6). A center Xis the CENTROID of its own PEDAL TRIANGLE IFF it is the symmedian point. The symme- dian point is the perspectivity center of a TRIANGLE and its TANGENTIAL TRIANGLE . In the above diagram with Kthe symmedian point, AK KKA/C30b2/C27c2 a2(3) (Honsberger 1995, p. 76). The symmedian point lies on the B ROCARD AXIS , and its distances from Kto the sides of the TRIANGLE are KKi/C301 2aitanv; (4) where vis the B ROCARD ANGLE . One B ROCARD LINE ,MEDIAN , and SYMMEDIAN (out of the three of each) are CONCURRENT , with AV;CK, and BGmeeting at a point, where Vis the first B ROCARD POINT andGis the CENTROID . Similarly, AV?;BG, and CK, where V?is the second B ROCARD POINT , meet at a point which is the ISOGONAL CONJUGATE of the first (Johnson 1929, pp. 268 /C1/69). The line joining the MIDPOINT of any side to the midpoint of the ALTITUDE on that side passes through K (left figure). In particular, the symmedian point of a RIGHT TRIANGLE is the MIDPOINT of the ALTITUDE to the HYPOTENUSE (right figure; Honsberger 1995, p. 59). The symmedian point K is the STEINER POINT of the first BROCARD TRIANGLE . Given a triangle DABC ; construct the triangle DA?B ?C? obtained as the intersection of the lines extended from each vertex though the symmedian point K of DABC with the CIRCUMCIRCLE of DABC : Then the symmedian point of DA?B ?C? is again K (Honsberger 1995, p. 77). The tangents to the CIRCUMCIRCLE of a triangle at two of its vertices meet on the SYMMEDIAN from the third vertex (Honsberger 1995, pp. 60 /C1/1). The GERGONNE POINT of a triangle is the symmedian point of its CONTACT TRIANGLE (Honsberger 1995, pp. 62 /C1/3). The symmedian point of a triangle is the CENTROID of its PEDAL TRIANGLE . And finally, the lengths of the sides of the PEDAL TRIANGLE of the symmedian point are proportional to the lengths of the MEDIANS of the original triangle (Honsberger 1995, p. 77) See also ANGLE BISECTOR ,BROCARD ANGLE ,BROCARD AXIS,B ROCARD DIAMETER ,C ENTROID (TRIANGLE ), COSYMMEDIAN TRIANGLES ,GREBE POINT ,ISOGONAL CONJUGATE ,LEMOINE CIRCLE ,LEMOINE LINE,LINE AT INFINITY ,M ITTENPUNKT ,PEDAL TRIANGLE ,STEI- NER POINTS ,SYMMEDIAN ,TANGENTIAL TRIANGLE References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 170, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 65, 1971. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 86, 1913. Honsberger, R. "The Symmedian Point." Ch. 7 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 53 /C1/7, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 217, 268 /C1/69, and 271 /C1/72, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/87, 1994. Kimberling, C. "Symmedian Point." http://cedar.evansvil- le.edu/~ck6/tcenters/class/sympt.html. Mackay, J. S. "Early History of the Symmedian Point." Proc. Edinburgh Math. Soc. 11,92/C1/03, 1892 /C1/893.Mackay, J. S. "Symmedians of a Triangle and Their Con- comitant Circles." Proc. Edinburgh Math. Soc. 14,37/C1/03, 1896. Symmetric A mathematical object is said to be symmetric if it is invariant ("looks the same") under a symmetry transformation. A function, matrix, etc., is symmetric if it remains unchanged in SIGN when indices are reversed. For example, Aij/C13ai/C27ajis symmetric since Aij/C30Aji:/ See also ANTISYMMETRIC ,S YMMETRIC FUNCTION , SYMMETRY Symmetric Bilinear Form A symmetric bilinear form on a VECTOR SPACE Vis a BILINEAR FUNCTION Q:V/C29V0R (1) which satisfies Q(v;w)/C30Q(w;v):/ For example, if Ais an/C29nSYMMETRIC MATRIX , then Q(v;w)/C30vTAw/C30/C142v;Aw/C143 (2) is a symmetric bilinear form. Consider A/C3012 2/C283YrtvYrtu ; (3) then Qa1;a2 ðÞ ;b1;b2 ðÞ ðÞ /C30a1b1/C272a1b2/C272a2b1/C283a2b2: (4) Here is a Mathematica function which takes a matrix to a bilinear form. MatrixToForm[a_List?MatrixQ][v_, w_] : /C30v.a.w For example, q/C30MatrixToForm[{{0, 1}, {1, -2}}]; q[{1, 0}, {1, 7}] yields 7. AQUADRATIC FORM may also be labeled Q, because quadratic forms are in a one-to-one correspondence with symmetric bilinear forms. Note that Q(a)/C30 Q(a;a)i sa QUADRATIC FORM .I fQ(a) is a quadratic form then it defines a symmetric bilinear form by Q(a;b)/C301 2[Q(a/C27b)/C28Q(a)/C28Q(b)]: (5) The kernel, or radical, of a symmetric bilinear form is the set of vectors kerQ/C30fv:Q(v;w)/C300for all w /C23Vg: (6) A quadratic form is called nondegenerate if its kernel is zero. That is, if for all v/C23V;there is a w/C23Vwith Q(v;w)"0:The rank of Qis the rank of the matrix aijYrvYru /C30Qei;ejYrvYru :/ The form Q is diagonalized if there is a basis vi ; called an orthogonal basis, such that bijYrvYru /C30Qvi ; vjYrvYru is a DIAGONAL MATRIX . Alternatively, there is a matrix C such that Q Cv; C w ðÞ /C30 CvðÞTAC wðÞ/C30vT CTACYrvYru w (7) is a DIAGONAL QUADRATIC FORM . The jth column of the matrix C is the vector vj :/ A nondegenerate symmetric bilinear form can be DIAGONALIZED , using GRAM- SCHMIDT ORTHONORMALI- ZATION to find the vi ; so that the diagonal matrix CTAC has entries either 1 or /C281. If there are p 1s and q -1s, then Q is said to have SIGNATURE (p, q), or if the dimension is understood then just signature p. Real nondegenerate symmetric bilinear forms are classi- fied by their signature, in the sense that given two vector spaces with forms of signature (p, q), there is an isomorphism of the vector spaces which takes one form to the other. A symmetric bilinear form with Q(v ; v) > 0 ; for all nonzero v, is called POSITIVE DEFINITE . For example, the usual inner product is positive definite. A positive definite form has signature (n; 0): A negative definite form is the negative of a positive form and has signature (0; n) : If the form is neither positive definite nor negative definite, then there must exist vectors w "0 such that Q(w; w) /C300; called isotropic vectors. A general symmetric bilinear form Q can be diag- onalized with diagonal entries 1, /C281, or 0, because the form Q is always nondegenerate on the QUOTIENT VECTOR SPACE V =ker Q : If V is a COMPLEX VECTOR SPACE , then a symmetric bilinear form can be diag- onalized to have entries 1 or 0. For other FIELDS , there are more SYMMETRIC BILINEAR FORMS than in the real or complex case. For instance, if the FIELD has CHARACTERISTIC 2, then it is not possible to divide by 2 since 2 /C300. Hence there is no correspondence between quadratic forms and symmetric bilinear forms in characteristic 2. See also DIAGONAL QUADRATIC FORM,FIELD,INDEX (MATRIX ), INNER PRODUCT ,QUADRATIC FORM,SIGNA- TURE ,SYMMETRIC BILINEAR FORM (GENERAL FIELDS ), VECTOR SPACE References Serre, J. P. A Course in Arithmetic. New York: Springer- Verlag, pp. 27 /C1/5, 1973. Symmetric Bilinear Form (General Fields) The symmetric bilinear forms on a VECTOR SPACE , whose FIELD k is not real, have been classified for some FIELDS . There are also theorems about sym- metric bilinear forms on free Abelian groups, for example Zn :/A SYMMETRIC BILINEAR FORM Q corresponds to a matrix A by giving a basis eiand setting aij /C30 Qei ; ejYrvYru : Two symmetric bilinear forms are consid- ered equivalent if a change of basis takes one to the other. Hence, A /C2CACT ; where C is any invertible matrix. Therefore, the rank of the symmetric bilinear form is an invariant. Also, det A can change by (det C)2det A: The coset of det A in k /C31=k/C312 is a WELL DEFINED invariant of Q, called the discriminant. For real forms, it is either 1 or /C281. For Q; the discriminant can be any RATIONAL NUMBER a=b where a and b are SQUAREFREE .A symmetric bilinear form on a FINITE FIELD is deter- mined by its rank and its discriminant. A symmetric bilinear form on the P-ADIC NUMBERS Qp is characterized by its rank, discriminant, and an- other invariant e(Q) : Given a basis ei ; orthogonal for Q, define ai /C30Qe1 ; e2 ðÞ ; then e(Q) /C30Y iBjai ; ajYrvYru where ai ; ajYrvYru is the HILBERT SYMBOL . Two symmetric bilinear forms are equivalent on the RATIONALS iff they are equivalent in every Qp as well as the reals (also called Q/C12:/) The data in Qpcan be thought of as "local" information, which can be patched together to yield "global" information in Q: So rational forms have a countable number of distinct invariants, three for every PRIME NUMBER , and two for the reals. See also HILBERT SYMBOL , P-ADIC NUMBER ,Q UAD- RATIC FORM,SYMMETRIC BILINEAR FORM,V ECTOR SPACE References Serre, J. P. A Course in Arithmetic. New York: Springer- Verlag, pp. 27 /C1/5, 1973. Symmetric Block Design A symmetric design is a BLOCK DESIGN (v, k, l ; r, b) with the same number of blocks as points, so b /C30v (or, equivalently, r /C30k). An example of a symmetric block design is a PROJECTIVE PLANE . See also BLOCK DESIGN ,PROJECTIVE PLANE References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1 /C1/2, 1992. Symmetric Design SYMMETRIC BLOCK DESIGN Symmetric Difference The set of elements belonging to one but not both of two given sets. It is therefore the UNION of the COMPLEMENT of A with respect to B and B with respect to A, and corresponds to the XOR operation in Boolean logic. The symmetric difference can be implemented in Mathematica as SymmetricDifference[a_, b_] : /C30 Union[Complement[a, b], Complement[b, a]] The symmetric difference of sets A and B is variously written as A /C155B ; A9B ; or A /C27B : The latter two notations are deprecated since these symbols have common meanings in other areas of mathematics. For example, for A /C30f1 ; 2; 3; 4g and B /C30f1; 4; 5g; A /C155B /C30f2; 3; 5g; since 2, 3, and 5 are each in one, but not both, sets. See also COMPLEMENT SET,DIFFERENCE ,SET DIFFER- ENCE ,UNION , XOR Symmetric Function A symmetric function on n variables x1 ; ..., xnis a function that is unchanged by any PERMUTATION of its variables. In most contexts, the term "symmetric function" refers to a polynomial on n variables with this feature (more properly called a "SYMMETRIC POLYNOMIAL "). Another type of symmetric functions is symmetric rational functions, which are the RA- TIONAL FUNCTIONS that are unchanged by PERMUTA- TION of variables. The SYMMETRIC POLYNOMIALS (respectively, sym- metric rational functions) can be expressed as poly- nomials (respectively, rational functions) in the SYMMETRIC POLYNOMIALS . This is called the FUNDA- MENTAL THEOREM OF SYMMETRIC FUNCTIONS . A function f(x) is sometimes said to be symmetric about the Y-AXIS if f(/C28x) /C30f(x): Examples of such functions include ½x½ (the ABSOLUTE VALUE ) and x2 (the PARABOLA ). See also FUNDAMENTAL THEOREM OF SYMMETRIC FUNCTIONS ,RATIONAL FUNCTION ,SYMMETRIC POLY- NOMIAL References Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999. Littlewood, J. E. A University Algebra, 2nd ed. London: Heinemann, 1958. Macdonald, I. G. Symmetric Functions and Hall Polyno- mials, 2nd ed. Oxford, England: Oxford University Press, 1995. Macdonald, I. G. Symmetric Functions and Orthogonal Polynomials. Providence, RI: Amer. Math. Soc., 1997. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Symmetric Function Identities." §1.7 in A /C30B. Wellesley, MA: A. K. Peters, pp. 12 /C1/3, 1996.Symmetric Group The symmetric group Snof degree n is the GROUP of all PERMUTATIONS on n symbols. Snis therefore of ORDER n! and contains as SUBGROUPS every GROUP of ORDER n. The number of CONJUGACY CLASSES of Sn is given by the PARTITION FUNCTION P. For example, let fabc g denote the permutation on three elements which takes the ath element to position 1, the bth element to position 2, and the cth element to position 3. Then the following table gives the MULTIPLICATION TABLE for Sn ; containing 3! /C306 elements with f123g the IDENTITY ELEMENT . The multiplication table can be generated using the Mathematica function SymmetricGroup[n_Integer?Positive] : /C30 Module[ {p /C30 Permutations[Range[n]], i, j}, Table[p[[i]][[p[[j]]]], {i, n!}, {j, n!}] ] /S3/ (123) (132) (213) (231) (312) (321) (123) (123) (132) (213) (231) (312) (321) (132) (132) (123) (312) (321) (213) (231) (213) (213) (231) (123) (132) (321) (312) (231) (231) (213) (321) (312) (123) (132) (312) (312) (321) (132) (123) (231) (213) (321) (321) (312) (231) (213) (132) (123) NETTO’S CONJECTURE states that the probability that two elements P1and P2of a symmetric group generate the entire group tends to 3u4as n 0/C12: This was proven by Dixon (1969). The probability that two elements generate Snforn/C301, 2, ... are 1, 3 u4, 1u2, 3u8, 19 u40, 53 u120, 103 u168, ... (Sloane’s A040173 and A040174). Finding a general formula for terms in the sequence is a famous UNSOLVED PROBLEM in GROUP THEORY . See also ALTERNATING GROUP ,C ONJUGACY CLASS , ERDOS- TURA´ N THEOREM ,F INITE GROUP ,JORDAN’S SYMMETRIC GROUP THEOREM ,NETTO’S CONJECTURE , PARTITION FUNCTION P,SIMPLE GROUP References Dixon, J. D. "The Probability of Generating the Symmetric Group." Math. Z. 110, 199/C1/05, 1969. Huang, J.-S. "Symmetric Groups." Ch. 3 in Lectures on Representation Theory. Singapore: World Scientific, pp. 15 /C1/5, 1999. Lomont, J. S. "Symmetric Groups." Ch. 7 in Applications of Finite Groups. New York: Dover, pp. 258 /C1/73, 1987. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 17, 1990. Sloane, N. J. A. Sequences A040173 and A040174 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#alt. Symmetric Matrix A symmetric matrix is a SQUARE MATRIX which satisfies AT /C30A where AT denotes the TRANSPOSE ,so aij /C30aji : This also implies A /C281AT /C30I; (1) where I is the IDENTITY MATRIX . For example, A /C3041 1 /C282YrtvYrtu (2) is a symmetric matrix. HERMITIAN MATRICES are a useful generalization of symmetric matrices for COM- PLEX MATRICES A matrix m can be tested to see if it is symmetric using the Mathematica function SymmetricQ[m_List?MatrixQ] : /C30 (m /C30/C30/C30 Transpose[m]) Written explicitly, the elements of a symmetric matrix A have the form a11a12/C1/C1/C1 a1n a21a22/C1/C1/C1 a2n nn::: n an1an2/C1/C1/C1 ann2 6643 775 (3) The symmetric part of any MATRIX may be obtained from As /C301 2A /C27ATYrvYru : (4) A MATRIX A is symmetric if it can be expressed in the form A /C30QDQT ; (5) where Q is an ORTHOGONAL MATRIX and D is a DIAGONAL MATRIX . This is equivalent to the MATRIX EQUATION AQ /C30QD ; (6) which is equivalent to AQn /C30 lnQn (7) for all n, where ln /C30Dnn : Therefore, the diagonal elements of D are the EIGENVALUES of A ; and the columns of Q are the corresponding EIGENVECTORS . The numbers of symmetric matrices of order n on s symbols are s, s3 ; s6 ; s10 ; ..., sk(k /C281)=2 : Therefore, for (0,1)-MATRICES , the numbers of distinct symmetric matrices of orders n /C301, 2, ... are 2, 8, 64, 1024, ... (Sloane’s A006125).See also ADJOINT MATRIX ,ANTISYMMETRIC MATRIX , BISYMMETRIC MATRIX ,HERMITIAN MATRIX ,PERSYM- METRIC MATRIX ,SKEW SYMMETRIC MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 12 and 115 /C1/17, 1962. Nash, J. C. "Real Symmetric Matrices." Ch. 10 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 119 /C1/34, 1990. Sloane, N. J. A. Sequences A006125/M1897 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Symmetric Points Two points z and zS /C23C are symmetric with respect to a CIRCLE or straight LINE L if all CIRCLES and straight LINES passing through z and zS are orthogonal to L. MO¨ BIUS TRANSFORMATIONS preserve symmetry. Let a straight line be given by a point z0and a unit VECTOR eiu;then zS/C30e2iuz/C28z0/C27z0; where ¯zis the COMPLEX CONJUGATE . Let a CIRCLE be given by center z0and RADIUS r, then zS/C30z0/C27r2 z/C28z0: See also MO¨ BIUS TRANSFORMATION Symmetric Polynomial A symmetric polynomial on nvariables x1;...,xnis a function that is unchanged by any PERMUTATION of its variables. Symmetric polynomials are always HOMO- GENEOUS POLYNOMIALS . The nelementary symmetric functions Pn(sometimes denoted sn)o n nvariables x1;...;xn fg are defined by P1/C30X 15i5nxi (1) P2/C30X 15iBj5nxixj (2) P3/C30X 15iBjBk5nxixjxk (3) P4/C30X 15iBjBkBl5nxixjxkxl (4) n Pn/C30X 15i5nxi: (5) The kth symmetric polynomial is defined as Symme- tricPolynomial [{x1, ...,xn},k] in the Mathematica add-on package Algebra‘SymmetricPolyno- mials‘ (which can be loaded with the command BBAlgebra‘ ). SymmetricReduction [f,{x1, ..., xn}] in the Mathematica add-on packageAlgebra‘- SymmetricPolynomials‘ (which can be loaded with the command BBAlgebra‘ ) gives a pair of polynomials fp; qg in x1 ; ..., xnwhere p is the symmetric part and q is the remainder. Alternatively, Pjx1 ; ...; xn ðÞ can be defined as the coefficient of xn /C28j in the GENERATING FUNCTION Y 1 5i5nx /C27xi ðÞ : (6) For example, on four variables x1 ; ..., x4 ; the elemen- tary symmetric functions are P1 /C30x1 /C27x2 /C27x3 /C27x4 (7) P2 /C30x1x2 /C27x1x3 /C27x1x4 /C27x2x3 /C27x2x4 /C27x3x4 (8) P3 /C30x1x2x3 /C27x1x2x4 /C27x1x3x4 /C27x2x3x4 (9) P4 /C30x1x2x3x4 : (10) Define skh1 ; ...; hn ðÞ as the coefficients of the GEN- ERATING FUNCTION ln 1 /C27x1t /C27x2t2 /C27x3t3 /C27...YrvYru /C30X/C12 k /C301s1 ktk /C30h1t /C271 2 /C28h2 1 /C272h2YrvYru t2 /C271 3h3 1 /C283h1h2 /C273h3YrvYru t3 /C27... ð11Þ so the first few values are s1 /C30h1 (12) s2 /C30/C28h21 /C272h2 (13) s3 /C30h31 /C283h1h2 /C273h3 (14) s4 /C30/C28h41 /C274h21h2 /C282h22 /C284h2h3 /C274h4 : (15) In general, sncan be computed from the DETERMI- NANT sn /C30(/C281)n /C281h1 100 /C1/C1/C1 0 2h2 h1 10::: 0 3h3 h2 h1 1::: 0 4h4 h3 h2 h1::: 0 nnnn::: 1 nhnhn/C281hn /C282hn/C283/C1/C1/C1 h1YrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrutYrut(16) (Littlewood 1958, Cadogan 1971). Then the elemen- tary symmetric functions satisfy the relationship X n k/C301xp k /C30(/C281)p /C281spP1 ; ...;Pn ðÞ : (17) In particular, Xn k/C301xk /C30P1 (18)Xn k /C301x2 k /C30P2 1 /C282P2 (19) Xn k /C301x3 k /C30P3 1 /C28P1 P2 /C273 P3 (20) Xn k/C301x4 k /C30P4 1 /C284P21 P2 /C272P22 /C274P1 P3 /C284P4 (21) (Schroeppel 1972), as can be verified by plugging in and multiplying through. See also FUNDAMENTAL THEOREM OF SYMMETRIC FUNCTIONS ,N EWTON- GIRARD FORMULAS ,N EWTON’S RELATIONS ,SYMMETRIC FUNCTION References Borwein, P. and Erde´lyi, T. Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 5, 1995. Cadogan, C. C. "The Mo¨bius Function and Connected Graphs." J. Combin. Th. B 11, 193 /C1/00, 1971. Littlewood, J. E. A University Algebra, 2nd ed. London: Heinemann, 1958. Schroeppel, R. Item 6 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 4, Feb. 1972. Se´roul, R. "Newton-Girard Formulas." §10.12 in Program- ming for Mathematicians. Berlin: Springer-Verlag, pp. 278 /C1/79, 2000. Symmetric Quadratic Form See also QUADRATIC FORM Symmetric Relation A RELATION R on a SET S is symmetric provided that for every x and y in S we have xRy IFF yRx :/ See also RELATION Symmetric Tensor A second- RANK symmetric TENSOR is defined as a TENSOR Afor which Amn/C30Anm: (1) Any TENSOR can be written as a sum of symmetric and ANTISYMMETRIC parts Amn/C301 2Amn/C27AnmðÞ /C2712Amn/C28AnmðÞ /C301 2Bmn S/C27BmnA ðÞ : (2) The symmetric part of a TENSOR is denoted using parentheses as T(a;b)/C131 2Tab/C27Tba ðÞ (3) Ta1;a2;...;an ðÞ /C131 n!X permutationsTa1a2/C1/C1/C1an: (4) Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example T ðabÞc ½de/C138¼1 4ðTabc de þ Tbac de /C28Tabc ed /C28Tbac ed Þ:ð5Þ (Wald 1984, p. 26). The product of a symmetric and an ANTISYMMETRIC TENSOR is 0. This can be seen as follows. Let a ab be ANTISYMMETRIC ,so a11 /C30a22 /C300 (6) a21 /C30/C28a12 : (7) Let bab be symmetric, so b12 /C30b21 : (8) Then a abbab /C30a11b11 /C27a12b12 /C27a21b21 /C27a22b22 /C300 /C27a12b12 /C28a12b12 /C270 /C300: (9) A symmetric second- RANK TENSOR Amnhas SCALAR invariants s1 /C30A11 /C27A22 /C27A22 (10) s2 /C30A22A33 /C27A33A11 /C27A11A22 /C28A2 23 /C28A231 /C28A212 : (11) References Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984. Symmetric Top Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C28M2 /C281 4 /C27 K2 /C28 2MK cos x sin2 x /C27 s /C27K2 /C2714Yru*Yru+"# y /C300: References Infeld, L. and Hull, T. E. "The Factorization Method." Rev. Mod. Phys. 23,21/C1/8, 1951. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Symmetroid A QUARTIC SURFACE which is the locus of zeros of the DETERMINANT of a SYMMETRIC 4 /C294 matrix of linear forms. A general symmetroid has 10 ORDINARY DOUBLE POINTS (Jessop 1916, Hunt 1996). References Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Hunt, B. "Symmetroids and Weddle Surfaces." §B.5.3 in The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 315 /C1/19, 1996.Jessop, C. Quartic Surfaces with Singular Points. Cam- bridge, England: Cambridge University Press, p. 166, 1916. Symmetry An intrinsic property of a mathematical object which causes it to remain invariant under certain classes of transformations (such as ROTATION , REFLECTION , INVERSION , or more abstract operations). The mathe- matical study of symmetry is systematized and formalized in the extremely powerful and beautiful area of mathematics called GROUP THEORY . Symmetry can be present in the form of coefficients of equations as well as in the physical arrangement of objects. By classifying the symmetry of polynomial equations using the machinery of GROUP THEORY , for example, it is possible to prove the unsolvability of the general QUINTIC EQUATION . In physics, the extremely powerful N OETHER’S SYM- METRY THEOREM states that each symmetry of a system leads to a physically conserved quantity. Symmetry under TRANSLATION corresponds to mo- mentum conservation, symmetry under ROTATION to angular momentum conservation, symmetry in timeto energy conservation, etc. See also C RYSTALLOGRAPHY RESTRICTION ,G ROUP THEORY ,NOETHER’S SYMMETRY THEOREM References Eppstein, D. "Symmetry and Group Theory." http://www.ic- s.uci.edu/~eppstein/junkyard/sym.html. Britton, J. Symmetry and Tessellations: Investigating Pat- terns. Englewood Cliffs, NJ: Prentice-Hall, 1999. Farmer, D. Groups and Symmetry. Providence, RI: Amer. Math. Soc., 1995. Pappas, T. "Art & Dynamic Symmetry." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, pp. 154 /C1/ 55, 1989. Radin, C. "Symmetry." Ch. 4 in Miles of Tiles. Providence, RI: Amer. Math. Soc., pp. 69 /C1/7, 1999. Rosen, J. A Symmetry Primer for Scientists. New York: Wiley, 1983. Rosen, J. Symmetry in Science: An Introduction to the General Theory. New York: Springer-Verlag, 1995. Schattschneider, D. Visions of Symmetry: Notebooks, Peri- odic Drawings, and Related Work of M. C. Escher. New York: W. H. Freeman, 1990. Stewart, I. and Golubitsky, M. Fearful Symmetry. New York: Viking Penguin, 1993. Voisin, C. Mirror Symmetry. Providence, RI: Amer. Math. Soc., 1999. Weisstein, E. W. "Books about Symmetry." http://www.trea- sure-troves.com/books/Symmetry.html. Yale, P. B. Geometry and Symmetry. New York: Dover, 1988. Symmetry Group GROUP Symmetry Operation Symmetry operations include the IMPROPER ROTA- TION ,INVERSION OPERATION ,MIRROR PLANE , and ROTATION . Together, these operations create 32 crys- tal classes corresponding to the 32 POINT GROUPS . The INVERSION OPERATION takes (x; y; z) 0 (/C28x;/C28y ;/C28z) and is denoted i. When used in conjunction with a ROTATION , it becomes an IMPROPER ROTATION .An IMPROPER ROTATION by 360/C14=n is denoted ¯n (or Sn): For periodic crystals, the CRYSTALLOGRAPHY RESTRIC- TION allows only the IMPROPER ROTATIONS ¯1; ¯2; ¯3; ¯4; and ¯6:/ The MIRROR PLANE symmetry operation takes (x; y; z) 0 (x; y;/C28z); (x;/C28y; z) 0 (x;/C28y; z); etc., which is equivalent to ¯2: Invariance under reflection can be denoted nsvor n sh : The ROTATION symmetry operation for 360/C14=n is denoted n (or Cn): For periodic crystals, CRYSTALLOGRAPHY RESTRICTION allows only 1, 2, 3, 4, and 6. Symmetry operations can be indicated with symbols such as Cn ; Sn ; E, i, nsv ; and nsh : 1. Cnindicates ROTATION about an n-fold symme- try axis. 2. Sn indicates IMPROPER ROTATION about an n-fold symmetry axis. 3. E (or I) indicates invariance under TRANSLA- TION . 4. i indicates a center of symmetry under INVER- SION. 5. nsvindicates invariance under n vertical RE- FLECTIONS . 6. nshindicates invariance under n horizontal REFLECTIONS . See also CRYSTALLOGRAPHY RESTRICTION ,G LIDE , IMPROPER ROTATION ,INVERSION OPERATION ,MIRROR PLANE ,POINT GROUPS ,ROTATION ,SYMMETRY References Addington, S. "The Four Types of Symmetry in the Plane." http://forum.swarthmore.edu/sum95/suzanne/symsu- san.html. Symmetry Principle SYMMETRIC POINTS are preserved under a MO¨ BIUS TRANSFORMATION . The SCHWARZ REFLECTION PRINCI- PLE is sometimes called the symmetry principle (Needham 2000, p. 252). See also MO¨ BIUS TRANSFORMATION ,S YMMETR IC POINTS References Needham, T. "Analytic Continuation." §5.XI in Visual Com- plex Analysis. New York: Clarendon Press, pp. 247 /C1/57, 2000.Symplectic Diffeomorphism A MAP (M1 ; v1) 0 (M2 ; v2) between the SYMPLECTIC MANIFOLDS (M1 ; v1) and (M2 ; v2) which is a DIFFEO- MORPHISM and T /C31( v2) /C30( v1) ; where T /C31 is the PULL- BACK MAP induced by T (i.e., the derivative of the DIFFEOMORPHISM T acting on tangent vectors). A symplectic diffeomorphism is also known as a SYM- PLECTOMORPHISM or CANONICAL TRANSFORMATION . See also DIFFEOMORPHISM ,PULLBACK MAP,SYMPLEC- TIC MANIFOLD References Guillemin, V. and Sternberg, S. Symplectic Techniques in Physics. New York: Cambridge University Press, p. 34, 1984. Symplectic Form A symplectic form on a SMOOTH MANIFOLD M is a smooth closed 2-FORM v on M which is nondegenerate such that at every point m, the alternating bilinear form vmon the TANGENT SPACE TmM is nondegene- rate. A symplectic form on a VECTOR SPACE V over Fq is a function f(x; y) (defined for all x; y /C23 V and taking values in Fq) which satisfies f( l1x1 /C27 l2x2 ; y) /C30 l1f(x1 ; y) /C27 l2f(x2 ; y) ; f(y; x) /C30/C28f(x; y) ; and f(x; x) /C300: f is called non-degenerate if f(x; y) /C300 for all y implies that x /C300. Symplectic forms can exist on M (or V) only if M (or V)is EVEN -dimensional. An example of a symplectic form over a vector space is the complex HILBERT SPACE with INNER PRODUCT /C1/C1/C1hi given by f(x;y)/C30Ix;yhi : See also SYMPLECTIC SPACE ,VECTOR SPACE Symplectic Geometry References Berndt, R. Einfu ¨hrung in die Symplektische Geometrie. Braunschweig, Germany: Vieweg, 1998. Symplectic Group For every even DIMENSION 2n;the symplectic group Sp(2n) is the GROUP of 2n/C292nMATRICES which preserve a nondegenerate skew symmetric BILINEAR FORM v;i.e., a SYMPLECTIC FORM . Every symplectic form can be put into a canonical form by finding a SYMPLECTIC BASIS . So, up to conjugation, there is only one symplectic group, in contrast to the ORTHOGONAL GROUP which preserves a nondegenerate SYMMETRIC BILINEAR FORM . As with the ORTHOGONAL GROUP , the columns of a symplectic matrix form a SYMPLECTIC BASIS . Since vn is a VOLUME FORM , the symplectic group preserves volume and ORIENTATION . Hence, Sp(2n) ƒ SL(2n) : In fact, Sp(2) is just the group of matrices with DETERMINANT 1. The three symplectic (0,1)- MATRICES are therefore 10 01YrtvYrtu ;1011YrtvYrtu ;1101YrtvYrtu : (1) The matrices 100 s 01 s 0 0010 00012 6643 775 (2) and cosh t sinh t 0 sinh t sinh t cosh t sinh t 0 0 0 cosh t /C28sinh t 00 /C28sinh t cosh t2 6643 775 (3) are in Sp(4) ; where v /C30e 1 ffle3 /C27e2 ffle4 : (4) In fact, both of these examples are 1-parameter subgroups. Here is a Mathematica function that tests whether a matrix is a symplectic matrix. SymplecticForm[n_Integer]: /C30 Join[PadLeft[IdentityMatrix[n],{n,2n}], PadRight[-IdentityMatrix[n],{n,2n}]] SymplecticQ[a_List]: /C30 EvenQ[Length[a]] && Transpose[a].SymplecticForm[Length[a]/2].a /C30/C30 SymplecticForm[Length[a]/2] Thinking of a matrix as given by (2n)2 coordinate functions, the set of matrices is identified with R(2n)2 : The symplectic matrices are the solutions to the (2n)2 equations AT JA /C30J ; (5) where J is defined by v(x; y) /C30 x ; Jy hi : (6) Note that these equations are redundant, since only 2n2 /C28n of these are independent, leaving 2n2 /C27n "free" variables. In fact, the symplectic group is a smooth 2n2 /C27n ðÞ /-dimensional SUBMANIFOLD of R2n :/Because the symplectic group is a GROUP and a MANIFOLD ,itisaL IE GROUP . Its TANGENT SPACE at the identity is the SYMPLECTIC LIE ALGEBRA sp(2n): The symplectic group is not COMPACT . Instead of using real numbers for the coefficients, it is possible to use coefficients from any FIELD F: The symplectic group Spn(q) for n EVEN is the GROUP of elements of the GENERAL LINEAR GROUP GLnthat preserve a given nonsingular SYMPLECTIC FORM . Any such MATRIX has DETERMINANT 1. See also DETERMINANT ,F IELD,G ENERAL LINEAR GROUP ,GROUP ,LIE ALGEBRA ,LIE GROUP ,LIE-TYPE GROUP ,LINEAR ALGEBRAIC GROUP ,PROJECTIVE SYM- PLECTIC GROUP ,QUADRATIC FORM,SIEGEL’S UPPER HALF-SPACE ,SUBMANIFOLD ,SYMPLECTIC BASIS,SYM- PLECTIC FORM,UNITARY GROUP ,VECTOR SPACE References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups Spn(q) and PSpn(q) /C30Sn(q):/ " §2.3 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. x-xi, 1985. Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#symp. Symplectic Manifold A pair (M ; v) ; where M is a MANIFOLD and v is a SYMPLECTIC FORM on M. The PHASE SPACE R2n /C30 Rn /C29Rn is a symplectic manifold. Near every point on a symplectic manifold, it is possible to find a set of local "Darboux coordinates" in which the SYMPLECTIC FORM has the simple form v/C30X kdqkffldpk (Sjamaar 1996), where dqkffldpkis a WEDGE PRO- DUCT . See also MANIFOLD ,SYMPLECTIC DIFFEOMORPHISM , SYMPLECTIC FORM References Sjamaar, R. "Symplectic Reduction and Riemann-Roch For- mulas for Multiplicities." Bull. Amer. Math. Soc. 33, 327/C1/ 38, 1996. Symplectic Map Informally, a symplectic map is a MAP which pre- serves the sum of AREAS projected onto the set of p2;q2 ðÞ planes. It is the generalization of an AREA- PRESERVING MAP . Formally, a symplectic map is a real-linear map T that preserves a SYMPLECTIC FORM f, i.e., for which f(Tx;Ty)/C30f(x;y) for all x,y. Every symplectic map Ton a complex HILBERT SPACE Hmay be written as U(cosh S/C27 J sinh S) ; where U is unitary, S is positive, and J is an anti-linear involution (i.e., complex conjugation). See also AREA-PRESERVING MAP,LIOUVILLE’S PHASE SPACE THEOREM Symplectic Space A real-linear VECTOR SPACE H equipped with a SYMPLECTIC FORM s. Symplectomorphism SYMPLECTIC DIFFEOMORPHISM Synclastic A surface on which the GAUSSIAN CURVATURE K is everywhere POSITIVE . When K is everywhere NEGA- TIVE, a surface is called ANTICLASTIC . A point at which the GAUSSIAN CURVATURE is POSITIVE is called an ELLIPTIC POINT . See also ANTICLASTIC ,E LLIPTIC POINT ,G AUSSIAN QUADRATURE ,HYPERBOLIC POINT ,PARABOLIC POINT , PLANAR POINT Synergetics Synergetics deals with systems composed of many subsystems which may each be of a very different nature. In particular, synergetics treats systems in which cooperation among subsystems creates orga- nized structure on macroscopic scales (Haken 1993). Examples of problems treated by synergetics include BIFURCATIONS , phase transitions in physics, convec- tive instabilities, coherent oscillations in lasers, non- linear oscillations in electrical circuits, population dynamics, etc. See also BIFURCATION ,CHAOS ,DYNAMICAL SYSTEM References Haken, H. Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology, 3rd rev. enl. ed. New York: Springer-Verlag, 1983. Haken, H. Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and Devices. New York: Springer-Verlag, 1993. Mikhailov, A. S. Foundations of Synergetics: Distributed Active Systems, 2nd ed. New York: Springer-Verlag, 1994. Mikhailov, A. S. and Loskutov, A. Y. Foundations of Syner- getics II: Complex Patterns, 2nd ed., enl. rev. New York: Springer-Verlag, 1996. Weisstein, E. W. "Books about Synergetics." http:// www.treasure-troves.com/books/Synergetics.html. Tschacher, W. and Dauwalder, J.-P. (Eds.). Dynamics, Synergetics, Autonomous Agents: Nonlinear Systems Ap- proaches to Cognitive Psychology and Cognitive Science. Singapore: World Scientific, 1999. Syntonic Comma COMMA OF DIDYMUSSyracuse Algorithm COLLATZ PROBLEM Syracuse Problem COLLATZ PROBLEM System of Differential Equations ORDINARY DIFFERENTIAL EQUATION System of Equations A linear system of equations may be denoted AX /C30Y (1) where A is a MATRIX and X and Y are VECTORS .As shown by CRAMER’S RULE , there is a unique solution if A has a MATRIX INVERSE A /C281 : In this case, X /C30A /C281Y (2) If Y /C300 ; then the solution is X /C300 : If A has no MATRIX INVERSE , then the solution SUBSPACE is either a LINE or the EMPTY SET. If two equations are multiples of each other, solutions are OF THE FORM X /C30A /C27tB (3) for t a REAL NUMBER . See also CRAMER’S RULE,D ETERMINANT ,M ATRIX INVERSE Syzygies Problem The problem of finding all independent irreducible algebraic relations among any finite set of QUANTICS . See also QUANTIC Syzygy A technical mathematical object defined in terms of a POLYNOMIAL RING of n variables over a FIELD k. Syzygies occur in TENSORS at rank 5, 7, 8, and all higher ranks, and play a role in restricting the number of independent ISOTROPIC TENSORS . An ex- ample of a rank-5 syzygy is eijk dlm /C28 ejkl dim /C27 ekli djm /C28 elij dkm /C300 ; where eijkis the PERMUTATION TENSOR anddijis the KRONECKER DELTA . See also FUNDAMENTAL SYSTEM ,H ILBERT BASIS THEOREM ,ISOTROPIC TENSOR ,K RONECKER DELTA , SYZYGIES PROBLEM ,TENSOR References Hilbert, D. "U ¨ber die Theorie der algebraischen Formen." Math. Ann. 36, 473/C1/34, 1890. Iyanaga, S. and Kawada, Y. (Eds.). "Syzygy Theory." §364F inEncyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1140, 1980. Olver, P. J. "Syzygies." Classical Invariant Theory. Cam- bridge, England: Cambridge University Press, pp. 110 /C1/ 12, 1999. Sylvester, J. J. "On a Theory of Syzygetic Relations of Two Rational Integral Functions, Comprising an Application of the Theory of Sturm’s Functions, and that of the Greatest Algebraic Common Measure." Philos. Trans. Roy. Soc. London 143, 407 /C1/48, 1853. Sze´kely Identity X/C12 k/C30/C28/C12A /C27B /C27C /C27D /C27E /C28k E /C28kYru$Yru% A /C27D k /C27DYru$Yru% B /C27C k /C27CYru$Yru% /C30A /C27C /C27D /C27E A /C27CYru$Yru% B /C27C /C27D /C27E C /C27EYru$Yru% : See also BINOMIAL SUMS References Koepf, W. "Hypergeometric Database." Ch. 3 in Hypergeo- metric Summation: An Algorithmic Approach to Summa- tion and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 35 /C1/6, 1998. Sze´kely, L. A. "Common Origin of Cubic Binomial Identities; A Generalization of Sura´nyi’s Proof of the Le Jen Shoo’s Formula." J. Combin. Th. Ser. A 40, 171 /C1/74, 1985. Szemere ´di’s Regularity Lemma A fundamental structural result in EXTREMAL GRAPH THEORY due to Szemere ´di (1978). The regularity lemma essentially says that every graph can be well-approximated by the union of a constant number of random-like BIPARTITE GRAPHS , called regular pairs. See also BLOW- UP LEMMA ,EXTREMAL GRAPH THEORY , SEYMOUR CONJECTURE ,SZEMERE ´ DI’S THEOREM References Komlo ´s, J. and Simonovitas, M. "Szemere ´di Regularity Lemma and Its Applications in Graph Theory." In Combi- natorics, Paul Erdos is Eighty, Vol. 1 (Ed. D. Miklo ´s, V. T. So´s, and T. Szonyi). Budapest: Ja´nos Bolyai Mathematical Society, pp. 295 /C1/52, 1993. Komlo ´s, J.; Sa´rkozy, G. N.; and Szemere ´di, E. "Proof of the Seymour Conjecture for Large Graphs." Ann. Comb. 2, 43 /C1/0, 1998. Szemere ´di, E. "Regular Partitions of Graphs." In Proble `mes combinatoires et the´orie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Paris: E´ ditions du Centre National de la Recherche Scientifique (CNRS), pp. 399 /C1/01, 1978.> Szemere ´di’s Theorem This entry contributed by KEVIN O’BRYANT Every sequence of integers with positive density contains arbitrarily long ARITHMETIC SEQUENCES . A corollary states that, for any positive integer k and positive real number d ; there exists a threshold number n(k ; d) such that for n ]n(k; r) every subsetof f1 ; 2 ; ...; ng with CARDINALITY larger than dn contains a k-term ARITHMETIC SEQUENCE . VAN DER WAERDEN’S THEOREM follows immediately by setting d /C30n=r : The best bounds for VAN DER WAERDEN NUMBERS are derived from bounds for n(k; r)in Szemere ´di’s Theorem. Szemere ´di’s theorem was conjectured by Erdos and Tura´n (1936). Roth (1953) proved the case k /C303, and was mentioned in his FIELDS MEDAL citation. Sze- mere´di (1969) proved the case k /C304, and the general theorem in 1975 as a consequence of SZEMERE ´ DI’S REGULARITY LEMMA (Szemere ´di 1975a), for which he collected a $1000 prize from Erdos. Fu¨rstenberg and Katznelson (1979) proved Szemere ´di’s theorem using ERGODIC THEORY . Gowers (1998ab) subsequently gave a new proof, with a better bound on n(k ; r) ; for the case k /C30 4 (mentioned in his FIELDS MEDAL citation; Lepowsky et al. 1999). Erdos offered a $3,000 prize for a proof of the proposition that "If the sum of reciprocals of a set of integers diverges, then that set contains arbitrarilylong arithmetic progressions." This conjecture is still open (unsolved), even for 3-term arithmetic progres- sions. Erdos also offered $10,000 for an asymptoticformula for r 3(n);the largest possible cardinality of a subset of f1;2;...;ngthat does not contain a 3-term arithmetic progression. See also ARITHMETIC SEQUENCE ,SZEMERE ´ DI’S REG- ULARITY LEMMA , VAN DER WAERDEN NUMBER , VAN DER WAERDEN’S THEOREM References Erdos, P. and Tura ´n, P. "On Some Sequences of Integers." J. London Math. Soc. 11, 261/C1/64, 1936. Fu¨rstenberg, H. and Katznelson, Y. "An Ergodic Szemere ´di Theorem for Commuting Transformations." J. Analyse Math. 34, 275/C1/91, 1979. Gowers, W. T. "Fourier Analysis and Szemere ´di’s Theorem." InProceedings of the International Congress of Mathema- ticians, Vol. I (Berlin, 1998). Doc. Math. , Extra Vol. I, 617/C1/29, 1998a. Gowers, W. T. "A New Proof of Szemere ´di’s Theorem for Arithmetic Progressions of Length Four." Geom. Funct. Anal. 8, pp. 529 /C1/51, 1998b. Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey Theory, 2nd ed. New York: Wiley, 1990. Guy, R. K. "Theorem of van der Waerden, Szemere ´di’s Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 204 /C1/09, 1994. Lepowsky, J.; Lindenstrauss, J.; Manin, Y.; and Milnor, J. "The Mathematical Work of the 1998 Fields Medalists."Not. Amer. Math. Soc. 46,1 7/C1 /6, 1999. Roth, K. "Sur quelques ensembles d’entiers." C. R. Acad. Sci. Paris 234, 388/C1/90, 1952. Roth, K. F. "On Certain Sets of Integers." J. London Math. Soc. 28, 104/C1/09, 1953. Szemere ´di, E. "On Sets of Integers Containing No Four Elements in Arithmetic Progression." Acta Math. Acad. Sci. Hungar. 20,8 9/C1/04, 1969. Szemere ´di, E. "On Sets of Integers Containing No k Elements in Arithmetic Progression." Acta Arith. 27, 199 /C1/45, 1975a. Szemere ´di, E. "On Sets of Integers Containing No k Elements in Arithmetic Progression." In Proceedings of the International Congress of Mathematicians, Volume 2, Held in Vancouver, B.C., August 21 /C1/9, 1974. Montreal, Quebec: Canad. Math. Congress, pp. 503 /C1/05, 1975b. Szilassi Polyhedron A HEPTAHEDRON which is topologically equivalent to a TORUS and for which every pair of faces has an EDGE in common. The Szilassi polyhedron has 14 VERTICES , seven faces, and 21 EDGES , and is the DUAL POLYHE- DRON of the CSA´ SZA´ R POLYHEDRON . This polyhedron was discovered by L. Szilassi in 1977. In the above illustration of the net, sides indicated by letters areconnected with the corresponding side indicated by the same letter but with a different number of primes. Like the TETRAHEDRON , each face of the Szilassi polyhedron touches all other faces. The SKELETON of the Szilassi polyhedron is equiva- lent to the HEAWOOD GRAPH , shown above. See also CSA´ SZA´ R POLYHEDRON ,H EAWOOD GRAPH , TOROIDAL POLYHEDRON References Ace, T. "Szilassi Polyhedron." http://www.qnet.com/~crux/ szilassi.html. Eppstein, D. "Polyhedra and Polytopes." http://www.ics.u- ci.edu/~eppstein/junkyard/polytope.html. Gardner, M. "Mathematical Games: In Which a Mathema- tical Aesthetic is Applied to Modern Minimal Art." Sci. Amer. 239,22/C1/2, Nov. 1978. Gardner, M. Fractal Music, Hypercards, and More Mathe- matical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 118 /C1/20, 1992. Hart, G. "Toroidal Polyhedra." http://www.georgehart.com/ virtual-polyhedra/toroidal.html. Weisstein, E. W. "Polyhedra." MATHEMATICA NOTEBOOK POLYHEDRA.M . Szpiro’s Conjecture A conjecture which relates the minimal DISCRIMINANT of an ELLIPTIC CURVE to the CONDUCTOR . If true, it would imply F ERMAT’S LAST THEOREM for sufficiently large exponents. See also CONDUCTOR ,D ISCRIMINANT (ELLIPTIC CURVE ), ELLIPTIC CURVE References Cox, D. A. "Introduction to Fermat’s Last Theorem." Amer. Math. Monthly 101,3/C1/4, 1994. T T2-Separation Axiom Given any two distinct points x, y, there exist neighborhoods u and v of x and y, respectively, with u S v /C30¥: It then follows that finite SUBSETS are CLOSED . See also CLOSURE (SET) Tableau YOUNG TABLEAU Tableau Class When a YOUNG TABLEAU is constructed using the so- called insertion algorithm, an element starts in some position on the first row, from which it may later be bumped. In contrast, the elements that start out in the ith column are said to belong to the ith class (Skiena 1990, p. 73). Tableau classes may be computed using TableauClasses [p] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). See also BUMPING ALGORITHM ,YOUNG TABLEAU References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Tabu Search A heuristic procedure which has proven efficient at solving COMBINATORIAL optimization problems. References Glover, F.; Taillard, E.; and De Werra, D. "A User’s Guide to Tabu Search." Ann. Oper. Res. 41,3/C1/28, 1993. Piwakowski, K. "Applying Tabu Search to Determine New Ramsey Numbers." Electronic J. Combinatorics 3,R61/C1/4, 1996. http://www.combinatorics.org/Volume_3/volu- me3.html#R6. Tacnode A DOUBLE POINT at which two OSCULATING CURVES are TANGENT . The above plot shows the tacnode of thecurve 2x4 /C283x2y /C27y2 /C282y3 /C27y4 /C300: The LINKS CURVE also has a tacnode at the origin. See also ACNODE ,CRUNODE ,DOUBLE POINT ,OSCU- LATING CURVES ,SPINODE References Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 57 /C1/58, 1978. Tacpoint A tangent point of two similar curves. Tactix NIM Tail Probability Define T as the set of all points t with probabilities P(x) such that a > t [P(a 5x 5a /C27da) BP0 or a Bt [ P(a 5x 5a /C27da) BP0 ; where P0is a POINT PROBABIL- ITY (often, the likelihood of an observed event). Then the associated tail probability is given by fTP(x) dx:/ See also P-VALUE ,POINT PROBABILITY Tait Coloring A 3-coloring of GRAPH EDGES so that no two EDGES of the same color meet at a VERTEX (Ball and Coxeter 1987, pp. 265 /C1/266). See also EDGE (GRAPH ), TAIT CYCLE ,VERTEX (GRAPH ) References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, 1987. Tait Cycle A set of circuits going along the EDGES of a GRAPH , each with an EVEN number of EDGES , such that just one of the circuits passes through each VERTEX (Ball and Coxeter 1987, pp. 265 /C1/266). See also EDGE (GRAPH ), EULERIAN CYCLE ,HAMILTO- NIAN CYCLE ,TAIT COLORING ,VERTEX (GRAPH ) References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, 1987. Tait Flyping Conjecture FLYPING CONJECTURE Tait’s Hamiltonian Graph Conjecture Every 3-connected CUBIC GRAPH has a HAMILTONIAN CIRCUIT . Proposed by Tait in 1880 and refuted by Tutte (1946) with the counterexample now known as TUTTE’S GRAPH . Had the conjecture been true, it would have implied the FOUR-COLOR THEOREM .A simpler counterexample was later given by Kozyrev and Grinberg. See also CONNECTED GRAPH ,CUBIC GRAPH ,FOUR- COLOR THEOREM ,H AMILTONIAN CIRCUIT ,H AMILTO- NIAN GRAPH ,TUTTE CONJECTURE ,TUTTE’S GRAPH , VERTEX (GRAPH ) References Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 82 /C1/89, 1973. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 198, 1990. Tait, P. G. "Remarks on the Colouring of Maps." Proc. Royal Soc. Edinburgh 10, 729, 1880. Tutte, W. T. "On Hamiltonian Circuits." J. London Math. Soc. 21,98/C1/101, 1946. Tutte, W. T. "Non-Hamiltonian Planar Maps." In Graph Theory and Computing (Ed. R. Read). New York: Aca- demic Press, pp. 295 /C1/301, 1972. Tait’s Knot Conjectures P. G. Tait undertook a study of KNOTS in response to Kelvin’s conjecture that the atoms were composed of knotted vortex tubes of ether (Thomson 1869). He categorized KNOTS in terms of the number of crossings in a plane projection. He also made some conjectures which remained unproven until the discovery of JONES POLYNOMIALS : 1. Reduced alternating diagrams have minimal CROSSING NUMBER , 2. Any two reduced alternating diagrams of a given knot have equal WRITHE , 3. The FLYPING CONJECTURE , which states that the number of crossings is the same for any diagram of an ALTERNATING KNOT . Conjectures (1) and (2) were proved by Kauffman (1987), Murasugi (1987ab), and Thistlethwaite (1987, 1988) using properties of the JONES POLYNOMIAL orKAUFFMAN POLYNOMIAL F (Hoste et al. 1998). Con- jecture (3) was proved true by Menasco and Thistle- thwaite (1991, 1993) using properties of the JONES POLYNOMIAL (Hoste et al. 1998). See also ALTERNATING KNOT,C ROSSING NUMBER (LINK), FLYPING CONJECTURE ,JONES POLYNOMIAL , KNOT,W RITHE References Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Kauffman, L. H. "State Models and the Jones Polynomial." Topology 26, 395 /C1/407, 1987. Menasco, W. and Thistlethwaite, M. "The Tait Flyping Conjecture." Bull. Amer. Math. Soc. 25, 403 /C1/412, 1991. Menasco, W. and Thistlethwaite, M. "The Classification of Alternating Links." Ann. Math. 138, 113 /C1/171, 1993. Murasugi, K. "The Jones Polynomial and Classical COn- jectures in Knot Theory." Topology 26, 187 /C1/194, 1987a. Murasugi, K. "Jones Polynomials and Classical Conjectures in Knot Theory II." Math. Proc. Cambridge Philos. Soc. 102, 317 /C1/318, 1987. Tait, P. G. "On Knots I, II, III." Scientific Papers, Vol. 1. London: Cambridge University Press, pp. 273 /C1/347, 1900. Thistlethwaite, M. B. "A Spanning Tree Expansion of the Jones Polynomial." Topology 26, 297 /C1/309, 1987. Thistlethwaite, M. B. "Kauffman’s Polynomial and Alternat- ing Links." Topology 27, 311 /C1/318, 1988. Thomson, W. H. "On Vortex Motion." Trans. Roy. Soc. Edinburgh 25, 217 /C1/260, 1869. TAK Function A RECURSIVE FUNCTION devised by I. Takeuchi. For INTEGERS x, y, and z, and a function h,itis TAKh(x; y; z) The number of function calls F0(a; b) required to compute TAK0(a;b;0) for a>b>0i s F0(a;b)/C304Xb k/C300a/C28b a/C27b/C282ka/C27b/C282k b/C28kiCkniCko /C283 /C301/C274Xb/C281 k/C300a/C28b a/C27b/C282ka/C27b/C282k b/C28kiCkniCko (Vardi 1991). The TAK function is also connected with the BALLOT PROBLEM (Vardi 1991). See also ACKERMANN FUNCTION ,BALLOT PROBLEM References Gabriel, R. P. Performance and Implementation of Lisp Systems. Cambridge, MA: MIT Press, 1985. Knuth, D. E. Textbook Examples of Recursion. Preprint 1990. Vardi, I. "The Running Time of TAK." Ch. 9 in Computa- tional Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 179 /C1/199, 1991. Takagi Fractal Curve BLANCMANGE FUNCTION Take-Away Game NIM-HEAP Takens-Bogdanov Bifurcation References Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373 /C1/ 388, 1981. Kuznetsov, Y. A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995. Takens, F. "Forced Oscillations and Bifurcations." Comm. Math. Inst. Rijksuniv. Utrecht 2,1/C1/111, 1974. Takeuchi Function TAK FUNCTION Talbot’s Curve A curve investigated by Talbot which is the NEGATIVE PEDAL CURVE of an ELLIPSE with respect to its center. It has four CUSPS and two NODES , provided the ECCENTRICITY of the ELLIPSE is greater than 1=ffiffiffi 2p : Its CARTESIAN EQUATION is x /C30a2 /C27 f2 sin2 tiCjiCk cos t a y /C30a2 /C28 2f2 /C27 f2 sin2 tiCjiCk sin t b ; where f is a constant. References Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967. MacTutor History of Mathematics Archive. "Talbot’s Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Tal- bots.html.Talisman Hexagon An (n, k)-talisman hexagon is an arrangement of nested hexagons containing the integers 1, 2, ..., Hn /C30 3n(n /C281) /C271; where Hn is the nth HEX NUMBER , such that the difference between all adjacent hexagons is at least as large as a number k. The hexagon illustrated above is a (3, 4)-talisman hexagon. See also HEX NUMBER ,M AGIC SQUARE ,TALISMAN SQUARE References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 111 /C1/112, 1979. Talisman Square Ann/C29nARRAY of the integers from 1 to n2such that the difference between any one integer and its neighbor (horizontally, vertically, or diagonally, with-out wrapping around) is greater than or equal to some value kis called a ( n, k)-talisman square. The above illustrations show (4, 2)-, (4, 3)-, (5, 4)-, and (6, 8)- talisman squares. See also ANTIMAGIC SQUARE ,HETEROSQUARE ,MAGIC SQUARE ,TALISMAN HEXAGON References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 110 /C1/113, 1979. Weisstein, E. W. "Magic Squares." M ATHEMATICA NOTEBOOK MAGICSQUARES.M . Tame Algebra LetAdenote an R/-algebra, so that Ais a VECTOR SPACE over Rand A/C29A0A (x;y)/C2x/C215y; where x/C215yisVECTOR MULTIPLICATION which is assumed to be BILINEAR . Now define Z/C13fx/C23a:x/C215y/C280 for some nonzero y/C23Ag; where 0 /C23Z:Ais said to be tame if Zis a finite union ofSUBSPACES ofA. A 2-D 0- ASSOCIATIVE algebra is tame, but a 4-D 4- ASSOCIATIVE algebra and a 3-D 1- ASSOCIATIVE algebra need not be tame. It is conjec- tured that a 3-D 2- ASSOCIATIVE algebra is tame, and proven that a 3-D 3- ASSOCIATIVE algebra is tame if it possesses a multiplicative IDENTITY ELEMENT . References Finch, S. "Zero Structures in Real Algebras." http:// www.mathsoft.com/asolve/zerodiv/zerodiv.html. Tame Knot AKNOT equivalent to a POLYGONAL KNOT . Knots which are not tame are called WILD KNOTS . References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 49, 1976. Tan TANGENT Tangency Theorem The external (internal) SIMILARITY POINT of two fixed CIRCLES is the point at which all the CIRCLES homo- geneously (nonhomogeneously) tangent to the fixed CIRCLES have the same POWER and at which all the tangency secants intersect. References Do¨rrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 157, 1965.Tangent The tangent function is defined by tanx/C13sinx cosx; (1) where sin xis the SINE function and cos xis the COSINE function. The notation tg xis sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). The word "tangent" also has an important relatedmeaning as a LINE orPLANE which touches a given curve or solid at a single point. These geometricalobjects are then called a TANGENT LINE orTANGENT PLANE , respectively. The definition of the tangent function can be ex- tended to complex arguments zusing the definition tanz/C30eiz/C28e/C28iz i(eiz/C27e/C28iz); (2) where Eis the base of the NATURAL LOGARITHM and I is the IMAGINARY NUMBER . A related function known as the HYPERBOLIC TANGENT is similarly defined, tanh z/C30ez/C28e/C28z ez/C27e/C28z: (3) Important tangent identities include tan2u/C271/C30sec2u (4) tan(a/C27b)/C30tana/C27tanb 1/C28tanatanb(5) tan(a/C28b)/C30tana/C28tanb 1/C27tanatanb(6) tan(2 a) /C302 tan a 1 /C28 tan2 a : (7) tan(na) /C30tan[( n /C28 1)a] /C27 tan a 1 /C28 tan[( n /C28 1)a] tan a(8) tana 2 ! /C30sin a 1 /C27 cos a (9) /C301 /C28 cos a sin a (10) /C301 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 tan2 ap tan a (11) /C30tan a sin a tan a /C27 sin a (12) in addition to the beautiful identity tan(a /C27 b /C27 g) /C30tan a /C27 tan b /C27 tan g /C28 tan a tan b tan g 1 /C28 tan b tan g /C28 tan g tan a /C28 tan a tan b : (13) There are a number of simple but interesting tangent identities based on those given above, including tan(A /C2760 /C14) tan(A /C2860/C14) /C27tan A tan(A /C2760/C14) /C27tan A tan(A /C2860 /C14) /C30/C283 (14) (Borchardt and Perrott 1930). The MACLAURIN SERIES valid for /C28p=2 Bx B p=2 for the tangent function is tan x /C30X/C12 n/C300(/C281)n /C28122n(22n /C28 1)B2n (2n)! x2n/C281 /C27... /C30x /C271 3 x3 /C272 15 x5 /C2717 315 x7 /C2762 2835 x9 /C27... ; (15) where Bn is a BERNOULLI NUMBER . /tan x is IRRATIONAL for any RATIONAL x "0; which can be proved by writing tan x as a CONTINUED FRACTION tan x /C30x 1 /C28x2 3 /C28x2 5 /C28x2 7 /C28 ...: (16) Lambert derived another CONTINUED FRACTION ex- pression for the tangent,tan x /C301 1 x/C281 3 x/C281 5 x/C281 7 x/C28 ...: (17) An interesting identity involving the PRODUCT of tangents is Y/C28(n/C281)=2 /C29 k /C301tankp n ! /C30ffiffiffinpfor n odd 1 for n even ;iC0C (18) where xbcis the FLOOR FUNCTION . Another tangent identity is tan(n tan/C281 x) /C301 i(1 /C27 ix)n /C28 (1 /C28 ix)n (1 /C27 ix)n /C27 (1 /C28 ix)m (19) (Beeler et al. 1972). The equation x/C30tanx (20) does not have simple closed-form solutions, but the first few approximate numerical solutions are 0, 4.49341, 7.72525, 10.9041, 14.0662, .... The differencebetween consecutive solutions gets closer and closer topfor higher order solutions. See also A LTERNATING PERMUTATION ,C OSINE ,C O- TANGENT ,INVERSE TANGENT ,M ORRIE’S LAW,SINE, TANGENT LINE,TANGENT PLANE References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/79, 1972. Beeler, M. et al. Item 16 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 9, Feb. 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 226, 1987. Borchardt, W. G. and Perrott, A. D. Ex. 33 in A New Trigonometry for Schools. London: G. Bell, 1930. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000. Spanier, J. and Oldham, K. B. "The Tangent tan( x) and Cotangent cot( x) Functions." Ch. 34 in An Atlas of Func- tions. Washington, DC: Hemisphere, pp. 319 /C1/330, 1987. Tangent Bifurcation FOLD BIFURCATION Tangent Bundle Every smooth manifold Mhas a tangent bundle TM, which consists of the TANGENT SPACE TMpat all points pinM. Since a tangent space TMpis the set of all tangent vectors to Matp, the tangent bundle is the collection of all tangent vectors, along with the information of the point to which they are tangent. TM /C30f(p ; v):p /C23 M ; v /C23 TMp g The tangent bundle is a special case of a VECTOR BUNDLE . As a bundle it has RANK n, where n is the dimension of M.A COORDINATE CHART on M provides a TRIVIALIZATION for TM. In the coordinates, x1 ; ...; xn ðÞ ; the vector fields v1 ; ...; vn ðÞ ; where vi /C30 @=@xi ; span the tangent vectors at every point (in the COORDINATE CHART ). The transition function from these coordinates to another set of coordinates is given by the JACOBIAN of the coordinate change. For example, on the UNIT SPHERE , at the point (1; 0; 0) there are two different coordinate charts defined on the same HEMISPHERE , f : U1 0 S2 and c : U2 0 S2 ; f x1 ; x2 ðÞ /C30 cos x1 cos x2 ; sin x1 cos x2 ; sin x2 ðÞ (1) c y1 ; y2 ðÞ /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28y2 1 /C28y22q ; y1 ; y1iCkniCko (2) with U1 /C30(/C28p=2; p=2) /C29(/C28p=2; p=2) and U2 /C30 y1 ; y2 ðÞ : y2 1 /C27y22 B1 fg : The map between the coordi- nate charts is a /C30 c/C281( f: y1 ; y2 ðÞ /C30 a x1 ; x2 ðÞ /C30 sin x1 ; cos x2 ; sin x2 ðÞ (3) The JACOBIAN of a : U1 0 U2is given by the matrix- valued function cos x1 cos x2sin x1 sin x2 0 cos x2iC0jiC0k (4) which has DETERMINANT cos x1 cos2 x2and so is invertible on U1:/ The tangent vectors transform by the Jacobian. At the point x1;x2 ðÞ inU1;a tangent vector vcorre- sponds to the tangent vector Jvatax1;x2 ðÞ inU2: These two are just different versions of the same element of the tangent bundle. See also CALCULUS ,COORDINATE CHART ,COTANGENT BUNDLE ,D IRECTIONAL DERIVATIVE ,E UCLIDEAN SPACE ,JACOBIAN ,M ANIFOLD ,T ANGENT BUNDLE , TANGENT SPACE ,TANGENT VECTOR ,VECTOR FIELD, VECTOR SPACETangent Circles Two circles with centers at xi;yi ðÞ with radii rifor i/C301;2 are mutually tangent if x1/C28x2 ðÞ2/C27y1/C28y2iCjiCk2/C30r19r2 ðÞ2: If the center of the second circle is inside the first, then the /C28and/C27signs both correspond to internally tangent circles. If the center of the second circle is outside the first, then the /C28sign corresponds to externally tangent circles and the /C27sign to internally tangent circles.Finding the circles tangent to three given circles is known as A POLLONIUS’ PROBLEM . There are four CIRCLES that are tangent all three sides (or their extensions) of a given TRIANGLE : the INCIRCLE Iand three EXCIRCLES J1;J2;andJ3:These four circles are, in turn, all touched by the NINE-POINT CIRCLE N. If two circles C1andC2of radii r1andr2are mutually tangent to each other and a line, then their centers are separated by a horizontal distance given by solving x2 2 /C27 r1 /C28r2 ðÞ2/C30 r1 /C27r2 ðÞ2(1) for x2 ; giving x2 /C302ffiffiffiffiffiffiffiffiffir1r2p: (2) The position and radius of a third circle tangent to the first two and the line can be found by solving the simultaneous equations x2 3 /C27 r1 /C28r3 ðÞ2/C30 r1 /C27r3 ðÞ2(3) x3 /C28x2 ðÞ2/C27 r2 /C28r3 ðÞ2/C30 r2 /C27r3 ðÞ2(4) for x3 and r3 ; giving x3 /C302r1ffiffiffiffiffir2p ffiffiffiffiffir 1p/C27ffiffiffiffiffir 2p (5) r3 /C30r1r2ffiffiffiffiffir 1p/C27ffiffiffiffiffir 2piCjiCk2 : (6) The latter equation can be written in the form 1 ffiffiffiffiffir 3p/C301 ffiffiffiffiffir 1p/C271 ffiffiffiffiffir 2p : (7) This problem was given as a Japanese temple problem on a tablet from 1824 in the Gumma Prefecture (Rothman 1998). See also APOLLONIUS’ PROBLEM ,CASEY’S THEOREM , CHAIN OF CIRCLES ,CIRCLE PACKING ,CIRCLE TAN- GENTS ,D ESCARTES CIRCLE THEOREM ,E XCIRCLE , FOUR COINS PROBLEM ,INCIRCLE ,M ALFATTI’S TAN- GENT TRIANGLE PROBLEM ,P APPUS CHAIN ,S ODDY CIRCLES ,TANGENT CURVES ,TANGENT SPHERES References Coolidge, J. L. "Mutually Tangent Circles." §1.3 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 31 /C1/44, 1971. Fukagawa, H. and Pedoe, D. "Two Circles," "Three Circles," "Four Circles," and "Many Circles." §1.1 /C1/1.5 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Ca- nada: Charles Babbage Research Foundation, pp. 3 /C1/13 and 79 /C1/88, 1989. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85 /C1/91, May 1998. Tangent Curves See also OSCULATING CURVES ,T ANGENT CIRCLES , TACNODE ,TANGENT LINE Tangent Developable A RULED SURFACE M is a tangent developable of a curve y if M can be parameterized by x(u; v) /C30y(u) /C27 vy ?(u) : A tangent developable is a FLAT SURFACE .See also BINORMAL DEVELOPABLE ,N ORMAL DEVEL- OPABLE References Gray, A. "Tangent Developables." §19.3 in Modern Differ- ential Geometry of Curves and Surfaces with Mathema- tica, 2nd ed. Boca Raton, FL: CRC Press, pp. 441 /C1/444, 1997. Tangent Externally Two curves are tangent externally at a point P if they lie on opposite sides of their common tangent at P See also TANGENT INTERNALLY Tangent Figures See also INCIDENT Tangent Hyperbolas Method HALLEY’S METHOD Tangent Indicatrix Let the SPEED s of a closed curve on the unit sphere S2 never vanish. Then the tangent indicatrix t /C13˙s ˙sjj is another closed curve on S2 : It is sometimes called the TANTRIX .Ifs IMMERSES in S2 ; then so will t:/ References Solomon, B. "Tantrices of Spherical Curves." Amer. Math. Monthly 103,30/C1/39, 1996. Tangent Internally Two curves are tangent internally at a point P if they lie on the same side of their common tangent at P See also TANGENT EXTERNALLY Tangent Line A straight line is tangent to a given curve f(x)ata point x0on the curve if the line passes through the point (x0 ; f(x0)) on the curve and has slope f ?(x0); where f ?(x) is the DERIVATIVE of f(x) :/ See also CIRCLE TANGENTS ,SECANT LINE,TANGENT , TANGENT PLANE ,TANGENT SPACE ,TANGENT VECTOR References Yates, R. C. "Instantaneous Center of Rotation and the Construction of Some Tangents." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 119 /C1/122, 1952. Tangent Map If f : M 0 N ; then the tangent map Tf associated to f is a VECTOR BUNDLE HOMEOMORPHISM Tf : TM 0 TN (i.e., a MAP between the TANGENT BUNDLES of M and N respectively). The tangent map corresponds to DIFFERENTIATION by the formula Tf(v) /C30(f(f)?(0) ; (1) where f?(0) /C30v (i.e., f is a curve passing through the base point to v in TM at time 0 with velocity v). In this case, if f : M 0 N and g : N 0 O ; then the CHAIN RULE is expressed as T(f(g) /C30Tf(Tg: (2) In other words, with this way of formalizing differ- entiation, the CHAIN RULE can be remembered by saying that "the process of taking the tangent map of a map is functorial." To a topologist, the form (f(g) ?(a) /C30f ?(g(a))(g ?(a) ; (3) for all a, is more intuitive than the usual form of the CHAIN RULE . See also DIFFEOMORPHISM References Gray, A. "Tangent Maps." §11.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 250 /C1/255, 1997. Tangent Number A number also called a ZAG NUMBER giving the number of ODD ALTERNATING PERMUTATIONS . The first few are 1, 2, 16, 272, 7936, ... (Sloane’s A000182). See also ALTERNATING PERMUTATION ,E NTRINGER NUMBER ,EULER ZIGZAG NUMBER ,SECANT NUMBER References Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663 /C1/ 688, 1967. Sloane, N. J. A. Sequences A000182/M2096 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Tangent Plane Let (x0 ; y0) be any point of a surface function z /C30 f(x; y) : Then the surface has a nonvertical tangent plane at (x0 ; y0) with equation z /C30f(x0 ; y0) /C27fx(x0 ; y0)(x /C28x0) /C27fy(x0 ; y0)(y /C28y0) : See also NORMAL VECTOR ,PLANE ,TANGENT ,TAN- GENT LINE,TANGENT SPACE ,TANGENT VECTORTangent Space Let x be a point in an n-dimensional COMPACT MANIFOLD M, and attach at x a copy of Rn tangential to M. The resulting structure is called the TANGENT SPACE of M at x and is denoted TxM : If g is a smooth curve passing through x, then the derivative of g at x is a VECTOR in TxM :/ See also TANGENT ,T ANGENT BUNDLE ,T ANGENT PLANE ,TANGENT SPACE (CHART ), TANGENT SPACE (SUBMANIFOLD ), TANGENT VECTOR Tangent Space (Chart) From the point of view of COORDINATE CHARTS , the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In an open set U in Rnthere are no constraints, so the tangent space at a point p is another copy of Rn : The set U could be a COORDINATE CHART for an n-dimensional MANIFOLD . The tangent space at p, denoted TMp ; is the set of possible VELOCITY VECTORS of paths through p. Hence there is a CANONICAL BASIS :if( x1 ; ...; xn) are the coordinates, then v1 ; ...; vnare a basis for the tangent space, where viis the velocity vector of a particle with unit speed moving inward along the coordinate xi:The collection of tangent vectors, called the TANGENT BUNDLE , is the PHASE SPACE of a single particle moving in the manifold M. It seems as if the tangent space at pis the same as the tangent space at all other points in the chart U. However, while they do share the same dimension and are ISOMORPHIC , in a change of coordinates, they lose their canonical isomorphism. For example, let U/C30(0;1) and V/C30(0;3) be coordi- nate charts for the unit interval I. We can change coordinates with f:U0Vdefined by f(x)/C30x/C272x2: This is a change of coordinates because the derivativedoes not vanish on U. But this change is not linear, and stretches out Imore near 1 than it does near 0 : The tangent vectors transform by the derivative. Atx/C301=4;they are stretched by a factor of df=dx/C302: While at x/C303=4;they are stretched out by a factor of df=dx/C304: / In general, the tangent vectors transform accordingto the J ACOBIAN . The tangent vector vatqcan also be considered as the tangent vector Jfvatf(q)i n another coordinate chart, where fis the DIFFEO- MORPHISM from one chart to the other. The linear transformation determined by the J ACOBIAN offis invertible, since fis a DIFFEOMORPHISM . Not only does the J ACOBIAN , and the CHAIN RULE , show that the tangent space is WELL DEFINED , independent of coordinate chart, but it also shows that tangent vectors "push forward." That is, given any smooth map f : X 0 Y between manifolds, it makes sense to map the tangent vectors of X to tangent vectors of Y. Writing ˜f as the function f between a coordinate chart in X and one in Y, then f/C31(v) /C30J˜f (v) maps v from TXp to TYf(p) : Another notation for f/C31is df, the DIFFERENTIAL of f. In the language of TENSORS , the tangent vector’s pushing forward means that a vector field is a COVARIANT TENSOR . See also CALCULUS ,COORDINATE CHART ,DIFFEREN- TIAL FORM,D IRECTIONAL DERIVATIVE ,E UCLIDEAN SPACE ,E XTERIOR ALGEBRA ,JACOBIAN ,M ANIFOLD , SUBMANIFOLD ,TANGENT BUNDLE ,TANGENT SPACE , VECTOR FIELD,VELOCITY VECTOR Tangent Space (Intrinsic) The tangent space at a point p in an ABSTRACT MANIFOLD M can be described without the use of embeddings or COORDINATE CHARTS . The elements of the tangent space are called tangent vectors, and the collection of tangent spaces forms the TANGENT BUNDLE . One description is to put an equivalence relation on smooth paths through the point p. More precisely, consider all smooth maps f : I 0 M where I /C30(/C281 ; 1) and f(0) /C30p: We say that two maps f and g are equivalent if they agree to first order. That is, in any coordinate chart around p, f ?(0) /C30g?(0) : If they are similar in one chart then they are similar in any other chart, by the CHAIN RULE . The notion of agreeing to first order depends on coordinate charts, but this cannot be completely eliminated since that is how manifolds are defined. Another way is to first define a VECTOR FIELD as a DERIVATION of the ring of smooth functions f : M 0 R: Then a tangent vector at a point p is an equivalence class of vector fields which agree at p. That is, X /C2Y if Xf(p) /C30Yf(p) for every smooth function f. Of course, the tangent space at pis the vector space of tangent vectors at p. The only drawback to this version is that aCOORDINATE CHART is required to show that the tangent space is an n-dimensional vector space. See also CHAIN RULE,COORDINATE CHART ,DERIVA- TION ALGEBRA ,D IFFERENTIAL FORM,D IRECTIONAL DERIVATIVE ,EXTERIOR ALGEBRA ,EUCLIDEAN SPACE , JACOBIAN ,LIE GROUP ,M ANIFOLD ,SHEAF ,TANGENT BUNDLE ,TANGENT SPACE ,VECTOR FIELD,VELOCITY VECTOR Tangent Space (Submanifold) The TANGENT PLANE to a surface at a point pis the tangent space at p(after translating to the origin). The elements of the tangent space are called TANGENT VECTORS , and they are CLOSED under addition and scalar multiplication. In particular, the tangent space is a VECTOR SPACE .Any SUBMANIFOLD of E UCLIDEAN SPACE , and more generally any SUBMANIFOLD of an ABSTRACT MANI- FOLD , has a tangent space at each point. The collec- tion of tangent spaces TMptoMforms the TANGENT BUNDLE TM/C30@p/C23Mp;TMpiCjiCk :AVECTOR FIELD as- signs to every point paTANGENT VECTOR in the tangent space at p. There are two ways of defining a submanifold, and each way gives rise to a different way of defining the tangent space. The first way uses a PARAMETERIZA- TION , and the second way uses a system of equations. Suppose that f/C30f1;...;fn ðÞ is a local PARAMETERIZA- TION of a SUBMANIFOLD Min E UCLIDEAN SPACE Rn: Say, f:U0Rn; (1) where Uis the open UNIT BALL inRk;andf(U)ƒM: At the point p/C30f(0);the tangent space is the image of the J ACOBIAN off, as a linear transformation from Rk toRn:For example, consider the UNIT SPHERE S2/C30y1;y2;y3 ðÞ :y2 1/C27y22/C27y23/C301iCniCo (2) inR3:Then the function (with the domain U/C30 x1;x2 ðÞ :x2 1/C27x22B1 fg ) f/C30x1;x2;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28x2 1/C28x22qiCkniCko (3) parameterizes a NEIGHBORHOOD of the north pole. Its Jacobian at (0 ;0) is given by the matrix 10 01 002 435 (4) whose IMAGE is the tangent space at p, TS2 (0;0;1)/C30f(a;b;0)g:iCk0iCk0 (5) An alternative description of a SUBMANIFOLD Mas the set of solutions to a system of equations leads to another description of tangent vectors. Consider a SUBMANIFOLD Mwhich is the set of solutions to the system of equations f1x1;...;xn ðÞ /C300 n (6) frx1;...;xn ðÞ /C300; where k/C27r/C30nand the J ACOBIAN off:Rn0Rr;with f/C30f1;...fn ðÞ ;has rank rat the solutions Mtof/C300. A tangent vector vat a solution pis an infinitesimal solution to the above equations (at p). The tangent vector v/C30v1;...;vn ðÞ is a solution of the derivative (linearization) of f, i.e., it is in the NULLSPACE of the JACOBIAN . Consider this method in the recomputation the tangent space of the sphere at the North Pole. The sphere is two-dimensional and is described as the solution to single equation (3 /C282 /C301) x2 1 /C27x22 /C27x23 /C301: Set f1 /C30x21 /C27x22 /C27x23 /C281: We want to compute the tangent space at the solution f1(0; 0; 1) /C300 (at the north pole). The JACOBIAN at this point is the 1 /C293 matrix [0; 0; 2]; and its nullspace is the tangent space TS2 (0; 0; 1) /C30f(a; b; 0)g:iCk0iCk0 (7) It appears that the tangent space depends either on the choice of parametrization, or on the choice of system of equations. Because the Jacobian of a composition of functions obeys the CHAIN RULE , the tangent space is WELL DEFINED . Note that the JACOBIAN of a DIFFEOMORPHISM is an INVERTIBLE LINEAR MAP, and these correspond to the ways the equations can be changed. The basic facts from LINEAR ALGEBRA used to show that the tangent space is WELL DEFINED are the following. 1. If A : Rk 0 Rk is invertible, then the image of B : Rk 0 Rn is the same as the image of AB. 2. If A : Rn 0 Rn is invertible, then the nullspace of B : Rn 0 Rr is the same as the nullspace of BA. More precisely, Null( BA) /C30A/C281(Null( B)) :/ These techniques work in any dimension. In addition, they generalize to submanifolds of an ABSTRACT MANIFOLD , because tangent vectors depend on local properties. In particular, the tangent space can be computed in any coordinate chart, because any change in COORDINATE CHART corresponds to a DIF- FEOMORPHISM in Euclidean space. The tangent space can give some geometric insight to higher-dimensional phenomena. For example, to compute the tangent space to the FLAT TORUS (donut) M in R4 ; note that it can be parametrized, by fx1 ; x2 ðÞ /C30 sin x1 ; cos x1 ; sin x2 ; cos x2 ðÞ (8) with domain U /C30 x1 ; x2 ðÞ : x2 1 /C27x22 B1 fg ; near the point p /C30f(0; 0) /C30(0; 1; 0; 1): Its JACOBIAN at p is the matrix 10 00 01002 6643 775; (9) whose image is the tangent space TM p/C30f(a;0;b;0)g:iCk0iCk0/ Alternatively, Mis the set of solutions to equations f1x1;x2;x3;x4 ðÞ /C30x2 1/C27x22/C281/C300 (10) f2x1;x2;x3;x4 ðÞ /C30x23/C27x24/C281/C300: (11) The Jacobian at the solution p/C30(0;1;0;1) is the matrix0200 0002iC0jiC0k ; (12) whose NULLSPACE is the tangent space TMp/C30f(a;0;b;0)g:iCk0iCk0/ See also CALCULUS ,COORDINATE CHART ,DIFFEREN- TIAL FORM,D IRECTIONAL DERIVATIVE ,E UCLIDEAN SPACE ,EXTERIOR ALGEBRA ,JACOBIAN ,LINEAR ALGE- BRA,MANIFOLD ,NULLSPACE ,TANGENT BUNDLE ,TAN- GENT PLANE ,T ANGENT SPACE (CHART ), TANGENT SPACE (INTRINSIC ), TANGENT VECTOR ,VECTOR FIELD, VECTOR SPACE ,VELOCITY VECTOR Tangent Spheres A special case of tangent spheres is given by Soddy’s hexlet, which consists of a chain of six spheres externally tangent to two mutually tangent spheres and internally tangent to a circumsphere. The bendsof the circles in the chain obey the relationship 1 r1/C271 r4/C301 r2/C271 r3/C301 r3/C271 r6: (1) AS ANGAKU PROBLEM from 1798 asks to distribute 30 identical spheres of radius rsuch that they are tangent to a single central sphere of radius Rand to four other small spheres. This can be accomplished(left figure) by placing the spheres at the vertices ofan ICOSIDODECAHEDRON (right figure) of side length a, where the radii randRare given by r/C301 2a (2) R/C301 2ffiffiffi 5p a (3) (Rothman 1998). In general, the BENDS of five mutually tangent spheres are related by 3 k2 1 /C27 k22 /C27 k23 /C27 k24 /C27 k25iCjiCk /C30 k1 /C27 k2 /C27 k3 /C27 k4 /C27 k5 ðÞ2: (4) Solving for k5 gives k 95/C301 2k1 /C27 k2 /C27 k3 /C27 k4 f 96 k1 k2 /C27 k1 k3 /C27 k1 k4 /C27 k2 k3 /C27 k2 k4 /C27 k3 k4 ðÞ½ /C283 k2 1 /C27 k22 /C27 k23 /C27 k24iCjiCk /C1381 =2 g: (5) (Soddy 1937a). Gosset (1937) pointed out that the expression under the square root sign is given by 6 k1 k2 /C27 k1 k3 /C27 k1 k4 /C27 k2 k3 /C27 k2 k4 /C27 k3 k4 ðÞf /C283 k21 /C27 k22 /C27 k23 /C27 k24iCjiCk g1 =2 /C303ffiffiffi 3p V k1 k2 k3 k4 ; (6) where V is the VOLUME of the TETRAHEDRON having vertices at the centers of the corresponding four spheres. Therefore, the equation for k5 can be written simplify as k5 /C301 2 s2 /C27ffiffiffi 3p e ; (7) where s /C30 k1 /C27 k2 /C27 k3 /C27 k4 (8) e /C303 2 V k1 k2 k3 k4 : (9) (Soddy 1937b). In addition, the tetrahedra formed by joining the four points of contact of any one sphere with the other four (when all five are in mutual contact) have opposite edges whose product is the constant 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 /C27 k5 ðÞ k2 /C27 k5 ðÞ k3 /C27 k5 ðÞ k4 /C27 k5 ðÞp (10) and the volume of these tetrahedra is V /C302ffiffiffi 3pk5 k1 /C27 k5 ðÞ k2 /C27 k5 ðÞ k3 /C27 k5 ðÞ k4 /C27 k5 ðÞ(11) (Soddy 1937b). Gosper has further extended this result to n /C272 mutually tangent n-D HYPERSPHERES , whose CURVATURES satisfy Xn/C271 i/C300ki ! 2 /C28nXn/C271 i /C300k2 i /C300: (12) Solving for kn/C271 gives kn/C271 /C30ffiffiffinpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i /C300kiiCjiCk2/C28(n /C28 1)Pni/C300k2 i/C27Pn i /C300kiq n /C28 1 : (13) For (at least) n /C302 and 3, the RADICAL equals f(n)V k0 k1 /C1/C1/C1kn ; (14) where V is the CONTENT of the SIMPLEX whose vertices are the centers of the n/C271 independentHYPERSPHERES . The RADICAND can also become NEGA- TIVE, yielding an IMAGINARY kn/C271:For n/C303, this corresponds to a sphere touching three large bowling balls and a small BB, all mutually tangent, which isan impossibility. See also B OWL OF INTEGERS ,HEXLET ,SODDY CIRCLES , SPHERE ,TANGENT CIRCLES ,TETRAHEDRON References Gosset, T. "The Hexlet." Nature 139, 251/C1/252, 1937. Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85/C1/91, May 1998. Soddy, F. "The Kiss Precise." Nature 137, 1021, 1936. Soddy, F. "The Bowl of Integers and the Hexlet." Nature 139,7 7/C1/79, 1937a. Soddy, F. Nature 139, 252, 1937b. Tangent Vector For a curve with POSITION VECTOR r(t);the unit tangent vector ˆT(t) is defined by ˆT(t)/C13r?(t) r?(t)jj/C30dr dt dr dtiCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0(1) /C30dr dt ds dt(2) /C30dr ds; (3) where tis a parameterization variable and sis the ARC LENGTH . For a function given parametrically by (f(t);g(t));the tangent vector relative to the point (f(t);g(t)) is therefore given by x(t)/C30f?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f?2/C27g?2p (4) y(t)/C30g?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif?2/C27g?2p : (5) To actually place the vector tangent to the curve, it must be displaced by ( f(t);g(t)):It is also true that dˆT ds/C30kˆN (6) dˆT dt/C30kds dtˆN (7) [˙T;¨T; /C5T]/C30k5d dst k ! ; (8) where Nis the NORMAL VECTOR ,kis the CURVATURE , andtis the TORSION . See also CURVATURE ,N ORMAL VECTOR ,T ANGENT , TANGENT BUNDLE ,TANGENT PLANE ,TANGENT SPACE , TANGENT VECTOR (MANIFOLD ), TORSION (DIFFEREN- TIAL GEOMETRY ) References Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5 inModern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 108 /C1/111, 1997. Tangent Vector (Manifold) Roughly speaking, a tangent vector is an infinitesi- mal displacement at a specific point on a MANIFOLD . The set of tangent vectors at a point Pforms a VECTOR SPACE called the TANGENT SPACE atP, and the collection of tangent spaces on a manifold forms a VECTOR BUNDLE called the TANGENT BUNDLE . A tangent vector at a point Pon a manifold is a tangent vector at Pin a COORDINATE CHART . A change in coordinates near Pcauses an INVERTIBLE LINEAR MAP of the tangent vector’s representations in the coordinates. This transformation is given by theJ ACOBIAN , which must be nonsingular in a change of coordinates. Hence the tangent vectors at Pare WELL DEFINED .A VECTOR FIELD is an assignment of a tangent vector for each point. The collection oftangent vectors forms the TANGENT BUNDLE , and a vector field is a SECTION of this bundle. Tangent vectors are used to do CALCULUS onMANI- FOLDS . Since manifolds are locally Euclidean, the usual notions of differentiation and integration makesense in any COORDINATE CHART , and they can be carried over to manifolds. More specifically, a tangent vector is the manifold version of a DIRECTIONALDERIVATIVE (at a point). An alternative analogy with calculus is the related notion of a VELOCITY VECTOR . There are at least three different points of view on tangent vectors. Each has its own pluses and minuses. The extrinsic points of view use the vectorspace structure of E UCLIDEAN SPACE . Thinking of a manifold as a SUBMANIFOLD of Euclidean space, a tangent vector can be thought of as an element in a TANGENT PLANE , or (submanifold) TANGENT SPACE .I n aCOORDINATE CHART , a tangent vector is a vector in a (chart) TANGENT SPACE , which is just a copy of EUCLIDEAN SPACE . The problem with the extrinsic points of view is that they depend on a choice of EMBEDDING orCOORDINATE CHART . There are a couple of ways to think about a tangent vector intrinsically, as an element of an abstract (intrinsic) TANGENT SPACE . These are more satisfying from an abstract point of view, but some-times it is necessary to do calculations in coordinatecharts. It is important to distinguish tangent vectors at P from tangent vectors at any other point Q, although they may seem parallel. On a L IE GROUP , there is a notion of parallelism, and there exist nonvanishing vector fields. In general, this is far from being true. On the sphere S2;for instance, any smooth vector field must vanish somewhere. A more intrinsic geometric definition of a tangent vector is to take a tangent vector at Pto be an EQUIVALENCE CLASS of paths through Pwhich agree to first order. An extrinsic geometric definition, for a submanifold, is to view the tangent vectors as a subspace of the tangent vectors of the ambient space, Algebraically, a vector field on a manifold is a DERIVATION on the RING of smooth functions. That is, a vector field acts on smooth functions and satisfies the PRODUCT RULE . A vector field Xacts on a function by the DIRECTIONAL DERIVATIVE on the function, dX(f)/C30X /C2159f: (1) It is more precise to say that the tangent bundle is the SHEAF of derivations on the sheaf of smooth functions, in which case the tangent vectors at Pare in the STALK of the sheaf at P. In fact, in coordinates ( x1;...;xn);the notation for the standard basis of tangent vectors at 0 is @ @xi; (2) where the derivation @=@xioffis the usual PARTIAL DERIVATIVE @f @xi: (3) Letting the base point vary in the coordinate chart, @=@xiare vector fields, but are only defined in this COORDINATE CHART . See also CALCULUS ,COORDINATE CHART ,DERIVATION ALGEBRA ,D IFFERENTIAL FORM,D IRECTIONAL DERI- VATIVE ,EUCLIDEAN SPACE ,EXTERIOR ALGEBRA ,LIE GROUP MANIFOLD ,SHEAF (TOPOLOGY ), STALK ,TAN- GENT BUNDLE ,TANGENT VECTOR ,TANGENT SPACE , TANGENT SPACE (SUBMANI FOLD ), VECTOR FIELD , VELOCITY VECTOR Tangential Angle For a PLANE CURVE , the tangential angle f is defined by r d f /C30ds ; (1) where s is the ARC LENGTH and r is the RADIUS OF CURVATURE . The tangential angle is therefore given by f /C30gt 0s ?(t) k(t) dt; (2) where k(t) is the CURVATURE . For a plane curve r(t); the tangential angle f(t) can also be defined by r?(t) r?(t)jj/C30cos[f(t)] sin[f(t)]iC0jiC0k : (3) Gray (1997) calls f the TURNING ANGLE instead of the tangential angle. See also ARC LENGTH ,CURVATURE ,PLANE CURVE , RADIUS OF CURVATURE ,T ORSION (DIFFERENTIAL GEOMETRY ) References Gray, A. "The Turning Angle." §1.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 19 /C1/20, 1997. Tangential Polygon The polygon formed by the lines tangent to the CIRCUMCIRCLE of a polygon. The tangential polygon of an n-gon is itself an n-gon. See also DUAL POLYHEDRON ,TANGENTIAL QUADRI- LATERAL ,TANGENTIAL TRIANGLETangential Quadrilateral A QUADRILATERAL which has an INCIRCLE , i.e., one for which a single circle can be constructed which is tangent to all four sides. Opposite sides of such a quadrilateral satisfy s /C30a /C27c /C30b /C27d; (1) where s /C301 2(a /C27b /C27c /C27d) (2) is the SEMIPERIMETER , and the AREA is A /C30rs; (3) where r is the INRADIUS . See also BICENTRIC QUADRILATERAL ,CYCLIC QUAD- RILATERAL ,INCIRCLE ,Q UADRILATERAL ,TANGENTIAL TRIANGLE References Harris, J. W. and Stocker, H. "Quadrilateral of Tangents." §3.6.8 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 86, 1998. Tangential Tetrahedron The planes passing through the vertices of a TETRA- HEDRON ABCD and tangent to the CIRCUMSPHERE at these points form another tetrahedron called the tangential tetrahedron. The four lines of intersection of the faces of a tetrahedron with the corresponding faces of its tangential tetrahedron form a hyperbolic group (Alt- shiller-Court 1979, p. 102). See also TETRAHEDRON References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 102, 1979. Tangential Triangle The TRIANGLE DT1T2T3 formed by the lines tangent to the CIRCUMCIRCLE of a given TRIANGLE DA1A2A3 at its VERTICES . It is the PEDAL TRIANGLE of DA1A2A3with the CIRCUMCENTER as the PEDAL POINT . The TRI- LINEAR COORDINATES of the VERTICES of the tangen- tial triangle are T1 /C30/C28a : b : c T2 /C30a : /C28b : c T3 /C30a : b : /C28c : The CONTACT TRIANGLE and tangential triangle are perspective from the GERGONNE POINT . Given a TRIANGLE DA1A2A3and its tangential trian- gle DT1T2T3 ; the extensions of the sides of the two triangles intersect in three points L1 ; L2 ; and L3 ; which are collinear (Honsberger 1995). The CIRCUMCENTER of the tangential triangle has TRIANGLE CENTER FUNCTION a/C30ab2cos(2 B)/C27c2cos(2 C)/C28a2cos(2 ;4)iC0iCB and lies on the E ULER LINE (Kimberling 1994) See also CIRCUMCIRCLE ,C ONTACT TRIANGLE ,G ER- GONNE POINT ,PEDAL TRIANGLE ,PERSPECTIVE ,TAN- GENTIAL QUADRILATERALReferences Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 151 /C1/153, 1995. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/187, 1994. Tangents Law LAW OF TANGENTS Tangent-Sphere Coordinates A coordinate system ( m;n;c) given by the coordinate transformation x/C30mcosc m2/C27n2(1) y/C30msinc m2/C27n2(2) z/C30n m2/C27n2(3) and defined for m>0;n/C23(/C28/C12;/C12);and c/C23[0;2p): Surfaces of constant mare given by the TOROIDS x2/C27y2/C27z2/C301 mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p ; (4) surface of constant nby the spheres tangent to the xy- plane x2/C27y2/C27z/C281 2n !2 /C301 4n2; (5) and surfaces of constant cby the half-planes tanc/C30y x: (6) The metric coefficients are gxx /C301 m2 /C27 n2 ðÞ2 (7) gyy /C301 m2 /C27 n2 ðÞ2 (8) gzz /C30m2 m2 /C27 n2 ðÞ2 : (9) References Moon, P. and Spencer, D. E. "Tangent-Sphere Coordinate ( m; n ; c) :/" Fig. 4.01 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 104 /C1/ 106, 1988. Tangle A region in a KNOT or LINK projection plane sur- rounded by a CIRCLE such that the KNOT or LINK crosses the circle exactly four times. Two tangles are equivalent if a sequence of REIDEMEISTER MOVES can be used to transform one into the other while keeping the four string endpoints fixed and not allowing strings to pass outside the CIRCLE . The simplest tangles are the /C12/-tangle and 0-tangle, shown above. A tangle with n left-handed twists is called an n-tangle, and one with n right-handed twists is called a /C28n/-tangle. By placing tangles side by side, more complicated tangles can be built up such as ( /C282, 3, 2), etc. The link created by connecting the ends of the tangles is now described by the sequence of tangle symbols, known as CONWAY’S KNOT NOTATION . If tangles are multiplied by 0 and then added, the resulting tangle symbols are sepa- rated by commas. Additional symbols which are used are the period, colon, and asterisk. Amazingly enough, two tangles described in this NOTATION are equivalent IFF the CONTINUED FRAC- TIONS OF THE FORM 2 /C271 3 /C271 /C282 are equal (Burde and Zieschang 1985)! an ALGEBRAIC TANGLE is any tangle obtained by ADDITIONS andMULTIPLICATIONS of rational tangles (Adams 1994). Not all tangles are ALGEBRAIC . See also ALGEBRAIC LINK,FLYPE ,PRETZEL KNOT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman pp. 41 /C1/51, 1994. Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, 1985. Murasugi, K. and Kurpita, B. I. A Study of Braids. Dor- drecht, Netherlands: Kluwer, 1999. Tanglecube A QUARTIC SURFACE given by the implicit equation x4 /C285x2 /C27y4 /C285y2 /C27z4 /C285z2 /C2711:8 /C300: References Banchoff, T. "The Best Homework Ever?" http://www.brow- n.edu/Administration/Brown_Alumni_Magazine/97/12 /C1/ 96/features/homework.html. Nordstrand, T. "Tangle." http://www.uib.no/people/nfytn/ tangltxt.htm. Tangled Hierarchy A system in which a STRANGE LOOP appears. See also STRANGE LOOP References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 10, 1989. Tangram A combination of the above plane polygonal pieces such that the EDGES are coincident. There are 13 convex tangrams (where a "convex tangram" is a set of tangram pieces arranged into a CONVEX POLYGON ). See also ORIGAMI ,STOMACHION References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 19 /C1/20, 1989. Gardner, M. "Tangrams, Part 1" and "Tangrams, Part 2." Chs. 3 /C1/4i n Time Travel and Other Mathematical Bewil- derments. New York: W. H. Freeman, pp. 27 /C1/54, 1988. Johnston, S. Fun with Tangrams Kit: 120 Puzzles with Two Complete Sets of Tangram Pieces. New York: Dover, 1977. Johnston, S. Tangrams ABC Kit. New York: Dover. Pappas, T. "Tangram Puzzle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 212, 1989. Read, R. C. Tangrams: 330 Puzzles. New York: Dover. Tanh HYPERBOLIC TANGENT Taniyama Conjecture TANIYAMA- SHIMURA CONJECTURE Taniyama-Shimura Conjecture A very general and important conjecture (and now theorem) connecting TOPOLOGY and NUMBER THEORY which arose from several problems proposed by Taniyama in a 1955 international mathematics sym-posium. Let Ebe an ELLIPTIC CURVE whose equation has INTEGER COEFFICIENTS , let Nbe the so-called CON- DUCTOR ofEand, for each n, let anbe the number appearing in the L-function of E. Then, in technical terms, the Taniyama-Shimura conjecture states that there exists a MODULAR FORM of weight two and level Nwhich is an EIGENFORM under the H ECKE OPERA- TORS and has a F OURIER SERIES aanqn:/In effect, the conjecture says that every rational ELLIPTIC CURVE is a MODULAR FORM in disguise. Or, more formally, the conjecture suggests that, for every ELLIPTIC CURVE y2/C30Ax3/C27Bx2/C27Cx/C27Dover the RA- TIONALS , there exist nonconstant MODULAR FUNC- TIONS f(z) and g(z) of the same level Nsuch that [f(z)]2/C30A[g(z)]2/C27Cg(z)/C27D: Equivalently, for every ELLIPTIC CURVE , there is a MODULAR FORM with the same D IRICHLET L-SERIES . In 1985, starting with a fictitious solution to F ER- MAT’S LAST THEOREM (the F REY CURVE ), G. Frey showed that he could create an unusual ELLIPTIC CURVE which appeared not to be modular. If the curve were not modular, then this would show that ifF ERMAT’S LAST THEOREM were false, then the Ta- niyama-Shimura conjecture would also be false.Furthermore, if the Taniyama-Shimura conjecturewere true, then so would be F ERMAT’S LAST THEOREM ! However, Frey did not actually prove that his curve was not modular. The conjecture that Frey’s curve was not modular came to be called the " EPSILON CONJECTURE ," and was quickly proved by Ribet (RIBET’S THEOREM ) in 1986, establishing a very close link between two mathematical structures (the Ta-niyama-Shimura conjecture and F ERMAT’S LAST THE- OREM ) which appeared previously to be completely unrelated. As of the early 1990s, most mathematicians believed that the Taniyama-Shimura conjecture was not ac-cessible to proof. However, A. Wiles was not one of these. He attempted to establish the correspondence between the set of ELLIPTIC CURVES and the set of modular elliptic curves by showing that the number of each was the same. Wiles accomplished this by "counting" Galois representations and comparingthem with the number of MODULAR FORMS . In 1993, after a monumental seven-year effort, Wiles (almost) proved the Taniyama-Shimura conjecture for special classes of curves called SEMISTABLE ELLIPTIC CURVES (which correspond to elliptic curves with SQUAREFREE CONDUCTORS ; Knapp 1999). Wiles had tried to use horizontal Iwasawa theory tocreate a so-called CLASS NUMBER FORMULA , but was initially unsuccessful and therefore used instead an extension of a result of Flach based on ideas from Kolyvagin. However, there was a problem with thisextension which was discovered during review of Wiles’ manuscript in September 1993. Former stu- dent Richard Taylor came to Princeton in early 1994to help Wiles patch up this error. After additionaleffort, Wiles discovered the reason that the Flach/ Kolyvagin approach was failing, and also discovered that it was precisely what had prevented Iwasawatheory from working. With this additional insight, Wiles was able to successfully complete the erroneous portion of the proof using Iwasawa theory, proving the SEMISTABLE case of the Taniyama-Shimura conjecture (Taylor and Wiles 1995, Wiles 1995) and, at the same time, establishing FERMAT’S LAST THEOREM as a true theorem. The existence of a proof of the full Taniyama- Shimura conjecture was announced at a conference by Kenneth Ribet on June, 21 1999 (Knapp 1999), and reported on National Public Radio’s Weekend Edition on July 31, 1999. The proof was completed by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, building on the earlier work of Wiles and Taylor (Mackenzie 1999, Morgan 1999). The best previous published result held for all CONDUCTORS except those divisible by 27 (Conrad et al. 1999; Knapp 1999). The general Breuil et al. proof for all elliptic curves removed this restriction, in the process relying on Wiles’ proof for rational ELLIPTIC CURVES . See also CONDUCTOR ,E LLIPTIC CURVE ,E PSILON CONJECTURE ,FERMAT’S LAST THEOREM ,LANGLANDS PROGRAM ,M ODULAR FORM,M ODULAR FUNCTION , RIBET’S THEOREM References --. Science 285, 178, 1999. American Mathematical Society. http://www.ams.org/new- in-math/10 /C1/1999-media.html#fermat. Conrad, B.; Diamond, F.; and Taylor, R. "Modularity of Certain Potentially Barsotti-Tate Galois Representa- tions." J. Amer. Math. Soc. 12, 521 /C1/567, 1999. Darmon, H. "A Proof of the Full Shimura-Taniyama-Weil Conjecture is Announced." Not. Amer. Math. Soc. 46, 1397 /C1/1406, 1999. Ekeland, I. "Curves and Numbers." Nature 405, 748 /C1/749, 2000. Knapp, A. W. "Proof Announced of Taniyama-Shimura-Weil Conjecture." Not. Amer. Math. Soc. 46, 863, 1999. Lang, S. "Some History of the Shimura-Taniyama Conjec- ture." Not. Amer. Math. Soc. 42, 1301 /C1/1307, 1995. Mackenzie, D. "Fermat’s Last Theorem Extended." Science 285, 178, 1999. Morgan, F. "Frank Morgan’s Math Chat." http:// www.maa.org/features/mathchat/mathchat_7_1_99.html. July 1, 1999. Peterson, I. "Curving Beyond Fermat’s Last Theorem." Sci. News 156, 221, Oct. 2, 1999. Shimura, G. and Taniyama, Y. Complex Multiplication of Abelian Varieties and Its Applications to Number Theory. Tokyo: Mathematical Society of Japan, 1961. Taylor, R. and Wiles, A. "Ring-Theoretic Properties of Certain Hecke Algebras." Ann. Math. 141, 553 /C1/572, 1995. Wiles, A. "Modular Elliptic-Curves and Fermat’s Last Theorem." Ann. Math. 141, 443 /C1/551, 1995. Taniyama-Shimura Theorem TANIYAMA- SHIMURA CONJECTURE Tank CYLINDRICAL SEGMENTTantrix TANGENT INDICATRIX Tapering Function APODIZATION FUNCTION Tarry Point The point T at which the lines through the VERTICES of a TRIANGLE PERPENDICULAR to the corresponding sides of the first BROCARD TRIANGLE , are CONCUR- RENT . The Tarry point lies on the CIRCUMCIRCLE opposite the STEINER POINT S. It has TRIANGLE CENTER FUNCTION a /C30bc b4 /C27 c4 /C28 a2b2 /C28 a2c2 /C30sec(A /C27 v) ; where v is the BROCARD ANGLE . The SIMSON LINE of the Tarry point is PERPENDICULAR to the line OK, when Ois the CIRCUMCENTER and Kis the SYMME- DIAN POINT (Lachlan 1893; Johnson 1929; Honsberger 1995, p. 121). The Tarry point of the first B ROCARD TRIANGLE of a TRIANGLE DABC is the CIRCUMCENTER ofDABC (Honsberger 1995, pp. 120 /C1/121). See also BROCARD ANGLE ,B ROCARD TRIANGLES , CIRCUMCIRCLE ,S YMMEDIAN POINT ,S IMSON LINE, STEINER POINTS References Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 77, 1971. Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 102, 1913. Honsberger, R. "The Steiner Point and the Tarry Point." §10.5 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 119 /C1/124, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 281 /C1/282, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/187, 1994. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 81, 1893. Tarry-Escott Problem PROUHET- TARRY- ESCOTT PROBLEM Tarski’s Recursive Definition of Satisfaction SATISFACTION Tarski’s Theorem Portions of this entry contributed by ADAM STRZE- BONSKI Tarski’s theorem says that the first-order theory of reals with /C27;+;/C30; and > allows QUANTIFIER ELIMINA- TION . This property is stronger than DECIDABILITY . For example, the first-order theory of reals with /C27;+; and /C30 is decidable, but does not allow QUANTIFIER ELIMINATION . Tarski’s theorem means that a QUANTIFIED SYSTEM of real algebraic equations and inequalities is a SEMI- ALGEBRAIC SET (Strzebonski 2000). Although Tarski proved that QUANTIFIER ELIMINA- TION was possible, his method was totally impractical (Davenport and Heintz 1988). A much more efficient procedure for implementing QUANTIFIER ELIMINATION is called CYLINDRICAL ALGEBRAIC DECOMPOSITION .It was developed by Collins (1975) and is implemented in Mathematica 4.0 asCylindricalAlgebraicDe- composition . See also CYLINDRICAL ALGEBRAIC DECOMPOSITION , DECIDABLE ,QUANTIFIED SYSTEM ,QUANTIFIER ,QUAN- TIFIER ELIMINATION ,SEMIALGEBRAIC SET References Collins, G. E. "Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition." In Proc. 2nd GI Conf. Automata Theory and Formal Languages. New York: Springer-Verlag, pp. 134 /C1/183, 1975. Davenport, J. and Heintz, J. "Real Quantifier Elimination if Doubly Exponential." J. Symb. Comput. 5,29/C1/35, 1988. Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753 /C1/759, 1996. Tarski, A. "Sur les ensembles de´finissables de nombres re´els." Fund. Math. 17, 210 /C1/239, 1931. Tarski, A. "A Decision Method for Elementary Algebra and Geometry." RAND Corp. monograph, 1948. Tarski, A. A Decision Method for Elementary Algebra and Geometry, 2nd ed. Berkeley, CA: University of California Press, 1951. Tate Conjecture See also HODGE CONJECTURE References Deligne, P. "The Hodge Conjecture." http://www.clay- math.org/prize_problems/hodge.pdf. Tate, J. T. "Algebraic Cycles and Poles of Zeta Functions." In Arithmetical Algebraic Geometry (Proc. Conf. PurdueUniv., 1963). New York: Harper and Row, pp. 93 /C1/110, 1965. Tau Conjecture Also known as RAMANUJAN’S HYPOTHESIS . Ramanu- jan proposed that t(n) /C2O n11=2/C27 eiCjiCk ; where t(n) is the TAU FUNCTION . This was proven by Deligne (1974) in the course of proving the more general PETERSSON CONJECTURE . Deligne was awarded the FIELDS MEDAL for his proof. See also PETERSSON CONJECTURE ,TAU FUNCTION References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 and 140, 1997. Deligne, P. "La conjecture de Weil. I." Inst. Hautes E ´tudes Sci. Publ. Math. 43, 273/C1/307, 1974. Deligne, P. "La conjecture de Weil. II." Inst. Hautes E ´tudes Sci. Publ. Math. 52, 137/C1/252, 1980. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 169, 1999. Tau Function A function t(n) related to the DIVISOR FUNCTION sk(n); also sometimes called R AMANUJAN’S TAU FUNCTION .I t is defined via the F OURIER SERIES of the MODULAR DISCRIMINANT D(t) for t/C23H;where His the UPPER HALF-PLANE ,b y D(t)/C30(2p)12X/C12 n/C301t(n)e2pint(1) (Apostol 1997, p. 20). The tau function is also given by the C AUCHY PRODUCT t(n)/C308000 s3(s3 ðÞ (s3 fg (n)/C28147s5(s5 ðÞ (n); (2) /C3065 756s11(n)/C27691 756s5(n)/C28691 3Xn/C281 k/C301s5(k)s5(n/C28k);(3) where sk(n) is the DIVISOR FUNCTION (Apostol 1997, pp. 24 and 140). The tau function has GENERATING FUNCTION X/C12 n/C301t(n)xn/C30xY/C12 n/C3011/C28xnðÞ24; (4) and the first few values are 1, /C2824, 252, /C281472, 4830, ... (Sloane’s A000594). The tau function is given by the Mathematica command RamanujanTau [n]i n theMathematica add-on package NumberTheory‘R- amanujan‘ (which can be loaded with the command BBNumberTheory‘ ). Lehmer conjectured that t(n)"0 for all nand verified this fact for nB214928639999 (Apostol 1997, p. 22). /t(n) is also given by g(/C28x)/C30X/C12 n/C301(/C281)nt(n)xn(5) g(x2)/C30X/C12 n/C301t1 2niCkCiCkA xn(6) X/C12 n/C301t(n)xn/C30x1/C283x/C275x3/C287x6/C27...iCjiCk8: (7) Ewell (1999) gave the beautiful formulas t(4n/C272)/C30/C283X2n/C271 k/C30123b(2k)s3(Od(2 k)) /C29X4n/C282k/C272 j/C300(/C281)jr8(4n/C272/C282k/C28j)r8(j) (8) Xn k/C30123b(2k)s3(Od2 k)) /C29X2n/C271/C282k j/C300(/C281)jr8(2n/C271/C282k/C28j)r8(j)/C300 (9) t(4m)/C30/C28211t(m)/C283X2m k/C30123b(2k)s3(Od2 k)) /C29X4m/C282k j/C300(/C281)jr8(4m/C282k/C28j)r8(j) (10) t(2n/C271)/C30X2n/C271 k/C30123[b(2k)/C281]s3(Od2 k)) /C29X2n/C272/C282k j/C300(/C281)jr8(3n/C272/C282k/C28j)r8(j); (11) where b(n) is the exponent of the exact power of 2 dividing n, Od( n) is the ODD PART ofn,sk(n) is the DIVISOR FUNCTION ofn, and rk(n) is the SUM OF SQUARES FUNCTION . For PRIME p, tpn/C271iCjiCk /C30t(p)tpnðÞ/C28p11tpn/C281iCjiCk (12) forn]1;and tpanðÞ/C30t(p)tpa/C281niCjiCk /C28p11tpa/C282niCjiCk (13) fora]2 and ( n;p)/C301 (Mordell 1917; Apostol 1997, p. 92). In O RE’S CONJECTURE , the tau function appears as the number of DIVISORS ofn. Ramanujan conjectured and Mordell (1917) proved that if ( n;n?)/C301;then t(nn?)/C30t(n)t(n?): (14) More generally,t(n)t(n?)/C30X dj(n;n?)d11tnn? d2 ! ; (15) which reduces to the first form if ( n;n?)/C301 (Mordell 1917; Apostol 1997, p. 93). Ramanujan conjectured and Watson proved that t(n) is divisible by 691 for almost all n, specifically t(n)/C13s11(n) (mod 691) ; (16) where sk(n) is the DIVISOR FUNCTION (Wilton 1930, Apostol 1997, pp. 93 and 140) and 691 is the NUMERATOR of the B ERNOULLI NUMBER B12:/ Ramanujan (1920) showed that t(2n)/C130 (mod 2) (17) t(3n)/C130 (mod 3) (18) t(5n)/C130 (mod 5) (19) (Darling 1921; Wilton 1930), t(7n/C27m)/C130 (mod 7) (20) form/C300 or one the quadratic non-residues of 7, i.e., 3, 5, 6, and t(23n/C27m)/C130 (mod 23) (21) form/C300 or one the quadratic non-residues of 23, i.e., 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 (Mordell 1922;Wilton 1930). Ewell (1999) showed that t(4n)/C13t(n) (mod 3) : (22) /t(n) is almost always divisible by 25/C21533/C21552/C21572/C21523 /C215 691 according to Ramanujan. In fact, Serre has shown that t(n) is almost always divisible by any integer (Andrews et al. 1988). Ramanujan also studied the D IRICHLET L-SERIES f(x)/C13X/C12 n/C301t(n)n/C28s; (23) which has properties analogous to the R IEMANN ZETA FUNCTION . It satisfies f(s)G(s) (2p)s/C30f(12/C28s) (2p)12/C28s: (24) It also has the Euler product representation X/C12 n/C301t(n) ns/C30Y p1 1/C28t(p)p/C28s/C27p11/C282s(25) fors/C30R[s]>7 (since t(n)/C30O(n6)) (Apostol 1997, p. 137). Ramanujan’s TAU-DIRICHLET SERIES conjec- ture alleges that all nontrivial zeros of f(s) lie on the lineR[s]/C306:fcan be split up into f(6/C27it)/C30z(t)e/C28iu(t); (26) where z(t) /C30G(6 /C27it)f(6 /C27it)(2 p) /C28it /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh( pt) pt 1 /C27 t2 ðÞ 4 /C27 t2 ðÞ 9 /C27 t2 ðÞ 16 /C27 t2 ðÞ 25 /C27 t2 ðÞs (27) u(t) /C30/C281 2 i lnG(6 /C27 it) G(6 /C28 it)"# /C28t ln(2p) : (28) The functions f(s) ; u(t) ; and z(t) are returned by the Mathematica commands RamanujanTauDiri- chletSeries [s] in the Mathematica add-on package NumberTheory‘Ramanujan‘ (which can be loaded with the command BBNumberTheory‘ ), Ramanu- janTauTheta [t] in the Mathematica add-on package NumberTheory‘Ramanujan‘ (which can be loaded with the command BBNumberTheory‘ ), and Ra- manujanTauZ [t] in the Mathematica add-on package NumberTheory‘Ramanujan‘ (which can be loaded with the command BBNumberTheory‘ ), respec- tively. The SUMMATORY tau function is given by T(n) /C30X n5xt(n) : (29) Here, the prime indicates that when x is an INTEGER , the last term t(x) should be replaced by1 2 t(x) :/ Ramanujan’s tau theta function Z(t)isa REAL func- tion for REAL t and is analogous to the RIEMANN- SIEGEL FUNCTION Z. The number of zeros in the critical strip from t /C300toT is given by N(t) /C30U(T) /C27T ln tDS(6 /C27 iT) ½/C138fg p ; (30) where U is the RIEMANN THETA FUNCTION and tDS is the TAU-DIRICHLET SERIES , defined by tDS(s) /C13X/C12 n/C301t(n) ns: (31) Ramanujan conjectured that the nontrivial zeros of the function are all real. Ramanujan’s tzfunction is defined by tz(t)/C30G(6/C27it)(2p)/C28it tDS(6/C27it)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh( pt) ptQ5 k/C301k2/C27t2s ; (32) where tDS(z) is the TAU-DIRICHLET SERIES . See also DEDEKIND ETA FUNCTION , J-FUNCTION , LEECH LATTICE ,O RE’S CONJECTURE ,P ARTITION FUNCTION P,TAU CONJECTURE ,TAU-DIRICHLET SER- IESReferences Andrews, G. E.; Berndt, B. C.; and Rankin, R. A. (Eds.). Ramanujan Revisited: Proceedings of the Centenary Con- ference New York: Academic Press, 1988. Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 20 /C1/21 and 51, 1997. Darling, H. B. C. Proc. London Math. Soc. 19, 350/C1/372, 1921. Ewell, J. A. "New Representations of Ramanujan’s Tau Function." Proc. Amer. Math. Soc. 128, 723/C1/726, 1999. Hardy, G. H. "Ramanujan’s Function t(n):/" Ch. 10 in Rama- nujan: Twelve Lectures on Subjects Suggested by His Lifeand Work, 3rd ed. New York: Chelsea, p. 63, 1999. Keiper, J. "On the Zeros of the Ramanujan t /-Dirichlet Series in the Critical Strip." Math. Comput. 65, 1613 /C1/1619, 1996. LeVeque, W. J. §F35 in Reviews in Number Theory 1940 /C1/ 1972. Providence, RI: Amer. Math. Soc., 1974. Lehmer, D. H. "Ramanujan’s Function t(n):/"Duke Math. J. 10, 483/C1/492, 1943. Moreno, C. J. "A Necessary and Sufficient Condition for the Riemann Hypothesis for Ramanujan’s Zeta Function."Illinois J. Math. 18, 107/C1 /114, 1974. Mordell, L. J. "On Mr. Ramanujan’s Empirical Expansions of Modular Functions." Proc. Cambridge Phil. Soc. 19, 117/C1/124, 1917. Mordell, L. J. "Note on Certain Modular Relations Consid- ered by Messrs Ramanujan, Darling, and Rogers." Proc. London Math. Soc. 20, 408/C1/416, 1922. Ramanujan, S. Proc. London Math. Soc. 18, 1920. Ramanujan, S. "Congruence Properties of Partitions." Math. Z.9, 147/C1/153, 1921. Sivaramakrishnan, R. Classical Theory of Arithmetic Func- tions. New York: Dekker, pp. 275 /C1/278, 1989. Sloane, N. J. A. Sequences A000594/M5153 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Spira, R. "Calculation of the Ramanujan Tau-Dirichlet Series." Math. Comput. 27, 379/C1 /385, 1973. Stanley, G. K. "Two Assertions Made by Ramanujan." J. London Math. Soc. 3, 232/C1/237, 1928. Stanley, G. K. Corrigendum to "Two Assertions Made by Ramanujan." J. London Math. Soc. 4, 32, 1929. Watson, G. N. "U ¨ber Ramanujansche Kongruenzeigenschaf- ten der Zerfa ¨llungsanzahlen." Math. Z. 39, 712/C1/731, 1935. Wilton, J. R. "Congruence Properties of Ramanujan’s Func- tiont(n):/"Proc. London Math. Soc. 31,1/C1/17, 1930. Yoshida, H. "On Calculations of Zeros of L-Functions Related with Ramanujan’s Discriminant Function on theCritical Line." J. Ramanujan Math. Soc. 3,8 7/C1 /95, 1988. Tauberian Theorem A Tauberian theorem is a theorem which deduces the convergence of an INFINITE SERIES on the basis of the properties of the function it defines and any kind of auxiliary HYPOTHESIS which prevents the general term of the series from converging to zero too slowly. Hardy (1999, p. 46) states that "a ‘Tauberian’ theo-rem may be defined as a corrected form of the false converse of an ‘A BELIAN THEOREM ’." Wiener’s Tauberian theorem states that if f/C23L1(R); then the translates of fspans a dense subspace IFF the F OURIER TRANSFORM is nonzero everywhere. This theorem is analogous with the theorem that if f/C23 L1(Z) (for a BANACH ALGEBRA with a unit), then f spans the whole space if and only if the GELFAND TRANSFORM is nonzero everywhere. See also ABELIAN THEOREM ,H ARDY- LITTLEWOOD TAUBERIAN THEOREM References Bromwich, T. J. I’a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 256, 1991. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 31 and 46, 1999. Katznelson, Y. An Introduction to Harmonic Analysis. New York: Dover, 1976. Wiener, N. The Fourier Integral and Certain of Its Applica- tions. New York: Dover, 1951. Tau-Dirichlet Series tDS(s) /C13X/C12 n/C301t(n) ns; where t(n) is the TAU FUNCTION . Ramanujan conjec- tured that all nontrivial zeros of tDS(s) lie on the line R[s] /C306:/ See also TAU FUNCTION References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1959. Keiper, J. "On the Zeros of the Ramanujan t/-Dirichlet Series in the Critical Strip." Math. Comput. 65, 1613 /C1/1619, 1996. Spira, R. "Calculation of the Ramanujan Tau-Dirichlet Series." Math. Comput. 27, 379 /C1/385, 1973. Yoshida, H. "On Calculations of Zeros of L-Functions Related with Ramanujan’s Discriminant Function on the Critical Line." J. Ramanujan Math. Soc. 3,87/C1/95, 1988. Tautochrone Problem The problem of finding the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a CYCLOID , a fact first discovered and published by Huygens in Horologium oscillatorium (1673). This property was also alluded to in the following passage from Moby Dick : "[The try- pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod , with the soapstone diligently circling round me, that I was first indirectly struck by the remark- able fact, that in geometry all bodies gliding along acycloid, my soapstone, for example, will descend from any point in precisely the same time" (Melville 1851). Huygens also constructed the first pendulum clock with a device to ensure that the pendulum was isochronous by forcing the pendulum to swing in an arc of a CYCLOID . This is accomplished by placing two evolutes of inverted cycloid arcs on each side of the pendulum’s point of suspension against which the pendulum is constrained to move (Wells 1991, p. 47; Gray 1997, p. 123). Unfortunately, friction along the arcs causes a greater error than that corrected by thecycloidal path (Gardner 1984). The PARAMETRIC EQUATIONS of the CYCLOID are x/C30a(u/C28sinu) (1) y/C30a(1/C28cosu): (2) To see that the CYCLOID satisfies the tautochrone property, consider the derivatives x?/C30a(1/C28cosu) (3) y?/C30asinu; (4) and x?2/C27y?2/C30a21/C282 cos u/C27cos2uiCjiCk /C27sin2uiC0iCB /C302a2(1/C28cosu): (5) Now 1 2mv2/C30mgy (6) v/C30ds dt/C30ffiffiffiffiffiffiffiffi 2gyp (7) dt/C30dsffiffiffiffiffiffiffiffi2gyp/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2/C27dy2p ffiffiffiffiffiffiffiffi2gyp /C30affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(1/C28cosu)p duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ga(1/C28cosu)p /C30ffiffiffi a gs du; (8) so the time required to travel from the top of the CYCLOID to the bottom is T/C30gp 0dt/C30ffiffiffi a gs p: (9) However, from an intermediate point u0; v/C30ds dt/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gy/C28y0 ðÞp ; (10) so T/C30gp u0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2(1/C28cosu) 2agcosu0/C28cosu ðÞs du /C30ffiffiffi a gs g p u0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 cos u cos u0 /C28 cos us du : (11) To integrate, rearrange this equation using the HALF- ANGLE FORMULAS sin1 2 xiCkCiCkA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 cos x 2s (12) cos12 xiCkCiCkA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 cos x 2s (13) with the latter rewritten in the form cos u /C302 cos212 uiCkCiCkA /C281 (14) to obtain T /C30ffiffiffiffiffi a gs g p u0sin12 uiCkCiCkA du ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos21 2 u0iCkCiCkA /C28 cos212 uiCkCiCkAr : (15) Now transform variables to u /C30cos12 uiCkCiCkA cos1 2 u0iCkCiCkA (16) du /C30/C28sin1 2 uiCkCiCkA du 2 cos1 2 u0iCkCiCkA ; (17) so T /C30/C282ffiffiffi a gs g0 1duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 u2p /C302ffiffiffi a gs sin/C281 uiC0iCB 1 0/C30 pffiffiffiffiffi a g ;s (18) and the amount of time is the same from any point. See also BRACHISTOCHRONE PROBLEM ,CYCLOID References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 129 /C1/130, 1984. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997. Lagrange, J. L. "Sue les courbes tautochrones." Me´m. de l’Acad. Roy. des Sci. et Belles-Lettres de Berlin 21, 1765. Reprinted in Oeuvres de Lagrange, tome 2, section deux- ie`me: Me´moires extraits des recueils de l’Academie royale des sciences et Belles-Lettres de Berlin. Paris: Gauthier- Villars, pp. 317 /C1/332, 1868. Melville, H. "The Tryworks." Ch. 96 in Moby Dick. New York: Bantam, 1981. Originally published in 1851. Muterspaugh, J.; Driver, T.; and Dick, J. E. "The Cycloid and Tautochronism." http://php.indiana.edu/~jedick/pro- ject/intro.html. Muterspaugh, J.; Driver, T.; and Dick, J. E. "P221 Tauto- chrone Problem." http://php.indiana.edu/~jedick/project/ project.html.Phillips, J. P. "Brachistochrone, Tautochrone, Cycloid--Ap- ple of Discord." Math. Teacher 60, 506 /C1/508, 1967. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 54 /C1/60 and 384 /C1/385, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 46 /C1/47, 1991. Tautology A logical statement in which the conclusion is equivalent to the premise. If p is a tautology, it is written ffip: A SENTENCE whose TRUTH TABLE contains only ‘T’ is called a tautology. The following SEN- TENCES are examples of tautologies: A fflB /C13!(!A /C150!B) (1) A /C150B /C13!A [B (2) A fflB /C13!(A [!B) (3) (Mendelson 1997, p. 26), where ffl denotes AND, /C13 denotes "is EQUIVALENT to," ! denotes NOT, /C150denotes OR, and [denotes implies. See also CONTINGENCY ,CONTRADICTION References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 13, 1958. Mendelson, E. "Tautology." §1.2 in Introduction to Mathe- matical Logic, 4th ed. London: Chapman & Hall, pp. 17 /C1/ 24, 1997. Taxicab Number Thenth taxicab number Ta( n) is the smallest number representable in nways as a sum of POSITIVE CUBES . The numbers derive their name from the H ARDY- RAMANUJAN NUMBER Ta(2)/C301729 /C3013/C27123 /C3093/C27103; (1) which is associated with a story told about Ramanu- jan by G. H. Hardy (Hofstadter 1989, Kanigel 1991, Snow 1993). However, this property was also known as early as 1657 by F. de Bessy (Berndt and Bhargava 1993, Guy 1994). Leech (1957) found Ta(3)/C3087539319 /C301673/C274363 /C302283/C274233 /C302553/C274143: (2) Rosenstiel et al. (1991) recently found Ta(4) /C306963472309248 /C3024213 /C27190833 /C3054363 /C27189483 /C30102003 /C27180723 /C30133223 /C27166303 : (3) D. Wilson found Ta(5) /C3048988659276962496 /C30387873 /C273657573 /C301078393 /C273627533 /C302052923 /C273429523 ¼ 2214243 þ 3365883 /C302315183 /C273319543 : (4) The first few taxicab numbers are therefore 2, 1729, 87539319, 6963472309248, ... (Sloane’s A011541). Hardy and Wright (Theorem 412, 1979) show that the number of such sums can be made arbitrarily large but, updating Guy (1994) with Wilson’s result, the least example is not known for six or more equal sums. Sloane defines a slightly different type of taxicab numbers, namely numbers which are sums of two cubes in two or more ways, the first few of which are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ... (Sloane’s A001235). See also DIOPHANTINE EQUATION–3RD POWERS , HARDY- RAMANUJAN NUMBER References Berndt, B. C. and Bhargava, S. "Ramanujan--For Low- brows." Am. Math. Monthly 100, 645 /C1/656, 1993. Guy, R. K. "Sums of Like Powers. Euler’s Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139 /C1/144, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 12 and 68, 1999. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 564, 1989. Kanigel, R. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Washington Square Press, p. 312, 1991. Leech, J. "Some Solutions of Diophantine Equations." Proc. Cambridge Phil. Soc. 53, 778 /C1/780, 1957.Plouffe, S. "Taxicab Numbers." http://www.lacim.uqam.ca/ pi/problem.html. Rosenstiel, E.; Dardis, J. A.; and Rosenstiel, C. R. "The Four Least Solutions in Distinct Positive Integers of the Diophantine Equation /s ¼ x3 þ y3 ¼ z3 þ w3 ¼ u3 þ v3/ /¼ m3 þ n3/." Bull. Inst. Math. Appl. 27, 155 /C1/157, 1991. Silverman, J. H. "Taxicabs and Sums of Two Cubes." Amer. Math. Monthly 100, 331 /C1/340, 1993. Sloane, N. J. A. Sequences A001235 and A011541 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Snow, C. P. Foreword to A Mathematician’s Apology, rep- rinted with a foreword by C. P. Snow (by G. H. Hardy). New York: Cambridge University Press, p. 37, 1993. Wooley, T. D. "Sums of Two Cubes." Internat. Math. Res. Not. No. 4, 181 /C1/184, 1995. Taylor Center The center of the TAYLOR CIRCLE , which is the SPIEKER CENTER of DH1H2H3;where Hiare the feet of the ALTITUDES . See also ALTITUDE ,SPIEKER CENTER ,TAYLOR CIRCLE References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 277, 1929. Taylor Circle From the feet HA;HB;andHCof each ALTITUDE of a TRIANGLE , draw lines PERPENDICULAR to the adjacent sides. Then the CIRCUMCIRCLE of the triangle formed by the PERPENDICULAR FEET is called the Taylor circle, and its center is called the T AYLOR CENTER . The Taylor circle is a T UCKER CIRCLE . There are a number of remarkable properties satis- fied by the figure obtained in the construction of the Taylor circle. These facts are probably well-known, but I have not seen them explicitly described else- where. 1. The feet of the perpendiculars from a given altitude foot are concyclic with the opposite vertex. 2. The two feet of the perpendiculars which are closest to a given vertex are concyclic with the feet of the altitudes on the corresponding sides. 3. The two feet of the perpendiculars which are closest to a give vertex are concyclic with thatvertex and with the intersection of the perpendi- culars. 4. The three circles through the ORTHOCENTER and the feet of the perpendiculars on a given side intersect pairwise along the altitudes. See also TAYLOR CENTER ,TUCKER CIRCLES References Casey, J. "Lemoine’s, Tucker’s, and Taylor’s Circle." Supp. Ch. §3i nA Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.Dublin: Hodges, Figgis, & Co., pp. 179 /C1 /189, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 71 /C1/73, 1971. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 277, 1929. Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 78, 1893. Taylor, H. M. Proc. London Math. Soc. 15. Taylor Expansion TAYLOR SERIES Taylor Polynomial TAYLOR SERIES Taylor Series A Taylor series is a SERIES EXPANSION of a FUNCTION about a point. A 1-D Taylor series is an expansion of a REAL FUNCTION f(x) about a point x/C30x0(sometimes written instead x/C30a). If x/C300, the expansion is known as a M ACLAURIN SERIES . To derive the Taylor series of a function f(x);note that the integral of the ( n/C271)/stDERIVATIVE f(n/C271)off(x) from the point x0to an arbitrary point xis given by gx x0f(n/C271)(x)dx/C30f(n)(x)iC0iCB x x0/C30f(n)(x)/C28f(n)x0ðÞ; (1) where f(n)x0ðÞis the nth derivative of f(x) evaluated at x0;and is therefore simply a constant. Now integrate a second time to obtain gx x0gx x0f(n/C271)(x)dx"# dx /C30gx x0f(n)(x)/C28f(n)x0ðÞiC0iCB dx /C30f(n/C281)(x)iC0iCB x x0/C28x/C28x0 ðÞ f(n)x0ðÞ /C30f(n/C281)(x)/C28f(n/C281)x0ðÞ/C28x/C28x0 ðÞ f(n)x0ðÞ; (2) where f(k)x0ðÞis again a constant. Integrating a third time, gggx x0f(n/C271)(x)(dx)3/C30f(n/C282)(x)/C28f(n/C282)(x0) /C28x/C28x0 ðÞ f(n/C281)x0ðÞ/C28x/C28x0 ðÞ2 2!f(n)x0ðÞ; (3) and continuing up to n/C271 integrations then gives g/C1/C1/C1gx x0|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n/C271f(n/C271)(x)(dx)n/C271 /C30f(x)/C28fx0ðÞ/C28x/C28x0 ðÞ f?x0ðÞ/C28x/C28x0 ðÞ2 2!fƒx0ðÞ /C28.../C28x/C28x0 ðÞn n!f(n)x0ðÞ: (4) Rearranging then gives the one-dimensional Taylor series f(x)/C30fx0ðÞ/C27x/C28x0 ðÞ f?x0ðÞ/C27x/C28x0 ðÞ2 2!fƒx0ðÞ/C27... /C27x/C28x0 ðÞn n!f(n)x0ðÞ/C27Rn; (5) /C30Xn k/C300x/C28x0 ðÞkf(k)x0ðÞ k!/C27Rn: (6) Here, Rnis a remainder term known as the L A- GRANGE REMAINDER , which is given by Rn/C30g/C1/C1/C1gx x0|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n/C271f(n/C271)(x)(dx)n/C271: (7) Rewriting the MULTIPLE INTEGRAL then gives Rn/C30gx x0f(n/C271)(t)(x/C28t)n n!dt: (8) Now, from the MEAN-VALUE THEOREM for a function g(x);it must be true that gx x0g(x)dx/C30x/C28x0 ðÞ gx/C31ðÞ (9) for some x/C31/C23x0;x ½/C138 :Therefore, integrating n/C271 times gives the result Rn/C30x/C27x0 ðÞn/C271 (n/C271)!f(n/C271)x/C31ðÞ ; (10) so the maximum error after nterms of the Taylor series is the maximum value of (10) running throughallx/C31/C23x 0;x ½/C138 :Note that the Lagrange remainder Rn is also sometimes taken to refer to the remainder when terms up to the ( n/C281)/st power are taken in theTaylor series (Whittaker and Watson 1990, pp. 95 /C1/ 96). An alternative form of the 1-D Taylor series may be obtained by letting x/C28x0/C13Dx (11) so that x/C13x0/C27Dx: (12) Substitute this result into (5) to give fx0/C27Dx ðÞ /C30fx0ðÞ/C27Dxf?(x0)/C271 2!(Dx)2fƒx0ðÞ/C27...:(13) A Taylor series of a REAL FUNCTION in two variables f(x;y) is given by f(x/C27Dx;y/C27Dy)/C30f(x;y)/C27[fx(x;y)Dx/C27fy(x;y)Dy] /C271 2![(Dx)2fxx(x;y)/C272DxDyfxy(x;y)/C27(Dy)2fyy(x;y)] /C271 3![(Dx)3fxxx(x;y)/C273(Dx)2Dyfxxy(x;y) /C273Dx(Dy)2fxyy(x;y)/C27(Dy)3fyyy(x;y)]/C27...: (14) This can be further generalized for a REAL FUNCTION innvariables, fx1;...;xn ðÞ /C30X/C12 j/C3001 j!Xn k/C301x?k/C28ak ðÞ@ @x?k"#j fx?1;...;x?n ðÞ8 < :9 = ; x?1/C30a1;...;x?n/C30an: ð15Þ Rewriting, fx1/C27a1;...;xn/C27an ðÞ /C30X/C12 j/C3001 j!Xn k/C301ak@ @x?k"#j fx?1;...;x?n ðÞ8 < :9 = ; x?1/C30a1;...;x?n/C30an: Taking n/C302 in (15) gives fx1;x2 ðÞ /C30X/C12 j/C300iC0C1 j!iC0j x?1/C28a1 ðÞ@ @x?1 /C27x?2/C28a2 ðÞ@ @x?2iC0kj fx?1;x?2 ðÞiC0A x?1/C30x1;x?2/C30x2 /C30fa1;a2 ðÞ /C27iC0j x1/C28a1 ðÞ@f @x1/C27x2/C28a2 ðÞ@f @x2iC0k /C271 2!iC0j x1/C28a1 ðÞ2@2f @x2 1/C272x1/C28a1 ðÞ x2/C28a2 ðÞ@2f @x1@x2 /C27 x2 /C28a2 ðÞ2@2f @x2 2iC0k /C27...: (17) Taking n /C303 in (16) gives fx1 /C27a1 ; x2 /C27x2 /C27a2 ; x3 /C27a3 ðÞ /C30X/C12 j/C300iC0C1 j!iCkn a1@ @x?1/C27a2@ @x?2/C27a3@ @x?3iCkoj /C2fx?1 ; x?2 ; x?3 ðÞiC0A x?1/C30x1 ; x?2/C30x2 ; x ?3/C30x3; (18) or, in VECTOR form f(r /C27a) /C30X/C12 j/C3001 j!a /C2159r? ðÞjf(r?)"# r?/C30r(19) The zeroth- and first-order terms are f(r) (20) and a /C2159r ? ðÞ f r?ðÞjr?/C30r ; (21) respectively. The second-order term is 1 2a /C2159r ? ðÞ a /C2159r ? ðÞ f(r?) jr?/C30r /C3012 a /C2159r ? a /C215 ( 9f(r?)) ½/C138r ?/C30r /C3012 a /C215 a /C2159r ?9r ?f r?ðÞ ðÞ ½/C138r?/C30r ; j (22) so the first few terms of the expansion are f(r /C27a) /C30f(r) /C27 a /C2159r ? ðÞ f(r?) jr ?/C30r /C2712 a /C215 a /C2159r ?9r ?f r?ðÞ ðÞ ½/C138 jr?/C30r : (23) Taylor series can also be defined for functions of a COMPLEX variable. By the CAUCHY INTEGRAL FOR- MULA , f(z) /C301 2pi gCf(z ?) dz z?/C28z/C301 2pi gCf(z ?) dz? z?/C28z0 ðÞ /C28 z /C28 z0 ðÞ /C301 2pi gCf(z?) dz ? z?/C28z0 ðÞ 1 /C28z /C28 z0 z?/C28z0 ! : (24) In the interior of C, z /C28 z0 jj z ?/C28z0 jjB1 (25) so, using 1 1 /C28 t /C30X/C12 n /C300tn ; (26) it follows thatf(z) /C301 2pi gCX/C12 n/C300z/C28z0 ðÞnf(z?)dz? z?/C28z0 ðÞn/C271 /C301 2piX/C12 n/C300z/C28z0 ðÞngCf(z?)dz z?/C28z0 ðÞn/C271: (27) Using the C AUCHY INTEGRAL FORMULA for deriva- tives, f(z)/C30X/C12 n/C300z/C28z0 ðÞnf(n)z0ðÞ n!: (28) See also CAUCHY REMAINDER ,LAGRANGE EXPANSION , LAGRANGE REMAINDER ,LAURENT SERIES ,LEGENDRE SERIES ,M ACLAURIN SERIES ,N EWTON’S FORWARD DIFFERENCE FORMULA ,TAYLOR’S THEOREM References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972. Arfken, G. "Taylor’s Expansion." §5.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 303 /C1/313, 1985. Comtet, L. "Calcul pratique des coefficients de Taylor d’une fonction alge ´brique." Enseign. Math. 10, 267/C1/270, 1964. Morse, P. M. and Feshbach, H. "Derivatives of Analytic Functions, Taylor and Laurent Series." §4.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 374 /C1/398, 1953. Whittaker, E. T. and Watson, G. N. "Forms of the Remain- der in Taylor’s Series." §5.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 95 /C1/96, 1990. Taylor-Greene-Chirikov Map STANDARD MAP Taylor’s Condition For a given POSITIVE INTEGER n, does there exist a WEIGHTED TREE with nVERTICES whose paths have weights 1, 2, ...,n 2iCjiCk ;wheren 2iCjiCk is a BINOMIAL COEFFICIENT ? Taylor showed that no such TREE can exist unless it is a PERFECT SQUARE or a PERFECT SQUARE plus 2. No such TREES are known except n /C302, 3, 4, and 6. See also GOLOMB RULER ,PERFECT DIFFERENCE SET, TREE References Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 56 /C1/60, 1985. Leech, J. "Another Tree Labeling Problem." Amer. Math. Monthly 82, 923 /C1/925, 1975. Taylor, H. "Odd Path Sums in an Edge-Labeled Tree." Math. Mag. 50, 258 /C1/259, 1977. Taylor’s Theorem The theorem that a function may be represented by a TAYLOR SERIES , f(x) /C30f(0) /C27xf ?(0) /C27x2 2!f ƒ(0) /C27.../C27xn /C281 (n /C28 1)!f(n/C281)(0) /C27gx 0(x /C28 u)n/C281 (n /C28 1)!f(n)(u) du : Taylor’s theorem without the remainder was first devised by Taylor in 1712 and published in 1915, but it was not until almost a century later than Lagrange and Cauchy derived approximations of the remainder term after a finite number of terms (Moritz 1937). These forms are now called the LAGRANGE REMAIN- DER and CAUCHY REMAINDER . Most modern proofs are based on Cox (1851), which is more elementary than that of Cauchy and Lagrange (Moritz 1923), and which Pringsheim (1900) referred to as "leaving hardly anything to wish for in terms of simplicity and strength" (Moritz 1923). See also CAUCHY REMAINDER ,LAGRANGE REMAINDER , TAYLOR SERIES References Cox, H. Cambridge and Dublin Math. J. 6, 80, 1851. Jeffreys, H. and Jeffreys, B. S. "Taylor’s Theorem." §1.133 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 50 /C1/51, 1988. Moritz, R. E. "A Note on Taylor’s Theorem." Amer. Math. Monthly 44,31/C1/33, 1937. Pringsheim. Bibliotheca Math. 1, 455, 1900. Todhunter, I. A Treatise on the Differential Calculus with Numerous Examples, 10th ed. London: Macmillan, p. 75, 1890. Tchebycheff CHEBYSHEV APPROXIMATION FORMULA ,C HEBYSHEV CONSTANTS ,CHEBYSHEV DEVIATION ,CHEBYSHEV DIF- FERENTIAL EQUATION ,CHEBYSHEV FUNCTIONS ,CHE- BYSHEV- GAUSS QUADRATURE ,C HEBYSHEV INEQUALITY ,C HEBYSHEV INEQUALITY ,C HEBYSHEV INTEGRAL ,C HEBYSHEV PHENOMENON ,C HEBYSHEV POLYNOMIAL OF THE FIRST KIND,CHEBYSHEV POLY-NOMIAL OF THE SECOND KIND,CHEBYSHEV QUADRA- TURE ,CHEBYSHEV- RADAU QUADRATURE ,CHEBYSHEV- SYLVESTER CONSTANT t-Design See also STEINER SYSTEM t-Distribution STUDENT’S T-DISTRIBUTION Teardrop Curve A plane curve given by the PARAMETRIC EQUATIONS x /C30cos t y /C30sin t sinm1 2 tiCkCiCkA : See also PEAR-SHAPED CURVE References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 174, 1993. Technique A specific method of performing an operation. The terms ALGORITHM , METHOD , and PROCEDURE are also used interchangeably. See also ALGORITHM ,METHOD ,PROCEDURE Teeko A game described by Scarne which is played on a 5 /C29 5 board by two players who alternate placing, one at a time, their four counters each, after which the counters are moved around (including diagonally).Four counters in a row or square wins (Beeler et al. 1972). In general, there are sixteen forms of the game, all of which were solved completely by Guy Steele in 1998 with the following results: standardteeko (44 winning configurations) is a draw, and advanced teeko (58 winning configurations) is a first-player win. Here is a more complete summary of the results. Variant Winner standard draw alternate draw one-move alternate drawtwo-move alternate drawthree-move alternate draw one-move standard draw two-move standard draw three-move standard draw standard, 58 positions first-player win (13 turns) alternate, 58 positions draw one-move alternate, 58 positionsdraw two-move alternate, 58 positionsdraw three-move alternate, 58 positionsdraw one-move standard, 58 positionsfirst-player win (25 turns) two-move standard, 58 positionsdraw three-move standard, 58 positionsdraw References Beeler, M. et al. Item 90 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 35, Feb. 1972. Teichmu ¨ ller Space TEICHMU ¨ LLER’S THEOREM asserts the EXISTENCE and UNIQUENESS of the extremal quasiconformal map between two compact RIEMANN SURFACES of the same GENUS modulo an EQUIVALENCE RELATION . The equivalence classes form the Teichmu ¨ller space Tp of compact RIEMANN SURFACES of GENUS p. See also RIEMANN’S MODULI PROBLEM Teichmu ¨ ller’s Principle See also JENKINS’ THEOREM References Jenkins, J. A. Univalent Functions and Conformal Map- ping. New York: Springer-Verlag, 1958. Jenkins, J. A. "Some Area Theorems and a Special Coeffi- cient Theorem." Illinois J. Math. 8,80/C1/99, 1964. Teichmu ¨ ller’s Theorem Asserts the EXISTENCE and UNIQUENESS of the ex- tremal quasiconformal map between two compact RIEMANN SURFACES of the same GENUS modulo an EQUIVALENCE RELATION . See also TEICHMU ¨ LLER SPACETeixeira’s Theorem An extended form of BU¨ RMANN’S THEOREM . Let f(z)be a function of z analytic in a ring-shaped region A, bounded by another curve C and an inner curve c. Let u(z) be a function analytic on and inside C having only one zero a (which is simple) within the contour. Further let x be a given point within A. Finally, let u(x)jjB u(z)jj (1) for all points z of C, and u(x)jj > u(z)jj (2) for all points z of c. Then f(x) /C30X/C12 n/C300An u(x)½/C138n/C27X/C12 n/C301Bn u(x)½/C138n ; (3) where An /C301 2pi gCf(z) u?(z) dz u(z)½/C138n/C271 (4) Bn /C301 2pi gcf(z) u(z)½/C138n/C281u ?(z) dz (5) (Whittaker and Watson 1990, pp. 131 /C1/132). See also BU¨ RMANN’S THEOREM ,LAGRANGE EXPANSION References Bateman, H. "An Extension of Lagrange’s Expansion." Trans. Amer. Math. Soc. 28, 346/C1/356, 1926. Teixeira, M. F. G. "Sur les se ´ries ordonne ´es suivant les puissance d’une fonction donne ´e."J. fu¨r Math. 122,9 7/C1/ 123, 1900. Whittaker, E. T. and Watson, G. N. "Teixeira’s Extended Form of Bu ¨rmann’s Theorem." §7.31 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cam- bridge University Press, pp. 131 /C1/132, 1990. Telegraph Equation The PARTIAL DIFFERENTIAL EQUATION uxx/C30autt/C27but/C27cu: References Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Telephone Problem GOSSIPING Telescoping Sum A sum in which subsequent terms cancel each other, leaving only initial and final terms. For example, S /C30Xn/C271 i/C301ai /C28ai /C271iCjiCk /C30 a1 /C28a2 ðÞ /C27 a2 /C28a3 ðÞ /C27.../C27 an/C282 /C28an /C281 ðÞ /C27 an/C281 /C28an ðÞ /C30 a1 /C28an ðÞ is a telescoping sum. See also ZEILBERGER’S ALGORITHM Temperature The "temperature" of a curve G is defined as T /C131 ln2l 2l /C28 h ! ; where l is the length of G and h is the length of the PERIMETER of the CONVEX HULL . The temperature of a curve is 0 only if the curve is a straight line, and increases as the curve becomes more "wiggly." See also CURLICUE FRACTAL References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 164 /C1/ 165, 1995. Templar Magic Square A MAGIC SQUARE -type arrangement of the words in the Latin sentence "Sator Arepo tenet opera rotas" ("the farmer Arepo keeps the world rolling"). This square has been found in excavations of ancient Pompeii. See also MAGIC SQUARE References Bouisson, S. M. La Magie: Ses Grands Rites, Son Histoire. Paris, pp. 147 /C1/148, 1958. Grosser, F. "Ein neuer Vorschlag zur Deutung der Sator- Formel." Archiv. f. Relig. 29, 165 /C1/169, 1926. Hocke, G. R. Manierismus in der Literatur: Sprach-Alchimie und esoterische Kombinationskunst. Hamburg, Germany: Rowohlt, p. 24, 1967. Temple Problem SANGAKU PROBLEMTennis Ball Theorem Any nontrivial, closed, simple, smooth SPHERICAL CURVE dividing the surface of a SPHERE into two parts of equal areas has at least four INFLECTION POINTS . See also BALL,BASEBALL COVER ,INFLECTION POINT , SPHERICAL CURVE References Arnold, V. I. Topological Invariants of Plane Curves and Caustics. Providence, RI: Amer. Math. Soc., 1994. Martinez-Maure, Y. "A Note on the Tennis Ball Theorem." Amer. Math. Monthly 103, 338 /C1/340, 1996. Tensegrity An ordered finite CONFIGURATION with certain pairs of points, called cables, which are constrained not to get further apart and certain other pairs of points, called struts, which are constrained not to get closer together. See also CONFIGURATION ,FRAMEWORK References Back, A. and Connelly, B. "Catalogue of Symmetric Tenseg- rities." http://mathlab.cit.cornell.edu/visualization/tenseg/ tenseg.html. Back, A. and Connelly, B. "Mathematics and Tensegrity." Amer. Sci. 86, 142/C1/151, 1998. Pugh, A. An Introduction to Tensegrity. Berkeley, CA: University of California Press, 1976. Tensor Annth-RANK tensor in m-space is a mathematical object in m-dimensional space that has nindices and mncomponents and obeys certain transformation rules. Each INDEX of a tensor ranges over the number of dimensions of SPACE . However, the dimension of the space is largely irrelevant in most tensor equa- tions (with the notable exception of the contractedK RONECKER DELTA ). The notation for a tensor is similar to that of a MATRIX (i.e., A/C30aijiCjiCk );except that a tensor ai;j;k;...may have an arbitrary number of INDICES . In addition, a tensor with RANK r/C27smay be of mixed type ( r, s), with rso- called "contravariant" INDICES and s"covariant" INDICES , denoted aj1;...;js i1;...;ir:Technically, a MATRIX is a tensor of type (1 ;1) and would be written aj iin tensor notation. InMathematica , a tensor of RANK nis represented using nested lists of depth n, and tensors can be generated using the command Array [a,{i,j, ...}]. Similarly, the dimensions of a tensor can be found usingDimensions [t], and the rank can be found usingRank [t]. Taking for example t/C30Array[a,{1,2,2,3}] gives the rank-4 tensor of dimensions {1, 2, 2, 3}, {{{{a[1,1,1,1],a[1,1,1,2],a[1,1,1,3]}, {a[1,1,2,1],a[1,1,2,2],a[1,1,2,3]}}, {{a[1,2,1,1],a[1,2,1,2],a[1,2,1,3]}, {a[1,2,2,1], a[1,2,2,2],a[1,2,2,3]}}}, {{{a[2,1,1,1],a[2,1,1,2],a[2,1,1,3]}, {a[2,1,2,1],a[2,1,2,2],a[2,1,2,3]}}, {{a[2,2,1,1],a[2,2,1,2],a[2,2,1,3]}, {a[2,2,2,1],a[2,2,2,2],a[2,2,2,3]}}}}. In n-dimensional space, each element aijklwould then represent an n-vector. A TENSOR SPACE of type (r, s) can be described as a TENSOR PRODUCT between r copies of VECTOR FIELDS and s copies of the dual vector fields, i.e., ONE-FORMS . For example, T(3; 1) /C30TM /C156TM /C156TM /C156T /C31M (1) is the VECTOR BUNDLE of (3; 1)/-tensors on a MANIFOLD M, where TM is the TANGENT BUNDLE of M and T /C31M is its dual. Tensors of type (r, s) form a VECTOR SPACE . This description generalized to any tensor type, and an INVERTIBLE LINEAR MAP J : V 0 W induces a map ˜J : V /C156V /C310 W /C156W /C31; where V /C31 is the DUAL VECTOR SPACE and J the JACOBIAN , defined by ˜Jv1 /C156v/C312 ðÞ /C30 Jv1 /C156 JTiCjiCk /C281v/C312iCkCiCkA ; (2) where JT is the PULLBACK MAP of a form is defined using the transpose of the JACOBIAN . This definition can be extended similarly to other TENSOR PRODUCTS of V and V /C31: When there is a change of COORDINATES , then tensors transform similarly, with J the JACO- BIAN of the linear transformation. Zeroth-rank tensors are called SCALARS , and first- rank tensors are called VECTORS . In tensor notation, a vector v would be written vi ; where i /C301, ..., m. Tensor notation can provide a very concise way of writing vector and more general identities. For example, in tensor notation, the DOT PRODUCT u /C215 v is simply written u /C215 v /C30uivi ; (3) where repeated indices are summed over (EINSTEIN SUMMATION ). Similarly, the CROSS PRODUCT can be concisely written as u /C29v /C30 eijkujvk ; (4) where eijk is the PERMUTATION TENSOR . CONTRAVARIANT second-rank tensors are objects which transform as A?ij /C30@x?i @xk@x?j @x ?lAkl : (5) COVARIANT second-rank tensors are objects which transform as C ?ij /C30@xk @x?i@xl @x?jCkl : (6)MIXED second-rank tensors are objects which trans- form as B ?ji/C30@x?i @xk@xl @x?jBk l : (7) If two tensors A and B have the same rank and the same COVARIANT and CONTRAVARIANT indices, then the can be added in the obvious way, Aij /C27Bij /C30Cij (8) Aij /C27Bij /C30Cij (9) Aij /C27Bij /C30Cij : (10) The indices of a tensor can be raised or lowered (INDEX RAISING and INDEX LOWERING , respectively) by multiplication by a so-called METRIC TENSOR , e.g., gijAj /C30Ai (11) gijAj /C30Ai (12) (Arfken 1985, p. 159). The generalization of the DOT PRODUCT applied to tensors is called CONTRACTION , and consists of setting two unlike indices equal to each other and then summing using the EINSTEIN SUMMATION convention. Various types of derivatives can be taken of tensors, the most common being the COMMA DERIVATIVE and COVARIANT DERIVATIVE . If the components of any tensor of any RANK vanish in one particular coordinate system, they vanish in all coordinate systems. A transformation of the variables of a tensor changes the tensor into another whose components are linear HOMOGENEOUS FUNCTIONS of the components of the original tensor. See also ANTISYMMETRIC TENSOR ,COMMA DERIVA- TIVE,CONTRACTION (TENSOR ), CONTRAVARIANT TEN- SOR,C OVARIANT DERIVATIVE ,C OVARIANT TENSOR , CURL,D IVERGENCE ,G RADIENT ,INDEX LOWERING , INDEX RAISING ,IRREDUCIBLE TENSOR ,ISOTROPIC TENSOR ,JACOBI TENSOR ,M IXED TENSOR ,R ICCI TENSOR ,R IEMANN TENSOR ,S CALAR ,S YMMETRIC TENSOR ,TENSOR SPACE ,TORSION TENSOR ,VECTOR , WEYL TENSOR References Abraham, R.; Marsden, J. E.; and Ratiu, T. S. Manifolds, Tensor Analysis, and Applications. New York: Springer- Verlag, 1991. Akivis, M. A. and Goldberg, V. V. An Introduction to Linear Algebra and Tensors. New York: Dover, 1972. Arfken, G. "Tensor Analysis." Ch. 3 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 118 /C1/167, 1985. Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, 1989. Bishop, R. and Goldberg, S. Tensor Analysis on Manifolds. New York: Dover, 1980. Jeffreys, H. Cartesian Tensors. Cambridge, England: Cam- bridge University Press, 1931. Jeffreys, H. and Jeffreys, B. S. "Tensors." Ch. 3 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 86 /C1/113, 1988. Joshi, A. W. Matrices and Tensors in Physics, 3rd ed. New York: Wiley, 1995. Lass, H. Vector and Tensor Analysis. New York: McGraw- Hill, 1950. Lawden, D. F. An Introduction to Tensor Calculus, Relativ- ity, and Cosmology, 3rd ed. Chichester, England: Wiley, 1982. McConnell, A. J. Applications of Tensor Analysis. New York: Dover, 1947. Morse, P. M. and Feshbach, H. "Vector and Tensor Formal- ism." §1.5 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44 /C1/54, 1953. Parker, L. and Christensen, S. M. MathTensor: A System for Doing Tensor Analysis by Computer. Reading, MA: Ad- dison-Wesley, 1994. Simmonds, J. G. A Brief on Tensor Analysis, 2nd ed. New York: Springer-Verlag, 1994. Sokolnikoff, I. S. Tensor Analysis--Theory and Applications, 2nd ed. New York: Wiley, 1964. Synge, J. L. and Schild, A. Tensor Calculus. New York: Dover, 1978. Weisstein, E. W. "Books about Tensors." http://www.trea- sure-troves.com/books/Tensors.html. Wrede, R. C. Introduction to Vector and Tensor Analysis. New York: Wiley, 1963. Tensor Calculus The set of rules for manipulating and calculating with TENSORS . Tensor Density A quantity which transforms like a TENSOR except for a scalar factor of a JACOBIAN . Tensor Direct Product Abstractly, the tensor direct product is the same as the TENSOR PRODUCT . However, it reflects an ap- proach toward calculation using coordinates, and indices in particular. The notion of tensor product is more algebraic, intrinsic, and abstract. For instance, up to ISOMORPHISM , the tensor product is commu- tative because V /C156W $W /C156V : Note this does not mean that the tensor product is symmetric. For two first- RANK TENSORS (i.e., VECTORS ), the tensor direct product is defined as a ?ib?j /C13@xk @x ?iak@x?j @xlbl /C30@xk @x?i@x?j @xlakbliCjiCk ; (1) which is a second- RANK TENSOR . The CONTRACTION of a direct product of first- RANK TENSORS is the SCALAR contr a?ib ?jiCjiCk /C30a ?ib?i /C30akbk : (2) For second- RANK TENSORS , Ai jBkl /C30Ciklj (3)Cikl? j/C30@x?i @xm@xn @x?j@x?k @xp@x?l @xqCmpqn: (4) In general, the direct product of two TENSORS is a TENSOR ofRANK equal to the sum of the two initial RANKS . The direct product is ASSOCIATIVE , but not COMMUTATIVE . The tensor direct product of two tensors aandbcan be implemented in Mathematica as TensorDirectProduct[a_List, b_List] : /C30 Outer[Times, a, b] See also DIRECT PRODUCT ,MATRIX DIRECT PRODUCT , TENSOR PRODUCT (VECTOR SPACE ) References Arfken, G. "Contraction, Direct Product." §3.2 in Mathema- tical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 124 /C1/126, 1985. Tensor Dual DUALTENSOR Tensor Product TENSOR DIRECT PRODUCT ,TENSOR PRODUCT (MOD- ULE), T ENSOR PRODUCT (VECTOR SPACE ) Tensor Product (Module) The tensor product between MODULES Aand Bis a more general notion than the TENSOR PRODUCT BETWEEN VECTOR SPACES . In this case, we replace "scalars" by a RING R. The familiar formulas hold, but nowais any element of R, a1/C27a2 ðÞ /C156b/C30a1/C156b/C27a2/C156b (1) a/C156b1/C27b2 ðÞ /C30a/C156b1/C27a/C156b2 (2) a(a/C156b)/C30(aa)/C156b/C30a/C156(ab): (3) This generalizes the definition of a tensor product for vector spaces since a VECTOR SPACE is a module over the scalar field. Also, VECTOR BUNDLES can be considered as PROJECTIVE MODULES over the ring of functions, and REPRESENTATIONS of a group Gcan be thought of as modules over CG. The generalizationcovers those kinds of tensor products as well. There are some interesting possibilities for the tensor product of modules that don’t occur in the case ofvector spaces. It is possible for A/C156 RBto be identi- cally zero. For example, the tensor product of Z2and Z3as modules over the integers, Z2/C156ZZ3;has no nonzero elements. It is enough to see that a/C156b/C300: Notice that 1 /C303/C282:Then (1)a/C156b/C30(3/C282)a/C156b/C30(/C282a)/C156b/C27a/C156(3b)/C300/C270 /C300; (4) since /C282a /C30/C28a /C28a /C300inZ2and 3b /C30b /C27b /C27b /C300in Z3 : In general, it is easier to show that elements are zero than to show they are not zero. Another interesting property of tensor products is that if f : A 0 B is ONTO , then so is the induced map g : A /C156C 0 B /C156C for any other module C. But if f : A 0 B is injective, then g : A /C156C 0 B /C156C may not be injective. For example, f : Z2 0 Z4 ; with f(1) /C302 is injective, but g : Z2 /C156Z Z2 0 Z4 /C156Z Z2 ; with g(1 /C1561) /C302 /C1561; is not injective. In Z4 /C156Z Z2 ; we have 2 /C1561 /C301 /C1562 /C301 /C1560 /C300::/ There is an algebraic description of this failure of injectivity, called the TOR module. Another way to think of the tensor product is in terms of its UNIVERSAL PROPERTY : Any BILINEAR MAP from A /C29B :0 C factors through the natural bilinear map A /C29B 0 A /C156B::/ See also MODULE ,MODULE DIRECT SUM,PROJECTIVE MODULE ,REPRESENTATION ,TENSOR PRODUCT (MOD- ULE), TENSOR PRODUCT (REPRESENTATION ), TENSOR PRODUCT (VECTOR SPACE ), TOR,U NIVERSAL PROP- ERTY ,VECTOR BUNDLE ,VECTOR SPACE Tensor Product (Representation) The TENSOR PRODUCT V /C156W of two REPRESENTATIONS of a GROUP G is also a REPRESENTATION of G.An element g of G acts on a basis element v /C156w by g(v /C156w) /C30gv /C156gw: If G is a FINITE GROUP and V is a FAITHFUL representation, then any representation is contained in /C156n V for some n.IfV1 is a representation of G1 and V2is a representation of G2 ; then V1 /C156V2is a representation of G1 /C29G2 ; called the EXTERNAL TEN- SOR PRODUCT . The regular tensor product is a special case, with the diagonal embedding of G in G /C29G :/ See also EXTERNAL TENSOR PRODUCT ,GROUP ,IRRE- DUCIBLE REPRESENTATION ,REPRESENTATION ,TENSOR PRODUCT (VECTOR SPACE ), VECTOR SPACE Tensor Product (Vector Space) The tensor product of two VECTOR SPACES V and W, denoted V /C156W and also called the TENSOR DIRECT PRODUCT , is a way of creating a new VECTOR SPACE analogous to multiplication of integers. For instance, Rn /C156Rk $Rnk : (1) In particular, R /C156Rn $Rn : (2) Also, the tensor product obeys a distributive law with the DIRECT SUM operation: U /C156(V /C154W) $(U /C156V) /C154(U /C156W) : (3)The analogy with an algebra is the motivation behind K-THEORY . The tensor product of two tensors a and b can be implemented in Mathematica as TensorProduct[a_List, b_List] : /C30 Outer[List, a, b] Algebraically, the vector space V /C156W is SPANNED by elements OF THE FORM v /C156w ; and the following rules are satisfied, for any scalar a: The definition is the same no matter which scalar FIELD is used. v1 /C27v2 ðÞ /C156w /C30v1 /C156w /C27v2 /C156w (4) v /C156 w1 /C27w2 ðÞ /C30v /C156w1 /C27v /C156w2 (5) a(v /C156w) /C30(av) /C156w /C30v /C156( aw) (6) One basic consequence of these formulas is that 0 /C156w /C30v /C1560 /C300 : (7) A VECTOR BASIS vi of V and wj of W gives a basis for V /C156W ; namely vi /C156wj ; for all pairs (i, j). An arbitrary element of V /C156W can be written uniquely as a ai ; jvi /C156wj ; where ai ; jare scalars. If V is n dimen- sional and W is k dimensional, then V /C156W has dimension nk. Using tensor products, one can define SYMMETRIC TENSORS , ANTISYMMETRIC TENSORS , as well as the EXTERIOR ALGEBRA . Moreover, the tensor product is generalized to the TENSOR PRODUCT OF VECTOR BUNDLES . In particular, tensor products of the TAN- GENT BUNDLE and its DUAL BUNDLE are studied in RIEMANNIAN GEOMETRY and physics. Sections of these bundles are often called TENSORS . In addition, it is possible to take the TENSOR PRODUCT OF REPRESENTA- TIONS to get another representation. All of these versions of tensor product can be under- stood as TENSOR PRODUCTS OF MODULES . The trick is to find the right way to think of these spaces as MODULES . See also ANTISYMMETRIC TENSOR ,EXTERIOR ALGE- BRA,FIELD, K-THEORY ,MODULE ,SYMMETRIC TENSOR , TENSOR ,TENSOR DIRECT PRODUCT ,TENSOR PRODUCT (MODULE ), TENSOR PRODUCT (REPRESENTATION ), VECTOR SPACE Tensor Space LetEbe a linear space over a FIELD K. Then the TENSOR PRODUCT /C156k l/C301Eis called a tensor space of degree k. More specifically, a tensor space of type ( r, s) can be described as a TENSOR PRODUCT between r copies of VECTOR FIELDS and scopies of the dual vector fields, i.e., ONE-FORMS . For example, T(3;1)/C30TM/C156TM/C156TM/C156T/C31M (1) is the VECTOR BUNDLE of (3 ;1) tensors on a MANIFOLD M. Tensors of type ( r, s) form a VECTOR SPACE . See also TENSOR ,VECTOR SPACE References Yokonuma, T. Tensor Spaces and Exterior Algebra. Provi- dence, RI: Amer. Math. Soc., 1992. Tensor Spherical Harmonic DOUBLE CONTRACTION RELATION Tensor Transpose TRANSPOSE Tent Map A piecewise linear, 1-D MAP on the interval [0; 1] exhibiting CHAOTIC dynamics and given by xn/C271 /C30 m 1 /C282 xn /C281 2iCk0iCk0iCk0iCk0iCk0iCk0iCkCiCkA : The case m /C301 is equivalent to the LOGISTIC EQUATION WITH R /C304. The NATURAL INVARIANT of the tent map is r /C301:/ See also 2X MOD 1 MAP,LOGISTIC EQUATION ,LOGISTIC EQUATION: R /C304 Tent Problem Consider a horse rider who wishes to feed his horse at a field, gather water from a river, and then return to his tent, all in the smallest overall distance possible. The path he should take is obtained by reflecting the tent across the near river bank, then reflecting this point about the field boundary, as illustrated above. References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 111 /C1/113, 1999. Terminal SINK (DIRECTED GRAPH )Ternary The BASE 3 method of counting in which only the digits 0, 1, and 2 are used. Ternary numbers arise in a number of problems in mathematics, including some problems of WEIGHING . According to Knuth (1981), "no substantial application of balanced ternary nota- tion has been made" (balanced ternary uses digits /C281, 0, and 1 instead of 0, 1, and 2). The following table gives the ternary equivalents of the first few decimal numbers. 1 1 11 102 21 210 2 2 12 110 22 211 3 10 13 111 23 212 4 11 14 112 24 220 5 12 15 120 25 221 6 20 16 121 26 222 7 21 17 122 27 1000 8 22 18 200 28 1001 9 100 19 201 29 1002 10 101 20 202 30 1010 Ternary digits have the following MULTIPLICATION TABLE . //C29/ 01 2 000 0 101 2 20211 Every EVEN NUMBER represented in ternary has an EVEN NUMBER (possibly 0) of 1s. This is true since a number is congruent mod (B /C281) to the sum of its base- B digits. In the case B /C303, there is only one digit (1) which is not a multiple of B /C281; so all we have to do is "cast out twos" and count the number of 1s in the base-3 representation. Erdos and Graham (1980) conjectured that no POWER of 2, 2n;is a SUM of distinct powers of 3 for n/C218. This is equivalent to the requirement that the ternary expansion of 2nalways contains a 2. This has been verified by Vardi (1991) up to n/C302/C215330:N. J. A. Sloane has conjectured that any POWER of 2 has a 0 in its ternary expansion (Vardi 1991, p. 28). See also BASE (NUMBER ), BINARY ,DECIMAL ,HEXADE- CIMAL ,OCTAL ,QUATERNARY References Erdos, P. and Graham, R. L. Old and New Problems and Results in Combinatorial Number Theory. Geneva, Swit- zerland: L’Enseignement Mathe ´matique Universite ´ de Gene`ve, Vol. 28, 1980. Gardner, M. "The Ternary System." Ch. 11 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 104 /C1/112, 1984. Knuth, D. E. The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addi- son-Wesley, pp. 173 /C1/175, 1998. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 10 /C1/ 11, 1991. Vardi, I. "The Digits of 2n in Base Three." Computational Recreations in Mathematica. Reading, MA: Addison-Wes- ley, pp. 20 /C1/25, 1991. Weisstein, E. W. "Bases." MATHEMATICA NOTEBOOK BASES.M . Ternary Goldbach Conjecture GOLDBACH CONJECTURE Ternary Tree See also BINARY TREE,COMPLETE TERNARY TREE Tessellation A regular TILING ofPOLYGONS (in 2-D), POLYHEDRA (3- D), or POLYTOPES (n-D) is called a tessellation. Tessellations can be specified using a S CHLA ¨FLI SYMBOL . The breaking up of self-intersecting polygons into simple polygons (illustrated above) is also called tessellation (Woo et al. 1999). Consider a 2-D tessellation with qregular p-gons at each VERTEX . In the PLANE , 1/C282 p ! p/C302p q(1) 1 p/C271 q/C3012; (2) so (p/C282q)(q/C282)/C304 (3) (Ball and Coxeter 1987), and the only factorizations are4/C304/C2151/C30(6/C282)(3/C282)[f6;3g (4) /C302/C2152/C30(4/C282)(4/C282)[f4;4g (5) /C301/C2154/C30(3/C282)(6/C282)[f3;6g: (6) Therefore, there are only three regular tessellations (composed of the HEXAGON ,SQUARE , and TRIANGLE ), illustrated as follows (Ghyka 1977, p. 76; Williams 1979, p. 36; Wells 1991, p. 213) There do not exist any regular STAR POLYGON tessel- lations in the PLANE . Regular tessellations of the SPHERE bySPHERICAL TRIANGLES are called TRIANGU- LAR SYMMETRY GROUPS . Regular tessellations of the plane by two or more convex regular POLYGONS such that the same POLY- GONS in the same order surround each VERTEX are called semiregular tessellations, or sometimes Archi-medean tessellations. In the plane, there are eightsuch tessellations, illustrated below (Ghyka 1977,pp. 76 /C1 /78; Williams 1979, pp. 37 /C1/41; Steinhaus 1983, pp. 78 /C1/82; Wells 1991, pp. 226 /C1/227). Williams (1979, pp. 37 /C1/41) also illustrates the DUAL TESSELLA- TIONS of the semiregular tessellations. The DUAL TESSELLATION of the tessellation of squares and equilateral triangles is called the CAIRO TESSELLA- TION (Williams 1979, p. 38; Wells 1991, p. 23). There are 14 polymorph, or demiregular, tessellations which are orderly compositions of the three regular and eight semiregular tessellations (Critchlow 1970, pp. 62 /C1/67; Ghyka 1977, pp. 78 /C1/80; Williams 1979, p. 43; Steinhaus 1983, pp. 79 and 81 /C1/82). In 3-D, a POLYHEDRON which is capable of tessellating space is called a SPACE-FILLING POLYHEDRON . Exam- ples include the CUBE , RHOMBIC DODECAHEDRON , and TRUNCATED OCTAHEDRON . There is also a 16-sided space-filler and a convex POLYHEDRON known as the SCHMITT- CONWAY BIPRISM which fills space only aperiodically. A tessellation of n-D polytopes is called a HONEY- COMB . See also ARCHIMEDEAN SOLID ,CAIRO TESSELLATION , CELL,D UAL TESSELLATION ,H INGED TESSELLATION , HONEYCOMB ,H ONEYCOMB CONJECTURE ,S CHLA ¨ FLI SYMBOL ,SEMIREGULAR POLYHEDRON ,SPACE- FILLING POLYHEDRON ,S PIRAL- SIMILARITY TESSELLATION , SYMMETRY ,TILING ,TRIANGULAR SYMMETRY GROUP , TRIANGULATION References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 105 /C1/107, 1987. Bhushan, A.; Kay, K.; and Williams, E. "Totally Tessellated." http://library.thinkquest.org/16661/. Britton, J. Symmetry and Tessellations: Investigating Pat- terns. Englewood Cliffs, NJ: Prentice-Hall, 1999. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 60 /C1/63, 1989. Gardner, M. Martin Gardner’s New Mathematical Diver- sions from Scientific American. New York: Simon and Schuster, pp. 201 /C1/203, 1966.Gardner, M. "Tilings with Convex Polygons." Ch. 13 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 162 /C1/176, 1988. Ghyka, M. The Geometry of Art and Life. New York: Dover, 1977. Kraitchik, M. "Mosaics." §8.2 in Mathematical Recreations. New York: W. W. Norton, pp. 199 /C1/207, 1942. Kraus, M. "Polygon Triangulation." http://library.wolfram.- com/packages/polygontriangulation/. Lines, L. Solid Geometry. New York: Dover, pp. 199 and 204/C1/207 1965. Pappas, T. "Tessellations." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 120 /C1/122, 1989. Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, p. 75, 1988. Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., 1999. Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., 1997. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 75 /C1/76, 1999. Vichera, M. "Archimedean Polyhedra." http://alpha.ujep.cz/ ~vicher/puzzle/telesa/telesa.htm. Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods of Semi-Regular Polyhedra." Geometriae Dedicata 1, 117/C1/ 123, 1972. Weisstein, E. W. "Books about Tilings." http://www.trea- sure-troves.com/books/Tilings.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 121, 213, and 226 /C1/227, 1991. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, pp. 35 /C1/ 43, 1979. Woo, M.; Neider, J.; Davis, T.; and Shreiner, D. Ch. 11 in OpenGL 1.2 Programming Guide, 3rd ed.: The Official Guide to Learning OpenGL, Version 1.2. Reading, MA: Addison-Wesley, 1999. Tesseract The HYPERCUBE inR4;also called the 8-cell, is known as a tesseract. It has the S CHLA ¨FLI SYMBOL f4;3;3g; and VERTICES (91;91;91;91):The above figures show two visualizations of the tesseract. The figure on the left is a projection of the tesseract in 3-space(Gardner 1977), and the figure on the right is the GRAPH of the tesseract symmetrically projected into the PLANE (Coxeter 1973). A tesseract has 16 VER- TICES ,32 EDGES ,24 SQUARES , and 8 CUBES . See also CUBE,H YPERCUBE ,M AGIC TESSERACT , POLYTOPE ,SIMPLEX References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 123, 1973. Dewdney, A. K. "Computer Recreations: A Program for Rotating Hypercubes Induces Four-Dimensional Demen- tia." Sci. Amer. 254,14/C1/23, Mar. 1986. Gardner, M. "Hypercubes." Ch. 4 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scien- tific American. New York: Vintage Books, pp. 41 /C1/54, 1977. Smith, H. J. "The Tesseract: A Look into 4-Dimensional Space." http://pweb.netcom.com/~hjsmith/WireFrame4/ tesseract.html. Tesseral Harmonic A SPHERICAL HARMONIC OF THE FORMcos sin (mf)Pm(cos u) l : These harmonics are so named because the curves on which they vanish are l /C28m parallels of latitude and 2m meridians, which divide the surface of a sphere into quadrangles whose angles are right angles (Whittaker and Watson 1990, p. 392). Resolving Pl(cos u) into factors linear in cos2 u; multi- plied by cos u when l is ODD, then replacing cos u by z=r allows the tesseral harmonics to be expressed as products of factors linear in x2 ; y2 ; and z2 multiplied by one of 1, x, y, z, yz, zx, xy, and xyz (Whittaker and Watson 1990, p. 536). See also SECTORIAL HARMONIC ,SPHERICAL HARMO- NIC,ZONAL HARMONIC References Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 197, 1959. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Tethered Bull Problem Let a bull be tethered to a silo whose horizontal CROSS SECTION is a CIRCLE of RADIUS R by a leash of length L. Then the AREA which the bull can graze if L 5Rp is A /C30pL2 2/C27L3 3R : References Hoffman, M. E. "The Bull and the Silo: An Application of Curvature." Amer. Math. Monthly 105,55/C1/58, 1998.Tetrabolo One of the 14 4-POLYABOLOES . See also POLYABOLO Tetrachoric Function The function defined by Tn(x) /C30( /C281)n/C281 ffiffiffinp Z(n/C281)(x) ; where Z(x) /C301ffiffiffiffiffiffi 2pp e /C28x2 =2 and Z(k)(x) is the kth derivative of Z(x) :/ See also NORMAL DISTRIBUTION ,STANDARD NORMAL DISTRIBUTION References Kenney, J. F. and Keeping, E. S. "Tetrachoric Correlation." §8.5 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 205 /C1/207, 1951. Tetracontagon A 40-sided POLYGON . Tetracuspid HYPOCYCLOID–4- CUSPED Tetracyclic Plane The set of all points xthat can be put into one-to-one correspondence with sets of essentially distinct values of four homogeneous coordinates /x0:x1:x2:x3/, not all simultaneously zero, which are connected by the relation x /C215 x /C30x2 0 /C27x21 /C27x22 /C27x23 /C300 : (1) See also PENTASPHERICAL SPACE References Coolidge, J. L. "Pentaspherical Space." Ch. 7 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 282 /C1/305, 1971. Tetrad A SET of four, also called a QUARTET . See also HEXAD ,M ONAD ,PAIR,QUARTET ,QUINTET , TRIAD,TRIPLE ,TWINS Tetradecagon A 14-sided POLYGON , sometimes called a TETRAKAIDE- CAGON . Tetradecahedron A 14-sided POLYHEDRON , sometimes called a TETRA- KAIDECAHEDRON . See also CUBOCTAHEDRON ,TRUNCATED OCTAHEDRON References Ghyka, M. The Geometry of Art and Life. New York: Dover, p. 54, 1977. Tetradic Tetradics transform DYADICS in much the same way that DYADICS transform VECTORS . They are repre- sented using Hebrew characters and have 81 compo- nents (Morse and Feshbach 1953, pp. 72 /C1/73). The use of tetradics is archaic, since TENSORS perform the same function but are notationally simpler. References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part 1. New York: McGraw-Hill, 1953.Tetradyakis Hexahedron The DUAL POLYHEDRON of the CUBITRUNCATED CU- BOCTAHEDRON U16 and Wenninger dual W79 :/ See also DUAL POLYHEDRON ,CUBITRUNCATED CUBOC- TAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 92, 1983. Tetraflexagon AFLEXAGON made with SQUARE faces. Gardner (1961) shows how to construct a tri-tetraflexagon, tetra-tetraflexagon, and hexa-tetraflexagon. See also FLEXAGON ,FLEXATUBE ,HEXAFLEXAGON References Chapman, P. B. "Square Flexagons." Math. Gaz. 45, 192 /C1/ 194, 1961. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 207, 1989. Gardner, M. "Mathematical Games: About Tetraflexagons and Tetraflexigation." Sci. Amer. 198, 122 /C1/126, May 1958. Gardner, M. "Hexaflexagons." Ch. 1 in The Scientific Amer- ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 1 /C1/14, 1959. Gardner, M. "Tetraflexagons." Ch. 2 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 24 /C1/ 31, 1961. Pappas, T. "Making a Tri-Tetra Flexagon." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 107, 1989. Tetragon QUADRILATERAL Tetragram Lachlan’s term for a set of four lines, no three of which are CONCURRENT . See also TETRASTIGM References Lachlan, R. "Properties of a Tetragram." §147 /C1/155 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 90 /C1/97, 1893. Tetrahedral Coordinates Coordinates useful for plotting projective 3-D curves OF THE FORM /f ðx0 ;x1 ;x2 ; x3 Þ¼0/ which are defined by x0 /C301 /C28z /C28ffiffiffi 2p xx1 /C301 /C28z /C27ffiffiffi 2p x x2 /C301 /C27z /C27ffiffiffi2p y x 3 /C301 /C27z /C28ffiffiffi 2p y See also CAYLEY CUBIC ,KUMMER SURFACE Tetrahedral Graph The PLATONIC GRAPH that is the unique POLYHEDRAL GRAPH on four nodes which is also the COMPLETE GRAPH K4 : The tetrahedral graph has 4 nodes, 6 edges, VERTEX CONNECTIVITY 4, EDGE CONNECTIVITY 3, GRAPH DIAMETER 1, GRAPH RADIUS 1, and GIRTH 3. It has CHROMATIC POLYNOMIAL pG(z) /C30z4 /C286z3 /C2711z2 /C286z and CHROMATIC NUMBER 4. See also CUBICAL GRAPH ,D ODECAHEDRAL GRAPH , ICOSAHEDRAL GRAPH ,OCTAHEDRAL GRAPH ,PLATONIC GRAPH ,POLYHEDRAL GRAPH ,TETRAHEDRON References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976. Tetrahedral Group The POINT GROUP of symmetries of the TETRAHEDRON having order 12 and denoted Td : The tetrahedral group has symmetry operations E,8C3;3C2;6S4;and 6sd(Cotton 1990). See also ICOSAHEDRAL GROUP ,OCTAHEDRAL GROUP , POINT GROUPS ,POLYHEDRAL GROUP ,TETRAHEDRON References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, p. 47, 1990. Coxeter, H. S. M. "The Polyhedral Groups." §3.5 in Regular Polytopes, 3rd ed. New York: Dover, pp. 46 /C1/47, 1973. Lomont, J. S. "Icosahedral Group." §3.10.C in Applications of Finite Groups. New York: Dover, p. 81, 1987. Tetrahedral Number A FIGURATE NUMBER Ten OF THE FORM Ten /C30Xn i /C301Tn /C301 6 n(n /C271)(n /C272) /C30n /C272 3iCkniCko ; (1) where Tn is the nth TRIANGULAR NUMBER and n miCjiCk is a BINOMIAL COEFFICIENT . These numbers correspond to placing discrete points in the configuration of a TETRAHEDRON (triangular base pyramid). Tetrahe- dral numbers are PYRAMIDAL NUMBERS with r /C303, and are the sum of consecutive TRIANGULAR NUM- BERS . The first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (Sloane’s A000292). The GENERATING FUNCTION of the tetrahedral numbers is x (x /C28 1)4 /C30x /C274x2 /C2710x3 /C2720x4 /C27...: (2) Tetrahedral numbers are EVEN , except for every fourth tetrahedral number, which is ODD (Conway and Guy 1996). The only numbers which are simultaneously SQUARE and TETRAHEDRAL are Te1 /C301; Te2 /C304; and Te48 /C30 19600 (giving S1 /C301; S2 /C304 ; and S140 /C3019600) ; as proved by Meyl (1878; cited in Dickson 1952, p. 25). Numbers which are simultaneously TRIANGULAR and TETRAHEDRAL satisfy the BINOMIAL COEFFICIENT equation Tn /C30n /C271 2iCkniCko /C30m /C272 3iCkniCko /C30Tem ; (3) the only solutions of which are Te1 /C30T1 /C301 (4) Te3 /C30T4 /C3010 (5) Te8 /C30T15 /C30120 (6) Te20 /C30T55 /C301540 (7) Te34 /C30T119 /C307140 (8) (Sloane’s A027568; Avanesov 1966/1967; Mordell 1969, p. 258; Guy 1994, p. 147).Beukers (1988) has studied the problem of finding numbers which are simultaneously tetrahedral and PYRAMIDAL via INTEGER points on an ELLIPTIC CURVE , and finds that the only solution is the trivial Te1 /C30P1 /C301 :/ See also PYRAMIDAL NUMBER ,SQUARE PYRAMIDAL NUMBER ,TRIANGULAR NUMBER ,TRUNCATED TETRA- HEDRAL NUMBER References Avanesov, E. T. "Solution of a Problem on Figurate Num- bers" [Russian]. Acta Arith. 12, 409 /C1/420, 1966/1967. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 59, 1987. Beukers, F. "On Oranges and Integral Points on Certain Plane Cubic Curves." Nieuw Arch. Wisk. 6, 203 /C1/210, 1988. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 44 /C1/46, 1996. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952. Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 147 /C1/150, 1994. Meyl, A.-J.-J. "Solution de Question 1194." Nouv. Ann. Math. 17, 464 /C1/467, 1878. Mordell, L. J. Diophantine Equations. New York: Academic Press, p. 258, 1969. Sloane, N. J. A. Sequences A000292/M3382 and A027568 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Tetrahedral Surface A SURFACE given by the PARAMETRIC EQUATIONS x /C30A(u /C28a)m(v /C28a)n y /C30B(u /C28b)m(v /C28b)n z/C30C(u/C28c)m(v/C28c)n: References Eisenhart, L. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, p. 267, 1960. Tetrahedroid A special case of a quartic K UMMER SURFACE . See also KUMMER SURFACE References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 17 /C1/19, 1986. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 183, 1994. Tetrahedron The regular tetrahedron, often simply called "the" tetrahedron, is the P LATONIC SOLID P1with four VERTICES , six EDGES , and four equivalent EQUILAT- ERAL TRIANGULAR faces, 4 f3g:It is also UNIFORM POLYHEDRON U1and Wenninger model W1:It is described by the S CHLA ¨FLI SYMBOL f3;3gand the WYTHOFF SYMBOL is 3½23:/ It is the prototype of the TETRAHEDRAL GROUP Td:The connectivity of the vertices is given by the TETRAHE- DRAL GRAPH , equivalent to the CIRCULANT GRAPH Ci1;2;3(4) and the COMPLETE GRAPH K4:/ The tetrahedron is its own DUAL POLYHEDRON , and therefore the centers of the faces of a tetrahedron form another tetrahedron (Steinhaus 1983, p. 201).The tetrahedron is the only simple POLYHEDRON with no DIAGONALS , and it cannot be STELLATED .I fa regular tetrahedron is cut by six planes, each passing through an edge and bisecting the opposite edge, it issliced into 24 pieces (Gardner 1984, pp. 190 and 192; and Langman 1951). Alexander Graham Bell was a proponent of use of the tetrahedron in framework structures, including kites(Bell 1903; Lesage 1956, Gardner 1984, pp. 184 /C1 /185). The opposite edges of a tetrahedron are perpendicu-lar, and so can form a universal coupling if hingedappropriately. Eight regular tetrahedra can be placed in a ring which rotates freely, and the number can be reduced to six for squashed irregular tetrahedra(Wells 1975, 1991) Let a tetrahedron be length aon a side. The VERTICES are located at ( x, 0, 0), ( //C28d;9a=2;0), and (0, 0, h). From the figure, x/C30a 2 cosp 6 !/C301 3ffiffiffi 3p a: (1) dis then d/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C281 2aiCkCiCkA2r /C3016ffiffiffi 3p a: (2) This gives the AREA of the base as A/C301 2a(R/C27x)/C3014ffiffiffi 3p a2: (3) The height is h/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28x2p /C301 3ffiffiffi 6p a: (4) The CIRCUMRADIUS Ris found from x2/C27(h/C28R)2/C30R2(5) x2/C27h2/C282hR/C27R2/C30R2: (6) Solving gives R/C30x2/C27h2 2h/C301 4ffiffiffi 6p a:0:61237 a: (7) The INRADIUS ris r/C13h/C28R/C301 12ffiffiffi6p a:0:20412 a; (8) which is also r/C301 4h/C3013R: (9) The ANGLE between the bottom plane and center is then given by f/C30tan/C281r x ! /C30tan/C2811 4ffiffiffi 2piCkCiCkA : (10) Given a tetrahedron of edge length asituated with vertical apex and with the origin of coordinate system at the CENTROID of the vertices, the four VERTICES are located at ( x;0;/C28r);(/C28d;9a=2;/C28r);(0;0;R);with, as shown above x/C301 3ffiffiffi 3p a (11) r/C301 12ffiffiffi 6p a (12) R/C301 4ffiffiffi 6p a (13) d/C301 6ffiffiffi 3p a: (14) The vertices of a tetrahedron of side lengthffiffiffi 2p can also be given by a particularly simple form when the vertices are taken as corners of a cube (Gardner 1984, pp. 192 /C1/194). One such tetrahedron for a cube of side length 1 gives the tetrahedron of side lengthffiffiffi 2p having vertices (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0), and satisfies the inequalities xþyþz52 ð15Þ x/C28y/C28z50 (16) /C28x/C27y/C28z50 (17) /C28x/C28y/C27z50: (18)The following table gives polyhedra which can be constructed by CUMULATION of a tetrahedron by pyramids of given heights h. h /(r/C27h)=h/Result /1 15ffiffiffi 6p //7 5/ TRIAKIS TETRAHEDRON /16ffiffiffi 6p /2 CUBE /1 3ffiffiffi 6p /3 9-faced star DELTAHEDRON Connecting opposite pairs of edges with equally spaced lines gives a configuration like that shownabove which divides the tetrahedron into eight regions: four open and four closed (Steinhaus 1983, p. 246). The MIDRADIUS of the tetrahedron is r/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C27d2p /C30ffiffi 1 8q a/C3014ffiffiffi 2p a :0:35355 a: (19) Plugging in for the VERTICES gives affiffiffi 3p ;0;0iCkCiCkA ;/C281 6ffiffiffi 3p a;91 2a;0iCkCiCkA ;and 0 ;0;12ffiffiffi 6p aiCkCiCkA : (20) Since a tetrahedron is a PYRAMID with a triangular base, V/C301 3Abh;giving V/C301 12ffiffiffi 2p a3: (21) The DIHEDRAL ANGLE is a/C30tan/C2812ffiffiffi 2piCkCiCkA /C30sin/C2811 3ffiffiffi 3piCkCiCkA /C30cos/C2811 3iCkCiCkA :70:53/C14: (22) By slicing a tetrahedron as shown above, a SQUARE can be obtained. This cut divides the tetrahedron into two congruent solids rotated by 90 8. The projection of a tetrahedron can be an EQUILATERAL TRIANGLE or a SQUARE (Steinhaus 1983, pp. 191 /C1/192). Now consider a general (not necessarily regular)tetrahedron, defined as a convex POLYHEDRON con- sisting of four (not necessarily identical) TRIANGULAR faces. Let the tetrahedron be specified by its VERTICES atxi;yi;zi ðÞ where i/C301, ..., 4. Then the VOLUME is given by V/C301 3!x1y1z11 x2y2z21 x3y3z31 x4y4z41iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0: (23) Specifying the tetrahedron by the three EDGE vectors a,b, and cfrom a given VERTEX , the VOLUME is V/C301 3!a /C215(b/C29c) jj : (24) If the faces are congruent and the sides have lengths a,b, and c, then V/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27b2/C28c2 ðÞ a2/C27c2/C28b2 ðÞ b2/C27c2/C28a2 ðÞ 72s (25) (Klee and Wagon 1991, p. 205). In general, if the edgebetween vertices iand jare of length /dij/, then the volume Vis given by the C AYLEY- MENGER DETERMI- NANT 288V2/C3001 1 1 1 10 d2 12d213d214 1d221 0d223d224 1d231d232 0d234 1d241d242d243 0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0: (26) Consider an arbitrary TETRAHEDRON A1A2A3A4with triangles T1/C30DA2A3A4;T2/C30DA1A3A4;T3/C30DA1A2A4; and T4/C30A1A2A3:Let the areas of these triangles be s1;s2;s3;and s4;respectively, and denote the DIHEDRAL ANGLE with respect to TiandTjfori"j/C301;2;3;4b y uij:Then the four face areas are connected by s2 k/C30X j"k 15j54s2j/C282X i;j"k 15i;j54sisjcosuij (27) involving the six DIHEDRAL ANGLES (Dostor 1905, pp. 252 /C1/293; Lee 1997). This is a generalization of the LAW OF COSINES to the tetrahedron. Furthermore, for any i"j/C301;2;3;4; V/C302 3lijsisjsinuij; (28) where lijis the length of the common edge of TiandTj (Lee 1997). LetAbe the set of edges of a tetrahedron and P(A) the power set of A. Write ¯tfor the complement in Aof an element t/C23P(A):LetFbe the set of triples fx;y;zg/C23 P(A) such that x;y;zspan a face of the tetrahedron, and let Gbe the set of eSf ðÞ@e@fiCjiCk /C23P(A);so that e;f/C23Fand e"f:InG, there are therefore three elements which are the pairs of opposite edges. Now define D, which associates to an edge xof length L the quantity L=ffiffiffiffiffiffi 12piCjiCk 2;p, which associates to an element t/C23P(A) the product of D(x) for all x/C23t;ands, which associates to tthe sum of D(x) for all x/C23t:Then the VOLUME of a tetrahedron is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX t/C23G(s(¯t)/C28s(t))p(t)/C28X t/C23Fp(t)r (29) (P. Kaeser). The analog of G AUSS’S CIRCLE PROBLEM can be asked for tetrahedra: how many LATTICE POINTS lie within a tetrahedron centered at the ORIGIN with a given INRADIUS (Lehmer 1940, Granville 1991, Xu and Yau 1992, Guy 1994).There are a number of interesting and unexpected theorems on the properties of general (i.e., notnecessarily regular) tetrahedron (Altshiller-Court 1979). If a plane divides two opposite edges of a tetrahedron in a given ratio, then it divides thevolume of the tetrahedron in the same ratio (Altshil- ler-Court 1979, p. 89). It follows that any plane passing through a BIMEDIAN of a tetrahedron bisects the volume of the tetrahedron (Altshiller-Court 1979, p. 90). Let the vertices of a tetrahedron be denoted A,B,C, and D, and denote the side lengths BC/C30a,CA/C30b, AB/C30c,DA/C30a?;DB/C30b?;and DC/C30c?:Then if D denotes the area of the triangle with sides of lengths byaa?;bb?;andcc?;the VOLUME and CIRCUMRADIUS of the tetrahedron are related by the beautiful formula 6RV/C30D (30) (Crelle 1821, p. 117; von Staudt 1860; Rouche ´and Comberousse 1922, pp. 568 /C1/576 and 643 /C1/664; Alt- shiller-Court 1979, p. 250). See also AUGMENTED TRUNCATED TETRAHEDRON , BANG’S THEOREM ,C UBE TETRAHEDRON PICKING , EHRHART POLYNOMIAL ,H ERONIAN TETRAHEDRON , HILBERT’S 3RD PROBLEM ,ISOSCELES TETRAHEDRON , PENTATOPE ,R EULEAUX TETRAHEDRON ,S IERPINSKI TETRAHEDRON ,SPHERE TETRAHEDRON PICKING ,STEL- LA OCTANGULA ,T ANGENT SPHERES ,T ANGENTIAL TETRAHEDRON ,TETRAHEDRON 4-COMPOUND ,TETRA- HEDRON 5-COMPOUND ,TETRAHEDRON 10-COMPOUND , TRIRECTANGULAR TETRAHEDRON ,TRUNCATED TETRA- HEDRON References Altshiller-Court, N. "The Tetrahedron." Ch. 4 in Modern Pure Solid Geometry. New York: Chelsea, pp. 48 /C1/110, 1979. Balliccioni, A. Coordonne ´es barycentriques et ge´ome´trie. Claude Hermant, 1964. Bell, A. G. "The Tetrahedral Principle in Kite Structure." Nat. Geographic 44, 219 /C1/251, 1903. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 228, 1987. Couderc, P. and Balliccioni, A. Premier Livre du Te´trae`dre. Paris: Gauthier-Villars, 1935. Crelle, A. L. "Einige Bemerkungen u¨ber die dreiseitige Pyramide." Sammlung mathematischer Aufsa ¨tze u. Be- merkungen 1, 105 /C1/132, 1821. Cundy, H. and Rollett, A. "Tetrahedron. 33." §3.5.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 84, 1989. Davie, T. "The Tetrahedron." http://www.dcs.st-and.ac.uk/ ~ad/mathrecs/polyhedra/tetrahedron.html. Dostor, G. Ele´ments de la the´orie des de´terminants, avec application a` l’alge`bre, la trigonome ´trie et la ge´ome´trie analytique dans le plan et l’espace, 2e`me ed. Paris: Gauthier-Villars, pp. 252 /C1/293, 1905. Gardner, M. "Tetrahedrons." Ch. 19 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 183 /C1/194, 1984. Dostor, G. Ele´ments de la the´orie des de´terminants, avec application a` l’alge`bre, la trigonome ´trie et la ge´ome´trie analytique dans le plan et l’espace, 2e`me ed. Paris: Gauthier-Villars, 1905. Granville, A. "The Lattice Points of an n-Dimensional Tetrahedron." Aequationes Math. 41, 234 /C1/241, 1991. Guy, R. K. "Gauß’s Lattice Point Problem." §F1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 240 /C1/241, 1994. Harris, J. W. and Stocker, H. "Tetrahedron." §4.3.1 and 4.4.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 98 /C1/100, 1998. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory, rev. ed. Washington, DC: Math. Assoc. Amer., 1991. Langman, H. Scripta Math. , Mar.-Jun. 1951. Lee, J. R. "The Law of Cosines in a Tetrahedron." J. Korea Soc. Math. Ed. Ser. B: Pure Appl. Math. 4,1/C1/6, 1997. Lehmer, D. H. "The Lattice Points of an n-Dimensional Tetrahedron." Duke Math. J. 7, 341 /C1/353, 1940. Lesage, J. "Alexander Graham Bell Museum: Tribute to Genius." Nat. Geographic 60, 227 /C1/256, 1956. Rouche ´, E. and de Comberousse, C. Traite ´ de Ge´ome´trie, nouv. e´d., vol. 1: Ge´ome´trie plane. Paris: Gauthier-Villars, 1922.Rouche ´, E. and de Comberousse, C. Traite ´ de Ge´ome´trie, nouv. e´d., vol. 2: Ge´ome´trie dans l’espace. Paris: Gauthier- Villars, 1922. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 191 /C1/192 and 246 /C1/247, 1999. Trigg, C. W. "Geometry of Paper Folding. II. Tetrahedral Models." School Sci. and Math. 54, 683 /C1/689, 1954. von Staudt, K. G. C. "Ueber einige geometrische Sa¨tze." J. reine angew. Math. 57,88/C1/89, 1860. Wells, D. "Puzzle Page." Games and Puzzles. Sep. 1975. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 217 /C1/218, 1991. Wenninger, M. J. "The Tetrahedron." Model 1 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 14, 1989. Xu, Y. and Yau, S. "A Sharp Estimate of the Number of Integral Points in a Tetrahedron." J. reine angew. Math. 423, 199/C1/219, 1992. Tetrahedron 4-Compound See also TETRAHEDRON ,TETRAHEDRON 5-COMPOUND , TETRAHEDRON 10-COMPOUND Tetrahedron 5-Compound APOLYHEDRON COMPOUND composed of five TETRA- HEDRA which is also one of the ICOSAHEDRON STELLA- TIONS . The 5 /C294 vertices of the tetrahedron are then 20 vertices of the DODECAHEDRON . Two tetrahedron 5- compounds of opposite CHIRALITY combine to make a TETRAHEDRON 10-COMPOUND (Cundy and Rollett 1989). The diagram above shows pieces which can be assembled to form the tetrahedron 5-compound (Cundy and Rollett 1989). The construction itself is rather challenging, and involves constructing a base tetrahedron, placing a "cap" around one of the apexes, and affixing a triangular pyramid to the opposite face. Twelve pyramids with complicated bases are then constructed and attached edge-to-edge in chains of three. The four chains of pyramids are then arranged about the eight vertices of the original two tetrahe- dra, with the points of coincidence of the three pyramids in each chain attached such that they coincide with intersections of the original two tetra- hedra such that five pyramids touch at a single point. The position, size, and orientation of the pyramidal cap and pyramids are illustrated in the diagram above, where a /C30cos /C2811 83ffiffiffi 2p /C27ffiffiffiffiffiffi10piCkCiCkAhi :22 :2388/C14 (1) d /C301 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23 /C283ffiffiffi 5pq (2) h /C301 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 33/C27ffiffiffi 5piCkCiCkAr (3) l1 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 153 /C28ffiffiffi 5piCkCiCkAr (4)l2 /C301 2ffiffiffi 2p (5) l3 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C27ffiffiffi 5pq (6) s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C27ffiffiffi 5pq (7) s1 /C301 5ffiffiffiffiffiffi 10p (8) s2 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 57 /C273ffiffiffi 5piCkCiCkAr : (9) The edge lengths and angles of the cap are given by b /C30cos /C2811 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C283ffiffiffi 5pqiCkniCko :82 :2388/C14 (10) e1 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C28ffiffiffi 5pq (11) e2 /C301 25 /C28ffiffiffi 5piCkCiCkA (12) e3 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C283ffiffiffi 5pq (13) e4 /C30e1 (14) e5 /C30s : (15) See also ICOSAHEDRON STELLATIONS ,POLYHEDRON COMPOUND ,T ETRAHEDRON ,T ETRAHEDRON 4-COM- POUND ,TETRAHEDRON 10-COMPOUND References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 135, 1987. Cundy, H. and Rollett, A. "Five Tetrahedra in a Dodecahe- dron." §3.10.8 in Mathematical Models, 3rd ed. Strad- broke, England: Tarquin Pub., pp. 139 /C1/141, 1989. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, p. 44, 1989. Tetrahedron 10-Compound Two TETRAHEDRON 5-COMPOUNDS of opposite CHIRAL- ITYcombined. See also POLYHEDRON COMPOUND ,TETRAHEDRON 5- COMPOUND References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 135, 1987. Cundy, H. and Rollett, A. "Ten Tetrahedra in a Dodecahe- dron." §3.10.9 in Mathematical Models, 3rd ed. Strad- broke, England: Tarquin Pub., pp. 141 /C1/142, 1989. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, p. 45, 1989. Tetrahedron Circumscribing References Finch, S. "Circumscribing Tetrahedron of Least Volume." http://www.mathsoft.com/asolve/ecalabi.html. van der Burg, J. W. "An Accurate and Robust Algorithm for the In-Sphere Criterion for Automated Delaunay-Based Tetrahedral Grid Generation." Paper P 98212 presented at The 6th International Conference on Numerical Grid Generation for Computational Field Simulation, Univer- sity of Greenwich, London, July 1998. 1998. Tetrahedron Tetrahedron Picking The expected VOLUME of a TETRAHEDRON with ver- tices chosen at random inside another TETRAHEDRON of unit volume appears to be numerically close to 1/ 57, but the exact analytic value is not known (Croft et al. 1991, p. 54). According to Solomon (1978, p. 124), "Explicit values for random points in non-spherical regions such as tetrahedrons, parallelepipeds, etc., have apparently not yet been successfully calculated." See also BALL TETRAHEDRON PICKING ,SPHERE TET- RAHEDRON PICKING References Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Random Polygons and Polyhedra." §B5 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 54 /C1/57, 1991. Klee, V. "What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body." Amer. Math. Monthly 76, 286 /C1/288, 1969. Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, p. 124, 1978. Tetrahemihexacron The DUAL POLYHEDRON of the TETRAHEMIHEXAHE- DRON U4 and Wenninger dual W67 :/See also DUAL POLYHEDRON ,T ETRAHEMIHEXAHE- DRON ,UNIFORM POLYHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 101 /C1/103, 1983. Tetrahemihexahedron The UNIFORM POLYHEDRON U4whose DUAL POLYHE- DRON is the TETRAHEMIHEXACRON . It has S CHLA ¨FLI SYMBOL r?3 3iCniCo and W YTHOFF SYMBOL3 23½2:Its faces are 4 f3g/C273f4g:It is a faceted form of the OCTAHE- DRON . Its CIRCUMRADIUS is R/C301 2ffiffiffi 2p : The CONVEX HULL of the tetrahemihexahedron is the OCTAHEDRON . References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 101 /C1/102, 1971. Tetrakaidecagon TETRADECAGON Tetrakaidecahedron TETRADECAHEDRON Tetrakis Hexahedron The 24-faced DUAL POLYHEDRON of the TRUNCATED OCTAHEDRON A12and Wenninger dual W7 : It can be constructed by CUMULATION of a unit edge-length CUBE by a pyramid with height 1/6. The edge lengths for the tetrakis hexahedron con- structed as the dual of the TRUNCATED OCTAHEDRON with unit edge lengths are s1 /C309 8ffiffiffi 2p (1) s2 /C303 2ffiffiffi 2p : (2) Normalizing so that s1 /C301 gives a tetrakis hexahe- dron with SURFACE AREA and VOLUME S /C3016 3ffiffiffi5p (3) V /C3032 9 : (4) See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , ICOSITETRAHEDRON ,TRUNCATED OCTAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 14 /C1/16, 1983. Tetranacci Number The tetranacci numbers are a generalization of the FIBONACCI NUMBERS defined by T0 /C300; T1 /C301; T2 /C301; T3 /C302; and the RECURRENCE RELATION Tn /C30Tn/C281 /C27Tn/C282 /C27Tn/C283 /C27Tn/C284 for n ]4: They represent the n /C304 case of the FIBONACCI N-STEP NUMBERS . The first few terms are 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ... (Sloane’s A000078). The ratio of adjacent terms tends to 1.92756, which is the REAL ROOT of x5 /C282x4 /C271 /C300 :/ See also FIBONACCI N-STEP NUMBER ,F IBONACCI NUMBER ,TRIBONACCI NUMBERReferences Sloane, N. J. A. Sequences A000078/M1108 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Tetrastigm Lachlan’s term for a set of four points, no three of which are COLLINEAR . See also TETRAGRAM References Lachlan, R. "Properties of a Tetrastigm." §139 /C1/146 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 85 /C1/90, 1893. Tetration POWER TOWER Tetriamond The three 3-polyiamonds are called tetriamonds. See also POLYIAMOND Tetrix The 3-D analog of the S IERPINSKI SIEVE illustrated above, also called the S IERPINSKI SPONGE or S IER- PINSKI TETRAHEDRON . Let Nnbe the number of tetrahedra, Lnthe length of a side, and Anthe fractional VOLUME of tetrahedra after the nth itera- tion. Then Nn/C304n(1) Ln/C301 2iCkCiCkAn /C302/C28n(2) An/C30L3 nNn/C301 2iCkCiCkAn : (3) The CAPACITY DIMENSION is therefore dcap/C30/C28lim n0/C12lnNn lnLn/C30/C28lim n0/C12ln 4nðÞ ln 2/C28nðÞ /C30ln 4 ln 2 /C302ln2 ln 2/C302; (4) so the tetrix has an INTEGER CAPACITY DIMENSION (which is one less than the DIMENSION of the 3-D TETRAHEDRA from which it is built), despite the fact that it is a FRACTAL . The following illustrations demonstrate how the dimension of the tetrix can be the same as that of the PLANE by showing three stages of the rotation of a tetrix, viewed along one of its edges. In the last frame, the tetrix "looks" like the 2-D PLANE . See also MENGER SPONGE ,SIERPINSKI SIEVE References Allanson, B. "The Fractal Tetrahedron" java applet. http:// www.adelaide.net.au/~allanson/Fractet.html. Dickau, R. M. "Sierpinski Tetrahedron." http://forum.s- warthmore.edu/advanced/robertd/tetrahedron.html. Eppstein, D. "Sierpinski Tetrahedra and Other Fractal Sponges." http://www.ics.uci.edu/~eppstein/junkyard/sier- pinski.html. Weisstein, E. W. "Fractals." M ATHEMATICA NOTEBOOK FRAC- TAL.M . Tetromino The five 4- POLYOMINOES , known as STRAIGHT ,L-,T-, SQUARE , and SKEW . References Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150/C1/156, May 1957. Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer- ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 124 /C1/140, 1959. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 80 /C1/81, 1975. Tg TANGENT Th HYPERBOLIC TANGENTThaˆbit ibn Kurrah Rule A number OF THE FORM 3/C2152n/C281 which is PRIME is sometimes called a Tha ˆbit ibn Kurrah number. The indices for the first few such numbers are 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, ... (Sloane’s A002235). Riesel(1969) extended the search to n51000 ;and the largest known today is n/C3026459. The numbers arise in a beautiful result of Tha ˆbit ibn Kurrah dating back to the tenth century (Woepcke1852; Escott 1946; Dickson 1952, pp. 5 and 39; Borho1972). Take n]2 and suppose that h¼3/C2152 n/C281 ð1Þ t/C303/C2152n/C281/C281 (2) s/C309/C21522n/C281/C281 (3) are all PRIME . Then 2nht;2ns ðÞ are an AMICABLE PAIR . This form was rediscovered by Fermat (1636) andDescartes (1638) and generalized by Euler to E ULER’S RULE (Borho 1972). In order for such numbers to exist, there must beprime 3 /C2152 n/C281 for two consecutive n, leaving only the possibilities 1, 2, 3, 4, and 6, 7. Of these, sis prime forn/C302, 4, and 7, giving the amicable pairs (220, 284), (17296, 18416), and (9363584, 9437056). In fact, various rules can be found that are analogous to Tha ˆbit ibn Kurrah’s. Denote a "Tha ˆbit rule" by Tb1;b ðÞ 2;p;F1;F2for given natural numbers b1 andb2;a prime pnot dividing b1;b2;and polynomials F1(X);F2(X)/C23Z[X]:Then a necessary condition for the set of AMICABLE PAIRS m1;m2 ðÞ of the form mi/C30 pnbiqi(i/C301, 2) with q1;q2prime and na natural number to be infinite is that p p/C281/C30b1 sb1ðÞ/C27b2 sb2ðÞ; (4) where s(n) is the divisor function (Borho 1972). As a result, mi/C30pnbiqi(i/C301, 2) form an AMICABLE PAIR ,i f for some n]1;both qi¼pnðp/C281Þðb1þb2Þ sðbiÞ/C281 (5) fori/C301, 2 are prime integers not dividing bip(Borho 1972). The following table summarizes some of the known Thaˆbit ibn Kurrah rules T(au;p;(u/C271)X;(u/C27 1)s(u)X/C281) (Borho 1972, te Riele 1974). au /s(u)/ p 22/5/C21511/ 72 127 /32/C2157/C21513// 5/C21517/ 108 193 /32/C2155/C21513// 11 /C21519/ 240 449 /32/C21572/C21513// 5/C21541/ 252 457 /32 /C215 72 /C215 13 /C215 19// 5 /C215 193 / 1164 2129 /34 /C215 5 /C215 11// 29 /C215 89/ 2700 5281 /32 /C215 7 /C215 13 /C215 41 /C215 163 // 5 /C215 977 / 5868 10753 /32 /C215 5 /C215 19 /C215 37// 7 /C215 887 / 7104 13313 /34 /C215 7 /C215 11 /C215 29// 13 /C215 521 / 7308 14081 /32 /C215 72 /C215 13 /C215 19 /C215 29// 41 /C215 173 / 7308 14401 /32 /C215 5 /C215 13 /C215 19// 29 /C215 569 / 17100 33601 /32 /C215 72 /C215 13// 5 /C215 53 /C215 97/ 31752 57457 /32 /C215 52 /C215 13 /C215 31// 149 /C215 449 / 67500 134401 /33 /C215 53 /C215 13// 149 /C215 449 / 67500 134401 /2 /C215 72 /C215 19 /C215 23// 11 /C215 13523 / 162288 311041 /34 /C215 5 /C215 11 /C215 59// 89 /C215 5309 / 477900 950401 /34 /C215 5 /C215 112 /C215 71// 709 /C215 2129 / 1512300 3021761 /32 /C215 72 /C215 11 /C215 19 /C215 43 /C215 89//293 /C215 22961 / 6750828 13478401 /22 /C215 31// 17 /C215 107 /C215 4339 / 8436960 16329601 28 /257 /C215 33023 / 8520192 17007103 /23 /C215 19 /C215 137 // 83 /C215 218651 / 18366768 36514801 /27 /C215 263 // 4271 /C215 280883 / 1199936448 2399587741 See also AMICABLE PAIR,E ULER’S RULE,R IESEL NUMBER References Borho, W. "On Thabit ibn Kurrah’s Formula for Amicable Numbers." Math. Comput. 26, 571 /C1/578, 1972. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, 1952. Escott, E. B. E. "Amicable Numbers." Scripta Math. 12,61/C1/ 72, 1946. Riesel, H. "Lucasian Criteria for the Primality of N /C30h 2nðÞ/C281 :/" Math. Comput. 23, 869 /C1/875, 1969. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkha ¨user, p. 394, 1994. Sloane, N. J. A. Sequences A002235/M0545 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. te Riele, H. J. J. "Four Large Amicable Pairs." Math. Comput. 28, 309 /C1/312, 1974. Woepcke, F. J. Asiatique 20, 320 /C1/429, 1852. Thales’ Theorem An ANGLE inscribed in a SEMICIRCLE is a RIGHT ANGLE . See also RIGHT ANGLE ,SEMICIRCLETheorem A statement which can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a PROOF . According to the Nobel Prize-winning physicist Ri- chard Feynman (1985), any theorem, no matter how difficult to prove in the first place, is viewed as "TRIVIAL " by mathematicians once it has been proven. Therefore, there are exactly two types of mathema- tical objects: TRIVIAL ones, and those which have not yet been proven. The late mathematician P. Erdos described a math- ematician as "a machine for turning coffee into theorems" (Hoffman 1998, p. 7). R. Graham has estimated that upwards of 250,000 mathematical theorems are published each year (Hoffman 1998, p. 204). See also AXIOM ,A XIOMATIC SYSTEM ,C OROLLARY , DEEP THEOREM ,PORISM ,LEMMA ,POSTULATE ,PRIN- CIPLE ,PROOF ,PROPOSITION ,TRIVIAL References Feynman, R. P. and Leighton, R. Surely You’re Joking, Mr. Feynman! New York: Bantam Books, 1985. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. TH //C215/OREM //C214 Computer-Supported Mathematical Theorem Proving. http://www.theorema.org/. Theorema Egregium GAUSS’S THEOREMA EGREGIUM Theory A theory is a set of SENTENCES which is CLOSED under logical implication. That is, given any subset of SENTENCES s1 ; s2 ; ... fg in the theory, if SENTENCE r is a logical consequence of s1 ; s2 ; ... fg ; then r must also be in the theory. See also LOGIC ,SENTENCE References Enderton, H. B. Elements of Set Theory. New York: Aca- demic Press, 1977. Theta Functions See also ABELIAN FUNCTION ,JACOBI THETA FUNC- TIONS ,M OCK THETA FUNCTION ,N EVILLE THETA FUNCTIONS ,R AMANUJAN THETA FUNCTIONS ,R IE- MANN THETA FUNCTION ,SIEGEL THETA FUNCTION Theta Operator In the NOTATION of Watson (1966), q/C13zd dz : References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Theta Series See also EISENSTEIN SERIES ,LEECH LATTICE Theta Subgroup LAMBDA GROUP Theta-0 Graph The GRAPH on seven nodes illustrated above. See also 15 PUZZLE References Archer, A. F. "A Modern Treatment of the 15 Puzzle." Amer. Math. Monthly 106, 793 /C1/799, 1999. Wilson, R. M. "Graph Puzzles, Homotopy, and the Alternat- ing Group." J. Combin. Th. Ser. B 16,86/C1/96, 1974. Thickness GRAPH THICKNESS Thiele’s Interpolation Formula Let r be a RECIPROCAL DIFFERENCE . Then Thiele’s interpolation formula is the CONTINUED FRACTION f ðxÞ¼f ðx1 Þþx /C28 x1 p ðx1 ;x2 Þþx /C28 x2 p2 ðx1 ;x2 ;x3 /C28 f ðx1 Þþ x /C28 x3 r3x1 ; x2 ; x3 ; x4 ðÞ /C28 r x1 ; x2 ðÞ /C27 ... : References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 881, 1972. Milne-Thomson, L. M. The Calculus of Finite Differences . London: Macmillan, 1951.Thiessen Polytope VORONOI POLYGON Thin Plate Spline This entry contributed by SERGE BELONGIE The thin plate spline is the two-dimensional analog of the CUBIC SPLINE in 1-D. It is the fundamental solution to the BIHARMONIC EQUATION , and has the form U(r) /C30r2 ln r : Given a set of data points, a weighted combination of thin plate splines centered about each data point gives the interpolation function that passes through the points exactly while minimizing the so-called "bending energy." Bending energy is defined here as the integral over R2 of the squares of the second derivatives, If x; yðÞ½/C138 /C30gg R2f2 xx /C272f2 xy /C27f2 yy dx dy: Regularization may be used to relax the requirement that the interpolant pass through the data points exactly. The name "thin plate spline" refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the z direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transfor- mation, one interprets the lifting of the plate as a displacement of the x or y coordinates within the plane. Thus, in general, two thin plate splines are needed to specify a 2-D coordinate transformation. See also CUBIC SPLINE ,SPLINE References Bookstein, F. L. "Principal Warps: Thin Plate Splines and the Decomposition of Deformations." IEEE Trans. Pattern Anal. Mach. Intell. 11, June 1989. Duchon, J. "Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces." RAIRO Analyse Nume ´rique 10,5/C1/12, 1976. Meinguet, J. "Multivariate Interpolation at Arbitrary Points Made Simple." J. Appl. Math. Phys. 30, 292 /C1/304, 1979. Wahba, G. Spline Models for Observational Data. Philadel- phia, PA: SIAM, 1990. Third Curvature Also known as the TOTAL CURVATURE . The linear element of the INDICATRIX dsP/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds2 T/C27ds2 Bq : See also LANCRET EQUATION Third Fundamental Form Let M be a REGULAR SURFACE with vP ; wPpoints in the TANGENT SPACE Mpof M. Then the third funda- mental form is given by III vP ; wP ðÞ /C30S vPðÞ /C215 S wPðÞ ; where S is the SHAPE OPERATOR . See also FIRST FUNDAMENTAL FORM,FUNDAMENTAL FORMS ,SECOND FUNDAMENTAL FORM,SHAPE OPERA- TOR References Gray, A. "The Three Fundamental Forms." §16.6 in Modern Differential Geometry of Curves and Surfaces with Math- ematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 380 /C1/ 382, 1997. Third Kind In the theory of special functions, a class of functions is said to be "of the third kind" if it is similar to but distinct from previously defined functions already defined to be of the FIRST and SECOND KINDS . The only common functions of the third kind are the ELLIPTIC INTEGRAL OF THE THIRD KIND II(n; f; k) and the Bessel function of the third kind (more commonly called the HANKEL FUNCTION ). See also ELLIPTIC INTEGRAL OF THE THIRD KIND, FIRST KIND,H ANKEL FUNCTION ,S ECOND KIND, SPECIAL FUNCTION Thirteen 13 Thom Transversality Theorem References Pohl, W. F. "The Self-Linking Number of a Closed Space Curve." J. Math. Mech. 17, 975 /C1/985, 1968. Thomae’s Theorem G(x /C27 y /C27 s /C27 1) G(x /C27 s /C27 1)G(y /C27 s /C27 1) 3 F2/C28a;/C28b; x /C27y /C27s /C271 x /C27s /C271 ; y /C27s /C271;1iCkniCko /C30G(a /C27 b /C27 s /C27 1) G(a /C27 s /C27 1)G(b /C27 s /C27 1) 3 F2/C28x;/C28y; a /C27b /C27s /C271 a /C27s /C271; b /C27s /C271 ;1iCkniCko ; where G(z) is the GAMMA FUNCTION and the function 3F2(a ; b ; c; d; e; z)isa GENERALIZED HYPERGEO- METRIC FUNCTION . This theorem is equivalent to equation (1) from Bailey (1935, p. 14) (Hardy 1999, p. 111). See also GAUSS’S HYPERGEOMETRIC THEOREM ,GEN- ERALIZED HYPERGEOMETRIC FUNCTION References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: University Press, p. 14, 1935.Hardy, G. H. "A Chapter from Ramanujan’s Note-Book." Proc. Cambridge Philos. Soc. 21, 492 /C1/503, 1923. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 104 /C1/105, 1999. Thomae, J. "Ueber die Funktionen welche durch Reihen von der Form Dargestellt Werden: 1 /C27pp?pƒ 1qq ƒ/C27/C1/C1/C1:/" J. fu¨r Math. 87,26/C1/73, 1879. Thomas Equation The PARTIAL DIFFERENTIAL EQUATION uxy /C27 aux /C27 buy /C27 guxuy /C300: References Rosales, R. R. "Exact Solutions of a Certain Nonlinear Wave Equation." J. Math. Phys. 45, 235 /C1/265, 1966. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 132, 1997. Thomas-Fermi Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C30y3 =2x/C281 =2 : References Bender, C. M. and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill, p. 25, 1978. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Thomassen Graph The HYPOTRACEABLE GRAPH illustrated above. See also HYPOTRACEABLE GRAPH ,THOMSEN GRAPH References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 240, 1976. Thomassen, C. "Hypohamiltonian and Hypotraceable Graphs." Disc. Math. 9,9 1/C1/96, 1974. Thompson Group The SPORADIC GROUP Th. References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/Th.html. Thompson Lamp Paradox A lamp is turned on for 1/2 minute, off for 1/4 minute, on for 1/8 minute, etc. At the end of one minute, the lamp switch will have been moved /C2100 times, where /C2100 is ALEPH-0 . Will the lamp be on or off? This PARADOX is actually nonsensical, since it is equivalent to asking if the "last" INTEGER is EVEN or ODD. References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 106 /C1/107, 1998. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 19 /C1/ 23, 1995. Thompson’s Functions BEI,BER,KELVIN FUNCTIONS Thom’s Eggs EGG-shaped curves constructed using multiple CIR- CLES which Thom (1967) used to model Megalithic stone rings in Britain. See also EGG,OVAL References Dixon, R. Mathographics. New York: Dover, p. 6, 1991. Thom, A. "Mathematical Background." Ch. 4 in Megalithic Sites in Britain. Oxford, England: Oxford University Press, pp. 27 /C1/33, 1967.Thomsen Graph The COMPLETE BIPARTITE GRAPH K3 ; 3 ; which is equivalent to the UTILITY GRAPH . It has a CROSSING NUMBER 1. See also COMPLETE BIPARTITE GRAPH ,C ROSSING NUMBER (GRAPH ), THOMASSEN GRAPH ,U TILITY GRAPH References Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 93, 1984. Thomsen’s Figure Take any TRIANGLE with VERTICES A, B, and C. Pick a point A1on the side opposite A, and draw a line PARALLEL to AB. Upon reaching the side AC at B1 ; draw the line PARALLEL to BC. Continue (left figure). Then A3 /C30A1 for any TRIANGLE .IfA1 is the MIDPOINT of BC, then A2 /C30A1 (right figure). See also MIDPOINT ,TRIANGLE References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 234, 1979. Thomson Problem Determine the stable equilibrium positions of N classical electrons constrained to move on the surface of a SPHERE and repelling each other by an inverse square law. Exact solutions for N/C302 to 8 are known, butN/C309 and 11 are still unknown. In reality, Earnshaw’s theorem guarantees that no system of discrete electric charges can be held in stable equilibrium under the influence of their elec- trical interaction alone (Aspden 1987). See also FEJES TO´ TH’S PROBLEM References Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Dowla, F.; and Wooten, F. "Method of Constrained Global Optimiza- tion." Phys. Rev. Let. 72, 2671 /C1/2674, 1994. Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Dowla, F.; and Wooten, F. "Method of Constrained Global Optimiza- tion--Reply." Phys. Rev. Let. 74, 1483, 1995. Ashby, N. and Brittin, W. E. "Thomson’s Problem." Amer. J. Phys. 54, 776 /C1/777, 1986. Aspden, H. "Earnshaw’s Theorem." Amer. J. Phys. 55, 199 /C1/ 200, 1987. Berezin, A. A. "Spontaneous Symmetry Breaking in Classi- cal Systems." Amer. J. Phys. 53, 1037, 1985. Calkin, M. G.; Kiang, D.; and Tindall, D. A. "Minimum Energy Configurations." Nature 319, 454, 1986. Erber, T. and Hockney, G. M. "Comment on ‘Method of Constrained Global Optimization."’ Phys. Rev. Let. 74, 1482 /C1/1483, 1995. Marx, E. "Five Charges on a Sphere." J. Franklin Inst. 290, 71 /C1/74, Jul. 1970. Melnyk, T. W.; Knop, O.; and Smith, W. R. "Extremal Arrangements of Points and Unit Charges on a Sphere: Equilibrium Configurations Revisited." Canad. J. Chem. 55, 1745 /C1/1761, 1977. Whyte, L. L. "Unique Arrangement of Points on a Sphere." Amer. Math. Monthly 59, 606 /C1/611, 1952. Thomson’s Principle DIRICHLET’S PRINCIPLE Thousand /1;000 /C30103 : The word "thousand" appears in common expressions in a number of languages, for example, "a thousand pardons" in English and "tusen takk" ("a thousand thanks") in Norwegian. See also HUNDRED ,LARGE NUMBER ,MILLION Three 3 Three Conics Theorem If three conics pass through two given points Q andQ ?; then the lines joining the other two intersections of each pair of conics PijP ?ij are CONCURRENT at a point X (Evelyn 1974, p. 15). The converse states that if two conics E2 and E3 meet at four points Q, Q ?; P1 ; and Q1 ; and if P2Q2and P3Q3are chords of E3and E2 ; respectively, which meet on P1Q1 ; then the six points lie on a conic. The dual of the theorem states that if three conics share two common tangents, then their remaining pairs of common tangents intersect at three collinear points. If the points Q and Q ? are taken as the POINTS AT INFINITY , then the theorem reduces to the theorem that RADICAL LINES of three CIRCLES are CONCURRENT in a point known as the RADICAL CENTER (Evelyn 1974, p. 15). If two of the points Pij and P?ij are taken as the POINTS AT INFINITY , then the theorem becomes that if two circles C1 and C2 pass through two points Q and Q ? on a conic E, then the lines determined by the pair of intersections of each circle with the conic are parallel (Evelyn 1974, p. 15). See also CONIC SECTION ,FOUR CONICS THEOREM , RADICAL CENTER References Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Three-Conics Theorem." §2.2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 11 /C1/18, 1974. Three Curtain Problem MONTY HALLPROBLEM Three Dogs Problem MICEPROBLEM Three j-Symbol WIGNER 3 J-SYMBOL Three Jug Problem Given three jugs with xpints in the first, yin the second, and zin the third, obtain a desired amount in one of the vessels by completely filling up and/or emptying vessels into others. This problem can be solved with the aid of TRILINEAR COORDINATES . References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 28 and 40, 1987. Coxeter, H. S. M. and Greitzer, S. L. "The Three Jug Problem." §4.6 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 89 /C1/93, 1967. O’Beirne, T. H. Puzzles and Paradoxes. New York: Oxford University Press, pp. 49 /C1/75, 1965. Perel’man, A. I. Zanumatel’naya Geometria. Moscow, 1958. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 61 /C1/63, 1999. Tweedie, M. C. K. Math. Gaz. 23, 278 /C1/282, 1939. Three-Choice Polygon A LATTICE POLYGON formed by a THREE-CHOICE WALK . The anisotropic perimeter and area generating func- tion G(x; y; q) /C30X m ]1X n]1X a ]aC(m; n; a)xmynqa ; where C(m; n; a) is the number of polygons with 2m horizonal bonds, 2n vertical bonds, and area a, is not yet known in closed form, but it can be evaluated in polynomial time (Conway et al. 1997, Bousquet- Me´lou 1999). The perimeter-generating function G(x; x; 1) has a logarithmic singularity and so is not algebraic, but is known to be D-finite (Conway et al. 1997, Bousquet-Me ´lou 1999). The anisotropic area and perimeter generating func- tion G(x; y; q) satisfies an inversion relation OF THE FORM G(x; y; q) /C27y2G(x=y ; 1=y; 1 =q) (Bousquet-Me ´lou et al. 1999). References Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/ 9908123/. Conway, A.; Cuttmann, A. J.; and Delest, M. "On the Number of Three-Choice Polygons." Math. Comput. Model. 26,51/C1/58, 1997.Three-Choice Walk A SELF-AVOIDING WALK in which steps may be to the left, right, or straight ahead after a vertical step, but only straight ahead of to the left after a horizontal step. A LATTICE POLYGON formed by a three-choice walk is called a THREE-CHOICE POLYGON . References Bousquet-Me ´lou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. Inversion Relations, Reciprocity and Poly- ominoes. 23 Aug 1999. http://xxx.lanl.gov/abs/math.CO/ 9908123/. Three-Colorable COLORABLE Threefoil Knot TREFOIL KNOT Three-In-A-Row TIC-TAC-TOE ThreeJ Symbol WIGNER 3J-SYMBOL Three-Valued Logic A logical structure which does not assume the EXCLUDED MIDDLE LAW. Three truth values are possible: true, false, or undecided. There are 3072 such logics. See also EXCLUDED MIDDLE LAW,FUZZY LOGIC ,LOGIC Thue Constant The base-2 TRANSCENDENTAL NUMBER 0:11011011111011011111...2 ; where the nth bit is 1 if n is not divisible by 3 and is the complement of the (n=3)/th bit if n is divisible by 3. It is also given by the SUBSTITUTION MAP 00111 10110: In decimal, the Thue constant equals 0.8590997969.... See also RABBIT CONSTANT ,THUE- MORSE CONSTANT Thue Equation This entry contributed by K EVIN O’BRYANT A Thue equation is a D IOPHANTINE EQUATION of the form Anxn/C27An/C281xn/C281y/C27An/C282xn/C282y2/C27.../C27A0yn/C30M; with n]3;Ai/C23Z;M"0/C23Z;andx, yunknown integer variables. Thue (1909) proved that such an equation has only finitely many solutions, but it was not until much later that Tzanakis and de Weger (1989) gave a practical algorithm for finding bounds on xjjand yjj: Although these bounds can be astronomically large in some cases, they are typically small enough to allow an exhaustive search for all solutions. See also DIOPHANTINE EQUATION References Thue, A. "U¨ ber Anna¨herungswerte algebraischer Zahlen." J. reine angew. Math. 135, 284 /C1/305, 1909. Tzanakis, N. and de Weger, B. M. M. "On the Practical Solution of the Thue Equation." J. Number Th. 31,99/C1/ 132, 1989. Thue Sequence The SEQUENCE of BINARY DIGITS of the THUE CON- STANT ,0:110110111110110111110110110 ...2(Sloa- ne’s A014578). See also RABBIT CONSTANT ,THUE CONSTANT References Guy, R. K. "Thue Sequences." §E21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 223 /C1/224, 1994. Sloane, N. J. A. Sequences A014578 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Thue-Morse Constant The constant also called the PARITY CONSTANT and defined by P /C131 2X/C12 n/C300P(n)2/C28n /C300:4124540336401075977... (1) (Sloane’s A014571), where P(n) is the PARITY of n. Dekking (1977) proved that the Thue-Morse constant is TRANSCENDENTAL , and Allouche and Shallit give a complete proof correcting a minor error of Dekking. The Thue-Morse constant can be written in base 2 by stages by taking the previous iteration an ; taking the complement an ; and appending, producing a0 /C300:02 a1 /C300 :012 a2 /C300:01102 a3 /C300 :011010012 a4 /C300 :01101001100101102 : (2) This can be written symbolically as an/C271 /C30an /C27an/C215 2/C282n (3) with a0 /C300: Here, the complement is the number an such that an /C27an /C300 :11...12|fflfflfflfflffl{zfflfflfflfflffl} 2n; which can be foundfrom an /C27an /C30X2n k /C30112iCkCiCkAk /C301 /C2812iCkCiCkA2n 1 /C281 2/C281 /C301 /C282/C282n : (4) Therefore, an /C301 /C282/C282n /C28an ; (5) and an/C271 /C30an /C27 1 /C282/C282n /C28aniCjiCk 2/C282n (6) /C302/C282n/C27122n/C281iCjiCk 1 /C2822n aniCjiCk : (7) The regular CONTINUED FRACTION for the Thue-Morse constant is [0 2 2214352142154414124111 51415015511142141431412131612121 50124241252111552225111 1274 3 5 2111 41115154721221211501412 867374 1 1 1 5 5 1 1 6 1 2 7 2 1650 23 3 1 1 1 2 5 3 84 1 1 1 1284 ...] (Sloane’s A014572), and seems to continue with sporadic large terms in suspicious-looking patterns. A nonregular CONTINUED FRACTION is P /C301 3 /C281 2 /C281 4 /C283 16 /C2815 256 /C28255 65536 /C28 ...: (8) A related infinite product is 4P /C302 /C281 /C215 3 /C215 15 /C215 255 /C215 65535 /C1/C1/C1 2 /C215 4 /C215 16 /C215 256 /C215 65536 /C1/C1/C1: (9) The SEQUENCE a/C12/C300110100110010110100101100 . . . (Sloane’s A010060) is known as the T HUE- MORSE SEQUENCE . See also RABBIT CONSTANT ,THUE CONSTANT References Allouche, J. P.; Arnold, A.; Berstel, J.; Brlek, S.; Jockusch, W.; Plouffe, S.; and Sagan, B. "A Relative of the Thue- Morse Sequence." Discr. Math. 139, 455/C1/461, 1995. Allouche, J. P. and Shallit, J. "The Ubiquitous Prouhet- Thue-Morse Sequence." http://www.math.uwaterloo.ca/~shallit/Papers/ubiq.ps. Dekking, F. M. "Transcendence du nombre de Thue-Morse." Comptes Rendus de l’Academie des Sciences de Paris 285, 157/C1 /160, 1977. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html. Schroeppel, R. and Gosper, R. W. Item 122 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 56 /C1 /57, Feb. 1972. Sloane, N. J. A. Sequences A010060, A014571, and A014572 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Thue-Morse Sequence The INTEGER SEQUENCE (also called the MORSE- THUE SEQUENCE ) 01101001100101101001011001101001... (1) (Sloane’s A010060) which arises in the THUE- MORSE CONSTANT . It can be generated from the SUBSTITU- TION MAP 0 0 01 (2) 1 0 10 (3) starting with 0 as follows: 0 0 01 0 0110 0 01101001 0 ... (4) Writing the sequence as a POWER SERIES over the FINITE FIELD GF(2), F(x) /C300 /C271x /C271x2 /C270x3 /C271x4 /C27...; (5) then F satisfies the quadratic equation (1 /C27x)F2 /C27F /C30x 1 /C27 x2(mod 2): (6) This equation has two solutions, F and F ?; where F ? is the complement of F, i.e., F /C27F ?/C301 /C27x /C27x2 /C27x3 /C27.../C301 1 /C27 x ; (7) which is consistent with the formula for the sum of the roots of a quadratic. The equality (6) can be demonstrated as follows. Let (abcdef ...) be a short- hand for the POWER SERIES a /C27bx /C27cx2 /C27dx3 /C27...; (8) so F(x) is (0110100110010110...). To get F2 ; simply use the rule for squaring POWER SERIES over GF(2) (A /C27B)2 /C30A2 /C27B2 (mod 2); (9) which extends to the simple rule for squaring a POWER SERIES a0 /C27a1x /C27a2x2 /C27...iCjiCk2 /C30a0 /C27a1x2 /C27a2x4 /C27... (mod 2); (10) i.e., space the series out by a factor of 2, (0 1101001 ...), and insert zeros in the ODD places to get F2 /C30(0010100010000010...) : (11) Then multiply by x (which just adds a zero at the front) to get xF2 /C30(00010100010000010...) : (12) Adding to F2 gives (1 /C27x)F2 /C30(0011110011000011...) : (13)This is the first term of the quadratic equation, which is the Thue-Morse sequence with each term doubled up. The next term is F, so we have (1 /C27x)F2 /C30(0011110011000011...) (14) F /C30(0110100110010110...) : (15) The sum is the above two sequences XORed together (there are no CARRIES because we’re working over GF(2)), giving (1 /C27x)F2 /C27F /C30(0101010101010101...) : (16) We therefore have (1 /C27x)F2 /C27F /C30x 1 /C27 x2 /C30x /C27x3 /C27x5 /C27x7 /C27x9 /C27x11 /C27... (mod 2): (17) The Thue-Morse sequence is an example of a cube- free sequence on two symbols (Morse and Hedlund 1944), i.e., it contains no substrings OF THE FORM WWW , where W is any WORD . For example, it does not contain the WORDS 000, 010101 or 010010010. In fact, the following stronger statement is true: the Thue-Morse sequence does not contain any sub- strings OF THE FORM WWa , where a is the first symbol of W. We can obtain a SQUAREFREE sequence on three symbols by doing the following: take the Thue-Morse sequence 0110100110010110... and look at the sequence of WORDS of length 2 that appear: 01 11 10 01 10 00 01 11 10 .... Replace 01 by 0, 10 by 1, 00 by 2 and 11 by 2 to get the following: 021012021.... Then this SEQUENCE isSQUAREFREE (Morse and Hedlund 1944). The Thue-Morse sequence has important connections with the G RAY CODE . Kindermann generates fractal music using the SELF-SIMILARITY of the Thue-Morse sequence. See also GRAY CODE,P ARITY CONSTANT ,R ABBIT SEQUENCE ,THUE SEQUENCE References Kindermann, L. "MusiNum--The Music in the Numbers." http://www.forwiss.uni-erlangen.de/~ kinderma/musinum/. Morse, M. and Hedlund, G. A. "Unending Chess, Symbolic Dynamics, and a Problem in Semigroups." Duke Math. J. 11,1/C1/7, 1944. Schroeder, M. R. Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, 1991. Sloane, N. J. A. Sequences A010060 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Thue’s Remainder Theorem THUE’S THEOREM Thue’s Theorem If n /C211, (a ; n) /C301 (i.e., a and n are RELATIVELY PRIME ), and m is the least integer >ffiffiffinp; then there exist an x and y such that ay /C139x (mod n) where 0 Bx Bm and 0 By Bm (Nagell 1951, pp. 122 /C1/ 124; Shanks 1993, p. 161) References Nagell, T. "Thue’s Remainder Theorem and Its General- ization by Scholtz." §36 in Introduction to Number Theory. New York: Wiley, pp. 122 /C1/124, 1951. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 161, 1993. Thue-Siegel-Roth Theorem If a is a TRANSCENDENTAL NUMBER , it can be approxi- mated by infinitely many RATIONAL NUMBERS m=n to within n /C28r ; where r is any POSITIVE number. See also IRRATIONALITY MEASURE ,LIOUVILLE’S AP- PROXIMATION THEOREM ,ROTH’S THEOREM ,SIEGEL’S THEOREM Thue-Siegel-Schneider-Roth Theorem THUE- SIEGEL- ROTH THEOREM Thurston’s Geometrization Conjecture Thurston’s conjecture has to do with geometric structures on 3-D MANIFOLDS . Before stating Thur- ston’s conjecture, some background information is useful. 3-dimensional MANIFOLDS possess what is known as a standard 2-level DECOMPOSITION . First, there is the CONNECTED SUM DECOMPOSITION , which says that every COMPACT 3-MANIFOLD is the CON- NECTED SUM of a unique collection of PRIME 3-MANI- FOLDS . The second DECOMPOSITION is the JACO-SHALEN- JOHANNSON TORUS DECOMPOSITION , which states that irreducible orientable COMPACT 3-MANIFOLDS have a CANONICAL (up to ISOTOPY ) minimal collection of disjointly EMBEDDED incompressible TORI such that each component of the 3-MANIFOLD removed by the TORI is either "atoroidal" or "Seifert-fibered." Thurston’s conjecture is that, after you split a 3- MANIFOLD into its CONNECTED SUM and then JACO- SHALEN-JOHANNSON TORUS DECOMPOSITION , the re- maining components each admit exactly one of the following geometries:1. EUCLIDEAN GEOMETRY , 2. HYPERBOLIC GEOMETRY , 3. SPHERICAL GEOMETRY , 4. the GEOMETRY of S2 /C27R ;/ 5. the GEOMETRY of H2 /C27R ;/ 6. the GEOMETRY of SL2R;/ 7. NIL GEOMETRY ,or 8. SOL GEOMETRY . Here, S2 is the 2-SPHERE and H2 is the HYPERBOLIC PLANE . If Thurston’s conjecture is true, the truth of the POINCARE ´ CONJECTURE immediately follows. See also CONNECTED SUM DECOMPOSITION ,E UCLI- DEAN GEOMETRY ,HYPERBOLIC GEOMETRY ,JACO-SHA- LEN- JOHANNSON TORUS DECOMPOSITION ,N IL GEOMETRY ,POINCARE ´ CONJECTURE ,SOL GEOMETRY , SPHERICAL GEOMETRY Thwaites Conjecture COLLATZ PROBLEM Ticktacktoe TIC-TAC-TOE Tic-Tac-Toe The usual game of tic-tac-toe (also called TICKTACK- TOE) is 3-in-a-row on a 3 /C293 board. However, a generalized N-IN-A-ROW on an u /C29v board can also be considered. For n /C301 and 2 the first player can always win. If the board is at least 3 /C294; the first player can win for n /C303. However, for TIC-TAC-TOE which uses a 3 /C293 board, a draw can always be obtained. If the board is at least 4 /C2930 ; the first player can win for n /C304. For n /C305, a draw can always be obtained on a 5 /C295 board, but the first player can win if the board is at least 15 /C2915: The cases n /C306 and 7 have not yet been fully analyzed for an n /C29n board, although draws can always be forced for n /C308 and 9. On an /C12/C29/C12 board, the first player can win for n/C301, 2, 3, and 4, but a tie can always be forced for n]8:For 3/C293/C293 and 4 /C29 4/C294;the first player can always win (Gardner 1979). See also BOARD ,PONG HAU K’I References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 103 /C1/104, 1987. de Fouquie `res, B. Ch. 18 in Les Jeux des Anciens, 2nd ed. . Paris, 1873. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240,1 8/C1/28, Jan. 1979a. Gardner, M. "Ticktacktoe Games." Ch. 9 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Free- man, pp. 94 /C1/105, 1983. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 10 /C1/11, 1999. Stewart, I. "A Shepherd Takes A Sheep Shot." Sci. Amer. 269, 154 /C1/156, 1993. Tietze Graph The graph illustrated above that provides a 6-color coloring of the MO¨ BIUS STRIP . See also MO¨ BIUS STRIP References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 243, 1976. Tight Closure The application of characteristic p methods in COM- MUTATIVE ALGEBRA , which is a synthesis of some areas of COMMUTATIVE ALGEBRA and ALGEBRAIC GEO- METRY . See also ALGEBRAIC GEOMETRY ,COMMUTATIVE ALGE- BRA References Bruns, W. "Tight Closure." Bull. Amer. Math. Soc. 33, 447 /C1/ 457, 1996. Huneke, C. "An Algebraist Commuting in Berkeley." Math. Intell. 11,40/C1/52, 1989. Tightly Embedded Q is said to be tightly embedded if Q S Qgjj is ODD for all g /C23 G /C28NG(Q) ; where NG(Q) is the NORMALIZER of Q in G. Tilde The mark ~ placed on top of a symbol to indicate some special property. ˜x is voiced "x-tilde." The tilde symbol is commonly used to denote an operator, e.g., the DIFFERENTIAL OPERATOR ˜D: In informal usage, "tilde" is often instead voiced as "twiddle." It is also some- times used to denote a MEDIAN (Kenney and Keeping 1962, p. 211). See also MEDIAN (STATISTICS ), DIFFERENTIAL OPERA- TOR References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 284, 1997. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.Tiling A plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover the plane. Given a single tile, the so- called first CORONA is the set of all tiles that have a common boundary point with the tile (including the original tile itself). WANG’S CONJECTURE (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically. A periodic tiling of the PLANE by POLYGONS or SPACE by POLYHEDRA is called a TESSEL- LATION . The conjecture was refuted in 1966 when R. Berger showed that an aperiodic set of 20,426 tiles exists. By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose discovered an aper- iodic set (when color-matching rules are included) of two tiles: the so-called PENROSE TILES . (Penrose also sued the Kimberly Clark Corporation over their quilted toilet paper, which allegedly resembles a Penrose aperiodic tiling; Mirsky 1997.) It is not known if there is a single aperiodic tile. The number of tilings possible for convex irregular POLY- GONS are given in the above table. n name known tilings 3 TRIANGLE TILING all 4 QUADRILATERAL TILING all 5 PENTAGON TILING 14 6 HEXAGON TILING 3 There are no tilings for identical convex n-gons for n ]7; although non-identical convex heptagons can tile the plane (Steinhaus 1983, p. 77; Gardner 1984, pp. 248 /C1/249). See also ANISOHEDRAL TILING ,C ORONA (TILING ), GOSPER ISLAND ,HARBORTH’S TILING ,HEESCH NUM- BER,HEESCH’S PROBLEM ,HONEYCOMB CONJECTURE , ISOHEDRAL TILING ,KOCH SNOWFLAKE ,MONOHEDRAL TILING ,PENROSE TILES,POLYGON TILING ,POLYOMI- NO TILING ,S PACE- FILLING POLYHEDRO N,S QUARE TILING ,TESSELLATION ,TILING THEOREM ,TRIANGLE TILING References Eppstein, D. "Tiling." http://www.ics.uci.edu/~eppstein/junk- yard/tiling.html. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 248 /C1/249, 1984. Gardner, M. "Tilings with Convex Polygons." Ch. 13 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 162 /C1/176, 1988. Gardner, M. "Penrose Tiling" and "Penrose Tiling II." Chs. 1 /C1/2in Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 1 /C1/29, 1989. Gru¨nbaum, B. and Shepard, G. C. "Some Problems on Plane Tilings." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 167 /C1/196, 1981. Gru¨nbaum, B. and Sheppard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1986. Mirsky, S. "The Emperor’s New Toilet Paper." Sci. Amer. 277, 24, July 1997. Pappas, T. "Mathematics & Moslem Art." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 178, 1989. Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 82 /C1/85, 1988. Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., 1997. Schattschneider, D. "In Praise of Amateurs." In The Math- ematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 140 /C1/166, 1981. Seyd, J. A. and Salman, A. S. Symmetries of Islamic Geome- trical Patterns. River Edge, NJ: World Scientific, 1995. Stein, S. and Szabo ´,S.Algebra and Tiling: Homomorphisms in the Service of Geometry. Washington, DC: Math. Assoc. Amer., 1994. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Stevens, P. S. Handbook of Regular Patterns: An Introduc- tion to Symmetry in Two Dimensions. Cambridge, MA: MIT Press, 1992. Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Weisstein, E. W. "Books about Tilings." http://www.trea- sure-troves.com/books/Tilings.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 177 /C1/179, 208, and 211, 1991. Tiling Problem Maximize the amount of floor space which can be covered with a fixed tile (Hoffman 1998, p. 173). See also BIN-PACKING PROBLEM ,C OOKIE- CUTTER PROBLEM References Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998. Tiling Theorem Due to Lebesgue and Brouwer. If an n-D figure is covered in any way by sufficiently small subregions, then there will exist points which belong to at least n /C271 of these subareas. Moreover, it is always possible to find a covering by arbitrarily small regions for which no point will belong to more than n /C271 regions. See also TESSELLATION ,TILINGTime Series Analysis Analysis of data ordered by the time the data were collected (usually spaced at equal intervals), called a time series. Common examples of a time series are daily temperature measurements, monthly sales, and yearly population figures. The goals of time series analysis are to describe the process generating the data, and to forecast future values. See also ANOVA, ARITHMETIC MEAN,CORRELATION COEFFICIENT ,C OVARIANCE ,D IFFERENCE TABLE , LEAST SQUARES FITTING ,M AXIMUM LIKELIHOOD , MOVING AVERAGE ,PERIODOGRAM ,PREDICTION THEO- RY,RANDOM VARIABLE ,RANDOM WALK,RESIDUAL , VARIANCE References Chatfield, C. The Analysis of Time Series: An Introduction, 5th ed. Boca Raton, FL: Chapman & Hall, 1996. Cryer, J. D. Time Series Analysis. Boston, MA: PWS Pub- lishers, 1986. Miller, R. B. and Wichern, D. W. Ch. 9 /C1/11 in Intermediate Business Statistics: Analysis of Variance, Regression, and Time Series. New York: Holt, Rinehart and Winston, pp. 353 /C1/438, 1977. Rao, T. S.; Priestly, M. B.; and Lessi, O. Applications of Time Series Analysis in Astronomy and Meteorology. Boca Raton, FL: Chapman & Hall, 1997. Shumway, R. H. and Stoffer, D. S. Time Series Analysis and Its Applications. New York: Springer-Verlag, 2000. Whittaker, E. T. and Robinson, G. "The Search for Periodi- cities." Ch. 13 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 343 /C1/362, 1967. Times The operation of MULTIPLICATION , i.e., a times b. Various notations are a /C29b; a /C215 b; ab, and (a)(b) : The "multiplication sign" /C29 is based on SAINT ANDREW’S CROSS (Bergamini 1969). Floating point MULTIPLICA- TION is sometimes denoted /C156:/ See also CROSS PRODUCT ,D OT PRODUCT ,M INUS , MULTIPLICATION ,PLUS,PRODUCT References Bergamini, D. Mathematics. New York: Time-Life Books, p. 11, 1969. Cundy, H. M. "What Is /C29/?"Math. Gaz. 43, 101, 1959. T-Integration A fast, accurate, and numerically stable NUMERICAL INTEGRATION formula given by Xn/C30Xn/C281/C27TG PdX dt ! n/C27(1/C28P)dX dt ! n/C281"# ; where Xis the integral, dX=dtis the integrand, P andGare "phase " and "gain" tuning parameters, n refers to the number of the iteration being evaluated, andTis the integration step size. For G/C301, varying P from 0 to 2 gives many classical first-order integrators: 1. G /C301 and P /C300: Euler integrator, 2. G /C301 and P /C301 =2:TRAPEZOIDAL RULE , 3. G /C301 and P /C301: Rectangular rule, 4. G /C301 and P /C303 =2:A DAMS’ METHOD . See also NUMERICAL INTEGRATION References Fowler, M. "A New Numerical Method for Simulation." Simulation 6,90/C1/92, Feb. 1976. Smith, J. M. "Recent Developments in Numerical Integra- tion." J. Dynam. Sys., Measurement and Control. Mar. 1974. Smith, J. M. "Zero-Order T-Integration and Its Relation to the Mean Value Theorem." In Proceedings of the Sixth Annual Pittsburgh Modeling and Simulation Conference, Part 1, April 24 /C1/25, 1975. Smith, J. M. "Modern Numerical Integration Methods." In Mathematical Modeling and Digital Simulation, 2nd ed. New York: John Wiley, 1988. Smith, J. M. "Fast T-Integration." J. Mech. Eng. Sys. 1,27/C1/ 31, Jul./Aug. 1990. Smith, J. M. "Jon Michael Smith on T-Integration: Trade Secrets in Numerical Analysis." http://members.aol.com/ jsmith46ws/ni1.htm. Titanic Prime A PRIME with ]1000 DIGITS . As of 1990, there were more than 1400 known (Ribenboim 1990). The table below gives the number of known titanic primes as a function of year end. Year Titanic Primes 1992 2254 1993 9166 1994 9779 1995 12391 References Caldwell, C. "The Ten Largest Known Primes." http:// www.utm.edu/research/primes/largest.html#largest. Morain, F. "Elliptic Curves, Primality Proving and Some Titanic Primes." Aste´rique 198 /C1/200, 245 /C1/251, 1992. Ribenboim, P. The Little Book of Big Primes. Berlin: Springer-Verlag, p. 97, 1990. Yates, S. "Titanic Primes." J. Recr. Math. 16, 250 /C1/262, 1983 /C1/84. Yates, S. "Sinkers of the Titanics." J. Recr. Math. 17, 268 /C1/ 274, 1984 /C1/85. Titchmarsh Theorem If f( v)is SQUARE INTEGRABLE over the REAL v/-axis, then any one of the following implies the other two:1. The FOURIER TRANSFORM F(t) /C30F[f( v)] is 0 for t B1. 2. Replacing v by z /C13x /C27iy ; the function f(z)is analytic in the COMPLEX PLANE z for y /C210 and approaches f(x) almost everywhere as y 0 0: Furthermore, f/C12 /C28/C12f(x /C27iy) jj2dx Bk for some num- ber k and y /C210 (i.e., the integral is bounded). 3. The REAL and IMAGINARY PARTS of F(z) are HILBERT TRANSFORMS of each other (Bracewell 1999, Problem 8, p. 273). See also FOURIER TRANSFORM ,HILBERT TRANSFORM References Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999. Titchmarsh’s Differential Equation The ORDINARY DIFFERENTIAL EQUATION yƒ/C27 l /C28x2niCjiCk y /C300: References Hille, E. Lectures on Ordinary Differential Equations. Reading, MA: Addison-Wesley, p. 617, 1969. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997. Tit-for-Tat A strategy for the iterated PRISONER’S DILEMMA in which a prisoner cooperates on the first move, and thereafter copies the previous move of the other prisoner. Any better strategy has more complicatedrules. See also P RISONER’S DILEMMA References Goetz, P. "Phil’s Good Enough Complexity Dictionary." http://www.cs.buffalo.edu/~goetz/dict.html. Tits Group AFINITE SIMPLE GROUP which is a SUBGROUP of the TWISTED CHEVALLEY GROUP2F4(2):/ Toeplitz Matrix Given 2 n/C281 numbers ak;where k/C30/C28n/C271;...,/C281, 0, 1, ..., n/C281;a Toeplitz matrix is a MATRIX which has constant values along negative-sloping diagonals, i.e., a matrix OF THE FORM a0a/C281a/C282/C1/C1/C1 a/C28n/C271 a1 a0a/C281:::n a2 a1a0:::a/C282 n:::::::::a/C281 an/C281/C1/C1/C1 a2a1 a02 666643 77775: M ATRIX EQUATIONS OF THE FORM Xn j/C301ai/C28jxj /C30yi can be solved with O n2ðÞoperations. Typical problems modelled by Toeplitz matrices include the numerical solution of certain differential and integral equations (regularization of inverse problems), the computation of SPLINES , TIME SERIES ANALYSIS , signal and image processing, MARKOV CHAINS , and QUEUING THEORY (Bini 1995). See also TRIANGULAR MATRIX ,VANDERMONDE MATRIX References Bini, D. "Toeplitz Matrices, Algorithms and Applications." ECRIM News Online Edition, No. 22, July 1995. http:// www.ercim.org/publication/Ercim_News/enw22/toe- plitz.html. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Vandermonde Matrices and Toeplitz Ma- trices." §2.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 82 /C1/89, 1992. Togliatti Surface Togliatti (1940, 1949) showed that QUINTIC SURFACES having 31 ORDINARY DOUBLE POINTS exist, although he did not explicitly derive equations for such surfaces. Beauville (1978) subsequently proved that 31 double points are the maximum possible, and quintic surfaces having 31 ORDINARY DOUBLE POINTS are therefore sometimes called Togliatti surfaces. van Straten (1993) subsequently constructed a 3-D family of solutions and in 1994, Barth derived the example known as the DERVISH . See also DERVISH ,ORDINARY DOUBLE POINT ,QUINTIC SURFACE References Beauville, A. "Surfaces alge´briques complexes." Aste´risque 54,1/C1/172, 1978. Endraß, S. "Togliatti Surfaces." http://enriques.mathemati- k.uni-mainz.de/kon/docs/Etogliatti.shtml. Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Togliatti, E. G. "Una notevole superficie de 5/C14 ordine con soli punti doppi isolati." Vierteljschr. Naturforsch. Ges. Zu¨rich 85, 127 /C1/132, 1940. Togliatti, E. "Sulle superficie monoidi col massimo numero di punti doppi." Ann. Mat. Pura Appl. 30, 201 /C1/209, 1949. van Straten, D. "A Quintic Hypersurface in P4 with 130 Nodes." Topology 32, 857 /C1/864, 1993. Tomography Tomography is the study of the reconstruction of 2- and 3-dimensional objects from 1-dimensional slices. The RADON TRANSFORM is an important tool in tomography. Rather surprisingly, there exist certain sets of four directions in Euclidean n-space such that X-rays of aconvex body in these directions distinguish it from all other convex bodies. See also ALEKSANDROV’S UNIQUENESS THEOREM , BRUNN- MINKOWSKI INEQUALITY ,B USEMANN- PETTY PROBLEM ,DVORETZKY’S THEOREM ,HAMMER’S X-RAY PROBLEMS ,RADON TRANSFORM ,STEREOLOGY References Gardner, R. J. "Geometric Tomography." Not. Amer. Math. Soc. 42, 422 /C1/429, 1995. Gardner, R. J. Geometric Tomography. New York: Cam- bridge University Press, 1995. Herman, G. T. and Kuba, A. (Eds.). Discrete Tomography: Foundations, Algorithms, and Applications. Boston, MA: Birkha ¨user, 1999. Kak, A. C. and Slaney, M. Principles of Computerized Tomographic Imaging. IEEE Press, 1988. Weisstein, E. W. "Books about Tomography." http:// www.treasure-troves.com/books/Tomography.html. Tooth Surface The QUARTIC SURFACE given by the equation x4 /C27y4 /C27z4 /C28 x2 /C27y2 /C27z2iCjiCk /C300: See also GOURSAT’S SURFACE References Nordstrand, T. "Surfaces." http://www.uib.no/people/nfytn/ surfaces.htm. Top-Dimensional Form In an EXTERIOR ALGEBRA fflV ; a top-dimensional form has degree n where n /C30dim V : Any form of higher degree must be zero. For example, if V /C30R4 then a /C30e1 ffle2 ffle3 ffle4 is a top-dimensional form, and any other top-dimen- sional form is lafor some l:/ See also DIFFERENTIAL K-FORM,EXTERIOR ALGEBRA , ORIENTATION (VECTOR SPACE ), VOLUME FORM Topological Basis A topological basis is a SUBSET Bof a SETTin which all other OPEN SETS can be written as UNIONS or finite INTERSECTIONS of B. For the REAL NUMBERS , the SET of all OPEN INTERVALS is a basis. Topological Completion The topological completion C of a FIELD F with respect to the ABSOLUTE VALUE /C215jjis the smallest FIELD containing F for which all CAUCHY SEQUENCES or rationals converge. References Burger, E. B. and Struppeck, T. "Does a/C12 n/C3001 n!Really Con- verge? Infinite Series and p-adic Analysis." Amer. Math. Monthly 103, 565 /C1/577, 1996. Topological Dimension LEBESGUE COVERING DIMENSION Topological Entropy The topological entropy of a MAP M is defined as hT(M) /C30sup WifghM ; Wifg ðÞ ; where Wifg is a partition of a bounded region W containing a probability measure which is invariant under M, and sup is the SUPREMUM . References Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 143 /C1/144, 1993. Topological Graph A simple unlabeled graph whose connectivity is considered purely on the basis of topological equiva- lence, so that two edges v1 ; v2 ðÞ and v2 ; v3 ðÞ joined by a node v2of degree two are considered equivalent to the single edge v1 ; v3 ðÞ :/ See also MATCH PROBLEM References Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Topological Group A CONTINUOUS GROUP G which has a HAUSDORFF TOPOLOGY is a topological group. The simplest exam- ple is the group of real numbers under addition. The HOMEOMORPHISM GROUP of any COMPACT HAUS- DORFF SPACE is a topological group when given the COMPACT-OPEN TOPOLOGY . Also, any LIE GROUP is a topological group. See also EFFECTIVE ACTION ,FREE ACTION ,GROUP , ISOTROPY GROUP ,M ATRIX GROUP ,O RBIT (GROUP ), QUOTIENT SPACE ,R EPRESENTATION ,T OPOLOGICAL GROUP ,TRANSITIVEReferences Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 7 /C1/14, 1987. Pontriagin, L. S. Topological Groups, 2nd ed. New York: Gordon and Breach, 1986. Topological Groupoid A topological groupoid over B is a GROUPOID G such that B and G are TOPOLOGICAL SPACES and a; b; and multiplication are continuous maps. Here, a and b are maps from G onto R2with a :(x; g ; y) /C2x and b :(x; g; y) /C2y :/ See also GROUPOID ,TOPOLOGICAL SPACE References Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744 /C1/752, 1996. Topological Manifold A TOPOLOGICAL SPACE M satisfying some separability (i.e., it is a HAUSDORFF SPACE ) and countability (i.e., it is a PARACOMPACT SPACE ) conditions such that every point p /C23 M has a NEIGHBORHOOD homeomorphic to an OPEN SET in Rnfor some n ]0: Every SMOOTH MANIFOLD is a topological manifold, but not necessa- rily vice versa. The first nonsmooth topological manifold occurs in 4-D. Nonparacompact manifolds are of little use in mathe- matics, but non-Hausdorff manifolds do occasionally arise in research (Hawking and Ellis 1975). For manifolds, Hausdorff and second countable are equivalent to Hausdorff and paracompact, and both are equivalent to the manifold being embeddable in some large-dimensional Euclidean space. See also HAUSDORFF SPACE ,M ANIFOLD ,PARACOM- PACT SPACE ,SMOOTH MANIFOLD ,TOPOLOGICAL SPACE References Hawking, S. W. and Ellis, G. F. R. The Large Scale Struc- ture of Space-Time. New York: Cambridge University Press, 1975. Topological Sort A topological sort is a PERMUTATION pof the vertices of a GRAPH such that an edge fi;jgimplies that i appears before jinp(Skiena 1990, p. 208). Only DIRECTED ACYCLIC GRAPHS can be topologically sorted. The topological sort of a graph can be computed using TopologicalSort [g] in the Mathematica add-on packageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). References Skiena, S. "Topological Sorting." §5.4.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 208 /C1/209, 1990. Topological Space A SET X for which a TOPOLOGY T has been specified is called a topological space (Munkres 1975, p. 76). In the chapter "Point Sets in General Spaces" Haus- dorff (1914) defined his concept of a topological space based on the four HAUSDORFF AXIOMS . 1. To each point x there corresponds at least one neighborhood U(x) ; and U(x) contains x. 2. If U(x) and V(x) are neighborhoods of the same point x, then there exists a neighborhood W(x)ofx such that W(x) is a subset of the union of U(x) and V(x) :/ 3. If y is a point in U(x) ; then there exists a neighborhood U(y)ofy such that U(y) is a subset of U(x) :/ 4. For distinct points x and y, there exist two disjoint neighborhoods U(x) and U(y):/ See also HAUSDORFF AXIOMS ,H AUSDORFF SPACE , KURATOWSKI’S CLOSURE- COMPONENT PROBLEM , MANIFOLD ,OPEN SET,TOPOLOGICAL VECTOR SPACE References Berge, C. Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. New York: Dover, 1997. Hausdorff, F. Grundzu ¨ge der Mengenlehre. Leipzig, Ger- many: von Veit, 1914. Republished as Set Theory, 2nd ed. New York: Chelsea, 1962. Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975. Topological Tree SERIES- REDUCED TREE Topological Vector Space A VECTOR SPACE with a HAUSDORFF TOPOLOGY such that the operations of VECTOR ADDITION and SCALAR MULTIPLICATION are CONTINUOUS . The interesting examples are infinite-dimensional spaces, such as a space of functions. For example, a HILBERT SPACE and aB ANACH SPACE are topological vector spaces. The choice of topology reflects what is meant by convergence of functions. For instance, for functions whose integrals converge, the BANACH SPACE L1(X); one of the LP-SPACES , is used. But if one is interested in POINTWISE CONVERGENCE , then no norm will suffice. Instead, for each x /C23 X define the SEMINORM fkkx/C30 f(x)jj on the vector space of functions on X. The seminorms define a topology, the smallest one in which the seminorms are CONTINUOUS . So lim fn /C30f is equiva- lent to lim fn(x) /C30f(x) for all x /C23 X ; i.e., POINTWISE CONVERGENCE . In a similar way, it is possible todefine a topology for which CONVERGENCE means UNIFORM CONVERGENCE on COMPACT SETS. See also BANACH SPACE ,HILBERT SPACE ,SEMINORM , TOPOLOGICAL SPACE ,VECTOR SPACE References Ko¨the, G. Topological Vector Spaces. New York: Springer- Verlag, 1979. Zimmer, R. Essential Results in Functional Analysis. Chi- cago: University of Chicago Press, pp. 13 /C1/17, 1990. Topologically Conjugate Two MAPS f; c : M 0 M are said to be topologically conjugate if there EXISTS a HOMEOMORPHISM h : M 0 M such that f(h /C30h( c; i.e., h maps c/-orbits onto f/- orbits. Two maps which are topologically conjugate cannot be distinguished topologically. See also ANOSOV DIFFEOMORPHISM ,STRUCTURALLY STABLE Topologically Transitive A FUNCTION f is topologically transitive if, given any two intervals U and V, there is some POSITIVE INTEGER k such that fk(U) S V "¥: Vaguely, this means that neighborhoods of points eventually get flung out to "big" sets so that they don’t necessarily stick together in one localized clump. See also CHAOS Topology Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings. (Tearing, however, is notallowed.) A CIRCLE is topologically equivalent to an ELLIPSE (into which it can be deformed by stretching) and a SPHERE is equivalent to an ELLIPSOID . Continu- ing along these lines, the SPACE of all positions of the minute hand on a clock is topologically equivalent to a CIRCLE (where SPACE of all positions means "the collection of all positions"). Similarly, the SPACE of all positions of the minute and hour hands is equivalent to a TORUS . The SPACE of all positions of the hour, minute and second hands form a 4-D object that cannot be visualized quite as simply as the former objects since it cannot be placed in our 3-D world, although it can be visualized by other means. There is more to topology, though. Topology began with the study of curves, surfaces, and other objects in the plane and 3-space. One of the central ideas in topology is that spatial objects like CIRCLES and SPHERES can be treated as objects in their own right, and knowledge of objects is independent of how they are "represented" or "embedded" in space. For exam- ple, the statement "if you remove a point from a CIRCLE , you get a line segment" applies just as well to the CIRCLE as to an ELLIPSE , and even to tangled or knotted CIRCLES , since the statement involves only topological properties. Topology has to do with the study of spatial objects such as curves, surfaces, the space we call our universe, the space-time of general relativity, frac- tals, knots, manifolds (objects with some of the same basic spatial properties as our universe), phase spaces that are encountered in physics (such as the space of hand-positions of a clock), symmetry groups like the collection of ways of rotating a top, etc. The "objects" of topology are often formally defined as TOPOLOGICAL SPACES . If two objects have the same topological properties, they are said to be HOME- OMORPHIC (although, strictly speaking, properties that are not destroyed by stretching and distorting an object are really properties preserved by ISOTOPY , not HOMEOMORPHISM ; ISOTOPY has to do with distort- ing embedded objects, while HOMEOMORPHISM is intrinsic). Topology is divided into ALGEBRAIC TOPOLOGY (also called COMBINATORIAL TOPOLOGY ), DIFFERENTIAL TO- POLOGY , and LOW-DIMENSIONAL TOPOLOGY . There is also a formal definition for a topology defined in terms of set operations. A SET X along with a collection T of SUBSETS of it is said to be a topology if the SUBSETS in T obey the following properties: 1. The (trivial) subsets X and the EMPTY SET ¥ are in T. 2. Whenever sets A and B are in T, then so is A S B :/ 3. Whenever two or more sets are in T, then so is their UNION (Bishop and Goldberg 1980). This definition can be used to enumerate the topologies on n symbols in Mathematica using the following code snippet. BBDiscreteMath‘Combinatorica‘; Topolo- gyQ[x_List,t_List]: /C30Module[{}, MemberQ[t,x]&&MemberQ[t,{}]&& And@@(MemberQ[t,#]&/@Intersection@@@KSub- sets[t,2])&& And@@(MemberQ[t,#]&/@Union@@@Subsets[t]) ] Topologies[n_]: /C30Module[{r /C30Range[n]}, Select[Subsets[Subsets[r]],TopologyQ[r,#]&] ] For example, the unique topology of order 1 is f¥;f1 gg; which the four topologies of order 2 are f¥;f1 g;f1; 2gg;f¥;f1; 2gg;f¥;f1; 2g;f2gg; and f¥;f1 g;f2g;f1; 2gg: The numbers of topologies on sets of cardinalities n /C301, 2, ... are 1, 4, 29, 355, 6942, ... (Sloane’s A000798). A SET X for which a topology T has been specified is called a TOPOLOGICAL SPACE (Munkres 1975, p. 76). For example, the SETX/C30f1;2;3;4gtogether withthe SUBSETS T/C30f¥;f1g;f2;3;4g;f1;2;3;4ggcom- prises a topology, and Xis a TOPOLOGICAL SPACE . Topologies can be built up from TOPOLOGICAL BASES . For the REAL NUMBERS , the topology is the UNION of OPEN INTERVALS . See also ALGEBRAIC TOPOLOGY ,DIFFERENTIAL TOPOL- OGY,GENUS ,KLEIN BOTTLE ,KURATOWSKI REDUCTION THEOREM ,LEFSHETZ TRACE FORMULA ,LOW-DIMEN- SIONAL TOPOLOGY ,M O¨ BIUS STRIP,POINT- SET TOPOL- OGY,PRETZEL TRANSFORMATION ,SPHERE EVERSION , TOPOLOGICAL SPACE ,ZARISKI TOPOLOGY References Adamson, I. A General Topology Workbook. Boston, MA: Birkha ¨user, 1996. Alexandrov, P. S. Elementary Concepts of Topology. New York: Dover. Armstrong, M. A. Basic Topology, rev. ed. New York: Springer-Verlag, 1997. Arnold, B. H. Intuitive Concepts in Elementary Topology. New York: Prentice-Hall, 1962. Barr, S. Experiments in Topology. New York: Dover, 1964. Berge, C. Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. New York: Dover, 1997. Bishop, R. and Goldberg, S. Tensor Analysis on Manifolds. New York: Dover, 1980. Blackett, D. W. Elementary Topology: A Combinatorial and Algebraic Approach. New York: Academic Press, 1967. Bloch, E. A First Course in Geometric Topology and Differential Geometry. Boston, MA: Birkha ¨user, 1996. Brown, J. I. and Watson, S. "The Number of Complements of a Topology on nPoints is at Least 2n(Except for Some Special Cases)." Discr. Math. 154,2 7/C1/39, 1996. Chinn, W. G. and Steenrod, N. E. First Concepts of Topol- ogy: The Geometry of Mappings of Segments, Curves,Circles, and Disks. Washington, DC: Math. Assoc. Amer., 1966. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 229, 1974. Dugundji, J. Topology. Englewood Cliffs, NJ: Prentice-Hall, 1965. Eppstein, D. "Geometric Topology." http://www.ics.uci.edu/ ~eppstein/junkyard/topo.html. Erne’, M. and Stege, K. "Counting Finite Posets and Topologies." , , . Evans, J. W.; Harary, F.; and Lynn, M. S. "On the Computer Enumeration of Finite Topologies." Commun. ACM 10, 295/C1 /297 and 313, 1967. Francis, G. K. A Topological Picturebook. New York: Springer-Verlag, 1987. Gemignani, M. C. Elementary Topology. New York: Dover, 1990. Greever, J. Theory and Examples of Point-Set Topology. Belmont, CA: Brooks/Cole, 1967. Heitzig, J. and Reinhold, J. "The Number of Unlabeled Orders on Fourteen Elements." Preprint No. 299. Han-over, Germany: Universita ¨t Hannover Institut fu ¨r Math- ematik, 1999. Hirsch, M. W. Differential Topology. New York: Springer- Verlag, 1988. Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988. Kahn, D. W. Topology: An Introduction to the Point-Set and Algebraic Areas. New York: Dover, 1995. Kelley, J. L. General Topology. New York: Springer-Verlag, 1975. Kinsey, L. C. Topology of Surfaces. New York: Springer- Verlag, 1993. Kleitman, D. and Rothschild, B. L. "The Number of Finite Topologies." Proc. Amer. Math. Soc. 25, 276 /C1/282, 1970. Lietzmann, W. Visual Topology. London: Chatto and Windus, 1965. Lipschutz, S. Theory and Problems of General Topology. New York: Schaum, 1965. Mendelson, B. Introduction to Topology. New York: Dover, 1990. Munkres, J. R. Elementary Differential Topology. Princeton, NJ: Princeton University Press, 1963. Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975. Praslov, V. V. and Sossinsky, A. B. Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology. Providence, RI: Amer. Math. Soc., 1996. Rayburn, M. "On the Borel Fields of a Finite Set." Proc. Amer. Math.. Soc. 19, 885 /C1/889, 1968. Seifert, H. and Threlfall, W. A Textbook of Topology. New York: Academic Press, 1980. Shafaat, A. "On the Number of Topologies Definable for a Finite Set." J. Austral. Math. Soc. 8, 194 /C1/198, 1968. Shakhmatv, D. and Watson, S. "Topology Atlas." http:// www.unipissing.ca/topology/. Sloane, N. J. A. Sequences A000798/M3631 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Steen, L. A. and Seebach, J. A. Jr. Counterexamples in Topology. New York: Dover, 1996. Thurston, W. P. Three-Dimensional Geometry and Topology, Vol. 1. Princeton, NJ: Princeton University Press, 1997. Tucker, A. W. and Bailey, H. S. Jr. "Topology." Sci. Amer. 182,18/C1/24, Jan. 1950. van Mill, J. and Reed, G. M. (Eds.). Open Problems in Topology. New York: Elsevier, 1990. Veblen, O. Analysis Situs, 2nd ed. New York: Amer. Math. Soc., 1946. Weisstein, E. W. "Books about Topology." http://www.trea- sure-troves.com/books/Topology.html. Topology (Digraph) An unlabeled TRANSITIVE DIGRAPH with n nodes is called a "topology." The numbers of distinct topologies on n /C301, 2, ... nodes are 1, 3, 9, 33, 139, 718, 4545, ... (Sloane’s A001930). No larger values are known. See also DIRECTED GRAPH ,TRANSITIVE DIGRAPHReferences Sloane, N. J. A. Sequences A001930/M2817 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Topos A CATEGORY modeled after the properties of the CATEGORY of sets. See also CATEGORY ,LOGOS References Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster- dam, Netherlands: North-Holland, 1990. McLarty, C. Elementary Categories, Elementary Toposes. New York: Oxford University Press, 1992. Toric Section A curve obtained by slicing a TORUS (generally a HORN TORUS ) with a plane. A SPIRIC SECTION is a special case of a toric section in which the slicing plane is perpendicular to both the midplane of the torus and to the plane x/C300. For planes parallel to the xy-plane, the toric sections are a single circle (for z/C300) or two concentric circles (for 0Bzjj5a):For planes containing the Z-AXIS , the section is two equal circles. Toric sections at oblique angles can be more compli- cated, passing from a crescent shape, through a U- shape, and into two disconnected kidney-shaped curves. See also SPIRIC SECTION ,TORUS Toric Variety Let m1 ; m2 ; ..., mn be distinct primitive elements of a 2-D LATTICE M such that det mi ; mi/C271iCjiCk > 0 for i /C301, ..., n /C281: Each collection G/C30 m1 ; m2 ; ...; mn fg then forms a set of rays of a unique complete fan in M, and therefore determines a 2-D toric variety XG:/ See also ALGEBRAIC VARIETY References Danilov, V. I. "The Geometry of Toric Varieties." Russ. Math. Surv. 33,97/C1/154, 1978. Fulton, W. Introduction to Toric Varieties. Princeton, NJ: Princeton University Press, 1993. Morelli, R. "Pick’s Theorem and the Todd Class of a Toric Variety." Adv. Math. 100, 183 /C1/231, 1993. Oda, T. Convex Bodies and Algebraic Geometry. New York: Springer-Verlag, 1987. Pommersheim, J. E. "Toric Varieties, Lattice Points, and Dedekind Sums." Math. Ann. 295,1/C1/24, 1993. Torispherical Dome A torispherical dome is the surface obtained from the intersection of a SPHERICAL CAP with a tangent TORUS , as illustrated above. The radius of the sphere R is called the "crown radius," and the radius of the torus is called the "knuckle radius." Torispherical domes are used to construct pressure vessels. See also DOME,SPHERICAL CAP Torn Square Fractal CESA` RO FRACTAL Toroid A SURFACE OF REVOLUTION obtained by rotating a closed PLANE CURVE about an axis parallel to the plane which does not intersect the curve. The sim- plest toroid is the TORUS . The word is also used to refer to a TOROIDAL POLYHEDRON (Gardner 1975). See also PAPPUS’S CENTROID THEOREM ,SURFACE OF REVOLUTION ,TANGENT- SPHERE COORDINATES TOROI- DAL POLYHEDRON ,TORUS References Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102/C1/107, May 1975. Toroidal Coordinates A system of CURVILINEAR COORDINATES for which several different notations are commonly used. In this work ( u;v;f) is used, whereas Arfken (1970) uses ( j;h;8) and Moon and Spencer (1988) use (h;u;c):The toroidal coordinates are defined by x/C30asinh ucosf cosh u/C28cosv(1) y/C30asinh usinf cosh u/C28cosv(2) z/C30asinv cosh u/C28cosv; (3) where sinh zis the HYPERBOLIC SINE and cosh zis the HYPERBOLIC COSINE . Surfaces of constant uare given by the TOROIDS x2/C27y2/C27z2/C27a2/C302affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2p coth u; (4) surfaces of constant vby the spherical bowls x2/C27y2/C27(z/C28acotv)2/C30a2 sin2v; (5) and surfaces of constant fby tanf/C30y x: (6) The SCALE FACTORS are hu /C30a cosh u /C28 cos v (7) hv /C30a cosh u /C28 cos v (8) hf /C30a sinh u cosh u /C28 cos v : (9) The LAPLACIAN is 92f /C30sinh u (cosh u /C28 cos v)3@ @usinh u cosh u /C28 cos v@f @u !"# /C27@ @vsinh u cosh u /C28 cos v@f @v ! /C27@ @ f /C2csch u cosh u /C28 cos v@f @ f !iC0k (10) /C30(cos v /C28cosh u) /C2iC0j sin v@f @v /C27(cos v /C28cosh u) /C2 csch2 u@2f @ f2 /C27@2f @v2 !iC0k /C27(cos v cosh u /C281) csch u@f @u /C27(cos v /C28cosh u)@2f @u2iC0k : (11) The HELMHOLTZ DIFFERENTIAL EQUATION is not se- parable in toroidal coordinates, but LAPLACE’S EQUA- TION is. See also BISPHERICAL COORDINATES ,FLAT-RING CY- CLIDE COORDINATES ,LAPLACE’S EQUATION– TOROIDAL COORDINATES References Arfken, G. "Toroidal Coordinates (/j; h; f) :/" §2.13 in Mathe- matical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112 /C1/115, 1970. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 264, 1959. Moon, P. and Spencer, D. E. "Toroidal Coordinates ( h; u; c):/ " Fig. 4.04 in Field Theory Handbook, Including Coordi- nate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 112 /C1/115, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953. Toroidal Crossing Number The first few toroidal crossing numbers for a COM- PLETE GRAPH are 0, 0, 0, 0, 0, 0, 0, 4, 9, 23, 42, 70, 105,154, 226, 326, ... (Sloane’s A014543). The toroidal crossing numbers for a COMPLETE BIGRAPH are given in the following table. 100000 0 2 0000 0 3 000 0 42 55 8 61 2 7 See also CROSSING NUMBER (GRAPH ), RECTILINEAR CROSSING NUMBER References Gardner, M. "Crossing Numbers." Ch. 11 in Knotted Dough- nuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 133 /C1/144, 1986. Guy, R. K. and Jenkyns, T. "The Toroidal Crossing Number of Km ; n :/" J. Comb. Th. 6, 235 /C1/250, 1969. Guy, R. K.; Jenkyns, T.; and Schaer, J. "Toroidal Crossing Number of the Complete Graph." J. Comb. Th. 4, 376 /C1/ 390, 1968. Sloane, N. J. A. Sequences A014543 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Toroidal Field A VECTOR FIELD resembling a TORUS which is purely circular about the Z-AXIS of a SPHERE (i.e., follows lines of LATITUDE ). A toroidal field takes the form T/C300 1 sinu@T @f /C28@T @u2 6666643 777775: See also D IVERGENCELESS FIELD,POLOIDAL FIELD References Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley, p. 239, 1977. Toroidal Function A class of functions also called RING FUNCTIONS which appear in systems having toroidal symmetry. Toroi- dal functions can be expressed in terms of theL EGENDRE FUNCTIONS and SECOND KINDS (Abramo- witz and Stegun 1972, p. 336): Pm n/C281 =2(cosh h) /C30[G(1 /C28 m)] /C28122 m 1 /C28e /C282hiCjiCk/C28me /C28(n/C271 =2)h /C292F11 2 /C28 m;12 /C27 n /C28 m;1/C282m;1/C28e /C282 hiCkCiCkA Pm n /C281 =2(cosh h) /C30G n /C27 m /C271 2iCkCiCkA (sinh h)m G n /C28 m /C271 2iCkCiCkA 2mffiffiffippG m /C271 2iCkCiCkA /C2g p 0sin2m f df (cosh h /C27 cos f sin h)n/C27m/C271 =2 Q m n/C281 =2(cosh h) /C30[ G(1 /C27 n)]/C281 ffiffiffippeimp G1 2 /C27 n /C27 miCkCiCkA /C29 1 /C28e /C282hiCjiCkme /C28(n/C271 =2)h 2F1 /C212 /C28 m ;12 /C27 n /C27 m;1/C27 m;1/C28e /C282 hiCkCiCkA Qm n/C281 =2(cosh h) /C30( /C281)m G n /C271 2iCkCiCkA G n /C28 m /C271 2iCkCiCkA /C2g/C12 0cosh( mt) dt (cosh h /C27 cosh t sinh h)n/C271 =2 for n /C21m. Byerly (1959) identifies 1 in=2Pn m(coth x) /C30cschn xdnPm(coth x) d(coth x)n as a TOROIDAL HARMONIC . See also CONICAL FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Toroidal Func- tions (or Ring Functions)." §8.11 in Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 336, 1972. Byerly, W. E. An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 266, 1959. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1468, 1980. Toroidal Harmonic TOROIDAL FUNCTION Toroidal Polyhedron A toroidal polyhedron is a POLYHEDRON with GENUS g ]1 (i.e., having one or more HOLES ). Examples of toroidal polyhedra include the CSA´ SZA´ R POLYHEDRON and SZILASSI POLYHEDRON , both of which have GENUS 1 (i.e., the TOPOLOGY of a TORUS ). The only known TOROIDAL POLYHEDRON with no DIAGONALS is the CSA´ SZA´ R POLYHEDRON . If another exists, it must have 12 or more VERTICES and GENUS g ]6 (Gardner 1975). The smallest known single-holetoroidal polyhedron made up of only EQUILATERAL TRIANGLES is composed of 48 of them. See also CSA´ SZA´ R POLYHEDRON ,SZILASSI POLYHE- DRON ,TOROID References Gardner, M. "Mathematical Games: On the Remarkable Csa´sza´r Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102/C1/107, May 1975. Gardner, M. Time Travel and Other Mathematical Bewil- derments. New York: W. H. Freeman, p. 141, 1988. Hart, G. "Toroidal Polyhedra." http://www.georgehart.com/ virtual-polyhedra/toroidal.html. Stewart, B. M. Adventures Among the Toroids, 2nd rev. ed. Okemos, MI: B. M. Stewart, 1984. Toronto Function The function defined by T(m;n;r)/C13r2n/C28m/C271e/C28r2G1 2m/C2712iCkCiCkA n! /C21F112(m/C271);n/C271;r2iCkCiCkA (1) (Heatley 1943; Abramowitz and Stegun 1972, p. 509), where1F1(a;b;z)i sa CONFLUENT HYPERGEOMETRIC FUNCTION andG(z) is the GAMMA FUNCTION . Heatley originally defined the function in terms of the integral T(m;n;p;a)/C30g/C12 0t/C28ne/C28p2t2In(2at)dt; (2) where In(x)i sa MODIFIED BESSEL FUNCTION OF THE FIRST KIND , which is similar to an integral of Watson (1966, p. 394), with Watson’s Jn(at) changed to In(2at) and a few other minor changes of variables. In termsof this function, T(m;n;r)/C302r n/C28m/C271e/C28r2T(m;n;1;r) (3) (Heatley 1943). Heatley (1943) also gives a number ofrecurrences and other identities satisfied by T(m;n;r): / References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 509, 1972. Erde´lyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 268, 1981. Heatley, A. H. "A Short Table of the Toronto Function." Trans. Roy. Soc. Canada 37,1 3/C1/29, 1943. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Torricelli Point FERMAT POINTS Torsion (Differential Geometry) The rate of change of the OSCULATING PLANE of a SPACE CURVE . The torsion t is POSITIVE for a right- handed curve, and NEGATIVE for a left-handed curve. A curve with CURVATURE k "0 is planar IFF t /C300:/ The torsion can be defined by t /C13/C28N /C215 B?; where N is the unit NORMAL VECTOR and B is the unit BINORMAL VECTOR . Written explicitly in terms of a parameterized VECTOR FUNCTION x, t /C30j˙x¨x /C5x j ¨x /C215 ¨x /C30 r2 ˙x¨x /C5x j; j where abcjj denotes a SCALAR TRIPLE PRODUCT and r is the RADIUS OF CURVATURE . The quantity 1=t is called the RADIUS OF TORSION and is denoted s or f:/ See also CURVATURE ,RADIUS OF CURVATURE ,RADIUS OF TORSION References Gray, A. "Drawing Space Curves with Assigned Curvature." §10.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 222 /C1/224, 1993. Kreyszig, E. "Torsion." §14 in Differential Geometry. New York: Dover, pp. 37 /C1/40, 1991. Torsion (Group) If G is a GROUP , then the torsion elements Tor(G)ofG (also called the torsion of G) are defined to be the set of elements g in G such that gn /C30e for some NATURAL NUMBER n, where e is the IDENTITY ELEMENT of the GROUP G. In the case that G is ABELIAN , Tor(G)isa SUBGROUP and is called the torsion subgroup of G. If Tor(G) consists only of the IDENTITY ELEMENT , the GROUP G is called torsion-free. See also ABELIAN GROUP ,F REE ABELIAN GROUP , GROUP ,IDENTITY ELEMENT Torsion Number One of a set of numbers defined in terms of an invariant generated by the finite cyclic covering spaces of a KNOT complement. The torsion numbers for KNOTS up to 9 crossings were cataloged by Reidemeister (1948). See also KNOT INVARIANT References Reidemeister, K. Knotentheorie. New York: Chelsea, 1948. Rolfsen, D. "Torsion Numbers." §6A in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 145 /C1/146, 1976.Torsion Subgroup TORSION (GROUP ) Torsion Tensor The TENSOR defined by Tl jk /C13/C28Gl jk /C28Gl kjiCjiCk ; where Gl jk are CONNECTION COEFFICIENTS . See also CONNECTION COEFFICIENT Torus A torus is a surface having GENUS 1, and therefore possessing a single " HOLE ." The usual torus in 3-D space is shaped like a donut, but the concept of the torus is extremely useful in higher dimensional spaceas well. One of the more common uses of n-D tori is in DYNAMICAL SYSTEMS . A fundamental result states that the PHASE SPACE trajectories of a H AMILTONIAN SYSTEM with nDEGREES OF FREEDOM and possessing nINTEGRALS OF MOTION lie on an n-D MANIFOLD which is topologically equivalent to an n-torus (Tabor 1989). The usual 3-D "ring" torus is known in older litera- ture as an " ANCHOR RING ." It can be constructed from aRECTANGLE by gluing both pairs of opposite edges together with no twists. Let the radius from the center of the hole to the center of the torus tube be c, and the radius of the tube be a. Then the equation in C ARTESIAN COORDI- NATES for a torus azimuthally symmetric about the Z- AXIS is c/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C27y2piCkCiCkA2 /C27z2/C30a2; (1) and the PARAMETRIC EQUATIONS are x¼ðcþacosvÞcosu ð2Þ y¼ðcþacosuÞsinu ð3Þ z¼asinv ð4Þ foru;v/C230;2p ½Þ :Three types of torus, known as the STANDARD TORI , are possible, depending on the relative sizes of aand c.c/C21acorresponds to the RING TORUS (shown above), c/C30acorresponds to a HORN TORUS which is tangent to itself at the point (0, 0, 0), and cBacorresponds to a self-intersecting SPINDLE TORUS (Pinkall 1986). If no specification is made, "torus" is taken to mean RING TORUS . The three STANDARD TORI are illustrated below, where the first image shows the full torus, the second a cut-away of the bottom half, and the third a CROSS SECTION of a plane passing through the Z-AXIS . The STANDARD TORI and their inversions are CY- CLIDES . If the coefficient of sin vin the formula for z is changed to b"a;anELLIPTIC TORUS results. To compute the metric properties of the ring torus,define the inner and outer radii by r/C13c/C28a ð5Þ R/C13cþa: ð6Þ Solving for aandcgives a/C30 1 2(R/C28r) (7) c/C301 2(R/C27r): (8) Then the SURFACE AREA of this torus is S/C30(2pa)(2pc)/C304p2ac (9) /C30p2(R/C27r)(R/C28r); (10) and the VOLUME can be computed from P APPUS’S CENTROID THEOREM V/C30pa2iCjiCk (2pc)/C302p2a2c (11)/C3014p2(R/C27r)(R/C28r)2: (12) The coefficients of the coefficients of the FIRST FUNDAMENTAL FORM are E/C30(c/C27acosv)2(13) F¼0 ð14Þ G¼a2ð15Þ and the coefficients of the SECOND FUNDAMENTAL FORM are e/C30/C28(c/C27acosv) cos v (16) f/C300 (17) g/C30/C28a; (18) giving R IEMANNIAN METRIC ds2/C30(c/C27acosv)2du2/C27a2dv2; (19) AREA ELEMENT dA/C30a(c/C27acosv)duffldv (20) (where duffldvis a WEDGE PRODUCT ), and G AUSSIAN and MEAN CURVATURES as K/C30cosv a(c/C27acosv)(21) H¼/C28cþ2acosv 2aðcþacosvÞð22Þ (Gray 1997, pp. 384 /C1/386). A torus with a HOLE inits surface can be turned inside out to yield an identical torus. A torus can be knotted externally or internally, but not both. These two cases are AMBIENT ISOTOPIES , but not REGULAR ISOTOPIES . There are therefore three possible ways of embedding a torus with zero or one KNOT . An arbitrary point Pon a torus (not lying in the xy- plane) can have four CIRCLES drawn through it. The first circle is in the plane of the torus and the secondis PERPENDICULAR to it. The third and fourth CIRCLES are called V ILLARCEAU CIRCLES (Villarceau 1848, Schmidt 1950, Coxeter 1969, Melnick 1983). To see that two additional CIRCLES exist, consider a coordinate system with origin at the center of torus, with ˆzpointing up. Specify the position of Pby its ANGLE fmeasured around the tube of the torus. Define f/C300 for the circle of points farthest away from the center of the torus (i.e., the points with x2 /C27 y2 /C30R2) ; and draw the X-AXIS as the intersection of a plane through the Z-AXIS and passing through P with the xy-plane. Rotate about the Y-AXIS by an ANGLE u; where u /C30sin/C281a c ! : (23) In terms of the old coordinates, the new coordinates are x /C30x1 cos u /C28z1 sin u (24) z /C30x1 sin u /C27z1 cos u: (25) So in x1 ; y1 ; z1 ðÞ coordinates, equation (1) of the torus becomes ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 cos u /C28z1 sin u ðÞ2/C27y2 1q /C28ciC0jiC0k2 /C27 x1 sin u /C27z1 cos u ðÞ2/C30a2 : (26) Expanding the left side gives x1 cos u /C28z1 sin u ðÞ2/C27y2 1 /C27c2 /C282cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 cos u /C28z1 sin u ðÞ2/C27y2 1q /C27 x1 sin u /C27z1 cos u ðÞ2/C30a2 : (27) But x1 cos u /C28z1 sin u ðÞ2/C27 x1 sin u /C27z1 cos u ðÞ2 /C30x2 1 /C27z21 ; (28) so x21 /C27y21 /C27z21 /C27c2 /C282cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 cos u /C28z1 sin u ðÞ2/C27y2 1q /C30a2 : (29) In the z1 /C300 plane, plugging in (23) and factoring gives x2 1 /C27 y1 /C28a ðÞ2/C28c2hi x21 /C27 y1 /C27a ðÞ2/C28c2hi /C300 : (30) This gives the CIRCLES x21 /C27 y1 /C28a ðÞ2/C30c2 (31) and x21 /C27 y1 /C27a ðÞ2/C30c2 (32) in the z1plane. Written in MATRIX form with para- meter t /C23 0; 2p ½Þ ; these are C1 /C30c cos t c sin t /C27a 02 435 (33)C 2 /C30c cos t c sin t /C28a 02435 (34) In the original (x; y; z) coordinates, C 1 /C30cos u 0 /C28sin u 010 /C28sin u 0 cos u2435c cos t c sin t /C27a 02435 /C30c cos u cos t c sin t /C27a /C28c sin u cos t2435 (35) C 2 /C30cos u 0 sin u 010 /C28sin u 0 cos u2 435c cos t c sin t /C28a 02435 /C30c cos u cos t c sin t /C28a /C28c sin u cos t2 435: (36) The point P must satisfy z /C30a sin f /C30c sin u cos t; (37) so cos t /C30 a sin f c sin u: (38) Plugging this in for x1and y1gives the ANGLE c by which the CIRCLE must be rotated about the Z-AXIS in order to make it pass through P, c /C30tan/C281y x ! /C30c sin t /C27 a ccosucost/C30cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28cos2p t/C27a ccosucost:(39) The four CIRCLES passing through Pare therefore C1/C30coscsinc0 /C28sinccosc0 00 12 435ccosucost csint/C27a /C28csinucost2435 (40) C 2/C30coscsinc0 /C28sinccosc0 00 12435ccosucost csint/C28a /C28csinucost2435 (41) C 3/C30(c/C27acosf) cos t (c/C27acosf) sin t asinf2 435 (42) C 4/C30c/C27acost 0 asint2 435: (43) See also A PPLE ,CYCLIDE ,DOUBLE TORUS ,ELLIPTIC TORUS ,G ENUS (SURFACE ), HORN TORUS ,K LEIN QUARTIC ,L EMON ,R ING TORUS ,S PINDLE TORUS , SPIRIC SECTION ,STANDARD TORI,T ORIC SECTION , TOROID ,TORUS COLORING ,TORUS CUTTING ,TORUS DISSECTION ,TRIPLE TORUS References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 /C1/132, 1987. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 132 /C1/133, 1969. Gray, A. "Tori." §13.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 304 /C1/306 and 384 /C1/386, 1997. Harris, J. W. and Stocker, H. "Torus." §4.10.5 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 113, 1998. JavaView. "Classic Surfaces from Differential Geometry: Torus." http://www-sfb288.math.tu-berlin.de/vgp/java- view/demo/surface/common/PaSurface_Torus.html. Melzak, Z. A. Invitation to Geometry. New York: Wiley, pp. 63 /C1/72, 1983. Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28 /C1/30, 1986. Schmidt, H. Die Inversion und ihre Anwendungen. Munich: Oldenbourg, p. 82, 1950. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 71 /C1/74, 1989. Villarceau, M. "The´ore`me sur le tore." Nouv. Ann. Math. 7, 345 /C1/347, 1848. Torus Coloring The number of colors SUFFICIENT for MAP COLORING on a surface of GENUS g is given by the HEAWOOD CONJECTURE , x(g) /C301 27 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 48g /C271piCkCiCkAjk ; where xbcis the FLOOR FUNCTION . The fact that x(g) (which is called the CHROMATIC NUMBER ) is also NECESSARY was proved by Ringel and Youngs (1968) with two exceptions: the SPHERE (which requires the same number of colors as the PLANE ) and the KLEIN BOTTLE .A g-holed TORUS therefore requires x(g) colors. For g /C300, 1, ..., the first few values of x(g) are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, ... (Sloane’s A000934). A set of regions requiring the maximum of seven regions is shown above for a normal TORUS The above figure shows the relationship between the HEAWOOD GRAPH and the 7-color torus coloring. See also CHROMATIC NUMBER ,FOUR- COLOR THEO- REM,H EAWOOD CONJECTURE ,H EAWOOD GRAPH , KLEIN BOTTLE ,MAP COLORING ,TORUS References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 244, 1976. Cadwell, J. H. Ch. 8 in Topics in Recreational Mathematics. Cambridge, England: Cambridge University Press, 1966. Gardner, M. "Mathematical Games: The Celebrated Four- Color Map Problem of Topology." Sci. Amer. 203, 218 /C1/ 222, Sep. 1960. Ringel, G. Map Color Theorem. New York: Springer-Verlag, 1974. Ringel, G. and Youngs, J. W. T. "Solution of the Heawood Map-Coloring Problem." Proc. Nat. Acad. Sci. USA 60, 438 /C1/445, 1968. Sloane, N. J. A. Sequences A000934/M3292 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 274 /C1/275, 1999. Wagon, S. "Map Coloring on a Torus." §7.5 in Mathematica in Action. New York: W. H. Freeman, pp. 232 /C1/237, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 70, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 228 /C1/229, 1991. Torus Cutting With ncuts of a TORUS ofGENUS 1, the maximum number of pieces which can be obtained is N(n)/C301 6n3/C273n3/C278niCjiCk : The first few terms are 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... (Sloane’s A003600). See also CAKE CUTTING ,CIRCLE DIVISION BY LINES, CYLINDER CUTTING ,PANCAKE CUTTING ,PLANE CUT- TING ,PIE CUTTING ,SQUARE DIVISION BY LINES References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 149 /C1/150, 1978. Sloane, N. J. A. Sequences A003600/M1594 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Torus Dissection A ring TORUS constructed out of a square of side length c can be dissected into two squares of arbitrary side lengths a and b (as long as they are consistent with the size of the original square), as illustrated above. See also DISSECTION ,TORUS References Stewart, I. "Squaring the Square." Sci. Amer. 277,94/C1/96, July 1997. Torus Knot A(p, q)-torus KNOT is obtained by looping a string through the HOLE of a TORUS p times with q revolutions before joining its ends, where p and q are RELATIVELY PRIME .A(p, q)-torus knot is equiva- lent to a (q, p)-torus knot. All torus knots are PRIME (Burde and Zieschang 1985, Hoste et al. 1998). Torus knots are all chiral, invertible, and have symmetry group D1 (Schreier 1924, Hoste et al. 1998). The CROSSING NUMBER of a (p, q)-torus knot is c /C30min fp(q /C281); q(p /C281)g (1) (Williams 1988, Murasugi and Przytycki 1989, Mur- asugi 1991, Hoste et al. 1998). The UNKNOTTING NUMBER of a (p, q)-torus knot is u /C301 2(p /C281)(q /C281) (2) (Adams 1991). Torus knots with fewer than 11 crossings are the TREFOIL KNOT 03 /C1/001 (3, 2), SOLOMON’S SEAL KNOT 05 /C1/ 001 (5, 2), 07 /C1/001 (7, 2), 08 /C1/019 (4, 3), 09 /C1/001 (9, 2), and 10 /C1/124 (5, 3) (Adams et al. 1991). The torus knots with 16 or fewer crossings are (3; 2); (5; 2); (7; 2); (9; 2); (11; 2); (13; 2); (15; 2); (4; 3); (5; 3); (7; 3); (8; 3); and (5; 4) (Hoste et al. 1998). The numbers of torus knots with n crossings are 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, ... (Sloane’s A051764).The only KNOTS which are not HYPERBOLIC KNOTS are torus knots and SATELLITE KNOTS (including COMPO- SITE KNOTS ). The (q; 2); (4; 3); and (5; 4)/-torus knots are ALMOST ALTERNATING KNOTS (Adams 1994, p. 142). The JONES POLYNOMIAL of an (m, n)-TORUS KNOT is t(m/C281)(n/C281)=2 1 /C28 tm/C271 /C28 tn/C271 /C27 tm/C27nðÞ 1 /C28 t2 : (3) The BRACKET POLYNOMIAL for the torus knot Kn /C30 (2; n) is given by the RECURRENCE RELATION Knhi/C30AKn/C281 hi /C27(/C281)n/C281A/C283n/C272 ; (4) where K1hi/C30/C28A3 : (5) See also ALMOST ALTERNATING KNOT,H YPERBOLIC KNOT,K NOT,S ATELLITE KNOT,S OLOMON’S SEAL KNOT,TREFOIL KNOT References Adams, C.; Hildebrand, M.; and Weeks, J. "Hyperbolic Invariants of Knots and Links." Trans. Amer. Math. Soc. 326,1/C1/56, 1991. Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, 1985. Gray, A. "Torus Knots." §9.2 in Modern Differential Geome- try of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 209 /C1/215, 1997. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/48, Fall 1998. Murasugi, K. "On the Braid Index of Alternating Links." Trans. Amer. Math. Soc. 326, 237 /C1/260, 1991. Murasugi, L. and Przytycki, J. "The Skein Polynomial of a Planar Star Product of Two Links." Math. Proc. Cam- bridge Philos. Soc. 106, 273 /C1/276, 1989. Schreier, O. "U¨ ber die Gruppen AaBb /C301:/" Abh. Math. Sem. Univ. Hamburg 3, 167 /C1/169, 1924. Sloane, N. J. A. Sequences A051764 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 275 /C1/277, 1999. Williams, R. F. "The Braid Index of an Algebraic Link." Braids (Santa Cruz, CA, 1986) . Providence, RI: Amer. Math. Soc., 1988. Total Angular Defect DESCARTES TOTAL ANGULAR DEFECT Total Curvature The total curvature of a curve is the quantityffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2/C27k2p ;where tis the TORSION and kis the CURVATURE . The total curvature is also called the THIRD CURVATURE . See also CURVATURE ,TORSION (DIFFERENTIAL GEO- METRY ) Total Differential EXACT DIFFERENTIAL Total Exchange GOSSIPING Total Function A FUNCTION defined for all possible input values. Total Graph The total graph T(G)ofa GRAPH G has a vertex for each edge and vertex of G, and edge in T(G) for every edge-edge and vertex-edge adjacency in G (Capo- bianco and Molluzzo 1978; Skiena 1990, p. 162). Total graphs are generalizations of LINE GRAPHS . See also LINE GRAPH References Capobianco, M. and Molluzzo, J. Examples and Counter- examples in Graph Theory. New York: North-Holland, 1978. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Total Intersection Theorem If one part of the total intersection group of a curve of order n with a curve of order n1 /C27n2constitutes the total intersection with a curve of order n1 ; then the other part will constitute the total intersection with a curve of order n2 :/ References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 32, 1959. Total Order A RELATION on a TOTALLY ORDERED SET. See also TOTALLY ORDERED SET Total Probability Theorem Given n MUTUALLY EXCLUSIVE EVENTS A1 ; ..., An whose probabilities sum to unity, then P(B) /C30PBA1jÞPA1ðÞ/C27.../C27PBAnjÞPAnðÞ ; ð ð where B is an arbitrary event, and PBAijÞ ð is the CONDITIONAL PROBABILITY of B assuming Ai :/ See also BAYES’ THEOREM ,CONDITIONAL PROBABIL- ITY,MUTUALLY EXCLUSIVE EVENTS References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 37 /C1/38, 1984.Total Space The SPACE E of a FIBER BUNDLE given by the MAP f : E 0 B; where B is the BASE SPACE of the FIBER BUNDLE . See also BASE SPACE ,FIBER BUNDLE ,SPACE Total Variation Measure Given a COMPLEX MEASURE m; there exists a POSITIVE MEASURE denoted mjjwhich measures the total varia- tion of m; also sometimes called simply "total varia- tion." In particular, ½m ½(E)ona SUBSET E is the largest sum of "variations" for any subdivision of E. Roughly speaking, a total variation measure is an infinitesi- mal version of the ABSOLUTE VALUE . More precisely, mjj(E) /C30supX im EiðÞjj (1) where the SUPREMUM is taken over all partitions @ Ei of E into MEASURABLE SUBSETS Ei :/ Note that m(X) jj may not be the same as mjj(X): When m already is a POSITIVE MEASURE , then m /C30 mjj: More generally, if m is ABSOLUTELY CONTINUOUS , that is m(E) /C30gEfdx ; (2) then so is mjj; and the total variation measure can be written as mjj(E) /C30gEfjjdx: (3) The total variation measure can be used to rewrite the original measure, in analogy to the norm of a COMPLEX NUMBER . The measure m has a POLAR REPRESENTATION dm /C30hdmjj (4) with hjj/C301:/ See also JORDAN MEASURE DECOMPOSITION ,M EA- SURE ,P OLAR REPRESENTATION (MEASURE ), RIESZ REPRESENTATION THEOREM References Rudin, W. Real and Complex Analysis. New York: McGraw- Hill, pp. 116 /C1/120, 1987. Totalistic Cellular Automaton A totalistic cellular automaton is a 1-D cellular automata in which the rules depend only on the total of the values of the cells in a neighborhood. These automata were introduced by Stephen Wolfram in 1983. See also CELLULAR AUTOMATON Totally Ordered Set A total order (or "totally ordered set," or "linearly ordered set") is a SET plus a relation on the set (called a TOTAL ORDER ) that satisfies the conditions for a PARTIAL ORDER plus an additional condition known as the comparability condition. A RELATION 5is a partial order on a SET S ( if the following properties hold. 1. Reflexivity: a 5a for all a /C23 S:/ 2. Weak antisymmetry: a 5b and b 5a implies a /C30b. 3. Transitivity: a 5b and b 5c implies a 5c :/ 4. Comparability (TRICHOTOMY LAW): For any a; b /C23 S; either a 5b or b 5a:/ The first three are the axioms of a PARTIAL ORDER , while addition of the TRICHOTOMY LAW defines a total order. Every finite totally ordered set is WELL ORDERED . Any two totally ordered sets with k elements (for k a nonnegative integer) are ORDER ISOMORPHIC , and therefore have the same ORDER TYPE (which is also an ORDINAL NUMBER ). See also ORDER ISOMORPHIC ,ORDER TYPE,PARTIAL ORDER ,RELATION ,TRICHOTOMY LAW,WELL ORDERED SET References Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 23, 2000. Totally Symmetric Self-Complementary Plane Partition A PLANE PARTITION which is invariant under permu- tation of the three axes and which is equal to its complement (i.e., the collection of cubes that are in a given box but do not belong to the solid Young diagram). The number of totally symmetric self- complementary PLANE PARTITIONS is the same as that for ALTERNATING SIGN MATRICES and DESCEND- ING PLANE PARTITIONS . See also ALTERNATING SIGN MATRIX ,D ESCENDING PLANE PARTITION ,PLANE PARTITION References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637 /C1/646. Totative A POSITIVE INTEGER less than or equal to a number n which is also RELATIVELY PRIME to n, where 1 is counted as being RELATIVELY PRIME to all numbers. The number of totatives of n is the value of the TOTIENT FUNCTION f(n):/ See also RELATIVELY PRIME ,TOTIENT FUNCTIONTotient Function The totient function f(n);also called Euler’s totient function, is defined as the number of POSITIVE INTEGERS 5nwhich are RELATIVELY PRIME to (i.e., do not contain any factor in common with) n, where 1 is counted as being RELATIVELY PRIME to all numbers. Since a number less than or equal to and RELATIVELY PRIME to a given number is called a TOTATIVE , the totient function f(n) can be simply defined as the number of TOTATIVES ofn. For example, there are eight TOTATIVES of 24 (1, 5, 7, 11, 13, 17, 19, and 23), sof(24)/C308:/ /f(n) is always EVEN forn]3:By convention, f(0)/C301; although Mathematica definesEulerPhi [0] equal to 0 for consistency with its FactorInteger [0] com- mand. The first few values of f(n) for n/C301, 2, ... are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ... (Sloane’s A000010). The totient function is given by the M O¨BIUS TRANSFORM of 1, 2, 3, 4, ... (Sloane and Plouffe 1995, p. 22). f(n)i s plotted above for small n. For a PRIME p, f(p)/C30p/C281; (1) since all numbers less than pare RELATIVELY PRIME top.I f m/C30pais a POWER of a PRIME , then the numbers which have a common factor with mare the multiples of p:p,2p;...,pa/C281ðÞ p:There are pa/C281of these multiples, so the number of factors RELATIVELY PRIME topais f(pa)/C30pa/C28pa/C281/C30pa/C281(p/C281)/C30pa1/C281 p ! : (2) Now take a general mdivisible by p. Let fp(m) be the number of POSITIVE INTEGERS 5mnot DIVISIBLE byp. As before, p,2p;..., (m=p)phave common factors, so fp(m)/C30m/C28m p/C30m1/C281 p ! : (3) Now let qbe some other PRIME dividing m. The INTEGERS divisible by qareq,2q;..., (m=q)q:But these duplicate pq,2pq;..., (m=pq)pq:So the number of terms which must be subtracted from fpto obtain fpqis Dfp(m)/C30m q/C28m pq/C30m q1/C281 p ! ; (4) and fpqðmÞ/C13fpðmÞ/C28DfqðmÞ /C30m1/C281 p ! /C28m p1/C281 p ! /C30m1/C281 p ! 1/C281 q ! : (5) By induction, the general case is then f(n)/C30n1/C281 p1 ! 1/C281 p2 ! /C1/C1/C11/C281 pr ! : (6) An interesting identity relates f(n2)t of(n); f(n2)/C30nf(n): (7) Another identity relates the DIVISORS dofntonvia X df(d)/C30n: (8) The DIVISOR FUNCTION satisfies the CONGRUENCE ns(n)/C132 (mod f(n)) /C30ns(n)/C130 (mod f(n)) if f(n)/C302 ns(n)/C132 (mod f(n)) otherwiseiC0C (9) for all PRIMES p]5 and no COMPOSITE with the exception of 4, 6, and 22, where s(n) is the DIVISOR FUNCTION . This fact was proved by Subbarao (1974), despite the implication to the contrary, "is it true for infinitely many composite n?," stated in Guy (1994, p. 92). No COMPOSITE solution is currently known to n/C281/C130 (mod f(n)) (10) (Honsberger 1976, p. 35). If the G OLDBACH CONJECTURE is true, then for every number m, there are PRIMES pandqsuch that f(p)/C27f(q)/C302m (11) (Guy 1994, p. 105). Guy (1994, p. 99) discussed solutions to f(s(n))/C30n; (12) where s(n) is the DIVISOR FUNCTION . F. Helenius has found 365 such solutions, the first of which are 2, 8, 12, 128, 240, 720, 6912, 32768, 142560, 712800, ... (Sloane’s A001229). Curious equalities of consecutive values include f(5186) /C30f(5187) /C30f(5188) /C302534(13) f(25930) /C30f(25935) /C30f(25942) /C302734(14) f(404471) /C30f(404473) /C30f(404477) /C302832527 (15) (Guy 1994, p. 91). McCranie found an arithmeticprogression of six numbers with equal totient func- tions, f(583200) /C30f(583230) /C30f(583260) /C30f(583290) /C30f(583320) /C30f(583350) /C30155520 ;(16) as well as other progressions of six numbers startingat 583200, 1166400, 1749600, ... (Sloane’s A050518). The SUMMATORY totient function, plotted above, is defined by F(n)/C13Xn k/C301f(k): (17) The first values of F(n) are 1, 2, 4, 6, 10, 12, 18, 22, 28, ... (Sloane’s A002088). F(n) has the asymptotic series F(x)/C21 2z(2)x2/C27O(xlnx) (18) /C23 p2x2/C27O(xlnx); (19) where z(z) is the R IEMANN ZETA FUNCTION (Perrot 1881; Nagell 1951, p. 131). An improved asymptoticestimate due to Walfisz (1963) is given by X N n/C301f(n)/C303N2 p2/C27ON(lnN)2=3(ln ln N)4=3hi :(20) Landau (1900, quoted in Dickson 1952) showed thatthe asymptotic series of the summatory function of / 1=fðnÞ/is XN n/C3011 f(n)/C30AlnN/C27B/C27OlnN N ! ; (21) where A/C30X/C12 k/C301m(k)½/C1382 kf(k)/C30z(2)z(3) z(6)/C30315 2p4z(3) /C301:9435964368 . . . (22) B/C30g315 2p4z(3)/C28X/C12 k/C301m(k)½/C1382lnk kf(k) /C30/C280:0595536246 . . . ; (23) /m(k) is the MO¨ BIUS FUNCTION , z(z) is the RIEMANN ZETA FUNCTION , and g is the EULER- MASCHERONI CONSTANT (Dickson). A can also be written A /C30Y/C12 k /C3011 /C28 p6 k 1 /C28 p/C282 kiCjiCk 1 /C28 p /C283 kiCjiCk /C30Y/C12 k /C3011 /C271 pkpk /C28 1 ðÞ"# : (24) Note that this constant is similar to ARTIN’S CON- STANT . See also DEDEKIND FUNCTION ,E ULER’S TOTIENT RULE,FERMAT’S LITTLE THEOREM ,LEHMER’S PRO- BLEM ,LEUDESDORF THEOREM ,NONCOTOTIENT ,NON- TOTIENT ,SILVERMAN CONSTANT ,TOTATIVE ,TOTIENT VALENCE FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "The Euler Totient Function." §24.3.2 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 826, 1972. Beiler, A. H. Ch. 12 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Conway, J. H. and Guy, R. K. "Euler’s Totient Numbers." The Book of Numbers. New York: Springer-Verlag, pp. 154 /C1/156, 1996. Courant, R. and Robbins, H. "Euler’s 8Function. Fermat’s Theorem Again." §2.4.3 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas andMethods, 2nd ed. Oxford, England: Oxford University Press, pp. 48 /C1 /49, 1996. DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Func- tions: Asymptotic Formulae for Sums of Reciprocals ofArithmetical Functions and Related Fields. Amsterdam, Netherlands: North-Holland, 1980. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 113 /C1 / 158, 1952. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/totient/totient.html. Guy, R. K. "Euler’s Totient Function," "Does f(n) Properly Divide n/C281;/" "Solutions of f(m)/C30s(n);/" "Carmichael’s Conjecture," "Gaps Between Totatives," "Iterations of f and s;/" "Behavior of f(s(n)) and s(f(n)):/"§B36-B42 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 90 /C1/99, 1994. Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974. Helenius, F. Untitled. http://pweb.netcom.com/~fredh/phi- sigma/pslist.html. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 35, 1976. Nagell, T. "Relatively Prime Numbers. Euler’s 8/-Function." §8i n Introduction to Number Theory. New York: Wiley, pp. 23 /C1/26, 1951. Niven, I. M.; Zuckerman, H. S.; and Montgomery, H. L. An Introduction to the Theory of Numbers, 5th ed. New York: Wiley, p. 51, 1991. Perrot, J. 1811. Quoted in Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality.New York: Chelsea, p. 126, 1952. Shanks, D. "Euler’s fFunction." §2.27 in Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 68 /C1 /71, 1993.Se´roul, R. "The Euler Phi Function." §2.7 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 14 /C1/15, 2000. Sloane, N. J. A. Sequences A000010/M0299, A002088/ M1008, A001229, and A050518 in "An On-Line Versionof the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Subbarao, M. V. "On Two Congruences for Primality." Pacific J. Math. 52, 261/C1 /268, 1974. Totient Function Constants SILVERMAN CONSTANT ,TOTIENT FUNCTION Totient Valence Function /Nf(m) is the number of INTEGERS nfor which the TOTIENT FUNCTION f(n)/C30m;also called the MULTI- PLICITY ofm(Guy 1994). Erdos(1958) proved that is a multiplicity occurs once, it occurs infinitely often. The table below lists values for f(N)550:/ /f(N)/multiplicity N 1 2 1, 2 2 3 3, 4, 6 4 4 5, 8, 10, 12 6 4 7, 9, 14, 18 8 5 15, 16, 20, 24, 30 10 2 11, 22 12 6 13, 21, 26, 28, 36, 42 16 6 17, 32, 34, 40, 48, 60 18 4 19, 27, 38, 54 20 5 25, 33, 44, 50, 6622 2 23, 46 24 10 35, 39, 45, 52, 56, 70, 72, 78, 84, 90 28 2 29, 58 30 2 31, 62 32 7 51, 64, 68, 80, 96, 102, 12036 8 37, 57, 63, 74, 76, 108, 114, 126 40 9 41, 55, 75, 82, 88, 100, 110, 132, 150 42 4 43, 49, 86, 98 44 3 69, 92, 138 46 2 47, 9448 11 65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210 A table listing the first value of f(N) with multi- plicities up to 100 follows (Sloane’s A007374; Sloane’s A014573). M /f/ M /f/ M /f/ M / f/ 0 3 26 2560 51 4992 76 21840 2 1 27 384 52 17640 77 9072 3 2 28 288 53 2016 78 38640 4 4 29 1320 54 1152 79 9360 5 8 30 3696 55 6000 80 81216 6 12 31 240 56 12288 81 4032 7 32 32 768 57 4752 82 5280 8 36 33 9000 58 2688 83 4800 9 40 34 432 59 3024 84 4608 10 24 35 7128 60 13680 85 16896 11 48 36 4200 61 9984 86 3456 12 160 37 480 62 1728 87 3840 13 396 38 576 63 1920 88 10800 14 2268 39 1296 64 2400 89 9504 15 704 40 1200 65 7560 90 18000 16 312 41 15936 66 2304 91 23520 17 72 42 3312 67 22848 92 39936 18 336 43 3072 68 8400 93 5040 19 216 44 3240 69 29160 94 26208 20 936 45 864 70 5376 95 27360 21 144 46 3120 71 3360 96 6480 22 624 47 7344 72 1440 97 9216 23 1056 48 3888 73 13248 98 2880 24 1760 49 720 74 11040 99 26496 25 360 50 1680 75 27720 100 34272 It is thought that Nf(m) ]2 (i.e., the totient valence function never takes on the value 1), but this has not been proven. This assertion is called CARMICHAEL’S TOTIENT FUNCTION CONJECTURE and is equivalent to the statement that for all n, there exists m "n such that f(n) /C30 f(m) (Ribenboim 1996, pp. 39 /C1/40). Any counterexample must have more than 10,000,000 DIGITS (Schlafly and Wagon 1994, erroneously given as 10,000 in Conway and Guy 1996). See also CARMICHAEL’S TOTIENT FUNCTION CONJEC- TURE ,SIERPINSKI’S CONJECTURE ,TOTIENT FUNCTION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 155, 1996.Erdos, P. "Some Remarks on Euler’s f/-Function." Acta Math. 4,10/C1/19, 1958. Ford, K. "The Distribution of Totients." Ramanujan J. 2, 67 /C1/151, 1998. Ford, K. "The Distribution of Totients, Electron. Res. Announc. Amer. Math. Soc. 4,27/C1/34, 1998. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 94, 1994. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996. Schlafly, A. and Wagon, S. "Carmichael’s Conjecture on the Euler Function is Valid Below 1010 ;000;000 :/" Math. Comput. 63, 415 /C1/419, 1994. Sloane, N. J. A. Sequences A007374/M1093 and A014573 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Touchard’s Congruence Bp /C27k /C13Bk /C27Bk /C271(mod p) ; when p is PRIME and Bn is a BELL NUMBER . See also BELL NUMBER Tour A sequence of moves on a chessboard by a CHESS piece in which each square of a CHESSBOARD is visited exactly once. See also CHESS ,H AMILTONIAN CIRCUIT ,K NIGHT’S TOUR,M AGIC TOUR,T RAVELING SALESMAN CON- STANTS Tournament ACOMPLETE DIRECTED GRAPH (Skiena 1990, p. 175). A so-called SCORE SEQUENCE can be associated with every tournament. The number of nonisomorphic tournaments on 2, 3, 4, ... nodes are 1, 2, 4, ..., illustrated above. The first and second 3-node tourna-ments shown above are called a TRANSITIVE TRIPLE and CYCLIC TRIPLE , respectively (Harary 1994, p. 204).Every tournament contains an odd number of H A- MILTONIAN PATHS (Re´dei 1934; Szele 1943; Skiena 1990, p. 175). However, a tournament has a directedH AMILTONIAN CIRCUIT IFF it is STRONGLY CONNECTED (Foulkes 1960; Harary and Moser 1966; Skiena 1990, p. 175). The term "tournament" also refers to an arrangement by which teams or players play against certain other teams or players in order to determine who is the best. In a "cup" tournament of n /C282k teams, teams play pairwise in a sequence of 1=2k /C281/-finals, ..., 1/8- finals, quarter-finals, semi-finals, and finals, with winners from each round playing other winners in the next round and losers being eliminated at each round. The second-place prize is usually awarded to the team which loses in the finals. However, this practice is unfair since the second-place team has not been required to play against the teams which were eliminated by the first-place (and presumably best) team, and therefore might actually be worse than one of the teams eliminated earlier by the best team (Steinhaus 1983). In general, to fairly determine the best two players from n contestants, n /C281 /C27log2(n /C281) rounds are required (Steinhaus 1983, p. 55). See also COMPLETE GRAPH ,DIRECTED GRAPH ,HAMIL- TONIAN PATH,SCORE SEQUENCE ,TOURNAMENT MA- TRIX References Boesch, F. and Tindell, R. "Robbins’ Theorem for Mixed Graphs." Amer. Math. Monthly 87, 716 /C1/719, 1980. Chartrand, G. "Tournaments." §27.2 in Introductory Graph Theory. New York: Dover, pp. 155 /C1/161, 1985. Chva´tal, V. and Thomassen, C. "Distances in Orientations of Graphs." J. Combin. Th. B 24,61/C1/75, 1978. Foulkes, J. D. "Directed Graphs and Assembly Schedules." In Proc. Symp. Appl. Math. Providence, RI: Amer. Math. Soc., pp. 218 /C1/289, 1960. Harary, F. "Tournaments." Graph Theory. Reading, MA: Addison-Wesley, pp. 205 /C1/208, 1994. Harary, F. and Moser, L. "The Theory of Round Robin Tournaments." Amer. Math. Monthly 73, 231 /C1/246, 1966. Harary, F. and Palmer, E. M. "On the Problem of Recon- structing a Tournament from Subtournaments." Monatsh. fu¨r Math. 71,14/C1/23, 1967. Moon, J. W. Topics on Tournaments. New York: Holt, Rinehart, and Winston, 1968. Re´dei, L. "Ein Kombinatorischer Satz." Acta Litt. Szeged. 7, 39 /C1/43, 1934. Roberts, F. S. Graph Theory and Its Applications to Pro- blems of Society. Philadelphia, PA: SIAM, 1978. Ruskey, F. "Information on Score Sequences." http:// www.theory.csc.uvic.ca/~cos/inf/nump/ScoreSequen- ce.html. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 54 /C1/55, 1999. Szele, T. "Kombinatorische Untersuchungen u¨ber den ger- ichteten vollsta ¨ndigen Graphen." Mat. Fiz. Lapok 50, 223 /C1/256, 1943. Tournament Matrix A matrix for a round-robin TOURNAMENT involving n players competing in n(n /C281)=2 matches (no ties allowed) having entriesaij ¼1 if player i defeats player j /C281 if player i loses to player j 0i f i ¼ j:8 < : The MATRIX satisfies A /C27AT /C27I /C30J ; where I is the IDENTITY MATRIX , J is an n /C29n MATRIX of all 1s, and AT is the MATRIX TRANSPOSE of A:/ The tournament matrix for n players has zero DETERMINANT IFF n is ODD (McCarthy and Benjamin 1996). The dimension of the NULLSPACE of an n- player tournament matrix is dim[nullspace] /C300 for n even 1 for n oddiC0C (McCarthy 1996). References McCarthy, C. A. and Benjamin, A. T. "Determinants of the Tournaments." Math. Mag. 69, 133 /C1/135, 1996. Michael, T. S. "The Ranks of Tournament Matrices." Amer. Math. Monthly 102, 637/C1/639, 1995. Tournament Sequence A tournament sequence is an increasing sequence of positive integers ( /t1;t2;...) such that t1/C301 and ti/C2715 2ti:Cook and Kleber (2000) show that M EEUSSEN SEQUENCES are isomorphic to tournament sequences. See also MEEUSSEN SEQUENCE References Cook, M. and Kleber, M. "Tournament Sequences and Meeussen Sequences." Electronic J. Combinatorics 7, No. 1, R44, 1 /C1/16, 2000. http://www.combinatorics.org/ Volume_7/v7i1toc.html#R44. Tower of Power POWER TOWER Towers of Hanoi APUZZLE invented by E. Lucas in 1883. Given a stack ofndisks arranged from largest on the bottom to smallest on top placed on a rod, together with two empty rods, the towers of Hanoi puzzle asks for the minimum number of moves required to reverse theorder of the stack (where moves are allowed only if they place smaller disks on top of larger disks). The problem is ISOMORPHIC to finding a HAMILTONIAN PATH on an n-HYPERCUBE (Gardner 1957, 1959). For n disks, the number of moves hn required is given by the RECURRENCE RELATION hn /C302hn /C281 /C271: Solving gives hn /C302n /C281: The number of disks moved after the kth step is the same as the element which needs to be added or deleted in the kth ADDEND of the RYSER FORMULA (Gardner 1988, Vardi 1991). The number of disk to be moved at nth step of the optimal solution to the problem are 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, ... (Sloane’s A001511). Amazingly, this is exactly the BINARY CARRY SEQUENCE plus one. AH ANOI GRAPH can be constructed whose VERTICES correspond to legal configurations of n towers of Hanoi, where the VERTICES are adjacent if the corresponding configurations can be obtained by a legal move. It can be solved using a binary GRAY CODE . Poole (1994) gives Mathematica routines for solving an arbitrary disk configuration in the fewest possible moves. The proof of minimality is achieved using the LUCAS CORRESPONDENCE which relates PASCAL’S TRI- ANGLE to the HANOI GRAPH .ALGORITHMS are known for transferring disks for four pegs, but none has been proved minimal. For additional references, see Poole (1994). See also BINARY CARRY SEQUENCE ,G RAY CODE, RYSER FORMULA References Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences." Theoret. Comput. Sci. 98, 163 /C1/197, 1992. Bogomolny, A. "Towers of Hanoi." http://www.cut-the-knot.- com/recurrence/hanoi.html. Chartrand, G. "The Tower of Hanoi Puzzle." §6.3 in Intro- ductory Graph Theory. New York: Dover, pp. 135 /C1/139, 1985. Dubrovsky, V. "Nesting Puzzles, Part I: Moving Oriental Towers." Quantum 6,53/C1/57 (Jan.) and 49 /C1/51 (Feb.), 1996. Flajolet, P.; Raoult, J.-C.; and Vuillemin, J. " The Number of Registers Required for Evaluating Arithmetic Expres- sions." Theoret. Comput. Sci. 9,99/C1/125, 1979. Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150 /C1/156, May 1957. Gardner, M. "The Icosian Game and the Tower of Hanoi." Ch. 6 in The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 55 /C1/62, 1959. Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Tempus Books, pp. 169 /C1/171, 1989. Kolar, M. "Towers of Hanoi." http://www.pangea.ca/kolar/ javascript/Hanoi/Hanoi.html. Poole, D. G. "The Towers and Triangles of Professor Claus (or, Pascal Knows Hanoi)." Math. Mag. 67, 323 /C1/344, 1994.Poole, D. G. "Towers of Hanoi." MATHEMATICA NOTEBOOK HANOI.M . Ruskey, F. "Towers of Hanoi." http://www.theory.csc.uvic.ca/ ~cos/inf/comb/SubsetInfo.html#Hanoi. Schoutte, P. H. "De Ringen van Brahma." Eigen Haard 22, 274 /C1/276, 1884. Sloane, N. J. A. Sequences A001511/M0127 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Kraitchik, M. "The Tower of Hanoi." §3.12.4 in Mathematical Recreations. New York: W. W. Norton, pp. 91 /C1/93, 1942. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, pp. 111 /C1/112, 1991. T-Polyomino The order n T-polyomino consists of a vertical line of n /C283 squares capped by a horizontal line of three squares centered on the line. See also L-POLYOMINO ,SKEW POLYOMINO ,SQUARE POLYOMINO ,STRAIGHT POLYOMINO T-Puzzle The DISSECTION of the four pieces shown at left into the capital letter "T" shown at right. See also DISSECTION References Pappas, T. "The T Problem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 35 and 230, 1989. Trace (Group) CHARACTER (GROUP ) Trace (Map) Let a PATCH be given by the map x : U 0 Rn ; where U is an open subset of R2 ; or more generally by x : A 0 Rn ; where A is any SUBSET of R2 : Then x(U) (or more generally, x(A)) is called the trace of x. See also PATCH References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 269 /C1/270, 1997. Trace (Matrix) The trace of an n /C29n SQUARE MATRIX A is defined to be Tr(A) /C13Xn i /C301aii ; (1) i.e., the sum of the diagonal elements. The matrix trace is implemented in Mathematica as Tr[list]. In GROUP THEORY , traces are known as "CHARACTERS ." For SQUARE MATRICES A and B ; it is true that Tr(A) /C30Tr(AT) (2) Tr(A /C27B) /C30Tr(A) /C27Tr(B) (3) Tr( aA) /C30 aTr(A) (4) (Lange 1987, p. 40), where AT denotes the TRANS- POSE . The trace is also invariant under a SIMILARITY TRANSFORMATION A ?/C13BAB-1 (5) (Lange 1987, p. 64). Since (bab/C281)ij /C30bilalkb/C281 kj (6) (where EINSTEIN SUMMATION is used here to sum over repeated indices), it follows that Tr(BAB/C281) /C30bilalkb/C281 ki /C30(b/C281b)klalk /C30 dklalk /C30akk /C30Tr(A) ; (7) where dij is the KRONECKER DELTA . The trace of a product of two square matrices is independent of the order of the multiplication since Tr(AB) /C30(ab)ii /C30aijbji /C30bjiaij /C30(ba)jj /C30Tr(BA) (8) (again using EINSTEIN SUMMATION ). Therefore, the trace of the COMMUTATOR of A and B is given by Tr([A ; B]) /C13Tr(AB) /C28Tr(BA) /C300: (9) The trace of a product of three or more square matrices, on the other hand, is invariant only under CYCLIC PERMUTATIONS of the order of multiplication of the matrices, by a similar argument. The product of a SYMMETRIC and an ANTISYMMETRIC MATRIX has zero trace, Tr(ASBA) /C300 : (10)The value of the trace can be found using the fact that the matrix can always be transformed to a coordinate system where the Z-AXIS lies along the axis of rotation. In the new coordinate system (which is assumed to also have been appropriately rescaled), the MATRIX is A ?/C30cos f sin f 0 /C28sin f cos f 0 00 12 435; (11) so the trace is Tr(A ?) /C30Tr(A) /C13a ii /C301 /C272 cos f: (12) See also CHARACTER (GROUP ), CONTRACTION (TEN- SOR), MATRIX ,SQUARE MATRIX ,TRACE (TENSOR ) References Lang, S. Linear Algebra, 3rd ed. New York: Springer- Verlag, pp. 40 and 64, 1987. Munkres, J. R. Elements of Algebraic Topology. Perseus Press, p. 122, 1993. Trace (Path) The image of the path g in C under the FUNCTION f is called the trace. This usage of the term "trace" is unrelated to the same term applied to MATRICES or TENSORS . Trace (Tensor) The trace of a second- RANK TENSOR T is a SCALAR given by the CONTRACTED mixed TENSOR equal to Ti i : The trace is implemented in Mathematica asTr[list]. The trace satisfies Tr M /C281(x)@ @xlM(x)"# /C30@ @xlln[det( x)]; and dln[det M]/C30ln[det( M/C27dM)]/C28ln(det M) /C30lndet(M/C27dM) detM"# /C30ln[det M/C281(M/C27dM)] /C30ln[det(1 /C27M/C281dM)] :ln[1/C27Tr(M/C281dM)] :Tr(M/C281dM): See also CHARACTER (GROUP ), CONTRACTION (TEN- SOR), TRACE (MATRIX ) Traceable Graph A GRAPH G that possesses a HAMILTONIAN PATH . HAMILTONIAN GRAPHS are therefore traceable, but the converse is not necessarily true. The number of traceable graphs on n /C301, 2, ... are 0, 1, 2, 5, 18, 91, 734, ... (Sloane’s A057864), the first few of which are illustrated above. See also HAMILTON- CONNECTED GRAPH ,H AMILTO- NIAN GRAPH ,HYPOTRACEABLE GRAPH References Sloane, N. J. A. Sequences A057864 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Thomassen, C. "Hypohamiltonian and Hypotraceable Graphs." Disc. Math. 9,9 1/C1/96, 1974. Tractory TRACTRIX Tractrisoid PSEUDOSPHERE Tractrix The tractrix is the CATENARY INVOLUTE described by a point initially on the vertex (making the CATENARY the TRACTRIX EVOLUTE ). The tractrix is sometimes called the TRACTORY orEQUITANGENTIAL CURVE . The tractrix was first studied by Huygens in 1692, who gave it the name "tractrix." Later, Leibniz, Johann Bernoulli, and others studied the curve. The tractrix arises from the following problem posedto Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of constant length being pulled along a straight horizontal line (Steinhaus 1983, pp. 250 /C1/251)? By associating the object with a dog, the string with a leash, and the pull along a horizontal line with the dog’s master, the curve has the descriptive name HUNDKURVE (hound curve) in German. Leibniz found the curve using the fact that the axis is an asymptoteto the tractrix (MacTutor Archive).In C ARTESIAN COORDINATES the tractrix has equation x/C30asech/C281y a ! /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28y2p : (1) One parametric form is x(t)/C30a(t/C28tanh t) (2) y(t)/C30asech t: (3) The ARC LENGTH ,CURVATURE , and TANGENTIAL ANGLE in this parameterization are s(t)/C30ln(cosh t) (4) k(t)/C30csch t (5) f(t)/C302 tan/C281tanh1 2tiCkCiCkAhi : (6) A second parametric form in terms of the ANGLE uof the straight line tangent to the tractrix can be found by computing u(t)/C30tan/C281dy dt dx dt0 BBB@1 CCCA/C30tan /C281/C28sech ttanh t tanh2t ! /C30/C28tanh/C281(csch t); (7) then solving for tand plugging back in to obtain x/C30a5ln tan1 2uiCkCiCkAhi /C27cosuno (8) /C30a/C28csch/C281(tanu)/C27cosuiCniCo (9) y/C30asinu (10) (Gray 1997). This parameterization has CURVATURE k(u)/C30tanu jj : (11) In terms of the angle u?/C30p=2/C27u;the PARAMETRIC EQUATIONS can be written x/C30agd/C281u?/C28sinu (12) /C30a[ln(sec u?/C27tan u?) /C28sin u?] (13) /C30a ln tan1 2 u ?/C2714 piCkCiCkAhi /C28sin u?no (14) y /C30a cos u? (15) (Lockwood 1967, p. 123), where gd/C281 x is the inverse GUDERMANNIAN FUNCTION . A parameterization which traverses the tractrix with constant speed a is given by x(t) /C30ae/C28v=afor v /C23 [0;/C12) aev =afor v /C23 (/C28/C12; 0]iC0C (16) y(t) /C30a tanh /C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28e/C282v =apiCkCiCkA /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28e /C282v=ap hi for v /C23 [0;/C12) a /C28tanh /C281ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28e2v =apiCkCiCkA /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28e2v =ap hi for v /C23 (/C28/C12; 0]:8 >>>>< >>>>:ð17Þ When a tractrix is rotated around its asymptote, a PSEUDOSPHERE results. This is a surface of constant NEGATIVE CURVATURE . For a tractrix, the length of a TANGENT from its point of contact to an asymptote is constant. The AREA between the tractrix and its asymptote is finite. See also CURVATURE ,D INI’S SURFACE ,G UDERMAN- NIAN FUNCTION ,M ICE PROBLEM ,P SEUDOSPHERE , PURSUIT CURVE ,TRACTROID References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 226, 1987. Gray, A. "The Tractrix" and "The Evolute of a Tractrix is a Catenary." §3.6 and 5.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 61 /C1/64 and 102 /C1/103, 1997. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 199 /C1/200, 1972. Lockwood, E. H. "The Tractrix and Catenary." Ch. 13 in A Book of Curves. Cambridge, England: Cambridge Univer- sity Press, pp. 118 /C1/124, 1967. MacTutor History of Mathematics Archive. "Tractrix." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Trac- trix.html. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 249 /C1/251, 1999. Yates, R. C. "Tractrix." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 221 /C1/224, 1952.Tractrix Evolute The EVOLUTE of the TRACTRIX is the CATENARY . Tractrix Radial Curve The RADIAL CURVE of the TRACTRIX is the KAPPA CURVE . Tractroid The SURFACE OF REVOLUTION produced by revolving the TRACTRIX x /C30sech u (1) z /C30u /C28tanh u (2) about the Z-AXIS is a tractroid given by x /C30sech u cos v (3) y /C30sech u sin v (4) z /C30u /C28tanh u : (5) See also PSEUDOSPHERE ,SURFACE OF REVOLUTION , TRACTRIX Trail PATH,W ALK Transcendental Curve A curve which intersects some straight line in an infinity of points (but for which not every point lies on this curve). See also ALGEBRAIC CURVE References Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanu- jan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201 /C1/219, 1989. Transcendental Equation An equation or formula involving TRANSCENDENTAL FUNCTIONS . Transcendental Function A function which is not an ALGEBRAIC FUNCTION .In other words, a function which "transcends," i.e., cannot be expressed in terms of, algebra. Examples of transcendental functions include the EXPONENTIAL FUNCTION , the TRIGONOMETRIC FUNCTIONS , and the inverses functions of both. See also ALGEBRAIC FUNCTION ,ELEMENTARY FUNC- TION ,PAINLEVE ´ TRANSCENDENTS Transcendental Number A number which is not the ROOT ofany POLYNOMIAL equation with INTEGER COEFFICIENTS , meaning that it is not an ALGEBRAIC NUMBER of any degree, is said to be transcendental. This definition guarantees that every transcendental number must also be IRRA- TIONAL , since a RATIONAL NUMBER is, by definition, anALGEBRAIC NUMBER of degree one. A number xcan then be tested to see if it is transcendental using theMathematica command Not[Element[ x, Alge- braics]]. Transcendental numbers are important in the history of mathematics because their investigation providedthe first proof that CIRCLE SQUARING , one of the GEOMETRIC PROBLEMS OF ANTIQUITY which had baffled mathematicians for more than 2000 yearswas, in fact, insoluble. Specifically, in order for anumber to be produced by a GEOMETRIC CONSTRUC- TION using the ancient Greek rules, it must be either RATIONAL or a very special kind of ALGEBRAIC NUMBER known as a E UCLIDEAN NUMBER . Because the number pis transcendental, the construction cannot be done according to the Greek rules. Georg Cantor was the first to prove the EXISTENCE of transcendental numbers. Liouville subsequently showed how to construct special cases (such as LIOUVILLE’S CONSTANT ) using L IOUVILLE’S APPROXI- MATION THEOREM . In particular, he showed that any number which has a rapidly converging sequence of rational approximations must be transcendental. For many years, it was only known how to determine ifspecial classes of numbers were transcendental. The determination of the status of more general numberswas considered an important enough unsolved pro- blem that it was one of H ILBERT’S PROBLEMS . Great progress was subsequently made by G ELFOND’S THEOREM , which gives a general rule for determining if special cases of numbers OF THE FORM abare transcendental. Baker produced a further revolution by proving the transcendence of sums of numbers OF THE FORM alnbfor ALGEBRAIC NUMBERS aandb:/ The number Ewas proven to be transcendental by Hermite in 1873, and PI(/p) by Lindemann in 1882. ep is transcendental by G ELFOND’S THEOREM since (/C281)/C28i/C30(eip)/C28i/C30ep: The G ELFOND- SCHNEIDER CONSTANT 2ffiffi 2p is also trans- cendental (Hardy and Wright 1979, p. 162). Known transcendentals are summarized in the followingtable, where sin xis the SINE function, J0(x)i sa BESSEL FUNCTION OF THE FIRST KIND ,x(n) kis the nth zero of Jk(x);Pis the T HUE- MORSE CONSTANT ,G(x)i s the GAMMA FUNCTION , and where z(n) is the R IEMANN ZETA FUNCTION . e Hermite (1873) /p/ Lindemann (1882) /ep/ Gelfond /epffiffi dp ;d/C23Z/C31/ Nesterenko (1999) /2ffiffi 2p / Hardy and Wright (1979, p. 162) /sin 1 / Hardy and Wright (1979, p. 162) /J0(1)/ Hardy and Wright (1979, p. 162) /ln 2 / Hardy and Wright (1979, p. 162) /ln 3 =ln 2 / Hardy and Wright (1979, p. 162), /x(1) 0/C302:4048255 . . . / Le Lionnais (1983, p. 46) /p/C27ln 2/C27ffiffiffi 2p ln 3 / Borwein et al. (1989) /P/C300:4124540336 . . . /Dekking (1977), Allouche and Shallit CHAMPERNOWNE CONSTANT THUE CONSTANT /G1 3iCkCiCkA / Le Lionnais (1983, p. 46) /G14iCkCiCkA / Chudnovsky (1984, p. 308), Waldschmidt, Nesterenko (1999) /G1 6iCkCiCkA / Chudnovsky (1984, p. 308) /G14iCkCiCkA p/C281=4/ Davis (1959) /z(2n);n/C23Z>1/ APE´ RY’S CONSTANT z(3) has been proved to be IRRA- TIONAL , but it is not known if it is transcendental. At least one of pe and p /C27e (and probably both) are transcendental, but transcendence has not been proven for either number on its own. It is not known if ee ; pp ; pe ; g (the EULER- MASCHERONI CONSTANT ), I0(2) ; or I1(2) (where In(x)isa MODIFIED BESSEL FUNCTION OF THE FIRST KIND ) are transcendental. The "degree" of transcendence of a number can be characterized by a so-called IRRATIONALITY MEASURE . There are still many fundamental and outstanding problems in transcendental number theory, including the CONSTANT PROBLEM and SCHANUEL’S CONJEC- TURE . See also ALGEBRAIC NUMBER ,ALGEBRAICALLY INDE- PENDENT ,A LGEBRAICS ,C ONSTANT PROBLEM ,FOUR EXPONENTIALS CONJECTURE ,G ELFOND’S THEOREM , IRRATIONAL NUMBER ,IRRATIONALITY MEASURE ,LIN- DEMANN- WEIERSTRASS THEOREM ,ROTH’S THEOREM , SCHANUEL’S CONJECTURE ,SIX EXPONENTIALS THEO- REM,THUE- SIEGEL- ROTH THEOREM References Allouche, J. P. and Shallit, J. In preparation. Baker, A. "Approximations to the Logarithm of Certain Rational Numbers." Acta Arith. 10, 315 /C1/323, 1964. Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers I." Mathematika 13, 204 /C1/216, 1966. Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers II." Mathematika 14, 102 /C1/107, 1966. Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers III." Mathematika 14, 220 /C1/228, 1966. Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers IV." Mathematika 15, 204 /C1/216, 1966. Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanu- jan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201 /C1/219, 1989. Chudnovsky, G. V. Contributions to the Theory of Transcen- dental Numbers. Providence, RI: Amer. Math. Soc., 1984. Courant, R. and Robbins, H. "Algebraic and Transcendental Numbers." §2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 103 /C1/107, 1996. Davis, P. J. "Leonhard Euler’s Integral: A Historical Profile of the Gamma Function." Amer. Math. Monthly 66, 849 /C1/ 869, 1959. Dekking, F. M. "Transcendence du nombre de Thue-Morse." C. R. Acad. Sci. Paris 285, 157 /C1/160, 1977. Gray, R. "Georg Cantor and Transcendental Numbers." Amer. Math. Monthly 101, 819 /C1/832, 1994. Hardy, G. H. and Wright, E. M. "Algebraic and Transcen- dental Numbers," "The Existence of Transcendental Numbers," and "Liouville’s Theorem and the Construction of Transcendental Numbers." §11.5 /C1/11.6 in An Introduc- tion to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 159 /C1/164, 1985. Hermite, C. "Sur la fonction exponentielle." C. R. Acad. Sci. Paris 77,18/C1/24, 74 /C1/79, and 226 /C1/233, 1873. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1979. Lindemann, F. "U¨ ber die Zahl p:/" Math. Ann. 20, 213 /C1/225, 1882. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 35, 1951.Nesterenko, Yu. V. "On Algebraic Independence of the Components of Solutions of a System of Linear Differen- tial Equations." [Russian.] Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495 /C1/512, 1974. English translation in Math. USSR 8, 501 /C1/518, 1974. Nesterenko, Yu. V. "Modular Functions and Transcendence Questions." [Russian.] Mat. Sbornik 187,65/C1/96, 1996. English translation in Sbornik Math. 187, 1319 /C1/1348, 1996. Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. http://www.math.jussieu.fr/~neste- ren/. Ramachandra, K. Lectures on Transcendental Numbers. Madras, India: Ramanujan Institute, 1969. Shidlovskii, A. B. Transcendental Numbers. New York: de Gruyter, 1989. Siegel, C. L. Transcendental Numbers. New York: Chelsea, 1965. Tijdeman, R. "An Auxiliary Result in the Theory of Trans- cendental Numbers." J. Numb. Th. 5,80/C1/94, 1973. Transcritical Bifurcation Letf:R/C29R0Rbe a one-parameter family of C2 maps satisfying f(0;m)/C300 (1) @f @x"# m/C300;x/C300/C300 (2) @2f @x@m"# 0;0>0 (3) @2f @x2"# m/C300;x/C300B0: (4) (Actually, condition (1) can be relaxed slightly.) Then there are two branches, one stable and one unstable. This BIFURCATION is called a transcritical bifurcation. An example of an equation displaying a transcriticalbifurcation is ˙x/C30mx/C28x 2(5) (Guckenheimer and Holmes 1997, p. 145). See also BIFURCATION ,PITCHFORK BIFURCATION References Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed.New York: Springer-Verlag, pp. 145 and 149 /C1/150, 1997. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 27 /C1/28, 1990. Transfer Function The engineering terminology for one use of F OURIER TRANSFORMS . By breaking up a wave pulse into its frequency spectrum fn/C30F(n)e2pint; (1) the entire signal can be written as a sum of contribu- tions from each frequency, f(t) /C30g/C12 /C28/C12fn dn /C30g/C12 /C28/C12F( n)e2 pint dn : (2) If the signal is modified in some way, it will become gn(t) /C30 f( n)fn(t) /C30 f(n)F( n)e2 pi nt (3) g(t) /C30g/C12 /C28/C12gn(t) dt /C30g/C12 /C28/C12f(n)F( n)e2 pi nt d n; ð4Þ where f( n) is known as the "transfer function." FOURIER TRANSFORMING f and F, f(n) /C30g/C12 /C28/C12F(t)e /C282pint dt (5) F( n) /C30g/C12 /C28/C12f(t)e /C282 pint dt : (6) From the CONVOLUTION THEOREM , g(t) /C30f(t) +F(t) /C30g/C12 /C28/C12f(t) F(t /C28r) dr : (7) See also CONVOLUTION THEOREM ,FOURIER TRANS- FORM Transfer Principle In NONSTANDARD ANALYSIS , the transfer principle is the technical form of the following intuitive idea: "Anything provable about a given SUPERSTRUCTURE V by passing to a nonstandard enlargement +V of V is also provable without doing so, and vice versa." It is a result of LOS’ THEOREM and the completeness theo- rem for first-order predicate logic The transfer principle is stated as follows. Let V be a superstructure, let +V be an enlargement of V, let s be any sentence in the language for (V ;/C23) ; and let +s denote the +-transformof s:Then (V ;/C23) ffi s ifandonlyif ( +V ;+/C23) ffi+s:/ See also LOS’ THEOREM ,NONSTANDARD ANALYSIS Transfinite Diameter Let f(z) /C30cz /C27c0 /C27c1z /C281 /C27c2z /C282 /C27... be an ANALYTIC FUNCTION , REGULAR and UNIVALENT for zjj> 1; which maps zjj> 1 CONFORMALLY onto the region T preserving the POINT AT INFINITY and its direction. Then the function f(z) is uniquely deter- mined and c is called the transfinite diameter, some- times also known as ROBIN’S CONSTANT or the CAPACITY of f(z):/ See also ANALYTIC FUNCTION ,REGULAR FUNCTION , UNIVALENT FUNCTIONTransfinite Number One of Cantor’s ORDINAL NUMBERS v; v /C271; v /C272; ..., v /C27 v; v /C27 v /C271; ...which is "larger" than any WHOLE NUMBER . See also ALEPH-0 ,ALEPH-1 ,CARDINAL NUMBER ,CON- TINUUM ,ORDINAL NUMBER ,W HOLE NUMBER References Ferreiro ´s, J. "The Transfinite Ordinals and Cantor’s Mature Theory." Ch. 8 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 257 /C1/296, 1999. Pappas, T. "Transfinite Numbers." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 156 /C1/158, 1989. Transform A shortened term for INTEGRAL TRANSFORM . Geometrically, if S and T are two transformations, then the SIMILARITY TRANSFORMATION TST /C281 is some- times called the transform (Woods 1961). See also ABEL TRANSFORM ,BOUSTROPHEDON TRANS- FORM ,DISCRETE FOURIER TRANSFORM ,FAST FOURIER TRANSFORM ,F OURIER TRANSFORM ,F RACTIONAL FOURIER TRANSFORM ,HANKEL TRANSFORM ,HARTLEY TRANSFORM ,H ILBERT TRANSFORM ,L APLACE- STIELTJES TRANSFORM ,LAPLACE TRANSFORM ,MELLIN TRANSFORM ,NUMBER THEORETIC TRANSFORM ,PON- CELET TRANSFORM ,R ADON TRANSFORM ,W AVELET TRANSFORM , Z-TRANSFORM References Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 5, 1961. Transform Theory INTEGRAL TRANSFORM Transformation A transformation T(a.k.a., MAP,FUNCTION ) over a DOMAIN Dtakes the elements X/C23Dto elements Y/C23 T(D);where the RANGE (a.k.a., image) of Tis defined as Range( T)/C30T(D)/C30fT(X):X/C23Dg: Note that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system , with space and objects embedded in it being viewed as fixed (a so-called ALIAS TRANS- FORMATION ), or on the space itself relative to a fixed coordinate system (a so-called ALIBI TRANSFORMA- TION ). Examples of transformations are summarized in the following table. Transforma- tionCharacterization DILATION center of dilation, scale decrease factor EXPANSION center of expansion, scale in- crease factor REFLECTION mirror line or plane ROTATION center of rotation, rotation angle SHEAR invariant line and SHEAR FACTOR STRETCH (1- way)invariant line and scale factor STRETCH (2- way)invariant lines and scale factors TRANSLATION displacement vector See also AFFINE TRANSFORMATION ,ALIAS TRANSFOR- MATION ,ALIBI TRANSFORMATION ,DILATION ,EXPAN- SION ,F UNCTION ,M AP,R EFLECTION ,R OTATION , SHEAR ,STRETCH ,TRANSFORM ,TRANSLATION References Coxeter, H. S. M. and Greitzer, S. L. "Transformations." Ch. 4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 80 /C1/102, 1967. Graustein, W. C. "Transformation." Ch. 7 in Introduction to Higher Geometry. New York: Macmillan, pp. 84 /C1/114, 1930. Kapur, J. N. Transformation Geometry. New Delhi, India: Mathematical Sciences Trust Society, 1994 /C1/95. Transition Function A transition function describes the difference in the way an object is described in two separate, over- lapping COORDINATE CHARTS , where the description of the same set may change in different coordinates. This even occurs in EUCLIDEAN SPACE R3 ; where any rotation of the usual x, y, and z axes gives another set of coordinates. For example, on the sphere, person A at the equator can use the usual directions of north, south, east, and west, but person B at the North Pole must use something else. However, both A and B can describe the region in between them in their coordinate charts. A transition function would then describe how to go from the coordinate chart for A to the coordinate chart for B. In the case of a MANIFOLD , a transition function is a map from one coordinate chart to another. Therefore, in a sense, a manifold is composed of coordinate charts, and the glue that holds them together is the transition functions. In the case of a BUNDLE , the transition functions are the glue that holds togetherits TRIVIALIZATIONS . Specifically, in this case the transition function describes an invertible transfor- mation of the FIBER . Naturally, the type of invertible transformation depends on the type of bundle. For instance, a VECTOR BUNDLE , which could be the TANGENT BUNDLE , has INVERTIBLE LINEAR transition functions. More pre- cisely, a transition function for a vector bundle of RANK r, on overlapping coordinate charts U1 and U2 ; is given by a function g12 : U1 S U2 0 GL(r) ; where GL is the GENERAL LINEAR GROUP . The fiber at p /C23 U1 S U2has two descriptions, and g12(p) is the INVERTIBLE LINEAR MAP that takes one to the other. The transition functions have to be consistent in the sense that if one goes to another description of the same set, and then back again, then nothing has changed. A necessary and sufficient condition for consistency is the following: Given three overlapping charts, the product g12g23g31has to be the constant map to the identity in GL(r) :/ A consistent set of transition functions for a VECTOR BUNDLE of RANK r can be interpreted as an element of the first CECH COHOMOLOGY GROUP of a manifold with coefficients in GL(r) :/ See also BUNDLE ,CECH COHOMOLOGY ,COORDINATE CHART ,M ANIFOLD ,T ANGENT BUNDLE ,T RIVIALIZA- TION ,VECTOR BUNDLE Transitive A RELATION R on a SET S is transitive provided that for all x, y and z in S such that xRy and yRz ; we also have xRz:/ See also ASSOCIATIVE ,COMMUTATIVE ,RELATION Transitive Closure The transitive closure of a BINARY RELATION R on a SET X is the minimal TRANSITIVE relation R? on X that contains R. Thus aR ?b for any elements a and b of X provided that there exist c0 ; c1 ; ..., cn with c0 /C30a; cn /C30 b; and crRcr/C271 for all 0 5r 5n:/ The transitive closure C(G)ofa GRAPH is a graph which contains an edge fu; vg whenever there is a directed path from u to v (Skiena 1990, p. 203). The transitive closure of a graph can be computed using TransitiveClosure [g] in the Mathematica add-on packageDiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also REFLEXIVE CLOSURE ,TRANSITIVE GRAPH , TRANSITIVE REDUCTION References Aho, A.; Garey, M. R.; and Ullman, J. D. "The Transitive Reduction of a Directed Graph." SIAM J. Comput. 1, 131 /C1/ 137, 1972. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Transitive Digraph A GRAPH G is transitive if any three vertices /ðx; y;zÞ/ such that edges (x; y) ;(y; z) /C23 G imply (x; y) /C23 G: Un- labeled transitive digraphs are called TOPOLOGIES . See also TOPOLOGY (DIGRAPH ), TRANSITIVE GRAPH , TRANSITIVE REDUCTION Transitive Graph A GRAPH G is called n-transitive with n ]1 if it has an n-ROUTE and if there is always a GRAPH AUTO- MORPHISM of G sending each n-ROUTE onto any other n-ROUTE (Harary 1994, p. 173). There are no n- transitive CUBIC GRAPHS for n /C215 (Harary 1994, p. 175). See also ROUTE ,TRANSITIVE CLOSURE ,TRANSITIVE DIGRAPH ,T RANSITIVE REDUCTION ,U NITRANSITIVE GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 162 and 174, 1990. Transitive Group When a GROUP ACTION is implicitly understood, i.e., a subgroup of a PERMUTATION GROUP , then the SUB- GROUP is called transitive if its action is transitive. For example, the ALTERNATING GROUP is transitive. A group may also be called k-transitive if there is any set on which the group acts FAITHFULLY and k- transitively. Transitivity is a result of the symmetry in the group. For instance, the SYMMETRIC GROUP Sn is n-transitive and the ALTERNATING GROUP Anis (n /C282)/-transitive. However, multiply transitive finite groups are rare. In fact, they have been completely determined using the CLASSIFICATION THEOREM OF FINITE GROUPS . Except for some SPORADIC examples, the multiply transitive groups fall into infinite families. Certain subgroups of the AFFINE GROUP on a finite VECTOR SPACE , including the AFFINE GROUP itself, are 2- transitive. Some of these are summarized below. The multiply transitive groups fall into six infinite families, and four classes of SPORADIC GROUPS . In the following enumeration, q is a power of a prime number.1. Certain subgroups of the AFFINE GROUP on a finite VECTOR SPACE , including the AFFINE GROUP itself, are 2-transitive. 2. The PROJECTIVE SPECIAL LINEAR GROUPS PSL(d; q) are 2-transitive, and PSL(2; q) is actu- ally 3-transitive. 3. The SYMPLECTIC GROUPS defined over the FIELD of two elements have two distinct actions which are 2-transitive. 4. The field K of q2 elements has an INVOLUTION s(a) /C30aq ; so s2 /C301; which allows a HERMITIAN FORM to be defined on a VECTOR SPACE on K. The UNITARY GROUP on V /C30/C1543 K ; denoted U2(q); pre- serves the ISOTROPIC VECTORS in V. The action of the PROJECTIVE SPECIAL UNITARY GROUP PSU3(q)is 2-transitive on the ISOTROPIC VECTORS . 5. The SUZUKI GROUP Sz(q) is the AUTOMORPHISM GROUP of a S(3; q /C271; q2 /C271) STEINER SYSTEM ,an INVERSIVE PLANE of order q, and its action is 2- transitive. 6. The REE GROUP R(q) is the AUTOMORPHISM GROUP of a S(2; q /C271; q3 /C271) STEINER SYSTEM ,a UNITAL of order q, and its action is 2-transitive. 7. The MATHIEU GROUPS M12 and M24 are the only 5-transitive groups besides S5 and A7 : The groups M11and M23are 4-transitive, and M22is 3- transitive.8. The PROJECTIVE SPECIAL LINEAR GROUP PSL(2; 11) has another 2-transitive action related to the WITT GEOMETRY W11 :/ 9. The HIGMAN- SIMS GROUP is 2-transitive. 10. The CONWAY GROUP Co3 is 2-transitive. See also FINITE SIMPLE GROUP ,L EECH LATTICE , MATHIEU GROUPS ,S TEINER SYSTEM ,T RANSITIVE GROUP ACTION References Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Transitive Group Action A GROUP ACTION G /C29X 0 X is transitive if it pos- sesses only a single ORBIT , i.e., for every pair of elements x and y, there is a group element g such that gx /C30y. In this case, X is ISOMORPHIC to the left COSETS of the isotropy group, X /C2G =Gx : The space X, which has a transitive group action, is called a HOMOGENEOUS SPACE when the group is a LIE GROUP . If, for every two pairs of points x1 ; x2 and y1 ; y2 ; there is a group element g such that gxi /C30yi ; then the GROUP ACTION is called doubly transitive. Similarly, a group action can be triply transitive and, in general, a GROUP ACTION isk-transitive if every set x1;...;yk fg of 2kdistinct elements has a group element gsuch that gxi/C30yi:/ See also EFFECTIVE ACTION ,FAITHFUL GROUP AC- TION ,FREE ACTION ,G ROUP ,ISOTROPY GROUP ,M A- TRIX GROUP ,ORBIT (GROUP ), QUOTIENT SPACE (LIE GROUP ), REPRESENTATION ,T OPOLOGICAL GROUP , TRANSITIVE GROUP References Burnside, W. "On Transitive Groups of Degree n and Class n /C281:/" Proc. London Math. Soc. 32, 240 /C1/246, 1900. Hulpke, A. Konstruktion transitiver Permutationsgruppen. Ph.D. thesis. Aachen, Germany: RWTH, 1996. Also avail- able as Aachener Beitra ¨ge zur Mathematik , No. 18, 1996. Kawakubo, K. The Theory of Transformation Groups. Oxford, England: Oxford University Press, pp. 4 /C1/6 and 41 /C1/49, 1987. Rotman, J. Theory of Groups. New York: Allyn and Bacon, pp. 180 /C1/184, 1984. Transitive Points Two points on a surface which are opposite to each other but not farthest from each other (e.g., the midpoints of opposite edges of a CUBE ) are said to be transitive points. The SPHERE has no transitive points. References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 175, 1999. Transitive Reduction The transitive reduction of a BINARY RELATION R on a SET X is the minimum relation R? on X with the same TRANSITIVE CLOSURE as R. Thus aR?b for any ele- ments a and b of X, provided that aRb and there exists no element c of X such that aRc and cRb:/ The transitive reduction of a GRAPH G is the smallest graph R(G) such that C(G) /C30C(R(G)) ; where C(G)is the TRANSITIVE CLOSURE of G (Skiena 1990, p. 203). See also REFLEXIVE REDUCTION ,T RANSITIVE CLO- SURE ,TRANSITIVE GRAPH References Aho, A.; Garey, M. R.; and Ullman, J. D. "The Transitive Reduction of a Directed Graph." SIAM J. Comput. 1, 131 /C1/ 137, 1972. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Transitive Triple The 3-node TOURNAMENT (and DIRECTED GRAPH ) illustrated above (Harary 1994, p. 205). See also CYCLIC TRIPLE ,TOURNAMENT References Harary, F. "Tournaments." Graph Theory. Reading, MA: Addison-Wesley, 1994. Transitivity Class Let S(T) be the group of symmetries which map a MONOHEDRAL TILING T onto itself. The TRANSITIVITY CLASS of a given tile T is then the collection of all tiles to which T can be mapped by one of the symmetries of S(T) :/ See also MONOHEDRAL TILING References Berglund, J. "Is There a k-Anisohedral Tile for k ]5/?" Amer. Math. Monthly 100, 585 /C1/588, 1993. Translation A transformation consisting of a constant offset with no ROTATION or distortion. In n-D EUCLIDEAN SPACE , a translation may be specified simply as a VECTOR giving the offset in each of the n coordinates. See also AFFINE GROUP ,D ILATION ,E UCLIDEAN GROUP ,E XPANSION ,G LIDE ,IMPROPER ROTATION , INVERSION OPERATION ,M IRROR IMAGE ,REFLECTION , ROTATION References Addington, S. "The Four Types of Symmetry in the Plane." http://forum.swarthmore.edu/sum95/suzanne/symsu- san.html. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 211, 1987. Coxeter, H. S. M. and Greitzer, S. L. "Translation." §4.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 81 /C1/82, 1967. Translation Relation A mathematical relationship transforming a function f(x) to the form f(x /C27a) :/ See also ARGUMENT ADDITION RELATION ,ARGUMENT MULTIPLICATION RELATION ,RECURRENCE RELATION , REFLECTION RELATION Transpose The object obtained by replacing all elements aij with aji : For a second- RANK TENSOR aij ; the tensor trans- pose is simply aji : The matrix transpose, written AT ; is the MATRIX obtained by exchanging A/’s rows and columns, and satisfies the identity (AT) /C281 /C30(A /C281)T : (1) Several other notations are commonly used, including ˜A (Arfken 1985, p. 201; Griffiths 1987, p. 223) and A? (Ayres 1962, p. 11; Courant and Hilbert 1989, p. 9) The product of two transposes satisfies (BTAT)ij /C30(bT)ik(aT)kj /C30bkiajk /C30ajkbki /C30(AB)ji /C30(AB)T ij ; (2) where EINSTEIN SUMMATION has been used to im- plicitly sum over repeated indices. Therefore, (AB)T /C30BTAT : (3) See also ADJOINT MATRIX ,C ONGRUENT MATRICES , CONJUGATE MATRIX References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 201, 1985. Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, pp. 11 /C1/12, 1962. Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989. Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, p. 220, 1987. See also SKEW SYMMETR IC MATRIX ,S YMMETR IC MATRIX Transpose Map PULLBACK MAP Transpose Partition CONJUGATE PARTITIONTransposition An exchange of two elements of an ordered list with all others staying the same. A transposition is there- fore a PERMUTATION of two elements. For example, the swapping of 2 and 5 to take the list 123456 to 153426 is a transposition. The PERMUTATION SYMBOL eijk/C1/C1/C1is defined as (/C281)n ; where n is the number of transpositions of pairs of elements that must be composed to build up the PERMUTATION . See also INVERSION NUMBER ,PERMUTATION ,PERMU- TATION SYMBOL ,TRANSPOSITION GRAPH ,TRANSPOSI- TION ORDER References Skiena, S. "Permutations from Transpositions." §1.1.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 9 /C1/11, 1990. Transposition Graph A GRAPH in which nodes correspond to permutations and edges are placed between permutations that differ by exactly one transposition (Skiena 1990, p. 9). All cycles in transposition graphs are of even length, making them BIPARTITE . The transposition graph of a MULTISET is always HAMILTONIAN (Chase 1973). See also TRANSPOSITION References Chase, P. J. "Transposition Graphs." SIAM J. Comput. 2, 128 /C1/133, 1973. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 9 /C1/10, 1990. Transposition Group A PERMUTATION GROUP in which the PERMUTATIONS are limited to TRANSPOSITIONS . See also PERMUTATION GROUP Transposition Order An ordering of PERMUTATIONS in which each two adjacent permutations differ by the TRANSPOSITION of two elements. For the permutations of f1;2;3gthere are two listings which are in transposition order. One is 123, 132, 312, 321, 231, 213, and the other is 123, 321, 312, 213, 231, 132. See also LEXICOGRAPHIC ORDER ,PERMUTATION References Ruskey, F. "Information on Combinations of a Set." http:// www.theory.csc.uvic.ca/~cos/inf/comb/CombinationsIn- fo.html. Transversal Array A set of n cells in an n /C29n SQUARE such that no two come from the same row and no two come from the same column. The number of transversals of an n /C29n SQUARE is n!(n FACTORIAL ). A Latin transversal is a transversal such that no two cells contain the same element (Snevily 1999). References Alon, N. Additive Latin Transversals. Preprint. Snevily, H. S. "The Cayley Addition Table of Zn :/" Amer. Math. Monthly 106, 584 /C1/585, 1999. Transversal Design A transversal design TDl(k; n) of order n, block size k, and index l is a triple (V, G, B) such that 1. V is a set of kn elements, 2. G is a partition of V into k classes, each of size n (the "groups"), 3. B is a collection of k-subsets of V (the "blocks"), and 4. Every unordered pair of elements from V is contained in either exactly one group or in exactly l blocks, but not both. References Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 112, 1996. Transversal Intersection Two SUBMANIFOLDS X and Y in an ambient space M intersect transversally if, for all p /C23 X S Y ; TXp /C27TYp /C30 v /C27w : v /C23 TXp ; w /C23 TYpiCniCo /C30TMp ;where the addition is in TMp ; and TXpdenotes the TANGENT MAP of Xp : If two submanifolds do not intersect, then they are automatically transversal. For example, two curves in R3 are transversal only if they do not intersect at all. When X and Y meet transversally then X S Y is a smooth SUBMANIFOLD of the expected dimension dim X /C27dim Y /C28dim M :/ In some sense, two submanifolds "ought" to intersect transversally and, by SARD’S THEOREM , any intersec- tion can be perturbed to be transversal. Intersection in HOMOLOGY only makes sense because an intersec- tion can be made to be transversal. Transversality is a sufficient condition for an inter- section to be stable after a perturbation. For example, the lines y /C30x and y /C300 intersect transversally, as do the perturbed lines y /C30x /C27t; and they intersect at only one point. However, y /C30x2 does not intersect y /C300 transversally. It intersects in one point, while y /C30x2 /C27t intersects in either none or two points, depending on whether t is positive or negative. When dim X /C27dim Y /C30dim M ; then a transversal intersection is an ISOLATED POINT . If the three spaces have an ORIENTATION , then the transversal condition means it is possible to assign a sign to the intersec- tion. If e1 ; ...; ekare an oriented basis for TXpand ek /C271 ; ...; enare an oriented basis for TYp ; then the intersection is /C271ife1 ; ...; enis oriented in M and /C281 otherwise. More generally, two SMOOTH MAPS f : X 0 M and g : Y 0 M are transversal if whenever p /C30f(x) /C30g(y) then df TXxðÞ/C27dg TYyiCjiCk /C30TMp :/ See also HOMOLOGY ,INTERSECTION (HOMOLOG Y), ORIENTATION (VECTOR SPACE ), SARD’S THEOREM , SUBMERSION Transversal Line A transversal line is a LINE which intersects each of a given set of other lines. It is also called a semisecant. See also LINE Transversal Plane References Altshiller-Court, N. "Transversals." Ch. 5 in Modern Pure Solid Geometry. New York: Chelsea, pp. 111 /C1/122, 1979. Transylvania Lottery A lottery in which three numbers are picked at random from the INTEGERS 1/C1/14. See also FANO PLANE Trapdoor Function An easily computed function whose inverse is extre- mely difficult to compute. An example is the multi- plication of two large PRIMES . Finding and verifying two large PRIMES is easy, as is their multiplication. But factorization of the resultant product is very difficult. See also RSA ENCRYPTION References Gardner, M. "Trapdoor Ciphers" and "Trapdoor Ciphers II." Chs. 13 /C1/14 in Penrose Tiles and Trapdoor Ciphers...and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 183 /C1/204, 1989. Trapdoor One-Way Function Informally, a function f : f0; 1 gl(n) /C29f0; 1gn 0 (0; 1gm(n) is a trapdoor one-way function if 1. It is a ONE-WAY FUNCTION , and 2. For fixed public key y /C23f0; 1gl(n) ; f(x; y)is viewed as a function fy(x)ofx that maps n bits to m(n) bits. Then there is an efficient algorithm that, on input y; fy(x) ; ziCkjiCkk produces x? such that fyx?ðÞ/C30 fy(x); for some trapdoor key z /C23f0 ; 1 gk(n) :/ f is a TRAPDOOR ONE-WAY HASH FUNCTION if f is also a ONE-WAY HASH FUNCTION , i.e., if additionally 3. Given M and f(M) ; it is hard to find a message M ?"M such that fM?ðÞ"f(M) :/ It is not known if a trapdoor one-way function can be constructed from any one-way function. An example of a trapdoor one-way function is factor- ization of a product of two large PRIMES . While selecting and verifying two large PRIMES and multi- plying them together is easy, factoring the resulting product is (as far as is known) very difficult. This is the basis for RSA ENCRYPTION , which is conjectured to be trapdoor one-way. See also ONE-WAY FUNCTION ,RSA ENCRYPTION , TRAPDOOR ONE-WAY HASH FUNCTION References Gardner, M. "Trapdoor Ciphers" and "Trapdoor Ciphers II." Chs. 13 /C1/14 in Penrose Tiles and Trapdoor Ciphers...and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 183 /C1/204, 1989. Luby, M. Pseudorandomness and Cryptographic Applica- tions. Princeton, NJ: Princeton University Press, 1996. RSA Laboratories. † "What Is a One-Way Function?" http:// www.rsasecurity.com/rsalabs/faq/2 /C1/3-2.html. Trapdoor One-Way Hash Function A function f : f0; 1gl(n) /C29f0; 1gn 0 (0; 1gm(n)is a TRAPDOOR ONE-WAY HASH FUNCTION if f is a TRAPDOOR ONE-WAY FUNCTION and is also a one-way hash function, i.e. if, additionally given M and f(M); it ishard to find a message M ?"M such that fM?ðÞ/C30f(M) :/ See also TRAPDOOR ONE-WAY FUNCTION Trapezium There are two common definitions of the trapezium. The American definition is a QUADRILATERAL with no PARALLEL sides. The British definition for a trape- zium is a QUADRILATERAL with two sides PARALLEL . Such a trapezium is equivalent to a TRAPEZOID and therefore has AREA A /C301 2(a /C27b)h: See also DIAMOND ,KITE,LOZENGE ,PARALLELOGRAM , QUADRILATERAL ,RHOMBOID ,RHOMBUS ,SKEW QUAD- RILATERAL ,STROMBUS ,TRAPEZOID Trapezohedron The trapezohedra are the DUAL POLYHEDRA of the Archimedean ANTIPRISMS . However, the name for these solids is not particular well chosen since their faces are not TRAPEZOIDS . The CUBE oriented along a space diagonal is a trapezohedron. The trapezohedra generated by taking the duals of the ANTIPRISMS have side length sn;half-heights (half the peak-to-peak distance) hn;surface areas Sn;and volumes Vn(where the latter two are normalized so that the shortest edge has length 1) given by s3/C301 2ffiffiffi 2p (1) h3/C301 4ffiffiffi 6p (2) S3/C306 (3) V3/C301 (4) s4/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p /C281q ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21/C27ffiffiffi 3piCkCiCkAr (5) h4 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 4 /C273ffiffiffi 2piCkCiCkAr (6) S4 /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 22 /C2716ffiffiffi 2pq (7) V4 /C301 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 58 /C2741ffiffiffi 2pq (8) s5 /C301 2ffiffiffi 5p /C281iCkCiCkA ;1 21 /C27ffiffiffi 5piCkCiCkA (9) h5 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq (10) S5 /C305ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 225 /C2711ffiffiffi 5piCkCiCkAr (11) V5 /C305 1211 /C275ffiffiffi5piCkCiCkA (12) s 6 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ffiffiffi 3p /C281iCkCiCkAr ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 25 /C273ffiffiffi 3piCkCiCkAr (13) h6 /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 38 /C2722ffiffiffi 3pq (14) S6 /C306ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24 /C2714ffiffiffi 3pq (15) V6 /C307ffiffiffi 2p /C274ffiffiffi 6p (16) s8 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pqr ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 28 /C275ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 258/C2741ffiffiffi 2piCkCiCkAriC0jiC0ks (17) h8 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 230 /C2720ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 850 /C27601ffiffiffi 2piCkCiCkAr iC0jiC0ks (18) S8 /C304ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144 /C2798ffiffiffi 2p /C274ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2516 /C271778ffiffiffi 2pqr (19) V8 /C30 2 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1150 /C27812ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2641130 /C271867559ffiffiffi 2pq iCkniCko :s (20) See also ANTIPRISM ,CUBE,DIPYRAMID ,DUAL POLY- HEDRON ,HEXAGONAL SCALENOHEDRON ,PENTAGONAL DELTAHEDRON ,PRISM ,TRAPEZOID References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 117, 1989. Pedagoguery Software. Poly . http://www.peda.com/poly/.Trapezoid A QUADRILATERAL with two sides PARALLEL . The trapezoid is equivalent to the British definition of TRAPEZIUM . The trapezoid depicted has central med- ian m /C301 2(a /C27b) ; AREA A /C3012(a /C27b)h /C30mh: The CENTROID lies on the median m at a distance x /C30b /C27 2a 3(a /C27 b)h from the vertical position of the lower left vertex. See also ISOSCELES TRAPEZOID ,PYRAMIDAL FRUSTUM , STROMBUS ,TRAPEZIUM References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Harris, J. W. and Stocker, H. "Trapezoid." §3.6.2 in Hand- book of Mathematics and Computational Science. New York: Springer-Verlag, pp. 82 /C1/83, 1998. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948. Trapezoidal Hexecontahedron DELTOIDAL HEXECONTAHEDRON Trapezoidal Icositetrahedron DELTOIDAL ICOSITETRAHEDRON Trapezoidal Rule The 2-point N EWTON- COTES FORMULA gx2 x1f(x)dx/C301 2hf1/C27f2 ðÞ /C281 12h3fƒ(j); where fi/C13fxiðÞ;his the separation between the points, and j is a point satisfying x1 5 j 5x2 : Picking j to maximize f ƒ(j) gives an upper bound for the error in the trapezoidal approximation to the INTEGRAL . See also BODE’S RULE,HARDY’S RULE,NEWTON- COTES FORMULAS ,S IMPSON’S 3/8 RULE,S IMPSON’S RULE, WEDDLE’S RULE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 885, 1972. Whittaker, E. T. and Robinson, G. "The Trapezoidal and Parabolic Rules." The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 156 /C1/158, 1967. Traveler’s Problem HAMILTONIAN CIRCUIT Traveling Salesman Constants N.B. A detailed online essay by S. Finch was the starting point for this entry. LetL(n;d) be the smallest TOUR length for npoints in a d-D HYPERCUBE . Then there exists a smallest constant a(d) such that for all optimal TOURS in the HYPERCUBE , lim sup n0/C12L(n;d) n(d/C281)=dffiffiffi dp5a(d); (1) and a constant b(d) such that for almost all optimal tours in the HYPERCUBE , lim n0/C12L(n;d) n(d/C281)=dffiffiffidp/C30b(d): (2) These constants satisfy the inequalities 0:44194Bg 2/C305 16ffiffiffi 2p 5b(2) 5dB0:6508B0:75983B3/C281=45a(2) 5fB0:98398 (3) 0:37313Bg35b(3)5121=66/C281=2B0:61772B0:64805 B21=63/C281=25a(3)50:90422 (4) 0:34207Bg45b(4)5121=86/C281=2B0:55696 B0:59460B2/C283=45a(4)50:8364 (5) (Fejes To ´th 1940, Verblunsky 1951, Few 1955, Beard- wood et al. 1959), where gd/C13G3/C271 d ! G1 2d/C271iCkCiCkAhi1=d 2ffiffiffippd1=2/C27d/C281=2 ðÞ(6) /G(z) is the GAMMA FUNCTION ,dis an expressioninvolving S TRUVE FUNCTIONS and N EUMANN FUNC- TIONS , f/C13280 3/C28ffiffiffi 3piCjiCk 840/C28280ffiffiffi3p /C274ffiffiffi5p /C28ffiffiffiffiffiffi10p (7) (Karloff 1989), and c/C13 1 23/C282=34/C27ln 3 ðÞ2=3(8) (Goddyn 1990). In the LIMIT d0/C12;/ 0:24197Blim d0/C12gd/C301ffiffiffiffiffiffiffiffi 2pep5lim inf d0/C12b(d) 5lim sup d0/C12b(d)5lim d0/C12121=(2d)6/C281=2/C301ffiffiffi6pB0:40825 (9) and 0:24197B1ffiffiffiffiffiffiffiffi2pep5lim d0/C12a(d)523/C28ffiffiffi 3piCjiCk uffiffiffiffiffiffiffiffi 2pep B0:4502 ; (10) where 1 25u/C30lim d0/C12[u(d)]1=d50:6602 ; (11) andu(d) is the best SPHERE PACKING density in d-D space (Goddyn 1990, Moran 1984, Kabatyanskii and Levenshtein 1978). Steele and Snyder (1989) provedthat the limit a(d) exists. Now consider the constant k/C13lim n0/C12L(n;2)ffiffiffinp/C30b(2)ffiffiffi 2p ; (12) so 5 8/C30g2ffiffiffi 2p 5k5dffiffiffi2p B0:9204 : (13) The best current estimate is k:0:7124 : / A certain self-avoiding SPACE-FILLING CURVE is an optimal TOUR through a set of npoints, where ncan be arbitrarily large. It has length l/C13lim m0/C12Lmffiffiffiffiffiffinmp/C3041/C272ffiffiffi 2piCjiCkffiffiffiffiffiffi 51p 153/C300:7147827 . . . ;(14) where Lmis the length of the curve at the mth iteration and nmis the point-set size (Moscato and Norman). References Beardwood, J.; Halton, J. H.; and Hammersley, J. M. "The Shortest Path Through Many Points." Proc. Cambridge Phil. Soc. 55, 299/C1/327, 1959. Chartrand, G. "The Salesman’s Problem: An Introduction to Hamiltonian Graphs." §3.2 in Introductory Graph Theory. New York: Dover, pp. 67 /C1/76, 1985. Fejes To ´th, L. "U ¨ber einen geometrischen Satz." Math. Zeit. 46,8 3/C1/85, 1940. Few, L. "The Shortest Path and the Shortest Road Through n Points." Mathematika 2, 141 /C1/144, 1955. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/sales/sales.html. Flood, M. "The Travelling Salesman Problem." Operations Res. 4,61/C1/75, 1956. Friedman, E. "Longest Travelling Salesman Cycles." http:// www.stetson.edu/~efriedma/tsp/. Goddyn, L. A. "Quantizers and the Worst Case Euclidean Traveling Salesman Problem." J. Combin. Th. Ser. B 50, 65 /C1/81, 1990. Kabatyanskii, G. A. and Levenshtein, V. I. "Bounds for Packing on a Sphere and in Space." Problems Inform. Transm. 14,1/C1/17, 1978. Karloff, H. J. "How Long Can a Euclidean Traveling Sales- man Tour Be?" SIAM J. Disc. Math. 2,91/C1/99, 1989. Moran, S. "On the Length of Optimal TSP Circuits in Sets of Bounded Diameter." J. Combin. Th. Ser. B 37, 113 /C1/141, 1984. Moscato, P. "Fractal Instances of the Traveling Salesman Constant." http://www.ing.unlp.edu.ar/cetad/mos/FRAC- TAL_TSP_home.html Steele, J. M. and Snyder, T. L. "Worst-Case Growth Rates of Some Classical Problems of Combinatorial Optimization." SIAM J. Comput. 18, 278 /C1/287, 1989. Verblunsky, S. "On the Shortest Path Through a Number of Points." Proc. Amer. Math. Soc. 2, 904 /C1/913, 1951. Traveling Salesman Problem A problem in GRAPH THEORY requiring the most efficient (i.e., least total distance) HAMILTONIAN CIRCUIT a salesman can take through each of n cities. No general method of solution is known, and the problem is NP-HARD . Solution to the traveling sales- man problem is implemented in Mathematica as TravelingSalesman [g] in the Mathematica add- on package DiscreteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also CHINESE POSTMAN PROBLEM ,D ENDRITE , HAMILTONIAN CIRCUIT ,PLATEAU’S PROBLEM ,TRAVEL- ING SALESMAN CONSTANTS References Applegate, D.; Bixby, R.; Chvatal, V.; and Cook, W. "Finding Cuts in the TSP (a Preliminary Report)." Technical Report 95 /C1/05, DIMACS. Piscataway NJ: Rutgers University, 1995. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 168 /C1/169, 1998.Kruskal, J. B. "On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem." Proc. Amer. Math. Soc. 7,48/C1/50, 1956. Lawler, E.; Lenstra, J.; Rinnooy Kan, A.; and Shmoys, D. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. New York: Wiley, 1985. Lin, S. "Computer Solutions of the Traveling Salesman Problem." Bell System Tech. J. 44, 2245 /C1/2269, 1965. Platzman, L. K. and Bartholdi, J. J. "Spacefilling Curves and the Planar Travelling Salesman Problem." J. Assoc. Comput. Mach. 46, 719 /C1/737, 1989. Reinelt, G. "TSPLIB--A Traveling Salesman Problem Li- brary." ORSA J. Comput. 3, 376 /C1/384, 1991. Rosenkrantz, D. J.; Stearns, R. E.; and Lewis, P. M. "An Analysis of Several Heuristics for the Traveling Salesman Problem." SIAM J. Comput. 6, 563 /C1/581, 1977. Skiena, S. "Traveling Salesman Tours." §5.3.5 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 199 /C1/202, 1990. Skiena, S. S. "Traveling Salesman Problem." §8.5.4 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 319 /C1/322, 1997. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 120 /C1/121, 1999. Trawler Problem A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed. How should the pursuing vessel proceed in order to be sure of catching the other boat? The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a SPIRAL whose origin is the point where the slow boat was when the fog set in. The SPIRAL can be constructed in such a way that the two boats will intersect before a complete turn is made. References Ogilvy, C. S. Excursions in Mathematics. New York: Dover, pp. 84 and 148, 1994. Trebly Magic Square TRIMAGIC SQUARE Tredecillion In the American system, 1042. See also LARGE NUMBER Tree A tree is a mathematical structure which can be viewed as either a GRAPH or as a DATA STRUCTURE . The two views are equivalent, since a tree DATA STRUCTURE contains not only a set of elements, but also connections between elements, giving a tree graph. Trees were first studied by Cayley (1857). A tree graph is a set of straight line segments connected at their ends containing no closed loops (cycles). In other words, it is a simple, undirected, connected, acyclic graph (or, equivalently, a con- nected FOREST ). A tree with n nodes has n /C281 EDGES . Conversely, a CONNECTED GRAPH with n nodes and n /C281 edges is a tree. All trees are BIPARTITE GRAPHS (Skiena 1990, p. 213). The points of connection are known as FORKS and the segments as BRANCHES . Final segments and the nodes at their ends are called LEAVES . A tree with two BRANCHES at each FORK and with one or two LEAVES at the end of each branch is called a BINARY TREE . Trees find applications in many diverse fields, in- cluding computer science, the enumeration of satu- rated hydrocarbons, the study of electrical circuits, etc. (Harary 1994, p. 4). A tree T has either one node which is a GRAPH CENTER , in which case it is called a CENTRAL TREE ,or two adjacent nodes which are GRAPH CENTERS ,in which case it is called a BICENTRAL TREE (Harary 1994, p. 35).When a special node is designated to turn a tree into a ROOTED TREE , it is called the ROOT (or sometimes "EVE.") In such a tree, each of the nodes which is one EDGE further away from a given node is called a CHILD , and nodes connected to the same node which are the same distance from the ROOT NODE are called SIBLINGS . Note that two BRANCHES placed end-to-end are equivalent to a single BRANCH which means, for example, that there is only one tree of order 3. The number t(n) of nonisomorphic trees of order n /C301, 2, ... (where trees of orders 1, 2, ..., 6 are illustrated above), are 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, ... (Sloane’s A000055). Otter showed that lim n0/C12t(n)n5=2 an/C30b; (1) (Otter 1948, Harary and Palmer 1973, Knuth 1969). Write the GENERATING FUNCTION forROOTED TREES as f(z)/C30X/C12 i/C300fizi; (2) where the COEFFICIENTS are fi/C271/C301 iXi j/C301X d½jdfd ! fi/C28j/C271; (3) with f0/C300 and f1/C301:Then a/C302:955765 . . . (4) is the unique POSITIVE ROOT of f1 x ! /C301; (5) and b/C301ffiffiffiffiffiffi 2pp 1/C27X/C12 k/C302f?1 ak ! 1 ak"# 3=2 /C300:5349485 . . . (6) See also B-TREE,B ICENTRAL TREE,B INARY TREE, CATERPILLAR GRAPH ,CAYLEY TREE,CENTRAL TREE, CHILD ,D IJKSTRA TREE,E VE,FOREST ,FREE TREE, KRUSKAL’S ALGORITHM ,K RUSKAL’S TREE THEOREM , LABELED TREE,LEAF (TREE), MATRIX TREE THEOREM , ORCHARD- PLANTING PROBLEM ,O RDERED TREE,O T- TER’S THEOREM ,P ATH GRAPH ,P LANTED PLANAR TREE,PO´ LYA ENUMERATION THEOREM ,POLYNEMA , QUADTREE ,R AMUS TREE,R ED-BLACK TREE,R OOT NODE,R OOTED TREE,SERIES- REDUCED TREE,SIB- LING ,SPANNING TREE,STAR GRAPH ,STEINER TREE, STERN- BROCOT TREE,W EAKLY BINARY TREE, WEIGHTED TREE References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/otter/otter.html. Bergeron, F.; Leroux, P.; and Labelle, G. Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, p. 284, 1998. Cayley, A. "On the Theory of Analytic Forms Called Trees." Philos. Mag. 13,19/C1/30, 1857. Reprinted in Mathematical Papers, Vol. 3. Cambridge: pp. 242 /C1/246, 1891. Chauvin, B.; Cohen, S.; and Rouault, A. (Eds.). Trees: Workshop in Versailles, June 14 /C1/16, 1995. Basel, Swit- zerland: Birkha ¨user, 1996. Gardner, M. "Trees." Ch. 17 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 240 /C1/250, 1978. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Harary, F. "Trees." Ch. 4 in Graph Theory. Reading, MA: Addison-Wesley, pp. 32 /C1/42, 187 /C1/194, and 231 /C1/234, 1994. Harary, F. and Manvel, B. "Trees." Scripta Math. 28, 327 /C1/ 333, 1970. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Ko¨nig, D. Theorie der endlichen und unendlichen Graphen. New York: Chelsea, p. 48, 1950. Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Otter, R. "The Number of Trees." Ann. Math. 49, 583 /C1/599, 1948. Skiena, S. "Trees." Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 107 and 151 /C1/153, 1990. Sloane, N. J. A. Sequences A000055/M0791 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M0791 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Wilf, H. S. Combinatorial Algorithms: An Update. Philadel- phia, PA: SIAM, 1989. Tree Centroid The set of all CENTROID POINTS in a WEIGHTED TREE (Harary 1994, p. 36). See also CENTROID POINT ,W EIGHTED TREE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Tree Searching N.B. A detailed online essay by S. Finch was the starting point for this entry.In database structures, two quantities are generally of interest: the average number of comparisons required to 1. Find an existing random record, and2. Insert a new random record into a data structure. Some constants which arise in the theory of digital tree searching are a/C13X /C12 k/C3011 2k/C281/C301:6066951524 . . . (1) b/C13X/C12 k/C3011 2n/C281 ðÞ2/C301:1373387363 . . . (2) Erdos (1948) proved that aisIRRATIONAL . The expected number of comparisons for a successful search is E/C30lnn ln 2/C27g/C281 ln 2/C28a/C273 2/C27d(n)/C27On/C281=2iCjiCk (3) /C2lgn/C280:716644 . . . /C27d(n); (4) and for an unsuccessful search is E/C30lnn ln 2/C27g ln 2/C28a/C2712/C27d(n)/C27On/C281=2iCjiCk (5) /C2lgn/C280:273948 . . . /C27d(n); (6) Here d(n);e(s);andr(n) are small-amplitude periodic functions, and LGis the base 2 LOGARITHM . The VARIANCE for searching is V/C21 12/C27p2/C276 6(ln 2)2/C28a/C28b/C27e(s) /C22:844383 . . . /C27e(s) (7) and for inserting is V/C21 12/C27p2 6(ln 2)2/C28a/C28b/C27e(s) /C20:763014 . . . /C27e(s): (8) The expected number of pairs of twin vacancies in a digital search tree is Anhi/C30u/C271/C281 Q1 ln 2/C27a2/C28a ! /C27r(n)"# n/C27OffiffiffinpiCjiCk ; (9) where Q/C13Y/C12 k/C3011/C281 2k ! /C300:2887880950 . . . (10) /C301 3/C281 3 /C2157/C271 3 /C2155 /C21515/C281 3 /C2155 /C21515 /C21521/C27. . . (11) /C30exp/C28X/C12 n/C3011 n(2n/C281)"# (12) /C30ffiffiffiffiffiffiffiffiffi 2p ln 2s expln 2 24/C28p2 6l n2 !Y/C12 n/C3011/C28exp/C284p2n ln 2 !"# (13) and u/C30X/C12 k/C301k2k/C271 1 /C2153 /C2157 /C21516/C1/C1/C12k/C281 ðÞXk j/C3011 2j/C281 /C307:7431319855 . . . (14) (Flajolet and Sedgewick 1986). The linear COEFFI- CIENT ofAnhi fluctuates around c/C30u/C271/C281 Q1 ln 2/C27a2/C28a ! /C300:3720486812 . . . ;(15) which can also be written c/C301 ln 2 /C2g/C12 0x 1/C27xdx (1/C27x)1/C271 2xiCkCiCkA 1/C2714xiCkCiCkA 1/C2718xiCkCiCkA /C1/C1/C1: (16) (Flajolet and Richmond 1992). References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/bin/bin.html. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/dig/dig.html. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/qdt/qdt.html. Flajolet, P. and Richmond, B. "Generalized Digital Trees and their Difference-Differential Equations." Random Struc- tures and Algorithms 3, 305/C1/320, 1992. Flajolet, P. and Sedgewick, R. "Digital Search Trees Revis- ited." SIAM Review 15, 748/C1/767, 1986. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, pp. 21, 134, 156, 493 /C1/499, and 580, 1973. Tree-Planting Problem ORCHARD- PLANTING PROBLEMTrefoil Curve The plane curve given by the equation x4/C27x2y2/C27y4/C30xx2/C28y2iCjiCk : Trefoil Knot The knot 03/C1/001, also called the THREEFOIL KNOT , which is the unique PRIME KNOT of three crossings. It has BRAID WORD s3 1:The trefoil and its MIRROR IMAGE are not equivalent, as first proved by Dehn (1914). The trefoil has A LEXANDER POLYNOMIAL /C28x2/C27x/C281 and is a (3, 2)- TORUS KNOT . The BRACKET POLYNOMIAL can be computed as follows. /C142L/C143/C30A3d2/C281/C27A2Bd1/C281/C27A2Bd1/C281/C27AB2d2/C281 /C27A2Bd1/C281/C27AB2d2/C281/C27AB2d2/C281/C27B3d3/C281 /C30A3d1/C273A2Bd0/C273AB2d1/C27B3d2: Plugging in B/C30A/C281 d/C30/C28A2/C28A/C282 gives /C142L/C143/C30A/C287/C28A/C283/C28A5: The normalized one-variable K AUFFMAN POLYNOMIAL Xis then given by XL/C30/C28 A3iCjiCk/C28w(L)/C142L/C143/C30/C28 A3iCjiCk/C283A/C287/C28A/C283/C28A5iCjiCk /C30A/C284/C27A/C2812/C28A/C2816; where the WRITHE w(L)/C303:The J ONES POLYNOMIAL is therefore V(t)/C30LA/C30t/C281=4iCjiCk /C30t/C27t3/C28t4/C30t1/C27t2/C28t3iCjiCk : Since Vt/C281ðÞ"V(t) ; we have shown that the mirror images are not equivalent. References Claremont High School. "Trefoil_Knot Movie." Binary en- coded QuickTime movie. ftp://chs.cusd.claremont.edu/pub/ knot/trefoil.cpt.bin. Crandall, R. E. Mathematica for the Sciences. Redwood City, CA: Addison-Wesley, 1993. Dehn, M. "Die beiden Kleeblattschlingen." Math. Ann. 75, 402 /C1/413, 1914. Kauffman, L. H. Knots and Physics. Singapore: World Scientific, pp. 29 /C1/35, 1991. Nordstrand, T. "Threefoil Knot." http://www.uib.no/people/ nfytn/tknottxt.htm. Pappas, T. "The Trefoil Knot." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 96, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 265, 1999. Trench Diggers’ Constant BEAM DETECTOR Triabolo One of the four 3-POLYABOLOES . See also POLYABOLO Triacontagon A 30-sided POLYGON . The regular triacontagon with side length 1 has INRADIUS r, CIRCUMRADIUS R, and AREA A given by r /C301 4ffiffiffiffiffiffi 15p /C273ffiffiffi3p /C27ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 /C2711ffiffiffi 5pq iCkniCko R /C301 22 /C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C276ffiffiffi 5pqiCkniCkoA /C3015 4ffiffiffiffiffiffi 15p /C273ffiffiffi3p /C27ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 /C2711ffiffiffi 5pq iCkniCko : See also P OLYGON ,REGULAR POLYGON ,TRIGONOME- TRY VALUES PI/30 Triacontahedron A 30-faced POLYHEDRON . See also ICOSIDODECAHEDRON ,M EDIAL DISDYAKIS TRIACONTAHEDRON ,RHOMBIC TRIACONTAHEDRON Triad A SET with three elements. See also HEXAD ,MONAD ,QUARTET ,QUINTET ,TETRAD Triakis Icosahedron The 60-faced DUAL POLYHEDRON of the TRUNCATED DODECAHEDRON A10and Wenninger dual W10:Wen- ninger (1989, p. 46) calls the SMALL TRIAMBIC ICOSA- HEDRON the triakis octahedron. Taking the dual of a TRUNCATED DODECAHEDRON with unit edge lengths gives a triakis icosahedron with edge lengths s1/C305 227/C27ffiffiffi5piCkCiCkA (1) s 2/C301 25/C275ffiffiffi 5piCkCiCkA : (2) The SURFACE AREA and VOLUME are S/C3075 11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12313/C27117ffiffiffi 5piCkCiCkAr (3) V /C30125 4419 /C279ffiffiffi 5piCkCiCkA : (4) See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , HEXECONTAHEDRON ,SMALL TRIAMBIC ICOSAHEDRON References Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, p. 46, 1989. Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 19 /C1/20, 1983. Triakis Octahedron GREAT TRIAKIS OCTAHEDRON ,SMALL TRIAKIS OCTA- HEDRON Triakis Tetrahedron The DUAL POLYHEDRON of the TRUNCATED TETRAHE- DRON A13and Wenninger dual W6 : It can be con- structed by CUMULATION of a unit edge-length TETRAHEDRON by a pyramid with height1 15ffiffiffi 6p :/ The triakis tetrahedron formed by taking the dual of a truncated tetrahedron with unit edge lengths has side lengths s1 /C309 5 (1) s2 /C303: (2) Normalizing so that s1 /C301 gives SURFACE AREA and VOLUME S /C3053ffiffiffiffiffiffi 11p (3) V /C3025 36ffiffiffi 2p (4) See also ARCHIMEDEAN DUAL,ARCHIMEDEAN SOLID , TRIAKIS TETRAHEDR ON STELLATIONS ,T RUNCATED TETRAHEDRONReferences Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 14 /C1/15 and 33, 1983. Triakis Tetrahedron Stellations B. Chilton and R. Whorf have studied stellations of the TRIAKIS TETRAHEDRON (Wenninger 1983, p. 36). Whorf has found 138 stellations, 44 of which are fully symmetric and 94 of which are enantiomorphs (Wen- ninger 1983, p. 36). See also STELLATION ,TRIAKIS TETRAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, pp. 36 /C1/37, 1983. Trial In statistics, a trial is a single performance of well- defined experiment (Papoulis 1984, p. 25), such as the flipping of a COIN , the generation of a RANDOM NUMBER , the dropping of a ball down the apex of a triangular lattice and having it fall into a single bin at the bottom, etc. See also BERNOULLI TRIAL,E VENT ,E XPERIMENT , LEXIS TRIALS ,OUTCOME ,POISSON TRIALS References Papoulis, A. "Repeated Trials." Ch. 3 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 47 /C1/82, 1984. Trial Division A brute-force method of finding a DIVISOR of an INTEGER n by simply plugging in one or a set of INTEGERS and seeing if they DIVIDE n. Repeated application of trial division to obtain the complete PRIME FACTORIZATION of a number is called DIRECT SEARCH FACTORIZATION . An individual integer being tested is called a TRIAL DIVISOR . See also DIRECT SEARCH FACTORIZATION ,DIVISION , PRIME FACTORIZATION Trial Divisor An INTEGER n which is tested to see if it divides a given number. See also TRIAL DIVISION Triamond The unique 3-POLYIAMOND , illustrated above. See also POLYIAMOND ,TRAPEZOID Triangle A triangle is a 3-sided POLYGON sometimes (but not very commonly) called the TRIGON . All triangles are convex. An ACUTE TRIANGLE is a triangle whose three angles are all ACUTE . A triangle with all sides equal is called EQUILATERAL . A triangle with two sides equal is called ISOSCELES . A triangle having an OBTUSE ANGLE is called an OBTUSE TRIANGLE . A triangle with aRIGHT ANGLE is called RIGHT . A triangle with all sides a different length is called SCALENE . In 1816, while studying the B ROCARD POINTS of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaus- tible in properties. How many as yet unknown properties of other figures may there not be?" (Wells1991, p. 21). The sum of ANGLES in a triangle is 180/C14/C30pradians (at least in E UCLIDEAN GEOMETRY ; this statement does nothold in NON- EUCLIDEAN GEOMETRY ). This can be established as follows. Let DAE IBC(DAE be PARALLEL toBC) in the above diagram, then the angles aandbsatisfy a/C30/C218DAB/C30/C218ABC andb/C30/ //C218EAC/C30/C218ACB ;as indicated. Adding g;it follows that a/C27b/C27g/C30180/C14; (1) since the sum of angles for the line segment mustequal two RIGHT ANGLES . Therefore, the sum of angles in the triangle is also 180 8. LetSstand for a triangle side and Afor an angle, and let a set of Ss and As be concatenated such that adjacent letters correspond to adjacent sides andangles in a triangle. Triangles are uniquely deter-mined by specifying three sides (SSS THEOREM ), two angles and a side (AAS THEOREM ), or two sides with an adjacent angle (SAS THEOREM ). In each of these cases, the unknown three quantities (there are three sides and three angles total) can be uniquely deter-mined. Other combinations of sides and angles do not uniquely determine a triangle: three angles specify a triangle only modulo a scale size (AAA THEOREM ), and one angle and two sides not containing it may specify one, two, or no triangles (ASS THEOREM ). Allowable side lengths a,b, and cfor a triangle are given by the set of inequalities a/C210,b/C210,c/C210, and a/C27b>c;b/C27c>a;a/C27c>b:/ The STRAIGHTEDGE and COMPASS construction of the triangle can be accomplished as follows. In the above figure, take OP0as a RADIUS and draw OB/C222OP0: Then bisect OBand construct P2P1IOP0:Extending BOto locate P3then gives the EQUILATERAL TRIANGLE DP1P2P3:Another construction proceeds by drawing a CIRCLE of the desired RADIUS rcentered at a point O. Choose a point Bon the circle’s CIRCUMFERENCE and draw another CIRCLE of radius rcentered at B. The two circles intersect at two points, P1andP2;andP3 is the second point at which the line BOintersects the first CIRCLE . In Proposition IV.4 of the ELEMENTS , Euclid showed how to inscribe a CIRCLE (the INCIRCLE ) in a given triangle by locating the INCENTER Ias the point of intersection of ANGLE BISECTORS . In Proposition IV.5, he showed how to circumscribe a CIRCLE (the CIR- CUMCIRCLE ) about a given triangle by locating the CIRCUMCENTER Oas the point of intersection of the PERPENDICULAR BISECTORS . unlike a general POLYGON with n]4 sides, a triangle always has both a CIRCUMCIRCLE and an INCIRCLE . such polygons are called BICENTRIC POLYGONS . Casey (1888, pp. 10 /C1/11) illustrates how to inscribe a SQUARE in an arbitrary triangle DABC :Construct the PERPENDICULAR CD/C222AB and the line segment BE/C30AD. Bisect /C218BDC ;and let Fbe the intersection of the bisector with BC. Then draw FKand FH through F, perpendicular to and parallel to AB, respectively. Let Gbe the intersection of FH and BC, and then construct FKandHJthrough FandH perpendicular to AB. Then IGHJI is an inscribed SQUARE . Permuting the order in which the vertices are taken gives an additional two congruent squares. These squares, however, are not necessarily the largest inscribed squares. C ALABI’S TRIANGLE is the only triangle (besides the EQUILATERAL TRIANGLE ) for which the largest inscribed SQUARE can be inscribed in three different ways. If the coordinates of the triangle VERTICES are given byxi;yjiCjiCk where i/C301 2, 3, then the signed AREADis given by the DETERMINANT D/C301 2!x1y11 x2y21 x3y31iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0; (2) so the actual area is obtained by taking the ABSOLUTE VALUE of (2). If the triangle is embedded in three- dimensional space with the coordinates of the VER- TICES given by xi;xj;ziiCjiCk ;where i/C301, 2, 3, then D/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y1z11 y2z21 y3z31iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk02 /C27z1x11 z2x21 z3x31iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk02 /C27x1y11 x2y21 x3y31iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk02vuuut :(3) In the above figure, let the CIRCUMCIRCLE passing through a triangle’s VERTICES have RADIUS r, and denote the CENTRAL ANGLES from the first point to the second u1;and to the third point by u2:Then the AREA of the triangle is given by D/C302r2sin1 2u1iCkCiCkA sin12u2iCkCiCkA sin12u1/C28u2 ðÞhi iCk0iCk0iCk0iCk0iCk0iCk0: (4) If a triangle has sides a,b,c, call the angles opposite these sides A,B, and C, respectively. Also define the SEMIPERIMETER sasHALF the PERIMETER : s/C131 2p/C3012(a/C27b/C27c): (5) The AREA of a triangle is then given by H ERON’S FORMULA D/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s/C28a)(s/C28b)(s/C28c)p ; (6) as well by the FORMULAS D/C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C27b/C27c)(b/C27c/C28a)(c/C27a/C28b)(a/C27b/C28c)p (7) /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2b2/C27a2c2/C27b2c2 ðÞ /C28a4/C27b4/C27c4 ðÞp (8) /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a/C27b)2/C28c2hi c2/C28(a/C28b)2hir (9) /C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p(p/C282a)(p/C282b)(p/C282c)p ; (10) /C302R2sinAsinBsinC (11) /C30abc 4R/C30rs (12) /C301 2aha (13) /C301 2bcsinA: (14) In the above formulas, hiis the ALTITUDE on side i,R is the CIRCUMRADIUS , and ris the INRADIUS (Johnson 1929, p. 11). A triangle with sides a,b, and ccan be constructed by selecting vertices (0, 0), ( a;0);and ( x, y), then solving x2/C27y2/C30b2(15) (x/C28a)2/C27y2/C30c2(16) simultaneously to obtain x/C30a2/C27b2/C28c2 2a(17) y/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (/C28a/C27b/C27c)(a/C28b/C28c)(a/C28b/C27c)(a/C27b/C27c)p 2a: (18) Expressing the side lengths a,b, and cin terms of the radii a?;b?;and c?of the mutually TANGENT CIRCLES centered on the TRIANGLE vertices (which define the SODDY CIRCLES ), a/C30b?/C27c? (19) b/C30a?/C27c? (20) c/C30a?/C27b?; (21) gives the particularly pretty form D/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a?b?c?(a?/C27b?/C27c?)p : (22) For additional FORMULAS , see Beyer (1987) and Baker (1884), who gives 110 FORMULAS for the AREA of a triangle. The ANGLES of a triangle satisfy cotA/C30b2/C27c2/C28a2 4D(23) where Dis the AREA (Johnson 1929, p. 11, with missing squared symbol added). This gives the pretty identity cotA/C27cotB/C27cotC/C30a2/C27b2/C27c2 4D: (24) In addition, tanA/C27tanB/C27tanC/C30tanAtanBtanC (25) (F.J. n.d., p. 206; Borchardt and Perrott 1930) and cotBcotC/C27cotCcotA/C27cotAcotB/C301 (26) tanAcotBcotC/C27tanBcotCcotA /C27tanCcotAcotB /C30tanA/C27tanB/C27tanC /C272(cot A/C27cotB/C27cotC) (27) (Siddons and Hughes 1929). Let a triangle have ANGLES A,B, and C. Then sinAsinBsinC5kABC ; (28) wherek/C303ffiffiffi 3p 2p !3 (29) (Abi-Khuzam 1974, Le Lionnais 1983). This can be used to prove that 8v3BABC ; (30) where vis the B ROCARD ANGLE . Other inequalities include sinA/C27sinB/C27sinC53 2ffiffiffi 3p (31) 15cosA/C27cosB/C27cosC53 2(32) sin1 2AiCkCiCkA sin12BiCkCiCkA sin12CiCkCiCkA 518 (33) tan1 2AiCkCiCkA /C27tan12BiCkCiCkA /C27tan12CiCkCiCkA ]ffiffiffi 3p (34) cotAcotBcotC51 9ffiffiffi 3p (35) cotA/C27cotB/C27cotC]ffiffiffi3p (36) sinAsinBsinC cotA/C27cotB/C27cotC53 8(37) tan1 2AiCkCiCkA /C27tan12BiCkCiCkA /C27tan12CiCkCiCkA tan1 2AiCkCiCkA tan12BiCkCiCkA tan12CiCkCiCkA ]9 (38) cos1 2AiCkCiCkA cos12BiCkCiCkA cos12CiCkCiCkA ]sinAsinBsinC ]sin(2 A) sin(2 B) sin(2 C) (39) 25cos212AiCkCiCkA /C27cos212BiCkCiCkA /C27cos212CiCkCiCkA 594 (40) cot1 2AiCkCiCkA cot12BiCkCiCkA /C27cot12BiCkCiCkA cot12CiCkCiCkA /C27cot1 2CiCkCiCkA cot12AiCkCiCkA ]9 (41) (Siddons and Hughes 1929, p. 283), and sinA/C27sinB/C27sinC cotA/C27cotB/C27cotC53 2(42) (Weisstein). TRIGONOMETRIC FUNCTIONS of half angles can be expressed in terms of the triangle sides: cos1 2AiCkCiCkA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s(s/C28a) bcs (43) sin1 2AiCkCiCkA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s/C28b)(s/C28c) bcs (44) tan1 2AiCkCiCkA /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s/C28b)(s/C28c) s(s/C28a)s ; (45) where sis the SEMIPERIMETER . The number of different triangles which have INTE- GRAL sides and PERIMETER n is T(n) /C30P3(n) /C28X 1 5i5 n=2bcP2(j) /C30n2 12"# /C28n 4$% n /C27 2 4$% /C30n2 48"# for n even (n /C27 3)2 48"# for n odd;8 >>>>< >>>>:(46) where P 2and P3are PARTITION FUNCTIONS P,[x]is the NINT function, and xbcis the FLOOR FUNCTION (Jordan et al. 1979, Andrews 1979, Honsberger 1985). The values of T(n) for n /C301, 2, ... are 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, ... (Sloane’s A005044), which is also ALCUIN’S SEQUENCE padded with two initial 0s. T(n) also satisfies T(2n) /C30T(2n /C283) /C30P3(n): (47) It is not known if a triangle with INTEGER sides, MEDIANS , and AREA exists (although there are incor- rect PROOFS of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have shown that there are infinitely many triangles with RATIONAL sides (HERONIAN TRIANGLES ) with two RATIONAL MEDIANS (Guy 1994). In the following paragraph, assume the specified sides and angles are adjacent to each other. Specify- ing three ANGLES does not uniquely define a triangle, but any two triangles with the same ANGLES are similar (the AAA THEOREM ). Specifying two ANGLES A and B and a side a uniquely determines a triangle with AREA D/C30a2 sin B sin C 2 sin A/C30a2 sin B sin(p /C28 A /C28 B) 2 sin A (48) (the AAS THEOREM ). Specifying an ANGLE A, a side c, and an ANGLE B uniquely specifies a triangle with AREA D/C30c2 2(cot A /C27 cot B) (49) (the ASA THEOREM ). Given a triangle with two sides, a the smaller and c the larger, and one known ANGLE A, ACUTE and opposite a, if sin A Ba=c ; there are two possible triangles. If sin A /C30a=c ; there is one possible triangle. If sin A > a =c ; there are no possible trian- gles. This is the ASS THEOREM . Let a be the base length and h be the height. Then D/C301 2 ah /C3012 ac sin B (50) (the SAS THEOREM ). Finally, if all three sides arespecified, a unique triangle is determined with AREA given by HERON’S FORMULA or by D/C30abc 4R; (51) where R is the CIRCUMRADIUS . This is the SSS THEOREM . If squares are erected externally on the sides of a triangle as illustrated above, then BOB /C222OCOA ; and BOB /C30OCOA (52) (Coxeter and Greitzer 1967, pp. 96 /C1/97). Dividing the sides of a triangle in a constant ratio r B1=2 and then drawing lines parallel to the ad- jacent sides passing through each of these points gives line segments which intersect each other and one of the medians in three places. If r > 1=2 ; then the extensions of the side parallels intersect the extensions of the medians. The medians bisect the area of a triangle, as do the side parallels with ratio 1 /C27ffiffiffi 2p : The envelope of the lines which bisect the area a triangle forms three hyperbolic arcs. The envelope is somewhat more complicated, however, for lines dividing the area of a triangle into a constant but unequal ratio (Dunnand Petty 1972, Ball 1980, Wells 1991). There are four CIRCLES which are tangent to the sides of a triangle, one internal and the rest external. Their centers are the points of intersection of the ANGLE BISECTORS of the triangle. Any triangle can be positioned such that its shadow under an orthogonal projection is EQUILATERAL . See also AAA THEOREM , AAS THEOREM ,A CUTE TRIANGLE ,A LCUIN’S SEQUENCE ,A LTITUDE ,A NGLE BISECTOR ,ANTICEVIAN TRIANGLE ,ANTICOMPLEMEN- TARY TRIANGLE ,A NTIPEDAL TRIANGLE , ASS THEO- REM,A SSOCIATED TRIANGLES ,B ELL TRIANGLE , BRIANCHON POINT ,BROCARD ANGLE ,BROCARD CIR- CLE,BROCARD MIDPOINT ,BROCARD POINTS ,BUTTER- FLY THEOREM ,C ENTROID (TRIANGLE ), CEVA’S THEOREM ,C EVIAN ,C EVIAN TRIANGLE ,C HASLES’S THEOREM ,CIRCULAR TRIANGLE ,CIRCUMCENTER ,CIR- CUMCIRCLE ,C IRCUMRADIUS ,C OMEDIAN TRIANGLES , CONTACT TRIANGLE ,C OSYMMEDIAN TRIANGLES , CROSSED LADDERS PROBLEM ,C RUCIAL POINT ,D - TRIANGLE , DE LONGCHAMPS POINT ,DESARGUES’ THE- OREM ,D IAGONAL TRIANGLE ,D ISSECTION ,E LKIES POINT ,EQUAL DETOUR POINT ,EQUILATERAL TRIAN- GLE,EULER LINE,EULER’S TRIANGLE ,EULER TRIAN- GLE FORMULA ,E XCENTER ,E XCENTRAL TRIANGLE , EXCIRCLE ,E XETER POINT ,E XMEDIAN ,E XMEDIAN POINT ,EXRADIUS ,EXTERIOR ANGLE THEOREM ,FAG- NANO’S PROBLEM ,FAR-OUT POINT ,FERMAT POINTS , FERMAT’S PROBLEM ,FEUERBACH POINT ,FEUERBACH’S THEOREM ,FUHRMANN TRIANGLE ,GERGONNE POINT , GREBE POINT ,GRIFFITHS POINTS ,GRIFFITHS’ THEO- REM,H ARMONIC CONJUGATE POINT S,H EILBRONN TRIANGLE PROBLEM ,H ERON’S FORMULA ,H ERONIAN TRIANGLE ,HOFSTADTER TRIANGLE ,HOMOTHETIC TRI- ANGLES ,H EPTAGONAL TRIANGLE ,INCENTER ,INCIR- CLE,INRADIUS ,ISODYNAMIC POINTS ,ISOGONAL CONJUGATE ,ISOPERIMETRIC POINT ,ISOSCELES TRIAN- GLE,K ABON TRIANGLES ,K ANIZSA TRIANGLE ,K IE- PERT’S HYPERBOLA ,K IEPERT’S PARABOLA ,L AW OF COSINES ,LAW OF SINES,LAW OF TANGENTS ,LEIBNIZ HARMONIC TRIANGLE ,L EMOINE CIRCLE ,L INE AT INFINITY ,LOSSNITSCH’S TRIANGLE ,MALFATTI POINTS , MEDIAL TRIANGLE ,M EDIAN (TRIANGLE ), MEDIAN TRIANGLE ,M ENELAUS’ THEOREM ,M ID-ARC POINTS , MITTENPUNKT ,M OLLWEIDE’S FORMULAS ,M ORLEY CENTERS ,M ORLEY’S THEOREM ,NAGEL POINT ,NAPO- LEON’S THEOREM ,N APOLEON TRIANGLES ,N EWTON’S FORMULAS ,NINE-POINT CIRCLE ,NUMBER TRIANGLE , OBTUSE TRIANGLE ,ONO INEQUALITY ,ORTHIC TRIAN- GLE,ORTHOCENTER ,ORTHOLOGIC TRIANGLES ,PARA- LOGIC TRIANGLES ,P ASCAL’S TRIANGLE ,P ASCH’S AXIOM ,PEDAL TRIANGLE ,PERPENDICULAR BISECTOR , PERSPECTIVE TRIANGLES ,P ETERSEN- SHOUTE THEO- REM,PIVOT THEOREM ,POWER POINT ,POWER (TRIAN- GLE), PRIME TRIANGLE ,P URSER’S THEOREM , QUADRILATERAL ,RATIONAL TRIANGLE ,ROUTH’S THE- OREM , SAS THEOREM ,SCALENE TRIANGLE ,SCHIFFLER POINT ,S CHWARZ TRIANGLE ,S CHWARZ’S TRIANGLE PROBLEM ,S EIDEL- ENTRINGER- ARNOLD TRIANGLE , SEYDEWITZ’S THEOREM ,SIMSON LINE,SPIEKER CEN- TER, SSS THEOREM ,S TEINER- LEHMUS THEOREM , STEINER POINTS ,STEWART’S THEOREM ,SYMMEDIAN POINT ,TANGENTIAL TRIANGLE ,TARRY POINT ,THOM- SEN’S FIGURE ,TORRICELLI POINT ,TRIANGLE TILING , TRIANGLE TRANSFORMATION PRINCIPLE ,YFF CENTRAL TRIANGLE ,YFF POINTS ,YFF TRIANGLES References Abi-Khuzam, F. "Proof of Yff’s Conjecture on the Brocard Angle of a Triangle." Elem. Math. 29, 141/C1/142, 1974.Andrews, G. "A Note on Partitions and Triangles with Integer Sides." Amer. Math. Monthly 86, 477, 1979. Baker, M. "A Collection of Formulæ for the Area of a Plane Triangle." Ann. Math. 1, 134/C1/138, 1884. Ball, D. "Halving Envelopes." Math. Gaz. 64, 166/C1/172, 1980. Berkhan, G. and Meyer, W. F. "Neuere Dreiecksgeometrie." InEncyklopaedie der Mathematischen Wissenschaften, Vol. 3AB 10 (Ed. F. Klein). Leipzig: Teubner, pp. 1173 /C1/ 1276, 1914. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 123 /C1/124, 1987. Borchardt, W. G. and Perrott, A. D. §133 in A New Trigono- metry for Schools. London: G. Bell, 1930. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. and Greitzer, S. L. "Points and Lines Connected with a Triangle." Ch. 1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1 /C1/26 and 96 /C1/ 97, 1967. Davis, P. "The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-History." Amer. Math. Monthly 102, 204/C1/214, 1995. Dunn, J. A. and Petty, J. E. "Halving a Triangle." Math. Gaz. 56, 105/C1/108, 1972. Durell, C. V. "Properties of the Triangle." Ch. 3 in Modern Geometry: The Straight Line and Circle. London: Macmil- lan, pp. 19 /C1/31, 1928. Eppstein, D. "Triangles and Simplices." http://www.ics.u- ci.edu/~eppstein/junkyard/triangulation.html. Feuerbach, K. W. Eigenschaften einiger merkwu ¨rdigen Punkte des geradlinigen Dreiecks, und mehrerer durchdie bestimmten Linien und Figuren. Nu¨rnberg, Germany, 1822. F. J. Elements de trigonometrie rectiligne. Paris: J. de Gigord, n.d. Fukagawa, H. and Pedoe, D. "One or Two Circles and Triangles," "Three Circles and Triangles," "Four Circles and Triangle," "Five Circles and Triangles," "Many Circles and Triangles," "Triangles." §2.2/C1 /2.6 and 4.1 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Ca- nada: Charles Babbage Research Foundation, pp. 26 /C1/37, 46/C1/47, 102 /C1/116, 129 /C1/130, 1989. Guy, R. K. "Triangles with Integer Sides, Medians, and Area." §D21 in Unsolved Problems in Number Theory, 2nd ed.New York: Springer-Verlag, pp. 188 /C1/190, 1994. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 39 /C1/47, 1985. Honsberger, R. "On Triangles." Ch. 3 in Episodes in Nine- teenth and Twentieth Century Euclidean Geometry. Wa- shington, DC: Math. Assoc. Amer., pp. 27 /C1/33, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Jordan, J. H.; Walch, R.; and Wisner, R. J. "Triangles with Integer Sides." Amer. Math. Monthly 86, 686/C1/689, 1979. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/187, 1994. Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,1/C1/295, 1998. Lachlan, R. "Properties of Triangles." Ch. 6 in An Elemen- tary Treatise on Modern Pure Geometry. London: Macmil- lian, pp. 51 /C1/81, 1893. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983. Schroeder. Das Dreieck und seine Beruhungskreise. Siddons, A. W. and Hughes, R. T. Trigonometry, Parts III- IV.London: Cambridge University Press, 1929. Sloane, N. J. A. Sequences A005044/M0146 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091 /C1/1094, 1965. Weisstein, E. W. "Plane Geometry." MATHEMATICA NOTE- BOOK PLANE GEOMETRY.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 21, 1991. Triangle Arcs In the above figure, let DABC be a RIGHT TRIANGLE , arcs AP and AQ be segments of CIRCLES centered at C and B respectively, and define a /C30BC (1) b /C30CA /C30CP (2) c /C30BA /C30BQ: (3) Then PQ2 /C302BP /C215 QC: (4) The figure also yields the algebraic identity b /C27c /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 /C27c2piCkCiCkA2 /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 /C27c2p /C28biCkCiCkA ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 /C27c2p /C28ciCkCiCkA : (5) See also ARC,TRIANGLE References Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, pp. 8 /C1/9, 1994. Dharmarajan, T. and Srinivasan, P. K. An Introduction to Creativity of Ramanujan, Part III. Madras: Assoc. Math. Teachers, pp. 11 /C1/13, 1987. Triangle Center A triangle center is a point whose TRILINEAR COORDI- NATES are defined in terms of the side lengths and angles of a TRIANGLE . The function giving the co- ordinates a : b : g is called the TRIANGLE CENTER FUNCTION . The four ancient centers are the CEN- TROID , INCENTER , CIRCUMCENTER , and ORTHOCENTER . For a listing of these and other triangle centers, see Kimberling (1994). A triangle center is said to be REGULAR IFF there is a TRIANGLE CENTER FUNCTION which is a POLYNOMIALin D; a, b, and c (where D is the AREA of the TRIANGLE ) such that the TRILINEAR COORDINATES of the center are f(a; b; c):f(b; c ; a):f(c ; a; b) : A triangle center is said to be a MAJOR TRIANGLE CENTER if the TRIANGLE CENTER FUNCTION a is a function of ANGLE A alone, and therefore b and g of B and C alone, respectively. See also MAJOR TRIANGLE CENTER ,REGULAR TRIAN- GLE CENTER ,T RIANGLE ,T RIANGLE CENTER FUNC- TION ,TRILINEAR COORDINATES ,TRILINEAR POLAR References Davis, P. J. "The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-History." Amer. Math. Monthly 102, 204 /C1/214, 1995. Dixon, R. "The Eight Centres of a Triangle." §1.5 in Mathographics. New York: Dover, pp. 55 /C1/61, 1991. Gale, D. "From Euclid to Descartes to Mathematica to Oblivion?" Math. Intell. 14,68/C1/69, 1992. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/167, 1994. Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,1/C1/295, 1998. Triangle Center Function A HOMOGENEOUS FUNCTION f(a ; b ; c) ; i.e., a function f such that f(ta ; tb ; tc) /C30tnf(a; b; c) ; which gives the TRILINEAR COORDINATES of a TRIAN- GLE CENTER as a : b : g /C30f(a ; b ; c):f(b; c ; a):f(c ; a ; b) : The variables may correspond to angles ( A,B,C)o r side lengths ( a,b,c), since these can be intercon- verted using the LAW OF COSINES . See also MAJOR TRIANGLE CENTER ,REGULAR TRIAN- GLE CENTER ,TRIANGLE CENTER ,TRILINEAR COORDI- NATES References Kimberling, C. "Triangle Centers as Functions." Rocky Mtn. J. Math. 23, 1269/C1/1286, 1993. Kimberling, C. "Triangle Centers." http://cedar.evansvil- le.edu/~ck6/tcenters/. Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,1/C1/295, 1998. Lester, J. "Triangles III: Complex Triangle Functions." Aequationes Math. 53,4/C1/35, 1997. Triangle Coefficient A function of three variables written D(abc)/C13 D(a;b;c) and defined by D(abc) /C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a /C27 b /C28 c)!(a /C28 b /C27 c)!( /C28a /C27 b /C27 c)! (a /C27 b /C27 c /C27 1)!s : References Shore, B. W. and Menzel, D. H. Principles of Atomic Spec- tra. New York: Wiley, p. 273, 1968. Triangle Condition The condition that j takes on the values j /C30j1 /C27j2 ; j1 /C27j2 /C281; ... ;½j1 /C28j2 ½; denoted D j1 j2 j ðÞ :/ References Sobelman, I. I. Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer-Verlag, p. 60, 1992. Triangle Counting Given rods of length 1, 2, ..., n, how many distinct triangles T(n) can be made? Lengths for which li /C30lj /C27lk obviously do not give triangles, but all other combina- tions of three rods do. The answer is T(n) /C301 24 n(n /C282)(2n /C285) for n even 1 24(n /C281)(n /C283)(2n /C281) for n odd:( The values for n /C301, 2, ...are 0, 0, 0, 1, 3, 7, 13, 22, 34, 50, ... (Sloane’s A002623). Somewhat surprisingly, this sequence is also given by the GENERATING FUNCTION f(x) /C30x4 (1 /C28 x)3(1 /C28 x2) /C30x4 /C273x5 /C277x6 /C2713x7 /C27...: See also TRIANGLE TILING References Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 278 /C1/282, 1991. Sloane, N. J. A. Sequences A002623/M2640 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Triangle Cubic Curve A CUBIC CURVE on which 37 notable triangle centers lie. References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 42 /C1/43, 1991.Triangle Function L(x) /C130 ½x½> 1 1 /C28½x½½x½B1iC0C (1) /C30P(x) +P(x) (2) ¼P(x) + Hx/C271 2iCkCiCkA /C28P(x) + Hx/C2812iCkCiCkA ; (3) where P is the RECTANGLE FUNCTION and H is the HEAVISIDE STEP FUNCTION . An obvious generalization used as an APODIZATION FUNCTION goes by the name of the BARTLETT FUNCTION . There is also a three-argument function known as the triangle function: l(x; y; z) /C13x2 /C27y2 /C27z2 /C282xy /C282xz /C282yz : (4) It follows that l a2 ; b2 ; c2iCjiCk /C30(a /C27b /C27c)(a /C27b /C28c)(a /C28b /C27c)(a /C28b /C28c) : (5) See also ABSOLUTE VALUE ,B ARTLETT FUNCTION , HEAVISIDE STEP FUNCTION ,RAMP FUNCTION ,REC- TANGLE FUNCTION ,SGN,TRIANGLE COEFFICIENT References Bracewell, R. "The Triangle Function of Unit Height and Area, L(x) :/"In The Fourier Transform and Its Applica- tions, 3rd ed. New York: McGraw-Hill, p. 53, 1999. Triangle Graph The CYCLE GRAPH C3;which is also the COMPLETE GRAPH K3:/ See also COMPLETE GRAPH ,CYCLE GRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 144, 1990. Triangle Inequality Let x and y be vectors ½x½/C28½y ½5½x /C27y½5½x½/C27½y½: (1) Equivalently, for COMPLEX NUMBERS z1 and z2 ; z1jj/C28 z2jj5 z1 /C27z2 jj 5 z1jj/C27 z2jj: (2) A generalization is Xn k /C301akiCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk0iCk05X n k /C301akjj: (3) See also ONO INEQUALITY , P-ADIC NUMBER ,STRONG TRIANGLE INEQUALITY References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Wal- tham, MA: Blaisdell, p. 42, 1967. Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 12, 1999. Triangle Interior To determine if a given point v lies in the interior of a given triangle, consider an individual vertex, denoted v0 ; and let v1and v2be the vectors from v0to the other two vertices. Expressing the vector from v0 to v in terms of v1 and v2 then gives v /C30v0 /C27av1 /C27bv2 ; (1) where a and b are constants. Solving for a and b gives a /C30det vv2 ðÞ/C28 det v0 v2 ðÞ det(v1 v2) (2) b /C30/C28det vv1 ðÞ /C28 det v0 v1 ðÞ det v1 v2 ðÞ; (3)where det(uv) /C30u /C29v /C30uxvy /C28uyvx (4) is the DETERMINANT of the matrix formed from the COLUMN VECTORS u and v. The point v will be "inside" the angle formed at v0 if a; b > 0; and so will be in the interior of the triangle if the corresponding a; b > 0 for each of the three vertices. More generally, a point v is in the interior of a TRIANGLE if the CONVEX HULL of the three vertices plus the point v contains three points instead of four. This means the point v is inside the CONVEX HULL of the triangle, which is just the triangle itself. See also CONVEX HULL,TRIANGLE Triangle of Figurate Numbers FIGURATE NUMBER TRIANGLE Triangle Packing The best known packings of equilateral triangles into an equilateral triangle are illustrated above for thefirst few cases (Friedman). The best known packings of equilateral triangles intoa circle are illustrated above for the first few cases(Friedman). The best known packings of equilateral triangles into a square are illustrated above for the first few cases (Friedman). Stewart (1998, 1999) considered the problem of finding the largest convex area that can be nontrivi- ally tiled with equilateral triangles whose sides are integers for a given number of triangles and which have no overall common divisor. There is no upper limit if an arbitrary number of triangles are used. The following table gives the best known packings for small numbers of triangles. n max. areareference n max. areareference 1 1 Stewart 199711 495 Stewart 1997 2 2 Stewart 199712 860 Stewart 1998 3 3 Stewart 199713 1559 Stewart 1998 4 7 Stewart 199714 2831 Stewart 19985 11 Stewart 199715 4782 Stewart 1999 6 20 Stewart 199716 8559 Stewart 1998 7 36 Stewart 199717 14279 Stewart 1998 8 71 Stewart 1997 9 146 Stewart 1997 10 260 Stewart 1997 See also CIRCLE PACKING ,EQUILATERAL TRIANGLE , PACKING ,SQUARE PACKING References Friedman, E. "Circles in Triangles." http://www.stetson.edu/ ~efriedma/cirintri/. Friedman, E. "Squares in Triangles." http://www.stetso- n.edu/~efriedma/squintri/. Friedman, E. "Triangles in Triangles." http://www.stetso- n.edu/~efriedma/triintri/. Graham, R. L. and Lubachevsky, B. D. "Dense Packings of Equal Disks in an Equilateral Triangle: From 22 to 34 and Beyond." Electronic J. Combinatorics 2,A 11 /C1/39, 1995. http://www.combinatorics.org/Volume_2/volu-me2.html#A1. Stewart, I. "Squaring the Square." Sci. Amer. 277,9 4/C1 /96, July 1997. Stewart, I. "Mathematical Recreations: Monks, Blobs and Common Knowledge. Feedback." Sci. Amer. 279, 97, Aug. 1998. Stewart, I. "Mathematical Recreations: The Synchronicity of Firefly Flashing. Feedback." Sci. Amer. 280, 106, Mar. 1999. Triangle Point Picking Given a triangle with one vertex at the origin and the others at positions v1andv2;one might think that a random point inside the triangle would be given by x/C30a1v1/C271/C28a1 ðÞ a2v2; where a1anda2are uniform variates in the interval [0;1]:However, as can be seen in the plot above, this samples the triangle nonuniformly, concentrating points in the v1 corner. To pick points uniformly distributed inside the triangle, instead pick x /C30a1v1 /C27a2v2 ; where a1 and a2 are uniform variates in the interval [0; 1]; which gives points uniformly distributed in a QUADRILATERAL (left figure). The points not in the TRIANGLE INTERIOR can then either be discarded, or transformed into the corresponding point inside the triangle (right figure). Picking n points independently and uniformly from a triangle with unit area gives a CONVEX HULL with expected area of A(n) /C301 /C282 n /C27 1Xn k/C3011 k /C301 /C282Hn n /C27 1 ; where Hnis a HARMONIC NUMBER (Buchta 1984, 1986). This is a special case of SIMPLEX POINT PICKING . See also SIMPLEX POINT PICKING ,TRIANGLE TRIANGLE PICKING References Buchta, C. "Zufallspolygone in konvexen Vielecken." J. reine angew. Math. 347, 212 /C1/220, 1984. Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653 /C1/659, 1986. Triangle Postulate The sum of the ANGLES of a TRIANGLE is two RIGHT ANGLES . This POSTULATE is equivalent to the PARAL- LEL AXIOM . References Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 54, 1990.Triangle Squaring Let CD be the ALTITUDE of a TRIANGLE DABC and let E be its MIDPOINT . Then area(DABC ) /C301 2 AB /C215 CD /C30AB /C215 DE ; and /C176ABFG can be SQUARED by RECTANGLE SQUAR- ING. The general POLYGON can be treated by drawing diagonals, SQUARING the constituent TRIANGLES , and then combining the SQUARES together using the PYTHAGOREAN THEOREM . See also PYTHAGOREAN THEOREM ,RECTANGLE SQUAR- ING,SQUARING References Dunham, W. "Hippocrates’ Quadrature of the Lune." Ch. 1 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 14 /C1/15, 1990. Triangle Tiling Any triangle tiles the plane (Wells 1991, p. 208). The total number of triangles (including inverted ones) in the above figures are given by N(n) /C3018 n(n /C272)(2n /C271) for n even 18 n(n /C272)(2n /C271) /C281 ½/C138 for n odd:( The first few values are 1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411, 525, 658, 812, 988, 1188, 1413, 1665, ... (Sloane’s A002717). See also EQUILATERAL TRIANGLE ,RECTANGLE TILING , TRIANGLE COUNTING ,TRIANGLE PACKING References Conway, J. H. and Guy, R. K. "How Many Triangles." In The Book of Numbers. New York: Springer-Verlag, pp. 83 /C1/84, 1996. Sloane, N. J. A. Sequences A002717/M3827 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 68 /C1/69 and 208, 1991. Triangle Transformation Principle The triangle transformation principle gives rules for transforming equations involving an INCIRCLE to equations about EXCIRCLES . See also EXCIRCLE ,INCIRCLE References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 191 /C1/192, 1929. Triangle Triangle Picking The mean area of a triangle picked inside a triangle with unit area is ¯A /C301=12 (Pfiefer 1989). This was proposed by Watson (1865) and solved by Sylvester, and is a special case of the general formula for POLYGON TRIANGLE PICKING . See also DISK TRIANGLE PICKING ,HEXAGON TRIANGLE PICKING ,POLYGON TRIANGLE PICKING ,SQUARE TRI- ANGLE PICKING ,SYLVESTER’S FOUR- POINT PROBLEM , TETRAHEDRON TETRAHEDRON PICKING References Pfiefer, R. E. "The Historical Development of J. J. Sylves- ter’s Four Point Problem." Math. Mag. 62, 309 /C1/317, 1989. Watson, S. "Question 1229." Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 4. London: F. Hodgson and Son, p. 101, 1865. Triangular Antiprism See also ANTIPRISMTriangular Cupola JOHNSON SOLID J3 : The bottom six VERTICES are 91 2ffiffiffi 3p ;91 2 ; 0iCkCiCkA ; 0;91 ; 0 ðÞ ; and the top three VERTICES are 1ffiffiffi 3p; 0 ;ffiffiffi 2 3s ! ;/C281 2ffiffiffi 3p;91 2 ;ffiffiffi 23s ! : See also J OHNSON SOLID Triangular Dipyramid The triangular (or TRIGONAL ) dipyramid is one of the convex DELTAHEDRA , and JOHNSON SOLID J12 :/ See also DELTAHEDRON ,D IPYRAMID ,H EXAHEDRON , JOHNSON SOLID ,PENTAGONAL DIPYRAMID Triangular Graph The triangular graph with nnodes on a side is denoted T(n):Tutte (1970) showed that the CHRO- MATIC POLYNOMIALS of planar triangular graphs possess a ROOT close to f2/C302:618033 . . . ;where fis the GOLDEN MEAN . More precisely, if nis the number ofVERTICES ofG, then PGf2iCjiCk 5f5/C28n (Le Lionnais 1983, p. 46). Every planar triangular graph possesses a VERTEX of degree 3, 4, or 5 (Le Lionnais 1983, pp. 49 and 53). See also LATTICE GRAPH References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Tutte, W. T. "On Chromatic Polynomials and the Golden Ratio." J. Combin. Theory 9, 289 /C1/296, 1970. Triangular Hebesphenorotunda JOHNSON SOLID J92 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Triangular Matrix An UPPER TRIANGULAR MATRIX U is defined by Uij /C30aijfor i 5j 0 for i > j :iC0C (1) Written explicitly, U /C30a11a12/C1/C1/C1 a1n 0 a22/C1/C1/C1 a2n nn::: n 00 /C1/C1/C1 ann2 6643 775: (2) A LOWER TRIANGULAR MATRIX L is defined by Lij /C30aijfor i ]j 0 for i Bj:iC0C (3) Written explicitly, L/C30a11 0 /C1/C1/C1 0 a21a22/C1/C1/C1 0 nn:::0 an1an2/C1/C1/C1 ann2 6643 775: (4) See also H ANKEL MATRIX ,H ESSENBERG MATRIX , HILBERT MATRIX ,LOWER TRIANGULAR MATRIX ,M A-TRIX,U PPER TRIANGULAR MATRIX ,V ANDERMONDE MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 10, 1962. Triangular Number AFIGURATE NUMBER OF THE FORM Tn/C131 2n(n/C271)/C30n/C271 2iCkniCko ; (1) wheren kiCjiCk is a BINOMIAL COEFFICIENT , obtained by building up regular triangles out of dots. The first few triangle numbers are 1, 3, 6, 10, 15, 21, ... (Sloane’s A000217). The odd triangular numbers are given by 1, 3, 15, 21, 45, 55, ... (Sloane’s A014493), while theeven triangular numbers are 6, 10, 28, 36, 66, 78, ... (Sloane’s A014494). /T4/C3010 gives the number and arrangement of BOWL- ING pins, while T5/C3015 gives the number and ar- rangement of balls in BILLIARDS . Triangular numbers satisfy the RECURRENCE RELATION T2 n/C271/C28T2 n/C30(n/C271)3; (2) as well as 3Tn/C27Tn/C281/C30T2n (3) 3Tn/C27Tn/C271/C30T2n/C271 (4) 1/C273/C275/C27.../C27(2n/C281)/C30Tn/C27Tn/C281: (5) In addition, the triangle numbers can be related to the square numbers by (2n/C271)2/C308T/C271/C30Tn/C281/C276Tn/C27Tn/C271 (6) (Conway and Guy 1996), as illustrated above (Wells1991, p. 198). They have the ordinary GENERATING FUNCTION f(x)/C30x (1/C28x)3/C30x/C273x2/C276x3/C2710x4/C2715x5/C27. . . (7) and EXPONENTIAL GENERATING FUNCTION g(x)/C301/C272x/C271 2x2iCkCiCkA ex /C301/C273x/C273x2/C2753x3/C2758x4/C27... /C301/C273x 1!/C276x2 2!/C2710x3 3!/C2715x4 4!/C27... ( 8 ) (Sloane and Plouffe 1995, p. 9). Every triangular number is also a HEXAGONAL NUM- BER, since 1 2r(r/C271) /C30r/C271 2 ! 2r/C271 2 ! /C281"# forrodd /C28r 2 ! 2/C28r 2 ! /C281"# forreven :8 >>>>< >>>>:(9) Also, every PENTAGONAL NUMBER is 1/3 of a triangular number. The sum of consecutive triangular numbers is a SQUARE NUMBER , since Tr/C27Tr/C281/C301 2r(r/C271)/C2712(r/C281)r ¼12r(r/C271)/C27(r/C281) ½/C138 /C30r2: (10) Interesting identities involving triangular numbers and SQUARE NUMBERS are X2n/C281 k/C301(/C281)k/C271Tk/C30n2(11) T2 n/C30Xn k/C301k3/C301 4n2(n/C271)2(12) X k/C301;3;...;qk3/C30Tn (13) forqODD and n/C301 2(q2/C272q/C281): (14) Triangular numbers also unexpectedly appear in integrals involving the ABSOLUTE VALUE OF THE FORM g1 0g1 0x/C28y jjndx dy/C302 (n/C271)(n/C272): (15) All EVEN PERFECT NUMBERS are triangular Tpwith PRIME p. Furthermore, every EVEN PERFECT NUMBER P/C216i s OF THE FORM P/C301/C279Tn/C30T3n/C271; (16) where Tnis a triangular number with n/C308j/C272 (Eaton 1995, 1996). Therefore, the nested expression 9(9/C1/C1/C1(9(9(9(9 Tn/C271)/C271)/C271)/C271) . . ./C271)/C271 (17) generates triangular numbers for any Tn:An INTEGERkis a triangular number IFF8k/C271i sa SQUARE NUMBER >1:/ The numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (Sloane’s A001110) are SQUARE TRIANGULAR NUM- BERS , i.e., numbers which are simultaneously trian- gular and SQUARE (Pietenpol 1962). The corresponding square roots are 1, 6, 35, 204, 1189, 6930, ... (Sloane’s A001109), and the indices of thecorresponding triangular numbers T naren/C301, 8, 49, 288, 1681, ... (Sloane’s A001108). Numbers which are simultaneously triangular and TETRAHEDRAL satisfy the BINOMIAL COEFFICIENT equation Tn/C30n/C271 2iCkniCko /C30m/C272 3iCkniCko /C30Tem; (18) the only solutions of which are Te3/C30T4/C3010 (19) Te8/C30T15/C30120 (20) Te20/C30T55/C301540 (21) Te34/C30T119/C307140 (22) (Guy 1994, p. 147).The following table gives triangular numbers T p having prime indices p. /Tnwith prime indicesA034953 3, 6, 15, 28, 66, 91, 153, 190, 276, 435, 496, ... oddTnwith prime indicesA034954 3, 15, 91, 153, 435, 703, 861, 1431, 1891, 2701, ... even Tnwith prime indicesA034955 6, 28, 66, 190, 276, 496, 946, 1128, 1770, 2278, ... The smallest of two INTEGERS for which n3/C2813 is four times a triangular number is 5 (Cesaro 1886; LeLionnais 1983, p. 56). The only F IBONACCI NUMBERS which are triangular are 1, 3, 21, and 55 (Ming 1989),and the only P ELL NUMBER which is triangular is 1 (McDaniel 1996). The BEAST NUMBER 666 is triangu- lar, since T6 /C2156/C30T36/C30666: (23) In fact, it is the largest REPDIGIT triangular number (Bellew and Weger 1975 /C1/76). FERMAT’S POLYGONAL NUMBER THEOREM states that every POSITIVE INTEGER is a sum of most three TRIANGULAR NUMBERS , four SQUARE NUMBERS , five PENTAGONAL NUMBERS , and nn-POLYGONAL NUM- BERS . Gauss proved the triangular case (Wells 1986, p. 47), and noted the event in his diary on July 10, 1796, with the notation ++E YRHKA num /C30D/C27D/C27D: (24) This case is equivalent to the statement that every number OF THE FORM 8m /C273 is a sum of three ODD SQUARES (Duke 1997). Dirichlet derived the number of ways in which an INTEGER m can be expressed as the sum of three triangular numbers (Duke 1997). The result is particularly simple for a PRIME OF THE FORM 8m /C273; in which case it is the number of squares mod 8m /C273 minus the number of nonsquares mod 8m /C273 in the INTERVAL 4m /C271 (Deligne 1973). The only triangular numbers which are the PRODUCT of three consecutive INTEGERS are 6, 120, 210, 990, 185136, 258474216 (Sloane’s A001219; Guy 1994, p. 148). See also FIGURATE NUMBER ,HEPTAGONAL TRIANGU- LAR NUMBE R,O CTAGONAL TRIANGULAR NUMBER , PENTAGONAL TRIANGULAR NUMBER ,PRONIC NUMBER , SQUARE TRIANGULAR NUMBER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 59, 1987. Bellew, D. W. and Weger, R. C. "Repdigit Triangular Num- bers." J. Recr. Math. 8,96/C1/97, 1975 /C1/76. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33 /C1/38, 1996. Deligne, P. "La Conjecture de Weil." Inst. Hautes E´ tudes Sci. Pub. Math. 43, 273 /C1/308, 1973. Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970. Duke, W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer. Math. Soc. 44, 190 /C1/196, 1997. Eaton, C. F. "Problem 1482." Math. Mag. 68, 307, 1995. Eaton, C. F. "Perfect Number in Terms of Triangular Numbers." Solution to Problem 1482. Math. Mag. 69, 308 /C1/309, 1996. Guy, R. K. "Sums of Squares" and "Figurate Numbers." §C20 and §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 /C1/138 and 147 /C1/150, 1994. Hindin, H. "Stars, Hexes, Triangular Numbers and Pytha- gorean Triples." J. Recr. Math. 16, 191 /C1/193, 1983 /C1/1984. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. McDaniel, W. L. "Triangular Numbers in the Pell Se- quence." Fib. Quart. 34, 105 /C1/107, 1996. Ming, L. "On Triangular Fibonacci Numbers." Fib. Quart. 27,98/C1/108, 1989. Pappas, T. "Triangular, Square & Pentagonal Numbers." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989. Pietenpol, J. L "Square Triangular Numbers." Amer. Math. Monthly 169, 168 /C1/169, 1962. Ram, R. "Triangle Numbers that are Perfect Squares." http://users.tellurian.net/hsejar/maths/triangle/. Satyanarayana, U. V. "On the Representation of Numbers as the Sum of Triangular Numbers." Math. Gaz. 45,40/C1/ 43, 1961. Sloane, N. J. A. Sequences A000217/M2535, A001108/ M4536, A001109/M4217, A001110/M5259, A001219,A014493, A014494, A034953, A034955, and A034955 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 47 /C1/ 48, 1986. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 199, 1991. Triangular Orthobicupola JOHNSON SOLID J27 ; consisting of eight equilateral triangles and six squares. If a triangular orthobicu- pola is oriented with triangles on top and bottom, the two halves may be rotated one sixth of a turn with respect to each other to obtain the CUBOCTAHEDRON . In hexagonal close packing, layers of spheres are packed so that spheres in alternating layers overlieone another. As in cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above.Connecting the centers of the external 12 spheres gives J 27(Steinhaus 1983, pp. 203 /C1/205), which is therefore also a SPACE-FILLING POLYHEDRON . See also CUBOCTAHEDRON ,JOHNSON SOLID,SPACE- FILLING POLYHEDRON ,SPHERE PACKING References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 203 /C1/205, 1999. Triangular Prism A PRISM composed of triangular faces. The regular right triangular prism of unit edge length has SUR- FACE AREA and VOLUME S /C301 2(6 /C27ffiffiffi 3p ) V /C301 4ffiffiffi 3p : See also PRISM Triangular Pyramid A PYRAMID having a triangular base. The SLANT HEIGHT of a regular triangular pyramid is a special case of the formula for a regular n-gonal PYRAMID with n /C303, given by s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 /C271 3 a2q ; (1) where h is the height and a is the length of a side of the base. The TETRAHEDRON is a special case of the triangular pyramid. See also PYRAMID ,TETRAHEDRON Triangular Square Number SQUARE TRIANGULAR NUMBER Triangular Symmetry Group Given a TRIANGLE with angles (/p=p ; p=q ; p=r) ; the resulting symmetry GROUP is called a (p; q; r) trian-gle group (also known as a SPHERICAL TESSELLATION ). In 3-D, such GROUPS must satisfy 1 p /C271 q /C271 r> 1; and so the only solutions are (2 ; 2; n) ; (2; 3 ; 3); (2; 3; 4); and (2; 3; 5) (Ball and Coxeter 1987). The group (2; 3 ; 6) gives rise to the semiregular planar TESSELLATIONS of types 1, 2, 5, and 7. The group (2; 3; 7) gives hyperbolic tessellations. See also GEODESIC DOME References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 155 /C1/161, 1987. Coxeter, H. S. M. "The Partition of a Sphere According to the Icosahedral Group." Scripta Math 4, 156/C1/157, 1936. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Kraitchik, M. "A Mosaic on the Sphere." §7.3 in Mathema- tical Recreations. New York: W. W. Norton, pp. 208 /C1/209, 1942. Triangulation Triangulation is the division of a surface or plane polygon into a set of TRIANGLES , usually with the restriction that each TRIANGLE side is entirely shared by two adjacent TRIANGLES . It was proved in 1925 that every surface has a triangulation, but it mightrequire an infinite number of TRIANGLES and the proof is difficult (Francis and Weeks 1999). A surface with a finite number of triangles in its triangulation is called COMPACT . Wickham-Jones (1994) gives an On3ðÞ algorithm for triangulation ("otectomy"), and O’Rourke (1998,p. 47) sketches a method for improving this to On 2ðÞ ; as first done by Lennes (1911). Garey et al. (1978) gave an algorithmically straightforward O(nlnn) method for triangulation, which was for many years believed optimal. However, Tarjan and van Wyk(1988) produced an O(nlg lg n) algorithm. This was followed by an unexpected result due to Chazelle (1991), who showed that an arbitrary SIMPLE POLY- GON can be triangulated in O(n):However, according to Skiena (1997), "this algorithm is quite hopeless toimplement." See also ART GALLERY THEOREM ,COMPACT SURFACE , DELAUNAY TRIANGULATION ,JAPANESE THEOREM , SIMPLE POLYGON ,TESSELLATION References Chazelle, B. "Triangulating a Simple Polygon in Linear Time." Disc. Comput. Geom. 6, 485 /C1/524, 1991. de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. "Polygon Triangulation: Guarding an Art Gal- lery." Ch. 3 in Computational Geometry: Algorithms and Applications, 2nd rev. ed. Berlin: Springer-Verlag, pp. 45 /C1/61, 2000. Fournier, A. and Montuno, D. Y. "Triangulating Simple Polygons and Equivalent Problems." ACM Trans. Gra- phics 3, 153 /C1/174, 1984. Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393 /C1/399, 1999. Friedman, E. "Triangulating Triangles." http://www.stetso- n.edu/~efriedma/triang/. Garey, M. R.; Johnson, D. S.; Preparata, F. P.; and Tarjan, R. E. "Triangulating a Simple Polygon." Inform. Process. Lett. 7, 175 /C1/179, 1978. Kraus, M. "Polygon Triangulation." http://library.wolfram.- com/packages/polygontriangulation/. O’Rourke, J. §2.3 in Computational Geometry in C, 2nd ed. Cambridge, England: Cambridge University Press, 1998. Rado´,T."U ¨ber den Begriff der Riemannschen Fla¨che." Acta Litt. Sci. Reg. Univ. Hungar. Francisco-Josephinae 2, 101 /C1/121, 1924 /C1/1926. Skiena, S. S. "Triangulation." §8.6.3 in The Algorithm De- sign Manual. New York: Springer-Verlag, pp. 355 /C1/357, 1997. Tarjan, R. and van Wyk, C. "An O(n lg lg n) Algorithm for Triangulating a Simple Polygon." SIAM J. Computing 17, 143 /C1/178, 1988. Wickham-Jones, T. "ExtendGraphics Packages for Mathe- matica 3.0." http://www.mathsource.com/cgi-bin/ msitem?0208 /C1/976. Wickham-Jones, T. Mathematica Graphics: Techniques and Applications. New York: Springer-Verlag, pp. 406 and 448, 1994. Triaugmented Dodecahedron JOHNSON SOLID J61 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Triaugmented Hexagonal Prism JOHNSON SOLID J57 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Triaugmented Triangular Prism One of the convex DELTAHEDRA . It is composed of 14 equilateral triangles, and is JOHNSON SOLID J51 : The VERTICES are (91 =2;91=2 ; 0); 0 ; 0;ffiffiffi 2p =2iCjiCk ; 0;91=2;/C28ffiffiffi 3p =2iCjiCk ;91/C27ffiffiffi6piCjiCk =4;0;/C28ffiffiffi2p /C27ffiffiffi3piCjiCk =4iCjiCk ; where the xand zcoordinates of the last are found by solving x 2/C271 2iCkCiCkA2 /C27z/C27ffiffiffi 3p =2iCkCiCkA2 /C3012(1) x/C281 2iCkCiCkA2 /C2712iCkCiCkA2 /C27z2/C3012: (2) For a triaugmented triangular prism with unit side length, the SURFACE AREA and VOLUME are S/C307 2ffiffiffi 3p (3) V/C301 42ffiffiffi 2p /C27ffiffiffi 3piCkCiCkA : (4) See also DELTAHEDRON ,JOHNSON SOLID Triaugmented Truncated Dodecahedron JOHNSON SOLID J71 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Triaxial Ellipsoid ELLIPSOID Tri-Axial Ellipsoid ELLIPSOID Tribar An IMPOSSIBLE FIGURE published by R. Penrose (1958). It also exists as a TRIBOX . References Draper, S. W. "The Penrose Triangle and a Family of Related Figures." Perception 7, 283 /C1/296, 1978. Fineman, M. The Nature of Visual Illusion. New York: Dover, p. 119, 1996. Jablan, S. "Set of Modular Elements ‘Space Tiles’." http:// members.tripod.com/~modularity/space.htm. Pappas, T. "The Impossible Tribar." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 13, 1989. Penrose, R. "Impossible Objects: A Special Type of Visual Illusion." Brit. J. Psychology 49,31/C1/33, 1958. Tribonacci Number The tribonacci numbers are a generalization of the FIBONACCI NUMBERS defined by T1 /C301; T2 /C301; T3 /C302; and the RECURRENCE RELATION Tn /C30Tn/C281 /C27Tn/C282 /C27Tn /C283 (1) for n ]4: The represent the n /C303 case of the FIBO- NACCI N-STEP NUMBERS . The first few terms are 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (Sloane’s A000073). Theratio of adjacent terms tends to 1.83929, which is the REAL ROOT of x4 /C282x3 /C271 /C300: The Tribonacci num- bers can also be computed using the GENERATING FUNCTION 1 1 /C28 z /C28 z2 /C28 z3 /C301 /C27z /C272z2 /C274z3 /C277z4 /C2713z5 /C2724z6 /C2744z7 /C2781z8 /C27149z9 /C27... : (2) An explicit FORMULA for Tn is also given by 31 319 /C27 3ffiffiffiffiffiffi 33piCjiCk 1=3/C271 319 /C28 3ffiffiffiffiffiffi 33piCjiCk 1=3/C271 3non 586 /C27 102ffiffiffiffiffiffi 33piCjiCk 1=3 586 /C27 102ffiffiffiffiffiffi33piCjiCk 2=3/C274 /C28 2 586 /C27 102ffiffiffiffiffiffi33piCjiCk 1=32 435; (3) where [x] denotes the NINT function (Plouffe). The first part of a NUMERATOR is related to the REAL root of x3 /C28x2 /C28x /C281 ; but determination of the DENOMI- NATOR requires an application of the LLL ALGORITHM . The numbers increase asymptotically to Tn /C2cn ; (4) where c /C3019 27 /C2719ffiffiffiffiffiffi 33piCkCiCkA1 =3 /C274 91927 /C2719ffiffiffiffiffiffi 33piCkCiCkA/C281 =3 /C271 3 /C301 :83928675521 ... (5) (Plouffe). See also FIBONACCI N-STEP NUMBER ,F IBONACCI NUMBER ,TETRANACCI NUMBER References Plouffe, S. "Tribonacci Constant." http://www.lacim.u- qam.ca/piDATA/tribo.txt. Sloane, N. J. A. Sequences A000073/M1074 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Tribox An IMPOSSIBLE FIGURE . See also IMPOSSIBLE FIGURE ,TRIBAR References Jablan, S. "Are Impossible Figures Possible?" http://mem- bers.tripod.com/~modularity/kulpa.htm. Trichotomy Law Every REAL NUMBER is NEGATIVE ,0,or POSITIVE . The law is sometimes states as "For arbitrary real numbers x and y, exactly one of the relations a Bb, a /C30b, a /C21b holds" (Apostol 1967, p. 20). See also SCHRO ¨ DER- BERNSTEIN THEOREM ,T OTAL ORDER References Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Wal- tham, MA: Blaisdell, 1967. Tricolorable A projection of a LINK is tricolorable if each of the strands in the projection can be colored in one of three different colors such that, at each crossing, all three colors come together or only one does and at least two different colors are used. The TREFOIL KNOT and trivial 2-link are tricolorable, but the UNKNOT , WHITEHEAD LINK , and FIGURE-OF-EIGHT KNOT are not. If the projection of a knot is tricolorable, then REIDEMEISTER MOVES on the knot preserve tricolor- ability, so either every projection of a knot is tricolor- able or none is. Tricomi Equation The PARTIAL DIFFERENTIAL EQUATION uyy /C30yuxx : References Manwell, A. R. The Tricomi Equation with Applications to the Theory of Plane Transonic Flow. Marshfield, MA: Pitman, 1979. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, 1997. Tricomi Function CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND,GORDON FUNCTION Tricuspoid DELTOID Tricylinder STEINMETZ SOLID Tridecagon A 13-sided POLYGON , sometimes also called the TRISKAIDECAGON .Trident The plane curve given by the equation xy/C30x3/C28a3: See also TRIDENT OF DESCARTES ,TRIDENT OF NEW- TON Trident of Descartes The plane curve given by the equation (a/C27x)(a/C28x)(2a/C28x)/C30x3/C282ax2/C28a2x/C272a3/C30axy y/C30(a/C27x)(a/C28x)(2a/C28x) ax: The above plot has a/C302. Trident of Newton The CUBIC CURVE defined by ax3/C27bx2/C27cx/C27d/C30xy with a"0:The curve cuts the axis in either one or three points. It was the 66th curve in Newton’s classification of CUBICS . Newton stated that the curve has four infinite legs and that the Y-AXIS is an ASYMPTOTE to two tending toward contrary parts. References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 109 /C1/110, 1972. MacTutor History of Mathematics Archive. "Trident of Newton." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Trident.html. Tridiagonal Matrix A MATRIX with NONZERO elements only on the diagonal and slots horizontally or vertically adjacent the diagonal (i.e., along the SUBDIAGONAL and SUPER- DIAGONAL ). A general 4 /C294 tridiagonal MATRIX has the form a11a12 00 a21a22a23 0 0 a32a33a34 00 a43a442 6643 775: Inversion of such a matrix requires only O 7nðÞ (as opposed to O(n3 =3)) arithmetic operations (Acton 1990, p. 332). See also DIAGONAL MATRIX ,JACOBI ALGORITHM , SUBDIAGONAL ,SUPERDIAGONAL References Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 331 /C1/334, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Tridiagonal and Band Diagonal Systems of Equations." §2.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 42 /C1/47, 1992. Tridiminished Icosahedron JOHNSON SOLID J63 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT .Tridiminished Rhombicosidodecahedron JOHNSON SOLID J83 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Tridyakis Icosahedron The DUAL POLYHEDRON of the ICOSITRUNCATED DODE- CADODECAHEDRON U45and Wenninger dual W84:/ See also DUAL POLYHEDRON ,ICOSITRUNCATED DODE- CADODECAHEDRON References Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, p. 96, 1983. Trifolium Lawrence (1972) defines a trifolium as a FOLIUM with b /C23 (0; 4a) : However, the term "the" trifolium is some- times applied to the FOLIUM with b /C30a, which is then the 3-petalled ROSE with Cartesian equation x2 /C27y2iCjiCk y2 /C27x(x /C27a)iC0iCB /C304axy2 and polar equation r /C30a cos u 4 sin2 u /C281iCjiCk /C30/C28a cos(3 u) : The trifolium with b /C30a is the RADIAL CURVE of the DELTOID . See also BIFOLIUM ,FOLIUM ,QUADRIFOLIUM References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152 /C1/153, 1972. MacTutor History of Mathematics Archive. "Trifolium." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Tri- folium.html. Trigon TRIANGLE Trigonal Dipyramid TRIANGULAR DIPYRAMID Trigonal Dodecahedron An irregular DODECAHEDRON . See also DODECAHEDRON ,PYRITOHEDRON ,RHOMBIC DODECAHEDRON References Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.New York: Wiley, p. 62, 1990.Trigonometric Addition Formulas Angle addition FORMULAS express trigonometric func- tions of sums of angles a9bin terms of functions of a andb:The fundamental formulas of angle addition in trigonometry are given by sin(a/C27b)/C30sinacosb/C27sinbcosa (1) sin(a/C28b)/C30sinacosb/C28sinbcosa (2) cos(a/C27b)/C30cosacosb/C28sinasinb (3) cos(a/C28b)/C30cosacosb/C27sinasinb (4) tan(a/C27b)/C30tana/C27tanb 1/C28tanatanb(5) tan(a/C28b)/C30tana/C28tanb 1/C27tanatanb: (6) The sine and cosine angle addition identities can be compactly summarized by the MATRIX EQUATION cosasina /C28sinacosaiC0jiC0k cosbsinb /C28sinbcosbiC0jiC0k /C30cos(a/C27b) sin( a/C27b) /C28sin(a/C27b) cos( a/C27b)iC0jiC0k : (7) These formulas can be simply derived using COMPLEX EXPONENTIALS and the E ULER FORMULA as follows. cos(a/C27b)/C27isin(a/C27b)/C30ei(a/C27b)/C30eiaeib /C30(cosa/C27isina)(cos b/C27isinb) /C30(cosacosb/C28sinasinb) /C27i(sinacosb/C27cosasinb):ð8Þ Equating REAL and IMAGINARY PARTS then gives (1) and (3), and (2) and (4) follow immediately bysubstituting /C28bforb: / Taking the ratio of (1) and (3) gives the tangent angle addition FORMULA tan(a/C27b)/C13sin(a/C27b) cos(a/C27b)/C30sinacosb/C27sinbcosa cosacosb/C28sinasinb /C30sina cosa/C27sinb cosb 1/C28sinasinb cosacosab/C30tana/C27tanb 1/C28tanatanb:(9) The DOUBLE-ANGLE FORMULAS are sin(2 a)/C302 sin acosa (10) cos(2 a)/C30cos2a/C28sin2a (11) /C302 cos2a/C281 (12) /C301/C282 sin2a (13) tan(2 a)/C302 tan a 1/C28tan2a: (14) MULTIPLE-ANGLE FORMULAS are given by sin(nx)/C30Xn k/C300n kiCkniCko coskxsinn/C28kxsin1 2(n/C28k)phi :(15) cos(nx)/C30Xn k/C300n kiCkniCko coskxsinn/C28kxcos12(n/C28k)phi ;(16) and can also be written using the RECURRENCE RELATIONS sin(nx)/C302 sin[( n/C281)x] cos x/C28sin[(n/C282)x] (17) cos(nx)/C302 cos[( n/C281)x] cos x/C28cos[(n/C282)x] (18) tan(nx)/C30tan[( n/C281)x]/C27tanx 1/C28tan[( n/C281)x] tan x: (19) SIMPSON’S FORMULAS are given by sina/C27sinb/C302 sina/C27b 2 ! cosa/C28b 2 ! (20) sina/C28sinb/C302 sina/C28b 2 ! cosa/C27b 2 ! (21) cosa/C27cosb/C302 cosa/C27b 2 ! cosa/C28b 2 ! (22) cosa/C28cosb/C30/C282 sina/C28b 2 ! sina/C27b 2 ! :(23) The angle addition formulas can also be derived purely algebraically without the use of COMPLEX NUMBERS . Consider the small RIGHT TRIANGLE in the figure above, which gives a/C30sina cos(a/C27b)(24) b/C30sinatan(a/C27b): (25) Now, the usual trigonometric definitions applied to the large RIGHT TRIANGLE givesin(a/C27b)/C30sinb/C27a cosa/C27b /C30sinb/C27sina cos(a/C27b) cosa/C27sinasin(a/C27b) cos(a/C27b)(26) cos(a/C27b)/C30cosb cosa/C27b /C30cosb cosa/C27sinasin(a/C27b) cos(a/C27b): (27) Solving these two equations simultaneously for the variables sin( a/C27b) and cos( a/C27b) then immediately gives sin(a/C27b)/C30cosasina/C27cosbsinb cosacosb/C27sinasinb(28) cos(a/C27b)/C30cos2b/C28sin2a cosacosb/C27sinasinb: (29) These can be put into the familiar forms with the aidof the trigonometric identities (cosacosb/C27sinasinb)(cos acosb/C27sinbcosa) /C30cosbsinb/C27cosasina (30) and (cosacosb/C27sinasinb)(cos acosb/C28sinacosb) /C30cos 2acos2b/C28sin2asin2b (31) /C301/C28sin2asin2b (32) /C30cos2a/C28sin2b (33) /C30cos2b/C28sin2a; (34) which can be verified by direct multiplication. Plug-ging (30) into (28) and (34) into (29) then gives sin(a/C27b)/C30sinacosb/C27sinbcosa (35) cos(a/C27b)/C30cosacosb/C28sinasinb; (36) as before. A similar proof due to Smiley and Smiley uses the left figure above to obtain sin a /C30sin( a /C27 b) cos b /C27sin b cos a sin a; (37) from which it follows that sin( a /C27 b) /C30sin a cos b /C27sin b cos a : (38) Similarly, from the right figure, sin a cos a /C30cos b sin b /C27cos(a /C27 b) sin a; (39) so cos(a /C27 b) /C30cos a cos b /C28sin a sin b: (40) Similar diagrams can be used to prove the angle subtraction formulas (Smiley 1999, Smiley and Smi- ley). In the figure at left, h /C30cos a cos b (41) x /C30h sin( a /C28 b) /C30(sin a /C28h sin b) cos a; (42) giving sin( a /C28 b) /C30sin a cos b /C28cos a sin b: (43) Similarly, in the figure at right, h /C30cos a sin b (44) x /C30h cos(a /C28 b) /C30(sin a /C27h cos b) cos a; (45) giving cos(a /C28 b) /C30cos a cos b /C27sin a sin b: (46) A more complex diagram can be used to obtain a proof from the tan(a /C28 b) identity (Ren 1999). In the above figure, let BF =BE /C30AD=DE : Then tan(a /C28 b) /C30DE BE /C30AD BF /C30tan a /C28 tan b 1 /C27 tan a tan b : (47) An interesting identity relating the sum and differ- ence tangent formulas is given by tan(a/C28b) tan(a/C27b)/C30sin(a/C28b) cos( a/C27b) cos(a/C28b) sin( a/C27b) /C30(sinacosb/C28sinbcosa)(cos acosb/C28sinasinb) (cosacosb/C27sinasinb)(sin acosb/C27sinbcosa) /C30sinacosa/C28sinbcosb sinacosa/C27sinbcosb: (48) See also DOUBLE- ANGLE FORMULAS ,H ALF-ANGLE FORMULAS ,M ULTIPLE- ANGLE FORMULAS ,P ROSTHA- PHAERESIS FORMULAS ,SIMPSON’S FORMULAS ,TRIGO- NOMETRIC ANGLES ,T RIGONOMETRIC PRODUCT FORMULAS ,TRIGONOMETRY References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Nelson, R. To appear in College Math. J. , March 2000. Ren, G. "Proof without Words: tan( a/C28b):/"College Math. J. 30, 212, 1999. Smiley, L. M. "Proof without Words: Geometry of Subtrac- tion Formulas." Math. Mag. 72, 366, 1999. Smiley, L. and Smiley, D. "Geometry of Addition and Subtraction Formulas." http://saturn.math.uaa.alas- ka.edu/~smiley/trigproofs.html. Trigonometric Angles The ANGLES np=m(with m, n integers) for which the trigonometric function may be expressed in terms of finite ROOT EXTRACTION ofreal numbers are limited to values of mwhich are precisely those which produce constructible POLYGONS . Gauss showed these to be OF THE FORM m /C302kp1p2 /C1/C1/C1ps ; where k is an INTEGER ]0 and the piare distinct FERMAT PRIMES . The first few values are m /C301, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, ... (Sloane’s A003401). Where possible, analytic expressions for trigono- metric functions with arguments of this form can be obtained using the Mathematica command Func- tionExpand . Although formulas for trigonometric functions may be found analytically for other m as well, the expressions involve ROOTS of COMPLEX NUMBERS obtained by solving a CUBIC , QUARTIC , or higher order equation. The cases m /C307 and m /C309 involve the CUBIC EQUATION and QUARTIC EQUATION , respec- tively. A partial table of the analytic values of SINE, COSINE , and TANGENT for arguments p=m is given below. Derivations of these formulas appear in the following entries. x (//C14)/ x (rad)/sin x// cos x// tan x/ 0.0 0 0 1 0 15.0 /1 12 p//1 4ffiffiffi 6p /C28ffiffiffi 2piCjiCk //1 4ffiffiffi 6p /C27ffiffiffi 2piCjiCk // 2 /C28ffiffiffi 3p / 18.0 /1 10 p//1 4ffiffiffi 5p /C281iCjiCk //1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C272ffiffiffi 5pp //1 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 /C2810ffiffiffi 5pp / 22.5 /1 8p//12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pp //1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pp //ffiffiffi2p /C281 / 30.0 /1 6p//12//12ffiffiffi 3p //1 3ffiffiffi 3p / 36.0 /1 5p//14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C282ffiffiffi 5pp //1 41 /C27ffiffiffi 5piCjiCk //ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C282ffiffiffi 5pp / 45.0 /1 4p//12ffiffiffi 2p //1 2ffiffiffi 2p / 1 60.0 /1 3p//12ffiffiffi 3p //1 2//ffiffiffi 3p / 90.0 /1 2p/ 10 /C12 180.0 /p/ 0 /C2810 There is a nice mnemonic for remembering sines of common angles, sin(0/C14) /C301 2ffiffiffi 0p (1) sin(30/C14) /C301 2ffiffiffi 1p (2) sin(45/C14) /C301 2ffiffiffi 2p (3) sin(60/C14) /C301 2ffiffiffi 3p (4) sin(90/C14) /C301 2ffiffiffi 4p : (5) See also TRIGONOMETRY VALUES 0,TRIGONOMETRY VALUES PI,TRIGONOMETRY VALUES PI/2,TRIGONOME- TRY VALUES PI/3,TRIGONOMETRY VALUES PI/4,TRIGO-NOMETRY VALUES PI/5,TRIGONOMETRY VALUES PI/6, TRIGONOMETRY VALUES PI/7,TRIGONOMETRY VALUES PI/8,T RIGONOMETRY VALUES PI/9,T RIGONOMETRY VALUES PI/10,TRIGONOMETRY VALUES PI/11,TRIGO- NOMETRY VALUES PI/12,TRIGONOMETRY VALUES PI/15, TRIGONOMETRY VALUES PI/16,TRIGONOMETRY VALUES PI/17,TRIGONOMETRY VALUES PI/18,TRIGONOMETRY VALUES PI/20,TRIGONOMETRY VALUES PI/24,TRIGO- NOMETRY VALUES PI/30,TRIGONOMETRY VALUES PI/32 Trigonometric Functions The functions (also called the CIRCULAR FUNCTIONS ) comprising TRIGONOMETRY : the COSECANT csc x; CO- SINE cos x; COTANGENT cot x; SECANT sec x; SINE sin x; and TANGENT tan x: The inverses of these functions are denoted csc /C281 x; cos/C281 x; cot /C281 x; sec/C281 x; sin /C281 x; and tan/C281 x: Note that the f /C281 NOTATION here means INVERSE FUNCTION , not f to the -1 POWER . See also DOUBLE- ANGLE FORMULAS ,H ALF-ANGLE FORMULAS ,HYPERBOLIC FUNCTIONS ,TRIGONOMETRY Trigonometric Power Formulas Power formulas include sin2 x /C301 2[1 /C28cos(2 x)] (1) sin3 x /C301 4[3 sin x /C28sin(3 x)] (2) sin4 x /C301 8[3 /C284 cos(2 x) /C27cos(4 x)] (3) and cos2 x /C3012[1 /C27cos(2 x)] (4) cos3 x /C3014[3 cos x /C27cos(3 x)] (5) cos4 x /C3018[3 /C274 cos(2 x) /C27cos(4 x)] (6) (Beyer 1987, p. 140). Formulas of these types can also be given analytically as sin2n x /C301 22n2n niCkniCko /C27( /C281)n 22n/C281Xn/C281 k /C300(/C281)k 2n kiCkniCko cos[2( n /C28k)x] (7) sin2n/C271 /C30( /C281)n 4nXn k /C300(/C281)k 2n /C271 kiCkniCko sin[2 n /C271 /C282k)x] (8) cos2n x /C301 22n2n niCkniCko /C271 22n/C281Xn /C281 k/C3002n kiCkniCko cos[2( n /C28k)x] (9) cos2n/C271 x /C301 4nXn k/C3002n/C271 kiCkniCko cos[(2 n/C271/C282k)x] (10) (Kogan), wheren miCjiCk is a BINOMIAL COEFFICIENT . See also TRIGONOMETRY References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Kogan, S. "A Note on Definite Integrals Involving Trigono- metric Functions." http://www.mathsoft.com/asolve/con- stant/pi/sin/sin.html. Trigonometric Product Formulas Trigonometric product formulas for the sum of the cosines and sines of two angles can be derived using the above figure (Kung 1996). From the figure, define u /C301 2( a /C28 b) (1) g /C301 2(a /C27 b) : (2) Then we have the identity s /C3012(sin a /C27sin b) /C30cos12( a /C28 b)hi sin12( a /C27 b)hi ð3Þ t /C3012(cos a /C27cos b) /C30cos12( a /C28 b)hi cos12( a /C27 b)hi :ð4Þ Trigonometric product formulas for the difference of the cosines and sines of two angles can be derived using the similar figure illustrated above (Kung 1996). With u and g as previously defined, the above figure gives u /C30cos b /C28cos a /C302 sin12( a /C28 b)hi sin12( a /C27 b)hi ð5Þ v /C30sin a /C28sin b /C302 sin1 2(a /C28 b)hi cos12( a /C27 b)hi :ð6Þ See also DOUBLE- ANGLE FORMULAS ,H ALF-ANGLE FORMULAS ,PROSTHAPHAERESIS FORMULAS ,TRIGONO- METRIC ADDITION FORMULAS ,TRIGONOMETRYReferences Kung, S. H. "Proof without Words: The Difference-Product Identities" and "Proof without Words: The Sum-Product Identities." Math. Mag. 69, 269, 1996. Trigonometric Series FOURIER SERIES Trigonometric Series Formulas Trigonometric identities which prove useful in the construction of map projections include A sin(2f) /C27B sin(4f) /C27C sin(6f) /C27D sin(8f) /C30sin(2f) A?/C27cos(2 f) B?/C27cos(2 f) C?/C27D ? cos(2 f) ðÞ ðÞ ðÞ ; (1) where A?/C13A /C28C (2) B?/C132B /C284D (3) C?/C134C (4) D ?/C138D: (5) A sin f /C27B sin(3 f) /C27C sin(5f) /C27D sin(7f) /C30sin f A?/C27sin2 f B ?/C27sin2 f C ?/C27D? sin2 fiCjiCkiCjiCkiCjiCk ; (6) where A?/C13A /C273B /C275C /C277D (7) B?/C13/C284B /C2820C /C2856D (8) C?/C1316C /C27112D (9) D?/C13/C2864D : (10) A /C27B cos(2 f) /C27C cos(4 f) /C27D cos(6 f) /C27E cos(8 f) /C30A?/C27cos(2 f) B ?/C27cos(2 f) C ?/C27cos(2 f) ð ð /C2 D?/C27E ? cos(2 f) ðÞÞÞ ; (11) where A?/C13A /C28C /C27E (12) B?/C13B /C283D (13) C ?/C132C /C288E (14) D?/C134D (15) E?/C138E (16) (Snyder 1987). See also TRIGONOMETRY References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 19, 1987. Trigonometric Substitution INTEGRALS OF THE FORM g f(cos u; sin u) du (1) can be solved by making the substitution z /C30eiu so that dz /C30ieiu du and expressing cos u /C30eiu /C27 e /C28iu 2/C30z /C27 z /C281 2 (2) sin u /C30eiu /C28 e /C28iu 2i/C30z /C28 z/C281 2i: (3) The integral can then be solved by CONTOUR INTE- GRATION . Alternatively, making the substitution t /C13tan(u=2) transforms (1) into g f2t 1 /C27 t2 ;1 /C28 t2 1 /C27 t2 ! 2 dt 1 /C27 t2 : (4) The following table gives trigonometric substitutions which can be used to transform integrals involving square roots. Form Substitution /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C28x2p //x/C30asinu/ /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27x2p //x/C30atanu/ /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2/C28a2p //x/C30asecu/ See also HYPERBOLIC SUBSTITUTION Trigonometry The study of ANGLES and of the angular relationships of planar and 3-D figures is known as trigonometry. The TRIGONOMETRIC FUNCTIONS (also called the CIR- CULAR FUNCTIONS ) comprising trigonometry are the COSECANT cscx;COSINE cosx;COTANGENT cotx;SE- CANT secx;SINE sinx;and TANGENT tanx:The inverses of these functions are denoted csc/C281x; cos/C281x;cot/C281x;sec/C281x;sin/C281x;and tan/C281x:Note that the f/C281NOTATION here means INVERSE FUNC- TION ,not f to the /C281POWER . The trigonometric functions are most simply definedusing the UNIT CIRCLE . Let ube an ANGLE measuredcounterclockwise from the X-AXIS along an ARCof the CIRCLE . Then cos uis the horizontal coordinate of the ARC endpoint, and sin uis the vertical component. The RATIO sinu=cosuis defined as tan u:As a result of this definition, the trigonometric functions are periodic with period 2 p;so func(2 pn/C27u)/C30func( u); (1) where nis an INTEGER and func is a trigonometric function. ARIGHT TRIANGLE has three sides, which can be uniquely identified as the HYPOTENUSE , adjacent to a given angle u;or opposite u:A helpful mnemonic for remembering the definitions of the trigonometric functions is then given by "oh, ah, oh-ah," sinu/C30opposite hypotenuse(2) cosu/C30adjacent hypotenuse(3) tanu/C30opposite adjacent: (4) From the P YTHAGOREAN THEOREM , sin2u/C27cos2u/C301: (5) Therefore, it is also true that tan2u/C271/C30sec2u (6) 1/C27cot2u/C30csc2u: (7) The trigonometric functions can be defined algebrai-cally in terms of COMPLEX EXPONENTIALS (i.e., using the E ULER FORMULA )a s sinz/C13eiz/C28e/C28iz 2i(8) cscz/C131 sinz/C302i eiz/C28e/C28iz(9) cosz/C13eiz/C27e/C28iz 2(10) secz/C131 cosz/C302 eiz/C27e/C28iz(11) tanz/C13sinz cosz/C30eiz/C28e/C28iz ieiz/C27e/C28iz ðÞ(12) cot z /C131 tan z /C30ieiz /C27 e/C28izðÞ eiz /C28 e /C28iz/C30i 1 /C27 e /C282izðÞ 1 /C28 e /C282iz: (13) Hybrid trigonometric product/sum formulas are sin( a /C27 b) sin( a /C28 b) /C30sin2 a /C28sin2 b /C30cos2 b /C28cos2 a (14) cos(a /C27 b) cos(a /C28 b) /C30cos2 a /C28sin2 b /C30cos2 b /C28sin2 a: (15) OSBORNE’S RULE gives a prescription for converting trigonometric identities to analogous identities for HYPERBOLIC FUNCTIONS . For IMAGINARY arguments, sin(iz) /C30i sinh z (16) cos(iz) /C30cosh z: (17) For COMPLEX arguments, sin(x /C27iy) /C30sin x cosh y /C27i cos x sinh y (18) cos(x /C27iy) /C30cos x cosh y /C28i sin x sinh y: (19) For the ABSOLUTE SQUARE of COMPLEX arguments z /C30 x /C27iy ; ½sin(x /C27iy) ½2 /C30sin2 x /C27sinh2 y (20) ½cos(x /C27iy)½2 /C30cos2 x /C27sinh2 y: (21) The MODULUS also satisfies the curious identity ½sin(x /C27iy)½/C30½sin x /C27sin(iy)½: (22) The only functions satisfying identities of this form, ½f(x/C27iy)½/C30½f(x)/C27f(iy)½ (23) are f(z)/C30Az;f(z)/C30Asin(bz);and f(z)/C30Asinh( bz) (Robinson 1957). See also COSECANT ,COSINE ,COTANGENT ,D OUBLE- ANGLE FORMULAS ,EUCLIDEAN NUMBER ,HALF-ANGLE FORMULAS ,INVERSE COSECANT ,INVERSE COSINE , INVERSE COTAN GENT ,INVERSE SECANT ,INVERSE SINE,INVERSE TANGENT ,INVERSE TRIGONOMETRIC FUNCTIONS ,O SBORNE’S RULE,POLYGON ,PROSTHA- PHAERESIS FORMULAS ,SECANT ,SINE,TANGENT ,TRI- GONOMETRIC ADDITION FORMULAS ,TRIGONOMETRIC ANGLES ,T RIGONOMETRIC FUNCTIONS ,T RIGONO- METRIC POWER FORMULAS ,TRIGONOMETRIC SERIES FORMULAS ,W ERNER FORMULAS References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th print- ing. New York: Dover, pp. 71 /C1/79, 1972. Bahm, L. B. The New Trigonometry on Your Own. Patter- son, NJ: Littlefield, Adams & Co., 1964.Beyer, W. H. "Trigonometry." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 134 /C1/152, 1987. Borchardt, W. G. and Perrott, A. D. A New Trigonometry for Schools. London: G. Bell, 1930. Dixon, R. "The Story of Sine and Cosine." §4.4 in Matho- graphics. New York: Dover, pp. 102 /C1/106, 1991. Hobson, E. W. A Treatise on Plane Trigonometry. London: Cambridge University Press, 1925. Kells, L. M.; Kern, W. F.; and Bland, J. R. Plane and Spherical Trigonometry. New York: McGraw-Hill, 1940. Maor, E. Trigonometric Delights. Princeton, NJ: Princeton University Press, 1998. Morrill, W. K. Plane Trigonometry, rev. ed. Dubuque, IA: Wm. C. Brown, 1964. Robinson, R. M. "A Curious Mathematical Identity." Amer. Math. Monthly 64,8 3/C1/85, 1957. Siddons, A. W. and Hughes, R. T. Trigonometry, Parts I-IV. London: Cambridge University Press, 1929. Sloane, N. J. A. Sequences A003401/M0505 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Thompson, J. E. Trigonometry for the Practical Man. Prin- ceton, NJ: Van Nostrand. Weisstein, E. W. "Exact Values of Trigonometric Functions." M ATHEMATICA NOTEBOOK TRIGEXACT.M . Yates, R. C. "Trigonometric Functions." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Ed- wards, pp. 225 /C1/232, 1952. Weisstein, E. W. "Books about Trigonometry." http:// www.treasure-troves.com/books/Trigonometry.html. Zill, D. G. and Dewar, J. M. Trigonometry, 2nd ed. New York: McGraw-Hill 1990. Trigonometry Values Pi By the definition of the trigonometric functions, cosp/C30/C281 (1) cosp/C30/C12 (2) cscp¼/C12 ð3Þ secp/C30/C281 (4) sinp/C300 (5) tanp/C300: (6) Trigonometry Values Pi/2 By the definition of the trigonometric functions, cosp 2 ! /C300 (1) cotp 2 ! /C300 (2) cscp 2 ! /C301 (3) secp 2 ! /C30/C12 (4) sinp 2 ! /C301 (5) tanp 2 ! /C30/C12: (6) See also DIGON Trigonometry Values Pi/3 cosp 3 ! /C301 2 (1) cotp 3 ! /C301 3ffiffiffi 3p (2) cscp 3 ! /C302 3ffiffiffi 3p (3) secp 3 ! /C302 (4) sinp 3 ! /C301 2ffiffiffi 3p (5) tanp 3 ! /C30ffiffiffi3p : (6) These formulas can be derived from knowledge of the TRIGONOMETRY VALUES FOR PI/6 sinp 6 ! /C301 2 (7) cosp 6 ! /C3012ffiffiffi 3p (8) together with the trigonometric identity sin(2a) /C302 sin a cos a; (9) giving sinp 3 ! /C302 sinp 6 ! cosp 6 ! /C3021 2iCkCiCkA 12ffiffiffi 3piCkCiCkA /C301 2ffiffiffi 3p (10) is obtained. Using the identity cos(2 a) /C301 /C282 sin2 a; (11) then gives cosp 3 ! /C301 /C282 sin2p 6 ! /C301 /C2821 2iCkCiCkA2 /C3012 : (12) See also EQUILATERAL TRIANGLETrigonometry Values Pi/4 cosp 4 ! /C301 2ffiffiffi 2p (1) cotp 4 ! /C301 (2) cscp 4 ! /C30ffiffiffi 2p (3) secp 4 ! /C30ffiffiffi 2p (4) sinp 4 ! /C301 2ffiffiffi 2p (5) tanp 4 ! /C301: (6) For a RIGHT ISOSCELES TRIANGLE , symmetry requires that the angle at each VERTEX be given by 1 2 p /C272a /C30 p; (7) so a /C30 p=4: The sides are equal, so sin2 a /C27cos2 a /C302 sin2 a /C301: (8) Solving gives the above equations. See also SQUARE Trigonometry Values Pi/5 cosp 5 ! /C30141/C27ffiffiffi 5piCkCiCkA (1) cos2p 5 ! /C301 4/C281/C27ffiffiffi 5piCkCiCkA (2) cotp 5 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pq (3) cot2p 5 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2810ffiffiffi 5pq (4) cscp 5 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2710ffiffiffi 5pq (5) csc2p 5 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2810ffiffiffi 5pq (6) secp 5 ! /C30ffiffiffi 5p /C281 (7) sec2 p 5 ! /C301 /C27ffiffiffi 5p (8) sinp 5 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C282ffiffiffi 5pq (9) sin2p 5 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C272ffiffiffi 5pq (10) tanp 5 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C282ffiffiffi 5pq (11) tan2 p 5 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq : (12) These formulas can be derived using the identity sin(5 a) /C305 sin a /C2820 sin3 a /C2716 sin5 a: (13) Now, let a /C13 p=5 and x /C13sin a: Then sin p /C300 /C305x /C2820x3 /C2716x5 (14) 16x4 /C2820x2 /C275 /C300 : (15) Solving the QUADRATIC EQUATION for x2 gives sin2p 5 ! /C30x2 /C3020 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( /C2820)2 /C28 4 /C215 16 /C215 5p 2 /C215 16 /C3020 9ffiffiffiffiffiffi 80p 32/C301 85 9ffiffiffi 5piCkCiCkA : (16) Now, sin p=5ðÞ must be less than sinp 4 ! /C301 2ffiffiffi 2p ; (17) so taking the MINUS SIGN and simplifying gives sinp 5 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C28ffiffiffi 5p 8s /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C282ffiffiffi 5pq : (18) /cos(p=5) can be computed from cosp 5 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28sin2p 5 !vuut/C301 41 /C27ffiffiffi 5piCkCiCkA : (19) See also DODECAHEDRON ,GOLDEN RATIO,ICOSAHE- DRON ,PENTAGON ,PENTAGRAMTrigonometry Values Pi/6 cosp 6 ! /C301 2ffiffiffi 3p (1) cotp 6 ! /C30ffiffiffi 3p (2) cscp 6 ! /C302 (3) secp 6 ! /C302 3ffiffiffi 3p (4) sinp 6 ! /C301 2 (5) tanp 6 ! /C3013ffiffiffi 3p : (6) Given a RIGHT TRIANGLE with angles defined to be a and 2a; it must be true that a /C272a /C271 2 p /C30 p; (7) so a /C30 p=6: Define the hypotenuse to have length 1 and the side opposite a to have length x, then the side opposite 2a has lengthffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p : This gives sin a /C13x and sin(2a) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p : (8) But sin(2a) /C302 sin a cos a /C302xffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28x2p ; (9) so we have ffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28x 2p /C302xffiffiffiffiffiffiffiffiffiffiffiffiffi1 /C28x 2p : (10) This gives 2x /C301 ; or sinp 6 ! /C301 2 : (11) /cos(p=6) is then computed from cosp 6 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28sin2p 6 !vuut/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C281 2iCkCiCkA2r /C3012ffiffiffi 3p :(12) See also HEXAGON ,HEXAGRAM Trigonometry Values Pi/7 Trigonometric functions of np=7 for nan integer cannot be expressed in terms of sums, products, and finite ROOT EXTRACTIONS onreal rational numbers because 7 is not a F ERMAT PRIME . This also means that the HEPTAGON is not a CONSTRUCTIBLE POLYGON . However, exact expressions involving roots of com- plex numbers can still be derived using the trigono- metric identity sin(na) /C302 sin[(n /C281)a] cos a /C28sin[(n /C282)a] : (1) The case n /C307 gives sin(7a) /C302 sin(6 a) cos a /C28sin(5a) /C302(32 cos5 a sin a /C2832 cos3 a sin a /C276 cos a sin a) cos a /C28 5 sin a /C2820 sin3 a /C2716 sin5 aiCjiCk /C3064 cos6 a sin a /C2864 cos4 a sin a /C2712 cos2 a sin a /C285 sin a /C2720 1 /C28cos2 aiCjiCk sin a /C2816 1 /C282 cos2 a /C27cos4 aiCjiCk sin a /C30sin a 64 cos6 a /C2880 cos4 a þ 24 cos2 a /C281iCjiCk : (2) Rewrite this using the identity cos2 a /C301 /C28sin2 a; sinp 7 ! /C30sin a(7 /C2856 sin2 a /C27112 sin4 a /C2864 sin6 a) /C30/C2864 sin a sin6 a /C28112 64sin4 a /C2756 64sin2 a /C287 64iCkCiCkA : (3) Now, let a /C13 p=7 and x /C13sin2 a; then sin( p) /C300 /C30x3 /C287 4 x2 /C2778 x /C287 64; (4) which is a CUBIC EQUATION in x. The ROOTS are numerically found to be x :0 :188255 ; 0 :611260 ; 0:950484 : But sin a /C30ffiffiffixp; so these ROOTS correspond to sin a :0:4338 ; sin(2a) :0:7817 ; sin(3a) :0:9749 : By NEWTON’S RELATION Y iri /C30/C28a0 (5) we have x1x2x3 /C307 64; (6) or sinp 7 ! sin2p 7 ! sin3p 7 ! /C30ffiffiffiffiffiffi 7 64s /C301 8ffiffiffi 7p : (7) Similarly, cosp 7 ! cos2p 7 ! cos3p 7 ! /C3018 (8) andcos 2p 7 ! /C28cosp 7 ! cos2 p 7 ! /C301 4 (9) (Bankoff and Garfunkel 1973). The constants of the CUBIC EQUATION are given by Q /C131 93a1 /C28a2 2iCjiCk /C301 93 /C21578 /C28/C2874iCkCiCkA2iC0jiC0k /C30/C287 144 (10) R /C131 549a2a1 /C282a3 2 /C2827a0iCjiCk /C301 549 /C287 4iCkCiCkA 17 8iCkCiCkA /C282 /C2874iCkCiCkA3 /C2827 /C287 64iCkCiCkAiC0jiC0k /C30/C28f73456 : (11) The DISCRIMINANT is then D /C13Q3 /C27R3 /C30/C28343 2 ;985;984 /C2749 11;943;936 /C30/C2849 442;368 B0; (12) so there are three distinct REAL ROOTS . Finding the first one, x /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R /C27ffiffiffiffi Dpq /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R /C28ffiffiffiffi Dpq /C281 3 a2 : (13) Writing ffiffiffiffi Dp /C303/C283 =27 128 i; (14) plugging in from above, and anticipating that the solution we have picked corresponds to sin(3p=7);/ sin3p 7 ! /C30ffiffiffixp /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C287 3456 /C273/C283=27 128 iq /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C287 3456 /C283/C283 =27 128 i /C281 3(/C2874)qr /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C287 3456 /C273/C283=27 128iq /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C287 3456/C283/C283=27 128iq /C277 12r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 3456/C281/C2733=2i ðÞq /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 34561/C2733=2i ðÞq /C277 12r See also HEPTAGON ,SILVER CONSTANT References Bankoff, L. and Garfunkel, J. "The Heptagonal Triangle." Math. Mag. 46,7/C1/19, 1973. Trigonometry Values Pi/8 cosp 8 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq (1) cos3 p 8 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pq (2) cotp 8 ! /C301 /C27ffiffiffi 2p (3) cot3p 8 ! /C30ffiffiffi2p /C281 (4) cscp 8 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C272ffiffiffi 2pq (5) csc3p 8 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C282ffiffiffi 2pq (6) secp 8 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C282ffiffiffi 2pq (7) sec3p 8 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C272ffiffiffi 2pq (8) secp 8 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pq (9) sin3p 8 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq (10) tanp 8 ! /C30ffiffiffi 2p /C281 (11) tan3p 8 ! /C301 /C27ffiffiffi2p : (12) sinp 8 ! /C30sin1 2/C215p 4 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 121 /C28cosp 4 !vuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21 /C2812ffiffiffi 2piCkCiCkAr /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pq : (13) Now, checking to see if the SQUARE ROOT can be simplified gives a2 /C28b2c /C3022 /C2812 /C215 2 /C304 /C282 /C302; (14) which is not a PERFECT SQUARE , so the above expres- sion cannot be simplified. Similarly, cosp 8 ! /C30cos1 2p 4 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 121 /C27cosp 4 !vuut/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21 /C27ffiffiffi 2p 3 !vuut/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pq (15) tanp 8 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2p 2 /C27ffiffiffi2ps /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi2piCjiCk 2 4 /C28 2vuut/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C27 2 /C28 4ffiffiffi 2p 2s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 /C28 4ffiffiffi 2p 2s /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C282ffiffiffi 2pq : (16) But a2 /C28b2c /C3032 /C28222 /C309 /C288 /C301 (17) is a PERFECT SQUARE , so we can find d /C301 2(3 91) /C301 ; 2 : (18) Rewrite the above as tanp 8 ! /C30ffiffiffi 2p /C281 (19) cotp 8 ! /C301ffiffiffi 2p /C281/C30ffiffiffi 2p /C271 2/C281/C30ffiffiffi 2p /C271: (20) See also OCTAGON Trigonometry Values Pi/9 Trigonometric functions of np=9 radians for nan integer not divisible by 3 (e.g., 40 8and 80 8) cannot be expressed in terms of sums, products, and finite ROOT EXTRACTIONS onRATIONAL NUMBERS because 9 is not a product of distinct F ERMAT PRIMES . This also means that the regular NONAGON is not a CONSTRUCTIBLE POLYGON . However, exact expressions involving roots of com- plex numbers can still be derived using the trigono- metric identity sin(3a)/C303 sin a/C284 sin3a: (1) Leta/C13p=9 and x/C13sina:Then the above identity gives the CUBIC EQUATION 4x3/C283x/C271 2ffiffiffi 3p /C300 (2) x3/C283 4x/C30/C2818ffiffiffi 3p : (3) This cubic is OF THE FORM x3/C27px/C30q; (4) where p/C30/C283 4(5) q/C30/C2818ffiffiffi 3p : (6) The DISCRIMINANT is then D /C13p 3 !3 /C27q 2 !2 /C30/C281 4 !3 /C27ffiffiffi 3p 16 !2 /C30/C281 16 /C215 4 /C273 16 /C215 16 /C30/C284 /C27 3 256 /C30/C281 256 B0: (7) There are therefore three REAL distinct roots, which are approximately /C280:9848 ; 0.3240, and 0.6428. We want the one in the first QUADRANT , which is 0.3240. sinp 9 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28ffiffiffi3p 16 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffi /C281 256svuut/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28ffiffiffi 3p 16/C28ffiffiffiffiffiffiffiffiffiffiffiffiffi /C281 256svuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C28ffiffiffi 3p 16/C271 16is /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3p 16/C271 16is /C302 /C284 =3ffiffiffiffiffiffiffiffiffiffiffiffiffiffi i /C28ffiffiffi 3pq /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi i /C27ffiffiffi 3pqiCkniCko :0:34202 (8) Similarly, cosp 9 ! /C302/C284 =3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27iffiffiffi 3pq /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28iffiffiffi 3pq iCkniCko :0:93969 : (9) Because of the NEWTON’S RELATIONS , we have the identities sinp 9 ! sin2p 9 ! sin4p 9 ! /C301 8ffiffiffi 3p (10) cosp 9 ! cos2p 9 ! cos4 p 9 ! /C301 8 (11) tanp 9 ! tan2p 9 ! tan4p 9 ! /C30ffiffiffi 3p : (12) (11) is known as M ORRIE’S LAW . See also MORRIE’S LAW,NONAGON ,STAR OF GOLIATH Trigonometry Values Pi/10 cosp 10 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pq (1) cos3p 10 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pq (2)cosp 10 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pq (3) cot3p 10 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 5pq (4) cscp 10 ! /C301þffiffiffi 5p (5) csc3p 10 ! /C30ffiffiffi5p /C281 (6) sec p 10 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2810ffiffiffi 5pq (7) sec3p 10 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2810ffiffiffi 5pq (8) sinp 10 ! /C301 4ffiffiffi 5p /C281iCkCiCkA (9) sin3p 10 ! /C301 41/C27ffiffiffi 5piCkCiCkA (10) tanp 10 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2810ffiffiffi 5pq (11) tan3p 10 ! /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2710ffiffiffi 5pq (12) To derive these formulas, start with sinp 10 ! /C30sin1 2/C215p 5 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21/C28cosp 5 !"#vuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21/C2814(1/C27ffiffiffi 5p )hir /C301 4ffiffiffi 5p /C281iCkCiCkA : (13) So we have cosp 10 ! /C30cos1 2/C215p 5 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 121/C27cosp 5 !"#vuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21/C2714(1/C27ffiffiffi 5p )hir /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pq (14) and tanp 10 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3/C28ffiffiffi 5p 5/C27ffiffiffi5ps /C301 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25/C2810ffiffiffi 5pq : (15) An interesting near-identity is given by 1 4cos1 10iCkCiCkA /C27cosh1 10iCkCiCkA /C272 cos1 20ffiffiffi 2piCkCiCkA cosh1 20ffiffiffi2piCkCiCkA hi :1 : (16) In fact, the left-hand side is approximately equal to / 1 þ 2 :480 /C2910/C2813 /. See also DECAGON ,DECAGRAM Trigonometry Values Pi/11 Trigonometric functions of np=11 for n an integer cannot be expressed in terms of sums, products, and finite ROOT EXTRACTIONS on real rational numbers because 11 is not a FERMAT PRIME . This also means that the UNDECAGON is not a CONSTRUCTIBLE POLY- GON. However, exact expressions involving roots of com- plex numbers can still be derived using the MULTIPLE- ANGLE FORMULA sin(na) /C30(/C281)(n/C281)=2Tn(sin a) ; (1) where Tnis a CHEBYSHEV POLYNOMIAL OF THE FIRST KIND . Plugging in n /C3011 gives sin(11 a) /C30sin a 11 /C28220 sin2 a /C271232 sin4 aiCj /C282816 sin6 a /C272816 sin8 /C281024 sin10 aÞ: (2) Letting a /C13 p=11 and x /C13sin2 a then gives sin p /C300 /C3011 /C28220x /C271232 x2 /C282816 x3 /C272816 x4 /C281024 x5 : (3) This equation is an irreducible QUINTIC EQUATION ,so an analytic solution involving FINITE ROOT EXTRAC- TIONS does not exist. The numerical ROOTS are x /C30 0:07937 ; 0.29229, 0.57115, 0.82743, 0.97974. So sin a /C300 :2817 ; sin(2a) /C300:5406 ; sin(3a) /C300:7557 ; sin(4a) /C300:9096 ; sin(5a) /C300:9898 : From one of NEW- TON’S IDENTITIES , sinp 11 ! sin2p 11 ! sin3p 11 ! sin4 p 11 ! sin5p 11 ! /C30ffiffiffiffiffiffiffiffiffiffiffi 11 1024s /C30ffiffiffiffiffiffi 11p 32 (4) cosp 11 ! cos2 p 11 ! cos3p 11 ! cos4p 11 ! cos5 p 11 ! /C301 32(5) tanp 11 ! tan2 p 11 ! tan3 p 11 ! tan4p 11 ! tan5p 11 ! /C30ffiffiffiffiffiffi 11p : (6) The trigonometric functions of p=11 also obey the identitytan3p 11 ! /C274 sin2 p 11 ! /C30ffiffiffiffiffiffi11p : (7) See also U NDECAGON References Beyer, W. H. "Trigonometry." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Trigonometry Values Pi/12 cosp 12 ! /C301 4ffiffiffi 6p /C27ffiffiffi2piCkCiCkA (1) cos 5p 12 ! /C301 4ffiffiffi 6p /C28ffiffiffi2piCkCiCkA (2) cot p 12 ! /C302/C27ffiffiffi3p (3) cot 5p 12 ! /C302/C28ffiffiffi 3p (4) cotp 12 ! /C30ffiffiffi6p /C27ffiffiffi2p (5) csc 5p 12 ! /C30ffiffiffi6p /C28ffiffiffi2p (6) sec p 12 ! /C30ffiffiffi6p /C28ffiffiffi2p (7) sec 5p 12 ! /C30ffiffiffi6p /C27ffiffiffi2p (8) sin p 12 ! /C301 4ffiffiffi 6p /C28ffiffiffi2piCkCiCkA (9) sin 5p 12 ! /C301 4ffiffiffi 6p /C27ffiffiffi2piCkCiCkA (10) tan p 12 ! /C302/C28ffiffiffi3p (11) tan 5p 12 ! /C302/C27ffiffiffi3p : (12) These can be derived using sinp 12 ! /C30sinp 3/C28p 4 ! /C30/C28sinp 4 ! cosp 3 ! /C27sinp 3 ! cosp 4 ! /C30/C281 2ffiffiffi 2p 1 2iCkCiCkA /C2712ffiffiffi 3p 1 2ffiffiffi 2piCkCiCkA /C301 4ffiffiffi 6p /C28ffiffiffi 2piCkCiCkA : (13) Similarly, cosp 12 ! /C30cosp 3/C28p 4 ! /C30cosp 4 ! cosp3 ! /C28sinp 3 ! sinp 4 ! /C301 212ffiffiffi 2piCkCiCkA /C271 2ffiffiffi 3p /C281 2ffiffiffi 2piCkCiCkA /C301 4ffiffiffi 6p /C28ffiffiffi2piCkCiCkA : (14) Trigonometry Values Pi/15 cosp 15 ! /C301 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C276ffiffiffi 5pq /C27ffiffiffi5p /C281iCkniCko (1) cos 2p 15 ! /C301 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C286ffiffiffi 5pq /C27ffiffiffi5p /C271iCkniCko (2) cos 4p 15 ! /C301 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C276ffiffiffi 5pq /C28ffiffiffi5p /C271iCkniCko (3) cos 7p 15 ! /C301 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C286ffiffiffi 5pq /C28ffiffiffi5p /C281iCkniCko (4) cot p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C272ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C276ffiffiffi 5pqr (5) cot2p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C282ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C286ffiffiffi 5pqr (6) cot4p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C272ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C276ffiffiffi 5pqr (7) cot7p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C282ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C286ffiffiffi 5pqr (8) cscp 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C272ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C276ffiffiffi 5pqr (9)csc2p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C282ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C286ffiffiffi 5pqr (10) csc4p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C272ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C276ffiffiffi 5pqr (11) csc7p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C282ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C286ffiffiffi 5pqr (12) secp 15 ! /C30/C272/C28ffiffiffi5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C286ffiffiffi 5pq (13) sec 2p 15 ! /C30/C282/C28ffiffiffi5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C276ffiffiffi 5pq (14) sec 4p 15 ! /C30/C282/C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C286ffiffiffi 5pq (15) sec7p 15 ! /C30/C272/C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C276ffiffiffi 5pq (16) sinp 15 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C28ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C286ffiffiffi 5pqr (17) sin2p 15 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C27ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C276ffiffiffi 5pqr (18) sin4p 15 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C28ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C286ffiffiffi 5pqr (19) sin7p 15 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30/C276ffiffiffi 5pqr (20) tanp 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23/C2810ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255/C27114ffiffiffi 5pqr (21) tan2p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23/C2710ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255/C27114ffiffiffi 5pqr (22) tan4p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23/C2810ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255/C27114ffiffiffi 5pqr (23) tan7p 15 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23/C2710ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255/C27114ffiffiffi 5pqr :(24) These can be derived using the TRIGONOMETRIC ADDITION FORMULAS sinp 15 ! /C30sinp 6/C28p 10 ! /C30sinp 6 ! cosp 10 ! /C28sinp 10 ! cosp 6 ! /C3012ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 185 /C27ffiffiffi 5piCkCiCkAs /C28ffiffiffi 3p 21 4ffiffiffi 5p /C281iCkCiCkA /C301 162ffiffiffi3p /C282ffiffiffiffiffiffi15p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40 /C278ffiffiffi 5pq iCkniCko (25) and cos p 15 ! /C30cosp 6 /C28p 10 ! /C30cosp 6 ! cosp 10 ! /C27sinp 6 ! sinp 10 ! /C30ffiffiffi 3p 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 85 /C27ffiffiffi 5piCkCiCkAs /C271 214ffiffiffi 5p /C281iCkCiCkA /C30 1 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 /C276ffiffiffi 5pq /C27ffiffiffi5p /C281iCkniCko : (26) See also P ENTADECAGON Trigonometry Values Pi/16 cosp 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqr (1) cos3p 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqr (2) cos5p 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqr (3) cos7p 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqr (4) cotp 16 ! /C30/C271/C27ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C272ffiffiffi 2pq (5) cot3p 16 ! /C30/C281/C27ffiffiffi2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C282ffiffiffi 2pq (6) cot 5p 16 ! /C30/C271/C28ffiffiffi 2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C282ffiffiffi 2pq (7) cot7p 16 ! /C30/C281/C28ffiffiffi2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C272ffiffiffi 2pq (8) csc p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C274ffiffiffi 2p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2714ffiffiffi 2pqr (9)csc3p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C284ffiffiffi 2p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2814ffiffiffi 2pqr (10) csc5p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C284ffiffiffi 2p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2814ffiffiffi 2pqr (11) csc7p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C274ffiffiffi 2p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2714ffiffiffi 2pqr (12) secp 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C274ffiffiffi 2p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2714ffiffiffi 2pqr (13) sec3p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C284ffiffiffi 2p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2814ffiffiffi 2pqr (14) sec5p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C284ffiffiffi 2p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2814ffiffiffi 2pqr (15) sec7p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C274ffiffiffi 2p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20/C2714ffiffiffi 2pqr (16) sinp 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqr (17) sin3p 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqr (18) sin5p 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqr (19) sin7p 16 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqr (20) tanp 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C272ffiffiffi 2pq /C28ffiffiffi2p /C281 (21) tan 3p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C282ffiffiffi 2pq /C28ffiffiffi2p /C271 (22) tan 5p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C282ffiffiffi 2pq /C27ffiffiffi2p /C281 (23) tan 7p 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4/C272ffiffiffi 2pq /C27ffiffiffi2p /C271: (24) These can be derived from the HALF-ANGLE FORMULAS sinp 16 ! /C30sin1 2/C215p 8 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21/C28cosp 8 !vuut/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21/C2812ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqiCkniCkos /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 /C2814ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqr /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqr (25) cosp 16 ! /C30cos1 2/C215p 8 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21 /C27cosp8 !vuut/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21 /C2712ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqiCkniCkos /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 /C2714ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqr /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqr (26) tanp 16 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pp 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2ppvuut /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 /C272ffiffiffi 2pq /C28ffiffiffi2p /C281: (27) See also H EXADECAGON Trigonometry Values Pi/17 Rather surprisingly, trigonometric functions of np=17 fornan integer can be expressed in terms of sums, products, and finite ROOT EXTRACTIONS because 17 is aF ERMAT PRIME . This makes the HEPTADECAGON a CONSTRUCTIBLE , as first proved by Gauss. Although Gauss did not actually explicitly provide a construc- tion, he did derive the trigonometric formulas belowusing a series of intermediate variables from which the final expressions were then built up. Let e/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17/C27ffiffiffiffiffiffi 17pq (1) e/C31/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17/C28ffiffiffiffiffiffi 17pq (2) d/C13ffiffiffiffiffiffi 17p /C281 (3) a/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34/C276ffiffiffiffiffiffi 17p /C27ffiffiffi 2pffiffiffiffiffiffi 17p /C281iCkCiCkA e/C31/C288ffiffiffi 2p er (4) b/C132ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17/C273ffiffiffiffiffiffi 17p /C282ffiffiffi 2p e/C28ffiffiffi2p e/C31q ; (5) then sin p 17 ! /C301 8ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e/C312/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a/C27e/C31 ðÞpq :0:18375 (6) cosp 17 ! /C301 8ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C27ffiffiffiffiffiffi 17p /C27ffiffiffi 2p a/C27e/C31 ðÞq :0:98297 (7)sin2p 17 ! /C301 16ffiffiffi 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4e/C312/C282ffiffiffi 2p de/C31/C278ffiffiffi2p e/C28ffiffiffi2p d/C272e/C31iCkCiCkA ar :0:36124 (8) cos 2p 17 ! /C301 16d/C27ffiffiffi 2p a/C27e/C31 ðÞhi :0:93247 (9) sin4p 17 ! /C301 128ffiffiffi 2p d/C272a/C27e/C31 ðÞhi /C294e/C312/C282ffiffiffi2p de/C31/C278ffiffiffi2p e/C28ffiffiffi2p d/C272e/C31iCkCiCkA ahi 1=2 :0:67370 (10) sin8p 17 ! /C301 16[136/C288ffiffiffiffiffiffi 17p /C278ffiffiffi 2p e/C282(ffiffiffiffiffiffi 34p /C283ffiffiffi 2p )e/C31 /C272b(d/C27ffiffiffi2p e/C31)]1=2:0:99573 (11) cos8p 17 ! /C301 16d/C27ffiffiffi 2p e/C31/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17/C273ffiffiffiffiffiffi 17p /C28ffiffiffi2p e/C31/C282ffiffiffi2p eq iCkniCko :0:09227 : (12) There are some interesting analytic formulas invol- ving the trigonometric functions of np=17:Define P(x)/C13(x/C281)(x/C282)x2/C271iCjiCk (13) g1(x)/C132/C27ffiffiffiffiffiffiffiffiffi P(x)p 1/C28x(14) g4(x)/C132/C28ffiffiffiffiffiffiffiffiffiP(x)p 1/C28x(15) f i(x)/C131 4gi(x)/C281 ½/C138 (16) a/C131 4tan/C2814; (17) where i/C301 or 4. Then f1(tan a)/C30cos2p 17 ! (18) f4(tan a)/C30cos8p 17 ! : (19) Another interesting identity is given by tan14tan/C2814iCkCiCkA /C302 cos6p 17 ! /C27cos10p 17 ! "# ; (20) where both sides are equal to C/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 21 7/C27ffiffiffiffiffiffi 17piCjiCkq /C28ffiffiffiffiffiffi17p /C281 4(21) (Wickner 1999). See also CONSTRUCTIBLE POLYGON ,FERMAT PRIME , HEPTADECAGON References Casey, J. Plane Trigonometry. Dublin: Hodges, Figgis, & Co., p. 220, 1888. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 192 /C1/194 and 229 /C1/230, 1996. Do¨rrie, H. "The Regular Heptadecagon." §37 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 177 /C1/184, 1965. Ore, Ø.Number Theory and Its History. New York: Dover, 1988. Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 348, 1994. Wickner, J. "Solution to Problem 1562: A Tangent and Cosine Identity." Math. Mag. 72, pp. 412 /C1/413, 1999. Trigonometry Values Pi/18 The exact values of cos( p=18) and sin p=18ðÞ can be given by infinite NESTED RADICALS sinp 18 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28...;pqrs where the sequence of signs /C27,/C27,/C28repeats with period 3, and cosp 18 ! /C301 16ffiffiffi 3pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C28...pqrs /C2710 B@1 CA; where the sequence of signs /C28;/C28;/C27repeats with period 3. Trigonometry Values Pi/20 cosp 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pqr (1) cos3p 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pqr (2) cos7p 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pqr (3) cos9p 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pqr (4) cotp 20 ! /C30/C271/C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pq (5) cot3p 20 ! /C30/C281/C27ffiffiffi5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 5pq (6) cot 7p 20 ! /C30/C281/C27ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 5pq (7)cot9p 20 ! /C30/C271/C27ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pq (8) cscp 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C274ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2722ffiffiffi 5pqr (9) csc3p 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C284ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2822ffiffiffi 5pqr (10) csc5p 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C284ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2822ffiffiffi 5pqr (11) csc7p 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C274ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2722ffiffiffi 5pqr (12) secp 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C274ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2722ffiffiffi 5pqr (13) sec3p 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C284ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2822ffiffiffi 5pqr (14) sec5p 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C284ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2822ffiffiffi 5pqr (15) sec7p 20 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12/C274ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50/C2722ffiffiffi 5pqr (16) sinp 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C2820ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pqr (17) sin3p 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C2820ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pqr (18) sin7p 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C2720ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C282ffiffiffi 5pqr (19) sin9p 20 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8/C2720ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10/C272ffiffiffi 5pqr (20) tanp 20 ! /C30/C271/C27ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pq (21) tan3p 20 ! /C30/C281/C27ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 5pq (22) tan7p 20 ! /C30/C281/C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C282ffiffiffi 5pq (23) tan9p 20 ! /C30/C271/C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pq : (24) These can be derived from the HALF-ANGLE FORMULAS sinp 20 ! /C30sin1 2p 10 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 121 /C28cosp 10 !vuut ¼1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C272ffiffiffi 5pqr cosp 20 ! /C30cos1 2p 10 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 121 /C27cosp 10 !vuut ¼1 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 /C272ffiffiffi 5pqr tanp 20 ! /C301 /C27ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pq : An interesting near-identity is given by 1 4cos1 10iCkCiCkA /C27cosh1 10iCkCiCkA /C272 cos1 20ffiffiffi 2piCkCiCkA cosh1 20ffiffiffi2piCkCiCkA hi :1: (25) In fact, the left-hand side is approximately equal to / 1 þ 2 :480 /C2910/C2813 /. Trigonometry Values Pi/24 cosp 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 3pqr (1) cos5p 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 3pqr (2) cos7p 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 3pqr (3) cos11 p 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 3pqr (4) cotp 24 ! /C30/C272 /C27ffiffiffi2p /C27ffiffiffi3p /C27ffiffiffi6p (5) cot 5p 24 ! /C30/C272 /C28ffiffiffi2p /C28ffiffiffi 3p /C27ffiffiffi6p (6) cot 7p 24 ! /C30/C282 /C28ffiffiffi 2p /C27ffiffiffi 3p /C27ffiffiffi6p (7) cot 11 p 24 ! /C30/C282 /C27ffiffiffi 2p /C28ffiffiffi 3p /C27ffiffiffi6p (8) csc p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2710ffiffiffi 2p /C278ffiffiffi 3p /C276ffiffiffi6pq (9)csc 5p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2810ffiffiffi 2p /C288ffiffiffi 3p /C276ffiffiffi6pq (10) csc 7p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2710ffiffiffi 2p /C288ffiffiffi 3p /C286ffiffiffi6pq (11) csc 11 p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2810ffiffiffi 2p /C278ffiffiffi 3p /C286ffiffiffi6pq (12) sec p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2810ffiffiffi 2p /C278ffiffiffi3p /C286ffiffiffi6pq (13) sec 5p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2710ffiffiffi 2p /C288ffiffiffi 3p /C286ffiffiffi6pq (14) sec 7p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2810ffiffiffi 2p /C288ffiffiffi 3p /C276ffiffiffi6pq (15) sec 11 p 24 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 /C2710ffiffiffi 2p /C278ffiffiffi 3p /C276ffiffiffi6pq (16) sin p 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 3pqr (17) sin5p 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 3pqr (18) sin7p 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 3pqr (19) sin11 p 24 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 3pqr (20) tanp 24 ! /C30/C282 /C27ffiffiffi 2p /C28ffiffiffi 3p /C27ffiffiffi6p (21) tan p 24 ! /C30/C282 /C28ffiffiffi 2p /C27ffiffiffi 3p /C27ffiffiffi6p (22) tan p 24 ! /C30/C272/C28ffiffiffi 2p /C28ffiffiffi 3p /C27ffiffiffi6p (23) tan p 24 ! /C30/C272/C27ffiffiffi2p /C27ffiffiffi3p /C27ffiffiffi6p : (24) See also I COSITETRAGON Trigonometry Values Pi/30 cosp 30 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 65/C27ffiffiffi 5piCkCiCkArs (1) cos7p 30 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C28ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 65/C27ffiffiffi 5piCkCiCkArs (2) cos11 p 30 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C27ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 65/C27ffiffiffi 5piCkCiCkArs (3) cos13 p 30 ! /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C28ffiffiffi 5p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 65/C27ffiffiffi 5piCkCiCkArs (4) cotp 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23 /C2710ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255 /C27114ffiffiffi 5pqr (5) cot7p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23 /C2810ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255 /C28114ffiffiffi 5pqr (6) cot11p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23 /C2710ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255 /C27114ffiffiffi 5pqr (7) cot13p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23 /C2810ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 255 /C28114ffiffiffi 5pqr (8) cscp 30 ! /C30/C272 /C27ffiffiffi5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C276ffiffiffi 5pq (9) csc 7 p 30 ! /C30/C282 /C27ffiffiffi5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C286ffiffiffi 5pq (10) csc 11p 30 ! /C30/C282 /C28ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C276ffiffiffi 5pq (11) csc13p 30 ! /C30/C272 /C28ffiffiffi5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C286ffiffiffi 5pq (12) sec p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 /C282ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C286ffiffiffi 5pqr (13) sec7p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 /C272ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C276ffiffiffi 5pqr (14) sec11p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 /C282ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C286ffiffiffi 5pqr (15) sec13p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 /C272ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C276ffiffiffi 5pqr (16) sinp 30 ! /C301 8/C281 /C28ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 /C286ffiffiffi 5pqiCkniCko (17)sin7p 30 ! /C301 8/C271 /C28ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 /C276ffiffiffi 5pqiCkniCko (18) sin11 p 30 ! /C301 8/C271 /C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 /C286ffiffiffi 5pqiCkniCko (19) sin13 p 30 ! /C301 8/C281 /C27ffiffiffi 5p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 /C276ffiffiffi 5pqiCkniCko (20) tanp 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C282ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C286ffiffiffi 5pqr (21) tan7p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C272ffiffiffi 5p /C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C276ffiffiffi 5pqr (22) tan11 p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C282ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 /C286ffiffiffi 5pqr (23) tan13 p 30 ! /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7/C272ffiffiffi 5p /C272ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15/C276ffiffiffi 5pqr : (24) See also TRIACONTAGON Trigonometry Values Pi/32 cosp 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqrs (1) cos3p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqrs (2) cos5p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqrs (3) cos7p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqrs (4) cos9p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffi 2pqrs (5) cos11p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqrs (6) cos13p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C28ffiffiffi 2pqrs (7) cos15 p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqrs (8) sinp 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqrs (9) sin3 p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pqrs (10) sin5 p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pqrs (11) sin7 p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqrs (12) sin9 p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqrs (13) sin11p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pqrs (14) sin13p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C28ffiffiffi 2pqrs (15) sin15 p 32 ! /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /C27ffiffiffi 2pqrs : (16) The functions cot(np=32); csc(np=32) ; sec(np=32); and tan(np=32) are roots of 8th degree polynomials, but the explicit expressions in terms of radicals are rather complicated. See also ICOSIDODECAGON Trigonometry Values * /0 By the definition of the trigonometric functions, cos 0 /C301 cot 0 /C30/C12 csc 0 /C30/C12 sec 0 /C301 sin 0 /C300 tan 0 /C300 :Trigyrate Rhombicosidodecahedron JOHNSON SOLID J75 :/ References Weisstein, E. W. "Johnson Solids." MATHEMATICA NOTEBOOK JOHNSON SOLIDS.M . Weisstein, E. W. "Johnson Solid Netlib Database." MATHE- MATICA NOTEBOOK JOHNSON SOLIDS.DAT . Trihedral Angle TRIHEDRON Trihedron A TRIPLE of three arbitrary vectors with common vertex (Altshiller-Court 1979), often called a trihedral angle since it determines three planes. The vectors are often taken to be unit vectors, and the term trihedron is frequently encountered in the consideration of the unit ORTHOGONAL VECTORS given byT,N, and B(TANGENT VECTOR ,NORMAL VECTOR , and BINORMAL VECTOR ). See also BINORMAL VECTOR ,C ENTROIDAL LINE, DIHEDRAL ANGLE ,ISOCLINAL LINE,ISOCLINAL PLANE , NORMAL VECTOR ,O RTHOCENTRIC LINE,T ANGENT VECTOR References Altshiller-Court, N. "The Trihedral Angle." Ch. 2 in Modern Pure Solid Geometry. New York: Chelsea, pp. 27 /C1/41, 1979. Trilinear Coordinates Given a TRIANGLE DABC ;the trilinear coordinates of a point Pwith respect to DABC are an ordered TRIPLE of numbers, each of which is PROPORTIONAL to the directed distance from Pto one of the side lines. Trilinear coordinates are denoted a:b:gor (a;b;g) and also are known as homogeneous coordinates or "trilinears." Trilinear coordinates were introduced by Plu¨cker in 1835. Since it is only the ratio of distances that is significant, the triplet of trilinear coordinates obtained by multiplying a given triplet by any nonzero constant describes the same point, so a:b:g/C30ma:mb:mg: (1) For simplicity, the three VERTICES A,B, and Cof a triangle are commonly written as 1 : 0 : 0 ;0:1:0 ; and 0 : 0 : 1 ;respectively. Trilinear coordinates can be normalized so that theygive the actual directed distances from Pto each of the sides. To perform the normalization, let the pointPin the above diagram have trilinear coordinates a: b:gand lie at distances a?;b?;andc?from the sides BC,AC, and AB, respectively. Then the distances a?/C30ka;b?/C30kb;andc?/C30kgcan be found by writing D a for the AREA ofDBPC ;and similarly for DbandDc:We then have D/C30Da/C27Db/C27Dc/C301 2aa?/C2712bb?/C2712cc? /C301 2(aka/C27bkb/C27ckg)/C3012k(aa/C27bb/C27cg): (2) so k/C132D aa/C27bb/C27cg; (3) where Dis the AREA ofDABC anda,b, and care the lengths of its sides (Kimberling 1998, pp. 26 /C1/27). To obtain trilinear coordinates giving the actual dis- tances, take k/C301, so we have the coordinates a?:b?:c?: (4) These normalized trilinear coordinates are known as EXACT TRILINEAR COORDINATES . The trilinear coordinates of the line ux/C27vy/C27wz/C300 (5)are u:v:w/C30abA:bdB:cdC; (6) where diis the POINT-LINE DISTANCE from VERTEX ito the LINE. The homogeneous BARYCENTRIC COORDINATES corre- sponding to trilinear coordinates a:b:gare (aa;bb;cg);and the trilinear coordinates correspond- ing to homogeneous BARYCENTRIC COORDINATES t1;t2;t3 ðÞ aret1=a:t2=b:t3=c:/ Important points a:b:gof a triangle are called TRIANGLE CENTERS , and the vector functions describ- ing the location of the points in terms of side length,angles, or both, are called TRIANGLE CENTER FUNC- TIONS f(a;b;c):Since by symmetry, triangle center functions are of the form f(a;b;c)/C30f(a;b;c):f(b;c;a):f(c;a;b); (7) it is common to call the scalar function f(a;b;c) "the" triangle center function. Note also that side lengths and angles are interconvertible through the LAW OF COSINES , so a triangle center function may be given in terms of side lengths, angles, or both. Trilinearcoordinates for some common triangle centers aresummarized in the following table, where A,B, and C are the angles at the corresponding vertices and a,b, andcare the opposite side lengths. Here, the normal- izations have been chosen to give the simplestpossible form. Point Trilinear Center Function CENTROID M /cscA;1=a/ CIRCUMCENTER O /cosA/ DELONGCHAMPS POINT/cosA/C28cosBcosC/ EQUAL DETOUR POINT/sec1 2AiCkCiCkA cos12BiCkCiCkA cos12CiCkCiCkA /C271/ FEUERBACH POINT F/1/C28cos(B/C28C)/ INCENTER I 1 ISOPERIMETRIC POINT/sec1 2AiCkCiCkA cos12BiCkCiCkA cos12CiCkCiCkA /C281/ SYMMEDIAN POINT a NINE-POINT CEN- TERN/cos(B/C28C)/ ORTHOCENTER H /cosBcosC/ vertex A /1:0:0 / vertex B /0:1:0 / vertex C /0:0:1 / To convert trilinear coordinates to a vector position for a given triangle specified by the x- and y- coordinates of its axes, pick two UNIT VECTORS along the sides. For instance, pick ˆa /C30a1 a2iC0jiC0k (8) ˆc /C30c1 c2iC0jiC0k (9) where these are the UNIT VECTORS BC and AB. Assume the TRIANGLE has been labeled such that A /C30 x1is the lower rightmost VERTEX and C /C30x2 : Then the VECTORS obtained by traveling la and lc along the sides and then inward PERPENDICULAR to them must meet x1 y1iC0jiC0k /C27lcc1 c2iC0jiC0k /C28k gc2 /C28c1iC0jiC0k /C30x2 y2iC0jiC0k /C27laa1 a2iC0jiC0k /C28kaa2 /C28a1iC0jiC0k : (10) Solving the two equations x1 /C27lcc1 /C28k gc2 /C30x2 /C27laa1 /C28kaa2 (11) y1 /C27lcc2 /C27kgc1 /C30y2 /C27laa2 /C27kaa1 ; (12) gives la /C30 ka a1c1 /C27 a2c2 ðÞ /C28 gkc2 1 /C27 c22 ðÞ /C27 c2x1 /C28 x2 ðÞ /C27 c1y2 /C28 y1 ðÞ a1c2 /C28 a2c1 (13) lc /C30 ka a21c1 /C27 a22 ðÞ /C28 gka1c1 /C27 a2c2 ðÞ /C27 a2x1 /C28 x2 ðÞ /C27 a1y2 /C28 y1 ðÞ a1c2 /C28 a2c1: (14) But ˆa and ˆc are UNIT VECTORS ,so la /C30ka a1c1 /C27 a2c2 ðÞ /C28 gk /C27 c2x1 /C28 x2 ðÞ /C27 c1y2 /C28 y1 ðÞ a1c2 /C28 a2c1 (15) lc /C30k a /C28 gka1c1 /C27 a2c2 ðÞ /C27 a2x1 /C28 x2 ðÞ /C27 a1y2 /C28 y1 ðÞ a1c2 /C28 a2c1: (16) And the VECTOR coordinates of the point a : b : g are then x /C30x1 /C27lcc1 c2iC0jiC0k /C28k gc2 /C28c1iC0jiC0k : (17) See also AREAL COORDINATES ,BARYCENTRIC COORDI- NATES ,EXACT TRILINEAR COORDINATES ,M AJOR TRI- ANGLE CENTER ,ORTHOCENTRIC COORDINATES ,POWER CURVE ,QUADRIPLANAR COORDINATES ,REGULAR TRI-ANGLE CENTER ,TRIANGLE ,TRIANGLE CENTER ,TRIAN- GLE CENTER FUNCTION ,TRILINEAR POLAR References Boyer, C. B. History of Analytic Geometry. New York: Yeshiva University, 1956. Casey, J. "The General Equation--Trilinear Co-Ordinates." Ch. 10 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 333 /C1/348, 1893. Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 67 /C1/71, 1959. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. "Some Applications of Trilinear Coordi- nates." Linear Algebra Appl. 226 /C1/228, 375 /C1/388, 1995. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163 /C1/187, 1994. Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,1/C1/295, 1998. Wong, M. K. F. Int. J. Math. Educ. Sci. Tech. 27, 293 /C1/296, 1996. Wong, M. K. F. Int. J. Math. Educ. Sci. Tech. 29, 143 /C1/145, 1998. Trilinear Line A LINE is given in TRILINEAR COORDINATES by l a /C27mb /C27ng /C300: See also LINE,TRILINEAR COORDINATES Trilinear Polar Given a TRIANGLE CENTER X /C30l : m : n; the line la /C27mb /C27ng /C300 is called the trilinear polar of X /C281 and is denoted L. See also CHASLES’S POLARS THEOREM Trillion The word trillion denotes different numbers in Amer- ican and British usage. In the American system, one trillion equals 1012. In the British, French, and German systems, one trillion equals 1018. See also BILLION ,LARGE NUMBER ,MILLION Trilogarithm A special case of the POLYLOGARITHM Lin(z) for n /C303. It is denoted Li3(z) ; or sometimes L3(z) : The notation Li3(x) for the trilogarithm is unfortunately similar to that for the LOGARITHMIC INTEGRAL Li(x) : Functional equations for the trilogarithm include Li3(z) /C27Li3(/C28z) /C301 4 Li3z2iCjiCk (1) Li3(/C28z) /C28Li3 /C28z /C281iCjiCk /C30/C2816(ln z)3 /C2816 p2 ln z (2) Li3(z) /C27Li3(1 /C28z) /C27Li31 /C28z /C281iCjiCk ¼ zð3 Þþ1 6ðln z Þ3 þ16p2 ln z /C2812 ðln zÞ2 ln ð1 /C28z Þð 3Þ Analytic values for Li3(x) include Li3(/C281) /C30/C283 4 z(3) (4) Li3(0) /C300 (5) Li31 2iCkCiCkA /C301 24/C282p2 ln 2 /C274(ln 2)3 /C2721 z(3)hi (6) Li3(1) /C30 z(3) (7) Li31 23 /C28ffiffiffi 5piCkCiCkAiCkCiCkA /C304 5 z(3) /C2723(ln f)3 /C282 15 p2 ln f (8) where z(3) is APE´ RY’S CONSTANT and f is the GOLDEN RATIO . Bailey et al. showed that 35 2z(3) /C28 p2 ln 2 /C3036 Li312iCkCiCkA /C2818 Li314iCkCiCkA /C284Li318iCkCiCkA /C27Li31 64iCkCiCkA (9) 2(ln 2)3 /C287z(3) /C30/C2824 Li31 2iCkCiCkA /C2718 Li314iCkCiCkA /C274Li318iCkCiCkA /C28Li31 64iCkCiCkA (10)10(ln 2)3 /C282p2 ln 2 /C30/C2848 Li31 2iCkCiCkA /C2754 Li314iCkCiCkA /C2712 Li318iCkCiCkA /C283Li31 64iCkCiCkA ; (11) See also DILOGARITHM ,POLYLOGARITHM References Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps. Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, pp. 154 /C1/156, 1981. Trimagic Square If replacing each number by its square or cube in a MAGIC SQUARE produces another MAGIC SQUARE , the square is said to be a trimagic square. Trimagic squares of order 32, 64, 81, and 128 are known. Tarry gave a method for constructing a trimagic square of order 128, Cazalas a method for trimagic squares of orders 64 and 81, and R. V. Heath a method for constructing an order 64 trimagic square which is different from Cazalas’s (Kraitchik 1942). Trimagic squares are also called TREBLY MAGIC SQUARES , and are 3-MULTIMAGIC SQUARES . See also BIMAGIC SQUARE ,M AGIC SQUARE ,M ULTI- MAGIC SQUARE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 212 /C1/213, 1987. Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical Recreations. New York: W. W. Norton, pp. 144 and 176 /C1/ 178, 1942. Trimean The trimean is defined to be TM /C131 4 ðH1 þ 2M þ H2 Þ; where Hi are the HINGES and M is the MEDIAN . Press et al. (1992) call this T UKEY’S TRIMEAN .I ti sa n L- ESTIMATE . See also HINGE , L-ESTIMATE ,M EAN,M EDIAN (STA- TISTICS ) References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 694, 1992. Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, pp. 46 /C1/47, 1977. Trimorphic Number A number n such that the last digits of n3 are the same as n. 49 is trimorphic since 493 /C30117649 (Wells 1986, p. 124). The first few are 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, ... (Sloane’s A033819). See also AUTOMORPHIC NUMBER ,NARCISSISTIC NUM- BER,SUPER- D NUMBER References Sloane, N. J. A. Sequences A033819 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. Trinoid A MINIMAL SURFACE discovered by L. P. M. Jorge and W. Meeks III in 1983 with ENNEPER- WEIERSTRASS PARAMETERIZATION f /C301 z3 /C28 1iCjiCk 2 (1) g /C30 z2 (2) (Dickson 1990). Explicitly, it is given by x /C30RiC0jreiu 31/C27 reiu /C27 r2e2iu ðÞ/C284ln reiu /C28 1 ðÞ 9 /C272ln1 /C27 reiu /C27 r2e2iuðÞ 9iC0k (3) y /C30/C281 9 TiC0j /C283reiu(1 /C27 reiu) r3e3iu /C28 1 /C274ffiffiffi 3p r3e3i u /C28 1 ðÞ tan /C2811 /C27 2reiu ffiffiffi3p ! r3e3i u /C28 1iC0k (4)z /C30R/C282 3 /C282 3 r3e3i u /C28 1 ðÞ"# ; (5) for 0 /C23 [0; 2p) and r /C23 [0; 4]:/ See also ENNEPER- WEIERSTRASS PARAMETERIZATION , MINIMAL SURFACE References Dickson, S. "Minimal Surfaces." Mathematica J. 1,38/C1/40, 1990. Ogawa, A. "The Trinoid Revisited." Mathematica J. 2,59/C1/ 60, 1992. Wolfram Research "Mathematica Version 2.0 Graphics Gallery." http://www.mathsource.com/cgi-bin/ msitem22?0207 /C1/155. Trinomial A POLYNOMIAL with three terms. See also BINOMIAL ,MONOMIAL ,POLYNOMIAL Trinomial Coefficient A coefficient of the TRINOMIAL TRIANGLE . The trino- mial coefficientn kiCjiCk 2;with n]0 and /C28n5k5n;is given by the coefficient of xn/C27kin the expansion of 1/C27x/C27x2ðÞn:Therefore, n /C28kiCkniCko 2/C30n kiCkniCko 2: Equivalently, the trinomial coefficients are defined by 1/C27x/C27x/C281iCjiCkn/C30Xn j/C30/C28nn jiCkniCko 2xj: (1) The trinomial coefficients satisfy m jiCkniCko 2/C30m/C281 j/C281iCkniCko 2/C27m/C281 jiCkniCko 2/C27m/C281 j/C271iCkniCko 2: (2) An alternatives definition of the trinomial coefficients is as the coefficients in ( x/C27y/C27z)n(Andrews 1990). The (usual) trinomial coefficient is also given by thenumber of permutations of nsymbols, each /C281, 0, or 1, which sum to k. For example, there seven permu- tations of three symbols which sum to 0, f/C281;0;1g; f/C281;1;0g;f0;/C281;1g;f0;0;0g;and f0;1;/C281g; f1;/C281;0g;f1;0;/C281g /,s o3 0iCjiCk 2/C307:Explicit formulas forn kiCjiCk 2are given by n kiCkniCko 2/C30Xn j/C300n! j!(j/C27m)!(n/C282j/C28m)!(3) n kiCkniCko 2/C30Xn j/C300(/C281)jn jiCkniCko 2n/C282j n/C28m/C28jiCkniCko (4) (Andrews 1990). The following table gives the firstn kiCjiCk 2trinomial coefficients for k /C300, 1, ... and n /C30k, k /C271; .... k Sloane (n, k)-trinomial coefficients 0 Sloane’s A0024261, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, ... 1 Sloane’s A0057171, 2, 6, 16, 45, 126, 357, 1016, 2907, 8350, ... 2 Sloane’s A0145311, 3, 10, 30, 90, 266, 784, 2304, ... 4 1, 5, 21, 77, 266, 882, 2850, 9042, ... 5 1, 6, 28, 112, 414, 1452, 4917, ... See also BINOMIAL COEFFICIENT ,CENTRAL TRINOMIAL COEFFICIENT ,TRINOMIAL TRIANGLE References Andrews, G. "Euler’s ‘exemplum memorabile inductionis fallacis’ and q-Trinomial Coefficients." J. Amer. Math. Soc. 3, 653 /C1/669, 1990. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 78, 1974. Hoggatt, V. E. Jr., and Bicknell, M. "Diagonal Sums of Generalized Pascal Triangles." Fib. Quart. 7, 341 /C1/358 and 393, 1969. Euler, L. "Exemplum Memorabile Inductionis Fallacis." Opera Omnia, Vol. 15. Leipzig, Germany: Teubner, p. 59, 1911. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Read- ing, MA: Addison-Wesley, p. 575, 1990. Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63,3/C1/20, 1990. Henrici, P. Applied and Computational Complex Analysis, Vol. 1. New York: Wiley, p. 42, 1974. Riordan, J. Combinatorial Identities. New York: Wiley, p. 74, 1979. Shapiro, L. W.; Getu, S.; Woan, W.-J.; and Woodson, L. C. "The Riordan Group." Disc. Appl. Math. 34, 229 /C1/239, 1991. Sloane, N. J. A. Sequences A002426/M2673, A005717/ M1612, and A014531 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Trinomial Identity x2 /C27axy /C27by2iCjiCk t2 /C27atu /C27bu2iCjiCk /C30r2 /C27ars /C27bs2 ; (1) where r /C30xt /C28byu (2) s /C30yt /C27xu /C27ayu: (3)Trinomial Triangle The NUMBER TRIANGLE obtained by starting with a row containing a single "1" and the next row contain- ing three 1s and then letting subsequent row ele- ments be computed by summing the elements above to the left, directly above, and above to the right: 1 111 12321 1367631 1 4 10 16 19 16 10 4 1 (Sloane’s A027907). The nth row can also be obtained by expanding 1 /C27x /C27x2ðÞnand taking coefficients: 1 /C27x /C27x2iCjiCk0/C301 1 /C27x /C27x2iCjiCk1/C301 /C27x /C27x2 1 /C27x /C27x2iCjiCk2/C301 /C272x /C273x2 /C272x3 /C27x4 and so on. See also CENTRAL TRINOMIAL COEFFICIENT ,PASCAL’S TRIANGLE ,TRINOMIAL COEFFICIENT References Sloane, N. J. A. Sequences A027907 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Triomino The two 3-POLYOMINOES are called triominoes, and are also known as the TROMINOES . The left triomino above is "STRAIGHT ," while the right triomino is called "right" or L-. There is also a game called triomino consisting of 55 equilateral triangles, each containing three numbers from 0 to 5 at each vertex. Every combination of tiles is in the game, although those tiles with three different values are allowed to be arranged only in clockwise-increasing order. See also L-POLYOMINO ,POLYOMINO ,STRAIGHT POLY- OMINO References Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer- ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 124 /C1/140, 1959. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 80 /C1/81, 1975. Lei, A. "Tromino." http://www.cs.ust.hk/~philipl/omino/tro- mino.html485 Triple A group of three elements, also called a TRIAD . See also AMICABLE TRIPLE ,MONAD ,PAIR,PYTHAGOR- EAN TRIPLE ,Q UADRUPLET ,Q UINTUPLET ,T ETRAD , TRIAD,TWINS Triple Jacobi Product JACOBI TRIPLE PRODUCT Triple Point A point where a curve intersects itself along three arcs. The above plot shows the triple point at the ORIGIN of the TRIFOLIUM x2 /C27y2ðÞ2/C273x2y /C28y3 /C300 :/ See also DOUBLE POINT ,QUADRUPLE POINT References Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 57 /C1/58, 1978. Triple Product SCALAR TRIPLE PRODUCT ,VECTOR TRIPLE PRODUCT Triple Scalar Product SCALAR TRIPLE PRODUCT Triple Torus A SPHERE with three HANDLES , i.e., a genus-3 TORUS . See also DOUBLE TORUS ,HANDLE ,TORUS Triple Vector Product VECTOR TRIPLE PRODUCT Triple Yahtzee YAHTZEE Triple-Free Set A SET of POSITIVE integers is called weakly triple-free if, for any integer x, the SET fx; 2x; 3xg¢S : It is called strongly triple-free if x /C23 S IMPLIES 2x QS and 3x QS (i.e., the set is both DOUBLE-FREE and triple-free). For example, the subsets of f1; 2; 3 g which are weakly triple-free are ¥;f1 g;f1 ; 2 g;f2 g;f2 ; 3 g; and f3g; while f1; 2; 3g and f1; 3g are not. Of these weakly triple-free sets, ¥;f1g;f2 g;f2; 3 g; and f3g are also strongly triple-free. The number of weakly triple-free subsets of / f1; 2; ... ;ng/ for n /C301, 2, ... are 2, 4, 6, 12, 24, 36, 72, 144, 240, 480, ... (Sloane’s A050293). The number of strongly triple-free subsets for n /C301, 2, ... are 2, 3, 5, 8, 16, 24, 48, 76, 132, ... (Sloane’s A050295). Define p(n) /C30max f½S½ : S ƒ(1; 2; ...; ng is weakly triple -freeg q(n) /C30max f½S½ : S ƒ(1; 2; ... ; ng is strongly triple -free g; where ½S½ denotes the CARDINAL NUMBER of (number of members in) S. Then for n /C301, 2, ..., p(n) is given by 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, ... (Sloane’s A050294), and q(n)by1,1,2,2,3,4,5,5,6,6,7,7,8, 8, 9, ... (Sloane’s A050296). Asymptotic formulas are given by lim n0/C12p(n) n]4 5 and lim n0/C12q(n) n/C300 :6134752692 ... (Finch). See also A-SEQUENCE ,DOUBLE- FREE SET,SUM-FREE SET References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/triple/triple.html. Sloane, N. J. A. Sequences A050293, A050294, A050295, and A050296 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Trip-Let A 3-dimensional solid which is shaped in such a way that its projections along three mutually perpendicu- lar axes are three different letters of the alphabet. Hofstadter (1989) has constructed such a solid for the letters G, E, and B. See also CORK PLUG,ROTOR References Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, cover and pp. xiv, 1, and 273, 1989. Triplet TRIPLE Triplicate-Ratio Circle LEMOINE CIRCLE Triquetra This entry contributed by DANA MACKENZIE A "triquetra" is a figure consisting of three circular arcs of equal radius, and has seen extensive use in heraldry (i.e., coats of arms), specifically in the case of the so-called BORROMEAN RINGS . The term "Triquetra theorem" was coined by Mackenzie (1992) to describe the geometric theorem that if three circles are concurrent at a single point, then the other three intersection points lie on a circle of the same radius as the first three. This version was first proved in 1916. Mackenzie (1992) generalized this theorem to the case where the three circles do not coincide. In this case, they form six intersection points, and if you partition the points into any two groups of three and look at the CIRCUMRADII of the points in those groups, there is a nice formula relating them to the radii of the triquetra circles. This formula has some pretty geometric consequences (or "porisms"). Ultimately, the triquetra theorem turns out to be closely related to PONCELET’S PORISM . See also BORROMEAN RINGS,CIRCLE- CIRCLE INTER- SECTION ,CIRCULAR TRIANGLE ,H ARUKI’S THEOREM , PONCELET’S PORISM ,R EULEAUX TRIANGLE ,V ENN DIAGRAMReferences Mackenzie, D. "Triquetras and Porisms." College Math. J. pp. 118 /C1/131. March 1992. Trirectangular Tetrahedron A TETRAHEDRON having a TRIHEDRON all of the face angles of which are right angles. The face opposite the vertex of the right angles is called the base. If the edge lengths bounding the trihedral angle are a, b, and c, then the side lengths of the base are given byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2p ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 /C27c2p ; andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 /C27c2p ; and so has SEMI- PERIMETER s /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2 /C27c2p /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 /C27c2p :iCkC (1) The VOLUME of the trirectangular tetrahedron is V /C301 6abc : (2) Using HERON’S FORMULA , the SURFACE AREA is there- fore S /C301 2ab /C27ac /C27bc /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2b2 /C27a2c2 /C27b2c2p iCkCiCkA : (3) Let DXYZbe the AREA of the triangle with vertices X, Y, and Z. The remarkable DE GUA’S THEOREM D2 ABC /C30D2OAB /C27D2OAC /C27D2OAC : (4) then follows from the identity ss/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C27b2piCkCiCkA s /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2 /C27c2piCkCiCkA s /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 /C27c2piCkCiCkA /C301 4a2b2 /C27a2c2 /C27b2c2iCjiCk ; (5) with sdefined by (1). See also DE GUA’S THEOREM ,TRIHEDRON References Altshiller-Court, N. "The Trirectangular Tetrahedron." §4.6a inModern Pure Solid Geometry. New York: Chelsea, pp. 91 /C1/94, 1979. Trisected Perimeter Point A triangle center which has a TRIANGLE CENTER FUNCTION a/C30bc(v/C28c/C27a)(v/C28a/C27b); where vis the unique REAL ROOT of 2x3/C283(a/C27b/C27c)x2/C27a2/C27b2/C27c2/C278bc/C278ca/C278abiCjiCk x /C28b2c/C27c2a/C27a2b/C275bc2/C275ca2/C275ab2/C279abciCjiCk /C300: References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163/C1/187, 1994. Trisection Angle trisection is the division of an arbitrary ANGLE into three equal ANGLES . It was one of the three GEOMETRIC PROBLEMS OF ANTIQUITY for which solu- tions using only COMPASS and STRAIGHTEDGE were sought. The problem was algebraically proved im- possible by Wantzel (1836). Although trisection is not possible for a general ANGLE using a Greek construction, there are some specific angles, such as p=2 and pradians (90 8and 1808, respectively), which can be trisected. Further- more, some ANGLES are geometrically trisectable, but cannot be constructed in the first place, such as 3 p=7 (Honsberger 1991). In addition, trisection of anarbitrary angle canbe accomplished using a marked RULER (a N EUSIS CONSTRUCTION ) as illustrated above(Courant and Robbins 1996). An approximate trisection is described by Steinhaus(Wazewski 1945, Steinhaus 1983, p. 7). Given anangle u/C30/C218AOB ;draw the bisector u=2/C30/C218AOC ; with OC/C30OA/C30OB/C301;then divide BCsuch that BD/C302CD:From the SAS THEOREM DCOB ;the length sis given by the formula b2/C30a2/C27c2/C282accosB (1) with s/C30b,a/C30c/C301;B/C30u=2; s/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C282 cos1 2uiCkCiCkAr /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28cos12uiCkCiCkA 2vuut /C302 sin1 4uiCkCiCkA ; (2) andLis then L/C302 3s/C3043sin14uiCkCiCkA : (3) The angle 8can then be computed from the formula 8/C30sin/C281 asinBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2/C27c2/C282accosBp ! (4) to obtain 8/C30sin/C281sin1 2uiCkCiCkA ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C282 cos1 2uiCkCiCkAr2 6643 775 /C30sin /C2812 sin1 4uiCkCiCkA cos14uiCkCiCkA 2 sin14uiCkCiCkA2 435 /C30sin /C281cos1 4uiCkCiCkAhi : (5) /fis then given by the formula for an SAS triangle DBOD f /C30sin/C281 L sin 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27 L2 /C28 2L cos 8p ! /C30sin/C2812 3sin12 uiCkCiCkA 1 /C288 9sin214 uiCkCiCkA2 435 /C30sin /C2816 sin1 2 uiCkCiCkA 5 /C27 4 cos1 2 uiCkCiCkA2 435: (6) The Maclaurin series is then f /C30 1 3 u /C277 648 u3 /C2719 31104 u5 /C27/C1/C1/C1:13 u (7) to a very good approximation. An ANGLE can also be divided into three (or any WHOLE NUMBER ) of equal parts using the QUADRATRIX OF HIPPIAS or TRISECTRIX . See also ANGLE BISECTOR ,M ACLAURIN TRISECTRIX , QUADRATRIX OF HIPPIAS ,TRISECTRIX References Bogomolny, A. "Angle Trisection." http://www.cut-the-knot.- com/pythagoras/archi.html. Bold, B. "The Problem of Trisecting an Angle." Ch. 5 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 33 /C1/37, 1982. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190 /C1/191, 1996. Courant, R. and Robbins, H. "Trisecting the Angle." §3.3.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 137 /C1/138, 1996. Coxeter, H. S.M. "Angle Trisection." §2.2 in Introduction to Geometry, 2nd ed. New York: Wiley, p. 28, 1969. Dixon, R. Mathographics. New York: Dover, pp. 50 /C1/51, 1991. Do¨rrie, H. "Trisection of an Angle." §36 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 172 /C1/177, 1965. Dudley, U. The Trisectors. Washington, DC: Math. Assoc. Amer., 1994. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 25 /C1/26, 1991. Klein, F. "The Delian Problem and the Trisection of the Angle." Ch. 2 in "Famous Problems of Elementary Geo- metry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 13 /C1/15, 1980. Ogilvy, C. S. "Solution to Problem E 1153." Amer. Math. Monthly 62, 584, 1955. Ogilvy, C. S. "Angle Trisection." Excursions in Geometry. New York: Dover, pp. 135 /C1/141, 1990. Scudder, H. T. "How to Trisect and Angle with a Carpenter’s Square." Amer. Math. Monthly 35, 250 /C1/251, 1928. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wantzel, M. L. "Recherches sur les moyens de reconnaı ˆtre si un Proble `me de Ge´ome´trie peut se re´soudre avec la re`gle et le compas." J. Math. pures appliq. 1, 366 /C1/372, 1836. Wazewski, T. Ann. Soc. Polonaise Math. 18, 164, 1945. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 25, 1991.Trisectrix A curve which can be used to trisect an angle. Although an arbitrary angle cannot be trisected using only COMPASS and STRAIGHTEDGE (i.e., according to the strict rules of Greek GEOMETRIC CONSTRUCTION ), it can be trisected using certain curves (which are assumed to have been constructed using some other means). See also CATALAN’S TRISECTRIX ,LIMAC ¸ ON,M ACLAUR- IN TRISECTRIX ,TRISECTION ,TSCHIRNHAUSEN CUBIC Trisectrix of Catalan TSCHIRNHAUSEN CUBIC Trisectrix of Maclaurin MACLAURIN TRISECTRIX Triskaidecagon TRIDECAGON Triskaidekaphobia The number 13 is traditionally associated with bad luck. This superstition leads some people to fear or avoid anything involving this number, a condition known as triskaidekaphobia. Triskaidekaphobia leads to interesting practices such as the numbering of floors as 1, 2, ..., 11, 12, 14, 15, ..., omitting the number 13, in many high-rise hotels. See also 13,BAKER’S DOZEN Tristan Edwards Projection A CYLINDRICAL EQUAL-AREA PROJECTION which uses a standard parallel of fs/C3037:383/C14:/ See also BALTHASART PROJECTION ,BEHRMANN CY- LINDRICAL EQUAL- AREA PROJECTION ,C YLINDRICAL EQUAL- AREA PROJECTION ,GALL ORTHOGRAPHIC PRO- JECTION ,LAMBERT AZIMUTHAL EQUAL- AREA PROJEC- TION ,PETERS PROJECTION Tritangent The tritangent of a CUBIC SURFACE is a PLANE which intersects the surface in three mutually intersecting lines. Each intersection of two lines is then a tangent point of the surface. See also CUBIC SURFACE References Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni- kl.de/~wwwagag/E/Galerie.html. Tritangent Triangle EXCENTRAL TRIANGLE Trivalent Graph CUBIC GRAPH Trivalent Tree BINARY TREE Trivial Related to or being the mathematically most simple case. More generally, the word "trivial" is used to describe any result which requires little or no effort to derive or prove. The word originates from the Latin TRIVIUM , which was the lower division of the seven liberal arts in medieval universities (cf. QUADRIVIUM ). According to the Nobel Prize-winning physicist Ri- chard Feynman (Feynman 1997), mathematicians designate any THEOREM as "trivial" once a proof has been obtained–no matter how difficult the theorem was to prove in the first place. There are therefore exactly two types of true mathematical propositions: trivial ones, and those which have not yet been proven. The opposite of a trivial theorem is a "DEEP THEO- REM." See also DEEP THEOREM ,D EGENERACY ,FRIVOLOUS THEOREM OF ARITHMETIC ,PROOF ,THEOREM ,TRIVIUM References Feynman, R. P. and Leighton, R. "A Different Set of Tools." In ‘Surely You’re Joking, Mr. Feynman!’: Adventures of a Curious Character. New York: W. W. Norton, pp. 69 /C1/72, 1997. Trivial Basis Trivial Group The trivial group is the unique GROUP containing exactly one element. That is, it is G /C30fe g; where e is the IDENTITY ELEMENT (so that ee /C30e). See also CYCLIC GROUP ,F INITE GROUP ,G ROUP , IDENTITY ELEMENT Trivialization Over a small NEIGHBORHOOD U of a MANIFOLD ,a VECTOR BUNDLE is spanned by the local sections defined on U. For example, in a COORDINATE CHART U with coordinates x1 ; ... ; xn ðÞ ; every smooth VEC- TOR FIELD can be written as a sum ai fi @=@xi where fi are smooth functions. The n vector fields @=@xispan the space of vector fields, considered as a MODULE over the RING of smooth real-valued functions. On this COORDINATE CHART U, the tangent bundle can be written U /C29Rn : This is a trivialization of the tangent bundle. In general, a vector bundle of RANK r is spanned LOCALLY by r independent SECTIONS . Every point has a NEIGHBORHOOD U and r sections defined on U, such that over every point in U the fibers are spanned by those r sections. Similarly, for a FIBER BUNDLE , near every point p /C23 M ; there is a neighborhood U such that the bundle over U is U /C29F ; where F is the fiber. A bundle is a set of trivializations that cover the base manifold. The trivializations are put together to form a bundle with its TRANSITION FUNCTIONS . See also BUNDLE ,FIBER BUNDLE ,MANIFOLD ,TRANSI- TION FUNCTION ,VECTOR BUNDLE Trivium A word derived from the Latin roots tri- (three) and via (ways, roads), therefore a crossing of three roads. In medieval universities, the trivium consisted of the three subjects in the lower division of the seven liberal arts: grammar, rhetoric, and logic. The word TRIVIAL derives from the fact that the trivium contained the least complicated studies. See also QUADRIVIUM ,TRIVIAL Trochoid The curve described by a point at a distance b from the center of a rolling CIRCLE of RADIUS a. x /C30a f /C28b sin f y /C30a /C28b cos f: If b Ba, the curve is a CURTATE CYCLOID .Ifb /C30a, the curve is a CYCLOID .Ifb /C21a, the curve is a PROLATE CYCLOID . See also CURTATE CYCLOID ,CYCLOID ,EPITROCHOID , HYPOTROCHOID ,PROLATE CYCLOID References Hall, L. "Trochoids, Roses, and Thorns--Beyond the Spiro- graph." College Math. J. 23,20/C1/35, 1992. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 46 /C1/50, 1991. Yates, R. C. "Trochoids." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 233 /C1/236, 1952. Tromino TRIOMINO Trott’s Constant The constant /x ¼ 0 :010841015122311136151129... / whose decimal digits are equal to the constant’s own CONTINUED FRACTION [0, 1, 0, 8, 4, 1, 0, 1, 5, ...]. This constant was discovered by M. Trott of Wolfram Research in 1999. It appears to be unique, and all attempts to find other such numbers have failed. See also CONTINUED FRACTION True A statement which is rigorously known to be correct. A statement which is not true is called FALSE , although certain statements can be proved to be rigorously UNDECIDABLE within the confines of a given set of assumptions and definitions. Regular two-valued LOGIC allows statements to be only true or FALSE , but FUZZY LOGIC treats "truth" as a continuum which can have any value between 0 and 1. The symbol Y is sometimes used to denote "true,"although "T" is more commonly used in TRUTH TABLES . See also ALETHIC ,BOOLEANS ,FALSE ,FUZZY LOGIC , LOGIC ,TRUTH TABLE ,UNDECIDABLE Truncatable Prime Call a number ncontaining no zeros right truncata- ble if nand all numbers obtained by successively removing the rightmost DIGIT are PRIME . There are 83 right truncatable primes in base 10. The first few are 2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239,293, 311, 313, 317, 373, 379, 593, 599, ... (Sloane’s A024770), the largest being 73,939,133 (Angell and Godwin 1977). The numbers of left prime strings lessthan 10, 10 2,1 03, ... are 4, 9, 14, 16, 15, 12, 8, and 5 (Sloane’s A050986; Rivera puzzle 70). If zeros are permitted, the sequence of right trunca- table primes are 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67,73, 83, 97, 103, 107, 113, 137, 167, 173, 197, 223, 283, 307, ... (Sloane’s A033664). Similarly, call a number nleft truncatable if nand all numbers obtained by successively removing the left- most DIGIT are PRIME . There are 4260 right prime strings in base 10 when the digit zero is not allowed (otherwise, if zeros are permitted, the sequence is infinite). The first few are 2, 3, 5, 7, 13, 17, 23,37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, ... (Sloane’s A024785), with the largest being 357,686,312,646,216,567,629,137 (Angell and Godwin1977, Baillie 1995). The numbers of right prime strings less than 10, 10 2,1 03, ... are 4, 11, 39, 99, 192, 326, 429, ... (Sloane’s A050987; Rivera puzzle 70). J. Shallit has shown that in base 10, there is a finite, minimal list of primes that do not have any other primes as substrings (where digits do notneed to be consecutive). This result is a special case of a muchmore general theorem, whose proof is unfortunately nonconstructive. Call an n-digit prime p n(with n]2) is a restricted left truncatable prime if 1. If the leftmost digit of piis deleted, a prime number pi/C281is obtained for 2 5i5n;and 2. No prime with n/C271 digits can have its leftmost digit removed to produce pn:/ Kahan and Weintraub (1998) dub such primes "Henry VIII primes." Restricted left truncatable primes pnare therefore a subset of left truncatable primes for which there are no left truncatable primes of length n/C271 having the same nlast digits as pn: There are a total of 1440 such primes, and the first few are 773, 3373, 3947, 4643, 5113, 6397, 6967, 7937, ... (Sloane’s A055522), the largest being357686312646216567629137 (Kahan and Weintraub 1998). See also PRIME ARRAY ,PRIME NUMBER References Angell, I. O. and Godwin, H. J. "On Truncatable Primes." Math. Comput. 31, 265 /C1/267, 1977. Baillie, R. "Largest Left-Truncatable Prime." sci.math.- num-analysis posting, Aug. 7, 1995. De Geest, P. "List of the 4260 Left-Truncatable Primes (without the Zero Digit)." http://www.ping.be/~ping6758/ truncat.htm. Kahan, S. and Weintraub, S. "Left Truncatable Primes." J. Recr. Math. 29, 254 /C1/264, 1998. Rivera, C. "Problems & Puzzles: Puzzle Prime Strings.-002." http://www.primepuzzles.net/puzzles/puzz_002.htm. Rivera, C. "Problems & Puzzles: Puzzle Primes Double Tree (A Puzzle Suggested by Paul Leyland).-070." http:// www.primepuzzles.net/puzzles/puzz_070.htm. Schroeppel, R. Item 33 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 14, Feb. 1972. Sloane, N. J. A. Sequences A024770, A024785, A032437, A033664, A050986, A050987, and A055522 in "An On- Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonli- ne.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Weisstein, E. W. "Left Prime Strings." MATHEMATICA NOTE- BOOK LEFTPRIME STRINGS.TXT . Weisstein, E. W. "Right Prime Strings." MATHEMATICA NOTEBOOK RIGHT PRIME STRINGS.TXT . Truncate To truncate a REAL NUMBER is to discard its non- integer part. Truncation of a (positive) number x therefore corresponds to taking the FLOOR FUNCTION xbc:/ See also CEILING FUNCTION ,FLOOR FUNCTION ,NINT, ROUND ,TRUNCATION Truncated Cone CONICAL FRUSTUM Truncated Cube The 14-faced ARCHIMEDEAN SOLID A9with faces 8f3 g/C276f8 g: It is also UNIFORM POLYHEDRON U9and Wenninger model W8 : It has SCHLA ¨ FLI SYMBOL t{4, 3}/ /C272668511278169369879040000 S12and WYTHOFF SYMBOL 23½4:/ The DUAL POLYHEDRON of the truncated cube is the TRIAKIS OCTAHEDRON . The INRADIUS r of the dual, MIDRADIUS r of the solid and dual, and CIRCUMRADIUS R of the solid for a /C301 are r /C301 175 /C272ffiffiffi 2piCkCiCkAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C274ffiffiffi 2pq :1:63828 r /C301 22 /C27ffiffiffi 2piCkCiCkA :1 :70711 R /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 /C274ffiffiffi 2pq :1:77882 : The distances from the center of the solid to the centroids of the triangular and octagonal faces are r3 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1317 /C2712ffiffiffi 2piCkCiCkAr (1) r8/C301 21/C27ffiffiffi 2piCkCiCkA : (2) The SURFACE AREA and VOLUME are S/C3026/C276ffiffiffi 2p /C272ffiffiffi 3p iCkCiCkA (3) V/C301 321/C2714ffiffiffi 2piCkCiCkA : (4) See also ARCHIMEDEAN SOLID ,ICOSITETRAHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 138, 1987. Cundy, H. and Rollett, A. "Truncated Cube. 3.82."§3.7.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 103, 1989. Wenninger, M. J. "The Truncated Hexahedron (Cube)." Model 8 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 22, 1989. Truncated Cube-Small Triakis Octahedron Compound The POLYHEDRON COMPOUND of the TRUNCATED CUBE and its dual, the SMALL TRIAKIS OCTAHEDRON . The compound can be constructed from a TRUNCATED CUBE of unit edge length by midpoint CUMULATION with heights h3 /C301 6ffiffiffi 3p 3 /C282ffiffiffi 2piCkCiCkA (1) h8/C301 21/C27ffiffiffi 2piCkCiCkA : (2) See also CUMULATION ,P OLYHEDRON COMPOUND , SMALL TRIAKIS OCTAHEDRON ,TRUNCATED CUBE Truncated Cuboctahedron GREAT RHOMBICUBOCTAHEDRON (ARCHIMEDEAN ) Truncated Cylinder CYLINDRICAL WEDGE Truncated Dodecadodecahedron The UNIFORM POLYHEDRON U59;also called the QUASI- TRUNCATED DODECAHEDRON , whose DUAL POLYHE- DRON is the MEDIAL DISDYAKIS TRIACONTAHEDRON .I t has S CHLA ¨FLI SYMBOL t’f5 2 5gand W YTHOFF SYMBOL 25 3½5:Its faces are 12 f10g/C2730f4g/C2712f10 3g:Its CIR- CUMRADIUS fora/C301i s R/C301 2ffiffiffiffiffiffi 11p :References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 152 /C1/153, 1989. Truncated Dodecahedron The 32-faced A RCHIMEDEAN SOLID A10with faces 20f3g/C2712f10g:It is also UNIFORM POLYHEDRON U26 and Wenninger model W10:It has S CHLA ¨FLI SYMBOL t/f5;3gand W YTHOFF SYMBOL 23½5:/ The DUAL POLYHEDRON is the TRIAKIS ICOSAHEDRON . To construct the truncated dodecahedron by TRUNCA- TION , note that we want the INRADIUS r10of the truncated pentagon to correspond with that of the original pentagon, r5 ; of unit side length s5 /C301: This means that the side lengths s10 of the decagonal faces in the truncated dodecahedron satisfy 1 2 s5 cotp 5 ! /C3012 s10 cotp 10 ! ; (1) giving s10 /C301 5ffiffiffi 5p s5 /C301 5ffiffiffi 5p : (2) The length of the corner which is chopped off is therefore given by l /C301 2 /C2812 s10 /C301 105 /C28ffiffiffi 5piCkCiCkA : (3) The INRADIUS r of the dual, MIDRADIUS pi of the solid and dual, and CIRCUMRADIUS R of the solid for a /C301 are r /C305 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 6141 /C2718ffiffiffi 5piCkCiCkAr :2:88526 (4) r /C301 45 /C273ffiffiffi 5piCkCiCkA :2:92705 (5) R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 74 /C2730ffiffiffi 5pq :2:96945 : (6) The distances from the center of the solid to the centroids of the triangular and decagonal faces are given by r3 /C301 12ffiffiffi 3p 9 /C275ffiffiffi5piCkCiCkA (7) r 10 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1225/C2711ffiffiffi 5piCkCiCkAr : (8) The SURFACE AREA and VOLUME are S/C305ffiffiffi 3p /C276ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5/C272ffiffiffi 5pqiCkniCko (9)V/C305 1299/C2747ffiffiffi 5piCkCiCkA : (10) See also ARCHIMEDEAN SOLID,H EXECONTAHEDRON , TRIAKIS ICOSAHEDRON ,TRUNCATED DODECAHEDRON- TRIAKIS ICOSAHEDRON COMPOUND References Cundy, H. and Rollett, A. "Truncated Dodecahedron. 3.102." §3.7.9 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 109, 1989. Wenninger, M. J. "The Truncated Dodecahedron." Model 10 inPolyhedron Models. Cambridge, England: Cambridge University Press, p. 24, 1989. Truncated Dodecahedron-Triakis Icosahedron Compound The POLYHEDRON COMPOUND of the TRUNCATED DO- DECAHEDRON and its dual, the TRIAKIS ICOSAHEDRON . The compound can be constructed from a TRUNCATED DODECAHEDRON of unit edge length by midpoint CUMULATION with heights h3/C301 372ffiffiffi 3p 1/C275ffiffiffi5piCkCiCkA (1) h 10/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 125/C27ffiffiffi 5piCkCiCkAr : (2) The resulting solid has edge lengths s1/C301 62ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 6397/C27ffiffiffi 5piCkCiCkAr (3) s2/C301 2(4) s3/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 125/C27ffiffiffi 5piCkCiCkAr (5) s4/C301 45/C27ffiffiffi 5piCkCiCkA ; (6) CIRCUMRADIUS R/C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1237/C2715ffiffiffi 5piCkCiCkAr (7) SURFACE AREA given by a root of a 32nd order polynomial with large integer coefficients, and VO- LUME V /C305 148815997 /C277693ffiffiffi 5p iCkCiCkA : (8) See also POLYHEDRON COMPOUND ,TRIAKIS ICOSAHE- DRON ,TRUNCATED DODECAHEDRON Truncated Exponential Function EXPONENTIAL SUM FUNCTION Truncated Great Dodecahedron The UNIFORM POLYHEDRON U37whose DUAL POLYHE- DRON is the SMALL STELLAPENTAKIS DODECAHEDRON . It has SCHLA ¨ FLI SYMBOL tf5 ;5 2 g: It has WYTHOFF SYMBOL 252 5: Its faces are 12 f52 g/C2712 f10g: Its CIR- CUMRADIUS for a /C301is R /C3014ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34/C2710ffiffiffi 5pq : See also GREAT ICOSAHEDRON References Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 115, 1971. Truncated Great Icosahedron GREAT TRUNCATED ICOSAHEDRON Truncated Hexahedron TRUNCATED CUBETruncated Icosahedron The 32-faced A RCHIMEDEAN SOLID A11corresponding to the facial arrangement 20 f6g/C2712f5g:It is the shape used in the construction of SOCCER BALLS , and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonatorsin the Fat Man atomic bomb (Rhodes 1996, p. 195). The truncated icosahedron has 60 vertices, and is also theC 60structure of pure carbon known as buckyballs (a.k.a. fullerenes ). The truncated icosahedron is UNIFORM POLYHEDRON U25and Wenninger model W9:It has S CHLA ¨FLI SYMBOL t/f3;5gand W YTHOFF SYMBOL 25½3:/ The DUAL POLYHEDRON of the truncated icosahedron is the PENTAKIS DODECAHEDRON . The INRADIUS rof the dual, MIDRADIUS rof the solid and dual, and CIRCUMRADIUS Rof the solid for a/C301 are r /C309 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10917 /C276ffiffiffi 5piCkCiCkAr :2:37713 r /C303 41 /C27ffiffiffi 5piCkCiCkA :2 :42705 R /C301 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 58 /C2718ffiffiffi 5pq :2:47802 : The distances from the center of the solid to the centroids of the pentagonal and hexagonal faces are given by r5 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10125 /C2741ffiffiffi 5piCkCiCkAr (1) r6 /C301 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 327 /C273ffiffiffi 5piCkCiCkAr : (2) The SURFACE AREA and VOLUME are S /C30310ffiffiffi 3p /C27ffiffiffi5pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 /C272ffiffiffi 5pqiCkniCko (3) V /C301 4125 /C2743ffiffiffi 5piCkCiCkA : (4) See also ARCHIMEDEAN SOLID,HEXECONTAHEDRON References Aldersey-Williams, H. The Most Beautiful Molecule. New York: Wiley, 1997. Chung, F. and Sternberg, S. "Mathematics and the Bucky- ball." Amer. Sci. 81,56/C1/71, 1993. Cundy, H. and Rollett, A. "Truncated Icosahedron. 5.62." §3.7.10 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 110, 1989. Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 101, 1998. Rhodes, R. Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books, 1996. Trott, M. "Constructing a Buckyball with Mathematica ." http://library.wolfram.com/demos/v4/Buckyball.nb. Wenninger, M. J. "The Truncated Icosahedron." Model 9 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 23, 1989. Truncated Icosahedron-Pentakis Dodecahedron Compound The POLYHEDRON COMPOUND of the TRUNCATED ICO- SAHEDRON and its dual, the PENTAKIS DODECAHE-DRON . The compound can be constructed from a TRUNCATED ICOSAHEDRON of unit edge length by midpoint CUMULATION with heights h5 /C301 38ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 10305 /C27131ffiffiffi 5piCkCiCkAr (1) h6 /C301 4ffiffiffi 3pffiffiffi5p /C283iCkCiCkA : (2) The resulting solid has edge lengths s 1 /C301 2 (3) s2 /C303 767 /C275ffiffiffi 5piCkCiCkA (4) s3 /C301 41 /C27ffiffiffi 5piCkCiCkA (5) s4 /C301 2ffiffiffi 3p (6) s5 /C303 4ffiffiffi 5p /C281iCkCiCkA ; (7) CIRCUMRADIUS R /C303 2ffiffiffi 3p ; (8) SURFACE AREA S given by the fourth largest positive root of 5141016030764996667610951639493717193603515625 /C289291774385004510118161779667281494140625000 S2 /C2763419261142631991476189330253320312540000 S4 /C282162618355523996143839802656250000000 S6 /C27406990705888262016944967600000000 S8 /C2843785979422682649316768000000 S10 /C272668511278169369879040000 S12 /C2885420833678869299200 S14 /C271113034787454976 S16(9) and VOLUME V/C305 1521477/C27162ffiffiffi 5piCkCiCkA : (10) See also CUMULATION ,P OLYHEDRON COMPOUND , PENTAKIS DODECAHEDRON ,T RUNCATED ICOSAHE- DRON Truncated Icosidodecahedron GREAT RHOMBICOSIDODECAHEDRON (ARCHIMEDEAN ) Truncated Octahedral Number AFIGURATE NUMBER which is constructed as an OCTAHEDRAL NUMBER with a SQUARE PYRAMID re- moved from each of the six VERTICES , TOn ¼ O3n/C282 /C286Pn/C281 ¼ 16n3 /C2833n2 þ 24n /C286 ; where /On/ is an OCTAHEDRAL NUMBER and /Pn/ is a SQUARE PYRAMIDAL NUMBER . The first few are 1, 38, 201, 586, ... (Sloane’s A005910). The GENERATING FUNCTION for the truncated octahedral numbers is xð6x3þ55x2þ34xþ1 ðx/C281Þ4¼xþ38x2þ201x3þ... See also OCTAHEDRAL NUMBER ,SQUARE PYRAMIDAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 52, 1996. Sloane, N. J. A. Sequences A005910/M5266 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Truncated Octahedron The 14-faced A RCHIMEDEAN SOLID A12;also known as the MECON , with faces 8 f6g/C276f4g:It is also UNIFORM POLYHEDRON U8and Wenninger model W7:It has SCHLA ¨FLI SYMBOL t/f3;4gand W YTHOFF SYMBOL 24½3:/The DUAL POLYHEDRON of the truncated octahedron is the TETRAKIS HEXAHEDRON . The truncated octahe- dron has the OhOCTAHEDRAL GROUP of symmetries. The form of the fluorite /CaF2 ðÞ resembles the trun- cated octahedron (Steinhaus 1983, pp. 207 /C1/208). The solid of unit edge length can be formed from an OCTAHEDRON of edge length 3 via TRUNCATION by removing six SQUARE PYRAMIDS , each with edge slant height s/C301, base a/C301 on a side, and height h. The height and base area of the SQUARE PYRAMID are then h/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2/C281 4a2csc2p n !vuut/C301 2ffiffiffi 2p a (1) Ab/C30a2: (2) The SURFACE AREA of the truncated octahedron is S/C306/C2712ffiffiffi 3p : (3) The VOLUME of the truncated octahedron is then given by the VOLUME of the OCTAHEDRON Voctahedron /C301 3ffiffiffi 2p s3/C309ffiffiffi2p a3(4) minus six times the volume of the SQUARE PYRAMID , V/C30Voctahedron /C2861 3AbhiCkCiCkA /C309ffiffiffi 2p /C28ffiffiffi2piCkCiCkA a3 /C308ffiffiffi2p a3: (5) The truncated octahedron is a SPACE-FILLING POLY- HEDRON (Steinhaus 1983, pp. 187 /C1/190 and 207). The INRADIUS rof the dual, MIDRADIUS rof the solid and dual, and CIRCUMRADIUS Rof the solid for a/C301 are r/C309 20ffiffiffiffiffiffi 10p :1:42302 (6) r/C303 2/C301:5 (7) R/C301 2ffiffiffiffiffiffi 10p :1:58114 : (8) The distances from the center of the solid to the centroids of the square and hexagonal faces are given by r4/C30ffiffiffi 2p (9) r6/C301 2ffiffiffi 6p : (10) See also ARCHIMEDEAN SOLID,ICOSITETRAHEDRON , KELVIN’S CONJECTURE ,OCTAHEDRON ,RHOMBIC DO- DECAHEDRON STELLATIONS ,SQUARE PYRAMID ,TRUN- CATION References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29 /C1/30 and 257, 1973. Cundy, H. and Rollett, A. "Truncated Octahedron. 4.62." §3.7.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 104, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Wenninger, M. J. "Truncated Octahedron." Model 7 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 21, 1989. Truncated Octahedron-Tetrakis Hexahedron Compound The POLYHEDRON COMPOUND of the TRUNCATED OCTA- HEDRON and its dual, the TETRAKIS HEXAHEDRON . The compound can be constructed from a TRUNCATED OCTAHEDRON of unit edge length by midpoint CUMU- LATION with heights h4 /C301 8ffiffiffi 2p (1) h6 /C301 4ffiffiffi 6p (2) See also CUMULATION ,P OLYHEDRON COMPOUND , TETRAKIS HEXAHEDRON ,TRUNCATED OCTAHEDRON Truncated Polyhedron A polyhedron with truncated faces, given by the SCHLA ¨ FLI SYMBOL t/fp q g:/ See also FRUSTUM ,R HOMBIC POLYHEDRON ,S NUB POLYHEDRON References Harris, J. W. and Stocker, H. "Obliquely Truncated n-Sided Prism." §4.2.5 in Handbook of Mathematics and Computa- tional Science. New York: Springer-Verlag, p. 98, 1998.Truncated Power Function The function defined by ya /C27/C13y afor y > 0 0 for y B0:iC0C See also POWER References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 22, 1993. Truncated Pyramid PYRAMIDAL FRUSTUM Truncated Square Pyramid The truncated square pyramid is a special case of a PYRAMIDAL FRUSTUM for a SQUARE PYRAMID . Let the base and top side lengths of the truncated pyramid be a and b, and let the height be h. Then the VOLUME of the solid is V /C301 3a2 /C27ab /C27b2iCjiCk h : This FORMULA was known to the Egyptians ca. 1850 BC. The Egyptians cannot have proved it without calculus, however, since Dehn showed in 1900 that no proof of this equation exists which does not rely on the concept of continuity (and therefore some form of INTEGRATION ). See also FRUSTUM ,PYRAMID ,PYRAMIDAL FRUSTUM , SQUARE PYRAMID Truncated Tetrahedral Number AFIGURATE NUMBER constructed by taking the (3n/C282)/thTETRAHEDRAL NUMBER and removing the (n/C281)/thTETRAHEDRAL NUMBER from each of the four corners, Ttetn/C13Te3n/C283/C284Ten/C281/C301 6n23n2/C2827n/C2710iCjiCk : The first few are 1, 16, 68, 180, 375, ... (Sloane’s A005906). The GENERATING FUNCTION for the trun- cated tetrahedral numbers is x(10x2 /C27 12x /C27 1) (x /C28 1)4 /C30x /C2716x2 /C2768x3 /C27180x4 /C27...: References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 46 /C1/47, 1996. Sloane, N. J. A. Sequences A005906/M5002 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Truncated Tetrahedron The ARCHIMEDEAN SOLID A13with faces 4f3g/C274f6g: It is also UNIFORM POLYHEDRON U2and Wenninger model W6 : It has SCHLA ¨ FLI SYMBOL t/f3; 3 g and WYTHOFF SYMBOL 23 ½ 3:/ The dual of the truncated tetrahedron is the TRIAKIS TETRAHEDRON . The INRADIUS r of the dual, MIDRADIUS r of the solid and dual, and CIRCUMRADIUS R of the solid for a /C301 are r /C309 44ffiffiffiffiffiffi 22p :0:95940 r /C303 4ffiffiffi 2p :1 :06066 R /C301 4ffiffiffiffiffiffi 22p :1:17260 The distances from the center of the solid to the centroids of the triangular and hexagonal faces aregiven by r3 ¼1 12ffiffiffi 6p ð1Þ r6 /C301 4ffiffiffi 6p : (2) The SURFACE AREA and VOLUME are S /C307ffiffiffi 3p (3) V /C3023 12ffiffiffi 2p : (4) See also ARCHIMEDEAN SOLID ,T RIAKIS TETRAHE- DRON ,TRUNCATED TETRAHEDRON- TRIAKIS TETRAHE- DRON COMPOUND References Cundy, H. and Rollett, A. "Truncated Tetrahedron. 3.62." §3.7.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 101, 1989. Wenninger, M. J. "The Truncated Tetrahedron." Model 6 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 20, 1989. Truncated Tetrahedron-Triakis Tetrahedron Compound The compound of a TRUNCATED TETRAHEDRON and its dual, the TRIAKIS TETRAHEDRON . The compound can be constructed from a TRUNCATED OCTAHEDRON of unit edge length by midpoint CUMULATION with heights h3/C301 30ffiffiffi 6p (1) h6/C301 2ffiffiffi 6p : (2) See also CUMULATION ,P OLYHEDRON COMPOUND , TRIAKIS TETRAHEDRON ,TRUNCATED TETRAHEDRON Truncation The removal of portions of SOLIDS falling outside a set of symmetrically placed planes. The dual operation consists of replacing facial polygons with pyramids,and is sometimes known as CUMULATION . The five P LATONIC SOLIDS belong to one of the following three truncation series (which, in the firsttwo cases, carry the solid to its DUAL POLYHEDRON ). See also CUMULATION ,D U¨ RER’S SOLID ,P YRAMID , STELLATION ,TRUNCATED CUBE,TRUNCATED DODECA- HEDRON ,TRUNCATED ICOSAHEDRON ,TRUNCATED OC- TAHEDRON ,T RUNCATED TETRAHEDRON ,V ERTEX FIGURE Truth Table A truth table is a 2-D array with n /C271 columns. The first n columns correspond to the possible values of n inputs, and the last column to the operation being performed. The rows list all possible combinations of inputs together with the corresponding outputs. For example, the following truth table shows the result of the binary AND operator acting on two inputs A and B, each of which may be true or false. AB /A fflB/ FF F FT F TF F TT T The following Mathematica code can be used to generate a truth table for n levels of operator op. TruthTable[op_, n_] : /C30 Module[ { l /C30 Flatten[Outer[List, Sequence @@ Table[{True, False}, {n}]], n - 1], a /C30 Array[A, n] }, DisplayForm[ GridBox[Prepend[Append[#, op @@ #] & /@ l, Append[a, op @@ a]], RowLines - /C21 True, ColumnLines - /C21 True] ] ] See also AND, CONNECTIVE ,EQUIVALENT ,IMPLIES ,KARNAUGH MAP,M ULTIPLICATION TABLE ,NAND, NOR, NOT, OR, XNOR, XOR References Carnap, R. "Truth Tables." §4in Introduction to Symbolic Logic and Its Applications. New York: Dover, pp. 10 /C1/15, 1958. Tschebyshev An alternative spelling of the name "CHEBYSHEV ." See also CHEBYSHEV Tschebyshev System HAARCONDITION Tschirnhausen Cubic Caustic The CAUSTIC of the T SCHIRNHAUSEN CUBIC taking the RADIANT POINT as the pole is N EILE’S PARABOLA . Tschirnhausen Cubic The Tschirnhausen cubic is a plane curve given by the polar equation r/C30asec31 3uiCkCiCkA (1) or parametric equation x/C30a1/C283t2iCjiCk (2) y/C30at3/C28t2iCjiCk (3) or x/C303at2/C283iCjiCk (4) y/C30at t2/C283iCjiCk : (5) The curve is also known as C ATALAN’S TRISECTRIX and L’HOSPITAL’S CUBIC . The name Tschirnhaus’s cubic is given in R. C. Archibald’s 1900 paper attempting to classify curves (MacTutor Archive). Tschirnhaus’s cubic is the NEGATIVE PEDAL CURVE of a PARABOLA with respect to the FOCUS . References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 87 /C1/90, 1972. MacTutor History of Mathematics Archive. "Tschirnhaus’s Cubic." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Tschirnhaus.html. Tschirnhausen Cubic Pedal Curve The PEDAL CURVE to the TSCHIRNHAUSEN CUBIC for PEDAL POINT at the origin is the PARABOLA x /C301 /C28t2 y /C302t: See also PARABOLA ,P EDAL CURVE ,P EDAL POINT , TSCHIRNHAUSEN CUBIC Tschirnhausen Transformation A transformation of a POLYNOMIAL equation f(x) /C300 which is OF THE FORM y /C30g(x) =h(x) where g and h are POLYNOMIALS and h(x) does not vanish at a root of f(x) /C300: The CUBIC EQUATION is a special case of such a transformation. Tschirnhaus (1683) showed that a POLYNOMIAL of degree n /C212 can be reduced to a form in which the xn/C281 and xn/C282 terms have 0 COEFFI- CIENTS . In 1786, E. S. Bring showed that a general QUINTIC EQUATION can be reduced to the form x5 /C27px /C27q /C300: In 1834, G. B. Jerrard showed that a Tschirnhaus transformation can be used to eliminate the xn/C281 ; xn/C282 ; and xn/C283terms for a general POLYNOMIAL equation of degree n /C213. See also BRING QUINTIC FORM,CUBIC EQUATION References Boyer, C. B. A History of Mathematics. New York: Wiley, pp. 472 /C1/473, 1968. Tschirnhaus. Acta Eruditorum. 1683. Tubular Neighborhood This entry contributed by RYAN BUDNEY A tubular neighborhood of a SUBMANIFOLD N /C23 M is an embedding of the NORMAL BUNDLE (/nN)ofN into M, i.e., f : nN 0 M ; where the image of the ZERO SECTION of the NORMAL BUNDLE is equal to N /C23 M :/ See also BALL,EMBEDDING ,K NOT EXTERIOR ,PRO- DUCT NEIGHBORHOODReferences Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 258, 1994. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 34 /C1/35, 1976. Tucker Circles The Tucker circles are a generalization of the COSINE CIRCLE and LEMOINE CIRCLE which can be viewed as a family of circles obtained by parallel displacing sides of the corresponding COSINE or LEMOINE HEXAGON . No matter how the segments are displaced, the TUCKER HEXAGON will close, and the 12 vertices will be CONCYCLIC . The COSINE CIRCLE and LEMOINE CIRCLE correspond to the special case where three sides of the TUCKER HEXAGON concur. Let three equal lines /P1Q1/, P2Q2 ; and P3Q3 be drawn ANTIPARALLEL to the sides of a triangle so that two (say P2Q2 and P3Q3) are on the same side of the third line as A2P2Q3A3 : Then P2Q3P3Q2is an isosceles TRAPEZOID , i.e., P3Q2 ; P1Q3 ; and P2Q1are parallel to the respective sides. The MIDPOINTS C1 ; C2 ; and C3 of the antiparallels are on the respective symmedians and divide them proportionally. If T divides KO in the same ratio, TC1 ; TC2 ; TC3 are parallel to the radii OA1 ; OA2 ; and OA3 and equal. Since the antiparallels are perpendicular to the symmedians, they form equal chords of a circle, called a Tucker circle, which passes through the six given points and has center T on the line KO (Honsberger 1995, pp. 92 /C1/94). If c /C13KC1 KA1/C30KC2 KA2/C30KC3 KA3/C30KT KO ; then the radius of the Tucker circle is Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C27(1 /C28c)2 tan vq ; where v is the BROCARD ANGLE . The COSINE CIRCLE ,LEMOINE CIRCLE , and TAYLOR CIRCLE are Tucker circles. See also ANTIPARALLEL ,B ROCARD ANGLE ,C OSINE CIRCLE ,C OSINE HEXAGON ,L EMOINE CIRCLE ,L E- MOINE HEXAGON ,TAYLOR CIRCLE References Casey, J. "Lemoine’s, Tucker’s, and Taylor’s Circle." Supp. Ch. §3i nA Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 179 /C1/189, 1888. Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 68, 1971. Honsberger, R. "The Tucker Circles." Ch. 9 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 87 /C1/98, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 271 /C1/277 and 300 /C1/301, 1929. Lachlan, R. §133 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 77, 1893. Tucker Hexagon A closed, self-intersecting concyclic hexagon con- structed along the sides of a triangle. A CIRCUMCIR- CLE of any of these hexagons is called a TUCKER CIRCLE . See also HEXAGON ,TUCKER CIRCLES References Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 90 /C1/91, 1995. Tukey’s Biweight The function c(z) /C30z 1 /C28z2 c2 !2 for ½z½Bc 0 for ½z½> c8 >< >: sometimes used in ROBUST ESTIMATION . It has a minimum at z /C30/C28c =ffiffiffi 3p and a maximum at z /C30c =ffiffiffi3p ; where c?(z) /C301 /C283x2 c2/C300 ;and an inflection point at z /C300, where c??(z) /C30/C286z c2 /C300: References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 697, 1992. Tukey’s Trimean TRIMEAN Tunnel Number Let a KNOT K be n-EMBEDDABLE . Then its tunnel number is a KNOT INVARIANT which is related to n. See also EMBEDDABLE KNOT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 114, 1994. Tura´n Graph The (n, k)-Tura ´n graph is the EXTREMAL GRAPH on n VERTICES which contains no k-CLIQUE . In other words, the Tura´n graph has the maximum possible number of EDGES of any n-vertex graph not contain- ing a COMPLETE GRAPH Kk : TURA´ N’S THEOREM gives the maximum number of edges t(n ; k) for the (n, k)- Tura´n graph. For k /C303, t(n; 3) /C301 4 n2 ; so the Tura´n graph is given by the COMPLETE BIPARTITE GRAPHS Kn=2; n=2 n even K(n/C281)=2;(n /C271)=2n odd:iC0C Tura´n graphs cen be generated usingTuran [n, p]in theMathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also CLIQUE ,COMPLETE BIPARTITE GRAPH ,EX- TREMAL GRAPH ,EXTREMAL GRAPH THEORY ,TURA´ N’S THEOREM References Aigner, M. "Tura ´n’s Graph Theorem." Amer. Math. Monthly 102, 808/C1/816, 1995. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 143 and 218, 1990. Tura´n, P. "On an Extremal Problem in Graph Theory." Mat. Fiz. Lapok 48, 436/C1/452, 1941. Tura´n’s Inequalities For a set of POSITIVE gk ; k /C300, 1, 2..., Tura´n’s inequalities are given by g2 k /C28 gk /C281 gk /C271 ]0 for k /C301, 2, .... See also JENSEN POLYNOMIAL References Csordas, G.; Varga, R. S.; and Vincze, I. "Jensen Polyno- mials with Applications to the Riemann z/-Function." J. Math. Anal. Appl. 153, 112 /C1/135, 1990. Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 388, 1975. Tura´n’s Theorem Let G(V ; E)bea GRAPH with VERTICES V and EDGES E on n VERTICES without a k-CLIQUE . Then t(n; k) 5(k /C28 2)n2 2(k /C28 1); where t(n; k) /C30½E ½ is the EDGE NUMBER . More pre- cisely, the K-GRAPH Kn1 ; ... ; nk/C281with ½ni /C28nj ½51 for i " j is the unique GRAPH without a k-CLIQUE with the maximal number of EDGES t(n; k) :/ See also CLIQUE ,ERDOS- STONE THEOREM ,EXTREMAL GRAPH THEORY , K-GRAPH ,TURA´ N GRAPH References Aigner, M. "Tura ´n’s Graph Theorem." Amer. Math. Monthly 102, 808 /C1/816, 1995. Pach, J. and Agarwal, P. K. Combinatorial Geometry. New York: Wiley, 1995. Turbine A VECTOR FIELD on a CIRCLE in which the directions of the VECTORS are all at the same ANGLE to the CIRCLE . See also CIRCLE ,VECTOR FIELD Turing Machine A theoretical computing machine which consists of an infinitely long magnetic tape on which instructions can be written and erased, a finite register of memory, and a processor capable of carrying out the following instructions: move the tape right, move the tape left, change the state of the register based on its current value and a value on the tape, and write or erase a value on the tape. The machine keeps processing instructions until it reaches a particular state, causing it to halt. Determining whether a Turing machine will halt for a given input and set of rules is called the HALTING PROBLEM . See also AUTOMATA THEORY ,AUTOMATIC SET,BUSY BEAVER ,CELLULAR AUTOMATON ,CHAITIN’S OMEGA , CHURCH- TURING THESIS ,C OMPUTABLE NUMBER ,DETERMINISTIC ,HALTING PROBLEM ,UNIVERSAL TUR- ING MACHINE References Davis, M. Computability and Unsolvability. New York: Dover. Itoˆ, K. (Ed.). "Turing Machines." §31B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 136 /C1/137, 1987. Penrose, R. "Algorithms and Turing Machines." Ch. 2 in The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 30 /C1/73, 1989. Turing, A. M. "On Computable Numbers, with an Applica- tion to the Entscheidungsproblem." Proc. London Math. Soc. Ser. 2 42, 230 /C1/265, 1937. Turing, A. M. "Correction to: On Computable Numbers, with an Application to the Entscheidungsproblem." Proc. Lon- don Math. Soc. Ser. 2 43, 544 /C1/546, 1938. Turning Angle TANGENTIAL ANGLE Tutte Conjecture Tutte (1971/72) conjectured that there is no non- HAMILTONIAN 3-connected BICUBIC GRAPHS . However, a counterexample was found by J. D. Horton in 1976 (Gropp 1990). See also BICUBIC GRAPH ,CUBIC GRAPH ,HAMILTONIAN GRAPH ,TAIT’S HAMILTONIAN GRAPH CONJECTURE References Gropp, H. "Configurations and the Tutte Conjecture." Ars. Combin. A 29, 171/C1/177, 1990. Tutte, W. T. "On the 2 /-Factors of Bicubic Graphs." Disc. Math. 1, 203/C1/208, 1971/72. Tutte Polynomial LetGbe a GRAPH , and let ea( T) denote the cardinality of the set of externally active edges of a spanning tree TofGand ia( T) denote the cardinality of the set of internally active edges of T. Then tG(x;y)/C30X T⁄Gxia(T)yea(T): References Gessel, I. M. and Sagan, B. E. "The Tutte Polynomial of a Graph, Depth-First Search, and Simplicial Complex Parti- tions." Electronic J. Combinatorics 3, No. 2, R9, 1 /C1/36, 1996. http://www.combinatorics.org/Volume_3/volu-me3_2.html#R9. Tutte, W. T. "A Contribution to the Theory of Chromatic Polynomials." Canad. J. Math. 6,8 0/C1 /91, 1953. Tutte-Coxeter Graph LEVIGRAPH Tutte’s Graph A counterexample to TAIT’S HAMILTONIAN GRAPH CONJECTURE given by Tutte (1946). A simpler coun- terexample was later given by Kozyrev and Grinberg. The LEVI GRAPH is sometimes also called the Tutte graph (Royle). See also HAMILTONIAN CIRCUIT ,LEVI GRAPH ,TAIT’S HAMILTONIAN GRAPH CONJECTURE References Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 82 /C1/89, 1973. Royle, G. "Cubic Cages." http://www.cs.uwa.edu.au/~gordon/ cages/. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 112, 1986. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 198, 1990. Tait, P. G. "Remarks on the Colouring of Maps." Proc. Royal Soc. Edinburgh 10, 729, 1880. Tutte, W. T. "On Hamiltonian Circuits." J. London Math. Soc. 21,98/C1/101, 1946. Tutte, W. T. "Non-Hamiltonian Planar Maps." In Graph Theory and Computing (Ed. R. Read). New York: Aca- demic Press, pp. 295 /C1/301, 1972. Tutte’s Theorem Let G be a GRAPH and S a SUBGRAPH of G. Let the number of ODD components in G /C28S be denoted S ?; and ½S½ the number of VERTICES of S. The condition / jSj]S?/ for every SUBSET of VERTICES is NECESSARY and SUFFICIENT for G to have a 1-FACTOR . See also FACTOR (GRAPH ) References Honsberger, R. "Lova ´sz’ Proof of a Theorem of Tutte." Ch. 14 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 147 /C1/157, 1976. Tutte, W. T. "The Factorization of Linear Graphs." J. London Math. Soc. 22, 107 /C1/111, 1947. Twiddle TILDETwig Let a COTREE of a spanning tree T in a CONNECTED GRAPH G be denoted /T /C31/. Then the edges of G which are not in /T /C31/ are called its twigs (Harary 1994, p. 39). See also COTREE References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Twin Peaks For an INTEGER n]2;let lpf( x) denote the LEAST PRIME FACTOR ofn.A PAIR ofINTEGERS (x, y) is called a twin peak if 1.xBy, 2. lpf( x)/C30lpf(y);/ 3. For all z,xBzByIMPLIES lpf(z)Blpf(x):/ A broken-line graph of the least prime factor function resembles a jagged terrain of mountains. In terms ofthis terrain, a twin peak consists of two mountains ofequal height with no mountain of equal or greater height between them. Denote the height of twin peak (x, y)b yp/C30lpf(x)/C30lpf(y):By definition of the LEAST PRIME FACTOR function, pmust be PRIME . Call the distance between two twin peaks ( x, y) s/C13y/C28x: Then smust be an EVEN multiple of p; that is, s/C30kp where kisEVEN . A twin peak with s/C30kpis called a kp-twin peak. Thus we can speak of 2 p/-twin peaks, 4p/-twin peaks, etc. A kp-twin peak is fully specified byk,p, and x, from which we can easily compute y/C13x/C27kp:/ The set of kp-twin peaks is periodic with period q/C30 p#;where p# is the PRIMORIAL ofp. That is, if ( x, y)i s akp-twin peak, then so is ( x/C27q;y/C27q):A funda- mental kp-twin peak is a twin peak having xin the fundamental period [0 ;q):The set of fundamental kp-twin peaks is symmetric with respect to the fundamental period; that is, if ( x, y) is a twin peak on [0 ;q);then so is ( q/C28y;q/C28x):/ The question of the EXISTENCE of twin peaks was first raised by David Wilson in the math-fun mailing list on Feb. 10, 1997. Wilson already had privately showed the EXISTENCE of twin peaks of height p5 13 to be unlikely, but was unable to rule them out altogether. Later that same day, John H. Conway, Johan de Jong, Derek Smith, and Manjul Bhargavacollaborated to discover the first twin peak. Two hours at the blackboard revealed that p/C30113 admits the 2 p /-twin peak x/C30126972592296404970720882679404584182254788131 which settled the EXISTENCE question. Immediately thereafter, Fred Helenius found the smaller 2 p/-twin peak with p /C3089 and x /C309503844926749390990454854843625839 : The effort now shifted to finding the least PRIME p admitting a 2p/-twin peak. On Feb. 12, 1997, Fred Helenius found p /C3071, which admits 240 fundamen- tal 2p/-twin peaks, the least being x /C307310131732015251470110369 : Helenius’s results were confirmed by Dan Hoey, who also computed the least 2p/-twin peak L(2p) and number of fundamental 2p/-twin peaks N(2p) for p /C3073, 79, and 83. His results are summarized in the following table (Sloane’s A009190). p /L(2p)//N(2p)/ 71 7310131732015251470110369 240 73 2061519317176132799110061 40296 79 3756800873017263196139951 164440 83 6316254452384500173544921 6625240 The 2p/-twin peak of height p /C3073 is the smallest known twin peak. Wilson found the smallest known 4p/-twin peak with p /C301327, as well as another very large 4p/-twin peak with p /C303203. Richard Schroeppel noted that the latter twin peak is at the high end of its fundamental period and that its reflection within the fundamental period [0; p#) is smaller. Many open questions remain concerning twin peaks, e.g., 1. What is the smallest twin peak (smallest n)? 2. What is the least PRIME p admitting a 4p/-twin peak? 3. Do 6p/-twin peaks exist? 4. Is there, as Conway has argued, an upper bound on the span of twin peaks? 5. Let /p Bq Br/ be PRIME .Ifp and r each admit kp- twin peaks, does q then necessarily admit a kp- twin peak? See also ANDRICA’S CONJECTURE ,DIVISOR FUNCTION , LEAST COMMON MULTIPLE ,LEAST PRIME FACTOR References Sloane, N. J. A. Sequences A009190 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Twin Prime Conjecture There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of TWIN PRIMES (Guy 1994, p. 19). It is not known if there are aninfinite number of such PRIMES (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true (Hardy and Wright 1979, p. 5). In the words of Shanks (1993, p. 219), "the evidence is overwhelm- ing." The conjecture that there are infinitely many integers n such that n /C271 is prime and n is twice a prime is very closely related (Shanks 1993, p. 30). A second twin prime conjecture states that adding a correction proportional to 1 =ln p to a computation of BRUN’S CONSTANT ending with ... /C271=p /C271=(p /C272) will give an estimate with error less than cffiffiffippln piCjiCk/C281: An extended form of this conjecture, sometimes called the strong twin prime conjecture (Shanks 1993, p. 30) states that Px(p; p /C272) /C22 P2gx 2dx (ln x)2 ; where P2is the TWIN PRIMES CONSTANT (Hardy and Littlewood 1922). This conjecture is a special case of the more general PRIME PATTERNS CONJECTURE cor- responding to the set S/C30f0;2g:/ See also BRUN’S CONSTANT ,P RIME ARITHMETIC PROGRESSION ,P RIME CONSTELLATION ,P RIME PAT- TERNS CONJECTURE ,TWIN PRIMES References Guy, R. K. "Gaps between Primes. Twin Primes." §A8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19 /C1/23, 1994. Hardy, G. H. and Littlewood, J. E. "Some Problems of ‘Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes." Acta Math. 44,1/C1/70, 1922. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 261 /C1/265, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 30, 1993. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 41, 1986. Twin Primes Twin primes are pairs of PRIMES OF THE FORM (p,p/C27 2):The term "twin prime" was coined by Paul Sta ¨ckel (1892/C1/1919; Tietze 1965, p. 19). The first few twin primes are n91 for n/C304, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (Sloane’s A014574). Explicitly, these are (3, 5), (5,7), (11, 13), (17, 19), (29, 31), (41, 43), ... (Sloane’s A001359 and A006512). The following table gives the first few pfor the twin primes ( p,p/C272); COUSIN PRIMES (p,p/C274);SEXY PRIMES (p,p/C276);etc. Triplet Sloane First Member (p,p/C272)/ Sloane’s A0013593, 5, 11, 17, 29, 41, 59,71, ... (p,p/C274)/ Sloane’s A0232003, 7, 13, 19, 37, 43, 67,79, ... (p,p/C276) / Sloane’sA0232015, 7, 11, 13, 17, 23, 31,37, ... (p,p/C278) / Sloane’s A0232023, 5, 11, 23, 29, 53, 59,71, ... (p,p/C2710) /Sloane’s A0232033, 7, 13, 19, 31, 37, 43,61, ... (p,p/C2712) /Sloane’sA0461335, 7, 11, 17, 19, 29, 31,41, ... Letp 2(n) be the number of twin primes pand p/C272 such that p5n:It is not known if there are an infinite number of such PRIMES (Wells 1986, p. 41; Shanks 1993), but it seems almost certain to be true (Hardyand Wright 1979, p. 5). All twin primes except (3, 5) are OF THE FORM 6n91:J. R. Chen has shown there exists an INFINITE number of PRIMES psuch that p/C272 has at most two factors (Le Lionnais 1983, p. 49). Bruns proved that there exists a computable INTEGER x0such that if x]x0;then p2ðxÞB100x ðlnxÞ2ð1Þ (Ribenboim 1996, p. 261). It has been shown that p2(x)5cY p>21/C281 (p/C281)2"# x (lnx)2 /C21/C27Oln ln x lnx !"# ; (2) written more concisely as p2(x)5cP2x (lnx)21/C27Oln ln x lnx !"# ; (3) where P2is known as the TWIN PRIMES CONSTANT and cis another constant. The constant chas been reduced to 68 =9:7:5556 (Fouvry and Iwaniec 1983), 128 =17:7:5294 (Fouvry 1984), 7 (Bombieri et al.1986), 6.9075 (Fouvry and Grupp 1986), and 6.8354 (Wu 1990). The bound on cis further reduced to 6.8325 (Haugland 1999). This calculation involvedevaluation of 7-fold integrals and fitting of threedifferent parameters. Hardy and Littlewood conjec- tured that c/C302 (Ribenboim 1996, p. 262). Wolf notes that the formula p 2(x)/C2P2[p(x)]2 x; (4) which increases as P2x=(lnx)2for large x, agrees with numerical data much better than does P2x=(lnx)2; although not as well as P2Li2(x):/Extending the search done by Brent in 1974 or 1975, Wolf has searched for the analog of the S KEWES NUMBER for twins, i.e., an xsuch that p2(x)/C28 P2Li2(x) changes sign. Wolf checked numbers up to 242and found more than 90,000 sign changes. From this data, Wolf conjectured that the number of signchanges n(n) for xBnofp 2(x)/C28P2Li2(x) is given by n(n)/C2ffiffiffinp lnn: (5) Proof of this conjecture would also imply the existence an infinite number of twin primes. Define E/C13lim inf n0/C12pn/C271/C28pn lnpn: (6) If there are an infinite number of twin primes, then E/C300. The best upper limit to date is E51 4/C27p=16/C30 0:44634 . . . (Huxley 1973, 1977). The best previous values were 15/16 (Ricci), 2 /C27ffiffiffi 3piCjiCk =8/C300:46650 . . . (Bombieri and Davenport 1966), and 2ffiffiffi2p /C281iCjiCk =4/C30 0:45706 . . . (Pil’Tai 1972), as quoted in Le Lionnais (1983, p. 26). Some large twin primes are 10 ;006;42891; 1;706;595/C2921123591;and 571 ;305/C292770191:An up-to-date table of known twin primes with 2000 or more digits follows. An extensive list is maintained byC. Caldwell at http://www.utm.edu/cgi-bin/caldwell/ primes.cgi/twin. /(p;p/C271)/ Digits Reference /260;497;545/C292662591/ 2003 Atkin and Rickert 1984 /43;690;485;351;513/C2910199591/2009 Dubner, Atkin 1985 /2;846!!!!91/ 2151 Dubner 1992 /10;757;0463/C2910225091/ 2259 Dubner, Atkin 1985 /663;777/C292765091/ 2309 Brown et al. 1989 /75;188;117;004/C2910229891/ 2309 Dubner 1989 /571305 /C292770191/ 2324 Brown et al. 1989 /1;171;452;282/C2910249091/ 2500 Dubner 1991 /459 /C2152852991/ 2571 Dubner 1993 /1;706;595 /C21521123591/ 3389 Noll et al. 1989 /4;655;478;828 /C21510342991/ 3439 Dubner 1993 /1;692;923;232 /C21510402091/ 4030 Dubner 1993 /6;797;727 /C21521532891/ 4622 Forbes 1995 /697;053; 813216352 91/ 4932 Indlekofer and Ja’rai 1994 /570;918; 348 /C215 105120 91/ 5129 Dubner 1995 /242;206; 083 /C215 238880 91/ 11713 Indlekofer and Ja’rai 1995 The last of these is the largest known twin prime pair. In 1995, Nicely discovered a flaw in the Intel † Pentium /TM microprocessor by computing the recipro- cals of 824,633,702,441 and 824,633,702,443, which should have been accurate to 19 decimal places but were incorrect from the tenth decimal place on (Cipra 1995, 1996; Nicely 1996). If n ]2; the INTEGERS n and n /C272 form a pair of twin primes IFF 4 ½ðn /C281Þ! þ 1 /C138þn /C130 ðmod nðn þ 2ÞÞ: ð7Þ /n /C30pp? where (p; p?) is a pair of twin primes IFF f(n)s(n) /C30(n /C283)(n /C271) (8) (Ribenboim 1996, p. 259). S. M. Ruiz has found the unexpected result that (n; n /C272) are twin primes IFF Xn i/C301ian /C27 2 i$% /C27n i$% ! /C302 /C27na /C27Xn i /C301ian /C27 1 i$% /C27n /C28 1 i$% ! (9) for a ]0; where xbcis the FLOOR FUNCTION . The values of p2(n) were found by Brent (1976) up to n /C301011 : T. Nicely calculated them up to 1014 in his calculation of BRUN’S CONSTANT . The following table gives the number less than increasing powers of 10 (Sloane’s A007508; Nicely 1998, 1999). Using a dis- tributed computation, Fry et al. obtained p2(1016)in 2000, although this value has not yet been made public. The following table gives p(10n) for various values of n, and extends a similar table with early references given by Ribenboim (1996, p. 263). n / p2(n)/ 103 35 104 205 105 1224 106 8,169 107 58,980 108 440,312109 3,424,506 1010 27,412,679 1011 224,376,048 1012 1,870,585,220 101315,834,664,872 1014135,780,321,665 10151,177,209,242,304 It is conjectured that every even number is a sum of a pair of twin primes except a finite number of excep- tions whose first few terms are 2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, ... (Sloane’s A007534; Wells1986, p. 132). See also B ITWIN CHAIN ,BRUN’S CONSTANT ,COUSIN PRIMES , DE POLIGNAC’S CONJECTURE, PRIME CONSTEL- LATION ,S EXY PRIMES ,T WIN PRIME CONJECTURE , TWIN PRIMES CONSTANT References Bombieri, E. and Davenport, H. "Small Differences Between Prime Numbers." Proc. Roy. Soc. Ser. A 293,1/C1/8, 1966. Bombieri, E.; Friedlander, J. B.; and Iwaniec, H. "Primes in Arithmetic Progression to Large Moduli." Acta Math. 156, 203/C1/251, 1986. Bradley, C. J. "The Location of Twin Primes." Math. Gaz. 67, 292/C1/294, 1983. Brent, R. P. "Irregularities in the Distribution of Primes and Twin Primes." Math. Comput. 29,4 3/C1/56, 1975. Brent, R. P. "UMT 4." Math. Comput. 29, 221, 1975. Brent, R. P. "Tables Concerning Irregularities in the Dis- tribution of Primes and Twin Primes to 1011."Math. Comput. 30, 379, 1976. Caldwell, C. http://www.utm.edu/cgi-bin/caldwell/pri- mes.cgi/twin. Caldwell, C. K. "The Top Twenty: Twin Primes." http:// www.utm.edu/research/primes/lists/top20/twin.html. Cipra, B. "How Number Theory Got the Best of the Pentium Chip." Science 267, 175, 1995. Cipra, B. "Divide and Conquer." What’s Happening in the Mathematical Sciences, 1995 /C1/1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 38 /C1/47, 1996. Fouvry, E ´. "Autour du the ´ore`me de Bombieri-Vinogradov." Acta. Math. 152, 219/C1/244, 1984. Fouvry, E ´. and Grupp, F. "On the Switching Principle in Sieve Theory." J. reine angew. Math. 370, 101/C1/126, 1986. Fouvey, E ´. and Iwaniec, H. "Primes in Arithmetic Progres- sion." Acta Arith. 42, 197/C1/218, 1983. Fry, P.; Nesheiwat, J.; and Szymanski, B. K. "Rensselaer’s Twin Prime Computing Effort." http://www.cs.rpi.edu/ research/twinp/. Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243,1 8/C1/28, Dec. 1980. Guy, R. K. "Gaps between Primes. Twin Primes." §A8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19 /C1/23, 1994. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Haugland, J. K. Application of Sieve Methods to Prime Numbers. Ph.D. thesis. Oxford, England: Oxford Univer- sity, 1999. Huxley, M. N. "Small Differences between Consecutive Primes." Mathematica 20, 229 /C1/232, 1973. Huxley, M. N. "Small Differences between Consecutive Primes. II." Mathematica 24, 142 /C1/152, 1977. Indlekofer, K. H. and Ja´rai, A. "Largest Known Twin Primes." Math. Comput. 65, 427 /C1/428, 1996. Indlekofer, K. H. and Ja´rai, A. "Largest Known Twin Primes and Sophie Germain Primes." Math. Comput. 68, 1317 /C1/ 1324, 1999. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Nicely, T. "Enumeration to 1014 of the Twin Primes and Brun’s Constant." Virginia J. Sci. 46, 195 /C1/204, 1996. Nicely, T. "Enumeration to 1 :6 /C291015 of the Twin Primes and Brun’s Constant." Submitted to Math. Comput. Parady, B. K.; Smith, J. F.; and Zarantonello, S. E. "Largest Known Twin Primes." Math. Comput. 55, 381 /C1/382, 1990. Ribenboim, P. "Twin Primes." §4.3 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 259 /C1/ 265, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 30, 1993. Sloane, N. J. A. Sequences A001359/M2476, A006512/ M3763, A007508/M1855, A007534, and A014574 in "An On-Line Version of the Encyclopedia of Integer Se- quences." http://www.research.att.com/~njas/sequences/ eisonline.html. Tietze, H. "Prime Numbers and Prime Twins." Ch. 1 in Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, pp. 1 /C1/20, 1965. Weintraub, S. "A Prime Gap of 864." J. Recr. Math. 25,42/C1/ 43, 1993. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 41, 1986. Wu, J. "Sur la suite des nombres premiers jumeaux." Acta. Arith. 55, 365 /C1/394, 1990. Twin Primes Constant The twin primes constant P2 (sometimes also denoted C2) is defined by P2 /C13Y p>2 p prime1 /C281 (p /C28 1)2"# (1) ln1 2 P2iCkCiCkA /C30X p ]3 p primelnp(p /C28 2) (p /C28 1)2"# /C30X p ]3 p primeln 1 /C282 p ! /C282ln 1/C281 p ! "# /C30/C28X/C12 j/C3022j /C28 2 jX p ]3 p primep /C28j ; (2) where the ps in sums and products are taken overPRIMES only. Flajolet and Vardi (1996) give series with accelerated convergence P2 /C30Y/C12 n/C302z(n)1/C282 /C28nðÞ ½/C138/C28In(3) /C303 415163536Y/C12 n /C302z(n)1/C282/C28nðÞ 1 /C283/C28nðÞ ½ /C2 1 /C285 /C28nðÞ 1 /C287/C28nðÞ /C138/C28In; (4) with In /C131 nX d ½nm(d)2n=d ; (5) where m(x) is the MO¨ BIUS FUNCTION . (4) has conver- gence like /C2(11=2)/C28n :/ /P2was computed to 45 digits by Wrench (1961) and Gourdon and Sebah list 60 digits. P2 /C300 :6601618158... : (6) Le Lionnais (1983, p. 30) calls P2the SHAH- WILSON CONSTANT , and 2P2the twin prime constant (Le Lionnais 1983, p. 37). See also BRUN’S CONSTANT ,GOLDBACH CONJECTURE , MERTENS CONSTANT ,TWIN PRIMES References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps. Gourdon, X. and Sebah, P. "Some Constants from Number Theory." http://xavier.gourdon.free.fr/Constants/Miscella- neous/constantsNumTheory.html. Hardy, G. H. and Littlewood, J. E. "Some Problems of ‘Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes." Acta Math. 44,1/C1/70, 1922. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 202, 1989. Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, p. 147, 1991. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, pp. 61 /C1/ 66, 1994. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 30, 1993. Wrench, J. W. "Evaluation of Artin’s Constant and the Twin Prime Constant." Math. Comput. 15, 396/C1/398, 1961. Twins BROTHERS ,PAIR Twirl AROTATION combined with an EXPANSION orCON- TRACTION . See also SCREW ,SHIFT Twist The twist of a ribbon measures how much it twists around its axis and is defined as the integral of the incremental twist around the ribbon. A formula for the twist is given by Tw(K) /C301 2p gKds omnadxm dsn ndn a ds; (1) where K is parameterized by xm(s) for 0 5s 5L along the length of the knot by parameter s, and the FRAME Kfassociated with K is ym /C30xm(s) /C27 onm(s); (2) where o is a small parameter and nm(s) is a unit VECTOR FIELD normal to the curve at s (Kaul 1999). Letting Lk be the linking number of the two compo- nents of a ribbon, Tw be the twist, and Wr be the WRITHE , then the CALUGAREANU THEOREM states that Lk(R) /C30Tw(R) /C27Wr(R) (3) (Adams 1994, p. 187). See also CALUGAREANU THEOREM ,SCREW ,W RITHE References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. Kaul, R. K. Topological Quantum Field Theories--A Meeting Ground for Physicists and Mathematicians. 15 Jul 1999. http://xxx.lanl.gov/abs/hep-th/9907119/. Twist Map A class of AREA-PRESERVING MAPS OF THE FORM ui /C271 /C30 ui /C272pa riðÞ ri /C271 /C30ri ; which maps CIRCLES into CIRCLES but with a twist resulting from the a /C30 a riðÞterm. Twist Move The REIDEMEISTER MOVE of type II. See also KNOT MOVE,REIDEMEISTER MOVES Twist Number WRITHE Twisted Chevalley Groups FINITE SIMPLE GROUPS of LIE-TYPE of ORDERS 14, 52, 78, 133, and 248. They are denoted 3D4(q) ; E6(q); E7(q) ; E8(q); F4(q) ; 2F4(2n)?; G2(q); 2G2(3n) ; 2B(2n):/See also CHEVALLEY GROUPS ,FINITE GROUP ,SIMPLE GROUP ,TITS GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#twi. Twisted Conic SKEW CONIC Twisted Sphere CORKSCREW SURFACE Twistor This entry contributed by EDGAR VAN TUYLL A twistor in MINKOWSKI SPACE may be defined as a pair consisting of a SPINOR FIELD and a complex conjugate SPINOR FIELD satisfying the TWISTOR EQUA- TION . See also MINKOWSKI SPACE ,SPINOR ,SPINOR FIELD, TWISTOR CORRESPONDENCE ,T WISTOR EQUATION , TWISTOR SPACE References Penrose, R. and Rindler, W. Spinors and Space-Time, Vol. 2: Spinor and Twistor Methods in Space-Time Geometry Cambridge, England: Cambridge University Press, 1987. Twistor Correspondence This entry contributed by EDGAR VAN TUYLL Oriented spheres in complex Euclidean 3-space can be represented as lines in complex projective 3-space ("Lie correspondence"), and the spheres may be thought of as the t /C300 representation of the light cones of events in MINKOWSKI SPACE . In effect, the Lie correspondence represents the points of (complexified compactified) MINKOWSKI SPACE by lines in complex projective 3-space, where meeting lines describe null- separated Minkowski points. This is the twistor correspondence. See also MINKOWSKI SPACE ,TWISTOR References Penrose, R. "The Central Programme of Twistor Theory." Chaos, Solitons and Fractals 10, 581 /C1/611, 1999. Twistor Space This entry contributed by EDGAR VAN TUYLL The collection of TWISTORS in MINKOWSKI SPACE that forms a four-dimensional COMPLEX VECTOR SPACE . See also COMPLEX SPACE ,M INKOWSKI SPACE ,TWIS- TOR Twist-Spun Knot A generalization of SPUN KNOTS due to Zeeman. This method produces 4-D KNOT types that cannot be produced by ordinary spinning. See also SPUN KNOT Two 2 Two Triangle Theorem DESARGUES’ THEOREM Two-Colorable Graph BIPARTITE GRAPH Two-Ears Theorem Except for TRIANGLES , every SIMPLE POLYGON has at least two nonoverlapping EARS . See also EAR,O NE-MOUTH THEOREM ,P RINCIPAL VERTEX References de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. Computational Geometry: Algorithms and Appli- cations, 2nd rev. ed. Berlin: Springer-Verlag, p. 59, 2000. Meisters, G. H. "Principal Vertices, Exposed Points, and Ears." Amer. Math. Monthly 87, 284 /C1/285, 1980. Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122,31/C1/35, 1991. Two-Form See also DIFFERENTIAL K-FORM,O NE-FORM,ZERO- FORM Two-Graph A two-graph (V ;D)isa GRAPH on nodes V with a collection D of unordered triples of the vertices (the so-called "odd triples") such that each 4-tuple of V contains an even number of elements of D as subsets. See also EULERIAN GRAPH References Bussemaker, F. C.; Mathon, R. A.; and Seidel, J. J. "Tables of Two-Graphs." In Combinatorics and Graph Theory (Ed. S. B. Rao). Berlin: Springer-Verlag, pp. 70 /C1/112, 1981. Mallows, C. L. and Sloane, N. J. A. "Two-Graphs, Switching Classes, and Euler Graphs are Equal in Number." SIAM J. Appl. Math. 28, 876 /C1/880, 1975. Spence, E. "Two-Graphs." Ch. VI.6 in Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, pp. 686 /C1/694, 1996. Two-Point Distance POINT- POINT DISTANCE–1- D, POINT- POINT DISTANCE– 2-D, POINT- POINT DISTANCE–3- D, SPHERE POINT PICK- INGTwo-Scale Expansion c /C30 A0 /C27 a1A1 /C27 a2A2 /C27... ðÞ eiS = a : Two-Sheeted Hyperboloid A HYPERBOLOID consisting of two distinct sheets. See also HYPERBOLOID Tychonof Compactness Theorem The topological product of any number of COMPACT SPACES is COMPACT . Type Whitehead and Russell (1927) devised a hierarchy of "types" in order to eliminate self-referential state- ments from Principia Mathematica , which purported to derive all of mathematics from logic. A set of the lowest type contained only objects (not sets), a set of the next higher type could contain only objects or sets of the lower type, and so on. Unfortunately, GO¨ DEL’S INCOMPLETENESS THEOREM showed that both Princi- pia Mathematica and all consistent formal systems must be incomplete. See also CLASS (SET), GO¨ DEL’S INCOMPLETENESS THEOREM References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 21 /C1/22, 1977. Ferreiro ´s, J. "Russell’s Theory of Types." §9.5 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 325 /C1/ 333, 1999. Gonseth, F. "La The´orie des types." §107 in Les mathe ´ma- tiques et la re´alite´: Essai sur la me´thode axiomatique. Paris: Fe´lix Alcan, pp. 257 /C1/259, 1936. Hofstadter, D. R. Go¨del, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 21 /C1/22, 1989. Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, 1927. Type I Error An error in a STATISTICAL TEST which occurs when a true hypothesis is rejected (a false negative in terms of the NULL HYPOTHESIS ). See also NULL HYPOTHESIS ,SENSITIVITY ,SPECIFICITY , STATISTICAL TEST,TYPE II ERROR Type II Error An error in a STATISTICAL TEST which occurs when a false hypothesis is accepted (a false positive in terms of the NULL HYPOTHESIS ). See also NULL HYPOTHESIS ,SENSITIVITY ,SPECIFICITY , STATISTICAL TEST,TYPE IERROR U U(n) Basic Hypergeometric Series Multiple series generalizations of basic hypergeo- metric series over the UNITARY GROUPS U(n/C271): The fundamental theorem of U(n) series takes c1;..., cnandx1;...,xnas indeterminates and n]1:Then c1/C1/C1/C1cn;q ðÞ N (q;q)N /C30X y1;y2;...;yn]0 ½y½/C30N/C26 Y 15rBs5n1/C28xr xsqyr/C28ys 1/C28xr xs2 66643 7775 /C29Y n r;s/C301xr xscs;q ! yr qxr xs;q ! yr2 6666643 777775q y2/C272y3/C27.../C27(n/C281)yn/C2/C3/C27 ; where it is assumed that none of the denominators vanish (Bhatnagar 1995, p. 22). The series in this theorem is called an SU(n) series (Milne 1985; Bhatnagar 1995, p. 22). Many other q-results, including the Q-BINOMIAL THEOREM and Q-SAALSCHU ¨TZ SUM , can be generalized toU(n/C271) series. References Bhatnagar, G. " /U(n/C271) Basic Hypergeometric Series." Ch. 2 inInverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, pp. 20 /C1/8, 1995. Biedenharn, L. C. and Louck, J. D. Angular Momentum in Quantum Physics: Theory and Applications. Reading, MA: Addison-Wesley, 1981. Biedenharn, L. C. and Louck, J. D. The Racah-Wigner Algebra in Quantum Theory. Reading, MA: Addison- Wesley, 1981. Denis, R. Y. and Gustafson, R. A. "An SU(n)q-Beta Integral Transformation and Multiple Hypergeometric Series Iden- tities." SIAM J. Math. Anal. 23, 552/C1/61, 1992. Gustafson, R. A. "Multilateral Summation Theorems for Ordinary and Basic Hypergeometric Series in U(n):/" SIAM J. Math. Anal. 18, 1576 /C1/596, 1987. Gustafson, R. A. and Krattenthaler, C. "Heine Transforma- tions for a New Kind of Basic Hypergeometric Series inU(n): /"J. Comput. Appl. Math. 68, 151/C1/58, 1996. Gustafson, R. A. and Krattenthaler, C. "Determinants Eva- luations and U(n) Extensions of Heine’s2f1Transforma- tions." In Special Functions, q -Series, and Related Topics (Ed. M. E. H. Ismail, D. R. Masson, and M. Rahman).Providence, RI: Amer. Math. Soc., pp. 83 /C1 /9, 1997. Holman, W. J. III. "Summation Theorems for Hypergeo- metric Series in U(n):/"SIAM J. Math. Anal. 11, 523/C1/32, 1980. Holman, W. J. III.; Biedenharn, L. C.; and Louck, J. D. "On Hypergeometric Series Well-Poised in SU(n):/"SIAM J. Math. Anal. 7, 529/C1/41, 1976.Milne, S. C. "An Elementary Proof of the Macdonald Identities for A(1) l:/"Adv. Math. 57,3 4/C1/0, 1985. Milne, S. C. "Basic Hypergeometric Series Very Well-Poised inU(n):/"J. Math. Anal. Appl. 122, 223/C1/56, 1987. Milne, S. C. "Balanced3f2Summation for U(n) Basic Hypergeometric Series." Adv. Math. 131,9 3/C1/87, 1997. Ulam Map f(x)/C301/C282x2 forx/C23[/C281;1]:Fixed points occur at x/C30/C28 1, 1/2, and order 2 fixed points at x/C3019ffiffiffi 5p/C0/C1 =4:The NATURAL DENSITY of the map is r(y)/C301 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28y2p : References Beck, C. and Schlo ¨gl, F. Thermodynamics of Chaotic Systems: An Introduction. Cambridge, England: Cam- bridge University Press, p. 194, 1995. Ulam Number ULAM SEQUENCE Ulam Sequence The Ulam sequence aifg/C30(u;v) is defined by a1/C30u; a2/C30v;with the general term anforn/C212 given by the least INTEGER expressible uniquely as the SUM of two distinct earlier terms. The numbers so produced are sometimes called U-NUMBERS or U LAM NUMBERS . The first few numbers in the (1, 2)-Ulam sequence are1, 2, 3, 4, 6, 8, 11, 13, 16, ... (Sloane’s A002858). Here, the first term after the initial (1, 2) is obviously 3 since 3 /C301/C272:The next term is 4 /C301/C273:(We don’t have to worry about 4 /C302/C272 since it is a sum of a single term instead of distinct terms.) 5 is not a member of the sequence since it is representable intwoways, 5 /C301/C274/C302/C273;but 6/C302/C274 is a member. Proceeding in the manner, we can generate Ulamsequences for any ( u, v), examples of which are given in the table below. (u, v) Sloane Sequence (1, 2) Sloane’s A0028581, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... (1, 3) Sloane’s A0028591, 3, 4, 5, 6, 8, 10, 12, 17, 21, ... (1, 4) Sloane’s A0036661, 4, 5, 6, 7, 8, 10, 16, 18, 19, ... (1, 5) Sloane’s A0036671, 5, 6, 7, 8, 9, 10, 12, 20, 22, ... (2, 3) Sloane’s A0018572, 3, 5, 7, 8, 9, 13, 14, 18, 19, ... (2, 4) Sloane’s A0489512, 4, 6, 8, 12, 16, 22, 26, 32, 36, ... (2, 5) Sloane’s A0073002, 5, 7, 9, 11, 12, 13, 15, 19, 23, ... Schmerl and Spiegel (1994) proved that Ulam se- quences (2; v) for ODD v ]5 have exactly two EVEN terms. Ulam sequences with only finitely many EVEN terms eventually must have periodic successive dif- ferences (Finch 1991, 1992abc). Cassaigne and Finch (1995) proved that the Ulam sequences (4; v) for 5 5 v /C131 (mod 4) have exactly three EVEN terms. The Ulam sequence can be generalized by the S- ADDITIVE SEQUENCE . See also GREEDY ALGORITHM , S-ADDITIVE SEQUENCE , STO¨ HR SEQUENCE References Cassaigne, J. and Finch, S. "A Class of 1-Additive Sequences and Quadratic Recurrences." Exper. Math 4,49/C1/0, 1995. Finch, S. "Conjectures About 1-Additive Sequences." Fib. Quart. 29, 209 /C1/14, 1991. Finch, S. "Are 0-Additive Sequences Always Regular?" Amer. Math. Monthly 99, 671 /C1/73, 1992a. Finch, S. "On the Regularity of Certain 1-Additive Se- quences." J. Combin. Th. Ser. A 60, 123 /C1/30, 1992b. Finch, S. "Patterns in 1-Additive Sequences." Exper. Math. 1,57/C1/3, 1992c. Finch, S. "Ulam s-Additive Sequences." http://www.math- soft.com/asolve/sadd/sadd.html. Guy, R. K. "A Quarter Century of Monthly Unsolved Problems, 1969 /C1/993." Amer. Math. Monthly 100, 945 /C1/ 49, 1993. Guy, R. K. "Ulam Numbers." §C4 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 109 /C1/10, 1994. Guy, R. K. and Nowakowski, R. J. "Monthly Unsolved Problems, 1969 /C1/995." Amer. Math. Monthly 102, 921 /C1/ 26, 1995. Recaman, B. "Questions on a Sequence of Ulam." Amer. Math. Monthly 80, 919 /C1/20, 1973. Schmerl, J. and Spiegel, E. "The Regularity of Some 1- Additive Sequences." J. Combin. Theory Ser. A 66, 172 /C1/ 75, 1994.Sloane, N. J. A. Sequences A001857/M0634, A002858/ M0557, A002859/M2303, A003666/M3237, A003667/ M3746, and A007300/M1328 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Ulam’s Conjecture Let graph G have p points viand graph H have p points ui ; where p ]3: Then if for each i, the SUBGRAPHS Gi /C30G /C28viand Hi /C30H /C28uiare ISO- MORPHIC , then the graphs G and H are ISOMORPHIC . See also ISOMORPHIC GRAPHS ,SUBGRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 12, 1994. Ulam’s Problem COLLATZ PROBLEM Ulam’s Spiral PRIME SPIRAL Ultrafactorial The function defined by U(n) /C30(n!)n! : The values for n /C300, 1, ..., are 1, 1, 4, 46656, 1333735776850284124449081472843776, ... (Sloane’s A046882). See also FACTORIAL References Sloane, N. J. A. Sequences A046882 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Ultrafilter This entry contributed by VIKTOR BENGTSSON Let S be a nonempty set, then an ultrafilter on S is a nonempty collection F of subsets of S having the following properties: 1. fiQF :/ 2. If A; B /C23 F then A S B /C23 F :/ 3. If A /C23 F and A ⁄B ⁄S then B /C23 F :/ 4. For any subset A of S, either A /C23 F or its complement A?/C30S /C28A /C23 F :/ An ultrafilter FonSis said to be free if it contains the COFINITE FILTER FSofS. See also COFINITE FILTER ,FILTER Ultrametric An ultrametric is a METRIC which satisfies the following strengthened version of the TRIANGLE IN- EQUALITY , d(x;z)5max( d(x;y);d(y;z)) for all x; y; z: At least two of d(x; y) ; d(y; z) ; and d(x; z) are the same. Let X be a SET, and let XN (where N is the SET of NATURAL NUMBERS ) denote the collection of sequences of elements of X (i.e., all the possible sequences x1 ; x2 ; x3 ; ...). For sequences a /C30 a1 ; a2 ; ... ðÞ ; b /C30 b1 ; b2 ; ... ðÞ ; let n be the number of initial places where the sequences agree, i.e., a1 /C30b1 ; a2 /C30b2 ; ..., an /C30bn ; but an/C271 "bn /C271 : Take n /C300ifa1 "b1 : Then defining d(a; b) /C302/C28n gives an ultrametric. The P-ADIC NORM metric is another example of an ultrametric. See also METRIC , P-ADIC NUMBER Ultrapower This entry contributed by MATT INSALL A specific type of ULTRAPRODUCT that can be used to construct nonstandard universes and obtain the TRANSFER PRINCIPLE as a corollary of LOS’ THEOREM for ultraproducts. See also LOS’ THEOREM ,N ONSTANDARD ANALYSIS , ULTRAPRODUCT Ultraproduct See also ULTRAPOWER Ultraradical A symbol which can be used to express solutions not obtainable by finite ROOT EXTRACTION . The solution to the irreducible QUINTIC EQUATION x5 /C27x /C30a is written . See also RADICAL Ultraspherical Differential Equation GEGENBAUER DIFFERENTIAL EQUATION Ultraspherical Function GEGENBAUER FUNCTION Ultraspherical Polynomial GEGENBAUER POLYNOMIAL Umbilic Point A point on a surface at which the CURVATURE is the same in any direction. Umbral Algebra The algebra structure of linear functionals on poly- nomials of a single variable (Roman 1984, pp. 2 /C1/).See also UMBRAL CALCULUS References Roman, S. "The Umbral Algebra." §2.1 in The Umbral Calculus. New York: Academic Press, pp. 6 /C1/2, 1984. Umbral Calculus Roman (1984, p. 2) describes umbral calculus as the study of the class of SHEFFER SEQUENCES . Umbral calculus provides a formalism for the systematic derivation and classification of almost all classical combinatorial identities for polynomial sequences, along with associated GENERATING FUNCTIONS , ex- pansions, duplication formulas, RECURRENCE RELA- TIONS , inversions, RODRIGUES FORMULA , etc., (e.g., the EULER- MACLAURIN INTEGRATION FORMULAS , Boo- le’s summation formula, the CHU-VANDERMONDE IDENTITY ,NEWTON’S DIVIDED DIFFERENCE INTERPOLA- TION FORMULA ,GREGORY’S FORMULA ,LAGRANGE IN- VERSION ). The term "umbral calculus" was coined by Sylvester from the word "umbra" (meaning "shadow" in Latin), and reflects the fact that for many types of identities involving sequences of polynomials with POWERS an ; "shadow" identities are obtained when the polyno- mials are changed to discrete values and the expo- nent in anis changed to the FALLING FACTORIAL (a)n /C13a(a /C281) /C1/C1/C1(a /C28n /C271):/ For example, NEWTON’S FORWARD DIFFERENCE FOR- MULA written in the form f(x /C27a) /C30X/C12 n/C300(a)n Dnf(x) n! (1) with f(x /C27a) /C13fx /C27alooks suspiciously like a finite analog of the TAYLOR SERIES expansion f(x /C27a) /C30X/C12 n/C300an ˜Dnf(x) n!; (2) where ˜D is the DIFFERENTIAL OPERATOR . Similarly, the CHU-VANDERMONDE IDENTITY (x/C27a)n/C30X/C12 k/C300n k/C18/C19 (a)k(x)n/C28k (3) withn k/C0/C1 aBINOMIAL COEFFICIENT , looks suspiciously like an analog of the BINOMIAL THEOREM (x/C27a)n/C30X/C12 k/C300n k/C18/C19 akxn/C28k(4) (Di Bucchianico and Loeb). See also APPELL SEQUENCE ,B INOMIAL THEOREM , CHU-VANDERMONDE IDENTITY ,COMBINATORICS ,FAA´ DI BRUNO’S FORMULA ,FINITE DIFFERENCE ,SHEFFER SEQUENCE References Bell, E. T. "Postulational Basis for the Umbral Calculus." Amer. J. Math. 62, 717 /C1/24, 1940. Roman, S. and Rota, G.-C. "The Umbral Calculus." Adv. Math. 27,95/C1/88, 1978. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Umbral Operator An operator T which maps some BASIC POLYNOMIAL SEQUENCE pn(x) into another BASIC POLYNOMIAL SEQUENCE qn(x) :/ See also BASIC POLYNOMIAL SEQUENCE References Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684 /C1/60, 1973. Umbrella WHITNEY UMBRELLA Unambiguous WELL DEFINED Unbiased Estimator A quantity which does not exhibit BIAS.An ESTIMATOR ˆu is an unbiased estimator of u if ˆu/C10/C11 /C30 u: See also BIAS (ESTIMATOR ), BIASED ESTIMATOR , ESTIMATOR , K-STATISTIC Unbounded See also BOUNDED Uncia 1 uncia /C131 12: The word uncia was Latin for a unit equal to 1/12 of another unit called the as. The words "inch" (1/12 of a foot) and "ounce" (originally 1/12 of a pound and still 1/12 of a "Troy pound," now used primarily to weigh precious metals) are derived from the word uncia. See also CALCUS ,H ALF,Q UARTER ,SCRUPLE ,U NIT FRACTION References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 4, 1996.Uncorrelated Variables xi and xj are said to be uncorrelated if their COVARIANCE is zero: cov xi ; xj/C0/C1 /C300: INDEPENDENT STATISTICS are always uncorrelated, but the converse is not necessarily true. See also COVARIANCE ,INDEPENDENT STATISTICS , UNCORRELATED NUMBERS Uncorrelated Numbers A sequence of numbers an is said to be uncorrelated if it satisfies lim n 0/C121 2nXn m/C30/C28na2 m /C301 lim n0/C121 2nXn m/C30/C28nam ak /C27m /C300 for k "0 :/ See also WIENER NUMBERS References Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. Uncountable Set UNCOUNTABLY INFINITE Uncountably Infinite An INFINITE SET, such as the real numbers, which is not COUNTABLY INFINITE . See also ALEPH-0 ,ALEPH-1 ,COUNTABLE SET,COUN- TABLY INFINITE ,FINITE ,INFINITE References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991. Undecagon HENDECAGON Undecidable Not DECIDABLE as a result of being neither formally provable nor unprovable. See also GO¨ DEL’S INCOMPLETENESS THEOREM ,R I- CHARDSON’S THEOREM Undecillion In the American system, 1036. See also LARGE NUMBER Undefined An expression in mathematics which does not have meaning and so which is not assigned an interpreta- tion. For example, DIVISION BY ZERO is undefined in the FIELD of REAL NUMBERS . See also AMBIGUOUS ,DIVISION BY ZERO,ILL DEFINED , INDETERMINATE ,W ELL DEFINED Underbar UNDERSCORE Underbrace BRACE Underdamping DAMPED SIMPLE HARMONIC MOTION– UNDERDAMPING Underdot A dot placed under a symbol to indicate a DUMMY VARIABLE , e.g., ˙c1 (Comtet 1974, p. 32). This notation, however, is not very common. See also DUMMY VARIABLE References Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 32, 1974. Underlying Space The space ½K ½ which is the subset of Rn that is the union of the simplices in a SIMPLICIAL COMPLEX K. The term POLYTOPE is sometimes used as a synonym for underlying space (Munkres 1991, p. 8). See also POLYHEDRON ,POLYTOPE References Munkres, J. R. Analysis on Manifolds. Reading, MA: Ad- dison-Wesley, 1991. Underscore A horizontal line placed under a symbol to indicate some special property. Underscores are sometimes used instead of over-arrows or bold typeface to indicate a VECTOR , for example x /C30¯x:/ References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 286, 1997. Undetermined Coefficients Method Given a nonhomogeneous ORDINARY DIFFERENTIAL EQUATION , select a differential operator which will annihilate the right side, and apply it to both sides. Find the solution to the homogeneous equation, plug it into the left side of the original equation, and solvefor constants by setting it equal to the right side. The solution is then obtained by plugging the determined constants into the homogeneous equation. See also ORDINARY DIFFERENTIAL EQUATION Undirected Graph A GRAPH for which the relations between pairs of vertices are symmetric, so that each edge has no directional character (as opposed to a DIRECTED GRAPH ). Unless otherwise indicated by context, the term "graph" can usually be taken to mean "undir- ected graph." See also DEGREE SEQUENCE ,D IRECTED GRAPH , GRAPH References Skiena, S. "Undirected Graphs." §3.2.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 92 /C1/3, 1990. Undulating Number A number OF THE FORM aba /C1/C1/C1;abab /C1/C1/C1;etc. The first few nontrivial undulants (with the stipulation that a"b) are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, ... (Sloane’s A046075). Including the trivial1- and 2-digit undulants and dropping the require-ment that a"bgives Sloane’s A033619. The first few undulating SQUARES are 121, 484, 676, 69696, ... (Sloane’s A016073), with no larger suchnumbers of fewer than a million digits (Pickover1995). Several tricks can be used to speed the search for square undulating numbers, especially by exam- ining the possible patterns of ending digits. Forexample, the only possible sets of four trailing digits for undulating SQUARES are 0404, 1616, 2121, 2929, 3636, 6161, 6464, 6969, 8484, and 9696. The only undulating POWER np/C30aba /C1/C1/C1for 35p531 and up to 100 digits is 73/C30343 (Pickover 1995). A large undulating prime is given by 7 /C27 720 10049/C281 ðÞ =99 (Pickover 1995). A binary undulant is a POWER of 2 whose base-10 representation contains one or both of the sequences 010 /C1/C1/C1and 101 /C1/C1/C1:The first few are 2nforn/C30103, 107, 138, 159, 179, 187, 192, 199, 205, ... (Sloane’sA046076). The smallest nfor which an undulating sequence of exactly d -digit occurs for d/C303, 4, ... are n/C30103,138,875,949,6617,1802,14545, ... (Sloane’s A046077). An undulating binary sequence of length10 occurs for n/C301;748;219 (Pickover 1995). References Pickover, C. A. "Is There a Double Smoothly Undulating Integer?" In Computers, Pattern, Chaos and Beauty. New York: St. Martin’s Press, 1990. Pickover, C. A. "The Undulation of the Monks." Ch. 20 in Keys to Infinity. New York: W. H. Freeman, pp. 159 /C1/61 1995. Sloane, N. J. A. Sequences A016073, A033619, A046075, A046076, and A046077 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.- att.com/~njas/sequences/eisonline.html. Unduloid A SURFACE OF REVOLUTION with constant NONZERO MEAN CURVATURE also called an ONDULOID .Itisa ROULETTE obtained from the path described by the FOCI of a CONIC SECTION when rolled on a LINE. This curve then generates an unduloid when revolved about the LINE. These curves are special cases of the shapes assumed by soap film spanning the gap between prescribed boundaries. The unduloid of a PARABOLA gives a CATENOID . See also CALCULUS OF VARIATIONS ,CATENOID ,ROUL- ETTE ,SURFACE OF REVOLUTION References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 48, 1989. Delaunay, C. "Sur la surface de re´volution dont la courbure moyenne est constante." J. math. pures appl. 6, 309 /C1/20, 1841. do Carmo, M. P. "The Onduloid." §3.5G in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 47 /C1/8, 1986. Fischer, G. (Ed.). Plate 97 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 93, 1986. Thompson, D’A. W. On Growth and Form, 2nd ed., compl. rev. ed. New York: Cambridge University Press, 1992. Yates, R. C. A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, p. 184, 1952. Unequal Two quantities a and b which are not equal are said to be unequal, and this relationship can be denoted a "b :/ See also EQUAL ,INEQUALITY References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 286, 1997. Unexpected Hanging Paradox A PARADOX also known as the SURPRISE EXAMINATION PARADOX or PREDICTION PARADOX . A prisoner is told that he will be hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur before it happens. He cannot be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason,and the same argument shows that he cannot be hanged on any other day. Nevertheless, the execu- tioner unexpectedly arrives on some day other than Friday, surprising the prisoner. This PARADOX is similar to that in Robert Louis Stevenson’s "BOTTLE IMP PARADOX ," in which you are offered the opportunity to buy, for whatever price you wish, a bottle containing a genie who will fulfill your every desire. The only catch is that the bottle must thereafter be resold for a price smaller than what you paid for it, or you will be condemned to live out the rest of your days in excruciating torment. Obviously, no one would buy the bottle for 1¢ since he would have to give the bottle away, but no one would accept the bottle knowing he would be unable to get rid of it. Similarly, no one would buy it for 2¢, and so on. However, for some reasonably large amount, it will always be possible to find a next buyer, so the bottle will be bought (Paulos 1995). See also BOTTLE IMP PARADOX ,SORITES PARADOX References Chow, T. Y. "The Surprise Examination or Unexpected Hanging Paradox." Amer. Math. Monthly 105,41/C1/1, 1998. Clark, D. "How Expected is the Unexpected Hanging?" Math. Mag. 67,55/C1/8, 1994. Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 158 /C1/59, 1998. Gardner, M. "The Paradox of the Unexpected Hanging." Ch. 1 in The Unexpected Hanging and Other Mathema- tical Diversions. Chicago, IL: Chicago University Press, pp. 11 /C1/3, 1991. Margalit, A. and Bar-Hillel, M. "Expecting the Unexpected." Philosophia 13, 263 /C1/88, 1983. Pappas, T. "The Paradox of the Unexpected Exam." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 147, 1989. Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, p. 97, 1995. Quine, W. V. O. "On a So-Called Paradox." Mind 62,65/C1/7, 1953. Unfair Game AGAME in which a certain player can always win when he plays properly. All CATEGORICAL GAMES are unfair (Steinhaus 1983, p. 16). See also CATEGORICAL GAME,GAME References Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Unfinished Game SHARING PROBLEM Unfolding In 1987, K. Fukuda conjectured that no convex polyhedra admit a self-overlapping unfolding The above figure show a counterexample to conjecture 1 found by M. Namiki. A tetrahedron which is also ununfoldable was subsequently found. Fukuda also conjectured that every CONVEX POLYHE- DRON can be uniquely constructed from any of its unfolding. The counterexample shown above was found by T. Matsui. The question of whether every CONVEX POLYHEDRON admits a self-unoverlapping unfolding is still un- settled. See also NET,POLYHEDRON References Bern, M.; Demaine, E. D.; Eppstein, D.; and Kuo, E. Ununfoldable Polyhedra. 3 Aug 1999. http://xxx.lanl.gov/ abs/cs.CG/9908003/. Unhappy Number A number which is not HAPPY is said to be unhappy. See also HAPPY NUMBERUnicursal Circuit A CIRCUIT in which an entire GRAPH is traversed in one route. An example of a curve which can be traced unicursally is the MOHAMMED SIGN. See also CIRCUIT ,EULERIAN CIRCUIT ,K O¨ NIGSBERG BRIDGE PROBLEM References Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 223 /C1/24, 1930. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 256 /C1/57, 1999. Unicyclic Graph ACONNECTED GRAPH containing exactly one cycle (Harary 1994, p. 41). References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Unidecagon HENDECAGON Uniform Apodization Function An APODIZATION FUNCTION f(x)/C301; (1) having INSTRUMENT FUNCTION I(x)/C30ga /C28ae/C282pikxdx/C30/C281 2pike/C282pika/C28e2pikx/C0/C1 /C30sin(2 pka) pk/C302asinc(2 pka): (2) The peak (in units of a) is 2. The extrema are given by letting b/C132pkaand solving d db(bsinb)/C30sinb/C28bcosb b2/C300 (3) sinb/C28bcosb/C300 (4) tanb/C30b: (5) Solving this numerically gives b0/C300;b1/C304:49341 ; b2/C307:72525 ;...for the first few solutions. The second of these is the peak POSITIVE sidelobe, and the third is the peak NEGATIVE sidelobe. As a fraction of the peak, they are 0.128375 and /C280:217234 :The FULL WIDTH AT HALF MAXIMUM is found by setting /I ðxÞ¼1 sinc( x) /C301 2 ; (6) and solving for x1=2 ; yielding x1 =2 /C302 pk1 =2a /C301:89549 : (7) Therefore, with L /C132a; FWHM /C302k1 =2 /C300 :603353 a/C301:20671 L: (8) See also APODIZATION FUNCTION Uniform Boundedness Principle A "pointwise-bounded" family of continuous linear OPERATORS from a BANACH SPACE to a NORMED SPACE is "uniformly bounded." Symbolically, if sup Ti(x) kk is FINITE for each x in the unit BALL , then sup Tikk is FINITE . The theorem is also called the BANACH- STEINHAUS THEOREM . References Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995. Uniform Convergence A SERIES a/C12 n/C301 un(x) is uniformly convergent to S(x) for a set E of values of x if, for each e > 0 ; an INTEGER N can be found such that Sn(x) /C28S(x) jj B e (1) for n ]N and all x /C23 E : To test for uniform conver- gence, use ABEL’S UNIFORM CONVERGENCE TEST or the WEIERSTRASS M-TEST . If individual terms un(x)ofa uniformly converging series are continuous, then 1. The series sum f(x) /C30X/C12 n/C301un(x) (2) is continuous, 2. The series may be integrated term by term gb af(x)dx/C30X/C12 n/C301gb aun(x)dx; (3) and 3. The series may be differentiated term by term d dxf(x)/C30X/C12 n/C301d dxun(x): (4) See also ABEL’S CONVERGENCE THEOREM ,A BEL’S UNIFORM CONVERGENCE TEST,W EIERSTRASS M-TESTReferences Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 299 /C1/01, 1985. Jeffreys, H. and Jeffreys, B. S. "Uniform Convergence of Sequences and Series" et seq. §1.112/C1/.1155 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cam- bridge University Press, pp. 37 /C1/3, 1988. Knopp, K. "Uniform Convergence." §18 in Theory of Func- tions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 71 /C1/3, 1996. Uniform Convexity This entry contributed by R ONALD M.AARTS To each e>0;there corresponds a dsuch that ½½f/C28 g½½Bewhenever ½½f½½/C30½½g½½/C301 and ½½(f/C27g)=2½½>1/C28d: This is a geometric property of the UNIT SPHERE of space: if the MIDPOINT of a LINE SEGMENT with endpoints on the surface of the sphere approaches the surface, then the endpoints must come closertogether (Cheney 1999). References Cheney, E. W. Introduction to Approximation Theory, 2nd ed.Providence, RI: Amer. Math. Soc., 1999. Uniform Distribution A distribution which has constant probability is called a uniform distribution, sometimes also called aRECTANGULAR DISTRIBUTION . The probability density function and cumulativedistribution function for a continuous uniform dis- tribution are P(x)/C30 1 b/C28aforaBxBb 0 for xBa;x>b8 < :(1) D(x)/C300 for xBa x/C28a b/C28afora5xBb 1 for x]b:8 >>< >>:(2) With a/C300 and b/C301, these can be written P(x)/C30Px/C271 2/C16/C17 (3) /C301 2[sgn( x)/C28sgn(x/C281)] (4) /C30H(x)/C28H(x/C281) (5) D(x)/C30xH(x)/C27(x/C281)H(x/C281); (6) where P(x) is the RECTANGLE FUNCTION and H(x)i s the H EAVISIDE STEP FUNCTION . For a continuous uniform distribution, the CHARAC- TERISTIC FUNCTION is f(t)/C302 (b/C28a)tsin1 2(b/C28a)thi ei(a/C27b)t=2; (7) and the MOMENT-GENERATING FUNCTION is M(t)/C30exthi/C30gb aext b/C28adx/C30ext t(b/C28a)"#b a; (8) so M(t)/C30etb/C28eta t(b/C28a)fort"0 0 for t/C300;8 < :(9) and M?(t)/C301 b/C28a1 tbebt/C28aeat/C0/C1 /C281 t2ebt/C28eat/C0/C1"# /C30ebt(bt/C281)/C28eat(at/C281) (b/C28a)t2: (10) Ifa/C300 and b/C301, the CHARACTERISTIC FUNCTION simplifies to f(t)/C302 sin1 2t/C16/C17 eit=2 t/C30i/C28icost/C27sint t: (11) The MOMENT-GENERATING FUNCTION is not differenti- able at zero, but the MOMENTS can be calculated by differentiating and then taking limt00:The RAW MOMENTS are given by m?1/C301 2(a/C27b) (12) m?2/C3013a2/C27ab/C27b2/C0/C1 (13) m?3/C3014(a/C27b)a2/C27b2/C0/C1 (14) m?4/C3015a4/C27a3b/C27a2b2/C27ab3/C27b4/C0/C1 : (15) The CENTRAL MOMENTS are then m1/C300 (16) m2/C301 12(b/C28a)2(17) m3/C300 (18) m4/C301 80(b/C28a)4; (19) so the MEAN ,VARIANCE ,SKEWNESS , and KURTOSIS are m/C3012(a/C27b) (20)s2¼m2¼1 12ðb/C28aÞ2ð21Þ g1/C30m3 s3=2/C300 (22) g2/C30/C286 5: (23) The distribution for the sum of nuniform variates on the interval [0 ;1] get be found using the CHARACTER- ISTIC FUNCTION as Pn(x)/C30F/C281i/C28cost/C27sint t !n "# (24) /C301 2(n/C281)!Xn k/C300(/C281)kn k/C18/C19 (x/C28k)n/C281sgn(x/C28k);(25) where the Fourier parameters are taken as (1 ;1): The first few values of Pn(x) then give P1(x)/C301 2[sgn(1 /C28x)/C27sgnx] (26) P2(x)/C3012[(/C282/C27x) sgn(/C282/C27x) /C282(/C281/C27x) sgn(/C281/C27x)/C27xsgnx] (27) P3(x)/C301 4[/C28(/C283/C27x)2sgn(/C283/C27x) /C273(/C282/C27x)2sgn(/C282/C27x) /C283(/C281/C27x)2sgn(/C281/C27x)/C27x2sgnx] (28) P4(x)/C301 12[(/C284/C27x)3sgn(/C284/C27x) /C284(/C283/C27x)3sgn(/C283/C27x) /C276(/C282/C27x)3sgn(/C282/C27x) /C284(/C281/C27x)3sgn(/C281/C27x)/C27x3sgnx]; (29) illustrated above. The probability distribution function and cumulative distributions function for a discrete uniform distribu- tion are P(n)/C301 N(30) D(n)/C30n N(31) forn/C301, ..., N. The MOMENT-GENERATING FUNCTION is M(t) /C30 enthi/C30XN n/C3011 Nent /C301 Net /C28 et(N /C271) 1 /C28 et /C30et 1 /C28 eNtðÞ N 1 /C28 et ðÞ: (32) The MOMENTS about 0 are m?m /C301 NXN n/C301nm ; (33) so m ?1 /C301 2(N /C271) (34) m?2 /C3016(N /C271)(2N /C271) (35) m?3 /C3014 N(N /C271)2 (36) m?4 /C301 30(N /C271)(2N /C271) 3N2 /C273N /C281/C0/C1 ; (37) and the MOMENTS about the MEAN are m2 /C301 12(N /C281)(N /C271) (38) m3 /C300 (39) m4 /C301 240(N /C281)(N /C271) 3N2 /C287/C0/C1 : (40) The MEAN , VARIANCE , SKEWNESS , and KURTOSIS are m /C301 2(N /C271) (41) s2 /C30 m2 /C301 12(N /C281)(N /C271) (42) g1 /C30m3 s3 =2 /C300 (43) g2 /C306 N2 /C27 1 ðÞ 5(N /C28 1)(N /C27 1) : (44) See also EQUIDISTRIBUTED SEQUENCE ,RANDOM NUM- BER,RECTANGLE FUNCTION References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 531 and 533, 1987. Uniform Polychoron A 4-D analog of the UNIFORM POLYHEDRA . In fact, the UNIFORM POLYHEDRA are cells of the uniform poly- chora. There are more than 8000 known uniform polychora. The vertex figures of uniform polychora are always vertex-inscriptable in hyperspheres. See also POLYCHORONReferences Olshevsky, G. "Uniform Polytopes in Four Dimensions." http://members.aol.com/Polycell/uniform.html. Uniform Polyhedron The uniform polyhedra are POLYHEDRA with identical VERTICES . Badoureau discovered 37 nonconvex uni- form polyhedra in the late nineteenth century, many previously unknown (Wenninger 1983, p. 55). Cox-eter et al. (1954) conjectured that there are 75 such polyhedra in which only two faces are allowed to meetat an EDGE , and this was subsequently proven. (However, when any EVEN number of faces may meet, there are 76 polyhedra.) If the five pentagonal PRISMS are included, the number rises to 80. The VERTICES of a uniform polyhedron all lie on a SPHERE whose center is their CENTROID . The VERTICES joined to another VERTEX lie on a CIRCLE . Except for a single non-Wythoffian case, uniformpolyhedra can be generated by Wythoff’s kaleido-scopic method of construction. In this construction, an initial vertex inside a special SPHERICAL TRIANGLE PQR is mapped to all the other vertices by repeated reflections across the three planar sides of this triangle. Similarly, PQR and its kaleidoscopic images must cover the sphere an integral number of timeswhich is referred to as the density dofPQR . The density d/C211 is dependent on the choice of angles p=p;p=q;p=ratP,Q,Rrespectively, where p,q,r are reduced rational numbers greater than one. Sucha spherical triangle is called a S CHWARZ TRIANGLE , conveniently denoted ( pqr):Except for the infinite dihedral family of ( p22) for p/C302, 3, 4, ..., there are only 44 kinds of Schwarz triangles (Coxeter et al. 1954, Coxeter 1973). It has been shown that thenumerators of p,q,rare limited to 2, 3, 4, 5 (4 and 5 cannot occur together) and so the nine choices forrational numbers are: 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, 5/4 (Messer 1999). The names of the uniform polyhedra were first formalized in Wenninger (1971), based on a listprepared by N. Johnson a few years earlier, as slightly modified by D. Luke. The names of the uniform duals appeared in Wenninger (1983), againbased on nomenclature suggested by Johnson. John- son also suggested a few modifications in the original nomenclature to incorporate some additionalthoughts, as well as to undo some of Luke’s less felicitous changes. The "List of polyhedra and dual models" in Wenninger (1983) gives revised names forseveral of the uniform polyhedra. Source code and binary programs for generating and viewing the uniform polyhedra are also available at http://www.math.technion.ac.il/~rl/kaleido/. The fol- lowing depictions of the polyhedra were produced byR. Maeder’s UniformPolyhedra.m package for Mathematica . In this package, uniform polyhedra are computed to the desired numerical precision by numerically solving the definition fundamental equa- tion, and lengths are normalized to give a MIDRADIUS of /r¼1/. Due to a limitation in Mathematica ’s renderer, uniform polyhedra 69, 72, 74, and 75 cannot be displayed using this package (Maeder 1993). The following table gives the names of the uniform polyhedra and their duals as given in Wenninger (1971). Coxeter et al. (1954) give many properties of the uniform solids, and Coxeter et al. (1953), Johnson (2000) and Messer give the quartic equation for determining the central angle subtending half an edge. The single non-Wythoffian case is the GREAT DIRHOMBICOSIDODECAHEDRON U75which has pseudo- WYTHOFF SYMBOL ½3=25=335 =2:/ nWYTHOFF SYMBOLName DUAL POLYHEDRON 1/3½23/ TETRAHEDRON TETRAHEDRON 2/23½3/ TRUNCATED TETRAHEDRON TRIAKIS TETRAHEDRON 3/3=23½3/ OCTAHEMIOCTAHEDRON OCTAHEMIOCTACRON 4/3=23½2/ TETRAHEMIHEXAHEDRON TETRAHEMIHEXACRON 5/4½23/ OCTAHEDRON CUBE 6/3½24/ CUBE OCTAHEDRON 7/2½34/ CUBOCTAHEDRON RHOMBIC DODECAHEDRON 8/24½3/ TRUNCATED OCTAHEDRON TETRAKIS HEXAHEDRON 9/23½4/ TRUNCATED CUBE TRIAKIS OCTAHEDRON 10 /34½2/ SMALL RHOMBICUBOCTAHEDRON DELTOIDAL ICOSITETRAHEDRON 11 /234 ½/ TRUNCATED CUBOCTAHEDRON DISDYAKIS DODECAHEDRON 12 /½234 / SNUB CUBE PENTAGONAL ICOSITETRAHE- DRON 13 /3=24½4/ SMALL CUBICUBOCTAHEDRON SMALL HEXACRONIC ICOSI- TETRAHEDRON 14 /34½4=3/ GREAT CUBICUBOCTAHEDRON GREAT HEXACRONIC ICOSI- TETRAHEDRON 15 /4=34½3/ CUBOHEMIOCTAHEDRON HEXAHEMIOCTACRON 16 /4=334 ½/ CUBITRUNCATED CUBOCTAHE- DRONTETRADYAKIS HEXAHEDRON 17 /3=24½2/ GREAT RHOMBICUBOCTAHEDRON GREAT DELTOIDAL ICOSITETRA- HEDRON 18 /3=224 ½/ SMALL RHOMBIHEXAHEDRON SMALL RHOMBIHEXACRON 19 /23½4=3/ STELLATED TRUNCATED HEXAHE-DRONGREAT TRIAKIS OCTAHEDRON 20 /4=323 ½/ GREAT TRUNCATED CUBOCTAHE-DRONGREAT DISDYAKIS DODECAHE-DRON 21 /4=33=22½/ GREAT RHOMBIHEXAHEDRON GREAT RHOMBIHEXACRON 22 /5½23/ ICOSAHEDRON DODECAHEDRON 23 /3½25/ DODECAHEDRON ICOSAHEDRON 24 /2½35/ ICOSIDODECAHEDRON RHOMBIC TRIACONTAHEDRON 25 /25½3/ TRUNCATED ICOSAHEDRON PENTAKIS DODECAHEDRON 26 /23½5/ TRUNCATED DODECAHEDRON TRIAKIS ICOSAHEDRON 27 /35½2/ SMALL RHOMBICOSIDODECAHE-DRONDELTOIDAL HEXECONTAHE-DRON 28 /235 ½/ TRUNCATED ICOSIDODECAHEDRON DISDYAKIS TRIACONTAHEDRON 29 /½235 / SNUB DODECAHEDRON PENTAGONAL HEXECONTAHE- DRON30 /3½5=23/ SMALL DITRIGONAL ICOSIDODECA-HEDRONSMALL TRIAMBIC ICOSAHEDRON 31 /5=23½3/ SMALL ICOSICOSIDODECAHEDRON SMALL ICOSACRONIC HEXECON- TAHEDRON 32 /½5=233 / SMALL SNUB ICOSICOSIDODECA-HEDRONSMALL HEXAGONAL HEXECON-TAHEDRON 33 /3=25½5/ SMALL DODECICOSIDODECAHE-DRONSMALL DODECACRONIC HEXE-CONTAHEDRON 34 /5½25=2/ SMALL STELLATED DODECAHE-DRONGREAT DODECAHEDRON 35 /5=2½25/ GREAT DODECAHEDRON SMALL STELLATED DODECAHE- DRON 36 /2½5=25/ DODECADODECAHEDRON MEDIAL RHOMBIC TRIACONTA- HEDRON 37 /25=2½5/ TRUNCATED GREAT DODECAHE-DRONSMALL STELLAPENTAKIS DO-DECAHEDRON 38 /5=25½2/ RHOMBIDODECADODECAHEDRON MEDIAL DELTOIDAL HEXECON- TAHEDRON 39 /25=25½/ SMALL RHOMBIDODECAHEDRON SMALL RHOMBIDODECACRON 40 /½25=25/ SNUB DODECADODECAHEDRON MEDIAL PENTAGONAL HEXE- CONTAHEDRON 41 /3½5=35/ DITRIGONAL DODECADODECAHE-DRONMEDIAL TRIAMBIC ICOSAHE-DRON 42 /35½5=3/ GREAT DITRIGONAL DODECICOSI-DODECAHEDRONGREAT DITRIGONAL DODECA-CRONIC HEXECONTAHEDRON 43 /5=33½5/ SMALL DITRIGONAL DODECICOSI-DODECAHEDRONSMALL DITRIGONAL DODECA-CRONIC HEXECONTAHEDRON 44 /5=35½3/ ICOSIDODECADODECAHEDRON MEDIAL ICOSACRONIC HEXE- CONTAHEDRON 45 /5=335 ½/ ICOSITRUNCATED DODECADODE-CAHEDRONTRIDYAKIS ICOSAHEDRON 46 /½5=335 / SNUB ICOSIDODECADODECAHE-DRONMEDIAL HEXAGONAL HEXECON-TAHEDRON 47 /3=2½35/ GREAT DITRIGONAL ICOSIDODECA-HEDRONGREAT TRIAMBIC ICOSAHEDRON 48 /3=25½3/ GREAT ICOSICOSIDODECAHEDRON GREAT ICOSACRONIC HEXECON- TAHEDRON 49 /3=23½5/ SMALL ICOSIHEMIDODECAHEDRON SMALL ICOSIHEMIDODECACRON 50 /3=235 ½/ SMALL DODECICOSAHEDRON SMALL DODECICOSACRON 51 /5=45½5/ SMALL DODECAHEMIDODECAHE-DRONSMALL DODECAHEMIDODECA-CRON 52 /3½25=2/ GREAT STELLATED DODECAHE-DRONGREAT ICOSAHEDRON 53 /5=2½23/ GREAT ICOSAHEDRON GREAT STELLATED DODECAHE- DRON 54 /2½5=23/ GREAT ICOSIDODECAHEDRON GREAT RHOMBIC TRIACONTAHE- DRON 55 /25=2½3/ GREAT TRUNCATED ICOSAHEDRON GREAT STELLAPENTAKIS DO- DECAHEDRON 56 /25=23½/ RHOMBICOSAHEDRON RHOMBICOSACRON 57 /½25=23/ GREAT SNUB ICOSIDODECAHE-DRONGREAT PENTAGONAL HEXECON-TAHEDRON 58 /25½5=3/ SMALL STELLATED TRUNCATEDDODECAHEDRONGREAT PENTAKIS DODECAHE-DRON 59 /5=325 ½/ TRUNCATED DODECADODECAHE-DRONMEDIAL DISDYAKIS TRIACONTA-HEDRON 60 /½5=325 / INVERTED SNUB DODECADODECA-HEDRONMEDIAL INVERTED PENTAGO-NAL HEXECONTAHEDRON 61 /5=23½5=3/ GREAT DODECICOSIDODECAHE-DRONGREAT DODECACRONIC HEXE-CONTAHEDRON 62 /5=35=2½3/ SMALL DODECAHEMICOSAHEDRON SMALL DODECAHEMICOSACRON 63 /5=35=23½/ GREAT DODECICOSAHEDRON GREAT DODECICOSACRON 64 /½5=35=23/ GREAT SNUB DODECICOSIDODECA- HEDRONGREAT HEXAGONAL HEXECON-TAHEDRON 65 /5=45½3/ GREAT DODECAHEMICOSAHEDRON GREAT DODECAHEMICOSACRON 66 /23½5=3/ GREAT STELLATED TRUNCATEDDODECAHEDRONGREAT TRIAKIS ICOSAHEDRON 67 /5=33½2/ GREAT RHOMBICOSIDODECAHE-DRONGREAT DELTOIDAL HEXECONTA-HEDRON 68 /5=323 ½/ GREAT TRUNCATED ICOSIDODECA-HEDRONGREAT DISDYAKIS TRIACONTA-HEDRON 69 /½5=323 / GREAT INVERTED SNUB ICOSIDO-DECAHEDRONGREAT INVERTED PENTAGONALHEXECONTAHEDRON 70 /5=35=2½5=3/ GREAT DODECAHEMIDODECAHE-DRONGREAT DODECAHEMIDODECA-CRON 71 /3=23½5=3/ GREAT ICOSIHEMIDODECAHEDRON GREAT ICOSIHEMIDODECACRON 72 /½3=23=25=2/ SMALL RETROSNUB ICOSICOSIDO-DECAHEDRONSMALL HEXAGRAMMIC HEXE-CONTAHEDRON 73 /3=25=32½/ GREAT RHOMBIDODECAHEDRON GREAT RHOMBIDODECACRON 74 /½3=25=32/ GREAT RETROSNUB ICOSIDODECA-HEDRONGREAT PENTAGRAMMIC HEXE-CONTAHEDRON 75 /½3=25=33/5/ 2GREAT DIRHOMBICOSIDODECAHE-DRONGREAT DIRHOMBICOSIDODECA-CRON 76 /25½2/ PENTAGONAL PRISM PENTAGONAL DIPYRAMID 77 /½225 / PENTAGONAL ANTIPRISM PENTAGONAL DELTAHEDRON 78 /25=2½2/ PENTAGRAMMIC PRISM PENTAGRAMMIC DIPYRAMID 79 /½225 =2/ PENTAGRAMMIC ANTIPRISM PENTAGRAMMIC DELTAHEDRON 80 /½225 =3/ PENTAGRAMMIC CROSSED ANTI-PRISMPENTAGRAMMIC CONCAVEDELTAHEDRON Johnson (2000) proposed a further revision of the "official" names of the uniform polyhedra and their duals and, at the same time, devised a literal symbol for each uniform polyhedron. For each uniformpolyhedron, Johnson (2000) gives its number inWenninger (1971), a modified S CHLA ¨FLI SYMBOL (following Coxeter), a literal symbol, and its newdesignated name. Not every uniform polyhedron hasa dual that is free from anomalies like coincident vertices or faces extending to infinity. For those that do, Johnson gives the name of the dual polyhedron. InJohnson’s new system, the uniform polyhedra are classified as follows: 1. Regular (regular polygonal vertex figures), 2. Quasi-regular (rectangular or ditrigonal vertex figures), 3. Versi-regular (orthodiagonal vertex figures),4. Truncated regular (isosceles triangular vertex figures), 5. Quasi-quasi-regular (trapezoidal vertex figures),6. Versi-quasi-regular (dipteroidal vertex figures), 7. Truncated quasi-regular (scalene triangular vertex figures),8. Snub quasi-regular (pentagonal, hexagonal, or octagonal vertex figures), 9. Prisms (truncated hosohedra),10. Antiprisms and crossed antiprisms (snubdihedra) Here is a brief description of Johnson’s symbols for the uniform polyhedra (Johnson). The star operator + appended to "D" or "E" replaces pentagons f5gby pentagrams f5=2g:The bar operator ½indicates the removal from a related figure of a set (or sets) of faces, leaving "holes" so that a different set of faces takes their place. Thus, C /½/O is obtained from the cubocta- hedron CO by replacing the eight triangles by four hexagons. In like manner, rR’ /½/CO has the twelve squares of the rhombicuboctahedron rCO and the six octagons of the small cubicuboctahedron R’CO but has holes in place of their six squares and eight triangles. The operator "r" stands for "rectified": apolyhedron is truncated to the midpoints of the edges. Operators "a", "b", and "c" in the S CHLA ¨FLI SYMBOLS for the ditrigonary (i.e., having ditrigonal vertex figures) polyhedra stand for "altered," "blended," and "converted." The operator "o" stands for "ossified" (after S. L. van Oss). Operators "s" and "t" stand for "simiated" (snub) and "truncated." Primes and capital letters are used for certain operators analogous to those just mentioned. Forinstance, rXY is the "rhombi-XY," with the faces of the quasi-regular XY supplemented by a set of square "rhombical" faces. The isomorphic r’XY has a crossedvertex figure. The operators "R" and "R’" denote a supplementary set of faces of a different kind– hexagons, octagons or octagrams, decagons or deca-grams. Likewise, the operators "T" and "S" indicate the presence of faces other than, or in addition to, those produced by the simpler operators "t" and "s".The vertex figure of s’XY, the "vertisnub XY", is a crossed polygon, and that of s*XY, the "retrosnub XY", has density 2 relative to its circumcenter. Regular polyhedra: p q/ 1 /f3;3g/T Tetrahedron Tetrahedron 2 /f3;4g/O Octahedron Cube 3 /f4;3g/C Cube Octahedron 4 /f3;5g/I Icosahedron Dodecahedron5 /f5;3g/D Dodecahedron Icosahedron 20 /f5=2;5g/D* Small stellated dodecahedronGreat dodeca- hedron 21 /f5;5=2g/E Great dodecahe- dronSmall stellateddodecahedron 22 /f5=2;3g/E* Great stellated dodecahedronGreat icosa- hedron 41 /f3;5=2g/J Great icosahe- dronGreat stellated dodecahedron Quasi-regular polyhedra: ( p:q)r / 11 r /f3;4g/ CO Cuboctahedron Rhombic dodeca- hedron 12 r /f3;5g/ ID Icosidodecahedron Rhombic triacon- tahedron 73 r /f5=2;5g/ED* Dodecadodecahe- dronMiddle rhombic triacontahedron 94 r /f5=2;3g/JE* Great icosidodeca- hedronGreat rhombic triacontahedron 70 a /f5;3g/ ID* Small ditrigonary icosidodecahedronSmall triambic icosahedron 80 b /f5;5=2g/DE* Ditrigonary dode- cadodecahedronMiddle triambic icosahedron 87 c /f3;5=2g/JE Great ditrigonary icosidodecahedronGreat triambic icosahedron Versi-regular polyhedra: q:h:q:h/ 67 o /f3;3g/ T/½/T Tetrahemihexahedron no dual 78 o /f3;4g/ C/½/O Cubohemioctahedron no dual 68 o /f4;3g/ O/½/C Octahemioctahedron no dual 91 o /f3;5g/ D/½/I Small dodecahemidodeca- hedronno dual 89 o /f5;3g/ I/½/D Small icosahemidodeca- hedronno dual 102 o /f5=2;5g/E/½/D* Small dodecahemiicosa- hedronno dual 100 o /f5;5=2g/D*/½/E Great dodecahemiicosa- hedronno dual 106 o /f5=2;3g/J/½/E* Great icosahemidodeca- hedronno dual 107 o /f3;5=2g/E*/½/J Great dodecahemidodeca- hedronno dual Truncated regular polyhedra: q:2p:2p/ 6t /f3;3g/ tT Truncated tetra- hedronTriakis tetra-hedron 7t /f3;4g/ tO Truncated octa- hedronTetrakis hexa-hedron 8t /f4;3g/ tC Truncated cube Triakis octa- hedron 92 t’ /f4;3g/ t’C stellatruncated cubeGreat triakis octahedron 9t /f3;5g/ tI Truncated icosa- hedronPentakis do-decahedron 10 t /f5;3g/ tD Truncated do- decahedronTriakis icosa- hedron 97 t’ /f5=2;5g/t’D* Small stellatrun- cated dodecahe-dronGreat pentakis dodecahedron 75 t /f5;5=2g/tE Great truncated dodecahedronSmall stellapenta- kis dodecahedron 104 t’ /f5=2;3g/t’E* Great stellatrun- cated dodecahe- dronGreat triakis icosahedron 95 t /f3;5=2g/tJ Great truncated icosahedronGreat stellapenta- kis dodecahedron Quasi-quasi-regular polyhedra: p:2r:q:2rand p:2s:q:2s/ 13 rr /f3;4g/ rCO Rhombicubocta- hedronStrombic disdodeca- hedron 69 R’r /f3;4g/ R’CO Small cubicubocta- hedronSmall sagittal disdo- decahedron 77 Rr /f3;4g/ RCO Great cubicubocta- hedronGreat strombic dis- dodecahedron 85 r’r /f3;4g/ r’CO Great rhombicub- octahedronGreat sagittal disdo- decahedron 14 rr /f3;5g/ rID Rhombicosidodeca- hedronStrombic hexecon- tahedron 72 R’r /f3;5g/ R’ID Small dodekicosido- decahedronSmall sagittalhexecontahedron 71 ra /f5;3g/ rID* Small icosified icosi- dodecahedronSmall strombic trisicosahedron 82 R’a /f5;3g/ R’ID* Small dodekified ico- sidodecahedronSmall sagittal trisico- sahedron 76 rr /f5=2;5g/ rED* Rhombidodecado- decahedronMiddle strombic tri- sicosahedron 83 R’r /f5=2;5g/R’ED* Icosified dodecado- decahedronMiddle sagittal trisi- cosahedron 81 Rc /f3;5=2g/RJE Great dodekified ico- sidodecahedronGreat strombic trisi-cosahedron 88 r’c /f3;5=2g/r’JE Great icosified icosi- dodecahedronGreat sagittal trisico- sahedron 99 Rr /f5=2;3g/RJE* Great dodekicosido- decahedronGreat strombic hexe- contahedron 105 r’r /f5=2;3g/r’JE* Great rhombicosido- decahedronGreat sagittal hexe- contahedron Versi-quasi-regular polyhedra: 2 r:2s:2r:2s/ 86 or /f3;4g/ rR’/½/CO Small rhombi- cubeSmall dipteral disdodecahedron 103 Or /f3;4g/Rr’/½/CO Great rhombi- cubeGreat dipteraldisdodecahedron74 or /f3;5g/ rR’/½/ID Small rhombido- decahedronSmall dipteral hexecontahedron 90 oa /f5;3g/rR’/½/ID* Small dodekico- sahedronSmall dipteral trisicosahedron 96 or /f5=2;5g/rR’/½/ED* Rhombicosahe- dronMiddle dipteraltrisicosahedron 101 Oc /f3;5=2g/Rr’/½/JE Great dodekico- sahedronGreat dipteral trisicosahedron 109 Or /f5=2;3g/Rr’/½/JE* Great rhombido- decahedronGreat dipteral hexecontahedron Truncated quasi-regular polyhedra: 2 p:2q:2r/ 15 tr /f3;4g/ tCO Truncated cuboc- tahedronDisdyakis dode- cahedron 93 t’r /f3;4g/t’CO Stellatruncated cuboctahedronGreat disdyakis dodecahedron 79 Tr /f3;4g/TCO Cubitruncated cuboctahedronTrisdyakis octa- hedron 16 tr /f3;5g/ tID Truncated icosi- dodecahedronDisdyakis tria-contahedron 98 t’r /f5=2;5g/t’ED* Stellatruncated dodecadodecahe- dronMiddle disdyakis triacontahedron 84 T’r /f5=2;5g/T’ED* Icositruncated dodecadodecahe- dronTrisdyakis icosa- hedron 108 t’r /f5=2;3g/t’JE* Stellatruncated icosidodecahe-dronGreat disdyakis triacontahedron Snub quasi-regular polyhedra: p:3:q:3:3o r p:3:q:3:r:3/ 17 sr /f3;4g/ sCO Snub cuboctahedron Petaloidal disdodeca- hedron 18 sr /f3;5g/ sID Snub icosidodecahe- dronPetaloidal hexeconta- hedron 110 sa /f5;3g/ sID* Snub disicosidodeca- hedronno dual 118 s*a /f5;3g/ s*ID* Retrosnub disicosido- decahedronno dual 111 sr /f5=2;5g/ sED* Snub dodecadodeca- hedronPetaloidal trisicosa- hedron 114 s’r /f5=2;5g/s’ED* Vertisnub dodecado- decahedronVertipetaloidal trisi- cosahedron 112 S’r /f5=2;5g/S’ED* Snub icosidodecado- decahedronHexaloidal trisicosa- hedron 113 sr /f5=2;3g/ sJE* Great snub icosido- decahedronGreat petaloidal hex- econtahedron 116 s’r /f5=2;3g/s’JE* Great vertisnub ico- sidodecahedronGreat vertipetaloidal hexecontahedron 117 s*r /f5=2;3g/s*JE* Great retrosnub ico- sidodecahedronGreat retropetaloidal hexecontahedron Snub quasi-regular polyhedron: (p :4:q:4)2 / 119 SSr /f5 =2; 3g/ SSJE* Great disnub disicosidisdode- cahedronno dual Prisms: p:4 :4/ /fpgxfg/ P(p) p-gonal prism, p /C303, 5, 6, ...p-gonal bipyra- mid /fp =dgxfg/ P(p/d) d-fold p-gonal prism, p=d > 2/d-fold p-gonal bipyramid Antiprisms and crossed antiprisms: 3 :3 :3:p/ s/fpg/h/fg/ Q(p) p-gonal anti- prism, p /C304, 5, 6, ...p-gonal antibi- pyramid s/ fp=dg/h/fg/ Q(p/d) d-fold p-gonal antiprism, p=d > 2/d-fold p-gonal antibipyramid s’/ fp=dg/h/fg/ Q’(p/d) d-fold p-gonal crossed anti- prism, 2 Bp=d B3/d-fold p-gonal crossed antibi- pyramid See also ARCHIMEDEAN SOLID ,AUGMENTED POLYHE- DRON ,DUAL POLYHEDRON ,JOHNSON SOLID ,KEPLER- POINSOT SOLID,MO¨ BIUS TRIANGLES ,PLATONIC SOLID , POLYHEDRON ,SCHWARZ TRIANGLE ,U NIFORM POLY- CHORON ,VERTEX FIGURE ,W YTHOFF SYMBOL References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 136, 1987. Bru¨ckner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900. Bulatov, V. "Compounds of Uniform Polyhedra." http:// www.physics.orst.edu/~bulatov/polyhedra/uniform_com- pounds/. Bulatov, V. "Dual Uniform Polyhedra." http://www.physic- s.orst.edu/~bulatov/polyhedra/dual/. Bulatov, V. "Uniform Polyhedra." http://www.physics.or- st.edu/~bulatov/polyhedra/uniform/. Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401 /C1/50, 1954. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Har’El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47,57/C1/10, 1993. Har’El, Z. "Kaleido." http://www.math.technion.ac.il/~rl/ka- leido/. Har’El, Z. "Eighty Dual Polyhedra Generated by Kaleido." http://www.math.technion.ac.il/~rl/kaleido/dual.html. Har’El, Z. "Eighty Uniform Polyhedra Generated by Ka- leido." http://www.math.technion.ac.il/~rl/kaleido/ poly.html.Hume, A. "Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals." Computing Science Tech. - Rept. No. 130. Murray Hill, NJ: AT&T Bell Lab., 1986. Hume, A. Information files on polyhedra. http://netlib.bell- labs.com/netlib/polyhedra/. Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169 /C1/00, 1966. Johnson, N. W. Uniform Polytopes. Cambridge, England: Cambridge University Press, 2000. Maeder, R. E. "Uniform Polyhedra." Mathematica J. 3, 1993. ftp://ftp.inf.ethz.ch/doc/papers/ti/scs/unipoly.ps.gz. Maeder, R. E. Polyhedra.m and PolyhedraExamples Mathematica notebooks. http://www.inf.ethz.ch/depart- ment/TI/rm/programs.html. Maeder, R. E. "The Uniform Polyhedra." http://www.in- f.ethz.ch/department/TI/rm/unipoly/. Messer, P. W. "Closed-Form Expressions for Uniform Poly- hedra and Their Duals." Unpublished manuscript. Messer, P. W. "Problem 1094." Crux Math. 11, 325, 1985. Messer, P. W. "Solution to Problem 1094." Crux Math. 13, 133, 1987. Skilling, J. "The Complete Set of Uniform Polyhedron." Phil. Trans. Roy. Soc. London, Ser. A 278, 111 /C1/36, 1975. Sopov, S. P. "Proof of the Completeness of the Enumeration of Uniform Polyhedra." Ukrain. Geom. Sbornik 8, 139 /C1/56, 1970. Virtual Image. The Uniform Polyhedra CD-ROM. 1997. http://ourworld.compuserve.com/homepages/vir_image/html/uniformpolyhedra.html. Weisstein, E. W. "Polyhedron Duals." M ATHEMATICA NOTE- BOOK DUALS.M . Weisstein, E. W. "Uniform Polyhedra." MATHEMATICA NOTE- BOOK UNIFORM POLYHEDRA.M . Wenninger, M. J. Dual Models. Cambridge, England: Cam- bridge University Press, 1983. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, pp. 1 /C1/0 and 98, 1989. Zalgaller, V. Convex Polyhedra with Regular Faces. New York: Consultants Bureau, 1969. Ziegler, G. M. Lectures on Polytopes. Berlin: Springer- Verlag, 1995. Uniform Variate A RANDOM NUMBER which lies within a specified range (which can, without loss of generality, be taken as [0, 1]), with a UNIFORM DISTRIBUTION . References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Uniform Deviates." §7.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 267 /C1/77, 1992. Uniformization See also UNIFORMIZATION THEOREM Uniformization Theorem See also UNIFORMIZATION Uniformly Cauchy The series a/C12 j/C301 fj(z) is said to be uniformly Cauchy on compact sets if, for each compact K ⁄U and each e > 0; there exists an N /C210 such that for all M ]L > N ; XM j/C30Lfj(z)/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12B e holds (Krantz 1999, p. 104). References Krantz, S. G. "The Cauchy Condition for a Series." §8.1.5 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 104, 1999. Uniformly Distributed Sequence EQUIDISTRIBUTED SEQUENCE Unimodal Distribution A STATISTICAL DISTRIBUTION such as the GAUSSIAN DISTRIBUTION which has a single "peak." See also BIMODAL DISTRIBUTION Unimodal Sequence A finite SEQUENCE which first increases and then decreases. A SEQUENCE s1 ; s2 ; ...; sn fg is unimodal if there exists a t such that s1 5s2 5...5st and st ]st /C271 ]...]sn : Unimodular Group A GROUP whose left HAAR MEASURE equals its right HAAR MEASURE . See also HAAR MEASURE ,M ODULAR GROUP GAMMA , MODULAR GROUP GAMMA0 ,MODULAR GROUP LAMBDA References Knapp, A. W. "Group Representations and Harmonic Ana- lysis, Part II." Not. Amer. Math. Soc. 43, 537 /C1/49, 1996. Unimodular Matrix A MATRIX A with INTEGER elements and DETERMINANT det(A) /C3091 ; also called a UNIT MATRIX . The inverse of a unimodular matrix is another unimodular matrix. A POSITIVE unimodular matrix has det (A) /C30/C271: The nth POWER of a POSITIVE UNIMODULAR MATRIX M /C30m11m12 m21m22/C20/C21 (1) isMn /C30m11Un/C281(a) /C28Un/C282(a) m12Un/C281(a) m21Un/C281(a) m22Un/C281(a) /C28Un /C282(a)/C20/C21 ; (2) where a /C131 2m11 /C27m22 ðÞ (3) and the Unare CHEBYSHEV POLYNOMIALS OF THE SECOND KIND , Um(x) /C30sin (m /C27 1) cos/C281 x ½/C138ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 x2p : (4) See also CHEBYSHEV POLYNOMIAL OF THE SECOND KIND References Born, M. and Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. New York: Pergamon Press, p. 67, 1980. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 149, 1980. Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 162, 2000. Unimodular Transformation A transformation x ?/C30Ax is unimodular if the DETER- MINANT of the MATRIX A satisfies det(A) /C3091: A NECESSARY and SUFFICIENT condition that a linear transformation transform a lattice to itself is that the transformation be unimodular. If z is a COMPLEX NUMBER , then the transformation z ?/C30az /C27 b cz /C27 d is called a unimodular if a, b, c, and d are integers with ad/C28bc/C301:The set of all unimodular transfor- mations forms a GROUP called the MODULAR GROUP . See also MODULAR GROUP ,MODULAR GROUP GAMMA Union The union of two sets AandBis the set obtained by combining the members of each. This is written A@B; and is pronounced " Aunion B"o r" AcupB." The union of sets A1through Anis written @n i/C301Ai:/ Let A,B,C, ... be sets, and let P(S) denote the probability of S. Then P(A@B)/C30P(A)/C27P(B)/C28P(ASB): (1) Similarly, P(A@B@C)/C30P[A@(B@C)] /C30P(A)/C27P(B@C)/C28P[AS(B@C)] /C30P(A) /C27[P(B) /C27P(C) /C28P(B S C)] /C28P[(A S B) @ (A S C)] /C30P(A) /C27P(B) /C27P(C) /C28P(B S C) /C28fP(A S B) /C27P(A S C) /C28P[(A S B) S (A S C)] g /C30P(A) /C27P(B) /C27P(C) /C28P(A S B) /C28P(A S C) /C28P(B S C) /C27P(A S B S C): (2) If A and B are DISJOINT SETS, then by definition P(A S B) /C300; so P(A @ B) /C30P(A) /C27P(B): (3) Continuing, for a set of n disjoint elements E1 ; E2 ; ..., En P /C160n i/C301Ei/C18/C19 /C30Xn i/C301PEiðÞ ; (4) which is the COUNTABLE ADDITIVITY PROBABILITY AXIOM . Now let Ei /C13A S Bi ; (5) then P /C160n i/C301E S Bi/C18/C19 /C30Xn i/C301PES Bi ðÞ : (6) See also DISJOINT UNION ,INTERSECTION , OR, UNION- CLOSED SET Union-Closed Set A union-closed set is a nonempty finite collection of distinct nonempty finite sets which is CLOSED under UNION . See also UNION- CLOSED SETS CONJECTURE Union-Closed Sets Conjecture Let A /C30 A1 ; A2 ; ...; An fg be a UNION-CLOSED SET, then the union-closed set conjecture states that an element exists which belongs to at least n=2 of the sets in A. Sarvate and Renaud (1989) showed that the conjecture is true if A1jj52; where A1 is the smallest set in A,orif n B11. They also showed that if the conjecture fails, then A1jjB Anjj=2; where Anis the largest set of A. The proof for the case where A has a 2-set can be effected as follows. Write A1 /C30fx; y g; then partition the sets of A into four disjoint families B0 ; Bx ; By ; and Bxy ; according to whether their intersection with A1 is ¥;fxg;fyg; or fx; yg; respectively. It follows that Bxy/C12/C12/C12/C12] B 0jjby taking unions with A1 ; where ½B ½ is the CARDINALITY of B. Now compare Bxjj with By/C12/C12/C12/C12: If B xjj] By/C12/C12/C12/C12; then B xjj/C27 Bxyjj] B0jj/C27 By/C12/C12/C12/C12; so x is in atleast half the sets of A. Similarly, if B xjj5 By/C12/C12/C12/C12; then y is in at least half the sets (Hoey). Unfortunately, this method of proof does extend to A 1jj/C303 ; since Sarvate and Renaud show an example of a UNION-CLOSED SET with A1 /C30fx; y; zg where none of x, y, z is in half the sets. However, in these cases, there are other elements which do appear in half the sets, so this is not a counterexample to the conjecture, but only a limitation to the method of proof given above (Hoey). See also UNION- CLOSED SET References Sarvate, D. G. and Renaud, J.-C. "On the Union-Closed Sets Conjecture." Ars Combin. 27, 149 /C1/53, 1989. Sarvate, D. G. and Renaud, J.-C. "Improved Bounds for the Union-closed Sets Conjecture." Ars Combin. 29, 181 /C1/85, 1990. Uniplanar Double Point ISOLATED SINGULARITY Unipotent A P-ELEMENT x of a GROUP G is unipotent if F /C31 CG(x) ðÞ is a P-GROUP , where F /C31 is the generalized FITTING SUBGROUP . See also FITTING SUBGROUP , P-ELEMENT , P-GROUP Unique The property of being the only possible solution (perhaps modulo a constant, class of transformation, etc.). See also ALEKSANDROV’S UNIQUENESS THEOREM , EXISTENCE ,M AY-THOMASON UNIQUENESS THEOREM , UNIQUE FACTORIZATION Unique Factorization See also FUNDAMENTAL THEOREM OF ARITHMETIC , UNIQUE FACTORIZATION DOMAIN Unique Factorization Domain See also FUNDAMENTAL THEOREM OF ARITHMETIC , UNIQUE FACTORIZATION Unique Factorization Theorem FUNDAMENTAL THEOREM OF ARITHMETIC Unit A unit is an element in a RING that has a multi- plicative inverse. If nis an ALGEBRAIC INTEGER which divides every ALGEBRAIC INTEGER in the FIELD ,nis called a unit in that FIELD . A given FIELD may contain an infinity of units. The units of Zn are the elements RELATIVELY PRIME to n. The units in Znwhich are SQUARES are called QUADRATIC RESIDUES . See also EISENSTEIN UNIT,F UNDAMENTAL UNIT, IMAGINARY UNIT,PRIME UNIT,QUADRATIC RESIDUE Unit Ball A BALL of RADIUS 1. See also SPHERE ,BALL,UNIT CUBE,UNIT SPHERE Unit Cell A parallelogram (parallelepiped) containing the mini- mum repeatable elements of a circle (sphere) packing. See also CIRCLE PACKING ,PACKING DENSITY ,SPHERE PACKING References Williams, R. "The Unit Cell Concept." §2 /C1/ in The Geome- trical Foundation of Natural Structure: A Source Book of Design. New York: Dover, pp. 48 /C1/1, 1979. Unit Circle A CIRCLE of RADIUS 1, such as the one used to defined the functions of TRIGONOMETRY . See also CIRCLE ,UNIT DISK,UNIT SQUARE References Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 3, 1996. Unit Cube A CUBE whose edge lengths are 1. The unit cube therefore has unit volume. See also CUBE,UNIT SQUARE ,UNIT SPHERE Unit Disk A DISK with RADIUS 1. See also FIVE DISKS PROBLEM ,LOWER HALF-DISK, SEMICIRCLE ,U NIT CIRCLE ,U NIT SQUARE ,U PPER HALF-DISK Unit Element IDENTITY ELEMENTUnit Fraction A unit fraction is a FRACTION with NUMERATOR 1. Examples of unit fractions include 1/2, 1/3, 1/12, and 1/123456. Unit fractions are also known as Egyptian fractions as a result of their extensive use by ancientEgyptians as a way of representing other fractions.The famous Rhind papyrus, dated to around 1650 BC, discusses unit fractions and contains a table of representations of 2 =nas a sum of distinct unit fractions for ODD nbetween 5 and 101. The reason the Egyptians chose this method for representingfractions is not clear, although Andre ´Weil character- ized the decision as "a wrong turn" (Hoffman 1998, pp. 153 /C1 /54). The unique fraction that the Egyptians did not represent using unit fractions was 2/3 (Wells 1986, p. 29). Unit fractions are almost always required to exclude repeated terms, since representations such as 1 =5/C27 1=5/C271=5 are trivial. Any RATIONAL NUMBER has representations as a sum of distinct unit fractions with arbitrarily many terms and with arbitrarily large DENOMINATORS , although for a given fixed number of terms, there are only finitely many. Fibonacci proved that any fraction can be REPRE- SENTED AS a sum of distinct unit fractions (Hoffman 1998, p. 154). An infinite chain of unit fractions canbe constructed using the identity 1 a/C301 a/C271/C271 a(a/C271): (1) Martin (1999) showed that for every positive RA- TIONAL NUMBER , there exist representations as unit fractions whose largest DENOMINATOR is at most N and whose DENOMINATORS form a positive proportion of the integers up to Nfor sufficiently large N. Each FRACTION x=ywith yODD has a unit fraction repre- sentation in which each DENOMINATOR is ODD (Breusch 1954; Guy 1994, p. 160). Every x=yhas a t-term representation where t/C30O(ffiffiffiffiffiffiffiffiffiffiffi logyp ) (Vose 1985). No algorithm is known for producing unit fraction representations having either a minimum number of terms or smallest possible denominator (Hoffman 1998, p. 155). However, there are a number of ALGORITHMS (including the BINARY REMAINDER METHOD ,CONTINUED FRACTION UNIT FRACTION ALGO- RITHM ,GENERALIZED REMAINDER METHOD ,GREEDY ALGORITHM ,REVERSE GREEDY ALGORITHM ,SMALL MULTIPLE METHOD , and SPLITTING ALGORITHM ) for decomposing an arbitrary FRACTION into unit frac- tions. In 1202, Fibonacci published an algorithm forconstructing unit fraction representations, and this algorithm was subsequently rediscovered by Sylve- ster (Hoffman 1998, p. 154; Martin 1999). Taking the fractions 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, ... (the numerators of which are Sloane’s A002260, and the denominators of which are n/C281 copies of the integer n), the unit fraction representations using the GREEDY ALGORITHM are 1 2 /C3012 1 3 /C3013 23 /C3012 /C271 6 14 /C3014 24 /C3012 3 4 /C3012 /C271 4 1 5 /C3015 25 /C3013 /C271 15 35 /C3012 /C271 10 45 /C3012 /C271 4/C271 20 : The number of terms in these representations are 1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, ... (Sloane’s A050205). The minimum denominators for each representation are given by 2, 3, 2, 4, 2, 2, 5, 3, 2, 2, 6, 3, 2, ... (Sloane’s A050206), and the maximum denominators are 2, 3, 6, 4, 2, 4, 5, 15, 10, 20, 6, 3, 2, ... (Sloane’s A050210). Wilf posed as a problem that any fraction with odd denominator can be REPRESENTED AS a sum of unit fractions, each having an odd denominator, and Graham proved that infinitely many fractions with a certain range can be represented as a sum of units fractions with square denominators (Hoffman 1998, p. 156). Paul Erdos and E. G. Straus have conjectured that the DIOPHANTINE EQUATION 4 n /C301 a /C271 b/C271 c (2) always can be solved (Obla´th 1950, Rosati 1954, Bernstein 1962, Yamamoto 1965, Vaughan 1970, Guy 1994), and Sierpinski (1956) conjectured that 5 n /C301 a /C271 b/C271 c (3) can be solved (Guy 1994). The HARMONIC NUMBER Hnis never an INTEGER except for H1 : This result was proved im 1915 by Taeisinger, and the more general results that any number of consecutive terms not necessarily starting with 1 never sum to an integer was proved byKu¨rscha´k in 1918 (Hoffman 1998, p. 157). In 1932, Erdos proved that the sum of the reciprocals of any number of equally spaced integers is never a recipro- cal. See also CALCUS ,EGYPTIAN NUMBER ,HALF,HARMO- NIC NUMBER ,QUARTER ,SCRUPLE ,UNCIA References Beck, A.; Bleicher, M. N.; and Crowe, D. W. Excursions into Mathematics. New York: Worth Publishers, 1970. Beeckmans, L. "The Splitting Algorithm for Egyptian Frac- tions." J. Number Th. 43, 173/C1/85, 1993. Bernstein, L "Zur Lo ¨sung der diophantischen Gleichung / m=n¼1=xþ1=yþ1=z/insbesondere im Falle m/C304."J. reine angew. Math. 211,1/C1/0, 1962. Bleicher, M. N. "A New Algorithm for the Expansion of Continued Fractions." J. Number Th. 4, 342/C1/82, 1972. Breusch, R. "A Special Case of Egyptian Fractions." Solution to advanced problem 4512. Amer. Math. Monthly 61, 200/C1/ 01, 1954. Brown, K. S. "Egyptian Unit Fractions." http://www.seanet.- com/~ksbrown/iegypt.htm. Eppstein, D. "Ten Algorithms for Egyptian Fractions." Mathematica Educ. Res. 4,5/C1/5, 1995. Eppstein, D. "Egyptian Fractions." http://www.ics.uci.edu/ ~eppstein/numth/egypt/. Eppstein, D. Egypt.ma Mathematica notebook. http:// www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma. Gardner, M. "Mathematical Games: In Which a Mathema- tical Aesthetic is Applied to Modern Minimal Art." Sci. Amer. 239,2 2/C1/2, Nov. 1978. Golomb, S. W. "An Algebraic Algorithm for the Representa- tion Problems of the Ahmes Papyrus." Amer. Math. Monthly 69, 785/C1/86, 1962. Graham, R. "On Finite Sums of Unit Fractions." Proc. London Math. Soc. 14, 193/C1/07, 1964. Guy, R. K. "Egyptian Fractions." §D11 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 87 /C1/3 and 158 /C1/66, 1994. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 153 /C1/57, 1998. Ke, Z. and Sun, Q. "On the Representation of 1 by Unit Fractions." Sichuan Daxue Xuebao 1,1 3/C1/9, 1964. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., pp. 175 /C1/77 and 206 /C1/08, 1991. Martin, G. "Dense Egyptian Fractions." Trans. Amer. Math. Soc. 351, 3641 /C1/657, 1999. Niven, I. and Zuckerman, H. S. An Introduction to the Theory of Numbers, 5th ed. New York: Wiley, p. 200, 1991. Obla´th, R. "Sur l’equation diophantienne / 4=n¼1=x1þ1=x2þ1=x3/."Mathesis 59, 308/C1/16, 1950. Rosati, L. A. "Sull’equazione diofantea / 4=n¼1=x1þ1=x2þ1=x3/."Boll. Un. Mat. Ital. 9,5 9/C1/3, 1954. Se´roul, R. "Egyptian Fractions." §8.8 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 181 /C1/87, 2000. Sierpinski, W. "Sur les de ´compositiones de nombres ratio- nelles en fractions primaires." Mathesis 65,1 6/C1/2, 1956. Sloane, N. J. A. Sequences A002260, A050205, A050206, and A050210 in "An On-Line Version of the Encyclopediaof Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Stewart, I. "The Riddle of the Vanishing Camel." Sci. Amer. 266, 122/C1 /24, June 1992. Tenenbaum, G. and Yokota, H. "Length and Denominators of Egyptian Fractions." J. Number Th. 35, 150/C1/56, 1990. Vaughan, R. C. "On a Problem of Erdos, Straus and Schinzel." Mathematika 17, 193 /C1/98, 1970. Vose, M. "Egyptian Fractions." Bull. London Math. Soc. 17, 21, 1985. Wagon, S. "Egyptian Fractions." §8.6 in Mathematica in Action. New York: W. H. Freeman, pp. 271 /C1/77, 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 29, 1986. Yamamoto, K. "On the Diophantine Equation / 4=n ¼ 1=x þ 1=y þ 1=z/." Mem. Fac. Sci. Kyushu U. Ser. A 19,37/C1/7, 1965. Unit Lattice A POINT LATTICE which can be constructed from an arbitrary PARALLELOGRAM of unit area. For any such planar lattice, the minimum distance c between any two points is a quantity characteristic of the lattice. This distance satisfies c 5ffiffiffiffiffiffiffi 2ffiffiffi 3ps (Hilbert and Cohn-Vossen 1999, p. 36). For a lattice in 3-D, c 521 =6 (Hilbert and Cohn-Vossen 1999, p. 45). See also HYPERSPHERE PACKING ,P OINT LATTICE , SPHERE PACKING References Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Unit Matrix An INTEGER MATRIX consisting of all 1s. The m /C29n unit matrix is often denoted Jmn ; or Jnif m /C30n. Square unit matrices have DETERMINANT 0. See also IDENTITY MATRIX ,UNIMODULAR MATRIX References Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626 /C1/30, 1972. Unit Neighborhood Graph A DISTANCE GRAPH with distance set 0; 1ð/C138 :/ See also DISTANCE GRAPH ,UNIT-DISTANCE GRAPH References Fishburn, P. C. "On the Sphericity and Cubicity of Graphs." J. Combin. Th. B 35, 309 /C1/18, 1983. Frankl, P. and Maehara, H. "Embedding the n-Cube in Lower Dimensions." European J. Combin. 7, 221 /C1/25, 1986. Frankl, P. and Maehara, H. "Open-Interval Graphs versus Closed-Interval Graphs." Discr. Math. 63,97/C1/00, 1987.Frankl, P. and Maehara, H. "The Johnson-Lindenstrauss Lemma and the Sphericity of Some Graphs." J. Combin. Th. B 44, 355 /C1/61, 1988. Maehara, H. "Independent Balls and Unit Neighborhood Graphs." Ryukyu Math. J. 1,38/C1/5, 1988. Maehara, H. and Ro¨dl, V. "On the Dimension to Represent a Graph by a Unit Distance Graph." Graphs Combin. 6, 365 /C1/67, 1990. Maehara, H. "Distance Graphs in Euclidean Space." Ryukyu Math. J. 5,33/C1/1, 1992. Unit Point The point in the PLANE with Cartesian coordinates (1, 1). References Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 9, 1961. Unit Ring A unit ring is a set together with two BINARY OPERATORS S(/C27;+) satisfying the following condi- tions: 1. Additive associativity: For all a; b; c /C23 S; ða þ bÞþc ¼ a þðb þ c Þ/, 2. Additive commutativity: For all a ; b /C23 S; a /C27b /C30b /C27a ;/ 3. Additive identity: There exists an element 0 /C23 S such that for all a /C23 S :0/C27a /C30a /C270 /C30a;/ 4. Additive inverse: For every a /C23 S; there exists a /C28a /C23 S such that /a þð/C28aÞ¼ð/C28a Þþa ¼ 0/, 5. Multiplicative associativity: For all a; b; c /C23 S; ða + b Þ+ c ¼ a +ðb + c Þ/, 6. Multiplicative identity: There exists an element 1 /C23 S such that for all a /C23 S; 1 + a /C30a + 1 /C30a ;/ 7. Left and right distributivity: For all a; b; c /C23 S; a +ðb þ c Þ¼ða + bÞþða + c Þ/ and / ðb þ c Þ+ a ¼/ / ðb + a Þþðc + a Þ/. Thus, a unit ring is a RING with a multiplicative identity. See also BINARY OPERATOR ,RING References Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968. Unit Sphere A SPHERE of RADIUS 1. See also SPHERE ,BALL,UNIT CIRCLE Unit Square A SQUARE with side lengths 1. The unit square usually means the one with coordinates (0, 0), (1, 0), (1, 1), (0, 1) in the real plane, or 0, 1, 1 /C27i; and i in the COMPLEX PLANE . See also HEILBRONN TRIANGLE PROBLEM ,U NIT CIRCLE ,UNIT CUBE,UNIT DISK Unit Vector A VECTOR of unit length, sometimes also called a DIRECTION VECTOR (Jeffreys and Jeffreys 1988). The unit vector ˆv having the same direction as a given (nonzero) vector v is defined by ˆv /C13v vjj; where vjjdenotes the NORM of v, is the unit vector in the same direction as the (finite) VECTOR v. A unit vector in the xn direction is given by ˆxn /C13@r @xn @r @xn/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12; where r is the RADIUS VECTOR . See also NORM,R ADIUS VECTOR ,V ECTOR ,Z ERO VECTOR References Jeffreys, H. and Jeffreys, B. S. "Direction Vectors." §2.034 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 64, 1988. Stephens, M. A. "The Testing of Unit Vectors for Random- ness." J. Amer. Stat. Assoc. 59, 160 /C1/67, 1964. Unital A BLOCK DESIGN OF THE FORM (/q3 /C271 ; q /C271; 1). References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1 /C1/2, 1992. Unitary An OPERATOR U satisfying U /C31U /C301 UU /C31/C301; where U /C31 is the ADJOINT .See also ANTIUNITARY References Sakurai, J. J. Modern Quantum Mechanics. Menlo Park, CA: Benjamin/Cummings, 1985. Unitary Aliquot Sequence An ALIQUOT SEQUENCE computed using the analog of the RESTRICTED DIVISOR FUNCTION s /C31(n) in which only UNITARY DIVISORS are included. See also ALIQUOT SEQUENCE ,U NITARY AMICABLE PAIR,UNITARY SOCIABLE NUMBERS References Guy, R. K. "Unitary Aliquot Sequences." §B8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 63 /C1/5, 1994. Unitary Amicable Pair A PAIR of numbers m and n such that s/C31(m) /C30 s /C31(n) /C30m /C27n; where s/C31(n) is the sum of UNITARY DIVISORS . Hagis (1971) and Garcı ´a (1987) give 82 such pairs. The first few are (114, 126), (1140, 1260), (18018, 22302), (32130, 40446), ... (Sloane’s A002952 and A002953). The largest known unitary amicable pair, each member of which has 192 digits, 22 /C215 32 /C215 59 /C215 73 /C215 11 /C215 13 /C215 172 /C215 19 /C215 29 /C215 41 /C215 43 /C215 47 /C21579 /C215 157 /C215 163 /C215 223 /C215 433 /C215 1303 /C215 1399 /C215 2053 /C2152719 /C215 5167 /C215 13187 /C215 16787 /C215 52747 /C215 98543 /C215284337 /C215 500739672615943 /C2157010355416623201 /C21516506961423173486727453 /C21510109028245165675006759491729 /C21553 /C215 9163813886186194062277465733355041 494845949854054479362983149601172267/C20/C21 (Y. Kohmoto). Kohmoto calls a unitary amicable pair whose mem- bers are squareful a proper unitary amicable pair. See also AMICABLE PAIR,SUPER UNITARY AMICABLE PAIR,UNITARY ALIQUOT SEQUENCE ,UNITARY DIVISOR References Garcı ´a, M. "New Unitary Amicable Couples." J. Recr. Math. 19,1 2/C1/4, 1987. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 57, 1994. Hagis, P. "Relatively Prime Amicable Numbers of Opposite Parity." Math. Comput. 25, 915/C1/18, 1971. Sloane, N. J. A. Sequences A002952/M5372 and A002953/ M5389 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Unitary Divisor A DIVISOR d of n for which GCD( d; n=d) /C301; (1) where GCD( m; n) is the GREATEST COMMON DIVISOR . For example, the divisors of 12 are f1; 2; 3; 4; 6; 12g; so the unitary divisors are f1; 3; 4; 12g:/ Given the PRIME FACTORIZATION n /C30Yk i/C301pai i; (2) then d /C30productpci i (3) is a unitary divisor of n if each ciis 0 or ai : For a PRIME POWER py ; the unitary divisors are 1 and py (Cohen 1990). The numbers of unitary divisors of n /C301, 2, ... are 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, ... (Sloane’s A034444). These numbers are also the numbers of squarefree divisors of n. The number of unitary divisors of n is also given by 2q ; where q is the number of different primes dividing n. The symbol s/C31(n) is used to denote to the UNITARY DIVISOR FUNCTION . See also BIUNITARY DIVISOR ,D IVISOR ,G REATEST COMMON DIVISOR , K-ARY DIVISOR ,SUPER UNITARY AMICABLE PAIR,SUPER UNITARY PERFECT NUMBER , UNITARY DIVISOR FUNCTION ,UNITARY PERFECT NUM- BER References Cohen, G. L. "On an Integer’s Infinary Divisors." Math. Comput. 54, 395 /C1/11, 1990. Guy, R. K. "Unitary Perfect Numbers." §B3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 53 /C1/9, 1994. Sloane, N. J. A. Sequences A034444 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Unitary Divisor Function The symbol s/C31(n) is used to denote to the sum-of- UNITARY DIVISORS function. If n is SQUAREFREE , then s(n) /C30 s/C31(n): For n /C301, 2, ..., the first few values of s/C31(n) are given by 1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, ... (Sloane’s A034448). See also UNITARY DIVISOR References Sloane, N. J. A. Sequences A034448 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html.Unitary Group The unitary group Un(q) is the set of n /C29n UNITARY MATRICES . See also LIE-TYPE GROUP ,UNITARY MATRIX References Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#unit. Unitary Matrix A SQUARE MATRIX U is a unitary matrix if U/C31/C30U/C281 ; (1) where U /C31 denotes the ADJOINT MATRIX and U/C281 is the MATRIX INVERSE . For example, A /C302/C281 =22/C281 =20 /C282/C281 =2i 2/C281=2i 0 00 i2 435 (2) is a unitary matrix. A matrix mcan be tested to see if it is unitary using the Mathematica function UnitaryQ[m_List?MatrixQ] : /C30 (Conjugate@[email protected] /C30/C30 IdentityMatrix@Length@m) The definition of a unitary matrix guarantees that U/C31U/C30I; (3) where Iis the IDENTITY MATRIX . In particular, a unitary matrix is always invertible, and U/C281/C30U/C31: Note that TRANSPOSE is a much simpler computation than inverse. Unitary matrices leave the length of a COMPLEX VECTOR unchanged. A SIMILARITY TRANS- FORMATION of a H ERMITIAN MATRIX with a unitary matrix gives uau/C281/C0/C1 /C31/C30(ua)u/C281/C0/C1/C2/C3 /C31/C30u/C281/C0/C1 /C31(ua)/C31/C30(u/C31)/C31(a/C31u/C31) /C30uau/C31/C30uau/C281: (4) Unitary matrices are NORMAL MATRICES .I fMis a unitary matrix, then the PERMANENT ½perm( M)½51 (5) (Minc 1978, p. 25, Vardi 1991). For REAL MATRICES , unitary is the same as ORTHOGO- NAL. In fact, there are some similarities between ORTHOGONAL MATRICES and unitary matrices. The rows of a unitary matrix are a UNITARY BASIS . That is, each row has length one, and their H ERMITIAN INNER PRODUCT is zero. Similarly, the columns are also a unitary basis. In fact, given any unitary basis, the matrix whose rows are that basis is a unitary matrix. It is automatically the case that the columns areanother unitary basis. The unitary matrices are precisely those matrices which preserve the H ERMITIAN INNER PRODUCT v; whi /C30 Uv ; Uw hi : (6) Also, the norm of the determinant of U is ½det U ½/C301: Unlike the ORTHOGONAL MATRICES , the unitary ma- trices are CONNECTED . If det U /C301 then U is a SPECIAL UNITARY MATRIX . The product of two unitary matrices is another unitary matrix. The inverse of a unitary matrix is another unitary matrix, and IDENTITY MATRICES are unitary. Hence the set of unitary matrices form a GROUP , called the UNITARY GROUP . See also ADJOINT MATRIX ,CLIFFORD ALGEBRA ,HER- MITIAN INNER PRODUCT ,HERMITIAN MATRIX ,NORMAL MATRIX ,O RTHOGONAL GROUP ,PERMANENT ,REPRE- SENTATION ,SKEW HERMITIAN MATRIX ,SPECIAL UNI- TARY MATRIX ,S PIN GROUP ,S YMMETRIC MATRIX UNITARY GROUP References Arfken, G. "Hermitian Matrices, Unitary Matrices." §4.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 209 /C1/17, 1985. Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 112, 1962. Minc, H. Permanents. Reading, MA: Addison-Wesley, 1978. Vardi, I. "Permanents." §6.1 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 108 and 110 /C1/12, 1991. Unitary Multiperfect Number A number n which is an INTEGER multiple k of the SUM of its UNITARY DIVISORS s/C31(n) is called a unitary k-multiperfect number. There are no ODD unitary multiperfect numbers. References Guy, R. K. "Unitary Perfect Numbers." §B3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 53 /C1/9, 1994. Suryanarayana, D. "The Number of Bi-Unitary Divisors of an Integer." The Theory of Arithmetic Functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1971. New York: Springer-Verlag, pp. 273 /C1/82, 1972. Suryanarayana, D. and Rao, R. S. R. C. "The Number of Bi- Unitary Divisors of an Integer. II." J. Indian Math. Soc. 39, 261 /C1/80, 1975. Wall, C. R. "Bi-Unitary Perfect Numbers." Proc. Amer. Math. Soc. 33,39/C1/2, 1972. Unitary Multiplicative Character A MULTIPLICATIVE CHARACTER is called unitary if it has ABSOLUTE VALUE 1 everywhere. See also MULTIPLICATIVE CHARACTER Unitary Operator An OPERATOR U satisfying l1 > l2 > 0 See also ANTIUNITARY OPERATORReferences Sakurai, J. J. Modern Quantum Mechanics. Menlo Park, CA: Benjamin/Cummings, 1985. Unitary Perfect Number A number n which is the sum of its UNITARY DIVISORS with the exception of n itself. There are no ODD unitary perfect numbers, and it has been conjec- tured that there are only a FINITE number of EVEN ones. The first few are 6, 60, 90, 87360, 146361946186458562560000, ... (Sloane’s A002827). References Guy, R. K. "Unitary Perfect Numbers." §B3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 53 /C1/9, 1994. Sloane, N. J. A. Sequences A002827/M4268 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Subbarao, M. V. and Warren, L. J. "Unitary Perfect Num- bers." Canad. Math. Bull. 9, 147 /C1/53, 1966. Wall, C. R. "The Fifth Unitary Perfect Number." Canad. Math. Bull. 18, 115 /C1/22, 1975. Wall, C. R. "On the Largest Odd Component of a Unitary Perfect Number." Fib. Quart. 25, 312 /C1/16, 1987. Unitary Sociable Numbers SOCIABLE NUMBERS computed using the analog of the RESTRICTED DIVISOR FUNCTION s /C31(n) in which only UNITARY DIVISORS are included. See also SOCIABLE NUMBERS References Guy, R. K. "Unitary Aliquot Sequences." §B8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 63 /C1/5, 1994. Unitary Transformation A transformation OF THE FORM A?/C30UAU /C31; where U /C31 denotes the ADJOINT operator. See also ADJOINT ,TRANSFORMATION Unitary Unimodular Group SPECIAL UNITARY GROUP Unit-Distance Graph A DISTANCE GRAPH in which all edges are of length 1. See also DISTANCE GRAPH ,U NIT NEIGHBORHOOD GRAPH References Anning, N. H. and Erdos, P. "Integral Distances." Bull. Amer. Math. Soc. 51, 598/C1/00, 1945. Buckley, F. and Harary, F. "On the Euclidean Dimension of a Wheel." Graphs and Combin. 4,2 3/C1/0, 1988. Chilakamarri, K. B. "Unit Distance Graphs in Rational n- Space." Discr. Math. 69, 213/C1/18, 1988. Erdos, P.; Harary, F.; and Tutte, W. T. "One on the Dimension of a Graph." Mathematika 12, 118 /C1/22, 1965. Maehara, H. "On Euclidean Dimension of a Complete Multipartite Graph." Discr. Math. 72, 285 /C1/89, 1988. Maehara, H. "Note on Induced Subgraphs of the Unit Distance Graph." Discr. Comput. Geom. 4,15/C1/8, 1989. Maehara, H. "Distances in a Rigid Unit-Distance Graph in the Plane." Discr. Appl. Math. 31, 193 /C1/00, 1991. Maehara, H. "Distance Graphs in Euclidean Space." Ryukyu Math. J. 5,33/C1/1, 1992. Maehara, H. and Ro¨dl, V. "On the Dimension to Represent a Graph by a Unit Distance Graph." Graphs Combin. 6, 365 /C1/67, 1990. Moser, L. and Moser, W. "Problem 10." Canad. Math. Bull. 4, 187 /C1/89, 1961. Unitransitive Graph A GRAPH G is n-unitransitive if it is CONNECTED , CUBIC , n-TRANSITIVE , and if for any two n-ROUTES W1 and W2 ; there is exactly one automorphism a of G such that aW1 /C30W2 :/ Because there are no n-transitive CUBIC GRAPHS for n /C215, there are also no n-unitransitive ones (Harary 1994, p. 175). However, there are n-unitransitive graphs for n 55 which are not CAGE GRAPHS (Harary 1994, p. 175). These include the 1-univariate graph of girth 12 on 432 nodes discovered by Frucht (1952), the 2-unitransitive CUBICAL and DODECAHEDRAL GRAPHS , and a set of 3-unitransitive graphs found by Coxeter (1950), one of which is illustrated above (Harary 1994, p. 175). See also CAGE GRAPH ,TRANSITIVE GRAPH References Coxeter, H. S. M. "Self-Dual Configurations and Regular Graphs." Bull. Amer. Math. Soc. 56, 413 /C1/55, 1950. Frucht, R. "A One-Regular Graph of Degree Three." Canad. J. Math. 4, 240 /C1/47, 1952. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 174 /C1/75, 1994. Tutte, W. T. "A Family of Cubical Graphs." Proc. Cambridge Philos. Soc. 43 459 /C1/74, 1947. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . UnitStep HEAVISIDE STEP FUNCTION Unity The number 1. There are nnth ROOTS OF UNITY , known as the DE MOIVRE NUMBERS .See also 1,PRIMITIVE ROOT OF UNITY Univalent Capable of taking on exactly one possible value. See also BIVALENT Univalent Function A function or transformation f in which f(z) does not overlap z. In MODULAR FUNCTION theory, a function is called univalent on a subgroup G if it is automorphic under G and VALENCE 1 (Apostol 1997). See also VALENCE References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 84, 1997. Univariate Function A FUNCTION of a single variable (e.g., f(x); g(z); u( j); etc.). See also MULTIVARIATE FUNCTION ,UNIVARIATE POLY- NOMIAL Univariate Polynomial A POLYNOMIAL in a single variable, e.g., /PðxÞ¼ / /a2x2 þ a1x þ a0/, as opposed to a MULTIVARIATE POLY- NOMIAL , e.g., P(x; y) /C30a22x2y2 /C27a21x2y /C27a12xy2 /C27a11xy /C27a10x /C27a01y /C27a00 : In common usage, if the word "univariate" is not used when describing a POLYNOMIALS , the POLYNOMIALS can assumed to be univariate. See also MULTIVARIATE POLYNOMIAL ,POLYNOMIAL , UNIVARIATE FUNCTION Universal Algebra A system of algebra having an empty set of relations. A universal algebra is often simply called an "alge- bra". Universal Category UNIVERSAL PREDICATE Universal Cover The universal cover of a CONNECTED TOPOLOGICAL SPACE Xis a SIMPLY CONNECTED space Ywith a map f:Y0Xthat is a COVER .I fXisSIMPLY CONNECTED , i.e., has a trivial FUNDAMENTAL GROUP , then it is its own universal cover. For instance, the sphere S2is its own universal cover. The universal cover is always unique, and always exists, as long as X is LOCALLY PATHWISE-CONNECTED (a very mild assumption). Any property of X can be lifted to its universal cover, as long as it is defined locally. Sometimes, the universal covers with special structures can be classified. For example, a RIEMANNIAN METRIC on X defines a metric on its universal cover. If the metric is FLAT , then its universal cover is EUCLIDEAN SPACE . Another example is the COMPLEX STRUCTURE of a RIEMANN SURFACE X, which also lifts to its universal cover. By the UNIFORMIZATION THEOREM , the only possible universal covers for X are the open unit disk, the complex plane C; or the RIEMANN SPHERE S2 :/ p : A 0 X The above left diagram shows the universal cover of the torus, i.e., the plane. A fundamental domain, shaded orange, can be identified with the torus. The REAL PROJECTIVE PLANE is the set of lines through the origin, and its universal cover is the sphere, shown in the right figure above. The only nontrivial DECK TRANSFORMATION is the ANTIPODAL MAP. The compact RIEMANN SURFACES with GENUSES g /C211 are g-holed TORI, and their universal covers are the UNIT DISK. The figure above shows a hyperbolic regular octagon in the disk. With the colored edges identified, it is a FUNDAMENTAL DOMAIN for the DOUBLE TORUS . Each hole has two loops, and cutting along each loop yields two edges per loop, or eight edges in total. Each loop is also shown in a different color, and arrows are drawn to provide instructions for lining them up. The FUNDAMENTAL DOMAIN is in gray and can be identified with the DOUBLE TORUS illustrated below. The above animation shows some translations of the fundamental domain by DECK TRANSFORMATIONS , which form a FUCHSIAN GROUP . They tile the disk by analogy with the square tilingthe plane for the SQUARE TORUS . Although it is difficult to visualize a hyperbolic regular octagon in the disk as a cut-up DOUBLE TORUS , the illustration above attempts to portray this. It is unfortunate that no hyperbolic compact manifold with constant negative curvature, can be embedded in R3 : As a result, this picture is not isometric to the hyperbolic regular octagon. However, the generators for the fundamental group are drawn in the same colors, and are examples of so-called cuts of a RIEMANN SURFACE . Roughly speaking, the universal cover of a space is obtained by the following procedure. First, the space is cut open to make a simply connected space with edges, which then becomes a fundamental domain, as the DOUBLE TORUS is cut to become a hyperbolic octagon or the SQUARE TORUS is cut open to become a square. Then a copy of the fundamental domain is added across an edge. The rule for adding a copy across an edge is that every point has to look the same as the original space, at least nearby. So the copies of the fundamental domain line up along edges which are identified in the original space, but more edges may also line up. Copies of the fundamental domain are added to the resulting space recursively, as long as there remains any edges. The result is a cover, with possibly infinitely many copies of a fundamentaldomain, which is simply connected. Any other COVER ofXis in turn covered by the universal cover of X,˜X:In this sense, the universal cover is the largest possible cover. In rigorous language, the universal cover has a UNIVERSAL PROPERTY .I f p?:˜X0Ais a COVERING MAP , then there exists a covering map p(˜psuch that the composition of pand ˜pis the projection from the universal cover to X. See also COVER ,DECK TRANSFORMATION ,FUNDAMEN- TAL GROUP ,SIMPLY CONNECTED ,U NIFORMIZATION , UNIVERSAL PROPERTY References Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, pp. 186 /C1/96, 1995. Massey, W. S. A Basic Course in Algebraic Topology. New York: Springer-Verlag, p. 132, 1991. Universal Formula Also called an existential formula. References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 34, 1958. Universal Graph COMPLETE GRAPH Universal Hash Function Let h : f0; 1gl(n) /C29f0 ; 1 gn 0f0; 1gm(n) be efficiently computable by an algorithm (solving a P-PROBLEM ). For fixed y /C23f0; 1gl(n) ; view h(x; y) as a function hy(x) of x that maps (or hashes) n bits to m(n) bits. Let Y /C23R f0; 1gl(n) ; then h is said to be a (pairwise independent) universal hash function if, for distinct x; x?/C23f0; 1gn and for all a; a ?/C23f0; 1gm(n) ; Pr YhY (x) /C30a ðÞ and hY (x?) /C30a? ðÞ ½/C138 /C301 22m(n); i.e., hYmaps all distinct x; x? independently and uniformly. These functions are easily constructible (Wegman and Carter 1981, Luby 1996). See also HASH FUNCTION References Luby, M. Pseudorandomness and Cryptographic Applica- tions. Princeton, NJ: Princeton University Press, 1996. Wegman, M. N. and Carter, J. L. "New Hash Functions and Their Use in Authentication and Set Equality." J. Comput. System Sci. 22, 265 /C1/79, 1981. Universal Metric Space UNIVERSAL SPACE Universal Predicate If the property of being an object is expressed by a basic predicate of the system, then such a predicate (if it exists) is called a universal predicate, or universal category. References Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 113, 1977. Universal Product Code UPC Universal Property A property of individuals which is shared by every individual.References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 107, 1958. Universal Quantifier A logical operator which forms propositions using the expression "FOR ALL x." See also FOR ALL References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 34, 1958. Universal Quantor UNIVERSAL QUANTIFIER Universal Sentence A sentence dealing with individual constants in which some constant, say a, appears one or more times and which is true for every individual in the domain of individuals to which a belongs. See also EXISTENTIAL SENTENCE References Carnap, R. Introduction to Symbolic Logic and Its Applica- tions. New York: Dover, p. 34, 1958. Universal Set A set fixed within the framework of a theory and consisting of all objects considered in this theory. References Fraenkel, A. A. and Bar-Hillel, Y. Foundations of Set Theory. Amsterdam, Netherlands, 1958. Universal Space A TOPOLOGICAL SPACE that contains a homeomorphic image of every topological space of a certain class. A METRIC SPACE U is said to be universal for a family of METRIC SPACES M if any space from M is isome- trically embeddable in U. Fre´chet (1910) proves that l/C12; the space of all bounded sequences of real numbers endowed with a supremum norm, is a universal space for the family M of all separable metric spaces. Ovchinnikov (2000) proved that there exists a metric d/C23R;inducing the usual topology, such that every finite METRIC SPACE embeds in ( R;d):/ See also METRIC SPACE References Fre´chet, M. "Les dimensions d’un ensemble abstrait." Math. Ann. 68, 145/C1/68, 1910. Holsztynski, W. " /Rnas a Universal Metric Space." Not. Amer. Math. Soc. 25, A-367, 1978. Ovchinnikov, S. Universal Metric Spaces According to W. Holsztynski. 13 Apr 2000. http://xxx.lanl.gov/abs/ math.GN/0004091/. Uryson, P. S. "Sur un espace me´trique universel." Bull. de Sciences Math. 5,1/C1/8, 1927. Universal Turing Machine AT URING MACHINE which, by appropriate program- ming using a finite length of input tape, can act as any TURING MACHINE whatsoever. See also CHAITIN’S CONSTANT ,H ALTING PROBLEM , TURING MACHINE References Penrose, R. The Emperor’s New Mind: Concerning Compu- ters, Minds, and the Laws of Physics. Oxford: Oxford University Press, pp. 51 /C1/7, 1989. Universal Vassiliev Invariant See also VASSILIEV INVARIANT Universe UNIVERSAL SET Unknot A closed loop which is not KNOTTED . In the 1930s, by making use of REIDEMEISTER MOVES , Reidemeister first proved that KNOTS exist which are distinct from the unknot. He proved this by COLORING each part of a knot diagram with one of three colors. The KNOT SUM of two unknots is another unknot. The JONES POLYNOMIAL of the unknot is defined to give the normalization V(t) /C301 : Haken (1961) devised an ALGORITHM to tell if a knot projection is the unknot. The ALGORITHM is so complicated, however, that it has never been imple- mented. Although it is not immediately obvious, the unknot is a PRIME KNOT . See also COLORABLE ,K NOT,K NOT THEORY ,LINK, REIDEMEISTER MOVES ,UNKNOTTING NUMBER References Haken, W. "Theorie der Normalflachen." Acta Math. 105, 245 /C1/75, 1961. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 264 /C1/65, 1999. Unknotting Number The smallest number of times a KNOT must be passed through itself to untie it. Lower bounds can be computed using relatively straightforward techni- ques, but it is in general difficult to determine exact values. Many unknotting numbers can be determined from a knot’s SIGNATURE .A KNOT with unknotting number 1 is a PRIME KNOT (Scharlemann 1985). It is not always true that the unknotting number isachieved in a projection with the minimal number of crossings. The following table is from Kirby (1997, pp. 88 /C1/9), with the values for 10 /C1/39 and 10 /C1/52 taken from Kawamura. The unknotting numbers for 10 /C1/54 and 10 /C1/61 can be found using MENASCO’S THEOREM (Stoi- menow 1998). 03 /C1/01 1 08 /C1/09 1 09 /C1/10 2or3 09 /C1/32 1or2 04 /C1/01 1 08 /C1/10 1or2 09 /C1/11 2 09 /C1/33 1 05 /C1/01 2 08 /C1/11 1 09 /C1/12 1 09 /C1/34 1 05 /C1/02 1 08 /C1/12 2 09 /C1/13 2or3 09 /C1/35 2or3 06 /C1/01 1 08 /C1/13 1 09 /C1/14 1 09 /C1/36 2 06 /C1/02 1 08 /C1/14 1 09 /C1/15 2 09 /C1/37 2 06 /C1/03 1 08 /C1/15 2 09 /C1/16 3 09 /C1/38 2or3 07 /C1/01 3 08 /C1/16 2 09 /C1/17 2 09 /C1/39 1 07 /C1/02 1 08 /C1/17 1 09 /C1/18 2 09 /C1/40 2 07 /C1/03 2 08 /C1/18 2 09 /C1/19 1 09 /C1/41 2 07 /C1/04 2 08 /C1/19 3 09 /C1/20 2 09 /C1/42 1 07 /C1/05 2 08 /C1/20 1 09 /C1/21 1 09 /C1/43 2 07 /C1/06 1 08 /C1/21 1 09 /C1/22 1 09 /C1/44 1 07/C1/07109/C1/014 09/C1/232 09/C1/451 08/C1/01109/C1/021 09/C1/241 09/C1/462 08/C1/02209/C1/033 09/C1/252 09/C1/472 08/C1/03209/C1/042 09/C1/261 09/C1/482 08/C1/04209/C1/052 09/C1/271 09/C1/492o r3 08/C1/05209/C1/063 09/C1/281 10/C1/394 08/C1/06209/C1/072 09/C1/291 10/C1/524 08/C1/07109/C1/082 09/C1/301 10/C1/543 08/C1/08209/C1/093 09/C1/312 10/C1/613 See also ALGEBRAIC UNKNOTTING NUMBER ,BENNE- QUIN’S CONJECTURE ,MENASCO’S THEOREM ,MILNOR’S CONJECTURE ,SIGNATURE (KNOT) References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 57 /C1/4, 1994. Cipra, B. "From Knot to Unknot." What’s Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Soc., pp. 8 /C1/3, 1994. Kawamura, T. "The Unknotting Numbers of 10139and 10152 are 4." Osaka J. Math. 35, 539 /C1/46, 1998. http:// ms421sun.ms.u-tokyo.ac.jp/~kawamura/worke.html. Kirby, R. (Ed.). "Problems in Low-Dimensional Topology." AMS/IP Stud. Adv. Math., 2.2, Geometric Topology (Athens, GA, 1993). Providence, RI: Amer. Math. Soc., pp. 35 /C1/73, 1997. Scharlemann, M. "Unknotting Number One Knots are Prime." Invent. Math. 82,37/C1/5, 1985. Stoimenow, A. "Positive Knots, Closed Braids and the Jones Polynomial." Rev. May, 1997. http://guests.mpim- bonn.mpg.de/alex/pos.ps.gz. Weisstein, E. W. "Knots and Links." MATHEMATICA NOTE- BOOK KNOTS.M . Unlabeled Graph A GRAPH in which individual nodes have no distinct identifications except through their interconnectivity. Graphs in which labels (which are most commonly numbers) are assigned to nodes are called LABELED GRAPHS . Unless indicated otherwise by context, the unmodified term "graph" generally refers to an unlabeled graph. See also GRAPH ,LABELED GRAPH ,SIMPLE GRAPH Unless If A is true unless B, then not-B IMPLIES A, but B does not necessarily imply not-A. See also IMPLIES ,PRECISELY UNLESS Unlesss PRECISELY UNLESS Unmixed A homogeneous IDEAL defining a projective ALGE- BRAIC VARIETY is unmixed if it has no embedded PRIME divisors. Unpoke Move POKE MOVE Unprojected Map EQUIRECTANGULAR PROJECTION Unsafe A position in a GAME is unsafe for player A if the person who plays next (player B) can win. Every unsafe position can be made SAFE by at least one move. See also GAME,SAFE Unsolved Problems There are many unsolved PROBLEMS in mathematics. Several famous problems which have recently been solved include1. The P O´LYA CONJECTURE (disproven by Hasel- grove 1958, smallest counterexample found by Tanaka in 1980), 2. The FOUR-COLOR THEOREM (by Appel and Haken in 1977 using a computer-assisted proof), 3. The B IEBERBACH CONJECTURE (by L. de Branges in 1985), 4. Tait’s FLYPING CONJECTURE (by Menasco and Thistlethwaite in 1991) and the other two of T AIT’S KNOT CONJECTURES (by various authors in 1987), 5. F ERMAT’S LAST THEOREM (by A. Wiles and R. Taylor in 1995),6. The K EPLER CONJECTURE (by T. C. Hales in 1998), and7. The T ANIYAMA- SHIMURA CONJECTURE (by Breuil, Conrad, Diamond, and Taylor in 1999). Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include 1. The G OLDBACH CONJECTURE , 2. The R IEMANN HYPOTHESIS , 3. The P OINCARE ´CONJECTURE , 4. The conjecture that there exists a H ADAMARD MATRIX for every positive multiple of 4, 5. The TWIN PRIME CONJECTURE (i.e., the conjecture that there are an infinite number of TWIN PRIMES ), 6. Determination of whether NP -PROBLEMS are actually P -PROBLEMS , 7. The C OLLATZ PROBLEM , 8. Proof that the 196-ALGORITHM does not terminate when applied to the number 196, 9. Proof that 10 is a SOLITARY NUMBER , 10. Finding a formula for the probability that two elements chosen at random generate the SYM- METRIC GROUP Sn;/ 11. Solving the HAPPY END PROBLEM for arbitrary n, 12. Finding an E ULER BRICK whose space diagonal is also an integer,13. Proving which numbers can be represented asa sum of three or four (positive or negative) CUBIC NUMBERS , 14. L EHMER’S MAHLER MEASURE PROBLEM and LEHMER’S TOTIENT PROBLEM on the existence of COMPOSITE NUMBERS nsuch that f(n)(n/C281); j where f(n) is the TOTIENT FUNCTION . The Clay Mathematics Institute of Cambridge, Mas- sachusetts (CMI) has named seven "Millennium Prize Problems," selected by focusing on important classic questions in mathematics that have resisted solutionover the years. A $7 million prize fund has been established for the solution to these problems, with $1 million allocated to each. The problems consist of the R IEMANN HYPOTHESIS ,P OINCARE ´ CONJECTURE , HODGE CONJECTURE ,S WINNERTON- DYER CONJEC- TURE , solution of the Navier-Stokes equation, formu- lation of Yang-Mills theory, and determination of whether NP-PROBLEMS are actually P-PROBLEMS . In 1900, David Hilbert proposed a list of 23 out- standing problems in mathematics (HILBERT’S PRO- BLEMS , a number of which have now been solved, but some of which remain open. In 1912, Landau pro- posed four simply stated problems, now known as LANDAU’S PROBLEMS , which continue to defy attack even today. One hundred years after Hilbert, Smale (2000) proposed a list of 18 outstanding problems. K. S. Brown, D. Eppstein, S. Finch, and C. Kimber- ling maintain webpages of unsolved problems in mathematics. Classic texts on unsolved problems in various areas of mathematics are Croft et al. (1991), in GEOMETRY , and Guy (1994), in NUMBER THEORY . See also BEAL’S CONJECTURE ,FERMAT’S LAST THEO- REM,H ILBERT’S PROBLEMS ,K EPLER CONJECTURE , LANDAU’S PROBLEMS ,M ATHEMATICS CONTESTS , MATHEMATICS PRIZES ,POINCARE ´ CONJECTURE ,PRO- BLEM ,SZEMERE ´ DI’S THEOREM ,TWIN PRIMES References Brown, K. S. "Most Wanted List of Elementary Unsolved Problems." http://www.seanet.com/~ksbrown/mwlist.htm. Clay Mathematics Institute. "Millennium Prize Problems." http://www.claymath.org/prize_problems/. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991. Emden-Weinert, T. "Graphs: Theory-Algorithms-Complex- ity." http://people.freenet.de/Emden-Weinert/graphs.html. Eppstein, D. "Open Problems." http://www.ics.uci.edu/~epp- stein/junkyard/open.html. Finch, S. "Unsolved Mathematical Problems." http:// www.mathsoft.com/asolve/. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 21, 1994. Kimberling, C. "Unsolved Problems and Rewards." http:// cedar.evansville.edu/~ck6/integer/unsolved.html. Klee, V. "Some Unsolved Problems in Plane Geometry." Math. Mag. 52, 131 /C1/45, 1979. Meschkowski, H. Unsolved and Unsolvable Problems in Geometry. London: Oliver & Boyd, 1966. Ogilvy, C. S. Tomorrow’s Math: Unsolved Problems for the Amateur. New York: Oxford University Press, 1962. Ogilvy, C. S. "Some Unsolved Problems of Modern Geome- try." Ch. 11 in Excursions in Geometry. New York: Dover, pp. 143 /C1/53, 1990. Ramachandra, K. "Many Famous Conjectures on Primes; Meagre But Precious Progress of a Deep Nature." Proc. Indian Nat. Sci. Acad. Part A 64, 643 /C1/50, 1998. Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 0821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Provi- dence, RI: Amer. Math. Soc., 2000. van Mill, J. and Reed, G. M. (Eds.). Open Problems in Topology. New York: Elsevier, 1990. Weisstein, E. W. "Books about Mathematics Problems." http://www.treasure-troves.com/books/MathematicsPro- blems.html. Unstable Improper Node A FIXED POINT for which the STABILITY MATRIX has equal POSITIVE EIGENVALUES .See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,STABLE NODE,STABLE SPIRAL POINT ,U N- STABLE NODE,U NSTABLE SPIRAL POINT ,U NSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Unstable Node A FIXED POINT for which the STABILITY MATRIX has both EIGENVALUES POSITIVE ,so /l1 /C21 l2 /C210/. See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,STABLE NODE,STABLE SPIRAL POINT ,STABLE STAR,UNSTABLE IMPROPER NODE,UNSTABLE SPIRAL POINT ,UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Unstable Spiral Point A FIXED POINT for which the STABILITY MATRIX has EIGENVALUES OF THE FORM l9/C30 a 9i b (with a; b > 0):/ See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,STABLE NODE,STABLE SPIRAL POINT ,STABLE STAR,UNSTABLE IMPROPER NODE,UNSTABLE NODE, UNSTABLE STAR References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Unstable Star A FIXED POINT for which the STABILITY MATRIX has one zero EIGENVECTOR with POSITIVE EIGENVALUE l>0:/ See also ELLIPTIC FIXED POINT (DIFFERENTIAL EQUA- TIONS ), FIXED POINT ,H YPERBOLIC FIXED POINT (DIFFERENTIAL EQUATIONS ), STABLE IMPROPER NODE,STABLE NODE,STABLE SPIRAL POINT ,STABLE STAR,UNSTABLE IMPROPER NODE,UNSTABLE NODE, UNSTABLE SPIRAL POINT References Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion. New York: Wiley, pp. 22 /C1/5, 1989. Untouchable Number An untouchable number is an INTEGER which is not the sum of the PROPER DIVISORS of any other number. The first few are 2, 5, 52, 88, 96, 120, 124, 146, ... (Sloane’s A005114). Erdos has proven that there are infinitely many. It is thought that 5 is the only ODD untouchable number. References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 840, 1972. Guy, R. K. "Untouchable Numbers." §B10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 66 /C1/7, 1994. Sloane, N. J. A. Sequences A005114/M1552 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 60, 1986. U-Number ULAM SEQUENCE UPC The universal product code (UPC) is a 12-digit number and associated machine-readable bar code used to identify products being purchased in grocery stores. UPCs encode an individual product, but not its price (this part is done by a store’s computer after reading the product identifier). The UPC is main- tained by the Uniform Code Council of Dayton, Ohio. The first and last digits are separated from the others and written in a smaller font size. The first six digits are a manufacturer identifier, and the next five digits identify a specific product. The last digit is a check digit obtained from a12 /C3010 /C28/C20/C18 3X11 i /C301 i oddai /C27X10 i/C302 i evenai/C19 (mod 10)/C21 (mod 10) ; where (mod 10) indicates taking the REMAINDER after dividing by 10. For example, the UPC for Tropicana Pure Premium orange juice is 0 48500 00102 8 where the check digit isa12 /C3010 /C28[3(0 /C278 /C270 /C270 /C271 /C272) /C27(4 /C275 /C270 /C270 /C270) (mod 10)] (mod 10) /C3010 /C28[42 (mod 10)] (mod 10) /C3010 /C282 (mod 10) /C308; as expected. See also CHECKSUM ,CODING THEORY , ISBN Upper Bound A function f is said to have a upper bound C if f(x) 5C for all x in its DOMAIN . The LEAST UPPER BOUND is called the SUPREMUM . See also INEQUALITY ,INFIMUM ,LEAST UPPER BOUND , LOWER BOUND ,SUPREMUM Upper Half-Disk The unit upper half-disk is the portion of the COM- PLEX PLANE satisfying zjj51;I[z] > 0 fg :/ See also DISK,LOWER HALF-DISK,REAL AXIS,SEMI- CIRCLE ,UNIT DISK,UPPER HALF-PLANE Upper Half-Plane The portion, often denoted H, of the COMPLEX PLANE fx /C27iy : x; y /C23 (/C28/C12;/C12) satisfying y /C30I[z] > 0 i.e., H /C30fx /C27iy : x /C23 (/C28/C12;/C12g; y /C23 (0;/C12) g:/ See also COMPLEX PLANE ,HALF-PLANE ,LEFT HALF- PLANE ,L OWER HALF-PLANE ,M ODULAR FUNCTION , RIGHT HALF-PLANE ,UPPER HALF-DISK References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 14, 1997. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 112, 1987. Upper Integral The limit of an UPPER SUM, when it exists, as the MESH SIZE approaches 0. See also LOWER INTEGRAL ,R IEMANN INTEGRAL , UPPER SUM Upper Limit Let the greatest term H of a SEQUENCE be a term which is greater than all but a finite number of the terms which are equal to H. Then H is called the upper limit of the SEQUENCE . An upper limit of a SERIES upper lim n0/C12Sn /C30lim n0/C12Sn /C30k is said to exist if, for every e > 0 ; Sn /C28k jj B e for infinitely many values of n and if no number larger than k has this property. See also LIMIT,LOWER LIMIT,SUPREMUM LIMIT References Bromwich, T. J. I’a and MacRobert, T. M. "Upper and Lower Limits of a Sequence." §5.1 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 40, 1991.Upper Sum For a given function f(x) over a partition of a given interval, the upper sum is the sum of box areas fx/C31 kðÞDxk using the greatest value of the function fx/C31kðÞ) in each subinterval Dxk :/ See also LOWER SUM,R IEMANN INTEGRAL ,U PPER INTEGRAL Upper Triangular Matrix A TRIANGULAR MATRIX U OF THE FORM Uij /C30aijfor i 5j 0 for i > j :/C26 Written explicitly, U /C30a11a12 /C1/C1/C1 a1n 0 a22 /C1/C1/C1 a2n nn::: n 00 /C1/C1/C1 ann2 6643 775: An upper triangular matrix with elements f[i,j] above the diagonal can be formed using Upper- DiagonalMatrix [f, n] in the Mathematica add-on package LinearAlgebra‘MatrixMultiplica- tion‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also TRIANGULAR MATRIX ,LOWER TRIANGULAR MATRIX References Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, p. 10, 1962. Upper-Trimmed Subsequence The upper-trimmed subsequence of x /C30 xnfg is the sequence l(x) obtained by dropping the first occur- rence of n for each n.Ifx is a FRACTAL SEQUENCE , then l(x)/C30x:/ See also LOWER- TRIMMED SUBSEQUENCE References Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157/C1/68, 1997. Upward Drawing HASSE DIAGRAM Urchin Kepler’s original name for the SMALL STELLATED DODECAHEDRON . Urelement An urelement contains no elements, belongs to some set, and is not identical with the EMPTY SET (Moore 1982, p. 3; Rubin 1967, p. 23). "Ur" is a German prefix which is difficult to translate literally, but has a meaning close to "primeval." Urelements are also called "atoms" (Rubin 1967, Moore 1982) or "indivi- duals" (Moore 1982). In "pure" set theory, all elements are sets and there are no urelements. Often, the axioms of set theory are modified to allow the presence of urelements for ease in representing something. In fact, before Paul Cohen developed the method of forcing, some of the inde- pendence theorems in set theory were shown if urelements were allowed. See also EMPTY SET,SET THEORY References Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. U-Statistic References Hoeffding, W. "The Strong Law of Large Numbers for U- Statistics." Univ. North Carolina Inst. Statistics Mimeo Series, No. 302, 1961. Serfling, R. J. Approximation Theorems of Mathematical Statistics. New York: Wiley, 1980. Utility Graph The utility problem posits three houses and threeutility companies–say, gas, electric, and water–and asks if each utility can be connected to each house without having any of the gas/water/electric lines/ pipes pass over any other. This is equivalent to the equation "Can a PLANAR GRAPH be constructed from each of three nodes (‘houses’) to each of three other nodes (‘utilities’)?" This problem was first posed in this form by H. E. Dudeney in 1917 (Gardner 1984, p. 92). The answer is that no such PLANAR GRAPH exists, and the proof can be effected using the JORDAN CURVE THEOREM , while a more general result encompassing this one is the KURATOWSKI REDUCTION THEOREM . The utility graph UG is the graph showing the relationships described above, also known as the THOMSEN GRAPH and, in the more formal parlance of GRAPH THEORY , is known as the COMPLETE BIPARTITE GRAPH K3 ; 3 :/ A simple proof of the nonplanarity of the utility graph can be effected by nothing that the graph consists of a GRAPH CYCLE G /C28A /C28W /C28B /C28E /C28C ; to which the three edges A /C28E; B /C28G ; and C /C28W must be added. Now, for each of the edges, we have choose whether to draw the edge inside or outside the GRAPH CYCLE , and so for two of the edges, we must make the same choice. But two lines can’t be drawn on the same side without crossing, hence the graph is not planar. See also COMPLETE BIPARTITE GRAPH ,KURATOWSKI REDUCTION THEOREM ,P LANAR GRAPH ,T HOMSEN GRAPH References Chartrand, G. "The Three Houses and Three Utilities Problem: An Introduction to Planar Graphs." §9.1 in Introductory Graph Theory. New York: Dover, pp. 191 /C1/ 02, 1985. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 92 /C1/4, 1984. Ore, Ø.Graphs and Their Uses. New York: Random House, pp. 14 /C1/7, 1963. Pappas, T. "Wood, Water, Grain Problem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 175 and 233, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 262 /C1/63, 1999. Utility Problem UTILITY GRAPH V Valence The number of POLES of an AUTOMORPHIC FUNCTION in the closure of its FUNDAMENTAL REGION . See also FUNDAMENTAL REGION ,U NIVALENT FUNC- TION ,VERTEX DEGREE References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 84, 1997. Valency VERTEX DEGREE Valle’s Two-Thirds Factorization Method References Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673 /C1/715, 1990. Valuation A generalization of the P-ADIC NORM first proposed by Ku¨rscha´k in 1913. A valuation ½/C215½on a FIELD Kis a FUNCTION from Kto the REAL NUMBERS Rsuch that the following properties hold for all x;y/C23K: 1.xjj]0;/ 2.xjj/C300IFFx/C300, 3.xyjj/C30xjjyjj;/ 4.xjj51IMPLIES 1/C27x jj5Cfor some constant C] 1 (independent of x). If (4) is satisfied for C/C302, then ½/C215½satisfies the TRIANGLE INEQUALITY , 4a.x/C27y jj5xjj/C27yjjfor all x;y/C23K:/ If (4) is satisfied for C/C301 then ½/C215½satisfies the stronger ULTRAMETRIC inequality 4b.x/C27y jj5max xjj;yjj ðÞ :/ The simplest valuation is the ABSOLUTE VALUE for REAL NUMBERS . A valuation satisfying (4b) is called non-A RCHIMEDEAN VALUATION ; otherwise, it is called ARCHIMEDEAN . If½/C215½1is a valuation on Kandl]1;then we can define a new valuation ½/C215½2by xjj2/C30xjjl 1: (1) This does indeed give a valuation, but possibly with a different constant CinAXIOM 4. If two valuations arerelated in this way, they are said to be equivalent, and this gives an equivalence relation on the collec- tion of all valuations on K. Any valuation is equiva- lent to one which satisfies the triangle inequality (4a). In view of this, we need only to study valuations satisfying (4a), and we often view axioms (4) and (4a) as interchangeable (although this is not strictly true). If two valuations are equivalent, then they are both non-A RCHIMEDEAN or both A RCHIMEDEAN .Q;R;and Cwith the usual Euclidean norms are Archimedean valuated fields. For any PRIME p, the P-ADIC NUMBERS Qpwith the p-adic valuation ½/C215½pis a NON- ARCHIME- DEAN FIELD . IfKis any FIELD , we can define the trivial valuation onKbyxjj/C301 for all x"0 and 0 jj/C300;which is a NON- ARCHIMEDEAN VALUATION .I fKis a FINITE FIELD , then the only possible valuation over Kis the trivial one. It can be shown that any valuation on Qis equivalent to one of the following: the trivial valuation, Euclideanabsolute norm ½/C215½;orp-adic valuation ½/C215½ p:/ The equivalence of any nontrivial valuation of Qto either the usual ABSOLUTE VALUE or to a P-ADIC NORM was proved by Ostrowski (1935). Equivalent valua-tions give rise to the same topology. Conversely, if two valuations have the same topology, then they are equivalent. A stronger result is the following: Let ½/C215½ 1; ½/C215½2;...,½/C215½kbe valuations over Kwhich are pairwise inequivalent and let a1;a2;...,akbe elements of K. Then there exists an infinite sequence ( /x1;x2;...) of elements of Ksuch that lim n0/C12w:r:t:½/C215½1xn/C30a1 (2) lim n0/C12w:r:t:½/C215½2xn/C30a2; (3) etc. This says that inequivalent valuations are, in some sense, completely independent of each other. For example, consider the rationals Qwith the 3-adic and 5-adic valuations ½/C215½3and½/C215½5;and consider the sequence of numbers given by xn/C3043 /C2155n/C2792 /C2153n 3n/C275n: (4) Then xn043 as n0/C12with respect to ½/C215½3;butxn0 92 as n0/C12with respect to ½/C215½5;illustrating that a sequence of numbers can tend to two different limits under two different valuations. A discrete valuation is a valuation for which the VALUATION GROUP is a discrete subset of the REAL NUMBERS R:Equivalently, a valuation (on a FIELD K) is discrete if there exists a REAL NUMBER o>0 such that xjj/C23(1/C28o;1/C27o)[½x½/C301 for all x/C23K: (5) The p-adic valuation on Qis discrete, but the ordinary absolute valuation is not. If ½/C215½ is a valuation on K, then it induces a metric d(x;y) /C30 x /C28y jj (6) on K, which in turn induces a TOPOLOGY on K.If ½/C215½ satisfies (4b), then the metric is an ULTRAMETRIC .We say that K ;½/C215½ ðÞ is a complete valuated field if the METRIC SPACE is complete. See also ABSOLUTE VALUE ,LOCAL FIELD,M ETRIC SPACE , P-ADIC NUMBER ,S TRASSMAN’S THEOREM , ULTRAMETRIC ,VALUATION GROUP References Cassels, J. W. S. Local Fields. Cambridge, England: Cam- bridge University Press, 1986. Koch, H. "Valuations." Ch. 4 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 103 /C1/139, 2000. Ostrowski, A. "Untersuchungen zur aritmetischen Theorie der Ko¨rper." Math. Zeit. 39, 269 /C1/404, 1935. van der Waerden, B. L. Algebra, 2 vols. New York: Springer- Verlag, 1991. Weiss, E. Algebraic Number Theory. New York: Dover, 1998. Valuation Group Let K ;½/C215½ ðÞ be a valuated FIELD . The valuation group G is defined to be the set G /C30 xjj: x /C23 K ;x "0 fg ; with the group operation being multiplication. It is a SUBGROUP of the POSITIVE REAL NUMBERS , under multiplication. Valuation Ring Let K ;½/C215½ ðÞ be a NON- ARCHIMEDEAN FIELD . Its valua- tion ring R is defined to be R /C30 x /C23 K : xjj51 fg : The valuation ring has maximal IDEAL M /C30 x /C23 K : xjj51 fg ; and the FIELD R=M is called the residue field, class field, or field of digits. For example, if K /C30Qp (P-ADIC NUMBERS ), then R /C30Zp (p-adic integers), M /C30pZp (p- adic integers congruent to 0 mod p), and R=M/ /C30GF(p), the FINITE FIELD of order p. See also P-ADIC NUMBER Valuation Theory The study of VALUATIONS which simplifies class field theory and the theory of FUNCTION FIELDS . See also FUNCTION FIELD,VALUATION References Iyanaga, S. and Kawada, Y. (Eds.). "Valuations." §425 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1350 /C1/1353, 1980.Value The quantity which a FUNCTION f takes upon applica- tion to a given quantity. See also VALUE (GAME) Value (Game) The solution to a GAME in GAME THEORY . When a SADDLE POINT is present max i 5mmin j5naij /C30min j5nmax i5maij /C13v; and v is the value for pure strategies. See also ABSOLUTE VALUE ,GAME THEORY ,M INIMAX THEOREM ,VALUATION Vampire Number A number v/C30xywith an EVEN number nofDIGITS formed by multiplying a pair of n=2/-DIGIT numbers (where the DIGITS are taken from the original number in any order) xandytogether. Pairs of trailing zeros are not allowed. If vis a vampire number, then xand yare called its "fangs." Examples of vampire numbers include 1260/C3021/C2960 1395/C3015/C2993 1435/C3035/C2941 1530/C3030/C2951 1827/C3021/C2987 2187/C3027/C2981 6880/C3080/C2986 (Sloane’s A014575). The 8-digit vampire numbers are 10025010, 10042510, 10052010, 10052064, 10081260, ... (Sloane’s A048938) and the 10-digit vampire numbers are 1000174288, 1000191991, 1000198206,1000250010, ... (Sloane’s A048939). The numbers of 2n-digit vampires are 0, 7, 148, 3228, ... (Sloane’s A048935). Vampire numbers having twodistinct pairs of fangs include 125460 /C30204/C29615/C30246/C29510 11930170 /C301301/C299170/C301310/C299107 12054060 /C302004/C296015/C302406/C295010 (Sloane’s A048936).Vampire numbers having three distinct pairs of fangs include 13078260 /C301620/C298073/C301863/C297020 /C302070/C296318 : (Sloane’s A048937).General formulas can be constructed for special classes of vampires, such as the fangs x/C3025 /C21510k/C271 y/C30100 10k/C271/C2752/C0/C1 =25; giving the vampire v/C30xy/C3010k/C271/C2752/C0/C1 10k/C272/C27100 10k/C271/C2752/C0/C1 =25 /C30x/C31/C21510k/C272/C27t /C3082 6/C275/C21510k/C0/C1 1/C2725 /C21510k/C0/C1 ; where x+denotes xwith the DIGITS reversed (Roushe and Rogers). Pickover (1995) also defines pseudovampire numbers, in which the multiplicands have different numbers ofdigits. References Pickover, C. A. "Vampire Numbers." Ch. 30 in Keys to Infinity. New York: Wiley, pp. 227 /C1/231, 1995. Pickover, C. A. "Vampire Numbers." Theta 9,1 1/C1/13, Spring 1995. Pickover, C. A. "Interview with a Number." Discover 16, 136, June 1995. Roushe, F. W. and Rogers, D. G. "Tame Vampires." Undated manuscript. Sloane, N. J. A. Sequences A014575, A048933, A048934, A048935, A048936, A048937, A048938, and A048939 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. van der Grinten Projection AMAP PROJECTION given by the transformation x/C30sgnl/C28l0 ðÞ /C2pAG/C28P2ðÞ /C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2G/C28P2 ðÞ2/C28P2/C27A2 ðÞ G2/C28P2 ðÞq /C20/C21 P2/C27A2 (1) y/C30sgnfðÞpPQ/C28Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2/C271 ðÞ P2/C27A2 ðÞ /C28Q2p/C2/C3 P2/C27A2;(2)where A/C301 2p l/C28l0/C28l/C28l0 p/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12(3) G/C30cosu sinu/C27cosu/C281(4) P/C30G2 sinu/C281 ! (5) u/C30sin/C2812f p/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12(6) Q/C30A 2/C27G: (7) The inverse FORMULAS are f/C30sgnyðÞp/C28m1cosu1/C271 3p/C16/C17 /C28c2 3c3"# (8) l/C30pX2/C27Y2/C281/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C272X2/C28Y2 ðÞ /C27X2/C27Y2 ðÞ2q /C12/C12/C12/C12/C12/C12/C12/C12 2X /C27l 0; (9) where X/C30x p(10) Y/C30y p(11) c1/C30/C28Yjj1/C27X2/C27Y2/C0/C1 (12) c2/C30c1/C282Y2/C27X2(13) c3/C30/C282c1/C271/C272Y2/C27X2/C27Y2/C0/C12(14) d/C30Y2 c3/C271 272c3 2 c3 3/C289c1c2 c2 3 ! (15) a1/C301 c3c1/C28c2 2 3c3 ! (16) m1/C302ffiffiffiffiffiffiffiffiffiffi /C281 3a1q (17) u1/C301 3cos/C2813d a1m1 ! : (18) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 239 /C1/242, 1987. van der Pol Equation An ORDINARY DIFFERENTIAL EQUATION which can be derived from the RAYLEIGH DIFFERENTIAL EQUATION by differentiating and setting y /C30y?: It is an equation describing self-sustaining oscillations in which en- ergy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given by yƒ/C28 m 1 /C28y2/C0/C1 y?/C27y /C300: See also RAYLEIGH DIFFERENTIAL EQUATION References Birkhoff, G. and Rota, G.-C. Ordinary Differential Equa- tions, 3rd ed. New York: Wiley, p. 134, 1978. Kreyszig, E. Advanced Engineering Mathematics, 6th ed. New York: Wiley, pp. 165 /C1/166, 1988. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. van der Waerden Number This entry contributed by KEVIN O’BRYANT One form of VAN DER WAERDEN’S THEOREM states that for every POSITIVE INTEGERS k and r, there exists a constant n(k;r) such that if n0 ]n(k ;r) and 1; 2;...;n0 fg ƒC1 @ C2 ...@ Cr ; the some set Cicon- tains an ARITHMETIC SEQUENCE of length k. The least possible value of n(k ;r) is known as a van der Waerden number. The only nontrivial van der Waer- den numbers that are known exactly are summarized in the following table. As shown in the table, the first few values of n(2;k) for k /C301, 2, ... are 1, 3, 9, 35, 178, ... (Sloane’s A005346). /r_k/ k /C303 k /C304 k /C305 r /C302 9 35 178 r /C30327 r /C30476 Shelah (1988) proved that van der Waerden’s num- bers are PRIMITIVE RECURSIVE . It is known that n(3;r) 5erc1 (1) and that n(4;r) 5eeerc2 (2) for some constants c1and c2 : In 1998, T. Gowers announced that he has proved the general result n(n;k) 5ee 1=rðÞ eek/C27110 ; (3) but this work has not yet been published. Berlekamp(1968) showed that for p a prime, n(p /C271 ;2) > p /C215 2p ; (4) and that probabilistic arguments using the LOVA´ SZ LOCAL LEMMA show that n(k;r)>rk erk ! 1/C27X(1) ðÞ : (5) See also SZEMERE ´ DI’S THEOREM , VAN DER WAERDEN’S THEOREM References Berlekamp, E. A "Construction for Partitions Which Avoid Long Arithmetic Progressions." Canad. Math. Bull. 11, 409/C1/414, 1968. Goodman, J. E. and O’Rourke, J. (Eds.). Handbook of Discrete & Computational Geometry. Boca Raton, FL: CRC Press, p. 159, 1997. Gowers, W. T. "Fourier Analysis and Szemere ´di’s Theorem." InProceedings of the International Congress of Mathema- ticians, Vol. 1. Doc. Math. 1998, Extra Vol. I . Berlin, 617 /C1/ 629, 1998. Available electronically from http:// www.mathematik.uni-bielefeld.de/documenta/xvol-icm/ Fields/Fields.html. Gowers, W. T."A New Proof of Szemere ´di’s Theorem for Arithmetic Progressions of Length Four." Geom. Funct. Anal. 8, 529/C1/551, 1998. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., p. 29, 1991. Shelah, S. "Primitive Recursive Bounds for van der Waerden Numbers." J. Amer. Math. Soc. 1, 683/C1/697, 1988. Sloane, N. J. A. Sequences A005346/M2819 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. van der Waerden’s Theorem This entry contributed by K EVIN O’BRYANT van der Waerden’s theorem is a theorem about the existence of arithmetic sequences in sets. The theo-rem can be stated in three equivalent forms. 1. For every POSITIVE INTEGERS kand r, there exists a constant n(k;r) such that if n0]n(k;r) and 1;2;...;n0 fg ƒC1@C2...@Cr;the some set Ci contains an ARITHMETIC SEQUENCE of length k. 2. If a0;a1;... fg is an infinite sequence of integers satisfying 0 Bak/C271/C28akBrfor some r, then the sequence contains arbitrarily long arithmetic pro- gressions. 3. For every positive integers kand r, there is a constant g(k;r) such that if g0]gk;rðÞ anda1;a2; ...,ag0satisfies 0 Bai/C271/C28ai5r;then kof the numbers a1;a2;...,ag0are in arithmetic progres- sion. The constants n(k;r) are called VAN DER WAERDEN NUMBERS , and no FORMULA forn(k;r) is known. van der Waerden’s Theorem is a COROLLARY of S ZEMER- E´DI’S THEOREM . See also ARITHMETIC SEQUENCE ,BAUDET’S CONJEC- TURE ,S ZEMERE ´ DI’S THEOREM , VAN DER WAERDEN NUMBER References Guy, R. K. "Theorem of van der Waerden, Szemere ´di’s Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 204 /C1/209, 1994. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., p. 29, 1991. Khinchin, A. Y. "Van der Waerden’s Theorem on Arithmetic Progressions." Ch. 1 in Three Pearls of Number Theory. New York: Dover, pp. 11 /C1/17, 1998. van der Waerden, B. L. "Beweis einer Baudetschen Vermu- tung." Nieuw Arch. Wiskunde 15, 212 /C1/216, 1927. van Kampen’s Theorem In the usual diagram of inclusion homeomorphisms, if the upper two maps are injective, then so are the other two. References Dodson, C. T. J. and Parker, P. E. A User’s Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, p. 88, 1997. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 74 /C1/75 and 369 /C1/373, 1976. van Wijngaarden-Deker-Brent Method BRENT’S METHOD Vandermonde Determinant D x1 ;... ; xn ðÞ /C131 x1x2 1/C1/C1/C1 xn /C281 1 1 x2x22/C1/C1/C1 xn /C281 2 nnn ::: n 1 xnx2 n/C1/C1/C1 xn /C281 n/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12 /C30Y i ;j i/C21jxi /C28xj/C0/C1 (Sharpe 1987). For INTEGERS a1 ; ..., an ;D a1 ; ... ;an ðÞ is divisible byQn i/C301(i /C281)! (Chapman 1996), the first few values of which are the SUPERFACTORIALS 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, ... (Sloane’s A000178). See also SUPERFACTORIAL ,VANDERMONDE MATRIX References Chapman, R. "A Polynomial Taking Integer Values." Math. Mag. 69, 121, 1996. Fletcher, A.; Miller, J. C. P.; Rosenhead, L.; and Comrie, L. J. An Index of Mathematical Tables, Vol. 1. Reading, MA: p. 50, 1962. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1111, 2000.Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial Coefficients." Ch. 5 in Concrete Mathematics: A Founda- tion for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, p. 231, 1994. Radoux, C. "Query 145." Not. Amer. Math. Soc. 25, 197, 1978. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., p. 53, 1963. Sharpe, D. §2.9 in Rings and Factorization. Cambridge, England: Cambridge University Press, 1987. Sloane, N. J. A. Sequences A000178/M2049 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Vandermonde Identity CHU-VANDERMONDE IDENTITY Vandermonde Matrix A type of matrix which arises in the LEAST SQUARES FITTING of POLYNOMIALS and the reconstruction of a STATISTICAL DISTRIBUTION from the distribution’s MOMENTS . The solution of an n /C29n Vandermonde matrix equation requires O n2ðÞ operations. A Van- dermonde matrix of order n is OF THE FORM 1 x1x2 1/C1/C1/C1 xn/C281 1 1 x2x2 2/C1/C1/C1 xn/C281 2 nnn:::n 1xnx2 n/C1/C1/C1 xn/C281 n2 6643 775: See also TOEPLITZ MATRIX ,T RIDIAGONAL MATRIX , VANDERMONDE DETERMINANT References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Vandermonde Matrices and Toeplitz Ma- trices."§2.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 82 /C1/89, 1992. Vandermonde Theorem CHU-VANDERMONDE IDENTITY Vandermonde’s Convolution Formula CHU-VANDERMONDE IDENTITY Vandermonde’s Sum CHU-VANDERMONDE IDENTITY Vandiver’s Criteria Letpbe an IRREGULAR PRIME , and let P/C30rp/C271b ea PRIME with PBp2/C28p:Also let tbe an INTEGER such that t3f1 (mod P). For an IRREGULAR PAIR (p;2k); form the product Q2k/C30t/C28rd=2Ym b/C301trb/C281/C0/C1 bp/C281/C282k ; where m /C301 2p1 /C281 ðÞ d /C30Xm n/C301np /C282k : If Qr 2k f1 (mod P) for all such IRREGULAR PAIRS , then FERMAT’S LAST THEOREM holds for exponent p. See also FERMAT’S LAST THEOREM ,IRREGULAR PAIR, IRREGULAR PRIME References Johnson, W. "Irregular Primes and Cyclotomic Invariants." Math. Comput. 29, 113 /C1/120, 1975. Vanish A quantity which takes on the value zero is said to vanish. For example, the function f(z) /C30z2 vanishes at the point z /C300. See also ROOT Vanishing Point The point or points to which the extensions of PARALLEL lines appear to converge in a PERSPECTIVE drawing. See also DESARGUES’ THEOREM ,PERSPECTIVE ,PRO- JECTIVE GEOMETRY References Dixon, R. "Perspective Drawings." Ch. 3 in Mathographics. New York: Dover, pp. 79 /C1/88, 1991. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 19 /C1/20, 1930. Varga’s Constant V /C131 L/C309 :2890254919... ; where L is the ONE-NINTH CONSTANT . See also ONE-NINTH CONSTANT Variance ForNsamples of a variate having a distribution with known MEAN m;the "population variance" (usually called "variance" for short, although the word "popu- lation" should be added when needed to distinguish it from the SAMPLE VARIANCE ) is defined byvar(x)/C131 NX x/C28m ðÞ2/C30x2/C282mx/C27m2/C10/C11 /C30x2/C10/C11 /C282mxhi/C27m2/C10/C11 /C30x2/C10/C11 /C282mxhi/C27m2; (1) where xhi/C131 NXN i/C301xi: (2) But since xhiis an UNBIASED ESTIMATOR for the MEAN m/C13xhi; (3) it follows that the variance s2/C13var(x)/C30x2/C10/C11 /C28m2: (4) The population STANDARD DEVIATION is then defined as s/C13ffiffiffiffiffiffiffiffiffiffiffiffiffi var(x)p /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2hi/C28m2p : (5) A useful identity involving the variance is var(f(x)/C27g(x))/C30var(f(x))/C27var(g(x)): (6) Therefore, var(ax/C27b)/C30(ax/C27b)/C28ax/C27b hi ½/C1382DE /C30ax/C27b/C28axhi/C28b ðÞ2DE /C30ax/C28am ðÞ2DE /C30a2(x/C28m)2DE /C30a2x/C28m ðÞ2DE /C30a2var(x) (7) var(b)/C300: (8) If the population MEAN is not known, using the sample mean ¯xinstead of the population mean mto compute s2/C13ˆs2 N/C131 NXN i/C301xi/C28¯x ðÞ2(9) gives a BIASED ESTIMATOR of the population variance. In such cases, it is appropriate to use a S TUDENT’S T- DISTRIBUTION instead of a G AUSSIAN DISTRIBUTION . However, it turns out (as discussed below) that an UNBIASED ESTIMATOR for the population variance is given by s?2/C13ˆs?2N/C131 N/C281XN i/C301xi/C28¯x ðÞ2: (10) For multiple variables, the variance is given using the definition of COVARIANCE , varXn i/C301xi ! /C30covXn i/C301xi;Xm j/C301xj ! /C30Xn i/C301Xm j/C301covxi;xj/C0/C1 /C30Xn i/C301Xm j/C301 j/C30icovxi;xj/C0/C1 /C27Xn i/C301Xm j/C301 j"icovxi;xj/C0/C1 /C30Xn i/C301covxi;xj/C0/C1 /C27Xn i/C301Xm j/C301 j"icovxi;xj/C0/C1 /C30Xn i/C301varxiðÞ/C272Xn i/C301Xm j/C30i/C271covxi;xj/C0/C1 : (11) A linear sum has a similar form: varXn i/C301aixi ! /C30covXn i/C301aixi;Xm j/C301ajxj ! /C30Xn i/C301Xm j/C301aiajcovxi;xj/C0/C1 /C30Xn i/C301a2 ivarxiðÞ/C272Xn i/C301Xm j/C30i/C271aiajcovxi;xj/C0/C1 : (12) These equations can be expressed using the COVAR- IANCE MATRIX . To estimate the POPULATION VARIANCE s2from a sample of Nelements with a priori unknown MEAN (i.e., the MEAN is estimated from the sample itself), we need an UNBIASED ESTIMATOR fors2:This is given by the K-STATISTIC k2;where vars2/C0/C1 /C30k2/C30N N/C281s2(13) andm2/C13s2is the SAMPLE VARIANCE , defined by s2/C131 NXN i/C301xi/C28¯x ðÞ2: (14) The quantity Ns2=s2has a CHI-SQUARED DISTRIBU- TION . Note that some authors prefer the definition s?2/C131 N/C281XN i/C301xi/C28¯x ðÞ2; (15) since this makes the sample variance an UNBIASED ESTIMATOR for the population variance. To find the variance of the SAMPLE VARIANCE s2; remember that vars2/C0/C1 /C13s4/C10/C11 /C28s2/C10/C112; (16)and s2/C10/C11 /C30N/C281 Nm2: (17) Now find s4hi: s4/C10/C11 /C30(s2)2DE /C30 x2/C10/C11 /C28xhi2/C16/C172/C28/C29 /C301 NX x2i/C281 NX xi !22 4352 *+ /C301 N2X xi/C16/C172/C28/C29 /C282 N3X x2 iX xi/C16/C172/C28/C29 /C271 N4 /C2X xi/C16/C174/C28/C29 : (18) Working on the first term of (18), X x2i/C16/C172/C28/C29 /C30X x4i/C27X x2ix2jDE /C30X x4iDE /C27X x2ix2jDE /C30Nx4i/C10/C11 /C27N(N/C281)x2i/C10/C11 x2jDE /C30Nm?4/C27N(N/C281)m?22: (19) The second term of (18) is known from K-STATISTIC , X x2iX xj/C16/C172/C28/C29 /C30Nm?4/C27N(N/C281)m?22; (20) as is the third term, X xi/C16/C174/C28/C29 /C30NX x4iDE /C273N(N/C281)X x2ix2jDE /C30Nm?4/C273N(N/C281)m?22: (21) Combining (18)-(21) gives s4/C10/C11 /C301 N2Nm?4/C27N(N/C281)m?22/C2/C3 /C282 N3 /C2Nm?4/C27N(N/C281)m?22/C2/C3 /C271 N4Nm?4/C273N(N/C281)m?22/C2/C3 /C301 N/C282 N2/C271 N3 ! m?4 /C27N/C281 N/C282(N/C281) N2/C273(N/C281) N3"# m?22 /C30N2/C282N/C271 N3 ! m?4/C27(N/C281)(N2/C282N/C273) N3m?22 /C30(N/C281)(N/C281)m?4/C27(N2/C282N/C273)m?22 N3(22) (Kenney and Keeping 1951, p. 164), so plugging in (17) and (22) gives vars2/C0/C1 /C30s4/C10/C11 /C28s2/C10/C112 /C30(N/C281) (N/C281)m?4/C27N2/C282N/C273 ðÞ m?22½/C138 N3 /C28(N/C281)2N N3m?22 /C30(N/C281) (N/C281)m?4/C28(N/C283)m?22½/C138 N3: (23) (Kenney and Keeping 1951, p. 164). Student calculated the SKEWNESS and KURTOSIS of the distribution of s2as g1/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 N/C281s (24) g2/C3012 N/C281(25) and conjectured that the true distribution is P EARSON TYPE IIIDISTRIBUTION fs2/C0/C1 /C30Cs2/C0/C1N/C283 ðÞ =2e/C28Ns2=2s2; (26) where s2/C30Ns2 N/C281(27) C/C30N 2s2 !N/C281 ðÞ =2 GN/C281 2 ! : (28) This was proven by R. A. Fisher.The distribution of sitself is given by f(s)/C302N 2s2 !N/C281 ðÞ =2 GN/C281 2 ! e/C28ns2=2s2sN/C282(29) shi/C30ffiffiffiffiffi 2 Ns GN 2 ! GN/C281 2 ! s/C13b(N)s; (30) whereb(N)/C13ffiffiffiffiffi 2 Ns GN 2 ! GN/C281 2 ! : (31) The MOMENTS are given by mr/C302 N !r=2GN/C281/C27r 2 ! GN/C281 2 ! sr; (32) and the variance is var(s)/C30n2/C28n2 1/C30N/C281 Ns2/C28b(N)s ½/C1382 /C301 NN/C281/C282G2N 2 ! G2N/C281 2 ! s22 666643 77775(33) An UNBIASED ESTIMATOR ofsiss=b(N):Romanovsky showed that b(N)/C301/C283 4N/C287 32N2/C28139 51849 N3/C27/C1/C1/C1 (34) When computing numerically, the MEAN must be computed before s2can be determined. This requires storing the set of sample values. It is possible to calculate s?2using a recursion relationship involving only the last sample as follows. Here, use mjto denote mcalculated from the first jsamples ( not thejth MOMENT ) mj/C13Pj i/C301xi j; (35) and s2 jdenotes the value for the sample variance s?2 calculated from the first jsamples. The first few values calculated for the MEAN are m1/C30x1 (36) m2/C301/C215m1/C27x2 2(37) m3/C302m2/C27x3 3: (38) Therefore, for j/C302, 3 it is true that mj/C30(j/C281)mj/C281/C27xj j(39) Therefore, by induction, mj/C271 /C30(j /C27 1) /C28 1 ½/C138 mj /C271 ðÞ/C281 /C27 xj /C271 j /C27 1 /C30jmj /C27 xj/C271 j /C27 1 (40) mj/C271(j /C271) /C30(j /C271)mj /C27 xj /C271 /C28 mj/C0/C1 (41) mj/C271 /C30 mj /C27xj/C271 /C28 mj j /C27 1; (42) and s2 j /C30Pj i/C301xi /C28 mj/C0/C12 j /C28 1 (43) for j ]2 ; so jsj/C2712 /C30jPj/C271 i/C301xi /C28 mj/C271/C0/C12 j/C30Xj/C271 i/C301xi /C28 mj/C271/C0/C12 /C30Xj/C271 i/C301xi /C28 mj/C0/C1 /C27 mi /C28 mj/C271/C0/C1/C2/C32 /C30Xj/C271 i/C301xi /C28 mj/C0/C12/C27Xj/C271 i/C301mj /C28 mj/C271/C0/C12/C272Xj/C271 i/C301xi /C28 mj/C0/C1 /C2 mj /C28 mj/C271/C0/C1 : (44) Working on the first term, Xj/C271 i/C301xi /C28 mj/C0/C12/C30Xj i/C301xi /C28 mj/C0/C12/C27 xj/C271 /C28 mj/C0/C12 /C30(j /C281)s2 j /C27 xj/C271 /C28 mj/C0/C12: (45) Use (41) to write xj/C271 /C28 mj /C30 j /C271 ðÞ mj/C271 /C28 mj/C0/C1 ; (46) so Xj/C271 i/C301xi /C28 mj/C0/C12/C30(j /C281)s2j /C27(j /C271)2 mj/C271 /C28 mj/C0/C12: (47) Now work on the second term in (44), Xj/C271 i/C301mj /C28 mj/C271/C0/C12/C30(j /C271) mj /C28 mj/C271/C0/C12: (48) Considering the third term in (44), Xj/C271 i/C301xi /C28 mj/C0/C1 mj /C28 mj/C271/C0/C1 /C30 mj /C28 mj/C271/C0/C1Xj /C271 i /C301xi /C28 mj/C0/C1 /C30(mj /C28 mj/C271)Xj i/C301(xi /C28 mj) /C27(xj /C271 /C28 mj)"#/C30 mj /C28 mj/C271/C0/C1 xj /C271 /C28 mj /C28j mj /C27Xj i/C301xi ! : (49) But Xj i/C301xi /C30j mj ; (50) so mj /C28 mj/C271/C0/C1 xj/C271 /C28 mj/C0/C1 /C30 mj /C28 mj/C271/C0/C1 (j /C271) mj/C271 /C28 mj/C0/C1 /C30/C28(j /C271) mj /C28 mj /C271/C0/C12: (51) Plugging (47), (48), and (51) into (44), jsj/C2712 /C30 (j /C281)s2j /C27 j /C271 ðÞ2mj/C271 /C28 mj/C0/C12hi /C27 (j /C271) mj/C28mj/C271/C0/C12hi /C272/C28(j/C271)mj/C28mj/C271/C0/C12hi /C30(j/C281)s2j/C27j/C271 ðÞ2mj/C271/C28mj/C0/C12 /C28(j/C271)(mj/C28mj/C271)2 /C30(j/C281)s2j/C27(j/C271) (j/C271)/C281 ½/C138 mj/C271/C28mj/C0/C12 /C30(j/C281)s2j/C27j(j/C271)mj/C271/C28mj/C0/C12; (52) so s2j/C271/C301/C281 j ! s2j/C27j/C271 ðÞ mj/C271/C28mj/C0/C12: (53) See also CENTRAL MOMENT ,C HARLIER’S CHECK , CORRELATION (STATISTICAL ), COVARIANCE ,C OVAR- IANCE MATRIX ,E RROR PROPAGATION , K-STATISTIC , MEAN,M OMENT ,RAW MOMENT ,SAMPLE VARIANCE , STANDARD ERROR References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 144 /C1/145, 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Moments of a Distribution: Mean, Variance, Skewness, and So Forth." §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.Cambridge, England: Cambridge University Press,pp. 604 /C1 /609, 1992. Roberts, M. J. and Riccardo, R. A Student’s Guide to Analysis of Variance. London: Routledge, 1999. Variate ARANDOM VARIABLE in statistics. References Kenney, J. F. and Keeping, E. S. "Variates." §1.2 in Mathe- matics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 5 /C1/6, 1962. Variation The D/-variation is a variation in which the varied path over which an integral is evaluated may end at different times than the correct path, and there may be variation in the coordinates at the endpoints. The d/-variation is a variation in which the varied path in configuration space terminates at the end- points representing the system configuration at the same time t1 and t2 as the correct path; i.e., the varied path always returns to the same endpoints in config- uration space, so dqit1ðÞ/C30dqit2ðÞ/C300: See also CALCULUS OF VARIATIONS ,V ARIATION OF ARGUMENT ,VARIATION OF PARAMETERS Variation Coefficient Ifsxis the STANDARD DEVIATION of a set of samples xi and ¯xitsMEAN , then V/C13sx ¯x: Variation of Argument Let arg f(z) ½/C138 denote the change in argument of a function f(z) around a CONTOUR g:Also let Ndenote the number of ROOTS off(z)i ngand Pdenote the number of POLES off(z)i ng:Then argf(z) ½/C138 /C301 2pN/C28P ðÞ : (1) To find arg f(z) ½/C138 in a given region R, break Rinto paths and find arg f(z) ½/C138 for each path. On a circular ARC z/C30Reiu; (2) letf(z)b ea POLYNOMIAL P(z) of degree n. Then argP(z) ½/C138 /C30arg znPzðÞ zn !"# /C30argzn½/C138 /C27argPzðÞ zn !"# : (3) Plugging in z/C30Reiugives argP(z) ½/C138 /C30argReiun/C2/C3 /C27argPR eiuðÞ Reiun"# (4)lim R0/C12PR eiuðÞ Reiun/C30constant½/C138 ; (5) so PR eiuðÞ Reiun"# /C300; (6) and argP(z) ½/C138 /C30argeiun/C2/C3 /C30nu2/C28u1 ðÞ : (7) For a REAL segment z/C30x, argf(x) ½/C138 /C30tan/C2810 f(x)"# /C300: (8) For an IMAGINARY segment z/C30iy, argf(iy) ½/C138 /C30tan/C281IP(iy) ½/C138 RP(iy) ½/C138()u2 u1: (9) Note that the ARGUMENT must change continuously, so "jumps" occur across inverse tangent asymptotes. Variation of Parameters For a second-order ORDINARY DIFFERENTIAL EQUA- TION , yƒ/C27p(x)y?/C27q(x)y/C30g(x): (1) Assume that linearly independent solutions y1(x) and y2(x) are known and seek v1(x) and v2(x) such that y/C31/C30v1y1/C27v2y2 (2) y?/C31/C30 v?1y1/C27v?2y2 ðÞ /C27v1y?1/C27v2y?2 ðÞ : (3) Now, impose the additional condition that v?1y1/C27v?2y2/C300 (4) so that y?/C31(x)/C30v1y?1/C27v2y?2 (5) yƒ/C31(x)/C30v?1y?1/C27v?2y?2/C27v1yƒ1/C27v2y?2: (6) Plug y/C31;y/C31?;andy/C31ƒback into the original equation to obtain v1yƒ1/C27py?1/C27qy1 ðÞ /C27v2yƒ2/C27py?2/C27qy2 ðÞ /C27v?1y?1/C27v?2y?2 /C30g(x) (7) v?1y? 1/C27v?2y?2/C30g(x): (8) Therefore, v?1y1/C27v?2y2/C300 (9) v?1y?1/C27v?2y?2/C30g(x): (10) Generalizing to an nth degree ODE, let y1;...,ynbe the solutions to the homogeneous ODE and let v?1(x); ...,v?n(x) be chosen such that y1v ?1 /C27y2v?2 /C27.../C27ynv ?n /C300 y?1v ?1 /C27y?2v?2 /C27.../C27y ?nv ?n /C300 n y(n/C281) 1v?1 /C27y(n/C281) 2v?2 /C27.../C27y(n/C281) nv ?n /C30g(x) :8 >>< >>:(11) Then the particular solution is then y+(x) /C30v1(x)y1(x) /C27.../C27vn(x)yn(x) : (12) Variational Calculus CALCULUS OF VARIATIONS Variety ALGEBRAIC VARIETY Varignon Parallelogram The figure formed when the MIDPOINTS of the sides of a convex QUADRILATERAL are joined. VARIGNON’S THEOREM demonstrated that this figure is a PARALLE- LOGRAM . The center of the Varignon parallelogram is the CENTROID of four point masses placed on the VERTICES of the QUADRILATERAL . See also BIMEDIAN ,M IDPOINT ,M IDPOINT POLYGON , PARALLELOGRAM ,QUADRILATERAL ,VARIGNON’S THE- OREM References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 53, 1967. Varignon’s Theorem The figure formed when the MIDPOINTS of the sides ofa convex QUADRILATERAL are joined in order is a PARALLELOGRAM . Equivalently, the BIMEDIANS bisect each other. The AREA of this VARIGNON PARALLELO- GRAM is half that of the QUADRILATERAL . The PERI- METER is equal to the sum of the diagonals of the original QUADRILATERAL . See also BIMEDIAN ,M IDPOINT ,M IDPOINT POLYGON , QUADRILATERAL ,VARIGNON PARALLELOGRAM References Coxeter, H. S. M. and Greitzer, S. L. "Quadrangles; Var- ignon’s Theorem." §3.1 in Geometry Revisited. Washing- ton, DC: Math. Assoc. Amer., pp. 51 /C1/56, 1967. Vassiliev Invariant This entry contributed by S ERGEI DUZHIN Vassiliev invariants, discovered around 1989, pro- vided a radically new way of looking at KNOTS . The notion of finite type (a.k.a. Vassiliev) KNOT INVAR- IANTS was independently invented by V. Vassiliev and M. Goussarov around 1989. Vassiliev’s approach is based on the study of discriminants in the (infinite- dimensional) spaces of SMOOTH MAPS from one MANI- FOLD into another. By definition, the discriminant consists of all maps with SINGULARITIES . For example, consider the space of all smooth maps from the circle into 3-space M/C30f:S10R3/C8/C9 :Iffis an EMBEDDING (i.e., has no singular points), then it represents a knot. The complement of the set of allknots is the discriminant SƒM:It consists of all smooth maps from S 1intoR3that have singularities, either local , where f?/C300;ornonlocal , where fis not injective. Two knots are equivalent IFFthey can be joined by a path in the space Mthat does not intersect the discriminant. Therefore, knot types arein one-to-one correspondence with the connected components of the complement M_S;and KNOT INVARIANTS with values in an A BELIAN GROUP Gare nothing but COHOMOLOGY CLASSES from H0M_S;G ðÞ : The FILTRATION ofSby subspaces corresponding to SINGULAR KNOTS with a given number of ORDINARY DOUBLE POINTS gives rise to a SPECTRAL SEQUENCE , which contains, in particular, the spaces of finite typeinvariants. Birman and Lin (1993) have contributed significantly to the simplification of the Vassiliev’s original tech- niques. In particular, they explained the relation between J ONES POLYNOMIALS and finite type invar- iants (Peterson 1992, Birman and Lin 1993, Bar- Natan 1995) and emphasized the role of the algebra of CHORD DIAGRAMS . In fact, substituting the POWER SERIES forexas the variable in the J ONES POLYNOMIAL yields a POWER SERIES whose COEFFICIENTS are Vassiliev invariants (Birman and Lin 1993). Kontse- vich (1993) proved the first difficult theorem aboutVassiliev invariants with the help of the K ONTSEVICH INTEGRAL . Bar-Natan undertook a thorough study of Vassiliev invariants; in particular, he showed the importance of the algebra of Feynman diagrams and diagrams with uni- and tri-valent vertices (Bar- Natan 1995). Bar-Natan (1995) remains the most authoritative source on the subject. Expressed in simple terms, Vassiliev’s fundamental idea is to study the prolongation of KNOT INVARIANTS to SINGULAR KNOTS –immersions f : S1 0 R3 having a finite number of ORDINARY DOUBLE POINTS . Let Xn denote the set of EQUIVALENCE CLASSES of SINGULAR KNOTS with n double points and no other singula- rities. The following definition is based on a recursion which allows to extend a KNOT INVARIANT from X0 to X1 ; then to X2 ; etc., and thus finally to the whole of X /C30@n Xn : Given a knot invariant v : X0 0 Q; its Vassiliev prolongation ˆv : X 0 Q is defined as by the rules 1. ˆvX /C13v ; j and 2. ˆv(/ /) /C30 ˆv(/ /) /C28 ˆv(/ /) (Vassiliev’s skein relation). The right-hand side of Vassiliev’s skein relation refers to the two resolutions of the double point– positive and negative. A crucial observation is that each of them is well-defined (does not depend on the plane projection used to express this relation). A KNOT INVARIANT v is called a Vassiliev invariant of order 5n/ if its prolongation ˆv vanishes on all knots with more than n double points. For example, the simplest nontrivial Vassiliev invariant v2has the following explicit description. Let D be an arbitrary KNOT DIAGRAM of the given knot K and w an arbitrary distinguished point on D, different from all crossings. Then v2(K) /C30X ijij UOOUoi oj ; where the summation spreads over all pairs of cross- ing points i, j such that (1) during one complete turn of the diagram in the positive direction starting from point w the points i and j are encountered in the order i ;j ;i ;j; and (2) the four corresponding passages through these crossing points are underpass, over- pass, overpass, and underpass, respectively. The numbers oi ; ojstand for the local WRITHE at points iand j, defined according to the above illustration. It turns out that the nth coefficient of the CONWAY POLYNOMIAL is a Vassiliev invariant of order n and, in particular, the second coefficient coincides with v2 :/ Vassiliev invariants are at least as strong as all known polynomial knot invariants: ALEXANDER , JONES ,K AUFFMAN , and HOMFLY POLYNOMIALS . This means that if two knots K1and K2can be distinguished by such a polynomial, then there is a Vassiliev invariant that takes different values for K1 and K2 :/ The set of all Q/-valued Vassiliev invariants V /C30@n Vn forms a VECTOR SPACE over the rationals, with the increasing FILTRATION Q /C30V0 ƒV1 ƒV2 ƒ...: The ASSOCIATED GRADED SPACE /C154n Vn =Vn/C281has a struc- ture of a HOPF ALGEBRA and can be interpreted as the algebra of CHORD DIAGRAMS . The numbers of independent Vassiliev invariants of a given degree n (i.e., the dimension of Vn) are known for n /C300 to 12 (Kneissler 1997) and are summarized in following table (A007473). n 0123 45678 91 01 11 2 /dim Vn/ 1123610193360104184316548 The totality of all Vassiliev invariants is equivalent to one UNIVERSAL VASSILIEV INVARIANT defined through the KONTSEVICH INTEGRAL . Two of the most important problems about Vassiliev invariants were raised in 1990 and remain unan- swered today. 1. Is it true that Vassiliev invariants distinguishknots? In other words, given two nonequivalent knots K 1andK2;is it always possible to indicate a finite type invariant vsuch that vK1ðÞ"vK2ðÞ?/ 2. Is it true that Vassiliev invariants can detect knot orientation? More specifically, is there a knot Kand a finite type invariant vsuch that v(K)" v(¯K);where ¯Kdiffers from Kby a change of parameterization that reverses the orientation? See also CHORD DIAGRAM ,H ABIRO MOVE,K NOT INVARIANT ,KONTSEVICH INTEGRAL ,UNIVERSAL VAS- SILIEV INVARIANT References Bar-Natan, D. "Bibliography of Vassiliev Invariants." http:// www.ma.huji.ac.il/~drorbn/VasBib/VasBib.html. Bar-Natan, D. "On the Vassiliev Knot Invariants." Topology 34, 423/C1/472, 1995. Birman, J. S. "New Points of View in Knot Theory." Bull. Amer. Math. Soc. 28, 253/C1/287, 1993. Birman, J. S. and Lin, X.-S. "Knot Polynomials and Vassi- liev’s Invariants." Invent. Math. 111, 225 /C1/270, 1993. Duzhin, S. V. "Vassiliev invariants and combinatorial structures." Online lecture notes, 1999 /C1/2000. http:// www.botik.ru/~duzhin/Vics. Goussarov, M. "On n-Equivalence of Knots and Invariants of Finite Degree." In Topology of Manifolds and Varieties (Ed. O. Viro). Providence, RI: Amer. Math. Soc., pp. 173 /C1/ 192, 1994. Kneissler, J. "The Number of Primitive Vassiliev Invariants up to Degree Twelve." 1997. http://www.math.uni- bonn.de/people/jk/pappvi12.cgi. Kontsevich, M. "Vassiliev’s Knot Invariants." Adv. Soviet Math. 16, Part 2, pp. 137 /C1/150, 1993. Peterson, I. "Knotty Views: Tying Together Different Ways of Looking at Knots." Sci. News 141, 186 /C1/187, 1992. Prasolov, V. V. and Sossinsky, A. B. Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology. Providence, RI: Amer. Math. Soc., 1996. Sloane, N. J. A. Sequences A007473/M0765 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Stoimenow, A. "Degree-3 Vassiliev Invariants." http://guest- s.mpim-bonn.mpg.de/alex/ptab/vas3.html. Vassiliev, V. A. "Cohomology of Knot Spaces." In Theory of Singularities and Its Applications (Ed. V. I. Arnold). Providence, RI: Amer. Math. Soc., pp. 23 /C1/69, 1990. Vassiliev, V. A. Complements of Discriminants of Smooth Maps: Topology and Applications. Providence, RI: Amer. Math. Soc., 1992. Vassiliev Polynomial Vassiliev (1990) introduced a radically new way of looking at KNOTS by considering a multidimensional space in which each point represents a possible 3-D knot configuration. If two KNOTS are equivalent, a path then exists in this space from one to the other. The paths can be associated with polynomial invar- iants. Birman and Lin (1993) subsequently found a way to translate this scheme into a set of rules and list of potential starting points, which makes analysis of Vassiliev polynomials much simpler. Bar-Natan (1995) and Birman and Lin (1993) proved that JONES POLYNOMIALS and several related expressions are directly connected (Peterson 1992). In fact, substitut- ing the POWER SERIES forR3as the variable in the JONES POLYNOMIAL yields a POWER SERIES whose COEFFICIENTS are Vassiliev polynomials (Birman and Lin 1993). Bar-Natan (1995) also discovered a link with Feynman diagrams (Peterson 1992). See also HABIRO MOVE References Bar-Natan, D. "On the Vassiliev Knot Invariants." Topology 34, 423/C1/472, 1995. Birman, J. S. "New Points of View in Knot Theory." Bull. Amer. Math. Soc. 28, 253/C1/287, 1993. Birman, J. S. and Lin, X.-S. "Knot Polynomials and Vassi- liev’s Invariants." Invent. Math. 111, 225/C1/270, 1993.Peterson, I. "Knotty Views: Tying Together Different Ways of Looking at Knots." Sci. News 141, 186/C1/187, 1992. Praslov, V. V. and Sossinsky, A. B. Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology. Providence, RI: Amer. Math. Soc., 1996. Stoimenow, A. "Degree-3 Vassiliev Invariants." http://guest- s.mpim-bonn.mpg.de/alex/ptab/vas3.html. Vassiliev, V. A. "Cohomology of Knot Spaces." In Theory of Singularities and Its Applications (Ed. V. I. Arnold). Providence, RI: Amer. Math. Soc., pp. 23 /C1/69, 1990. Vassiliev, V. A. Complements of Discriminants of Smooth Maps: Topology and Applications. Providence, RI: Amer. Math. Soc., 1992. Vault Let a vault consist of two equal half- CYLINDERS of radius rwhich intersect at RIGHT ANGLES so that the lines of their intersections (the "groins") terminate in the VERTICES of a SQUARE . Two vaults placed bottom- to-top form a S TEINMETZ SOLID on two cylinders. Solving the equations x2/C27z2/C30r2(1) y2/C27z2/C30r2(2) simultaneously gives x/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p (3) y/C309ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p : (4) One quarter of the vault can therefore be described by the PARAMETRIC EQUATIONS x/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p (5) y/C30/C28uffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p (6) z/C30z: (7) The SURFACE AREA of the vault is therefore given by A/C304gl(z)rdu; (8) where l(z) is the length of a cross section at height z anduis the angle a point on the center of this line makes with the origin. But z/C30rsinu;so dz/C30rcosudu/C30rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C28sin2up du/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2/C28z2p du; and l(z) /C302ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28x2p (9) A /C304gr 02rffiffiffiffiffiffiffiffiffiffiffiffiffiffir 2 /C28z2p dzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28 z2p /C304gr 02rdz/C308r2 : (10) The VOLUME of the vault is V /C30gr 02ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C28z2p/C16/C172 dz/C308 3r3: (11) See also CYLINDER ,DOME,STEINMETZ SOLID References Lines, L. Solid Geometry. New York: Dover, pp. 112 /C1/113, 1965. Moore, M. "Symmetrical Intersections of Right Circular Cylinders." Math. Gaz. 58, 181/C1/185, 1974. Vector A vector is formally defined as an element of a VECTOR SPACE . In the commonly encountered VECTOR SPACE Rn(i.e., Euclidean n-space), a vector is given by n coordinates and can be specified as A1;A2;...;An ðÞ : Vectors can be added together ( VECTOR ADDITION ) and multiplied by SCALARS (SCALAR MULTIPLICATION ). VECTOR MULTIPLICATION is not uniquely defined, but a number of different types of products, such as the DOT PRODUCT ,CROSS PRODUCT ,TENSOR DIRECT PRO- DUCT can be defined for pairs of vectors. A vector from a point Ato a point Bis denoted AB/C131!; and a vector vmay be denoted /C0v;or more commonly, v. The point Ais often called the "tail" of the vector, andBis called the vector’s "head." A vector with unit length is called a UNIT VECTOR and is denoted using a HAT,ˆv:An arbitrary vector may be converted to a UNIT VECTOR by dividing by its NORM (i.e., length), vjj/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 1/C27v22/C27.../C27v2 nq ; (1) giving ˆv/C13v vjj(2) AZERO VECTOR , denoted 0 ;is a vector of length 0, and thus has all components equal to zero. Since vectors remain unchanged under TRANSLATION , it is often convenient to consider the tail Aas located at the origin when, for example, defining VECTOR ADDITION and SCALAR MULTIPLICATION . A vector may also be defined as a set of nnumbers A0; ...,Anthat transform according to the ruleA?i/C30aijAj; (3) where E INSTEIN SUMMATION notation has been used, aij/C30@x?i @xj/C30@xj @x?i(4) are constants (corresponding to the DIRECTION CO- SINES ), with partial derivatives taken with respect to the original and transformed coordinate axes, and i;j/C301;...,n(Arfken 1985, p. 10). This makes a vector aTENSOR ofRANK one. A vector with ncomponents in called an n-vector, and a SCALAR may therefore be thought of as a 1-vector (or a 0- RANK TENSOR ). Vectors are invariant under TRANSLATION , and they reverse sign upon inversion. Objects which resemble vectors but do not reverse sign upon inversion are known as PSEUDOVECTORS . A vector is represented in Mathematica as a list of numbers { a1,a2, ..., an}. V ECTOR ADDITION is then simply written using a plus sign, e.g., { a1,a2, ..., an}/C27{b1,b2, ...,bn}, and SCALAR MULTIPLICATION is indicated by placing a scalar next to a vector (with orwithout an optional asterisk), s{a1,a2, ...,an}. Let ˆnbe the UNIT VECTOR defined in SPHERICAL COORDINATES by ˆn/C13cosusinf sinusinf cosf2 435: (5) Then the average value of the x-component of the ˆn over the surface of the UNIT SPHERE is given by nxhi/C30g2p 0gp 0cosusinf ðÞ sinfdfdu f2p 0fp 0sinfdfdu /C301 4psinu ½/C1382p 0g2p 0sin2fdf/C300: (6) More generally, nihi/C300 (7) fori/C30x,y,o rz(indexed as 1, 2, 3), and ninj/C10/C11 /C301 3dij (8) ninjnk/C10/C11 /C300 (9) ninknlnm hi /C301 15dikdlm/C27dildkm/C27dimdkl ðÞ : (10) Given vectors a,b,c,d, the average values of a number of quantities over the UNIT SPHERE are given by a/C215ˆnðÞ2DE /C301 3a2(11) a/C215ˆnðÞ b/C215ˆnðÞ hi /C301 3a/C215b (12) a /C215ˆnðÞ ˆn hi /C301 3a (13) a /C29ˆn ðÞ2DE /C302 3a2 (14) a /C29ˆn ðÞ /C215 b /C29ˆn ðÞ hi /C302 3a /C215b; (15) and a /C215ˆnðÞ b /C215ˆnðÞ c /C215ˆnðÞ d /C215ˆnðÞ hi /C301 15 (a /C215d)(c /C215d) /C27(a /C215c)(b /C215d) /C27(a /C215d)(b /C215c) ½/C138 (16) where dijis the KRONECKER DELTA , a /C215b is a DOT PRODUCT , and EINSTEIN SUMMATION has been used. A MAP f : Rn /C2Rn which assigns each x a VECTOR FUNCTION f(x) is called a VECTOR FIELD . See also COLUMN VECTOR ,CONTRAVARIANT VECTOR , COVARIANT VECTOR ,F OUR- VECTOR ,H ELMHOLTZ’S THEOREM ,N ORM,N ULL VECTOR ,O NE-FORM,PSEU- DOVECTOR ,R OW VECTOR ,S CALAR ,T ENSOR ,U NIT VECTOR ,V ECTOR BASIS,V ECTOR BUNDLE ,V ECTOR FIELD ,V ECTOR FUNCTION ,V ECTOR SPACE ,Z ERO VECTOR References Arfken, G. "Vector Analysis." Ch. 1 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 1 /C1/84, 1985. Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, 1989. Crowe, M. J. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. New York: Dover, 1985. Gibbs, J. W. and Wilson, E. B. Vector Analysis: A Text-Book for the Use of Students of Mathematics and Physics, Founded Upon the Lectures of J. Willard Gibbs. New York: Dover, 1960. Jeffreys, H. and Jeffreys, B. S. "Scalars and Vectors." Ch. 2 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 56 /C1/85, 1988. Marsden, J. E. and Tromba, A. J. Vector Calculus, 4th ed. New York: W. H. Freeman, 1996. Morse, P. M. and Feshbach, H. "Vector and Tensor Formal- ism."§1.5 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44 /C1/54, 1953. Schey, H. M. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton, 1973. Schwartz, M.; Green, S.; and Rutledge, W. A. Vector Analy- sis with Applications to Geometry and Physics. New York: Harper Brothers, 1960. Spiegel, M. R. Schaum’s Outline of Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis. New York: Schaum, 1959. Weisstein, E. W. "Books about Vectors." http://www.trea- sure-troves.com/books/Vectors.html.Vector Addition The so-called PARALLELOGRAM LAW gives the rule for vector addition of vectors A and B. The sum A /C27B of the vectors is obtained by placing them head to tail and drawing the vector from the free tail to the free head. Vector addition is indicated in Mathematica using a plus sign, e.g., {a1, a2, ..., an} /C27{b1, b2, ..., bn }. See also COMPLEX ADDITION ,CROSS PRODUCT ,DOT PRODUCT ,PARALLELOGRAM LAW,SCALAR MULTIPLI- CATION ,VECTOR ,VECTOR MULTIPLICATION Vector Basis A vector basis is any SET of n LINEARLY INDEPENDENT VECTORS capable of generating an n-dimensional SUBSPACE of Rn : Given a HYPERPLANE defined by x1 /C27x2 /C27x3 /C27x4 /C27x5 /C300; a basis is found by solving for x1 in terms of x2 ; x3 ; x4 ; and x5 : Carrying out this procedure, x1 /C30/C28x2 /C28x3 /C28x4 /C28x5 ; so x1 x2 x3 x4 x52 666643 77775/C30x 2/C281 1 0002 666643 77775/C27x 3/C281 0 1002 666643 77775/C27x 4/C281 0 0102 666643 77775/C27x 5/C281 0 0012 666643 77775; and the above VECTOR form an (unnormalized) BASIS . Given a MATRIX A with an orthonormal basis, the MATRIX corresponding to a new basis, expressed in terms of the original ˆx1;...;ˆxnis A?/C30Aˆx1... Aˆxn ½/C138 See also BASIS,BILINEAR BASIS,M ODULAR SYSTEM BASIS,ORTHONORMAL BASIS,TOPOLOGICAL BASIS Vector Bundle A special class of FIBER BUNDLE in which the FIBER is aVECTOR SPACE V. Technically, a little more is required; namely, if f:E0Bis a BUNDLE with FIBER Rn;to be a vector bundle, all of the FIBERS f/C281(x) for x /C23 B need to have a coherent VECTOR SPACE structure. One way to say this is that the "TRIVIALIZATIONS " h : f /C281(U) 0 U /C29Rn ; are FIBER -for- FIBER VECTOR SPACE ISOMORPHISMS . A vector bundle is a TOTAL SPACE E along with a SURJECTIVE map p : E 0 B to a base manifold B. Any FIBER p/C281(b)isa VECTOR SPACE ISOMORPHIC to V. The simplest nontrivial vector bundle is a LINE BUNDLE on the circle, and is analogous to the MO¨ BIUS STRIP . One use for vector bundles is a generalization of VECTOR FUNCTIONS . For instance, the tangent vectors of an n-dimensional manifold are isomorphic to Rn at a point p in a COORDINATE CHART . But the isomorph- ism with Rn depends on the choice of COORDINATE CHART . Nearby p, the vector fields look like functions. To define vector fields on the whole manifold requires the TANGENT BUNDLE , which is a special case of a vector bundle. A SECTION of a vector bundle E is a map s : B 0 E whose projection, p(s is the identity map on B. For instance, on a TRIVIAL BUNDLE E /C30B /C29V ; a section s corresponds to a function f : B 0 V by s(b) /C30(b;f(b)) :/ Near every point in a vector bundle, there is a TRIVIALIZATION . The structure of the vector bundle, as in all BUNDLES , is that it is LOCALLY TRIVIAL . In the case of a vector bundle, the TRANSITION FUNCTIONS between the trivializations take values in linear invertible transformations of the fiber. Since the element zero in V is fixed by any linear transformation, the zero section always exists. By "nontrivial section," it is meant that it is not the zero section. There are several adjectives that can specify proper- ties of a vector bundle. A COMPLEX VECTOR BUNDLE has a fiber V which is a COMPLEX VECTOR SPACE .A REAL VECTOR BUNDLE has a fiber which is a real VECTOR SPACE , which is the default kind of vector bundle. A LINE BUNDLE has a fiber which is one dimensional. A CONTINUOUS VECTOR BUNDLE is a manifold E with a CONTINUOUS projection map p: A SMOOTH VECTORBUNDLE is a smooth manifold E with a smooth projection p:Finally, a HOLOMORPHIC VECTOR BUNDLE is a COMPLEX MANIFOLD Ewith a HOLOMORPHIC projection p:In this last case, the fiber must be a complex vector space. So there could be a smooth complex vector bundle, but not a holomorphic real vector bundle. Vector bundles can have metrics on their fibers, either R IEMANNIAN or H ERMITIAN , and CONNECTIONS . See also CONNECTION (VECTOR BUNDLE ), FIBER , FIBER BUNDLE ,H ERMITIAN METRIC , K-THEORY ,LIE ALGEBROID ,L INEAR ALGEBRA ,P RINCIPAL BUNDLE , RANK (BUNDLE ), REAL VECTOR BUNDLE ,RIEMANNIAN METRIC ,STABLE EQUIVALENCE ,TANGENT BUNDLE , TANGENT MAP,T RIVIAL BUNDLE ,V ECTOR SPACE , WHITNEY SUM Vector Cross Product CROSS PRODUCT Vector Derivative The basic types of derivatives operating on a VECTOR FIELD are the CURL 9/C29;DIVERGENCE 9/C215;and GRADI- ENT9:/ Vector derivative identities involving the CURL in- clude 9/C29(kA)/C30k9/C29A (1) 9/C29(fA)/C30f(9/C29A)/C27(9f)/C29A (2) 9/C29(A/C29B) /C30(B /C2159)A/C28(A /C2159)B/C27A(9 /C215B)/C28B9 /C215A ðÞ (3) 9/C29A f ! /C30f(9/C29A)/C27A/C299fðÞ f2(4) 9/C29(A/C27B)/C309/C29A/C279/C29B: (5) In C ARTESIAN COORDINATES 9/C29x/C309/C29y/C309/C29z/C300 (6) 9/C29ˆx/C309/C29ˆy/C309/C29ˆz/C300: (7) InSPHERICAL COORDINATES , 9/C29r/C300 (8) 9/C29ˆr/C300 (9) 9/C29rf(r) ½/C138/C30f(r)(9/C29r)/C279f(r) ½/C138 /C29r /C30f(r)(0)/C27df drˆr/C29r/C300/C270/C300: (10) Vector derivative identities involving the DIVERGENCE include 9 /C215kAðÞ/C30k9 /C215A (11) 9 /C215 (fA) /C30f( 9 /C215 A) /C27(9f) /C215 A (12) 9 /C215 (A /C29B) /C30B /C215 (9/C29A) /C28A /C215 ( 9/C29B) (13) 9 /C215A f ! /C30f 9 /C215 A ðÞ /C289fðÞ /C215 A f2 (14) 9 /C215 (A /C27B) /C309 /C215 A /C279 /C215 B (15) In CARTESIAN COORDINATES , 9 /C215 x /C309 /C215 y /C309 /C215 z /C301 (16) 9 /C215 ˆx /C309 /C215 ˆy /C309 /C215 ˆz /C300 : (17) In SPHERICAL COORDINATES , 9 /C215 r /C303 (18) 9 /C215 ˆr /C302 r (19) 9 /C215 rf(r) ½/C138/C30@ @xxf(r) ½/C138/C27@ @yyf(r) ½/C138/C27@ @zzf(r) ½/C138 (20) @ @x [xf(r)] /C30x@f @x /C27f /C30x@f @r@r @x /C27f (21) @r @x /C30@ @xx2 /C27y2 /C27z2/C0/C11 =2/C30xx2 /C27y2 /C27z2/C0/C1/C281=2/C30x r(22) @ @x[xf(r)] /C30x2 rdf dr /C27f : (23) By symmetry, 9 /C215 [rf(r)] /C303f(r) /C271 r(x2 /C27y2 /C27z2)df dr /C303f(r) /C27rdf dr (24) 9 /C215 (ˆrf(r)) /C303 r(r) /C27df dr (25) 9 /C215 ˆrrnðÞ/C303rn/C281 /C27(n /C281)rn/C281 /C30(n /C272)rn/C281 : (26) Vector derivative identities involving the GRADIENT include 9(kf) /C30k9f (27) 9(fg) /C30f 9g /C27g9f (28) 9(A /C215 B) /C30A /C29( 9/C29B) /C27B /C29( 9/C29A) /C27(A /C2159)B /C27(B /C2159)A (29) 9(A /C2159f) /C30A /C29( 9/C299f) /C279f /C29( 9/C29A) /C27A /C2159( 9f) /C279f /C2159A /C309f /C29( 9/C29A) /C27A /C2159( 9f) /C279f /C2159A (30) 9f g ! /C30g 9f /C28 f 9g g2 (31)9(f /C27g) /C309f /C279g (32) 9(A /C215 A) /C302A /C29( 9/C29A) /C272(A /C2159)A (33) (A /C2159)A /C3091 2A2/C16/C17 /C28A /C29( 9/C29A) : (34) Vector second derivative identities include 92t /C139 /C215 ( 9t) /C30@2t @x2 /C27@2t @y2 /C27@2t @z2 (35) 92A /C309( 9 /C215 A) /C289/C29( 9/C29A): (36) This very important second derivative is known as the LAPLACIAN . 9/C29( 9t) /C300 (37) 9(9 /C215 A) /C3092A /C279/C29( 9/C29A) (38) 9 /C215 (9/C29A) /C300 (39) 9/C29( 9/C29A) /C309( 9 /C215 A) /C2892A 9/C2992A/C0/C1 /C309/C2999 /C215 A ðÞ½/C138 /C289/C299/C299/C29A ðÞ ½/C138 /C30/C289/C299/C299/C29A ðÞ ½/C138 /C30/C2899 /C2159/C29A ðÞ ½/C138 /C2892 9/C29A ðÞ/C8/C3 g /C3092 9/C29A ðÞ (40) 92 9 /C215 A ðÞ /C309 /C21599 /C215 A ðÞ½/C138 /C309 /C21592A /C279/C299/C29A ðÞ/C2/C3 /C309 /C21592A/C0/C1 (41) 92 9/C299/C29A ðÞ ½/C138 /C3092 99 /C215 A ðÞ /C2892A/C2/C3 /C3092 99 /C215 A ðÞ½/C138 /C2894A (42) 9/C2992 9/C29A ðÞ/C2/C3 /C3092 99 /C215 A ðÞ½/C138 /C2894A (43) 94A /C30/C2892 9/C299/C29A ðÞ ½/C138 /C2792 99 /C215 A ðÞ½/C138 /C309/C2992 9/C29A ðÞ/C2/C3 /C2892 9/C299/C29A ðÞ ½/C138 : (44) Identities involving combinations of vector deriva- tives include A/C299AðÞ/C30129A /C215A ðÞ /C28A /C2159 ðÞ A (45) 9/C29f9fðÞ /C30f9/C299fðÞ/C279fðÞ/C299fðÞ/C300 (46) A /C2159 ðÞ ˆr/C30A/C28ˆrA /C215ˆr ðÞ r(47) 9f/C215A/C309 /C215fAðÞ/C28f9 /C215A ðÞ (48) f9 /C215A ðÞ /C309 /C215fAðÞ/C28A9f; (49) where (48) and (49) follow from divergence rule (2). See also CURL,DIVERGENCE ,GRADIENT ,LAPLACIAN , VECTOR INTEGRAL ,V ECTOR QUADRUPLE PRODUCT , VECTOR TRIPLE PRODUCT References Gradshteyn, I. S. and Ryzhik, I. M. "Vector Field Theorem." Ch. 10 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1081 /C1/1092, 2000. Morse, P. M. and Feshbach, H. "Table of Useful Vector and Dyadic Equations." Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 50 /C1/54 and 114 /C1/115, 1953. Vector Direct Product Given VECTORS u and v, the vector direct product is uv /C13u /C156vT where /C156is the MATRIX DIRECT PRODUCT and vT is the matrix TRANSPOSE . For 3 /C293 vectors uv /C30u1vT u1vT u1vT2 435/C30u 1v1u1v2u1v3 u2v1u2v2u2v3 u3v1u3v2u3v32435: Note that if u /C30ˆx i ; then uj /C30 dij ; where dijis the KRONECKER DELTA . See also MATRIX DIRECT PRODUCT ,SHERMAN- MORRI- SON FORMULA ,W OODBURY FORMULA Vector Division There is no unique solution A to the MATRIX equation y /C30Ax unless x is PARALLEL to y, in which case A is a SCALAR . Therefore, vector division is not defined. See also MATRIX ,SCALAR Vector Field A MAP f : Rn /C2Rn which assigns each x a VECTOR FUNCTION f(x): In French, a vector field is called "un champ." Several vector fields are illustrated above. A vector field is uniquely specified by giving its DIVER- GENCE and CURL within a region and its normalcomponent over the boundary, a result known as HELMHOLTZ’S THEOREM (Arfken 1985, p. 79). FLOWS are generated by vector fields and vice versa. A vector field is a SECTION of its TANGENT BUNDLE . See also FLOW,SCALAR FIELD,SEIFERT CONJECTURE , TANGENT BUNDLE ,VECTOR ,W ILSON PLUG References Arfken, G. "Vector Analysis." Ch. 1 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 1 /C1/84, 1985. Gray, A. "Vector Fields on Rn /" and "Derivatives of Vector Fields on Rn :/" §11.4 and 11.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 255 /C1/258, 1997. Morse, P. M. and Feshbach, H. "Vector Fields." §1.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 8 /C1/21, 1953. Vector Function A function of one or more variables whose RANGE is 3- dimensional (or, in general, n-dimensional), as com- pared to a SCALAR FUNCTION , whose RANGE is 1- dimensional. Vector functions are also called vector- valued functions. See also COMPLEX FUNCTION ,REAL FUNCTION ,SCA- LAR FUNCTION ,VECTOR Vector Harmonic VECTOR SPHERICAL HARMONIC Vector Helmholtz Equation HELMHOLTZ DIFFERENTIAL EQUATION Vector Integral The following vector integrals are related to the CURL THEOREM .I f F/C13c/C29Px;y;x ðÞ ; (1) then gCds/C29P/C30gsda/C299 ðÞ /C29P (2) If F/C13cF; (3) then gCFd s/C30gsda/C299F (4) The following are related to the DIVERGENCE THEO- REM.I f F/C13c/C29Px;y;x ðÞ ; (5) then gV9/C29F dV /C30gsda /C29F : (6) Finally, if F /C13cF ; (7) then gV9FdV/C30gsFda: (8) See also CURL THEOREM ,D IVERGENCE THEOREM , GRADIENT THEOREM ,G REEN’S IDENTITIES ,LINE IN- TEGRAL ,S URFACE INTEGRAL ,V ECTOR DERIVATIVE , VOLUME INTEGRAL Vector Laplacian A vector Laplacian can be defined for a VECTOR A by 92A /C3099 /C215 A ðÞ /C289/C299/C29A ðÞ (1) in vector notation. The notation Aissometimesalso usedforavectorLaplacian(MoonandSpencer1988, p.3).Intensornotation, A is written Am ; and the identity becomes 92Am /C30A;l m; l/C30 g lkAm; l/C0/C1 ; k /C30gl k;kAm; l /C27glkA m; lk : (2) Similarly, a TENSOR Laplacian can be given by 92Aab /C30A; l ab; l (3) See also LAPLACIAN ,VECTOR POISSON EQUATION References Moon, P. and Spencer, D. E. "The Meaning of the Vector Laplacian." J. Franklin Inst. 256, 551 /C1/558, 1953. Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, 1988. Vector Multiplication Although the multiplication of one vector by another is not uniquely defined (cf. SCALAR MULTIPLICATION , which is multiplication of a VECTOR by a SCALAR ), several types of useful vector products can be defined, as summarized in the following table. product name symbol result DOT PRODUCT /u /C215 v/ SCALAR CROSS PRODUCT /u /C29v/ PSEUDOVECTOR VECTOR DIRECT PRODUCT/uv / TENSORVector multiplication can also be defined for vectors taken three at a time, as summarized in the following table. product name symbol result VECTOR TRIPLE PRODUCT/u /C29(v /C29w)/ VECTOR SCALAR TRIPLE PRODUCT/[u ; v; w]/ PSEUDOSCALAR A number of VECTOR QUADRUPLE PRODUCTS can also be defined. See also CROSS PRODUCT ,D OT PRODUCT ,SCALAR MULTIPLICATION ,SCALAR TRIPLE PRODUCT ,VECTOR , VECTOR ADDITION ,VECTOR DIRECT PRODUCT ,VECTOR QUADRUPLE PRODUCT ,V ECTOR TRIPLE PRODUCT , VECTOR TRIPLE PRODUCT Vector Norm Given an n-D VECTOR x /C30x1 x2 n xn2 6643 775; a vector norm xkk(sometimes written simply xkk)isa NONNEGATIVE number satisfying 1. xkk> 0 when x "0 and xkk/C300 IFF x /C300;/ 2. kxkk/C30 kjjxjjjjfor any SCALAR k, 3. x /C27y kk5 xkk/C27 ykk / The vector norm xjjpis implemented as Vector- Norm [m, p] in the Mathematica add-on package LinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ), where 1 5p B/C12 :/ See also COMPATIBLE ,L1-NORM,L2-NORM, L-INFI- NITY- NORM,MATRIX NORM,NATURAL NORM,NORM References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1114, 2000. Vector Ordering If the first NONZERO component of the vector differ- ence A /C28B is > 0; then A cB: If the first NONZERO component of A/C28BisB0;then /A)B: See also PRECEDES ,SUCCEEDS Vector Poisson Equation The PARTIAL DIFFERENTIAL EQUATION AA /C30/C289/C29E; where AistheVECTORLAPLACIAN. See also POISSON’S EQUATION ,VECTOR LAPLACIAN References Moon, P. and Spencer, D. E. "The Meaning of the Vector Laplacian." J. Franklin Inst. 256, 551 /C1/558, 1953. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Vector Potential A function A such that B /C139/C29A : The most common use of a vector potential is the representation of a magnetic field. If a VECTOR FIELD has zero DIVERGENCE , it may be represented by a vector potential. See also DIVERGENCE ,H ELMHOLTZ’S THEOREM ,PO- TENTIAL FUNCTION ,S OLENOIDAL FIELD ,V ECTOR FIELD Vector Product CROSS PRODUCT ,SCALAR TRIPLE PRODUCT ,VECTOR MULTIPLICATION ,VECTOR DIRECT PRODUCT ,VECTOR QUADRUPLE PRODUCT ,VECTOR TRIPLE PRODUCT Vector Quadruple Product There are a number of algebraic identities involving sets of four VECTORS .LAGRANGE’S IDENTITY is given by (A /C29B) /C215 (C /C29D) /C30(A /C215 C)(A /C215 D) /C28(A /C215 D)(B /C215 C) : (1) A number of other useful identities include (A /C29B)2 /C13(A /C29B) /C215 (A /C29B) /C30(A /C215 A)(B /C215 B) /C28(A /C215 B)(B /C215 A) /C30A2B2 /C28(A /C215 B)2 (2) A /C29(B /C29(C /C29D)) /C30B(A /C215 (C /C29D)) /C28(A /C215 B)(C /C29D) (3) (A /C29B) /C29(C /C29D) /C30(C /C29D) /C29(B /C29A) (4) /C30[A ;B;D]C /C28[A ;B ;C]D (5) [C ;D ;A]B /C28[C ;D ;B]A ; (6) where A ;B;C ½/C138 denotes the SCALAR TRIPLE PRODUCT . See also LAGRANGE’S IDENTITY ,SCALAR TRIPLE PRO- DUCT ,V ECTOR MULTIPLICATION ,V ECTOR TRIPLE PRODUCTVector Space A vector space over Rn is a set of VECTORS for which any VECTORS X ; Y, and Z /C23 Rn and any SCALARS r, s /C23R have the following properties: 1. COMMUTATIVITY : X /C27Y /C30Y /C27X : 2. ASSOCIATIVITY of VECTOR ADDITION : (X /C27Y) /C27Z /C30X /C27(Y /C27Z) : 3. Additive identity: For all X, 0 /C27X /C30X /C270 /C30X : 4. Existence of additive inverse: For any X, there exists a /C28X such that X /C27(/C28X) /C300 : 5. ASSOCIATIVITY of scalar multiplication: r(sX) /C30(rs)X : 6. DISTRIBUTIVITY of scalar sums: (r /C27s)X /C30rX /C27sX : 7. DISTRIBUTIVITY of vector sums: r(X /C27Y) /C30rX /C27rY : 8. Scalar multiplication identity: 1X /C30X : Let V be a vector space of dimension n over the FIELD of q elements (where q is necessarily a power of a prime number). Then the number of distinct non- singular linear operators on V is M(n;q) /C30 qn /C28q0/C0/C1 qn /C28q1/C0/C1 qn /C28q2/C0/C1 /C1/C1/C1 qn /C28qn /C281/C0/C1 (1) and the number of distinct k-dimensional subspaces of V is S(k; n;q) /C30qn /C28 q0ðÞ qn /C28 q1ðÞ qn /C28 q2ðÞ /C1 /C1 /C1 qn /C28 qk/C281/C0/C1 M(k ;q) (2) /C30qn /C28 1 ðÞ qn/C281 /C28 1 ðÞ qn /C282 /C28 1 ðÞ/C1/C1/C1 qn/C28k /C271 /C28 1/C0/C1 qk /C28 1 ðÞ qk/C281 /C28 1 ðÞ qk/C282 /C28 1 ðÞ/C1/C1/C1 q /C28 1 ðÞ: (3) A consequence of the AXIOM OF CHOICE is that every vector space has a BASIS . AMODULE is abstractly similar to a vector space, but it uses a RING to define COEFFICIENTS instead of the FIELD used for vector spaces. M ODULES have COEFFI- CIENTS in much more general algebraic objects. See also BANACH SPACE ,B ASIS (VECTOR SPACE ), FIELD ,F UNCTION SPACE ,H ILBERT SPACE ,INNER PRODUCT SPACE ,MODULE ,QUOTIENT VECTOR SPACE , RING,S YMPLECTIC SPACE ,T OPOLOGICAL VECTOR SPACE ,VECTOR References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 530 /C1/534, 1985. Vector Spherical Harmonic The SPHERICAL HARMONICS can be generalized to vector spherical harmonics by looking for a SCALAR FUNCTION cand a constant VECTOR csuch that M/C139/C29(cc)/C30c(9/C29c)/C27(9c)/C29c /C309cðÞ/C29c/C30/C28c/C299c; (1) so 9 /C215M/C300: (2) Now use the vector identities 92M/C30929/C29M ðÞ /C309/C2992M/C0/C1 /C30992cc/C0/C1 /C309/C29c92c/C0/C1 (3) k2M/C30k29/C29(cc)/C309/C29c92c/C0/C1 (4) so 92M/C27k2M/C309/C29c92c/C27k2c/C0/C1/C2/C3 ; (5) andMsatisfies the vector H ELMHOLTZ DIFFERENTIAL EQUATION ifcsatisfies the scalar H ELMHOLTZ DIF- FERENTIAL EQUATION 92c/C27k2c/C300: (6) Construct another vector function N/C309/C29M k; (7) which also satisfies the vector H ELMHOLTZ DIFFER- ENTIAL EQUATION since 92N/C301 k929/C29M ðÞ /C301k9/C299 2M/C0/C1 /C301k9/C29/C28 k 2M/C0/C1 /C30/C28k9/C29M/C30/C28k2N; (8) which gives 92N/C27k2N/C300: (9) We have the additional identity 9/C29N/C301k9/C29(9/C29M)/C301k9/C29(9/C215M) /C301k9 2M/C281k9 2M/C30/C2892M k/C30kM: (10) In this formalism, cis called the generating functionand cis called the PILOT VECTOR . The choice of generating function is determined by the symmetry of the scalar equation, i.e., it is chosen to solve the desired scalar differential equation. If Mis taken as M/C309/C29(rc); (11) where ris the radius vector, then Mis a solution to the vector wave equation in spherical coordinates. If we want vector solutions which are tangential to the radius vector, M /C215r/C30r/C215(9c/C29c)/C30(9c)(c/C29r)/C300; (12) so c/C29r/C300 (13) and we may take c/C30r (14) (Arfken 1985, pp. 707 /C1/711; Bohren and Huffman 1983, p. 88). A number of conventions are in use. Hill (1954) defines Vm l/C13/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffi l/C271 2l/C271s Ym lˆr/C271ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (l/C271)(2l/C271)p@Ym l @uˆu /C27iMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (l/C271)(2l/C271)p sinuYm lˆf (15) Wm l/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2l/C271s Ym lˆr/C271ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l(2l/C271)p@Ym l @uˆu /C27iMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil(2l/C271)p sinuYm lˆf (16) Xm l/C30/C28Mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil(l/C271)p sinuYm lˆu/C28iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil(l/C271)p @Ym l @uˆf(17) Morse and Feshbach (1953) define vector harmonics called B,C, and Pusing rather complicated expres- sions. References Arfken, G. "Vector Spherical Harmonics." §12.11 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 707 /C1/711, 1985. Blatt, J. M. and Weisskopf, V. "Vector Spherical Harmo- nics." Appendix B, §1i nTheoretical Nuclear Physics. New York: Wiley, pp. 796 /C1/799, 1952. Bohren, C. F. and Huffman, D. R. Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. Hill, E. H. "The Theory of Vector Spherical Harmonics." Amer. J. Phys. 22, 211/C1/214, 1954. Jackson, J. D. Classical Electrodynamics, 2nd ed. New York: Wiley, pp. 744 /C1/755, 1975. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part II. New York: McGraw-Hill, pp. 1898 /C1/ 1901, 1953. Vector Transformation Law The set of n quantities vjare components of an n-D VECTOR v IFF, under ROTATION , v?i /C30aijvj for i /C301, 2, ..., n. The DIRECTION COSINES between x?i and xj are aij /C13@x?i @xj/C30@xj @x?i: They satisfy the orthogonality condition aijaik /C30@xj @x?i@x?i @xk/C30@xj @xk/C30 djk ; where djk is the KRONECKER DELTA . See also TENSOR ,VECTOR Vector Triple Product The vector triple product identity is also known as the BAC -CAB IDENTITY , and can be written in the form A /C29(B /C29C) /C30B(A /C215 C) /C28C(A /C215 B) (1) (A /C29B) /C29C /C30/C28C /C29(A /C29B) /C30/C28A(B /C215 C) /C27B(A /C215 C) (2) See also BAC -CAB IDENTITY ,CROSS PRODUCT ,DOT PRODUCT ,P ERMUTATION SYMBOL ,S CALAR TRIPLE PRODUCT ,VECTOR MULTIPLICATION ,VECTOR QUAD- RUPLE PRODUCT References Arfken, G. "Triple Scalar Product, Triple Vector Product." §1.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 26 /C1/33, 1985. Jeffreys, H. and Jeffreys, B. S. "The Triple Vector Product." §2.092 /C1/2.094 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 75 /C1/76, 1988. Vector-Valued Function VECTOR FUNCTION Vee The symbol /C150variously means "disjunction" (i.e., OR in LOGIC ) or "join" (for a LATTICE ). See also OR, WEDGE Velocity v /C13dr dt; where r is the POSITION VECTOR and d=dt is the derivative with respect to time. Expressed in terms ofthe ARC LENGTH , v /C30ds dtˆT ; where ˆT is the unit TANGENT VECTOR , so the SPEED (which is the magnitude of the velocity) is v /C13 vjj/C30ds dt/C30r?(t)jj : See also ANGULAR VELOCITY ,P OSITION VECTOR , SPEED Velocity Vector The idea of a velocity vector comes from classical physics. By representing the position and motion of asingle particle using vectors, the equations for motion are simpler and more intuitive. Suppose the position of a particle at time tis given by the position vector s(t)/C30(s 1(t);s2(t);s3(t)):Then the velocity vector v(t)i s the derivative of the position, v/C30ds dt/C30ds1 dt;ds2 dt;ds3 dt ! : For example, suppose a particle is confined to the plane and its position is given by s/C30( cos t;sint): Then it travels along the unit circle at constant speed. Its velocity vector is v/C30(/C28sint;cost):In a diagram, it makes sense to translate the velocity vector so itoriginates at s. In particular, it is drawn as an arrow from stos/C27v: / Another example is a particle traveling along a HYPERBOLA specified parametrically by s(t)/C30 ( sinh( t) ; cosh( t)): Its velocity vector is then given by v /C30(cosh( t) ; sinh( t)) ; illustrated above. Travel down the same path, but using a different function is called a REPARAMETRIZATION , and the CHAIN RULE describes the change in velocity. For example, the HYPERBOLA can also be parametrized by r(t) /C30 t;ffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27t2p/C0/C1 : Note that r(sinh( t)) /C30s(t); and by the CHAIN RULE , dr =dt(cosh t) /C30ds =dt:/ Note that the set of possible velocity vectors forms a VECTOR SPACE .Ifr and s are two paths through the origin, then so is r /C27s and the velocity vector of this path is dr =dt /C27ds =dt: Similarly, if a is a scalar, then the path as has velocity vector av: It makes sense to distinguish the velocity vectors at different points. In physics, the set of all velocity vectors gives all possible combinations of position and momentum, and is called phase space. In mathematics, the velocity vectors form the tangent space, and the collection of tangent spaces forms the TANGENT BUNDLE . See also CALCULUS ,C OORDINATE CHART ,D IREC- TIONAL DERIVATIVE ,E UCLIDEAN SPACE ,JACOBIAN , MANIFOLD ,TANGENT BUNDLE ,TANGENT SPACE ,TAN- GENT VECTOR ,VECTOR FIELD,VECTOR SPACE Venn Diagram A schematic diagram used in LOGIC theory to depict collections of sets and represent their relationships. The Venn diagrams on two and three sets are illustrated above. The order-two diagram (left) con- sists of two intersecting circles, producing a total of four regions, A, B, A S B ; and ¥ (the EMPTY SET, represented by none of the regions occupied). Here, A S B denotes the INTERSECTION of sets A and B. The order-three diagram (right) consists of three symmetrically placed mutually intersecting CIRCLES comprising a total of eight regions. The regions labeled A, B, and C consist of members which areonly in one set and no others, the three regions labelled A S B ; A S C ; and B S C consist of members which are in two sets but not the third, the region A S B S C consists of members which are simulta- neously in all three, and no regions occupied repre- sents ¥:/ In general, an order- n Venn diagram is a collection of n simple closed curves in the PLANE such that 1. The curves partition the PLANE into 2n con- nected regions, and 2. Each SUBSET S of 1;2; ... ;n fg corresponds to a unique region formed by the intersection of the interiors of the curves in S (Ruskey). Since there aren k/C0/C1 (the BINOMIAL COEFFICIENT ) ways to pick k members from a total of n, the number of regions in an order n Venn diagram is N /C30Xn k/C300n k/C18/C19 /C302n ; (where the region outside the diagram is included in the count). The region of INTERSECTION of the three CIRCLES A S B S C in the order three Venn diagram in the special case of the center of each being located at the INTERSECTION of the other two is a geometric shape known as a REULEAUX TRIANGLE . See also CIRCLE ,FLOWER OF LIFE,H ARUKI’S THEO- REM,INTERSECTION ,L ENS,M AGIC CIRCLES ,R EU- LEAUX TRIANGLE ,S EED OF LIFE, Ogilvy, C. S. "Solution to Problem E 1154." Amer. Math. Monthly 62, 584 /C1/585, 1955. References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 255 /C1/256, 1989. Ruskey, F. "A Survey of Venn Diagrams." Elec. J. Combin. 4, DS#5, 1997. http://www.combinatorics.org/Surveys/ds5/ VennEJC.html. Ruskey, F. "Venn Diagrams." http://www.theory.csc.uvic.ca/ ~cos/inf/comb/SubsetInfo.html#Venn. Verging Construction NEUSIS CONSTRUCTION Verhulst Model LOGISTIC MAP Verma Module See also MODULE References Huang, J.-S. "Verma Modules." §5.4 in Lectures on Repre- sentation Theory. Singapore: World Scientific, pp. 52 /C1/53, 1999. Veronese Surface A smooth 2-D surface given by embedding the PROJECTIVE PLANE into projective 5-space by the homogeneous parametric equations v(x;y;z) /C30 x2 ; y2 ;z2 ; xy;xz; yz/C0/C1 : The surface can be projected smoothly into 4-space, but all 3-D projections have singularities (Coffman). The projections of these surfaces in 3-D are called STEINER SURFACES . The VOLUME of the Veronese surface is 2p2 :/ See also STEINER SURFACE References Coffman, A. "Steiner Surfaces." http://www.ipfw.edu/math/ Coffman/steinersurface.html. Veronese Variety VERONESE SURFACE Versed Sine VERSINE Versiera WITCH OF AGNESI Versine vers( z) /C131 /C28cos z ; where cos z is the COSINE . Using a trigonometric identity, the versine is equal to vers( z) /C302 sin21 2z/C16/C17 : See also COSINE ,COVERSINE ,EXSECANT ,HAVERSINE References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 78, 1972. Vertex (Graph) A point of a GRAPH , also called a NODE . See also EDGE (GRAPH ), NULL GRAPH ,TAIT COLOR- ING,TAIT CYCLE ,TAIT’S HAMILTONIAN GRAPH CON- JECTURE ,VERTEX (POLYGON ) Vertex (Parabola) For a PARABOLA oriented vertically and opening upwards, the vertex is the point where the curve reaches a minimum.Vertex (Polygon) A point at which two EDGES of a POLYGON meet. See also PRINCIPAL VERTEX ,VERTEX (GRAPH ), VERTEX (POLYHEDRON ) Vertex (Polyhedron) A point at which three of more EDGES of a POLYHE- DRON meet. The concept can also be generalized to a POLYTOPE . See also VERTEX (GRAPH ), VERTEX (POLYGON ) Vertex (Polytope) The vertex of a POLYTOPE is a point where edges of the POLYTOPE meet. Vertex Angle The point about which an ANGLE is measured is called the angle’s vertex, and the angle associated with a given vertex is called the vertex angle. See also ANGLE Vertex Coloring A vertex coloring is an assignment of labels or colors to each vertex of a graph such that no edge connects two identically colored vertices. The most common type of vertex coloring seeks to minimize the number of colors for a given graph. BRELAZ’S HEURISTIC ALGORITHM can be used to find a good, but not necessarily minimal, vertex coloring of a GRAPH . Finding a minimal coloring can be done using brute- force search (Christofides 1971; Wilf 1984; Skiena 1990, p. 214). The minimum number of colors which with the vertices of a graph G may be colored is called the CHROMATIC NUMBER , denoted x(G) :/ The only one-colorable graphs are EMPTY GRAPHS , and two-colorable graphs are exactly BIPARTITE GRAPHS . The FOUR-COLOR THEOREM establishes that all PLA- NAR GRAPHS are 4-colorable. See also BRELAZ’S HEURISTIC ALGORITHM ,BROOKS’ THEOREM ,C HROMATIC NUMBER ,C HROMATIC POLY- NOMIAL ,C OLORING ,E DGE CHROMATIC NUMBER , FOUR- COLOR THEOREM , K-COLORING References Christofides, N. "An Algorithm for the Chromatic Number of a Graph." Computer J. 14,38/C1/39, 1971. Gould, R. (Ed.). Graph Theory. Menlo Park, CA: Benjamin- Cummings, 1988. Manvel, B. "Extremely Greedy Coloring Algorithms." In Graphs and Applications (Ed. F. Harary and J. Maybee). New York: Wiley, pp. 257 /C1/270, 1985. Matula D. W.; Marble, G.; and Isaacson, J. D. "Graph Coloring Algorithms." In Graph Theory and Computing (Ed. R. Read). New York: Academic Press, pp. 109 /C1/122, 1972. Skiena, S. "Finding a Vertex Coloring."§5.5.3 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 214 /C1/215, 1990. Wilf, H. "Backtrack: An X(1) Expected Time Algorithm for the Graph Coloring Problem." Info. Proc. Let. 18, 119 /C1/ 121, 1984. Vertex Connectivity The minimum number of nodes k(G) whose deletion from a GRAPH G disconnects it. Vertex connectivity is sometimes called "point connectivity" or simply "con- nectivity". Let l(G) be the EDGE CONNECTIVITY of a graph G and d(G) its minimum degree, then for any graph, k(G) 5 l(G) 5 d(G) (Whitney 1932, Harary 1994, p. 43). The vertex connectivity of a graph can be determined with the command VertexConnectivity [g] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). See also DISCONNECTED GRAPH ,EDGE CONNECTIVITY , K-CONNECTED GRAPH ,MENGER’S THEOREMReferences Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 43, 1994. Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 178 /C1/179, 1990. Whitney, H. "Congruent Graphs and the Connectivity of Graphs." Amer. J. Math. 54, 150 /C1/168, 1932. Vertex Cover Let S be a collection of subsets of a finite set X. The smallest subset Y of X that meets every member of S is called the vertex cover, or hitting set. However, some authors call any such set a vertex cover, and then refer to the minimum vertex cover (Skiena 1990, p. 218). Finding the hitting set is an NP-COMPLETE PROBLEM . Vertex covers, indicated with red coloring, are shown above for a number of graphs. In a COMPLETE K- PARTITE GRAPH , and vertex cover contains vertices from at least k /C281 stages. The minimum vertex cover of a GRAPH can be computed usingMinimumVertex- Cover [g] in the Mathematica add-on package Dis- creteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). See also CLIQUE ,EDGE COVER ,INDEPENDENT SET References Skiena, S. "Minimum Vertex Cover." §5.6.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 218, 1990. Skiena, S. S. "Vertex Cover." §8.5.3 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 317 /C1/318, 1997. Vertex Degree The degree of a VERTEX vof a GRAPH Gis the number ofEDGES which touch v. The vertex degrees are illustrated above for a random graph. The vertex degree is also called the local degree or valency. The ordered list of vertex degrees in a given graph is called its DEGREE SEQUENCE . A list of vertex degrees of a graph can be given byVertexDegrees [g] in the Mathematica add-on package DiscreteMath‘Com- binatorica‘ (which can be loaded with the com- mand BBDiscreteMath‘ ). The minimum vertex degree in a GRAPH G is denoted d(G); and the maximum degree is denoted D(G) (Skiena 1990, p. 157). The VERTEX degree of a point v in a GRAPH , denoted r(v); satisfies Xn i/C301r viðÞ/C302E ; where E is the total number of EDGES . DIRECTED GRAPHS have two types of degrees, known as the INDEGREE and the OUTDEGREE . See also DEGREE SEQUENCE ,DIRECTED GRAPH ,EDGE (GRAPH ), EVEN NODE,G RAPH ,INDEGREE ,L OCAL DEGREE ,O DD NODE,O UTDEGREE ,PLANTED TREE, VERTEX (GRAPH ) References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Vertex Enumeration A CONVEX POLYHEDRON is defined as the set of solutions to a system of linear inequalities mx 5b; where m is a REAL s /C29d MATRIX and b is a REAL s- VECTOR . Given m and b, vertex enumeration is the determination of the polyhedron’s VERTICES . See also COMPUTATIONAL GEOMETRY ,CONVEX POLY- HEDRON ,POLYHEDRON References Avis, D. and Fukuda, K. "A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra." In Proceedings of the 7th ACM Symposium on Computational Geometry, North Conway, NH, 1991, pp. 98 /C1/104, 1991. Fukada, K. and Mizukosh, I. "Vertex Enumeration Package for Convex Polytopes and Arrangements, Version 0.41 Beta." http://www.mathsource.com/cgi-bin/msitem?0202 /C1/ 633.Vertex Figure The vertex figure at a vertex V of a POLYGON is the line segment joining the MIDPOINTS of the two adjacent sides meeting at V. For a regular n-gon with side length a, the length v of the vertex figure is v /C30a cosp n ! The vertex figure at a vertex V of a POLYHEDRON is the polygon whose sides are the vertex figures of the faces surrounding V. The faces that join at a VERTEX form a SOLID ANGLE whose section by the plane is the vertex figure. See also MIDPOINT ,RECTIFICATION ,TRUNCATION References Coxeter, H. S. M. "The Polytopes with Regular-Prismatic Vertex Figures." Phil. Trans. Roy. Soc. 229, 330 /C1/425, 1930. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 16, 1973. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 76, 1989. Vertex Scheme If K is a SIMPLICIAL COMPLEX , let V be the VERTEX SET of K. Furthermore, let K be the collection of all subsets a0 ;...; an fg of V such that the vertices a0 ; ..., anspan a SIMPLEX of K. Then the collection K is called the vertex scheme of K(Munkres 1993, p. 15). See also GEOMETRIC REALIZATION ,VERTEX SET References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Vertex Set The vertex set of a GRAPH is simply a set of all vertices of the graph. The vertex set V of an ABSTRACT SIMPLICIAL COMPLEX S is the union of one-point elements of S (Munkres 1993, p. 15). See also DOMINATION NUMBER ,EDGE SET,VERTEX SCHEME References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Vertex-Transitive Graph A GRAPH such that every pair of vertices is equivalent under some element of its automorphism group. Every nontrivial graph that is EDGE-TRANSITIVE but not vertex-transitive contains at least 20 vertices (Skiena 1990, p. 186). The smallest known CUBIC GRAPH that is EDGE- but not vertex-transitive is the GRAY GRAPH . See also EDGE-TRANSITIVE GRAPH ,FOLKMAN GRAPH , GRAY GRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Vertical Oriented in an up-down position. See also HORIZONTAL Vertical Perspective Projection A MAP PROJECTION given by the transformation equations x /C30k? cos f sin( l /C28 l0) (1) y /C30k? cos f1 sin f /C28sin f1 cos f cos l /C28 l0 ðÞ ½/C138 ; (2) where P is the distance of the point of perspective inunits of SPHERE RADII and k?/C30P /C28 1 P /C28 cos c (3) cos c /C30sin f1 sin f /C27cos f1 cos f cos l /C28 l0 ðÞ (4) References Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 173 /C1/178, 1987. Vertical Rule BAR,MACRON Vertical Tangent A function f(x) has a vertical tangent line at x0 if f is continuous at x0 and lim x 0x0f ?(x) /C309/C12 Vertical-Horizontal Illusion The HORIZONTAL line segment in the above figure appears to be shorter than the VERTICAL line segment, despite the fact that it has the same length. See also ILLUSION ,M U¨ LLER- LYER ILLUSION ,POGGEN- DORFF ILLUSION ,PONZO’S ILLUSION References Fineman, M. The Nature of Visual Illusion. New York: Dover, p. 153, 1996. Vertically Convex Polyomino COLUMN- CONVEX POLYOMINO Veryprime APOSITIVE INTEGER nis a veryprime IFFall primes p5ffiffiffinpsatisfy 2nmod p ðÞ½/C138 /C28p jj 51 very strong 2nmod p ðÞ½/C138 /C28p jj 5ffiffiffippstrong 2nmod p ðÞ½/C138 /C28p jj 5p=2 weak8 < : The weak veryprimes are then 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 103, 107, 137, 157, 173, 227, 347, 487, 773, ... (Sloane’s A050264), the strong veryprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47,53, 67, 73, 137, 227, ..., and the very strong very- primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 137, ..., with no others in the first 100,000 primes. See also QUITEPRIME References Ferry, J. "RE: Veryprimes defined." sci.math posting, 09 Sep 1999. Sloane, N. J. A. Sequences A050264 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Veselov-Novikov Equation The system of PARTIAL DIFFERENTIAL EQUATIONS @t /C27@3 z /C27@3 ¯z/C0/C1 v /C27@z(uv) /C27@¯z(vw) (1) @¯zu /C303@zv (2) @zw /C303@¯zv (3) where ¯z is the COMPLEX CONJUGATE of z. References Bogdanov, L. V. "Veselov-Novikov Equation as a Natural Two-Dimensional Generalization of the Korteweg-de Vries Equation." Theor. Math. Phys. 70, 219 /C1/233, 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. Vesica Piscis LENS Vibration Problem Solution of a system of second-order homogeneous ordinary differential equations with constant COEFFI- CIENTS OF THE FORM d2x dt2 /C27bx /C300; where b is a POSITIVE DEFINITE MATRIX . To solve the vibration problem, 1. Solve the CHARACTERISTIC EQUATION of b to get EIGENVALUES l1 ; ..., ln : Define vi /C13ffiffiffiffi lip :/ 2. Compute the corresponding EIGENVECTORS e1 ; ..., en :/ 3. The normal modes of oscillation are given by x1 /C30A1 sin v1t /C27 a1 ðÞ e1 ; ..., xn /C30Ansin vnt /C27 an ðÞ en ; where A1 ; ..., Anand a1 ; ..., anare arbitrary constants. 4. The general solution is x /C30an i /C301xi :/Vickrey Auction An AUCTION in which the highest bidder wins but pays only the second-highest bid. This variation over the normal bidding procedure is supposed to encou- rage bidders to bid the largest amount they are willing to pay. See also AUCTION References Vickrey, W. "Counterspeculation, Auctions, and Competitive Sealed Tenders." J. Finance 16,8/C1/27, 1961. Reprinted in The Economics of Information, Vol. 1 (Ed. D. K. Levine and S. A. Lippman). Aldershot, Hants, England: Elgar, pp. 8 /C1/44, 1995. Viergruppe The mathematical group Z2 /C29Z2 ; also denoted D2 : Its multiplication table is VI /V1//V2//V3/ II /V1//V2//V3/ /V1//V1/ I /V3//V2/ /V2//V2//V3/ I /V1/ /V3//V3//V2//V1/ I See also DIHEDRAL GROUP ,FINITE GROUP Z4 References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 184 /C1/185 and 239 /C1/240, 1985. Vieta’s Substitution The substitution of x /C30w /C28p 3w (1) into the standard form CUBIC EQUATION x3 /C27px /C30q: (2) The result reduces the cubic to the equation w3 /C28p3 27w3 /C28q /C300; (3) which is easily turned into a QUADRATIC EQUATION in w3 by multiplying through by w3 to obtain w3/C0/C12/C28qw3/C0/C1 /C281 27p3/C300 (4) See also CUBIC EQUATION ,QUADRATIC EQUATION Vigesimal The base-20 notational system for representing REAL NUMBERS . The digits used to represent numbers using vigesimal NOTATION are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, and J. A base-20 number system was used by the Aztecs and Mayans. The Mayans compiled extensive observations of planetary posi- tions in base-20 notation. See also BASE (NUMBER ), BINARY ,DECIMAL ,HEXADE- CIMAL ,OCTAL ,QUATERNARY ,TERNARY References Weisstein, E. W. "Bases." MATHEMATICA NOTEBOOK BASES.M . Vigintillion In the American system, 1063. See also LARGE NUMBER Villarceau Circles Given an arbitrary point on a TORUS , four CIRCLES can be drawn through it. The first is in the plane of the TORUS and the second is PERPENDICULAR to it. The third and fourth CIRCLES are called Villarceau circles. See also TORUS References Melzak, Z. A. Invitation to Geometry. New York: Wiley, pp. 63 /C1/72, 1983. Villarceau, M. "The´ore`me sur le tore." Nouv. Ann. Math. 7, 345 /C1/347, 1848. Vinculum A horizontal line placed above multiple quantities to indicate that they form a unit. It is most commonly used to denote 1. A RADICAL (/ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12345p ) ;/ 2. Repeating decimals (/0 :111);/ 3. The distance between two points AB;/ 4. The COMPLEX CONJUGATE z1 /C27z2 ; or 5N EGATION of a logical expression, A fflB /C30!(A fflB) :/ See also BAR,MACRON ,RADICAL ,SOLIDUS References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 286, 1997. Vinogradov’s Theorem Every sufficiently large ODD number is a sum of three PRIMES (Vinogradov 1937). Ramachandra and San- karanarayanan (1997) have shown that for suffi- ciently large n, the error term is /C10n= ln n ðÞ4: This theorem is closely related to WARING’S PRIME NUMBER CONJECTURE .See also GOLDBACH CONJECTURE ,SCHNIRELMANN’S THEOREM ,W ARING’S PRIME NUMBER CONJECTURE References Ramachandra, K. and Sankaranarayanan, A. "Vinogradov’s Three Primes Theorem." Math. Student 66,1/C1/4 and 27 /C1/ 72, 1997. Vaughan, R. C. The Hardy-Littlewood Method. Cambridge, England: Cambridge University Press, 1981. Vinogradov, I. M. The Method of Trigonometrical Sums in the Theory of Numbers (Russian). Trav. Inst. Math. Stekloff, Vol. 10, 1937. Vinogradov, I. M. The Method of Trigonometrical Sums in the Theory of Numbers (Russian). Trav. Inst. Math. Stekloff, Vol. 23, 1947. Vinogradov, I. M. The Method of Trigonometrical Sums in the Theory of Numbers. London: Interscience, no year given. Virgule A diagonal slash resembling the SOLIDUS , but with slightly less slant, used to denote DIVISION for in-line equations such as a =b; 1= x /C281 ðÞ2; etc. See also SOLIDUS References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 286, 1997. Virtual Group GROUPOID Visibility VISIBLE POINT Visibility Graph LetSbe a set of simple polygonal obstacles in the plane, then the nodes of the visibility graph of Sare just the vertices of S, and there is an edge (called a visibility edge) between vertices vand wif these vertices are mutually visible. References de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. "Visibility Graphs: Finding the Shortest Route." Ch. 15 in Computational Geometry: Algorithms and Ap- plications, 2nd rev. ed. Berlin: Springer-Verlag, pp. 307 /C1/ 317, 2000. Visible Point Two LATTICE POINTS (x, y) and (x ?; y?) are mutually visible if the line segment joining them contains no further LATTICE POINTS . This corresponds to the requirement that (x?/C28x;y?/C28y) /C301; where (m, n) denotes the GREATEST COMMON DIVISOR . The plots above show the first few points visible from the ORIGIN . If a LATTICE POINT is selected at random in 2-D, the probability that it is visible from the origin is 6=p2 : This is also the probability that two INTEGERS picked at random are RELATIVELY PRIME .Ifa LATTICE POINT is picked at random in n-D, the probability that it is visible from the ORIGIN is 1=z(n) ; where z(n) is the RIEMANN ZETA FUNCTION . An invisible figure is a POLYGON all of whose corners are invisible. There are invisible sets of every finite shape. The lower left-hand corner of the invisible squares with smallest x coordinate of AREAS 2 and 3 are (14, 20) and (104, 6200). See also LATTICE POINT ,O RCHARD VISIBILITY PRO- BLEM ,RIEMANN ZETA FUNCTION References Apostol, T.§3.8 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976. Asano, T.; Ghosh, S. K.; and Shermer, T. C. "Visibility in the Plane." Ch. 19 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 829 /C1/876, 2000. Baake, M.; Grimm, U.; and Warrington, D. H. "Some Re- marks on the Visible Points of a Lattice." J. Phys. A: Math. General 27, 2669 /C1/2674, 1994.Baake, M.; Moody, R. V.; and Pleasants, P. A. B. Diffraction from Visible Lattice Points and kth Power Free Integers. 19 Jun 1999. http://xxx.lanl.gov/abs/math.MG/9906132/. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 208 /C1/210, 1984. Gosper, R. W. and Schroeppel, R. Item 48 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM- 239, p. 17, Feb. 1972. Herzog, F. and Stewart, B. M. "Patterns of Visible and Nonvisible Lattice Points." Amer. Math. Monthly 78, 487 /C1/ 496, 1971. Mosseri, R. "Visible Points in a Lattice." J. Phys. A: Math. Gen. 25, L25-L29, 1992. Schroeder, M. R. "A Simple Function and Its Fourier Trans- form." Math. Intell. 4, 158 /C1/161, 1982. Schroeder, M. R. Number Theory in Science and Commu- nication, 2nd ed. New York: Springer-Verlag, 1990 Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 100 /C1/101, 1999. Visible Point Vector Identity A set of identities involving n-D visible lattice points was discovered by Campbell (1994). Examples include Y (a ;b) /C301 a]0;b511 /C28yazb/C0/C1 /C281 =b/C30 1 /C28z ðÞ/C281 = 1 /C28y ðÞ for yzjj; zjjB1 and Y (a ;b;c) /C301 a;b]0;c 511 /C28xaybzc/C0/C1 /C281 =c/C30 1 /C28z ðÞ/C281= 1 /C28x ðÞ 1 /C28y ðÞ ½/C138 for xyzjj; xzjj; yzjj; zjjB1 :/ References Campbell, G. B. "Infinite Products Over Visible Lattice Points." Internat. J. Math. Math. Sci. 17, 637 /C1/654, 1994. Campbell, G. B. "Visible Point Vector Identities." http:// www.geocities.com/CapeCanaveral/Launchpad/9416/ vpv.html. Vitali’s Convergence Theorem Letfn(z) be a sequence of functions, each regular in a region D, let fn(z) jj5Mfor every nandzinD, and letfn(z) tend to a limit as n0/C12at a set of points having a LIMIT POINT inside D. Then fn(z) tends uniformly to a limit in any region bounded by a contour interior to D, the limit therefore being an analytic function of z. See also MONTEL’S THEOREM References Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 168, 1960. Viviani’s Curve The SPACE CURVE giving the intersection of the CYLINDER x /C28a ðÞ2/C27y2 /C30a2 (1) and the SPHERE x2 /C27y2 /C27z2 /C304a2 : (2) It is given by the PARAMETRIC EQUATIONS x /C30a 1 /C27cos t ðÞ (3) y /C30a sin t (4) z /C302a sin1 2t/C16/C17 : (5) The CURVATURE and TORSION are given by k(t) /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 /C27 3 cos tp a 3 /C27 cos t ðÞ3=2 (6) t(t) /C306 cos1 2t/C16/C17 a(13 /C27 3 cos t) : (7) See also CYLINDER ,CYLINDER- SPHERE INTERSECTION , SPHERE ,STEINMETZ SOLID References Gray, A. "Viviani’s Curve." §8.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 201 /C1/202, 1997. Kenison, E. and Bradley, H. C. Descriptive Geometry. New York: Macmillan, p. 284, 1935. von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 270, 1993. Viviani’s Theorem For a point P inside an EQUILATERAL TRIANGLE DABC ; the sum of the perpendiculars pifrom P to the sides of the TRIANGLE is equal to the ALTITUDE h. This result is simply proved as follows, DABC /C30DPBC /C27DPCA /C27DPAB : (1) With s the side length,1 2sh /C3012spa /C2712spb /C2712spc ; (2) so h /C30pa /C27pb /C27pc : (3) See also ALTITUDE ,EQUILATERAL TRIANGLE Vizing Conjecture Let g(G) denote the DOMINATION NUMBER of a SIMPLE GRAPH G. Then Vizing (1963) conjectured that g(G) g(H) 5 g(G /C29H) ; where G /C29H is the GRAPH PRODUCT . While the full conjecture remains open, Clark and Suen (2000) have proved the looser result g(G)g(H) 52 g(G /C29H) : See also DOMINATION NUMBER References Clark, W. E. and Suen, S. "An Inequality Related to Vizing’s Conjecture." Electronic J. Combinatorics 7, No. 1, N4, 1 /C1/ 3, 2000. http://www.combinatorics.org/Volume_7/ v7i1toc.html#N4. Hartnell, B. and Rall, D. F. "Domination in Cartesian Products: Vizing’s Conjecture." In Domination in Graphs--Advanced Topics (Ed. T. W. Haynes, S. T. He- detniemi, and P. J. Slater). New York: Dekker, pp. 163 /C1/ 189, 1998. Vizing, V. G. "The Cartesian Product of Graphs." Vycisl. Sistemy 9,30/C1/43, 1963. Vojta’s Conjecture A conjecture which treats the heights of points relative to a canonical class of a curve defined over the INTEGERS . References Cox, D. A. "Introduction to Fermat’s Last Theorem." Amer. Math. Monthly 101,3/C1/14, 1994. Volterra Integral Equation of the First Kind An INTEGRAL EQUATION OF THE FORM f(x)/C30gx ak(x;t)f(t)dt: See also FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND,FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND,INTEGRAL EQUATION ,VOLTERRA INTE- GRAL EQUATION OF THE SECOND KIND References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 865, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Volterra Equations." §18.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 786 /C1/788, 1992. Volterra Integral Equation of the Second Kind An INTEGRAL EQUATION OF THE FORM f(x) /C30f(x) /C27gx ak(x;t) f(t)dt See also FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND,FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND,INTEGRAL EQUATION ,VOLTERRA INTE- GRAL EQUATION OF THE FIRST KIND References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 865, 1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Volterra Equations." §18.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 786 /C1/788, 1992. Volume The volume of a solid body is the amount of "space" it occupies. Volume has units of LENGTH cubed (i.e., cm3 ; m3 ; in3 ; etc.) For example, the volume of a box (RECTANGULAR PARALLELEPIPED )of LENGTH L, WIDTH W, and HEIGHT H is given by V /C30L /C29W /C29H : The volume can also be computed for irregularly- shaped and curved solids such as the CYLINDER and CUBE . The volume of a SURFACE OF REVOLUTION is particularly simple to compute due to its symmetry. The following table gives volumes for some common SURFACES . Here r denotes the RADIUS , h the height, and A the base AREA , and, in the case of the TORUS , R the distance from the torus center to the center of the tube (Beyer 1987). SURFACE Volume CONE /1 3 pr2h/ CONICAL FRUSTUM /13 phR2 1 /C27R22 /C27R1R2 ðÞ / CUBE /a3 / CYLINDER / pr2h/ ELLIPSOID /4 3pabc /OBLATE SPHEROID /43 pa2b/ PROLATE SPHEROID /43 pab2/ PYRAMID /1 3Ah / PYRAMIDAL FRUSTUM /13hA1 /C27A2 /C27ffiffiffiffiffiffiffiffiffiffiffi A1A2p/C0/C1 / SPHERE /4 3 pr3/ SPHERICAL CAP /13ph2(3r /C28h)/ SPHERICAL SECTOR /2 3 pr2h/ SPHERICAL SEGMENT /1 6 ph 3a2 /C273b2 /C27h2ðÞ / TORUS /2 p2Rr2/ Even simple SURFACES can display surprisingly coun- terintuitive properties. For instance, the SURFACE OF REVOLUTION ofy/C301=xaround the X-AXIS forx]1i s called G ABRIEL’S HORN , and has finite volume, but infinite SURFACE AREA . The generalization of volume to nDIMENSIONS forn] 4 is known as CONTENT . See also ARC LENGTH ,AREA,BELLOWS CONJECTURE , CONTENT ,H EIGHT ,LENGTH (SIZE), SURFACE AREA, SURFACE OF REVOLUTION ,V OLUME ELEMENT ,V O- LUME THEOREM ,W IDTH (SIZE) References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127 /C1/132, 1987. Volume Element A volume element is the differential element dV whose VOLUME INTEGRAL over some range in a given coordinate system gives the VOLUME of a solid, V/C30gggGdx dy dz : (1) InRn;the volume of the infinitesimal n-HYPERCUBE bounded by dx1;...,dxnhas volume given by the WEDGE PRODUCT dV/C30dx1ffl:::ffldxn (2) (Gray 1997). The use of the antisymmetric WEDGE PRODUCT in- stead of the symmetric product dx1:::dxnis a technical refinement often omitted in informal usage. Dropping the wedges, the volume element for CURVILINEAR COORDINATES inR3is given by dV/C30h1ˆu1du1 ðÞ /C215h2ˆu2du2 ðÞ /C29h3ˆu3du3 ðÞ jj (3) /C30h1h2h3du1du2du3 (4) /C30@r @u1/C215@r @u2/C29@r @u3/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12du 1du2du3 (5) /C30@x @u1@x @u2@x @u3 @y @u1@y @u2@y @u3 @z @u1@z @u2@z @u3/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12du 1 du2 du3 (6) /C30@(x; y;z) @ u1 ;u2 ;u3 ðÞ/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12 du 1 du2 du3 ; (7) where the latter is the JACOBIAN and the hi are SCALE FACTORS . See also AREA ELEMENT ,JACOBIAN ,LINE ELEMENT , RIEMANNIAN METRIC ,SCALE FACTOR ,SURFACE AREA, SURFACE INTEGRAL ,VOLUME INTEGRAL References Gray, A. "Isometries and Conformal Maps of Surfaces."§15.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 346 /C1/351, 1997. Volume Integral A triple integral over three coordinates giving the VOLUME within some region G, V /C30gggGdx dy dz : See also AREA INTEGRAL ,INTEGRAL ,LINE INTEGRAL , MULTIPLE INTEGRAL ,SURFACE INTEGRAL ,VOLUME , VOLUME ELEMENT References Leathem, J. G. Volume and Surface Integrals Used in Physics. 1905. Volume Theorem If the top and bottom bases of a solid are equal in area, lie in PARALLEL PLANES , and every SECTION of the solid parallel to the bases is equal in area to that of the base, then the VOLUME of the solid is the product of base and altitude. See also CAVALIERI’S PRINCIPLE ,VOLUME References Kern, W. F. and Bland, J. R. "Volume Theorem." §12 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 27 /C1/28, 1948.von Aubel’s Theorem Given an arbitrary QUADRILATERAL , place a SQUARE outwardly on each side, and connect the centers of opposite SQUARES . Then the two lines are of equal length and cross at a RIGHT ANGLE . See also QUADRILATERAL ,RIGHT ANGLE ,SQUARE References Kitchen, E. "Do¨rrie Tiles and Related Miniatures." Math. Mag. 67, 128 /C1/130, 1994. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 11, 1991. von Dyck’s Theorem Let a GROUP G have a presentation G /C30 x1 ;...;xn ðj rjx1 ;...; xn ðÞ ;j /C23 J Þ so that G /C30F =R; where F is the FREE GROUP with basis x1 ; ... ;xn fg and R is the NORMAL SUBGROUP generated by the rj : If H is a GROUP with H /C30 y1 ;...;yn hi and if rjy1 ;...; yn ðÞ /C301 for all j, then there is a surjective homomorphism G 0 H with xi /C2yi for alli. See also DYCK’S THEOREM ,FREE GROUP ,N ORMAL SUBGROUP References Rotman, J. J. An Introduction to the Theory of Groups, 4th ed.New York: Springer-Verlag, p. 346, 1995. von Ka ´rma´n Equations The system of PARTIAL DIFFERENTIAL EQUATIONS 94u/C30Ev2 xy/C28vxxvyy/C16/C17 94v/C30a/C27buyyvxx/C27uxxvyy/C282uxyvxy/C0/C1 ; where 94is the BIHARMONIC OPERATOR . References Ames, K. A. and Ames, W. F. "On Group Analysis of the Von Ka´rma´n Equation." Nonlinear Anal. 6, 845/C1/853, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. von Mangoldt Function MANGOLDT FUNCTION von Mises Distribution References Evans, M.; Hastings, N.; and Peacock, B. "von Mises Distribution." Ch. 41 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 189 /C1/191, 2000. von Neumann Algebra A GROUP "with bells and whistles." It was while studying von Neumann algebras that Jones discov- ered the amazing and highly unexpected connections with KNOT THEORY which led to the formulation of the JONES POLYNOMIAL . References Iyanaga, S. and Kawada, Y. (Eds.). "Von Neumann Alge- bras." §430 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1358 /C1/1363, 1980. von Neumann-Bernays-Go ¨del Set Theory This entry contributed by MATTHEW SZUDZIK von Neumann-Bernays-Go ¨del set theory (abbreviated "NBG") is a version of SET THEORY which was designed to give the same results as ZERMELO- FRAENKEL SET THEORY , but in a more logically elegant fashion. It can be viewed as a conservative extension of ZERMELO- FRAENKEL SET THEORY in the sense that a statement about sets is provable in NBG if and only if it is provable in ZERMELO- FRAENKEL SET THEORY . ZERMELO- FRAENKEL SET THEORY is not finitely axio- matized. For example, the AXIOM OF REPLACEMENT is not really a single axiom, but an infinite family of axioms, since it is preceded by the stipulation that it is true "for any set-theoretic formula A(u;v):/" Mon- tague (1961) proved that ZERMELO- FRAENKEL SET THEORY is not finitely axiomatizable, i.e., there is no finite set of axioms which is logically equivalent to the infinite set of ZERMELO- FRAENKEL AXIOMS . In con- trast, von Neumann-Bernays-Go ¨del set theory has only finitely many axioms, and this was the main motivation in its construction. This was accomplished by extending the language of ZERMELO- FRAENKEL SET THEORY to be capable of talking about CLASSES . See also CLASS (SET), SET THEORY ,ZERMELO- FRAEN- KEL AXIOMS ,ZERMELO- FRAENKEL SET THEORY References Itoˆ, K. (Ed.). "Bernays-Go ¨del Set Theory."§33C in Encyclo- pedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cam- bridge, MA: MIT Press, p. 148, 1986.Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997. Montague, R. "Semantic Closure and Non-Finite Axiomatiz- ability. I." In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2 /C1/9 September 1959). Oxford, England: Pergamon, pp. 45 /C1/69, 1961. von Staudt Theorem VON STAUDT- CLAUSEN THEOREM von Staudt-Clausen Theorem B2n /C30An /C28X pk (pk /C281)j2n1 pk; where B2n is a BERNOULLI NUMBER , An is an INTEGER , and the pk/s are the PRIMES satisfying pk /C281j2k : For example, for k /C301, the primes included in the sum are 2 and 3, since (2 /C281)j2 and (3 /C281)j2: Similarly, for k /C306, the included primes are (2, 3, 5, 7, 13), since (1, 2, 3, 6, 12) divide 12 /C302 /C2156: The first few values of An for n /C301, 2, ... are 1, 1, 1, 1, 1, 1, 2, /C286, 56, /C28528, ... (Sloane’s A000146). The theorem was rediscovered by Ramanujan (Hardy 1999, p. 11) and can be proved using P-ADIC NUMBERS . See also BERNOULLI NUMBER , P-ADIC NUMBER References Clausen, T. "Theorem." Astron. Nach. 17, 351 /C1/352, 1840. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 109, 1996. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Wright, E. M. "The Theorem of von Staudt" and "Proof of von Staudt’s Theorem." §7.9 /C1/7.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 90 /C1/93, 1979. Rado, R. "A New Proof of a Theorem of V. Staudt." J. London Math. Soc. 9,85/C1/88, 1934. Rado, R. "A Note on the Bernoullian Numbers." J. London Math. Soc. 9,88/C1/90, 1934. Sloane, N. J. A. Sequences A000146/M1717 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Staudt, K. G. C. von. "Beweis eines Lehrsatzes, die Bernoul- lischen Zahlen betreffend." J. reine angew. Math. 21, 372/C1/374, 1840. Voronoi Cell The generalization of a V ORONOI POLYGON ton-D, for n/C212. See also DODECAHEDRAL CONJECTURE ,V ORONOI POLYGON Voronoi Diagram The partitioning of a plane with n points into n convex POLYGONS such that each POLYGON contains exactly one point and every point in a given POLYGON is closer to its central point than to any other. A Voronoi diagram is sometimes also known as a DIRICHLET TESSELLATION . The cells are called DIRICH- LET REGIONS ,T HIESSEN POLYTOPES ,orV ORONOI POLYGONS . The Mathematica command Diagram- Plot [pts] in the Mathematica add-on packageDis- creteMath‘ComputationalGeometry‘ (which can be loaded with the command BBDiscreteMath‘ ) plots the Voronoi diagram of the given list of points. The DELAUNAY TRIANGULATION and Voronoi diagram in R2 are dual to each other. See also ART GALLERY THEOREM ,C OMPUTATIONAL GEOMETRY ,D ELAUNAY TRIANGULATION ,M EDIAL AXIS,TRIANGULATION ,VORONOI POLYGON References Aurenhammer, F. and Klein, R. "Voronoi Diagrams." Ch. 5 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Hol- land, pp. 201 /C1/290, 2000. Eppstein, D. "Nearest Neighbors and Voronoi Diagrams." http://www.ics.uci.edu/~eppstein/junkyard/nn.html. de Berg, M.; van Kreveld, M.; Overmans, M.; and Schwarz- kopf, O. "Voronoi Diagrams: The Post Office Problem." Ch. 7 in Computational Geometry: Algorithms and Appli- cations, 2nd rev. ed. Berlin: Springer-Verlag, pp. 147 /C1/ 163, 2000. Guibas, L. and Stolfi, J. "Primitives for the Manipulation of General Subdivisions and the Computations of Voronoi Diagrams." ACM Trans. Graphics 4,74/C1/123, 1985. Klee, V. "On the Complexity of d-Dimensional Voronoi Diagrams." Archiv. Math. 34,75/C1/80, 1980.Okabe, A.; Boots, B.; and Sugihara, K. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed. New York: Wiley, 2000. Preparata, F. R. and Shamos, M. I. Computational Geome- try: An Introduction. New York: Springer-Verlag, 1985. Skiena, S. S. "Voronoi Diagrams." §8.6.4 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 358 /C1/ 360, 1997. Voronoi Polygon A POLYGON whose interior consists of all points in the plane which are closer to a particular LATTICE POINT than to any other. The generalization to n-D is called aD IRICHLET REGION ,THIESSEN POLYTOPE ,orV OR- ONOI CELL . References Dirichlet, G. L. "U¨ ber die Reduktion der positiven quad- ratischen Formen mit drei unbestimmten ganzen Zahlen." J. reine angew. Math. 40, 209 /C1/227, 1850. Voronoi, G. "Recherches sur les paralle ´loe`dres Primitives." J. reine angew. Math. 134, 198 /C1/287, 1908. Williams, R. The Geometrical Foundation of Natural Struc- ture: A Source Book of Design. New York: Dover, p. 43, 1979. Voting The simple process of voting leads to surprisingly counterintuitive paradoxes. For example, if three people vote for three candidates, giving the rankings A, B, C; B, C, A; and C, A, B. A majority prefers A to B, B to C, but also C to A (Gardner 1984, p. 25)! It is also possible to conduct a secret ballot even if the votes are sent in to a central polling station (Lipton and Widgerson, Honsberger 1985). See also ARROW’S PARADOX ,BALLOT PROBLEM ,CAKE CUTTING ,M AY’S THEOREM ,Q UOTA SYSTEM ,SOCIAL CHOICE THEORY References Black, D. Theory of Committees and Elections. Cambridge, England: Cambridge University Press, 1958. Black, D. A Mathematical Approach to Proportional Repre- sentation: Duncan Black on Lewis Carroll. Boston, MA: Kluwer, 1995. Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, pp. 317 /C1/330, 1997. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 25, 1984. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 157 /C1/162, 1985. Huntington, E. V. "A Paradox in the Scoring of Completing Teams." Science 88, 287/C1/288, 1938. Lipton, R. G.; and Widgerson, A. "Multi-Party Crypto- graphic Protocols." Niemi, R. G. and Riker, W. H. Sci. Amer. 234,2 1/C1/27, Jun. 1976. Riker, W. H. "Voting and the Summation of Preferences." Amer. Political Sci. Rev. , Dec. 1961. Saari, D. G. Math. Intell. 10, 32, 1988. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 72 /C1/74, 1999. VR Number A "visual representation" number which is a sum of some simple function of its digits. For example, 1233/C30122/C27332 2661653 /C3016532/C282662 221859 /C30223/C27183/C27593 40585 /C304!/C270!/C275!/C278!/C275! 148349 /C30!1/C27!4/C27!8/C27!3/C27!4/C27!94913/C30(4/C279/C271/C273)3 are all VR numbers given by Madachy (1979). References Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 165 /C1/171, 1979. Vulgar Fraction COMMON FRACTION Vulgar Series FAREY SERIES W W2-Constant W2 /C301 :529954037... : References Plouffe, S. "W2 Constant." http://www.lacim.uqam.ca/pi- DATA/w2.txt. Wada Basin A BASIN OF ATTRACTION in which every point on the common boundary of that basin and another basin is also a boundary of a third basin. In other words, no matter how closely a boundary point is zoomed into, all three basins appear in the picture. See also BASIN OF ATTRACTION References Nusse, H. E. and Yorke, J. A. "Basins of Attraction." Science 271, 1376 /C1/380, 1996. Wadati-Konno-Ichikawa-Shimizu Equation The PARTIAL DIFFERENTIAL EQUATION iut /C27 1 /C27 ujj2u/C1;/C17/C281=2 u/C2Q/C21 xx/C300: References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 53, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 135, 1997. Wagstaff’s Conjecture A modification of the EBERHART’S CONJECTURE pro- posed by Wagstaff (1983) which proposes that if qn is the nth prime such that Mqnis a MERSENNE PRIME , then qn /C2 2e/C28g/CQ/C1n; where g is the EULER- MASCHERONI CONSTANT . See also EBERHART’S CONJECTURE References Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 412, 1996. Wagstaff, S. S. "Divisors of Mersenne Numbers." Math. Comput. 40, 385 /C1/97, 1983.Wald’s Equation For a sequence of independent identically distributed random variates X1 ; ..., XNand a random positive integer N, the EXPECTATION VALUES satisfy X1 /C27.../C27XN hi /C30 X1hi Nhi: See also EXPECTATION VALUE Walk A sequence of VERTICES and EDGES such that the VERTICES and EDGES are adjacent. A walk is therefore equivalent to a graph CYCLE , but with the VERTICES along the walk enumerated as well as the EDGES . See also CIRCUIT ,G RAPH CYCLE ,P ATH,R ANDOM WALK Wallace-Bolyai-Gerwein Theorem Two POLYGONS are congruent by DISSECTION IFF they have the same AREA . In particular, any POLYGON is congruent by DISSECTION to a SQUARE of the same AREA . Laczkovich (1988) also proved that a CIRCLE is congruent by DISSECTION to a SQUARE (furthermore, the DISSECTION can be accomplished using TRANSLA- TIONS only). See also DISSECTION References Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., pp. 50 /C1/1, 1991. Laczkovich, M. "Von Neumann’s Paradox with Translation." Fund. Math. 131,1/C1/2, 1988. Wallace-Simson Line SIMSON LINE Wallace-Simson Theorem SIMSON LINE Wallis Cosine Formula gp=2 0cosn xdx /C30p 21 /C215 3 /C215 5 /C1/C1/C1(n /C28 1) 2 /C215 4 /C215 6 /C1/C1/C1nfor n /C302;4;... 2/C2154/C2156/C1/C1/C1(n/C281) 1/C2153/C2155/C1/C1/C1nforn/C303;5;...:8 >>>< >>>: See also WALLIS FORMULA ,W ALLIS SINE FORMULA Wallis Formula The Wallis formula follows from the INFINITE PRO- DUCT representation of the SINE sin x /C30xY/C12 n/C3011 /C28x2 p2n2 ! : (1) Taking x /C30p=2 gives 1 /C30p 2Y/C12 n/C3011 /C281 2nðÞ2"# /C30p2Y /C12 n/C3012nðÞ2/C281 2nðÞ2"# ; (2) so p2 /C30Y /C12 n/C301(2n)2 (2n /C28 1)(2n /C27 1)"# /C302 /C215 2 1 /C215 34 /C215 4 3 /C215 56 /C215 6 5 /C215 7 /C1/C1/C1: (3) A derivation due to Y. L. Yung uses the RIEMANN ZETA FUNCTION . Define F(s) /C13/C28Lis(/C281) /C30X/C12 n /C301/C281ðÞn ns /C30 1 /C2821 /C28s/CQ/C1 z(s) (4) F ? sðÞ/C30X/C12 n/C301/C281ðÞnln n ns; (5) so F ?(0) /C30X/C12 n/C301/C281ðÞnln n /C30/C28ln 1 /C27ln 2 /C28ln 3 /C27... /C30ln2 /C215 4 /C215 6 /C1/C1/C1 1 /C215 3 /C215 5 /C1/C1/C1 ! : (6) Taking the derivative of the zeta function expression gives d ds1 /C2821/C28s/CQ/C1 z(s) /C3021 /C28s(ln 2)z(s) /C27 1 /C2821 /C28s/CQ/C1 z?(s) (7) d ds1 /C2821 /C28s/CQ/C1 z(s)"# s/C300/C30/C28ln 2 /C28 z?(0) /C30/C28ln 2 /C2712ln(2 p) /C30lnffiffiffiffiffiffi 2pp 2 ! /C30lnffiffiffi p 2s ! : (8) Equating and squaring then gives the Wallis formula, which can also be expressed p2 /C30 4 z(0)e /C28 zt(0)hi2 : (9) The Q-ANALOG of the Wallis formula for q /C302isY/C12 k /C3011 /C28q/C28k/CQ/C1 /C281/C303:4627466194... (10) (Finch). See also WALLIS COSINE FORMULA ,W ALLIS SINE FORMULA References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 258, 1972. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/dig/dig.html. Jeffreys, H. and Jeffreys, B. S. "Wallis’s Formula for p:/" §15.07 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 468, 1988. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 63 /C1/4, 1951. Wallis Sieve A compact set W/C12 with AREA m W/C12ðÞ/C308 924254849/C1/C1/C1/C30p4 created by punching a square hole of length 1=3 in the center of a square. In each of the eight squares remaining, punch out another hole of length 1 =(3 /C2155); and so on. Wallis Sine Formula gp=2 0sinnxd x /C30p 21/C2153/C2155/C1/C1/C1(n/C281) 2/C2154/C2156/C1/C1/C1nforn/C302;4;... 2/C2154/C2156/C1/C1/C1(n/C281) 1/C2153/C2155/C1/C1/C1nforn/C303;5;...:8 >>>< >>>: See also WALLIS COSINE FORMULA ,W ALLIS FORMULA Wallis’s Conical Edge The RIGHT CONOID surface given by the PARAMETRIC EQUATIONS xu;vðÞ/C30v cos u yu; vðÞ/C30v sin u zu;vðÞ/C30cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28b2 cos2 u:p See also RIGHT CONOID References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 454 /C1/55, 1997. Wallis’s Problem Find nontrivial solutions to s x2ðÞ/C30 s y2ðÞ other than (x;y) /C30(4; 5); where s(n) is the DIVISOR FUNCTION . Nontrivial solutions means that solutions which are multiples of smaller solutions are not considered. For example, multiples m of (x;y) /C30(4;5) are solutions for m /C303, 7, 9, 11, 13, 17, 19, 23, 21, .... Nontrivial solutions to Wallis’s equation include (x;y) /C30(4; 5); (326, 407), (406, 489), (627, 749), (740, 878), (880, 1451), (888, 1102), (1026, 1208), (1110, 1943), (1284, 1528, 1605), (1510, 1809), (1628, 1630, 2035), (1956, 2030, 2445), (2013, 2557), (2072, 3097), (2508, 2996, 3135, 3745), .... See also DIVISOR FUNCTION ,FERMAT’S DIVISOR PRO- BLEM References Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 54 /C1/6, 1952. Wallpaper Groups The 17 PLANE SYMMETRY GROUPS . Their symbols are p1, p2, pm, pg, cm, pmm, pmg, pgg, cmm, p4, p4m, p4g, p3, p31m, p3m1, p6, and p6m. For a descriptionof the symmetry elements present in each space group, see Coxeter (1969, p. 413). References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina- tion. New York: Chelsea, 1999. Joyce, D. E. "Wallpaper Groups (Plane Symmetry Groups)." http://aleph0.clarku.edu/~djoyce/wallpaper/. Schattschneider, D. "The Plane Symmetry Groups: Their Recognition and Notation." Amer. Math. Monthly 85, 439 /C1/50, 1978. Weyl, H. Symmetry. Princeton, NJ: Princeton University Press, 1952. Zwillinger, D. (Ed.). "Crystallographic Groups." §4.2.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 259 /C1/64, 1995. Walsh Function Functions consisting of a number of fixed-amplitude square pulses interposed with zeros. Following Har- muth (1969), designate those with EVEN symmetry Cal(k; t) and those with ODD symmetry Sal(k; t): Define the SEQUENCY k as half the number of zero crossings in the time base. Walsh functions with nonidentical SEQUENCIES are ORTHOGONAL , as are the functions Cal(k ;t) and Sal(k ;t): The product of two Walsh functions is also a Walsh function. The Walsh functions are then given by Wal( k;t) /C30Cal k=2; t ðÞ for k /C300;2 ;4;... Sal (k /C271)=2; t ðÞ for k /C301;3 ;5;...:/C2; The Walsh functions Cal(k, t) for k /C300, 1, ..., n=2 /C281 and Sal(k ;t) for k /C301, 2, ..., n=2 are given by the rows of the HADAMARD MATRIX Hn :/ See also HADAMARD MATRIX ,SEQUENCY References Beauchamp, K. G. Walsh Functions and Their Applications. London: Academic Press, 1975. Harmuth, H. F. "Applications of Walsh Functions in Com- munications." IEEE Spectrum 6,82/C1/1, 1969. Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, p. 204, 1986. Tzafestas, S. G. Walsh Functions in Signal and Systems Analysis and Design. New York: Van Nostrand Reinhold, 1985. Walsh, J. L. "A Closed Set of Normal Orthogonal Functions." Amer. J. Math. 45,5/C1/4, 1923. Walsh Index The statistical INDEX Pw /C30Pffiffiffiffiffiffiffiffiffiffiq0qnppnPffiffiffiffiffiffiffiffiffiffiq 0qnpp0; where pnis the price per unit in period nandqnis the quantity produced in period n. See also INDEX References Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962. Wangerin Differential Equation The ORDINARY DIFFERENTIAL EQUATION y ƒ/C271 21 x /C28 a1/C271 x /C28 a2/C271 x /C28 a3"# y? /C2714A0 /C27 A1x /C27 A2x2 x /C28 a1 ðÞ x /C28 a2 ðÞ x /C28 a3 ðÞ"# y /C300: See also LAME´ ’S DIFFERENTIAL EQUATION References Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, p. 157, 1961. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Wang’s Conjecture Wang’s conjecture states that if a set of tiles can tile the plane, then they can always be arranged to do so periodically (Wang 1961). The CONJECTURE was refuted when Berger (1966) showed that an aperiodic set of tiles existed. Berger used 20,426 tiles, but the number has subsequently been greatly reduced. In fact, Culik (1996) has reduced the number of tiles to 13. See also TILING References Adler, A. and Holroyd, F. C. "Some Results on One-Dimen- sional Tilings." Geom. Dedicata 10,49/C1/8, 1981. Berger, R. "The Undecidability of the Domino Problem." Mem. Amer. Math. Soc. No. 66,1/C1/2, 1966. Culik, K. II "An Aperiodic Set of 13 Wang Tiles." Disc. Math. 160, 245 /C1/51, 1996. Gru¨nbaum, B. and Sheppard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1986. Hanf, W. "Nonrecursive Tilings of the Plane. I." J. Symbolic Logic 39, 283 /C1/85, 1974. Kari, J. "A Small Aperiodic Set of Wang Tiles." Disc. Math. 160, 259 /C1/64, 1996. Mozes, S. "Tilings, Substitution Systems, and Dynamical Systems Generated by Them." J. Analyse Math. 53, 139 /C1/ 86, 1989. Myers, D. "Nonrecursive Tilings of the Plane. II." J. Sym- bolic Logic 39, 286 /C1/94, 1974. Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., pp. 6 /C1/, 1999. Robinson, R. M. "Undecidability and Nonperiodicity for Tilings of the Plane." Invent. Math. 12, 177 /C1/09, 1971. Smith, T. "Penrose Tilings and Wang Tilings." http:// www.innerx.net/personal/tsmith/pwtile.html. Wang, H. "Proving Theorems by Pattern Recognition. II." Bell Systems Tech. J. 40,1/C1/1, 1961.Ward’s Primality Test Let N be an ODD INTEGER , and assume there exists a LUCAS SEQUENCE Unfg with associated SYLVESTER CYCLOTOMIC NUMBERS Qnfg such that there is an n >ffiffiffiffiffi Np (with n and N RELATIVELY PRIME ) for which N DIVIDES Qn : Then N is a PRIME unless it has one of the following two forms: 1. N /C30 n /C281 ðÞ2; with n /C281 PRIME and n /C214, or 2. N /C30n2 /C281; with n /C281 and n /C271 PRIME . See also LUCAS SEQUENCE ,SYLVESTER CYCLOTOMIC NUMBER References Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 69 /C1/0, 1989. Waring Formula An /C27Bn /C30Xn=2½/C138 j/C300(/C281)jn n /C28 jn /C28j j/C18/C19 ABðÞjA /C27B ðÞn /C282j; where xbc is the FLOOR FUNCTION andn k/CQ/C1 is a BINOMIAL COEFFICIENT . See also FERMAT’S LAST THEOREM Waring’s Conjecture WARING’S PRIME NUMBER CONJECTURE ,W ARING’S PROBLEM Waring’s Prime Number Conjecture Every ODD INTEGER n is a PRIME or the sum of three PRIMES . This problem is closely related to VINOGRA- DOV’S THEOREM . See also GOLDBACH CONJECTURE ,SCHNIRELMANN’S THEOREM ,VINOGRADOV’S THEOREM Waring’s Problem In his Meditationes algebraicae , Waring (1770, 1782) proposed a generalization of L AGRANGE’S FOUR- SQUARE THEOREM , stating that every RATIONAL IN- TEGER is the sum of a fixed number g(n)o f nth POWERS ofINTEGERS , where nis any given POSITIVE INTEGER andg(n) depends only on n. Waring origin- ally speculated that g(2)/C304;g(3)/C309;and g(4)/C3019: In 1909, Hilbert proved the general conjecture using an identity in 25-fold multiple integrals (Rademacherand Toeplitz 1957, pp. 52 /C1 /1). In L AGRANGE’S FOUR-SQUARE THEOREM , Lagrange proved that g(2)/C304;where 4 may be reduced to 3 except for numbers OF THE FORM 4n(8k/C277) (as proved by Legendre; Hardy 1999, p. 12). In the early twen- tieth century, Dickson, Pillai, and Niven proved that g(3)/C309:Hilbert, Hardy, and Vinogradov proved g(4)521;and this was subsequently reduced to g(4)/C3019 by Balasubramanian et al. (1986). Liouville proved (using L AGRANGE’S FOUR-SQUARE THEOREM and L IOUVILLE POLYNOMIAL IDENTITY ) that g(5)553; and this was improved to 47, 45, 41, 39, 38, and finally g(5)537 by Wieferich. See Rademacher and Toeplitz (1957, p. 56) for a simple proof. J.-J. Chen (1964) proved that g(5)/C3037:/ Dickson (1936), Pillai (1936), and Niven also conjec- tured an explicit formula for g(s) for s/C216 (Bell 1945, pp. 318 and 602), based on the relationship 3 2 !n /C2832 ! n$% /C301/C2812 ! n32 ! n /C272$%() : (1) If the D IOPHANTINE (i.e., nis restricted to being an INTEGER ) inequality frac32 ! n"# 51/C2834 ! n (2) is true, where frac( x) is the FRACTIONAL PART ofx, then g(n)/C302n/C2732 ! n$% /C282: (3) This was given as a lower bound by Euler, and has been verified to be correct for 6 5n5471;600;000 (Kubina and Wunderlich 1990, extending Stemmler1990). Furthermore, Mahler (1957) proved that atmost a FINITE number of nexceed Euler’s lower bound. Unfortunately, the proof is nonconstructive. There is also a related (but more difficult) problem of finding the least INTEGER nsuch that every POSITIVE INTEGER beyond a certain point (i.e., all but a FINITE number) is the SUM ofGnnthPOWERS . From 1920 /C1/ 928, Hardy and Littlewood showed that G(n)5(n/C282)2n/C281/C275 (4) and conjectured that GkðÞB2k/C271 for knot a power of 2 4k forka power of 2 :/C2; (5) The best currently known bound is GkðÞBcklnk (6) for some constant c. Heilbronn (1936) improved Vinogradov’s results to obtain GnðÞ56nlnn/C274/C273l n 3 /C272 n !"# n/C273: (7) It has long been known that G(2)/C304:/ Dickson and Landau proved that the only INTEGERS requiring nine CUBES are 23 and 239, thus establish- ingG(3)58:Wieferich proved that only 15 INTEGERS require eight CUBES : 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, and 454 (Sloane’sA018889), establishing G(3)57 (Wells 1986, p. 70). The largest number known requiring seven CUBES is 8042. In 1933, Hardy and Littlewood showed that G(4)519; but this was improved in 1936 to 16 or 17, and shown to be exactly 16 by Davenport (1939b). Vaughan (1986) greatly improved on the method of Hardy and Littlewood, obtaining improved results for n] 5:These results were then further improved by Bru¨dern (1990), who gave G(5)518;and Wooley (1992), who gave Gnforn/C306 to 20. Vaughan and Wooley (1993) showed G(8)542:/ Let G/C27(n) denote the smallest number such that almost all sufficiently large INTEGERS are the sum of G/C27(n)nth POWERS . Then G/C27(3)/C304 (Davenport 1939a), G/C27(4)/C3015 (Hardy and Littlewood 1925), G/C27(8)/C3032 (Vaughan 1986), and G/C27(16)/C3064 (Wooley 1992). If the negatives of POWERS are permitted in addition to the powers themselves, the largest num-ber of nth POWERS needed to represent an arbitrary integer are denoted eg(n) and EG(n) (Wright 1934, Hunter 1941, Gardner 1986). In general, these valuesare much harder to calculate than are g(n) and G n:/ The following table gives g(n);Gn;G/C27(n);eg(n);and EG(n) for n520:The sequence of g(n) is Sloane’s A002804. n /g(n)// Gn//G/C27(n)// eg(n)//EG(n)/ 24 4 33 39 /57//54/ [4, 5] 41 9 1 6 /515/[9, 10] 53 7 /518/ 67 3 /527/ 7 143 /536/ 8 279 /542//532/ 9 548 /555/ 10 1079 /563/ 11 2132 /570/ 12 4223 /579/ 13 8384 /587/ 14 16673 /595/ 15 33203 /5103 / 16 66190 /5112 //564/ 17 132055 /5120 / 18 263619 /n/C29n/ 19 526502 /5138 / 20 1051899 /5146 / See also EULER’S CONJECTURE ,SCHNIRELMANN CON- STANT ,S CHNIRELMANN’S THEOREM ,V INOGRADOV’S THEOREM References Archibald, R. G. "Waring’s Problem: Squares." Scripta Math. 7,33/C1/8, 1940. Balasubramanian, R.; Deshouillers, J.-M.; and Dress, F. "Proble `me de Waring pour les bicarre ´s 1, 2." C. R. Acad. Sci. Paris Se´r. I Math. 303,85/C1/8 and 161 /C1/63, 1986. Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, 1945. Bru¨dern, J. "On Waring’s Problem for Fifth Powers and Some Related Topics." Proc. London Math. Soc. 61, 457 /C1/ 79, 1990. Davenport, H. "On Waring’s Problem for Cubes." Acta Math. 71, 123 /C1/43, 1939a. Davenport, H. "On Waring’s Problem for Fourth Powers." Ann. Math. 40, 731 /C1/47, 1939b. Dickson, L. E. "Waring’s Problem and Related Results." Ch. 25 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 717 /C1/29, 1952. Gardner, M. "Waring’s Problems." Ch. 18 in Knotted Dough- nuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 222 /C1/31, 1986. Guy, R. K. "Sums of Squares." §C20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 /C1/38, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Hardy, G. H. and Littlewood, J. E. "Some Problems of Partitio Numerorum (VI): Further Researches in Waring’s Problem." Math. Z. 23,1/C1/7, 1925. Hardy, G. H. and Wright, E. M. "The Representation of a Number by Two or Four Squares" and "Representation by Cubes and Higher Powers." Chs. 20 /C1/1inAn Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 297 /C1/39, 1979. Hunter, W. "The Representation of Numbers by Sums of Fourth Powers." J. London Math. Soc. 16, 177 /C1/79, 1941. Khinchin, A. Y. "An Elementary Solution of Waring’s Pro- blem." Ch. 3 in Three Pearls of Number Theory. New York: Dover, pp. 37 /C1/4, 1998. Kubina, J. M. and Wunderlich, M. C. "Extending Waring’s Conjecture to 471,600,000." Math. Comput. 55, 815 /C1/20, 1990. Mahler, K. "On the Fractional Parts of the Powers of a Rational Number (II)." Mathematica 4, 122 /C1/24, 1957. Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, 1957. Sloane, N. J. A. Sequences A018889 and A002804/M3361 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html. Small, C. "Waring’s Problem." Math. Mag. 50,12/C1/6, 1977. Stemmler, R. M. "The Ideal Waring Theorem for Exponents 401 /C1/00,000." Math. Comput. 55, 815 /C1/20, 1990. Stewart, I. "The Waring Experience." Nature 323, 674, 1986. Vaughan, R. C. "On Waring’s Problem for Smaller Expo- nents." Proc. London Math. Soc. 52, 445 /C1/63, 1986. Vaughan, R. C. and Wooley, T. D. "On Waring’s Problem: Some Refinements." Proc. London Math. Soc. 63,35/C1/8, 1991. Vaughan, R. C. and Wooley, T. D. "Further Improvements in Waring’s Problem." Phil. Trans. Roy. Soc. London A 345, 363 /C1/76, 1993a.Vaughan, R. C. and Wooley, T. D. "Further Improvements in Waring’s Problem III. Eighth Powers." Phil. Trans. Roy. Soc. London A 345, 385 /C1/96, 1993b. Waring, E. Meditationes algebraicae. Cambridge, England: pp. 204 /C1/05, 1770. Waring, E. Meditationes algebraicae, 3rd ed. Cambridge, England: pp. 349 /C1/50, 1782. Waring, E. Meditationes Algebraicae: An English Transla- tion of the Work of Edward Waring. Providence, RI: Amer. Math. Soc., 1991. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 70 and 75, 1986. Wooley, T. D. "Large Improvements in Waring’s Problem." Ann. Math. 135, 131 /C1/64, 1992. Wright, E. M. "An Easier Waring’s Problem." J. London Math. Soc. 9, 267 /C1/72, 1934. Waring’s Sum Conjecture WARING’S PROBLEM Waring’s Theorem If each of two curves meets the LINE AT INFINITY in distinct, nonsingular points, and if all their intersec- tions are finite, then if to each common point there is attached a weight equal to the number of intersec- tions absorbed therein, the CENTER OF MASS of these points is the center of gravity of the intersections ofthe asymptotes. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 166, 1959. Wasteful Number A number nis called wasteful if the number of digits in the prime factorization of n(including powers) uses more digits than the number of digits in n. The first few wasteful numbers are 4, 6, 8, 9, 12, 18, 20,22, 24 ... (Sloane’s A046760). Pinch calls thesenumbers "frugal" and includes 1 as a frugal number. See also E CONOMICAL NUMBER ,EQUIDIGITAL NUMBER References Pinch, R. G. E. "Economical Numbers." http://www.chalce- don.demon.co.uk/publish.html#62. Rivera, C. "Problems & Puzzles: Puzzle Sequences of Con- secutive Economical Numbers.-053." http://www.prime- puzzles.net/puzzles/puzz_053.htm. Santos, B. R. "Problem 2204. Equidigital Representation." J. Recr. Math. 27,5 8/C1/9, 1995. Sloane, N. J. A. Sequences A046760 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re-search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." M ATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Watchman Theorem ARTGALLERY THEOREM Watson Identities Let a;/C28b; and /C28g /C281be the roots of the CUBIC EQUATION t3 /C272t2 /C28t /C281 /C300 ; (1) then the normalized DILOGARITHM LxðÞsatisfies L( a) /C28L a2/CQ/C1 /C301 7 (2) L( b) /C2712L b2/CQ/C1 /C3057 (3) L( g) /C2712L g2/CQ/C1 /C3047 : References Bytsko, A. G. Two-Term Dilogarithm Identities Related to Conformal Field Theory. 9 Nov 1999. http://xxx.lanl.gov/ abs/math-ph/9911012/. Watson, G. N. Quart. J. Math. Oxford Ser. 8, 39, 1937. Watson Quintuple Product Identity QUINTUPLE PRODUCT IDENTITY Watson-Nicholson Formula Let H iðÞ n(x)beaH ANKEL FUNCTION OF THE FIRST or SECOND KIND , let x; n > 0; and define w /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x n !2 /C281vuut: Then H iðÞ n(x) /C303/C281=2w exp f(/C281)i /C271i[ p=6 /C27 n(w /C281 3w3 /C28tan /C281 w)] gH(i) 1 =3(13nw) /C27O n /C281/C12/C12/C12/C12: References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1475, 1980. Watson’s Formula Let Jn(z)beaB ESSEL FUNCTION OF THE FIRST KIND , Yn(z)aB ESSEL FUNCTION OF THE SECOND KIND , and Kn(z)a MODIFIED BESSEL FUNCTION OF THE FIRST KIND . Also let R[z] > 0 and require R[m /C28 n] B1: Then Jm(z)Y n(z) /C28Jn(z)Y m(z) /C304 sin (m /C28 n) p ½/C138 p2 g/C12 0Kn/C28m(2z sinh t)e/C28 m/C27 n ðÞ t dt: The fourth edition of Gradshteyn and Ryzhik (2000), Iyanaga and Kawada (1980), and Ito (1987) erro- neously give the exponential with a PLUS SIGN.A related integral is given byJn(z)@YnzðÞ @ n/C28Yn(z)@Jn(z) @ n/C30/C284 pg/C12 0K0(2z sinh t)e /C282nt dt for R[z] > 0:/ See also DIXON- FERRAR FORMULA ,NICHOLSON’S FOR- MULA References Gradshteyn, I. S. and Ryzhik, I. M. Eqns. 6.617.1 and 6.617.2 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 710, 2000. Itoˆ, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed. Cambridge, MA: MIT Press, p. 1806, 1987. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1476, 1980. Watson’s Theorem 3F2a ;b ;c 1 2(a /C27b /C27c) ; c;1/C2Q/C21 /C30G1 2/C1;/C17 G12 /C27 c/C1;/C17 G121 /C27 a /C27 b ðÞhi G12 /C2812a /C2812b /C27 c/C1;/C17 G1 21 /C27 a ðÞhi G121 /C27 b ðÞhi G12 /C2812a /C27 c/C1;/C17 G12 /C2812b /C27 c/C1;/C17 ; where3F2(a; b;c;d;e;z)isa GENERALIZED HYPERGEO- METRIC FUNCTION andG(z) is the GAMMA FUNCTION (Bailey 1935, p. 16; Koepf 1998, p. 32). See also GENERALIZED HYPERGEOMETRIC FUNCTION , WATSON- WHIPPLE TRANSFORMATION ,W HIPPLE’S IDENTITY References Bailey, W. N. "Watson’s Theorem." §3.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 16, 1935. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Watson-Whipple Transformation If at least one of d,e,o rfhas the form q/C28Nfor some nonnegative integer N(in which case both sums terminate after N/C271 terms), then 8f7a;qa1=2;/C28qa1=2;b;c;d;e;f a1=2;/C28a1=2;aq b;aq c;aq d;aq e;aq f;q;a2q2 bcdef2 435 /C30 aq;aq de;aq df;aq ef ! /C12 aq d;aq c;aq f;aq def ! /C124f3aq bc;d;e;f aq b;aq c;def a;q;q266643 7775; where a 1;a2;...;ar;q ðÞ/C12is a generalized Q-POCHHAM- MER SYMBOL a1;a2;...;ar;q ðÞ/C12/C30a1;q ðÞ/C12a2;q ðÞ/C12...ar;qðÞ/C12; and each of8 f7and4 f3is a Q-HYPERGEOMETRIC FUNCTION . See also Q-HYPERGEOMETRIC FUNCTION , Q-POCHHAM- MER SYMBOL , Q-SERIES References Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 242, 1990. Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." To appear in J. London Math. Soc. 2000. Watt’s Curve A curve named after James Watt (1736 /C1/819), the Scottish engineer who developed the steam engine (MacTutor Archive). The curve is produced by a LINKAGE of rods connecting two wheels of equal diameter. Let the two wheels have RADIUS b and let their centers be located a distance 2a apart. Further suppose that a rod of length 2c is fixed at each end to the CIRCUMFERENCE of the two wheels. Let P be the MIDPOINT of the rod. Then Watt’s curve C is the LOCUS of P. The POLAR equation of Watt’s curve is r2 /C30b2 /C28 a sin u 9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28a2 cos2 up/C1;/C172 : If a /C30c, then C is a CIRCLE of RADIUS b with a figure of eight inside it. See also WATT’S PARALLELOGRAM References Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 162, 1967. MacTutor History of Mathematics Archive. "Watt’s Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/ Watts.html. Watt’s Parallelogram A LINKAGE used in the original steam engine to turn back-and-forth motion into approximately straight- line motion.See also LINKAGE ,W ATT’S CURVE References Rademacher, H. and Toeplitz, O. The Enjoyment of Mathe- matics: Selections from Mathematics for the Amateur.Princeton, NJ: Princeton University Press, pp. 119 /C1 /21, 1957. Wave A4 - POLYHEX . References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other MathematicalSleight-of-Mind from Scientific American. New York: Vintage, p. 147, 1978. Wave Equation The wave equation is the important PARTIAL DIFFER- ENTIAL EQUATION 92c/C301 v2@2c @t2; (1) which can also be written v292c/C30ctt; (2) where 92is the L APLACIAN ,o r I2c/C300; (3) where I2is the D’ALEMBERTIAN . The 1-D wave equation is @2c @x2/C301 v2@2c @t2: (4) In order to specify a wave, the equation is subject to boundary conditions c(0;t)/C300 (5) c(L;t)/C300; (6) and initial conditions c(x;0)/C30f(x) (7) @c @t(x;0)/C30g(x): (8) The wave equation can be solved using the so-calledd’Alembert’s solution, a F OURIER TRANSFORM method, orSEPARATION OF VARIABLES . d’Alembert devised his solution in 1746, and Euler subsequently expanded the method in 1748. Let j/C13x/C28at (9) h/C13x/C27at: (10) By the CHAIN RULE , @2c @x2/C30@2c @j2/C272@2c @j@h/C27@2c @h2(11) 1 v2@2c @t2/C30@2c @j2/C282@2c @j@h/C27@2c @h2: (12) The wave equation then becomes @2c @j@h/C300: (13) Any solution of this equation is OF THE FORM c(j;h)/C30f(h)/C27g(j)/C30f(x/C27vt)/C27g(x/C28vt); (14) where fandgareanyfunctions. They represent two waveforms traveling in opposite directions, fin the NEGATIVE xdirection and gin the POSITIVE x direction. The 1-D wave equation can also be solved by applying aFOURIER TRANSFORM to each side, g/C12 /C28/C12@2c(x;t) @x2e/C282pikxdx/C301 v2g/C12 /C28/C12@2c(x;t) @t2e/C282pikxdx;(15) which is given, with the help of the F OURIER TRANS- FORM DERIVATIVE identity, by 2pikðÞ2C(k;t)/C301 v2@2C(k;t) @t2; (16) where C(k;t)/C13Fc(x;t) ½/C138 /C30g/C12 /C28/C12cx;tðÞe/C282pikxdx: (17) This has solution C(k;t)/C13A(k)e2pikvt/C27B(k)e/C282pikvt: (18) Taking the inverse F OURIER TRANSFORM gives c(x;t)/C13g/C12 /C28/C12C(k;t)e2pikxdx /C30g/C12 /C28/C12A(k)e2pikvt/C27B(k)e/C282pikvt/C2/C3 e/C282pikxdk /C30g/C12 /C28/C12A(k)e/C282pik x/C28vt ðÞdk/C27g/C12 /C28/C12B(k)e/C282pik x/C27vt ðÞdk /C30f1(x/C28vt)/C27f2(x/C27vt); (19) where f1(u)/C13FA(k) ½/C138/C30g/C12 /C28/C12A(k)e/C282pikudk (20)f2(u)/C13FB(k) ½/C138/C30g/C12 /C28/C12B(k)e/C282pikudk: (21) This solution is still subject to all other initial and boundary conditions. The 1-D wave equation can be solved by SEPARATION OF VARIABLES using a trial solution c(x;t)/C30X(x)T(t): (22) This gives Td2X dx2/C301 v2Xd2T dt2(23) 1 Xd2X dx2/C301 v21 Td2T dt2/C30/C28k2: (24) So the solution for Xis X(x)/C30Ccos (kx)/C27Dsin (kx): (25) Rewriting (24) gives 1 Td2T dt2/C30/C28v2k2/C13/C28v2; (26) so the solution for Tis T(t)/C30Ecos (vt)/C27Fsin (vt); (27) where v/C13v=k:Applying the boundary conditions c(0;t)/C30c(L;t)/C300 to (25) gives C/C300kL/C30mp; (28) where mis an INTEGER . Plugging (25), (27) and (28) back in for cin (23) gives, for a particular value of m, cm(x;t)/C30EmsinvmtðÞ/C27FmcosvmtðÞ ½/C138 Dmsinmpx L ! /C13AmcosvmtðÞ/C27BmsinvmtðÞ ½/C138 sinmpx L ! : ð29Þ The initial condition c(x;0)/C300 then gives Bm/C300;so (29) becomes cm(x;t)/C30AmcosvmtðÞ sinmpx L ! : (30) The general solution is a sum over all possible values ofm,s o c(x;t)/C30X/C12 m/C301AmcosvmtðÞ sinmpx L ! : (31) Using ORTHOGONALITY of sines again, gL 0sinlpx L ! sinmpx L ! dx/C301 2Ldlm; (32) where dlmis the K RONECKER DELTA defined by dmn/C131m/C30n 0m"n;/C2; (33) gives gL 0c(x;0) sinmpx L ! dx /C30X/C12 l/C301Alsinlpx L ! sinmpx L ! dx /C30X/C12 l/C301Al1 2Ldlm/C3012LAm; (34) so we have Am/C302 LgL 0c(x;0) sinmpx L ! dx: (35) The computation of Am/s for specific initial distortions is derived in the F OURIER SINE SERIES section. We already have found that Bm/C300;so the equation of motion for the string (31), with vm/C13vkm/C30vmp L; (36) is c(x;t)/C30X/C12 m/C301Amcosvmpt L ! sinmpx L ! ; (37) where the AmCOEFFICIENTS are given by (35). A damped 1-D wave @2c @x2/C301 v2@2c @t2/C27b@c @t; (38) given boundary conditions c(0;t)/C300 (39) c(L;t)/C300; (40) initial conditions c(x;0)/C30f(x) (41) @c @t(x;0)/C30g(x) (42) and the additional constraint 0BbB2p Lv; (43) can also be solved as a F OURIER SERIES .c(x;t)/C30X/C12 n/C301sinnpx L ! e/C28v2bt=2ansinmntðÞ/C27bncosmntðÞ ½/C138 ; (44) where mn/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4v2n2p2/C28b2L2v4p 2L/C30vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4n2p2/C28b2L2v2p 2L(45) bn/C302 LgL 0sinnpx L ! f(x)dx (46) an/C302 LmngL 0sinnpx L ! g(x)/C27v2b 2f(x)"# dx() :ð47Þ To find the motion of a rectangular membrane with sides of length LxandLy(in the absence of gravity), use the 2-D wave equation @2z @x2/C27@2z @y2/C301 v2@2z @t2; (48) where z(x;y;t) is the vertical displacement of a point on the membrane at position ( x, y) and time t. Use SEPARATION OF VARIABLES to look for solutions OF THE FORM z(x;y;t)/C30X(x)Y(y)T(t): (49) Plugging (49) into (48) gives YTd2X dx2/C27XTd2Y dy2/C301 v2XYd2T dt2; (50) where the partial derivatives have now becomecomplete derivatives. Multiplying (50) by v 2=XYT gives v2 Xd2X dx2/C27v2 Yd2Y dy2/C301 Td2T dt2: (51) The left and right sides must both be equal to aconstant, so we can separate the equation by writing the right side as 1 Td2T dt2/C30/C28v2: (52) This has solution T(t)/C30Cvcos (vt)/C27Dvsin (vt): (53) Plugging (52) back into (51), v2 Xd2X dx2/C27v2 Yd2Y dy2/C30/C28v2; (54) which we can rewrite as 1 Xd2X dx2/C30/C281 Yd2Y dy2/C28v2 v2/C30/C28k2 x (55) since the left and right sides again must both be equal to a constant. We can now separate out the equation 1 Yd2Y dy2/C30k2 x/C28v2 v2/C13/C28k2y; (56) where we have defined a new constant kysatisfying k2x/C27k2y/C30v2 v2: (57) Equations (55) and (56) have solutions X(x)/C30EcoskxxðÞ/C27FsinkxxðÞ (58) Y(y)/C30Gcoskyy/CQ/C1 /C27Hsinkyy/CQ/C1 : (59) We now apply the boundary conditions to (58) and (59). The conditions z(0;y;t)/C300 and z(x;0;t)/C300 mean that E/C300G/C300: (60) Similarly, the conditions zLx;y;t ðÞ /C300 and zx;Ly;t/CQ/C1 /C300 give sin kxLx ðÞ /C300 and sin kyLy/CQ/C1 /C300;so Lxkx/C30ppandLyky/C30qp;where pandqare INTEGERS . Solving for the allowed values of kxandkythen gives kx/C30pp Lxky/C30qp Ly: (61) Plugging (54), (58), (59), (60), and (61) back into (24) gives the solution for particular values of pandq, zpq(x;y;t)/C30Cvcos(vt)/C27Dvsin(vt) ½/C138 Fpsinppx Lx !"# /C2Hqsinqpy Ly !"# : (62) Lumping the constants together by writing Apq/C13 CvFpHq(we can do this since vis a function of pand q,s oCvcan be written as Cpq) and Bpq/C13DvFpHq;we obtain zpq(x;y;t)/C30Apqcosvpqt/CQ/C1 /C27Bpqsinvpqt/CQ/C1 /C2/C3 /C2sinppx Lx ! sinqpy Ly ! : (63) Plots of the spatial part for modes (1, 1), (1, 2), (2, 1),and (2, 2) follow. The general solution is a sum over all possible values ofpandq, so the final solution is z(x;y;t)/C30X/C12 p/C301X/C12 q/C301Apqcosvpqt/CQ/C1/C2/C27Bpqsinvpqt/CQ/C1 /C138sinppx Lx ! sinqpy Ly ! ; (64) where vis defined by combining (57) and (61) to yield vpq/C13pvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Lz !2 /C27q Ly !2vuut: (65) Given the initial conditions z(x;y;0) and @z @t(x;y;0);we can compute the Apq/s and Bpq/s explicitly. To accom- plish this, we make use of the orthogonality of the SINE function in the form I/C13gL 0sinmpx L ! sinnpx L ! dx/C301 2Ldmn; (66) where dmnis the K RONECKER DELTA . This can be demonstrated by direct INTEGRATION . Let u/C13px=Lso du/C30(p=L)dxin (66), then I/C30L pgp 0sin(mu) sin( nu)du: (67) Now use the trigonometric identity sinasinb/C3012cos(a/C28b)/C28cos(a/C27b) ½/C138 (68) to write I/C30L 2pgp 0cos (m/C28n)u ½/C138 du/C27gp 0cos (m/C27n)u ½/C138 du:(69) Note that for an INTEGER l"0;the following INTE- GRAL vanishes gp 0cos(lu)du/C301 lsin(lu) ½/C138p 0/C301 lsin(lu)/C28sin 0 ½/C138 /C301 lsin(lp)/C300; (70) since sin( lp)/C300 when lis an INTEGER . Therefore, I/C300 when l/C13m/C28n"0:However, Idoes notvanish when l/C300, since gp 0cos(0 /C215u)du/C30gp 0du/C30p: (71) We therefore have that I/C30Ldmn=2;so we have derived (66). Now we multiply z(x;y;0) by two sine terms and integrate between 0 and Lxand between 0 andLy; I/C30gLy 0gLx 0z(x;y;0) sinppx Lx ! dx"# sinqpy Ly ! dy:(72) Now plug in z(x;y;t);sett/C300, and prime the indices to distinguish them from the pandqin (72), I /C30X/C12 q?/C301 gLy 0X/C12 p ?/C301Ap ?q?gLx 0sinp px Lx ! sinp ?px Lx ! dx"# /C29sinqpy Ly ! sinq?py Ly ! dy: (73) Making use of (66) in (73), I /C30X/C12 q ?/C301 gLy 0X/C12 p ?/C301Ap ?q ?Lx 2dp ;p ?qpy Ly ! sinq?py Ly ! dy; (74) so the sums over p ? and q ? collapse to a single term I /C30Lx 2X/C12 p /C301Apq ?Ly 2dq ;q ?LxLy 4Apq : (75) Equating (74) and (75) and solving for Apq then gives Apq /C304 LxLygLy 0gLx 0z(x; y;0) sinp px Lx ! dx"# sinqpx Ly ! dy: (76) An analogous derivation gives the Bpq/sas Bpq /C304 vpqLxLygLy 0gLx 0@z @t(x;y;0) sinppx Lx ! dx"# /C29sinqpx Ly ! dy : (77) The equation of motion for a membrane shaped as a RIGHT ISOSCELES TRIANGLE of length c on a side and with the sides oriented along the POSITIVE x and y axes is given by c(x;y;t) /C30 Cpqcos(vpqt) /C27Dpq sin( vpqt)/C2/C3 /C29 sinppx c ! sinqpy c ! /C28sinqpx c ! sinp py c ! "# ; (78) where vpq /C30pv cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 /C27q2p (79) and p, q INTEGERS with p /C21q. This solution can be obtained by subtracting two wave solutions for a square membrane with the indices reversed. Since points on the diagonal which are equidistant from the center must have the same wave equation solution (by symmetry), this procedure gives a wavefunction which will vanish along the diagonal as long as p and q are both EVEN or ODD. We must further restrict the modes since those with p Bq give wavefunctions which are just the NEGATIVE of (q, p) and (p, p) give an identically zero wavefunction. The following plots show (3, 1), (4, 2), (5, 1), and (5,3). See also D’ALEMBERTIAN ,TELEGRAPH EQUATION References Abramowitz, M. and Stegun, C. A. (Eds.). "Wave Equation in Prolate and Oblate Spheroidal Coordinates." §21.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 752 /C1/53, 1972. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 124 /C1/25 and 271, 1953. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, 1997. Wave Operator An OPERATOR relating the asymptotic state of a DYNAMICAL SYSTEM governed by the Schro ¨dinger equation id dtc(t)/C30Hc(t) to its original asymptotic state. See also SCATTERING OPERATOR Wave Surface ASURFACE represented parametrically by ELLIPTIC FUNCTIONS . Wavelet Wavelets are a class of a functions used to localize a given function in both space and scaling. A family ofwavelets can be constructed from a function c(x); sometimes known as a "mother wavelet," which isconfined in a finite interval. "Daughter wavelets"c a;b(x) are then formed by translation ( b) and con- traction ( a). Wavelets are especially useful for com- pressing image data, since a WAVELET TRANSFORM has properties which are in some ways superior to a conventional F OURIER TRANSFORM . An individual wavelet can be defined by ca;b(x)/C30ajj/C281=2cx/C28b a ! : (1) Then Wc(f)(a;b)/C301ffiffiffiapg/C12 /C28/C12f(t)ct/C28b a ! dt; (2) and C ALDERO ´N’S FORMULA gives f(x) /C30Ccg/C12 /C28/C12g/C12 /C28/C12f ; ca ;b/C1Q/C11 ca;b(x)a /C282 da db: (3) A common type of wavelet is defined using HAAR FUNCTIONS . See also FOURIER TRANSFORM ,H AAR FUNCTION , LEMARIE ´ ’S WAVELET ,W AVELET TRANSFORM References Benedetto, J. J. and Frazier, M. (Eds.). Wavelets: Mathe- matics and Applications. Boca Raton, FL: CRC Press, 1994. Chui, C. K. An Introduction to Wavelets. San Diego, CA: Academic Press, 1992. Chui, C. K. (Ed.). Wavelets: A Tutorial in Theory and Applications. San Diego, CA: Academic Press, 1992. Chui, C. K.; Montefusco, L.; and Puccio, L. (Eds.). Wavelets: Theory, Algorithms, and Applications. San Diego, CA: Academic Press, 1994. Daubechies, I. Ten Lectures on Wavelets. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992. Erlebacher, G. H.; Hussaini, M. Y.; and Jameson, L. M. (Eds.). Wavelets: Theory and Applications. New York: Oxford University Press, 1996. Foufoula-Georgiou, E. and Kumar, P. (Eds.). Wavelets in Geophysics. San Diego, CA: Academic Press, 1994. Herna ´ndez, E. and Weiss, G. A First Course on Wavelets. Boca Raton, FL: CRC Press, 1996. Hubbard, B. B. The World According to Wavelets: The Story of a Mathematical Technique in the Making, 2nd rev. upd. ed. New York: A. K. Peters, 1998. Jawerth, B. and Sweldens, W. "An Overview of Wavelet Based Multiresolution Analysis." SIAM Rev. 36, 377 /C1/12, 1994. Kaiser, G. A Friendly Guide to Wavelets. Cambridge, MA: Birkha ¨user, 1994. Massopust, P. R. Fractal Functions, Fractal Surfaces, and Wavelets. San Diego, CA: Academic Press, 1994. Meyer, Y. Wavelets: Algorithms and Applications. Philadel- phia, PA: SIAM Press, 1993. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Wavelet Transforms." §13.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 584 /C1/99, 1992. Resnikoff, H. L. and Wells, R. O. J. Wavelet Analysis: The Scalable Structure of Information. New York: Springer- Verlag, 1998. Schumaker, L. L. and Webb, G. (Eds.). Recent Advances in Wavelet Analysis. San Diego, CA: Academic Press, 1993. Stollnitz, E. J.; DeRose, T. D.; and Salesin, D. H. "Wavelets for Computer Graphics: A Primer, Part 1." IEEE Compu- ter Graphics and Appl. 15, No. 3, 76 /C1/4, 1995. Stollnitz, E. J.; DeRose, T. D.; and Salesin, D. H. "Wavelets for Computer Graphics: A Primer, Part 2." IEEE Compu- ter Graphics and Appl. 15, No. 4, 75 /C1/5, 1995. Strang, G. "Wavelets and Dilation Equations: A Brief Introduction." SIAM Rev. 31, 614 /C1/27, 1989. Strang, G. "Wavelets." Amer. Sci. 82, 250 /C1/55, 1994. Taswell, C. Handbook of Wavelet Transform Algorithms. Boston, MA: Birkha ¨user, 1996. Teolis, A. Computational Signal Processing with Wavelets. Boston, MA: Birkha ¨user, 1997. Vidakovic, B. Statistical Modeling by Wavelets. New York: Wiley, 1999. Walker, J. S. A Primer on Wavelets and their Scientific Applications. Boca Raton, FL: CRC Press, 1999. Walter, G. G. Wavelets and Other Orthogonal Systems with Applications. Boca Raton, FL: CRC Press, 1994."Wavelet Digest." http://www.wavelet.org/wavelet/. Weisstein, E. W. "Books about Wavelets." http://www.trea- sure-troves.com/books/Wavelets.html. Wickerhauser, M. V. Adapted Wavelet Analysis from Theory to Software. Wellesley, MA: Peters, 1994. Wavelet Matrix Any discrete finite WAVELET TRANSFORM can be REPRESENTED AS a matrix, and such a wavelet matrix can be computed in O(n) steps, compared to O(n lg n) for the FOURIER MATRIX , where lg x /C30log2 x is the base-2 LOGARITHM . A single wavelet matrix can be built using HAAR FUNCTIONS . See also FOURIER MATRIX ,H AAR FUNCTION ,W AVE- LET,W AVELET TRANSFORM Wavelet Transform A transform which localizes a function both in space and scaling and has some desirable properties com- pared to the FOURIER TRANSFORM . The transform is based on a WAVELET MATRIX , which can be computed more quickly than the analogous F OURIER MATRIX . See also DAUBECHIES WAVELET FILTER ,LEMARIE’S WAVELET ,W AVELET MATRIX References Blair, D. and MathSoft, Inc. "Wavelet Resources." http:// www.mathsoft.com/wavelets.html. Daubechies, I. Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992. DeVore, R.; Jawerth, B.; and Lucier, B. "Images Compres- sion through Wavelet Transform Coding." IEEE Trans. Information Th. 38, 719/C1/46, 1992. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Wavelet Transforms." §13.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 584 /C1/99, 1992. Strang, G. "Wavelet Transforms Versus Fourier Trans- forms." Bull. Amer. Math. Soc. 28, 288/C1/05, 1993. Weak Convergence Weak convergence is usually either denoted xn0wxor xnDx:ASEQUENCE xnfg ofVECTORS in an INNER PRODUCT SPACE Eis called weakly convergent to a VECTOR inEif xn;yhi 0x;yhi as n0/C12;for all y/C23E: Every STRONGLY CONVERGENT sequence is also weakly convergent (but the opposite does not usually hold). This can be seen as follows. Consider the sequence xnfg that converges strongly to x, i.e., xn/C28x kk 00a s n0/C12:SCHWARZ’S INEQUALITY now gives xn/C28x;y hijj 5xn/C28x kk ykk as n0/C12: The definition of weak convergence is therefore satisfied. See also INNER PRODUCT SPACE ,SCHWARZ’S INEQUAL- ITY,STRONG CONVERGENCE Weak Law of Large Numbers A result in probability theory also known as BER- NOULLI’S THEOREM or the weak law of large numbers (in contrast to the STRONG LAW OF LARGE NUMBERS ). Let X1 ; ..., Xnbe a sequence of independent and identically distributed random variables, each having a MEAN /C142Xi /C143/C30 m and STANDARD DEVIATION s: Define a new variable X /C13X1 /C27 ... /C27 Xn n: (1) Then, as n 0/C12; the sample mean xhiequals the population MEAN m of each variable. Xhi/C30X1 /C27 ... /C27 Xn n*+ /C301 nX1hi/C27.../C27 Xnhi ðÞ /C30n m n/C30 m: (2) In addition, var XðÞ/C30varX1 /C27 ... /C27 X2 n ! /C30varX1 n ! /C27.../C27varXn n ! /C30s2 n2 /C27.../C27s2 n2 /C30s2 n: (3) Therefore, by the CHEBYSHEV INEQUALITY , for all e > 0; PX/C28 m jj] e ðÞ 5var XðÞ e2/C30s2 n e2 : (4) As n 0/C12; it then follows that lim n0/C12PX/C28 m jj] e ðÞ /C300 : (5) (Khintchine 1929). Stated another way, the probabil- ity that the average X1/C27.../C27Xn ðÞ =n/C28m jj Beforean arbitrary POSITIVE quantity approaches 1 as n0/C12 (Feller 1968, pp. 228 /C1/29). See also ASYMPTOTIC EQUIPARTITION PROPERTY , CENTRAL LIMIT THEOREM ,CHEBYSHEV INEQUALITY , FRIVOLOUS THEOREM OF ARITHMETIC ,LAW OF TRULY LARGE NUMBERS ,STRONG LAW OF LARGE NUMBERS References Feller, W. "Laws of Large Numbers." Ch. 10 in An Introduc- tion to Probability Theory and Its Applications, Vol. 1, 3rd ed.New York: Wiley, pp. 228 /C1/47, 1968. Feller, W. "Law of Large Numbers for Identically Distrib- uted Variables." §7.7 in An Introduction to ProbabilityTheory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 231 /C1/34, 1971. Khintchine, A. "Sur la loi des grands nombres." Comptes rendus de l’Acade ´mie des Sciences 189, 477/C1/79, 1929. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 69 /C1/1, 1984. Weakly Binary Tree N.B. A detailed online essay by S. Finch was the starting point for this entry. AROOTED TREE for which the ROOT NODE is adjacent to at most two VERTICES , and all nonroot VERTICES are adjacent to at most three VERTICES . Let b(n) be the number of weakly binary trees of order n, then b(5)/C30 6:Let g(z)/C30X/C12 i/C300gizi; (1) where g0/C300 (2) g1/C30g2/C30g3/C301 (3) g2i/C271/C30Xi j/C301g2i/C271/C28jgj (4) g2i/C301 2gigi/C271 ðÞXi/C281 j/C301g2i/C28jgj: (5) Otter (Otter 1948, Harary and Palmer 1973, Knuth 1969) showed that lim n0/C12b(n)n3=2 jn/C30h; (6) where j/C302:48325 . . . (7) is the unique POSITIVE ROOT of g1 x ! /C301; (8) and h/C300:7916032 . . . : (9) /j1is also given by j/C30lim n0/C12cnðÞ2/C28n; (10) where cnis given by c0¼2 (11) cn/C30cn/C281 ðÞ2/C272; (12) giving h /C301 2ffiffiffi j psffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 /C271 c1/C271 c1c2/C271 c1c2c3/C27...s : (13) References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/otter/otter.html. Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1969. Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, 1997. Otter, R. "The Number of Trees." Ann. Math. 49, 583 /C1/99, 1948. Weakly Complete Sequence A SEQUENCE of numbers V /C30 nnfg is said to be weakly complete if every POSITIVE INTEGER n beyond a certain point N is the sum of some SUBSEQUENCE of V (Honsberger 1985). Dropping two terms from the FIBONACCI NUMBERS produces a SEQUENCE which is not even weakly complete. However, the SEQUENCE F ?n /C13Fn /C28(/C281)n is weakly complete, even with any finite subsequence deleted (Graham 1964). See also COMPLETE SEQUENCE References Graham, R. "A Property of Fibonacci Numbers." Fib. Quart. 2,1/C1/0, 1964. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 128, 1985. Weakly Connected Component A weakly connected component is a maximal SUB- GRAPH of a DIRECTED GRAPH such that for every pair of vertices u, v in the SUBGRAPH , there is an undirected path from u to v and a directed path from v to u. Weakly connected components can be found usingStronglyConnectedComponents [g]in the Mathematica add-on package DiscreteMath‘- Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ) (Skiena 1990, p. 172). See also WEAKLY CONNECTED DIGRAPH References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Weakly Connected Digraph A DIRECTED GRAPH in which it is possible to reach any node starting from any other node by traversing edges in some direction (i.e., not necessarily in the direction they point). The nodes in a strongly con- nected digraph therefore must all have either OUT- DEGREE or INDEGREE of at least 1. The numbers of nonisomorphic simple weakly connected digraphs on n /C301, 2, ... nodes are 1, 2, 13, 199, 9364, ... (Sloane’s A003085). See also CONNECTED DIGRAPH ,S TRONGLY CON- NECTED DIGRAPH ,W EAKLY CONNECTED COMPONENT References Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, p. 218, 1973. Skiena, S. "Strong and Weak Connectivity." §5.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, pp. 172 /C1/74, 1990. Sloane, N. J. A. Sequences A003085/M2067 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Weakly Differentiable See also DIFFERENTIABLE Weakly Independent An infinite sequence aifg of POSITIVE INTEGERS is called weakly independent if any relation aeiaiwith ei /C300or 91 and ei /C300; except finitely often, IMPLIES ei /C300 for all i. See also STRONGLY INDEPENDENT References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 136, 1994. Weakly Prime APRIME NUMBER is said to be weakly prime if changing a single digit to every other possible digit produces a COMPOSITE NUMBER when performed on each digit. The first few such numbers are 294001, 505447, 584141, 604171, 971767, 1062599, ... (Sloa- ne’s A050249). See also COMPOSITE NUMBER ,PRIME NUMBER References --. "Problem #12." http://math.smsu.edu/~les/POW12.html. Rivera, C. "Problems & Puzzles: Puzzle Weakly Primes.- 017." http://www.primepuzzles.net/puzzles/puzz_017.htm. Sloane, N. J. A. Sequences A050249 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTE- BOOK INTEGER SEQUENCES.M . Weakly Triple-Free Set TRIPLE- FREE SET Web Graph A graph formed by connecting several concentric WHEEL GRAPHS along spokes. See also WHEEL GRAPH Weber Differential Equations Consider the differential equation satisfied by w/C30z/C281=2Wk;/C281=41 2z2/C1;/C17 ; (1) where Wis a W HITTAKER FUNCTION , which is given by d zd zdw z1=2/CQ/C1 zd z"# /C27/C281 4/C272k z2/C273 4z4 ! wz1=2/C300 (2) d2w dz2/C272k/C281 4z2/C1;/C17 w/C300 (3) (Moon and Spencer 1961, p. 153; Zwillinger 1997, p. 128). This is usually rewritten d2Dn(z) dz2/C27n/C271 2/C2814z2/C1;/C17 DnzðÞ/C300: (4) The solutions are PARABOLIC CYLINDER FUNCTIONS . The equations d2U du2/C28c/C27k2u2/CQ/C1 U/C300 (5) d2V du2/C28c/C28k2v2/CQ/C1 V/C300; (6) which arise by separating variables in L APLACE’S EQUATION inPARABOLIC CYLINDRICAL COORDINATES , are also known as the Weber differential equations. As above, the solutions are known as P ARABOLIC CYLINDER FUNCTIONS . Zwillinger (1997, p. 127) calls yn/C27y? x/C271/C28n2 x2 ! y/C30/C281 px2x/C27n/C27(x/C28n) cos( np) ½/C138 (7)the Weber differential equation (Gradshteyn and Ryzhik 2000, p. 989). References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 989, 2000. Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997. Weber Functions Although B ESSEL FUNCTIONS OF THE SECOND KIND are sometimes called Weber functions, Abramowitz andStegun (1972) define a separate Weber function as E n(z)/C301 pgp 0sinnu/C28zsinu ðÞ du: (1) Letting zn/C30e2pi=nbe a ROOT OF UNITY , another set of Weber functions is defined as f(z)/C30h1 2z/C271 ðÞ/C1;/C17 z48h(z)(2) f1(z)/C30h1 2z/C1;/C17 h(z)(3) f2(z)/C30ffiffiffi 2ph(2z) h(z)(4) g2/C30f24(z)/C2816 f8(z)(5) g3/C30f24(z)/C278 ½/C138 f8 1(z)/C28f8 2(z) ½/C138 f8(z)(6) (Weber 1902, Atkin and Morain 1993), where h(z)i s the D EDEKIND ETA FUNCTION . The Weber functions satisfy the identities f(z/C271)/C30f1(z) z48(7) f1(z/C271)/C30f(z) z48(8) f2(z/C271)/C30z24f2(z) (9) f/C281 z ! /C30f(z) (10) f1/C281 z ! /C30f2(z) (11) f2/C281 z ! /C30f1(z) (12) (Weber 1902, Atkin and Morain 1993). See also ANGER FUNCTION ,BESSEL FUNCTION OF THE SECOND KIND,D EDEKIND ETA FUNCTION , J-FUNC- TION ,JACOBI IDENTITIES ,JACOBI TRIPLE PRODUCT , MODIFIED STRUVE FUNCTION , Q-FUNCTION ,STRUVE FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Anger and Weber Functions." §12.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 498 /C1/99, 1972. Atkin, A. O. L. and Morain, F. "Elliptic Curves and Prim- ality Proving." Math. Comput. 61,29/C1/8, 1993. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 68 /C1/9, 1987. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Anger Function Jn(x) and Weber Function En(x):/" §1.5 in Integrals and Series, Vol. 3: More Special Func- tions. Newark, NJ: Gordon and Breach, p. 28, 1990. Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, pp. 113 /C1/14, 1902. Weber’s Discontinuous Integrals g/C12 0J0(ax) cos(cx) dx /C300 a Bc 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 /C28 c2p a > c8 < : g/C12 0J0(ax) sin(cx) dx /C301ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 /C28 a2p a Bc 0 a /C21c ;8 < : where J0(z) is a zeroth order BESSEL FUNCTION OF THE FIRST KIND . References Bowman, F. Introduction to Bessel Functions. New York: Dover, pp. 59 /C1/0, 1958. Weber’s Formula 1 2p2 e /C28 a2/C27b2ðÞ = 4p2ðÞInab 2p2 ! /C30g/C12 0e /C28p2t2 Jn(at)Jn(bt)tdt; where R[n] >/C281 ; arg p jjBp=4; and a, b /C210, Jn(z)isa BESSEL FUNCTION OF THE FIRST KIND , and In(z)isa MODIFIED BESSEL FUNCTION OF THE FIRST KIND . See also BESSEL FUNCTION OF THE FIRST KIND, MODIFIED BESSEL FUNCTION OF THE FIRST KIND References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1476, 1980.Weber’s Theorem If two curves of the same GENUS (CURVE ) > 1 are in rational correspondence, then that correspondence is BIRATIONAL . References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 135, 1959. Weber-Sonine Formula For R[ m /C27nu] > 0; arg p jjBp=4 ; and a /C210, g/C12 0Jn(at)e /C28p2t2 tm/C281dt /C30a 2p !nG1 2n /C27 m ðÞhi 2pm G n /C27 1 ðÞ1 F11 2( n /C27 m); n /C271; /C28a2 2p2 ! ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND , G(z) is the GAMMA FUNCTION , and1F1(a;b;z)isa CONFLUENT HYPERGEOMETRIC FUNCTION . References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1474, 1980. Wedderburn’s Theorem A FINITE DIVISION RING is a FIELD . Weddle’s Rule Let the values of a function f(x) be tabulated at points xiequally spaced by h /C30xi/C271 /C28xi ; so f1 /C30f(x1) ; f2 /C30 f(x2) ; ..., f7 /C30f(x7) : Then Weddle’s rule approximating the integral of f(x) is given by the NEWTON- COTES -like formula gx6n x1f(x) dx /C303 10hf1/C275f2/C27f3/C276f4/C275f5/C27f6 ð /C27.../C275f6n/C281/C27f6nÞ See also BODE’S RULE,HARDY’S RULE,NEWTON- COTES FORMULAS ,SHOVELTON’S RULE,SIMPSON’S 3/8 RULE, SIMPSON’S RULE,TRAPEZOIDAL RULE References King, A. E. "Approximate Integration. Note on Quadrature Formulae: Their Construction and Application to Actuar- ial Functions." Trans. Faculty of Actuaries 9, 218/C1/31, 1923. Sheppard, W. F. "Some Quadrature-Formulæ." Proc. Lon- don Math. Soc. 32, 258/C1/77, 1900. Whittaker, E. T. and Robinson, G. The Calculus of Observa- tions: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 151, 1967. Wedge The term "wedge" has a number of meanings in mathematics. It is sometimes used as another name for the CARET symbol, as well as being the notation (/ffl) for logical AND. In SOLID GEOMETRY , a wedge is a right triangular PRISM turned so that it rests on one of its lateral rectangular faces (left figure). Harris and Stocker (1998) define a more general type of wedge in which the top edge is symmetrically shortened, causing the end triangles to slant obliquely (right figure). For a wedge of base lengths a and b, height h, and top edge length c, the VOLUME of the wedge is V /C301 6h 2a /C27c ðÞ : In the case c /C30a, this simplifies to V /C30ha =2: The CENTROID is located at a height ¯z/C30a/C27c ðÞ h 22a/C27c ðÞ above the base, which simplifies to h3forc/C30a. See also AND, CARET ,CONICAL WEDGE ,CYLINDRICAL WEDGE ,PRISM ,SPHERICAL WEDGE References Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 286, 1997. Harris, J. W. and Stocker, H. "Wedge." §4.5.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 101, 1998. Weisstein, E. W. "SolidGeometry." M ATHEMATICA NOTEBOOK SOLIDGEOMETRY.M . Wedge Product The wedge product is the product in an EXTERIOR ALGEBRA .I faandbare DIFFERENTIAL K-FORMS of degrees pandq, respectively, then afflb/C30(/C281)pqbffla: (1) It is not (in general) COMMUTATIVE , but it is ASSOCIA-TIVE, (afflb)fflu/C30affl(bfflu); (2) and BILINEAR c1a1/C27c2a2 ðÞ fflb/C30c1a1fflb ðÞ /C27c2a2fflb ðÞ (3) afflc1b1/C27c2b2 ðÞ /C30c1afflb1 ðÞ /C27c2afflb2 ðÞ (4) (Spivak 1999, p. 203), where c1andc2are constants. The alternating algebra is generated by elements of degree one, and so the wedge product can be defined using a basis eiforV: ei1ffl...ffleip/C1;/C17 fflej1ffl...fflejq/C1;/C17 /C30ei1ffl...ffleip fflej1ffl...fflejq(5) when the indices i1;...;ip;i1;...;iq;are distinct, and the product is zero otherwise. While the formula affla/C300 holds when ahas degree one, it does not hold in general. For example, consider a/C30e1ffle2/C27e3ffle4: affla/C30e1ffle2 ðÞffle1ffle2 ðÞ /C27e1ffle2 ðÞffle3ffle4 ðÞ /C27e3ffle4 ðÞffle1ffle2 ðÞ /C27e3ffle4 ðÞffle3ffle4 ðÞ /C300/C27e1ffle2ffle3ffle4/C27e3ffle4ffle1ffle2/C270 /C302e1ffle2ffle3ffle4 (6) Ifa1;...;akhave degree one, then they are linearly independent IFFa1ffl...fflak"0:/ The wedge product is the "correct" type of product touse in computing a VOLUME ELEMENT dV/C30dx1ffl...ffldxn: (7) The wedge product can therefore be used to calculate DETERMINANTS and volumes of PARALLELEPIPEDS . For example, write det A/C30detc1;...;cn ðÞ where ciare the columns of A. Then c1ffl...fflcn/C30detc1;...;cn ðÞ e1ffl...fflen (8) and det c1;...;cn ðÞjj is the volume of the PARALLELE- PIPED spanned by c1;...;cn:/ InMathematica ,a k-form can be written as an ANTISYMMETRIC k-tensor. Using this format, the following Mathematica function computes the wedge product.vars . Alt[x_List] : /C30Module[ { p/C30TensorRank[x], perms }, perms /C30Permutations[Range[p]]; Sum[Signature[perms[[i]]] Transpose[x, perms[[i]]],{i, p!}]/p! ] Wedge1[a_List, b_List] : /C30Alt[Outer[Times, a, b]] It is also possible to use an n-nested binary tree to represent the algebra of differential forms. Using this format, the following Mathematica function computes the wedge product recursively. Wedge2[{a_?(! ListQ[#1] &), b_?(! ListQ[#1] &)}, {c_?(! ListQ[#1] &), d_?(! ListQ[#1] &)}] : /C30 {a d /C27 b c, b d} sgn2[a_?ListQ] : /C30 MapIndexed[(Times[#1, Power[-1, Tr[#2]]] &), a, {TensorRank[a]}]; Wedge2[{a_List, b_List}, {c_List, d_List}] : /C30 {Wedge2[a, d] /C27 Wedge2[sgn2[b], c], Wedge2[b, d]} See also COHOMOLOGY ,CUP PRODUCT ,DETERMINANT , DIFFERENTIAL K-FORM,EXTERIOR ALGEBRA ,EXTER- IOR DERIVATIVE ,EXTERIOR POWER ,INNER PRODUCT , TENSOR PRODUCT (MODULE ), VECTOR SPACE ,V O- LUME ,VOLUME ELEMENT References Berger, M. Differential Geometry. New York: Springer- Verlag, 1988. Flanders, H. Differential Forms with Applications to the Physical Sciences. New York: Academic Press, 1963. Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd ed. Houston, TX: Publish or Perish, pp. 275 /C1/80, 1999. Sternberg, S. Differential Geometry. New York: Chelsea, pp. 14 /C1/0, 1983. Weibull Distribution The Weibull distribution is given by P(x) /C30 ab /C28axa/C281e /C28 x =bðÞa(1) D(x) /C301 /C28e/C28 x= bðÞa(2) for x /C23 0 ;/C12½Þ ; and is implemented in Mathematica as WeibullDistribution [a, b] in the Mathematica add-on package Statistics‘ContinuousDistri- butions‘ (which can be loaded with the command BBStatistics‘ ). The RAW MOMENTS of the dis- tribution are m?1 /C30b G 1 /C27 a/C281/CQ/C1 (3) m ?2 /C30b2 G 1 /C272a/C281/CQ/C1 (4) m ?3 /C30b3 G 1 /C273a/C281/CQ/C1 (5) m?4 /C30b4 G 1 /C274a/C281/CQ/C1 ; (6) and the MEAN , VARIANCE , SKEWNESS , and KURTOSIS of are m /C30 bG 1 /C27 a/C281/CQ/C1 (7) s2 /C30 b2 G 1 /C272a/C281/CQ/C1 /C28G2 1 /C27 a/C281/CQ/C1 /C2/C3 (8) g1 /C302G3 1 /C27 a/C281ðÞ /C28 3G 1 /C27 a/C281ðÞ G 1 /C27 2a/C281ðÞ G 1 /C27 2a/C281 ðÞ /C28G2 1 /C27 a/C281 ðÞ/C2/C3 3 =2/C27G 1 /C27 3a/C281ðÞ G 1 /C27 2 a/C281 ðÞ /C28G2 1 /C27 a/C281 ðÞ/C2/C3 3 =2 (9) g2 /C30f(a) G 1 /C27 2a/C281 ðÞ /C28G2 1 /C27 a/C281 ðÞ/C2/C3 2 ; (10) where G(z) is the GAMMA FUNCTION and f(a) /C13/C286G4 1 /C27 a/C281/CQ/C1 /C2712G4 1 /C27 a/C281/CQ/C1 G 1 /C272 a/C281/CQ/C1 /C283 G2 1 /C272a/C281/CQ/C1 /C284G 1 /C27 a/C281/CQ/C1 G 1 /C273a/C281/CQ/C1 /C27G 1 /C274a/C281/CQ/C1 : (11) A slightly different form of the distribution is defined by P(x) /C30a bxa /C281e /C28xa = b (12) D(x) /C301 /C28e/C28x a = b (13) (Mendenhall and Sincich 1995). This has RAW MO- MENTS m1 /C30 b1 =a G 1 /C27 a/C281/CQ/C1 (14) m2 /C30 b2 =a G 1 /C272a/C281/CQ/C1 (15) m3 /C30 b3 =a G 1 /C273a/C281/CQ/C1 (16) m4 /C30 b4 =a G 1 /C274a/C281/CQ/C1 (17) so the MEAN and VARIANCE for this form are m/C30b1=aG1/C27a/C281/CQ/C1 (18) s2/C30b2=aG1/C272a/C281/CQ/C1 /C28G21/C27a/C281/CQ/C1 /C2/C3 (19) The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed toquantify fatigue data, but it is also used in analysis of systems involving a "weakest link." See also F ISHER- TIPPETT DISTRIBUTION References Mendenhall, W. and Sincich, T. Statistics for Engineering and the Sciences, 4th ed. Englewood Cliffs, NJ: Prentice Hall, 1995. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992. Weierstrass Approximation Theorem Iffis a continuous real-valued function on [ a, b] and if any e>0 is given, then there exists a POLYNOMIAL pon [a, b] such that f(x)/C28P(x) jj Be for all x/C23a;b½/C138 :In words, any continuous function on a closed and bounded interval can be uniformly ap- proximated on that interval by POLYNOMIALS to any degree of accuracy. See also MU¨ NTZ’S THEOREM References Jeffreys, H. and Jeffreys, B. S. "Weierstrass’s Theorem on Approximation by Polynomials" and "Extension of Weier- strass’s Approximation Theory." §14.08/C1/4.081 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 446 /C1/48, 1988. Weierstrass Constant s1 2/C1;/C17 /C3012Y m;nðÞ" 0;0ðÞ1/C281 2(m/C27ni)"# e1=2(m/C27ni) ½/C138 /C271=8(m/C27ni)2½/C138 /C3025=4ffiffiffippep=8 G21 4/C1;/C17/C300:4749493799 . . . : References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 62, 1983. Plouffe, S. "Weierstrass Constant." http://www.lacim.u- qam.ca/piDATA/weier.txt. Waldschmidt, M. "Fonctions entie `res et nombres transcen- dants." Cong. Nat. Soc. Sav. Nancy 5, 1978. Waldschmidt, M. "Nombres transcendants et fonctions sigma de Weierstrass." C. R. Math. Rep. Acad. Sci. Canada 1, 111/C1/14, 1978/79. Weierstrass Elliptic Function The Weierstrass elliptic functions (or Weierstrass /C212/- functions, voiced " p-functions") are elliptic functions which, unlike the J ACOBI ELLIPTIC FUNCTIONS , have a second-order POLE atz/C300. The above plots show the Weierstrass elliptic function /C212(z) and its derivative /C212?(z) for invariants (defined below) of g2/C300 and g3/C30 0:The Weierstrass elliptic function is implemented inMathematica asWeierstrassP [u,{g1,g2}]. The plots above show the derivatives of the Weier- strass /C212/-function. Weierstrass elliptic functions are denoted /C212(z) and can be defined by /C212(z)/C301 z2/C27X ?/C12 m;n/C30/C28/C121 z/C282mv1/C282nv2 ðÞ2/C281 2mv1/C272nv2 ðÞ2"# (1) (Whittaker and Watson 1990, p. 434). Write Vmn/C13 2mv1/C272nv2:Then this can be written /C212(z)/C30z/C282/C27X ? m;nz/C28Vmn ðÞ/C282/C28V/C282 mnhi : (2) An equivalent definition which converges more ra- pidly is /C212(z)/C30p 2v1 !2/C2Q /C281 3/C27X/C12 n/C30/C28/C12csc2/C18z/C282nv2 2v1p/C19 /C28X ?/C12 n/C30/C28/C12csc2/C18nv2 v1p/C19/C21 (3) (Whittaker and Watson 1990, p. 434). /C212(z)i sa n EVEN FUNCTION since /C212(/C28z) gives the same terms in a different order. To specify /C212completely, its periods or invariants, written /C212zðjv1;v2Þand/C212z;g1;g2 ðÞ ;re- spectively, must also be specified. The series expansion of /C212(z) is given by /C212(z)/C30z/C282/C28X/C12 k/C302ckz2k/C282; (4) where c2/C30g2 20(5) c3/C30g3 28(6) and ck/C303 2k/C271 ðÞ k/C283 ðÞXk/C282 m/C302cmck/C28m (7) fork]4 (Abramowitz and Stegun 1972, p. 635). The first few values for ckfork]4 in terms of c2andc3are given by c4/C301 3c2 2 (8) c5/C301 113c2c3 ðÞ (9) c6/C301 392c32/C273c23/CQ/C1 (10) c7/C302 332c22c3 (11) c8/C305 729311c42/C2736c2c23/CQ/C1 (12) c9/C3029 2717c32c3/C2711c23/CQ/C1 (13) c10/C301 240669242c52/C271455 c22c23/CQ/C1 (14) (Abramowitz and Stegun 1972, p. 636). The Weierstrass elliptic function describes how to get from a TORUS giving the solutions of an ELLIPTIC CURVE to the algebraic form of the ELLIPTIC CURVE . The differential equation from which Weierstrass elliptic functions arise can be found by expanding about the origin the function f(z)/C13/C212(z)/C28z/C282: /C212(z)/C28z/C282/C30f(0)/C27f?(0)z/C271 2!fƒ(0)z2/C271 3!f§(0)z3 /C271 4f(4)(0)z4/C27...: (15) Butf(0)/C300 and the function is even, so f?(0)/C30f§(0)/C30 0 and f(z)/C30/C212(z)/C28z/C282/C301 2!fƒ(0)z2/C271 4f(4)(0)z4/C27...:(16) Taking the derivatives f?/C30/C28 2S?z/C28Vmn ðÞ/C283(17) fƒ/C306S?z/C28Vmn ðÞ/C284(18) f§/C30/C2824S?z/C28Vmn ðÞ/C285(19) f(4)/C30120S?z/C28Vmn ðÞ/C286: (20) So fƒ(0)/C306S?V/C284 mn (21) f(4)(0)/C30120S?V/C286 mn: (22) Plugging in, /C212(z)/C28z/C282/C303S?V/C284 mnz2/C275S?V/C286 mnz4/C27Oz6/CQ/C1 (23) Define the INVARIANTSg2/C1360S?V/C284 mn (24) g3/C13140S?V/C286 mn; (25) then /C212(z)/C30z/C282/C271 20g2z2/C271 28g3z4/C27Oz6/CQ/C1 (26) /C212?(z)/C30/C282z/C283/C271 10g2z/C271 7g3z3/C27Oz5/CQ/C1 : (27) Now cube (26) and square (27) /C2123(z)/C30z/C286/C273 20g2z/C282/C273 28g3/C27Oz2/CQ/C1 (28) /C212?2(z)/C304z/C286/C2825g2z/C282/C2847g3/C27Oz2/CQ/C1 : (29) Taking (29) minus 4 /C29(28) cancels out the z/C286term, giving /C212?2(z)/C284/C2123(z)/C30/C2825/C2835/C1;/C17 g2z/C282/C27/C2847/C2837/C1;/C17 g3/C27Oz2/CQ/C1 /C30/C28g2z/C282/C28g3/C27Oz2/CQ/C1 (30) /C212?2(z)/C284/C2123(z)/C27g2z/C282/C27g3/C30Oz2/CQ/C1 : (31) But, from (16) /C212(z)/C30z/C282/C271 2!fƒ(0)z2/C2714f(4)(0)z4/C27...; (32) so //C212ðzÞ¼z/C282þOðz2Þ/and (31) can be written /C212?2(z)/C284/C2123(z)/C27g2/C212(z)/C27g3/C30Oz2/CQ/C1 : (33) But the Weierstrass elliptic function is analytic at the origin and therefore at all points congruent to the origin. There are no other places where a singularity can occur, so this function is an ELLIPTIC FUNCTION with no SINGULARITIES .B yL IOUVILLE’S ELLIPTIC FUNCTION THEOREM , it is therefore a constant. But asz00;Oz2ðÞ00;so /C212?2(z)/C304/C2123(z)/C28g2/C212(z)/C28g3 (34) (Whittaker and Watson 1990, pp. 436 /C1/37). The solution to the differential equation y?2/C304y3/C28g2y/C28g3 (35) is therefore given by y/C30/C212(z/C27a);providing that numbers v1andv2exist which satisfy the equations defining the INVARIANTS . Writing the differential equation in terms of its roots e1;e2;ande3; y?2/C304y3/C28g2y/C28g3/C304y/C28e1 ðÞ y/C28e2 ðÞ y/C28e3 ðÞ (36) (Rainville 1971, p. 312), 2l n y?ðÞ/C30ln 4/C27X3 r/C301lny/C28er ðÞ (37) 2yƒ y?/C30y?X3 r/C301y/C28er ðÞ/C281(38) 2yƒ y?2/C30X3 r/C301y/C28er ðÞ/C281(39) 2y?2y§/C28yƒ2y?yƒ ðÞ y?4/C30/C28y?X3 r/C301y/C28er ðÞ/C282(40) 2y§ y?3/C284yƒ2 y?4/C30/C28X3 r/C301y/C28er ðÞ/C282: (41) Now take (41) divided by 4 plus [(41) divided by 4] quantity squared, y§ 2y?3/C28yƒ2 y?4 ! /C27yƒ2 4y?4 ! /C30/C281 4X3 r/C301y/C28er ðÞ/C282/C271 16X3 r/C301y/C28er ðÞ/C281"# 2 (42) 3yƒ2 4y?4/C28y§ 2y?3/C303 16X3 r/C301y/C28er ðÞ/C282/C283 8yY3 r/C301y/C28er ðÞ/C281:(43) The term on the right is half the S CHWARZIAN DERIVATIVE . The DERIVATIVE of the Weierstrass elliptic function is given by /C212?(z)/C30d dz/C212(z)/C30/C282X m;n1 z/C28Vmn ðÞ3 /C30/C282z/C283/C282X ? m;nz/C28Vmn ðÞ/C283: (44) This is an ODD FUNCTION which is itself an elliptic function with pole of order 3 at z/C300. T heINTEGRAL is given by z/C30g/C12 /C212(z)4t3/C28g2t/C28g3/CQ/C1/C281=2dt: (45) The second derivative satisfies /C212ƒ1 2v1/C1;/C17 /C302e1/C28e2 ðÞ e1/C28e3 ðÞ (46) (Apostol 1997, p. 23). A duplication formula is obtained as follows. /C212(2z)/C30lim y0z/C212(y/C27z) /C301 4lim y0z/C212?(z)/C28/C212?(y) /C212(z)/C28/C212(y)"#2 /C28/C212(z)/C28lim y0z/C212(y) /C3014lim h00/C212(z)/C28/C212?(z/C27h) /C212(z)/C28/C212(z/C27h)"#2 /C282/C212(z) /C301 4lim h00/C212?(z)/C28/C212?(z/C27h) h"#(/C2lim h00h /C212(z)/C28/C212(z/C27h)"# g2/C282/C212(z) /C3014/C212ƒ(z) /C212?(z)"#2 /C282/C212(z) (47) (Apostol 1997, p. 24). A general addition theorem is obtained as follows. Given /C212?(z)/C30A/C212(z)/C27B (48) /C212?(y)/C30A/C212(y)/C27B (49) with zero yand zwhere zf9ymod 2 v1;2v2 ðÞ ;find the third zero z:Consider /C212?zðÞ/C28A/C212zðÞ/C28B:This has a pole of order three at z/C300;but the sum of zeros (/C300) equals the sum of poles for an ELLIPTIC FUNC- TION ,s oz/C27y/C27z/C300 and z/C30/C28z/C28y: /C212?(/C28z/C28y)/C30A/C212(/C28z/C28y)/C27B (50) /C28/C212?(z/C27y)/C30A/C212(z/C27y)/C27B: (51) Combining (48), (49), and (51) gives /C212(z) /C212?(z)1 /C212(y) /C212?(y)1 /C212(z/C27y)/C28/C212(z/C27y)12 435A /C281 B2435/C300 0 02 435; (52) so /C212(z) /C212?(z)1 /C212(y) /C212?(y)1 /C212(z/C27y)/C28/C212(z/C27y)1/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C300: (53) Defining u/C27v/C27w/C300 where /u/C13z/andv/C13ygives the symmetric form /C212(u)/C212?(u)1 /C212(v)/C212?(v)1 /C212(w)/C212(w)1/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C300 (54) (Whittaker and Watson 1990, p. 440). To get the expression explicitly, start again with /C212?(z)/C28A/C212(z)/C28B/C300; (55) where z/C30z;y;/C28z/C28y: /C212? 2(z)/C28A/C212(z)/C27B ½/C1382/C300: (56) But from (34), /C212?2(z)/C304/C2123(z)/C28g2/C212(z)/C28g3;so 4/C2123(z)/C28A2/C2122(z)/C28(2AB/C27g2)/C212(z)/C28B2/C27g3/CQ/C1 /C300: (57) The solutions /C212zðÞ/C13zare given by 4z3/C28A2z2/C282AB/C27g2 ðÞ z/C28B2/C27g3/CQ/C1 /C300: (58) But the sum of roots equals the COEFFICIENT of the squared term, so /C212(z)/C27/C212(y)/C27/C212(z/C27y)/C301 4A2(59) /C212?(z)/C28/C212?(y)/C30A/C212(z)/C28/C212(y) ½/C138 (60) A/C30/C212?(z)/C28/C212?(y) /C212(z)/C28/C212(y)(61) /C212(z/C27y)/C301 4/C212?(z)/C28/C212?(y) /C212(z)/C28/C212(y)"#2 /C28/C212(z)/C28/C212(y) (62) (Whittaker and Watson 1990, p. 441). Half-period identities include x/C13/C2121 2v1/C1;/C17 /C30/C212/C28hv1/C27v1 ðÞ /C30e1/C27e1/C28e2 ðÞ e1/C28e3 ðÞ /C212/C2812v1/C1;/C17 /C28e1 /C30e1/C27e1/C28e2 ðÞ e1/C28e3 ðÞ x/C28e1: (63) Multiplying through, x2/C28e1x/C30e1x/C28e2 1/C27e1/C28e2 ðÞ e1/C28e3 ðÞ (64) x2/C282e1/C27e21/C28e1/C28e2 ðÞ e1/C28e3 ðÞ/C2/C3 /C300; (65) which gives /C2121 2v1/C1;/C17 /C30122e19ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4e2 1/C284e21/C28e1/C28e2 ðÞ e1/C28e3 ðÞ ½/C138q/C2;/C27 /C30e19ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e1/C28e2 ðÞ e1/C28e3 ðÞp : (66) From Whittaker and Watson (1990, p. 445), /C212?1 2v1/C1;/C17 /C30/C282ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e1/C28e2 ðÞ e1/C28e3 ðÞp /C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1/C28e2p/C27ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1/C28e3p/CQ/C1 : (67) The function is HOMOGENEOUS , /C212lzðjlv1;lv2Þ/C30l/C282/C212zðjv1;v2Þ (68) /C212lz;l/C284g2;l/C286g3/CQ/C1 /C30l/C282/C212z;g2;g3 ðÞ : (69) To invert the function, find 2 v1and 2 v2of/C212zðjv1;v2Þ when given /C212z;g1;g2 ðÞ :Lete1;e2;ande3be the roots such that e1/C28e2 ðÞ =e1/C28e3 ðÞ is not a REAL NUMBER >1 orB0:Determine the PARAMETER tfrom e1/C28e2 e1/C28e3/C30q4 40ðjtÞ q430ðjtÞ: (70) Now pick A/C13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie1/C28e2p q2 40ðjtÞ: (71) As long as g3 2"27g3;the periods are then 2v1/C30pA (72) 2v2/C30pt A: (73) Weierstrass elliptic functions can be expressed interms of J ACOBI ELLIPTIC FUNCTIONS by /C212u;g2;g3 ðÞ /C30e3/C27e1/C28e3 ðÞ ns2uffiffiffiffiffiffiffiffiffiffiffiffiffiffie1/C28e3p;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2/C28e3 e1/C28e3s ! ; (74) where /C212v1ðÞ/C30e1 (75) /C212v2ðÞ/C30e2 (76) /C212v3ðÞ/C30/C28/C212/C28v1/C28v2 ðÞ /C30e3; (77) and the INVARIANTS are g2/C1360X ? m;nV/C284 mn (78) g3/C13140X ? m;nV/C286 mn: (79) Here, Vmn/C132mv1/C282nv2:/ An addition formula for the Weierstrass elliptic function can be derived as follows. /C212z/C27v1 ðÞ /C27/C212zðÞ/C27/C212v1ðÞ /C301 4/C212?(z)/C28/C212?(v1) /C212(z)/C28/C212(v1)"#2 /C3014/C212?2(z) /C212zðÞ/C28e1 ½/C1382: (80) Use /C212?(z)/C304Y3 r/C301/C212(z)/C28er ½/C138 ; (81) so /C212z/C27v1 ðÞ /C30/C28/C212(z)/C28e1/C27144Q3 r/C301/C212(z)/C28er ½/C138 /C212(z)/C28e1 ½/C1382 /C30/C28/C212(z)/C28e1/C27/C212(z)/C28e2 ½/C138 /C212(z)/C28e3 ½/C138 /C212(z)/C28e1:ð82Þ Use a3 r/C301er/C300; /C212z/C27v1 ðÞ /C30e1/C27/C282e1/C28/C212(z) ½/C138 /C212(z)/C28e1 ½/C138 /C212(z)/C28e1 /C27/C2122(z)/C28/C212(z)e2/C27e3 ðÞ /C27e2e3 /C212(z)/C28e1 /C30e1/C27/C28/C212(z)e1/C27e2/C27e3 ðÞ /C27e2e3/C272e2 1 /C212(z)/C28e1: (83) But a3 r/C301er/C300 and 2e2 1/C27e2e3/C30e21/C28e1e2/C27e3 ðÞ /C27e2e3 /C30e1/C28e2 ðÞ e1/C28e3 ðÞ ; (84) so /C212 z /C27 v1 ðÞ /C30e1 /C27e1 /C28 e2 ðÞ e1 /C28 e3 ðÞ /C212(z) /C28 e1: (85) The periods of the Weierstrass elliptic function are given as follows. When g2and g3are REAL and g3 2 /C28 27g23 > 0; then e1 ; e2 ; and e3 are REAL and defined such that e1 > e2 > e3 : v1 /C30g/C12 e14t3 /C28g2t /C28g3/CQ/C1/C281=2dt (86) v3 /C30/C28ige2 /C28/C12g3 /C27g2t /C284t3/CQ/C1/C281=2dt (87) v2 /C30/C28v1 /C28 v3 : (88) The roots of the Weierstrass elliptic function satisfy e1 /C30/C212 v1ðÞ (89) e2 /C30/C212 v2ðÞ (90) e3 /C30/C212 v3ðÞ ; (91) where v3 /C13/C28v1 /C28 v2 : The ei/s are ROOTS of 4t3 /C28g2t /C28 g3and are unequal so that e1 "e2 "e3 :: They can be found from the relationships e1/C27e2/C27e3/C30/C28a2/C300 (92) e2e3/C27e3e1/C27e1e2/C30a1/C30/C281 4g2 (93) e1e2e3/C30/C28a0/C301 4g3: (94) See also ELLIPTIC CURVE ,ELLIPTIC FUNCTION ,EISEN- STEIN SERIES ,EQUIANHARMONIC CASE,JACOBI ELLIP- TIC F UNCTIONS ,L EMNISCATE C ASE , PSEUDOLEMNISCATE CASE,W EIERSTRASS SIGMA FUNCTION ,W EIERSTRASS ZETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Weierstrass Elliptic and Related Functions." Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627 /C1/71, 1972. Apostol, T. M. "The Weierstrass /C212Function," "The Laurent Expansion of /C212Near the Origin," "Differential Equation Satisfied by /C212;/" "The Eisenstein Series and the Invariants g2and g3;/" "The Numbers e1;e2;and e3;/" and "The Discriminant D:/"§1.6/C1/.11 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 9 /C1/4, 1997. Eichler, M. and Zagier, D. "On the Zeros of the Weierstrass /C212/-Function." Math. Ann. 258, 399/C1/07, 1982. Fischer, G. (Ed.). Plates 129 /C1/31 in Mathematische Modelle/ Mathematical Models, Bildband/Photograph Volume.Braunschweig, Germany: Vieweg, pp. 126 /C1 /28, 1986. Huang, J. "Integral Representation of Harmonic Lattice Sums." J. Math. Phys. 40, 5240 /C1/246, 1999. Rainville, E. D. Special Functions. New York: Chelsea, 1971. To¨lke, F. "Spezielle Weierstraßsche /C212/-Funktionen." Ch. 4 in Praktische Funktionenlehre, zweiter Band: Theta-Funk-tionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 115 /C1/44, 1966. To¨lke, F. Praktische Funktionenlehre, fu ¨nfter Band: Allge- meine Weierstraßsche Funktionen und Ableitungen nachdem Parameter. Integrale der Theta-Funktionen und Bi-linear-Entwicklungen. Berlin: Springer-Verlag, 1968. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Woods, F. S. "The Function p(u): /"§160 in Advanced Calcu- lus: A Course Arranged with Special Reference to theNeeds of Students of Applied Mathematics. Boston, MA: Ginn, pp. 381 /C1 /82, 1926. Weierstrass Extreme Value Theorem EXTREME VALUE THEOREM Weierstrass Factor Theorem Let any finite or infinite set of points having no finite LIMIT POINT be prescribed, and associate with each of its points a definite positive integer as its order. Then there exists an ENTIRE FUNCTION which has zeros to the prescribed orders at precisely the prescribed points, and is otherwise different from zero. More- over, this function can be REPRESENTED AS a product from which one can read off again the positions and orders of the zeros. Furthermore, if G0(z) is one such function, then G(z)/C30eh(z)G0(z) is the most general function satisfying the conditions of the problem, where h(z) denotes an arbitrary ENTIRE FUNCTION . References Knopp, K. "Weierstrass’s Factor-Theorem." §1i n Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 1 /C1/, 1996. Krantz, S. G. "The Weierstrass Factorization Theorem." §8.2 inHandbook of Complex Analysis. Boston, MA: Birkha ¨u- ser, pp. 109 /C1/10, 1999. Weierstrass Factorization Theorem WEIERSTRASS FACTOR THEOREM Weierstrass Form A general form into which an ELLIPTIC CURVE over any FIELD Kcan be transformed is called the Weierstrass form, and is given by y2/C27ay/C30x3/C27bx2/C27cxy/C27dx/C27e; where a,b,c,d, and eare elements of K. Weierstrass Function A CONTINUOUS FUNCTION which is nowhere DIFFER- ENTIABLE . It is given by f(x) /C30X/C12 n/C301bn cos an px ðÞ where a is an ODD NUMBER , b /C23 (0;1); and ab > 1 /C27 3p=2: The above plot is for a /C3019 and b /C301=2:/ See also BLANCMANGE FUNCTION ,CONTINUOUS FUNC- TION ,DIFFERENTIABLE References Berry, M. V. and Lewis, Z. V. "On the Weierstrass-Mandel- brot Function." Proc. Roy. Soc. London Ser. A 370, 459 /C1/ 84, 1980. Darboux, G. "Me´moire sur les fonctions discontinues." Ann. l’E´ cole Normale, Ser. 2 4,57/C1/12, 1875. Darboux, G. "Me´moire sur les fonctions discontinues." Ann. l’E´ cole Normale, Ser. 2 8, 195 /C1/02, 1879. du Bois-Reymond, P. "Versuch einer Klassification der will- ku¨rlichen Functionen reeller Argumente nach ihren A¨ n- derungen in den kleinsten Intervallen." J. fu¨r Math. 79, 21 /C1/7, 1875. Faber, G. "Einfaches Beispiel einer stetigen nirgends differ- entiierbaren [sic] Funktion." Jahresber. Deutschen Math. Verein. 16 538 /C1/40, 1907. Hardy, G. H. "Weierstrass’s Non-Differentiable Function." Trans. Amer. Math. Soc. 17, 301 /C1/25, 1916. Landsberg, G. "U¨ ber Differentziierbarkeit stetiger Funktio- nen." Jahresber. Deutschen Math. Verein. 17,46/C1/1, 1908. Lerch, M. "Ueber die Nichtdifferentiirbarkeit [sic] gewisser Functionen." J. reine angew. Math. 13, 126 /C1/38, 1888. Mandelbrot, B. B. "Weierstrass Functions and Kin. Ultra- violet and Infrared Catastrophe." The Fractal Geometry of Nature. New York: W. H. Freeman, pp. 388 /C1/90, 1983. Pickover, C. A. Keys to Infinity. New York: Wiley, p. 190, 1995. Weierstrass, K. Abhandlungen aus der Functionenlehre. Berlin: J. Springer, p. 97, 1886. Weierstrass Intermediate Value Theorem If a continuous function defined on an interval is sometimes POSITIVE and sometimes NEGATIVE , it must be 0 at some point. Weierstrass M-Test Let a/C12 k /C301un(x)bea SERIES of functions all defined for a set E of values of x. If there is a CONVERGENT series of constantsX/C12 n/C301Mn ; such that un(x) jj5Mn for all x /C23 E; then the series exhibits ABSOLUTE CONVERGENCE for each x /C23 E as well as UNIFORM CONVERGENCE inE. See also ABSOLUTE CONVERGENCE ,U NIFORM CON- VERGENCE References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 301 /C1/03, 1985. Jeffreys, H. and Jeffreys, B. S. " MTest" and "Extension of theMTest." §1.1151 /C1/.1152 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 40 /C1/1, 1988. Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 73, 1996. Weierstrass Operator The operator ent2=2which satisfies ent2=2p(x)/C301ffiffiffiffiffiffiffiffi 2pnpg/C12 /C28/C12e/C28u2=(2n)p(x/C27u)du forn>0:/ References Roman, S. The Umbral Calculus. New York: Academic Press, p. 88, 1984. Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684/C1/60, 1973. Weierstrass Point APOLE of multiplicity less than p/C271:/ References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 290 /C1/91, 1959. Weierstrass Product Inequality If 05a;b;c;d51;then (1/C28a)(1/C28b)(1/C28c)(1/C28d)/C27a/C27b/C27c/C27d]1: References Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 244 /C1/45, 1985. Weierstrass Sigma Function The QUASIPERIODIC FUNCTION defined by d dzlns(z)/C30z(z); (1) where z(z) is the WEIERSTRASS ZETA FUNCTION and lim z0/C12s(z) z/C301: (2) Then s(z) /C30zY ?/C12 m;n/C30/C28/C121 /C28z Vmn ! expz Vmn/C27z2 2V2 mn ! "# ; (3) where the term with m /C30n /C300 is omitted from the product. In addition, s(z) satisfies s z /C272v1 ðÞ /C30/C28e2 h1 z/C27v1 ðÞs(z) (4) s z /C272v2 ðÞ /C30/C28e2 h2 z/C27v2 ðÞs(z) (5) and sr(z) /C30e /C28hrz s z /C27 vr ðÞ svrðÞ (6) for r /C301, 2, 3. /s(z) can be expressed in terms of JACOBI THETA FUNCTIONS using the expression s zðj v1 ; v2 Þ/C302 v1 pq?1exp /C28n2 q§1 6q?1 ! q1nv2 v1/C12/C12/C12/C12/C12! ; (7) where n /C13pz = 2v 1ðÞ ; and h1 /C30/C28p2 q§1 12 v1 q?1(8) h2 /C30/C28p2 v2 q§1 12 v2 1 q?1/C28pi 2v1: (9) There is a beautiful series expansion for s(z) ; given by the DOUBLE SUM s(z) /C30X/C12 m;n/C300amn1 2g2/C1;/C17m 2g3ðÞn z4m/C276n/C271 4m /C27 6n /C27 1 ðÞ ! ; (10) where a00 /C301; amn /C300 for either subscript negative, and other values are gives by the RECURRENCE RELATION amn /C303(m /C271)am/C271 ;n/C271 /C2716 3 (n /C271)am/C282;n/C271 /C2813(2m /C273n /C281)(4m /C276n /C281)am/C281 ;n(11) (Abramowitz and Stegun 1972, pp. 635 /C1/36). The following table gives the values of the amn coefficients for small m and n. n /C300 n /C301 n /C302 n /C303 /a0n/ 1 /C283 /C2854 14904 /a1n/ /C281 /C2818 4968 502200 /a2n/ /C289 513 257580 162100440 /a3n/ 69 33588 20019960 /C289465715080/a4n/ 321 2808945 /C28376375410 /C284582619446320 /a5n/160839 /C2841843142 /C28210469286736 /C281028311276281264 See also WEIERSTRASS ELLIPTIC FUNCTION ,W EIER- STRASS ZETA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Weierstrass Elliptic and Related Functions." Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627 /C1/71, 1972. Knopp, K. "Example: Weierstrass’s s/-Function." §2d in Theory of Functions Parts I and II, Two Volumes Boundas One, Part II. New York: Dover, pp. 27 /C1 /0, 1996. To¨lke, F. "Spezielle Weierstraßsche Sigma-Funktionen." Ch. 9 in Praktische Funktionenlehre, dritter Band: Jaco- bische elliptische Funktionen, Legendresche elliptischeNormalintegrale und spezielle Weierstraßsche Zeta- undSigma Funktionen. Berlin: Springer-Verlag, pp. 164 /C1 /80, 1967. Whittaker, E. T. and Watson, G. N. "The Function s(z):/" §20.42 in A Course in Modern Analysis, 4th ed. Cam- bridge, England: Cambridge University Press, pp. 447 /C1/ 48, 450 /C1/52, and 458 /C1/61, 1990. Weierstrass Zeta Function The QUASIPERIODIC FUNCTION defined by dz(z) dz/C13/C28/C212(z) (1) with lim z00z(z)/C28z/C281/C12/C12/C12/C12/C300: (2) Then z(z)/C28z/C281/C30/C28gz 0/C212(z)/C28z/C282/C2/C3 dz /C30/C28S?gz 0z/C28Vmn ðÞ/C282/C28V/C282 mnhi dz (3) zzðÞ/C30z/C281/C27X ?/C12 m;n/C30/C28/C12z/C28Vmn ðÞ/C281/C27V/C281 mn/C27zV/C282 mnhi (4) soz(z)i sa n ODD FUNCTION . Integrating /C212z/C272v1 ðÞ /C30 /C212(z) gives zz/C272v1 ðÞ /C30zzðÞ/C272h1: (5) Letting z/C30/C28v1gives z/C28v1 ðÞ/C272h1/C30/C28zv1ðÞ/C272h1;so / h1¼zðv1Þ/. Similarly, h2/C30zv2ðÞ :From Whittaker and Watson (1990), h1v2/C28h2v1/C301 2pi (6) Ifx/C27y/C27z/C300;then z(x)/C27z(y)/C27z(z) ½/C1382/C27z?(x)/C27z?(y)/C27z?(z)/C300 (7) (Whittaker and Watson 1990, p. 446). Also, 21 /C212(x) /C2122(x) 1 /C212(y) /C2122(y) 1 /C212(z) /C2122(z)/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12 1 /C212(x) /C212?(x) 1 /C212(y) /C212?(y) 1 /C212(z) /C212?(z)/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C30 z(x /C27y /C27z) /C28 z(x) /C28 z(y) /C28 z(z) (8) (Whittaker and Watson 1990, p. 446). The series expansion of z(z) is given by z(z) /C30z /C281 /C28X/C12 k/C302ckz2k /C281 2k /C28 1 ; (9) where c2 /C30g2 20 (10) c3 /C30g3 28 (11) and ck /C303 2k /C27 1 ðÞ k /C28 3 ðÞXk /C282 m/C302cmck /C28m (12) for k ]4 (Abramowitz and Stegun 1972, p. 635). See also WEIERSTRASS ELLIPTIC FUNCTION ,W EIER- STRASS SIGMA FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). "Weierstrass Elliptic and Related Functions." Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627 /C1/71, 1972. To¨lke, F. "Spezielle Weierstraßsche Zeta-Funktionen." Ch. 8 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normal- integrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 145 /C1/63, 1967. Whittaker, E. T. and Watson, G. N. "Quasi-Periodic Func- tions. The Function z(z)/" and "The Quasi-Periodicity of the Function z(z):/" §20.4 and 20.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, pp. 445 /C1/47 and 449 /C1/51, 1990. Weierstrass-Casorati Theorem An ANALYTIC FUNCTION approaches any given value arbitrarily closely in any e/-NEIGHBORHOOD of an ESSENTIAL SINGULARITY . See also ANALYTIC FUNCTION ,ESSENTIAL SINGULAR- ITY References Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 114 /C1/15 and 124 /C1/25, 1996. Krantz, S. G. "The Casorati-Weierstrass Theorem." §4.1.6 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 43, 1999.Weierstrass-Erdman Corner Condition In the CALCULUS OF VARIATIONS , the condition fy?x;y;y? x/C28ðÞ ðÞ /C30fy?x; y;y? x/C27/CQ/C1/CQ/C1 must hold at a corner (x, y) of a minimizing arc E12 :/ WeierstrassHalfPeriods WEIERSTRASS ELLIPTIC FUNCTION WeierstrassInvariants WEIERSTRASS ELLIPTIC FUNCTION Weierstrass-Mandelbrot Function WEIERSTRASS FUNCTION WeierstrassP WEIERSTRASS ELLIPTIC FUNCTION WeierstrassPPrime WEIERSTRASS ELLIPTIC FUNCTION Weierstrass’s Double Series Theorem Let all of the functions fn(z) /C30X/C12 k/C300a(n) kz /C28z0 ðÞk with n /C300, 1, 2, ..., be regular at least for z /C28z0 jjBr; and let F(z) /C30X/C12 n /C300fn(z) ¼½að0 Þ 0þ a ð0 Þ 1ðz /C28z0 Þþ... þ að0 Þ kðz /C28z0 Þk þ .../C138 /C27 a(1) 0/C27a(1)1z /C28z0 ðÞ /C27.../C27a(1) kz /C28z0 ðÞk/C27...hi /C27... /C27 a(n) 0/C27a(n) 1z /C28z0 ðÞ /C27.../C27a(n) kz /C28z0 ðÞk/C27...hi /C27... be uniformly convergent for z /C28z0 5 r Br for every r Br : Then the coefficients in any column form a convergent series. Furthermore, setting a(0)k/C27a(1)k/C27.../C27a(n) k/C27.../C30X/C12 n/C300a(n) k/C30Ak fork/C300, 1, 2, ..., it then follows that X/C12 k/C300Akz/C28z0 ðÞk is the POWER SERIES forF(z);which converges at least forz/C28z0 jjBr:/ See also DOUBLE SERIES References Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 83, 1996. Weierstrass’s Gap Theorem Given a succession of nonsingular points which are on a nonhyperelliptic curve of GENUS p, but are not a group of the canonical series, the number of groups of the first k which cannot constitute the group of simple POLES of a RATIONAL FUNCTION is p. If points next to each other are taken, then the theorem becomes: Given a nonsingular point of a nonhyper- elliptic curve of GENUS p, then the orders which it cannot possess as the single pole of a RATIONAL FUNCTION are p in number. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 290, 1959. Weierstrass’s Polynomial Theorem A function, continuous in a finite close interval, can be approximated with a preassigned accuracy by POLYNOMIALS . A function of a REAL variable which is continuous and has period 2 p can be approximated by trigonometric POLYNOMIALS . References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 5, 1975. Weierstrass’s Theorem There are at least two theorems known as Weier- strass’s theorem. The first states that the only HYPERCOMPLEX NUMBER systems with commutative multiplication and addition are the algebra with one unit such that e /C30e2 and the GAUSSIAN INTEGERS . In harmonic analysis, let U ⁄C be any OPEN SET, and let a1 ; a2 ; ..., be a finite or infinite sequence in U (possibly with repetitions) that has no ACCUMULATION POINT in U. There there exists an ANALYTIC FUNCTION f on U whose zero set is precisely aj/C8/C9 (Krantz 1999, p. 111). See also GAUSSIAN INTEGER ,H YPERCOMPLEX NUM- BER,PEIRCE’S THEOREM References Krantz, S. G. "Weierstrass’s Theorem" §8.3.2 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 111, 1999. WeierstrassSigma WEIERSTRASS SIGMA FUNCTION WeierstrassZeta WEIERSTRASS ZETA FUNCTIONWeighing n weighings are SUFFICIENT to find a bad COIN among 3n /C281 ðÞ =2 COINS (Steinhaus 1983, p. 61). vos Savant (1993) gives an algorithm for finding a bad ball among 12 balls in three weighings (which, in addi- tion, determines if the bad ball is heavier or lighter than the other 11), and Steinhaus (1983, pp. 58 /C1/1) gives an algorithm for 13 balls. Bachet’s weights problem asks for the minimum number of weights (which can be placed in either pan of a two-arm balance) required to weigh any integral number of pounds from 1 to 40 (Steinhaus 1983, p. 52). The solution is 1, 3, 9, and 27: 1, 2 /C30 /C281 /C273; 3, 4 /C301 /C273; 5 /C30/C281 /C283 /C279; 6 /C30/C283 /C279 ; 7 /C301 /C28 3 /C279 ; 8 /C30/C281 /C279; 9, 10 /C301 /C279; 11 /C30/C281 /C273 /C279 ; 12 /C30 3 /C279 ; 13 /C301 /C273 /C279; 14 /C30/C281 /C283 /C289 /C2727; 15 /C30/C283 /C28 9/C2727;16/C301/C283/C289/C2727;17/C30/C281/C289/C2727;and so on. See also GOLOMB RULER ,PERFECT DIFFERENCE SET, SORTING ,THREE JUG PROBLEM References Bachet, C. G. Problem 5, Appendix in Proble `mes plaisants et de´lectables, 2nd ed. p. 215, 1624. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 50 /C1/2, 1987. Bellman, R. and Gluss, B. "On Various Versions of the Defective Coin Problem." Information and Control 4, 118/C1/ 31, 1961. Descartes, B. Eureka, No. 13, Oct. 1950. Dyson, F. J. "The Problem of the Pennies." Math. Gaz. 30, 231/C1/34, 1946. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 29 /C1/3 and 106 /C1/09, 1984. Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 52 /C1/5, 1942. O’Beirne, T. H. Chs. 2 and 3 in Puzzles and Paradoxes. Oxford, England: Oxford University Press, 1965. Pappas, T. "Counterfeit Coin Puzzle." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 181, 1989. Smith, C. A. B. "The Counterfeit Coin Problem." Math. Gaz. 31,3 1/C1/9, 1947. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. Strong, C. L. "The Amateur Scientist." Sci. Amer. , May 1955. Tartaglia. Book 1, Ch. 16, §32 in Trattato de’ numeri e misure, Vol. 2. Venice, 1556. Tweedle, M. C. K. Math. Gaz. 23, 278/C1/82, 1938. vos Savant, M. The World’s Most Famous Math Problem. New York: St. Martin’s Press, pp. 39 /C1/2, 1993. Weight The word weight has many uses in mathematics. It can refer to a function w(x) (also called a WEIGHTING FUNCTION orWEIGHT FUNCTION ) used to normalize ORTHOGONAL FUNCTIONS . It can also be used to indicate one of a set of a multiplicative constantsplaced in front of terms in a MOVING AVERAGE , NEWTON- COTES FORMULAS , edge or vertex of a GRAPH or TREE , etc. It also refers to the power k in the multiplicative factor c t /C27d ðÞkdefining a MODULAR FORM . The weight of a TREE at a point u is the maximum number of edges in any BRANCH at u (Harary 1994, p. 35). See also MODULAR FORM,M OVING AVERAGE ,N EW- TON-COTES FORMULAS ,W EIGHTED TREE,W EIGHTING FUNCTION References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 35, 1994. Weight (Lie Algebra) Consider a collection of DIAGONAL MATRICES H1 ;...;Hk ; which SPAN a subspace h: Then the ith EIGENVALUE , i.e., the ith entry along the diagonal, is a LINEAR FUNCTIONAL on h; and is called a weight. The general setting for weights occurs in a REPRE- SENTATION of a SEMISIMPLE LIE ALGEBRA , in which case the CARTAN SUBALGEBRA h is ABELIAN and can be put into diagonal form. For example, consider the standard representation of the SPECIAL LINEAR LIE ALGEBRA sl3(C)onC3 : Then H1 /C30100 0 /C2810 0002 435 (1) and H 2 /C30100 0 /C2810 0012 435 (2) span the C ARTAN SUBALGEBRA h: There are three weights, a1hij/CQ/C1 /C30h11 (3) a2hij/CQ/C1 /C30h22 (4) and a3hij/CQ/C1 /C30h33 ; (5) corresponding to the decomposition of C3 /C30 e1hi/C154 e2hi/C154 e3hi (6) into its eigenspaces. Note that a1 /C27 a2 /C27 a3 /C300 ; be- cause the matrices have zero TRACE . The eigenvectors e1 ; e2 ;e3are called WEIGHT VECTORS , and the corre- sponding eigenspaces are called WEIGHT SPACES . In the important special case of the ADJOINT REPRE- SENTATION of a SEMISIMPLE LIE ALGEBRA , the weights are called ROOTS and the WEIGHT SPACE is called the ROOT SPACE . The roots generate a DISCRETE LATTICE , called the ROOT LATTICE , in the DUAL SPACE h+: The set of all possible weights forms a WEIGHT LATTICE ,which contains the ROOT LATTICE . The REPRESENTA- TIONS of g can be classified using the WEIGHT LATTICE . See also CARTAN MATRIX ,LIE ALGEBRA ,ROOT (LIE ALGEBRA ), ROOT SYSTEM ,SEMISIMPLE LIE ALGEBRA , WEIGHT (LIE ALGEBRA ), WEYL CHAMBER ,W EYL GROUP References Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991. Jacobson, N. Lie Algebras. New York: Dover, 1979. Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkha ¨user, 1996. Weight Function WEIGHTING FUNCTION Weighted Graph A TREE in which each branch is given a numerical WEIGHT . A weighted graph is therefore a special type of LABELED GRAPH in which the labels are numbers (which are usually taken to be positive). See also LABELED GRAPH ,T AYLOR’S CONDITION , WEIGHTED TREE Weighted Inversion Statistic A STATISTIC w on the SYMMETRIC GROUP Sn is called a weighted inversion statistic if there exists an UPPER TRIANGULAR MATRIX W /C30 wij/CQ/C1 such that w( s) /C30X i Bjxsi > sj/CQ/C1 wij ; where x is the CHARACTERISTIC FUNCTION . The inversion count (/wij /C301 for i Bj) defined by Cramer (1750) and the major index (/wi ;i /C271 /C30i; wij /C30 0 otherwise) defined by MacMahon (1913) are both weighted inversion statistics (Degenhardt and Milne). See also INVERSION STATISTIC ,SYMMETRIC GROUP References Cramer, G. "Intr. a `l’analyse de lignes courbes alge ´briques." Geneva, 657 /C1/59, 1750. Degenhardt, S. L. and Milne, S. C. "Weighted Inversion Statistics and Their Symmetry Groups." Preprint. MacMahon, P. A. "The Indices of Permutations." Amer. J. Math. 35, 281/C1/22, 1913. Weighted Tree ATREE to whose nodes and/or edges labels (usually number) are assigned. The word "weight" also has a more specific meaning when applied to trees, namely the weight of a TREE at a point u is the maximum number of edges in any BRANCH at u (Harary 1994, p. 35), as illustrated above. A point having minimal weight for the tree is called a CENTROID POINT , and the TREE CENTROID is the set of all CENTROID POINTS . See also CENTROID POINT ,LABELED GRAPH ,TAYLOR’S CONDITION ,TREE,TREE CENTROID ,WEIGHTED GRAPH References Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994. Weisstein, E. W. "Graphs." MATHEMATICA NOTEBOOK GRAPHS.M . Weighting Function A function w(x) used to normalize ORTHONORMAL FUNCTIONS g fn(x) ½/C1382w(x) dx /C30Nn : See also WEIGHT Weil-Brezin Map ZAK TRANSFORM Weill’s Theorem Given the INCIRCLE and CIRCUMCIRCLE of a BICENTRIC POLYGON of n sides, the centroid of the tangent points on the INCIRCLE is a fixed point independent of theparticular polygon. More generally, the LOCUS of the centroid of any number of the n points is a CIRCLE (Casey 1888). See also BICENTRIC POLYGON ,PONCELET’S PORISM References Casey, J. Quart. J. Pure Appl. Math. 5, 44, 1862. Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 164, 1888. Weill. Liouville’s J. (Ser. 3) 4, 270, 1878. Weingarten Equations The Weingarten equations express the derivatives of the NORMAL to a surface using derivatives of the position vector. Let x : U 0 R3 be a REGULAR PATCH , then the SHAPE OPERATOR S of x is given in terms of the basis xu ;xv fg by /C28S xuðÞ/C30Nu /C30fF /C28 eG EG /C28 F2 xu /C27eF /C28 fE EG /C28 F2 xv (1) /C28S xvðÞ/C30Nv /C30gF /C28 fG EG /C28 F2 xu /C27fF /C28 gE EG /C28 F2 xv ; (2) where N is the NORMAL VECTOR , E, F, and G the coefficients of the first FUNDAMENTAL FORM ds2 /C30Edu2 /C272Fdudv /C27Gdv2 ; (3) ande,f, and gthe coefficients of the second FUNDA- MENTAL FORM given by e/C30/C28Nu/C215xu/C30N/C215xuu (4) f/C30/C28Nv/C215xu/C30N/C215xuv /C30Nvu/C215xvu/C30/C28Nu/C215xv (5) g/C30/C28Nv/C215xv/C30N/C215xvv (6) See also FUNDAMENTAL FORMS ,SHAPE OPERATOR References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 369 /C1/71, 1997. Weingarten Map SHAPE OPERATOR Weird Number A number which is ABUNDANT without being SEMI- PERFECT .(A SEMIPERFECT NUMBER is the sum of any set of its own DIVISORS .) The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ... (Sloane’s A006037). No ODD weird numbers are known, but an infinite number of weird numbers are known to exist. The SEQUENCE of weird numbers has POSITIVE SCHNIRELMANN DENSITY . See also ABUNDANT NUMBER ,SCHNIRELMANN DEN- SITY,SEMIPERFECT NUMBER References Benkoski, S. "Are All Weird Numbers Even?" Amer. Math. Monthly 79, 774, 1972. Benkoski, S. J. and Erdos, P. "On Weird and Pseudoperfect Numbers." Math. Comput. 28, 617 /C1/23, 1974. Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45 /C1/3, 1994. Sloane, N. J. A. Sequences A006037/M5339 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Welch Apodization Function The APODIZATION FUNCTION A(x) /C301 /C28x2 a2 : Its FULL WIDTH AT HALF MAXIMUM isffiffiffi 2p a : Its INSTRU- MENT FUNCTION is I(k) /C302affiffiffiffiffiffi 2ppJ3=2(2pka) 2pkaðÞ3 =2 /C30asin(2pka) /C28 2 pak cos(2 pak) 2a3k3 p3 ; where Jn(z)isaB ESSEL FUNCTION OF THE FIRST KIND . It has a width of 1.59044, a maximum of4 3 ; maximum NEGATIVE sidelobe of /C280:0861713 times the peak, and maximum POSITIVE sidelobe of 0.356044 times the peak. See also APODIZATION FUNCTION ,INSTRUMENT FUNC- TION References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art ofScientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 547, 1992. Well Defined An expression is called "well defined" (or UNAMBIG- UOUS ) if its definition assigns it a unique interpreta- tion or value. Otherwise, the expression is said to not be well defined or to be AMBIGUOUS . For example, the expression abc (the PRODUCT )is well defined if a, b, and c are integers. Because integers are ASSOCIATIVE , abc has the same value whether it is interpreted to mean (ab)c or a(bc): However, if a, b, and c are CAYLEY NUMBERS , then the expression abc is not well defined, since CAYLEY NUMBER are not, in general, ASSOCIATIVE , so that the two interpretations (ab)c and a(bc) can be different. Sometimes, ambiguities are implicitly resolved by notational convention. For example, the conventional interpretation of afflbfflc /C30abcis a bcðÞ; never ab/CQ/C1c; so that the expression afflbfflc is well defined even though exponentiation is nonassociative. The term "well defined" also has a technical meaning in field of PARTIAL DIFFERENTIAL EQUATIONS . A solu- tion to a PARTIAL DIFFERENTIAL EQUATION that is a continuous function of its values on the boundary is said to be well defined. Otherwise, a solution is called ILL DEFINED . See also AMBIGUOUS ,ILL DEFINED ,UNDEFINED Well Order WELL ORDERED SET Well Ordered Set A TOTALLY ORDERED SET A;5ðÞ is said to be well ordered IFF every nonempty SUBSET of A has a least element (Ciesielski 1997, p. 38; Moore 1982, p. 2; Rubin 1967, p. 159; Suppes 1972, p. 75). Every finite TOTALLY ORDERED SET is well ordered. The set of integers Z, which has no least element, is an example of a set that is not well ordered. An ORDINAL NUMBER is the ORDER TYPE of a well ordered set. See also AXIOM OF CHOICE ,H ILBERT’S PROBLEMS , INITIAL SEGMENT ,MONOMIAL ORDER ,ORDINAL NUM- BER,ORDER TYPE,SUBSET ,W ELL ORDERING PRINCI- PLE References Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. Ferreiro ´s, J. "Well-Ordered Sets." §8.4 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkha ¨user, pp. 274 /C1/ 78, 1999. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer-Verlag, 1982. Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967. Se´roul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. 22 /C1/3, 2000. Suppes, P. Axiomatic Set Theory. New York: Dover, 1972. Well Ordering Principle Every nonempty set of POSITIVE INTEGERS contains a smallest member. See also AXIOM OF CHOICE ,W ELL ORDERED SET References Apostol, T. M. "The Well-Ordering Principle." §I 4.3 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blais- dell, pp. 34 /C1/5, 1967. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 149, 1993. Well-Poised A GENERALIZED HYPERGEOMETRIC FUNCTION pFqa1 ;a2 ;...;ap b1 ; b2 ;...; bq;z/C2Q/C21 is said to be well-poised if p /C30q /C271 and 1 /C27a1 /C30 b1 /C27a2 /C30.../C30 bq /C27ap /C271 See also GENERALIZED HYPERGEOMETRIC FUNCTION , K-BALANCED ,NEARLY- POISED ,SAALSCHU ¨ TZIAN References Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, p. 11, 1935. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 43, 1998. Whipple, F. J. W. "On Well-Poised Series, Generalized Hypergeometric Series Having Parameters in Pairs, Each Pair with the Same Sum." Proc. London Math. Soc. 24, 247 /C1/63, 1926. Whipple, F. J. W. "Well-Poised Series and Other General- ized Hypergeometric Series." Proc. London Math. Soc. Ser. 2 25, 525 /C1/44, 1926. Werner Formulas 2 sin a cos b /C30sin( a/C28 b) /C27sin( a/C27 b) (1) 2 cos a cos b /C30cos(a/C28 b) /C27cos(a/C27 b) (2) 2 cos a sin b /C30sin( a/C27 b) /C28sin( a/C28 b) (3) 2 sin a sin b /C30cos(a/C28 b) /C28cos(a/C27 b) (4) See also TRIGONOMETRIC ADDITION FORMULAS Werner Projection A nonconformal, equal-area projection which is a special case of the BONNE PROJECTION where one ofthe poles is taken as the standard parallel. Because of its heart shape, this projection is sometimes also called "cordiform." See also BONNE PROJECTION ,MAP PROJECTION References MathWorks. "Mapping Toolbox: Bonne Projection." http:// www.mathworks.com/access/helpdesk/help/toolbox/map/ wernerprojection.shtml. Weyl Character Formula References Hsiang, W. Y. "Weyl Character Formula and the Classifica- tion of Complex Irreducible Representations." Lec. 4, §4in Lectures on Lie Groups. Singapore: World Scientific, pp. 74 /C1/7, 2000. Weyl Group Let L be a finite-dimensional split SEMISIMPLE LIE ALGEBRA over a FIELD of CHARACTERISTIC 0, H a splitting CARTAN SUBALGEBRA , and a weight of H in a representation of L : Then L?/C30LSa /C30l/C282 L; aðÞ ( a; a)( a) is also a weight. Furthermore, the reflections Sa with a a root, generate a group of linear transformations in H/C310called the Weyl group W of L relative to H; where H/C31/ is the CONJUGATE SPACE of H and H/C310is the Q-SPACE spanned by the roots (Jacobson 1979, pp. 112, 117, and 119). The Weyl group acts on the roots of a semisimple Lie algebra, and it is a finite group. The animations above illustrate this action for Weyl Group acting on the roots of a homotopy from one Weyl matrix to the next one (i.e., it slides the arrows from g to h) in the first two figures, while the third figure shows the Weyl Group acting on the roots of the C ARTAN MATRIX of the infinite family of semisimple lie algebras A3(cf. DYNKIN DIAGRAM ), which is the SPECIAL LINEAR LIE ALGEBRA ,sl4:/ See also CARTAN MATRIX ,D YNKIN DIAGRAM ,L IE ALGEBRA ,L IE GROUP ,M ACDONALD’S CONSTANT- TERM CONJECTURE ,R OOT (LIE ALGEBRA ), ROOT SYSTEM ,ROOT LATTICE ,SEMISIMPLE LIE ALGEBRA , WEIGHT LATTICE ,W EYL CHAMBER References Andrews, G. E. "The Macdonald Conjectures." q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 41, 1986. Huang, J.-S. "The Weyl Group." §4.5 in Lectures on Repre- sentation Theory. Singapore: World Scientific, pp. 36 /C1/8, 1999. Jacobson, N. Lie Algebras. New York: Dover, pp. 112 /C1/19 and 240 /C1/43, 1979. Weyl Reduction References Hsiang, W. Y. "Coxeter Groups, Weyl Reduction, and Weyl Formulas." Lec. 4 in Lectures on Lie Groups. Singapore: World Scientific, pp. 46 /C1/7 and 58 /C1/7, 2000. Weyl Tensor The TENSOR /Cabcd/ defined by Rabcd /C30Cabcd /C272 n /C28 2ga[cRd]b /C28gb[cRd]a/CQ/C1 /C282 (n /C28 1)(n /C28 2)Rga[cgd]b ; (1) where Rabcd is the RIEMANN TENSOR , R is the SCALAR CURVATURE , gabis the METRIC TENSOR , and Ta1...an ½/C138 denotes the ANTISYMMETRIC TENSOR part (Wald 1984, p. 40). The Weyl tensor is defined so that every CONTRAC- TION between indices gives 0. In particular, Cl mlk/C300 (2) (Weinberg 1972, p. 146). The number of independent components for a Weyl tensor in N-D for N ]3is given by CN /C301 12N(N /C271)(N /C272)(N /C283) (3) (Weinberg 1972, p. 146). For N /C303, 4, ..., this gives 0, 10, 35, 84, 168, ... (Sloane’s A052472). See also CURVATURE SCALAR ,RIEMANN TENSOR References Eisenhart, L. P. Riemannian Geometry. Princeton, NJ: Princeton University Press, 1964. Sloane, N. J. A. Sequences A052472 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972. Weyl, H. "Reine Infinitesimalgeometrie." Math. Z. 2, 384 /C1/ 11, 1918.Weyl’s Criterion A SEQUENCE x1 ; x2 ;... fg is EQUIDISTRIBUTED IFF lim N 0/C121 NX nBNe2pimxn /C300 for each m /C301, 2, .... A consequence of this result is that the sequence frac( nx) fg is dense and EQUIDIS- TRIBUTED in the interval 0 ;1½/C138 for irrational x, where n /C301, 2, ... and frac( x) is the FRACTIONAL PART of x (Finch). See also EQUIDISTRIBUTED SEQUENCE ,RAMANUJAN’S SUM References Cassels, J. W. S. An Introduction to Diophantine Analysis. Cambridge, England: Cambridge University Press, 1965. Finch, S. "Powers of 3/2 Modulo One." http://www.mathsoft.- com/asolve/pwrs32/pwrs32.html. Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, p. 226, 1974. Po´lya, G. and Szego, G. Problems and Theorems in Analysis I. New York: Springer-Verlag, 1972. Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., pp. 79 /C1/0, 1999. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison-Wesley, pp. 155 /C1/56 and 254, 1991. Weyl’s Denominator Formula See also ROOT SYSTEM References Simpson, T. "Three Generalizations of Weyl’s Denominator Formula." Electronic J. Combinatorics 3, R12 1 /C1/1, 1996. http://www.combinatorics.org/Volume_3/volu- me3.html#R12. Weyrich’s Formula For r and x real, with 0 5argffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 /C28r2p/C1;/C17 Bp and 0 5 arg k Bp; 1 2ig/C12 /C28/C12H(1) 0rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 /C28r2p/C1;/C17 eirxdr /C30eikffiffiffiffiffiffiffiffiffi r2 /C27x2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 /C27 x2p ; where H(1) 0(x)isaH ANKEL FUNCTION OF THE FIRST KIND . See also HANKEL FUNCTION OF THE FIRST KIND References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1474, 1980. W-Function LAMBERT’S W-FUNCTION Wheat and Chessboard Problem Let one grain of wheat be placed on the first square of a CHESSBOARD , two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an 8 /C298 CHESSBOARD ? Since this is a GEOMETRIC SERIES , the answer for n squares is Xn/C281 i/C3002i /C302n /C281; aM ERSENNE NUMBER . Plugging in n ¼ 8 /C298 ¼ 84 then gives 264 /C281/C28/C3018446744073709551615 :/ See also MERSENNE NUMBER References Pappas, T. "The Wheat & Chessboard." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 17, 1989. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 23 /C1/4, 1999. Wheel ARISTOTLE’S WHEEL PARADOX ,B ENHAM’S WHEEL , WHEEL GRAPH Wheel Graph A GRAPH Wnof order n which contains a CYCLE of order n /C281 ; and for which every NODE in the cycle is connected to one other NODE (which is known as the HUB). The edges of a wheel which include the HUB are called spokes (Skiena 1990, p. 146). The wheel Wn can be defined as the graph K1 þ Cn/C281 ; where K1is the (trivial) COMPLETE GRAPH on 1 node and Cnis the CYCLE GRAPH . Wheel graphs can be constructed using Wheel [n] in the Mathematica add-on package Dis- creteMath‘Combinatorica‘ (which can be loaded with the command BBDiscreteMath‘ ). In a wheel graph, the HUB has DEGREE n /C281; and other nodes have degree 3. Wheel graphs are 3- connected. W4 /C30K4 ; where K4 is the COMPLETE GRAPH of order four. The CHROMATIC NUMBER of Wn is x WnðÞ/C304 for n odd 3 for n even :/C2; See also COMPLETE GRAPH ,GEAR GRAPH ,HUB,W EB GRAPHReferences Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 46, 1994. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 148, 1986. Skiena, S. "Cycles, Stars, and Wheels." §4.2.3 in Implement- ing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 91 and 144 /C1/47, 1990. Wheel Paradox ARISTOTLE’S WHEEL PARADOX Whewell Equation An INTRINSIC EQUATION which expresses a curve in terms of its ARC LENGTH s and TANGENTIAL ANGLE f:/ See also ARC LENGTH ,CESA` RO EQUATION ,INTRINSIC EQUATION ,NATURAL EQUATION ,TANGENTIAL ANGLE References Yates, R. C. "Intrinsic Equations." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 123 /C1/26, 1952. Whipple’s Identity Whipple derived a great many identities for GENERAL- IZED HYPERGEOMETRIC FUNCTIONS , many of which are consequently known as Whipple’s identities (trans- formations, etc.). Among Whipple’s identities include 3F2a ;1 /C28a ;c e ;1 /C272c /C28e;1/C2Q/C21 /C3021/C282c pG(e) G(1 /C27 2c /C28 e) G1 2(a /C27 e)hi G12(a /C27 1 /C27 2c /C28 e)hi /C291 G12(1 /C28 a /C27 e)hi G12(2 /C27 2c /C28 a /C28 e)hi (Bailey 1935, p. 15; Koepf 1998, p. 32), where 3F2(a ;b;c;d;e;z)isa GENERALIZED HYPERGEOMETRIC FUNCTION and G(z)isa GAMMA FUNCTION , and 6F5a; 1 /C271 2a ; b ; c; d; e 12a ; 1 /C27a /C28b ; 1 /C28a /C27c; 1 /C27a /C28d; 1 /C27a /C28e;1"# /C30G(1 /C27 a /C28 d)G(1 /C27 a /C28 e) G(1 /C27 a)G(1 /C27 a /C28 d /C28 e)3 F21 /C27a /C28b /C28c; d; e; 1/C27a/C28b;1/C27a/C28c/C2Q/C21 (Bailey 1935, p. 28). See also GENERALIZED HYPERGEOMETRIC FUNCTION , WATSON’S THEOREM References Bailey, W. N. "Whipple’s Theorem on the Sum of a /3F2/."§3.4 inGeneralised Hypergeometric Series. Cambridge, Eng- land: Cambridge University Press, p. 16, 1935. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. Whipple, F. J. W. "Well-Poised Series and Other General- ized Hypergeometric Series." Proc. London Math. Soc. Ser. 225, 525/C1/44, 1926. Whipple’s Transformation 7F6a;1 /C271 2a;b ;c ;d;e ;/C28m 12a ;1 /C27a /C28b; 1 /C27a /C28c ; 1 /C27a /C28d;1 /C27a /C28e ;1 /C27a /C27m2 643 75 /C30(1 /C27 a)m(1 /C27 a /C28 d /C28 e)m (1 /C27 a /C28 d)m(1 /C27 a /C28 e)m /C24F31 /C27a /C28b /C28c ;d;e ;/C28m 1 /C27a /C28b;1 /C27a /C28c; d /C27e /C28a /C28m/C2Q/C21 ; where7F6and4F3are GENERALIZED HYPERGEO- METRIC FUNCTIONS and G(z) is the GAMMA FUNCTION . Another transformation due to Whipple (1926) is given by 4F3a ;b;/C28z ;/C28n u;v ;w;1/C2Q/C21 /C30G(u /C27 z /C27 n) G(w /C27 z /C27 n) G(v) G(w) G(v /C27 z) G(v /C27 n) G(w /C27 n) G(w /C27 z) /C294F3u /C28a;u /C28b;/C28z ;/C28n 1 /C28v /C28z /C28n ;1 /C28w /C28z /C28n;u;1/C2Q/C21 (1) for one of z and n a NONNEGATIVE INTEGER (Andrews and Burge 1993). See also GENERALIZED HYPERGEOMETRIC FUNCTION , WATSON- WHIPPLE TRANSFORMATION References Andrews, G. E. and Burge, W. H. "Determinant Identities." Pacific J. Math. 158,1/C1/4, 1993. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, pp. 25 and 29, 1935. Whipple, F. J. W. "Well-Poised Series and Other General- ized Hypergeometric Series." Proc. London Math. Soc. Ser. 2 25, 525 /C1/44, 1926. Whipple, F. J. W. "On Well-Poised Series, Generalized Hypergeometric Series Having Parameters in Pairs, Each Pair with the Same Sum." Proc. London Math. Soc. 24, 247 /C1/63, 1926. Whipple, F. J. W. "A Fundamental Relation Between Gen- eralized Hypergeometric Series." Proc. London Math. Soc. 26, 257 /C1/72, 1927. Whirl Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the n mice in the MICE PROBLEM , and form n LOGARITHMIC SPIRALS .See also DAISY,D ERIVED POLYGON ,L OGARITHMIC SPIRAL ,MICE PROBLEM ,SWIRL References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, p. 66, 1991. Pappas, T. "Spider & Spirals." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 228, 1989. Weisstein, E. W. "Fractals." MATHEMATICA NOTEBOOK FRAC- TAL.M . Weisstein, E. W. "Mice Problem." MATHEMATICA NOTEBOOK MICEPROBLEM.M . Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 201 /C1/02, 1991. Whisker Plot BOX-AND- WHISKER PLOT Whitehead Double The SATELLITE KNOT of an UNKNOT twisted inside a TORUS . See also SATELLITE KNOT,TORUS ,UNKNOT References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 115 /C1/16, 1994. Whitehead Link The LINK 05 /C1/2 /C1/1, illustrated above, with BRAID WORD s2 1 s22 s/C281 1s/C282 2and JONES POLYNOMIAL V(t) /C30t/C283=2 /C281 /C27t /C282t2 /C27t3 /C282t4 /C27t5/CQ/C1 The Whitehead link has LINKING NUMBER 0. It was discovered by Whitehead in 1934 (Whitehead 1962, pp. 21 /C1/0) as a counterexample to a piece of an attempted proof of the P OINCARE ´CONJECTURE (Mil- nor). See also POINCARE ´ CONJECTURE . References Milnor, J. "The Poincare ´Conjecture." http://www.clay- math.org/prize_problems/poincare.pdf. Whitehead, J. H. C. Mathematical Works, Vol. 2. London: Pergamon Press, 1962. Whitehead Manifold An open 3- MANIFOLD which is simply connected but is topologically distinct from Euclidean 3-space. References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 82, 1976. Whitehead’s Theorem MAPS between CW -COMPLEXES that induce ISOMORPH- ISMS on all HOMOTOPY GROUPS are actually HOMOTOPY equivalences. See also CW -COMPLEX ,HOMOTOPY GROUP ,ISOMORPH- ISM Whitney Singularity PINCH POINT Whitney Sum An operation that takes two VECTOR BUNDLES over a fixed SPACE and produces a new VECTOR BUNDLE over the same SPACE .IfE1and E2are VECTOR BUNDLES over B, then the Whitney sum E1 /C154E2 is the VECTOR BUNDLE over B such that each FIBER over B is naturally the DIRECT SUM of the E1and E2FIBERS over B. The Whitney sum is therefore the FIBER for FIBER DIRECT SUM of the two BUNDLES E1and E2 : An easy formal definition of the Whitney sum is that E1 /C154E2 is the pull-back BUNDLE of the diagonal map from B toB/C29B;where the BUNDLE over B/C29BisE1/C29E2:/ See also BUNDLE ,FIBER,VECTOR BUNDLE Whitney Umbrella A surface which can be interpreted as a self-inter- secting RECTANGLE in 3-D. It is given by the para- metric equations x/C30uv (1) y/C30u (2) z¼v2ð3Þ foru;v/C23/C281;1 ½/C138 :The center of the "plus" shape which is the end of the line of self-intersection is a PINCHPOINT . The coefficients of the FIRST FUNDAMENTAL FORM are E/C301/C27v2(4) F/C30uv (5) G¼u2þ4v2(6) and the SECOND FUNDAMENTAL FORM are e¼0 (7) f¼2uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2þ4v2þ4v4p (8) g/C30/C282uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2/C274v2/C274v4p (9) giving AREA ELEMENT dA/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2/C274v21/C27v2 ðÞp (10) and G AUSSIAN CURVATURE and MEAN CURVATURE K/C30/C284v2 u2/C274v2/C274v4 ðÞ2(11) H/C30/C28u1/C273v2ðÞ u2/C274v2/C274v4 ðÞ3=2 (12) References Francis, G. K. A Topological Picturebook. New York: Springer-Verlag, pp. 8 /C1/, 1987. Gray, A. "The Whitney Umbrella." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.Boca Raton, FL: CRC Press, pp. 311 and 401 /C1/02, 1997. Whitney-Graustein Theorem A 1937 theorem which classified planar regular closed curves up to regular HOMOTOPY by their WIND- ING NUMBERS . In his thesis, S. Smale generalized this result to regular closed curves on an n-MANIFOLD . Whitney-Mikhlin Extension Constants N.B. A detailed online essay by S. Finch was thestarting point for this entry. Let B n(r) be the n-D closed BALL ofRADIUS r/C211 centered at the ORIGIN . A function which is defined on B(r) is called an extension to B(r) of a function f defined on B(1) if F(x)/C30f(x)/C214x/C23B(1) (1) Given 2 B ANACH SPACES of functions defined on B(1) andB(r);find the extension operator from one to the other of minimal norm. Mikhlin (1986) found the best constants xsuch that this condition, corresponding to the Sobolev W(1;2) integral norm, is satisfied, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gB(1)f(x)½/C1382/C27Xn j/C301@f @xj !22 435dxvuuut 5xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gB(r)F(x) ½/C1382/C27Xn j/C301@F @xj !22 435dxvuuut : (2) /x(1;r)/C301:Let n/C301 2(n/C282); (3) then for n/C212, x(n;r)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27In(1) In/C271(1)In(r)Kn/C271(1)/C27Kn(r)In/C271(1) In(r)Kn(1)/C28Kn(r)In(1)s ; (4) where In(z)i sa MODIFIED BESSEL FUNCTION OF THE FIRST KIND andKn(z)i sa MODIFIED BESSEL FUNCTION OF THE SECOND KIND . For n/C302, x(2;r)/C30max/C2;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27 In(1) In/C271(1)In(r)Kn/C271(1)/C27Kn(r)In/C271(1) In(r)Kn(1)/C28Kn(r)In(1)s ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27 I1(1) I1(1)/C27I2(1)/C2Q 1/C27I1(r)K0(1)/C27K1(r)I0(1) I1(r)K1(1)/C28K1(r)I1(1)/C21s/C27 ;(5) Forr0/C12; x(n;/C12)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/C27 In(1) In/C271(1)Kn(1) Kn(1)s ; (6) which is bounded by n/C281Bx(n;/C12)Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n/C281)2/C274q (7) For ODD n, the RECURRENCE RELATIONS ak/C271/C30ak/C281/C28(2k/C281)ak (8) bk/C271/C30bk/C281/C28(2k/C281)bk (9) with a0/C30e/C27e/C281(10) a1/C30e/C28e/C281(11) b0/C30e/C281(12) b1/C30e/C281(13) where Eis the constant 2.71828..., give x(2k/C271;/C12)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C27ak ak/C271bk/C271 bks : (14) The first few are x(3;/C12)/C30e (15)x(5;/C12)/C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 e2/C287s (16) x(7;/C12)/C30ffiffiffi 2 7s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 37/C285e2s (17) x(9;/C12)/C301ffiffiffiffiffiffi 37pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 18e2/C28133s (18) x(11;/C12)/C301ffiffiffiffiffiffiffiffi133pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 2431/C28329e2s (19) x(13;/C12)/C30ffiffiffiffiffiffiffiffiffiffiffi 2 2431s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 3655 e2/C2827007s : (20) Similar formulas can be given for even nin terms of I0(1);I1(1);K0(1);K1(1):/ References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/mkhln/mkhln.html. Mikhlin, S. G. Constants in Some Inequalities of Analysis. New York: Wiley, 1986. Whittaker Differential Equation d2u dz2/C27du dz/C27k z/C271 4/C28m2 z2 ! u/C300 (1) Letu/C13e/C28z=2Wk;m(z);where Wk;m(z) denotes a W HIT- TAKER FUNCTION . Then (1) becomes d dz/C2812e/C28z=2W/C27e/C28z=2W?/C1;/C17 /C27/C2812e/C28z=2W/C27e/C28z=2W?/C1;/C17 /C27k z/C271 4/C28m2 z2 ! e/C28z=2W/C300: (2) Rearranging, 14e/C28z=2W/C2812e/C28z=2W?/C2812e/C28z=2W?/C27e/C28z=2Wƒ/C1;/C17 p /C27/C281 2e/C28z=2W/C27e/C28z=2W?/C1;/C17 /C27k z/C2714/C28m2 z2 ! e/C28z=2W/C300 (3) /C281 4e/C28z=2W/C27e/C28z=2Wƒ/C27k z/C2714/C28m2 z2 ! e/C28z=2W/C300;(4) so Wƒ/C27/C281 4/C27k z/C271 4/C28m2 z2 ! W/C300; (5) where W?/C13dW=dz(Abramowitz and Stegun 1972, p. 505; Zwillinger 1997, p. 128). The solutions are known as W HITTAKER FUNCTIONS . Replacing W(z)b y y(x);the solutions can also be written in the form y /C30e/C28x =2xm/C271 =2[C1U(1 2 /C28k /C27m;2m /C271;xÞ /C27C2L2m /C281=2/C27k /C28m(x) /C138; (6) where U(a ;b;z)isa CONFLUENT HYPERGEOMETRIC FUNCTION OF THE SECOND KIND and La n(x)isa generalized LAGUERRE POLYNOMIAL . See also WHITTAKER FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 505, 1972. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997. Whittaker Function Solutions to the WHITTAKER DIFFERENTIAL EQUATION . The linearly independent solutions are Mk;m(z) /C13z1 =2 /C27me /C28z=2 /C29/C2Q 1 /C271 2 /C27 m /C28 k 1! 2m /C27 1 ðÞz /C2712 /C27 m /C28 k/C1;/C17 32 /C27 m /C28 k/C1;/C17 2! 2m /C27 1 ðÞ 2m /C27 2 ðÞz2 /C27.../C21 ; (1) and Mk ;/C28m(z); where Mk;m(z)isa CONFLUENT HYPER- GEOMETRIC FUNCTION . In terms of CONFLUENT HYPER- GEOMETRIC FUNCTIONS , the Whittaker functions are Mk ;m(z) /C30e /C28z=2zm/C271 =2 1F11 2 /C27m /C28k;1 /C272m;z/C1;/C17 (2) Wk ;m(z) /C30e /C28z=2zm/C271 =2U1 2 /C27m /C28k; 1 /C272m;z/C1;/C17 (3) (Abramowitz and Stegun 1972, p. 505; Whittaker and Watson 1990, pp. 339 /C1/51). However, the CONFLUENT HYPERGEOMETRIC FUNCTION disappears when 2m is an INTEGER , so Whittaker functions are often defined instead. The Whittaker functions are related to the PARABOLIC CYLINDER FUNCTIONS . When argz jjB3 p=2 and 2m is not an INTEGER , Wk;mzðÞ/C30G/C282m ðÞ G12 /C28 m /C28 k/C1;/C17 Mk ;m(z) /C27G 2mðÞ G1 2 /C27 m /C28 k/C1;/C17 Mk ;/C28m(z): (4) When arg(/C28z) jj B3p=2 and 2m is not an INTEGER , W/C28k ;m /C28zðÞ/C30G/C282m ðÞ G1 2 /C28 m /C28 k/C1;/C17 M/C28k ;m(/C28z) /C27G 2mðÞ G12 /C27 m /C27 k/C1;/C17 M/C28k ;/C28m(/C28z) : (5)Whittaker functions satisfy the RECURRENCE RELA- TIONS Wk ;m(z) /C30z1 =2Wk/C281 =2 ;m/C281=2(z) /C2712 /C28k /C27m/C1;/C17 Wk /C281;m(z) (6) Wk ;m(z) /C30z1 =2Wk/C281 =2 ;m/C271=2(z) /C2712 /C28k /C28m/C1;/C17 Wk /C281;m(z) (7) zW ?k ;m(z) /C30 k /C281 2z/C1;/C17 Wk ;m(z) /C28 m2 /C28 k /C281 2/C1;/C172/C2Q/C21 Wk /C281 ;m(z) : (8) See also CONFLUENT HYPERGEOMETRIC FUNCTION , KUMMER’S FORMULAS ,PEARSON- CUNNINGHAM FUNC- TION ,SCHLO ¨ MILCH’S FUNCTION ,SONINE POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.). "Confluent Hy- pergeometric Functions." Ch. 13 in Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 503 /C1/15, 1972. Iyanaga, S. and Kawada, Y. (Eds.). "Whittaker Functions." Appendix A, Table 19.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1469 /C1/471, 1980. Meijer, C. S. "U¨ ber die Integraldarstellungen der Whitta- kerschen Funktion Wk;m(z) und der Hankelschen und Besselschen Funktionen." Nieuw Arch. Wisk. 18,35/C1/7, 1936. Whittaker, E. T. Bull. Amer. Math. Soc. 10, 125 /C1/34, 1904. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Whittaker-Hill Differential Equation The second-order ORDINARY DIFFERENTIAL EQUATION yƒ/C27A/C27Bcos(2 x)/C27Ccos(4 x) ½/C138 y/C300: See also HILL’S DIFFERENTIAL EQUATION ,M ATHIEU DIFFERENTIAL EQUATION References Urwin, K. M. and Arscott, F. M. "Theory of the Whittaker- Hill Equation." Proc. Roy. Soc. Edinburgh 69,2 8/C1/4, 1970. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997. Whole Number One of the numbers 1, 2, 3, ... (Sloane’s A000027), also called the COUNTING NUMBERS orNATURAL NUMBERS . 0 is sometimes included in the list of "whole" numbers(Bourbaki 1968, Halmos 1974), but there seems to beno general agreement. Some authors also interpret "whole number" to mean "a number having FRAC- TIONAL PART of zero," making the whole numbers equivalent to the integers. Due to lack of standard terminology, the following terms are recommended in preference to "COUNTING NUMBER ," "NATURAL NUMBER ," and "whole number." set name symbol ..., /C282, /C281, 0, 1, 2, ... INTEGERS Z 1, 2, 3, 4, ... POSITIVE INTEGERS Z/C27 0, 1, 2, 3, 4, ... NONNEGATIVE INTE- GERSZ* 0, /C281, /C282, /C283, /C284, ...NONPOSITIVE INTEGERS /C281, /C282, /C283, /C284, ... NEGATIVE INTEGERS Z/C28 See also COUNTING NUMBER ,F RACTIONAL PART, INTEGER ,N,N ATURAL NUMBER ,Z,Z /C27,Z/C27,Z* References Bourbaki, N. Elements of Mathematics: Theory of Sets. Paris, France: Hermann, 1968. Halmos, P. R. Naive Set Theory. New York: Springer- Verlag, 1974. Sloane, N. J. A. Sequences A000027/M0472 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Width (Partial Order) For a PARTIAL ORDER , the size of the longest ANTIC- HAIN is called the width. See also ANTICHAIN ,L ENGTH (PARTIAL ORDER ), PARTIAL ORDER Width (Size) The width of a box is the horizontal distance from side to side (usually defined to be greater than the DEPTH , the horizontal distance from front to back). See also DEPTH (SIZE), HEIGHT References Eppstein, D. "Width, Diameter, and Geometric Inequalities." http://www.ics.uci.edu/~eppstein/junkyard/diam.html. Wiedersehen Manifold The only Wiedersehen manifolds are the standard round spheres, as was established by proof of the BLASCHKE CONJECTURE . See also BLASCHKE CONJECTURE Wieferich Prime A Wieferich prime is a PRIME p which is a solution to the CONGRUENCE equation2p /C281 /C131 mod p2/CQ/C1 : Note the similarity of this expression to the special case of FERMAT’S LITTLE THEOREM 2p /C281 /C131 mod p ðÞ ; which holds for all ODD PRIMES . However, the only Wieferich primes less than 4 /C291012 are p /C301093 and 3511 (Lehmer 1981, Crandall 1986, Crandall et al. 1997). Interestingly, one less than these numbers have suggestive periodic BINARY representations 1092 /C30100010001002 3510 /C301101101101102 : A PRIME factor p of a MERSENNE NUMBER Mq /C302q /C281 is a Wieferich prime IFF p2 j2q /C281: Therefore, MERS- ENNE PRIMES are not Wieferich primes. If the first case of FERMAT’S LAST THEOREM is false for exponent p, then p must be a Wieferich prime (Wieferich 1909). If p j2n 91 with p and n RELATIVELY PRIME , then p is a Wieferich prime IFF p2 also divides 2n 91 : The CONJECTURE that there are no three POWERFUL NUMBERS implies that there are infinitely many Wieferich primes (Granville 1986, Vardi 1991). In addition, the ABC CONJECTURE implies that there are at least C ln x Wieferich primes 5x for some constant C(Silverman 1988, Vardi 1991). See also ABC CONJECTURE ,FERMAT’S LAST THEOREM , FERMAT QUOTIENT ,M ERSENNE NUMBER ,M IRIMA- NOFF’S CONGRUENCE ,POWERFUL NUMBER References Brillhart, J.; Tonascia, J.; and Winberger, P. "On the Fermat Quotient." In Computers and Number Theory (Ed. A. O. L. Atkin and B. J. Birch). New York: Academic Press, pp. 213 /C1/22, 1971. Crandall, R. Projects in Scientific Computation. New York: Springer-Verlag, 1986. Crandall, R.; Dilcher, K; and Pomerance, C. "A search for Wieferich and Wilson Primes." Math. Comput. 66, 433/C1/ 49, 1997. Granville, A. "Powerful Numbers and Fermat’s Last Theo- rem." C. R. Math. Rep. Acad. Sci. Canada 8, 215/C1/18, 1986. Lehmer, D. H. "On Fermat’s Quotient, Base Two." Math. Comput. 36, 289/C1/90, 1981. Montgomery, P. "New Solutions of ap/C281/C131 mod p2ðÞ :/"Math. Comput. 61, 361/C1/63, 1991. Ribenboim, P. "Wieferich Primes." §5.3 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 333 /C1/46, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 116 and 157, 1993. Silverman, J. "Wieferich’s Criterion and the abc Conjecture." J. Number Th. 30, 226/C1/37, 1988. Vardi, I. "Wieferich." §5.4 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 59 /C1/2 and 96 /C1/03, 1991. Wieferich, A. "Zum letzten Fermat’schen Theorem." J. reine angew. Math. 136, 293/C1/02, 1909. Wielandt’s Theorem Let the n /C29n MATRIX A satisfy the conditions of the PERRON- FROBENIUS THEOREM and the n /C29n MATRIX C /C30cij satisfy cij/C12/C12/C12/C125a ij for i; j /C301; 2, ..., n. Then any EIGENVALUE l0of C satisfies the inequality l0jj5R with the equality sign holding only when there exists an n /C29n MATRIX D /C30 dij (where dij is the KRONECKER DELTA ) and C /C30l0 RDAD /C281 : References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000. Wiener Filter An optimal FILTER used for the removal of noise from a signal which is corrupted by the measuring process itself. See also FILTER References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Optimal (Wiener) Filtering with the FFT." §13.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 539 /C1/42, 1992. Wiener Function BROWN FUNCTION Wiener Measure The probability law on the space of continuous functions g with g(0) /C300; induced by the WIENER PROCESS . See also WIENER PROCESS References Karatsas, I. and Shreve, S. Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer-Verlag, 1997. Wiener Numbers A sequence of UNCORRELATED NUMBERS an developed by Wiener (1926 /C1/927). The numbers are constructed by beginning with f1;/C281 g; then forming the outer product with f1;/C281g to obtain f1;1 g;f1;/C281 g fg ;f/C281;1 g;f/C281;/C281 g fg fg : This row is repeated twice, and its outer product is then taken to givef1 ;1;1 g;f1;1 ;/C281 fg ; 1 ;/C281;1 g;f1;/C281 ;/C281 fg fg ; f/C281 ;1;1 g;f/C281;1 ;/C281g;f/C281 ;/C281;1g;f/C281;/C281;/C281 g fg g : This is then repeated four times. The procedure is repeated, and the result repeated eight times, and so on. The sequences from each stage are then concate- nated to form the sequence 1, /C281, 1, 1, 1, /C281, /C281, 1, /C281, /C281, 1, 1, 1, /C281, /C281, 1, /C281, /C281, .... See also UNCORRELATED NUMBERS References Papoulis, A. "The Wiener Numbers." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 258 /C1/ 59, 1962. Wiener, N. "The Spectrum of an Array and Its Applications to the Study of the Translation Properties of a Simple Class of Arithmetical Functions." J. Math. Phys. 6, 1926 /C1/ 927. Wiener Process A continuous-time stochastic process W(t) for t ]0 with W(0) /C300 and such that the increment W(t) /C28 W(s) is Gaussian with mean 0 and variance t /C28s for any 0 5s Bt; and increments for nonoverlapping time intervals are independent. Brownian motion (i.e., random walk with random step sizes) is the most common example of a Wiener process. See also ITOˆ ’S LEMMA ,R ANDOM WALK,W IENER PROCESS References Karatsas, I. and Shreve, S. Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer-Verlag, 1997. Papoulis, A. "Wiener-Le ´vy Process." §15/C1/inProbability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 292 /C1/93, 1984. Wiener Space MALLIAVIN CALCULUS ,W IENER MEASURE Wiener-Khintchine Theorem Recall the definition of the AUTOCORRELATION func- tion C(t) of a function E(t); C(t)/C13g/C12 /C28/C12¯E(t)E(t/C27t)dt: (1) Also recall that the F OURIER TRANSFORM ofE(t)i s defined by E(t)/C13g/C12 /C28/C12Ene/C282pintdn; (2) giving a COMPLEX CONJUGATE of ¯E(t)/C13g/C12 /C28/C12¯Ene2pintdn (3) Plugging ¯E(t) and E(t/C27t) into the AUTOCORRELATION function therefore gives C(t) /C30g/C12 /C28/C12g/C12 /C28/C12¯Ene2pintdn/C2Q/C21g/C12 /C28/C12¯En?e /C282 pi n?(t/C27 t)dn ?/C2Q/C21 dt /C30g/C12 /C28/C12g/C12 /C28/C12g/C12 /C28/C12¯EnE n?e /C282 pi t(n?/C28 n)e /C282 pin?t dt dn dn ? /C30g/C12 /C28/C12g/C12 /C28/C12¯EnE n? dn?/C28 n ðÞ e /C282 piv ?tdn dn ? /C30g/C12 /C12¯EnE ne /C282 pintdn /C30g/C12 /C28/C12Enjj2e /C282 pi ntdn /C30F Enjj2hi ; (4) so, amazingly, the AUTOCORRELATION is simply given by the FOURIER TRANSFORM of the ABSOLUTE SQUARE of E( n); C(t) /C30F E( n) jj2hi : (5) The Wiener-Khintchine theorem is a special case of the CROSS-CORRELATION THEOREM with f /C30g. See also AUTOCORRELATION ,C ROSS- CORRELATION THEOREM ,FOURIER TRANSFORM Wiener-Lee Transform The integral transform obtained by defining v /C13/C28tan1 2 d/C1;/C17 ; (1) and writing H( v) /C30R( v) /C27iX( v); (2) where R(v) and X( v) are a HILBERT TRANSFORM pair as H( v) /C30 r( d) /C28ix(d) (3) (Papoulis 1962, p. 201). See also HILBERT TRANSFORM ,INTEGRAL TRANSFORM References Papoulis, A. "Wiener-Lee Transforms." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 201 /C1/ 03, 1962. Wiener-Le ´vy Process WIENER PROCESS Wigner 3j-Symbol The Wigner 3 j/-symbols are written j1j2j3 m1m2m3/C18/C19 (1) and are sometimes expressed using the relatedCLEBSCH- GORDAN COEFFICIENTS Cj m1m2/C30j1j2m1m2 ð jj1j2jmÞ (2) (Condon and Shortley 1951, pp. 74 /C1/5; Wigner 1959, p. 206), or R ACAH V-COEFFICIENTS Vj1j2j;m1m2m ðÞ : (3) The allowed values of j1;j2;j3;m1;m2;and m3are given by the constraints placed on C LEBSCH- GORDAN COEFFICIENTS . The Wigner 3 j/-symbols are returned by the Mathematica function ThreeJSymbol [{j1, m1}, {j2,m2}, {j3,m3}]. Connections among the Wigner 3 j;Clebsch-Gordan, and Racah Vsymbols are given by j1j2m1m2 ð jj1j2jmÞ /C30(/C281)m/C27j1/C28j2ffiffiffiffiffiffiffiffiffiffiffiffiffi 2j/C271pj1j2 j m1m2/C28m/C18/C19 (4) j1j2m1m2 ðj j1j2jmÞ /C30(/C281)j/C27mffiffiffiffiffiffiffiffiffiffiffiffiffi 2j/C271p Vj1j2j;m1m2/C28m ðÞ (5) Vj1j2j;m1m2/C28m ðÞ /C30(/C281)/C28j1/C27j2/C27jj1j2j1 m2mm2/C18/C19 :(6) The Wigner 3 j/-symbols have the symmetries j1j2j1 m1m2m/C18/C19 /C30j1jj1 m2mm1/C18/C19 /C30jj2j2 mm1m2/C18/C19 /C30(/C281)j1/C27j2/C27jj2j1j m2m1m/C18/C19 ¼(/C281)j1/C27j2/C27jj1jj2 m1mm2/C18/C19 ¼(/C281)j1/C27j2/C27jjj2j1 mm2m1/C18/C19 /C30(/C281)j1/C27j2/C27jjj2 j /C28m1/C28m2/C28m/C18/C19 :(7) The symbols obey the orthogonality relations X j;m(2j/C271)j1j2j m1m2m/C18/C19 j1j2j m?1m?2m/C18/C19 /C30dm1m?1dm2m?2(8) X m1;m2(2j/C271)j1j2j m1m2m/C18/C19 j1j2j? m1m2m?/C18/C19 /C30djj?dmm?; (9) where dijis the K RONECKER DELTA . General formulas are very complicated, but some specific cases are j1 j2 j1 /C27j2 m1m2/C28m1 /C28m2/C18/C19 /C30(/C281)j1/C28j2/C27m1/C27m2 /C29/C2Q2j1ðÞ!2j2ðÞ! 2j1 /C27 2j2 /C27 1 ðÞ j1 /C27 m1 ðÞ /C2j1 /C27 j2 /C27 m1 /C27 m2 ðÞ ! j1 /C27 j2 /C28 m1 /C28 m2 ðÞ ! j1 /C28 m1 ðÞ j2 /C27 m2 ðÞ j2 /C28 m2 ðÞ !/C211 =2 (10) j1j2 j j1/C28j1/C28m/C18/C19 /C30(/C281)/C28j1/C27j2/C27m /C29/C2Q2j1ðÞ! /C28j1 /C27 j2 /C27 j ðÞ ! j1 /C27 j2 /C27 j /C27 1 ðÞ ! j1 /C28 j2 /C27 j ðÞ ! /C29j1 /C27 j2 /C27 m1 /C27 m2 ðÞ ! j1 /C27 j2 /C28 m1 /C28 m2 ðÞ ! j1 /C27 j2 /C28 j ðÞ ! j1 /C27 j2 /C28 j ðÞ ! /C28j1 /C27 j2 /C28 m ðÞ !(j /C27 m)!/C21 (11) j1j2j 000/C18/C19 /C30(/C281)gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2g /C28 2j1 ðÞ 2g /C28 2j2 ðÞ !2g /C28 2j ðÞ ! 2g /C27 1 ðÞ !s /C29g! g /C28 j1 ðÞ ! g /C28 j2 ðÞ ! g /C28 j ðÞ ! if J /C302g 0 if J /C302g /C271;8 >>>>>>>>>>< >>>>>>>>>>:(12) for J /C13j 1 /C27j2 /C27j:/ For SPHERICAL HARMONICS Ym l(u ; f) ;/ Ym1 l1(u ; f)Ym2 l2( u; f) /C30X l ;mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l1 /C27 1 ðÞ 2l2 /C27 1 ðÞ 2l /C27 1 ðÞ 4ps /C2l1 l2 l m1m2m/C18/C19 ¯Ym lu; fðÞl1l2l 000/C18/C19 : (13) For values of l3obeying the TRIANGLE CONDITION D l1l2l3 ðÞ ;/ gYm1 l1( u; f)Ym2 l2( u; f)Ym3 l3( u; f) sinu du d f /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l1/C271 ðÞ 2l2/C271 ðÞ 2l3/C271 ðÞ 4ps l1l2l3 000/C18/C19 /C2l1l2l3 m1m2m3/C18/C19 (14) and 1 2gPl1(cosu)Pl2(cosu) sin udu/C30l1l2l3 000/C18/C192 :(15) See also CLEBSCH- GORDAN COEFFICIENT ,RACAH V-COEFFICIENT ,R ACAH W-COEFFICIENT ,W IGNER 6J- SYMBOL ,W IGNER 9J-SYMBOL References Abramowitz, M. and Stegun, C. A. (Eds.). "Vector-Addition Coefficients." §27.9 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1006 /C1/010, 1972. Condon, E. U. and Shortley, G. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 1951. de Shalit, A. and Talmi, I. Nuclear Shell Theory. New York: Academic Press, 1963. Gordy, W. and Cook, R. L. Microwave Molecular Spectra, 3rd ed. New York: Wiley, pp. 804 /C1/11, 1984. Messiah, A. "Clebsch-Gordan (C.-G.) Coefficients and ‘ /3j/’ Symbols." Appendix C.I in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 1054 /C1/060, 1962. Rose, M. E. Elementary Theory of Angular Momentum. New York: Dover, 1995. Rotenberg, M.; Bivens, R.; Metropolis, N.; and Wooten, J. K. The3j and 6j Symbols. Cambridge, MA: MIT Press, 1959. Shore, B. W. and Menzel, D. H. Principles of Atomic Spec- tra. New York: Wiley, pp. 275 /C1/76, 1968. Sobel’man, I. I. "Angular Momenta." Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer- Verlag, 1992. Wigner, E. P. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, expanded andimproved ed. New York: Academic Press, 1959. Wigner 6j-Symbol A generalization of C LEBSCH- GORDAN COEFFICIENTS and W IGNER 3 J-SYMBOL which arises in the coupling of three angular momenta. The Wigner 6 j/-symbols are returned by the Mathematica functionSixJSym- bol[{j1,j2,j3}, {j4,j5,j6}]. Let tensor operators T(k)andU(k)act, respectively, on subsystems 1 and 2 of a system, with subsystem 1 characterized by angular momentum j1and subsys- tem 2 by the angular momentum j2:Then the matrix elements of the scalar product of these two tensoroperators in the coupled basis J /C30j 1/C27j2are given by t?1j?1t?2j?2J?M?T(k)/C215U(k)/C12/C12/C12/C12t 1j1t2j2JM/CQ/C1 /C30dJJ?dMM?(/C281)j1/C27j?2/C27JJj?2j?1 kj1j2/C2;/C27 /C29t?1j?1T(k)/C13/C13/C13/C13t 1j1/CQ/C1 t?1j?2U(k)/C13/C13/C13/C13t 2j2/CQ/C1 ; (1) where Jj?2j?1 kj1j2/C2;/C27 is the Wigner 6 j/-symbol and t1and t2represent additional pertinent quantum numbers characteriz- ing subsystems 1 and 2 (Gordy and Cook 1984). Edmonds (1968) gives analytic forms of the 6 j/-symbol for simple cases, and Shore and Menzel (1968) and Gordy and Cook (1984) give abc 0 cb/C2;/C27 /C30( /C281)s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2b /C27 1)(2c /C27 1)p (2) abc 1 cb/C2;/C27 /C302(/C281)s/C271Xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b(2b /C27 1)(2b /C27 2)2c(2c /C27 1)(2c /C27 2)p (3) abc 2 cb/C2;/C27 /C30 2(/C281)s 3X(X /C28 1) /C28 4b(b /C27 1)c(c /C27 1) ½/C138ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2b /C28 1)2b(2b /C27 1)(2b /C27 2)(2b /C27 3)(2c /C28 1)2c(2c /C27 1)(2c /C27 2)(2c /C27 3)p ; (4) where s /C13a /C27b /C27c (5) X /C13b(b /C271) /C27c(c /C271) /C28a(a /C271): (6) See also CLEBSCH- GORDAN COEFFICIENT ,RACAH V- COEFFICIENT ,R ACAH W-COEFFICIENT ,W IGNER 3J- SYMBOL ,W IGNER 9J-SYMBOL References Carter, J. S.; Flath, D. E.; and Saito, M. The Classical and Quantum 6j/-Symbols. Princeton, NJ: Princeton Univer- sity Press, 1995. Edmonds, A. R. Angular Momentum in Quantum Me- chanics, 2nd ed., rev. printing. Princeton, NJ: Princeton University Press, 1968. Gordy, W. and Cook, R. L. Microwave Molecular Spectra, 3rd ed. New York: Wiley, pp. 807 /C1/09, 1984. Messiah, A. "Racah Coefficients and ‘/6j/’ Symbols." Appendix C.II in Quantum Mechanics, Vol. 2. Amsterdam, Nether- lands: North-Holland, pp. 567 /C1/69 and 1061 /C1/066, 1962. Rotenberg, M.; Bivens, R.; Metropolis, N.; and Wooten, J. K. The 3j and 6j Symbols. Cambridge, MA: MIT Press, 1959. Shore, B. W. and Menzel, D. H. Principles of Atomic Spec- tra. New York: Wiley, pp. 279 /C1/84, 1968. Wigner 9j-Symbol A generalization of CLEBSCH- GORDAN COEFFICIENTS and WIGNER 3J- and WIGNER 6J-SYMBOLS which arises in the coupling of four angular momenta and can be written in terms of the WIGNER 3J- and WIGNER 6J-SYMBOLS . Let tensor operators T k1ðÞand U k2ðÞact, respectively, on subsystems 1 and 2. Then the reduced matrix element of the product T k1ðÞ/C29U k2ðÞ of these two irreducible operators in the coupled representation is given in terms of the reduced matrix elements of the individual operators in the uncoupled representation by t? t ?j?1 t ?2 j?2J ?jj T k1ðÞ/C29U k2ðÞ/C2/C3 (k)jj tt1 j1 t2 j2J/C1;/C17 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2J /C271)(2J ?/C271)(2k /C271)p X tƒj?1j1k1 j?2j2k2 J ? Jk8 < :9 = ;/C29 t ? t ?1 j?1T k1ðÞ/C13/C13/C13/C13t ƒt 1 j1/CQ/C1 t ƒt?2 j ?2U k2ðÞ/C13/C13/C13/C13tt 2 j2/CQ/C1 ; (1) where j?1j1k1 j?2j2k2 J ? Jk8 < :9 = ; is a Wigner 9j/-symbol (Gordy and Cook 1984). Shore and Menzel (1968) give the explicit formulas abC deF GHJ8 < :9 = ;/C30X x(/C281)2x(2x/C271) /C29abC FJx/C2;/C27 deF bxH/C2;/C27 GHJ xad/C2;/C27 (2) abJ cd J KK 08 < :9 = ;/C30(/C281)b/C27c/C27J/C27K ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2J/C271)(2K/C271)pabJ dcK/C2;/C27 (3) SS 1 LL 2 JJ 18 < :9 = ;/C30SLJ LS 1/C2;/C27 JLS LJ 1/C2;/C27 52LL L11/C2;/C27 /C27(/C281)S/C27L/C27J/C271 15(2L/C271)SLJ LS 1/C2;/C27 2LL L11/C2;/C27 : (4) See also CLEBSCH- GORDAN COEFFICIENT ,RACAH V- COEFFICIENT ,R ACAH W-COEFFICIENT ,W IGNER 3J- SYMBOL ,W IGNER 6J-SYMBOL References Gordy, W. and Cook, R. L. Microwave Molecular Spectra, 3rd ed. New York: Wiley, pp. 807 /C1/09, 1984. Messiah, A. "‘ /9j/’ Symbols." Appendix C.III in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Hol- land, pp. 567 /C1/69 and 1066 /C1/068, 1962. Shore, B. W. and Menzel, D. H. Principles of Atomic Spec- tra. New York: Wiley, pp. 279 /C1/84, 1968. Wigner-Eckart Theorem A theorem of fundamental importance in spectro- scopy and angular momentum theory which provides both (1) an explicit form for the dependence of all matrix elements of irreducible tensors on the projec-tion quantum numbers and (2) a formal expression ofthe conservation laws of angular momentum (Rose 1995). The theorem states that the dependence of the matrix element /ðj?m?jTLMjjmÞ/on the projection quantum numbers is entirely contained in the W IGNER 3 J- SYMBOL (or, equivalently, the C LEBSCH- GORDAN COEF- FICIENT ), given by ðj?m?jTLMjjmÞ¼CðjLj?;mMm?Þðj?jjTLjjjÞ; where /CðjLj ?;mMm ?Þ/ is a CLEBSCH- GORDAN COEFFI- CIENT and /TLM/ is a set of tensor operators (Rose 1995, p. 85). See also CLEBSCH- GORDAN COEFFICIENT ,W IGNER 3J- SYMBOL References Cohen-Tannoudji, C.; Diu, B.; and Laloe¨, F. "Vector Opera- tors: The Wigner-Eckart Theorem." Complement /DX/ in Quantum Mechanics, Vol. 2. New York: Wiley, pp. 1048 /C1/ 058, 1977. Eckart, C. "The Application of Group Theory to the Quan- tum Dynamics of Monatomic Systems." Rev. Mod. Phys. 2, 305 /C1/80, 1930. Edmonds, A. R. Angular Momentum in Quantum Me- chanics, 2nd ed., rev. printing. Princeton, NJ: Princeton University Press, 1968. Gordy, W. and Cook, R. L. Microwave Molecular Spectra, 3rd ed. New York: Wiley, p. 807, 1984. Messiah, A. "Representation of Irreducible Tensor Opera- tors: Wigner-Eckart Theorem." §32 in Quantum Me- chanics, Vol. 2. Amsterdam, Netherlands: North- Holland, pp. 573 /C1/75, 1962. Rose, M. E. "The Wigner-Eckart Theorem." §19 in Elemen- tary Theory of Angular Momentum. New York: Dover, pp. 85 /C1/4, 1995. Shore, B. W. and Menzel, D. H. "Tensor Operators and the Wigner-Eckart Theorem." §6.4 in Principles of Atomic Spectra. New York: Wiley, pp. 285 /C1/94, 1968. Wigner, E. P. "Einige Folgerungen aus der Schro ¨din- gerschen Theorie fu¨r die Termstrukturen." Z. Physik 43, 624 /C1/52, 1927. Wigner, E. P. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, expanded and improved ed. New York: Academic Press, 1959. Wybourne, B. G. Symmetry Principles and Atomic Spectro- scopy. New York: Wiley, pp. 89 and 93 /C1/6, 1970. Wilbraham-Gibbs Constant N.B. A detailed online essay by S. Finch was the starting point for this entry. Let a piecewise smooth function f with only finitely many discontinuities (which are all jumps) be defined on /½/C28p; p/C138/ with FOURIER SERIES ak /C301 pgp - pf(t) cos(kt) dt (1) bk /C301pgp -pf(t) sin(kt) dt; (2) Sn(f ;x) /C301 2a0 /C27Xn k /C301ak cos(kx) /C27bk sin(kx) ½/C138() : (3) Let a discontinuity be at x /C30c, with lim x0c/C28f(x) > lim x0c/C27f(x) ; (4) so D /C13 lim x 0c /C28f(x)hi /C28 lim x 0c/C27f(x)/C2Q/C21 > 0: (5) Definef(c) /C3012lim x0c/C28f(x) /C27 lim x 0c/C27f(x)/C2Q/C21 ; (6) and let x /C30xn Bc be the first local minimum and x /C30 jn > c the first local maximum of Sn(f ;x) on either side of xn : Then lim n 0/C12Snf ;xn ðÞ/C30 f(c) /C27D pG? (7) lim n0/C12Snf ; jn ðÞ /C30 f(c) /C28D pG?; (8) where G?/C13gp 0sinc u du /C301:851937052... (9) Here, sinc x /C13sin x=x is the SINC FUNCTION . The FOURIER SERIES of y /C30x therefore does not converge to /C28p and p at the ends, but to /C282G ? and 2G?: This phenomenon was observed by Wilbraham (1848) and Gibbs (1899). Although Wilbraham was the first to note the phenomenon, the constant G ? is frequently (and unfairly) credited to Gibbs and known as the GIBBS CONSTANT . A related constant sometimes also called the GIBBS CONSTANT is G /C132 pG ?/C302pgp 0sincxdx /C301:17897974447216727... (10) (Le Lionnais 1983). References Carslaw, H. S. Introduction to the Theory of Fourier’s Series and Integrals, 3rd ed. New York: Dover, 1930. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/gibbs/gibbs.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 36 and 43, 1983. Zygmund, A. G. Trigonometric Series 1, 2nd ed. Cambridge, England: Cambridge University Press, 1959. Wilcoxon Rank Sum Test A nonparametric alternative to the two-sample t-test. See also PAIRED T-TEST,PARAMETRIC TEST Wilcoxon Signed Rank Test A nonparametric alternative to the PAIRED T-TEST which is similar to the F ISHER SIGN TEST . This test assumes that there is information in the magnitudes of the differences between paired observations, as well as the signs. Take the paired observations,calculate the differences, and rank them from smal-lest to largest by ABSOLUTE VALUE . Add all the ranks associated with POSITIVE differences, giving the /T/C27/ statistic. Finally, the P-VALUE associated with this statistic is found from an appropriate table. The Wilcoxon test is an R-ESTIMATE . See also FISHER SIGN TEST,H YPOTHESIS TESTING , PAIRED T-TEST,PARAMETRIC TEST Wild Knot A KNOT which is not a TAME KNOT . See also TAME KNOT References Milnor, J. "Most Knots are Wild." Fund. Math. 54, 335 /C1/38, 1964. Wild Point For any point P on the boundary of an ordinary BALL , find a NEIGHBORHOOD of P in which the intersection with the BALL ’s boundary cuts the NEIGHBORHOOD into two parts, each HOMEOMORPHIC to a BALL . A wild point is a point on the boundary that has no such NEIGHBORHOOD . See also BALL,HOMEOMORPHIC ,NEIGHBORHOOD Wilf Class Two sets T1and T2belong to the same Wilf class if SnT1ðÞjj /C30 SnT2ðÞjj for all n, where SnTðÞdenotes the set of permutations on f1 ;...;n g that AVOID the pattern T. Two sets having the same Wilf class are said to be WILF EQUIVALENT . See also AVOIDED PATTERN ,W ILF EQUIVALENT , PERMUTATION PATTERN References Mansour, T. Permutations Avoiding a Pattern from Skand at Least Two Patterns from S3 : 31 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007194/. Wilf Equivalent Two sets T1 and T2 are called Wilf equivalent if they belong to the same W ILF CLASS . See also WILF CLASS ,PERMUTATION PATTERN References Mansour, T. Permutations Avoiding a Pattern from Skand at Least Two Patterns from S3:31 Jul 2000. http:// xxx.lanl.gov/abs/math.CO/0007194/. Wilf-Zeilberger Pair A pair of CLOSED FORM functions ( F, G ) is said to be a Wilf-Zeilberger pair if F(n/C271;k)/C28F(n;k)/C30G(n;k/C271)/C28G(n;k): (1) The Wilf-Zeilberger formalism provides succinct proofs of known identities and allows new identities to be discovered whenever it succeeds in finding aproof certificate for a known identity. However, if thestarting point is an unknown hypergeometric sum, then the Wilf-Zeilberger method cannot discover aclosed form solution, while Z EILBERGER’S ALGORITHM can. Wilf-Zeilberger pairs are very useful in proving HYPERGEOMETRIC IDENTITIES OF THE FORM X kt(n;k)/C30rhs(n) (2) for which the SUMMAND t(n;k) vanishes for all k outside some finite interval. Now divide by the right- hand side to obtain X kF(n;k)/C301; (3) where F(n;k)/C13t(n;k) rhs(n): (4) Now use a RATIONAL FUNCTION R(n;k) provided by ZEILBERGER’S ALGORITHM , define G(n;k)/C13R(n;k)F(n;k): (5) The identity (1) then results. Summing the relationover all integers then telescopes the right side to 0, giving X kF(n/C271;k)/C30X kF(n;k): (6) Therefore, akF(n;k) is independent of n, and so must be a constant. If Fis properly normalized, then it will be true that akF(0;k)/C301:/ For example, consider the BINOMIAL COEFFICIENT identity Xn k/C300n k/C18/C19 /C302n; (7) the function R(n;k) returned by Z EILBERGER’S ALGO- RITHM is R(n;k)/C30k 2(k/C28n/C281): (8) Therefore, F(n;k)/C30n k/C18/C19 2/C28n(9) and G(n;k)/C13R(n;k)F(n;k)/C30k 2(k/C28n/C281)n k/C18/C19 2/C28n /C30/C28kn!2/C28n 2(n/C271/C28k)!k!(n/C28k)!/C30/C28n k/C281/C18/C19 2/C28n/C281: ð10Þ Taking F(n /C271 ;k) /C28F(n;k) /C30G(n;k /C271) /C28G(n ;k) (11) then gives the alleged identity n /C271 k/C18/C19 2/C28n/C281 /C28n k/C18/C19 2 /C28n /C30/C28n k/C18/C19 2/C28n/C281 /C27n k /C281/C18/C19 2/C28n/C281? (12) Expanding and evaluating shows that the identity does actually hold, and it can also be verified that F(0;k) /C300 k/C18/C19 /C301 for k /C300 0 otherwise ;/C2; (13) so ak F(0;k) /C301 (Petkovsek et al. 1996, pp. 25 /C1/7). For any Wilf-Zeilberger pair (F, G), X/C12 n/C300G(n;0) /C30X/C12 n/C301F(n;n /C281) /C27G(n /C281 ;n /C281) ½/C138 (14) whenever either side converges (Zeilberger 1993). In addition, X/C12 n/C300G(n;0) /C30X/C12 n/C300Fs(n /C271);n ðÞ /C27Xs/C281 i/C300G(sn /C27i ;n)"# /C28lim n0/C12Xn /C281 k/C300F(sn ;k); (15) X/C12 k /C300F(0;k) /C30X/C12 n/C300G(n;0) /C28lim k 0/C12X/C12 n/C300G(n;k) ; (16) and X/C12 n/C300G(n ;0) /C30X/C12 n/C300/C2QXt /C281 n/C300F(s(n /C271);tn /C27j) /C27Xs/C281 n/C300G(sn /C27i; tn)/C21 /C28lim n0/C12Xn/C281 k /C300Fs;t(n;k); (17) where Fs ;t(n; k) /C30Xt/C281 j/C300F(sn ;tk /C27j) (18) Gs;t(n;k) /C30Xs/C281 i/C300G(sn /C27i; tk) (19) (Amdeberhan and Zeilberger 1997). The latter iden- tity has been used to compute APE´ RY’S CONSTANT to a large number of decimal places (Wedeniwski). See also APE´ RY’S CONSTANT ,CONVERGENCE IMPROVE- MENT ,GOSPER’S ALGORITHM ,SISTER CELINE’S METH- OD,ZEILBERGER’S ALGORITHM References Amdeberhan, T. and Zeilberger, D. "Hypergeometric Series Acceleration via the WZ Method." Electronic J. Combina-torics 4, No. 2, R3, 1 /C1/, 1997. http://www.combinatoric- s.org/Volume_4/wilftoc.html#R03. Also available at http:// www.math.temple.edu/~zeilberg/mamarim/mamar- imhtml/accel.html. Cipra, B. A. "How the Grinch Stole Mathematics." Science 245, 595, 1989. Koepf, W. "Algorithms for m-fold Hypergeometric Summa- tion." J. Symb. Comput. 20, 399 /C1/17, 1995. Koepf, W. "The Wilf-Zeilberger Method." Ch. 6 in Hypergeo- metric Summation: An Algorithmic Approach to Summa- tion and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 80 /C1/2, 1998. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "The WZ Phenomenon." Ch. 7 in A /C30B. Wellesley, MA: A. K. Peters, pp. 121 /C1/40, 1996. Wilf, H. S. and Zeilberger, D. "Rational Functions Certify Combinatorial Identities." J. Amer. Math. Soc. 3, 147 /C1/58, 1990. Zeilberger, D. "The Method of Creative Telescoping." J. Symb. Comput. 11, 195 /C1/04, 1991. Zeilberger, D. "Closed Form (Pun Intended!)." Contemporary Math. 143, 579 /C1/07, 1993. Wilkie’s Theorem Let f x1 ;...;xm ðÞ be an Lexpformula, where Lexp /C13 L @ exfg and L is the language of ordered rings L /C30 /C27;/C28;/C215;B; 0;1 fg : Then there exist n ]m and f1 ;...; fs /C23 Z x1 ;...xn ; ex1 ;...exn ½/C138 such that f x1 ; ... ;xn ðÞ is equiva- lent to /C215xm/C271 /C1/C1/C1/C215xnf1x1 ;...;xn ;ex1 ;...;exn ðÞ /C30...:/C30fsx1 ;...; xn ;ex1 ;...;exn ðÞ /C300 (Marker 1996, Wilkie 1996). In other words, every formula is equivalent to an existential formula and every definable set is the projection of an exponential variety (Marker 1996). References Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753 /C1/59, 1996. Wilkie, A. J. "Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Restricted Pfaffian Functions and the Exponential Function." J. Amer. Math. Soc. 9, 1051 /C1/094, 1996. Williams p/C271 Factorization Method A variant of the POLLARD P-1 FACTORIZATION METHOD which uses LUCAS SEQUENCES to achieve rapid factor- ization if some factor p of N has a decomposition of p/C271 in small PRIME FACTORS . See also LUCAS SEQUENCE ,POLLARD P-1 FACTORIZA- TION METHOD ,PRIME FACTORIZATION ALGORITHMS References Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkha ¨user, p. 177, 1994. Williams, H. C. "A p/C271 Method of Factoring." Math. Comput. 39, 225/C1/34, 1982. Wilson Plug A 3-D surface with constant VECTOR FIELD on its boundary which traps at least one trajectory which enters it. See also VECTOR FIELD Wilson Polynomial The orthogonal polynomial defined by pn(x;a ;b; c; d) /C30(a /C27b)n(a /C27c)n(a /C27d)n /C294F3/C28n;a /C27b /C27c /C27d /C27n /C281 ;a /C28x;a /C27x a /C27b;a /C27c ;a /C27d ;1/C18/C19 : The first few are p0(x;a ;b; c; d) /C301 p1(x;a ;b;c ;d) /C30abc /C27abd /C27acd /C27bcd /C27(a /C27b /C27c /C27d)x2 : The Wilson polynomials obey the identity pn(x;a;b ;c ;d) /C30pn(x;b;a ;c ;d) : References Koekoek, R. and Swarttouw, R. F. "Wilson." §1.1 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Uni- versiteit Delft, Faculty of Technical Mathematics and Informatics Report 98 /C1/7, pp. 24 /C1/6, 1998. ftp://www.twi.- tudelft.nl/publications/tech-reports/1998/DUT-TWI-98 /C1/ 7.ps.gz. Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 116, 1998. Wilson, J. A. "Some Hypergeometric Orthogonal Polyno- mials." SIAM J. Math. Anal. 11, 690 /C1/01, 1980. Wilson Prime A PRIME satisfying W(p) /C130 (mod p) ; where W(p) is the WILSON QUOTIENT , or equivalently, (p /C281)! /C13/C281 (mod p2) : 5, 13, and 563 (Sloane’s A007540) are the only Wilson primes less than 5 /C29108 (Crandall et al. 1997). See also BROWN NUMBERS References Crandall, R.; Dilcher, K; and Pomerance, C. "A search for Wieferich and Wilson Primes." Math. Comput. 66, 433 /C1/ 49, 1997. Gonter, R. H. and Kundert, E. G. "All Numbers Up to 18,876,041 Have Been Tested without Finding a New Wilson Prime." Preprint, 1994. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983.Ribenboim, P. "Wilson Primes." §5.4 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 346 /C1/50, 1996. Sloane, N. J. A. Sequences A007540/M3838 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 73, 1991. Wilson Quotient W(p) /C13(p /C28 1)! /C28 1 p: References Crandall, R.; Dilcher, K; and Pomerance, C. "A search for Wieferich and Wilson Primes." Math. Comput. 66, 433 /C1/ 49, 1997. Lehmer, E. "On Congruences Involving Bernoulli Numbers and the Quotients of Fermat and Wilson." Ann. Math. 39, 350 /C1/60, 1938. Wilson’s Primality Test WILSON’S THEOREM Wilson’s Theorem IFF p is a PRIME , then (p /C281)! /C271 is a multiple of p, that is (p /C281)! /C13/C281 (mod p): This theorem was proposed by John Wilson in 1770 (although it was previously known to Leibniz) and proved by Lagrange in 1773. Unlike FERMAT’S LITTLE THEOREM , Wilson’s theorem is both NECESSARY and SUFFICIENT for primality. For a COMPOSITE NUMBER , (n/C281)!/C130 (mod n) except when n/C304. See also FERMAT’S LITTLE THEOREM ,W ILSON’S THE- OREM COROLLARY ,WILSON’S THEOREM (GAUSS’S GEN- ERALIZATION ) References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, p. 61, 1987. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 142 /C1/43 and 168 /C1/69, 1996. Hilton, P.; Holton, D.; and Pedersen, J. Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 41 /C1/2, 1997. Nagell, T. "Wilson’s Theorem and Its Generalizations." Introduction to Number Theory. New York: Wiley, pp. 99 /C1/01, 1951. Ore, Ø.Number Theory and Its History. New York: Dover, pp. 259 /C1/61, 1988. Se´roul, R. "Wilson’s Theorem." §2.9 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 16 /C1/7, 2000. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 37 /C1/8, 1993. Wilson’s Theorem (Gauss’s Generalization) Let P(n) be the product of INTEGERS that are less than or equal to and RELATIVELY PRIME to an integer n. Then P(n) /C13Yn k/C302 k ½nk /C30/C281 (mod n) for n /C304;p a ;2p a 1 (mod n) otherwise :/C2; When m /C302, this reduces to P /C131 mod 2 ðÞ which is equivalent to P /C13/C281 mod 2 ðÞ :/ See also WILSON’S THEOREM ,W ILSON’S THEOREM COROLLARY Wilson’s Theorem Corollary Iff a PRIME p is OF THE FORM 4x /C271; then 2xðÞ! ½/C1382/C13/C281 mod p ðÞ : Wimp Transform The INTEGRAL TRANSFORM defined by (K f)(x) /C30g/C12 /C28/C12Gm;n/C272 p /C272 ;q/C18 tj1 /C28 n /C27ix ;1 /C28 n /C28ix ; ap/CQ/C1 bp/CQ/C1/C19 f(t) dt; where Ga ;b c;dis MEIJER’S G-FUNCTION . References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 24, 1993. Winding Number (Contour) The winding number of a CONTOUR g about a point z0 ; denoted n g ;z0ðÞ ; is defined by n( g; a) /C301 2pi G gdz z /C28 z0 and gives the number of times g curve passes around a point. The winding number is also called the index, and denoted Indgz0ðÞ:/ The contour winding number was part of the inspira- tion for the idea of the DEGREE of a MAP between two COMPACT , oriented MANIFOLDS of the same DIMEN-SION. In the language of the DEGREE of a MAP,ifg : 0;1½/C1380 C is a closed curve (i.e., g(0) /C30 g(1)) ; then it can be considered as a FUNCTION from S1 to C : In that context, the winding number of g around a point p in C is given by the degree of the MAP g/C28p g/C28p jj from the CIRCLE to the CIRCLE . See also RESIDUE (COMPLEX ANALYSIS ) References Krantz, S. G. "The Index or Winding Number of a Curve about a Point." §4.4.4 in Handbook of Complex Analysis. Boston, MA: Birkha ¨user, pp. 49 /C1/0, 1999. Winding Number (Map) The winding number W(u) of a map f(u) with initial value uis defined by W(u)/C13lim n0/C12fn(u)/C28u n; which represents the average increase in the angle u per unit time (average frequency). A system with a RATIONAL winding number W/C30p=qisMODE-LOCKED , whereas a system with an IRRATIONAL winding number is QUASIPERIODIC . Note that since the RA- TIONALS are a set of zero MEASURE on any finite interval, almost all winding numbers will be irra- tional, so almost all maps will be QUASIPERIODIC . References Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 129, 1990. Windmill One name for the figure used by Euclid to prove theP YTHAGOREAN THEOREM . BRIDE’S CHAIR ,PEACOCK’S TAIL Window Function RECTANGLE FUNCTION Winkler Conditions Conditions arising in the study of the R OBBINS AXIOM and its connection with B OOLEAN ALGEBRA . Winkler studied Boolean conditions (such as idempotence orexistence of a zero) which would make a R OBBINS ALGEBRA become a B OOLEAN ALGEBRA . Winkler showed that each of the conditions /C215C;/C215D;C/C150D/C30C /C215C;/C215D;!(C/C150D)/C30!C where A/C150Bdenotes OR and ! Adenotes NOT, known as the first and second Winkler conditions, SUFFICES . A computer proof demonstrated that every ROBBINS ALGEBRA satisfies the second Winkler condition, from which it follows immediately that all ROBBINS ALGE- BRAS are BOOLEAN . See also BOOLEAN ALGEBRA ,H UNTINGTON AXIOM , ROBBINS ALGEBRA ,ROBBINS AXIOM References McCune, W. "Robbins Algebras are Boolean." http://www- unix.mcs.anl.gov/~mccune/papers/robbins/. Winkler, S. "Robbins Algebra: Conditions that Make a Near- Boolean Algebra Boolean." J. Automated Reasoning 6, 465 /C1/89, 1990. Winkler, S. "Absorption and Idempotency Criteria for a Problem in Near-Boolean Algebra." J. Algebra 153, 414 /C1/ 23, 1992. Winograd Transform A discrete FAST FOURIER TRANSFORM ALGORITHM which can be implemented for N /C302, 3, 4, 5, 7, 8, 11, 13, and 16 points. See also FAST FOURIER TRANSFORM Wirtinger’s Inequality If y has period 2p; y? is L2 ; and g2 p 0ydx/C300; (1) then g2 p 0y2 dx Bg2p 0y?2 dx (2) unless y /C30A cos x /C27B sin x (3) (Hardy et al. 1988). Another inequality attributed to Wirtinger involves the KA¨ HLER FORM , which in Cn can be written v /C30/C281 2iX dzkffld¯zk : (4) Given 2k vectors X1 ; ... ;X2kin R2n #Cn ; let X /C30 X1ffl/C1/C1/C1fflX2kdenote the oriented k-dimensional PAR- ALLELEPIPED and Xjjits k-dimensional volume. Then vk(X) 5k! Xjj; (5) with equality IFF the vectors span a k-dimensional complex subspace of Cn ; and they are positively oriented. Here, vk is the kth EXTERIOR POWER for 1 5 k 5n; and the orientation of a COMPLEX SUBSPACE is determined by its COMPLEX STRUCTURE . See also KA¨ HLER FORM References Blaschke, W. Kreis und Kugel. Leipzig, Germany: p. 105, 1916.Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "Further Examples: Wirtinger’s Inequality." §7.7 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 184 /C1/87, 1988. Wirtinger-Sobolev Isoperimetric Constants Constants g such that gVfjjqdx/C2Q/C21 1 =q 5 ggVXN i/C301@f @xi/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12p dx"# 1 =p ; where f is a real-valued smooth function on a region V satisfying some BOUNDARY CONDITIONS . References Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/ws/ws.html. Wishart Distribution If Xifor i /C301, ..., m has a GAUSSIAN MULTIVARIATE DISTRIBUTION with mean vector m /C300 and COVAR- IANCE MATRIX S; and X denotes the m /C29p matrix composed of the row vectors Xi ; then the p /C29p matrix XTX has a Wishart distribution with scale matrix S and degrees of freedom parameter m. The Wishart distribution is most typically used when describing the COVARIANCE MATRIX of multinormal samples. See also F-DISTRIBUTION ,G AUSSIAN MULTIVARIATE DISTRIBUTION ,HOTELLING T-SQUARED DISTRIBUTION Witch of Agnesi A curve studied and named "versiera" (Italian for "she-devil" or "witch") by Maria Agnesi in 1748 in herbook Istituzioni Analitiche (MacTutor Archive). It is also known as cubique d’Agnesi or agne ´sienne. Some suggest that Agnesi confused an old Italian wordmeaning "free to move" with another meaning "witch." The curve had been studied earlier by Fermat and Guido Grandi in 1703.It is the curve obtained by drawing a line from theorigin through the CIRCLE of radius a(OB), then picking the point with the ycoordinate of the intersection with the circle and the xcoordinate of the intersection of the extension of line OBwith the liney/C302a:The curve has INFLECTION POINTS aty/C30 3a=2:The line y/C300i sa n ASYMPTOTE to the curve. In parametric form, x/C302acotu (1) y /C30a 1 /C28cos(2 u) ½/C138 ; (2) or x /C302at (3) y /C302a 1 /C27 t2 : (4) In rectangular coordinates, y /C308a3 x2 /C27 4a2 : (5) See also LAME´ CURVE References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 226, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 90 /C1/3, 1972. MacTutor History of Mathematics Archive. "Witch of Ag- nesi." http://www-groups.dcs.st-and.ac.uk/~history/ Curves/Witch.html. Yates, R. C. "Witch of Agnesi." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 237 /C1/ 38, 1952. Witness A witness is a number which, as a result of its number theoretic properties, guarantees either the composite- ness or primality of a number n. Witnesses are most commonly used in connection with FERMAT’S LITTLE THEOREM CONVERSE .AP RATT CERTIFICATE uses witnesses to prove primality, and MILLER’S PRIMALITY TEST uses witnesses to prove compositeness. See also ADLEMAN- POMERANCE- RUMELY PRIMALITY TEST,FERMAT’S LITTLE THEOREM CONVERSE ,M ILL- ER’S PRIMALITY TEST,PRATT CERTIFICATE ,PRIMALITY CERTIFICATE Witt Geometry References Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996. Wittenbauer’s Parallelogram Divide the sides of a QUADRILATERAL into three equal parts. The figure formed by connecting and extendingadjacent points on either side of a VERTEX is a PARALLELOGRAM known as Wittenbauer’s parallelo- gram. See also QUADRILATERAL ,W ITTENBAUER’S THEOREM Wittenbauer’s Theorem The CENTROID of a QUADRILATERAL LAMINA is the center of its WITTENBAUER’S PARALLELOGRAM . See also CENTROID (GEOMETRIC ), LAMINA ,Q UADRI- LATERAL ,W ITTENBAUER’S PARALLELOGRAM Witten’s Equations Also called the SEIBERG- WITTEN INVARIANTS . For a connection A and a POSITIVE SPINOR f /C23G V/C27/CQ/C1 ; DA f /C300 FA /C27/C30i s( f; f) : The solutions are called monopoles and are the minima of the functional gXFA /C27/C28i s( f; f)/C12/C12/C12/C122/C27DA fjj2/C1;/C17 : See also LICHNEROWICZ FORMULA ,L ICHNEROWICZ- WEITZENBOCK FORMULA ,S EIBERG- WITTEN EQUA- TIONS References Cipra, B. "A Tale of Two Theories." What’s Happening in the Mathematical Sciences, 1995 /C1/996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 14 /C1/5, 1996. Donaldson, S. K. "The Seiberg-Witten Equations and 4- Manifold Topology." Bull. Amer. Math. Soc. 33,45/C1/0, 1996. Kotschick, D. "Gauge Theory is Dead!--Long Live Gauge Theory!" Not. Amer. Math. Soc. 42, 335 /C1/38, 1995. Seiberg, N. and Witten, E. "Monopoles, Duality, and Chiral Symmetry Breaking in N /C302 Supersymmetric QCD." Nucl. Phys. B 431, 581 /C1/40, 1994. Witten, E. "Monopoles and 4-Manifolds." Math. Res. Let. 1, 769 /C1/96, 1994. Wolfskehl Prize A prize of 100,000 German marks offered for the first valid proof of FERMAT’S LAST THEOREM (Ball and Coxeter 1987, p. 72; Barner 1997; Hoffman 1998, pp. 193 /C1/94 and 199). The prize was collected by Andrew Wiles after his successful proof of the theorem in the years 1993 /C1/995. See also FERMAT’S LAST THEOREM ,M ATHEMATICS PRIZES References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 69 /C1/3, 1987. Barner, K. "Paul Wolfskehl and the Wolfskehl Prize." Not. Amer. Math. Soc. 44, 1294 /C1/303, 1997. Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 193 /C1/99, 1998. Wolstenholme’s Theorem If p is a PRIME > 3 ; then the NUMERATOR of 1 /C271 2 /C2713 /C27.../C271 p /C28 1 is divisible by p2 and the NUMERATOR of 1 /C271 22 /C271 32 /C27.../C271 (p /C28 1)2 is divisible by p. These imply that if p ]5is PRIME , then 2p /C281 p /C281/C18/C19 /C131 (mod p3) : References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 85, 1994. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 21, 1989. Woodall Number Numbers OF THE FORM Wn /C302nn /C281: The first few are 1, 7, 23, 63, 159, 383, ... (Sloane’s A003261). The only Woodall numbers Wnfor n B 100;000 which are PRIME are for n /C305312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, ... (Sloane’s A014617; Ballinger). See also CULLEN NUMBER ,CUNNINGHAM NUMBER , FERMAT NUMBER ,M ERSENNE NUMBER ,SIERPINSKI NUMBER OF THE FIRST KIND References Ballinger, R. "Cullen Primes: Definition and Status." http:// vamri.xray.ufl.edu/proths/cullen.html. Caldwell, C. K. "The Top Twenty: Woodall Primes." http:// www.utm.edu/research/primes/lists/top20/Woodall.html. Guy, R. K. "Cullen Numbers." §B20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 77, 1994. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/woodall/. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 360 /C1/61, 1996. Sloane, N. J. A. Sequences A003261/M4379 and A014617 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/se- quences/eisonline.html.Woodbury Formula A /C27UVT/CQ/C1 /C281/C30A /C281 /C28 A/C281U 1 /C27VTA /C281U/CQ/C1 /C281VTA/C281hi : See also SHERMAN- MORRISON FORMULA References Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, p. 51, 1996. Woolhouse’s Formulas Let the values of a function f(x) be tabulated at points xiequally spaced by h /C30xi/C271 /C28xi ; so f1 /C30f(x1) ; f2 /C30 f(x2) ; ..., fn /C30fxnðÞ: Then Woolhouse’s formulas ap- proximating the integral of f(x) are given by the NEWTON- COTES -like formulas gx11 x1f(x) dx /C305/C2Q 223 3909f1 /C27f11 ðÞ /C275875 18144f2 /C27f10 ðÞ /C274625 10584f4 /C27f8 ðÞ /C2741 112 f5/C21 gx29 x1f(x) dx /C3014/C2Q 7 195f1 /C27f29 ðÞ /C2716807 66690f3 /C27f27 ðÞ /C27128285f8 /C27f22 ðÞ /C2771 135 f15/C21 : References King, A. E. "Approximate Integration. Note on Quadrature Formulae: Their Construction and Application to Actuar- ial Functions." Trans. Faculty of Actuaries 9, 218 /C1/31, 1923. Sheppard, W. F. "Some Quadrature-Formulæ." Proc. Lon- don Math. Soc. 32, 258 /C1/77, 1900. Whittaker, E. T. and Robinson, G. "Woolhouse’s Formulae." The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 158, 1967. Woolhouse, W. S. B. "On Integration by Means of Selected Values of the Function." J. Inst. Act. 27, 122/C1/55, 1888. Word A finite sequence of nletters from some ALPHABET is said to be an n-ary word. See also CUBEFREE WORD,O VERLAPFREE WORD, SQUAREFREE WORD Word Sequence An INTEGER SEQUENCE whose terms are defined in terms of number-related words in some language. For example, the following table gives the sequences of numbers having digits whose English names (zero, one, two, three, four, five, six, seven, eight, nine) arein alphabetical order and also satisfy some other property. property Sloane sequence ordered A053432 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ... distinct, orderedA053433 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, ... prime, ordered A053434 2, 3, 5, 7, 11, 13, 17, 41, 43, 47, 53, 59, ... distinct,prime, orderedA053435 2, 3, 5, 7, 13, 17, 41, 43, 47, 53, 59, 73, ... See also L OOK AND SAY SEQUENCE References Sloane, N. J. A. Sequences A053432, A053433, A053434, and A053435 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. World Line The path of an object through PHASE SPACE . Worm One of the seven 4-POLYHEXES . S. Kim has observed that four worms solve the puzzle of finding a non- three- COLORABLE map with only four congruent countries (as long as no lakes are allowed). See also COLORABLE References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, p. 147, 1978. Gosper, R. W. G. "Quattroslabia." http://www.ippi.com/rwg/ Quattroslabia.htm. Worpitzky’s Identity xn /C30Xn k /C301n k/C28/C29 x /C27k /C281 n/C18/C19 ; wheren k/C1Q/C11 is an EULERIAN NUMBER andn k/CQ/C1 is a BINOMIAL COEFFICIENT (Worpitzky 1883; Comtet 1974, p. 242). See also BINOMIAL SUMS,EULERIAN NUMBERReferences Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974. Worpitzky. "Studien u ¨ber die Bernoullischen und Euler- ischen Zahlen." J. reine angew. Math. 94, 203/C1/32, 1883. Wright Function The ENTIRE FUNCTION f(r;b;z)/C30X/C12 k/C300zk k!G(rk/C27b); where r>/C281 and b/C23C;named after the British mathematician E. M. Wright. References Gorenflo, R.; Luchko, Yu.; and Mainardi, F. "Analytical Properties and Applications of the Wright Function." Fractional Calc. Appl. Anal. 2, 383/C1/15, 1999. Writhe Also called the TWIST NUMBER . The sum of crossings p of a LINK L, w(L)/C30X p/C23C(L)e(p); (1) where e(p) defined to be 91 if the overpass slants from top left to bottom right or bottom left to top right and C(L) is the set of crossings of an oriented LINK . The writhe of a minimal knot diagram is notaKNOT INVARIANT , as exemplified by the P ERKO PAIR , which have differing writhes (Hoste et al. 1998). If a KNOT KisAMPHICHIRAL , then w(K)/C300 (Thistle- thwaite). A formula for the writhe is given by Wr(K)/C301 4pgKdsgKdt emdem dsdea dt(2) where Kis parameterized by xm(s) for 05s5Lalong the length of the knot by parameter s, and the FRAME Kfassociated with Kis ym/C30xm(s)/C27enm(s); (3) where eis a small parameter, nm(s) is a unit VECTOR FIELD normal to the curve at s, and the vector field em is given by e m(s ;t) /C30ym(t) /C28 xm(s) y(t) /C28 x(s) jj(4) (Kaul 1999). Letting Lk be the LINKING NUMBER of the two components of a ribbon, Tw be the TWIST , and Wr be the writhe, then the CALUGAREANU THEOREM states that Lk(K) /C30Tw(K) /C27Wr(K): (5) (Adams 1994, p. 187). See also CALUGAREANU THEOREM ,SCREW ,TWIST References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20,33/C1/8, Fall 1998. Kaul, R. K. Topological Quantum Field Theories--A Meeting Ground for Physicists and Mathematicians. 15 Jul 1999. http://xxx.lanl.gov/abs/hep-th/9907119/. Wronskian W f1 ;...; fn ðÞ /C13f1 f2 /C1/C1/C1 fn f ?1 f?2 /C1/C1/C1 f ?n nn::: n f(n/C281) 1 f(n/C281) 2 /C1/C1/C1 f(n/C281) n/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12: If the Wronskian is NONZERO in some region, the functions fiare LINEARLY INDEPENDENT .If W /C300 over some range, the functions are linearly dependent somewhere in the range. See also ABEL’S DIFFERENTIAL EQUATION IDENTITY , GRAM DETERMINANT ,LINEARLY DEPENDENT FUNC- TIONS References Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 524 /C1/25, 1953. W-Transform The W-transform of a function f(x) is defined by the integral (Wf)(x) /C30 Wmn pqn ;( a)p bq/CQ/C1/C12/C12/C12/C12/C12/C12/C12/C12f(t)/C18/C19 (x) (1) /C301 2pi g sG n /C28ix /C28s ; n /C27ix /C28s ðÞ /C29GbmðÞ/C27s ; 1 /C28 anðÞ/C28s an/C271 p/C1;/C17 /C27s ; 1 /C28 bm/C271 q/C1;/C17 /C28s"# f /C31(1 /C28s) ds ; (2) whereGbmðÞ/C27s ; 1 /C28 anðÞ/C28s an/C271 p/C1;/C17 /C27s ; 1 /C28 bm/C271 q/C1;/C17 /C28s"# /C30Gb1 /C27s; ...; bm /C27s ; 1 /C28 a1 /C28s ; ...; 1 /C28 an /C28s an/C271 /C27s; ...; ap /C27s 1 /C28 bm /C271 /C28s; ... 1/C28 bq /C28s/C2Q/C21 (3) /C30Qm j/C301 G bj/C27s/CQ/C1Qnj/C301 G 1 /C28 aj /C28 s/CQ/C1 Qp j /C30n/C271 G aj /C27s/CQ/C1Qqj /C30m/C271 G 1 /C28 bj /C28 s/CQ/C1 ; (4) /R[ n] > 1=2 ; n and the components of the vectors ap/CQ/C1 and bq/CQ/C1 are complex numbers satisfying the condi- tions R ap/C2/C3 Þ"1=2 ;3=2; 5=2; ... ; 3/2, 5/2, ... and R bq/C2/C3 "/C281=2 ;/C283=2;/C285=2;...;/C283/2, /C285/2, ..., f /C31(s)is the MELLIN TRANSFORM of a function f(x) and s is the CONTOUR s /C30 1=2 /C28i /C12;1 =2 /C27i /C12 fg :/ See also G-TRANSFORM References Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. "The W- Transform and Its Inversion." §37.5 in Fractional Inte- grals and Derivatives. Yverdon, Switzerland: Gordon and Breach, pp. 752 /C1/58, 1993. Wulff Shape An equilibrium MINIMAL SURFACE for a crystal or drop which has the least anisotropic surface free energy for a given volume. It is the anisotropic analog of a SPHERE . In the case of a sessile drop, the Wulff shapes becomes the Winterbottom shape (Dunlop and Mag- nen 1999, p. 31). See also SPHERE References Dunlop, F. and Magnen, J. "A Wulff Shape from Construc- tive Field Theory." In Mathematical Results in Statistical Mechanics, Marseilles, France, July 27 /C1/1 1998 (Ed. S. Miracle-Sole ´, J. Ruis, and V. Zagrebnov). Singapore: World Scientific, pp. 31 /C1/2, 1999. Winterbottom, W. L. "Equilibrium Shape of a Small Particle in Contact with a Foreign Substrate." Acta Metal. 15, 303 /C1/10, 1967. Wulff, G. "Zur Frage der Geschwindigkeit des Wachstums und der Auflo¨sung der Krystallflagen." Z. Kryst. Mineral. 34, 449, 1901. Wynn’s Epsilon Method A method for numerical evaluation of SUMS and PRODUCTS which samples a number of additional terms in the series and then tries to fit them to a POLYNOMIAL multiplied by a decaying exponential. Wynn’s epsilon method can be applied to the terms of a series using the Mathematica command Sequen- ceLimit [l]. See also EULER- MACLAURIN INTEGRATION FORMULAS Wythoff Array A INTERSPERSION array given by 1235 8 1 32 13 4 55 /C1/C1/C1 4 7 11 18 29 47 76 123 199 /C1/C1/C1 6 10 16 26 42 68 110 178 288 /C1/C1/C1 9 15 24 39 63 102 165 267 432 /C1/C1/C1 12 20 32 52 84 136 220 356 576 /C1/C1/C1 14 23 37 60 97 157 254 411 665 /C1/C1/C1 17 28 45 73 118 191 309 500 809 /C1/C1/C1 19 31 50 81 131 212 343 555 898 /C1/C1/C1 22 36 58 94 152 246 398 644 1042 /C1/C1/C1 nnnn nnnnn::: the first row of which is the FIBONACCI NUMBERS . See also BEATTY SEQUENCE ,F IBONACCI NUMBER , INTERSPERSION ,STOLARSKY ARRAY References Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157 /C1/68, 1997. Sloane, N. J. A. "The Wythoff Array and the Para-Fibonacci Sequence." http://www.research.att.com/~njas/sequences/ classic.html. Wythoff Construction A method of constructing UNIFORM POLYHEDRA . See also UNIFORM POLYHEDRON References Har’El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47,57/C1/10, 1993. Wythoff Symbol A symbol consisting of three rational numbers that can be used to describe UNIFORM POLYHEDRA based on how a point C in a spherical triangle can be selected so as to trace the vertices of regular polygonal faces. For example, the Wythoff symbol for the TETRAHE- DRON is 3 j23: There are four types of Wythoff symbols, pqr ; j p j qr; pqj r and pqr j; and one exceptional symbol,3 253 352/C12/C12/C12 (which is used for the GREAT DIRHOM- BICOSIDODECAHEDRON ). The meaning of the bars ½ may be summarized as follows (Wenninger 1989, p. 10; Messer). Consider a SPHERICAL TRIANGLE PQR whose angles are p=p;p=q; and p=r : 1. pqr : j C is a special point within PQR that traces snub polyhedra by even reflections . 2. p j qr(or p j rq):C is the vertex P. 3. qrj p (or rq j p):C lies on the are PQ and the bisector of the opposite angle R. 4. pqr j (or any permutation of the three letters): C is the incenter of the triangle PQR . Some special cases in terms of SCHLA ¨ FLI SYMBOLS are p j q 2 /C30p j 2 q /C30 q;pfg2 j pq/C30p q/C2;/C27 pqj 2 /C30rp q/C2;/C27 2 qj p /C30tp;qfg 2 pqj t /C30p q/C2;/C27 j 2 pq/C30sp q/C2;/C27 See also SCHLA ¨ FLI SYMBOL ,S CHWARZ TRIANGLE , UNIFORM POLYHEDRON References Har’El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47,57/C1/10, 1993. Messer, P. W. "Closed-Form Expressions for Uniform Poly- hedra and Their Duals." Unpublished manuscript. Wenninger, M. J. Polyhedron Models. New York: Cam- bridge University Press, pp. 8 /C1/0, 1989. Wythoff’s Game A game played with two heaps of counters in which a player may take any number from either heap or the same number from both. The player taking the last counter wins. The rth SAFE combination is (x; x /C27r); where x /C30 frbc ; with f the GOLDEN RATIO and xbcthe FLOOR FUNCTION . It is also true that x /C27r /C30 f2r/C9/C= : The first few SAFE combinations are (1, 2), (3, 5), (4, 7), (6, 10), ... (Sloane’s A000201 and A001950), which are the pairs of elements from the complementary BEATTY SEQUENCES forfandf2(Wells 1986, p. 40). See also BEATTY SEQUENCE ,NIM,SAFE References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 39 /C1/0, 1987. Coxeter, H. S. M. "The Golden Section, Phyllotaxis, and Wythoff’s Game." Scripta Math. 19, 135/C1/43, 1953. O’Beirne, T. H. Puzzles and Paradoxes. Oxford, England: Oxford University Press, pp. 109 and 134 /C1/38, 1965. Sloane, N. J. A. Sequences A000201/M2322 and A001950/ M1332 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986. Wythoff, W. A. "A Modification of the Game of Nim." Nieuw Arch. Wiskunde 8, 199/C1 /02, 1907/1909. X x-Axis The horizontal axis of a 2-D plot in CARTESIAN COORDINATES . Physicists and astronomers sometimes call this axis the ABSCISSA , although that term is more commonly used to refer to coordinates along the X- AXIS. See also ABSCISSA ,ORDINATE , Y-AXIS, Z-AXIS Xi Function j(z) /C131 2 z(z /C281)G1 2 z/C16/C17 pz=2z(z) /C30(z /C28 1)G1 2 z /C27 1/C16/C17 z(z) ffiffiffiffiffipzp ; (1) where z(z) is the RIEMANN ZETA FUNCTION and G(z)is the GAMMA FUNCTION (Gradshteyn and Ryzhik 2000, p. 1076; Hardy 1999, p. 41). The j function satisfies the identity j(1 /C28z) /C30 j(z) : (2)The zeros of j(z) and of its DERIVATIVES are all located on the CRITICAL STRIP z /C30 s /C27it ; where 0 B s B1: Therefore, the nontrivial zeros of the RIEMANN ZETA FUNCTION exactly correspond to those of j(z): The function j(z) is related to what Gradshteyn and Ryzhik (2000, p. 1074) call J(t)by J(t) /C13 j(z); (3) where z /C131 2 /C27it : This function can also be defined as J(it) /C131 2t2 /C2814/C16/C17 p/C28t=2 /C281 =4 G12 t /C2714/C16/C17 z t /C2712/C16/C17 ; (4) giving J(t) /C30/C2812t2 /C2714/C16/C17 pit =2 /C281=4 G14 /C2812 it/C16/C17 z12 /C28it/C16/C17 : (5) The DE BRUIJN- NEWMAN CONSTANT is defined in terms of the J(t) function. See also DE BRUIJN- NEWMAN CONSTANT ,R IEMANN HYPOTHESIS ,RIEMANN- SIEGEL FUNCTIONS ,RIEMANN ZETA FUNCTION References Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Com- putational Strategies for the Riemann Zeta Function." CECM-98:118, 23 Jun 1999. http://www.cecm.sfu.ca/pre- prints/1999pp.html#98:118. Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361 /C1/372, 1979. Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. II." Math. Comput. 39, 681 /C1/88, 1982. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, corr. enl. 4th ed. San Diego, CA: Academic Press, 2000. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999. Titchmarsh, E. C. and Heath-Brown, D. R. The Theory of the Riemann Zeta-Function, 2nd ed. Oxford, England: Oxford University Press, 1986. x-Intercept The point at which a curve or function crosses the X- AXIS (i.e., when y/C300 in 2-D). See also LINE, Y-INTERCEPT XNOR The CONNECTIVE in logic corresponding to the ex- clusive nor operation. A XNOR B is equivalent to (A fflB) /C150(!A ffl!B) ; where ffl denotes AND, /C150 denotes OR, and !A denotes NOT. The circuit diagram symbol for an XNOR gate is illustrated above, and the XNOR TRUTH TABLE is given below. ABA XNOR B TTT TFF FTFFFT See also AND, B INARY OPERATOR ,BOOLEAN ALGEBRA , CONNECTIVE ,LOGIC , NAND, NOR, NOT, OR, PAS- CAL’S TRIANGLE ,TRUTH TABLE , XOR References Simpson, R. E. "The Exclusive NOR (XNOR) Gate." §12.5.7 inIntroductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 539 and 554, 1987. XOR Portions of this entry contributed by R OGER GER- MUNDSSON ACONNECTIVE inLOGIC known as the "exclusive or," orEXCLUSIVE DISJUNCTION . It yields true if exactly one (but not both) of two conditions is true. The XOR operation does not have a standard symbol, but issometimes denoted A¯/C150B(this work) or A/C154B(Simp- son 1987, pp. 539 and 550 /C1 /54).A¯/C150Bis read " AAUT B," where "aut" is Latin for "or, but not both." The circuit diagram symbol for an XOR gate is illustratedabove. In SET THEORY ,A¯/C150Bis typically called the SYMMETRIC DIFFERENCE . The XOR function is imple- mented in Mathematica 4.1 asXOR. The binary XOR operation A¯/C150Bis identical to NONEQUIVALENCE AfB:A¯/C150Bcan be implemented using AND and OR gates asA¯/C150B/C30(Affl!B)/C150(!AfflB) (1) /C30(A/C150)ffl!(AfflB); (2) whereffldenotes AND and /C150denotes OR, and can be implemented using only NOT and NAND gates as A¯/C150B/C30(A¯ffl!B)¯ffl(!A¯fflB) (3) (Simpson 1987), where ¯ffldenotes NAND. The BINARY XOR operator has the following TRUTH TABLE . AB /A¯/C150B/ TTF TFTFTTFFF The BINOMIAL COEFFICIENTm n/C0/C1 mod 2 can be com- puted using the XOR operation nXOR m, making PASCAL’S TRIANGLE mod 2 very easy to construct. For multiple arguments, XOR is defined to be true if an odd number of its arguments are true, and false otherwise. This definition is quite common in compu-ter science, where XOR is usually thought of as addition modulo 2. In this context, it arises in polynomial algebra modulo 2, arithmetic circuitswith a full adder, and in parity generating or checking. While this means that the multiargument "XOR" can no longer be thought of as "the exclusiveOR" operation, this form is rarely used in mathema-tical logic and so does not cause very much confusion. The XOR operation is associative, so a¯/C150(b¯/C150c) is the same as ( a¯/C150b)¯/C150c:Computation of the multiargu- ment XOR requires evaluation of all its arguments todetermine the truth value, and hence there is no "lazy" special evaluation form (as there is for ANDand OR). The ternary XOR operator therefore has the following truth table. ABC /A¯/C150B¯/C150C/ TTTT TTFFTFTFTFFTFTTFFTFTFFTTFFFF See also AND, AUT,B INARY OPERATOR ,B OOLEAN ALGEBRA ,CONNECTIVE ,LOGIC , NAND, NOR, NOT, OR, PASCAL’S TRIANGLE ,S YMMETRIC DIFFERENCE , TRUTH TABLE , XNORReferences Simpson, R. E. "The Exclusive OR (XOR) Gate." §12.5.6 in Introductory Electronics for Scientists and Engineers, 2nd ed.Boston, MA: Allyn and Bacon, pp. 550 /C1/54, 1987. Y Yacht A6- POLYIAMOND . References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton Univer- sity Press, p. 92, 1994. Yahtzee Yahtzee is a game played with five 6-sided DICE. Players take turns rolling the dice, and trying to get certain types of rolls, each with an assigned point value, as summarized in the following table. Players are allowed a total of three rolls, with any subset of dice capable of being set aside at each roll. In addition to runs of a single number, other rolls include 3 of a kind (three of the same number), 4 of a kind (four of the same number), full house (two of one number and three of another), small straight (4 numbers in a row), large straight (5 numbers in a row), Yahtzee (five of the same number), and chance (any roll). aces sum of 1s twos sum of 2s threes sum of 3s fours sum of 4s fives sum of 5s sixes sum of 6s 3 of a kind sum of all dice 4 of a kind sum of all dice full house 25 sm. straight 30 lg. straight 40 Yahtzee 50 chance sum of all dice In a variant of the game known as triple Yahtzee, players try to get each type of roll three times over the course of the game instead of just once, with pointvalues for each roll being placed in a single, double, or triple column, whose values are multiplied by the stated weight when scores are totaled. The following tables summarizes the probability of obtaining var- ious rolls. In this table, lower-value rolls are excluded from the results, so, for example, the probability of obtaining a three of a kind excludes rolls that are actually fours of a kind or Yahtzees. Similarly, the three of a kind probability excludes rolls that are full houses, and the two of a kind probability excludes rolls that are small straights. type 1 2 3 overall 2 of a kind /65 108//65 108//65 108//1180205 1259712/ 3 of a kind /25 162/ 4 of a kind /25 1296/ full house /25 648/ sm. straight /1081/ lg. straight /5 162/ Yahtzee /1 1296//83 6993/ type 1 2 3 overall 2 of a kind 60.19% 60.19% 60.19% 93.69% 3 of a kind 15.43% 4 of a kind 1.93%full house 3.86%sm. straight 12.35% lg. straight 3.09% Yahtzee 0.08% 1.19% See also D ICE Yanghui Triangle PASCAL’S TRIANGLE Yang-Mills Equation The anti-self-dual Yang-Mills equation is the system ofPARTIAL DIFFERENTIAL EQUATIONS @ @¯x1V/C281@V @x1 ! /C27@ @¯x2V/C281@V @x2 ! /C300: References Ablowitz, M. J.; Costa, D. G.; and Tenenblat, K. "Solutions of Multidimensional Extensions of the Anti-Self-Dual Yang-Mills Equation." Stud. Appl. Math. 77,37/C1/46 1987. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997. y-Axis The vertical axis of a 2-D plot in CARTESIAN COORDI- NATES . Physicists and astronomers sometimes call this axis the ORDINATE , although that term is more commonly used to refer to coordinates along the Y- AXIS. See also ABSCISSA ,ORDINATE , X-AXIS, Z-AXIS Yff Center of Congruence Let three ISOSCELIZERS be constructed on a TRIANGLE , one for each side. Now parallel-displace these ISO- SCELIZERS until they concur in a single point. This point is called the Yff center of congruence and has TRIANGLE CENTER FUNCTION a /C30sec1 2 A/C16/C17 : By analogy with the determination of the YFF CENTRAL TRIANGLE , the angle a1is related to the isoscelizer distance l1and the inner triangle sidelengths ti are given by sin1 2 a1/C16/C17 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 cos a1 2s /C3012t2 /C27 t3 ðÞ l1 and so on. Therefore, the length liand tican be determined by solving the six simultaneous equations l2 /C27l3 /C28t1 /C30s1 l1 /C27l3 /C28t2 /C30s2 l1 /C27l2 /C28t3 /C30s3 t2/C27t3 l1 !2 /C3021/C28s2 2/C27s23/C28s21 2s2s3 ! t1/C27t3 l2 !2 /C3021/C28s21/C27s23/C28s22 2s1s3 ! t1/C27t2 l3 !2 /C3021/C28s21/C27s22/C28s23 2s1s2 ! : See also CONGRUENT ISOSCELIZERS POINT ,ISOSCELI- ZER,YFF CENTRAL TRIANGLE References Kimberling, C. "Yff Center of Congruence." http://cedar.e- vansville.edu/~ck6/tcenters/recent/yffcc.html. Yff Central Triangle Let three ISOSCELIZERS be constructed on a TRIANGLE , one for each side. This makes all of the inner triangles SIMILAR to each other. However, there is a unique set of three isoscelizers for which the four interior triangles are congruent. The innermost triangle iscalled the Yff central triangle. Let the side lengths be denoted si;the side lengths of the Yff central triangle ti;and the distances of the ISOSCELIZERS from the vertices li (for i /C301, 2, 3), then the LAW OF COSINES gives cos a1 /C30s2 2 /C27 s23 /C28 s21 2s2s3 and so on, and trigonometry gives sin1 2 a1/C16/C17 /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C28 cos a1 2s /C3012t1 /C27 t2 /C27 t3 ðÞ l1 and so on. Three more equations are obtained by noting that the sums of lengths along each side must sum to that side length. Therefore, the size of the Yff central triangle and the positions of the ISOSCELIZERS can be determined by solving the six simultaneous equations l2 /C27l3 /C28t1 /C30s1 l1 /C27l3 /C28t2 /C30s2 l1 /C27l2 /C28t3 /C30s3 t1 /C27 t2 /C27 t3 l1 !2 /C3021/C28s2 2 /C27 s23 /C28 s21 2s2s3 ! t1 /C27 t2 /C27 t3 l2 !2 /C3021/C28s2 1 /C27 s23 /C28 s22 2s1s3 ! t1 /C27 t2 /C27 t3 l3 !2 /C3021/C28s2 1 /C27 s22 /C28 s23 2s1s2 ! : See also ISOSCELIZER ,YFF CENTER OF CONGRUENCE Yff Points Let points A?; B ?; and C? be marked off some fixed distance x along each of the sides BC, CA, and AB. Then the lines AA?; BB?; and CC? concur in a point U known as the first Yff point if x3 /C30(a /C28x)(b /C28x)(c /C28x) : (1) This equation has a single real root u, which can by obtained by solving the CUBIC EQUATION f(x) /C302x3 /C28px2 /C27qx /C28r /C300 ; (2)where p /C30a /C27b /C27c (3) q /C30ab /C27ac /C27bc (4) r /C30abc : (5) The ISOTOMIC CONJUGATE POINT U ? is called the second Yff point. The TRIANGLE CENTER FUNCTIONS of the first and second points are given by a /C301 ac /C28 u b /C28 u !1=3 (6) and a?/C301 ab /C28 u c /C28 u !1 =3 ; (7) respectively. Analogous to the inequality v 5 p=6 for the BROCARD ANGLE v; u 5p =6 holds for the Yff points, with equality in the case of an EQUILATERAL TRIANGLE . Analogous to v B ai B p /C283 v (8) for i /C301, 2, 3, the Yff points satisfy u Bai Bp /C283u : (9) Yff (1963) gives a number of other interesting proper- ties. The line UU?isPERPENDICULAR to the line containing the INCENTER Iand CIRCUMCENTER O, and its length is given by UU?/C304uIOD u3/C27abc; (10) where Dis the AREA of the TRIANGLE . See also BROCARD POINTS ,YFF TRIANGLES References Yff, P. "An Analog of the Brocard Points." Amer. Math. Monthly 70, 495/C1/501, 1963. Yff Triangles The TRIANGLE DA?B?C?formed by connecting the points used to construct the Y FF POINTS is called the first Yff triangle. The AREA of the triangle is D/C30u3 2R ; where R is the CIRCUMRADIUS of the original TRIAN- GLE DABC : The second Yff triangle is formed by connecting the ISOTOMIC CONJUGATE POINTS of A?; B ?; and C?:/ See also YFF POINTS References Yff, P. "An Analog of the Brocard Points." Amer. Math. Monthly 70, 495 /C1/501, 1963. y-Intercept The point at which a curve or function crosses the Y- AXIS (i.e., when x /C300 in 2-D). See also LINE, X-INTERCEPT Yin-Yang A figure used in many Asian cultures to symbolize the unity of the two "opposite" male and female elements, the "yin" and "yang." The solid and hollow parts composing the symbol are similar and combine to make a CIRCLE . Each part consists of two equal oppositely oriented SEMICIRCLES of radius 1/2 joined at their edges, plus a SEMICIRCLE of radius 1 joining the other edges. See also BASEBALL COVER ,CIRCLE ,PIECEWISE CIR- CULAR CURVE ,SEMICIRCLE References Dixon, R. Mathographics. New York: Dover, p. 11, 1991. Gardner, M. "Mathematical Games: A New Collection of ‘Brain-Teasers."’ Sci. Amer. 203, 172 /C1/180, Oct. 1960. Gardner, M. "Mathematical Games: More About the Shapes that Can Be Made with Complex Dominoes." Sci. Amer. 203, 186 /C1/198, Nov. 1960.Young Diagram FERRERS DIAGRAM ,YOUNG TABLEAU Young Girl-Old Woman Illusion A perceptual ILLUSION in which the brain switches between seeing a young girl and an old woman. See also RABBIT- DUCK ILLUSION References Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 173, 1989. Young Tableau The Young tableau (plural, "tableaux") of a F ERRERS DIAGRAM is obtained by placing the numbers 1, ..., n in the nboxes of the diagram. A "standard" Young tableau is a Young tableau in which the numbers form a nondecreasing sequence along each line andalong each column. For example, the standard Youngtableaux of size n/C303 are given by ff1;2;3gg; ff1;3g;f2gg;ff1;2g;f3gg;and ff1g;f2g;f3gg;illu- strated above. The BUMPING ALGORITHM is used to construct a standard Young tableau from a permuta- tion of f1;...;ng;and the number of standard Young tableaux of size 1, 2, 3, ... are 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (Sloane’s A000085). These num- bers can be generated by the RECURRENCE RELATION a(n)/C30a(n/C281)/C27(n/C281)a(n/C282) with a(1)/C301 and a(2)/C302:This is the same as the number of INVOLUTIONS onnelements (Skiena 1990, p. 32). The number of all possible standard Young tableaux of a given shape can also be considered, and can be calculated with the HOOK LENGTH FORMULA . For example, the illustration above shows the 35 stan- dard tableaux of shape f3; 2; 1; 1g:/ The partitions of integers less than or equal to mn in which there are at most n parts and in which no part is larger than m correspond (1) to Young tableaux which fit inside and m /C29n rectangle and (2) to lattice paths which travel from the upper right corner of the rectangle to the lower left in /m þ n/ leftward and downward steps. The number of Young diagrams fitting inside an m /C29n rectangle is given by the BINOMIAL COEFFICIENTm/C27n m/C0/C1 /C30 m/C27n n/C0/C1 : The above exam- ple shows the 2 /C272 2/C18/C19 /C304 2/C18/C19 /C304! 2!2! /C3024 4/C306 Young 2 /C292 diagrams. There is a correspondence between a PERMUTATION and a pair of Young tableaux, known as the SCHENSTED CORRESPONDENCE . See also BUMPING ALGORITHM ,D URFEE SQUARE , HOOK LENGTH FORMULA ,INVOLUTION (PERMUTA- TION ), PARTITION ,PARTITION FUNCTION P,RANDOMTABLEAU SCHENSTED CORRESPONDENCE ,T ABLEAU CLASS References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637/C1/646. Comtet, L. "Standard Tableaux." Ch. 2, Exercise 26 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 125 /C1/126, 1974. Fulton, W. Young Tableaux with Applications to Representa- tion Theory and Geometry. New York: Cambridge Uni- versity Press, 1997. Kreweras, G. "Sur une class de proble `mes de de ´nombrement lie´s au treillis des partitions d’entiers." Cahiers Buro 6,2/C1/ 107, 1965. Kreweras, G. "De ´nombrements de chemins minimaux a ` sauts impose ´s."Comptes rendus 263,1/C1/3, 1966. Kreweras, G. "Sur une extension du proble `me dir ‘de Simon Newcomb’." Comptes rendus 263,4 3/C1/45, 1966. Kreweras, G. "Traitement simultane ´du ‘proble `me de Young’ et du ‘proble `me de Simon Newcomb’." Cahiers Buro 10, 23/C1/31, 1967. Messiah, A. Appendix D in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, p. 1113, 1961 /C1/ 62. Ruskey, F. "Information on Permutations." http:// www.theory.csc.uvic.ca/~cos/inf/perm/PermInfo.html#Ta-bleau. Skiena, S. "Young Tableaux." §2.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory withMathematica. Reading, MA: Addison-Wesley, pp. 63 /C1 /76, 1990. Skiena, S. S. The Algorithm Design Manual. New York: Springer-Verlag, pp. 254 /C1/255, 1997. Sloane, N. J. A. Sequences A000085/M1221 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam- bridge, England: Cambridge University Press, 1999. Wilf, H. "The Computer-Aided Discovery of a Theorem about Young Tableaux." J. Symb. Comput. 20, 731/C1 /735, 1995. Young’s Inequality Let fbe a real-valued, continuous, and strictly increasing function on [0 ;c] with c/C210. Iff(0)/C300;a/C23 [0;c];andb/C23[0;f(c)];then ga 0f(x)dx/C27gb 0f/C281(x)dx]ab; (1) where f/C281is the INVERSE FUNCTION off. Equality holds IFFb/C30f(a):/ Taking the particular function f(x)/C30xp/C281gives the special case ap p/C27p/C281 p ! bp=(p/C281)]ab; (2) which is often written in the symmetric form ap p/C27bq q]ab; (3) where a;b]0;p/C211, and 1 p /C271 q /C301: (4) References Cooper, R. "Notes on Certain Inequalities. I." J. London Math. Soc. 2,17/C1/21, 1927. Cooper, R. "Notes on Certain Inequalities. II." J. London Math. Soc. 2, 159 /C1/163, 1927. Hardy, G. H.; Littlewood, J. E.; and Po´lya, G. "A Theorem of W. H. Young." §8.3 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 198 /C1/200, 1988. Mitrinovic, D. S. "Young’s Inequality." §2.7 in Analytic Inequalities. New York: Springer-Verlag, pp. 48 /C1/50, 1970. Oppenheim, A. "Note on Mr. Cooper’s Generalization of Young’s Inequality." J. London Math. Soc. 2,21/C1/23, 1927. Riesz, F. "Su alcune disuguaglianze." Boll. Un. Mat. Ital. 7, 77 /C1/79, 1928. Takahashi, T. "Remarks on Some Inequalities." Toˆhoku Math. J. 36,99/C1/106, 1932. Young, W. H. "On Classes of Summable Functions and Their Fourier Series." Proc. Roy. Soc. London Ser. A 87, 225 /C1/ 229, 1912. Young’s Integral Let f(x)bea REAL continuous monotonic strictly increasing function on the interval [0 ; a] with f(0) /C300 and b 5f(a); then ab 5ga 0f(x) dx /C27gb 0f /C281(y) dy; where f /C281(y) is the INVERSE FUNCTION . Equality holds IFF b /C30f(a) :/ References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000. Young’s Lattice Young’s lattice Ypis the PARTIAL ORDER of partitions CONTAINED within a PARTITION pordered by contain- ment (Stanton and White 1986; Skiena 1990, p. 77). See also CONTAINED PARTITION ,PARTITION References Skiena, S. Implementing Discrete Mathematics: Combinato- rics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990. Stanton, D. and White, D. Constructive Combinatorics. New York: Springer-Verlag, 1986. Z Z The DOUBLESTRUCK capital letter Z, Z, denotes the RING of INTEGERS ..., /C282, /C281, 0, 1, 2, .... The symbol derives from the German word Zahl , meaning "num- ber" (Dummit and Foote 1998, p. 1). The RING of integers is sometimes also denoted using the double- struck capital I, I. See also C, C*,COUNTING NUMBER ,I,N,N ATURAL NUMBER ,Q,R,W HOLE NUMBER ,Z/C28,Z/C27 References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 1, 1998. Z/C28 The NEGATIVE INTEGERS ..., /C283, /C282, /C281. See also COUNTING NUMBE R,N ATURAL NUMBER , NEGATIVE ,W HOLE NUMBER ,Z,Z /C27,Z* Z/C27 The POSITIVE INTEGERS 1, 2, 3, ..., equivalent to N. See also COUNTING NUMBER ,N,N ATURAL NUMBER , POSITIVE ,W HOLE NUMBER ,Z,Z -,Z* References Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 1, 1998. Zag Number An EVEN ALTERNATING PERMUTATION number, more commonly called a TANGENT NUMBER . See also ALTERNATING PERMUTATION ,TANGENT NUM- BER,ZIG NUMBER Zak Transform This entry contributed by RONALD M. AARTS The Zak transform is a signal transform relevant to time-continuous signals sampled at a uniform rate and an arbitrary clock phase (Janssen 1988). The Zak transform of a signal can be considered as a mixed time-frequency representation of f (Zf)T(t; n) /C30T1 =2X/C12 k/C30/C28/C12f(t /C27kT)e /C282 piknT for 0 5t 5T and 0 5 n 5T /C281 : The Zak transform is sometimes also known as the Weil-Brezin map. References Brezin, J. "Function Theory on Metabelian Solvmanifolds." J. Funct. Analysis 10,33/C1/51, 1972.Janssen, A. J. E. M. "The Zak Transform: A Signal Trans- form for Sampled Time-Continuous Signals." Philips J. Res. 43,23/C1/69, 1988. Weil, A. "Sur certains groupes d’ope´rateurs unitaires." Acta Math. 111, 143 /C1/211, 1964. Zak, J. Phys. Rev. Lett. 19, 1385, 1967. Zak, J. Phys. Rev. 168, 686, 1968. Zalcman’s Lemma Let f be a family of MEROMORPHIC FUNCTIONS on the UNIT DISK D which are not normal at 0. Then there exist sequences fnin F, zn ; rn ; and a nonconstant function f meromorphic in the plane such that fnzn /C27 rnz ðÞ 0 f(z); locally and uniformly (in the spherical sense) in the COMPLEX PLANE C (Schwick 2000), where zn 0 0 and rn 0 0 :/ References Schwick, W. "A Note on Zalcman’s Lemma." New Zealand J. Math. 29,71/C1/72, 2000. Zalcman, L. "A Heuristic Principle in Complex Function Theory." Amer. Math. Monthly 82, 813 /C1/817, 1975. Zarankiewicz’s Conjecture The CROSSING NUMBER for a COMPLETE BIGRAPH is n 2$% n /C28 1 2$% m 2$% m /C28 1 2$% ; where xbcis the FLOOR FUNCTION . The original proof by Zarankiewicz (1954) contained an error, but was subsequently solved in some special cases by Guy (1969). The conjecture has been shown to be true forallm;n57;and Zarankiewicz has shown that in general, the FORMULA provides an upper bound to the actual number. See also COMPLETE BIGRAPH ,C ROSSING NUMBER (GRAPH ) References Guy, R. K. "The Decline and Fall of Zarankiewicz’s Theo- rem." In Proof Techniques in Graph Theory, Proceedings of the Second Ann Arbor Graph Theory Conference, Ann Arbor, Michigan, 1968. New York: Academic Press, pp. 63 /C1/69, 1969. Zarankiewicz, K. "On a Problem of P. Tura ´n Concerning Graphs." Fund. Math. 41, 137/C1/145, 1954. Zariski Topology ATOPOLOGY of an infinite set whose OPEN SETS have finite complements. The Zariski topology is a TOPOL- OGY which is well-suited for the study of polynomial equations in ALGEBRAIC GEOMETRY , since in Zariski topology, there are many fewer OPEN SETS than in the usual METRIC TOPOLOGY . In fact, the only CLOSED SETS are the ALGEBRAIC SETS , which are the zeros of polynomials. For example, in C ; the only nontrivial closed sets are finite collections of points. In C2 ; there are also the zeros of polynomials such as lines ax /C27by and cusps x2 /C27y3 :/ The Zariski topology is not HAUSDORFF . In fact, any two open sets must intersect, and cannot be DISJOINT . Also, the open sets are DENSE , in the Zariski topology as well as in the usual METRIC TOPOLOGY . Because there are fewer open sets than in the usual topology, it is more difficult for a function to be continuous in Zariski topology. For example, a CON- TINUOUS FUNCTION ðCn ; Zariski) 0(C, metric) must be a constant function. Conversely, when the range has the Zariski topology, it is easier for a function to be CONTINUOUS . In particular, the polynomials are CON- TINUOUS FUNCTIONS Cn ; Zariski) 0 (C ; Zariski ðÞ :/ See also ALGEBRAIC VARIETY ,C ATEGORY THEORY , COMMUTATIVE ALGEBRA ,C ONIC SECTION ,IDEAL , PRIME IDEAL ,PROJECTIVE VARIETY ,SCHEME References Bump, D. Algebraic Geometry. Singapore: World Scientific, pp. 1 /C1/6, 1998. Hartshorne, R. Algebraic Geometry. New York: Springer- Verlag, 1977. Zaslavskii Map The 2-D map xn/C271 /C30 xn /C27 n 1 /C27 myn ðÞ /C27 enm cos 2pxn ðÞ ½/C138 (mod1) yn/C271 /C30e /C28G yn /C27 e cos 2pxn ðÞ ½/C138 ; where m /C131 /C28 e/C28G G (Zaslavskii 1978). It has CORRELATION EXPONENT n : 1:5 (Grassberger and Procaccia 1983) and CAPACITY DIMENSION 1.39 (Russell et al. 1980). References Grassberger, P. and Procaccia, I. "Measuring the Strange- ness of Strange Attractors." Physica D 9, 189 /C1/208, 1983. Russell, D. A.; Hanson, J. D.; and Ott, E. "Dimension of Strange Attractors." Phys. Rev. Let. 45, 1175 /C1/1178, 1980. Zaslavskii, G. M. "The Simplest Case of a Strange Attrac- tor." Phys. Let. 69A, 145 /C1/147, 1978. Zassenhaus-Berlekamp Algorithm A method for factoring POLYNOMIALS .z-Axis The axis in 3-D CARTESIAN COORDINATES which is usually oriented vertically. CYLINDRICAL COORDI- NATES are defined such that the z-axis is the axis about which the azimuthal coordinate u is measured. See also AXIS, X-AXIS, Y-AXIS z-Distribution FISHER’S Z-DISTRIBUTION ,STUDENT’S Z-DISTRIBUTION Zeckendorf Representation A number written as a sum of nonconsecutive FIBONACCI NUMBERS , n /C30XL k /C300ekFk ; where ek are 0 or 1 and ek ek/C271 /C300 : Every POSITIVE INTEGER can be written uniquely in such a form. See also ZECKENDORF’S THEOREM References Grabner, P. J.; Tichy, R. F.; Nemes, I.; and Petho, A. "On the Least Significant Digit of Zeckendorf Expansions." Fib. Quart. 34, 147 /C1/151, 1996. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 40, 1991. Zeckendorf, E. "Repre ´sentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas." Bull. Soc. Roy. Sci. Lie`ge 41, 179 /C1/182, 1972. Zeckendorf’s Theorem The SEQUENCE Fn /C281 fg is COMPLETE even if re- stricted to subsequences which contain no two con- secutive terms, where Fnis a F IBONACCI NUMBER . See also FIBONACCI DUAL THEOREM ,ZECKENDORF REPRESENTATION References Brown, J. L. Jr. "Zeckendorf’s Theorem and Some Applica- tions." Fib. Quart. 2, 163/C1/168, 1964. Keller, T. J. "Generalizations of Zeckendorf’s Theorem." Fib. Quart. 10,9 5/C1/112, 1972. Lekkerkerker, C. G. "Voorstelling van natuurlijke getallen door een som van Fibonacci." Simon Stevin 29, 190/C1/195, 1951/C1/52. Zeeman’s Paradox There is only one point in front of a PERSPECTIVE drawing where its three mutually PERPENDICULAR VANISHING POINTS appear in mutually PERPENDICU- LAR directions, but such a drawing nonetheless appears realistic from a variety of distances and angles. See also LEONARDO’S PARADOX ,PERSPECTIVE ,VAN- ISHING POINT References Dixon, R. Mathographics. New York: Dover, p. 82, 1991. Zeilberger-Bressoud Theorem Dyson (1962abc) conjectured that the constant term in the LAURENT SERIES Y 1 5i"j5n1 /C28xi xj !ai (1) is a1 /C27 a2 /C27 ... /C27 an ðÞ ! a1!a2!...an!; (2) based on a problem in particle physics. The theorem is called DYSON’S CONJECTURE , and was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970). A q-analog of this theorem (Andrews 1975) states that the coefficient of x0 1 x02...x0nin Y 1 5i"j5nxi xjeij; q ! ai(3) where eij /C131 for i Bj q for i > j/C26 (4) is given by (q; q)a1 /C27a2 /C27... /C27an (q; q)a1(q; q)a2/C1/C1/C1(q; q)an: (5) This can also be stated in the form that the constant term of Y 1 5i Bj5n1 /C28xi =xj/C0/C1 1 /C28qxi =qj/C0/C1 /C1/C1/C1 1 /C28qai/C281xi =xj/C0/C1 /C29 1 /C28qxj =xi/C0/C1 1 /C28q2xj =xi/C0/C1 /C1/C1/C1 1 /C28qaj xj =xi/C0/C1 ; (6) is the Q-MULTINOMIAL COEFFICIENT a1 /C27/C1/C1/C1/C27 an ½/C138 ! a1½/C138! /C1/C1/C1 an½/C138!; (7) wheren½/C138! /C13(1)(1 /C27q) /C1/C1/C1 1 /C27q /C27/C1/C1/C1/C27qn/C281/C0/C1 : (8) The amazing proof of this theorem was given by Zeilberger and Bressoud (1985). The full theorem reduces to Dyson’s version when q /C301. It also gives the Q-ANALOG of DIXON’S THEOREM as X/C12 k /C30/C28/C12(/C281)kqk(3k /C271)=2 b /C27c c /C27k/C18/C19 c /C27a a /C27k/C18/C19 a /C27b b /C27k/C18/C19 ¼(q;q)a/C27b/C27c (q;q)a(q;q)b(q;q)c(9) (Andrews 1975, 1986). With q/C301 and a/C30b/C30c/C30p;it gives the beautiful and well-known identity X2p k/C300(/C281)k2p k/C18/C193 /C30(/C281)p(3p)! (p!)3(10) (Andrews 1986). See also DIXON’S THEOREM , Q-MULTINOMIAL COEFFI- CIENT ,M ACDONALD’S CONSTANT- TERM CONJECTURE , MULTINOMIAL COEFFICIENT References Andrews, G. E. "Problems and Prospects for Basic Hyper- geometric Functions." In The Theory and Application of Special Functions (Ed. R. Askey). New York: Academic Press, pp. 191 /C1/224, 1975. Andrews, G. E. "The Zeilberger-Bressoud Theorem." §4.3 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 36 /C1/38, 1986. Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. I." J. Math. Phys. 3, 140/C1/156, 1962a. Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. II." J. Math. Phys. 3, 157/C1/165, 1962b. Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. III." J. Math. Phys. 3, 166/C1/175, 1962c. Good, I. J. "Short Proof of a Conjecture by Dyson." J. Math. Phys. 11, 1884, 1970. Gunson, J. "Proof of a Conjecture of Dyson in the Statistical Theory of Energy Levels." J. Math. Phys. 3, 752/C1/753, 1962. Wilson, K. G. "Proof of a Conjecture by Dyson." J. Math. Phys. 3, 1040 /C1/1043, 1962. Zeilberger, D. and Bressoud, D. M. "A Proof of Andrews’ q- Dyson Conjecture." Disc. Math. 54, 201/C1/224, 1985. Zeilberger’s Algorithm An ALGORITHM which finds a POLYNOMIAL recurrence for terminating HYPERGEOMETRIC IDENTITIES OF THE FORM X kn k/C18/C19QA i/C301(ain/C27a?ik/C27aƒi)!QB i/C301(bin/C27b?ik/C27bƒi)!zk /C30CQ¯A i/C301(¯ain/C27¯a?i)! Q¯B i/C301(¯bin/C27¯b?i)¯xn; wheren k/C0/C1 is a BINOMIAL COEFFICIENT ,ai;a?i;¯ai;bi;b?i; ¯bi are constant integers and a ƒi ; ¯a ?i ; bƒi ; ¯b?i ; C, x, and z are complex numbers (Zeilberger 1990). The method was called CREATIVE TELESCOPING by van der Poorten (1979), and led to the development of the amazing machinery of WILF-ZEILBERGER PAIRS . The also exists a q-analog of the algorithm, called the Q-ZEILBERGER ALGORITHM . See also BINOMIAL SERIES ,BINOMIAL SUMS,GOSPER’S ALGORITHM ,HYPERGEOMETRIC IDENTITY , Q-ZEILBER- GER ALGORITHM ,SISTER CELINE’S METHOD ,W ILF- ZEILBERGER PAIR References Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Koepf, W. "Algorithms for m-fold Hypergeometric Summa- tion." J. Symb. Comput. 20, 399/C1/417, 1995. Koepf, W. "Zeilberger’s Algorithm." Ch. 7 in Hypergeometric Summation: An Algorithmic Approach to Summation andSpecial Function Identities. Braunschweig, Germany: Vieweg, pp. 93 /C1 /123, 1998. Krattenthaler, C. "HYP and HYPQ: The Mathematica Package HYP." http://radon.mat.univie.ac.at/People/kratt/hyp_hypq/hyp.html. Paule, P. and Riese, A. "A Mathematica q -Analogue of Zeilberger’s Algorithm Based on an Algebraically Moti-vated Approach to q-Hypergeometric Telescoping." In Special Functions, q -Series and Related Topics, Fields Institute Communications 14, 179/C1 /210, 1997. Paule, P. and Schorn, M. "A Mathematica Version of Zeilberger’s Algorithm for Proving Binomial Coefficient Identities." J. Symb. Comput. 20, 673/C1/698, 1995. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Zeilberger’s Algorithm." Ch. 6 in A/C30B.Wellesley, MA: A. K. Peters, pp. 101 /C1/119, 1996. Riese, A. "A Generalization of Gosper’s Algorithm to Bibasic Hypergeometric Summation." Electronic J. Combinatorics 1, R19 1 /C1/16, 1996. http://www.combinatorics.org/Vo- lume_1/volume1.html#R19. van der Poorten, A. "A Proof that Euler Missed... Ape ´ry’s Proof of the Irrationality of z(3):/"Math. Intel. 1, 196/C1/203, 1979. Wegschaider, K. Computer Generated Proofs of Binomial Multi-Sum Identities. Diploma Thesis, RISC. Linz, Aus- tria: J. Kepler University, May 1997. Zeilberger, D. "Doron Zeilberger’s Maple Packages and Programs: EKHAD." http://www.math.temple.edu/~zeil- berg/programs.html. Zeilberger, D. "A Fast Algorithm for Proving Terminating Hypergeometric Series Identities." Discrete Math. 80, 207/C1/211, 1990. Zeilberger, D. "A Holonomic Systems Approach to Special Function Identities." J. Comput. Appl. Math. 32, 321/C1/368, 1990. Zeilberger, D. "The Method of Creative Telescoping." J. Symb. Comput. 11, 195/C1/204, 1991. Zeisel Number A number N/C30p1p2/C1/C1/C1pnwhere the pi/s are distinct PRIMES andn]3 such that pi/C30Api/C281/C27B fori/C301, 2, ..., n,p0taken as 1, and with Aand B some fixed integers. For example, 1885 /C301/C2155/C21513 /C21529 is a Zeisel number with ( A;B)/C30(2;3) since 5/C302/C2151/C273; 13/C302/C2155/C273; 29/C302/C21513/C273; as is 114985 /C301/C2155/C21513 /C21529 /C21561 since 5/C302/C2151/C273; 13/C302/C2155/C273; 29/C302/C21513/C273; 61/C302/C21529/C273: The first few Zeisel numbers are 105, 1419, 1729, 1885, 4505, ... (Sloane’s A051015), which correspondto constants (1, 2), (4, /C281), (1, 6), (2, 3), (3, 2), .... References Brown, K. S. "Zeisel Numbers." http://www.seanet.com/ ~ksbrown/kmath015.htm. Sloane, N. J. A. Sequences A051015 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.re- search.att.com/~njas/sequences/eisonline.html. Zenithal Projection AZIMUTHAL PROJECTION Zeno’s Paradoxes A set of four PARADOXES dealing with counterintuitive aspects of continuous space and time. 1. Dichotomy paradox: Before an object can travel a given distance d, it must travel a distance d=2: In order to travel d=2;it must travel d=4;etc. Since this sequence goes on forever, it therefore appears that the distance dcannot be traveled. The resolution of the paradox awaited CALCULUS and the proof that infinite GEOMETRIC SERIES such as a/C12 i/C301(1=2)i/C301 can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed totraverse the distances. 2. Achilles and the tortoise paradox: A fleet-of-foot Achilles is unable to catch a plodding tortoisewhich has been given a head start, since during the time it takes Achilles to catch up to a given position, the tortoise has moved forward somedistance. But this is obviously fallacious sinceAchilles will clearly pass the tortoise! The resolu- tion is similar to that of the dichotomy paradox. 3. Arrow paradox: An arrow in flight has aninstantaneous position at a given instant of time. At that instant, however, it is indistinguishable from a motionless arrow in the same position, so how is the motion of the arrow perceived? 4. Stade paradox: A paradox arising from the assumption that space and time can be divided only by a definite amount. References Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 218 /C1/220, 1998. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 163 /C1/166, 1984. Gru¨nbaum, A. Modern Science and Zeno’s Paradoxes. Middletown, CT: Wesleyan University Press, 1967. Pappas, T. "Zeno’s Paradox--Achilles & the Tortoise." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, pp. 116 /C1/117, 1989. Russell, B. Our Knowledge and the External World as a Field for Scientific Method in Philosophy. New York: Routledge, 1993. Salmon, W. (Ed.). Zeno’s Paradoxes. New York: Bobs- Merrill, 1970. Stewart, I. "Objections from Elea." In From Here to Infinity: A Guide to Today’s Mathematics. Oxford, England: Oxford University Press, p. 72, 1996. vos Savant, M. The World’s Most Famous Math Problem. New York: St. Martin’s Press, pp. 50 /C1/55, 1993. Zermelo Set Theory The version of set theory obtained if Axiom 6 of ZERMELO- FRAENKEL SET THEORY is replaced by 6’. Selection axiom (or "axiom of subsets"): for any set-theoretic formula A(u) ;/C214x /C215y /C214u(u /C23 y /C13/ /u /C23 x fflA(u));/ which can be deduced from Axiom 6. However, there seems to be some disagreement in the literature about just which axioms of ZERMELO- FRAENKEL SET THEORY constitute "Zermelo Set Theory." Mendelson (1997) does not include the AXIOMS OF CHOICE , FOUNDATION , REPLACEMENT In Zermelo set theory, but does includes 6’. However, Enderton (1977) includes the AXIOMS OF CHOICE and FOUNDATION , but does not include the AXIOMS OF REPLACEMENT or Selection. See also SET THEORY ,ZERMELO- FRAENKEL SET THE- ORY References Enderton, H. B. Elements of Set Theory. New York: Aca- demic Press, 1977. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997. Iyanaga, S. and Kawada, Y. (Eds.). "Zermelo-Fraenkel Set Theory." §35B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 135, 1980. Zermelo, E. "U¨ ber Grenzzahlen und Mengenbereiche." Fund. Math. 16,29/C1/47, 1930.Zermelo-Fraenkel Axioms The Zermelo-Fraenkel axioms are the basis for ZERMELO- FRAENKEL SET THEORY . In the following (Iyanaga and Kawada 1980), /C215 stands for EXISTS , ~ for does not exist, /C23 for "is an element of," ¥ for the EMPTY SET, /C214 for FOR ALL, [ for IMPLIES , ! for NOT (NEGATION ), fflfor AND, /C150for OR, /C13for "is EQUIVALENT to," and S denotes the union y of all the sets that are the elements of x. 1. AXIOM OF EXTENSIONALITY : /C214x(x /C23 a /C13/ /x /C23 b) [a /C30b:/ 2. AXIOM OF THE UNORDERED PAIR: /C215x /C214y(y /C23 x /C13/ /y /C30a /C150y /C30b) :/ 3. AXIOM OF THE SUM SET: /C215x /C214y(y /C23 x /C13/C215z /C23 a(y /C23 z)) :/ 4. AXIOM OF THE POWER SET: /C214x /C215yy /C23 x /C13/ //C214z /C23 y(z /C23 a)):/ 5. AXIOM OF THE EMPTY SET: /C215x /C214y(!y /C23 x) :/ 6. AXIOM OF INFINITY : /C215x( ¥/C23 x /C150/C214y /C23 x(y?/C23 x)):/ 7. AXIOM OF SEPARATION : /C215x /C214y(y /C23 x /C13y /C23 a fflA(y)):/ 8. AXIOM OF REPLACEMENT (or axiom of compre- hension, or axiom of subsets): /C215x/C214y/C23/ /a(/C215zA(y;z)[/C215z/C23xA(y;z)):/ 9. Axiom of regularity (or AXIOM OF FOUNDATION ): /C215xA(x)[/C215x(A(x)ffl/C214y/C23x(!A(y))):/ 10. A XIOM OF CHOICE :/C214x/C23a/C215A(x;y)[/ //C215y/C214x/C23aA(x;y(x)):/ The system of axioms 1 /C1/9 is called Z ERMELO- FRAEN- KEL SET THEORY , denoted "ZF." The system of axioms 1/C1/9 minus the AXIOM OF REPLACEMENT (i.e., axioms 1/C1/7 plus 8) is called Z ERMELO SET THEORY , denoted "Z." The set of axioms 1 /C1/9 plus the AXIOM OF CHOICE is usually denoted "ZFC." However, note that there seems to be some disagree- ment in the literature about just what axiomsconstitute "Z ERMELO SET THEORY ." Mendelson (1997) does notinclude the AXIOMS OF CHOICE ,FOUNDATION , orREPLACEMENT in Zermelo set theory, but does include the AXIOM OF REPLACEMENT . However, En- derton (1977) includes the AXIOMS OF CHOICE and FOUNDATION , but does not include the AXIOM OF REPLACEMENT . Abian (1969) proved CONSISTENCY and independence of four of the Zermelo-Fraenkel axioms. See also AXIOM OF CHOICE ,AXIOM OF FOUNDATION , AXIOM OF REPLACEMENT ,SET THEORY , VON NEU- MANN -BERNAYS- GO¨ DEL SET THEORY ,Z ERMELO- FRAENKEL SET THEORY ,ZERMELO SET THEORY References Abian, A. "On the Independence of Set Theoretical Axioms." Amer. Math. Monthly 76, 787/C1/790, 1969. Enderton, H. B. Elements of Set Theory. New York: Aca- demic Press, 1977. Itoˆ, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146 /C1/148, 1986. Iyanaga, S. and Kawada, Y. (Eds.). "Zermelo-Fraenkel Set Theory." §35B in Encyclopedic Dictionary of Mathematics, Vol. 1. Cambridge, MA: MIT Press, pp. 134 /C1/135, 1980. Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997. Zermelo, E. "U¨ ber Grenzzahlen und Mengenbereiche." Fund. Math. 16,29/C1/47, 1930. Zermelo-Fraenkel Set Theory A version of SET THEORY which is a formal system expressed in first-order predicate LOGIC . Zermelo- Fraenkel set theory is based on the ZERMELO- FRAEN- KEL AXIOMS . ZERMELO- FRAENKEL SET THEORY is not finitely axio- matized. For example, the AXIOM OF REPLACEMENT is not really a single axiom, but an infinite family of axioms, since it is preceded by the stipulation that it is true "For any set-theoretic formula A(u; v) :/" Mon- tague (1961) proved that ZERMELO- FRAENKEL SET THEORY is not finitely axiomatizable, i.e., there is no finite set of axioms which is logically equivalent to the infinite set of Z ERMELO- FRAENKEL AXIOMS .VON NEU- MANN- BERNAYS- GO¨DEL SET THEORY provides an equivalent finitely axiomized system. See also LOGIC ,SET THEORY , VON NEUMANN- BER- NAYS- GO¨ DEL SET THEORY ,ZERMELO- FRAENKEL AX- IOMS ,ZERMELO SET THEORY References Montague, R. "Semantic Closure and Non-Finite Axiomatiz- ability. I." In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2/C1/9 September 1959). Oxford, England: Pergamon, pp. 45 /C1/69, 1961. Zermelo, E. "U ¨ber Grenzzahlen und Mengenbereiche." Fund. Math. 16,2 9/C1/47, 1930. Zermelo’s Axiom of Choice AXIOM OF CHOICE Zernike Polynomial ORTHOGONAL POLYNOMIALS which arise in the expan- sion of a wavefront function for optical systems with circular pupils. The ODD and EVEN Zernike polyno- mials are given by oUm n(r;f) eUm n(r;f)/C30Rm n(r)sin cos(mf) (1) with radial function Rm n(r)/C30X(n/C28m)=2 i/C300(/C281)l(n/C28l)! l!1 2(n/C27m)/C28lhi !12(n/C28m)/C28lhi !rn/C282l ð2Þ fornandmintegers with n]m]0 and n/C28mEVEN . Otherwise, Rm n(r)/C300: (3)Here, fis the azimuthal angle with 0 5fB2pandr is the radial distance with 0 5r51 (Prata and Rusch 1989). The radial functions satisfy the orthogonality relation g1 0Rm n(r)Rmn?(r)rdr/C301 2(n/C271)dnn?; (4) where dijis the K RONECKER DELTA , and are related to the B ESSEL FUNCTION OF THE FIRST KIND by g1 0Rmn(r)Jm(vr)rdr/C30(/C281)(n/C28m)=2Jn/C271(v) v(5) (Born and Wolf 1989, p. 466). The radial Zernike polynomials have the GENERATING FUNCTION 1/C27z/C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C282z1/C282r2 ðÞ /C27z2p/C2/C3 m (2zr)mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C282z1/C282r2 ðÞ /C27z2p /C30X/C12 s/C300zsR9m m/C272s(r); (6) and are normalized so that R9m n(1)/C301 (7) (Born and Wolf 1989, p. 465). The first few NONZERO radial polynomials are R0 0(r)/C301 R11(r)/C30r R02(r)/C302r2/C281 R22(r)/C30r2 R13(r)/C303r3/C282r R33(r)/C30r3 R04(r)/C306r4/C286r2/C271 R24(r)/C304r4/C283r2 R44(r)/C30r4 (Born and Wolf 1989, p. 465). The Zernike polynomial is a special case of the J ACOBI POLYNOMIAL with P(a;b) n?(x)/C30(/C281)n?Rm n(r) ra(8) and x/C301/C282r2(9) b/C300 (10) a/C30m (11) n?/C301 2(n/C28m): (12) The Zernike polynomials also satisfy the RECURRENCE RELATIONS rRm n ( r) /C301 2(n /C27 1) /C2 (n /C27m /C272)Rm/C271 n /C271 (r) /C27(n /C28m)Rm/C281 n/C281 ( r)/C2/C3 (13) Rmn /C272( r) /C30n /C27 2 (n /C27 2)2 /C28 m2/C26/C20 4(n /C271)r2 /C28(n /C27 m)2 n /C28(n /C28 m /C27 2)2 n /C27 2/C21 Rmn ( r) /C28n2 /C28 m2 nRmn/C282( r) (14) Rmn (r) /C27Rm/C272 n( r) /C301 n /C27 1dRm/C271 n/C271 ( r) /C28 Rm/C271 n /C281 (r)/C2/C3 dr (15) (Prata and Rusch 1989). The coefficients Am nand Bmn in the expansion of an arbitrary radial function F( r; f) in terms of Zernike polynomials F(r ; f) /C30X/C12 m/C300X/C12 n/C30mAm noUm n ( r ; f) /C27Bmne Um n ( r ; f) ½/C138 (16) are given by Am n Bmn/C30(n /C27 1) e2 mn pg1 0g2 p 0F( r ; f)oUm n ( r ; f) eUm n (r ; f) r df dr; (17) where emn /C13e /C131ffiffiffi 2p for m /C300; n "0 1 otherwise8 < : (18) Let a "primary" aberration be given by F/C30a ?lmn¯Y2l/C27m 1( u; f) rn cosm u (19) with 2l /C27m /C27n /C304 and where ¯Y is the COMPLEX CONJUGATE of Y, and define A?lmn /C30a ?lmn¯Y2l /C27m 1( u; f) ; (20) giving F/C301 e2 nmAlmnRm n ( r) cos(mu) : (21) Then the types of primary aberrations are given in the following table (Born and Wolf 1989, p. 470). Aberration lmnA /A?/ spherical aberration04 0 /A?040 r4// eA040R0 4(r)/ coma 0 3 1 /A?031r3cosu//A031R13(r) cos u/ astigmatism 0 2 2 /A?022r2cos2u//A022R22(r) cos(2 u)/field curvature12 0 /A?120r2// eA120R0 2r/ distortion 1 1 1 /A?111rcosu//A111R1 1(r) cos u/ See also JACOBI POLYNOMIAL References Bezdidko, S. N. "The Use of Zernike Polynomials in Optics." Sov. J. Opt. Techn. 41, 425, 1974. Bhatia, A. B. and Wolf, E. "On the Circle Polynomials of Zernike and Related Orthogonal Sets." Proc. Cambridge Phil. Soc. 50, 40, 1954. Born, M. and Wolf, E. "The Diffraction Theory of Aberra- tions." Ch. 9 in Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. New York: Pergamon Press, pp. 459 /C1/490, 1989. Mahajan, V. N. "Zernike Circle Polynomials and Optical Aberrations of Systems with Circular Pupils." In Engi- neering and Lab. Notes 17(Ed. R. R. Shannon), p. S-21, Aug. 1994. Prata, A. and Rusch, W. V. T. "Algorithm for Computation of Zernike Polynomials Expansion Coefficients." Appl. Opt. 28, 749/C1/754, 1989. Wang, J. Y. and Silva, D. E. "Wave-Front Interpretation with Zernike Polynomials." Appl. Opt. 19, 1510 /C1/1518, 1980. Wyant, J. C. "Zernike Polynomials." http://wyant.opt-sci.ar- izona.edu/zernikes/zernikes.htm. Zernike, F. "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode."Physica 1, 689/C1 /704, 1934. Zhang, S. and Shannon, R. R. "Catalog of Spot Diagrams." Ch. 4 in Applied Optics and Optical Engineering, Vol. 11. New York: Academic Press, p. 201, 1992. Zero The INTEGER denoted 0 which, when used as a counting number, means that no objects are present. It is the only INTEGER (and, in fact, the only REAL NUMBER ) which is neither NEGATIVE nor POSITIVE .A number which is not zero is said to be NONZERO .A ROOT of a function fis also sometimes known as "a zero of f." Because the number of PERMUTATIONS of 0 elements is 1, 0! (zero FACTORIAL ) is defined as 1 (Wells 1986, p. 31). This definition is useful in expressing many mathematical identities in simple form. A number other than 0 taken to the POWER 0 is defined to be 1, but 00is undefined. Defining 00/C301 allows some formulas to be expressed simply (Knuth 1997,p. 56), although the same could be said for thealternate definition 0 0/C300 (Wells 1986, p. 26). An example of a formula which can be expressed con-cisely by defining 0 0/C301 is the beautiful analytical formula for the integral of the generalized SINC FUNCTION g/C12 0sinax xbdx/C30p1/C28c(/C281)(a/C28b)=2 bc 2a/C28c(b/C281)! /C29Xa =2bc/C28c k /C300(/C281)k a k/C18/C19 (a /C282k)b /C281 ln(a /C282k) ½/C138 c given by Kogan, where a ]b > c ; c /C13a /C28b (mod 2); and xbcis the FLOOR FUNCTION . The following table gives the first few numbers n such that the decimal expansion of kn contains no zeros, for small k. The largest known n for which 2n contain no zeros is 86 (Madachy 1979), with no other n 54:6 /C29107 (M. Cook), improving the 3:0739 /C29107 limit obtained by Beeler and Gosper (1972). The values a(n) such that the positions of the right-most zero in 2a(n) increases are 10, 20, 30, 40, 46, 68, 93, 95, 129, 176, 229, 700, 1757, 1958, 7931, 57356, 269518, ... (Sloane’s A031140). The positions in which the right-most zeros occur are 2, 5, 8, 11, 12, 13, 14, 23, 36, 38, 54, 57, 59, 93, 115, 119, 120, 121, 136, 138, 164, ... (Sloane’s A031141). The right-most zero of 2781 ;717 ;865occurs at the 217th decimal place, the farthest over for powers up to 2 :5 /C29109 :/ k Sloane n such that kn contains no 0s 2 Sloane’s A0073771, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, ... 3 Sloane’s A0307001, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, ... 4 Sloane’s A0307011, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43, ... 5 Sloane’s A0088391, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58, ... 6 Sloane’s A0307021, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44, ... 7 Sloane’s A0307031, 2, 3, 6, 7, 10, 11, 19, 35 8 Sloane’s A0307041, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27 9 Sloane’s A0307051, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34 11 Sloane’s A0307061, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 41, ... While it has not been proven that the numbers listed above are the only ones without zeros for a given base, the probability that any additional ones exist is vanishingly small. Under this assumption, the se- quence of largest n such that kn contains no zeros for k /C302, 3, ... is then given by 86, 68, 43, 58, 44, 35, 27, 34, 0, 41, ... (Sloane’s A020665). See also 10,APPROXIMATE ZERO,DIVISION BY ZERO, FALLACY ,N AUGHT ,N EGATIVE ,N ONNEGATIVE ,N ON- ZERO ,ONE,POSITIVE ,TWO,ZEROFREEReferences Beeler, M. and Gosper, R. W. Item 57 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 22, Feb. 1972. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison- Wesley, p. 56, 1997. Kogan, S. "A Note on Definite Integrals Involving Trigono- metric Functions." http://www.mathsoft.com/asolve/con- stant/pi/sin/sin.html. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 127 /C1/128, 1979. Pappas, T. "Zero-Where & When." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 162, 1989. Sloane, N. J. A. Sequences A007377/M0485 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 23 /C1/ 26, 1986. Zero (Root) ROOT Zero Divisor A NONZERO element x of a RING for which x /C215 y /C300; where y is some other NONZERO element and the multiplication x /C215 y is the multiplication of the RING . A RING with no zero divisors is known as an INTEGRAL DOMAIN . Let Adenote an R/-algebra, so that Ais a VECTOR SPACE over Rand A/C29A0A (x;y)/C2x/C215y: Now define Z/C13x/C23A:x/C215y/C300 for some nonzero y/C23A fg ; where 0 /C23Z:Ais said to be m-ASSOCIATIVE if there exists an m-dimensional SUBSPACE SofAsuch that (y/C215x)/C215z/C30y/C215(x/C215z) for all y;z/C23Aandx/C23S:Ais said to be TAME ifZis a finite union of SUBSPACES ofA. References Finch, S. "Unsolved Mathematics Problems: Zero Structures in Real Algebras." http://www.mathsoft.com/asolve/zero- div/zerodiv.html. Zero Irrelevancy Proof CLASSIFICATION THEOREM OF SURFACES Zero Map See also IDENTITY MAP Zero Matrix AMATRIX consisting of all 0s, denoted 0 :The MATRIX EXPONENTIAL of 0 is given by the IDENTITY MATRIX I: Anm/C29nzero matrix can be generated using Zer- oMatrix [m,n] in the Mathematica add-on pack- ageLinearAlgebra‘MatrixMultiplication‘ (which can be loaded with the command BBLinearAlgebra‘ ). See also IDENTITY MATRIX Zero Section This entry contributed by RYAN BUDNEY The zero section of a VECTOR BUNDLE is the SUBMANI- FOLD of the bundle that consists of all the ZERO VECTORS . See also BUNDLE ,M ANIFOLD ,S ECTION (BUNDLE ), VECTOR BUNDLE ,ZERO VECTOR Zero Set If f is a function on an OPEN SET U, then the zero set of f is the set Z /C30 z /C23 U : f(z) /C300 fg :/ References Krantz, S. G. Handbook of Complex Analysis. Boston, MA: Birkha ¨user, p. 268, 1999. Zero Vector A ZERO VECTOR , denoted 0; is a VECTOR of length 0, and thus has all components equal to zero. See also UNIT VECTOR ,ZERO VECTOR Zero-Form See also DIFFERENTIAL K-FORM,O NE-FORM,T WO- FORM Zerofree An integer whose decimal digits contain no zeros is said to be zerofree. Zerofree squares are easy to generate, e.g., 33333333333333342 /C3011111111111111115555555555555556 : (1) Around 1990, D. Hickerson considered the problem of finding large zerofree cubes. After some experimenta- tion, he found a formula that generated infinitely many of them. In March 1998, Bill Gosper asked about 0-free nth powers, pointing out that heuristi- cally we should expect there to be infinitely many zerofree squares, cubes, ..., 21st powers, but only finitely many 22nd powers, etc. At this point, Hick- erson couldn’t locate his formula for cubes, and so came up with the new formulaf(n) /C302 /C215 105n /C28 104n /C27 17 /C215 103n/C281 /C27 102n /C27 10n/C282 3 ; (2) which is 0-free if n /C132 ðmod 3 Þ and n ]5:/ In April 1999, Ed Pegg conjectured onsci.math that there are only finitely many zerofree cubes, so Hickerson posted his new counterexample, (mista- kenly claiming that it was the one he had found 10 years ago). A few days later, Lew Baxter posted the slightly simpler example f(n) /C301 3(2 /C215 105n /C28104n /C272 /C215 103n /C27102n /C2710n /C271); (3) known as the BAXTER- HICKERSON FUNCTION . There is apparently no proof that there exist infi- nitely many zerofree 4th powers, 5th powers, ..., or 21st powers. See also BAXTER- HICKERSON FUNCTION ,ZERO Zero-Sum Game AGAME in which players make payments only to each other. One player’s loss is the other player’s gain, so the total amount of "money" available remains con- stant. See also FINITE GAME,GAME References Dresher, M. The Mathematics of Games of Strategy: Theory and Applications. New York: Dover, p. 2, 1981. Zeta HURWITZ ZETAFUNCTION ,RIEMANN ZETAFUNCTION Zeta Fuchsian The zeta Fuchsians are class of functions discovered by Poincare ´which are related to the AUTOMORPHIC FUNCTIONS . See also AUTOMORPHIC FUNCTION Zeta Function A function satisfying certain properties which is computed as an INFINITE SUM ofNEGATIVE POWERS . The most commonly encountered zeta function is theR IEMANN ZETA FUNCTION , z(n)/C13X/C12 k/C3011 kn: See also DEDEKIND FUNCTION ,D IRICHLET BETA FUNCTION ,DIRICHLET ETA FUNCTION ,DIRICHLET L- SERIES ,DIRICHLET LAMBDA FUNCTION ,EPSTEIN ZETA FUNCTION ,JACOBI ZETA FUNCTION ,NINT ZETA FUNC- TION ,PERIODIC ZETA FUNCTION ,PRIME ZETA FUNC- TION ,R IEMANN ZETA FUNCTION ,S ELBERG ZETA FUNCTION References Ireland, K. and Rosen, M. "The Zeta Function." Ch. 11 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 151 /C1/171, 1990. Zeuthen’s Rule On an ALGEBRAIC CURVE , the sum of the number of coincidences at a noncuspidal point C is the sum of the orders of the infinitesimal distances from a nearby point P to the corresponding points when the distance PC is taken as the principal infinitesi- mal. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 131, 1959. Zeuthen’s Theorem If there is a ( n; n ?) correspondence between two curves of GENUS p and p ? and the number of BRANCH POINTS properly counted are b and b?; then b /C272n ?(p /C281) /C30 b?/C272n(p?/C281): See also CHASLES- CAYLEY- BRILL FORMULA References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 246, 1959. Zig Number An ODD ALTERNATING PERMUTATION number, more commonly called an EULER NUMBER or SECANT NUM- BER. See also ALTERNATING PERMUTATION ,EULER NUM- BER,ZAG NUMBER Zigzag Permutation ALTERNATING PERMUTATION Zig-Zag Triangle SEIDEL- ENTRINGER- ARNOLD TRIANGLE Zillion A generic word for a very LARGE NUMBER . The term has no WELL DEFINED mathematical meaning. Con- way and Guy (1996) define the nth zillion as 103n/C273 in the American system million /C30106 ; ð billion /C30109 ; trillion /C301012 ; ...); and 106n in the British systemmillion /C30106 ; ð billion /C301012 ; trillion /C301018 ; ...); Con- way and Guy (1996) also define the words N-PLEX and N-MINEX for 10n and 10/C28n ; respectively. See also LARGE NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 13 /C1/16, 1996. Zip Half a ZIP-PAIR . ZIP Proof CLASSIFICATION THEOREM OF SURFACES Zipf’s Law In the English language, the probability of encounter- ing the rth most common word is given roughly by P(r) /C300 :1 =r for r up to 1000 or so. The law breaks down for less frequent words, since the HARMONIC SERIES diverges. Pierce’s (1980, p. 87) statement that a P(r) > 1 for r /C308727 is incorrect. Goetz states the law as follows: The frequency of a word is inversely proportional to its RANK rsuch that P(r):1 rln(1 :78R); where Ris the number of different words. See also HARMONIC SERIES ,RANK (STATISTICS ) References Bogomolny, A. "Benford’s Law and Zipf’s Law." http:// www.cut-the-knot.com/do_you_know/zipfLaw.html. Goetz, P. "Phil’s Good Enough Complexity Dictionary." http://www.cs.buffalo.edu/~goetz/dict.html. Li, W. "Zipf’s Law." http://linkage.rockefeller.edu/wli/zipf/. Pierce, J. R. Introduction to Information Theory: Symbols, Signals, and Noise, 2nd rev. ed. New York: Dover, pp. 86 /C1/ 87 and 238 /C1/239, 1980. Zip-Pair A pair of zips, each ZIPbeing half a zipper, which can be zippered up to close a surface along a curve. The concept of a zip-pair can be extremely useful intopological arguments, and zips can be used to illustrate the construction of the CAP,CROSS-CAP , HANDLE , and CROSS-HANDLE . See also ZIP References Francis, G. K. and Weeks, J. R. "Conway’s ZIP Proof." Amer. Math. Monthly 106, 393/C1/399, 1999. Z-Number A Z-number is a REAL NUMBER z such that 0 5frac3 2 !k j2 435B 1 2 for all k /C301, 2, ..., where frac /(x) is the fractional part of x. Mahler (1968) showed that there is at most one Z-number in each interval [n ; n /C271) for integer n, and therefore concluded that it is unlikely that any Z- numbers exist. The Z-numbers arise in the analysis of the COLLATZ PROBLEM . See also COLLATZ PROBLEM References Flatto, L. "Z-Numbers and b/-Transformations." Symbolic Dynamics and its Applications, Contemporary Math. 135, 181 /C1/201, 1992. Guy, R. K. "Mahler’s Z-Numbers." §E18 in Unsolved Pro- blems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 220, 1994. Lagarias, J. C. "The 3x /C271 Problem and its Generalizations." Amer. Math. Monthly 92,3/C1/23, 1985. http://www.cecm.s- fu.ca/organics/papers/lagarias/. Mahler, K. "An Unsolved Problem on the Powers of 3/2." Austral. Math. Soc. 8, 313 /C1/321, 1968. Tijdman, R. "Note on Mahler’s3 2/-Problem." Kongel. Norske Vidensk Selsk. Skr. 16,1/C1/4, 1972. Zo¨llner’s Illusion In this ILLUSION , the VERTICAL lines in the above figure are PARALLEL , but appear to be tilted at an angle. In 1860, F. Zo¨llner sent his discovery in a letter to physicist and scholar J. C. Poggendorff, editor of Annalen der Physik und Chemie , who subsequently discovered the related POGGENDORFF ILLUSION . See also ILLUSION ,POGGENDORFF ILLUSION References IllusionWorks. "Poggendorf [sic]." http://www.illusion- works.com/html/poggendorf.html. IllusionWorks. "Zollner." http://www.illusionworks.com/ html/zollner.html. Jablan, S. "Some Visual Illusions Occurring in Interrupted Systems." http://members.tripod.com/~modularity/in- terr.htm. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 172, 1989.Zome A kit consisting of rods and slotted balls that can be used to construct three-dimensional configurations. The balls into which the rods are placed resembles an"expanded" SMALL RHOMBICOSIDODECAHEDRON , with the squares replaced by rectangles, as illustrated above. The rods come in four colors, and there are three lengths for each color, as summarized in thetable below. Here, fis the GOLDEN RATIO . color lengths n blue /fn // n/C300;1;2/ yellow /cos1 6p/C16/C17 fn //n/C300;1;2/ red /cos1 10p/C16/C17 fn //n/C300;1;2/ green /cos14p/C16/C17 fn //n/C30/C281;0;1/ References Hart, G. W. and Picciotto, H. "Zome Geometry: Hands-on Learning with Zome Models." http://www.georgehart.com/ zomebook/zomebook.html. Zome System. http://www.zometool.com/. Zonal Harmonic ASPHERICAL HARMONIC OF THE FORM Pl(cosu);i.e., one which reduces to a L EGENDRE POLYNOMIAL (Whit- taker and Watson 1990, p. 302). These harmonics are termed "zonal" since the curves on a UNIT SPHERE (with center at the origin) on which Pl(cosu) vanishes arelparallels of latitude which divide the surface into zones (Whittaker and Watson 1990, p. 392). Resolving Pl(cosu) into factors linear in cos2u ðÞ ; multiplied by (cos u) when lisODD, then replacing (cosu)b yz=rallows the zonal harmonic rlPl(cosu)t o be expressed as a product of factors linear in x2;y2; and z2;with the product multiplied by zwhen nis ODD (Whittaker and Watson 1990, p. 1990). See also LEGENDRE POLYNOMIAL ,SECTORIAL HARMO- NIC,SPHERICAL HARMONIC ,TESSERAL HARMONIC References Byerly, W. E. "Zonal Harmonics." Ch. 5 in An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 144 /C1/194, 1959. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Zone The SURFACE AREA of a SPHERICAL SEGMENT . Call the RADIUS of the SPHERE R, the upper and lower RADII b and a, respectively, and the height of the SPHERICAL SEGMENT h. The zone is a SURFACE OF REVOLUTION about the Z-AXIS , so the SURFACE AREA is given by S /C302pg xffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27x?2p dz : (1) In the xz-plane, the equation of the zone is simply that of a CIRCLE , x /C30ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28z2p ; (2) so x?/C30/C28 zR2 /C28z2/C0/C1/C281 =2(3) x ?2 /C30z2 R2 /C28 z2 ; (4) and S /C302pgffiffiffiffiffiffiffiffiffiffiffi R2 /C28b2p ffiffiffiffiffiffiffiffiffiffiffi R2 /C28a2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2 /C28z2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /C27z2 R2 /C28 z2s dz /C302pRgffiffiffiffiffiffiffiffiffiffiffi R2 /C28b2p ffiffiffiffiffiffiffiffiffiffiffi R2 /C28a2p dz /C302 pRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 /C28b2p /C28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2 /C28a2p/C16/C17 /C302 pRh: (5) This result is somewhat surprising since it depends only on the height of the zone, not its vertical position with respect to the SPHERE . See also SPHERE ,SPHERICAL CAP,SPHERICAL SEG- MENT ,ZONOHEDRON References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 130, 1987.Kern, W. F. and Bland, J. R. "Zone." §35 in Solid Mensura- tion with Proofs, 2nd ed. New York: Wiley, pp. 95 /C1/97, 1948. Zonohedron A CONVEX POLYHEDRON whose faces all possess a central symmetry (Coxeter 1973, pp. 27 /C1/30). Equiva- lently, a convex polyhedron whose faces are PARAL- LEL-sided 2m/-gons. There exist n(n /C281) PARALLELOGRAMS in a nonsingu- lar zonohedron, where n is the number of different directions in which EDGES occur (Ball and Coxeter 1987, pp. 141 /C1/144). Zonohedra include the CUBE , ENNEACONTAHEDRON , GREAT RHOMBIC TRIACONTAHE- DRON , GREAT RHOMBICUBOCTAHEDRON , MEDIAL RHOM- BIC TRIACONTAHEDRON , RHOMBIC DODECAHEDRON , RHOMBIC ICOSAHEDRON , RHOMBIC TRIACONTAHEDRON , and RHOMBOHEDRON , as well as the entire class of PARALLELEPIPEDS . Regular zonohedra have bands of PARALLELOGRAMS which form equators and are called "ZONES ." Every convex polyhedron bounded solely by PARALLELO- GRAMS is a zonohedron (Coxeter 1973, p. 27). Plate II (following p. 32 of Coxeter 1973) illustrates some equilateral zonohedra. Equilateral zonohedra can be regarded as 3-dimensional projections of n-D HYPER- CUBES (Ball and Coxeter 1987). See also CUBE,ENNEACONTAHEDRON ,GREAT RHOM- BIC TRIACONTAHEDRON ,G REAT RHOMBICUBOCTAHE- DRON (ARCHIMEDEAN ), HYPERCUBE ,MEDIAL RHOMBIC TRIACONTAHEDRON ,R HOMBIC DODECAHED RON, RHOMBIC ICOSAHEDRON ,R HOMBIC TRIACONTAHE- DRON ,RHOMBOHEDRON References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recrea- tions and Essays, 13th ed. New York: Dover, pp. 141 /C1/144, 1987. Coxeter, H. S. M. "Zonohedra." §2.8 in Regular Polytopes, 3rd ed. New York: Dover, pp. 27 /C1/30, 1973. Coxeter, H. S. M. Ch. 4 in The Beauty of Geometry: Twelve Essays. New York: Dover, 1999. Eppstein, D. "Ukrainian Easter Egg." http://www.ics.u- ci.edu/~eppstein/junkyard/ukraine/. Fedorov, E. S. Zeitschr. Krystallographie und Mineralogie 21, 689, 1893. Fedorov, E.W. Nachala Ucheniya o Figurakh. Leningrad, 1953. Hart, G. "Zonohedra." http://www.georgehart.com/virtual- polyhedra/zonohedra-info.html. Harp, G. W. "Zonohedrification." Mathematica J. 7, 374/C1/ 383, 1999. Kelly, L. M. and Moser, W. O. J. "On the Number of Ordinary Lines Determined by nPoints." Canad. J. Math. 1, 210/C1/219, 1958. Zonotype The M INKOWSKI SUM of line segments. Zoomeron Equation The PARTIAL DIFFERENTIAL EQUATION d2 dt2/C28d2 dx2 ! uxy u ! /C272u2/C0/C1 xt/C300: References Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolu- tion Equations. New York: North-Holland, p. 58, 1982. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 135, 1997. Zorn’s Lemma IfSis any nonempty PARTIALLY ORDERED SET in which every CHAIN has an upper bound, then Shas a maximal element. This statement is equivalent to the AXIOM OF CHOICE . See also AXIOM OF CHOICE z-Score The z-score associated with the ith observation of a random variable xis given by zi/C13xi/C28¯x s; where ¯xis the MEAN andsthe STANDARD DEVIATION of all observations x1;...,xn:/ Zsigmondy Theorem If 15bBaand ( a;b)/C301 (i.e., aand bare RELA- TIVELY PRIME ), then an/C28bnhas a PRIMITIVE PRIME FACTOR with the following two possible exceptions: 1. 26/C2816:/ 2.n/C302 and a/C27bis a POWER of 2. Similarly, if a>b]1;then an/C27bnhas a PRIMITIVE PRIME FACTOR with the exception 23/C2713/C309:/ References Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, p. 27, 1991. Z-Transform The Z-transform of F(t) is defined by ZF(t)½/C138/C30LF/C31(t) ½/C138 ; (1) where F/C31(t)/C30F(t)dT(t)/C30X/C12 n/C300F(nT)d(t/C28nT); (2) /d(t) is the DELTA FUNCTION ,Tis the sampling period, andLf½/C138is the L APLACE TRANSFORM . An alternative definition isZF(t)½/C138/C30X residues1 1/C28eTzz/C281 ! f(z); (3) where f(z)/C30X/C12 n/C300F(nT)z/C28n: (4) The inverse Z-transform is Z/C281[f(z)]/C30F/C31(t)/C301 2piGf(z)zn/C281dz: (5) The GENERATING FUNCTION ofG(t) of a sequence of numbers f(n) given by the Z-transform of f(n) in the variable 1 =t(Germundsson 2000). It satisfies Z[aF(t)/C27bG(t)]/C30aZ[F(t)]/C27bZ[F(t)] (6) Z[F(t/C27T)]/C30zZ[F(t)]/C28zF(0) (7) Z[F(t/C272T)]/C30z2Z[F(t)]/C28z2F(0)/C28zF(t) (8) Z[F(t/C27mT)]/C30zmZ[F(t)]/C28Xm/C281 r/C300zm/C28rF(rt) (9) Z[F(t/C28mT)]/C30z/C28mZ[F(t)] (10) ZeatF(t) ½/C138 /C30Ze/C28aTz/C2/C3 (11) Ze/C28atFtðÞ ½/C138 /C30ZeaTz/C2/C3 (12) tF(t)/C30/C28Tzd dzZF(t)½/C138 (13) t/C281F(t)/C30/C281 Tgz 0f(z) zdz: (14) Transforms of special functions (Beyer 1987, pp. 426 /C1/ 427) include Zd(t)½/C138/C301 (15) Zd(t/C28mT) ½/C138 /C30z/C28m(16) ZH(t) ½/C138/C30z z/C281(17) ZH(t/C28mT) ½/C138 /C30z zm(z/C281)(18) Zt½/C138/C30Tz (z/C281)2(19) Zt2/C2/C3 /C30T2z(z/C271) z/C281 ðÞ3(20) Zt3/C2/C3 /C30T3zz2/C274z/C271 ðÞ (z/C281)4(21) Zavt½/C138/C30z z /C28 a vT (22) Z cos(vt) ½/C138 /C30z sin( vT) z2 /C28 2z cos(vT) /C27 1(23) Z sin( vt) ½/C138 /C30zz/C28 cos(vT) ½/C138 Z2 /C28 2z cos(vT) /C27 1 ; (24) where H(t) is the HEAVISIDE STEP FUNCTION . In general, Ztn½/C138/C30(/C281)n lim x00dn dxnz z /C28 e /C28xT ! (25) TnzPn k/C301n k/C28/C29 zk /C281 (z /C28 1)n /C271 ; (26) where then k/C10/C11 are EULERIAN NUMBERS . Amazingly, the Z-transforms of tn are therefore generators for EULER’S TRIANGLE . The discrete z-transform of a sequence aj/C8/C9/C12 j/C30/C28/C12is defined as A(z) /C30Za½/C138/C30X/C12 k/C30/C28/C12akz/C28k (27)(Krantz 1999, p. 214). The DISCRETE FOURIER TRANS- FORM is therefore a special case of the z-transform with z /C13e /C282pi=N : (28) Az-transform with z/C13e/C282pia=N(29) fora"91 is called a FRACTIONAL FOURIER TRANS- FORM . See also DISCRETE FOURIER TRANSFORM ,E ULER’S TRIANGLE ,EULERIAN NUMBER ,FRACTIONAL FOURIER TRANSFORM References Arndt, J. "The z-Transform (ZT)." Ch. 3 in "Remarks on FFT Algorithms." http://www.jjj.de/fxt/. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 424 /C1/428, 1987. Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 257 /C1/262, 1999. Germundsson, R. " Mathematica Version 4." Mathematica J. 7, 497/C1/524, 2000. z-Transform (Population) POPULATION COMPARISON